{"nodes":[{"data_url":"../dependencies.json","id":"roadmap","kind":"scope","result":false,"route":"../roadmap/index.html","scope":"roadmap","status":"planned","title":"ATLAS roadmap"},{"area":"Foundations & Logic","data_url":"chapters/Logic.json","id":"Logic","kind":"scope","parent":0,"result":false,"route":"../roadmap/Logic/index.html","scope":"Logic","status":"can_state","summary":"Logic studies formal languages, proof, models, semantics, consistency, and the foundations of mathematical reasoning.","title":"Logic"},{"data_url":"scopes/Logic/modal-logic.json","id":"Logic/modal-logic","kind":"scope","parent":1,"result":false,"route":"../roadmap/Logic/modal-logic/index.html","scope":"Logic/modal-logic","status":"fully_proved","summary":"Logics of necessity and possibility interpreted on Kripke models, their axiomatizations, completeness and decidability, and extensions such as epistemic logic, dynamic logic and the modal \u03bc-calculus.","title":"Modal logic"},{"data_url":"scopes/Logic/modal-logic/epistemic-logic.json","id":"Logic/modal-logic/epistemic-logic","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","title":"Epistemic logic"},{"id":"Logic/modal-logic/epistemic-logic/model","kind":"node","lean":"Cslib.Epistemic.Model","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#model","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Given a type of worlds $W$, a type of atoms and a type of agents, an epistemic model consists of a binary accessibility relation $R_i$ on $W$ for each agent $i$ and a valuation assigning to each world the atoms true there; the world type $W$ is required to be nonempty.","title":"Epistemic models"},{"id":"Logic/modal-logic/epistemic-logic/formula","kind":"node","lean":"Cslib.Epistemic.Formula","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#formula","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Given a type of atoms and a type of agents, the formulas of epistemic logic are built inductively: every atom $p$ is a formula, falsity $\\bot$ is a formula, if $\\varphi$ and $\\psi$ are formulas then so is the implication $\\varphi \\to \\psi$, and if $\\varphi$ is a formula and $i$ is an agent then $K_i \\varphi$ (read \"agent $i$ knows $\\varphi$\") is a formula.","title":"Epistemic formulas"},{"id":"Logic/modal-logic/epistemic-logic/not","kind":"node","lean":"Cslib.Epistemic.Formula.not","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#not","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"The negation $\\neg\\varphi$ of an epistemic formula $\\varphi$ is the implication $\\varphi \\to \\bot$.","title":"Negation of epistemic formulas"},{"id":"Logic/modal-logic/epistemic-logic/or","kind":"node","lean":"Cslib.Epistemic.Formula.or","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#or","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"The disjunction $\\varphi \\lor \\psi$ of epistemic formulas $\\varphi$ and $\\psi$ is the implication $\\neg\\varphi \\to \\psi$.","title":"Disjunction of epistemic formulas"},{"id":"Logic/modal-logic/epistemic-logic/sat","kind":"node","lean":"Cslib.Epistemic.Sat","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#sat","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Satisfaction of an epistemic formula at a world $w$ of a model $M$ is defined by recursion on the formula, writing $R_i$ for the accessibility relation of agent $i$ in $M$ and $K_i \\varphi$ for the formula \"agent $i$ knows $\\varphi$\": an atom $a$ holds at $w$ when the valuation of $M$ makes $a$ true at $w$; $\\bot$ holds at no world; $\\varphi \\to \\psi$ holds at $w$ when $\\psi$ holds at $w$ whenever $\\varphi$ does;\u2026","title":"Satisfaction in an epistemic model"},{"id":"Logic/modal-logic/epistemic-logic/valid","kind":"node","lean":"Cslib.Epistemic.Valid","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#valid","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"A formula $\\varphi$ is valid in an epistemic model $M$ when $\\varphi$ is satisfied at every world of $M$.","title":"Validity in an epistemic model"},{"id":"Logic/modal-logic/epistemic-logic/not-valid-bot","kind":"node","lean":"Cslib.Epistemic.not_valid_bot","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#not-valid-bot","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"In every epistemic model, the formula $\\bot$ is not valid: it fails at some world.","title":"Falsity is not valid in an epistemic model"},{"id":"Logic/modal-logic/epistemic-logic/sat-or","kind":"node","lean":"Cslib.Epistemic.sat_or","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#sat-or","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"In an epistemic model $M$, the disjunction $\\varphi \\lor \\psi$ of two formulas, defined as $\\neg \\varphi \\to \\psi$, is satisfied at a world $w$ if and only if $\\varphi$ is satisfied at $w$ or $\\psi$ is satisfied at $w$.","title":"Satisfaction of a disjunction"},{"id":"Logic/modal-logic/epistemic-logic/valid-know-dist","kind":"node","lean":"Cslib.Epistemic.valid_know_dist","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#valid-know-dist","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Write $K_i \\varphi$ for \"agent $i$ knows $\\varphi$\". For every epistemic model $M$, every agent $i$ and all formulas $\\varphi$ and $\\psi$, the formula $K_i (\\varphi \\to \\psi) \\to (K_i \\varphi \\to K_i \\psi)$ is valid in $M$: knowledge distributes over implication.","title":"Distribution axiom K"},{"id":"Logic/modal-logic/epistemic-logic/valid-know-truth","kind":"node","lean":"Cslib.Epistemic.valid_know_truth","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#valid-know-truth","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Write $K_i \\varphi$ for \"agent $i$ knows $\\varphi$\". Let $M$ be an epistemic model in which the accessibility relation $R_i$ of agent $i$ is reflexive. Then for every formula $\\varphi$, the formula $K_i \\varphi \\to \\varphi$ is valid in $M$.","title":"Truth axiom T"},{"id":"Logic/modal-logic/epistemic-logic/valid-know-pos","kind":"node","lean":"Cslib.Epistemic.valid_know_pos","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#valid-know-pos","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Write $K_i \\varphi$ for \"agent $i$ knows $\\varphi$\". Let $M$ be an epistemic model in which the accessibility relation $R_i$ of agent $i$ is transitive. Then for every formula $\\varphi$, the formula $K_i \\varphi \\to K_i K_i \\varphi$ is valid in $M$.","title":"Positive introspection axiom 4"},{"id":"Logic/modal-logic/epistemic-logic/valid-know-neg","kind":"node","lean":"Cslib.Epistemic.valid_know_neg","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#valid-know-neg","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Write $K_i \\varphi$ for \"agent $i$ knows $\\varphi$\". Let $M$ be an epistemic model in which the accessibility relation $R_i$ of agent $i$ is euclidean: for all worlds $a$, $b$, $c$, if $a \\mathrel{R_i} b$ and $a \\mathrel{R_i} c$ then $b \\mathrel{R_i} c$. Then for every formula $\\varphi$, the formula $\\neg K_i \\varphi \\to K_i \\neg K_i \\varphi$ is valid in $M$.","title":"Negative introspection axiom 5"},{"id":"Logic/modal-logic/epistemic-logic/muddy","kind":"node","lean":"Cslib.Epistemic.muddy","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#muddy","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"The epistemic model whose worlds are the two booleans, with $\\mathrm{true}$ read as heads and $\\mathrm{false}$ as tails, with a single atom $p$ true exactly at the heads world, and with two agents $0$ and $1$: agent $0$ sees the coin, so its accessibility relation is equality of worlds, and agent $1$ does not, so its accessibility relation relates every pair of worlds.","title":"The two-agent coin model"},{"id":"Logic/modal-logic/epistemic-logic/sat-know-zero-atom","kind":"node","lean":"Cslib.Epistemic.sat_know_zero_atom","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#sat-know-zero-atom","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Let $p$ be the atom \"the coin is heads\" of the two-agent coin model. At a world $v$ of that model, the formula $K_0 p$ holds if and only if $v$ is the heads world: agent $0$ sees the coin.","title":"Agent 0 knows the coin is heads exactly at heads"},{"id":"Logic/modal-logic/epistemic-logic/sat-know-zero-not-atom","kind":"node","lean":"Cslib.Epistemic.sat_know_zero_not_atom","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#sat-know-zero-not-atom","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Let $p$ be the atom \"the coin is heads\" of the two-agent coin model. At a world $v$ of that model, the formula $K_0 \\neg p$ holds if and only if $v$ is the tails world.","title":"Agent 0 knows the coin is tails exactly at tails"},{"id":"Logic/modal-logic/epistemic-logic/muddy-not-know-one-p-true","kind":"node","lean":"Cslib.Epistemic.muddy_not_know_one_p_true","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#muddy-not-know-one-p-true","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Let $p$ be the atom \"the coin is heads\" of the two-agent coin model. The formula $K_1 p$ is false at the heads world: agent $1$ considers the tails world possible.","title":"Agent 1 does not know that the coin is heads"},{"id":"Logic/modal-logic/epistemic-logic/muddy-know-one-knows-whether","kind":"node","lean":"Cslib.Epistemic.muddy_know_one_knows_whether","parent":3,"result":true,"route":"../roadmap/Logic/modal-logic/epistemic-logic/index.html#muddy-know-one-knows-whether","scope":"Logic/modal-logic/epistemic-logic","status":"fully_proved","summary":"Let $p$ be the atom \"the coin is heads\" of the two-agent coin model. At every world of that model the formula $K_1(K_0 p \\lor K_0 \\neg p)$ holds: agent $1$ knows that agent $0$ either knows $p$ or knows $\\neg p$.","title":"Agent 1 knows that agent 0 knows whether the coin is heads"},{"data_url":"scopes/Logic/modal-logic/kripke-semantics.json","id":"Logic/modal-logic/kripke-semantics","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","title":"Kripke semantics"},{"id":"Logic/modal-logic/kripke-semantics/formula","kind":"node","lean":"Cslib.Modal.Formula","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#formula","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For a type $\\alpha$ of atoms, the formulas of basic modal logic over $\\alpha$ are generated as follows: every atom $p \\in \\alpha$ is a formula; falsity $\\bot$ is a formula; if $\\varphi$ and $\\psi$ are formulas then so is the implication $\\varphi \\to \\psi$; and if $\\varphi$ is a formula then so is the necessity formula $\\Box\\varphi$.","title":"Modal formulas"},{"id":"Logic/modal-logic/kripke-semantics/neg","kind":"node","lean":"Cslib.Modal.Formula.neg","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#neg","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"The negation of a modal formula $\\varphi$ is the formula $\\neg\\varphi := \\varphi \\to \\bot$.","title":"Negation of modal formulas"},{"id":"Logic/modal-logic/kripke-semantics/dia","kind":"node","lean":"Cslib.Modal.Formula.dia","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#dia","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For a modal formula $\\varphi$, the formula $\\Diamond\\varphi$ (\"possibly $\\varphi$\") is $\\neg\\Box\\neg\\varphi$, where $\\neg\\chi$ stands for $\\chi \\to \\bot$.","title":"The possibility operator"},{"id":"Logic/modal-logic/kripke-semantics/top","kind":"node","lean":"Cslib.Modal.Formula.top","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#top","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"The formula $\\top$ is the negation $\\neg\\bot$ of falsity, that is, the implication $\\bot \\to \\bot$.","title":"The truth constant"},{"id":"Logic/modal-logic/kripke-semantics/and","kind":"node","lean":"Cslib.Modal.Formula.and","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#and","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For modal formulas $\\varphi$ and $\\psi$, the conjunction $\\varphi \\wedge \\psi$ is the formula $\\neg(\\varphi \\to \\neg\\psi)$, where $\\neg\\chi$ stands for $\\chi \\to \\bot$.","title":"Conjunction"},{"id":"Logic/modal-logic/kripke-semantics/or","kind":"node","lean":"Cslib.Modal.Formula.or","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#or","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"The disjunction of modal formulas $\\varphi$ and $\\psi$ is the formula $\\varphi \\lor \\psi := \\neg\\varphi \\to \\psi$, where $\\neg\\varphi$ is $\\varphi \\to \\bot$.","title":"Disjunction of modal formulas"},{"id":"Logic/modal-logic/kripke-semantics/axiom-k","kind":"node","lean":"Cslib.Modal.Formula.axiomK","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#axiom-k","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For modal formulas $\\varphi$ and $\\psi$, the distribution axiom K is the formula $\\Box(\\varphi \\to \\psi) \\to (\\Box\\varphi \\to \\Box\\psi)$.","title":"Axiom K"},{"id":"Logic/modal-logic/kripke-semantics/axiom-t","kind":"node","lean":"Cslib.Modal.Formula.axiomT","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#axiom-t","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For a modal formula $\\varphi$, the reflexivity axiom T is the formula $\\Box\\varphi \\to \\varphi$.","title":"Axiom T"},{"id":"Logic/modal-logic/kripke-semantics/axiom-four","kind":"node","lean":"Cslib.Modal.Formula.axiomFour","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#axiom-four","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For a modal formula $\\varphi$, the transitivity axiom 4 is the formula $\\Box\\varphi \\to \\Box\\Box\\varphi$.","title":"Axiom 4"},{"id":"Logic/modal-logic/kripke-semantics/axiom-d","kind":"node","lean":"Cslib.Modal.Formula.axiomD","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#axiom-d","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For a modal formula $\\varphi$, the seriality axiom D is the formula $\\Box\\varphi \\to \\Diamond\\varphi$, where $\\Diamond\\psi$ is $\\neg\\Box\\neg\\psi$ and $\\neg\\chi$ stands for $\\chi \\to \\bot$.","title":"Axiom D"},{"id":"Logic/modal-logic/kripke-semantics/axiom-b","kind":"node","lean":"Cslib.Modal.Formula.axiomB","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#axiom-b","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For a modal formula $\\varphi$, the symmetry axiom B is the formula $\\varphi \\to \\Box\\Diamond\\varphi$, where $\\Diamond\\psi$ is $\\neg\\Box\\neg\\psi$ and $\\neg\\chi$ stands for $\\chi \\to \\bot$.","title":"Axiom B"},{"id":"Logic/modal-logic/kripke-semantics/axiom-five","kind":"node","lean":"Cslib.Modal.Formula.axiomFive","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#axiom-five","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For a modal formula $\\varphi$, the euclidean axiom 5 is the formula $\\Diamond\\varphi \\to \\Box\\Diamond\\varphi$, where $\\Diamond\\psi$ is $\\neg\\Box\\neg\\psi$ and $\\neg\\chi$ stands for $\\chi \\to \\bot$.","title":"Axiom 5"},{"id":"Logic/modal-logic/kripke-semantics/kripke-model","kind":"node","lean":"Cslib.Modal.KripkeModel","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#kripke-model","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"A Kripke model consists of a nonempty collection of worlds, an accessibility relation, and an atom valuation.","title":"Kripke model"},{"id":"Logic/modal-logic/kripke-semantics/satisfies","kind":"node","lean":"Cslib.Modal.KripkeModel.Satisfies","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#satisfies","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ be a Kripke model with accessibility relation $R$. Satisfaction of a modal formula at a world $w$ of $M$ is defined by recursion on the formula: $w$ satisfies an atom $p$ if and only if the valuation of $M$ makes $p$ true at $w$; $w$ never satisfies $\\bot$; $w$ satisfies $\\varphi \\to \\psi$ if and only if $w$ satisfies $\\psi$ whenever it satisfies $\\varphi$; and $w$ satisfies $\\Box\\varphi$ if and only if\u2026","title":"Satisfaction in a Kripke model"},{"id":"Logic/modal-logic/kripke-semantics/valid","kind":"node","lean":"Cslib.Modal.KripkeModel.Valid","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#valid","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"A formula $\\varphi$ is globally valid in a Kripke model $M$ when $\\varphi$ is satisfied at every world of $M$.","title":"Global validity in a Kripke model"},{"id":"Logic/modal-logic/kripke-semantics/not-valid-bot","kind":"node","lean":"Cslib.Modal.KripkeModel.not_valid_bot","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#not-valid-bot","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For every Kripke model $M$, the formula $\\bot$ is not valid in $M$: since $M$ has at least one world and no world satisfies $\\bot$, some world of $M$ fails to satisfy $\\bot$.","title":"Falsity is valid in no Kripke model"},{"id":"Logic/modal-logic/kripke-semantics/satisfies-dia","kind":"node","lean":"Cslib.Modal.KripkeModel.satisfies_dia","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#satisfies-dia","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ be a Kripke model with accessibility relation $R$, and let $\\Diamond\\varphi$ denote the formula $\\neg\\Box\\neg\\varphi$, where $\\neg\\chi$ is $\\chi \\to \\bot$. For every world $w$ and modal formula $\\varphi$, $w$ satisfies $\\Diamond\\varphi$ in $M$ if and only if there is a world $v$ with $w \\mathrel{R} v$ such that $v$ satisfies $\\varphi$ in $M$.","title":"Semantics of possibility"},{"id":"Logic/modal-logic/kripke-semantics/valid-axiom-k","kind":"node","lean":"Cslib.Modal.KripkeModel.valid_axiomK","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#valid-axiom-k","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"The K axiom is valid on every Kripke model.","title":"Validity of axiom K on Kripke models"},{"id":"Logic/modal-logic/kripke-semantics/valid-axiom-t","kind":"node","lean":"Cslib.Modal.KripkeModel.valid_axiomT","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#valid-axiom-t","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ be a Kripke model whose accessibility relation $R$ is reflexive: $R\\,w\\,w$ for every world $w$. Then for every formula $\\varphi$ the T axiom $\\Box\\varphi \\to \\varphi$ is globally valid in $M$.","title":"Axiom T is valid on reflexive Kripke models"},{"id":"Logic/modal-logic/kripke-semantics/valid-axiom-four","kind":"node","lean":"Cslib.Modal.KripkeModel.valid_axiomFour","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#valid-axiom-four","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ be a Kripke model whose accessibility relation $R$ is transitive: $R\\,u\\,v$ and $R\\,v\\,w$ imply $R\\,u\\,w$ for all worlds $u, v, w$. Then for every formula $\\varphi$ the 4 axiom $\\Box\\varphi \\to \\Box\\Box\\varphi$ is globally valid in $M$.","title":"Axiom 4 is valid on transitive Kripke models"},{"id":"Logic/modal-logic/kripke-semantics/valid-axiom-d","kind":"node","lean":"Cslib.Modal.KripkeModel.valid_axiomD","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#valid-axiom-d","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ be a Kripke model whose accessibility relation $R$ is serial: every world $w$ has some world $v$ with $R\\,w\\,v$. Then for every formula $\\varphi$ the D axiom $\\Box\\varphi \\to \\Diamond\\varphi$, where $\\Diamond\\varphi$ abbreviates $\\neg\\Box\\neg\\varphi$, is globally valid in $M$.","title":"Axiom D is valid on serial Kripke models"},{"id":"Logic/modal-logic/kripke-semantics/valid-axiom-b","kind":"node","lean":"Cslib.Modal.KripkeModel.valid_axiomB","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#valid-axiom-b","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ be a Kripke model whose accessibility relation $R$ is symmetric: $R\\,u\\,v$ implies $R\\,v\\,u$ for all worlds $u, v$. Then for every formula $\\varphi$ the B axiom $\\varphi \\to \\Box\\Diamond\\varphi$, where $\\Diamond\\varphi$ abbreviates $\\neg\\Box\\neg\\varphi$, is globally valid in $M$.","title":"Axiom B is valid on symmetric Kripke models"},{"id":"Logic/modal-logic/kripke-semantics/valid-axiom-five","kind":"node","lean":"Cslib.Modal.KripkeModel.valid_axiomFive","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#valid-axiom-five","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ be a Kripke model whose accessibility relation $R$ is euclidean: $R\\,u\\,v$ and $R\\,u\\,w$ imply $R\\,v\\,w$ for all worlds $u, v, w$. Then for every formula $\\varphi$ the 5 axiom $\\Diamond\\varphi \\to \\Box\\Diamond\\varphi$, where $\\Diamond\\varphi$ abbreviates $\\neg\\Box\\neg\\varphi$, is globally valid in $M$.","title":"Axiom 5 is valid on euclidean Kripke models"},{"id":"Logic/modal-logic/kripke-semantics/bounded-morphism","kind":"node","lean":"Cslib.Modal.BoundedMorphism","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#bounded-morphism","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ and $N$ be Kripke models over the same atoms, with sets of worlds $W$ and $V$ and accessibility relations $R_M$ and $R_N$. A bounded morphism from $M$ to $N$ is a map $f : W \\to V$ such that every atom holds at a world $w$ in $M$ if and only if it holds at $f(w)$ in $N$; whenever $w \\mathrel{R_M} w'$, also $f(w) \\mathrel{R_N} f(w')$ (forth); and whenever $f(w) \\mathrel{R_N} v'$, there is a world $w'$ with\u2026","title":"Bounded morphism"},{"id":"Logic/modal-logic/kripke-semantics/ext","kind":"node","lean":"Cslib.Modal.BoundedMorphism.ext","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#ext","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Two bounded morphisms $f$ and $g$ between the same Kripke models are equal if $f(w) = g(w)$ for every world $w$.","title":"Extensionality for bounded morphisms"},{"id":"Logic/modal-logic/kripke-semantics/id","kind":"node","lean":"Cslib.Modal.BoundedMorphism.id","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#id","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"For every Kripke model $M$, the identity map on the worlds of $M$ is a bounded morphism from $M$ to itself.","title":"Identity bounded morphism"},{"id":"Logic/modal-logic/kripke-semantics/comp","kind":"node","lean":"Cslib.Modal.BoundedMorphism.comp","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#comp","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M_1$, $M_2$, $M_3$ be Kripke models over the same atoms. If $f$ is a bounded morphism from $M_1$ to $M_2$ and $g$ is a bounded morphism from $M_2$ to $M_3$, then the composite map $g \\circ f$ on worlds is a bounded morphism from $M_1$ to $M_3$.","title":"Composition of bounded morphisms"},{"id":"Logic/modal-logic/kripke-semantics/satisfies-iff","kind":"node","lean":"Cslib.Modal.BoundedMorphism.satisfies_iff","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#satisfies-iff","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $f$ be a bounded morphism from a Kripke model $M$ to a Kripke model $N$ over the same atoms. Then for every modal formula $\\varphi$ and every world $w$ of $M$, $w$ satisfies $\\varphi$ in $M$ if and only if $f(w)$ satisfies $\\varphi$ in $N$.","title":"Invariance of modal satisfaction under bounded morphisms"},{"id":"Logic/modal-logic/kripke-semantics/valid-iff-of-surjective","kind":"node","lean":"Cslib.Modal.BoundedMorphism.valid_iff_of_surjective","parent":21,"result":true,"route":"../roadmap/Logic/modal-logic/kripke-semantics/index.html#valid-iff-of-surjective","scope":"Logic/modal-logic/kripke-semantics","status":"fully_proved","summary":"Let $M$ and $N$ be Kripke models over the same atoms and let $f$ be a bounded morphism from $M$ to $N$ whose map on worlds is surjective. Then for every formula $\\varphi$, $\\varphi$ is globally valid in $M$ if and only if $\\varphi$ is globally valid in $N$.","title":"Surjective bounded morphisms preserve and reflect validity"},{"data_url":"scopes/Logic/modal-logic/bisimulation.json","id":"Logic/modal-logic/bisimulation","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/bisimulation/index.html","scope":"Logic/modal-logic/bisimulation","status":"fully_proved","title":"Bisimulation"},{"id":"Logic/modal-logic/bisimulation/bisimulation","kind":"node","lean":"Cslib.Modal.Bisimulation","parent":51,"result":true,"route":"../roadmap/Logic/modal-logic/bisimulation/index.html#bisimulation","scope":"Logic/modal-logic/bisimulation","status":"fully_proved","summary":"Let $M$ and $N$ be Kripke models over the same atoms, with sets of worlds $W$ and $V$ and accessibility relations $R_M$ and $R_N$. A relation $Z \\subseteq W \\times V$ is a bisimulation between $M$ and $N$ when, for all $w \\in W$ and $v \\in V$ with $w \\mathrel{Z} v$: each atom $a$ is true at $w$ in $M$ if and only if $a$ is true at $v$ in $N$; for every $w'$ with $w \\mathrel{R_M} w'$ there is $v'$ with\u2026","title":"Bisimulation"},{"id":"Logic/modal-logic/bisimulation/satisfies-iff","kind":"node","lean":"Cslib.Modal.Bisimulation.satisfies_iff","parent":51,"result":true,"route":"../roadmap/Logic/modal-logic/bisimulation/index.html#satisfies-iff","scope":"Logic/modal-logic/bisimulation","status":"fully_proved","summary":"Let $Z$ be a bisimulation between Kripke models $M$ and $N$ over the same atoms. If $Z$ relates a world $w$ of $M$ to a world $v$ of $N$, then every modal formula $\\varphi$ is satisfied at $w$ in $M$ if and only if $\\varphi$ is satisfied at $v$ in $N$.","title":"Bisimulations preserve modal formulas"},{"id":"Logic/modal-logic/bisimulation/bisimilar","kind":"node","lean":"Cslib.Modal.Bisimulation.Bisimilar","parent":51,"result":true,"route":"../roadmap/Logic/modal-logic/bisimulation/index.html#bisimilar","scope":"Logic/modal-logic/bisimulation","status":"fully_proved","summary":"Let $M$ and $N$ be Kripke models over the same atoms. A world $w$ of $M$ and a world $v$ of $N$ are bisimilar if some bisimulation between $M$ and $N$ relates $w$ to $v$.","title":"Bisimilar worlds"},{"id":"Logic/modal-logic/bisimulation/satisfies-iff-2","kind":"node","lean":"Cslib.Modal.Bisimulation.Bisimilar.satisfies_iff","parent":51,"result":true,"route":"../roadmap/Logic/modal-logic/bisimulation/index.html#satisfies-iff-2","scope":"Logic/modal-logic/bisimulation","status":"fully_proved","summary":"If a world $w$ of a Kripke model $M$ and a world $v$ of a Kripke model $N$ over the same atoms are bisimilar, then every modal formula $\\varphi$ is satisfied at $w$ in $M$ if and only if $\\varphi$ is satisfied at $v$ in $N$.","title":"Bisimilar worlds are modally equivalent"},{"data_url":"scopes/Logic/modal-logic/modal-calculus.json","id":"Logic/modal-logic/modal-calculus","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","title":"Modal \u03bc-calculus"},{"id":"Logic/modal-logic/modal-calculus/mu-formula","kind":"node","lean":"Cslib.Modal.MuCalculus.MuFormula","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-formula","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"The formulas over a type of atoms are built from atoms $a$, negated atoms $\\neg a$, variables $X_n$ for $n \\in \\mathbb{N}$ (de Bruijn indices for the fixed-point binders), $\\bot$ and $\\top$ by conjunction $\\varphi \\wedge \\psi$, disjunction $\\varphi \\vee \\psi$, the modalities $\\Box\\varphi$ and $\\Diamond\\varphi$, and the fixed-point binders $\\mu\\varphi$ and $\\nu\\varphi$.","title":"Formulas of the modal \u03bc-calculus"},{"id":"Logic/modal-logic/modal-calculus/neg","kind":"node","lean":"Cslib.Modal.MuCalculus.MuFormula.neg","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#neg","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"The meta-level negation $\\neg\\varphi$ of a \u03bc-calculus formula $\\varphi$ swaps each connective with its dual. It is defined by recursion: an atom $p$ becomes the negated atom $\\overline{p}$ and $\\overline{p}$ becomes $p$, each variable $x_n$ stays $x_n$, $\\neg\\bot = \\top$, $\\neg\\top = \\bot$, $\\neg(\\varphi \\wedge \\psi) = \\neg\\varphi \\vee \\neg\\psi$, $\\neg(\\varphi \\vee \\psi) = \\neg\\varphi \\wedge \\neg\\psi$,\u2026","title":"Negation of \u03bc-calculus formulas"},{"id":"Logic/modal-logic/modal-calculus/neg-neg","kind":"node","lean":"Cslib.Modal.MuCalculus.MuFormula.neg_neg","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#neg-neg","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Write $\\neg\\varphi$ for the meta-level negation of a \u03bc-calculus formula $\\varphi$, which swaps each connective with its dual. For every \u03bc-calculus formula $\\varphi$, negating twice gives back $\\varphi$: $\\neg\\neg\\varphi = \\varphi$.","title":"Meta-level negation is an involution"},{"id":"Logic/modal-logic/modal-calculus/extend-env","kind":"node","lean":"Cslib.Modal.MuCalculus.extendEnv","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#extend-env","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"An environment $\\rho$ assigns to each variable index $n \\in \\mathbb{N}$ a predicate $\\rho(n)$ on worlds. Its extension by a predicate $S$ on worlds is the environment that assigns $S$ to index $0$ and $\\rho(n)$ to index $n + 1$.","title":"Extending an environment under a binder"},{"id":"Logic/modal-logic/modal-calculus/sat","kind":"node","lean":"Cslib.Modal.MuCalculus.sat","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model with accessibility relation $R$ and valuation $V$, and let $\\rho$ be an environment assigning a set of worlds to each variable index $n \\in \\mathbb{N}$. For a set of worlds $S$, write $\\rho[0 \\mapsto S]$ for the environment sending index $0$ to $S$ and index $n + 1$ to $\\rho(n)$. The set $[\\![\\varphi]\\!]_\\rho$ of worlds satisfying a formula $\\varphi$ in $M$ under $\\rho$ is defined by\u2026","title":"Satisfaction in the modal \u03bc-calculus"},{"id":"Logic/modal-logic/modal-calculus/sat-mono-env","kind":"node","lean":"Cslib.Modal.MuCalculus.sat_mono_env","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat-mono-env","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and let $\\rho$ and $\\rho'$ be environments (each assigns a set of worlds of $M$ to every variable $n$) with $\\rho(n) \\subseteq \\rho'(n)$ for every $n$. Then every \u03bc-calculus formula $\\varphi$ holds under $\\rho'$ at every world of $M$ where it holds under $\\rho$: enlarging every variable's denotation can only enlarge a formula's denotation.","title":"Satisfaction is monotone in the environment"},{"id":"Logic/modal-logic/modal-calculus/sat-mono-extend","kind":"node","lean":"Cslib.Modal.MuCalculus.sat_mono_extend","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat-mono-extend","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment assigning a set of worlds to each variable index $n \\in \\mathbb{N}$, and $\\varphi$ a \u03bc-calculus formula. For a set of worlds $S$, write $\\rho[0 \\mapsto S]$ for the environment sending index $0$ to $S$ and index $n + 1$ to $\\rho(n)$. If $S_1 \\subseteq S_2$ are sets of worlds, then every world that satisfies $\\varphi$ in $M$ under $\\rho[0 \\mapsto S_1]$ also satisfies\u2026","title":"Monotonicity in the bound variable"},{"id":"Logic/modal-logic/modal-calculus/mu-pre-fixed","kind":"node","lean":"Cslib.Modal.MuCalculus.mu_pre_fixed","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-pre-fixed","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment and $\\varphi$ a \u03bc-calculus formula. Write $\\llbracket\\psi\\rrbracket_\\sigma$ for the set of worlds of $M$ at which $\\psi$ holds under the environment $\\sigma$, and $\\rho[S]$ for the extension of $\\rho$ under a binder by a set of worlds $S$ (variable $0$ denotes $S$ and variable $n + 1$ denotes $\\rho(n)$). For $T = \\llbracket\\mu\\varphi\\rrbracket_\\rho$, where\u2026","title":"The least fixed point is a pre-fixed point"},{"id":"Logic/modal-logic/modal-calculus/mu-le-fixed","kind":"node","lean":"Cslib.Modal.MuCalculus.mu_le_fixed","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-le-fixed","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For every Kripke model $M$, environment $\\rho$ and \u03bc-calculus formula $\\varphi$, every world satisfying $\\mu\\varphi$ in $\\rho$ also satisfies $\\varphi$ in the extension of $\\rho$ by the set of worlds satisfying $\\mu\\varphi$ in $\\rho$.","title":"The least fixed point is contained in its unfolding"},{"id":"Logic/modal-logic/modal-calculus/sat-mu-unfold","kind":"node","lean":"Cslib.Modal.MuCalculus.sat_mu_unfold","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat-mu-unfold","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment assigning a set of worlds to each variable index $n \\in \\mathbb{N}$, and $\\varphi$ a \u03bc-calculus formula. Let $A$ be the set of worlds satisfying $\\mu\\varphi$ in $M$ under $\\rho$. Then $A$ is exactly the set of worlds satisfying $\\varphi$ in $M$ under the environment that sends index $0$ to $A$ and index $n + 1$ to $\\rho(n)$.","title":"The \u03bc unfolding"},{"id":"Logic/modal-logic/modal-calculus/nu-post-fixed","kind":"node","lean":"Cslib.Modal.MuCalculus.nu_post_fixed","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#nu-post-fixed","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment and $\\varphi$ a \u03bc-calculus formula. Write $\\llbracket\\psi\\rrbracket_\\sigma$ for the set of worlds of $M$ at which $\\psi$ holds under the environment $\\sigma$, and $\\rho[S]$ for the extension of $\\rho$ under a binder by a set of worlds $S$ (variable $0$ denotes $S$ and variable $n + 1$ denotes $\\rho(n)$). For $T = \\llbracket\\nu\\varphi\\rrbracket_\\rho$, where\u2026","title":"The greatest fixed point is a post-fixed point"},{"id":"Logic/modal-logic/modal-calculus/nu-le-post-fixed","kind":"node","lean":"Cslib.Modal.MuCalculus.nu_le_post_fixed","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#nu-le-post-fixed","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment and $\\varphi$ a \u03bc-calculus formula. Write $\\llbracket\\psi\\rrbracket_\\sigma$ for the set of worlds of $M$ at which $\\psi$ holds under the environment $\\sigma$, and $\\rho[S]$ for the extension of $\\rho$ under a binder by a set of worlds $S$ (variable $0$ denotes $S$ and variable $n + 1$ denotes $\\rho(n)$). For $T = \\llbracket\\nu\\varphi\\rrbracket_\\rho$, where\u2026","title":"The greatest fixed point is a pre-fixed point"},{"id":"Logic/modal-logic/modal-calculus/sat-nu-unfold","kind":"node","lean":"Cslib.Modal.MuCalculus.sat_nu_unfold","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat-nu-unfold","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment assigning a set of worlds to each variable index $n \\in \\mathbb{N}$, and $\\varphi$ a \u03bc-calculus formula. Let $A$ be the set of worlds satisfying $\\nu\\varphi$ in $M$ under $\\rho$. Then $A$ is exactly the set of worlds satisfying $\\varphi$ in $M$ under the environment that sends index $0$ to $A$ and index $n + 1$ to $\\rho(n)$.","title":"The \u03bd unfolding"},{"id":"Logic/modal-logic/modal-calculus/mu-induction","kind":"node","lean":"Cslib.Modal.MuCalculus.mu_induction","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-induction","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment, $\\varphi$ a \u03bc-calculus formula and $P$ a predicate on worlds. If every world satisfying $\\varphi$ in the extension of $\\rho$ by $P$ satisfies $P$, then every world satisfying $\\mu\\varphi$ in $\\rho$ satisfies $P$.","title":"Park induction"},{"id":"Logic/modal-logic/modal-calculus/nu-coinduction","kind":"node","lean":"Cslib.Modal.MuCalculus.nu_coinduction","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#nu-coinduction","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment, $\\varphi$ a \u03bc-calculus formula and $P$ a set of worlds of $M$. Write $\\llbracket\\psi\\rrbracket_\\sigma$ for the set of worlds at which $\\psi$ holds under the environment $\\sigma$, and $\\rho[P]$ for the extension of $\\rho$ under a binder by $P$ (variable $0$ denotes $P$ and variable $n + 1$ denotes $\\rho(n)$). If $P \\subseteq \\llbracket\\varphi\\rrbracket_{\\rho[P]}$,\u2026","title":"Coinduction for the greatest fixed point"},{"id":"Logic/modal-logic/modal-calculus/mu-op","kind":"node","lean":"Cslib.Modal.MuCalculus.muOp","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-op","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Given a Kripke model $M$, an environment $\\rho$ (a set of worlds $\\rho(n)$ for each variable $n$) and a \u03bc-calculus formula $\\varphi$, this is the monotone map on sets of worlds of $M$ sending $S$ to the set of worlds at which $\\varphi$ holds under $\\rho[S]$, the extension of $\\rho$ under a binder by $S$ (variable $0$ denotes $S$ and variable $n + 1$ denotes $\\rho(n)$).","title":"The fixed-point operator of a formula"},{"id":"Logic/modal-logic/modal-calculus/sat-mu-eq-lfp","kind":"node","lean":"Cslib.Modal.MuCalculus.sat_mu_eq_lfp","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat-mu-eq-lfp","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment assigning a set of worlds to each variable index $n \\in \\mathbb{N}$, and $\\varphi$ a \u03bc-calculus formula. For a set of worlds $S$, write $\\rho[0 \\mapsto S]$ for the environment sending index $0$ to $S$ and index $n + 1$ to $\\rho(n)$. Then the set of worlds satisfying $\\mu\\varphi$ in $M$ under $\\rho$ is the least fixed point of the monotone operator on sets of worlds\u2026","title":"The \u03bc formula denotes the least fixed point"},{"id":"Logic/modal-logic/modal-calculus/sat-nu-eq-gfp","kind":"node","lean":"Cslib.Modal.MuCalculus.sat_nu_eq_gfp","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat-nu-eq-gfp","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment assigning a set of worlds to each variable index $n \\in \\mathbb{N}$, and $\\varphi$ a \u03bc-calculus formula. For a set of worlds $S$, write $\\rho[0 \\mapsto S]$ for the environment sending index $0$ to $S$ and index $n + 1$ to $\\rho(n)$. Then the set of worlds satisfying $\\nu\\varphi$ in $M$ under $\\rho$ is the greatest fixed point of the monotone operator on sets of\u2026","title":"The \u03bd formula denotes the greatest fixed point"},{"id":"Logic/modal-logic/modal-calculus/extend-env-compl","kind":"node","lean":"Cslib.Modal.MuCalculus.extendEnv_compl","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#extend-env-compl","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For every environment $\\rho$ (assigning to each variable index $n$ a predicate $\\rho(n)$ on worlds) and every predicate $S$ on worlds, complementing every predicate of the extension of $\\rho$ by $S$ gives the extension of the complemented environment $n \\mapsto \\neg\\rho(n)$ by the predicate $w \\mapsto \\neg S(w)$.","title":"Complementing an extended environment"},{"id":"Logic/modal-logic/modal-calculus/compl-compl","kind":"node","lean":"Cslib.Modal.MuCalculus.compl_compl","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#compl-compl","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For every predicate $S$ on worlds, the predicate $w \\mapsto \\neg\\neg S(w)$ equals $S$.","title":"Double complement of a predicate"},{"id":"Logic/modal-logic/modal-calculus/sat-neg","kind":"node","lean":"Cslib.Modal.MuCalculus.sat_neg","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat-neg","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment assigning a set of worlds to each variable index $n \\in \\mathbb{N}$, and $\\varphi$ a \u03bc-calculus formula. Let $\\neg\\varphi$ be the dual formula of $\\varphi$, obtained by swapping atoms with negated atoms, $\\bot$ with $\\top$, $\\wedge$ with $\\vee$, $\\Box$ with $\\Diamond$ and $\\mu$ with $\\nu$, and keeping variables. Write $\\overline{\\rho}$ for the environment sending\u2026","title":"Duality"},{"id":"Logic/modal-logic/modal-calculus/ef","kind":"node","lean":"Cslib.Modal.MuCalculus.EF","parent":56,"reading_title":{"format":"inline-parts-v1","parts":[["text","The \u03bc-calculus formula "],["math","\\(\\mathsf{EF}\\,p\\)"]],"source":"The \u03bc-calculus formula $\\mathsf{EF}\\,p$"},"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#ef","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For an atom $p$, $\\mathsf{EF}\\,p$ is the least fixed point formula $\\mu X.\\,(p \\vee \\Diamond X)$. It expresses that some state reachable from the current one satisfies $p$.","title":"The \u03bc-calculus formula $\\mathsf{EF}\\,p$"},{"id":"Logic/modal-logic/modal-calculus/ag","kind":"node","lean":"Cslib.Modal.MuCalculus.AG","parent":56,"reading_title":{"format":"inline-parts-v1","parts":[["text","The \u03bc-calculus formula "],["math","\\(\\mathsf{AG}\\,p\\)"]],"source":"The \u03bc-calculus formula $\\mathsf{AG}\\,p$"},"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#ag","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For an atom $p$, $\\mathsf{AG}\\,p$ is the greatest fixed point formula $\\nu X.\\,(p \\wedge \\Box X)$. It expresses that every state reachable from the current one satisfies $p$.","title":"The \u03bc-calculus formula $\\mathsf{AG}\\,p$"},{"id":"Logic/modal-logic/modal-calculus/ef-iff-reachable","kind":"node","lean":"Cslib.Modal.MuCalculus.EF_iff_reachable","parent":56,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathsf{EF}\\)"],["text"," is reachability"]],"source":"$\\mathsf{EF}$ is reachability"},"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#ef-iff-reachable","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For every Kripke model $M$, every environment $\\rho$ (an assignment of a predicate on worlds to each variable index), every atom $p$ and every world $w$, the \u03bc-calculus formula $\\mathsf{EF}\\,p$ holds at $w$ in $\\rho$ if and only if $p$ is true at some world reachable from $w$ in zero or more steps of the accessibility relation of $M$.","title":"$\\mathsf{EF}$ is reachability"},{"id":"Logic/modal-logic/modal-calculus/ag-iff-forall-reachable","kind":"node","lean":"Cslib.Modal.MuCalculus.AG_iff_forall_reachable","parent":56,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathsf{AG}\\)"],["text"," is universal reachability"]],"source":"$\\mathsf{AG}$ is universal reachability"},"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#ag-iff-forall-reachable","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For every Kripke model $M$, every environment $\\rho$ (an assignment of a predicate on worlds to each variable index), every atom $p$ and every world $w$, the \u03bc-calculus formula $\\mathsf{AG}\\,p$ holds at $w$ in $\\rho$ if and only if $p$ is true at every world reachable from $w$ in zero or more steps of the accessibility relation of $M$.","title":"$\\mathsf{AG}$ is universal reachability"},{"id":"Logic/modal-logic/modal-calculus/mu-mu-var-one-eq-bot","kind":"node","lean":"Cslib.Modal.MuCalculus.mu_mu_var_one_eq_bot","parent":56,"reading_title":{"format":"inline-parts-v1","parts":[["text","The formula "],["math","\\(\\mu X.\\, \\mu Y.\\, X\\)"],["text"," is false everywhere"]],"source":"The formula $\\mu X.\\, \\mu Y.\\, X$ is false everywhere"},"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-mu-var-one-eq-bot","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For every Kripke model $M$ and every environment $\\rho$, the formula $\\mu X.\\, \\mu Y.\\, X$ holds at no world of $M$ under $\\rho$. In de Bruijn form this formula is $\\mu\\, \\mu\\, x_1$: inside the inner binder, variable $1$ refers to the outer binder, not to the inner one.","title":"The formula $\\mu X.\\, \\mu Y.\\, X$ is false everywhere"},{"id":"Logic/modal-logic/modal-calculus/nu-nu-var-one-eq-top","kind":"node","lean":"Cslib.Modal.MuCalculus.nu_nu_var_one_eq_top","parent":56,"reading_title":{"format":"inline-parts-v1","parts":[["text","The formula "],["math","\\(\\nu X.\\, \\nu Y.\\, X\\)"],["text"," is true everywhere"]],"source":"The formula $\\nu X.\\, \\nu Y.\\, X$ is true everywhere"},"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#nu-nu-var-one-eq-top","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"For every Kripke model $M$ and every environment $\\rho$, the formula $\\nu X.\\, \\nu Y.\\, X$ holds at every world of $M$ under $\\rho$. In de Bruijn form this formula is $\\nu\\, \\nu\\, x_1$: inside the inner binder, variable $1$ refers to the outer binder. This is the dual of the fact that $\\mu X.\\, \\mu Y.\\, X$ is false everywhere.","title":"The formula $\\nu X.\\, \\nu Y.\\, X$ is true everywhere"},{"id":"Logic/modal-logic/modal-calculus/closed-at","kind":"node","lean":"Cslib.Modal.MuCalculus.MuFormula.ClosedAt","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#closed-at","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"A formula is closed at depth `d` when every free de Bruijn variable is `< d`: only `\u03bc` / `\u03bd` increase the depth; `\u25a1` / `\u25c7` do not bind variables and preserve it.","title":"\u03bc-calculus formula closed at a given binding depth"},{"id":"Logic/modal-logic/modal-calculus/closed","kind":"node","lean":"Cslib.Modal.MuCalculus.MuFormula.Closed","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#closed","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"A formula is closed when it is closed at depth `0`: every variable is bound.","title":"Closed modal \u03bc-calculus formula"},{"id":"Logic/modal-logic/modal-calculus/depth","kind":"node","lean":"Cslib.Modal.MuCalculus.MuFormula.depth","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#depth","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"The depth of a formula: the length of its longest root-to-leaf path, counting each connective once.","title":"Depth of a modal \u03bc-calculus formula"},{"id":"Logic/modal-logic/modal-calculus/sat-closed-env","kind":"node","lean":"Cslib.Modal.MuCalculus.sat_closed_env","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#sat-closed-env","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"A closed formula does not depend on the environment: environments agreeing below the depth give the same denotation.","title":"Denotation of a closed \u03bc-calculus formula is environment-independent"},{"id":"Logic/modal-logic/modal-calculus/mu-code","kind":"node","lean":"Cslib.Modal.MuCalculus.muCode","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-code","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"An injective encoding of \u03bc-calculus formulas over `\u2115` atoms as natural numbers, via the Cantor pairing function `Nat.pair`. This is a Nat code only, not a complexity input encoding: repeated pairing squares the code at every unary connective, so the binary length of the code of `box^d (atom 0)` doubles at each `box` and is exponential in `d`. Complexity statements use `muTokens` / `muDecodeList` instead.","title":"Cantor-pairing number code of a \u03bc-calculus formula"},{"id":"Logic/modal-logic/modal-calculus/mu-decode-fuel","kind":"node","lean":"Cslib.Modal.MuCalculus.muDecodeFuel","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-decode-fuel","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Fuelled decoding of a `muCode` number back to a formula. The fuel bounds the recursion depth; `muDecode` supplies fuel `c + 1`, which exceeds every formula's depth (see `depth_le_code`).","title":"Fuelled decoding of a \u03bc-calculus number code"},{"id":"Logic/modal-logic/modal-calculus/mu-decode","kind":"node","lean":"Cslib.Modal.MuCalculus.muDecode","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-decode","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Decode a `muCode` number back to the formula it encodes.","title":"Decoding a \u03bc-calculus number code to a formula"},{"id":"Logic/modal-logic/modal-calculus/depth-le-code","kind":"node","lean":"Cslib.Modal.MuCalculus.depth_le_code","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#depth-le-code","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"A formula's depth is at most its code: pairing dominates each component (`Nat.add_le_pair`).","title":"Depth of a \u03bc-calculus formula is at most its number code"},{"id":"Logic/modal-logic/modal-calculus/mu-decode-fuel-mu-code","kind":"node","lean":"Cslib.Modal.MuCalculus.muDecodeFuel_muCode","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-decode-fuel-mu-code","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Decoding inverts encoding, with enough fuel: a formula's own code carries more fuel than its depth needs.","title":"Fuelled decoding inverts the \u03bc-calculus number code"},{"id":"Logic/modal-logic/modal-calculus/mu-decode-mu-code","kind":"node","lean":"Cslib.Modal.MuCalculus.muDecode_muCode","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-decode-mu-code","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Decoding inverts encoding.","title":"Decoding inverts the \u03bc-calculus number code"},{"id":"Logic/modal-logic/modal-calculus/mu-tokens","kind":"node","lean":"Cslib.Modal.MuCalculus.muTokens","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-tokens","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"A polynomial-size token-list encoding of \u03bc-calculus formulas over `\u2115` atoms, used for the complexity statements in `WantedExt`. Each connective contributes one tag token (`Nat.pair` of a small tag with `0`), and each atom or variable contributes one token `Nat.pair tag value`, whose binary size is `O(log value)`. The canonical bitstring encoding of a `List \u2115` delimits each token, so the bitstring for a formula has\u2026","title":"Polynomial-size token-list encoding of \u03bc-calculus formulas"},{"id":"Logic/modal-logic/modal-calculus/mu-parse-fuel","kind":"node","lean":"Cslib.Modal.MuCalculus.muParseFuel","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-parse-fuel","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Fuelled parsing of a `muTokens` token list back to a formula, returning the formula and the unconsumed suffix. The fuel bounds the recursion depth; `muDecodeList` supplies fuel `l.length + 1`.","title":"Fuelled parser for modal \u03bc-calculus token lists"},{"id":"Logic/modal-logic/modal-calculus/mu-decode-list","kind":"node","lean":"Cslib.Modal.MuCalculus.muDecodeList","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-decode-list","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Decode a `muTokens` token list back to the formula it encodes, rejecting trailing tokens.","title":"Decoding a modal \u03bc-calculus token list to a formula"},{"id":"Logic/modal-logic/modal-calculus/mu-parse-fuel-mu-tokens","kind":"node","lean":"Cslib.Modal.MuCalculus.muParseFuel_muTokens","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-parse-fuel-mu-tokens","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Parsing inverts token encoding, with enough fuel.","title":"Fuelled parsing inverts modal \u03bc-calculus token encoding"},{"id":"Logic/modal-logic/modal-calculus/mu-decode-list-mu-tokens","kind":"node","lean":"Cslib.Modal.MuCalculus.muDecodeList_muTokens","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-decode-list-mu-tokens","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"Decoding inverts token encoding.","title":"Token-list decoding inverts modal \u03bc-calculus token encoding"},{"id":"Logic/modal-logic/modal-calculus/mu-code-injective","kind":"node","lean":"Cslib.Modal.MuCalculus.muCode_injective","parent":56,"result":true,"route":"../roadmap/Logic/modal-logic/modal-calculus/index.html#mu-code-injective","scope":"Logic/modal-logic/modal-calculus","status":"fully_proved","summary":"The encoding is injective.","title":"Injectivity of the \u03bc-calculus number code"},{"data_url":"scopes/Logic/modal-logic/model-checking-game-for-the-modal-calculus.json","id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","title":"Model-checking game for the modal \u03bc-calculus"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/game-position","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.GamePosition","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#game-position","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"A position of the \u03bc-calculus model-checking game: the formula, the world, and the stack of enclosing fixpoint binders, innermost first.","title":"Position of the \u03bc-calculus model-checking game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/player","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.Player","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#player","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The player to move: `true` is Verifier, `false` is Refuter.","title":"Player of the \u03bc-calculus model-checking game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/move","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.Move","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#move","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Fix a Kripke model $M$ with accessibility relation $R$. A game position $(\\varphi, w, s)$ consists of a \u03bc-calculus formula $\\varphi$ (with de Bruijn variables), a world $w$ and a stack $s$ of fixpoint binders, innermost first. The moves are exactly the following. From $(\\varphi \\wedge \\psi, w, s)$ and from $(\\varphi \\vee \\psi, w, s)$ to $(\\varphi, w, s)$ and to $(\\psi, w, s)$. From $(\\Box \\varphi, w, s)$ and from\u2026","title":"Moves of the \u03bc-calculus model-checking game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/owner","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.Owner","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#owner","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The owner of a game position $(\\varphi, w, s)$ is Verifier when $\\varphi$ is a disjunction, a diamond formula or a greatest fixed point, and Refuter for every other formula.","title":"Owner of a game position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/prio","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.prio","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#prio","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Only fixpoint-binder positions carry a priority. For a formula $\\varphi$, a world $w$ and a binder stack $s$ of length $k$, the position $(\\mu X.\\,\\varphi, w, s)$, whose formula is the least fixpoint with body $\\varphi$, has priority $2k+1$, the position $(\\nu X.\\,\\varphi, w, s)$, whose formula is the greatest fixpoint with body $\\varphi$, has priority $2k+2$, and every other position has no priority. Priorities\u2026","title":"Priority of a game position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/no-moves","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.NoMoves","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#no-moves","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"In a Kripke model $M$, a game position $p$ has no moves when there is no position $q$ such that going from $p$ to $q$ is a move of the model-checking game in $M$.","title":"Positions with no moves"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/terminal-win","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.TerminalWin","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#terminal-win","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\mathit{env}$ an environment assigning a predicate on worlds to every index. At a game position $(\\varphi, w, st)$, where $st$ is the stack of enclosing fixpoint binders and $|st|$ its length, Verifier wins exactly when $\\varphi$ is an atom $a$ true at $w$, a negated atom $\\neg a$ with $a$ false at $w$, $\\top$, a box $\\Box\\psi$, or a variable with de Bruijn index $n \\ge |st|$ such\u2026","title":"Winner at a terminal position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/parity-win","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ParityWin","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#parity-win","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For an infinite sequence of game positions $p_0, p_1, \\dots$, Verifier wins by the parity condition when there is an even priority $e$ such that infinitely many positions $p_n$ have priority $e$, and every priority $e' < e$ is the priority of only finitely many positions $p_n$. Equivalently, some priority occurs infinitely often and the least such priority is even; in particular, if no priority occurs infinitely\u2026","title":"Parity winning condition"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/strategy","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.Strategy","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#strategy","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"A Verifier strategy for the model-checking game on a Kripke model $M$ is a map $\\sigma$ from game positions to game positions such that, at every Verifier-owned position $p$ from which some move exists, there is a move from $p$ to $\\sigma(p)$.","title":"Verifier strategy"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/play-from","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.PlayFrom","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#play-from","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Fix a Kripke model $M$ and a Verifier strategy $\\sigma$. The finite plays from a position $p$ that follow $\\sigma$ are the lists of positions defined inductively: the one-element list $[p]$ is such a play; and if the game has a move from $p$ to $q$, where $q$ is the position chosen by $\\sigma$ at $p$ whenever Verifier owns $p$, then $p$ followed by any such play from $q$ is such a play from $p$.","title":"Finite plays consistent with a strategy"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/play-last","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.playLast","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#play-last","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The last position of a finite list of game positions is its final entry when the list is nonempty; the empty list has no last position.","title":"Last position of a play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/strategy-wins","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.StrategyWins","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#strategy-wins","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment assigning a predicate on worlds to every index, $\\sigma$ a Verifier strategy for the model-checking game on $M$ and $p$ a game position. A play is consistent with $\\sigma$ when each step is a move and, at every Verifier-owned position, goes to the position chosen by $\\sigma$. The strategy $\\sigma$ wins from $p$ in $\\mathit{env}$ when every finite play from\u2026","title":"Winning strategy from a position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/verifier-wins","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.VerifierWins","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#verifier-wins","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For a Kripke model $M$, an environment $\\mathit{env}$ and a game position $p$, Verifier wins from $p$ when some Verifier strategy $\\sigma$ in the model-checking game on $M$ wins from $p$ in environment $\\mathit{env}$: every complete finite $\\sigma$-consistent play from $p$ ends at a terminal winning position, and every infinite $\\sigma$-consistent play from $p$ satisfies the parity condition.","title":"Verifier has a winning strategy"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/closed-at","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ClosedAt","parent":100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formulas closed at depth "],["math","\\(d\\)"]],"source":"Formulas closed at depth $d$"},"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#closed-at","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Closedness of a \u03bc-calculus formula $\\varphi$ (with de Bruijn variables) at a depth $d \\in \\mathbb{N}$ is defined by recursion on $\\varphi$: atoms, negated atoms, $\\bot$ and $\\top$ are closed at every depth; the variable with index $n$ is closed at depth $d$ exactly when $n < d$; $\\varphi \\wedge \\psi$ and $\\varphi \\vee \\psi$ are closed at depth $d$ when both $\\varphi$ and $\\psi$ are; $\\Box\\varphi$ and\u2026","title":"Formulas closed at depth $d$"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/closed","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.Closed","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#closed","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"A \u03bc-calculus formula $\\varphi$ is closed when it is closed at depth $0$, that is, when it has no free variables.","title":"Closed formulas"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/mono","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ClosedAt.mono","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#mono","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $\\varphi$ be a \u03bc-calculus formula and let $d \\le d'$ be natural numbers. If $\\varphi$ is closed at depth $d$ (every free variable of $\\varphi$ is less than $d$), then $\\varphi$ is closed at depth $d'$: more enclosing binders can only bind more variables.","title":"Closedness is monotone in the depth"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/sat-eq-of-closed-at","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.sat_eq_of_closedAt","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#sat-eq-of-closed-at","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ a \u03bc-calculus formula and $d$ a natural number such that $\\varphi$ is closed at depth $d$: every occurrence of a variable with de Bruijn index $n$ lying under $k$ fixpoint binders of $\\varphi$ satisfies $n < d + k$. If two environments $\\mathit{env}$ and $\\mathit{env}'$, each assigning a predicate on worlds to every index, agree at every index $n < d$, then for every world $w$,\u2026","title":"Satisfaction depends only on the environment below the closure depth"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/sat-closed","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.sat_closed","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#sat-closed","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a closed \u03bc-calculus formula, that is, one with no free variables. For any two environments $\\mathrm{env}$ and $\\mathrm{env}'$, each assigning a set of worlds to every variable index, and every world $w$, $\\varphi$ is satisfied at $w$ in $M$ under $\\mathrm{env}$ if and only if it is satisfied at $w$ in $M$ under $\\mathrm{env}'$.","title":"Closed formulas are environment-independent"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/counterexample-model","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.counterexampleModel","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#counterexample-model","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The Kripke model with a single world, which is related to itself, and a single atom, which is false at that world.","title":"The one-world reflexive Kripke model"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/counterexample-formula","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.counterexampleFormula","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#counterexample-formula","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The \u03bc-calculus formula $\\mu X.\\, \\nu Y.\\, X$ over a one-element type of atoms: a least fixpoint whose body is a greatest fixpoint whose body is the variable $X$ bound by the outer least fixpoint, so the variable $Y$ bound by the inner greatest fixpoint never occurs.","title":"A least fixpoint over a vacuous greatest fixpoint"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-pos0","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.cePos0","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-pos0","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The game position consisting of the formula $\\mu X.\\,\\nu Y.\\,X$, the unique world of the one-world model, and the empty binder stack.","title":"Starting position of the regression play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-pos1","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.cePos1","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-pos1","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The game position consisting of the formula $\\nu Y.\\,X$, the unique world of the one-world model, and the binder stack $[\\mu X.\\,\\nu Y.\\,X]$. Here $X$ is the de Bruijn variable $1$, which refers to the binder $\\mu X$ on the stack.","title":"Second position of the regression play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-pos2","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.cePos2","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-pos2","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The game position consisting of the variable $X$, the unique world of the one-world model, and the binder stack $[\\nu Y.\\,X,\\ \\mu X.\\,\\nu Y.\\,X]$, listed innermost first. Here $X$ is the de Bruijn variable $1$, which refers to the outer binder $\\mu X$.","title":"Third position of the regression play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-next","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ceNext","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-next","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $\\Phi = \\mu X.\\, \\nu Y.\\, X$ be the regression formula over the one-element type of atoms, and write game positions over the one-element type of worlds as (formula, world, stack of enclosing binders, innermost first). For every world $w$, the step map sends $(\\Phi, w, [\\,])$ to $(\\nu Y.\\, X, w, [\\Phi])$, sends $(\\nu Y.\\, X, w, [\\Phi])$ to $(X, w, [\\nu Y.\\, X, \\Phi])$, and sends $(X, w, [\\nu Y.\\, X, \\Phi])$ to\u2026","title":"Step map of the regression play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-play","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.cePlay","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-play","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Write $\\ast$ for the unique world, and consider the game positions (formula, world, binder stack listed innermost first) $P_0 = (\\mu X.\\,\\nu Y.\\,X,\\ \\ast,\\ [\\,])$, $P_1 = (\\nu Y.\\,X,\\ \\ast,\\ [\\mu X.\\,\\nu Y.\\,X])$ and $P_2 = (X,\\ \\ast,\\ [\\nu Y.\\,X,\\ \\mu X.\\,\\nu Y.\\,X])$. Let $s$ be the step map sending $P_0$ to $P_1$, $P_1$ to $P_2$ and $P_2$ to $P_0$, and fixing every other position. The regression play is the\u2026","title":"The regression play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-next-ce-pos0","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ceNext_cePos0","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-next-ce-pos0","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $\\Phi = \\mu X.\\, \\nu Y.\\, X$ be the regression formula, $\\ast$ the unique world, and write game positions as (formula, world, stack of enclosing binders, innermost first). The step map of the regression play sends the first cycle position $(\\Phi, \\ast, [\\,])$ to the second cycle position $(\\nu Y.\\, X, \\ast, [\\Phi])$.","title":"Regression cycle, first step"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-next-ce-pos1","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ceNext_cePos1","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-next-ce-pos1","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $\\Phi = \\mu X.\\, \\nu Y.\\, X$ be the regression formula, $\\ast$ the unique world, and write game positions as (formula, world, stack of enclosing binders, innermost first). The step map of the regression play sends the second cycle position $(\\nu Y.\\, X, \\ast, [\\Phi])$ to the third cycle position $(X, \\ast, [\\nu Y.\\, X, \\Phi])$, where $X$ is the variable bound by the outer $\\mu$.","title":"Regression cycle, second step"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-next-ce-pos2","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ceNext_cePos2","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-next-ce-pos2","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $\\Phi = \\mu X.\\, \\nu Y.\\, X$ be the regression formula, $\\ast$ the unique world, and write game positions as (formula, world, stack of enclosing binders, innermost first). The step map of the regression play sends the third cycle position $(X, \\ast, [\\nu Y.\\, X, \\Phi])$, where $X$ is the variable bound by the outer $\\mu$, back to the first cycle position $(\\Phi, \\ast, [\\,])$.","title":"Regression cycle, third step"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-play-mem","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.cePlay_mem","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-play-mem","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the position at step $n$ of the regression play is the first, the second or the third cycle position.","title":"The regression play stays on the cycle"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-play-period","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.cePlay_period","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-play-period","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Number the positions of the regression play from $0$. For every natural number $n$, its position number $n + 3$ equals its position number $n$.","title":"The regression play has period three"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-play-mul-three","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.cePlay_mul_three","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-play-mul-three","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Number the positions of the regression play of $\\mu X.\\,\\nu Y.\\,X$ from $0$. For every natural number $k$, its position number $3k$ is its starting position: the formula $\\mu X.\\,\\nu Y.\\,X$ at the unique world, with the empty binder stack.","title":"The regression play returns to its start every three steps"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-play-moves","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.cePlay_moves","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-play-moves","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, in the model-checking game on the one-world regression model there is a move from the position at step $n$ of the regression play to the position at step $n + 1$.","title":"The regression play is a play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-prio0","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ce_prio0","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-prio0","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The starting position of the regression play, consisting of the formula $\\mu X.\\,\\nu Y.\\,X$, the unique world and the empty binder stack, has priority $1$: a $\\mu$-binder position with $k$ binders on the stack has priority $2k + 1$, and here $k = 0$.","title":"Priority of the outer binder in the regression play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ce-prio1","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.ce_prio1","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ce-prio1","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The second position of the regression play, consisting of the formula $\\nu Y.\\,X$, the unique world and the binder stack $[\\mu X.\\,\\nu Y.\\,X]$, has priority $4$: a $\\nu$-binder position with $k$ binders on the stack has priority $2k + 2$, and here $k = 1$.","title":"Priority of the inner binder in the regression play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/counterexample-parity-win-false","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.counterexample_parityWin_false","parent":100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Verifier loses the infinite play of "],["math","\\(\\mu X.\\, \\nu Y.\\, X\\)"]],"source":"Verifier loses the infinite play of $\\mu X.\\, \\nu Y.\\, X$"},"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#counterexample-parity-win-false","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Consider the infinite sequence of positions of the model-checking game for $\\mu X.\\, \\nu Y.\\, X$ (with one-element types of worlds and atoms) that starts at this formula with empty binder stack and repeats three steps: unfold $\\mu X$, unfold $\\nu Y$, and follow the variable $X$ back to the formula with empty stack. This play does not satisfy the parity condition: it is not the case that the least binder priority\u2026","title":"Verifier loses the infinite play of $\\mu X.\\, \\nu Y.\\, X$"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/counterexample-sat-false","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.counterexample_sat_false","parent":100,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mu X.\\, \\nu Y.\\, X\\)"],["text"," is false in the one-world model"]],"source":"$\\mu X.\\, \\nu Y.\\, X$ is false in the one-world model"},"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#counterexample-sat-false","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"In the Kripke model with a single world, which is related to itself, and a single atom, which is false there, under the environment that assigns the empty set to every variable, the formula $\\mu X.\\, \\nu Y.\\, X$ is not satisfied at the unique world.","title":"$\\mu X.\\, \\nu Y.\\, X$ is false in the one-world model"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/fix-free-formula","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FixFreeFormula","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#fix-free-formula","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The fixpoint-free formulas over a type of atoms are built from atoms $p$, negated atoms $\\neg p$, $\\bot$ and $\\top$ by conjunction $\\varphi \\land \\psi$, disjunction $\\varphi \\lor \\psi$, necessity $\\Box\\varphi$ and possibility $\\Diamond\\varphi$.","title":"Fixpoint-free formulas"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/to-mu","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FixFreeFormula.toMu","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#to-mu","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The map from fixpoint-free formulas to \u03bc-calculus formulas defined by recursion: atoms, negated atoms, $\\bot$ and $\\top$ go to the same atom, negated atom, $\\bot$ and $\\top$, and $\\varphi \\wedge \\psi$, $\\varphi \\vee \\psi$, $\\Box \\varphi$ and $\\Diamond \\varphi$ go to the conjunction, disjunction, box and diamond of the images of $\\varphi$ and $\\psi$.","title":"Embedding of the fixpoint-free fragment into the \u03bc-calculus"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.Wins","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model with accessibility relation $R$. For a fixpoint-free formula $\\varphi$ and a world $w$, \"$\\varphi$ is winning at $w$\" is defined by recursion on $\\varphi$: an atom $a$ is winning at $w$ iff $a$ is true at $w$ in $M$; a negated atom $\\lnot a$ iff $a$ is not true at $w$; $\\bot$ is never winning and $\\top$ is always winning; $\\varphi \\land \\psi$ is winning at $w$ iff both $\\varphi$ and $\\psi$\u2026","title":"Backward-induction winning region of the fixpoint-free fragment"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-iff-sat","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.wins_iff_sat","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-iff-sat","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every Kripke model $M$, environment $\\mathit{env}$, fixpoint-free formula $\\varphi$ and world $w$, $w$ lies in the backward-induction winning region of $\\varphi$ if and only if $w$ satisfies the \u03bc-calculus embedding of $\\varphi$ in $M$ under $\\mathit{env}$.","title":"Backward-induction winning equals satisfaction on the fixpoint-free fragment"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/size","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.MuFormula.size","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#size","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The size $|\\varphi|$ of a \u03bc-calculus formula $\\varphi$ is the number of nodes of its syntax tree: an atom, a negated atom, a variable, $\\bot$ and $\\top$ have size $1$; $|\\varphi \\wedge \\psi| = |\\varphi \\vee \\psi| = |\\varphi| + |\\psi| + 1$; and $|\\Box\\varphi| = |\\Diamond\\varphi| = |\\mu\\varphi| = |\\nu\\varphi| = |\\varphi| + 1$.","title":"Size of a \u03bc-calculus formula"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/to-mu-inv","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.toMuInv","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#to-mu-inv","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The partial map from \u03bc-calculus formulas to fixpoint-free formulas defined by recursion: an atom, a negated atom, $\\bot$ and $\\top$ are sent to the same formula; a conjunction or disjunction is sent to the conjunction or disjunction of the values at its two parts, and $\\Box\\psi$, $\\Diamond\\psi$ to the box or diamond of the value at $\\psi$, whenever those values are defined; the map is undefined at variables and at\u2026","title":"Partial inverse of the fixpoint-free embedding"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/to-mu-inv-to-mu","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.toMuInv_toMu","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#to-mu-inv-to-mu","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every fixpoint-free formula $\\varphi$, the partial inverse of the embedding of fixpoint-free formulas into \u03bc-calculus formulas is defined at the image of $\\varphi$ under the embedding, with value $\\varphi$.","title":"The partial inverse recovers fixpoint-free formulas"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/to-mu-to-mu-inv","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.toMu_toMuInv","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#to-mu-to-mu-inv","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every \u03bc-calculus formula $\\psi$ and every fixpoint-free formula $\\varphi$: if the partial inverse of the embedding of fixpoint-free formulas into \u03bc-calculus formulas is defined at $\\psi$ with value $\\varphi$, then the embedding sends $\\varphi$ to $\\psi$.","title":"The embedding inverts the partial inverse"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/to-mu-injective","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FixFreeFormula.toMu_injective","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#to-mu-injective","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The embedding of fixpoint-free formulas into \u03bc-calculus formulas is injective: distinct fixpoint-free formulas have distinct images.","title":"The fixpoint-free embedding is injective"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-move","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.winsMove","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-move","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, and identify each fixpoint-free formula with its embedding into the \u03bc-calculus. For a \u03bc-calculus formula $\\psi$, a world $w$ and a stack $s$ of fixpoint binders, the Wins-guided move at the game position $(\\psi, w, s)$ is the following position. If $s$ is empty and $\\psi$ is a fixpoint-free disjunction $\\psi_1 \\lor \\psi_2$, it is $(\\psi_1, w, [])$ when $\\psi_1$ is winning at $w$ (in the\u2026","title":"Wins-guided move"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-move-or","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.winsMove_or","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-move-or","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi_1$ and $\\varphi_2$ fixpoint-free formulas and $w$ a world in the backward-induction winning region of $\\varphi_1 \\vee \\varphi_2$. Then for some $i \\in \\{1, 2\\}$, $w$ lies in the backward-induction winning region of $\\varphi_i$ and the Wins-guided move at the position made of the embedding of $\\varphi_1 \\vee \\varphi_2$, the world $w$ and the empty binder stack is the position made\u2026","title":"The Wins-guided move at a disjunction"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-move-dia","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.winsMove_dia","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-move-dia","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ a fixpoint-free formula and $w$ a world in the backward-induction winning region of $\\Diamond \\varphi$. Then there is a successor $v$ of $w$ in $M$ lying in the backward-induction winning region of $\\varphi$ such that the Wins-guided move at the position made of the embedding of $\\Diamond \\varphi$, the world $w$ and the empty binder stack is the position made of the embedding of\u2026","title":"The Wins-guided move at a diamond"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-move-is-move","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.winsMove_isMove","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-move-is-move","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For a Kripke model $M$ and a game position $p$ owned by Verifier from which some move exists, the Wins-guided move at $p$ is a legal move of the model-checking game on $M$ from $p$.","title":"The Wins-guided move is legal"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-strategy","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.winsStrategy","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-strategy","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For a Kripke model $M$, the Wins-guided strategy is the Verifier strategy of the model-checking game on $M$ that chooses, at each game position $(\\psi, w, s)$, the Wins-guided move at $(\\psi, w, s)$. At every Verifier-owned position that has a move, this choice is a legal move.","title":"Wins-guided strategy"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/ne-nil","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.PlayFrom.ne_nil","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#ne-nil","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\sigma$ a Verifier strategy and $p$ a game position. Every finite play from $p$ consistent with $\\sigma$ is a nonempty list of positions.","title":"Finite plays are nonempty"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/play-last-cons","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.playLast_cons","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#play-last-cons","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For a game position $p$ and a nonempty list $ps$ of game positions, the list obtained by prepending $p$ to $ps$ has the same last position as $ps$.","title":"Prepending to a nonempty play keeps the last position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/move-of-to-mu","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.move_of_toMu","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#move-of-to-mu","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ a fixpoint-free formula and $w$ a world. Every move of the model-checking game in $M$ from the position $(\\varphi, w, [\\,])$, with $\\varphi$ embedded in the \u03bc-calculus and the empty stack, leads to a position $(\\varphi', w', [\\,])$ for some fixpoint-free formula $\\varphi'$ and world $w'$, and the size of $\\varphi'$ is strictly smaller than the size of $\\varphi$ (sizes of the\u2026","title":"Fragment moves decrease the formula size"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/no-infinite-play-of-fix-free","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.no_infinite_play_of_fixFree","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#no-infinite-play-of-fix-free","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ a fixpoint-free formula and $w$ a world. There is no sequence of game positions $p_0, p_1, p_2, \\dots$ with $p_0 = (\\varphi, w, [\\,])$ (the embedded formula with the empty stack) such that each step from $p_n$ to $p_{n+1}$ is a move of the model-checking game in $M$. The reason is that every move strictly decreases the formula size.","title":"No infinite play from a fragment position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/strategy-wins-of-move","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.strategyWins_of_move","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#strategy-wins-of-move","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment and $\\sigma$ a Verifier strategy for the model-checking game on $M$. If $\\sigma$ wins from a position $p$ in $\\mathit{env}$, and $q$ is a position reachable from $p$ by one move that is the position chosen by $\\sigma$ at $p$ whenever Verifier owns $p$, then $\\sigma$ also wins from $q$ in $\\mathit{env}$.","title":"Winning strategies stay winning along consistent moves"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-strategy-wins-finite-aux","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.winsStrategy_wins_finite_aux","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-strategy-wins-finite-aux","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The Wins-guided strategy wins every complete finite play from a winning fragment position.","title":"Wins-guided strategy wins finite plays (auxiliary induction)"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-strategy-wins-finite","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.winsStrategy_wins_finite","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-strategy-wins-finite","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The Wins-guided strategy wins every complete finite play from a winning fragment position.","title":"Wins-guided strategy wins every complete finite play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-implies-verifier-wins","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.wins_implies_verifierWins","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-implies-verifier-wins","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment assigning a set of worlds to each variable index, $\\varphi$ a fixpoint-free formula and $w$ a world in the backward-induction winning region of $\\varphi$. Then Verifier has a winning strategy, in environment $\\mathit{env}$, in the model-checking game on $M$ from the position made of the \u03bc-calculus embedding of $\\varphi$, the world $w$ and the empty binder\u2026","title":"Game-vs-satisfaction, strategy direction"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/or-cases","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.Move.or_cases","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#or-cases","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ and $\\psi$ \u03bc-calculus formulas, $w$ a world and $s$ a stack of fixpoint binders. If going from the game position $(\\varphi \\vee \\psi, w, s)$ to a position $q$ is a move of the model-checking game in $M$, then $q$ is $(\\varphi, w, s)$ or $(\\psi, w, s)$: the world and the stack are unchanged.","title":"Inversion for disjunction moves"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/dia-cases","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.Move.dia_cases","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#dia-cases","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"In the model-checking game on a Kripke model $M$, every move from the position with formula $\\Diamond\\varphi$, world $w$ and binder stack $s$ goes to the position with formula $\\varphi$, world $v$ and the same stack $s$, for some world $v$ accessible from $w$ in $M$.","title":"Moves from a diamond position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/verifier-wins-implies-wins","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.verifierWins_implies_wins","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#verifier-wins-implies-wins","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment assigning a set of worlds to each variable index, $\\varphi$ a fixpoint-free formula and $w$ a world. If Verifier has a winning strategy, in environment $\\mathit{env}$, in the model-checking game on $M$ from the position made of the \u03bc-calculus embedding of $\\varphi$, the world $w$ and the empty binder stack, then $w$ lies in the backward-induction winning\u2026","title":"Game-vs-satisfaction, converse direction"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/wins-iff-verifier-wins","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.wins_iff_verifierWins","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#wins-iff-verifier-wins","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every Kripke model $M$, environment $\\mathit{env}$, fixpoint-free formula $\\varphi$ and world $w$, $w$ lies in the backward-induction winning region of $\\varphi$ if and only if Verifier has a winning strategy, in environment $\\mathit{env}$, in the model-checking game on $M$ from the position made of the \u03bc-calculus embedding of $\\varphi$, the world $w$ and the empty binder stack.","title":"Backward-induction winning equals game winning on the fixpoint-free fragment"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-pos","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FragPos","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-pos","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"A position of the totalized fragment game, over a type of atoms and a type of worlds, is either an embedded position $(\\varphi, w)$ of a fixpoint-free formula $\\varphi$ and a world $w$, standing for the model-checking game position $(\\varphi, w, [])$ with $\\varphi$ embedded as a \u03bc-calculus formula and the empty fixpoint stack, or one of two absorbing states: the winning state, where Verifier has won, and the losing\u2026","title":"Positions of the totalized fragment game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-of-game-pos","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragOfGamePos","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-of-game-pos","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"A game position with empty binder stack whose formula is the \u03bc-calculus reading of a fixpoint-free formula $\\varphi$, at world $w$, is sent to the embedded position $(\\varphi, w)$. Every other game position (one whose formula contains a fixpoint operator or a variable, or one with a nonempty stack) is sent to the losing state.","title":"Projection of game positions onto the fragment arena"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-of-game-pos-to-mu","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragOfGamePos_toMu","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-of-game-pos-to-mu","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every fixpoint-free formula $\\varphi$ and world $w$, projecting the game position with formula $\\varphi$ (read as a \u03bc-calculus formula), world $w$ and empty binder stack onto the fragment arena gives the embedded position $(\\varphi, w)$.","title":"Projection of an embedded position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-move","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FragMove","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-move","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $e$ an environment, and read each fixpoint-free formula $\\varphi$ as a \u03bc-calculus formula. The totalized fragment game of $M$ and $e$ has these moves: from an embedded position $(\\varphi, w)$ to $(\\varphi', w')$ whenever the model-checking game of $M$ has a move from $(\\varphi, w)$ to $(\\varphi', w')$ with empty binder stacks; from $(\\varphi, w)$ to the winning state when the game\u2026","title":"Moves of the totalized fragment game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-owner","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragOwner","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-owner","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"In the totalized fragment game, an embedded position $(\\varphi, w)$ of a fixpoint-free formula $\\varphi$ at a world $w$ is owned by the even player if Verifier owns the model-checking game position $(\\varphi, w, [])$ (with $\\varphi$ embedded as a \u03bc-calculus formula and the empty fixpoint stack), and by the odd player otherwise, that is, when Refuter owns it. The absorbing winning and losing states are both owned by\u2026","title":"Owners in the totalized fragment game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-priority","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragPriority","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-priority","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"In the totalized fragment game, every embedded position $(\\varphi, w)$ of a fixpoint-free formula $\\varphi$ at a world $w$ has priority $0$, the absorbing winning state has priority $0$, and the absorbing losing state has priority $1$.","title":"Priorities in the totalized fragment game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/fragment-game","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragmentGame","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#fragment-game","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\rho$ an environment, and write $(\\varphi, w, [])$ for the game position with formula $\\varphi$, world $w$ and empty binder stack. The fragment game of $M$ and $\\rho$ is the parity game on the totalized fragment positions defined as follows. An embedded position $(\\varphi, w)$ moves to $(\\varphi', w')$ when $(\\varphi, w, [])$ has a move to $(\\varphi', w', [])$ in the model-checking\u2026","title":"Fragment parity game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/win-eq","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FragMove.win_eq","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#win-eq","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"In the totalized fragment game on a Kripke model $M$ with environment $\\mathit{env}$, every move from the absorbing Verifier-wins state $\\mathrm{win}$ leads back to it: if $\\mathrm{win}$ moves to $v$, then $v = \\mathrm{win}$.","title":"The absorbing win state"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/lose-eq","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FragMove.lose_eq","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#lose-eq","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"In the fragment game of a Kripke model $M$ and an environment $\\rho$, every move from the losing state goes to the losing state.","title":"The losing state is absorbing"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/pos-move-iff","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FragMove.pos_move_iff","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#pos-move-iff","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathrm{env}$ an environment, $\\varphi, \\varphi'$ fixpoint-free formulas and $w, w'$ worlds. If the totalized fragment game has a move from the embedded position $(\\varphi, w)$ to the embedded position $(\\varphi', w')$, then the model-checking game has a move from $(\\varphi, w, [])$ to $(\\varphi', w', [])$, where the fixpoint-free formulas are viewed as \u03bc-calculus formulas and $[]$ is\u2026","title":"Fragment moves between embedded positions are game moves"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/pos-to-win","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FragMove.pos_toWin","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#pos-to-win","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathrm{env}$ an environment, $\\varphi$ a fixpoint-free formula and $w$ a world. If the totalized fragment game has a move from the embedded position $(\\varphi, w)$ to the absorbing winning state, then the game position $(\\varphi, w, [])$, with $\\varphi$ viewed as a \u03bc-calculus formula and an empty binder stack, has no moves, and Verifier wins there as a terminal position in the\u2026","title":"Moves to the winning state come from terminal winning positions"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/pos-to-lose","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.FragMove.pos_toLose","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#pos-to-lose","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathrm{env}$ an environment, $\\varphi$ a fixpoint-free formula and $w$ a world. If the totalized fragment game has a move from the embedded position $(\\varphi, w)$ to the absorbing losing state, then the game position $(\\varphi, w, [])$, with $\\varphi$ viewed as a \u03bc-calculus formula and an empty binder stack, has no moves, and Verifier does not win there as a terminal position in the\u2026","title":"Moves to the losing state come from terminal losing positions"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-owner-pos-eq-even","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragOwner_pos_eq_even","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-owner-pos-eq-even","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every fixpoint-free formula $\\varphi$ and world $w$, the even player owns the embedded position $(\\varphi, w)$ of the totalized fragment game if and only if Verifier owns the model-checking game position $(\\varphi, w, [])$, with $\\varphi$ embedded as a \u03bc-calculus formula and the empty fixpoint stack.","title":"Owner of an embedded position in the totalized fragment game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/canonical-next","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.canonicalNext","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#canonical-next","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment and $\\sigma$ a Verifier strategy for the model-checking game on $M$. Write $\\langle \\psi, v \\rangle$ for the game position with the embedding of a fixpoint-free formula $\\psi$, the world $v$ and the empty binder stack. The canonical next move is the map on positions of the fragment arena defined as follows. The absorbing states $\\mathrm{win}$ and\u2026","title":"Canonical next move induced by a bespoke strategy"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/canonical-next-pos-of-owner","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.canonicalNext_pos_of_owner","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#canonical-next-pos-of-owner","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment and $\\sigma$ a Verifier strategy for the model-checking game on $M$. Write $\\langle \\psi, v \\rangle$ for the game position with the embedding of a fixpoint-free formula $\\psi$, the world $v$ and the empty binder stack. Let $\\varphi$ be a fixpoint-free formula and $w$ a world such that Verifier owns $\\langle \\varphi, w \\rangle$ and $\\langle \\varphi, w \\rangle$\u2026","title":"The canonical move follows the bespoke strategy at Verifier positions with a move"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-strategy","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragStrategy","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-strategy","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"The canonical positional strategy induced by a bespoke strategy.","title":"Positional strategy induced by a bespoke strategy"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-to-pos","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragToPos","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-to-pos","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Given a default game position $p_0$, an embedded fragment position $(\\varphi, w)$ is sent to the game position with formula $\\varphi$, world $w$ and empty binder stack, and the absorbing winning and losing states are both sent to $p_0$.","title":"Game position of a fragment position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-segment","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragSegment","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-segment","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $p_0$ be a game position and let $\\pi = \\pi_0 \\pi_1 \\pi_2 \\dots$ be an infinite sequence of positions of the totalized fragment game. For natural numbers $s$ and $k$, the segment of $\\pi$ from $s$ of length $k + 1$ is the list of game positions obtained from $\\pi_s, \\pi_{s+1}, \\dots, \\pi_{s+k}$, in this order, by sending each embedded position $(\\varphi, w)$ to the game position with formula $\\varphi$, world\u2026","title":"Segment of a fragment play as game positions"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-segment-cons","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragSegment_cons","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-segment-cons","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $p_0$ be a model-checking game position and $\\pi$ an infinite sequence of positions of the totalized fragment game. For each $i$, let $\\hat\\pi_i$ be $p_0$ if $\\pi_i$ is an absorbing state; if $\\pi_i$ is the embedded position $(\\varphi, w)$ of a fixpoint-free formula $\\varphi$ and a world $w$, let $\\hat\\pi_i$ be the game position $(\\varphi, w, [])$, with $\\varphi$ embedded as a \u03bc-calculus formula and the empty\u2026","title":"Splitting off the head of a segment"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/canonical-play-from-segment","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.canonical_playFrom_segment","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#canonical-play-from-segment","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment and $\\sigma$ a Verifier strategy for the model-checking game on $M$. Write $\\langle \\psi, v \\rangle$ for the game position with the embedding of a fixpoint-free formula $\\psi$, the world $v$ and the empty binder stack. Let $\\pi$ be a play of the fragment game of $M$ and $\\mathit{env}$ that conforms to the canonical positional strategy induced by $\\sigma$, and\u2026","title":"Segments of canonical plays through fragment positions are bespoke plays"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-play-eq-win-of-le","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragPlay_eq_win_of_le","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-play-eq-win-of-le","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\mathit{env}$ an environment assigning a set of worlds to each free-variable index. If $\\pi$ is a play of the totalized fragment game of $M$ and $\\mathit{env}$ and $\\pi_k$ is the absorbing winning state, then $\\pi_n$ is the winning state for every $n \\ge k$.","title":"Plays of the totalized fragment game stay at the winning state"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-play-eq-lose-of-le","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragPlay_eq_lose_of_le","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-play-eq-lose-of-le","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\mathit{env}$ an environment assigning a set of worlds to each free-variable index. If $\\pi$ is a play of the totalized fragment game of $M$ and $\\mathit{env}$ and $\\pi_k$ is the absorbing losing state, then $\\pi_n$ is the losing state for every $n \\ge k$.","title":"Plays of the totalized fragment game stay at the losing state"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/won-by-even-of-eventually-win","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.wonBy_even_of_eventually_win","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#won-by-even-of-eventually-win","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\mathit{env}$ an environment for free variables. If an infinite sequence $\\pi_0 \\pi_1 \\pi_2 \\dots$ of positions of the totalized fragment game of $M$ and $\\mathit{env}$ has a $k_0$ such that $\\pi_n$ is the absorbing winning state for all $n \\ge k_0$, then $\\pi$ is won by the even player in that game.","title":"Sequences that stay at the win state are won by the even player"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/won-by-odd-of-eventually-lose","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.wonBy_odd_of_eventually_lose","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#won-by-odd-of-eventually-lose","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\mathit{env}$ an environment for free variables. If an infinite sequence $\\pi_0 \\pi_1 \\pi_2 \\dots$ of positions of the totalized fragment game of $M$ and $\\mathit{env}$ has a $k_0$ such that $\\pi_n$ is the absorbing losing state for all $n \\ge k_0$, then $\\pi$ is won by the odd player in that game.","title":"Sequences that stay at the lose state are won by the odd player"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-segment-ne-nil","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragSegment_ne_nil","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-segment-ne-nil","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $p_0$ be a model-checking game position and $\\pi$ an infinite sequence of positions of the totalized fragment game. For all natural numbers $s$ and $k$, the segment of $\\pi$ from index $s$ to index $s + k$, read as a list of model-checking game positions with $p_0$ standing for the absorbing states, is not the empty list.","title":"Segments are nonempty"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-segment-play-last","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragSegment_playLast","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-segment-play-last","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $p_0$ be a model-checking game position and $\\pi$ an infinite sequence of positions of the totalized fragment game. For each $i$, let $\\hat\\pi_i$ be $p_0$ if $\\pi_i$ is an absorbing state; if $\\pi_i$ is the embedded position $(\\varphi, w)$ of a fixpoint-free formula $\\varphi$ and a world $w$, let $\\hat\\pi_i$ be the game position $(\\varphi, w, [])$, with $\\varphi$ embedded as a \u03bc-calculus formula and the empty\u2026","title":"Last position of a segment"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/fragment-bespoke-to-canonical","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragment_bespoke_to_canonical","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#fragment-bespoke-to-canonical","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment, $\\sigma$ a Verifier strategy of the model-checking game on $M$, $\\varphi$ a fixpoint-free formula and $w$ a world. If $\\sigma$ wins in environment $\\rho$ from the game position with formula $\\varphi$, world $w$ and empty binder stack, then the positional strategy of the even player induced by $\\sigma$ wins from the embedded position $(\\varphi, w)$ in the fragment\u2026","title":"Winning Verifier strategies transfer to the fragment game"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-move-cases","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragMove_cases","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-move-cases","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model and $e$ an environment, and read each fixpoint-free formula $\\varphi$ as a \u03bc-calculus formula. Every move from $a$ to $b$ in the totalized fragment game of $M$ and $e$ is of one of five kinds: $a$ and $b$ are embedded positions $(\\varphi, w)$ and $(\\varphi', w')$ and the model-checking game of $M$ has a move from $(\\varphi, w)$ to $(\\varphi', w')$ with empty binder stacks; or\u2026","title":"Moves of the totalized fragment game, by cases"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/canonical-next-pos","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.canonical_next_pos","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#canonical-next-pos","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment, and $\\sigma$ any positional strategy of the even player (Verifier) in the fragment game of $M$ and $\\mathit{env}$. Write $\\langle \\psi, v \\rangle$ for the game position with the embedding of a fixpoint-free formula $\\psi$, the world $v$ and the empty binder stack. Let $\\varphi$ be a fixpoint-free formula and $w$ a world such that the even player owns the\u2026","title":"Verifier's positional strategies make a game move at Verifier positions that have one"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/lose-play","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.losePlay","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#lose-play","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For a fixpoint-free formula $\\varphi$ and a world $w$, the lose play is the infinite sequence of positions of the fragment arena that is at the position $(\\varphi, w)$ at time $0$ and at the absorbing losing state at every later time. It is used for terminal losing positions $(\\varphi, w)$.","title":"The lose play"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/lose-play-zero","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.losePlay_zero","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#lose-play-zero","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For a fixpoint-free formula $\\varphi$ and a world $w$, the lose play from $(\\varphi, w)$ is at the fragment-arena position $(\\varphi, w)$ at time $0$.","title":"The lose play starts at its fragment position"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/lose-play-succ","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.losePlay_succ","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#lose-play-succ","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"For every fixpoint-free formula $\\varphi$, world $w$ and natural number $n$, the lose play from $(\\varphi, w)$ is at the losing state at time $n + 1$.","title":"The lose play stays at the losing state"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/frag-pos-not-mem-winning-region-of-termi","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragPos_not_mem_winningRegion_of_terminal_lose","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#frag-pos-not-mem-winning-region-of-termi","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\mathit{env}$ an environment assigning a set of worlds to each free-variable index, $\\varphi$ a fixpoint-free formula and $w$ a world. If the model-checking game position $(\\varphi, w, [])$, with $\\varphi$ embedded as a \u03bc-calculus formula and the empty fixpoint stack, has no moves in $M$ and is not a terminal win for Verifier in $M$ under $\\mathit{env}$, then the embedded position\u2026","title":"Terminal losing positions are not won by the even player"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/fragment-canonical-to-wins","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragment_canonical_to_wins","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#fragment-canonical-to-wins","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment, $\\varphi$ a fixpoint-free formula and $w$ a world. If the embedded position $(\\varphi, w)$ lies in the winning region of the even player in the fragment game of $M$ and $\\rho$, then the backward-induction winning condition $\\mathrm{Wins}_M(\\varphi, w)$ holds.","title":"The fragment game's winning region is contained in Wins"},{"id":"Logic/modal-logic/model-checking-game-for-the-modal-calculus/fragment-wins-iff-winning-region","kind":"node","lean":"Cslib.Modal.MuCalculus.Game.fragmentWins_iff_winningRegion","parent":100,"result":true,"route":"../roadmap/Logic/modal-logic/model-checking-game-for-the-modal-calculus/index.html#fragment-wins-iff-winning-region","scope":"Logic/modal-logic/model-checking-game-for-the-modal-calculus","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\rho$ an environment, $\\varphi$ a fixpoint-free formula and $w$ a world. Then the backward-induction winning condition $\\mathrm{Wins}_M(\\varphi, w)$ holds if and only if the embedded position $(\\varphi, w)$ lies in the winning region of the even player in the fragment game of $M$ and $\\rho$.","title":"Wins is the winning region of the fragment game"},{"data_url":"scopes/Logic/modal-logic/normal-modal-logics.json","id":"Logic/modal-logic/normal-modal-logics","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","title":"Normal modal logics"},{"id":"Logic/modal-logic/normal-modal-logics/derivable","kind":"node","lean":"Cslib.Modal.Derivable","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#derivable","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas. The theorems of the normal modal logic $\\mathbf{K} \\oplus \\Lambda$ form the least set of modal formulas that contains, for all modal formulas $\\varphi$, $\\psi$, $\\chi$, the propositional axioms $\\varphi \\to (\\psi \\to \\varphi)$, $(\\varphi \\to (\\psi \\to \\chi)) \\to ((\\varphi \\to \\psi) \\to (\\varphi \\to \\chi))$ and $\\neg\\neg\\varphi \\to \\varphi$ (where $\\neg\\varphi$ is\u2026","title":"Theorems of a normal modal logic"},{"id":"Logic/modal-logic/normal-modal-logics/deduces","kind":"node","lean":"Cslib.Modal.Deduces","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#deduces","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas. The local consequences of the hypotheses $\\Gamma$ in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$ form the least set of modal formulas that contains every member of $\\Gamma$ and every theorem of $\\mathbf{K} \\oplus \\Lambda$ and is closed under modus ponens: whenever it contains $\\varphi \\to \\psi$ and $\\varphi$, it contains $\\psi$.","title":"Local consequence in a normal modal logic"},{"id":"Logic/modal-logic/normal-modal-logics/k","kind":"node","lean":"Cslib.Modal.Axioms.K","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#k","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"The set of extra axioms of the normal modal logic K is the empty set of modal formulas over a given type of atoms; that is, K adds no axioms to the propositional axioms and the distribution axiom.","title":"Extra axioms of the modal logic K"},{"id":"Logic/modal-logic/normal-modal-logics/t","kind":"node","lean":"Cslib.Modal.Axioms.T","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#t","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"The extra axioms of the modal logic T are the formulas $\\Box\\varphi \\to \\varphi$ (axiom T), for all modal formulas $\\varphi$.","title":"Extra axioms of T"},{"id":"Logic/modal-logic/normal-modal-logics/s4","kind":"node","lean":"Cslib.Modal.Axioms.S4","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#s4","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"The extra axioms of the modal logic S4 are the formulas $\\Box\\varphi \\to \\varphi$ (axiom T) and $\\Box\\varphi \\to \\Box\\Box\\varphi$ (axiom 4), for all modal formulas $\\varphi$.","title":"Extra axioms of S4"},{"id":"Logic/modal-logic/normal-modal-logics/s5","kind":"node","lean":"Cslib.Modal.Axioms.S5","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#s5","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"For a modal formula $\\varphi$, write $\\Diamond\\varphi$ for $\\neg\\Box\\neg\\varphi$. The extra axioms of the modal logic S5 are the formulas $\\Box\\varphi \\to \\varphi$ (axiom T) and $\\Diamond\\varphi \\to \\Box\\Diamond\\varphi$ (axiom 5), for all modal formulas $\\varphi$.","title":"Extra axioms of S5"},{"id":"Logic/modal-logic/normal-modal-logics/imp-self","kind":"node","lean":"Cslib.Modal.Derivable.imp_self","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#imp-self","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas. For every modal formula $\\varphi$, the formula $\\varphi \\to \\varphi$ is a theorem of the normal modal logic $\\mathbf{K} \\oplus \\Lambda$.","title":"Self-implication is a theorem"},{"id":"Logic/modal-logic/normal-modal-logics/mono","kind":"node","lean":"Cslib.Modal.Derivable.mono","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#mono","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda \\subseteq \\Lambda'$ be sets of modal formulas. Every theorem of the normal modal logic $\\mathrm{K} \\oplus \\Lambda$ is a theorem of $\\mathrm{K} \\oplus \\Lambda'$.","title":"Monotonicity of theorems in the extra axioms"},{"id":"Logic/modal-logic/normal-modal-logics/mono-2","kind":"node","lean":"Cslib.Modal.Deduces.mono","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#mono-2","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$, $\\Gamma$ and $\\Delta$ be sets of modal formulas with $\\Gamma \\subseteq \\Delta$. If a formula $\\varphi$ is a local consequence of the hypotheses $\\Gamma$ in the normal modal logic $\\mathrm{K} \\oplus \\Lambda$, then $\\varphi$ is a local consequence of the hypotheses $\\Delta$ in $\\mathrm{K} \\oplus \\Lambda$.","title":"Monotonicity of local consequence in the hypotheses"},{"id":"Logic/modal-logic/normal-modal-logics/trans","kind":"node","lean":"Cslib.Modal.Deduces.trans","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#trans","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas and let $\\Gamma$ and $\\Delta$ be sets of formulas such that every formula of $\\Delta$ is a local consequence of $\\Gamma$ in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$. If a formula $\\varphi$ is a local consequence of $\\Delta$ in $\\mathbf{K} \\oplus \\Lambda$, then $\\varphi$ is a local consequence of $\\Gamma$ in $\\mathbf{K} \\oplus \\Lambda$.","title":"Transitivity of local consequence"},{"id":"Logic/modal-logic/normal-modal-logics/imp-of-insert","kind":"node","lean":"Cslib.Modal.Deduces.imp_of_insert","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#imp-of-insert","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas and let $\\varphi$, $\\psi$ be modal formulas. If $\\varphi$ is a local consequence of $\\Gamma \\cup \\{\\psi\\}$ in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, then $\\psi \\to \\varphi$ is a local consequence of $\\Gamma$ in $\\mathbf{K} \\oplus \\Lambda$.","title":"The deduction theorem for local modal consequence, forward direction"},{"id":"Logic/modal-logic/normal-modal-logics/deduces-insert-iff","kind":"node","lean":"Cslib.Modal.deduces_insert_iff","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#deduces-insert-iff","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas and let $\\varphi$, $\\psi$ be modal formulas. In the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, $\\varphi$ is a local consequence of $\\Gamma \\cup \\{\\psi\\}$ if and only if $\\psi \\to \\varphi$ is a local consequence of $\\Gamma$.","title":"The deduction theorem for local modal consequence"},{"id":"Logic/modal-logic/normal-modal-logics/deduces-empty-iff","kind":"node","lean":"Cslib.Modal.deduces_empty_iff","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#deduces-empty-iff","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas. A modal formula $\\varphi$ is a local consequence of the empty set of hypotheses in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$ if and only if $\\varphi$ is a theorem of $\\mathbf{K} \\oplus \\Lambda$.","title":"Consequences of no hypotheses are the theorems"},{"id":"Logic/modal-logic/normal-modal-logics/exists-finset","kind":"node","lean":"Cslib.Modal.Deduces.exists_finset","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#exists-finset","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas. If a modal formula $\\varphi$ is a local consequence of the hypotheses $\\Gamma$ in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, then there is a finite set $\\Delta \\subseteq \\Gamma$ such that $\\varphi$ is a local consequence of $\\Delta$ in $\\mathbf{K} \\oplus \\Lambda$.","title":"Compactness of local modal consequence"},{"id":"Logic/modal-logic/normal-modal-logics/of-bot","kind":"node","lean":"Cslib.Modal.Deduces.of_bot","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#of-bot","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas and $\\Gamma$ a set of hypotheses. If $\\bot$ is a local consequence of $\\Gamma$ in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, then every formula $\\varphi$ is a local consequence of $\\Gamma$ in $\\mathbf{K} \\oplus \\Lambda$.","title":"Ex falso quodlibet"},{"id":"Logic/modal-logic/normal-modal-logics/bot-of-neg","kind":"node","lean":"Cslib.Modal.Deduces.bot_of_neg","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#bot-of-neg","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas and let $\\varphi$ be a modal formula. If both $\\varphi$ and its negation $\\neg\\varphi$ are local consequences of the hypotheses $\\Gamma$ in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, then so is $\\bot$.","title":"Contradiction yields falsity"},{"id":"Logic/modal-logic/normal-modal-logics/of-insert-neg","kind":"node","lean":"Cslib.Modal.Deduces.of_insert_neg","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#of-insert-neg","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas, $\\Gamma$ a set of hypotheses and $\\varphi$ a formula. If $\\bot$ is a local consequence of $\\Gamma \\cup \\{\\neg\\varphi\\}$ in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, then $\\varphi$ is a local consequence of $\\Gamma$ in $\\mathbf{K} \\oplus \\Lambda$.","title":"Proof by contradiction"},{"id":"Logic/modal-logic/normal-modal-logics/neg-of-insert","kind":"node","lean":"Cslib.Modal.Deduces.neg_of_insert","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#neg-of-insert","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas and $\\varphi$ a formula. If $\\bot$ is a local consequence of the hypotheses $\\Gamma \\cup \\{\\varphi\\}$ in the normal modal logic $\\mathrm{K} \\oplus \\Lambda$, then $\\neg\\varphi$ is a local consequence of $\\Gamma$ in $\\mathrm{K} \\oplus \\Lambda$, where $\\neg\\varphi$ abbreviates $\\varphi \\to \\bot$.","title":"Negation introduction"},{"id":"Logic/modal-logic/normal-modal-logics/imp-trans","kind":"node","lean":"Cslib.Modal.Deduces.imp_trans","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#imp-trans","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas and let $\\varphi$, $\\psi$, $\\chi$ be modal formulas. If $\\varphi \\to \\psi$ and $\\psi \\to \\chi$ are local consequences of the hypotheses $\\Gamma$ in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, then so is $\\varphi \\to \\chi$.","title":"Hypothetical syllogism"},{"id":"Logic/modal-logic/normal-modal-logics/neg-neg","kind":"node","lean":"Cslib.Modal.Deduces.neg_neg","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#neg-neg","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas. If a formula $\\varphi$ is a local consequence of the hypotheses $\\Gamma$ in the normal modal logic $\\mathrm{K} \\oplus \\Lambda$, then so is $\\neg\\neg\\varphi$, where $\\neg\\psi$ abbreviates $\\psi \\to \\bot$.","title":"Double-negation introduction"},{"id":"Logic/modal-logic/normal-modal-logics/of-deduces","kind":"node","lean":"Cslib.Modal.Derivable.of_deduces","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#of-deduces","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas and let $\\Gamma$ be a set of formulas each of which is a theorem of the normal modal logic $\\mathbf{K} \\oplus \\Lambda$. If a formula $\\varphi$ is a local consequence of $\\Gamma$ in $\\mathbf{K} \\oplus \\Lambda$, then $\\varphi$ is a theorem of $\\mathbf{K} \\oplus \\Lambda$.","title":"Local consequences of theorems are theorems"},{"id":"Logic/modal-logic/normal-modal-logics/box-imp-box","kind":"node","lean":"Cslib.Modal.Derivable.box_imp_box","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#box-imp-box","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas and let $\\varphi$, $\\psi$ be modal formulas. If $\\varphi \\to \\psi$ is a theorem of the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, then so is $\\Box\\varphi \\to \\Box\\psi$.","title":"The monotonicity rule for necessity"},{"id":"Logic/modal-logic/normal-modal-logics/valid","kind":"node","lean":"Cslib.Modal.Derivable.valid","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#valid","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas and $M$ a Kripke model in which every formula of $\\Lambda$ is valid. Then every formula derivable in $K \\oplus \\Lambda$ (from the propositional axioms, the distribution axiom and the members of $\\Lambda$ by modus ponens and necessitation) is valid in $M$, that is, it holds at every world of $M$.","title":"Soundness of modal Hilbert systems for Kripke models"},{"id":"Logic/modal-logic/normal-modal-logics/satisfies","kind":"node","lean":"Cslib.Modal.Deduces.satisfies","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#satisfies","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\Lambda$ a set of modal formulas each of which is valid in $M$. If a formula $\\varphi$ is a local consequence of a set $\\Gamma$ of formulas in the normal modal logic $\\mathbf{K} \\oplus \\Lambda$, then $\\varphi$ holds at every world of $M$ at which every formula of $\\Gamma$ holds.","title":"Local soundness"},{"id":"Logic/modal-logic/normal-modal-logics/valid-k","kind":"node","lean":"Cslib.Modal.Derivable.valid_K","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#valid-k","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Every theorem of the modal logic K is valid in every Kripke model: if a formula $\\varphi$ is derivable in K, then $\\varphi$ holds at every world of every Kripke model $M$.","title":"Soundness of K for Kripke models"},{"id":"Logic/modal-logic/normal-modal-logics/valid-t","kind":"node","lean":"Cslib.Modal.Derivable.valid_T","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#valid-t","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"If the accessibility relation of a Kripke model $M$ is reflexive, then every formula derivable in the modal logic T is valid in $M$, that is, it holds at every world of $M$.","title":"Soundness of T for reflexive Kripke models"},{"id":"Logic/modal-logic/normal-modal-logics/valid-s4","kind":"node","lean":"Cslib.Modal.Derivable.valid_S4","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#valid-s4","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"If the accessibility relation of a Kripke model $M$ is reflexive and transitive, then every formula derivable in the modal logic S4 is valid in $M$, that is, it holds at every world of $M$.","title":"Soundness of S4 for reflexive transitive Kripke models"},{"id":"Logic/modal-logic/normal-modal-logics/valid-s5","kind":"node","lean":"Cslib.Modal.Derivable.valid_S5","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#valid-s5","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"If the accessibility relation of a Kripke model $M$ is an equivalence relation, then every formula derivable in the modal logic S5 is valid in $M$, that is, it holds at every world of $M$.","title":"Soundness of S5 for Kripke models with an equivalence relation"},{"id":"Logic/modal-logic/normal-modal-logics/unit","kind":"node","lean":"Cslib.Modal.KripkeModel.unit","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#unit","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"For any type of atoms, the one-world Kripke model has a single world, which sees itself, and every atom is false at that world.","title":"The one-world Kripke model"},{"id":"Logic/modal-logic/normal-modal-logics/unit-equivalence","kind":"node","lean":"Cslib.Modal.KripkeModel.unit_equivalence","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#unit-equivalence","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"The accessibility relation of the one-world Kripke model, whose single world sees itself and at which every atom is false, is an equivalence relation.","title":"The one-world model has an equivalence relation"},{"id":"Logic/modal-logic/normal-modal-logics/not-derivable-bot","kind":"node","lean":"Cslib.Modal.not_derivable_bot","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#not-derivable-bot","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas each of which is valid in the one-world Kripke model whose world sees itself and in which every atom is false. Then $\\bot$ is not a theorem of the normal modal logic $\\mathrm{K} \\oplus \\Lambda$.","title":"Consistency of normal modal logics"},{"id":"Logic/modal-logic/normal-modal-logics/not-derivable-bot-s5","kind":"node","lean":"Cslib.Modal.not_derivable_bot_S5","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#not-derivable-bot-s5","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"For every type of atoms, $\\bot$ is not a theorem of the normal modal logic S5, whose extra axioms are the formulas $\\Box\\varphi \\to \\varphi$ and $\\Diamond\\varphi \\to \\Box\\Diamond\\varphi$, where $\\Diamond\\varphi$ abbreviates $\\neg\\Box\\neg\\varphi$ and $\\neg\\psi$ abbreviates $\\psi \\to \\bot$.","title":"Consistency of S5"},{"id":"Logic/modal-logic/normal-modal-logics/not-derivable-bot-s4","kind":"node","lean":"Cslib.Modal.not_derivable_bot_S4","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#not-derivable-bot-s4","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"For every type of atoms, $\\bot$ is not a theorem of the normal modal logic S4, whose extra axioms are the formulas $\\Box\\varphi \\to \\varphi$ and $\\Box\\varphi \\to \\Box\\Box\\varphi$.","title":"Consistency of S4"},{"id":"Logic/modal-logic/normal-modal-logics/not-derivable-bot-t","kind":"node","lean":"Cslib.Modal.not_derivable_bot_T","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#not-derivable-bot-t","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"For every type of atoms, $\\bot$ is not a theorem of the normal modal logic T, whose extra axioms are the formulas $\\Box\\varphi \\to \\varphi$.","title":"Consistency of T"},{"id":"Logic/modal-logic/normal-modal-logics/not-derivable-bot-k","kind":"node","lean":"Cslib.Modal.not_derivable_bot_K","parent":198,"result":true,"route":"../roadmap/Logic/modal-logic/normal-modal-logics/index.html#not-derivable-bot-k","scope":"Logic/modal-logic/normal-modal-logics","status":"fully_proved","summary":"For every type of atoms, $\\bot$ is not a theorem of the normal modal logic K, the logic with no extra axioms.","title":"Consistency of K"},{"data_url":"scopes/Logic/modal-logic/propositional-dynamic-logic.json","id":"Logic/modal-logic/propositional-dynamic-logic","kind":"scope","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/index.html","scope":"Logic/modal-logic/propositional-dynamic-logic","status":"fully_proved","title":"Propositional dynamic logic"},{"data_url":"scopes/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl.json","id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","kind":"scope","navigation":"module","parent":234,"result":false,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","title":"Propositional dynamic logic (PDL)"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/model","kind":"node","lean":"Cslib.PDL.Model","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#model","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"A model of propositional dynamic logic with set of worlds $W$ consists of a binary accessibility relation $R_a$ on $W$ for each atomic action label $a$, and a valuation that says which atoms are true at each world.","title":"PDL model"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/formula","kind":"node","lean":"Cslib.PDL.Formula","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#formula","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"The formulas of propositional dynamic logic over a type of atoms and a type of atomic action labels are the atoms $p$, falsity $\\bot$, implications $\\varphi \\to \\psi$ of formulas, and boxes $[\\alpha]\\varphi$ of a program $\\alpha$ and a formula $\\varphi$. Formulas and programs are defined by simultaneous induction.","title":"PDL formulas"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/program","kind":"node","lean":"Cslib.PDL.Program","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#program","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"The programs of propositional dynamic logic over a type of atoms and a type of atomic action labels are the atomic actions $a$, sequential compositions $\\alpha ; \\beta$, nondeterministic choices $\\alpha \\cup \\beta$, iterations $\\alpha^*$, and tests $\\varphi?$ of a formula $\\varphi$. Programs and formulas are defined by simultaneous induction.","title":"PDL programs"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/dia","kind":"node","lean":"Cslib.PDL.Formula.dia","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#dia","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"For a program $\\alpha$ and a formula $\\varphi$, the formula $\\langle\\alpha\\rangle\\varphi$ is defined as $\\alpha \\to \\bot$, that is, as $\\neg[\\alpha]\\neg\\varphi$: possibility is the dual of necessity.","title":"Possibility in PDL"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/prog-rel","kind":"node","lean":"Cslib.PDL.ProgRel","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#prog-rel","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"In a model $M$, each program $\\alpha$ denotes a binary relation $R_\\alpha$ on worlds: an atomic action $a$ denotes its accessibility relation $R_a$; $(w, v) \\in R_{\\alpha;\\beta}$ iff $(w, u) \\in R_\\alpha$ and $(u, v) \\in R_\\beta$ for some world $u$; $R_{\\alpha \\cup \\beta} = R_\\alpha \\cup R_\\beta$; $R_{\\alpha^*}$ is the reflexive-transitive closure of $R_\\alpha$; and $(w, v) \\in R_{\\varphi?}$ iff $w = v$ and\u2026","title":"Relation denoted by a PDL program"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/sat","kind":"node","lean":"Cslib.PDL.Sat","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#sat","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"Truth of a formula at a world $w$ of a model $M$ is defined by recursion: an atom $p$ holds at $w$ iff the valuation of $M$ makes $p$ true at $w$; $\\bot$ holds nowhere; $\\varphi \\to \\psi$ holds at $w$ iff $\\psi$ holds at $w$ whenever $\\varphi$ does; and $[\\alpha]\\varphi$ holds at $w$ iff $\\varphi$ holds at every world $v$ with $(w, v) \\in R_\\alpha$, where $R_\\alpha$ is the relation denoted by $\\alpha$ in $M$.","title":"Satisfaction of PDL formulas"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/sat-box-seq","kind":"node","lean":"Cslib.PDL.sat_box_seq","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#sat-box-seq","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"For every model $M$, world $w$, programs $\\alpha$ and $\\beta$, and formula $\\varphi$, the formula $[\\alpha;\\beta]\\varphi$ holds at $w$ if and only if $[\\alpha][\\beta]\\varphi$ holds at $w$.","title":"Box of a sequential composition"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/sat-box-choice","kind":"node","lean":"Cslib.PDL.sat_box_choice","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#sat-box-choice","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"For every model $M$, world $w$, programs $\\alpha$ and $\\beta$, and formula $\\varphi$, the formula $[\\alpha \\cup \\beta]\\varphi$ holds at $w$ if and only if both $[\\alpha]\\varphi$ and $[\\beta]\\varphi$ hold at $w$.","title":"Box of a choice"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/sat-box-star-unfold","kind":"node","lean":"Cslib.PDL.sat_box_star_unfold","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#sat-box-star-unfold","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"For every model $M$, world $w$, program $\\alpha$ and formula $\\varphi$, the formula $[\\alpha^*]\\varphi$ holds at $w$ if and only if $\\varphi$ holds at $w$ and $[\\alpha][\\alpha^*]\\varphi$ holds at $w$.","title":"Unfolding the box of an iteration"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/sat-box-test","kind":"node","lean":"Cslib.PDL.sat_box_test","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#sat-box-test","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"For every model $M$, world $w$ and formulas $\\psi$ and $\\varphi$, the formula $[\\psi?]\\varphi$ holds at $w$ if and only if $\\varphi$ holds at $w$ whenever $\\psi$ holds at $w$.","title":"Box of a test"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/toggle-model","kind":"node","lean":"Cslib.PDL.toggleModel","parent":235,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#toggle-model","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"The PDL model whose worlds are the two Booleans $\\mathsf{true}$ and $\\mathsf{false}$, with a single atomic action $a$ and a single atom $p$: the action $a$ relates each world $s$ exactly to its negation $\\lnot s$, and $p$ holds exactly at $\\mathsf{true}$.","title":"The toggle model"},{"id":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/toggle-not-box-seq-atom","kind":"node","lean":"Cslib.PDL.toggle_not_box_seq_atom","parent":235,"reading_title":{"format":"inline-parts-v1","parts":[["text","Failure of "],["math","\\((a;a)\\,p\\)"],["text"," in the toggle model"]],"source":"Failure of $(a;a)\\,p$ in the toggle model"},"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl/index.html#toggle-not-box-seq-atom","scope":"Logic/modal-logic/propositional-dynamic-logic/propositional-dynamic-logic-pdl","status":"fully_proved","summary":"In the toggle model, with atomic action $a$ and atom $p$, the formula $[a;a]\\,p$ is not satisfied at the world $\\mathsf{false}$.","title":"Failure of $(a;a)\\,p$ in the toggle model"},{"data_url":"scopes/Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs.json","id":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs","kind":"scope","navigation":"module","parent":234,"result":false,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/index.html","scope":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs","status":"fully_proved","title":"Relational interpretation of PDL programs"},{"id":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/to-relation","kind":"node","lean":"Cslib.PDL.Program.toRelation","parent":248,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/index.html#to-relation","scope":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs","status":"fully_proved","summary":"Interpret a PDL program in CSLib's canonical relation algebra.","title":"Relational interpretation of a PDL program"},{"id":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/mem-to-relation","kind":"node","lean":"Cslib.PDL.Program.mem_toRelation","parent":248,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/index.html#mem-to-relation","scope":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs","status":"fully_proved","summary":"The representation conversion preserves exactly the executions.","title":"Relational interpretation preserves executions"},{"id":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/to-relation-seq","kind":"node","lean":"Cslib.PDL.Program.toRelation_seq","parent":248,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/index.html#to-relation-seq","scope":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs","status":"fully_proved","summary":"The interpretation of sequencing uses canonical composition.","title":"Sequencing is interpreted by relational composition"},{"id":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/to-relation-choice","kind":"node","lean":"Cslib.PDL.Program.toRelation_choice","parent":248,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/index.html#to-relation-choice","scope":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs","status":"fully_proved","summary":"The interpretation of choice uses canonical union.","title":"Choice is interpreted by relational union"},{"id":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/to-relation-test","kind":"node","lean":"Cslib.PDL.Program.toRelation_test","parent":248,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/index.html#to-relation-test","scope":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs","status":"fully_proved","summary":"Formula tests use the existing PDL satisfaction predicate.","title":"Tests are interpreted by PDL satisfaction"},{"id":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/to-relation-star","kind":"node","lean":"Cslib.PDL.Program.toRelation_star","parent":248,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs/index.html#to-relation-star","scope":"Logic/modal-logic/propositional-dynamic-logic/relational-interpretation-of-pdl-programs","status":"fully_proved","summary":"Iteration in the canonical carrier is the same reflexive-transitive closure as in `ProgRel`, including its identity boundary.","title":"Iteration is interpreted by reflexive-transitive closure"},{"data_url":"scopes/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl.json","id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","kind":"scope","navigation":"module","parent":234,"result":false,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","title":"Partial correctness and loop invariants for PDL"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/hoare","kind":"node","lean":"Cslib.PDL.Hoare","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#hoare","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Every execution of `a` from `P` ends in `Q`, without a termination requirement.","title":"Partial-correctness Hoare triple for PDL programs"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/hoare-iff-subset-core","kind":"node","lean":"Cslib.PDL.hoare_iff_subset_core","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#hoare-iff-subset-core","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Partial correctness agrees with the canonical relation's core.","title":"Partial correctness as inclusion in the relational core"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/hoare-iff-test-le","kind":"node","lean":"Cslib.PDL.hoare_iff_test_le","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#hoare-iff-test-le","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Hoare verification is equivalent to the KAT test inequality.","title":"Hoare triples as KAT test inequalities"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/hoare-iff-test-eq","kind":"node","lean":"Cslib.PDL.hoare_iff_test_eq","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#hoare-iff-test-eq","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"The equivalent KAT absorption equation for verification.","title":"Hoare triples as KAT absorption equations"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/hoare-iff-box","kind":"node","lean":"Cslib.PDL.hoare_iff_box","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#hoare-iff-box","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Formula assertions connect Hoare verification to global PDL validity, not just an atomic-action modality.","title":"Hoare triples with formula assertions as box validity"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/consequence","kind":"node","lean":"Cslib.PDL.Hoare.consequence","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#consequence","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Strengthen the precondition and weaken the postcondition.","title":"Hoare rule of consequence"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/seq","kind":"node","lean":"Cslib.PDL.Hoare.seq","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#seq","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Sequential verification uses the first postcondition as the second precondition.","title":"Hoare rule for sequential composition"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/choice-iff","kind":"node","lean":"Cslib.PDL.Hoare.choice_iff","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#choice-iff","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Nondeterministic choice is safe exactly when both branches are.","title":"Hoare rule for nondeterministic choice"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/test-iff","kind":"node","lean":"Cslib.PDL.Hoare.test_iff","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#test-iff","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"A formula test requires the postcondition only where the test passes.","title":"Hoare rule for formula tests"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/star","kind":"node","lean":"Cslib.PDL.Hoare.star","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#star","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"A one-step invariant holds after every finite number of iterations, including zero, via CSLib's canonical closure-preservation theorem.","title":"Hoare invariant rule for iteration"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/while-do","kind":"node","lean":"Cslib.PDL.Program.whileDo","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#while-do","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"A guarded loop in the existing PDL syntax: iterate guarded body steps, then test that the guard is false.","title":"Guarded while loop in PDL"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/while-do-2","kind":"node","lean":"Cslib.PDL.Hoare.whileDo","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#while-do-2","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"The guarded-loop invariant rule is partial correctness, including zero-step exits; it does not guarantee termination.","title":"Hoare invariant rule for guarded while loops"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/while-do-consequence","kind":"node","lean":"Cslib.PDL.Hoare.whileDo_consequence","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#while-do-consequence","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Verify a guarded loop using initialization, invariant preservation, and the exit implication to the desired postcondition.","title":"While-loop rule with initialization and exit implication"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/while-do-iff-exists-invariant","kind":"node","lean":"Cslib.PDL.Hoare.whileDo_iff_exists_invariant","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#while-do-iff-exists-invariant","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"The guarded-loop rule is semantically complete for arbitrary set assertions. The witness is guarded-body reachability from `P`, independent of `Q`; this is not a completeness claim for a syntactic assertion language.","title":"Semantic completeness of the while-loop invariant rule"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/sat-box-star-induction","kind":"node","lean":"Cslib.PDL.sat_box_star_induction","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#sat-box-star-induction","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"PDL induction restricted to states reachable from the initial world. This proves `[a*](\u03c6 \u2192 [a]\u03c6) \u2192 (\u03c6 \u2192 [a*]\u03c6)` without assuming preservation on unreachable states.","title":"PDL induction restricted to reachable states"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/test-mul-star-absorption","kind":"node","lean":"Cslib.PDL.test_mul_star_absorption","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#test-mul-star-absorption","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"Concrete relational test absorption lifts through PDL iteration, connecting the algebraic verification equation to closure invariance.","title":"Test absorption lifts through iteration"},{"id":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/test-mul-while-do-absorption","kind":"node","lean":"Cslib.PDL.test_mul_whileDo_absorption","parent":255,"result":true,"route":"../roadmap/Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl/index.html#test-mul-while-do-absorption","scope":"Logic/modal-logic/propositional-dynamic-logic/partial-correctness-and-loop-invariants-for-pdl","status":"fully_proved","summary":"A guarded-body verification equation implies the loop's exit verification equation over the same canonical relational carrier.","title":"Loop exit absorption from a guarded-body absorption equation"},{"data_url":"scopes/Logic/modal-logic/sahlqvist-formulas.json","id":"Logic/modal-logic/sahlqvist-formulas","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","title":"Sahlqvist formulas"},{"id":"Logic/modal-logic/sahlqvist-formulas/is-positive","kind":"node","lean":"Cslib.Modal.IsPositive","parent":273,"result":true,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html#is-positive","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","summary":"Positive and negative modal formulas are defined by simultaneous recursion: every atom is positive, $\\bot$ is positive, $\\varphi \\to \\psi$ is positive if $\\varphi$ is negative and $\\psi$ is positive, and $\\Box\\varphi$ is positive if $\\varphi$ is positive.","title":"Positive modal formulas"},{"id":"Logic/modal-logic/sahlqvist-formulas/is-negative","kind":"node","lean":"Cslib.Modal.IsNegative","parent":273,"result":true,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html#is-negative","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","summary":"Negative and positive modal formulas are defined by simultaneous recursion: no atom is negative, $\\bot$ is negative, $\\varphi \\to \\psi$ is negative if $\\varphi$ is positive and $\\psi$ is negative, and $\\Box\\varphi$ is negative if $\\varphi$ is negative.","title":"Negative modal formulas"},{"id":"Logic/modal-logic/sahlqvist-formulas/box-pow","kind":"node","lean":"Cslib.Modal.boxPow","parent":273,"result":true,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html#box-pow","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","summary":"For a natural number $k$ and a modal formula $\\varphi$, the iterated box $\\Box^k \\varphi$ is defined by $\\Box^0 \\varphi = \\varphi$ and $\\Box^{k+1} \\varphi = \\Box\\,\\Box^k \\varphi$.","title":"Iterated box"},{"id":"Logic/modal-logic/sahlqvist-formulas/is-sahlqvist-antecedent","kind":"node","lean":"Cslib.Modal.IsSahlqvistAntecedent","parent":273,"result":true,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html#is-sahlqvist-antecedent","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","summary":"The Sahlqvist antecedents form the smallest class of modal formulas that contains every boxed atom $\\Box^k p$ (an atom $p$ under $k \\ge 0$ boxes) and every negative formula, and is closed under $\\wedge$, $\\vee$ and $\\Diamond$.","title":"Sahlqvist antecedents"},{"id":"Logic/modal-logic/sahlqvist-formulas/atoms","kind":"node","lean":"Cslib.Modal.Formula.atoms","parent":273,"result":true,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html#atoms","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","summary":"The set of atoms (propositional variables) occurring in a modal formula.","title":"Atoms of a modal formula"},{"id":"Logic/modal-logic/sahlqvist-formulas/is-sahlqvist-implication","kind":"node","lean":"Cslib.Modal.IsSahlqvistImplication","parent":273,"result":true,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html#is-sahlqvist-implication","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","summary":"A modal formula is a Sahlqvist implication if it is an implication $\\alpha \\to \\beta$ whose antecedent $\\alpha$ is a Sahlqvist antecedent and whose consequent $\\beta$ is a positive formula.","title":"Sahlqvist implications"},{"id":"Logic/modal-logic/sahlqvist-formulas/is-sahlqvist","kind":"node","lean":"Cslib.Modal.IsSahlqvist","parent":273,"result":true,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html#is-sahlqvist","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","summary":"The Sahlqvist formulas form the smallest class of modal formulas that contains every Sahlqvist implication, is closed under $\\wedge$ and $\\Box$, and contains $\\varphi \\vee \\psi$ whenever $\\varphi$ and $\\psi$ are Sahlqvist formulas with no atom in common.","title":"Sahlqvist formulas"},{"id":"Logic/modal-logic/sahlqvist-formulas/is-sahlqvist-axiom-b","kind":"node","lean":"Cslib.Modal.isSahlqvist_axiomB","parent":273,"result":true,"route":"../roadmap/Logic/modal-logic/sahlqvist-formulas/index.html#is-sahlqvist-axiom-b","scope":"Logic/modal-logic/sahlqvist-formulas","status":"fully_proved","summary":"For every atom $p$, the instance $p \\to \\Box\\Diamond p$ of the B axiom is a Sahlqvist formula: its antecedent $p$ is a boxed atom with zero boxes and its consequent $\\Box\\Diamond p$ is positive.","title":"The B axiom is a Sahlqvist formula"},{"data_url":"scopes/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic.json","id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","title":"Soundness and completeness of normal modal logics"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/of-proposition","kind":"node","lean":"Cslib.Modal.Formula.ofProposition","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#of-proposition","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"CSLib's unimodal propositions are translated into modal formulas in the $\\bot$, $\\to$, $\\Box$ basis by recursion, writing $\\varphi'$ for the translation of $\\varphi$: an atom $p$ goes to $p$, falsehood to $\\bot$, $\\neg\\varphi$ to $\\varphi' \\to \\bot$, $\\varphi \\lor \\psi$ to $(\\varphi' \\to \\bot) \\to \\psi'$, and the diamond (the unimodal triangle) applied to $\\varphi$ goes to $\\Diamond\\varphi'$, that is, to\u2026","title":"Translation of CSLib's modal propositions"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/of-model","kind":"node","lean":"Cslib.Modal.KripkeModel.ofModel","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#of-model","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For a CSLib unimodal model $m$ on a nonempty type of worlds, the associated Kripke model has the same worlds, the binary accessibility relation of $m$ as its accessibility relation, and the valuation of $m$ as its valuation.","title":"Kripke model of a CSLib model"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/satisfies-of-proposition","kind":"node","lean":"Cslib.Modal.satisfies_ofProposition","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#satisfies-of-proposition","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $m$ be a CSLib unimodal model on a nonempty type of worlds, $w$ a world and $\\varphi$ a CSLib unimodal proposition. Then the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis holds at $w$ in the Kripke model of $m$ if and only if $\\varphi$ holds at $w$ in $m$.","title":"The translation preserves satisfaction"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/to-model","kind":"node","lean":"Cslib.Modal.KripkeModel.toModel","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#to-model","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"A Kripke model $k$ on worlds $W$ gives the CSLib unimodal model on $W$ whose frame is the unimodal frame built from the accessibility relation of $k$ and whose valuation is the valuation of $k$.","title":"The CSLib unimodal model of a Kripke model"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/of-model-to-model","kind":"node","lean":"Cslib.Modal.KripkeModel.ofModel_toModel","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#of-model-to-model","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $k$ be a Kripke model on a nonempty type of worlds. Converting $k$ to a CSLib unimodal model and converting that model back to a Kripke model gives $k$.","title":"Kripke to CSLib and back is the identity"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/to-model-of-model","kind":"node","lean":"Cslib.Modal.KripkeModel.toModel_ofModel","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#to-model-of-model","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For a nonempty type of worlds $W$, converting a CSLib unimodal model $m$ on $W$ into a Kripke model and then back into a CSLib unimodal model returns $m$: a unimodal frame is determined by its binary accessibility relation.","title":"Round trip from CSLib models through Kripke models"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/to-proposition","kind":"node","lean":"Cslib.Modal.Formula.toProposition","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#to-proposition","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Each modal formula built from atoms, $\\bot$, $\\to$ and $\\Box$ is translated into a CSLib unimodal proposition by recursion: an atom $p$ goes to the atom $p$, $\\bot$ goes to false, $\\varphi \\to \\psi$ goes to $\\neg\\varphi' \\lor \\psi'$, and $\\Box\\varphi$ goes to $\\neg\\Diamond\\neg\\varphi'$, where $\\varphi'$ and $\\psi'$ are the translations of $\\varphi$ and $\\psi$ and $\\Diamond$ is the possibility operator of CSLib's\u2026","title":"Translation of modal formulas into CSLib propositions"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/satisfies-to-proposition","kind":"node","lean":"Cslib.Modal.satisfies_toProposition","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#satisfies-to-proposition","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $k$ be a Kripke model, $w$ a world and $\\varphi$ a modal formula in the $\\bot$, $\\to$, $\\Box$ basis. Then $\\varphi$ holds at $w$ in $k$ if and only if its translation into a CSLib unimodal proposition holds at $w$ in the CSLib model associated with $k$.","title":"The reverse translation preserves satisfaction"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/valid-to-proposition","kind":"node","lean":"Cslib.Modal.KripkeModel.valid_toProposition","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#valid-to-proposition","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"A modal formula $\\varphi$ is valid in a Kripke model $k$ (true at every world of $k$) if and only if its translation into a CSLib unimodal proposition is satisfied at every world of the CSLib unimodal model associated with $k$.","title":"Validity agrees with its CSLib translation"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/mem-logic-of-valid","kind":"node","lean":"Cslib.Modal.mem_logic_of_valid","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#mem-logic-of-valid","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $S$ be a class of CSLib unimodal models on a type of worlds and $\\varphi$ a CSLib unimodal proposition. Suppose that, if the type of worlds is nonempty, the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is valid in the Kripke model of every model in $S$. Then $\\varphi$ belongs to CSLib's semantic logic of $S$: it holds at every world of every model in $S$.","title":"Validity of the translation gives membership in CSLib's logic"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/mem-k","kind":"node","lean":"Cslib.Modal.Derivable.mem_K","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#mem-k","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CSLib unimodal proposition. If the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of the Hilbert system K, then, for every type of worlds, $\\varphi$ belongs to CSLib's modal logic K: it holds at every world of every CSLib unimodal model on that type.","title":"Soundness for CSLib's K"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/mem-t","kind":"node","lean":"Cslib.Modal.Derivable.mem_T","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#mem-t","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CSLib unimodal proposition. If the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of the Hilbert system T, then, for every type of worlds, $\\varphi$ belongs to CSLib's modal logic T: it holds at every world of every CSLib unimodal model on that type whose accessibility relation is reflexive.","title":"Soundness for CSLib's T"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/mem-s4","kind":"node","lean":"Cslib.Modal.Derivable.mem_S4","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#mem-s4","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CSLib unimodal proposition. If the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of the Hilbert system S4, then, for every type of worlds, $\\varphi$ belongs to CSLib's modal logic S4: it holds at every world of every CSLib unimodal model on that type whose accessibility relation is reflexive and transitive.","title":"Soundness for CSLib's S4"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/mem-s5","kind":"node","lean":"Cslib.Modal.Derivable.mem_S5","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#mem-s5","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CSLib unimodal proposition. If the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of the Hilbert system S5, then, for every type of worlds, $\\varphi$ belongs to CSLib's modal logic S5: it holds at every world of every CSLib unimodal model on that type whose accessibility relation $R$ is reflexive, transitive and Euclidean ($R(w, u)$ and $R(w, v)$ imply $R(u, v)$).","title":"Soundness for CSLib's S5"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/consistent","kind":"node","lean":"Cslib.Modal.Consistent","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#consistent","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For sets of formulas $\\Lambda$ and $\\Gamma$, the set $\\Gamma$ is consistent relative to $K \\oplus \\Lambda$ when $\\bot$ is not deducible from $\\Gamma$ in $K \\oplus \\Lambda$.","title":"Consistency relative to a normal modal logic"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/consistent-is-of-finite-character","kind":"node","lean":"Cslib.Modal.consistent_isOfFiniteCharacter","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#consistent-is-of-finite-character","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every set $\\Lambda$ of extra axioms, a set of formulas is consistent relative to $K \\oplus \\Lambda$ iff all its finite subsets are.","title":"Consistency is of finite character"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/exists-maximal-consistent","kind":"node","lean":"Cslib.Modal.exists_maximal_consistent","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#exists-maximal-consistent","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas and $\\mathrm{K} \\oplus \\Lambda$ the normal modal logic K extended by the axioms $\\Lambda$. Every set $\\Gamma$ of modal formulas that is consistent relative to $\\mathrm{K} \\oplus \\Lambda$ is contained in a set $M$ of modal formulas that is maximal among the sets consistent relative to $\\mathrm{K} \\oplus \\Lambda$.","title":"Lindenbaum's lemma for normal modal logics"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/mem-of-deduces","kind":"node","lean":"Cslib.Modal.MaximalConsistent.mem_of_deduces","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#mem-of-deduces","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas and $M$ a set of modal formulas that is maximal among the sets consistent relative to the normal modal logic K extended by the axioms $\\Lambda$. Every formula that is a local consequence of $M$ in that logic belongs to $M$.","title":"Maximal consistent sets of modal formulas are deductively closed"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/mem-of-foldr-imp-mem","kind":"node","lean":"Cslib.Modal.MaximalConsistent.mem_of_foldr_imp_mem","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#mem-of-foldr-imp-mem","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas, $M$ a set of modal formulas that is maximal among the sets consistent relative to the normal modal logic K extended by the axioms $\\Lambda$, and $\\varphi$ and $\\gamma_1, \\dots, \\gamma_n$ modal formulas. If $\\gamma_1 \\to (\\gamma_2 \\to \\cdots \\to (\\gamma_n \\to \\varphi) \\cdots) \\in M$ and $\\gamma_1, \\dots, \\gamma_n \\in M$, then $\\varphi \\in M$.","title":"Iterated modus ponens in maximal consistent sets"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/foldr-imp","kind":"node","lean":"Cslib.Modal.Deduces.foldr_imp","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#foldr-imp","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas, $\\varphi$ a modal formula and $\\gamma_1, \\dots, \\gamma_n$ a list of modal formulas. If $\\varphi$ is a local consequence of $\\Gamma \\cup \\{\\gamma_1, \\dots, \\gamma_n\\}$ in the normal modal logic K extended by the axioms $\\Lambda$, then $\\gamma_1 \\to (\\gamma_2 \\to \\cdots \\to (\\gamma_n \\to \\varphi) \\cdots)$ is a local consequence of $\\Gamma$ in the same logic.","title":"Discharging a list of hypotheses"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/box-foldr-imp","kind":"node","lean":"Cslib.Modal.Derivable.box_foldr_imp","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#box-foldr-imp","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of extra axioms, let $\\gamma_1, \\dots, \\gamma_n$ be formulas (with $n \\ge 0$) and let $\\varphi$ be a formula. Then $\\Box(\\gamma_1 \\to \\cdots \\to \\gamma_n \\to \\varphi) \\to (\\Box\\gamma_1 \\to \\cdots \\to \\Box\\gamma_n \\to \\Box\\varphi)$ is a theorem of $K \\oplus \\Lambda$.","title":"Boxing an implication chain"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/canonical-world","kind":"node","lean":"Cslib.Modal.CanonicalWorld","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#canonical-world","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For a set $\\Lambda$ of extra axioms, the worlds of the canonical model of $K \\oplus \\Lambda$ are the maximal consistent sets of $K \\oplus \\Lambda$: the sets of formulas that are consistent relative to $K \\oplus \\Lambda$ and have no consistent proper superset.","title":"Worlds of the canonical model"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/rel","kind":"node","lean":"Cslib.Modal.CanonicalWorld.rel","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#rel","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For a set $\\Lambda$ of modal formulas and maximal $(K \\oplus \\Lambda)$-consistent sets $M$ and $N$ (worlds of the canonical model), $M$ is related to $N$ iff every formula $\\varphi$ with $\\Box\\varphi \\in M$ belongs to $N$.","title":"The canonical accessibility relation"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/canonical-model","kind":"node","lean":"Cslib.Modal.canonicalModel","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#canonical-model","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of extra axioms such that $\\bot$ is not a theorem of $K \\oplus \\Lambda$. The canonical model of $K \\oplus \\Lambda$ is the Kripke model whose worlds are the maximal consistent sets of $K \\oplus \\Lambda$ (there is at least one, by Lindenbaum's lemma), in which $M$ sees $N$ iff $\\varphi \\in N$ for every formula $\\varphi$ with $\\Box\\varphi \\in M$, and in which an atom $p$ is true at $M$ iff\u2026","title":"The canonical model"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/exists-rel-not-mem","kind":"node","lean":"Cslib.Modal.CanonicalWorld.exists_rel_not_mem","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#exists-rel-not-mem","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas, $\\mathrm{K} \\oplus \\Lambda$ the normal modal logic K extended by the axioms $\\Lambda$, $M$ a canonical world, that is, a set of modal formulas maximal among the sets consistent relative to $\\mathrm{K} \\oplus \\Lambda$, and $\\varphi$ a modal formula. If $\\Box\\varphi \\notin M$, then there is a canonical world $N$ with $\\varphi \\notin N$ that is accessible from $M$ in the\u2026","title":"Existence lemma"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/satisfies-canonical-iff","kind":"node","lean":"Cslib.Modal.satisfies_canonical_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#satisfies-canonical-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"A formula holds at a world of the canonical model iff it belongs to it.","title":"Truth lemma"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/of-canonical","kind":"node","lean":"Cslib.Modal.Deduces.of_canonical","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#of-canonical","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ and $\\Gamma$ be sets of modal formulas and $\\varphi$ a formula. If $\\varphi$ belongs to every maximal $(K \\oplus \\Lambda)$-consistent set that contains $\\Gamma$, then $\\Gamma$ deduces $\\varphi$ in $K \\oplus \\Lambda$.","title":"Deducibility from maximal consistent extensions"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/of-canonical-model","kind":"node","lean":"Cslib.Modal.Deduces.of_canonicalModel","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#of-canonical-model","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas such that $\\bot$ is not a theorem of $K \\oplus \\Lambda$, let $\\Gamma$ be a set of formulas and $\\varphi$ a formula. If $\\varphi$ holds at every world of the canonical model of $K \\oplus \\Lambda$ at which every formula of $\\Gamma$ holds, then $\\Gamma$ deduces $\\varphi$ in $K \\oplus \\Lambda$.","title":"Completeness relative to the canonical model"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-box-imp-box-neg-neg","kind":"node","lean":"Cslib.Modal.CanonicalWorld.derivable_box_imp_box_neg_neg","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-box-imp-box-neg-neg","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every set $\\Lambda$ of extra axioms and every modal formula $\\psi$, the formula $\\Box\\psi \\to \\Box\\neg\\neg\\psi$ is a theorem of the normal modal logic $K \\oplus \\Lambda$, where $\\neg\\chi$ abbreviates $\\chi \\to \\bot$.","title":"Double-negation introduction under a box"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/locally-entails","kind":"node","lean":"Cslib.Modal.LocallyEntails","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#locally-entails","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $C$ be a class of Kripke models, given as a predicate on models. A set $\\Gamma$ of modal formulas locally entails a formula $\\varphi$ over $C$ when, for every Kripke model $M$ in $C$ whose type of worlds lies in the universe of the atoms and every world $w$ of $M$, if every formula of $\\Gamma$ is satisfied at $w$ in $M$, then $\\varphi$ is satisfied at $w$ in $M$.","title":"Local semantic consequence over a class of Kripke models"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/completeness-k","kind":"node","lean":"Cslib.Modal.completeness_K","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#completeness-k","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Gamma$ be a set of formulas and $\\varphi$ a formula. Suppose that for every Kripke model whose worlds form a type in the universe of the atoms, and every world $w$ of it at which all formulas of $\\Gamma$ hold, $\\varphi$ holds at $w$. Then $\\varphi$ is deducible from $\\Gamma$ in K.","title":"Strong completeness of K"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/completeness-t","kind":"node","lean":"Cslib.Modal.completeness_T","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#completeness-t","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Gamma$ be a set of formulas and $\\varphi$ a formula. Suppose that for every Kripke model with reflexive accessibility relation whose worlds form a type in the universe of the atoms, and every world $w$ of it at which all formulas of $\\Gamma$ hold, $\\varphi$ holds at $w$. Then $\\varphi$ is deducible from $\\Gamma$ in T.","title":"Strong completeness of T"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/completeness-s4","kind":"node","lean":"Cslib.Modal.completeness_S4","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#completeness-s4","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Gamma$ be a set of formulas and $\\varphi$ a formula. Suppose that for every Kripke model with reflexive and transitive accessibility relation whose worlds form a type in the universe of the atoms, and every world $w$ of it at which all formulas of $\\Gamma$ hold, $\\varphi$ holds at $w$. Then $\\varphi$ is deducible from $\\Gamma$ in S4.","title":"Strong completeness of S4"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/completeness-s5","kind":"node","lean":"Cslib.Modal.completeness_S5","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#completeness-s5","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Gamma$ be a set of formulas and $\\varphi$ a formula. Suppose that for every Kripke model whose accessibility relation is an equivalence relation and whose worlds form a type in the universe of the atoms, and every world $w$ of it at which all formulas of $\\Gamma$ hold, $\\varphi$ holds at $w$. Then $\\varphi$ is deducible from $\\Gamma$ in S5.","title":"Strong completeness of S5"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-k-iff","kind":"node","lean":"Cslib.Modal.derivable_K_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-k-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of K if and only if $\\varphi$ is valid, that is, true at every world, in every Kripke model whose worlds live in the universe of the atoms.","title":"Soundness and completeness of K"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-t-iff","kind":"node","lean":"Cslib.Modal.derivable_T_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-t-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of T if and only if $\\varphi$ is valid in every Kripke model whose accessibility relation is reflexive and whose type of worlds lies in the same universe as the type of atoms.","title":"Soundness and completeness of T"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-s4-iff","kind":"node","lean":"Cslib.Modal.derivable_S4_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-s4-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of S4 if and only if $\\varphi$ is valid in every Kripke model whose accessibility relation is reflexive and transitive and whose type of worlds lies in the same universe as the type of atoms.","title":"Soundness and completeness of S4"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-s5-iff","kind":"node","lean":"Cslib.Modal.derivable_S5_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-s5-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of S5 if and only if $\\varphi$ is valid in every Kripke model whose accessibility relation is an equivalence relation and whose type of worlds lies in the same universe as the type of atoms.","title":"Soundness and completeness of S5"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/satisfiable-of-consistent","kind":"node","lean":"Cslib.Modal.satisfiable_of_consistent","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#satisfiable-of-consistent","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of modal formulas and $C$ a class of Kripke models whose worlds lie in the universe of the atoms, such that $C$ contains the canonical model of $K \\oplus \\Lambda$ whenever $\\bot$ is not a theorem of $K \\oplus \\Lambda$. Then every $(K \\oplus \\Lambda)$-consistent set $\\Gamma$ of formulas is satisfiable in $C$: some model in $C$ has a world at which every formula of $\\Gamma$ holds.","title":"Model existence from the canonical model"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/satisfiable-k","kind":"node","lean":"Cslib.Modal.satisfiable_K","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#satisfiable-k","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every K-consistent set $\\Gamma$ of modal formulas there are a Kripke model, whose worlds lie in the universe of the atoms, and a world of it at which every formula of $\\Gamma$ holds.","title":"Model existence for K"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/satisfiable-t","kind":"node","lean":"Cslib.Modal.satisfiable_T","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#satisfiable-t","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every T-consistent set $\\Gamma$ of modal formulas there are a reflexive Kripke model, whose worlds lie in the universe of the atoms, and a world of it at which every formula of $\\Gamma$ holds.","title":"Model existence for T"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/satisfiable-s4","kind":"node","lean":"Cslib.Modal.satisfiable_S4","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#satisfiable-s4","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every S4-consistent set $\\Gamma$ of modal formulas there are a reflexive transitive Kripke model, whose worlds lie in the universe of the atoms, and a world of it at which every formula of $\\Gamma$ holds.","title":"Model existence for S4"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/satisfiable-s5","kind":"node","lean":"Cslib.Modal.satisfiable_S5","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#satisfiable-s5","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every S5-consistent set $\\Gamma$ of modal formulas there are a Kripke model whose accessibility relation is an equivalence and whose worlds lie in the universe of the atoms, and a world of it at which every formula of $\\Gamma$ holds.","title":"Model existence for S5"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/deduces-k-iff","kind":"node","lean":"Cslib.Modal.deduces_K_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#deduces-k-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every set $\\Gamma$ of modal formulas and every modal formula $\\varphi$, $\\varphi$ is a local consequence of $\\Gamma$ in K if and only if $\\Gamma$ locally entails $\\varphi$ over all Kripke models: $\\varphi$ holds at every world of every Kripke model, with worlds in the universe of the atoms, at which every formula of $\\Gamma$ holds.","title":"Strong soundness and completeness of K"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/deduces-t-iff","kind":"node","lean":"Cslib.Modal.deduces_T_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#deduces-t-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every set $\\Gamma$ of modal formulas and every modal formula $\\varphi$, $\\varphi$ is a local consequence of $\\Gamma$ in T if and only if $\\Gamma$ locally entails $\\varphi$ over reflexive Kripke models: $\\varphi$ holds at every world of every Kripke model with reflexive accessibility relation, with worlds in the universe of the atoms, at which every formula of $\\Gamma$ holds.","title":"Strong soundness and completeness of T"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/deduces-s4-iff","kind":"node","lean":"Cslib.Modal.deduces_S4_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#deduces-s4-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every set $\\Gamma$ of modal formulas and every modal formula $\\varphi$, $\\varphi$ is a local consequence of $\\Gamma$ in S4 if and only if $\\Gamma$ locally entails $\\varphi$ over reflexive transitive Kripke models: $\\varphi$ holds at every world of every Kripke model with reflexive and transitive accessibility relation, with worlds in the universe of the atoms, at which every formula of $\\Gamma$ holds.","title":"Strong soundness and completeness of S4"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/deduces-s5-iff","kind":"node","lean":"Cslib.Modal.deduces_S5_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#deduces-s5-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For every set $\\Gamma$ of modal formulas and every modal formula $\\varphi$, $\\varphi$ is a local consequence of $\\Gamma$ in S5 if and only if $\\Gamma$ locally entails $\\varphi$ over Kripke models whose accessibility relation is an equivalence relation: $\\varphi$ holds at every world of every such Kripke model, with worlds in the universe of the atoms, at which every formula of $\\Gamma$ holds.","title":"Strong soundness and completeness of S5"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/canonical-cslib-model","kind":"node","lean":"Cslib.Modal.canonicalCslibModel","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#canonical-cslib-model","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For a set $\\Lambda$ of extra axioms, this is the unimodal model of CSLib's modal semantics whose worlds are the maximal consistent sets of $K \\oplus \\Lambda$, in which $M$ sees $N$ iff $\\varphi \\in N$ for every formula $\\varphi$ with $\\Box\\varphi \\in M$, and in which an atom $p$ is true at $M$ iff $p \\in M$.","title":"The canonical model as a CSLib model"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-of-proposition-of-mem-logic","kind":"node","lean":"Cslib.Modal.derivable_ofProposition_of_mem_logic","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-of-proposition-of-mem-logic","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\Lambda$ be a set of extra axioms, and let $S$ be a set of CSLib unimodal models whose worlds are the maximal consistent sets of $K \\oplus \\Lambda$. If $S$ contains the canonical model of $K \\oplus \\Lambda$, viewed as a CSLib model, and a CSLib proposition $\\varphi$ holds at every world of every model in $S$, then the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of\u2026","title":"Validity in a class containing the canonical model gives derivability"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-of-proposition-k-iff","kind":"node","lean":"Cslib.Modal.derivable_ofProposition_K_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-of-proposition-k-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\varphi$ be a proposition of CSLib's unimodal language over a type of atoms $A$. Then the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of K if and only if, for every type of worlds $W$ in the universe of $A$, $\\varphi$ belongs to CSLib's semantic logic K over $W$ and $A$, that is, $\\varphi$ holds at every world of every CSLib model with worlds $W$.","title":"Completeness for CSLib's K"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-of-proposition-t-iff","kind":"node","lean":"Cslib.Modal.derivable_ofProposition_T_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-of-proposition-t-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"For a CSLib unimodal proposition $\\varphi$, the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of the Hilbert system T if and only if, for every type of worlds in the universe of the atoms, $\\varphi$ belongs to CSLib's modal logic T over that type, that is, $\\varphi$ holds at every world of every CSLib model on that type whose accessibility relation is reflexive.","title":"Completeness for CSLib's T"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-of-proposition-s4-iff","kind":"node","lean":"Cslib.Modal.derivable_ofProposition_S4_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-of-proposition-s4-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\varphi$ be a proposition of CSLib's unimodal language over a type of atoms $A$. Then the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of S4 if and only if, for every type of worlds $W$ in the universe of $A$, $\\varphi$ belongs to CSLib's semantic logic S4 over $W$ and $A$, that is, $\\varphi$ holds at every world of every CSLib model with worlds $W$ whose accessibility relation is\u2026","title":"Completeness for CSLib's S4"},{"id":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/derivable-of-proposition-s5-iff","kind":"node","lean":"Cslib.Modal.derivable_ofProposition_S5_iff","parent":282,"result":true,"route":"../roadmap/Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic/index.html#derivable-of-proposition-s5-iff","scope":"Logic/modal-logic/soundness-and-completeness-of-normal-modal-logic","status":"fully_proved","summary":"Let $\\varphi$ be a proposition of CSLib's unimodal language over a type of atoms $A$. Then the translation of $\\varphi$ into the $\\bot$, $\\to$, $\\Box$ basis is a theorem of S5 if and only if, for every type of worlds $W$ in the universe of $A$, $\\varphi$ belongs to CSLib's semantic logic S5 over $W$ and $A$, that is, $\\varphi$ holds at every world of every CSLib model with worlds $W$ whose accessibility relation\u2026","title":"Completeness for CSLib's S5"},{"data_url":"scopes/Logic/modal-logic/filtration-and-the-finite-model-property.json","id":"Logic/modal-logic/filtration-and-the-finite-model-property","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","title":"Filtration and the finite model property"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/sub","kind":"node","lean":"Cslib.Modal.Formula.sub","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#sub","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"The finite set $\\mathrm{sub}(\\varphi)$ of subformulas of a modal formula $\\varphi$, including $\\varphi$ itself, is defined by recursion: $\\mathrm{sub}(p) = \\{p\\}$ for an atom $p$, $\\mathrm{sub}(\\bot) = \\{\\bot\\}$, $\\mathrm{sub}(\\varphi \\to \\psi) = \\{\\varphi \\to \\psi\\} \\cup \\mathrm{sub}(\\varphi) \\cup \\mathrm{sub}(\\psi)$, and $\\mathrm{sub}(\\Box\\varphi) = \\{\\Box\\varphi\\} \\cup \\mathrm{sub}(\\varphi)$.","title":"Subformulas"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/sub-subset-of-mem","kind":"node","lean":"Cslib.Modal.Formula.sub_subset_of_mem","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#sub-subset-of-mem","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"If $\\psi$ is a subformula of a modal formula $\\varphi$, then every subformula of $\\psi$ is a subformula of $\\varphi$.","title":"The subformula relation is transitive"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/fin-model","kind":"node","lean":"Cslib.Modal.finModel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#fin-model","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $U$ be a nonempty finite set of finite sets of formulas and $R$ a finite set of pairs of finite sets of formulas. The Kripke model on $U$ with relation $R$ has the elements of $U$ as worlds, a world $s$ sees a world $t$ iff $(s, t) \\in R$, and an atom $p$ is true at $s$ iff $p \\in s$.","title":"Finite Kripke model on sets of formulas"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/eval-fin","kind":"node","lean":"Cslib.Modal.evalFin","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#eval-fin","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Given a finite set $U$ of finite sets of formulas, a finite set $R$ of pairs of such sets and a finite set $s$ of formulas, the Boolean value of a formula at $s$ is defined by recursion on the formula: an atom $p$ is true iff $p \\in s$; $\\bot$ is false; $\\varphi \\to \\psi$ is true iff $\\varphi$ is false or $\\psi$ is true at $s$; and $\\Box\\varphi$ is true iff $\\varphi$ is true at every $t \\in U$ with $(s, t) \\in R$.","title":"Boolean evaluation in a finite model"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/sub-type","kind":"node","lean":"Cslib.Modal.subType","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#sub-type","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Given a Kripke model $M$ and a modal formula $\\varphi$, the type (or $\\Sigma$-type, for $\\Sigma$ the set of subformulas of $\\varphi$) of a world $w$ of $M$ is the finite set of subformulas of $\\varphi$ that hold at $w$ in $M$.","title":"Type of a world"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/filtration-worlds","kind":"node","lean":"Cslib.Modal.filtrationWorlds","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#filtration-worlds","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a modal formula. The worlds of the filtration of $M$ through the subformulas of $\\varphi$ form the finite set of types occurring in $M$: the subsets $s$ of the set $\\mathrm{sub}(\\varphi)$ of subformulas of $\\varphi$ such that $s$ is the set of subformulas of $\\varphi$ satisfied at some world of $M$.","title":"Worlds of the filtration"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/filtration-rel","kind":"node","lean":"Cslib.Modal.filtrationRel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#filtration-rel","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a modal formula, and call the set of subformulas of $\\varphi$ satisfied at a world $w$ of $M$ the type of $w$. The smallest filtration relation is the finite set of pairs $(s, t)$ of types such that there are worlds $w$ and $w'$ of $M$ of types $s$ and $t$ with $w'$ accessible from $w$.","title":"Smallest filtration relation"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/exists-sub-type-eq","kind":"node","lean":"Cslib.Modal.exists_subType_eq","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#exists-sub-type-eq","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a modal formula, and call the set of subformulas of $\\varphi$ satisfied at a world $w$ of $M$ the type of $w$. Every world $s$ of the filtration of $M$ through the subformulas of $\\varphi$, that is, every element of the finite set of types occurring in $M$, is the type of some world of $M$.","title":"Filtration worlds are realized types"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/satisfies-iff-of-sub-type-eq","kind":"node","lean":"Cslib.Modal.satisfies_iff_of_subType_eq","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#satisfies-iff-of-sub-type-eq","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a modal formula, and call the set of subformulas of $\\varphi$ true at a world $w$ of $M$ the type of $w$. If two worlds $w$ and $v$ of $M$ have the same type, then every subformula $\\psi$ of $\\varphi$ holds at $w$ iff it holds at $v$.","title":"Worlds of the same type agree on subformulas"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/is-filtration","kind":"node","lean":"Cslib.Modal.IsFiltration","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#is-filtration","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ a modal formula, $\\Sigma$ the set of subformulas of $\\varphi$, and for a world $w$ let $[w]$, the type of $w$, be the set of formulas of $\\Sigma$ true at $w$ in $M$. A relation $R$ on finite sets of formulas is a filtration of $M$ through $\\Sigma$ when (min) for all worlds $w, v$ related in $M$, the pair $([w], [v])$ is in $R$, and (max) for all worlds $w, v$ with\u2026","title":"Filtration relations"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/satisfies-fin-model-filtration-worlds-if","kind":"node","lean":"Cslib.Modal.satisfies_finModel_filtrationWorlds_iff","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#satisfies-fin-model-filtration-worlds-if","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ a modal formula, and for a world $w$ let $[w]$, the type of $w$, be the set of subformulas of $\\varphi$ true at $w$ in $M$. Let $R$ be a filtration of $M$ through the subformulas of $\\varphi$, and let $M^R$ be the Kripke model whose worlds are the types of the worlds of $M$, with accessibility relation $R$ and an atom true at a type iff the type contains it. Then for every\u2026","title":"Filtration lemma"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/is-filtration-filtration-rel","kind":"node","lean":"Cslib.Modal.isFiltration_filtrationRel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#is-filtration-filtration-rel","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a modal formula, and call the set of subformulas of $\\varphi$ true at a world $w$ the type of $w$. The relation that relates two types when some worlds of those types are related in $M$ is a filtration of $M$ through the set of subformulas of $\\varphi$.","title":"The smallest filtration is a filtration"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/filtration","kind":"node","lean":"Cslib.Modal.filtration","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#filtration","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a modal formula, and call the set of subformulas of $\\varphi$ satisfied at a world $w$ of $M$ the type of $w$. The filtration of $M$ through the subformulas of $\\varphi$ is the Kripke model whose worlds are the types occurring in $M$, in which a type $s$ sees a type $t$ iff some world of type $s$ sees some world of type $t$ in $M$, and in which an atom $p$ is true at $s$ iff\u2026","title":"Smallest filtration of a Kripke model"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/satisfies-filtration-iff","kind":"node","lean":"Cslib.Modal.satisfies_filtration_iff","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#satisfies-filtration-iff","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ a modal formula, and for a world $w$ let $[w]$, the type of $w$, be the set of subformulas of $\\varphi$ true at $w$ in $M$. For every subformula $\\psi$ of $\\varphi$ and every world $w$ of $M$, $\\psi$ holds at $[w]$ in the smallest filtration of $M$ through the subformulas of $\\varphi$ iff $\\psi$ holds at $w$ in $M$.","title":"Filtration lemma for the smallest filtration"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/exists-not-satisfies-of-not-derivable-k","kind":"node","lean":"Cslib.Modal.exists_not_satisfies_of_not_derivable_K","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#exists-not-satisfies-of-not-derivable-k","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"A modal formula that is not a theorem of K fails at some world of some Kripke model.","title":"Completeness of K, countermodel form"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/exists-finite-countermodel","kind":"node","lean":"Cslib.Modal.exists_finite_countermodel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#exists-finite-countermodel","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"A formula that is not a K-theorem fails in a Kripke model with at most `2 ^ |sub \u03c6|` worlds.","title":"Finite model property of K"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/fin-valid","kind":"node","lean":"Cslib.Modal.FinValid","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#fin-valid","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"A modal formula $\\varphi$ passes the finite check when, for every nonempty set $U$ of subsets of the set $\\mathrm{sub}(\\varphi)$ of subformulas of $\\varphi$, every relation $R \\subseteq U \\times U$ and every $s \\in U$, the formula $\\varphi$ is true at $s$ in the Kripke model with set of worlds $U$, accessibility relation $R$, and an atom $p$ true at a world $t$ iff $p \\in t$; truth is computed by recursion on\u2026","title":"Finite validity check for K"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/derivable-k-iff-fin-valid","kind":"node","lean":"Cslib.Modal.derivable_K_iff_finValid","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#derivable-k-iff-fin-valid","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of K iff it passes the following finite check: for every nonempty set $U$ of subsets of the set $\\mathrm{sub}(\\varphi)$ of subformulas of $\\varphi$, every relation $R \\subseteq U \\times U$ and every $s \\in U$, $\\varphi$ evaluates to true at $s$ in the finite Kripke model on $U$ with relation $R$ in which an atom $p$ is true at $s$ iff $p \\in s$.","title":"Finite-check criterion for K-theorems"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/derivable-k-iff-forall-fin-model","kind":"node","lean":"Cslib.Modal.derivable_K_iff_forall_finModel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#derivable-k-iff-forall-fin-model","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of K iff for every nonempty set $U$ of subsets of the set $\\mathrm{sub}(\\varphi)$ of subformulas of $\\varphi$ and every finite set $R$ of pairs of finite sets of formulas, $\\varphi$ holds at every world of the Kripke model whose worlds are the elements of $U$, in which $s$ sees $t$ iff $(s, t) \\in R$, and in which an atom $p$ is true at $s$ iff $p \\in s$.","title":"K-theorems via finite models on sets of subformulas"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/not-valid-fin-model-of-is-filtration","kind":"node","lean":"Cslib.Modal.not_valid_finModel_of_isFiltration","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#not-valid-fin-model-of-is-filtration","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model, $\\varphi$ a formula and $R$ a filtration of $M$ through the subformulas of $\\varphi$. If $\\varphi$ fails at some world of $M$, then $\\varphi$ is not valid in the finite model whose worlds are the types of worlds of $M$ (the sets of subformulas of $\\varphi$ satisfied at a world), with relation $R$ and an atom true at a type iff it belongs to it.","title":"A filtration refutes what the model refutes"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/lemmon-rel","kind":"node","lean":"Cslib.Modal.lemmonRel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#lemmon-rel","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a formula, and let $U_\\varphi$ be the set of types of worlds of $M$ (the sets of subformulas of $\\varphi$ satisfied at a world). The Lemmon relation is the relation $R$ on $U_\\varphi$ such that $s \\mathrel{R} t$ iff for every formula $\\psi$ with $\\Box\\psi$ a subformula of $\\varphi$, $\\Box\\psi \\in s$ implies $\\Box\\psi \\in t$ and $\\psi \\in t$.","title":"Lemmon filtration relation"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/s5-rel","kind":"node","lean":"Cslib.Modal.s5Rel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#s5-rel","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model and $\\varphi$ a formula, and let $U_\\varphi$ be the set of types of worlds of $M$ (the sets of subformulas of $\\varphi$ satisfied at a world). The S5 relation is the relation $R$ on $U_\\varphi$ such that $s \\mathrel{R} t$ iff for every formula $\\psi$ with $\\Box\\psi$ a subformula of $\\varphi$, $\\Box\\psi \\in s$ iff $\\Box\\psi \\in t$.","title":"S5 filtration relation"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/is-filtration-lemmon-rel","kind":"node","lean":"Cslib.Modal.isFiltration_lemmonRel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#is-filtration-lemmon-rel","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"For every Kripke model $M$ with a transitive accessibility relation and every formula $\\varphi$, the Lemmon relation of $M$ and $\\varphi$ is a filtration of $M$ through the subformulas of $\\varphi$.","title":"The Lemmon relation is a filtration of a transitive model"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/is-filtration-s5-rel","kind":"node","lean":"Cslib.Modal.isFiltration_s5Rel","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#is-filtration-s5-rel","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"For every Kripke model $M$ whose accessibility relation is an equivalence relation and every formula $\\varphi$, the S5 relation of $M$ and $\\varphi$ is a filtration of $M$ through the subformulas of $\\varphi$.","title":"The S5 relation is a filtration of an equivalence model"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/fin-refl","kind":"node","lean":"Cslib.Modal.FinRefl","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#fin-refl","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $U$ be a finite set of finite sets of modal formulas and $R$ a finite set of pairs of such sets. Then $R$ is reflexive on $U$ when $(s, s) \\in R$ for every $s \\in U$.","title":"Reflexivity on a finite set of worlds"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/fin-trans","kind":"node","lean":"Cslib.Modal.FinTrans","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#fin-trans","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $U$ be a finite set of finite sets of modal formulas and $R$ a finite set of pairs of such sets. Then $R$ is transitive on $U$ when $(s, r) \\in R$ for all $s, t, r \\in U$ with $(s, t) \\in R$ and $(t, r) \\in R$.","title":"Transitivity on a finite set of worlds"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/fin-symm","kind":"node","lean":"Cslib.Modal.FinSymm","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#fin-symm","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $U$ be a finite set of finite sets of modal formulas and $R$ a finite set of pairs of such sets. Then $R$ is symmetric on $U$ when $(t, s) \\in R$ for all $s, t \\in U$ with $(s, t) \\in R$.","title":"Symmetry on a finite set of worlds"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/fin-refl-of-is-filtration","kind":"node","lean":"Cslib.Modal.finRefl_of_isFiltration","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#fin-refl-of-is-filtration","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $M$ be a Kripke model whose accessibility relation is reflexive, let $\\varphi$ be a modal formula, and let $R$ be a filtration of $M$ through the subformulas of $\\varphi$. Then $(s, s) \\in R$ for every set $s$ that is the set of subformulas of $\\varphi$ true at some world of $M$.","title":"Filtrations of reflexive models are reflexive"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/exists-finite-countermodel-t","kind":"node","lean":"Cslib.Modal.exists_finite_countermodel_T","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#exists-finite-countermodel-t","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"If a modal formula $\\varphi$ is not a theorem of T, then there is a Kripke model with finitely many worlds, at most $2^{|\\mathrm{sub}(\\varphi)|}$ of them, where $\\mathrm{sub}(\\varphi)$ is the set of subformulas of $\\varphi$, whose accessibility relation is reflexive and in which $\\varphi$ is false at some world.","title":"Finite model property of T"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/exists-finite-countermodel-s4","kind":"node","lean":"Cslib.Modal.exists_finite_countermodel_S4","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#exists-finite-countermodel-s4","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"If a modal formula $\\varphi$ is not a theorem of S4, then there is a Kripke model with finitely many worlds, at most $2^{|\\mathrm{sub}(\\varphi)|}$ of them, where $\\mathrm{sub}(\\varphi)$ is the set of subformulas of $\\varphi$, whose accessibility relation is reflexive and transitive and in which $\\varphi$ is false at some world.","title":"Finite model property of S4"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/exists-finite-countermodel-s5","kind":"node","lean":"Cslib.Modal.exists_finite_countermodel_S5","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#exists-finite-countermodel-s5","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"If a modal formula $\\varphi$ is not a theorem of S5, then there is a Kripke model with finitely many worlds, at most $2^{|\\mathrm{sub}(\\varphi)|}$ of them, where $\\mathrm{sub}(\\varphi)$ is the set of subformulas of $\\varphi$, whose accessibility relation is an equivalence relation and in which $\\varphi$ is false at some world.","title":"Finite model property of S5"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/fin-valid-with","kind":"node","lean":"Cslib.Modal.FinValidWith","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#fin-valid-with","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Given a condition $C$ on pairs $(U, R)$ of a finite set $U$ of sets of formulas and a finite set $R$ of pairs of such sets, a formula $\\varphi$ passes the finite check for $C$ when, for every nonempty set $U$ of subsets of the subformulas of $\\varphi$ and every relation $R \\subseteq U \\times U$ with $C(U, R)$, $\\varphi$ holds at every $s \\in U$ in the Kripke model with worlds $U$, accessibility relation $R$, and an\u2026","title":"Finite validity check for a frame condition"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/fin-valid-with-apply","kind":"node","lean":"Cslib.Modal.finValidWith_apply","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#fin-valid-with-apply","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"Let $C$ be a condition on pairs $(U, R)$ of a finite set $U$ of sets of formulas and a finite relation $R$ on such sets, and let $\\varphi$ be a formula that passes the finite check for $C$. Let $M$ be a Kripke model, let $U_\\varphi$ be the set of types of worlds of $M$ (the sets of subformulas of $\\varphi$ satisfied at a world), and let $R \\subseteq U_\\varphi \\times U_\\varphi$ satisfy $C(U_\\varphi, R)$. Then\u2026","title":"The finite check covers the filtration worlds"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/derivable-t-iff-fin-valid-with","kind":"node","lean":"Cslib.Modal.derivable_T_iff_finValidWith","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#derivable-t-iff-fin-valid-with","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of T if and only if, for every nonempty set $U$ of subsets of $\\mathrm{sub}(\\varphi)$, the set of subformulas of $\\varphi$, and every relation $R \\subseteq U \\times U$ that is reflexive on $U$, the formula $\\varphi$ is true at every world $s \\in U$ of the finite model with worlds $U$, accessibility relation $R$, and each atom true at $s$ exactly when it belongs to $s$.","title":"T-provability via finite reflexive models"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/derivable-s4-iff-fin-valid-with","kind":"node","lean":"Cslib.Modal.derivable_S4_iff_finValidWith","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#derivable-s4-iff-fin-valid-with","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of S4 if and only if, for every nonempty set $U$ of subsets of $\\mathrm{sub}(\\varphi)$, the set of subformulas of $\\varphi$, and every relation $R \\subseteq U \\times U$ that is reflexive and transitive on $U$, the formula $\\varphi$ is true at every world $s \\in U$ of the finite model with worlds $U$, accessibility relation $R$, and each atom true at $s$ exactly when it belongs\u2026","title":"S4-provability via finite reflexive transitive models"},{"id":"Logic/modal-logic/filtration-and-the-finite-model-property/derivable-s5-iff-fin-valid-with","kind":"node","lean":"Cslib.Modal.derivable_S5_iff_finValidWith","parent":336,"result":true,"route":"../roadmap/Logic/modal-logic/filtration-and-the-finite-model-property/index.html#derivable-s5-iff-fin-valid-with","scope":"Logic/modal-logic/filtration-and-the-finite-model-property","status":"fully_proved","summary":"A modal formula $\\varphi$ is a theorem of S5 if and only if, for every nonempty set $U$ of subsets of $\\mathrm{sub}(\\varphi)$, the set of subformulas of $\\varphi$, and every relation $R \\subseteq U \\times U$ that is reflexive, symmetric and transitive on $U$, the formula $\\varphi$ is true at every world $s \\in U$ of the finite model with worlds $U$, accessibility relation $R$, and each atom true at $s$ exactly when\u2026","title":"S5-provability via finite equivalence models"},{"data_url":"scopes/Logic/modal-logic/standard-translation-of-modal-logic.json","id":"Logic/modal-logic/standard-translation-of-modal-logic","kind":"scope","navigation":"module","parent":2,"result":false,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","title":"Standard translation of modal logic"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/modal-relations","kind":"node","lean":"Cslib.Modal.ModalRelations","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#modal-relations","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"Relation symbols of the modal first-order language: accessibility `R` (binary) and a unary predicate `P_a` for each atom `a`.","title":"Relation symbols of the modal first-order language"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/modal-language","kind":"node","lean":"Cslib.Modal.modalLanguage","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#modal-language","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"For a set of atoms, the first-order language of modal logic has no function symbols; its relation symbols are a binary symbol $R$ (accessibility) and a unary symbol $P_a$ for each atom $a$.","title":"The first-order modal language"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/frame-relations","kind":"node","lean":"Cslib.Modal.FrameRelations","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#frame-relations","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"The relation symbols of the first-order frame language, indexed by arity: a single binary symbol $R$ for accessibility, and no symbols of any other arity.","title":"Relation symbols of the frame language"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/frame-language","kind":"node","lean":"Cslib.Modal.frameLanguage","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#frame-language","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"The first-order language of frames: no function symbols, one binary relation `R`.","title":"First-order language of Kripke frames"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/kripke-structure","kind":"node","lean":"Cslib.Modal.kripkeStructure","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#kripke-structure","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"A Kripke model $M$ with set of worlds $W$ makes $W$ a structure for the modal language: the accessibility symbol $R$ is interpreted as the accessibility relation of $M$, and the predicate $P_a$ for an atom $a$ holds at exactly the worlds where $a$ is true under the valuation of $M$.","title":"A Kripke model as a first-order structure"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/frame-structure","kind":"node","lean":"Cslib.Modal.frameStructure","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#frame-structure","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"A binary relation $r$ on a set $W$ (a bare Kripke frame) makes $W$ a structure for the frame language, in which the accessibility symbol $R$ is interpreted as $r$: $R(w, w')$ holds if and only if $r(w, w')$.","title":"A frame as a first-order structure"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/standard-translation","kind":"node","lean":"Cslib.Modal.standardTranslation","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#standard-translation","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"For $n$ variables $x_0, \\dots, x_{n-1}$ in scope and a term $t$ of the modal language for the current world, the standard translation $ST_t(\\varphi)$ of a modal formula $\\varphi$ is the first-order formula over the modal language in these variables defined by recursion on $\\varphi$: $ST_t(a) = P_a(t)$ for an atom $a$; $ST_t(\\bot) = \\bot$; $ST_t(\\varphi \\to \\psi) = ST_t(\\varphi) \\to ST_t(\\psi)$; and\u2026","title":"Standard translation"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/realize-standard-translation","kind":"node","lean":"Cslib.Modal.realize_standardTranslation","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#realize-standard-translation","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"Let $M$ be a Kripke model with set of worlds $W$, viewed as a structure for the modal language. For every $n$, every term $t$ of the modal language in the variables $x_0, \\dots, x_{n-1}$, every modal formula $\\varphi$ and every assignment of worlds in $W$ to $x_0, \\dots, x_{n-1}$, the standard translation of $\\varphi$ with $n$ variables in scope and current-world term $t$ holds in $M$ under this assignment if and\u2026","title":"Correctness of the standard translation"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/bisimulation-invariant","kind":"node","lean":"Cslib.Modal.BisimulationInvariant","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#bisimulation-invariant","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"A first-order formula $\\alpha$ over the modal language, with one variable $x$ and no other free variables, is bisimulation-invariant if for all Kripke models $M$ and $N$ over the same atoms, every bisimulation $Z$ between $M$ and $N$ and all worlds $w$ of $M$ and $v$ of $N$ with $Z(w, v)$, the formula $\\alpha$ holds in $M$ (viewed as a structure for the modal language) at $x = w$ if and only if it holds in $N$ at\u2026","title":"Bisimulation-invariant formulas"},{"id":"Logic/modal-logic/standard-translation-of-modal-logic/bisimulation-invariant-standard-translat","kind":"node","lean":"Cslib.Modal.bisimulationInvariant_standardTranslation","parent":373,"result":true,"route":"../roadmap/Logic/modal-logic/standard-translation-of-modal-logic/index.html#bisimulation-invariant-standard-translat","scope":"Logic/modal-logic/standard-translation-of-modal-logic","status":"fully_proved","summary":"For every modal formula $\\varphi$, its standard translation $ST_x(\\varphi)$ (the first-order formula over the modal language whose single variable $x$ stands for the current world) is bisimulation-invariant: for all Kripke models $M$ and $N$, every bisimulation $Z$ between $M$ and $N$ and all worlds $w$ of $M$ and $v$ of $N$ with $Z(w, v)$, the formula $ST_x(\\varphi)$ holds in $M$ at $x = w$ if and only if it holds\u2026","title":"Modal formulas are bisimulation-invariant under the standard translation"},{"data_url":"scopes/Logic/propositional-logic.json","id":"Logic/propositional-logic","kind":"scope","parent":1,"result":false,"route":"../roadmap/Logic/propositional-logic/index.html","scope":"Logic/propositional-logic","status":"fully_proved","summary":"Classical propositional logic: truth-table semantics, natural deduction and its completeness, functional completeness, normal forms and CNF transformations, resolution, interpolation, SAT-solving procedures (Davis\u2013Putnam, DPLL, 2-SAT, Horn-SAT) and binary decision diagrams.","title":"Propositional logic and satisfiability"},{"data_url":"scopes/Logic/propositional-logic/propositional-natural-deduction.json","id":"Logic/propositional-logic/propositional-natural-deduction","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","title":"Propositional natural deduction"},{"id":"Logic/propositional-logic/propositional-natural-deduction/eval-bool","kind":"node","lean":"Cslib.Logic.PL.Proposition.evalBool","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#eval-bool","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Given a Boolean valuation $v$ of the atoms, the truth value of a proposition is computed recursively: an atom $x$ has value $v(x)$, $A \\wedge B$ is true iff both $A$ and $B$ are, $A \\vee B$ is true iff at least one of them is, and $A \\to B$ is true iff $A$ is false or $B$ is true. Falsum $\\bot$, when present, is itself an atom, so its value $v(\\bot)$ is part of the valuation and need not be false.","title":"Truth-table evaluation"},{"id":"Logic/propositional-logic/propositional-natural-deduction/eval-bool-neg","kind":"node","lean":"Cslib.Logic.PL.Proposition.evalBool_neg","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#eval-bool-neg","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Suppose the atoms include a distinguished atom $\\bot$, and let $v$ be a Boolean valuation of the atoms with $v(\\bot) = \\mathrm{false}$. Then for every proposition $A$, the truth-table value under $v$ of $\\neg A$, defined as $A \\to \\bot$, is the Boolean negation of the truth-table value of $A$ under $v$.","title":"Truth value of a negation"},{"id":"Logic/propositional-logic/propositional-natural-deduction/bool-entails","kind":"node","lean":"Cslib.Logic.PL.Theory.BoolEntails","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#bool-entails","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory (a set of propositions), $\\Gamma$ a context (a finite set of propositions) and $A$ a proposition. Then $A$ is a truth-table consequence of $\\Gamma$ in $T$ when every Boolean valuation of the atoms under which all members of $T$ and all members of $\\Gamma$ evaluate to true also makes $A$ evaluate to true.","title":"Truth-table consequence"},{"id":"Logic/propositional-logic/propositional-natural-deduction/bool-sound","kind":"node","lean":"Cslib.Logic.PL.Theory.Derivation.boolSound","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#bool-sound","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory, $\\Gamma$ a context and $A$ a proposition. For every natural-deduction derivation of $\\Gamma \\vdash A$ with axioms from $T$, every Boolean valuation of the atoms that makes all members of $T$ and of $\\Gamma$ true makes $A$ true.","title":"Soundness of natural deduction for truth tables"},{"id":"Logic/propositional-logic/propositional-natural-deduction/bool-sound-2","kind":"node","lean":"Cslib.Logic.PL.DerivableIn.boolSound","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#bool-sound-2","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory, $\\Gamma$ a context and $A$ a proposition. If the sequent $\\Gamma \\vdash A$ is derivable in natural deduction with axioms from $T$, then $A$ is a truth-table consequence of $\\Gamma$ in $T$: every Boolean valuation of the atoms that makes all members of $T$ and of $\\Gamma$ true makes $A$ true.","title":"Truth-table soundness of derivability"},{"id":"Logic/propositional-logic/propositional-natural-deduction/eval-bool-of-mem-ipl","kind":"node","lean":"Cslib.Logic.PL.evalBool_of_mem_ipl","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#eval-bool-of-mem-ipl","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Suppose the atoms include a distinguished atom $\\bot$, and let $v$ be a Boolean valuation of the atoms with $v(\\bot) = \\mathrm{false}$. Then every instance $\\bot \\to B$ of ex falso, that is every axiom of intuitionistic propositional logic, is true under $v$.","title":"Ex falso is true when falsum is false"},{"id":"Logic/propositional-logic/propositional-natural-deduction/eval-bool-of-mem-cpl","kind":"node","lean":"Cslib.Logic.PL.evalBool_of_mem_cpl","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#eval-bool-of-mem-cpl","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Suppose the atoms include a distinguished atom $\\bot$, and let $v$ be a Boolean valuation of the atoms with $v(\\bot) = \\mathrm{false}$. Then every instance $\\neg\\neg B \\to B$ of double-negation elimination, that is every axiom of classical propositional logic, is true under $v$.","title":"Double-negation elimination is true when falsum is false"},{"id":"Logic/propositional-logic/propositional-natural-deduction/eval-bool-of-cpl","kind":"node","lean":"Cslib.Logic.PL.DerivableIn.evalBool_of_cpl","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#eval-bool-of-cpl","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Suppose the atoms include a distinguished atom $\\bot$, and let $v$ be a Boolean valuation of the atoms with $v(\\bot) = \\mathrm{false}$. If the sequent $\\Gamma \\vdash A$ is derivable in classical propositional logic (natural deduction whose axioms are all instances $\\neg\\neg B \\to B$ of double-negation elimination) and every member of $\\Gamma$ is true under $v$, then $A$ is true under $v$.","title":"Classical derivability implies truth under valuations with false falsum"},{"id":"Logic/propositional-logic/propositional-natural-deduction/eval-bool-of-ipl","kind":"node","lean":"Cslib.Logic.PL.DerivableIn.evalBool_of_ipl","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#eval-bool-of-ipl","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Suppose the atoms include a distinguished atom $\\bot$, and let $v$ be a Boolean valuation of the atoms with $v(\\bot) = \\mathrm{false}$. If the sequent $\\Gamma \\vdash A$ is derivable in intuitionistic propositional logic (natural deduction whose axioms are all instances $\\bot \\to B$ of ex falso) and every member of $\\Gamma$ is true under $v$, then $A$ is true under $v$.","title":"Intuitionistic derivability implies truth under valuations with false falsum"},{"id":"Logic/propositional-logic/propositional-natural-deduction/not-derivable-in-cpl-bot","kind":"node","lean":"Cslib.Logic.PL.not_derivableIn_cpl_bot","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#not-derivable-in-cpl-bot","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"For every type of atoms with a designated atom $\\bot$, the proposition $\\bot$ is not derivable from the empty context in natural deduction over the classical theory, whose axioms are all instances $\\neg\\neg A \\to A$ of double-negation elimination, where $\\neg A$ abbreviates $A \\to \\bot$.","title":"Consistency of classical propositional logic"},{"id":"Logic/propositional-logic/propositional-natural-deduction/not-derivable-in-cpl-atom","kind":"node","lean":"Cslib.Logic.PL.not_derivableIn_cpl_atom","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#not-derivable-in-cpl-atom","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Suppose the atoms include a distinguished atom $\\bot$. For every atom $x$, including $\\bot$, the proposition $x$ is not derivable from the empty context in classical propositional logic (natural deduction whose axioms are all instances $\\neg\\neg B \\to B$ of double-negation elimination).","title":"Classical logic proves no atom"},{"id":"Logic/propositional-logic/propositional-natural-deduction/bool-satisfiable","kind":"node","lean":"Cslib.Logic.PL.BoolSatisfiable","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#bool-satisfiable","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"A set $S$ of propositions is satisfiable when some Boolean valuation $v$ of the atoms with $v(\\bot) = \\mathrm{false}$ makes every member of $S$ true.","title":"Satisfiable sets of propositions"},{"id":"Logic/propositional-logic/propositional-natural-deduction/consistent-with","kind":"node","lean":"Cslib.Logic.PL.Theory.ConsistentWith","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#consistent-with","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory, that is, a set of propositions. A set $S$ of propositions is consistent relative to $T$ when there is no finite set $\\Gamma \\subseteq S$ such that the sequent $\\Gamma \\vdash \\bot$ is derivable in natural deduction from $T$.","title":"Consistency relative to a theory"},{"id":"Logic/propositional-logic/propositional-natural-deduction/exists-maximal-consistent-with","kind":"node","lean":"Cslib.Logic.PL.Theory.exists_maximal_consistentWith","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#exists-maximal-consistent-with","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory. Every set $S$ of propositions that is consistent relative to $T$ is contained in a set $M$ that is consistent relative to $T$ and not properly contained in any other set consistent relative to $T$.","title":"Lindenbaum's lemma for propositional natural deduction"},{"id":"Logic/propositional-logic/propositional-natural-deduction/exists-refutation","kind":"node","lean":"Cslib.Logic.PL.MaximalConsistent.exists_refutation","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#exists-refutation","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory and $M$ a maximal set of propositions consistent relative to $T$. If a proposition $A$ is not in $M$, then there is a finite set $\\Gamma \\subseteq M$ such that the sequent $\\Gamma \\cup \\{A\\} \\vdash \\bot$ is derivable in natural deduction from $T$.","title":"Propositions outside a maximal consistent set are refuted by it"},{"id":"Logic/propositional-logic/propositional-natural-deduction/mem-of-derivable-in","kind":"node","lean":"Cslib.Logic.PL.MaximalConsistent.mem_of_derivableIn","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#mem-of-derivable-in","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory and $M$ a maximal set of propositions consistent relative to $T$. If $\\Gamma$ is a finite subset of $M$ and the sequent $\\Gamma \\vdash A$ is derivable in natural deduction from $T$, then $A \\in M$.","title":"Maximal consistent sets are deductively closed in propositional natural deduction"},{"id":"Logic/propositional-logic/propositional-natural-deduction/subset-theory","kind":"node","lean":"Cslib.Logic.PL.MaximalConsistent.subset_theory","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#subset-theory","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory and $M$ a maximal set of propositions consistent relative to $T$. Then every proposition of $T$ belongs to $M$, that is, $T \\subseteq M$.","title":"A maximal consistent set contains its theory"},{"id":"Logic/propositional-logic/propositional-natural-deduction/mem-or-neg-mem","kind":"node","lean":"Cslib.Logic.PL.MaximalConsistent.mem_or_neg_mem","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#mem-or-neg-mem","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory and $M$ a maximal set of propositions consistent relative to $T$. Then for every proposition $A$, either $A \\in M$ or $\\neg A \\in M$, where $\\neg A$ is the proposition $A \\to \\bot$.","title":"Each proposition or its negation lies in a maximal consistent set"},{"id":"Logic/propositional-logic/propositional-natural-deduction/eval-bool-eq-true-iff","kind":"node","lean":"Cslib.Logic.PL.MaximalConsistent.evalBool_eq_true_iff","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#eval-bool-eq-true-iff","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be an intuitionistic theory, that is, one that derives $\\bot \\to B$ for every proposition $B$. Let $M$ be a set of propositions that is consistent relative to $T$ and not properly contained in any other set consistent relative to $T$, and let $v_M$ be the valuation that makes an atom $x$ true exactly when the atomic proposition $x$ belongs to $M$. Then a proposition $A$ evaluates to true under $v_M$ iff\u2026","title":"Truth lemma for propositional natural deduction"},{"id":"Logic/propositional-logic/propositional-natural-deduction/bool-satisfiable-2","kind":"node","lean":"Cslib.Logic.PL.Theory.ConsistentWith.boolSatisfiable","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#bool-satisfiable-2","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be an intuitionistic theory, that is, one that derives $\\bot \\to B$ for every proposition $B$. If a set $S$ of propositions is consistent relative to $T$, then $T \\cup S$ is satisfiable: some Boolean valuation $v$ of the atoms with $v(\\bot) = \\mathrm{false}$ makes every member of $T$ and every member of $S$ true.","title":"Model existence for propositional natural deduction"},{"id":"Logic/propositional-logic/propositional-natural-deduction/derivable-in-of-forall-eval-bool","kind":"node","lean":"Cslib.Logic.PL.Theory.derivableIn_of_forall_evalBool","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#derivable-in-of-forall-eval-bool","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a classical theory, that is, one that derives $\\neg\\neg B \\to B$ for every proposition $B$, where $\\neg B$ abbreviates $B \\to \\bot$. Let $\\Gamma$ be a finite set of propositions and $A$ a proposition. If every Boolean valuation $v$ of the atoms with $v(\\bot) = \\mathrm{false}$ that makes every member of $T$ and every member of $\\Gamma$ true also makes $A$ true, then the sequent $\\Gamma \\vdash A$ is\u2026","title":"Strong completeness of classical propositional natural deduction"},{"id":"Logic/propositional-logic/propositional-natural-deduction/derivable-in-iff-forall-eval-bool","kind":"node","lean":"Cslib.Logic.PL.Theory.derivableIn_iff_forall_evalBool","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#derivable-in-iff-forall-eval-bool","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a classical theory, that is, one that derives $\\neg\\neg B \\to B$ for every proposition $B$, where $\\neg B$ abbreviates $B \\to \\bot$. For every finite set $\\Gamma$ of propositions and every proposition $A$, the sequent $\\Gamma \\vdash A$ is derivable from $T$ iff every Boolean valuation $v$ of the atoms with $v(\\bot) = \\mathrm{false}$ that makes every member of $T$ and every member of $\\Gamma$ true also\u2026","title":"Soundness and completeness for classical theories, with \u22a5 false"},{"id":"Logic/propositional-logic/propositional-natural-deduction/derivable-in-iff-bool-entails","kind":"node","lean":"Cslib.Logic.PL.Theory.derivableIn_iff_boolEntails","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#derivable-in-iff-bool-entails","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $T$ be a classical theory, that is, one that derives $\\neg\\neg B \\to B$ for every proposition $B$, where $\\neg B$ abbreviates $B \\to \\bot$. For every finite set $\\Gamma$ of propositions and every proposition $A$, the sequent $\\Gamma \\vdash A$ is derivable from $T$ iff every Boolean valuation of the atoms that makes every member of $T$ and every member of $\\Gamma$ true also makes $A$ true.","title":"Soundness and completeness for classical theories"},{"id":"Logic/propositional-logic/propositional-natural-deduction/derivable-in-cpl-iff","kind":"node","lean":"Cslib.Logic.PL.derivableIn_cpl_iff","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#derivable-in-cpl-iff","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"Let $\\Gamma$ be a finite set of propositions and $A$ a proposition. The sequent $\\Gamma \\vdash A$ is derivable in classical propositional logic (natural deduction with every instance of $\\neg\\neg B \\to B$ as an axiom, where $\\neg B$ abbreviates $B \\to \\bot$) iff every Boolean valuation $v$ of the atoms with $v(\\bot) = \\mathrm{false}$ that makes every member of $\\Gamma$ true also makes $A$ true.","title":"Completeness for truth tables"},{"id":"Logic/propositional-logic/propositional-natural-deduction/bool-satisfiable-iff-forall-finset","kind":"node","lean":"Cslib.Logic.PL.boolSatisfiable_iff_forall_finset","parent":385,"result":true,"route":"../roadmap/Logic/propositional-logic/propositional-natural-deduction/index.html#bool-satisfiable-iff-forall-finset","scope":"Logic/propositional-logic/propositional-natural-deduction","status":"fully_proved","summary":"A set of propositions is satisfiable, with `\u22a5` false, iff each of its finite subsets is.","title":"Compactness"},{"data_url":"scopes/Logic/propositional-logic/binary-decision-diagrams.json","id":"Logic/propositional-logic/binary-decision-diagrams","kind":"scope","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/index.html","scope":"Logic/propositional-logic/binary-decision-diagrams","status":"fully_proved","title":"Binary decision diagrams"},{"data_url":"scopes/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic.json","id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","kind":"scope","navigation":"module","parent":411,"result":false,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","title":"Reduced ordered binary decision diagrams: canonicity and apply"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/bdd","kind":"node","lean":"Cslib.Logic.PL.BDD","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#bdd","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"A binary decision diagram, in tree form, is either a leaf carrying a Boolean value, or a node consisting of a variable $v \\in \\mathbb{N}$ and two diagrams: the low child, for $v$ false, and the high child, for $v$ true.","title":"Binary decision diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/eval","kind":"node","lean":"Cslib.Logic.PL.BDD.eval","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#eval","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"Under an assignment $\\sigma$ of a Boolean value to each variable $v \\in \\mathbb{N}$, the value of a leaf is its Boolean value, and the value of a node testing $v$ is the value of its high child if $\\sigma(v)$ is true and the value of its low child if $\\sigma(v)$ is false.","title":"Evaluation of a binary decision diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/size","kind":"node","lean":"Cslib.Logic.PL.BDD.size","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#size","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"The size of a diagram is its number of leaves and nodes, counted in tree form: a leaf has size $1$, and a node with children $\\ell$ and $h$ has size one more than the sum of the sizes of $\\ell$ and $h$.","title":"Size of a diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/ordered-from","kind":"node","lean":"Cslib.Logic.PL.BDD.OrderedFrom","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#ordered-from","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"For a natural number $n$, a diagram is ordered from $n$ when every variable it tests is at least $n$ and the tested variables strictly increase along every path. Recursively: every leaf is ordered from $n$, and the node testing $v$ with children $\\ell$ and $h$ is ordered from $n$ when $n \\le v$ and both $\\ell$ and $h$ are ordered from $v + 1$.","title":"Diagrams ordered from a variable"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/reduced","kind":"node","lean":"Cslib.Logic.PL.BDD.Reduced","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#reduced","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"A diagram is reduced when no node in it has two equal children: every leaf is reduced, and the node testing $v$ with children $\\ell$ and $h$ is reduced when $\\ell \\ne h$ and both $\\ell$ and $h$ are reduced.","title":"Reduced diagrams"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/is-robdd","kind":"node","lean":"Cslib.Logic.PL.BDD.IsROBDD","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#is-robdd","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"A binary decision diagram is a reduced ordered binary decision diagram (ROBDD), for the natural order on variables, when it is ordered from $0$ (the tested variables strictly increase along every path) and reduced (no node has two equal children).","title":"Reduced ordered binary decision diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/eval-update-of-lt","kind":"node","lean":"Cslib.Logic.PL.BDD.eval_update_of_lt","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#eval-update-of-lt","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"Let $t$ be a binary decision diagram ordered from $n \\in \\mathbb{N}$ and let $v < n$ be a variable. For every assignment $\\sigma$ of Boolean values to the variables and every Boolean $b$, the value of $t$ under $\\sigma[v \\mapsto b]$ (the assignment $\\sigma$ with $v$ set to $b$) equals the value of $t$ under $\\sigma$.","title":"Independence from variables below the order"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/eval-update-node","kind":"node","lean":"Cslib.Logic.PL.BDD.eval_update_node","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#eval-update-node","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"Let $t$ be a node testing the variable $v$ with low child $\\ell$ and high child $h$, and suppose $t$ is ordered from some $n \\in \\mathbb{N}$. For every assignment $\\sigma$ of Boolean values to the variables and every Boolean $b$, the value of $t$ under $\\sigma[v \\mapsto b]$ (the assignment $\\sigma$ with $v$ set to $b$) is the value of $h$ under $\\sigma$ if $b$ is true, and the value of $\\ell$ under $\\sigma$ if $b$\u2026","title":"Fixing the top variable of an ordered node"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/size-pos","kind":"node","lean":"Cslib.Logic.PL.BDD.size_pos","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#size-pos","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"For every binary decision diagram $t$, the size of $t$ (its number of leaves and nodes, counted in tree form) is strictly positive.","title":"Binary decision diagrams have positive size"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/eq-of-eval-eq","kind":"node","lean":"Cslib.Logic.PL.BDD.eq_of_eval_eq","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#eq-of-eval-eq","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"Canonicity (Bryant): two reduced ordered diagrams with the same function are equal.","title":"Canonicity"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/eq-iff-eval-eq","kind":"node","lean":"Cslib.Logic.PL.BDD.eq_iff_eval_eq","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#eq-iff-eval-eq","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"If $t_1$ and $t_2$ are reduced ordered binary decision diagrams, then $t_1 = t_2$ if and only if $t_1$ and $t_2$ have the same value under every assignment of Boolean values to the variables.","title":"Reduced ordered diagrams are equal iff they have the same values"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/eq-leaf-false-iff","kind":"node","lean":"Cslib.Logic.PL.BDD.eq_leaf_false_iff","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#eq-leaf-false-iff","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"A reduced ordered binary decision diagram $t$ is the leaf with value false if and only if $t$ has value false under every assignment of Boolean values to the variables.","title":"Unsatisfiable reduced ordered diagrams"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/mk","kind":"node","lean":"Cslib.Logic.PL.BDD.mk","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#mk","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"For a variable $v \\in \\mathbb{N}$ and diagrams $\\ell$ and $h$, the diagram $\\mathrm{mk}(v, \\ell, h)$ is $\\ell$ if $\\ell = h$, and otherwise the node that tests $v$ with $\\ell$ as its false-child and $h$ as its true-child.","title":"Node constructor removing a redundant test"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/apply","kind":"node","lean":"Cslib.Logic.PL.BDD.apply","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#apply","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"For a binary Boolean operation $\\mathrm{op}$ and binary decision diagrams $t_1, t_2$, the diagram $A_{\\mathrm{op}}(t_1, t_2)$ is defined by recursion. If $t_1$ and $t_2$ are leaves with values $a$ and $b$, it is the leaf with value $\\mathrm{op}(a, b)$. Otherwise let $v$ be the smallest variable tested at the root of $t_1$ or of $t_2$ (a leaf tests none), and for $i = 1, 2$ let $\\ell_i$ and $h_i$ be the low and high\u2026","title":"Bryant's apply operation"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/eval-apply","kind":"node","lean":"Cslib.Logic.PL.BDD.eval_apply","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#eval-apply","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"For every binary Boolean operation $\\mathrm{op}$, all binary decision diagrams $t_1$ and $t_2$, and every assignment $\\sigma$ of Boolean values to the variables, the value under $\\sigma$ of the diagram $\\mathrm{apply}(\\mathrm{op}, t_1, t_2)$ obtained from $t_1$ and $t_2$ by Bryant's apply operation under $\\mathrm{op}$ is $\\mathrm{op}(a_1, a_2)$, where $a_i$ is the value of $t_i$ under $\\sigma$.","title":"Apply computes the operation pointwise"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/ordered-from-apply","kind":"node","lean":"Cslib.Logic.PL.BDD.orderedFrom_apply","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#ordered-from-apply","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"Let $\\mathrm{op}$ be a binary Boolean operation and $n$ a natural number. If the diagrams $t_1$ and $t_2$ are both ordered from $n$, then the result of apply with $\\mathrm{op}$ on $t_1$ and $t_2$ is ordered from $n$.","title":"Apply preserves orderedness"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/reduced-apply","kind":"node","lean":"Cslib.Logic.PL.BDD.reduced_apply","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#reduced-apply","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"For every binary Boolean operation $\\mathrm{op}$ and all diagrams $t_1$ and $t_2$, the result of apply with $\\mathrm{op}$ on $t_1$ and $t_2$ is reduced.","title":"Apply returns reduced diagrams"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/of-proposition","kind":"node","lean":"Cslib.Logic.PL.BDD.ofProposition","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#of-proposition","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"Each proposition $A$ whose atoms are natural numbers is sent to a binary decision diagram by recursion on $A$: an atom $x$ goes to the node testing $x$ with the leaf false as its false-child and the leaf true as its true-child, and a conjunction $A \\wedge B$, disjunction $A \\vee B$ or implication $A \\to B$ goes to the result of apply on the diagrams of $A$ and $B$ under the Boolean operation and, or, or\u2026","title":"Diagram of a proposition"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/of-proposition-eq-iff","kind":"node","lean":"Cslib.Logic.PL.BDD.ofProposition_eq_iff","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#of-proposition-eq-iff","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"For propositions $A$ and $B$ over natural-number atoms, the binary decision diagrams of $A$ and $B$ are equal if and only if $A$ and $B$ have the same truth value under every valuation $\\sigma : \\mathbb{N} \\to \\mathrm{Bool}$.","title":"Equal BDDs iff equal truth tables"},{"id":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/of-proposition-ne-leaf-false-iff","kind":"node","lean":"Cslib.Logic.PL.BDD.ofProposition_ne_leaf_false_iff","parent":412,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic/index.html#of-proposition-ne-leaf-false-iff","scope":"Logic/propositional-logic/binary-decision-diagrams/reduced-ordered-binary-decision-diagrams-canonic","status":"fully_proved","summary":"For a proposition $A$ over natural-number atoms, the binary decision diagram of $A$ is not the leaf false if and only if some valuation $\\sigma : \\mathbb{N} \\to \\mathrm{Bool}$ makes $A$ true.","title":"Satisfiable iff the BDD is not the false leaf"},{"data_url":"scopes/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd.json","id":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","kind":"scope","navigation":"module","parent":411,"result":false,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/index.html","scope":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","status":"fully_proved","title":"Counting satisfying assignments of a BDD"},{"id":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/vars-lt","kind":"node","lean":"Cslib.Logic.PL.BDD.VarsLT","parent":433,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/index.html#vars-lt","scope":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","status":"fully_proved","summary":"For a natural number $m$, a diagram has its variables below $m$ when every variable tested at a node of the diagram is less than $m$.","title":"Variables below a bound"},{"id":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/sat-count-from","kind":"node","lean":"Cslib.Logic.PL.BDD.satCountFrom","parent":433,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/index.html#sat-count-from","scope":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","status":"fully_proved","summary":"For natural numbers $m$ and $n$ and a diagram $t$, the satisfy-count $c_m(n, t)$ of $t$ from $n$ with bound $m$ is defined by recursion on $t$: a leaf $\\mathrm{true}$ gives $2^{m - n}$, a leaf $\\mathrm{false}$ gives $0$, and a node testing the variable $v$ with child $t_0$ for false and child $t_1$ for true gives $2^{v - n} (c_m(v + 1, t_0) + c_m(v + 1, t_1))$. Here $m - n$ and $v - n$ are truncated subtraction,\u2026","title":"One-pass satisfy-count from a variable"},{"id":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/sat-count","kind":"node","lean":"Cslib.Logic.PL.BDD.satCount","parent":433,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/index.html#sat-count","scope":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","status":"fully_proved","summary":"For a natural number $m$ and a diagram $t$, the satisfy-count of $t$ below $m$ is the one-pass satisfy-count $c_m(0, t)$ of $t$ from the variable $0$ with bound $m$.","title":"Satisfy-count"},{"id":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/count-models","kind":"node","lean":"Cslib.Logic.PL.BDD.countModels","parent":433,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/index.html#count-models","scope":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","status":"fully_proved","summary":"An assignment is a map $\\sigma$ from the variables, which are natural numbers, to the truth values $\\mathrm{false}$ and $\\mathrm{true}$. For a list $x_1, \\dots, x_k$ of variables, a function $f$ from assignments to truth values, and an assignment $\\sigma$, the model count of $f$ over $x_1, \\dots, x_k$ from $\\sigma$ is the number of tuples $(b_1, \\dots, b_k) \\in \\{\\mathrm{false}, \\mathrm{true}\\}^k$ such that $f$ is\u2026","title":"Model count by enumeration"},{"id":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/count-models-congr","kind":"node","lean":"Cslib.Logic.PL.BDD.countModels_congr","parent":433,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/index.html#count-models-congr","scope":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","status":"fully_proved","summary":"Let $x_1, \\dots, x_k$ be a list of variables, $f$ and $g$ functions from assignments to truth values, and $\\sigma$ an assignment. If $f(\\tau) = g(\\tau)$ for every assignment $\\tau$ such that $\\tau(x) = \\sigma(x)$ for every variable $x$ not in the list, then the model counts of $f$ and of $g$ over $x_1, \\dots, x_k$ from $\\sigma$ are equal.","title":"Congruence for the model count"},{"id":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/sat-count-from-eq-count-models","kind":"node","lean":"Cslib.Logic.PL.BDD.satCountFrom_eq_countModels","parent":433,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/index.html#sat-count-from-eq-count-models","scope":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","status":"fully_proved","summary":"Let $m$ and $n$ be natural numbers and $t$ a diagram ordered from $n$ (every tested variable is at least $n$, and tested variables strictly increase along every path) in which every tested variable is below $m$. Then for every assignment $\\sigma$, the one-pass satisfy-count $c_m(n, t)$ of $t$ from $n$ with bound $m$ equals the model count of the function $\\tau \\mapsto t(\\tau)$ over the variables\u2026","title":"The satisfy-count from a variable counts models"},{"id":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/sat-count-eq-count-models","kind":"node","lean":"Cslib.Logic.PL.BDD.satCount_eq_countModels","parent":433,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd/index.html#sat-count-eq-count-models","scope":"Logic/propositional-logic/binary-decision-diagrams/counting-satisfying-assignments-of-a-bdd","status":"fully_proved","summary":"Let $m$ be a natural number and $t$ a diagram ordered from $0$ (tested variables strictly increase along every path) in which every tested variable is below $m$. Then for every assignment $\\sigma$, the satisfy-count of $t$ below $m$ equals the model count of the function $\\tau \\mapsto t(\\tau)$ over the variables $0, 1, \\dots, m - 1$ from $\\sigma$, where $t(\\tau)$ is the value of $t$ under $\\tau$. In other words, it\u2026","title":"Correctness of the satisfy-count"},{"data_url":"scopes/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths.json","id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","kind":"scope","navigation":"module","parent":411,"result":false,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","title":"Restriction, quantification, and satisfying paths for BDDs"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/restrict","kind":"node","lean":"Cslib.Logic.PL.BDD.restrict","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#restrict","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"For a variable $x$ and a Boolean $b$, the restriction (cofactor) $t[x := b]$ of a binary decision diagram $t$ is defined recursively: a leaf is kept; a node testing $x$ is replaced by the restriction of its $\\mathrm{hi}$ child if $b$ is true and of its $\\mathrm{lo}$ child if $b$ is false; and a node testing any other variable $v$ is rebuilt from the restrictions of its two children with the reducing node\u2026","title":"Restriction of a binary decision diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/reduced-restrict","kind":"node","lean":"Cslib.Logic.PL.BDD.reduced_restrict","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#reduced-restrict","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"For every binary decision diagram $t$, every variable $x$ and every Boolean $b$, the cofactor $t[x := b]$ is reduced: no node has two equal children.","title":"Restriction yields a reduced diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/is-robdd-restrict","kind":"node","lean":"Cslib.Logic.PL.BDD.isROBDD_restrict","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#is-robdd-restrict","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"If $t$ is an ordered binary decision diagram (tested variables strictly increase along every path), then for every variable $x$ and every Boolean $b$ the cofactor $t[x := b]$ is a reduced ordered binary decision diagram.","title":"Restriction yields an ROBDD"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/exists-var","kind":"node","lean":"Cslib.Logic.PL.BDD.existsVar","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#exists-var","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"For a variable $x$ and a binary decision diagram $t$, the existential quantification of $t$ over $x$ is the diagram obtained by combining the cofactors $t[x := \\mathrm{false}]$ and $t[x := \\mathrm{true}]$ with Bryant's apply under disjunction.","title":"Existential quantification of a diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/forall-var","kind":"node","lean":"Cslib.Logic.PL.BDD.forallVar","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#forall-var","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"For a variable $x$ and a binary decision diagram $t$, the universal quantification of $t$ over $x$ is the diagram obtained by combining the cofactors $t[x := \\mathrm{false}]$ and $t[x := \\mathrm{true}]$ with Bryant's apply under conjunction.","title":"Universal quantification of a diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/is-robdd-exists-var","kind":"node","lean":"Cslib.Logic.PL.BDD.isROBDD_existsVar","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#is-robdd-exists-var","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"If $t$ is an ordered binary decision diagram (tested variables strictly increase along every path), then for every variable $x$ the existential quantification of $t$ over $x$ is a reduced ordered binary decision diagram.","title":"Existential quantification yields an ROBDD"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/sat-path","kind":"node","lean":"Cslib.Logic.PL.BDD.satPath","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#sat-path","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"The satisfying path of a diagram is found by a depth-first search for a leaf $\\mathrm{true}$ that tries the $\\mathrm{hi}$ (true) child of each node before its $\\mathrm{lo}$ (false) child. It is returned as the list of pairs $(v, c)$ of the variables $v$ tested along the path with the branches $c$ taken (empty at a leaf $\\mathrm{true}$), and the search returns nothing when the diagram has no leaf $\\mathrm{true}$.","title":"Satisfying path of a binary decision diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/of-path","kind":"node","lean":"Cslib.Logic.PL.BDD.ofPath","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#of-path","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"For a path $p$, given as a list of pairs $(v, b)$ of a variable $v \\in \\mathbb{N}$ and a Boolean $b$, the assignment read off $p$ sends a variable $x$ to the Boolean of the first pair in $p$ whose variable is $x$, and to $\\mathrm{false}$ if $x$ occurs in no pair of $p$.","title":"Assignment read off a path"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/eval-of-path-sat-path","kind":"node","lean":"Cslib.Logic.PL.BDD.eval_ofPath_satPath","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#eval-of-path-sat-path","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"Let $t$ be a binary decision diagram ordered from some $n$ (every tested variable is at least $n$, and tested variables strictly increase along every path). If the satisfying-path search on $t$ returns a path $p$, then $t$ evaluates to true under the assignment read off $p$.","title":"The path assignment satisfies an ordered diagram"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/sat-path-is-some-of-ne","kind":"node","lean":"Cslib.Logic.PL.BDD.satPath_isSome_of_ne","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#sat-path-is-some-of-ne","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"If a binary decision diagram $t$ is reduced and is not the leaf $\\mathrm{false}$, then the search for a satisfying path in $t$ succeeds: it returns a path from the root of $t$ to a leaf $\\mathrm{true}$.","title":"Reduced diagrams other than false have a satisfying path"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/exists-eval-bool-eq-false-at-iff","kind":"node","lean":"Cslib.Logic.PL.BDD.exists_evalBool_eq_false_at_iff","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#exists-eval-bool-eq-false-at-iff","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"Let $A$ be a proposition over atoms indexed by natural numbers, let $t$ be its ROBDD, and let $f$ be a natural number. Then $A$ evaluates to true under some valuation $\\sigma$ with $\\sigma(f) = \\mathrm{false}$ if and only if the cofactor $t[f := \\mathrm{false}]$ is not the leaf $\\mathrm{false}$.","title":"Satisfiability with one variable fixed to false"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/sat-path-restrict-of-proposition","kind":"node","lean":"Cslib.Logic.PL.BDD.satPath_restrict_ofProposition","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#sat-path-restrict-of-proposition","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"Let $A$ be a proposition over natural-number atoms and let $f$ be a natural number. If some assignment $\\sigma$ with $\\sigma(f) = \\mathrm{false}$ satisfies $A$, then the search for a satisfying path in the restriction at $f := \\mathrm{false}$ of the ROBDD of $A$ returns a path $p$, and the assignment read off $p$ (the recorded value on each variable of $p$, false elsewhere), changed to send $f$ to $\\mathrm{false}$,\u2026","title":"Satisfy-one with a variable fixed to false"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/eval-bool-all-true","kind":"node","lean":"Cslib.Logic.PL.BDD.evalBool_allTrue","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#eval-bool-all-true","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"Every proposition $A$ over atoms indexed by natural numbers, built from atoms with $\\land$, $\\lor$ and $\\to$ only, evaluates to true under the valuation that makes every atom true.","title":"The all-true valuation satisfies every proposition"},{"id":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/of-proposition-sat-path","kind":"node","lean":"Cslib.Logic.PL.BDD.ofProposition_satPath","parent":441,"result":true,"route":"../roadmap/Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths/index.html#of-proposition-sat-path","scope":"Logic/propositional-logic/binary-decision-diagrams/restriction-quantification-and-satisfying-paths","status":"fully_proved","summary":"For every proposition $A$ over atoms indexed by natural numbers, the satisfying-path search on the ROBDD of $A$ returns a path $p$, and $A$ evaluates to true under the assignment read off $p$.","title":"Satisfy-one for propositions"},{"data_url":"scopes/Logic/propositional-logic/conjunctive-normal-form.json","id":"Logic/propositional-logic/conjunctive-normal-form","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","title":"Conjunctive normal form"},{"id":"Logic/propositional-logic/conjunctive-normal-form/cnfclause","kind":"node","lean":"Cslib.Logic.PL.CNFClause","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#cnfclause","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"A CNF clause is a finite list of literals, each a pair $(x, b)$ of a variable index $x \\in \\mathbb{N}$ and a polarity $b \\in \\{\\mathrm{true}, \\mathrm{false}\\}$.","title":"CNF clause"},{"id":"Logic/propositional-logic/conjunctive-normal-form/cnf","kind":"node","lean":"Cslib.Logic.PL.CNF","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#cnf","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"A CNF formula is a finite list of CNF clauses, read as the conjunction of its clauses.","title":"CNF formula"},{"id":"Logic/propositional-logic/conjunctive-normal-form/cnfsat","kind":"node","lean":"Cslib.Logic.PL.CNFSat","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#cnfsat","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"A CNF formula $\\varphi$ is satisfiable when there is an assignment $v : \\mathbb{N} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ such that every clause $C$ of $\\varphi$ contains a literal $(x, b)$ with $v(x) = b$.","title":"Satisfiability of a CNF formula"},{"id":"Logic/propositional-logic/conjunctive-normal-form/satisfies","kind":"node","lean":"Cslib.Logic.PL.CNFLiteral.Satisfies","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#satisfies","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"An assignment $v : \\mathbb{N} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ satisfies a literal $(x, b)$, with variable $x \\in \\mathbb{N}$ and polarity $b \\in \\{\\mathrm{true}, \\mathrm{false}\\}$, when $v(x) = b$.","title":"Satisfaction of a literal"},{"id":"Logic/propositional-logic/conjunctive-normal-form/satisfies-2","kind":"node","lean":"Cslib.Logic.PL.CNFClause.Satisfies","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#satisfies-2","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"An assignment $v : \\mathbb{N} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ satisfies a CNF clause $C$ when at least one literal of $C$ is satisfied by $v$, that is, $C$ contains a literal $(x, b)$ with $v(x) = b$.","title":"Satisfaction of a clause"},{"id":"Logic/propositional-logic/conjunctive-normal-form/satisfies-3","kind":"node","lean":"Cslib.Logic.PL.CNF.Satisfies","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#satisfies-3","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"An assignment $v : \\mathbb{N} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ satisfies a CNF formula $\\varphi$ when $v$ satisfies every clause of $\\varphi$.","title":"Satisfaction of a CNF formula"},{"id":"Logic/propositional-logic/conjunctive-normal-form/cnf-sat-iff-exists-satisfies","kind":"node","lean":"Cslib.Logic.PL.cnfSat_iff_exists_satisfies","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#cnf-sat-iff-exists-satisfies","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"A CNF formula $\\varphi$ is satisfiable if and only if there is an assignment $v : \\mathbb{N} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ that satisfies $\\varphi$.","title":"Satisfiability means having a satisfying assignment"},{"id":"Logic/propositional-logic/conjunctive-normal-form/disj","kind":"node","lean":"Cslib.Logic.PL.CNF.disj","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#disj","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"For CNFs $\\varphi = C_1 \\land \\dots \\land C_m$ and $\\psi = D_1 \\land \\dots \\land D_n$, their disjunction is the CNF with one clause $C_i \\lor D_j$ (the literals of $C_i$ followed by those of $D_j$) for each pair $(i, j)$, listed with $i$ as the outer and $j$ as the inner index. It expresses $(\\bigwedge_i C_i) \\lor (\\bigwedge_j D_j)$ as $\\bigwedge_{i,j} (C_i \\lor D_j)$.","title":"Disjunction of two CNFs"},{"id":"Logic/propositional-logic/conjunctive-normal-form/cnf-acc","kind":"node","lean":"Cslib.Logic.PL.Proposition.cnfAcc","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#cnf-acc","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $\\iota$ number the atoms by natural numbers, read a literal $(n, p)$ as variable $n$ with polarity $p$, and write $1$ and $0$ for the polarities true and false. For a polarity $p$, a proposition $\\varphi$ and a CNF $S$, the CNF $\\mathcal{A}_p(\\varphi, S)$ is defined by recursion on $\\varphi$, together with the disjoining builder $\\mathcal{O}_p$. For an atom $a$, $\\mathcal{A}_p(a, S)$ is the unit clause\u2026","title":"Accumulating CNF builder"},{"id":"Logic/propositional-logic/conjunctive-normal-form/cnf-or","kind":"node","lean":"Cslib.Logic.PL.Proposition.cnfOr","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#cnf-or","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $\\iota$ number the atoms by natural numbers, read a literal $(n, p)$ as variable $n$ with polarity $p$, and write $1$ and $0$ for the polarities true and false. For a polarity $p$, a proposition $\\varphi$ and a CNF $S$, the CNF $\\mathcal{O}_p(\\varphi, S)$ is defined by recursion on $\\varphi$, together with the accumulating CNF builder $\\mathcal{A}_p$. Write $X \\otimes S$ for the disjunction of two CNFs $X$ and\u2026","title":"Disjoining CNF builder"},{"id":"Logic/propositional-logic/conjunctive-normal-form/cnf-of","kind":"node","lean":"Cslib.Logic.PL.Proposition.cnfOf","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#cnf-of","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $\\iota$ number the atoms by natural numbers. For a polarity $p$ and a proposition $\\varphi$, the CNF of $\\varphi$ at polarity $p$ is the result of the accumulating CNF builder at polarity $p$ on $\\varphi$ with the empty CNF as accumulator; the atom $a$ becomes the variable $\\iota(a)$.","title":"CNF of a proposition at a polarity"},{"id":"Logic/propositional-logic/conjunctive-normal-form/to-cnf","kind":"node","lean":"Cslib.Logic.PL.Proposition.toCNF","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#to-cnf","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $\\iota$ number the atoms by natural numbers. The CNF of a proposition $A$ is its CNF at polarity true, with the atom $a$ as the variable $\\iota(a)$.","title":"CNF of a proposition"},{"id":"Logic/propositional-logic/conjunctive-normal-form/to-neg-cnf","kind":"node","lean":"Cslib.Logic.PL.Proposition.toNegCNF","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#to-neg-cnf","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $\\iota$ number the atoms by natural numbers. The complement CNF of a proposition $A$ is its CNF at polarity false, with the atom $a$ as the variable $\\iota(a)$.","title":"CNF of the complement of a proposition"},{"id":"Logic/propositional-logic/conjunctive-normal-form/satisfies-to-cnf-iff","kind":"node","lean":"Cslib.Logic.PL.Proposition.satisfies_toCNF_iff","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#satisfies-to-cnf-iff","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $\\iota$ number the atoms by natural numbers. For every proposition $A$ and every assignment $\\sigma$ of Boolean values to the natural-number variables, the CNF of $A$ built with atom $a$ as variable $\\iota(a)$ is satisfied by $\\sigma$ if and only if $A$ evaluates to true under the valuation $\\sigma \\circ \\iota$ of the atoms.","title":"Equivalent CNF"},{"id":"Logic/propositional-logic/conjunctive-normal-form/cnf-sat-to-cnf-bot-iff","kind":"node","lean":"Cslib.Logic.PL.Proposition.cnfSat_toCNF_bot_iff","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#cnf-sat-to-cnf-bot-iff","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let the atoms have a distinguished atom $\\bot$ and let $\\iota$ be an injective numbering of the atoms by natural numbers. For every proposition $A$, the CNF consisting of the unit clause with the single negative literal on variable $\\iota(\\bot)$ followed by the clauses of the CNF of $A$ (atom $a$ read as variable $\\iota(a)$) is satisfiable if and only if some valuation $v$ of the atoms with $v(\\bot)$ false makes\u2026","title":"Satisfiability of a CNF with falsum false"},{"id":"Logic/propositional-logic/conjunctive-normal-form/vars","kind":"node","lean":"Cslib.Logic.PL.CNFClause.vars","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#vars","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"The variables of a CNF clause $C$ form the finite set of natural numbers $x$ such that $C$ contains a literal $(x, b)$ for some polarity $b$.","title":"Variables of a clause"},{"id":"Logic/propositional-logic/conjunctive-normal-form/restrict","kind":"node","lean":"Cslib.Logic.PL.CNFClause.restrict","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#restrict","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Given a literal $\\ell = (x, b)$, the restriction of a clause by $\\ell$ is the clause obtained by deleting every literal whose variable is $x$, of either polarity, keeping the other literals in order.","title":"Restriction of a clause by a literal"},{"id":"Logic/propositional-logic/conjunctive-normal-form/satisfies-restrict-iff","kind":"node","lean":"Cslib.Logic.PL.CNFClause.satisfies_restrict_iff","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#satisfies-restrict-iff","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $C$ be a clause, $\\ell = (x, b)$ a literal that does not occur in $C$, and $v$ an assignment of Booleans to variables with $v(x) = b$. Then $v$ satisfies the restriction of $C$ by $\\ell$ if and only if $v$ satisfies $C$.","title":"Satisfaction of a restricted clause"},{"id":"Logic/propositional-logic/conjunctive-normal-form/vars-2","kind":"node","lean":"Cslib.Logic.PL.CNF.vars","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#vars-2","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"The variables of a CNF formula $\\varphi$ form the union of the sets of variables of its clauses, that is, the finite set of natural numbers $x$ such that some clause of $\\varphi$ contains a literal $(x, b)$ for some polarity $b$.","title":"Variables of a CNF formula"},{"id":"Logic/propositional-logic/conjunctive-normal-form/restrict-2","kind":"node","lean":"Cslib.Logic.PL.CNF.restrict","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#restrict-2","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Given a literal $\\ell = (x, b)$, the restriction of a CNF formula $\\varphi$ by $\\ell$ is obtained by deleting every clause of $\\varphi$ that contains $\\ell$ and replacing every other clause by its restriction by $\\ell$, the clause obtained by deleting every literal whose variable is $x$, of either polarity; the remaining clauses keep their order.","title":"Restriction of a CNF formula by a literal"},{"id":"Logic/propositional-logic/conjunctive-normal-form/vars-restrict-subset","kind":"node","lean":"Cslib.Logic.PL.CNF.vars_restrict_subset","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#vars-restrict-subset","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"For every CNF formula $\\varphi$ and every literal $\\ell$, every variable of the restriction of $\\varphi$ by $\\ell$ is a variable of $\\varphi$.","title":"Restriction does not add variables"},{"id":"Logic/propositional-logic/conjunctive-normal-form/not-mem-vars-restrict","kind":"node","lean":"Cslib.Logic.PL.CNF.not_mem_vars_restrict","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#not-mem-vars-restrict","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"For every CNF formula $F$ and every literal $\\ell = (x, b)$, the variable $x$ does not occur in the restriction of $F$ by $\\ell$.","title":"The restricted variable no longer occurs"},{"id":"Logic/propositional-logic/conjunctive-normal-form/vars-restrict-ssubset","kind":"node","lean":"Cslib.Logic.PL.CNF.vars_restrict_ssubset","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#vars-restrict-ssubset","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula and $(x, b)$ a literal. If $x$ is a variable of $\\varphi$, then the set of variables of the restriction of $\\varphi$ by $(x, b)$ is a proper subset of the set of variables of $\\varphi$.","title":"Restricting by an occurring variable shrinks the variable set"},{"id":"Logic/propositional-logic/conjunctive-normal-form/card-vars-restrict-lt","kind":"node","lean":"Cslib.Logic.PL.CNF.card_vars_restrict_lt","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#card-vars-restrict-lt","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $F$ be a CNF formula and let $\\ell = (x, b)$ be a literal whose variable $x$ occurs in $F$. Then the restriction of $F$ by $\\ell$ has strictly fewer distinct variables than $F$.","title":"Restriction reduces the number of variables"},{"id":"Logic/propositional-logic/conjunctive-normal-form/satisfies-restrict-iff-2","kind":"node","lean":"Cslib.Logic.PL.CNF.satisfies_restrict_iff","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#satisfies-restrict-iff-2","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula, $\\ell = (x, b)$ a literal, and $v$ an assignment of Booleans to variables with $v(x) = b$. Then $v$ satisfies the restriction of $\\varphi$ by $\\ell$ if and only if $v$ satisfies $\\varphi$.","title":"Satisfaction of a restricted CNF formula"},{"id":"Logic/propositional-logic/conjunctive-normal-form/satisfies-congr","kind":"node","lean":"Cslib.Logic.PL.CNF.satisfies_congr","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#satisfies-congr","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $F$ be a CNF formula and let $v_1, v_2$ be assignments of Booleans to variables with $v_1(x) = v_2(x)$ for every variable $x$ occurring in $F$. If $v_1$ satisfies $F$, then $v_2$ satisfies $F$.","title":"Satisfaction depends only on the occurring variables"},{"id":"Logic/propositional-logic/conjunctive-normal-form/satisfies-of-satisfies-restrict","kind":"node","lean":"Cslib.Logic.PL.CNF.satisfies_of_satisfies_restrict","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#satisfies-of-satisfies-restrict","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $F$ be a CNF formula, $\\ell = (x, b)$ a literal and $v$ an assignment of Booleans to variables. If $v$ satisfies the restriction of $F$ by $\\ell$, then the assignment that agrees with $v$ except that it sends $x$ to $b$ satisfies $F$.","title":"Lifting a satisfying assignment from a restriction"},{"id":"Logic/propositional-logic/conjunctive-normal-form/sat-iff-restrict-true-or-false","kind":"node","lean":"Cslib.Logic.PL.CNF.sat_iff_restrict_true_or_false","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#sat-iff-restrict-true-or-false","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"For every CNF formula $F$ and every variable $x$, the formula $F$ is satisfiable if and only if the restriction of $F$ by the literal $(x, \\mathrm{true})$ is satisfiable or the restriction of $F$ by the literal $(x, \\mathrm{false})$ is satisfiable.","title":"Splitting on a variable"},{"id":"Logic/propositional-logic/conjunctive-normal-form/sat-iff-restrict-of-unit","kind":"node","lean":"Cslib.Logic.PL.CNF.sat_iff_restrict_of_unit","parent":456,"result":true,"route":"../roadmap/Logic/propositional-logic/conjunctive-normal-form/index.html#sat-iff-restrict-of-unit","scope":"Logic/propositional-logic/conjunctive-normal-form","status":"fully_proved","summary":"Let $F$ be a CNF formula that contains the unit clause whose only literal is $\\ell$. Then $F$ is satisfiable if and only if the restriction of $F$ by $\\ell$ is satisfiable.","title":"Unit propagation preserves satisfiability"},{"data_url":"scopes/Logic/propositional-logic/craig-interpolation.json","id":"Logic/propositional-logic/craig-interpolation","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","title":"Craig interpolation"},{"id":"Logic/propositional-logic/craig-interpolation/atoms","kind":"node","lean":"Cslib.Logic.PL.Proposition.atoms","parent":486,"result":true,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html#atoms","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","summary":"The atoms of a proposition $A$ form the finite set of atoms occurring in $A$: an atom $x$ has the single atom $x$, and the atoms of $B \\wedge C$, $B \\vee C$ and $B \\to C$ are the atoms of $B$ together with the atoms of $C$.","title":"Atoms of a proposition"},{"id":"Logic/propositional-logic/craig-interpolation/eval-bool-congr","kind":"node","lean":"Cslib.Logic.PL.Proposition.evalBool_congr","parent":486,"result":true,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html#eval-bool-congr","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","summary":"If two Boolean valuations $v$ and $w$ agree on every atom of a proposition $A$, then $A$ has the same truth-table value under $v$ as under $w$.","title":"The truth value depends only on the atoms"},{"id":"Logic/propositional-logic/craig-interpolation/fix-atom","kind":"node","lean":"Cslib.Logic.PL.Proposition.fixAtom","parent":486,"result":true,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html#fix-atom","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","summary":"For an atom $p$, a Boolean $b$ and a proposition $A$, the proposition $A[p := b]$ is obtained from $A$ by replacing every occurrence of the atom $p$ by the constant $\\top$, written $\\bot \\to \\bot$, if $b$ is true, and by $\\bot$ if $b$ is false.","title":"Substituting a truth value for an atom"},{"id":"Logic/propositional-logic/craig-interpolation/elim-atom","kind":"node","lean":"Cslib.Logic.PL.Proposition.elimAtom","parent":486,"result":true,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html#elim-atom","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","summary":"For an atom $p$ and a proposition $A$, the Boolean quantification $\\exists p.\\, A$ is the disjunction $A[p := \\top] \\vee A[p := \\bot]$, where $A[p := \\top]$ and $A[p := \\bot]$ replace the atom $p$ in $A$ by the constant $\\top = (\\bot \\to \\bot)$ and by $\\bot$ respectively.","title":"Boolean quantification over an atom"},{"id":"Logic/propositional-logic/craig-interpolation/elim-atoms","kind":"node","lean":"Cslib.Logic.PL.Proposition.elimAtoms","parent":486,"result":true,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html#elim-atoms","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","summary":"For a list of atoms $p_1, \\dots, p_n$ and a proposition $A$, the iterated Boolean quantification $\\exists p_1 \\cdots p_n.\\, A$ is $\\exists p_1.\\, \\exists p_2.\\, \\cdots \\exists p_n.\\, A$: $A$ is quantified over $p_n$ first and over $p_1$ last, and for the empty list the result is $A$ itself.","title":"Boolean quantification over a list of atoms"},{"id":"Logic/propositional-logic/craig-interpolation/eval-bool-elim-atoms","kind":"node","lean":"Cslib.Logic.PL.Proposition.evalBool_elimAtoms","parent":486,"result":true,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html#eval-bool-elim-atoms","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","summary":"Let $p_1, \\dots, p_n$ be a list of atoms that does not contain $\\bot$, let $A$ be a proposition and let $v$ be a Boolean valuation with $v(\\bot)$ false. Then the iterated Boolean quantification $\\exists p_1 \\cdots p_n.\\, A$ is true under $v$ if and only if $A$ is true under some valuation $w$ that agrees with $v$ on every atom not in the list.","title":"Truth of a Boolean quantification over a list of atoms"},{"id":"Logic/propositional-logic/craig-interpolation/exists-interpolant","kind":"node","lean":"Cslib.Logic.PL.exists_interpolant","parent":486,"result":true,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html#exists-interpolant","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","summary":"Let $A$ and $B$ be propositions such that every Boolean valuation $v$ with $v(\\bot)$ false that makes $A$ true also makes $B$ true. Then there is a proposition $I$ whose atoms all lie among the atoms common to $A$ and $B$ together with $\\bot$, such that every valuation $v$ with $v(\\bot)$ false that makes $A$ true makes $I$ true, and every valuation $v$ with $v(\\bot)$ false that makes $I$ true makes $B$ true.","title":"Craig interpolation, semantic form"},{"id":"Logic/propositional-logic/craig-interpolation/exists-interpolant-cpl","kind":"node","lean":"Cslib.Logic.PL.exists_interpolant_cpl","parent":486,"result":true,"route":"../roadmap/Logic/propositional-logic/craig-interpolation/index.html#exists-interpolant-cpl","scope":"Logic/propositional-logic/craig-interpolation","status":"fully_proved","summary":"Let $A$ and $B$ be propositions such that $A \\to B$ is derivable without assumptions in classical propositional logic $\\mathrm{CPL}$ (natural deduction with the axioms $\\neg\\neg C \\to C$). Then there is a proposition $I$ whose atoms all lie among the atoms common to $A$ and $B$ together with $\\bot$, such that $A \\to I$ and $I \\to B$ are both derivable without assumptions in $\\mathrm{CPL}$.","title":"Craig interpolation for classical natural deduction"},{"data_url":"scopes/Logic/propositional-logic/dpll-algorithm.json","id":"Logic/propositional-logic/dpll-algorithm","kind":"scope","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/index.html","scope":"Logic/propositional-logic/dpll-algorithm","status":"fully_proved","title":"DPLL algorithm"},{"data_url":"scopes/Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver.json","id":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver","kind":"scope","navigation":"module","parent":495,"result":false,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/index.html","scope":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver","status":"fully_proved","title":"A deterministic DPLL solver"},{"id":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/first-unit","kind":"node","lean":"Cslib.Algorithms.SAT.DPLL.firstUnit","parent":496,"result":true,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/index.html#first-unit","scope":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver","status":"fully_proved","summary":"For a CNF formula $\\varphi$, read as a list of clauses, this returns the literal $\\ell$ of the first clause of $\\varphi$ that consists of the single literal $\\ell$, and returns nothing when no clause of $\\varphi$ has exactly one literal.","title":"First unit literal"},{"id":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/first-variable","kind":"node","lean":"Cslib.Algorithms.SAT.DPLL.firstVariable","parent":496,"result":true,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/index.html#first-variable","scope":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver","status":"fully_proved","summary":"For a CNF formula $\\varphi$, read as a list of clauses, this returns the variable of the first literal of the first nonempty clause of $\\varphi$, that is, the first variable occurring in $\\varphi$ in source order, and returns nothing when every clause of $\\varphi$ is empty (in particular when $\\varphi$ has no clauses).","title":"First variable of a CNF formula"},{"id":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/clause-satisfied","kind":"node","lean":"Cslib.Algorithms.SAT.DPLL.clauseSatisfied","parent":496,"result":true,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/index.html#clause-satisfied","scope":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver","status":"fully_proved","summary":"For an assignment $v$ of Boolean values to the variables $0, 1, 2, \\ldots$ and a clause $C$, whose literals are pairs $(x, b)$ of a variable $x$ and a polarity $b$, this Boolean is true exactly when some literal $(x, b)$ of $C$ satisfies $v(x) = b$.","title":"Boolean clause satisfaction check"},{"id":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/formula-satisfied","kind":"node","lean":"Cslib.Algorithms.SAT.DPLL.formulaSatisfied","parent":496,"result":true,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/index.html#formula-satisfied","scope":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver","status":"fully_proved","summary":"For an assignment $v$ of Boolean values to the variables $0, 1, 2, \\ldots$ and a CNF formula $\\varphi$, this Boolean is true exactly when every clause of $\\varphi$ passes the Boolean clause satisfaction check for $v$, that is, every clause contains a literal $(x, b)$ with $v(x) = b$.","title":"Boolean formula satisfaction check"},{"id":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/decide-with-fuel","kind":"node","lean":"Cslib.Algorithms.SAT.DPLL.decideWithFuel","parent":496,"result":true,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/index.html#decide-with-fuel","scope":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver","status":"fully_proved","summary":"Given a natural number $n$ (the fuel) and a CNF formula $\\varphi$, the procedure returns a Boolean value. Literals are pairs $(x, b)$ of a variable $x$ and a polarity $b$. With fuel $0$ the procedure returns true exactly when $\\varphi$ has no clauses. With fuel $n + 1$ it returns true if $\\varphi$ has no clauses, and otherwise returns false if some clause of $\\varphi$ is empty. Otherwise, if some clause of\u2026","title":"DPLL with a fuel budget"},{"id":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/dpll","kind":"node","lean":"Cslib.Algorithms.SAT.DPLL.dpll","parent":496,"result":true,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver/index.html#dpll","scope":"Logic/propositional-logic/dpll-algorithm/a-deterministic-dpll-solver","status":"fully_proved","summary":"On a CNF formula $\\varphi$, the DPLL procedure runs DPLL with fuel equal to the number of distinct variables occurring in $\\varphi$ and returns its Boolean result.","title":"The DPLL procedure"},{"data_url":"scopes/Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver.json","id":"Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver","kind":"scope","navigation":"module","parent":495,"result":false,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver/index.html","scope":"Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver","status":"fully_proved","title":"Correctness of the DPLL solver"},{"id":"Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver/decide-with-fuel-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.SAT.DPLL.decideWithFuel_eq_true_iff","parent":503,"result":true,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver/index.html#decide-with-fuel-eq-true-iff","scope":"Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula and let $n$ be a natural number at least the number of distinct variables occurring in $\\varphi$. Then DPLL with fuel $n$ returns true on $\\varphi$ if and only if $\\varphi$ is satisfiable.","title":"Correctness of DPLL with sufficient fuel"},{"id":"Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver/dpll-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.SAT.DPLL.dpll_eq_true_iff","parent":503,"result":true,"route":"../roadmap/Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver/index.html#dpll-eq-true-iff","scope":"Logic/propositional-logic/dpll-algorithm/correctness-of-the-dpll-solver","status":"fully_proved","summary":"For every CNF formula $\\varphi$, the DPLL procedure returns true on $\\varphi$ if and only if $\\varphi$ is satisfiable.","title":"Correctness of DPLL"},{"data_url":"scopes/Logic/propositional-logic/functional-completeness.json","id":"Logic/propositional-logic/functional-completeness","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","title":"Functional completeness"},{"id":"Logic/propositional-logic/functional-completeness/shannon","kind":"node","lean":"Cslib.Logic.PL.Proposition.shannon","parent":506,"result":true,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html#shannon","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","summary":"Let $f$ assign a truth value to every valuation of the atoms. The Shannon expansion $S(\\ell, f)$ of $f$ over a list $\\ell$ of atoms is defined by recursion on $\\ell$. For the empty list it is $\\bot \\to \\bot$ if $f$ is true at the valuation that makes every atom false, and $\\bot$ otherwise. For the list $x \\cdot \\ell$ with head $x$ and tail $\\ell$,","title":"Shannon expansion"},{"id":"Logic/propositional-logic/functional-completeness/eval-bool-shannon","kind":"node","lean":"Cslib.Logic.PL.Proposition.evalBool_shannon","parent":506,"result":true,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html#eval-bool-shannon","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","summary":"Let $x_1, \\dots, x_n$ be a list of atoms and let $f$ assign a truth value to every valuation of the atoms, such that $f(v) = f(w)$ whenever $v$ and $w$ agree on $x_1, \\dots, x_n$. Then for every valuation $v$ that makes $\\bot$ false, the Shannon expansion of $f$ over $x_1, \\dots, x_n$ has truth value $f(v)$ under $v$.","title":"Correctness of the Shannon expansion"},{"id":"Logic/propositional-logic/functional-completeness/exists-eval-bool-eq","kind":"node","lean":"Cslib.Logic.PL.exists_evalBool_eq","parent":506,"result":true,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html#exists-eval-bool-eq","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","summary":"Every Boolean function of the atoms in a finite list is the truth table, on valuations with `\u22a5` false, of some proposition.","title":"Functional completeness"},{"id":"Logic/propositional-logic/functional-completeness/is-imp-bot","kind":"node","lean":"Cslib.Logic.PL.Proposition.IsImpBot","parent":506,"reading_title":{"format":"inline-parts-v1","parts":[["text","Propositions built from "],["math","\\(\\to\\)"],["text"," and "],["math","\\(\\bot\\)"]],"source":"Propositions built from $\\to$ and $\\bot$"},"result":true,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html#is-imp-bot","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","summary":"A proposition is built from $\\to$ and $\\bot$ when it contains no conjunction and no disjunction: every atom (including $\\bot$, which is an atom) is, and $A \\to B$ is exactly when both $A$ and $B$ are.","title":"Propositions built from $\\to$ and $\\bot$"},{"id":"Logic/propositional-logic/functional-completeness/to-imp-bot","kind":"node","lean":"Cslib.Logic.PL.Proposition.toImpBot","parent":506,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation into "],["math","\\(\\to\\)"],["text"," and "],["math","\\(\\bot\\)"]],"source":"Translation into $\\to$ and $\\bot$"},"result":true,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html#to-imp-bot","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","summary":"The translation $A^*$ of a proposition $A$ is defined by recursion on $A$: each atom is unchanged, $(A \\to B)^* = A^* \\to B^*$, $(A \\wedge B)^* = (A^* \\to (B^* \\to \\bot)) \\to \\bot$ and $(A \\vee B)^* = (A^* \\to \\bot) \\to B^*$.","title":"Translation into $\\to$ and $\\bot$"},{"id":"Logic/propositional-logic/functional-completeness/eval-bool-to-imp-bot","kind":"node","lean":"Cslib.Logic.PL.Proposition.evalBool_toImpBot","parent":506,"reading_title":{"format":"inline-parts-v1","parts":[["text","The translation into "],["math","\\(\\to\\)"],["text"," and "],["math","\\(\\bot\\)"],["text"," preserves truth values"]],"source":"The translation into $\\to$ and $\\bot$ preserves truth values"},"result":true,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html#eval-bool-to-imp-bot","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","summary":"For every proposition $A$ and every valuation $v$ that makes $\\bot$ false, the translation of $A$ into $\\to$ and $\\bot$ has the same truth value under $v$ as $A$.","title":"The translation into $\\to$ and $\\bot$ preserves truth values"},{"id":"Logic/propositional-logic/functional-completeness/exists-is-imp-bot-eval-bool-eq","kind":"node","lean":"Cslib.Logic.PL.exists_isImpBot_evalBool_eq","parent":506,"result":true,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html#exists-is-imp-bot-eval-bool-eq","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","summary":"Every Boolean function of the atoms in a finite list is the truth table, on valuations with `\u22a5` false, of a proposition built with `\u2192` alone.","title":"Functional completeness of `\u2192` and `\u22a5`"},{"id":"Logic/propositional-logic/functional-completeness/derivable-in-cpl-to-imp-bot-iff","kind":"node","lean":"Cslib.Logic.PL.derivableIn_cpl_toImpBot_iff","parent":506,"reading_title":{"format":"inline-parts-v1","parts":[["text","The translation into "],["math","\\(\\to\\)"],["text"," and "],["math","\\(\\bot\\)"],["text"," preserves and reflects classical derivability"]],"source":"The translation into $\\to$ and $\\bot$ preserves and reflects classical derivability"},"result":true,"route":"../roadmap/Logic/propositional-logic/functional-completeness/index.html#derivable-in-cpl-to-imp-bot-iff","scope":"Logic/propositional-logic/functional-completeness","status":"fully_proved","summary":"For every proposition $A$, let $A^*$ be its translation into $\\to$ and $\\bot$, which keeps atoms and implications and recursively replaces each $B \\wedge C$ by $(B \\to (C \\to \\bot)) \\to \\bot$ and each $B \\vee C$ by $(B \\to \\bot) \\to C$. Then $A^*$ is derivable from no hypotheses in classical propositional logic if and only if $A$ is.","title":"The translation into $\\to$ and $\\bot$ preserves and reflects classical derivability"},{"data_url":"scopes/Logic/propositional-logic/horn-satisfiability.json","id":"Logic/propositional-logic/horn-satisfiability","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","title":"Horn satisfiability"},{"id":"Logic/propositional-logic/horn-satisfiability/is-horn","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.IsHorn","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#is-horn","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"A CNF formula is a Horn formula when every clause contains at most one distinct positive literal.","title":"Horn formula"},{"id":"Logic/propositional-logic/horn-satisfiability/body-in","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.bodyIn","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#body-in","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"For a finite set $T$ of variables and a CNF clause $C$, this Boolean test is true exactly when every negative literal of $C$ has its variable in $T$.","title":"Body of a clause contained in a set"},{"id":"Logic/propositional-logic/horn-satisfiability/derive","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.derive","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#derive","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and a finite set $T$ of variables, the derived set consists of the variables $x$ such that some clause of $\\varphi$ contains the positive literal $x$ and has every one of its negative literals on a variable in $T$.","title":"Forward-chaining step"},{"id":"Logic/propositional-logic/horn-satisfiability/close","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.close","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#close","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and a finite set $T$ of variables, the closure of $T$ is computed by iterating the forward-chaining step of $\\varphi$: if every variable derived from $T$ already lies in $T$, the closure is $T$; otherwise it is the closure of the union of $T$ with the derived variables.","title":"Forward-chaining closure"},{"id":"Logic/propositional-logic/horn-satisfiability/least-model","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.leastModel","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#least-model","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, the least model candidate is the assignment that makes a variable true exactly when it lies in the forward-chaining closure of the empty set under $\\varphi$.","title":"Least model candidate"},{"id":"Logic/propositional-logic/horn-satisfiability/least-model-le","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.leastModel_le","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#least-model-le","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"Let $\\varphi$ be a Horn CNF formula and $\\sigma$ an assignment satisfying $\\varphi$. Every variable that the least model candidate of $\\varphi$ makes true is also true under $\\sigma$.","title":"On Horn formulas, the least model candidate lies below every model"},{"id":"Logic/propositional-logic/horn-satisfiability/satisfies-least-model-iff","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.satisfies_leastModel_iff","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#satisfies-least-model-iff","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"A Horn CNF formula $\\varphi$ is satisfiable if and only if its least model candidate satisfies $\\varphi$.","title":"Horn satisfiability via the least model"},{"id":"Logic/propositional-logic/horn-satisfiability/horn-sat","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.hornSat","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#horn-sat","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, the Horn satisfiability test returns the Boolean truth value of $\\varphi$ under its least model candidate.","title":"Horn satisfiability test"},{"id":"Logic/propositional-logic/horn-satisfiability/horn-sat-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.SAT.Horn.hornSat_eq_true_iff","parent":515,"result":true,"route":"../roadmap/Logic/propositional-logic/horn-satisfiability/index.html#horn-sat-eq-true-iff","scope":"Logic/propositional-logic/horn-satisfiability","status":"fully_proved","summary":"If $\\varphi$ is a Horn CNF formula, then the Horn satisfiability test returns true on $\\varphi$ if and only if $\\varphi$ is satisfiable.","title":"Correctness of the Horn satisfiability test"},{"data_url":"scopes/Logic/propositional-logic/model-count-preserving-cnf-transformations.json","id":"Logic/propositional-logic/model-count-preserving-cnf-transformations","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","title":"Model-count-preserving CNF transformations"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/model-count","kind":"node","lean":"Cslib.Logic.PL.CNF.modelCount","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#model-count","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"The number of Boolean assignments to the variables occurring in a formula that satisfy it.","title":"Model count of a CNF formula"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/compact","kind":"node","lean":"Cslib.Logic.PL.CNF.compact","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#compact","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Rename every occurring variable by its position in the last-occurrence list.","title":"Exact variable compaction of a CNF formula"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/length-compact","kind":"node","lean":"Cslib.Logic.PL.CNF.length_compact","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#length-compact","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Compaction preserves the number of clauses.","title":"Compaction preserves the number of clauses"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/clause-lengths-compact","kind":"node","lean":"Cslib.Logic.PL.CNF.clauseLengths_compact","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#clause-lengths-compact","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Compaction preserves every clause length, in the original clause order.","title":"Compaction preserves clause lengths"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/bounded-clause-length-compact-iff","kind":"node","lean":"Cslib.Logic.PL.CNF.boundedClauseLength_compact_iff","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#bounded-clause-length-compact-iff","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Compaction preserves every upper bound on clause length.","title":"Compaction preserves clause-length bounds"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/nodup-compact-iff","kind":"node","lean":"Cslib.Logic.PL.CNF.nodup_compact_iff","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#nodup-compact-iff","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Compaction preserves the absence of duplicate clauses.","title":"Compaction preserves absence of duplicate clauses"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/vars-compact","kind":"node","lean":"Cslib.Logic.PL.CNF.vars_compact","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#vars-compact","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"The variables of a compacted formula are exactly its dense initial range.","title":"Variables of a compacted formula form an initial range"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/satisfies-compact-iff","kind":"node","lean":"Cslib.Logic.PL.CNF.satisfies_compact_iff","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#satisfies-compact-iff","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Satisfaction after compaction is satisfaction before compaction with names translated by their exact last-occurrence positions.","title":"Satisfaction is preserved by variable compaction"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/model-count-compact","kind":"node","lean":"Cslib.Logic.PL.CNF.modelCount_compact","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#model-count-compact","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Exact variable compaction preserves the number of assignments to the occurring variables.","title":"Variable compaction preserves the model count"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/card-satisfying-fin","kind":"node","lean":"Cslib.Logic.PL.CNF.card_satisfying_fin","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#card-satisfying-fin","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"If every occurring name is below `N`, the count over all `Fin N` assignments is the support count times one Boolean choice for each unused ambient variable.","title":"Counting satisfying assignments over a larger variable range"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/num-fresh-parsimonious3","kind":"node","lean":"Cslib.Logic.PL.CNF.numFreshParsimonious3","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#num-fresh-parsimonious3","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"The number of fresh variables used by the parsimonious width-three reduction.","title":"Number of fresh variables in the parsimonious 3-CNF reduction"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/to3-parsimonious","kind":"node","lean":"Cslib.Logic.PL.CNF.to3Parsimonious","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#to3-parsimonious","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Reduce a CNF to clauses of length at most three, allocating fresh variables consecutively from `start`. Repeated output clauses are removed in last-occurrence order.","title":"Parsimonious reduction of CNF to 3-CNF"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/satisfies-of-satisfies-to3-parsimonious","kind":"node","lean":"Cslib.Logic.PL.CNF.satisfies_of_satisfies_to3Parsimonious","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#satisfies-of-satisfies-to3-parsimonious","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Satisfaction of the reduced formula implies satisfaction of the original formula under the same values of the original variable names.","title":"Soundness of the parsimonious 3-CNF reduction"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/is-three-cnf-to3-parsimonious","kind":"node","lean":"Cslib.Logic.PL.CNF.isThreeCNF_to3Parsimonious","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#is-three-cnf-to3-parsimonious","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Every output clause has length at most three, and duplicate clauses have been removed.","title":"The parsimonious reduction outputs a duplicate-free 3-CNF"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/size-to3-parsimonious","kind":"node","lean":"Cslib.Logic.PL.CNF.size_to3Parsimonious","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#size-to3-parsimonious","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"A width bound on the input gives the source bounds on allocated fresh variables and output clauses.","title":"Size bounds for the parsimonious 3-CNF reduction"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/exists-unique-satisfies-to3-parsimonious","kind":"node","lean":"Cslib.Logic.PL.CNF.existsUnique_satisfies_to3Parsimonious","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#exists-unique-satisfies-to3-parsimonious","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"Every satisfying assignment of the original formula has exactly one satisfying extension that can differ only on the allocated fresh-variable interval.","title":"Unique satisfying extension in the parsimonious 3-CNF reduction"},{"id":"Logic/propositional-logic/model-count-preserving-cnf-transformations/model-count-to3-parsimonious","kind":"node","lean":"Cslib.Logic.PL.CNF.modelCount_to3Parsimonious","parent":525,"result":true,"route":"../roadmap/Logic/propositional-logic/model-count-preserving-cnf-transformations/index.html#model-count-to3-parsimonious","scope":"Logic/propositional-logic/model-count-preserving-cnf-transformations","status":"fully_proved","summary":"The parsimonious reduction preserves the number of assignments to the variables occurring in the formula.","title":"The parsimonious 3-CNF reduction preserves the model count"},{"data_url":"scopes/Logic/propositional-logic/resolution.json","id":"Logic/propositional-logic/resolution","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/resolution/index.html","scope":"Logic/propositional-logic/resolution","status":"fully_proved","title":"Resolution"},{"id":"Logic/propositional-logic/resolution/resolvent","kind":"node","lean":"Cslib.Logic.PL.CNFClause.resolvent","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#resolvent","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"Let $x$ be a variable and $C$, $D$ clauses. The resolvent of $C$ and $D$ on $x$ is the clause consisting of the literals of $C$ other than $(x, \\mathrm{true})$, in order, followed by the literals of $D$ other than $(x, \\mathrm{false})$, in order.","title":"Resolvent of two clauses"},{"id":"Logic/propositional-logic/resolution/satisfies-resolvent","kind":"node","lean":"Cslib.Logic.PL.CNFClause.satisfies_resolvent","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#satisfies-resolvent","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"Let $x$ be a variable and $C$, $D$ clauses. Every assignment that satisfies both $C$ and $D$ satisfies the resolvent of $C$ and $D$ on $x$.","title":"Resolution rule soundness"},{"id":"Logic/propositional-logic/resolution/resolution-derivable","kind":"node","lean":"Cslib.Logic.PL.CNF.ResolutionDerivable","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#resolution-derivable","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"Given a CNF formula $\\varphi$, the clauses derivable from $\\varphi$ by resolution form the smallest set $S$ of clauses that contains every clause of $\\varphi$ and, for every variable $x$ and all clauses $C, D \\in S$ such that $C$ contains the literal $(x, \\mathrm{true})$ and $D$ contains the literal $(x, \\mathrm{false})$, contains the resolvent of $C$ and $D$ on $x$.","title":"Resolution derivability"},{"id":"Logic/propositional-logic/resolution/satisfies","kind":"node","lean":"Cslib.Logic.PL.CNF.ResolutionDerivable.satisfies","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#satisfies","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula. Every clause derivable by resolution from $\\varphi$ is satisfied by every assignment that satisfies $\\varphi$.","title":"Soundness of resolution"},{"id":"Logic/propositional-logic/resolution/not-cnf-sat-of-resolution-derivable-nil","kind":"node","lean":"Cslib.Logic.PL.CNF.not_cnfSat_of_resolutionDerivable_nil","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#not-cnf-sat-of-resolution-derivable-nil","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"If the empty clause is derivable by resolution from a CNF formula $\\varphi$, then $\\varphi$ is not satisfiable.","title":"Refutable formulas are unsatisfiable"},{"id":"Logic/propositional-logic/resolution/mem-restrict-iff","kind":"node","lean":"Cslib.Logic.PL.CNF.mem_restrict_iff","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#mem-restrict-iff","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula and $\\ell$ a literal. A clause $C$ belongs to the restriction of $\\varphi$ by $\\ell$ if and only if there is a clause $C_0$ of $\\varphi$ that does not contain $\\ell$ such that $C$ is the restriction of $C_0$ by $\\ell$.","title":"Clauses of a restricted formula"},{"id":"Logic/propositional-logic/resolution/lift-restrict","kind":"node","lean":"Cslib.Logic.PL.CNF.ResolutionDerivable.lift_restrict","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#lift-restrict","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula, $x$ a variable and $b$ a Boolean value. If a clause $D$ is derivable by resolution from the restriction of $\\varphi$ by the literal $(x, b)$, then some clause $D_0$ is derivable by resolution from $\\varphi$ such that every literal of $D_0$ is either a literal of $D$ or the complementary literal $(x, \\lnot b)$.","title":"Lifting lemma"},{"id":"Logic/propositional-logic/resolution/nil-mem-of-vars-eq-empty","kind":"node","lean":"Cslib.Logic.PL.CNF.nil_mem_of_vars_eq_empty","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#nil-mem-of-vars-eq-empty","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"If a CNF formula $\\varphi$ has no variables and is not satisfiable, then the empty clause is one of the clauses of $\\varphi$.","title":"Unsatisfiable formulas without variables contain the empty clause"},{"id":"Logic/propositional-logic/resolution/resolution-derivable-nil-of-not-cnf-sat","kind":"node","lean":"Cslib.Logic.PL.CNF.resolutionDerivable_nil_of_not_cnfSat","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#resolution-derivable-nil-of-not-cnf-sat","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"Refutation completeness of resolution: every unsatisfiable CNF formula has a resolution derivation of the empty clause.","title":"Refutation completeness"},{"id":"Logic/propositional-logic/resolution/resolution-derivable-nil-iff-not-cnf-sat","kind":"node","lean":"Cslib.Logic.PL.CNF.resolutionDerivable_nil_iff_not_cnfSat","parent":543,"result":true,"route":"../roadmap/Logic/propositional-logic/resolution/index.html#resolution-derivable-nil-iff-not-cnf-sat","scope":"Logic/propositional-logic/resolution","status":"fully_proved","summary":"A CNF formula has a resolution refutation iff it is unsatisfiable.","title":"Soundness and refutation completeness"},{"data_url":"scopes/Logic/propositional-logic/2-satisfiability.json","id":"Logic/propositional-logic/2-satisfiability","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","title":"2-satisfiability"},{"id":"Logic/propositional-logic/2-satisfiability/is-two-cnf","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.IsTwoCNF","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#is-two-cnf","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"A CNF formula is a 2-CNF formula when each of its clauses, viewed as a list of literals, has length at most $2$ (repeated literals are counted with multiplicity).","title":"2-CNF formulas"},{"id":"Logic/propositional-logic/2-satisfiability/neg","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.neg","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#neg","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"The complement of a literal $(x, b)$, with variable index $x \\in \\mathbb{N}$ and polarity $b$, is the literal $(x, \\lnot b)$: same variable, opposite polarity.","title":"Complement of a literal"},{"id":"Logic/propositional-logic/2-satisfiability/eq-or-eq-of-length-le-two","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.eq_or_eq_of_length_le_two","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#eq-or-eq-of-length-le-two","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"In a clause with at most two literals, any two different members $a$ and $b$ are all of its members: every member $c$ of the clause equals $a$ or $b$.","title":"Clauses with at most two literals"},{"id":"Logic/propositional-logic/2-satisfiability/imp-edge","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.ImpEdge","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#imp-edge","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and literals $l$ and $m$, there is an implication edge from $l$ to $m$ when some clause of $\\varphi$ contains both $\\neg l$ and $m$, either as two different literals or with $m$ as its only literal (so a unit clause $m$ gives the edge from $\\neg m$ to $m$).","title":"Implication edge"},{"id":"Logic/propositional-logic/2-satisfiability/imp-reach","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.ImpReach","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#imp-reach","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, a literal $l$ reaches a literal $m$ when there is a path of zero or more implication edges of $\\varphi$ from $l$ to $m$; that is, reachability is the reflexive-transitive closure of the implication edge relation.","title":"Reachability in the implication graph"},{"id":"Logic/propositional-logic/2-satisfiability/skew","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.ImpEdge.skew","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#skew","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and literals $l$ and $m$, if the implication graph of $\\varphi$ has an edge from $l$ to $m$, then it also has the contrapositive edge from $\\lnot m$ to $\\lnot l$, where $\\lnot$ denotes the complementary literal.","title":"Skew symmetry of the implication graph"},{"id":"Logic/propositional-logic/2-satisfiability/satisfies-of-imp-edge","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.satisfies_of_impEdge","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#satisfies-of-imp-edge","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"Let $\\varphi$ be a 2-CNF formula and $\\sigma$ an assignment of truth values to the variables that satisfies $\\varphi$. If the implication graph of $\\varphi$ has an edge from a literal $l$ to a literal $m$ and $\\sigma$ satisfies $l$, then $\\sigma$ satisfies $m$.","title":"Satisfying assignments respect implication edges"},{"id":"Logic/propositional-logic/2-satisfiability/inv","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.Inv","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#inv","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"A clause $C$ satisfies the invariant with respect to a CNF formula $\\varphi$ when for every literal $a$ of $C$ there is a literal $b$ of $C$ such that the complementary literal $\\lnot a$ reaches $b$ in the implication graph of $\\varphi$ (by a path of zero or more edges).","title":"Resolution invariant of the implication graph"},{"id":"Logic/propositional-logic/2-satisfiability/exists-contradictory-of-resolution-deriv","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.exists_contradictory_of_resolutionDerivable_nil","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#exists-contradictory-of-resolution-deriv","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"If a CNF formula $\\varphi$ does not contain the empty clause and the empty clause is derivable from $\\varphi$ by resolution, then there is a variable $x$ such that the literals $x$ and $\\neg x$ reach each other in the implication graph of $\\varphi$.","title":"Resolution refutations give contradictory cycles"},{"id":"Logic/propositional-logic/2-satisfiability/cnf-sat-iff-of-is-two-cnf","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.cnfSat_iff_of_isTwoCNF","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#cnf-sat-iff-of-is-two-cnf","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"2-SAT characterization (Aspvall, Plass, Tarjan): a 2-CNF formula is satisfiable iff it has no empty clause and no variable `x` such that `x` and `\u00acx` reach each other in the implication graph.","title":"2-SAT characterization"},{"id":"Logic/propositional-logic/2-satisfiability/lit","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.Lit","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#lit","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, a literal over $\\varphi$ is a pair $(x, b)$ of a variable index $x \\in \\mathbb{N}$ and a polarity $b \\in \\{\\mathrm{true}, \\mathrm{false}\\}$ such that the variable $x$ occurs in $\\varphi$. These literals are the vertices of the implication graph of $\\varphi$.","title":"Literals over the variables of a formula"},{"id":"Logic/propositional-logic/2-satisfiability/neg-2","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.Lit.neg","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#neg-2","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, the complement of a literal $(x, b)$ over the variables of $\\varphi$ is the literal $(x, \\lnot b)$, which is again a literal over the variables of $\\varphi$.","title":"Complement of a literal over a formula"},{"id":"Logic/propositional-logic/2-satisfiability/var-list","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.varList","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#var-list","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, this is the list of the variables occurring in the literals of $\\varphi$, each exactly once, in the order of their last occurrences when the clauses of $\\varphi$ and their literals are read from first to last.","title":"Duplicate-free list of the variables of a CNF formula"},{"id":"Logic/propositional-logic/2-satisfiability/lit-list","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.litList","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#lit-list","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, this is the list of all literals over the variables of $\\varphi$, each appearing exactly once, arranged variable by variable: for each variable $x$ of $\\varphi$, the positive literal $(x, \\mathrm{true})$ is immediately followed by the negative literal $(x, \\mathrm{false})$.","title":"List of the literals of a CNF formula"},{"id":"Logic/propositional-logic/2-satisfiability/successors","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.successors","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#successors","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula and $l$ a literal over the variables of $\\varphi$, and write $\\neg l$ for the complementary literal. The successors of $l$ are the literals $m$ over the variables of $\\varphi$ such that the implication graph of $\\varphi$ has an edge from $l$ to $m$: some clause of $\\varphi$ contains $\\neg l$ and $m$, either as two different literals or as its only literal. They are listed without\u2026","title":"Successors in the implication graph"},{"id":"Logic/propositional-logic/2-satisfiability/adjacency-table","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.adjacencyTable","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#adjacency-table","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, the hash map that sends each literal over the variables of $\\varphi$ to the list of its implication-graph successors, that is, of the literals $m$ over the variables of $\\varphi$ with an implication edge from it to $m$.","title":"Adjacency table of the implication graph"},{"id":"Logic/propositional-logic/2-satisfiability/implication-graph","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.implicationGraph","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#implication-graph","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, the finite directed graph whose vertices are the literals over the variables of $\\varphi$, listed without repetition, and whose out-neighbours of a literal $l$ are the literals $m$ over the variables of $\\varphi$ with an implication edge from $l$ to $m$, read from the adjacency table of $\\varphi$.","title":"Implication graph as a finite directed graph"},{"id":"Logic/propositional-logic/2-satisfiability/mem-vars-of-imp-edge","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.mem_vars_of_impEdge","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#mem-vars-of-imp-edge","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and literals $l$ and $m$, if the implication graph of $\\varphi$ has an edge from $l$ to $m$, then the variable of $m$ occurs in $\\varphi$.","title":"Implication edges end over variables of the formula"},{"id":"Logic/propositional-logic/2-satisfiability/implication-graph-reachable-iff","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.implicationGraph_reachable_iff","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#implication-graph-reachable-iff","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and literals $l$ and $m$ over the variables of $\\varphi$, $m$ is reachable from $l$ in the implication graph of $\\varphi$, viewed as a finite directed graph on these literals, if and only if $l$ reaches $m$ by implication edges of $\\varphi$.","title":"Reachability in the finite implication graph"},{"id":"Logic/propositional-logic/2-satisfiability/decide2-sat","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.decide2SAT","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#decide2-sat","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, the procedure returns false if $\\varphi$ contains the empty clause; otherwise it returns true if and only if no block of the strongly connected components computed for the implication graph of $\\varphi$ contains a literal together with its complementary literal.","title":"SCC-based 2-SAT decision procedure"},{"id":"Logic/propositional-logic/2-satisfiability/mem-vars-of-imp-reach-true-false","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.mem_vars_of_impReach_true_false","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#mem-vars-of-imp-reach-true-false","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and a variable $x$, if the positive literal $(x, \\mathrm{true})$ reaches the negative literal $(x, \\mathrm{false})$ in the implication graph of $\\varphi$, then $x$ occurs in $\\varphi$.","title":"A positive literal reaching its complement lies over a variable of the formula"},{"id":"Logic/propositional-logic/2-satisfiability/decide2-sat-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.SAT.TwoSAT.decide2SAT_eq_true_iff","parent":554,"result":true,"route":"../roadmap/Logic/propositional-logic/2-satisfiability/index.html#decide2-sat-eq-true-iff","scope":"Logic/propositional-logic/2-satisfiability","status":"fully_proved","summary":"For every CNF formula $\\varphi$ in which every clause has at most two literals, the SCC-based 2-SAT decision procedure returns true on $\\varphi$ if and only if $\\varphi$ is satisfiable.","title":"Correctness of the SCC-based 2-SAT procedure"},{"data_url":"scopes/Logic/propositional-logic/davis-putnam-algorithm.json","id":"Logic/propositional-logic/davis-putnam-algorithm","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","title":"Davis\u2013Putnam algorithm"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/pos-clauses","kind":"node","lean":"Cslib.Logic.PL.CNF.posClauses","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#pos-clauses","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"For a variable $x$ and a CNF formula $\\varphi$, the formula consisting of the clauses of $\\varphi$, in their order, that contain the literal $x$ and do not contain the literal $\\neg x$.","title":"Clauses containing a variable only positively"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/neg-clauses","kind":"node","lean":"Cslib.Logic.PL.CNF.negClauses","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#neg-clauses","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"For a variable $x$ and a CNF formula $\\varphi$, the formula consisting of the clauses of $\\varphi$, in their order, that contain the literal $\\neg x$ and do not contain the literal $x$.","title":"Clauses containing a variable only negatively"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/eliminate","kind":"node","lean":"Cslib.Logic.PL.CNF.eliminate","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#eliminate","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"Eliminating a variable $x$ from a CNF formula $\\varphi$ gives the formula consisting of the clauses of $\\varphi$ that contain neither $x$ nor $\\neg x$, followed by the resolvents on $x$ of $C$ and $D$, for every clause $C$ of $\\varphi$ that contains $x$ but not $\\neg x$ and every clause $D$ of $\\varphi$ that contains $\\neg x$ but not $x$. Clauses containing both $x$ and $\\neg x$ are dropped.","title":"Davis-Putnam elimination of a variable"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/fst-ne-of-mem-resolvent","kind":"node","lean":"Cslib.Logic.PL.CNF.fst_ne_of_mem_resolvent","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#fst-ne-of-mem-resolvent","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"Let $x$ be a variable and let $C$ and $D$ be clauses such that $C$ does not contain $\\neg x$ and $D$ does not contain $x$. Then no literal of the resolvent of $C$ and $D$ on $x$ has variable $x$.","title":"Resolvents do not mention the resolved variable"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/satisfies-eliminate","kind":"node","lean":"Cslib.Logic.PL.CNF.satisfies_eliminate","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#satisfies-eliminate","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"Let $x$ be a variable and $\\varphi$ a CNF formula. Every assignment that satisfies $\\varphi$ also satisfies the formula obtained from $\\varphi$ by eliminating $x$.","title":"Soundness of a Davis-Putnam step"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/exists-satisfies-update-of-satisfies-eli","kind":"node","lean":"Cslib.Logic.PL.CNF.exists_satisfies_update_of_satisfies_eliminate","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#exists-satisfies-update-of-satisfies-eli","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"Let $x$ be a variable, $\\varphi$ a CNF formula and $\\sigma$ an assignment that satisfies the formula obtained from $\\varphi$ by eliminating $x$. Then there is a truth value $b$ such that the assignment that agrees with $\\sigma$ except that it sends $x$ to $b$ satisfies $\\varphi$.","title":"Completeness of a Davis-Putnam step"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/cnf-sat-eliminate-iff","kind":"node","lean":"Cslib.Logic.PL.CNF.cnfSat_eliminate_iff","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#cnf-sat-eliminate-iff","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"For every variable $x$ and every CNF formula $\\varphi$, the formula obtained from $\\varphi$ by eliminating $x$ is satisfiable if and only if $\\varphi$ is satisfiable.","title":"Davis-Putnam elimination preserves satisfiability"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/eliminate-all","kind":"node","lean":"Cslib.Logic.PL.CNF.eliminateAll","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#eliminate-all","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"Given a list $x_1, \\dots, x_n$ of variables and a CNF formula $\\varphi$, eliminate $x_1$ from $\\varphi$, then $x_2$ from the result, and so on up to $x_n$. For the empty list the result is $\\varphi$.","title":"Successive elimination of a list of variables"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/var-list","kind":"node","lean":"Cslib.Logic.PL.CNF.varList","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#var-list","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"For a CNF formula $\\varphi$, the list of the variables occurring in the literals of $\\varphi$, each listed exactly once, in order of first occurrence when the clauses of $\\varphi$, and the literals within each clause, are read from left to right.","title":"Variable list for Davis\u2013Putnam elimination"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/davis-putnam","kind":"node","lean":"Cslib.Logic.PL.CNF.davisPutnam","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#davis-putnam","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"Given a CNF formula $\\varphi$, eliminate the variables that occur in $\\varphi$ one after another, each exactly once, in the order of their first occurrence in $\\varphi$, each elimination being applied to the result of the previous one. The procedure returns true if the resulting formula does not contain the empty clause, and false if it does.","title":"The Davis-Putnam procedure"},{"id":"Logic/propositional-logic/davis-putnam-algorithm/davis-putnam-eq-true-iff","kind":"node","lean":"Cslib.Logic.PL.CNF.davisPutnam_eq_true_iff","parent":577,"result":true,"route":"../roadmap/Logic/propositional-logic/davis-putnam-algorithm/index.html#davis-putnam-eq-true-iff","scope":"Logic/propositional-logic/davis-putnam-algorithm","status":"fully_proved","summary":"For every CNF formula $\\varphi$, the Davis-Putnam procedure returns true on $\\varphi$ if and only if $\\varphi$ is satisfiable.","title":"Correctness of the Davis-Putnam procedure"},{"data_url":"scopes/Logic/propositional-logic/tseitin-transformation.json","id":"Logic/propositional-logic/tseitin-transformation","kind":"scope","navigation":"module","parent":384,"result":false,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","title":"Tseitin transformation"},{"id":"Logic/propositional-logic/tseitin-transformation/neg-lit","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.negLit","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#neg-lit","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"The complement of a literal $(x, b)$ with variable $x$ and polarity $b$ is the literal $(x, \\lnot b)$: the same variable with the opposite polarity.","title":"Complementary literal"},{"id":"Logic/propositional-logic/tseitin-transformation/and-clauses","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.andClauses","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#and-clauses","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For literals $g$, $a$ and $b$, the defining clauses of the gate literal $g$ for the conjunction of $a$ and $b$ are, in this order, the three clauses $\\lnot g \\lor a$, $\\lnot g \\lor b$ and $g \\lor \\lnot a \\lor \\lnot b$, where $\\lnot \\ell$ denotes the complementary literal of $\\ell$ (same variable, opposite polarity).","title":"Tseitin clauses for a conjunction"},{"id":"Logic/propositional-logic/tseitin-transformation/or-clauses","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.orClauses","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#or-clauses","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For literals $g$, $a$ and $b$, the defining clauses of the gate literal $g$ for the disjunction of $a$ and $b$ are the three clauses $\\lnot g \\lor a \\lor b$, $g \\lor \\lnot a$ and $g \\lor \\lnot b$, where $\\lnot \\ell$ denotes the complementary literal of $\\ell$.","title":"Tseitin clauses for a disjunction"},{"id":"Logic/propositional-logic/tseitin-transformation/imp-clauses","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.impClauses","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#imp-clauses","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For literals $g$, $a$ and $b$, the defining clauses of the gate literal $g$ for the implication from $a$ to $b$ are, in this order, the three clauses $\\lnot g \\lor \\lnot a \\lor b$, $g \\lor a$ and $g \\lor \\lnot b$, where $\\lnot \\ell$ denotes the complementary literal of $\\ell$ (same variable, opposite polarity).","title":"Tseitin clauses for an implication"},{"id":"Logic/propositional-logic/tseitin-transformation/tseitin-step","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.tseitinStep","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#tseitin-step","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Given a rule $\\mathit{cl}$ assigning to literals $g$, $a$, $b$ a list of defining clauses, and the results $(a, C_A, k_A)$ and $(b, C_B, k_B)$ of encoding the two operands of a binary connective (each a literal, a clause list and a next unused gate number), the step returns the positive literal $g$ of variable $2k_B+1$, the clause list $C_A$, then $C_B$, then $\\mathit{cl}(g, a, b)$, and the next gate number $k_B+1$.","title":"One step of the Tseitin encoding"},{"id":"Logic/propositional-logic/tseitin-transformation/acc-step","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.accStep","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#acc-step","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Let $c$ map a gate literal and two operand literals to a list of clauses, and let $(a, \\varphi_A, k_A)$ and $(b, \\varphi_B, k_B)$ be the results for the two operands, each a literal, a clause list and a next free gate number. The step returns the positive literal $g$ of the variable $2k_B + 1$, the clause list $c(g, a, b)$ followed by $\\varphi_B$, and the next free gate number $k_B + 1$. Neither $\\varphi_A$ nor\u2026","title":"One step of the linear-time Tseitin builder"},{"id":"Logic/propositional-logic/tseitin-transformation/connective-count","kind":"node","lean":"Cslib.Logic.PL.Proposition.connectiveCount","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#connective-count","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For a propositional formula $A$, built from atoms with the binary connectives $\\wedge$, $\\vee$ and $\\to$, the connective count of $A$ is the number of occurrences of these connectives in $A$. It is $0$ for an atom, and for $B \\circ C$, with $\\circ$ one of $\\wedge$, $\\vee$ and $\\to$, it is the count of $B$ plus the count of $C$ plus $1$.","title":"Connective count of a propositional formula"},{"id":"Logic/propositional-logic/tseitin-transformation/tseitin-aux","kind":"node","lean":"Cslib.Logic.PL.Proposition.tseitinAux","parent":589,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tseitin encoding with gates numbered from "],["math","\\(k\\)"]],"source":"Tseitin encoding with gates numbered from $k$"},"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#tseitin-aux","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Fix a map $\\iota$ from atoms to natural numbers. Given a formula $A$ and a gate number $k$, the encoding returns a literal naming $A$, a list of defining clauses and the next unused gate number. For an atom $a$ it returns the positive literal of variable $2\\iota(a)$, no clauses, and $k$. For $A \\circ B$ with $\\circ$ one of $\\wedge$, $\\vee$, $\\to$, it encodes $A$ from gate $k$, obtaining a literal $a$, clauses $C_A$\u2026","title":"Tseitin encoding with gates numbered from $k$"},{"id":"Logic/propositional-logic/tseitin-transformation/tseitin-acc","kind":"node","lean":"Cslib.Logic.PL.Proposition.tseitinAcc","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#tseitin-acc","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Fix a map $\\iota$ from atoms to natural numbers. Given a formula $A$, a gate number $k$ and a clause list $\\mathit{acc}$, the builder returns a literal naming $A$, a clause list and the next unused gate number. For an atom $a$ it returns the positive literal of variable $2\\iota(a)$, the list $\\mathit{acc}$ unchanged, and $k$. For $A \\circ B$ with $\\circ$ one of $\\wedge$, $\\vee$, $\\to$, it runs on $A$ from gate $k$\u2026","title":"Linear-time Tseitin builder"},{"id":"Logic/propositional-logic/tseitin-transformation/tseitin-acc-eq","kind":"node","lean":"Cslib.Logic.PL.Proposition.tseitinAcc_eq","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#tseitin-acc-eq","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For every map $\\iota$ from atoms to natural numbers, every formula $A$, every gate number $k$ and every clause list $\\mathit{acc}$, the linear-time Tseitin builder run on $A$ from gate $k$ with accumulator $\\mathit{acc}$ returns the same literal naming $A$ and the same next unused gate number as the reference Tseitin encoding of $A$ with gates numbered from $k$, and its clause list is $X$ followed by $\\mathit{acc}$\u2026","title":"The linear-time Tseitin builder agrees with the reference encoding"},{"id":"Logic/propositional-logic/tseitin-transformation/tseitin","kind":"node","lean":"Cslib.Logic.PL.Proposition.tseitin","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#tseitin","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For a map $\\iota$ from atoms to natural numbers and a formula $A$, run the linear-time Tseitin builder on $A$ with gates numbered from $0$ and the empty accumulator, obtaining a literal $\\ell$ naming $A$ and a clause list $C$. The Tseitin CNF of $A$ is the unit clause $[\\ell]$ followed by $C$.","title":"Tseitin CNF"},{"id":"Logic/propositional-logic/tseitin-transformation/tseitin-eq","kind":"node","lean":"Cslib.Logic.PL.Proposition.tseitin_eq","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#tseitin-eq","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For every map $\\iota$ from atoms to natural numbers and every formula $A$, the Tseitin CNF of $A$ is the unit clause whose literal is the literal naming $A$ in the reference Tseitin encoding of $A$ with gates numbered from $0$, followed by a list of clauses that is a permutation of the defining clauses of that encoding.","title":"The Tseitin CNF reorders the reference encoding"},{"id":"Logic/propositional-logic/tseitin-transformation/tseitin-bot","kind":"node","lean":"Cslib.Logic.PL.Proposition.tseitinBot","parent":589,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tseitin CNF with "],["math","\\(\\bot\\)"],["text"," false"]],"source":"Tseitin CNF with $\\bot$ false"},"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#tseitin-bot","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For an atom type with a distinguished atom $\\bot$, a map $\\iota$ from atoms to natural numbers and a formula $A$, this is the CNF consisting of the unit clause whose literal is the negative literal of variable $2\\iota(\\bot)$, followed by the Tseitin CNF of $A$ with respect to $\\iota$.","title":"Tseitin CNF with $\\bot$ false"},{"id":"Logic/propositional-logic/tseitin-transformation/length-tseitin-aux","kind":"node","lean":"Cslib.Logic.PL.Proposition.length_tseitinAux","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#length-tseitin-aux","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For every map $\\iota$ from atoms to natural numbers, every formula $A$ and every gate number $k$, the list of defining clauses produced by the Tseitin encoding of $A$ with gates numbered from $k$ has length $3c$, where $c$ is the number of binary connectives of $A$.","title":"Number of defining clauses in the Tseitin encoding"},{"id":"Logic/propositional-logic/tseitin-transformation/length-tseitin","kind":"node","lean":"Cslib.Logic.PL.Proposition.length_tseitin","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#length-tseitin","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For every proposition $A$ and every map $\\iota$ from atoms to natural numbers, the Tseitin CNF of $A$ with respect to $\\iota$, as a list of clauses counted with repetition, has length $3c + 1$, where $c$ is the number of binary connectives of $A$: three defining clauses per connective plus the unit clause asserting $A$.","title":"Number of clauses of the Tseitin CNF"},{"id":"Logic/propositional-logic/tseitin-transformation/length-le-three-of-mem-tseitin","kind":"node","lean":"Cslib.Logic.PL.Proposition.length_le_three_of_mem_tseitin","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#length-le-three-of-mem-tseitin","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For every proposition $A$ and every map $\\iota$ from atoms to natural numbers, every clause of the Tseitin CNF of $A$ with respect to $\\iota$ has at most three literals.","title":"Tseitin clauses have at most three literals"},{"id":"Logic/propositional-logic/tseitin-transformation/lit-val","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.litVal","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#lit-val","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Let $\\sigma$ assign a Boolean value to every natural-number variable, and let $\\ell = (x, b)$ be a literal with variable $x$ and polarity $b$. The truth value of $\\ell$ under $\\sigma$ is true exactly when $\\sigma(x) = b$.","title":"Truth value of a literal"},{"id":"Logic/propositional-logic/tseitin-transformation/satisfies-append","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.satisfies_append","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#satisfies-append","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For every assignment $\\sigma$ of Boolean values to the natural-number variables and all CNF formulas $\\varphi$ and $\\psi$, $\\sigma$ satisfies the formula consisting of the clauses of $\\varphi$ followed by those of $\\psi$ if and only if $\\sigma$ satisfies both $\\varphi$ and $\\psi$.","title":"Satisfying a concatenation of CNF formulas"},{"id":"Logic/propositional-logic/tseitin-transformation/is-gate","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.IsGate","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#is-gate","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For natural numbers $k$ and $k'$, a variable $x \\in \\mathbb{N}$ is a gate in range $[k, k')$ if $x = 2j + 1$ for some $j \\in \\mathbb{N}$ with $k \\le j < k'$.","title":"Gate variables in a range"},{"id":"Logic/propositional-logic/tseitin-transformation/allowed","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.Allowed","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#allowed","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"For natural numbers $k$ and $k'$, a variable $x \\in \\mathbb{N}$ is allowed for the gate range $[k, k')$ if $x$ is even (it names an atom) or $x$ is a gate in range $[k, k')$, that is, $x = 2j + 1$ for some $j \\in \\mathbb{N}$ with $k \\le j < k'$.","title":"Allowed variables for a gate range"},{"id":"Logic/propositional-logic/tseitin-transformation/atom-dependent","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.AtomDependent","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#atom-dependent","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"A function $f$ from assignments $\\sigma : \\mathbb{N} \\to \\mathrm{Bool}$ to Booleans depends only on the atom variables if $f(\\sigma) = f(\\tau)$ for all assignments $\\sigma$ and $\\tau$ with $\\sigma(2n) = \\tau(2n)$ for every $n \\in \\mathbb{N}$.","title":"Functions of the atom variables"},{"id":"Logic/propositional-logic/tseitin-transformation/spec","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.Spec","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#spec","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Let $k$ be a natural number, let $r = (\\ell, \\Phi, k')$ consist of a literal $\\ell$, a CNF formula $\\Phi$ and a natural number $k'$, and let $v$ map each assignment of Boolean values to the natural-number variables to a Boolean value. Call a variable a gate in range if it equals $2j + 1$ for some $j$ with $k \\le j < k'$. The invariant holds for $k$, $r$ and $v$ when $k \\le k'$; the variable of $\\ell$ and every\u2026","title":"Tseitin encoding invariant"},{"id":"Logic/propositional-logic/tseitin-transformation/satisfies-of-agree","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.satisfies_of_agree","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#satisfies-of-agree","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula and let $\\sigma$ and $\\tau$ be assignments of Boolean values to the natural-number variables that agree on every variable occurring in a literal of $\\varphi$. If $\\sigma$ satisfies $\\varphi$, then $\\tau$ satisfies $\\varphi$.","title":"Satisfaction depends only on the occurring variables"},{"id":"Logic/propositional-logic/tseitin-transformation/atom-val","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.atomVal","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#atom-val","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Given a map $\\iota$ from atoms to natural numbers, a proposition $A$ and an assignment $\\sigma : \\mathbb{N} \\to \\mathrm{Bool}$, this is the truth value of $A$ under the valuation that gives each atom $a$ the value $\\sigma(2\\iota(a))$.","title":"Value of a proposition read off the atom variables"},{"id":"Logic/propositional-logic/tseitin-transformation/spec-tseitin-aux","kind":"node","lean":"Cslib.Algorithms.SAT.Tseitin.spec_tseitinAux","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#spec-tseitin-aux","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Let $\\iota$ map atoms to natural numbers, let $A$ be a proposition and $k$ a natural number, and let $(\\ell, \\Phi, k')$ be the reference Tseitin encoding of $A$ with gates numbered from $k$: the literal naming $A$, its defining clauses and the next unused gate number. For an assignment $\\sigma$ of Boolean values to the natural-number variables, let $A[\\sigma]$ be the truth value of $A$ when each atom $a$ gets the\u2026","title":"Correctness invariant of the Tseitin encoding"},{"id":"Logic/propositional-logic/tseitin-transformation/eval-bool-iff-exists-satisfies-tseitin","kind":"node","lean":"Cslib.Logic.PL.Proposition.evalBool_iff_exists_satisfies_tseitin","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#eval-bool-iff-exists-satisfies-tseitin","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Let $\\iota$ be an injective map from atoms to natural numbers, let $A$ be a proposition and let $v$ be a valuation of the atoms. Then $A$ is true under $v$ if and only if some assignment $\\sigma : \\mathbb{N} \\to \\mathrm{Bool}$ with $\\sigma(2\\iota(a)) = v(a)$ for every atom $a$ satisfies the Tseitin CNF of $A$ with respect to $\\iota$.","title":"Tseitin correctness"},{"id":"Logic/propositional-logic/tseitin-transformation/cnf-sat-tseitin-bot-iff","kind":"node","lean":"Cslib.Logic.PL.Proposition.cnfSat_tseitinBot_iff","parent":589,"result":true,"route":"../roadmap/Logic/propositional-logic/tseitin-transformation/index.html#cnf-sat-tseitin-bot-iff","scope":"Logic/propositional-logic/tseitin-transformation","status":"fully_proved","summary":"Let the atoms have a distinguished atom $\\bot$, let $\\iota$ be an injective map from atoms to natural numbers and let $A$ be a proposition. The Tseitin CNF of $A$ with respect to $\\iota$, together with the unit clause making the variable $2\\iota(\\bot)$ false, is satisfiable if and only if $A$ is true under some valuation $v$ of the atoms with $v(\\bot)$ false.","title":"Equisatisfiability of the Tseitin transformation"},{"data_url":"scopes/Logic/first-order-logic.json","id":"Logic/first-order-logic","kind":"scope","parent":1,"result":false,"route":"../roadmap/Logic/first-order-logic/index.html","scope":"Logic/first-order-logic","status":"fully_proved","summary":"Syntax, proof calculi and semantics of first-order predicate logic, including soundness and G\u00f6del's completeness theorem.","title":"First-order logic"},{"data_url":"scopes/Logic/first-order-logic/embedding-propositional-logic-into-first-order-l.json","id":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l","kind":"scope","navigation":"module","parent":617,"result":false,"route":"../roadmap/Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/index.html","scope":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l","status":"fully_proved","title":"Embedding propositional logic into first-order logic"},{"id":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/prop-relations","kind":"node","lean":"Cslib.Logic.FOL.PropRelations","parent":618,"result":true,"route":"../roadmap/Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/index.html#prop-relations","scope":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l","status":"fully_proved","summary":"For a type of atoms, the family of relation symbols indexed by arity that consists of one nullary relation symbol for each atom $a$ and has no relation symbols of positive arity.","title":"Relation symbols of the propositional language"},{"id":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/prop-language","kind":"node","lean":"Cslib.Logic.FOL.propLanguage","parent":618,"result":true,"route":"../roadmap/Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/index.html#prop-language","scope":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l","status":"fully_proved","summary":"For a type of atoms, the first-order language whose relation symbols are exactly one nullary relation symbol for each atom, with no relation symbols of positive arity and no function symbols.","title":"First-order language of propositional atoms"},{"id":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/embed","kind":"node","lean":"Cslib.Logic.FOL.Proposition.embed","parent":618,"result":true,"route":"../roadmap/Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/index.html#embed","scope":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l","status":"fully_proved","summary":"The translation of propositions over a type of atoms into sentences of the first-order language of propositional atoms, defined by recursion: an atom $a$ becomes the atomic sentence given by the nullary relation symbol of $a$, and $A \\wedge B$, $A \\vee B$ and $A \\to B$ become the conjunction, the disjunction and the implication of the translations of $A$ and $B$.","title":"Embedding propositions as first-order sentences"},{"id":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/bool-structure","kind":"node","lean":"Cslib.Logic.FOL.Proposition.boolStructure","parent":618,"result":true,"route":"../roadmap/Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/index.html#bool-structure","scope":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l","status":"fully_proved","summary":"Given a valuation $v$ assigning a Boolean to each atom, the Boolean structure of $v$ is the structure for the first-order language of propositional atoms whose domain is the set of Booleans $\\{\\mathrm{false}, \\mathrm{true}\\}$ and in which the nullary relation symbol of an atom $a$ holds exactly when $v(a)$ is true. The language has no function symbols, so nothing else needs to be interpreted.","title":"Boolean structure of a valuation"},{"id":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/realize-embed-iff-eval-bool","kind":"node","lean":"Cslib.Logic.FOL.Proposition.realize_embed_iff_evalBool","parent":618,"result":true,"route":"../roadmap/Logic/first-order-logic/embedding-propositional-logic-into-first-order-l/index.html#realize-embed-iff-eval-bool","scope":"Logic/first-order-logic/embedding-propositional-logic-into-first-order-l","status":"fully_proved","summary":"For every valuation $v$ assigning a Boolean to each atom and every proposition $A$, the first-order sentence obtained by embedding $A$ is true in the Boolean structure of $v$ if and only if the truth-table value of $A$ under $v$ is true.","title":"The embedding of propositions preserves truth-table semantics"},{"data_url":"scopes/Logic/first-order-logic/first-order-syntax.json","id":"Logic/first-order-logic/first-order-syntax","kind":"scope","navigation":"module","parent":617,"result":false,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","title":"First-order syntax"},{"id":"Logic/first-order-logic/first-order-syntax/leaf-count","kind":"node","lean":"FirstOrder.Language.Term.leafCount","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#leaf-count","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"The number of leaves of a first-order term.","title":"Leaf count of a first-order term"},{"id":"Logic/first-order-logic/first-order-syntax/leaf-count-relabel","kind":"node","lean":"FirstOrder.Language.Term.leafCount_relabel","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#leaf-count-relabel","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Relabeling variables does not change the number of leaves.","title":"Relabeling variables preserves the leaf count"},{"id":"Logic/first-order-logic/first-order-syntax/term-symbols","kind":"node","lean":"Cslib.Logic.FOL.termSymbols","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#term-symbols","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"The set of symbols of a first-order language $L$ that occur in a term $t$ (with free variables and possibly bound variables), defined by recursion on $t$: a variable contributes no symbols, and $f(t_1, \\dots, t_k)$ contributes the function symbol $f$ together with the symbols of the terms $t_i$. The result lies in the set of all symbols of $L$, but only function symbols can occur in a term.","title":"Symbols of a term"},{"id":"Logic/first-order-logic/first-order-syntax/formula-symbols","kind":"node","lean":"Cslib.Logic.FOL.formulaSymbols","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#formula-symbols","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"The set of function and relation symbols of a first-order language $L$ that occur in a formula $\\varphi$ (with free variables and possibly bound variables), defined by recursion on $\\varphi$: falsum $\\bot$ contributes no symbols, an equation $t_1 = t_2$ contributes the symbols of the terms $t_1$ and $t_2$, an atomic formula $R(t_1, \\dots, t_k)$ contributes $R$ together with the symbols of the terms $t_i$, an\u2026","title":"Symbols of a formula"},{"id":"Logic/first-order-logic/first-order-syntax/instantiate-term","kind":"node","lean":"Cslib.Logic.FOL.instantiateTerm","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#instantiate-term","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language, $t$ an $L$-term with free variables in $\\alpha$, and $s$ an $L$-term with free variables in $\\alpha$ and $m + 1$ bound variables $0, \\dots, m$. The instantiation of $t$ in $s$ is the term with free variables in $\\alpha$ and $m$ bound variables obtained from $s$ by substituting $t$ for the bound variable $0$, replacing each other bound variable $i + 1$ by $i$, and leaving the free\u2026","title":"Instantiating the outermost bound variable of a term"},{"id":"Logic/first-order-logic/first-order-syntax/instantiate","kind":"node","lean":"Cslib.Logic.FOL.instantiate","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#instantiate","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language, $t$ an $L$-term with free variables in $\\alpha$, and $\\varphi$ a bounded $L$-formula with free variables in $\\alpha$ and $m + 1$ bound variables $0, \\dots, m$. The instantiation of $t$ in $\\varphi$ is the bounded formula with free variables in $\\alpha$ and $m$ bound variables obtained from $\\varphi$ by instantiating $t$ in every term of its atomic subformulas: in each term, $t$ is\u2026","title":"Instantiating the outermost bound variable of a formula"},{"id":"Logic/first-order-logic/first-order-syntax/instantiate-term-var-zero","kind":"node","lean":"Cslib.Logic.FOL.instantiateTerm_var_zero","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#instantiate-term-var-zero","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language, $t$ an $L$-term with free variables in $\\alpha$, and $m$ a natural number. Instantiating $t$ in the term consisting of the bound variable $0$ (a term with free variables in $\\alpha$ and $m + 1$ bound variables) gives $t$, regarded as a term with free variables in $\\alpha$ and $m$ bound variables.","title":"Instantiation at the outermost bound variable"},{"id":"Logic/first-order-logic/first-order-syntax/instantiate-term-var-succ","kind":"node","lean":"Cslib.Logic.FOL.instantiateTerm_var_succ","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#instantiate-term-var-succ","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language, $t$ an $L$-term with free variables in $\\alpha$, and $i < m$. Instantiating $t$ in the term consisting of the bound variable $i + 1$ (a term with free variables in $\\alpha$ and $m + 1$ bound variables) gives the term consisting of the bound variable $i$ (with $m$ bound variables).","title":"Instantiation shifts the remaining bound variables down"},{"id":"Logic/first-order-logic/first-order-syntax/instantiate-all","kind":"node","lean":"Cslib.Logic.FOL.instantiate_all","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#instantiate-all","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language, $t$ an $L$-term with free variables in $\\alpha$, and $\\varphi$ a bounded $L$-formula with free variables in $\\alpha$ and $m + 2$ bound variables. Instantiating $t$ in the universal formula $\\forall \\varphi$ (which has $m + 1$ bound variables) gives the universal quantification of the instantiation of $t$ in $\\varphi$.","title":"Instantiation commutes with the universal quantifier"},{"id":"Logic/first-order-logic/first-order-syntax/realize-instantiate-formula","kind":"node","lean":"Cslib.Logic.FOL.realize_instantiate_formula","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#realize-instantiate-formula","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language, $M$ an $L$-structure, $v : \\alpha \\to M$ an assignment of the free variables, $t$ an $L$-term with free variables in $\\alpha$, and $\\varphi$ a bounded $L$-formula with free variables in $\\alpha$ and one bound variable. Then the instantiation of $t$ in $\\varphi$, a formula with free variables in $\\alpha$, holds in $M$ under $v$ if and only if $\\varphi$ holds in $M$ under $v$ with\u2026","title":"Truth of an instance"},{"id":"Logic/first-order-logic/first-order-syntax/realize-all-formula","kind":"node","lean":"Cslib.Logic.FOL.realize_all_formula","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#realize-all-formula","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language, $M$ an $L$-structure, $v : \\alpha \\to M$ an assignment of the free variables, and $\\varphi$ a bounded $L$-formula with free variables in $\\alpha$ and one bound variable. Then the formula $\\forall \\varphi$ holds in $M$ under $v$ if and only if, for every $a \\in M$, $\\varphi$ holds in $M$ under $v$ with its bound variable set to $a$.","title":"Truth of a universal formula"},{"id":"Logic/first-order-logic/first-order-syntax/realize-congr","kind":"node","lean":"Cslib.Logic.FOL.realize_congr","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#realize-congr","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language, $M$ an $L$-structure, and $\\varphi$ a bounded $L$-formula with free variables in $\\alpha$ and $n$ bound variables. If two assignments $v, v' : \\alpha \\to M$ agree on every free variable of $\\varphi$, then for every choice of values in $M$ for the $n$ bound variables, $\\varphi$ holds in $M$ under $v$ and these values if and only if it holds under $v'$ and these values.","title":"Truth depends only on the free variables"},{"id":"Logic/first-order-logic/first-order-syntax/rename-term","kind":"node","lean":"Cslib.Logic.FOL.renameTerm","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#rename-term","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Given a map $f : \\alpha \\to \\beta$ and $n \\in \\mathbb{N}$, a term of a first-order language over the variables $\\alpha \\sqcup \\mathrm{Fin}\\,n$ (free variables from $\\alpha$, bound variables from $\\mathrm{Fin}\\,n$) becomes a term over $\\beta \\sqcup \\mathrm{Fin}\\,n$ by replacing each free variable $a$ with $f(a)$ and leaving the bound variables unchanged.","title":"Renaming free variables of a term"},{"id":"Logic/first-order-logic/first-order-syntax/rename","kind":"node","lean":"Cslib.Logic.FOL.rename","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#rename","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Given a map $f : \\alpha \\to \\beta$, a bounded formula $\\varphi$ of a first-order language $L$ with free variables in $\\alpha$ and $n$ bound variables becomes a bounded formula of $L$ with free variables in $\\beta$ and the same $n$ bound variables, by replacing each free variable $a$ with $f(a)$ in every term of $\\varphi$ and leaving the bound variables, the relation symbols and the logical connectives and\u2026","title":"Renaming free variables of a formula"},{"id":"Logic/first-order-logic/first-order-syntax/realize-rename","kind":"node","lean":"Cslib.Logic.FOL.realize_rename","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#realize-rename","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $M$ be a structure for a first-order language $L$, let $f : \\alpha \\to \\beta$, and let $\\varphi$ be a bounded formula of $L$ with free variables in $\\alpha$ and $n$ bound variables. For every assignment $v : \\beta \\to M$ of the free variables and every assignment $x : \\mathrm{Fin}\\,n \\to M$ of the bound variables, the renaming of $\\varphi$ along $f$ holds in $M$ at $v$ and $x$ if and only if $\\varphi$ holds in\u2026","title":"Truth of a renamed formula"},{"id":"Logic/first-order-logic/first-order-syntax/relabel-inl-comp-eq-cast-le-rename","kind":"node","lean":"Cslib.Logic.FOL.relabel_inl_comp_eq_castLE_rename","parent":624,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relabeling along "],["math","\\(f\\)"],["text"," agrees with renaming"]],"source":"Relabeling along $f$ agrees with renaming"},"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#relabel-inl-comp-eq-cast-le-rename","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $f : \\alpha \\to \\beta$ and let $\\varphi$ be a bounded formula of a first-order language with free variables in $\\alpha$ and $k$ bound variables. Mathlib's relabeling of $\\varphi$ along the map $\\alpha \\to \\beta \\sqcup \\mathrm{Fin}\\,0$ sending $a$ to the free variable $f(a)$ replaces each free variable $a$ by $f(a)$ and produces a formula with $0 + k$ bound variables. This formula equals the renaming of\u2026","title":"Relabeling along $f$ agrees with renaming"},{"id":"Logic/first-order-logic/first-order-syntax/formula-relabel-eq-rename","kind":"node","lean":"Cslib.Logic.FOL.formula_relabel_eq_rename","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#formula-relabel-eq-rename","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language and $f : \\alpha \\to \\beta$ a map. For every $L$-formula $\\varphi$ with free variables in $\\alpha$ (a bounded formula with no bound variables), Mathlib's relabelling of $\\varphi$ along $f$ equals the renaming of the free variables of $\\varphi$ along $f$.","title":"Relabelling a formula is renaming"},{"id":"Logic/first-order-logic/first-order-syntax/rename-congr","kind":"node","lean":"Cslib.Logic.FOL.rename_congr","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#rename-congr","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"If a map $f : \\alpha \\to \\alpha$ fixes every free variable of a bounded formula $\\varphi$, then renaming the free variables of $\\varphi$ along $f$ leaves $\\varphi$ unchanged.","title":"Renaming that fixes the free variables"},{"id":"Logic/first-order-logic/first-order-syntax/free-var-finset-rename","kind":"node","lean":"Cslib.Logic.FOL.freeVarFinset_rename","parent":624,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-syntax/index.html#free-var-finset-rename","scope":"Logic/first-order-logic/first-order-syntax","status":"fully_proved","summary":"Let $L$ be a first-order language and $f : \\alpha \\to \\beta$ a map. For every bounded $L$-formula $\\varphi$ with free variables in $\\alpha$ and $n$ bound variables, the set of free variables of the renaming of $\\varphi$ along $f$ is the image under $f$ of the set of free variables of $\\varphi$.","title":"Free variables of a renamed formula"},{"data_url":"scopes/Logic/first-order-logic/first-order-natural-deduction.json","id":"Logic/first-order-logic/first-order-natural-deduction","kind":"scope","navigation":"module","parent":617,"result":false,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","title":"First-order natural deduction"},{"id":"Logic/first-order-logic/first-order-natural-deduction/derivable","kind":"node","lean":"Cslib.Logic.FOL.Derivable","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#derivable","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Fix a first-order language $L$ and a type $\\alpha$ of variables; formulas and terms below are those of $L$ with free variables in $\\alpha$. For a formula $\\varphi$ with one additional bound variable and a term $t$, write $\\varphi[t]$ for $\\varphi$ with $t$ substituted for that bound variable and $\\forall \\varphi$ for its universal quantification. Write $\\neg \\psi$ for $\\psi \\to \\bot$, and $\\psi, \\Gamma$ for the\u2026","title":"Natural deduction"},{"id":"Logic/first-order-logic/first-order-natural-deduction/sound","kind":"node","lean":"Cslib.Logic.FOL.Derivable.sound","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#sound","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $\\Gamma$ be a finite list of formulas of a first-order language $L$ with free variables in $\\alpha$, and let $\\varphi$ be such a formula. If $\\Gamma \\vdash \\varphi$ in natural deduction, then for every $L$-structure $M$ and every assignment $v \\colon \\alpha \\to M$ under which all members of $\\Gamma$ hold, $\\varphi$ holds under $v$.","title":"Soundness of first-order natural deduction"},{"id":"Logic/first-order-logic/first-order-natural-deduction/rename","kind":"node","lean":"Cslib.Logic.FOL.Derivable.rename","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#rename","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $f \\colon \\alpha \\to \\beta$ be an injective map between types of variables. For a formula $\\psi$ of a first-order language $L$ with free variables in $\\alpha$, write $f(\\psi)$ for the renaming of the free variables of $\\psi$ by $f$, and for a finite list $\\Gamma$ of such formulas write $f(\\Gamma)$ for the list obtained by renaming each member of $\\Gamma$. If $\\Gamma$ is such a list, $\\varphi$ is such a formula\u2026","title":"Injective renaming preserves derivability"},{"id":"Logic/first-order-logic/first-order-natural-deduction/relabel","kind":"node","lean":"Cslib.Logic.FOL.Derivable.relabel","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#relabel","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $f \\colon \\alpha \\to \\beta$ be an injective map between types of variables. For a formula $\\psi$ of a first-order language $L$ with free variables in $\\alpha$, write $f(\\psi)$ for Mathlib's relabeling of $\\psi$ by $f$, which replaces each free variable $x$ by $f(x)$, and for a finite list $\\Gamma$ of such formulas write $f(\\Gamma)$ for the list obtained by applying this to each member of $\\Gamma$. If $\\Gamma$\u2026","title":"Injective relabeling preserves derivability"},{"id":"Logic/first-order-logic/first-order-natural-deduction/mono","kind":"node","lean":"Cslib.Logic.FOL.Derivable.mono","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#mono","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $\\alpha$ be an infinite type of variables, let $\\Gamma$ and $\\Delta$ be finite lists of formulas of a first-order language $L$ with free variables in $\\alpha$ such that every member of $\\Gamma$ is a member of $\\Delta$, and let $\\varphi$ be such a formula. If $\\Gamma \\vdash \\varphi$ in natural deduction, then $\\Delta \\vdash \\varphi$.","title":"Weakening"},{"id":"Logic/first-order-logic/first-order-natural-deduction/cut","kind":"node","lean":"Cslib.Logic.FOL.Derivable.cut","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#cut","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $\\Gamma$ be a finite list of formulas of a first-order language $L$ with free variables in $\\alpha$, let $\\varphi$ and $\\psi$ be such formulas, and write $\\varphi, \\Gamma$ for the list $\\Gamma$ with $\\varphi$ added in front. If $\\Gamma \\vdash \\varphi$ and $\\varphi, \\Gamma \\vdash \\psi$ in natural deduction, then $\\Gamma \\vdash \\psi$.","title":"Cut"},{"id":"Logic/first-order-logic/first-order-natural-deduction/ex-i","kind":"node","lean":"Cslib.Logic.FOL.Derivable.exI","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#ex-i","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $\\Gamma$ be a finite list of formulas of a first-order language $L$ with free variables in $\\alpha$, let $\\varphi$ be such a formula with one additional bound variable, let $t$ be a term, and write $\\varphi[t]$ for $\\varphi$ with $t$ substituted for that bound variable. If $\\Gamma \\vdash \\varphi[t]$ in natural deduction, then $\\Gamma \\vdash \\exists \\varphi$, where $\\exists \\varphi$ abbreviates\u2026","title":"Existential introduction"},{"id":"Logic/first-order-logic/first-order-natural-deduction/symm","kind":"node","lean":"Cslib.Logic.FOL.Derivable.symm","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#symm","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $\\Gamma$ be a list of first-order formulas and let $s$ and $t$ be terms. If the equation $s = t$ is derivable from the hypotheses $\\Gamma$, then the equation $t = s$ is derivable from $\\Gamma$.","title":"Symmetry of equality in natural deduction"},{"id":"Logic/first-order-logic/first-order-natural-deduction/trans","kind":"node","lean":"Cslib.Logic.FOL.Derivable.trans","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#trans","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $\\Gamma$ be a list of first-order formulas and let $r$, $s$ and $t$ be terms. If the equations $r = s$ and $s = t$ are both derivable from the hypotheses $\\Gamma$, then the equation $r = t$ is derivable from $\\Gamma$.","title":"Transitivity of equality in natural deduction"},{"id":"Logic/first-order-logic/first-order-natural-deduction/provable","kind":"node","lean":"Cslib.Logic.FOL.Provable","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#provable","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"A set $T$ of formulas of a first-order language $L$ with free variables in $\\alpha$ proves such a formula $\\varphi$ when there is a finite list $\\Gamma$ of members of $T$ with $\\Gamma \\vdash \\varphi$ in natural deduction.","title":"Provability from a set of formulas"},{"id":"Logic/first-order-logic/first-order-natural-deduction/models","kind":"node","lean":"Cslib.Logic.FOL.Provable.models","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#models","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"Let $T$ be a theory, that is, a set of sentences of a first-order language $L$, and regard each sentence of $T$ as a formula with free variables in $\\alpha$. If a formula $\\varphi$ with free variables in $\\alpha$ is provable from these formulas, then $\\varphi$ holds in every model of $T$ under every assignment of elements of the model to the variables in $\\alpha$.","title":"Soundness for theories"},{"id":"Logic/first-order-logic/first-order-natural-deduction/not-derivable-nil-bot","kind":"node","lean":"Cslib.Logic.FOL.not_derivable_nil_bot","parent":644,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-natural-deduction/index.html#not-derivable-nil-bot","scope":"Logic/first-order-logic/first-order-natural-deduction","status":"fully_proved","summary":"The empty context does not derive `\u22a5`.","title":"Consistency of the calculus"},{"data_url":"scopes/Logic/first-order-logic/first-order-unification.json","id":"Logic/first-order-logic/first-order-unification","kind":"scope","navigation":"module","parent":617,"result":false,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","title":"First-order unification"},{"id":"Logic/first-order-logic/first-order-unification/term","kind":"node","lean":"Cslib.Unification.Term","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#term","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For a type $C$ of constants, a term over $C$ is either a variable $x$, indexed by a natural number $x \\in \\mathbb{N}$, or a constant $c \\in C$, or a binary node built from two terms $l$ and $r$.","title":"Terms for unification"},{"id":"Logic/first-order-logic/first-order-unification/subst","kind":"node","lean":"Cslib.Unification.Subst","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#subst","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"A substitution is a map assigning a term to every variable, that is, a function from the set $\\mathbb{N}$ of variables to terms.","title":"Substitution"},{"id":"Logic/first-order-logic/first-order-unification/subst-2","kind":"node","lean":"Cslib.Unification.Term.subst","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#subst-2","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For a substitution $\\sigma$ and a term $t$, the term $t\\sigma$ is obtained from $t$ by replacing every variable $x$ by $\\sigma(x)$: a variable $x$ becomes $\\sigma(x)$, a constant is unchanged, and a node with subterms $l$ and $r$ becomes the node with subterms $l\\sigma$ and $r\\sigma$.","title":"Applying a substitution to a term"},{"id":"Logic/first-order-logic/first-order-unification/size","kind":"node","lean":"Cslib.Unification.Term.size","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#size","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"The size of a term is the number of symbols in it: a variable and a constant have size $1$, and a node with subterms $l$ and $r$ has size equal to the size of $l$ plus the size of $r$ plus $1$.","title":"Size of a term"},{"id":"Logic/first-order-logic/first-order-unification/vars","kind":"node","lean":"Cslib.Unification.Term.vars","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#vars","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"The variables of a term form a finite set of natural numbers, defined recursively: the variable $x$ has the variable set $\\{x\\}$, a constant has no variables, and the variables of a node with subterms $l$ and $r$ are those of $l$ together with those of $r$.","title":"Variables of a term"},{"id":"Logic/first-order-logic/first-order-unification/subst-congr","kind":"node","lean":"Cslib.Unification.Term.subst_congr","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#subst-congr","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"If two substitutions $\\sigma$ and $\\tau$ agree on every variable occurring in a term $u$, then applying $\\sigma$ to $u$ gives the same term as applying $\\tau$ to $u$.","title":"Substitution depends only on the variables of the term"},{"id":"Logic/first-order-logic/first-order-unification/exists-of-mem-vars-subst","kind":"node","lean":"Cslib.Unification.Term.exists_of_mem_vars_subst","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#exists-of-mem-vars-subst","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $s$ be a substitution and $t$ a term. Every variable of the term obtained by applying $s$ to $t$ is a variable of $s(y)$ for some variable $y$ of $t$.","title":"Variables of a substituted term"},{"id":"Logic/first-order-logic/first-order-unification/vars-subset-vars-subst","kind":"node","lean":"Cslib.Unification.Term.vars_subset_vars_subst","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#vars-subset-vars-subst","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For every substitution $s$, term $t$ and variable $y$ occurring in $t$, every variable of the term $s(y)$ occurs in the term obtained from $t$ by applying $s$.","title":"Variables of a substituted variable occur in the substituted term"},{"id":"Logic/first-order-logic/first-order-unification/size-subst-lt-of-mem","kind":"node","lean":"Cslib.Unification.Term.size_subst_lt_of_mem","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#size-subst-lt-of-mem","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $x$ be a variable and $t$ a term in which $x$ occurs, with $t$ not the variable $x$ itself. Then for every substitution $\\sigma$, the term $\\sigma(x)$ has strictly smaller size than $t\\sigma$, the result of applying $\\sigma$ to $t$.","title":"Occurs check: size bound"},{"id":"Logic/first-order-logic/first-order-unification/vars-subst-single-subset","kind":"node","lean":"Cslib.Unification.Term.vars_subst_single_subset","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#vars-subst-single-subset","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $x$ be a variable and $t$ a term in which $x$ does not occur. For every term $u$, every variable of the term obtained from $u$ by replacing each occurrence of $x$ with $t$ is a variable of $u$ or of $t$, and is different from $x$.","title":"Substituting for a variable that does not occur in the substituted term"},{"id":"Logic/first-order-logic/first-order-unification/comp","kind":"node","lean":"Cslib.Unification.Subst.comp","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#comp","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For substitutions $s$ and $t$, the composite $s \\gg t$ (first $s$, then $t$) sends each variable $x$ to the term obtained from $s(x)$ by applying $t$ to every variable of $s(x)$.","title":"Composition of substitutions on first-order terms"},{"id":"Logic/first-order-logic/first-order-unification/single","kind":"node","lean":"Cslib.Unification.Subst.single","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#single","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For a variable $x$ and a term $t$, the substitution $[x \\mapsto t]$ sends $x$ to $t$ and every other variable $y$ to the term $y$.","title":"Single-variable substitution on first-order terms"},{"id":"Logic/first-order-logic/first-order-unification/single-comp-eq","kind":"node","lean":"Cslib.Unification.Subst.single_comp_eq","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#single-comp-eq","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $x$ be a variable, $t$ a term, and $[x \\mapsto t]$ the substitution replacing $x$ by $t$ and fixing every other variable. If a substitution $\\sigma$ satisfies $\\sigma(x) = t\\sigma$, where $t\\sigma$ is the result of applying $\\sigma$ to $t$, then the composite substitution that applies $[x \\mapsto t]$ first and $\\sigma$ second equals $\\sigma$.","title":"Absorbing a single-variable substitution"},{"id":"Logic/first-order-logic/first-order-unification/subst-single-of-not-mem","kind":"node","lean":"Cslib.Unification.Subst.subst_single_of_not_mem","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#subst-single-of-not-mem","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"If the variable $x$ does not occur in the term $t$, then applying to $t$ the substitution $[x \\mapsto t]$, which replaces $x$ by $t$ and fixes every other variable, leaves $t$ unchanged.","title":"A term is fixed by substituting it for a variable it avoids"},{"id":"Logic/first-order-logic/first-order-unification/equations","kind":"node","lean":"Cslib.Unification.Equations","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#equations","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"A list of equations over constants $C$ is a finite list of pairs $(t, u)$ of terms over $C$, each pair standing for the equation $t = u$.","title":"Lists of equations"},{"id":"Logic/first-order-logic/first-order-unification/unifies","kind":"node","lean":"Cslib.Unification.Unifies","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#unifies","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"A substitution $s$ unifies a list $E$ of equations between terms when, for every equation $(t, u)$ in $E$, applying $s$ to $t$ and applying $s$ to $u$ give the same term.","title":"Unifier of a list of equations"},{"id":"Logic/first-order-logic/first-order-unification/is-mgu","kind":"node","lean":"Cslib.Unification.IsMGU","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#is-mgu","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"A substitution $s$ is a most general unifier of a list of equations $E$ when $s$ unifies $E$ and every unifier $s'$ of $E$ factors through $s$, that is, $s' = s \\gg r$ for some substitution $r$, where $s \\gg r$ applies first $s$ and then $r$.","title":"Most general unifier"},{"id":"Logic/first-order-logic/first-order-unification/subst-eqs","kind":"node","lean":"Cslib.Unification.substEqs","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#subst-eqs","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For a substitution $\\sigma$ and a list $E$ of equations between terms, the result is the list obtained by replacing each equation $u = v$ of $E$, in order, by $u\\sigma = v\\sigma$, where $u\\sigma$ is the result of applying $\\sigma$ to $u$.","title":"Applying a substitution to a list of equations"},{"id":"Logic/first-order-logic/first-order-unification/eq-vars","kind":"node","lean":"Cslib.Unification.eqVars","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#eq-vars","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"The variables of a list of equations form the finite set of all variables that occur on either side of any of its equations.","title":"Variables of a list of equations"},{"id":"Logic/first-order-logic/first-order-unification/eq-size","kind":"node","lean":"Cslib.Unification.eqSize","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#eq-size","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"The total size of a list of equations is the sum, over all its equations, of the sizes of the two sides, where the size of a term is its number of variable, constant and node symbols.","title":"Total size of a list of equations"},{"id":"Logic/first-order-logic/first-order-unification/card-eq-vars-elim-lt","kind":"node","lean":"Cslib.Unification.card_eqVars_elim_lt","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#card-eq-vars-elim-lt","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $x$ be a variable, $t$ a term in which $x$ does not occur, and $E$ a list of equations. Applying the substitution that replaces $x$ by $t$ (and fixes every other variable) to both sides of every equation of $E$ gives a list of equations with strictly fewer distinct variables than the list obtained by putting the equation $x = t$ in front of $E$.","title":"Eliminating a solved variable lowers the number of variables"},{"id":"Logic/first-order-logic/first-order-unification/unify","kind":"node","lean":"Cslib.Unification.unify","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#unify","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"The algorithm takes a list of equations between terms and returns a substitution or fails. On the empty list it returns the identity substitution. Otherwise it looks at the first equation $t = u$ and the remaining list $E$. If $t$ and $u$ are the same variable, or the same constant, the equation is dropped and the algorithm continues on $E$; if they are two different constants, it fails. If $t$ is a variable $x$\u2026","title":"Martelli\u2013Montanari unification"},{"id":"Logic/first-order-logic/first-order-unification/unify-spec","kind":"node","lean":"Cslib.Unification.UnifySpec","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#unify-spec","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $E$ be a list of equations between terms and $r$ either a substitution or failure. The specification holds for $E$ and $r$ when, if $r$ is a substitution $s$, then $s$ unifies $E$ and every substitution $s'$ that unifies $E$ equals the composite substitution that applies first $s$ and then $s'$; and, if $r$ is failure, then no substitution unifies $E$.","title":"Specification of a unification result"},{"id":"Logic/first-order-logic/first-order-unification/congr","kind":"node","lean":"Cslib.Unification.UnifySpec.congr","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#congr","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $E$ and $E'$ be lists of equations such that a substitution unifies $E$ exactly when it unifies $E'$. If an optional substitution $r$ satisfies the unification specification for $E'$ (a returned substitution $s$ unifies the list and every unifier $s'$ of it satisfies $s' = s \\gg s'$, where $s \\gg s'$ applies first $s$ and then $s'$, and failure means the list has no unifier), then $r$ satisfies the unification\u2026","title":"Lists of equations with the same unifiers satisfy the same specification"},{"id":"Logic/first-order-logic/first-order-unification/none-of-head","kind":"node","lean":"Cslib.Unification.UnifySpec.none_of_head","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#none-of-head","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $E$ be a list of equations between terms and let $u = v$ be a further equation, where $u\\sigma$ denotes the result of applying a substitution $\\sigma$ to $u$. If $u\\sigma \\ne v\\sigma$ for every substitution $\\sigma$, then returning no substitution meets the unification specification for the list obtained by putting $u = v$ in front of $E$: no substitution unifies that list.","title":"Failure is correct when the first equation has no unifier"},{"id":"Logic/first-order-logic/first-order-unification/elim","kind":"node","lean":"Cslib.Unification.UnifySpec.elim","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#elim","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $x$ be a variable, $t$ a term in which $x$ does not occur, $e$ an equation and $E$ a list of equations. Suppose that for every substitution $s$, the two sides of $e$ become equal under $s$ exactly when $s(x)$ equals the result of applying $s$ to $t$. Let $\\sigma$ be the substitution that replaces $x$ by $t$ and fixes every other variable, and let $E'$ be the list obtained by applying $\\sigma$ to both sides of\u2026","title":"The elimination step preserves the unification specification"},{"id":"Logic/first-order-logic/first-order-unification/unifies-unify","kind":"node","lean":"Cslib.Unification.unifies_unify","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#unifies-unify","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For every list $E$ of equations between terms, if the Martelli\u2013Montanari algorithm returns a substitution $s$ on $E$, then $s$ unifies $E$: applying $s$ to the two sides of any equation of $E$ gives the same term.","title":"Soundness of unification"},{"id":"Logic/first-order-logic/first-order-unification/unify-eq-some-comp","kind":"node","lean":"Cslib.Unification.unify_eq_some_comp","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#unify-eq-some-comp","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For every list $E$ of equations between terms, if the Martelli\u2013Montanari algorithm returns a substitution $s$ on $E$, then every substitution $s'$ that unifies $E$ equals the composite substitution that applies first $s$ and then $s'$.","title":"Most generality of unification"},{"id":"Logic/first-order-logic/first-order-unification/is-mgu-unify","kind":"node","lean":"Cslib.Unification.isMGU_unify","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#is-mgu-unify","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"If the Martelli-Montanari unification algorithm, run on a list of equations $E$, returns a substitution $s$, then $s$ is a most general unifier of $E$.","title":"Unification returns a most general unifier"},{"id":"Logic/first-order-logic/first-order-unification/unify-comp-self","kind":"node","lean":"Cslib.Unification.unify_comp_self","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#unify-comp-self","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For every list $E$ of equations between terms, if the Martelli\u2013Montanari algorithm returns a substitution $s$ on $E$, then $s$ is idempotent: the composite substitution that applies first $s$ and then $s$ equals $s$.","title":"Unification returns an idempotent substitution"},{"id":"Logic/first-order-logic/first-order-unification/unify-eq-none-iff","kind":"node","lean":"Cslib.Unification.unify_eq_none_iff","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#unify-eq-none-iff","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"For every list $E$ of equations between terms, the Martelli\u2013Montanari algorithm fails on $E$ if and only if no substitution unifies $E$.","title":"Completeness of unification"},{"id":"Logic/first-order-logic/first-order-unification/vars-single-comp-subset","kind":"node","lean":"Cslib.Unification.vars_single_comp_subset","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#vars-single-comp-subset","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Write $V(F)$ for the set of variables of a list $F$ of equations. Let $x_0$ be a variable, $t$ a term in which $x_0$ does not occur, $e = (e_1, e_2)$ an equation such that every variable of $t$ occurs in $e_1$ or $e_2$, and $E$ a list of equations, and let $E'$ be obtained from $E$ by substituting $t$ for $x_0$ on both sides of every equation. Let $s_0$ be a substitution such that, for every variable $x$, every\u2026","title":"The elimination step of unification introduces no variables"},{"id":"Logic/first-order-logic/first-order-unification/vars-unify-subset","kind":"node","lean":"Cslib.Unification.vars_unify_subset","parent":657,"result":true,"route":"../roadmap/Logic/first-order-logic/first-order-unification/index.html#vars-unify-subset","scope":"Logic/first-order-logic/first-order-unification","status":"fully_proved","summary":"Let $E$ be a list of equations between terms. If the unification algorithm succeeds on $E$ and returns the substitution $s$, then for every variable $x$, each variable of the term $s(x)$ is either $x$ itself or a variable occurring in some equation of $E$.","title":"Unification introduces no new variables"},{"data_url":"scopes/Logic/first-order-logic/g-del-s-completeness-theorem.json","id":"Logic/first-order-logic/g-del-s-completeness-theorem","kind":"scope","navigation":"module","parent":617,"result":false,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","title":"G\u00f6del's completeness theorem"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/mp","kind":"node","lean":"Cslib.Logic.FOL.Provable.mp","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#mp","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, let $\\alpha$ be an infinite type of variables, and let $S$ be a set of formulas of $L$ with free variables in $\\alpha$. For formulas $\\varphi$ and $\\psi$, if $S$ proves $\\varphi \\to \\psi$ and $S$ proves $\\varphi$, then $S$ proves $\\psi$.","title":"Modus ponens for provability from sets"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/of-bot","kind":"node","lean":"Cslib.Logic.FOL.Provable.of_bot","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#of-bot","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, let $\\alpha$ be an infinite type of variables, and let $S$ be a set of formulas of $L$ with free variables in $\\alpha$. If $S$ proves $\\bot$, then $S$ proves every formula $\\varphi$.","title":"Ex falso for provability from sets"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/imp-of-insert","kind":"node","lean":"Cslib.Logic.FOL.Provable.imp_of_insert","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#imp-of-insert","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\alpha$ be an infinite set of variables, let $S$ be a set of formulas with free variables in $\\alpha$, and let $\\varphi$, $\\psi$ be such formulas. If $S \\cup \\{\\varphi\\}$ proves $\\psi$, then $S$ proves $\\varphi \\to \\psi$.","title":"Deduction theorem"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/consistent","kind":"node","lean":"Cslib.Logic.FOL.Consistent","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#consistent","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language and $\\alpha$ a type of variables. A set $S$ of formulas of $L$ with free variables in $\\alpha$ is consistent when $S$ does not prove $\\bot$.","title":"Consistent set of formulas"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/provable-iff-not-consistent-insert-not","kind":"node","lean":"Cslib.Logic.FOL.provable_iff_not_consistent_insert_not","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#provable-iff-not-consistent-insert-not","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, let $\\alpha$ be an infinite type of variables, let $S$ be a set of formulas of $L$ with free variables in $\\alpha$, and let $\\varphi$ be a formula. Then $S$ proves $\\varphi$ if and only if $S \\cup \\{\\neg\\varphi\\}$ is not consistent.","title":"Provability as inconsistency with the negation"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/provable-not-iff-not-consistent-insert","kind":"node","lean":"Cslib.Logic.FOL.provable_not_iff_not_consistent_insert","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#provable-not-iff-not-consistent-insert","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, let $\\alpha$ be an infinite type of variables, let $S$ be a set of formulas of $L$ with free variables in $\\alpha$, and let $\\varphi$ be a formula. Then $S$ proves $\\neg\\varphi$ if and only if $S \\cup \\{\\varphi\\}$ is not consistent.","title":"Refutability as inconsistency"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/consistent-s-union-of-is-chain","kind":"node","lean":"Cslib.Logic.FOL.consistent_sUnion_of_isChain","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#consistent-s-union-of-is-chain","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"If $C$ is a nonempty family of sets of formulas that is totally ordered by inclusion and every member of $C$ is consistent, then the union $\\bigcup C$ is consistent.","title":"Unions of chains of consistent sets are consistent"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/maximal-consistent","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#maximal-consistent","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"A set $\\Delta$ of formulas is maximal consistent when it is consistent and every consistent set of formulas containing $\\Delta$ is contained in $\\Delta$, that is, equals $\\Delta$.","title":"Maximal consistent set"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/mem-of-consistent-insert","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent.mem_of_consistent_insert","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#mem-of-consistent-insert","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\Delta$ be a maximal consistent set of first-order formulas of a language $L$ with free variables in $\\alpha$, and let $\\varphi$ be a formula. If $\\Delta \\cup \\{\\varphi\\}$ is consistent, then $\\varphi \\in \\Delta$.","title":"Maximal consistent sets absorb consistent extensions"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/exists-maximal-consistent","kind":"node","lean":"Cslib.Logic.FOL.exists_maximalConsistent","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#exists-maximal-consistent","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Every consistent set $S$ of first-order formulas is contained in a maximal consistent set.","title":"Lindenbaum's lemma for first-order logic"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/mem-of-provable","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent.mem_of_provable","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#mem-of-provable","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, let $\\alpha$ be an infinite type of variables, and let $\\Delta$ be a maximal consistent set of formulas of $L$ with free variables in $\\alpha$. Every formula $\\varphi$ that $\\Delta$ proves lies in $\\Delta$.","title":"Maximal consistent sets of first-order formulas are deductively closed"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/not-mem-iff","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent.not_mem_iff","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#not-mem-iff","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, let $\\alpha$ be an infinite type of variables, and let $\\Delta$ be a maximal consistent set of formulas of $L$ with free variables in $\\alpha$. For every formula $\\varphi$, $\\neg\\varphi \\in \\Delta$ if and only if $\\varphi \\notin \\Delta$; so exactly one of $\\varphi$ and $\\neg\\varphi$ lies in $\\Delta$.","title":"Maximal consistent sets decide every formula"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/instantiate-mem-of-all-mem","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent.instantiate_mem_of_all_mem","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#instantiate-mem-of-all-mem","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\alpha$ be an infinite set of variables and let $\\Delta$ be a maximal consistent set of formulas with free variables in $\\alpha$. Let $\\varphi(x)$ be a formula with one bound variable $x$ left open, and for a term $t$ write $\\varphi(t)$ for the result of substituting $t$ for $x$. If $\\Delta$ contains $\\forall x\\, \\varphi$, then $\\Delta$ contains $\\varphi(t)$ for every term $t$.","title":"Universal instantiation in maximal consistent sets"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/instantiate-mem-of-equal","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent.instantiate_mem_of_equal","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#instantiate-mem-of-equal","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\alpha$ be an infinite set of variables and let $\\Delta$ be a maximal consistent set of formulas with free variables in $\\alpha$. Let $\\varphi(x)$ be a formula with one bound variable $x$ left open, and for a term $u$ write $\\varphi(u)$ for the result of substituting $u$ for $x$. If $s$ and $t$ are terms such that $\\Delta$ contains the equation $s = t$ and the formula $\\varphi(s)$, then $\\Delta$ contains\u2026","title":"Substitution of equals in maximal consistent sets"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/henkin-axiom","kind":"node","lean":"Cslib.Logic.FOL.henkinAxiom","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#henkin-axiom","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\varphi(x)$ be a formula with one bound variable $x$ left open and let $c$ be a variable; write $\\varphi(c)$ for the result of substituting $c$ for $x$. The Henkin axiom for $\\forall x\\, \\varphi$ with witness $c$ is the formula $\\neg \\forall x\\, \\varphi \\to \\neg \\varphi(c)$.","title":"Henkin axiom"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/insert-henkin-axiom","kind":"node","lean":"Cslib.Logic.FOL.Consistent.insert_henkinAxiom","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#insert-henkin-axiom","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\alpha$ be an infinite set of variables and let $S$ be a consistent set of formulas with free variables in $\\alpha$. Let $\\varphi(x)$ be a formula with one bound variable $x$ left open, and let $c \\in \\alpha$ be a variable that occurs free neither in $\\varphi$ nor in any formula of $S$. Then $S$ together with the Henkin axiom for $\\forall x\\, \\varphi$ with witness $c$ is consistent.","title":"Adding a Henkin axiom preserves consistency"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/all-mem-of-henkin","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent.all_mem_of_henkin","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#all-mem-of-henkin","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\alpha$ be an infinite set of variables and let $\\Delta$ be a maximal consistent set of formulas with free variables in $\\alpha$. Let $\\varphi(x)$ be a formula with one bound variable $x$ left open, let $c \\in \\alpha$, and write $\\varphi(c)$ for the result of substituting $c$ for $x$. If $\\Delta$ contains the Henkin axiom for $\\forall x\\, \\varphi$ with witness $c$ and contains $\\varphi(c)$, then $\\Delta$\u2026","title":"Universal formulas from Henkin witnesses"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/free-var-finset-instantiate-subset","kind":"node","lean":"Cslib.Logic.FOL.freeVarFinset_instantiate_subset","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#free-var-finset-instantiate-subset","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $t$ be a term and $\\varphi$ a first-order formula with $m + 1$ bound variables, and let $\\varphi(t)$ be the instance obtained by substituting $t$ for the outermost bound variable. Every free variable of $\\varphi(t)$ is a free variable of $\\varphi$ or a variable of $t$.","title":"Free variables of an instance"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/depth","kind":"node","lean":"Cslib.Logic.FOL.depth","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#depth","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"The depth $d(\\varphi)$ of a first-order formula $\\varphi$ is the nesting height of its implications and universal quantifiers: $\\bot$, equations and relation atoms have depth $0$, the implication $\\varphi \\to \\psi$ has depth $\\max(d(\\varphi), d(\\psi)) + 1$, and $\\forall \\varphi$ has depth $d(\\varphi) + 1$.","title":"Depth of a formula"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/rename-congr-of-eq-on","kind":"node","lean":"Cslib.Logic.FOL.rename_congr_of_eqOn","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#rename-congr-of-eq-on","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types of variables, let $n \\in \\mathbb{N}$, and let $\\varphi$ be a formula with free variables in $\\alpha$ and $n$ further variables reserved for binding by enclosing quantifiers. If two maps $f, g : \\alpha \\to \\beta$ agree on every variable from $\\alpha$ that occurs free in $\\varphi$, then renaming the variables of $\\varphi$ from $\\alpha$ along $f$ and along $g$ gives the same formula.","title":"Renaming depends only on the free variables"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/relabel-of-infinite","kind":"node","lean":"Cslib.Logic.FOL.Derivable.relabel_of_infinite","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#relabel-of-infinite","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types of variables with $\\beta$ infinite, and let $f : \\alpha \\to \\beta$ be any map. If $\\varphi$ is derivable from the list of hypotheses $\\Gamma$ in natural deduction, then the formula obtained from $\\varphi$ by renaming each free variable $x$ to $f(x)$ is derivable from the list obtained by renaming the free variables of each formula of $\\Gamma$ in the same way.","title":"Renaming free variables along an arbitrary map preserves derivability"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/relabel-double","kind":"node","lean":"Cslib.Logic.FOL.Consistent.relabel_double","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#relabel-double","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"If $S$ is a consistent set of formulas with free variables in $\\mathbb{N}$, then the set obtained from $S$ by renaming every free variable $x$ to $2x$ in each of its formulas is consistent.","title":"Doubling the variables preserves consistency"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/update-chain","kind":"node","lean":"Cslib.Logic.FOL.update_chain","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#update-chain","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $r$ be a relation on a type $\\beta$, let $n$ be a natural number and let $P$ be a property of $n$-tuples $x = (x_i)_{i < n}$ in $\\beta$. Suppose that for every tuple $x$, every position $i$ and all $a, b \\in \\beta$ with $r(a, b)$, if $P$ holds of $x$ with its $i$-th entry replaced by $a$, then $P$ holds of $x$ with its $i$-th entry replaced by $b$. Then for all tuples $x, y$ with $r(x_i, y_i)$ for every $i$,\u2026","title":"Replacing arguments one at a time"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/func-update","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent.func_update","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#func-update","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\Delta$ be a maximal consistent set of first-order formulas with free variables in $\\mathbb{N}$, let $f$ be an $n$-ary function symbol, let $x = (x_0, \\dots, x_{n-1})$ be terms over $\\mathbb{N}$ and let $i < n$. For a term $u$, write $x[i := u]$ for $x$ with its $i$-th entry replaced by $u$. If the equation $a = b$ lies in $\\Delta$, then the equation $f(x[i := a]) = f(x[i := b])$ lies in $\\Delta$.","title":"Congruence for function symbols in a maximal consistent set"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/rel-update","kind":"node","lean":"Cslib.Logic.FOL.MaximalConsistent.rel_update","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#rel-update","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\Delta$ be a maximal consistent set of formulas with free variables in $\\mathbb{N}$, let $R$ be an $n$-ary relation symbol, let $x_0, \\dots, x_{n-1}$, $a$ and $b$ be terms with variables in $\\mathbb{N}$, and let $i < n$. If $\\Delta$ contains $a = b$ and contains $R(x_0, \\dots, x_{n-1})$ with $x_i$ replaced by $a$, then $\\Delta$ contains $R(x_0, \\dots, x_{n-1})$ with $x_i$ replaced by $b$.","title":"Replacing an argument of a relation symbol by an equal term"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-structure","kind":"node","lean":"Cslib.Logic.FOL.termStructure","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-structure","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\Delta$ be a set of first-order formulas in a language $L$ with free variables in $\\mathbb{N}$. The term structure of $\\Delta$ is the $L$-structure whose domain is the set of $L$-terms with variables in $\\mathbb{N}$, in which an $n$-ary function symbol $f$ sends terms $t_1, \\dots, t_n$ to the term $f(t_1, \\dots, t_n)$, and an $n$-ary relation symbol $R$ holds of $t_1, \\dots, t_n$ exactly when the atomic\u2026","title":"Term structure of a set of formulas"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-setoid","kind":"node","lean":"Cslib.Logic.FOL.termSetoid","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-setoid","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\Delta$ be a maximal consistent set of first-order formulas in a language $L$ with free variables in $\\mathbb{N}$. Provable equality in $\\Delta$ is the equivalence relation on $L$-terms with variables in $\\mathbb{N}$ under which $s \\sim t$ holds exactly when the equation $s = t$ belongs to $\\Delta$.","title":"Provable equality of terms"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-model","kind":"node","lean":"Cslib.Logic.FOL.TermModel","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-model","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For a maximal consistent set $\\Delta$ of formulas with free variables in $\\mathbb{N}$, the term model of $\\Delta$ is the set of terms with variables in $\\mathbb{N}$ modulo the equivalence relation that identifies $s$ and $t$ when the formula $s = t$ belongs to $\\Delta$.","title":"Term model of a maximal consistent set"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-assignment","kind":"node","lean":"Cslib.Logic.FOL.termAssignment","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-assignment","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For a maximal consistent set $\\Delta$ of formulas with free variables in $\\mathbb{N}$, the canonical assignment sends each variable $x \\in \\mathbb{N}$ to $[x]$, the class of the variable term $x$ in the term model of $\\Delta$.","title":"Canonical assignment of the term model"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/lower-term","kind":"node","lean":"Cslib.Logic.FOL.lowerTerm","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#lower-term","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"A term whose variables are free variables from $\\mathbb{N}$ together with bound variables from the empty index set is lowered to the term over $\\mathbb{N}$ of the same shape, in which every free variable $x$ stays $x$.","title":"Lowering a term with no bound variables"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/realize-term-model-iff","kind":"node","lean":"Cslib.Logic.FOL.realize_termModel_iff","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#realize-term-model-iff","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\Delta$ be a maximal consistent set of formulas with free variables in $\\mathbb{N}$ such that, for every formula $\\varphi$ with one bound variable, if $\\Delta$ contains $\\varphi(t)$ for every term $t$ (where $\\varphi(t)$ is the result of substituting $t$ for the bound variable), then $\\Delta$ contains $\\forall x\\,\\varphi$. Then a formula $\\psi$ with free variables in $\\mathbb{N}$ holds in the term model of\u2026","title":"Truth lemma for the first-order term model"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/enum-body","kind":"node","lean":"Cslib.Logic.FOL.enumBody","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#enum-body","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For a language with countably many symbols, the formulas $\\varphi_0, \\varphi_1, \\dots$ are a fixed enumeration, chosen by choice, of all first-order formulas with free variables in $\\mathbb{N}$ and one bound variable: every such formula is $\\varphi_k$ for some $k \\in \\mathbb{N}$.","title":"Enumeration of formulas with one bound variable"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/var-bound","kind":"node","lean":"Cslib.Logic.FOL.varBound","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#var-bound","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\varphi$ be a first-order formula whose variables are natural numbers together with $n$ variables reserved for quantifiers. Its variable bound $b(\\varphi)$ is one more than the largest natural number occurring as a free variable of $\\varphi$, or $1$ when there is none; it is a strict upper bound on these free variables.","title":"Variable bound of a formula"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/witness-base","kind":"node","lean":"Cslib.Logic.FOL.witnessBase","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#witness-base","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language with countably many symbols, let $\\varphi_0, \\varphi_1, \\dots$ be the fixed surjective enumeration of the $L$-formulas with free variables in $\\mathbb{N}$ and one variable reserved for a quantifier, and let $b(\\varphi)$ be one more than the largest free variable of $\\varphi$, or $1$ if it has none. The witness bases are defined by $w_0 = b(\\varphi_0)$ and\u2026","title":"Witness bases"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/witness","kind":"node","lean":"Cslib.Logic.FOL.witness","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#witness","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language with countably many symbols and let $w_0 < w_1 < \\dots$ be its witness bases, built from the fixed enumeration $\\varphi_0, \\varphi_1, \\dots$ of the $L$-formulas with one variable reserved for a quantifier. The $k$-th Henkin witness is the variable $c_k = 2 w_k + 1$. It is odd, so it is not a free variable of any formula whose free variables are all even, and it exceeds every free\u2026","title":"Henkin witnesses"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/henkin-axiom-at","kind":"node","lean":"Cslib.Logic.FOL.henkinAxiomAt","parent":691,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-th Henkin axiom"]],"source":"The $k$-th Henkin axiom"},"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#henkin-axiom-at","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For a language with countably many symbols and $k \\in \\mathbb{N}$, the $k$-th Henkin axiom is the Henkin axiom $\\neg \\forall \\varphi_k \\to \\neg \\varphi_k(c_k)$, where $\\varphi_k$ is the $k$-th formula in the enumeration of formulas with one bound variable and $c_k$ is the $k$-th Henkin witness.","title":"The $k$-th Henkin axiom"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/henkin-stage","kind":"node","lean":"Cslib.Logic.FOL.henkinStage","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#henkin-stage","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $S$ be a set of first-order formulas with free variables in $\\mathbb{N}$, in a language with countably many symbols, and let $H_k$ be the $k$-th Henkin axiom. The stages $S_k$ are defined by $S_0 = S$ and $S_{k+1} = S_k \\cup \\{H_k\\}$, so $S_k$ is $S$ together with $H_0, \\dots, H_{k-1}$.","title":"Stages of the Henkin extension"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/henkin-set","kind":"node","lean":"Cslib.Logic.FOL.henkinSet","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#henkin-set","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For a language with countably many symbols, the Henkin extension of a set $S$ of first-order formulas with free variables in $\\mathbb{N}$ is $S$ together with all Henkin axioms, the $k$-th Henkin axiom for every $k \\in \\mathbb{N}$.","title":"Henkin extension"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/exists-maximal-consistent-henkin","kind":"node","lean":"Cslib.Logic.FOL.exists_maximalConsistent_henkin","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#exists-maximal-consistent-henkin","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a language with countably many symbols and let $S$ be a consistent set of first-order formulas with free variables in $\\mathbb{N}$ such that every free variable of every formula in $S$ is even. Then $S$ is contained in a maximal consistent set $\\Delta$ with the Henkin property: for every formula $\\varphi$ with one bound variable, if every instance $\\varphi(t)$, for $t$ a term over $\\mathbb{N}$, lies in\u2026","title":"Henkin extension of a consistent set"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/exists-model-of-consistent","kind":"node","lean":"Cslib.Logic.FOL.exists_model_of_consistent","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#exists-model-of-consistent","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a language with countably many symbols. Every consistent set $S$ of first-order formulas with free variables in $\\mathbb{N}$ is satisfiable: there are a nonempty $L$-structure $M$, in the universe of the function symbols of $L$, and an assignment $v : \\mathbb{N} \\to M$ under which every formula of $S$ holds.","title":"Model existence for first-order logic"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/provable-of-entails","kind":"node","lean":"Cslib.Logic.FOL.provable_of_entails","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#provable-of-entails","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a language with countably many function and relation symbols, whose function symbols live in universe $u$. Let $S$ be a set of formulas of $L$ and $\\varphi$ a formula of $L$, all with free variables in $\\mathbb{N}$. If for every $L$-structure $M$ with carrier in universe $u$ and every assignment $v : \\mathbb{N} \\to M$ under which all formulas of $S$ hold, $\\varphi$ also holds under $v$, then $\\varphi$ is\u2026","title":"Strong completeness of first-order natural deduction"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/provable-of-models","kind":"node","lean":"Cslib.Logic.FOL.provable_of_models","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#provable-of-models","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a language with countably many function and relation symbols, whose function symbols live in universe $u$ and relation symbols in universe $v$. Let $T$ be a theory of $L$ and $\\varphi$ a formula of $L$ with free variables in $\\mathbb{N}$. If $\\varphi$ holds under every assignment of the variables in every nonempty model of $T$ whose carrier lies in universe $\\max(u, v)$, then $\\varphi$ is provable from\u2026","title":"Completeness of first-order natural deduction for theories"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/provable-iff-models","kind":"node","lean":"Cslib.Logic.FOL.provable_iff_models","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#provable-iff-models","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a language with countably many function and relation symbols, whose function symbols live in universe $u$ and relation symbols in universe $v$. For every theory $T$ of $L$ and every formula $\\varphi$ of $L$ with free variables in $\\mathbb{N}$, $\\varphi$ is provable from the sentences of $T$, viewed as formulas over $\\mathbb{N}$, if and only if $\\varphi$ holds under every assignment of the variables in\u2026","title":"Soundness and completeness of first-order natural deduction"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-symbol-finset","kind":"node","lean":"Cslib.Logic.FOL.termSymbolFinset","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-symbol-finset","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"The finite set of symbols occurring in a term $t$ of a first-order language $L$, a finite subset of the function and relation symbols of $L$, is defined by recursion on $t$: it is empty when $t$ is a variable, and when $t = f(t_0, \\dots, t_{l-1})$ for an $l$-ary function symbol $f$ it consists of $f$ together with the symbols occurring in each of $t_0, \\dots, t_{l-1}$.","title":"Finite set of symbols of a term"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/mem-term-symbol-finset-func","kind":"node","lean":"Cslib.Logic.FOL.mem_termSymbolFinset_func","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#mem-term-symbol-finset-func","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, $f$ an $l$-ary function symbol of $L$, and $t_1, \\dots, t_l$ $L$-terms with variables in $\\alpha$. Then $f$ belongs to the finite set of symbols occurring in the term $f(t_1, \\dots, t_l)$.","title":"The head function symbol of a term occurs in it"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-symbol-finset-func-subset","kind":"node","lean":"Cslib.Logic.FOL.termSymbolFinset_func_subset","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-symbol-finset-func-subset","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $f$ be an $l$-ary function symbol of a first-order language $L$ and let $t_0, \\dots, t_{l-1}$ be terms of $L$ in variables from $\\alpha$. For every $i < l$, every symbol occurring in $t_i$ occurs in the term $f(t_0, \\dots, t_{l-1})$: the finite set of symbols of $t_i$ is contained in that of $f(t_0, \\dots, t_{l-1})$.","title":"Symbols of an argument of a function application"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/formula-symbol-finset","kind":"node","lean":"Cslib.Logic.FOL.formulaSymbolFinset","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#formula-symbol-finset","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"The finite set of function and relation symbols occurring in a bounded formula of a first-order language $L$, defined by recursion on the formula: $\\bot$ has no symbols; an equation $t_1 = t_2$ has the function symbols of $t_1$ and $t_2$; an atomic formula $R(t_1, \\dots, t_l)$ has $R$ together with the function symbols of $t_1, \\dots, t_l$; an implication $\\varphi \\to \\psi$ has the symbols of $\\varphi$ and of\u2026","title":"Finite set of symbols of a formula"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-symbol-finset-equal-subset-left","kind":"node","lean":"Cslib.Logic.FOL.termSymbolFinset_equal_subset_left","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-symbol-finset-equal-subset-left","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language and let $t_1$, $t_2$ be terms of $L$ in free variables from $\\alpha$ and $n$ bound variables. Every symbol occurring in $t_1$ occurs in the atomic formula $t_1 = t_2$, viewed as a formula with $n$ bound variables: the finite set of symbols of $t_1$ is contained in that of $t_1 = t_2$.","title":"Symbols of the left side of an equation"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-symbol-finset-equal-subset-right","kind":"node","lean":"Cslib.Logic.FOL.termSymbolFinset_equal_subset_right","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-symbol-finset-equal-subset-right","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language and let $t_1$, $t_2$ be terms of $L$ in free variables from $\\alpha$ and $n$ bound variables. Every symbol occurring in $t_2$ occurs in the atomic formula $t_1 = t_2$, viewed as a formula with $n$ bound variables: the finite set of symbols of $t_2$ is contained in that of $t_1 = t_2$.","title":"Symbols of the right side of an equation"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/mem-formula-symbol-finset-rel","kind":"node","lean":"Cslib.Logic.FOL.mem_formulaSymbolFinset_rel","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#mem-formula-symbol-finset-rel","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, $R$ an $l$-ary relation symbol of $L$, and $t_1, \\dots, t_l$ $L$-terms with free variables in $\\alpha$ and $n$ bound variables. Then $R$ belongs to the finite set of symbols occurring in the atomic bounded formula $R(t_1, \\dots, t_l)$.","title":"The relation symbol of an atomic formula occurs in it"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/term-symbol-finset-rel-subset","kind":"node","lean":"Cslib.Logic.FOL.termSymbolFinset_rel_subset","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#term-symbol-finset-rel-subset","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $R$ be an $l$-ary relation symbol of a first-order language $L$ and let $t_0, \\dots, t_{l-1}$ be terms of $L$ in free variables from $\\alpha$ and $n$ bound variables. For every $i < l$, every symbol occurring in $t_i$ occurs in the atomic formula $R(t_0, \\dots, t_{l-1})$, viewed as a formula with $n$ bound variables: the finite set of symbols of $t_i$ is contained in that of $R(t_0, \\dots, t_{l-1})$.","title":"Symbols of an argument of a relational atomic formula"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/formula-symbol-finset-imp-subset-left","kind":"node","lean":"Cslib.Logic.FOL.formulaSymbolFinset_imp_subset_left","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#formula-symbol-finset-imp-subset-left","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For bounded formulas $\\varphi$ and $\\psi$ of a first-order language $L$ with free variables in $\\alpha$ and $n$ bound variables, every symbol occurring in $\\varphi$ occurs in $\\varphi \\to \\psi$.","title":"Symbols of the antecedent occur in an implication"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/formula-symbol-finset-imp-subset-right","kind":"node","lean":"Cslib.Logic.FOL.formulaSymbolFinset_imp_subset_right","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#formula-symbol-finset-imp-subset-right","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For bounded formulas $\\varphi$ and $\\psi$ of a first-order language $L$ with free variables in $\\alpha$ and $n$ bound variables, every symbol occurring in $\\psi$ occurs in $\\varphi \\to \\psi$.","title":"Symbols of the consequent occur in an implication"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/symbols-of","kind":"node","lean":"Cslib.Logic.FOL.symbolsOf","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#symbols-of","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For a finite set $F$ of first-order formulas of a language $L$ with free variables indexed by $\\mathbb{N}$, the symbols of $F$ are the finite set of function and relation symbols of $L$ that occur in some formula of $F$, that is, the union over $\\psi \\in F$ of the finite sets of symbols occurring in $\\psi$.","title":"Symbols of a finite set of formulas"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/mem-symbols-of","kind":"node","lean":"Cslib.Logic.FOL.mem_symbolsOf","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#mem-symbols-of","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, $F$ a finite set of $L$-formulas with free variables in $\\mathbb{N}$, and $\\psi \\in F$. Every symbol occurring in $\\psi$ belongs to the finite set of symbols occurring in some formula of $F$.","title":"A symbol of a member of a set of formulas is a symbol of the set"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/sub-language","kind":"node","lean":"Cslib.Logic.FOL.subLanguage","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#sub-language","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"For a finite set $F$ of formulas of a first-order language $L$ with free variables in $\\mathbb{N}$, the sublanguage of $L$ on the symbols occurring in $F$ has as its $n$-ary function symbols the $n$-ary function symbols of $L$ that occur in some formula of $F$, and as its $n$-ary relation symbols the $n$-ary relation symbols of $L$ that occur in some formula of $F$.","title":"Sublanguage of a finite set of formulas"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/sub-language-incl","kind":"node","lean":"Cslib.Logic.FOL.subLanguageIncl","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#sub-language-incl","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language. For a finite set $F$ of formulas of $L$ with free variables in $\\mathbb{N}$, this is the language homomorphism from the sublanguage of $L$ on the symbols occurring in $F$ into $L$ that sends each function symbol and each relation symbol to itself.","title":"Inclusion of the sublanguage of a finite set of formulas"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/sub-language-incl-injective","kind":"node","lean":"Cslib.Logic.FOL.subLanguageIncl_injective","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#sub-language-incl-injective","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language. For every finite set $F$ of formulas of $L$ with free variables in $\\mathbb{N}$, the inclusion into $L$ of the sublanguage of $L$ on the symbols occurring in $F$ is injective on function symbols and on relation symbols.","title":"The inclusion of the sublanguage of a finite set of formulas is injective"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/exists-term-preimage","kind":"node","lean":"Cslib.Logic.FOL.exists_term_preimage","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#exists-term-preimage","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $F$ be a finite set of formulas of a first-order language $L$ with free variables in $\\mathbb{N}$, and let $L_F$ be the sublanguage of $L$ on the symbols occurring in $F$, with inclusion $\\iota : L_F \\to L$. Every term $t$ of $L$ whose variables are drawn from $\\mathbb{N}$ and from $n$ bound variables, each of whose symbols occurs in some formula of $F$, is the image under $\\iota$ of a term of $L_F$ with\u2026","title":"Terms lift to the finite-symbol sublanguage"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/exists-bounded-formula-preimage","kind":"node","lean":"Cslib.Logic.FOL.exists_boundedFormula_preimage","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#exists-bounded-formula-preimage","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $F$ be a finite set of formulas of a first-order language $L$ with free variables in $\\mathbb{N}$, and let $L_F$ be the sublanguage of $L$ on the symbols occurring in $F$, with inclusion $\\iota : L_F \\to L$. Every bounded formula $\\varphi$ of $L$ with free variables in $\\mathbb{N}$ and $n$ bound variables, each of whose symbols occurs in some formula of $F$, is the image under $\\iota$ of a bounded formula of\u2026","title":"Bounded formulas lift to the finite-symbol sublanguage"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/on-term-relabel","kind":"node","lean":"Cslib.Logic.FOL.onTerm_relabel","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#on-term-relabel","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $h \\colon L \\to L'$ be a homomorphism of first-order languages, and for an $L$-term $u$ write $h(u)$ for the $L'$-term obtained by replacing every function symbol by its image under $h$. Let $t$ be an $L$-term with variables in $\\alpha$ and $g \\colon \\alpha \\to \\beta$ a map. Then $h$ applied to the term obtained from $t$ by renaming each variable $a$ to $g(a)$ equals the term obtained from $h(t)$ by the same\u2026","title":"Language homomorphisms commute with renaming variables"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/on-term-subst","kind":"node","lean":"Cslib.Logic.FOL.onTerm_subst","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#on-term-subst","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $h \\colon L \\to L'$ be a homomorphism of first-order languages, and for an $L$-term $u$ write $h(u)$ for the $L'$-term obtained by replacing every function symbol by its image under $h$. Let $s$ be an $L$-term with variables in $\\beta$ and $g$ a map assigning to each $b \\in \\beta$ an $L$-term $g(b)$ with variables in $\\gamma$. Then $h$ applied to the term obtained from $s$ by substituting $g(b)$ for each\u2026","title":"Language homomorphisms commute with substitution in terms"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/subst-congr","kind":"node","lean":"Cslib.Logic.FOL.subst_congr","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#subst-congr","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, let $s$ be a term of $L$ with variables in a type $\\beta$, and let $g$ and $g'$ assign to each variable in $\\beta$ a term of $L$ with variables in a type $\\gamma$. If $g(a) = g'(a)$ for every $a \\in \\beta$, then substituting $g$ into $s$ gives the same term as substituting $g'$ into $s$.","title":"Term substitution depends only on the values of the substitution"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/on-term-instantiate-term","kind":"node","lean":"Cslib.Logic.FOL.onTerm_instantiateTerm","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#on-term-instantiate-term","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $h \\colon L \\to L'$ be a homomorphism of first-order languages, and for an $L$-term $u$ write $h(u)$ for the $L'$-term obtained by replacing every function symbol by its image under $h$. Let $t$ be an $L$-term with free variables in $\\alpha$ and $s$ an $L$-term with free variables in $\\alpha$ and $m + 1$ bound variables. Then $h$ applied to the instantiation of $t$ for the outermost bound variable (the one with\u2026","title":"Language homomorphisms commute with instantiation in terms"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/on-bounded-formula-instantiate","kind":"node","lean":"Cslib.Logic.FOL.onBoundedFormula_instantiate","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#on-bounded-formula-instantiate","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $h \\colon L \\to L'$ be a homomorphism of first-order languages. For an $L$-term or bounded $L$-formula $x$, write $h(x)$ for the term or bounded formula of $L'$ obtained by replacing every symbol by its image under $h$. Let $t$ be an $L$-term with free variables in $\\alpha$ and $\\varphi$ a bounded $L$-formula with free variables in $\\alpha$ and $m + 1$ bound variables. Then $h$ applied to the instantiation of\u2026","title":"Language homomorphisms commute with instantiation in formulas"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/var-finset-left-on-term","kind":"node","lean":"Cslib.Logic.FOL.varFinsetLeft_onTerm","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#var-finset-left-on-term","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $\\phi : L \\to L'$ be a homomorphism of first-order languages and let $t$ be a term of $L$ in free variables from $\\alpha$ and $n$ bound variables. The finite set of free variables from $\\alpha$ occurring in the image $\\phi(t)$ of $t$ under $\\phi$ equals the finite set of those occurring in $t$.","title":"Language homomorphisms preserve the free variables of a term"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/free-var-finset-on-bounded-formula","kind":"node","lean":"Cslib.Logic.FOL.freeVarFinset_onBoundedFormula","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#free-var-finset-on-bounded-formula","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $h : L \\to L'$ be a homomorphism of first-order languages, and let $\\varphi$ be a bounded formula of $L$ with free variables in $\\alpha$ and $n$ bound variables. The formula of $L'$ obtained from $\\varphi$ by replacing each symbol by its image under $h$ has the same free variables as $\\varphi$.","title":"Language homomorphisms preserve free variables"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/on-formula-equal","kind":"node","lean":"Cslib.Logic.FOL.onFormula_equal","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#on-formula-equal","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $h \\colon L \\to L'$ be a homomorphism of first-order languages. For an $L$-term or $L$-formula $x$, write $h(x)$ for the term or formula of $L'$ obtained by replacing every symbol by its image under $h$. If $s$ and $t$ are $L$-terms with free variables in $\\alpha$, then $h$ applied to the formula $s = t$ is the formula $h(s) = h(t)$.","title":"Language homomorphisms preserve equations"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/map-lhom","kind":"node","lean":"Cslib.Logic.FOL.Derivable.mapLHom","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#map-lhom","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $h : L \\to L'$ be a homomorphism of first-order languages, and write $h(\\varphi)$ for the formula of $L'$ obtained from a formula $\\varphi$ of $L$ by replacing each symbol by its image under $h$. Let $\\Gamma$ be a finite list of formulas of $L$ with free variables in $\\alpha$, and let $\\varphi$ be a formula of $L$ with free variables in $\\alpha$. If $\\Gamma \\vdash \\varphi$ in natural deduction, then\u2026","title":"Derivations survive language homomorphisms"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/sub-language-preimage","kind":"node","lean":"Cslib.Logic.FOL.subLanguagePreimage","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#sub-language-preimage","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language. For a finite set $F$ of formulas of $L$ with free variables in $\\mathbb{N}$, this is the set of those formulas $\\psi_0$ of the sublanguage of $L$ on the symbols occurring in $F$, with free variables in $\\mathbb{N}$, whose image under the inclusion of that sublanguage into $L$ belongs to $F$.","title":"Preimage of a finite set of formulas in its sublanguage"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/consistent-sub-language-preimage","kind":"node","lean":"Cslib.Logic.FOL.consistent_subLanguagePreimage","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#consistent-sub-language-preimage","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $F$ be a finite set of formulas of a first-order language $L$ with free variables in $\\mathbb{N}$, and let $L_F$ be the sublanguage of $L$ on the symbols occurring in $F$, with inclusion $\\iota : L_F \\to L$. If $F$ is consistent, then so is the preimage of $F$ in $L_F$: the set of formulas $\\psi_0$ of $L_F$ with free variables in $\\mathbb{N}$ whose image $\\iota(\\psi_0)$ under the inclusion belongs to $F$.","title":"Consistency passes to the finite-symbol sublanguage"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/sub-language-symbols-to","kind":"node","lean":"Cslib.Logic.FOL.subLanguageSymbolsTo","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#sub-language-symbols-to","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language. For a finite set $F$ of formulas of $L$ with free variables in $\\mathbb{N}$, this map sends each function symbol (respectively relation symbol) of the sublanguage of $L$ on the symbols occurring in $F$ to the same function symbol (respectively relation symbol) of $L$, with the same arity.","title":"Symbols of the sublanguage of a finite set of formulas as symbols of the language"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/mem-sub-language-symbols-to","kind":"node","lean":"Cslib.Logic.FOL.mem_subLanguageSymbolsTo","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#mem-sub-language-symbols-to","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language and $F$ a finite set of $L$-formulas with free variables in $\\mathbb{N}$. The map from the symbols of the sublanguage of $L$ on the symbols occurring in $F$ to the symbols of $L$ sends every symbol of that sublanguage to a symbol occurring in some formula of $F$.","title":"Symbols of a generated sublanguage occur in the generating formulas"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/sub-language-symbols-to-injective","kind":"node","lean":"Cslib.Logic.FOL.subLanguageSymbolsTo_injective","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#sub-language-symbols-to-injective","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language. For every finite set $F$ of formulas of $L$ with free variables in $\\mathbb{N}$, the map sending each symbol of the sublanguage of $L$ on the symbols occurring in $F$ to the same symbol of $L$ is injective.","title":"The symbol map of the sublanguage of a finite set of formulas is injective"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/sym-embed","kind":"node","lean":"Cslib.Logic.FOL.symEmbed","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#sym-embed","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language. For a finite set $F$ of formulas of $L$ with free variables in $\\mathbb{N}$, this map sends each symbol of the sublanguage of $L$ on the symbols occurring in $F$ to the same symbol of $L$, regarded as an element of the finite set of symbols occurring in $F$.","title":"Symbols of the sublanguage as symbols occurring in a finite set of formulas"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/exists-model-of-consistent-finset","kind":"node","lean":"Cslib.Logic.FOL.exists_model_of_consistent_finset","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#exists-model-of-consistent-finset","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, with no countability hypothesis on its symbols. Every finite consistent set $F$ of formulas of $L$ with free variables in $\\mathbb{N}$ is satisfiable: there are a nonempty $L$-structure $M$, in the universe of the function symbols of $L$, and an assignment $v : \\mathbb{N} \\to M$ under which every formula of $F$ holds.","title":"Finite satisfiability"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/exists-model-of-consistent-general","kind":"node","lean":"Cslib.Logic.FOL.exists_model_of_consistent_general","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#exists-model-of-consistent-general","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, with no countability hypothesis on its symbols. Every consistent set $S$ of formulas of $L$ with free variables in $\\mathbb{N}$ is satisfiable: there are a nonempty $L$-structure $M$, in the larger of the universes of the function symbols and the relation symbols of $L$, and an assignment $v : \\mathbb{N} \\to M$ under which every formula of $S$ holds.","title":"Model existence for arbitrary languages"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/provable-of-entails-general","kind":"node","lean":"Cslib.Logic.FOL.provable_of_entails_general","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#provable-of-entails-general","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language whose function symbols live in universe $u$ and relation symbols in universe $v$, with no restriction on the number of symbols. Let $S$ be a set of formulas of $L$ and $\\varphi$ a formula of $L$, all with free variables in $\\mathbb{N}$. If for every $L$-structure $M$ with carrier in universe $\\max(u, v)$ and every assignment $e : \\mathbb{N} \\to M$ under which all formulas of $S$\u2026","title":"Strong completeness for arbitrary languages"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/provable-of-models-general","kind":"node","lean":"Cslib.Logic.FOL.provable_of_models_general","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#provable-of-models-general","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language whose function symbols live in universe $u$ and relation symbols in universe $v$, with no restriction on the number of symbols. Let $T$ be a theory of $L$ and $\\varphi$ a formula of $L$ with free variables in $\\mathbb{N}$. If $\\varphi$ holds under every assignment of the variables in every nonempty model of $T$ whose carrier lies in universe $\\max(u, v)$, then $\\varphi$ is provable\u2026","title":"Completeness for arbitrary languages"},{"id":"Logic/first-order-logic/g-del-s-completeness-theorem/provable-iff-models-general","kind":"node","lean":"Cslib.Logic.FOL.provable_iff_models_general","parent":691,"result":true,"route":"../roadmap/Logic/first-order-logic/g-del-s-completeness-theorem/index.html#provable-iff-models-general","scope":"Logic/first-order-logic/g-del-s-completeness-theorem","status":"fully_proved","summary":"Let $L$ be a first-order language, with no countability hypothesis, $T$ a theory of $L$ (a set of sentences), and $\\varphi$ an $L$-formula with free variables in $\\mathbb{N}$. Then $\\varphi$ is provable from the sentences of $T$, each regarded as a formula with free variables in $\\mathbb{N}$, if and only if $\\varphi$ holds in every nonempty model of $T$ under every assignment of elements of the model to the\u2026","title":"Soundness and completeness for arbitrary languages"},{"data_url":"scopes/Logic/temporal-logic.json","id":"Logic/temporal-logic","kind":"scope","parent":1,"result":false,"route":"../roadmap/Logic/temporal-logic/index.html","scope":"Logic/temporal-logic","status":"fully_proved","summary":"Linear and branching-time temporal logics interpreted over infinite words and transition systems, with automata-theoretic and fixed-point model-checking procedures.","title":"Temporal logic and model checking"},{"data_url":"scopes/Logic/temporal-logic/computation-tree-logic.json","id":"Logic/temporal-logic/computation-tree-logic","kind":"scope","navigation":"module","parent":772,"result":false,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","title":"Computation tree logic"},{"id":"Logic/temporal-logic/computation-tree-logic/formula","kind":"node","lean":"Cslib.CTL.Formula","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#formula","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"The formulas of computation tree logic over a type $A$ of atoms are generated as follows: every atom $a \\in A$ is a formula, $\\top$ is a formula, and if $\\varphi$ and $\\psi$ are formulas then so are the negation $\\neg\\varphi$, the conjunction $\\varphi \\land \\psi$, the existential next $\\mathsf{EX}\\,\\varphi$, the existential until $\\mathsf{E}[\\varphi\\,\\mathsf{U}\\,\\psi]$, the existential globally\u2026","title":"CTL formulas"},{"id":"Logic/temporal-logic/computation-tree-logic/bot","kind":"node","lean":"Cslib.CTL.Formula.bot","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#bot","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"The CTL formula $\\bot$ (falsity) is the negation $\\neg\\top$ of the formula $\\top$ (truth).","title":"Falsity in CTL"},{"id":"Logic/temporal-logic/computation-tree-logic/or","kind":"node","lean":"Cslib.CTL.Formula.or","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#or","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"For CTL formulas $\\varphi$ and $\\psi$, the disjunction $\\varphi \\lor \\psi$ is defined as $\\neg(\\neg\\varphi \\land \\neg\\psi)$.","title":"CTL disjunction"},{"id":"Logic/temporal-logic/computation-tree-logic/imp","kind":"node","lean":"Cslib.CTL.Formula.imp","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#imp","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"For CTL formulas $\\varphi$ and $\\psi$, the implication $\\varphi \\to \\psi$ is defined as the disjunction $\\neg\\varphi \\lor \\psi$.","title":"CTL implication"},{"id":"Logic/temporal-logic/computation-tree-logic/iff","kind":"node","lean":"Cslib.CTL.Formula.iff","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#iff","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"For CTL formulas $\\varphi$ and $\\psi$, the bi-implication $\\varphi \\leftrightarrow \\psi$ is defined as the conjunction $(\\varphi \\to \\psi) \\land (\\psi \\to \\varphi)$ of the two implications.","title":"CTL bi-implication"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-finally","kind":"node","lean":"Cslib.CTL.Formula.existsFinally","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-finally","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"For a CTL formula $\\varphi$, the formula $\\mathsf{EF}\\,\\varphi$ is the existential until $\\mathsf{E}[\\top \\mathbin{\\mathsf{U}} \\varphi]$ with $\\top$ on the left.","title":"Exists finally"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-next","kind":"node","lean":"Cslib.CTL.Formula.forallNext","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","The CTL operator "],["math","\\(\\mathsf{AX}\\)"]],"source":"The CTL operator $\\mathsf{AX}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-next","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"For a CTL formula $\\varphi$, the formula $\\mathsf{AX}\\,\\varphi$ (\"on all next states, $\\varphi$\") is defined as $\\neg\\,\\mathsf{EX}\\,\\neg\\varphi$.","title":"The CTL operator $\\mathsf{AX}$"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-finally","kind":"node","lean":"Cslib.CTL.Formula.forallFinally","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-finally","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"`AF \u03c6`: every maximal path reaches a state satisfying `\u03c6`, as `A[\u22a4 U \u03c6]`.","title":"The CTL operator AF (inevitably)"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-globally","kind":"node","lean":"Cslib.CTL.Formula.forallGlobally","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","The CTL operator "],["math","\\(\\mathrm{AG}\\)"]],"source":"The CTL operator $\\mathrm{AG}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-globally","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"For a CTL formula $\\varphi$, the formula $\\mathrm{AG}\\,\\varphi$ is $\\neg\\,\\mathrm{EF}\\,\\neg\\varphi$. It holds at a state exactly when every state reachable from it satisfies $\\varphi$.","title":"The CTL operator $\\mathrm{AG}$"},{"id":"Logic/temporal-logic/computation-tree-logic/satisfies","kind":"node","lean":"Cslib.CTL.Satisfies","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#satisfies","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Fix a labelled transition system and a valuation saying which atoms hold at which states. Satisfaction of a CTL formula at a state $s$ is defined by recursion on the formula. An atom $a$ holds at $s$ when the valuation makes $a$ true at $s$; $\\top$ always holds; $\\neg\\varphi$ holds when $\\varphi$ does not; $\\varphi \\land \\psi$ holds when both $\\varphi$ and $\\psi$ hold; $\\mathsf{EX}\\,\\varphi$ holds when there is a\u2026","title":"Satisfaction of CTL formulas"},{"id":"Logic/temporal-logic/computation-tree-logic/pre-exists","kind":"node","lean":"Cslib.CTL.PreExists","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#pre-exists","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"For a labelled transition system and a predicate $Z$ on its states, a state $s$ satisfies $\\mathrm{Pre}_\\exists(Z)$ when some successor of $s$ satisfies $Z$: there are a label $a$ and a state $t$ satisfying $Z$ with a transition from $s$ to $t$ labelled $a$.","title":"Existential predecessor operator"},{"id":"Logic/temporal-logic/computation-tree-logic/serial","kind":"node","lean":"Cslib.CTL.Serial","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#serial","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"A labelled transition system is serial (total) when every state $s$ has a successor: there are a label $a$ and a state $t$ with a transition from $s$ to $t$ labelled $a$.","title":"Serial transition systems"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-omega-run-extending-of-serial","kind":"node","lean":"Cslib.CTL.exists_omegaRun_extending_of_serial","parent":773,"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-omega-run-extending-of-serial","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"In a serial labelled transition system, if $s_0, s_1, s_2, \\dots$ is a finite run of length $k$, then there is an infinite run $s'_0, s'_1, s'_2, \\dots$ with $s'_i = s_i$ for every $i \\le k$.","title":"Finite runs extend to infinite runs in serial systems"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-finally-iff","kind":"node","lean":"Cslib.CTL.Satisfies.existsFinally_iff","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathsf{EF}\\)"],["text"," via finite runs"]],"source":"$\\mathsf{EF}$ via finite runs"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-finally-iff","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CTL formula. In a labelled transition system with a valuation of the atoms at its states, a state $s$ satisfies $\\mathsf{EF}\\,\\varphi$ if and only if there are a sequence of states $s_0, s_1, s_2, \\dots$ with $s_0 = s$ and a natural number $k$ such that the sequence is a finite run of length $k$ and $s_k$ satisfies $\\varphi$.","title":"$\\mathsf{EF}$ via finite runs"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-until-expansion","kind":"node","lean":"Cslib.CTL.Satisfies.existsUntil_expansion","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-step expansion of "],["math","\\(\\mathrm{EU}\\)"]],"source":"One-step expansion of $\\mathrm{EU}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-until-expansion","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ and $\\psi$ be CTL formulas and $s$ a state of a labelled transition system with a valuation of the atoms. Then $s$ satisfies $\\mathrm{E}[\\varphi \\mathbin{\\mathrm{U}} \\psi]$ if and only if either $s$ satisfies $\\psi$, or $s$ satisfies $\\varphi$ and some successor of $s$ satisfies $\\mathrm{E}[\\varphi \\mathbin{\\mathrm{U}} \\psi]$.","title":"One-step expansion of $\\mathrm{EU}$"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-finally-iff-reachable","kind":"node","lean":"Cslib.CTL.Satisfies.existsFinally_iff_reachable","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathsf{EF}\\)"],["text"," means reachability"]],"source":"$\\mathsf{EF}$ means reachability"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-finally-iff-reachable","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CTL formula. In a labelled transition system with a valuation of the atoms at its states, a state $s$ satisfies $\\mathsf{EF}\\,\\varphi$ if and only if some state $t$ reachable from $s$ by a finite, possibly empty, sequence of transitions satisfies $\\varphi$.","title":"$\\mathsf{EF}$ means reachability"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-globally-iff-reachable","kind":"node","lean":"Cslib.CTL.Satisfies.forallGlobally_iff_reachable","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{AG}\\)"],["text"," holds iff every reachable state satisfies the formula"]],"source":"$\\mathrm{AG}$ holds iff every reachable state satisfies the formula"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-globally-iff-reachable","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CTL formula and $s$ a state of a labelled transition system with a valuation of the atoms. Then $s$ satisfies $\\mathrm{AG}\\,\\varphi$ if and only if every state $t$ reachable from $s$ (through a finite sequence of transitions, possibly empty) satisfies $\\varphi$.","title":"$\\mathrm{AG}$ holds iff every reachable state satisfies the formula"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-next-iff","kind":"node","lean":"Cslib.CTL.Satisfies.forallNext_iff","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(\\mathsf{AX}\\)"]],"source":"Characterization of $\\mathsf{AX}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-next-iff","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $s$ be a state of a labelled transition system with an atom valuation, and let $\\varphi$ be a CTL formula. Then $s$ satisfies $\\mathsf{AX}\\,\\varphi$ if and only if every state $t$ with a transition from $s$ to $t$, under any label, satisfies $\\varphi$.","title":"Characterization of $\\mathsf{AX}$"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-finally-iff-omega-and-finite","kind":"node","lean":"Cslib.CTL.Satisfies.forallFinally_iff_omega_and_finite","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal-path form of "],["math","\\(\\mathrm{AF}\\)"]],"source":"Maximal-path form of $\\mathrm{AF}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-finally-iff-omega-and-finite","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CTL formula and $s$ a state of a labelled transition system with a valuation of the atoms. Then $s$ satisfies $\\mathrm{AF}\\,\\varphi$ if and only if both of the following hold: for every infinite run $s_0, s_1, s_2, \\dots$ with $s_0 = s$ there is some $k$ such that $s_k$ satisfies $\\varphi$; and for every finite run $s_0, s_1, s_2, \\dots$ of length $k$ with $s_0 = s$ whose endpoint $s_k$ has no\u2026","title":"Maximal-path form of $\\mathrm{AF}$"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-until-deadlock","kind":"node","lean":"Cslib.CTL.Satisfies.forallUntil_deadlock","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathsf{A}(\\varphi\\,\\mathsf{U}\\,\\psi)\\)"],["text"," at a deadlocked state"]],"source":"$\\mathsf{A}(\\varphi\\,\\mathsf{U}\\,\\psi)$ at a deadlocked state"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-until-deadlock","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $s$ be a state of a labelled transition system with an atom valuation, and suppose that $s$ is deadlocked: there is no transition from $s$ under any label. Then for all CTL formulas $\\varphi$ and $\\psi$, $s$ satisfies $\\mathsf{A}[\\varphi\\,\\mathsf{U}\\,\\psi]$ if and only if $s$ satisfies $\\psi$.","title":"$\\mathsf{A}(\\varphi\\,\\mathsf{U}\\,\\psi)$ at a deadlocked state"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-finally-deadlock","kind":"node","lean":"Cslib.CTL.Satisfies.forallFinally_deadlock","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{AF}\\)"],["text"," at a deadlocked state"]],"source":"$\\mathrm{AF}$ at a deadlocked state"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-finally-deadlock","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CTL formula and $s$ a state of a labelled transition system with a valuation of the atoms. If $s$ has no outgoing transition, then $s$ satisfies $\\mathrm{AF}\\,\\varphi$ if and only if $s$ satisfies $\\varphi$.","title":"$\\mathrm{AF}$ at a deadlocked state"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-finally-iff-not-exists-globally-n","kind":"node","lean":"Cslib.CTL.Satisfies.forallFinally_iff_not_existsGlobally_not_of_serial","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","On serial systems "],["math","\\(\\mathrm{AF}\\)"],["text"," is "],["math","\\(\\neg\\mathrm{EG}\\,\\neg\\)"]],"source":"On serial systems $\\mathrm{AF}$ is $\\neg\\mathrm{EG}\\,\\neg$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-finally-iff-not-exists-globally-n","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CTL formula and $s$ a state of a serial labelled transition system with a valuation of the atoms. Then $s$ satisfies $\\mathrm{AF}\\,\\varphi$ if and only if $s$ does not satisfy $\\mathrm{EG}\\,\\neg\\varphi$.","title":"On serial systems $\\mathrm{AF}$ is $\\neg\\mathrm{EG}\\,\\neg$"},{"id":"Logic/temporal-logic/computation-tree-logic/forall-until-iff-classical-of-serial","kind":"node","lean":"Cslib.CTL.Satisfies.forallUntil_iff_classical_of_serial","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classical form of "],["math","\\(\\mathsf{A}(\\varphi\\,\\mathsf{U}\\,\\psi)\\)"],["text"," on serial systems"]],"source":"Classical form of $\\mathsf{A}(\\varphi\\,\\mathsf{U}\\,\\psi)$ on serial systems"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#forall-until-iff-classical-of-serial","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $s$ be a state of a serial labelled transition system with an atom valuation, and let $\\varphi$ and $\\psi$ be CTL formulas. Then $s$ satisfies $\\mathsf{A}[\\varphi\\,\\mathsf{U}\\,\\psi]$ if and only if $s$ satisfies neither $\\mathsf{E}[\\neg\\psi\\,\\mathsf{U}\\,(\\neg\\varphi \\land \\neg\\psi)]$ nor $\\mathsf{EG}\\,\\neg\\psi$. That is, no finite run from $s$ ends at a state satisfying $\\neg\\varphi \\land \\neg\\psi$ with\u2026","title":"Classical form of $\\mathsf{A}(\\varphi\\,\\mathsf{U}\\,\\psi)$ on serial systems"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-finally-expansion","kind":"node","lean":"Cslib.CTL.Satisfies.existsFinally_expansion","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-step expansion of "],["math","\\(\\mathsf{EF}\\)"]],"source":"One-step expansion of $\\mathsf{EF}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-finally-expansion","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CTL formula. In a labelled transition system with a valuation of the atoms at its states, a state $s$ satisfies $\\mathsf{EF}\\,\\varphi$ if and only if $s$ satisfies $\\varphi$ or $s$ satisfies $\\mathsf{EX}\\,\\mathsf{EF}\\,\\varphi$, that is, some successor of $s$ satisfies $\\mathsf{EF}\\,\\varphi$.","title":"One-step expansion of $\\mathsf{EF}$"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-globally-expansion","kind":"node","lean":"Cslib.CTL.Satisfies.existsGlobally_expansion","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-step expansion of "],["math","\\(\\mathsf{EG}\\)"]],"source":"One-step expansion of $\\mathsf{EG}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-globally-expansion","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ be a CTL formula. In a labelled transition system with a valuation of the atoms at its states, a state $s$ satisfies $\\mathsf{EG}\\,\\varphi$ if and only if $s$ satisfies $\\varphi$ and $s$ satisfies $\\mathsf{EX}\\,\\mathsf{EG}\\,\\varphi$, that is, some successor of $s$ satisfies $\\mathsf{EG}\\,\\varphi$.","title":"One-step expansion of $\\mathsf{EG}$"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-finally-fun","kind":"node","lean":"Cslib.CTL.Satisfies.existsFinallyFun","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\mathsf{EF}\\)"],["text"," functional"]],"source":"The $\\mathsf{EF}$ functional"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-finally-fun","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Given a labelled transition system, a valuation of the atoms at its states, and a CTL formula $\\varphi$, this is the map sending a set of states $Z$ to the set of states that satisfy $\\varphi$ or have some successor in $Z$.","title":"The $\\mathsf{EF}$ functional"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-finally-order-hom","kind":"node","lean":"Cslib.CTL.Satisfies.existsFinallyOrderHom","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\mathsf{EF}\\)"],["text"," operator"]],"source":"The $\\mathsf{EF}$ operator"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-finally-order-hom","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Given a labelled transition system, a valuation of the atoms at its states, and a CTL formula $\\varphi$, this is the monotone operator on sets of states, ordered by inclusion, sending a set $Z$ to the set of states that satisfy $\\varphi$ or have some successor in $Z$.","title":"The $\\mathsf{EF}$ operator"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-globally-fun","kind":"node","lean":"Cslib.CTL.Satisfies.existsGloballyFun","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","The functional for "],["math","\\(\\mathrm{EG}\\)"]],"source":"The functional for $\\mathrm{EG}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-globally-fun","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Given a labelled transition system, a valuation of the atoms and a CTL formula $\\varphi$, this is the operator that sends a predicate $Z$ on states to the predicate holding at a state $s$ when $s$ satisfies $\\varphi$ and some successor of $s$ satisfies $Z$, that is, there are a label $a$ and a state $t$ with a transition from $s$ to $t$ labelled $a$ such that $Z$ holds at $t$.","title":"The functional for $\\mathrm{EG}$"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-globally-order-hom","kind":"node","lean":"Cslib.CTL.Satisfies.existsGloballyOrderHom","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\mathrm{EG}\\)"],["text"," functional as a monotone operator"]],"source":"The $\\mathrm{EG}$ functional as a monotone operator"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-globally-order-hom","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Given a labelled transition system, a valuation of the atoms and a CTL formula $\\varphi$, this is the monotone map on predicates on states, ordered pointwise by implication, that sends a predicate $Z$ to the predicate holding at a state $s$ when $s$ satisfies $\\varphi$ and some successor of $s$ satisfies $Z$.","title":"The $\\mathrm{EG}$ functional as a monotone operator"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-finally-eq-lfp","kind":"node","lean":"Cslib.CTL.Satisfies.existsFinally_eq_lfp","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathsf{EF}\\)"],["text"," is a least fixed point"]],"source":"$\\mathsf{EF}$ is a least fixed point"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-finally-eq-lfp","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Fix a labelled transition system, a valuation of the atoms at its states, and a CTL formula $\\varphi$. Write $[\\![\\varphi]\\!]$ for the set of states satisfying $\\varphi$ and $\\mathrm{Pre}_\\exists(Z)$ for the set of states having some successor in a set of states $Z$. Then the set of states satisfying $\\mathsf{EF}\\,\\varphi$ is the least fixed point of the monotone operator\u2026","title":"$\\mathsf{EF}$ is a least fixed point"},{"id":"Logic/temporal-logic/computation-tree-logic/exists-globally-eq-gfp","kind":"node","lean":"Cslib.CTL.Satisfies.existsGlobally_eq_gfp","parent":773,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathsf{EG}\\)"],["text"," is a greatest fixed point"]],"source":"$\\mathsf{EG}$ is a greatest fixed point"},"result":true,"route":"../roadmap/Logic/temporal-logic/computation-tree-logic/index.html#exists-globally-eq-gfp","scope":"Logic/temporal-logic/computation-tree-logic","status":"fully_proved","summary":"Fix a labelled transition system, a valuation of the atoms at its states, and a CTL formula $\\varphi$. Write $[\\![\\varphi]\\!]$ for the set of states satisfying $\\varphi$ and $\\mathrm{Pre}_\\exists(Z)$ for the set of states having some successor in a set of states $Z$. Then the set of states satisfying $\\mathsf{EG}\\,\\varphi$ is the greatest fixed point of the monotone operator\u2026","title":"$\\mathsf{EG}$ is a greatest fixed point"},{"data_url":"scopes/Logic/temporal-logic/linear-temporal-logic.json","id":"Logic/temporal-logic/linear-temporal-logic","kind":"scope","navigation":"module","parent":772,"result":false,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","title":"Linear temporal logic"},{"id":"Logic/temporal-logic/linear-temporal-logic/formula","kind":"node","lean":"Cslib.LTL.Formula","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#formula","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For a type $A$ of atoms, the formulas of linear temporal logic (LTL) over $A$ are generated inductively: every atom $a \\in A$ is a formula, the truth constant $\\top$ is a formula, and if $\\varphi$ and $\\psi$ are formulas then so are the negation $\\neg\\varphi$, the conjunction $\\varphi \\wedge \\psi$, the next formula $\\mathbf{X}\\,\\varphi$ and the until formula $\\varphi \\mathbin{\\mathbf{U}} \\psi$.","title":"Formulas of linear temporal logic"},{"id":"Logic/temporal-logic/linear-temporal-logic/bot","kind":"node","lean":"Cslib.LTL.Formula.bot","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#bot","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"The LTL formula $\\bot$ (falsity) is defined as $\\neg\\top$, the negation of the truth constant $\\top$.","title":"Falsity in LTL"},{"id":"Logic/temporal-logic/linear-temporal-logic/or","kind":"node","lean":"Cslib.LTL.Formula.or","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#or","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For LTL formulas $\\varphi$ and $\\psi$, the disjunction $\\varphi \\vee \\psi$ is defined as $\\neg(\\neg\\varphi \\wedge \\neg\\psi)$.","title":"Disjunction in LTL"},{"id":"Logic/temporal-logic/linear-temporal-logic/imp","kind":"node","lean":"Cslib.LTL.Formula.imp","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#imp","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For LTL formulas $\\varphi$ and $\\psi$, the implication $\\varphi \\to \\psi$ is defined as the disjunction $\\neg\\varphi \\vee \\psi$.","title":"Implication in LTL"},{"id":"Logic/temporal-logic/linear-temporal-logic/iff","kind":"node","lean":"Cslib.LTL.Formula.iff","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#iff","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For LTL formulas $\\varphi$ and $\\psi$, the bi-implication $\\varphi \\leftrightarrow \\psi$ is defined as the conjunction $(\\varphi \\to \\psi) \\wedge (\\psi \\to \\varphi)$ of the two implications.","title":"Bi-implication in LTL"},{"id":"Logic/temporal-logic/linear-temporal-logic/eventually","kind":"node","lean":"Cslib.LTL.Formula.eventually","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#eventually","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For an LTL formula $\\varphi$, the formula $\\mathbf{F}\\,\\varphi$ (\"eventually $\\varphi$\") is defined as the until formula $\\top \\mathbin{\\mathbf{U}} \\varphi$.","title":"The eventually operator of LTL"},{"id":"Logic/temporal-logic/linear-temporal-logic/always","kind":"node","lean":"Cslib.LTL.Formula.always","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#always","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For an LTL formula $\\varphi$, the formula $\\mathbf{G}\\,\\varphi$ (\"always $\\varphi$\") is defined as $\\neg\\,\\mathbf{F}\\,\\neg\\varphi$, the negation of \"eventually $\\neg\\varphi$\".","title":"The always operator of LTL"},{"id":"Logic/temporal-logic/linear-temporal-logic/release","kind":"node","lean":"Cslib.LTL.Formula.release","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#release","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For LTL formulas $\\varphi$ and $\\psi$, the release formula $\\varphi \\mathbin{\\mathbf{R}} \\psi$ is defined as $\\neg(\\neg\\varphi \\mathbin{\\mathbf{U}} \\neg\\psi)$, the dual of until.","title":"The release operator of LTL"},{"id":"Logic/temporal-logic/linear-temporal-logic/satisfies","kind":"node","lean":"Cslib.LTL.Word.Satisfies","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#satisfies","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"Let $v$ be a valuation that says which atoms are true at each letter, and for an \u03c9-word $w$ and a natural number $k$ let $w^{(k)}$ be the suffix $i \\mapsto w(k + i)$. Whether $w$ satisfies an LTL formula $\\varphi$ under $v$, written $w \\models \\varphi$, is defined by recursion on $\\varphi$: an atom $a$ holds when $v$ makes $a$ true at the first letter $w(0)$; truth always holds; $\\neg \\varphi$ holds when $\\varphi$\u2026","title":"Satisfaction of LTL formulas on \u03c9-words"},{"id":"Logic/temporal-logic/linear-temporal-logic/satisfies-at","kind":"node","lean":"Cslib.LTL.Word.SatisfiesAt","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#satisfies-at","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"Let $v$ be a valuation of atoms on letters, $w$ an \u03c9-word, $k$ a natural number and $\\varphi$ an LTL formula. Then $\\varphi$ holds at position $k$ of $w$ under $v$ when the suffix of $w$ starting at position $k$, the \u03c9-word $i \\mapsto w(k + i)$, satisfies $\\varphi$ under $v$.","title":"Satisfaction at a position"},{"id":"Logic/temporal-logic/linear-temporal-logic/language","kind":"node","lean":"Cslib.LTL.Word.language","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#language","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"Given a valuation $v$ of atoms on letters, the \u03c9-language of an LTL formula $\\varphi$ is the set of \u03c9-words that satisfy $\\varphi$ under $v$.","title":"Language of an LTL formula"},{"id":"Logic/temporal-logic/linear-temporal-logic/satisfies-at-zero","kind":"node","lean":"Cslib.LTL.Word.satisfiesAt_zero","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#satisfies-at-zero","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation of atoms on letters, \u03c9-word $w$ and LTL formula $\\varphi$, $\\varphi$ holds at position $0$ of $w$ if and only if $w$ satisfies $\\varphi$.","title":"Satisfaction at position zero"},{"id":"Logic/temporal-logic/linear-temporal-logic/satisfies-at-drop","kind":"node","lean":"Cslib.LTL.Word.satisfiesAt_drop","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#satisfies-at-drop","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"Let $w$ be an \u03c9-word and, for a natural number $k$, let $w^{(k)}$ be its suffix starting at position $k$, the \u03c9-word $i \\mapsto w(k + i)$. For every valuation of atoms on letters, every LTL formula $\\varphi$ and every natural number $j$, $\\varphi$ holds at position $j$ of $w^{(k)}$ if and only if $\\varphi$ holds at position $k + j$ of $w$.","title":"Satisfaction on a suffix"},{"id":"Logic/temporal-logic/linear-temporal-logic/atom-iff","kind":"node","lean":"Cslib.LTL.Word.Satisfies.atom_iff","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#atom-iff","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters, every \u03c9-word $w$ and every atom $a$, the word $w$ satisfies the atomic formula $a$ under $v$ if and only if $v$ makes $a$ true at the first letter $w_0$ of $w$.","title":"Satisfaction of an atom"},{"id":"Logic/temporal-logic/linear-temporal-logic/bot-iff","kind":"node","lean":"Cslib.LTL.Word.Satisfies.bot_iff","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#bot-iff","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters and every \u03c9-word $w$, the word $w$ does not satisfy the LTL formula $\\bot$ (falsity) under $v$.","title":"Falsity is never satisfied"},{"id":"Logic/temporal-logic/linear-temporal-logic/eventually-iff","kind":"node","lean":"Cslib.LTL.Word.Satisfies.eventually_iff","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#eventually-iff","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters, every \u03c9-word $w$ and every LTL formula $\\varphi$, the word $w$ satisfies $\\mathbf{F}\\,\\varphi$ (eventually $\\varphi$) under $v$ if and only if for some $k \\in \\mathbb{N}$ the suffix of $w$ starting at position $k$ satisfies $\\varphi$ under $v$.","title":"Semantics of eventually"},{"id":"Logic/temporal-logic/linear-temporal-logic/release-iff","kind":"node","lean":"Cslib.LTL.Word.Satisfies.release_iff","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#release-iff","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For a valuation of atoms on letters, an \u03c9-word $w$ and LTL formulas $\\varphi$ and $\\psi$, $w$ satisfies $\\varphi \\mathbin{\\mathsf{R}} \\psi$ ($\\varphi$ releases $\\psi$) if and only if for every position $k$, either $\\psi$ holds at position $k$ of $w$ or $\\varphi$ holds at some position $j < k$ of $w$.","title":"Semantics of release"},{"id":"Logic/temporal-logic/linear-temporal-logic/not-next","kind":"node","lean":"Cslib.LTL.Word.Satisfies.not_next","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#not-next","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters, every \u03c9-word $w$ and every LTL formula $\\varphi$, the word $w$ satisfies $\\neg\\mathbf{X}\\,\\varphi$ under $v$ if and only if it satisfies $\\mathbf{X}\\,\\neg\\varphi$ under $v$, where $\\mathbf{X}$ is the next operator.","title":"Next is self-dual"},{"id":"Logic/temporal-logic/linear-temporal-logic/until-expansion","kind":"node","lean":"Cslib.LTL.Word.Satisfies.until_expansion","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#until-expansion","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation of atoms on letters, \u03c9-word $w$ and LTL formulas $\\varphi$ and $\\psi$, $w$ satisfies $\\varphi \\mathbin{\\mathsf{U}} \\psi$ ($\\varphi$ until $\\psi$) if and only if either $w$ satisfies $\\psi$, or $w$ satisfies $\\varphi$ and also satisfies $\\mathsf{X}(\\varphi \\mathbin{\\mathsf{U}} \\psi)$, that is, next ($\\varphi$ until $\\psi$).","title":"Expansion law for until"},{"id":"Logic/temporal-logic/linear-temporal-logic/release-expansion","kind":"node","lean":"Cslib.LTL.Word.Satisfies.release_expansion","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#release-expansion","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters, every \u03c9-word $w$ and all LTL formulas $\\varphi$ and $\\psi$, the word $w$ satisfies $\\varphi \\mathbin{\\mathcal{R}} \\psi$ ($\\varphi$ releases $\\psi$) under $v$ if and only if it satisfies $\\psi$ and it satisfies $\\varphi$ or $\\mathbf{X}(\\varphi \\mathbin{\\mathcal{R}} \\psi)$, all under $v$, where $\\mathbf{X}$ is the next operator.","title":"Expansion law for release"},{"id":"Logic/temporal-logic/linear-temporal-logic/always-always","kind":"node","lean":"Cslib.LTL.Word.Satisfies.always_always","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#always-always","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters, every \u03c9-word $w$ and every LTL formula $\\varphi$, the word $w$ satisfies $\\mathbf{G}\\,\\mathbf{G}\\,\\varphi$ under $v$ if and only if it satisfies $\\mathbf{G}\\,\\varphi$ under $v$, where $\\mathbf{G}$ is the always operator.","title":"Idempotence of always"},{"id":"Logic/temporal-logic/linear-temporal-logic/always-eventually-iff","kind":"node","lean":"Cslib.LTL.Word.Satisfies.always_eventually_iff","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#always-eventually-iff","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters, every \u03c9-word $w$ and every LTL formula $\\varphi$, the word $w$ satisfies $\\mathbf{G}\\,\\mathbf{F}\\,\\varphi$ (always eventually $\\varphi$) under $v$ if and only if there are infinitely many $k \\in \\mathbb{N}$ such that the suffix of $w$ starting at position $k$ satisfies $\\varphi$ under $v$.","title":"Always eventually means infinitely often"},{"id":"Logic/temporal-logic/linear-temporal-logic/eventually-always-iff","kind":"node","lean":"Cslib.LTL.Word.Satisfies.eventually_always_iff","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#eventually-always-iff","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters, every \u03c9-word $w$ and every LTL formula $\\varphi$, the word $w$ satisfies $\\mathbf{F}\\,\\mathbf{G}\\,\\varphi$ (eventually always $\\varphi$) under $v$ if and only if there is $n \\in \\mathbb{N}$ such that for every $k \\ge n$ the suffix of $w$ starting at position $k$ satisfies $\\varphi$ under $v$.","title":"Eventually always means from some point on"},{"id":"Logic/temporal-logic/linear-temporal-logic/satisfies-at-atom","kind":"node","lean":"Cslib.LTL.Word.satisfiesAt_atom","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#satisfies-at-atom","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"Let $v$ be a valuation of atoms on letters, $w$ an \u03c9-word, $a$ an atom and $k$ a natural number. The atomic formula $a$ holds at position $k$ of $w$ under $v$ if and only if $v$ makes $a$ true at the letter $w(k)$.","title":"Satisfaction of an atom at a position"},{"id":"Logic/temporal-logic/linear-temporal-logic/satisfies-map","kind":"node","lean":"Cslib.LTL.Word.satisfies_map","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#satisfies-map","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"Let $v$ be a valuation of atoms on letters of an alphabet $\\Sigma$, let $f \\colon \\Sigma' \\to \\Sigma$ be a map, and let $w$ be an \u03c9-word over $\\Sigma'$. For every LTL formula $\\varphi$, the relabelled word $i \\mapsto f(w(i))$ satisfies $\\varphi$ under $v$ if and only if $w$ satisfies $\\varphi$ under the valuation that makes an atom $a$ true at a letter $s$ of $\\Sigma'$ exactly when $v$ makes $a$ true at $f(s)$.","title":"Relabelling the letters of an \u03c9-word"},{"id":"Logic/temporal-logic/linear-temporal-logic/mem-language","kind":"node","lean":"Cslib.LTL.Word.mem_language","parent":805,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-temporal-logic/index.html#mem-language","scope":"Logic/temporal-logic/linear-temporal-logic","status":"fully_proved","summary":"For every valuation $v$ of atoms on letters, every \u03c9-word $w$ and every LTL formula $\\varphi$, the word $w$ belongs to the \u03c9-language of $\\varphi$ under $v$ if and only if $w$ satisfies $\\varphi$ under $v$.","title":"Membership in the language of a formula"},{"data_url":"scopes/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal.json","id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","kind":"scope","navigation":"module","parent":772,"result":false,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","title":"B\u00fcchi automata for a fragment of linear temporal logic"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/prop-formula","kind":"node","lean":"Cslib.LTL.PropFormula","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#prop-formula","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"The propositional formulas over a set of atoms are generated by the atoms, the constant truth $\\top$, negation $\\neg\\varphi$ and conjunction $\\varphi \\wedge \\psi$.","title":"Propositional formulas"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/eval","kind":"node","lean":"Cslib.LTL.PropFormula.eval","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#eval","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"A letter is a set $s$ of atoms (the atoms true at it). A propositional formula is evaluated at $s$ by recursion: an atom $a$ is true iff $a \\in s$, truth $\\top$ is true, $\\neg\\varphi$ is true iff $\\varphi$ is not, and $\\varphi \\wedge \\psi$ is true iff both $\\varphi$ and $\\psi$ are true.","title":"Evaluating a propositional formula at a letter"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/to-ltl","kind":"node","lean":"Cslib.LTL.PropFormula.toLTL","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#to-ltl","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"The translation of propositional formulas into LTL formulas is defined by recursion: an atom $a$ goes to the LTL atom $a$, truth goes to the LTL truth constant $\\top$, $\\neg\\varphi$ goes to the LTL negation of the translation of $\\varphi$, and $\\varphi \\wedge \\psi$ goes to the LTL conjunction of the translations of $\\varphi$ and $\\psi$.","title":"Propositional formulas as LTL formulas"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/satisfies-to-ltl","kind":"node","lean":"Cslib.LTL.satisfies_toLTL","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#satisfies-to-ltl","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"Let $\\varphi$ be a propositional formula and let $w = w_0 w_1 w_2 \\dots$ be an \u03c9-word whose letters are sets of atoms. Under the valuation in which an atom holds at a letter exactly when it belongs to that letter, $w$ satisfies the LTL translation of $\\varphi$ if and only if $\\varphi$ evaluates to true on the first letter $w_0$.","title":"Propositional formulas are evaluated at the first letter"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/satisfies-at-to-ltl","kind":"node","lean":"Cslib.LTL.satisfiesAt_toLTL","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#satisfies-at-to-ltl","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"Let $\\varphi$ be a propositional formula, let $w = w_0 w_1 w_2 \\dots$ be an \u03c9-word whose letters are sets of atoms, and let $k \\in \\mathbb{N}$. Under the valuation in which an atom holds at a letter exactly when it belongs to that letter, the LTL translation of $\\varphi$ holds at position $k$ of $w$ if and only if $\\varphi$ evaluates to true on the letter $w_k$.","title":"Propositional formulas are evaluated at the current letter"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/always-na","kind":"node","lean":"Cslib.LTL.alwaysNa","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#always-na","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"The one-state B\u00fcchi automaton for `G \u03c6` with `\u03c6` propositional: it stays forever, requiring every letter to satisfy `\u03c6`.","title":"One-state B\u00fcchi automaton for always \u03c6"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/language-always-na","kind":"node","lean":"Cslib.LTL.language_alwaysNa","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#language-always-na","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"For a propositional formula $\\varphi$, the one-state B\u00fcchi automaton for $\\mathbf{G}\\,\\varphi$ accepts exactly the infinite words of letters (sets of atoms) that satisfy the LTL formula $\\mathbf{G}\\,\\varphi$, where an atom $a$ holds at a letter $s$ iff $a \\in s$.","title":"The always automaton recognizes G \u03c6"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/language-always-eventually","kind":"node","lean":"Cslib.LTL.language_always_eventually","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#language-always-eventually","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"Let $\\varphi$ be a propositional formula and $p$ the set of letters (sets of atoms) at which $\\varphi$ evaluates to true. The two-state B\u00fcchi automaton for \u201cinfinitely often $p$\u201d accepts exactly the infinite words that satisfy the LTL formula $\\mathbf{G}\\,\\mathbf{F}\\,\\varphi$, where an atom $a$ holds at a letter $s$ iff $a \\in s$. Equivalently, a word is accepted iff $\\varphi$ holds at infinitely many positions.","title":"The infinitely-often automaton recognizes G F \u03c6"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/language-eventually-always","kind":"node","lean":"Cslib.LTL.language_eventually_always","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#language-eventually-always","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"Let $\\varphi$ be a propositional formula and $p$ the set of letters (sets of atoms) at which $\\varphi$ evaluates to true. The two-state B\u00fcchi automaton for \u201ceventually always $p$\u201d accepts exactly the infinite words that satisfy the LTL formula $\\mathbf{F}\\,\\mathbf{G}\\,\\varphi$, where an atom $a$ holds at a letter $s$ iff $a \\in s$. Equivalently, a word is accepted iff $\\varphi$ holds at all but finitely many\u2026","title":"The eventually-always automaton recognizes F G \u03c6"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/prop-na","kind":"node","lean":"Cslib.LTL.propNa","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#prop-na","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"For a propositional formula $\\varphi$, this B\u00fcchi automaton over letters that are sets of atoms has a start state $u$ and two verdict states $t$ and $f$, of which only $t$ is accepting. From $u$, on a letter $s$ it moves to $t$ if $\\varphi$ evaluates to true at $s$ and to $f$ otherwise; from $t$ and from $f$ it stays in the same state on every letter. So it records the verdict of $\\varphi$ on the first letter.","title":"B\u00fcchi automaton for a propositional formula"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/language-prop-na","kind":"node","lean":"Cslib.LTL.language_propNa","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#language-prop-na","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"For a propositional formula $\\varphi$, the three-state B\u00fcchi automaton for $\\varphi$ accepts exactly the infinite words of letters (sets of atoms) that satisfy $\\varphi$ read as an LTL formula, where an atom $a$ holds at a letter $s$ iff $a \\in s$.","title":"The propositional automaton recognizes \u03c6"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/shift-buchi","kind":"node","lean":"Cslib.LTL.shiftBuchi","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#shift-buchi","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"Let $A$ be a B\u00fcchi automaton over the alphabet of sets of atoms. Its shift is the B\u00fcchi automaton whose states are the states of $A$ together with one new state $\\bot$, whose only start state is $\\bot$, and whose accepting states are the accepting states of $A$. Its transitions are exactly the following: on every letter, from $\\bot$ to each start state of $A$; and on a letter $x$, from a state $q$ of $A$ to a state\u2026","title":"The shift of a B\u00fcchi automaton"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/language-shift-buchi","kind":"node","lean":"Cslib.LTL.language_shiftBuchi","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#language-shift-buchi","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"For a B\u00fcchi automaton $A$ over letters that are sets of atoms, the shift automaton of $A$ accepts an infinite word $x_0 x_1 x_2 \\ldots$ iff $A$ accepts its tail $x_1 x_2 x_3 \\ldots$.","title":"The shift automaton reads the tail"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/safety","kind":"node","lean":"Cslib.LTL.Safety","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#safety","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"The safety formulas over a type of atoms are generated by the following rules: every propositional formula $\\varphi$ is a safety formula; if $\\psi$ and $\\rho$ are safety formulas, then so are $\\psi \\wedge \\rho$, $\\psi \\vee \\rho$ and $X\\,\\psi$ (next $\\psi$); and for every propositional formula $\\varphi$, $G\\,\\varphi$ (always $\\varphi$) is a safety formula.","title":"The restricted safety fragment of LTL"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/to-ltl-2","kind":"node","lean":"Cslib.LTL.Safety.toLTL","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#to-ltl-2","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"The translation $t$ of safety formulas into LTL formulas is defined by recursion: $t(\\varphi)$ is the LTL translation of $\\varphi$ for a propositional formula $\\varphi$; $t(\\psi \\wedge \\rho) = t(\\psi) \\wedge t(\\rho)$; $t(\\psi \\vee \\rho) = \\neg(\\neg t(\\psi) \\wedge \\neg t(\\rho))$, the derived LTL disjunction; $t(X\\,\\psi) = X\\,t(\\psi)$, where $X$ is next; and $t(G\\,\\varphi) = G\\,t(\\varphi)$ for a propositional formula\u2026","title":"Safety formulas as LTL formulas"},{"id":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/exists-buchi-of-safety","kind":"node","lean":"Cslib.LTL.exists_buchi_of_safety","parent":832,"result":true,"route":"../roadmap/Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal/index.html#exists-buchi-of-safety","scope":"Logic/temporal-logic/b-chi-automata-for-a-fragment-of-linear-temporal","status":"fully_proved","summary":"Take as letters the sets of atoms, an atom $a$ holding at a letter $s$ iff $a \\in s$. For every formula $\\psi$ of the restricted safety fragment there is a nondeterministic B\u00fcchi automaton with finitely many states whose \u03c9-language is exactly the set of infinite words satisfying $\\psi$, read as an LTL formula.","title":"B\u00fcchi automata for the restricted safety fragment"},{"data_url":"scopes/Logic/temporal-logic/ctl.json","id":"Logic/temporal-logic/ctl","kind":"scope","parent":772,"result":false,"route":"../roadmap/Logic/temporal-logic/ctl/index.html","scope":"Logic/temporal-logic/ctl","status":"fully_proved","title":"CTL*"},{"data_url":"scopes/Logic/temporal-logic/ctl/ctl-state-and-path-formulas.json","id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","kind":"scope","navigation":"module","parent":849,"result":false,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","title":"CTL*: state and path formulas"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/state-formula","kind":"node","lean":"Cslib.CTLStar.StateFormula","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["text","State formulas of CTL"],["math","\\(^{\\ast}\\)"]],"source":"State formulas of CTL$^{\\ast}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#state-formula","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"State formulas of CTL*: atoms, truth, negation, conjunction, and the path quantifiers `E \u03c0` and `A \u03c0`.","title":"State formulas of CTL$^{\\ast}$"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/path-formula","kind":"node","lean":"Cslib.CTLStar.PathFormula","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Path formulas of CTL"],["math","\\(^{\\ast}\\)"]],"source":"Path formulas of CTL$^{\\ast}$"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#path-formula","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"Path formulas of CTL*: a state formula at the first position, negation, conjunction, `next`, and `until`.","title":"Path formulas of CTL$^{\\ast}$"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/always-path","kind":"node","lean":"Cslib.CTLStar.PathFormula.alwaysPath","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Always operator on CTL"],["math","\\(^{\\ast}\\)"],["text"," path formulas"]],"source":"Always operator on CTL$^{\\ast}$ path formulas"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#always-path","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"`G \u03c0`: `\u03c0` holds at every position, as `\u00ac(\u22a4 U \u00ac\u03c0)`.","title":"Always operator on CTL$^{\\ast}$ path formulas"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/eventually-path","kind":"node","lean":"Cslib.CTLStar.PathFormula.eventuallyPath","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventually operator on CTL"],["math","\\(^{\\ast}\\)"],["text"," path formulas"]],"source":"Eventually operator on CTL$^{\\ast}$ path formulas"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#eventually-path","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"`F \u03c0`: `\u03c0` holds at some position, as `\u22a4 U \u03c0`.","title":"Eventually operator on CTL$^{\\ast}$ path formulas"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/state-sat","kind":"node","lean":"Cslib.CTLStar.StateSat","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Satisfaction of CTL"],["math","\\(^{\\ast}\\)"],["text"," state formulas"]],"source":"Satisfaction of CTL$^{\\ast}$ state formulas"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#state-sat","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"Satisfaction of a CTL* state formula at a state: `E \u03c0` holds when some infinite run from the state satisfies `\u03c0`; `A \u03c0` when every infinite run does.","title":"Satisfaction of CTL$^{\\ast}$ state formulas"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/path-sat","kind":"node","lean":"Cslib.CTLStar.PathSat","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Satisfaction of CTL"],["math","\\(^{\\ast}\\)"],["text"," path formulas on infinite runs"]],"source":"Satisfaction of CTL$^{\\ast}$ path formulas on infinite runs"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#path-sat","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"Satisfaction of a CTL* path formula on an infinite run: `next` shifts the run, `until` asks for a target position with the guard before it.","title":"Satisfaction of CTL$^{\\ast}$ path formulas on infinite runs"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/path-sat-state","kind":"node","lean":"Cslib.CTLStar.pathSat_state","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#path-sat-state","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"Unfolding: a state formula on a run is satisfaction at its first state.","title":"A state formula on a run holds at its first state"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/path-sat-next","kind":"node","lean":"Cslib.CTLStar.pathSat_next","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#path-sat-next","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"Unfolding: `next` shifts the run.","title":"Next on a run is satisfaction on the shifted run"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/shift-seq-apply-zero","kind":"node","lean":"Cslib.CTLStar.shiftSeq_apply_zero","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#shift-seq-apply-zero","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"`(shiftSeq run k) 0 = run k`: the `0 + k` is discharged by `Nat.zero_add`.","title":"A run shifted by k starts at its k-th state"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/path-sat-state-shift","kind":"node","lean":"Cslib.CTLStar.pathSat_state_shift","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#path-sat-state-shift","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"A state formula on a shifted run is satisfaction at the shifted position.","title":"A state formula on a shifted run holds at the shifted state"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/path-sat-always","kind":"node","lean":"Cslib.CTLStar.pathSat_always","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#path-sat-always","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"`G \u03c0` holds on a run iff `\u03c0` holds on every suffix.","title":"Always holds on a run iff its argument holds on every suffix"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/path-sat-eventually","kind":"node","lean":"Cslib.CTLStar.pathSat_eventually","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#path-sat-eventually","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"`F \u03c0` holds on a run iff `\u03c0` holds on some suffix.","title":"Eventually holds on a run iff its argument holds on some suffix"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/embed-ltl","kind":"node","lean":"Cslib.CTLStar.embedLTL","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Embedding of LTL formulas as CTL"],["math","\\(^{\\ast}\\)"],["text"," path formulas"]],"source":"Embedding of LTL formulas as CTL$^{\\ast}$ path formulas"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#embed-ltl","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The LTL embedding: every LTL formula is a CTL* path formula.","title":"Embedding of LTL formulas as CTL$^{\\ast}$ path formulas"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/embed-ltl-sat","kind":"node","lean":"Cslib.CTLStar.embedLTL_sat","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#embed-ltl-sat","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The LTL embedding preserves semantics on every state sequence, including sequences that are not valid runs of `lts`.","title":"The LTL embedding preserves semantics on every sequence"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/embed-ltl-sat-at","kind":"node","lean":"Cslib.CTLStar.embedLTL_sat_at","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#embed-ltl-sat-at","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The structural LTL embedding also agrees at every position, without assuming the sequence is a valid transition run.","title":"The LTL embedding preserves semantics at every position"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/embed-ctl","kind":"node","lean":"Cslib.CTLStar.embedCTL","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Embedding of CTL formulas as CTL"],["math","\\(^{\\ast}\\)"],["text"," state formulas"]],"source":"Embedding of CTL formulas as CTL$^{\\ast}$ state formulas"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#embed-ctl","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The CTL embedding: every CTL formula is a CTL* state formula. `EG \u03c6` is `E G (state \u03c6)`; `AU` is `A (state \u03c6 U state \u03c8)`.","title":"Embedding of CTL formulas as CTL$^{\\ast}$ state formulas"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/embed-ctl-sat","kind":"node","lean":"Cslib.CTLStar.embedCTL_sat","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#embed-ctl-sat","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The CTL embedding preserves semantics on serial systems.","title":"The CTL embedding preserves semantics on serial systems"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/toggle-lts","kind":"node","lean":"Cslib.CTLStar.toggleLTS","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#toggle-lts","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The toggle system: every transition flips the state.","title":"Toggle transition system"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/toggle-valuation","kind":"node","lean":"Cslib.CTLStar.toggleValuation","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#toggle-valuation","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The valuation: `p` holds exactly at `true`.","title":"Valuation of the toggle system"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/toggle-serial","kind":"node","lean":"Cslib.CTLStar.toggle_serial","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#toggle-serial","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The toggle system is serial: every state has a successor.","title":"The toggle system is serial"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/toggle-run","kind":"node","lean":"Cslib.CTLStar.toggleRun","parent":850,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#toggle-run","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"The alternating run of the toggle system: `false, true, false, true, ...`, defined by recursion so each step flips the state.","title":"Alternating run of the toggle system"},{"id":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas/toggle-egf-p","kind":"node","lean":"Cslib.CTLStar.toggle_egf_p","parent":850,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(E\\,G\\,F\\,p\\)"],["text"," holds at false in the toggle system"]],"source":"$E\\,G\\,F\\,p$ holds at false in the toggle system"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-and-path-formulas/index.html#toggle-egf-p","scope":"Logic/temporal-logic/ctl/ctl-state-and-path-formulas","status":"fully_proved","summary":"`E G F p` holds at `false` in the toggle system: the alternating run visits `true` at every other position. This example makes no claim of an expressiveness separation.","title":"$E\\,G\\,F\\,p$ holds at false in the toggle system"},{"data_url":"scopes/Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl.json","id":"Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl","kind":"scope","navigation":"module","parent":849,"result":false,"route":"../roadmap/Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl/index.html","scope":"Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl","status":"fully_proved","title":"Deadlock boundary of infinite-run CTL*"},{"id":"Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl/not-state-sat-exists-of-terminal","kind":"node","lean":"Cslib.CTLStar.not_stateSat_exists_of_terminal","parent":873,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl/index.html#not-state-sat-exists-of-terminal","scope":"Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl","status":"fully_proved","summary":"No existential infinite path formula holds at a terminal state.","title":"No existential path formula holds at a terminal state"},{"id":"Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl/state-sat-forall-of-terminal","kind":"node","lean":"Cslib.CTLStar.stateSat_forall_of_terminal","parent":873,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl/index.html#state-sat-forall-of-terminal","scope":"Logic/temporal-logic/ctl/deadlock-boundary-of-infinite-run-ctl","status":"fully_proved","summary":"Universal infinite path quantification is vacuous at a terminal state.","title":"Universal path formulas hold vacuously at terminal states"},{"data_url":"scopes/Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex.json","id":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex","kind":"scope","navigation":"module","parent":849,"result":false,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/index.html","scope":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex","status":"fully_proved","title":"CTL* state-run semantics and canonical labeled executions"},{"id":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/upstream-state-sat","kind":"node","lean":"Cslib.CTLStar.UpstreamStateSat","parent":876,"reading_title":{"format":"inline-parts-v1","parts":[["text","CTL"],["math","\\(^{\\ast}\\)"],["text"," state satisfaction over labeled executions"]],"source":"CTL$^{\\ast}$ state satisfaction over labeled executions"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/index.html#upstream-state-sat","scope":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex","status":"fully_proved","summary":"Satisfaction of a CTL* state formula at a state over canonical `Cslib.LTS`: `E \u03c0` holds when some infinite labeled execution from the state satisfies `\u03c0`; `A \u03c0` when every infinite labeled execution does.","title":"CTL$^{\\ast}$ state satisfaction over labeled executions"},{"id":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/upstream-path-sat","kind":"node","lean":"Cslib.CTLStar.UpstreamPathSat","parent":876,"reading_title":{"format":"inline-parts-v1","parts":[["text","CTL"],["math","\\(^{\\ast}\\)"],["text"," path satisfaction on labeled executions"]],"source":"CTL$^{\\ast}$ path satisfaction on labeled executions"},"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/index.html#upstream-path-sat","scope":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex","status":"fully_proved","summary":"Satisfaction of a CTL* path formula on a canonical infinite labeled execution: `next` drops the first state, while `until` asks for a target position with the guard before it.","title":"CTL$^{\\ast}$ path satisfaction on labeled executions"},{"id":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/state-sat-iff-upstream","kind":"node","lean":"Cslib.CTLStar.stateSat_iff_upstream","parent":876,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/index.html#state-sat-iff-upstream","scope":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex","status":"fully_proved","summary":"CTL* state-sequence/execution bridge, state formulas. CTL* state satisfaction over `lts` agrees with satisfaction quantified over canonical `OmegaExecution`s.","title":"CTL* state-sequence/execution bridge, state formulas"},{"id":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/path-sat-iff-upstream","kind":"node","lean":"Cslib.CTLStar.pathSat_iff_upstream","parent":876,"result":true,"route":"../roadmap/Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex/index.html#path-sat-iff-upstream","scope":"Logic/temporal-logic/ctl/ctl-state-run-semantics-and-canonical-labeled-ex","status":"fully_proved","summary":"CTL* state-sequence/execution bridge, path formulas. CTL* path satisfaction on a state run agrees with the canonical labeled-execution presentation on the same state sequence.","title":"CTL* state-sequence/execution bridge, path formulas"},{"data_url":"scopes/Logic/temporal-logic/linear-time-calculus.json","id":"Logic/temporal-logic/linear-time-calculus","kind":"scope","navigation":"module","parent":772,"result":false,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","title":"Linear-time \u03bc-calculus"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-formula","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.LinearMuFormula","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-formula","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Formulas of the linear-time \u03bc-calculus over atoms `Atom`, with de Bruijn variables for the fixed-point binders.","title":"Formulas of the linear-time \u03bc-calculus"},{"id":"Logic/temporal-logic/linear-time-calculus/neg","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.LinearMuFormula.neg","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#neg","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Meta-level negation, swapping each connective with its dual.","title":"Dual negation of linear \u03bc-calculus formulas"},{"id":"Logic/temporal-logic/linear-time-calculus/neg-neg","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.LinearMuFormula.neg_neg","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#neg-neg","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Negation is an involution.","title":"Negation of linear \u03bc-calculus formulas is an involution"},{"id":"Logic/temporal-logic/linear-time-calculus/closed-at","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.LinearMuFormula.ClosedAt","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#closed-at","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"`\u03c6` is closed at depth `d` when every free variable is below `d` (de Bruijn): `var n` needs `n < d`, and binders increase the depth.","title":"Closedness of a \u03bc-calculus formula at a binder depth"},{"id":"Logic/temporal-logic/linear-time-calculus/closed","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.LinearMuFormula.Closed","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#closed","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"A formula with no free variables.","title":"Closed linear \u03bc-calculus formula"},{"id":"Logic/temporal-logic/linear-time-calculus/depth","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.LinearMuFormula.depth","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#depth","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The depth of a formula: the length of its longest root-to-leaf path, counting each connective once.","title":"Depth of a linear \u03bc-calculus formula"},{"id":"Logic/temporal-logic/linear-time-calculus/extend-env","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.extendEnv","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#extend-env","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Extend an environment under a binder (de Bruijn).","title":"Extending an environment under a binder"},{"id":"Logic/temporal-logic/linear-time-calculus/sat","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Satisfaction of a linear \u03bc-calculus formula at a position of the word `val`. `\u03bc X. \u03c6` is the least pre-fixed point of `S \u21a6 \u27e6\u03c6\u27e7[X \u21a6 S]`, stated directly as an infimum over pre-fixed points; `\u03bd X. \u03c6` is the greatest post-fixed point, stated as a supremum over post-fixed points.","title":"Satisfaction of linear \u03bc-calculus formulas"},{"id":"Logic/temporal-logic/linear-time-calculus/sat-mono-env","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat_mono_env","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat-mono-env","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Satisfaction is monotone in the environment.","title":"Satisfaction is monotone in the environment"},{"id":"Logic/temporal-logic/linear-time-calculus/sat-mono-extend","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat_mono_extend","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat-mono-extend","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Monotonicity in the bound variable's denotation.","title":"Satisfaction is monotone in the bound variable"},{"id":"Logic/temporal-logic/linear-time-calculus/mu-pre-fixed","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.mu_pre_fixed","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#mu-pre-fixed","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The least fixed point is a pre-fixed point.","title":"Least fixed point is a pre-fixed point"},{"id":"Logic/temporal-logic/linear-time-calculus/mu-le-fixed","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.mu_le_fixed","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#mu-le-fixed","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The least fixed point is a fixed point (the lower half).","title":"Least fixed point is a fixed point: lower inclusion"},{"id":"Logic/temporal-logic/linear-time-calculus/sat-mu-unfold","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat_mu_unfold","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat-mu-unfold","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"`\u03bc X. \u03c6 \u2261 \u03c6[\u03bc X. \u03c6 / X]`.","title":"The \u03bc unfolding"},{"id":"Logic/temporal-logic/linear-time-calculus/nu-post-fixed","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.nu_post_fixed","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#nu-post-fixed","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The greatest fixed point is a post-fixed point (the upper half).","title":"Greatest fixed point is a post-fixed point"},{"id":"Logic/temporal-logic/linear-time-calculus/nu-le-post-fixed","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.nu_le_post_fixed","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#nu-le-post-fixed","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The greatest fixed point is a fixed point (the upper half).","title":"Greatest fixed point is a fixed point: upper inclusion"},{"id":"Logic/temporal-logic/linear-time-calculus/sat-nu-unfold","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat_nu_unfold","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat-nu-unfold","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"`\u03bd X. \u03c6 \u2261 \u03c6[\u03bd X. \u03c6 / X]`.","title":"The \u03bd unfolding"},{"id":"Logic/temporal-logic/linear-time-calculus/mu-induction","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.mu_induction","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#mu-induction","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"To show `\u03bc X. \u03c6 \u2264 P`, it suffices that `P` is a pre-fixed point.","title":"Park induction"},{"id":"Logic/temporal-logic/linear-time-calculus/nu-coinduction","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.nu_coinduction","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#nu-coinduction","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Every post-fixed point is below `\u03bd X. \u03c6`.","title":"Coinduction"},{"id":"Logic/temporal-logic/linear-time-calculus/mu-op","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.muOp","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#mu-op","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The \u03bc operator as a monotone operator on predicates.","title":"The \u03bc operator as a monotone operator on predicates"},{"id":"Logic/temporal-logic/linear-time-calculus/sat-mu-eq-lfp","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat_mu_eq_lfp","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat-mu-eq-lfp","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The \u03bc fixed point is Mathlib's least fixed point of the operator.","title":"\u03bc-formulas denote least fixed points"},{"id":"Logic/temporal-logic/linear-time-calculus/sat-nu-eq-gfp","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat_nu_eq_gfp","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat-nu-eq-gfp","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The \u03bd fixed point is Mathlib's greatest fixed point of the operator.","title":"\u03bd-formulas denote greatest fixed points"},{"id":"Logic/temporal-logic/linear-time-calculus/sat-closed-env","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat_closed_env","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat-closed-env","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"A closed formula does not depend on the environment: environments agreeing below the depth give the same denotation.","title":"Closed formulas do not depend on the environment"},{"id":"Logic/temporal-logic/linear-time-calculus/extend-env-compl","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.extendEnv_compl","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#extend-env-compl","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Complementing an environment commutes with extending it.","title":"Complementing an environment commutes with extension"},{"id":"Logic/temporal-logic/linear-time-calculus/compl-compl","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.compl_compl","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#compl-compl","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Complementing a predicate twice returns it.","title":"Double complement of a predicate"},{"id":"Logic/temporal-logic/linear-time-calculus/sat-neg","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.sat_neg","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#sat-neg","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"`\u27e6\u00ac\u03c6\u27e7 = (\u27e6\u03c6\u27e7)\u1d9c` when the environment is complemented.","title":"Duality"},{"id":"Logic/temporal-logic/linear-time-calculus/f","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.F","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#f","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"`F p`: `p` holds at some future position.","title":"Eventually operator in the linear \u03bc-calculus"},{"id":"Logic/temporal-logic/linear-time-calculus/g","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.G","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#g","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"`G p`: `p` holds at every future position.","title":"Always operator in the linear \u03bc-calculus"},{"id":"Logic/temporal-logic/linear-time-calculus/f-iff","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.F_iff","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#f-iff","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The linear \u03bc-calculus `F` is eventuality.","title":"The linear \u03bc-calculus `F` is eventuality"},{"id":"Logic/temporal-logic/linear-time-calculus/g-iff","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.G_iff","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#g-iff","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The linear \u03bc-calculus `G` is universality over the future.","title":"The linear \u03bc-calculus `G` is universality over the future"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-code","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuCode","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-code","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"An injective encoding of linear \u03bc-calculus formulas over `\u2115` atoms as natural numbers, via the Cantor pairing function `Nat.pair`. This is a Nat code only, not a complexity input encoding: repeated pairing squares the code at every unary connective, so the binary length of the code of `next^d (atom 0)` doubles at each `next` and is exponential in `d`. Complexity statements use `linearMuTokens` /\u2026","title":"Cantor-pairing code of linear \u03bc-calculus formulas"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-code-injective","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuCode_injective","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-code-injective","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"The encoding is injective.","title":"Injectivity of the linear \u03bc-calculus code"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-decode-fuel","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuDecodeFuel","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-decode-fuel","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Fuelled decoding of a `linearMuCode` number back to a formula. The fuel bounds the recursion depth; `linearMuDecode` supplies fuel `c + 1`, which exceeds every formula's depth (see `linearDepth_le_code`).","title":"Fuelled decoding of linear \u03bc-calculus codes"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-decode","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuDecode","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-decode","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Decode a `linearMuCode` number back to the formula it encodes.","title":"Decoding linear \u03bc-calculus codes"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-depth-le-code","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearDepth_le_code","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-depth-le-code","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"A formula's depth is at most its code: pairing dominates each component (`Nat.add_le_pair`).","title":"Formula depth is bounded by its code"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-decode-fuel-linear-mu-code","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuDecodeFuel_linearMuCode","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-decode-fuel-linear-mu-code","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Decoding inverts encoding, with enough fuel.","title":"Fuelled decoding inverts encoding"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-decode-linear-mu-code","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuDecode_linearMuCode","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-decode-linear-mu-code","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Decoding inverts encoding.","title":"Decoding inverts the linear \u03bc-calculus encoding"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-tokens","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuTokens","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-tokens","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"A polynomial-size token-list encoding of linear \u03bc-calculus formulas over `\u2115` atoms, used for the complexity statements in `WantedExt`. Each connective contributes one tag token (`Nat.pair` of a small tag with `0`), and each atom or variable contributes one token `Nat.pair tag value`, whose binary size is `O(log value)`. The canonical bitstring encoding of a `List \u2115` delimits each token, so the bitstring for a\u2026","title":"Token-list encoding of linear \u03bc-calculus formulas"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-parse-fuel","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuParseFuel","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-parse-fuel","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Fuelled parsing of a `linearMuTokens` token list back to a formula, returning the formula and the unconsumed suffix. The fuel bounds the recursion depth; `linearMuDecodeList` supplies fuel `l.length + 1`, which exceeds the token length of every formula encoded in `l`.","title":"Fuelled parser for linear \u03bc-calculus token lists"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-decode-list","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuDecodeList","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-decode-list","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Decode a `linearMuTokens` token list back to the formula it encodes, rejecting trailing tokens.","title":"Decoding a linear \u03bc-calculus token list to a formula"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-parse-fuel-tokens","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuParseFuel_tokens","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-parse-fuel-tokens","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Parsing inverts token encoding, with enough fuel.","title":"Fuelled parsing inverts linear \u03bc-calculus token encoding"},{"id":"Logic/temporal-logic/linear-time-calculus/linear-mu-decode-list-tokens","kind":"node","lean":"Cslib.Modal.LinearMuCalculus.linearMuDecodeList_tokens","parent":881,"result":true,"route":"../roadmap/Logic/temporal-logic/linear-time-calculus/index.html#linear-mu-decode-list-tokens","scope":"Logic/temporal-logic/linear-time-calculus","status":"fully_proved","summary":"Decoding inverts token encoding.","title":"Token-list decoding inverts linear \u03bc-calculus token encoding"},{"data_url":"scopes/Logic/temporal-logic/model-checking-for-computation-tree-logic.json","id":"Logic/temporal-logic/model-checking-for-computation-tree-logic","kind":"scope","navigation":"module","parent":772,"result":false,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","title":"Model checking for computation tree logic"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/exists-maximal-path","kind":"node","lean":"Cslib.CTL.exists_maximalPath","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#exists-maximal-path","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"In a labelled transition system, for every state $s$ there is an infinite run $s_0, s_1, s_2, \\dots$ with $s_0 = s$, or there are a sequence of states $s_0, s_1, s_2, \\dots$ with $s_0 = s$ and a $k$ such that $s_0, \\dots, s_k$ is a finite run of length $k$ and $s_k$ is deadlocked, that is, has no outgoing transition.","title":"Every state starts a maximal path"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/not-exists-next-top-iff","kind":"node","lean":"Cslib.CTL.Satisfies.not_existsNext_top_iff","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#not-exists-next-top-iff","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"A state $s$ of a labelled transition system satisfies the CTL formula $\\neg\\mathsf{EX}\\,\\top$, under any valuation of the atoms, if and only if $s$ is deadlocked: there is no label $a$ and state $t$ with a transition from $s$ to $t$ labelled $a$.","title":"Deadlocked states are exactly those satisfying \u00acEX \u22a4"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/forall-until-bad-iff","kind":"node","lean":"Cslib.CTL.Satisfies.forallUntil_bad_iff","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#forall-until-bad-iff","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ and $\\psi$ be CTL formulas, and write $t \\models \\chi$ when the state $t$ of a labelled transition system satisfies the formula $\\chi$ under a given valuation of the atoms. Then $t \\models \\neg\\psi \\land (\\neg\\varphi \\lor \\neg\\mathsf{EX}\\,\\top)$ if and only if $t \\not\\models \\psi$ and either $t \\not\\models \\varphi$ or $t$ is deadlocked, that is, there is no label $a$ and state $u$ with a transition\u2026","title":"Bad target of the deadlock-correct until duality"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/forall-until-expansion-forward","kind":"node","lean":"Cslib.CTL.Satisfies.forallUntil_expansion_forward","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#forall-until-expansion-forward","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ and $\\psi$ be CTL formulas, and write $s \\models \\chi$ when the state $s$ of a labelled transition system satisfies the formula $\\chi$ under a given valuation of the atoms. If $s \\models \\mathsf{A}[\\varphi \\mathbin{\\mathsf{U}} \\psi]$, then either $s \\models \\psi$, or $s \\models \\varphi$, $s$ has at least one outgoing transition, and every state $t$ reached from $s$ by a single transition with any\u2026","title":"One-step unfolding of universal until, forward direction"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/forall-until-iff-not","kind":"node","lean":"Cslib.CTL.Satisfies.forallUntil_iff_not","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#forall-until-iff-not","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let $\\varphi$ and $\\psi$ be CTL formulas, and write $s \\models \\chi$ when the state $s$ of a labelled transition system satisfies the formula $\\chi$ under a given valuation of the atoms. Then $s \\models \\mathsf{A}[\\varphi \\mathbin{\\mathsf{U}} \\psi]$ if and only if neither $s \\models \\mathsf{E}[\\neg\\psi \\mathbin{\\mathsf{U}} (\\neg\\psi \\land (\\neg\\varphi \\lor \\neg\\mathsf{EX}\\,\\top))]$ nor\u2026","title":"Deadlock-correct duality for universal until"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/exists-until-congr","kind":"node","lean":"Cslib.CTL.Satisfies.existsUntil_congr","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#exists-until-congr","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"In a labelled transition system with a valuation of atoms, let $\\rho$, $\\alpha$ and $\\beta$ be CTL formulas such that every state satisfies $\\alpha$ if and only if it satisfies $\\beta$. Then every state $s$ satisfies $\\mathsf{E}[\\rho \\,\\mathsf{U}\\, \\alpha]$ if and only if it satisfies $\\mathsf{E}[\\rho \\,\\mathsf{U}\\, \\beta]$.","title":"Congruence of existential until in its target"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/forall-finally-iff-not","kind":"node","lean":"Cslib.CTL.Satisfies.forallFinally_iff_not","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#forall-finally-iff-not","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"In a labelled transition system with a valuation of atoms, let $\\varphi$ be a CTL formula. A state $s$ satisfies $\\mathsf{AF}\\,\\varphi$ if and only if $s$ satisfies neither $\\mathsf{E}[\\neg\\varphi \\,\\mathsf{U}\\, (\\neg\\varphi \\wedge \\neg\\mathsf{EX}\\,\\top)]$ nor $\\mathsf{EG}\\,\\neg\\varphi$.","title":"Deadlock-correct duality for AF"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/exists-next-iff-pre-exists","kind":"node","lean":"Cslib.CTL.Satisfies.existsNext_iff_preExists","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#exists-next-iff-pre-exists","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"In a labelled transition system with a valuation of atoms, let $\\varphi$ be a CTL formula and write $\\mathrm{Sat}(\\varphi)$ for the set of states satisfying $\\varphi$. A state $s$ satisfies $\\mathsf{EX}\\,\\varphi$ if and only if $s$ belongs to the existential predecessor set of $\\mathrm{Sat}(\\varphi)$, that is, some transition leads from $s$ to a state satisfying $\\varphi$.","title":"Existential next via the predecessor operator"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/exists-until-fun","kind":"node","lean":"Cslib.CTL.Satisfies.existsUntilFun","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#exists-until-fun","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"In a labelled transition system with a valuation of atoms, let $\\varphi$ and $\\psi$ be CTL formulas. The existential-until functional sends a set $Z$ of states to the set of states $s$ such that either $s$ satisfies $\\psi$, or $s$ satisfies $\\varphi$ and some transition leads from $s$ to a state in $Z$.","title":"The existential-until functional"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/exists-until-order-hom","kind":"node","lean":"Cslib.CTL.Satisfies.existsUntilOrderHom","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#exists-until-order-hom","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"In a labelled transition system with a valuation of atoms, let $\\varphi$ and $\\psi$ be CTL formulas. The map sending a set $Z$ of states to the set of states $s$ such that either $s$ satisfies $\\psi$, or $s$ satisfies $\\varphi$ and some transition leads from $s$ to a state in $Z$, is monotone for inclusion; this is that map, regarded as an order-preserving map on sets of states.","title":"The existential-until functional as a monotone map"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/exists-until-eq-lfp","kind":"node","lean":"Cslib.CTL.Satisfies.existsUntil_eq_lfp","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#exists-until-eq-lfp","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"In a labelled transition system with a valuation of atoms, let $\\varphi$ and $\\psi$ be CTL formulas, write $\\mathrm{Sat}(\\chi)$ for the set of states satisfying a formula $\\chi$, and write $\\mathrm{Pre}_\\exists(Z)$ for the set of states having a transition into a set $Z$ of states. Then the set of states satisfying $\\mathsf{E}[\\varphi \\,\\mathsf{U}\\, \\psi]$ is the least fixed point of the monotone map\u2026","title":"Existential until as a least fixed point"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/pre-exists-fin","kind":"node","lean":"Cslib.CTL.preExistsFin","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#pre-exists-fin","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let a labelled transition system have finitely many states and finitely many labels, with decidable transitions. For a finite set $Z$ of states, its existential predecessor set is the finite set of states $s$ for which there are a label $a$ and a state $t \\in Z$ with a transition from $s$ to $t$ labelled $a$.","title":"Finite existential predecessor set"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/e-until-fin","kind":"node","lean":"Cslib.CTL.eUntilFin","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#e-until-fin","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let $\\mathcal{T}$ be a labelled transition system with finitely many states and finitely many labels, decidable transitions and decidable equality of states, and let $A$ and $B$ be sets of states. For a set $Z$ of states write $\\mathrm{Pre}_\\exists(Z)$ for the set of states having a transition into $Z$. The existential-until set with side set $A$ and target set $B$ is obtained by applying the map\u2026","title":"Finite existential-until set"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/e-globally-fin","kind":"node","lean":"Cslib.CTL.eGloballyFin","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#e-globally-fin","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let $\\mathcal{T}$ be a labelled transition system with finitely many states and finitely many labels, decidable transitions and decidable equality of states, and let $A$ be a set of states. For a set $Z$ of states write $\\mathrm{Pre}_\\exists(Z)$ for the set of states having a transition into $Z$. The existential-globally set of $A$ is obtained by applying the map $Z \\mapsto A \\cap \\mathrm{Pre}_\\exists(Z)$ to the\u2026","title":"Finite existential-globally set"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/mem-e-until-fin-iff","kind":"node","lean":"Cslib.CTL.mem_eUntilFin_iff","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#mem-e-until-fin-iff","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let a labelled transition system have finitely many states and finitely many labels, with decidable transitions and decidable equality of states, and fix a valuation of the atoms. For a finite set $Z$ of states let $\\mathrm{Pre}_\\exists(Z)$ be the set of states with a transition into $Z$, and for finite sets $A$ and $B$ of states let $\\mathrm{EU}_{\\mathrm{fin}}(A, B)$ be the least fixed point of\u2026","title":"Correctness of the finite fixed-point computation of EU"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/mem-e-globally-fin-iff","kind":"node","lean":"Cslib.CTL.mem_eGloballyFin_iff","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#mem-e-globally-fin-iff","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let a labelled transition system have finitely many states and finitely many labels, with decidable transitions and decidable equality of states, and fix a valuation of the atoms. For a finite set $Z$ of states let $\\mathrm{Pre}_\\exists(Z)$ be the set of states with a transition into $Z$, and for a finite set $A$ of states let $\\mathrm{EG}_{\\mathrm{fin}}(A)$ be the greatest fixed point of\u2026","title":"Correctness of the finite fixed-point computation of EG"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/sat","kind":"node","lean":"Cslib.CTL.sat","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#sat","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"The set of states satisfying a CTL formula in a finite system, computed by structural recursion: atoms by filtering the valuation, `EX` by `preExistsFin`, `EU` and `EG` by the finite fixed points `eUntilFin` and `eGloballyFin`, and `AU` by complementing the deadlock-correct duality of `Satisfies.forallUntil_iff_not` (the bad-target set is built from the complement sets of the subformulas and\u2026","title":"Finite-state CTL model checking by structural recursion"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/mem-sat","kind":"node","lean":"Cslib.CTL.mem_sat","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#mem-sat","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let a labelled transition system have finitely many states and finitely many labels, with decidable transitions and decidable equality of states, and let the valuation of the atoms be decidable. For every CTL formula $\\varphi$ and every state $s$, the state $s$ lies in the set $\\mathrm{sat}(\\varphi)$ computed by the model checker if and only if $s$ satisfies $\\varphi$.","title":"Correctness of the CTL model checker"},{"id":"Logic/temporal-logic/model-checking-for-computation-tree-logic/check","kind":"node","lean":"Cslib.CTL.check","parent":923,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-computation-tree-logic/index.html#check","scope":"Logic/temporal-logic/model-checking-for-computation-tree-logic","status":"fully_proved","summary":"Let $\\mathcal{T}$ be a labelled transition system with finitely many states and finitely many labels, decidable transitions and decidable equality of states, and let $V$ be a decidable valuation of atoms at states. For a state $s$ and a CTL formula $\\varphi$, the model checker returns true exactly when $s$ belongs to the set of states that the satisfaction-set algorithm computes for $\\varphi$ in $\\mathcal{T}$ under\u2026","title":"The CTL model checker as a Boolean decision procedure"},{"data_url":"scopes/Logic/temporal-logic/model-checking-for-linear-temporal-logic.json","id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","kind":"scope","navigation":"module","parent":772,"result":false,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","title":"Model checking for linear temporal logic"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/state","kind":"node","lean":"Cslib.LTL.Tableau.State","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#state","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"One Boolean for each syntactic occurrence in an LTL formula.","title":"Truth-annotation states of the LTL tableau"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/value","kind":"node","lean":"Cslib.LTL.Tableau.value","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#value","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The truth bit at the root of an annotation.","title":"Root truth bit of a tableau annotation"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/obligation-count","kind":"node","lean":"Cslib.LTL.Tableau.obligationCount","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#obligation-count","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The number of until occurrences, hence of fairness requirements.","title":"Number of until fairness obligations"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/step","kind":"node","lean":"Cslib.LTL.Tableau.step","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#step","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Local equations at every occurrence. The symbol labels the source state.","title":"Local transition relation of the LTL tableau"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/obligation","kind":"node","lean":"Cslib.LTL.Tableau.obligation","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#obligation","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The Boolean test for each until fairness requirement.","title":"Boolean test for an until fairness obligation"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/fair","kind":"node","lean":"Cslib.LTL.Tableau.Fair","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#fair","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Every until occurrence is absent or fulfilled at arbitrarily late times.","title":"Fairness of a tableau annotation sequence"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/automaton","kind":"node","lean":"Cslib.LTL.Tableau.automaton","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#automaton","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Generalized B\u00fcchi tableau: the root is initially true.","title":"Generalized B\u00fcchi tableau for an LTL formula"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/fair-iff","kind":"node","lean":"Cslib.LTL.Tableau.fair_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#fair-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Structural fairness is exactly generalized B\u00fcchi acceptance.","title":"Tableau fairness is generalized B\u00fcchi acceptance"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/mem-language-iff","kind":"node","lean":"Cslib.LTL.Tableau.mem_language_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#mem-language-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Membership exposes exactly the initial, local, and fairness conditions.","title":"Tableau acceptance via initial, local and fairness conditions"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/mem-automaton-iff","kind":"node","lean":"Cslib.LTL.Tableau.mem_automaton_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#mem-automaton-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The full tableau recognizes exactly the existing LTL satisfaction relation.","title":"LTL tableau accepts exactly the satisfying words"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/language-automaton","kind":"node","lean":"Cslib.LTL.Tableau.language_automaton","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#language-automaton","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Exact omega-language equality for the generalized tableau.","title":"Generalized LTL tableau language equals the formula language"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/buchi","kind":"node","lean":"Cslib.LTL.Tableau.buchi","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#buchi","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Ordinary B\u00fcchi recognition, using the existing fairness counter.","title":"B\u00fcchi automaton for an LTL formula"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/language-buchi","kind":"node","lean":"Cslib.LTL.Tableau.language_buchi","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#language-buchi","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Every LTL formula has precisely its existing language as a B\u00fcchi language.","title":"B\u00fcchi automaton language equals the LTL formula language"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/mem-buchi-iff","kind":"node","lean":"Cslib.LTL.Tableau.mem_buchi_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#mem-buchi-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The ordinary B\u00fcchi recognizer accepts exactly satisfaction of the formula.","title":"B\u00fcchi automaton accepts exactly the satisfying words"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/exists-buchi","kind":"node","lean":"Cslib.LTL.exists_buchi","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#exists-buchi","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Every existing LTL formula has a finite-state B\u00fcchi recognizer under the same valuation. Neither atoms nor the alphabet need be finite or decidable.","title":"Every LTL formula has a finite-state B\u00fcchi automaton"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/is-run","kind":"node","lean":"Cslib.LTL.ModelChecking.IsRun","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#is-run","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"An initial infinite state run, witnessed by a canonical action execution.","title":"Initial infinite run of a transition system"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/is-run-iff","kind":"node","lean":"Cslib.LTL.ModelChecking.isRun_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#is-run-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Equivalently, every adjacent state pair has a real model transition.","title":"Runs are sequences with a transition between adjacent states"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/product","kind":"node","lean":"Cslib.LTL.ModelChecking.product","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#product","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Synchronized state-reading product, with the recognizer's B\u00fcchi condition.","title":"Product of a transition system with a B\u00fcchi recognizer"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/mem-product-iff","kind":"node","lean":"Cslib.LTL.ModelChecking.mem_product_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#mem-product-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"A product accepts precisely the model runs accepted by its recognizer.","title":"Product accepts exactly the runs accepted by the recognizer"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/counterexample","kind":"node","lean":"Cslib.LTL.ModelChecking.counterexample","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#counterexample","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The finite-state product for runs that violate the full LTL formula.","title":"Counterexample product for an LTL formula"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/models","kind":"node","lean":"Cslib.LTL.ModelChecking.Models","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#models","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"All initial infinite canonical executions satisfy the formula.","title":"Satisfaction of an LTL formula by all initial runs"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/mem-counterexample-iff","kind":"node","lean":"Cslib.LTL.ModelChecking.mem_counterexample_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#mem-counterexample-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"A counterexample word is exactly an infinite model run violating LTL.","title":"Counterexample words are runs violating the formula"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/models-iff-no-lasso","kind":"node","lean":"Cslib.LTL.ModelChecking.models_iff_no_lasso","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#models-iff-no-lasso","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Full-LTL model checking reduces to the proved finite B\u00fcchi lasso criterion. A cycle must be nonempty; neither seriality nor finite action/atom types are needed.","title":"Finite-model LTL satisfaction iff no counterexample lasso"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/models-iff-language-empty","kind":"node","lean":"Cslib.LTL.ModelChecking.models_iff_language_empty","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#models-iff-language-empty","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Universal infinite-run satisfaction is emptiness of the counterexample language.","title":"LTL satisfaction iff the counterexample language is empty"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/advance","kind":"node","lean":"Cslib.LTL.Tableau.advance","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#advance","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Computable fairness-counter update, reading the destination annotation.","title":"Computable fairness-counter update"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/next-count-eq-advance","kind":"node","lean":"Cslib.LTL.Tableau.nextCount_eq_advance","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#next-count-eq-advance","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The computable update is exactly the existing generic degeneralization.","title":"Computable counter update equals generic degeneralization"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/buchi-tr-iff","kind":"node","lean":"Cslib.LTL.Tableau.buchi_tr_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#buchi-tr-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The existing B\u00fcchi transition can be tested using the computable update.","title":"B\u00fcchi transition test via the computable counter update"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/check-b","kind":"node","lean":"Cslib.LTL.ModelChecking.checkB","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#check-b","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"Full-LTL checking by the existing finite B\u00fcchi reachability algorithm. The finite/decidable data requirements are only for execution, not recognition.","title":"Executable full-LTL model checker"},{"id":"Logic/temporal-logic/model-checking-for-linear-temporal-logic/check-b-iff","kind":"node","lean":"Cslib.LTL.ModelChecking.checkB_iff","parent":943,"result":true,"route":"../roadmap/Logic/temporal-logic/model-checking-for-linear-temporal-logic/index.html#check-b-iff","scope":"Logic/temporal-logic/model-checking-for-linear-temporal-logic","status":"fully_proved","summary":"The executable checker is correct for all initial infinite model runs.","title":"Correctness of the executable LTL model checker"},{"data_url":"scopes/Logic/proof-theory.json","id":"Logic/proof-theory","kind":"scope","parent":1,"result":false,"route":"../roadmap/Logic/proof-theory/index.html","scope":"Logic/proof-theory","status":"fully_proved","summary":"Structural study of formal proofs and provability: sequent calculi, linear logic and translations into it, and G\u00f6del's incompleteness theorems.","title":"Proof theory"},{"data_url":"scopes/Logic/proof-theory/cut-elimination-for-classical-linear-logic.json","id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic","kind":"scope","parent":973,"result":false,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/index.html","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic","status":"fully_proved","title":"Cut elimination for classical linear logic"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/scan-to-list","kind":"node","lean":"Multiset.scanToList","parent":974,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/index.html#scan-to-list","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic","status":"fully_proved","summary":"Compute a list representative of a submultiset by scanning a supplied list. The resulting list preserves the order of the selected occurrences in the supplied list.","title":"List representative of a submultiset by scanning a list"},{"data_url":"scopes/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination.json","id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","kind":"scope","navigation":"module","parent":974,"result":false,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","title":"Prerequisite APIs for the frozen cut-elimination source port"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/all-quest","kind":"node","lean":"List.allQuest","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#all-quest","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"Checks that all propositions in a list are question marks (`\u0294`).","title":"Check that every proposition in a list is a why-not proposition"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/all-quest-coe","kind":"node","lean":"Cslib.Logic.CLL.Sequent.allQuest_coe","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#all-quest-coe","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"Checking a list context agrees with checking its multiset coercion.","title":"Why-not check agrees on a list and its multiset"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/all-quest-zero","kind":"node","lean":"Cslib.Logic.CLL.Sequent.allQuest_zero","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#all-quest-zero","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"The empty sequent consists of `\u0294`-propositions.","title":"Empty sequent consists of question-mark propositions"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/all-quest-cons-iff","kind":"node","lean":"Cslib.Logic.CLL.Sequent.allQuest_cons_iff","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#all-quest-cons-iff","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"A cons sequent consists of `\u0294`-propositions exactly when its head and tail do.","title":"A cons sequent is all question marks iff its head and tail are"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/all-quest-quest-cons","kind":"node","lean":"Cslib.Logic.CLL.Sequent.allQuest_quest_cons","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#all-quest-quest-cons","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"Adding a `\u0294`-proposition preserves and reflects `allQuest`.","title":"A question-mark head does not change the all-question-mark property"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/not-all-quest-bang-cons","kind":"node","lean":"Cslib.Logic.CLL.Sequent.not_allQuest_bang_cons","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#not-all-quest-bang-cons","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"A sequent containing a `!`-proposition cannot satisfy `allQuest`.","title":"A sequent headed by a bang proposition is not all question marks"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/all-quest-add","kind":"node","lean":"Cslib.Logic.CLL.Sequent.allQuest_add","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#all-quest-add","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"A sum consists of `\u0294`-propositions exactly when both summands do.","title":"A sum of sequents is all question marks iff both summands are"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/height","kind":"node","lean":"Cslib.Logic.CLL.Proof.height","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#height","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"Height of a proof.","title":"Height of a classical linear logic proof"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/rw-conclusion-height","kind":"node","lean":"Cslib.Logic.CLL.Proof.rwConclusion_height","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#rw-conclusion-height","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"Transporting a conclusion preserves proof height.","title":"Conclusion transport preserves proof height"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/cut-free-rw-conclusion","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutFree_rwConclusion","parent":976,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination/index.html#cut-free-rw-conclusion","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/prerequisite-apis-for-the-frozen-cut-elimination","status":"fully_proved","summary":"Conclusion transport preserves the pinned Boolean cut-free check.","title":"Conclusion transport preserves the Boolean cut-free check"},{"data_url":"scopes/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin.json","id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","kind":"scope","navigation":"module","parent":974,"result":false,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","title":"Cut-free proofs, cut admissibility, and cut elimination for classical linear logic"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/is-cut-free","kind":"node","lean":"Cslib.Logic.CLL.Proof.IsCutFree","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#is-cut-free","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"A proof is cut-free if it does not contain any applications of rule cut.","title":"Cut-freeness of a proof"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/is-cut-free-iff-cut-free","kind":"node","lean":"Cslib.Logic.CLL.Proof.isCutFree_iff_cutFree","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#is-cut-free-iff-cut-free","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"The source predicate agrees exactly with the pinned Boolean predicate.","title":"Cut-freeness predicate agrees with the Boolean check"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/rec","kind":"node","lean":"Cslib.Logic.CLL.Proof.IsCutFree.rec","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#rec","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Recursor for cut-free proofs: never sees the `cut` case.","title":"Recursor for cut-free proofs"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/is-cut-free-rw-conclusion","kind":"node","lean":"Cslib.Logic.CLL.Proof.isCutFree_rwConclusion","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#is-cut-free-rw-conclusion","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Transporting a conclusion preserves cut-freeness.","title":"Conclusion transport preserves cut-freeness"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/cf","kind":"node","lean":"Cslib.Logic.CLL.CF","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#cf","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Tag for the cut-free fragment of CLL.","title":"Cut-free fragment of classical linear logic"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/proof","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#proof","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Proofs in the CF inference system.","title":"Proofs of the cut-free inference system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/rw-conclusion","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.rwConclusion","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#rw-conclusion","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rewrites the conclusion of a cut-free proof. If `h` is omitted, tries to prove the equality using `grind only [multiset]`.","title":"Rewriting the conclusion of a cut-free proof"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/ax","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.ax","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#ax","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Axiom for CF.","title":"Axiom rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/one","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.one","parent":987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit rule "],["math","\\(1\\)"],["text"," of the cut-free system"]],"source":"Unit rule $1$ of the cut-free system"},"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#one","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `1` for CF.","title":"Unit rule $1$ of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/top","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.top","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#top","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `\u22a4` for CF.","title":"Top rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/bot","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.bot","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#bot","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `\u22a5` for CF.","title":"Bottom rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/parr","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.parr","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#parr","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `\u214b` for CF.","title":"Par rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/tensor","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.tensor","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#tensor","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `\u2297` for CF.","title":"Tensor rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/oplus","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.oplus\u2081","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#oplus","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `\u2295\u2081` for CF.","title":"First plus rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/oplus-2","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.oplus\u2082","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#oplus-2","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `\u2295\u2082` for CF.","title":"Second plus rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/with","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.with","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#with","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `&` for CF.","title":"With rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/quest","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.quest","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#quest","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `?` (dereliction) for CF.","title":"Dereliction rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/weaken","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.weaken","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#weaken","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `?` (weaken) for CF.","title":"Weakening rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/contract","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.contract","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#contract","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `?` (contract) for CF.","title":"Contraction rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/bang","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.bang","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#bang","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Rule `!` (bang/promotion) for CF.","title":"Promotion rule of the cut-free system"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/weaken-context","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.weakenContext","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#weaken-context","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Given a list of `\u0294`-propositions `\u0393`, transforms a cut-free proof of `\u0394` into a proof of `\u0393 + \u0394` via weakening.","title":"Weakening by a list of question-mark propositions"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/contract-context","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.contractContext","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#contract-context","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Given a list of `\u0294`-propositions `\u0393`, transforms a proof of `\u0393 + \u0393 + \u0394` into a proof of `\u0393 + \u0394` via contraction.","title":"Contraction of a duplicated question-mark context"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/eq-zero-of-zero-eq-add","kind":"node","lean":"Cslib.Logic.CLL.Sequent.eq_zero_of_zero_eq_add","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#eq-zero-of-zero-eq-add","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"An empty sum has an empty right residual context.","title":"Empty sum of sequents has an empty right summand"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/split-cons","kind":"node","lean":"Cslib.Logic.CLL.Sequent.splitCons","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#split-cons","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Distinguish a principal selected occurrence using decidable equality. Given `a ::\u2098 \u0393 = b ::\u2098 \u0394`, either `a = b` and `\u0393 = \u0394`, or `a \u2260 b` and the propositions are swapped: `\u0393 = b ::\u2098 \u039b` and `\u0394 = a ::\u2098 \u039b` for some `\u039b`. The tail `\u0394` can be an arbitrary multiset; the residual list is obtained from `\u0393`.","title":"Case split on an equation of cons sequents"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/split-cons-add","kind":"node","lean":"Cslib.Logic.CLL.Sequent.splitConsAdd","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#split-cons-add","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Given `a ::\u2098 \u0393 = (\u0394 : Sequent) + \u0398`, locate `a` in either `\u0394` or `\u0398` and erase it from that summand, returning the residual list.","title":"Locating a head proposition in a sum of sequents"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/split-replicate-cons","kind":"node","lean":"Cslib.Logic.CLL.Sequent.splitReplicateCons","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#split-replicate-cons","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Given `Multiset.replicate n a + \u0393 = b ::\u2098 \u0394`, either the head `b` is one of the `n` copies of `a` (so `b = a` and the tail is `replicate (n-1) a + \u0393`), or `b` belongs to `\u0393` (so `\u0394 = Multiset.replicate n a + \u0393'` where `\u0393 = b ::\u2098 \u0393'`). The tail `\u0394` can be an arbitrary multiset; the residual list is obtained from `\u0393`.","title":"Case split of a cons against replicated copies"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/split-replicate-add","kind":"node","lean":"Cslib.Logic.CLL.Sequent.splitReplicateAdd","parent":987,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin/index.html#split-replicate-add","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/cut-free-proofs-cut-admissibility-and-cut-elimin","status":"fully_proved","summary":"Given `Multiset.replicate n a + \u0393 = (\u0394 : Sequent) + \u0398`, split the `n` copies of `a` between `\u0394` and `\u0398`, returning counts `k` and `l` and residual lists.","title":"Splitting replicated copies across a sum of sequents"},{"data_url":"scopes/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c.json","id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c","kind":"scope","navigation":"module","parent":974,"result":false,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/index.html","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c","status":"fully_proved","title":"Source exponential multicut for the locked CLL calculus"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/cut-admissible-at","kind":"node","lean":"Cslib.Logic.CLL.CutAdmissibleAt","parent":1015,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/index.html#cut-admissible-at","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c","status":"fully_proved","summary":"Cut admissibility at a fixed proposition `a`: for any list sequents `\u0393`, `\u0394`, given cut-free proofs of `a ::\u2098 \u0393` and `a\u2ae0 ::\u2098 \u0394`, produce a cut-free proof of `(\u0393 : Sequent Atom) + \u0394`.","title":"Cut admissibility at a fixed proposition"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/cut-admissible-below","kind":"node","lean":"Cslib.Logic.CLL.CutAdmissibleBelow","parent":1015,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/index.html#cut-admissible-below","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c","status":"fully_proved","summary":"The outer induction hypothesis for cut admissibility: for all propositions `a` with `sizeOf a < bound`, cut is admissible at `a`.","title":"Cut admissibility below a size bound"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/split-head","kind":"node","lean":"Cslib.Logic.CLL.splitHead","parent":1015,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/index.html#split-head","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c","status":"fully_proved","summary":"Locate a selected question mark or a residual head using `Sequent.splitReplicateCons`.","title":"Locating a selected question mark or a residual head"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/split-add","kind":"node","lean":"Cslib.Logic.CLL.splitAdd","parent":1015,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/index.html#split-add","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c","status":"fully_proved","summary":"Split replicated question marks across two arbitrary multiset summands.","title":"Splitting replicated question marks across two summands"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/quest-multicut-ax","kind":"node","lean":"Cslib.Logic.CLL.Proof.questMulticutAx","parent":1015,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/index.html#quest-multicut-ax","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c","status":"fully_proved","summary":"Ax case for `questMulticut`: the selected dual question mark is one of the copies of `a\u2ae0`. This isolates the recursive branch from the plain-term outer proof.","title":"Axiom case of the exponential multicut"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/quest-multicut","kind":"node","lean":"Cslib.Logic.CLL.Proof.questMulticut","parent":1015,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c/index.html#quest-multicut","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-exponential-multicut-for-the-locked-cll-c","status":"fully_proved","summary":"Exponential/multicut helper: given `p : CF\u21d3(a ::\u2098 \u0393)` (the \"substitutend\") and `q : CF\u21d3(Multiset.replicate n (\u0294(a\u2ae0)) + \u0394)` (the \"supply\" with `n` selected dual question marks), produce `CF\u21d3((\u0393 : Sequent) + \u0394)`.","title":"Exponential multicut against replicated question marks"},{"data_url":"scopes/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul.json","id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","kind":"scope","navigation":"module","parent":974,"result":false,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","title":"Source cut elimination for the locked CLL calculus"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/present-sum-for-cut","kind":"node","lean":"Cslib.Logic.CLL.Sequent.presentSumForCut","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#present-sum-for-cut","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Present both summands using only the supplied list.","title":"List presentations of two summands from a supplied list"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/present-tail-for-cut","kind":"node","lean":"Cslib.Logic.CLL.Sequent.presentTailForCut","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#present-tail-for-cut","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Present a tail without choosing a multiset representative.","title":"List presentation of a tail without choosing a representative"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/split-cut-head","kind":"node","lean":"Cslib.Logic.CLL.Sequent.splitCutHead","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#split-cut-head","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Split a selected occurrence against a rule head; the rule tail can be a multiset.","title":"Splitting a selected occurrence against a rule head"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-principal","kind":"node","lean":"Cslib.Logic.CLL.Proof.CutPrincipal","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-principal","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"The shape of a cut-free proof when the cut proposition is in principal position. Each constructor corresponds to a rule whose conclusion matches the cut proposition. Axioms are handled separately in `cutStep`. Tensor carries explicit presentations of both premises' residual contexts.","title":"Principal shapes of a cut-free proof at the cut proposition"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-step","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutStep","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-step","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Decompose a cut against a cut-free proof: either expose a principal rule matching the cut proposition, or commute the cut through one inference (non-principal case). Every recursive call is on a strictly shorter supplying derivation.","title":"Cut step exposing a principal rule or commuting the cut"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/bang-cut-adm","kind":"node","lean":"Cslib.Logic.CLL.Proof.bangCutAdm","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#bang-cut-adm","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Cut a bang proposition by commuting through its supplying proof to an axiom or promotion. At promotion, delegates to `questMulticut` with one selected occurrence.","title":"Cut admissibility for a bang proposition"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/quest-cut-adm","kind":"node","lean":"Cslib.Logic.CLL.Proof.questCutAdm","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#quest-cut-adm","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Reverse the exponential cut orientation using `bangCutAdm`, duality, and exchange. This is a wrapper, not a recursive call that swaps premises at the same measure.","title":"Question-mark cut via the bang case, duality and exchange"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-principal-2","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutPrincipal","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-principal-2","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Reduce a pair of principal rules whose conclusions match the cut proposition. Every recursive call is on a proper subproposition of the original cut proposition, so termination is guaranteed by proposition size.","title":"Reduction of a cut between matching principal rules"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-adm-non-exponential","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutAdmNonExponential","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-adm-non-exponential","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Cut admissibility for non-exponential propositions: uses height induction on the sum of premise heights, with commuting cases for non-principal rules. Only callable when neither `a` nor `a\u2ae0` is a question mark.","title":"Cut admissibility for non-exponential propositions"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-adm-of-smaller","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutAdmOfSmaller","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-adm-of-smaller","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"The proposition-size induction step for cut admissibility. Dispatches to `bangCutAdm`, `questCutAdm`, or `cutAdmNonExponential` depending on the top-level connective.","title":"Proposition-size induction step for cut admissibility"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-adm-impl","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutAdmImpl","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-adm-impl","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"The core cut admissibility implementation: given decidable atom equality and explicit list sequents, combine cut-free proofs of dual distinguished propositions into a cut-free proof of the sum of their residual sequents. Recurses on proposition size.","title":"Core cut admissibility procedure on list sequents"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-adm","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutAdm","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-adm","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Public entry point for cut admissibility: given cut-free proofs of `a ::\u2098 \u0393` and `a\u2ae0 ::\u2098 \u0394`, produce a cut-free proof of `(\u0393 : Sequent) + \u0394`.","title":"Cut admissibility for cut-free classical linear logic proofs"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-elim-aux","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutElimAux","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-elim-aux","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Compute cut-free proofs by structural recursion on the original (possibly non-cut-free) proof. Each case recursively normalises the premises, then uses `cutAdm` at each cut. The list `\u0393` is the explicit sequent for the current subgoal.","title":"Cut elimination by structural recursion on the proof"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-elim-list","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutElimList","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-elim-list","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"List-based cut elimination with the concrete list presentation explicit.","title":"List-based cut elimination in the source calculus"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/cut-elim","kind":"node","lean":"Cslib.Logic.CLL.Proof.cutElim","parent":1022,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul/index.html#cut-elim","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/source-cut-elimination-for-the-locked-cll-calcul","status":"fully_proved","summary":"Cut elimination, using a noncomputable list presentation.","title":"Cut elimination with a noncomputable list presentation"},{"data_url":"scopes/Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el.json","id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el","kind":"scope","navigation":"module","parent":974,"result":false,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/index.html","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el","status":"fully_proved","title":"Canonical pinned API for the upstream CLL cut-elimination port"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/equiv-cut-free-proof","kind":"node","lean":"Cslib.Logic.CLL.CF.Proof.equivCutFreeProof","parent":1038,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/index.html#equiv-cut-free-proof","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el","status":"fully_proved","summary":"Source and pinned cut-free proof bundles contain the same proof values.","title":"Equivalence of source and pinned cut-free proof bundles"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/cut-elim-list","kind":"node","lean":"Cslib.Logic.CLL.Proof.cut_elim_list","parent":1038,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/index.html#cut-elim-list","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el","status":"fully_proved","summary":"Compute a cut-free original CLL proof using an explicit list presentation.","title":"Cut-free original proof from an explicit list presentation"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/cut-elim","kind":"node","lean":"Cslib.Logic.CLL.Proof.cut_elim","parent":1038,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/index.html#cut-elim","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el","status":"fully_proved","summary":"Every original CLL proof has a cut-free original proof of the same sequent. This wrapper uses classical atom equality and a classical list representative.","title":"Cut elimination for original classical linear logic proofs"},{"id":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/cut-adm-list","kind":"node","lean":"Cslib.Logic.CLL.CutFreeProof.cutAdmList","parent":1038,"result":true,"route":"../roadmap/Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el/index.html#cut-adm-list","scope":"Logic/proof-theory/cut-elimination-for-classical-linear-logic/canonical-pinned-api-for-the-upstream-cll-cut-el","status":"fully_proved","summary":"Cut admissibility in the existing pinned bundle, with explicit list tails.","title":"Cut admissibility in the pinned bundle with list tails"},{"data_url":"scopes/Logic/proof-theory/girard-s-translation-into-linear-logic.json","id":"Logic/proof-theory/girard-s-translation-into-linear-logic","kind":"scope","navigation":"module","parent":973,"result":false,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","title":"Girard's translation into linear logic"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/all-quest-cons-quest","kind":"node","lean":"Cslib.Logic.CLL.Sequent.allQuest_cons_quest","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#all-quest-cons-quest","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"For every CLL proposition $a$ and every CLL sequent $\\Delta$, every proposition of the sequent $?a, \\Delta$ has the form $?b$ for some $b$ if and only if every proposition of $\\Delta$ does.","title":"Adding a why-not proposition keeps a sequent all why-not"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/translate","kind":"node","lean":"Cslib.Logic.CLL.Girard.translate","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#translate","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Given a map $f$ from propositional atoms to CLL propositions, the translation $A \\mapsto A^\\circ$ of propositional formulas into CLL is defined by recursion: $a^\\circ = f(a)$ for an atom $a$, $(A \\wedge B)^\\circ = A^\\circ \\mathbin{\\&} B^\\circ$, $(A \\vee B)^\\circ = {!A^\\circ} \\oplus {!B^\\circ}$, and $(A \\to B)^\\circ$ is the multiplicative disjunction (par) of $?\\big((A^\\circ)^\\perp\\big)$ and $B^\\circ$, that is, the\u2026","title":"Girard's translation"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/translate-imp","kind":"node","lean":"Cslib.Logic.CLL.Girard.translate_imp","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#translate-imp","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"For every map $f$ from propositional atoms to CLL propositions and all propositional formulas $A$ and $B$, $(A \\to B)^\\circ = {!A^\\circ} \\multimap B^\\circ$, where $C^\\circ$ denotes Girard's translation of a formula $C$ with atoms translated by $f$ and $\\multimap$ is linear implication in CLL.","title":"Girard's translation of an implication"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/hyp","kind":"node","lean":"Cslib.Logic.CLL.Girard.hyp","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#hyp","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Given a map $f$ from atoms to CLL propositions and a propositional formula $A$, this is the CLL proposition $?((A^\\circ)^\\perp)$, where $A^\\circ$ is Girard's translation of $A$ with atoms translated by $f$ and $(\\cdot)^\\perp$ is linear negation.","title":"Girard's translation of a reusable hypothesis"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/hyps","kind":"node","lean":"Cslib.Logic.CLL.Girard.hyps","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#hyps","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Given a map $f$ from propositional atoms to CLL propositions and a multiset $\\Gamma$ of propositional formulas, $?\\Gamma^{\\circ\\perp}$ is the CLL sequent consisting of the propositions $?\\big((A^\\circ)^\\perp\\big)$ for $A \\in \\Gamma$, counted with multiplicity, where $A^\\circ$ is Girard's translation of $A$ with atoms translated by $f$.","title":"Translated hypotheses of a context"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/weaken-hyps","kind":"node","lean":"Cslib.Logic.CLL.Girard.weaken_hyps","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#weaken-hyps","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map propositional atoms to CLL propositions. If a CLL sequent $\\vdash \\Delta$ is provable, then for every multiset $\\Gamma$ of propositional formulas the sequent $\\vdash ?\\Gamma^{\\circ\\perp}, \\Delta$ is provable, where $?\\Gamma^{\\circ\\perp}$ is the multiset of the propositions $?\\big((A^\\circ)^\\perp\\big)$ for $A \\in \\Gamma$ and $A^\\circ$ is Girard's translation of $A$ with atoms translated by $f$.","title":"Weakening by translated hypotheses"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/contract-hyps","kind":"node","lean":"Cslib.Logic.CLL.Girard.contract_hyps","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#contract-hyps","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map atoms to CLL propositions, let $\\Gamma$ be a multiset of propositional formulas, and write $?\\Gamma^{\\circ\\perp}$ for the CLL sequent of the propositions $?((B^\\circ)^\\perp)$, one for each $B \\in \\Gamma$ counted with multiplicity, where $B^\\circ$ is Girard's translation of $B$ with atoms translated by $f$. For every CLL sequent $\\Delta$, if $\\vdash ?\\Gamma^{\\circ\\perp}, ?\\Gamma^{\\circ\\perp}, \\Delta$ is\u2026","title":"Contraction of a translated hypothesis context"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/bang-hyps","kind":"node","lean":"Cslib.Logic.CLL.Girard.bang_hyps","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#bang-hyps","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map atoms to CLL propositions, let $\\Gamma$ be a multiset of propositional formulas, and write $?\\Gamma^{\\circ\\perp}$ for the CLL sequent of the propositions $?((B^\\circ)^\\perp)$, one for each $B \\in \\Gamma$ counted with multiplicity, where $B^\\circ$ is Girard's translation of $B$ with atoms translated by $f$. For every CLL proposition $a$, if the sequent $\\vdash a, ?\\Gamma^{\\circ\\perp}$ is provable in CLL,\u2026","title":"Promotion in a translated hypothesis context"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/cut-closed","kind":"node","lean":"Cslib.Logic.CLL.Girard.cut_closed","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#cut-closed","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map atoms to CLL propositions, let $\\Gamma$ be a multiset of propositional formulas, and write $?\\Gamma^{\\circ\\perp}$ for the CLL sequent of the propositions $?((B^\\circ)^\\perp)$, one for each $B \\in \\Gamma$ counted with multiplicity, where $B^\\circ$ is Girard's translation of $B$ with atoms translated by $f$. For all CLL propositions $a$ and $c$, if $\\vdash a, ?\\Gamma^{\\circ\\perp}$ is provable in CLL and\u2026","title":"Cut of a translated hypothesis context against a closed proof"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/cut-hyps","kind":"node","lean":"Cslib.Logic.CLL.Girard.cut_hyps","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#cut-hyps","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map atoms to CLL propositions, let $\\Gamma$ be a multiset of propositional formulas, and write $?\\Gamma^{\\circ\\perp}$ for the CLL sequent of the propositions $?((B^\\circ)^\\perp)$, one for each $B \\in \\Gamma$ counted with multiplicity, where $B^\\circ$ is Girard's translation of $B$ with atoms translated by $f$. For all CLL propositions $a$ and $c$, if both $\\vdash a, ?\\Gamma^{\\circ\\perp}$ and\u2026","title":"Cut within a translated hypothesis context"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/hyp-mem","kind":"node","lean":"Cslib.Logic.CLL.Girard.hyp_mem","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#hyp-mem","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map atoms to CLL propositions, let $\\Gamma$ be a multiset of propositional formulas, and write $?\\Gamma^{\\circ\\perp}$ for the CLL sequent of the propositions $?((B^\\circ)^\\perp)$, one for each $B \\in \\Gamma$ counted with multiplicity, where $B^\\circ$ is Girard's translation of $B$ with atoms translated by $f$. For every propositional formula $A \\in \\Gamma$, the sequent $\\vdash A^\\circ, ?\\Gamma^{\\circ\\perp}$\u2026","title":"Translated hypotheses are provable from their context"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/of-insert","kind":"node","lean":"Cslib.Logic.CLL.Girard.of_insert","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#of-insert","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map propositional atoms to CLL propositions, let $\\Gamma$ be a finite set of propositional formulas, $A$ a propositional formula and $c$ a CLL proposition. Write $B^\\circ$ for Girard's translation of a formula $B$ with atoms translated by $f$, and $?\\Delta^{\\circ\\perp}$ for the multiset of the propositions $?\\big((B^\\circ)^\\perp\\big)$ for $B \\in \\Delta$. If the CLL sequent\u2026","title":"Separating a new hypothesis in the translation"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/derivable-of-derivation","kind":"node","lean":"Cslib.Logic.CLL.Girard.derivable_of_derivation","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#derivable-of-derivation","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map atoms to CLL propositions and write $B^\\circ$ for Girard's translation of a propositional formula $B$ with atoms translated by $f$. Let $T$ be a propositional theory such that the one-formula sequent $\\vdash B^\\circ$ is provable in CLL for every $B \\in T$. If $\\Gamma$ is a finite set of propositional formulas and $\\Gamma \\vdash A$ has a natural-deduction derivation (in minimal propositional logic) using\u2026","title":"Girard's translation preserves provability"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/mpl","kind":"node","lean":"Cslib.Logic.CLL.Girard.mpl","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#mpl","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"For every map $f$ from propositional atoms to CLL propositions, if a propositional formula $A$ is derivable in natural deduction from the empty context with no axioms (minimal logic), then the one-sided CLL sequent $\\vdash A^\\circ$ is provable, where $A^\\circ$ is Girard's translation of $A$ with atoms translated by $f$.","title":"Theorems of minimal logic translate to CLL theorems"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/bot-to-zero","kind":"node","lean":"Cslib.Logic.CLL.Girard.botToZero","parent":1043,"reading_title":{"format":"inline-parts-v1","parts":[["text","The atom map sending "],["math","\\(\\bot\\)"],["text"," to "],["math","\\(0\\)"]],"source":"The atom map sending $\\bot$ to $0$"},"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#bot-to-zero","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"For a type of atoms with a distinguished atom $\\bot$ and decidable equality, this is the map from atoms to CLL propositions that sends $\\bot$ to the additive unit $0$ and every other atom $a$ to the CLL atom $a$.","title":"The atom map sending $\\bot$ to $0$"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/efq","kind":"node","lean":"Cslib.Logic.CLL.Girard.efq","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#efq","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map atoms to CLL propositions and send the atom $\\bot$ to the additive unit $0$. Then for every propositional formula $B$, the one-formula CLL sequent $\\vdash (\\bot \\to B)^\\circ$ is provable, where $(\\cdot)^\\circ$ is Girard's translation with atoms translated by $f$.","title":"Ex falso translates to a CLL theorem"},{"id":"Logic/proof-theory/girard-s-translation-into-linear-logic/ipl","kind":"node","lean":"Cslib.Logic.CLL.Girard.ipl","parent":1043,"result":true,"route":"../roadmap/Logic/proof-theory/girard-s-translation-into-linear-logic/index.html#ipl","scope":"Logic/proof-theory/girard-s-translation-into-linear-logic","status":"fully_proved","summary":"Let $f$ map propositional atoms to CLL propositions and send the atom $\\bot$ to the additive unit $0$. If a propositional formula $A$ is derivable in natural deduction from the empty context in intuitionistic propositional logic (no axioms other than ex falso $\\bot \\to B$ for every formula $B$), then the one-sided CLL sequent $\\vdash A^\\circ$ is provable, where $A^\\circ$ is Girard's translation of $A$ with atoms\u2026","title":"Theorems of intuitionistic logic translate to CLL theorems"},{"data_url":"scopes/Logic/proof-theory/g-del-s-incompleteness-theorems.json","id":"Logic/proof-theory/g-del-s-incompleteness-theorems","kind":"scope","navigation":"module","parent":973,"result":false,"route":"../roadmap/Logic/proof-theory/g-del-s-incompleteness-theorems/index.html","scope":"Logic/proof-theory/g-del-s-incompleteness-theorems","status":"fully_proved","title":"G\u00f6del's incompleteness theorems"},{"id":"Logic/proof-theory/g-del-s-incompleteness-theorems/godel-rosser-abstract-schema","kind":"node","lean":"MetaMathlibExt.godel_rosser_abstract_schema","parent":1061,"result":true,"route":"../roadmap/Logic/proof-theory/g-del-s-incompleteness-theorems/index.html#godel-rosser-abstract-schema","scope":"Logic/proof-theory/g-del-s-incompleteness-theorems","status":"fully_proved","summary":"Let $\\mathcal{L}$ be a set of formulas with operations $\\neg$ and $\\to$, a substitution map sending $A \\in \\mathcal{L}$ and $n \\in \\mathbb{N}$ to $A[n] \\in \\mathcal{L}$, a coding map $\\ulcorner \\cdot \\urcorner : \\mathcal{L} \\to \\mathbb{N}$, a formula $R \\in \\mathcal{L}$, predicates \"provable\" (written $\\vdash A$), \"Rosser provable\" and \"true\" on $\\mathcal{L}$, and a proof relation $\\mathrm{Prf}(A, n)$ between\u2026","title":"Abstract G\u00f6del\u2013Rosser incompleteness theorem"},{"id":"Logic/proof-theory/g-del-s-incompleteness-theorems/hilbert-bernays-lob-second-incompletenes","kind":"node","lean":"MetaMathlibExt.hilbert_bernays_lob_second_incompleteness","parent":1061,"result":true,"route":"../roadmap/Logic/proof-theory/g-del-s-incompleteness-theorems/index.html#hilbert-bernays-lob-second-incompletenes","scope":"Logic/proof-theory/g-del-s-incompleteness-theorems","status":"fully_proved","summary":"Let $\\mathcal{L}$ be a set of sentences with an implication $\\to$, a sentence $\\bot$, a negation satisfying $\\neg A = (A \\to \\bot)$ for all $A$, and a provability operator $\\Box : \\mathcal{L} \\to \\mathcal{L}$. Let $U \\vdash A$ be a relation between theories $U \\subseteq \\mathcal{L}$ and sentences $A \\in \\mathcal{L}$ such that for every theory $U$ and all sentences $A$, $B$, $C$: $U \\vdash A \\to (B \\to A)$,\u2026","title":"G\u00f6del's second incompleteness theorem, abstract Hilbert\u2013Bernays\u2013L\u00f6b form"},{"data_url":"scopes/Logic/model-theory.json","id":"Logic/model-theory","kind":"scope","parent":1,"result":false,"route":"../roadmap/Logic/model-theory/index.html","scope":"Logic/model-theory","status":"can_state","summary":"Relations between formal theories and their structures: first-order languages of algebraic structures, quantifier elimination, categoricity and o-minimality.","title":"Model theory"},{"id":"Logic/model-theory/morley-categoricity-theorem","kind":"node","lean":"MathlibExt.ModelTheory.MorleyCategoricityWanted.ModelTheory.morley_categoricity_theorem","parent":1064,"result":true,"route":"../roadmap/Logic/model-theory/index.html#morley-categoricity-theorem","scope":"Logic/model-theory","status":"can_prove","summary":"Let $L$ be a first-order language with at most countably many symbols, and let $T$ be a complete $L$-theory all of whose models are infinite. Call $T$ $\\kappa$-categorical if any two models of $T$ of cardinality $\\kappa$ are isomorphic. If $T$ is $\\kappa$-categorical for some uncountable cardinal $\\kappa$, then $T$ is $\\mu$-categorical for every uncountable cardinal $\\mu$.","title":"Morley's categoricity theorem"},{"data_url":"scopes/Logic/model-theory/first-order-language-of-groups.json","id":"Logic/model-theory/first-order-language-of-groups","kind":"scope","navigation":"module","parent":1064,"result":false,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","title":"First-order language of groups"},{"id":"Logic/model-theory/first-order-language-of-groups/group-function","kind":"node","lean":"FirstOrder.Language.GroupFunction","parent":1066,"result":true,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html#group-function","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","summary":"The function symbols of the first-order language of groups, graded by arity, are a constant symbol for the identity (arity $0$), a symbol for inversion (arity $1$) and a symbol for multiplication (arity $2$); there are no others.","title":"Group function symbols"},{"id":"Logic/model-theory/first-order-language-of-groups/group-language","kind":"node","lean":"FirstOrder.Language.groupLanguage","parent":1066,"result":true,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html#group-language","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","summary":"The first-order language of groups has the group function symbols (a constant for the identity, a unary symbol for inversion and a binary symbol for multiplication) as its function symbols and no relation symbols.","title":"The first-order language of groups"},{"id":"Logic/model-theory/first-order-language-of-groups/group-language-structure","kind":"node","lean":"FirstOrder.Language.groupLanguageStructure","parent":1066,"result":true,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html#group-language-structure","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","summary":"For a group $G$, the canonical structure on $G$ for the first-order language of groups interprets the identity constant as the identity $1$ of $G$, inversion as $x \\mapsto x^{-1}$ and multiplication as $(x, y) \\mapsto xy$; there are no relation symbols to interpret.","title":"The canonical structure on a group"},{"id":"Logic/model-theory/first-order-language-of-groups/group-language-functions","kind":"node","lean":"FirstOrder.Language.groupLanguage_functions","parent":1066,"result":true,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html#group-language-functions","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the $n$-ary function symbols of the first-order language of groups are exactly the group function symbols of arity $n$.","title":"The function symbols of the language of groups"},{"id":"Logic/model-theory/first-order-language-of-groups/group-language-relations","kind":"node","lean":"FirstOrder.Language.groupLanguage_relations","parent":1066,"result":true,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html#group-language-relations","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the first-order language of groups has no $n$-ary relation symbols.","title":"The language of groups has no relation symbols"},{"id":"Logic/model-theory/first-order-language-of-groups/group-language-structure-mul","kind":"node","lean":"FirstOrder.Language.groupLanguageStructure_mul","parent":1066,"result":true,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html#group-language-structure-mul","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","summary":"For every group $G$ and every pair $(x_0, x_1)$ of elements of $G$, the multiplication symbol of the language of groups, interpreted in the canonical structure on $G$, sends $(x_0, x_1)$ to $x_0 x_1$.","title":"Multiplication in the canonical structure"},{"id":"Logic/model-theory/first-order-language-of-groups/group-language-structure-inv","kind":"node","lean":"FirstOrder.Language.groupLanguageStructure_inv","parent":1066,"result":true,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html#group-language-structure-inv","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","summary":"For every group $G$ and every one-element tuple $(x_0)$ of elements of $G$, the inversion symbol of the language of groups, interpreted in the canonical structure on $G$, sends $(x_0)$ to $x_0^{-1}$.","title":"Inversion in the canonical structure"},{"id":"Logic/model-theory/first-order-language-of-groups/group-language-structure-one","kind":"node","lean":"FirstOrder.Language.groupLanguageStructure_one","parent":1066,"result":true,"route":"../roadmap/Logic/model-theory/first-order-language-of-groups/index.html#group-language-structure-one","scope":"Logic/model-theory/first-order-language-of-groups","status":"fully_proved","summary":"For every group $G$, the identity constant symbol of the language of groups, interpreted in the canonical structure on $G$, takes the value $1$ at the empty tuple, where $1$ is the identity of $G$.","title":"The identity in the canonical structure"},{"data_url":"scopes/Logic/model-theory/quantifier-elimination-for-real-closed-fields.json","id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","kind":"scope","navigation":"module","parent":1064,"result":false,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"can_state","title":"Quantifier elimination for real closed fields"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/term","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Term","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#term","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"Terms are built from variables $x_i$, one for each natural number $i$ (a de Bruijn index), and integer constants $k \\in \\mathbb{Z}$, using addition $s + t$, multiplication $s \\cdot t$ and negation $-t$.","title":"Terms of the language of ordered rings"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/formula","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Formula","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#formula","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"Formulas are built from the atomic formulas $s < t$ and $s = t$, for terms $s$ and $t$, and falsity $\\bot$, by implication $\\varphi \\to \\psi$ and universal quantification $\\forall \\varphi$, which binds de Bruijn index $0$ in $\\varphi$. Negation, truth, disjunction, conjunction and existential quantification are defined from these.","title":"Formulas of the language of ordered rings"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/not","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Formula.not","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#not","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"The negation $\\neg\\varphi$ of a formula $\\varphi$ is the formula $\\varphi \\to \\bot$.","title":"Negation of a formula"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/tru","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Formula.tru","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#tru","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"The formula $\\top$ is the negation $\\neg\\bot$ of falsity, that is, the formula $\\bot \\to \\bot$.","title":"The true formula"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/or","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Formula.or","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#or","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"The disjunction $\\varphi \\lor \\psi$ of formulas $\\varphi$ and $\\psi$ is the formula $\\neg\\varphi \\to \\psi$.","title":"Disjunction of formulas"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/and","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Formula.and","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#and","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"For formulas $\\varphi$ and $\\psi$, the conjunction $\\varphi \\wedge \\psi$ is the formula $\\neg(\\varphi \\to \\neg\\psi)$.","title":"Conjunction of formulas"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/ex","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Formula.ex","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#ex","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"For a formula $\\varphi$, the formula $\\exists \\varphi$ is $\\neg \\forall \\neg \\varphi$; it binds de Bruijn index $0$ in $\\varphi$.","title":"Existential quantification of formulas"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/cons","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.cons","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#cons","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"An environment is a map $e : \\mathbb{N} \\to \\mathbb{R}$ that assigns a real number to each de Bruijn index. For $x \\in \\mathbb{R}$, the extension of $e$ by $x$ sends index $0$ to $x$ and index $i + 1$ to $e(i)$ for every $i \\in \\mathbb{N}$.","title":"Extending an environment"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/eval","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Term.eval","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#eval","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"Given an environment $e : \\mathbb{N} \\to \\mathbb{R}$, the value of a term in $\\mathbb{R}$ is defined by recursion on the term: the variable with de Bruijn index $i$ has value $e(i)$, the integer constant $k$ has value $k$, and the value of a sum, product or negation of terms is the sum, product or negation in $\\mathbb{R}$ of the values of its parts.","title":"Evaluation of terms"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/holds","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Formula.Holds","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#holds","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"Let $e : \\mathbb{N} \\to \\mathbb{R}$ be an environment. Whether a formula holds under $e$ is defined by recursion on the formula: $s < t$ holds when the value of $s$ under $e$ is less than the value of $t$ under $e$; $s = t$ holds when these values are equal; $\\bot$ never holds; $\\varphi \\to \\psi$ holds when $\\varphi$ holding under $e$ implies $\\psi$ holding under $e$; and $\\forall \\varphi$ holds when, for every\u2026","title":"Satisfaction of formulas over the reals"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/is-qf","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.Formula.IsQF","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#is-qf","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"fully_proved","summary":"A formula is quantifier free if it contains no universal quantifier $\\forall$.","title":"Quantifier-free formulas"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/qe","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.qe","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#qe","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"can_state","summary":"Quantifier elimination is a map sending each formula $\\varphi$ of the language of ordered rings to a formula $\\operatorname{qe}(\\varphi)$, subject to two requirements: $\\operatorname{qe}(\\varphi)$ is quantifier free, and for every environment $e : \\mathbb{N} \\to \\mathbb{R}$ assigning a real number to each variable, $\\operatorname{qe}(\\varphi)$ holds in $\\mathbb{R}$ under $e$ if and only if $\\varphi$ does.","title":"Quantifier elimination over the reals"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/is-qf-qe","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.isQF_qe","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#is-qf-qe","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"stated","summary":"Let $\\mathrm{qe}$ be the quantifier elimination map on formulas. For every formula $\\varphi$, the formula $\\mathrm{qe}(\\varphi)$ is quantifier free.","title":"Quantifier elimination yields quantifier-free formulas"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/holds-qe","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.holds_qe","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#holds-qe","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"stated","summary":"Let $\\mathrm{qe}$ be the quantifier elimination map on formulas. For every formula $\\varphi$ and every environment $e : \\mathbb{N} \\to \\mathbb{R}$, the formula $\\mathrm{qe}(\\varphi)$ holds in $\\mathbb{R}$ under $e$ if and only if $\\varphi$ holds in $\\mathbb{R}$ under $e$.","title":"Quantifier elimination preserves truth"},{"id":"Logic/model-theory/quantifier-elimination-for-real-closed-fields/holds-ex-sq","kind":"node","lean":"MathlibExt.TheoryOfComputation.RealClosedFieldQEWanted.ProgramVerification.RealClosedFieldQE.holds_ex_sq","parent":1075,"result":true,"route":"../roadmap/Logic/model-theory/quantifier-elimination-for-real-closed-fields/index.html#holds-ex-sq","scope":"Logic/model-theory/quantifier-elimination-for-real-closed-fields","status":"can_prove","summary":"For every environment $e : \\mathbb{N} \\to \\mathbb{R}$, the formula $\\exists x.\\ x \\cdot x = a$ holds under $e$ if and only if the formula $\\neg(a < 0)$ holds under $e$. In the first formula the bound variable $x$ is de Bruijn index $0$ and $a$ is index $1$; in the second $a$ is index $0$; in both, $a$ denotes $e(0)$. Thus $e(0)$ is a square in $\\mathbb{R}$ exactly when $e(0) \\ge 0$.","title":"A real number is a square exactly when it is nonnegative"},{"data_url":"scopes/Logic/model-theory/wilkie-s-conjecture.json","id":"Logic/model-theory/wilkie-s-conjecture","kind":"scope","navigation":"module","parent":1064,"result":false,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html","scope":"Logic/model-theory/wilkie-s-conjecture","status":"can_prove","title":"Wilkie's conjecture"},{"id":"Logic/model-theory/wilkie-s-conjecture/exp-func","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.expFunc","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#exp-func","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"The function symbols of the language of exponentiation, sorted by arity: there is exactly one, the unary symbol $\\exp$, and there are none of any other arity.","title":"Function symbols of the language of exponentiation"},{"id":"Logic/model-theory/wilkie-s-conjecture/exp","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.Language.exp","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#exp","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"The first-order language with a single function symbol, the unary symbol $\\exp$, and no relation symbols.","title":"Language of exponentiation"},{"id":"Logic/model-theory/wilkie-s-conjecture/exp-ring","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.expRing","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#exp-ring","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"The first-order language whose function symbols are those of the language of rings, namely the binary symbols $+$ and $\\cdot$, the unary symbol $-$ and the constants $0$ and $1$, together with the unary symbol $\\exp$ of the language of exponentiation; it has no relation symbols.","title":"Language of rings with exponentiation"},{"id":"Logic/model-theory/wilkie-s-conjecture/is-semialgebraic","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.IsSemialgebraic","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#is-semialgebraic","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"A set $X \\subseteq \\mathbb{R}^n$ is semialgebraic if it is definable, with parameters from $\\mathbb{R}$, in the real numbers viewed as a structure for the language of rings.","title":"Semialgebraic set"},{"id":"Logic/model-theory/wilkie-s-conjecture/is-rexp-definable","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.IsRExpDefinable","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#is-rexp-definable","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"A set $X \\subseteq \\mathbb{R}^n$ is definable in $\\mathbb{R}_{\\exp}$ if it is definable, with parameters from $\\mathbb{R}$, in the real numbers viewed as a structure for the language of rings with exponentiation, where the ring symbols have their usual meaning and $\\exp$ is interpreted as the real exponential function.","title":"Definable sets of the real exponential field"},{"id":"Logic/model-theory/wilkie-s-conjecture/alg-part","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.algPart","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#alg-part","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"For a set $X \\subseteq \\mathbb{R}^n$, the algebraic part $X^{\\mathrm{alg}}$ of $X$ is the union of all connected, infinite semialgebraic sets $Y$ with $Y \\subseteq X$.","title":"Algebraic part of a set"},{"id":"Logic/model-theory/wilkie-s-conjecture/trans-part","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.transPart","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#trans-part","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"For a set $X \\subseteq \\mathbb{R}^n$, the transcendental part of $X$ is $X^{\\mathrm{trans}} := X \\setminus X^{\\mathrm{alg}}$, where $X^{\\mathrm{alg}}$ is the algebraic part of $X$.","title":"Transcendental part of a set"},{"id":"Logic/model-theory/wilkie-s-conjecture/height","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.height","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#height","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"For $a = (a_1, \\dots, a_n) \\in \\mathbb{R}^n$, the height $H(a)$ is the maximum of the absolute multiplicative Weil heights of the coordinates $a_1, \\dots, a_n$. A coordinate that is not algebraic over $\\mathbb{Q}$ counts with height $1$, and for $n = 0$ the height is $0$.","title":"Height of a point"},{"id":"Logic/model-theory/wilkie-s-conjecture/degree-height-le","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.degreeHeightLE","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#degree-height-le","scope":"Logic/model-theory/wilkie-s-conjecture","status":"fully_proved","summary":"For a set $X \\subseteq \\mathbb{R}^n$ and natural numbers $g$ and $H$, the set $X(g, H)$ consists of the points $x = (x_1, \\dots, x_n) \\in X$ whose coordinates are all algebraic over $\\mathbb{Q}$, such that the field $\\mathbb{Q}(x) = \\mathbb{Q}(x_1, \\dots, x_n)$ satisfies $[\\mathbb{Q}(x) : \\mathbb{Q}] \\le g$ and the height $H(x)$ of $x$, the maximum of the absolute multiplicative Weil heights of $x_1, \\dots, x_n$\u2026","title":"Points of bounded degree and height"},{"id":"Logic/model-theory/wilkie-s-conjecture/corollary-1","kind":"node","lean":"MathlibExt.ModelTheory.WilkiesConjectureWanted.WilkiesConjecture.corollary_1","parent":1091,"result":true,"route":"../roadmap/Logic/model-theory/wilkie-s-conjecture/index.html#corollary-1","scope":"Logic/model-theory/wilkie-s-conjecture","status":"can_prove","summary":"Let $X \\subseteq \\mathbb{R}^n$ be definable in $\\mathbb{R}_{\\exp}$. Then there is a polynomial $f$ in two variables with real coefficients such that the following holds for all natural numbers $g$ and $H$. Let $X^{\\mathrm{trans}}(g, H)$ be the set of points $x = (x_1, \\dots, x_n)$ of the transcendental part $X^{\\mathrm{trans}}$ of $X$ whose coordinates are all algebraic over $\\mathbb{Q}$, such that the field\u2026","title":"Wilkie's conjecture for the real exponential field"},{"area":"Foundations & Logic","data_url":"chapters/SetTheory.json","id":"SetTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/SetTheory/index.html","scope":"SetTheory","status":"can_prove","summary":"Set theory studies membership, cardinality, ordinals, infinity, and foundational constructions for mathematics.","title":"Set Theory"},{"data_url":"scopes/SetTheory/cardinal-arithmetic.json","id":"SetTheory/cardinal-arithmetic","kind":"scope","parent":1102,"result":false,"route":"../roadmap/SetTheory/cardinal-arithmetic/index.html","scope":"SetTheory/cardinal-arithmetic","status":"fully_proved","summary":"The arithmetic of infinite cardinals, including the continuum hypothesis and the singular cardinals hypothesis, together with cardinal invariants of the continuum and the cofinal structure of directed sets.","title":"Cardinal arithmetic and cardinal characteristics"},{"id":"SetTheory/cardinal-arithmetic/singular-cardinals-hypothesis","kind":"node","lean":"MetaMathlibExt.SingularCardinalsHypothesis","parent":1103,"result":true,"route":"../roadmap/SetTheory/cardinal-arithmetic/index.html#singular-cardinals-hypothesis","scope":"SetTheory/cardinal-arithmetic","status":"fully_proved","summary":"For all singular cardinal $\\kappa$, if $2^{cof(\\kappa)}\\lt \\kappa, then $\\kappa^{cof(\\kappa)} = \\kappa^+$.","title":"Singular cardinals hypothesis"},{"data_url":"scopes/SetTheory/cardinal-arithmetic/bounding-number.json","id":"SetTheory/cardinal-arithmetic/bounding-number","kind":"scope","navigation":"module","parent":1103,"result":false,"route":"../roadmap/SetTheory/cardinal-arithmetic/bounding-number/index.html","scope":"SetTheory/cardinal-arithmetic/bounding-number","status":"fully_proved","title":"Bounding number"},{"id":"SetTheory/cardinal-arithmetic/bounding-number/is-unbounded-family","kind":"node","lean":"MetaMathlibExt.IsUnboundedFamily","parent":1105,"result":true,"route":"../roadmap/SetTheory/cardinal-arithmetic/bounding-number/index.html#is-unbounded-family","scope":"SetTheory/cardinal-arithmetic/bounding-number","status":"fully_proved","summary":"A family $F$ of functions $\\mathbb{N} \\to \\mathbb{N}$ is unbounded if no function $g : \\mathbb{N} \\to \\mathbb{N}$ bounds every member of $F$, where $g$ bounds $f$ if there is an $N$ such that $f(n) \\le g(n)$ for all $n \\ge N$.","title":"Unbounded family"},{"id":"SetTheory/cardinal-arithmetic/bounding-number/bounding-number","kind":"node","lean":"MetaMathlibExt.boundingNumber","parent":1105,"result":true,"route":"../roadmap/SetTheory/cardinal-arithmetic/bounding-number/index.html#bounding-number","scope":"SetTheory/cardinal-arithmetic/bounding-number","status":"fully_proved","summary":"The bounding number $\\mathfrak{b}$ is the infimum of the cardinalities of all unbounded families of functions $\\mathbb{N} \\to \\mathbb{N}$.","title":"Bounding number"},{"data_url":"scopes/SetTheory/cardinal-arithmetic/continuum-hypothesis.json","id":"SetTheory/cardinal-arithmetic/continuum-hypothesis","kind":"scope","navigation":"module","parent":1103,"result":false,"route":"../roadmap/SetTheory/cardinal-arithmetic/continuum-hypothesis/index.html","scope":"SetTheory/cardinal-arithmetic/continuum-hypothesis","status":"fully_proved","title":"Continuum hypothesis"},{"id":"SetTheory/cardinal-arithmetic/continuum-hypothesis/continuum-hypothesis","kind":"node","lean":"MetaMathlibExt.ContinuumHypothesis","parent":1108,"result":true,"route":"../roadmap/SetTheory/cardinal-arithmetic/continuum-hypothesis/index.html#continuum-hypothesis","scope":"SetTheory/cardinal-arithmetic/continuum-hypothesis","status":"fully_proved","summary":"CH: 2^{\u2135\u2080}=\u2135\u2081","title":"Continuum hypothesis"},{"id":"SetTheory/cardinal-arithmetic/continuum-hypothesis/equivalent-forms-of-the-continuum-hypoth","kind":"node","lean":"MetaMathlibExt.continuumHypothesis_iff_continuum_eq_aleph_one","parent":1108,"result":true,"route":"../roadmap/SetTheory/cardinal-arithmetic/continuum-hypothesis/index.html#equivalent-forms-of-the-continuum-hypoth","scope":"SetTheory/cardinal-arithmetic/continuum-hypothesis","status":"fully_proved","summary":"1. The continuum hypothesis holds if and only if the continuum equals \u2135\u2081. 2. The continuum hypothesis holds if and only if the continuum is at most \u2135\u2081. 3. The continuum hypothesis holds if and only if the reals have cardinality \u2135\u2081, i.e. |\u211d| = \u2135\u2081. 4. The continuum hypothesis holds if and only if there is no cardinal strictly between \u2135\u2080 and the continuum. 5.","title":"Equivalent forms of the continuum hypothesis"},{"data_url":"scopes/SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types.json","id":"SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types","kind":"scope","navigation":"module","parent":1103,"result":false,"route":"../roadmap/SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types/index.html","scope":"SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types","status":"fully_proved","title":"Tukey reducibility and cofinal types"},{"id":"SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types/tukey-le","kind":"node","lean":"MetaMathlibExt.TukeyLE","parent":1111,"result":true,"route":"../roadmap/SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types/index.html#tukey-le","scope":"SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types","status":"fully_proved","summary":"For preordered sets $D$ and $E$, $D$ is Tukey reducible to $E$, written $D \\le_T E$, if there is a map $f \\colon D \\to E$ that carries unbounded sets to unbounded sets: for every $A \\subseteq D$ with no upper bound in $D$, the image $f[A]$ has no upper bound in $E$.","title":"Tukey reducibility"},{"id":"SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types/tukey-equiv","kind":"node","lean":"MetaMathlibExt.TukeyEquiv","parent":1111,"result":true,"route":"../roadmap/SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types/index.html#tukey-equiv","scope":"SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types","status":"fully_proved","summary":"Preordered sets $D$ and $E$ are Tukey equivalent (cofinally equivalent), written $D \\equiv_T E$, if $D \\le_T E$ and $E \\le_T D$, where $\\le_T$ is Tukey reducibility.","title":"Tukey equivalence"},{"id":"SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types/cofinal-type","kind":"node","lean":"MetaMathlibExt.CofinalType","parent":1111,"result":true,"route":"../roadmap/SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types/index.html#cofinal-type","scope":"SetTheory/cardinal-arithmetic/tukey-reducibility-and-cofinal-types","status":"fully_proved","summary":"A cofinal type is an equivalence class of preordered sets under Tukey equivalence $\\equiv_T$, where $D \\equiv_T E$ means $D \\le_T E$ and $E \\le_T D$ (Tukey reducibility in both directions).","title":"Cofinal type"},{"data_url":"scopes/SetTheory/large-cardinals.json","id":"SetTheory/large-cardinals","kind":"scope","parent":1102,"result":false,"route":"../roadmap/SetTheory/large-cardinals/index.html","scope":"SetTheory/large-cardinals","status":"fully_proved","summary":"Cardinals with strong closure or reflection properties, such as inaccessible, Mahlo, weakly compact and measurable cardinals, whose existence cannot be proved in ZFC.","title":"Large cardinals"},{"id":"SetTheory/large-cardinals/measurable-cardinal","kind":"node","lean":"MetaMathlibExt.IsMeasurable","parent":1115,"result":true,"route":"../roadmap/SetTheory/large-cardinals/index.html#measurable-cardinal","scope":"SetTheory/large-cardinals","status":"fully_proved","summary":"A cardinal \u03ba is *measurable* if \u03ba > \u2135\u2080 and there is a set I with |I| = \u03ba carrying an ultrafilter U on I which is","title":"Measurable cardinal"},{"id":"SetTheory/large-cardinals/is-weakly-compact","kind":"node","lean":"MetaMathlibExt.IsWeaklyCompact","parent":1115,"result":true,"route":"../roadmap/SetTheory/large-cardinals/index.html#is-weakly-compact","scope":"SetTheory/large-cardinals","status":"fully_proved","summary":"A cardinal $\\kappa$ is *weakly compact* if $\\aleph_0 < \\kappa$ and $\\kappa \\to (\\kappa)^2_2$: for every set $I$ of cardinality $\\kappa$ and every colouring $c$ of the finite subsets of $I$ with two colours, there are a subset $H \\subseteq I$ of cardinality $\\kappa$ and a colour $k$ such that $c(s) = k$ for every two-element subset $s \\subseteq H$.","title":"Weakly compact cardinal"},{"data_url":"scopes/SetTheory/large-cardinals/inaccessible-cardinals.json","id":"SetTheory/large-cardinals/inaccessible-cardinals","kind":"scope","navigation":"module","parent":1115,"result":false,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","title":"Inaccessible cardinals"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/weakly-inaccessible-cardinal","kind":"node","lean":"MetaMathlibExt.IsWeaklyInaccessible","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#weakly-inaccessible-cardinal","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"A cardinal \u03ba is *weakly inaccessible* if it is an uncountable, regular limit cardinal.","title":"Weakly inaccessible cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/set-of-inaccessible-cardinals-below-a-ca","kind":"node","lean":"MetaMathlibExt.inaccessibleBelow","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#set-of-inaccessible-cardinals-below-a-ca","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"InAccBelow \u03ba is the set of all inaccessible cardinals below \u03ba.","title":"Set of inaccessible cardinals below a cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/inaccessible-fixed-point","kind":"node","lean":"MetaMathlibExt.IsInaccessibleFixedPoint","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#inaccessible-fixed-point","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"A cardinal \u03ba is an *inaccessible fixed point* if \u03ba is inaccessible and the inaccessibles below \u03ba have order type \u03ba (i.e. \u03ba is the \u03ba-th inaccessible).","title":"Inaccessible fixed point"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/the-beth-hierarchy-stays-below-an-inacce","kind":"node","lean":"Cardinal.IsInaccessible.beth_lt","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#the-beth-hierarchy-stays-below-an-inacce","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"Below an inaccessible `\u03ba`, the beth hierarchy stays below `\u03ba`: `\u03ba` is a beth fixed point (`IsInaccessible.beth_ord`) and `beth` is strictly monotone.","title":"The beth hierarchy stays below an inaccessible cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/the-aleph-hierarchy-stays-below-an-inacc","kind":"node","lean":"MetaMathlibExt.Cardinal.IsInaccessible.aleph_lt","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#the-aleph-hierarchy-stays-below-an-inacc","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"Below an inaccessible `\u03ba`, the aleph hierarchy stays below `\u03ba`: same argument via `IsInaccessible.aleph_ord`.","title":"The aleph hierarchy stays below an inaccessible cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/at-most-kappa-inaccessible-cardinals-lie","kind":"node","lean":"MetaMathlibExt.mk_inaccessibleBelow_le","parent":1118,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most "],["math","\\(\\kappa\\)"],["text"," inaccessible cardinals lie below "],["math","\\(\\kappa\\)"]],"source":"At most $\\kappa$ inaccessible cardinals lie below $\\kappa$"},"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#at-most-kappa-inaccessible-cardinals-lie","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"There are at most `\u03ba` inaccessibles below `\u03ba`: the set injects into `\u21a5(Set.Iio \u03ba.ord)`, which has size `lift \u03ba`. The `\u2264` half of the counting needed for the Mahlo-implies-fixed-point benchmark item.","title":"At most $\\kappa$ inaccessible cardinals lie below $\\kappa$"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/regular-cardinals-are-unbounded-below-an","kind":"node","lean":"MetaMathlibExt.exists_regular_between","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#regular-cardinals-are-unbounded-below-an","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"Regular cardinals are unbounded below an uncountable limit cardinal `\u03ba` (in particular a weakly inaccessible one): above any `a < \u03ba` sits `succ (max a \u2135\u2080)`, regular by `isRegular_succ` and below `\u03ba` since `\u03ba` is a limit.","title":"Regular cardinals are unbounded below an uncountable limit cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/under-gch-weakly-inaccessible-cardinals","kind":"node","lean":"MetaMathlibExt.IsWeaklyInaccessible.isInaccessible_of_gch","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#under-gch-weakly-inaccessible-cardinals","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"Under (local) GCH, weakly inaccessible implies strongly inaccessible: for infinite `\u03bc < \u03ba`, `2 ^ \u03bc = succ \u03bc < \u03ba` by the limit property, and the finite case is `power_lt_aleph0`.","title":"Under GCH, weakly inaccessible cardinals are inaccessible"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/strong-limit-cardinals-are-closed-under","kind":"node","lean":"Cardinal.IsStrongLimit.power_lt","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#strong-limit-cardinals-are-closed-under","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"A strong limit `\u03ba` is closed under cardinal exponentiation: `x ^ y \u2264 2 ^ (x * y)` (via `cantor` and `power_mul`), and `x * y < \u03ba` by `mul_lt_of_lt`. No infiniteness hypothesis is needed \u2014 it is automatic by `IsStrongLimit.aleph0_le`.","title":"Strong limit cardinals are closed under exponentiation"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/existence-of-an-inaccessible-with-no-ina","kind":"node","lean":"MetaMathlibExt.exists_isInaccessible_inaccessibleBelow_eq_empty","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#existence-of-an-inaccessible-with-no-ina","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"If any inaccessible exists, some inaccessible has no inaccessibles below it: minimize over the initial ordinals of inaccessible cardinals (`WellFounded.has_min`), and any member of `inaccessibleBelow` would give a smaller one.","title":"Existence of an inaccessible with no inaccessibles below it"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/weak-power-below-a-strong-limit-cardinal","kind":"node","lean":"MetaMathlibExt.Cardinal.IsStrongLimit.powerlt_eq_self","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#weak-power-below-a-strong-limit-cardinal","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"For `2 \u2264 x < \u03ba` with `\u03ba` a strong limit, `x ^< \u03ba = \u03ba`: every smaller power is below `\u03ba` (`IsStrongLimit.power_lt`), while every `y < \u03ba` sits below `2 ^ y \u2264 x ^< \u03ba`. No infiniteness hypothesis is needed \u2014 it is automatic by `IsStrongLimit.aleph0_le`.","title":"Weak power below a strong limit cardinal equals the cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/strong-limits-are-unbounded-below-an-ina","kind":"node","lean":"MetaMathlibExt.Cardinal.IsInaccessible.exists_isStrongLimit_between","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#strong-limits-are-unbounded-below-an-ina","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"Strong limits are unbounded below an inaccessible `\u03ba`: above any `a < \u03ba` sits `\u2136_{a.ord + \u03c9}`, a strong limit by `isStrongLimit_beth` since `a.ord + \u03c9` is a limit ordinal (`isSuccLimit_add`), and below `\u03ba` since `(a.ord + \u03c9).card = a.ord.card + \u2135\u2080 < \u03ba` (`add_lt_of_lt`).","title":"Strong limits are unbounded below an inaccessible cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/set-of-regular-cardinals-below-a-cardina","kind":"node","lean":"MetaMathlibExt.regularBelow","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#set-of-regular-cardinals-below-a-cardina","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"The regular cardinals below `\u03ba`, carried as initial ordinals \u2014 parallel to `inaccessibleBelow`, and definitionally the stationary set in `IsWeaklyMahlo`. Spelled with Mathlib's `Ordinal.IsInitial`.","title":"Set of regular cardinals below a cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/a-regular-beth-fixed-point-is-uncountabl","kind":"node","lean":"MetaMathlibExt.aleph0_lt_of_isRegular_beth_ord","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#a-regular-beth-fixed-point-is-uncountabl","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"A regular beth fixed point is uncountable: regularity gives `\u2135\u2080 \u2264 \u03ba`, and `\u03ba = \u2135\u2080` fails the fixed-point equation since `\u2136_\u03c9 > \u2136_0 = \u2135\u2080`.","title":"A regular beth fixed point is uncountable"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/a-regular-beth-fixed-point-is-inaccessib","kind":"node","lean":"MetaMathlibExt.Cardinal.IsInaccessible.of_isRegular_beth_ord","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#a-regular-beth-fixed-point-is-inaccessib","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"Converse to `IsInaccessible.beth_ord`: a regular beth fixed point is inaccessible. Uncountability is derived, not assumed (`aleph0_lt_of_isRegular_beth_ord`); for `x < \u03ba`, `2 ^ x \u2264 2 ^ \u2136_{x.ord} = \u2136_{succ x.ord} < \u03ba`, using that `\u03ba.ord` is a limit ordinal.","title":"A regular beth fixed point is inaccessible"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/inaccessible-cardinals-are-the-regular-b","kind":"node","lean":"MetaMathlibExt.Cardinal.isInaccessible_iff_isRegular_beth_ord","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#inaccessible-cardinals-are-the-regular-b","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"`\u03ba` is inaccessible iff it is regular and a beth fixed point (uncountability is automatic by `aleph0_lt_of_isRegular_beth_ord`): this packages `of_isRegular_beth_ord` with Mathlib's `IsInaccessible.beth_ord`.","title":"Inaccessible cardinals are the regular beth fixed points"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/exactly-kappa-inaccessibles-lie-below-an","kind":"node","lean":"MetaMathlibExt.IsInaccessibleFixedPoint.mk_inaccessibleBelow","parent":1118,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exactly "],["math","\\(\\kappa\\)"],["text"," inaccessibles lie below an inaccessible fixed point"]],"source":"Exactly $\\kappa$ inaccessibles lie below an inaccessible fixed point"},"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#exactly-kappa-inaccessibles-lie-below-an","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"The fixed-point equation forces `#(inaccessibleBelow \u03ba) = lift \u03ba` (`card_type`): the companion to the upper bound `mk_inaccessibleBelow_le`.","title":"Exactly $\\kappa$ inaccessibles lie below an inaccessible fixed point"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/inaccessibles-are-unbounded-below-an-ina","kind":"node","lean":"MetaMathlibExt.IsInaccessibleFixedPoint.exists_gt","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#inaccessibles-are-unbounded-below-an-ina","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"Inaccessibles below an inaccessible fixed point are unbounded: if they were bounded by `a`, the set would inject into `Iic a.val`, hence have size at most `lift (a.val + 1).card < lift \u03ba` \u2014 contradicting `mk_inaccessibleBelow`.","title":"Inaccessibles are unbounded below an inaccessible fixed point"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/limit-cardinals-are-unbounded-below-a-we","kind":"node","lean":"MetaMathlibExt.IsWeaklyInaccessible.exists_isSuccLimit_between","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#limit-cardinals-are-unbounded-below-a-we","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"Limit cardinals are unbounded below a weakly inaccessible `\u03ba`: the sup of the `\u03c9`-chain `\u03bc 0 = succ (max a \u2135\u2080)`, `\u03bc (n+1) = succ (\u03bc n)` stays below `\u03ba` (`sum_lt_of_isRegular`), exceeds `a`, and is a limit since every `x < \u03c3` lies below some `\u03bc n` (`exists_lt_of_lt_ciSup`).","title":"Limit cardinals are unbounded below a weakly inaccessible cardinal"},{"id":"SetTheory/large-cardinals/inaccessible-cardinals/supremum-of-a-bounded-set-of-initial-ord","kind":"node","lean":"MetaMathlibExt.sSup_initial_of_forall_initial","parent":1118,"result":true,"route":"../roadmap/SetTheory/large-cardinals/inaccessible-cardinals/index.html#supremum-of-a-bounded-set-of-initial-ord","scope":"SetTheory/large-cardinals/inaccessible-cardinals","status":"fully_proved","summary":"The sup of a bounded set of initial ordinals is initial: if `\u03c3.card.ord < \u03c3` then some member exceeds `\u03c3.card.ord`, trapping it strictly between itself and its own initial ordinal.","title":"Supremum of a bounded set of initial ordinals is initial"},{"data_url":"scopes/SetTheory/large-cardinals/mahlo-cardinals.json","id":"SetTheory/large-cardinals/mahlo-cardinals","kind":"scope","navigation":"module","parent":1115,"result":false,"route":"../roadmap/SetTheory/large-cardinals/mahlo-cardinals/index.html","scope":"SetTheory/large-cardinals/mahlo-cardinals","status":"fully_proved","title":"Mahlo cardinals"},{"id":"SetTheory/large-cardinals/mahlo-cardinals/mahlo-cardinal","kind":"node","lean":"MetaMathlibExt.IsMahlo","parent":1139,"result":true,"route":"../roadmap/SetTheory/large-cardinals/mahlo-cardinals/index.html#mahlo-cardinal","scope":"SetTheory/large-cardinals/mahlo-cardinals","status":"fully_proved","summary":"A cardinal \u03ba is *Mahlo* if \u03ba is inaccessible and the set { \u03bb < \u03ba : \u03bb is inaccessible } of inaccessibles below \u03ba is stationary in \u03ba.","title":"Mahlo cardinal"},{"id":"SetTheory/large-cardinals/mahlo-cardinals/weakly-mahlo-cardinal","kind":"node","lean":"MetaMathlibExt.IsWeaklyMahlo","parent":1139,"result":true,"route":"../roadmap/SetTheory/large-cardinals/mahlo-cardinals/index.html#weakly-mahlo-cardinal","scope":"SetTheory/large-cardinals/mahlo-cardinals","status":"fully_proved","summary":"A cardinal \u03ba is *weakly Mahlo* if \u03ba is weakly inaccessible and { \u03bb < \u03ba : \u03bb is a regular cardinal } is stationary in \u03ba.","title":"Weakly Mahlo cardinal"},{"data_url":"scopes/SetTheory/infinitary-combinatorics.json","id":"SetTheory/infinitary-combinatorics","kind":"scope","parent":1102,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/index.html","scope":"SetTheory/infinitary-combinatorics","status":"can_prove","summary":"Combinatorics of infinite and uncountable sets: clubs and stationary sets, partition relations, \u0394-systems, almost disjoint families, combinatorial principles such as diamond and square, set-theoretic trees, Ramsey ultrafilters and the structure of P(\u03c9)/fin.","title":"Infinitary combinatorics"},{"id":"SetTheory/infinitary-combinatorics/is-ramsey-ultrafilter","kind":"node","lean":"MetaMathlibExt.IsRamseyUltrafilter","parent":1142,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/index.html#is-ramsey-ultrafilter","scope":"SetTheory/infinitary-combinatorics","status":"fully_proved","summary":"An ultrafilter $U$ on $\\mathbb{N}$ is *Ramsey* if for every sequence $(A_n)_{n \\in \\mathbb{N}}$ of pairwise disjoint subsets of $\\mathbb{N}$ whose union is $\\mathbb{N}$ and such that $A_n \\notin U$ for all $n$, there is $X \\in U$ such that $X \\cap A_n$ has at most one element for every $n$.","title":"Ramsey ultrafilter"},{"id":"SetTheory/infinitary-combinatorics/is-delta-system","kind":"node","lean":"MetaMathlibExt.IsDeltaSystem","parent":1142,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/index.html#is-delta-system","scope":"SetTheory/infinitary-combinatorics","status":"fully_proved","summary":"A family $A$ of subsets of a type $\\alpha$ is a \u0394-system with root $r \\subseteq \\alpha$ if $s \\cap t = r$ for any two distinct members $s$ and $t$ of $A$.","title":"\u0394-system"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/almost-disjoint-families.json","id":"SetTheory/infinitary-combinatorics/almost-disjoint-families","kind":"scope","navigation":"module","parent":1142,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/almost-disjoint-families/index.html","scope":"SetTheory/infinitary-combinatorics/almost-disjoint-families","status":"fully_proved","title":"Almost disjoint families"},{"id":"SetTheory/infinitary-combinatorics/almost-disjoint-families/kappa-almost-disjoint-family-of-sets","kind":"node","lean":"MetaMathlibExt.IsAlmostDisjointFamilyKappa","parent":1145,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\kappa\\)"],["text","-almost disjoint family of sets"]],"source":"$\\kappa$-almost disjoint family of sets"},"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/almost-disjoint-families/index.html#kappa-almost-disjoint-family-of-sets","scope":"SetTheory/infinitary-combinatorics/almost-disjoint-families","status":"fully_proved","summary":"A family F \u2286 \ud835\udcab(I) is *\u03ba-almost disjoint* if |A| = \u03ba for every A \u2208 F and |A \u2229 B| < \u03ba for all distinct A, B \u2208 F.","title":"$\\kappa$-almost disjoint family of sets"},{"id":"SetTheory/infinitary-combinatorics/almost-disjoint-families/functions-almost-disjoint-modulo-the-non","kind":"node","lean":"MetaMathlibExt.AlmostDisjointModNS","parent":1145,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/almost-disjoint-families/index.html#functions-almost-disjoint-modulo-the-non","scope":"SetTheory/infinitary-combinatorics/almost-disjoint-families","status":"fully_proved","summary":"Functions f, g \u2208 \u220f_{\u03b1<\u03c9\u2081} A_\u03b1 are *almost disjoint mod the nonstationary ideal* if the set { \u03b1 < \u03c9\u2081 : f(\u03b1) = g(\u03b1) } on which they agree is nonstationary in \u03c9\u2081.","title":"Functions almost disjoint modulo the nonstationary ideal"},{"id":"SetTheory/infinitary-combinatorics/almost-disjoint-families/almost-disjoint-family-of-functions-on-a","kind":"node","lean":"MetaMathlibExt.IsAlmostDisjointFunFamily","parent":1145,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/almost-disjoint-families/index.html#almost-disjoint-family-of-functions-on-a","scope":"SetTheory/infinitary-combinatorics/almost-disjoint-families","status":"fully_proved","summary":"Let \u03ba be a regular cardinal. A family of functions \u03ba \u2192 \u03ba is *almost disjoint* if for all f and g in the family with f\u2260 g, |{ \u03b1 \u2223 f(\u03b1 ) = g(\u03b1 ) }|<\u03ba..","title":"Almost disjoint family of functions on a regular cardinal"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees.json","id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees","kind":"scope","parent":1142,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/index.html","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees","status":"fully_proved","title":"Aronszajn, Suslin and special trees"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory.json","id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory","kind":"scope","navigation":"module","parent":1149,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory/index.html","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory","status":"fully_proved","title":"Trees in set theory"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory/tree-set-theory","kind":"node","lean":"MetaMathlibExt.IsTree","parent":1150,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory/index.html#tree-set-theory","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory","status":"fully_proved","summary":"A *tree* is a partially ordered set (T, <) iwith the property that for each x \u2208 T, the set { y \u2208 T : y < x } of all predecessors of x is well-ordered by <).","title":"Tree (set theory)"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory/level-of-a-well-founded-partial-order","kind":"node","lean":"MetaMathlibExt.level","parent":1150,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory/index.html#level-of-a-well-founded-partial-order","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory","status":"fully_proved","summary":"For an ordinal \u03b1, the *\u03b1-th level* of a well-founded partial order (T, <) is Lev_\u03b1(T) = { x \u2208 T : ht(x) = \u03b1 }, the set of elements whose height equals \u03b1.","title":"Level of a well-founded partial order"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory/normal-tree-at-a-cardinal","kind":"node","lean":"MetaMathlibExt.IsNormalTreeAt","parent":1150,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory/index.html#normal-tree-at-a-cardinal","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/trees-in-set-theory","status":"fully_proved","summary":"Let \u03ba be a cardinal and (\u03b1,\u2264) a well-founded partial order; for x\u2208\u03b1 let rank(x) denote the rank of x in the well-founded relation <. \u03b1 is a *normal tree at \u03ba* if:","title":"Normal tree at a cardinal"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees.json","id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees","kind":"scope","navigation":"module","parent":1149,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees/index.html","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees","status":"fully_proved","title":"Aronszajn trees"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees/kappa-aronszajn-tree","kind":"node","lean":"MetaMathlibExt.IsAronszajnTreeAt","parent":1154,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\kappa\\)"],["text","-Aronszajn tree"]],"source":"$\\kappa$-Aronszajn tree"},"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees/index.html#kappa-aronszajn-tree","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees","status":"fully_proved","summary":"A \u03ba-Aronszajn tree is a \u03ba-tree such that there are no cofinal branch through the tree.","title":"$\\kappa$-Aronszajn tree"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees/aronszajn-tree","kind":"node","lean":"MetaMathlibExt.IsAronszajnTree","parent":1154,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees/index.html#aronszajn-tree","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees","status":"fully_proved","summary":"An *Aronszajn tree* is a tree of height $\\aleph_1}$ all of whose levels are at most countable and which has no uncountable branches.","title":"Aronszajn tree"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees/omega-1-tree","kind":"node","lean":"MetaMathlibExt.IsOmega1Tree","parent":1154,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\omega_1\\)"],["text","-tree"]],"source":"$\\omega_1$-tree"},"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees/index.html#omega-1-tree","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/aronszajn-trees","status":"fully_proved","summary":"An *\u03c9\u2081-tree* is a tree of height \u03c9\u2081 all of whose levels are countable: every node has height < \u03c9\u2081, every level Lev_\u03b1(T) with \u03b1 < \u03c9\u2081 is nonempty, and every such level is countable.","title":"$\\omega_1$-tree"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders.json","id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders","kind":"scope","navigation":"module","parent":1149,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders/index.html","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders","status":"fully_proved","title":"Special partial orders"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders/partial-order-that-is-a-union-of-countab","kind":"node","lean":"MetaMathlibExt.IsUnionOfOmegaAntichains","parent":1158,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders/index.html#partial-order-that-is-a-union-of-countab","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders","status":"fully_proved","summary":"A partial order (P, \u2264) is a *union of \u03c9 antichains* if P = \u22c3_{n<\u03c9} A\u2099 where each A\u2099 \u2286 P is an antichain (a set of pairwise \u2264-incomparable elements).","title":"Partial order that is a union of countably many antichains"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders/special-partial-order","kind":"node","lean":"MetaMathlibExt.IsSpecial","parent":1158,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders/index.html#special-partial-order","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/special-partial-orders","status":"fully_proved","summary":"A partial order (\u03b1,\u2264) is *special* if there is a map f : \u03b1 \u2192 \u211a that is strictly increasing, i.e.\\ x < y implies f(x) < f(y). Note that f need not be injective: nothing is required of incomparable elements.","title":"Special partial order"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees.json","id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees","kind":"scope","navigation":"module","parent":1149,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees/index.html","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees","status":"fully_proved","title":"Suslin trees"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees/kappa-suslin-tree","kind":"node","lean":"MetaMathlibExt.IsSuslinTreeAt","parent":1161,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\kappa\\)"],["text","-Suslin tree"]],"source":"$\\kappa$-Suslin tree"},"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees/index.html#kappa-suslin-tree","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees","status":"fully_proved","summary":"A \u03ba-Suslin tree is a \u03ba-Aronszajn tree with no antichains of size \u03ba.","title":"$\\kappa$-Suslin tree"},{"id":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees/suslin-tree","kind":"node","lean":"MetaMathlibExt.IsSuslinTree","parent":1161,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees/index.html#suslin-tree","scope":"SetTheory/infinitary-combinatorics/aronszajn-suslin-and-special-trees/suslin-trees","status":"fully_proved","summary":"A *Suslin tree* is a tree of height $\\aleph_1}$ with no uncountable branches and no uncountable antichains.","title":"Suslin tree"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/automorphisms-of-p-fin.json","id":"SetTheory/infinitary-combinatorics/automorphisms-of-p-fin","kind":"scope","navigation":"module","parent":1142,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/automorphisms-of-p-fin/index.html","scope":"SetTheory/infinitary-combinatorics/automorphisms-of-p-fin","status":"fully_proved","title":"Automorphisms of P(\u03c9)/fin"},{"id":"SetTheory/infinitary-combinatorics/automorphisms-of-p-fin/eq-mod-finite","kind":"node","lean":"MetaMathlibExt.EqModFinite","parent":1164,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/automorphisms-of-p-fin/index.html#eq-mod-finite","scope":"SetTheory/infinitary-combinatorics/automorphisms-of-p-fin","status":"fully_proved","summary":"Subsets $A$ and $B$ of $\\mathbb{N}$ are *equal modulo finite*, written $A =^{\\ast} B$, if their symmetric difference $A \\mathbin{\\triangle} B$ is finite.","title":"Equality modulo finite sets"},{"id":"SetTheory/infinitary-combinatorics/automorphisms-of-p-fin/is-pow-mod-finite-automorphism","kind":"node","lean":"MetaMathlibExt.IsPowModFiniteAutomorphism","parent":1164,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/automorphisms-of-p-fin/index.html#is-pow-mod-finite-automorphism","scope":"SetTheory/infinitary-combinatorics/automorphisms-of-p-fin","status":"fully_proved","summary":"For subsets $A$, $B$ of $\\mathbb{N}$, write $A =^{\\ast} B$ when their symmetric difference is finite. A map $\\Phi$ from subsets of $\\mathbb{N}$ to subsets of $\\mathbb{N}$ presents an automorphism of $P(\\omega)/\\mathrm{fin}$ if for all subsets $A$, $B$ of $\\mathbb{N}$: it respects $=^{\\ast}$ ($A =^{\\ast} B$ implies $\\Phi(A) =^{\\ast} \\Phi(B)$), it preserves meets ($\\Phi(A \\cap B) =^{\\ast} \\Phi(A) \\cap \\Phi(B)$) and\u2026","title":"Automorphisms of P(\u03c9)/fin on representatives"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals.json","id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","kind":"scope","navigation":"module","parent":1142,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","title":"Clubs, stationary sets and normal ideals"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/countable-subsets-of-a-set","kind":"node","lean":"MetaMathlibExt.CountableSubsets","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#countable-subsets-of-a-set","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"[\u03b1]^\u03c9 or P_{\u03c9\u2081}(\u03b1) is the set of countable subsets of \u03b1","title":"Countable subsets of a set"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/is-club-omega","kind":"node","lean":"MetaMathlibExt.IsClubOmega","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#is-club-omega","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"Let $X$ be a set. A family $C$ of subsets of $X$ is club in the countable subsets of $X$ when every member of $C$ is countable (finite or countably infinite), every countable set $x \\subseteq X$ is contained in some member of $C$, and for every sequence $x_0 \\subseteq x_1 \\subseteq x_2 \\subseteq \\cdots$ of members of $C$ the union $\\bigcup_{n \\in \\mathbb{N}} x_n$ belongs to $C$.","title":"Club families of countable sets"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/is-stationary-omega","kind":"node","lean":"MetaMathlibExt.IsStationaryOmega","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#is-stationary-omega","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"Let $X$ be a set. A family $S$ of subsets of $X$ is stationary in the countable subsets of $X$ when $S$ meets every family $C$ that is club in the countable subsets of $X$, that is, some set belongs to both $S$ and $C$.","title":"Stationary families of countable sets"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/closure-points","kind":"node","lean":"MetaMathlibExt.ClosurePoints","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#closure-points","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"Let $X$ be a set and let $F$ be a function assigning an element of $X$ to every finite subset of $X$. The closure points of $F$ are the countable (finite or countably infinite) sets $a \\subseteq X$ such that $F(s) \\in a$ for every finite set $s \\subseteq a$.","title":"Closure points of a finitary function"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/closure-points-seq","kind":"node","lean":"MetaMathlibExt.ClosurePointsSeq","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#closure-points-seq","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"Let $X$ be a set and let $(F_n)_{n \\in \\mathbb{N}}$ be a sequence of functions, each assigning an element of $X$ to every finite subset of $X$. The closure points of $(F_n)$ are the countable (finite or countably infinite) sets $x \\subseteq X$ such that $F_n(s) \\in x$ for every $n \\in \\mathbb{N}$ and every finite set $s \\subseteq x$.","title":"Closure points of a sequence of finitary functions"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/closure-points-and-clubs-of-countable-su","kind":"node","lean":"MetaMathlibExt.closurePointsSeq_const","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#closure-points-and-clubs-of-countable-su","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"1. Let F : Finset(\u03b1) \u2192 \u03b1. Then ClosurePointsSeq(\u03bb n : \u2115,, F) = ClosurePoints(F). 2. For every finitary operation F, \u2205 \u2209 ClosurePoints(F). 3. Let f : \u2115 \u2192 \ud835\udcab(\u03b1) with each f(n) countable. Then \u22c3_{n \u2208 \u2115} f(n) is countable. 4. [\u03b1]^\u03c9 is club in [\u03b1]^\u03c9. 5. \u2205 is not stationary in [\u03b1]^\u03c9. 6. If S is stationary in [\u03b1]^\u03c9 then S \u2260 \u2205. 7.","title":"Closure points and clubs of countable subsets"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/subsets-of-lambda-of-size-less-than-kapp","kind":"node","lean":"MetaMathlibExt.powersetCardLT","parent":1167,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subsets of "],["math","\\(\\lambda\\)"],["text"," of size less than "],["math","\\(\\kappa\\)"]],"source":"Subsets of $\\lambda$ of size less than $\\kappa$"},"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#subsets-of-lambda-of-size-less-than-kapp","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"P_\u03ba(\u03bb) := { x \u2286 \u03bb \u2223 |x| < \u03ba }","title":"Subsets of $\\lambda$ of size less than $\\kappa$"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/club-subset-of-p-kappa-lambda","kind":"node","lean":"MetaMathlibExt.IsClubPowersetCardLT","parent":1167,"reading_title":{"format":"inline-parts-v1","parts":[["text","Club subset of "],["math","\\(P_\\kappa(\\lambda)\\)"]],"source":"Club subset of $P_\\kappa(\\lambda)$"},"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#club-subset-of-p-kappa-lambda","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"C \u2286 P_\u03ba(\u03bb) is club iff","title":"Club subset of $P_\\kappa(\\lambda)$"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/family-containing-all-bounded-subsets-of","kind":"node","lean":"MetaMathlibExt.ContainsAllBounded","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#family-containing-all-bounded-subsets-of","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"A set X\u2286\u03ba is *bounded in \u03ba* if there is some \u03b2<\u03ba with \u03b1<\u03b2 for every \u03b1\u2208 X; equivalently, X\u2286\u03b2 for some \u03b2<\u03ba.","title":"Family containing all bounded subsets of a cardinal"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/kappa-complete-ideal-on-a-cardinal","kind":"node","lean":"MetaMathlibExt.IsKappaCompleteIdeal","parent":1167,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\kappa\\)"],["text","-complete ideal on a cardinal"]],"source":"$\\kappa$-complete ideal on a cardinal"},"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#kappa-complete-ideal-on-a-cardinal","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"A *\u03ba-complete ideal on \u03ba* is a collection I \u2286 \ud835\udcab(\u03ba) such that:","title":"$\\kappa$-complete ideal on a cardinal"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/normal-ideal-on-a-cardinal","kind":"node","lean":"MetaMathlibExt.IsNormalIdeal","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#normal-ideal-on-a-cardinal","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"Let \u03ba be a cardinal, identified with \u03ba = Iio(\u03ba.ord) = {\u03b1 : Ordinal \u2223 \u03b1 < \u03ba.ord}, and let I\u2286\ud835\udcab(\u03ba).","title":"Normal ideal on a cardinal"},{"id":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/is-regressive","kind":"node","lean":"MetaMathlibExt.IsRegressive","parent":1167,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals/index.html#is-regressive","scope":"SetTheory/infinitary-combinatorics/clubs-stationary-sets-and-normal-ideals","status":"fully_proved","summary":"Let $\\kappa$ be a cardinal, identified with its initial ordinal, let $S$ be a set of ordinals less than $\\kappa$, and let $f$ be a function from $S$ to the ordinals less than $\\kappa$. Then $f$ is regressive if $f(\\alpha) < \\alpha$ for every nonzero $\\alpha \\in S$.","title":"Regressive functions"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/diamond-principle.json","id":"SetTheory/infinitary-combinatorics/diamond-principle","kind":"scope","navigation":"module","parent":1142,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/diamond-principle/index.html","scope":"SetTheory/infinitary-combinatorics/diamond-principle","status":"fully_proved","title":"Diamond principle"},{"id":"SetTheory/infinitary-combinatorics/diamond-principle/diamond-principle","kind":"node","lean":"MetaMathlibExt.Diamond","parent":1180,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/diamond-principle/index.html#diamond-principle","scope":"SetTheory/infinitary-combinatorics/diamond-principle","status":"fully_proved","summary":"\u25c7_\u03ba holds iff there is a sequence \u27e8 S_\u03b1 : \u03b1 < \u03ba\u27e9 with S_\u03b1 \u2286 \u03b1 for all \u03b1 < \u03ba, such that for every X \u2286 \u03ba, {\u03b1 < \u03ba : X \u2229 \u03b1 = S_\u03b1} is stationary in \u03ba.","title":"Diamond principle"},{"id":"SetTheory/infinitary-combinatorics/diamond-principle/diamond-minus-principle","kind":"node","lean":"MetaMathlibExt.DiamondMinus","parent":1180,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/diamond-principle/index.html#diamond-minus-principle","scope":"SetTheory/infinitary-combinatorics/diamond-principle","status":"fully_proved","summary":"\u25c7\u207b_\u03ba holds iff there is a sequence \u27e8 \ud835\udc9c_\u03b1 : \u03b1 < \u03ba\u27e9 with each \ud835\udc9c_\u03b1 \u2286 \ud835\udcab(\u03b1) and |\ud835\udc9c_\u03b1| \u2264 |\u03b1|, such that for every X \u2286 \u03ba the set {\u03b1 < \u03ba : X \u2229 \u03b1 \u2208 \ud835\udc9c_\u03b1} is stationary in \u03ba.","title":"Diamond-minus principle"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/partition-calculus.json","id":"SetTheory/infinitary-combinatorics/partition-calculus","kind":"scope","navigation":"module","parent":1142,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"can_prove","title":"Partition calculus"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/ordinal-cardinal-ramsey","kind":"node","lean":"OrdinalCardinalRamsey","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#ordinal-cardinal-ramsey","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"fully_proved","summary":"For ordinals $\\alpha$, $\\beta$ and a cardinal $c$, the partition property $\\alpha \\to (\\beta, c)^2$ holds if for every coloring of the edges of the complete graph on the ordinals less than $\\alpha$ in red and blue, either some set of vertices whose order type is $\\beta$ has all its edges red, or some set of vertices of cardinality exactly $c$ has all its edges blue.","title":"Ordinal\u2013cardinal Ramsey property"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/is-homogeneous-for","kind":"node","lean":"MetaMathlibExt.IsHomogeneousFor","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#is-homogeneous-for","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"fully_proved","summary":"Let $c$ assign a colour in $\\gamma$ to each finite subset of a type $I$, and let $n \\in \\mathbb{N}$. A set $H \\subseteq I$ is homogeneous for $c$ with colour $k \\in \\gamma$ if $c(s) = k$ for every $n$-element subset $s$ of $H$.","title":"Homogeneous set"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/arrows-pairs","kind":"node","lean":"MathlibExt.Combinatorics.AlephOmegaPartitionSumErdosProblem1219Wanted.ArrowsPairs","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#arrows-pairs","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"fully_proved","summary":"For cardinals $\\kappa$ and $\\mu$, the relation $\\kappa \\to (\\mu)^2_2$ holds if for every set $X$ of cardinality $\\kappa$ and every symmetric colouring $c$ of the ordered pairs of elements of $X$ with two colours, there are a subset $H \\subseteq X$ of cardinality $\\mu$ and a colour $b$ such that $c(x, y) = b$ for all distinct $x, y \\in H$.","title":"Partition relation for pairs with two colours"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/aleph-omega-partition-of-cardinal-sum","kind":"node","lean":"MathlibExt.Combinatorics.AlephOmegaPartitionSumErdosProblem1219Wanted.alephOmegaPartitionOfCardinalSum","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#aleph-omega-partition-of-cardinal-sum","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"fully_proved","summary":"The statement that for every strictly increasing sequence $n_0 < n_1 < \\cdots$ of natural numbers such that the cardinals $2^{\\aleph_{n_k}}$ are strictly increasing in $k$ and $2^{\\aleph_{n_0}} > \\aleph_\\omega$, the partition relation $\\sum_k 2^{\\aleph_{n_k}} \\to (\\aleph_\\omega)^2_2$ holds: for every set $X$ of cardinality $\\sum_k 2^{\\aleph_{n_k}}$ and every symmetric colouring of the pairs of elements of $X$ with\u2026","title":"Erd\u0151s Problem 1219: a partition relation for sums of powers of two"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/aleph-omega-partition-of-cardinal-sum-ho","kind":"node","lean":"MathlibExt.Combinatorics.AlephOmegaPartitionSumErdosProblem1219Wanted.alephOmegaPartitionOfCardinalSum_holds","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#aleph-omega-partition-of-cardinal-sum-ho","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"can_prove","summary":"For every strictly increasing sequence $n_0 < n_1 < \\cdots$ of natural numbers such that the cardinals $2^{\\aleph_{n_k}}$ are strictly increasing in $k$ and $2^{\\aleph_{n_0}} > \\aleph_\\omega$, the partition relation $\\sum_k 2^{\\aleph_{n_k}} \\to (\\aleph_\\omega)^2_2$ holds: for every set $X$ of cardinality $\\sum_k 2^{\\aleph_{n_k}}$ and every symmetric colouring of the pairs of elements of $X$ with two colours, some\u2026","title":"Shelah's theorem (Erd\u0151s Problem 1219)"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem591Wanted.conjecture","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#conjecture","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"fully_proved","summary":"The statement that, for $\\alpha = \\omega^{\\omega^2}$, every red/blue colouring of the edges of the complete graph on the ordinals less than $\\alpha$ (that is, every pair of complementary simple graphs, red and blue, on this set) has a red clique whose order type is $\\alpha$ or a blue clique with exactly $3$ elements. In partition notation, $\\alpha \\to (\\alpha, 3)^2$.","title":"Erd\u0151s Problem 591"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem591Wanted.conjecture_holds","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#conjecture-holds","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"can_prove","summary":"Let $\\alpha = \\omega^{\\omega^2}$. For every red/blue colouring of the edges of the complete graph on the ordinals less than $\\alpha$, that is, every pair of complementary simple graphs (red and blue) on this set, there is a red clique whose order type is $\\alpha$ or a blue clique with exactly $3$ elements. In partition notation, $\\alpha \\to (\\alpha, 3)^2$.","title":"Erd\u0151s Problem 591 is true"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/partition-rel","kind":"node","lean":"MathlibExt.Combinatorics.PartitionOrdinalTriangleCliqueErdosProblem118Wanted.PartitionRel","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#partition-rel","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"fully_proved","summary":"For an ordinal $\\alpha$ and a natural number $n$, the partition relation $\\alpha \\to (\\alpha, n)^2$ holds when every colouring of the pairs $x < y$ of ordinals below $\\alpha$ in red and blue has a red set of order type $\\alpha$ or a blue set of size $n$: either there is a strictly increasing map $f$ from the ordinals below $\\alpha$ to the ordinals below $\\alpha$ such that the pair $\\{f(x), f(y)\\}$ is red for all\u2026","title":"Ordinal partition relation"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/partition-ordinal-triangle-implies-cliqu","kind":"node","lean":"MathlibExt.Combinatorics.PartitionOrdinalTriangleCliqueErdosProblem118Wanted.partitionOrdinalTriangleImpliesClique","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#partition-ordinal-triangle-implies-cliqu","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"fully_proved","summary":"The statement that for every ordinal $\\alpha$ satisfying the partition relation $\\alpha \\to (\\alpha, 3)^2$ and every natural number $n \\ge 3$, the ordinal $\\alpha$ also satisfies $\\alpha \\to (\\alpha, n)^2$.","title":"Erd\u0151s Problem 118 for ordinals"},{"id":"SetTheory/infinitary-combinatorics/partition-calculus/partition-ordinal-triangle-implies-cliqu-2","kind":"node","lean":"MathlibExt.Combinatorics.PartitionOrdinalTriangleCliqueErdosProblem118Wanted.partitionOrdinalTriangleImpliesClique_refuted","parent":1183,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/partition-calculus/index.html#partition-ordinal-triangle-implies-cliqu-2","scope":"SetTheory/infinitary-combinatorics/partition-calculus","status":"can_prove","summary":"There exist an ordinal $\\alpha$ satisfying the partition relation $\\alpha \\to (\\alpha, 3)^2$ and a natural number $n \\ge 3$ such that $\\alpha$ does not satisfy $\\alpha \\to (\\alpha, n)^2$.","title":"Erd\u0151s Problem 118 fails for ordinals"},{"data_url":"scopes/SetTheory/infinitary-combinatorics/square-principle.json","id":"SetTheory/infinitary-combinatorics/square-principle","kind":"scope","navigation":"module","parent":1142,"result":false,"route":"../roadmap/SetTheory/infinitary-combinatorics/square-principle/index.html","scope":"SetTheory/infinitary-combinatorics/square-principle","status":"fully_proved","title":"Square principle"},{"id":"SetTheory/infinitary-combinatorics/square-principle/square-sequence","kind":"node","lean":"MetaMathlibExt.IsSquareSequence","parent":1194,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/square-principle/index.html#square-sequence","scope":"SetTheory/infinitary-combinatorics/square-principle","status":"fully_proved","summary":"Jensen's square principle sequence: C = (C_\u03b1 : \u03b1 < \u03ba\u207a) is a \u25a1_\u03ba-sequence: for every limit \u03b1 < \u03ba\u207a, C_\u03b1 is club in \u03b1 of order type at most \u03ba, and the family coheres \u2014 C_\u03b2 = C_\u03b1 \u2229 \u03b2 whenever \u03b2 is a limit point of C_\u03b1.","title":"Square sequence"},{"id":"SetTheory/infinitary-combinatorics/square-principle/jensen-s-square-principle","kind":"node","lean":"MetaMathlibExt.Square","parent":1194,"result":true,"route":"../roadmap/SetTheory/infinitary-combinatorics/square-principle/index.html#jensen-s-square-principle","scope":"SetTheory/infinitary-combinatorics/square-principle","status":"fully_proved","summary":"\u25a1_\u03ba: a \u25a1_\u03ba-sequence exists","title":"Jensen's square principle"},{"data_url":"scopes/SetTheory/forcing.json","id":"SetTheory/forcing","kind":"scope","parent":1102,"result":false,"route":"../roadmap/SetTheory/forcing/index.html","scope":"SetTheory/forcing","status":"fully_proved","summary":"The forcing method and the axioms that assert the existence of generic filters, such as Martin's axiom and the proper forcing axiom, together with combinatorial dichotomies like the open coloring axiom and the P-ideal dichotomy that follow from them.","title":"Forcing and forcing axioms"},{"data_url":"scopes/SetTheory/forcing/martin-s-axiom.json","id":"SetTheory/forcing/martin-s-axiom","kind":"scope","navigation":"module","parent":1197,"result":false,"route":"../roadmap/SetTheory/forcing/martin-s-axiom/index.html","scope":"SetTheory/forcing/martin-s-axiom","status":"fully_proved","title":"Martin's axiom"},{"id":"SetTheory/forcing/martin-s-axiom/is-ccc","kind":"node","lean":"MetaMathlibExt.IsCCC","parent":1198,"result":true,"route":"../roadmap/SetTheory/forcing/martin-s-axiom/index.html#is-ccc","scope":"SetTheory/forcing/martin-s-axiom","status":"fully_proved","summary":"A partial order $(P, \\le)$ has the countable chain condition (c.c.c.) if every antichain of $P$ is countable, where an antichain is a set of pairwise incompatible elements: no two distinct members of it have a common lower bound in $P$.","title":"Countable chain condition"},{"id":"SetTheory/forcing/martin-s-axiom/is-sigma-centered","kind":"node","lean":"MetaMathlibExt.IsSigmaCentered","parent":1198,"result":true,"route":"../roadmap/SetTheory/forcing/martin-s-axiom/index.html#is-sigma-centered","scope":"SetTheory/forcing/martin-s-axiom","status":"fully_proved","summary":"A partial order $P$ is $\\sigma$-centered if there is a map $c : P \\to \\mathbb{N}$ such that for every $n \\in \\mathbb{N}$, every finite set of elements $p$ with $c(p) = n$ has a common lower bound in $P$. Thus $P$ splits into countably many centered pieces $c^{-1}(n)$; the empty set is among the finite sets, so $P$ must be nonempty.","title":"\u03c3-centered partial orders"},{"id":"SetTheory/forcing/martin-s-axiom/martin-axiom-sigma-centered","kind":"node","lean":"MetaMathlibExt.MartinAxiomSigmaCentered","parent":1198,"result":true,"route":"../roadmap/SetTheory/forcing/martin-s-axiom/index.html#martin-axiom-sigma-centered","scope":"SetTheory/forcing/martin-s-axiom","status":"fully_proved","summary":"For a cardinal $\\kappa$, this is the statement that for every $\\sigma$-centered partial order $P$, every family $\\mathcal{D}$ of at most $\\kappa$ dense subsets of $P$ and every $p \\in P$, there is a filter $G$ on $P$ with $p \\in G$ that meets every member of $\\mathcal{D}$. Here $D \\subseteq P$ is dense if every element of $P$ has an element of $D$ below it, and a filter is a nonempty upward closed subset of $P$ any\u2026","title":"Martin's axiom for \u03c3-centered partial orders"},{"id":"SetTheory/forcing/martin-s-axiom/martin-axiom","kind":"node","lean":"MetaMathlibExt.MartinAxiom","parent":1198,"result":true,"route":"../roadmap/SetTheory/forcing/martin-s-axiom/index.html#martin-axiom","scope":"SetTheory/forcing/martin-s-axiom","status":"fully_proved","summary":"For a cardinal $\\kappa$, $\\mathrm{MA}(\\kappa)$ is the statement that for every partial order $P$ with the countable chain condition, every family $\\mathcal{D}$ of at most $\\kappa$ dense subsets of $P$ and every $p \\in P$, there is a filter $G$ on $P$ with $p \\in G$ that meets every member of $\\mathcal{D}$. Here $D \\subseteq P$ is dense if every element of $P$ has an element of $D$ below it, and a filter is a\u2026","title":"Martin's axiom"},{"data_url":"scopes/SetTheory/forcing/open-coloring-axiom.json","id":"SetTheory/forcing/open-coloring-axiom","kind":"scope","navigation":"module","parent":1197,"result":false,"route":"../roadmap/SetTheory/forcing/open-coloring-axiom/index.html","scope":"SetTheory/forcing/open-coloring-axiom","status":"fully_proved","title":"Open coloring axiom"},{"id":"SetTheory/forcing/open-coloring-axiom/is-open-coloring","kind":"node","lean":"MetaMathlibExt.IsOpenColoring","parent":1203,"result":true,"route":"../roadmap/SetTheory/forcing/open-coloring-axiom/index.html#is-open-coloring","scope":"SetTheory/forcing/open-coloring-axiom","status":"fully_proved","summary":"Let $X$ be a topological space. A set $K \\subseteq X \\times X$ of ordered pairs is an open coloring of $X$ if $K$ is open in the product topology, symmetric ($(x, y) \\in K$ implies $(y, x) \\in K$) and irreflexive ($(x, x) \\notin K$ for every $x \\in X$).","title":"Open coloring of a topological space"},{"id":"SetTheory/forcing/open-coloring-axiom/is-coloring-homogeneous","kind":"node","lean":"MetaMathlibExt.IsColoringHomogeneous","parent":1203,"result":true,"route":"../roadmap/SetTheory/forcing/open-coloring-axiom/index.html#is-coloring-homogeneous","scope":"SetTheory/forcing/open-coloring-axiom","status":"fully_proved","summary":"Let $X$ be a set and $K \\subseteq X \\times X$. A set $H \\subseteq X$ is $K$-homogeneous if $(x, y) \\in K$ for all distinct $x, y \\in H$.","title":"Homogeneous set for a coloring"},{"id":"SetTheory/forcing/open-coloring-axiom/is-coloring-free","kind":"node","lean":"MetaMathlibExt.IsColoringFree","parent":1203,"result":true,"route":"../roadmap/SetTheory/forcing/open-coloring-axiom/index.html#is-coloring-free","scope":"SetTheory/forcing/open-coloring-axiom","status":"fully_proved","summary":"Let $X$ be a set and $K \\subseteq X \\times X$. A set $H \\subseteq X$ is $K$-free if $(x, y) \\notin K$ for all $x, y \\in H$.","title":"Free set for a coloring"},{"id":"SetTheory/forcing/open-coloring-axiom/open-coloring-axiom","kind":"node","lean":"MetaMathlibExt.OpenColoringAxiom","parent":1203,"result":true,"route":"../roadmap/SetTheory/forcing/open-coloring-axiom/index.html#open-coloring-axiom","scope":"SetTheory/forcing/open-coloring-axiom","status":"fully_proved","summary":"Todorcevic's Open Coloring Axiom: for every open coloring of a second-countable space, either there is an uncountable homogeneous set, or the space is a countable union of free sets.","title":"Open coloring axiom"},{"data_url":"scopes/SetTheory/forcing/p-ideal-dichotomy.json","id":"SetTheory/forcing/p-ideal-dichotomy","kind":"scope","navigation":"module","parent":1197,"result":false,"route":"../roadmap/SetTheory/forcing/p-ideal-dichotomy/index.html","scope":"SetTheory/forcing/p-ideal-dichotomy","status":"fully_proved","title":"P-ideal dichotomy"},{"id":"SetTheory/forcing/p-ideal-dichotomy/subset-mod-finite","kind":"node","lean":"MetaMathlibExt.SubsetModFinite","parent":1208,"result":true,"route":"../roadmap/SetTheory/forcing/p-ideal-dichotomy/index.html#subset-mod-finite","scope":"SetTheory/forcing/p-ideal-dichotomy","status":"fully_proved","summary":"For subsets $a$ and $b$ of a set $\\alpha$, $a \\subseteq^* b$ ($a$ is almost contained in $b$) if the difference $a \\setminus b$ is finite.","title":"Inclusion modulo finite"},{"id":"SetTheory/forcing/p-ideal-dichotomy/is-pideal","kind":"node","lean":"MetaMathlibExt.IsPIdeal","parent":1208,"result":true,"route":"../roadmap/SetTheory/forcing/p-ideal-dichotomy/index.html#is-pideal","scope":"SetTheory/forcing/p-ideal-dichotomy","status":"fully_proved","summary":"A family $I$ of subsets of a set $\\alpha$ is a P-ideal of countable sets if every member of $I$ is countable, every subset of a member of $I$ is in $I$, every finite subset of $\\alpha$ is in $I$, $I$ is closed under binary unions, and for every sequence $(a_n)_{n \\in \\mathbb{N}}$ of members of $I$ there is $b \\in I$ with $a_n \\subseteq^* b$ (that is, $a_n \\setminus b$ is finite) for every $n$.","title":"P-ideal of countable sets"},{"id":"SetTheory/forcing/p-ideal-dichotomy/is-orthogonal-to","kind":"node","lean":"MetaMathlibExt.IsOrthogonalTo","parent":1208,"result":true,"route":"../roadmap/SetTheory/forcing/p-ideal-dichotomy/index.html#is-orthogonal-to","scope":"SetTheory/forcing/p-ideal-dichotomy","status":"fully_proved","summary":"Let $I$ be a family of subsets of a set $\\alpha$. A set $X \\subseteq \\alpha$ is orthogonal to $I$ if $X \\cap a$ is finite for every $a \\in I$.","title":"Orthogonality to a family of sets"},{"id":"SetTheory/forcing/p-ideal-dichotomy/p-ideal-dichotomy","kind":"node","lean":"MetaMathlibExt.PIdealDichotomy","parent":1208,"result":true,"route":"../roadmap/SetTheory/forcing/p-ideal-dichotomy/index.html#p-ideal-dichotomy","scope":"SetTheory/forcing/p-ideal-dichotomy","status":"fully_proved","summary":"The P-ideal dichotomy (PID): for every P-ideal I of countable subsets of an uncountable set, either some uncountable set has all its countable subsets in I, or the whole set is a countable union of sets orthogonal to I.","title":"P-ideal dichotomy"},{"data_url":"scopes/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom.json","id":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","kind":"scope","navigation":"module","parent":1197,"result":false,"route":"../roadmap/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/index.html","scope":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","status":"fully_proved","title":"Proper forcing and the proper forcing axiom"},{"id":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/model-conds","kind":"node","lean":"MetaMathlibExt.modelConds","parent":1213,"result":true,"route":"../roadmap/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/index.html#model-conds","scope":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","status":"fully_proved","summary":"Let $\\mathbb{P}$ be a set and $M$ a subset of the disjoint union $\\mathbb{P} \\sqcup \\mathcal{P}(\\mathbb{P})$. The conditions in $M$ form the set $M_c$ of those $p \\in \\mathbb{P}$ whose copy in the disjoint union belongs to $M$; informally, $M_c = M \\cap \\mathbb{P}$.","title":"Conditions in a model"},{"id":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/model-sets","kind":"node","lean":"MetaMathlibExt.modelSets","parent":1213,"result":true,"route":"../roadmap/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/index.html#model-sets","scope":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","status":"fully_proved","summary":"Let $\\mathbb{P}$ be a set and $M$ a subset of the disjoint union $\\mathbb{P} \\sqcup \\mathcal{P}(\\mathbb{P})$. The sets of conditions in $M$ form the family $M_s$ of those $D \\subseteq \\mathbb{P}$ whose copy in the disjoint union belongs to $M$; informally, $M_s = M \\cap \\mathcal{P}(\\mathbb{P})$.","title":"Sets of conditions in a model"},{"id":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/is-compatible","kind":"node","lean":"MetaMathlibExt.IsCompatible","parent":1213,"result":true,"route":"../roadmap/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/index.html#is-compatible","scope":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","status":"fully_proved","summary":"Two elements $p$ and $q$ of a partial order $\\mathbb{P}$ are compatible if there is $r \\in \\mathbb{P}$ with $r \\le p$ and $r \\le q$.","title":"Compatible conditions"},{"id":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/is-predense-below","kind":"node","lean":"MetaMathlibExt.IsPredenseBelow","parent":1213,"result":true,"route":"../roadmap/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/index.html#is-predense-below","scope":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","status":"fully_proved","summary":"A set $A \\subseteq \\mathbb{P}$ of elements of a partial order $\\mathbb{P}$ is predense below $q \\in \\mathbb{P}$ if for every $r \\le q$ there is $a \\in A$ such that $r$ and $a$ are compatible.","title":"Predense below a condition"},{"id":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/is-generic-for","kind":"node","lean":"MetaMathlibExt.IsGenericFor","parent":1213,"result":true,"route":"../roadmap/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/index.html#is-generic-for","scope":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","status":"fully_proved","summary":"Let $\\mathbb{P}$ be a partial order and $M$ a subset of the disjoint union $\\mathbb{P} \\sqcup \\mathcal{P}(\\mathbb{P})$. Write $M_c$ for the set of $p \\in \\mathbb{P}$ whose copy in the disjoint union lies in $M$, and $M_s$ for the set of $D \\subseteq \\mathbb{P}$ whose copy lies in $M$. A condition $q \\in \\mathbb{P}$ is $(M, \\mathbb{P})$-generic if for every $D \\in M_s$ that is dense in $\\mathbb{P}$ (every element of\u2026","title":"(M, \u2119)-generic conditions"},{"id":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/is-proper-forcing","kind":"node","lean":"MetaMathlibExt.IsProperForcing","parent":1213,"result":true,"route":"../roadmap/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/index.html#is-proper-forcing","scope":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","status":"fully_proved","summary":"Let $\\mathbb{P}$ be a partial order and let $X$ be the disjoint union $\\mathbb{P} \\sqcup \\mathcal{P}(\\mathbb{P})$; for $M \\subseteq X$ write $M_c$ for the set of $p \\in \\mathbb{P}$ whose copy in $X$ lies in $M$. Then $\\mathbb{P}$ is proper if there is a club $C$ in $[X]^\\omega$ such that for every $M \\in C$ and every $p \\in M_c$ there is an $(M, \\mathbb{P})$-generic condition $q \\le p$. Here a club in $[X]^\\omega$\u2026","title":"Proper partial orders"},{"id":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/proper-forcing-axiom","kind":"node","lean":"MetaMathlibExt.ProperForcingAxiom","parent":1213,"result":true,"route":"../roadmap/SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom/index.html#proper-forcing-axiom","scope":"SetTheory/forcing/proper-forcing-and-the-proper-forcing-axiom","status":"fully_proved","summary":"For every proper partial order $\\mathbb{P}$, every family $\\mathcal{D}$ of at most $\\aleph_1$ dense subsets of $\\mathbb{P}$, and every condition $p \\in \\mathbb{P}$, there is a filter $G \\subseteq \\mathbb{P}$ with $p \\in G$ that meets every member of $\\mathcal{D}$, that is, $G \\cap D \\neq \\emptyset$ for every $D \\in \\mathcal{D}$.","title":"Proper Forcing Axiom"},{"area":"Foundations & Logic","data_url":"chapters/Order.json","id":"Order","kind":"scope","parent":0,"result":false,"route":"../roadmap/Order/index.html","scope":"Order","status":"fully_proved","summary":"Order theory studies comparison relations, lattices, intervals, monotonicity, and order-complete constructions.","title":"Order Theory"},{"data_url":"scopes/Order/ordered-sets.json","id":"Order/ordered-sets","kind":"scope","parent":1221,"result":false,"route":"../roadmap/Order/ordered-sets/index.html","scope":"Order/ordered-sets","status":"fully_proved","summary":"Partial, strict and total orders, dense linear orders, monotone maps and sequences, and representations of finite posets.","title":"Ordered sets"},{"id":"Order/ordered-sets/is-dense-linear-order-without-endpoints","kind":"node","lean":"MetaMathlibExt.IsDenseLinearOrderWithoutEndpoints","parent":1222,"result":true,"route":"../roadmap/Order/ordered-sets/index.html#is-dense-linear-order-without-endpoints","scope":"Order/ordered-sets","status":"fully_proved","summary":"A relation $R$ on a type $\\alpha$ is a *dense linear order without endpoints* if it is a strict total order (irreflexive, transitive, and trichotomous: for all $a, b$, at least one of $R(a,b)$, $a = b$, $R(b,a)$ holds), it is dense (whenever $R(a,b)$, there is $c$ with $R(a,c)$ and $R(c,b)$), and it has no endpoints (for every $a$ there are $b$ and $b'$ with $R(b,a)$ and $R(a,b')$).","title":"Dense linear order without endpoints"},{"data_url":"scopes/Order/ordered-sets/monotone-sequences.json","id":"Order/ordered-sets/monotone-sequences","kind":"scope","navigation":"module","parent":1222,"result":false,"route":"../roadmap/Order/ordered-sets/monotone-sequences/index.html","scope":"Order/ordered-sets/monotone-sequences","status":"fully_proved","title":"Monotone sequences"},{"id":"Order/ordered-sets/monotone-sequences/monotone-nat-iff-le-succ","kind":"node","lean":"monotone_nat_iff_le_succ","parent":1224,"result":true,"route":"../roadmap/Order/ordered-sets/monotone-sequences/index.html#monotone-nat-iff-le-succ","scope":"Order/ordered-sets/monotone-sequences","status":"fully_proved","summary":"Let $\\alpha$ be a preordered set. A sequence $f : \\mathbb{N} \\to \\alpha$ is monotone (that is, $f(m) \\le f(n)$ whenever $m \\le n$) if and only if $f(n) \\le f(n+1)$ for every $n \\in \\mathbb{N}$.","title":"Successor criterion for monotone sequences"},{"id":"Order/ordered-sets/monotone-sequences/antitone-nat-iff-succ-le","kind":"node","lean":"antitone_nat_iff_succ_le","parent":1224,"result":true,"route":"../roadmap/Order/ordered-sets/monotone-sequences/index.html#antitone-nat-iff-succ-le","scope":"Order/ordered-sets/monotone-sequences","status":"fully_proved","summary":"Let $\\alpha$ be a preordered set. A sequence $f : \\mathbb{N} \\to \\alpha$ is antitone (that is, $f(n) \\le f(m)$ whenever $m \\le n$) if and only if $f(n+1) \\le f(n)$ for every $n \\in \\mathbb{N}$.","title":"Successor criterion for antitone sequences"},{"id":"Order/ordered-sets/monotone-sequences/monotone-or-antitone-nat-iff","kind":"node","lean":"monotone_or_antitone_nat_iff","parent":1224,"result":true,"route":"../roadmap/Order/ordered-sets/monotone-sequences/index.html#monotone-or-antitone-nat-iff","scope":"Order/ordered-sets/monotone-sequences","status":"fully_proved","summary":"Let $\\alpha$ be a preordered set and let $f : \\mathbb{N} \\to \\alpha$ be a sequence. Then $f$ is monotone ($f(m) \\le f(n)$ whenever $m \\le n$) or antitone ($f(n) \\le f(m)$ whenever $m \\le n$) if and only if either $f(n) \\le f(n+1)$ for every $n \\in \\mathbb{N}$, or $f(n+1) \\le f(n)$ for every $n \\in \\mathbb{N}$.","title":"Successor criterion for monotone or antitone sequences"},{"data_url":"scopes/Order/ordered-sets/strict-total-orders.json","id":"Order/ordered-sets/strict-total-orders","kind":"scope","navigation":"module","parent":1222,"result":false,"route":"../roadmap/Order/ordered-sets/strict-total-orders/index.html","scope":"Order/ordered-sets/strict-total-orders","status":"fully_proved","title":"Strict total orders"},{"id":"Order/ordered-sets/strict-total-orders/strict-total-order","kind":"node","lean":"MathlibExt.Order.StrictTotalOrder","parent":1228,"result":true,"route":"../roadmap/Order/ordered-sets/strict-total-orders/index.html#strict-total-order","scope":"Order/ordered-sets/strict-total-orders","status":"fully_proved","summary":"A strict total order on a type $\\alpha$ consists of a binary relation $<$ on $\\alpha$ that is irreflexive (no $a < a$), transitive ($a < b$ and $b < c$ imply $a < c$) and total on distinct elements (if $a \\ne b$, then $a < b$ or $b < a$). It models strict preference.","title":"Strict total order"},{"id":"Order/ordered-sets/strict-total-orders/ext","kind":"node","lean":"MathlibExt.Order.StrictTotalOrder.ext","parent":1228,"result":true,"route":"../roadmap/Order/ordered-sets/strict-total-orders/index.html#ext","scope":"Order/ordered-sets/strict-total-orders","status":"fully_proved","summary":"Two strict total orders on a type $\\alpha$ with the same relation are equal.","title":"Strict total orders are determined by their relation"},{"data_url":"scopes/Order/lattices-boolean-algebras.json","id":"Order/lattices-boolean-algebras","kind":"scope","parent":1221,"result":false,"route":"../roadmap/Order/lattices-boolean-algebras/index.html","scope":"Order/lattices-boolean-algebras","status":"fully_proved","summary":"Lattices and Boolean algebras, their representation theorems, and Boolean functions viewed as maps between Boolean lattices.","title":"Lattices and Boolean algebras"},{"id":"Order/lattices-boolean-algebras/stone-representation-theorem","kind":"node","lean":"MathlibExt.Order.BooleanAlgebra.StoneRepresentation.stone_representation_theorem","parent":1231,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/index.html#stone-representation-theorem","scope":"Order/lattices-boolean-algebras","status":"fully_proved","summary":"Every Boolean algebra $B$ embeds in a power set algebra: there are a set $X$ and an injective map $f$ from $B$ to the power set of $X$ such that $f(a \\sqcup b) = f(a) \\cup f(b)$ and $f(a \\sqcap b) = f(a) \\cap f(b)$ for all $a, b \\in B$, $f(\\top) = X$ and $f(\\bot) = \\emptyset$.","title":"Stone representation theorem"},{"data_url":"scopes/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance.json","id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","kind":"scope","navigation":"module","parent":1231,"result":false,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","title":"Lattice valuations and the Jaccard distance"},{"id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/is-modular-valuation","kind":"node","lean":"JaccardLattice.IsModularValuation","parent":1233,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html#is-modular-valuation","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","summary":"A function $f \\colon L \\to \\mathbb{R}$ on a lattice $L$, with join $\\vee$ and meet $\\wedge$, is a modular valuation when $f(X) + f(Y) = f(X \\vee Y) + f(X \\wedge Y)$ for all $X, Y \\in L$.","title":"Modular valuation"},{"id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/is-strictly-positive","kind":"node","lean":"JaccardLattice.IsStrictlyPositive","parent":1233,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html#is-strictly-positive","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","summary":"A function $f \\colon L \\to \\mathbb{R}$ on a lattice $L$ is strictly positive when $f(X) > 0$ for every $X \\in L$.","title":"Strictly positive valuation"},{"id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/gilbert-condition","kind":"node","lean":"JaccardLattice.GilbertCondition","parent":1233,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html#gilbert-condition","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","summary":"A function $f \\colon L \\to \\mathbb{R}$ on a lattice $L$, with join $\\vee$ and meet $\\wedge$, satisfies the Gilbert condition when $f(X \\vee Z) + f(Z \\vee Y) + f(X \\wedge Y) \\ge f(X \\wedge Z) + f(Z \\wedge Y) + f(X \\vee Y \\vee Z)$ for all $X, Y, Z \\in L$.","title":"Gilbert condition"},{"id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/jaccard-dist","kind":"node","lean":"JaccardLattice.jaccardDist","parent":1233,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html#jaccard-dist","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","summary":"For a function $f \\colon L \\to \\mathbb{R}$ on a lattice $L$ with join $\\vee$ and meet $\\wedge$, the Jaccard distance of $X, Y \\in L$ is $d_f(X, Y) = 1 - f(X \\wedge Y) / f(X \\vee Y)$.","title":"Generalized Jaccard distance"},{"id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/gilbert-condition-iff","kind":"node","lean":"JaccardLattice.gilbertCondition_iff","parent":1233,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html#gilbert-condition-iff","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","summary":"A function $f \\colon L \\to \\mathbb{R}$ on a lattice $L$ satisfies the Gilbert condition if and only if $f$ is a modular valuation and is monotone, that is, $X \\le Y$ implies $f(X) \\le f(Y)$.","title":"Characterization of the Gilbert condition"},{"id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/jaccard-dist-eq-div","kind":"node","lean":"JaccardLattice.jaccardDist_eq_div","parent":1233,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html#jaccard-dist-eq-div","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","summary":"Let $f \\colon L \\to \\mathbb{R}$ be a strictly positive function on a lattice $L$, with join $\\vee$ and meet $\\wedge$. Then for all $X, Y \\in L$ the Jaccard distance of $X$ and $Y$ with respect to $f$ is $d_f(X, Y) = (f(X \\vee Y) - f(X \\wedge Y)) / f(X \\vee Y)$.","title":"Jaccard distance as a single quotient"},{"id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/jaccard-dist-triangle-of-gilbert-conditi","kind":"node","lean":"JaccardLattice.jaccardDist_triangle_of_gilbertCondition","parent":1233,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html#jaccard-dist-triangle-of-gilbert-conditi","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","summary":"Let $f \\colon L \\to \\mathbb{R}$ be a strictly positive function on a lattice $L$ that satisfies the Gilbert condition, and let $d_f$ be the Jaccard distance of $f$. Then $d_f(U, W) + d_f(W, V) \\ge d_f(U, V)$ for all $U, V, W \\in L$.","title":"Jaccard triangle inequality under the Gilbert condition"},{"id":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/jaccard-dist-triangle","kind":"node","lean":"JaccardLattice.jaccardDist_triangle","parent":1233,"result":true,"route":"../roadmap/Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance/index.html#jaccard-dist-triangle","scope":"Order/lattices-boolean-algebras/lattice-valuations-and-the-jaccard-distance","status":"fully_proved","summary":"Let $f \\colon L \\to \\mathbb{R}$ be a strictly positive, monotone modular valuation on a lattice $L$, and let $d_f$ be the Jaccard distance of $f$. Then $d_f(U, W) + d_f(W, V) \\ge d_f(U, V)$ for all $U, V, W \\in L$.","title":"Triangle inequality for the generalized Jaccard distance"},{"data_url":"scopes/Order/fixed-points-domain-theory.json","id":"Order/fixed-points-domain-theory","kind":"scope","parent":1221,"result":false,"route":"../roadmap/Order/fixed-points-domain-theory/index.html","scope":"Order/fixed-points-domain-theory","status":"fully_proved","summary":"Least and greatest fixed points of monotone and Scott-continuous maps, Kleene iteration, and the complete partial orders that underlie denotational semantics.","title":"Fixed points and domain theory"},{"data_url":"scopes/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets.json","id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","kind":"scope","navigation":"module","parent":1242,"result":false,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","title":"Least and greatest fixed points on finite sets"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/lfp-fin","kind":"node","lean":"Finset.lfpFin","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#lfp-fin","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with $|\\alpha|$ elements, let $f$ map subsets of $\\alpha$ to subsets of $\\alpha$, and write $f^n$ for the $n$-fold iterate of $f$. Then $\\mathrm{lfp}_{\\mathrm{fin}}(f)$ is the set $f^{|\\alpha|}(\\emptyset)$, obtained by applying $f$ to the empty set $|\\alpha|$ times.","title":"Least fixed point of a finset map by finite iteration"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/gfp-fin","kind":"node","lean":"Finset.gfpFin","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#gfp-fin","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with $|\\alpha|$ elements, let $f$ map subsets of $\\alpha$ to subsets of $\\alpha$, and write $f^n$ for the $n$-fold iterate of $f$. Then $\\mathrm{gfp}_{\\mathrm{fin}}(f)$ is the set $f^{|\\alpha|}(\\alpha)$, obtained by applying $f$ to the whole of $\\alpha$ $|\\alpha|$ times.","title":"Greatest fixed point of a finset map by finite iteration"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/lfp-fin-fixed","kind":"node","lean":"Finset.lfpFin_fixed","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#lfp-fin-fixed","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with $|\\alpha|$ elements, let $f$ be a map from subsets of $\\alpha$ to subsets of $\\alpha$ that is monotone for inclusion, and let $\\mathrm{lfp}_{\\mathrm{fin}}(f) = f^{|\\alpha|}(\\emptyset)$ be the result of applying $f$ to $\\emptyset$ $|\\alpha|$ times. Then $\\mathrm{lfp}_{\\mathrm{fin}}(f)$ is a fixed point of $f$, that is,\u2026","title":"Finite iteration from the empty set reaches a fixed point"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/gfp-fin-fixed","kind":"node","lean":"Finset.gfpFin_fixed","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#gfp-fin-fixed","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with $|\\alpha|$ elements, let $f$ be a map from subsets of $\\alpha$ to subsets of $\\alpha$ that is monotone for inclusion, and let $\\mathrm{gfp}_{\\mathrm{fin}}(f) = f^{|\\alpha|}(\\alpha)$ be the result of applying $f$ to $\\alpha$ $|\\alpha|$ times. Then $\\mathrm{gfp}_{\\mathrm{fin}}(f)$ is a fixed point of $f$, that is, $f(\\mathrm{gfp}_{\\mathrm{fin}}(f)) = \\mathrm{gfp}_{\\mathrm{fin}}(f)$.","title":"Finite iteration from the whole type reaches a fixed point"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/lfp-fin-le","kind":"node","lean":"Finset.lfpFin_le","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#lfp-fin-le","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with $|\\alpha|$ elements, let $f$ be a map from subsets of $\\alpha$ to subsets of $\\alpha$ that is monotone for inclusion, and let $\\mathrm{lfp}_{\\mathrm{fin}}(f) = f^{|\\alpha|}(\\emptyset)$ be the result of applying $f$ to $\\emptyset$ $|\\alpha|$ times. If $Z \\subseteq \\alpha$ satisfies $f(Z) \\subseteq Z$, then $\\mathrm{lfp}_{\\mathrm{fin}}(f) \\subseteq Z$.","title":"The finite least fixed point lies in every pre-fixed point"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/le-gfp-fin","kind":"node","lean":"Finset.le_gfpFin","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#le-gfp-fin","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with $|\\alpha|$ elements, let $f$ be a map from subsets of $\\alpha$ to subsets of $\\alpha$ that is monotone for inclusion, and let $\\mathrm{gfp}_{\\mathrm{fin}}(f) = f^{|\\alpha|}(\\alpha)$ be the result of applying $f$ to $\\alpha$ $|\\alpha|$ times. If $Z \\subseteq \\alpha$ satisfies $Z \\subseteq f(Z)$, then $Z \\subseteq \\mathrm{gfp}_{\\mathrm{fin}}(f)$.","title":"Every post-fixed point lies in the finite greatest fixed point"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/mem-finset-iterate-iff","kind":"node","lean":"Finset.mem_finset_iterate_iff","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#mem-finset-iterate-iff","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a type, let $f$ map finite subsets of $\\alpha$ to finite subsets of $\\alpha$, and let $F$ map predicates on $\\alpha$ to predicates on $\\alpha$. For a finite set $Z \\subseteq \\alpha$ write $P_Z$ for the predicate $t \\mapsto t \\in Z$, and suppose that for every finite $Z \\subseteq \\alpha$ and every $s \\in \\alpha$, $s \\in f(Z)$ if and only if $F(P_Z)(s)$ holds. Write $f^n$ and $F^n$ for the $n$-fold\u2026","title":"Finset and predicate iterations from the bottom agree"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/mem-finset-iterate-univ-iff","kind":"node","lean":"Finset.mem_finset_iterate_univ_iff","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#mem-finset-iterate-univ-iff","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type, let $f$ map subsets of $\\alpha$ to subsets of $\\alpha$, and let $F$ map predicates on $\\alpha$ to predicates on $\\alpha$. For $Z \\subseteq \\alpha$ write $P_Z$ for the predicate $t \\mapsto t \\in Z$, and suppose that for every $Z \\subseteq \\alpha$ and every $s \\in \\alpha$, $s \\in f(Z)$ if and only if $F(P_Z)(s)$ holds. Write $f^n$ and $F^n$ for the $n$-fold iterates and $\\top$ for the\u2026","title":"Finset and predicate iterations from the top agree"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/mem-lfp-fin-iff-lfp","kind":"node","lean":"Finset.mem_lfpFin_iff_lfp","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#mem-lfp-fin-iff-lfp","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with $|\\alpha|$ elements, let $f$ be a map from subsets of $\\alpha$ to subsets of $\\alpha$ that is monotone for inclusion, and let $\\mathrm{lfp}_{\\mathrm{fin}}(f) = f^{|\\alpha|}(\\emptyset)$ be the result of applying $f$ to $\\emptyset$ $|\\alpha|$ times. Let $F$ be a map from predicates on $\\alpha$ to predicates on $\\alpha$ that is monotone for the pointwise order. For\u2026","title":"Finite iteration computes the least fixed point"},{"id":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/mem-gfp-fin-iff-gfp","kind":"node","lean":"Finset.mem_gfpFin_iff_gfp","parent":1243,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets/index.html#mem-gfp-fin-iff-gfp","scope":"Order/fixed-points-domain-theory/least-and-greatest-fixed-points-on-finite-sets","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with $|\\alpha|$ elements, let $f$ be a map from subsets of $\\alpha$ to subsets of $\\alpha$ that is monotone for inclusion, and let $\\mathrm{gfp}_{\\mathrm{fin}}(f) = f^{|\\alpha|}(\\alpha)$ be the result of applying $f$ to $\\alpha$ $|\\alpha|$ times. Let $F$ be a map from predicates on $\\alpha$ to predicates on $\\alpha$ that is monotone for the pointwise order. For $Z \\subseteq \\alpha$\u2026","title":"Finite iteration computes the greatest fixed point"},{"data_url":"scopes/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points.json","id":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","kind":"scope","navigation":"module","parent":1242,"result":false,"route":"../roadmap/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/index.html","scope":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","status":"fully_proved","title":"Scott-continuous maps and least fixed points"},{"id":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/scott-continuous-of-apply","kind":"node","lean":"OmegaCompletePartialOrder.ContinuousHom.\u03c9ScottContinuous_of_apply","parent":1254,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/index.html#scott-continuous-of-apply","scope":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","status":"fully_proved","summary":"Let $\\alpha$, $\\beta$, $\\gamma$ be \u03c9-complete partial orders, and let $g$ be a map from $\\alpha$ to the \u03c9-complete partial order of Scott-continuous maps $\\beta \\to \\gamma$, ordered pointwise. If for every $b \\in \\beta$ the map $a \\mapsto g(a)(b)$ is Scott-continuous, then $g$ is Scott-continuous. Here a map is Scott-continuous when it is monotone and preserves suprema of \u03c9-chains.","title":"Scott continuity into a function space is checked pointwise"},{"id":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/curry","kind":"node","lean":"OmegaCompletePartialOrder.ContinuousHom.curry","parent":1254,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/index.html#curry","scope":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","status":"fully_proved","summary":"Let $\\alpha$, $\\beta$, $\\gamma$ be \u03c9-complete partial orders and let $f : \\alpha \\times \\beta \\to \\gamma$ be a Scott-continuous map, that is, a map that is monotone and preserves suprema of \u03c9-chains. Its currying is the Scott-continuous map from $\\alpha$ to the \u03c9-complete partial order of Scott-continuous maps $\\beta \\to \\gamma$, ordered pointwise, that sends $a$ to the map $b \\mapsto f(a, b)$.","title":"Currying of Scott-continuous maps"},{"id":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/lfp","kind":"node","lean":"OmegaCompletePartialOrder.ContinuousHom.lfp","parent":1254,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/index.html#lfp","scope":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","status":"fully_proved","summary":"Let $D$ be an \u03c9-complete partial order with a least element $\\bot$, and let $f : D \\to D$ be Scott-continuous, that is, monotone and preserving suprema of \u03c9-chains. The least fixed point $\\operatorname{lfp} f$ of $f$ is the supremum of the Kleene chain $\\bot \\le f(\\bot) \\le f(f(\\bot)) \\le \\cdots$, whose $n$-th term is the $n$-fold iterate $f^n(\\bot)$.","title":"Least fixed point of a Scott-continuous map"},{"id":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/lfp-eq","kind":"node","lean":"OmegaCompletePartialOrder.ContinuousHom.lfp_eq","parent":1254,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/index.html#lfp-eq","scope":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","status":"fully_proved","summary":"Let $D$ be an \u03c9-complete partial order with a least element $\\bot$, and let $f : D \\to D$ be Scott-continuous, that is, monotone and preserving suprema of \u03c9-chains. Then the least fixed point $\\operatorname{lfp} f$ of $f$ satisfies $f(\\operatorname{lfp} f) = \\operatorname{lfp} f$.","title":"The least fixed point is a fixed point"},{"id":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/lfp-le","kind":"node","lean":"OmegaCompletePartialOrder.ContinuousHom.lfp_le","parent":1254,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/index.html#lfp-le","scope":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","status":"fully_proved","summary":"Let $D$ be an \u03c9-complete partial order with a least element $\\bot$, let $f : D \\to D$ be Scott-continuous, that is, monotone and preserving suprema of \u03c9-chains, and let $\\operatorname{lfp} f$ be the supremum of the Kleene chain $\\bot \\le f(\\bot) \\le f(f(\\bot)) \\le \\cdots$. If $a \\in D$ satisfies $f(a) \\le a$, then $\\operatorname{lfp} f \\le a$.","title":"The least fixed point lies below every prefixed point"},{"id":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/lfp-scott-continuous","kind":"node","lean":"OmegaCompletePartialOrder.ContinuousHom.lfp_\u03c9ScottContinuous","parent":1254,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/index.html#lfp-scott-continuous","scope":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","status":"fully_proved","summary":"Let $D$ be an \u03c9-complete partial order with a least element $\\bot$. For a Scott-continuous map $f : D \\to D$, that is, one that is monotone and preserves suprema of \u03c9-chains, let $\\operatorname{lfp} f$ be its least fixed point. The map $f \\mapsto \\operatorname{lfp} f$, from the \u03c9-complete partial order of Scott-continuous maps $D \\to D$, ordered pointwise, to $D$, is Scott-continuous.","title":"Scott continuity of the least-fixed-point operator"},{"id":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/lfp-hom","kind":"node","lean":"OmegaCompletePartialOrder.ContinuousHom.lfpHom","parent":1254,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points/index.html#lfp-hom","scope":"Order/fixed-points-domain-theory/scott-continuous-maps-and-least-fixed-points","status":"fully_proved","summary":"Let $D$ be an \u03c9-complete partial order with a least element $\\bot$. The least-fixed-point operator, sending a Scott-continuous map $f : D \\to D$ to its least fixed point $\\operatorname{lfp} f$, bundled as a Scott-continuous map from the \u03c9-complete partial order of Scott-continuous maps $D \\to D$, ordered pointwise, to $D$.","title":"The least-fixed-point operator as a Scott-continuous map"},{"data_url":"scopes/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations.json","id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","kind":"scope","navigation":"module","parent":1242,"result":false,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","title":"Transitive closure of finite relations"},{"id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/post-fin","kind":"node","lean":"Relation.postFin","parent":1262,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html#post-fin","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","summary":"Let $\\alpha$ be a finite type and let $r$ be a decidable relation on $\\alpha$. For a finite set $Z \\subseteq \\alpha$, the set of $r$-successors of $Z$ is the finite set $\\mathrm{post}_r(Z) = \\{t \\in \\alpha \\mid r(z, t) \\text{ for some } z \\in Z\\}$.","title":"Successors of a finite set"},{"id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/trans-gen-step","kind":"node","lean":"Relation.transGenStep","parent":1262,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html#trans-gen-step","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with decidable equality, let $r$ be a decidable relation on $\\alpha$, and write $\\mathrm{post}_r(Z)$ for the set of $r$-successors of elements of a finite set $Z \\subseteq \\alpha$. For a finite set $A \\subseteq \\alpha$, the step map sends a finite set $Z \\subseteq \\alpha$ to $\\mathrm{post}_r(A) \\cup \\mathrm{post}_r(Z)$.","title":"One step of the reachability iteration"},{"id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/trans-gen-fin-from","kind":"node","lean":"Relation.transGenFinFrom","parent":1262,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html#trans-gen-fin-from","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with decidable equality, let $r$ be a decidable relation on $\\alpha$, and let $A \\subseteq \\alpha$ be a finite set. Write $\\mathrm{post}_r(Z)$ for the set of $r$-successors of elements of a finite set $Z \\subseteq \\alpha$, and let $f$ be the step map $f(Z) = \\mathrm{post}_r(A) \\cup \\mathrm{post}_r(Z)$. The finite set of elements reachable from $A$ is $f^{|\\alpha|}(\\emptyset)$, obtained\u2026","title":"Elements reachable from a finite set of sources"},{"id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/trans-gen-fin","kind":"node","lean":"Relation.transGenFin","parent":1262,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html#trans-gen-fin","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with decidable equality, let $r$ be a decidable relation on $\\alpha$, and let $a \\in \\alpha$. The finite set of elements reachable from $a$ is the finite set of elements reachable from the source set $\\{a\\}$, that is, the elements reachable from $a$ in one or more $r$-steps.","title":"Elements reachable from a point"},{"id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/mem-trans-gen-fin-from","kind":"node","lean":"Relation.mem_transGenFinFrom","parent":1262,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html#mem-trans-gen-fin-from","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with decidable equality, let $r$ be a decidable relation on $\\alpha$, and let $A \\subseteq \\alpha$ be a finite set. An element $t \\in \\alpha$ lies in the finite set of elements reachable from $A$ in one or more $r$-steps (computed as a finite least fixed point) if and only if there is some $a \\in A$ such that $(a, t)$ lies in the transitive closure of $r$.","title":"Membership in the set reachable from finitely many sources"},{"id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/mem-trans-gen-fin","kind":"node","lean":"Relation.mem_transGenFin","parent":1262,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html#mem-trans-gen-fin","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with decidable equality, let $r$ be a decidable relation on $\\alpha$, and let $a \\in \\alpha$. An element $t \\in \\alpha$ lies in the finite set of elements reachable from $a$ in one or more $r$-steps (computed as a finite least fixed point) if and only if $(a, t)$ lies in the transitive closure of $r$.","title":"Membership in the set reachable from a point"},{"id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/decidable-trans-gen","kind":"node","lean":"Relation.decidableTransGen","parent":1262,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html#decidable-trans-gen","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with decidable equality and let $r$ be a decidable relation on $\\alpha$. Then the transitive closure of $r$ is decidable: for all $a, t \\in \\alpha$ it is decidable whether $t$ is reachable from $a$ in one or more $r$-steps.","title":"Decidability of the transitive closure on a finite type"},{"id":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/decidable-refl-trans-gen","kind":"node","lean":"Relation.decidableReflTransGen","parent":1262,"result":true,"route":"../roadmap/Order/fixed-points-domain-theory/transitive-closure-of-finite-relations/index.html#decidable-refl-trans-gen","scope":"Order/fixed-points-domain-theory/transitive-closure-of-finite-relations","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with decidable equality and let $r$ be a decidable relation on $\\alpha$. Then the reflexive-transitive closure of $r$ is decidable: for all $a, t \\in \\alpha$ it is decidable whether $t$ is reachable from $a$ in zero or more $r$-steps.","title":"Decidability of the reflexive-transitive closure on a finite type"},{"area":"Foundations & Logic","data_url":"chapters/CategoryTheory.json","id":"CategoryTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/CategoryTheory/index.html","scope":"CategoryTheory","status":"can_prove","summary":"Category theory organizes mathematical structures through morphisms, functors, universal properties, limits, and adjunctions.","title":"Category Theory"},{"data_url":"scopes/CategoryTheory/monoidal-categories.json","id":"CategoryTheory/monoidal-categories","kind":"scope","parent":1271,"result":false,"route":"../roadmap/CategoryTheory/monoidal-categories/index.html","scope":"CategoryTheory/monoidal-categories","status":"fully_proved","summary":"Monoidal categories equip a category with a tensor product and unit object, with braided and symmetric variants governing how tensor factors may be exchanged.","title":"Monoidal categories"},{"id":"CategoryTheory/monoidal-categories/nonempty-iso-of-tensor-iso-tensor-unit","kind":"node","lean":"CategoryTheory.MonoidalCategory.nonempty_iso_of_tensor_iso_tensorUnit","parent":1272,"result":true,"route":"../roadmap/CategoryTheory/monoidal-categories/index.html#nonempty-iso-of-tensor-iso-tensor-unit","scope":"CategoryTheory/monoidal-categories","status":"fully_proved","summary":"In a braided monoidal category, tensoring with the unit is cancellative up to isomorphism: if `M \u2297 N \u2245 \ud835\udfd9`, `M' \u2297 N' \u2245 \ud835\udfd9`, and `M \u2245 M'`, then `N \u2245 N'`.","title":"Tensor inverses are unique up to isomorphism in braided categories"},{"data_url":"scopes/CategoryTheory/topos-theory.json","id":"CategoryTheory/topos-theory","kind":"scope","parent":1271,"result":false,"route":"../roadmap/CategoryTheory/topos-theory/index.html","scope":"CategoryTheory/topos-theory","status":"can_prove","summary":"Topos theory studies elementary and Grothendieck toposes, categories with finite limits, exponentials and a subobject classifier that behave like universes of sets.","title":"Topos theory"},{"data_url":"scopes/CategoryTheory/topos-theory/elementary-toposes.json","id":"CategoryTheory/topos-theory/elementary-toposes","kind":"scope","navigation":"module","parent":1274,"result":false,"route":"../roadmap/CategoryTheory/topos-theory/elementary-toposes/index.html","scope":"CategoryTheory/topos-theory/elementary-toposes","status":"can_prove","title":"Elementary toposes"},{"id":"CategoryTheory/topos-theory/elementary-toposes/is-topos","kind":"node","lean":"MathlibExt.CategoryTheory.FundamentalToposWanted.ToposTheory.IsTopos","parent":1275,"result":true,"route":"../roadmap/CategoryTheory/topos-theory/elementary-toposes/index.html#is-topos","scope":"CategoryTheory/topos-theory/elementary-toposes","status":"fully_proved","summary":"A category $E$ is an elementary topos if it has all finite limits, it admits a cartesian monoidal structure for which it is monoidal closed (that is, $E$ is cartesian closed), and it has a subobject classifier.","title":"Elementary topos"},{"id":"CategoryTheory/topos-theory/elementary-toposes/fundamental-topos-theory","kind":"node","lean":"MathlibExt.CategoryTheory.FundamentalToposWanted.ToposTheory.fundamental_topos_theory","parent":1275,"result":true,"route":"../roadmap/CategoryTheory/topos-theory/elementary-toposes/index.html#fundamental-topos-theory","scope":"CategoryTheory/topos-theory/elementary-toposes","status":"can_prove","summary":"If $E$ is an elementary topos and $X$ is an object of $E$, then the slice category $E/X$ of objects of $E$ over $X$ is again an elementary topos.","title":"Fundamental theorem of topos theory"},{"area":"Algebra & Number Theory","data_url":"chapters/Algebra.json","id":"Algebra","kind":"scope","parent":0,"result":false,"route":"../roadmap/Algebra/index.html","scope":"Algebra","status":"can_prove","summary":"Algebra studies structures defined by operations and the laws, morphisms, and symmetries that organize them.","title":"Algebra"},{"data_url":"scopes/Algebra/polynomials.json","id":"Algebra/polynomials","kind":"scope","parent":1278,"result":false,"route":"../roadmap/Algebra/polynomials/index.html","scope":"Algebra/polynomials","status":"can_prove","summary":"Univariate and multivariate polynomials and formal power series as algebraic objects: coefficients, derivations, symmetric polynomials, irreducibility, polynomial maps and rational series.","title":"Polynomials and power series"},{"id":"Algebra/polynomials/symmetric-polynomial-in-esymm","kind":"node","lean":"Polynomial.symmetric_polynomial_in_esymm","parent":1279,"result":true,"route":"../roadmap/Algebra/polynomials/index.html#symmetric-polynomial-in-esymm","scope":"Algebra/polynomials","status":"fully_proved","summary":"A symmetric polynomial is a polynomial in the elementary symmetric polynomials, with a single witness `q` depending only on `f`; consequently, for every monic split polynomial `P`, that same `q` evaluated at the signed coefficients of `P` equals `f` evaluated at the roots of `P`. This is the book's corollary that a symmetric polynomial in the roots of `P` can be written as a polynomial in the coefficients of `P`;\u2026","title":"Symmetric polynomials in the roots are polynomials in the coefficients"},{"data_url":"scopes/Algebra/polynomials/alternating-binomial-sums-of-rational-functions.json","id":"Algebra/polynomials/alternating-binomial-sums-of-rational-functions","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/alternating-binomial-sums-of-rational-functions/index.html","scope":"Algebra/polynomials/alternating-binomial-sums-of-rational-functions","status":"can_prove","title":"Alternating binomial sums of rational functions"},{"id":"Algebra/polynomials/alternating-binomial-sums-of-rational-functions/x-sub-one-pow-dvd-alternating-sum-prod-x","kind":"node","lean":"MetaMathlibExt.X_sub_one_pow_dvd_alternating_sum_prod_X_pow_add_C","parent":1281,"reading_title":{"format":"inline-parts-v1","parts":[["text","The factor "],["math","\\((x - 1)^n\\)"],["text"," of an alternating binomial sum"]],"source":"The factor $(x - 1)^n$ of an alternating binomial sum"},"result":true,"route":"../roadmap/Algebra/polynomials/alternating-binomial-sums-of-rational-functions/index.html#x-sub-one-pow-dvd-alternating-sum-prod-x","scope":"Algebra/polynomials/alternating-binomial-sums-of-rational-functions","status":"can_prove","summary":"For every real number $a$ and every natural number $n$, the polynomial $(x - 1)^n$ divides, in $\\mathbb{R}[x]$, the polynomial","title":"The factor $(x - 1)^n$ of an alternating binomial sum"},{"id":"Algebra/polynomials/alternating-binomial-sums-of-rational-functions/x-sub-one-pow-dvd-alternating-sum-prod-c","kind":"node","lean":"MetaMathlibExt.X_sub_one_pow_dvd_alternating_sum_prod_comp_X_pow","parent":1281,"reading_title":{"format":"inline-parts-v1","parts":[["text","The factor "],["math","\\((x - 1)^n\\)"],["text"," for polynomials with distinct real roots"]],"source":"The factor $(x - 1)^n$ for polynomials with distinct real roots"},"result":true,"route":"../roadmap/Algebra/polynomials/alternating-binomial-sums-of-rational-functions/index.html#x-sub-one-pow-dvd-alternating-sum-prod-c","scope":"Algebra/polynomials/alternating-binomial-sums-of-rational-functions","status":"can_prove","summary":"Let $c$ be a real number, let $s$ be a finite set of real numbers, and let $f(x) = c \\prod_{r \\in s} (x - r)$ in $\\mathbb{R}[x]$. For every natural number $n$, the polynomial $(x - 1)^n$ divides the polynomial","title":"The factor $(x - 1)^n$ for polynomials with distinct real roots"},{"data_url":"scopes/Algebra/polynomials/binomial-convolutions-of-rational-power-series.json","id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","title":"Binomial convolutions of rational power series"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/binomial-product","kind":"node","lean":"MetaMathlibExt.binomialProduct","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#binomial-product","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"fully_proved","summary":"Let $K$ be a commutative ring. The binomial product of formal power series $A = \\sum_n a_n X^n$ and $B = \\sum_n b_n X^n$ in $K[[X]]$ is the power series $\\sum_n c_n X^n$ with $c_n = \\sum_{k=0}^{n} \\binom{n}{k} a_k b_{n-k}$.","title":"Binomial (Hurwitz) product of power series"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/coeff-binomial-product","kind":"node","lean":"MetaMathlibExt.coeff_binomialProduct","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#coeff-binomial-product","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"fully_proved","summary":"Let $K$ be a commutative ring and let $A = \\sum_n a_n X^n$ and $B = \\sum_n b_n X^n$ be formal power series over $K$. For every natural number $n$, the coefficient of $X^n$ in the binomial product of $A$ and $B$ is $\\sum_{k=0}^{n} \\binom{n}{k} a_k b_{n-k}$.","title":"Coefficients of the binomial product"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/constant-coeff-binomial-product","kind":"node","lean":"MetaMathlibExt.constantCoeff_binomialProduct","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#constant-coeff-binomial-product","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"fully_proved","summary":"Let $K$ be a commutative ring. For formal power series $A$ and $B$ over $K$, the constant coefficient of the binomial product of $A$ and $B$ is the product of the constant coefficients of $A$ and $B$.","title":"Constant coefficient of the binomial product"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/exists-polynomial-quotient-of-extension","kind":"node","lean":"MetaMathlibExt.exists_polynomial_quotient_of_extension","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#exists-polynomial-quotient-of-extension","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $F$ be a field, let $G$ be a field extension of $F$ (a field with an $F$-algebra structure), and let $A$ be a power series over $F$. If there are polynomials $p$, $q$ over $G$ with $q \\ne 0$ and $A q = p$, where $A$ is viewed as a power series over $G$, then there are polynomials $P$, $Q$ over $F$ with $Q \\ne 0$ and $A Q = P$.","title":"Rationality of power series descends from field extensions"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/exists-orthogonal-of-orthogonal-over-ext","kind":"node","lean":"MetaMathlibExt.exists_orthogonal_of_orthogonal_over_extension","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#exists-orthogonal-of-orthogonal-over-ext","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $F$ be a field, let $G$ be a field extension of $F$ (a field with an $F$-algebra structure), let $n \\in \\mathbb{N}$ and let $S \\subseteq F^n$. If some nonzero $t \\in G^n$ satisfies $\\sum_{i=1}^n s_i t_i = 0$ in $G$ for every $s \\in S$, then some nonzero $u \\in F^n$ satisfies $\\sum_{i=1}^n s_i u_i = 0$ for every $s \\in S$.","title":"Orthogonal vectors descend from field extensions"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/binomial-product-x-pow-mul-inv-pow","kind":"node","lean":"MetaMathlibExt.binomialProduct_X_pow_mul_inv_pow","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#binomial-product-x-pow-mul-inv-pow","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"For every field $K$ of characteristic $0$, all $j, k \\in \\mathbb{N}$ and all $\\alpha, \\beta \\in K$,","title":"Binomial product of shifted powers of geometric series"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/binomial-product-inv-one-sub","kind":"node","lean":"MetaMathlibExt.binomialProduct_inv_one_sub","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#binomial-product-inv-one-sub","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"For every field $K$ of characteristic $0$, every power series $A(x)$ over $K$ and every $\\beta \\in K$,","title":"Binomial product with a geometric series"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/binomial-product-rational","kind":"node","lean":"MetaMathlibExt.binomialProduct_rational","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#binomial-product-rational","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $K$ be a field of characteristic $0$, let $\\alpha_1, \\dots, \\alpha_m, \\beta_1, \\dots, \\beta_n \\in K$, not necessarily distinct or nonzero, and let $R$, $S$ be polynomials over $K$ with $\\deg R < m$ and $\\deg S < n$. If $A$, $B$ are power series over $K$ with $A(x) \\prod_{i=1}^m (1 - \\alpha_i x) = R(x)$ and $B(x) \\prod_{j=1}^n (1 - \\beta_j x) = S(x)$, then there is a polynomial $T$ over $K$ with $\\deg T < mn$\u2026","title":"Binomial products of proper rational series"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/binomial-product-rational-general","kind":"node","lean":"MetaMathlibExt.binomialProduct_rational_general","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#binomial-product-rational-general","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $K$ be a field of characteristic $0$, let $\\alpha_1, \\dots, \\alpha_m$ and $\\beta_1, \\dots, \\beta_n$ be nonzero elements of $K$, and let $R$, $S$ be polynomials over $K$. Let $A$, $B$ be power series over $K$ with $A(x) \\prod_{i=1}^m (1 - \\alpha_i x) = R(x)$ and $B(x) \\prod_{j=1}^n (1 - \\beta_j x) = S(x)$. Put $u = \\max(d_R + 1 - m, 0)$ and $v = \\max(d_S + 1 - n, 0)$, where $d_R$ and $d_S$ are the degrees of $R$\u2026","title":"Binomial products of arbitrary rational series"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/binomial-product-x-pow","kind":"node","lean":"MetaMathlibExt.binomialProduct_X_pow","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#binomial-product-x-pow","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"For every field $K$ of characteristic $0$, every $m \\in \\mathbb{N}$ and every power series $A(x)$ over $K$,","title":"Binomial product with a monomial"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/prod-one-sub-add-eq-resultant","kind":"node","lean":"MetaMathlibExt.prod_one_sub_add_eq_resultant","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#prod-one-sub-add-eq-resultant","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $K$ be a field of characteristic $0$, let $m, n \\ge 0$ be integers, and let $\\alpha_1, \\dots, \\alpha_m$ and $\\beta_1, \\dots, \\beta_n$ be elements of $K$. Regard $f(y) = \\prod_{i=1}^{m} (1 - y - \\alpha_i x)$ and $g(y) = \\prod_{j=1}^{n} (y - \\beta_j x)$ as polynomials in $y$ with coefficients in $K[x]$, and let $\\operatorname{Res}_{m,n}(f, g)$ be the determinant of their Sylvester matrix formed with the formal\u2026","title":"Resultant identity for pairwise sums"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/binomial-product-lucas-v-generating-func","kind":"node","lean":"MetaMathlibExt.binomialProduct_lucasV_generatingFunction","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#binomial-product-lucas-v-generating-func","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $K$ be a field of characteristic $0$, let $a, b \\in K$, and let $g(x) = \\sum_{n \\ge 0} G_n x^n$ be a power series over $K$ with $g(x)(1 - ax - bx^2) = 2 - ax$. Then","title":"Binomial square of the Lucas generating function"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/binomial-product-cubic-denominator","kind":"node","lean":"MetaMathlibExt.binomialProduct_cubic_denominator","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#binomial-product-cubic-denominator","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $K$ be a field of characteristic $0$, let $A, B, C \\in K$ with $C \\ne 0$, and let $D(x) = 1 + Ax + Bx^2 + Cx^3$. Suppose that $D(x) = \\prod_{i=1}^{3} (1 - \\alpha_i x)$ with $\\alpha_1, \\alpha_2, \\alpha_3 \\in K$ and that $2\\alpha_i \\ne \\alpha_j + \\alpha_k$ whenever $i, j, k \\in \\{1, 2, 3\\}$ are not all equal. Let $r$ and $s$ be power series over $K$ such that $r D$ and $s D$ are polynomials of degree less than\u2026","title":"Binomial products of series with a cubic denominator"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/hadamard-product-rational","kind":"node","lean":"MetaMathlibExt.hadamardProduct_rational","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#hadamard-product-rational","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $K$ be a field of characteristic $0$, let $\\alpha_1, \\dots, \\alpha_m, \\beta_1, \\dots, \\beta_n \\in K$, and let $A = \\sum_k a_k x^k$ and $B = \\sum_k b_k x^k$ be power series over $K$ such that $A(x) \\prod_{i=1}^m (1 - \\alpha_i x)$ and $B(x) \\prod_{j=1}^n (1 - \\beta_j x)$ are polynomials. Then their Hadamard product $A * B = \\sum_k a_k b_k x^k$ has the property that\u2026","title":"Hadamard products of rational series"},{"id":"Algebra/polynomials/binomial-convolutions-of-rational-power-series/prod-one-sub-mul-eq-resultant","kind":"node","lean":"MetaMathlibExt.prod_one_sub_mul_eq_resultant","parent":1284,"result":true,"route":"../roadmap/Algebra/polynomials/binomial-convolutions-of-rational-power-series/index.html#prod-one-sub-mul-eq-resultant","scope":"Algebra/polynomials/binomial-convolutions-of-rational-power-series","status":"can_prove","summary":"Let $K$ be a field of characteristic $0$, let $m, n \\ge 0$ be integers, and let $\\alpha_1, \\dots, \\alpha_m$ and $\\beta_1, \\dots, \\beta_n$ be elements of $K$. Regard $f(y) = \\prod_{i=1}^{m} (1 - \\alpha_i y)$ and $g(y) = \\prod_{j=1}^{n} (y - \\beta_j x)$ as polynomials in $y$ with coefficients in $K[x]$, and let $\\operatorname{Res}_{m,n}(f, g)$ be the determinant of their Sylvester matrix formed with the formal\u2026","title":"Resultant identity for pairwise products"},{"data_url":"scopes/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials.json","id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","title":"Coefficient L1 norm of multivariate polynomials"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm","kind":"node","lean":"MvPolynomial.coeffL1Norm","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"The rational sum of the absolute values of a polynomial's coefficients.","title":"Coefficient L1 norm of a multivariate polynomial"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-eq-sum","kind":"node","lean":"MvPolynomial.coeffL1Norm_eq_sum","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-eq-sum","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"Enlarging the summation set beyond the support does not change the sum.","title":"Coefficient L1 norm as a sum over a larger index set"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-nonneg","kind":"node","lean":"MvPolynomial.coeffL1Norm_nonneg","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-nonneg","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"The coefficient L1 norm is nonnegative.","title":"Nonnegativity of the coefficient L1 norm"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-zero","kind":"node","lean":"MvPolynomial.coeffL1Norm_zero","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-zero","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"The zero polynomial has coefficient L1 norm zero.","title":"Coefficient L1 norm of the zero polynomial"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-add-le","kind":"node","lean":"MvPolynomial.coeffL1Norm_add_le","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-add-le","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"Adding polynomials increases the coefficient L1 norm by at most the sum.","title":"Triangle inequality for the coefficient L1 norm"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-sum-le","kind":"node","lean":"MvPolynomial.coeffL1Norm_sum_le","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-sum-le","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"The norm of a finite sum is bounded by the sum of the coefficient norms.","title":"Coefficient L1 norm of a finite sum"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-smul","kind":"node","lean":"MvPolynomial.coeffL1Norm_smul","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-smul","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"Rational scalar multiplication scales the norm by the scalar's absolute value.","title":"Coefficient L1 norm of a scalar multiple"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-monomial","kind":"node","lean":"MvPolynomial.coeffL1Norm_monomial","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-monomial","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"A monomial's coefficient L1 norm is the absolute value of its coefficient, including when the coefficient is zero.","title":"Coefficient L1 norm of a monomial"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-monomial-le","kind":"node","lean":"MvPolynomial.coeffL1Norm_monomial_le","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-monomial-le","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"The coefficient L1 norm of a monomial is at most its absolute coefficient.","title":"Upper bound on the coefficient L1 norm of a monomial"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/coeff-l1-norm-mul-le","kind":"node","lean":"MvPolynomial.coeffL1Norm_mul_le","parent":1300,"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#coeff-l1-norm-mul-le","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"The coefficient convolution in a product is bounded by the product of the coefficient L1 norms; cancellation can make this inequality strict.","title":"Submultiplicativity of the coefficient L1 norm"},{"id":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/dist-coeff-zero-eq-coeff-l1-norm","kind":"node","lean":"MvPolynomial.dist_coeff_zero_eq_coeffL1Norm","parent":1300,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficient L1 norm as an "],["math","\\(\\ell^1\\)"],["text"," distance"]],"source":"Coefficient L1 norm as an $\\ell^1$ distance"},"result":true,"route":"../roadmap/Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials/index.html#dist-coeff-zero-eq-coeff-l1-norm","scope":"Algebra/polynomials/coefficient-l1-norm-of-multivariate-polynomials","status":"fully_proved","summary":"The real cast of the coefficient L1 norm agrees with the explicit exponent-one metric on the exact rational coefficient Finsupp carrier.","title":"Coefficient L1 norm as an $\\ell^1$ distance"},{"data_url":"scopes/Algebra/polynomials/coefficients-of-products-with-linear-factors.json","id":"Algebra/polynomials/coefficients-of-products-with-linear-factors","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","title":"Coefficients of products with linear factors"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/coeff-mul-eq-sum-support","kind":"node","lean":"MvPolynomial.IsHomogeneous.coeff_mul_eq_sum_support","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#coeff-mul-eq-sum-support","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"The coefficient of `p * q`, when `p` is homogeneous of degree one, is the sum obtained by removing one occurrence of each variable in the monomial.","title":"Coefficients of a product with a homogeneous linear polynomial"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/mul-linear-coeff-list","kind":"node","lean":"Polynomial.mulLinearCoeffList","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#mul-linear-coeff-list","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"The dense little-endian coefficients after left multiplication by `C r + X * C c`.","title":"Coefficient list after multiplication by a linear polynomial"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/length-mul-linear-coeff-list","kind":"node","lean":"Polynomial.length_mulLinearCoeffList","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#length-mul-linear-coeff-list","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"A linear-coefficient update always adds one dense coefficient.","title":"Length of the coefficient list after a linear multiplication"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/get-d-mul-linear-coeff-list","kind":"node","lean":"Polynomial.getD_mulLinearCoeffList","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#get-d-mul-linear-coeff-list","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"The coefficient at every index, including beyond the dense output, follows the source recurrence `r * l[j] + c * l[j - 1]`.","title":"Coefficient recurrence for multiplication by a linear polynomial"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/of-coeff-list-mul-linear-coeff-list","kind":"node","lean":"Polynomial.ofCoeffList_mulLinearCoeffList","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#of-coeff-list-mul-linear-coeff-list","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"Interpreting the dense output as a little-endian polynomial is left multiplication by the linear polynomial `C r + X * C c`.","title":"Coefficient-list multiplication agrees with polynomial multiplication"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/prod-c-add-x-mul-c-eq-sum","kind":"node","lean":"Finset.prod_C_add_X_mul_C_eq_sum","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#prod-c-add-x-mul-c-eq-sum","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"Expand a product of affine linear polynomials by the subset of factors from which the linear term is selected.","title":"Expansion of a product of affine linear polynomials"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/prod-c-add-x-mul-c-coeff","kind":"node","lean":"Finset.prod_C_add_X_mul_C_coeff","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#prod-c-add-x-mul-c-coeff","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"The coefficient of degree `i` in a product of affine linear polynomials is the sum over the `i` factors from which the linear term is selected.","title":"Coefficients of a product of affine linear polynomials"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/prod-c-add-x-mul-c-nat-degree-le","kind":"node","lean":"Finset.prod_C_add_X_mul_C_natDegree_le","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#prod-c-add-x-mul-c-nat-degree-le","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"A product of affine linear polynomials has degree at most the number of factors.","title":"Degree bound for a product of affine linear polynomials"},{"id":"Algebra/polynomials/coefficients-of-products-with-linear-factors/abs-prod-c-add-x-mul-c-coeff-le-choose","kind":"node","lean":"Finset.abs_prod_C_add_X_mul_C_coeff_le_choose","parent":1312,"result":true,"route":"../roadmap/Algebra/polynomials/coefficients-of-products-with-linear-factors/index.html#abs-prod-c-add-x-mul-c-coeff-le-choose","scope":"Algebra/polynomials/coefficients-of-products-with-linear-factors","status":"fully_proved","summary":"If every constant and linear coefficient is at most one in absolute value, then the absolute value of coefficient `i` is at most `S.card.choose i`.","title":"Binomial bound on coefficients of a product of affine polynomials"},{"data_url":"scopes/Algebra/polynomials/deep-power-sum-congruences.json","id":"Algebra/polynomials/deep-power-sum-congruences","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","title":"Deep power-sum congruences"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-divided-power-ideal","kind":"node","lean":"MetaMathlibExt.IsDividedPowerIdeal","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-divided-power-ideal","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"An ideal $I$ of a commutative ring $A$ is a *divided-power ideal* when for every $a \\in I$ and every integer $k \\ge 1$ there is $c \\in I$ with $a^k = k! \\, c$, that is, $a^k \\in k! \\, I$. For a prime $p$, in a torsion-free $\\mathbb{Z}_{(p)}$-algebra this says $a^k / k! \\in I$.","title":"Divided-power ideal"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-divided-power-ideal-bot","kind":"node","lean":"MetaMathlibExt.isDividedPowerIdeal_bot","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-divided-power-ideal-bot","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"In every commutative ring, the zero ideal is a divided-power ideal.","title":"The zero ideal is a divided-power ideal"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-divided-power-ideal-2","kind":"node","lean":"MetaMathlibExt.DividedPowers.isDividedPowerIdeal","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-divided-power-ideal-2","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"Let $I$ be an ideal of a commutative ring $A$. If $I$ carries a divided-power structure $(\\gamma_k)_{k \\ge 0}$ in the usual sense, then $I$ is a divided-power ideal: for every $a \\in I$ and every $k \\ge 1$ there is $c \\in I$ with $a^k = k!\\, c$.","title":"Ideals with divided powers are divided-power ideals"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-elem-symm","kind":"node","lean":"MetaMathlibExt.polyElemSymm","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-elem-symm","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"Let $P$ be a polynomial of degree $d$ over a commutative ring, with $d = 0$ for $P = 0$. For a natural number $n$, $e_n(P)$ is $(-1)^n$ times the coefficient of $X^{d - n}$ in $P$ if $n \\le d$, and $0$ if $n > d$. In particular $e_0(P) = 1$ when $P$ is monic.","title":"Elementary symmetric functions of a polynomial"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-elem-symm-zero","kind":"node","lean":"MetaMathlibExt.polyElemSymm_zero","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-elem-symm-zero","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"For every monic polynomial $P$ over a commutative ring, $e_0(P) = 1$.","title":"The zeroth elementary symmetric function of a monic polynomial"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-elem-symm-of-nat-degree-lt","kind":"node","lean":"MetaMathlibExt.polyElemSymm_of_natDegree_lt","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-elem-symm-of-nat-degree-lt","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"Let $P$ be a polynomial of degree $d$ over a commutative ring, with $d = 0$ for $P = 0$. Then $e_n(P) = 0$ for every $n > d$.","title":"Elementary symmetric functions vanish above the degree"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-power-sum","kind":"node","lean":"MetaMathlibExt.polyPowerSum","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-power-sum","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"For a polynomial $P$ over a commutative ring $A$, with elementary symmetric functions $e_i = e_i(P)$, the power sums $p_n = p_n(P) \\in A$ are defined recursively: $p_0$ is the degree of $P$ as a natural number ($0$ for $P = 0$), and for $n \\ge 1$ Newton's identity holds:","title":"Power sums of a polynomial"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-power-sum-zero","kind":"node","lean":"MetaMathlibExt.polyPowerSum_zero","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-power-sum-zero","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"For every polynomial $P$ over a commutative ring $A$, the power sum $p_0(P)$ equals the degree of $P$ as a natural number (taken to be $0$ for $P = 0$), viewed as an element of $A$.","title":"The zeroth power sum of a polynomial is its degree"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-power-sum-one","kind":"node","lean":"MetaMathlibExt.polyPowerSum_one","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-power-sum-one","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"For every polynomial $P$ over a commutative ring, $p_1(P) = e_1(P)$.","title":"The first power sum is the first elementary symmetric function"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-power-sum-succ","kind":"node","lean":"MetaMathlibExt.polyPowerSum_succ","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-power-sum-succ","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"For every polynomial $P$ over a commutative ring and every $n \\ge 0$, writing $e_i = e_i(P)$ and $p_i = p_i(P)$ for the elementary symmetric functions and power sums of $P$,","title":"Newton's recurrence for the power sums of a polynomial"},{"id":"Algebra/polynomials/deep-power-sum-congruences/pow-mem-prime-pow-padic-val-nat-factoria","kind":"node","lean":"MetaMathlibExt.pow_mem_prime_pow_padicValNat_factorial_iff","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#pow-mem-prime-pow-padic-val-nat-factoria","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime, $v_p$ the $p$-adic valuation, and $\\mathfrak{b}$ an ideal of a commutative ring $B$. The following are equivalent: for every $n \\ge 1$ and every $a \\in \\mathfrak{b}$ there is $c \\in \\mathfrak{b}$ with $a^n = p^{v_p(n!)} c$; and for every $a \\in \\mathfrak{b}$ there is $c \\in \\mathfrak{b}$ with $a^p = p c$.","title":"A prime-exponent criterion for powers in an ideal"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-divided-power-ideal-iff-pow-prime-mem","kind":"node","lean":"MetaMathlibExt.isDividedPowerIdeal_iff_pow_prime_mem","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-th power criterion for divided-power ideals"]],"source":"The $p$-th power criterion for divided-power ideals"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-divided-power-ideal-iff-pow-prime-mem","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime and let $A$ be a commutative ring in which every natural number not divisible by $p$ is a unit (a $\\mathbb{Z}_{(p)}$-algebra). An ideal $\\mathfrak{a}$ of $A$ is a divided-power ideal if and only if for every $a \\in \\mathfrak{a}$ there is $c \\in \\mathfrak{a}$ with $a^p = p\\,c$, that is, $a^p \\in p\\mathfrak{a}$.","title":"The $p$-th power criterion for divided-power ideals"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-divided-power-ideal-span-iff","kind":"node","lean":"MetaMathlibExt.isDividedPowerIdeal_span_iff","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-divided-power-ideal-span-iff","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime and $A$ a commutative ring in which every natural number not divisible by $p$ is a unit (a $\\mathbb{Z}_{(p)}$-algebra). For a subset $S \\subseteq A$, the ideal $\\mathfrak{a}$ generated by $S$ is a divided-power ideal if and only if for every $a \\in S$ there is $c \\in \\mathfrak{a}$ with $a^p = pc$, that is, $a^p \\in p\\mathfrak{a}$.","title":"Divided-power ideals generated by a set"},{"id":"Algebra/polynomials/deep-power-sum-congruences/mul","kind":"node","lean":"MetaMathlibExt.IsDividedPowerIdeal.mul","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#mul","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $A$ be a commutative ring. If $\\mathfrak{a}$ is a divided-power ideal of $A$, then for every ideal $\\mathfrak{b}$ of $A$ the product $\\mathfrak{a}\\mathfrak{b}$ is a divided-power ideal.","title":"Product of a divided-power ideal with an ideal"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-divided-power-ideal-maximal-ideal-pow","kind":"node","lean":"MetaMathlibExt.isDividedPowerIdeal_maximalIdeal_pow_iff","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-divided-power-ideal-maximal-ideal-pow","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime and let $O$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ such that $pO = \\mathfrak{m}^e$ for some integer $e \\ge 1$. Then for every $k \\ge 0$, the ideal $\\mathfrak{m}^k$ is a divided-power ideal if and only if $e \\le (p - 1)k$.","title":"Powers of the maximal ideal of a DVR that are divided-power ideals"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-divided-power-ideal-maximal-ideal-iff","kind":"node","lean":"MetaMathlibExt.isDividedPowerIdeal_maximalIdeal_iff","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-divided-power-ideal-maximal-ideal-iff","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime and let $O$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ such that $pO = \\mathfrak{m}^e$ for some integer $e \\ge 1$. Then $\\mathfrak{m}$ is a divided-power ideal if and only if $e \\le p - 1$.","title":"When the maximal ideal of a DVR is a divided-power ideal"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-divided-power-ideal-span-prime-padic","kind":"node","lean":"MetaMathlibExt.isDividedPowerIdeal_span_prime_padicInt","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((p)\\)"],["text"," is a divided-power ideal of "],["math","\\(\\mathbb{Z}_p\\)"]],"source":"$(p)$ is a divided-power ideal of $\\mathbb{Z}_p$"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-divided-power-ideal-span-prime-padic","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"For every prime $p$, the ideal generated by $p$ in the ring $\\mathbb{Z}_p$ of $p$-adic integers is a divided-power ideal.","title":"$(p)$ is a divided-power ideal of $\\mathbb{Z}_p$"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-power-sum-sub-mem-of-poly-elem-symm","kind":"node","lean":"MetaMathlibExt.polyPowerSum_sub_mem_of_polyElemSymm_sub_mem","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-power-sum-sub-mem-of-poly-elem-symm","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime and let $A$ be a commutative ring without additive torsion in which every natural number not divisible by $p$ is a unit (a torsion-free $\\mathbb{Z}_{(p)}$-algebra). Let $\\mathfrak{a}$ be a divided-power ideal of $A$, let $P, Q \\in A[X]$ be monic and let $N \\ge 1$; write $e_n$ and $p_n$ for the elementary symmetric functions and power sums of a polynomial. If $e_n(P) - e_n(Q) \\in \\mathfrak{a}$ for\u2026","title":"Congruent elementary symmetric functions give deep power-sum congruences"},{"id":"Algebra/polynomials/deep-power-sum-congruences/poly-elem-symm-sub-mem-of-poly-power-sum","kind":"node","lean":"MetaMathlibExt.polyElemSymm_sub_mem_of_polyPowerSum_sub_mem","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#poly-elem-symm-sub-mem-of-poly-power-sum","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime and $A$ a torsion-free commutative ring in which every natural number not divisible by $p$ is a unit (a torsion-free $\\mathbb{Z}_{(p)}$-algebra). Let $\\mathfrak{a}$ be a divided-power ideal of $A$, let $P, Q \\in A[X]$ be monic and let $N$ be a natural number. Suppose that for every $n$ with $1 \\le n \\le N$ there is $c \\in \\mathfrak{a}$ with $p_n(P) - p_n(Q) = n c$, that is,\u2026","title":"Power-sum congruences imply elementary symmetric congruences"},{"id":"Algebra/polynomials/deep-power-sum-congruences/exp-signed-power-sum-series-eq-sum-parti","kind":"node","lean":"MetaMathlibExt.exp_signedPowerSumSeries_eq_sum_partitions","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#exp-signed-power-sum-series-eq-sum-parti","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $\\sigma$ be a finite set of variables, and in $\\mathbb{Q}[x_i : i \\in \\sigma]$ let $p_s = \\sum_{i \\in \\sigma} x_i^s$ be the $s$-th power sum. Let $S$ be a set of natural numbers and let $P_S(t) = \\sum_{s \\in S,\\ s \\ge 1} \\frac{(-1)^{s-1}}{s} p_s t^s$. For a partition $\\lambda$ of $n$ with parts $\\lambda_1, \\dots, \\lambda_\\ell$, write $p_\\lambda = p_{\\lambda_1} \\cdots p_{\\lambda_\\ell}$,\u2026","title":"Exponential of a power-sum series restricted to a set of degrees"},{"id":"Algebra/polynomials/deep-power-sum-congruences/exp-signed-power-sum-series-eq-esymm-ser","kind":"node","lean":"MetaMathlibExt.exp_signedPowerSumSeries_eq_esymmSeries","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#exp-signed-power-sum-series-eq-esymm-ser","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $\\sigma$ be a finite set of variables, and in $\\mathbb{Q}[x_i : i \\in \\sigma]$ let $e_k$ be the $k$-th elementary symmetric polynomial and $p_n = \\sum_{i \\in \\sigma} x_i^n$ the $n$-th power sum. Then, as formal power series in $t$ with coefficients in $\\mathbb{Q}[x_i : i \\in \\sigma]$,","title":"The generating function of the elementary symmetric polynomials"},{"id":"Algebra/polynomials/deep-power-sum-congruences/esymm-eq-sum-partitions","kind":"node","lean":"MetaMathlibExt.esymm_eq_sum_partitions","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#esymm-eq-sum-partitions","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $\\sigma$ be a finite set of variables, and in $\\mathbb{Q}[x_i : i \\in \\sigma]$ let $e_n$ be the $n$-th elementary symmetric polynomial and $p_k = \\sum_{i \\in \\sigma} x_i^k$ the $k$-th power sum. For a partition $\\lambda$ of $n$ with parts $\\lambda_1, \\dots, \\lambda_\\ell$, write $p_\\lambda = p_{\\lambda_1} \\cdots p_{\\lambda_\\ell}$, $\\varepsilon_\\lambda = (-1)^{n - \\ell}$ and $z_\\lambda = \\prod_a a^{r_a} \\, r_a!$,\u2026","title":"Elementary symmetric polynomials in terms of power sums"},{"id":"Algebra/polynomials/deep-power-sum-congruences/padic-val-nat-multinomial-eq-sum-carries","kind":"node","lean":"MetaMathlibExt.padicValNat_multinomial_eq_sum_carries","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#padic-val-nat-multinomial-eq-sum-carries","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime and let $r_1, \\dots, r_k$ be natural numbers with sum $m$. For $i \\ge 1$ let $c_i = \\lfloor ((r_1 \\bmod p^i) + \\dots + (r_k \\bmod p^i)) / p^i \\rfloor$. Then","title":"Kummer's theorem for multinomial coefficients"},{"id":"Algebra/polynomials/deep-power-sum-congruences/padic-val-nat-sum-le-add-padic-val-nat-m","kind":"node","lean":"MetaMathlibExt.padicValNat_sum_le_add_padicValNat_multinomial","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#padic-val-nat-sum-le-add-padic-val-nat-m","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime, let $r_1, \\dots, r_k$ be natural numbers with sum $m$, and let $j$ be an index with $r_j \\ne 0$. Then","title":"A valuation bound from Kummer's theorem"},{"id":"Algebra/polynomials/deep-power-sum-congruences/pow-sub-pow-mem-of-sub-mem","kind":"node","lean":"MetaMathlibExt.pow_sub_pow_mem_of_sub_mem","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#pow-sub-pow-mem-of-sub-mem","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime, let $A$ be a commutative ring in which every natural number not divisible by $p$ is a unit (a $\\mathbb{Z}_{(p)}$-algebra), and let $\\mathfrak{a}$ be a divided-power ideal of $A$. If $x, y \\in A$ satisfy $x - y \\in \\mathfrak{a}$, then for every $m \\ge 0$ there is $c \\in \\mathfrak{a}$ with $x^{p^m} - y^{p^m} = p^m c$, and for every $n \\ge 0$ there is $c \\in \\mathfrak{a}$ with $x^n - y^n = n c$.","title":"Powers of elements congruent modulo a divided-power ideal"},{"id":"Algebra/polynomials/deep-power-sum-congruences/multiset-prod-sub-prod-mem-of-count-dvd","kind":"node","lean":"MetaMathlibExt.multiset_prod_sub_prod_mem_of_count_dvd","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruence of products over a partition with multiplicities divisible by "],["math","\\(p^v\\)"]],"source":"Congruence of products over a partition with multiplicities divisible by $p^v$"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#multiset-prod-sub-prod-mem-of-count-dvd","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime, $A$ a commutative ring and $\\mathfrak{a}$ a divided-power ideal of $A$. Let $f, g : \\mathbb{N} \\to A$ satisfy $f_n - g_n \\in \\mathfrak{a}$ for every $n \\ge 1$. For a partition $\\lambda = (\\lambda_1, \\dots, \\lambda_k)$ of a natural number $m$, write $f_\\lambda = f_{\\lambda_1} \\cdots f_{\\lambda_k}$ and $g_\\lambda = g_{\\lambda_1} \\cdots g_{\\lambda_k}$. If $v$ is a natural number such that every\u2026","title":"Congruence of products over a partition with multiplicities divisible by $p^v$"},{"id":"Algebra/polynomials/deep-power-sum-congruences/sum-moebius-divisors-not-dvd-prime","kind":"node","lean":"MetaMathlibExt.sum_moebius_divisors_not_dvd_prime","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#sum-moebius-divisors-not-dvd-prime","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime, $\\mu$ the M\u00f6bius function and $n \\ge 1$. The sum of $\\mu(d)$ over the divisors $d$ of $n$ with $p \\nmid d$ is $1$ if $n$ is a power of $p$ (including $n = 1 = p^0$) and $0$ otherwise.","title":"M\u00f6bius sum over the divisors prime to a given prime"},{"id":"Algebra/polynomials/deep-power-sum-congruences/artin-hasse-coeff-den-not-dvd","kind":"node","lean":"MetaMathlibExt.artinHasse_coeff_den_not_dvd","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Artin\u2013Hasse exponential has "],["math","\\(p\\)"],["text","-integral coefficients"]],"source":"The Artin\u2013Hasse exponential has $p$-integral coefficients"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#artin-hasse-coeff-den-not-dvd","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime and let $F(z) = \\exp\\big(\\sum_{j \\ge 0} z^{p^j} / p^j\\big) \\in \\mathbb{Q}[[z]]$ be the Artin\u2013Hasse exponential. For every $n \\ge 0$, the coefficient of $z^n$ in $F$, written in lowest terms, has a denominator not divisible by $p$.","title":"The Artin\u2013Hasse exponential has $p$-integral coefficients"},{"id":"Algebra/polynomials/deep-power-sum-congruences/artin-hasse-eq-prod-one-sub-pow","kind":"node","lean":"MetaMathlibExt.artinHasse_eq_prod_one_sub_pow","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#artin-hasse-eq-prod-one-sub-pow","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime, let $\\mu$ be the M\u00f6bius function and let $F(z) = \\exp\\big(\\sum_{j \\ge 0} z^{p^j} / p^j\\big) \\in \\mathbb{Q}[[z]]$ be the Artin\u2013Hasse exponential. For $d \\ge 1$ and $a \\in \\mathbb{Q}$ write $(1 - z^d)^a = \\exp\\big(a \\log(1 - z^d)\\big)$, where $\\log(1 - z^d) = -\\sum_{k \\ge 1} z^{dk}/k$. Then for every $N \\ge 0$, $F$ has the same coefficients of $z^0, \\dots, z^N$ as the finite product of the series\u2026","title":"The Artin\u2013Hasse exponential as a product"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-psplitting","kind":"node","lean":"MetaMathlibExt.IsPSplitting","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-splitting of multisets of parts"]],"source":"$p$-splitting of multisets of parts"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-psplitting","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"Let $p$ be a natural number and $s$, $t$ multisets of natural numbers (multisets of parts). Then $s$ is a *$p$-splitting* of $t$ when there is $u \\ge 1$ such that $t$ contains the part $pu$ and $s$ is obtained from $t$ by removing one instance of $pu$ and adding $p$ copies of $u$.","title":"$p$-splitting of multisets of parts"},{"id":"Algebra/polynomials/deep-power-sum-congruences/is-psplitting-replicate-singleton","kind":"node","lean":"MetaMathlibExt.isPSplitting_replicate_singleton","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text"," copies of "],["math","\\(u\\)"],["text"," form a "],["math","\\(p\\)"],["text","-splitting of "],["math","\\(\\{pu\\}\\)"]],"source":"$p$ copies of $u$ form a $p$-splitting of $\\{pu\\}$"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#is-psplitting-replicate-singleton","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"For every natural number $p$ and every integer $u \\ge 1$, the multiset consisting of $p$ copies of $u$ is a $p$-splitting of the one-element multiset $\\{pu\\}$.","title":"$p$ copies of $u$ form a $p$-splitting of $\\{pu\\}$"},{"id":"Algebra/polynomials/deep-power-sum-congruences/p-equiv-class","kind":"node","lean":"MetaMathlibExt.pEquivClass","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-equivalence class of a multiset of parts"]],"source":"The $p$-equivalence class of a multiset of parts"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#p-equiv-class","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"Let $p$ be a natural number and $\\lambda$ a multiset of natural numbers whose elements sum to $n$. The $p$-equivalence class $C_\\lambda$ is the set of partitions $\\mu$ of $n$ with $\\mu \\sim_p \\lambda$, where $\\sim_p$ ($p$-equivalence) is the equivalence relation on multisets of natural numbers generated by $p$-splitting.","title":"The $p$-equivalence class of a multiset of parts"},{"id":"Algebra/polynomials/deep-power-sum-congruences/p-equiv-class-sum","kind":"node","lean":"MetaMathlibExt.pEquivClassSum","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-equivalence class sum"]],"source":"The $p$-equivalence class sum"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#p-equiv-class-sum","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"Let $\\sigma$ be a finite set of variables, let $p$ be a natural number and let $\\lambda$ be a multiset of natural numbers whose elements sum to $n$. In $\\mathbb{Q}[x_i : i \\in \\sigma]$ let $p_k = \\sum_{i \\in \\sigma} x_i^k$ be the $k$-th power sum. For a partition $\\mu$ of $n$ with parts $\\mu_1, \\dots, \\mu_\\ell$, write $p_\\mu = p_{\\mu_1} \\cdots p_{\\mu_\\ell}$, $\\varepsilon_\\mu = (-1)^{n - \\ell}$ and\u2026","title":"The $p$-equivalence class sum"},{"id":"Algebra/polynomials/deep-power-sum-congruences/p-equiv-class-sum-zero","kind":"node","lean":"MetaMathlibExt.pEquivClassSum_zero","parent":1322,"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#p-equiv-class-sum-zero","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"fully_proved","summary":"For every finite set of variables $\\sigma$ and every natural number $p$, the $p$-equivalence class sum of the empty multiset is $g_\\emptyset = 1$.","title":"The class sum of the empty partition"},{"id":"Algebra/polynomials/deep-power-sum-congruences/p-equiv-class-sum-add-of-disjoint","kind":"node","lean":"MetaMathlibExt.pEquivClassSum_add_of_disjoint","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicativity of "],["math","\\(g_\\lambda\\)"],["text"," over disjoint parts"]],"source":"Multiplicativity of $g_\\lambda$ over disjoint parts"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#p-equiv-class-sum-add-of-disjoint","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime, $\\sigma$ a finite set of variables, and for a multiset $\\lambda$ of parts let $g_\\lambda$ be its $p$-equivalence class sum in $\\mathbb{Q}[x_i : i \\in \\sigma]$. Let $\\lambda$ and $\\mu$ be multisets of natural numbers, none of whose elements is divisible by $p$ (so no part is $0$), with no part in common. Then $g_{\\lambda + \\mu} = g_\\lambda \\, g_\\mu$, where $\\lambda + \\mu$ is the union of\u2026","title":"Multiplicativity of $g_\\lambda$ over disjoint parts"},{"id":"Algebra/polynomials/deep-power-sum-congruences/p-equiv-class-sum-eq-neg-one-pow-smul","kind":"node","lean":"MetaMathlibExt.pEquivClassSum_eq_neg_one_pow_smul","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["text","For odd "],["math","\\(p\\)"],["text"," the signs in "],["math","\\(g_\\lambda\\)"],["text"," are constant"]],"source":"For odd $p$ the signs in $g_\\lambda$ are constant"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#p-equiv-class-sum-eq-neg-one-pow-smul","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be an odd prime, $\\sigma$ a finite set of variables and $\\lambda$ a partition of $n$ with $\\ell(\\lambda)$ parts. Then the $p$-equivalence class sum $g_\\lambda$ in $\\mathbb{Q}[x_i : i \\in \\sigma]$ satisfies","title":"For odd $p$ the signs in $g_\\lambda$ are constant"},{"id":"Algebra/polynomials/deep-power-sum-congruences/p-equiv-class-sum-coeff-den-not-dvd","kind":"node","lean":"MetaMathlibExt.pEquivClassSum_coeff_den_not_dvd","parent":1322,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(g_\\lambda\\)"],["text"," is "],["math","\\(p\\)"],["text","-integral"]],"source":"$g_\\lambda$ is $p$-integral"},"result":true,"route":"../roadmap/Algebra/polynomials/deep-power-sum-congruences/index.html#p-equiv-class-sum-coeff-den-not-dvd","scope":"Algebra/polynomials/deep-power-sum-congruences","status":"can_prove","summary":"Let $p$ be a prime, $\\sigma$ a finite set of variables and $\\lambda$ a partition of $n$. Every coefficient of the $p$-equivalence class sum $g_\\lambda$ in $\\mathbb{Q}[x_i : i \\in \\sigma]$ is a rational number whose denominator in lowest terms is not divisible by $p$.","title":"$g_\\lambda$ is $p$-integral"},{"data_url":"scopes/Algebra/polynomials/denominators-of-rational-polynomials.json","id":"Algebra/polynomials/denominators-of-rational-polynomials","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/denominators-of-rational-polynomials/index.html","scope":"Algebra/polynomials/denominators-of-rational-polynomials","status":"fully_proved","title":"Denominators of rational polynomials"},{"id":"Algebra/polynomials/denominators-of-rational-polynomials/polynomial-denominator","kind":"node","lean":"MetaMathlibExt.polynomialDenominator","parent":1360,"result":true,"route":"../roadmap/Algebra/polynomials/denominators-of-rational-polynomials/index.html#polynomial-denominator","scope":"Algebra/polynomials/denominators-of-rational-polynomials","status":"fully_proved","summary":"For a polynomial $P \\in \\mathbb{Q}[x]$, the denominator $\\operatorname{denom}(P)$ is the least common multiple of the denominators (in lowest terms) of the nonzero coefficients of $P$; in particular $\\operatorname{denom}(0) = 1$. Thus $\\operatorname{denom}(P) = 1$ exactly when $P \\in \\mathbb{Z}[x]$.","title":"Denominator of a rational polynomial"},{"id":"Algebra/polynomials/denominators-of-rational-polynomials/coeff-den-dvd-polynomial-denominator","kind":"node","lean":"MetaMathlibExt.coeff_den_dvd_polynomialDenominator","parent":1360,"result":true,"route":"../roadmap/Algebra/polynomials/denominators-of-rational-polynomials/index.html#coeff-den-dvd-polynomial-denominator","scope":"Algebra/polynomials/denominators-of-rational-polynomials","status":"fully_proved","summary":"For every polynomial $P \\in \\mathbb{Q}[x]$ and every $i \\in \\mathbb{N}$, the denominator (in lowest terms) of the coefficient of $x^i$ in $P$ divides the denominator $\\operatorname{denom}(P)$ of $P$.","title":"Coefficient denominators divide the denominator of a polynomial"},{"id":"Algebra/polynomials/denominators-of-rational-polynomials/polynomial-denominator-eq-one-iff","kind":"node","lean":"MetaMathlibExt.polynomialDenominator_eq_one_iff","parent":1360,"result":true,"route":"../roadmap/Algebra/polynomials/denominators-of-rational-polynomials/index.html#polynomial-denominator-eq-one-iff","scope":"Algebra/polynomials/denominators-of-rational-polynomials","status":"fully_proved","summary":"For every polynomial $P \\in \\mathbb{Q}[x]$, its denominator $\\operatorname{denom}(P)$ equals $1$ if and only if, for every $i \\in \\mathbb{N}$, the coefficient of $x^i$ in $P$ is an integer.","title":"Denominator one characterizes integer polynomials"},{"data_url":"scopes/Algebra/polynomials/irreducibility-criteria-for-integer-polynomials.json","id":"Algebra/polynomials/irreducibility-criteria-for-integer-polynomials","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/irreducibility-criteria-for-integer-polynomials/index.html","scope":"Algebra/polynomials/irreducibility-criteria-for-integer-polynomials","status":"fully_proved","title":"Irreducibility criteria for integer polynomials"},{"id":"Algebra/polynomials/irreducibility-criteria-for-integer-polynomials/cohn-criterion","kind":"node","lean":"MetaMathlibExt.cohn_criterion","parent":1364,"result":true,"route":"../roadmap/Algebra/polynomials/irreducibility-criteria-for-integer-polynomials/index.html#cohn-criterion","scope":"Algebra/polynomials/irreducibility-criteria-for-integer-polynomials","status":"fully_proved","summary":"Let $b \\ge 2$ be an integer and let $p$ be a prime whose base-$b$ digits are $a_0, a_1, \\dots, a_n$, with units digit $a_0$ and leading digit $a_n \\ne 0$. Then the polynomial $f(x) = a_0 + a_1 x + \\dots + a_n x^n$ is irreducible in $\\mathbb{Z}[x]$.","title":"Cohn's irreducibility criterion"},{"id":"Algebra/polynomials/irreducibility-criteria-for-integer-polynomials/irreducible-of-dumas-padic-criterion","kind":"node","lean":"MetaMathlibExt.irreducible_of_dumas_padic_criterion","parent":1364,"result":true,"route":"../roadmap/Algebra/polynomials/irreducibility-criteria-for-integer-polynomials/index.html#irreducible-of-dumas-padic-criterion","scope":"Algebra/polynomials/irreducibility-criteria-for-integer-polynomials","status":"fully_proved","summary":"Let $p$ be a prime, write $v_p$ for the $p$-adic valuation on $\\mathbb{Q}$, and let $f = a_0 + a_1 x + \\dots + a_n x^n \\in \\mathbb{Z}[x]$ have degree $n \\ge 1$, with $a_0 \\ne 0$ and $a_n \\ne 0$. Suppose that no integer $k > 1$ dividing $n$ divides $v_p(a_0) - v_p(a_n)$, that is, $\\gcd(v_p(a_0) - v_p(a_n), n) = 1$, and that for every $0 \\le i \\le n$ with $a_i \\ne 0$,\u2026","title":"Dumas irreducibility criterion"},{"data_url":"scopes/Algebra/polynomials/mestre-s-square-decomposition.json","id":"Algebra/polynomials/mestre-s-square-decomposition","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/mestre-s-square-decomposition/index.html","scope":"Algebra/polynomials/mestre-s-square-decomposition","status":"fully_proved","title":"Mestre's square decomposition"},{"id":"Algebra/polynomials/mestre-s-square-decomposition/mestre-monic-square-decomposition-genera","kind":"node","lean":"MetaMathlibExt.mestre_monic_square_decomposition_general","parent":1367,"result":true,"route":"../roadmap/Algebra/polynomials/mestre-s-square-decomposition/index.html#mestre-monic-square-decomposition-genera","scope":"Algebra/polynomials/mestre-s-square-decomposition","status":"fully_proved","summary":"Let $K$ be a field of characteristic different from $2$ and let $d \\ge 0$ be an integer. Every monic polynomial $P \\in K[X]$ of degree $2d + 2$ can be written as $P = Q^2 - R$ with $Q, R \\in K[X]$, $Q$ monic of degree $d + 1$ and $R$ of degree at most $d$ (possibly $R = 0$), and the pair $(Q, R)$ is unique.","title":"Mestre's square decomposition, including the quadratic case"},{"id":"Algebra/polynomials/mestre-s-square-decomposition/mestre-monic-square-decomposition","kind":"node","lean":"MetaMathlibExt.mestre_monic_square_decomposition","parent":1367,"result":true,"route":"../roadmap/Algebra/polynomials/mestre-s-square-decomposition/index.html#mestre-monic-square-decomposition","scope":"Algebra/polynomials/mestre-s-square-decomposition","status":"fully_proved","summary":"Let $K$ be a field of characteristic different from $2$ and let $d \\ge 1$ be an integer. Every monic polynomial $P \\in K[X]$ of degree $2d + 2$ can be written as $P = Q^2 - R$ with $Q, R \\in K[X]$, $Q$ monic of degree $d + 1$ and $R$ of degree at most $d$ (possibly $R = 0$), and the pair $(Q, R)$ is unique.","title":"Mestre's square decomposition"},{"data_url":"scopes/Algebra/polynomials/mixed-partial-derivatives-of-polynomials.json","id":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/mixed-partial-derivatives-of-polynomials/index.html","scope":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials","status":"fully_proved","title":"Mixed partial derivatives of polynomials"},{"id":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials/mixed-differential","kind":"node","lean":"MvPolynomial.mixedDifferential","parent":1370,"result":true,"route":"../roadmap/Algebra/polynomials/mixed-partial-derivatives-of-polynomials/index.html#mixed-differential","scope":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\sigma$ be an index type, and write $\\partial_s$ for the partial derivative with respect to $x_s$ on the polynomial ring $R[x_s : s \\in \\sigma]$. For a finite list $(i_1, \\dots, i_k)$ of indices in $\\sigma$ (repetitions allowed) and a polynomial $p \\in R[x_s : s \\in \\sigma]$, the mixed differential of $p$ along the list is $(1 - \\partial_{i_k}) \\cdots (1 - \\partial_{i_1})\\, p$:\u2026","title":"Mixed differential operator"},{"id":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials/mixed-differential-nil","kind":"node","lean":"MvPolynomial.mixedDifferential_nil","parent":1370,"result":true,"route":"../roadmap/Algebra/polynomials/mixed-partial-derivatives-of-polynomials/index.html#mixed-differential-nil","scope":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\sigma$ be an index type, and let $p \\in R[x_s : s \\in \\sigma]$. The mixed differential of $p$ along the empty list of indices is $p$ itself.","title":"Mixed differential along the empty list"},{"id":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials/mixed-differential-cons","kind":"node","lean":"MvPolynomial.mixedDifferential_cons","parent":1370,"result":true,"route":"../roadmap/Algebra/polynomials/mixed-partial-derivatives-of-polynomials/index.html#mixed-differential-cons","scope":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\sigma$ be an index type, let $i \\in \\sigma$, let $L$ be a finite list of indices in $\\sigma$, and let $p \\in R[x_s : s \\in \\sigma]$. The mixed differential of $p$ along the list that starts with $i$ and continues with $L$ equals the mixed differential along $L$ of $p - \\partial_i p$, where $\\partial_i$ is the partial derivative with respect to $x_i$.","title":"Mixed differential along a list with a given first index"},{"id":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials/mixed-differential-eq-of-perm","kind":"node","lean":"MvPolynomial.mixedDifferential_eq_of_perm","parent":1370,"result":true,"route":"../roadmap/Algebra/polynomials/mixed-partial-derivatives-of-polynomials/index.html#mixed-differential-eq-of-perm","scope":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\sigma$ be an index type, and let $p \\in R[x_s : s \\in \\sigma]$. If two finite lists $L$ and $L'$ of indices in $\\sigma$ are permutations of each other, then the mixed differentials of $p$ along $L$ and along $L'$ are equal.","title":"The mixed differential ignores the order of the indices"},{"id":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials/mixed-differential-append","kind":"node","lean":"MvPolynomial.mixedDifferential_append","parent":1370,"result":true,"route":"../roadmap/Algebra/polynomials/mixed-partial-derivatives-of-polynomials/index.html#mixed-differential-append","scope":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\sigma$ be an index type, let $p \\in R[x_s : s \\in \\sigma]$, and let $L$ and $L'$ be finite lists of indices in $\\sigma$. The mixed differential of $p$ along the concatenation of $L$ followed by $L'$ equals the mixed differential along $L'$ of the mixed differential of $p$ along $L$.","title":"Mixed differential along a concatenated list"},{"id":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials/pderiv-mixed-differential","kind":"node","lean":"MvPolynomial.pderiv_mixedDifferential","parent":1370,"result":true,"route":"../roadmap/Algebra/polynomials/mixed-partial-derivatives-of-polynomials/index.html#pderiv-mixed-differential","scope":"Algebra/polynomials/mixed-partial-derivatives-of-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $i \\in \\sigma$, let $L$ be a finite list of indices in $\\sigma$, and let $p \\in R[x_s : s \\in \\sigma]$. Writing $\\partial_i$ for the partial derivative with respect to $x_i$, the partial derivative $\\partial_i$ of the mixed differential of $p$ along $L$ equals the mixed differential along $L$ of $\\partial_i p$.","title":"Partial derivatives commute with the mixed differential"},{"data_url":"scopes/Algebra/polynomials/newton-polygon.json","id":"Algebra/polynomials/newton-polygon","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","title":"Newton polygon"},{"id":"Algebra/polynomials/newton-polygon/exponent-to-real","kind":"node","lean":"NewtonPolygon.exponentToReal","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#exponent-to-real","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"The map from exponent vectors of monomials in two variables to $\\mathbb{R}^2$ sends $e = (e_0, e_1) \\in \\mathbb{N}^2$ to the point $(e_0, e_1) \\in \\mathbb{R}^2$.","title":"Exponent vectors as points of the plane"},{"id":"Algebra/polynomials/newton-polygon/newton-support","kind":"node","lean":"NewtonPolygon.newtonSupport","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#newton-support","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"Let $f$ be a polynomial in two variables $x_0, x_1$ over a commutative semiring $k$. The Newton support of $f$ is the set of exponent vectors $(e_0, e_1) \\in \\mathbb{N}^2$ of the monomials $x_0^{e_0} x_1^{e_1}$ with nonzero coefficient in $f$, viewed as points of $\\mathbb{R}^2$.","title":"Newton support"},{"id":"Algebra/polynomials/newton-polygon/newton-polygon","kind":"node","lean":"NewtonPolygon.newtonPolygon","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#newton-polygon","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"Let $f$ be a polynomial in two variables over a commutative semiring $k$. The Newton polygon of $f$ is the convex hull in $\\mathbb{R}^2$ of the Newton support of $f$, the set of exponent vectors of the monomials with nonzero coefficient in $f$, viewed as points of $\\mathbb{R}^2$.","title":"Newton polygon"},{"id":"Algebra/polynomials/newton-polygon/newton-polygon-interior","kind":"node","lean":"NewtonPolygon.newtonPolygonInterior","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#newton-polygon-interior","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"For a polynomial $f$ in two variables over a commutative semiring $k$, the interior of the Newton polygon of $f$ is its topological interior as a subset of $\\mathbb{R}^2$.","title":"Interior of the Newton polygon"},{"id":"Algebra/polynomials/newton-polygon/newton-polygon-boundary","kind":"node","lean":"NewtonPolygon.newtonPolygonBoundary","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#newton-polygon-boundary","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"For a polynomial $f$ in two variables over a commutative semiring $k$, the boundary of the Newton polygon of $f$ is its topological boundary in $\\mathbb{R}^2$: the closure of the Newton polygon minus its interior.","title":"Boundary of the Newton polygon"},{"id":"Algebra/polynomials/newton-polygon/edge-restriction","kind":"node","lean":"NewtonPolygon.edgeRestriction","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#edge-restriction","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"Let $f$ be a polynomial in two variables over a commutative semiring $k$, and let $\\Gamma \\subseteq \\mathbb{R}^2$. The restriction of $f$ to $\\Gamma$ is the polynomial formed by the terms of $f$ whose exponent vector, viewed as a point of $\\mathbb{R}^2$, lies in $\\Gamma$; the other terms are dropped.","title":"Restriction of a polynomial to a set of exponents"},{"id":"Algebra/polynomials/newton-polygon/mem-newton-support","kind":"node","lean":"NewtonPolygon.mem_newtonSupport","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#mem-newton-support","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"Let $f$ be a polynomial in two variables over a commutative semiring $k$. A point $x \\in \\mathbb{R}^2$ lies in the Newton support of $f$ if and only if $x$ is the exponent vector $e \\in \\mathbb{N}^2$ of some monomial with nonzero coefficient in $f$, viewed as a point of $\\mathbb{R}^2$.","title":"Membership in the Newton support"},{"id":"Algebra/polynomials/newton-polygon/newton-support-zero","kind":"node","lean":"NewtonPolygon.newtonSupport_zero","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#newton-support-zero","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"For every commutative semiring $k$, the Newton support of the zero polynomial in two variables over $k$ is the empty subset of $\\mathbb{R}^2$.","title":"The Newton support of the zero polynomial is empty"},{"id":"Algebra/polynomials/newton-polygon/newton-polygon-zero","kind":"node","lean":"NewtonPolygon.newtonPolygon_zero","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#newton-polygon-zero","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"For every commutative semiring $k$, the Newton polygon of the zero polynomial in two variables over $k$ is the empty subset of $\\mathbb{R}^2$.","title":"The Newton polygon of the zero polynomial is empty"},{"id":"Algebra/polynomials/newton-polygon/newton-support-subset-newton-polygon","kind":"node","lean":"NewtonPolygon.newtonSupport_subset_newtonPolygon","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#newton-support-subset-newton-polygon","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"For every polynomial $f$ in two variables over a commutative semiring $k$, the Newton support of $f$ is contained in the Newton polygon of $f$, the convex hull of the Newton support in $\\mathbb{R}^2$.","title":"The Newton support lies in the Newton polygon"},{"id":"Algebra/polynomials/newton-polygon/newton-polygon-interior-subset-newton-po","kind":"node","lean":"NewtonPolygon.newtonPolygonInterior_subset_newtonPolygon","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#newton-polygon-interior-subset-newton-po","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"For every polynomial $f$ in two variables over a commutative semiring $k$, the interior of the Newton polygon of $f$ in $\\mathbb{R}^2$ is contained in the Newton polygon of $f$.","title":"The interior of the Newton polygon lies in the polygon"},{"id":"Algebra/polynomials/newton-polygon/edge-restriction-empty","kind":"node","lean":"NewtonPolygon.edgeRestriction_empty","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#edge-restriction-empty","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"For every polynomial $f$ in two variables over a commutative semiring $k$, the restriction of $f$ to the empty set, that is, the sum of the terms of $f$ whose exponent vector, viewed as a point of $\\mathbb{R}^2$, lies in $\\emptyset$, is the zero polynomial.","title":"Edge restriction to the empty set"},{"id":"Algebra/polynomials/newton-polygon/edge-restriction-univ","kind":"node","lean":"NewtonPolygon.edgeRestriction_univ","parent":1377,"result":true,"route":"../roadmap/Algebra/polynomials/newton-polygon/index.html#edge-restriction-univ","scope":"Algebra/polynomials/newton-polygon","status":"fully_proved","summary":"For every polynomial $f$ in two variables over a commutative semiring $k$, the restriction of $f$ to all of $\\mathbb{R}^2$, that is, the sum of the terms of $f$ whose exponent vector, viewed as a point of $\\mathbb{R}^2$, lies in $\\mathbb{R}^2$, is $f$ itself.","title":"Edge restriction to the whole plane"},{"data_url":"scopes/Algebra/polynomials/partial-fraction-decomposition.json","id":"Algebra/polynomials/partial-fraction-decomposition","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/partial-fraction-decomposition/index.html","scope":"Algebra/polynomials/partial-fraction-decomposition","status":"can_prove","title":"Partial fraction decomposition"},{"id":"Algebra/polynomials/partial-fraction-decomposition/rat-func-partial-fractions-complex","kind":"node","lean":"MathlibExt.FieldTheory.ComplexPartialFractionsWanted.ratFunc_partial_fractions_complex","parent":1391,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial fraction decomposition over "],["math","\\(\\mathbb{C}\\)"]],"source":"Partial fraction decomposition over $\\mathbb{C}$"},"result":true,"route":"../roadmap/Algebra/polynomials/partial-fraction-decomposition/index.html#rat-func-partial-fractions-complex","scope":"Algebra/polynomials/partial-fraction-decomposition","status":"can_prove","summary":"For every complex rational function $f \\in \\mathbb{C}(X)$ there exist a polynomial $q \\in \\mathbb{C}[X]$, a finite set $S$ of monic polynomials of degree $1$ in $\\mathbb{C}[X]$, and, for each $p \\in S$, a natural number $m_p$ and polynomials $g_{p,k} \\in \\mathbb{C}[X]$ for $k < m_p$, each of degree less than the degree of $p$, such that, in $\\mathbb{C}(X)$,","title":"Partial fraction decomposition over $\\mathbb{C}$"},{"id":"Algebra/polynomials/partial-fraction-decomposition/rat-func-partial-fractions-real","kind":"node","lean":"MathlibExt.FieldTheory.RealPartialFractionsWanted.ratFunc_partial_fractions_real","parent":1391,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial fraction decomposition over "],["math","\\(\\mathbb{R}\\)"]],"source":"Partial fraction decomposition over $\\mathbb{R}$"},"result":true,"route":"../roadmap/Algebra/polynomials/partial-fraction-decomposition/index.html#rat-func-partial-fractions-real","scope":"Algebra/polynomials/partial-fraction-decomposition","status":"can_prove","summary":"For every real rational function $f \\in \\mathbb{R}(X)$ there exist a polynomial $q \\in \\mathbb{R}[X]$, a finite set $S$ of monic polynomials in $\\mathbb{R}[X]$, each of degree $1$ or irreducible of degree $2$, and, for each $p \\in S$, a natural number $m_p$ and polynomials $g_{p,k} \\in \\mathbb{R}[X]$ for $k < m_p$, each of degree less than the degree of $p$, such that, in $\\mathbb{R}(X)$,","title":"Partial fraction decomposition over $\\mathbb{R}$"},{"data_url":"scopes/Algebra/polynomials/polynomial-automorphisms-of-the-plane.json","id":"Algebra/polynomials/polynomial-automorphisms-of-the-plane","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/polynomial-automorphisms-of-the-plane/index.html","scope":"Algebra/polynomials/polynomial-automorphisms-of-the-plane","status":"can_prove","title":"Polynomial automorphisms of the plane"},{"id":"Algebra/polynomials/polynomial-automorphisms-of-the-plane/scottish-book-problem79-claim","kind":"node","lean":"MathlibExt.Algebra.Polynomial.ScottishBook79Wanted.ScottishBookProblem79Claim","parent":1394,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-automorphisms-of-the-plane/index.html#scottish-book-problem79-claim","scope":"Algebra/polynomials/polynomial-automorphisms-of-the-plane","status":"fully_proved","summary":"The statement that every polynomial map of the plane with a polynomial inverse is affine: for all pairs $P = (P_1, P_2)$ and $Q = (Q_1, Q_2)$ of real polynomials in two variables such that $Q(P(x)) = x$ for every $x \\in \\mathbb{R}^2$ and $P(Q(y)) = y$ for every $y \\in \\mathbb{R}^2$, there are a real $2 \\times 2$ matrix $A$ with $\\det A \\neq 0$ and a vector $c \\in \\mathbb{R}^2$ such that $P(x) = Ax + c$ for every\u2026","title":"Mazur\u2013Orlicz claim on polynomial automorphisms of the plane"},{"id":"Algebra/polynomials/polynomial-automorphisms-of-the-plane/mazur-orlicz79-refuted","kind":"node","lean":"MathlibExt.Algebra.Polynomial.ScottishBook79Wanted.mazurOrlicz79_refuted","parent":1394,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-automorphisms-of-the-plane/index.html#mazur-orlicz79-refuted","scope":"Algebra/polynomials/polynomial-automorphisms-of-the-plane","status":"can_prove","summary":"The claim of Problem 79 of the Scottish Book for the plane is false: there is a polynomial map $P$ of $\\mathbb{R}^2$ to itself with a polynomial two-sided inverse that is not of the form $x \\mapsto Ax + c$ with $A$ an invertible real $2 \\times 2$ matrix and $c \\in \\mathbb{R}^2$. For example, the map $(x_1, x_2) \\mapsto (x_1, x_2 + x_1^2)$ has the inverse $(y_1, y_2) \\mapsto (y_1, y_2 - y_1^2)$.","title":"Mazur\u2013Orlicz Problem 79 has a negative answer"},{"data_url":"scopes/Algebra/polynomials/polynomial-maps-between-affine-spaces.json","id":"Algebra/polynomials/polynomial-maps-between-affine-spaces","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","title":"Polynomial maps between affine spaces"},{"id":"Algebra/polynomials/polynomial-maps-between-affine-spaces/regular-function","kind":"node","lean":"MvPolynomial.RegularFunction","parent":1397,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html#regular-function","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","summary":"For a commutative semiring $k$ and index types $\\sigma$ and $\\tau$, a regular function from $k^\\sigma$ to $k^\\tau$ is a family $F = (F_t)_{t \\in \\tau}$ of polynomials in the variables $X_s$, $s \\in \\sigma$, with coefficients in $k$.","title":"Regular functions"},{"id":"Algebra/polynomials/polynomial-maps-between-affine-spaces/jacobian","kind":"node","lean":"MvPolynomial.RegularFunction.Jacobian","parent":1397,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html#jacobian","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","summary":"The Jacobian of a regular function $F = (F_j)_{j \\in \\tau}$ from $k^\\sigma$ to $k^\\tau$ is the $\\tau \\times \\sigma$ matrix, with entries in the polynomial ring $k[X_i : i \\in \\sigma]$, whose entry in row $j$ and column $i$ is the formal partial derivative $\\partial F_j / \\partial X_i$.","title":"Jacobian matrix of a regular function"},{"id":"Algebra/polynomials/polynomial-maps-between-affine-spaces/comp","kind":"node","lean":"MvPolynomial.RegularFunction.comp","parent":1397,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html#comp","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","summary":"For regular functions $G$ from $k^\\tau$ to $k^\\iota$ and $F$ from $k^\\sigma$ to $k^\\tau$, the composition $G \\circ F$ is the regular function from $k^\\sigma$ to $k^\\iota$ whose $i$-th component, for $i \\in \\iota$, is the polynomial obtained from $G_i$ by substituting $F_t$ for the variable $X_t$ for every $t \\in \\tau$.","title":"Composition of regular functions"},{"id":"Algebra/polynomials/polynomial-maps-between-affine-spaces/id-comp","kind":"node","lean":"MvPolynomial.RegularFunction.id_comp","parent":1397,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html#id-comp","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","summary":"For every regular function $F$ from $k^\\sigma$ to $k^\\tau$, composing the identity regular function on $k^\\tau$, whose components are the variables $X_t$ for $t \\in \\tau$, with $F$ gives $F$, that is, $\\mathrm{id} \\circ F = F$.","title":"The identity is a left unit for composition"},{"id":"Algebra/polynomials/polynomial-maps-between-affine-spaces/comp-id","kind":"node","lean":"MvPolynomial.RegularFunction.comp_id","parent":1397,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html#comp-id","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","summary":"For every regular function $F$ from $k^\\sigma$ to $k^\\tau$, composing $F$ with the identity regular function on $k^\\sigma$, whose components are the variables $X_s$ for $s \\in \\sigma$, gives $F$, that is, $F \\circ \\mathrm{id} = F$.","title":"The identity is a right unit for composition"},{"id":"Algebra/polynomials/polynomial-maps-between-affine-spaces/comp-assoc","kind":"node","lean":"MvPolynomial.RegularFunction.comp_assoc","parent":1397,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html#comp-assoc","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","summary":"For regular functions $H$ from $k^\\iota$ to $k^\\kappa$, $G$ from $k^\\tau$ to $k^\\iota$ and $F$ from $k^\\sigma$ to $k^\\tau$, we have $(H \\circ G) \\circ F = H \\circ (G \\circ F)$.","title":"Associativity of composition of regular functions"},{"id":"Algebra/polynomials/polynomial-maps-between-affine-spaces/aeval","kind":"node","lean":"MvPolynomial.RegularFunction.aeval","parent":1397,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html#aeval","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","summary":"Let $k$ be a commutative semiring and let $S$ be a commutative $k$-algebra. For a regular function $F$ from $k^\\sigma$ to $k^\\tau$ and a point $a \\in S^\\sigma$, the evaluation of $F$ at $a$ is the point $F(a) \\in S^\\tau$ whose $t$-th coordinate, for $t \\in \\tau$, is the value of the polynomial $F_t$ at $a$, obtained by substituting $a_s$ for $X_s$ for every $s \\in \\sigma$.","title":"Evaluation of a regular function at a point"},{"id":"Algebra/polynomials/polynomial-maps-between-affine-spaces/comp-aeval","kind":"node","lean":"MvPolynomial.RegularFunction.comp_aeval","parent":1397,"result":true,"route":"../roadmap/Algebra/polynomials/polynomial-maps-between-affine-spaces/index.html#comp-aeval","scope":"Algebra/polynomials/polynomial-maps-between-affine-spaces","status":"fully_proved","summary":"Let $k$ be a commutative semiring and let $S$ be a commutative $k$-algebra. For regular functions $G$ from $k^\\tau$ to $k^\\iota$ and $F$ from $k^\\sigma$ to $k^\\tau$ and every point $a \\in S^\\sigma$, write $F(a) \\in S^\\tau$ for the evaluation of $F$ at $a$. Then evaluating the composition $G \\circ F$ at $a$ gives $G(F(a))$, the evaluation of $G$ at the point $F(a)$.","title":"Evaluation respects composition"},{"data_url":"scopes/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog.json","id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","title":"Polynomials with coefficients in arithmetic progression"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/perron-irreducible","kind":"node","lean":"MetaMathlibExt.perron_irreducible","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#perron-irreducible","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $f = x^n + a_{n-1} x^{n-1} + \\cdots + a_0$ be a monic polynomial in $\\mathbb{Z}[x]$ of degree $n \\ge 1$ with $a_0 \\neq 0$ and $|a_{n-1}| > 1 + \\sum_{i=0}^{n-2} |a_i|$. Then $f$ is irreducible in $\\mathbb{Z}[x]$.","title":"Perron's irreducibility criterion"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-irreducible","kind":"node","lean":"MetaMathlibExt.arithProgPoly_irreducible","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-irreducible","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m \\ge 2$ and let $a$, $d$ be coprime positive integers with $m^2 d^2 < 8a$. Then the polynomial $a x^m + (a + d) x^{m-1} + \\cdots + (a + (m - 1)d) x + (a + md)$ is irreducible in $\\mathbb{Z}[x]$.","title":"Irreducibility of polynomials with coefficients in an arithmetic progression"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-four-not-mul-quadratics","kind":"node","lean":"MetaMathlibExt.arithProgPoly_four_not_mul_quadratics","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-four-not-mul-quadratics","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $a$, $d$ be coprime integers with $0 < d \\le a$. Then the polynomial $a x^4 + (a + d) x^3 + (a + 2d) x^2 + (a + 3d) x + (a + 4d)$ is not the product of two polynomials of degree $2$ in $\\mathbb{Z}[x]$.","title":"The arithmetic-progression quartic is not a product of two quadratics"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-irreducible-of-primitive","kind":"node","lean":"MetaMathlibExt.arithProgPoly_irreducible_of_primitiveRoot","parent":1406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Irreducibility when "],["math","\\(d\\)"],["text"," has a primitive-root prime factor"]],"source":"Irreducibility when $d$ has a primitive-root prime factor"},"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-irreducible-of-primitive","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m \\ge 2$ with $m + 1$ prime, and let $a$, $d$ be coprime positive integers. Suppose $d$ has a prime divisor $p$ that is not divisible by $m + 1$ and whose residue modulo $m + 1$ has multiplicative order $m$ (that is, $p$ is a primitive root modulo $m + 1$). Then the polynomial $a x^m + (a + d) x^{m-1} + \\cdots + (a + (m - 1)d) x + (a + md)$ is irreducible in $\\mathbb{Z}[x]$.","title":"Irreducibility when $d$ has a primitive-root prime factor"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-mul-sub-one-sq","kind":"node","lean":"MetaMathlibExt.arithProgPoly_mul_sub_one_sq","parent":1406,"reading_title":{"format":"inline-parts-v1","parts":[["text","The arithmetic-progression polynomial times "],["math","\\((x - 1)^2\\)"]],"source":"The arithmetic-progression polynomial times $(x - 1)^2$"},"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-mul-sub-one-sq","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"For all natural numbers $m$, $a$, $d$, the polynomial $A_m(x) = \\sum_{i=0}^{m} (a + id) x^{m-i}$ satisfies, in $\\mathbb{Z}[x]$,","title":"The arithmetic-progression polynomial times $(x - 1)^2$"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-root-pow-eq","kind":"node","lean":"MetaMathlibExt.arithProgPoly_root_pow_eq","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-root-pow-eq","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m$ and $a$ be natural numbers, let $d$ be a positive integer, and let $A_m(x) = a x^m + (a + d) x^{m-1} + \\cdots + (a + md)$, that is, $A_m(x) = \\sum_{i=0}^{m} (a + id) x^{m-i} \\in \\mathbb{Z}[x]$. If $z \\in \\mathbb{C}$ is a root of $A_m$ or $z = 1$, then","title":"Power identity at the roots of an arithmetic-progression polynomial"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-eq3-exists-unique-pos","kind":"node","lean":"MetaMathlibExt.arithProgPoly_eq3_existsUnique_pos","parent":1406,"reading_title":{"format":"inline-parts-v1","parts":[["text","The auxiliary equation has a unique positive root, which exceeds "],["math","\\(1\\)"]],"source":"The auxiliary equation has a unique positive root, which exceeds $1$"},"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-eq3-exists-unique-pos","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m \\ge 2$ and let $a$, $d$ be integers with $0 < d \\le a$. Then the equation $a x^{m+2} + (a - d) x^{m+1} - (a + (m + 1)d) x - (a + md) = 0$ has exactly one positive real solution, and every positive real solution $x$ satisfies $x > 1$.","title":"The auxiliary equation has a unique positive root, which exceeds $1$"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-root-norm-le","kind":"node","lean":"MetaMathlibExt.arithProgPoly_root_norm_le","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-root-norm-le","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m \\ge 2$, let $a$, $d$ be integers with $0 < d \\le a$, and let $\\lambda > 0$ be a real number with $a \\lambda^{m+2} + (a - d) \\lambda^{m+1} - (a + (m + 1)d) \\lambda - (a + md) = 0$. Then every complex zero $z$ of $A_m(x) = a x^m + (a + d) x^{m-1} + \\cdots + (a + md)$ satisfies $|z| \\le \\lambda$, and $|z| = \\lambda$ only if $z = -\\lambda$.","title":"Bound on the zeros of the arithmetic-progression polynomial"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-lam-pow-gt","kind":"node","lean":"MetaMathlibExt.arithProgPoly_lam_pow_gt","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-lam-pow-gt","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m \\ge 2$, let $a$, $d$ be integers with $0 < d \\le a$, and let $\\lambda > 0$ be a real number with $a \\lambda^{m+2} + (a - d) \\lambda^{m+1} - (a + (m + 1)d) \\lambda - (a + md) = 0$. Then $a + md < a \\lambda^m$.","title":"Lower bound on a power of the auxiliary root"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/one-add-mul-rpow-inv-gt","kind":"node","lean":"MetaMathlibExt.one_add_mul_rpow_inv_gt","parent":1406,"reading_title":{"format":"inline-parts-v1","parts":[["text","A rational lower bound for "],["math","\\((1 + mx)^{1/m}\\)"]],"source":"A rational lower bound for $(1 + mx)^{1/m}$"},"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#one-add-mul-rpow-inv-gt","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"For every natural number $m \\ge 2$ and every real $x > 0$,","title":"A rational lower bound for $(1 + mx)^{1/m}$"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-floor-lam-pow-lt","kind":"node","lean":"MetaMathlibExt.arithProgPoly_floor_lam_pow_lt","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-floor-lam-pow-lt","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m \\ge 2$, let $a$, $d$ be natural numbers with $d > 0$ and $(m + 1)d < 2a$, and let $\\lambda > 0$ be a real number with $a \\lambda^{m+2} + (a - d) \\lambda^{m+1} - (a + (m + 1)d) \\lambda - (a + md) = 0$. Then $\\lfloor a \\lambda^p \\rfloor < a + pd$ for every integer $p$ with $1 \\le p \\le m - 1$.","title":"Strict upper bound on floors at powers of the auxiliary root"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-floor-lam-pow-eq","kind":"node","lean":"MetaMathlibExt.arithProgPoly_floor_lam_pow_eq","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-floor-lam-pow-eq","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m \\ge 2$, let $a$, $d$ be positive integers, and let $p$ be an integer with $1 \\le p \\le m - 1$ and $p(m - p)d^2 < 2a$. If $\\lambda > 0$ is a real number with $a \\lambda^{m+2} + (a - d) \\lambda^{m+1} - (a + (m + 1)d) \\lambda - (a + md) = 0$, then $\\lfloor a \\lambda^p \\rfloor = a + pd - 1$.","title":"Exact floor at a power of the auxiliary root (single exponent)"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-floor-lam-pow-eq-of-eigh","kind":"node","lean":"MetaMathlibExt.arithProgPoly_floor_lam_pow_eq_of_eight","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-floor-lam-pow-eq-of-eigh","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $m \\ge 2$, let $a$, $d$ be positive integers with $m^2 d^2 < 8a$, and let $\\lambda > 0$ be a real number with $a \\lambda^{m+2} + (a - d) \\lambda^{m+1} - (a + (m + 1)d) \\lambda - (a + md) = 0$. Then $\\lfloor a \\lambda^p \\rfloor = a + pd - 1$ for every integer $p$ with $1 \\le p \\le m - 1$.","title":"Exact floors at powers of the auxiliary root (all exponents)"},{"id":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/arith-prog-poly-two-reducible-iff","kind":"node","lean":"MetaMathlibExt.arithProgPoly_two_reducible_iff","parent":1406,"result":true,"route":"../roadmap/Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog/index.html#arith-prog-poly-two-reducible-iff","scope":"Algebra/polynomials/polynomials-with-coefficients-in-arithmetic-prog","status":"can_prove","summary":"Let $d \\ge 2$ be an integer and, for a positive integer $a$, let $A_2(x) = ax^2 + (a + d)x + (a + 2d)$ in $\\mathbb{Z}[x]$. There is a positive integer $a$ coprime to $d$ for which $A_2$ is not irreducible in $\\mathbb{Z}[x]$ if and only if every prime divisor of $d$ is congruent to $1$ modulo $3$. In that case the number of positive integers $a$ coprime to $d$ for which $A_2$ is not irreducible is exactly $2^{n-1}$,\u2026","title":"Reducible quadratics with coefficients in an arithmetic progression"},{"data_url":"scopes/Algebra/polynomials/reciprocal-0-1-polynomials.json","id":"Algebra/polynomials/reciprocal-0-1-polynomials","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"can_prove","title":"Reciprocal 0,1-polynomials"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/is-reciprocal-poly","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.IsReciprocalPoly","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#is-reciprocal-poly","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"fully_proved","summary":"A polynomial $f(x) = a_0 + a_1 x + \\dots + a_d x^d$ in $\\mathbb{Z}[x]$ of degree $d$ (with $d = 0$ for the zero polynomial) is *reciprocal* (palindromic) if it equals its reversal $x^d f(1/x)$, that is, if $a_i = a_{d-i}$ for all $0 \\le i \\le d$.","title":"Reciprocal polynomial"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/is-reciprocal-poly-iff","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.isReciprocalPoly_iff","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#is-reciprocal-poly-iff","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"fully_proved","summary":"A polynomial $f \\in \\mathbb{Z}[x]$ of degree $d$ (with $d = 0$ for the zero polynomial) is reciprocal if and only if its reversal $x^d f(1/x)$ equals $f$.","title":"Characterization of reciprocal polynomials"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/is-zero-one-poly","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.IsZeroOnePoly","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#is-zero-one-poly","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"fully_proved","summary":"A polynomial $f \\in \\mathbb{Z}[x]$ is a *$0,1$-polynomial* if every coefficient of $f$ is $0$ or $1$.","title":"0,1-polynomial"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/is-zero-one-poly-iff","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.isZeroOnePoly_iff","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#is-zero-one-poly-iff","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"fully_proved","summary":"A polynomial $f \\in \\mathbb{Z}[x]$ is a $0,1$-polynomial if and only if, for every $n \\ge 0$, the coefficient of $x^n$ in $f$ is $0$ or $1$.","title":"Characterization of 0,1-polynomials"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/is-noncyclotomic","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.IsNoncyclotomic","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#is-noncyclotomic","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"fully_proved","summary":"A polynomial $f \\in \\mathbb{Z}[x]$ is *non-cyclotomic* if $f \\ne \\Phi_n$ for every positive integer $n$, where $\\Phi_n$ denotes the $n$-th cyclotomic polynomial.","title":"Non-cyclotomic polynomial"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/is-noncyclotomic-iff","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.isNoncyclotomic_iff","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#is-noncyclotomic-iff","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"fully_proved","summary":"A polynomial $f \\in \\mathbb{Z}[x]$ is non-cyclotomic if and only if $f \\ne \\Phi_n$ for every positive integer $n$, where $\\Phi_n$ denotes the $n$-th cyclotomic polynomial.","title":"Characterization of non-cyclotomic polynomials"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/exists-unique-even-odd-reciprocal","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.existsUnique_evenOdd_reciprocal","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#exists-unique-even-odd-reciprocal","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"can_prove","summary":"Let $h \\in \\mathbb{Z}[x]$ be an irreducible reciprocal polynomial of degree $2k$ such that $h \\ne g(x^2)$ for every $g \\in \\mathbb{Z}[x]$. Then there is exactly one $h_e \\in \\mathbb{Z}[x]$ for which there is exactly one $h_o \\in \\mathbb{Z}[x]$ for which there is exactly one natural number $\\ell$ such that $h_e$ and $h_o$ are reciprocal, $\\ell$ is odd and positive, $\\deg h_e = k$, $\\deg h_o = \\max(k - \\ell, 0)$,\u2026","title":"Even\u2013odd decomposition of a reciprocal polynomial"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/exists-unique-even-odd-zero-one","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.existsUnique_evenOdd_zeroOne","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#exists-unique-even-odd-zero-one","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x]$ be an irreducible reciprocal $0,1$-polynomial of degree $2m$ such that $f(x^2)$ is not irreducible. Then there is exactly one $f_e \\in \\mathbb{Z}[x]$ for which there is exactly one $f_o \\in \\mathbb{Z}[x]$ for which there is exactly one natural number $\\ell$ such that $f_e$ and $f_o$ are reciprocal $0,1$-polynomials, $\\ell$ is odd and positive, $\\deg f_e = m$,\u2026","title":"Even\u2013odd decomposition of a reciprocal 0,1-polynomial"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/l-eq-min-support-and-lt","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.l_eq_min_support_and_lt","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#l-eq-min-support-and-lt","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x]$ be an irreducible, reciprocal, non-cyclotomic $0,1$-polynomial of degree $2m$ such that $f(x^2)$ is not irreducible. Let $f_e, f_o \\in \\mathbb{Z}[x]$ and let $\\ell$ be a positive odd integer with $f(x) = f_e(x^2) + x^\\ell f_o(x^2)$, $\\deg f_e = m$ and $\\deg f_o = \\max(m - \\ell, 0)$, where the degree of the zero polynomial is taken to be $0$. Then the coefficient of $x^e$ in $f$ is $0$ for\u2026","title":"Position of the first odd exponent"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/irreducible-comp-pow-two-pow-of-card-lt","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.irreducible_comp_pow_two_pow_of_card_lt","parent":1421,"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#irreducible-comp-pow-two-pow-of-card-lt","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x]$ be an irreducible, reciprocal, non-cyclotomic $0,1$-polynomial with fewer than $15$ nonzero coefficients. Then $f(x^{2^n})$ is irreducible in $\\mathbb{Z}[x]$ for every natural number $n$.","title":"Irreducibility under repeated squaring of the variable"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/not-irreducible-comp-pow-iff","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.not_irreducible_comp_pow_iff","parent":1421,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capelli's theorem for "],["math","\\(f(x^r)\\)"]],"source":"Capelli's theorem for $f(x^r)$"},"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#not-irreducible-comp-pow-iff","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x]$ be irreducible in $\\mathbb{Z}[x]$ and let $r \\ge 2$ be an integer. Then $f(x^r)$ is reducible if and only if either $f(x^p)$ is reducible for some prime $p$ dividing $r$, or $4$ divides $r$ and $f(x^4)$ is reducible. Here reducible means not irreducible in $\\mathbb{Z}[x]$.","title":"Capelli's theorem for $f(x^r)$"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/map-comp-sq-eq-mul","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.map_comp_sq_eq_mul","parent":1421,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square compositions of "],["math","\\(0,1\\)"],["text","-polynomials modulo "],["math","\\(4\\)"]],"source":"Square compositions of $0,1$-polynomials modulo $4$"},"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#map-comp-sq-eq-mul","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x]$ be irreducible in $\\mathbb{Z}[x]$, with every coefficient equal to $0$ or $1$ and with constant term $f(0) = 1$. If $f(x^2)$ is not irreducible in $\\mathbb{Z}[x]$, then $f(x^2) \\equiv f(x)\\,f(-x) \\pmod 4$ as polynomials, that is, the two polynomials have the same image in $(\\mathbb{Z}/4\\mathbb{Z})[x]$.","title":"Square compositions of $0,1$-polynomials modulo $4$"},{"id":"Algebra/polynomials/reciprocal-0-1-polynomials/not-irreducible-comp-four-iff","kind":"node","lean":"MathlibExt.Algebra.PalindromicZeroOneWanted.not_irreducible_comp_four_iff","parent":1421,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reducibility of "],["math","\\(f(x^4)\\)"],["text"," and "],["math","\\(f(x^2)\\)"],["text"," for "],["math","\\(0,1\\)"],["text","-polynomials"]],"source":"Reducibility of $f(x^4)$ and $f(x^2)$ for $0,1$-polynomials"},"result":true,"route":"../roadmap/Algebra/polynomials/reciprocal-0-1-polynomials/index.html#not-irreducible-comp-four-iff","scope":"Algebra/polynomials/reciprocal-0-1-polynomials","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x]$ be irreducible in $\\mathbb{Z}[x]$ and of positive degree, with every coefficient equal to $0$ or $1$ and with constant term $f(0) = 1$. Then $f(x^4)$ is not irreducible in $\\mathbb{Z}[x]$ if and only if $f(x^2)$ is not irreducible in $\\mathbb{Z}[x]$.","title":"Reducibility of $f(x^4)$ and $f(x^2)$ for $0,1$-polynomials"},{"data_url":"scopes/Algebra/polynomials/squares-of-sparse-polynomials.json","id":"Algebra/polynomials/squares-of-sparse-polynomials","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/squares-of-sparse-polynomials/index.html","scope":"Algebra/polynomials/squares-of-sparse-polynomials","status":"can_prove","title":"Squares of sparse polynomials"},{"id":"Algebra/polynomials/squares-of-sparse-polynomials/conjecture","kind":"node","lean":"MathlibExt.Analysis.SparsePolynomialSquareTermBoundWanted.conjecture","parent":1435,"result":true,"route":"../roadmap/Algebra/polynomials/squares-of-sparse-polynomials/index.html#conjecture","scope":"Algebra/polynomials/squares-of-sparse-polynomials","status":"fully_proved","summary":"There is a function $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every integer $k \\ge 1$, every complex polynomial $p$ with at least $f(k)$ nonzero coefficients has a square $p^2$ with at least $k$ nonzero coefficients.","title":"Erd\u0151s\u2013R\u00e9nyi conjecture on squares of sparse polynomials"},{"id":"Algebra/polynomials/squares-of-sparse-polynomials/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.SparsePolynomialSquareTermBoundWanted.conjecture_holds","parent":1435,"result":true,"route":"../roadmap/Algebra/polynomials/squares-of-sparse-polynomials/index.html#conjecture-holds","scope":"Algebra/polynomials/squares-of-sparse-polynomials","status":"can_prove","summary":"The Erd\u0151s\u2013R\u00e9nyi conjecture holds: there is a function $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every integer $k \\ge 1$, every complex polynomial $p$ with at least $f(k)$ nonzero coefficients has a square $p^2$ with at least $k$ nonzero coefficients.","title":"Schinzel's theorem on squares of sparse polynomials"},{"data_url":"scopes/Algebra/polynomials/weighted-derivations-of-monoid-algebras.json","id":"Algebra/polynomials/weighted-derivations-of-monoid-algebras","kind":"scope","navigation":"module","parent":1279,"result":false,"route":"../roadmap/Algebra/polynomials/weighted-derivations-of-monoid-algebras/index.html","scope":"Algebra/polynomials/weighted-derivations-of-monoid-algebras","status":"fully_proved","title":"Weighted derivations of monoid algebras"},{"id":"Algebra/polynomials/weighted-derivations-of-monoid-algebras/weighted-derivation","kind":"node","lean":"AddMonoidAlgebra.weightedDerivation","parent":1438,"result":true,"route":"../roadmap/Algebra/polynomials/weighted-derivations-of-monoid-algebras/index.html#weighted-derivation","scope":"Algebra/polynomials/weighted-derivations-of-monoid-algebras","status":"fully_proved","summary":"Multiply the coefficient at each exponent by an additive weight of that exponent.","title":"Weighted derivation of an additive monoid algebra"},{"id":"Algebra/polynomials/weighted-derivations-of-monoid-algebras/coeff-weighted-derivation","kind":"node","lean":"AddMonoidAlgebra.coeff_weightedDerivation","parent":1438,"result":true,"route":"../roadmap/Algebra/polynomials/weighted-derivations-of-monoid-algebras/index.html#coeff-weighted-derivation","scope":"Algebra/polynomials/weighted-derivations-of-monoid-algebras","status":"fully_proved","summary":"The coefficient at `g` is multiplied by the weight of `g`.","title":"Coefficients of a weighted derivation"},{"id":"Algebra/polynomials/weighted-derivations-of-monoid-algebras/weighted-derivation-single","kind":"node","lean":"AddMonoidAlgebra.weightedDerivation_single","parent":1438,"result":true,"route":"../roadmap/Algebra/polynomials/weighted-derivations-of-monoid-algebras/index.html#weighted-derivation-single","scope":"Algebra/polynomials/weighted-derivations-of-monoid-algebras","status":"fully_proved","summary":"A basis term keeps its exponent and has its coefficient multiplied by the weight.","title":"Weighted derivation of a basis term"},{"id":"Algebra/polynomials/weighted-derivations-of-monoid-algebras/weighted-derivation-zero","kind":"node","lean":"AddMonoidAlgebra.weightedDerivation_zero","parent":1438,"result":true,"route":"../roadmap/Algebra/polynomials/weighted-derivations-of-monoid-algebras/index.html#weighted-derivation-zero","scope":"Algebra/polynomials/weighted-derivations-of-monoid-algebras","status":"fully_proved","summary":"The zero weight determines the zero derivation.","title":"Weighted derivation with zero weight"},{"data_url":"scopes/Algebra/orthogonal-polynomials.json","id":"Algebra/orthogonal-polynomials","kind":"scope","parent":1278,"result":false,"route":"../roadmap/Algebra/orthogonal-polynomials/index.html","scope":"Algebra/orthogonal-polynomials","status":"can_prove","summary":"Classical and combinatorial families of polynomials, their recurrences and generating functions, and orthogonality via Favard's theorem, J-fractions and Riordan arrays.","title":"Orthogonal polynomials and polynomial sequences"},{"id":"Algebra/orthogonal-polynomials/asc-pochhammer-eval-add","kind":"node","lean":"MetaMathlibExt.ascPochhammer_eval_add","parent":1443,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/index.html#asc-pochhammer-eval-add","scope":"Algebra/orthogonal-polynomials","status":"fully_proved","summary":"For every natural number $n$ and all real numbers $a$ and $b$, writing $x^{\\overline{m}} = x(x+1)\\cdots(x+m-1)$ for the rising factorial,","title":"Binomial theorem for rising factorials"},{"id":"Algebra/orthogonal-polynomials/classical-orthogonal-polynomial-family","kind":"node","lean":"MetaMathlibExt.ClassicalOrthogonalPolynomialFamily","parent":1443,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/index.html#classical-orthogonal-polynomial-family","scope":"Algebra/orthogonal-polynomials","status":"fully_proved","summary":"A classical orthogonal polynomial family consists of real polynomials $P_0, P_1, P_2, \\dots$, each $P_n$ monic of degree exactly $n$, a measure $\\mu$ on $\\mathbb{R}$, a measurable set $D \\subseteq \\mathbb{R}$ that is an interval with nonempty interior, and a function $w : \\mathbb{R} \\to \\mathbb{R}$ (the weight), such that $w(x) \\ge 0$ for every $x \\in D$; $\\mu$ is the measure with density $w$ with respect to\u2026","title":"Classical orthogonal polynomial family"},{"id":"Algebra/orthogonal-polynomials/riordan-coeff-orth-equiv","kind":"node","lean":"MetaMathlibExt.riordan_coeff_orth_equiv","parent":1443,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/index.html#riordan-coeff-orth-equiv","scope":"Algebra/orthogonal-polynomials","status":"fully_proved","summary":"Let $d$ and $h$ be real formal power series with $[x^0]\\,h = 0$, $[x^1]\\,h \\ne 0$ and $[x^0]\\,d \\ne 0$, where $[x^n]$ denotes the coefficient of $x^n$, and let $L$ be the Riordan array of $(d, h)$, with entries $L_{n,k} = [x^n]\\,(d h^k)$ for $n, k \\ge 0$. Call $L$ a monic orthogonal coefficient array if there are real polynomials $P_0, P_1, \\dots$ with $[x^k]\\,P_n = L_{n,k}$ for all $n, k$, each $P_n$ monic, and\u2026","title":"Meixner-type characterization of Riordan orthogonal polynomial arrays"},{"data_url":"scopes/Algebra/orthogonal-polynomials/bessel-polynomials.json","id":"Algebra/orthogonal-polynomials/bessel-polynomials","kind":"scope","navigation":"module","parent":1443,"result":false,"route":"../roadmap/Algebra/orthogonal-polynomials/bessel-polynomials/index.html","scope":"Algebra/orthogonal-polynomials/bessel-polynomials","status":"fully_proved","title":"Bessel polynomials"},{"id":"Algebra/orthogonal-polynomials/bessel-polynomials/bessel-polynomial","kind":"node","lean":"MetaMathlibExt.besselPolynomial","parent":1447,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/bessel-polynomials/index.html#bessel-polynomial","scope":"Algebra/orthogonal-polynomials/bessel-polynomials","status":"fully_proved","summary":"For a natural number $n$, the $n$-th classical Bessel polynomial $y_n$ is the polynomial with rational coefficients $y_n(x) = \\sum_{k=0}^{n} \\frac{(n+k)!}{2^k \\, k! \\, (n-k)!} \\, x^k$.","title":"Bessel polynomial"},{"id":"Algebra/orthogonal-polynomials/bessel-polynomials/bessel-polynomial-eq-sum","kind":"node","lean":"MetaMathlibExt.besselPolynomial_eq_sum","parent":1447,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/bessel-polynomials/index.html#bessel-polynomial-eq-sum","scope":"Algebra/orthogonal-polynomials/bessel-polynomials","status":"fully_proved","summary":"For every natural number $n$, the Bessel polynomial $y_n \\in \\mathbb{Q}[x]$ equals $\\sum_{k=0}^{n} \\frac{(n+k)!}{2^k \\, k! \\, (n-k)!} \\, x^k$.","title":"Explicit formula for the Bessel polynomials"},{"id":"Algebra/orthogonal-polynomials/bessel-polynomials/bessel-polynomial-zero","kind":"node","lean":"MetaMathlibExt.besselPolynomial_zero","parent":1447,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/bessel-polynomials/index.html#bessel-polynomial-zero","scope":"Algebra/orthogonal-polynomials/bessel-polynomials","status":"fully_proved","summary":"The Bessel polynomial $y_0$ is the constant polynomial $1$.","title":"The zeroth Bessel polynomial"},{"id":"Algebra/orthogonal-polynomials/bessel-polynomials/bessel-polynomial-one","kind":"node","lean":"MetaMathlibExt.besselPolynomial_one","parent":1447,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/bessel-polynomials/index.html#bessel-polynomial-one","scope":"Algebra/orthogonal-polynomials/bessel-polynomials","status":"fully_proved","summary":"The Bessel polynomial $y_1$ equals $1 + x$.","title":"The first Bessel polynomial"},{"id":"Algebra/orthogonal-polynomials/bessel-polynomials/bessel-polynomial-two","kind":"node","lean":"MetaMathlibExt.besselPolynomial_two","parent":1447,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/bessel-polynomials/index.html#bessel-polynomial-two","scope":"Algebra/orthogonal-polynomials/bessel-polynomials","status":"fully_proved","summary":"The Bessel polynomial $y_2$ equals $1 + 3x + 3x^2$.","title":"The second Bessel polynomial"},{"data_url":"scopes/Algebra/orthogonal-polynomials/catalan-type-polynomials.json","id":"Algebra/orthogonal-polynomials/catalan-type-polynomials","kind":"scope","navigation":"module","parent":1443,"result":false,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"can_prove","title":"Catalan-type polynomials"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/trinomial-neg-pow-series","kind":"node","lean":"MetaMathlibExt.trinomialNegPowSeries","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#trinomial-neg-pow-series","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"fully_proved","summary":"For real numbers $\\lambda$, $c$, $\\alpha$, $\\beta$ and a natural number $m$, this is the formal power series in $t$ with real coefficients","title":"Negative real power of a trinomial as a power series"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/constant-coeff-trinomial-neg-pow-series","kind":"node","lean":"MetaMathlibExt.constantCoeff_trinomialNegPowSeries","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#constant-coeff-trinomial-neg-pow-series","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"fully_proved","summary":"For all real numbers $\\lambda, c, \\alpha, \\beta$ and every natural number $m \\ne 0$, the negative real power of the trinomial $c - \\alpha t + \\beta t^m$ with exponent $\\lambda$ as a power series in $t$, that is, the real power $c^{-\\lambda}$ times the binomial series $(1 + u)^{-\\lambda}$ with $u = (-\\alpha t + \\beta t^m)/c$ substituted (where $u = 0$ when $c = 0$), has constant term $c^{-\\lambda}$.","title":"Constant term of the trinomial series"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/trinomial-neg-pow-series-neg-one","kind":"node","lean":"MetaMathlibExt.trinomialNegPowSeries_neg_one","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#trinomial-neg-pow-series-neg-one","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"fully_proved","summary":"For all real numbers $\\alpha, \\beta$ and every natural number $m \\ne 0$, the formal expansion of $(1 - \\alpha t + \\beta t^m)^{-\\lambda}$ in powers of $t$ at $\\lambda = -1$ is the polynomial $1 - \\alpha t + \\beta t^m$ itself.","title":"The trinomial series at exponent one"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/kampe-de-feriet-hermite","kind":"node","lean":"MetaMathlibExt.kampeDeFerietHermite","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#kampe-de-feriet-hermite","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"fully_proved","summary":"For natural numbers $m, n$ and elements $x, y$ of a commutative $\\mathbb{R}$-algebra, the Kamp\u00e9 de F\u00e9riet (two-variable Hermite) polynomial is","title":"Kamp\u00e9 de F\u00e9riet polynomials"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/kampe-de-feriet-hermite-zero","kind":"node","lean":"MetaMathlibExt.kampeDeFerietHermite_zero","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#kampe-de-feriet-hermite-zero","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"fully_proved","summary":"For every natural number $m$ and all elements $x, y$ of a commutative $\\mathbb{R}$-algebra, $H^m_0(x, y) = 1$.","title":"The zeroth Kamp\u00e9 de F\u00e9riet polynomial"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/kampe-de-feriet-hermite-two-two","kind":"node","lean":"MetaMathlibExt.kampeDeFerietHermite_two_two","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#kampe-de-feriet-hermite-two-two","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"fully_proved","summary":"For all real numbers $x$ and $y$, $H^2_2(x, y) = x^2 + 2y$.","title":"The second Kamp\u00e9 de F\u00e9riet polynomial"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/coeff-hermite-shift-mul-trinomial-neg-po","kind":"node","lean":"MetaMathlibExt.coeff_hermiteShift_mul_trinomialNegPowSeries","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#coeff-hermite-shift-mul-trinomial-neg-po","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"can_prove","summary":"Write $H_n(x, y) = H^2_n(x, y)$ for the Kamp\u00e9 de F\u00e9riet polynomials. For a natural number $m$ and real numbers $\\lambda, z$, write $(1 - m t + z t^m)^{-\\lambda}$ and $(1 - m z t + t^m)^{-\\lambda}$ for the negative real powers of the trinomials $1 - m t + z t^m$ and $1 - m z t + t^m$ with exponent $\\lambda$ as power series in $t$ (for $m \\ge 1$ these are their expansions in powers of $t$), and let\u2026","title":"Hermite-shifted Catalan-type polynomials as convolutions"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/coeff-nieto-hermite-series-mul-trinomial","kind":"node","lean":"MetaMathlibExt.coeff_nietoHermiteSeries_mul_trinomialNegPowSeries","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#coeff-nieto-hermite-series-mul-trinomial","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"can_prove","summary":"Let $j, k, m, r$ be natural numbers with $k \\le r$ and let $\\lambda, x, y$ be real numbers. Let $E(t) = e^{2yt - t^2}$ be the exponential series, so that its coefficient of $t^n$ is $H_n(y) / n!$ with $H_n$ the Hermite polynomials, and let $F(t)$ be the power series whose coefficient of $t^n$ is $H_n(y) / n!$ when $n \\ge k$ and $j \\mid n - k$, and $0$ otherwise (for $j \\ge 1$ this is the series\u2026","title":"Nieto\u2013Hermite Catalan-type polynomials as convolutions"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/coeff-catalan-type-kernel-decomposition","kind":"node","lean":"MetaMathlibExt.coeff_catalanTypeKernel_decomposition","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#coeff-catalan-type-kernel-decomposition","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"can_prove","summary":"Let $j, k, m, d, n$ be natural numbers, let $a, b, c, \\lambda, x, y$ be real numbers and let $A$ be a polynomial with integer coefficients. Let $K(t)$ be the negative real power of the trinomial $c - a x t + b (2x - 1)^d t^m$ with exponent $\\lambda$ as a power series in $t$ (for $c > 0$ and $m \\ge 1$ this is the expansion of $(c - a x t + b (2x - 1)^d t^m)^{-\\lambda}$ in powers of $t$), and let\u2026","title":"Splitting the numerator of a unified Catalan-type generating function"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/det-catalan-type-lower-triangular","kind":"node","lean":"MetaMathlibExt.det_catalanTypeLowerTriangular","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#det-catalan-type-lower-triangular","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"can_prove","summary":"Let $m, N$ be natural numbers, let $\\lambda, x$ be real numbers, let $A$ be a polynomial with integer coefficients, and let $P_n$ be the coefficient of $t^n$ in $(1 + A(x)\\, t)(1 - m t + x t^m)^{-\\lambda}$, where $(1 - m t + x t^m)^{-\\lambda}$ denotes the negative real power of the trinomial $1 - m t + x t^m$ with exponent $\\lambda$ as a power series in $t$ (for $m \\ge 1$, its expansion in powers of $t$). Then the\u2026","title":"Determinant of the lower-triangular Catalan-type matrix"},{"id":"Algebra/orthogonal-polynomials/catalan-type-polynomials/det-catalan-type-hankel-eq-zero","kind":"node","lean":"MetaMathlibExt.det_catalanTypeHankel_eq_zero","parent":1453,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/catalan-type-polynomials/index.html#det-catalan-type-hankel-eq-zero","scope":"Algebra/orthogonal-polynomials/catalan-type-polynomials","status":"can_prove","summary":"Let $m, N, r$ be natural numbers with $r \\le N$, let $\\lambda, x$ be real numbers, let $A$ be a polynomial with integer coefficients, and let $P_n$ be the coefficient of $t^n$ in $(1 + A(x)\\, t)(1 - m t + x t^m)^{-\\lambda}$, where $(1 - m t + x t^m)^{-\\lambda}$ denotes the negative real power of the trinomial $1 - m t + x t^m$ with exponent $\\lambda$ as a power series in $t$ (for $m \\ge 1$, its expansion in powers\u2026","title":"Hankel determinants of recurrent Catalan-type sequences vanish"},{"data_url":"scopes/Algebra/orthogonal-polynomials/charlier-polynomials.json","id":"Algebra/orthogonal-polynomials/charlier-polynomials","kind":"scope","navigation":"module","parent":1443,"result":false,"route":"../roadmap/Algebra/orthogonal-polynomials/charlier-polynomials/index.html","scope":"Algebra/orthogonal-polynomials/charlier-polynomials","status":"fully_proved","title":"Charlier polynomials"},{"id":"Algebra/orthogonal-polynomials/charlier-polynomials/unsigned-charlier-order0","kind":"node","lean":"MetaMathlibExt.unsignedCharlierOrder0","parent":1465,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/charlier-polynomials/index.html#unsigned-charlier-order0","scope":"Algebra/orthogonal-polynomials/charlier-polynomials","status":"fully_proved","summary":"For a semiring $R$ and a natural number $n$, the $n$-th unsigned Charlier polynomial of order $0$ is","title":"Unsigned Charlier polynomials of order 0"},{"id":"Algebra/orthogonal-polynomials/charlier-polynomials/unsigned-charlier-coeff","kind":"node","lean":"MetaMathlibExt.unsignedCharlierCoeff","parent":1465,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/charlier-polynomials/index.html#unsigned-charlier-coeff","scope":"Algebra/orthogonal-polynomials/charlier-polynomials","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the unsigned Charlier coefficient is the natural number","title":"Unsigned Charlier coefficient"},{"id":"Algebra/orthogonal-polynomials/charlier-polynomials/unsigned-charlier-coeff-eq","kind":"node","lean":"MetaMathlibExt.unsignedCharlier_coeff_eq","parent":1465,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/charlier-polynomials/index.html#unsigned-charlier-coeff-eq","scope":"Algebra/orthogonal-polynomials/charlier-polynomials","status":"fully_proved","summary":"For all natural numbers $n$ and $m$, the coefficient of $x^m$ in the $n$-th unsigned Charlier polynomial of order $0$ over $\\mathbb{N}$ equals the unsigned Charlier coefficient $c(n, m)$.","title":"Coefficients of the unsigned Charlier polynomials"},{"data_url":"scopes/Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind.json","id":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind","kind":"scope","navigation":"module","parent":1443,"result":false,"route":"../roadmap/Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/index.html","scope":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind","status":"fully_proved","title":"Chebyshev polynomials of the second kind"},{"id":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/chebyshev-u-coeff-eq-binomial-sum","kind":"node","lean":"MetaMathlibExt.chebyshevU_coeff_eq_binomial_sum","parent":1469,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/index.html#chebyshev-u-coeff-eq-binomial-sum","scope":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind","status":"fully_proved","summary":"Let $U_n \\in \\mathbb{Z}[x]$ denote the $n$-th Chebyshev polynomial of the second kind, and let $k \\le n$ be natural numbers. If $n$ and $k$ have the same parity, the coefficient of $x^k$ in $U_n$ is $(-1)^{(n-k)/2} \\sum_{i=0}^{k} \\binom{(n+k)/2}{i} \\binom{(n+k)/2 - i}{(n-k)/2}$; otherwise it is $0$.","title":"Coefficients of the Chebyshev polynomials of the second kind"},{"id":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/p","kind":"node","lean":"MetaMathlibExt.GeneralizedChebyshevSecondKind.P","parent":1469,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/index.html#p","scope":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind","status":"fully_proved","summary":"For a commutative ring $R$ and $r, s \\in R$, the polynomials $P_n \\in R[x]$, $n \\ge 0$, are defined by $P_0 = 1$, $P_1 = x - r$ and $P_{n+2} = (x - r) P_{n+1} - s P_n$ for every $n \\ge 0$.","title":"Modified Chebyshev base polynomials of the second kind"},{"id":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/q","kind":"node","lean":"MetaMathlibExt.GeneralizedChebyshevSecondKind.Q","parent":1469,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/index.html#q","scope":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind","status":"fully_proved","summary":"For a commutative ring $R$ and $r, s, \\lambda, \\mu \\in R$, let $P_n \\in R[x]$ be the modified Chebyshev base polynomials with parameters $r, s$. The polynomials $Q_n \\in R[x]$ are defined by $Q_0 = 1$, $Q_1 = P_1 - \\lambda P_0$ and $Q_{n+2} = P_{n+2} - \\lambda P_{n+1} - \\mu P_n$ for every $n \\ge 0$.","title":"Generalized Chebyshev polynomials of the second kind"},{"id":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/q-recurrence","kind":"node","lean":"MetaMathlibExt.GeneralizedChebyshevSecondKind.Q_recurrence","parent":1469,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind/index.html#q-recurrence","scope":"Algebra/orthogonal-polynomials/chebyshev-polynomials-of-the-second-kind","status":"fully_proved","summary":"For a commutative ring $R$, all $r, s, \\lambda, \\mu \\in R$ and every $n \\ge 0$, the generalized Chebyshev polynomials of the second kind $Q_n \\in R[x]$ with parameters $r, s, \\lambda, \\mu$ satisfy $Q_{n+3} = (x - r) Q_{n+2} - s Q_{n+1}$.","title":"Three-term recurrence for generalized Chebyshev polynomials"},{"data_url":"scopes/Algebra/orthogonal-polynomials/favard-s-theorem.json","id":"Algebra/orthogonal-polynomials/favard-s-theorem","kind":"scope","navigation":"module","parent":1443,"result":false,"route":"../roadmap/Algebra/orthogonal-polynomials/favard-s-theorem/index.html","scope":"Algebra/orthogonal-polynomials/favard-s-theorem","status":"fully_proved","title":"Favard's theorem"},{"id":"Algebra/orthogonal-polynomials/favard-s-theorem/is-formally-orthogonal","kind":"node","lean":"MetaMathlibExt.Polynomial.IsFormallyOrthogonal","parent":1474,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/favard-s-theorem/index.html#is-formally-orthogonal","scope":"Algebra/orthogonal-polynomials/favard-s-theorem","status":"fully_proved","summary":"Let $R$ be a field. A family $(p_n)_{n \\ge 0}$ of polynomials over $R$ is formally orthogonal if there is an $R$-linear functional $L \\colon R[x] \\to R$ such that each $p_n$ has degree $n$, $L(p_n p_m) = 0$ whenever $n \\ne m$, and $L(p_n^2) \\ne 0$ for every $n$.","title":"Formally orthogonal polynomial family"},{"id":"Algebra/orthogonal-polynomials/favard-s-theorem/favard-forward","kind":"node","lean":"MetaMathlibExt.favard_forward","parent":1474,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/favard-s-theorem/index.html#favard-forward","scope":"Algebra/orthogonal-polynomials/favard-s-theorem","status":"fully_proved","summary":"Let $R$ be a field and let $(p_n)_{n \\ge 0}$ be a formally orthogonal family of monic polynomials over $R$. Then there are sequences $(\\alpha_n)_{n \\ge 0}$ and $(\\beta_n)_{n \\ge 0}$ in $R$ with $\\beta_n \\ne 0$ for all $n \\ge 1$ such that $p_0 = 1$, $p_1 = x - \\alpha_0$, and $p_{n+1} = (x - \\alpha_n) \\, p_n - \\beta_n \\, p_{n-1}$ for all $n \\ge 1$.","title":"Favard's theorem, forward direction"},{"id":"Algebra/orthogonal-polynomials/favard-s-theorem/favard-iff","kind":"node","lean":"MetaMathlibExt.favard_iff","parent":1474,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/favard-s-theorem/index.html#favard-iff","scope":"Algebra/orthogonal-polynomials/favard-s-theorem","status":"fully_proved","summary":"Let $R$ be a field and let $(p_n)_{n \\ge 0}$ be a family of monic polynomials in $R[x]$. Then $(p_n)$ is formally orthogonal if and only if there are sequences $(\\alpha_n)_{n \\ge 0}$ and $(\\beta_n)_{n \\ge 0}$ in $R$ with $\\beta_n \\ne 0$ for all $n \\ge 1$ such that $p_0 = 1$, $p_1 = x - \\alpha_0$, and $p_{n+1} = (x - \\alpha_n) \\, p_n - \\beta_n \\, p_{n-1}$ for all $n \\ge 1$.","title":"Favard's theorem"},{"data_url":"scopes/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials.json","id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","kind":"scope","navigation":"module","parent":1443,"result":false,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","title":"Kassel\u2013Reutenauer polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-coeff","kind":"node","lean":"MetaMathlibExt.kasselReutenauerCoeff","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Kassel\u2013Reutenauer coefficients "],["math","\\(a_{n,i}\\)"]],"source":"The Kassel\u2013Reutenauer coefficients $a_{n,i}$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-coeff","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"For natural numbers $n$ and $i$, $a_{n,i}$ is the number of positive divisors $d$ of $n$ with","title":"The Kassel\u2013Reutenauer coefficients $a_{n,i}$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"For a natural number $n$, the Kassel\u2013Reutenauer polynomial $\\bar P_n(X) \\in \\mathbb{Z}[X]$ is defined by","title":"Kassel\u2013Reutenauer polynomial"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"For $n \\ge 0$, $F_n(X)$ is the polynomial with integer coefficients","title":"Partial sums of the monic Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-coeff-small","kind":"node","lean":"MetaMathlibExt.kasselReutenauerCoeff_small","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small values of the coefficients "],["math","\\(a_{n,i}\\)"]],"source":"Small values of the coefficients $a_{n,i}$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-coeff-small","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"The Kassel\u2013Reutenauer coefficients satisfy $a_{1,0} = 1$, $a_{2,0} = a_{2,1} = 1$, $a_{3,0} = 0$ and $a_{3,1} = a_{3,2} = 1$.","title":"Small values of the coefficients $a_{n,i}$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-one","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_one","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-one","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"The Kassel\u2013Reutenauer polynomial $\\bar P_1$ is the constant polynomial $1$.","title":"The first Kassel\u2013Reutenauer polynomial"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-two","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_two","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Kassel\u2013Reutenauer polynomial "],["math","\\(\\bar P_2\\)"]],"source":"The Kassel\u2013Reutenauer polynomial $\\bar P_2$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-two","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"The Kassel\u2013Reutenauer polynomial $\\bar P_2(X) \\in \\mathbb{Z}[X]$ equals $X + 1$.","title":"The Kassel\u2013Reutenauer polynomial $\\bar P_2$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-three","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_three","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-three","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"The Kassel\u2013Reutenauer polynomial $\\bar P_3$ is $X^2 + X - 2$.","title":"The third Kassel\u2013Reutenauer polynomial"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum-zero","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum_zero","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum-zero","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"Let $F_n = 1 + \\sum_{k=1}^{n} \\bar T_k \\in \\mathbb{Z}[X]$ be the partial sums of the monic Chebyshev polynomials $\\bar T_k$ ($\\bar T_0 = 2$, $\\bar T_1 = X$, $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$). Then $F_0 = 1$.","title":"Zeroth partial sum of the Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum-succ","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum_succ","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum-succ","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"Let $F_n = 1 + \\sum_{k=1}^{n} \\bar T_k \\in \\mathbb{Z}[X]$ be the partial sums of the monic Chebyshev polynomials $\\bar T_k$ ($\\bar T_0 = 2$, $\\bar T_1 = X$, $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$). For every natural number $n$, $F_{n+1} = F_n + \\bar T_{n+1}$.","title":"Successive partial sums of the Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum-one-two","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum_one_two","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum-one-two","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"fully_proved","summary":"Let $F_n = 1 + \\sum_{k=1}^{n} \\bar T_k \\in \\mathbb{Z}[X]$ be the partial sums of the monic Chebyshev polynomials $\\bar T_k$ ($\\bar T_0 = 2$, $\\bar T_1 = X$, $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$). Then $F_1 = X + 1$ and $F_2 = X^2 + X - 1$.","title":"First and second partial sums of the Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-eval-pos","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_eval_pos","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positivity of the Kassel\u2013Reutenauer polynomials at integers "],["math","\\(N \\ge 2\\)"]],"source":"Positivity of the Kassel\u2013Reutenauer polynomials at integers $N \\ge 2$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-eval-pos","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"For every $n \\ge 1$ and every integer $N \\ge 2$, the Kassel\u2013Reutenauer polynomial $\\bar P_n$ satisfies $\\bar P_n(N) > 0$.","title":"Positivity of the Kassel\u2013Reutenauer polynomials at integers $N \\ge 2$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-norm-multiplicati","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_norm_multiplicative_rootOfUnity","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-norm-multiplicati","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $q \\in \\mathbb{C}$ be a primitive $d$-th root of unity with $d \\in \\{1, 2, 3, 4\\}$, and let $\\bar P_n$ be the Kassel\u2013Reutenauer polynomials. Then for all coprime integers $m, n \\ge 1$,","title":"Multiplicativity at roots of unity of order at most 4"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-norm-almost-multi","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_norm_almost_multiplicative_sixth","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-norm-almost-multi","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $\\omega \\in \\mathbb{C}$ be a primitive sixth root of unity, and for $n \\ge 1$ let $v_n = |\\bar P_n(\\omega + \\omega^{-1})|$, where $\\bar P_n$ is the Kassel\u2013Reutenauer polynomial. Let $m, n \\ge 1$ be coprime. Then $v_m v_n = v_{mn}$ if $(m \\bmod 3, n \\bmod 3)$ is $(0, 1)$, $(1, 1)$ or $(1, 2)$; $v_m v_n = 2 v_{mn}$ if $(m \\bmod 3, n \\bmod 3) = (0, 2)$; and $v_m v_n = 4 v_{mn}$ if\u2026","title":"Almost multiplicativity at a primitive sixth root of unity"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-sq-sub-dvd","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_sq_sub_dvd","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicativity of "],["math","\\(\\bar P_n^2\\)"],["text"," modulo "],["math","\\(X (X + 1) (X^2 - 4)\\)"]],"source":"Multiplicativity of $\\bar P_n^2$ modulo $X (X + 1) (X^2 - 4)$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-sq-sub-dvd","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $\\bar P_n$ be the Kassel\u2013Reutenauer polynomials. For coprime integers $m, n \\ge 1$, the polynomial $\\bar P_{mn}(X)^2 - \\bar P_m(X)^2 \\bar P_n(X)^2$ is divisible in $\\mathbb{Z}[X]$ by $X (X + 1) (X^2 - 4)$; if moreover $(m \\bmod 3, n \\bmod 3)$ is $(0, 1)$, $(1, 1)$ or $(1, 2)$, it is divisible by $X (X + 1) (X^2 - 4) (X - 1)$.","title":"Multiplicativity of $\\bar P_n^2$ modulo $X (X + 1) (X^2 - 4)$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-abs-eval-multipli","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_abs_eval_multiplicative_iff","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integers at which "],["math","\\(|\\bar P_n|\\)"],["text"," is multiplicative"]],"source":"Integers at which $|\\bar P_n|$ is multiplicative"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-abs-eval-multipli","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $\\bar P_n$ be the Kassel\u2013Reutenauer polynomial and let $N$ be an integer. Then $|\\bar P_{mn}(N)| = |\\bar P_m(N)|\\,|\\bar P_n(N)|$ for all coprime positive integers $m$ and $n$ if and only if $N \\in \\{-2, -1, 0, 2\\}$.","title":"Integers at which $|\\bar P_n|$ is multiplicative"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-c-eval-pos-of-two-le","kind":"node","lean":"MetaMathlibExt.chebyshevC_eval_pos_of_two_le","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-c-eval-pos-of-two-le","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $\\bar T_k \\in \\mathbb{R}[X]$ be the monic Chebyshev polynomials, given by $\\bar T_0 = 2$, $\\bar T_1 = X$ and $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$. Then $\\bar T_k(x) > 0$ for every natural number $k$ and every real number $x \\ge 2$.","title":"Positivity of the monic Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-c-eq-trace-pow","kind":"node","lean":"MetaMathlibExt.chebyshevC_eq_trace_pow","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-c-eq-trace-pow","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $\\bar T_k \\in \\mathbb{Z}[X]$ be the monic Chebyshev polynomials, given by $\\bar T_0 = 2$, $\\bar T_1 = X$ and $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$, and let $M$ be the $2 \\times 2$ matrix over $\\mathbb{Z}[X]$ with rows $(X, -1)$ and $(1, 0)$. Then $\\bar T_k = \\operatorname{tr}(M^k)$ for every natural number $k$.","title":"Trace formula for the monic Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum-generating-functi","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum_generatingFunction","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum-generating-functi","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $F_n = 1 + \\sum_{k=1}^{n} \\bar T_k \\in \\mathbb{Z}[X]$ be the partial sums of the monic Chebyshev polynomials $\\bar T_k$ ($\\bar T_0 = 2$, $\\bar T_1 = X$, $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$). In the ring $\\mathbb{Z}[X][[t]]$ of formal power series in $t$ over $\\mathbb{Z}[X]$,","title":"Generating function of the partial sums of Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum-eq-binomial-sum","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum_eq_binomial_sum","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum-eq-binomial-sum","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $F_n = 1 + \\sum_{k=1}^{n} \\bar T_k \\in \\mathbb{Z}[X]$ be the partial sums of the monic Chebyshev polynomials $\\bar T_k$ ($\\bar T_0 = 2$, $\\bar T_1 = X$, $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$). For every $n \\ge 1$,","title":"Binomial expansion of the partial sums of Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum-top-coeffs","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum_top_coeffs","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum-top-coeffs","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $F_n = 1 + \\sum_{k=1}^{n} \\bar T_k \\in \\mathbb{Z}[X]$ be the partial sums of the monic Chebyshev polynomials $\\bar T_k$ ($\\bar T_0 = 2$, $\\bar T_1 = X$, $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$). For every $n \\ge 5$, the polynomial","title":"Leading coefficients of the partial sums of Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum-recurrence","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum_recurrence","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum-recurrence","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $F_n = 1 + \\sum_{k=1}^{n} \\bar T_k \\in \\mathbb{Z}[X]$ be the partial sums of the monic Chebyshev polynomials $\\bar T_k$ ($\\bar T_0 = 2$, $\\bar T_1 = X$, $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$). For every $k \\ge 1$, $F_{k+1} = X F_k - F_{k-1}$.","title":"Three-term recurrence for the partial sums of Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-sub-chebyshev-cpa","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_sub_chebyshevCPartialSum_coeff","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Approximation of "],["math","\\(\\bar P_n\\)"],["text"," by "],["math","\\(F_{n-1}\\)"]],"source":"Approximation of $\\bar P_n$ by $F_{n-1}$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-sub-chebyshev-cpa","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $\\bar P_n$ be the Kassel\u2013Reutenauer polynomials, and for $r \\ge 0$ let $F_r(X) = 1 + \\sum_{j=1}^{r} \\bar T_j(X)$ be the partial sum of the monic Chebyshev polynomials $\\bar T_j$. For $n \\ge 2$, every nonzero coefficient of $\\bar P_n(X) - F_{n-1}(X)$ sits in a degree $k$ with $2k + 2 < n$; that is, the difference has degree less than $n/2 - 1$.","title":"Approximation of $\\bar P_n$ by $F_{n-1}$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-top-coeffs","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_top_coeffs","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-top-coeffs","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"For $n \\ge 10$, the polynomial","title":"Top coefficients of the Kassel\u2013Reutenauer polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-coeff-eq-one","kind":"node","lean":"MetaMathlibExt.kasselReutenauerCoeff_eq_one","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","The top coefficients "],["math","\\(a_{n,i}\\)"],["text"," equal "],["math","\\(1\\)"]],"source":"The top coefficients $a_{n,i}$ equal $1$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-coeff-eq-one","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $n \\ge 1$ and let $i$ be a natural number with $n \\le 2i + 2$ and $i \\le n - 1$; equivalently $n/2 - 1 \\le i \\le n - 1$, where $n/2$ is the rational number, not the integer quotient. Then the Kassel\u2013Reutenauer coefficient $a_{n,i}$ equals $1$.","title":"The top coefficients $a_{n,i}$ equal $1$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-eq-odd-divisor-su","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_eq_oddDivisor_sum","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-eq-odd-divisor-su","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, and for each odd divisor $d$ of $n$ put $r_n(d) = n/d - (d+1)/2$, an integer. Then the Kassel\u2013Reutenauer polynomial $\\bar P_n$ satisfies","title":"Odd-divisor formula for the Kassel\u2013Reutenauer polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-eq-chebyshev-cpar","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_eq_chebyshevCPartialSum_iff","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\bar P_n\\)"],["text"," equals "],["math","\\(F_{n-1}\\)"],["text"," exactly for powers of two"]],"source":"$\\bar P_n$ equals $F_{n-1}$ exactly for powers of two"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-eq-chebyshev-cpar","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, let $\\bar P_n$ be the Kassel\u2013Reutenauer polynomial, and let $F_{n-1} = 1 + \\sum_{k=1}^{n-1} \\bar T_k$ be the partial sum of the monic Chebyshev polynomials. Then $\\bar P_n = F_{n-1}$ if and only if $n = 2^k$ for some natural number $k$.","title":"$\\bar P_n$ equals $F_{n-1}$ exactly for powers of two"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-c-div-pow-eq-sum","kind":"node","lean":"MetaMathlibExt.kasselReutenauer_C_div_pow_eq_sum","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(C_n(q)/q^n\\)"],["text"," as a sum over the solutions of "],["math","\\(h(h + 1 + 2a) = 2n\\)"]],"source":"$C_n(q)/q^n$ as a sum over the solutions of $h(h + 1 + 2a) = 2n$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-c-div-pow-eq-sum","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, let $\\bar P_n$ be the Kassel\u2013Reutenauer polynomial, and write $C_n(q)/q^n$ for the Laurent polynomial $(q - 2 + q^{-1})\\,\\bar P_n(q + q^{-1})$ in $\\mathbb{Z}[q, q^{-1}]$. Then","title":"$C_n(q)/q^n$ as a sum over the solutions of $h(h + 1 + 2a) = 2n$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-c-div-pow-increasing-s","kind":"node","lean":"MetaMathlibExt.kasselReutenauer_C_div_pow_increasingSequences","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Increasing-sequence and odd-divisor formulas for "],["math","\\(C_n(q)/q^n\\)"]],"source":"Increasing-sequence and odd-divisor formulas for $C_n(q)/q^n$"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-c-div-pow-increasing-s","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, let $\\bar P_n$ be the Kassel\u2013Reutenauer polynomial, and write $C_n(q)/q^n$ for the Laurent polynomial $(q - 2 + q^{-1})\\,\\bar P_n(q + q^{-1})$ in $\\mathbb{Z}[q, q^{-1}]$. For integers $h \\ge 1$ and $a$ let $IS(a, h) = \\{a + 1, \\dots, a + h\\}$.","title":"Increasing-sequence and odd-divisor formulas for $C_n(q)/q^n$"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-c-abs-coeff-sum","kind":"node","lean":"MetaMathlibExt.kasselReutenauer_C_abs_coeff_sum","parent":1478,"reading_title":{"format":"inline-parts-v1","parts":[["text","The absolute coefficient sum of "],["math","\\(C_n\\)"],["text"," counts odd divisors"]],"source":"The absolute coefficient sum of $C_n$ counts odd divisors"},"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-c-abs-coeff-sum","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, let $\\bar P_n$ be the Kassel\u2013Reutenauer polynomial, and write $C_n(q)/q^n$ for the Laurent polynomial $(q - 2 + q^{-1})\\,\\bar P_n(q + q^{-1})$ in $\\mathbb{Z}[q, q^{-1}]$. Then the sum of the absolute values of the coefficients of $C_n(q)/q^n$ equals four times the number of odd divisors of $n$.","title":"The absolute coefficient sum of $C_n$ counts odd divisors"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/chebyshev-cpartial-sum-laurent-identity","kind":"node","lean":"MetaMathlibExt.chebyshevCPartialSum_laurent_identity","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#chebyshev-cpartial-sum-laurent-identity","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $F_r = 1 + \\sum_{k=1}^{r} \\bar T_k \\in \\mathbb{Z}[X]$ be the partial sums of the monic Chebyshev polynomials $\\bar T_k$ ($\\bar T_0 = 2$, $\\bar T_1 = X$, $\\bar T_{k+2} = X \\bar T_{k+1} - \\bar T_k$). For every natural number $r$, in the ring $\\mathbb{Z}[q, q^{-1}]$ of Laurent polynomials,","title":"Laurent polynomial identity for the partial sums of Chebyshev polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/kassel-reutenauer-poly-laurent-increasin","kind":"node","lean":"MetaMathlibExt.kasselReutenauerPoly_laurent_increasingSequences","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#kassel-reutenauer-poly-laurent-increasin","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $n \\ge 1$ and work in the ring $\\mathbb{Z}[q, q^{-1}]$ of Laurent polynomials. For integers $a$ and $h$ let $IS(a, h) = \\{a+1, \\dots, a+h\\}$. For an odd divisor $d$ of $n$ put $r_n(d) = n/d - (d+1)/2$, $A_d = IS(r_n(d), d)$ and $\\check A_d = IS(-r_n(d) - 1, 2n/d)$. Then the Kassel\u2013Reutenauer polynomial $\\bar P_n$ satisfies","title":"Increasing-sequence formula for the Kassel\u2013Reutenauer polynomials"},{"id":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/hilbert-scheme-torus-zeta-eq-prod","kind":"node","lean":"MetaMathlibExt.hilbertSchemeTorus_zeta_eq_prod","parent":1478,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials/index.html#hilbert-scheme-torus-zeta-eq-prod","scope":"Algebra/orthogonal-polynomials/kassel-reutenauer-polynomials","status":"can_prove","summary":"Let $p$ be a prime, $k \\ge 1$, $q = p^k$ and $n \\ge 1$. For $m \\ge 1$ let $\\mathbb{F}_{q^m}$ be the field with $q^m$ elements and let $N_m$ be the number of ideals $I$ of the Laurent polynomial ring $\\mathbb{F}_{q^m}[x^{\\pm 1}, y^{\\pm 1}]$ (the group algebra of $\\mathbb{Z} \\times \\mathbb{Z}$) such that the quotient by $I$ has dimension $n$ over $\\mathbb{F}_{q^m}$; this is the number of $\\mathbb{F}_{q^m}$-points of\u2026","title":"Zeta function of the Hilbert scheme of points on the two-dimensional torus"},{"data_url":"scopes/Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions.json","id":"Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions","kind":"scope","navigation":"module","parent":1443,"result":false,"route":"../roadmap/Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions/index.html","scope":"Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions","status":"fully_proved","title":"Moment generating functions as J-fractions"},{"id":"Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions/moment-generating-function-as-j-fraction","kind":"node","lean":"MetaMathlibExt.moment_generating_function_as_J_fraction_general","parent":1511,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions/index.html#moment-generating-function-as-j-fraction","scope":"Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions","status":"fully_proved","summary":"Let $p_0, p_1, \\dots$ be real polynomials, let $L$ be an $\\mathbb{R}$-linear functional on $\\mathbb{R}[x]$, and let $(\\alpha_n)$, $(\\beta_n)$ and $(\\mu_k)$ be real sequences such that $p_0 = 1$, $p_1 = x - \\alpha_0$, $p_{n+1} = (x - \\alpha_n) p_n - \\beta_n p_{n-1}$ for $n \\ge 1$, $\\mu_k = L(x^k)$ for every $k$, and $L(p_n) = 0$ for $n \\ge 1$. Then there are formal power series $S_0, S_1, \\dots$ over $\\mathbb{R}$,\u2026","title":"Moment generating function as a J-fraction, general form"},{"id":"Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions/moment-generating-function-as-j-fraction-2","kind":"node","lean":"MetaMathlibExt.moment_generating_function_as_J_fraction","parent":1511,"result":true,"route":"../roadmap/Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions/index.html#moment-generating-function-as-j-fraction-2","scope":"Algebra/orthogonal-polynomials/moment-generating-functions-as-j-fractions","status":"fully_proved","summary":"Let $p_0, p_1, \\dots$ be real polynomials, each $p_n$ monic of degree $n$, let $L$ be an $\\mathbb{R}$-linear functional on $\\mathbb{R}[x]$, and let $(\\alpha_n)$, $(\\beta_n)$ and $(\\mu_k)$ be real sequences. Suppose that $p_0 = 1$, $p_1 = x - \\alpha_0$ and $p_{n+1} = (x - \\alpha_n) p_n - \\beta_n p_{n-1}$ for $n \\ge 1$, that $\\beta_n \\ne 0$ for $n \\ge 1$, that the $p_n$ are orthogonal for $L$, meaning\u2026","title":"Moment generating function as a J-fraction"},{"data_url":"scopes/Algebra/zeros-of-polynomials.json","id":"Algebra/zeros-of-polynomials","kind":"scope","parent":1278,"result":false,"route":"../roadmap/Algebra/zeros-of-polynomials/index.html","scope":"Algebra/zeros-of-polynomials","status":"can_prove","summary":"Location, counting and multiplicity of the zeros of polynomials: real-rootedness, stability, interlacing, root bounds and Sturm sequences.","title":"Zeros of polynomials"},{"id":"Algebra/zeros-of-polynomials/auerbach-polynomial-zero-set-separates-i","kind":"node","lean":"MathlibExt.Algebra.Polynomial.ScottishBook148Wanted.auerbachPolynomialZeroSetSeparatesIffFactorsSignDefinite","parent":1514,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/index.html#auerbach-polynomial-zero-set-separates-i","scope":"Algebra/zeros-of-polynomials","status":"can_prove","summary":"Let `P` be a nonzero real polynomial in `n` variables. Its zero set does not cut (separate) Euclidean space, i.e. the complement of the zero set is connected, if and only if every irreducible factor of `P` over the reals is always nonnegative or always nonpositive (as a function). The complement of a real algebraic hypersurface is semialgebraic, so connectedness of the complement is equivalent to the\u2026","title":"Auerbach's problem on zero sets of real polynomials"},{"data_url":"scopes/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials.json","id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","kind":"scope","navigation":"module","parent":1514,"result":false,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","title":"L1 norm and root bounds for polynomials"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/l1-norm","kind":"node","lean":"Polynomial.l1Norm","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#l1-norm","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ be a semiring and $v$ a real-valued absolute value on $K$. The L1 norm of a polynomial $p = \\sum_i a_i X^i \\in K[X]$ with respect to $v$ is the real number $\\|p\\|_v = \\sum_i v(a_i)$, where the sum runs over the indices $i$ with $a_i \\neq 0$.","title":"L1 norm of a polynomial"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/l1-norm-def","kind":"node","lean":"Polynomial.l1Norm_def","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#l1-norm-def","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ be a semiring and $v$ a real-valued absolute value on $K$. For every polynomial $p = \\sum_i a_i X^i \\in K[X]$, the L1 norm $\\|p\\|_v$ of $p$ with respect to $v$ equals $\\sum_{i \\in S} v(a_i)$, where $S = \\{ i : a_i \\neq 0 \\}$ is the support of $p$.","title":"L1 norm as a sum over the support"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/l1-norm-zero","kind":"node","lean":"Polynomial.l1Norm_zero","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#l1-norm-zero","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ be a semiring and $v$ a real-valued absolute value on $K$. The L1 norm of the zero polynomial in $K[X]$ with respect to $v$ is $0$.","title":"L1 norm of the zero polynomial"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/l1-norm-nonneg","kind":"node","lean":"Polynomial.l1Norm_nonneg","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#l1-norm-nonneg","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ be a semiring and $v$ a real-valued absolute value on $K$. For every polynomial $p \\in K[X]$, the L1 norm of $p$ with respect to $v$ satisfies $\\|p\\|_v \\ge 0$.","title":"Nonnegativity of the L1 norm"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/l1-norm-eq-sum-range","kind":"node","lean":"Polynomial.l1Norm_eq_sum_range","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#l1-norm-eq-sum-range","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ be a semiring and $v$ a real-valued absolute value on $K$. Let $p = \\sum_i a_i X^i \\in K[X]$ and let $n$ be the degree of $p$, with $n = 0$ when $p = 0$. Then the L1 norm of $p$ with respect to $v$ is $\\|p\\|_v = \\sum_{i=0}^{n} v(a_i)$.","title":"L1 norm as a sum up to the degree"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/l1-norm-eq-sum-range-add-one-of-monic","kind":"node","lean":"Polynomial.l1Norm_eq_sum_range_add_one_of_monic","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#l1-norm-eq-sum-range-add-one-of-monic","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ be a nontrivial semiring (one with $0 \\neq 1$) and $v$ a real-valued absolute value on $K$. If $f = \\sum_i a_i X^i \\in K[X]$ is monic of degree $n$, then the L1 norm of $f$ with respect to $v$ is $\\|f\\|_v = \\sum_{i=0}^{n-1} v(a_i) + 1$.","title":"L1 norm of a monic polynomial"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/one-le-l1-norm-of-monic","kind":"node","lean":"Polynomial.one_le_l1Norm_of_monic","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#one-le-l1-norm-of-monic","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ be a nontrivial semiring (one with $0 \\neq 1$) and $v$ a real-valued absolute value on $K$. If $f \\in K[X]$ is monic, then the L1 norm of $f$ with respect to $v$ satisfies $\\|f\\|_v \\ge 1$.","title":"Monic polynomials have L1 norm at least one"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/nat-degree-eq-of-monic-of-l1-norm-sub-lt","kind":"node","lean":"Polynomial.natDegree_eq_of_monic_of_l1Norm_sub_lt_one","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#nat-degree-eq-of-monic-of-l1-norm-sub-lt","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ be a nontrivial ring and $v$ a real-valued absolute value on $K$. If $f, g \\in K[x]$ are monic and the L1 norm of $f - g$ with respect to $v$ is strictly less than $1$, then $f$ and $g$ have the same degree.","title":"Monic polynomials close in L1 norm have the same degree"},{"id":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/aeval-lt-l1-norm-of-monic-of-aeval-eq-ze","kind":"node","lean":"Polynomial.aeval_lt_l1Norm_of_monic_of_aeval_eq_zero","parent":1516,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials/index.html#aeval-lt-l1-norm-of-monic-of-aeval-eq-ze","scope":"Algebra/zeros-of-polynomials/l1-norm-and-root-bounds-for-polynomials","status":"fully_proved","summary":"Let $K$ and $L$ be fields with $L$ a $K$-algebra, let $v$ be a real-valued absolute value on $K$, and let $w$ be a real-valued absolute value on $L$ lying over $v$, that is, $w(x) = v(x)$ for every $x \\in K$ viewed in $L$ through the algebra map. If $f$ is a monic polynomial over $K$ and $\\alpha \\in L$ is a root of $f$, then $w(\\alpha)$ is strictly less than the L1 norm of $f$ with respect to $v$.","title":"Root bound by the L1 norm"},{"data_url":"scopes/Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros.json","id":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros","kind":"scope","navigation":"module","parent":1514,"result":false,"route":"../roadmap/Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/index.html","scope":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros","status":"can_prove","title":"Parity-change bound on real zeros"},{"id":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/c-tilde","kind":"node","lean":"MathlibExt.Algebra.RealZerosParityChangeBoundWanted.cTilde","parent":1526,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/index.html#c-tilde","scope":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros","status":"fully_proved","summary":"For a real polynomial $f = \\sum_k a_k x^k$ and a natural number $k$, $\\tilde c_k(f) = (k+1)\\,a_k^2 - k\\,a_{k-1}a_{k+1}$, where $a_j = 0$ for $j$ above the degree of $f$; for $k = 0$ the second term vanishes, so $\\tilde c_0(f) = a_0^2$.","title":"Newton-type coefficient differences"},{"id":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/pos-indices","kind":"node","lean":"MathlibExt.Algebra.RealZerosParityChangeBoundWanted.posIndices","parent":1526,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/index.html#pos-indices","scope":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros","status":"fully_proved","summary":"For a real polynomial $f = a_0 + a_1 x + \\dots + a_n x^n$ of degree $n$ (with $n = 0$ for $f = 0$), set $a_{n+1} = 0$, $\\tilde c_0(f) = a_0^2$ and $\\tilde c_k(f) = (k+1)\\,a_k^2 - k\\,a_{k-1}a_{k+1}$ for $1 \\le k \\le n$. This is the list of the indices $k \\in \\{0, \\dots, n\\}$ with $\\tilde c_k(f) > 0$, in increasing order.","title":"Indices with positive Newton-type difference"},{"id":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/v-val","kind":"node","lean":"MathlibExt.Algebra.RealZerosParityChangeBoundWanted.vVal","parent":1526,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/index.html#v-val","scope":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros","status":"fully_proved","summary":"For a real polynomial $f = a_0 + a_1 x + \\dots + a_n x^n$ of degree $n$ (with $n = 0$ for $f = 0$), set $a_{n+1} = 0$, $\\tilde c_0(f) = a_0^2$ and $\\tilde c_k(f) = (k+1)\\,a_k^2 - k\\,a_{k-1}a_{k+1}$ for $1 \\le k \\le n$. Let $k_1 < k_2 < \\dots < k_m$ be the indices $k \\le n$ with $\\tilde c_k(f) > 0$. Then $v(f)$ is the number of $i < m$ such that $k_i$ and $k_{i+1}$ have different parities.","title":"Parity changes of the positive-difference indices"},{"id":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/conjecture","kind":"node","lean":"MathlibExt.Algebra.RealZerosParityChangeBoundWanted.conjecture","parent":1526,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/index.html#conjecture","scope":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros","status":"fully_proved","summary":"For a real polynomial $f = a_0 + a_1 x + \\dots + a_n x^n$ of degree $n$, set $a_{n+1} = 0$, $\\tilde c_0(f) = a_0^2$ and $\\tilde c_k(f) = (k+1)\\,a_k^2 - k\\,a_{k-1}a_{k+1}$ for $1 \\le k \\le n$, and let $v(f)$ be the number of parity changes between consecutive terms of the increasing list of indices $k \\le n$ with $\\tilde c_k(f) > 0$. The conjecture asserts that every such $f$ with $a_k > 0$ for all $0 \\le k \\le n$\u2026","title":"Parity-change bound on real zeros (conjecture)"},{"id":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/conjecture-refuted","kind":"node","lean":"MathlibExt.Algebra.RealZerosParityChangeBoundWanted.conjecture_refuted","parent":1526,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros/index.html#conjecture-refuted","scope":"Algebra/zeros-of-polynomials/parity-change-bound-on-real-zeros","status":"can_prove","summary":"For a real polynomial $f = a_0 + a_1 x + \\dots + a_n x^n$ of degree $n$, set $a_{n+1} = 0$, $\\tilde c_0(f) = a_0^2$ and $\\tilde c_k(f) = (k+1)\\,a_k^2 - k\\,a_{k-1}a_{k+1}$ for $1 \\le k \\le n$, and let $v(f)$ be the number of parity changes between consecutive terms of the increasing list of indices $k \\le n$ with $\\tilde c_k(f) > 0$. There is such an $f$ with $a_k > 0$ for all $0 \\le k \\le n$ whose number of\u2026","title":"The parity-change bound on real zeros is false"},{"data_url":"scopes/Algebra/zeros-of-polynomials/real-rooted-polynomials.json","id":"Algebra/zeros-of-polynomials/real-rooted-polynomials","kind":"scope","navigation":"module","parent":1514,"result":false,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","title":"Real-rooted polynomials"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/is-real-rooted","kind":"node","lean":"Polynomial.IsRealRooted","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#is-real-rooted","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"A complex polynomial $p$ is real-rooted if every element $z$ of its multiset of roots is real, that is, $\\operatorname{Im} z = 0$.","title":"Real-rooted polynomials"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/is-real-rooted-of-map-star-eq-self-of-ro","kind":"node","lean":"Polynomial.isRealRooted_of_map_star_eq_self_of_roots_im_nonpos","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#is-real-rooted-of-map-star-eq-self-of-ro","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"Let $p$ be a nonzero complex polynomial that is unchanged when complex conjugation is applied to each of its coefficients. If every root $z$ of $p$ satisfies $\\operatorname{Im} z \\le 0$, then $p$ is real-rooted.","title":"Conjugation-invariant polynomials with roots in the closed lower half-plane are real-rooted"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/max-real-root","kind":"node","lean":"Polynomial.maxRealRoot","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#max-real-root","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"For a complex polynomial $p$, let $\\rho(p)$ be the largest real part $\\operatorname{Re} z$ of an element $z$ of the multiset of roots of $p$, with $\\rho(p) = 0$ when that multiset is empty (as it is for nonzero constants and for $p = 0$).","title":"Largest real part of a root"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/root-re-le-max-real-root","kind":"node","lean":"Polynomial.root_re_le_maxRealRoot","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#root-re-le-max-real-root","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"Let $p$ be a complex polynomial and let $\\rho(p)$ be the largest real part of an element of its multiset of roots, or $0$ if that multiset is empty. Then every element $z$ of the multiset of roots of $p$ satisfies $\\operatorname{Re} z \\le \\rho(p)$.","title":"Real parts of roots are bounded by the largest one"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/exists-root-re-eq-max-real-root","kind":"node","lean":"Polynomial.exists_root_re_eq_maxRealRoot","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#exists-root-re-eq-max-real-root","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"Let $p$ be a complex polynomial whose multiset of roots is nonempty, and let $\\rho(p)$ be the largest real part of a root of $p$. Then some root $z$ of $p$ satisfies $\\operatorname{Re} z = \\rho(p)$.","title":"The largest real part of a root is attained"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/roots-ne-zero-of-nat-degree-pos","kind":"node","lean":"Polynomial.roots_ne_zero_of_natDegree_pos","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#roots-ne-zero-of-nat-degree-pos","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"Every complex polynomial of positive degree has at least one root.","title":"Nonconstant complex polynomials have a root"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/coe-max-real-root-mem-roots","kind":"node","lean":"Polynomial.coe_maxRealRoot_mem_roots","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#coe-max-real-root-mem-roots","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"Let $p$ be a real-rooted complex polynomial of positive degree, and let $\\rho(p)$ be the largest real part of a root of $p$. Then the real number $\\rho(p)$, viewed as a complex number, is a root of $p$.","title":"For a real-rooted polynomial, the largest real part of a root is a root"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/eval-re-pos-of-max-real-root-lt","kind":"node","lean":"Polynomial.eval_re_pos_of_maxRealRoot_lt","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#eval-re-pos-of-max-real-root-lt","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"Let $p$ be a monic real-rooted complex polynomial, and let $\\rho(p)$ be the largest real part of a root of $p$, or $0$ if $p$ has no roots. Then $\\operatorname{Re} p(x) > 0$ for every real number $x > \\rho(p)$.","title":"Monic real-rooted polynomials have positive real part right of the largest root"},{"id":"Algebra/zeros-of-polynomials/real-rooted-polynomials/eval-re-nonneg-of-max-real-root-le","kind":"node","lean":"Polynomial.eval_re_nonneg_of_maxRealRoot_le","parent":1532,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/real-rooted-polynomials/index.html#eval-re-nonneg-of-max-real-root-le","scope":"Algebra/zeros-of-polynomials/real-rooted-polynomials","status":"fully_proved","summary":"Let $p$ be a monic real-rooted complex polynomial, and let $\\rho(p)$ be the largest real part of a root of $p$, or $0$ if $p$ has no roots. Then $\\operatorname{Re} p(x) \\ge 0$ for every real number $x \\ge \\rho(p)$.","title":"Monic real-rooted polynomials have nonnegative real part from the largest root on"},{"data_url":"scopes/Algebra/zeros-of-polynomials/interlacing-families-of-polynomials.json","id":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials","kind":"scope","navigation":"module","parent":1514,"result":false,"route":"../roadmap/Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/index.html","scope":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials","status":"fully_proved","title":"Interlacing families of polynomials"},{"id":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/has-upper-interlacing-point","kind":"node","lean":"Polynomial.HasUpperInterlacingPoint","parent":1542,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/index.html#has-upper-interlacing-point","scope":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials","status":"fully_proved","summary":"Let $p$ be a complex polynomial and let $\\operatorname{maxroot}(p)$ be the largest real part of an element of the multiset of roots of $p$, with $\\operatorname{maxroot}(p) = 0$ when that multiset is empty (as it is for nonzero constants and for $p = 0$). A real number $a$ is an upper interlacing point of $p$ if $a \\le \\operatorname{maxroot}(p)$ and, after removing one copy of $\\operatorname{maxroot}(p)$ from the\u2026","title":"Upper interlacing point"},{"id":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/has-common-upper-interlacing","kind":"node","lean":"Polynomial.HasCommonUpperInterlacing","parent":1542,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/index.html#has-common-upper-interlacing","scope":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials","status":"fully_proved","summary":"A family $(P_i)_{i \\in I}$ of complex polynomials, indexed by an arbitrary type $I$, has a common upper interlacing point if some real number $a$ is an upper interlacing point of $P_i$ for every $i \\in I$.","title":"Common upper interlacing point"},{"id":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/eval-re-nonpos-of-upper-interlacing","kind":"node","lean":"Polynomial.eval_re_nonpos_of_upperInterlacing","parent":1542,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/index.html#eval-re-nonpos-of-upper-interlacing","scope":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials","status":"fully_proved","summary":"Let $p$ be a monic complex polynomial of positive degree all of whose roots are real, let $\\operatorname{maxroot}(p)$ be its largest root, and let $a$ be an upper interlacing point of $p$. Then for every real $x$ with $a \\le x \\le \\operatorname{maxroot}(p)$, the real part of $p(x)$ is at most $0$.","title":"Nonpositivity between an upper interlacing point and the largest root"},{"id":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/exists-max-real-root-le-weighted-sum-of","kind":"node","lean":"Polynomial.exists_maxRealRoot_le_weightedSum_of_commonUpperInterlacing","parent":1542,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/index.html#exists-max-real-root-le-weighted-sum-of","scope":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials","status":"fully_proved","summary":"Let $I$ be a nonempty finite set and, for each $i \\in I$, let $P_i$ be a monic complex polynomial of positive degree all of whose roots are real, such that the family $(P_i)_{i \\in I}$ has a common upper interlacing point. Let $w_i \\ge 0$ be real weights such that $Q = \\sum_{i \\in I} w_i P_i$ is monic and all of its roots are real. Then there is $i \\in I$ with\u2026","title":"Common upper interlacing bounds some member's largest root by a weighted sum's"},{"id":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/exists-max-real-root-le-weighted-sum-of-2","kind":"node","lean":"Polynomial.exists_maxRealRoot_le_weightedSum_of_segment_realRooted","parent":1542,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/interlacing-families-of-polynomials/index.html#exists-max-real-root-le-weighted-sum-of-2","scope":"Algebra/zeros-of-polynomials/interlacing-families-of-polynomials","status":"fully_proved","summary":"Let $I$ be a nonempty finite set, let $n \\ge 1$, and for each $i \\in I$ let $P_i$ be a monic complex polynomial of degree $n$ and $w_i \\ge 0$ a real weight. Suppose that $Q = \\sum_{i \\in I} w_i P_i$ is monic of degree $n$ and that, for every $i \\in I$ and every $t \\in [0, 1]$, all roots of $(1 - t) Q + t P_i$ are real. Then there is $i \\in I$ with $\\operatorname{maxroot}(P_i) \\le \\operatorname{maxroot}(Q)$, where\u2026","title":"Some member's largest root is at most that of a weighted sum, via real-rooted segments"},{"data_url":"scopes/Algebra/zeros-of-polynomials/stable-polynomials.json","id":"Algebra/zeros-of-polynomials/stable-polynomials","kind":"scope","navigation":"module","parent":1514,"result":false,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","title":"Stable polynomials"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/is-stable","kind":"node","lean":"MvPolynomial.IsStable","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#is-stable","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"A complex multivariate polynomial is stable if it has no zero when every variable lies in the open upper half-plane.","title":"Stable polynomials"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/is-real-stable","kind":"node","lean":"MvPolynomial.IsRealStable","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#is-real-stable","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"A complex multivariate polynomial is real stable if it is stable and has real coefficients.","title":"Real stable polynomials"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/univariate-specialization","kind":"node","lean":"MvPolynomial.univariateSpecialization","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#univariate-specialization","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $p$ be a multivariate polynomial over $R$ in variables $X_j$ indexed by $j \\in \\sigma$, and let $i \\in \\sigma$ and $x : \\sigma \\to R$. The univariate specialization of $p$ at $i$ and $x$ is the polynomial in $R[X]$ obtained from $p$ by substituting $X$ for $X_i$ and the constant $x_j$ for every $X_j$ with $j \\ne i$; the value $x_i$ is not used.","title":"Univariate specialization"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/derivative-univariate-specialization","kind":"node","lean":"MvPolynomial.derivative_univariateSpecialization","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#derivative-univariate-specialization","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $p$ be a multivariate polynomial over $R$ in variables $X_j$ indexed by $j \\in \\sigma$, let $i \\in \\sigma$ and $x : \\sigma \\to R$, and for a polynomial $q$ write $q_{i,x} \\in R[X]$ for its univariate specialization that keeps $X_i$ as the variable $X$ and sets every other $X_j$ to $x_j$. Then $(\\partial p / \\partial X_i)_{i,x} = (p_{i,x})'$, the derivative of $p_{i,x}$.","title":"Partial derivatives under univariate specialization"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/eval-univariate-specialization","kind":"node","lean":"MvPolynomial.eval_univariateSpecialization","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#eval-univariate-specialization","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $p$ be a multivariate polynomial over $R$ in variables $X_j$ indexed by $j \\in \\sigma$, and let $i \\in \\sigma$, $x : \\sigma \\to R$ and $t \\in R$. Evaluating at $t$ the univariate specialization of $p$ that keeps $X_i$ as the variable and sets every other $X_j$ to $x_j$ gives $p(y)$, where $y_i = t$ and $y_j = x_j$ for $j \\ne i$.","title":"Evaluating a univariate specialization"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/map-star-univariate-specialization","kind":"node","lean":"MvPolynomial.map_star_univariateSpecialization","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#map-star-univariate-specialization","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $p$ be a complex multivariate polynomial with real coefficients in variables $X_j$ indexed by $j \\in \\sigma$, let $i \\in \\sigma$, and let $a : \\sigma \\to \\mathbb{C}$ with $a_j$ real for every $j \\ne i$. Then the univariate specialization of $p$ that keeps $X_i$ as the variable and sets every other $X_j$ to $a_j$ has real coefficients.","title":"Univariate specialization at real values preserves real coefficients"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/roots-im-nonpos-univariate-specializatio","kind":"node","lean":"MvPolynomial.IsStable.roots_im_nonpos_univariateSpecialization","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#roots-im-nonpos-univariate-specializatio","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $p$ be a stable complex multivariate polynomial in variables $X_j$ indexed by $j \\in \\sigma$, let $i \\in \\sigma$, and let $a : \\sigma \\to \\mathbb{C}$ with $\\operatorname{Im} a_j \\ge 0$ for every $j \\ne i$. If the univariate specialization of $p$ that keeps $X_i$ as the variable and sets every other $X_j$ to $a_j$ is nonzero, then every root $w$ of it satisfies $\\operatorname{Im} w \\le 0$.","title":"Roots of specializations of stable polynomials lie in the closed lower half-plane"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/is-real-rooted-univariate-specialization","kind":"node","lean":"MvPolynomial.isRealRooted_univariateSpecialization_of_isRealStable","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#is-real-rooted-univariate-specialization","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $p$ be a real stable complex multivariate polynomial in variables $X_j$ indexed by $j \\in \\sigma$, let $i \\in \\sigma$, and for $x : \\sigma \\to \\mathbb{C}$ write $p_{i,x} \\in \\mathbb{C}[X]$ for the univariate specialization of $p$ that keeps $X_i$ as the variable $X$ and sets every other $X_j$ to $x_j$. Suppose there is a natural number $d$ such that for every $x : \\sigma \\to \\mathbb{C}$ the polynomial $p_{i,x}$\u2026","title":"Real-rootedness of univariate specializations of real stable polynomials"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/sub-pderiv","kind":"node","lean":"MvPolynomial.IsStable.sub_pderiv","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#sub-pderiv","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $p$ be a stable complex multivariate polynomial in variables $X_j$ indexed by $j \\in \\sigma$, let $i \\in \\sigma$, and write $\\partial_i p$ for the partial derivative of $p$ with respect to $X_i$. Then $p - \\partial_i p$ is stable.","title":"Subtracting a partial derivative preserves stability"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/sub-pderiv-2","kind":"node","lean":"MvPolynomial.IsRealStable.sub_pderiv","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#sub-pderiv-2","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $p$ be a real stable complex multivariate polynomial in variables $X_j$ indexed by $j \\in \\sigma$, let $i \\in \\sigma$, and write $\\partial_i p$ for the partial derivative of $p$ with respect to $X_i$. Then $p - \\partial_i p$ is real stable.","title":"Subtracting a partial derivative preserves real stability"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/mixed-differential","kind":"node","lean":"MvPolynomial.IsStable.mixedDifferential","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#mixed-differential","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $p$ be a stable complex multivariate polynomial in variables $X_s$ indexed by $s \\in \\sigma$, and write $\\partial_s$ for the partial derivative with respect to $X_s$. For every finite list $(i_1, \\dots, i_k)$ of indices in $\\sigma$ (repetitions allowed, possibly empty), the mixed differential of $p$ along the list, $(1 - \\partial_{i_k}) \\cdots (1 - \\partial_{i_1})\\, p$, is stable.","title":"The mixed differential preserves stability"},{"id":"Algebra/zeros-of-polynomials/stable-polynomials/mixed-differential-2","kind":"node","lean":"MvPolynomial.IsRealStable.mixedDifferential","parent":1548,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/stable-polynomials/index.html#mixed-differential-2","scope":"Algebra/zeros-of-polynomials/stable-polynomials","status":"fully_proved","summary":"Let $p$ be a real stable complex multivariate polynomial in variables $X_s$ indexed by $s \\in \\sigma$, and write $\\partial_s$ for the partial derivative with respect to $X_s$. For every finite list $(i_1, \\dots, i_k)$ of indices in $\\sigma$ (repetitions allowed, possibly empty), the mixed differential of $p$ along the list, $(1 - \\partial_{i_k}) \\cdots (1 - \\partial_{i_1})\\, p$, is real stable.","title":"The mixed differential preserves real stability"},{"data_url":"scopes/Algebra/zeros-of-polynomials/sturm-s-theorem.json","id":"Algebra/zeros-of-polynomials/sturm-s-theorem","kind":"scope","navigation":"module","parent":1514,"result":false,"route":"../roadmap/Algebra/zeros-of-polynomials/sturm-s-theorem/index.html","scope":"Algebra/zeros-of-polynomials/sturm-s-theorem","status":"can_prove","title":"Sturm's theorem"},{"id":"Algebra/zeros-of-polynomials/sturm-s-theorem/sturm-aux","kind":"node","lean":"MathlibExt.Algebra.SturmWanted.Algebra.sturmAux","parent":1561,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/sturm-s-theorem/index.html#sturm-aux","scope":"Algebra/zeros-of-polynomials/sturm-s-theorem","status":"fully_proved","summary":"For real polynomials $a$, $b$ and a natural number $n$, this is the list of polynomials defined recursively by: for $n = 0$ it is the one-entry list $(a)$; for fuel $n + 1$ it is $(a)$ if $b = 0$, and otherwise it is $a$ followed by the list for the pair $b$, $-(a \\bmod b)$ with fuel $n$, where $a \\bmod b$ is the remainder of $a$ on division by $b$.","title":"Negated-remainder sequence with fuel"},{"id":"Algebra/zeros-of-polynomials/sturm-s-theorem/sturm-chain","kind":"node","lean":"MathlibExt.Algebra.SturmWanted.Algebra.sturmChain","parent":1561,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/sturm-s-theorem/index.html#sturm-chain","scope":"Algebra/zeros-of-polynomials/sturm-s-theorem","status":"fully_proved","summary":"The Sturm chain of a real polynomial $p$ is the list $p_0, p_1, \\dots, p_m$ with $p_0 = p$, $p_1 = p'$ and $p_{k+1} = -(p_{k-1} \\bmod p_k)$, the negated remainder of $p_{k-1}$ on division by $p_k$, where the list stops at $p_m$ as soon as the next entry would be $0$; thus $p_m$ is its last entry and, for $p \\ne 0$, every entry is nonzero. For $p = 0$ the chain is the one-entry list $(0)$.","title":"Sturm chain"},{"id":"Algebra/zeros-of-polynomials/sturm-s-theorem/sign-changes","kind":"node","lean":"MathlibExt.Algebra.SturmWanted.Algebra.signChanges","parent":1561,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/sturm-s-theorem/index.html#sign-changes","scope":"Algebra/zeros-of-polynomials/sturm-s-theorem","status":"fully_proved","summary":"For a list $x_1, \\dots, x_n$ of real numbers, let $y_1, \\dots, y_m$ be the entries of the list that are nonzero, in their original order. The number of sign changes of the list is the number of indices $i$ with $1 \\le i < m$ and $y_i y_{i+1} < 0$, that is, the number of adjacent pairs of opposite sign once the zeros are removed.","title":"Number of sign changes in a list of reals"},{"id":"Algebra/zeros-of-polynomials/sturm-s-theorem/sigma","kind":"node","lean":"MathlibExt.Algebra.SturmWanted.Algebra.sigma","parent":1561,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/sturm-s-theorem/index.html#sigma","scope":"Algebra/zeros-of-polynomials/sturm-s-theorem","status":"fully_proved","summary":"For a real polynomial $p$ with Sturm chain $p_0, p_1, \\dots, p_m$ and a real number $x$, $\\sigma_p(x)$ is the number of sign changes, ignoring zeros, in the sequence of values $p_0(x), p_1(x), \\dots, p_m(x)$.","title":"Sturm sign-variation function"},{"id":"Algebra/zeros-of-polynomials/sturm-s-theorem/sturm","kind":"node","lean":"MathlibExt.Algebra.SturmWanted.Algebra.sturm","parent":1561,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/sturm-s-theorem/index.html#sturm","scope":"Algebra/zeros-of-polynomials/sturm-s-theorem","status":"can_prove","summary":"Let $p$ be a squarefree real polynomial and let $a < b$ be real numbers with $p(a) \\ne 0$ and $p(b) \\ne 0$. Then the number of distinct real roots $x$ of $p$ with $a < x < b$ equals $\\sigma_p(a) - \\sigma_p(b)$, where $\\sigma_p$ is the Sturm sign-variation function of $p$ and the subtraction is truncated subtraction of natural numbers.","title":"Sturm's theorem"},{"data_url":"scopes/Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots.json","id":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots","kind":"scope","navigation":"module","parent":1514,"result":false,"route":"../roadmap/Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/index.html","scope":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots","status":"fully_proved","title":"Taylor expansion and double roots"},{"id":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/exists-unique-taylor-remainder","kind":"node","lean":"Polynomial.existsUnique_taylor_remainder","parent":1567,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/index.html#exists-unique-taylor-remainder","scope":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $f \\in R[X]$ and $a \\in R$, and let $f'$ be the formal derivative of $f$. There is a unique polynomial $g \\in R[X]$ such that","title":"First-order Taylor expansion of a polynomial"},{"id":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/eval-and-derivative-eval-eq-zero-iff","kind":"node","lean":"Polynomial.eval_and_derivative_eval_eq_zero_iff","parent":1567,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/index.html#eval-and-derivative-eval-eq-zero-iff","scope":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $f \\in R[x]$ and $a \\in R$, and let $f'$ be the formal derivative of $f$. Then $f(a) = 0$ and $f'(a) = 0$ if and only if $f = (x - a)^2 g$ for some $g \\in R[x]$.","title":"Double root criterion"},{"id":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/sq-dvd-iff-eval-and-derivative-eval-eq-z","kind":"node","lean":"Polynomial.sq_dvd_iff_eval_and_derivative_eval_eq_zero","parent":1567,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/index.html#sq-dvd-iff-eval-and-derivative-eval-eq-z","scope":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $f \\in R[x]$ and $a \\in R$, and let $f'$ be the formal derivative of $f$. Then $(x - a)^2$ divides $f$ in $R[x]$ if and only if $f(a) = 0$ and $f'(a) = 0$.","title":"Double root criterion in divisibility form"},{"id":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/not-sq-dvd-of-derivative-eval-ne-zero","kind":"node","lean":"Polynomial.not_sq_dvd_of_derivative_eval_ne_zero","parent":1567,"result":true,"route":"../roadmap/Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots/index.html#not-sq-dvd-of-derivative-eval-ne-zero","scope":"Algebra/zeros-of-polynomials/taylor-expansion-and-double-roots","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $f \\in R[x]$ and $a \\in R$, and let $f'$ be the formal derivative of $f$. If $f'(a) \\neq 0$, then $(x - a)^2$ does not divide $f$ in $R[x]$.","title":"A nonvanishing derivative excludes a double root"},{"data_url":"scopes/Algebra/nonassociative-algebras.json","id":"Algebra/nonassociative-algebras","kind":"scope","parent":1278,"result":false,"route":"../roadmap/Algebra/nonassociative-algebras/index.html","scope":"Algebra/nonassociative-algebras","status":"can_prove","summary":"Lie algebras and their structure and representations, free Lie algebras, Poisson algebras, and racks and quandles.","title":"Lie algebras and nonassociative structures"},{"data_url":"scopes/Algebra/nonassociative-algebras/levi-decomposition.json","id":"Algebra/nonassociative-algebras/levi-decomposition","kind":"scope","navigation":"module","parent":1572,"result":false,"route":"../roadmap/Algebra/nonassociative-algebras/levi-decomposition/index.html","scope":"Algebra/nonassociative-algebras/levi-decomposition","status":"fully_proved","title":"Levi decomposition"},{"id":"Algebra/nonassociative-algebras/levi-decomposition/exists-is-compl-of-is-semisimple","kind":"node","lean":"LieModule.exists_isCompl_of_isSemisimple","parent":1573,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/levi-decomposition/index.html#exists-is-compl-of-is-semisimple","scope":"Algebra/nonassociative-algebras/levi-decomposition","status":"fully_proved","summary":"Let $L$ be a finite-dimensional semisimple Lie algebra over a field $K$ of characteristic zero, and let $M$ be a finite-dimensional Lie module over $L$ and $K$. Then every Lie submodule $W$ of $M$ has a complement: there is a Lie submodule $U$ of $M$ with $W \\cap U = 0$ and $W + U = M$.","title":"Weyl's complete reducibility theorem"},{"id":"Algebra/nonassociative-algebras/levi-decomposition/levi-decomposition","kind":"node","lean":"MathlibExt.Algebra.Lie.LandmarkWanted.levi_decomposition","parent":1573,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/levi-decomposition/index.html#levi-decomposition","scope":"Algebra/nonassociative-algebras/levi-decomposition","status":"fully_proved","summary":"Let $L$ be a finite-dimensional Lie algebra over a field $K$ of characteristic zero, and let $\\operatorname{rad}(L)$ be its radical, the largest solvable ideal of $L$. Then there is a semisimple Lie subalgebra $S$ of $L$ (a Levi factor) such that $L = \\operatorname{rad}(L) \\oplus S$ as $K$-vector spaces, that is, $\\operatorname{rad}(L) + S = L$ and $\\operatorname{rad}(L) \\cap S = 0$.","title":"Levi decomposition"},{"data_url":"scopes/Algebra/nonassociative-algebras/ado-s-theorem.json","id":"Algebra/nonassociative-algebras/ado-s-theorem","kind":"scope","navigation":"module","parent":1572,"result":false,"route":"../roadmap/Algebra/nonassociative-algebras/ado-s-theorem/index.html","scope":"Algebra/nonassociative-algebras/ado-s-theorem","status":"can_prove","title":"Ado's theorem"},{"id":"Algebra/nonassociative-algebras/ado-s-theorem/ado","kind":"node","lean":"MathlibExt.Algebra.Lie.LandmarkWanted.ado","parent":1576,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/ado-s-theorem/index.html#ado","scope":"Algebra/nonassociative-algebras/ado-s-theorem","status":"fully_proved","summary":"Every finite-dimensional Lie algebra $L$ over a field $K$ of characteristic zero has a faithful finite-dimensional representation: there are a natural number $n$ and an injective homomorphism of Lie algebras over $K$ from $L$ into the Lie algebra of $K$-linear endomorphisms of $K^n$ with the commutator bracket.","title":"Ado's theorem"},{"id":"Algebra/nonassociative-algebras/ado-s-theorem/ado-iwasawa","kind":"node","lean":"MathlibExt.GroupTheory.AdoIwasawaWanted.RepresentationTheory.AdoIwasawa.adoIwasawa","parent":1576,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/ado-s-theorem/index.html#ado-iwasawa","scope":"Algebra/nonassociative-algebras/ado-s-theorem","status":"can_prove","summary":"Let $K$ be a field and let $L$ be a finite-dimensional Lie algebra over $K$. Then $L$ has a faithful finite-dimensional representation: there are a finite-dimensional vector space $V$ over $K$ and an injective homomorphism of Lie algebras over $K$ from $L$ to the Lie algebra of $K$-linear endomorphisms of $V$ with the commutator bracket.","title":"Ado\u2013Iwasawa theorem"},{"data_url":"scopes/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals.json","id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","kind":"scope","navigation":"module","parent":1572,"result":false,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","title":"Poisson algebras and Poisson ideals"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/poisson-bracket","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.PoissonBracket","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#poisson-bracket","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra. A Poisson bracket on $A$ is a map assigning to $x, y \\in A$ an element $\\lbrace x, y \\rbrace \\in A$ such that for all $x, y, z \\in A$ and $c \\in \\mathbb{C}$: $\\lbrace x + y, z \\rbrace = \\lbrace x, z \\rbrace + \\lbrace y, z \\rbrace$ and $\\lbrace x, y + z \\rbrace = \\lbrace x, y \\rbrace + \\lbrace x, z \\rbrace$; $\\lbrace x, y \\rbrace = -\\lbrace y, x \\rbrace$; the Leibniz\u2026","title":"Poisson bracket"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/is-poisson-ideal","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.IsPoissonIdeal","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#is-poisson-ideal","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket $\\lbrace \\cdot, \\cdot \\rbrace$. An ideal $P$ of $A$ is a Poisson ideal when $\\lbrace a, p \\rbrace \\in P$ for all $a \\in A$ and all $p \\in P$.","title":"Poisson ideal"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/is-poisson-prime","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.IsPoissonPrime","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#is-poisson-prime","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket. An ideal $P$ of $A$ is a Poisson prime when it is both a prime ideal and a Poisson ideal.","title":"Poisson prime ideal"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/poisson-core","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.poissonCore","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#poisson-core","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket. For an ideal $J$ of $A$, the Poisson core of $J$ is the supremum, in the lattice of ideals of $A$, of the set of all Poisson ideals of $A$ contained in $J$.","title":"Poisson core"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/bracket-zero","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.bracket_zero","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#bracket-zero","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket $\\lbrace \\cdot, \\cdot \\rbrace$. For every $a \\in A$, $\\lbrace a, 0 \\rbrace = 0$.","title":"Bracket with zero"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/bot-poisson","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.bot_poisson","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#bot-poisson","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket. The zero ideal of $A$ is a Poisson ideal.","title":"The zero ideal is a Poisson ideal"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/sup-poisson","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.sup_poisson","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#sup-poisson","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket, and let $I$, $K$ and $J$ be ideals of $A$. If $I$ and $K$ are Poisson ideals contained in $J$, then their sum $I + K$ is a Poisson ideal contained in $J$.","title":"The sum of two Poisson ideals inside an ideal"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/poisson-core-poisson","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.poissonCore_poisson","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#poisson-core-poisson","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket. For every ideal $J$ of $A$, the Poisson core of $J$ is a Poisson ideal.","title":"The Poisson core is a Poisson ideal"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/poisson-core-le","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.poissonCore_le","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#poisson-core-le","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket. For every ideal $J$ of $A$, the Poisson core of $J$ is contained in $J$.","title":"The Poisson core is contained in the ideal"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/le-poisson-core","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.le_poissonCore","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#le-poisson-core","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket, and let $J$ and $P$ be ideals of $A$. If $P \\subseteq J$ and $P$ is a Poisson ideal, then $P$ is contained in the Poisson core of $J$.","title":"Poisson ideals inside an ideal lie in its Poisson core"},{"id":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/poisson-core-is-greatest","kind":"node","lean":"MathlibExt.Algebra.Lie.PoissonBracket.poissonCore_isGreatest","parent":1579,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals/index.html#poisson-core-is-greatest","scope":"Algebra/nonassociative-algebras/poisson-algebras-and-poisson-ideals","status":"fully_proved","summary":"Let $A$ be a commutative $\\mathbb{C}$-algebra with a Poisson bracket. For every ideal $J$ of $A$, the Poisson core of $J$ is the greatest element of the set of Poisson ideals of $A$ contained in $J$: it is a Poisson ideal contained in $J$, and it contains every Poisson ideal of $A$ contained in $J$.","title":"Universal property of the Poisson core"},{"data_url":"scopes/Algebra/nonassociative-algebras/rota-baxter-operators-on-groups.json","id":"Algebra/nonassociative-algebras/rota-baxter-operators-on-groups","kind":"scope","navigation":"module","parent":1572,"result":false,"route":"../roadmap/Algebra/nonassociative-algebras/rota-baxter-operators-on-groups/index.html","scope":"Algebra/nonassociative-algebras/rota-baxter-operators-on-groups","status":"can_prove","title":"Rota\u2013Baxter operators on groups"},{"id":"Algebra/nonassociative-algebras/rota-baxter-operators-on-groups/is-rota-baxter-operator","kind":"node","lean":"MathlibExt.GroupTheory.SurjectiveRotaBaxterOperatorKourovka20Problem125Wanted.IsRotaBaxterOperator","parent":1591,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/rota-baxter-operators-on-groups/index.html#is-rota-baxter-operator","scope":"Algebra/nonassociative-algebras/rota-baxter-operators-on-groups","status":"fully_proved","summary":"A Rota\u2013Baxter operator (of weight $1$) on a group $G$ is a map $B \\colon G \\to G$ such that $B(g)\\,B(h) = B\\bigl(g\\,B(g)\\,h\\,B(g)^{-1}\\bigr)$ for all $g, h \\in G$.","title":"Rota\u2013Baxter operators on groups"},{"id":"Algebra/nonassociative-algebras/rota-baxter-operators-on-groups/surjective-non-injective-rota-baxter-ope","kind":"node","lean":"MathlibExt.GroupTheory.SurjectiveRotaBaxterOperatorKourovka20Problem125Wanted.surjectiveNonInjectiveRotaBaxterOperatorExists","parent":1591,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/rota-baxter-operators-on-groups/index.html#surjective-non-injective-rota-baxter-ope","scope":"Algebra/nonassociative-algebras/rota-baxter-operators-on-groups","status":"fully_proved","summary":"The statement, asked by V. G. Bardakov, that there exist a non-abelian group $G$ and a Rota\u2013Baxter operator $B$ on $G$ such that $B$ is surjective but not injective.","title":"Kourovka Problem 20.125 (surjective non-injective Rota\u2013Baxter operators)"},{"id":"Algebra/nonassociative-algebras/rota-baxter-operators-on-groups/surjective-non-injective-rota-baxter-ope-2","kind":"node","lean":"MathlibExt.GroupTheory.SurjectiveRotaBaxterOperatorKourovka20Problem125Wanted.surjectiveNonInjectiveRotaBaxterOperatorExists_holds","parent":1591,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/rota-baxter-operators-on-groups/index.html#surjective-non-injective-rota-baxter-ope-2","scope":"Algebra/nonassociative-algebras/rota-baxter-operators-on-groups","status":"can_prove","summary":"There exist a non-abelian group $G$ and a Rota\u2013Baxter operator $B$ on $G$ that is surjective but not injective.","title":"A surjective, non-injective Rota\u2013Baxter operator on a non-abelian group"},{"data_url":"scopes/Algebra/nonassociative-algebras/virtual-racks-and-quandles.json","id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","kind":"scope","navigation":"module","parent":1572,"result":false,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"can_prove","title":"Virtual racks and quandles"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/rack-aut","kind":"node","lean":"MetaMathlibExt.rackAut","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#rack-aut","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"For a rack $R$ with operation $\\triangleleft$, the automorphism group $\\operatorname{Aut} R$ is the subgroup of the group of permutations of $R$ consisting of the permutations $f$ with $f(x \\triangleleft y) = f(x) \\triangleleft f(y)$ for all $x, y \\in R$.","title":"Automorphism group of a rack"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/virtual-rack-iso","kind":"node","lean":"MetaMathlibExt.VirtualRackIso","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#virtual-rack-iso","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"Write $x \\triangleleft y$ for the operation of a rack. A virtual rack $(R, f)$ is a rack $R$ together with a rack automorphism $f$ of $R$, its virtual structure. Let $R_1$ and $R_2$ be racks and let $f_1$ and $f_2$ be permutations of $R_1$ and $R_2$. Then $(R_1, f_1)$ and $(R_2, f_2)$ are isomorphic when there is a bijection $\\psi \\colon R_1 \\to R_2$ such that\u2026","title":"Isomorphism of virtual racks"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/virtual-rack-class-count","kind":"node","lean":"MetaMathlibExt.virtualRackClassCount","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#virtual-rack-class-count","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"For a rack $R$, let $\\operatorname{Aut} R$ be its group of rack automorphisms; a virtual structure on $R$ is an element $f \\in \\operatorname{Aut} R$, and $(R, f)$ is the corresponding virtual rack. The number $v(R)$ is the number of classes of virtual structures on $R$, where $f, g \\in \\operatorname{Aut} R$ are identified when the virtual racks $(R, f)$ and $(R, g)$ are isomorphic; that is, the number of\u2026","title":"Number of isomorphism classes of virtual racks on a rack"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/dihedral-affine-perm","kind":"node","lean":"MetaMathlibExt.dihedralAffinePerm","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#dihedral-affine-perm","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"For a natural number $n$, an element $a \\in \\mathbb{Z}/n\\mathbb{Z}$ and a unit $u \\in (\\mathbb{Z}/n\\mathbb{Z})^\\times$, $T_{a,u}$ is the affine map $x \\mapsto a + u x$ of $\\mathbb{Z}/n\\mathbb{Z}$, viewed as a permutation of the dihedral quandle $R_n$ (the set $\\mathbb{Z}/n\\mathbb{Z}$ with $x \\triangleleft y = 2x - y$).","title":"Affine permutations of the dihedral quandle"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/virtual-rack-iso-refl","kind":"node","lean":"MetaMathlibExt.virtualRackIso_refl","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#virtual-rack-iso-refl","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"For every rack $R$ and every permutation $f$ of $R$, the pair $(R, f)$ is isomorphic to itself in the sense of virtual rack isomorphism. The permutation $f$ need not be a rack automorphism.","title":"Reflexivity of virtual rack isomorphism"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/dihedral-affine-perm-mem-rack-aut","kind":"node","lean":"MetaMathlibExt.dihedralAffinePerm_mem_rackAut","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#dihedral-affine-perm-mem-rack-aut","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"For every natural number $n$, every $a \\in \\mathbb{Z}/n\\mathbb{Z}$ and every unit $u \\in (\\mathbb{Z}/n\\mathbb{Z})^\\times$, the affine map $T_{a,u} \\colon x \\mapsto a + u x$ is a rack automorphism of the dihedral quandle $R_n$, the set $\\mathbb{Z}/n\\mathbb{Z}$ with $x \\triangleleft y = 2x - y$.","title":"Affine maps are automorphisms of the dihedral quandle"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/zmod-units-mul-aut","kind":"node","lean":"MetaMathlibExt.zmodUnitsMulAut","parent":1595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplication action of the units of "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"]],"source":"Multiplication action of the units of $\\mathbb{Z}/n\\mathbb{Z}$"},"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#zmod-units-mul-aut","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"For a natural number $n$, the group homomorphism from the unit group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ to the automorphism group of the additive group $\\mathbb{Z}/n\\mathbb{Z}$ that sends a unit $u$ to multiplication by $u$, the automorphism $x \\mapsto u x$.","title":"Multiplication action of the units of $\\mathbb{Z}/n\\mathbb{Z}$"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/cyclic-holomorph","kind":"node","lean":"MetaMathlibExt.cyclicHolomorph","parent":1595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holomorph of "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"]],"source":"Holomorph of $\\mathbb{Z}/n\\mathbb{Z}$"},"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#cyclic-holomorph","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"For a natural number $n$, let $A = \\mathbb{Z}/n\\mathbb{Z}$. The holomorph of $A$ is the semidirect product $A \\rtimes A^\\times$ in which the unit group $A^\\times$ acts on $A$ by multiplication. Its elements are the pairs $(a, u)$ with $a \\in A$ and $u \\in A^\\times$, with product $(a, u)(b, v) = (a + u b, u v)$.","title":"Holomorph of $\\mathbb{Z}/n\\mathbb{Z}$"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/cyclic-holomorph-mk-mul-mk","kind":"node","lean":"MetaMathlibExt.cyclicHolomorph_mk_mul_mk","parent":1595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplication in the holomorph of "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"]],"source":"Multiplication in the holomorph of $\\mathbb{Z}/n\\mathbb{Z}$"},"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#cyclic-holomorph-mk-mul-mk","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"fully_proved","summary":"For a natural number $n$, let $A = \\mathbb{Z}/n\\mathbb{Z}$ and write $(a, u)$ for the element of the holomorph $A \\rtimes A^\\times$ with $a \\in A$ and $u \\in A^\\times$. For all $a, b \\in A$ and $u, v \\in A^\\times$, $(a, u)(b, v) = (a + u b, u v)$.","title":"Multiplication in the holomorph of $\\mathbb{Z}/n\\mathbb{Z}$"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/virtual-rack-iso-iff-is-conj","kind":"node","lean":"MetaMathlibExt.virtualRackIso_iff_isConj","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#virtual-rack-iso-iff-is-conj","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"can_prove","summary":"Let $R$ be a rack and let $f_1$ and $f_2$ be elements of its automorphism group $\\operatorname{Aut} R$. Then the virtual racks $(R, f_1)$ and $(R, f_2)$ are isomorphic if and only if $f_1$ and $f_2$ are conjugate in $\\operatorname{Aut} R$.","title":"Isomorphism of virtual structures is conjugacy"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/center-eq-bot-iff-mul-aut-eq-rack-aut-co","kind":"node","lean":"MetaMathlibExt.center_eq_bot_iff_mulAut_eq_rackAut_conj","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#center-eq-bot-iff-mul-aut-eq-rack-aut-co","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"can_prove","summary":"For a group $G$, let $\\operatorname{Conj}(G)$ be the conjugation quandle on $G$, with $x \\triangleleft y = x y x^{-1}$. Then $G$ has trivial center if and only if the permutations of $G$ that are group automorphisms are exactly the rack automorphisms of $\\operatorname{Conj}(G)$.","title":"Trivial center and automorphisms of the conjugation quandle"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/dihedral-virtual-rack-iso-iff","kind":"node","lean":"MetaMathlibExt.dihedral_virtualRackIso_iff","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#dihedral-virtual-rack-iso-iff","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"can_prove","summary":"Let $n > 0$, let $R_n$ be the dihedral quandle on $\\mathbb{Z}/n\\mathbb{Z}$ (with $x \\triangleleft y = 2x - y$), and for $a \\in \\mathbb{Z}/n\\mathbb{Z}$ and $u \\in (\\mathbb{Z}/n\\mathbb{Z})^\\times$ let $T_{a,u}$ be the permutation $x \\mapsto a + u x$ of $R_n$. For all $a, b \\in \\mathbb{Z}/n\\mathbb{Z}$ and $u, v \\in (\\mathbb{Z}/n\\mathbb{Z})^\\times$, the virtual racks $(R_n, T_{a,u})$ and $(R_n, T_{b,v})$ are isomorphic\u2026","title":"Isomorphism of affine virtual dihedral quandles"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/cyclic-holomorph-is-conj-iff","kind":"node","lean":"MetaMathlibExt.cyclicHolomorph_isConj_iff","parent":1595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conjugacy in the holomorph of "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"]],"source":"Conjugacy in the holomorph of $\\mathbb{Z}/n\\mathbb{Z}$"},"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#cyclic-holomorph-is-conj-iff","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"can_prove","summary":"Let $n > 0$, let $A = \\mathbb{Z}/n\\mathbb{Z}$, and write $(a, u)$ for the element of the holomorph $A \\rtimes A^\\times$ with $a \\in A$ and $u \\in A^\\times$. Two elements $(a, u)$ and $(b, v)$ are conjugate if and only if $u = v$ and there is a unit $y \\in A^\\times$ with $b \\equiv y a$ modulo the ideal $(1 - u)A$ generated by $1 - u$.","title":"Conjugacy in the holomorph of $\\mathbb{Z}/n\\mathbb{Z}$"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/card-conj-classes-cyclic-holomorph","kind":"node","lean":"MetaMathlibExt.card_conjClasses_cyclicHolomorph","parent":1595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Class number of the holomorph of "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"]],"source":"Class number of the holomorph of $\\mathbb{Z}/n\\mathbb{Z}$"},"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#card-conj-classes-cyclic-holomorph","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"can_prove","summary":"For every integer $n > 0$, the number of conjugacy classes of the holomorph $\\mathbb{Z}/n\\mathbb{Z} \\rtimes (\\mathbb{Z}/n\\mathbb{Z})^\\times$ of $\\mathbb{Z}/n\\mathbb{Z}$ equals $\\varphi(n) \\sum_{d \\mid n} 1/\\varphi(d)$, where $\\varphi$ is Euler's totient function.","title":"Class number of the holomorph of $\\mathbb{Z}/n\\mathbb{Z}$"},{"id":"Algebra/nonassociative-algebras/virtual-racks-and-quandles/virtual-rack-class-count-dihedral","kind":"node","lean":"MetaMathlibExt.virtualRackClassCount_dihedral","parent":1595,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/virtual-racks-and-quandles/index.html#virtual-rack-class-count-dihedral","scope":"Algebra/nonassociative-algebras/virtual-racks-and-quandles","status":"can_prove","summary":"For every $n > 0$, the number $v(R_n)$ of isomorphism classes of virtual racks whose underlying rack is the dihedral quandle $R_n$ (the set $\\mathbb{Z}/n\\mathbb{Z}$ with $x \\triangleleft y = 2x - y$) equals $\\varphi(n) \\sum_{d \\mid n} 1/\\varphi(d)$, where $\\varphi$ is Euler's totient function.","title":"Virtual quandle structures on the dihedral quandle"},{"data_url":"scopes/Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras.json","id":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras","kind":"scope","navigation":"module","parent":1572,"result":false,"route":"../roadmap/Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/index.html","scope":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras","status":"fully_proved","title":"Witt's formula for free Lie algebras"},{"id":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/is-free-lie-monomial","kind":"node","lean":"MetaMathlibExt.IsFreeLieMonomial","parent":1611,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/index.html#is-free-lie-monomial","scope":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras","status":"fully_proved","summary":"Let $K$ be a field, $r$ a natural number, and $L$ the free Lie algebra over $K$ on generators $x_0, \\dots, x_{r-1}$. The relation \"$x$ is a homogeneous Lie monomial of multidegree $d$\", for $d \\in \\mathbb{N}^r$ and $x \\in L$, is defined inductively: each generator $x_i$ is a monomial of multidegree $e_i$, which is $1$ at $i$ and $0$ elsewhere; and if $x$ is a monomial of multidegree $a$ and $y$ is a monomial of\u2026","title":"Free Lie monomials"},{"id":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/free-lie-multihomogeneous-component","kind":"node","lean":"MetaMathlibExt.freeLieMultihomogeneousComponent","parent":1611,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/index.html#free-lie-multihomogeneous-component","scope":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras","status":"fully_proved","summary":"Let $K$ be a field, $r$ a natural number, and $L$ the free Lie algebra over $K$ on generators $x_0, \\dots, x_{r-1}$. For a multidegree $d \\in \\mathbb{N}^r$, the multihomogeneous component of $L$ of multidegree $d$ is the $K$-linear span of the homogeneous Lie monomials of multidegree $d$, that is, of the iterated brackets of the generators in which each $x_i$ occurs exactly $d_i$ times.","title":"Multihomogeneous components of a free Lie algebra"},{"id":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/witt-multigraded-value","kind":"node","lean":"MetaMathlibExt.wittMultigradedValue","parent":1611,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/index.html#witt-multigraded-value","scope":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras","status":"fully_proved","summary":"For natural numbers $n_1, \\dots, n_r$, let $n = n_1 + \\dots + n_r$ and $g = \\gcd(n_1, \\dots, n_r)$. The rational number $M(n_1, \\dots, n_r)$ is $0$ if $n = 0$, and otherwise $M(n_1, \\dots, n_r) = \\frac{1}{n} \\sum_{d \\mid g} \\mu(d) \\binom{n/d}{n_1/d, \\ldots, n_r/d}$, where the sum runs over the positive divisors $d$ of $g$, $\\mu$ is the M\u00f6bius function, and the last factor is the multinomial coefficient of\u2026","title":"Witt's multigraded value"},{"id":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/finrank-free-lie-multihomogeneous-compon","kind":"node","lean":"MetaMathlibExt.finrank_freeLieMultihomogeneousComponent","parent":1611,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/index.html#finrank-free-lie-multihomogeneous-compon","scope":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras","status":"fully_proved","summary":"Let $K$ be a field and let $n_1, \\dots, n_r$ be natural numbers with $n_1 + \\dots + n_r > 0$. Then the multihomogeneous component of multidegree $(n_1, \\dots, n_r)$ of the free Lie algebra over $K$ on $r$ generators has dimension over $K$ equal to Witt's multigraded value $M(n_1, \\dots, n_r)$.","title":"Dimension of a multihomogeneous component of a free Lie algebra"},{"id":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/witt-multigraded-dimension-formula","kind":"node","lean":"MetaMathlibExt.witt_multigraded_dimension_formula","parent":1611,"result":true,"route":"../roadmap/Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras/index.html#witt-multigraded-dimension-formula","scope":"Algebra/nonassociative-algebras/witt-s-formula-for-free-lie-algebras","status":"fully_proved","summary":"Let $K$ be a field, let $r > 0$, and let $n_1, \\dots, n_r$ be natural numbers with $n_1 + \\dots + n_r > 0$. Then the multihomogeneous component of multidegree $(n_1, \\dots, n_r)$ of the free Lie algebra over $K$ on $r$ generators has dimension over $K$ equal to Witt's multigraded value $M(n_1, \\dots, n_r)$.","title":"Witt's multigraded dimension formula"},{"data_url":"scopes/Algebra/associative-algebras.json","id":"Algebra/associative-algebras","kind":"scope","parent":1278,"result":false,"route":"../roadmap/Algebra/associative-algebras/index.html","scope":"Algebra/associative-algebras","status":"can_prove","summary":"Noncommutative rings and finite-dimensional associative algebras, including division and central simple algebras, quaternion algebras, polynomial identities, nil ideals and Hopf algebras. Group rings are under Group Theory.","title":"Associative rings and algebras"},{"id":"Algebra/associative-algebras/amitsur-levitzki","kind":"node","lean":"MathlibExt.Algebra.Algebra.AmitsurLevitzkiWanted.amitsur_levitzki","parent":1617,"result":true,"route":"../roadmap/Algebra/associative-algebras/index.html#amitsur-levitzki","scope":"Algebra/associative-algebras","status":"fully_proved","summary":"Let $R$ be a commutative ring and $n$ a natural number. For all $n \\times n$ matrices $M_1, \\dots, M_{2n}$ over $R$, the sum over all permutations $\\sigma$ of $\\{1, \\dots, 2n\\}$ of $\\mathrm{sgn}(\\sigma) M_{\\sigma(1)} M_{\\sigma(2)} \\cdots M_{\\sigma(2n)}$ is the zero matrix, where $\\mathrm{sgn}(\\sigma) \\in \\{1, -1\\}$ is the sign of $\\sigma$. In other words, the standard polynomial of degree $2n$ vanishes on\u2026","title":"Amitsur\u2013Levitzki theorem"},{"id":"Algebra/associative-algebras/burnside-matrix","kind":"node","lean":"MathlibExt.Algebra.Algebra.BurnsideMatrixWanted.burnside_matrix","parent":1617,"result":true,"route":"../roadmap/Algebra/associative-algebras/index.html#burnside-matrix","scope":"Algebra/associative-algebras","status":"fully_proved","summary":"Let $k$ be an algebraically closed field, $n$ a natural number, and $A$ a $k$-subalgebra of the algebra $M_n(k)$ of $n \\times n$ matrices over $k$. Suppose that the only subspaces $W$ of $k^n$ with $Mv \\in W$ for all $M \\in A$ and all $v \\in W$ are $0$ and $k^n$. Then $A = M_n(k)$.","title":"Burnside's theorem on matrix algebras"},{"id":"Algebra/associative-algebras/frobenius-real-division-algebra","kind":"node","lean":"MathlibExt.Algebra.Algebra.FrobeniusDivisionWanted.frobenius_real_division_algebra","parent":1617,"result":true,"route":"../roadmap/Algebra/associative-algebras/index.html#frobenius-real-division-algebra","scope":"Algebra/associative-algebras","status":"fully_proved","summary":"Let $A$ be a finite-dimensional associative unital division algebra over $\\mathbb{R}$. Then $A$ is isomorphic as an $\\mathbb{R}$-algebra to $\\mathbb{R}$, to $\\mathbb{C}$, or to the algebra $\\mathbb{H}$ of real quaternions.","title":"Frobenius theorem on real division algebras"},{"id":"Algebra/associative-algebras/skolem-noether","kind":"node","lean":"MathlibExt.Algebra.Central.SkolemNoether.skolem_noether","parent":1617,"result":true,"route":"../roadmap/Algebra/associative-algebras/index.html#skolem-noether","scope":"Algebra/associative-algebras","status":"fully_proved","summary":"Let $k$ be a field and let $A$ be a finite-dimensional central simple algebra over $k$. Then every $k$-algebra automorphism $\\sigma$ of $A$ is inner: there is a unit $u \\in A^\\times$ such that $\\sigma(x) = u x u^{-1}$ for all $x \\in A$.","title":"Skolem\u2013Noether theorem"},{"data_url":"scopes/Algebra/associative-algebras/antipode-of-a-hopf-algebra.json","id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra","kind":"scope","parent":1617,"result":false,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/index.html","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra","status":"fully_proved","title":"Antipode of a Hopf algebra"},{"data_url":"scopes/Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution.json","id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution","kind":"scope","navigation":"module","parent":1622,"result":false,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/index.html","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution","status":"fully_proved","title":"Antipode properties and a bialgebra convolution-power criterion"},{"id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/antipode-antipode","kind":"node","lean":"HopfAlgebra.antipode_antipode","parent":1623,"result":true,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/index.html#antipode-antipode","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution","status":"fully_proved","summary":"The antipode of a commutative Hopf algebra is an involution.","title":"Antipode of a commutative Hopf algebra is an involution"},{"id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/is-unit-with-conv-alg-hom","kind":"node","lean":"HopfAlgebra.isUnit_withConv_algHom","parent":1623,"result":true,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/index.html#is-unit-with-conv-alg-hom","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution","status":"fully_proved","summary":"Algebra homomorphisms from a possibly noncommutative Hopf algebra to a commutative algebra are convolution units.","title":"Algebra homomorphisms from a Hopf algebra are convolution units"},{"id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/exists-comp-antipode-conv-mul-eq-one","kind":"node","lean":"HopfAlgebra.exists_comp_antipode_convMul_eq_one","parent":1623,"result":true,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/index.html#exists-comp-antipode-conv-mul-eq-one","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution","status":"fully_proved","summary":"Every algebra homomorphism from a possibly noncommutative Hopf algebra to a commutative algebra has a convolution inverse given by precomposition with the antipode.","title":"Convolution inverse of an algebra homomorphism via the antipode"},{"id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/forall-with-conv-pow-eq-one-iff-to-conv","kind":"node","lean":"Bialgebra.forall_withConv_pow_eq_one_iff_toConv_id_pow_eq_one","parent":1623,"result":true,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/index.html#forall-with-conv-pow-eq-one-iff-to-conv","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution","status":"fully_proved","summary":"A convolution power is trivial on all points iff it is trivial on the universal point.","title":"Convolution-power criterion for bialgebras"},{"id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/forall-with-conv-pow-eq-one-iff-to-conv-2","kind":"node","lean":"HopfAlgebra.forall_withConv_pow_eq_one_iff_toConv_id_pow_eq_one","parent":1623,"result":true,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution/index.html#forall-with-conv-pow-eq-one-iff-to-conv-2","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/antipode-properties-and-a-bialgebra-convolution","status":"fully_proved","summary":"A compatibility corollary for the convolution-power criterion on Hopf algebras.","title":"Convolution-power criterion for Hopf algebras"},{"data_url":"scopes/Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode.json","id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode","kind":"scope","navigation":"module","parent":1622,"result":false,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode/index.html","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode","status":"fully_proved","title":"Comultiplication of the antipode"},{"id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode/comul-antipode","kind":"node","lean":"HopfAlgebra.comul_antipode","parent":1629,"result":true,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode/index.html#comul-antipode","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode","status":"fully_proved","summary":"Comultiplication anticommutes with the antipode up to the tensor swap.","title":"Comultiplication of the antipode"},{"id":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode/map-antipode-comul-of-is-cocomm","kind":"node","lean":"HopfAlgebra.map_antipode_comul_of_isCocomm","parent":1629,"result":true,"route":"../roadmap/Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode/index.html#map-antipode-comul-of-is-cocomm","scope":"Algebra/associative-algebras/antipode-of-a-hopf-algebra/comultiplication-of-the-antipode","status":"fully_proved","summary":"In a cocommutative Hopf algebra, comultiplication commutes with the antipode.","title":"Comultiplication commutes with the antipode when cocommutative"},{"data_url":"scopes/Algebra/associative-algebras/jacobson-s-commutativity-theorem.json","id":"Algebra/associative-algebras/jacobson-s-commutativity-theorem","kind":"scope","navigation":"module","parent":1617,"result":false,"route":"../roadmap/Algebra/associative-algebras/jacobson-s-commutativity-theorem/index.html","scope":"Algebra/associative-algebras/jacobson-s-commutativity-theorem","status":"fully_proved","title":"Jacobson's commutativity theorem"},{"id":"Algebra/associative-algebras/jacobson-s-commutativity-theorem/is-prime-ideal","kind":"node","lean":"MathlibExt.Algebra.Ring.JacobsonCommutativity.IsPrimeIdeal","parent":1632,"result":true,"route":"../roadmap/Algebra/associative-algebras/jacobson-s-commutativity-theorem/index.html#is-prime-ideal","scope":"Algebra/associative-algebras/jacobson-s-commutativity-theorem","status":"fully_proved","summary":"A two-sided ideal $P$ of a ring $R$ is prime if $P \\ne R$ and, for all $a, b \\in R$, if $a r b \\in P$ for every $r \\in R$, then $a \\in P$ or $b \\in P$.","title":"Prime two-sided ideal"},{"id":"Algebra/associative-algebras/jacobson-s-commutativity-theorem/is-msystem","kind":"node","lean":"MathlibExt.Algebra.Ring.JacobsonCommutativity.IsMSystem","parent":1632,"result":true,"route":"../roadmap/Algebra/associative-algebras/jacobson-s-commutativity-theorem/index.html#is-msystem","scope":"Algebra/associative-algebras/jacobson-s-commutativity-theorem","status":"fully_proved","summary":"A subset $M$ of a ring $R$ is an $m$-system if it is nonempty, does not contain $0$, and for all $a, b \\in M$ there is some $r \\in R$ with $a r b \\in M$.","title":"m-system"},{"id":"Algebra/associative-algebras/jacobson-s-commutativity-theorem/jacobson-commutativity","kind":"node","lean":"MathlibExt.Algebra.Ring.JacobsonCommutativity.jacobson_commutativity","parent":1632,"result":true,"route":"../roadmap/Algebra/associative-algebras/jacobson-s-commutativity-theorem/index.html#jacobson-commutativity","scope":"Algebra/associative-algebras/jacobson-s-commutativity-theorem","status":"fully_proved","summary":"Let $R$ be a ring. If for every $x \\in R$ there is a natural number $n > 1$ with $x^n = x$, then $R$ is commutative, that is, $xy = yx$ for all $x, y \\in R$.","title":"Jacobson's commutativity theorem"},{"data_url":"scopes/Algebra/associative-algebras/k-the-s-conjecture.json","id":"Algebra/associative-algebras/k-the-s-conjecture","kind":"scope","navigation":"module","parent":1617,"result":false,"route":"../roadmap/Algebra/associative-algebras/k-the-s-conjecture/index.html","scope":"Algebra/associative-algebras/k-the-s-conjecture","status":"can_prove","title":"K\u00f6the's conjecture"},{"id":"Algebra/associative-algebras/k-the-s-conjecture/is-nil-left-ideal","kind":"node","lean":"MathlibExt.Algebra.KoetheNilLeftIdealsWanted.IsNilLeftIdeal","parent":1636,"result":true,"route":"../roadmap/Algebra/associative-algebras/k-the-s-conjecture/index.html#is-nil-left-ideal","scope":"Algebra/associative-algebras/k-the-s-conjecture","status":"fully_proved","summary":"A left ideal $I$ of a ring $R$ is nil if each of its elements is nilpotent: for every $x \\in I$ there is a natural number $n$ with $x^n = 0$.","title":"Nil left ideal"},{"id":"Algebra/associative-algebras/k-the-s-conjecture/conjecture","kind":"node","lean":"MathlibExt.Algebra.KoetheNilLeftIdealsWanted.conjecture","parent":1636,"result":true,"route":"../roadmap/Algebra/associative-algebras/k-the-s-conjecture/index.html#conjecture","scope":"Algebra/associative-algebras/k-the-s-conjecture","status":"fully_proved","summary":"The statement that for every ring $R$ (associative, with identity) and all nil left ideals $I$ and $J$ of $R$, the sum $I + J$ is again a nil left ideal.","title":"K\u00f6the's conjecture"},{"id":"Algebra/associative-algebras/k-the-s-conjecture/conjecture-refuted","kind":"node","lean":"MathlibExt.Algebra.KoetheNilLeftIdealsWanted.conjecture_refuted","parent":1636,"result":true,"route":"../roadmap/Algebra/associative-algebras/k-the-s-conjecture/index.html#conjecture-refuted","scope":"Algebra/associative-algebras/k-the-s-conjecture","status":"can_prove","summary":"There exist a ring $R$ (associative, with identity) and nil left ideals $I$ and $J$ of $R$ (left ideals all of whose elements are nilpotent) such that the sum $I + J$ is not nil.","title":"K\u00f6the's conjecture is false"},{"data_url":"scopes/Algebra/associative-algebras/splitting-of-quaternion-algebras.json","id":"Algebra/associative-algebras/splitting-of-quaternion-algebras","kind":"scope","navigation":"module","parent":1617,"result":false,"route":"../roadmap/Algebra/associative-algebras/splitting-of-quaternion-algebras/index.html","scope":"Algebra/associative-algebras/splitting-of-quaternion-algebras","status":"fully_proved","title":"Splitting of quaternion algebras"},{"id":"Algebra/associative-algebras/splitting-of-quaternion-algebras/forall-quaternary-norm-form-eq-zero-iff","kind":"node","lean":"QuadraticForm.forall_quaternary_normForm_eq_zero_iff_forall_ternary_eq_zero","parent":1640,"result":true,"route":"../roadmap/Algebra/associative-algebras/splitting-of-quaternion-algebras/index.html#forall-quaternary-norm-form-eq-zero-iff","scope":"Algebra/associative-algebras/splitting-of-quaternion-algebras","status":"fully_proved","summary":"The quaternary norm form `x\u2080\u00b2 - a\u00b7x\u2081\u00b2 - b\u00b7x\u2082\u00b2 + a\u00b7b\u00b7x\u2083\u00b2` of the quaternion algebra `\u210d[K, a, b]` is anisotropic if and only if the ternary form `z\u00b2 - a\u00b7x\u00b2 - b\u00b7y\u00b2` is. This equivalence also holds when either parameter is zero; both forms are then isotropic.","title":"Anisotropy of the quaternion norm form reduces to a ternary form"},{"id":"Algebra/associative-algebras/splitting-of-quaternion-algebras/nonempty-alg-equiv-matrix-of-norm-form-e","kind":"node","lean":"QuaternionAlgebra.nonempty_algEquiv_matrix_of_normForm_eq_zero","parent":1640,"result":true,"route":"../roadmap/Algebra/associative-algebras/splitting-of-quaternion-algebras/index.html#nonempty-alg-equiv-matrix-of-norm-form-e","scope":"Algebra/associative-algebras/splitting-of-quaternion-algebras","status":"fully_proved","summary":"If the norm form of a quaternion algebra over a field of characteristic not two is isotropic, then the algebra splits: it is isomorphic to the `2 \u00d7 2` matrix algebra.","title":"Quaternion algebras with isotropic norm form split"},{"data_url":"scopes/Algebra/algebraic-identities.json","id":"Algebra/algebraic-identities","kind":"scope","parent":1278,"result":false,"route":"../roadmap/Algebra/algebraic-identities/index.html","scope":"Algebra/algebraic-identities","status":"can_prove","summary":"Polynomial and product identities, sum-of-squares identities, functional equations on rings, and elementary operations on functions and sets.","title":"Algebraic identities and operations"},{"id":"Algebra/algebraic-identities/candido-identity","kind":"node","lean":"MetaMathlibExt.candido_identity","parent":1643,"result":true,"route":"../roadmap/Algebra/algebraic-identities/index.html#candido-identity","scope":"Algebra/algebraic-identities","status":"fully_proved","summary":"For all elements $x$ and $y$ of a commutative semiring, $2\\left(x^4 + y^4 + (x + y)^4\\right) = \\left(x^2 + y^2 + (x + y)^2\\right)^2$.","title":"Candido's identity"},{"id":"Algebra/algebraic-identities/indicator-one","kind":"node","lean":"Set.indicatorOne","parent":1643,"result":true,"route":"../roadmap/Algebra/algebraic-identities/index.html#indicator-one","scope":"Algebra/algebraic-identities","status":"fully_proved","summary":"For a set $A \\subseteq \\alpha$ and a type $R$ with a zero and a one, the indicator function $\\mathbb{1}_A : \\alpha \\to R$ takes the value $1$ at every point of $A$ and the value $0$ at every point outside $A$.","title":"Indicator function of a set"},{"id":"Algebra/algebraic-identities/is-ladditive","kind":"node","lean":"MetaMathlibExt.IsLAdditive","parent":1643,"result":true,"route":"../roadmap/Algebra/algebraic-identities/index.html#is-ladditive","scope":"Algebra/algebraic-identities","status":"fully_proved","summary":"A function $f \\colon K \\to K$ on a ring $K$ is L-additive when $f(mn) = f(m)\\,n + f(n)\\,m$ for all $m, n \\in K$.","title":"L-additive functions on a ring"},{"id":"Algebra/algebraic-identities/prod-comp-eq-prod-pow","kind":"node","lean":"Fintype.prod_comp_eq_prod_pow","parent":1643,"result":true,"route":"../roadmap/Algebra/algebraic-identities/index.html#prod-comp-eq-prod-pow","scope":"Algebra/algebraic-identities","status":"fully_proved","summary":"Let $\\beta$ and $\\gamma$ be finite types, let $M$ be a commutative monoid, and let $g : \\beta \\to \\gamma$ and $f : \\gamma \\to M$. Then","title":"Products regrouped by the values of a function"},{"data_url":"scopes/Algebra/algebraic-identities/associative-operations-on-self-maps.json","id":"Algebra/algebraic-identities/associative-operations-on-self-maps","kind":"scope","navigation":"module","parent":1643,"result":false,"route":"../roadmap/Algebra/algebraic-identities/associative-operations-on-self-maps/index.html","scope":"Algebra/algebraic-identities/associative-operations-on-self-maps","status":"can_prove","title":"Associative operations on self-maps"},{"id":"Algebra/algebraic-identities/associative-operations-on-self-maps/is-associative","kind":"node","lean":"MathlibExt.Algebra.ScottishBook171Wanted.IsAssociative","parent":1648,"result":true,"route":"../roadmap/Algebra/algebraic-identities/associative-operations-on-self-maps/index.html#is-associative","scope":"Algebra/algebraic-identities/associative-operations-on-self-maps","status":"fully_proved","summary":"Let $A$ be a set and let $U$ be an operation that assigns to any two maps $f, g \\colon A \\to A$ a map $U(f, g) \\colon A \\to A$. We say $U$ is associative (condition (1) of Problem 171) if $U(f, U(g, h)) = U(U(f, g), h)$ for all maps $f, g, h \\colon A \\to A$.","title":"Associative operations on self-maps"},{"id":"Algebra/algebraic-identities/associative-operations-on-self-maps/is-permutation-invariant","kind":"node","lean":"MathlibExt.Algebra.ScottishBook171Wanted.IsPermutationInvariant","parent":1648,"result":true,"route":"../roadmap/Algebra/algebraic-identities/associative-operations-on-self-maps/index.html#is-permutation-invariant","scope":"Algebra/algebraic-identities/associative-operations-on-self-maps","status":"fully_proved","summary":"Let $A$ be a set and let $U$ be an operation that assigns to any two maps $f, g \\colon A \\to A$ a map $U(f, g) \\colon A \\to A$. We say $U$ is invariant under permutations of $A$ (condition (2) of Problem 171) if conjugating the arguments by a permutation conjugates the result: for every permutation $p$ of $A$ and all maps $f, g \\colon A \\to A$,\u2026","title":"Permutation-invariant operations on self-maps"},{"id":"Algebra/algebraic-identities/associative-operations-on-self-maps/scottish-book171-refuted","kind":"node","lean":"MathlibExt.Algebra.ScottishBook171Wanted.scottishBook171_refuted","parent":1648,"result":true,"route":"../roadmap/Algebra/algebraic-identities/associative-operations-on-self-maps/index.html#scottish-book171-refuted","scope":"Algebra/algebraic-identities/associative-operations-on-self-maps","status":"can_prove","summary":"Not every associative, permutation-invariant operation on self-maps is composition: there are a set $A$, an operation $U$ that assigns to any two maps $f, g \\colon A \\to A$ a map $U(f, g) \\colon A \\to A$ and is both associative (condition (1) of Problem 171) and invariant under permutations of $A$ (condition (2)), and maps $f, g \\colon A \\to A$ with $U(f, g) \\ne f \\circ g$, where $(f \\circ g)(x) = f(g(x))$.","title":"Scottish Book Problem 171 is false"},{"data_url":"scopes/Algebra/algebraic-identities/degen-s-eight-square-identity.json","id":"Algebra/algebraic-identities/degen-s-eight-square-identity","kind":"scope","navigation":"module","parent":1643,"result":false,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","title":"Degen's eight-square identity"},{"id":"Algebra/algebraic-identities/degen-s-eight-square-identity/basis-sign","kind":"node","lean":"Octonion.basisSign","parent":1652,"result":true,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html#basis-sign","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","summary":"The sign of the product of basis elements indexed by `p` and `q`.","title":"Sign of a product of octonion basis elements"},{"id":"Algebra/algebraic-identities/degen-s-eight-square-identity/structure-constant","kind":"node","lean":"Octonion.structureConstant","parent":1652,"result":true,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html#structure-constant","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","summary":"The signed structure constant for the basis product indexed by `p` and `q`.","title":"Signed structure constant of the eight-square algebra"},{"id":"Algebra/algebraic-identities/degen-s-eight-square-identity/basis-sign-mul-self","kind":"node","lean":"Octonion.basisSign_mul_self","parent":1652,"result":true,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html#basis-sign-mul-self","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","summary":"Every basis multiplication sign squares to one.","title":"Basis multiplication signs square to one"},{"id":"Algebra/algebraic-identities/degen-s-eight-square-identity/basis-sign-mul-add-swap","kind":"node","lean":"Octonion.basisSign_mul_add_swap","parent":1652,"result":true,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html#basis-sign-mul-add-swap","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","summary":"The cancellation rule of signs for two distinct decompositions of one xor index.","title":"Sign cancellation for two decompositions of an xor index"},{"id":"Algebra/algebraic-identities/degen-s-eight-square-identity/sum-structure-constant-mul","kind":"node","lean":"Octonion.sum_structureConstant_mul","parent":1652,"result":true,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html#sum-structure-constant-mul","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","summary":"Contracting two structure constants leaves their common xor component.","title":"Contraction of two structure constants"},{"id":"Algebra/algebraic-identities/degen-s-eight-square-identity/crossing-identity","kind":"node","lean":"Octonion.crossing_identity","parent":1652,"result":true,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html#crossing-identity","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","summary":"The two contractions in the eight-square crossing decomposition.","title":"Eight-square crossing identity"},{"id":"Algebra/algebraic-identities/degen-s-eight-square-identity/abs-structure-constant-le-one","kind":"node","lean":"Octonion.abs_structureConstant_le_one","parent":1652,"result":true,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html#abs-structure-constant-le-one","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","summary":"A structure constant has absolute value at most one.","title":"Structure constants are bounded by one"},{"id":"Algebra/algebraic-identities/degen-s-eight-square-identity/abs-sum-structure-constant-mul-le-one","kind":"node","lean":"Octonion.abs_sum_structureConstant_mul_le_one","parent":1652,"result":true,"route":"../roadmap/Algebra/algebraic-identities/degen-s-eight-square-identity/index.html#abs-sum-structure-constant-mul-le-one","scope":"Algebra/algebraic-identities/degen-s-eight-square-identity","status":"fully_proved","summary":"A contraction of two structure constants has absolute value at most one.","title":"Contractions of structure constants are bounded by one"},{"area":"Algebra & Number Theory","data_url":"chapters/GroupTheory.json","id":"GroupTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/GroupTheory/index.html","scope":"GroupTheory","status":"can_prove","summary":"Group theory studies symmetry through groups, actions, subgroups, quotients, and structural decompositions.","title":"Group Theory"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups.json","id":"GroupTheory/characters-of-finite-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/index.html","scope":"GroupTheory/characters-of-finite-groups","status":"can_prove","summary":"Complex characters and representations used to study finite groups: class functions and orthogonality, irreducible character bases, character degrees and values, McKay conjecture, rational groups, permutation characters, blocks of defect zero, Frobenius group determinant, convolution on finite groups, and Burnside's p^a q^b theorem; general representation theory of algebras and Lie groups belongs to\u2026","title":"Characters of finite groups"},{"id":"GroupTheory/characters-of-finite-groups/m23-irrep-tensor-square-decomp","kind":"node","lean":"MathlibExt.GroupTheory.M23TensorSquareWanted.RepresentationTheory.m23_irrep_tensor_square_decomp","parent":1662,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/index.html#m23-irrep-tensor-square-decomp","scope":"GroupTheory/characters-of-finite-groups","status":"can_prove","summary":"There exist a finite simple group $G$ of order $10200960$ and an irreducible complex representation $\\rho$ of $G$ on a vector space $V$ of dimension $22$ such that the tensor square $V \\otimes V$, regarded as a module over the group algebra $\\mathbb{C}[G]$ through the diagonal action $g \\mapsto \\rho(g) \\otimes \\rho(g)$, has exactly $4$ isotypic components.","title":"Four isotypic components in a tensor square for a simple group of order 10200960"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero.json","id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"can_prove","title":"Blocks of defect zero"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/is-psoluble","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlockCentralizerKourovka12Problem26Wanted.IsPSoluble","parent":1664,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-soluble groups"]],"source":"$p$-soluble groups"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#is-psoluble","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"fully_proved","summary":"Let $p$ be a natural number. A group $G$ is $p$-soluble when it has a subnormal series $1 = H_0 \\le H_1 \\le \\dots \\le H_n = G$, each $H_i$ normal in $H_{i+1}$, whose factors $H_{i+1}/H_i$ are $p$-groups or have order prime to $p$. Here an infinite factor is taken to have order $0$.","title":"$p$-soluble groups"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/has-defect-zero-block","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlockCentralizerKourovka12Problem26Wanted.HasDefectZeroBlock","parent":1664,"reading_title":{"format":"inline-parts-v1","parts":[["text","Groups with a "],["math","\\(p\\)"],["text","-block of defect zero"]],"source":"Groups with a $p$-block of defect zero"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#has-defect-zero-block","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"fully_proved","summary":"Let $p$ be a natural number. A group $G$ has a $p$-block of defect zero when some irreducible finite-dimensional complex representation of $G$ has degree divisible by $p^{v_p(|G|)}$, where $v_p(|G|)$ is the exponent of $p$ in the prime factorization of the order $|G|$ of $G$ (this exponent is $0$ when $p$ is not prime or $G$ is infinite). For a prime $p$ and a finite group $G$, the number $p^{v_p(|G|)}$ is $|G|_p$,\u2026","title":"Groups with a $p$-block of defect zero"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/defect-zero-block-iff-pregular-centraliz","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlockCentralizerKourovka12Problem26Wanted.defectZeroBlockIffPRegularCentralizer","parent":1664,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#defect-zero-block-iff-pregular-centraliz","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"fully_proved","summary":"This is the proposition (Shi Shengming) that for every prime $p$ and every finite $p$-soluble group $G$, the group $G$ has a $p$-block of defect zero if and only if some normal subgroup $N$ of $G$ of order prime to $p$ contains an element $x$ whose centralizer $C_G(x)$ in $G$ has order prime to $p$.","title":"Kourovka Problem 12.26"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/defect-zero-block-iff-pregular-centraliz-2","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlockCentralizerKourovka12Problem26Wanted.defectZeroBlockIffPRegularCentralizer_refuted","parent":1664,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#defect-zero-block-iff-pregular-centraliz-2","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"can_prove","summary":"The proposition of Kourovka Problem 12.26 is false: it is not the case that for every prime $p$ and every finite $p$-soluble group $G$, the group $G$ has a $p$-block of defect zero if and only if some normal subgroup $N$ of $G$ of order prime to $p$ contains an element $x$ whose centralizer $C_G(x)$ in $G$ has order prime to $p$.","title":"Refutation of Kourovka Problem 12.26"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/defect-zero-characters","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlocksCommutatorCountKourovka12Problem80Wanted.defectZeroCharacters","parent":1664,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characters of "],["math","\\(p\\)"],["text","-defect zero"]],"source":"Characters of $p$-defect zero"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#defect-zero-characters","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"fully_proved","summary":"For a natural number $p$ and a group $G$, the set of characters $\\chi \\colon G \\to \\mathbb{C}$ of irreducible finite-dimensional complex representations of $G$ whose degree is divisible by $p^a$, where $a$ is the exponent of $p$ in the prime factorization of the order $|G|$ (and $a = 0$ when $G$ is infinite or $p$ is not prime).","title":"Characters of $p$-defect zero"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/commutator-solution-count","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlocksCommutatorCountKourovka12Problem80Wanted.commutatorSolutionCount","parent":1664,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#commutator-solution-count","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"fully_proved","summary":"For an element $g$ of a group $G$, the number of pairs $(x, y) \\in G \\times G$ with $[x, y] = x y x^{-1} y^{-1} = g$, taken to be $0$ when there are infinitely many such pairs.","title":"Number of solutions of a commutator equation"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/nondivisible-commutator-classes","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlocksCommutatorCountKourovka12Problem80Wanted.nondivisibleCommutatorClasses","parent":1664,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conjugacy classes with commutator count not divisible by "],["math","\\(p\\)"]],"source":"Conjugacy classes with commutator count not divisible by $p$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#nondivisible-commutator-classes","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"fully_proved","summary":"For a natural number $p$ and a group $G$, the set of conjugacy classes of $G$ containing an element $g$ such that the number of pairs $(x, y) \\in G \\times G$ with $[x, y] = x y x^{-1} y^{-1} = g$ (taken to be $0$ when there are infinitely many) is not divisible by $p$.","title":"Conjugacy classes with commutator count not divisible by $p$"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/defect-zero-blocks-eq-nondivisible-class","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlocksCommutatorCountKourovka12Problem80Wanted.defectZeroBlocksEqNondivisibleClasses","parent":1664,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#defect-zero-blocks-eq-nondivisible-class","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"fully_proved","summary":"For every prime $p$ and every finite group $G$ whose order is divisible by $p$, the number of irreducible complex characters of $G$ of $p$-defect zero equals the number of conjugacy classes of $G$ containing an element $g$ for which the number of pairs $(x, y) \\in G \\times G$ with $[x, y] = x y x^{-1} y^{-1} = g$ is not divisible by $p$.","title":"Kourovka Problem 12.80(a): blocks of defect zero and commutator equations"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/simple-defect-zero-blocks-eq-nondivisibl","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlocksCommutatorCountKourovka12Problem80Wanted.simpleDefectZeroBlocksEqNondivisibleClasses","parent":1664,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#simple-defect-zero-blocks-eq-nondivisibl","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"fully_proved","summary":"For every prime $p$ and every finite simple group $G$ whose order is divisible by $p$, the number of irreducible complex characters of $G$ of $p$-defect zero equals the number of conjugacy classes of $G$ containing an element $g$ for which the number of pairs $(x, y) \\in G \\times G$ with $[x, y] = x y x^{-1} y^{-1} = g$ is not divisible by $p$.","title":"Kourovka Problem 12.80(b): the case of simple groups"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/defect-zero-blocks-eq-nondivisible-class-2","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlocksCommutatorCountKourovka12Problem80Wanted.defectZeroBlocksEqNondivisibleClasses_refuted","parent":1664,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#defect-zero-blocks-eq-nondivisible-class-2","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"can_prove","summary":"It is not true that for every prime $p$ and every finite group $G$ whose order is divisible by $p$, the number of irreducible complex characters of $G$ of $p$-defect zero equals the number of conjugacy classes of $G$ containing an element $g$ for which the number of pairs $(x, y) \\in G \\times G$ with $[x, y] = x y x^{-1} y^{-1} = g$ is not divisible by $p$.","title":"Kourovka Problem 12.80(a) has a negative answer"},{"id":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/simple-defect-zero-blocks-eq-nondivisibl-2","kind":"node","lean":"MathlibExt.GroupTheory.DefectZeroBlocksCommutatorCountKourovka12Problem80Wanted.simpleDefectZeroBlocksEqNondivisibleClasses_refuted","parent":1664,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/blocks-of-defect-zero/index.html#simple-defect-zero-blocks-eq-nondivisibl-2","scope":"GroupTheory/characters-of-finite-groups/blocks-of-defect-zero","status":"can_prove","summary":"It is not true that for every prime $p$ and every finite simple group $G$ whose order is divisible by $p$, the number of irreducible complex characters of $G$ of $p$-defect zero equals the number of conjugacy classes of $G$ containing an element $g$ for which the number of pairs $(x, y) \\in G \\times G$ with $[x, y] = x y x^{-1} y^{-1} = g$ is not divisible by $p$.","title":"Kourovka Problem 12.80(b) has a negative answer"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem.json","id":"GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem/index.html","scope":"GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem","status":"can_prove","title":"Brauer's splitting field theorem"},{"id":"GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem/brauer-character-in-cyclotomic","kind":"node","lean":"MathlibExt.GroupTheory.BrauerCharacterInCyclotomicWanted.RepresentationTheory.brauer_character_in_cyclotomic","parent":1676,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem/index.html#brauer-character-in-cyclotomic","scope":"GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem","status":"can_prove","summary":"Let $G$ be a finite group of exponent $n$, and let $\\mathbb{Q}(\\zeta_n)$ be the $n$-th cyclotomic field. There is a ring homomorphism $\\varphi \\colon \\mathbb{Q}(\\zeta_n) \\to \\mathbb{C}$ such that for every finite-dimensional complex representation $\\rho$ of $G$ on a vector space $V$ and every $g \\in G$, the trace of $\\rho(g)$ lies in the image of $\\varphi$.","title":"Character values of a finite group lie in a cyclotomic field"},{"id":"GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem/brauer-splitting-field","kind":"node","lean":"MathlibExt.GroupTheory.BrauerSplittingFieldWanted.RepresentationTheory.brauer_splitting_field","parent":1676,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem/index.html#brauer-splitting-field","scope":"GroupTheory/characters-of-finite-groups/brauer-s-splitting-field-theorem","status":"can_prove","summary":"Let $G$ be a finite group of exponent $n$, let $K = \\mathbb{Q}(\\zeta_n)$ be the $n$-th cyclotomic field, and let $\\rho$ be a representation of $G$ on a finite-dimensional complex vector space $V$. Then there are a ring homomorphism $\\varphi \\colon K \\to \\mathbb{C}$, a $K$-vector space $W$ with a representation $\\sigma$ of $G$ on $W$, and a $\\mathbb{C}$-linear isomorphism $f \\colon \\mathbb{C} \\otimes_K W \\to V$,\u2026","title":"Brauer's splitting field theorem"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem.json","id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","title":"Burnside's p^a q^b theorem"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/simple-fdrep-of-irreducible","kind":"node","lean":"BurnsidePaqb.simple_fdrep_of_irreducible","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#simple-fdrep-of-irreducible","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group and $\\sigma$ an irreducible representation of $G$ on a finite-dimensional complex vector space $V$. Then $V$ with the action $\\sigma$ is a simple object of the category of finite-dimensional complex representations of $G$.","title":"Irreducible representations are simple"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/irreducible-of-simple-fdrep-aux","kind":"node","lean":"BurnsidePaqb.irreducible_of_simple_fdrep_aux","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#irreducible-of-simple-fdrep-aux","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group and $V$ a simple object of the category of finite-dimensional complex representations of $G$. If $\\sigma$ is a representation of $G$ on the underlying vector space of $V$ that equals the action of $G$ on $V$, then $\\sigma$ is irreducible.","title":"Simple finite-dimensional representations are irreducible (auxiliary form)"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/irreducible-of-simple-fdrep","kind":"node","lean":"BurnsidePaqb.irreducible_of_simple_fdrep","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#irreducible-of-simple-fdrep","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. If $V$ is a simple object of the category of finite-dimensional complex representations of $G$, then the representation of $G$ on the underlying vector space of $V$ is irreducible.","title":"Simple finite-dimensional complex representations of a finite group are irreducible"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/iso-setoid","kind":"node","lean":"BurnsidePaqb.Simple.isoSetoid","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#iso-setoid","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For a finite group $G$, two simple finite-dimensional complex representations of $G$ are related when they are isomorphic; this is an equivalence relation on the simple finite-dimensional complex representations of $G$.","title":"Isomorphism of simple representations"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/irr","kind":"node","lean":"BurnsidePaqb.Irr","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#irr","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For a finite group $G$, $\\mathrm{Irr}(G)$ is the set of isomorphism classes of simple finite-dimensional complex representations of $G$.","title":"Isomorphism classes of simple representations"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-fun","kind":"node","lean":"BurnsidePaqb.Irr.charFun","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-fun","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a group. For an isomorphism class $c$ of simple finite-dimensional complex representations of $G$, the character of $c$ is the function $\\chi_c \\colon G \\to \\mathbb{C}$ given by $\\chi_c(g) = \\operatorname{tr} \\rho(g)$ for any representative $(V, \\rho)$ of $c$. It does not depend on the representative, since isomorphic representations have the same character.","title":"Character of an isomorphism class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/rep-bundle","kind":"node","lean":"BurnsidePaqb.Irr.repBundle","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#rep-bundle","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For a finite group $G$ and an isomorphism class $c$ of simple finite-dimensional complex representations of $G$, this is a simple finite-dimensional complex representation of $G$ in the class $c$, fixed by the axiom of choice.","title":"Chosen representative of an isomorphism class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-fun-inj","kind":"node","lean":"BurnsidePaqb.Irr.charFun_inj","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-fun-inj","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. Two isomorphism classes of simple finite-dimensional complex representations of $G$ that have the same character are equal.","title":"Characters determine irreducible representations"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/triv","kind":"node","lean":"BurnsidePaqb.Irr.triv","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#triv","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. The trivial representation of $G$ on $\\mathbb{C}$, where every element of $G$ acts as the identity, is simple, and this is its class among the isomorphism classes of simple finite-dimensional complex representations of $G$.","title":"The trivial irreducible class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-fun-triv","kind":"node","lean":"BurnsidePaqb.Irr.charFun_triv","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-fun-triv","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. The character of the isomorphism class of the trivial representation of $G$ on $\\mathbb{C}$ takes the value $1$ at every $g \\in G$.","title":"Character of the trivial representation"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-fun-one","kind":"node","lean":"BurnsidePaqb.Irr.charFun_one","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-fun-one","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a group and let $c$ be an isomorphism class of simple finite-dimensional complex representations of $G$. The character of $c$ at the identity $1 \\in G$ equals the dimension over $\\mathbb{C}$ of the chosen representative of $c$.","title":"Character value at the identity"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-fun-eq-rep-bundle-char","kind":"node","lean":"BurnsidePaqb.Irr.charFun_eq_repBundle_char","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-fun-eq-rep-bundle-char","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a group and let $c$ be an isomorphism class of simple finite-dimensional complex representations of $G$, with chosen representative $(V, \\rho)$. For every $x \\in G$, the character of $c$ at $x$ equals the character of the representation $\\rho$ at $x$, namely $\\operatorname{tr} \\rho(x)$.","title":"Character of a class via its chosen representative"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-inner-orth","kind":"node","lean":"BurnsidePaqb.charInner_orth","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-inner-orth","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group and let $\\chi_i$ and $\\chi_j$ be the characters of isomorphism classes $i$ and $j$ of simple finite-dimensional complex representations of $G$. Then","title":"Orthogonality of irreducible characters"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/is-class-fun","kind":"node","lean":"BurnsidePaqb.IsClassFun","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#is-class-fun","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"A function $f \\colon G \\to \\mathbb{C}$ on a finite group $G$ is a class function when $f(hgh^{-1}) = f(g)$ for all $g, h \\in G$, that is, when $f$ is constant on conjugacy classes.","title":"Class function"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/card-bound","kind":"node","lean":"BurnsidePaqb.cardBound","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#card-bound","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. Every finite set of isomorphism classes of simple finite-dimensional complex representations of $G$ has at most $|G|$ elements.","title":"Bound on the number of irreducible representations"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/zimm-op","kind":"node","lean":"BurnsidePaqb.zimmOp","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#zimm-op","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $M$ be a module over the group algebra $\\mathbb{C}[G]$ with a compatible $\\mathbb{C}$-module structure, and write $\\rho(g)$ for the action of $g \\in G$ on $M$. For a function $f : G \\to \\mathbb{C}$, the averaging operator of $f$ is the $\\mathbb{C}$-linear endomorphism $T_f = \\sum_{g \\in G} f(g)\\,\\rho(g^{-1})$ of $M$.","title":"Zimmerman's averaging operator"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/zimm-op-comm","kind":"node","lean":"BurnsidePaqb.zimmOp_comm","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The averaging operator of a class function is "],["math","\\(G\\)"],["text","-equivariant"]],"source":"The averaging operator of a class function is $G$-equivariant"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#zimm-op-comm","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $M$ be a module over the group algebra $\\mathbb{C}[G]$ with a compatible $\\mathbb{C}$-module structure, and write $\\rho(h)$ for the action of $h \\in G$ on $M$. If $f : G \\to \\mathbb{C}$ is a class function, then its averaging operator $T_f = \\sum_{g \\in G} f(g)\\,\\rho(g^{-1})$ satisfies $\\rho(h)(T_f x) = T_f(\\rho(h) x)$ for all $h \\in G$ and $x \\in M$.","title":"The averaging operator of a class function is $G$-equivariant"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/classfun-eq-zero-of-orth","kind":"node","lean":"BurnsidePaqb.classfun_eq_zero_of_orth","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#classfun-eq-zero-of-orth","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group and let $f \\colon G \\to \\mathbb{C}$ be a class function, that is, $f(hgh^{-1}) = f(g)$ for all $g, h \\in G$. Suppose that for every isomorphism class $c$ of simple finite-dimensional complex representations of $G$, with character $\\chi_c$,","title":"Class functions orthogonal to all irreducible characters vanish"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/classfun-expand","kind":"node","lean":"BurnsidePaqb.classfun_expand","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#classfun-expand","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $f \\colon G \\to \\mathbb{C}$ be a class function, and for each of the finitely many isomorphism classes $c$ of simple finite-dimensional complex representations of $G$ let $\\chi_c$ be its character. Then","title":"Expansion of class functions in irreducible characters"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/delta-one","kind":"node","lean":"BurnsidePaqb.deltaOne","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#delta-one","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For a finite group $G$, $\\delta_1 \\colon G \\to \\mathbb{C}$ is the function with $\\delta_1(1) = 1$ and $\\delta_1(g) = 0$ for every $g \\ne 1$.","title":"Delta function at the identity"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/delta-one-inner","kind":"node","lean":"BurnsidePaqb.deltaOne_inner","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#delta-one-inner","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $\\delta \\colon G \\to \\mathbb{C}$ be the indicator function of the identity, so $\\delta(1) = 1$ and $\\delta(g) = 0$ for $g \\neq 1$, and let $\\chi_c$ be the character of an isomorphism class $c$ of simple finite-dimensional complex representations of $G$. Then","title":"Pairing of the identity indicator with a character"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/regular-character-vanishes","kind":"node","lean":"BurnsidePaqb.regular_character_vanishes","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#regular-character-vanishes","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group and, for each isomorphism class $c$ of simple finite-dimensional complex representations of $G$, let $\\chi_c$ be its character. For every $g \\in G$ with $g \\ne 1$, $\\sum_c \\chi_c(1)\\,\\chi_c(g) = 0$, where the sum runs over all such classes $c$.","title":"Column orthogonality at the identity"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/sum-deg-sq","kind":"node","lean":"BurnsidePaqb.sum_deg_sq","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#sum-deg-sq","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group and, for each isomorphism class $c$ of simple finite-dimensional complex representations of $G$, let $\\chi_c$ be its character. Then $\\sum_c \\chi_c(1)^2 = |G|$, where the sum runs over all such classes $c$.","title":"Sum of squared degrees"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/cyc-subfield","kind":"node","lean":"BurnsidePaqb.cycSubfield","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cyclotomic subfield of "],["math","\\(\\mathbb{C}\\)"]],"source":"Cyclotomic subfield of $\\mathbb{C}$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#cyc-subfield","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For a natural number $m$, the $m$-th cyclotomic field is the intermediate field of $\\mathbb{C}/\\mathbb{Q}$ generated over $\\mathbb{Q}$ by the $m$-th roots of unity, that is, by the complex roots of $X^m - 1$.","title":"Cyclotomic subfield of $\\mathbb{C}$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/map-x-pow-sub-c-rat","kind":"node","lean":"BurnsidePaqb.map_X_pow_sub_C_rat","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(X^m - 1\\)"],["text"," over "],["math","\\(\\mathbb{Q}\\)"],["text"," and over "],["math","\\(\\mathbb{C}\\)"]],"source":"$X^m - 1$ over $\\mathbb{Q}$ and over $\\mathbb{C}$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#map-x-pow-sub-c-rat","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For every natural number $m$, mapping the coefficients of the rational polynomial $X^m - 1$ along the inclusion $\\mathbb{Q} \\to \\mathbb{C}$ gives the complex polynomial $X^m - 1$.","title":"$X^m - 1$ over $\\mathbb{Q}$ and over $\\mathbb{C}$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/cyc-root-set","kind":"node","lean":"BurnsidePaqb.cyc_rootSet","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Complex roots of "],["math","\\(X^m - 1\\)"]],"source":"Complex roots of $X^m - 1$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#cyc-root-set","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For every natural number $m \\neq 0$, the set of complex roots of the rational polynomial $X^m - 1$ is exactly the set of $m$-th roots of unity in $\\mathbb{C}$.","title":"Complex roots of $X^m - 1$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/cyc-is-splitting","kind":"node","lean":"BurnsidePaqb.cyc_isSplitting","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(m\\)"],["text","-th cyclotomic field is a splitting field of "],["math","\\(X^m - 1\\)"]],"source":"The $m$-th cyclotomic field is a splitting field of $X^m - 1$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#cyc-is-splitting","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For every natural number $m \\neq 0$, the subfield of $\\mathbb{C}$ generated over $\\mathbb{Q}$ by the $m$-th roots of unity is a splitting field over $\\mathbb{Q}$ of the polynomial $X^m - 1 \\in \\mathbb{Q}[X]$.","title":"The $m$-th cyclotomic field is a splitting field of $X^m - 1$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/cyc-subfield-zero","kind":"node","lean":"BurnsidePaqb.cycSubfield_zero","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zeroth cyclotomic field is "],["math","\\(\\mathbb{Q}\\)"]],"source":"The zeroth cyclotomic field is $\\mathbb{Q}$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#cyc-subfield-zero","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For $m = 0$ the cyclotomic subfield of $\\mathbb{C}$ is the bottom intermediate field, that is $\\mathbb{Q}$ itself: the polynomial $X^0 - 1$ is zero and is taken to have no roots, so nothing is adjoined.","title":"The zeroth cyclotomic field is $\\mathbb{Q}$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/conj-class","kind":"node","lean":"BurnsidePaqb.conjClass","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#conj-class","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For an element $g$ of a finite group $G$, the conjugacy class of $g$ is the finite set $\\{h g h^{-1} : h \\in G\\}$, the orbit of $g$ under the action of $G$ on itself by conjugation.","title":"Conjugacy class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/conj-class-card-mul-stabilizer-card","kind":"node","lean":"BurnsidePaqb.conjClass_card_mul_stabilizer_card","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#conj-class-card-mul-stabilizer-card","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For every element $g$ of a finite group $G$, the size of the conjugacy class of $g$ times the order of the stabilizer of $g$ under the action of $G$ on itself by conjugation, $x \\cdot y = x y x^{-1}$, equals the order of $G$.","title":"Orbit\u2013stabilizer for a conjugacy class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/centralizer-equiv-stabilizer","kind":"node","lean":"BurnsidePaqb.centralizerEquivStabilizer","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#centralizer-equiv-stabilizer","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For an element $g$ of a finite group $G$, this is the bijection from the centralizer $C_G(g) = \\{x \\in G : xg = gx\\}$ onto the stabilizer of $g$ for the action of $G$ on itself by conjugation, $x \\cdot y = x y x^{-1}$, that sends each element $x$ to itself.","title":"Centralizer as a conjugation stabilizer"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/conj-class-card-mul-centralizer-card","kind":"node","lean":"BurnsidePaqb.conjClass_card_mul_centralizer_card","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#conj-class-card-mul-centralizer-card","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $g$ be an element of a finite group $G$, let $g^G$ be its conjugacy class and let $C_G(g) = \\{x \\in G : xg = gx\\}$ be its centralizer. Then $|g^G| \\cdot |C_G(g)| = |G|$.","title":"Class size times centralizer order"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/conj-mem-conj-class","kind":"node","lean":"BurnsidePaqb.conj_mem_conjClass","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#conj-mem-conj-class","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $g$, $h$ and $x$ be elements of a finite group $G$. If $x$ lies in the conjugacy class of $g$, then so does $h x h^{-1}$.","title":"Conjugacy classes are closed under conjugation"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/exists-ne-one-class-prime-pow","kind":"node","lean":"BurnsidePaqb.exists_ne_one_class_prime_pow","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","A nontrivial element whose class size is a power of "],["math","\\(p\\)"]],"source":"A nontrivial element whose class size is a power of $p$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#exists-ne-one-class-prime-pow","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $p$ and $q$ be primes, let $a$ and $b$ be natural numbers with $b \\ge 1$, and let $G$ be a finite group of order $p^a q^b$. Then there are an element $g \\ne 1$ of $G$ and a natural number $k \\le a$ such that the conjugacy class of $g$ has exactly $p^k$ elements.","title":"A nontrivial element whose class size is a power of $p$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/class-sum","kind":"node","lean":"BurnsidePaqb.classSum","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#class-sum","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a group. The class sum of a finite subset $C \\subseteq G$ is the element $\\widehat{C} = \\sum_{x \\in C} x$ of the complex group algebra $\\mathbb{C}[G]$.","title":"Class sum of a finite subset"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/class-sum-central","kind":"node","lean":"BurnsidePaqb.classSum_central","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#class-sum-central","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $g \\in G$, and let $\\widehat{C_g} = \\sum_{x \\in C_g} x$ be the sum, in the complex group algebra $\\mathbb{C}[G]$, of the elements of the conjugacy class $C_g$ of $g$. Then $\\widehat{C_g} h = h \\widehat{C_g}$ for every $h \\in G$, where $h$ is viewed as a basis element of $\\mathbb{C}[G]$.","title":"Class sums commute with group elements"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/class-sum-central-2","kind":"node","lean":"BurnsidePaqb.classSum_central'","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#class-sum-central-2","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $g \\in G$, and let $\\widehat{C_g} = \\sum_{x \\in C_g} x$ be the sum, in the complex group algebra $\\mathbb{C}[G]$, of the elements of the conjugacy class $C_g$ of $g$. Then $w \\widehat{C_g} = \\widehat{C_g} w$ for every $w \\in \\mathbb{C}[G]$.","title":"The class sum of the conjugacy class of an element is central"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/class-rep","kind":"node","lean":"BurnsidePaqb.classRep","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#class-rep","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For each conjugacy class $K$ of a group $G$, the representative of $K$ is an element $g_K \\in G$, fixed once and for all by choice, whose conjugacy class is $K$.","title":"Chosen representative of a conjugacy class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/carrier-to-finset","kind":"node","lean":"BurnsidePaqb.carrier_toFinset","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#carrier-to-finset","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. For every conjugacy class $K$ of $G$, the set of elements of $K$ equals the conjugacy class of the chosen representative of $K$.","title":"Conjugacy class of the chosen representative"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/struct-const","kind":"node","lean":"BurnsidePaqb.structConst","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#struct-const","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. For finite subsets $C_i, C_j \\subseteq G$ and an element $z \\in G$, the structure constant is the number of pairs $(x, y) \\in C_i \\times C_j$ with $xy = z$.","title":"Structure constants of a pair of subsets"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/struct-const-conj","kind":"node","lean":"BurnsidePaqb.structConst_conj","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#struct-const-conj","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $g_i, g_j \\in G$ with conjugacy classes $C_{g_i}$ and $C_{g_j}$, and for $z \\in G$ let $a(z)$ be the number of pairs $(x, y) \\in C_{g_i} \\times C_{g_j}$ with $x y = z$. If $u, z \\in G$ are conjugate, then $a(u) = a(z)$.","title":"Class structure constants are class functions"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/coeff-class-sum-mul","kind":"node","lean":"BurnsidePaqb.coeff_classSum_mul","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#coeff-class-sum-mul","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a group, let $C_i$ and $C_j$ be finite subsets of $G$, and write $\\widehat{C} = \\sum_{x \\in C} x$ in the complex group algebra $\\mathbb{C}[G]$. For every $u \\in G$, the coefficient of $u$ in $\\widehat{C_i} \\widehat{C_j}$ is the number of pairs $(x, y) \\in C_i \\times C_j$ with $x y = u$.","title":"Coefficients of a product of class sums"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/class-sum-mul","kind":"node","lean":"BurnsidePaqb.classSum_mul","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#class-sum-mul","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. For a finite subset $C$ of $G$ write $\\widehat{C} = \\sum_{x \\in C} x$ in the complex group algebra $\\mathbb{C}[G]$, and for $g \\in G$ let $C_g$ be its conjugacy class. Let $g_i, g_j \\in G$, and for each conjugacy class $K$ of $G$ with chosen representative $z_K$ let $a_K$ be the number of pairs $(x, y) \\in C_{g_i} \\times C_{g_j}$ with $x y = z_K$. Then\u2026","title":"Products of class sums expand in class sums"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/conj-class-eq-of-is-conj","kind":"node","lean":"BurnsidePaqb.conjClass_eq_of_isConj","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#conj-class-eq-of-is-conj","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. If $g_1, g_2 \\in G$ are conjugate, then the conjugacy class of $g_1$ equals the conjugacy class of $g_2$.","title":"Conjugate elements have the same conjugacy class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/class-sum-carrier-central","kind":"node","lean":"BurnsidePaqb.classSum_carrier_central","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#class-sum-carrier-central","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group. For every conjugacy class $K$ of $G$ and every element $w$ of the complex group algebra $\\mathbb{C}[G]$, the class sum $\\widehat{K} = \\sum_{x \\in K} x$ satisfies $w \\widehat{K} = \\widehat{K} w$.","title":"Class sums of conjugacy classes are central"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/exists-scalar-of-central","kind":"node","lean":"BurnsidePaqb.exists_scalar_of_central","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#exists-scalar-of-central","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a group $G$ on a finite-dimensional complex vector space $V$, extended linearly to the group algebra $\\mathbb{C}[G]$. If $z \\in \\mathbb{C}[G]$ commutes with every element of $\\mathbb{C}[G]$, then $z$ acts on $V$ as a scalar: there is $c \\in \\mathbb{C}$ with $\\rho(z) v = c\\, v$ for all $v \\in V$.","title":"Schur's lemma for central elements of the group algebra"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/central-scalar","kind":"node","lean":"BurnsidePaqb.centralScalar","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#central-scalar","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$. For a conjugacy class $C$ of $G$, the class sum $\\sum_{x \\in C} x$ is central in the group algebra $\\mathbb{C}[G]$, so it acts on $V$ as multiplication by a complex number. The central scalar $\\omega_\\rho(C)$ is that number: $\\rho\\bigl(\\sum_{x \\in C} x\\bigr) v = \\omega_\\rho(C)\\, v$ for all $v \\in V$.","title":"Central scalar of a conjugacy class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/central-scalar-mul","kind":"node","lean":"BurnsidePaqb.centralScalar_mul","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#central-scalar-mul","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and let $\\omega_\\rho(C)$ be the scalar by which the class sum of a conjugacy class $C$ acts on $V$. Fix a representative $g_C$ of each conjugacy class $C$. For conjugacy classes $C_1$ and $C_2$,","title":"Central scalars multiply via the class structure constants"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/central-scalar-trace","kind":"node","lean":"BurnsidePaqb.centralScalar_trace","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#central-scalar-trace","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$ with character $\\chi$, and let $\\omega_\\rho(C)$ be the scalar by which the class sum of a conjugacy class $C$ acts on $V$. Then $\\omega_\\rho(C) \\cdot \\dim V = \\sum_{x \\in C} \\chi(x)$.","title":"Trace formula for central scalars"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/scalar-gens","kind":"node","lean":"BurnsidePaqb.scalarGens","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#scalar-gens","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and let $\\omega_\\rho(C)$ be the scalar by which the class sum of a conjugacy class $C$ acts on $V$. The set $S_\\rho$ is the finite set of complex numbers consisting of $1$ and the values $\\omega_\\rho(C)$ for all conjugacy classes $C$ of $G$.","title":"Generators of the central-scalar ring"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/scalar-gens-mul","kind":"node","lean":"BurnsidePaqb.scalarGens_mul","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#scalar-gens-mul","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and let $S_\\rho$ be the finite set of complex numbers consisting of $1$ and the scalars by which the class sums of the conjugacy classes of $G$ act on $V$. If $a, b \\in S_\\rho$, then $ab$ lies in the $\\mathbb{Z}$-linear span of $S_\\rho$.","title":"Products of central-scalar generators stay in their span"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/scalar-subalg","kind":"node","lean":"BurnsidePaqb.scalarSubalg","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#scalar-subalg","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and let $S_\\rho$ be the finite set of complex numbers consisting of $1$ and the scalars by which the class sums of the conjugacy classes of $G$ act on $V$. The $\\mathbb{Z}$-linear span of $S_\\rho$ is closed under multiplication, and this is that span as a $\\mathbb{Z}$-subalgebra of $\\mathbb{C}$.","title":"Ring generated by the central scalars"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/central-scalar-is-integral","kind":"node","lean":"BurnsidePaqb.centralScalar_isIntegral","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#central-scalar-is-integral","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$. For every conjugacy class $C$ of $G$, the complex number $\\omega_\\rho(C)$ by which the class sum $\\sum_{x \\in C} x$ acts on $V$ is an algebraic integer, that is, integral over $\\mathbb{Z}$.","title":"Central scalars are algebraic integers"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/class-sum-central-scalar-is-integral","kind":"node","lean":"BurnsidePaqb.classSum_centralScalar_isIntegral","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#class-sum-central-scalar-is-integral","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$, with character $\\chi$. For every $g \\in G$, if $C$ is the conjugacy class of $g$, then $|C|\\,\\chi(g)/\\chi(1)$ is an algebraic integer.","title":"Class size times character ratio is an algebraic integer"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/character-eq-sum-roots","kind":"node","lean":"BurnsidePaqb.character_eq_sum_roots","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#character-eq-sum-roots","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a group $G$ on a finite-dimensional complex vector space, with character $\\chi$, and let $g \\in G$. Then $\\chi(g)$ is the sum of the roots of the characteristic polynomial of $\\rho(g)$, counted with multiplicity.","title":"Character value as the sum of characteristic roots"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/charpoly-root-pow-order-of","kind":"node","lean":"BurnsidePaqb.charpoly_root_pow_orderOf","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#charpoly-root-pow-order-of","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a group, $V$ a finite-dimensional complex vector space, $\\rho$ a representation of $G$ on $V$, and $g \\in G$ an element of order $n$, where $n = 0$ if $g$ has infinite order. Then every root $\\mu$ of the characteristic polynomial of $\\rho(g)$ satisfies $\\mu^n = 1$; in particular, when $g$ has finite order, $\\mu$ is a root of unity.","title":"Characteristic roots are killed by the order of the element"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/charpoly-root-norm-eq-one","kind":"node","lean":"BurnsidePaqb.charpoly_root_norm_eq_one","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#charpoly-root-norm-eq-one","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, $V$ a finite-dimensional complex vector space and $\\rho$ a representation of $G$ on $V$. For every $g \\in G$, every root $\\mu$ of the characteristic polynomial of $\\rho(g)$ satisfies $|\\mu| = 1$.","title":"Eigenvalues of a finite group representation have modulus one"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/re-mul-conj","kind":"node","lean":"BurnsidePaqb.re_mul_conj","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#re-mul-conj","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For complex numbers $a$ and $b$, $\\operatorname{Re} a \\operatorname{Re} b + \\operatorname{Im} a \\operatorname{Im} b = \\operatorname{Re}(a \\bar{b})$.","title":"Real part of a product with a conjugate"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/norm-mul-conj-eq-one","kind":"node","lean":"BurnsidePaqb.norm_mul_conj_eq_one","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#norm-mul-conj-eq-one","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For complex numbers $a$ and $b$ with $|a| = |b| = 1$, we have $|a \\bar{b}| = 1$.","title":"Product of a unit complex number with a conjugate unit"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/eq-of-mul-conj-eq-one","kind":"node","lean":"BurnsidePaqb.eq_of_mul_conj_eq_one","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#eq-of-mul-conj-eq-one","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $a, b \\in \\mathbb{C}$ with $|b| = 1$. If $a \\bar{b} = 1$, then $a = b$.","title":"Cancelling a unit-modulus conjugate"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/re-lt-one-of-ne-one","kind":"node","lean":"BurnsidePaqb.re_lt_one_of_ne_one","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Real part of a unit complex number other than "],["math","\\(1\\)"]],"source":"Real part of a unit complex number other than $1$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#re-lt-one-of-ne-one","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"If $w \\in \\mathbb{C}$ satisfies $|w| = 1$ and $w \\neq 1$, then $\\operatorname{Re} w < 1$.","title":"Real part of a unit complex number other than $1$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/abs-sum-roots-of-unity-lt","kind":"node","lean":"BurnsidePaqb.abs_sum_roots_of_unity_lt","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#abs-sum-roots-of-unity-lt","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\mu_1, \\dots, \\mu_n$ be complex numbers with $|\\mu_i| = 1$ for every $i$, not all equal. Then $\\left|\\sum_{i=1}^{n} \\mu_i\\right| < n$.","title":"Strict triangle inequality for unit complex numbers"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/character-is-integral","kind":"node","lean":"BurnsidePaqb.character_isIntegral","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#character-is-integral","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space, with character $\\chi$. Then $\\chi(g)$ is an algebraic integer for every $g \\in G$.","title":"Character values are algebraic integers"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-div-deg-is-integral-of-coprime","kind":"node","lean":"BurnsidePaqb.charDivDeg_isIntegral_of_coprime","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-div-deg-is-integral-of-coprime","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $\\rho$ be an irreducible representation of $G$ on a nonzero finite-dimensional complex vector space $V$, with character $\\chi$, and let $g \\in G$. If the size of the conjugacy class of $g$ is coprime to $\\dim V$, then $\\chi(g)/\\chi(1)$ is an algebraic integer.","title":"Integrality of the normalized character value at a coprime class"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/abs-add-two-ne-unit-lt-two","kind":"node","lean":"BurnsidePaqb.abs_add_two_ne_unit_lt_two","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#abs-add-two-ne-unit-lt-two","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"If $a$ and $b$ are distinct complex numbers with $|a| = |b| = 1$, then $|a + b| < 2$.","title":"Sum of two distinct unit complex numbers"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/abs-multiset-sum-lt","kind":"node","lean":"BurnsidePaqb.abs_multiset_sum_lt","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#abs-multiset-sum-lt","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $s$ be a finite multiset of complex numbers, each of absolute value $1$, containing two distinct elements. Then the absolute value of the sum of $s$ is strictly less than the number of elements of $s$, counted with multiplicity.","title":"Strict triangle inequality for unit complex numbers, multiset form"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/charpoly-roots-card","kind":"node","lean":"BurnsidePaqb.charpoly_roots_card","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#charpoly-roots-card","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a group, $V$ a finite-dimensional complex vector space, $\\rho$ a representation of $G$ on $V$ and $g \\in G$. The characteristic polynomial of $\\rho(g)$ has exactly $\\dim_{\\mathbb{C}} V$ roots in $\\mathbb{C}$, counted with multiplicity.","title":"Number of characteristic roots of a representation matrix"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-roots-not-all-equal","kind":"node","lean":"BurnsidePaqb.char_roots_not_all_equal","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-roots-not-all-equal","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $\\rho$ be a representation of $G$ on a nonzero finite-dimensional complex vector space $V$, with character $\\chi$, and let $g \\in G$. If $|\\chi(g)| < \\dim V$, then the characteristic polynomial of $\\rho(g)$ has two distinct roots.","title":"Characteristic roots are not all equal below the degree"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-mem-cyc-subfield","kind":"node","lean":"BurnsidePaqb.char_mem_cycSubfield","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-mem-cyc-subfield","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a group $G$ on a finite-dimensional complex vector space, with character $\\chi$, and let $g \\in G$ have finite order $m$. Then $\\chi(g)$ lies in the subfield of $\\mathbb{C}$ generated over $\\mathbb{Q}$ by the $m$-th roots of unity.","title":"Character values lie in a cyclotomic field"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/conj-root-elem","kind":"node","lean":"BurnsidePaqb.conjRootElem","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#conj-root-elem","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $m$ be a positive integer, let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and let $g \\in G$ be such that every root $\\mu$ of the characteristic polynomial of $\\rho(g)$ satisfies $\\mu^m = 1$. Each such root $\\mu$ is regarded as an element of the subfield $K \\subseteq \\mathbb{C}$ obtained by adjoining the $m$-th roots of unity to $\\mathbb{Q}$.","title":"Characteristic roots as elements of the cyclotomic field"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/embedding-char-eq-conj-sum","kind":"node","lean":"BurnsidePaqb.embedding_char_eq_conj_sum","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#embedding-char-eq-conj-sum","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a group, $\\rho$ a representation of $G$ on a finite-dimensional complex vector space $V$ with character $\\chi$, let $m$ be a positive integer, and let $g \\in G$ be such that every root $\\mu$ of the characteristic polynomial of $\\rho(g)$ satisfies $\\mu^m = 1$. Let $K \\subseteq \\mathbb{C}$ be the field obtained by adjoining the $m$-th roots of unity to $\\mathbb{Q}$, and suppose $\\chi(g) \\in K$. Then for\u2026","title":"Field embeddings applied to character values"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-eq-zero-of-coprime-of-norm-lt","kind":"node","lean":"BurnsidePaqb.char_eq_zero_of_coprime_of_norm_lt","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-eq-zero-of-coprime-of-norm-lt","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $\\rho$ be an irreducible representation of $G$ on a nonzero finite-dimensional complex vector space $V$, with character $\\chi$, and let $g \\in G$. If the size of the conjugacy class of $g$ is coprime to $\\dim V$ and $|\\chi(g)| < \\dim V$, then $\\chi(g) = 0$.","title":"Character vanishing from coprimality and a strict norm bound"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/burnside-coprime-vanish","kind":"node","lean":"BurnsidePaqb.burnside_coprime_vanish","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#burnside-coprime-vanish","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $\\rho$ be an irreducible representation of $G$ on a nonzero finite-dimensional complex vector space $V$, with character $\\chi$, and let $g \\in G$. If the size of the conjugacy class of $g$ is coprime to $\\dim V$, then $|\\chi(g)| = \\dim V$ or $\\chi(g) = 0$.","title":"Burnside's vanishing theorem for coprime class sizes"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/self-pair-sum","kind":"node","lean":"BurnsidePaqb.self_pair_sum","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#self-pair-sum","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space, with character $\\chi$. Then $\\sum_{g \\in G} \\chi(g)\\chi(g^{-1}) = |G|$.","title":"Self-orthogonality of an irreducible character"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/sum-over-classes","kind":"node","lean":"BurnsidePaqb.sum_over_classes","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#sum-over-classes","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"For a finite group $G$ and every function $f : G \\to \\mathbb{C}$, $\\sum_{g \\in G} f(g) = \\sum_{l} \\sum_{x \\in l} f(x)$, where $l$ runs over the conjugacy classes of $G$.","title":"Splitting a sum over conjugacy classes"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/conj-to-rep","kind":"node","lean":"BurnsidePaqb.conj_to_rep","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#conj-to-rep","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $l$ be a conjugacy class of a finite group $G$ and $g_l$ its chosen representative. For every $x \\in l$ there is $c \\in G$ with $c x c^{-1} = g_l$.","title":"Members of a conjugacy class are conjugate to its representative"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-eq-of-mem-carrier","kind":"node","lean":"BurnsidePaqb.char_eq_of_mem_carrier","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-eq-of-mem-carrier","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space, with character $\\chi$, and let $g_l$ be the chosen representative of a conjugacy class $l$ of $G$. Then $\\chi(x) = \\chi(g_l)$ for every $x \\in l$.","title":"Characters are constant on conjugacy classes"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-inv-eq-of-mem-carrier","kind":"node","lean":"BurnsidePaqb.char_inv_eq_of_mem_carrier","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-inv-eq-of-mem-carrier","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space, with character $\\chi$, and let $g_l$ be the chosen representative of a conjugacy class $l$ of $G$. Then $\\chi(x^{-1}) = \\chi(g_l^{-1})$ for every $x \\in l$.","title":"Characters at inverses are constant on conjugacy classes"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/class-inner-sum","kind":"node","lean":"BurnsidePaqb.class_inner_sum","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#class-inner-sum","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a finite-dimensional complex vector space $V$, with character $\\chi$. For a conjugacy class $l$ of $G$, let $g_l$ be its chosen representative and $\\omega(l)$ the central scalar of $l$, the scalar by which the class sum of $l$ acts on $V$. Then $\\sum_{x \\in l} \\chi(x)\\chi(x^{-1}) = (\\dim_{\\mathbb{C}} V)\\,\\omega(l)\\,\\chi(g_l^{-1})$.","title":"Contribution of a conjugacy class to the self-pairing sum"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/degree-dvd-card","kind":"node","lean":"BurnsidePaqb.degree_dvd_card","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#degree-dvd-card","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a finite group $G$ on a nonzero finite-dimensional complex vector space $V$. Then $\\dim_{\\mathbb{C}} V$ divides $|G|$.","title":"Degrees of irreducible representations divide the group order"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/all-roots-eq-of-norm-eq","kind":"node","lean":"BurnsidePaqb.all_roots_eq_of_norm_eq","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#all-roots-eq-of-norm-eq","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, $V$ a nonzero finite-dimensional complex vector space and $\\rho$ an irreducible representation of $G$ on $V$ with character $\\chi$. If $g \\in G$ satisfies $|\\chi(g)| = \\dim V$, then any two roots of the characteristic polynomial of $\\rho(g)$ are equal.","title":"Characters of full absolute value have equal eigenvalues"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/eq-smul-one-of-all-roots-eq","kind":"node","lean":"BurnsidePaqb.eq_smul_one_of_all_roots_eq","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#eq-smul-one-of-all-roots-eq","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, $V$ a nonzero finite-dimensional complex vector space, $\\rho$ a representation of $G$ on $V$, $g \\in G$ and $\\mu_0 \\in \\mathbb{C}$. If every root of the characteristic polynomial of $\\rho(g)$ equals $\\mu_0$, then $\\rho(g) = \\mu_0 \\cdot \\mathrm{id}_V$.","title":"Elements with a single eigenvalue act as scalars"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-center","kind":"node","lean":"BurnsidePaqb.charCenter","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-center","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a group $G$ on a nonzero complex vector space $V$. The scalar center of $\\rho$ is the subgroup of $G$ consisting of the elements $g$ such that $\\rho(g) = c \\cdot \\mathrm{id}_V$ for some $c \\in \\mathbb{C}$.","title":"Scalar center of a representation"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/mem-char-center-iff","kind":"node","lean":"BurnsidePaqb.mem_charCenter_iff","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#mem-char-center-iff","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a group $G$ on a nonzero complex vector space $V$. An element $g \\in G$ lies in the scalar center of $\\rho$ if and only if $\\rho(g) = c \\cdot \\mathrm{id}_V$ for some $c \\in \\mathbb{C}$.","title":"Membership in the scalar center"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/mem-char-center-of-eq-smul","kind":"node","lean":"BurnsidePaqb.mem_charCenter_of_eq_smul","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#mem-char-center-of-eq-smul","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a group $G$ on a nonzero complex vector space $V$. If $g \\in G$ and $c \\in \\mathbb{C}$ satisfy $\\rho(g) = c \\cdot \\mathrm{id}_V$, then $g$ lies in the scalar center of $\\rho$.","title":"Elements acting by a scalar lie in the scalar center"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/exists-eq-smul-of-mem-char-center","kind":"node","lean":"BurnsidePaqb.exists_eq_smul_of_mem_charCenter","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#exists-eq-smul-of-mem-char-center","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a group $G$ on a nonzero complex vector space $V$. If $g$ lies in the scalar center of $\\rho$, then there is $c \\in \\mathbb{C}$ with $\\rho(g) = c \\cdot \\mathrm{id}_V$.","title":"Elements of the scalar center act by scalars"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-center-proper-of-nonlinear","kind":"node","lean":"BurnsidePaqb.charCenter_proper_of_nonlinear","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-center-proper-of-nonlinear","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be an irreducible representation of a group $G$ on a finite-dimensional complex vector space $V$ with $\\dim V > 1$. Then the scalar center of $\\rho$ is a proper subgroup of $G$.","title":"The scalar center of a nonlinear irreducible representation is proper"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/norm-eq-of-mem-char-center","kind":"node","lean":"BurnsidePaqb.norm_eq_of_mem_charCenter","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#norm-eq-of-mem-char-center","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, $V$ a nonzero finite-dimensional complex vector space and $\\rho$ a representation of $G$ on $V$ with character $\\chi$. If $g$ lies in the scalar center of $\\rho$, then $|\\chi(g)| = \\dim V$.","title":"Elements of the scalar center have character of full absolute value"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/false-of-one-add-prime-mul-is-integral","kind":"node","lean":"BurnsidePaqb.false_of_one_add_prime_mul_isIntegral","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","An algebraic integer cannot solve "],["math","\\(1 + p\\beta = 0\\)"],["text"," for a prime "],["math","\\(p\\)"]],"source":"An algebraic integer cannot solve $1 + p\\beta = 0$ for a prime $p$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#false-of-one-add-prime-mul-is-integral","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $p$ be a prime number. There is no algebraic integer $\\beta \\in \\mathbb{C}$ with $1 + p\\beta = 0$, since such a $\\beta$ would make $-1/p$ an algebraic integer.","title":"An algebraic integer cannot solve $1 + p\\beta = 0$ for a prime $p$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/eq-smul-one-of-finrank-one","kind":"node","lean":"BurnsidePaqb.eq_smul_one_of_finrank_one","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#eq-smul-one-of-finrank-one","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $\\rho$ be a representation of a group $G$ on a complex vector space $V$ of dimension $1$, and let $\\chi$ be the character of $\\rho$. Then for every $g \\in G$ the operator $\\rho(g)$ is multiplication by the scalar $\\chi(g)$, that is, $\\rho(g) = \\chi(g) \\cdot \\mathrm{id}_V$.","title":"One-dimensional representations act by their character"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/linear-char-hom","kind":"node","lean":"BurnsidePaqb.linearCharHom","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","A linear character as a homomorphism to "],["math","\\(\\mathbb{C}^\\times\\)"]],"source":"A linear character as a homomorphism to $\\mathbb{C}^\\times$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#linear-char-hom","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group and let $\\rho$ be a representation of $G$ on a complex vector space $V$ of dimension $1$, with character $\\chi$. This is the group homomorphism from $G$ to the multiplicative group $\\mathbb{C}^\\times$ of nonzero complex numbers that sends $g$ to $\\chi(g)$.","title":"A linear character as a homomorphism to $\\mathbb{C}^\\times$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/card-eq-prime-of-nontrivial-linear","kind":"node","lean":"BurnsidePaqb.card_eq_prime_of_nontrivial_linear","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#card-eq-prime-of-nontrivial-linear","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite simple group and let $\\rho$ be a representation of $G$ on a complex vector space $V$ of dimension $1$. If the character of $\\rho$ is not the constant function $1$, then the order of $G$ is a prime number.","title":"A simple group with a nontrivial linear character has prime order"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/char-zero-of-nonlinear-of-coprime","kind":"node","lean":"BurnsidePaqb.char_zero_of_nonlinear_of_coprime","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#char-zero-of-nonlinear-of-coprime","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Nonlinear irreducible characters of a simple group vanish on nonidentity classes of coprime size. Let $G$ be a finite simple group, let $\\rho$ be an irreducible representation of $G$ on a finite-dimensional complex vector space $V$ of dimension $d > 1$, and let $\\chi$ be the character of $\\rho$. If $g \\ne 1$ is an element of $G$ whose conjugacy class has size coprime to $d$, then $\\chi(g) = 0$.","title":"Nonlinear irreducible characters of a simple group vanish on nonidentity classes of coprime size"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/cor6-of-prime-pow-class","kind":"node","lean":"BurnsidePaqb.cor6_of_prime_pow_class","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#cor6-of-prime-pow-class","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite simple group and let $g \\ne 1$ be an element of $G$ whose conjugacy class has size $r^k$ for some prime $r$ and some $k \\ge 1$. Then the order of $G$ is a prime number.","title":"A simple group with a prime-power conjugacy class has prime order"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/eq-prime-pow-mul-of-dvd","kind":"node","lean":"BurnsidePaqb.eq_prime_pow_mul_of_dvd","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisors of "],["math","\\(p^a q^b\\)"]],"source":"Divisors of $p^a q^b$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#eq-prime-pow-mul-of-dvd","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $p$ and $q$ be primes and let $a$ and $b$ be natural numbers. Every natural number $d$ dividing $p^a q^b$ has the form $d = p^{a'} q^{b'}$ for some natural numbers $a'$ and $b'$.","title":"Divisors of $p^a q^b$"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/is-solvable-of-prime-card","kind":"node","lean":"BurnsidePaqb.isSolvable_of_prime_card","parent":1679,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#is-solvable-of-prime-card","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Every finite group whose order is a prime number is solvable.","title":"Groups of prime order are solvable"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/burnside-paqb-of-simple-prime","kind":"node","lean":"BurnsidePaqb.burnside_paqb_of_simple_prime","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Burnside's "],["math","\\(p^a q^b\\)"],["text"," theorem from the prime-power class corollary"]],"source":"Burnside's $p^a q^b$ theorem from the prime-power class corollary"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#burnside-paqb-of-simple-prime","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Suppose that every finite simple group $H$ having an element $h \\ne 1$ whose conjugacy class has size $r^k$ for some prime $r$ and some $k \\ge 1$ has prime order. Then for all primes $p$ and $q$ and all natural numbers $a$ and $b$, every finite group $G$ of order $p^a q^b$ is solvable.","title":"Burnside's $p^a q^b$ theorem from the prime-power class corollary"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/burnside-paqb-zero","kind":"node","lean":"BurnsidePaqb.burnside_paqb_zero","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Burnside's "],["math","\\(p^a q^b\\)"],["text"," theorem in the lowest universe"]],"source":"Burnside's $p^a q^b$ theorem in the lowest universe"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#burnside-paqb-zero","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group whose underlying type lies in the lowest type universe, let $p$ and $q$ be primes and let $a$ and $b$ be natural numbers. If $G$ has order $p^a q^b$, then $G$ is solvable.","title":"Burnside's $p^a q^b$ theorem in the lowest universe"},{"id":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/burnside-paqb","kind":"node","lean":"BurnsidePaqb.burnside_paqb","parent":1679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Burnside's "],["math","\\(p^a q^b\\)"],["text"," theorem"]],"source":"Burnside's $p^a q^b$ theorem"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem/index.html#burnside-paqb","scope":"GroupTheory/characters-of-finite-groups/burnside-s-p-a-q-b-theorem","status":"fully_proved","summary":"Let $G$ be a finite group, let $p$ and $q$ be primes and let $a$ and $b$ be natural numbers. If $G$ has order $p^a q^b$, then $G$ is solvable.","title":"Burnside's $p^a q^b$ theorem"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups.json","id":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/index.html","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups","status":"can_prove","title":"Character degrees of Sylow subgroups"},{"id":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/irreducible-degrees","kind":"node","lean":"MathlibExt.GroupTheory.SylowCharacterDegreePPartKourovka15Problem62Wanted.irreducibleDegrees","parent":1779,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/index.html#irreducible-degrees","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups","status":"fully_proved","summary":"For a group $G$, the set of natural numbers $d$ such that $G$ has an irreducible complex representation of finite dimension $d$; for a finite group these are the degrees $\\chi(1)$ of its irreducible characters.","title":"Irreducible complex representation degrees"},{"id":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/max-ppart-exponent","kind":"node","lean":"MathlibExt.GroupTheory.SylowCharacterDegreePPartKourovka15Problem62Wanted.maxPPartExponent","parent":1779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Largest "],["math","\\(p\\)"],["text","-adic exponent of the irreducible degrees"]],"source":"Largest $p$-adic exponent of the irreducible degrees"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/index.html#max-ppart-exponent","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups","status":"fully_proved","summary":"For a natural number $p$ and a group $G$, $e_p(G)$ is the supremum in $\\mathbb{N}$ of the exponents of $p$ in the prime factorizations of the dimensions of the finite-dimensional irreducible complex representations of $G$, where the supremum of an empty or unbounded set is $0$. For a prime $p$ and a finite group $G$, it is the largest $e$ such that $p^e$ divides the degree $\\chi(1)$ of some irreducible character\u2026","title":"Largest $p$-adic exponent of the irreducible degrees"},{"id":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/sylow-character-degree-exponent-bounded","kind":"node","lean":"MathlibExt.GroupTheory.SylowCharacterDegreePPartKourovka15Problem62Wanted.sylowCharacterDegreeExponentBounded","parent":1779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 15.62 ("],["math","\\(p\\)"],["text","-parts of character degrees of Sylow subgroups)"]],"source":"Kourovka Problem 15.62 ($p$-parts of character degrees of Sylow subgroups)"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/index.html#sylow-character-degree-exponent-bounded","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups","status":"fully_proved","summary":"The statement, asked by A. Moret\u00f3, that there is a function $f \\colon \\mathbb{N} \\to \\mathbb{N}$ such that $e_p(P) \\le f(e_p(G))$ for every prime $p$, every finite group $G$ and every Sylow $p$-subgroup $P$ of $G$, where $e_p(H)$ is the largest exponent of $p$ dividing an irreducible character degree of $H$.","title":"Kourovka Problem 15.62 ($p$-parts of character degrees of Sylow subgroups)"},{"id":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/sylow-character-degree-exponent-bounded-2","kind":"node","lean":"MathlibExt.GroupTheory.SylowCharacterDegreePPartKourovka15Problem62Wanted.sylowCharacterDegreeExponentBounded_holds","parent":1779,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups/index.html#sylow-character-degree-exponent-bounded-2","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-sylow-subgroups","status":"can_prove","summary":"There is a function $f \\colon \\mathbb{N} \\to \\mathbb{N}$ such that for every prime $p$, every finite group $G$ and every Sylow $p$-subgroup $P$ of $G$, $e_p(P) \\le f(e_p(G))$, where $e_p(H)$ is the largest exponent of $p$ dividing an irreducible character degree of $H$.","title":"Character degrees of Sylow subgroups are bounded in terms of those of the group"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups.json","id":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/index.html","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups","status":"can_prove","title":"Character degrees of symmetric groups"},{"id":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/is-irreducible-rep","kind":"node","lean":"MathlibExt.GroupTheory.SymmetricGroupEqualCharacterDegreesKourovka16Problem67Wanted.IsIrreducibleRep","parent":1784,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/index.html#is-irreducible-rep","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups","status":"fully_proved","summary":"Let $G$ be a group and $n$ a natural number. A homomorphism $\\rho \\colon G \\to \\mathrm{GL}_n(\\mathbb{C})$ is irreducible when $n \\ge 1$ and every subspace $W$ of $\\mathbb{C}^n$ such that $\\rho(g) v \\in W$ for all $g \\in G$ and all $v \\in W$ is either $0$ or $\\mathbb{C}^n$.","title":"Irreducible complex matrix representations (Kourovka 16.67)"},{"id":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/is-equivalent-rep","kind":"node","lean":"MathlibExt.GroupTheory.SymmetricGroupEqualCharacterDegreesKourovka16Problem67Wanted.IsEquivalentRep","parent":1784,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/index.html#is-equivalent-rep","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups","status":"fully_proved","summary":"Let $G$ be a group and $n$ a natural number. Two homomorphisms $\\rho, \\sigma \\colon G \\to \\mathrm{GL}_n(\\mathbb{C})$ are equivalent when there is a single invertible matrix $P \\in \\mathrm{GL}_n(\\mathbb{C})$ with $\\sigma(g) = P \\rho(g) P^{-1}$ for every $g \\in G$.","title":"Equivalent complex matrix representations (Kourovka 16.67)"},{"id":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/symmetric-group-many-characters-of-equal","kind":"node","lean":"MathlibExt.GroupTheory.SymmetricGroupEqualCharacterDegreesKourovka16Problem67Wanted.symmetricGroupManyCharactersOfEqualDegree","parent":1784,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/index.html#symmetric-group-many-characters-of-equal","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups","status":"fully_proved","summary":"For every natural number $k$ there is a natural number $n_0$ such that for every $n \\ge n_0$ there are a natural number $d$ and irreducible representations $\\rho_1, \\dots, \\rho_k$ of the symmetric group $S_n$ into $\\mathrm{GL}_d(\\mathbb{C})$ that are pairwise inequivalent: $\\rho_i$ and $\\rho_j$ are not equivalent whenever $i \\ne j$.","title":"Moret\u00f3's conjecture on equal character degrees of symmetric groups"},{"id":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/symmetric-group-many-characters-of-equal-2","kind":"node","lean":"MathlibExt.GroupTheory.SymmetricGroupEqualCharacterDegreesKourovka16Problem67Wanted.symmetricGroupManyCharactersOfEqualDegree_holds","parent":1784,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups/index.html#symmetric-group-many-characters-of-equal-2","scope":"GroupTheory/characters-of-finite-groups/character-degrees-of-symmetric-groups","status":"can_prove","summary":"Moret\u00f3's conjecture holds: for every natural number $k$ there is a natural number $n_0$ such that for every $n \\ge n_0$ there are a natural number $d$ and irreducible representations $\\rho_1, \\dots, \\rho_k$ of the symmetric group $S_n$ into $\\mathrm{GL}_d(\\mathbb{C})$ that are pairwise inequivalent.","title":"Symmetric groups have many irreducible characters of equal degree"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/character-values-of-simple-groups.json","id":"GroupTheory/characters-of-finite-groups/character-values-of-simple-groups","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-values-of-simple-groups/index.html","scope":"GroupTheory/characters-of-finite-groups/character-values-of-simple-groups","status":"can_prove","title":"Character values of simple groups"},{"id":"GroupTheory/characters-of-finite-groups/character-values-of-simple-groups/irr-char-values","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupCharacterValuesKourovka11Problem97Wanted.irrCharValues","parent":1789,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-values-of-simple-groups/index.html#irr-char-values","scope":"GroupTheory/characters-of-finite-groups/character-values-of-simple-groups","status":"fully_proved","summary":"For an element $g$ of a group $G$, the set of complex numbers $\\chi(g)$, where $\\chi$ ranges over the characters of the irreducible finite-dimensional complex representations of $G$.","title":"Values of the irreducible characters at an element"},{"id":"GroupTheory/characters-of-finite-groups/character-values-of-simple-groups/finitely-many-simple-given-character-val","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupCharacterValuesKourovka11Problem97Wanted.finitelyManySimpleGivenCharacterValues","parent":1789,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-values-of-simple-groups/index.html#finitely-many-simple-given-character-val","scope":"GroupTheory/characters-of-finite-groups/character-values-of-simple-groups","status":"fully_proved","summary":"The statement that for every finite set $S$ of complex numbers there is a bound $N$ such that every finite non-abelian simple group $G$ having an element $g$ at which the set of values of the irreducible complex characters of $G$ is exactly $S$ has order at most $N$. The problem is due to S. P. Strunkov.","title":"Kourovka Problem 11.97 (simple groups with a given set of character values)"},{"id":"GroupTheory/characters-of-finite-groups/character-values-of-simple-groups/finitely-many-simple-given-character-val-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupCharacterValuesKourovka11Problem97Wanted.finitelyManySimpleGivenCharacterValues_refuted","parent":1789,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/character-values-of-simple-groups/index.html#finitely-many-simple-given-character-val-2","scope":"GroupTheory/characters-of-finite-groups/character-values-of-simple-groups","status":"can_prove","summary":"There is a finite set $S$ of complex numbers such that for every natural number $N$ there is a finite non-abelian simple group $G$ with more than $N$ elements and an element $g \\in G$ at which the set of values of the irreducible complex characters of $G$ is exactly $S$.","title":"Kourovka Problem 11.97 has a negative answer"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/convolution-on-finite-groups.json","id":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/index.html","scope":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups","status":"fully_proved","title":"Convolution on finite groups"},{"id":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/conv","kind":"node","lean":"MathlibExt.GroupTheory.FiniteGroupConvolutionWanted.conv","parent":1793,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/index.html#conv","scope":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups","status":"fully_proved","summary":"For a finite group $G$ and functions $f, g : G \\to \\mathbb{C}$, the convolution $f \\ast g : G \\to \\mathbb{C}$ is defined by $(f \\ast g)(x) = \\sum_{y \\in G} f(y)\\, g(y^{-1} x)$ for $x \\in G$.","title":"Convolution on a finite group"},{"id":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/conv-assoc","kind":"node","lean":"MathlibExt.GroupTheory.FiniteGroupConvolutionWanted.conv_assoc","parent":1793,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/index.html#conv-assoc","scope":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups","status":"fully_proved","summary":"Let $G$ be a finite group and write $f \\ast g$ for the convolution $(f \\ast g)(x) = \\sum_{y \\in G} f(y)\\, g(y^{-1} x)$ of functions on $G$. For all $f, g, h : G \\to \\mathbb{C}$ we have $(f \\ast g) \\ast h = f \\ast (g \\ast h)$.","title":"Associativity of convolution"},{"id":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/conv-single-one","kind":"node","lean":"MathlibExt.GroupTheory.FiniteGroupConvolutionWanted.conv_single_one","parent":1793,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/index.html#conv-single-one","scope":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups","status":"fully_proved","summary":"Let $G$ be a finite group, write $f \\ast g$ for the convolution $(f \\ast g)(x) = \\sum_{y \\in G} f(y)\\, g(y^{-1} x)$ of functions on $G$, and let $\\delta_1 : G \\to \\mathbb{C}$ be given by $\\delta_1(1) = 1$ and $\\delta_1(x) = 0$ for $x \\neq 1$. Then $\\delta_1 \\ast f = f$ for every $f : G \\to \\mathbb{C}$.","title":"The delta function at the identity is a left unit for convolution"},{"id":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/conv-comm-of-comm","kind":"node","lean":"MathlibExt.GroupTheory.FiniteGroupConvolutionWanted.conv_comm_of_comm","parent":1793,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/index.html#conv-comm-of-comm","scope":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group and write $f \\ast g$ for the convolution $(f \\ast g)(x) = \\sum_{y \\in G} f(y)\\, g(y^{-1} x)$ of functions on $G$. For all $f, g : G \\to \\mathbb{C}$ we have $f \\ast g = g \\ast f$.","title":"Convolution on a finite abelian group is commutative"},{"id":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/char-pair-conv","kind":"node","lean":"MathlibExt.GroupTheory.FiniteGroupConvolutionWanted.charPair_conv","parent":1793,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/convolution-on-finite-groups/index.html#char-pair-conv","scope":"GroupTheory/characters-of-finite-groups/convolution-on-finite-groups","status":"fully_proved","summary":"Let $G$ be a finite group, let $\\varphi : G \\to \\mathbb{C}$ satisfy $\\varphi(1) = 1$ and $\\varphi(xy) = \\varphi(x)\\varphi(y)$ for all $x, y \\in G$ (a one-dimensional character), and let $f, g : G \\to \\mathbb{C}$. Writing $f \\ast g$ for the convolution $(f \\ast g)(x) = \\sum_{y \\in G} f(y)\\, g(y^{-1} x)$, we have","title":"Convolution theorem for one-dimensional characters"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/frobenius-group-determinant.json","id":"GroupTheory/characters-of-finite-groups/frobenius-group-determinant","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/frobenius-group-determinant/index.html","scope":"GroupTheory/characters-of-finite-groups/frobenius-group-determinant","status":"can_prove","title":"Frobenius group determinant"},{"id":"GroupTheory/characters-of-finite-groups/frobenius-group-determinant/group-matrix","kind":"node","lean":"MathlibExt.GroupTheory.FrobeniusDeterminantWanted.RepresentationTheory.FrobeniusDeterminant.groupMatrix","parent":1799,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/frobenius-group-determinant/index.html#group-matrix","scope":"GroupTheory/characters-of-finite-groups/frobenius-group-determinant","status":"fully_proved","summary":"For a finite group $G$, the group matrix of Dedekind and Frobenius is the $G \\times G$ matrix over the polynomial ring $\\mathbb{C}[x_g : g \\in G]$ whose entry at $(g, h)$ is the variable $x_{gh}$.","title":"Group matrix"},{"id":"GroupTheory/characters-of-finite-groups/frobenius-group-determinant/group-determinant","kind":"node","lean":"MathlibExt.GroupTheory.FrobeniusDeterminantWanted.RepresentationTheory.FrobeniusDeterminant.groupDeterminant","parent":1799,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/frobenius-group-determinant/index.html#group-determinant","scope":"GroupTheory/characters-of-finite-groups/frobenius-group-determinant","status":"fully_proved","summary":"For a finite group $G$, the group determinant $\\Theta(G)$ is the determinant of the group matrix of $G$, the $G \\times G$ matrix over $\\mathbb{C}[x_g : g \\in G]$ whose entry at $(g, h)$ is the variable $x_{gh}$; it is a polynomial in these variables.","title":"Group determinant"},{"id":"GroupTheory/characters-of-finite-groups/frobenius-group-determinant/frobenius-group-determinant","kind":"node","lean":"MathlibExt.GroupTheory.FrobeniusDeterminantWanted.RepresentationTheory.FrobeniusDeterminant.frobenius_group_determinant","parent":1799,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/frobenius-group-determinant/index.html#frobenius-group-determinant","scope":"GroupTheory/characters-of-finite-groups/frobenius-group-determinant","status":"can_prove","summary":"Let $G$ be a finite group and let $r$ be the number of conjugacy classes of $G$. Then there exist polynomials $p_1, \\dots, p_r$ in $\\mathbb{C}[x_g : g \\in G]$ such that each $p_j$ is irreducible, $p_i$ and $p_j$ are not associated for $i \\neq j$, and the group determinant of $G$ equals $\\prod_{j=1}^{r} p_j^{\\deg p_j}$, where $\\deg p_j$ is the total degree of $p_j$.","title":"Frobenius determinant theorem"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/integer-group-determinants.json","id":"GroupTheory/characters-of-finite-groups/integer-group-determinants","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","title":"Integer group determinants"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/integer-group-determinants","kind":"node","lean":"MetaMathlibExt.integerGroupDeterminants","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#integer-group-determinants","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For a finite group $G$, written additively and not necessarily commutative, $S(G)$ is the set of integers $\\det(x_{g - h})_{g, h \\in G}$, where $x$ ranges over all functions $G \\to \\mathbb{Z}$.","title":"Integer group determinants"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/one-mem-integer-group-determinants","kind":"node","lean":"MetaMathlibExt.one_mem_integerGroupDeterminants","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(1\\)"],["text"," is an integer group determinant"]],"source":"$1$ is an integer group determinant"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#one-mem-integer-group-determinants","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For every finite group $G$, written additively, $1 \\in S(G)$, where $S(G)$ is the set of integer group determinants of $G$: the indicator function of the identity element gives the identity matrix.","title":"$1$ is an integer group determinant"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/circulant-det-four","kind":"node","lean":"MetaMathlibExt.circulantDetFour","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The group determinant of "],["math","\\(C_4\\)"]],"source":"The group determinant of $C_4$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#circulant-det-four","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For a commutative ring $R$ and $x = (x_0, x_1, x_2, x_3) \\in R^4$, $D_4(x_0, x_1, x_2, x_3)$ is the determinant of the $4 \\times 4$ circulant matrix whose $(i, j)$ entry is $x_{i - j}$, indices taken mod $4$.","title":"The group determinant of $C_4$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/circulant-det-four-single-one","kind":"node","lean":"MetaMathlibExt.circulantDetFour_single_one","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#circulant-det-four-single-one","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"Over every commutative ring $R$, the $4 \\times 4$ circulant determinant satisfies $D_4(1, 0, 0, 0) = 1$.","title":"The 4 \u00d7 4 circulant determinant of (1, 0, 0, 0)"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-b","kind":"node","lean":"MetaMathlibExt.c4SqB","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sums "],["math","\\(b_i\\)"]],"source":"The sums $b_i$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-b","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ and $i \\in \\mathbb{Z}/4$, $b_i = \\bigl(a(i, 0) + a(i, 2)\\bigr) + \\bigl(a(i, 1) + a(i, 3)\\bigr)$. In the paper's indexing $a_{r + 4s} = a(r, s)$ this is $b_i = (a_i + a_{i+8}) + (a_{i+4} + a_{i+12})$ for $0 \\le i \\le 3$.","title":"The sums $b_i$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-c","kind":"node","lean":"MetaMathlibExt.c4SqC","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The alternating sums "],["math","\\(c_i\\)"]],"source":"The alternating sums $c_i$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-c","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ and $i \\in \\mathbb{Z}/4$, $c_i = \\bigl(a(i, 0) + a(i, 2)\\bigr) - \\bigl(a(i, 1) + a(i, 3)\\bigr)$. In the paper's indexing $a_{r + 4s} = a(r, s)$ this is $c_i = (a_i + a_{i+8}) - (a_{i+4} + a_{i+12})$ for $0 \\le i \\le 3$.","title":"The alternating sums $c_i$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-d","kind":"node","lean":"MetaMathlibExt.c4SqD","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The differences "],["math","\\(d_i\\)"],["text"," of a function on "],["math","\\(C_4^2\\)"]],"source":"The differences $d_i$ of a function on $C_4^2$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-d","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$, the integers $d_0, \\dots, d_7$ are $d_i = a(i, 0) - a(i, 2)$ and $d_{i+4} = a(i, 1) - a(i, 3)$ for $0 \\le i \\le 3$. Writing $a_{r + 4s}$ for $a(r, s)$, this is the paper's $d_i = a_i - a_{i+8}$ for $0 \\le i \\le 7$.","title":"The differences $d_i$ of a function on $C_4^2$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-alpha","kind":"node","lean":"MetaMathlibExt.c4SqAlpha","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Gaussian integers "],["math","\\(\\alpha_i\\)"]],"source":"The Gaussian integers $\\alpha_i$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-alpha","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ with differences $d_0, \\dots, d_7$ (so $d_i = a(i, 0) - a(i, 2)$ and $d_{i+4} = a(i, 1) - a(i, 3)$ for $0 \\le i \\le 3$), the Gaussian integers $\\alpha_0, \\dots, \\alpha_3 \\in \\mathbb{Z}[\\sqrt{-1}]$ are $\\alpha_i = d_i + d_{i+4} \\sqrt{-1}$.","title":"The Gaussian integers $\\alpha_i$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-beta","kind":"node","lean":"MetaMathlibExt.c4SqBeta","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Gaussian integer "],["math","\\(\\beta\\)"]],"source":"The Gaussian integer $\\beta$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-beta","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ with Gaussian integers $\\alpha_0, \\dots, \\alpha_3$, $\\beta = (\\alpha_0 + \\alpha_2)^2 - (\\alpha_1 + \\alpha_3)^2 \\in \\mathbb{Z}[\\sqrt{-1}]$.","title":"The Gaussian integer $\\beta$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-gamma","kind":"node","lean":"MetaMathlibExt.c4SqGamma","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Gaussian integer "],["math","\\(\\gamma\\)"],["text"," of a function on "],["math","\\(C_4^2\\)"]],"source":"The Gaussian integer $\\gamma$ of a function on $C_4^2$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-gamma","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"For $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$, let $\\alpha_i = (a(i, 0) - a(i, 2)) + (a(i, 1) - a(i, 3))\\sqrt{-1} \\in \\mathbb{Z}[\\sqrt{-1}]$ for $0 \\le i \\le 3$. Then $\\gamma = (\\alpha_0 - \\alpha_2)^2 + (\\alpha_1 - \\alpha_3)^2 \\in \\mathbb{Z}[\\sqrt{-1}]$.","title":"The Gaussian integer $\\gamma$ of a function on $C_4^2$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-notation-single","kind":"node","lean":"MetaMathlibExt.c4Sq_notation_single","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-notation-single","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"Let $\\delta$ be the integer-valued function on $C_4^2 = \\mathbb{Z}/4 \\times \\mathbb{Z}/4$ that is $1$ at $(0, 0)$ and $0$ elsewhere. For $a = \\delta$ the associated integers are $b = c = (1, 0, 0, 0)$ and $d = (1, 0, 0, 0, 0, 0, 0, 0)$, and the associated Gaussian integers are $\\beta = \\gamma = 1$.","title":"The C\u2084\u00b2 notation at the indicator of the identity"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/primes-five-mod-eight","kind":"node","lean":"MetaMathlibExt.primesFiveModEight","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes congruent to "],["math","\\(-3\\)"],["text"," modulo "],["math","\\(8\\)"]],"source":"Primes congruent to $-3$ modulo $8$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#primes-five-mod-eight","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"$P$ is the set of positive prime integers $p$ with $p \\equiv -3 \\pmod 8$, that is, $p \\equiv 5 \\pmod 8$.","title":"Primes congruent to $-3$ modulo $8$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/five-thirteen-mem-primes-five-mod-eight","kind":"node","lean":"MetaMathlibExt.five_thirteen_mem_primesFiveModEight","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(5\\)"],["text"," and "],["math","\\(13\\)"],["text"," lie in "],["math","\\(P\\)"]],"source":"$5$ and $13$ lie in $P$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#five-thirteen-mem-primes-five-mod-eight","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"The integers $5$ and $13$ both belong to the set $P$ of positive primes $p \\equiv -3 \\pmod 8$.","title":"$5$ and $13$ lie in $P$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-exceptional-set","kind":"node","lean":"MetaMathlibExt.c4SqExceptionalSet","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The exceptional set "],["math","\\(A\\)"],["text"," for "],["math","\\(C_4^2\\)"]],"source":"The exceptional set $A$ for $C_4^2$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-exceptional-set","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"Let $P$ be the set of positive primes $p \\equiv 5 \\pmod 8$. The set $A \\subseteq \\mathbb{Z}$ consists of all products $(8k - 3)(8l - 3)(8m - 3)(8n - 3)$ with $k, l, m, n \\in \\mathbb{Z}$ such that $8l - 3$, $8m - 3$ and $8n - 3$ lie in $P$ and $k + l \\not\\equiv m + n \\pmod 2$.","title":"The exceptional set $A$ for $C_4^2$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/mem-c4-sq-exceptional-set-1625","kind":"node","lean":"MetaMathlibExt.mem_c4SqExceptionalSet_1625","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The exceptional set "],["math","\\(A\\)"],["text"," contains "],["math","\\(1625\\)"]],"source":"The exceptional set $A$ contains $1625$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#mem-c4-sq-exceptional-set-1625","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"fully_proved","summary":"Let $P$ be the set of positive primes $p \\equiv -3 \\pmod 8$ and $A$ the set of products $(8k-3)(8l-3)(8m-3)(8n-3)$ with $k, l, m, n \\in \\mathbb{Z}$, $8l-3, 8m-3, 8n-3 \\in P$ and $k + l \\not\\equiv m + n \\pmod 2$. Then $1625 \\in A$: take $k = l = m = 1$ and $n = 2$, so that $1625 = 5 \\cdot 5 \\cdot 5 \\cdot 13$.","title":"The exceptional set $A$ contains $1625$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/circulant-det-four-eq","kind":"node","lean":"MetaMathlibExt.circulantDetFour_eq","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#circulant-det-four-eq","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"For every commutative ring $R$ and all $x_0, x_1, x_2, x_3 \\in R$, the $4 \\times 4$ circulant determinant $D_4$ satisfies","title":"Factorization of the 4 \u00d7 4 circulant determinant"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-group-det-eq-prod-circulant-det-fo","kind":"node","lean":"MetaMathlibExt.c4Sq_groupDet_eq_prod_circulantDetFour","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Factorization of the complex group determinant of "],["math","\\(C_4^2\\)"]],"source":"Factorization of the complex group determinant of $C_4^2$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-group-det-eq-prod-circulant-det-fo","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"For every $y \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{C}$,","title":"Factorization of the complex group determinant of $C_4^2$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/sq-add-sq-eq-mul-of-mod-eight","kind":"node","lean":"MetaMathlibExt.sq_add_sq_eq_mul_of_mod_eight","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#sq-add-sq-eq-mul-of-mod-eight","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a$ and $b$ be integers with $a^2 + b^2 \\equiv -3 \\pmod 8$. Then there are integers $k$ and $l$ such that $8l - 3$ lies in the set $P$ of positive primes congruent to $5$ modulo $8$, and $a^2 + b^2 = (8k + 1)(8l - 3)$.","title":"Sums of two squares congruent to \u22123 modulo 8"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-norm-beta-gamma-eq","kind":"node","lean":"MetaMathlibExt.c4Sq_norm_beta_gamma_eq","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-norm-beta-gamma-eq","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a$ be an integer-valued function on $C_4^2 = \\mathbb{Z}/4 \\times \\mathbb{Z}/4$, with associated integers $d_0, \\dots, d_7$ and Gaussian integers $\\beta$, $\\gamma$, and write $\\beta\\bar\\beta$, $\\gamma\\bar\\gamma$ for the norms of $\\beta$ and $\\gamma$. Then","title":"Norms of \u03b2 and \u03b3 as products of sums of two squares (C\u2084\u00b2)"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-group-det-congr-mod-two","kind":"node","lean":"MetaMathlibExt.c4Sq_groupDet_congr_mod_two","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The group determinant of "],["math","\\(C_4^2\\)"],["text"," modulo "],["math","\\(2\\)"]],"source":"The group determinant of $C_4^2$ modulo $2$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-group-det-congr-mod-two","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$. For $0 \\le i \\le 3$ let $b_i = a(i, 0) + a(i, 2) + a(i, 1) + a(i, 3)$, $c_i = a(i, 0) + a(i, 2) - a(i, 1) - a(i, 3)$ and $\\alpha_i = (a(i, 0) - a(i, 2)) + (a(i, 1) - a(i, 3))\\sqrt{-1} \\in \\mathbb{Z}[\\sqrt{-1}]$, and let $\\beta = (\\alpha_0 + \\alpha_2)^2 - (\\alpha_1 + \\alpha_3)^2$ and $\\gamma = (\\alpha_0 - \\alpha_2)^2 + (\\alpha_1 - \\alpha_3)^2$, with\u2026","title":"The group determinant of $C_4^2$ modulo $2$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-norm-beta-gamma-eq-sq-sub","kind":"node","lean":"MetaMathlibExt.c4Sq_norm_beta_gamma_eq_sq_sub","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-norm-beta-gamma-eq-sq-sub","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a$ be an integer-valued function on $C_4^2 = \\mathbb{Z}/4 \\times \\mathbb{Z}/4$, with associated integers $d_0, \\dots, d_7$ and Gaussian integers $\\beta$, $\\gamma$, and write $\\beta\\bar\\beta$, $\\gamma\\bar\\gamma$ for the norms of $\\beta$ and $\\gamma$. Then","title":"Norms of \u03b2 and \u03b3 as differences of squares (C\u2084\u00b2)"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-d-sum-sq-sq-sub-sq","kind":"node","lean":"MetaMathlibExt.c4SqD_sum_sq_sq_sub_sq","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Difference of the squared sums of squares of the "],["math","\\(d_i\\)"]],"source":"Difference of the squared sums of squares of the $d_i$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-d-sum-sq-sq-sub-sq","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$, and let $d_i = a(i, 0) - a(i, 2)$ and $d_{i+4} = a(i, 1) - a(i, 3)$ for $0 \\le i \\le 3$. Then","title":"Difference of the squared sums of squares of the $d_i$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-d-cross-terms-mod-four","kind":"node","lean":"MetaMathlibExt.c4SqD_cross_terms_mod_four","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cross terms of the "],["math","\\(d_i\\)"],["text"," modulo "],["math","\\(4\\)"]],"source":"Cross terms of the $d_i$ modulo $4$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-d-cross-terms-mod-four","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ have sums $b_i$, alternating sums $c_i$ and differences $d_0, \\dots, d_7$. Then the following five congruences hold modulo $4$: $2(d_0 d_2 + d_4 d_6 + d_1 d_3 + d_5 d_7) \\equiv b_0 b_2 + b_1 b_3 + c_0 c_2 + c_1 c_3$; $2(d_0 d_7 + d_2 d_5 + d_4 d_3 + d_6 d_1) \\equiv b_0 b_3 + b_2 b_1 - c_0 c_3 - c_2 c_1$;\u2026","title":"Cross terms of the $d_i$ modulo $4$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/circulant-det-four-mod-sixteen","kind":"node","lean":"MetaMathlibExt.circulantDetFour_mod_sixteen","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#circulant-det-four-mod-sixteen","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"For all integers $k$, $l$, $m$, $n$, the $4 \\times 4$ circulant determinant $D_4$ satisfies $D_4(2k + 1, 2l, 2m, 2n) \\equiv 8m + 1 \\pmod{16}$ and $D_4(2k, 2l + 1, 2m + 1, 2n + 1) \\equiv 8(k + l + n) - 3 \\pmod{16}$.","title":"The 4 \u00d7 4 circulant determinant modulo 16 at two parity patterns"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/circulant-det-four-two-power-cases","kind":"node","lean":"MetaMathlibExt.circulantDetFour_two_power_cases","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of "],["math","\\(2\\)"],["text"," dividing the "],["math","\\(C_4\\)"],["text"," group determinant at four parity patterns"]],"source":"Powers of $2$ dividing the $C_4$ group determinant at four parity patterns"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#circulant-det-four-two-power-cases","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $D(x_0, x_1, x_2, x_3)$ be the group determinant of $C_4$, that is the determinant of the $4 \\times 4$ circulant matrix with $(i, j)$ entry $x_{i - j}$ (indices mod $4$). For integers $k, l, m, n$ put $D_1 = D(2k, 2l, 2m, 2n)$, $D_2 = D(2k+1, 2l+1, 2m+1, 2n+1)$, $D_3 = D(2k, 2l+1, 2m, 2n+1)$, $D_4 = D(2k, 2l, 2m+1, 2n+1)$, $w_3 = (2k+2l+1)(2m+2n+1)$ and $w_4 = (2k+2m+1)(2l+2n+1)$, and write\u2026","title":"Powers of $2$ dividing the $C_4$ group determinant at four parity patterns"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/odd-mem-integer-group-determinants-c4-sq","kind":"node","lean":"MetaMathlibExt.odd_mem_integerGroupDeterminants_c4Sq","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd integer group determinants of "],["math","\\(C_4 \\times C_4\\)"]],"source":"Odd integer group determinants of $C_4 \\times C_4$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#odd-mem-integer-group-determinants-c4-sq","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $C_4$ be the cyclic group of order $4$, $S(C_4^2)$ the set of integer group determinants of $C_4 \\times C_4$ and $P$ the set of positive primes $p \\equiv -3 \\pmod 8$, and let $A$ be the set of products $(8k-3)(8l-3)(8m-3)(8n-3)$ with $k, l, m, n \\in \\mathbb{Z}$, $8l-3, 8m-3, 8n-3 \\in P$ and $k + l \\not\\equiv m + n \\pmod 2$. Every odd $z \\in S(C_4^2)$ is of the form $16m + 1$ for some integer $m$ or lies in $A$.","title":"Odd integer group determinants of $C_4 \\times C_4$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-circulant-det-four-product-classes","kind":"node","lean":"MetaMathlibExt.c4Sq_circulantDetFour_product_classes","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Factorization shapes of "],["math","\\(D_4(b) D_4(c)\\)"]],"source":"Factorization shapes of $D_4(b) D_4(c)$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-circulant-det-four-product-classes","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ have sums $b_i$ and alternating sums $c_i$, let $D_4$ be the $4 \\times 4$ circulant determinant, and suppose $b_0 + b_2 \\not\\equiv b_1 + b_3 \\pmod 2$. Put $x = b_0 b_2 + b_1 b_3$, $y = c_0 c_2 + c_1 c_3$ and $D = D_4(b) D_4(c)$. If $(x, y) \\equiv (0, 0)$ or $(2, 2) \\pmod 4$, then $D = (8k + 1)(8l + 1)$ for some integers $k \\equiv l \\pmod 2$. If\u2026","title":"Factorization shapes of $D_4(b) D_4(c)$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-circulant-det-four-product-mod-six","kind":"node","lean":"MetaMathlibExt.c4Sq_circulantDetFour_product_mod_sixteen","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(D_4(b) D_4(c)\\)"],["text"," modulo "],["math","\\(16\\)"]],"source":"$D_4(b) D_4(c)$ modulo $16$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-circulant-det-four-product-mod-six","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ have sums $b_i$ and alternating sums $c_i$, and let $D_4$ be the $4 \\times 4$ circulant determinant. If $b_0 + b_2 \\not\\equiv b_1 + b_3 \\pmod 2$, then $D_4(b) D_4(c) \\equiv 1 - 4(b_0 b_2 + b_1 b_3 + c_0 c_2 + c_1 c_3) \\pmod{16}$.","title":"$D_4(b) D_4(c)$ modulo $16$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-circulant-det-four-product-prime-f","kind":"node","lean":"MetaMathlibExt.c4Sq_circulantDetFour_product_prime_factor","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prime factor "],["math","\\(8l - 3\\)"],["text"," of "],["math","\\(D_4(b) D_4(c)\\)"]],"source":"A prime factor $8l - 3$ of $D_4(b) D_4(c)$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-circulant-det-four-product-prime-f","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ have sums $b_i$, alternating sums $c_i$ and Gaussian integer $\\beta$, let $D_4$ be the $4 \\times 4$ circulant determinant, and let $P$ be the set of positive primes $p \\equiv 5 \\pmod 8$. Suppose $b_0 + b_2 \\not\\equiv b_1 + b_3 \\pmod 2$ and that the norm $\\beta \\bar\\beta \\equiv -3 \\pmod 8$, and put $Q = b_0 b_2 + b_1 b_3 + c_0 c_2 + c_1 c_3$. If\u2026","title":"A prime factor $8l - 3$ of $D_4(b) D_4(c)$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-norm-beta-sub-norm-gamma-mod-sixte","kind":"node","lean":"MetaMathlibExt.c4Sq_norm_beta_sub_norm_gamma_mod_sixteen","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-norm-beta-sub-norm-gamma-mod-sixte","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a$ be an integer-valued function on $C_4^2 = \\mathbb{Z}/4 \\times \\mathbb{Z}/4$, with associated integers $b_i$, $c_i$ $(0 \\le i \\le 3)$ and Gaussian integers $\\beta$, $\\gamma$, and write $\\beta\\bar\\beta$, $\\gamma\\bar\\gamma$ for the norms of $\\beta$ and $\\gamma$. If $b_0 + b_2 \\not\\equiv b_1 + b_3 \\pmod 2$, then $\\beta\\bar\\beta - \\gamma\\bar\\gamma \\equiv 4(b_0b_2 + b_1b_3 + c_0c_2 + c_1c_3) \\pmod{16}$.","title":"The difference of the norms of \u03b2 and \u03b3 modulo 16 (C\u2084\u00b2)"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-norm-beta-gamma-prime-factors","kind":"node","lean":"MetaMathlibExt.c4Sq_norm_beta_gamma_prime_factors","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-norm-beta-gamma-prime-factors","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a$ be an integer-valued function on $C_4^2 = \\mathbb{Z}/4 \\times \\mathbb{Z}/4$, with associated integers $b_i$, $c_i$ $(0 \\le i \\le 3)$ and Gaussian integers $\\beta$, $\\gamma$, write $\\beta\\bar\\beta$, $\\gamma\\bar\\gamma$ for the norms of $\\beta$ and $\\gamma$, let $P$ be the set of primes $p \\equiv -3 \\pmod 8$, and put $Q = b_0b_2 + b_1b_3 + c_0c_2 + c_1c_3$. Suppose that $b_0 + b_2 \\not\\equiv b_1 + b_3 \\pmod 2$\u2026","title":"Two prime factors \u2261 \u22123 mod 8 of \u03b2\u03b2\u0304\u03b3\u03b3\u0304 (C\u2084\u00b2)"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/even-mem-integer-group-determinants-c4-s","kind":"node","lean":"MetaMathlibExt.even_mem_integerGroupDeterminants_c4Sq","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even integer group determinants of "],["math","\\(C_4 \\times C_4\\)"]],"source":"Even integer group determinants of $C_4 \\times C_4$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#even-mem-integer-group-determinants-c4-s","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $C_4$ be the cyclic group of order $4$, $S(C_4^2)$ the set of integer group determinants of $C_4 \\times C_4$ and $P$ the set of positive primes $p \\equiv -3 \\pmod 8$, and let $z \\in S(C_4^2)$. If $z$ is even, then $2^{15} \\mid z$. If $z = 2^{15}(2t + 1)$ for some integer $t$, then $z = 2^{15} p (2m + 1)$ for some $p \\in P$ and some integer $m$.","title":"Even integer group determinants of $C_4 \\times C_4$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-circulant-det-four-product-two-pow","kind":"node","lean":"MetaMathlibExt.c4Sq_circulantDetFour_product_two_power","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The power of "],["math","\\(2\\)"],["text"," in "],["math","\\(D_4(b) D_4(c)\\)"]],"source":"The power of $2$ in $D_4(b) D_4(c)$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-circulant-det-four-product-two-pow","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ have sums $b_i$ and alternating sums $c_i$, let $D_4$ be the $4 \\times 4$ circulant determinant, put $D = D_4(b) D_4(c)$, and say that $D \\in 2^e \\mathbb{Z}_{\\mathrm{odd}}$ when $D = 2^e (2t + 1)$ for some integer $t$. If every $b_i$ is even, then $D \\in 2^8 \\mathbb{Z}_{\\mathrm{odd}}$ when\u2026","title":"The power of $2$ in $D_4(b) D_4(c)$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-norm-beta-gamma-two-power","kind":"node","lean":"MetaMathlibExt.c4Sq_norm_beta_gamma_two_power","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The power of "],["math","\\(2\\)"],["text"," in "],["math","\\(\\beta\\bar\\beta\\gamma\\bar\\gamma\\)"]],"source":"The power of $2$ in $\\beta\\bar\\beta\\gamma\\bar\\gamma$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-norm-beta-gamma-two-power","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ have sums $b_i$, alternating sums $c_i$, differences $d_0, \\dots, d_7$ and Gaussian integers $\\beta$, $\\gamma$, and put $N = \\beta \\bar\\beta \\gamma \\bar\\gamma$, the product of the norms of $\\beta$ and $\\gamma$. If $b_0 + b_2$ and $b_1 + b_3$ are both even, then $N = 2^4 (2t + 1)$ for some integer $t$ when\u2026","title":"The power of $2$ in $\\beta\\bar\\beta\\gamma\\bar\\gamma$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-cross-sum-mod-four-of-two-power","kind":"node","lean":"MetaMathlibExt.c4Sq_cross_sum_mod_four_of_two_power","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two conditions forcing "],["math","\\(b_0 b_2 + b_1 b_3 + c_0 c_2 + c_1 c_3 \\equiv 2 \\pmod 4\\)"]],"source":"Two conditions forcing $b_0 b_2 + b_1 b_3 + c_0 c_2 + c_1 c_3 \\equiv 2 \\pmod 4$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-cross-sum-mod-four-of-two-power","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$ have sums $b_i$, alternating sums $c_i$ and Gaussian integers $\\beta$, $\\gamma$; let $D_4$ be the $4 \\times 4$ circulant determinant, and put $D = D_4(b) D_4(c)$, $N = \\beta \\bar\\beta \\gamma \\bar\\gamma$ and $Q = b_0 b_2 + b_1 b_3 + c_0 c_2 + c_1 c_3$. If every $b_i$ is odd and $D = 2^{11} (2t + 1)$ for some integer $t$, then $Q \\equiv 2 \\pmod 4$. If\u2026","title":"Two conditions forcing $b_0 b_2 + b_1 b_3 + c_0 c_2 + c_1 c_3 \\equiv 2 \\pmod 4$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-norm-beta-gamma-prime-factor","kind":"node","lean":"MetaMathlibExt.c4Sq_norm_beta_gamma_prime_factor","parent":1803,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-norm-beta-gamma-prime-factor","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a$ be an integer-valued function on $C_4^2 = \\mathbb{Z}/4 \\times \\mathbb{Z}/4$, with associated integers $b_i$, $c_i$ $(0 \\le i \\le 3)$ and Gaussian integers $\\beta$, $\\gamma$, write $\\beta\\bar\\beta$ and $\\gamma\\bar\\gamma$ for the norms of $\\beta$ and $\\gamma$, and let $P$ be the set of primes $p \\equiv -3 \\pmod 8$. Suppose that $b_0 + b_2 \\equiv b_1 + b_3 \\equiv 0 \\pmod 2$, that\u2026","title":"A prime factor p \u2261 \u22123 mod 8 of \u03b2\u03b2\u0304\u03b3\u03b3\u0304 in the even case (C\u2084\u00b2)"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-group-det-factors-prime-factor","kind":"node","lean":"MetaMathlibExt.c4Sq_groupDet_factors_prime_factor","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prime factor "],["math","\\(p \\equiv 5 \\pmod 8\\)"],["text"," of "],["math","\\(D_4(b) D_4(c) \\beta\\bar\\beta \\gamma\\bar\\gamma\\)"]],"source":"A prime factor $p \\equiv 5 \\pmod 8$ of $D_4(b) D_4(c) \\beta\\bar\\beta \\gamma\\bar\\gamma$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-group-det-factors-prime-factor","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $a \\colon \\mathbb{Z}/4 \\times \\mathbb{Z}/4 \\to \\mathbb{Z}$. For $0 \\le i \\le 3$ let $b_i = a(i, 0) + a(i, 2) + a(i, 1) + a(i, 3)$, $c_i = a(i, 0) + a(i, 2) - a(i, 1) - a(i, 3)$ and $\\alpha_i = (a(i, 0) - a(i, 2)) + (a(i, 1) - a(i, 3))\\sqrt{-1} \\in \\mathbb{Z}[\\sqrt{-1}]$, and let $\\beta = (\\alpha_0 + \\alpha_2)^2 - (\\alpha_1 + \\alpha_3)^2$ and $\\gamma = (\\alpha_0 - \\alpha_2)^2 + (\\alpha_1 - \\alpha_3)^2$, with\u2026","title":"A prime factor $p \\equiv 5 \\pmod 8$ of $D_4(b) D_4(c) \\beta\\bar\\beta \\gamma\\bar\\gamma$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-integer-group-determinants-basic-m","kind":"node","lean":"MetaMathlibExt.c4Sq_integerGroupDeterminants_basic_members","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Basic integer group determinants of "],["math","\\(C_4^2\\)"]],"source":"Basic integer group determinants of $C_4^2$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-integer-group-determinants-basic-m","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"For every integer $m$, the integers $16m + 1$, $2^{16} (4m + 1)$, $2^{16} (4m - 1)$ and $2^{16} \\cdot 2m$ all lie in the set $S(C_4^2)$ of integer group determinants of $C_4^2 = \\mathbb{Z}/4 \\times \\mathbb{Z}/4$.","title":"Basic integer group determinants of $C_4^2$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-integer-group-determinants-four-fa","kind":"node","lean":"MetaMathlibExt.c4Sq_integerGroupDeterminants_four_factor_members","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Four-factor integer group determinants of "],["math","\\(C_4^2\\)"]],"source":"Four-factor integer group determinants of $C_4^2$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-integer-group-determinants-four-fa","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $P$ be the set of positive primes $p \\equiv 5 \\pmod 8$, and let $k, l, m, n$ be integers such that $16l - 3$, $16l + 5$, $16m - 3$, $16m + 5$, $16n - 3$ and $16n + 5$ all lie in $P$. Then the four integers $(16k - 3)(16l + 5)(16m - 3)(16n - 3)$, $(16k - 3)(16l + 5)(16m + 5)(16n + 5)$, $(16k + 5)(16l - 3)(16m - 3)(16n - 3)$ and $(16k + 5)(16l - 3)(16m + 5)(16n + 5)$ all lie in the set $S(C_4^2)$ of integer group\u2026","title":"Four-factor integer group determinants of $C_4^2$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/c4-sq-integer-group-determinants-two-pow","kind":"node","lean":"MetaMathlibExt.c4Sq_integerGroupDeterminants_two_pow_fifteen","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer group determinants of "],["math","\\(C_4^2\\)"],["text"," of the form "],["math","\\(2^{15} p (2m + 1)\\)"]],"source":"Integer group determinants of $C_4^2$ of the form $2^{15} p (2m + 1)$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#c4-sq-integer-group-determinants-two-pow","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $P$ be the set of positive primes $p \\equiv 5 \\pmod 8$. For every integer $m$ and every $p \\in P$, the integer $2^{15} p (2m + 1)$ lies in the set $S(C_4^2)$ of integer group determinants of $C_4^2 = \\mathbb{Z}/4 \\times \\mathbb{Z}/4$.","title":"Integer group determinants of $C_4^2$ of the form $2^{15} p (2m + 1)$"},{"id":"GroupTheory/characters-of-finite-groups/integer-group-determinants/integer-group-determinants-c4-sq","kind":"node","lean":"MetaMathlibExt.integerGroupDeterminants_c4Sq","parent":1803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer group determinants of "],["math","\\(C_4 \\times C_4\\)"]],"source":"Integer group determinants of $C_4 \\times C_4$"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/integer-group-determinants/index.html#integer-group-determinants-c4-sq","scope":"GroupTheory/characters-of-finite-groups/integer-group-determinants","status":"can_prove","summary":"Let $C_4$ be the cyclic group of order $4$, let $S(C_4^2)$ be the set of integer group determinants of $C_4 \\times C_4$, let $P$ be the set of positive primes $p \\equiv -3 \\pmod 8$, and let $A$ be the set of products $(8k-3)(8l-3)(8m-3)(8n-3)$ with $k, l, m, n \\in \\mathbb{Z}$, $8l-3, 8m-3, 8n-3 \\in P$ and $k + l \\not\\equiv m + n \\pmod 2$. Then $A$ is a proper subset of $\\{16m - 7 \\mid m \\in \\mathbb{Z}\\}$, and","title":"Integer group determinants of $C_4 \\times C_4$"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions.json","id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","title":"Irreducible characters and class functions"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/is-class-function","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.IsClassFunction","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#is-class-function","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a group and $k$ a set. A function $f : G \\to k$ is a class function when it is constant on conjugacy classes, that is, $f(h g h^{-1}) = f(g)$ for all $g, h \\in G$.","title":"Class functions on a group"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/class-functions","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.classFunctions","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#class-functions","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a group and $k$ a field. The space of class functions $G \\to k$ is the $k$-linear subspace of the vector space of all functions $G \\to k$, under pointwise addition and scalar multiplication, consisting of the class functions.","title":"The space of class functions"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/character-is-class-function","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.character_isClassFunction","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#character-is-class-function","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a group and $k$ a field. For every finite-dimensional representation $V$ of $G$ over $k$, the character of $V$, the function sending $g \\in G$ to the trace of the linear map by which $g$ acts on $V$, is a class function on $G$.","title":"Characters are class functions"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/eq-of-is-conj","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.IsClassFunction.eq_of_isConj","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#eq-of-is-conj","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a group and let $f : G \\to k$ be a class function with values in a set $k$. If $x, y \\in G$ are conjugate, then $f(x) = f(y)$.","title":"Class functions agree on conjugate elements"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/is-class-function-iff-factor-conj-classe","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.isClassFunction_iff_factor_conjClasses","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#is-class-function-iff-factor-conj-classe","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a group and $k$ a set, and write $[x]$ for the conjugacy class of $x \\in G$. A function $f : G \\to k$ is a class function if and only if there is a function $q$ from the set of conjugacy classes of $G$ to $k$ such that $f(x) = q([x])$ for all $x \\in G$.","title":"Class functions factor through conjugacy classes"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/class-functions-equiv-conj-classes","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.classFunctionsEquivConjClasses","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#class-functions-equiv-conj-classes","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a group and $k$ a field, and write $[x]$ for the conjugacy class of $x \\in G$. This is the $k$-linear isomorphism from the space of class functions $G \\to k$ to the vector space of all functions from the set of conjugacy classes of $G$ to $k$ that sends a class function $f$ to the function $[x] \\mapsto f(x)$; its inverse sends a function $q$ on conjugacy classes to the class function $x \\mapsto q([x])$.","title":"Class functions as functions on conjugacy classes"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/class-functions-equiv-conj-classes-apply","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.classFunctionsEquivConjClasses_apply_mk","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#class-functions-equiv-conj-classes-apply","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a group and $k$ a field. For every class function $f : G \\to k$ and every $x \\in G$, the function on conjugacy classes that corresponds to $f$ under the linear isomorphism between class functions and functions on conjugacy classes takes the value $f(x)$ at the conjugacy class of $x$.","title":"Evaluating a class function on conjugacy classes"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/class-functions-equiv-conj-classes-symm","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.classFunctionsEquivConjClasses_symm_apply","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#class-functions-equiv-conj-classes-symm","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a group and $k$ a field. For every function $q$ from the set of conjugacy classes of $G$ to $k$ and every $x \\in G$, the class function that corresponds to $q$ under the inverse of the linear isomorphism between class functions and functions on conjugacy classes takes at $x$ the value of $q$ at the conjugacy class of $x$.","title":"Lifting a function on conjugacy classes"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/finrank-class-functions-eq-card-conj-cla","kind":"node","lean":"MathlibExt.GroupTheory.ClassFunctionWanted.finrank_classFunctions_eq_card_conjClasses","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#finrank-class-functions-eq-card-conj-cla","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a finite group and $k$ a field. The space of class functions $G \\to k$ is a $k$-vector space whose dimension equals the number of conjugacy classes of $G$.","title":"Dimension of the space of class functions"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/simple-of-is-irreducible","kind":"node","lean":"FDRep.simple_of_isIrreducible","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#simple-of-is-irreducible","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $k$ be a field, $G$ a group and $\\rho$ an irreducible representation of $G$ on a finite-dimensional $k$-vector space $V$. Then $\\rho$, viewed as an object of the category of finite-dimensional representations of $G$ over $k$, is a simple object.","title":"Irreducible representations are simple objects"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/linear-independent-character","kind":"node","lean":"FDRep.linearIndependent_character","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#linear-independent-character","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a finite group and let $(V_i)_{i \\in I}$ be a family of simple finite-dimensional complex representations of $G$ such that $V_i \\cong V_j$ implies $i = j$. Then the characters of the $V_i$ are linearly independent over $\\mathbb{C}$.","title":"Characters of nonisomorphic simple representations are linearly independent"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/eq-zero-of-is-class-function-of-orthogon","kind":"node","lean":"FDRep.eq_zero_of_isClassFunction_of_orthogonal","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#eq-zero-of-is-class-function-of-orthogon","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a finite group and let $f : G \\to \\mathbb{C}$ satisfy $f(hgh^{-1}) = f(g)$ for all $g, h \\in G$. Suppose that for every simple finite-dimensional complex representation $W$ of $G$, with character $\\chi_W$,","title":"A class function orthogonal to all irreducible characters vanishes"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/linear-independent-irr-characters","kind":"node","lean":"MathlibExt.GroupTheory.IrreducibleCharacterBasisWanted.linearIndependent_irrCharacters","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#linear-independent-irr-characters","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a finite group. The irreducible complex characters of $G$, that is, the functions $G \\to \\mathbb{C}$ that are the character of some simple finite-dimensional complex representation of $G$, are linearly independent over $\\mathbb{C}$ in the space of functions $G \\to \\mathbb{C}$.","title":"Irreducible characters are linearly independent"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/mem-span-irr-characters-of-is-class-func","kind":"node","lean":"MathlibExt.GroupTheory.IrreducibleCharacterBasisWanted.mem_span_irrCharacters_of_isClassFunction","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#mem-span-irr-characters-of-is-class-func","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a finite group and let $f : G \\to \\mathbb{C}$ satisfy $f(hgh^{-1}) = f(g)$ for all $g, h \\in G$. Then $f$ is a $\\mathbb{C}$-linear combination of irreducible complex characters of $G$, that is, of functions $G \\to \\mathbb{C}$ that are the character of some simple finite-dimensional complex representation of $G$.","title":"Irreducible characters span the class functions"},{"id":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/card-irr-characters-eq-card-conj-classes","kind":"node","lean":"MathlibExt.GroupTheory.IrreducibleCharacterBasisWanted.card_irrCharacters_eq_card_conjClasses","parent":1845,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions/index.html#card-irr-characters-eq-card-conj-classes","scope":"GroupTheory/characters-of-finite-groups/irreducible-characters-and-class-functions","status":"fully_proved","summary":"Let $G$ be a finite group. The number of irreducible complex characters of $G$, that is, of functions $G \\to \\mathbb{C}$ that are the character of some simple finite-dimensional complex representation of $G$, equals the number of conjugacy classes of $G$.","title":"Number of irreducible characters"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/mckay-conjecture.json","id":"GroupTheory/characters-of-finite-groups/mckay-conjecture","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/mckay-conjecture/index.html","scope":"GroupTheory/characters-of-finite-groups/mckay-conjecture","status":"can_prove","title":"McKay conjecture"},{"id":"GroupTheory/characters-of-finite-groups/mckay-conjecture/p-prime-characters","kind":"node","lean":"MetaMathlibExt.pPrimeCharacters","parent":1861,"reading_title":{"format":"inline-parts-v1","parts":[["text","Irreducible characters of "],["math","\\(p'\\)"],["text","-degree"]],"source":"Irreducible characters of $p'$-degree"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/mckay-conjecture/index.html#p-prime-characters","scope":"GroupTheory/characters-of-finite-groups/mckay-conjecture","status":"fully_proved","summary":"For a group $G$ and a natural number $p$, $\\mathrm{Irr}_{p'}(G)$ is the set of functions $G \\to \\mathbb{C}$ that are characters of simple finite-dimensional complex representations of $G$ whose dimension is not divisible by $p$.","title":"Irreducible characters of $p'$-degree"},{"id":"GroupTheory/characters-of-finite-groups/mckay-conjecture/mckay-irr-equicardinal","kind":"node","lean":"MetaMathlibExt.mckay_irr_equicardinal","parent":1861,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/mckay-conjecture/index.html#mckay-irr-equicardinal","scope":"GroupTheory/characters-of-finite-groups/mckay-conjecture","status":"can_prove","summary":"For every finite group $G$, every prime $p$ and every Sylow $p$-subgroup $P$ of $G$, with normalizer $N_G(P)$ in $G$, the sets $\\mathrm{Irr}_{p'}(G)$ and $\\mathrm{Irr}_{p'}(N_G(P))$ of irreducible complex characters of degree not divisible by $p$ have the same cardinality.","title":"McKay conjecture (Cabanes\u2013Sp\u00e4th theorem)"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/permutation-characters.json","id":"GroupTheory/characters-of-finite-groups/permutation-characters","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"can_prove","title":"Permutation characters"},{"id":"GroupTheory/characters-of-finite-groups/permutation-characters/perm-char","kind":"node","lean":"MathlibExt.GroupTheory.PermutationCharacterGenerationKourovka12Problem79Wanted.permChar","parent":1864,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html#perm-char","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"fully_proved","summary":"For a subgroup $H$ of a group $G$, the permutation character $1_H^G : G \\to \\mathbb{C}$ sends $g \\in G$ to the number of left cosets $xH$ of $H$ with $gxH = xH$, and to $0$ when $g$ fixes infinitely many cosets. For finite $G$ and the trivial subgroup $H = 1$ this is the regular character $1^G$.","title":"Complex permutation character on the cosets of a subgroup"},{"id":"GroupTheory/characters-of-finite-groups/permutation-characters/strunkov-function","kind":"node","lean":"MathlibExt.GroupTheory.PermutationCharacterGenerationKourovka12Problem79Wanted.strunkovFunction","parent":1864,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html#strunkov-function","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"fully_proved","summary":"For elements $a, b$ of a group $G$, Strunkov's function $\\varphi : G \\to \\mathbb{C}$ is","title":"Strunkov's class function"},{"id":"GroupTheory/characters-of-finite-groups/permutation-characters/is-character","kind":"node","lean":"MathlibExt.GroupTheory.PermutationCharacterGenerationKourovka12Problem79Wanted.IsCharacter","parent":1864,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html#is-character","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"fully_proved","summary":"A function $\\varphi : G \\to \\mathbb{C}$ on a group $G$ is a character of $G$ when there are a finite-dimensional complex vector space $V$ and a representation $\\rho$ of $G$ on $V$ whose character $g \\mapsto \\operatorname{tr} \\rho(g)$ equals $\\varphi$.","title":"Characters of a group"},{"id":"GroupTheory/characters-of-finite-groups/permutation-characters/strunkov-character-implies-generation","kind":"node","lean":"MathlibExt.GroupTheory.PermutationCharacterGenerationKourovka12Problem79Wanted.strunkovCharacterImpliesGeneration","parent":1864,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html#strunkov-character-implies-generation","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"fully_proved","summary":"The assertion that for every finite group $G$ and all $a, b \\in G$, if Strunkov's class function $1^G - 1_{\\langle a \\rangle}^G - 1_{\\langle b \\rangle}^G - 1_{\\langle ab \\rangle}^G + 2$ is a character of $G$, then $a$ and $b$ generate $G$. Here $1_H^G$ is the permutation character of $G$ on the cosets of a subgroup $H$, $1^G = 1_1^G$, $\\langle x \\rangle$ is the cyclic subgroup generated by $x$, and $2$ is the\u2026","title":"Strunkov's two-generation criterion (Kourovka 12.79)"},{"id":"GroupTheory/characters-of-finite-groups/permutation-characters/strunkov-character-implies-generation-re","kind":"node","lean":"MathlibExt.GroupTheory.PermutationCharacterGenerationKourovka12Problem79Wanted.strunkovCharacterImpliesGeneration_refuted","parent":1864,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html#strunkov-character-implies-generation-re","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"can_prove","summary":"It is not true that for every finite group $G$ and all $a, b \\in G$ such that Strunkov's class function $1^G - 1_{\\langle a \\rangle}^G - 1_{\\langle b \\rangle}^G - 1_{\\langle ab \\rangle}^G + 2$ is a character of $G$, the elements $a$ and $b$ generate $G$. Here $1_H^G$ is the permutation character of $G$ on the cosets of a subgroup $H$, $1^G = 1_1^G$, $\\langle x \\rangle$ is the cyclic subgroup generated by $x$, and\u2026","title":"Strunkov's two-generation criterion fails (Kourovka 12.79)"},{"id":"GroupTheory/characters-of-finite-groups/permutation-characters/perm-char-2","kind":"node","lean":"MathlibExt.GroupTheory.PermutationCharactersIndependentKourovka11Problem9Wanted.permChar","parent":1864,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html#perm-char-2","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"fully_proved","summary":"For a subgroup $H$ of a group $G$, the permutation character $1_H^G : G \\to \\mathbb{Q}$ sends $g \\in G$ to the number of left cosets $xH$ of $H$ with $gxH = xH$, and to $0$ when $g$ fixes infinitely many cosets.","title":"Rational permutation character on the cosets of a subgroup"},{"id":"GroupTheory/characters-of-finite-groups/permutation-characters/insoluble-independent-permutation-charac","kind":"node","lean":"MathlibExt.GroupTheory.PermutationCharactersIndependentKourovka11Problem9Wanted.insolubleIndependentPermutationCharacters","parent":1864,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html#insoluble-independent-permutation-charac","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"fully_proved","summary":"The assertion that there is a finite non-soluble group $G$ such that, choosing a representative $H$ of each conjugacy class of subgroups of $G$, the rational-valued permutation characters $1_H^G$ of $G$ on the left cosets of these subgroups $H$ are linearly independent over $\\mathbb{Q}$.","title":"Finite insoluble groups with independent permutation characters (Kourovka 11.9)"},{"id":"GroupTheory/characters-of-finite-groups/permutation-characters/insoluble-independent-permutation-charac-2","kind":"node","lean":"MathlibExt.GroupTheory.PermutationCharactersIndependentKourovka11Problem9Wanted.insolubleIndependentPermutationCharacters_refuted","parent":1864,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/permutation-characters/index.html#insoluble-independent-permutation-charac-2","scope":"GroupTheory/characters-of-finite-groups/permutation-characters","status":"can_prove","summary":"There is no finite non-soluble group $G$ such that, choosing a representative $H$ of each conjugacy class of subgroups of $G$, the rational-valued permutation characters $1_H^G$ of $G$ on the left cosets of these subgroups $H$ are linearly independent over $\\mathbb{Q}$.","title":"No insoluble group has independent permutation characters (Kourovka 11.9)"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/rational-groups.json","id":"GroupTheory/characters-of-finite-groups/rational-groups","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/rational-groups/index.html","scope":"GroupTheory/characters-of-finite-groups/rational-groups","status":"can_prove","title":"Rational groups"},{"id":"GroupTheory/characters-of-finite-groups/rational-groups/is-rational-group","kind":"node","lean":"MathlibExt.GroupTheory.RationalGroupSylowTwoKourovka16Problem36Wanted.IsRationalGroup","parent":1873,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/rational-groups/index.html#is-rational-group","scope":"GroupTheory/characters-of-finite-groups/rational-groups","status":"fully_proved","summary":"A group $G$ is rational if every element $g \\in G$ is conjugate to $g^k$ for every natural number $k$ coprime to the order of $g$, where an element of infinite order has order $0$, so that only $k = 1$ is coprime to it. For finite groups this is the classical equivalent form of the condition that all ordinary characters of $G$ are rational-valued.","title":"Rational groups, conjugacy form"},{"id":"GroupTheory/characters-of-finite-groups/rational-groups/sylow-two-of-rational-is-rational","kind":"node","lean":"MathlibExt.GroupTheory.RationalGroupSylowTwoKourovka16Problem36Wanted.sylowTwoOfRationalIsRational","parent":1873,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/rational-groups/index.html#sylow-two-of-rational-is-rational","scope":"GroupTheory/characters-of-finite-groups/rational-groups","status":"fully_proved","summary":"The conjecture that for every finite rational group $G$, every Sylow $2$-subgroup of $G$, viewed as a group in its own right, is rational. Here a group is rational if every element $g$ is conjugate to $g^k$ for every natural number $k$ coprime to the order of $g$.","title":"Darafsheh's question on Sylow 2-subgroups of rational groups (Kourovka 16.36)"},{"id":"GroupTheory/characters-of-finite-groups/rational-groups/sylow-two-of-rational-is-rational-refute","kind":"node","lean":"MathlibExt.GroupTheory.RationalGroupSylowTwoKourovka16Problem36Wanted.sylowTwoOfRationalIsRational_refuted","parent":1873,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/rational-groups/index.html#sylow-two-of-rational-is-rational-refute","scope":"GroupTheory/characters-of-finite-groups/rational-groups","status":"can_prove","summary":"There is a finite rational group with a Sylow $2$-subgroup that is not rational as a group in its own right. Here a group is rational if every element $g$ is conjugate to $g^k$ for every natural number $k$ coprime to the order of $g$.","title":"Sylow 2-subgroups of rational groups need not be rational (Kourovka 16.36)"},{"id":"GroupTheory/characters-of-finite-groups/rational-groups/is-rational-group-2","kind":"node","lean":"MathlibExt.GroupTheory.SylowTwoSymmetricRationalKourovka15Problem25Wanted.IsRationalGroup","parent":1873,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/rational-groups/index.html#is-rational-group-2","scope":"GroupTheory/characters-of-finite-groups/rational-groups","status":"fully_proved","summary":"A group $G$ is rational when every element $g \\in G$ is conjugate in $G$ to $g^k$ for every natural number $k$ coprime to the order of $g$ (an element of infinite order counts as having order $0$, so for it only $k = 1$ qualifies).","title":"Rational group (conjugacy form, Kourovka 15.25)"},{"id":"GroupTheory/characters-of-finite-groups/rational-groups/sylow-two-subgroup-symmetric-rational","kind":"node","lean":"MathlibExt.GroupTheory.SylowTwoSymmetricRationalKourovka15Problem25Wanted.sylowTwoSubgroupSymmetricRational","parent":1873,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/rational-groups/index.html#sylow-two-subgroup-symmetric-rational","scope":"GroupTheory/characters-of-finite-groups/rational-groups","status":"fully_proved","summary":"The statement that for every natural number $n$, every Sylow $2$-subgroup $P$ of the symmetric group $S_{2^n}$ on $2^n$ points is rational as a group in its own right: every $g \\in P$ is conjugate within $P$ to $g^k$ for every natural number $k$ coprime to the order of $g$.","title":"Kourovka Problem 15.25 (statement)"},{"id":"GroupTheory/characters-of-finite-groups/rational-groups/sylow-two-subgroup-symmetric-rational-ho","kind":"node","lean":"MathlibExt.GroupTheory.SylowTwoSymmetricRationalKourovka15Problem25Wanted.sylowTwoSubgroupSymmetricRational_holds","parent":1873,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sylow 2-subgroups of the symmetric groups of degree "],["math","\\(2^n\\)"],["text"," are rational (Kourovka 15.25)"]],"source":"Sylow 2-subgroups of the symmetric groups of degree $2^n$ are rational (Kourovka 15.25)"},"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/rational-groups/index.html#sylow-two-subgroup-symmetric-rational-ho","scope":"GroupTheory/characters-of-finite-groups/rational-groups","status":"can_prove","summary":"For every natural number $n$, every Sylow $2$-subgroup $P$ of the symmetric group $S_{2^n}$ on $2^n$ points is rational as a group in its own right: every $g \\in P$ is conjugate within $P$ to $g^k$ for every natural number $k$ coprime to the order of $g$.","title":"Sylow 2-subgroups of the symmetric groups of degree $2^n$ are rational (Kourovka 15.25)"},{"data_url":"scopes/GroupTheory/characters-of-finite-groups/simply-reducible-groups.json","id":"GroupTheory/characters-of-finite-groups/simply-reducible-groups","kind":"scope","navigation":"module","parent":1662,"result":false,"route":"../roadmap/GroupTheory/characters-of-finite-groups/simply-reducible-groups/index.html","scope":"GroupTheory/characters-of-finite-groups/simply-reducible-groups","status":"can_prove","title":"Simply reducible groups"},{"id":"GroupTheory/characters-of-finite-groups/simply-reducible-groups/is-simply-reducible","kind":"node","lean":"MathlibExt.GroupTheory.SimplyReducibleSolubleKourovka11Problem94Wanted.IsSimplyReducible","parent":1880,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/simply-reducible-groups/index.html#is-simply-reducible","scope":"GroupTheory/characters-of-finite-groups/simply-reducible-groups","status":"fully_proved","summary":"A group $G$ is simply reducible when the character of every irreducible finite-dimensional complex representation of $G$ is real-valued, and for all irreducible finite-dimensional complex representations $U$, $V$ and $W$ of $G$, the space $\\mathrm{Hom}_G(U, V \\otimes W)$ of $G$-equivariant linear maps from $U$ to the tensor product $V \\otimes W$ has complex dimension at most $1$.","title":"Simply reducible group"},{"id":"GroupTheory/characters-of-finite-groups/simply-reducible-groups/simply-reducible-soluble","kind":"node","lean":"MathlibExt.GroupTheory.SimplyReducibleSolubleKourovka11Problem94Wanted.simplyReducibleSoluble","parent":1880,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/simply-reducible-groups/index.html#simply-reducible-soluble","scope":"GroupTheory/characters-of-finite-groups/simply-reducible-groups","status":"fully_proved","summary":"The statement, asked by S. P. Strunkov as Kourovka Problem 11.94, that every finite simply reducible group is soluble.","title":"Solubility of finite simply reducible groups (Kourovka 11.94)"},{"id":"GroupTheory/characters-of-finite-groups/simply-reducible-groups/simply-reducible-soluble-holds","kind":"node","lean":"MathlibExt.GroupTheory.SimplyReducibleSolubleKourovka11Problem94Wanted.simplyReducibleSoluble_holds","parent":1880,"result":true,"route":"../roadmap/GroupTheory/characters-of-finite-groups/simply-reducible-groups/index.html#simply-reducible-soluble-holds","scope":"GroupTheory/characters-of-finite-groups/simply-reducible-groups","status":"can_prove","summary":"Every finite simply reducible group is soluble; that is, the statement of Kourovka Problem 11.94 holds.","title":"Finite simply reducible groups are soluble"},{"data_url":"scopes/GroupTheory/finite-simple-groups.json","id":"GroupTheory/finite-simple-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/finite-simple-groups/index.html","scope":"GroupTheory/finite-simple-groups","status":"can_prove","summary":"Finite simple and almost simple groups and the local analysis behind their classification: specific simple groups (Mathieu, Suzuki, PSL), strongly embedded subgroups (Bender\u2013Suzuki), classification by Sylow 2-subgroups (Gorenstein\u2013Walter, Brauer\u2013Suzuki), Baer\u2013Suzuki, Glauberman Z*, Brauer\u2013Fowler, involution centralizers and 3-transpositions; generation questions go to Generation of groups and spectrum recognition\u2026","title":"Finite simple groups and involutions"},{"id":"GroupTheory/finite-simple-groups/brauer-fowler","kind":"node","lean":"MathlibExt.GroupTheory.BrauerFowlerWanted.GroupTheory.brauer_fowler","parent":1884,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/index.html#brauer-fowler","scope":"GroupTheory/finite-simple-groups","status":"can_prove","summary":"There is a function $f \\colon \\mathbb{N} \\to \\mathbb{N}$ such that for every finite nonabelian simple group $G$ and every element $t \\in G$ of order $2$, with centralizer $C_G(t)$ in $G$, the order of $G$ satisfies $|G| \\le f(|C_G(t)|)$.","title":"Brauer\u2013Fowler theorem"},{"id":"GroupTheory/finite-simple-groups/gorenstein-walter","kind":"node","lean":"MathlibExt.GroupTheory.GorensteinWalterWanted.GroupTheory.gorenstein_walter","parent":1884,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/index.html#gorenstein-walter","scope":"GroupTheory/finite-simple-groups","status":"can_prove","summary":"Let $G$ be a finite nonabelian simple group, and let $P$ be a Sylow $2$-subgroup of $G$ that is isomorphic to the dihedral group $D_n$ for some natural number $n$ (of order $2n$ when $n \\ge 1$; $D_0$ is the infinite dihedral group). Then $G$ is isomorphic to the alternating group $A_7$, or to the projective special linear group $\\mathrm{PSL}_2(\\mathbb{F}_q)$ for some power $q = p^k$ of an odd prime $p$ with\u2026","title":"Gorenstein\u2013Walter theorem"},{"data_url":"scopes/GroupTheory/finite-simple-groups/3-transposition-groups.json","id":"GroupTheory/finite-simple-groups/3-transposition-groups","kind":"scope","navigation":"module","parent":1884,"result":false,"route":"../roadmap/GroupTheory/finite-simple-groups/3-transposition-groups/index.html","scope":"GroupTheory/finite-simple-groups/3-transposition-groups","status":"can_prove","title":"3-transposition groups"},{"id":"GroupTheory/finite-simple-groups/3-transposition-groups/is-three-transposition-set","kind":"node","lean":"MathlibExt.GroupTheory.ThreeTranspositionLocallyFiniteKourovka12Problem76Wanted.IsThreeTranspositionSet","parent":1887,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/3-transposition-groups/index.html#is-three-transposition-set","scope":"GroupTheory/finite-simple-groups/3-transposition-groups","status":"fully_proved","summary":"A subset $D$ of a group $G$ is a set of 3-transpositions when it is closed under conjugation ($g d g^{-1} \\in D$ for all $g \\in G$ and $d \\in D$), every $d \\in D$ is an involution ($d \\ne 1$ and $d^2 = 1$), and for all $d, e \\in D$ the product $de$ has order at most $3$, that is, $de = 1$, $(de)^2 = 1$ or $(de)^3 = 1$.","title":"Set of 3-transpositions"},{"id":"GroupTheory/finite-simple-groups/3-transposition-groups/is-locally-finite","kind":"node","lean":"MathlibExt.GroupTheory.ThreeTranspositionLocallyFiniteKourovka12Problem76Wanted.IsLocallyFinite","parent":1887,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/3-transposition-groups/index.html#is-locally-finite","scope":"GroupTheory/finite-simple-groups/3-transposition-groups","status":"fully_proved","summary":"A group $G$ is locally finite when every finite subset $S \\subseteq G$ generates a finite subgroup; equivalently, every finitely generated subgroup of $G$ is finite.","title":"Locally finite group, via finite subsets"},{"id":"GroupTheory/finite-simple-groups/3-transposition-groups/three-transposition-group-locally-finite","kind":"node","lean":"MathlibExt.GroupTheory.ThreeTranspositionLocallyFiniteKourovka12Problem76Wanted.threeTranspositionGroupLocallyFinite","parent":1887,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/3-transposition-groups/index.html#three-transposition-group-locally-finite","scope":"GroupTheory/finite-simple-groups/3-transposition-groups","status":"fully_proved","summary":"The statement that for every group $G$ and every set $D$ of 3-transpositions of $G$ that generates $G$, the group $G$ is locally finite.","title":"Sozutov's question on 3-transposition groups (Kourovka 12.76)"},{"id":"GroupTheory/finite-simple-groups/3-transposition-groups/three-transposition-group-locally-finite-2","kind":"node","lean":"MathlibExt.GroupTheory.ThreeTranspositionLocallyFiniteKourovka12Problem76Wanted.threeTranspositionGroupLocallyFinite_holds","parent":1887,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/3-transposition-groups/index.html#three-transposition-group-locally-finite-2","scope":"GroupTheory/finite-simple-groups/3-transposition-groups","status":"can_prove","summary":"If a group $G$ is generated by a set $D$ of 3-transpositions (a conjugation-invariant set of involutions whose pairwise products have order at most $3$), then $G$ is locally finite: every finite subset of $G$ generates a finite subgroup of $G$.","title":"Groups generated by 3-transpositions are locally finite"},{"data_url":"scopes/GroupTheory/finite-simple-groups/baer-suzuki-theorem.json","id":"GroupTheory/finite-simple-groups/baer-suzuki-theorem","kind":"scope","navigation":"module","parent":1884,"result":false,"route":"../roadmap/GroupTheory/finite-simple-groups/baer-suzuki-theorem/index.html","scope":"GroupTheory/finite-simple-groups/baer-suzuki-theorem","status":"can_prove","title":"Baer\u2013Suzuki theorem"},{"id":"GroupTheory/finite-simple-groups/baer-suzuki-theorem/baer-suzuki-periodic-two","kind":"node","lean":"MathlibExt.GroupTheory.BaerSuzukiPeriodicBinaryFiniteKourovka11Problem11Wanted.baerSuzukiPeriodicTwo","parent":1892,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/baer-suzuki-theorem/index.html#baer-suzuki-periodic-two","scope":"GroupTheory/finite-simple-groups/baer-suzuki-theorem","status":"fully_proved","summary":"The proposition that for every group $G$ in which every element has finite order and every element $a \\in G$ such that any two conjugates of $a$ generate a finite $2$-subgroup, there is a normal $2$-subgroup of $G$ containing $a$. Here a $2$-subgroup is a subgroup in which every element has order a power of $2$.","title":"Baer\u2013Suzuki property for periodic groups at the prime 2"},{"id":"GroupTheory/finite-simple-groups/baer-suzuki-theorem/baer-suzuki-binary-finite","kind":"node","lean":"MathlibExt.GroupTheory.BaerSuzukiPeriodicBinaryFiniteKourovka11Problem11Wanted.baerSuzukiBinaryFinite","parent":1892,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/baer-suzuki-theorem/index.html#baer-suzuki-binary-finite","scope":"GroupTheory/finite-simple-groups/baer-suzuki-theorem","status":"fully_proved","summary":"The proposition that for every prime $p$, every group $G$ in which any two elements generate a finite subgroup, and every element $a \\in G$ such that any two conjugates of $a$ generate a $p$-subgroup, there is a normal $p$-subgroup of $G$ containing $a$. Here a $p$-subgroup is a subgroup in which every element has order a power of $p$.","title":"Baer\u2013Suzuki property for binary finite groups"},{"id":"GroupTheory/finite-simple-groups/baer-suzuki-theorem/baer-suzuki-periodic-two-refuted","kind":"node","lean":"MathlibExt.GroupTheory.BaerSuzukiPeriodicBinaryFiniteKourovka11Problem11Wanted.baerSuzukiPeriodicTwo_refuted","parent":1892,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/baer-suzuki-theorem/index.html#baer-suzuki-periodic-two-refuted","scope":"GroupTheory/finite-simple-groups/baer-suzuki-theorem","status":"can_prove","summary":"There is a group $G$ in which every element has finite order, together with an element $a \\in G$ such that any two conjugates of $a$ generate a finite $2$-subgroup, but no normal $2$-subgroup of $G$ contains $a$. Here a $2$-subgroup is a subgroup in which every element has order a power of $2$.","title":"Failure of the Baer\u2013Suzuki theorem for periodic groups at the prime 2"},{"id":"GroupTheory/finite-simple-groups/baer-suzuki-theorem/baer-suzuki-binary-finite-holds","kind":"node","lean":"MathlibExt.GroupTheory.BaerSuzukiPeriodicBinaryFiniteKourovka11Problem11Wanted.baerSuzukiBinaryFinite_holds","parent":1892,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/baer-suzuki-theorem/index.html#baer-suzuki-binary-finite-holds","scope":"GroupTheory/finite-simple-groups/baer-suzuki-theorem","status":"can_prove","summary":"For every prime $p$, every group $G$ in which any two elements generate a finite subgroup, and every element $a \\in G$ such that any two conjugates of $a$ generate a $p$-subgroup, there is a normal $p$-subgroup of $G$ containing $a$. Here a $p$-subgroup is a subgroup in which every element has order a power of $p$.","title":"Baer\u2013Suzuki theorem for binary finite groups"},{"id":"GroupTheory/finite-simple-groups/baer-suzuki-theorem/p-core","kind":"node","lean":"MathlibExt.GroupTheory.BaerSuzukiWanted.GroupTheory.Defs.pCore","parent":1892,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-core of a group"]],"source":"The $p$-core of a group"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/baer-suzuki-theorem/index.html#p-core","scope":"GroupTheory/finite-simple-groups/baer-suzuki-theorem","status":"fully_proved","summary":"For a natural number $p$ and a group $G$, the $p$-core $O_p(G)$ is the supremum, in the lattice of subgroups of $G$, of all normal subgroups $N$ of $G$ that are $p$-groups, that is, such that every $y \\in N$ satisfies $y^{p^k} = 1$ for some $k \\ge 0$.","title":"The $p$-core of a group"},{"id":"GroupTheory/finite-simple-groups/baer-suzuki-theorem/baer-suzuki","kind":"node","lean":"MathlibExt.GroupTheory.BaerSuzukiWanted.GroupTheory.baer_suzuki","parent":1892,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/baer-suzuki-theorem/index.html#baer-suzuki","scope":"GroupTheory/finite-simple-groups/baer-suzuki-theorem","status":"can_prove","summary":"Let $G$ be a finite group, $p$ a prime and $x \\in G$. Then $x$ lies in the $p$-core $O_p(G)$ of $G$ if and only if, for every $g \\in G$, the subgroup of $G$ generated by $x$ and $g x g^{-1}$ is a $p$-group, that is, every element of it has order a power of $p$.","title":"Baer\u2013Suzuki theorem"},{"data_url":"scopes/GroupTheory/finite-simple-groups/bender-suzuki-theorem.json","id":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","kind":"scope","navigation":"module","parent":1884,"result":false,"route":"../roadmap/GroupTheory/finite-simple-groups/bender-suzuki-theorem/index.html","scope":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","status":"can_prove","title":"Bender\u2013Suzuki theorem"},{"id":"GroupTheory/finite-simple-groups/bender-suzuki-theorem/is-strongly-embedded","kind":"node","lean":"MathlibExt.GroupTheory.BenderSuzukiWanted.GroupTheory.Defs.IsStronglyEmbedded","parent":1899,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/bender-suzuki-theorem/index.html#is-strongly-embedded","scope":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","status":"fully_proved","summary":"A subgroup $M$ of a group $X$ is strongly embedded in $X$ if $M \\ne X$, the order of $M$ is even, and for every $g \\in X$ outside $M$ the order of $M \\cap g M g^{-1}$ is odd. Here an infinite group is assigned order $0$, so an infinite $M$ counts as having even order and an infinite intersection never has odd order.","title":"Strongly embedded subgroups"},{"id":"GroupTheory/finite-simple-groups/bender-suzuki-theorem/psl2-model","kind":"node","lean":"MathlibExt.GroupTheory.BenderSuzukiWanted.GroupTheory.Defs.PSL2Model","parent":1899,"reading_title":{"format":"inline-parts-v1","parts":[["text","The group "],["math","\\(\\mathrm{PSL}(2, 2^n)\\)"]],"source":"The group $\\mathrm{PSL}(2, 2^n)$"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/bender-suzuki-theorem/index.html#psl2-model","scope":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","status":"fully_proved","summary":"For a natural number $n \\ge 1$, $\\mathrm{PSL}(2, 2^n)$ is the projective special linear group of $2 \\times 2$ matrices over the Galois field $\\mathrm{GF}(2^n)$. For $n = 0$ the same construction is taken over $\\mathrm{GF}(2)$ instead.","title":"The group $\\mathrm{PSL}(2, 2^n)$"},{"id":"GroupTheory/finite-simple-groups/bender-suzuki-theorem/sz-generators","kind":"node","lean":"MathlibExt.GroupTheory.BenderSuzukiWanted.GroupTheory.Defs.SzGenerators","parent":1899,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/bender-suzuki-theorem/index.html#sz-generators","scope":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","status":"fully_proved","summary":"Let $n$ be a natural number, $q = 2^{2n+1}$, $\\theta = 2^{n+1}$ and $F = \\mathrm{GF}(q)$. The generating set of $\\mathrm{Sz}(q)$ is the set of $4 \\times 4$ matrices over $F$ of the following three kinds: for $a, b \\in F$, the root element with rows $(1, a, b, a^{2+\\theta} + ab + b^{\\theta})$, $(0, 1, a^{\\theta}, a^{1+\\theta} + b)$, $(0, 0, 1, a)$ and $(0, 0, 0, 1)$; for nonzero $x \\in F$, the torus element with\u2026","title":"Generators of the Suzuki group"},{"id":"GroupTheory/finite-simple-groups/bender-suzuki-theorem/sz-model","kind":"node","lean":"MathlibExt.GroupTheory.BenderSuzukiWanted.GroupTheory.Defs.SzModel","parent":1899,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Suzuki group "],["math","\\(\\mathrm{Sz}(2^{2n+1})\\)"]],"source":"The Suzuki group $\\mathrm{Sz}(2^{2n+1})$"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/bender-suzuki-theorem/index.html#sz-model","scope":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","status":"fully_proved","summary":"For a natural number $n$, the Suzuki group $\\mathrm{Sz}(2^{2n+1})$ is the subgroup of $\\mathrm{GL}(4, \\mathrm{GF}(2^{2n+1}))$ generated by the matrices in the generating set of the Suzuki group (all of which are invertible).","title":"The Suzuki group $\\mathrm{Sz}(2^{2n+1})$"},{"id":"GroupTheory/finite-simple-groups/bender-suzuki-theorem/psu3-model","kind":"node","lean":"MathlibExt.GroupTheory.BenderSuzukiWanted.GroupTheory.Defs.PSU3Model","parent":1899,"reading_title":{"format":"inline-parts-v1","parts":[["text","The group "],["math","\\(\\mathrm{PSU}(3, 2^n)\\)"]],"source":"The group $\\mathrm{PSU}(3, 2^n)$"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/bender-suzuki-theorem/index.html#psu3-model","scope":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","status":"fully_proved","summary":"Let $n \\ge 1$, let $q = 2^n$ and let $J$ be the $3 \\times 3$ matrix with ones on the antidiagonal and zeros elsewhere. Then $\\mathrm{PSU}(3, q)$ is the image in the projective general linear group $\\mathrm{PGL}(3, \\mathrm{GF}(q^2))$ of the group of matrices $A \\in \\mathrm{GL}(3, \\mathrm{GF}(q^2))$ with $\\det A = 1$ and $\\bar{A}^{\\mathsf{T}} J A = J$, where $\\bar{A}$ is obtained from $A$ by applying $z \\mapsto z^q$\u2026","title":"The group $\\mathrm{PSU}(3, 2^n)$"},{"id":"GroupTheory/finite-simple-groups/bender-suzuki-theorem/is-simple-bender-group","kind":"node","lean":"MathlibExt.GroupTheory.BenderSuzukiWanted.GroupTheory.Defs.IsSimpleBenderGroup","parent":1899,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/bender-suzuki-theorem/index.html#is-simple-bender-group","scope":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","status":"fully_proved","summary":"A group $G$ is a simple Bender group if it is isomorphic to $\\mathrm{PSL}(2, 2^n)$ for some $n \\ge 2$, to the Suzuki group $\\mathrm{Sz}(2^{2n+1})$ for some $n \\ge 1$, or to $\\mathrm{PSU}(3, 2^n)$ for some $n \\ge 2$.","title":"Simple Bender groups"},{"id":"GroupTheory/finite-simple-groups/bender-suzuki-theorem/bender-suzuki","kind":"node","lean":"MathlibExt.GroupTheory.BenderSuzukiWanted.GroupTheory.bender_suzuki","parent":1899,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/bender-suzuki-theorem/index.html#bender-suzuki","scope":"GroupTheory/finite-simple-groups/bender-suzuki-theorem","status":"can_prove","summary":"Let $X$ be a finite simple group that has a strongly embedded subgroup $M$. Then $X$ is a simple Bender group: $X$ is isomorphic to $\\mathrm{PSL}(2, 2^n)$ for some $n \\ge 2$, to the Suzuki group $\\mathrm{Sz}(2^{2n+1})$ for some $n \\ge 1$, or to $\\mathrm{PSU}(3, 2^n)$ for some $n \\ge 2$.","title":"Bender\u2013Suzuki theorem"},{"data_url":"scopes/GroupTheory/finite-simple-groups/brauer-suzuki-theorem.json","id":"GroupTheory/finite-simple-groups/brauer-suzuki-theorem","kind":"scope","navigation":"module","parent":1884,"result":false,"route":"../roadmap/GroupTheory/finite-simple-groups/brauer-suzuki-theorem/index.html","scope":"GroupTheory/finite-simple-groups/brauer-suzuki-theorem","status":"can_prove","title":"Brauer\u2013Suzuki theorem"},{"id":"GroupTheory/finite-simple-groups/brauer-suzuki-theorem/odd-core","kind":"node","lean":"MathlibExt.GroupTheory.BrauerSuzukiWanted.GroupTheory.Defs.oddCore","parent":1907,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/brauer-suzuki-theorem/index.html#odd-core","scope":"GroupTheory/finite-simple-groups/brauer-suzuki-theorem","status":"fully_proved","summary":"The odd core $O(G)$ of a group $G$ is the supremum, in the lattice of subgroups of $G$, of all normal subgroups of $G$ whose order is finite and odd.","title":"Odd core of a group"},{"id":"GroupTheory/finite-simple-groups/brauer-suzuki-theorem/brauer-suzuki","kind":"node","lean":"MathlibExt.GroupTheory.BrauerSuzukiWanted.GroupTheory.brauer_suzuki","parent":1907,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/brauer-suzuki-theorem/index.html#brauer-suzuki","scope":"GroupTheory/finite-simple-groups/brauer-suzuki-theorem","status":"can_prove","summary":"Let $G$ be a finite group, let $O(G)$, the odd core of $G$, be the subgroup generated by all normal subgroups of $G$ of odd order, let $n \\ge 3$, and let $P$ be a Sylow $2$-subgroup of $G$ isomorphic to the generalized quaternion group $Q_{2^n}$ of order $2^n$. If $t \\in P$ has order $2$, then the image of $t$ in $G / O(G)$ lies in the center of $G / O(G)$.","title":"Brauer\u2013Suzuki theorem"},{"data_url":"scopes/GroupTheory/finite-simple-groups/glauberman-z-theorem.json","id":"GroupTheory/finite-simple-groups/glauberman-z-theorem","kind":"scope","navigation":"module","parent":1884,"result":false,"route":"../roadmap/GroupTheory/finite-simple-groups/glauberman-z-theorem/index.html","scope":"GroupTheory/finite-simple-groups/glauberman-z-theorem","status":"can_prove","title":"Glauberman Z*-theorem"},{"id":"GroupTheory/finite-simple-groups/glauberman-z-theorem/glauberman-z-star","kind":"node","lean":"MathlibExt.GroupTheory.GlaubermanZStarWanted.GroupTheory.glauberman_zStar","parent":1910,"reading_title":{"format":"inline-parts-v1","parts":[["text","Glauberman's "],["math","\\(Z^*\\)"],["text","-theorem"]],"source":"Glauberman's $Z^*$-theorem"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/glauberman-z-theorem/index.html#glauberman-z-star","scope":"GroupTheory/finite-simple-groups/glauberman-z-theorem","status":"can_prove","summary":"Glauberman's $Z^*$-theorem. Let $G$ be a finite group and let $t \\in G$ be an involution ($t \\neq 1$ and $t^2 = 1$) such that every conjugate $g t g^{-1}$ of $t$ that commutes with $t$ equals $t$. Then there is a normal subgroup $N$ of $G$ of odd order such that $g t g^{-1} t^{-1} \\in N$ for every $g \\in G$.","title":"Glauberman's $Z^*$-theorem"},{"id":"GroupTheory/finite-simple-groups/glauberman-z-theorem/odd-core","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionOddProductCentralKourovka11Problem13Wanted.oddCore","parent":1910,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/glauberman-z-theorem/index.html#odd-core","scope":"GroupTheory/finite-simple-groups/glauberman-z-theorem","status":"fully_proved","summary":"For a group $G$, $O(G)$ is the join, in the lattice of subgroups of $G$, of all normal subgroups $N$ of $G$ such that every element of $N$ has finite odd order. In a periodic group this join is again a normal subgroup consisting of elements of odd order, so it is the largest such subgroup.","title":"Odd core from normal subgroups of odd-order elements"},{"id":"GroupTheory/finite-simple-groups/glauberman-z-theorem/periodic-involution-central-mod-odd-core","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionOddProductCentralKourovka11Problem13Wanted.periodicInvolutionCentralModOddCore","parent":1910,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/glauberman-z-theorem/index.html#periodic-involution-central-mod-odd-core","scope":"GroupTheory/finite-simple-groups/glauberman-z-theorem","status":"fully_proved","summary":"For a group $G$, let $O(G)$ be the join of all normal subgroups of $G$ all of whose elements have odd order. This is the statement that for every periodic group $G$ and every involution $i \\in G$ such that $g^{-1} i g i$ has odd order for every $g \\in G$, the image of $i$ in $G / O(G)$ is central, that is, $g i g^{-1} i^{-1} \\in O(G)$ for every $g \\in G$.","title":"Borovik's question on involutions with odd-order products (Kourovka 11.13)"},{"id":"GroupTheory/finite-simple-groups/glauberman-z-theorem/periodic-involution-central-mod-odd-core-2","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionOddProductCentralKourovka11Problem13Wanted.periodicInvolutionCentralModOddCore_refuted","parent":1910,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/glauberman-z-theorem/index.html#periodic-involution-central-mod-odd-core-2","scope":"GroupTheory/finite-simple-groups/glauberman-z-theorem","status":"can_prove","summary":"For a group $G$, let $O(G)$ be the join of all normal subgroups of $G$ all of whose elements have odd order. There exist a periodic group $G$ and an involution $i \\in G$ such that $g^{-1} i g i$ has odd order for every $g \\in G$, but the image of $i$ in $G / O(G)$ is not central: $g i g^{-1} i^{-1} \\notin O(G)$ for some $g \\in G$. This answers Kourovka Problem 11.13 in the negative.","title":"A periodic counterexample to Borovik's question (Kourovka 11.13)"},{"data_url":"scopes/GroupTheory/finite-simple-groups/involution-centralizers.json","id":"GroupTheory/finite-simple-groups/involution-centralizers","kind":"scope","navigation":"module","parent":1884,"result":false,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"can_prove","title":"Involution centralizers"},{"id":"GroupTheory/finite-simple-groups/involution-centralizers/is-locally-finite-field","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionCentralizerPSLTwoKourovka18Problem59Wanted.IsLocallyFiniteField","parent":1915,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html#is-locally-finite-field","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"fully_proved","summary":"A field $P$ is locally finite when, for every finite subset $s$ of $P$, the subfield of $P$ generated by $s$ is finite.","title":"Locally finite fields (Kourovka 18.59)"},{"id":"GroupTheory/finite-simple-groups/involution-centralizers/periodic-group-with-pslinvolution-centra","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionCentralizerPSLTwoKourovka18Problem59Wanted.periodicGroupWithPSLInvolutionCentralizers","parent":1915,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html#periodic-group-with-pslinvolution-centra","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"fully_proved","summary":"The proposition (Kourovka 18.59, D. V. Lytkina) that there are a periodic group $G$ which contains an involution and in which all involutions are conjugate, and an infinite locally finite field $P$ of characteristic $2$, such that the centralizer of every involution $i$ of $G$ is isomorphic to $\\langle i \\rangle \\times \\mathrm{PSL}_2(P)$. Here $\\langle i \\rangle$ is the cyclic subgroup generated by $i$ and\u2026","title":"Kourovka Problem 18.59 on involution centralizers"},{"id":"GroupTheory/finite-simple-groups/involution-centralizers/periodic-group-with-pslinvolution-centra-2","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionCentralizerPSLTwoKourovka18Problem59Wanted.periodicGroupWithPSLInvolutionCentralizers_refuted","parent":1915,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html#periodic-group-with-pslinvolution-centra-2","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"can_prove","summary":"There is no periodic group $G$ that contains an involution, in which all involutions are conjugate, and for which there is an infinite locally finite field $P$ of characteristic $2$ such that the centralizer of every involution $i$ of $G$ is isomorphic to $\\langle i \\rangle \\times \\mathrm{PSL}_2(P)$. Here $\\langle i \\rangle$ is the cyclic subgroup generated by $i$ and $\\mathrm{PSL}_2(P)$ is the quotient of\u2026","title":"Negative answer to Kourovka Problem 18.59"},{"id":"GroupTheory/finite-simple-groups/involution-centralizers/commuting-involutions","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionFewCommutingInvolutionsKourovka11Problem96Wanted.commutingInvolutions","parent":1915,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html#commuting-involutions","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"fully_proved","summary":"For an element $t$ of a group $G$, this is the set of involutions of $G$ that commute with $t$: the elements $s \\in G$ of order $2$ with $st = ts$.","title":"Involutions commuting with an element"},{"id":"GroupTheory/finite-simple-groups/involution-centralizers/finitely-many-simple-few-commuting-invol","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionFewCommutingInvolutionsKourovka11Problem96Wanted.finitelyManySimpleFewCommutingInvolutions","parent":1915,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html#finitely-many-simple-few-commuting-invol","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"fully_proved","summary":"This is the statement that for every natural number $n$ there is a bound $N$ such that every finite simple group $G$ containing an involution $t$ that commutes with at most $n$ involutions of $G$ (counting $t$ itself) has order at most $N$. Bounding the order expresses \"only finitely many such groups up to isomorphism\".","title":"Strunkov's finiteness question for simple groups (Kourovka 11.96(a))"},{"id":"GroupTheory/finite-simple-groups/involution-centralizers/no-infinite-simple-few-commuting-involut","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionFewCommutingInvolutionsKourovka11Problem96Wanted.noInfiniteSimpleFewCommutingInvolutions","parent":1915,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html#no-infinite-simple-few-commuting-involut","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"fully_proved","summary":"This is the statement that for every natural number $n$, no infinite simple group $G$ contains an involution $t$ that commutes with at most $n$ involutions of $G$ (counting $t$ itself).","title":"Strunkov's question on infinite simple groups (Kourovka 11.96(b))"},{"id":"GroupTheory/finite-simple-groups/involution-centralizers/finitely-many-simple-few-commuting-invol-2","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionFewCommutingInvolutionsKourovka11Problem96Wanted.finitelyManySimpleFewCommutingInvolutions_holds","parent":1915,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html#finitely-many-simple-few-commuting-invol-2","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"can_prove","summary":"For every natural number $n$ there is a bound $N$ such that every finite simple group $G$ containing an involution $t$ that commutes with at most $n$ involutions of $G$ (counting $t$ itself) has order at most $N$.","title":"Finitely many finite simple groups with an involution commuting with few involutions"},{"id":"GroupTheory/finite-simple-groups/involution-centralizers/no-infinite-simple-few-commuting-involut-2","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionFewCommutingInvolutionsKourovka11Problem96Wanted.noInfiniteSimpleFewCommutingInvolutions_refuted","parent":1915,"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/involution-centralizers/index.html#no-infinite-simple-few-commuting-involut-2","scope":"GroupTheory/finite-simple-groups/involution-centralizers","status":"can_prove","summary":"There exist a natural number $n$ and an infinite simple group $G$ containing an involution $t$ that commutes with at most $n$ involutions of $G$ (counting $t$ itself). This answers Kourovka Problem 11.96(b) in the negative.","title":"An infinite simple group with an involution commuting with few involutions"},{"data_url":"scopes/GroupTheory/finite-simple-groups/mathieu-group-m.json","id":"GroupTheory/finite-simple-groups/mathieu-group-m","kind":"scope","navigation":"module","parent":1884,"result":false,"route":"../roadmap/GroupTheory/finite-simple-groups/mathieu-group-m/index.html","scope":"GroupTheory/finite-simple-groups/mathieu-group-m","status":"fully_proved","title":"Mathieu group M\u2082\u2083"},{"id":"GroupTheory/finite-simple-groups/mathieu-group-m/m23-cycle","kind":"node","lean":"MathieuGroup.m23Cycle","parent":1924,"reading_title":{"format":"inline-parts-v1","parts":[["text","The standard 23-cycle in "],["math","\\(M_{23}\\)"]],"source":"The standard 23-cycle in $M_{23}$"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/mathieu-group-m/index.html#m23-cycle","scope":"GroupTheory/finite-simple-groups/mathieu-group-m","status":"fully_proved","summary":"The permutation of $\\{0, 1, \\dots, 22\\}$ that sends $i$ to $i + 1$ for $i < 22$ and sends $22$ to $0$, that is, the 23-cycle $(0\\ 1\\ 2\\ \\cdots\\ 22)$. It is the first of the two standard generators of the Mathieu group $M_{23}$.","title":"The standard 23-cycle in $M_{23}$"},{"id":"GroupTheory/finite-simple-groups/mathieu-group-m/m23-five-cycles","kind":"node","lean":"MathieuGroup.m23FiveCycles","parent":1924,"reading_title":{"format":"inline-parts-v1","parts":[["text","The standard product of four 5-cycles in "],["math","\\(M_{23}\\)"]],"source":"The standard product of four 5-cycles in $M_{23}$"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/mathieu-group-m/index.html#m23-five-cycles","scope":"GroupTheory/finite-simple-groups/mathieu-group-m","status":"fully_proved","summary":"The permutation of the points $\\{0, 1, \\dots, 22\\}$ given by the product of the four disjoint 5-cycles $(2\\ 16\\ 9\\ 6\\ 8)(3\\ 12\\ 13\\ 18\\ 4)(7\\ 17\\ 10\\ 11\\ 22)(14\\ 19\\ 21\\ 20\\ 15)$, where a cycle $(a_1\\ a_2\\ \\cdots\\ a_5)$ sends $a_k$ to $a_{k+1}$ for $k < 5$ and sends $a_5$ to $a_1$. It fixes the three remaining points $0$, $1$ and $5$, and it is the second of the two standard generators of the Mathieu group $M_{23}$.","title":"The standard product of four 5-cycles in $M_{23}$"},{"id":"GroupTheory/finite-simple-groups/mathieu-group-m/m23","kind":"node","lean":"MathieuGroup.M23","parent":1924,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Mathieu group "],["math","\\(M_{23}\\)"]],"source":"The Mathieu group $M_{23}$"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/mathieu-group-m/index.html#m23","scope":"GroupTheory/finite-simple-groups/mathieu-group-m","status":"fully_proved","summary":"The subgroup of the symmetric group on $\\{0, 1, \\dots, 22\\}$ generated by the standard 23-cycle $(0\\ 1\\ 2\\ \\cdots\\ 22)$ and the standard product of four disjoint 5-cycles $(2\\ 16\\ 9\\ 6\\ 8)(3\\ 12\\ 13\\ 18\\ 4)(7\\ 17\\ 10\\ 11\\ 22)(14\\ 19\\ 21\\ 20\\ 15)$.","title":"The Mathieu group $M_{23}$"},{"id":"GroupTheory/finite-simple-groups/mathieu-group-m/m23-le-iff","kind":"node","lean":"MathieuGroup.m23_le_iff","parent":1924,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subgroups containing "],["math","\\(M_{23}\\)"]],"source":"Subgroups containing $M_{23}$"},"result":true,"route":"../roadmap/GroupTheory/finite-simple-groups/mathieu-group-m/index.html#m23-le-iff","scope":"GroupTheory/finite-simple-groups/mathieu-group-m","status":"fully_proved","summary":"A subgroup $H$ of the symmetric group on $\\{0, 1, \\dots, 22\\}$ contains the Mathieu group $M_{23}$ if and only if $H$ contains both the standard 23-cycle and the standard product of four 5-cycles that generate $M_{23}$.","title":"Subgroups containing $M_{23}$"},{"data_url":"scopes/GroupTheory/soluble-finite-groups.json","id":"GroupTheory/soluble-finite-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/index.html","scope":"GroupTheory/soluble-finite-groups","status":"can_prove","summary":"Structure of finite soluble groups: supersolubility criteria, p-solubility and p-length, chief factors and their complements, \u03c3-solubility and \u03c3-nilpotency, Schmidt subgroups, derived length bounds for finite groups of special type, and factorized groups; classes of groups closed under operations go to Formations and Fitting classes.","title":"Soluble and supersoluble finite groups"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations.json","id":"GroupTheory/soluble-finite-groups/formations","kind":"scope","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/index.html","scope":"GroupTheory/soluble-finite-groups/formations","status":"can_prove","summary":"Classes of finite groups: formations, local and hereditary formations, Fitting and radical classes, injectors, subgroup functors, residuals, and lattices of formations; varieties defined by laws go to Varieties, quasivarieties and equations.","title":"Formations and Fitting classes"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups.json","id":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/index.html","scope":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups","status":"can_prove","title":"Fitting classes of supersoluble groups"},{"id":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/group-class","kind":"node","lean":"MathlibExt.GroupTheory.FittingSupersolubleCentralKourovka12Problem91Wanted.GroupClass","parent":1931,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/index.html#group-class","scope":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups","status":"fully_proved","summary":"A class of groups is a property of groups (in the lowest type universe) that is closed under isomorphism: whenever a group $H$ has the property and $H$ is isomorphic to a group $K$, then $K$ has the property. Only the finite members of a class are used in Kourovka Problem 12.91.","title":"Isomorphism-closed classes of groups (Kourovka 12.91)"},{"id":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/is-fitting","kind":"node","lean":"MathlibExt.GroupTheory.FittingSupersolubleCentralKourovka12Problem91Wanted.IsFitting","parent":1931,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/index.html#is-fitting","scope":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups","status":"fully_proved","summary":"A class $\\mathfrak{X}$ of groups is a Fitting class (of finite groups) if, for every finite group $G$: whenever $G \\in \\mathfrak{X}$, every normal subgroup of $G$ belongs to $\\mathfrak{X}$; and whenever $N_1$ and $N_2$ are normal subgroups of $G$ that belong to $\\mathfrak{X}$, the subgroup $N_1 N_2$ they generate belongs to $\\mathfrak{X}$.","title":"Fitting classes (Kourovka 12.91)"},{"id":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/is-supersoluble","kind":"node","lean":"MathlibExt.GroupTheory.FittingSupersolubleCentralKourovka12Problem91Wanted.IsSupersoluble","parent":1931,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/index.html#is-supersoluble","scope":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups","status":"fully_proved","summary":"A group $G$ is supersoluble if it has a chain of subgroups $1 = G_0 \\le G_1 \\le \\cdots \\le G_k = G$, each normal in $G$, such that for every $i < k$ there is an element $x \\in G_{i+1}$ with $G_{i+1} = \\langle x \\rangle G_i$; that is, every factor $G_{i+1}/G_i$ is cyclic.","title":"Supersoluble groups (Kourovka 12.91)"},{"id":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/is-minimal-normal","kind":"node","lean":"MathlibExt.GroupTheory.FittingSupersolubleCentralKourovka12Problem91Wanted.IsMinimalNormal","parent":1931,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/index.html#is-minimal-normal","scope":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups","status":"fully_proved","summary":"A subgroup $N$ of a group $G$ is minimal normal if $N$ is a nontrivial normal subgroup of $G$ and every nontrivial normal subgroup $M$ of $G$ with $M \\le N$ equals $N$.","title":"Minimal normal subgroups (Kourovka 12.91)"},{"id":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/fitting-supersoluble-minimal-normal-cent","kind":"node","lean":"MathlibExt.GroupTheory.FittingSupersolubleCentralKourovka12Problem91Wanted.fittingSupersolubleMinimalNormalCentral","parent":1931,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/index.html#fitting-supersoluble-minimal-normal-cent","scope":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups","status":"fully_proved","summary":"The proposition that for every class $\\mathfrak{X}$ of groups that is a Fitting class and all of whose finite members are supersoluble, every minimal normal subgroup of every finite group $G \\in \\mathfrak{X}$ is contained in the center of $G$.","title":"Heineken's question on Fitting classes of supersoluble groups"},{"id":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/fitting-supersoluble-minimal-normal-cent-2","kind":"node","lean":"MathlibExt.GroupTheory.FittingSupersolubleCentralKourovka12Problem91Wanted.fittingSupersolubleMinimalNormalCentral_refuted","parent":1931,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups/index.html#fitting-supersoluble-minimal-normal-cent-2","scope":"GroupTheory/soluble-finite-groups/formations/fitting-classes-of-supersoluble-groups","status":"can_prove","summary":"There exist a class $\\mathfrak{X}$ of groups that is a Fitting class and all of whose finite members are supersoluble, a finite group $G \\in \\mathfrak{X}$, and a minimal normal subgroup $N$ of $G$ that is not contained in the center of $G$.","title":"A Fitting class of supersoluble groups with a non-central minimal normal subgroup"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties.json","id":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/index.html","scope":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties","status":"can_prove","title":"Formations of p-groups and varieties"},{"id":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/group-class","kind":"node","lean":"MathlibExt.GroupTheory.FormationVarietyPGroupsKourovka11Problem21Wanted.GroupClass","parent":1938,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/index.html#group-class","scope":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties","status":"fully_proved","summary":"A class of groups is a predicate on groups: it assigns to every group $G$ (in the lowest universe of types) a proposition, read as \"$G$ belongs to the class\". No closure under isomorphism is required.","title":"Classes of groups"},{"id":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/is-formation","kind":"node","lean":"MathlibExt.GroupTheory.FormationVarietyPGroupsKourovka11Problem21Wanted.IsFormation","parent":1938,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/index.html#is-formation","scope":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties","status":"fully_proved","summary":"A class $\\mathfrak F$ of groups is a formation when every group in $\\mathfrak F$ is finite, $\\mathfrak F$ contains at least one group, $\\mathfrak F$ is closed under isomorphism, $\\mathfrak F$ is closed under quotients (if $G \\in \\mathfrak F$ and $N$ is a normal subgroup of $G$, then $G/N \\in \\mathfrak F$), and $\\mathfrak F$ is closed under subdirect products: if $G$ is a finite group with normal subgroups $M$ and\u2026","title":"Formations of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/satisfies-laws","kind":"node","lean":"MathlibExt.GroupTheory.FormationVarietyPGroupsKourovka11Problem21Wanted.SatisfiesLaws","parent":1938,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/index.html#satisfies-laws","scope":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties","status":"fully_proved","summary":"Let $W$ be a set of words in the free group on countably many generators $x_0, x_1, x_2, \\dots$. A group $G$ satisfies the laws in $W$ when every homomorphism from this free group to $G$ sends every $w \\in W$ to $1$. The groups satisfying $W$ form the variety defined by $W$, and every variety arises this way.","title":"Groups satisfying a set of laws"},{"id":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/subformation-of-pgroups-is-varietal","kind":"node","lean":"MathlibExt.GroupTheory.FormationVarietyPGroupsKourovka11Problem21Wanted.subformationOfPGroupsIsVarietal","parent":1938,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 11.21 (Vasil'yev's question on formations of "],["math","\\(p\\)"],["text","-groups)"]],"source":"Kourovka Problem 11.21 (Vasil'yev's question on formations of $p$-groups)"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/index.html#subformation-of-pgroups-is-varietal","scope":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties","status":"fully_proved","summary":"The proposition that for every prime $p$ and every formation $\\mathfrak F$ all of whose members are $p$-groups (a subformation of the formation $\\mathfrak N_p$ of all finite $p$-groups), there is a set $W$ of laws, words in the free group on countably many generators, such that a finite $p$-group lies in $\\mathfrak F$ if and only if it satisfies every law in $W$. In other words, every such $\\mathfrak F$ equals\u2026","title":"Kourovka Problem 11.21 (Vasil'yev's question on formations of $p$-groups)"},{"id":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/subformation-of-pgroups-is-varietal-refu","kind":"node","lean":"MathlibExt.GroupTheory.FormationVarietyPGroupsKourovka11Problem21Wanted.subformationOfPGroupsIsVarietal_refuted","parent":1938,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties/index.html#subformation-of-pgroups-is-varietal-refu","scope":"GroupTheory/soluble-finite-groups/formations/formations-of-p-groups-and-varieties","status":"can_prove","summary":"There are a prime $p$ and a formation $\\mathfrak F$ all of whose members are $p$-groups such that for every set $W$ of laws, words in the free group on countably many generators, some finite $p$-group lies in $\\mathfrak F$ and fails a law in $W$, or satisfies every law in $W$ and does not lie in $\\mathfrak F$. Thus $\\mathfrak F$ is not of the form $\\mathfrak N_p \\cap \\mathfrak M$ for any variety $\\mathfrak M$,\u2026","title":"Kourovka Problem 11.21 has a negative answer"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups.json","id":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/index.html","scope":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","status":"can_prove","title":"Hereditary formations and Shmidt groups"},{"id":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/group-class","kind":"node","lean":"MathlibExt.GroupTheory.HereditaryFormationShmidtLocalKourovka11Problem91Wanted.GroupClass","parent":1944,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/index.html#group-class","scope":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","status":"fully_proved","summary":"A class of groups is a property of groups: for each group $G$ it gives a proposition, read as \"$G$ belongs to the class\". No closure under isomorphism is required.","title":"Classes of groups"},{"id":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/is-hereditary-soluble-formation","kind":"node","lean":"MathlibExt.GroupTheory.HereditaryFormationShmidtLocalKourovka11Problem91Wanted.IsHereditarySolubleFormation","parent":1944,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/index.html#is-hereditary-soluble-formation","scope":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","status":"fully_proved","summary":"A class $\\mathfrak F$ of groups is a hereditary formation of finite soluble groups if every group in $\\mathfrak F$ is finite and soluble; $\\mathfrak F$ is closed under isomorphisms, under taking subgroups and under taking quotients by normal subgroups; and for every finite group $G$ with normal subgroups $M$ and $N$ such that $G/M$ and $G/N$ lie in $\\mathfrak F$, also $G/(M \\cap N)$ lies in $\\mathfrak F$.","title":"Hereditary formations of finite soluble groups"},{"id":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/is-saturated","kind":"node","lean":"MathlibExt.GroupTheory.HereditaryFormationShmidtLocalKourovka11Problem91Wanted.IsSaturated","parent":1944,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/index.html#is-saturated","scope":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","status":"fully_proved","summary":"A class $\\mathfrak F$ of groups is saturated if every finite soluble group $G$ such that $G/\\Phi(G)$ lies in $\\mathfrak F$, where $\\Phi(G)$ is the Frattini subgroup of $G$, itself lies in $\\mathfrak F$.","title":"Saturated classes of groups"},{"id":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/is-minimal-non","kind":"node","lean":"MathlibExt.GroupTheory.HereditaryFormationShmidtLocalKourovka11Problem91Wanted.IsMinimalNon","parent":1944,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/index.html#is-minimal-non","scope":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","status":"fully_proved","summary":"Let $\\mathfrak F$ be a class of groups. A group $G$ is a minimal non-$\\mathfrak F$-group if $G$ does not lie in $\\mathfrak F$ but every proper subgroup of $G$ lies in $\\mathfrak F$.","title":"Minimal non-members of a class of groups"},{"id":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/is-shmidt-group","kind":"node","lean":"MathlibExt.GroupTheory.HereditaryFormationShmidtLocalKourovka11Problem91Wanted.IsShmidtGroup","parent":1944,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/index.html#is-shmidt-group","scope":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","status":"fully_proved","summary":"A group $G$ is a Shmidt group if it is finite and not nilpotent, but every proper subgroup of $G$ is nilpotent.","title":"Shmidt groups"},{"id":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/hereditary-formation-local-of-shmidt","kind":"node","lean":"MathlibExt.GroupTheory.HereditaryFormationShmidtLocalKourovka11Problem91Wanted.hereditaryFormationLocalOfShmidt","parent":1944,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/index.html#hereditary-formation-local-of-shmidt","scope":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","status":"fully_proved","summary":"The statement that every hereditary formation $\\mathfrak F$ of finite soluble groups such that every finite soluble non-simple minimal non-$\\mathfrak F$-group (a group outside $\\mathfrak F$ all of whose proper subgroups lie in $\\mathfrak F$) is a Shmidt group is saturated, that is, every finite soluble group $G$ whose quotient $G/\\Phi(G)$ by its Frattini subgroup $\\Phi(G)$ lies in $\\mathfrak F$ itself lies in\u2026","title":"Kourovka Problem 11.91 (statement)"},{"id":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/hereditary-formation-local-of-shmidt-hol","kind":"node","lean":"MathlibExt.GroupTheory.HereditaryFormationShmidtLocalKourovka11Problem91Wanted.hereditaryFormationLocalOfShmidt_holds","parent":1944,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups/index.html#hereditary-formation-local-of-shmidt-hol","scope":"GroupTheory/soluble-finite-groups/formations/hereditary-formations-and-shmidt-groups","status":"can_prove","summary":"Let $\\mathfrak F$ be a hereditary formation of finite soluble groups: every group in $\\mathfrak F$ is finite and soluble, $\\mathfrak F$ is closed under isomorphisms, subgroups and quotients, and whenever $M$ and $N$ are normal subgroups of a finite group $G$ with $G/M$ and $G/N$ in $\\mathfrak F$, also $G/(M \\cap N)$ lies in $\\mathfrak F$. Suppose that every finite soluble group that is not simple and is a minimal\u2026","title":"Skiba's theorem (Kourovka Problem 11.91)"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/indecomposable-formations.json","id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"can_prove","title":"Indecomposable formations"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/group-class","kind":"node","lean":"MathlibExt.GroupTheory.PGroupFormationIndecomposableKourovka10Problem72Wanted.GroupClass","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#group-class","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"A class of finite groups is given by a membership predicate: a proposition $X(G)$ for every finite group $G$. It is not required to be closed under isomorphisms.","title":"Class of finite groups (Kourovka 10.72)"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/is-formation","kind":"node","lean":"MathlibExt.GroupTheory.PGroupFormationIndecomposableKourovka10Problem72Wanted.IsFormation","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#is-formation","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"A class $X$ of finite groups is a formation when it is closed under isomorphisms, contains the trivial group, contains the quotient $G/N$ of each of its members $G$ by every normal subgroup $N$, and contains $G/(N_1 \\cap N_2)$ whenever $N_1$ and $N_2$ are normal subgroups of a finite group $G$ with $G/N_1$ and $G/N_2$ in $X$. The last condition is closure under subdirect products.","title":"Formation of finite groups (Kourovka 10.72)"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/residual","kind":"node","lean":"MathlibExt.GroupTheory.PGroupFormationIndecomposableKourovka10Problem72Wanted.residual","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#residual","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"For a class $X$ of finite groups and a finite group $G$, the $X$-residual of $G$ is the intersection of all normal subgroups $N$ of $G$ with $G/N \\in X$. When $X$ is a formation, it is the smallest such $N$.","title":"Residual of a finite group with respect to a class (Kourovka 10.72)"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/p-group-formation-indecomposable","kind":"node","lean":"MathlibExt.GroupTheory.PGroupFormationIndecomposableKourovka10Problem72Wanted.pGroupFormationIndecomposable","parent":1952,"reading_title":{"format":"inline-parts-v1","parts":[["text","Indecomposability of the formation of finite "],["math","\\(p\\)"],["text","-groups (Kourovka 10.72)"]],"source":"Indecomposability of the formation of finite $p$-groups (Kourovka 10.72)"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#p-group-formation-indecomposable","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"The statement that for every prime $p$, the class $\\mathfrak{N}_p$ of all finite $p$-groups is not the Gasch\u00fctz product $\\mathfrak{F} \\circ \\mathfrak{G}$ of two formations $\\mathfrak{F}$ and $\\mathfrak{G}$ that consist of $p$-groups, each contain a nontrivial group and each omit some finite $p$-group. Here a finite group $G$ lies in $\\mathfrak{F} \\circ \\mathfrak{G}$ when its $\\mathfrak{G}$-residual lies in\u2026","title":"Indecomposability of the formation of finite $p$-groups (Kourovka 10.72)"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/p-group-formation-indecomposable-holds","kind":"node","lean":"MathlibExt.GroupTheory.PGroupFormationIndecomposableKourovka10Problem72Wanted.pGroupFormationIndecomposable_holds","parent":1952,"reading_title":{"format":"inline-parts-v1","parts":[["text","The formation of finite "],["math","\\(p\\)"],["text","-groups is indecomposable"]],"source":"The formation of finite $p$-groups is indecomposable"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#p-group-formation-indecomposable-holds","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"can_prove","summary":"For every prime $p$, the class $\\mathfrak{N}_p$ of all finite $p$-groups is not the Gasch\u00fctz product $\\mathfrak{F} \\circ \\mathfrak{G}$ of two formations $\\mathfrak{F}$ and $\\mathfrak{G}$ that consist of $p$-groups, each contain a nontrivial group and each omit some finite $p$-group. Here a finite group $G$ lies in $\\mathfrak{F} \\circ \\mathfrak{G}$ when its $\\mathfrak{G}$-residual lies in $\\mathfrak{F}$.","title":"The formation of finite $p$-groups is indecomposable"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/group-class-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleCompositionFactorFormationKourovka12Problem97Wanted.GroupClass","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#group-class-2","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"A class of groups is given by a membership predicate on groups such that every group isomorphic to a member is again a member. Only the finite members matter in the formulation of Kourovka Problem 12.97.","title":"Isomorphism-closed classes of groups (Kourovka 12.97)"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/is-formation-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleCompositionFactorFormationKourovka12Problem97Wanted.IsFormation","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#is-formation-2","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"A class $X$ of groups is a formation when it contains the trivial group, the quotient $G/N$ lies in $X$ for every finite group $G$ in $X$ and every normal subgroup $N$ of $G$, and for every finite group $G$ with normal subgroups $N_1$ and $N_2$ such that $G/N_1$ and $G/N_2$ lie in $X$, the quotient $G/(N_1 \\cap N_2)$ lies in $X$.","title":"Formations of finite groups (Kourovka 12.97)"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/nontrivial-class","kind":"node","lean":"MathlibExt.GroupTheory.SimpleCompositionFactorFormationKourovka12Problem97Wanted.NontrivialClass","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#nontrivial-class","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"A class $X$ of groups is nontrivial when it contains a nontrivial finite group; for a formation this means that $X$ is not the trivial formation.","title":"Classes containing a nontrivial finite group"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/residual-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleCompositionFactorFormationKourovka12Problem97Wanted.residual","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#residual-2","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"For a class $X$ of groups and a group $G$, the $X$-residual of $G$ is the intersection of all normal subgroups $N$ of $G$ such that the quotient $G/N$ lies in $X$.","title":"The residual of a group with respect to a class (Kourovka 12.97)"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/all-composition-factors-iso","kind":"node","lean":"MathlibExt.GroupTheory.SimpleCompositionFactorFormationKourovka12Problem97Wanted.AllCompositionFactorsIso","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#all-composition-factors-iso","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"For groups $T$ and $G$, the group $G$ lies in the class $\\mathfrak{F}_T$ when it has a subnormal series $1 = H_0 \\le H_1 \\le \\dots \\le H_n = G$, each $H_i$ normal in $H_{i+1}$, all of whose factors $H_{i+1}/H_i$ are isomorphic to $T$. For simple $T$ these are the groups all of whose composition factors are isomorphic to $T$.","title":"Groups with a subnormal series whose factors are all isomorphic to a given group"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/is-proper-in","kind":"node","lean":"MathlibExt.GroupTheory.SimpleCompositionFactorFormationKourovka12Problem97Wanted.IsProperIn","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#is-proper-in","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"`X` is a proper subclass of `\ud835\udd09_T`: some finite group of `\ud835\udd09_T` does not belong to `X`.","title":"Proper subclass of a formation"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/composition-factor-formation-indecomposa","kind":"node","lean":"MathlibExt.GroupTheory.SimpleCompositionFactorFormationKourovka12Problem97Wanted.compositionFactorFormationIndecomposable","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#composition-factor-formation-indecomposa","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"fully_proved","summary":"For a group $T$, write $\\mathfrak{F}_T$ for the class of groups having a subnormal series all of whose factors are isomorphic to $T$. This is the proposition that, for every finite non-abelian simple group $T$ and all formations $\\mathfrak{M}$ and $\\mathfrak{H}$ whose finite members lie in $\\mathfrak{F}_T$, each containing a nontrivial finite group and each missing some finite group of $\\mathfrak{F}_T$, it is not\u2026","title":"Kourovka Problem 12.97 (Shemetkov)"},{"id":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations/composition-factor-formation-indecomposa-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleCompositionFactorFormationKourovka12Problem97Wanted.compositionFactorFormationIndecomposable_holds","parent":1952,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/indecomposable-formations/index.html#composition-factor-formation-indecomposa-2","scope":"GroupTheory/soluble-finite-groups/formations/indecomposable-formations","status":"can_prove","summary":"Let $T$ be a finite non-abelian simple group, and write $\\mathfrak{F}_T$ for the class of groups having a subnormal series all of whose factors are isomorphic to $T$. Let $\\mathfrak{M}$ and $\\mathfrak{H}$ be formations whose finite members lie in $\\mathfrak{F}_T$, each containing a nontrivial finite group and each missing some finite group of $\\mathfrak{F}_T$. Then it is not true that, for every finite group $G$,\u2026","title":"Mel'nikov's theorem on Kourovka Problem 12.97"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes.json","id":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/index.html","scope":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes","status":"can_prove","title":"Injectors of Fitting classes"},{"id":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/group-class","kind":"node","lean":"MathlibExt.GroupTheory.FittingClassWithoutInjectorKourovka12Problem96Wanted.GroupClass","parent":1966,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/index.html#group-class","scope":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes","status":"fully_proved","summary":"A class of groups is a property of groups that is closed under isomorphism: if a group $H$ has the property and $K$ is a group isomorphic to $H$, then $K$ has it too.","title":"Isomorphism-closed classes of groups (Kourovka 12.96)"},{"id":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/is-fitting","kind":"node","lean":"MathlibExt.GroupTheory.FittingClassWithoutInjectorKourovka12Problem96Wanted.IsFitting","parent":1966,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/index.html#is-fitting","scope":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes","status":"fully_proved","summary":"A class $\\mathfrak{X}$ of groups is a Fitting class of finite groups if, for every finite group $G$, every normal subgroup of $G$ lies in $\\mathfrak{X}$ whenever $G$ does, and the product $N_1 N_2$ of two normal subgroups $N_1$ and $N_2$ of $G$ lies in $\\mathfrak{X}$ whenever $N_1$ and $N_2$ do.","title":"Fitting classes of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/is-xmaximal-in","kind":"node","lean":"MathlibExt.GroupTheory.FittingClassWithoutInjectorKourovka12Problem96Wanted.IsXMaximalIn","parent":1966,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/index.html#is-xmaximal-in","scope":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes","status":"fully_proved","summary":"Let $\\mathfrak{X}$ be a class of groups and let $V$ and $K$ be subgroups of a group $G$. Then $V$ is $\\mathfrak{X}$-maximal in $K$ if $V \\le K$, $V$ lies in $\\mathfrak{X}$, and no subgroup $W$ of $G$ with $V < W \\le K$ lies in $\\mathfrak{X}$.","title":"Maximal subgroups within a class of groups"},{"id":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/is-injector","kind":"node","lean":"MathlibExt.GroupTheory.FittingClassWithoutInjectorKourovka12Problem96Wanted.IsInjector","parent":1966,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/index.html#is-injector","scope":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes","status":"fully_proved","summary":"Let $\\mathfrak{X}$ be a class of groups and $G$ a group. A subgroup $V$ of $G$ is an $\\mathfrak{X}$-injector of $G$ if, for every subnormal subgroup $N$ of $G$, the intersection $V \\cap N$ is $\\mathfrak{X}$-maximal in $N$.","title":"Injectors"},{"id":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/fitting-class-without-injectors","kind":"node","lean":"MathlibExt.GroupTheory.FittingClassWithoutInjectorKourovka12Problem96Wanted.fittingClassWithoutInjectors","parent":1966,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/index.html#fitting-class-without-injectors","scope":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes","status":"fully_proved","summary":"The statement that there are a Fitting class $\\mathfrak{X}$ of finite groups that contains at least one finite group and a finite non-soluble group $G$ such that no subgroup of $G$ is an $\\mathfrak{X}$-injector of $G$.","title":"Kourovka Problem 12.96: a Fitting class without injectors"},{"id":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/fitting-class-without-injectors-holds","kind":"node","lean":"MathlibExt.GroupTheory.FittingClassWithoutInjectorKourovka12Problem96Wanted.fittingClassWithoutInjectors_holds","parent":1966,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes/index.html#fitting-class-without-injectors-holds","scope":"GroupTheory/soluble-finite-groups/formations/injectors-of-fitting-classes","status":"can_prove","summary":"There are a Fitting class $\\mathfrak{X}$ of finite groups that contains at least one finite group and a finite non-soluble group $G$ such that no subgroup of $G$ is an $\\mathfrak{X}$-injector of $G$.","title":"A Fitting class with no injectors in a non-soluble group"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/lattices-of-formations.json","id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"can_prove","title":"Lattices of formations"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/group-class","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.GroupClass","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#group-class","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A class of groups $\\mathfrak{X}$ is given by a membership predicate on groups whose underlying type lies in the lowest universe, such that whenever a group $H$ belongs to $\\mathfrak{X}$ and $K$ is a group isomorphic to $H$, the group $K$ also belongs to $\\mathfrak{X}$.","title":"Isomorphism-closed classes of groups (Kourovka 18.92)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-formation","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.IsFormation","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-formation","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A class $\\mathfrak{X}$ of groups is a formation if every group in $\\mathfrak{X}$ is finite, $\\mathfrak{X}$ contains the trivial group, for every finite group $G$ in $\\mathfrak{X}$ and every normal subgroup $N$ of $G$ the quotient $G/N$ lies in $\\mathfrak{X}$, and for every finite group $G$ with normal subgroups $N_1$ and $N_2$ such that $G/N_1$ and $G/N_2$ lie in $\\mathfrak{X}$, the quotient $G/(N_1 \\cap N_2)$ lies\u2026","title":"Formations of finite groups (Kourovka 18.92)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/incl","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.Incl","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#incl","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"For classes of groups $\\mathfrak{X}$ and $\\mathfrak{Y}$, the class $\\mathfrak{X}$ is included in $\\mathfrak{Y}$, written $\\mathfrak{X} \\subseteq \\mathfrak{Y}$, if every group that belongs to $\\mathfrak{X}$ also belongs to $\\mathfrak{Y}$.","title":"Inclusion of group classes"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-complete-lattice-of-formations","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.IsCompleteLatticeOfFormations","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-complete-lattice-of-formations","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A set $\\Theta$ of classes of groups is a complete lattice of formations if it is nonempty, every member of $\\Theta$ is a formation, for every nonempty subset $S \\subseteq \\Theta$ the intersection of the classes in $S$ (the class of groups belonging to every member of $S$) belongs to $\\Theta$, and $\\Theta$ has a member that contains every member of $\\Theta$.","title":"Complete lattices of formations"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-join-in","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.IsJoinIn","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-join-in","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"Let $\\Theta$ and $S$ be sets of classes of groups. A class $\\mathfrak{J}$ is the join of $S$ in $\\Theta$ if $\\mathfrak{J} \\in \\Theta$, every member of $S$ is contained in $\\mathfrak{J}$, and $\\mathfrak{J}$ is contained in every member of $\\Theta$ that contains every member of $S$.","title":"Joins in a set of group classes"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-compact-in","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.IsCompactIn","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-compact-in","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"Let $\\Theta$ be a set of classes of groups and $\\mathfrak{A}$ a class of groups. Then $\\mathfrak{A}$ is compact in $\\Theta$ if, for every subset $S \\subseteq \\Theta$ and every join $\\mathfrak{J}$ of $S$ in $\\Theta$ with $\\mathfrak{A} \\subseteq \\mathfrak{J}$, there are a finite subset $S_1 \\subseteq S$ and a join $\\mathfrak{J}_1$ of $S_1$ in $\\Theta$ with $\\mathfrak{A} \\subseteq \\mathfrak{J}_1$.","title":"Compact elements of a set of group classes"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-algebraic-in","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.IsAlgebraicIn","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-algebraic-in","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A set $\\Theta$ of classes of groups is algebraic if every member $\\mathfrak{A}$ of $\\Theta$ is the join in $\\Theta$ of some subset $S \\subseteq \\Theta$ all of whose members are compact in $\\Theta$.","title":"Algebraic sets of group classes"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-modular-in","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.IsModularIn","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-modular-in","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A set $\\Theta$ of classes of groups is modular if for all $\\mathfrak{A}, \\mathfrak{B}, \\mathfrak{C} \\in \\Theta$ with $\\mathfrak{A} \\subseteq \\mathfrak{C}$ the following holds: whenever $\\mathfrak{J}_1$ is the join in $\\Theta$ of the set $\\{\\mathfrak{A}, \\mathfrak{B} \\cap \\mathfrak{C}\\}$ and $\\mathfrak{D}$ is the join in $\\Theta$ of the set $\\{\\mathfrak{A}, \\mathfrak{B}\\}$, the classes $\\mathfrak{J}_1$ and\u2026","title":"Modular sets of group classes"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/non-algebraic-complete-lattice-of-format","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.nonAlgebraicCompleteLatticeOfFormations","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#non-algebraic-complete-lattice-of-format","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"The proposition that some set $\\Theta$ of classes of groups is a complete lattice of formations of finite groups and is not algebraic.","title":"Kourovka Problem 18.92(a) (A. N. Skiba)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/non-modular-complete-lattice-of-formatio","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.nonModularCompleteLatticeOfFormations","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#non-modular-complete-lattice-of-formatio","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"The proposition that there is a complete lattice of formations of finite groups that is not modular, joins being least upper bounds in the lattice and meets being intersections.","title":"The non-modular formation lattice problem (Kourovka 18.92(b), A. N. Skiba)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/non-algebraic-complete-lattice-of-format-2","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.nonAlgebraicCompleteLatticeOfFormations_holds","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#non-algebraic-complete-lattice-of-format-2","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"can_prove","summary":"There is a set $\\Theta$ of classes of groups that is a complete lattice of formations of finite groups and is not algebraic; that is, Kourovka Problem 18.92(a) has a positive answer.","title":"Non-algebraic complete lattices of formations exist"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/non-modular-complete-lattice-of-formatio-2","kind":"node","lean":"MathlibExt.GroupTheory.CompleteLatticeOfFormationsKourovka18Problem92Wanted.nonModularCompleteLatticeOfFormations_holds","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#non-modular-complete-lattice-of-formatio-2","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"can_prove","summary":"There is a complete lattice of formations of finite groups that is not modular: a nonempty set $\\Theta$ of formations of finite groups, ordered by inclusion, that contains the intersection of each of its nonempty subsets and has a largest element, in which the modular law $A \\vee (B \\wedge C) = (A \\vee B) \\wedge C$ for $A \\le C$ fails, joins being least upper bounds in $\\Theta$ and meets intersections.","title":"A non-modular complete lattice of formations exists (Kourovka 18.92(b))"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/finite-group-class","kind":"node","lean":"MathlibExt.GroupTheory.FiniteModularLatticeFormationsKourovka13Problem51Wanted.FiniteGroupClass","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#finite-group-class","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A class of finite groups is a property of finite groups: for each finite group $G$ it states whether $G$ belongs to the class.","title":"Class of finite groups (Kourovka 13.51)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-formation-2","kind":"node","lean":"MathlibExt.GroupTheory.FiniteModularLatticeFormationsKourovka13Problem51Wanted.IsFormation","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-formation-2","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A class $\\mathfrak F$ of finite groups is a formation when it is closed under homomorphic images (if $G \\in \\mathfrak F$ and $\\varphi : G \\to H$ is a surjective homomorphism of finite groups, then $H \\in \\mathfrak F$) and, for every finite group $G$ and normal subgroups $N_1$ and $N_2$ of $G$, $G/N_1 \\in \\mathfrak F$ and $G/N_2 \\in \\mathfrak F$ imply $G/(N_1 \\cap N_2) \\in \\mathfrak F$ (closure under finite\u2026","title":"Formation of finite groups (Kourovka 13.51)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/class-meet","kind":"node","lean":"MathlibExt.GroupTheory.FiniteModularLatticeFormationsKourovka13Problem51Wanted.classMeet","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#class-meet","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"For classes $\\mathfrak F_1$ and $\\mathfrak F_2$ of finite groups, their intersection $\\mathfrak F_1 \\cap \\mathfrak F_2$ is the class of finite groups $G$ with $G \\in \\mathfrak F_1$ and $G \\in \\mathfrak F_2$. For formations $\\mathfrak F_1$ and $\\mathfrak F_2$ it is their meet in the lattice of formations.","title":"Intersection of classes of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/formation-join","kind":"node","lean":"MathlibExt.GroupTheory.FiniteModularLatticeFormationsKourovka13Problem51Wanted.formationJoin","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#formation-join","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"For classes $\\mathfrak F_1$ and $\\mathfrak F_2$ of finite groups, their formation join is the class of finite groups $G$ that belong to every formation containing all groups of $\\mathfrak F_1$ and of $\\mathfrak F_2$: the intersection of all formations containing $\\mathfrak F_1 \\cup \\mathfrak F_2$, which is the smallest such formation. For formations $\\mathfrak F_1$ and $\\mathfrak F_2$ it is their join in the\u2026","title":"Join of formations (Kourovka 13.51)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/finite-modular-lattice-embeds-in-formati","kind":"node","lean":"MathlibExt.GroupTheory.FiniteModularLatticeFormationsKourovka13Problem51Wanted.finiteModularLatticeEmbedsInFormations","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#finite-modular-lattice-embeds-in-formati","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"For every finite modular lattice $L$ there is an injective map $f$ from $L$ to the formations of finite groups such that, for all $a, b \\in L$, $f(a \\wedge b) = f(a) \\cap f(b)$ and $f(a \\vee b)$ is the smallest formation containing $f(a) \\cup f(b)$.","title":"Kourovka Problem 13.51 (Skiba): finite modular lattices embed in the lattice of formations"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/finite-modular-lattice-embeds-in-formati-2","kind":"node","lean":"MathlibExt.GroupTheory.FiniteModularLatticeFormationsKourovka13Problem51Wanted.finiteModularLatticeEmbedsInFormations_refuted","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#finite-modular-lattice-embeds-in-formati-2","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"can_prove","summary":"There is a finite modular lattice $L$ that admits no injective map $f$ from $L$ to the formations of finite groups such that, for all $a, b \\in L$, $f(a \\wedge b) = f(a) \\cap f(b)$ and $f(a \\vee b)$ is the smallest formation containing $f(a) \\cup f(b)$. This answers Kourovka Problem 13.51 (Skiba) in the negative.","title":"Not every finite modular lattice embeds in the lattice of formations"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/class-le","kind":"node","lean":"MathlibExt.GroupTheory.FiniteGroupFormationShared.ClassLE","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#class-le","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"For classes $\\mathfrak{F}$ and $\\mathfrak{G}$ of finite groups, $\\mathfrak{F} \\subseteq \\mathfrak{G}$ means that every finite group in $\\mathfrak{F}$ belongs to $\\mathfrak{G}$.","title":"Inclusion of classes of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-local-formation","kind":"node","lean":"MathlibExt.GroupTheory.FiniteGroupFormationShared.IsLocalFormation","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-local-formation","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A class $\\mathfrak{F}$ of finite groups is a local formation when it is a formation, contains the trivial group, and is saturated: a finite group $G$ belongs to $\\mathfrak{F}$ whenever $G/\\Phi(G)$ does, where $\\Phi(G)$ is the Frattini subgroup of $G$.","title":"Local formation of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/residual","kind":"node","lean":"MathlibExt.GroupTheory.FiniteGroupFormationShared.residual","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#residual","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"For a class $\\mathfrak{F}$ of finite groups and a finite group $G$, the $\\mathfrak{F}$-residual $G^{\\mathfrak{F}}$ is the intersection of all normal subgroups $N$ of $G$ with $G/N \\in \\mathfrak{F}$.","title":"Residual of a finite group with respect to a class"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/group-class-2","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentFormationLatticeDistributiveKourovka10Problem56Wanted.GroupClass","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#group-class-2","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A class of finite groups is given by its membership predicate: a proposition about each finite group, true exactly for the members of the class.","title":"Classes of finite groups (Kourovka 10.56)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-formation-3","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentFormationLatticeDistributiveKourovka10Problem56Wanted.IsFormation","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-formation-3","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A class $X$ of finite groups is a formation if it is closed under isomorphism, contains the trivial group, contains $G/N$ whenever it contains $G$ and $N$ is a normal subgroup of $G$, and is closed under subdirect products: if $N_1$ and $N_2$ are normal subgroups of a finite group $G$ with $G/N_1 \\in X$ and $G/N_2 \\in X$, then $G/(N_1 \\cap N_2) \\in X$.","title":"Formations of finite groups (Kourovka 10.56)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/is-nilpotent-class-four-formation","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentFormationLatticeDistributiveKourovka10Problem56Wanted.IsNilpotentClassFourFormation","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#is-nilpotent-class-four-formation","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"A class $X$ of finite groups is a formation of nilpotent groups of class at most $4$ if $X$ is a formation and every group $G$ in $X$ is nilpotent of class at most $4$, that is, the fourth term $\\zeta_4(G)$ of the upper central series of $G$ is all of $G$.","title":"Formations of nilpotent groups of class at most 4"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/meet","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentFormationLatticeDistributiveKourovka10Problem56Wanted.meet","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#meet","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"The meet of two classes $X$ and $Y$ of finite groups is their intersection $X \\cap Y$: the class of finite groups that belong to both $X$ and $Y$.","title":"Intersection of classes of finite groups (Kourovka 10.56)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/formation-join-2","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentFormationLatticeDistributiveKourovka10Problem56Wanted.formationJoin","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#formation-join-2","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"For classes $X$ and $Y$ of finite groups, the formation generated by $X \\cup Y$ is the class of finite groups that belong to every formation containing both $X$ and $Y$. For formations $X$ and $Y$ it is their join in the lattice of formations.","title":"Formation generated by two classes (Kourovka 10.56)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/nilpotent-class-four-formations-distribu","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentFormationLatticeDistributiveKourovka10Problem56Wanted.nilpotentClassFourFormationsDistributive","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#nilpotent-class-four-formations-distribu","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"fully_proved","summary":"For all formations $A$, $B$, $C$ of finite nilpotent groups of class at most $4$, a finite group lies in $A \\cap (B \\vee C)$ if and only if it lies in $(A \\cap B) \\vee (A \\cap C)$, where $X \\vee Y$ denotes the formation generated by $X \\cup Y$. That is, the lattice of formations of finite nilpotent groups of class at most $4$ is distributive (A. N. Skiba).","title":"The distributivity question for nilpotent class-4 formations (Kourovka 10.56)"},{"id":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations/nilpotent-class-four-formations-distribu-2","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentFormationLatticeDistributiveKourovka10Problem56Wanted.nilpotentClassFourFormationsDistributive_refuted","parent":1973,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/lattices-of-formations/index.html#nilpotent-class-four-formations-distribu-2","scope":"GroupTheory/soluble-finite-groups/formations/lattices-of-formations","status":"can_prove","summary":"There exist formations $A$, $B$, $C$ of finite nilpotent groups of class at most $4$ such that the classes $A \\cap (B \\vee C)$ and $(A \\cap B) \\vee (A \\cap C)$ differ, that is, some finite group lies in exactly one of them; here $X \\vee Y$ denotes the formation generated by $X \\cup Y$, the intersection of all formations containing $X$ and $Y$.","title":"The lattice of nilpotent class-4 formations is not distributive (Kourovka 10.56)"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations.json","id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"can_prove","title":"Radical hereditary formations"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/group-class","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.GroupClass","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#group-class","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"fully_proved","summary":"A class of groups $X$ is a property of groups that is invariant under isomorphism: if a group $H$ belongs to $X$ and $K$ is a group isomorphic to $H$, then $K$ belongs to $X$.","title":"Isomorphism-closed class of groups"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/quot-mem","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.QuotMem","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#quot-mem","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"fully_proved","summary":"For a class of groups $X$, a group $G$ and a subgroup $N$ of $G$, this holds when $N$ is normal in $G$ and the quotient group $G/N$ belongs to $X$.","title":"Quotient lying in a class"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/is-formation","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.IsFormation","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#is-formation","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"fully_proved","summary":"A class of groups $X$ is a formation if it contains the trivial group, for every finite group $G$ in $X$ and every normal subgroup $N$ of $G$ the quotient $G/N$ belongs to $X$, and for every finite group $G$ with normal subgroups $N_1, N_2$ such that $G/N_1$ and $G/N_2$ belong to $X$, the quotient $G/(N_1 \\cap N_2)$ belongs to $X$.","title":"Formation of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/is-radical","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.IsRadical","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#is-radical","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"fully_proved","summary":"A class of groups $X$ is radical if, for every finite group $G$, every normal subgroup of $G$ belongs to $X$ whenever $G$ does, and whenever $N_1, N_2$ are normal subgroups of $G$ that belong to $X$, their product $N_1 N_2$ belongs to $X$.","title":"Radical (Fitting) class"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/is-hereditary","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.IsHereditary","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#is-hereditary","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"fully_proved","summary":"A class of groups $X$ is hereditary if every subgroup of a finite group in $X$ belongs to $X$.","title":"Hereditary class of groups"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/is-soluble-radical","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.IsSolubleRadical","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#is-soluble-radical","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"fully_proved","summary":"A subgroup $R$ of a group $G$ is the soluble radical of $G$ if $R$ is a soluble normal subgroup of $G$ that contains every soluble normal subgroup of $G$.","title":"Soluble radical"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/is-composition","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.IsComposition","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#is-composition","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"fully_proved","summary":"A class of groups $X$ is solubly saturated (a composition class) if, for every finite group $G$ with soluble radical $R$, whenever the Frattini subgroup $\\Phi(R)$ of $R$ is normal in $G$ and $G/\\Phi(R)$ belongs to $X$, the group $G$ belongs to $X$.","title":"Solubly saturated class"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/radical-hereditary-formation-is-composit","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.radicalHereditaryFormationIsComposition","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#radical-hereditary-formation-is-composit","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"fully_proved","summary":"The statement that every radical hereditary formation of finite groups is a composition formation: every class of groups that is a formation, a radical class and a hereditary class is solubly saturated.","title":"Kourovka Problem 12.7 (Vasil'ev)"},{"id":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/radical-hereditary-formation-is-composit-2","kind":"node","lean":"MathlibExt.GroupTheory.RadicalHereditaryFormationKourovka12Problem7Wanted.radicalHereditaryFormationIsComposition_refuted","parent":2002,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations/index.html#radical-hereditary-formation-is-composit-2","scope":"GroupTheory/soluble-finite-groups/formations/radical-hereditary-formations","status":"can_prove","summary":"It is not true that every radical hereditary formation of finite groups is a composition formation: some class of groups that is a formation, a radical class and a hereditary class is not solubly saturated.","title":"Radical hereditary formations need not be composition formations"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr.json","id":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/index.html","scope":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr","status":"can_prove","title":"Subformations of the formation generated by a group"},{"id":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/finite-group-class","kind":"node","lean":"MathlibExt.GroupTheory.FormationSubformationsFiniteKourovka16Problem31Wanted.FiniteGroupClass","parent":2012,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/index.html#finite-group-class","scope":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr","status":"fully_proved","summary":"A class of finite groups is a predicate on finite groups: it assigns to every finite group $G$ a proposition, read as \"$G$ belongs to the class\". The class need not be closed under isomorphism.","title":"Classes of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/is-formation","kind":"node","lean":"MathlibExt.GroupTheory.FormationSubformationsFiniteKourovka16Problem31Wanted.IsFormation","parent":2012,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/index.html#is-formation","scope":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr","status":"fully_proved","summary":"A class $\\mathfrak{F}$ of finite groups is a formation if it is closed under homomorphic images (whenever $f \\colon G \\to H$ is a surjective homomorphism of finite groups and $G \\in \\mathfrak{F}$, then $H \\in \\mathfrak{F}$) and under finite subdirect products (whenever $N_1$ and $N_2$ are normal subgroups of a finite group $G$ with $G / N_1 \\in \\mathfrak{F}$ and $G / N_2 \\in \\mathfrak{F}$, then\u2026","title":"Formations of finite groups, possibly empty"},{"id":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/formation-generated-by","kind":"node","lean":"MathlibExt.GroupTheory.FormationSubformationsFiniteKourovka16Problem31Wanted.formationGeneratedBy","parent":2012,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/index.html#formation-generated-by","scope":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr","status":"fully_proved","summary":"For a finite group $G$, the formation generated by $G$ is the class of finite groups $H$ such that $H$ belongs to every formation that contains $G$; that is, it is the intersection of all formations containing $G$.","title":"Formation generated by a finite group"},{"id":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/finitely-many-subformations","kind":"node","lean":"MathlibExt.GroupTheory.FormationSubformationsFiniteKourovka16Problem31Wanted.finitelyManySubformations","parent":2012,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/index.html#finitely-many-subformations","scope":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr","status":"fully_proved","summary":"The statement (V. A. Vedernikov) that for every finite group $G$ only finitely many formations of finite groups are contained in the formation generated by $G$. Here formations may be empty, and a formation $\\mathfrak{F}$ is contained in the formation generated by $G$ when every finite group in $\\mathfrak{F}$ belongs to the formation generated by $G$.","title":"Kourovka Problem 16.31 for finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/finitely-many-subformations-refuted","kind":"node","lean":"MathlibExt.GroupTheory.FormationSubformationsFiniteKourovka16Problem31Wanted.finitelyManySubformations_refuted","parent":2012,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr/index.html#finitely-many-subformations-refuted","scope":"GroupTheory/soluble-finite-groups/formations/subformations-of-the-formation-generated-by-a-gr","status":"can_prove","summary":"There is a finite group $G$ such that infinitely many formations are contained in the formation generated by $G$. That is, the answer to Kourovka Problem 16.31 is negative, even for finite $G$.","title":"The formation generated by a finite group can have infinitely many subformations"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups.json","id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","kind":"scope","navigation":"module","parent":1930,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"can_prove","title":"Subgroup functors and closed classes of groups"},{"id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/finite-group-class","kind":"node","lean":"MathlibExt.GroupTheory.ClosedSubgroupFunctorLatticeWidthKourovka16Problem82Wanted.FiniteGroupClass","parent":2018,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html#finite-group-class","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"fully_proved","summary":"A class $\\mathfrak{X}$ of finite groups is a predicate on finite groups $G$ (with carrier in the lowest type universe, together with their group structure); the groups satisfying it are the members of the class, written $G \\in \\mathfrak{X}$.","title":"Classes of finite groups (Kourovka 16.82 setting)"},{"id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/is-closed-class","kind":"node","lean":"MathlibExt.GroupTheory.ClosedSubgroupFunctorLatticeWidthKourovka16Problem82Wanted.IsClosedClass","parent":2018,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html#is-closed-class","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"fully_proved","summary":"A class $\\mathfrak{X}$ of finite groups is closed when it contains at least one group and is closed under homomorphic images, subgroups and direct products: if $G \\in \\mathfrak{X}$ and $f : G \\to H$ is a surjective homomorphism onto a finite group $H$, then $H \\in \\mathfrak{X}$; every subgroup of a group in $\\mathfrak{X}$ lies in $\\mathfrak{X}$; and if $G, H \\in \\mathfrak{X}$, then $G \\times H \\in \\mathfrak{X}$.","title":"Closed classes of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/subgroup-assignment","kind":"node","lean":"MathlibExt.GroupTheory.ClosedSubgroupFunctorLatticeWidthKourovka16Problem82Wanted.SubgroupAssignment","parent":2018,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html#subgroup-assignment","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"fully_proved","summary":"A subgroup assignment $\\tau$ assigns to every finite group $G$ (with carrier in the lowest type universe) a set $\\tau(G)$ of subgroups of $G$.","title":"Subgroup assignments"},{"id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/is-closed-subgroup-functor","kind":"node","lean":"MathlibExt.GroupTheory.ClosedSubgroupFunctorLatticeWidthKourovka16Problem82Wanted.IsClosedSubgroupFunctor","parent":2018,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html#is-closed-subgroup-functor","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"fully_proved","summary":"Let $\\mathfrak{X}$ be a class of finite groups. A subgroup assignment $\\tau$ is a closed subgroup functor on $\\mathfrak{X}$ (A. N. Skiba) when the following three conditions hold. For every $G \\in \\mathfrak{X}$, $G \\in \\tau(G)$. For every surjective homomorphism $\\varphi : A \\to B$ with $A, B \\in \\mathfrak{X}$, $\\varphi(H) \\in \\tau(B)$ for every $H \\in \\tau(A)$ and $\\varphi^{-1}(T) \\in \\tau(A)$ for every\u2026","title":"Closed subgroup functors"},{"id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/functor-le","kind":"node","lean":"MathlibExt.GroupTheory.ClosedSubgroupFunctorLatticeWidthKourovka16Problem82Wanted.FunctorLE","parent":2018,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html#functor-le","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"fully_proved","summary":"Let $\\mathfrak{X}$ be a class of finite groups. For subgroup assignments $\\tau_1$ and $\\tau_2$, $\\tau_1 \\le \\tau_2$ on $\\mathfrak{X}$ when $\\tau_1(G) \\subseteq \\tau_2(G)$ for every $G \\in \\mathfrak{X}$. This is the order of the lattice $F(\\mathfrak{X})$ of closed subgroup functors on $\\mathfrak{X}$.","title":"Order on subgroup functors"},{"id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/primes-of","kind":"node","lean":"MathlibExt.GroupTheory.ClosedSubgroupFunctorLatticeWidthKourovka16Problem82Wanted.primesOf","parent":2018,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html#primes-of","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"fully_proved","summary":"For a class $\\mathfrak{X}$ of finite groups, $\\pi(\\mathfrak{X})$ is the set of primes $p$ that divide the order of some group in $\\mathfrak{X}$.","title":"Prime divisors of a class of finite groups"},{"id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/exists-nonnilpotent-class-of-small-width","kind":"node","lean":"MathlibExt.GroupTheory.ClosedSubgroupFunctorLatticeWidthKourovka16Problem82Wanted.existsNonnilpotentClassOfSmallWidth","parent":2018,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html#exists-nonnilpotent-class-of-small-width","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"fully_proved","summary":"For a class $\\mathfrak{X}$ of finite groups, let $\\pi(\\mathfrak{X})$ be the set of primes dividing the order of some group in $\\mathfrak{X}$, and for subgroup assignments $\\tau_1, \\tau_2$ write $\\tau_1 \\le \\tau_2$ on $\\mathfrak{X}$ when $\\tau_1(G) \\subseteq \\tau_2(G)$ for every $G \\in \\mathfrak{X}$. There is a class $\\mathfrak{X}$ of finite groups that contains at least one group, is closed under homomorphic\u2026","title":"Kourovka Problem 16.82 (Skiba)"},{"id":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/exists-nonnilpotent-class-of-small-width-2","kind":"node","lean":"MathlibExt.GroupTheory.ClosedSubgroupFunctorLatticeWidthKourovka16Problem82Wanted.existsNonnilpotentClassOfSmallWidth_refuted","parent":2018,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups/index.html#exists-nonnilpotent-class-of-small-width-2","scope":"GroupTheory/soluble-finite-groups/formations/subgroup-functors-and-closed-classes-of-groups","status":"can_prove","summary":"For a class $\\mathfrak{X}$ of finite groups, let $\\pi(\\mathfrak{X})$ be the set of primes dividing the order of some group in $\\mathfrak{X}$, and for subgroup assignments $\\tau_1, \\tau_2$ write $\\tau_1 \\le \\tau_2$ on $\\mathfrak{X}$ when $\\tau_1(G) \\subseteq \\tau_2(G)$ for every $G \\in \\mathfrak{X}$. There is no class $\\mathfrak{X}$ of finite groups that contains at least one group, is closed under homomorphic\u2026","title":"Kourovka Problem 16.82 has a negative answer"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations.json","id":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/index.html","scope":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations","status":"can_prove","title":"Abnormal supersoluble factorizations"},{"id":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/is-supersoluble","kind":"node","lean":"MathlibExt.GroupTheory.AbnormalSupersolubleFactorizationKourovka19Problem100Wanted.IsSupersoluble","parent":2027,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/index.html#is-supersoluble","scope":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations","status":"fully_proved","summary":"A group $G$ is supersoluble if there are a natural number $k$ and subgroups $1 = S_0 \\le S_1 \\le \\cdots \\le S_k = G$, each normal in $G$, such that for every $i < k$ we have $S_i \\le S_{i+1}$ and there is an element $x \\in S_{i+1}$ such that $S_{i+1}$ is generated by $x$ and $S_i$; that is, every factor $S_{i+1}/S_i$ is cyclic.","title":"Supersoluble groups via cyclic normal series"},{"id":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/is-abnormal","kind":"node","lean":"MathlibExt.GroupTheory.AbnormalSupersolubleFactorizationKourovka19Problem100Wanted.IsAbnormal","parent":2027,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/index.html#is-abnormal","scope":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is abnormal in $G$ if every element $g \\in G$ lies in the subgroup generated by $H$ and its conjugate $gHg^{-1}$.","title":"Abnormal subgroup"},{"id":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/abnormal-supersoluble-tri-factorization","kind":"node","lean":"MathlibExt.GroupTheory.AbnormalSupersolubleFactorizationKourovka19Problem100Wanted.abnormalSupersolubleTriFactorizationSupersoluble","parent":2027,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/index.html#abnormal-supersoluble-tri-factorization","scope":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations","status":"fully_proved","summary":"The proposition that for every finite group $G$ and all subgroups $A$, $B$, $C$ of $G$ that are abnormal in $G$ and supersoluble, if $G = AB = BC = CA$ as sets, then $G$ is supersoluble.","title":"Abnormal supersoluble trifactorizations (Kourovka 19.100)"},{"id":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/abnormal-supersoluble-tri-factorization-2","kind":"node","lean":"MathlibExt.GroupTheory.AbnormalSupersolubleFactorizationKourovka19Problem100Wanted.abnormalSupersolubleTriFactorizationSupersoluble_holds","parent":2027,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations/index.html#abnormal-supersoluble-tri-factorization-2","scope":"GroupTheory/soluble-finite-groups/abnormal-supersoluble-factorizations","status":"can_prove","summary":"Let $G$ be a finite group and let $A$, $B$, $C$ be subgroups of $G$, each abnormal in $G$ and supersoluble, such that $G = AB = BC = CA$ as sets. Then $G$ is supersoluble.","title":"Kourovka Problem 19.100 has a positive answer"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/alperin-groups.json","id":"GroupTheory/soluble-finite-groups/alperin-groups","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/alperin-groups/index.html","scope":"GroupTheory/soluble-finite-groups/alperin-groups","status":"can_prove","title":"Alperin groups"},{"id":"GroupTheory/soluble-finite-groups/alperin-groups/is-alperin-group","kind":"node","lean":"MathlibExt.GroupTheory.AlperinGroupDerivedLengthKourovka18Problem33Wanted.IsAlperinGroup","parent":2032,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/alperin-groups/index.html#is-alperin-group","scope":"GroupTheory/soluble-finite-groups/alperin-groups","status":"fully_proved","summary":"A group $G$ is an Alperin group when, for all $x, y \\in G$, the derived subgroup $[H, H]$ of the subgroup $H = \\langle x, y \\rangle$ generated by $x$ and $y$ is cyclic.","title":"Alperin group"},{"id":"GroupTheory/soluble-finite-groups/alperin-groups/alperin-group-derived-length-at-most-six","kind":"node","lean":"MathlibExt.GroupTheory.AlperinGroupDerivedLengthKourovka18Problem33Wanted.alperinGroupDerivedLengthAtMostSix","parent":2032,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/alperin-groups/index.html#alperin-group-derived-length-at-most-six","scope":"GroupTheory/soluble-finite-groups/alperin-groups","status":"fully_proved","summary":"This proposition asserts that every finite Alperin group $G$ has derived length at most $6$, that is, its sixth derived subgroup $G^{(6)}$ is trivial. Here $G$ is an Alperin group when, for all $x, y \\in G$, the derived subgroup of the subgroup $\\langle x, y \\rangle$ generated by $x$ and $y$ is cyclic.","title":"Kourovka Problem 18.33 on the derived length of finite Alperin groups"},{"id":"GroupTheory/soluble-finite-groups/alperin-groups/alperin-group-derived-length-at-most-six-2","kind":"node","lean":"MathlibExt.GroupTheory.AlperinGroupDerivedLengthKourovka18Problem33Wanted.alperinGroupDerivedLengthAtMostSix_holds","parent":2032,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/alperin-groups/index.html#alperin-group-derived-length-at-most-six-2","scope":"GroupTheory/soluble-finite-groups/alperin-groups","status":"can_prove","summary":"Let $G$ be a finite group in which, for all $x, y \\in G$, the derived subgroup of the subgroup $\\langle x, y \\rangle$ generated by $x$ and $y$ is cyclic (an Alperin group). Then the sixth derived subgroup $G^{(6)}$ of $G$ is trivial, that is, $G$ has derived length at most $6$.","title":"Finite Alperin groups have derived length at most six"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/products-of-maximal-subgroups.json","id":"GroupTheory/soluble-finite-groups/products-of-maximal-subgroups","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/products-of-maximal-subgroups/index.html","scope":"GroupTheory/soluble-finite-groups/products-of-maximal-subgroups","status":"can_prove","title":"Products of maximal subgroups"},{"id":"GroupTheory/soluble-finite-groups/products-of-maximal-subgroups/nonsoluble-product-of-nonconjugate-maxim","kind":"node","lean":"MathlibExt.GroupTheory.NonconjugateMaximalSubgroupProductsKourovka10Problem34Wanted.nonsolubleProductOfNonconjugateMaximals","parent":2036,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/products-of-maximal-subgroups/index.html#nonsoluble-product-of-nonconjugate-maxim","scope":"GroupTheory/soluble-finite-groups/products-of-maximal-subgroups","status":"fully_proved","summary":"The statement that there is a finite group $G$ that is not soluble such that $G = MN$ for every two maximal subgroups $M$ and $N$ of $G$ that are not conjugate in $G$. Here $MN$ is the set of products $mn$ with $m \\in M$ and $n \\in N$.","title":"Non-soluble products of non-conjugate maximal subgroups (Kourovka 10.34, Monakhov)"},{"id":"GroupTheory/soluble-finite-groups/products-of-maximal-subgroups/nonsoluble-product-of-nonconjugate-maxim-2","kind":"node","lean":"MathlibExt.GroupTheory.NonconjugateMaximalSubgroupProductsKourovka10Problem34Wanted.nonsolubleProductOfNonconjugateMaximals_refuted","parent":2036,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/products-of-maximal-subgroups/index.html#nonsoluble-product-of-nonconjugate-maxim-2","scope":"GroupTheory/soluble-finite-groups/products-of-maximal-subgroups","status":"can_prove","summary":"There is no finite group $G$ that is not soluble such that $G = MN$ for every two maximal subgroups $M$ and $N$ of $G$ that are not conjugate in $G$; here $MN$ is the set of products $mn$ with $m \\in M$ and $n \\in N$. This answers Kourovka Problem 10.34 in the negative.","title":"Finite groups that are the product of every two non-conjugate maximal subgroups are soluble"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/supersolubility-criteria.json","id":"GroupTheory/soluble-finite-groups/supersolubility-criteria","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersolubility-criteria/index.html","scope":"GroupTheory/soluble-finite-groups/supersolubility-criteria","status":"can_prove","title":"Supersolubility criteria"},{"id":"GroupTheory/soluble-finite-groups/supersolubility-criteria/is-supersoluble","kind":"node","lean":"MathlibExt.GroupTheory.NonSupersolubleUncomplementedMaximalKourovka15Problem81Wanted.IsSupersoluble","parent":2039,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersolubility-criteria/index.html#is-supersoluble","scope":"GroupTheory/soluble-finite-groups/supersolubility-criteria","status":"fully_proved","summary":"A group $G$ is supersoluble when it has a series $1 = H_0 \\le H_1 \\le \\cdots \\le H_n = G$ of subgroups, each normal in $G$, such that every factor $H_{i+1}/H_i$ is cyclic.","title":"Supersoluble group (Kourovka 15.81)"},{"id":"GroupTheory/soluble-finite-groups/supersolubility-criteria/non-supersoluble-has-uncomplemented-sylo","kind":"node","lean":"MathlibExt.GroupTheory.NonSupersolubleUncomplementedMaximalKourovka15Problem81Wanted.nonSupersolubleHasUncomplementedSylowMaximal","parent":2039,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersolubility-criteria/index.html#non-supersoluble-has-uncomplemented-sylo","scope":"GroupTheory/soluble-finite-groups/supersolubility-criteria","status":"fully_proved","summary":"The statement that every finite group $G$ that is not supersoluble has, for some prime $p$, a non-cyclic Sylow $p$-subgroup $P$ with a maximal subgroup $M$ of $P$ that has no proper complement in $G$, that is, there is no subgroup $K \\ne G$ with $MK = G$ and $M \\cap K = 1$.","title":"Uncomplemented maximal subgroups of Sylow subgroups (Kourovka 15.81, Skiba)"},{"id":"GroupTheory/soluble-finite-groups/supersolubility-criteria/non-supersoluble-has-uncomplemented-sylo-2","kind":"node","lean":"MathlibExt.GroupTheory.NonSupersolubleUncomplementedMaximalKourovka15Problem81Wanted.nonSupersolubleHasUncomplementedSylowMaximal_holds","parent":2039,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersolubility-criteria/index.html#non-supersoluble-has-uncomplemented-sylo-2","scope":"GroupTheory/soluble-finite-groups/supersolubility-criteria","status":"can_prove","summary":"For every finite group $G$ that is not supersoluble, there are a prime $p$ and a non-cyclic Sylow $p$-subgroup $P$ of $G$ with a maximal subgroup $M$ of $P$ that has no proper complement in $G$: there is no subgroup $K \\ne G$ with $MK = G$ and $M \\cap K = 1$. This answers Kourovka Problem 15.81 in the affirmative.","title":"Finite non-supersoluble groups have uncomplemented maximal subgroups of Sylow subgroups"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality.json","id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"can_prove","title":"\u03c3-solubility, \u03c3-nilpotency and \u03c3-subnormality"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-chief-factor","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsChiefFactor","parent":2043,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-chief-factor","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"For subgroups $A$ and $B$ of a group $G$, $A / B$ is a chief factor of $G$ if $A$ and $B$ are normal in $G$, $B < A$, and every normal subgroup $N$ of $G$ with $B \\le N \\le A$ equals $B$ or $A$.","title":"Chief factors"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/fitting-height-le","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.FittingHeightLE","parent":2043,"reading_title":{"format":"inline-parts-v1","parts":[["text","Groups of Fitting height at most "],["math","\\(h\\)"]],"source":"Groups of Fitting height at most $h$"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#fitting-height-le","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"For a natural number $h$, a group $G$ has Fitting height (nilpotent length) at most $h$ if there are normal subgroups $1 = G_0 \\le G_1 \\le \\cdots \\le G_h = G$ of $G$ such that for every $i < h$ some term of the lower central series of $G_{i+1}$ lies in $G_i$, that is, $G_{i+1} / G_i$ is nilpotent.","title":"Groups of Fitting height at most $h$"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-composition-factor","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsCompositionFactor","parent":2043,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-composition-factor","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"A group $S$ is a composition factor of a group $K$ if $S$ is simple and some subnormal subgroup $H$ of $K$ has a surjective homomorphism onto $S$.","title":"Composition factors via subnormal subgroups"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-conjugacy-separable","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsConjugacySeparable","parent":2043,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-conjugacy-separable","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"A group $G$ is conjugacy separable if any two elements $x, y \\in G$ whose images are conjugate in every quotient $G / N$ by a normal subgroup $N$ of finite index (that is, $g x g^{-1} y^{-1} \\in N$ for some $g \\in G$) are conjugate in $G$.","title":"Conjugacy separable groups"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-residually-finite-p","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsResiduallyFiniteP","parent":2043,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residually finite "],["math","\\(p\\)"],["text","-groups"]],"source":"Residually finite $p$-groups"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-residually-finite-p","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"For a natural number $p$, a group $G$ is residually a finite $p$-group if every element $g \\ne 1$ of $G$ lies outside some normal subgroup of $G$ whose index is $p^k$ for some $k \\ge 0$, where an infinite index counts as $0$.","title":"Residually finite $p$-groups"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/restricted-wreath","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.restrictedWreath","parent":2043,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#restricted-wreath","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"For groups $A$ and $B$, let $A \\wr_r B = A^B \\rtimes B$ be the regular (unrestricted) wreath product, where $A^B$ is the group of all functions $f : B \\to A$ and $b \\in B$ acts on it by $(b \\cdot f)(x) = f(b^{-1} x)$. The restricted wreath product $A \\wr B$ is the subgroup of $A \\wr_r B$ generated by the elements $(f, b)$ whose function part $f$ has finite support, that is, $f(x) \\ne 1$ for only finitely many\u2026","title":"Restricted wreath product"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-prime-partition","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsPrimePartition","parent":2043,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-prime-partition","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"A family $\\sigma = (\\sigma_i)_{i \\in I}$ of sets of natural numbers is a partition of the primes if every $\\sigma_i$ is a nonempty set of primes, $\\sigma_i$ and $\\sigma_j$ are disjoint whenever $i \\ne j$, and every prime lies in some $\\sigma_i$.","title":"Partitions of the primes"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-sigma-primary","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsSigmaPrimary","parent":2043,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\sigma\\)"],["text","-primary numbers"]],"source":"$\\sigma$-primary numbers"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-sigma-primary","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"For a family $\\sigma = (\\sigma_i)_{i \\in I}$ of sets of natural numbers, a natural number $n$ is $\\sigma$-primary if there is an index $i$ such that every prime divisor of $n$ lies in $\\sigma_i$. For $n = 0$ this means that some $\\sigma_i$ contains every prime.","title":"$\\sigma$-primary numbers"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-sigma-subnormal","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsSigmaSubnormal","parent":2043,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\sigma\\)"],["text","-subnormal subgroups"]],"source":"$\\sigma$-subnormal subgroups"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-sigma-subnormal","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"For a family $\\sigma = (\\sigma_i)_{i \\in I}$ of sets of natural numbers, a subgroup $A$ of a group $G$ is $\\sigma$-subnormal in $G$ if there is a chain of subgroups $A = A_0 \\le A_1 \\le \\cdots \\le A_n = G$ such that for every $i < n$ either $A_i$ is normal in $A_{i+1}$, or the index in $A_{i+1}$ of the normal core of $A_i$ in $A_{i+1}$ is $\\sigma$-primary (an infinite index counts as $0$).","title":"$\\sigma$-subnormal subgroups"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-sigma-nilpotent","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsSigmaNilpotent","parent":2043,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\sigma\\)"],["text","-nilpotent groups"]],"source":"$\\sigma$-nilpotent groups"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-sigma-nilpotent","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"For a family $\\sigma = (\\sigma_i)_{i \\in I}$ of sets of natural numbers, a group $G$ is $\\sigma$-nilpotent if it is the internal direct product of finitely many normal subgroups $H_1, \\dots, H_t$ whose orders are $\\sigma$-primary (the order of an infinite subgroup counts as $0$): the $H_k$ generate $G$, and each $H_k$ meets the subgroup generated by the others trivially.","title":"$\\sigma$-nilpotent groups"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/is-sigma-soluble","kind":"node","lean":"MathlibExt.GroupTheory.KourovkaNotebookShared.IsSigmaSoluble","parent":2043,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\sigma\\)"],["text","-soluble groups"]],"source":"$\\sigma$-soluble groups"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#is-sigma-soluble","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"For a family $\\sigma = (\\sigma_i)_{i \\in I}$ of sets of natural numbers, a group $G$ is $\\sigma$-soluble if for every chief factor $A / B$ of $G$ the index $|A : B|$ is $\\sigma$-primary (an infinite index counts as $0$).","title":"$\\sigma$-soluble groups"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/sigma-subnormal-from-joins-with-conjugat","kind":"node","lean":"MathlibExt.GroupTheory.SigmaSubnormalJoinConjugateKourovka19Problem84Wanted.sigmaSubnormalFromJoinsWithConjugates","parent":2043,"reading_title":{"format":"inline-parts-v1","parts":[["text","Skiba's "],["math","\\(\\sigma\\)"],["text","-subnormality question (Kourovka 19.84)"]],"source":"Skiba's $\\sigma$-subnormality question (Kourovka 19.84)"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#sigma-subnormal-from-joins-with-conjugat","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"fully_proved","summary":"The statement that for every partition $\\sigma$ of the set of primes, every finite group $G$ and every subgroup $A$ of $G$, if $A$ is $\\sigma$-subnormal in the subgroup $\\langle A, A^x \\rangle$ generated by $A$ and $A^x = xAx^{-1}$ for every $x \\in G$, then $A$ is $\\sigma$-subnormal in $G$.","title":"Skiba's $\\sigma$-subnormality question (Kourovka 19.84)"},{"id":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/sigma-subnormal-from-joins-with-conjugat-2","kind":"node","lean":"MathlibExt.GroupTheory.SigmaSubnormalJoinConjugateKourovka19Problem84Wanted.sigmaSubnormalFromJoinsWithConjugates_refuted","parent":2043,"reading_title":{"format":"inline-parts-v1","parts":[["text","A counterexample to Skiba's "],["math","\\(\\sigma\\)"],["text","-subnormality question"]],"source":"A counterexample to Skiba's $\\sigma$-subnormality question"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality/index.html#sigma-subnormal-from-joins-with-conjugat-2","scope":"GroupTheory/soluble-finite-groups/solubility-nilpotency-and-subnormality","status":"can_prove","summary":"The statement of Kourovka Problem 19.84 is false: there are a partition $\\sigma$ of the set of primes, a finite group $G$ and a subgroup $A$ of $G$ such that $A$ is $\\sigma$-subnormal in the subgroup $\\langle A, A^x \\rangle$ generated by $A$ and $A^x = xAx^{-1}$ for every $x \\in G$, but $A$ is not $\\sigma$-subnormal in $G$.","title":"A counterexample to Skiba's $\\sigma$-subnormality question"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors.json","id":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/index.html","scope":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors","status":"can_prove","title":"Frattini and complemented chief factors"},{"id":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/is-frattini-chief-factor","kind":"node","lean":"MathlibExt.GroupTheory.CoverAvoidIntersectionCoreKourovka19Problem38Wanted.IsFrattiniChiefFactor","parent":2057,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/index.html#is-frattini-chief-factor","scope":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors","status":"fully_proved","summary":"For subgroups $A$ and $B$ of a group $G$, $A/B$ is a Frattini chief factor of $G$ when $A/B$ is a chief factor of $G$ and $A$ is contained in every maximal subgroup of $G$ that contains $B$; equivalently, $A/B \\le \\Phi(G/B)$, where $\\Phi$ denotes the Frattini subgroup.","title":"Frattini chief factor"},{"id":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/is-complemented-chief-factor","kind":"node","lean":"MathlibExt.GroupTheory.CoverAvoidIntersectionCoreKourovka19Problem38Wanted.IsComplementedChiefFactor","parent":2057,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/index.html#is-complemented-chief-factor","scope":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors","status":"fully_proved","summary":"For subgroups $A$ and $B$ of a group $G$, $A/B$ is a complemented chief factor of $G$ when $A/B$ is a chief factor of $G$ and some subgroup $M$ of $G$ satisfies $\\langle M, A \\rangle = G$ and $M \\cap A = B$ (as $A$ is normal, the first condition says $MA = G$).","title":"Complemented chief factor"},{"id":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/cover-avoid-triple-intersection-core","kind":"node","lean":"MathlibExt.GroupTheory.CoverAvoidIntersectionCoreKourovka19Problem38Wanted.coverAvoidTripleIntersectionCore","parent":2057,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/index.html#cover-avoid-triple-intersection-core","scope":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors","status":"fully_proved","summary":"The statement that for every finite soluble group $G$ and every subgroup $H$ of $G$ that covers every Frattini chief factor $A/B$ of $G$, that is, $A \\le \\langle H, B \\rangle$ (which is $HB$, as $B$ is normal), and avoids every complemented chief factor $A/B$ of $G$, that is, $H \\cap A \\le B$, there are $x, y \\in G$ such that $H \\cap H^x \\cap H^y = H_G$, where $H^x = xHx^{-1}$ and $H_G$ is the normal core of $H$,\u2026","title":"Kourovka Problem 19.38"},{"id":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/cover-avoid-triple-intersection-core-hol","kind":"node","lean":"MathlibExt.GroupTheory.CoverAvoidIntersectionCoreKourovka19Problem38Wanted.coverAvoidTripleIntersectionCore_holds","parent":2057,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors/index.html#cover-avoid-triple-intersection-core-hol","scope":"GroupTheory/soluble-finite-groups/frattini-and-complemented-chief-factors","status":"can_prove","summary":"Let $H$ be a subgroup of a finite soluble group $G$. Suppose that $H$ covers every Frattini chief factor $A/B$ of $G$, that is, $A \\le \\langle H, B \\rangle$ (which is $HB$, as $B$ is normal), and avoids every complemented chief factor $A/B$ of $G$, that is, $H \\cap A \\le B$. Then there are $x, y \\in G$ such that $H \\cap H^x \\cap H^y = H_G$, where $H^x = xHx^{-1}$ and $H_G$ is the normal core of $H$, the largest\u2026","title":"Kamornikov\u2013Shemetkova theorem (Kourovka Problem 19.38)"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/schmidt-subgroups.json","id":"GroupTheory/soluble-finite-groups/schmidt-subgroups","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"can_prove","title":"Schmidt subgroups"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/is-minimal-non-nilpotent","kind":"node","lean":"MathlibExt.GroupTheory.MinimalNonNilpotentPermutableCoreKourovka17Problem56Wanted.IsMinimalNonNilpotent","parent":2062,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#is-minimal-non-nilpotent","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"fully_proved","summary":"A subgroup $M$ of a group $G$ is minimal non-nilpotent if $M$ is not nilpotent but every proper subgroup of $M$ is nilpotent.","title":"Minimal non-nilpotent subgroup"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/permutes-minimal-non-nilpotent-core-quot","kind":"node","lean":"MathlibExt.GroupTheory.MinimalNonNilpotentPermutableCoreKourovka17Problem56Wanted.permutesMinimalNonNilpotentCoreQuotientNilpotent","parent":2062,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#permutes-minimal-non-nilpotent-core-quot","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"fully_proved","summary":"The statement that if $H$ is a subgroup of a finite group $G$ with $HM = MH$ (as sets of products) for every minimal non-nilpotent subgroup $M$ of $G$, then the quotient $H/H_G$ is nilpotent, where the normal core $H_G$ is the largest normal subgroup of $G$ contained in $H$.","title":"Subgroups permuting with all minimal non-nilpotent subgroups (Kourovka 17.56)"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/permutes-minimal-non-nilpotent-core-quot-2","kind":"node","lean":"MathlibExt.GroupTheory.MinimalNonNilpotentPermutableCoreKourovka17Problem56Wanted.permutesMinimalNonNilpotentCoreQuotientNilpotent_refuted","parent":2062,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#permutes-minimal-non-nilpotent-core-quot-2","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"can_prove","summary":"There exist a finite group $G$ and a subgroup $H$ of $G$ with $HM = MH$ (as sets of products) for every minimal non-nilpotent subgroup $M$ of $G$ such that the quotient $H/H_G$ is not nilpotent, where the normal core $H_G$ is the largest normal subgroup of $G$ contained in $H$.","title":"Permutability with minimal non-nilpotent subgroups does not force a nilpotent core quotient"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/is-schmidt-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.SchmidtSubgroupsPrimeSubnormalKourovka18Problem30Wanted.IsSchmidtSubgroup","parent":2062,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#is-schmidt-subgroup","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is a Schmidt subgroup when $H$ is not nilpotent but every proper subgroup of $H$ is nilpotent.","title":"Schmidt subgroup"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/is-psubnormal","kind":"node","lean":"MathlibExt.GroupTheory.SchmidtSubgroupsPrimeSubnormalKourovka18Problem30Wanted.IsPSubnormal","parent":2062,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathbb{P}\\)"],["text","-subnormal subgroup"]],"source":"$\\mathbb{P}$-subnormal subgroup"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#is-psubnormal","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is $\\mathbb{P}$-subnormal in $G$ when $H = G$ or there is a chain of subgroups $H = H_0 < H_1 < \\cdots < H_n = G$ in which every index $|H_i : H_{i-1}|$ is a prime number.","title":"$\\mathbb{P}$-subnormal subgroup"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/schmidt-subgroups-psubnormal-implies-sol","kind":"node","lean":"MathlibExt.GroupTheory.SchmidtSubgroupsPrimeSubnormalKourovka18Problem30Wanted.schmidtSubgroupsPSubnormalImpliesSoluble","parent":2062,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#schmidt-subgroups-psubnormal-implies-sol","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"fully_proved","summary":"The statement that every finite group in which every Schmidt subgroup is $\\mathbb{P}$-subnormal is soluble. The problem is due to A. F. Vasil'ev, T. I. Vasil'eva and V. N. Tyutyanov.","title":"Claim of Kourovka Problem 18.30"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/schmidt-subgroups-psubnormal-implies-sol-2","kind":"node","lean":"MathlibExt.GroupTheory.SchmidtSubgroupsPrimeSubnormalKourovka18Problem30Wanted.schmidtSubgroupsPSubnormalImpliesSoluble_holds","parent":2062,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite groups with "],["math","\\(\\mathbb{P}\\)"],["text","-subnormal Schmidt subgroups are soluble"]],"source":"Finite groups with $\\mathbb{P}$-subnormal Schmidt subgroups are soluble"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#schmidt-subgroups-psubnormal-implies-sol-2","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"can_prove","summary":"Every finite group in which every Schmidt subgroup is $\\mathbb{P}$-subnormal is soluble. This answers Kourovka Notebook Problem 18.30 in the affirmative.","title":"Finite groups with $\\mathbb{P}$-subnormal Schmidt subgroups are soluble"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/schmidt-sigma-subnormal-cyclic-quotient","kind":"node","lean":"MathlibExt.GroupTheory.SchmidtSubgroupsSigmaSubnormalKourovka19Problem85Wanted.schmidtSigmaSubnormalCyclicQuotient","parent":2062,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#schmidt-sigma-subnormal-cyclic-quotient","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"fully_proved","summary":"The statement that for every partition $\\sigma$ of the set of all primes into nonempty pairwise disjoint blocks and every finite group $G$ in which every Schmidt subgroup is $\\sigma$-subnormal, $G$ has a normal $\\sigma$-nilpotent subgroup $N$ such that $G / N$ is cyclic: there is $g \\in G$ such that every $x \\in G$ satisfies $x g^{-k} \\in N$ for some integer $k$. The problem is due to A. N. Skiba.","title":"Claim of Kourovka Problem 19.85"},{"id":"GroupTheory/soluble-finite-groups/schmidt-subgroups/schmidt-sigma-subnormal-cyclic-quotient-2","kind":"node","lean":"MathlibExt.GroupTheory.SchmidtSubgroupsSigmaSubnormalKourovka19Problem85Wanted.schmidtSigmaSubnormalCyclicQuotient_holds","parent":2062,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite groups with "],["math","\\(\\sigma\\)"],["text","-subnormal Schmidt subgroups are "],["math","\\(\\sigma\\)"],["text","-nilpotent-by-cyclic"]],"source":"Finite groups with $\\sigma$-subnormal Schmidt subgroups are $\\sigma$-nilpotent-by-cyclic"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/schmidt-subgroups/index.html#schmidt-sigma-subnormal-cyclic-quotient-2","scope":"GroupTheory/soluble-finite-groups/schmidt-subgroups","status":"can_prove","summary":"Let $\\sigma$ be a partition of the set of all primes into nonempty pairwise disjoint blocks, and let $G$ be a finite group in which every Schmidt subgroup is $\\sigma$-subnormal. Then $G$ has a normal $\\sigma$-nilpotent subgroup $N$ such that $G / N$ is cyclic: there is $g \\in G$ such that every $x \\in G$ satisfies $x g^{-k} \\in N$ for some integer $k$. This answers Kourovka Notebook Problem 19.85 in the affirmative.","title":"Finite groups with $\\sigma$-subnormal Schmidt subgroups are $\\sigma$-nilpotent-by-cyclic"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups.json","id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"can_prove","title":"Supersoluble maximal subgroups"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/is-supersoluble","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.IsSupersoluble","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#is-supersoluble","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"fully_proved","summary":"A group $G$ is supersoluble when there are a natural number $n$ and elements $g_0, \\dots, g_{n-1}$ of $G$ that generate $G$ such that, for every $i \\le n$, the subgroup $G_i = \\langle g_j : j < i \\rangle$ is normal in $G$.","title":"Supersoluble group (generator form)"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/maximal-full-prime-supersoluble","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.MaximalFullPrimeSupersoluble","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#maximal-full-prime-supersoluble","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"fully_proved","summary":"A group $G$ satisfies Monakhov's condition when every maximal subgroup $M$ of $G$ with $\\pi(M) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of prime factors of the order of $X$ if $X$ is finite, and is empty if $X$ is infinite.","title":"Monakhov's condition"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/has-chief-factor-rank-gt","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.HasChiefFactorRankGT","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#has-chief-factor-rank-gt","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"fully_proved","summary":"A group $G$ has a chief factor of rank greater than a natural number $r$ when there are a chief factor $A / B$ of $G$, a prime $p$ and a natural number $k > r$ such that the index $[A : B]$ equals $p^k$.","title":"Chief factor of rank above a bound"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/plength-leone","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.PLengthLEOne","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#plength-leone","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"fully_proved","summary":"For a natural number $p$, a group $G$ has $p$-length at most $1$ when there are normal subgroups $N \\le M$ of $G$ such that $N$ is finite of order not divisible by $p$, the index $[M : N]$ equals $p^k$ for some natural number $k$ (an infinite index counting as $0$), and the index $[G : M]$ is finite and not divisible by $p$.","title":"Groups of p-length at most one"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/every-nonabelian-simple-group-occurs","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.everyNonabelianSimpleGroupOccurs","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#every-nonabelian-simple-group-occurs","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"fully_proved","summary":"The proposition that for every finite non-abelian simple group $S$ there is a finite group $G$ such that $S$ is a composition factor of $G$ and every maximal subgroup $M$ of $G$ with $\\pi(M) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of primes dividing the order of $X$.","title":"Kourovka 19.55(a) (Monakhov): every non-abelian simple group occurs"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/fitting-height-unbounded","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.fittingHeightUnbounded","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#fitting-height-unbounded","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"fully_proved","summary":"The proposition that for every natural number $h$ there is a finite soluble group $G$ whose Fitting height is greater than $h$ and in which every maximal subgroup $M$ with $\\pi(M) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of primes dividing the order of $X$.","title":"Kourovka 19.55(b) (Monakhov): unbounded Fitting height"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/rank-unbounded","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.rankUnbounded","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#rank-unbounded","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"fully_proved","summary":"The statement that for every natural number $r$ there is a finite soluble group $G$ satisfying Monakhov's condition (every maximal subgroup $M$ of $G$ with $\\pi(M) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of prime divisors of $|X|$) that has a chief factor $A/B$ of order $p^k$ for some prime $p$ and some $k > r$. This is the answer to Kourovka 19.55(b) for the rank.","title":"Unbounded chief-factor rank under Monakhov's condition (the claim)"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/p-length-at-most-one","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.pLengthAtMostOne","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#p-length-at-most-one","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"fully_proved","summary":"The proposition that for every finite soluble group $G$ in which every maximal subgroup $H$ with $\\pi(H) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of primes dividing the order of $X$, and for every prime $p$, the group $G$ has $p$-length at most $1$: there are normal subgroups $N \\le M$ of $G$ such that $p$ does not divide $|N|$, the index $[M : N]$ is a power of $p$, and $p$ does not divide $[G : M]$.","title":"Kourovka 19.55(b) (Monakhov): p-length at most one"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/every-nonabelian-simple-group-occurs-hol","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.everyNonabelianSimpleGroupOccurs_holds","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#every-nonabelian-simple-group-occurs-hol","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"can_prove","summary":"For every finite non-abelian simple group $S$ there is a finite group $G$ such that $S$ is a composition factor of $G$ and every maximal subgroup $M$ of $G$ with $\\pi(M) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of primes dividing the order of $X$.","title":"Moret\u00f3: every non-abelian simple group occurs in Kourovka 19.55(a)"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/fitting-height-unbounded-holds","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.fittingHeightUnbounded_holds","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#fitting-height-unbounded-holds","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"can_prove","summary":"For every natural number $h$ there is a finite soluble group $G$ whose Fitting height is greater than $h$ and in which every maximal subgroup $M$ with $\\pi(M) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of primes dividing the order of $X$.","title":"Moret\u00f3: soluble groups in Kourovka 19.55 have unbounded Fitting height"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/rank-unbounded-holds","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.rankUnbounded_holds","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#rank-unbounded-holds","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"can_prove","summary":"For every natural number $r$ there is a finite soluble group $G$ such that every maximal subgroup $M$ of $G$ with $\\pi(M) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of prime divisors of $|X|$, and $G$ has a chief factor $A/B$ (normal subgroups $B < A$ of $G$ with no normal subgroup of $G$ strictly between them) of order $p^k$ for some prime $p$ and some $k > r$.","title":"Unbounded chief-factor rank under Monakhov's condition"},{"id":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/p-length-at-most-one-holds","kind":"node","lean":"MathlibExt.GroupTheory.FullPrimeMaximalSupersolubleKourovka19Problem55Wanted.pLengthAtMostOne_holds","parent":2072,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups/index.html#p-length-at-most-one-holds","scope":"GroupTheory/soluble-finite-groups/supersoluble-maximal-subgroups","status":"can_prove","summary":"Let $G$ be a finite soluble group in which every maximal subgroup $H$ with $\\pi(H) = \\pi(G)$ is supersoluble, where $\\pi(X)$ is the set of primes dividing the order of $X$. Then for every prime $p$ the group $G$ has $p$-length at most $1$: there are normal subgroups $N \\le M$ of $G$ such that $p$ does not divide $|N|$, the index $[M : N]$ is a power of $p$, and $p$ does not divide $[G : M]$.","title":"Moret\u00f3: soluble groups in Kourovka 19.55 have p-length at most one"},{"data_url":"scopes/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro.json","id":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","kind":"scope","navigation":"module","parent":1929,"result":false,"route":"../roadmap/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/index.html","scope":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","status":"can_prove","title":"Supplements of maximal subgroups of Sylow subgroups"},{"id":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/is-psoluble","kind":"node","lean":"MathlibExt.GroupTheory.SylowMaximalPSolubleSupplementKourovka17Problem111Wanted.IsPSoluble","parent":2085,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-soluble groups (series with "],["math","\\(p\\)"],["text","- and "],["math","\\(p'\\)"],["text","-factors)"]],"source":"$p$-soluble groups (series with $p$- and $p'$-factors)"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/index.html#is-psoluble","scope":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","status":"fully_proved","summary":"A group $G$ is $p$-soluble when it has a chain of subgroups $1 = H_0 \\le H_1 \\le \\cdots \\le H_n = G$, each normal in the next, such that for every $i < n$ the relative index $|H_{i+1} : H_i|$ (taken to be $0$ when it is infinite) is a power of $p$ or is coprime to $p$. For a finite group this says that every factor $H_{i+1}/H_i$ is a $p$-group or a $p'$-group.","title":"$p$-soluble groups (series with $p$- and $p'$-factors)"},{"id":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/sylow-maximal-supplement-psoluble","kind":"node","lean":"MathlibExt.GroupTheory.SylowMaximalPSolubleSupplementKourovka17Problem111Wanted.sylowMaximalSupplementPSoluble","parent":2085,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/index.html#sylow-maximal-supplement-psoluble","scope":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","status":"fully_proved","summary":"The statement, asked by A. N. Skiba, that for every prime $p$, every finite group $G$ whose order is divisible by $p$ and every Sylow $p$-subgroup $P$ of $G$, if every maximal subgroup $M$ of $P$ has a $p$-soluble supplement in $G$, that is, a $p$-soluble subgroup $K$ of $G$ with $MK = G$, then $G$ is $p$-soluble.","title":"Kourovka Problem 17.111 (supplements of maximal subgroups of Sylow subgroups)"},{"id":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/sylow-maximal-supplement-psoluble-holds","kind":"node","lean":"MathlibExt.GroupTheory.SylowMaximalPSolubleSupplementKourovka17Problem111Wanted.sylowMaximalSupplementPSoluble_holds","parent":2085,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-soluble supplements of maximal subgroups of a Sylow subgroup force "],["math","\\(p\\)"],["text","-solubility"]],"source":"$p$-soluble supplements of maximal subgroups of a Sylow subgroup force $p$-solubility"},"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/index.html#sylow-maximal-supplement-psoluble-holds","scope":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","status":"can_prove","summary":"Let $p$ be a prime, $G$ a finite group whose order is divisible by $p$, and $P$ a Sylow $p$-subgroup of $G$. If every maximal subgroup $M$ of $P$ has a $p$-soluble supplement in $G$, that is, a $p$-soluble subgroup $K$ of $G$ with $MK = G$, then $G$ is $p$-soluble.","title":"$p$-soluble supplements of maximal subgroups of a Sylow subgroup force $p$-solubility"},{"id":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/sylow-maximal-sigma-nilpotent-supplement","kind":"node","lean":"MathlibExt.GroupTheory.SylowMaximalSigmaNilpotentSupplementKourovka19Problem88Wanted.sylowMaximalSigmaNilpotentSupplement","parent":2085,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/index.html#sylow-maximal-sigma-nilpotent-supplement","scope":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","status":"fully_proved","summary":"The statement that for every partition $\\sigma$ of the set of all primes and every finite group $G$: if for every prime $p$, every Sylow $p$-subgroup $P$ of $G$ and every maximal subgroup $V$ of $P$ there is a $\\sigma$-nilpotent subgroup $T$ of $G$ with $VT = G$, that is, every element of $G$ is a product $vt$ with $v \\in V$ and $t \\in T$, then $G$ is $\\sigma$-nilpotent. Here a partition of the primes is a family\u2026","title":"Kourovka Problem 19.88 (statement)"},{"id":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/sylow-maximal-sigma-nilpotent-supplement-2","kind":"node","lean":"MathlibExt.GroupTheory.SylowMaximalSigmaNilpotentSupplementKourovka19Problem88Wanted.sylowMaximalSigmaNilpotentSupplement_holds","parent":2085,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/index.html#sylow-maximal-sigma-nilpotent-supplement-2","scope":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","status":"can_prove","summary":"Let $\\sigma$ be a partition of the set of all primes and let $G$ be a finite group. If for every prime $p$, every Sylow $p$-subgroup $P$ of $G$ and every maximal subgroup $V$ of $P$ there is a $\\sigma$-nilpotent subgroup $T$ of $G$ with $VT = G$, that is, every element of $G$ is a product $vt$ with $v \\in V$ and $t \\in T$, then $G$ is $\\sigma$-nilpotent. Here a partition of the primes is a family of nonempty,\u2026","title":"\u03c3-nilpotent supplements to the maximal subgroups of Sylow subgroups (Kourovka 19.88)"},{"id":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/sylow-maximal-sigma-soluble-supplement","kind":"node","lean":"MathlibExt.GroupTheory.SylowMaximalSigmaSolubleSupplementKourovka19Problem87Wanted.sylowMaximalSigmaSolubleSupplement","parent":2085,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/index.html#sylow-maximal-sigma-soluble-supplement","scope":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","status":"fully_proved","summary":"The statement that for every partition $\\sigma$ of the set of all primes and every finite group $G$: if for every prime $p$, every Sylow $p$-subgroup $P$ of $G$ and every maximal subgroup $V$ of $P$ there is a $\\sigma$-soluble subgroup $T$ of $G$ with $VT = G$, that is, every element of $G$ is a product $vt$ with $v \\in V$ and $t \\in T$, then $G$ is $\\sigma$-soluble. Here a partition of the primes is a family of\u2026","title":"Kourovka Problem 19.87 (statement)"},{"id":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/sylow-maximal-sigma-soluble-supplement-h","kind":"node","lean":"MathlibExt.GroupTheory.SylowMaximalSigmaSolubleSupplementKourovka19Problem87Wanted.sylowMaximalSigmaSolubleSupplement_holds","parent":2085,"result":true,"route":"../roadmap/GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro/index.html#sylow-maximal-sigma-soluble-supplement-h","scope":"GroupTheory/soluble-finite-groups/supplements-of-maximal-subgroups-of-sylow-subgro","status":"can_prove","summary":"Let $\\sigma$ be a partition of the set of all primes and let $G$ be a finite group. If for every prime $p$, every Sylow $p$-subgroup $P$ of $G$ and every maximal subgroup $V$ of $P$ there is a $\\sigma$-soluble subgroup $T$ of $G$ with $VT = G$, that is, every element of $G$ is a product $vt$ with $v \\in V$ and $t \\in T$, then $G$ is $\\sigma$-soluble. Here a partition of the primes is a family of nonempty, pairwise\u2026","title":"\u03c3-soluble supplements to the maximal subgroups of Sylow subgroups (Kourovka 19.87)"},{"data_url":"scopes/GroupTheory/geometric-group-theory.json","id":"GroupTheory/geometric-group-theory","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/index.html","scope":"GroupTheory/geometric-group-theory","status":"can_prove","summary":"Large-scale and analytic properties of finitely generated groups: growth, word-hyperbolic groups and their quotients, amenability and F\u00f8lner sets, elementary amenable groups, property (T), soficity and permutation stability.","title":"Geometric group theory"},{"data_url":"scopes/GroupTheory/geometric-group-theory/branch-and-thompson-groups.json","id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups","kind":"scope","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/index.html","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups","status":"can_prove","summary":"Groups acting on rooted trees, branch and just-infinite groups of intermediate growth, finite-state (automaton) groups, and Thompson's group F and its piecewise-linear relatives.","title":"Branch, automaton and Thompson groups"},{"data_url":"scopes/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups.json","id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","kind":"scope","navigation":"module","parent":2094,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"can_prove","title":"Amenability and growth of branch groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/tree-vertex","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.TreeVertex","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#tree-vertex","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"For a sequence $m$ of natural numbers, the vertices of the spherically homogeneous rooted tree with branching sequence $m$ are the finite words $l_0 l_1 \\cdots l_{k-1}$ of natural numbers with $l_i < m(i)$ for every $i < k$. The length of a word is its level and its prefixes are its ancestors.","title":"Vertices of a spherically homogeneous rooted tree (Kourovka 15.14 version)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/rigid-vertex-stabilizer","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.rigidVertexStabilizer","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#rigid-vertex-stabilizer","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"Let $G$ be a group acting by permutations $\\rho$ on the vertices of the rooted tree with branching sequence $m$, and let $v$ be a vertex. The rigid vertex stabilizer of $v$ is the set of elements $g \\in G$ whose support lies in the subtree rooted at $v$: $\\rho(g)$ fixes every vertex $w$ of which $v$ is not a prefix.","title":"Rigid vertex stabilizers (Kourovka 15.14 version)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/rigid-level-stabilizer","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.rigidLevelStabilizer","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#rigid-level-stabilizer","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"Let $G$ be a group acting by permutations $\\rho$ on the vertices of the rooted tree with branching sequence $m$, and let $n$ be a natural number. The $n$-th rigid level stabilizer of $\\rho$ is the subgroup of $G$ generated by the rigid vertex stabilizers of all vertices at level $n$, the words of length $n$.","title":"Rigid level stabilizers (Kourovka 15.14 version)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/is-branch-group","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.IsBranchGroup","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#is-branch-group","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"A group $G$ is a branch group (Kourovka 15.12) if there are a sequence $m$ of natural numbers with $m(i) \\ge 2$ for every $i$ and an action $\\rho$ of $G$ by permutations of the vertices of the spherically homogeneous rooted tree with branching sequence $m$ such that: $\\rho$ is faithful; each $\\rho(g)$ is a tree automorphism, that is, for all vertices $v$ and $w$, $v$ is a prefix of $w$ if and only if $\\rho(g)(v)$\u2026","title":"Branch groups (Kourovka 15.14 version)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/growth-rate","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.growthRate","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#growth-rate","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"The growth rate `\u03c9(G, P) = limsup \u03b2_P(n)^(1/n)` of a finite generating family `P`, where `\u03b2_P(n)` is the number of elements that are products of at most `n` generators and their inverses.","title":"Growth rate of a finitely generated group"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/has-exponential-growth","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.HasExponentialGrowth","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#has-exponential-growth","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"A group $G$ has exponential growth when it is finitely generated and every finite generating family $P$ of $G$ has growth rate $\\omega(G, P) > 1$.","title":"Groups of exponential growth"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/contains-free-group-rank-two","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.ContainsFreeGroupRankTwo","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#contains-free-group-rank-two","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"A group $G$ contains a free subgroup of rank $2$ when there is an injective group homomorphism from the free group $F_2$ on two generators into $G$.","title":"Groups containing a free subgroup of rank two"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/fg-nonamenable-branch-group","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.fgNonamenableBranchGroup","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#fg-nonamenable-branch-group","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"The statement that there exists a finitely generated branch group that is not amenable.","title":"Kourovka Problem 15.14(a): non-amenable branch groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/fg-nonamenable-branch-group-without-free","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.fgNonamenableBranchGroupWithoutFreeSubgroup","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#fg-nonamenable-branch-group-without-free","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"The statement that there exists a finitely generated branch group that is not amenable and contains no free subgroup of rank $2$.","title":"Kourovka Problem 15.14(b): non-amenable branch groups without free subgroups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/fg-branch-group-containing-free-group","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.fgBranchGroupContainingFreeGroup","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#fg-branch-group-containing-free-group","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"The statement that there exists a finitely generated branch group that contains a free subgroup of rank $2$.","title":"Kourovka Problem 15.14(c): branch groups containing free subgroups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/fg-branch-group-of-exponential-growth","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.fgBranchGroupOfExponentialGrowth","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#fg-branch-group-of-exponential-growth","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"fully_proved","summary":"The statement that there exists a finitely generated branch group of exponential growth.","title":"Kourovka Problem 15.14(d): branch groups of exponential growth"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/fg-nonamenable-branch-group-holds","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.fgNonamenableBranchGroup_holds","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#fg-nonamenable-branch-group-holds","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"can_prove","summary":"There is a finitely generated branch group that is not amenable.","title":"Non-amenable branch groups exist (Kourovka 15.14(a))"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/fg-nonamenable-branch-group-without-free-2","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.fgNonamenableBranchGroupWithoutFreeSubgroup_holds","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#fg-nonamenable-branch-group-without-free-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"can_prove","summary":"There is a finitely generated branch group that is not amenable and contains no free subgroup of rank $2$.","title":"Non-amenable branch groups without free subgroups exist (Kourovka 15.14(b))"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/fg-branch-group-containing-free-group-ho","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.fgBranchGroupContainingFreeGroup_holds","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#fg-branch-group-containing-free-group-ho","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"can_prove","summary":"There is a finitely generated branch group that contains a free subgroup of rank $2$.","title":"Branch groups containing free subgroups exist (Kourovka 15.14(c))"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/fg-branch-group-of-exponential-growth-ho","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyGeneratedBranchGroupsKourovka15Problem14Wanted.fgBranchGroupOfExponentialGrowth_holds","parent":2095,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups/index.html#fg-branch-group-of-exponential-growth-ho","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/amenability-and-growth-of-branch-groups","status":"can_prove","summary":"There is a finitely generated branch group of exponential growth.","title":"Branch groups of exponential growth exist (Kourovka 15.14(d))"},{"data_url":"scopes/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees.json","id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","kind":"scope","navigation":"module","parent":2094,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"can_prove","title":"Finite-state automorphisms of rooted trees"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/is-tree-aut","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateConjugacyProblemKourovka16Problem34Wanted.IsTreeAut","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#is-tree-aut","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, the vertices of the $d$-regular rooted tree are the finite words over the alphabet $\\{0, \\dots, d-1\\}$. A permutation $f$ of this set of words is a tree automorphism when, for all words $u$ and $v$, writing $|u|$ for the length of $u$ and $uv$ for the concatenation of $u$ and $v$, the word $f(u)$ has length $|u|$ and the first $|u|$ letters of $f(uv)$ form $f(u)$.","title":"Automorphism of the regular rooted tree"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/tree-section","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateConjugacyProblemKourovka16Problem34Wanted.treeSection","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#tree-section","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, let $f$ be a permutation of the finite words over $\\{0, \\dots, d-1\\}$ (the vertices of the $d$-regular rooted tree) and let $u$ be a word, of length $|u|$. The section (state) of $f$ at $u$ is the map sending each word $w$ to the word obtained from $f(uw)$ by deleting its first $|u|$ letters, where $uw$ is the concatenation of $u$ and $w$.","title":"Section of a map of the regular rooted tree at a vertex"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/is-finite-state-aut","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateConjugacyProblemKourovka16Problem34Wanted.IsFiniteStateAut","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#is-finite-state-aut","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, a permutation $f$ of the finite words over $\\{0, \\dots, d-1\\}$ is a finite-state automorphism of the $d$-regular rooted tree when it is an automorphism of the tree and the sections of $f$ at all vertices $u$ form a finite set of maps.","title":"Finite-state automorphism of the regular rooted tree"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/word-value","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateConjugacyProblemKourovka16Problem34Wanted.wordValue","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#word-value","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For $d, k \\in \\mathbb{N}$, let $g_0, \\dots, g_{k-1}$ be permutations of the finite words over $\\{0, \\dots, d-1\\}$, and let $w$ be a word whose letters are pairs $(i, b)$ with $i < k$ and $b$ a boolean. The value of $w$ is the element of the group generated by $g_0, \\dots, g_{k-1}$ obtained by multiplying, from left to right, $g_i$ for each letter $(i, \\mathrm{true})$ and $g_i^{-1}$ for each letter\u2026","title":"Value of a word in the generators"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/finite-state-conjugacy-decidable","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateConjugacyProblemKourovka16Problem34Wanted.finiteStateConjugacyDecidable","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#finite-state-conjugacy-decidable","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"The proposition that every finitely generated group of finite-state automorphisms of a regular rooted tree has decidable conjugacy problem: for all $d, k \\in \\mathbb{N}$ and all finite-state automorphisms $g_0, \\dots, g_{k-1}$ of the $d$-regular rooted tree, some algorithm decides, given two words in the $g_i$ and their inverses, whether the elements they spell are conjugate in the group generated by\u2026","title":"Kourovka Problem 16.34: conjugacy in finite-state automorphism groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/finite-state-conjugacy-decidable-refuted","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateConjugacyProblemKourovka16Problem34Wanted.finiteStateConjugacyDecidable_refuted","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#finite-state-conjugacy-decidable-refuted","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"can_prove","summary":"The answer to Kourovka Problem 16.34 is negative: there exist $d, k \\in \\mathbb{N}$ and finite-state automorphisms $g_0, \\dots, g_{k-1}$ of the $d$-regular rooted tree such that no algorithm decides, given two words in the $g_i$ and their inverses, whether the elements they spell are conjugate in the group generated by $g_0, \\dots, g_{k-1}$.","title":"Finite-state tree automorphism groups with undecidable conjugacy problem"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/is-tree-aut-2","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateFreeProductKourovka16Problem85Wanted.IsTreeAut","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#is-tree-aut-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"Let $d$ be a natural number and let $X^*$ be the set of finite words over the alphabet $X = \\{0, \\dots, d-1\\}$, the vertices of the $d$-regular rooted tree. A permutation $f$ of $X^*$ is a tree automorphism when, for all words $u$ and $v$, the word $f(u)$ has the same length $|u|$ as $u$ and the first $|u|$ letters of $f(uv)$ form the word $f(u)$, where $uv$ denotes concatenation.","title":"Automorphisms of the regular rooted tree (Kourovka 16.85)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/tree-section-2","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateFreeProductKourovka16Problem85Wanted.treeSection","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#tree-section-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For a natural number $d$, let $X^*$ be the set of finite words over $X = \\{0, \\dots, d-1\\}$, let $f$ be a permutation of $X^*$ and let $u \\in X^*$, of length $|u|$. The section (state) of $f$ at $u$ is the map $X^* \\to X^*$ sending a word $w$ to the word obtained from $f(uw)$ by deleting its first $|u|$ letters (all of them if it has fewer), where $uw$ denotes concatenation.","title":"Section of a permutation of the tree at a vertex (Kourovka 16.85)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/is-finite-state-aut-2","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateFreeProductKourovka16Problem85Wanted.IsFiniteStateAut","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#is-finite-state-aut-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For a natural number $d$, let $X^*$ be the set of finite words over $X = \\{0, \\dots, d-1\\}$. A permutation $f$ of $X^*$ is a finite-state automorphism when it is an automorphism of the $d$-regular rooted tree and its sections at the vertices $u \\in X^*$ form a finite set of maps $X^* \\to X^*$.","title":"Finite-state automorphisms of the regular rooted tree (Kourovka 16.85)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/acts-faithfully-by-finite-state-auts","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateFreeProductKourovka16Problem85Wanted.ActsFaithfullyByFiniteStateAuts","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#acts-faithfully-by-finite-state-auts","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"A group $G$ acts faithfully on a regular rooted tree by finite-state automorphisms when there are a natural number $d \\ge 2$ and an injective group homomorphism $\\varphi$ from $G$ to the group of permutations of the set of finite words over $\\{0, \\dots, d-1\\}$ such that $\\varphi(g)$ is a finite-state automorphism of the $d$-regular rooted tree for every $g \\in G$.","title":"Faithful actions by finite-state tree automorphisms"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/finite-state-free-product","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateFreeProductKourovka16Problem85Wanted.finiteStateFreeProduct","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#finite-state-free-product","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"The statement that for all groups $G$ and $H$, if $G$ and $H$ each act faithfully on a regular rooted tree by finite-state automorphisms (the two trees may have different degrees), then so does their free product $G \\ast H$.","title":"Finite-state actions of free products (Kourovka Problem 16.85)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/finite-state-free-product-holds","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateFreeProductKourovka16Problem85Wanted.finiteStateFreeProduct_holds","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#finite-state-free-product-holds","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"can_prove","summary":"For all groups $G$ and $H$, if $G$ and $H$ each act faithfully on a regular rooted tree by finite-state automorphisms, then their free product $G \\ast H$ acts faithfully on some regular rooted tree by finite-state automorphisms.","title":"Fedorova\u2013Oliynyk theorem (Kourovka Problem 16.85)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/is-tree-automorphism","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateIrreducibleGeneratorsKourovka16Problem86Wanted.IsTreeAutomorphism","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#is-tree-automorphism","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For a natural number $d$, the $d$-regular rooted tree has as vertices the finite words over the alphabet $\\{0, \\dots, d-1\\}$; write $vw$ for the concatenation of words $v$ and $w$. The root is the empty word and the children of a word $v$ are the words $vi$ for letters $i$. A permutation $g$ of the words is a tree automorphism when for all words $v$ and $w$ there is a word $u$ of the same length as $w$ with\u2026","title":"Automorphisms of the regular rooted tree"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/tree-section-3","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateIrreducibleGeneratorsKourovka16Problem86Wanted.treeSection","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#tree-section-3","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For a natural number $d$, let $g$ be a permutation of the finite words over the alphabet $\\{0, \\dots, d-1\\}$ and let $v$ be a word. The section (state) of $g$ at $v$ is the map sending a word $w$ to the word obtained from $g(vw)$ by deleting its first $|g(v)|$ letters, where $vw$ is the concatenation of $v$ and $w$ and $|g(v)|$ is the length of $g(v)$. When $g$ is a tree automorphism, the image of $w$ is the word\u2026","title":"Section of a word permutation at a vertex"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/finite-state-automorphisms","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateIrreducibleGeneratorsKourovka16Problem86Wanted.finiteStateAutomorphisms","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#finite-state-automorphisms","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"For a natural number $d$, consider the $d$-regular rooted tree whose vertices are the finite words over the alphabet $\\{0, \\dots, d-1\\}$. The group of finite-state automorphisms of this tree is the subgroup of the group of all permutations of the words generated by the tree automorphisms that have only finitely many distinct sections.","title":"Finite-state automorphisms of a regular rooted tree"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/finite-state-automorphisms-irreducible-g","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateIrreducibleGeneratorsKourovka16Problem86Wanted.finiteStateAutomorphismsIrreducibleGenerators","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#finite-state-automorphisms-irreducible-g","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"fully_proved","summary":"Kourovka Problem 16.86 (Sushchanski\u012d): irreducible generators of finite-state automorphism groups. The statement that for every integer $d \\ge 2$ the group of finite-state automorphisms of the $d$-regular rooted tree has an irreducible system of generators: a set $S$ of elements of the group that generates the group and no proper subset of which generates the group.","title":"Kourovka Problem 16.86 (Sushchanski\u012d): irreducible generators of finite-state automorphism groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/finite-state-automorphisms-irreducible-g-2","kind":"node","lean":"MathlibExt.GroupTheory.FiniteStateIrreducibleGeneratorsKourovka16Problem86Wanted.finiteStateAutomorphismsIrreducibleGenerators_holds","parent":2111,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees/index.html#finite-state-automorphisms-irreducible-g-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/finite-state-automorphisms-of-rooted-trees","status":"can_prove","summary":"For every integer $d \\ge 2$, the group of finite-state automorphisms of the $d$-regular rooted tree has an irreducible system of generators: a set $S$ of elements of the group that generates the group and no proper subset of which generates the group.","title":"Lavrenyuk's theorem on irreducible generators of finite-state automorphism groups"},{"data_url":"scopes/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group.json","id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","kind":"scope","navigation":"module","parent":2094,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/index.html","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","status":"can_prove","title":"Golden-ratio Thompson group"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/is-dyadic","kind":"node","lean":"MathlibExt.GroupTheory.GoldenRatioThompsonEmbedWanted.IsDyadic","parent":2129,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/index.html#is-dyadic","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","status":"fully_proved","summary":"A real number $x$ is dyadic when $x = a / 2^n$ for some integer $a$ and some natural number $n$, that is, when $x$ lies in $\\mathbb{Z}[1/2]$.","title":"Dyadic rationals"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/is-golden-integer","kind":"node","lean":"MathlibExt.GroupTheory.GoldenRatioThompsonEmbedWanted.IsGoldenInteger","parent":2129,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/index.html#is-golden-integer","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","status":"fully_proved","summary":"Let $\\tau = (1 + \\sqrt{5})/2$ be the golden ratio. A real number $x$ is a golden integer when $x = a + b\\tau$ for some integers $a$ and $b$. Since $\\tau^2 = \\tau + 1$, these are exactly the elements of $\\mathbb{Z}[\\tau]$.","title":"Golden integers"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/is-plhomeo-with","kind":"node","lean":"MathlibExt.GroupTheory.GoldenRatioThompsonEmbedWanted.IsPLHomeoWith","parent":2129,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/index.html#is-plhomeo-with","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","status":"fully_proved","summary":"Let $B$ and $S$ be sets of real numbers. A permutation $f$ of $\\mathbb{R}$ is piecewise linear with breakpoints in $B$ and slopes in $S$ when $f(x) = x$ for every $x \\notin [0, 1]$, and there are a natural number $n$ and points $0 = t_0 < t_1 < \\dots < t_{n+1} = 1$, all in $B$, such that for each $i \\le n$ there is $s \\in S$ with $f(x) = f(t_i) + s (x - t_i)$ for all $x \\in [t_i, t_{i+1}]$.","title":"Piecewise-linear maps with prescribed breakpoints and slopes"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/thompson-f","kind":"node","lean":"MathlibExt.GroupTheory.GoldenRatioThompsonEmbedWanted.thompsonF","parent":2129,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/index.html#thompson-f","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","status":"fully_proved","summary":"Thompson's group $F$ is the set of permutations $f$ of $\\mathbb{R}$ such that $f(x) = x$ for every $x \\notin [0, 1]$, and there are a natural number $n$ and points $0 = t_0 < t_1 < \\dots < t_{n+1} = 1$, each of the form $a / 2^m$ with $a \\in \\mathbb{Z}$ and $m \\in \\mathbb{N}$, such that on each interval $[t_i, t_{i+1}]$ the map $f$ is affine with slope $2^k$ for some integer $k$, that is,\u2026","title":"Thompson's group F"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/cleary-ftau","kind":"node","lean":"MathlibExt.GroupTheory.GoldenRatioThompsonEmbedWanted.clearyFTau","parent":2129,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/index.html#cleary-ftau","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","status":"fully_proved","summary":"Let $\\tau = (1 + \\sqrt{5})/2$ be the golden ratio. Cleary's group $F_\\tau$ is the set of permutations $f$ of $\\mathbb{R}$ such that $f(x) = x$ for every $x \\notin [0, 1]$, and there are a natural number $n$ and points $0 = t_0 < t_1 < \\dots < t_{n+1} = 1$, each of the form $a + b\\tau$ with $a, b \\in \\mathbb{Z}$, such that on each interval $[t_i, t_{i+1}]$ the map $f$ is affine with slope $\\tau^k$ for some integer\u2026","title":"Cleary's golden-ratio Thompson group"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/conjecture","kind":"node","lean":"MathlibExt.GroupTheory.GoldenRatioThompsonEmbedWanted.conjecture","parent":2129,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/index.html#conjecture","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","status":"fully_proved","summary":"The proposition that there is a map $\\varphi$ from permutations of $\\mathbb{R}$ to permutations of $\\mathbb{R}$ that sends every element of Cleary's golden-ratio Thompson group $F_\\tau$ into Thompson's group $F$, is injective on $F_\\tau$, and satisfies $\\varphi(g h) = \\varphi(g) \\varphi(h)$ for all $g, h \\in F_\\tau$, where the product of permutations is composition.","title":"Golden-ratio Thompson embedding conjecture"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/conjecture-refuted","kind":"node","lean":"MathlibExt.GroupTheory.GoldenRatioThompsonEmbedWanted.conjecture_refuted","parent":2129,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group/index.html#conjecture-refuted","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/golden-ratio-thompson-group","status":"can_prove","summary":"There is no map $\\varphi$ from permutations of $\\mathbb{R}$ to permutations of $\\mathbb{R}$ that sends every element of Cleary's golden-ratio Thompson group $F_\\tau$ into Thompson's group $F$, is injective on $F_\\tau$, and satisfies $\\varphi(g h) = \\varphi(g) \\varphi(h)$ for all $g, h \\in F_\\tau$, where the product of permutations is composition. In other words, the golden-ratio Thompson embedding conjecture is\u2026","title":"Cleary's golden-ratio group does not embed in Thompson's group F"},{"data_url":"scopes/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups.json","id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","kind":"scope","navigation":"module","parent":2094,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"can_prove","title":"Just infinite groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/is-just-infinite","kind":"node","lean":"MathlibExt.GroupTheory.HereditarilyJustInfiniteGroupsKourovka15Problem18Wanted.IsJustInfinite","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#is-just-infinite","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"A group $G$ is just infinite if it is infinite and every nontrivial normal subgroup of $G$ has finite index in $G$.","title":"Just infinite groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/is-residually-finite","kind":"node","lean":"MathlibExt.GroupTheory.HereditarilyJustInfiniteGroupsKourovka15Problem18Wanted.IsResiduallyFinite","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#is-residually-finite","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"A group $G$ is residually finite if every element $g \\ne 1$ of $G$ lies outside some normal subgroup of finite index in $G$.","title":"Residually finite groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/is-hereditarily-just-infinite","kind":"node","lean":"MathlibExt.GroupTheory.HereditarilyJustInfiniteGroupsKourovka15Problem18Wanted.IsHereditarilyJustInfinite","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#is-hereditarily-just-infinite","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"A group $G$ is hereditarily just infinite (in the sense of Kourovka 15.18) if it is infinite, residually finite, and every nontrivial normal subgroup of $G$, viewed as a group in its own right, is just infinite.","title":"Hereditarily just infinite groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/fg-hereditarily-just-infinite-torsion-gr","kind":"node","lean":"MathlibExt.GroupTheory.HereditarilyJustInfiniteGroupsKourovka15Problem18Wanted.fgHereditarilyJustInfiniteTorsionGroup","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#fg-hereditarily-just-infinite-torsion-gr","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"The proposition that there exists a finitely generated hereditarily just infinite group $G$ that is a torsion group, i.e. every element of $G$ has finite order.","title":"Kourovka 15.18(a): hereditarily just infinite torsion groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/fg-hereditarily-just-infinite-is-linear","kind":"node","lean":"MathlibExt.GroupTheory.HereditarilyJustInfiniteGroupsKourovka15Problem18Wanted.fgHereditarilyJustInfiniteIsLinear","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#fg-hereditarily-just-infinite-is-linear","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"The proposition that every finitely generated hereditarily just infinite group $G$ is linear, i.e. that there are a field $K$, a natural number $n$ and an injective group homomorphism $G \\to \\mathrm{GL}_n(K)$.","title":"Kourovka 15.18(b): linearity of hereditarily just infinite groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/fg-hereditarily-just-infinite-torsion-gr-2","kind":"node","lean":"MathlibExt.GroupTheory.HereditarilyJustInfiniteGroupsKourovka15Problem18Wanted.fgHereditarilyJustInfiniteTorsionGroup_holds","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#fg-hereditarily-just-infinite-torsion-gr-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"can_prove","summary":"There is a finitely generated hereditarily just infinite group $G$ in which every element has finite order.","title":"Finitely generated hereditarily just infinite torsion groups exist"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/fg-hereditarily-just-infinite-is-linear-2","kind":"node","lean":"MathlibExt.GroupTheory.HereditarilyJustInfiniteGroupsKourovka15Problem18Wanted.fgHereditarilyJustInfiniteIsLinear_refuted","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#fg-hereditarily-just-infinite-is-linear-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"can_prove","summary":"There is a finitely generated hereditarily just infinite group $G$ such that for no field $K$ and no natural number $n$ is there an injective group homomorphism $G \\to \\mathrm{GL}_n(K)$.","title":"Not every finitely generated hereditarily just infinite group is linear"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/tree-vertex","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.TreeVertex","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#tree-vertex","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"For a sequence $m : \\mathbb{N} \\to \\mathbb{N}$ of branching indices, the vertices of the spherically homogeneous rooted tree with branching sequence $m$ are the finite words $x_0 x_1 \\cdots x_{\\ell-1}$ of natural numbers with $x_i < m(i)$ for every $i < \\ell$. The length of a word is its level and its prefixes are its ancestors.","title":"Vertices of a spherically homogeneous rooted tree"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/rigid-vertex-stabilizer","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.rigidVertexStabilizer","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#rigid-vertex-stabilizer","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"Let $\\rho$ be a homomorphism from a group $G$ to the group of permutations of the vertices of the rooted tree with branching sequence $m$, and let $v$ be a vertex. The rigid vertex stabilizer of $v$ is the set of elements $g \\in G$ such that $\\rho(g)$ fixes every vertex that does not have $v$ as a prefix, that is, the elements whose support (the set of vertices they move) lies in the subtree rooted at $v$.","title":"Rigid vertex stabilizer"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/rigid-level-stabilizer","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.rigidLevelStabilizer","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#rigid-level-stabilizer","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"Let $\\rho$ be a homomorphism from a group $G$ to the group of permutations of the vertices of the rooted tree with branching sequence $m$. For a natural number $n$, the $n$-th rigid level stabilizer is the subgroup of $G$ generated by the union of the rigid vertex stabilizers of all vertices at level $n$, that is, of all words of length $n$.","title":"Rigid level stabilizer"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/is-branch-group","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.IsBranchGroup","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#is-branch-group","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"A group $G$ is a branch group (Kourovka 15.12) if there are a sequence $m : \\mathbb{N} \\to \\mathbb{N}$ with $m(i) \\ge 2$ for all $i$ and a homomorphism $\\rho$ from $G$ to the group of permutations of the vertices of the rooted tree with branching sequence $m$ such that $\\rho$ is injective; every $\\rho(g)$ is a tree automorphism, that is, for all vertices $v$ and $w$, $v$ is a prefix of $w$ if and only if\u2026","title":"Branch groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/growth-rate","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.growthRate","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#growth-rate","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"Let $P$ be a generating family of a group $G$ indexed by $\\{0, \\dots, k-1\\}$, and let $\\beta_P(n)$ be the number of elements of $G$ whose word length with respect to $P$ is at most $n$, that is, which are products of at most $n$ members of $P$ and their inverses. The growth rate of $P$ is the real number $\\omega(G, P) = \\limsup_{n \\to \\infty} \\beta_P(n)^{1/n}$.","title":"Growth rate of a finite generating family"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/is-just-infinite-2","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.IsJustInfinite","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#is-just-infinite-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"A group $G$ is just infinite if it is infinite and every nontrivial normal subgroup of $G$ has finite index, that is, all proper quotients of $G$ are finite.","title":"Just infinite groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/has-intermediate-growth","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.HasIntermediateGrowth","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#has-intermediate-growth","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"A group $G$ has intermediate growth if it is finitely generated and, for every natural number $k$ and every generating family $P$ of $G$ indexed by $\\{0, \\dots, k-1\\}$, the growth rate of $P$ is at most $1$ (subexponential growth) and $P$ does not have polynomial growth.","title":"Groups of intermediate growth"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/just-infinite-intermediate-growth-is-bra","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.justInfiniteIntermediateGrowthIsBranch","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#just-infinite-intermediate-growth-is-bra","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"fully_proved","summary":"Kourovka Problem 15.17 (L. Bartholdi, R. I. Grigorchuk, Z. \u0160uni\u0107) is the assertion that every finitely generated just infinite group of intermediate growth is a branch group.","title":"The branch question for just infinite groups of intermediate growth"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/just-infinite-intermediate-growth-is-bra-2","kind":"node","lean":"MathlibExt.GroupTheory.JustInfiniteIntermediateGrowthBranchKourovka15Problem17Wanted.justInfiniteIntermediateGrowthIsBranch_refuted","parent":2137,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups/index.html#just-infinite-intermediate-growth-is-bra-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/just-infinite-groups","status":"can_prove","summary":"It is not the case that every finitely generated just infinite group of intermediate growth is a branch group: the Kourovka Archive answers Problem 15.17 \"No, not every\".","title":"Just infinite groups of intermediate growth need not be branch groups"},{"data_url":"scopes/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups.json","id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","kind":"scope","navigation":"module","parent":2094,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"can_prove","title":"Maximal subgroups of branch groups"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/tree-vertex","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupInfiniteIndexMaximalKourovka15Problem15Wanted.TreeVertex","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#tree-vertex","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"For a branching sequence $m \\colon \\mathbb{N} \\to \\mathbb{N}$, the vertices of the spherically homogeneous rooted tree are the finite words $(x_0, \\dots, x_{k-1})$ of natural numbers with $x_i < m(i)$ for every $i < k$. The length $k$ of a word is its level and its prefixes are its ancestors.","title":"Vertices of the spherically homogeneous rooted tree (Kourovka 15.15)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/rigid-vertex-stabilizer","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupInfiniteIndexMaximalKourovka15Problem15Wanted.rigidVertexStabilizer","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#rigid-vertex-stabilizer","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"Let $\\rho$ be a homomorphism from a group $G$ to the permutations of the vertices of the tree with branching sequence $m$, and let $v$ be a vertex. The rigid vertex stabilizer of $v$ is the set of $g \\in G$ such that $\\rho(g)$ fixes every vertex $w$ of which $v$ is not a prefix, that is, every vertex outside the subtree rooted at $v$.","title":"Rigid vertex stabilizer (Kourovka 15.15)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/rigid-level-stabilizer","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupInfiniteIndexMaximalKourovka15Problem15Wanted.rigidLevelStabilizer","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#rigid-level-stabilizer","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"Let $\\rho$ be a homomorphism from a group $G$ to the permutations of the vertices of the tree with branching sequence $m$, and let $n$ be a natural number. The $n$-th rigid level stabilizer is the subgroup of $G$ generated by the rigid vertex stabilizers of all vertices at level $n$.","title":"Rigid level stabilizer (Kourovka 15.15)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/is-branch-group","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupInfiniteIndexMaximalKourovka15Problem15Wanted.IsBranchGroup","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#is-branch-group","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"A group $G$ is a branch group if there are a sequence $m \\colon \\mathbb{N} \\to \\mathbb{N}$ with $m(i) \\ge 2$ for all $i$ and an injective homomorphism $\\rho$ from $G$ to the permutations of the vertices of the spherically homogeneous rooted tree with branching sequence $m$ such that each $\\rho(g)$ is a tree automorphism (for all vertices $v$ and $w$, $v$ is a prefix of $w$ if and only if $\\rho(g)(v)$ is a prefix of\u2026","title":"Branch group (Kourovka 15.15)"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/branch-group-maximal-subgroups-finite-in","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupInfiniteIndexMaximalKourovka15Problem15Wanted.branchGroupMaximalSubgroupsFiniteIndex","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#branch-group-maximal-subgroups-finite-in","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"The statement (L. Bartholdi, R. I. Grigorchuk, Z. \u0160uni\u0107) that every maximal subgroup of every finitely generated branch group has finite index.","title":"Kourovka 15.15: maximal subgroups of finitely generated branch groups have finite index"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/branch-group-maximal-subgroups-finite-in-2","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupInfiniteIndexMaximalKourovka15Problem15Wanted.branchGroupMaximalSubgroupsFiniteIndex_refuted","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#branch-group-maximal-subgroups-finite-in-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"can_prove","summary":"The Kourovka 15.15 statement that every maximal subgroup of a finitely generated branch group has finite index is false: some finitely generated branch group has a maximal subgroup of infinite index.","title":"A finitely generated branch group with a maximal subgroup of infinite index"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/rooted-tree","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupShared.RootedTree","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#rooted-tree","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"A locally finite rooted tree is given by finite sets $V_n$ for $n \\ge 0$, the vertices at level $n$, where $V_0$ consists of the root alone, together with parent maps $V_{n+1} \\to V_n$, such that every vertex has at least two children: for every $v \\in V_n$ there are distinct $x, y \\in V_{n+1}$ whose parent is $v$.","title":"Rooted tree given by levels"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/is-descendant","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupShared.RootedTree.IsDescendant","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#is-descendant","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"For vertices $v$ at level $n$ and $w$ at level $m$ of a rooted tree $T$, $w$ is a descendant of $v$ when $w$ is reached from $v$ by moving from a vertex to one of its children zero or more times: $v$ is a descendant of itself, and a vertex at level $k + 1$ is a descendant of $v$ whenever its parent is.","title":"Descendants in a rooted tree"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/level-transitive-faithful-action","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupShared.LevelTransitiveFaithfulAction","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#level-transitive-faithful-action","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"Let $T$ be a rooted tree with vertex sets $V_n$ at the levels $n \\ge 0$ and parent maps $V_{n+1} \\to V_n$, and let $G$ be a group. A faithful level-transitive action of $G$ on $T$ consists of an action of $G$ on each level $V_n$ by permutations, that is, a homomorphism $G \\to \\mathrm{Sym}(V_n)$ for every $n$, such that the actions commute with the parent maps (so $G$ acts by tree automorphisms), only the identity\u2026","title":"Faithful level-transitive action on a rooted tree"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/is-rigid-at","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupShared.LevelTransitiveFaithfulAction.IsRigidAt","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#is-rigid-at","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"Given a faithful level-transitive action of a group $G$ on a rooted tree $T$ and a vertex $v$ of $T$, an element $g \\in G$ lies in the rigid stabilizer of $v$ when $g$ fixes every vertex of $T$ that is not a descendant of $v$.","title":"Rigid vertex stabilizer"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/rigid-level-stabilizer-2","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupShared.LevelTransitiveFaithfulAction.rigidLevelStabilizer","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#rigid-level-stabilizer-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"Given a faithful level-transitive action of a group $G$ on a rooted tree $T$ and a level $n$, the rigid stabilizer of level $n$ is the subgroup of $G$ generated by the rigid stabilizers of all vertices of $T$ at level $n$.","title":"Rigid stabilizer of a level"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/is-branch-action","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupShared.LevelTransitiveFaithfulAction.IsBranchAction","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#is-branch-action","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"A faithful level-transitive action of a group $G$ on a rooted tree $T$ is a branch action when, for every level $n$, the rigid stabilizer of level $n$ has finite index in $G$.","title":"Branch action"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/is-branch-group-2","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupShared.IsBranchGroup","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#is-branch-group-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"A group $G$ is a branch group when there exist a rooted tree $T$ (locally finite, with every vertex having at least two children) and a faithful level-transitive action of $G$ on $T$ by tree automorphisms that is a branch action.","title":"Branch group"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/branch-group-finite-index-maximal-subgro","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupMaximalSubgroupsNormalKourovka20Problem101Wanted.branchGroupFiniteIndexMaximalSubgroupsNormal","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#branch-group-finite-index-maximal-subgro","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"fully_proved","summary":"Kourovka 20.101: finite-index maximal subgroups of finitely generated branch groups are normal. The statement (A. Thillaisundaram) that in every finitely generated branch group every maximal subgroup of finite index is normal.","title":"Kourovka 20.101: finite-index maximal subgroups of finitely generated branch groups are normal"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/branch-group-finite-index-maximal-subgro-2","kind":"node","lean":"MathlibExt.GroupTheory.BranchGroupMaximalSubgroupsNormalKourovka20Problem101Wanted.branchGroupFiniteIndexMaximalSubgroupsNormal_refuted","parent":2154,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups/index.html#branch-group-finite-index-maximal-subgro-2","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/maximal-subgroups-of-branch-groups","status":"can_prove","summary":"The Kourovka 20.101 statement that every maximal subgroup of finite index of a finitely generated branch group is normal is false: some finitely generated branch group has a maximal subgroup of finite index that is not normal.","title":"A finitely generated branch group with a non-normal maximal subgroup of finite index"},{"data_url":"scopes/GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f.json","id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f","kind":"scope","navigation":"module","parent":2094,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/index.html","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f","status":"can_prove","title":"Thompson's group F"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/thompson-relators","kind":"node","lean":"MathlibExt.GroupTheory.ThompsonGroupHowsonKourovka19Problem40Wanted.thompsonRelators","parent":2170,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relators of Thompson's group "],["math","\\(F\\)"]],"source":"Relators of Thompson's group $F$"},"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/index.html#thompson-relators","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f","status":"fully_proved","summary":"In the free group on two generators $x_0$ and $x_1$, this is the set of the two elements $[x_0 x_1^{-1}, x_0^{-1} x_1 x_0]$ and $[x_0 x_1^{-1}, x_0^{-2} x_1 x_0^2]$, where $[a, b] = a b a^{-1} b^{-1}$. They are the relators of the standard presentation of Thompson's group $F$.","title":"Relators of Thompson's group $F$"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/thompson-f","kind":"node","lean":"MathlibExt.GroupTheory.ThompsonGroupHowsonKourovka19Problem40Wanted.ThompsonF","parent":2170,"reading_title":{"format":"inline-parts-v1","parts":[["text","Thompson's group "],["math","\\(F\\)"],["text"," by its standard presentation"]],"source":"Thompson's group $F$ by its standard presentation"},"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/index.html#thompson-f","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f","status":"fully_proved","summary":"Thompson's group $F$ is the group with generators $x_0, x_1$ and relations $[x_0 x_1^{-1}, x_0^{-1} x_1 x_0] = 1$ and $[x_0 x_1^{-1}, x_0^{-2} x_1 x_0^2] = 1$, where $[a, b] = a b a^{-1} b^{-1}$; that is, the quotient of the free group on $x_0, x_1$ by the normal subgroup generated by these two relators.","title":"Thompson's group $F$ by its standard presentation"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/thompson-fhowson","kind":"node","lean":"MathlibExt.GroupTheory.ThompsonGroupHowsonKourovka19Problem40Wanted.thompsonFHowson","parent":2170,"reading_title":{"format":"inline-parts-v1","parts":[["text","Howson property for Thompson's group "],["math","\\(F\\)"]],"source":"Howson property for Thompson's group $F$"},"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/index.html#thompson-fhowson","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f","status":"fully_proved","summary":"The statement that Thompson's group $F$ has the Howson property: for all finitely generated subgroups $H$ and $K$ of $F$, the intersection $H \\cap K$ is finitely generated.","title":"Howson property for Thompson's group $F$"},{"id":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/thompson-fhowson-refuted","kind":"node","lean":"MathlibExt.GroupTheory.ThompsonGroupHowsonKourovka19Problem40Wanted.thompsonFHowson_refuted","parent":2170,"reading_title":{"format":"inline-parts-v1","parts":[["text","Thompson's group "],["math","\\(F\\)"],["text"," does not have the Howson property"]],"source":"Thompson's group $F$ does not have the Howson property"},"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f/index.html#thompson-fhowson-refuted","scope":"GroupTheory/geometric-group-theory/branch-and-thompson-groups/thompson-s-group-f","status":"can_prove","summary":"It is not the case that the intersection of any two finitely generated subgroups of Thompson's group $F$ is finitely generated: there are finitely generated subgroups $H$ and $K$ of $F$ such that $H \\cap K$ is not finitely generated.","title":"Thompson's group $F$ does not have the Howson property"},{"data_url":"scopes/GroupTheory/geometric-group-theory/amenable-groups.json","id":"GroupTheory/geometric-group-theory/amenable-groups","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/amenable-groups/index.html","scope":"GroupTheory/geometric-group-theory/amenable-groups","status":"fully_proved","title":"Amenable groups"},{"id":"GroupTheory/geometric-group-theory/amenable-groups/is-amenable","kind":"node","lean":"IsAmenable","parent":2175,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/amenable-groups/index.html#is-amenable","scope":"GroupTheory/geometric-group-theory/amenable-groups","status":"fully_proved","summary":"A discrete group $G$ is *amenable* if it satisfies the F\u00f8lner condition: for every finite subset $K \\subseteq G$ and every real $\\varepsilon > 0$ there is a finite nonempty subset $F \\subseteq G$ such that $|sF \\mathbin{\\triangle} F| / |F| < \\varepsilon$ for every $s \\in K$. Here $sF = \\{sf : f \\in F\\}$ is the left translate of $F$ by $s$, $\\triangle$ is symmetric difference and $|\\cdot|$ is cardinality.","title":"Amenable group (F\u00f8lner condition)"},{"id":"GroupTheory/geometric-group-theory/amenable-groups/is-amenable-of-finite","kind":"node","lean":"isAmenable_of_finite","parent":2175,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/amenable-groups/index.html#is-amenable-of-finite","scope":"GroupTheory/geometric-group-theory/amenable-groups","status":"fully_proved","summary":"Every finite group is amenable.","title":"Finite groups are amenable"},{"id":"GroupTheory/geometric-group-theory/amenable-groups/is-amenable-of-subsingleton","kind":"node","lean":"isAmenable_of_subsingleton","parent":2175,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/amenable-groups/index.html#is-amenable-of-subsingleton","scope":"GroupTheory/geometric-group-theory/amenable-groups","status":"fully_proved","summary":"Every group with at most one element is amenable; the set $F = \\{1\\}$ witnesses the F\u00f8lner condition.","title":"Trivial groups are amenable"},{"data_url":"scopes/GroupTheory/geometric-group-theory/elementary-amenable-groups.json","id":"GroupTheory/geometric-group-theory/elementary-amenable-groups","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/elementary-amenable-groups/index.html","scope":"GroupTheory/geometric-group-theory/elementary-amenable-groups","status":"can_prove","title":"Elementary amenable groups"},{"id":"GroupTheory/geometric-group-theory/elementary-amenable-groups/is-elementary-amenable","kind":"node","lean":"MathlibExt.GroupTheory.ElementaryAmenableVirtuallySolvableKourovka16Problem52Wanted.IsElementaryAmenable","parent":2179,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/elementary-amenable-groups/index.html#is-elementary-amenable","scope":"GroupTheory/geometric-group-theory/elementary-amenable-groups","status":"fully_proved","summary":"The class of elementary amenable groups is the smallest class of groups that contains every finite group and every abelian group and is closed under isomorphisms, subgroups, quotients, extensions and directed unions. Closure under extensions means that a group $G$ with a normal subgroup $N$ is elementary amenable whenever $N$ and $G/N$ are; closure under directed unions means that $G$ is elementary amenable\u2026","title":"Elementary amenable groups"},{"id":"GroupTheory/geometric-group-theory/elementary-amenable-groups/finitely-presented-elementary-amenable-v","kind":"node","lean":"MathlibExt.GroupTheory.ElementaryAmenableVirtuallySolvableKourovka16Problem52Wanted.finitelyPresentedElementaryAmenableVirtuallySolvable","parent":2179,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/elementary-amenable-groups/index.html#finitely-presented-elementary-amenable-v","scope":"GroupTheory/geometric-group-theory/elementary-amenable-groups","status":"fully_proved","summary":"The statement that every finitely presented elementary amenable group $G$ is solvable-by-finite, that is, $G$ has a solvable subgroup of finite index.","title":"Kourovka Problem 16.52 (Linnell\u2013Schick)"},{"id":"GroupTheory/geometric-group-theory/elementary-amenable-groups/finitely-presented-elementary-amenable-v-2","kind":"node","lean":"MathlibExt.GroupTheory.ElementaryAmenableVirtuallySolvableKourovka16Problem52Wanted.finitelyPresentedElementaryAmenableVirtuallySolvable_refuted","parent":2179,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/elementary-amenable-groups/index.html#finitely-presented-elementary-amenable-v-2","scope":"GroupTheory/geometric-group-theory/elementary-amenable-groups","status":"can_prove","summary":"There is a finitely presented elementary amenable group $G$ that is not solvable-by-finite: no subgroup of finite index in $G$ is solvable. This answers Kourovka Problem 16.52 (P. Linnell, T. Schick) in the negative.","title":"Finitely presented elementary amenable groups need not be virtually solvable"},{"data_url":"scopes/GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups.json","id":"GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups/index.html","scope":"GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups","status":"can_prove","title":"Finitely generated simple amenable groups"},{"id":"GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups/infinite-fgsimple-amenable-group","kind":"node","lean":"MathlibExt.GroupTheory.InfiniteSimpleAmenableGroupKourovka15Problem5Wanted.infiniteFGSimpleAmenableGroup","parent":2183,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups/index.html#infinite-fgsimple-amenable-group","scope":"GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups","status":"fully_proved","summary":"The statement that there exists a group $G$ that is infinite, finitely generated, simple and amenable, where amenable means that $G$ satisfies the F\u00f8lner condition.","title":"Kourovka Problem 15.5 (statement): infinite finitely generated simple amenable groups"},{"id":"GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups/infinite-fgsimple-amenable-group-holds","kind":"node","lean":"MathlibExt.GroupTheory.InfiniteSimpleAmenableGroupKourovka15Problem5Wanted.infiniteFGSimpleAmenableGroup_holds","parent":2183,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups/index.html#infinite-fgsimple-amenable-group-holds","scope":"GroupTheory/geometric-group-theory/finitely-generated-simple-amenable-groups","status":"can_prove","summary":"There exists a group $G$ that is infinite, finitely generated, simple and amenable, where amenable means that $G$ satisfies the F\u00f8lner condition.","title":"Juschenko\u2013Monod theorem (Kourovka Problem 15.5)"},{"data_url":"scopes/GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth.json","id":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/index.html","scope":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth","status":"can_prove","title":"F\u00f8lner sets and polynomial growth"},{"id":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/word-ball","kind":"node","lean":"MathlibExt.GroupTheory.FolnerPolynomialSequencePolynomialGrowthKourovka14Problem27Wanted.wordBall","parent":2186,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/index.html#word-ball","scope":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth","status":"fully_proved","summary":"For a finite subset $S$ of a group $\\Gamma$ and a natural number $n$, the ball $B_S(n)$ of radius $n$ is the set of products of at most $n$ elements of $S \\cup S^{-1}$, the empty product $1$ included.","title":"Ball in the word metric"},{"id":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/has-polynomial-growth","kind":"node","lean":"MathlibExt.GroupTheory.FolnerPolynomialSequencePolynomialGrowthKourovka14Problem27Wanted.HasPolynomialGrowth","parent":2186,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/index.html#has-polynomial-growth","scope":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth","status":"fully_proved","summary":"Let $S$ be a finite subset of a group $\\Gamma$, and let $B_S(n)$ be the ball of radius $n$ for $S$, the set of products of at most $n$ elements of $S \\cup S^{-1}$. Then $\\Gamma$ has polynomial growth with respect to $S$ if there are natural numbers $C$ and $d$ such that for every natural number $n$ the ball $B_S(n)$ has at most $C\\,(n + 1)^d$ elements.","title":"Polynomial growth with respect to a finite subset"},{"id":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/boundary","kind":"node","lean":"MathlibExt.GroupTheory.FolnerPolynomialSequencePolynomialGrowthKourovka14Problem27Wanted.boundary","parent":2186,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/index.html#boundary","scope":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth","status":"fully_proved","summary":"For a group $\\Gamma$ and finite subsets $S$ and $F$ of $\\Gamma$, the boundary of $F$ with respect to $S$ is the set $\\partial F = \\{\\gamma \\in \\Gamma \\setminus F : \\gamma s \\in F \\text{ for some } s \\in S\\}$.","title":"Boundary of a finite subset of a group"},{"id":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/polynomial-folner-exhaustion-implies-pol","kind":"node","lean":"MathlibExt.GroupTheory.FolnerPolynomialSequencePolynomialGrowthKourovka14Problem27Wanted.polynomialFolnerExhaustionImpliesPolynomialGrowth","parent":2186,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/index.html#polynomial-folner-exhaustion-implies-pol","scope":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth","status":"fully_proved","summary":"The statement that for every group $\\Gamma$ generated by a finite subset $S$ the following holds: if there are finite subsets $F_0 \\subsetneq F_1 \\subsetneq F_2 \\subsetneq \\cdots$ of $\\Gamma$ whose union is $\\Gamma$ such that $|\\partial F_k| / |F_k| \\to 0$ as $k \\to \\infty$, where $\\partial F_k$ is the boundary of $F_k$ with respect to $S$, and such that for some real $c \\ge 0$ and some natural number $d \\ge 1$ we\u2026","title":"Vaillant's question on F\u00f8lner exhaustions of polynomial size (Kourovka 14.27)"},{"id":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/polynomial-folner-exhaustion-implies-pol-2","kind":"node","lean":"MathlibExt.GroupTheory.FolnerPolynomialSequencePolynomialGrowthKourovka14Problem27Wanted.polynomialFolnerExhaustionImpliesPolynomialGrowth_refuted","parent":2186,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth/index.html#polynomial-folner-exhaustion-implies-pol-2","scope":"GroupTheory/geometric-group-theory/f-lner-sets-and-polynomial-growth","status":"can_prove","summary":"There exist a group $\\Gamma$ generated by a finite subset $S$ and finite subsets $F_0 \\subsetneq F_1 \\subsetneq F_2 \\subsetneq \\cdots$ of $\\Gamma$ whose union is $\\Gamma$ such that $|\\partial F_k| / |F_k| \\to 0$ as $k \\to \\infty$, where $\\partial F_k$ is the boundary of $F_k$ with respect to $S$, and such that for some real $c \\ge 0$ and some natural number $d \\ge 1$ we have $|F_k| \\le c\\,(k + 1)^d$ for every $k$,\u2026","title":"F\u00f8lner exhaustions of polynomial size need not give polynomial growth"},{"data_url":"scopes/GroupTheory/geometric-group-theory/growth-of-groups.json","id":"GroupTheory/geometric-group-theory/growth-of-groups","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-of-groups/index.html","scope":"GroupTheory/geometric-group-theory/growth-of-groups","status":"fully_proved","title":"Growth of groups"},{"id":"GroupTheory/geometric-group-theory/growth-of-groups/growth","kind":"node","lean":"Group.Generators.growth","parent":2192,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-of-groups/index.html#growth","scope":"GroupTheory/geometric-group-theory/growth-of-groups","status":"fully_proved","summary":"Let $P$ be a generating family of a group $G$. The growth function $\\gamma_P$ of $P$ assigns to each natural number $n$ the number of elements $g \\in G$ whose word length with respect to $P$ is at most $n$, that is, the cardinality of the ball of radius $n$ in the word metric of $P$; when this ball is infinite, $\\gamma_P(n)$ is taken to be $0$.","title":"Growth function of a generating family"},{"id":"GroupTheory/geometric-group-theory/growth-of-groups/has-polynomial-growth","kind":"node","lean":"Group.Generators.HasPolynomialGrowth","parent":2192,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-of-groups/index.html#has-polynomial-growth","scope":"GroupTheory/geometric-group-theory/growth-of-groups","status":"fully_proved","summary":"Let $P$ be a generating family of a group $G$, with growth function $\\gamma_P$. Then $P$ has polynomial growth if, for every natural number $n$, the ball of radius $n$ (the set of elements of $G$ of word length at most $n$ with respect to $P$) is finite, and there are natural numbers $C$ and $d$ such that $\\gamma_P(n) \\le C (n + 1)^d$ for every natural number $n$.","title":"Polynomial growth"},{"id":"GroupTheory/geometric-group-theory/growth-of-groups/growth-zero","kind":"node","lean":"Group.Generators.growth_zero","parent":2192,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-of-groups/index.html#growth-zero","scope":"GroupTheory/geometric-group-theory/growth-of-groups","status":"fully_proved","summary":"For every generating family $P$ of a group $G$, the growth function of $P$ takes the value $1$ at $0$: the ball of radius $0$ consists of the identity element alone.","title":"The growth function at radius zero"},{"id":"GroupTheory/geometric-group-theory/growth-of-groups/growth-pos","kind":"node","lean":"Group.Generators.growth_pos","parent":2192,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-of-groups/index.html#growth-pos","scope":"GroupTheory/geometric-group-theory/growth-of-groups","status":"fully_proved","summary":"For every generating family $P$ of a finite group $G$ and every natural number $n$, the growth function of $P$ is positive at $n$: the ball of radius $n$ contains the identity.","title":"The growth function of a finite group is positive"},{"id":"GroupTheory/geometric-group-theory/growth-of-groups/growth-le-card","kind":"node","lean":"Group.Generators.growth_le_card","parent":2192,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-of-groups/index.html#growth-le-card","scope":"GroupTheory/geometric-group-theory/growth-of-groups","status":"fully_proved","summary":"For every generating family $P$ of a finite group $G$ and every natural number $n$, the value of the growth function of $P$ at $n$ is at most the order of $G$, since every ball is a subset of $G$.","title":"The growth function is bounded by the order of the group"},{"id":"GroupTheory/geometric-group-theory/growth-of-groups/has-polynomial-growth-of-finite","kind":"node","lean":"Group.Generators.hasPolynomialGrowth_of_finite","parent":2192,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-of-groups/index.html#has-polynomial-growth-of-finite","scope":"GroupTheory/geometric-group-theory/growth-of-groups","status":"fully_proved","summary":"Every generating family of a finite group has polynomial growth: its growth function is bounded by the order of the group, a polynomial of degree $0$.","title":"Finite groups have polynomial growth"},{"id":"GroupTheory/geometric-group-theory/growth-of-groups/growth-pos-2","kind":"node","lean":"Group.Generators.HasPolynomialGrowth.growth_pos","parent":2192,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-of-groups/index.html#growth-pos-2","scope":"GroupTheory/geometric-group-theory/growth-of-groups","status":"fully_proved","summary":"If a generating family $P$ of a group $G$ has polynomial growth, then the growth function of $P$ is positive at every natural number $n$.","title":"Polynomial growth makes the growth function positive"},{"data_url":"scopes/GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups.json","id":"GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups/index.html","scope":"GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups","status":"can_prove","title":"Growth rates of amenable groups"},{"id":"GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups/growth-rate","kind":"node","lean":"MathlibExt.GroupTheory.AmenableLargeGrowthRateKourovka15Problem4Wanted.growthRate","parent":2200,"reading_title":{"format":"inline-parts-v1","parts":[["text","Growth rate of a "],["math","\\(k\\)"],["text","-element generating family (Kourovka 14.7)"]],"source":"Growth rate of a $k$-element generating family (Kourovka 14.7)"},"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups/index.html#growth-rate","scope":"GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups","status":"fully_proved","summary":"Let $\\Gamma$ be a group and $S = (s_1, \\dots, s_k)$ a family of $k$ elements generating $\\Gamma$. For $n \\in \\mathbb{N}$ let $\\beta_S(n)$ be the number of elements of $\\Gamma$ that are products of at most $n$ elements of $S \\cup S^{-1}$. The growth rate of $\\Gamma$ with respect to $S$ is the real number $\\omega(\\Gamma, S) = \\limsup_{n \\to \\infty} \\beta_S(n)^{1/n}$.","title":"Growth rate of a $k$-element generating family (Kourovka 14.7)"},{"id":"GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups/large-growth-rate-implies-nonamenable","kind":"node","lean":"MathlibExt.GroupTheory.AmenableLargeGrowthRateKourovka15Problem4Wanted.largeGrowthRateImpliesNonamenable","parent":2200,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups/index.html#large-growth-rate-implies-nonamenable","scope":"GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups","status":"fully_proved","summary":"The proposition, asked by P. de la Harpe, that there are $k \\ge 2$ and a real $\\varepsilon > 0$ such that every group $\\Gamma$ generated by a family $S$ of $k$ distinct elements whose growth rate satisfies $\\omega(\\Gamma, S) \\ge 2k - 1 - \\varepsilon$ is non-amenable (amenability being the F\u00f8lner condition).","title":"Kourovka Problem 15.4 (large growth rate implies non-amenability)"},{"id":"GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups/large-growth-rate-implies-nonamenable-re","kind":"node","lean":"MathlibExt.GroupTheory.AmenableLargeGrowthRateKourovka15Problem4Wanted.largeGrowthRateImpliesNonamenable_refuted","parent":2200,"reading_title":{"format":"inline-parts-v1","parts":[["text","Amenable groups with growth rate close to "],["math","\\(2k - 1\\)"],["text"," (Kourovka 15.4 refuted)"]],"source":"Amenable groups with growth rate close to $2k - 1$ (Kourovka 15.4 refuted)"},"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups/index.html#large-growth-rate-implies-nonamenable-re","scope":"GroupTheory/geometric-group-theory/growth-rates-of-amenable-groups","status":"can_prove","summary":"For a group $\\Gamma$ generated by a family $S$ of $k$ elements and $n \\in \\mathbb{N}$, let $\\beta_S(n)$ be the number of elements of $\\Gamma$ that are products of at most $n$ elements of $S \\cup S^{-1}$, and let $\\omega(\\Gamma, S) = \\limsup_{n \\to \\infty} \\beta_S(n)^{1/n}$ be the growth rate. For every $k \\ge 2$ and every real $\\varepsilon > 0$ there are an amenable group $\\Gamma$ (amenability being the F\u00f8lner\u2026","title":"Amenable groups with growth rate close to $2k - 1$ (Kourovka 15.4 refuted)"},{"data_url":"scopes/GroupTheory/geometric-group-theory/hyperbolic-groups.json","id":"GroupTheory/geometric-group-theory/hyperbolic-groups","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/hyperbolic-groups/index.html","scope":"GroupTheory/geometric-group-theory/hyperbolic-groups","status":"fully_proved","title":"Hyperbolic groups"},{"id":"GroupTheory/geometric-group-theory/hyperbolic-groups/is-hyperbolic","kind":"node","lean":"Group.IsHyperbolic","parent":2204,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/hyperbolic-groups/index.html#is-hyperbolic","scope":"GroupTheory/geometric-group-theory/hyperbolic-groups","status":"fully_proved","summary":"A group $G$ is word-hyperbolic if there are a finite index set $\\iota$ and a family $(s_i)_{i \\in \\iota}$ of elements generating $G$ such that $G$, equipped with the word metric of this generating family, is $\\delta$-hyperbolic for some real number $\\delta$.","title":"Word-hyperbolic groups"},{"id":"GroupTheory/geometric-group-theory/hyperbolic-groups/fg","kind":"node","lean":"Group.IsHyperbolic.fg","parent":2204,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/hyperbolic-groups/index.html#fg","scope":"GroupTheory/geometric-group-theory/hyperbolic-groups","status":"fully_proved","summary":"Every word-hyperbolic group is finitely generated.","title":"Word-hyperbolic groups are finitely generated"},{"id":"GroupTheory/geometric-group-theory/hyperbolic-groups/of-subsingleton","kind":"node","lean":"Group.IsHyperbolic.of_subsingleton","parent":2204,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/hyperbolic-groups/index.html#of-subsingleton","scope":"GroupTheory/geometric-group-theory/hyperbolic-groups","status":"fully_proved","summary":"Every trivial group is word-hyperbolic: the empty family generates it, and its word metric is $0$-hyperbolic.","title":"The trivial group is word-hyperbolic"},{"data_url":"scopes/GroupTheory/geometric-group-theory/permutation-stability.json","id":"GroupTheory/geometric-group-theory/permutation-stability","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","title":"Permutation stability"},{"id":"GroupTheory/geometric-group-theory/permutation-stability/normalized-hamming-distance","kind":"node","lean":"PermutationStability.normalizedHammingDistance","parent":2208,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html#normalized-hamming-distance","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","summary":"For a natural number $n$ and permutations $\\sigma$, $\\tau$ of $\\{0, \\dots, n-1\\}$, the normalized Hamming distance $d_n(\\sigma, \\tau)$ is the number of points $i < n$ with $\\sigma(i) \\ne \\tau(i)$, divided by $n$, as a real number. For $n = 0$ it is $0$; this single value does not affect statements about limits as $n \\to \\infty$.","title":"Normalized Hamming distance"},{"id":"GroupTheory/geometric-group-theory/permutation-stability/is-almost-homomorphism","kind":"node","lean":"PermutationStability.IsAlmostHomomorphism","parent":2208,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html#is-almost-homomorphism","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","summary":"Let $G$ be a group and, for each natural number $n$, let $f_n \\colon G \\to S_n$ be an arbitrary map, where $S_n$ is the group of permutations of $\\{0, \\dots, n-1\\}$. The sequence $(f_n)$ is an almost-homomorphism when for all $g, h \\in G$ the normalized Hamming distance $d_n(f_n(g) f_n(h), f_n(gh))$ tends to $0$ as $n \\to \\infty$.","title":"Almost-homomorphisms to symmetric groups"},{"id":"GroupTheory/geometric-group-theory/permutation-stability/is-close-to-homomorphisms","kind":"node","lean":"PermutationStability.IsCloseToHomomorphisms","parent":2208,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html#is-close-to-homomorphisms","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","summary":"Let $G$ be a group and let $f_n \\colon G \\to S_n$ ($n \\in \\mathbb{N}$) be maps into the symmetric groups $S_n$ of permutations of $\\{0, \\dots, n-1\\}$. The sequence $(f_n)$ is close to homomorphisms when there are group homomorphisms $\\rho_n \\colon G \\to S_n$ such that for every $g \\in G$ the normalized Hamming distance $d_n(\\rho_n(g), f_n(g))$ tends to $0$ as $n \\to \\infty$.","title":"Closeness to homomorphisms"},{"id":"GroupTheory/geometric-group-theory/permutation-stability/permutation-stable","kind":"node","lean":"PermutationStability.PermutationStable","parent":2208,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html#permutation-stable","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","summary":"A group $G$ is permutation stable when every almost-homomorphism $(f_n)$ from $G$ to the symmetric groups $S_n$ is close to homomorphisms.","title":"Permutation stability"},{"id":"GroupTheory/geometric-group-theory/permutation-stability/flexible-hamming-distance","kind":"node","lean":"PermutationStability.flexibleHammingDistance","parent":2208,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html#flexible-hamming-distance","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","summary":"Let $n \\le m$ be natural numbers, $\\sigma$ a permutation of $\\{0, \\dots, n-1\\}$ and $\\tau$ a permutation of $\\{0, \\dots, m-1\\}$. The flexible Hamming distance of $\\sigma$ and $\\tau$ is the real number","title":"Flexible Hamming distance"},{"id":"GroupTheory/geometric-group-theory/permutation-stability/is-flexibly-close-to-homomorphisms","kind":"node","lean":"PermutationStability.IsFlexiblyCloseToHomomorphisms","parent":2208,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html#is-flexibly-close-to-homomorphisms","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","summary":"Let $G$ be a group and let $f_n \\colon G \\to S_n$ ($n \\in \\mathbb{N}$) be maps into the symmetric groups $S_n$ of permutations of $\\{0, \\dots, n-1\\}$. The sequence $(f_n)$ is flexibly close to homomorphisms when there are natural numbers $m_n \\ge n$ and group homomorphisms $\\rho_n \\colon G \\to S_{m_n}$ such that for every $g \\in G$ the flexible Hamming distance $d^{\\mathrm{flex}}_n(f_n(g), \\rho_n(g))$ tends to $0$\u2026","title":"Flexible closeness to homomorphisms"},{"id":"GroupTheory/geometric-group-theory/permutation-stability/flexible-permutation-stable","kind":"node","lean":"PermutationStability.FlexiblePermutationStable","parent":2208,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html#flexible-permutation-stable","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","summary":"A group $G$ is flexibly permutation stable when every almost-homomorphism $(f_n)$ from $G$ to the symmetric groups is flexibly close to homomorphisms.","title":"Flexible permutation stability"},{"id":"GroupTheory/geometric-group-theory/permutation-stability/metabelian","kind":"node","lean":"PermutationStability.Metabelian","parent":2208,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/permutation-stability/index.html#metabelian","scope":"GroupTheory/geometric-group-theory/permutation-stability","status":"fully_proved","summary":"Let $G$ be a group and let $G' = [G, G]$ be its commutator subgroup. Then $G$ is metabelian when its second derived subgroup $G'' = [G', G']$ is trivial.","title":"Metabelian groups"},{"data_url":"scopes/GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups.json","id":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/index.html","scope":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups","status":"can_prove","title":"Power quotients of hyperbolic groups"},{"id":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/power-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicPowerSubgroupsKourovka11Problem74Wanted.powerSubgroup","parent":2217,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/index.html#power-subgroup","scope":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups","status":"fully_proved","summary":"For a group $G$ and a natural number $n$, $G^n$ is the subgroup of $G$ generated by the $n$-th powers $g^n$ of all elements $g \\in G$.","title":"Power subgroup"},{"id":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/power-quotient-infinite","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicPowerSubgroupsKourovka11Problem74Wanted.powerQuotientInfinite","parent":2217,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/index.html#power-quotient-infinite","scope":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups","status":"fully_proved","summary":"For every word-hyperbolic group $G$ that is not virtually cyclic (non-elementary), there is an integer $n \\ge 1$ such that the subgroup $G^n$ generated by the $n$-th powers of the elements of $G$ has infinite index in $G$, i.e. the quotient $G / G^n$ is infinite.","title":"Kourovka Problem 11.74(a) (infinite power quotients of hyperbolic groups)"},{"id":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/power-subgroups-trivial-intersection","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicPowerSubgroupsKourovka11Problem74Wanted.powerSubgroupsTrivialIntersection","parent":2217,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/index.html#power-subgroups-trivial-intersection","scope":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups","status":"fully_proved","summary":"For every word-hyperbolic group $G$ that is not virtually cyclic (non-elementary), the only element of $G$ lying in the subgroup $G^n$ generated by the $n$-th powers for every integer $n \\ge 1$ is the identity, i.e. $\\bigcap_{n \\ge 1} G^n = 1$.","title":"Kourovka Problem 11.74(b) (trivial intersection of power subgroups)"},{"id":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/power-quotient-infinite-holds","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicPowerSubgroupsKourovka11Problem74Wanted.powerQuotientInfinite_holds","parent":2217,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/index.html#power-quotient-infinite-holds","scope":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups","status":"can_prove","summary":"For every word-hyperbolic group $G$ that is not virtually cyclic, there is an integer $n \\ge 1$ such that the subgroup $G^n$ generated by the $n$-th powers of the elements of $G$ has infinite index in $G$, i.e. the quotient $G / G^n$ is infinite.","title":"Non-elementary hyperbolic groups have infinite power quotients"},{"id":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/power-subgroups-trivial-intersection-hol","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicPowerSubgroupsKourovka11Problem74Wanted.powerSubgroupsTrivialIntersection_holds","parent":2217,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups/index.html#power-subgroups-trivial-intersection-hol","scope":"GroupTheory/geometric-group-theory/power-quotients-of-hyperbolic-groups","status":"can_prove","summary":"For every word-hyperbolic group $G$ that is not virtually cyclic, the only element of $G$ lying in the subgroup $G^n$ generated by the $n$-th powers for every integer $n \\ge 1$ is the identity, i.e. $\\bigcap_{n \\ge 1} G^n = 1$.","title":"Power subgroups of a non-elementary hyperbolic group intersect trivially"},{"data_url":"scopes/GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group.json","id":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/index.html","scope":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group","status":"can_prove","title":"Property (T) for automorphism groups of free groups"},{"id":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/laplacian","kind":"node","lean":"MathlibExt.GroupTheory.OnPropertyTWanted.OnPropertyT.Laplacian","parent":2223,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/index.html#laplacian","scope":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group","status":"fully_proved","summary":"For a group $G$ and a finite subset $S \\subseteq G$, the Laplacian of $S$ is the element $\\Delta_S = |S| - \\sum_{s \\in S} s$ of the real group ring $\\mathbb{R}G$, where $|S|$ is the number of elements of $S$, taken as $|S|$ times the identity element of $\\mathbb{R}G$. When $S$ is a symmetric generating set of $G$, this is the Laplacian of Equation (1) in the paper of Kaluba, Kielak and Nowak on property (T) for\u2026","title":"Laplacian of a finite subset of a group"},{"id":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/group-ring-involution","kind":"node","lean":"MathlibExt.GroupTheory.OnPropertyTWanted.OnPropertyT.groupRingInvolution","parent":2223,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/index.html#group-ring-involution","scope":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group","status":"fully_proved","summary":"For a group $G$, the involution of the real group ring $\\mathbb{R}G$ is the $\\mathbb{R}$-linear map $\\mathbb{R}G \\to \\mathbb{R}G$ that sends $\\sum_{g \\in G} a_g g$ to $\\sum_{g \\in G} a_g g^{-1}$, that is, the linear extension of inversion on $G$.","title":"Involution of the real group ring"},{"id":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/property-t","kind":"node","lean":"MathlibExt.GroupTheory.OnPropertyTWanted.OnPropertyT.PropertyT","parent":2223,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/index.html#property-t","scope":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group","status":"fully_proved","summary":"Let $G$ be a group. For a finite subset $S \\subseteq G$, write $\\Delta_S = |S| - \\sum_{s \\in S} s$ for the Laplacian of $S$ in the real group ring $\\mathbb{R}G$, and write $\\xi \\mapsto \\xi^{\\star}$ for the involution of $\\mathbb{R}G$ that extends inversion on $G$ linearly. The group $G$ has Kazhdan's property (T) when there exist a finite subset $S \\subseteq G$ with $S^{-1} = S$ that generates $G$, a real number\u2026","title":"Property (T)"},{"id":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/theorem-1","kind":"node","lean":"MathlibExt.GroupTheory.OnPropertyTWanted.OnPropertyT.theorem_1","parent":2223,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group/index.html#theorem-1","scope":"GroupTheory/geometric-group-theory/property-t-for-automorphism-groups-of-free-group","status":"can_prove","summary":"For every natural number $n \\ge 6$, the automorphism group $\\operatorname{Aut}(F_n)$ of the free group $F_n$ on $n$ generators has property (T).","title":"Property (T) for automorphism groups of free groups of rank at least 6"},{"data_url":"scopes/GroupTheory/geometric-group-theory/sofic-groups.json","id":"GroupTheory/geometric-group-theory/sofic-groups","kind":"scope","navigation":"module","parent":2093,"result":false,"route":"../roadmap/GroupTheory/geometric-group-theory/sofic-groups/index.html","scope":"GroupTheory/geometric-group-theory/sofic-groups","status":"can_prove","title":"Sofic groups"},{"id":"GroupTheory/geometric-group-theory/sofic-groups/sofic-hamming-dist","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem86Wanted.soficHammingDist","parent":2228,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/sofic-groups/index.html#sofic-hamming-dist","scope":"GroupTheory/geometric-group-theory/sofic-groups","status":"fully_proved","summary":"For a natural number $n$ and permutations $\\sigma$, $\\tau$ of $\\{0, \\dots, n-1\\}$, the normalized Hamming distance $d(\\sigma, \\tau)$ is the number of points $i < n$ with $\\sigma(i) \\ne \\tau(i)$, divided by $n$, as a real number; for $n = 0$ it is $0$. It is used in the definition of sofic groups [KOU-21.86].","title":"Hamming distance for sofic approximations"},{"id":"GroupTheory/geometric-group-theory/sofic-groups/is-sofic","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem86Wanted.IsSofic","parent":2228,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/sofic-groups/index.html#is-sofic","scope":"GroupTheory/geometric-group-theory/sofic-groups","status":"fully_proved","summary":"Let $d$ denote the normalized Hamming distance on permutations. A group $G$ is sofic if for every finite subset $F \\subseteq G$ with $1 \\in F$ and every real $\\varepsilon > 0$ there are an integer $n \\ge 1$ and a map $\\varphi$ from $F$ to the permutations of $\\{0, \\dots, n-1\\}$ such that $\\varphi(1)$ is the identity, $d(\\varphi(gh), \\varphi(g)\\varphi(h)) < \\varepsilon$ for all $g, h \\in F$ with $gh \\in F$, and\u2026","title":"Sofic groups"},{"id":"GroupTheory/geometric-group-theory/sofic-groups/conjecture","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem86Wanted.conjecture","parent":2228,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/sofic-groups/index.html#conjecture","scope":"GroupTheory/geometric-group-theory/sofic-groups","status":"fully_proved","summary":"The statement that every group $G$ is sofic [KOU-21.86].","title":"Soficity problem (Kourovka Notebook Problem 21.86)"},{"id":"GroupTheory/geometric-group-theory/sofic-groups/conjecture-refuted","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem86Wanted.conjecture_refuted","parent":2228,"result":true,"route":"../roadmap/GroupTheory/geometric-group-theory/sofic-groups/index.html#conjecture-refuted","scope":"GroupTheory/geometric-group-theory/sofic-groups","status":"can_prove","summary":"The soficity problem (Kourovka Notebook Problem 21.86) has a negative answer: there exists a group that is not sofic.","title":"Not every group is sofic"},{"data_url":"scopes/GroupTheory/free-groups.json","id":"GroupTheory/free-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/free-groups/index.html","scope":"GroupTheory/free-groups","status":"can_prove","summary":"Free groups and free metabelian groups: subgroups (inert, compressed, retracts, fixed subgroups), elements (primitive, test, orbit-blocking words, normal closures), and the groups Aut(F_n), Out(F_n) and IA-automorphisms; equations over free groups go to Varieties, quasivarieties and equations, and commutator length to Commutators, word maps and Engel conditions.","title":"Free groups and their automorphisms"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups","kind":"scope","parent":2233,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups","status":"can_prove","summary":"Subgroup and element structure inside free and free metabelian groups: rank, inertness and compressedness, derived subgroups, retracts, primitive and equally distributed elements, independent sets, separation, Cayley\u2013Hamilton analogues and characteristic quotients.","title":"Subgroups and elements of free groups"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups","status":"can_prove","title":"Cayley\u2013Hamilton analogue for free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups/free-group-cayley-hamilton","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupCayleyHamiltonKourovka17Problem32Wanted.freeGroupCayleyHamilton","parent":2235,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups/index.html#free-group-cayley-hamilton","scope":"GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups","status":"fully_proved","summary":"The assertion that for every natural number $n$, every element $w$ of the free group $F_n$ of rank $n$ and every automorphism $\\varphi$ of $F_n$, if $w, \\varphi(w), \\dots, \\varphi^n(w)$ generate $F_n$, then already $w, \\varphi(w), \\dots, \\varphi^{n-1}(w)$ generate $F_n$.","title":"Bogopolski's Cayley\u2013Hamilton question for free groups (Kourovka 17.32)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups/free-group-cayley-hamilton-refuted","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupCayleyHamiltonKourovka17Problem32Wanted.freeGroupCayleyHamilton_refuted","parent":2235,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups/index.html#free-group-cayley-hamilton-refuted","scope":"GroupTheory/free-groups/subgroups-of-free-groups/cayley-hamilton-analogue-for-free-groups","status":"can_prove","summary":"There are a natural number $n$, an element $w$ of the free group $F_n$ of rank $n$ and an automorphism $\\varphi$ of $F_n$ such that $w, \\varphi(w), \\dots, \\varphi^n(w)$ generate $F_n$ but $w, \\varphi(w), \\dots, \\varphi^{n-1}(w)$ do not.","title":"Bogopolski's Cayley\u2013Hamilton analogue fails for free groups (Kourovka 17.32)"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups","status":"can_prove","title":"Characteristic quotients of free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups/characteristic-quotient-fewer-generators","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupCharacteristicQuotientRankKourovka17Problem62Wanted.characteristicQuotientFewerGenerators","parent":2238,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups/index.html#characteristic-quotient-fewer-generators","scope":"GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups","status":"fully_proved","summary":"The statement that for every natural number $n \\ge 3$ the free group $F_n$ on $n$ generators has a characteristic subgroup $C \\ne F_n$ such that the quotient $F_n / C$ is generated by two elements, hence by fewer than $n$ elements. This is the positive answer to Problem 17.62 (J. Conrad) in every rank where two generators are fewer than $n$.","title":"Kourovka Problem 17.62 (two-generated characteristic quotients of free groups)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups/characteristic-quotient-fewer-generators-2","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupCharacteristicQuotientRankKourovka17Problem62Wanted.characteristicQuotientFewerGenerators_holds","parent":2238,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups/index.html#characteristic-quotient-fewer-generators-2","scope":"GroupTheory/free-groups/subgroups-of-free-groups/characteristic-quotients-of-free-groups","status":"can_prove","summary":"For every natural number $n \\ge 3$, the free group $F_n$ on $n$ generators has a characteristic subgroup $C \\ne F_n$ such that the quotient $F_n / C$ is generated by two elements; since $2 < n$, the quotient is generated by fewer than $n$ elements.","title":"Free groups of rank at least 3 have proper two-generated characteristic quotients"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups","status":"can_prove","title":"Derived subgroups of subgroups of free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups/derived-inclusion-implies-inclusion","kind":"node","lean":"MathlibExt.GroupTheory.DerivedSubgroupInclusionFreeGroupKourovka13Problem62Wanted.derivedInclusionImpliesInclusion","parent":2241,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups/index.html#derived-inclusion-implies-inclusion","scope":"GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups","status":"fully_proved","summary":"The conjecture of V. P. Shaptala that for every free group $F$ and all non-cyclic subgroups $U$ and $V$ of $F$, if the commutator subgroup $[U, U]$ is contained in the commutator subgroup $[V, V]$, then $U$ is contained in $V$.","title":"Inclusion of derived subgroups in free groups (Kourovka 13.62)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups/derived-inclusion-implies-inclusion-refu","kind":"node","lean":"MathlibExt.GroupTheory.DerivedSubgroupInclusionFreeGroupKourovka13Problem62Wanted.derivedInclusionImpliesInclusion_refuted","parent":2241,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups/index.html#derived-inclusion-implies-inclusion-refu","scope":"GroupTheory/free-groups/subgroups-of-free-groups/derived-subgroups-of-subgroups-of-free-groups","status":"can_prove","summary":"There exist a free group $F$ and non-cyclic subgroups $U$ and $V$ of $F$ such that the commutator subgroup $[U, U]$ is contained in the commutator subgroup $[V, V]$ but $U$ is not contained in $V$. That is, the conjecture of Kourovka Problem 13.62 is false; the Kourovka Archive credits this answer to M. J. Dunwoody.","title":"Inclusion of derived subgroups does not imply inclusion in free groups"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian","status":"can_prove","title":"Equally distributed elements of free metabelian groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/free-metabelian","kind":"node","lean":"MathlibExt.GroupTheory.FreeMetabelianSameDistributionKourovka19Problem94Wanted.FreeMetabelian","parent":2244,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free metabelian group of rank "],["math","\\(n\\)"],["text"," (Kourovka 19.94)"]],"source":"Free metabelian group of rank $n$ (Kourovka 19.94)"},"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/index.html#free-metabelian","scope":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian","status":"fully_proved","summary":"For a natural number $n$, let $F_n$ be the free group on $n$ generators. The free metabelian group of rank $n$ is the quotient $M_n = F_n / F_n''$ of $F_n$ by its second derived subgroup $F_n'' = [[F_n, F_n], [F_n, F_n]]$.","title":"Free metabelian group of rank $n$ (Kourovka 19.94)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/same-distribution","kind":"node","lean":"MathlibExt.GroupTheory.FreeMetabelianSameDistributionKourovka19Problem94Wanted.SameDistribution","parent":2244,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/index.html#same-distribution","scope":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian","status":"fully_proved","summary":"Let $n$ be a natural number and $F_n$ the free group on generators $x_1, \\dots, x_n$. Words $w_1, w_2 \\in F_n$ have the same distribution when, for every finite metabelian group $G$ (that is, $G'' = 1$) and every $g \\in G$, the number of tuples $(a_1, \\dots, a_n) \\in G^n$ with $w_1(a_1, \\dots, a_n) = g$ equals the number of tuples with $w_2(a_1, \\dots, a_n) = g$. Here $w(a_1, \\dots, a_n)$ is the image of $w$ under\u2026","title":"Words with the same distribution"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/same-distribution-iff-automorphic","kind":"node","lean":"MathlibExt.GroupTheory.FreeMetabelianSameDistributionKourovka19Problem94Wanted.sameDistributionIffAutomorphic","parent":2244,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/index.html#same-distribution-iff-automorphic","scope":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian","status":"fully_proved","summary":"The statement that for every natural number $n$ and all words $w_1, w_2$ in the free group $F_n$ of rank $n$, the words $w_1$ and $w_2$ have the same distribution if and only if some automorphism of the free metabelian group $M_n = F_n / F_n''$ maps the image of $w_1$ in $M_n$ to the image of $w_2$.","title":"Timoshenko's question on equally distributed elements (Kourovka 19.94)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/same-distribution-iff-automorphic-refute","kind":"node","lean":"MathlibExt.GroupTheory.FreeMetabelianSameDistributionKourovka19Problem94Wanted.sameDistributionIffAutomorphic_refuted","parent":2244,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian/index.html#same-distribution-iff-automorphic-refute","scope":"GroupTheory/free-groups/subgroups-of-free-groups/equally-distributed-elements-of-free-metabelian","status":"can_prove","summary":"It is not the case that for every natural number $n$ and all words $w_1, w_2$ in the free group $F_n$ of rank $n$, the words $w_1$ and $w_2$ have the same distribution if and only if some automorphism of the free metabelian group $M_n = F_n / F_n''$ maps the image of $w_1$ in $M_n$ to the image of $w_2$.","title":"Equally distributed elements need not be automorphic (Kourovka 19.94)"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"can_prove","title":"Inert and compressed subgroups of free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/subgroup-rank","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupInertKourovka19Problem102Wanted.subgroupRank","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#subgroup-rank","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"For a subgroup $H$ of a group $G$, the rank $r(H) \\in \\mathbb{N} \\cup \\{\\infty\\}$ is the least size of a finite subset of $G$ that generates $H$, with $r(H) = \\infty$ when $H$ is not finitely generated.","title":"Rank of a subgroup"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/is-inert","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupInertKourovka19Problem102Wanted.IsInert","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#is-inert","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"Let $F_n$ be the free group on $n$ generators and $r(K) \\in \\mathbb{N} \\cup \\{\\infty\\}$ the rank of a subgroup $K$. A subgroup $H$ of $F_n$ is inert when $r(H \\cap K) \\le r(K)$ for every subgroup $K$ of $F_n$.","title":"Inert subgroup of a free group of finite rank"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/is-compressed","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupInertKourovka19Problem102Wanted.IsCompressed","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#is-compressed","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"Let $F_n$ be the free group on $n$ generators and $r(K) \\in \\mathbb{N} \\cup \\{\\infty\\}$ the rank of a subgroup $K$. A subgroup $H$ of $F_n$ is compressed when $r(H) \\le r(K)$ for every subgroup $K$ of $F_n$ containing $H$.","title":"Compressed subgroup of a free group of finite rank"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/compressed-subgroups-inert","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupInertKourovka19Problem102Wanted.compressedSubgroupsInert","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#compressed-subgroups-inert","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"The proposition that for every $n \\in \\mathbb{N}$, every compressed subgroup of the free group $F_n$ on $n$ generators is inert.","title":"Ventura's compressed-implies-inert problem in finite rank (Kourovka 19.102)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/compressed-subgroups-inert-holds","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupInertKourovka19Problem102Wanted.compressedSubgroupsInert_holds","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#compressed-subgroups-inert-holds","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$, every compressed subgroup $H$ of the free group $F_n$ on $n$ generators is inert.","title":"Compressed subgroups of free groups of finite rank are inert (Kourovka 19.102)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/subgroup-rank-2","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupIntersectionKourovka19Problem103Wanted.subgroupRank","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#subgroup-rank-2","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"For a subgroup $H$ of a group $G$, the rank $r(H) \\in \\mathbb{N} \\cup \\{\\infty\\}$ is the least number of elements of a finite subset of $G$ that generates $H$, with $r(H) = \\infty$ when $H$ is not finitely generated.","title":"Rank of a subgroup (Kourovka 19.103)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/is-compressed-2","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupIntersectionKourovka19Problem103Wanted.IsCompressed","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#is-compressed-2","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"A subgroup $H$ of the free group $F_n$ on $n$ generators is compressed when $r(H) \\le r(K)$ for every subgroup $K$ of $F_n$ that contains $H$, where $r$ denotes the rank of a subgroup.","title":"Compressed subgroup of a free group (Kourovka 19.103)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/intersection-of-compressed-compressed","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupIntersectionKourovka19Problem103Wanted.intersectionOfCompressedCompressed","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#intersection-of-compressed-compressed","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"The statement that for every natural number $n$ and all subgroups $H$ and $K$ of the free group $F_n$ on $n$ generators, if $H$ and $K$ are compressed, then $H \\cap K$ is compressed.","title":"Kourovka Problem 19.103 for two compressed subgroups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/intersection-of-compressed-compressed-ho","kind":"node","lean":"MathlibExt.GroupTheory.CompressedSubgroupIntersectionKourovka19Problem103Wanted.intersectionOfCompressedCompressed_holds","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#intersection-of-compressed-compressed-ho","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"can_prove","summary":"For every natural number $n$ and all subgroups $H$ and $K$ of the free group $F_n$ on $n$ generators, if $H$ and $K$ are compressed, then $H \\cap K$ is compressed. That is, Kourovka Problem 19.103 has a positive answer for the intersection of two subgroups of a free group of finite rank.","title":"The intersection of two compressed subgroups of a finite-rank free group is compressed"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/subgroup-rank-3","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismFixedSubgroupInertKourovka19Problem105Wanted.subgroupRank","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#subgroup-rank-3","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"For a subgroup $H$ of a group $G$, the rank $r(H)$ is the least size of a finite subset of $G$ that generates $H$, or $\\infty$ if $H$ is not finitely generated.","title":"Rank of a subgroup"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/is-inert-2","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismFixedSubgroupInertKourovka19Problem105Wanted.IsInert","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#is-inert-2","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"Let $F_n$ be the free group on $n$ generators and let $r(K)$ denote the rank of a subgroup $K$. A subgroup $H \\le F_n$ is inert if $r(H \\cap K) \\le r(K)$ for every subgroup $K \\le F_n$.","title":"Inert subgroups of a free group"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/endomorphism-fixed-subgroup-inert","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismFixedSubgroupInertKourovka19Problem105Wanted.endomorphismFixedSubgroupInert","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#endomorphism-fixed-subgroup-inert","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"The statement that for every $n \\in \\mathbb{N}$ and every endomorphism $\\varphi$ of the free group $F_n$ on $n$ generators, the fixed subgroup $\\{x \\in F_n \\mid \\varphi(x) = x\\}$ of $\\varphi$ is inert in $F_n$.","title":"Kourovka Problem 19.105 (Ventura) for free groups of finite rank"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/endomorphism-fixed-subgroup-inert-holds","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismFixedSubgroupInertKourovka19Problem105Wanted.endomorphismFixedSubgroupInert_holds","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#endomorphism-fixed-subgroup-inert-holds","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"can_prove","summary":"Let $n \\in \\mathbb{N}$, let $F_n$ be the free group on $n$ generators, and for a subgroup $H \\le F_n$ let $r(H)$ be its rank, the least size of a finite generating set of $H$, or $\\infty$ if $H$ is not finitely generated. For every endomorphism $\\varphi$ of $F_n$, the fixed subgroup $\\{x \\in F_n \\mid \\varphi(x) = x\\}$ of $\\varphi$ is inert in $F_n$: its intersection with every subgroup $K \\le F_n$ has rank at most\u2026","title":"Fixed subgroups of endomorphisms of free groups are inert"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/subgroup-rank-4","kind":"node","lean":"MathlibExt.GroupTheory.InertSubgroupDecidableKourovka19Problem104Wanted.subgroupRank","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#subgroup-rank-4","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"For a subgroup $H$ of a group $G$, the rank $r(H) \\in \\mathbb{N} \\cup \\{\\infty\\}$ is the least size of a finite subset of $G$ that generates $H$, and $r(H) = \\infty$ if $H$ is not finitely generated.","title":"Rank of a subgroup, possibly infinite"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/is-inert-3","kind":"node","lean":"MathlibExt.GroupTheory.InertSubgroupDecidableKourovka19Problem104Wanted.IsInert","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#is-inert-3","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"A subgroup $H$ of the free group $F_n$ on $n$ generators is inert if $r(H \\cap K) \\le r(K)$ for every subgroup $K$ of $F_n$, where the rank $r(L)$ of a subgroup $L$ is the least size of a finite generating set of $L$, and $\\infty$ when $L$ is not finitely generated.","title":"Inert subgroup of a free group of finite rank"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/generated-by","kind":"node","lean":"MathlibExt.GroupTheory.InertSubgroupDecidableKourovka19Problem104Wanted.generatedBy","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#generated-by","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"Let $F_n$ be the free group on generators $x_0, \\dots, x_{n-1}$. A word is a finite list of letters $(i, b)$ with $i < n$ and $b$ a boolean, where $(i, \\mathrm{true})$ stands for $x_i$ and $(i, \\mathrm{false})$ for $x_i^{-1}$. Given a finite list of words, this is the subgroup of $F_n$ generated by the elements of $F_n$ that the words represent, each word representing the product of its letters.","title":"Subgroup of a free group generated by a list of words"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/inert-subgroup-decidable","kind":"node","lean":"MathlibExt.GroupTheory.InertSubgroupDecidableKourovka19Problem104Wanted.inertSubgroupDecidable","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#inert-subgroup-decidable","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"fully_proved","summary":"The statement that for every natural number $n$ there is an algorithm deciding, for a finite list of words in the generators $x_0, \\dots, x_{n-1}$ of the free group $F_n$ and their inverses, whether the subgroup of $F_n$ they generate is inert; that is, the predicate on lists of words is computable.","title":"Kourovka Problem 19.104 (statement): deciding inertness"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/inert-subgroup-decidable-holds","kind":"node","lean":"MathlibExt.GroupTheory.InertSubgroupDecidableKourovka19Problem104Wanted.inertSubgroupDecidable_holds","parent":2249,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups/index.html#inert-subgroup-decidable-holds","scope":"GroupTheory/free-groups/subgroups-of-free-groups/inert-and-compressed-subgroups-of-free-groups","status":"can_prove","summary":"For every natural number $n$, let $F_n$ be the free group on generators $x_0, \\dots, x_{n-1}$, and give a finitely generated subgroup of $F_n$ by a finite list of generating words in the letters $x_i^{\\pm 1}$. Then there is an algorithm deciding, for such a list, whether the subgroup $H$ it generates is inert, that is, whether $r(H \\cap K) \\le r(K)$ for every subgroup $K$ of $F_n$. Here the rank $r(L)$ of a\u2026","title":"Inertness is decidable in free groups of finite rank (Kourovka Problem 19.104)"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups","status":"can_prove","title":"Normal closures in free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups/minimal-normal-closure-element-conj","kind":"node","lean":"MathlibExt.GroupTheory.NormalClosureMinimalElementKourovka11Problem82Wanted.minimalNormalClosureElementConj","parent":2268,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups/index.html#minimal-normal-closure-element-conj","scope":"GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups","status":"fully_proved","summary":"The statement that for every free group $F$ and every $r \\ne 1$ in $F$, if $s$ is a nontrivial element of the normal closure $R$ of $r$ in $F$ whose reduced word length is minimal among the nontrivial elements of $R$, then $s$ is conjugate to $r$, to $r^{-1}$, or, for some $f \\in F$, to $[r, f] = r^{-1} f^{-1} r f$ or to $[r^{-1}, f] = r f^{-1} r^{-1} f$.","title":"Shortest elements of normal closures in free groups (Kourovka 11.82, Remeslennikov)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups/minimal-normal-closure-element-conj-refu","kind":"node","lean":"MathlibExt.GroupTheory.NormalClosureMinimalElementKourovka11Problem82Wanted.minimalNormalClosureElementConj_refuted","parent":2268,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups/index.html#minimal-normal-closure-element-conj-refu","scope":"GroupTheory/free-groups/subgroups-of-free-groups/normal-closures-in-free-groups","status":"can_prove","summary":"There are a free group $F$, an element $r \\ne 1$ of $F$, and a nontrivial element $s$ of the normal closure $R$ of $r$ in $F$ whose reduced word length is minimal among the nontrivial elements of $R$, such that $s$ is conjugate neither to $r$ nor to $r^{-1}$, nor, for any $f \\in F$, to $[r, f] = r^{-1} f^{-1} r f$ or to $[r^{-1}, f] = r f^{-1} r^{-1} f$.","title":"Negative answer to Kourovka 11.82 on shortest elements of normal closures"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups","status":"can_prove","title":"Normally independent sets in free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups/is-normally-independent","kind":"node","lean":"MathlibExt.GroupTheory.MassiveIndependentSetFreeGroupKourovka10Problem13Wanted.IsNormallyIndependent","parent":2271,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups/index.html#is-normally-independent","scope":"GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups","status":"fully_proved","summary":"A subset $E$ of the free group $F_2$ on two generators is normally independent when no element $w \\in E$ lies in the normal closure of $E \\setminus \\{w\\}$, the smallest normal subgroup of $F_2$ containing $E \\setminus \\{w\\}$.","title":"Normally independent sets in the free group of rank two"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups/exists-massive-normally-independent-set","kind":"node","lean":"MathlibExt.GroupTheory.MassiveIndependentSetFreeGroupKourovka10Problem13Wanted.existsMassiveNormallyIndependentSet","parent":2271,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups/index.html#exists-massive-normally-independent-set","scope":"GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups","status":"fully_proved","summary":"The statement that some subset $E$ of the free group $F_2$ on two generators is normally independent (no $w \\in E$ lies in the normal closure of $E \\setminus \\{w\\}$) and massive: for every real number $c$ with $0 < c < 3$ and every natural number $N$ there is $n \\ge N$ such that the set $E_n$ of elements of $E$ of reduced word length $n$ has at least $c^n$ elements.","title":"Grigorchuk's problem on massive independent sets in the free group of rank two"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups/exists-massive-normally-independent-set-2","kind":"node","lean":"MathlibExt.GroupTheory.MassiveIndependentSetFreeGroupKourovka10Problem13Wanted.existsMassiveNormallyIndependentSet_refuted","parent":2271,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups/index.html#exists-massive-normally-independent-set-2","scope":"GroupTheory/free-groups/subgroups-of-free-groups/normally-independent-sets-in-free-groups","status":"can_prove","summary":"There is no subset $E$ of the free group $F_2$ on two generators such that no element $w \\in E$ lies in the normal closure of $E \\setminus \\{w\\}$ and, for every real number $c$ with $0 < c < 3$ and every natural number $N$, there is $n \\ge N$ for which at least $c^n$ elements of $E$ have reduced word length $n$.","title":"No massive normally independent set in the free group of rank two"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words","status":"can_prove","title":"Orbit-blocking words"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words/exists-orbit-blocking-word","kind":"node","lean":"MathlibExt.GroupTheory.OrbitBlockingWordKourovka20Problem88Wanted.existsOrbitBlockingWord","parent":2275,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words/index.html#exists-orbit-blocking-word","scope":"GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words","status":"fully_proved","summary":"The statement that for every $r \\ge 1$ and every element $u$ of the free group $F_r$ on $r$ generators there is $v \\in F_r$ such that, for every automorphism $\\varphi$ of $F_r$ for which the reduced word of $\\varphi(u)$ is cyclically reduced, the reduced word of $v$ is not a subword (a contiguous factor) of the reduced word of $\\varphi(u)$.","title":"Kourovka Problem 20.88 (statement)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words/exists-orbit-blocking-word-holds","kind":"node","lean":"MathlibExt.GroupTheory.OrbitBlockingWordKourovka20Problem88Wanted.existsOrbitBlockingWord_holds","parent":2275,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words/index.html#exists-orbit-blocking-word-holds","scope":"GroupTheory/free-groups/subgroups-of-free-groups/orbit-blocking-words","status":"can_prove","summary":"For every $r \\ge 1$ and every element $u$ of the free group $F_r$ on $r$ generators there is $v \\in F_r$ such that, for every automorphism $\\varphi$ of $F_r$ for which the reduced word of $\\varphi(u)$ is cyclically reduced, the reduced word of $v$ is not a subword (a contiguous factor) of the reduced word of $\\varphi(u)$.","title":"Orbit-blocking words exist (Kourovka Problem 20.88)"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups","status":"can_prove","title":"Palindromic width of free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups/is-almost-palindrome","kind":"node","lean":"MathlibExt.GroupTheory.AlmostPalindromicWidthKourovka19Problem8Wanted.IsAlmostPalindrome","parent":2278,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups/index.html#is-almost-palindrome","scope":"GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups","status":"fully_proved","summary":"For a natural number $m$, a word $w$ in the letters $x^{\\pm 1}$ and $y^{\\pm 1}$ is an almost $m$-palindrome if there is a palindrome $p$ in the same letters with the same length as $w$ such that $w$ and $p$ differ in at most $m$ positions. The word $w$ need not be reduced.","title":"Almost palindromes"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups/almost-palindromic-width-finite","kind":"node","lean":"MathlibExt.GroupTheory.AlmostPalindromicWidthKourovka19Problem8Wanted.almostPalindromicWidthFinite","parent":2278,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite almost-palindromic width of "],["math","\\(F_2\\)"],["text"," (Kourovka 19.8)"]],"source":"Finite almost-palindromic width of $F_2$ (Kourovka 19.8)"},"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups/index.html#almost-palindromic-width-finite","scope":"GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups","status":"fully_proved","summary":"The proposition that there are positive integers $m$ and $c$ such that every element of the free group $F_2$ on generators $x, y$ is a product of at most $c$ elements, each represented by an almost $m$-palindrome, that is, by a word in $x^{\\pm 1}, y^{\\pm 1}$ (not necessarily reduced) that differs from a palindrome of the same length in at most $m$ positions.","title":"Finite almost-palindromic width of $F_2$ (Kourovka 19.8)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups/almost-palindromic-width-finite-refuted","kind":"node","lean":"MathlibExt.GroupTheory.AlmostPalindromicWidthKourovka19Problem8Wanted.almostPalindromicWidthFinite_refuted","parent":2278,"reading_title":{"format":"inline-parts-v1","parts":[["text","The almost-palindromic width of "],["math","\\(F_2\\)"],["text"," is infinite (Kourovka 19.8)"]],"source":"The almost-palindromic width of $F_2$ is infinite (Kourovka 19.8)"},"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups/index.html#almost-palindromic-width-finite-refuted","scope":"GroupTheory/free-groups/subgroups-of-free-groups/palindromic-width-of-free-groups","status":"can_prove","summary":"There are no positive integers $m$ and $c$ such that every element of the free group $F_2$ on generators $x, y$ is a product of at most $c$ elements, each represented by an almost $m$-palindrome, that is, by a word in $x^{\\pm 1}, y^{\\pm 1}$ (not necessarily reduced) that differs from a palindrome of the same length in at most $m$ positions. So the question of Kourovka Problem 19.8 has a negative answer.","title":"The almost-palindromic width of $F_2$ is infinite (Kourovka 19.8)"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups","status":"can_prove","title":"Primitive elements of free metabelian groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/free-metabelian-group","kind":"node","lean":"MathlibExt.GroupTheory.FreeMetabelianPrimitivePreservingKourovka14Problem85Wanted.FreeMetabelianGroup","parent":2282,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free metabelian group "],["math","\\(M_n\\)"],["text"," (Kourovka 14.85)"]],"source":"Free metabelian group $M_n$ (Kourovka 14.85)"},"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/index.html#free-metabelian-group","scope":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups","status":"fully_proved","summary":"For a natural number $n$, the free metabelian group $M_n = F_n / F_n''$ of rank $n$: the quotient of the free group $F_n$ on $n$ generators by its second derived subgroup $F_n'' = [[F_n, F_n], [F_n, F_n]]$.","title":"Free metabelian group $M_n$ (Kourovka 14.85)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/is-primitive","kind":"node","lean":"MathlibExt.GroupTheory.FreeMetabelianPrimitivePreservingKourovka14Problem85Wanted.IsPrimitive","parent":2282,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/index.html#is-primitive","scope":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups","status":"fully_proved","summary":"Let $F_n$ be the free group on $n$ generators, let $F_n''$ be its second derived subgroup, and let $M_n = F_n / F_n''$ be the free metabelian group of rank $n$. Let $x_0, \\dots, x_{n-1}$ be the images in $M_n$ of the free generators of $F_n$. An element $y$ of $M_n$ is primitive when $y = \\sigma(x_i)$ for some automorphism $\\sigma$ of $M_n$ and some index $i$; that is, $y$ is the image of a standard basis element\u2026","title":"Primitive elements of the free metabelian group"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/primitive-preserving-endomorphism-is-aut","kind":"node","lean":"MathlibExt.GroupTheory.FreeMetabelianPrimitivePreservingKourovka14Problem85Wanted.primitivePreservingEndomorphismIsAutomorphism","parent":2282,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/index.html#primitive-preserving-endomorphism-is-aut","scope":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups","status":"fully_proved","summary":"The statement, posed by E. I. Timoshenko and V. Shpilrain, that for every natural number $n$, every endomorphism $\\varphi$ of the free metabelian group $M_n$ of rank $n$ that maps every primitive element of $M_n$ to a primitive element is bijective, hence an automorphism.","title":"Primitive-preserving endomorphism problem for free metabelian groups (Kourovka 14.85)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/primitive-preserving-endomorphism-is-aut-2","kind":"node","lean":"MathlibExt.GroupTheory.FreeMetabelianPrimitivePreservingKourovka14Problem85Wanted.primitivePreservingEndomorphismIsAutomorphism_holds","parent":2282,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups/index.html#primitive-preserving-endomorphism-is-aut-2","scope":"GroupTheory/free-groups/subgroups-of-free-groups/primitive-elements-of-free-metabelian-groups","status":"can_prove","summary":"For every natural number $n$, every endomorphism $\\varphi$ of the free metabelian group $M_n$ of rank $n$ that maps every primitive element of $M_n$ to a primitive element is bijective, hence an automorphism of $M_n$. This answers Kourovka Problem 14.85 positively.","title":"Primitive-preserving endomorphisms of free metabelian groups are automorphisms"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups","status":"can_prove","title":"Residual finiteness growth of free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups/log-index-separation","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupLogIndexSeparationKourovka15Problem35Wanted.logIndexSeparation","parent":2287,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups/index.html#log-index-separation","scope":"GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups","status":"fully_proved","summary":"The statement, posed by O. V. Bogopolski, that for some integer $r \\ge 2$ there is a real number $C$ such that every element $w$ of the free group $F_r$ of rank $r$ whose reduced word length $n$ satisfies $n > 1$ lies outside some subgroup of $F_r$ of finite index at most $C \\log n$, where $\\log$ is the natural logarithm.","title":"Logarithmic-index separation in free groups (Kourovka 15.35)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups/log-index-separation-refuted","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupLogIndexSeparationKourovka15Problem35Wanted.logIndexSeparation_refuted","parent":2287,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups/index.html#log-index-separation-refuted","scope":"GroupTheory/free-groups/subgroups-of-free-groups/residual-finiteness-growth-of-free-groups","status":"can_prove","summary":"For every integer $r \\ge 2$ and every real number $C$, there is an element $w$ of the free group $F_r$ of rank $r$ whose reduced word length $n$ satisfies $n > 1$ and such that every subgroup of finite index in $F_r$ that does not contain $w$ has index greater than $C \\log n$, where $\\log$ is the natural logarithm. That is, the logarithmic-index separation property of Kourovka Problem 15.35 holds for no rank\u2026","title":"Bogopolski's logarithmic-index separation fails in free groups of rank at least 2"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups","status":"can_prove","title":"Retracts of free groups"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups/is-retract","kind":"node","lean":"MathlibExt.GroupTheory.RetractIntersectionFreeGroupKourovka17Problem19Wanted.IsRetract","parent":2290,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups/index.html#is-retract","scope":"GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups","status":"fully_proved","summary":"A subgroup $R$ of a group $F$ is a retract of $F$ when there is an endomorphism $r \\colon F \\to F$ whose image is $R$ and which fixes every element of $R$.","title":"Retract of a group"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups/retract-intersection-is-retract","kind":"node","lean":"MathlibExt.GroupTheory.RetractIntersectionFreeGroupKourovka17Problem19Wanted.retractIntersectionIsRetract","parent":2290,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups/index.html#retract-intersection-is-retract","scope":"GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups","status":"fully_proved","summary":"The statement that for every natural number $n$, every retract $R$ of the free group $F_n$ on $n$ generators and every finitely generated subgroup $H$ of $F_n$, the intersection $H \\cap R$ is a retract of $H$.","title":"Bergman's retract intersection question (Kourovka Problem 17.19)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups/retract-intersection-is-retract-refuted","kind":"node","lean":"MathlibExt.GroupTheory.RetractIntersectionFreeGroupKourovka17Problem19Wanted.retractIntersectionIsRetract_refuted","parent":2290,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups/index.html#retract-intersection-is-retract-refuted","scope":"GroupTheory/free-groups/subgroups-of-free-groups/retracts-of-free-groups","status":"can_prove","summary":"It is not the case that for every natural number $n$, every retract $R$ of the free group $F_n$ on $n$ generators and every finitely generated subgroup $H$ of $F_n$, the intersection $H \\cap R$ is a retract of $H$.","title":"Negative answer to Bergman's retract intersection question (Kourovka Problem 17.19)"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub","status":"can_prove","title":"Subgroups of free groups with normal derived subgroup"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub/derived-normal-implies-normal","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupDerivedNormalSubgroupKourovka11Problem124Wanted.derivedNormalImpliesNormal","parent":2294,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub/index.html#derived-normal-implies-normal","scope":"GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub","status":"fully_proved","summary":"The statement that for every set $X$ such that the free group $F$ on $X$ is not cyclic, every subgroup $R$ of $F$ that is not cyclic and whose derived subgroup $[R, R]$ is normal in $F$ is itself normal in $F$.","title":"Kourovka Problem 11.124 (free-group subgroups with normal derived subgroup)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub/derived-normal-implies-normal-holds","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupDerivedNormalSubgroupKourovka11Problem124Wanted.derivedNormalImpliesNormal_holds","parent":2294,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub/index.html#derived-normal-implies-normal-holds","scope":"GroupTheory/free-groups/subgroups-of-free-groups/subgroups-of-free-groups-with-normal-derived-sub","status":"can_prove","summary":"Let $F$ be a free group on an arbitrary set of generators, and suppose that $F$ is not cyclic. Then every subgroup $R$ of $F$ that is not cyclic and whose derived subgroup $[R, R]$ is normal in $F$ is itself normal in $F$.","title":"A non-cyclic subgroup of a free group with normal derived subgroup is normal"},{"data_url":"scopes/GroupTheory/free-groups/subgroups-of-free-groups/test-elements.json","id":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements","kind":"scope","navigation":"module","parent":2234,"result":false,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/test-elements/index.html","scope":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements","status":"can_prove","title":"Test elements"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements/free-soluble-group-rank-two","kind":"node","lean":"MathlibExt.GroupTheory.FreeSolubleTestElementKourovka14Problem88Wanted.FreeSolubleGroupRankTwo","parent":2297,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/test-elements/index.html#free-soluble-group-rank-two","scope":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements","status":"fully_proved","summary":"The free soluble group of rank `2` and derived length `d`: the quotient `F\u2082 / F\u2082\u207d\u1d48\u207e` of the free group on two generators by the `d`-th term of its derived series.","title":"Free soluble group of rank two"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements/is-test-element","kind":"node","lean":"MathlibExt.GroupTheory.FreeSolubleTestElementKourovka14Problem88Wanted.IsTestElement","parent":2297,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/test-elements/index.html#is-test-element","scope":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements","status":"fully_proved","summary":"An element $u$ of a group $G$ is a test element of $G$ if every endomorphism $\\varphi$ of $G$ with $\\varphi(u) = u$ is bijective, that is, an automorphism of $G$.","title":"Test element"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements/free-soluble-rank-two-has-test-element","kind":"node","lean":"MathlibExt.GroupTheory.FreeSolubleTestElementKourovka14Problem88Wanted.freeSolubleRankTwoHasTestElement","parent":2297,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/test-elements/index.html#free-soluble-rank-two-has-test-element","scope":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements","status":"fully_proved","summary":"The statement that for every integer $d > 2$ the free soluble group $F_2 / F_2^{(d)}$ of rank $2$ and derived length $d$ has a test element, where $F_2$ is the free group on two generators and $F_2^{(d)}$ is the $d$-th term of its derived series.","title":"Kourovka Problem 14.88 (test elements of free soluble groups of rank 2)"},{"id":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements/free-soluble-rank-two-has-test-element-h","kind":"node","lean":"MathlibExt.GroupTheory.FreeSolubleTestElementKourovka14Problem88Wanted.freeSolubleRankTwoHasTestElement_holds","parent":2297,"result":true,"route":"../roadmap/GroupTheory/free-groups/subgroups-of-free-groups/test-elements/index.html#free-soluble-rank-two-has-test-element-h","scope":"GroupTheory/free-groups/subgroups-of-free-groups/test-elements","status":"can_prove","summary":"Resolved true: the Kourovka Archive answers 14.88 \"Yes, it does\" (Timoshenko): for every `d > 2` the free soluble group `F\u2082/F\u2082\u207d\u1d48\u207e` of rank `2` and derived length `d` has a test element. Source: E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 14.88 (B. Fine, V. Shpilrain); resolution: E. I.\u2026","title":"Free soluble groups of rank two have test elements"},{"data_url":"scopes/GroupTheory/free-groups/automorphisms-of-free-groups.json","id":"GroupTheory/free-groups/automorphisms-of-free-groups","kind":"scope","parent":2233,"result":false,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/index.html","scope":"GroupTheory/free-groups/automorphisms-of-free-groups","status":"can_prove","summary":"The groups Aut(F_n) and Out(F_n): fixed points and fixed subgroups of automorphisms, word problem, automaticity, linearity of IA-automorphisms, embeddings of free products into Aut(F_n).","title":"Automorphisms of free groups"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/out-free-group","kind":"node","lean":"MathlibExt.GroupTheory.OutFreeGroupShared.OutFreeGroup","parent":2302,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/index.html#out-free-group","scope":"GroupTheory/free-groups/automorphisms-of-free-groups","status":"fully_proved","summary":"For a natural number $n$, let $F_n$ be the free group on $n$ generators. Its outer automorphism group $\\mathrm{Out}(F_n)$ is the quotient of the automorphism group $\\mathrm{Aut}(F_n)$ by the normal subgroup generated by the inner automorphisms. Since the inner automorphisms already form a normal subgroup $\\mathrm{Inn}(F_n)$, this is $\\mathrm{Aut}(F_n) / \\mathrm{Inn}(F_n)$.","title":"Outer automorphism group of a free group"},{"data_url":"scopes/GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou.json","id":"GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou","kind":"scope","navigation":"module","parent":2302,"result":false,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou/index.html","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou","status":"can_prove","title":"Automaticity of automorphism groups of free groups"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou/is-automatic-group","kind":"node","lean":"MathlibExt.GroupTheory.AutFnAutomaticQuestion73Wanted.IsAutomaticGroup","parent":2304,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou/index.html#is-automatic-group","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou","status":"fully_proved","summary":"A group $G$ is automatic if there are a finite alphabet $A$, an involution $\\iota$ of $A$, a map $\\mathrm{ev} \\colon A \\to G$ with $\\mathrm{ev}(\\iota(a)) = \\mathrm{ev}(a)^{-1}$ for all $a \\in A$, a language $L$ of words over $A$ that is exactly the set of words accepted by some deterministic finite automaton, and a natural number $k$ with the following properties. Write $\\overline{w} \\in G$ for the product of the\u2026","title":"Automatic group"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou/conjecture","kind":"node","lean":"MathlibExt.GroupTheory.AutFnAutomaticQuestion73Wanted.conjecture","parent":2304,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou/index.html#conjecture","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou","status":"fully_proved","summary":"The statement that for every natural number $n > 3$ the automorphism group $\\mathrm{Aut}(F_n)$ of the free group $F_n$ on $n$ generators is an automatic group.","title":"Automaticity of automorphism groups of free groups (Farb's Question 7.3)"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou/conjecture-refuted","kind":"node","lean":"MathlibExt.GroupTheory.AutFnAutomaticQuestion73Wanted.conjecture_refuted","parent":2304,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou/index.html#conjecture-refuted","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/automaticity-of-automorphism-groups-of-free-grou","status":"can_prove","summary":"It is not the case that for every natural number $n > 3$ the automorphism group $\\mathrm{Aut}(F_n)$ of the free group on $n$ generators is automatic.","title":"Negative answer to Farb's Question 7.3"},{"data_url":"scopes/GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups.json","id":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups","kind":"scope","navigation":"module","parent":2302,"result":false,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/index.html","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups","status":"can_prove","title":"Endomorphisms of free and free metabelian groups"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/free-metabelian-group","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismsDeterminedByTwoElementsKourovka13Problem66Wanted.FreeMetabelianGroup","parent":2308,"reading_title":{"format":"inline-parts-v1","parts":[["text","The free metabelian group "],["math","\\(M_n\\)"]],"source":"The free metabelian group $M_n$"},"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/index.html#free-metabelian-group","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups","status":"fully_proved","summary":"For a natural number $n$, the free metabelian group of rank $n$ is the quotient $F_n / F_n''$ of the free group $F_n$ on $n$ generators by its second derived subgroup $F_n'' = [F_n', F_n']$, where $F_n' = [F_n, F_n]$ is the derived subgroup of $F_n$.","title":"The free metabelian group $M_n$"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/determined-by-two-elements","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismsDeterminedByTwoElementsKourovka13Problem66Wanted.DeterminedByTwoElements","parent":2308,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/index.html#determined-by-two-elements","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups","status":"fully_proved","summary":"A group $F$ has this property when there exist elements $g, h \\in F$ such that any two endomorphisms $\\varphi, \\psi$ of $F$ whose images are not cyclic subgroups and which satisfy $\\varphi(g) = \\psi(g)$ and $\\varphi(h) = \\psi(h)$ are equal.","title":"Groups whose endomorphisms with non-cyclic images are determined by two elements"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/free-group-endomorphisms-determined-by-t","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismsDeterminedByTwoElementsKourovka13Problem66Wanted.freeGroupEndomorphismsDeterminedByTwoElements","parent":2308,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/index.html#free-group-endomorphisms-determined-by-t","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups","status":"fully_proved","summary":"For every natural number $n$, the free group $F_n$ of rank $n$ contains two elements $g$ and $h$ such that any two endomorphisms of $F_n$ whose images are not cyclic subgroups and which agree at $g$ and at $h$ are equal; that is, $F_n$ has its endomorphisms with non-cyclic images determined by two elements.","title":"Shpil'rain's two-element problem for free groups (Kourovka 13.66(a))"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/free-metabelian-endomorphisms-determined","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismsDeterminedByTwoElementsKourovka13Problem66Wanted.freeMetabelianEndomorphismsDeterminedByTwoElements","parent":2308,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/index.html#free-metabelian-endomorphisms-determined","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups","status":"fully_proved","summary":"For every natural number $n$, the free metabelian group $M_n = F_n / F_n''$ of rank $n$ (the quotient of the free group $F_n$ of rank $n$ by its second derived subgroup) contains two elements $g$ and $h$ such that any two endomorphisms of $M_n$ whose images are not cyclic subgroups and which agree at $g$ and at $h$ are equal.","title":"Shpil'rain's two-element problem for free metabelian groups (Kourovka 13.66(b))"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/free-group-endomorphisms-determined-by-t-2","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismsDeterminedByTwoElementsKourovka13Problem66Wanted.freeGroupEndomorphismsDeterminedByTwoElements_holds","parent":2308,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/index.html#free-group-endomorphisms-determined-by-t-2","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups","status":"can_prove","summary":"For every natural number $n$, the free group $F_n$ of rank $n$ contains two elements $g$ and $h$ such that any two endomorphisms $\\varphi, \\psi$ of $F_n$ whose images are not cyclic subgroups and which satisfy $\\varphi(g) = \\psi(g)$ and $\\varphi(h) = \\psi(h)$ are equal.","title":"Endomorphisms of free groups with non-cyclic images are determined by two elements"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/free-metabelian-endomorphisms-determined-2","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismsDeterminedByTwoElementsKourovka13Problem66Wanted.freeMetabelianEndomorphismsDeterminedByTwoElements_refuted","parent":2308,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups/index.html#free-metabelian-endomorphisms-determined-2","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/endomorphisms-of-free-and-free-metabelian-groups","status":"can_prove","summary":"For a natural number $n$, let $M_n = F_n / F_n''$ be the free metabelian group of rank $n$, the quotient of the free group $F_n$ of rank $n$ by its second derived subgroup. There is an $n$ such that for all elements $g, h \\in M_n$ there are endomorphisms $\\varphi \\ne \\psi$ of $M_n$ whose images are not cyclic subgroups and which satisfy $\\varphi(g) = \\psi(g)$ and $\\varphi(h) = \\psi(h)$.","title":"Endomorphisms of free metabelian groups are not always determined by two elements"},{"data_url":"scopes/GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of.json","id":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of","kind":"scope","navigation":"module","parent":2302,"result":false,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/index.html","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of","status":"can_prove","title":"Finite-index subgroups of automorphism groups of free groups"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/first-betti-rank","kind":"node","lean":"MathlibExt.Topology.AutFnFiniteIndexPositiveBettiWanted.firstBettiRank","parent":2315,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/index.html#first-betti-rank","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of","status":"fully_proved","summary":"The first Betti number of a group $G$ is the rank, as a cardinal, of its abelianization $G^{\\mathrm{ab}}$ viewed as a $\\mathbb{Z}$-module.","title":"First Betti number of a group"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/conjecture","kind":"node","lean":"MathlibExt.Topology.AutFnFiniteIndexPositiveBettiWanted.conjecture","parent":2315,"reading_title":{"format":"inline-parts-v1","parts":[["text","Farb's positive first Betti number question for "],["math","\\(\\mathrm{Aut}(F_n)\\)"]],"source":"Farb's positive first Betti number question for $\\mathrm{Aut}(F_n)$"},"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/index.html#conjecture","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of","status":"fully_proved","summary":"The proposition that for every natural number $n > 3$, the automorphism group $\\mathrm{Aut}(F_n)$ of the free group $F_n$ on $n$ generators has a finite-index subgroup $H$ whose first Betti number, the $\\mathbb{Z}$-rank of the abelianization of $H$, is positive.","title":"Farb's positive first Betti number question for $\\mathrm{Aut}(F_n)$"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/conjecture-refuted","kind":"node","lean":"MathlibExt.Topology.AutFnFiniteIndexPositiveBettiWanted.conjecture_refuted","parent":2315,"reading_title":{"format":"inline-parts-v1","parts":[["text","Some "],["math","\\(\\mathrm{Aut}(F_n)\\)"],["text"," with "],["math","\\(n > 3\\)"],["text"," has no finite-index subgroup of positive first Betti number"]],"source":"Some $\\mathrm{Aut}(F_n)$ with $n > 3$ has no finite-index subgroup of positive first Betti number"},"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/index.html#conjecture-refuted","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of","status":"can_prove","summary":"Some $\\mathrm{Aut}(F_n)$ with $n > 3$ has no finite-index subgroup of positive first Betti number. It is not the case that for every natural number $n > 3$ the automorphism group $\\mathrm{Aut}(F_n)$ of the free group $F_n$ on $n$ generators has a finite-index subgroup $H$ whose first Betti number, the $\\mathbb{Z}$-rank of the abelianization of $H$, is positive.","title":"Some $\\mathrm{Aut}(F_n)$ with $n > 3$ has no finite-index subgroup of positive first Betti number"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/has-property-fa","kind":"node","lean":"MathlibExt.Topology.AutFnFiniteIndexPropertyFAWanted.HasPropertyFA","parent":2315,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/index.html#has-property-fa","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of","status":"fully_proved","summary":"A group $G$ has Property FA when every action of $G$ on a tree without edge inversion has a global fixed vertex: for every tree $T$ with vertex set $V$ and every homomorphism $\\alpha$ from $G$ to the group of permutations of $V$ such that each $\\alpha(g)$ satisfies $\\alpha(g)(u) \\sim \\alpha(g)(v) \\iff u \\sim v$ for all vertices $u, v$ (where $\\sim$ is adjacency in $T$), and such that no $\\alpha(g)$ swaps the two\u2026","title":"Serre's Property FA"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/conjecture-2","kind":"node","lean":"MathlibExt.Topology.AutFnFiniteIndexPropertyFAWanted.conjecture","parent":2315,"reading_title":{"format":"inline-parts-v1","parts":[["text","Farb's question on Property FA for finite-index subgroups of "],["math","\\(\\operatorname{Aut}(F_n)\\)"]],"source":"Farb's question on Property FA for finite-index subgroups of $\\operatorname{Aut}(F_n)$"},"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/index.html#conjecture-2","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of","status":"fully_proved","summary":"The proposition that for every natural number $n \\ge 4$, every subgroup of finite index in the automorphism group $\\operatorname{Aut}(F_n)$ of the free group $F_n$ on $n$ generators has Serre's Property FA.","title":"Farb's question on Property FA for finite-index subgroups of $\\operatorname{Aut}(F_n)$"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/conjecture-holds","kind":"node","lean":"MathlibExt.Topology.AutFnFiniteIndexPropertyFAWanted.conjecture_holds","parent":2315,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite-index subgroups of "],["math","\\(\\operatorname{Aut}(F_n)\\)"],["text"," have Property FA for "],["math","\\(n \\ge 4\\)"]],"source":"Finite-index subgroups of $\\operatorname{Aut}(F_n)$ have Property FA for $n \\ge 4$"},"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of/index.html#conjecture-holds","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/finite-index-subgroups-of-automorphism-groups-of","status":"can_prove","summary":"For every natural number $n \\ge 4$, every subgroup of finite index in the automorphism group $\\operatorname{Aut}(F_n)$ of the free group $F_n$ on $n$ generators has Serre's Property FA.","title":"Finite-index subgroups of $\\operatorname{Aut}(F_n)$ have Property FA for $n \\ge 4$"},{"data_url":"scopes/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms.json","id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","kind":"scope","navigation":"module","parent":2302,"result":false,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"can_prove","title":"Fixed subgroups of free group automorphisms"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/aut-norm","kind":"node","lean":"MathlibExt.GroupTheory.FixedSubgroupBasisLengthKourovka14Problem23Wanted.autNorm","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#aut-norm","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"fully_proved","summary":"Let $F_n$ be the free group on the basis $x_0, \\dots, x_{n-1}$, and for $w \\in F_n$ let $|w|$ be the length of the reduced word representing $w$. The norm of an automorphism $\\alpha$ of $F_n$ is $\\|\\alpha\\| = \\max(|\\alpha(x_0)|, \\dots, |\\alpha(x_{n-1})|)$, taken to be $0$ when $n = 0$.","title":"Norm of an automorphism of a free group"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/fixed-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.FixedSubgroupBasisLengthKourovka14Problem23Wanted.fixedSubgroup","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#fixed-subgroup","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"fully_proved","summary":"For an automorphism $\\alpha$ of the free group $F_n$ on $x_0, \\dots, x_{n-1}$, the fixed subgroup of $\\alpha$ is the subgroup $\\mathrm{Fix}(\\alpha) = \\{x \\in F_n \\mid \\alpha(x) = x\\}$ of $F_n$.","title":"Fixed subgroup of a free group automorphism"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/has-short-fixed-basis","kind":"node","lean":"MathlibExt.GroupTheory.FixedSubgroupBasisLengthKourovka14Problem23Wanted.HasShortFixedBasis","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#has-short-fixed-basis","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"fully_proved","summary":"Let $\\alpha$ be an automorphism of the free group $F_n$ on $x_0, \\dots, x_{n-1}$, let $|w|$ be the length of the reduced word representing $w \\in F_n$, and let $L$ be a natural number. The fixed subgroup $\\mathrm{Fix}(\\alpha)$ has a short free basis with bound $L$ when, for some $k \\ge 0$, it has a free basis $y_1, \\dots, y_k$ with $|y_i| \\le L$ for every $i$.","title":"Fixed subgroups with a short free basis"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/fixed-subgroup-basis-recursive-bound-eac","kind":"node","lean":"MathlibExt.GroupTheory.FixedSubgroupBasisLengthKourovka14Problem23Wanted.fixedSubgroupBasisRecursiveBoundEachRank","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#fixed-subgroup-basis-recursive-bound-eac","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"fully_proved","summary":"For every $n$ there is a computable function $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every automorphism $\\alpha$ of the free group $F_n$ on $x_0, \\dots, x_{n-1}$, the fixed subgroup $\\mathrm{Fix}(\\alpha)$ has a finite free basis all of whose elements have reduced word length at most $f(\\|\\alpha\\|)$, where $\\|\\alpha\\|$ is the maximum of the reduced word lengths of $\\alpha(x_0), \\dots, \\alpha(x_{n-1})$.","title":"Kourovka Problem 14.23, rank-dependent form"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/fixed-subgroup-basis-recursive-bound-uni","kind":"node","lean":"MathlibExt.GroupTheory.FixedSubgroupBasisLengthKourovka14Problem23Wanted.fixedSubgroupBasisRecursiveBoundUniform","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#fixed-subgroup-basis-recursive-bound-uni","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"fully_proved","summary":"There is a computable function $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every $n$ and every automorphism $\\alpha$ of the free group $F_n$ on $x_0, \\dots, x_{n-1}$, the fixed subgroup $\\mathrm{Fix}(\\alpha)$ has a finite free basis all of whose elements have reduced word length at most $f(\\|\\alpha\\|)$, where $\\|\\alpha\\|$ is the maximum of the reduced word lengths of $\\alpha(x_0), \\dots, \\alpha(x_{n-1})$.","title":"Kourovka Problem 14.23, rank-uniform form"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/fixed-subgroup-basis-recursive-bound-eac-2","kind":"node","lean":"MathlibExt.GroupTheory.FixedSubgroupBasisLengthKourovka14Problem23Wanted.fixedSubgroupBasisRecursiveBoundEachRank_holds","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#fixed-subgroup-basis-recursive-bound-eac-2","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"can_prove","summary":"For every $n$ there is a computable function $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every automorphism $\\alpha$ of the free group $F_n$ on $x_0, \\dots, x_{n-1}$, the fixed subgroup $\\mathrm{Fix}(\\alpha)$ has a finite free basis all of whose elements have reduced word length at most $f(\\|\\alpha\\|)$, where $\\|\\alpha\\|$ is the maximum of the reduced word lengths of $\\alpha(x_0), \\dots, \\alpha(x_{n-1})$.","title":"Kourovka Problem 14.23 holds for each rank"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/fixed-subgroup-basis-recursive-bound-uni-2","kind":"node","lean":"MathlibExt.GroupTheory.FixedSubgroupBasisLengthKourovka14Problem23Wanted.fixedSubgroupBasisRecursiveBoundUniform_refuted","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#fixed-subgroup-basis-recursive-bound-uni-2","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"can_prove","summary":"There is no computable function $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every $n$ and every automorphism $\\alpha$ of the free group $F_n$ on $x_0, \\dots, x_{n-1}$, the fixed subgroup $\\mathrm{Fix}(\\alpha)$ has a finite free basis all of whose elements have reduced word length at most $f(\\|\\alpha\\|)$, where $\\|\\alpha\\|$ is the maximum of the reduced word lengths of $\\alpha(x_0), \\dots, \\alpha(x_{n-1})$.","title":"Kourovka Problem 14.23 fails uniformly in the rank"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/word","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupAutomorphismFixedPointKourovka10Problem25Wanted.Word","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#word","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"fully_proved","summary":"A word of rank $r$ is a finite list of pairs $(i, \\varepsilon)$ with $i < r$ and $\\varepsilon$ a Boolean. It encodes the element of the free group $F_r$ of rank $r$ that is the product of its letters, where $(i, \\mathrm{true})$ stands for the $i$-th free generator and $(i, \\mathrm{false})$ for its inverse.","title":"Encoded words in a free group of finite rank"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/endo","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupAutomorphismFixedPointKourovka10Problem25Wanted.endo","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#endo","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"fully_proved","summary":"Let $F_r$ be the free group on generators $e_0, \\dots, e_{r-1}$. Given, for each $i < r$, a word $u_i$ in these generators and their inverses, this is the endomorphism of $F_r$ sending each $e_i$ to the element of $F_r$ represented by $u_i$.","title":"Endomorphism of a free group given by generator images"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/automorphism-fixed-point-decidable","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupAutomorphismFixedPointKourovka10Problem25Wanted.automorphismFixedPointDecidable","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#automorphism-fixed-point-decidable","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"fully_proved","summary":"The assertion that for every natural number $r$ there is a computable function $f$ which takes the images of the $r$ free generators of the free group $F_r$, each encoded as a word in the generators and their inverses, and returns a Boolean, such that whenever the endomorphism $\\varphi$ of $F_r$ sending each generator to its image is bijective, $f$ returns true exactly when $\\varphi(w) = w$ for some $w \\neq 1$ in\u2026","title":"Decidability of fixed points of free group automorphisms (Kourovka 10.25)"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/automorphism-fixed-point-decidable-holds","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupAutomorphismFixedPointKourovka10Problem25Wanted.automorphismFixedPointDecidable_holds","parent":2322,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms/index.html#automorphism-fixed-point-decidable-holds","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/fixed-subgroups-of-free-group-automorphisms","status":"can_prove","summary":"For every natural number $r$ there is a computable function $f$ which takes the images of the $r$ free generators of the free group $F_r$, each encoded as a word in the generators and their inverses, and returns a Boolean, such that whenever the endomorphism $\\varphi$ of $F_r$ sending each generator to its image is bijective, $f$ returns true exactly when $\\varphi(w) = w$ for some $w \\neq 1$ in $F_r$.","title":"Fixed points of free group automorphisms are decidable (Kourovka 10.25)"},{"data_url":"scopes/GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro.json","id":"GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro","kind":"scope","navigation":"module","parent":2302,"result":false,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro/index.html","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro","status":"can_prove","title":"Free products in automorphism groups of free groups"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro/free-product-aut-free-group-embeds","kind":"node","lean":"MathlibExt.GroupTheory.FreeProductAutFreeGroupEmbeddingKourovka18Problem11Wanted.freeProductAutFreeGroupEmbeds","parent":2334,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bardakov's question on free products in "],["math","\\(\\mathrm{Aut}(F_n)\\)"],["text"," (Kourovka 18.11)"]],"source":"Bardakov's question on free products in $\\mathrm{Aut}(F_n)$ (Kourovka 18.11)"},"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro/index.html#free-product-aut-free-group-embeds","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro","status":"fully_proved","summary":"The statement that for every natural number $n \\ge 2$ and all subgroups $G$ and $H$ of the automorphism group $\\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$, there are a natural number $m$ and an injective group homomorphism from the free product $G \\ast H$ to $\\mathrm{Aut}(F_m)$.","title":"Bardakov's question on free products in $\\mathrm{Aut}(F_n)$ (Kourovka 18.11)"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro/free-product-aut-free-group-embeds-holds","kind":"node","lean":"MathlibExt.GroupTheory.FreeProductAutFreeGroupEmbeddingKourovka18Problem11Wanted.freeProductAutFreeGroupEmbeds_holds","parent":2334,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free products of subgroups of "],["math","\\(\\mathrm{Aut}(F_n)\\)"],["text"," embed in some "],["math","\\(\\mathrm{Aut}(F_m)\\)"]],"source":"Free products of subgroups of $\\mathrm{Aut}(F_n)$ embed in some $\\mathrm{Aut}(F_m)$"},"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro/index.html#free-product-aut-free-group-embeds-holds","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/free-products-in-automorphism-groups-of-free-gro","status":"can_prove","summary":"For every natural number $n \\ge 2$ and all subgroups $G$ and $H$ of the automorphism group $\\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$, there are a natural number $m$ and an injective group homomorphism from the free product $G \\ast H$ to $\\mathrm{Aut}(F_m)$.","title":"Free products of subgroups of $\\mathrm{Aut}(F_n)$ embed in some $\\mathrm{Aut}(F_m)$"},{"data_url":"scopes/GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups.json","id":"GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups","kind":"scope","navigation":"module","parent":2302,"result":false,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups/index.html","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups","status":"can_prove","title":"Linearity of IA-automorphism groups"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups/ia-aut","kind":"node","lean":"MathlibExt.GroupTheory.IAAutomorphismsLinearKourovka15Problem10Wanted.iaAut","parent":2337,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups/index.html#ia-aut","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups","status":"fully_proved","summary":"For a natural number $n$, let $F_n$ be the free group on $n$ generators. $\\mathrm{IA}(F_n)$ is the subgroup of the automorphism group of $F_n$ consisting of the automorphisms $\\varphi$ that act trivially on the abelianization $F_n/[F_n, F_n]$, that is, such that $\\varphi(x)$ and $x$ have the same image in $F_n/[F_n, F_n]$ for every $x \\in F_n$.","title":"IA-automorphisms of a free group"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups/ia-aut-linear","kind":"node","lean":"MathlibExt.GroupTheory.IAAutomorphismsLinearKourovka15Problem10Wanted.iaAutLinear","parent":2337,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups/index.html#ia-aut-linear","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups","status":"fully_proved","summary":"The statement that for some $n \\ge 3$ the group $\\mathrm{IA}(F_n)$ of automorphisms of the free group $F_n$ on $n$ generators that act trivially on its abelianization $F_n/[F_n, F_n]$ is linear, that is, there are a field $K$, a natural number $m$ and an injective group homomorphism $\\mathrm{IA}(F_n) \\to \\mathrm{GL}_m(K)$.","title":"Kourovka Problem 15.10 (statement): linearity of IA-automorphism groups of free groups"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups/ia-aut-linear-refuted","kind":"node","lean":"MathlibExt.GroupTheory.IAAutomorphismsLinearKourovka15Problem10Wanted.iaAutLinear_refuted","parent":2337,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups/index.html#ia-aut-linear-refuted","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/linearity-of-ia-automorphism-groups","status":"can_prove","summary":"Resolved false: the Kourovka Archive answers 15.10 No: `IA(F\u2099)` is not linear for `n \u2265 5` (Pettet) and for every `n \u2265 3` (Bardakov and Mikhailov). Source: E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 15.10 (Yu. I. Merzlyakov; V. G. Bardakov); resolution: V. G. Bardakov and R. Mikhailov,\u2026","title":"IA-automorphism group of a free group is not linear"},{"data_url":"scopes/GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro.json","id":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro","kind":"scope","navigation":"module","parent":2302,"result":false,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/index.html","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro","status":"can_prove","title":"Word problem for automorphism groups of free groups"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/eval-aut-word","kind":"node","lean":"MathlibExt.GroupTheory.AutFreeGroupWordProblemPolyTimeKourovka16Problem109Wanted.evalAutWord","parent":2341,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/index.html#eval-aut-word","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro","status":"fully_proved","summary":"Let $F_n$ be the free group on $n$ generators and let $g_0, \\dots, g_{k-1}$ be automorphisms of $F_n$. A word $w = ((i_1, b_1), \\dots, (i_r, b_r))$ is a finite list of letters $(i, b)$ with $i \\in \\mathbb{N}$ and $b$ a boolean. The automorphism spelled by $w$ over $(g_i)$ is the product $a_1 a_2 \\cdots a_r$ in $\\operatorname{Aut}(F_n)$, whose group operation is composition, where for $i_j < k$ the factor $a_j$ is\u2026","title":"The automorphism spelled by a word"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/aut-word-problem","kind":"node","lean":"MathlibExt.GroupTheory.AutFreeGroupWordProblemPolyTimeKourovka16Problem109Wanted.autWordProblem","parent":2341,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/index.html#aut-word-problem","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro","status":"fully_proved","summary":"Let $F_n$ be the free group on $n$ generators and let $g_0, \\dots, g_{k-1}$ be automorphisms of $F_n$. A word is a finite list of letters $(i, b)$ with $i \\in \\mathbb{N}$ and $b$ a boolean. The word problem for $(g_i)$ is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of lists of pairs of a natural number and a boolean, to a word $w$ such that\u2026","title":"The word problem of Aut(F\u2099) for a family of automorphisms"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/aut-free-group-word-problem-in-p","kind":"node","lean":"MathlibExt.GroupTheory.AutFreeGroupWordProblemPolyTimeKourovka16Problem109Wanted.autFreeGroupWordProblemInP","parent":2341,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/index.html#aut-free-group-word-problem-in-p","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro","status":"fully_proved","summary":"The proposition that for every $n \\ge 2$, writing $F_n$ for the free group on $n$ generators, for every natural number $k$ and every family $g_0, \\dots, g_{k-1}$ of automorphisms of $F_n$ that generates $\\operatorname{Aut}(F_n)$, the word problem of $\\operatorname{Aut}(F_n)$ for the family $(g_i)$, a decision problem on bitstrings, belongs to the complexity class P.","title":"Polynomial-time word problem for Aut(F\u2099) (Kourovka 16.109)"},{"id":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/aut-free-group-word-problem-in-p-holds","kind":"node","lean":"MathlibExt.GroupTheory.AutFreeGroupWordProblemPolyTimeKourovka16Problem109Wanted.autFreeGroupWordProblemInP_holds","parent":2341,"result":true,"route":"../roadmap/GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro/index.html#aut-free-group-word-problem-in-p-holds","scope":"GroupTheory/free-groups/automorphisms-of-free-groups/word-problem-for-automorphism-groups-of-free-gro","status":"can_prove","summary":"Let $n \\ge 2$ and let $F_n$ be the free group on $n$ generators. For every natural number $k$ and every family $g_0, \\dots, g_{k-1}$ of automorphisms of $F_n$ that generates $\\operatorname{Aut}(F_n)$, the word problem of $\\operatorname{Aut}(F_n)$ for this family belongs to the complexity class P. Here a word is a finite list of letters $(i, b)$ with $i \\in \\mathbb{N}$ and $b$ a boolean, the letter\u2026","title":"The word problem for Aut(F\u2099) is solvable in polynomial time"},{"data_url":"scopes/GroupTheory/free-groups/cartesian-powers-of-free-groups.json","id":"GroupTheory/free-groups/cartesian-powers-of-free-groups","kind":"scope","navigation":"module","parent":2233,"result":false,"route":"../roadmap/GroupTheory/free-groups/cartesian-powers-of-free-groups/index.html","scope":"GroupTheory/free-groups/cartesian-powers-of-free-groups","status":"can_prove","title":"Cartesian powers of free groups"},{"id":"GroupTheory/free-groups/cartesian-powers-of-free-groups/cartesian-power-abelianization-torsion-f","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupPowerAbelianizationKourovka13Problem18Wanted.cartesianPowerAbelianizationTorsionFree","parent":2346,"result":true,"route":"../roadmap/GroupTheory/free-groups/cartesian-powers-of-free-groups/index.html#cartesian-power-abelianization-torsion-f","scope":"GroupTheory/free-groups/cartesian-powers-of-free-groups","status":"fully_proved","summary":"The statement, posed by A. M. Gaglione and D. Spellman, that for every integer $n \\ge 2$ the abelianization of the unrestricted (Cartesian) product $F_n^{\\mathbb{N}}$ of countably many copies of the free group $F_n$ of rank $n$ is torsion-free: none of its nontrivial elements has finite order.","title":"Torsion-freeness of abelianized Cartesian powers of free groups (Kourovka 13.18)"},{"id":"GroupTheory/free-groups/cartesian-powers-of-free-groups/cartesian-power-abelianization-torsion-f-2","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupPowerAbelianizationKourovka13Problem18Wanted.cartesianPowerAbelianizationTorsionFree_refuted","parent":2346,"result":true,"route":"../roadmap/GroupTheory/free-groups/cartesian-powers-of-free-groups/index.html#cartesian-power-abelianization-torsion-f-2","scope":"GroupTheory/free-groups/cartesian-powers-of-free-groups","status":"can_prove","summary":"For some integer $n \\ge 2$, the abelianization of the unrestricted (Cartesian) product $F_n^{\\mathbb{N}}$ of countably many copies of the free group $F_n$ of rank $n$ contains a nontrivial element of finite order. That is, the torsion-freeness asked for in Kourovka Problem 13.18 fails.","title":"Abelianized Cartesian powers of free groups can have torsion"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems.json","id":"GroupTheory/presentations-and-decision-problems","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/index.html","scope":"GroupTheory/presentations-and-decision-problems","status":"can_prove","summary":"Combinatorial group theory: presentations, one-relator groups, HNN extensions and embeddings, SQ-universality, finitely presented simple groups, the word and conjugacy problems (Novikov\u2013Boone, Boone\u2013Higman), Andrews\u2013Curtis moves, generic properties of presentations and torsion quotients.","title":"Presentations and decision problems"},{"id":"GroupTheory/presentations-and-decision-problems/higman-neumann-neumann-embedding","kind":"node","lean":"MathlibExt.GroupTheory.HNNEmbeddingWanted.higman_neumann_neumann_embedding","parent":2349,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/index.html#higman-neumann-neumann-embedding","scope":"GroupTheory/presentations-and-decision-problems","status":"fully_proved","summary":"Every countable group $G$ embeds in a group generated by two elements: there are a group $H$, elements $a, b \\in H$ that generate $H$, and an injective group homomorphism $f \\colon G \\to H$.","title":"Higman\u2013Neumann\u2013Neumann embedding theorem"},{"id":"GroupTheory/presentations-and-decision-problems/higman-infinite-simple","kind":"node","lean":"MathlibExt.GroupTheory.HigmanInfiniteSimpleWanted.GroupTheory.HigmanInfiniteSimpleProblem.higman_infinite_simple","parent":2349,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/index.html#higman-infinite-simple","scope":"GroupTheory/presentations-and-decision-problems","status":"can_prove","summary":"There exist a natural number $n$ and a finite set $R$ of elements of the free group on generators $x_1, \\dots, x_n$ such that the group $\\langle x_1, \\dots, x_n \\mid R \\rangle$ presented by these generators and relators is simple and infinite.","title":"Infinite finitely presented simple groups exist"},{"id":"GroupTheory/presentations-and-decision-problems/is-squniversal","kind":"node","lean":"MathlibExt.GroupTheory.SQUniversalShared.IsSQUniversal","parent":2349,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/index.html#is-squniversal","scope":"GroupTheory/presentations-and-decision-problems","status":"fully_proved","summary":"A group $G$ is SQ-universal if every countable group is isomorphic to a subgroup of a quotient of $G$: for every countable group $C$ there are a group $Q$, a surjective homomorphism $G \\to Q$ and an injective homomorphism $C \\to Q$.","title":"SQ-universal group"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/braid-and-artin.json","id":"GroupTheory/presentations-and-decision-problems/braid-and-artin","kind":"scope","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/index.html","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin","status":"can_prove","summary":"Braid groups, pure braid groups and Artin\u2013Tits groups of Coxeter matrices: endomorphisms, roots, conjugators and conjugacy questions.","title":"Braid and Artin groups"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups.json","id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","kind":"scope","navigation":"module","parent":2353,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","title":"Artin groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/alternating-word","kind":"node","lean":"CoxeterMatrix.alternatingWord","parent":2354,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html#alternating-word","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","summary":"For elements $a$ and $b$ of a monoid and $n \\in \\mathbb{N}$, the alternating word of length $n$ starting with $a$ is the product $a b a b \\cdots$ of $n$ factors that alternate between $a$ and $b$, beginning with $a$; for $n = 0$ it is the identity.","title":"Alternating word"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/map-alternating-word","kind":"node","lean":"CoxeterMatrix.map_alternatingWord","parent":2354,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html#map-alternating-word","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","summary":"Let $f : G \\to H$ be a homomorphism of monoids. For all $a, b \\in G$ and $n \\in \\mathbb{N}$, $f$ maps the alternating word $a b a \\cdots$ of length $n$ starting with $a$ to the alternating word $f(a) f(b) f(a) \\cdots$ of length $n$ starting with $f(a)$.","title":"Monoid homomorphisms preserve alternating words"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/artin-relation","kind":"node","lean":"CoxeterMatrix.artinRelation","parent":2354,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html#artin-relation","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","summary":"Let $M = (m_{ij})$ be a Coxeter matrix over an index type $B$, and write $s_i$ for the free generators of the free group on $B$. For $i, j \\in B$, let $u$ and $v$ be the alternating words $s_i s_j s_i \\cdots$ and $s_j s_i s_j \\cdots$ of length $m_{ij}$. The Artin relator of the ordered pair $(i, j)$ is the element $u v^{-1}$ of the free group; setting it equal to $1$ imposes the braid relation $u = v$.","title":"Artin relator"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/artin-relations-set","kind":"node","lean":"CoxeterMatrix.artinRelationsSet","parent":2354,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html#artin-relations-set","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","summary":"Let $M = (m_{ij})$ be a Coxeter matrix over an index type $B$, where $m_{ij} = 0$ encodes $m_{ij} = \\infty$, and write $s_i$ for the free generators of the free group on $B$. The set of Artin relators of $M$ consists of the elements $u v^{-1}$ of the free group, one for each ordered pair of distinct $i, j \\in B$ with $m_{ij} \\neq 0$, where $u$ and $v$ are the alternating words $s_i s_j s_i \\cdots$ and\u2026","title":"Artin relators of a Coxeter matrix"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/artin-group","kind":"node","lean":"CoxeterMatrix.ArtinGroup","parent":2354,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html#artin-group","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","summary":"Let $M = (m_{ij})_{i, j \\in B}$ be a Coxeter matrix over an index type $B$, where $m_{ij} = 0$ encodes $m_{ij} = \\infty$. The Artin group (Artin\u2013Tits group) of $M$ is the group presented by generators $s_i$, one for each $i \\in B$, subject only to the braid relations: for every ordered pair of distinct $i, j \\in B$ with $m_{ij} \\neq 0$, the alternating words $s_i s_j s_i \\cdots$ and $s_j s_i s_j \\cdots$ of length\u2026","title":"Artin group"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/of","kind":"node","lean":"CoxeterMatrix.ArtinGroup.of","parent":2354,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html#of","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","summary":"For a Coxeter matrix $M$ over an index type $B$ and $i \\in B$, the generator $s_i$ of the Artin group of $M$ is the image of the free generator $i$ under the quotient map from the free group on $B$ onto the Artin group.","title":"Generators of the Artin group"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/closure-range-of","kind":"node","lean":"CoxeterMatrix.ArtinGroup.closure_range_of","parent":2354,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html#closure-range-of","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","summary":"For a Coxeter matrix $M$ over an index type $B$, the subgroup of the Artin group of $M$ generated by its generators $s_i$, $i \\in B$, is the whole group.","title":"The generators generate the Artin group"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/braid-relation","kind":"node","lean":"CoxeterMatrix.ArtinGroup.braid_relation","parent":2354,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups/index.html#braid-relation","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/artin-groups","status":"fully_proved","summary":"Let $M = (m_{ij})$ be a Coxeter matrix over an index type $B$, where $m_{ij} = 0$ encodes $m_{ij} = \\infty$, and let $s_i$ ($i \\in B$) be the generators of the Artin group of $M$. For distinct $i, j \\in B$ with $m_{ij} \\neq 0$, the alternating words $s_i s_j s_i \\cdots$ and $s_j s_i s_j \\cdots$ of length $m_{ij}$ are equal in the Artin group.","title":"Braid relations in the Artin group"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou.json","id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou","kind":"scope","navigation":"module","parent":2353,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou/index.html","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou","status":"can_prove","title":"Braid groups in automorphism groups of free groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou/braid-group-four","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupFourAutFreeGroupTwoKourovka17Problem14Wanted.BraidGroupFour","parent":2363,"reading_title":{"format":"inline-parts-v1","parts":[["text","The braid group "],["math","\\(B_4\\)"]],"source":"The braid group $B_4$"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou/index.html#braid-group-four","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou","status":"fully_proved","summary":"The braid group on four strands, given by Artin's presentation: the group generated by $\\sigma_1, \\sigma_2, \\sigma_3$ subject to the relations $\\sigma_1 \\sigma_2 \\sigma_1 = \\sigma_2 \\sigma_1 \\sigma_2$, $\\sigma_2 \\sigma_3 \\sigma_2 = \\sigma_3 \\sigma_2 \\sigma_3$ and $\\sigma_1 \\sigma_3 = \\sigma_3 \\sigma_1$.","title":"The braid group $B_4$"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou/braid-group-four-embeds-in-aut-free-grou","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupFourAutFreeGroupTwoKourovka17Problem14Wanted.braidGroupFourEmbedsInAutFreeGroupTwo","parent":2363,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bardakov's question on embedding "],["math","\\(B_4\\)"],["text"," in "],["math","\\(\\operatorname{Aut}(F_2)\\)"]],"source":"Bardakov's question on embedding $B_4$ in $\\operatorname{Aut}(F_2)$"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou/index.html#braid-group-four-embeds-in-aut-free-grou","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou","status":"fully_proved","summary":"The statement that there is an injective group homomorphism from the braid group $B_4$ on four strands to the automorphism group $\\operatorname{Aut}(F_2)$ of the free group $F_2$ of rank $2$. Kourovka Problem 17.14 (V. G. Bardakov) asks whether, for $n \\ge 4$, the braid group $B_n$ embeds in the automorphism group of the free group $F_{n-2}$ of rank $n - 2$; this is the case $n = 4$.","title":"Bardakov's question on embedding $B_4$ in $\\operatorname{Aut}(F_2)$"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou/braid-group-four-embeds-in-aut-free-grou-2","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupFourAutFreeGroupTwoKourovka17Problem14Wanted.braidGroupFourEmbedsInAutFreeGroupTwo_refuted","parent":2363,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bardakov\u2013Bellingeri: "],["math","\\(B_4\\)"],["text"," does not embed in "],["math","\\(\\operatorname{Aut}(F_2)\\)"]],"source":"Bardakov\u2013Bellingeri: $B_4$ does not embed in $\\operatorname{Aut}(F_2)$"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou/index.html#braid-group-four-embeds-in-aut-free-grou-2","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/braid-groups-in-automorphism-groups-of-free-grou","status":"can_prove","summary":"There is no injective group homomorphism from the braid group $B_4$ on four strands to the automorphism group $\\operatorname{Aut}(F_2)$ of the free group $F_2$ of rank $2$. Kourovka Problem 17.14 (V. G. Bardakov) asks whether, for $n \\ge 4$, the braid group $B_n$ embeds in the automorphism group of the free group $F_{n-2}$ of rank $n - 2$; this answers the case $n = 4$ negatively.","title":"Bardakov\u2013Bellingeri: $B_4$ does not embed in $\\operatorname{Aut}(F_2)$"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups.json","id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups","kind":"scope","navigation":"module","parent":2353,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups","status":"can_prove","title":"Conjugators in braid groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups/braid-of","kind":"node","lean":"MathlibExt.GroupTheory.BraidConjugatorLengthKourovka19Problem81Wanted.braidOf","parent":2367,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups/index.html#braid-of","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, let $B_{n+1}$ be the braid group on $n + 1$ strands, realized as the Artin group of the Coxeter matrix of type $A_n$, with standard generators $\\sigma_1, \\dots, \\sigma_n$. A braid word is an element of the free group on $\\sigma_1, \\dots, \\sigma_n$; the braid it represents is its image under the group homomorphism from this free group to $B_{n+1}$ that sends each $\\sigma_i$ to the generator\u2026","title":"The braid represented by a braid word"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups/braid-polynomial-conjugator-length","kind":"node","lean":"MathlibExt.GroupTheory.BraidConjugatorLengthKourovka19Problem81Wanted.braidPolynomialConjugatorLength","parent":2367,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups/index.html#braid-polynomial-conjugator-length","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups","status":"fully_proved","summary":"The proposition (the stronger question of Kourovka Problem 19.81, V. Shpilrain) that for every $n \\in \\mathbb{N}$ there are natural numbers $C$ and $d$ with the following property. Let $B_{n+1}$ be the braid group on $n + 1$ strands, realized as the Artin group of the Coxeter matrix of type $A_n$, with standard generators $\\sigma_1, \\dots, \\sigma_n$; a braid word is an element of the free group on\u2026","title":"Polynomially short conjugators in braid groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups/braid-polynomial-conjugator-length-holds","kind":"node","lean":"MathlibExt.GroupTheory.BraidConjugatorLengthKourovka19Problem81Wanted.braidPolynomialConjugatorLength_holds","parent":2367,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups/index.html#braid-polynomial-conjugator-length-holds","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/conjugators-in-braid-groups","status":"can_prove","summary":"For $n \\in \\mathbb{N}$, let $B_{n+1}$ be the braid group on $n + 1$ strands, realized as the Artin group of the Coxeter matrix of type $A_n$, with standard generators $\\sigma_1, \\dots, \\sigma_n$. A braid word is an element of the free group on $\\sigma_1, \\dots, \\sigma_n$, and its length $|u|$ is the length of its freely reduced form. For every $n$ there are natural numbers $C$ and $d$ such that for all braid words\u2026","title":"Braid groups have polynomially short conjugators"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups.json","id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","kind":"scope","navigation":"module","parent":2353,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"can_prove","title":"Endomorphisms of braid groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/braid-relators","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupEndomorphismsKourovka14Problem102Wanted.braidRelators","parent":2371,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html#braid-relators","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"fully_proved","summary":"Artin's relators for the braid group on `n` strands, in the free group on the generators `\u03c3\u2080, \u2026, \u03c3_{n-2}` (indexed by `Fin (n - 1)`): `\u03c3\u1d62 \u03c3\u2c7c = \u03c3\u2c7c \u03c3\u1d62` when `j \u2265 i + 2`, and `\u03c3\u1d62 \u03c3\u1d62\u208a\u2081 \u03c3\u1d62 = \u03c3\u1d62\u208a\u2081 \u03c3\u1d62 \u03c3\u1d62\u208a\u2081`.","title":"Artin's braid relators"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/braid-group","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupEndomorphismsKourovka14Problem102Wanted.BraidGroup","parent":2371,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html#braid-group","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"fully_proved","summary":"The braid group `B\u2099` on `n` strands, by Artin's presentation.","title":"Braid group on n strands"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/braid-non-injective-noncyclic-endomorphi","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupEndomorphismsKourovka14Problem102Wanted.braidNonInjectiveNoncyclicEndomorphism","parent":2371,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html#braid-non-injective-noncyclic-endomorphi","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"fully_proved","summary":"The statement that for some $n > 4$ the braid group $B_n$ on $n$ strands has a non-injective endomorphism whose image is not cyclic (in particular not trivial). Posed by V. Lin and V. Shpilrain.","title":"Kourovka Problem 14.102(a): non-injective braid endomorphisms with non-cyclic image"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/braid-commutator-endomorphism-automorphi","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupEndomorphismsKourovka14Problem102Wanted.braidCommutatorEndomorphismAutomorphism","parent":2371,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html#braid-commutator-endomorphism-automorphi","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"fully_proved","summary":"The statement that for every $n > 4$, every endomorphism of the commutator subgroup $[B_n, B_n]$ of the braid group $B_n$ on $n$ strands other than the trivial endomorphism is bijective, that is, an automorphism. Posed by V. Lin and V. Shpilrain.","title":"Kourovka Problem 14.102(b): endomorphisms of the braid commutator subgroup"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/braid-torsion-free-nonabelian-proper-quo","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupEndomorphismsKourovka14Problem102Wanted.braidTorsionFreeNonabelianProperQuotient","parent":2371,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html#braid-torsion-free-nonabelian-proper-quo","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"fully_proved","summary":"The statement that for every $n > 4$, the braid group $B_n$ on $n$ strands has a nontrivial normal subgroup $N$ such that the quotient $B_n / N$ is non-abelian and torsion-free (no element other than the identity has finite order). Posed by V. Lin and V. Shpilrain.","title":"Kourovka Problem 14.102(c): torsion-free quotients of braid groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/braid-non-injective-noncyclic-endomorphi-2","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupEndomorphismsKourovka14Problem102Wanted.braidNonInjectiveNoncyclicEndomorphism_refuted","parent":2371,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html#braid-non-injective-noncyclic-endomorphi-2","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"can_prove","summary":"The answer to Kourovka Problem 14.102(a) is negative: there is no $n > 4$ for which the braid group $B_n$ on $n$ strands has a non-injective endomorphism whose image is not cyclic. Equivalently, for every $n \\ge 5$, every non-injective endomorphism of $B_n$ has cyclic image.","title":"Non-injective endomorphisms of braid groups on five or more strands have cyclic image"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/braid-commutator-endomorphism-automorphi-2","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupEndomorphismsKourovka14Problem102Wanted.braidCommutatorEndomorphismAutomorphism_holds","parent":2371,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html#braid-commutator-endomorphism-automorphi-2","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"can_prove","summary":"Nontrivial endomorphisms of the braid commutator subgroup on five or more strands are automorphisms. For every $n > 4$, every endomorphism of the commutator subgroup $[B_n, B_n]$ of the braid group $B_n$ on $n$ strands other than the trivial endomorphism is bijective, that is, an automorphism.","title":"Nontrivial endomorphisms of the braid commutator subgroup on five or more strands are automorphisms"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/braid-torsion-free-nonabelian-proper-quo-2","kind":"node","lean":"MathlibExt.GroupTheory.BraidGroupEndomorphismsKourovka14Problem102Wanted.braidTorsionFreeNonabelianProperQuotient_holds","parent":2371,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups/index.html#braid-torsion-free-nonabelian-proper-quo-2","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/endomorphisms-of-braid-groups","status":"can_prove","summary":"For every $n > 4$, the braid group $B_n$ on $n$ strands has a nontrivial normal subgroup $N$ such that the quotient $B_n / N$ is non-abelian and torsion-free (no element other than the identity has finite order).","title":"Braid groups on five or more strands have proper non-abelian torsion-free quotients"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups.json","id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups","kind":"scope","navigation":"module","parent":2353,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups","status":"can_prove","title":"Pure braid groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/pure-braid-group","kind":"node","lean":"MathlibExt.GroupTheory.PureBraidConjugateInverseKourovka11Problem54Wanted.pureBraidGroup","parent":2380,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/index.html#pure-braid-group","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the pure (coloured) braid group $P_{n+1}$ on $n + 1$ strands is the normal closure, in the braid group $B_{n+1}$, of the squares $\\sigma_i^2$ of the standard generators $\\sigma_1, \\dots, \\sigma_n$ of $B_{n+1}$, where $B_{n+1}$ is the Artin group of the Coxeter matrix of type $A_n$.","title":"Pure braid group"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/pure-braid-conj-inv-trivial","kind":"node","lean":"MathlibExt.GroupTheory.PureBraidConjugateInverseKourovka11Problem54Wanted.pureBraidConjInvTrivial","parent":2380,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/index.html#pure-braid-conj-inv-trivial","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups","status":"fully_proved","summary":"The proposition that for every $n \\in \\mathbb{N}$, every pure braid $\\beta$ on $n + 1$ strands that is conjugate to its inverse inside the pure braid group, that is, $\\gamma \\beta \\gamma^{-1} = \\beta^{-1}$ for some pure braid $\\gamma$ on $n + 1$ strands, is the identity braid.","title":"Kourovka 11.54: pure braids conjugate to their inverses"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/pure-braid-conj-inv-trivial-holds","kind":"node","lean":"MathlibExt.GroupTheory.PureBraidConjugateInverseKourovka11Problem54Wanted.pureBraidConjInvTrivial_holds","parent":2380,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/index.html#pure-braid-conj-inv-trivial-holds","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$ and every pure braid $\\beta$ on $n + 1$ strands, if $\\gamma \\beta \\gamma^{-1} = \\beta^{-1}$ for some pure braid $\\gamma$ on $n + 1$ strands, then $\\beta = 1$.","title":"Kourovka 11.54: only the trivial pure braid is conjugate to its inverse"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/pure-braid-group-2","kind":"node","lean":"MathlibExt.GroupTheory.PureBraidUniqueRootsKourovka11Problem55Wanted.pureBraidGroup","parent":2380,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/index.html#pure-braid-group-2","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the pure (coloured) braid group $P_{n+1}$ on $n + 1$ strands is the normal closure, in the braid group $B_{n+1}$, of the squares $\\sigma_i^2$ of the standard generators $\\sigma_1, \\dots, \\sigma_n$ of $B_{n+1}$, where $B_{n+1}$ is the Artin group of the Coxeter matrix of type $A_n$.","title":"Pure braid group"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/pure-braid-unique-roots","kind":"node","lean":"MathlibExt.GroupTheory.PureBraidUniqueRootsKourovka11Problem55Wanted.pureBraidUniqueRoots","parent":2380,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/index.html#pure-braid-unique-roots","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups","status":"fully_proved","summary":"The proposition that for every $n \\in \\mathbb{N}$ and every integer $k \\ge 1$, if pure braids $\\alpha$ and $\\beta$ on $n + 1$ strands satisfy $\\alpha^k = \\beta^k$, then $\\alpha = \\beta$.","title":"Kourovka 11.55: unique roots in pure braid groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/pure-braid-unique-roots-holds","kind":"node","lean":"MathlibExt.GroupTheory.PureBraidUniqueRootsKourovka11Problem55Wanted.pureBraidUniqueRoots_holds","parent":2380,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups/index.html#pure-braid-unique-roots-holds","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/pure-braid-groups","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$ and every integer $k \\ge 1$, if pure braids $\\alpha$ and $\\beta$ on $n + 1$ strands satisfy $\\alpha^k = \\beta^k$, then $\\alpha = \\beta$.","title":"Kourovka 11.55: roots in pure braid groups are unique"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups.json","id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups","kind":"scope","navigation":"module","parent":2353,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups","status":"can_prove","title":"Roots in braid groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/braid-relators","kind":"node","lean":"MathlibExt.GroupTheory.BraidRootsConjugateKourovka10Problem23Wanted.braidRelators","parent":2387,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/index.html#braid-relators","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups","status":"fully_proved","summary":"For a natural number $n$, the set of Artin's relators for the braid group on $n$ strands, as elements of the free group on the $n - 1$ generators $\\sigma_0, \\dots, \\sigma_{n-2}$ (no generators when $n \\le 1$): the commutators $\\sigma_i \\sigma_j \\sigma_i^{-1} \\sigma_j^{-1}$ for $0 \\le i$ and $i + 2 \\le j \\le n - 2$, and the words $\\sigma_i \\sigma_{i+1} \\sigma_i (\\sigma_{i+1} \\sigma_i \\sigma_{i+1})^{-1}$ for\u2026","title":"Artin's braid relators"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/braid-group","kind":"node","lean":"MathlibExt.GroupTheory.BraidRootsConjugateKourovka10Problem23Wanted.BraidGroup","parent":2387,"reading_title":{"format":"inline-parts-v1","parts":[["text","The braid group "],["math","\\(B_n\\)"],["text"," by Artin's presentation"]],"source":"The braid group $B_n$ by Artin's presentation"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/index.html#braid-group","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups","status":"fully_proved","summary":"For a natural number $n$, the braid group $B_n$ on $n$ strands is the group generated by $\\sigma_0, \\dots, \\sigma_{n-2}$ subject to the relations $\\sigma_i \\sigma_j = \\sigma_j \\sigma_i$ for $j \\ge i + 2$ and $\\sigma_i \\sigma_{i+1} \\sigma_i = \\sigma_{i+1} \\sigma_i \\sigma_{i+1}$ for $0 \\le i \\le n - 3$; it is the trivial group when $n \\le 1$.","title":"The braid group $B_n$ by Artin's presentation"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/braid-roots-unique-up-to-conjugacy","kind":"node","lean":"MathlibExt.GroupTheory.BraidRootsConjugateKourovka10Problem23Wanted.braidRootsUniqueUpToConjugacy","parent":2387,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/index.html#braid-roots-unique-up-to-conjugacy","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups","status":"fully_proved","summary":"The statement that for every natural number $n$ and every integer $k \\ge 1$, any two elements $\\alpha, \\beta$ of the braid group $B_n$ on $n$ strands with $\\alpha^k = \\beta^k$ are conjugate in $B_n$. This is Kourovka Problem 10.23 (G. S. Makanin).","title":"Makanin's root-uniqueness question for braid groups"},{"id":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/braid-roots-unique-up-to-conjugacy-holds","kind":"node","lean":"MathlibExt.GroupTheory.BraidRootsConjugateKourovka10Problem23Wanted.braidRootsUniqueUpToConjugacy_holds","parent":2387,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups/index.html#braid-roots-unique-up-to-conjugacy-holds","scope":"GroupTheory/presentations-and-decision-problems/braid-and-artin/roots-in-braid-groups","status":"can_prove","summary":"For every natural number $n$ and every integer $k \\ge 1$, if two elements $\\alpha, \\beta$ of the braid group $B_n$ on $n$ strands satisfy $\\alpha^k = \\beta^k$, then $\\alpha$ and $\\beta$ are conjugate in $B_n$. This answers Kourovka Problem 10.23 (G. S. Makanin) affirmatively.","title":"Roots in braid groups are unique up to conjugacy (Gonz\u00e1lez-Meneses)"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/andrews-curtis-moves.json","id":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/index.html","scope":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves","status":"can_prove","title":"Andrews\u2013Curtis moves"},{"id":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/is-acmove","kind":"node","lean":"MathlibExt.GroupTheory.AndrewsCurtisGroupPresentationKourovka18Problem89Wanted.IsACMove","parent":2392,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/index.html#is-acmove","scope":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves","status":"fully_proved","summary":"Let $F_n$ be the free group on $n$ generators. A map $f \\colon F_n^n \\to F_n^n$ on $n$-tuples $(u_1, \\dots, u_n)$ of elements of $F_n$ is an elementary Andrews\u2013Curtis move if there is an index $i$ such that $f$ replaces the entry $u_i$ of every tuple by $u_i u_j$, $u_i u_j^{-1}$, $u_j u_i$ or $u_j^{-1} u_i$ for a fixed index $j \\ne i$, or by $u_i^{-1}$, or by $w u_i w^{-1}$ for a fixed $w \\in F_n$, and leaves all\u2026","title":"Elementary Andrews\u2013Curtis moves"},{"id":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/andrews-curtis-group","kind":"node","lean":"MathlibExt.GroupTheory.AndrewsCurtisGroupPresentationKourovka18Problem89Wanted.andrewsCurtisGroup","parent":2392,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Andrews\u2013Curtis group "],["math","\\(AC_n\\)"]],"source":"The Andrews\u2013Curtis group $AC_n$"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/index.html#andrews-curtis-group","scope":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ let $F_n$ be the free group on $n$ generators. The Andrews\u2013Curtis group $AC_n$ is the subgroup of the group of permutations of the set $F_n^n$ of $n$-tuples $(u_1, \\dots, u_n)$ of elements of $F_n$ generated by the elementary Andrews\u2013Curtis moves that are permutations: $u_i \\mapsto u_i u_j^{\\pm 1}$ and $u_i \\mapsto u_j^{\\pm 1} u_i$ for indices $i \\ne j$, $u_i \\mapsto u_i^{-1}$, and\u2026","title":"The Andrews\u2013Curtis group $AC_n$"},{"id":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/andrews-curtis-two-finitely-presented","kind":"node","lean":"MathlibExt.GroupTheory.AndrewsCurtisGroupPresentationKourovka18Problem89Wanted.andrewsCurtisTwoFinitelyPresented","parent":2392,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 18.89 for "],["math","\\(n = 2\\)"]],"source":"Kourovka Problem 18.89 for $n = 2$"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/index.html#andrews-curtis-two-finitely-presented","scope":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves","status":"fully_proved","summary":"The proposition, asked by J. Swan and A. Lisitsa, that the Andrews\u2013Curtis group $AC_2$ (the subgroup of the permutation group of the set of pairs of elements of the free group $F_2$ generated by the elementary Andrews\u2013Curtis moves) is finitely presented.","title":"Kourovka Problem 18.89 for $n = 2$"},{"id":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/andrews-curtis-finitely-presented-of-thr","kind":"node","lean":"MathlibExt.GroupTheory.AndrewsCurtisGroupPresentationKourovka18Problem89Wanted.andrewsCurtisFinitelyPresentedOfThreeLe","parent":2392,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 18.89 for "],["math","\\(n \\ge 3\\)"]],"source":"Kourovka Problem 18.89 for $n \\ge 3$"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/index.html#andrews-curtis-finitely-presented-of-thr","scope":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves","status":"fully_proved","summary":"The proposition that for every $n \\ge 3$ the Andrews\u2013Curtis group $AC_n$ (the subgroup of the permutation group of the set of $n$-tuples of elements of the free group $F_n$ generated by the elementary Andrews\u2013Curtis moves) is finitely presented.","title":"Kourovka Problem 18.89 for $n \\ge 3$"},{"id":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/andrews-curtis-two-finitely-presented-re","kind":"node","lean":"MathlibExt.GroupTheory.AndrewsCurtisGroupPresentationKourovka18Problem89Wanted.andrewsCurtisTwoFinitelyPresented_refuted","parent":2392,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Andrews\u2013Curtis group "],["math","\\(AC_2\\)"],["text"," is not finitely presented (Kourovka 18.89)"]],"source":"The Andrews\u2013Curtis group $AC_2$ is not finitely presented (Kourovka 18.89)"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/index.html#andrews-curtis-two-finitely-presented-re","scope":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves","status":"can_prove","summary":"The Andrews\u2013Curtis group $AC_2$, the subgroup of the permutation group of the set $F_2^2$ of pairs of elements of the free group $F_2$ generated by the elementary Andrews\u2013Curtis moves, is not finitely presented.","title":"The Andrews\u2013Curtis group $AC_2$ is not finitely presented (Kourovka 18.89)"},{"id":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/andrews-curtis-finitely-presented-of-thr-2","kind":"node","lean":"MathlibExt.GroupTheory.AndrewsCurtisGroupPresentationKourovka18Problem89Wanted.andrewsCurtisFinitelyPresentedOfThreeLe_holds","parent":2392,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Andrews\u2013Curtis groups "],["math","\\(AC_n\\)"],["text"," are finitely presented for "],["math","\\(n \\ge 3\\)"],["text"," (Kourovka 18.89)"]],"source":"The Andrews\u2013Curtis groups $AC_n$ are finitely presented for $n \\ge 3$ (Kourovka 18.89)"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/andrews-curtis-moves/index.html#andrews-curtis-finitely-presented-of-thr-2","scope":"GroupTheory/presentations-and-decision-problems/andrews-curtis-moves","status":"can_prove","summary":"For every $n \\ge 3$, the Andrews\u2013Curtis group $AC_n$, the subgroup of the permutation group of the set $F_n^n$ of $n$-tuples of elements of the free group $F_n$ generated by the elementary Andrews\u2013Curtis moves, is finitely presented.","title":"The Andrews\u2013Curtis groups $AC_n$ are finitely presented for $n \\ge 3$ (Kourovka 18.89)"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/baumslag-gersten-group.json","id":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/index.html","scope":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group","status":"can_prove","title":"Baumslag\u2013Gersten group"},{"id":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/baumslag-group","kind":"node","lean":"MathlibExt.GroupTheory.BaumslagGroupGermsFaithfulKourovka17Problem99Wanted.BaumslagGroup","parent":2399,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/index.html#baumslag-group","scope":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group","status":"fully_proved","summary":"Baumslag's group is the one-relator group $B = \\langle a, b \\mid (bab^{-1})\\, a\\, (bab^{-1})^{-1} = a^2 \\rangle$, the quotient of the free group on the two generators $a$ and $b$ by the normal subgroup generated by $(bab^{-1})\\, a\\, (bab^{-1})^{-1} a^{-2}$.","title":"Baumslag's group"},{"id":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/letter-function","kind":"node","lean":"MathlibExt.GroupTheory.BaumslagGroupGermsFaithfulKourovka17Problem99Wanted.letterFunction","parent":2399,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/index.html#letter-function","scope":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group","status":"fully_proved","summary":"To each letter of the free group on $a, b$ attach a function $\\mathbb{R} \\to \\mathbb{R}$: $a$ gives $f(x) = 2x$, $a^{-1}$ gives $x/2$, $b$ gives $g(x) = 2^x$ and $b^{-1}$ gives $\\log_2 x$.","title":"Letter functions for Baumslag's group"},{"id":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/word-function","kind":"node","lean":"MathlibExt.GroupTheory.BaumslagGroupGermsFaithfulKourovka17Problem99Wanted.wordFunction","parent":2399,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/index.html#word-function","scope":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group","status":"fully_proved","summary":"To an element $w$ of the free group on $a, b$ with reduced word $l_1 l_2 \\cdots l_n$ attach the composite $\\varphi_w = \\varphi_{l_1} \\circ \\varphi_{l_2} \\circ \\cdots \\circ \\varphi_{l_n}$, where $\\varphi_l$ is the function attached to the letter $l$; the first letter is applied last, and $\\varphi_1$ is the identity. With this order products of germs are composites, and $f(x) = 2x$ and $g(x) = 2^x$ satisfy the\u2026","title":"Word functions for Baumslag's group"},{"id":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/baumslag-group-germ-representation-faith","kind":"node","lean":"MathlibExt.GroupTheory.BaumslagGroupGermsFaithfulKourovka17Problem99Wanted.baumslagGroupGermRepresentationFaithful","parent":2399,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/index.html#baumslag-group-germ-representation-faith","scope":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group","status":"fully_proved","summary":"Let $B$ be Baumslag's group $\\langle a, b \\mid (bab^{-1})\\, a\\, (bab^{-1})^{-1} = a^2 \\rangle$, let $f(x) = 2x$ and $g(x) = 2^x$, and for $w$ in the free group on $a, b$ let $\\varphi_w$ be its word function, the composite of $f$, $g$ and their inverses along the reduced word of $w$. The representation $a \\mapsto f$, $b \\mapsto g$ of $B$ by germs at $+\\infty$ is faithful when every $w$ with $\\varphi_w(x) = x$ for\u2026","title":"Faithfulness of Baumslag's germ representation (Kourovka 17.99)"},{"id":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/baumslag-group-germ-representation-faith-2","kind":"node","lean":"MathlibExt.GroupTheory.BaumslagGroupGermsFaithfulKourovka17Problem99Wanted.baumslagGroupGermRepresentationFaithful_refuted","parent":2399,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/baumslag-gersten-group/index.html#baumslag-group-germ-representation-faith-2","scope":"GroupTheory/presentations-and-decision-problems/baumslag-gersten-group","status":"can_prove","summary":"Let $B$ be Baumslag's group $\\langle a, b \\mid (bab^{-1})\\, a\\, (bab^{-1})^{-1} = a^2 \\rangle$, let $f(x) = 2x$ and $g(x) = 2^x$, and for $w$ in the free group on $a, b$ with reduced word $l_1 l_2 \\cdots l_k$ let $\\varphi_w = \\varphi_{l_1} \\circ \\varphi_{l_2} \\circ \\cdots \\circ \\varphi_{l_k}$, where $\\varphi_a = f$, $\\varphi_{a^{-1}}(x) = x/2$, $\\varphi_b = g$ and $\\varphi_{b^{-1}}(x) = \\log_2 x$. The\u2026","title":"Baumslag's germ representation is not faithful"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups.json","id":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups","status":"can_prove","title":"Coherence of one-relator groups"},{"id":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/coherent","kind":"node","lean":"MathlibExt.GroupTheory.OnCoherenceOfOneRelatorGroupsWanted.Group.Coherent","parent":2405,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/index.html#coherent","scope":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups","status":"fully_proved","summary":"A group is coherent if all of its finitely generated subgroups are finitely presented.","title":"Coherent group"},{"id":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/coherent-2","kind":"node","lean":"MathlibExt.GroupTheory.OnCoherenceOfOneRelatorGroupsWanted.Ring.Coherent","parent":2405,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/index.html#coherent-2","scope":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups","status":"fully_proved","summary":"A semiring $R$ is (left) coherent if every finitely generated left ideal of $R$ is finitely presented as an $R$-module.","title":"Coherent ring"},{"id":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/one-relator-group","kind":"node","lean":"MathlibExt.GroupTheory.OnCoherenceOfOneRelatorGroupsWanted.Group.OneRelator","parent":2405,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/index.html#one-relator-group","scope":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups","status":"fully_proved","summary":"A group `G` is called one-relator if it has a presentation with a single relation.","title":"One-relator group"},{"id":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/theorem-1-1","kind":"node","lean":"MathlibExt.GroupTheory.OnCoherenceOfOneRelatorGroupsWanted.OnCoherenceOfOneRelatorGroups.theorem_1_1","parent":2405,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups/index.html#theorem-1-1","scope":"GroupTheory/presentations-and-decision-problems/coherence-of-one-relator-groups","status":"can_prove","summary":"Let $G$ be a one-relator group and $K$ a field. Then $G$ is a coherent group and the group algebra $K[G]$ is a (left) coherent ring.","title":"Jaikin-Zapirain\u2013Linton theorem: one-relator groups are coherent"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions.json","id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"can_prove","title":"Decision problems in free abelianized extensions"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/is-recursively-enumerable","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.IsRecursivelyEnumerable","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#is-recursively-enumerable","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"fully_proved","summary":"Let $n$ be a natural number and let $F$ be the free group on the generators $x_0, \\dots, x_{n-1}$. A subgroup $N$ of $F$ is recursively enumerable when the set of words $w$ (finite lists of letters, each a generator together with a sign) whose image in $F$ lies in $N$ is recursively enumerable.","title":"Recursively enumerable subgroup of a free group"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/word-problem-solvable","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.WordProblemSolvable","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#word-problem-solvable","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"fully_proved","summary":"Let $n$ be a natural number, let $F$ be the free group on the generators $x_0, \\dots, x_{n-1}$ and let $N$ be a subgroup of $F$. The word problem for $F / N$ is solvable when membership in $N$ is decidable: the predicate on words $w$ (finite lists of letters, each a generator together with a sign) that the image of $w$ in $F$ lies in $N$ is computable.","title":"Solvable word problem for a quotient of a free group"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/conjugacy-problem-solvable","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.ConjugacyProblemSolvable","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#conjugacy-problem-solvable","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"fully_proved","summary":"Let $n$ be a natural number, let $F$ be the free group on the generators $x_0, \\dots, x_{n-1}$ and let $N$ be a subgroup of $F$. The conjugacy problem for $F / N$ is solvable when the predicate on pairs of words $(w, v)$ (finite lists of letters, each a generator together with a sign) that some $u \\in F$ satisfies $u w u^{-1} v^{-1} \\in N$, with $w$ and $v$ read as elements of $F$, is computable.","title":"Solvable conjugacy problem for a quotient of a free group"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/word-problem-iff-metabelianization","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.wordProblemIffMetabelianization","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#word-problem-iff-metabelianization","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"fully_proved","summary":"The statement that for every natural number $n$ and every recursively enumerable normal subgroup $R$ of the free group $F$ on $n$ generators, with commutator subgroup $[R, R]$, the word problem for $F / R$ is solvable if and only if the word problem for $F / [R, R]$ is solvable.","title":"Kourovka Problem 12.98(a) (V. E. Shpil'rain)"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/conjugacy-problem-iff-metabelianization","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.conjugacyProblemIffMetabelianization","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#conjugacy-problem-iff-metabelianization","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"fully_proved","summary":"The statement that for every natural number $n$ and every recursively enumerable normal subgroup $R$ of the free group $F$ on $n$ generators, with commutator subgroup $[R, R]$, the conjugacy problem for $F / R$ is solvable if and only if the conjugacy problem for $F / [R, R]$ is solvable.","title":"Kourovka Problem 12.98(b) (V. E. Shpil'rain)"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/metabelianization-word-problem-solves-co","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.metabelianizationWordProblemSolvesConjugacy","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#metabelianization-word-problem-solves-co","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"fully_proved","summary":"The statement that for every natural number $n$ and every recursively enumerable normal subgroup $R$ of the free group $F$ on $n$ generators, with commutator subgroup $[R, R]$, if the word problem for $F / [R, R]$ is solvable then the conjugacy problem for $F / [R, R]$ is solvable.","title":"Kourovka Problem 12.98(c) (V. E. Shpil'rain)"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/word-problem-iff-metabelianization-holds","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.wordProblemIffMetabelianization_holds","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#word-problem-iff-metabelianization-holds","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"can_prove","summary":"For every natural number $n$ and every recursively enumerable normal subgroup $R$ of the free group $F$ on $n$ generators, with commutator subgroup $[R, R]$, the word problem for $F / R$ is solvable if and only if the word problem for $F / [R, R]$ is solvable.","title":"Kourovka 12.98(a) has a positive answer"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/conjugacy-problem-iff-metabelianization-2","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.conjugacyProblemIffMetabelianization_refuted","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#conjugacy-problem-iff-metabelianization-2","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"can_prove","summary":"It is not the case that for every natural number $n$ and every recursively enumerable normal subgroup $R$ of the free group $F$ on $n$ generators, with commutator subgroup $[R, R]$, the conjugacy problem for $F / R$ is solvable if and only if the conjugacy problem for $F / [R, R]$ is solvable: for some such $n$ and $R$, exactly one of these two conjugacy problems is solvable.","title":"Kourovka 12.98(b) has a negative answer"},{"id":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/metabelianization-word-problem-solves-co-2","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianizationDecisionProblemsKourovka12Problem98Wanted.metabelianizationWordProblemSolvesConjugacy_refuted","parent":2410,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions/index.html#metabelianization-word-problem-solves-co-2","scope":"GroupTheory/presentations-and-decision-problems/decision-problems-in-free-abelianized-extensions","status":"can_prove","summary":"It is not the case that for every natural number $n$ and every recursively enumerable normal subgroup $R$ of the free group $F$ on $n$ generators, with commutator subgroup $[R, R]$, solvability of the word problem for $F / [R, R]$ implies solvability of the conjugacy problem for $F / [R, R]$: for some such $n$ and $R$, the word problem for $F / [R, R]$ is solvable but its conjugacy problem is not.","title":"Kourovka 12.98(c) has a negative answer"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups.json","id":"GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups","status":"can_prove","title":"Finite quotients of one-relator groups"},{"id":"GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups/soluble-group-image-of-one-relator-group","kind":"node","lean":"MathlibExt.GroupTheory.SolubleOneRelatorQuotientKourovka18Problem75Wanted.solubleGroupImageOfOneRelatorGroups","parent":2420,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups/index.html#soluble-group-image-of-one-relator-group","scope":"GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups","status":"fully_proved","summary":"The statement, posed by A. Yu. Olshanskii, that every finite soluble group $G$ has a natural number $d$ such that for every element $w$ of the free group on $x_1, \\dots, x_d$ there is a surjective homomorphism from the one-relator group $\\langle x_1, \\dots, x_d \\mid w \\rangle$ onto $G$.","title":"Finite soluble groups as images of one-relator groups (Kourovka 18.75)"},{"id":"GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups/soluble-group-image-of-one-relator-group-2","kind":"node","lean":"MathlibExt.GroupTheory.SolubleOneRelatorQuotientKourovka18Problem75Wanted.solubleGroupImageOfOneRelatorGroups_holds","parent":2420,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups/index.html#soluble-group-image-of-one-relator-group-2","scope":"GroupTheory/presentations-and-decision-problems/finite-quotients-of-one-relator-groups","status":"can_prove","summary":"Every finite soluble group $G$ has a natural number $d$ such that for every element $w$ of the free group on $x_1, \\dots, x_d$ there is a surjective homomorphism from the one-relator group $\\langle x_1, \\dots, x_d \\mid w \\rangle$ onto $G$. That is, Kourovka Problem 18.75 has a positive answer.","title":"Finite soluble groups are images of one-relator groups"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups.json","id":"GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups","status":"can_prove","title":"Finitely presented soluble groups"},{"id":"GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups/is-metanilpotent-by-finite","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyPresentedSolubleMetanilpotentKourovka16Problem35Wanted.IsMetanilpotentByFinite","parent":2423,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups/index.html#is-metanilpotent-by-finite","scope":"GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups","status":"fully_proved","summary":"A group $G$ is (nilpotent-by-nilpotent)-by-finite when it has a normal subgroup $M$ of finite index and a normal subgroup $N$ of $M$ such that $N$ and $M/N$ are both nilpotent.","title":"Metanilpotent-by-finite groups"},{"id":"GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups/finitely-presented-soluble-metanilpotent","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyPresentedSolubleMetanilpotentKourovka16Problem35Wanted.finitelyPresentedSolubleMetanilpotentByFinite","parent":2423,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups/index.html#finitely-presented-soluble-metanilpotent","scope":"GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups","status":"fully_proved","summary":"The proposition that every finitely presented soluble group $G$ is (nilpotent-by-nilpotent)-by-finite, that is, $G$ has a normal subgroup $M$ of finite index and a normal subgroup $N$ of $M$ such that $N$ and $M/N$ are both nilpotent.","title":"Groves's question (Kourovka 16.35)"},{"id":"GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups/finitely-presented-soluble-metanilpotent-2","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyPresentedSolubleMetanilpotentKourovka16Problem35Wanted.finitelyPresentedSolubleMetanilpotentByFinite_refuted","parent":2423,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups/index.html#finitely-presented-soluble-metanilpotent-2","scope":"GroupTheory/presentations-and-decision-problems/finitely-presented-soluble-groups","status":"can_prove","summary":"Not every finitely presented soluble group is (nilpotent-by-nilpotent)-by-finite: there is a finitely presented soluble group $G$ such that for no normal subgroup $M$ of finite index in $G$ is there a normal subgroup $N$ of $M$ with $N$ and $M/N$ both nilpotent.","title":"Groves's question (Kourovka 16.35) has a negative answer"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups.json","id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"can_prove","title":"Free subgroups of one-relator groups"},{"id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/one-relator-infinite-index-free-is-free","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorInfiniteIndexFreeKourovka20Problem41Wanted.oneRelatorInfiniteIndexFreeIsFreeOrSurface","parent":2427,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html#one-relator-infinite-index-free-is-free","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"fully_proved","summary":"The proposition that every non-cyclic residually finite group $G$ such that every subgroup of finite index in $G$ (including $G$ itself) is isomorphic to a one-relator group $\\langle x_0, \\dots, x_{n-1} \\mid r \\rangle$ for some natural number $n$ and some element $r$ of the free group on $x_0, \\dots, x_{n-1}$, and every subgroup of infinite index in $G$ is free, is free, or is isomorphic to\u2026","title":"Kourovka Problem 20.41: one-relator groups with free infinite-index subgroups"},{"id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/one-relator-infinite-index-free-is-free-2","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorInfiniteIndexFreeKourovka20Problem41Wanted.oneRelatorInfiniteIndexFreeIsFreeOrSurface_holds","parent":2427,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html#one-relator-infinite-index-free-is-free-2","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"can_prove","summary":"Let $G$ be a non-cyclic residually finite group such that every subgroup of finite index in $G$ (including $G$ itself) is isomorphic to a one-relator group $\\langle x_0, \\dots, x_{n-1} \\mid r \\rangle$ for some natural number $n$ and some element $r$ of the free group on $x_0, \\dots, x_{n-1}$, and every subgroup of infinite index in $G$ is free. Then $G$ is free, or $G$ is isomorphic to\u2026","title":"Kourovka Problem 20.41 has a positive answer"},{"id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/is-one-relator-group","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorSurfaceGroupKourovka20Problem22Wanted.IsOneRelatorGroup","parent":2427,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html#is-one-relator-group","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"fully_proved","summary":"A group $G$ is a one-relator group if, for some $n \\ge 0$ and some element $r$ of the free group on $x_1, \\dots, x_n$, $G$ is isomorphic to the group $\\langle x_1, \\dots, x_n \\mid r \\rangle$ with generators $x_1, \\dots, x_n$ and the single relator $r$.","title":"One-relator groups with finitely many generators"},{"id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/orientable-surface-relator","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorSurfaceGroupKourovka20Problem22Wanted.orientableSurfaceRelator","parent":2427,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html#orientable-surface-relator","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"fully_proved","summary":"For $g \\ge 0$, the orientable surface relator of genus $g$ is the element $[a_1, b_1] [a_2, b_2] \\cdots [a_g, b_g]$ of the free group on the $2g$ generators $a_1, \\dots, a_g, b_1, \\dots, b_g$, where $[a, b] = a b a^{-1} b^{-1}$; for $g = 0$ it is the identity.","title":"Orientable surface relator"},{"id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/nonorientable-surface-relator","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorSurfaceGroupKourovka20Problem22Wanted.nonorientableSurfaceRelator","parent":2427,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html#nonorientable-surface-relator","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"fully_proved","summary":"For $k \\ge 0$, the non-orientable surface relator of genus $k$ is the element $a_1^2 a_2^2 \\cdots a_k^2$ of the free group on the $k$ generators $a_1, \\dots, a_k$; for $k = 0$ it is the identity.","title":"Non-orientable surface relator"},{"id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/is-surface-group","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorSurfaceGroupKourovka20Problem22Wanted.IsSurfaceGroup","parent":2427,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html#is-surface-group","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"fully_proved","summary":"A group $G$ is a surface group if either $G$ is isomorphic to $\\langle a_1, b_1, \\dots, a_g, b_g \\mid [a_1, b_1] \\cdots [a_g, b_g] \\rangle$ for some $g \\ge 1$, where $[a, b] = a b a^{-1} b^{-1}$, or $G$ is isomorphic to $\\langle a_1, \\dots, a_k \\mid a_1^2 \\cdots a_k^2 \\rangle$ for some $k \\ge 2$.","title":"Surface groups"},{"id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/one-relator-infinite-index-free-is-surfa","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorSurfaceGroupKourovka20Problem22Wanted.oneRelatorInfiniteIndexFreeIsSurfaceGroup","parent":2427,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html#one-relator-infinite-index-free-is-surfa","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"fully_proved","summary":"The statement that every group $G$ with the following properties is a surface group: $G$ is isomorphic to $\\langle x_1, \\dots, x_n \\mid r \\rangle$ for some $n \\ge 0$ and some element $r$ of the free group on $x_1, \\dots, x_n$; $G$ is neither free nor cyclic; and every subgroup of infinite index in $G$ is free. Here $G$ is a surface group if it is isomorphic to\u2026","title":"Kourovka Problem 20.22 (statement)"},{"id":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/one-relator-infinite-index-free-is-surfa-2","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorSurfaceGroupKourovka20Problem22Wanted.oneRelatorInfiniteIndexFreeIsSurfaceGroup_holds","parent":2427,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups/index.html#one-relator-infinite-index-free-is-surfa-2","scope":"GroupTheory/presentations-and-decision-problems/free-subgroups-of-one-relator-groups","status":"can_prove","summary":"Let $G$ be a group isomorphic to $\\langle x_1, \\dots, x_n \\mid r \\rangle$ for some $n \\ge 0$ and some element $r$ of the free group on $x_1, \\dots, x_n$. If $G$ is neither free nor cyclic and every subgroup of infinite index in $G$ is free, then $G$ is a surface group: $G$ is isomorphic to $\\langle a_1, b_1, \\dots, a_g, b_g \\mid [a_1, b_1] \\cdots [a_g, b_g] \\rangle$ for some $g \\ge 1$, where\u2026","title":"Wilton's theorem (Kourovka Problem 20.22)"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations.json","id":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/index.html","scope":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations","status":"can_prove","title":"Generic properties of group presentations"},{"id":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/presentations","kind":"node","lean":"MathlibExt.GroupTheory.GenericFreeSubgroupsKourovka11Problem75Wanted.presentations","parent":2436,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/index.html#presentations","scope":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations","status":"fully_proved","summary":"For natural numbers $m$, $n$ and $d$, this is the set of $n$-tuples $(r_1, \\dots, r_n)$ of elements of the free group $F_m$ on generators $x_1, \\dots, x_m$ such that the reduced word of each $r_i$ is cyclically reduced (the empty word counts as cyclically reduced) and the reduced word lengths satisfy $|r_1| + \\dots + |r_n| = d$. Each tuple stands for the presentation\u2026","title":"Presentations counted by Arzhantseva and Olshanskii"},{"id":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/subgroups-with-fewer-generators-free","kind":"node","lean":"MathlibExt.GroupTheory.GenericFreeSubgroupsKourovka11Problem75Wanted.SubgroupsWithFewerGeneratorsFree","parent":2436,"reading_title":{"format":"inline-parts-v1","parts":[["text","Groups whose subgroups with fewer than "],["math","\\(m\\)"],["text"," generators are free"]],"source":"Groups whose subgroups with fewer than $m$ generators are free"},"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/index.html#subgroups-with-fewer-generators-free","scope":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations","status":"fully_proved","summary":"For a natural number $m$, a group $G$ has this property if, for every finite set $S \\subseteq G$ with fewer than $m$ elements, the subgroup of $G$ generated by $S$ is a free group.","title":"Groups whose subgroups with fewer than $m$ generators are free"},{"id":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/fewer-generator-subgroups-free-generic","kind":"node","lean":"MathlibExt.GroupTheory.GenericFreeSubgroupsKourovka11Problem75Wanted.fewerGeneratorSubgroupsFreeGeneric","parent":2436,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/index.html#fewer-generator-subgroups-free-generic","scope":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations","status":"fully_proved","summary":"This is the proposition, in the form proved by Arzhantseva and Olshanskii, that the following holds for all natural numbers $m \\ge 2$ and $n \\ge 1$. Let $F_m$ be the free group on $x_1, \\dots, x_m$. For each $d$, let $N_d$ be the number of $n$-tuples $(r_1, \\dots, r_n)$ of elements of $F_m$ whose reduced words are cyclically reduced and have total length $|r_1| + \\dots + |r_n| = d$, and let $N_d^f$ be the number of\u2026","title":"Olshanskii's genericity problem (Kourovka 11.75)"},{"id":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/fewer-generator-subgroups-free-generic-h","kind":"node","lean":"MathlibExt.GroupTheory.GenericFreeSubgroupsKourovka11Problem75Wanted.fewerGeneratorSubgroupsFreeGeneric_holds","parent":2436,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations/index.html#fewer-generator-subgroups-free-generic-h","scope":"GroupTheory/presentations-and-decision-problems/generic-properties-of-group-presentations","status":"can_prove","summary":"Let $m \\ge 2$ and $n \\ge 1$ be natural numbers and let $F_m$ be the free group on $x_1, \\dots, x_m$. For each $d$, let $N_d$ be the number of $n$-tuples $(r_1, \\dots, r_n)$ of elements of $F_m$ whose reduced words are cyclically reduced and have total length $|r_1| + \\dots + |r_n| = d$, and let $N_d^f$ be the number of those tuples for which every subgroup of $\\langle x_1, \\dots, x_m \\mid r_1, \\dots, r_n \\rangle$\u2026","title":"Generically, subgroups with fewer generators are free (Kourovka 11.75)"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups.json","id":"GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups","status":"can_prove","title":"Involutions in quotient groups"},{"id":"GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups/involution-free-quotient-by-non-square-p","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionFreeQuotientKourovka18Problem94Wanted.involutionFreeQuotientByNonSquarePower","parent":2441,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups/index.html#involution-free-quotient-by-non-square-p","scope":"GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups","status":"fully_proved","summary":"This is the statement that whenever $G$ is a group without involutions (if $g^2 = 1$ then $g = 1$), $a \\in G$ is not the square of any element of $G$, and $n$ is an odd natural number, the quotient of $G$ by the normal closure of $a^n$ has no involutions: every element $y$ of the quotient with $y^2 = 1$ is $1$.","title":"Sozutov's question on involutions in quotients (Kourovka 18.94)"},{"id":"GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups/involution-free-quotient-by-non-square-p-2","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionFreeQuotientKourovka18Problem94Wanted.involutionFreeQuotientByNonSquarePower_refuted","parent":2441,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups/index.html#involution-free-quotient-by-non-square-p-2","scope":"GroupTheory/presentations-and-decision-problems/involutions-in-quotient-groups","status":"can_prove","summary":"There exist a group $G$ without involutions (if $g^2 = 1$ then $g = 1$), an element $a \\in G$ that is not the square of any element of $G$, and an odd natural number $n$ such that the quotient of $G$ by the normal closure of $a^n$ contains an involution, that is, an element $y \\ne 1$ with $y^2 = 1$. This answers Kourovka Problem 18.94 in the negative.","title":"Quotients by the normal closure of an odd power of a non-square can have involutions"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/just-finite-presentations.json","id":"GroupTheory/presentations-and-decision-problems/just-finite-presentations","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/just-finite-presentations/index.html","scope":"GroupTheory/presentations-and-decision-problems/just-finite-presentations","status":"can_prove","title":"Just finite presentations"},{"id":"GroupTheory/presentations-and-decision-problems/just-finite-presentations/finite-group-has-just-finite-presentatio","kind":"node","lean":"MathlibExt.GroupTheory.JustFinitePresentationKourovka21Problem10Wanted.finiteGroupHasJustFinitePresentation","parent":2444,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/just-finite-presentations/index.html#finite-group-has-just-finite-presentatio","scope":"GroupTheory/presentations-and-decision-problems/just-finite-presentations","status":"fully_proved","summary":"The statement that every finite group $G$ has a just finite presentation: a presentation $\\langle X \\mid R \\rangle$ with $X$ and $R$ finite that defines $G$ and such that removing any single relator from $R$ gives a presentation of an infinite group.","title":"Barnea's question on just finite presentations (Kourovka 21.10)"},{"id":"GroupTheory/presentations-and-decision-problems/just-finite-presentations/finite-group-has-just-finite-presentatio-2","kind":"node","lean":"MathlibExt.GroupTheory.JustFinitePresentationKourovka21Problem10Wanted.finiteGroupHasJustFinitePresentation_holds","parent":2444,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/just-finite-presentations/index.html#finite-group-has-just-finite-presentatio-2","scope":"GroupTheory/presentations-and-decision-problems/just-finite-presentations","status":"can_prove","summary":"For every finite group $G$ there are a natural number $n$ and a finite subset $R$ of the free group on $n$ generators $x_1, \\dots, x_n$ such that the group $\\langle x_1, \\dots, x_n \\mid R \\rangle$ is isomorphic to $G$ and, for every $r \\in R$, the group $\\langle x_1, \\dots, x_n \\mid R \\setminus \\{r\\} \\rangle$ is infinite. That is, the answer to Kourovka Problem 21.10 is positive.","title":"Every finite group has a just finite presentation"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups.json","id":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups","status":"can_prove","title":"Torsion length of finitely presented groups"},{"id":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/torsion-series","kind":"node","lean":"MathlibExt.GroupTheory.SolubleTorsionLengthKourovka19Problem23Wanted.torsionSeries","parent":2447,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/index.html#torsion-series","scope":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups","status":"fully_proved","summary":"For a group $G$, the torsion series of Chiodo and Vyas is the sequence of subgroups $\\mathrm{Tor}_i(G)$ of $G$ defined by $\\mathrm{Tor}_0(G) = 1$, the trivial subgroup, and, for each $i$, $\\mathrm{Tor}_{i+1}(G)$ is the normal closure in $G$ of the set of elements $g$ such that $g^n \\in \\mathrm{Tor}_i(G)$ for some integer $n > 0$.","title":"Torsion series"},{"id":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/torsion-length-gt-two","kind":"node","lean":"MathlibExt.GroupTheory.SolubleTorsionLengthKourovka19Problem23Wanted.TorsionLengthGtTwo","parent":2447,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/index.html#torsion-length-gt-two","scope":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups","status":"fully_proved","summary":"A group $G$ has torsion length greater than $2$ when $G / \\mathrm{Tor}_2(G)$ is not torsion-free, where $\\mathrm{Tor}_2(G)$ is the second term of the torsion series of $G$: some element $g \\in G$ outside $\\mathrm{Tor}_2(G)$ has a power $g^n$ with $n > 0$ in $\\mathrm{Tor}_2(G)$.","title":"Torsion length greater than two"},{"id":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/exists-finitely-presented-soluble-torsio","kind":"node","lean":"MathlibExt.GroupTheory.SolubleTorsionLengthKourovka19Problem23Wanted.existsFinitelyPresentedSolubleTorsionLengthGtTwo","parent":2447,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/index.html#exists-finitely-presented-soluble-torsio","scope":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups","status":"fully_proved","summary":"The statement that there is a finitely presented soluble group $G$ of torsion length greater than $2$, that is, such that $G / \\mathrm{Tor}_2(G)$ is not torsion-free, where $\\mathrm{Tor}_2(G)$ is the second term of the torsion series of $G$.","title":"Kourovka Problem 19.23 (Chiodo\u2013Vyas), finitely presented form"},{"id":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/exists-finitely-presented-soluble-torsio-2","kind":"node","lean":"MathlibExt.GroupTheory.SolubleTorsionLengthKourovka19Problem23Wanted.existsFinitelyPresentedSolubleTorsionLengthGtTwo_holds","parent":2447,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups/index.html#exists-finitely-presented-soluble-torsio-2","scope":"GroupTheory/presentations-and-decision-problems/torsion-length-of-finitely-presented-groups","status":"can_prove","summary":"Resolved true: the Kourovka Archive answers 19.23 \"Yes, such groups exist\"; Leary and Minasyan (Theorem 2.14) construct polycyclic groups, which are finitely presented and soluble, of every torsion length `n \u2265 1`, in particular of torsion length `3`. Source: E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved\u2026","title":"Finitely presented soluble groups of torsion length greater than two"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups.json","id":"GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups","status":"can_prove","title":"Torsion quotients of finitely presented groups"},{"id":"GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups/torsion-closure","kind":"node","lean":"MathlibExt.GroupTheory.TorsionQuotientFinitelyPresentedKourovka19Problem24Wanted.torsionClosure","parent":2452,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups/index.html#torsion-closure","scope":"GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups","status":"fully_proved","summary":"For a group $G$, $\\mathrm{Tor}(G)$ is the normal closure in $G$ of the set of elements of finite order of $G$.","title":"Torsion closure"},{"id":"GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups/exists-finitely-presented-torsion-quotie","kind":"node","lean":"MathlibExt.GroupTheory.TorsionQuotientFinitelyPresentedKourovka19Problem24Wanted.existsFinitelyPresentedTorsionQuotientNotFinitelyPresented","parent":2452,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups/index.html#exists-finitely-presented-torsion-quotie","scope":"GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups","status":"fully_proved","summary":"The statement, asked by M. Chiodo and R. Vyas, that there exists a finitely presented group $G$ such that $G / \\mathrm{Tor}(G)$ is not finitely presented, where $\\mathrm{Tor}(G)$ is the normal closure of the set of elements of finite order of $G$.","title":"Kourovka Problem 19.24 (torsion quotients of finitely presented groups)"},{"id":"GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups/exists-finitely-presented-torsion-quotie-2","kind":"node","lean":"MathlibExt.GroupTheory.TorsionQuotientFinitelyPresentedKourovka19Problem24Wanted.existsFinitelyPresentedTorsionQuotientNotFinitelyPresented_holds","parent":2452,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups/index.html#exists-finitely-presented-torsion-quotie-2","scope":"GroupTheory/presentations-and-decision-problems/torsion-quotients-of-finitely-presented-groups","status":"can_prove","summary":"There exists a finitely presented group $G$ such that $G / \\mathrm{Tor}(G)$ is not finitely presented, where $\\mathrm{Tor}(G)$ is the normal closure of the set of elements of finite order of $G$.","title":"Torsion quotients of finitely presented groups need not be finitely presented"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/word-problem-for-groups.json","id":"GroupTheory/presentations-and-decision-problems/word-problem-for-groups","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/word-problem-for-groups/index.html","scope":"GroupTheory/presentations-and-decision-problems/word-problem-for-groups","status":"can_prove","title":"Word problem for groups"},{"id":"GroupTheory/presentations-and-decision-problems/word-problem-for-groups/word-problem-solvable","kind":"node","lean":"MathlibExt.GroupTheory.WordProblemSolvableShared.WordProblemSolvable","parent":2456,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/word-problem-for-groups/index.html#word-problem-solvable","scope":"GroupTheory/presentations-and-decision-problems/word-problem-for-groups","status":"fully_proved","summary":"Let $G$ be a group, $n$ a natural number and $\\varphi$ a homomorphism from the free group on $n$ generators to $G$. The word problem is solvable for $\\varphi$ when the predicate on words $w$ (finite lists of letters, each a generator together with a sign) given by $\\varphi(w) = 1$ in $G$ is computable: some algorithm decides, given a word $w$, whether $\\varphi(w) = 1$.","title":"Solvable word problem"},{"id":"GroupTheory/presentations-and-decision-problems/word-problem-for-groups/novikov-unsolvable","kind":"node","lean":"MathlibExt.GroupTheory.NovikovUnsolvableWanted.GroupTheory.NovikovUnsolvableProblem.novikov_unsolvable","parent":2456,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/word-problem-for-groups/index.html#novikov-unsolvable","scope":"GroupTheory/presentations-and-decision-problems/word-problem-for-groups","status":"can_prove","summary":"There exist a natural number $n$ and a finite set $R$ of elements of the free group $F_n$ on generators $g_1, \\dots, g_n$ such that the word problem of the canonical surjection from $F_n$ onto the presented group $\\langle g_1, \\dots, g_n \\mid R \\rangle$ is not solvable: no algorithm decides, for a word $w$ in the $g_i$ and their inverses, whether $w$ represents the identity of\u2026","title":"Novikov\u2013Boone theorem (unsolvability of the word problem)"},{"data_url":"scopes/GroupTheory/presentations-and-decision-problems/boone-higman-theorem.json","id":"GroupTheory/presentations-and-decision-problems/boone-higman-theorem","kind":"scope","navigation":"module","parent":2349,"result":false,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/boone-higman-theorem/index.html","scope":"GroupTheory/presentations-and-decision-problems/boone-higman-theorem","status":"can_prove","title":"Boone\u2013Higman theorem"},{"id":"GroupTheory/presentations-and-decision-problems/boone-higman-theorem/boone-higman-embedding","kind":"node","lean":"MathlibExt.GroupTheory.BooneHigmanEmbeddingWanted.GroupTheory.BooneHigmanEmbedding.boone_higman_embedding","parent":2459,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/boone-higman-theorem/index.html#boone-higman-embedding","scope":"GroupTheory/presentations-and-decision-problems/boone-higman-theorem","status":"can_prove","summary":"Let $\\varphi \\colon F_n \\to G$ be a surjective homomorphism from the free group $F_n$ on $n$ generators onto a group $G$. If there are injective homomorphisms $G \\to H$ and $H \\to K$, where $H$ is a simple group and $K$ is a finitely presented group, then the word problem of $G$ with respect to $\\varphi$ is solvable.","title":"Boone\u2013Higman theorem (easy direction)"},{"id":"GroupTheory/presentations-and-decision-problems/boone-higman-theorem/boone-higman-simple","kind":"node","lean":"MathlibExt.GroupTheory.BooneHigmanSimpleWanted.GroupTheory.BooneHigmanSimpleProblem.boone_higman_simple","parent":2459,"result":true,"route":"../roadmap/GroupTheory/presentations-and-decision-problems/boone-higman-theorem/index.html#boone-higman-simple","scope":"GroupTheory/presentations-and-decision-problems/boone-higman-theorem","status":"can_prove","summary":"Let $G$ be a simple group, let $F_n$ be the free group on $n$ generators, and let $\\varphi \\colon F_n \\to G$ be a surjective homomorphism whose kernel is the normal closure of finitely many elements. Then the word problem of $G$ with respect to $\\varphi$ is solvable: there is an algorithm that decides, for every word $w$ in the $n$ generators and their inverses, whether $\\varphi(w) = 1$.","title":"Kuznetsov's theorem on finitely presented simple groups"},{"data_url":"scopes/GroupTheory/permutation-groups.json","id":"GroupTheory/permutation-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/index.html","scope":"GroupTheory/permutation-groups","status":"can_prove","summary":"Groups acting on sets: orbits, stabilizers, fixed-point counts, transitivity, bases, closures, Frobenius groups, symmetric and alternating groups as permutation groups, and infinite permutation groups (highly homogeneous, cofinitary, oligomorphic, subdegree-finite).","title":"Permutation groups and group actions"},{"data_url":"scopes/GroupTheory/permutation-groups/finite-permutation-groups.json","id":"GroupTheory/permutation-groups/finite-permutation-groups","kind":"scope","parent":2462,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/index.html","scope":"GroupTheory/permutation-groups/finite-permutation-groups","status":"can_prove","summary":"Actions of finite groups on finite sets: Burnside-type counting, stabilizers, multiply and doubly transitive finite groups, Frobenius groups, bases and independent sets, 2-closures, regular orbits of linear actions, coverings and products in finite symmetric groups.","title":"Finite permutation groups"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/card-stabilizer-dvd-card","kind":"node","lean":"Subgroup.card_stabilizer_dvd_card","parent":2463,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/index.html#card-stabilizer-dvd-card","scope":"GroupTheory/permutation-groups/finite-permutation-groups","status":"fully_proved","summary":"Let $G$ be a finite group, let $H$ be a subgroup of $G$ and let $U$ be a subset of $G$ such that $hu \\in U$ for all $h \\in H$ and $u \\in U$. Then $|H|$ divides $|U|$.","title":"A subgroup stabilizing a subset has order dividing its size"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/card-add-mul-card-sub-one-eq-card-orbits","kind":"node","lean":"MulAction.card_add_mul_card_sub_one_eq_card_orbits_mul_card","parent":2463,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/index.html#card-add-mul-card-sub-one-eq-card-orbits","scope":"GroupTheory/permutation-groups/finite-permutation-groups","status":"fully_proved","summary":"Let a finite group $G$ act on a finite set $X$, and let $c$ be a natural number such that every element $g \\ne 1$ of $G$ fixes exactly $c$ points of $X$. Writing $N$ for the number of orbits of $G$ on $X$, we have $|X| + c\\,(|G| - 1) = N \\cdot |G|$.","title":"Burnside's lemma with a uniform fixed-point count"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/frobenius-kernel-is-normal","kind":"node","lean":"MathlibExt.GroupTheory.FrobeniusWanted.GroupTheory.frobenius_kernel_isNormal","parent":2463,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/index.html#frobenius-kernel-is-normal","scope":"GroupTheory/permutation-groups/finite-permutation-groups","status":"can_prove","summary":"Let a finite group $G$ act faithfully and transitively on a finite set $X$ with at least two elements, such that every point stabiliser is nontrivial and no nonidentity element of $G$ fixes two distinct points of $X$. Then the set consisting of the identity together with the elements of $G$ that fix no point of $X$ is a normal subgroup of $G$.","title":"Frobenius's theorem on the Frobenius kernel"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/five-transitive-card-classification","kind":"node","lean":"MathlibExt.GroupTheory.MultiplyTransitiveWanted.GroupTheory.five_transitive_card_classification","parent":2463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Orders of finite "],["math","\\(5\\)"],["text","-transitive permutation groups"]],"source":"Orders of finite $5$-transitive permutation groups"},"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/index.html#five-transitive-card-classification","scope":"GroupTheory/permutation-groups/finite-permutation-groups","status":"can_prove","summary":"Let a finite group $G$ act faithfully on a finite set $X$ with $n = |X| \\ge 5$, and suppose that the action is $5$-transitive: for any two injective maps $a, b \\colon \\{0, 1, 2, 3, 4\\} \\to X$ there is $g \\in G$ with $g \\cdot a(i) = b(i)$ for all $i$. Then $|G| = n!$, or $n \\ge 7$ and $|G| = n!/2$, or $n = 12$ and $|G| = 95040$, or $n = 24$ and $|G| = 244823040$.","title":"Orders of finite $5$-transitive permutation groups"},{"data_url":"scopes/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups.json","id":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","kind":"scope","navigation":"module","parent":2463,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/index.html","scope":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","status":"can_prove","title":"2-closures of permutation groups"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/is-two-closed-action","kind":"node","lean":"MathlibExt.GroupTheory.TotallyTwoClosedInsolubleKourovka20Problem1Wanted.IsTwoClosedAction","parent":2468,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/index.html#is-two-closed-action","scope":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","status":"fully_proved","summary":"An action of a group $G$ on a set $\\Omega$ is $2$-closed when every permutation $\\sigma$ of $\\Omega$ that preserves each $G$-orbit on ordered pairs (that is, for all $\\alpha, \\beta \\in \\Omega$ some $g \\in G$ satisfies $\\sigma(\\alpha) = g \\alpha$ and $\\sigma(\\beta) = g \\beta$) is induced by an element of $G$: there is $g \\in G$ with $\\sigma(\\omega) = g \\omega$ for all $\\omega \\in \\Omega$.","title":"2-closed actions"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/is-totally-two-closed","kind":"node","lean":"MathlibExt.GroupTheory.TotallyTwoClosedInsolubleKourovka20Problem1Wanted.IsTotallyTwoClosed","parent":2468,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/index.html#is-totally-two-closed","scope":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","status":"fully_proved","summary":"A group $G$ is totally $2$-closed when every faithful action of $G$ on a set $\\Omega$ is $2$-closed; equivalently, in each faithful permutation representation of $G$ on a set $\\Omega$, $G$ is the largest subgroup of $\\mathrm{Sym}(\\Omega)$ preserving each of its orbits on $\\Omega^2$.","title":"Totally 2-closed groups"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/insoluble-totally-two-closed-with-nontri","kind":"node","lean":"MathlibExt.GroupTheory.TotallyTwoClosedInsolubleKourovka20Problem1Wanted.insolubleTotallyTwoClosedWithNontrivialFitting","parent":2468,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/index.html#insoluble-totally-two-closed-with-nontri","scope":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","status":"fully_proved","summary":"The statement, asked by M. Arezoomand, M. A. Iranmanesh, C. E. Praeger and G. Tracey, that there exists a finite group $G$ that is not soluble, is totally $2$-closed, and has a nontrivial nilpotent normal subgroup (equivalently, a nontrivial Fitting subgroup).","title":"Kourovka Problem 20.1 (insoluble totally 2-closed groups)"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/insoluble-totally-two-closed-with-nontri-2","kind":"node","lean":"MathlibExt.GroupTheory.TotallyTwoClosedInsolubleKourovka20Problem1Wanted.insolubleTotallyTwoClosedWithNontrivialFitting_holds","parent":2468,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/index.html#insoluble-totally-two-closed-with-nontri-2","scope":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","status":"can_prove","summary":"There exists a finite group $G$ that is not soluble, is totally $2$-closed, and has a nontrivial nilpotent normal subgroup (equivalently, a nontrivial Fitting subgroup).","title":"A finite insoluble totally 2-closed group with nontrivial Fitting subgroup"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/two-closure","kind":"node","lean":"MathlibExt.GroupTheory.TwoClosureSolvableCompositionFactorsKourovka19Problem67Wanted.twoClosure","parent":2468,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(2\\)"],["text","-closure of a permutation group"]],"source":"The $2$-closure of a permutation group"},"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/index.html#two-closure","scope":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","status":"fully_proved","summary":"For a subgroup $G$ of the symmetric group $\\mathrm{Sym}(\\Omega)$, the $2$-closure $G^{(2)}$ of $G$ is the subgroup of $\\mathrm{Sym}(\\Omega)$ generated by the permutations $\\sigma$ that map every pair of points into its $G$-orbit, that is, such that for all $\\alpha, \\beta \\in \\Omega$ there is $g \\in G$ with $g(\\alpha) = \\sigma(\\alpha)$ and $g(\\beta) = \\sigma(\\beta)$.","title":"The $2$-closure of a permutation group"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/solvable-two-closure-factors-cyclic-or-a","kind":"node","lean":"MathlibExt.GroupTheory.TwoClosureSolvableCompositionFactorsKourovka19Problem67Wanted.solvableTwoClosureFactorsCyclicOrAlternating","parent":2468,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/index.html#solvable-two-closure-factors-cyclic-or-a","scope":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","status":"fully_proved","summary":"The statement that for every finite set $\\Omega$ and every solvable subgroup $G$ of the symmetric group $\\mathrm{Sym}(\\Omega)$, every composition factor of the $2$-closure $G^{(2)}$ of $G$ is cyclic or isomorphic to the alternating group $A_n$ for some natural number $n$.","title":"Kourovka Problem 19.67"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/solvable-two-closure-factors-cyclic-or-a-2","kind":"node","lean":"MathlibExt.GroupTheory.TwoClosureSolvableCompositionFactorsKourovka19Problem67Wanted.solvableTwoClosureFactorsCyclicOrAlternating_refuted","parent":2468,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups/index.html#solvable-two-closure-factors-cyclic-or-a-2","scope":"GroupTheory/permutation-groups/finite-permutation-groups/2-closures-of-permutation-groups","status":"can_prove","summary":"There exist a finite set $\\Omega$, a solvable subgroup $G$ of the symmetric group $\\mathrm{Sym}(\\Omega)$, and a composition factor $S$ of the $2$-closure $G^{(2)}$ of $G$ such that $S$ is not cyclic and $S$ is not isomorphic to the alternating group $A_n$ for any $n$. Here $G^{(2)}$ is the subgroup of $\\mathrm{Sym}(\\Omega)$ generated by the permutations $\\sigma$ such that for all $\\alpha, \\beta \\in \\Omega$ some\u2026","title":"Skresanov's counterexample to Kourovka Problem 19.67"},{"data_url":"scopes/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups.json","id":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","kind":"scope","navigation":"module","parent":2463,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/index.html","scope":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","status":"can_prove","title":"Bases of permutation groups"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/is-base","kind":"node","lean":"MathlibExt.GroupTheory.MinimalBaseIndependentSetKourovka16Problem45Wanted.IsBase","parent":2476,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/index.html#is-base","scope":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","status":"fully_proved","summary":"For a group $G$ acting on a set $\\Omega$, a finite set $B \\subseteq \\Omega$ is a base when its pointwise stabilizer is trivial: the only $g \\in G$ with $g \\cdot \\omega = \\omega$ for all $\\omega \\in B$ is $g = 1$.","title":"Base of a group action"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/is-minimal-base","kind":"node","lean":"MathlibExt.GroupTheory.MinimalBaseIndependentSetKourovka16Problem45Wanted.IsMinimalBase","parent":2476,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/index.html#is-minimal-base","scope":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","status":"fully_proved","summary":"For a group $G$ acting on a set $\\Omega$, a finite set $B \\subseteq \\Omega$ is a minimal base when it is a base, that is, its pointwise stabilizer in $G$ is trivial, and for every $\\omega \\in B$ the set $B \\setminus \\{\\omega\\}$ is not a base.","title":"Minimal base"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/max-minimal-base-size","kind":"node","lean":"MathlibExt.GroupTheory.MinimalBaseIndependentSetKourovka16Problem45Wanted.maxMinimalBaseSize","parent":2476,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/index.html#max-minimal-base-size","scope":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","status":"fully_proved","summary":"For a group $G$, the number $b(G)$ is the supremum in $\\mathbb{N}$ of the sizes of the minimal bases of $G$ over all actions of $G$ on a set $\\Omega$, where a minimal base is a finite set $B \\subseteq \\Omega$ whose pointwise stabilizer in $G$ is trivial while that of $B \\setminus \\{\\omega\\}$ is not, for every $\\omega \\in B$. It is the largest such size when these sizes are bounded, as they are for finite $G$, and\u2026","title":"Maximal size of a minimal base"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/is-independent","kind":"node","lean":"MathlibExt.GroupTheory.MinimalBaseIndependentSetKourovka16Problem45Wanted.IsIndependent","parent":2476,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/index.html#is-independent","scope":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","status":"fully_proved","summary":"A finite subset $S$ of a group $G$ is independent when no element $s \\in S$ lies in the subgroup of $G$ generated by $S \\setminus \\{s\\}$.","title":"Independent subset of a group"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/max-independent-size","kind":"node","lean":"MathlibExt.GroupTheory.MinimalBaseIndependentSetKourovka16Problem45Wanted.maxIndependentSize","parent":2476,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/index.html#max-independent-size","scope":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","status":"fully_proved","summary":"For a group $G$, a finite subset $S \\subseteq G$ is independent when no $s \\in S$ lies in the subgroup generated by $S \\setminus \\{s\\}$. The number $\\mu'(G)$ is the supremum in $\\mathbb{N}$ of the sizes of the finite independent subsets of $G$: their largest size when these sizes are bounded, as they are for finite $G$, and $0$ when they are unbounded.","title":"Maximal size of an independent set"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/max-minimal-base-size-eq-max-independent","kind":"node","lean":"MathlibExt.GroupTheory.MinimalBaseIndependentSetKourovka16Problem45Wanted.maxMinimalBaseSizeEqMaxIndependentSize","parent":2476,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/index.html#max-minimal-base-size-eq-max-independent","scope":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","status":"fully_proved","summary":"The statement that $b(G) = \\mu'(G)$ for every finite group $G$, where $b(G)$ is the largest size of a minimal base of $G$ over all actions of $G$ on a set, and $\\mu'(G)$ is the largest size of an independent subset of $G$ (a subset none of whose elements lies in the subgroup generated by the others).","title":"Cameron's question on minimal bases and independent sets"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/max-minimal-base-size-eq-max-independent-2","kind":"node","lean":"MathlibExt.GroupTheory.MinimalBaseIndependentSetKourovka16Problem45Wanted.maxMinimalBaseSizeEqMaxIndependentSize_refuted","parent":2476,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups/index.html#max-minimal-base-size-eq-max-independent-2","scope":"GroupTheory/permutation-groups/finite-permutation-groups/bases-of-permutation-groups","status":"can_prove","summary":"It is not true that $b(G) = \\mu'(G)$ for every finite group $G$, where $b(G)$ is the largest size of a minimal base of $G$ over all actions of $G$ on a set, and $\\mu'(G)$ is the largest size of an independent subset of $G$.","title":"Kourovka Problem 16.45 has a negative answer"},{"data_url":"scopes/GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions.json","id":"GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions","kind":"scope","navigation":"module","parent":2463,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions/index.html","scope":"GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions","status":"can_prove","title":"Largest orbits of linear group actions"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions/two-largest-orbits-implies-dihedral-eigh","kind":"node","lean":"MathlibExt.GroupTheory.TwoLargestOrbitsDihedralKourovka18Problem44Wanted.twoLargestOrbitsImpliesDihedralEight","parent":2484,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions/index.html#two-largest-orbits-implies-dihedral-eigh","scope":"GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions","status":"fully_proved","summary":"Keller's question on two largest orbits and the dihedral group of order eight (Kourovka Problem 18.44). The statement that for every finite non-abelian group $G$, with derived subgroup $G'$, and every finite faithful irreducible $G$-module $V$ (a nonzero finite abelian group on which $G$ acts faithfully by automorphisms with no $G$-invariant subgroups other than $0$ and $V$), writing $M = |G/G'|$: if every orbit of\u2026","title":"Keller's question on two largest orbits and the dihedral group of order eight (Kourovka Problem 18.44)"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions/two-largest-orbits-implies-dihedral-eigh-2","kind":"node","lean":"MathlibExt.GroupTheory.TwoLargestOrbitsDihedralKourovka18Problem44Wanted.twoLargestOrbitsImpliesDihedralEight_holds","parent":2484,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions/index.html#two-largest-orbits-implies-dihedral-eigh-2","scope":"GroupTheory/permutation-groups/finite-permutation-groups/largest-orbits-of-linear-group-actions","status":"can_prove","summary":"Let $G$ be a finite non-abelian group, let $G'$ be its derived subgroup, and let $V$ be a finite faithful irreducible $G$-module, that is, a nonzero finite abelian group on which $G$ acts faithfully by automorphisms with no $G$-invariant subgroups other than $0$ and $V$. Write $M = |G/G'|$. If every orbit of $G$ on $V$ has at most $M$ elements and exactly two orbits have exactly $M$ elements, then $G$ is isomorphic\u2026","title":"Two largest orbits force the dihedral group of order eight (Kourovka 18.44)"},{"data_url":"scopes/GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups.json","id":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups","kind":"scope","navigation":"module","parent":2463,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/index.html","scope":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups","status":"can_prove","title":"Normal coverings of symmetric groups"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/normal-covering-number","kind":"node","lean":"MathlibExt.GroupTheory.NormalCoveringSymmetricKourovka18Problem23Wanted.normalCoveringNumber","parent":2487,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/index.html#normal-covering-number","scope":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups","status":"fully_proved","summary":"For a natural number $n$, the normal covering number $\\gamma(S_n)$ is the least $k$ for which there are proper subgroups $H_1, \\dots, H_k$ of the symmetric group $S_n$ such that every element of $S_n$ is conjugate to an element of some $H_i$. When no such subgroups exist, as for $n \\le 2$, it is $0$.","title":"Normal covering number of the symmetric group"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/conjectured-value","kind":"node","lean":"MathlibExt.GroupTheory.NormalCoveringSymmetricKourovka18Problem23Wanted.conjecturedValue","parent":2487,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/index.html#conjectured-value","scope":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups","status":"fully_proved","summary":"For a natural number $n \\ge 2$, write $n = p_1^{\\alpha_1} \\cdots p_r^{\\alpha_r}$ with primes $p_1 < \\cdots < p_r$ and exponents $\\alpha_i \\ge 1$, and let $p_2$ be the least prime factor of $n / p_1^{\\alpha_1}$. The conjectured value of $n$ is the rational number $\\frac{n}{2}(1 - \\frac{1}{p_1})$ if $r = 1$ and $\\alpha_1 = 1$; $\\frac{n}{2}(1 - \\frac{1}{p_1}) + 1$ if $r = 1$ and $\\alpha_1 \\ge 2$;\u2026","title":"Bubboloni\u2013Praeger\u2013Spiga conjectured value"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/normal-covering-number-eventually-conjec","kind":"node","lean":"MathlibExt.GroupTheory.NormalCoveringSymmetricKourovka18Problem23Wanted.normalCoveringNumberEventuallyConjectured","parent":2487,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/index.html#normal-covering-number-eventually-conjec","scope":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups","status":"fully_proved","summary":"There is a natural number $N$ such that, for every $n \\ge N$, the normal covering number $\\gamma(S_n)$ of the symmetric group $S_n$ equals the Bubboloni\u2013Praeger\u2013Spiga conjectured value of $n$.","title":"Bubboloni\u2013Praeger\u2013Spiga normal covering conjecture (eventual form)"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/normal-covering-number-eventually-conjec-2","kind":"node","lean":"MathlibExt.GroupTheory.NormalCoveringSymmetricKourovka18Problem23Wanted.normalCoveringNumberEventuallyConjectured_refuted","parent":2487,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups/index.html#normal-covering-number-eventually-conjec-2","scope":"GroupTheory/permutation-groups/finite-permutation-groups/normal-coverings-of-symmetric-groups","status":"can_prove","summary":"It is not the case that there is a natural number $N$ such that, for every $n \\ge N$, the normal covering number $\\gamma(S_n)$ of the symmetric group $S_n$ equals the Bubboloni\u2013Praeger\u2013Spiga conjectured value of $n$. In other words, the conjecture fails for arbitrarily large $n$.","title":"Refutation of the Bubboloni\u2013Praeger\u2013Spiga normal covering conjecture"},{"data_url":"scopes/GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups.json","id":"GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups","kind":"scope","navigation":"module","parent":2463,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups/index.html","scope":"GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups","status":"can_prove","title":"Orders of products in symmetric groups"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups/symmetric-group-has-product-with-orders","kind":"node","lean":"MathlibExt.GroupTheory.SymmetricGroupProductOrdersKourovka18Problem49Wanted.symmetricGroupHasProductWithOrders","parent":2492,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups/index.html#symmetric-group-has-product-with-orders","scope":"GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups","status":"fully_proved","summary":"For every natural number $n$ and all natural numbers $a, b, c$ with $1 < a \\le n - 2$, $1 < b \\le n - 2$ and $1 < c \\le n - 2$, the symmetric group $S_n$ contains permutations $x$ and $y$ such that $x$ has order $a$, $y$ has order $b$ and $xy$ has order $c$.","title":"Kohl's question on orders of products in symmetric groups"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups/symmetric-group-has-product-with-orders-2","kind":"node","lean":"MathlibExt.GroupTheory.SymmetricGroupProductOrdersKourovka18Problem49Wanted.symmetricGroupHasProductWithOrders_holds","parent":2492,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups/index.html#symmetric-group-has-product-with-orders-2","scope":"GroupTheory/permutation-groups/finite-permutation-groups/orders-of-products-in-symmetric-groups","status":"can_prove","summary":"For every natural number $n$ and all natural numbers $a, b, c$ with $1 < a \\le n - 2$, $1 < b \\le n - 2$ and $1 < c \\le n - 2$, the symmetric group $S_n$ contains permutations $x$ and $y$ of orders $a$ and $b$ whose product $xy$ has order $c$.","title":"Permutations of prescribed orders with product of prescribed order"},{"data_url":"scopes/GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions.json","id":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions","kind":"scope","navigation":"module","parent":2463,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/index.html","scope":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions","status":"fully_proved","title":"Poles of group actions"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/poles","kind":"node","lean":"MulAction.poles","parent":2495,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/index.html#poles","scope":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions","status":"fully_proved","summary":"A pole of a group action is a point fixed by some nonidentity element.","title":"Poles of a group action"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/mem-poles","kind":"node","lean":"MulAction.mem_poles","parent":2495,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/index.html#mem-poles","scope":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions","status":"fully_proved","summary":"Let a group $G$ act on a set $X$. A point $p \\in X$ is a pole of the action if and only if there is some $g \\in G$ with $g \\neq 1$ and $g \\cdot p = p$.","title":"Characterization of poles"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/smul-mem-poles","kind":"node","lean":"MulAction.smul_mem_poles","parent":2495,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/index.html#smul-mem-poles","scope":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions","status":"fully_proved","summary":"Let a group $G$ act on a set $X$. If $p \\in X$ is a pole of the action, then $g \\cdot p$ is a pole for every $g \\in G$.","title":"The action sends poles to poles"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/image-smul-poles","kind":"node","lean":"MulAction.image_smul_poles","parent":2495,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/index.html#image-smul-poles","scope":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions","status":"fully_proved","summary":"Let a group $G$ act on a set $X$ and let $P \\subseteq X$ be the set of poles of the action. For every $g \\in G$, the image of $P$ under the map $x \\mapsto g \\cdot x$ is $P$.","title":"The poles are invariant under the action"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/poles-eq-i-union-fixed-by","kind":"node","lean":"MulAction.poles_eq_iUnion_fixedBy","parent":2495,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions/index.html#poles-eq-i-union-fixed-by","scope":"GroupTheory/permutation-groups/finite-permutation-groups/poles-of-group-actions","status":"fully_proved","summary":"Let a group $G$ act on a set $X$. The set of poles equals $\\bigcup_{g \\in G,\\, g \\neq 1} \\{x \\in X : g \\cdot x = x\\}$, the union of the fixed-point sets of the nonidentity elements of $G$.","title":"The poles as a union of fixed-point sets"},{"data_url":"scopes/GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups.json","id":"GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups","kind":"scope","navigation":"module","parent":2463,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups/index.html","scope":"GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups","status":"can_prove","title":"Regular orbits of linear groups"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups/p-regular-orbits-imply-regular-orbit","kind":"node","lean":"MathlibExt.GroupTheory.PRegularOrbitsRegularOrbitKourovka12Problem25Wanted.pRegularOrbitsImplyRegularOrbit","parent":2501,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zhang's question on regular orbits from "],["math","\\(p\\)"],["text","-regular orbits"]],"source":"Zhang's question on regular orbits from $p$-regular orbits"},"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups/index.html#p-regular-orbits-imply-regular-orbit","scope":"GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups","status":"fully_proved","summary":"The statement that for every finite field $F$, every nonzero finite-dimensional vector space $V$ over $F$ and every finite group $G$ acting faithfully on $V$ by $F$-linear maps, if the only $G$-invariant subspaces of $V$ are $0$ and $V$, and for every prime $p$ some vector of $V$ has a stabilizer in $G$ of order prime to $p$ (a $p$-regular orbit), then some vector of $V$ has trivial stabilizer in $G$ (a regular\u2026","title":"Zhang's question on regular orbits from $p$-regular orbits"},{"id":"GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups/p-regular-orbits-imply-regular-orbit-ref","kind":"node","lean":"MathlibExt.GroupTheory.PRegularOrbitsRegularOrbitKourovka12Problem25Wanted.pRegularOrbitsImplyRegularOrbit_refuted","parent":2501,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-regular orbits for every prime do not force a regular orbit"]],"source":"$p$-regular orbits for every prime do not force a regular orbit"},"result":true,"route":"../roadmap/GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups/index.html#p-regular-orbits-imply-regular-orbit-ref","scope":"GroupTheory/permutation-groups/finite-permutation-groups/regular-orbits-of-linear-groups","status":"can_prove","summary":"There exist a finite field $F$, a nonzero finite-dimensional vector space $V$ over $F$ and a finite group $G$ acting faithfully on $V$ by $F$-linear maps such that the only $G$-invariant subspaces of $V$ are $0$ and $V$ and, for every prime $p$, some vector of $V$ has a stabilizer in $G$ of order prime to $p$, but no vector of $V$ has trivial stabilizer in $G$.","title":"$p$-regular orbits for every prime do not force a regular orbit"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups","kind":"scope","parent":2462,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups","status":"can_prove","summary":"Permutation groups on infinite sets: highly homogeneous groups of orders, circular orders and betweenness (Cameron), the pointwise topology on permutations, cofinitary and subdegree-finite groups, infinite sharply or doubly transitive groups, embeddings into the infinite symmetric group, class transpositions of the integers.","title":"Infinite permutation groups"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers","status":"can_prove","title":"Class transpositions of the integers"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/res-class","kind":"node","lean":"MathlibExt.GroupTheory.ClassTranspositionProductOrdersKourovka18Problem48Wanted.resClass","parent":2505,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/index.html#res-class","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers","status":"fully_proved","summary":"For integers $r$ and $m$, the residue class $r(m)$ is the set $\\{x \\in \\mathbb{Z} : x \\equiv r \\pmod{m}\\} = \\{r + k m : k \\in \\mathbb{Z}\\}$.","title":"Residue class"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/is-class-trans","kind":"node","lean":"MathlibExt.GroupTheory.ClassTranspositionProductOrdersKourovka18Problem48Wanted.IsClassTrans","parent":2505,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/index.html#is-class-trans","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers","status":"fully_proved","summary":"For integers $r_1, m_1, r_2, m_2$, write $r_1(m_1) = r_1 + m_1\\mathbb{Z}$ and $r_2(m_2) = r_2 + m_2\\mathbb{Z}$ for the residue classes. A permutation $\\sigma$ of $\\mathbb{Z}$ is the class transposition $\\tau_{r_1(m_1), r_2(m_2)}$ when $0 \\le r_1 < m_1$ and $0 \\le r_2 < m_2$, the residue classes $r_1(m_1)$ and $r_2(m_2)$ are disjoint, $\\sigma$ interchanges $r_1 + k m_1$ and $r_2 + k m_2$ for every integer $k$, and\u2026","title":"Class transposition of two residue classes"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/is-class-transposition","kind":"node","lean":"MathlibExt.GroupTheory.ClassTranspositionProductOrdersKourovka18Problem48Wanted.IsClassTransposition","parent":2505,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/index.html#is-class-transposition","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers","status":"fully_proved","summary":"A permutation $\\sigma$ of $\\mathbb{Z}$ is a class transposition when it is the class transposition $\\tau_{r_1(m_1), r_2(m_2)}$ for some integers $r_1, m_1, r_2, m_2$.","title":"Class transpositions of the integers"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/class-transposition-product-orders-finit","kind":"node","lean":"MathlibExt.GroupTheory.ClassTranspositionProductOrdersKourovka18Problem48Wanted.classTranspositionProductOrdersFinite","parent":2505,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/index.html#class-transposition-product-orders-finit","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers","status":"fully_proved","summary":"Only finitely many positive integers occur as orders of products $\\sigma\\tau$ of two class transpositions $\\sigma$ and $\\tau$ of $\\mathbb{Z}$.","title":"Claim of Kourovka Problem 18.48"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/class-transposition-product-orders-finit-2","kind":"node","lean":"MathlibExt.GroupTheory.ClassTranspositionProductOrdersKourovka18Problem48Wanted.classTranspositionProductOrdersFinite_holds","parent":2505,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers/index.html#class-transposition-product-orders-finit-2","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/class-transpositions-of-the-integers","status":"can_prove","summary":"The claim of Kourovka Problem 18.48 holds: the set of positive integers $k$ such that $\\sigma\\tau$ has order $k$ for some class transpositions $\\sigma$ and $\\tau$ of $\\mathbb{Z}$ is finite.","title":"Products of two class transpositions have only finitely many orders"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group","status":"can_prove","title":"Free groups in the infinite symmetric group"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group/free-group-embeds-finite-orbits","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupPermutationFiniteOrbitsKourovka12Problem10Wanted.freeGroupEmbedsFiniteOrbits","parent":2511,"reading_title":{"format":"inline-parts-v1","parts":[["text","Embedding the free group of rank 2 in "],["math","\\(\\mathrm{Sym}(\\mathbb{N})\\)"],["text"," with finitely many orbits (Kourovka 12.10)"]],"source":"Embedding the free group of rank 2 in $\\mathrm{Sym}(\\mathbb{N})$ with finitely many orbits (Kourovka 12.10)"},"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group/index.html#free-group-embeds-finite-orbits","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group","status":"fully_proved","summary":"Embedding the free group of rank 2 in $\\mathrm{Sym}(\\mathbb{N})$ with finitely many orbits (Kourovka 12.10). The statement, posed by P. Neumann and A. M. W. Glass, that there is an injective group homomorphism $f$ from the free group $F_2$ of rank $2$ into the symmetric group $\\mathrm{Sym}(\\mathbb{N})$ of all permutations of $\\mathbb{N}$ such that, for every non-identity element $w$ of $F_2$, the cyclic subgroup\u2026","title":"Embedding the free group of rank 2 in $\\mathrm{Sym}(\\mathbb{N})$ with finitely many orbits (Kourovka 12.10)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group/free-group-embeds-finite-orbits-holds","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupPermutationFiniteOrbitsKourovka12Problem10Wanted.freeGroupEmbedsFiniteOrbits_holds","parent":2511,"reading_title":{"format":"inline-parts-v1","parts":[["text","The free group of rank 2 embeds in "],["math","\\(\\mathrm{Sym}(\\mathbb{N})\\)"],["text"," with finitely many orbits"]],"source":"The free group of rank 2 embeds in $\\mathrm{Sym}(\\mathbb{N})$ with finitely many orbits"},"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group/index.html#free-group-embeds-finite-orbits-holds","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/free-groups-in-the-infinite-symmetric-group","status":"can_prove","summary":"There is an injective group homomorphism $f$ from the free group $F_2$ of rank $2$ into the symmetric group $\\mathrm{Sym}(\\mathbb{N})$ of all permutations of $\\mathbb{N}$ such that, for every non-identity element $w$ of $F_2$, the cyclic subgroup generated by $f(w)$ has only finitely many orbits on $\\mathbb{N}$. This answers Kourovka Problem 12.10 positively.","title":"The free group of rank 2 embeds in $\\mathrm{Sym}(\\mathbb{N})$ with finitely many orbits"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","title":"Highly homogeneous permutation groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/is-between","kind":"node","lean":"MetaMathlibExt.isBetween","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#is-between","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $<$ and let $a, b, c \\in \\alpha$. Then $b$ lies strictly between $a$ and $c$ when either $a < b$ and $b < c$, or $c < b$ and $b < a$.","title":"Strict betweenness"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/betweenness-aut-subgroup","kind":"node","lean":"MetaMathlibExt.betweennessAutSubgroup","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#betweenness-aut-subgroup","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered set, and say that $b$ lies strictly between $a$ and $c$ when $a < b < c$ or $c < b < a$. The betweenness automorphism group of $\\alpha$ is the subgroup of the group of permutations of $\\alpha$ consisting of the permutations $\\sigma$ such that, for all $a, b, c \\in \\alpha$, $b$ lies strictly between $a$ and $c$ if and only if $\\sigma(b)$ lies strictly between $\\sigma(a)$ and\u2026","title":"The betweenness automorphism group of a linear order"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/circular-order-aut-subgroup","kind":"node","lean":"MetaMathlibExt.circularOrderAutSubgroup","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#circular-order-aut-subgroup","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"Let $\\alpha$ be a circularly ordered set with non-strict circular betweenness relation $\\operatorname{btw}$. The automorphism group of $\\alpha$ is the subgroup of the group of permutations of $\\alpha$ consisting of the permutations $\\sigma$ such that, for all $a, b, c \\in \\alpha$, $\\operatorname{btw}(a, b, c)$ holds if and only if $\\operatorname{btw}(\\sigma(a), \\sigma(b), \\sigma(c))$ holds.","title":"The automorphism group of a circular order"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/is-highly-homogeneous","kind":"node","lean":"MetaMathlibExt.IsHighlyHomogeneous","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#is-highly-homogeneous","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"Let $G$ be a group of permutations of a set $\\alpha$. Then $G$ is highly homogeneous when, for every natural number $k$, it acts transitively on the $k$-element subsets of $\\alpha$: for any two $k$-element subsets $X$ and $Y$ of $\\alpha$ there is some $g \\in G$ with $g(X) = Y$.","title":"Highly homogeneous permutation groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/is-highly-homogeneous-top","kind":"node","lean":"MetaMathlibExt.isHighlyHomogeneous_top","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#is-highly-homogeneous-top","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"For every set $\\alpha$, the group of all permutations of $\\alpha$ is highly homogeneous: for every natural number $k$, any $k$-element subset of $\\alpha$ can be carried to any other $k$-element subset of $\\alpha$ by some permutation of $\\alpha$.","title":"The full symmetric group is highly homogeneous"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/order-aut-subgroup","kind":"node","lean":"MetaMathlibExt.orderAutSubgroup","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#order-aut-subgroup","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"For a type $\\alpha$ equipped with a relation $<$, the order automorphism group of $\\alpha$ is the subgroup of the group of permutations of $\\alpha$ consisting of the permutations $\\sigma$ such that for all $a, b \\in \\alpha$, $a < b$ holds if and only if $\\sigma(a) < \\sigma(b)$ holds.","title":"Order automorphism group"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/pointwise-topology","kind":"node","lean":"Equiv.Perm.pointwiseTopology","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#pointwise-topology","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"For a type $\\alpha$, the pointwise convergence topology on the group of permutations of $\\alpha$ is the topology induced by the map sending a permutation $\\sigma$ to the underlying function $\\sigma \\colon \\alpha \\to \\alpha$, where $\\alpha$ carries the discrete topology and the space of functions $\\alpha \\to \\alpha$ carries the product topology.","title":"Pointwise convergence topology on permutations"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/betweenness-aut-subgroup-is-closed-and-h","kind":"node","lean":"MetaMathlibExt.CameronBetweennessGroupWanted.betweennessAutSubgroup_isClosed_and_highlyHomogeneous","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#betweenness-aut-subgroup-is-closed-and-h","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"Let $G$ be the group of permutations $\\sigma$ of $\\mathbb{Q}$ preserving strict betweenness in both directions: for all $a, b, c \\in \\mathbb{Q}$, $b$ lies strictly between $a$ and $c$ (that is, $a < b < c$ or $c < b < a$) if and only if $\\sigma(b)$ lies strictly between $\\sigma(a)$ and $\\sigma(c)$. Then $G$ is closed in the topology of pointwise convergence on the permutations of $\\mathbb{Q}$, where $\\mathbb{Q}$\u2026","title":"The betweenness group of the rationals is closed and highly homogeneous"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/circular-order-aut-subgroup-is-closed-an","kind":"node","lean":"MetaMathlibExt.CameronCircularOrderGroupWanted.circularOrderAutSubgroup_isClosed_and_highlyHomogeneous","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#circular-order-aut-subgroup-is-closed-an","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"The automorphism group of a countably infinite dense circular order is closed and highly homogeneous. Let $\\alpha$ be a countably infinite set with a circular order, and write $\\operatorname{btw}(a, b, c)$ when $b$ lies between $a$ and $c$ going around from $a$ to $c$. Suppose the circular order is dense: for all distinct $a, b \\in \\alpha$ there is some $c \\in \\alpha$, different from $a$ and $b$, with\u2026","title":"The automorphism group of a countably infinite dense circular order is closed and highly homogeneous"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/order-aut-subgroup-is-closed-and-highly","kind":"node","lean":"MetaMathlibExt.CameronOrderGroupWanted.orderAutSubgroup_isClosed_and_highlyHomogeneous","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#order-aut-subgroup-is-closed-and-highly","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"The group of order automorphisms of $\\mathbb{Q}$, that is, of the permutations $\\sigma$ of $\\mathbb{Q}$ such that $a < b$ if and only if $\\sigma(a) < \\sigma(b)$ for all $a, b \\in \\mathbb{Q}$, is closed in the topology of pointwise convergence on the permutations of $\\mathbb{Q}$, where $\\mathbb{Q}$ carries the discrete topology, and it is highly homogeneous: for every natural number $k$ it has a single orbit on the\u2026","title":"The order-automorphism group of the rationals is closed and highly homogeneous"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/cameron-symmetric-group-closed-highly-ho","kind":"node","lean":"MetaMathlibExt.cameron_symmetricGroup_closed_highlyHomogeneous","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#cameron-symmetric-group-closed-highly-ho","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"Let $\\operatorname{Sym}(\\mathbb{Q})$ be the group of all permutations of the set $\\mathbb{Q}$. Then $\\operatorname{Sym}(\\mathbb{Q})$ is closed in the topology of pointwise convergence on the permutations of $\\mathbb{Q}$, and it is highly homogeneous: for every natural number $k$ it has a single orbit on the $k$-element subsets of $\\mathbb{Q}$.","title":"The symmetric group of the rationals is closed and highly homogeneous"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/circular-separates","kind":"node","lean":"MetaMathlibExt.circularSeparates","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#circular-separates","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"Let $\\alpha$ be a circular order, and write $\\operatorname{sbtw}(x, y, z)$ for its strict betweenness relation ($y$ lies strictly between $x$ and $z$ going around the circle). For $a, b, c, d \\in \\alpha$, the pairs $a, b$ and $c, d$ separate each other when either $\\operatorname{sbtw}(a, c, b)$ and $\\operatorname{sbtw}(b, d, a)$ both hold, or $\\operatorname{sbtw}(a, d, b)$ and $\\operatorname{sbtw}(b, c, a)$ both\u2026","title":"Separation of pairs in a circular order"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/separation-aut-subgroup","kind":"node","lean":"MetaMathlibExt.separationAutSubgroup","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#separation-aut-subgroup","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"For a circular order $\\alpha$, the separation-preserving group of $\\alpha$ is the subgroup of the group of permutations of $\\alpha$ consisting of the permutations $\\sigma$ such that for all $a, b, c, d \\in \\alpha$, the pairs $a, b$ and $c, d$ separate each other if and only if the pairs $\\sigma(a), \\sigma(b)$ and $\\sigma(c), \\sigma(d)$ separate each other.","title":"Separation-preserving permutation group"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/separation-aut-subgroup-is-closed-and-hi","kind":"node","lean":"MetaMathlibExt.CameronSeparationGroupWanted.separationAutSubgroup_isClosed_and_highlyHomogeneous","parent":2514,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups/index.html#separation-aut-subgroup-is-closed-and-hi","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/highly-homogeneous-permutation-groups","status":"fully_proved","summary":"The separation group of a countably infinite dense circular order is closed and highly homogeneous. Let $\\alpha$ be a countably infinite set with a circular order, and write $\\operatorname{btw}(a, b, c)$ when $b$ lies between $a$ and $c$ going around from $a$ to $c$. Suppose the circular order is dense: for all distinct $a, b \\in \\alpha$ there is some $c \\in \\alpha$, different from $a$ and $b$, with\u2026","title":"The separation group of a countably infinite dense circular order is closed and highly homogeneous"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups","status":"can_prove","title":"Cofinitary permutation groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups/is-cofinitary","kind":"node","lean":"MathlibExt.GroupTheory.ClosedCofinitaryLocallyCompactKourovka13Problem28Wanted.IsCofinitary","parent":2529,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups/index.html#is-cofinitary","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups","status":"fully_proved","summary":"A subgroup $G$ of the group of permutations of a set $\\Omega$ is cofinitary when every non-identity element of $G$ fixes only finitely many points of $\\Omega$.","title":"Cofinitary permutation groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups/closed-cofinitary-locally-compact","kind":"node","lean":"MathlibExt.GroupTheory.ClosedCofinitaryLocallyCompactKourovka13Problem28Wanted.closedCofinitaryLocallyCompact","parent":2529,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups/index.html#closed-cofinitary-locally-compact","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups","status":"fully_proved","summary":"For every infinite set $\\Omega$, every cofinitary subgroup $G$ of the group of permutations of $\\Omega$ that is closed in the topology of pointwise convergence is locally compact in the subspace topology.","title":"Claim of Kourovka Problem 13.28"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups/closed-cofinitary-locally-compact-refute","kind":"node","lean":"MathlibExt.GroupTheory.ClosedCofinitaryLocallyCompactKourovka13Problem28Wanted.closedCofinitaryLocallyCompact_refuted","parent":2529,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups/index.html#closed-cofinitary-locally-compact-refute","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/cofinitary-permutation-groups","status":"can_prove","summary":"The claim of Kourovka Problem 13.28 is false: there are an infinite set $\\Omega$ and a subgroup $G$ of the group of permutations of $\\Omega$ such that $G$ is cofinitary (every non-identity element of $G$ fixes only finitely many points of $\\Omega$) and closed in the topology of pointwise convergence, but $G$ is not locally compact in the subspace topology.","title":"Closed cofinitary permutation groups need not be locally compact"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups","status":"can_prove","title":"Distinguishing sets of subdegree-finite groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups/closed-subdegree-finite-distinguishing-s","kind":"node","lean":"MathlibExt.GroupTheory.SubdegreeFiniteDistinguishingSetKourovka19Problem89Wanted.closedSubdegreeFiniteDistinguishingSet","parent":2533,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups/index.html#closed-subdegree-finite-distinguishing-s","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups","status":"fully_proved","summary":"The statement that for every countably infinite set $\\Omega$ and every subgroup $G$ of the symmetric group $\\mathrm{Sym}(\\Omega)$ that is closed in the topology of pointwise convergence and subdegree-finite (for all $a, b \\in \\Omega$ the orbit of $b$ under the stabilizer of $a$ in $G$ is finite), if every non-identity element of $G$ moves infinitely many points of $\\Omega$, then there is a subset\u2026","title":"Kourovka Problem 19.89: Smith's conjecture on distinguishing sets"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups/closed-subdegree-finite-distinguishing-s-2","kind":"node","lean":"MathlibExt.GroupTheory.SubdegreeFiniteDistinguishingSetKourovka19Problem89Wanted.closedSubdegreeFiniteDistinguishingSet_holds","parent":2533,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups/index.html#closed-subdegree-finite-distinguishing-s-2","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/distinguishing-sets-of-subdegree-finite-groups","status":"can_prove","summary":"Let $\\Omega$ be a countably infinite set and let $G$ be a subgroup of the symmetric group $\\mathrm{Sym}(\\Omega)$ that is closed in the topology of pointwise convergence and subdegree-finite: for all $a, b \\in \\Omega$ the orbit of $b$ under the stabilizer of $a$ in $G$ is finite. If every non-identity element of $G$ moves infinitely many points of $\\Omega$, then there is a subset $X \\subseteq \\Omega$ whose setwise\u2026","title":"Lehner's theorem on distinguishing sets of subdegree-finite groups"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups","status":"can_prove","title":"Orbit algebras of permutation groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/inc-mul","kind":"node","lean":"MathlibExt.GroupTheory.ReducedIncidenceAlgebraDomainKourovka13Problem29Wanted.incMul","parent":2536,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/index.html#inc-mul","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups","status":"fully_proved","summary":"Let $\\Omega$ be a set. For functions $f$ and $g$ from the finite subsets of $\\Omega$ to $\\mathbb{Q}$, their product $fg$ is the function on finite subsets given by $(fg)(X) = \\sum_{Y \\subseteq X} f(Y)\\, g(X \\setminus Y)$.","title":"Product of the reduced incidence algebra"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/is-in-a","kind":"node","lean":"MathlibExt.GroupTheory.ReducedIncidenceAlgebraDomainKourovka13Problem29Wanted.IsInA","parent":2536,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/index.html#is-in-a","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups","status":"fully_proved","summary":"Let $\\Omega$ be a set. A function $f$ from the finite subsets of $\\Omega$ to $\\mathbb{Q}$ lies in the algebra $A = \\bigoplus_n V_n$, where $V_n$ consists of the functions vanishing off the $n$-element subsets, when there is a natural number $N$ such that $f(X) = 0$ for every finite subset $X$ of $\\Omega$ with more than $N$ elements.","title":"Elements of the reduced incidence algebra"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/is-invariant","kind":"node","lean":"MathlibExt.GroupTheory.ReducedIncidenceAlgebraDomainKourovka13Problem29Wanted.IsInvariant","parent":2536,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/index.html#is-invariant","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups","status":"fully_proved","summary":"Let $G$ be a group of permutations of a set $\\Omega$. A function $f$ from the finite subsets of $\\Omega$ to $\\mathbb{Q}$ is $G$-invariant when $f(\\sigma(X)) = f(X)$ for every $\\sigma \\in G$ and every finite subset $X$ of $\\Omega$, where $\\sigma(X)$ is the image of $X$ under $\\sigma$.","title":"Invariant functions on finite subsets"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/invariant-incidence-algebra-is-domain","kind":"node","lean":"MathlibExt.GroupTheory.ReducedIncidenceAlgebraDomainKourovka13Problem29Wanted.invariantIncidenceAlgebraIsDomain","parent":2536,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/index.html#invariant-incidence-algebra-is-domain","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups","status":"fully_proved","summary":"The statement that for every infinite set $\\Omega$ and every group $G$ of permutations of $\\Omega$ all of whose orbits are infinite, the following holds: if $f$ and $g$ are nonzero functions from the finite subsets of $\\Omega$ to $\\mathbb{Q}$ that are $G$-invariant ($f(\\sigma(X)) = f(X)$ for all $\\sigma \\in G$ and all finite $X$, and likewise for $g$) and that each vanish on all subsets of size larger than some\u2026","title":"Cameron's conjecture on invariant incidence algebras (Kourovka 13.29)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/invariant-incidence-algebra-is-domain-ho","kind":"node","lean":"MathlibExt.GroupTheory.ReducedIncidenceAlgebraDomainKourovka13Problem29Wanted.invariantIncidenceAlgebraIsDomain_holds","parent":2536,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups/index.html#invariant-incidence-algebra-is-domain-ho","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/orbit-algebras-of-permutation-groups","status":"can_prove","summary":"Let $\\Omega$ be an infinite set and let $G$ be a group of permutations of $\\Omega$ all of whose orbits are infinite. Let $f$ and $g$ be nonzero functions from the finite subsets of $\\Omega$ to $\\mathbb{Q}$ that are $G$-invariant ($f(\\sigma(X)) = f(X)$ for all $\\sigma \\in G$ and all finite $X$, and likewise for $g$) and that each vanish on all subsets of size larger than some bound. Then their product in the reduced\u2026","title":"Cameron's conjecture on invariant incidence algebras holds (Pouzet)"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals","status":"can_prove","title":"Permutation groups isomorphic to the rationals"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals/rational-permutation-group-infinite-disp","kind":"node","lean":"MathlibExt.GroupTheory.RationalPermutationGroupDisplacementKourovka20Problem98Wanted.rationalPermutationGroupInfiniteDisplacement","parent":2542,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals/index.html#rational-permutation-group-infinite-disp","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals","status":"fully_proved","summary":"Suchkov's question on displacements in permutation groups isomorphic to the rationals (Kourovka 20.98). The statement that for every subgroup $G$ of the group of permutations of $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$ that is isomorphic to the additive group $(\\mathbb{Q}, +)$, there is an element $g \\in G$ such that the set of displacements $\\{a - g(a) : a \\in \\mathbb{N}\\}$, computed in $\\mathbb{Z}$, is infinite.","title":"Suchkov's question on displacements in permutation groups isomorphic to the rationals (Kourovka 20.98)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals/rational-permutation-group-infinite-disp-2","kind":"node","lean":"MathlibExt.GroupTheory.RationalPermutationGroupDisplacementKourovka20Problem98Wanted.rationalPermutationGroupInfiniteDisplacement_holds","parent":2542,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals/index.html#rational-permutation-group-infinite-disp-2","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/permutation-groups-isomorphic-to-the-rationals","status":"can_prove","summary":"For every subgroup $G$ of the group of permutations of $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$ that is isomorphic to the additive group $(\\mathbb{Q}, +)$, there is an element $g \\in G$ such that the set of displacements $\\{a - g(a) : a \\in \\mathbb{N}\\}$, computed in $\\mathbb{Z}$, is infinite. That is, the answer to Kourovka Problem 20.98 is yes.","title":"Permutation groups isomorphic to the rationals have infinite displacement (Kourovka 20.98)"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive","status":"can_prove","title":"Point stabilizers of infinite doubly transitive groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/doubly-transitive-infinite-cyclic-stabil","kind":"node","lean":"MathlibExt.GroupTheory.DoublyTransitiveCyclicStabilizerKourovka10Problem63Wanted.doublyTransitiveInfiniteCyclicStabilizer","parent":2545,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/index.html#doubly-transitive-infinite-cyclic-stabil","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive","status":"fully_proved","summary":"The statement that there exist a group $G$ and a faithful action of $G$ on a set $X$ such that for any two ordered pairs of distinct points of $X$ some element of $G$ maps the first to the second, and the stabilizer of some point $x \\in X$ is isomorphic to the infinite cyclic group $\\mathbb{Z}$.","title":"Doubly transitive group with infinite cyclic point stabilizer (Kourovka 10.63)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/doubly-transitive-infinite-cyclic-stabil-2","kind":"node","lean":"MathlibExt.GroupTheory.DoublyTransitiveCyclicStabilizerKourovka10Problem63Wanted.doublyTransitiveInfiniteCyclicStabilizer_refuted","parent":2545,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/index.html#doubly-transitive-infinite-cyclic-stabil-2","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive","status":"can_prove","summary":"If a group $G$ acts faithfully on a set $X$ such that for any two ordered pairs of distinct points of $X$ some element of $G$ maps the first to the second, then no point stabilizer of the action is isomorphic to the infinite cyclic group $\\mathbb{Z}$.","title":"No doubly transitive group has an infinite cyclic point stabilizer (Kourovka 10.63)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/doubly-transitive-nonperiodic-periodic-s","kind":"node","lean":"MathlibExt.GroupTheory.DoublyTransitivePeriodicStabilizerKourovka10Problem64Wanted.doublyTransitiveNonperiodicPeriodicStabilizer","parent":2545,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/index.html#doubly-transitive-nonperiodic-periodic-s","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive","status":"fully_proved","summary":"The statement that there exist a group $G$ and a faithful action of $G$ on a set $X$ such that for any two ordered pairs of distinct points of $X$ some element of $G$ maps the first to the second, $G$ has an element of infinite order, and every element of the stabilizer of some point $x \\in X$ has finite order.","title":"Non-periodic doubly transitive group with periodic point stabilizer (Kourovka 10.64)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/doubly-transitive-nonperiodic-periodic-s-2","kind":"node","lean":"MathlibExt.GroupTheory.DoublyTransitivePeriodicStabilizerKourovka10Problem64Wanted.doublyTransitiveNonperiodicPeriodicStabilizer_holds","parent":2545,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive/index.html#doubly-transitive-nonperiodic-periodic-s-2","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/point-stabilizers-of-infinite-doubly-transitive","status":"can_prove","summary":"There exist a group $G$ with an element of infinite order and a faithful action of $G$ on a set $X$ such that for any two ordered pairs of distinct points of $X$ some element of $G$ maps the first to the second, and every element of the stabilizer of some point $x \\in X$ has finite order.","title":"A non-periodic doubly transitive group with a periodic point stabilizer (Kourovka 10.64)"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups","status":"can_prove","title":"Sharply 2-transitive groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups/is-sharply-two-transitive","kind":"node","lean":"MathlibExt.GroupTheory.SharplyTwoTransitiveAbelianNormalKourovka11Problem52Wanted.IsSharplyTwoTransitive","parent":2550,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups/index.html#is-sharply-two-transitive","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups","status":"fully_proved","summary":"An action of a group $G$ on a set $\\Omega$ is sharply doubly transitive when for all points $\\alpha, \\beta, \\gamma, \\delta \\in \\Omega$ with $\\alpha \\ne \\beta$ and $\\gamma \\ne \\delta$ there is exactly one $g \\in G$ with $g \\cdot \\alpha = \\gamma$ and $g \\cdot \\beta = \\delta$.","title":"Sharply doubly transitive action"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups/sharply-two-transitive-has-abelian-norma","kind":"node","lean":"MathlibExt.GroupTheory.SharplyTwoTransitiveAbelianNormalKourovka11Problem52Wanted.sharplyTwoTransitiveHasAbelianNormal","parent":2550,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups/index.html#sharply-two-transitive-has-abelian-norma","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups","status":"fully_proved","summary":"The statement that every group $G$ acting sharply doubly transitively on a set $\\Omega$ with at least two points has a nontrivial abelian normal subgroup.","title":"Mazurov's question on sharply 2-transitive groups (Kourovka 11.52)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups/sharply-two-transitive-has-abelian-norma-2","kind":"node","lean":"MathlibExt.GroupTheory.SharplyTwoTransitiveAbelianNormalKourovka11Problem52Wanted.sharplyTwoTransitiveHasAbelianNormal_refuted","parent":2550,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups/index.html#sharply-two-transitive-has-abelian-norma-2","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/sharply-2-transitive-groups","status":"can_prove","summary":"The statement of Kourovka Problem 11.52 is false: there is a group $G$ acting sharply doubly transitively on a set $\\Omega$ with at least two points such that no nontrivial normal subgroup of $G$ is abelian.","title":"A sharply 2-transitive group without nontrivial abelian normal subgroups"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group","status":"can_prove","title":"Subgroups of the infinite symmetric group"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group/real-add-group-embeds-in-perm-nat","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook95Wanted.real_addGroup_embeds_in_perm_nat","parent":2554,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group/index.html#real-add-group-embeds-in-perm-nat","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group","status":"can_prove","summary":"There is an injective group homomorphism from the additive group $(\\mathbb{R}, +)$ of real numbers into the group $S_\\infty$ of all permutations of the natural numbers.","title":"The additive group of the reals embeds in the infinite symmetric group"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group/separable-metrizable-group-embeds-in-sym","kind":"node","lean":"MathlibExt.GroupTheory.SeparableMetrizableGroupSymEmbeddingKourovka18Problem36Wanted.separableMetrizableGroupEmbedsInSym","parent":2554,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group/index.html#separable-metrizable-group-embeds-in-sym","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group","status":"fully_proved","summary":"The claim (J. D. Dixon) that every group $G$ that admits a metrizable topology making it a separable topological group admits an injective group homomorphism into the group $\\mathrm{Sym}(\\mathbb{N})$ of all permutations of the natural numbers.","title":"Claim of Kourovka Problem 18.36"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group/separable-metrizable-group-embeds-in-sym-2","kind":"node","lean":"MathlibExt.GroupTheory.SeparableMetrizableGroupSymEmbeddingKourovka18Problem36Wanted.separableMetrizableGroupEmbedsInSym_refuted","parent":2554,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group/index.html#separable-metrizable-group-embeds-in-sym-2","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/subgroups-of-the-infinite-symmetric-group","status":"can_prove","summary":"The claim of Kourovka Problem 18.36 is false: there is a group $G$ that admits a metrizable topology making it a separable topological group but admits no injective group homomorphism into the group $\\mathrm{Sym}(\\mathbb{N})$ of all permutations of the natural numbers.","title":"Separable metrizable groups need not embed in the infinite symmetric group"},{"data_url":"scopes/GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe.json","id":"GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe","kind":"scope","navigation":"module","parent":2504,"result":false,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe/index.html","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe","status":"can_prove","title":"Transitive automorphism groups of elementary abelian groups"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe/is-locally-finite-field","kind":"node","lean":"MathlibExt.GroupTheory.ElementaryAbelianTransitiveFieldActionKourovka18Problem60Wanted.IsLocallyFiniteField","parent":2558,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe/index.html#is-locally-finite-field","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe","status":"fully_proved","summary":"A field $P$ is locally finite when the subfield generated by any finite subset of $P$ is finite.","title":"Locally finite field (Kourovka 18.60)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe/no-transitive-field-like-automorphism-gr","kind":"node","lean":"MathlibExt.GroupTheory.ElementaryAbelianTransitiveFieldActionKourovka18Problem60Wanted.noTransitiveFieldLikeAutomorphismGroup","parent":2558,"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe/index.html#no-transitive-field-like-automorphism-gr","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe","status":"fully_proved","summary":"The conjecture of D. V. Lytkina, as a proposition: for every countably infinite elementary abelian $2$-group $V$ (an abelian group with $v + v = 0$ for all $v \\in V$) there is no subgroup $G$ of the symmetric group $\\mathrm{Sym}(V)$ consisting of additive bijections such that (a) $G$ is transitive on $V \\setminus \\{0\\}$, and (b) for every nonzero $v \\in V$, writing $H = G_v$ for the stabilizer of $v$ in $G$, there\u2026","title":"Kourovka Problem 18.60 (field-like transitive automorphism groups)"},{"id":"GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe/no-transitive-field-like-automorphism-gr-2","kind":"node","lean":"MathlibExt.GroupTheory.ElementaryAbelianTransitiveFieldActionKourovka18Problem60Wanted.noTransitiveFieldLikeAutomorphismGroup_holds","parent":2558,"reading_title":{"format":"inline-parts-v1","parts":[["text","No field-like transitive automorphism group of a countably infinite elementary abelian "],["math","\\(2\\)"],["text","-group"]],"source":"No field-like transitive automorphism group of a countably infinite elementary abelian $2$-group"},"result":true,"route":"../roadmap/GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe/index.html#no-transitive-field-like-automorphism-gr-2","scope":"GroupTheory/permutation-groups/infinite-permutation-groups/transitive-automorphism-groups-of-elementary-abe","status":"can_prove","summary":"No field-like transitive automorphism group of a countably infinite elementary abelian $2$-group. The conjecture of Kourovka Problem 18.60 holds: for every countably infinite elementary abelian $2$-group $V$ (an abelian group with $v + v = 0$ for all $v \\in V$) there is no subgroup $G$ of the symmetric group $\\mathrm{Sym}(V)$ consisting of additive bijections such that (a) $G$ is transitive on $V \\setminus \\{0\\}$,\u2026","title":"No field-like transitive automorphism group of a countably infinite elementary abelian $2$-group"},{"data_url":"scopes/GroupTheory/group-rings.json","id":"GroupTheory/group-rings","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/group-rings/index.html","scope":"GroupTheory/group-rings","status":"can_prove","summary":"Group rings and group algebras: unit and zero-divisor questions (Kaplansky unit conjecture), Zassenhaus conjectures, augmentation and dimension subgroups, Ore conditions, Jordan decomposition, projective modules over group rings.","title":"Group rings"},{"id":"GroupTheory/group-rings/augmentation-right-ideal","kind":"node","lean":"MathlibExt.GroupTheory.GroupRingRightIdealShared.augmentationRightIdeal","parent":2562,"result":true,"route":"../roadmap/GroupTheory/group-rings/index.html#augmentation-right-ideal","scope":"GroupTheory/group-rings","status":"fully_proved","summary":"Let $G$ be a group, $S \\subseteq G$ a subset and $\\mathbb{Z}G$ the integral group ring of $G$. This is the right ideal of $\\mathbb{Z}G$ generated by the elements $g - 1$ with $g \\in S$, regarded as a right $\\mathbb{Z}G$-submodule of $\\mathbb{Z}G$.","title":"Augmentation right ideal of a subset"},{"data_url":"scopes/GroupTheory/group-rings/dimension-subgroups.json","id":"GroupTheory/group-rings/dimension-subgroups","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/dimension-subgroups/index.html","scope":"GroupTheory/group-rings/dimension-subgroups","status":"can_prove","title":"Dimension subgroups"},{"id":"GroupTheory/group-rings/dimension-subgroups/augmentation-ideal","kind":"node","lean":"MathlibExt.GroupTheory.DimensionSubgroupLowerCentralKourovka12Problem22Wanted.augmentationIdeal","parent":2564,"result":true,"route":"../roadmap/GroupTheory/group-rings/dimension-subgroups/index.html#augmentation-ideal","scope":"GroupTheory/group-rings/dimension-subgroups","status":"fully_proved","summary":"For a group $G$, the augmentation ideal $\\Delta(G)$ of the integral group ring $\\mathbb{Z}G$ is the $\\mathbb{Z}$-submodule spanned by the elements $g - 1$ for $g \\in G$. It is the kernel of the augmentation map $\\mathbb{Z}G \\to \\mathbb{Z}$, $\\sum_g a_g g \\mapsto \\sum_g a_g$.","title":"Augmentation ideal of the integral group ring"},{"id":"GroupTheory/group-rings/dimension-subgroups/mem-dimension-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.DimensionSubgroupLowerCentralKourovka12Problem22Wanted.MemDimensionSubgroup","parent":2564,"result":true,"route":"../roadmap/GroupTheory/group-rings/dimension-subgroups/index.html#mem-dimension-subgroup","scope":"GroupTheory/group-rings/dimension-subgroups","status":"fully_proved","summary":"Let $\\Delta(G)$ be the augmentation ideal of the integral group ring $\\mathbb{Z}G$ of a group $G$, and for $n \\in \\mathbb{N}$ let $\\Delta(G)^n$ be its $n$-th power, the $\\mathbb{Z}$-span of all products of $n$ elements of $\\Delta(G)$. An element $g \\in G$ lies in the $n$-th dimension subgroup $D_n(G) = G \\cap (1 + \\Delta(G)^n)$ when $g - 1 \\in \\Delta(G)^n$.","title":"Dimension subgroups"},{"id":"GroupTheory/group-rings/dimension-subgroups/dimension-quotient-central","kind":"node","lean":"MathlibExt.GroupTheory.DimensionSubgroupLowerCentralKourovka12Problem22Wanted.dimensionQuotientCentral","parent":2564,"result":true,"route":"../roadmap/GroupTheory/group-rings/dimension-subgroups/index.html#dimension-quotient-central","scope":"GroupTheory/group-rings/dimension-subgroups","status":"fully_proved","summary":"The statement that for every group $G$, every $n \\ge 1$, every element $x$ of the $n$-th dimension subgroup $D_n(G)$ and every $g \\in G$, the commutator $x g x^{-1} g^{-1}$ lies in $\\gamma_n(G)$, the $n$-th term of the lower central series of $G$, indexed so that $\\gamma_1(G) = G$ and $\\gamma_{k+1}(G) = [\\gamma_k(G), G]$. Equivalently, the image of $D_n(G)$ in $G/\\gamma_n(G)$ is central.","title":"Gupta\u2013Kuz'min centrality question (Kourovka 12.22(a))"},{"id":"GroupTheory/group-rings/dimension-subgroups/dimension-quotient-exponent-two","kind":"node","lean":"MathlibExt.GroupTheory.DimensionSubgroupLowerCentralKourovka12Problem22Wanted.dimensionQuotientExponentTwo","parent":2564,"result":true,"route":"../roadmap/GroupTheory/group-rings/dimension-subgroups/index.html#dimension-quotient-exponent-two","scope":"GroupTheory/group-rings/dimension-subgroups","status":"fully_proved","summary":"The statement that for every group $G$, every $n \\ge 1$ and every element $x$ of the $n$-th dimension subgroup $D_n(G)$, the square $x^2$ lies in $\\gamma_n(G)$, the $n$-th term of the lower central series of $G$, indexed so that $\\gamma_1(G) = G$ and $\\gamma_{k+1}(G) = [\\gamma_k(G), G]$. Equivalently, the image of $D_n(G)$ in $G/\\gamma_n(G)$ has exponent dividing $2$.","title":"Gupta\u2013Kuz'min exponent question (Kourovka 12.22(b))"},{"id":"GroupTheory/group-rings/dimension-subgroups/dimension-quotient-central-refuted","kind":"node","lean":"MathlibExt.GroupTheory.DimensionSubgroupLowerCentralKourovka12Problem22Wanted.dimensionQuotientCentral_refuted","parent":2564,"result":true,"route":"../roadmap/GroupTheory/group-rings/dimension-subgroups/index.html#dimension-quotient-central-refuted","scope":"GroupTheory/group-rings/dimension-subgroups","status":"can_prove","summary":"There exist a group $G$, an integer $n \\ge 1$, an element $x$ of the $n$-th dimension subgroup $D_n(G)$ and an element $g \\in G$ such that the commutator $x g x^{-1} g^{-1}$ does not lie in $\\gamma_n(G)$, the $n$-th term of the lower central series of $G$ (indexed so that $\\gamma_1(G) = G$). That is, the answer to Kourovka Problem 12.22(a) is negative: the image of $D_n(G)$ in $G/\\gamma_n(G)$ need not be central.","title":"Dimension quotients need not be central"},{"id":"GroupTheory/group-rings/dimension-subgroups/dimension-quotient-exponent-two-refuted","kind":"node","lean":"MathlibExt.GroupTheory.DimensionSubgroupLowerCentralKourovka12Problem22Wanted.dimensionQuotientExponentTwo_refuted","parent":2564,"result":true,"route":"../roadmap/GroupTheory/group-rings/dimension-subgroups/index.html#dimension-quotient-exponent-two-refuted","scope":"GroupTheory/group-rings/dimension-subgroups","status":"can_prove","summary":"There exist a group $G$, an integer $n \\ge 1$ and an element $x$ of the $n$-th dimension subgroup $D_n(G)$ such that $x^2$ does not lie in $\\gamma_n(G)$, the $n$-th term of the lower central series of $G$ (indexed so that $\\gamma_1(G) = G$). That is, the answer to Kourovka Problem 12.22(b) is negative: the image of $D_n(G)$ in $G/\\gamma_n(G)$ need not have exponent dividing $2$.","title":"Dimension quotients need not have exponent dividing 2"},{"data_url":"scopes/GroupTheory/group-rings/group-ring-powers-in-fields.json","id":"GroupTheory/group-rings/group-ring-powers-in-fields","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/group-ring-powers-in-fields/index.html","scope":"GroupTheory/group-rings/group-ring-powers-in-fields","status":"can_prove","title":"Group-ring powers in fields"},{"id":"GroupTheory/group-rings/group-ring-powers-in-fields/subfield-contains-infinitely-many-invari","kind":"node","lean":"MathlibExt.GroupTheory.FieldAutomorphismGroupRingKourovka11Problem79Wanted.subfieldContainsInfinitelyManyInvariants","parent":2571,"result":true,"route":"../roadmap/GroupTheory/group-rings/group-ring-powers-in-fields/index.html#subfield-contains-infinitely-many-invari","scope":"GroupTheory/group-rings/group-ring-powers-in-fields","status":"fully_proved","summary":"The proposition that the following holds for every prime $p$, every infinite field $F$ of characteristic $p$, every finite group $G$ of ring automorphisms of $F$, and every element $\\xi = \\sum_{g \\in G} \\xi_g \\, g$ of the group ring $\\mathbb{Z}G$ with $\\xi \\notin p\\mathbb{Z}G$, that is, with some integer coefficient $\\xi_g$ not divisible by $p$. For nonzero $x \\in F$ put $x^\\xi = \\prod_{g \\in G} g(x)^{\\xi_g}$. If a\u2026","title":"Group-ring powers and invariant elements (Kourovka 11.79)"},{"id":"GroupTheory/group-rings/group-ring-powers-in-fields/subfield-contains-infinitely-many-invari-2","kind":"node","lean":"MathlibExt.GroupTheory.FieldAutomorphismGroupRingKourovka11Problem79Wanted.subfieldContainsInfinitelyManyInvariants_holds","parent":2571,"result":true,"route":"../roadmap/GroupTheory/group-rings/group-ring-powers-in-fields/index.html#subfield-contains-infinitely-many-invari-2","scope":"GroupTheory/group-rings/group-ring-powers-in-fields","status":"can_prove","summary":"Let $p$ be a prime, $F$ an infinite field of characteristic $p$, $G$ a finite group of ring automorphisms of $F$, and $\\xi = \\sum_{g \\in G} \\xi_g \\, g$ an element of the group ring $\\mathbb{Z}G$ with $\\xi \\notin p\\mathbb{Z}G$, that is, with some integer coefficient $\\xi_g$ not divisible by $p$. For nonzero $x \\in F$ put $x^\\xi = \\prod_{g \\in G} g(x)^{\\xi_g}$. If a subfield $K$ of $F$ contains $x^\\xi$ for every\u2026","title":"Subfields containing all group-ring powers have infinitely many invariants"},{"data_url":"scopes/GroupTheory/group-rings/inversion-on-commutative-group-rings.json","id":"GroupTheory/group-rings/inversion-on-commutative-group-rings","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/inversion-on-commutative-group-rings/index.html","scope":"GroupTheory/group-rings/inversion-on-commutative-group-rings","status":"fully_proved","title":"Inversion on commutative group rings"},{"id":"GroupTheory/group-rings/inversion-on-commutative-group-rings/inversion","kind":"node","lean":"GroupRing.inversion","parent":2574,"result":true,"route":"../roadmap/GroupTheory/group-rings/inversion-on-commutative-group-rings/index.html#inversion","scope":"GroupTheory/group-rings/inversion-on-commutative-group-rings","status":"fully_proved","summary":"Let $R$ be a commutative ring and $G$ a commutative group. The inversion of the group ring $R[G]$ is the ring automorphism of $R[G]$ induced by the group automorphism $g \\mapsto g^{-1}$ of $G$: it sends $\\sum_g r_g\\, g$ to $\\sum_g r_g\\, g^{-1}$.","title":"Inversion on a group ring"},{"id":"GroupTheory/group-rings/inversion-on-commutative-group-rings/inversion-single","kind":"node","lean":"GroupRing.inversion_single","parent":2574,"result":true,"route":"../roadmap/GroupTheory/group-rings/inversion-on-commutative-group-rings/index.html#inversion-single","scope":"GroupTheory/group-rings/inversion-on-commutative-group-rings","status":"fully_proved","summary":"Let $R$ be a commutative ring and $G$ a commutative group. For every $g \\in G$ and $r \\in R$, the inversion of the group ring $R[G]$ sends $r\\, g$ to $r\\, g^{-1}$.","title":"Inversion on basis elements"},{"id":"GroupTheory/group-rings/inversion-on-commutative-group-rings/inversion-involution","kind":"node","lean":"GroupRing.inversion_involution","parent":2574,"result":true,"route":"../roadmap/GroupTheory/group-rings/inversion-on-commutative-group-rings/index.html#inversion-involution","scope":"GroupTheory/group-rings/inversion-on-commutative-group-rings","status":"fully_proved","summary":"Let $R$ be a commutative ring, $G$ a commutative group and $\\iota$ the inversion of the group ring $R[G]$. For every $x \\in R[G]$, $\\iota(\\iota(x)) = x$.","title":"Group-ring inversion is an involution"},{"id":"GroupTheory/group-rings/inversion-on-commutative-group-rings/inversion-twist","kind":"node","lean":"GroupRing.InversionTwist","parent":2574,"result":true,"route":"../roadmap/GroupTheory/group-rings/inversion-on-commutative-group-rings/index.html#inversion-twist","scope":"GroupTheory/group-rings/inversion-on-commutative-group-rings","status":"fully_proved","summary":"Let $R$ be a commutative ring, $G$ a commutative group and $A$ a module over the group ring $R[G]$. The inversion twist of $A$ is a copy of $A$ with the same addition, on which $r \\in R[G]$ acts by $a \\mapsto \\iota(r) \\cdot a$, where $\\iota$ is the inversion of $R[G]$ and $\\cdot$ is the original action on $A$.","title":"Inversion twist of a group-ring module"},{"id":"GroupTheory/group-rings/inversion-on-commutative-group-rings/equiv","kind":"node","lean":"GroupRing.InversionTwist.equiv","parent":2574,"result":true,"route":"../roadmap/GroupTheory/group-rings/inversion-on-commutative-group-rings/index.html#equiv","scope":"GroupTheory/group-rings/inversion-on-commutative-group-rings","status":"fully_proved","summary":"For every type $A$, the map sending an element of the inversion twist of $A$ to its underlying element of $A$ is a bijection, whose inverse sends $a \\in A$ to its copy in the twist.","title":"Underlying bijection of the inversion twist"},{"id":"GroupTheory/group-rings/inversion-on-commutative-group-rings/val-smul","kind":"node","lean":"GroupRing.InversionTwist.val_smul","parent":2574,"result":true,"route":"../roadmap/GroupTheory/group-rings/inversion-on-commutative-group-rings/index.html#val-smul","scope":"GroupTheory/group-rings/inversion-on-commutative-group-rings","status":"fully_proved","summary":"Let $R$ be a commutative ring, $G$ a commutative group and $A$ a module over the group ring $R[G]$. For every $r \\in R[G]$ and every element $a$ of the inversion twist of $A$, with underlying element $\\bar a \\in A$, the underlying element of $r \\cdot a$ is $\\iota(r) \\cdot \\bar a$, where $\\iota$ is the inversion of $R[G]$ and $\\cdot$ on the right is the original action on $A$.","title":"Scalar action on the inversion twist"},{"id":"GroupTheory/group-rings/inversion-on-commutative-group-rings/twist-twist-equiv","kind":"node","lean":"GroupRing.InversionTwist.twistTwistEquiv","parent":2574,"result":true,"route":"../roadmap/GroupTheory/group-rings/inversion-on-commutative-group-rings/index.html#twist-twist-equiv","scope":"GroupTheory/group-rings/inversion-on-commutative-group-rings","status":"fully_proved","summary":"Let $R$ be a commutative ring, $G$ a commutative group and $A$ a module over the group ring $R[G]$. The map sending an element of the inversion twist of the inversion twist of $A$ to its underlying element of $A$ is an $R[G]$-linear isomorphism.","title":"Twisting twice by inversion recovers the module"},{"data_url":"scopes/GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings.json","id":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/index.html","scope":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings","status":"can_prove","title":"Jordan decomposition in integral group rings"},{"id":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/has-integral-coeffs","kind":"node","lean":"MathlibExt.GroupTheory.IntegralGroupRingJordanDecompositionKourovka16Problem104Wanted.HasIntegralCoeffs","parent":2582,"result":true,"route":"../roadmap/GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/index.html#has-integral-coeffs","scope":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings","status":"fully_proved","summary":"An element $x$ of the monoid algebra $\\mathbb{Q}[G]$ lies in $\\mathbb{Z}[G]$ when all its coefficients are integers: for every $g \\in G$, the coefficient of $x$ at $g$ is an integer.","title":"Elements with integer coefficients"},{"id":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/has-ajd","kind":"node","lean":"MathlibExt.GroupTheory.IntegralGroupRingJordanDecompositionKourovka16Problem104Wanted.HasAJD","parent":2582,"result":true,"route":"../roadmap/GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/index.html#has-ajd","scope":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings","status":"fully_proved","summary":"For a group $G$, let $\\mathbb{Z}[G]$ be the set of elements of the rational group algebra $\\mathbb{Q}[G]$ all of whose coefficients are integers. Then $\\mathbb{Z}[G]$ has the additive Jordan decomposition property (AJD) when, for all $a, s, n \\in \\mathbb{Q}[G]$ such that $a \\in \\mathbb{Z}[G]$, $a = s + n$, $sn = ns$, $n$ is nilpotent and the minimal polynomial of $s$ over $\\mathbb{Q}$ is squarefree, both $s$ and\u2026","title":"Additive Jordan decomposition property"},{"id":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/has-mjd","kind":"node","lean":"MathlibExt.GroupTheory.IntegralGroupRingJordanDecompositionKourovka16Problem104Wanted.HasMJD","parent":2582,"result":true,"route":"../roadmap/GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/index.html#has-mjd","scope":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings","status":"fully_proved","summary":"For a group $G$, let $\\mathbb{Z}[G]$ be the set of elements of the rational group algebra $\\mathbb{Q}[G]$ all of whose coefficients are integers. Then $\\mathbb{Z}[G]$ has the multiplicative Jordan decomposition property (MJD) when, for all $a, b, s, u \\in \\mathbb{Q}[G]$ such that $a, b \\in \\mathbb{Z}[G]$, $ab = ba = 1$, $a = su$, $su = us$, $u - 1$ is nilpotent and the minimal polynomial of $s$ over $\\mathbb{Q}$ is\u2026","title":"Multiplicative Jordan decomposition property"},{"id":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/finitely-many-two-groups-mjdnot-ajd","kind":"node","lean":"MathlibExt.GroupTheory.IntegralGroupRingJordanDecompositionKourovka16Problem104Wanted.finitelyManyTwoGroupsMJDNotAJD","parent":2582,"result":true,"route":"../roadmap/GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/index.html#finitely-many-two-groups-mjdnot-ajd","scope":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings","status":"fully_proved","summary":"The proposition that there is a natural number $B$ such that every finite $2$-group $G$ whose integral group ring $\\mathbb{Z}[G]$ has the multiplicative Jordan decomposition property (MJD) but not the additive Jordan decomposition property (AJD) has order at most $B$.","title":"Kourovka Problem 16.104"},{"id":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/finitely-many-two-groups-mjdnot-ajd-hold","kind":"node","lean":"MathlibExt.GroupTheory.IntegralGroupRingJordanDecompositionKourovka16Problem104Wanted.finitelyManyTwoGroupsMJDNotAJD_holds","parent":2582,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finitely many "],["math","\\(2\\)"],["text","-groups have MJD but not AJD"]],"source":"Finitely many $2$-groups have MJD but not AJD"},"result":true,"route":"../roadmap/GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings/index.html#finitely-many-two-groups-mjdnot-ajd-hold","scope":"GroupTheory/group-rings/jordan-decomposition-in-integral-group-rings","status":"can_prove","summary":"There is a natural number $B$ such that every finite $2$-group $G$ whose integral group ring $\\mathbb{Z}[G]$ has the multiplicative Jordan decomposition property (MJD) but not the additive Jordan decomposition property (AJD) has order at most $B$. In particular, only finitely many isomorphism classes of finite $2$-groups have MJD but not AJD, which answers Kourovka Problem 16.104 in the affirmative.","title":"Finitely many $2$-groups have MJD but not AJD"},{"data_url":"scopes/GroupTheory/group-rings/kaplansky-s-unit-conjecture.json","id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"can_prove","title":"Kaplansky's unit conjecture"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/generators","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.generators","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#generators","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"This is the two-element set consisting of the distinct formal symbols $a$ and $b$, which serve as the generators of the group presentation in Theorem A of Gardam's counterexample to the unit conjecture.","title":"Generators of Gardam's presentation"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/a","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.a","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#a","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"$a$ is the generator $a$, viewed as an element of the free group on the two generators $a$ and $b$.","title":"First free generator of Gardam's presentation"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/b","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.b","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#b","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"$b$ is the generator $b$, viewed as an element of the free group on the two generators $a$ and $b$.","title":"Second free generator of Gardam's presentation"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/relations","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.relations","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#relations","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"In the free group on the generators $a$ and $b$, the relators are the two elements $b^{-1}a^2ba^2$ and $a^{-1}b^2ab^2$.","title":"Relators of Gardam's group"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/p","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.P","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#p","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"$P$ is the group with generators $a$ and $b$ and relators $b^{-1}a^2ba^2$ and $a^{-1}b^2ab^2$, that is, $P = \\langle a, b \\mid b^{-1}a^2b = a^{-2}, a^{-1}b^2a = b^{-2} \\rangle$.","title":"Gardam's group"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/x","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.x","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#x","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"In the group $P$ with presentation $\\langle a, b \\mid b^{-1}a^2b = a^{-2},\\ a^{-1}b^2a = b^{-2} \\rangle$, $x = a^2$.","title":"The element x of Gardam's group"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/y","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.y","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#y","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"In the group $P$ with presentation $\\langle a, b \\mid b^{-1}a^2b = a^{-2},\\ a^{-1}b^2a = b^{-2} \\rangle$, $y = b^2$.","title":"The element y of Gardam's group"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/z","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.z","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#z","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"In the group $P$ with presentation $\\langle a, b \\mid b^{-1}a^2b = a^{-2},\\ a^{-1}b^2a = b^{-2} \\rangle$, $z = (ab)^2$.","title":"The element z of Gardam's group"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/r","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.R","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#r","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"$\\mathbb{F}_2[P]$ is the group ring of Gardam's group $P$ over the field $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$.","title":"Group ring of Gardam's group over the field with two elements"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/p-2","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.p","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#p-2","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"In the group ring $\\mathbb{F}_2[P]$ of Gardam's group $P$ with generators $a$ and $b$, let $x = a^2$, $y = b^2$ and $z = (ab)^2$ in $P$, viewed as elements of $\\mathbb{F}_2[P]$. Then $p$ is the element $(1 + x)(1 + y)(1 + z^{-1})$ of $\\mathbb{F}_2[P]$.","title":"Gardam's element p"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/q","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.q","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#q","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"In the group ring $\\mathbb{F}_2[P]$ of Gardam's group $P$ with generators $a$ and $b$, let $x = a^2$, $y = b^2$ and $z = (ab)^2$ in $P$, viewed as elements of $\\mathbb{F}_2[P]$. Then $q$ is the element $x^{-1}y^{-1} + x + y^{-1}z + z$ of $\\mathbb{F}_2[P]$.","title":"Gardam's element q"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/r-2","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.r","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#r-2","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"In the group ring $\\mathbb{F}_2[P]$ of Gardam's group $P$ with generators $a$ and $b$, let $x = a^2$, $y = b^2$ and $z = (ab)^2$ in $P$, viewed as elements of $\\mathbb{F}_2[P]$. Then $r$ is the element $1 + x + y^{-1}z + xyz$ of $\\mathbb{F}_2[P]$.","title":"Gardam's element r"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/s","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.s","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#s","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"In the group ring $\\mathbb{F}_2[P]$ of Gardam's group $P$ with generators $a$ and $b$, let $x = a^2$, $y = b^2$ and $z = (ab)^2$ in $P$, viewed as elements of $\\mathbb{F}_2[P]$. Then $s$ is the element $1 + (x + x^{-1} + y + y^{-1})z^{-1}$ of $\\mathbb{F}_2[P]$.","title":"Gardam's element s"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/u","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.u","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#u","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"fully_proved","summary":"Let $P$ be the group with presentation $\\langle a, b \\mid b^{-1}a^2b = a^{-2},\\ a^{-1}b^2a = b^{-2} \\rangle$ and set $x = a^2$, $y = b^2$, $z = (ab)^2$. In the group ring $\\mathbb{F}_2[P]$, $u = p + qa + rb + sab$, where $p = (1 + x)(1 + y)(1 + z^{-1})$, $q = x^{-1}y^{-1} + x + y^{-1}z + z$, $r = 1 + x + y^{-1}z + xyz$ and $s = 1 + (x + x^{-1} + y + y^{-1})z^{-1}$.","title":"Gardam's unit"},{"id":"GroupTheory/group-rings/kaplansky-s-unit-conjecture/theorem-a","kind":"node","lean":"MathlibExt.GroupTheory.UnitConjectureWanted.UnitConjecture.theorem_A","parent":2588,"result":true,"route":"../roadmap/GroupTheory/group-rings/kaplansky-s-unit-conjecture/index.html#theorem-a","scope":"GroupTheory/group-rings/kaplansky-s-unit-conjecture","status":"can_prove","summary":"Let $P$ be the group with presentation $\\langle a, b \\mid b^{-1}a^2b = a^{-2},\\ a^{-1}b^2a = b^{-2} \\rangle$, and set $x = a^2$, $y = b^2$ and $z = (ab)^2$. In the group ring $\\mathbb{F}_2[P]$ set $p = (1 + x)(1 + y)(1 + z^{-1})$, $q = x^{-1}y^{-1} + x + y^{-1}z + z$, $r = 1 + x + y^{-1}z + xyz$ and $s = 1 + (x + x^{-1} + y + y^{-1})z^{-1}$, and let $u = p + qa + rb + sab$. Then $P$ is torsion-free (if $g \\in P$\u2026","title":"Gardam's counterexample to the unit conjecture"},{"data_url":"scopes/GroupTheory/group-rings/ore-condition-for-group-algebras.json","id":"GroupTheory/group-rings/ore-condition-for-group-algebras","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/ore-condition-for-group-algebras/index.html","scope":"GroupTheory/group-rings/ore-condition-for-group-algebras","status":"can_prove","title":"Ore condition for group algebras"},{"id":"GroupTheory/group-rings/ore-condition-for-group-algebras/shift","kind":"node","lean":"MathlibExt.GroupTheory.LamplighterGroupAlgebraOreKourovka12Problem47Wanted.shift","parent":2604,"result":true,"route":"../roadmap/GroupTheory/group-rings/ore-condition-for-group-algebras/index.html#shift","scope":"GroupTheory/group-rings/ore-condition-for-group-algebras","status":"fully_proved","summary":"For a natural number $p$, the shift is the automorphism $f \\mapsto f(\\cdot - 1)$ of the base group $\\bigoplus_{i \\in \\mathbb{Z}} \\mathbb{Z}/p$, the additive group of finitely supported functions $\\mathbb{Z} \\to \\mathbb{Z}/p$, viewed as a multiplicative group.","title":"Shift of the lamplighter base group"},{"id":"GroupTheory/group-rings/ore-condition-for-group-algebras/lamplighter","kind":"node","lean":"MathlibExt.GroupTheory.LamplighterGroupAlgebraOreKourovka12Problem47Wanted.Lamplighter","parent":2604,"result":true,"route":"../roadmap/GroupTheory/group-rings/ore-condition-for-group-algebras/index.html#lamplighter","scope":"GroupTheory/group-rings/ore-condition-for-group-algebras","status":"fully_proved","summary":"For a natural number $p$, the lamplighter group $\\mathbb{Z}/p \\wr \\mathbb{Z}$ is the semidirect product $\\left(\\bigoplus_{i \\in \\mathbb{Z}} \\mathbb{Z}/p\\right) \\rtimes \\mathbb{Z}$, in which $n \\in \\mathbb{Z}$ acts on the base group by the $n$-th power of the shift $f \\mapsto f(\\cdot - 1)$.","title":"Lamplighter group"},{"id":"GroupTheory/group-rings/ore-condition-for-group-algebras/is-non-zero-divisor","kind":"node","lean":"MathlibExt.GroupTheory.LamplighterGroupAlgebraOreKourovka12Problem47Wanted.IsNonZeroDivisor","parent":2604,"result":true,"route":"../roadmap/GroupTheory/group-rings/ore-condition-for-group-algebras/index.html#is-non-zero-divisor","scope":"GroupTheory/group-rings/ore-condition-for-group-algebras","status":"fully_proved","summary":"An element $s$ of a ring $R$ is a non-zero-divisor when it is neither a left nor a right zero divisor: for every $z \\in R$, $zs = 0$ implies $z = 0$ and $sz = 0$ implies $z = 0$.","title":"Non-zero-divisor"},{"id":"GroupTheory/group-rings/ore-condition-for-group-algebras/is-ore-ring","kind":"node","lean":"MathlibExt.GroupTheory.LamplighterGroupAlgebraOreKourovka12Problem47Wanted.IsOreRing","parent":2604,"result":true,"route":"../roadmap/GroupTheory/group-rings/ore-condition-for-group-algebras/index.html#is-ore-ring","scope":"GroupTheory/group-rings/ore-condition-for-group-algebras","status":"fully_proved","summary":"A ring $R$ satisfies the right Ore condition for its non-zero-divisors, so that it has a classical right ring of quotients, when for all $s, t \\in R$ with $s$ a non-zero-divisor there are $x, y \\in R$ with $x$ a non-zero-divisor and $sy = tx$.","title":"Right Ore ring"},{"id":"GroupTheory/group-rings/ore-condition-for-group-algebras/lamplighter-group-algebra-ore","kind":"node","lean":"MathlibExt.GroupTheory.LamplighterGroupAlgebraOreKourovka12Problem47Wanted.lamplighterGroupAlgebraOre","parent":2604,"result":true,"route":"../roadmap/GroupTheory/group-rings/ore-condition-for-group-algebras/index.html#lamplighter-group-algebra-ore","scope":"GroupTheory/group-rings/ore-condition-for-group-algebras","status":"fully_proved","summary":"There are a field $k$ and a prime $p$ such that the group algebra $k[\\mathbb{Z}/p \\wr \\mathbb{Z}]$ of the lamplighter group $\\mathbb{Z}/p \\wr \\mathbb{Z}$ satisfies the right Ore condition for its non-zero-divisors, that is, has a classical right ring of quotients.","title":"Kourovka Problem 12.47: some lamplighter group algebra is Ore"},{"id":"GroupTheory/group-rings/ore-condition-for-group-algebras/lamplighter-group-algebra-ore-refuted","kind":"node","lean":"MathlibExt.GroupTheory.LamplighterGroupAlgebraOreKourovka12Problem47Wanted.lamplighterGroupAlgebraOre_refuted","parent":2604,"result":true,"route":"../roadmap/GroupTheory/group-rings/ore-condition-for-group-algebras/index.html#lamplighter-group-algebra-ore-refuted","scope":"GroupTheory/group-rings/ore-condition-for-group-algebras","status":"can_prove","summary":"For every field $k$ and every prime $p$, the group algebra $k[\\mathbb{Z}/p \\wr \\mathbb{Z}]$ of the lamplighter group $\\mathbb{Z}/p \\wr \\mathbb{Z}$ does not satisfy the right Ore condition for its non-zero-divisors: there are $s$ and $t$ in it, with $s$ a non-zero-divisor, such that $sy = tx$ holds for no $x$ and $y$ with $x$ a non-zero-divisor.","title":"Lamplighter group algebras are not Ore"},{"data_url":"scopes/GroupTheory/group-rings/projective-modules-over-group-rings.json","id":"GroupTheory/group-rings/projective-modules-over-group-rings","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/projective-modules-over-group-rings/index.html","scope":"GroupTheory/group-rings/projective-modules-over-group-rings","status":"can_prove","title":"Projective modules over group rings"},{"id":"GroupTheory/group-rings/projective-modules-over-group-rings/is-polycyclic","kind":"node","lean":"MathlibExt.GroupTheory.ProjectiveGroupRingModuleFreeKourovka11Problem5Wanted.IsPolycyclic","parent":2611,"result":true,"route":"../roadmap/GroupTheory/group-rings/projective-modules-over-group-rings/index.html#is-polycyclic","scope":"GroupTheory/group-rings/projective-modules-over-group-rings","status":"fully_proved","summary":"A group $G$ is polycyclic if for some $n \\ge 0$ there are subgroups $1 = H_0 \\le H_1 \\le \\cdots \\le H_n = G$ such that for each $i < n$, $H_i$ is normal in $H_{i+1}$ and the quotient $H_{i+1} / H_i$ is cyclic.","title":"Polycyclic group"},{"id":"GroupTheory/group-rings/projective-modules-over-group-rings/is-almost-polycyclic","kind":"node","lean":"MathlibExt.GroupTheory.ProjectiveGroupRingModuleFreeKourovka11Problem5Wanted.IsAlmostPolycyclic","parent":2611,"result":true,"route":"../roadmap/GroupTheory/group-rings/projective-modules-over-group-rings/index.html#is-almost-polycyclic","scope":"GroupTheory/group-rings/projective-modules-over-group-rings","status":"fully_proved","summary":"A group $G$ is almost polycyclic (polycyclic-by-finite) if it has a subgroup of finite index that is polycyclic.","title":"Almost polycyclic group"},{"id":"GroupTheory/group-rings/projective-modules-over-group-rings/projective-with-two-independent-free","kind":"node","lean":"MathlibExt.GroupTheory.ProjectiveGroupRingModuleFreeKourovka11Problem5Wanted.projectiveWithTwoIndependentFree","parent":2611,"result":true,"route":"../roadmap/GroupTheory/group-rings/projective-modules-over-group-rings/index.html#projective-with-two-independent-free","scope":"GroupTheory/group-rings/projective-modules-over-group-rings","status":"fully_proved","summary":"This is the proposition that, for every commutative ring $k$, every torsion-free almost polycyclic group $G$ (no element $g \\ne 1$ of $G$ has finite order) and every finitely generated projective module $P$ over the group ring $kG$, if $P$ contains two elements that are linearly independent over $kG$, then $P$ is a free $kG$-module.","title":"Artamonov's freeness question (Kourovka 11.5)"},{"id":"GroupTheory/group-rings/projective-modules-over-group-rings/projective-with-two-independent-free-ref","kind":"node","lean":"MathlibExt.GroupTheory.ProjectiveGroupRingModuleFreeKourovka11Problem5Wanted.projectiveWithTwoIndependentFree_refuted","parent":2611,"result":true,"route":"../roadmap/GroupTheory/group-rings/projective-modules-over-group-rings/index.html#projective-with-two-independent-free-ref","scope":"GroupTheory/group-rings/projective-modules-over-group-rings","status":"can_prove","summary":"It is not the case that for every commutative ring $k$, every torsion-free almost polycyclic group $G$ (no element $g \\ne 1$ of $G$ has finite order) and every finitely generated projective module $P$ over the group ring $kG$ containing two elements that are linearly independent over $kG$, the module $P$ is free over $kG$.","title":"Artamonov's freeness question has a negative answer"},{"data_url":"scopes/GroupTheory/group-rings/right-ideals-of-free-group-algebras.json","id":"GroupTheory/group-rings/right-ideals-of-free-group-algebras","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/right-ideals-of-free-group-algebras/index.html","scope":"GroupTheory/group-rings/right-ideals-of-free-group-algebras","status":"can_prove","title":"Right ideals of free group algebras"},{"id":"GroupTheory/group-rings/right-ideals-of-free-group-algebras/twist-character","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupAlgebraTwistedIdealsKourovka12Problem46Wanted.twistCharacter","parent":2616,"result":true,"route":"../roadmap/GroupTheory/group-rings/right-ideals-of-free-group-algebras/index.html#twist-character","scope":"GroupTheory/group-rings/right-ideals-of-free-group-algebras","status":"fully_proved","summary":"Let $F$ be the free group on $x, y$. For nonzero complex numbers $a, b$, this is the homomorphism $\\chi \\colon F \\to \\mathbb{C}^\\times$ with $\\chi(x) = a$ and $\\chi(y) = b$.","title":"Twisting character of the free group of rank two"},{"id":"GroupTheory/group-rings/right-ideals-of-free-group-algebras/theta","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupAlgebraTwistedIdealsKourovka12Problem46Wanted.theta","parent":2616,"result":true,"route":"../roadmap/GroupTheory/group-rings/right-ideals-of-free-group-algebras/index.html#theta","scope":"GroupTheory/group-rings/right-ideals-of-free-group-algebras","status":"fully_proved","summary":"Let $F$ be the free group on $x, y$, let $\\mathbb{C}F$ be its complex group algebra, and for nonzero complex numbers $a, b$ let $\\chi$ be the character of $F$ with $\\chi(x) = a$ and $\\chi(y) = b$. The map $\\vartheta_{a,b}$ sends $\\alpha = \\sum_{g \\in F} c_g\\, g \\in \\mathbb{C}F$ to $\\sum_{g \\in F} \\chi(g)\\, c_g\\, g$.","title":"Twisting automorphism of the free group algebra"},{"id":"GroupTheory/group-rings/right-ideals-of-free-group-algebras/twisted-right-ideals-intersect","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupAlgebraTwistedIdealsKourovka12Problem46Wanted.twistedRightIdealsIntersect","parent":2616,"result":true,"route":"../roadmap/GroupTheory/group-rings/right-ideals-of-free-group-algebras/index.html#twisted-right-ideals-intersect","scope":"GroupTheory/group-rings/right-ideals-of-free-group-algebras","status":"fully_proved","summary":"Let $F$ be the free group on $x, y$, let $\\mathbb{C}F$ be its complex group algebra, and for nonzero complex numbers $a, b$ let $\\vartheta_{a,b}$ be the twisting automorphism of $\\mathbb{C}F$ with $\\vartheta_{a,b}(x) = a x$ and $\\vartheta_{a,b}(y) = b y$. The assertion is that for every nonzero $\\alpha \\in \\mathbb{C}F$ there are complex numbers $a, b \\neq 1$ with $|a| = |b| = 1$ such that the right ideals\u2026","title":"Linnell's twisted right ideal question (Kourovka 12.46)"},{"id":"GroupTheory/group-rings/right-ideals-of-free-group-algebras/twisted-right-ideals-intersect-refuted","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupAlgebraTwistedIdealsKourovka12Problem46Wanted.twistedRightIdealsIntersect_refuted","parent":2616,"result":true,"route":"../roadmap/GroupTheory/group-rings/right-ideals-of-free-group-algebras/index.html#twisted-right-ideals-intersect-refuted","scope":"GroupTheory/group-rings/right-ideals-of-free-group-algebras","status":"can_prove","summary":"Let $F$ be the free group on $x, y$, let $\\mathbb{C}F$ be its complex group algebra, and for nonzero complex numbers $a, b$ let $\\vartheta_{a,b}$ be the twisting automorphism of $\\mathbb{C}F$ with $\\vartheta_{a,b}(x) = a x$ and $\\vartheta_{a,b}(y) = b y$. There is a nonzero $\\alpha \\in \\mathbb{C}F$ such that for all complex numbers $a, b \\neq 1$ with $|a| = |b| = 1$ the right ideals $\\alpha\\,\\mathbb{C}F$ and\u2026","title":"Linnell's twisted right ideal question has a negative answer (Kourovka 12.46)"},{"data_url":"scopes/GroupTheory/group-rings/units-of-integral-group-rings.json","id":"GroupTheory/group-rings/units-of-integral-group-rings","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/units-of-integral-group-rings/index.html","scope":"GroupTheory/group-rings/units-of-integral-group-rings","status":"can_prove","title":"Units of integral group rings"},{"id":"GroupTheory/group-rings/units-of-integral-group-rings/owrgroup","kind":"node","lean":"MathlibExt.Algebra.ExoticUnitsC2C2C2C6Wanted.OWRGroup","parent":2621,"reading_title":{"format":"inline-parts-v1","parts":[["text","The group "],["math","\\(C_2 \\times C_2 \\times C_2 \\times C_6\\)"]],"source":"The group $C_2 \\times C_2 \\times C_2 \\times C_6$"},"result":true,"route":"../roadmap/GroupTheory/group-rings/units-of-integral-group-rings/index.html#owrgroup","scope":"GroupTheory/group-rings/units-of-integral-group-rings","status":"fully_proved","summary":"The abelian group $G = C_2 \\times C_2 \\times C_2 \\times C_6$, the product of cyclic groups of orders $2$, $2$, $2$ and $6$, written additively as $\\mathbb{Z}/2 \\times \\mathbb{Z}/2 \\times \\mathbb{Z}/2 \\times \\mathbb{Z}/6$.","title":"The group $C_2 \\times C_2 \\times C_2 \\times C_6$"},{"id":"GroupTheory/group-rings/units-of-integral-group-rings/owrgroup-ring","kind":"node","lean":"MathlibExt.Algebra.ExoticUnitsC2C2C2C6Wanted.OWRGroupRing","parent":2621,"reading_title":{"format":"inline-parts-v1","parts":[["text","The integral group ring of "],["math","\\(C_2 \\times C_2 \\times C_2 \\times C_6\\)"]],"source":"The integral group ring of $C_2 \\times C_2 \\times C_2 \\times C_6$"},"result":true,"route":"../roadmap/GroupTheory/group-rings/units-of-integral-group-rings/index.html#owrgroup-ring","scope":"GroupTheory/group-rings/units-of-integral-group-rings","status":"fully_proved","summary":"The integral group ring $\\mathbb{Z}G$ of $G = C_2 \\times C_2 \\times C_2 \\times C_6$: finitely supported formal $\\mathbb{Z}$-linear combinations of elements of $G$, with the product extending the group operation of $G$ bilinearly.","title":"The integral group ring of $C_2 \\times C_2 \\times C_2 \\times C_6$"},{"id":"GroupTheory/group-rings/units-of-integral-group-rings/conjecture","kind":"node","lean":"MathlibExt.Algebra.ExoticUnitsC2C2C2C6Wanted.conjecture","parent":2621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exotic unit conjecture for "],["math","\\(C_2 \\times C_2 \\times C_2 \\times C_6\\)"]],"source":"Exotic unit conjecture for $C_2 \\times C_2 \\times C_2 \\times C_6$"},"result":true,"route":"../roadmap/GroupTheory/group-rings/units-of-integral-group-rings/index.html#conjecture","scope":"GroupTheory/group-rings/units-of-integral-group-rings","status":"fully_proved","summary":"Let $G = C_2 \\times C_2 \\times C_2 \\times C_6$. The statement that the integral group ring $\\mathbb{Z}G$ contains an exotic unit: a unit $u$ with $u \\ne g$ and $u \\ne -g$ for every $g \\in G$.","title":"Exotic unit conjecture for $C_2 \\times C_2 \\times C_2 \\times C_6$"},{"id":"GroupTheory/group-rings/units-of-integral-group-rings/conjecture-refuted","kind":"node","lean":"MathlibExt.Algebra.ExoticUnitsC2C2C2C6Wanted.conjecture_refuted","parent":2621,"reading_title":{"format":"inline-parts-v1","parts":[["text","No exotic units in the integral group ring of "],["math","\\(C_2 \\times C_2 \\times C_2 \\times C_6\\)"]],"source":"No exotic units in the integral group ring of $C_2 \\times C_2 \\times C_2 \\times C_6$"},"result":true,"route":"../roadmap/GroupTheory/group-rings/units-of-integral-group-rings/index.html#conjecture-refuted","scope":"GroupTheory/group-rings/units-of-integral-group-rings","status":"can_prove","summary":"Let $G = C_2 \\times C_2 \\times C_2 \\times C_6$. Every unit $u$ of the integral group ring $\\mathbb{Z}G$ is trivial: $u = g$ or $u = -g$ for some $g \\in G$.","title":"No exotic units in the integral group ring of $C_2 \\times C_2 \\times C_2 \\times C_6$"},{"data_url":"scopes/GroupTheory/group-rings/zassenhaus-conjectures.json","id":"GroupTheory/group-rings/zassenhaus-conjectures","kind":"scope","navigation":"module","parent":2562,"result":false,"route":"../roadmap/GroupTheory/group-rings/zassenhaus-conjectures/index.html","scope":"GroupTheory/group-rings/zassenhaus-conjectures","status":"can_prove","title":"Zassenhaus conjectures"},{"id":"GroupTheory/group-rings/zassenhaus-conjectures/zassenhaus-conjecture","kind":"node","lean":"MathlibExt.GroupTheory.ZassenhausConjectureKourovka17Problem67Wanted.zassenhausConjecture","parent":2626,"result":true,"route":"../roadmap/GroupTheory/group-rings/zassenhaus-conjectures/index.html#zassenhaus-conjecture","scope":"GroupTheory/group-rings/zassenhaus-conjectures","status":"fully_proved","summary":"Kourovka 17.67 (H. Zassenhaus's conjecture, recorded by V. D. Mazurov): every unit of finite order of the integral group ring `\u2124G` of a finite group `G` is conjugate, by a unit of the rational group ring `\u211aG`, to an element `\u00b1g` with `g \u2208 G`.","title":"Zassenhaus conjecture on torsion units"},{"id":"GroupTheory/group-rings/zassenhaus-conjectures/zassenhaus-conjecture-refuted","kind":"node","lean":"MathlibExt.GroupTheory.ZassenhausConjectureKourovka17Problem67Wanted.zassenhausConjecture_refuted","parent":2626,"result":true,"route":"../roadmap/GroupTheory/group-rings/zassenhaus-conjectures/index.html#zassenhaus-conjecture-refuted","scope":"GroupTheory/group-rings/zassenhaus-conjectures","status":"can_prove","summary":"There exist a finite group $G$ and a unit $u$ of finite order of the integral group ring $\\mathbb{Z}G$ such that for every unit $v$ of the rational group ring $\\mathbb{Q}G$, every $g \\in G$ and every sign $\\varepsilon \\in \\{1, -1\\}$, the conjugate $v u v^{-1}$, computed in $\\mathbb{Q}G$, is not equal to $\\varepsilon g$. That is, the first Zassenhaus conjecture is false.","title":"Counterexample to the first Zassenhaus conjecture (Kourovka 17.67)"},{"data_url":"scopes/GroupTheory/abelian-groups.json","id":"GroupTheory/abelian-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/index.html","scope":"GroupTheory/abelian-groups","status":"can_prove","summary":"Structure of abelian groups: finite and finitely generated abelian groups (invariant factors, elementary divisors, torsion, ranks, dual groups), torsion-free and infinite abelian groups (Baer\u2013Specker group, pure subgroups, direct-sum cancellation, ultraproducts of abelian groups); colourings and partitions of abelian groups go to Combinatorics in groups.","title":"Abelian groups"},{"id":"GroupTheory/abelian-groups/nat-card-torsion-by-quotient-eq-pow-finr","kind":"node","lean":"Submodule.natCard_torsionBy_quotient_eq_pow_finrank","parent":2629,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cardinality of the "],["math","\\(n\\)"],["text","-torsion of a vector space modulo a lattice"]],"source":"Cardinality of the $n$-torsion of a vector space modulo a lattice"},"result":true,"route":"../roadmap/GroupTheory/abelian-groups/index.html#nat-card-torsion-by-quotient-eq-pow-finr","scope":"GroupTheory/abelian-groups","status":"fully_proved","summary":"The `n`-torsion of the quotient of a vector space by a free finite lattice has cardinality `n ^ finrank L`.","title":"Cardinality of the $n$-torsion of a vector space modulo a lattice"},{"id":"GroupTheory/abelian-groups/is-pure-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.PureSubgroupShared.IsPureSubgroup","parent":2629,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/index.html#is-pure-subgroup","scope":"GroupTheory/abelian-groups","status":"fully_proved","summary":"A subgroup $B$ of an abelian group $A$ is pure when $nA \\cap B = nB$ for every integer $n$: for every integer $n$ and every $a \\in A$ with $na \\in B$ there is some $b \\in B$ with $nb = na$.","title":"Pure subgroups of abelian groups"},{"data_url":"scopes/GroupTheory/abelian-groups/baer-specker-group.json","id":"GroupTheory/abelian-groups/baer-specker-group","kind":"scope","navigation":"module","parent":2629,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/baer-specker-group/index.html","scope":"GroupTheory/abelian-groups/baer-specker-group","status":"can_prove","title":"Baer\u2013Specker group"},{"id":"GroupTheory/abelian-groups/baer-specker-group/baer-specker-subgroup-dual-free-uncounta","kind":"node","lean":"MathlibExt.GroupTheory.BaerSpeckerSubgroupDualFreeKourovka17Problem24Wanted.baerSpeckerSubgroupDualFreeUncountable","parent":2632,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/baer-specker-group/index.html#baer-specker-subgroup-dual-free-uncounta","scope":"GroupTheory/abelian-groups/baer-specker-group","status":"fully_proved","summary":"The proposition that the Baer\u2013Specker group $\\mathbb{Z}^{\\mathbb{N}}$ of all integer sequences has a subgroup $H$ whose dual $\\operatorname{Hom}(H, \\mathbb{Z})$ is a free abelian group of rank greater than $\\aleph_0$.","title":"Bergman's question on subgroups of the Baer\u2013Specker group (Kourovka 17.24)"},{"id":"GroupTheory/abelian-groups/baer-specker-group/baer-specker-subgroup-dual-free-uncounta-2","kind":"node","lean":"MathlibExt.GroupTheory.BaerSpeckerSubgroupDualFreeKourovka17Problem24Wanted.baerSpeckerSubgroupDualFreeUncountable_holds","parent":2632,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/baer-specker-group/index.html#baer-specker-subgroup-dual-free-uncounta-2","scope":"GroupTheory/abelian-groups/baer-specker-group","status":"can_prove","summary":"The Baer\u2013Specker group $\\mathbb{Z}^{\\mathbb{N}}$ of all integer sequences has a subgroup $H$ whose dual $\\operatorname{Hom}(H, \\mathbb{Z})$ is a free abelian group of rank greater than $\\aleph_0$; this answers Kourovka 17.24 positively.","title":"A subgroup of the Baer\u2013Specker group with free dual of uncountable rank"},{"data_url":"scopes/GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups.json","id":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups","kind":"scope","navigation":"module","parent":2629,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/index.html","scope":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups","status":"fully_proved","title":"Cyclic sublattices of abelian groups"},{"id":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/is-cyclic-sublattice","kind":"node","lean":"AddSubgroup.IsCyclicSublattice","parent":2635,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/index.html#is-cyclic-sublattice","scope":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups","status":"fully_proved","summary":"Let $A$ be an abelian group and let $L'$ and $L$ be subgroups of $A$. Then $L'$ is a cyclic sublattice of $L$ when $L' \\subseteq L$ and the quotient group $L/L'$ is cyclic; formally the quotient is taken by $L' \\cap L$, regarded as a subgroup of $L$.","title":"Cyclic sublattice of a subgroup"},{"id":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/is-cyclic-sublattice-iff","kind":"node","lean":"AddSubgroup.isCyclicSublattice_iff","parent":2635,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/index.html#is-cyclic-sublattice-iff","scope":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups","status":"fully_proved","summary":"Let $L'$ and $L$ be subgroups of an abelian group $A$. Then $L'$ is a cyclic sublattice of $L$ if and only if $L' \\subseteq L$ and the quotient group $L/(L' \\cap L)$, with $L' \\cap L$ regarded as a subgroup of $L$, is cyclic.","title":"Characterization of cyclic sublattices"},{"id":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/le","kind":"node","lean":"AddSubgroup.IsCyclicSublattice.le","parent":2635,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/index.html#le","scope":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups","status":"fully_proved","summary":"Let $L'$ and $L$ be subgroups of an abelian group $A$. If $L'$ is a cyclic sublattice of $L$, then $L' \\subseteq L$.","title":"A cyclic sublattice is contained in its ambient subgroup"},{"id":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/is-add-cyclic-quotient","kind":"node","lean":"AddSubgroup.IsCyclicSublattice.isAddCyclic_quotient","parent":2635,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/index.html#is-add-cyclic-quotient","scope":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups","status":"fully_proved","summary":"Let $L'$ and $L$ be subgroups of an abelian group $A$. If $L'$ is a cyclic sublattice of $L$, then the quotient group $L/L'$ is cyclic.","title":"The quotient by a cyclic sublattice is cyclic"},{"id":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/mk","kind":"node","lean":"AddSubgroup.IsCyclicSublattice.mk","parent":2635,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups/index.html#mk","scope":"GroupTheory/abelian-groups/cyclic-sublattices-of-abelian-groups","status":"fully_proved","summary":"Let $L'$ and $L$ be subgroups of an abelian group $A$. If $L' \\subseteq L$ and the quotient group $L/L'$ is cyclic, then $L'$ is a cyclic sublattice of $L$.","title":"Criterion for a cyclic sublattice"},{"data_url":"scopes/GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian.json","id":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian","kind":"scope","navigation":"module","parent":2629,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/index.html","scope":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian","status":"can_prove","title":"Direct-sum cancellation for torsion-free abelian groups"},{"id":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/is-torsion-free-add-group","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeAbelianDirectSumCancellationKourovka17Problem21Wanted.IsTorsionFreeAddGroup","parent":2641,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/index.html#is-torsion-free-add-group","scope":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian","status":"fully_proved","summary":"An abelian group $A$, written additively, is torsion-free when for every natural number $n \\ne 0$ and every $a \\in A$, $n a = 0$ implies $a = 0$.","title":"Torsion-free abelian group"},{"id":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/torsion-free-absorbs-direct-summand","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeAbelianDirectSumCancellationKourovka17Problem21Wanted.torsionFreeAbsorbsDirectSummand","parent":2641,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/index.html#torsion-free-absorbs-direct-summand","scope":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian","status":"fully_proved","summary":"The statement that for all torsion-free abelian groups $A$, $B$, $C$, if $A \\cong A \\oplus B \\oplus C$ then $A \\cong A \\oplus B$.","title":"Bergman's absorption question (Kourovka 17.21(a))"},{"id":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/torsion-free-absorbs-direct-summand-of-i","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeAbelianDirectSumCancellationKourovka17Problem21Wanted.torsionFreeAbsorbsDirectSummandOfIso","parent":2641,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/index.html#torsion-free-absorbs-direct-summand-of-i","scope":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian","status":"fully_proved","summary":"The statement that for all torsion-free abelian groups $A$, $B$, $C$ with $B \\cong C$, if $A \\cong A \\oplus B \\oplus C$ then $A \\cong A \\oplus B$.","title":"Bergman's absorption question for isomorphic summands (Kourovka 17.21(b))"},{"id":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/torsion-free-absorbs-direct-summand-refu","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeAbelianDirectSumCancellationKourovka17Problem21Wanted.torsionFreeAbsorbsDirectSummand_refuted","parent":2641,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/index.html#torsion-free-absorbs-direct-summand-refu","scope":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian","status":"can_prove","summary":"There exist torsion-free abelian groups $A$, $B$, $C$ with $A \\cong A \\oplus B \\oplus C$ but $A \\not\\cong A \\oplus B$; thus the answer to Kourovka Problem 17.21(a) is negative.","title":"Failure of direct-summand absorption for torsion-free abelian groups"},{"id":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/torsion-free-absorbs-direct-summand-of-i-2","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeAbelianDirectSumCancellationKourovka17Problem21Wanted.torsionFreeAbsorbsDirectSummandOfIso_refuted","parent":2641,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian/index.html#torsion-free-absorbs-direct-summand-of-i-2","scope":"GroupTheory/abelian-groups/direct-sum-cancellation-for-torsion-free-abelian","status":"can_prove","summary":"There exist torsion-free abelian groups $A$, $B$, $C$ with $B \\cong C$ and $A \\cong A \\oplus B \\oplus C$ but $A \\not\\cong A \\oplus B$; thus the answer to Kourovka Problem 17.21(b) is negative.","title":"Failure of direct-summand absorption with isomorphic summands"},{"data_url":"scopes/GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups.json","id":"GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups","kind":"scope","navigation":"module","parent":2629,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups/index.html","scope":"GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups","status":"fully_proved","title":"Dual groups of finite abelian groups"},{"id":"GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups/finite-abelian-dual-iso","kind":"node","lean":"MathlibExt.GroupTheory.FiniteAbelianDualWanted.finite_abelian_dual_iso","parent":2647,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups/index.html#finite-abelian-dual-iso","scope":"GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups","status":"fully_proved","summary":"Every finite abelian group $A$ is isomorphic to its dual group, the group of characters of $A$ under pointwise multiplication, where a character of $A$ is a map $\\chi : A \\to \\mathbb{C}$ with $\\chi(0) = 1$ and $\\chi(a + a') = \\chi(a)\\chi(a')$ for all $a, a' \\in A$.","title":"A finite abelian group is isomorphic to its dual group"},{"id":"GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups/dual-prod","kind":"node","lean":"MathlibExt.GroupTheory.FiniteAbelianDualWanted.dual_prod","parent":2647,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups/index.html#dual-prod","scope":"GroupTheory/abelian-groups/dual-groups-of-finite-abelian-groups","status":"fully_proved","summary":"For all abelian groups $A$ and $B$, the group of characters of $A \\times B$ is isomorphic to the product of the character groups of $A$ and $B$. Here a character of $A$ is a map $\\chi : A \\to \\mathbb{C}$ with $\\chi(0) = 1$ and $\\chi(a + a') = \\chi(a)\\chi(a')$ for all $a, a' \\in A$, and the characters of $A$ form a group under pointwise multiplication.","title":"Characters of a product of abelian groups"},{"data_url":"scopes/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups.json","id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","kind":"scope","navigation":"module","parent":2629,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"can_prove","title":"Homomorphic images of ultraproducts of groups"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/ultraproduct-kernel","kind":"node","lean":"MathlibExt.GroupTheory.AbelianImagesUltraproductsKourovka20Problem13Wanted.ultraproductKernel","parent":2650,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#ultraproduct-kernel","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"fully_proved","summary":"For an ultrafilter $U$ on $\\mathbb{N}$ and a sequence $(G_i)_{i \\in \\mathbb{N}}$ of groups, the normal subgroup of $\\prod_i G_i$ generated by the sequences $(g_i)_i$ with $\\{i : g_i = 1\\} \\in U$. These sequences already form a normal subgroup, so the kernel consists of exactly them.","title":"Ultraproduct kernel"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/ultraproduct","kind":"node","lean":"MathlibExt.GroupTheory.AbelianImagesUltraproductsKourovka20Problem13Wanted.Ultraproduct","parent":2650,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#ultraproduct","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"fully_proved","summary":"For an ultrafilter $U$ on $\\mathbb{N}$ and a sequence $(G_i)_{i \\in \\mathbb{N}}$ of groups, the ultraproduct $\\left(\\prod_i G_i\\right)/U$ is the quotient of the direct product $\\prod_i G_i$ by the ultraproduct kernel, the normal subgroup of sequences $(g_i)_i$ with $\\{i : g_i = 1\\} \\in U$.","title":"Ultraproduct of a sequence of groups"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/abelian-image-of-ultraproduct-is-image-o","kind":"node","lean":"MathlibExt.GroupTheory.AbelianImagesUltraproductsKourovka20Problem13Wanted.abelianImageOfUltraproductIsImageOfAbelianUltraproduct","parent":2650,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#abelian-image-of-ultraproduct-is-image-o","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"fully_proved","summary":"The proposition that for every abelian group $A$, every sequence $(G_i)_{i \\in \\mathbb{N}}$ of (not necessarily abelian) groups and every nonprincipal ultrafilter $U$ on $\\mathbb{N}$ (one containing every cofinite set), if there is a surjective homomorphism from the ultraproduct $\\left(\\prod_i G_i\\right)/U$ onto $A$, then there are a sequence $(H_i)_{i \\in \\mathbb{N}}$ of abelian groups and a nonprincipal\u2026","title":"Kourovka Problem 20.13(a) (Bergman)"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/abelian-image-of-product-of-finite-is-im","kind":"node","lean":"MathlibExt.GroupTheory.AbelianImagesUltraproductsKourovka20Problem13Wanted.abelianImageOfProductOfFiniteIsImageOfAbelianProduct","parent":2650,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#abelian-image-of-product-of-finite-is-im","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"fully_proved","summary":"The proposition that every abelian group $A$ that is a homomorphic image of a direct product $\\prod_{i \\in I} G_i$ of an infinite family $(G_i)_{i \\in I}$ of finite (not necessarily abelian) groups is also a homomorphic image of the direct product $\\prod_{j \\in J} H_j$ of some family $(H_j)_{j \\in J}$ of finite abelian groups.","title":"Kourovka Problem 20.13(b) (Bergman)"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/abelian-image-of-ultraproduct-is-image-o-2","kind":"node","lean":"MathlibExt.GroupTheory.AbelianImagesUltraproductsKourovka20Problem13Wanted.abelianImageOfUltraproductIsImageOfAbelianUltraproduct_holds","parent":2650,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#abelian-image-of-ultraproduct-is-image-o-2","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"can_prove","summary":"Let $A$ be an abelian group, let $(G_i)_{i \\in \\mathbb{N}}$ be a sequence of (not necessarily abelian) groups, and let $U$ be a nonprincipal ultrafilter on $\\mathbb{N}$, that is, one containing every cofinite set. If there is a surjective homomorphism from the ultraproduct $\\left(\\prod_i G_i\\right)/U$ onto $A$, then there are a sequence $(H_i)_{i \\in \\mathbb{N}}$ of abelian groups and a nonprincipal ultrafilter $V$\u2026","title":"Abelian images of ultraproducts are images of abelian ultraproducts"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/abelian-image-of-product-of-finite-is-im-2","kind":"node","lean":"MathlibExt.GroupTheory.AbelianImagesUltraproductsKourovka20Problem13Wanted.abelianImageOfProductOfFiniteIsImageOfAbelianProduct_holds","parent":2650,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#abelian-image-of-product-of-finite-is-im-2","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"can_prove","summary":"Let $A$ be an abelian group. If $A$ is a homomorphic image of a direct product $\\prod_{i \\in I} G_i$ of an infinite family $(G_i)_{i \\in I}$ of finite (not necessarily abelian) groups, then $A$ is a homomorphic image of the direct product $\\prod_{j \\in J} H_j$ of some family $(H_j)_{j \\in J}$ of finite abelian groups.","title":"Abelian images of products of finite groups"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/is-locally-ultrapower-image","kind":"node","lean":"MathlibExt.GroupTheory.UltraproductImageKourovka21Problem12Wanted.IsLocallyUltrapowerImage","parent":2650,"reading_title":{"format":"inline-parts-v1","parts":[["text","Groups locally an image of the ultrapower "],["math","\\(\\mathbb{Z}^{\\mathbb{N}}/U\\)"]],"source":"Groups locally an image of the ultrapower $\\mathbb{Z}^{\\mathbb{N}}/U$"},"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#is-locally-ultrapower-image","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"fully_proved","summary":"Let $U$ be an ultrafilter on $\\mathbb{N}$ and $B$ a group. Then $B$ is locally an image of the ultrapower $\\mathbb{Z}^{\\mathbb{N}}/U$ when every element $b \\in B$ lies in a subgroup $H$ of $B$ that is a homomorphic image of $\\mathbb{Z}^{\\mathbb{N}}/U$: there is a surjective homomorphism $\\psi$ from the additive group $\\mathbb{Z}^{\\mathbb{N}}$ of integer sequences onto $H$ such that $\\psi(f) = 1$ for every sequence\u2026","title":"Groups locally an image of the ultrapower $\\mathbb{Z}^{\\mathbb{N}}/U$"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/locally-ultrapower-image-is-ultraproduct","kind":"node","lean":"MathlibExt.GroupTheory.UltraproductImageKourovka21Problem12Wanted.locallyUltrapowerImageIsUltraproductImage","parent":2650,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#locally-ultrapower-image-is-ultraproduct","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"fully_proved","summary":"There is a nonprincipal ultrafilter $U$ on $\\mathbb{N}$ such that every group $B$ that is locally an image of the ultrapower $\\mathbb{Z}^{\\mathbb{N}}/U$ (every element of $B$ lies in a subgroup of $B$ that is a homomorphic image of $\\mathbb{Z}^{\\mathbb{N}}/U$) is a homomorphic image of an ultraproduct $(\\prod_{i \\in \\mathbb{N}} G_i)/U$: there are groups $G_i$, $i \\in \\mathbb{N}$, and a surjective homomorphism\u2026","title":"Bergman's question on images of countable ultraproducts (Kourovka 21.12)"},{"id":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/locally-ultrapower-image-is-ultraproduct-2","kind":"node","lean":"MathlibExt.GroupTheory.UltraproductImageKourovka21Problem12Wanted.locallyUltrapowerImageIsUltraproductImage_refuted","parent":2650,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups/index.html#locally-ultrapower-image-is-ultraproduct-2","scope":"GroupTheory/abelian-groups/homomorphic-images-of-ultraproducts-of-groups","status":"can_prove","summary":"For every nonprincipal ultrafilter $U$ on $\\mathbb{N}$ there is a group $B$ that is locally an image of the ultrapower $\\mathbb{Z}^{\\mathbb{N}}/U$ (every element of $B$ lies in a subgroup of $B$ that is a homomorphic image of $\\mathbb{Z}^{\\mathbb{N}}/U$) but is not a homomorphic image of any ultraproduct $(\\prod_{i \\in \\mathbb{N}} G_i)/U$ of groups: for no groups $G_i$, $i \\in \\mathbb{N}$, is there a surjective\u2026","title":"Bergman's question 21.12 has a negative answer"},{"data_url":"scopes/GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups.json","id":"GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups","kind":"scope","navigation":"module","parent":2629,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups/index.html","scope":"GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups","status":"can_prove","title":"Increasing unions of isomorphic subgroups"},{"id":"GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups/scottish-book-problem62-claim","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook62Wanted.ScottishBookProblem62Claim","parent":2660,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups/index.html#scottish-book-problem62-claim","scope":"GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups","status":"fully_proved","summary":"The claim that for every abelian group $G$ and every sequence of subgroups $H_0 \\le H_1 \\le H_2 \\le \\cdots$ of $G$ whose union is $G$ and each of which is isomorphic to $H_1$, the group $G$ is itself isomorphic to $H_1$.","title":"Claim of Scottish Book Problem 62"},{"id":"GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups/mazur-ulam62-refuted","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook62Wanted.mazurUlam62_refuted","parent":2660,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups/index.html#mazur-ulam62-refuted","scope":"GroupTheory/abelian-groups/increasing-unions-of-isomorphic-subgroups","status":"can_prove","summary":"The claim of Scottish Book Problem 62 is false: there are an abelian group $G$ and subgroups $H_0 \\le H_1 \\le H_2 \\le \\cdots$ of $G$ whose union is $G$, each isomorphic to $H_1$, such that $G$ is not isomorphic to $H_1$.","title":"Refutation of Scottish Book Problem 62"},{"data_url":"scopes/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups.json","id":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","kind":"scope","navigation":"module","parent":2629,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/index.html","scope":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","status":"fully_proved","title":"Structure of finitely generated abelian groups"},{"id":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/elementary-divisors-unique","kind":"node","lean":"elementary_divisors_unique","parent":2663,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/index.html#elementary-divisors-unique","scope":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","status":"fully_proved","summary":"Let $p$ be a prime and let $A$ be an abelian group. If $A \\cong \\bigoplus_{i=1}^{n} \\mathbb{Z}/p^{e_i}\\mathbb{Z}$ and $A \\cong \\bigoplus_{j=1}^{m} \\mathbb{Z}/p^{f_j}\\mathbb{Z}$ for positive integers $e_1, \\dots, e_n$ and $f_1, \\dots, f_m$, then $n = m$ and the multisets $\\{e_1, \\dots, e_n\\}$ and $\\{f_1, \\dots, f_m\\}$ are equal.","title":"Uniqueness of elementary divisors"},{"id":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/fg-torsion","kind":"node","lean":"AddCommGroup.fg_torsion","parent":2663,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/index.html#fg-torsion","scope":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","status":"fully_proved","summary":"If $A$ is a finitely generated abelian group, then its torsion subgroup is finitely generated.","title":"Torsion subgroup of a finitely generated abelian group is finitely generated"},{"id":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/finite-torsion-of-fg","kind":"node","lean":"AddCommGroup.finite_torsion_of_fg","parent":2663,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/index.html#finite-torsion-of-fg","scope":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","status":"fully_proved","summary":"If $A$ is a finitely generated abelian group, then its torsion subgroup is finite.","title":"Torsion subgroup of a finitely generated abelian group is finite"},{"id":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/torsion-equiv-direct-sum-zmod-of-fg","kind":"node","lean":"AddCommGroup.torsion_equiv_directSum_zmod_of_fg","parent":2663,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/index.html#torsion-equiv-direct-sum-zmod-of-fg","scope":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","status":"fully_proved","summary":"If $A$ is a finitely generated abelian group, then there exist a natural number $n$ and positive integers $d_1, \\dots, d_n$ such that the torsion subgroup of $A$ is isomorphic to the direct sum $\\mathbb{Z}/d_1\\mathbb{Z} \\oplus \\cdots \\oplus \\mathbb{Z}/d_n\\mathbb{Z}$ of finite cyclic groups.","title":"Torsion subgroup of a finitely generated abelian group as a sum of cyclic groups"},{"id":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/equiv-free-prod-direct-sum-invariant-fac","kind":"node","lean":"AddCommGroup.equiv_free_prod_directSum_invariant_factors","parent":2663,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/index.html#equiv-free-prod-direct-sum-invariant-fac","scope":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","status":"fully_proved","summary":"Every finitely generated abelian group $A$ is isomorphic to $\\mathbb{Z}^a \\times \\bigoplus_{i < k} \\mathbb{Z}/d_i\\mathbb{Z}$ for some $a, k \\in \\mathbb{N}$ and natural numbers $d_0, \\dots, d_{k-1}$ with $d_i > 1$ for every $i$ and $d_i \\mid d_j$ whenever $i \\le j$.","title":"Fundamental theorem of finitely generated abelian groups, invariant factor form"},{"id":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/equiv-direct-sum-invariant-factors-of-fi","kind":"node","lean":"AddCommGroup.equiv_directSum_invariant_factors_of_finite","parent":2663,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/index.html#equiv-direct-sum-invariant-factors-of-fi","scope":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","status":"fully_proved","summary":"Every finite abelian group $A$ is isomorphic to $\\bigoplus_{i < k} \\mathbb{Z}/d_i\\mathbb{Z}$ for some $k \\in \\mathbb{N}$ and natural numbers $d_0, \\dots, d_{k-1}$ with $d_i > 1$ for every $i$ and $d_i \\mid d_j$ whenever $i \\le j$.","title":"Invariant factor decomposition of finite abelian groups"},{"id":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/elementary-divisors-equiv-invariant-fact","kind":"node","lean":"AddCommGroup.elementary_divisors_equiv_invariant_factors","parent":2663,"result":true,"route":"../roadmap/GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups/index.html#elementary-divisors-equiv-invariant-fact","scope":"GroupTheory/abelian-groups/structure-of-finitely-generated-abelian-groups","status":"fully_proved","summary":"Let $I$ be a finite set, and for each $i \\in I$ let $p_i$ be a prime and $e_i$ a natural number. Then there are $k \\in \\mathbb{N}$ and natural numbers $d_0, \\dots, d_{k-1}$ with $d_j > 1$ for every $j$ and $d_j \\mid d_l$ whenever $j \\le l$, such that $\\bigoplus_{i \\in I} \\mathbb{Z}/p_i^{e_i}\\mathbb{Z}$ is isomorphic to $\\bigoplus_{j < k} \\mathbb{Z}/d_j\\mathbb{Z}$.","title":"From elementary divisors to invariant factors"},{"data_url":"scopes/GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups.json","id":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups","kind":"scope","navigation":"module","parent":2629,"result":false,"route":"../roadmap/GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/index.html","scope":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups","status":"fully_proved","title":"Two-power ranks of finite abelian groups"},{"id":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/two-power-image","kind":"node","lean":"AbelianGroup.twoPowerImage","parent":2671,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subgroup "],["math","\\(2^kA\\)"]],"source":"The subgroup $2^kA$"},"result":true,"route":"../roadmap/GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/index.html#two-power-image","scope":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups","status":"fully_proved","summary":"For an abelian group $A$, written additively, and a natural number $k$, $2^kA$ is the subgroup $\\{2^k a : a \\in A\\}$ of $A$, the image of multiplication by $2^k$.","title":"The subgroup $2^kA$"},{"id":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/higher-two-power-layer","kind":"node","lean":"AbelianGroup.higherTwoPowerLayer","parent":2671,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-power layer "],["math","\\(2^{j-1}A / 2^jA\\)"]],"source":"The two-power layer $2^{j-1}A / 2^jA$"},"result":true,"route":"../roadmap/GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/index.html#higher-two-power-layer","scope":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups","status":"fully_proved","summary":"For an abelian group $A$, written additively, and a natural number $j$, this is the quotient group $2^{j-1}A / 2^jA$, where $2^kA$ denotes the subgroup $\\{2^k a : a \\in A\\}$ and $2^jA \\subseteq 2^{j-1}A$. For $j = 0$, where $j - 1$ is read as $0$, it is the trivial quotient $A / A$.","title":"The two-power layer $2^{j-1}A / 2^jA$"},{"id":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/higher-two-power-rank","kind":"node","lean":"AbelianGroup.higherTwoPowerRank","parent":2671,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(2^j\\)"],["text","-rank of a finite abelian group"]],"source":"The $2^j$-rank of a finite abelian group"},"result":true,"route":"../roadmap/GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/index.html#higher-two-power-rank","scope":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups","status":"fully_proved","summary":"Let $A$ be a finite abelian group, written additively, and let $j \\ge 1$ be an integer. Writing $2^kA$ for the subgroup $\\{2^k a : a \\in A\\}$, the $2^j$-rank of $A$ is the dimension over $\\mathbb{F}_2$ of the quotient $2^{j-1}A / 2^jA$, which is an $\\mathbb{F}_2$-vector space because $2$ kills it.","title":"The $2^j$-rank of a finite abelian group"},{"id":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/four-rank","kind":"node","lean":"AbelianGroup.fourRank","parent":2671,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(4\\)"],["text","-rank of a finite abelian group"]],"source":"The $4$-rank of a finite abelian group"},"result":true,"route":"../roadmap/GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups/index.html#four-rank","scope":"GroupTheory/abelian-groups/two-power-ranks-of-finite-abelian-groups","status":"fully_proved","summary":"For a finite abelian group $A$, written additively, the $4$-rank of $A$ is the dimension over $\\mathbb{F}_2$ of the quotient $2A / 4A$, where $2A = \\{2a : a \\in A\\}$ and $4A = \\{4a : a \\in A\\}$; it is the $2^j$-rank of $A$ for $j = 2$.","title":"The $4$-rank of a finite abelian group"},{"data_url":"scopes/GroupTheory/orders-and-spectra.json","id":"GroupTheory/orders-and-spectra","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/index.html","scope":"GroupTheory/orders-and-spectra","status":"can_prove","summary":"Arithmetic of finite groups through orders: cyclic numbers and groups of a given order (groups of order 10, dihedral groups from an index-two cyclic subgroup), counts of elements of each order, the spectrum (set of element orders), recognition of groups by spectrum or by order and spectrum, Thompson's type, and totient sums; conjugacy-class sizes go to Conjugacy classes.","title":"Group orders, element orders and spectra"},{"id":"GroupTheory/orders-and-spectra/nonempty-mul-equiv-dihedral-group-of-ind","kind":"node","lean":"MetaMathlibExt.nonempty_mulEquiv_dihedralGroup_of_index_eq_two","parent":2676,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/index.html#nonempty-mul-equiv-dihedral-group-of-ind","scope":"GroupTheory/orders-and-spectra","status":"fully_proved","summary":"Let $G$ be a finite group and let $H$ be a cyclic subgroup of $G$ of index $2$. If $g^2 = 1$ for every element $g$ of $G$ that is not in $H$, then $G$ is isomorphic to the dihedral group $D_n$ of order $2n$, where $n = |H|$.","title":"Dihedral groups from a cyclic subgroup of index two"},{"data_url":"scopes/GroupTheory/orders-and-spectra/counts-of-elements-of-each-order.json","id":"GroupTheory/orders-and-spectra/counts-of-elements-of-each-order","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/counts-of-elements-of-each-order/index.html","scope":"GroupTheory/orders-and-spectra/counts-of-elements-of-each-order","status":"can_prove","title":"Counts of elements of each order"},{"id":"GroupTheory/orders-and-spectra/counts-of-elements-of-each-order/element-order-count-prime-power-orders","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderCountPrimePowerKourovka18Problem112Wanted.elementOrderCountPrimePowerOrders","parent":2678,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/counts-of-elements-of-each-order/index.html#element-order-count-prime-power-orders","scope":"GroupTheory/orders-and-spectra/counts-of-elements-of-each-order","status":"fully_proved","summary":"The proposition that if a finite group $G$ has the property that, for every divisor $d > 1$ of $|G|$ and every subgroup $H$ of $G$ whose order is coprime to $d$, $|H|$ divides the number of elements of $G$ of order $d$, then every non-identity element of $G$ has prime-power order.","title":"Shi's question on element-order counts (Kourovka 18.112)"},{"id":"GroupTheory/orders-and-spectra/counts-of-elements-of-each-order/element-order-count-prime-power-orders-h","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderCountPrimePowerKourovka18Problem112Wanted.elementOrderCountPrimePowerOrders_holds","parent":2678,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/counts-of-elements-of-each-order/index.html#element-order-count-prime-power-orders-h","scope":"GroupTheory/orders-and-spectra/counts-of-elements-of-each-order","status":"can_prove","summary":"Let $G$ be a finite group such that for every divisor $d > 1$ of $|G|$ and every subgroup $H$ of $G$ whose order is coprime to $d$, the order $|H|$ divides the number of elements of $G$ of order $d$. Then every non-identity element of $G$ has prime-power order.","title":"Counting elements of each order forces prime-power orders"},{"data_url":"scopes/GroupTheory/orders-and-spectra/cyclic-numbers.json","id":"GroupTheory/orders-and-spectra/cyclic-numbers","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/cyclic-numbers/index.html","scope":"GroupTheory/orders-and-spectra/cyclic-numbers","status":"fully_proved","title":"Cyclic numbers"},{"id":"GroupTheory/orders-and-spectra/cyclic-numbers/of-squarefree-nat-card","kind":"node","lean":"IsAddCyclic.of_squarefree_natCard","parent":2681,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/cyclic-numbers/index.html#of-squarefree-nat-card","scope":"GroupTheory/orders-and-spectra/cyclic-numbers","status":"fully_proved","summary":"An additive commutative group of squarefree cardinality is additively cyclic.","title":"Additive groups of squarefree order are cyclic"},{"id":"GroupTheory/orders-and-spectra/cyclic-numbers/szele-cyclic-number-criterion","kind":"node","lean":"MetaMathlibExt.szele_cyclic_number_criterion","parent":2681,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/cyclic-numbers/index.html#szele-cyclic-number-criterion","scope":"GroupTheory/orders-and-spectra/cyclic-numbers","status":"fully_proved","summary":"Let $n$ be a positive integer. Every group of order $n$ is cyclic if and only if $n$ is relatively prime to $\\varphi(n)$, that is, $\\gcd(n, \\varphi(n)) = 1$, where $\\varphi$ is Euler's totient function.","title":"Szele's criterion for cyclic numbers"},{"data_url":"scopes/GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea.json","id":"GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea/index.html","scope":"GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea","status":"can_prove","title":"Element orders of extensions of projective linear groups"},{"id":"GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea/psl","kind":"node","lean":"MathlibExt.GroupTheory.PSLExtensionNewElementOrderKourovka14Problem60Wanted.PSL","parent":2684,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea/index.html#psl","scope":"GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea","status":"fully_proved","summary":"For a natural number $n$ and a field $F$, $\\mathrm{PSL}_n(F)$ is the quotient $\\mathrm{SL}_n(F)/Z(\\mathrm{SL}_n(F))$ of the special linear group $\\mathrm{SL}_n(F)$ of $n \\times n$ matrices over $F$ with determinant $1$ by its centre.","title":"Projective special linear group"},{"id":"GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea/extension-of-pslhas-new-element-order","kind":"node","lean":"MathlibExt.GroupTheory.PSLExtensionNewElementOrderKourovka14Problem60Wanted.extensionOfPSLHasNewElementOrder","parent":2684,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea/index.html#extension-of-pslhas-new-element-order","scope":"GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea","status":"fully_proved","summary":"The proposition that for every $n \\ge 3$, every prime $p$ and every $k \\ge 1$, with $q = p^k$ and $\\mathbb{F}_q$ the field with $q$ elements, the following holds: whenever $H$ is a nontrivial normal subgroup of a finite group $G$ and $G/H$ is isomorphic to the projective special linear group $\\mathrm{PSL}_n(\\mathbb{F}_q)$, some element of $G$ has an order that is not the order of any element of $G/H$.","title":"Kourovka 14.60: new element orders in extensions of projective special linear groups"},{"id":"GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea/extension-of-pslhas-new-element-order-ho","kind":"node","lean":"MathlibExt.GroupTheory.PSLExtensionNewElementOrderKourovka14Problem60Wanted.extensionOfPSLHasNewElementOrder_holds","parent":2684,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea/index.html#extension-of-pslhas-new-element-order-ho","scope":"GroupTheory/orders-and-spectra/element-orders-of-extensions-of-projective-linea","status":"can_prove","summary":"Resolved true: the Kourovka Archive answers 14.60 \"Yes, it is true\": Zavarnitsine proved it for `n \u2260 4` and Grechkoseeva and Skresanov for `n = 4` (the editors note that an earlier claim of failure for `n = 4` was erroneous). Source: E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 14.60 (V.\u2026","title":"Extensions of PSL have a new element order"},{"data_url":"scopes/GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum.json","id":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/index.html","scope":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum","status":"can_prove","title":"Finite groups with the same spectrum"},{"id":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/element-orders","kind":"node","lean":"MathlibExt.GroupTheory.SameSpectrumIncomparableGroupsKourovka16Problem106Wanted.elementOrders","parent":2688,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/index.html#element-orders","scope":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum","status":"fully_proved","summary":"The spectrum $\\pi_e(G)$ of a group $G$ is the set of orders of the elements of $G$, where an element of infinite order counts as having order $0$.","title":"Element-order spectrum of a group"},{"id":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/is-subgroup-or-normal-section-of","kind":"node","lean":"MathlibExt.GroupTheory.SameSpectrumIncomparableGroupsKourovka16Problem106Wanted.IsSubgroupOrNormalSectionOf","parent":2688,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/index.html#is-subgroup-or-normal-section-of","scope":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum","status":"fully_proved","summary":"For groups $A$ and $B$, the statement that $A$ is isomorphic to a subgroup of $B$ or to a quotient of a normal subgroup of $B$: there is an injective homomorphism $A \\to B$, or there are a normal subgroup $N$ of $B$ and a surjective homomorphism $N \\to A$.","title":"Subgroup or quotient of a normal subgroup"},{"id":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/same-spectrum-incomparable-groups","kind":"node","lean":"MathlibExt.GroupTheory.SameSpectrumIncomparableGroupsKourovka16Problem106Wanted.sameSpectrumIncomparableGroups","parent":2688,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/index.html#same-spectrum-incomparable-groups","scope":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum","status":"fully_proved","summary":"Finite groups with equal spectra, neither a subgroup or normal-subgroup quotient of the other (problem). The statement that there exist finite groups $G_1$ and $G_2$ with the same spectrum $\\pi_e(G_1) = \\pi_e(G_2)$ (the set of orders of elements), such that every finite group with spectrum $\\pi_e(G_1)$ has order at most some fixed bound $B$, and such that neither of $G_1$, $G_2$ is isomorphic to a subgroup or to a\u2026","title":"Finite groups with equal spectra, neither a subgroup or normal-subgroup quotient of the other (problem)"},{"id":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/same-spectrum-incomparable-groups-holds","kind":"node","lean":"MathlibExt.GroupTheory.SameSpectrumIncomparableGroupsKourovka16Problem106Wanted.sameSpectrumIncomparableGroups_holds","parent":2688,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum/index.html#same-spectrum-incomparable-groups-holds","scope":"GroupTheory/orders-and-spectra/finite-groups-with-the-same-spectrum","status":"can_prove","summary":"Finite groups with equal spectra, neither a subgroup or normal-subgroup quotient of the other (Kourovka 16.106). There exist finite groups $G_1$ and $G_2$ such that:","title":"Finite groups with equal spectra, neither a subgroup or normal-subgroup quotient of the other (Kourovka 16.106)"},{"data_url":"scopes/GroupTheory/orders-and-spectra/groups-of-order-10.json","id":"GroupTheory/orders-and-spectra/groups-of-order-10","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/groups-of-order-10/index.html","scope":"GroupTheory/orders-and-spectra/groups-of-order-10","status":"fully_proved","title":"Groups of order 10"},{"id":"GroupTheory/orders-and-spectra/groups-of-order-10/nonempty-mul-equiv-dihedral-group-of-car","kind":"node","lean":"MetaMathlibExt.nonempty_mulEquiv_dihedralGroup_of_card_eq_ten_of_not_isCyclic","parent":2693,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/groups-of-order-10/index.html#nonempty-mul-equiv-dihedral-group-of-car","scope":"GroupTheory/orders-and-spectra/groups-of-order-10","status":"fully_proved","summary":"Every finite group of order $10$ that is not cyclic is isomorphic to the dihedral group $D_5$ of order $10$.","title":"Noncyclic groups of order 10 are dihedral"},{"id":"GroupTheory/orders-and-spectra/groups-of-order-10/groups-of-order-10-classification","kind":"node","lean":"MetaMathlibExt.groups_of_order_10_classification","parent":2693,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/groups-of-order-10/index.html#groups-of-order-10-classification","scope":"GroupTheory/orders-and-spectra/groups-of-order-10","status":"fully_proved","summary":"Every finite group of order $10$ is isomorphic either to the cyclic group $C_{10}$ of order $10$ or to the dihedral group $D_5$ of order $10$.","title":"Classification of groups of order 10"},{"id":"GroupTheory/orders-and-spectra/groups-of-order-10/groups-of-order-10-classification-xor","kind":"node","lean":"MetaMathlibExt.groups_of_order_10_classification_xor","parent":2693,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/groups-of-order-10/index.html#groups-of-order-10-classification-xor","scope":"GroupTheory/orders-and-spectra/groups-of-order-10","status":"fully_proved","summary":"Let $G$ be a finite group of order $10$. Then exactly one of the following holds: $G$ is isomorphic to the cyclic group $C_{10}$ of order $10$, or $G$ is isomorphic to the dihedral group $D_5$ of order $10$.","title":"Exclusive classification of groups of order 10"},{"id":"GroupTheory/orders-and-spectra/groups-of-order-10/groups-of-order-10-classification-prod","kind":"node","lean":"MetaMathlibExt.groups_of_order_10_classification_prod","parent":2693,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/groups-of-order-10/index.html#groups-of-order-10-classification-prod","scope":"GroupTheory/orders-and-spectra/groups-of-order-10","status":"fully_proved","summary":"Let $C_n$ denote the cyclic group of order $n$ and let $D_5$ denote the dihedral group of order $10$. Every finite group of order $10$ is isomorphic to the direct product $C_5 \\times C_2$ or to $D_5$.","title":"Classification of groups of order 10, product form"},{"id":"GroupTheory/orders-and-spectra/groups-of-order-10/groups-of-order-10-classification-prod-x","kind":"node","lean":"MetaMathlibExt.groups_of_order_10_classification_prod_xor","parent":2693,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/groups-of-order-10/index.html#groups-of-order-10-classification-prod-x","scope":"GroupTheory/orders-and-spectra/groups-of-order-10","status":"fully_proved","summary":"Let $G$ be a finite group of order $10$, let $C_n$ denote the cyclic group of order $n$, and let $D_5$ denote the dihedral group of order $10$. Then exactly one of the following holds: $G$ is isomorphic to the direct product $C_5 \\times C_2$, or $G$ is isomorphic to $D_5$.","title":"Exclusive classification of groups of order 10, product form"},{"data_url":"scopes/GroupTheory/orders-and-spectra/isospectral-finite-simple-groups.json","id":"GroupTheory/orders-and-spectra/isospectral-finite-simple-groups","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/isospectral-finite-simple-groups/index.html","scope":"GroupTheory/orders-and-spectra/isospectral-finite-simple-groups","status":"can_prove","title":"Isospectral finite simple groups"},{"id":"GroupTheory/orders-and-spectra/isospectral-finite-simple-groups/spectrum","kind":"node","lean":"MathlibExt.GroupTheory.ThreeIsospectralSimpleGroupsKourovka16Problem25Wanted.spectrum","parent":2699,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/isospectral-finite-simple-groups/index.html#spectrum","scope":"GroupTheory/orders-and-spectra/isospectral-finite-simple-groups","status":"fully_proved","summary":"The spectrum $\\omega(G)$ of a group $G$ is the set of orders of its elements, where an element of infinite order is counted as having order $0$.","title":"Spectrum of a group"},{"id":"GroupTheory/orders-and-spectra/isospectral-finite-simple-groups/three-isospectral-simple-groups","kind":"node","lean":"MathlibExt.GroupTheory.ThreeIsospectralSimpleGroupsKourovka16Problem25Wanted.threeIsospectralSimpleGroups","parent":2699,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/isospectral-finite-simple-groups/index.html#three-isospectral-simple-groups","scope":"GroupTheory/orders-and-spectra/isospectral-finite-simple-groups","status":"fully_proved","summary":"Kourovka 16.25 (A. V. Vasil'ev): there are three pairwise non-isomorphic finite non-abelian simple groups with the same spectrum.","title":"Three isospectral finite simple groups"},{"id":"GroupTheory/orders-and-spectra/isospectral-finite-simple-groups/three-isospectral-simple-groups-refuted","kind":"node","lean":"MathlibExt.GroupTheory.ThreeIsospectralSimpleGroupsKourovka16Problem25Wanted.threeIsospectralSimpleGroups_refuted","parent":2699,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/isospectral-finite-simple-groups/index.html#three-isospectral-simple-groups-refuted","scope":"GroupTheory/orders-and-spectra/isospectral-finite-simple-groups","status":"can_prove","summary":"There do not exist three pairwise non-isomorphic finite non-abelian simple groups $G_1, G_2, G_3$ with $\\omega(G_1) = \\omega(G_2) = \\omega(G_3)$, where the spectrum $\\omega(G)$ of a group $G$ is the set of orders of its elements.","title":"No three non-isomorphic finite non-abelian simple groups are isospectral"},{"data_url":"scopes/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum.json","id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"can_prove","title":"Number of groups with a given spectrum"},{"id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/element-orders","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderSetRealizationsKourovka12Problem84Wanted.elementOrders","parent":2703,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html#element-orders","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"fully_proved","summary":"The spectrum of a group $G$ is the set $\\{\\operatorname{ord}(g) : g \\in G\\}$ of the orders of its elements, where an element of infinite order is counted as having order $0$.","title":"The spectrum of a group (Kourovka 12.84)"},{"id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/two-realizations-imply-infinitely-many","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderSetRealizationsKourovka12Problem84Wanted.twoRealizationsImplyInfinitelyMany","parent":2703,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html#two-realizations-imply-infinitely-many","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"fully_proved","summary":"The proposition that for every set $\\Omega$ of natural numbers, if two non-isomorphic finite groups both have set of element orders $\\Omega$, then for every natural number $k$ there are $k$ pairwise non-isomorphic finite groups whose set of element orders is $\\Omega$; that is, infinitely many.","title":"Syskin's question on groups with a given spectrum (Kourovka 12.84)"},{"id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/two-realizations-imply-infinitely-many-r","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderSetRealizationsKourovka12Problem84Wanted.twoRealizationsImplyInfinitelyMany_refuted","parent":2703,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html#two-realizations-imply-infinitely-many-r","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"can_prove","summary":"There are a set $\\Omega$ of natural numbers and a natural number $k$ such that two non-isomorphic finite groups have set of element orders $\\Omega$, but there are no $k$ pairwise non-isomorphic finite groups whose set of element orders is $\\Omega$.","title":"Two isospectral finite groups do not force infinitely many"},{"id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/element-orders-2","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderSpectrumCountKourovka13Problem63Wanted.elementOrders","parent":2703,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html#element-orders-2","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"fully_proved","summary":"For a group $G$, the spectrum $\\pi_e(G)$ is the set of orders of elements of $G$, where an element of infinite order counts as having order $0$.","title":"Spectrum of a group (Kourovka 13.63)"},{"id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/at-most-iso-classes","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderSpectrumCountKourovka13Problem63Wanted.AtMostIsoClasses","parent":2703,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most "],["math","\\(k\\)"],["text"," isomorphism types with spectrum "],["math","\\(\\Gamma\\)"]],"source":"At most $k$ isomorphism types with spectrum $\\Gamma$"},"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html#at-most-iso-classes","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"fully_proved","summary":"For a set $\\Gamma$ of natural numbers and a natural number $k$, this property holds when, for any finite groups $G_0, \\dots, G_k$ (not necessarily distinct) whose spectra $\\pi_e(G_i)$ (the sets of orders of elements of $G_i$) all equal $\\Gamma$, there are indices $i \\ne j$ with $G_i \\cong G_j$.","title":"At most $k$ isomorphism types with spectrum $\\Gamma$"},{"id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/infinitely-many-iso-classes","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderSpectrumCountKourovka13Problem63Wanted.InfinitelyManyIsoClasses","parent":2703,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many isomorphism types with spectrum "],["math","\\(\\Gamma\\)"]],"source":"Infinitely many isomorphism types with spectrum $\\Gamma$"},"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html#infinitely-many-iso-classes","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"fully_proved","summary":"For a set $\\Gamma$ of natural numbers, this property holds when, for every natural number $n$, there are $n$ pairwise non-isomorphic finite groups $G_0, \\dots, G_{n-1}$ whose spectra $\\pi_e(G_i)$ (the sets of orders of elements of $G_i$) all equal $\\Gamma$.","title":"Infinitely many isomorphism types with spectrum $\\Gamma$"},{"id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/spectrum-group-count-bounded-or-infinite","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderSpectrumCountKourovka13Problem63Wanted.spectrumGroupCountBoundedOrInfinite","parent":2703,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html#spectrum-group-count-bounded-or-infinite","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"fully_proved","summary":"The proposition, asked about by W. J. Shi, that there is a natural number $k$ such that for every set $\\Gamma$ of natural numbers, either $h(\\Gamma) \\le k$ or $h(\\Gamma) = \\infty$, where $h(\\Gamma)$ is the number of isomorphism classes of finite groups $G$ whose spectrum $\\pi_e(G)$ (set of element orders) is $\\Gamma$. Here $h(\\Gamma) \\le k$ means that among any $k + 1$ finite groups with spectrum $\\Gamma$ two are\u2026","title":"Kourovka Problem 13.63 (bounded or infinite number of groups per spectrum)"},{"id":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/spectrum-group-count-bounded-or-infinite-2","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrderSpectrumCountKourovka13Problem63Wanted.spectrumGroupCountBoundedOrInfinite_refuted","parent":2703,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum/index.html#spectrum-group-count-bounded-or-infinite-2","scope":"GroupTheory/orders-and-spectra/number-of-groups-with-a-given-spectrum","status":"can_prove","summary":"For a set $\\Gamma$ of natural numbers, let $h(\\Gamma)$ be the number of isomorphism classes of finite groups $G$ whose spectrum $\\pi_e(G)$ (set of element orders) is $\\Gamma$; $h(\\Gamma) \\le k$ means that among any $k + 1$ finite groups with spectrum $\\Gamma$ two are isomorphic, and $h(\\Gamma) = \\infty$ means that for every $n$ there are $n$ pairwise non-isomorphic finite groups with spectrum $\\Gamma$. Kourovka\u2026","title":"No uniform bound on the number of groups with a given spectrum"},{"data_url":"scopes/GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups.json","id":"GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups/index.html","scope":"GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups","status":"can_prove","title":"Order-divisibility bijections onto cyclic groups"},{"id":"GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups/order-divisibility-bijection-to-cyclic","kind":"node","lean":"MathlibExt.GroupTheory.OrderDivisibilityBijectionCyclicKourovka18Problem1Wanted.orderDivisibilityBijectionToCyclic","parent":2712,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups/index.html#order-divisibility-bijection-to-cyclic","scope":"GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups","status":"fully_proved","summary":"The statement that for every finite group $G$, of order $n$ say, there is a bijection $f$ from $G$ onto the cyclic group $\\mathbb{Z}/n\\mathbb{Z}$ such that for every $x \\in G$ the order of $x$ divides the additive order of $f(x)$.","title":"Kourovka Problem 18.1 (statement)"},{"id":"GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups/order-divisibility-bijection-to-cyclic-h","kind":"node","lean":"MathlibExt.GroupTheory.OrderDivisibilityBijectionCyclicKourovka18Problem1Wanted.orderDivisibilityBijectionToCyclic_holds","parent":2712,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups/index.html#order-divisibility-bijection-to-cyclic-h","scope":"GroupTheory/orders-and-spectra/order-divisibility-bijections-onto-cyclic-groups","status":"can_prove","summary":"For every finite group $G$, of order $n$ say, there is a bijection $f$ from $G$ onto the cyclic group $\\mathbb{Z}/n\\mathbb{Z}$ such that for every $x \\in G$ the order of $x$ divides the additive order of $f(x)$.","title":"Amiri's theorem (Kourovka Problem 18.1)"},{"data_url":"scopes/GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou.json","id":"GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou/index.html","scope":"GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou","status":"can_prove","title":"Prime divisors of element orders in soluble groups"},{"id":"GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou/soluble-prime-count-linear-in-element-pr","kind":"node","lean":"MathlibExt.GroupTheory.SolublePrimeDivisorsLinearBoundKourovka14Problem66Wanted.solublePrimeCountLinearInElementPrimeCount","parent":2715,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou/index.html#soluble-prime-count-linear-in-element-pr","scope":"GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou","status":"fully_proved","summary":"The statement, posed by A. Moret\u00f3, that there are natural numbers $a$ and $b$ such that for every finite soluble group $G$ and every natural number $m$, if the order of every element of $G$ has at most $m$ distinct prime divisors, then $|G|$ has at most $am + b$ distinct prime divisors.","title":"Linear bound on the prime divisors of a finite soluble group (Kourovka 14.66)"},{"id":"GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou/soluble-prime-count-linear-in-element-pr-2","kind":"node","lean":"MathlibExt.GroupTheory.SolublePrimeDivisorsLinearBoundKourovka14Problem66Wanted.solublePrimeCountLinearInElementPrimeCount_holds","parent":2715,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou/index.html#soluble-prime-count-linear-in-element-pr-2","scope":"GroupTheory/orders-and-spectra/prime-divisors-of-element-orders-in-soluble-grou","status":"can_prove","summary":"There are natural numbers $a$ and $b$ such that for every finite soluble group $G$ and every natural number $m$, if the order of every element of $G$ has at most $m$ distinct prime divisors, then $|G|$ has at most $am + b$ distinct prime divisors. That is, Kourovka Problem 14.66 has a positive answer.","title":"Finite soluble groups satisfy a linear prime-divisor bound"},{"data_url":"scopes/GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum.json","id":"GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum/index.html","scope":"GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum","status":"can_prove","title":"Recognition by order and spectrum"},{"id":"GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum/element-orders","kind":"node","lean":"MathlibExt.GroupTheory.OrderSpectrumSimpleGroupKourovka12Problem39Wanted.elementOrders","parent":2718,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum/index.html#element-orders","scope":"GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum","status":"fully_proved","summary":"For a group $G$, the set of natural numbers that occur as the order of some element of $G$, where an element of infinite order is counted as having order $0$.","title":"Set of element orders (Kourovka 12.39)"},{"id":"GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum/simple-group-recognized-by-order-and-spe","kind":"node","lean":"MathlibExt.GroupTheory.OrderSpectrumSimpleGroupKourovka12Problem39Wanted.simpleGroupRecognizedByOrderAndSpectrum","parent":2718,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum/index.html#simple-group-recognized-by-order-and-spe","scope":"GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum","status":"fully_proved","summary":"The statement that for every finite group $G$ and every finite simple group $S$, if $G$ and $S$ have the same order and the same set of element orders, then $G$ is isomorphic to $S$.","title":"Shi's conjecture on recognition by order and element orders (Kourovka 12.39)"},{"id":"GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum/simple-group-recognized-by-order-and-spe-2","kind":"node","lean":"MathlibExt.GroupTheory.OrderSpectrumSimpleGroupKourovka12Problem39Wanted.simpleGroupRecognizedByOrderAndSpectrum_holds","parent":2718,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum/index.html#simple-group-recognized-by-order-and-spe-2","scope":"GroupTheory/orders-and-spectra/recognition-by-order-and-spectrum","status":"can_prove","summary":"For every finite group $G$ and every finite simple group $S$, if $G$ and $S$ have the same order and the same set of element orders, then $G$ is isomorphic to $S$. That is, Shi's conjecture (Kourovka Problem 12.39) holds.","title":"Finite simple groups are recognized by order and element orders"},{"data_url":"scopes/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum.json","id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"can_prove","title":"Recognition of alternating groups by spectrum"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/element-orders","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingGroupSpectrumRecognitionKourovka16Problem107Wanted.elementOrders","parent":2722,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#element-orders","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"fully_proved","summary":"The spectrum $\\pi_e(G)$ of a group $G$ is the set of orders of the elements of $G$, where an element of infinite order is assigned order $0$.","title":"Spectrum of a group"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/alternating-group-recognizable-by-spectr","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingGroupSpectrumRecognitionKourovka16Problem107Wanted.alternatingGroupRecognizableBySpectrum","parent":2722,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#alternating-group-recognizable-by-spectr","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"fully_proved","summary":"This proposition asserts that there is a natural number $N$ such that for every $n \\ge N$, every finite group $G$ with the same spectrum (set of element orders) as the alternating group $A_n$ is isomorphic to $A_n$.","title":"Kourovka Problem 16.107 on recognition of alternating groups by spectrum"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/alternating-group-recognizable-by-spectr-2","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingGroupSpectrumRecognitionKourovka16Problem107Wanted.alternatingGroupRecognizableBySpectrum_holds","parent":2722,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#alternating-group-recognizable-by-spectr-2","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"can_prove","summary":"There is a natural number $N$ such that for every $n \\ge N$, every finite group $G$ whose set of element orders equals that of the alternating group $A_n$ is isomorphic to $A_n$.","title":"Large alternating groups are recognizable by spectrum"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/spectrum","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingSpectrumCompositionFactorsKourovka16Problem27Wanted.spectrum","parent":2722,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#spectrum","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"fully_proved","summary":"The spectrum $\\omega(G)$ of a group $G$ is the set of natural numbers that are orders of elements of $G$, where an element of infinite order is assigned order $0$.","title":"Spectrum of a group"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/is-composition-series","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingSpectrumCompositionFactorsKourovka16Problem27Wanted.IsCompositionSeries","parent":2722,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#is-composition-series","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"fully_proved","summary":"Let $G$ be a group and $m$ a natural number. A sequence $s_0, s_1, \\dots, s_m$ of subgroups of $G$ is a composition series of length $m$ if $s_0$ is the trivial subgroup, $s_m = G$, and for each $i < m$ the subgroup $s_i$ is a normal subgroup of $s_{i+1}$ and the quotient $s_{i+1}/s_i$ is a simple group.","title":"Finite composition series"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/is-nonabelian-factor","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingSpectrumCompositionFactorsKourovka16Problem27Wanted.IsNonabelianFactor","parent":2722,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#is-nonabelian-factor","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"fully_proved","summary":"Let $G$ be a group, let $s_0, s_1, \\dots, s_m$ be subgroups of $G$, and let $i < m$. The $i$-th factor of the chain is non-abelian if $s_i \\cap s_{i+1}$ is a normal subgroup of $s_{i+1}$ and the quotient group $s_{i+1}/(s_i \\cap s_{i+1})$ is non-abelian, that is, it has elements $x, y$ with $xy \\ne yx$. When $s_i \\le s_{i+1}$, this quotient is $s_{i+1}/s_i$.","title":"Non-abelian factor of a chain of subgroups"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/alternating-spectrum-one-nonabelian-fact","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingSpectrumCompositionFactorsKourovka16Problem27Wanted.alternatingSpectrumOneNonabelianFactor","parent":2722,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#alternating-spectrum-one-nonabelian-fact","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"fully_proved","summary":"The statement that for every natural number $n$ and every finite group $G$ whose spectrum, the set of orders of its elements, equals the spectrum of the alternating group $A_n$, every composition series of $G$ has at most one non-abelian factor.","title":"Kourovka Problem 16.27 (statement)"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/alternating-spectrum-one-nonabelian-fact-2","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingSpectrumCompositionFactorsKourovka16Problem27Wanted.alternatingSpectrumOneNonabelianFactor_holds","parent":2722,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#alternating-spectrum-one-nonabelian-fact-2","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"can_prove","summary":"Let $n$ be a natural number and let $G$ be a finite group whose spectrum, the set of orders of its elements, equals the spectrum of the alternating group $A_n$. Then every composition series of $G$ has at most one non-abelian factor.","title":"Groups isospectral to an alternating group (Kourovka Problem 16.27)"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/oc-seven-group-is-alternating","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrdersOneToSevenAlternatingKourovka19Problem80Wanted.ocSevenGroupIsAlternating","parent":2722,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 19.80 ("],["math","\\(OC_7\\)"],["text","-groups)"]],"source":"Kourovka Problem 19.80 ($OC_7$-groups)"},"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#oc-seven-group-is-alternating","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"fully_proved","summary":"The proposition, asked by W. J. Shi, that every $OC_7$-group, that is, every group whose set of element orders is exactly $\\{1, 2, \\dots, 7\\}$ (such a group is periodic), is isomorphic to the alternating group $A_7$.","title":"Kourovka Problem 19.80 ($OC_7$-groups)"},{"id":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/oc-seven-group-is-alternating-holds","kind":"node","lean":"MathlibExt.GroupTheory.ElementOrdersOneToSevenAlternatingKourovka19Problem80Wanted.ocSevenGroupIsAlternating_holds","parent":2722,"reading_title":{"format":"inline-parts-v1","parts":[["text","Groups with element orders "],["math","\\(1\\)"],["text"," to "],["math","\\(7\\)"],["text"," are isomorphic to "],["math","\\(A_7\\)"]],"source":"Groups with element orders $1$ to $7$ are isomorphic to $A_7$"},"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum/index.html#oc-seven-group-is-alternating-holds","scope":"GroupTheory/orders-and-spectra/recognition-of-alternating-groups-by-spectrum","status":"can_prove","summary":"Every group, finite or infinite, whose set of element orders is exactly $\\{1, 2, \\dots, 7\\}$ is isomorphic to the alternating group $A_7$. This answers Kourovka Problem 19.80 positively (Mamontov and Jabara).","title":"Groups with element orders $1$ to $7$ are isomorphic to $A_7$"},{"data_url":"scopes/GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum.json","id":"GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum/index.html","scope":"GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum","status":"can_prove","title":"Recognition of PSL(2, 7) by spectrum"},{"id":"GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum/spectrum-recognizes-psl27","kind":"node","lean":"MathlibExt.GroupTheory.SpectrumRecognitionPSL27Kourovka16Problem57Wanted.spectrumRecognizesPSL27","parent":2733,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum/index.html#spectrum-recognizes-psl27","scope":"GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum","status":"fully_proved","summary":"The statement that every group $G$, finite or infinite, whose set of element orders $\\{\\operatorname{ord}(g) : g \\in G\\}$ equals $\\{1, 2, 3, 4, 7\\}$ is isomorphic to $L_2(7) = \\mathrm{PSL}(2, 7)$, the projective special linear group of $2 \\times 2$ matrices over the field $\\mathbb{Z}/7\\mathbb{Z}$.","title":"Kourovka Problem 16.57: recognition of PSL(2, 7) by spectrum"},{"id":"GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum/spectrum-recognizes-psl27-holds","kind":"node","lean":"MathlibExt.GroupTheory.SpectrumRecognitionPSL27Kourovka16Problem57Wanted.spectrumRecognizesPSL27_holds","parent":2733,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum/index.html#spectrum-recognizes-psl27-holds","scope":"GroupTheory/orders-and-spectra/recognition-of-psl-2-7-by-spectrum","status":"can_prove","summary":"Every group $G$, finite or infinite, whose set of element orders $\\{\\operatorname{ord}(g) : g \\in G\\}$ equals $\\{1, 2, 3, 4, 7\\}$ is isomorphic to $L_2(7) = \\mathrm{PSL}(2, 7)$, the projective special linear group of $2 \\times 2$ matrices over the field $\\mathbb{Z}/7\\mathbb{Z}$.","title":"Lytkina\u2013Kuznetsov theorem: PSL(2, 7) is recognizable by spectrum"},{"data_url":"scopes/GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr.json","id":"GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr/index.html","scope":"GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr","status":"can_prove","title":"Subgroups with all prime divisors of a simple group"},{"id":"GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr/finitely-many-simple-with-full-spectrum","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupPrimeSpectrumSubgroupKourovka11Problem64Wanted.finitelyManySimpleWithFullSpectrumSubgroup","parent":2736,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr/index.html#finitely-many-simple-with-full-spectrum","scope":"GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr","status":"fully_proved","summary":"The statement that there is a natural number $N$ such that every finite simple group $G$ that is not isomorphic to the alternating group $A_n$ for any natural number $n$ and has a proper subgroup $H$ with $\\pi(H) = \\pi(G)$ satisfies $|G| \\le N$, where $\\pi(X)$ denotes the set of prime divisors of the order $|X|$.","title":"Kourovka Problem 11.64 (Monakhov)"},{"id":"GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr/finitely-many-simple-with-full-spectrum-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupPrimeSpectrumSubgroupKourovka11Problem64Wanted.finitelyManySimpleWithFullSpectrumSubgroup_refuted","parent":2736,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr/index.html#finitely-many-simple-with-full-spectrum-2","scope":"GroupTheory/orders-and-spectra/subgroups-with-all-prime-divisors-of-a-simple-gr","status":"can_prove","summary":"Resolved false: the Kourovka Archive records (V. I. Zenkov) that the symplectic groups `G = S_{4k}(2^s)` have proper subgroups `H \u2245 \u03a9\u207b_{4k}(2^s)` with `\u03c0(H) = \u03c0(G)`, an infinite family of non-alternating finite simple groups. Source: E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 11.64 (V.\u2026","title":"Infinitely many simple groups with proper subgroups of equal prime set"},{"data_url":"scopes/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type.json","id":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/index.html","scope":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","status":"can_prove","title":"Thompson's problem on groups of the same type"},{"id":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/same-power-equation-type","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem134bWanted.SamePowerEquationType","parent":2739,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/index.html#same-power-equation-type","scope":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","status":"fully_proved","summary":"Two groups $G$ and $H$ have the same power-equation type when, for every positive integer $n$, the equation $x^n = 1$ has the same number of solutions in $G$ as in $H$. An infinite set of solutions is counted as $0$, so the notion is meant for finite groups.","title":"Same power-equation type"},{"id":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/is-almost-simple","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem134bWanted.IsAlmostSimple","parent":2739,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/index.html#is-almost-simple","scope":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","status":"fully_proved","summary":"A group $G$ is almost simple when there is a nonabelian finite simple group $S$ such that $G$ is isomorphic to a subgroup $A$ of the automorphism group $\\mathrm{Aut}(S)$ that contains the inner automorphism group $\\mathrm{Inn}(S)$.","title":"Almost simple group"},{"id":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/conjecture","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem134bWanted.conjecture","parent":2739,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/index.html#conjecture","scope":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","status":"fully_proved","summary":"The conjecture that for all finite groups $G$ and $H$, if $G$ is almost simple and $G$ and $H$ have the same power-equation type, then $G$ is isomorphic to $H$.","title":"Characterization of almost simple groups by power-equation type (Kourovka 21.134(b))"},{"id":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/conjecture-refuted","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem134bWanted.conjecture_refuted","parent":2739,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/index.html#conjecture-refuted","scope":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","status":"can_prove","summary":"There exist finite groups $G$ and $H$ such that $G$ is almost simple, $G$ and $H$ have the same power-equation type, and $G$ is not isomorphic to $H$. That is, the conjecture of Kourovka Problem 21.134(b) is false.","title":"Power-equation type does not characterize almost simple groups"},{"id":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/same-type","kind":"node","lean":"MathlibExt.GroupTheory.SolubleTypeRecognitionKourovka12Problem37Wanted.SameType","parent":2739,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/index.html#same-type","scope":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","status":"fully_proved","summary":"Two groups $G$ and $H$ have the same type when, for every integer $n \\ge 1$, the equation $x^n = 1$ has the same number of solutions in $G$ as in $H$, where an infinite set of solutions counts as having $0$ elements.","title":"Groups of the same type"},{"id":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/soluble-of-same-type-as-soluble","kind":"node","lean":"MathlibExt.GroupTheory.SolubleTypeRecognitionKourovka12Problem37Wanted.solubleOfSameTypeAsSoluble","parent":2739,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/index.html#soluble-of-same-type-as-soluble","scope":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","status":"fully_proved","summary":"Every finite group of the same type as a finite soluble group is soluble: if $G$ and $H$ are finite groups of the same type and $H$ is soluble, then $G$ is soluble.","title":"Kourovka Problem 12.37 (J. G. Thompson)"},{"id":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/soluble-of-same-type-as-soluble-refuted","kind":"node","lean":"MathlibExt.GroupTheory.SolubleTypeRecognitionKourovka12Problem37Wanted.solubleOfSameTypeAsSoluble_refuted","parent":2739,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type/index.html#soluble-of-same-type-as-soluble-refuted","scope":"GroupTheory/orders-and-spectra/thompson-s-problem-on-groups-of-the-same-type","status":"can_prove","summary":"The answer to Kourovka Problem 12.37 is negative: there are finite groups $G$ and $H$ of the same type such that $H$ is soluble and $G$ is not soluble.","title":"Solubility is not determined by the type"},{"data_url":"scopes/GroupTheory/orders-and-spectra/totient-sums-of-element-orders.json","id":"GroupTheory/orders-and-spectra/totient-sums-of-element-orders","kind":"scope","navigation":"module","parent":2676,"result":false,"route":"../roadmap/GroupTheory/orders-and-spectra/totient-sums-of-element-orders/index.html","scope":"GroupTheory/orders-and-spectra/totient-sums-of-element-orders","status":"can_prove","title":"Totient sums of element orders"},{"id":"GroupTheory/orders-and-spectra/totient-sums-of-element-orders/same-order-totient-sum-simple","kind":"node","lean":"MathlibExt.GroupTheory.TotientSumSimplicityKourovka19Problem25Wanted.sameOrderTotientSumSimple","parent":2747,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/totient-sums-of-element-orders/index.html#same-order-totient-sum-simple","scope":"GroupTheory/orders-and-spectra/totient-sums-of-element-orders","status":"fully_proved","summary":"The statement that for all finite groups $G$ and $H$ of the same order with $\\sum_{g \\in G} \\varphi(|g|) = \\sum_{h \\in H} \\varphi(|h|)$, where $|g|$ is the order of $g$ and $\\varphi$ is Euler's totient function, if $G$ is simple then $H$ is simple.","title":"Curtin\u2013Pourgholi question on totient sums and simplicity (Kourovka 19.25)"},{"id":"GroupTheory/orders-and-spectra/totient-sums-of-element-orders/same-order-totient-sum-simple-refuted","kind":"node","lean":"MathlibExt.GroupTheory.TotientSumSimplicityKourovka19Problem25Wanted.sameOrderTotientSumSimple_refuted","parent":2747,"result":true,"route":"../roadmap/GroupTheory/orders-and-spectra/totient-sums-of-element-orders/index.html#same-order-totient-sum-simple-refuted","scope":"GroupTheory/orders-and-spectra/totient-sums-of-element-orders","status":"can_prove","summary":"There are finite groups $G$ and $H$ of the same order with $\\sum_{g \\in G} \\varphi(|g|) = \\sum_{h \\in H} \\varphi(|h|)$, where $|g|$ is the order of $g$ and $\\varphi$ is Euler's totient function, such that $G$ is simple and $H$ is not simple. So the Curtin\u2013Pourgholi question has a negative answer.","title":"Order and totient sum do not determine simplicity (Kourovka 19.25)"},{"data_url":"scopes/GroupTheory/conjugacy-classes.json","id":"GroupTheory/conjugacy-classes","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/index.html","scope":"GroupTheory/conjugacy-classes","status":"can_prove","summary":"Conjugacy classes of groups as a structural invariant: class sizes and breadth, the class number, commuting probability, Thompson's conjecture on class sizes, products of conjugacy classes, and relations between class sizes and group structure; class sizes of p-groups go to Finite p-groups, and character-theoretic class arguments to Characters of finite groups.","title":"Conjugacy classes"},{"id":"GroupTheory/conjugacy-classes/comm-prob-closed","kind":"node","lean":"MathlibExt.GroupTheory.CommProbClosedWanted.GroupTheory.commProb_closed","parent":2750,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/index.html#comm-prob-closed","scope":"GroupTheory/conjugacy-classes","status":"can_prove","summary":"Let $\\Pr(G)$ denote the commuting probability of a group $G$: $\\Pr(G) = |\\{(x, y) \\in G \\times G : xy = yx\\}| / |G|^2$ if $G$ is finite, and $\\Pr(G) = 0$ if $G$ is infinite. Then the set $\\{\\Pr(G) : G \\text{ a group}\\}$ is a closed subset of $\\mathbb{R}$.","title":"The set of commuting probabilities is closed"},{"data_url":"scopes/GroupTheory/conjugacy-classes/breadth-of-a-finite-group.json","id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","kind":"scope","navigation":"module","parent":2750,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","title":"Breadth of a finite group"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/element-breadth","kind":"node","lean":"ClassBreadth.elementBreadth","parent":2752,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#element-breadth","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"Let $G$ be a finite group, $p$ a natural number and $g \\in G$, and write $g^G$ for the conjugacy class of $g$. When $p \\ge 2$, the breadth $\\operatorname{br}_p(g)$ of $g$ is $\\lfloor \\log_p |g^G| \\rfloor$, the largest natural number $k$ with $p^k \\le |g^G|$; when $p \\le 1$, it is $0$.","title":"Breadth of an element"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/breadth","kind":"node","lean":"ClassBreadth.breadth","parent":2752,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#breadth","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"For a finite group $G$ and a natural number $p$, the breadth $\\operatorname{br}_p(G)$ of $G$ is the maximum of the element breadths $\\operatorname{br}_p(g)$ over all $g \\in G$.","title":"Breadth of a finite group"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/card-carrier-dvd-card","kind":"node","lean":"ClassBreadth.card_carrier_dvd_card","parent":2752,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#card-carrier-dvd-card","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"In a finite group $G$, the number of elements of the conjugacy class of any $g \\in G$ divides the order $|G|$.","title":"Conjugacy class sizes divide the group order"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/exists-card-carrier-eq-pow","kind":"node","lean":"ClassBreadth.exists_card_carrier_eq_pow","parent":2752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conjugacy classes of a finite "],["math","\\(p\\)"],["text","-group have "],["math","\\(p\\)"],["text","-power size"]],"source":"Conjugacy classes of a finite $p$-group have $p$-power size"},"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#exists-card-carrier-eq-pow","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"Let $p$ be a prime and $G$ a finite $p$-group. For every $g \\in G$ there is a natural number $k$ such that the conjugacy class of $g$ has exactly $p^k$ elements.","title":"Conjugacy classes of a finite $p$-group have $p$-power size"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/element-breadth-eq-of-card-eq","kind":"node","lean":"ClassBreadth.elementBreadth_eq_of_card_eq","parent":2752,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#element-breadth-eq-of-card-eq","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"Let $G$ be a finite group, $p$ a prime, $k$ a natural number and $g \\in G$. If the conjugacy class of $g$ has exactly $p^k$ elements, then the breadth $\\operatorname{br}_p(g)$ of $g$ equals $k$.","title":"Breadth from an exact class size"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/pow-element-breadth-eq-card","kind":"node","lean":"ClassBreadth.pow_elementBreadth_eq_card","parent":2752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Class sizes in a "],["math","\\(p\\)"],["text","-group via breadth"]],"source":"Class sizes in a $p$-group via breadth"},"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#pow-element-breadth-eq-card","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"Let $p$ be a prime, $G$ a finite $p$-group and $g \\in G$, and write $g^G$ for the conjugacy class of $g$. Then the breadth $\\operatorname{br}_p(g)$ of $g$ satisfies $p^{\\operatorname{br}_p(g)} = |g^G|$.","title":"Class sizes in a $p$-group via breadth"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/carrier-mk-one","kind":"node","lean":"ClassBreadth.carrier_mk_one","parent":2752,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#carrier-mk-one","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"In every group, the conjugacy class of the identity element $1$ is the singleton $\\{1\\}$.","title":"The conjugacy class of the identity"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/element-breadth-one","kind":"node","lean":"ClassBreadth.elementBreadth_one","parent":2752,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#element-breadth-one","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"In a finite group, the breadth $\\operatorname{br}_p(1)$ of the identity element $1$ is $0$ for every natural number $p$.","title":"Breadth of the identity"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/element-breadth-le-breadth","kind":"node","lean":"ClassBreadth.elementBreadth_le_breadth","parent":2752,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#element-breadth-le-breadth","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"Let $G$ be a finite group and $p$ a natural number. For every $g \\in G$, the breadth $\\operatorname{br}_p(g)$ of $g$ is at most the breadth $\\operatorname{br}_p(G)$ of $G$.","title":"Element breadth is at most the group breadth"},{"id":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group/exists-breadth-eq-element-breadth","kind":"node","lean":"ClassBreadth.exists_breadth_eq_elementBreadth","parent":2752,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/breadth-of-a-finite-group/index.html#exists-breadth-eq-element-breadth","scope":"GroupTheory/conjugacy-classes/breadth-of-a-finite-group","status":"fully_proved","summary":"Let $G$ be a finite group and $p$ a natural number. There is an element $g \\in G$ whose breadth $\\operatorname{br}_p(g)$ equals the breadth $\\operatorname{br}_p(G)$ of $G$.","title":"The group breadth is attained"},{"data_url":"scopes/GroupTheory/conjugacy-classes/centralizers-of-maximal-order.json","id":"GroupTheory/conjugacy-classes/centralizers-of-maximal-order","kind":"scope","navigation":"module","parent":2750,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/centralizers-of-maximal-order/index.html","scope":"GroupTheory/conjugacy-classes/centralizers-of-maximal-order","status":"can_prove","title":"Centralizers of maximal order"},{"id":"GroupTheory/conjugacy-classes/centralizers-of-maximal-order/maximal-centralizer-contains-normalized","kind":"node","lean":"MathlibExt.GroupTheory.MaximalCentralizerNilpotentSubgroupKourovka14Problem67Wanted.maximalCentralizerContainsNormalizedNilpotent","parent":2763,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/centralizers-of-maximal-order/index.html#maximal-centralizer-contains-normalized","scope":"GroupTheory/conjugacy-classes/centralizers-of-maximal-order","status":"fully_proved","summary":"For an element $x$ of a group $G$ write $C_G(x)$ for its centralizer. This is the statement that for every finite group $G$ and every element $a \\ne 1$ of $G$ with $|C_G(x)| \\le |C_G(a)|$ for every $x \\ne 1$ in $G$, every nilpotent subgroup $H$ of $G$ normalized by $C_G(a)$ (that is, $C_G(a)$ is contained in the normalizer of $H$) is contained in $C_G(a)$.","title":"Mukhamet'yanov\u2013Fomin problem on centralizers of maximal order"},{"id":"GroupTheory/conjugacy-classes/centralizers-of-maximal-order/maximal-centralizer-contains-normalized-2","kind":"node","lean":"MathlibExt.GroupTheory.MaximalCentralizerNilpotentSubgroupKourovka14Problem67Wanted.maximalCentralizerContainsNormalizedNilpotent_refuted","parent":2763,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/centralizers-of-maximal-order/index.html#maximal-centralizer-contains-normalized-2","scope":"GroupTheory/conjugacy-classes/centralizers-of-maximal-order","status":"can_prove","summary":"For an element $x$ of a group $G$ write $C_G(x)$ for its centralizer. There exist a finite group $G$, an element $a \\ne 1$ of $G$ with $|C_G(x)| \\le |C_G(a)|$ for every $x \\ne 1$ in $G$, and a nilpotent subgroup $H$ of $G$ normalized by $C_G(a)$ (that is, $C_G(a)$ is contained in the normalizer of $H$) such that $H$ is not contained in $C_G(a)$.","title":"Centralizers of maximal order need not contain normalized nilpotent subgroups"},{"data_url":"scopes/GroupTheory/conjugacy-classes/class-number-of-a-finite-group.json","id":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group","kind":"scope","navigation":"module","parent":2750,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/class-number-of-a-finite-group/index.html","scope":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group","status":"can_prove","title":"Class number of a finite group"},{"id":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group/maximal-order-class-number-insoluble","kind":"node","lean":"MathlibExt.GroupTheory.ClassNumberMaximalOrderInsolubleKourovka17Problem77Wanted.maximalOrderClassNumberInsoluble","parent":2766,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/class-number-of-a-finite-group/index.html#maximal-order-class-number-insoluble","scope":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group","status":"fully_proved","summary":"The proposition that for every integer $k \\ge 1$ such that some insoluble finite group has exactly $k$ conjugacy classes, every finite group $G$ with exactly $k$ conjugacy classes whose order is at least the order of every finite group with exactly $k$ conjugacy classes is insoluble. This is Kourovka Problem 17.77 (R. Heffernan, D. MacHale).","title":"Heffernan\u2013MacHale problem on groups of maximal order with a given class number"},{"id":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group/maximal-order-class-number-insoluble-ref","kind":"node","lean":"MathlibExt.GroupTheory.ClassNumberMaximalOrderInsolubleKourovka17Problem77Wanted.maximalOrderClassNumberInsoluble_refuted","parent":2766,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/class-number-of-a-finite-group/index.html#maximal-order-class-number-insoluble-ref","scope":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group","status":"can_prove","summary":"There are an integer $k \\ge 1$, an insoluble finite group with exactly $k$ conjugacy classes, and a soluble finite group $G$ with exactly $k$ conjugacy classes whose order is at least the order of every finite group with exactly $k$ conjugacy classes. So it is not true that whenever some insoluble finite group has exactly $k \\ge 1$ conjugacy classes, every finite group of maximal order with exactly $k$ conjugacy\u2026","title":"A finite group of maximal order with a given class number can be soluble"},{"id":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group/class-number-at-most-nilpotent-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.ClassNumberNilpotentSubgroupKourovka14Problem54Wanted.classNumberAtMostNilpotentSubgroupOrder","parent":2766,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/class-number-of-a-finite-group/index.html#class-number-at-most-nilpotent-subgroup","scope":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group","status":"fully_proved","summary":"The proposition that every finite group $G$ has a nilpotent subgroup $N$ whose order $|N|$ is at least the number $k(G)$ of conjugacy classes of $G$. This is Kourovka Problem 14.54 (M. W. Liebeck, L. Pyber).","title":"Liebeck\u2013Pyber problem on the class number and nilpotent subgroups"},{"id":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group/class-number-at-most-nilpotent-subgroup-2","kind":"node","lean":"MathlibExt.GroupTheory.ClassNumberNilpotentSubgroupKourovka14Problem54Wanted.classNumberAtMostNilpotentSubgroupOrder_refuted","parent":2766,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/class-number-of-a-finite-group/index.html#class-number-at-most-nilpotent-subgroup-2","scope":"GroupTheory/conjugacy-classes/class-number-of-a-finite-group","status":"can_prove","summary":"There is a finite group $G$ such that the number $k(G)$ of conjugacy classes of $G$ is greater than $|N|$ for every nilpotent subgroup $N$ of $G$.","title":"The class number can exceed the order of every nilpotent subgroup"},{"data_url":"scopes/GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups.json","id":"GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups","kind":"scope","navigation":"module","parent":2750,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups/index.html","scope":"GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups","status":"can_prove","title":"Largest conjugacy class of perfect groups"},{"id":"GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups/perfect-centreless-order-le-max-class-sq","kind":"node","lean":"MathlibExt.GroupTheory.PerfectCentrelessOrderMaxClassKourovka20Problem30Wanted.perfectCentrelessOrderLeMaxClassSq","parent":2771,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups/index.html#perfect-centreless-order-le-max-class-sq","scope":"GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups","status":"fully_proved","summary":"The proposition that every finite group $G$ that is perfect ($[G, G] = G$) and has trivial centre contains an element $x$ with $|G| \\le |x^G|^2$, where $x^G$ is the conjugacy class of $x$. Equivalently, $|G| \\le n^2$, where $n$ is the largest size of a conjugacy class of $G$.","title":"Kourovka 20.30: the order of a perfect centreless group and its largest conjugacy class"},{"id":"GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups/perfect-centreless-order-le-max-class-sq-2","kind":"node","lean":"MathlibExt.GroupTheory.PerfectCentrelessOrderMaxClassKourovka20Problem30Wanted.perfectCentrelessOrderLeMaxClassSq_refuted","parent":2771,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups/index.html#perfect-centreless-order-le-max-class-sq-2","scope":"GroupTheory/conjugacy-classes/largest-conjugacy-class-of-perfect-groups","status":"can_prove","summary":"It is not true that every finite group $G$ that is perfect ($[G, G] = G$) and has trivial centre contains an element $x$ with $|G| \\le |x^G|^2$, where $x^G$ is the conjugacy class of $x$. That is, some perfect centreless finite group $G$ satisfies $|G| > n^2$, where $n$ is the largest size of a conjugacy class of $G$.","title":"Perfect centreless groups need not have order at most the square of the largest class size"},{"data_url":"scopes/GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes.json","id":"GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes","kind":"scope","navigation":"module","parent":2750,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes/index.html","scope":"GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes","status":"can_prove","title":"p-parts of conjugacy class sizes"},{"id":"GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes/has-normal-pcomplement","kind":"node","lean":"MathlibExt.GroupTheory.MaximalClassSizePElementNormalComplementKourovka20Problem27Wanted.HasNormalPComplement","parent":2774,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes/index.html#has-normal-pcomplement","scope":"GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes","status":"fully_proved","summary":"A group $G$ has a normal $p$-complement when it has a normal subgroup $N$ and a Sylow $p$-subgroup $P$ that are complements of each other: every element of $G$ can be written uniquely as $nq$ with $n \\in N$ and $q \\in P$.","title":"Groups with a normal p-complement"},{"id":"GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes/maximal-class-size-pelement-gives-normal","kind":"node","lean":"MathlibExt.GroupTheory.MaximalClassSizePElementNormalComplementKourovka20Problem27Wanted.maximalClassSizePElementGivesNormalComplement","parent":2774,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes/index.html#maximal-class-size-pelement-gives-normal","scope":"GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes","status":"fully_proved","summary":"For a prime $p$, the $p$-part of a positive integer $m$ is the largest power of $p$ dividing $m$. This is the statement that for every finite group $G$, every prime $p$ and every $p$-element $g$ of $G$ (that is, $g^{p^k} = 1$ for some natural number $k$), if for every $x \\in G$ the $p$-part of the size of the conjugacy class of $x$ is at most the $p$-part of the size of the conjugacy class of $g$, then $G$ has a\u2026","title":"Gorshkov's problem on maximal p-parts of class sizes"},{"id":"GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes/maximal-class-size-pelement-gives-normal-2","kind":"node","lean":"MathlibExt.GroupTheory.MaximalClassSizePElementNormalComplementKourovka20Problem27Wanted.maximalClassSizePElementGivesNormalComplement_refuted","parent":2774,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes/index.html#maximal-class-size-pelement-gives-normal-2","scope":"GroupTheory/conjugacy-classes/p-parts-of-conjugacy-class-sizes","status":"can_prove","summary":"For a prime $p$, the $p$-part of a positive integer $m$ is the largest power of $p$ dividing $m$. There exist a finite group $G$, a prime $p$ and a $p$-element $g$ of $G$ (that is, $g^{p^k} = 1$ for some natural number $k$) such that for every $x \\in G$ the $p$-part of the size of the conjugacy class of $x$ is at most the $p$-part of the size of the conjugacy class of $g$, but $G$ has no normal $p$-complement.","title":"A maximal p-part of a class size does not force a normal p-complement"},{"data_url":"scopes/GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro.json","id":"GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro","kind":"scope","navigation":"module","parent":2750,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro/index.html","scope":"GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro","status":"can_prove","title":"Products of conjugacy classes in alternating groups"},{"id":"GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro/product-of-three-large-classes-is-altern","kind":"node","lean":"MathlibExt.GroupTheory.ProductThreeConjugacyClassesAlternatingKourovka20Problem23Wanted.productOfThreeLargeClassesIsAlternatingGroup","parent":2778,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro/index.html#product-of-three-large-classes-is-altern","scope":"GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro","status":"fully_proved","summary":"The statement that there are a real number $\\delta$ with $0 < \\delta < 1$ and a natural number $N$ such that for every $n \\ge N$ and all conjugacy classes $A$, $B$, $C$ of the alternating group $\\mathrm{Alt}(n)$, each of size at least $|\\mathrm{Alt}(n)|^{\\delta}$, the product set $ABC$ is all of $\\mathrm{Alt}(n)$.","title":"Products of three large conjugacy classes in alternating groups (Kourovka 20.23)"},{"id":"GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro/product-of-three-large-classes-is-altern-2","kind":"node","lean":"MathlibExt.GroupTheory.ProductThreeConjugacyClassesAlternatingKourovka20Problem23Wanted.productOfThreeLargeClassesIsAlternatingGroup_holds","parent":2778,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro/index.html#product-of-three-large-classes-is-altern-2","scope":"GroupTheory/conjugacy-classes/products-of-conjugacy-classes-in-alternating-gro","status":"can_prove","summary":"There are a real number $\\delta$ with $0 < \\delta < 1$ and a natural number $N$ such that for every $n \\ge N$ and all conjugacy classes $A$, $B$, $C$ of the alternating group $\\mathrm{Alt}(n)$, each of size at least $|\\mathrm{Alt}(n)|^{\\delta}$, the product set $ABC$ is all of $\\mathrm{Alt}(n)$. This is the positive answer to Kourovka Problem 20.23.","title":"Three large conjugacy classes multiply to the whole alternating group"},{"data_url":"scopes/GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes.json","id":"GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes","kind":"scope","navigation":"module","parent":2750,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes/index.html","scope":"GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes","status":"can_prove","title":"Thompson's conjecture on conjugacy class sizes"},{"id":"GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes/class-sizes","kind":"node","lean":"MathlibExt.GroupTheory.ThompsonClassSizesKourovka12Problem38Wanted.classSizes","parent":2781,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes/index.html#class-sizes","scope":"GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes","status":"fully_proved","summary":"For a group $G$, $N(G)$ is the set of natural numbers $|\\{y \\in G : y \\text{ is conjugate to } x\\}|$ for $x \\in G$, that is, the set of sizes of the conjugacy classes of $G$, where an infinite class counts as size $0$.","title":"The set of conjugacy class sizes"},{"id":"GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes/simple-group-recognized-by-class-sizes","kind":"node","lean":"MathlibExt.GroupTheory.ThompsonClassSizesKourovka12Problem38Wanted.simpleGroupRecognizedByClassSizes","parent":2781,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes/index.html#simple-group-recognized-by-class-sizes","scope":"GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes","status":"fully_proved","summary":"Write $N(G)$ for the set of sizes of the conjugacy classes of a group $G$. Thompson's conjecture is the statement that for all finite groups $G$ and $H$, if $G$ is simple and non-abelian, $H$ has trivial centre, and $N(G) = N(H)$, then $G$ and $H$ are isomorphic.","title":"Thompson's conjecture on conjugacy class sizes"},{"id":"GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes/simple-group-recognized-by-class-sizes-h","kind":"node","lean":"MathlibExt.GroupTheory.ThompsonClassSizesKourovka12Problem38Wanted.simpleGroupRecognizedByClassSizes_holds","parent":2781,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes/index.html#simple-group-recognized-by-class-sizes-h","scope":"GroupTheory/conjugacy-classes/thompson-s-conjecture-on-conjugacy-class-sizes","status":"can_prove","summary":"For all finite groups $G$ and $H$, if $G$ is simple and non-abelian, $H$ has trivial centre, and $G$ and $H$ have the same set of conjugacy class sizes, then $G$ and $H$ are isomorphic.","title":"Thompson's conjecture on conjugacy class sizes is true"},{"data_url":"scopes/GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes.json","id":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes","kind":"scope","navigation":"module","parent":2750,"result":false,"route":"../roadmap/GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/index.html","scope":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes","status":"can_prove","title":"Torsion-free groups with three conjugacy classes"},{"id":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/torsion-free-three-classes-index-two","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeThreeClassesIndexTwoKourovka11Problem42Wanted.torsionFreeThreeClassesIndexTwo","parent":2785,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/index.html#torsion-free-three-classes-index-two","scope":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes","status":"fully_proved","summary":"The statement that there is a torsion-free group $G$ (no element of $G$ other than the identity has finite order) that has exactly three conjugacy classes and has a subgroup of index $2$.","title":"Kourovka Problem 11.42 (statement)"},{"id":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/torsion-free-three-classes-index-two-hol","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeThreeClassesIndexTwoKourovka11Problem42Wanted.torsionFreeThreeClassesIndexTwo_holds","parent":2785,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/index.html#torsion-free-three-classes-index-two-hol","scope":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes","status":"can_prove","summary":"There is a torsion-free group $G$ (no element of $G$ other than the identity has finite order) that has exactly three conjugacy classes and has a subgroup of index $2$.","title":"Minasyan's theorem (Kourovka Problem 11.42)"},{"id":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/torsion-free-three-classes-no-inverse-co","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeThreeClassesNoRealKourovka11Problem67Wanted.torsionFreeThreeClassesNoInverseConj","parent":2785,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/index.html#torsion-free-three-classes-no-inverse-co","scope":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes","status":"fully_proved","summary":"The statement that there is a torsion-free group $G$ (no element of $G$ other than the identity has finite order) with exactly three conjugacy classes in which no element $g \\neq 1$ is conjugate to its inverse $g^{-1}$, that is, no nontrivial conjugacy class contains a pair of mutually inverse elements.","title":"Kourovka Problem 11.67 (statement)"},{"id":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/torsion-free-three-classes-no-inverse-co-2","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeThreeClassesNoRealKourovka11Problem67Wanted.torsionFreeThreeClassesNoInverseConj_holds","parent":2785,"result":true,"route":"../roadmap/GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes/index.html#torsion-free-three-classes-no-inverse-co-2","scope":"GroupTheory/conjugacy-classes/torsion-free-groups-with-three-conjugacy-classes","status":"can_prove","summary":"There is a torsion-free group $G$ (no element of $G$ other than the identity has finite order) with exactly three conjugacy classes in which no element $g \\neq 1$ is conjugate to its inverse $g^{-1}$.","title":"Bagayoko's theorem (Kourovka Problem 11.67)"},{"data_url":"scopes/GroupTheory/generation.json","id":"GroupTheory/generation","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/generation/index.html","scope":"GroupTheory/generation","status":"can_prove","summary":"Minimal and special generating sets: generator numbers of almost simple and soluble groups, (2,3)-generation, generation by elements of prime-power order, spread, coprime generation, two-generated direct powers, Eulerian functions and probabilistic generation.","title":"Generation of groups"},{"id":"GroupTheory/generation/closure-eq-i-union-pow","kind":"node","lean":"Subgroup.closure_eq_iUnion_pow","parent":2790,"result":true,"route":"../roadmap/GroupTheory/generation/index.html#closure-eq-i-union-pow","scope":"GroupTheory/generation","status":"fully_proved","summary":"Let $G$ be a group and let $X \\subseteq G$ satisfy $X^{-1} = X$, where $X^{-1} = \\{x^{-1} : x \\in X\\}$. Then the subgroup of $G$ generated by $X$ equals $\\bigcup_{n \\in \\mathbb{N}} X^n$, where $X^0 = \\{1\\}$ and $X^{n+1} = X^n X = \\{yx : y \\in X^n,\\ x \\in X\\}$.","title":"Subgroup generated by a symmetric set"},{"data_url":"scopes/GroupTheory/generation/2-3-generation-of-groups.json","id":"GroupTheory/generation/2-3-generation-of-groups","kind":"scope","navigation":"module","parent":2790,"result":false,"route":"../roadmap/GroupTheory/generation/2-3-generation-of-groups/index.html","scope":"GroupTheory/generation/2-3-generation-of-groups","status":"can_prove","title":"(2,3)-generation of groups"},{"id":"GroupTheory/generation/2-3-generation-of-groups/sl3-int-generated-by-order-two-and-three","kind":"node","lean":"MathlibExt.GroupTheory.SL3IntTwoThreeGenerationKourovka14Problem49Wanted.sl3IntGeneratedByOrderTwoAndThree","parent":2792,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((2,3)\\)"],["text","-generation of "],["math","\\(\\mathrm{SL}_3(\\mathbb{Z})\\)"],["text"," (Kourovka 14.49)"]],"source":"$(2,3)$-generation of $\\mathrm{SL}_3(\\mathbb{Z})$ (Kourovka 14.49)"},"result":true,"route":"../roadmap/GroupTheory/generation/2-3-generation-of-groups/index.html#sl3-int-generated-by-order-two-and-three","scope":"GroupTheory/generation/2-3-generation-of-groups","status":"fully_proved","summary":"The statement that there are $a, b \\in \\mathrm{SL}_3(\\mathbb{Z})$ such that $a$ has order $2$, $b$ has order $3$, and $a$ and $b$ generate $\\mathrm{SL}_3(\\mathbb{Z})$.","title":"$(2,3)$-generation of $\\mathrm{SL}_3(\\mathbb{Z})$ (Kourovka 14.49)"},{"id":"GroupTheory/generation/2-3-generation-of-groups/sl3-int-generated-by-order-two-and-three-2","kind":"node","lean":"MathlibExt.GroupTheory.SL3IntTwoThreeGenerationKourovka14Problem49Wanted.sl3IntGeneratedByOrderTwoAndThree_refuted","parent":2792,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{SL}_3(\\mathbb{Z})\\)"],["text"," is not "],["math","\\((2,3)\\)"],["text","-generated"]],"source":"$\\mathrm{SL}_3(\\mathbb{Z})$ is not $(2,3)$-generated"},"result":true,"route":"../roadmap/GroupTheory/generation/2-3-generation-of-groups/index.html#sl3-int-generated-by-order-two-and-three-2","scope":"GroupTheory/generation/2-3-generation-of-groups","status":"can_prove","summary":"There are no $a, b \\in \\mathrm{SL}_3(\\mathbb{Z})$ such that $a$ has order $2$, $b$ has order $3$, and $a$ and $b$ generate $\\mathrm{SL}_3(\\mathbb{Z})$. Equivalently, $\\mathrm{SL}_3(\\mathbb{Z})$ is not a quotient of the modular group $\\mathrm{PSL}_2(\\mathbb{Z}) \\cong C_2 \\ast C_3$.","title":"$\\mathrm{SL}_3(\\mathbb{Z})$ is not $(2,3)$-generated"},{"data_url":"scopes/GroupTheory/generation/coprime-generation.json","id":"GroupTheory/generation/coprime-generation","kind":"scope","navigation":"module","parent":2790,"result":false,"route":"../roadmap/GroupTheory/generation/coprime-generation/index.html","scope":"GroupTheory/generation/coprime-generation","status":"can_prove","title":"Coprime generation"},{"id":"GroupTheory/generation/coprime-generation/coprime-generated-rank-bound","kind":"node","lean":"MathlibExt.GroupTheory.CoprimeGenerationRankKourovka16Problem53Wanted.coprimeGeneratedRankBound","parent":2795,"result":true,"route":"../roadmap/GroupTheory/generation/coprime-generation/index.html#coprime-generated-rank-bound","scope":"GroupTheory/generation/coprime-generation","status":"fully_proved","summary":"The statement that for every finite group $G$, all subgroups $A$ and $B$ of $G$ of coprime orders that together generate $G$, and every natural number $d$, if each of $A$ and $B$ is generated by at most $d$ elements, then $G$ is generated by at most $d + 1$ elements.","title":"Lucchini's coprime generation bound (Kourovka 16.53)"},{"id":"GroupTheory/generation/coprime-generation/coprime-generated-rank-bound-refuted","kind":"node","lean":"MathlibExt.GroupTheory.CoprimeGenerationRankKourovka16Problem53Wanted.coprimeGeneratedRankBound_refuted","parent":2795,"result":true,"route":"../roadmap/GroupTheory/generation/coprime-generation/index.html#coprime-generated-rank-bound-refuted","scope":"GroupTheory/generation/coprime-generation","status":"can_prove","summary":"There exist a finite group $G$, subgroups $A$ and $B$ of $G$ of coprime orders that together generate $G$, and a natural number $d$ such that each of $A$ and $B$ is generated by at most $d$ elements, but no set of at most $d + 1$ elements generates $G$. That is, Kourovka Problem 16.53 has a negative answer.","title":"Lucchini's coprime generation bound fails"},{"data_url":"scopes/GroupTheory/generation/eulerian-polynomials-of-groups.json","id":"GroupTheory/generation/eulerian-polynomials-of-groups","kind":"scope","navigation":"module","parent":2790,"result":false,"route":"../roadmap/GroupTheory/generation/eulerian-polynomials-of-groups/index.html","scope":"GroupTheory/generation/eulerian-polynomials-of-groups","status":"can_prove","title":"Eulerian polynomials of groups"},{"id":"GroupTheory/generation/eulerian-polynomials-of-groups/order-polynomial","kind":"node","lean":"MathlibExt.GroupTheory.EulerianPolynomialFactorizationKourovka11Problem104Wanted.orderPolynomial","parent":2798,"result":true,"route":"../roadmap/GroupTheory/generation/eulerian-polynomials-of-groups/index.html#order-polynomial","scope":"GroupTheory/generation/eulerian-polynomials-of-groups","status":"fully_proved","summary":"For a subgroup $H$ of a group, the order polynomial of $H$ is $f(H) = \\prod_p x_p^{v_p(|H|)}$ in the integer polynomial ring $\\mathbb{Z}[x_0, x_1, \\dots]$, where $p$ runs over the primes, $x_p$ is the variable indexed by $p$, and $v_p(|H|)$ is the exponent of $p$ in the order $|H|$ of $H$. If $H$ is infinite, $f(H) = 1$.","title":"Order polynomial of a subgroup"},{"id":"GroupTheory/generation/eulerian-polynomials-of-groups/is-eulerian-function","kind":"node","lean":"MathlibExt.GroupTheory.EulerianPolynomialFactorizationKourovka11Problem104Wanted.IsEulerianFunction","parent":2798,"result":true,"route":"../roadmap/GroupTheory/generation/eulerian-polynomials-of-groups/index.html#is-eulerian-function","scope":"GroupTheory/generation/eulerian-polynomials-of-groups","status":"fully_proved","summary":"Let $G$ be a group and let $f$ be the order polynomial. A function $\\varphi$ from the subgroups of $G$ to the integer polynomial ring $\\mathbb{Z}[x_0, x_1, \\dots]$ is an Eulerian function of $G$ if $\\sum_{K \\le H} \\varphi(K) = f(H)$ for every subgroup $H$ of $G$, the sum running over the subgroups $K$ of $H$. The value $\\varphi(G)$ is the Eulerian polynomial of $G$.","title":"Eulerian function of a group (G. E. Wall)"},{"id":"GroupTheory/generation/eulerian-polynomials-of-groups/is-psolvable","kind":"node","lean":"MathlibExt.GroupTheory.EulerianPolynomialFactorizationKourovka11Problem104Wanted.IsPSolvable","parent":2798,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-solvable group"]],"source":"$p$-solvable group"},"result":true,"route":"../roadmap/GroupTheory/generation/eulerian-polynomials-of-groups/index.html#is-psolvable","scope":"GroupTheory/generation/eulerian-polynomials-of-groups","status":"fully_proved","summary":"For a natural number $p$, a group $G$ is $p$-solvable if it has a chain of subgroups $1 = N_0 \\le N_1 \\le \\dots \\le N_n = G$ in which each $N_i$ is normal in $N_{i+1}$ and each index $[N_{i+1} : N_i]$, taken to be $0$ when it is infinite, is a power $p^k$ with $k \\ge 0$ or is coprime to $p$. For finite $G$ and prime $p$ this says that every factor $N_{i+1}/N_i$ is a $p$-group or has order prime to $p$.","title":"$p$-solvable group"},{"id":"GroupTheory/generation/eulerian-polynomials-of-groups/eulerian-polynomial-splits-solvable","kind":"node","lean":"MathlibExt.GroupTheory.EulerianPolynomialFactorizationKourovka11Problem104Wanted.eulerianPolynomialSplitsSolvable","parent":2798,"result":true,"route":"../roadmap/GroupTheory/generation/eulerian-polynomials-of-groups/index.html#eulerian-polynomial-splits-solvable","scope":"GroupTheory/generation/eulerian-polynomials-of-groups","status":"fully_proved","summary":"This is the statement that for every finite group $G$ and every Eulerian function $\\varphi$ of $G$, with values in the integer polynomial ring $\\mathbb{Z}[x_0, x_1, \\dots]$: if there are a finite set $S$ of primes and, for each $q \\in S$, an integer polynomial $P_q$ involving no variable other than $x_q$, such that the Eulerian polynomial $\\varphi(G)$ equals $\\prod_{q \\in S} P_q$, then $G$ is solvable.","title":"Kourovka Problem 11.104(a) (Eulerian polynomials and solvability, G. E. Wall)"},{"id":"GroupTheory/generation/eulerian-polynomials-of-groups/eulerian-polynomial-separates-psolvable","kind":"node","lean":"MathlibExt.GroupTheory.EulerianPolynomialFactorizationKourovka11Problem104Wanted.eulerianPolynomialSeparatesPSolvable","parent":2798,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 11.104(b) (Eulerian polynomials and "],["math","\\(p\\)"],["text","-solvability)"]],"source":"Kourovka Problem 11.104(b) (Eulerian polynomials and $p$-solvability)"},"result":true,"route":"../roadmap/GroupTheory/generation/eulerian-polynomials-of-groups/index.html#eulerian-polynomial-separates-psolvable","scope":"GroupTheory/generation/eulerian-polynomials-of-groups","status":"fully_proved","summary":"This is the statement that for every finite group $G$, every prime $p$ and every Eulerian function $\\varphi$ of $G$, with values in the integer polynomial ring $\\mathbb{Z}[x_0, x_1, \\dots]$: if the Eulerian polynomial $\\varphi(G)$ is a product $PQ$ of integer polynomials where $P$ involves no variable other than $x_p$ and $x_p$ does not occur in $Q$, then $G$ is $p$-solvable.","title":"Kourovka Problem 11.104(b) (Eulerian polynomials and $p$-solvability)"},{"id":"GroupTheory/generation/eulerian-polynomials-of-groups/eulerian-polynomial-splits-solvable-hold","kind":"node","lean":"MathlibExt.GroupTheory.EulerianPolynomialFactorizationKourovka11Problem104Wanted.eulerianPolynomialSplitsSolvable_holds","parent":2798,"result":true,"route":"../roadmap/GroupTheory/generation/eulerian-polynomials-of-groups/index.html#eulerian-polynomial-splits-solvable-hold","scope":"GroupTheory/generation/eulerian-polynomials-of-groups","status":"can_prove","summary":"Let $G$ be a finite group and let $\\varphi$ be an Eulerian function of $G$, with values in the integer polynomial ring $\\mathbb{Z}[x_0, x_1, \\dots]$. If there are a finite set $S$ of primes and, for each $q \\in S$, an integer polynomial $P_q$ involving no variable other than $x_q$, such that the Eulerian polynomial $\\varphi(G)$ equals $\\prod_{q \\in S} P_q$, then $G$ is solvable.","title":"Solvability from one-variable factorizations of the Eulerian polynomial"},{"id":"GroupTheory/generation/eulerian-polynomials-of-groups/eulerian-polynomial-separates-psolvable-2","kind":"node","lean":"MathlibExt.GroupTheory.EulerianPolynomialFactorizationKourovka11Problem104Wanted.eulerianPolynomialSeparatesPSolvable_holds","parent":2798,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eulerian polynomial factorization implies "],["math","\\(p\\)"],["text","-solvability"]],"source":"Eulerian polynomial factorization implies $p$-solvability"},"result":true,"route":"../roadmap/GroupTheory/generation/eulerian-polynomials-of-groups/index.html#eulerian-polynomial-separates-psolvable-2","scope":"GroupTheory/generation/eulerian-polynomials-of-groups","status":"can_prove","summary":"Let $G$ be a finite group, let $p$ be a prime, and let $\\varphi$ be an Eulerian function of $G$, with values in the integer polynomial ring $\\mathbb{Z}[x_0, x_1, \\dots]$. If the Eulerian polynomial $\\varphi(G)$ is a product $PQ$ of integer polynomials where $P$ involves no variable other than $x_p$ and $x_p$ does not occur in $Q$, then $G$ is $p$-solvable.","title":"Eulerian polynomial factorization implies $p$-solvability"},{"data_url":"scopes/GroupTheory/generation/generation-by-elements-of-prime-power-order.json","id":"GroupTheory/generation/generation-by-elements-of-prime-power-order","kind":"scope","navigation":"module","parent":2790,"result":false,"route":"../roadmap/GroupTheory/generation/generation-by-elements-of-prime-power-order/index.html","scope":"GroupTheory/generation/generation-by-elements-of-prime-power-order","status":"can_prove","title":"Generation by elements of prime-power order"},{"id":"GroupTheory/generation/generation-by-elements-of-prime-power-order/simple-group-generated-by-prime-power-or","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupPrimePowerGeneratorsKourovka18Problem52Wanted.simpleGroupGeneratedByPrimePowerOrderElements","parent":2806,"result":true,"route":"../roadmap/GroupTheory/generation/generation-by-elements-of-prime-power-order/index.html#simple-group-generated-by-prime-power-or","scope":"GroupTheory/generation/generation-by-elements-of-prime-power-order","status":"fully_proved","summary":"The statement that every finite simple group $G$ is generated by two elements $x, y \\in G$ whose orders are prime powers, that is, each of the form $p^k$ with $p$ prime and $k \\ge 1$ (the primes and exponents may differ between $x$ and $y$ and depend on the group). The problem is due to J. Krempa.","title":"Kourovka Problem 18.52 (generation by two elements of prime-power order)"},{"id":"GroupTheory/generation/generation-by-elements-of-prime-power-order/simple-group-generated-by-prime-power-or-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupPrimePowerGeneratorsKourovka18Problem52Wanted.simpleGroupGeneratedByPrimePowerOrderElements_holds","parent":2806,"result":true,"route":"../roadmap/GroupTheory/generation/generation-by-elements-of-prime-power-order/index.html#simple-group-generated-by-prime-power-or-2","scope":"GroupTheory/generation/generation-by-elements-of-prime-power-order","status":"can_prove","summary":"Every finite simple group $G$ is generated by two elements $x, y \\in G$ whose orders are prime powers, that is, each of the form $p^k$ with $p$ prime and $k \\ge 1$, where the prime and the exponent may differ between $x$ and $y$. This answers Kourovka Problem 18.52 positively.","title":"Finite simple groups are generated by two elements of prime-power order"},{"data_url":"scopes/GroupTheory/generation/generator-number-of-almost-simple-groups.json","id":"GroupTheory/generation/generator-number-of-almost-simple-groups","kind":"scope","navigation":"module","parent":2790,"result":false,"route":"../roadmap/GroupTheory/generation/generator-number-of-almost-simple-groups/index.html","scope":"GroupTheory/generation/generator-number-of-almost-simple-groups","status":"can_prove","title":"Generator number of almost simple groups"},{"id":"GroupTheory/generation/generator-number-of-almost-simple-groups/rel-gen-num","kind":"node","lean":"MathlibExt.GroupTheory.AlmostSimpleGeneratorNumberKourovka11Problem27Wanted.relGenNum","parent":2809,"result":true,"route":"../roadmap/GroupTheory/generation/generator-number-of-almost-simple-groups/index.html#rel-gen-num","scope":"GroupTheory/generation/generator-number-of-almost-simple-groups","status":"fully_proved","summary":"For subgroups $G$ and $N$ of a group $A$, the relative generator number of $G$ modulo $N$ is the least $k$ such that some $k$ elements of $A$ together with $N$ generate $G$, and $0$ if no finite set of elements does. For a normal subgroup $N \\le G$ with $G / N$ finitely generated it is $d(G / N)$, the minimal number of generators of $G / N$.","title":"Relative generator number"},{"id":"GroupTheory/generation/generator-number-of-almost-simple-groups/almost-simple-gen-num-eq","kind":"node","lean":"MathlibExt.GroupTheory.AlmostSimpleGeneratorNumberKourovka11Problem27Wanted.almostSimpleGenNumEq","parent":2809,"result":true,"route":"../roadmap/GroupTheory/generation/generator-number-of-almost-simple-groups/index.html#almost-simple-gen-num-eq","scope":"GroupTheory/generation/generator-number-of-almost-simple-groups","status":"fully_proved","summary":"The proposition that for every finite non-abelian simple group $S$, identified with its inner automorphism group $\\mathrm{Inn}(S) \\le \\mathrm{Aut}(S)$, and every subgroup $G$ of $\\mathrm{Aut}(S)$ containing $S$, $d(G) = \\max(2, d(G / S))$. Here $d(G)$ is the minimal number of generators of $G$, and $d(G / S)$, the minimal number of generators of $G / S$, is taken to be the least number of elements of $G$ that\u2026","title":"The generator-number formula for almost simple groups (Kourovka 11.27)"},{"id":"GroupTheory/generation/generator-number-of-almost-simple-groups/almost-simple-gen-num-le-three","kind":"node","lean":"MathlibExt.GroupTheory.AlmostSimpleGeneratorNumberKourovka11Problem27Wanted.almostSimpleGenNumLeThree","parent":2809,"result":true,"route":"../roadmap/GroupTheory/generation/generator-number-of-almost-simple-groups/index.html#almost-simple-gen-num-le-three","scope":"GroupTheory/generation/generator-number-of-almost-simple-groups","status":"fully_proved","summary":"The proposition that for every finite non-abelian simple group $S$, identified with its inner automorphism group $\\mathrm{Inn}(S) \\le \\mathrm{Aut}(S)$, every subgroup $G$ of $\\mathrm{Aut}(S)$ containing $S$ is generated by at most $3$ elements.","title":"The 3-generator bound for almost simple groups (Kourovka 11.27)"},{"id":"GroupTheory/generation/generator-number-of-almost-simple-groups/almost-simple-gen-num-eq-holds","kind":"node","lean":"MathlibExt.GroupTheory.AlmostSimpleGeneratorNumberKourovka11Problem27Wanted.almostSimpleGenNumEq_holds","parent":2809,"result":true,"route":"../roadmap/GroupTheory/generation/generator-number-of-almost-simple-groups/index.html#almost-simple-gen-num-eq-holds","scope":"GroupTheory/generation/generator-number-of-almost-simple-groups","status":"can_prove","summary":"Let $S$ be a finite non-abelian simple group, identified with its inner automorphism group $\\mathrm{Inn}(S) \\le \\mathrm{Aut}(S)$. For every subgroup $G$ of $\\mathrm{Aut}(S)$ containing $S$, $d(G) = \\max(2, d(G / S))$, where $d(G)$ is the minimal number of generators of $G$ and $d(G / S)$, the minimal number of generators of $G / S$, is taken to be the least number of elements of $G$ that together with $S$ generate\u2026","title":"Generator number of almost simple groups (Kourovka 11.27)"},{"id":"GroupTheory/generation/generator-number-of-almost-simple-groups/almost-simple-gen-num-le-three-holds","kind":"node","lean":"MathlibExt.GroupTheory.AlmostSimpleGeneratorNumberKourovka11Problem27Wanted.almostSimpleGenNumLeThree_holds","parent":2809,"result":true,"route":"../roadmap/GroupTheory/generation/generator-number-of-almost-simple-groups/index.html#almost-simple-gen-num-le-three-holds","scope":"GroupTheory/generation/generator-number-of-almost-simple-groups","status":"can_prove","summary":"Let $S$ be a finite non-abelian simple group, identified with its inner automorphism group $\\mathrm{Inn}(S) \\le \\mathrm{Aut}(S)$. Every subgroup $G$ of $\\mathrm{Aut}(S)$ containing $S$ is generated by at most $3$ elements.","title":"Almost simple groups are generated by 3 elements (Kourovka 11.27)"},{"id":"GroupTheory/generation/generator-number-of-almost-simple-groups/is-almost-simple","kind":"node","lean":"MathlibExt.GroupTheory.AlmostSimpleShared.IsAlmostSimple","parent":2809,"result":true,"route":"../roadmap/GroupTheory/generation/generator-number-of-almost-simple-groups/index.html#is-almost-simple","scope":"GroupTheory/generation/generator-number-of-almost-simple-groups","status":"fully_proved","summary":"A group $G$ is almost simple when there is a nonabelian finite simple group $S$ such that $G$ is isomorphic to a subgroup $A$ of the automorphism group $\\mathrm{Aut}(S)$ that contains the inner automorphism group $\\mathrm{Inn}(S)$.","title":"Almost simple groups"},{"data_url":"scopes/GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups.json","id":"GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups","kind":"scope","navigation":"module","parent":2790,"result":false,"route":"../roadmap/GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups/index.html","scope":"GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups","status":"can_prove","title":"Rank of a group generated by two subgroups"},{"id":"GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups/rank-additive-generation","kind":"node","lean":"MathlibExt.GroupTheory.GenerationBySubgroupsRankKourovka12Problem71Wanted.rankAdditiveGeneration","parent":2816,"result":true,"route":"../roadmap/GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups/index.html#rank-additive-generation","scope":"GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups","status":"fully_proved","summary":"Write $d(X)$ for the minimal number of generators of a finite group $X$. This is the proposition that for all finite groups $A$ and $B$ there are a finite group $G$ and subgroups $H, K \\le G$ with $H \\cong A$, $K \\cong B$ and $\\langle H, K \\rangle = G$ such that $d(G) = d(A) + d(B)$.","title":"Ribes' question on generator counts (Kourovka 12.71)"},{"id":"GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups/rank-additive-generation-refuted","kind":"node","lean":"MathlibExt.GroupTheory.GenerationBySubgroupsRankKourovka12Problem71Wanted.rankAdditiveGeneration_refuted","parent":2816,"result":true,"route":"../roadmap/GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups/index.html#rank-additive-generation-refuted","scope":"GroupTheory/generation/rank-of-a-group-generated-by-two-subgroups","status":"can_prove","summary":"Write $d(X)$ for the minimal number of generators of a finite group $X$. There are finite groups $A$ and $B$ such that for every finite group $G$ and all subgroups $H, K \\le G$ with $H \\cong A$, $K \\cong B$ and $\\langle H, K \\rangle = G$, we have $d(G) \\ne d(A) + d(B)$.","title":"Generator counts of groups generated by two subgroups are not always additive (Kourovka 12.71)"},{"data_url":"scopes/GroupTheory/generation/spread-of-a-group.json","id":"GroupTheory/generation/spread-of-a-group","kind":"scope","navigation":"module","parent":2790,"result":false,"route":"../roadmap/GroupTheory/generation/spread-of-a-group/index.html","scope":"GroupTheory/generation/spread-of-a-group","status":"can_prove","title":"Spread of a group"},{"id":"GroupTheory/generation/spread-of-a-group/has-spread-at-least","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem38Wanted.HasSpreadAtLeast","parent":2819,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spread at least "],["math","\\(k\\)"]],"source":"Spread at least $k$"},"result":true,"route":"../roadmap/GroupTheory/generation/spread-of-a-group/index.html#has-spread-at-least","scope":"GroupTheory/generation/spread-of-a-group","status":"fully_proved","summary":"A group $G$ has spread at least $k$, for a natural number $k$, when for every family $x_1, \\dots, x_k$ of nontrivial elements of $G$, not necessarily distinct, there is an element $y \\in G$ such that the subgroup $\\langle x_i, y \\rangle$ generated by $x_i$ and $y$ is all of $G$ for every $i$.","title":"Spread at least $k$"},{"id":"GroupTheory/generation/spread-of-a-group/has-spread-exactly","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem38Wanted.HasSpreadExactly","parent":2819,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spread exactly "],["math","\\(k\\)"]],"source":"Spread exactly $k$"},"result":true,"route":"../roadmap/GroupTheory/generation/spread-of-a-group/index.html#has-spread-exactly","scope":"GroupTheory/generation/spread-of-a-group","status":"fully_proved","summary":"A group $G$ has spread exactly $k$, for a natural number $k$, when $G$ has spread at least $k$ and, for every natural number $m > k$, $G$ does not have spread at least $m$.","title":"Spread exactly $k$"},{"id":"GroupTheory/generation/spread-of-a-group/conjecture","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem38Wanted.conjecture","parent":2819,"result":true,"route":"../roadmap/GroupTheory/generation/spread-of-a-group/index.html#conjecture","scope":"GroupTheory/generation/spread-of-a-group","status":"fully_proved","summary":"The conjecture that there exists a group of spread exactly $1$.","title":"Existence of a group of spread exactly one (Kourovka 21.38)"},{"id":"GroupTheory/generation/spread-of-a-group/conjecture-holds","kind":"node","lean":"MathlibExt.GroupTheory.Kourovka21Problem38Wanted.conjecture_holds","parent":2819,"result":true,"route":"../roadmap/GroupTheory/generation/spread-of-a-group/index.html#conjecture-holds","scope":"GroupTheory/generation/spread-of-a-group","status":"can_prove","summary":"There exists a group $G$ of spread exactly $1$: for every nontrivial $x \\in G$ there is $y \\in G$ with $\\langle x, y \\rangle = G$, but for every natural number $m > 1$ there are nontrivial elements $x_1, \\dots, x_m \\in G$ such that no single $y \\in G$ satisfies $\\langle x_i, y \\rangle = G$ for all $i$. This answers Kourovka Problem 21.38 affirmatively.","title":"There is a group of spread exactly one"},{"id":"GroupTheory/generation/spread-of-a-group/s","kind":"node","lean":"MathlibExt.GroupTheory.SpreadOfAFiniteGroupWanted.SpreadOfAFiniteGroup.s","parent":2819,"result":true,"route":"../roadmap/GroupTheory/generation/spread-of-a-group/index.html#s","scope":"GroupTheory/generation/spread-of-a-group","status":"fully_proved","summary":"Let $G$ be a group. The spread $s(G)$ of $G$ is the supremum in $\\mathbb{N} \\cup \\{\\infty\\}$ of the natural numbers $k$ such that for any non-trivial elements $x_1, \\dots, x_k$ of $G$ (not necessarily distinct) there exists $y \\in G$ such that the subgroup $\\langle x_i, y \\rangle$ generated by $x_i$ and $y$ is all of $G$ for every $i$. In particular $s(G) = \\infty$ when there is no largest such $k$.","title":"Spread of a group"},{"id":"GroupTheory/generation/spread-of-a-group/theorem-1","kind":"node","lean":"MathlibExt.GroupTheory.SpreadOfAFiniteGroupWanted.SpreadOfAFiniteGroup.theorem_1","parent":2819,"result":true,"route":"../roadmap/GroupTheory/generation/spread-of-a-group/index.html#theorem-1","scope":"GroupTheory/generation/spread-of-a-group","status":"can_prove","summary":"Let $G$ be a finite group. Then the spread $s(G)$ is at least $2$ if and only if every proper quotient of $G$ is cyclic, that is, $G/N$ is cyclic for every non-trivial normal subgroup $N$ of $G$.","title":"Burness\u2013Guralnick\u2013Harper spread theorem"},{"data_url":"scopes/GroupTheory/generation/two-generated-direct-powers-of-simple-groups.json","id":"GroupTheory/generation/two-generated-direct-powers-of-simple-groups","kind":"scope","navigation":"module","parent":2790,"result":false,"route":"../roadmap/GroupTheory/generation/two-generated-direct-powers-of-simple-groups/index.html","scope":"GroupTheory/generation/two-generated-direct-powers-of-simple-groups","status":"can_prove","title":"Two-generated direct powers of simple groups"},{"id":"GroupTheory/generation/two-generated-direct-powers-of-simple-groups/direct-power-two-generated-bound","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupDirectPowerTwoGeneratedKourovka17Problem116Wanted.directPowerTwoGeneratedBound","parent":2826,"result":true,"route":"../roadmap/GroupTheory/generation/two-generated-direct-powers-of-simple-groups/index.html#direct-power-two-generated-bound","scope":"GroupTheory/generation/two-generated-direct-powers-of-simple-groups","status":"fully_proved","summary":"The statement, in the form of its resolution, that for every non-abelian finite simple group $G$ there is a natural number $n$ with $n^2 > 4|G|$, that is, $n > 2\\sqrt{|G|}$, such that the direct power $G^n$ is generated by two elements. The problem is due to A. Erfanian and J. Wiegold.","title":"Kourovka Problem 17.116 (two-generated direct powers of simple groups)"},{"id":"GroupTheory/generation/two-generated-direct-powers-of-simple-groups/direct-power-two-generated-bound-holds","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupDirectPowerTwoGeneratedKourovka17Problem116Wanted.directPowerTwoGeneratedBound_holds","parent":2826,"result":true,"route":"../roadmap/GroupTheory/generation/two-generated-direct-powers-of-simple-groups/index.html#direct-power-two-generated-bound-holds","scope":"GroupTheory/generation/two-generated-direct-powers-of-simple-groups","status":"can_prove","summary":"For every non-abelian finite simple group $G$ there is a natural number $n$ with $n^2 > 4|G|$, that is, $n > 2\\sqrt{|G|}$, such that the direct power $G^n$ is generated by two elements. This answers Kourovka Problem 17.116 positively.","title":"Mar\u00f3ti\u2013Tamburini bound for two-generated direct powers"},{"data_url":"scopes/GroupTheory/sylow-and-hall-subgroups.json","id":"GroupTheory/sylow-and-hall-subgroups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/index.html","scope":"GroupTheory/sylow-and-hall-subgroups","status":"can_prove","summary":"Sylow theory and its generalization to sets of primes: Hall \u03c0-subgroups, the classes E_\u03c0, C_\u03c0 and D_\u03c0, pronormality and intersections of Hall subgroups, groups whose maximal subgroups are Hall; questions on supplements of Sylow maximal subgroups forcing solubility go to Soluble and supersoluble finite groups.","title":"Sylow and Hall subgroups"},{"data_url":"scopes/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups.json","id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","kind":"scope","navigation":"module","parent":2829,"result":false,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"can_prove","title":"Groups with Hall maximal subgroups"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/is-hall-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.HallMaximalPronormalSecondMaximalKourovka19Problem109Wanted.IsHallSubgroup","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#is-hall-subgroup","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is a Hall subgroup when its order $|H|$ is coprime to its index $|G : H|$, where an infinite order or index is counted as $0$.","title":"Hall subgroup (Kourovka Problem 19.109)"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/is-pronormal","kind":"node","lean":"MathlibExt.GroupTheory.HallMaximalPronormalSecondMaximalKourovka19Problem109Wanted.IsPronormal","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#is-pronormal","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is pronormal in $G$ when for every $g \\in G$ there is an element $x$ of the subgroup $\\langle H, gHg^{-1} \\rangle$ generated by $H$ and $gHg^{-1}$ such that $xHx^{-1} = gHg^{-1}$.","title":"Pronormal subgroup"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/is-second-maximal","kind":"node","lean":"MathlibExt.GroupTheory.HallMaximalPronormalSecondMaximalKourovka19Problem109Wanted.IsSecondMaximal","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#is-second-maximal","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"A subgroup $K$ of a group $G$ is second maximal when it is a maximal subgroup of a maximal subgroup of $G$: there is a maximal subgroup $M$ of $G$ with $K < M$ and no subgroup $L$ of $G$ with $K < L < M$.","title":"Second maximal subgroup"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/maximal-hall-iff-second-maximal-pronorma","kind":"node","lean":"MathlibExt.GroupTheory.HallMaximalPronormalSecondMaximalKourovka19Problem109Wanted.maximalHallIffSecondMaximalPronormal","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#maximal-hall-iff-second-maximal-pronorma","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"For every non-abelian finite simple group $G$, all maximal subgroups of $G$ are Hall subgroups if and only if every second maximal subgroup of $G$ is pronormal in $G$.","title":"Zenkov's question on Hall maximal subgroups (Kourovka Problem 19.109)"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/maximal-hall-iff-second-maximal-pronorma-2","kind":"node","lean":"MathlibExt.GroupTheory.HallMaximalPronormalSecondMaximalKourovka19Problem109Wanted.maximalHallIffSecondMaximalPronormal_refuted","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#maximal-hall-iff-second-maximal-pronorma-2","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"can_prove","summary":"It is not true that for every non-abelian finite simple group $G$, all maximal subgroups of $G$ are Hall subgroups if and only if every second maximal subgroup of $G$ is pronormal in $G$.","title":"Negative answer to Zenkov's question (Kourovka Problem 19.109)"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/is-hall-subgroup-2","kind":"node","lean":"MathlibExt.GroupTheory.MaximalHallCompositionFactorsKourovka17Problem92Wanted.IsHallSubgroup","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#is-hall-subgroup-2","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is a Hall subgroup when its order $|H|$ is coprime to its index $|G : H|$.","title":"Hall subgroup (Kourovka 17.92)"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/is-composition-factor","kind":"node","lean":"MathlibExt.GroupTheory.MaximalHallCompositionFactorsKourovka17Problem92Wanted.IsCompositionFactor","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#is-composition-factor","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"For groups $G$ and $S$, $S$ is a composition factor of $G$ when there are subgroups $H$ and $K$ of $G$ such that $H$ is subnormal in $G$, $H \\cap K$ is normal in $H$, and the quotient $H/(H \\cap K)$ is a simple group isomorphic to $S$.","title":"Composition factor of a group (Kourovka 17.92)"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/is-maximal-hall-nonsoluble","kind":"node","lean":"MathlibExt.GroupTheory.MaximalHallCompositionFactorsKourovka17Problem92Wanted.IsMaximalHallNonsoluble","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#is-maximal-hall-nonsoluble","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"A group $G$ has this property when $G$ is not soluble and every maximal subgroup of $G$ is a Hall subgroup.","title":"Non-soluble groups with Hall maximal subgroups"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/maximal-hall-composition-factors-in-list","kind":"node","lean":"MathlibExt.GroupTheory.MaximalHallCompositionFactorsKourovka17Problem92Wanted.maximalHallCompositionFactorsInList","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#maximal-hall-composition-factors-in-list","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"The proposition that every non-abelian composition factor of a finite non-soluble group all of whose maximal subgroups are Hall subgroups is isomorphic to $L_2(7)$, $L_2(11)$ or $L_5(2)$. Here $L_n(q)$ is the projective special linear group $\\mathrm{PSL}_n(\\mathbb{Z}/q\\mathbb{Z})$, the quotient of $\\mathrm{SL}_n(\\mathbb{Z}/q\\mathbb{Z})$ by its centre.","title":"Kourovka 17.92, first half: the possible composition factors"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/maximal-hall-composition-factors-occur","kind":"node","lean":"MathlibExt.GroupTheory.MaximalHallCompositionFactorsKourovka17Problem92Wanted.maximalHallCompositionFactorsOccur","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#maximal-hall-composition-factors-occur","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"fully_proved","summary":"The proposition that each of $L_2(7)$, $L_2(11)$ and $L_5(2)$ is a composition factor of some finite non-soluble group all of whose maximal subgroups are Hall subgroups. Here $L_n(q)$ is the projective special linear group $\\mathrm{PSL}_n(\\mathbb{Z}/q\\mathbb{Z})$, the quotient of $\\mathrm{SL}_n(\\mathbb{Z}/q\\mathbb{Z})$ by its centre.","title":"Kourovka 17.92, second half: the three factors occur"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/maximal-hall-composition-factors-in-list-2","kind":"node","lean":"MathlibExt.GroupTheory.MaximalHallCompositionFactorsKourovka17Problem92Wanted.maximalHallCompositionFactorsInList_holds","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#maximal-hall-composition-factors-in-list-2","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"can_prove","summary":"Every non-abelian composition factor of a finite non-soluble group all of whose maximal subgroups are Hall subgroups is isomorphic to $L_2(7)$, $L_2(11)$ or $L_5(2)$, so the first half of the answer to Kourovka 17.92 holds. Here $L_n(q)$ is the projective special linear group $\\mathrm{PSL}_n(\\mathbb{Z}/q\\mathbb{Z})$, the quotient of $\\mathrm{SL}_n(\\mathbb{Z}/q\\mathbb{Z})$ by its centre.","title":"Maslova's theorem on groups with Hall maximal subgroups"},{"id":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/maximal-hall-composition-factors-occur-h","kind":"node","lean":"MathlibExt.GroupTheory.MaximalHallCompositionFactorsKourovka17Problem92Wanted.maximalHallCompositionFactorsOccur_holds","parent":2830,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups/index.html#maximal-hall-composition-factors-occur-h","scope":"GroupTheory/sylow-and-hall-subgroups/groups-with-hall-maximal-subgroups","status":"can_prove","summary":"Each of $L_2(7)$, $L_2(11)$ and $L_5(2)$ is a composition factor of some finite non-soluble group all of whose maximal subgroups are Hall subgroups, so the second half of the answer to Kourovka 17.92 holds. Here $L_n(q)$ is the projective special linear group $\\mathrm{PSL}_n(\\mathbb{Z}/q\\mathbb{Z})$, the quotient of $\\mathrm{SL}_n(\\mathbb{Z}/q\\mathbb{Z})$ by its centre.","title":"The three answer groups of Kourovka 17.92 occur"},{"data_url":"scopes/GroupTheory/sylow-and-hall-subgroups/hall-subgroups.json","id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","kind":"scope","navigation":"module","parent":2829,"result":false,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"can_prove","title":"Hall \u03c0-subgroups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-pi-number","kind":"node","lean":"MathlibExt.GroupTheory.DPiHallIntersectionKourovka18Problem31Wanted.IsPiNumber","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-pi-number","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A natural number $n$ is a $\\pi$-number when every prime divisor of $n$ lies in $\\pi$.","title":"\u03c0-numbers (Kourovka 18.31)"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-pi-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.DPiHallIntersectionKourovka18Problem31Wanted.IsPiSubgroup","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-pi-subgroup","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A subgroup $H$ of a group $G$ is a $\\pi$-subgroup when its order is a $\\pi$-number, that is, every prime divisor of $|H|$ lies in $\\pi$.","title":"\u03c0-subgroups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-hall-pi-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.DPiHallIntersectionKourovka18Problem31Wanted.IsHallPiSubgroup","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-hall-pi-subgroup","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A subgroup $H$ of a group $G$ is a Hall $\\pi$-subgroup when it is a $\\pi$-subgroup and no prime divisor of its index $|G : H|$ lies in $\\pi$.","title":"Hall \u03c0-subgroups (Kourovka 18.31)"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-maximal-pi-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.DPiHallIntersectionKourovka18Problem31Wanted.IsMaximalPiSubgroup","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-maximal-pi-subgroup","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A subgroup $H$ of a group $G$ is a maximal $\\pi$-subgroup when it is a $\\pi$-subgroup and every $\\pi$-subgroup $K$ of $G$ containing $H$ equals $H$.","title":"Maximal \u03c0-subgroups (Kourovka 18.31)"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-dpi-group","kind":"node","lean":"MathlibExt.GroupTheory.DPiHallIntersectionKourovka18Problem31Wanted.IsDPiGroup","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-dpi-group","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A group $G$ is a $D_\\pi$-group when any two maximal $\\pi$-subgroups $H$ and $K$ of $G$ are conjugate, that is, $K = gHg^{-1}$ for some $g \\in G$.","title":"D\u03c0-groups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/pi-core","kind":"node","lean":"MathlibExt.GroupTheory.DPiHallIntersectionKourovka18Problem31Wanted.piCore","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#pi-core","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. For a group $G$, the $\\pi$-core $O_\\pi(G)$ is the subgroup of $G$ generated by all normal $\\pi$-subgroups of $G$.","title":"The \u03c0-core"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/three-hall-subgroups-intersect-in-core","kind":"node","lean":"MathlibExt.GroupTheory.DPiHallIntersectionKourovka18Problem31Wanted.threeHallSubgroupsIntersectInCore","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#three-hall-subgroups-intersect-in-core","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"For every set $\\pi$ of natural numbers and every finite $D_\\pi$-group $G$, there are Hall $\\pi$-subgroups $H_1$, $H_2$, $H_3$ of $G$, not necessarily distinct, with $H_1 \\cap H_2 \\cap H_3 = O_\\pi(G)$, the $\\pi$-core of $G$.","title":"Three Hall \u03c0-subgroups meeting in the \u03c0-core (Kourovka 18.31)"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/three-hall-subgroups-intersect-in-core-r","kind":"node","lean":"MathlibExt.GroupTheory.DPiHallIntersectionKourovka18Problem31Wanted.threeHallSubgroupsIntersectInCore_refuted","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#three-hall-subgroups-intersect-in-core-r","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"can_prove","summary":"It is not the case that for every set $\\pi$ of natural numbers every finite $D_\\pi$-group $G$ has Hall $\\pi$-subgroups $H_1$, $H_2$, $H_3$ with $H_1 \\cap H_2 \\cap H_3 = O_\\pi(G)$, the $\\pi$-core of $G$: the answer to Kourovka 18.31 is negative.","title":"Three Hall \u03c0-subgroups need not meet in the \u03c0-core"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-pi-number-2","kind":"node","lean":"MathlibExt.GroupTheory.DPiNormalSubgroupKourovka13Problem33Wanted.IsPiNumber","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\pi\\)"],["text","-numbers"]],"source":"$\\pi$-numbers"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-pi-number-2","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A natural number $n$ is a $\\pi$-number when every prime dividing $n$ lies in $\\pi$.","title":"$\\pi$-numbers"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-hall-pi-subgroup-2","kind":"node","lean":"MathlibExt.GroupTheory.DPiNormalSubgroupKourovka13Problem33Wanted.IsHallPiSubgroup","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hall "],["math","\\(\\pi\\)"],["text","-subgroups"]],"source":"Hall $\\pi$-subgroups"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-hall-pi-subgroup-2","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A subgroup $H$ of a group $G$ is a Hall $\\pi$-subgroup when its order $|H|$ is a $\\pi$-number and no prime dividing its index $|G : H|$ lies in $\\pi$. Here an infinite order or index is taken to be $0$, which every prime divides.","title":"Hall $\\pi$-subgroups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-dpi","kind":"node","lean":"MathlibExt.GroupTheory.DPiNormalSubgroupKourovka13Problem33Wanted.IsDPi","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(D_\\pi\\)"],["text","-groups"]],"source":"$D_\\pi$-groups"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-dpi","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers (only the primes in $\\pi$ matter). A group $G$ is a $D_\\pi$-group (Kourovka Archive 3.62) when $G$ has a Hall $\\pi$-subgroup, any two Hall $\\pi$-subgroups $H$ and $K$ of $G$ are conjugate ($K = gHg^{-1}$ for some $g \\in G$), and every subgroup of $G$ whose order is a $\\pi$-number lies in some Hall $\\pi$-subgroup. Here the order of an infinite subgroup is taken to be $0$, which\u2026","title":"$D_\\pi$-groups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/normal-subgroup-of-dpi-is-dpi","kind":"node","lean":"MathlibExt.GroupTheory.DPiNormalSubgroupKourovka13Problem33Wanted.normalSubgroupOfDPiIsDPi","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#normal-subgroup-of-dpi-is-dpi","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"This is the proposition (F. Gross) that for every set $\\pi$ of natural numbers, every normal subgroup of a finite $D_\\pi$-group is itself a $D_\\pi$-group.","title":"Kourovka Problem 13.33"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/normal-subgroup-of-dpi-is-dpi-holds","kind":"node","lean":"MathlibExt.GroupTheory.DPiNormalSubgroupKourovka13Problem33Wanted.normalSubgroupOfDPiIsDPi_holds","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normal subgroups of finite "],["math","\\(D_\\pi\\)"],["text","-groups are "],["math","\\(D_\\pi\\)"],["text","-groups"]],"source":"Normal subgroups of finite $D_\\pi$-groups are $D_\\pi$-groups"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#normal-subgroup-of-dpi-is-dpi-holds","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"can_prove","summary":"Kourovka Problem 13.33 has a positive answer: for every set $\\pi$ of natural numbers, every normal subgroup of a finite $D_\\pi$-group is itself a $D_\\pi$-group.","title":"Normal subgroups of finite $D_\\pi$-groups are $D_\\pi$-groups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-pi-number-3","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.IsPiNumber","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\pi\\)"],["text","-number"]],"source":"$\\pi$-number"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-pi-number-3","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers (only the primes in it matter). A natural number $n$ is a $\\pi$-number when every prime divisor of $n$ lies in $\\pi$.","title":"$\\pi$-number"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-hall-pi-subgroup-3","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.IsHallPiSubgroup","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hall "],["math","\\(\\pi\\)"],["text","-subgroup"]],"source":"Hall $\\pi$-subgroup"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-hall-pi-subgroup-3","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers and $G$ a group. A subgroup $H$ of $G$ is a Hall $\\pi$-subgroup when its order $|H|$ is a $\\pi$-number and no prime divisor of its index $|G : H|$ lies in $\\pi$.","title":"Hall $\\pi$-subgroup"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-maximal-pi-subgroup-2","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.IsMaximalPiSubgroup","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal "],["math","\\(\\pi\\)"],["text","-subgroup"]],"source":"Maximal $\\pi$-subgroup"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-maximal-pi-subgroup-2","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers and $G$ a group. A subgroup $H$ of $G$ is a maximal $\\pi$-subgroup when its order $|H|$ is a $\\pi$-number and every subgroup $L$ of $G$ that contains $H$ and whose order is a $\\pi$-number equals $H$.","title":"Maximal $\\pi$-subgroup"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-cpi-group","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.IsCPiGroup","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(C_\\pi\\)"],["text","-group"]],"source":"$C_\\pi$-group"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-cpi-group","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A group $G$ is a $C_\\pi$-group when it has a Hall $\\pi$-subgroup and any two Hall $\\pi$-subgroups $H$ and $K$ of $G$ are conjugate, that is, $K = gHg^{-1}$ for some $g \\in G$.","title":"$C_\\pi$-group"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-dpi-group-2","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.IsDPiGroup","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(D_\\pi\\)"],["text","-group"]],"source":"$D_\\pi$-group"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-dpi-group-2","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A group $G$ is a $D_\\pi$-group when any two maximal $\\pi$-subgroups $H$ and $K$ of $G$ are conjugate, that is, $K = gHg^{-1}$ for some $g \\in G$.","title":"$D_\\pi$-group"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/hall-overgroup-cpi","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.hallOvergroupCPi","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 17.44(a): overgroups of Hall subgroups in "],["math","\\(C_\\pi\\)"],["text","-groups"]],"source":"Kourovka Problem 17.44(a): overgroups of Hall subgroups in $C_\\pi$-groups"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#hall-overgroup-cpi","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"The statement, posed by E. P. Vdovin and D. O. Revin, that for every set $\\pi$ of natural numbers and every finite $C_\\pi$-group $G$, every subgroup $K$ of $G$ that contains a Hall $\\pi$-subgroup of $G$ is itself a $C_\\pi$-group.","title":"Kourovka Problem 17.44(a): overgroups of Hall subgroups in $C_\\pi$-groups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/hall-overgroup-dpi","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.hallOvergroupDPi","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 17.44(b): overgroups of Hall subgroups in "],["math","\\(D_\\pi\\)"],["text","-groups"]],"source":"Kourovka Problem 17.44(b): overgroups of Hall subgroups in $D_\\pi$-groups"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#hall-overgroup-dpi","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"The statement, posed by E. P. Vdovin and D. O. Revin, that for every set $\\pi$ of natural numbers and every finite $D_\\pi$-group $G$, every subgroup $K$ of $G$ that contains a Hall $\\pi$-subgroup of $G$ is itself a $D_\\pi$-group.","title":"Kourovka Problem 17.44(b): overgroups of Hall subgroups in $D_\\pi$-groups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/hall-overgroup-cpi-holds","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.hallOvergroupCPi_holds","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overgroups of Hall subgroups in "],["math","\\(C_\\pi\\)"],["text","-groups (Vdovin\u2013Revin)"]],"source":"Overgroups of Hall subgroups in $C_\\pi$-groups (Vdovin\u2013Revin)"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#hall-overgroup-cpi-holds","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"can_prove","summary":"Let $\\pi$ be a set of natural numbers and let $G$ be a finite $C_\\pi$-group. Then every subgroup $K$ of $G$ that contains a Hall $\\pi$-subgroup of $G$ is itself a $C_\\pi$-group. This answers Kourovka Problem 17.44(a) in the affirmative.","title":"Overgroups of Hall subgroups in $C_\\pi$-groups (Vdovin\u2013Revin)"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/hall-overgroup-dpi-holds","kind":"node","lean":"MathlibExt.GroupTheory.HallOvergroupCPiDPiKourovka17Problem44Wanted.hallOvergroupDPi_holds","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overgroups of Hall subgroups in "],["math","\\(D_\\pi\\)"],["text","-groups (Manzaeva, Vdovin\u2013Manzaeva\u2013Revin)"]],"source":"Overgroups of Hall subgroups in $D_\\pi$-groups (Manzaeva, Vdovin\u2013Manzaeva\u2013Revin)"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#hall-overgroup-dpi-holds","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"can_prove","summary":"Let $\\pi$ be a set of natural numbers and let $G$ be a finite $D_\\pi$-group. Then every subgroup $K$ of $G$ that contains a Hall $\\pi$-subgroup of $G$ is itself a $D_\\pi$-group. This answers Kourovka Problem 17.44(b) in the affirmative.","title":"Overgroups of Hall subgroups in $D_\\pi$-groups (Manzaeva, Vdovin\u2013Manzaeva\u2013Revin)"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-pi-number-4","kind":"node","lean":"MathlibExt.GroupTheory.HallSubgroupPrimeSetIntersectionKourovka18Problem38Wanted.IsPiNumber","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-pi-number-4","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A natural number $n$ is a $\\pi$-number when every prime divisor of $n$ lies in $\\pi$.","title":"\u03c0-numbers"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/is-hall-pi-subgroup-4","kind":"node","lean":"MathlibExt.GroupTheory.HallSubgroupPrimeSetIntersectionKourovka18Problem38Wanted.IsHallPiSubgroup","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#is-hall-pi-subgroup-4","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers and let $H$ be a subgroup of a group $G$. Then $H$ is a Hall $\\pi$-subgroup when its order $|H|$ is a $\\pi$-number and no prime divisor of its index $|G : H|$ lies in $\\pi$.","title":"Hall \u03c0-subgroups"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/has-hall-pi-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.HallSubgroupPrimeSetIntersectionKourovka18Problem38Wanted.HasHallPiSubgroup","parent":2843,"reading_title":{"format":"inline-parts-v1","parts":[["text","The class "],["math","\\(E_\\pi\\)"],["text"," of groups with a Hall \u03c0-subgroup"]],"source":"The class $E_\\pi$ of groups with a Hall \u03c0-subgroup"},"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#has-hall-pi-subgroup","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"Let $\\pi$ be a set of natural numbers. A group $G$ belongs to the class $E_\\pi$ when $G$ has a Hall $\\pi$-subgroup.","title":"The class $E_\\pi$ of groups with a Hall \u03c0-subgroup"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/hall-subgroup-for-intersection-of-prime","kind":"node","lean":"MathlibExt.GroupTheory.HallSubgroupPrimeSetIntersectionKourovka18Problem38Wanted.hallSubgroupForIntersectionOfPrimeSets","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#hall-subgroup-for-intersection-of-prime","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"fully_proved","summary":"This is the statement that for all sets $\\pi_1$ and $\\pi_2$ of natural numbers, every finite group that has a Hall $\\pi_1$-subgroup and a Hall $\\pi_2$-subgroup also has a Hall $(\\pi_1 \\cap \\pi_2)$-subgroup; that is, writing $E_\\pi$ for the class of finite groups with a Hall $\\pi$-subgroup, $E_{\\pi_1} \\cap E_{\\pi_2}$ is contained in $E_{\\pi_1 \\cap \\pi_2}$.","title":"Zavarnitsine's question on Hall subgroups for intersections of prime sets"},{"id":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups/hall-subgroup-for-intersection-of-prime-2","kind":"node","lean":"MathlibExt.GroupTheory.HallSubgroupPrimeSetIntersectionKourovka18Problem38Wanted.hallSubgroupForIntersectionOfPrimeSets_holds","parent":2843,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/hall-subgroups/index.html#hall-subgroup-for-intersection-of-prime-2","scope":"GroupTheory/sylow-and-hall-subgroups/hall-subgroups","status":"can_prove","summary":"For all sets $\\pi_1$ and $\\pi_2$ of natural numbers, every finite group that has a Hall $\\pi_1$-subgroup and a Hall $\\pi_2$-subgroup also has a Hall $(\\pi_1 \\cap \\pi_2)$-subgroup; that is, writing $E_\\pi$ for the class of finite groups with a Hall $\\pi$-subgroup, $E_{\\pi_1} \\cap E_{\\pi_2}$ is contained in $E_{\\pi_1 \\cap \\pi_2}$.","title":"Hall subgroups exist for intersections of prime sets"},{"data_url":"scopes/GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups.json","id":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups","kind":"scope","navigation":"module","parent":2829,"result":false,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/index.html","scope":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups","status":"can_prove","title":"Pronormality of Hall subgroups"},{"id":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/is-hall-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.HallPronormalNormalClosureKourovka18Problem32Wanted.IsHallSubgroup","parent":2871,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/index.html#is-hall-subgroup","scope":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is a Hall subgroup when its order $|H|$ is coprime to its index $|G : H|$.","title":"Hall subgroups (order coprime to index)"},{"id":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/is-pronormal-in","kind":"node","lean":"MathlibExt.GroupTheory.HallPronormalNormalClosureKourovka18Problem32Wanted.IsPronormalIn","parent":2871,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/index.html#is-pronormal-in","scope":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups","status":"fully_proved","summary":"Let $H$ and $K$ be subgroups of a group $G$. Then $H$ is pronormal in $K$ when for every $g \\in K$ there is an element $x$ of the subgroup $\\langle H, gHg^{-1} \\rangle$ generated by $H$ and $gHg^{-1}$ such that $xHx^{-1} = gHg^{-1}$, that is, $H$ and its conjugate $gHg^{-1}$ are conjugate by an element of $\\langle H, gHg^{-1} \\rangle$.","title":"Pronormality in a subgroup"},{"id":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/hall-subgroup-pronormal-in-normal-closur","kind":"node","lean":"MathlibExt.GroupTheory.HallPronormalNormalClosureKourovka18Problem32Wanted.hallSubgroupPronormalInNormalClosure","parent":2871,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/index.html#hall-subgroup-pronormal-in-normal-closur","scope":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups","status":"fully_proved","summary":"This is the statement that for every finite group $G$ and every Hall subgroup $H$ of $G$ (a subgroup whose order $|H|$ is coprime to its index $|G : H|$), $H$ is pronormal in its normal closure $\\langle H^G \\rangle$, the smallest normal subgroup of $G$ containing $H$: for every $g \\in \\langle H^G \\rangle$ there is an element $x$ of the subgroup $\\langle H, gHg^{-1} \\rangle$ generated by $H$ and $gHg^{-1}$ with\u2026","title":"Vdovin\u2013Revin question on Hall subgroups and their normal closures"},{"id":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/hall-subgroup-pronormal-in-normal-closur-2","kind":"node","lean":"MathlibExt.GroupTheory.HallPronormalNormalClosureKourovka18Problem32Wanted.hallSubgroupPronormalInNormalClosure_refuted","parent":2871,"result":true,"route":"../roadmap/GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups/index.html#hall-subgroup-pronormal-in-normal-closur-2","scope":"GroupTheory/sylow-and-hall-subgroups/pronormality-of-hall-subgroups","status":"can_prove","summary":"There exist a finite group $G$ and a Hall subgroup $H$ of $G$ (that is, the order $|H|$ is coprime to the index $|G : H|$) such that $H$ is not pronormal in its normal closure $\\langle H^G \\rangle$, the smallest normal subgroup of $G$ containing $H$: for some $g \\in \\langle H^G \\rangle$, no element $x$ of the subgroup $\\langle H, gHg^{-1} \\rangle$ generated by $H$ and $gHg^{-1}$ satisfies $xHx^{-1} = gHg^{-1}$.","title":"Hall subgroups need not be pronormal in their normal closure"},{"data_url":"scopes/GroupTheory/subgroup-embedding.json","id":"GroupTheory/subgroup-embedding","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/index.html","scope":"GroupTheory/subgroup-embedding","status":"can_prove","summary":"How subgroups sit inside a group: subnormal, permutable, propermutable, pronormal, polynormal and paranormal subgroups, maximal subgroups and their conjugates, intersections of conjugates of nilpotent subgroups and injectors with the Fitting subgroup, centralizer lattices and complemented subgroup lattices.","title":"Subgroup embedding and subgroup lattices"},{"data_url":"scopes/GroupTheory/subgroup-embedding/block-lattices-of-groups.json","id":"GroupTheory/subgroup-embedding/block-lattices-of-groups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/block-lattices-of-groups/index.html","scope":"GroupTheory/subgroup-embedding/block-lattices-of-groups","status":"can_prove","title":"Block lattices of groups"},{"id":"GroupTheory/subgroup-embedding/block-lattices-of-groups/lattice-term","kind":"node","lean":"MathlibExt.GroupTheory.BlockLatticeVarietyKourovka11Problem93Wanted.LatticeTerm","parent":2877,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/block-lattices-of-groups/index.html#lattice-term","scope":"GroupTheory/subgroup-embedding/block-lattices-of-groups","status":"fully_proved","summary":"Lattice terms in the variables $x_0, x_1, x_2, \\dots$ are built inductively: each variable $x_n$ is a term, and for terms $s$ and $t$ their join $s \\vee t$ and their meet $s \\wedge t$ are terms.","title":"Lattice terms"},{"id":"GroupTheory/subgroup-embedding/block-lattices-of-groups/eval","kind":"node","lean":"MathlibExt.GroupTheory.BlockLatticeVarietyKourovka11Problem93Wanted.LatticeTerm.eval","parent":2877,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/block-lattices-of-groups/index.html#eval","scope":"GroupTheory/subgroup-embedding/block-lattices-of-groups","status":"fully_proved","summary":"Let $L$ be a lattice and $v$ an assignment of an element $v(n)$ of $L$ to each variable $x_n$. The value of a lattice term under $v$ is defined by recursion on the term: the variable $x_n$ has value $v(n)$, and a join $s \\vee t$ or meet $s \\wedge t$ has as value the join or meet in $L$ of the values of $s$ and $t$.","title":"Evaluation of lattice terms"},{"id":"GroupTheory/subgroup-embedding/block-lattices-of-groups/is-block","kind":"node","lean":"MathlibExt.GroupTheory.BlockLatticeVarietyKourovka11Problem93Wanted.IsBlock","parent":2877,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/block-lattices-of-groups/index.html#is-block","scope":"GroupTheory/subgroup-embedding/block-lattices-of-groups","status":"fully_proved","summary":"A subset $X$ of a group $G$ is a block if it is empty or a coset $gN$ of a normal subgroup $N$ of $G$ (Kourovka Notebook, Archive 9.62). Ordered by inclusion, the blocks of $G$ form the block lattice $C(G)$.","title":"Blocks of a group"},{"id":"GroupTheory/subgroup-embedding/block-lattices-of-groups/block-sup","kind":"node","lean":"MathlibExt.GroupTheory.BlockLatticeVarietyKourovka11Problem93Wanted.blockSup","parent":2877,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/block-lattices-of-groups/index.html#block-sup","scope":"GroupTheory/subgroup-embedding/block-lattices-of-groups","status":"fully_proved","summary":"For subsets $X$ and $Y$ of a group $G$, their block join is the intersection of all blocks of $G$ that contain $X \\cup Y$. Since blocks are closed under intersections and $G$ itself is a block, this is the smallest block containing $X \\cup Y$; for blocks $X$ and $Y$ it is their join in the block lattice of $G$.","title":"Join in the block lattice"},{"id":"GroupTheory/subgroup-embedding/block-lattices-of-groups/eval-block","kind":"node","lean":"MathlibExt.GroupTheory.BlockLatticeVarietyKourovka11Problem93Wanted.LatticeTerm.evalBlock","parent":2877,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/block-lattices-of-groups/index.html#eval-block","scope":"GroupTheory/subgroup-embedding/block-lattices-of-groups","status":"fully_proved","summary":"Let $G$ be a group and $v$ an assignment of a subset $v(n)$ of $G$ to each variable $x_n$. The value of a lattice term under $v$ is defined by recursion on the term: the variable $x_n$ has value $v(n)$, a meet $s \\wedge t$ has as value the intersection of the values of $s$ and $t$, and a join $s \\vee t$ has as value the block join of the values of $s$ and $t$, the intersection of all blocks of $G$ containing their\u2026","title":"Evaluation of lattice terms in the block lattice"},{"id":"GroupTheory/subgroup-embedding/block-lattices-of-groups/block-lattices-generate-all-lattices","kind":"node","lean":"MathlibExt.GroupTheory.BlockLatticeVarietyKourovka11Problem93Wanted.blockLatticesGenerateAllLattices","parent":2877,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/block-lattices-of-groups/index.html#block-lattices-generate-all-lattices","scope":"GroupTheory/subgroup-embedding/block-lattices-of-groups","status":"fully_proved","summary":"Kourovka Problem 11.93: do block lattices generate all lattices? The proposition, asked by D. M. Smirnov, that the block lattices of finite groups generate the variety of all lattices. Here the blocks of a group $G$ are the cosets of normal subgroups of $G$ and the empty set; ordered by inclusion they form the block lattice of $G$, whose meet is intersection and whose join of two blocks is the intersection of all\u2026","title":"Kourovka Problem 11.93: do block lattices generate all lattices?"},{"id":"GroupTheory/subgroup-embedding/block-lattices-of-groups/block-lattices-generate-all-lattices-ref","kind":"node","lean":"MathlibExt.GroupTheory.BlockLatticeVarietyKourovka11Problem93Wanted.blockLatticesGenerateAllLattices_refuted","parent":2877,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/block-lattices-of-groups/index.html#block-lattices-generate-all-lattices-ref","scope":"GroupTheory/subgroup-embedding/block-lattices-of-groups","status":"can_prove","summary":"The block lattices of finite groups do not generate the variety of all lattices: there are lattice terms $p$ and $q$ such that $p$ and $q$ take the same value in the block lattice of every finite group $G$ under every assignment of blocks of $G$ to the variables, but take different values in some lattice $L$ under some assignment of elements of $L$ to the variables. Here the blocks of $G$ are the cosets of normal\u2026","title":"Smirnov's negative answer to Kourovka Problem 11.93"},{"data_url":"scopes/GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups.json","id":"GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups","status":"can_prove","title":"Conjugacy of maximal subgroups"},{"id":"GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups/isomorphic-simple-maximal-subgroups-aut","kind":"node","lean":"MathlibExt.GroupTheory.IsomorphicSimpleMaximalSubgroupsKourovka18Problem115Wanted.isomorphicSimpleMaximalSubgroupsAutConjugate","parent":2885,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups/index.html#isomorphic-simple-maximal-subgroups-aut","scope":"GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups","status":"fully_proved","summary":"The statement that in every finite simple group $G$, any two isomorphic maximal subgroups $X$ and $Y$ that are themselves simple groups are conjugate in $\\operatorname{Aut} G$, that is, $\\varphi(X) = Y$ for some automorphism $\\varphi$ of $G$.","title":"Schmid's question on isomorphic simple maximal subgroups (Kourovka 18.115)"},{"id":"GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups/isomorphic-simple-maximal-subgroups-aut-2","kind":"node","lean":"MathlibExt.GroupTheory.IsomorphicSimpleMaximalSubgroupsKourovka18Problem115Wanted.isomorphicSimpleMaximalSubgroupsAutConjugate_refuted","parent":2885,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups/index.html#isomorphic-simple-maximal-subgroups-aut-2","scope":"GroupTheory/subgroup-embedding/conjugacy-of-maximal-subgroups","status":"can_prove","summary":"There are a finite simple group $G$ and isomorphic maximal subgroups $X$ and $Y$ of $G$, both simple groups, such that no automorphism $\\varphi$ of $G$ satisfies $\\varphi(X) = Y$. That is, the answer to Kourovka Problem 18.115 is negative.","title":"Isomorphic simple maximal subgroups need not be conjugate under automorphisms"},{"data_url":"scopes/GroupTheory/subgroup-embedding/fully-invariant-subgroups.json","id":"GroupTheory/subgroup-embedding/fully-invariant-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/fully-invariant-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/fully-invariant-subgroups","status":"can_prove","title":"Fully invariant subgroups"},{"id":"GroupTheory/subgroup-embedding/fully-invariant-subgroups/fully-invariant-subgroups-of-each-order","kind":"node","lean":"MathlibExt.GroupTheory.FullyInvariantSubgroupsCyclicKourovka16Problem61Wanted.fullyInvariantSubgroupsOfEachOrderCyclic","parent":2888,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/fully-invariant-subgroups/index.html#fully-invariant-subgroups-of-each-order","scope":"GroupTheory/subgroup-embedding/fully-invariant-subgroups","status":"fully_proved","summary":"The statement that every finite group $G$ having, for every divisor $d$ of $|G|$, a fully invariant subgroup of order $d$ (a subgroup $H$ with $\\varphi(H) \\le H$ for every endomorphism $\\varphi$ of $G$) is cyclic.","title":"MacHale's question on fully invariant subgroups (Kourovka 16.61)"},{"id":"GroupTheory/subgroup-embedding/fully-invariant-subgroups/fully-invariant-subgroups-of-each-order-2","kind":"node","lean":"MathlibExt.GroupTheory.FullyInvariantSubgroupsCyclicKourovka16Problem61Wanted.fullyInvariantSubgroupsOfEachOrderCyclic_refuted","parent":2888,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/fully-invariant-subgroups/index.html#fully-invariant-subgroups-of-each-order-2","scope":"GroupTheory/subgroup-embedding/fully-invariant-subgroups","status":"can_prove","summary":"Not every finite group $G$ having, for every divisor $d$ of $|G|$, a fully invariant subgroup of order $d$ (a subgroup $H$ with $\\varphi(H) \\le H$ for every endomorphism $\\varphi$ of $G$) is cyclic.","title":"Fully invariant subgroups of every order do not force cyclicity"},{"data_url":"scopes/GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups.json","id":"GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups","status":"can_prove","title":"Groups isomorphic to their normal subgroups"},{"id":"GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups/iso-to-normal-subgroups-free-simple-or-c","kind":"node","lean":"MathlibExt.GroupTheory.IsomorphicNormalSubgroupsKourovka12Problem45Wanted.isoToNormalSubgroupsFreeSimpleOrCyclic","parent":2891,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups/index.html#iso-to-normal-subgroups-free-simple-or-c","scope":"GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups","status":"fully_proved","summary":"The statement that every nontrivial group that is isomorphic to each of its nontrivial normal subgroups is free of infinite rank (isomorphic to the free group on an infinite set), simple, or infinite cyclic.","title":"Hall's question on groups isomorphic to their nontrivial normal subgroups (Kourovka 12.45)"},{"id":"GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups/iso-to-normal-subgroups-free-simple-or-c-2","kind":"node","lean":"MathlibExt.GroupTheory.IsomorphicNormalSubgroupsKourovka12Problem45Wanted.isoToNormalSubgroupsFreeSimpleOrCyclic_refuted","parent":2891,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups/index.html#iso-to-normal-subgroups-free-simple-or-c-2","scope":"GroupTheory/subgroup-embedding/groups-isomorphic-to-their-normal-subgroups","status":"can_prove","summary":"A group isomorphic to its nontrivial normal subgroups need not be free of infinite rank, simple or infinite cyclic. There is a nontrivial group $G$ that is isomorphic to each of its nontrivial normal subgroups and is neither free of infinite rank (isomorphic to the free group on an infinite set), nor simple, nor infinite cyclic. That is, the answer to Kourovka Problem 12.45 is negative (V. N. Obraztsov, 1997).","title":"A group isomorphic to its nontrivial normal subgroups need not be free of infinite rank, simple or infinite cyclic"},{"data_url":"scopes/GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices.json","id":"GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices/index.html","scope":"GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices","status":"can_prove","title":"Groups with complemented subgroup lattices"},{"id":"GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices/normal-subgroup-of-kgroup-is-kgroup","kind":"node","lean":"MathlibExt.GroupTheory.ComplementedSubgroupLatticeNormalKourovka11Problem128Wanted.normalSubgroupOfKGroupIsKGroup","parent":2894,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 11.128 on normal subgroups of "],["math","\\(K\\)"],["text","-groups"]],"source":"Kourovka Problem 11.128 on normal subgroups of $K$-groups"},"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices/index.html#normal-subgroup-of-kgroup-is-kgroup","scope":"GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices","status":"fully_proved","summary":"Call a group $G$ a $K$-group when its subgroup lattice is complemented: every subgroup $A$ of $G$ has a subgroup $B$ with $A \\cap B = 1$ and $\\langle A, B \\rangle = G$. This is the assertion (M. Emaldi) that every normal subgroup of a $K$-group, viewed as a group in its own right, is again a $K$-group.","title":"Kourovka Problem 11.128 on normal subgroups of $K$-groups"},{"id":"GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices/normal-subgroup-of-kgroup-is-kgroup-refu","kind":"node","lean":"MathlibExt.GroupTheory.ComplementedSubgroupLatticeNormalKourovka11Problem128Wanted.normalSubgroupOfKGroupIsKGroup_refuted","parent":2894,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normal subgroups of "],["math","\\(K\\)"],["text","-groups need not be "],["math","\\(K\\)"],["text","-groups"]],"source":"Normal subgroups of $K$-groups need not be $K$-groups"},"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices/index.html#normal-subgroup-of-kgroup-is-kgroup-refu","scope":"GroupTheory/subgroup-embedding/groups-with-complemented-subgroup-lattices","status":"can_prove","summary":"There is a group $G$ whose subgroup lattice is complemented (a $K$-group) with a normal subgroup $N$ whose own subgroup lattice is not complemented; so Kourovka Problem 11.128 has a negative answer.","title":"Normal subgroups of $K$-groups need not be $K$-groups"},{"data_url":"scopes/GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups.json","id":"GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups","status":"can_prove","title":"Groups with few maximal subgroups"},{"id":"GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups/three-maximal-subgroups-finite","kind":"node","lean":"MathlibExt.GroupTheory.ThreeMaximalSubgroupsInfiniteKourovka17Problem17Wanted.threeMaximalSubgroupsFinite","parent":2897,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups/index.html#three-maximal-subgroups-finite","scope":"GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups","status":"fully_proved","summary":"Kourovka 17.17 (G. M. Bergman) asks whether a finitely generated group with only finitely many maximal subgroups must be finite. This is the case of three maximal subgroups, which the problem singles out: every finitely generated group with exactly three maximal subgroups is finite.","title":"Bergman's question for three maximal subgroups"},{"id":"GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups/three-maximal-subgroups-finite-refuted","kind":"node","lean":"MathlibExt.GroupTheory.ThreeMaximalSubgroupsInfiniteKourovka17Problem17Wanted.threeMaximalSubgroupsFinite_refuted","parent":2897,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups/index.html#three-maximal-subgroups-finite-refuted","scope":"GroupTheory/subgroup-embedding/groups-with-few-maximal-subgroups","status":"can_prove","summary":"There is a finitely generated infinite group with exactly three maximal subgroups; that is, it is false that every finitely generated group with exactly three maximal subgroups is finite.","title":"Not every finitely generated group with three maximal subgroups is finite"},{"data_url":"scopes/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups.json","id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"can_prove","title":"Intersections of conjugate nilpotent subgroups"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/nilpotent-subgroups-trivial-conjugate-in","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentSubgroupsTrivialIntersectionKourovka15Problem40Wanted.nilpotentSubgroupsTrivialConjugateIntersection","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#nilpotent-subgroups-trivial-conjugate-in","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"The assertion, in the two-subgroup form of its resolution, that for every finite non-abelian simple group $G$ and all nilpotent subgroups $A$ and $B$ of $G$ there is $x \\in G$ with $A \\cap x B x^{-1} = 1$.","title":"Kourovka Problem 15.40"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/nilpotent-subgroups-trivial-conjugate-in-2","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentSubgroupsTrivialIntersectionKourovka15Problem40Wanted.nilpotentSubgroupsTrivialConjugateIntersection_holds","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#nilpotent-subgroups-trivial-conjugate-in-2","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"can_prove","summary":"Let $G$ be a finite non-abelian simple group and let $A$ and $B$ be nilpotent subgroups of $G$. Then there is $x \\in G$ with $A \\cap x B x^{-1} = 1$.","title":"Nilpotent subgroups of a finite simple group have conjugates meeting trivially"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/fitting-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentThreeConjugatesFittingKourovka17Problem40Wanted.fittingSubgroup","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#fitting-subgroup","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"The Fitting subgroup `F(G)`: the join of the nilpotent normal subgroups of `G` (for finite `G` the largest nilpotent normal subgroup).","title":"Fitting subgroup"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/nilpotent-three-conjugates-in-fitting","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentThreeConjugatesFittingKourovka17Problem40Wanted.nilpotentThreeConjugatesInFitting","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#nilpotent-three-conjugates-in-fitting","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"The assertion that for every finite group $G$ and every nilpotent subgroup $N$ of $G$ there are $x, y \\in G$ with $N \\cap x N x^{-1} \\cap y N y^{-1} \\le F(G)$, where $F(G)$ is the Fitting subgroup of $G$.","title":"Kourovka Problem 17.40"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/nilpotent-three-conjugates-in-fitting-ho","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentThreeConjugatesFittingKourovka17Problem40Wanted.nilpotentThreeConjugatesInFitting_holds","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#nilpotent-three-conjugates-in-fitting-ho","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"can_prove","summary":"For every finite group $G$ and every nilpotent subgroup $N$ of $G$ there are $x, y \\in G$ with $N \\cap x N x^{-1} \\cap y N y^{-1} \\le F(G)$, where $F(G)$ is the Fitting subgroup of $G$.","title":"Three conjugates of a nilpotent subgroup meet inside the Fitting subgroup"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/triple-intersections","kind":"node","lean":"MathlibExt.GroupTheory.MinimalIntersectionsFittingKourovka20Problem122Wanted.tripleIntersections","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#triple-intersections","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"For subgroups $A$, $B$, $C$ of a group $G$, this is the set of all subgroups of $G$ of the form $A \\cap xBx^{-1} \\cap yCy^{-1}$ with $x, y \\in G$, that is, the intersections of $A$ with a conjugate of $B$ and a conjugate of $C$.","title":"Triple intersections with conjugates"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/inclusion-minimal-intersection-join","kind":"node","lean":"MathlibExt.GroupTheory.MinimalIntersectionsFittingKourovka20Problem122Wanted.inclusionMinimalIntersectionJoin","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#inclusion-minimal-intersection-join","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"For subgroups $A$, $B$, $C$ of a group $G$, the subgroup $\\mathrm{Min}_G(A, B, C)$ is the subgroup of $G$ generated by those intersections $A \\cap xBx^{-1} \\cap yCy^{-1}$ ($x, y \\in G$) that are minimal by inclusion among all such intersections, that is, that contain no other intersection of this form.","title":"Subgroup generated by the inclusion-minimal triple intersections"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/minimal-order-intersection-join","kind":"node","lean":"MathlibExt.GroupTheory.MinimalIntersectionsFittingKourovka20Problem122Wanted.minimalOrderIntersectionJoin","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#minimal-order-intersection-join","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"For subgroups $A$, $B$, $C$ of a group $G$, the subgroup $\\min_G(A, B, C)$ is the subgroup of $G$ generated by those intersections $A \\cap xBx^{-1} \\cap yCy^{-1}$ ($x, y \\in G$) whose order is at most the order of every such intersection.","title":"Subgroup generated by the triple intersections of least order"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/minimal-order-intersections-in-fitting","kind":"node","lean":"MathlibExt.GroupTheory.MinimalIntersectionsFittingKourovka20Problem122Wanted.minimalOrderIntersectionsInFitting","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#minimal-order-intersections-in-fitting","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"The statement that for every finite soluble group $G$ and all nilpotent subgroups $A$, $B$, $C$ of $G$, the subgroup $\\min_G(A, B, C)$ generated by the intersections $A \\cap xBx^{-1} \\cap yCy^{-1}$ ($x, y \\in G$) of least order is contained in the Fitting subgroup $F(G)$, the join of all nilpotent normal subgroups of $G$.","title":"Minimal-order triple intersections and the Fitting subgroup (Kourovka 20.122(a), (c))"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/inclusion-minimal-intersections-in-fitti","kind":"node","lean":"MathlibExt.GroupTheory.MinimalIntersectionsFittingKourovka20Problem122Wanted.inclusionMinimalIntersectionsInFitting","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#inclusion-minimal-intersections-in-fitti","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"The statement that for every finite soluble group $G$ and all nilpotent subgroups $A$, $B$, $C$ of $G$, the subgroup $\\mathrm{Min}_G(A, B, C)$ generated by those intersections $A \\cap xBx^{-1} \\cap yCy^{-1}$ ($x, y \\in G$) that are minimal by inclusion among all such intersections is contained in the Fitting subgroup $F(G)$, the join of all nilpotent normal subgroups of $G$.","title":"Inclusion-minimal triple intersections and the Fitting subgroup (Kourovka 20.122(b), (c))"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/minimal-order-intersections-in-fitting-r","kind":"node","lean":"MathlibExt.GroupTheory.MinimalIntersectionsFittingKourovka20Problem122Wanted.minimalOrderIntersectionsInFitting_refuted","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#minimal-order-intersections-in-fitting-r","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"can_prove","summary":"It is not true that for every finite soluble group $G$ and all nilpotent subgroups $A$, $B$, $C$ of $G$, the subgroup $\\min_G(A, B, C)$ generated by the intersections $A \\cap xBx^{-1} \\cap yCy^{-1}$ ($x, y \\in G$) of least order is contained in the Fitting subgroup $F(G)$, the join of all nilpotent normal subgroups of $G$.","title":"Minimal-order triple intersections need not lie in the Fitting subgroup"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/inclusion-minimal-intersections-in-fitti-2","kind":"node","lean":"MathlibExt.GroupTheory.MinimalIntersectionsFittingKourovka20Problem122Wanted.inclusionMinimalIntersectionsInFitting_refuted","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#inclusion-minimal-intersections-in-fitti-2","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"can_prove","summary":"It is not true that for every finite soluble group $G$ and all nilpotent subgroups $A$, $B$, $C$ of $G$, the subgroup $\\mathrm{Min}_G(A, B, C)$ generated by those intersections $A \\cap xBx^{-1} \\cap yCy^{-1}$ ($x, y \\in G$) that are minimal by inclusion among all such intersections is contained in the Fitting subgroup $F(G)$, the join of all nilpotent normal subgroups of $G$.","title":"Inclusion-minimal triple intersections need not lie in the Fitting subgroup"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/is-nilpotent-injector","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentInjectorConjugatesKourovka19Problem37Wanted.IsNilpotentInjector","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#is-nilpotent-injector","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is a nilpotent injector when, for every subnormal subgroup $K$ of $G$, the intersection $H \\cap K$ is nilpotent and is a maximal nilpotent subgroup of $K$: every nilpotent subgroup $L$ of $G$ with $H \\cap K \\le L \\le K$ equals $H \\cap K$.","title":"Nilpotent injector"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/nilpotent-injector-bounded-conjugates","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentInjectorConjugatesKourovka19Problem37Wanted.nilpotentInjectorBoundedConjugates","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#nilpotent-injector-bounded-conjugates","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"The assertion that there is a natural number $k$ such that for every finite group $G$ and every nilpotent injector $H$ of $G$ there are elements $g_1, \\dots, g_k \\in G$, not necessarily distinct, with $g_1 H g_1^{-1} \\cap \\dots \\cap g_k H g_k^{-1} = F(G)$, where $F(G)$ is the Fitting subgroup of $G$.","title":"Kourovka Problem 19.37(a)"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/nilpotent-injector-three-conjugates","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentInjectorConjugatesKourovka19Problem37Wanted.nilpotentInjectorThreeConjugates","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#nilpotent-injector-three-conjugates","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"fully_proved","summary":"The assertion that for every finite group $G$ and every nilpotent injector $H$ of $G$ there are elements $g_1, g_2, g_3 \\in G$, not necessarily distinct, with $g_1 H g_1^{-1} \\cap g_2 H g_2^{-1} \\cap g_3 H g_3^{-1} = F(G)$, where $F(G)$ is the Fitting subgroup of $G$.","title":"Kourovka Problem 19.37(b)"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/nilpotent-injector-bounded-conjugates-ho","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentInjectorConjugatesKourovka19Problem37Wanted.nilpotentInjectorBoundedConjugates_holds","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#nilpotent-injector-bounded-conjugates-ho","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"can_prove","summary":"There is a natural number $k$ such that for every finite group $G$ and every nilpotent injector $H$ of $G$ there are elements $g_1, \\dots, g_k \\in G$, not necessarily distinct, with $g_1 H g_1^{-1} \\cap \\dots \\cap g_k H g_k^{-1} = F(G)$, where $F(G)$ is the Fitting subgroup of $G$.","title":"Boundedly many conjugates of a nilpotent injector meet in the Fitting subgroup"},{"id":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/nilpotent-injector-three-conjugates-hold","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentInjectorConjugatesKourovka19Problem37Wanted.nilpotentInjectorThreeConjugates_holds","parent":2900,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups/index.html#nilpotent-injector-three-conjugates-hold","scope":"GroupTheory/subgroup-embedding/intersections-of-conjugate-nilpotent-subgroups","status":"can_prove","summary":"For every finite group $G$ and every nilpotent injector $H$ of $G$ there are elements $g_1, g_2, g_3 \\in G$, not necessarily distinct, with $g_1 H g_1^{-1} \\cap g_2 H g_2^{-1} \\cap g_3 H g_3^{-1} = F(G)$, where $F(G)$ is the Fitting subgroup of $G$.","title":"Three conjugates of a nilpotent injector meet in the Fitting subgroup"},{"data_url":"scopes/GroupTheory/subgroup-embedding/lattices-of-centralizers.json","id":"GroupTheory/subgroup-embedding/lattices-of-centralizers","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/lattices-of-centralizers/index.html","scope":"GroupTheory/subgroup-embedding/lattices-of-centralizers","status":"can_prove","title":"Lattices of centralizers"},{"id":"GroupTheory/subgroup-embedding/lattices-of-centralizers/is-centralizer","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerLatticeModularKourovka11Problem4Wanted.IsCentralizer","parent":2918,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/lattices-of-centralizers/index.html#is-centralizer","scope":"GroupTheory/subgroup-embedding/lattices-of-centralizers","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is a centralizer if $H = C_G(X)$ for some subset $X \\subseteq G$, where $C_G(X)$ is the subgroup of elements of $G$ that commute with every element of $X$.","title":"Centralizer subgroups"},{"id":"GroupTheory/subgroup-embedding/lattices-of-centralizers/centralizer-sublattice-modular","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerLatticeModularKourovka11Problem4Wanted.centralizerSublatticeModular","parent":2918,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/lattices-of-centralizers/index.html#centralizer-sublattice-modular","scope":"GroupTheory/subgroup-embedding/lattices-of-centralizers","status":"fully_proved","summary":"Call a subgroup of a group $G$ a centralizer if it equals $C_G(X)$ for some subset $X \\subseteq G$. This is the proposition that for every group $G$ in which the join (in the subgroup lattice) of any two centralizers is again a centralizer, any three centralizers $A$, $B$, $C$ of $G$ with $A \\le C$ satisfy the modular law $(A \\sqcup B) \\sqcap C = A \\sqcup (B \\sqcap C)$, where $\\sqcup$ and $\\sqcap$ are the join and\u2026","title":"Modularity of centralizer sublattices (Kourovka 11.4)"},{"id":"GroupTheory/subgroup-embedding/lattices-of-centralizers/centralizer-sublattice-modular-refuted","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerLatticeModularKourovka11Problem4Wanted.centralizerSublatticeModular_refuted","parent":2918,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/lattices-of-centralizers/index.html#centralizer-sublattice-modular-refuted","scope":"GroupTheory/subgroup-embedding/lattices-of-centralizers","status":"can_prove","summary":"Call a subgroup of a group $G$ a centralizer if it equals $C_G(X)$ for some subset $X \\subseteq G$. Modularity of centralizer sublattices fails: there is a group $G$ in which the join (in the subgroup lattice) of any two centralizers is again a centralizer, but which has centralizers $A$, $B$, $C$ with $A \\le C$ and $(A \\sqcup B) \\sqcap C \\ne A \\sqcup (B \\sqcap C)$, where $\\sqcup$ and $\\sqcap$ are the join and meet\u2026","title":"Lattices of centralizers need not be modular (Kourovka 11.4)"},{"data_url":"scopes/GroupTheory/subgroup-embedding/maximal-soluble-subgroups.json","id":"GroupTheory/subgroup-embedding/maximal-soluble-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/maximal-soluble-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/maximal-soluble-subgroups","status":"can_prove","title":"Maximal soluble subgroups"},{"id":"GroupTheory/subgroup-embedding/maximal-soluble-subgroups/maximal-soluble-subgroup-normalizer-supp","kind":"node","lean":"MathlibExt.GroupTheory.MaximalSolubleSubgroupNormalizerKourovka14Problem62Wanted.maximalSolubleSubgroupNormalizerSupplement","parent":2922,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/maximal-soluble-subgroups/index.html#maximal-soluble-subgroup-normalizer-supp","scope":"GroupTheory/subgroup-embedding/maximal-soluble-subgroups","status":"fully_proved","summary":"The proposition that for every finite group $G$ and every non-soluble normal subgroup $H$ of $G$ there is a maximal soluble subgroup $S$ of $H$ (a soluble subgroup $S \\le H$ contained in no strictly larger soluble subgroup of $H$) such that $G = H\\,N_G(S)$, where $N_G(S)$ is the normalizer of $S$ in $G$; since $H$ is normal, this says that $H$ and $N_G(S)$ generate $G$.","title":"Kourovka 14.62: a maximal soluble subgroup whose normalizer supplements"},{"id":"GroupTheory/subgroup-embedding/maximal-soluble-subgroups/maximal-soluble-subgroup-normalizer-supp-2","kind":"node","lean":"MathlibExt.GroupTheory.MaximalSolubleSubgroupNormalizerKourovka14Problem62Wanted.maximalSolubleSubgroupNormalizerSupplement_holds","parent":2922,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/maximal-soluble-subgroups/index.html#maximal-soluble-subgroup-normalizer-supp-2","scope":"GroupTheory/subgroup-embedding/maximal-soluble-subgroups","status":"can_prove","summary":"For every finite group $G$ and every non-soluble normal subgroup $H$ of $G$ there is a maximal soluble subgroup $S$ of $H$ (a soluble subgroup $S \\le H$ contained in no strictly larger soluble subgroup of $H$) such that $G = H\\,N_G(S)$, where $N_G(S)$ is the normalizer of $S$ in $G$; since $H$ is normal, this says that $H$ and $N_G(S)$ generate $G$.","title":"Zenkov\u2013Monakhov\u2013Revin theorem (Kourovka 14.62)"},{"data_url":"scopes/GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups.json","id":"GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups/index.html","scope":"GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups","status":"can_prove","title":"Permutable subgroups of alternating groups"},{"id":"GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups/is-gpermutable","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingGroupPermutableSubgroupsKourovka17Problem37Wanted.IsGPermutable","parent":2925,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(G\\)"],["text","-permutable subgroups"]],"source":"$G$-permutable subgroups"},"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups/index.html#is-gpermutable","scope":"GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups","status":"fully_proved","summary":"A subgroup $A$ of a group $G$ is $G$-permutable when for every subgroup $B$ of $G$ there is $x \\in G$ such that $A B^x = B^x A$, where $B^x = x B x^{-1}$ is the conjugate of $B$ by $x$ and the products are products of subsets of $G$.","title":"$G$-permutable subgroups"},{"id":"GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups/alternating-group-no-proper-permutable-s","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingGroupPermutableSubgroupsKourovka17Problem37Wanted.alternatingGroupNoProperPermutableSubgroups","parent":2925,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups/index.html#alternating-group-no-proper-permutable-s","scope":"GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups","status":"fully_proved","summary":"This proposition asserts that there is a natural number $n$ such that for every $m > n$, every $A_m$-permutable subgroup of the alternating group $A_m$ is either the trivial subgroup or $A_m$ itself. Here a subgroup $A$ of a group $G$ is $G$-permutable when for every subgroup $B$ of $G$ there is $x \\in G$ with $A B^x = B^x A$, where $B^x = x B x^{-1}$ is the conjugate of $B$ by $x$ and products of subgroups are\u2026","title":"Kourovka Problem 17.37 on permutable subgroups of alternating groups"},{"id":"GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups/alternating-group-no-proper-permutable-s-2","kind":"node","lean":"MathlibExt.GroupTheory.AlternatingGroupPermutableSubgroupsKourovka17Problem37Wanted.alternatingGroupNoProperPermutableSubgroups_holds","parent":2925,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups/index.html#alternating-group-no-proper-permutable-s-2","scope":"GroupTheory/subgroup-embedding/permutable-subgroups-of-alternating-groups","status":"can_prove","summary":"There is a natural number $n$ such that for every $m > n$, every $A_m$-permutable subgroup of the alternating group $A_m$ is either the trivial subgroup or $A_m$ itself. Here a subgroup $A$ of a group $G$ is $G$-permutable when for every subgroup $B$ of $G$ there is $x \\in G$ with $A B^x = B^x A$, where $B^x = x B x^{-1}$.","title":"Permutable subgroups of large alternating groups are trivial or the whole group"},{"data_url":"scopes/GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups.json","id":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups","status":"can_prove","title":"Polynormal and paranormal subgroups"},{"id":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/conj-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.PolynormalParanormalSubgroupKourovka14Problem50Wanted.conjSubgroup","parent":2929,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/index.html#conj-subgroup","scope":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups","status":"fully_proved","summary":"For a subgroup $A$ of a group $G$ and an element $x \\in G$, the conjugate subgroup is $A^x = x^{-1} A x = \\{x^{-1} a x : a \\in A\\}$.","title":"Conjugate subgroup"},{"id":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/is-paranormal","kind":"node","lean":"MathlibExt.GroupTheory.PolynormalParanormalSubgroupKourovka14Problem50Wanted.IsParanormal","parent":2929,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/index.html#is-paranormal","scope":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups","status":"fully_proved","summary":"A subgroup $A$ of a group $G$ is paranormal (Z. I. Borevich) when, for every $x \\in G$, the conjugate $A^x = x^{-1} A x$ is contained in the subgroup generated by the conjugates $A^u$ with $u$ in the subgroup $\\langle A, A^x \\rangle$ generated by $A$ and $A^x$.","title":"Paranormal subgroup"},{"id":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/is-polynormal","kind":"node","lean":"MathlibExt.GroupTheory.PolynormalParanormalSubgroupKourovka14Problem50Wanted.IsPolynormal","parent":2929,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/index.html#is-polynormal","scope":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups","status":"fully_proved","summary":"A subgroup $A$ of a group $G$ is polynormal (Z. I. Borevich) when, for every $x \\in G$, the conjugate $A^x = x^{-1} A x$ is contained in the subgroup generated by the conjugates $A^u$ with $u \\in A^{\\langle x \\rangle}$, where $A^{\\langle x \\rangle}$ is the subgroup generated by the conjugates $A^{x^n}$ with $n \\in \\mathbb{Z}$.","title":"Polynormal subgroup"},{"id":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/polynormal-implies-paranormal","kind":"node","lean":"MathlibExt.GroupTheory.PolynormalParanormalSubgroupKourovka14Problem50Wanted.polynormalImpliesParanormal","parent":2929,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/index.html#polynormal-implies-paranormal","scope":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups","status":"fully_proved","summary":"Kourovka 14.50 (Z. I. Borevich; A. S. Kondratiev): every polynormal subgroup of a finite group is paranormal.","title":"Polynormal subgroups of finite groups are paranormal"},{"id":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/polynormal-implies-paranormal-refuted","kind":"node","lean":"MathlibExt.GroupTheory.PolynormalParanormalSubgroupKourovka14Problem50Wanted.polynormalImpliesParanormal_refuted","parent":2929,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups/index.html#polynormal-implies-paranormal-refuted","scope":"GroupTheory/subgroup-embedding/polynormal-and-paranormal-subgroups","status":"can_prove","summary":"There is a finite group $G$ with a subgroup $A$ that is polynormal but not paranormal. So the conjecture of Kourovka Problem 14.50 is false.","title":"A polynormal subgroup that is not paranormal (Kourovka 14.50 refuted)"},{"data_url":"scopes/GroupTheory/subgroup-embedding/pronormal-subgroups.json","id":"GroupTheory/subgroup-embedding/pronormal-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/pronormal-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/pronormal-subgroups","status":"can_prove","title":"Pronormal subgroups"},{"id":"GroupTheory/subgroup-embedding/pronormal-subgroups/exists-infinite-group-without-proper-pro","kind":"node","lean":"MathlibExt.GroupTheory.NoProperPronormalSubgroupsKourovka19Problem74Wanted.existsInfiniteGroupWithoutProperPronormal","parent":2935,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/pronormal-subgroups/index.html#exists-infinite-group-without-proper-pro","scope":"GroupTheory/subgroup-embedding/pronormal-subgroups","status":"fully_proved","summary":"The statement that there is an infinite group $G$ in which every pronormal subgroup is trivial or equal to $G$. Here a subgroup $H$ of $G$ is pronormal when, for every $g \\in G$, the subgroups $H$ and $H^g = gHg^{-1}$ are conjugate by an element of $\\langle H, H^g \\rangle$.","title":"Infinite groups without nontrivial proper pronormal subgroups (Kourovka 19.74, Revin)"},{"id":"GroupTheory/subgroup-embedding/pronormal-subgroups/exists-infinite-group-without-proper-pro-2","kind":"node","lean":"MathlibExt.GroupTheory.NoProperPronormalSubgroupsKourovka19Problem74Wanted.existsInfiniteGroupWithoutProperPronormal_holds","parent":2935,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/pronormal-subgroups/index.html#exists-infinite-group-without-proper-pro-2","scope":"GroupTheory/subgroup-embedding/pronormal-subgroups","status":"can_prove","summary":"There is an infinite group $G$ in which every pronormal subgroup is trivial or equal to $G$. Here a subgroup $H$ of $G$ is pronormal when, for every $g \\in G$, the subgroups $H$ and $H^g = gHg^{-1}$ are conjugate by an element of $\\langle H, H^g \\rangle$. This answers Kourovka Problem 19.74 in the affirmative.","title":"An infinite group without nontrivial proper pronormal subgroups exists"},{"data_url":"scopes/GroupTheory/subgroup-embedding/propermutable-subgroups.json","id":"GroupTheory/subgroup-embedding/propermutable-subgroups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/propermutable-subgroups/index.html","scope":"GroupTheory/subgroup-embedding/propermutable-subgroups","status":"can_prove","title":"Propermutable subgroups"},{"id":"GroupTheory/subgroup-embedding/propermutable-subgroups/permutes","kind":"node","lean":"MathlibExt.GroupTheory.PropermutableSubgroupKourovka18Problem91Wanted.Permutes","parent":2938,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/propermutable-subgroups/index.html#permutes","scope":"GroupTheory/subgroup-embedding/propermutable-subgroups","status":"fully_proved","summary":"Let $H$ and $X$ be subgroups of a group $G$. Then $H$ permutes with $X$ when $HX = XH$ as subsets of $G$, where $HX = \\{hx : h \\in H, x \\in X\\}$ and $XH = \\{xh : x \\in X, h \\in H\\}$.","title":"Permuting subgroups"},{"id":"GroupTheory/subgroup-embedding/propermutable-subgroups/is-propermutable","kind":"node","lean":"MathlibExt.GroupTheory.PropermutableSubgroupKourovka18Problem91Wanted.IsPropermutable","parent":2938,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/propermutable-subgroups/index.html#is-propermutable","scope":"GroupTheory/subgroup-embedding/propermutable-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is propermutable in $G$ when there is a subgroup $B$ of $G$ with $G = N_G(H)B$ (every element of $G$ is a product $nb$ with $n$ in the normalizer $N_G(H)$ of $H$ and $b \\in B$) such that $H$ permutes with every subgroup $X$ of $B$, that is, $HX = XH$.","title":"Propermutable subgroups"},{"id":"GroupTheory/subgroup-embedding/propermutable-subgroups/is-two-nilpotent","kind":"node","lean":"MathlibExt.GroupTheory.PropermutableSubgroupKourovka18Problem91Wanted.IsTwoNilpotent","parent":2938,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/propermutable-subgroups/index.html#is-two-nilpotent","scope":"GroupTheory/subgroup-embedding/propermutable-subgroups","status":"fully_proved","summary":"A group $G$ is $2$-nilpotent when it has a normal subgroup $N$ of finite odd order whose index in $G$ is $2^k$ for some natural number $k$.","title":"2-nilpotent groups"},{"id":"GroupTheory/subgroup-embedding/propermutable-subgroups/propermutable-not-in-intermediate","kind":"node","lean":"MathlibExt.GroupTheory.PropermutableSubgroupKourovka18Problem91Wanted.propermutableNotInIntermediate","parent":2938,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/propermutable-subgroups/index.html#propermutable-not-in-intermediate","scope":"GroupTheory/subgroup-embedding/propermutable-subgroups","status":"fully_proved","summary":"The statement of Kourovka Problem 18.91(a) (A. N. Skiba): there exist a finite group $G$ and subgroups $A \\le B \\le G$ such that $A$ is propermutable in $G$ but $A$, regarded as a subgroup of $B$, is not propermutable in $B$.","title":"Kourovka Problem 18.91(a) (statement)"},{"id":"GroupTheory/subgroup-embedding/propermutable-subgroups/propermutable-sylow-subgroups-two-nilpot","kind":"node","lean":"MathlibExt.GroupTheory.PropermutableSubgroupKourovka18Problem91Wanted.propermutableSylowSubgroupsTwoNilpotent","parent":2938,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/propermutable-subgroups/index.html#propermutable-sylow-subgroups-two-nilpot","scope":"GroupTheory/subgroup-embedding/propermutable-subgroups","status":"fully_proved","summary":"The statement of Kourovka Problem 18.91(b) (A. N. Skiba), with the range $1 \\le k < n$: let $G$ be a finite group and $P$ a non-abelian Sylow $2$-subgroup of $G$, with $|P| = 2^n$, and let $k$ be a natural number with $1 \\le k < n$. If every subgroup of $P$ of order $2^k$ is propermutable in $G$, and, when $k = 1$, every cyclic subgroup of $P$ of order $4$ is propermutable in $G$, then $G$ is $2$-nilpotent.","title":"Kourovka Problem 18.91(b) (statement)"},{"id":"GroupTheory/subgroup-embedding/propermutable-subgroups/propermutable-not-in-intermediate-holds","kind":"node","lean":"MathlibExt.GroupTheory.PropermutableSubgroupKourovka18Problem91Wanted.propermutableNotInIntermediate_holds","parent":2938,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/propermutable-subgroups/index.html#propermutable-not-in-intermediate-holds","scope":"GroupTheory/subgroup-embedding/propermutable-subgroups","status":"can_prove","summary":"There exist a finite group $G$ and subgroups $A \\le B \\le G$ such that $A$ is propermutable in $G$ but $A$, regarded as a subgroup of $B$, is not propermutable in $B$. This is the positive answer to Kourovka Problem 18.91(a).","title":"Propermutability is not inherited by intermediate subgroups"},{"id":"GroupTheory/subgroup-embedding/propermutable-subgroups/propermutable-sylow-subgroups-two-nilpot-2","kind":"node","lean":"MathlibExt.GroupTheory.PropermutableSubgroupKourovka18Problem91Wanted.propermutableSylowSubgroupsTwoNilpotent_holds","parent":2938,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/propermutable-subgroups/index.html#propermutable-sylow-subgroups-two-nilpot-2","scope":"GroupTheory/subgroup-embedding/propermutable-subgroups","status":"can_prove","summary":"Let $G$ be a finite group and $P$ a non-abelian Sylow $2$-subgroup of $G$, with $|P| = 2^n$, and let $k$ be a natural number with $1 \\le k < n$. If every subgroup of $P$ of order $2^k$ is propermutable in $G$, and, when $k = 1$, every cyclic subgroup of $P$ of order $4$ is propermutable in $G$, then $G$ is $2$-nilpotent. This is the positive answer to Kourovka Problem 18.91(b).","title":"Propermutable subgroups of a non-abelian Sylow 2-subgroup force 2-nilpotency"},{"data_url":"scopes/GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups.json","id":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/index.html","scope":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups","status":"can_prove","title":"Subgroup lattices of Chernikov groups"},{"id":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/is-quasicyclic","kind":"node","lean":"MathlibExt.GroupTheory.ChernikovSubgroupLatticeDimensionKourovka11Problem116Wanted.IsQuasicyclic","parent":2946,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/index.html#is-quasicyclic","scope":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups","status":"fully_proved","summary":"A group $Q$ is quasicyclic if it is abelian and infinite and every proper subgroup of $Q$ is finite.","title":"Quasicyclic group"},{"id":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/is-chernikov","kind":"node","lean":"MathlibExt.GroupTheory.ChernikovSubgroupLatticeDimensionKourovka11Problem116Wanted.IsChernikov","parent":2946,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/index.html#is-chernikov","scope":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups","status":"fully_proved","summary":"A group $G$ is a Chernikov group if it has a normal subgroup of finite index that is isomorphic to a direct product $Q_1 \\times \\dots \\times Q_r$ of finitely many quasicyclic groups $Q_1, \\dots, Q_r$. The case $r = 0$ is allowed, so every finite group is a Chernikov group.","title":"Chernikov group"},{"id":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/contains-quasicyclic-square","kind":"node","lean":"MathlibExt.GroupTheory.ChernikovSubgroupLatticeDimensionKourovka11Problem116Wanted.ContainsQuasicyclicSquare","parent":2946,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/index.html#contains-quasicyclic-square","scope":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups","status":"fully_proved","summary":"A group $G$ contains a direct product of two quasicyclic groups over the same prime when there are a quasicyclic group $Q$ and an injective group homomorphism $Q \\times Q \\to G$ (quasicyclic groups over the same prime are isomorphic).","title":"Groups containing the square of a quasicyclic group"},{"id":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/has-finite-order-dimension","kind":"node","lean":"MathlibExt.GroupTheory.ChernikovSubgroupLatticeDimensionKourovka11Problem116Wanted.HasFiniteOrderDimension","parent":2946,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/index.html#has-finite-order-dimension","scope":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups","status":"fully_proved","summary":"A partially ordered set has finite (order) dimension: its order relation is the intersection of finitely many linear orders on the same set.","title":"Poset of finite order dimension"},{"id":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/chernikov-subgroup-lattice-finite-dimens","kind":"node","lean":"MathlibExt.GroupTheory.ChernikovSubgroupLatticeDimensionKourovka11Problem116Wanted.chernikovSubgroupLatticeFiniteDimension","parent":2946,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/index.html#chernikov-subgroup-lattice-finite-dimens","scope":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups","status":"fully_proved","summary":"The proposition that for every Chernikov group $G$ that does not contain a direct product of two quasicyclic groups over the same prime (that is, there is no quasicyclic group $Q$ with an injective homomorphism $Q \\times Q \\to G$), the lattice of subgroups of $G$ has finite order dimension: the inclusion order on the subgroups of $G$ is the intersection of finitely many linear orders on that set. This is Kourovka\u2026","title":"Shevrin's problem on the dimension of subgroup lattices of Chernikov groups"},{"id":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/chernikov-subgroup-lattice-finite-dimens-2","kind":"node","lean":"MathlibExt.GroupTheory.ChernikovSubgroupLatticeDimensionKourovka11Problem116Wanted.chernikovSubgroupLatticeFiniteDimension_holds","parent":2946,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups/index.html#chernikov-subgroup-lattice-finite-dimens-2","scope":"GroupTheory/subgroup-embedding/subgroup-lattices-of-chernikov-groups","status":"can_prove","summary":"Let $G$ be a Chernikov group, that is, a group with a normal subgroup of finite index isomorphic to a direct product of finitely many quasicyclic groups (infinite abelian groups all of whose proper subgroups are finite). Suppose that $G$ does not contain a direct product of two quasicyclic groups over the same prime: there is no quasicyclic group $Q$ with an injective homomorphism $Q \\times Q \\to G$. Then the\u2026","title":"Subgroup lattices of Chernikov groups without a quasicyclic square have finite dimension"},{"data_url":"scopes/GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates.json","id":"GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates","kind":"scope","navigation":"module","parent":2876,"result":false,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates/index.html","scope":"GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates","status":"can_prove","title":"Subgroups permuting with their conjugates"},{"id":"GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates/permuting-conjugates-not-simple","kind":"node","lean":"MathlibExt.GroupTheory.PermutingConjugatesSimpleGroupKourovka10Problem66Wanted.permutingConjugatesNotSimple","parent":2953,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates/index.html#permuting-conjugates-not-simple","scope":"GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates","status":"fully_proved","summary":"The conjecture that a group $G$ containing nontrivial subgroups $A$ and $B$ such that the product set $AB$ is not all of $G$ and $A \\cdot gBg^{-1} = gBg^{-1} \\cdot A$ for every $g \\in G$ is not simple. Products of subgroups are products of subsets of $G$.","title":"Permuting conjugates and simplicity (Kourovka 10.66)"},{"id":"GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates/permuting-conjugates-not-simple-refuted","kind":"node","lean":"MathlibExt.GroupTheory.PermutingConjugatesSimpleGroupKourovka10Problem66Wanted.permutingConjugatesNotSimple_refuted","parent":2953,"result":true,"route":"../roadmap/GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates/index.html#permuting-conjugates-not-simple-refuted","scope":"GroupTheory/subgroup-embedding/subgroups-permuting-with-their-conjugates","status":"can_prove","summary":"There exist a simple group $G$ and nontrivial subgroups $A$ and $B$ of $G$ such that the product set $AB$ is not all of $G$ and $A \\cdot gBg^{-1} = gBg^{-1} \\cdot A$ for every $g \\in G$, where products of subgroups are products of subsets of $G$. So the conjecture of Kourovka Problem 10.66 is false.","title":"A simple group with permuting conjugate subgroups (Kourovka 10.66 refuted)"},{"data_url":"scopes/GroupTheory/finite-p-groups.json","id":"GroupTheory/finite-p-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/index.html","scope":"GroupTheory/finite-p-groups","status":"can_prove","summary":"Groups of prime-power order: composition series, Frattini subgroups and \u03a6-extensions, special, powerful, metacyclic and Camina p-groups, conjugacy class sizes of p-groups, generalized quaternion groups, coclass and rank analogues, derived length bounds, and orders of automorphism groups of p-groups only when the p-group structure is the point.","title":"Finite p-groups"},{"id":"GroupTheory/finite-p-groups/is-generalized-quaternion","kind":"node","lean":"MathlibExt.GroupTheory.GroupClassesShared.IsGeneralizedQuaternion","parent":2956,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/index.html#is-generalized-quaternion","scope":"GroupTheory/finite-p-groups","status":"fully_proved","summary":"A group $P$ is a generalized quaternion group if, for some natural number $k \\ge 1$, it is isomorphic to the quaternion group $Q_{2^{k+2}} = \\langle a, x \\mid a^{2^{k+1}} = 1,\\ x^2 = a^{2^k},\\ x^{-1} a x = a^{-1} \\rangle$ of order $2^{k+2}$. Thus $P$ has order $8$, $16$, $32$ or some larger power of $2$.","title":"Generalized quaternion group"},{"data_url":"scopes/GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj.json","id":"GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj/index.html","scope":"GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj","status":"can_prove","title":"2-groups in which normal closures determine conjugacy"},{"id":"GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj/normal-closure-conj-derived-abelian","kind":"node","lean":"MathlibExt.GroupTheory.TwoGroupNormalClosureConjugateKourovka12Problem15Wanted.normalClosureConjDerivedAbelian","parent":2958,"reading_title":{"format":"inline-parts-v1","parts":[["text","Golikova\u2013Starostin question on finite "],["math","\\(2\\)"],["text","-groups in which normal closures determine conjugacy (Kourovka 12.15)"]],"source":"Golikova\u2013Starostin question on finite $2$-groups in which normal closures determine conjugacy (Kourovka 12.15)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj/index.html#normal-closure-conj-derived-abelian","scope":"GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj","status":"fully_proved","summary":"Golikova\u2013Starostin question on finite $2$-groups in which normal closures determine conjugacy (Kourovka 12.15). The statement that for every finite $2$-group $G$ in which any two elements with the same normal closure are conjugate, the derived subgroup $G'$ of $G$ is abelian.","title":"Golikova\u2013Starostin question on finite $2$-groups in which normal closures determine conjugacy (Kourovka 12.15)"},{"id":"GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj/normal-closure-conj-derived-abelian-refu","kind":"node","lean":"MathlibExt.GroupTheory.TwoGroupNormalClosureConjugateKourovka12Problem15Wanted.normalClosureConjDerivedAbelian_refuted","parent":2958,"reading_title":{"format":"inline-parts-v1","parts":[["text","Equal normal closures forcing conjugacy does not make the derived subgroup of a finite "],["math","\\(2\\)"],["text","-group abelian (Kourovka 12.15)"]],"source":"Equal normal closures forcing conjugacy does not make the derived subgroup of a finite $2$-group abelian (Kourovka 12.15)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj/index.html#normal-closure-conj-derived-abelian-refu","scope":"GroupTheory/finite-p-groups/2-groups-in-which-normal-closures-determine-conj","status":"can_prove","summary":"Equal normal closures forcing conjugacy does not make the derived subgroup of a finite $2$-group abelian (Kourovka 12.15). There exists a finite $2$-group $G$ in which any two elements with the same normal closure are conjugate, but whose derived subgroup $G'$ is not abelian.","title":"Equal normal closures forcing conjugacy does not make the derived subgroup of a finite $2$-group abelian (Kourovka 12.15)"},{"data_url":"scopes/GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups.json","id":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/index.html","scope":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups","status":"can_prove","title":"Abelian subgroups of p-groups"},{"id":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/abelian-subgroup-generators-le-twice-nor","kind":"node","lean":"MathlibExt.GroupTheory.AbelianNormalGeneratorsPGroupKourovka20Problem75Wanted.abelianSubgroupGeneratorsLeTwiceNormal","parent":2961,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/index.html#abelian-subgroup-generators-le-twice-nor","scope":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups","status":"fully_proved","summary":"The proposition that for every prime $p$, every natural number $k$ and every finite $p$-group $G$, if every abelian normal subgroup of $G$ can be generated by at most $k$ elements, then every abelian subgroup of $G$ can be generated by at most $2k$ elements.","title":"Pyber's question on generators of abelian subgroups (Kourovka 20.75)"},{"id":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/abelian-subgroup-generators-le-twice-nor-2","kind":"node","lean":"MathlibExt.GroupTheory.AbelianNormalGeneratorsPGroupKourovka20Problem75Wanted.abelianSubgroupGeneratorsLeTwiceNormal_refuted","parent":2961,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/index.html#abelian-subgroup-generators-le-twice-nor-2","scope":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups","status":"can_prove","summary":"The statement of Kourovka Problem 20.75 is false: there are a prime $p$, a natural number $k$, a finite $p$-group $G$ all of whose abelian normal subgroups can be generated by at most $k$ elements, and an abelian subgroup of $G$ that cannot be generated by at most $2k$ elements.","title":"Kourovka Problem 20.75 has a negative answer"},{"id":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/abelian-subgroup-order-le-square-normal","kind":"node","lean":"MathlibExt.GroupTheory.AbelianNormalOrderPGroupKourovka20Problem76Wanted.abelianSubgroupOrderLeSquareNormal","parent":2961,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/index.html#abelian-subgroup-order-le-square-normal","scope":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups","status":"fully_proved","summary":"The proposition that for every prime $p$, every natural number $k$ and every finite $p$-group $G$, if every abelian normal subgroup of $G$ has order at most $p^k$, then every abelian subgroup of $G$ has order at most $p^{2k}$.","title":"Pyber's question on orders of abelian subgroups (Kourovka 20.76)"},{"id":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/abelian-subgroup-order-le-square-normal-2","kind":"node","lean":"MathlibExt.GroupTheory.AbelianNormalOrderPGroupKourovka20Problem76Wanted.abelianSubgroupOrderLeSquareNormal_refuted","parent":2961,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups/index.html#abelian-subgroup-order-le-square-normal-2","scope":"GroupTheory/finite-p-groups/abelian-subgroups-of-p-groups","status":"can_prove","summary":"The statement of Kourovka Problem 20.76 is false: there are a prime $p$, a natural number $k$, a finite $p$-group $G$ all of whose abelian normal subgroups have order at most $p^k$, and an abelian subgroup of $G$ of order greater than $p^{2k}$.","title":"Kourovka Problem 20.76 has a negative answer"},{"data_url":"scopes/GroupTheory/finite-p-groups/camina-groups.json","id":"GroupTheory/finite-p-groups/camina-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/camina-groups/index.html","scope":"GroupTheory/finite-p-groups/camina-groups","status":"can_prove","title":"Camina groups"},{"id":"GroupTheory/finite-p-groups/camina-groups/nilpotent-camina-class-bounded","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentCaminaClassUnboundedKourovka17Problem47Wanted.nilpotentCaminaClassBounded","parent":2966,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/camina-groups/index.html#nilpotent-camina-class-bounded","scope":"GroupTheory/finite-p-groups/camina-groups","status":"fully_proved","summary":"Kourovka Problem 17.47 (S. H. Ghate, A. S. Muktibodh) asks whether this proposition holds: there is a natural number $c$ such that every nilpotent group $G$ in which, for every $x \\notin [G, G]$, the coset $x[G, G]$ of the commutator subgroup $[G, G]$ equals the conjugacy class $x^G$ of $x$, has nilpotency class at most $c$.","title":"Bounded class for nilpotent Camina-type groups"},{"id":"GroupTheory/finite-p-groups/camina-groups/nilpotent-camina-class-bounded-refuted","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentCaminaClassUnboundedKourovka17Problem47Wanted.nilpotentCaminaClassBounded_refuted","parent":2966,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/camina-groups/index.html#nilpotent-camina-class-bounded-refuted","scope":"GroupTheory/finite-p-groups/camina-groups","status":"can_prove","summary":"There is no natural number $c$ such that every nilpotent group $G$ in which, for every $x \\notin [G, G]$, the coset $x[G, G]$ of the commutator subgroup $[G, G]$ equals the conjugacy class $x^G$ of $x$, has nilpotency class at most $c$. So Kourovka Problem 17.47 has a negative answer.","title":"Nilpotent Camina-type groups have unbounded class"},{"data_url":"scopes/GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups.json","id":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/index.html","scope":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups","status":"can_prove","title":"Coclass and rank of p-groups"},{"id":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/min-generators","kind":"node","lean":"MathlibExt.GroupTheory.RankAnalogueMaximalClassKourovka16Problem103Wanted.minGenerators","parent":2969,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/index.html#min-generators","scope":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups","status":"fully_proved","summary":"For a subgroup $H$ of a group $G$, the minimal number of generators of $H$ is the least natural number $n$ such that $H$ is generated by some finite set of $n$ elements of $G$; it is $0$ when $H$ is not finitely generated.","title":"Minimal number of generators of a subgroup"},{"id":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/group-rank","kind":"node","lean":"MathlibExt.GroupTheory.RankAnalogueMaximalClassKourovka16Problem103Wanted.groupRank","parent":2969,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/index.html#group-rank","scope":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups","status":"fully_proved","summary":"The rank of a group $G$ is the supremum, in the natural numbers, of the minimal numbers of generators of the subgroups of $G$, where an unbounded set of natural numbers has supremum $0$. For a finite group $G$ it is the largest minimal number of generators of a subgroup of $G$.","title":"Rank of a group"},{"id":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/has-cyclic-lower-central-factors-from-se","kind":"node","lean":"MathlibExt.GroupTheory.RankAnalogueMaximalClassKourovka16Problem103Wanted.HasCyclicLowerCentralFactorsFromSecond","parent":2969,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/index.html#has-cyclic-lower-central-factors-from-se","scope":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups","status":"fully_proved","summary":"Let $\\gamma_1(P) = P$ and $\\gamma_{i+1}(P) = [\\gamma_i(P), P]$ be the lower central series of a group $P$. The group $P$ has cyclic lower central factors from the second on if for every $i \\ge 2$ there is an element $g \\in \\gamma_i(P)$ such that every $x \\in \\gamma_i(P)$ satisfies $x g^{-k} \\in \\gamma_{i+1}(P)$ for some integer $k$, that is, the factor $\\gamma_i(P)/\\gamma_{i+1}(P)$ is cyclic.","title":"Cyclic lower central factors from the second on"},{"id":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/rank-analogue-maximal-class","kind":"node","lean":"MathlibExt.GroupTheory.RankAnalogueMaximalClassKourovka16Problem103Wanted.rankAnalogueMaximalClass","parent":2969,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/index.html#rank-analogue-maximal-class","scope":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups","status":"fully_proved","summary":"The statement that for some prime $p$ there is a natural number $r$ such that every finite $p$-group $P$ generated by two elements whose lower central factors $\\gamma_i(P)/\\gamma_{i+1}(P)$ are cyclic for all $i \\ge 2$ has a normal subgroup $N$ of nilpotency class at most $2$ with $P/N$ of rank at most $r$. Here $\\gamma_1(P) = P$ and $\\gamma_{i+1}(P) = [\\gamma_i(P), P]$, and the rank of a finite group is the largest\u2026","title":"Rank analogue of the Shepherd-Leedham-Green-McKay theorem (Kourovka 16.103)"},{"id":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/rank-analogue-maximal-class-refuted","kind":"node","lean":"MathlibExt.GroupTheory.RankAnalogueMaximalClassKourovka16Problem103Wanted.rankAnalogueMaximalClass_refuted","parent":2969,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups/index.html#rank-analogue-maximal-class-refuted","scope":"GroupTheory/finite-p-groups/coclass-and-rank-of-p-groups","status":"can_prove","summary":"For every prime $p$ and every natural number $r$ there is a finite $p$-group $P$ generated by two elements, whose lower central factors $\\gamma_i(P)/\\gamma_{i+1}(P)$ are cyclic for all $i \\ge 2$, such that for every normal subgroup $N$ of $P$ of nilpotency class at most $2$ the quotient $P/N$ has rank greater than $r$. Here $\\gamma_1(P) = P$ and $\\gamma_{i+1}(P) = [\\gamma_i(P), P]$, and the rank of a finite group\u2026","title":"No rank analogue of the Shepherd-Leedham-Green-McKay theorem (Kourovka 16.103)"},{"data_url":"scopes/GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup.json","id":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/index.html","scope":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup","status":"can_prove","title":"Cohomology of p-groups and the Frattini subgroup"},{"id":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/in-frattini-center","kind":"node","lean":"MathlibExt.GroupTheory.FrattiniCenterFirstCohomologyKourovka17Problem2Wanted.InFrattiniCenter","parent":2975,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/index.html#in-frattini-center","scope":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup","status":"fully_proved","summary":"An element $x$ of a group $G$ lies in the centre $Z(\\Phi(G))$ of the Frattini subgroup $\\Phi(G)$ when $x \\in \\Phi(G)$ and $xy = yx$ for every $y \\in \\Phi(G)$.","title":"Centre of the Frattini subgroup"},{"id":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/frattini-center-h1-trivial","kind":"node","lean":"MathlibExt.GroupTheory.FrattiniCenterFirstCohomologyKourovka17Problem2Wanted.FrattiniCenterH1Trivial","parent":2975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing first cohomology of "],["math","\\(G/\\Phi(G)\\)"],["text"," in "],["math","\\(Z(\\Phi(G))\\)"]],"source":"Vanishing first cohomology of $G/\\Phi(G)$ in $Z(\\Phi(G))$"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/index.html#frattini-center-h1-trivial","scope":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup","status":"fully_proved","summary":"Let $G$ be a group, let $\\Phi(G)$ be its Frattini subgroup and let $Z(\\Phi(G))$ be the centre of $\\Phi(G)$. The property holds when every map $\\delta \\colon G \\to Z(\\Phi(G))$ with $\\delta(gh) = \\delta(g)\\, g\\, \\delta(h)\\, g^{-1}$ for all $g, h \\in G$ and $\\delta(g) = 1$ for all $g \\in \\Phi(G)$ is a coboundary: there is an $a \\in Z(\\Phi(G))$ with $\\delta(g) = g a g^{-1} a^{-1}$ for all $g \\in G$.","title":"Vanishing first cohomology of $G/\\Phi(G)$ in $Z(\\Phi(G))$"},{"id":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/nonabelian-pgroup-frattini-center-h1-tri","kind":"node","lean":"MathlibExt.GroupTheory.FrattiniCenterFirstCohomologyKourovka17Problem2Wanted.nonabelianPGroupFrattiniCenterH1Trivial","parent":2975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 17.2 (a non-abelian "],["math","\\(p\\)"],["text","-group with trivial "],["math","\\(H^1\\)"],["text",")"]],"source":"Kourovka Problem 17.2 (a non-abelian $p$-group with trivial $H^1$)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/index.html#nonabelian-pgroup-frattini-center-h1-tri","scope":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup","status":"fully_proved","summary":"The statement, a question of P. Schmid recorded by A. Abdollahi, that there exist a prime $p$ and a finite non-abelian $p$-group $G$ such that, writing $\\Phi(G)$ for the Frattini subgroup of $G$ and $Z(\\Phi(G))$ for its centre, the first cohomology group $H^1(G/\\Phi(G), Z(\\Phi(G)))$ for the conjugation action of $G/\\Phi(G)$ on $Z(\\Phi(G))$ is trivial.","title":"Kourovka Problem 17.2 (a non-abelian $p$-group with trivial $H^1$)"},{"id":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/nonabelian-pgroup-frattini-center-h1-tri-2","kind":"node","lean":"MathlibExt.GroupTheory.FrattiniCenterFirstCohomologyKourovka17Problem2Wanted.nonabelianPGroupFrattiniCenterH1Trivial_holds","parent":2975,"reading_title":{"format":"inline-parts-v1","parts":[["text","A non-abelian "],["math","\\(p\\)"],["text","-group with "],["math","\\(H^1(G/\\Phi(G), Z(\\Phi(G))) = 0\\)"],["text"," (Kourovka 17.2)"]],"source":"A non-abelian $p$-group with $H^1(G/\\Phi(G), Z(\\Phi(G))) = 0$ (Kourovka 17.2)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup/index.html#nonabelian-pgroup-frattini-center-h1-tri-2","scope":"GroupTheory/finite-p-groups/cohomology-of-p-groups-and-the-frattini-subgroup","status":"can_prove","summary":"There exist a prime $p$ and a finite non-abelian $p$-group $G$ such that, writing $\\Phi(G)$ for the Frattini subgroup of $G$ and $Z(\\Phi(G))$ for its centre, the first cohomology group $H^1(G/\\Phi(G), Z(\\Phi(G)))$ for the conjugation action of $G/\\Phi(G)$ on $Z(\\Phi(G))$ is trivial.","title":"A non-abelian $p$-group with $H^1(G/\\Phi(G), Z(\\Phi(G))) = 0$ (Kourovka 17.2)"},{"data_url":"scopes/GroupTheory/finite-p-groups/composition-series-of-finite-p-groups.json","id":"GroupTheory/finite-p-groups/composition-series-of-finite-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/composition-series-of-finite-p-groups/index.html","scope":"GroupTheory/finite-p-groups/composition-series-of-finite-p-groups","status":"fully_proved","title":"Composition series of finite p-groups"},{"id":"GroupTheory/finite-p-groups/composition-series-of-finite-p-groups/exists-chain-zmod-p-quotients","kind":"node","lean":"IsPGroup.exists_chain_zmod_p_quotients","parent":2980,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/composition-series-of-finite-p-groups/index.html#exists-chain-zmod-p-quotients","scope":"GroupTheory/finite-p-groups/composition-series-of-finite-p-groups","status":"fully_proved","summary":"Let $p$ be a prime and let $G$ be a finite $p$-group of order $p^n$. Then there are subgroups $G = H_0 \\supseteq H_1 \\supseteq \\cdots \\supseteq H_n = \\{1\\}$ such that for each $i < n$, $H_{i+1}$ is a normal subgroup of $H_i$ and there is a surjective homomorphism $H_i \\to \\mathbb{Z}/p\\mathbb{Z}$ with kernel $H_{i+1}$. By the first isomorphism theorem, $H_i / H_{i+1} \\cong \\mathbb{Z}/p\\mathbb{Z}$ for each $i < n$.","title":"Composition series of finite p-groups"},{"id":"GroupTheory/finite-p-groups/composition-series-of-finite-p-groups/is-solvable","kind":"node","lean":"IsPGroup.isSolvable","parent":2980,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/composition-series-of-finite-p-groups/index.html#is-solvable","scope":"GroupTheory/finite-p-groups/composition-series-of-finite-p-groups","status":"fully_proved","summary":"For every prime $p$, every finite $p$-group is solvable.","title":"Finite p-groups are solvable"},{"data_url":"scopes/GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups.json","id":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/index.html","scope":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups","status":"can_prove","title":"Conjugacy class sizes of p-groups"},{"id":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/every-power-set-is-class-size-set","kind":"node","lean":"MathlibExt.GroupTheory.PGroupConjugacyClassSizesKourovka14Problem77Wanted.everyPowerSetIsClassSizeSet","parent":2983,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/index.html#every-power-set-is-class-size-set","scope":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups","status":"fully_proved","summary":"The statement that for every prime $p$ and every finite set $X$ of natural numbers that contains $1$ and consists of powers of $p$, there is a finite $p$-group $G$ whose set of conjugacy class sizes $\\{|C| : C \\text{ a conjugacy class of } G\\}$ equals $X$.","title":"Prescribed sets of conjugacy class sizes of finite p-groups (Kourovka 14.77)"},{"id":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/every-power-set-is-class-size-set-holds","kind":"node","lean":"MathlibExt.GroupTheory.PGroupConjugacyClassSizesKourovka14Problem77Wanted.everyPowerSetIsClassSizeSet_holds","parent":2983,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/index.html#every-power-set-is-class-size-set-holds","scope":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups","status":"can_prove","summary":"For every prime $p$ and every finite set $X$ of natural numbers that contains $1$ and consists of powers of $p$, there is a finite $p$-group whose set of conjugacy class sizes is exactly $X$. This answers Kourovka Problem 14.77 (Sangroniz) in the affirmative.","title":"Cossey\u2013Hawkes theorem on conjugacy class sizes of p-groups"},{"id":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/num-max-size-conj-classes","kind":"node","lean":"MathlibExt.GroupTheory.PGroupMaxSizeConjugacyClassesKourovka14Problem92Wanted.numMaxSizeConjClasses","parent":2983,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/index.html#num-max-size-conj-classes","scope":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups","status":"fully_proved","summary":"For a group $G$, the number of conjugacy classes $c$ of $G$ such that every conjugacy class of $G$ has size at most the size of $c$. It is meant for finite groups: sizes and counts are taken in $\\mathbb{N}$, where an infinite set has size $0$.","title":"Number of conjugacy classes of maximum size"},{"id":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/odd-pgroup-with-pred-max-size-conj-class","kind":"node","lean":"MathlibExt.GroupTheory.PGroupMaxSizeConjugacyClassesKourovka14Problem92Wanted.oddPGroupWithPredMaxSizeConjClasses","parent":2983,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd "],["math","\\(p\\)"],["text","-groups with "],["math","\\(p - 1\\)"],["text"," conjugacy classes of maximum size (Kourovka 14.92)"]],"source":"Odd $p$-groups with $p - 1$ conjugacy classes of maximum size (Kourovka 14.92)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/index.html#odd-pgroup-with-pred-max-size-conj-class","scope":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups","status":"fully_proved","summary":"The statement that there exist an odd prime $p$ and a finite $p$-group with exactly $p - 1$ conjugacy classes of maximum size.","title":"Odd $p$-groups with $p - 1$ conjugacy classes of maximum size (Kourovka 14.92)"},{"id":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/odd-pgroup-with-pred-max-size-conj-class-2","kind":"node","lean":"MathlibExt.GroupTheory.PGroupMaxSizeConjugacyClassesKourovka14Problem92Wanted.oddPGroupWithPredMaxSizeConjClasses_holds","parent":2983,"reading_title":{"format":"inline-parts-v1","parts":[["text","Some odd "],["math","\\(p\\)"],["text","-group has exactly "],["math","\\(p - 1\\)"],["text"," conjugacy classes of maximum size"]],"source":"Some odd $p$-group has exactly $p - 1$ conjugacy classes of maximum size"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups/index.html#odd-pgroup-with-pred-max-size-conj-class-2","scope":"GroupTheory/finite-p-groups/conjugacy-class-sizes-of-p-groups","status":"can_prove","summary":"There exist an odd prime $p$ and a finite $p$-group with exactly $p - 1$ conjugacy classes of maximum size.","title":"Some odd $p$-group has exactly $p - 1$ conjugacy classes of maximum size"},{"data_url":"scopes/GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups.json","id":"GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups/index.html","scope":"GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups","status":"can_prove","title":"Derived and Frattini subgroups of p-groups"},{"id":"GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups/derived-subgroup-is-frattini","kind":"node","lean":"MathlibExt.GroupTheory.DerivedSubgroupAsFrattiniKourovka16Problem11Wanted.derivedSubgroupIsFrattini","parent":2989,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups/index.html#derived-subgroup-is-frattini","scope":"GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups","status":"fully_proved","summary":"The conjecture of Ya. G. Berkovich that for every prime $p$ and every finite $p$-group $G$ there is a finite $p$-group $H$ whose Frattini subgroup $\\Phi(H)$, the intersection of the maximal subgroups of $H$, is isomorphic to the derived subgroup $[G, G]$ of $G$.","title":"Derived subgroups of p-groups as Frattini subgroups (Kourovka 16.11)"},{"id":"GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups/derived-subgroup-is-frattini-refuted","kind":"node","lean":"MathlibExt.GroupTheory.DerivedSubgroupAsFrattiniKourovka16Problem11Wanted.derivedSubgroupIsFrattini_refuted","parent":2989,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups/index.html#derived-subgroup-is-frattini-refuted","scope":"GroupTheory/finite-p-groups/derived-and-frattini-subgroups-of-p-groups","status":"can_prove","summary":"There exist a prime $p$ and a finite $p$-group $G$ such that for every finite $p$-group $H$, the Frattini subgroup $\\Phi(H)$ of $H$ is not isomorphic to the derived subgroup $[G, G]$ of $G$. That is, the conjecture of Kourovka Problem 16.11 is false.","title":"Derived subgroups of p-groups need not be Frattini subgroups of p-groups"},{"data_url":"scopes/GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr.json","id":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/index.html","scope":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr","status":"can_prove","title":"Derived length of p-groups via 2-generator subgroups"},{"id":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/two-generator-cyclic-derived-length-at-m","kind":"node","lean":"MathlibExt.GroupTheory.TwoGeneratorCyclicDerivedLengthKourovka17Problem46Wanted.twoGeneratorCyclicDerivedLengthAtMostFour","parent":2992,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 17.46, with the bound "],["math","\\(4\\)"]],"source":"Kourovka Problem 17.46, with the bound $4$"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/index.html#two-generator-cyclic-derived-length-at-m","scope":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr","status":"fully_proved","summary":"The statement that for every prime $p$ and every finite $p$-group $G$ such that for all $x, y \\in G$ the derived subgroup $[H, H]$ of $H = \\langle x, y \\rangle$ is cyclic, $G$ has derived length at most $4$, that is, the fourth term $G^{(4)}$ of the derived series of $G$ is trivial.","title":"Kourovka Problem 17.46, with the bound $4$"},{"id":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/two-generator-cyclic-derived-length-at-m-2","kind":"node","lean":"MathlibExt.GroupTheory.TwoGeneratorCyclicDerivedLengthKourovka17Problem46Wanted.twoGeneratorCyclicDerivedLengthAtMostFour_holds","parent":2992,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derived length of "],["math","\\(p\\)"],["text","-groups whose "],["math","\\(2\\)"],["text","-generator subgroups have cyclic derived subgroups (Kourovka Problem 17.46)"]],"source":"Derived length of $p$-groups whose $2$-generator subgroups have cyclic derived subgroups (Kourovka Problem 17.46)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/index.html#two-generator-cyclic-derived-length-at-m-2","scope":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr","status":"can_prove","summary":"Derived length of $p$-groups whose $2$-generator subgroups have cyclic derived subgroups (Kourovka Problem 17.46). Let $p$ be a prime and $G$ a finite $p$-group such that for all $x, y \\in G$ the derived subgroup $[H, H]$ of $H = \\langle x, y \\rangle$ is cyclic. Then $G$ has derived length at most $4$, that is, the fourth term $G^{(4)}$ of the derived series of $G$ is trivial.","title":"Derived length of $p$-groups whose $2$-generator subgroups have cyclic derived subgroups (Kourovka Problem 17.46)"},{"id":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/is-metacyclic","kind":"node","lean":"MathlibExt.GroupTheory.TwoGeneratorMetacyclicDerivedLengthKourovka15Problem33Wanted.IsMetacyclic","parent":2992,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/index.html#is-metacyclic","scope":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr","status":"fully_proved","summary":"A group $G$ is metacyclic if it has a cyclic normal subgroup $N$ such that the quotient $G/N$ is cyclic.","title":"Metacyclic group"},{"id":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/two-generator-metacyclic-metabelian","kind":"node","lean":"MathlibExt.GroupTheory.TwoGeneratorMetacyclicDerivedLengthKourovka15Problem33Wanted.twoGeneratorMetacyclicMetabelian","parent":2992,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 15.33, with the bound "],["math","\\(2\\)"]],"source":"Kourovka Problem 15.33, with the bound $2$"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/index.html#two-generator-metacyclic-metabelian","scope":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr","status":"fully_proved","summary":"The statement that every finite $2$-group $G$ such that for all $x, y \\in G$ the subgroup $\\langle x, y \\rangle$ generated by $x$ and $y$ is metacyclic has derived length at most $2$, that is, its second derived subgroup $G''$ is trivial.","title":"Kourovka Problem 15.33, with the bound $2$"},{"id":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/two-generator-metacyclic-metabelian-hold","kind":"node","lean":"MathlibExt.GroupTheory.TwoGeneratorMetacyclicDerivedLengthKourovka15Problem33Wanted.twoGeneratorMetacyclicMetabelian_holds","parent":2992,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite "],["math","\\(2\\)"],["text","-groups with metacyclic "],["math","\\(2\\)"],["text","-generator subgroups are metabelian (Kourovka Problem 15.33)"]],"source":"Finite $2$-groups with metacyclic $2$-generator subgroups are metabelian (Kourovka Problem 15.33)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr/index.html#two-generator-metacyclic-metabelian-hold","scope":"GroupTheory/finite-p-groups/derived-length-of-p-groups-via-2-generator-subgr","status":"can_prove","summary":"Finite $2$-groups with metacyclic $2$-generator subgroups are metabelian (Kourovka Problem 15.33). Let $G$ be a finite $2$-group such that for all $x, y \\in G$ the subgroup $\\langle x, y \\rangle$ generated by $x$ and $y$ is metacyclic, that is, has a cyclic normal subgroup with cyclic quotient. Then $G$ is metabelian: its second derived subgroup $G''$ is trivial.","title":"Finite $2$-groups with metacyclic $2$-generator subgroups are metabelian (Kourovka Problem 15.33)"},{"data_url":"scopes/GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order.json","id":"GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order/index.html","scope":"GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order","status":"can_prove","title":"Generating p-groups by elements of equal order"},{"id":"GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order/p-group-equal-order-generators","kind":"node","lean":"MathlibExt.GroupTheory.PGroupEqualOrderGeneratorsKourovka11Problem6Wanted.pGroupEqualOrderGenerators","parent":2998,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order/index.html#p-group-equal-order-generators","scope":"GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order","status":"fully_proved","summary":"The statement that for every odd prime $p$, every finite $p$-group $G$ has a generating set $S$ all of whose elements have the same order.","title":"Generating sets of equal order in finite p-groups (Kourovka 11.6)"},{"id":"GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order/p-group-equal-order-generators-refuted","kind":"node","lean":"MathlibExt.GroupTheory.PGroupEqualOrderGeneratorsKourovka11Problem6Wanted.pGroupEqualOrderGenerators_refuted","parent":2998,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order/index.html#p-group-equal-order-generators-refuted","scope":"GroupTheory/finite-p-groups/generating-p-groups-by-elements-of-equal-order","status":"can_prove","summary":"For some odd prime $p$ there is a finite $p$-group $G$ such that no generating set of $G$ consists of elements that all have the same order. This answers Kourovka Problem 11.6 (Bagi\u0144ski) in the negative (O'Brien, Scoppola and Vaughan-Lee).","title":"Finite p-groups without a generating set of elements of equal order"},{"data_url":"scopes/GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups.json","id":"GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups/index.html","scope":"GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups","status":"can_prove","title":"Large automorphism groups of p-groups"},{"id":"GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups/cyclic-subgroup-of-index-from-large-auto","kind":"node","lean":"MathlibExt.GroupTheory.LargeAutomorphismCyclicSubgroupKourovka15Problem32Wanted.cyclicSubgroupOfIndexFromLargeAutomorphism","parent":3001,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups/index.html#cyclic-subgroup-of-index-from-large-auto","scope":"GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups","status":"fully_proved","summary":"For every prime $p$ and all natural numbers $k$ and $m$ with $m \\ge 64k + 2$, every finite group $G$ of order $p^m$ that has an automorphism of order $p^{m-k}$ has a cyclic subgroup of index $p^k$.","title":"Kourovka Problem 15.32: large automorphisms and cyclic subgroups"},{"id":"GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups/cyclic-subgroup-of-index-from-large-auto-2","kind":"node","lean":"MathlibExt.GroupTheory.LargeAutomorphismCyclicSubgroupKourovka15Problem32Wanted.cyclicSubgroupOfIndexFromLargeAutomorphism_holds","parent":3001,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups/index.html#cyclic-subgroup-of-index-from-large-auto-2","scope":"GroupTheory/finite-p-groups/large-automorphism-groups-of-p-groups","status":"can_prove","summary":"For every prime $p$ and all natural numbers $k$ and $m$ with $m \\ge 64k + 2$, if a finite group $G$ of order $p^m$ has an automorphism of order $p^{m-k}$, then $G$ has a cyclic subgroup of index $p^k$.","title":"Large automorphisms of p-groups force cyclic subgroups of given index"},{"data_url":"scopes/GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups.json","id":"GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups/index.html","scope":"GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups","status":"can_prove","title":"Normal subgroups of exponent p in p-groups"},{"id":"GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups/normal-exponent-psubgroup-of-large-prime","kind":"node","lean":"MathlibExt.GroupTheory.NormalExponentPSubgroupKourovka15Problem24Wanted.normalExponentPSubgroupOfLargePrime","parent":3004,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups/index.html#normal-exponent-psubgroup-of-large-prime","scope":"GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups","status":"fully_proved","summary":"For every prime $p$ and every natural number $n < p$, if a finite $p$-group $G$ has a subgroup of order $p^n$ all of whose elements $x$ satisfy $x^p = 1$, then $G$ has a normal subgroup of order $p^n$ all of whose elements $x$ satisfy $x^p = 1$.","title":"Berkovich's question on normal subgroups of exponent p, for p > n"},{"id":"GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups/normal-exponent-psubgroup-of-large-prime-2","kind":"node","lean":"MathlibExt.GroupTheory.NormalExponentPSubgroupKourovka15Problem24Wanted.normalExponentPSubgroupOfLargePrime_holds","parent":3004,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups/index.html#normal-exponent-psubgroup-of-large-prime-2","scope":"GroupTheory/finite-p-groups/normal-subgroups-of-exponent-p-in-p-groups","status":"can_prove","summary":"For every prime $p$ and every natural number $n < p$, if a finite $p$-group $G$ has a subgroup $H$ of order $p^n$ with $x^p = 1$ for all $x \\in H$, then $G$ has a normal subgroup $N$ of order $p^n$ with $x^p = 1$ for all $x \\in N$.","title":"Normal subgroups of exponent p in p-groups when p > n"},{"data_url":"scopes/GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank.json","id":"GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank/index.html","scope":"GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank","status":"can_prove","title":"Orders of 2-groups of bounded abelian rank"},{"id":"GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank/two-group-order-bound-by-abelian-rank","kind":"node","lean":"MathlibExt.GroupTheory.TwoGroupAbelianRankOrderBoundKourovka19Problem75Wanted.twoGroupOrderBoundByAbelianRank","parent":3007,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sambale's question on the order of "],["math","\\(2\\)"],["text","-groups of bounded abelian rank (Kourovka 19.75)"]],"source":"Sambale's question on the order of $2$-groups of bounded abelian rank (Kourovka 19.75)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank/index.html#two-group-order-bound-by-abelian-rank","scope":"GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank","status":"fully_proved","summary":"The statement that for all natural numbers $e$ and $r$, every finite $2$-group $P$ of exponent $2^e$ in which every abelian subgroup is generated by at most $r$ elements has order $|P| \\le 2^{r(e+1)}$.","title":"Sambale's question on the order of $2$-groups of bounded abelian rank (Kourovka 19.75)"},{"id":"GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank/two-group-order-bound-by-abelian-rank-re","kind":"node","lean":"MathlibExt.GroupTheory.TwoGroupAbelianRankOrderBoundKourovka19Problem75Wanted.twoGroupOrderBoundByAbelianRank_refuted","parent":3007,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sambale's order bound for "],["math","\\(2\\)"],["text","-groups of bounded abelian rank fails (Kourovka 19.75)"]],"source":"Sambale's order bound for $2$-groups of bounded abelian rank fails (Kourovka 19.75)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank/index.html#two-group-order-bound-by-abelian-rank-re","scope":"GroupTheory/finite-p-groups/orders-of-2-groups-of-bounded-abelian-rank","status":"can_prove","summary":"There exist natural numbers $e$ and $r$ and a finite $2$-group $P$ of exponent $2^e$ in which every abelian subgroup is generated by at most $r$ elements, such that $|P| > 2^{r(e+1)}$.","title":"Sambale's order bound for $2$-groups of bounded abelian rank fails (Kourovka 19.75)"},{"data_url":"scopes/GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups.json","id":"GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups/index.html","scope":"GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups","status":"can_prove","title":"p-groups with special maximal subgroups"},{"id":"GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups/is-special-group","kind":"node","lean":"MathlibExt.GroupTheory.MaximalSubgroupsSpecialKourovka16Problem13Wanted.IsSpecialGroup","parent":3010,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups/index.html#is-special-group","scope":"GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups","status":"fully_proved","summary":"A group $H$ is special when its center, its commutator subgroup and its Frattini subgroup coincide: $Z(H) = [H, H] = \\Phi(H)$.","title":"Special group"},{"id":"GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups/large-pgroups-with-special-maximal-subgr","kind":"node","lean":"MathlibExt.GroupTheory.MaximalSubgroupsSpecialKourovka16Problem13Wanted.largePGroupsWithSpecialMaximalSubgroups","parent":3010,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups/index.html#large-pgroups-with-special-maximal-subgr","scope":"GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups","status":"fully_proved","summary":"The statement that for every prime $p$ and every natural number $n$ there is a finite $p$-group $G$ of order greater than $n$ all of whose maximal subgroups $H$ are special, that is, have center $Z(H)$, commutator subgroup $[H, H]$ and Frattini subgroup $\\Phi(H)$ all equal. This is the form in which the problem was resolved; the problem as posed asks only for one finite $p$-group all of whose maximal subgroups are\u2026","title":"Kourovka Problem 16.13, strong form (statement)"},{"id":"GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups/large-pgroups-with-special-maximal-subgr-2","kind":"node","lean":"MathlibExt.GroupTheory.MaximalSubgroupsSpecialKourovka16Problem13Wanted.largePGroupsWithSpecialMaximalSubgroups_holds","parent":3010,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arbitrarily large "],["math","\\(p\\)"],["text","-groups all of whose maximal subgroups are special"]],"source":"Arbitrarily large $p$-groups all of whose maximal subgroups are special"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups/index.html#large-pgroups-with-special-maximal-subgr-2","scope":"GroupTheory/finite-p-groups/p-groups-with-special-maximal-subgroups","status":"can_prove","summary":"For every prime $p$ and every natural number $n$ there is a finite $p$-group $G$ of order greater than $n$ such that every maximal subgroup $H$ of $G$ is special, that is, its center $Z(H)$, its commutator subgroup $[H, H]$ and its Frattini subgroup $\\Phi(H)$ coincide. This answers Kourovka Problem 16.13 positively.","title":"Arbitrarily large $p$-groups all of whose maximal subgroups are special"},{"data_url":"scopes/GroupTheory/finite-p-groups/partitions-of-p-groups.json","id":"GroupTheory/finite-p-groups/partitions-of-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/partitions-of-p-groups/index.html","scope":"GroupTheory/finite-p-groups/partitions-of-p-groups","status":"can_prove","title":"Partitions of p-groups"},{"id":"GroupTheory/finite-p-groups/partitions-of-p-groups/is-group-partition","kind":"node","lean":"MathlibExt.GroupTheory.PGroupPartitionAbelianComponentKourovka15Problem26Wanted.IsGroupPartition","parent":3014,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/partitions-of-p-groups/index.html#is-group-partition","scope":"GroupTheory/finite-p-groups/partitions-of-p-groups","status":"fully_proved","summary":"Let $G$ be a group. A set $C$ of subgroups of $G$ is a partition of $G$ when every member of $C$ is a proper subgroup of $G$, every element of $G$ lies in some member of $C$, and any two distinct members of $C$ intersect in the trivial subgroup. The members of $C$ are called its components.","title":"Partition of a group"},{"id":"GroupTheory/finite-p-groups/partitions-of-p-groups/p-group-partition-has-abelian-component","kind":"node","lean":"MathlibExt.GroupTheory.PGroupPartitionAbelianComponentKourovka15Problem26Wanted.pGroupPartitionHasAbelianComponent","parent":3014,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/partitions-of-p-groups/index.html#p-group-partition-has-abelian-component","scope":"GroupTheory/finite-p-groups/partitions-of-p-groups","status":"fully_proved","summary":"The statement that for every prime $p$, every finite $p$-group $G$ and every partition $C$ of $G$ (a set of proper subgroups of $G$ whose union is $G$ and any two distinct members of which intersect in the trivial subgroup), some member $H$ of $C$ is abelian, that is, $xy = yx$ for all $x, y \\in H$.","title":"Berkovich's question on abelian components of partitions"},{"id":"GroupTheory/finite-p-groups/partitions-of-p-groups/p-group-partition-has-abelian-component-2","kind":"node","lean":"MathlibExt.GroupTheory.PGroupPartitionAbelianComponentKourovka15Problem26Wanted.pGroupPartitionHasAbelianComponent_holds","parent":3014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every partition of a finite "],["math","\\(p\\)"],["text","-group has an abelian component"]],"source":"Every partition of a finite $p$-group has an abelian component"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/partitions-of-p-groups/index.html#p-group-partition-has-abelian-component-2","scope":"GroupTheory/finite-p-groups/partitions-of-p-groups","status":"can_prove","summary":"For every prime $p$, every finite $p$-group $G$ and every partition $C$ of $G$ (a set of proper subgroups of $G$ whose union is $G$ and any two distinct members of which intersect in the trivial subgroup), some member $H$ of $C$ is abelian, that is, $xy = yx$ for all $x, y \\in H$.","title":"Every partition of a finite $p$-group has an abelian component"},{"data_url":"scopes/GroupTheory/finite-p-groups/powerful-p-groups.json","id":"GroupTheory/finite-p-groups/powerful-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/powerful-p-groups/index.html","scope":"GroupTheory/finite-p-groups/powerful-p-groups","status":"can_prove","title":"Powerful p-groups"},{"id":"GroupTheory/finite-p-groups/powerful-p-groups/pth-powers-subgroup-powerful","kind":"node","lean":"MathlibExt.GroupTheory.PthPowersSubgroupPowerfulKourovka21Problem137Wanted.pthPowersSubgroupPowerful","parent":3018,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/powerful-p-groups/index.html#pth-powers-subgroup-powerful","scope":"GroupTheory/finite-p-groups/powerful-p-groups","status":"fully_proved","summary":"The proposition that there is a prime $p$ such that for every finite $p$-group $G$ and every subgroup $P$ of $G$ whose underlying set is exactly the set $\\{x^p : x \\in G\\}$ of $p$-th powers in $G$, the subgroup $P$ is powerful: its commutator subgroup $[P, P]$ is contained in the subgroup $P^m$ generated by the $m$-th powers of the elements of $P$, where $m = p$ for odd $p$ and $m = 4$ for $p = 2$.","title":"Kourovka 21.137, main question"},{"id":"GroupTheory/finite-p-groups/powerful-p-groups/pth-powers-subgroup-abelian-exponent-psq","kind":"node","lean":"MathlibExt.GroupTheory.PthPowersSubgroupPowerfulKourovka21Problem137Wanted.pthPowersSubgroupAbelianExponentPSq","parent":3018,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/powerful-p-groups/index.html#pth-powers-subgroup-abelian-exponent-psq","scope":"GroupTheory/finite-p-groups/powerful-p-groups","status":"fully_proved","summary":"The proposition that there is an odd prime $p$ such that for every finite $p$-group $G$ of exponent $p^2$ and every subgroup $P$ of $G$ whose underlying set is exactly the set $\\{x^p : x \\in G\\}$ of $p$-th powers in $G$, the subgroup $P$ is abelian.","title":"Kourovka 21.137, odd-prime form"},{"id":"GroupTheory/finite-p-groups/powerful-p-groups/pth-powers-subgroup-powerful-refuted","kind":"node","lean":"MathlibExt.GroupTheory.PthPowersSubgroupPowerfulKourovka21Problem137Wanted.pthPowersSubgroupPowerful_refuted","parent":3018,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/powerful-p-groups/index.html#pth-powers-subgroup-powerful-refuted","scope":"GroupTheory/finite-p-groups/powerful-p-groups","status":"can_prove","summary":"For every prime $p$ there is a finite $p$-group $G$ with a subgroup $P$ whose underlying set is exactly the set $\\{x^p : x \\in G\\}$ of $p$-th powers in $G$, such that $P$ is not powerful: the commutator subgroup $[P, P]$ is not contained in the subgroup $P^m$ generated by the $m$-th powers of the elements of $P$, where $m = p$ for odd $p$ and $m = 4$ for $p = 2$.","title":"Kourovka 21.137 fails for every prime"},{"id":"GroupTheory/finite-p-groups/powerful-p-groups/pth-powers-subgroup-abelian-exponent-psq-2","kind":"node","lean":"MathlibExt.GroupTheory.PthPowersSubgroupPowerfulKourovka21Problem137Wanted.pthPowersSubgroupAbelianExponentPSq_refuted","parent":3018,"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/powerful-p-groups/index.html#pth-powers-subgroup-abelian-exponent-psq-2","scope":"GroupTheory/finite-p-groups/powerful-p-groups","status":"can_prove","summary":"For every odd prime $p$ there is a finite $p$-group $G$ of exponent $p^2$ with a subgroup $P$ whose underlying set is exactly the set $\\{x^p : x \\in G\\}$ of $p$-th powers in $G$, such that $P$ is not abelian.","title":"Kourovka 21.137 fails for every odd prime"},{"data_url":"scopes/GroupTheory/finite-p-groups/extensions-of-p-groups.json","id":"GroupTheory/finite-p-groups/extensions-of-p-groups","kind":"scope","navigation":"module","parent":2956,"result":false,"route":"../roadmap/GroupTheory/finite-p-groups/extensions-of-p-groups/index.html","scope":"GroupTheory/finite-p-groups/extensions-of-p-groups","status":"can_prove","title":"\u03a6-extensions of p-groups"},{"id":"GroupTheory/finite-p-groups/extensions-of-p-groups/is-phi-extension","kind":"node","lean":"MathlibExt.GroupTheory.FrattiniExtensionSequenceKourovka16Problem12Wanted.IsPhiExtension","parent":3023,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\Phi\\)"],["text","-extension (Berkovich)"]],"source":"$\\Phi$-extension (Berkovich)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/extensions-of-p-groups/index.html#is-phi-extension","scope":"GroupTheory/finite-p-groups/extensions-of-p-groups","status":"fully_proved","summary":"For a natural number $p$ and groups $H$ and $G$, $H$ is a $\\Phi$-extension of $G$ when $H$ is a finite $p$-group with a normal subgroup $N$ of order $p$ such that $N \\le \\Phi(H)$, where $\\Phi(H)$ is the Frattini subgroup of $H$, and $H/N \\cong G$.","title":"$\\Phi$-extension (Berkovich)"},{"id":"GroupTheory/finite-p-groups/extensions-of-p-groups/phi-extension-sequence","kind":"node","lean":"MathlibExt.GroupTheory.FrattiniExtensionSequenceKourovka16Problem12Wanted.phiExtensionSequence","parent":3023,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 16.12 (infinite chains of "],["math","\\(\\Phi\\)"],["text","-extensions)"]],"source":"Kourovka Problem 16.12 (infinite chains of $\\Phi$-extensions)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/extensions-of-p-groups/index.html#phi-extension-sequence","scope":"GroupTheory/finite-p-groups/extensions-of-p-groups","status":"fully_proved","summary":"The statement, a question of Ya. G. Berkovich, that for every prime $p$ and every nontrivial finite $p$-group $G$ there are groups $G_0, G_1, G_2, \\dots$ with $G_0 \\cong G$ such that, for every $i \\ge 0$, $G_{i+1}$ is a $\\Phi$-extension of $G_i$ for the prime $p$.","title":"Kourovka Problem 16.12 (infinite chains of $\\Phi$-extensions)"},{"id":"GroupTheory/finite-p-groups/extensions-of-p-groups/phi-extension-sequence-holds","kind":"node","lean":"MathlibExt.GroupTheory.FrattiniExtensionSequenceKourovka16Problem12Wanted.phiExtensionSequence_holds","parent":3023,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinite chains of "],["math","\\(\\Phi\\)"],["text","-extensions (Kourovka 16.12)"]],"source":"Infinite chains of $\\Phi$-extensions (Kourovka 16.12)"},"result":true,"route":"../roadmap/GroupTheory/finite-p-groups/extensions-of-p-groups/index.html#phi-extension-sequence-holds","scope":"GroupTheory/finite-p-groups/extensions-of-p-groups","status":"can_prove","summary":"For every prime $p$ and every nontrivial finite $p$-group $G$ there are groups $G_0, G_1, G_2, \\dots$ with $G_0 \\cong G$ such that, for every $i \\ge 0$, $G_{i+1}$ is a $\\Phi$-extension of $G_i$ for the prime $p$. This answers Kourovka Problem 16.12 in the affirmative: the Kourovka Archive answers \"Yes, it is\", citing Kamornikov (2008).","title":"Infinite chains of $\\Phi$-extensions (Kourovka 16.12)"},{"data_url":"scopes/GroupTheory/orderable-groups.json","id":"GroupTheory/orderable-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/index.html","scope":"GroupTheory/orderable-groups","status":"can_prove","summary":"Left-, right- and bi-orderable groups: spaces of orders, generalized torsion, unique product groups, relatively convex subgroups, orderability of homeomorphism groups and amalgams, and decision problems for orderable groups.","title":"Orderable groups"},{"data_url":"scopes/GroupTheory/orderable-groups/generalized-torsion.json","id":"GroupTheory/orderable-groups/generalized-torsion","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/generalized-torsion/index.html","scope":"GroupTheory/orderable-groups/generalized-torsion","status":"can_prove","title":"Generalized torsion"},{"id":"GroupTheory/orderable-groups/generalized-torsion/is-generalized-torsion","kind":"node","lean":"MathlibExt.GroupTheory.GeneralizedTorsionFreeProductKourovka16Problem49Wanted.IsGeneralizedTorsion","parent":3028,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/generalized-torsion/index.html#is-generalized-torsion","scope":"GroupTheory/orderable-groups/generalized-torsion","status":"fully_proved","summary":"An element $h$ of a group $H$ is a generalized torsion element when $h \\ne 1$ and some nonempty product of conjugates of $h$ is trivial, that is, $x_1^{-1} h x_1 \\cdots x_k^{-1} h x_k = 1$ for some $k \\ge 1$ and some $x_1, \\dots, x_k \\in H$.","title":"Generalized torsion element (Kourovka 16.49)"},{"id":"GroupTheory/orderable-groups/generalized-torsion/free-product-no-generalized-torsion","kind":"node","lean":"MathlibExt.GroupTheory.GeneralizedTorsionFreeProductKourovka16Problem49Wanted.freeProductNoGeneralizedTorsion","parent":3028,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/generalized-torsion/index.html#free-product-no-generalized-torsion","scope":"GroupTheory/orderable-groups/generalized-torsion","status":"fully_proved","summary":"The statement that for every family of groups $(G_i)_{i \\in I}$ none of which contains a generalized torsion element (an element $h \\ne 1$ such that some nonempty product of conjugates of $h$ is trivial), the free product of the family contains no generalized torsion element either.","title":"Kopytov\u2013Medvedev question on generalized torsion in free products (Kourovka 16.49)"},{"id":"GroupTheory/orderable-groups/generalized-torsion/free-product-no-generalized-torsion-hold","kind":"node","lean":"MathlibExt.GroupTheory.GeneralizedTorsionFreeProductKourovka16Problem49Wanted.freeProductNoGeneralizedTorsion_holds","parent":3028,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/generalized-torsion/index.html#free-product-no-generalized-torsion-hold","scope":"GroupTheory/orderable-groups/generalized-torsion","status":"can_prove","summary":"For every family of groups $(G_i)_{i \\in I}$ none of which contains a generalized torsion element (an element $h \\ne 1$ such that some nonempty product of conjugates of $h$ is trivial), the free product of the family contains no generalized torsion element.","title":"Free products of groups without generalized torsion have no generalized torsion"},{"id":"GroupTheory/orderable-groups/generalized-torsion/is-generalized-torsion-2","kind":"node","lean":"MathlibExt.GroupTheory.GeneralizedTorsionRightOrderableKourovka16Problem48Wanted.IsGeneralizedTorsion","parent":3028,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/generalized-torsion/index.html#is-generalized-torsion-2","scope":"GroupTheory/orderable-groups/generalized-torsion","status":"fully_proved","summary":"An element $h$ of a group $H$ is a generalized torsion element if $h \\ne 1$ and there are $n \\ge 1$ elements $x_1, \\dots, x_n \\in H$, not necessarily distinct, such that $(x_1^{-1} h x_1)(x_2^{-1} h x_2) \\cdots (x_n^{-1} h x_n) = 1$.","title":"Generalized torsion elements (Kourovka 16.48)"},{"id":"GroupTheory/orderable-groups/generalized-torsion/right-orders","kind":"node","lean":"MathlibExt.GroupTheory.GeneralizedTorsionRightOrderableKourovka16Problem48Wanted.rightOrders","parent":3028,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/generalized-torsion/index.html#right-orders","scope":"GroupTheory/orderable-groups/generalized-torsion","status":"fully_proved","summary":"The right orders of a group $G$ are the binary relations $\\preceq$ on $G$ that are linear orders (reflexive, transitive, antisymmetric and total) and are invariant under right multiplication: $a \\preceq b$ implies $ac \\preceq bc$ for all $a, b, c \\in G$.","title":"Right orders on a group (Kourovka 16.48)"},{"id":"GroupTheory/orderable-groups/generalized-torsion/no-generalized-torsion-right-orderable","kind":"node","lean":"MathlibExt.GroupTheory.GeneralizedTorsionRightOrderableKourovka16Problem48Wanted.noGeneralizedTorsionRightOrderable","parent":3028,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/generalized-torsion/index.html#no-generalized-torsion-right-orderable","scope":"GroupTheory/orderable-groups/generalized-torsion","status":"fully_proved","summary":"This is the proposition that every group $H$ with no generalized torsion element is right-orderable, that is, admits a right order.","title":"The Kopytov\u2013Medvedev question (Kourovka 16.48)"},{"id":"GroupTheory/orderable-groups/generalized-torsion/no-generalized-torsion-right-orderable-r","kind":"node","lean":"MathlibExt.GroupTheory.GeneralizedTorsionRightOrderableKourovka16Problem48Wanted.noGeneralizedTorsionRightOrderable_refuted","parent":3028,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/generalized-torsion/index.html#no-generalized-torsion-right-orderable-r","scope":"GroupTheory/orderable-groups/generalized-torsion","status":"can_prove","summary":"The Kopytov\u2013Medvedev proposition is false: there is a group $H$ with no generalized torsion element that admits no right order.","title":"Some group without generalized torsion is not right-orderable (Kourovka 16.48)"},{"data_url":"scopes/GroupTheory/orderable-groups/isolated-orders-on-free-groups.json","id":"GroupTheory/orderable-groups/isolated-orders-on-free-groups","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/isolated-orders-on-free-groups/index.html","scope":"GroupTheory/orderable-groups/isolated-orders-on-free-groups","status":"can_prove","title":"Isolated orders on free groups"},{"id":"GroupTheory/orderable-groups/isolated-orders-on-free-groups/is-bi-order","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupIsolatedBiOrderKourovka12Problem51Wanted.IsBiOrder","parent":3036,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/isolated-orders-on-free-groups/index.html#is-bi-order","scope":"GroupTheory/orderable-groups/isolated-orders-on-free-groups","status":"fully_proved","summary":"A relation $r$ on a group $G$ is a bi-invariant order if $r$ is a strict total order on $G$ (irreflexive, transitive and trichotomous) and $r(a, b)$ implies $r(cad, cbd)$ for all $a, b, c, d \\in G$.","title":"Bi-invariant strict total orders on a group"},{"id":"GroupTheory/orderable-groups/isolated-orders-on-free-groups/free-group-has-isolated-bi-order","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupIsolatedBiOrderKourovka12Problem51Wanted.freeGroupHasIsolatedBiOrder","parent":3036,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/isolated-orders-on-free-groups/index.html#free-group-has-isolated-bi-order","scope":"GroupTheory/orderable-groups/isolated-orders-on-free-groups","status":"fully_proved","summary":"The statement that for some natural number $n \\ge 2$ there is a finite subset $S$ of the free group $F_n$ on $n$ generators such that exactly one bi-invariant strict total order $<$ on $F_n$ (a strict total order such that $a < b$ implies $cad < cbd$ for all $a, b, c, d \\in F_n$) makes every element of $S$ positive, that is, satisfies $1 < s$ for all $s \\in S$. Such an order is an isolated point of the space of\u2026","title":"Kourovka Problem 12.51 (isolated bi-orders of free groups)"},{"id":"GroupTheory/orderable-groups/isolated-orders-on-free-groups/free-group-has-isolated-bi-order-refuted","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupIsolatedBiOrderKourovka12Problem51Wanted.freeGroupHasIsolatedBiOrder_refuted","parent":3036,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/isolated-orders-on-free-groups/index.html#free-group-has-isolated-bi-order-refuted","scope":"GroupTheory/orderable-groups/isolated-orders-on-free-groups","status":"can_prove","summary":"For every natural number $n \\ge 2$ and every finite subset $S$ of the free group $F_n$ on $n$ generators, it is not the case that exactly one bi-invariant strict total order $<$ on $F_n$ (a strict total order such that $a < b$ implies $cad < cbd$ for all $a, b, c, d \\in F_n$) satisfies $1 < s$ for every $s \\in S$.","title":"No free group of finite rank at least 2 has an isolated bi-order"},{"data_url":"scopes/GroupTheory/orderable-groups/left-orderable-skew-braces.json","id":"GroupTheory/orderable-groups/left-orderable-skew-braces","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/left-orderable-skew-braces/index.html","scope":"GroupTheory/orderable-groups/left-orderable-skew-braces","status":"can_prove","title":"Left-orderable skew braces"},{"id":"GroupTheory/orderable-groups/left-orderable-skew-braces/left-orderable-mul","kind":"node","lean":"MathlibExt.GroupTheory.SkewBraceLeftOrderableKourovka19Problem49Wanted.LeftOrderableMul","parent":3040,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/left-orderable-skew-braces/index.html#left-orderable-mul","scope":"GroupTheory/orderable-groups/left-orderable-skew-braces","status":"fully_proved","summary":"The multiplication of a set $B$ is left-orderable when there is a strict total order $<$ on $B$ that is invariant under left multiplication: for all $a, b, c \\in B$, if $b < c$ then $a b < a c$.","title":"Left-orderable multiplication"},{"id":"GroupTheory/orderable-groups/left-orderable-skew-braces/left-orderable-add","kind":"node","lean":"MathlibExt.GroupTheory.SkewBraceLeftOrderableKourovka19Problem49Wanted.LeftOrderableAdd","parent":3040,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/left-orderable-skew-braces/index.html#left-orderable-add","scope":"GroupTheory/orderable-groups/left-orderable-skew-braces","status":"fully_proved","summary":"The addition of a set $B$ is left-orderable when there is a strict total order $<$ on $B$ that is invariant under left addition: for all $a, b, c \\in B$, if $b < c$ then $a + b < a + c$.","title":"Left-orderable addition"},{"id":"GroupTheory/orderable-groups/left-orderable-skew-braces/skew-brace-additive-left-orderable","kind":"node","lean":"MathlibExt.GroupTheory.SkewBraceLeftOrderableKourovka19Problem49Wanted.skewBraceAdditiveLeftOrderable","parent":3040,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/left-orderable-skew-braces/index.html#skew-brace-additive-left-orderable","scope":"GroupTheory/orderable-groups/left-orderable-skew-braces","status":"fully_proved","summary":"The statement that for every skew brace $B$, that is, a set with two group structures, an addition $+$ (not necessarily commutative) and a multiplication $\\cdot$, satisfying $a \\cdot (b + c) = a \\cdot b - a + a \\cdot c$ for all $a, b, c \\in B$: if the multiplication of $B$ is left-orderable, then the addition of $B$ is left-orderable.","title":"Left-orderability of the additive group of a skew brace (Kourovka 19.49)"},{"id":"GroupTheory/orderable-groups/left-orderable-skew-braces/skew-brace-additive-left-orderable-refut","kind":"node","lean":"MathlibExt.GroupTheory.SkewBraceLeftOrderableKourovka19Problem49Wanted.skewBraceAdditiveLeftOrderable_refuted","parent":3040,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/left-orderable-skew-braces/index.html#skew-brace-additive-left-orderable-refut","scope":"GroupTheory/orderable-groups/left-orderable-skew-braces","status":"can_prove","summary":"There is a skew brace $B$, that is, a set with two group structures, an addition $+$ and a multiplication $\\cdot$, satisfying $a \\cdot (b + c) = a \\cdot b - a + a \\cdot c$ for all $a, b, c \\in B$, whose multiplication is left-orderable but whose addition is not left-orderable. That is, the statement of Kourovka Problem 19.49 fails.","title":"A skew brace whose additive group is not left-orderable"},{"data_url":"scopes/GroupTheory/orderable-groups/orderability-of-amalgamated-free-products.json","id":"GroupTheory/orderable-groups/orderability-of-amalgamated-free-products","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-amalgamated-free-products/index.html","scope":"GroupTheory/orderable-groups/orderability-of-amalgamated-free-products","status":"can_prove","title":"Orderability of amalgamated free products"},{"id":"GroupTheory/orderable-groups/orderability-of-amalgamated-free-products/is-right-orderable","kind":"node","lean":"MathlibExt.GroupTheory.AmalgamatedOrderedGroupsRightOrderableKourovka15Problem34Wanted.IsRightOrderable","parent":3045,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-amalgamated-free-products/index.html#is-right-orderable","scope":"GroupTheory/orderable-groups/orderability-of-amalgamated-free-products","status":"fully_proved","summary":"A group $G$ is right-orderable if it admits a linear order $\\le$ that is invariant under right multiplication: $a \\le b$ implies $ac \\le bc$ for all $a, b, c \\in G$.","title":"Right-orderable group"},{"id":"GroupTheory/orderable-groups/orderability-of-amalgamated-free-products/amalgam-of-ordered-groups-right-orderabl","kind":"node","lean":"MathlibExt.GroupTheory.AmalgamatedOrderedGroupsRightOrderableKourovka15Problem34Wanted.amalgamOfOrderedGroupsRightOrderable","parent":3045,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-amalgamated-free-products/index.html#amalgam-of-ordered-groups-right-orderabl","scope":"GroupTheory/orderable-groups/orderability-of-amalgamated-free-products","status":"fully_proved","summary":"The statement that for every family $(G_i)_{i \\in I}$ of groups, each with a linear order invariant under multiplication on both sides, every group $H$ with such a linear order, and all strictly increasing group homomorphisms $\\varphi_i \\colon H \\to G_i$, the free product of the $G_i$ with $H$ amalgamated along the maps $\\varphi_i$ is right-orderable.","title":"Kourovka Problem 15.34 (statement)"},{"id":"GroupTheory/orderable-groups/orderability-of-amalgamated-free-products/amalgam-of-ordered-groups-right-orderabl-2","kind":"node","lean":"MathlibExt.GroupTheory.AmalgamatedOrderedGroupsRightOrderableKourovka15Problem34Wanted.amalgamOfOrderedGroupsRightOrderable_holds","parent":3045,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-amalgamated-free-products/index.html#amalgam-of-ordered-groups-right-orderabl-2","scope":"GroupTheory/orderable-groups/orderability-of-amalgamated-free-products","status":"can_prove","summary":"Let $(G_i)_{i \\in I}$ be a family of groups, each with a linear order invariant under multiplication on both sides ($a \\le b$ implies $ca \\le cb$ and $ac \\le bc$), let $H$ be a group with such a linear order, and for each $i$ let $\\varphi_i \\colon H \\to G_i$ be a strictly increasing group homomorphism. Then the free product of the $G_i$ with $H$ amalgamated along the maps $\\varphi_i$ is right-orderable: it admits a\u2026","title":"Bludov\u2013Glass theorem (Kourovka Problem 15.34)"},{"data_url":"scopes/GroupTheory/orderable-groups/orderability-of-homeomorphism-groups.json","id":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/index.html","scope":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups","status":"can_prove","title":"Orderability of homeomorphism groups"},{"id":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/square","kind":"node","lean":"MathlibExt.GroupTheory.BoundaryFixingHomeomorphismsOrderableKourovka17Problem20Wanted.Square","parent":3049,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/index.html#square","scope":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups","status":"fully_proved","summary":"The closed square $I^2 = [0, 1] \\times [0, 1]$, the product of two copies of the unit interval, used as a model of the closed disc.","title":"The closed square"},{"id":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/square-boundary","kind":"node","lean":"MathlibExt.GroupTheory.BoundaryFixingHomeomorphismsOrderableKourovka17Problem20Wanted.squareBoundary","parent":3049,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/index.html#square-boundary","scope":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups","status":"fully_proved","summary":"The boundary $\\partial I^2$ of the closed square $I^2 = [0, 1] \\times [0, 1]$ is the set of points $(x, y) \\in I^2$ with $x = 0$, $x = 1$, $y = 0$ or $y = 1$.","title":"Boundary of the square"},{"id":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/boundary-fixing-homeomorphisms","kind":"node","lean":"MathlibExt.GroupTheory.BoundaryFixingHomeomorphismsOrderableKourovka17Problem20Wanted.boundaryFixingHomeomorphisms","parent":3049,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/index.html#boundary-fixing-homeomorphisms","scope":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups","status":"fully_proved","summary":"Let $I^2 = [0, 1] \\times [0, 1]$ be the closed square and $\\partial I^2$ its boundary. $\\mathrm{Homeo}(I^2; \\partial I^2)$ is the subgroup of the group of homeomorphisms of $I^2$ consisting of those homeomorphisms $h$ with $h(x) = x$ for every $x \\in \\partial I^2$.","title":"Boundary-fixing homeomorphisms of the square"},{"id":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/right-orders","kind":"node","lean":"MathlibExt.GroupTheory.BoundaryFixingHomeomorphismsOrderableKourovka17Problem20Wanted.rightOrders","parent":3049,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/index.html#right-orders","scope":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups","status":"fully_proved","summary":"The right orders of `G`: linear orders (as relations) invariant under right multiplication.","title":"Right orders of a group"},{"id":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/disc-boundary-fixing-homeomorphisms-righ","kind":"node","lean":"MathlibExt.GroupTheory.BoundaryFixingHomeomorphismsOrderableKourovka17Problem20Wanted.discBoundaryFixingHomeomorphismsRightOrderable","parent":3049,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/index.html#disc-boundary-fixing-homeomorphisms-righ","scope":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups","status":"fully_proved","summary":"The proposition (Kourovka Problem 17.20, G. M. Bergman, for the closed disc) that the group $\\mathrm{Homeo}(I^2; \\partial I^2)$ of homeomorphisms of the closed square $I^2 = [0, 1] \\times [0, 1]$, a model of the closed disc, that fix the boundary $\\partial I^2$ pointwise is right-orderable: it carries a linear order $\\le$ such that $a \\le b$ implies $ac \\le bc$ for all $a, b, c$.","title":"Right-orderability of the boundary-fixing homeomorphisms of the disc"},{"id":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/disc-boundary-fixing-homeomorphisms-righ-2","kind":"node","lean":"MathlibExt.GroupTheory.BoundaryFixingHomeomorphismsOrderableKourovka17Problem20Wanted.discBoundaryFixingHomeomorphismsRightOrderable_refuted","parent":3049,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/orderability-of-homeomorphism-groups/index.html#disc-boundary-fixing-homeomorphisms-righ-2","scope":"GroupTheory/orderable-groups/orderability-of-homeomorphism-groups","status":"can_prove","summary":"Let $I^2 = [0, 1] \\times [0, 1]$ be the closed square, a model of the closed disc, and let $\\mathrm{Homeo}(I^2; \\partial I^2)$ be the group of homeomorphisms of $I^2$ that fix every point of the boundary $\\partial I^2$. There is no linear order $\\le$ on $\\mathrm{Homeo}(I^2; \\partial I^2)$ such that $a \\le b$ implies $ac \\le bc$ for all $a, b, c$; that is, this group is not right-orderable.","title":"Boundary-fixing homeomorphisms of the disc are not right-orderable"},{"data_url":"scopes/GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes.json","id":"GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes/index.html","scope":"GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes","status":"can_prove","title":"Ordered groups with three conjugacy classes"},{"id":"GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes/ordered-embeds-in-three-classes","kind":"node","lean":"MathlibExt.GroupTheory.OrderedGroupThreeClassesEmbeddingKourovka11Problem68Wanted.orderedEmbedsInThreeClasses","parent":3056,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes/index.html#ordered-embeds-in-three-classes","scope":"GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes","status":"fully_proved","summary":"The statement that for every fully ordered group $G$ there are a fully ordered group $H$ with exactly three conjugacy classes and a strictly increasing homomorphism $G \\to H$ (an order-preserving embedding). Here a fully ordered group is a group with a linear order $\\le$ such that $a \\le b$ implies $ca \\le cb$ and $ac \\le bc$ for all $a$, $b$, $c$.","title":"Kourovka Problem 11.68 (ordered groups with three conjugacy classes)"},{"id":"GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes/ordered-embeds-in-three-classes-refuted","kind":"node","lean":"MathlibExt.GroupTheory.OrderedGroupThreeClassesEmbeddingKourovka11Problem68Wanted.orderedEmbedsInThreeClasses_refuted","parent":3056,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes/index.html#ordered-embeds-in-three-classes-refuted","scope":"GroupTheory/orderable-groups/ordered-groups-with-three-conjugacy-classes","status":"can_prove","summary":"There is a fully ordered group $G$ such that for every fully ordered group $H$ with exactly three conjugacy classes, no homomorphism $G \\to H$ is strictly increasing. Here a fully ordered group is a group with a linear order $\\le$ such that $a \\le b$ implies $ca \\le cb$ and $ac \\le bc$ for all $a$, $b$, $c$. That is, the answer to Kourovka Problem 11.68 is negative.","title":"Not every fully ordered group embeds in one with three conjugacy classes"},{"data_url":"scopes/GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups.json","id":"GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups/index.html","scope":"GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups","status":"can_prove","title":"Right-orderability of residually finite p-groups"},{"id":"GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups/is-right-orderable","kind":"node","lean":"MathlibExt.GroupTheory.ResiduallyTwoPrimesNotRightOrderableKourovka10Problem30Wanted.IsRightOrderable","parent":3059,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups/index.html#is-right-orderable","scope":"GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups","status":"fully_proved","summary":"A group $G$ is right-orderable when it admits a linear order $\\le$ that is invariant under right multiplication: $a \\le b$ implies $ac \\le bc$ for all $a, b, c \\in G$.","title":"Right-orderable groups"},{"id":"GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups/residually-two-primes-not-right-orderabl","kind":"node","lean":"MathlibExt.GroupTheory.ResiduallyTwoPrimesNotRightOrderableKourovka10Problem30Wanted.residuallyTwoPrimesNotRightOrderable","parent":3059,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups/index.html#residually-two-primes-not-right-orderabl","scope":"GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups","status":"fully_proved","summary":"The statement that there exist a group $G$ and distinct primes $p$ and $q$ such that $G$ is residually a finite $p$-group and residually a finite $q$-group, but $G$ is not right-orderable.","title":"Medvedev's question on right-orderability (Kourovka 10.30)"},{"id":"GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups/residually-two-primes-not-right-orderabl-2","kind":"node","lean":"MathlibExt.GroupTheory.ResiduallyTwoPrimesNotRightOrderableKourovka10Problem30Wanted.residuallyTwoPrimesNotRightOrderable_holds","parent":3059,"reading_title":{"format":"inline-parts-v1","parts":[["text","A non-right-orderable group that is residually "],["math","\\(p\\)"],["text"," for two primes (Linnell)"]],"source":"A non-right-orderable group that is residually $p$ for two primes (Linnell)"},"result":true,"route":"../roadmap/GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups/index.html#residually-two-primes-not-right-orderabl-2","scope":"GroupTheory/orderable-groups/right-orderability-of-residually-finite-p-groups","status":"can_prove","summary":"There exist a group $G$ and distinct primes $p$ and $q$ such that $G$ is residually a finite $p$-group and residually a finite $q$-group but is not right-orderable.","title":"A non-right-orderable group that is residually $p$ for two primes (Linnell)"},{"data_url":"scopes/GroupTheory/orderable-groups/relatively-convex-subgroups.json","id":"GroupTheory/orderable-groups/relatively-convex-subgroups","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/relatively-convex-subgroups/index.html","scope":"GroupTheory/orderable-groups/relatively-convex-subgroups","status":"can_prove","title":"Relatively convex subgroups"},{"id":"GroupTheory/orderable-groups/relatively-convex-subgroups/is-right-invariant-order","kind":"node","lean":"MathlibExt.GroupTheory.RightRelativelyConvexSublatticeKourovka21Problem147Wanted.IsRightInvariantOrder","parent":3063,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/relatively-convex-subgroups/index.html#is-right-invariant-order","scope":"GroupTheory/orderable-groups/relatively-convex-subgroups","status":"fully_proved","summary":"A linear order $\\le$ on a group $G$ is right-invariant (a right order) when $a \\le b$ implies $ac \\le bc$ for all $a, b, c \\in G$.","title":"Right-invariant linear order"},{"id":"GroupTheory/orderable-groups/relatively-convex-subgroups/is-right-relatively-convex","kind":"node","lean":"MathlibExt.GroupTheory.RightRelativelyConvexSublatticeKourovka21Problem147Wanted.IsRightRelativelyConvex","parent":3063,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/relatively-convex-subgroups/index.html#is-right-relatively-convex","scope":"GroupTheory/orderable-groups/relatively-convex-subgroups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is right-relatively convex when there is a right-invariant linear order $\\le$ on $G$ for which $H$ is convex: whenever $h_1 \\le g \\le h_2$ with $h_1, h_2 \\in H$ and $g \\in G$, also $g \\in H$.","title":"Right-relatively convex subgroup"},{"id":"GroupTheory/orderable-groups/relatively-convex-subgroups/right-relatively-convex-sublattice","kind":"node","lean":"MathlibExt.GroupTheory.RightRelativelyConvexSublatticeKourovka21Problem147Wanted.rightRelativelyConvexSublattice","parent":3063,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/relatively-convex-subgroups/index.html#right-relatively-convex-sublattice","scope":"GroupTheory/orderable-groups/relatively-convex-subgroups","status":"fully_proved","summary":"Sublattice question for right-relatively convex subgroups (Kourovka Problem 21.147, non-abelian case). The statement that for every non-abelian right-orderable group $G$ and all right-relatively convex subgroups $H$ and $K$ of $G$, both the intersection $H \\cap K$ and the join $H \\vee K$ (the subgroup generated by $H$ and $K$) are right-relatively convex; that is, the right-relatively convex subgroups of $G$ form a\u2026","title":"Sublattice question for right-relatively convex subgroups (Kourovka Problem 21.147, non-abelian case)"},{"id":"GroupTheory/orderable-groups/relatively-convex-subgroups/right-relatively-convex-sublattice-refut","kind":"node","lean":"MathlibExt.GroupTheory.RightRelativelyConvexSublatticeKourovka21Problem147Wanted.rightRelativelyConvexSublattice_refuted","parent":3063,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/relatively-convex-subgroups/index.html#right-relatively-convex-sublattice-refut","scope":"GroupTheory/orderable-groups/relatively-convex-subgroups","status":"can_prove","summary":"It is not the case that for every non-abelian right-orderable group $G$ and all right-relatively convex subgroups $H$ and $K$ of $G$, both the intersection $H \\cap K$ and the join $H \\vee K$ (the subgroup generated by $H$ and $K$) are right-relatively convex.","title":"Negative answer to Kourovka Problem 21.147 for non-abelian groups"},{"data_url":"scopes/GroupTheory/orderable-groups/simple-right-orderable-groups.json","id":"GroupTheory/orderable-groups/simple-right-orderable-groups","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/simple-right-orderable-groups/index.html","scope":"GroupTheory/orderable-groups/simple-right-orderable-groups","status":"can_prove","title":"Simple right-orderable groups"},{"id":"GroupTheory/orderable-groups/simple-right-orderable-groups/right-orders","kind":"node","lean":"MathlibExt.GroupTheory.SimpleFinitelyGeneratedRightOrderableKourovka16Problem50Wanted.rightOrders","parent":3068,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/simple-right-orderable-groups/index.html#right-orders","scope":"GroupTheory/orderable-groups/simple-right-orderable-groups","status":"fully_proved","summary":"The right orders of a group $G$ are the linear orders $\\le$ on $G$, viewed as relations, that are invariant under right multiplication: $a \\le b$ implies $ac \\le bc$ for all $a, b, c \\in G$.","title":"Right orders on a group (Kourovka Problem 16.50)"},{"id":"GroupTheory/orderable-groups/simple-right-orderable-groups/exists-simple-fgright-orderable","kind":"node","lean":"MathlibExt.GroupTheory.SimpleFinitelyGeneratedRightOrderableKourovka16Problem50Wanted.existsSimpleFGRightOrderable","parent":3068,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/simple-right-orderable-groups/index.html#exists-simple-fgright-orderable","scope":"GroupTheory/orderable-groups/simple-right-orderable-groups","status":"fully_proved","summary":"The statement that there exists a finitely generated simple group $G$ admitting a right order, that is, a linear order $\\le$ on $G$ such that $a \\le b$ implies $ac \\le bc$ for all $a, b, c \\in G$. The problem is due to V. M. Kopytov and N. Ya. Medvedev.","title":"Kourovka Problem 16.50 (finitely generated simple right-orderable groups)"},{"id":"GroupTheory/orderable-groups/simple-right-orderable-groups/exists-simple-fgright-orderable-holds","kind":"node","lean":"MathlibExt.GroupTheory.SimpleFinitelyGeneratedRightOrderableKourovka16Problem50Wanted.existsSimpleFGRightOrderable_holds","parent":3068,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/simple-right-orderable-groups/index.html#exists-simple-fgright-orderable-holds","scope":"GroupTheory/orderable-groups/simple-right-orderable-groups","status":"can_prove","summary":"There is a finitely generated simple group $G$ that admits a right order, that is, a linear order $\\le$ on $G$ such that $a \\le b$ implies $ac \\le bc$ for all $a, b, c \\in G$. This answers Kourovka Problem 16.50 positively.","title":"Finitely generated simple right-orderable groups exist"},{"data_url":"scopes/GroupTheory/orderable-groups/spaces-of-right-orders.json","id":"GroupTheory/orderable-groups/spaces-of-right-orders","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/spaces-of-right-orders/index.html","scope":"GroupTheory/orderable-groups/spaces-of-right-orders","status":"can_prove","title":"Spaces of right orders"},{"id":"GroupTheory/orderable-groups/spaces-of-right-orders/right-orders","kind":"node","lean":"MathlibExt.GroupTheory.CountablyManyRightOrdersKourovka16Problem51Wanted.rightOrders","parent":3072,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/spaces-of-right-orders/index.html#right-orders","scope":"GroupTheory/orderable-groups/spaces-of-right-orders","status":"fully_proved","summary":"The right orders of a group $G$ are the linear orders $\\le$ on $G$ that are invariant under right multiplication: $a \\le b$ implies $ac \\le bc$ for all $a, b, c \\in G$.","title":"Right orders of a group (Kourovka 16.51)"},{"id":"GroupTheory/orderable-groups/spaces-of-right-orders/exists-countably-infinitely-right-ordera","kind":"node","lean":"MathlibExt.GroupTheory.CountablyManyRightOrdersKourovka16Problem51Wanted.existsCountablyInfinitelyRightOrderable","parent":3072,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/spaces-of-right-orders/index.html#exists-countably-infinitely-right-ordera","scope":"GroupTheory/orderable-groups/spaces-of-right-orders","status":"fully_proved","summary":"The statement that some group has a countably infinite set of right orders.","title":"Groups with countably infinitely many right orders (Kourovka 16.51)"},{"id":"GroupTheory/orderable-groups/spaces-of-right-orders/exists-countably-infinitely-right-ordera-2","kind":"node","lean":"MathlibExt.GroupTheory.CountablyManyRightOrdersKourovka16Problem51Wanted.existsCountablyInfinitelyRightOrderable_refuted","parent":3072,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/spaces-of-right-orders/index.html#exists-countably-infinitely-right-ordera-2","scope":"GroupTheory/orderable-groups/spaces-of-right-orders","status":"can_prove","summary":"For every group $G$, the set of right orders on $G$ (linear orders $\\le$ on $G$ such that $a \\le b$ implies $ac \\le bc$ for all $a, b, c \\in G$) is either finite or uncountable. That is, Kourovka Problem 16.51 has a negative answer.","title":"No group has countably infinitely many right orders"},{"data_url":"scopes/GroupTheory/orderable-groups/unique-product-groups.json","id":"GroupTheory/orderable-groups/unique-product-groups","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/unique-product-groups/index.html","scope":"GroupTheory/orderable-groups/unique-product-groups","status":"can_prove","title":"Unique product groups"},{"id":"GroupTheory/orderable-groups/unique-product-groups/is-left-orderable","kind":"node","lean":"MathlibExt.GroupTheory.UniqueProductNotLeftOrderableKourovka15Problem49Wanted.IsLeftOrderable","parent":3076,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/unique-product-groups/index.html#is-left-orderable","scope":"GroupTheory/orderable-groups/unique-product-groups","status":"fully_proved","summary":"A group $G$ is left-orderable if it admits a linear order $\\le$ such that $a \\le b$ implies $ca \\le cb$ for all $a, b, c \\in G$.","title":"Left-orderable groups"},{"id":"GroupTheory/orderable-groups/unique-product-groups/unique-product-group-not-left-orderable","kind":"node","lean":"MathlibExt.GroupTheory.UniqueProductNotLeftOrderableKourovka15Problem49Wanted.uniqueProductGroupNotLeftOrderable","parent":3076,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/unique-product-groups/index.html#unique-product-group-not-left-orderable","scope":"GroupTheory/orderable-groups/unique-product-groups","status":"fully_proved","summary":"The statement, asked by P. Linnell, that there exists a group $G$ with unique products that is not left-orderable. Here $G$ has unique products if for all finite nonempty subsets $X, Y \\subseteq G$ there are $x_0 \\in X$ and $y_0 \\in Y$ such that $(x_0, y_0)$ is the only pair $(x, y) \\in X \\times Y$ with $xy = x_0 y_0$.","title":"Kourovka Problem 15.49 (unique product groups and left orders)"},{"id":"GroupTheory/orderable-groups/unique-product-groups/unique-product-group-not-left-orderable-2","kind":"node","lean":"MathlibExt.GroupTheory.UniqueProductNotLeftOrderableKourovka15Problem49Wanted.uniqueProductGroupNotLeftOrderable_holds","parent":3076,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/unique-product-groups/index.html#unique-product-group-not-left-orderable-2","scope":"GroupTheory/orderable-groups/unique-product-groups","status":"can_prove","summary":"There exists a group $G$ that has unique products but is not left-orderable: for all finite nonempty subsets $X, Y \\subseteq G$ there are $x_0 \\in X$ and $y_0 \\in Y$ such that $(x_0, y_0)$ is the only pair $(x, y) \\in X \\times Y$ with $xy = x_0 y_0$, and yet $G$ admits no linear order invariant under left multiplication.","title":"A unique product group need not be left-orderable"},{"data_url":"scopes/GroupTheory/orderable-groups/word-problem-for-orderable-groups.json","id":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","kind":"scope","navigation":"module","parent":3027,"result":false,"route":"../roadmap/GroupTheory/orderable-groups/word-problem-for-orderable-groups/index.html","scope":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","status":"can_prove","title":"Word problem for orderable groups"},{"id":"GroupTheory/orderable-groups/word-problem-for-orderable-groups/bi-orders","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyPresentedOrderableWordProblemKourovka17Problem29Wanted.biOrders","parent":3080,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/word-problem-for-orderable-groups/index.html#bi-orders","scope":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","status":"fully_proved","summary":"For a group $G$, the bi-invariant orders of $G$ are the linear orders $\\le$ on $G$ (reflexive, antisymmetric, transitive and total relations) that are invariant under left and right multiplication: for all $a, b, c \\in G$, $a \\le b$ implies $ca \\le cb$ and $ac \\le bc$.","title":"Bi-invariant linear orders on a group"},{"id":"GroupTheory/orderable-groups/word-problem-for-orderable-groups/finitely-presented-orderable-insoluble-w","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyPresentedOrderableWordProblemKourovka17Problem29Wanted.finitelyPresentedOrderableInsolubleWordProblem","parent":3080,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/word-problem-for-orderable-groups/index.html#finitely-presented-orderable-insoluble-w","scope":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","status":"fully_proved","summary":"The statement that there is a finitely presented group $G$ that admits a bi-invariant linear order and has a finite generating family $g_1, \\dots, g_k$ for which the word problem is insoluble: no algorithm decides, given a word in the $g_i$ and their inverses, whether it equals $1$ in $G$.","title":"Kourovka Problem 17.29 (V. V. Bludov, A. M. W. Glass)"},{"id":"GroupTheory/orderable-groups/word-problem-for-orderable-groups/finitely-presented-orderable-insoluble-w-2","kind":"node","lean":"MathlibExt.GroupTheory.FinitelyPresentedOrderableWordProblemKourovka17Problem29Wanted.finitelyPresentedOrderableInsolubleWordProblem_holds","parent":3080,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/word-problem-for-orderable-groups/index.html#finitely-presented-orderable-insoluble-w-2","scope":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","status":"can_prove","summary":"There is a finitely presented group $G$ that admits a bi-invariant linear order and has a finite generating family $g_1, \\dots, g_k$ for which the word problem is insoluble: no algorithm decides, given a word in the $g_i$ and their inverses, whether it equals $1$ in $G$.","title":"Bludov\u2013Glass: a finitely presented orderable group with insoluble word problem"},{"id":"GroupTheory/orderable-groups/word-problem-for-orderable-groups/is-recursively-presented","kind":"node","lean":"MathlibExt.GroupTheory.SolubleRightOrderableWordProblemKourovka17Problem28Wanted.IsRecursivelyPresented","parent":3080,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/word-problem-for-orderable-groups/index.html#is-recursively-presented","scope":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","status":"fully_proved","summary":"Let $G$ be a group, $k$ a natural number and $\\varphi$ a homomorphism from the free group on $k$ generators to $G$. The group $G$ is recursively presented with respect to $\\varphi$ when the set of words $w$ (finite lists of letters $(i, \\varepsilon)$ with $i < k$ and $\\varepsilon$ true or false, the letter $(i, \\mathrm{false})$ standing for the inverse of the $i$-th generator) with $\\varphi(w) = 1$ in $G$ is\u2026","title":"Recursively presented group"},{"id":"GroupTheory/orderable-groups/word-problem-for-orderable-groups/right-orders","kind":"node","lean":"MathlibExt.GroupTheory.SolubleRightOrderableWordProblemKourovka17Problem28Wanted.rightOrders","parent":3080,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/word-problem-for-orderable-groups/index.html#right-orders","scope":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","status":"fully_proved","summary":"The right orders of `G`: linear orders (as relations) invariant under right multiplication.","title":"Right orders of a group"},{"id":"GroupTheory/orderable-groups/word-problem-for-orderable-groups/soluble-right-orderable-insoluble-word-p","kind":"node","lean":"MathlibExt.GroupTheory.SolubleRightOrderableWordProblemKourovka17Problem28Wanted.solubleRightOrderableInsolubleWordProblem","parent":3080,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/word-problem-for-orderable-groups/index.html#soluble-right-orderable-insoluble-word-p","scope":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","status":"fully_proved","summary":"The statement that there exist a soluble group $G$ that admits a right order, a natural number $k$ and elements $g_0, \\dots, g_{k-1}$ generating $G$ such that, for the homomorphism $\\varphi$ from the free group on $k$ generators to $G$ sending the $i$-th generator to $g_i$, the group $G$ is recursively presented with respect to $\\varphi$ and the word problem for $\\varphi$ is not solvable. In words: there is a\u2026","title":"The Bludov\u2013Glass question (Kourovka 17.28)"},{"id":"GroupTheory/orderable-groups/word-problem-for-orderable-groups/soluble-right-orderable-insoluble-word-p-2","kind":"node","lean":"MathlibExt.GroupTheory.SolubleRightOrderableWordProblemKourovka17Problem28Wanted.solubleRightOrderableInsolubleWordProblem_holds","parent":3080,"result":true,"route":"../roadmap/GroupTheory/orderable-groups/word-problem-for-orderable-groups/index.html#soluble-right-orderable-insoluble-word-p-2","scope":"GroupTheory/orderable-groups/word-problem-for-orderable-groups","status":"can_prove","summary":"There exist a soluble group $G$ that admits a right order, a natural number $k$ and elements $g_0, \\dots, g_{k-1}$ generating $G$ such that, for the homomorphism $\\varphi$ from the free group on $k$ generators to $G$ sending the $i$-th generator to $g_i$, the group $G$ is recursively presented with respect to $\\varphi$ and the word problem for $\\varphi$ is not solvable.","title":"A soluble right-orderable group with insoluble word problem (Kourovka 17.28)"},{"data_url":"scopes/GroupTheory/varieties-and-equations.json","id":"GroupTheory/varieties-and-equations","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/index.html","scope":"GroupTheory/varieties-and-equations","status":"can_prove","summary":"Groups defined by laws and quasi-identities: varieties, relatively free groups, test ranks, quasivarieties and their lattices, Hall's verbal conjectures, plus equations over groups, equationally Noetherian groups, geometric equivalence and elementary theories of groups.","title":"Varieties, quasivarieties and equations"},{"id":"GroupTheory/varieties-and-equations/is-locally-finite-variety","kind":"node","lean":"MathlibExt.GroupTheory.GroupVarietyShared.IsLocallyFiniteVariety","parent":3088,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/index.html#is-locally-finite-variety","scope":"GroupTheory/varieties-and-equations","status":"fully_proved","summary":"Let $F$ be the free group on countably many generators $x_0, x_1, x_2, \\dots$ and let $N$ be a subgroup of $F$, regarded as a set of laws. The variety defined by $N$ is locally finite if every finitely generated group satisfying all laws in $N$ is finite.","title":"Locally finite variety"},{"data_url":"scopes/GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g.json","id":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/index.html","scope":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g","status":"can_prove","title":"Automorphisms of categories of relatively free groups"},{"id":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/relatively-free-group","kind":"node","lean":"MathlibExt.GroupTheory.VarietyCategoryOuterAutomorphismKourovka20Problem72Wanted.relativelyFreeGroup","parent":3090,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/index.html#relatively-free-group","scope":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g","status":"fully_proved","summary":"Let $W$ be a set of words in the free group $F(\\mathbb{N})$ on $x_0, x_1, \\ldots$, defining a variety $\\Theta$, and let $X \\subseteq \\mathbb{N}$ be finite. The free group $F_\\Theta(X)$ of $\\Theta$ on $X$ is the quotient of the free group $F(X)$ on $X$ by its verbal subgroup, the normal closure of the set of values $\\varphi(w)$ with $w \\in W$ and $\\varphi \\colon F(\\mathbb{N}) \\to F(X)$ a homomorphism. This set of\u2026","title":"Free group of a variety on a finite set"},{"id":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/theta-zero","kind":"node","lean":"MathlibExt.GroupTheory.VarietyCategoryOuterAutomorphismKourovka20Problem72Wanted.ThetaZero","parent":3090,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/index.html#theta-zero","scope":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g","status":"fully_proved","summary":"Let $W$ be a set of words in the free group on $x_0, x_1, \\ldots$, defining a variety $\\Theta$. The category $\\Theta^0$ has one object for each finite subset $X$ of $X^0 = \\mathbb{N}$, namely the free group $F_\\Theta(X)$ of $\\Theta$ on $X$, and its morphisms are all group homomorphisms between these groups.","title":"Plotkin's category of free groups of a variety"},{"id":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/exists-variety-with-non-inner-category-a","kind":"node","lean":"MathlibExt.GroupTheory.VarietyCategoryOuterAutomorphismKourovka20Problem72Wanted.existsVarietyWithNonInnerCategoryAutomorphism","parent":3090,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/index.html#exists-variety-with-non-inner-category-a","scope":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g","status":"fully_proved","summary":"The statement that there is a set $W$ of words in the free group on $x_0, x_1, \\ldots$ such that, for the variety $\\Theta$ with laws $W$, the category $\\Theta^0$ of the free groups of $\\Theta$ on the finite subsets of $\\mathbb{N}$ has an automorphism that is not inner: functors $\\Phi, \\Psi \\colon \\Theta^0 \\to \\Theta^0$ whose composites $\\Psi \\circ \\Phi$ and $\\Phi \\circ \\Psi$ both equal the identity functor, with\u2026","title":"Plotkin's question on outer automorphisms of the category of free groups of a variety"},{"id":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/exists-variety-with-non-inner-category-a-2","kind":"node","lean":"MathlibExt.GroupTheory.VarietyCategoryOuterAutomorphismKourovka20Problem72Wanted.existsVarietyWithNonInnerCategoryAutomorphism_holds","parent":3090,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g/index.html#exists-variety-with-non-inner-category-a-2","scope":"GroupTheory/varieties-and-equations/automorphisms-of-categories-of-relatively-free-g","status":"can_prove","summary":"There is a set $W$ of words in the free group on $x_0, x_1, \\ldots$ such that, for the variety $\\Theta$ with laws $W$, the category $\\Theta^0$, whose objects are the free groups $F_\\Theta(X)$ of $\\Theta$ on the finite subsets $X \\subseteq \\mathbb{N}$ and whose morphisms are all group homomorphisms, has an automorphism that is not inner: there are functors $\\Phi, \\Psi \\colon \\Theta^0 \\to \\Theta^0$ whose composites\u2026","title":"Some variety of groups has an outer automorphism of its category of free groups"},{"data_url":"scopes/GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups.json","id":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/index.html","scope":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups","status":"can_prove","title":"Axiomatizable classes of groups"},{"id":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/group-class","kind":"node","lean":"MathlibExt.GroupTheory.AxiomatizableFormationVarietyKourovka14Problem32Wanted.GroupClass","parent":3095,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/index.html#group-class","scope":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups","status":"fully_proved","summary":"A class of groups, as a predicate on groups in `Type`.","title":"Class of groups"},{"id":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/is-formation","kind":"node","lean":"MathlibExt.GroupTheory.AxiomatizableFormationVarietyKourovka14Problem32Wanted.IsFormation","parent":3095,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/index.html#is-formation","scope":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups","status":"fully_proved","summary":"A class $\\mathfrak{F}$ of groups is a formation when it contains at least one group, it is closed under surjective homomorphic images (if $G \\in \\mathfrak{F}$ and $\\varphi \\colon G \\to H$ is a surjective homomorphism, then $H \\in \\mathfrak{F}$), and for every group $G$ and all normal subgroups $N_1$, $N_2$ of $G$, if $G/N_1$ and $G/N_2$ lie in $\\mathfrak{F}$, then so does $G/(N_1 \\cap N_2)$.","title":"Formations of groups in the sense of Gaglione and Spellman"},{"id":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/is-axiomatizable","kind":"node","lean":"MathlibExt.GroupTheory.AxiomatizableFormationVarietyKourovka14Problem32Wanted.IsAxiomatizable","parent":3095,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/index.html#is-axiomatizable","scope":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups","status":"fully_proved","summary":"A class $\\mathfrak{F}$ of groups is first-order axiomatizable when there is a theory $T$ in the first-order language of groups such that a group $G$ lies in $\\mathfrak{F}$ if and only if $G$, with its canonical structure, is a model of $T$.","title":"First-order axiomatizable classes of groups"},{"id":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/is-variety","kind":"node","lean":"MathlibExt.GroupTheory.AxiomatizableFormationVarietyKourovka14Problem32Wanted.IsVariety","parent":3095,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/index.html#is-variety","scope":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups","status":"fully_proved","summary":"A class $\\mathfrak{F}$ of groups is a variety when there is a set $W$ of elements of the free group on generators $x_0, x_1, x_2, \\dots$ such that a group $G$ lies in $\\mathfrak{F}$ if and only if every $w \\in W$ evaluates to $1$ in $G$ under every assignment of elements of $G$ to the generators.","title":"Varieties of groups"},{"id":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/axiomatizable-formation-is-variety","kind":"node","lean":"MathlibExt.GroupTheory.AxiomatizableFormationVarietyKourovka14Problem32Wanted.axiomatizableFormationIsVariety","parent":3095,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/index.html#axiomatizable-formation-is-variety","scope":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups","status":"fully_proved","summary":"The proposition that every first-order axiomatizable formation of groups is a variety: every class $\\mathfrak{F}$ of groups that is a formation and is first-order axiomatizable is a variety.","title":"The Gaglione\u2013Spellman question on axiomatizable formations (Kourovka 14.32)"},{"id":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/axiomatizable-formation-is-variety-refut","kind":"node","lean":"MathlibExt.GroupTheory.AxiomatizableFormationVarietyKourovka14Problem32Wanted.axiomatizableFormationIsVariety_refuted","parent":3095,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups/index.html#axiomatizable-formation-is-variety-refut","scope":"GroupTheory/varieties-and-equations/axiomatizable-classes-of-groups","status":"can_prove","summary":"The Gaglione\u2013Spellman question (Kourovka 14.32) has a negative answer: it is not the case that every first-order axiomatizable formation of groups is a variety, that is, some class of groups is a formation and is first-order axiomatizable but is not a variety.","title":"Not every axiomatizable formation of groups is a variety"},{"data_url":"scopes/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro.json","id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"can_prove","title":"Elementary theory of torsion-free hyperbolic groups"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/word","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.Word","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#word","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"fully_proved","summary":"A group word in the variables $x_0, x_1, \\dots$ is coded as a finite list of letters $(i, b)$, where $i$ is a natural number and $b$ is a Boolean: the letter $(i, \\mathrm{true})$ stands for $x_i$ and the letter $(i, \\mathrm{false})$ for $x_i^{-1}$. The word is read as the product of its letters in order, an element of the free group on $x_0, x_1, \\dots$.","title":"Group words in countably many variables"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/eval-word","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.evalWord","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#eval-word","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"fully_proved","summary":"Let $G$ be a group and $a \\colon \\mathbb{N} \\to G$ an assignment of values to the variables $x_0, x_1, \\dots$. The value of a group word $w$ under $a$ is the product in $G$, in the order of the letters of $w$, of $a(i)$ for each letter $x_i$ and of $a(i)^{-1}$ for each letter $x_i^{-1}$; equivalently, it is the image of $w$ under the homomorphism from the free group on $x_0, x_1, \\dots$ to $G$ that sends $x_i$ to\u2026","title":"Value of a group word"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/cnf","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.CNF","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#cnf","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"fully_proved","summary":"A quantifier-free formula of the language of groups in conjunctive normal form is coded as a finite list of clauses, each clause a finite list of literals. A literal is a pair $(b, w)$ of a Boolean $b$ and a group word $w$; it stands for the equation $w = 1$ when $b$ is true and for the inequation $w \\ne 1$ when $b$ is false. The formula is the conjunction of its clauses, and each clause is the disjunction of its\u2026","title":"Quantifier-free formulas in conjunctive normal form"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/holds-at","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.CNF.HoldsAt","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#holds-at","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"fully_proved","summary":"Let $G$ be a group, let $a \\colon \\mathbb{N} \\to G$ assign values to the variables $x_0, x_1, \\dots$, and let $\\varphi$ be a quantifier-free formula in conjunctive normal form. Then $\\varphi$ holds in $G$ under $a$ when every clause of $\\varphi$ contains a literal $(b, w)$ such that the value of $w$ under $a$ equals $1$ if and only if $b$ is true, that is, a literal $w = 1$ whose word has value $1$ or a literal\u2026","title":"Truth of a formula in conjunctive normal form"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/realize-prenex","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.realizePrenex","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#realize-prenex","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"fully_proved","summary":"Let $G$ be a group, $\\varphi$ a quantifier-free formula in conjunctive normal form, $q$ a finite list of Booleans read as a quantifier prefix ($\\mathrm{true}$ for $\\forall$, $\\mathrm{false}$ for $\\exists$), $i$ a natural number and $a \\colon \\mathbb{N} \\to G$ an assignment of values to the variables $x_0, x_1, \\dots$. Truth in $G$ under $a$ of the formula with prefix $q$, whose first quantifier binds $x_i$, and\u2026","title":"Truth of a prenex formula"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/prenex-sentence","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.PrenexSentence","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#prenex-sentence","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"fully_proved","summary":"A sentence of the language of groups in prenex form with matrix in conjunctive normal form is coded as a pair $(q, \\varphi)$ of a finite list $q$ of Booleans, the quantifier prefix ($\\mathrm{true}$ for $\\forall$, $\\mathrm{false}$ for $\\exists$), and a quantifier-free formula $\\varphi$ in conjunctive normal form. The quantifiers of $q$ bind the variables $x_0, x_1, \\dots$ in order, and variables of $\\varphi$ not\u2026","title":"Prenex sentences of the language of groups"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/holds-in","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.PrenexSentence.HoldsIn","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#holds-in","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"fully_proved","summary":"A prenex sentence $(q, \\varphi)$ holds in a group $G$ when the formula with quantifier prefix $q$ and matrix $\\varphi$, whose quantifiers bind the variables $x_0, x_1, \\dots$ in order, is true in $G$ under the assignment that gives every variable the value $1$; thus the variables not bound by $q$ are read as $1$.","title":"Truth of a prenex sentence in a group"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/torsion-free-hyperbolic-theory-decidable","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.torsionFreeHyperbolicTheoryDecidable","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#torsion-free-hyperbolic-theory-decidable","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"fully_proved","summary":"The proposition stating Kourovka Problem 16.71 (A. G. Myasnikov, O. G. Kharlampovich): for every torsion-free word-hyperbolic group $G$ (every element $g \\ne 1$ of $G$ has infinite order), the set of prenex sentences of the language of groups that hold in $G$ is computable, that is, some algorithm decides for each prenex sentence whether it holds in $G$.","title":"Decidability of the elementary theory of torsion-free hyperbolic groups"},{"id":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/torsion-free-hyperbolic-theory-decidable-2","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeHyperbolicElementaryTheoryKourovka16Problem71Wanted.torsionFreeHyperbolicTheoryDecidable_holds","parent":3102,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro/index.html#torsion-free-hyperbolic-theory-decidable-2","scope":"GroupTheory/varieties-and-equations/elementary-theory-of-torsion-free-hyperbolic-gro","status":"can_prove","summary":"For every torsion-free word-hyperbolic group $G$ (every element $g \\ne 1$ of $G$ has infinite order), the set of prenex sentences of the language of groups that hold in $G$ is computable, that is, some algorithm decides for each prenex sentence whether it holds in $G$.","title":"Torsion-free hyperbolic groups have decidable elementary theory"},{"data_url":"scopes/GroupTheory/varieties-and-equations/equationally-noetherian-groups.json","id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"can_prove","title":"Equationally Noetherian groups"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/eval-equation","kind":"node","lean":"MathlibExt.GroupTheory.FreeProductEquationallyNoetherianKourovka14Problem20Wanted.evalEquation","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#eval-equation","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"fully_proved","summary":"For a group $G$, let $F_n$ be the free group on the unknowns $t_0, \\dots, t_{n-1}$. For an assignment $x = (x_0, \\dots, x_{n-1}) \\in G^n$, evaluation at $x$ is the group homomorphism $G * F_n \\to G$ from the free product of $G$ and $F_n$ that is the identity on $G$ and sends $t_i$ to $x_i$ for each $i < n$; it sends an equation $w \\in G * F_n$ over $G$ to its value $w(x)$ at $x$.","title":"Evaluation of an equation at an assignment (Kourovka 14.20)"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/solutions","kind":"node","lean":"MathlibExt.GroupTheory.FreeProductEquationallyNoetherianKourovka14Problem20Wanted.solutions","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#solutions","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"fully_proved","summary":"For a group $G$, let $F_n$ be the free group on the unknowns $t_0, \\dots, t_{n-1}$. The solution set of a set $S \\subseteq G * F_n$ of equations over $G$ is the set of assignments $x = (x_0, \\dots, x_{n-1}) \\in G^n$ such that $w(x) = 1$ for every $w \\in S$, where $w(x)$ is the image of $w$ under the homomorphism $G * F_n \\to G$ that is the identity on $G$ and sends $t_i$ to $x_i$.","title":"Solution set of a system of equations (Kourovka 14.20)"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/equationally-noetherian","kind":"node","lean":"MathlibExt.GroupTheory.FreeProductEquationallyNoetherianKourovka14Problem20Wanted.EquationallyNoetherian","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#equationally-noetherian","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"fully_proved","summary":"A group $G$ is equationally Noetherian (has the Noetherian Equation Property) if for every $n \\in \\mathbb{N}$ and every set $S$ of equations over $G$ in the unknowns $t_0, \\dots, t_{n-1}$, that is, every subset $S$ of the free product $G * F_n$ of $G$ with the free group $F_n$ on $t_0, \\dots, t_{n-1}$, there is a finite subset $S_0 \\subseteq S$ with the same solution set as $S$. Here the solution set of a set of\u2026","title":"Equationally Noetherian groups (Kourovka 14.20)"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/free-product-equationally-noetherian","kind":"node","lean":"MathlibExt.GroupTheory.FreeProductEquationallyNoetherianKourovka14Problem20Wanted.freeProductEquationallyNoetherian","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#free-product-equationally-noetherian","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"fully_proved","summary":"The statement that for all groups $G$ and $H$, if $G$ and $H$ are equationally Noetherian (have the Noetherian Equation Property), then their free product $G * H$ is equationally Noetherian.","title":"Kourovka Problem 14.20 (free products of equationally Noetherian groups)"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/free-product-equationally-noetherian-hol","kind":"node","lean":"MathlibExt.GroupTheory.FreeProductEquationallyNoetherianKourovka14Problem20Wanted.freeProductEquationallyNoetherian_holds","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#free-product-equationally-noetherian-hol","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"can_prove","summary":"Resolved true: the Kourovka Notebook (21st issue) answers 14.20 \"Yes, it does\", citing Sela's preprint (Theorem 9.1): the free product of two equationally Noetherian groups is equationally Noetherian. Source: E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.20 (G. Baumslag, A. G. Myasnikov, V. N. Remeslennikov);\u2026","title":"Free products of equationally Noetherian groups are equationally Noetherian"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/eval-equation-2","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicEquationallyNoetherianKourovka14Problem19Wanted.evalEquation","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#eval-equation-2","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"fully_proved","summary":"Let $G$ be a group and $n$ a natural number. An equation over $G$ in the unknowns $x_0, \\dots, x_{n-1}$ is an element of the free product $G * F(x_0, \\dots, x_{n-1})$ of $G$ with the free group on $x_0, \\dots, x_{n-1}$. For an assignment $a = (a_0, \\dots, a_{n-1}) \\in G^n$, evaluation at $a$ is the group homomorphism $G * F(x_0, \\dots, x_{n-1}) \\to G$ that is the identity on $G$ and sends each unknown $x_i$ to $a_i$.","title":"Evaluating an equation at an assignment"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/solutions-2","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicEquationallyNoetherianKourovka14Problem19Wanted.solutions","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#solutions-2","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"fully_proved","summary":"Let $G$ be a group, $n$ a natural number and $S$ a set of equations over $G$ in the unknowns $x_0, \\dots, x_{n-1}$, that is, a subset of the free product $G * F(x_0, \\dots, x_{n-1})$ of $G$ with the free group on $x_0, \\dots, x_{n-1}$. The solution set of $S$ is the set of assignments $a \\in G^n$ at which every $w \\in S$ evaluates to $1$.","title":"Solution set of a system of equations"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/equationally-noetherian-2","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicEquationallyNoetherianKourovka14Problem19Wanted.EquationallyNoetherian","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#equationally-noetherian-2","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"fully_proved","summary":"A group $G$ has the Noetherian Equation Property (is equationally Noetherian) if for every natural number $n$ and every system $S$ of equations over $G$ in the unknowns $x_0, \\dots, x_{n-1}$, that is, every subset $S$ of the free product $G * F(x_0, \\dots, x_{n-1})$ of $G$ with the free group on $x_0, \\dots, x_{n-1}$, there is a finite subset $S_0 \\subseteq S$ whose solution set in $G^n$ equals the solution set of\u2026","title":"The Noetherian Equation Property"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/hyperbolic-group-equationally-noetherian","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicEquationallyNoetherianKourovka14Problem19Wanted.hyperbolicGroupEquationallyNoetherian","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#hyperbolic-group-equationally-noetherian","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"fully_proved","summary":"Every word-hyperbolic group $G$ has the Noetherian Equation Property: for every natural number $n$, every system of equations over $G$ in the unknowns $x_0, \\dots, x_{n-1}$ has the same solution set in $G^n$ as some finite subsystem.","title":"Kourovka Problem 14.19 (hyperbolic groups and the Noetherian Equation Property)"},{"id":"GroupTheory/varieties-and-equations/equationally-noetherian-groups/hyperbolic-group-equationally-noetherian-2","kind":"node","lean":"MathlibExt.GroupTheory.HyperbolicEquationallyNoetherianKourovka14Problem19Wanted.hyperbolicGroupEquationallyNoetherian_holds","parent":3112,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equationally-noetherian-groups/index.html#hyperbolic-group-equationally-noetherian-2","scope":"GroupTheory/varieties-and-equations/equationally-noetherian-groups","status":"can_prove","summary":"Every word-hyperbolic group $G$ has the Noetherian Equation Property: for every natural number $n$, every system of equations over $G$ in the unknowns $x_0, \\dots, x_{n-1}$ has the same solution set in $G^n$ as some finite subsystem.","title":"Hyperbolic groups are equationally Noetherian"},{"data_url":"scopes/GroupTheory/varieties-and-equations/equations-over-free-groups.json","id":"GroupTheory/varieties-and-equations/equations-over-free-groups","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/equations-over-free-groups/index.html","scope":"GroupTheory/varieties-and-equations/equations-over-free-groups","status":"can_prove","title":"Equations over free groups"},{"id":"GroupTheory/varieties-and-equations/equations-over-free-groups/unsolvable-equation-unsolvable-in-finite","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupEquationFiniteQuotientKourovka14Problem48Wanted.unsolvableEquationUnsolvableInFiniteQuotient","parent":3123,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equations-over-free-groups/index.html#unsolvable-equation-unsolvable-in-finite","scope":"GroupTheory/varieties-and-equations/equations-over-free-groups","status":"fully_proved","summary":"The statement that for every free group $F$ on an arbitrary set of generators, every natural number $n$ and every equation $w$ over $F$ in the unknowns $x_0, \\dots, x_{n-1}$ (an element of the free product of $F$ with the free group on $x_0, \\dots, x_{n-1}$): if no $g_0, \\dots, g_{n-1} \\in F$ are such that substituting $g_i$ for $x_i$ sends $w$ to $1$, then there is a normal subgroup $N$ of finite index in $F$ such\u2026","title":"Kourovka Problem 14.48 (equations over free groups in finite quotients)"},{"id":"GroupTheory/varieties-and-equations/equations-over-free-groups/unsolvable-equation-unsolvable-in-finite-2","kind":"node","lean":"MathlibExt.GroupTheory.FreeGroupEquationFiniteQuotientKourovka14Problem48Wanted.unsolvableEquationUnsolvableInFiniteQuotient_refuted","parent":3123,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/equations-over-free-groups/index.html#unsolvable-equation-unsolvable-in-finite-2","scope":"GroupTheory/varieties-and-equations/equations-over-free-groups","status":"can_prove","summary":"There are a free group $F$ on some set of generators, a natural number $n$ and an equation $w$ over $F$ in the unknowns $x_0, \\dots, x_{n-1}$, that is, an element $w$ of the free product of $F$ with the free group on $x_0, \\dots, x_{n-1}$, such that no $g_0, \\dots, g_{n-1} \\in F$ satisfy the equation (substituting $g_i$ for $x_i$ never sends $w$ to $1$), but for every normal subgroup $N$ of finite index in $F$\u2026","title":"An unsolvable equation over a free group that is solvable in every finite quotient"},{"data_url":"scopes/GroupTheory/varieties-and-equations/geometric-equivalence-of-groups.json","id":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/index.html","scope":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups","status":"can_prove","title":"Geometric equivalence of groups"},{"id":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/quasi-identity","kind":"node","lean":"MathlibExt.GroupTheory.QuasiIdentitiesGeometricEquivalenceKourovka14Problem71Wanted.QuasiIdentity","parent":3126,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/index.html#quasi-identity","scope":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups","status":"fully_proved","summary":"A quasi-identity consists of a natural number $n$, a finite list of premises $w_1, \\dots, w_k$ and a conclusion $w$, all elements of the free group $F_n$ on the unknowns $x_0, \\dots, x_{n-1}$. It stands for the formula $\\forall x_0, \\dots, x_{n-1}\\, (w_1 = 1 \\wedge \\dots \\wedge w_k = 1 \\to w = 1)$.","title":"Quasi-identities of groups"},{"id":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/holds-in","kind":"node","lean":"MathlibExt.GroupTheory.QuasiIdentitiesGeometricEquivalenceKourovka14Problem71Wanted.QuasiIdentity.HoldsIn","parent":3126,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/index.html#holds-in","scope":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups","status":"fully_proved","summary":"Let $q$ be a quasi-identity with unknowns $x_0, \\dots, x_{n-1}$, premises $w_1, \\dots, w_k$ and conclusion $w$, all in the free group $F_n$ on $x_0, \\dots, x_{n-1}$. For $g_0, \\dots, g_{n-1}$ in a group $G$ and $u \\in F_n$, write $u(g_0, \\dots, g_{n-1})$ for the image of $u$ under the homomorphism $F_n \\to G$ sending each $x_j$ to $g_j$. Then $q$ holds in $G$ when for all $g_0, \\dots, g_{n-1} \\in G$ with\u2026","title":"Satisfaction of a quasi-identity in a group"},{"id":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/g-closure","kind":"node","lean":"MathlibExt.GroupTheory.QuasiIdentitiesGeometricEquivalenceKourovka14Problem71Wanted.gClosure","parent":3126,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/index.html#g-closure","scope":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups","status":"fully_proved","summary":"Let $G$ be a group, let $F_n$ be the free group on the basis $x_0, \\dots, x_{n-1}$, and let $T \\subseteq F_n$. The $G$-closure $\\mathrm{cl}_G(T)$ is the set of $w \\in F_n$ such that $\\mu(w) = 1$ for every homomorphism $\\mu : F_n \\to G$ with $\\mu(t) = 1$ for all $t \\in T$, that is, the intersection of the kernels of all homomorphisms $F_n \\to G$ that vanish on $T$.","title":"Closure of a set of words with respect to a group"},{"id":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/geometrically-equivalent","kind":"node","lean":"MathlibExt.GroupTheory.QuasiIdentitiesGeometricEquivalenceKourovka14Problem71Wanted.GeometricallyEquivalent","parent":3126,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/index.html#geometrically-equivalent","scope":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups","status":"fully_proved","summary":"Two groups $G$ and $H$ are geometrically equivalent (B. I. Plotkin) when $\\mathrm{cl}_G(T) = \\mathrm{cl}_H(T)$ for every natural number $n$ and every subset $T$ of the free group $F_n$ of rank $n$, where $\\mathrm{cl}_G(T)$ is the $G$-closure of $T$: the intersection of the kernels of all homomorphisms $F_n \\to G$ that vanish on $T$.","title":"Geometrically equivalent groups"},{"id":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/same-quasi-identities-implies-geometrica","kind":"node","lean":"MathlibExt.GroupTheory.QuasiIdentitiesGeometricEquivalenceKourovka14Problem71Wanted.sameQuasiIdentitiesImpliesGeometricallyEquivalent","parent":3126,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/index.html#same-quasi-identities-implies-geometrica","scope":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups","status":"fully_proved","summary":"The proposition (Kourovka Problem 14.71, B. I. Plotkin) that any two groups $G$ and $H$ satisfying the same quasi-identities, meaning that every quasi-identity holds in $G$ if and only if it holds in $H$, are geometrically equivalent.","title":"Plotkin's problem on quasi-identities and geometric equivalence"},{"id":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/same-quasi-identities-implies-geometrica-2","kind":"node","lean":"MathlibExt.GroupTheory.QuasiIdentitiesGeometricEquivalenceKourovka14Problem71Wanted.sameQuasiIdentitiesImpliesGeometricallyEquivalent_refuted","parent":3126,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/geometric-equivalence-of-groups/index.html#same-quasi-identities-implies-geometrica-2","scope":"GroupTheory/varieties-and-equations/geometric-equivalence-of-groups","status":"can_prove","summary":"The statement of Kourovka Problem 14.71 is false: there are groups $G$ and $H$ such that every quasi-identity holds in $G$ if and only if it holds in $H$, but $G$ and $H$ are not geometrically equivalent.","title":"Groups with the same quasi-identities need not be geometrically equivalent"},{"data_url":"scopes/GroupTheory/varieties-and-equations/hall-s-verbal-conjectures.json","id":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/index.html","scope":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures","status":"can_prove","title":"Hall's verbal conjectures"},{"id":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/is-relatively-free","kind":"node","lean":"MathlibExt.GroupTheory.HallVerbalFinitenessRelativelyFreeKourovka12Problem31Wanted.IsRelativelyFree","parent":3133,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/index.html#is-relatively-free","scope":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures","status":"fully_proved","summary":"A group $G$ is relatively free if it has a family $(x_i)_{i \\in I}$ of elements that generates $G$ such that for every family $(y_i)_{i \\in I}$ of elements of $G$ there is an endomorphism $\\varphi$ of $G$ with $\\varphi(x_i) = y_i$ for all $i \\in I$.","title":"Relatively free group"},{"id":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/word-values","kind":"node","lean":"MathlibExt.GroupTheory.HallVerbalFinitenessRelativelyFreeKourovka12Problem31Wanted.wordValues","parent":3133,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/index.html#word-values","scope":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures","status":"fully_proved","summary":"For a word $v$ in the free group on $n$ generators and a group $G$, the set of values of $v$ on $G$ is the set of elements $v(g_1, \\dots, g_n)$ obtained by substituting elements $g_1, \\dots, g_n \\in G$ for the generators, that is, the set of images of $v$ under all homomorphisms from that free group to $G$.","title":"Values of a word"},{"id":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/relatively-free-finite-word-values-finit","kind":"node","lean":"MathlibExt.GroupTheory.HallVerbalFinitenessRelativelyFreeKourovka12Problem31Wanted.relativelyFreeFiniteWordValuesFiniteVerbal","parent":3133,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/index.html#relatively-free-finite-word-values-finit","scope":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures","status":"fully_proved","summary":"The statement that for every relatively free group $G$, every $n \\in \\mathbb{N}$ and every word $v$ in the free group on $n$ generators, if $v$ takes only finitely many values on $G$, then the verbal subgroup $v(G)$ of $G$ generated by these values is finite.","title":"Hall's verbal finiteness conjecture for relatively free groups (Kourovka 12.31)"},{"id":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/relatively-free-finite-word-values-finit-2","kind":"node","lean":"MathlibExt.GroupTheory.HallVerbalFinitenessRelativelyFreeKourovka12Problem31Wanted.relativelyFreeFiniteWordValuesFiniteVerbal_refuted","parent":3133,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/hall-s-verbal-conjectures/index.html#relatively-free-finite-word-values-finit-2","scope":"GroupTheory/varieties-and-equations/hall-s-verbal-conjectures","status":"can_prove","summary":"Hall's verbal finiteness conjecture for relatively free groups is false: there are a relatively free group $G$, a natural number $n$ and a word $v$ in the free group on $n$ generators such that $v$ takes only finitely many values on $G$ but the verbal subgroup $v(G)$ of $G$ generated by these values is infinite.","title":"Hall's conjecture fails for relatively free groups"},{"data_url":"scopes/GroupTheory/varieties-and-equations/lattices-of-quasivarieties.json","id":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/lattices-of-quasivarieties/index.html","scope":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties","status":"can_prove","title":"Lattices of quasivarieties"},{"id":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties/in-quasivariety","kind":"node","lean":"MathlibExt.GroupTheory.FinitePGroupQuasivarietiesSublatticeKourovka10Problem9Wanted.InQuasivariety","parent":3138,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/lattices-of-quasivarieties/index.html#in-quasivariety","scope":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties","status":"fully_proved","summary":"For a group $K$, a group $X$ lies in the class $Q(K)$ when $X$ embeds in a direct power of $K$: there are a set $I$ and an injective homomorphism $X \\to K^I$.","title":"Quasivariety generated by a finite group"},{"id":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties/in-quasivariety-join","kind":"node","lean":"MathlibExt.GroupTheory.FinitePGroupQuasivarietiesSublatticeKourovka10Problem9Wanted.InQuasivarietyJoin","parent":3138,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/lattices-of-quasivarieties/index.html#in-quasivariety-join","scope":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties","status":"fully_proved","summary":"For groups $G$ and $H$, a group $X$ lies in the class $Q(G, H)$ when $X$ embeds in the product of a direct power of $G$ and a direct power of $H$: there are sets $I$ and $J$ and an injective homomorphism $X \\to G^I \\times H^J$.","title":"Join of the quasivarieties generated by two finite groups"},{"id":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties/finite-pgroup-quasivarieties-sublattice","kind":"node","lean":"MathlibExt.GroupTheory.FinitePGroupQuasivarietiesSublatticeKourovka10Problem9Wanted.finitePGroupQuasivarietiesSublattice","parent":3138,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka Problem 10.9 (Budkin): quasivarieties generated by finite "],["math","\\(p\\)"],["text","-groups form a sublattice"]],"source":"Kourovka Problem 10.9 (Budkin): quasivarieties generated by finite $p$-groups form a sublattice"},"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/lattices-of-quasivarieties/index.html#finite-pgroup-quasivarieties-sublattice","scope":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties","status":"fully_proved","summary":"Kourovka Problem 10.9 (Budkin): quasivarieties generated by finite $p$-groups form a sublattice. For every prime $p$ and all finite $p$-groups $G$ and $H$, there is a finite $p$-group $K$ such that a group lies in the quasivariety generated by $K$ exactly when it lies in both the quasivariety generated by $G$ and the quasivariety generated by $H$, and there is a finite $p$-group $K'$ such that a group lies in the\u2026","title":"Kourovka Problem 10.9 (Budkin): quasivarieties generated by finite $p$-groups form a sublattice"},{"id":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties/finite-pgroup-quasivarieties-sublattice-2","kind":"node","lean":"MathlibExt.GroupTheory.FinitePGroupQuasivarietiesSublatticeKourovka10Problem9Wanted.finitePGroupQuasivarietiesSublattice_refuted","parent":3138,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quasivarieties generated by finite "],["math","\\(p\\)"],["text","-groups need not form a sublattice"]],"source":"Quasivarieties generated by finite $p$-groups need not form a sublattice"},"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/lattices-of-quasivarieties/index.html#finite-pgroup-quasivarieties-sublattice-2","scope":"GroupTheory/varieties-and-equations/lattices-of-quasivarieties","status":"can_prove","summary":"For some prime $p$ there are finite $p$-groups $G$ and $H$ such that no finite $p$-group generates the intersection of the quasivarieties generated by $G$ and by $H$, or no finite $p$-group generates their join. This answers Kourovka Problem 10.9 (Budkin) in the negative.","title":"Quasivarieties generated by finite $p$-groups need not form a sublattice"},{"data_url":"scopes/GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety.json","id":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/index.html","scope":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety","status":"can_prove","title":"Presentations in the torsion-free quasivariety"},{"id":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/torsion-free-relations","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeQuasivarietyPresentationKourovka18Problem17Wanted.torsionFreeRelations","parent":3143,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/index.html#torsion-free-relations","scope":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety","status":"fully_proved","summary":"Let $n$ be a natural number, let $F_n$ be the free group on the $n$ letters $0, 1, \\dots, n - 1$, and let $S$ be a subset of $F_n$. The relation subgroup of $S$ in the quasivariety of torsion-free groups is the intersection of all normal subgroups $N$ of $F_n$ that contain $S$ and satisfy the following: for every $x \\in F_n$ and every integer $k \\ge 1$, if $x^k \\in N$ then $x \\in N$. Equivalently, it is the least\u2026","title":"Relation subgroup of a torsion-free presentation"},{"id":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/is-torsion-free-finitely-presented","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeQuasivarietyPresentationKourovka18Problem17Wanted.IsTorsionFreeFinitelyPresented","parent":3143,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/index.html#is-torsion-free-finitely-presented","scope":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety","status":"fully_proved","summary":"A group $G$ is finitely presented in the quasivariety of torsion-free groups if there are a natural number $n$, a finite subset $S$ of the free group $F_n$ on $n$ letters, and a surjective homomorphism $f \\colon F_n \\to G$ whose kernel is the relation subgroup of $S$ in the quasivariety of torsion-free groups.","title":"Groups finitely presented in the torsion-free quasivariety"},{"id":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/torsion-free-presented-not-finitely-pres","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeQuasivarietyPresentationKourovka18Problem17Wanted.torsionFreePresentedNotFinitelyPresented","parent":3143,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/index.html#torsion-free-presented-not-finitely-pres","scope":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety","status":"fully_proved","summary":"The statement that there is a torsion-free group $G$ (no element of $G$ other than the identity has finite order) that is finitely presented in the quasivariety of torsion-free groups but is not a finitely presented group.","title":"Kourovka Problem 18.17 (statement)"},{"id":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/torsion-free-presented-not-finitely-pres-2","kind":"node","lean":"MathlibExt.GroupTheory.TorsionFreeQuasivarietyPresentationKourovka18Problem17Wanted.torsionFreePresentedNotFinitelyPresented_holds","parent":3143,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety/index.html#torsion-free-presented-not-finitely-pres-2","scope":"GroupTheory/varieties-and-equations/presentations-in-the-torsion-free-quasivariety","status":"can_prove","summary":"There is a torsion-free group $G$ (no element of $G$ other than the identity has finite order) that is finitely presented in the quasivariety of torsion-free groups but is not a finitely presented group.","title":"Budkin's theorem (Kourovka Problem 18.17)"},{"data_url":"scopes/GroupTheory/varieties-and-equations/quasivarieties-of-groups.json","id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"can_prove","title":"Quasivarieties of groups"},{"id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups/quasi-identity","kind":"node","lean":"MathlibExt.GroupTheory.QuasivarietyMaximalSubquasivarietiesKourovka14Problem25Wanted.QuasiIdentity","parent":3148,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html#quasi-identity","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"fully_proved","summary":"A quasi-identity in $n$ unknowns $x_0, \\dots, x_{n-1}$ is a formula $\\forall x\\,(w_1 = 1 \\wedge \\dots \\wedge w_k = 1 \\to w = 1)$, given by the number $n$, a finite list of premises $w_1, \\dots, w_k$ and a conclusion $w$, all of them elements of the free group on $x_0, \\dots, x_{n-1}$.","title":"Quasi-identities of groups"},{"id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups/holds-in","kind":"node","lean":"MathlibExt.GroupTheory.QuasivarietyMaximalSubquasivarietiesKourovka14Problem25Wanted.QuasiIdentity.HoldsIn","parent":3148,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html#holds-in","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"fully_proved","summary":"A quasi-identity $\\forall x\\,(w_1 = 1 \\wedge \\dots \\wedge w_k = 1 \\to w = 1)$ in the unknowns $x_0, \\dots, x_{n-1}$ holds in a group $G$ when every assignment of elements of $G$ to the unknowns under which each premise $w_1, \\dots, w_k$ evaluates to $1$ in $G$ also makes the conclusion $w$ evaluate to $1$.","title":"Quasi-identities holding in a group"},{"id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups/models","kind":"node","lean":"MathlibExt.GroupTheory.QuasivarietyMaximalSubquasivarietiesKourovka14Problem25Wanted.Models","parent":3148,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html#models","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"fully_proved","summary":"For a set $Q$ of quasi-identities, a group $H$ belongs to the quasivariety defined by $Q$ when every quasi-identity in $Q$ holds in $H$. Every quasivariety of groups has this form.","title":"The quasivariety defined by a set of quasi-identities"},{"id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups/quasi-theory","kind":"node","lean":"MathlibExt.GroupTheory.QuasivarietyMaximalSubquasivarietiesKourovka14Problem25Wanted.quasiTheory","parent":3148,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html#quasi-theory","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"fully_proved","summary":"The quasi-theory of a group $G$ is the set of all quasi-identities that hold in $G$. The quasivariety defined by this set is the quasivariety $qG$ generated by $G$.","title":"The quasi-identities of a group"},{"id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups/qvle","kind":"node","lean":"MathlibExt.GroupTheory.QuasivarietyMaximalSubquasivarietiesKourovka14Problem25Wanted.QVLe","parent":3148,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html#qvle","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"fully_proved","summary":"For sets $Q_1$ and $Q_2$ of quasi-identities, the quasivariety defined by $Q_1$ is contained in the quasivariety defined by $Q_2$: every group in which all quasi-identities of $Q_1$ hold also satisfies all quasi-identities of $Q_2$.","title":"Inclusion of quasivarieties"},{"id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups/is-maximal-proper-subquasivariety","kind":"node","lean":"MathlibExt.GroupTheory.QuasivarietyMaximalSubquasivarietiesKourovka14Problem25Wanted.IsMaximalProperSubquasivariety","parent":3148,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html#is-maximal-proper-subquasivariety","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"fully_proved","summary":"Let $G$ be a group and let $qG$ be the quasivariety generated by $G$, that is, the quasivariety defined by the set of all quasi-identities that hold in $G$. A set $Q$ of quasi-identities defines a maximal proper subquasivariety of $qG$ when the quasivariety defined by $Q$ is contained in $qG$ but does not contain $qG$, and every quasivariety $V$ defined by a set of quasi-identities with $V$ containing the\u2026","title":"Maximal proper subquasivarieties"},{"id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups/finitely-generated-group-infinitely-many","kind":"node","lean":"MathlibExt.GroupTheory.QuasivarietyMaximalSubquasivarietiesKourovka14Problem25Wanted.finitelyGeneratedGroupInfinitelyManyMaximalSubquasivarieties","parent":3148,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html#finitely-generated-group-infinitely-many","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"fully_proved","summary":"The statement that there is a finitely generated group $G$ such that the quasivariety $qG$ generated by $G$ has infinitely many maximal proper subquasivarieties: for every natural number $k$ there are sets of quasi-identities $Q_0, \\dots, Q_{k-1}$, each defining a maximal proper subquasivariety of $qG$, such that for $i \\ne j$ the quasivarieties defined by $Q_i$ and $Q_j$ are different.","title":"Budkin's problem on maximal subquasivarieties (Kourovka 14.25)"},{"id":"GroupTheory/varieties-and-equations/quasivarieties-of-groups/finitely-generated-group-infinitely-many-2","kind":"node","lean":"MathlibExt.GroupTheory.QuasivarietyMaximalSubquasivarietiesKourovka14Problem25Wanted.finitelyGeneratedGroupInfinitelyManyMaximalSubquasivarieties_holds","parent":3148,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/quasivarieties-of-groups/index.html#finitely-generated-group-infinitely-many-2","scope":"GroupTheory/varieties-and-equations/quasivarieties-of-groups","status":"can_prove","summary":"There is a finitely generated group $G$ such that the quasivariety $qG$ generated by $G$ has infinitely many maximal proper subquasivarieties: for every natural number $k$ there are sets of quasi-identities $Q_0, \\dots, Q_{k-1}$, each defining a maximal proper subquasivariety of $qG$, such that for $i \\ne j$ the quasivarieties defined by $Q_i$ and $Q_j$ are different.","title":"Some finitely generated group has infinitely many maximal subquasivarieties"},{"data_url":"scopes/GroupTheory/varieties-and-equations/independent-bases-of-laws.json","id":"GroupTheory/varieties-and-equations/independent-bases-of-laws","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/independent-bases-of-laws/index.html","scope":"GroupTheory/varieties-and-equations/independent-bases-of-laws","status":"fully_proved","title":"Independent bases of laws"},{"id":"GroupTheory/varieties-and-equations/independent-bases-of-laws/law-consequence","kind":"node","lean":"MathlibExt.GroupTheory.GroupLawsShared.LawConsequence","parent":3157,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/independent-bases-of-laws/index.html#law-consequence","scope":"GroupTheory/varieties-and-equations/independent-bases-of-laws","status":"fully_proved","summary":"Let $F$ be the free group on countably many generators $x_0, x_1, x_2, \\dots$, let $W \\subseteq F$ be a set of laws and let $w \\in F$. Then $w$ is a consequence of $W$ if $w$ holds in every group $G$ satisfying all laws in $W$: whenever every homomorphism $F \\to G$ sends each element of $W$ to $1$, every homomorphism $F \\to G$ sends $w$ to $1$.","title":"Consequence of a set of laws"},{"id":"GroupTheory/varieties-and-equations/independent-bases-of-laws/has-independent-law-basis","kind":"node","lean":"MathlibExt.GroupTheory.GroupLawsShared.HasIndependentLawBasis","parent":3157,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/independent-bases-of-laws/index.html#has-independent-law-basis","scope":"GroupTheory/varieties-and-equations/independent-bases-of-laws","status":"fully_proved","summary":"Let $F$ be the free group on countably many generators $x_0, x_1, x_2, \\dots$ and let $W \\subseteq F$ be a set of laws. The variety defined by $W$ has an independent basis of identities if there is a set $B \\subseteq F$ of laws such that a law is a consequence of $B$ exactly when it is a consequence of $W$, and no $b \\in B$ is a consequence of $B \\setminus \\{b\\}$.","title":"Independent basis of laws"},{"id":"GroupTheory/varieties-and-equations/independent-bases-of-laws/quasi-identity-consequence","kind":"node","lean":"MathlibExt.GroupTheory.GroupLawsShared.QuasiIdentityConsequence","parent":3157,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/independent-bases-of-laws/index.html#quasi-identity-consequence","scope":"GroupTheory/varieties-and-equations/independent-bases-of-laws","status":"fully_proved","summary":"Let $Q$ be a set of quasi-identities and $q$ a quasi-identity. Then $q$ is a consequence of $Q$ if $q$ holds in every group in which all members of $Q$ hold.","title":"Consequence of a set of quasi-identities"},{"id":"GroupTheory/varieties-and-equations/independent-bases-of-laws/has-independent-quasi-identity-basis","kind":"node","lean":"MathlibExt.GroupTheory.GroupLawsShared.HasIndependentQuasiIdentityBasis","parent":3157,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/independent-bases-of-laws/index.html#has-independent-quasi-identity-basis","scope":"GroupTheory/varieties-and-equations/independent-bases-of-laws","status":"fully_proved","summary":"The quasi-identities of a group $G$ have an independent basis if there is a set $B$ of quasi-identities such that a quasi-identity is a consequence of $B$ exactly when it holds in $G$, and no $b \\in B$ is a consequence of $B \\setminus \\{b\\}$.","title":"Independent basis of quasi-identities"},{"data_url":"scopes/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups.json","id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"can_prove","title":"Test ranks of relatively free groups"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/satisfies-laws","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.SatisfiesLaws","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#satisfies-laws","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"fully_proved","summary":"Let $F_\\mathbb{N}$ be the free group on countably many generators and $W \\subseteq F_\\mathbb{N}$ a set of words, the laws. A group $G$ satisfies the laws $W$ when $\\varphi(w) = 1$ for every $w \\in W$ and every homomorphism $\\varphi \\colon F_\\mathbb{N} \\to G$, that is, every law takes the value $1$ under every substitution of elements of $G$ for the variables. The variety defined by $W$ is the class of these groups.","title":"Groups satisfying a set of laws in countably many variables"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/verbal-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.verbalSubgroup","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#verbal-subgroup","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"fully_proved","summary":"Let $W$ be a set of laws in the free group $F_\\mathbb{N}$ on countably many generators, and let $F_r$ be the free group on $r$ generators. The verbal subgroup $V_W(F_r)$ of $F_r$ for $W$ is the normal closure in $F_r$ of all elements $\\varphi(w)$ with $w \\in W$ and $\\varphi \\colon F_\\mathbb{N} \\to F_r$ a homomorphism.","title":"Verbal subgroup of a free group of finite rank"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/relatively-free-group","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.RelativelyFreeGroup","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#relatively-free-group","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"fully_proved","summary":"Let $W$ be a set of laws in the free group $F_\\mathbb{N}$ on countably many generators, and let $r \\in \\mathbb{N}$. The free $r$-generator group $G_r$ of the variety defined by $W$ is the quotient $F_r / V_W(F_r)$ of the free group $F_r$ on $r$ generators by its verbal subgroup $V_W(F_r)$ for $W$.","title":"Relatively free group as a quotient of a free group"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/is-test-set","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.IsTestSet","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#is-test-set","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"fully_proved","summary":"A subset $S$ of a group $G$ is a test set when every endomorphism of $G$ that fixes every element of $S$ is bijective, that is, an automorphism of $G$.","title":"Test set of a group"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/has-test-rank","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.HasTestRank","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#has-test-rank","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"fully_proved","summary":"A group $G$ has test rank $r \\in \\mathbb{N}$ when $G$ has a test set of cardinality $r$ and every test set of $G$ has cardinality at least $r$ (an infinite set counting as having cardinality $\\infty$), that is, when the minimum of the cardinalities of the test sets of $G$ is $r$.","title":"Test rank of a group"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/test-rank-variety-abelian","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.testRankVarietyAbelian","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#test-rank-variety-abelian","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"fully_proved","summary":"The statement that for every set $W$ of laws in the free group $F_\\mathbb{N}$ on countably many generators, if for every $r \\ge 1$ the free $r$-generator group $G_r$ of the variety defined by $W$ has test rank $r$, then this variety is abelian: every group satisfying the laws $W$ is abelian.","title":"Timoshenko's test rank question, part (a)"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/test-rank-nonperiodic-variety-all-abelia","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.testRankNonperiodicVarietyAllAbelian","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#test-rank-nonperiodic-variety-all-abelia","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"fully_proved","summary":"The statement that for every set $W$ of laws in the free group $F_\\mathbb{N}$ on countably many generators, if for every $r \\ge 1$ the free $r$-generator group $G_r$ of the variety defined by $W$ has test rank $r$ and the variety is not periodic (there is no $n > 0$ with $g^n = 1$ for every element $g$ of every group satisfying $W$), then the variety is the variety of all abelian groups: a group satisfies the laws\u2026","title":"Timoshenko's test rank question, part (b)"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/test-rank-variety-abelian-refuted","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.testRankVarietyAbelian_refuted","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#test-rank-variety-abelian-refuted","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"can_prove","summary":"There is a set $W$ of laws in the free group $F_\\mathbb{N}$ on countably many generators such that for every $r \\ge 1$ the free $r$-generator group $G_r$ of the variety defined by $W$ has test rank $r$, but the variety is not abelian: some group satisfying the laws $W$ is not abelian.","title":"Negative answer to Timoshenko's test rank question, part (a)"},{"id":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/test-rank-nonperiodic-variety-all-abelia-2","kind":"node","lean":"MathlibExt.GroupTheory.TestRankNonabelianVarietyKourovka16Problem87Wanted.testRankNonperiodicVarietyAllAbelian_refuted","parent":3162,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups/index.html#test-rank-nonperiodic-variety-all-abelia-2","scope":"GroupTheory/varieties-and-equations/test-ranks-of-relatively-free-groups","status":"can_prove","summary":"Resolved false: the same variety (`[[x, y], z] = 1`, `[x, y]^p = 1`, `p` prime) is not periodic since it contains `\u2124`, its free groups of every rank `r \u2265 1` have test rank `r` (Kinyon-Urban, Proposition 3.9), and it is not the variety of all abelian groups since it contains the non-abelian group `UT\u2083(\ud835\udd3d_p)`. The Kourovka Notebook (21st issue) answers 16.87(b) \"No\" (AI solution confirmed by E. I. Timoshenko). Source:\u2026","title":"Non-periodic varieties with full test rank need not be abelian"},{"data_url":"scopes/GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v.json","id":"GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v","kind":"scope","navigation":"module","parent":3088,"result":false,"route":"../roadmap/GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v/index.html","scope":"GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v","status":"can_prove","title":"Torsion groups of unbounded exponent in proper varieties"},{"id":"GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v/torsion-unbounded-exponent-proper-variet","kind":"node","lean":"MathlibExt.GroupTheory.TorsionUnboundedExponentProperVarietyKourovka17Problem79Wanted.torsionUnboundedExponentProperVariety","parent":3172,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v/index.html#torsion-unbounded-exponent-proper-variet","scope":"GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v","status":"fully_proved","summary":"The statement, asked by O. Macedo\u0144ska, that there exists a finitely generated group $G$ in which every element has finite order, which has unbounded exponent (there is no $n \\ge 1$ with $g^n = 1$ for all $g \\in G$), and which satisfies a nontrivial law, that is, some element $w \\ne 1$ of the free group $F$ on countably infinitely many generators is mapped to $1$ by every homomorphism $F \\to G$ (equivalently, $G$\u2026","title":"Kourovka Problem 17.79 (torsion groups of unbounded exponent in proper varieties)"},{"id":"GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v/torsion-unbounded-exponent-proper-variet-2","kind":"node","lean":"MathlibExt.GroupTheory.TorsionUnboundedExponentProperVarietyKourovka17Problem79Wanted.torsionUnboundedExponentProperVariety_holds","parent":3172,"result":true,"route":"../roadmap/GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v/index.html#torsion-unbounded-exponent-proper-variet-2","scope":"GroupTheory/varieties-and-equations/torsion-groups-of-unbounded-exponent-in-proper-v","status":"can_prove","summary":"Resolved true: the Kourovka Archive answers 17.79 \"Yes, moreover, there is a continuum of such groups\" (Atabekyan, 1986), and records Osin's example: a free group of the variety defined by `x\u207f y = y x\u207f` is a central extension of a free Burnside group of exponent `n` with free abelian centre of countable rank (Ashmanov-Olshanskii), and adding the relations `x\u1d62^i = 1` for a basis `(x\u1d62)` of the centre gives a periodic\u2026","title":"Periodic groups of unbounded exponent in a proper variety"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions.json","id":"GroupTheory/commutators-and-engel-conditions","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/index.html","scope":"GroupTheory/commutators-and-engel-conditions","status":"can_prove","summary":"Commutator calculus and nilpotency in (often infinite) groups: basic commutators, Engel elements, locally nilpotent and hypercentral groups, metabelian and polycyclic groups, localization of nilpotent groups, commutator width and length, word maps and their images, and local word identities.","title":"Commutators, word maps and Engel conditions"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/basic-commutators.json","id":"GroupTheory/commutators-and-engel-conditions/basic-commutators","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/basic-commutators/index.html","scope":"GroupTheory/commutators-and-engel-conditions/basic-commutators","status":"can_prove","title":"Basic commutators"},{"id":"GroupTheory/commutators-and-engel-conditions/basic-commutators/bracket","kind":"node","lean":"MathlibExt.GroupTheory.BasicCommutatorEqualityKourovka11Problem20Wanted.Bracket","parent":3176,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/basic-commutators/index.html#bracket","scope":"GroupTheory/commutators-and-engel-conditions/basic-commutators","status":"fully_proved","summary":"The formal commutators (bracket arrangements) in an alphabet $\\alpha$ are the formal expressions generated by two rules: every letter $a \\in \\alpha$ is a formal commutator, and for formal commutators $u$ and $v$, the bracket $[u, v]$ is a formal commutator.","title":"Formal commutators"},{"id":"GroupTheory/commutators-and-engel-conditions/basic-commutators/weight","kind":"node","lean":"MathlibExt.GroupTheory.BasicCommutatorEqualityKourovka11Problem20Wanted.Bracket.weight","parent":3176,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/basic-commutators/index.html#weight","scope":"GroupTheory/commutators-and-engel-conditions/basic-commutators","status":"fully_proved","summary":"The weight of a formal commutator is the number of occurrences of letters in it: a letter has weight $1$, and $[u, v]$ has weight the weight of $u$ plus the weight of $v$.","title":"Weight of a formal commutator"},{"id":"GroupTheory/commutators-and-engel-conditions/basic-commutators/kcomm","kind":"node","lean":"MathlibExt.GroupTheory.BasicCommutatorEqualityKourovka11Problem20Wanted.kcomm","parent":3176,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/basic-commutators/index.html#kcomm","scope":"GroupTheory/commutators-and-engel-conditions/basic-commutators","status":"fully_proved","summary":"For elements $x$ and $y$ of a group, their commutator is $[x, y] = x^{-1} y^{-1} x y$ (the convention of the Kourovka Notebook and of Hall).","title":"Commutator, Kourovka\u2013Hall convention"},{"id":"GroupTheory/commutators-and-engel-conditions/basic-commutators/eval","kind":"node","lean":"MathlibExt.GroupTheory.BasicCommutatorEqualityKourovka11Problem20Wanted.Bracket.eval","parent":3176,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/basic-commutators/index.html#eval","scope":"GroupTheory/commutators-and-engel-conditions/basic-commutators","status":"fully_proved","summary":"Each formal commutator in the letters $\\alpha$ has a value in the free group $F(\\alpha)$ on $\\alpha$: a letter $a$ has the value of the generator $a$, and $[u, v]$ has the value $\\bar u^{-1} \\bar v^{-1} \\bar u \\bar v$, where $\\bar u$ and $\\bar v$ are the values of $u$ and $v$.","title":"Value of a formal commutator"},{"id":"GroupTheory/commutators-and-engel-conditions/basic-commutators/is-basic","kind":"node","lean":"MathlibExt.GroupTheory.BasicCommutatorEqualityKourovka11Problem20Wanted.Bracket.IsBasic","parent":3176,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/basic-commutators/index.html#is-basic","scope":"GroupTheory/commutators-and-engel-conditions/basic-commutators","status":"fully_proved","summary":"Let $<$ be a relation on the formal commutators in the letters $\\alpha$ (in applications, a linear order). Every letter is basic for $<$, and a formal commutator $[u, v]$ is basic for $<$ when $u$ and $v$ are basic for $<$, $v < u$, and, in case $u = [y, z]$ is itself a bracket, $v < z$ fails (for a linear order: $z \\le v$). This is M. Hall's definition.","title":"Basic commutators"},{"id":"GroupTheory/commutators-and-engel-conditions/basic-commutators/basic-commutator-equality-determines","kind":"node","lean":"MathlibExt.GroupTheory.BasicCommutatorEqualityKourovka11Problem20Wanted.basicCommutatorEqualityDetermines","parent":3176,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/basic-commutators/index.html#basic-commutator-equality-determines","scope":"GroupTheory/commutators-and-engel-conditions/basic-commutators","status":"fully_proved","summary":"The proposition (Kourovka Problem 11.20, A. Gaglione and D. Spellman) that the following holds. Write $[x, y] = x^{-1} y^{-1} x y$. For every linearly ordered type $\\alpha$ and every linear order $<$ on the formal commutators in the letters $\\alpha$ that orders the letters as in $\\alpha$ and puts formal commutators of smaller weight first (within one weight it is arbitrary), if $a$ and $b$ are formal commutators\u2026","title":"Gaglione\u2013Spellman claim on equal commutators of basic commutators"},{"id":"GroupTheory/commutators-and-engel-conditions/basic-commutators/basic-commutator-equality-determines-ref","kind":"node","lean":"MathlibExt.GroupTheory.BasicCommutatorEqualityKourovka11Problem20Wanted.basicCommutatorEqualityDetermines_refuted","parent":3176,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/basic-commutators/index.html#basic-commutator-equality-determines-ref","scope":"GroupTheory/commutators-and-engel-conditions/basic-commutators","status":"can_prove","summary":"The Gaglione\u2013Spellman claim on equal commutators of basic commutators (Kourovka Problem 11.20) is false: writing $[x, y] = x^{-1} y^{-1} x y$, there are a linearly ordered type $\\alpha$, a linear order $<$ on the formal commutators in the letters $\\alpha$ that orders the letters as in $\\alpha$ and puts formal commutators of smaller weight first, formal commutators $a$ and $b$ such that the formal commutator\u2026","title":"Equal commutators of basic commutators do not determine their entries"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/commutator-power-identities.json","id":"GroupTheory/commutators-and-engel-conditions/commutator-power-identities","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-power-identities/index.html","scope":"GroupTheory/commutators-and-engel-conditions/commutator-power-identities","status":"can_prove","title":"Commutator power identities"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-power-identities/commutator-identity-periodic-derived","kind":"node","lean":"MathlibExt.GroupTheory.CommutatorPowerPeriodicDerivedKourovka13Problem34Wanted.commutatorIdentityPeriodicDerived","parent":3184,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-power-identities/index.html#commutator-identity-periodic-derived","scope":"GroupTheory/commutators-and-engel-conditions/commutator-power-identities","status":"fully_proved","summary":"This is the proposition that for every integer $n \\ge 1$ and every group $G$ with $[x, y]^n = 1$ for all $x, y \\in G$, every element of the derived subgroup $[G, G]$ has finite order, that is, the derived subgroup of $G$ is periodic.","title":"Derived subgroups under a commutator power identity (Kourovka 13.34)"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-power-identities/commutator-identity-periodic-derived-ref","kind":"node","lean":"MathlibExt.GroupTheory.CommutatorPowerPeriodicDerivedKourovka13Problem34Wanted.commutatorIdentityPeriodicDerived_refuted","parent":3184,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-power-identities/index.html#commutator-identity-periodic-derived-ref","scope":"GroupTheory/commutators-and-engel-conditions/commutator-power-identities","status":"can_prove","summary":"There exist an integer $n \\ge 1$ and a group $G$ such that $[x, y]^n = 1$ for all $x, y \\in G$ but some element of the derived subgroup $[G, G]$ has infinite order. This answers Kourovka Notebook Problem 13.34 (I. D. Macdonald; V. D. Mazurov) in the negative, by results of G. S. Deryabina and P. A. Kozhevnikov from 1999 and of S. I. Adyan from 2000.","title":"A commutator power identity need not make the derived subgroup periodic"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/commutator-width.json","id":"GroupTheory/commutators-and-engel-conditions/commutator-width","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"can_prove","title":"Commutator width"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/is-product-of-commutators","kind":"node","lean":"MathlibExt.GroupTheory.AutFreeGroupCommutatorLengthKourovka14Problem15Wanted.IsProductOfCommutators","parent":3187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Products of at most "],["math","\\(n\\)"],["text"," commutators (Kourovka 14.15)"]],"source":"Products of at most $n$ commutators (Kourovka 14.15)"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#is-product-of-commutators","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"Let $G$ be a group and $n$ a natural number. An element $z \\in G$ is a product of at most $n$ commutators when there are $m \\le n$ and elements $x_1, y_1, \\dots, x_m, y_m \\in G$ with $z = [x_1, y_1] \\cdots [x_m, y_m]$, where $[x, y] = xyx^{-1}y^{-1}$; for $m = 0$ the empty product is the identity.","title":"Products of at most $n$ commutators (Kourovka 14.15)"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/aut-free-group-commutator-length-unbound","kind":"node","lean":"MathlibExt.GroupTheory.AutFreeGroupCommutatorLengthKourovka14Problem15Wanted.autFreeGroupCommutatorLengthUnbounded","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#aut-free-group-commutator-length-unbound","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"The proposition that for every $n \\ge 3$, writing $F_n$ for the free group on $n$ generators, and for every natural number $k$, some element of the derived subgroup of $\\operatorname{Aut}(F_n)$ is not a product of at most $k$ commutators $xyx^{-1}y^{-1}$ of elements of $\\operatorname{Aut}(F_n)$. This is the answer $k_n = \\infty$ to Bardakov's question, which asks for the supremum $k_n$ of the commutator lengths of\u2026","title":"Unbounded commutator length in Aut(F\u2099) (Kourovka 14.15)"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/aut-free-group-commutator-length-unbound-2","kind":"node","lean":"MathlibExt.GroupTheory.AutFreeGroupCommutatorLengthKourovka14Problem15Wanted.autFreeGroupCommutatorLengthUnbounded_holds","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#aut-free-group-commutator-length-unbound-2","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"can_prove","summary":"Let $n \\ge 3$ and let $F_n$ be the free group on $n$ generators. For every natural number $k$ there is an element of the derived subgroup of $\\operatorname{Aut}(F_n)$ that is not a product of at most $k$ commutators $xyx^{-1}y^{-1}$ with $x, y \\in \\operatorname{Aut}(F_n)$. Equivalently, the supremum $k_n$ of the commutator lengths of the elements of the derived subgroup of $\\operatorname{Aut}(F_n)$ is infinite.","title":"Commutator length in Aut(F\u2099) is unbounded"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/is-product-of-commutators-2","kind":"node","lean":"MathlibExt.GroupTheory.CommutatorWidthRatioKourovka16Problem8Wanted.IsProductOfCommutators","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#is-product-of-commutators-2","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"For a group $G$, a natural number $m$ and $g \\in G$, $g$ is a product of at most $m$ commutators when there are $k \\le m$ and $a_1, b_1, \\dots, a_k, b_k \\in G$ with $g = [a_1, b_1] \\cdots [a_k, b_k]$, where $[a, b] = aba^{-1}b^{-1}$; for $k = 0$ the empty product is the identity.","title":"Products of a bounded number of commutators"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/commutator-width","kind":"node","lean":"MathlibExt.GroupTheory.CommutatorWidthRatioKourovka16Problem8Wanted.commutatorWidth","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#commutator-width","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"For a group $G$ with derived subgroup $G'$, the commutator width $w(G')$ is the least positive integer $m$ such that every element of $G'$ is a product of at most $m$ commutators $aba^{-1}b^{-1}$ with $a, b \\in G$; it is $0$ when no such $m$ exists.","title":"Commutator width of the derived subgroup"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/max-commutator-width-ratio-is-one-sixth","kind":"node","lean":"MathlibExt.GroupTheory.CommutatorWidthRatioKourovka16Problem8Wanted.maxCommutatorWidthRatioIsOneSixth","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#max-commutator-width-ratio-is-one-sixth","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"The proposition that for every finite non-abelian group $G$ the commutator width $w(G')$ of its derived subgroup $G'$ satisfies $w(G')/|G| \\le 1/6$, and that for the symmetric group $S_3$ equality holds: $w(S_3')/|S_3| = 1/6$.","title":"Bardakov's commutator width ratio claim (Kourovka 16.8)"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/max-commutator-width-ratio-is-one-sixth-2","kind":"node","lean":"MathlibExt.GroupTheory.CommutatorWidthRatioKourovka16Problem8Wanted.maxCommutatorWidthRatioIsOneSixth_holds","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#max-commutator-width-ratio-is-one-sixth-2","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"can_prove","summary":"For every finite non-abelian group $G$, the commutator width $w(G')$ of its derived subgroup satisfies $w(G')/|G| \\le 1/6$, and for the symmetric group $S_3$ equality holds, $w(S_3')/|S_3| = 1/6$; so Kourovka Problem 16.8 has a positive answer.","title":"The maximal commutator width ratio is one sixth"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/is-product-of-commutators-3","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupUnboundedCommutatorLengthKourovka14Problem13Wanted.IsProductOfCommutators","parent":3187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Products of at most "],["math","\\(n\\)"],["text"," commutators"]],"source":"Products of at most $n$ commutators"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#is-product-of-commutators-3","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"An element $z$ of a group $G$ is a product of at most $n$ commutators when $z = [x_1, y_1] \\cdots [x_k, y_k]$ for some $k \\le n$ and some $x_1, y_1, \\dots, x_k, y_k \\in G$, where $[x, y] = x y x^{-1} y^{-1}$; for $k = 0$ the product is the identity.","title":"Products of at most $n$ commutators"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/commutator-length-unbounded","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupUnboundedCommutatorLengthKourovka14Problem13Wanted.CommutatorLengthUnbounded","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#commutator-length-unbounded","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"A group $G$ has unbounded commutator length when for every natural number $n$ some element of the derived subgroup $[G, G]$ is not a product of at most $n$ commutators $x y x^{-1} y^{-1}$ with $x, y \\in G$.","title":"Unbounded commutator length"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/simple-group-unbounded-commutator-length","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupUnboundedCommutatorLengthKourovka14Problem13Wanted.simpleGroupUnboundedCommutatorLength","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#simple-group-unbounded-commutator-length","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"The statement that there is a simple group $G$ such that for every natural number $n$ some element of the derived subgroup $[G, G]$ is not a product of at most $n$ commutators $x y x^{-1} y^{-1}$ with $x, y \\in G$.","title":"Kourovka Problem 14.13(a) (Bardakov)"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/finitely-presented-simple-group-unbounde","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupUnboundedCommutatorLengthKourovka14Problem13Wanted.finitelyPresentedSimpleGroupUnboundedCommutatorLength","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#finitely-presented-simple-group-unbounde","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"fully_proved","summary":"The statement that there is a finitely presented simple group $G$ such that for every natural number $n$ some element of the derived subgroup $[G, G]$ is not a product of at most $n$ commutators $x y x^{-1} y^{-1}$ with $x, y \\in G$.","title":"Kourovka Problem 14.13(b) (Bardakov)"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/simple-group-unbounded-commutator-length-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupUnboundedCommutatorLengthKourovka14Problem13Wanted.simpleGroupUnboundedCommutatorLength_holds","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#simple-group-unbounded-commutator-length-2","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"can_prove","summary":"There is a simple group $G$ such that for every natural number $n$ some element of the derived subgroup $[G, G]$ is not a product of at most $n$ commutators $x y x^{-1} y^{-1}$ with $x, y \\in G$. This answers Kourovka Problem 14.13(a) in the affirmative; the Kourovka Archive records examples constructed by Barge and Ghys, and finitely generated ones by Muranov.","title":"A simple group with unbounded commutator length"},{"id":"GroupTheory/commutators-and-engel-conditions/commutator-width/finitely-presented-simple-group-unbounde-2","kind":"node","lean":"MathlibExt.GroupTheory.SimpleGroupUnboundedCommutatorLengthKourovka14Problem13Wanted.finitelyPresentedSimpleGroupUnboundedCommutatorLength_holds","parent":3187,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/commutator-width/index.html#finitely-presented-simple-group-unbounde-2","scope":"GroupTheory/commutators-and-engel-conditions/commutator-width","status":"can_prove","summary":"There is a finitely presented simple group $G$ such that for every natural number $n$ some element of the derived subgroup $[G, G]$ is not a product of at most $n$ commutators $x y x^{-1} y^{-1}$ with $x, y \\in G$. This answers Kourovka Problem 14.13(b) in the affirmative; the Kourovka Archive cites Caprace and Fujiwara.","title":"A finitely presented simple group with unbounded commutator length"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions.json","id":"GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions/index.html","scope":"GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions","status":"can_prove","title":"Engel conditions on involutions"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions/is-involution-engel-two-group","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionEngelTwoGroupDerivedLengthKourovka18Problem57Wanted.IsInvolutionEngelTwoGroup","parent":3201,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions/index.html#is-involution-engel-two-group","scope":"GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions","status":"fully_proved","summary":"A group $G$ has this property when every element of $G$ has order a power of $2$, $G$ is generated by its involutions, and $[[x, u], u] = 1$ for every $x \\in G$ and every involution $u \\in G$, where $[x, u] = x^{-1} u^{-1} x u$.","title":"Involution-generated 2-groups with the Engel condition on involutions"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions/involution-engel-two-groups-bounded-deri","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionEngelTwoGroupDerivedLengthKourovka18Problem57Wanted.involutionEngelTwoGroupsBoundedDerivedLength","parent":3201,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions/index.html#involution-engel-two-groups-bounded-deri","scope":"GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions","status":"fully_proved","summary":"The proposition (Kourovka 18.57, D. V. Lytkina) that there is a natural number $d$ such that $G^{(d)} = 1$ for every finite group $G$ in which every element has order a power of $2$, which is generated by its involutions, and in which $[[x, u], u] = 1$ for every $x \\in G$ and every involution $u \\in G$; here $[x, u] = x^{-1} u^{-1} x u$ and $G^{(d)}$ is the $d$-th term of the derived series of $G$.","title":"Kourovka Problem 18.57 on the derived length of involution-generated 2-groups"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions/involution-engel-two-groups-bounded-deri-2","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionEngelTwoGroupDerivedLengthKourovka18Problem57Wanted.involutionEngelTwoGroupsBoundedDerivedLength_refuted","parent":3201,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions/index.html#involution-engel-two-groups-bounded-deri-2","scope":"GroupTheory/commutators-and-engel-conditions/engel-conditions-on-involutions","status":"can_prove","summary":"There is no natural number $d$ such that $G^{(d)} = 1$ for every finite group $G$ in which every element has order a power of $2$, which is generated by its involutions, and in which $[[x, u], u] = 1$ for every $x \\in G$ and every involution $u \\in G$; here $[x, u] = x^{-1} u^{-1} x u$ and $G^{(d)}$ is the $d$-th term of the derived series of $G$. Thus the derived length of such groups is unbounded.","title":"Negative answer to Kourovka Problem 18.57"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/engel-elements.json","id":"GroupTheory/commutators-and-engel-conditions/engel-elements","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"can_prove","title":"Engel elements"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/kcomm","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementLengthPolynomialKourovka11Problem88Wanted.kcomm","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#kcomm","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"For elements $x$ and $y$ of a group, the commutator is $[x, y] = x^{-1} y^{-1} x y$ (the convention of the Kourovka Notebook).","title":"Commutator for Engel lengths"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/engel-comm","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementLengthPolynomialKourovka11Problem88Wanted.engelComm","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#engel-comm","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"For elements $h$ and $g$ of a group and a natural number $l$, the iterated commutator $[h, g; l]$ is defined by $[h, g; 0] = h$ and $[h, g; l + 1] = [[h, g; l], g]$, where $[x, y] = x^{-1} y^{-1} x y$; in particular $[h, g; 1] = [h, g]$.","title":"Iterated commutators for Engel lengths"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/is-engel","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementLengthPolynomialKourovka11Problem88Wanted.IsEngel","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#is-engel","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"An element $g$ of a group $G$ is an Engel element if there is an $l \\ge 1$ such that $[h, g; l] = 1$ for every $h \\in G$, where $[h, g; 0] = h$, $[h, g; i + 1] = [[h, g; i], g]$ and $[x, y] = x^{-1} y^{-1} x y$.","title":"Bounded Engel element"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/engel-length","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementLengthPolynomialKourovka11Problem88Wanted.engelLength","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#engel-length","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"For an element $g$ of a group $G$, the Engel length $l(g)$ is the least $k \\ge 1$ such that $[h, g; k] = 1$ for every $h \\in G$, where $[h, g; 0] = h$, $[h, g; i + 1] = [[h, g; i], g]$ and $[x, y] = x^{-1} y^{-1} x y$. If no such $k$ exists, that is, if $g$ is not an Engel element, then $l(g) = 0$.","title":"Engel length"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/engel-length-polynomial-bound","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementLengthPolynomialKourovka11Problem88Wanted.engelLengthPolynomialBound","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#engel-length-polynomial-bound","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"The statement that every group $G$ has a polynomial $\\varphi(x, y)$ in two variables with natural-number coefficients such that $l(uv) \\le \\varphi(l(u), l(v))$ for all $u, v \\in G$ for which $u$, $v$ and $uv$ are Engel elements. Here $g \\in G$ is an Engel element if there is a $k \\ge 1$ with $[h, g; k] = 1$ for all $h \\in G$, and its Engel length $l(g)$ is the least such $k$, where $[h, g; 0] = h$,\u2026","title":"Roman'kov's problem on Engel lengths of products (Kourovka 11.88)"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/engel-length-polynomial-bound-refuted","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementLengthPolynomialKourovka11Problem88Wanted.engelLengthPolynomialBound_refuted","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#engel-length-polynomial-bound-refuted","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"can_prove","summary":"There is a group $G$ such that for every polynomial $\\varphi(x, y)$ in two variables with natural-number coefficients there are $u, v \\in G$ such that $u$, $v$ and $uv$ are Engel elements and $l(uv) > \\varphi(l(u), l(v))$. Here $g \\in G$ is an Engel element if there is a $k \\ge 1$ with $[h, g; k] = 1$ for all $h \\in G$, and its Engel length $l(g)$ is the least such $k$, where $[h, g; 0] = h$,\u2026","title":"Engel lengths of products admit no polynomial bound (Kourovka 11.88)"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/is-engel-element","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementSimpleGroupKourovka16Problem17Wanted.IsEngelElement","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#is-engel-element","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"An element $g$ of a group $G$ is an Engel element if for every $h \\in G$ there is an $n \\ge 0$ with $h_n = 1$, where $h_0 = h$ and $h_{i + 1} = [h_i, g]$ with $[x, g] = x^{-1} g^{-1} x g$; that is, some iterated commutator $[h, g, \\dots, g]$ is trivial.","title":"Engel element"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/simple-group-no-nontrivial-engel-element","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementSimpleGroupKourovka16Problem17Wanted.simpleGroupNoNontrivialEngelElement","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#simple-group-no-nontrivial-engel-element","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"The statement that no non-abelian simple group contains an Engel element other than the identity: for every simple group $G$ with elements $a$, $b$ such that $ab \\ne ba$, every Engel element $g$ of $G$ satisfies $g = 1$. Here $g$ is an Engel element if for every $h \\in G$ some iterated commutator $[h, g, \\dots, g]$ is trivial, where $[x, g] = x^{-1} g^{-1} x g$.","title":"Bludov's problem on Engel elements in simple groups (Kourovka 16.17)"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/simple-group-no-nontrivial-engel-element-2","kind":"node","lean":"MathlibExt.GroupTheory.EngelElementSimpleGroupKourovka16Problem17Wanted.simpleGroupNoNontrivialEngelElement_refuted","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#simple-group-no-nontrivial-engel-element-2","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"can_prove","summary":"There is a non-abelian simple group $G$ (one with elements $a$, $b$ such that $ab \\ne ba$) that contains an Engel element $g \\ne 1$. Here $g$ is an Engel element if for every $h \\in G$ some iterated commutator $[h, g, \\dots, g]$ is trivial, where $[x, g] = x^{-1} g^{-1} x g$.","title":"Non-abelian simple groups can have nontrivial Engel elements (Kourovka 16.17)"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/is-right-engel","kind":"node","lean":"MathlibExt.GroupTheory.RightEngelPowerUpperCentralKourovka17Problem12Wanted.IsRightEngel","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#is-right-engel","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"For a natural number $n$, an element $a$ of a group $G$ is a right $n$-Engel element when $[a, {}_n x] = 1$ for every $x \\in G$, where $[y, x] = y^{-1} x^{-1} y x$ and $[a, {}_n x] = [\\dots[[a, x], x], \\dots, x]$ is the left-normed commutator with $n$ copies of $x$ (so $[a, {}_0 x] = a$).","title":"Right Engel element"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/right-engel-power-upper-central","kind":"node","lean":"MathlibExt.GroupTheory.RightEngelPowerUpperCentralKourovka17Problem12Wanted.rightEngelPowerUpperCentral","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#right-engel-power-upper-central","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"fully_proved","summary":"The statement that there are functions $e, c \\colon \\mathbb{N} \\to \\mathbb{N}$ with $e(n) > 0$ for every $n$ such that, for every natural number $n$, every nilpotent group $G$ and every normal subgroup $H$ of $G$ all of whose elements are right $n$-Engel elements of $G$, the subgroup $H^{e(n)}$ generated by the $e(n)$-th powers of elements of $H$ is contained in the $c(n)$-th term $\\zeta_{c(n)}(G)$ of the upper\u2026","title":"Abdollahi's question on right Engel subgroups (Kourovka Problem 17.12)"},{"id":"GroupTheory/commutators-and-engel-conditions/engel-elements/right-engel-power-upper-central-holds","kind":"node","lean":"MathlibExt.GroupTheory.RightEngelPowerUpperCentralKourovka17Problem12Wanted.rightEngelPowerUpperCentral_holds","parent":3205,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/engel-elements/index.html#right-engel-power-upper-central-holds","scope":"GroupTheory/commutators-and-engel-conditions/engel-elements","status":"can_prove","summary":"There are functions $e, c \\colon \\mathbb{N} \\to \\mathbb{N}$ with $e(n) > 0$ for every $n$ such that, for every natural number $n$, every nilpotent group $G$ and every normal subgroup $H$ of $G$ all of whose elements are right $n$-Engel elements of $G$, the subgroup $H^{e(n)}$ generated by the $e(n)$-th powers of elements of $H$ is contained in the $c(n)$-th term $\\zeta_{c(n)}(G)$ of the upper central series of $G$.","title":"Positive answer to Abdollahi's question on right Engel subgroups (Kourovka Problem 17.12)"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms.json","id":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/index.html","scope":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms","status":"can_prove","title":"Groups with unique ordered-product normal forms"},{"id":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/is-polycyclic","kind":"node","lean":"MathlibExt.GroupTheory.UniqueProductPresentationPolycyclicKourovka13Problem11Wanted.IsPolycyclic","parent":3218,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/index.html#is-polycyclic","scope":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms","status":"fully_proved","summary":"A group $G$ is polycyclic if it has a finite series of subgroups $1 = H_0 \\le H_1 \\le \\cdots \\le H_n = G$ in which each $H_i$ is normal in $H_{i+1}$ and each quotient $H_{i+1} / H_i$ is cyclic.","title":"Polycyclic groups (Kourovka Problem 13.11)"},{"id":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/is-almost-polycyclic","kind":"node","lean":"MathlibExt.GroupTheory.UniqueProductPresentationPolycyclicKourovka13Problem11Wanted.IsAlmostPolycyclic","parent":3218,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/index.html#is-almost-polycyclic","scope":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms","status":"fully_proved","summary":"A group $G$ is almost polycyclic, or virtually polycyclic, if some subgroup of finite index in $G$ is polycyclic.","title":"Almost polycyclic groups (Kourovka Problem 13.11)"},{"id":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/has-unique-product-presentation","kind":"node","lean":"MathlibExt.GroupTheory.UniqueProductPresentationPolycyclicKourovka13Problem11Wanted.HasUniqueProductPresentation","parent":3218,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/index.html#has-unique-product-presentation","scope":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms","status":"fully_proved","summary":"A group $G$ has a unique product presentation if there are $n \\in \\mathbb{N}$ and elements $a_0, \\dots, a_{n-1} \\in G$ such that every element of $G$ equals $a_0^{k_0} a_1^{k_1} \\cdots a_{n-1}^{k_{n-1}}$ for exactly one tuple $(k_0, \\dots, k_{n-1}) \\in \\mathbb{Z}^n$.","title":"Groups with a unique product presentation"},{"id":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/torsion-free-unique-product-almost-polyc","kind":"node","lean":"MathlibExt.GroupTheory.UniqueProductPresentationPolycyclicKourovka13Problem11Wanted.torsionFreeUniqueProductAlmostPolycyclic","parent":3218,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/index.html#torsion-free-unique-product-almost-polyc","scope":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms","status":"fully_proved","summary":"The statement, asked by V. V. Bludov, that every group $G$ which is torsion-free (no element of $G$ other than $1$ has finite order) and has a unique product presentation is almost polycyclic.","title":"Kourovka Problem 13.11 (torsion-free groups with a unique product presentation)"},{"id":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/torsion-free-unique-product-almost-polyc-2","kind":"node","lean":"MathlibExt.GroupTheory.UniqueProductPresentationPolycyclicKourovka13Problem11Wanted.torsionFreeUniqueProductAlmostPolycyclic_refuted","parent":3218,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms/index.html#torsion-free-unique-product-almost-polyc-2","scope":"GroupTheory/commutators-and-engel-conditions/groups-with-unique-ordered-product-normal-forms","status":"can_prove","summary":"Not every torsion-free group with a unique product presentation is almost polycyclic: there is a group $G$ in which no element other than $1$ has finite order, which has a unique product presentation, and which is not almost polycyclic.","title":"A torsion-free group with a unique product presentation need not be almost polycyclic"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series.json","id":"GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series/index.html","scope":"GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series","status":"can_prove","title":"Intersection of the derived series"},{"id":"GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series/derived-series-intersection-perfect","kind":"node","lean":"MathlibExt.GroupTheory.DerivedSeriesIntersectionPerfectKourovka17Problem82Wanted.derivedSeriesIntersectionPerfect","parent":3224,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series/index.html#derived-series-intersection-perfect","scope":"GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series","status":"fully_proved","summary":"The conjecture of R. Mikhailov that for every finitely presented group $G$, the intersection $D = \\bigcap_{n} G^{(n)}$ of the terms $G^{(n)}$ of the derived series of $G$ satisfies $[D, D] = D$, that is, $D$ has trivial abelianization.","title":"Perfectness of the intersection of the derived series (Kourovka 17.82)"},{"id":"GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series/derived-series-intersection-perfect-refu","kind":"node","lean":"MathlibExt.GroupTheory.DerivedSeriesIntersectionPerfectKourovka17Problem82Wanted.derivedSeriesIntersectionPerfect_refuted","parent":3224,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series/index.html#derived-series-intersection-perfect-refu","scope":"GroupTheory/commutators-and-engel-conditions/intersection-of-the-derived-series","status":"can_prove","summary":"There exists a finitely presented group $G$ such that the intersection $D = \\bigcap_{n} G^{(n)}$ of the terms $G^{(n)}$ of the derived series of $G$ satisfies $[D, D] \\ne D$. That is, the conjecture of Kourovka Problem 17.82 is false; the Kourovka Archive credits this answer to V. A. Roman'kov (2011).","title":"The intersection of the derived series need not be perfect"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/local-word-identities.json","id":"GroupTheory/commutators-and-engel-conditions/local-word-identities","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/local-word-identities/index.html","scope":"GroupTheory/commutators-and-engel-conditions/local-word-identities","status":"can_prove","title":"Local word identities"},{"id":"GroupTheory/commutators-and-engel-conditions/local-word-identities/local-word-identity-holds-globally","kind":"node","lean":"MathlibExt.GroupTheory.LocalWordIdentityConnectedGroupKourovka18Problem71Wanted.localWordIdentityHoldsGlobally","parent":3227,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/local-word-identities/index.html#local-word-identity-holds-globally","scope":"GroupTheory/commutators-and-engel-conditions/local-word-identities","status":"fully_proved","summary":"The statement that for every connected Hausdorff topological group $G$ and every word $W(x, y)$ in the free group on two generators $x$, $y$, if $W(a, b) = 1$ for all $(a, b)$ in some nonempty open subset of $G \\times G$, then $W(a, b) = 1$ for all $a, b \\in G$.","title":"Local word identities in connected groups (Kourovka 18.71)"},{"id":"GroupTheory/commutators-and-engel-conditions/local-word-identities/local-word-identity-holds-globally-refut","kind":"node","lean":"MathlibExt.GroupTheory.LocalWordIdentityConnectedGroupKourovka18Problem71Wanted.localWordIdentityHoldsGlobally_refuted","parent":3227,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/local-word-identities/index.html#local-word-identity-holds-globally-refut","scope":"GroupTheory/commutators-and-engel-conditions/local-word-identities","status":"can_prove","summary":"There are a connected Hausdorff topological group $G$, a word $W(x, y)$ in the free group on two generators $x$, $y$ and a nonempty open subset $U$ of $G \\times G$ such that $W(a, b) = 1$ for all $(a, b) \\in U$ but $W(a, b) \\ne 1$ for some $a, b \\in G$. This answers Kourovka Notebook Problem 18.71 in the negative.","title":"Local word identities need not hold globally in connected groups"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups.json","id":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/index.html","scope":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups","status":"can_prove","title":"Localization of nilpotent groups"},{"id":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/is-plocal","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentExtensionSplitAtPrimesKourovka12Problem93Wanted.IsPLocal","parent":3230,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-local groups"]],"source":"$p$-local groups"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/index.html#is-plocal","scope":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups","status":"fully_proved","summary":"Let $p$ be a natural number. A group $H$ is $p$-local when, for every natural number $n$ coprime to $p$, the map $x \\mapsto x^n$ is a bijection of $H$.","title":"$p$-local groups"},{"id":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/is-plocalization","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentExtensionSplitAtPrimesKourovka12Problem93Wanted.IsPLocalization","parent":3230,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-localization of a group"]],"source":"$p$-localization of a group"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/index.html#is-plocalization","scope":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups","status":"fully_proved","summary":"Let $p$ be a natural number. A homomorphism of groups $e \\colon G \\to L$ is a $p$-localization when $L$ is nilpotent and $p$-local, and every homomorphism $f$ from $G$ to a nilpotent $p$-local group $H$ factors uniquely through $e$: there is exactly one homomorphism $g \\colon L \\to H$ with $g \\circ e = f$.","title":"$p$-localization of a group"},{"id":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/splits-at-prime","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentExtensionSplitAtPrimesKourovka12Problem93Wanted.SplitsAtPrime","parent":3230,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/index.html#splits-at-prime","scope":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups","status":"fully_proved","summary":"Let $p$ be a natural number and $\\pi \\colon G \\to Q$ a homomorphism of groups (in the intended use, the quotient map of an extension). Then $\\pi$ splits at $p$ when there exist $p$-localizations $e_G \\colon G \\to G_p$ and $e_Q \\colon Q \\to Q_p$ and homomorphisms $\\pi_p \\colon G_p \\to Q_p$ and $s \\colon Q_p \\to G_p$ with $\\pi_p \\circ e_G = e_Q \\circ \\pi$ and $\\pi_p \\circ s = \\mathrm{id}_{Q_p}$; that is, the\u2026","title":"Splitting at a prime"},{"id":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/nilpotent-extension-splits-if-splits-at","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentExtensionSplitAtPrimesKourovka12Problem93Wanted.nilpotentExtensionSplitsIfSplitsAtPrimes","parent":3230,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/index.html#nilpotent-extension-splits-if-splits-at","scope":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups","status":"fully_proved","summary":"The proposition that every extension of nilpotent groups with finitely generated quotient that splits at every prime splits: for all nilpotent groups $N$, $G$ and $Q$ with $Q$ finitely generated, every injective homomorphism $i \\colon N \\to G$ and every surjective homomorphism $\\pi \\colon G \\to Q$ with $\\operatorname{im} i = \\ker \\pi$ such that the extension splits at every prime $p$, there is a homomorphism\u2026","title":"Hilton's splitting question for nilpotent extensions (Kourovka 12.93)"},{"id":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/nilpotent-extension-splits-if-splits-at-2","kind":"node","lean":"MathlibExt.GroupTheory.NilpotentExtensionSplitAtPrimesKourovka12Problem93Wanted.nilpotentExtensionSplitsIfSplitsAtPrimes_refuted","parent":3230,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups/index.html#nilpotent-extension-splits-if-splits-at-2","scope":"GroupTheory/commutators-and-engel-conditions/localization-of-nilpotent-groups","status":"can_prove","summary":"It is not true that every extension of nilpotent groups with finitely generated quotient that splits at every prime splits: there are nilpotent groups $N$, $G$ and $Q$ with $Q$ finitely generated, an injective homomorphism $i \\colon N \\to G$ and a surjective homomorphism $\\pi \\colon G \\to Q$ with $\\operatorname{im} i = \\ker \\pi$, such that the extension splits at every prime $p$ (for some $p$-localizations $G_p$ of\u2026","title":"Nilpotent extensions that split at every prime need not split"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups.json","id":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/index.html","scope":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","status":"can_prove","title":"Locally nilpotent groups"},{"id":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/is-locally-nilpotent","kind":"node","lean":"MathlibExt.GroupTheory.LocallyNilpotentShared.IsLocallyNilpotent","parent":3236,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/index.html#is-locally-nilpotent","scope":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","status":"fully_proved","summary":"A subgroup $N$ of a group $G$ is locally nilpotent when every subgroup generated by finitely many elements of $N$ is nilpotent.","title":"Locally nilpotent subgroup"},{"id":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/hirsch-plotkin-radical","kind":"node","lean":"MathlibExt.GroupTheory.LocallyNilpotentShared.hirschPlotkinRadical","parent":3236,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/index.html#hirsch-plotkin-radical","scope":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","status":"fully_proved","summary":"The Hirsch\u2013Plotkin radical of a group $G$ is the subgroup generated by all locally nilpotent normal subgroups of $G$. By the Hirsch\u2013Plotkin theorem it is the largest locally nilpotent normal subgroup of $G$.","title":"Hirsch\u2013Plotkin radical"},{"id":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/is-locally-nilpotent-2","kind":"node","lean":"MathlibExt.GroupTheory.MinCentralizerHypercentralKourovka13Problem16Wanted.IsLocallyNilpotent","parent":3236,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/index.html#is-locally-nilpotent-2","scope":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","status":"fully_proved","summary":"A group $G$ is locally nilpotent if for every finite subset $S \\subseteq G$ the subgroup generated by $S$ is nilpotent, that is, every finitely generated subgroup of $G$ is nilpotent.","title":"Locally nilpotent group (Kourovka 13.16)"},{"id":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/has-min-centralizers","kind":"node","lean":"MathlibExt.GroupTheory.MinCentralizerHypercentralKourovka13Problem16Wanted.HasMinCentralizers","parent":3236,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/index.html#has-min-centralizers","scope":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","status":"fully_proved","summary":"For a subset $X$ of a group $G$, let $C_G(X)$ be its centralizer, the subgroup of elements of $G$ that commute with every element of $X$. The group $G$ satisfies the minimal condition on centralizers if every nonempty set of such centralizers has a minimal element with respect to inclusion; equivalently, there is no infinite strictly descending chain $C_G(X_0) > C_G(X_1) > \\cdots$ of centralizers.","title":"Minimal condition on centralizers (Kourovka 13.16)"},{"id":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/is-hypercentral","kind":"node","lean":"MathlibExt.GroupTheory.MinCentralizerHypercentralKourovka13Problem16Wanted.IsHypercentral","parent":3236,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/index.html#is-hypercentral","scope":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","status":"fully_proved","summary":"A group $G$ is hypercentral if for every proper normal subgroup $N$ of $G$ there is some $g \\in G$ with $g \\notin N$ and $g h g^{-1} h^{-1} \\in N$ for all $h \\in G$; that is, every nontrivial quotient $G/N$ has nontrivial centre (equivalently, the transfinite upper central series of $G$ reaches $G$).","title":"Hypercentral group (Kourovka 13.16)"},{"id":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/locally-nilpotent-min-centralizers-hyper","kind":"node","lean":"MathlibExt.GroupTheory.MinCentralizerHypercentralKourovka13Problem16Wanted.locallyNilpotentMinCentralizersHypercentral","parent":3236,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/index.html#locally-nilpotent-min-centralizers-hyper","scope":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","status":"fully_proved","summary":"The proposition that every locally nilpotent group satisfying the minimal condition on centralizers is hypercentral.","title":"Kourovka Problem 13.16 (Wagner)"},{"id":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/locally-nilpotent-min-centralizers-hyper-2","kind":"node","lean":"MathlibExt.GroupTheory.MinCentralizerHypercentralKourovka13Problem16Wanted.locallyNilpotentMinCentralizersHypercentral_holds","parent":3236,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups/index.html#locally-nilpotent-min-centralizers-hyper-2","scope":"GroupTheory/commutators-and-engel-conditions/locally-nilpotent-groups","status":"can_prove","summary":"Every locally nilpotent group satisfying the minimal condition on centralizers is hypercentral. This answers Kourovka Problem 13.16 (F. O. Wagner) in the affirmative (V. V. Bludov, 1998).","title":"Locally nilpotent groups with the minimal condition on centralizers are hypercentral"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups.json","id":"GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups/index.html","scope":"GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups","status":"can_prove","title":"Non-commutators in finite groups"},{"id":"GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups/exactly-one-noncommutator","kind":"node","lean":"MathlibExt.GroupTheory.ExactlyOneNoncommutatorKourovka17Problem76Wanted.exactlyOneNoncommutator","parent":3244,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups/index.html#exactly-one-noncommutator","scope":"GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups","status":"fully_proved","summary":"This is the statement that there is a finite group $G$ of order greater than $2$ in which exactly one element is not a commutator, that is, not of the form $x^{-1}y^{-1}xy$ with $x, y \\in G$.","title":"Kourovka Problem 17.76 (finite groups with exactly one non-commutator)"},{"id":"GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups/exactly-one-noncommutator-holds","kind":"node","lean":"MathlibExt.GroupTheory.ExactlyOneNoncommutatorKourovka17Problem76Wanted.exactlyOneNoncommutator_holds","parent":3244,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups/index.html#exactly-one-noncommutator-holds","scope":"GroupTheory/commutators-and-engel-conditions/non-commutators-in-finite-groups","status":"can_prove","summary":"There is a finite group $G$ of order greater than $2$ in which exactly one element is not a commutator, that is, not of the form $x^{-1}y^{-1}xy$ with $x, y \\in G$.","title":"A finite group with exactly one non-commutator"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups.json","id":"GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups/index.html","scope":"GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups","status":"can_prove","title":"Perfect locally nilpotent p-groups"},{"id":"GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups/is-locally-nilpotent","kind":"node","lean":"MathlibExt.GroupTheory.PerfectLocallyNilpotentPGroupKourovka14Problem86Wanted.IsLocallyNilpotent","parent":3247,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups/index.html#is-locally-nilpotent","scope":"GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups","status":"fully_proved","summary":"A group is locally nilpotent if every finitely generated subgroup is nilpotent.","title":"Locally nilpotent group"},{"id":"GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups/perfect-locally-nilpotent-pgroup-with-ni","kind":"node","lean":"MathlibExt.GroupTheory.PerfectLocallyNilpotentPGroupKourovka14Problem86Wanted.perfectLocallyNilpotentPGroupWithNilpotentProperSubgroups","parent":3247,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka 14.86: perfect locally nilpotent "],["math","\\(p\\)"],["text","-groups with nilpotent proper subgroups"]],"source":"Kourovka 14.86: perfect locally nilpotent $p$-groups with nilpotent proper subgroups"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups/index.html#perfect-locally-nilpotent-pgroup-with-ni","scope":"GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups","status":"fully_proved","summary":"The proposition that for some prime $p$ there is an infinite locally nilpotent $p$-group $G$ (every element of $G$ has order a power of $p$) such that $G = [G, G]$ and every proper subgroup of $G$ is nilpotent.","title":"Kourovka 14.86: perfect locally nilpotent $p$-groups with nilpotent proper subgroups"},{"id":"GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups/perfect-locally-nilpotent-pgroup-with-ni-2","kind":"node","lean":"MathlibExt.GroupTheory.PerfectLocallyNilpotentPGroupKourovka14Problem86Wanted.perfectLocallyNilpotentPGroupWithNilpotentProperSubgroups_refuted","parent":3247,"reading_title":{"format":"inline-parts-v1","parts":[["text","No infinite perfect locally nilpotent "],["math","\\(p\\)"],["text","-group has only nilpotent proper subgroups"]],"source":"No infinite perfect locally nilpotent $p$-group has only nilpotent proper subgroups"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups/index.html#perfect-locally-nilpotent-pgroup-with-ni-2","scope":"GroupTheory/commutators-and-engel-conditions/perfect-locally-nilpotent-p-groups","status":"can_prove","summary":"There is no prime $p$ and no infinite locally nilpotent $p$-group $G$ (every element of $G$ has order a power of $p$) such that $G = [G, G]$ and every proper subgroup of $G$ is nilpotent.","title":"No infinite perfect locally nilpotent $p$-group has only nilpotent proper subgroups"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups.json","id":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/index.html","scope":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups","status":"can_prove","title":"Residually nilpotent groups"},{"id":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/is-residually-nilpotent","kind":"node","lean":"MathlibExt.GroupTheory.FreeCentralExtensionResiduallyNilpotentKourovka16Problem65Wanted.IsResiduallyNilpotent","parent":3251,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/index.html#is-residually-nilpotent","scope":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups","status":"fully_proved","summary":"A group $G$ is residually nilpotent if for every element $g \\ne 1$ of $G$ there is a normal subgroup $N$ of $G$ with $g \\notin N$ such that the quotient $G/N$ is nilpotent.","title":"Residually nilpotent group"},{"id":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/is-residually-torsion-free-nilpotent","kind":"node","lean":"MathlibExt.GroupTheory.FreeCentralExtensionResiduallyNilpotentKourovka16Problem65Wanted.IsResiduallyTorsionFreeNilpotent","parent":3251,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/index.html#is-residually-torsion-free-nilpotent","scope":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups","status":"fully_proved","summary":"A group $G$ is residually torsion-free nilpotent if for every element $g \\ne 1$ of $G$ there is a normal subgroup $N$ of $G$ with $g \\notin N$ such that the quotient $G/N$ is nilpotent and torsion-free, that is, every nontrivial element of $G/N$ has infinite order.","title":"Residually torsion-free nilpotent group"},{"id":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/exists-presentation-with-non-residually","kind":"node","lean":"MathlibExt.GroupTheory.FreeCentralExtensionResiduallyNilpotentKourovka16Problem65Wanted.existsPresentationWithNonResiduallyNilpotentQuotient","parent":3251,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/index.html#exists-presentation-with-non-residually","scope":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups","status":"fully_proved","summary":"The assertion that there exist a free group $F$ and a normal subgroup $R$ of $F$ such that $F/R$ is finitely presented and residually torsion-free nilpotent, while $F/[F, R]$ is not residually nilpotent, where $[F, R]$ is the subgroup of $F$ generated by all commutators $[f, r]$ with $f \\in F$ and $r \\in R$.","title":"Mikhailov\u2013Passi question on free central extensions (Kourovka 16.65)"},{"id":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/exists-presentation-with-non-residually-2","kind":"node","lean":"MathlibExt.GroupTheory.FreeCentralExtensionResiduallyNilpotentKourovka16Problem65Wanted.existsPresentationWithNonResiduallyNilpotentQuotient_holds","parent":3251,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups/index.html#exists-presentation-with-non-residually-2","scope":"GroupTheory/commutators-and-engel-conditions/residually-nilpotent-groups","status":"can_prove","summary":"There exist a free group $F$ and a normal subgroup $R$ of $F$ such that $F/R$ is finitely presented and residually torsion-free nilpotent, while $F/[F, R]$ is not residually nilpotent; here $[F, R]$ is the subgroup of $F$ generated by all commutators $[f, r]$ with $f \\in F$ and $r \\in R$.","title":"Free central extensions need not be residually nilpotent"},{"data_url":"scopes/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups.json","id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","kind":"scope","navigation":"module","parent":3175,"result":false,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"can_prove","title":"Word maps on Lie groups"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/gl2-trace-range-contains-ray","kind":"node","lean":"MathlibExt.GroupTheory.WordMapTraceRangeKourovka16Problem69Wanted.gl2TraceRangeContainsRay","parent":3256,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mycielski's trace-range property for "],["math","\\(\\mathrm{GL}(2, \\mathbb{R})\\)"],["text"," (Kourovka 16.69(a))"]],"source":"Mycielski's trace-range property for $\\mathrm{GL}(2, \\mathbb{R})$ (Kourovka 16.69(a))"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#gl2-trace-range-contains-ray","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"fully_proved","summary":"For every nontrivial element $w$ of the free group on two generators $x, y$, the set of traces $\\operatorname{tr} w(A, B)$, where $A, B$ range over $\\mathrm{GL}(2, \\mathbb{R})$ and $w(A, B)$ is obtained by substituting $A$ for $x$ and $B$ for $y$, contains the interval $[-2, \\infty)$.","title":"Mycielski's trace-range property for $\\mathrm{GL}(2, \\mathbb{R})$ (Kourovka 16.69(a))"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/quaternion-real-part-range-contains-inte","kind":"node","lean":"MathlibExt.GroupTheory.WordMapTraceRangeKourovka16Problem69Wanted.quaternionRealPartRangeContainsInterval","parent":3256,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#quaternion-real-part-range-contains-inte","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"fully_proved","summary":"For every nontrivial element $w$ of the free group on two generators $x, y$, the set of real parts $\\operatorname{Re} w(p, q)$, where $p, q$ range over the nonzero real quaternions and $w(p, q)$ is obtained by substituting $p$ for $x$ and $q$ for $y$, contains the interval $[-5/27, 1]$.","title":"Mycielski's real-part property for quaternion word maps (Kourovka 16.69(b))"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/gl2-trace-range-contains-ray-refuted","kind":"node","lean":"MathlibExt.GroupTheory.WordMapTraceRangeKourovka16Problem69Wanted.gl2TraceRangeContainsRay_refuted","parent":3256,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mycielski's trace-range question for "],["math","\\(\\mathrm{GL}(2, \\mathbb{R})\\)"],["text"," has a negative answer"]],"source":"Mycielski's trace-range question for $\\mathrm{GL}(2, \\mathbb{R})$ has a negative answer"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#gl2-trace-range-contains-ray-refuted","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"can_prove","summary":"There is a nontrivial element $w$ of the free group on two generators $x, y$ such that some real number $t \\ge -2$ is not the trace of $w(A, B)$ for any $A, B \\in \\mathrm{GL}(2, \\mathbb{R})$, where $w(A, B)$ is obtained by substituting $A$ for $x$ and $B$ for $y$.","title":"Mycielski's trace-range question for $\\mathrm{GL}(2, \\mathbb{R})$ has a negative answer"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/quaternion-real-part-range-contains-inte-2","kind":"node","lean":"MathlibExt.GroupTheory.WordMapTraceRangeKourovka16Problem69Wanted.quaternionRealPartRangeContainsInterval_refuted","parent":3256,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#quaternion-real-part-range-contains-inte-2","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"can_prove","summary":"There is a nontrivial element $w$ of the free group on two generators $x, y$ such that some real number $t$ with $-5/27 \\le t \\le 1$ is not the real part of $w(p, q)$ for any nonzero real quaternions $p, q$, where $w(p, q)$ is obtained by substituting $p$ for $x$ and $q$ for $y$.","title":"Mycielski's real-part question for quaternion word maps has a negative answer"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/word-map","kind":"node","lean":"MathlibExt.GroupTheory.WordMapsSurjectivityKourovka16Problem68Wanted.wordMap","parent":3256,"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#word-map","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"fully_proved","summary":"For a group $G$ and an element $w$ of the free group $F_2$ on two generators $x_0, x_1$, the word map of $w$ is the map $G \\times G \\to G$ sending $(x, y)$ to $w(x, y)$, the image of $w$ under the group homomorphism $F_2 \\to G$ with $x_0 \\mapsto x$ and $x_1 \\mapsto y$.","title":"Word map"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/psl2-real-word-maps-surjective","kind":"node","lean":"MathlibExt.GroupTheory.WordMapsSurjectivityKourovka16Problem68Wanted.psl2RealWordMapsSurjective","parent":3256,"reading_title":{"format":"inline-parts-v1","parts":[["text","Surjectivity of word maps on "],["math","\\(\\mathrm{PSL}(2, \\mathbb{R})\\)"],["text"," (Kourovka 16.68(a))"]],"source":"Surjectivity of word maps on $\\mathrm{PSL}(2, \\mathbb{R})$ (Kourovka 16.68(a))"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#psl2-real-word-maps-surjective","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"fully_proved","summary":"The statement, asked about by J. Mycielski, that for every nontrivial element $w$ of the free group $F_2$ on two generators, the word map $\\mathrm{PSL}(2, \\mathbb{R}) \\times \\mathrm{PSL}(2, \\mathbb{R}) \\to \\mathrm{PSL}(2, \\mathbb{R})$, $(x, y) \\mapsto w(x, y)$, is surjective.","title":"Surjectivity of word maps on $\\mathrm{PSL}(2, \\mathbb{R})$ (Kourovka 16.68(a))"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/psl2-complex-word-maps-surjective","kind":"node","lean":"MathlibExt.GroupTheory.WordMapsSurjectivityKourovka16Problem68Wanted.psl2ComplexWordMapsSurjective","parent":3256,"reading_title":{"format":"inline-parts-v1","parts":[["text","Surjectivity of word maps on "],["math","\\(\\mathrm{PSL}(2, \\mathbb{C})\\)"],["text"," (Kourovka 16.68(b))"]],"source":"Surjectivity of word maps on $\\mathrm{PSL}(2, \\mathbb{C})$ (Kourovka 16.68(b))"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#psl2-complex-word-maps-surjective","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"fully_proved","summary":"The statement, asked about by J. Mycielski, that for every nontrivial element $w$ of the free group $F_2$ on two generators, the word map $\\mathrm{PSL}(2, \\mathbb{C}) \\times \\mathrm{PSL}(2, \\mathbb{C}) \\to \\mathrm{PSL}(2, \\mathbb{C})$, $(x, y) \\mapsto w(x, y)$, is surjective.","title":"Surjectivity of word maps on $\\mathrm{PSL}(2, \\mathbb{C})$ (Kourovka 16.68(b))"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/so3-word-maps-surjective","kind":"node","lean":"MathlibExt.GroupTheory.WordMapsSurjectivityKourovka16Problem68Wanted.so3WordMapsSurjective","parent":3256,"reading_title":{"format":"inline-parts-v1","parts":[["text","Surjectivity of word maps on "],["math","\\(\\mathrm{SO}(3, \\mathbb{R})\\)"],["text"," (Kourovka 16.68(c))"]],"source":"Surjectivity of word maps on $\\mathrm{SO}(3, \\mathbb{R})$ (Kourovka 16.68(c))"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#so3-word-maps-surjective","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"fully_proved","summary":"The statement, asked about by J. Mycielski, that for every nontrivial element $w$ of the free group $F_2$ on two generators, the word map $\\mathrm{SO}(3, \\mathbb{R}) \\times \\mathrm{SO}(3, \\mathbb{R}) \\to \\mathrm{SO}(3, \\mathbb{R})$, $(x, y) \\mapsto w(x, y)$, is surjective.","title":"Surjectivity of word maps on $\\mathrm{SO}(3, \\mathbb{R})$ (Kourovka 16.68(c))"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/psl2-real-word-maps-surjective-refuted","kind":"node","lean":"MathlibExt.GroupTheory.WordMapsSurjectivityKourovka16Problem68Wanted.psl2RealWordMapsSurjective_refuted","parent":3256,"reading_title":{"format":"inline-parts-v1","parts":[["text","Some nontrivial word map on "],["math","\\(\\mathrm{PSL}(2, \\mathbb{R})\\)"],["text"," is not surjective (Kourovka 16.68(a))"]],"source":"Some nontrivial word map on $\\mathrm{PSL}(2, \\mathbb{R})$ is not surjective (Kourovka 16.68(a))"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#psl2-real-word-maps-surjective-refuted","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"can_prove","summary":"There is a nontrivial element $w$ of the free group $F_2$ on two generators whose word map $\\mathrm{PSL}(2, \\mathbb{R}) \\times \\mathrm{PSL}(2, \\mathbb{R}) \\to \\mathrm{PSL}(2, \\mathbb{R})$, $(x, y) \\mapsto w(x, y)$, is not surjective. That is, Mycielski's question has a negative answer for $\\mathrm{PSL}(2, \\mathbb{R})$.","title":"Some nontrivial word map on $\\mathrm{PSL}(2, \\mathbb{R})$ is not surjective (Kourovka 16.68(a))"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/psl2-complex-word-maps-surjective-holds","kind":"node","lean":"MathlibExt.GroupTheory.WordMapsSurjectivityKourovka16Problem68Wanted.psl2ComplexWordMapsSurjective_holds","parent":3256,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nontrivial word maps on "],["math","\\(\\mathrm{PSL}(2, \\mathbb{C})\\)"],["text"," are surjective (Kourovka 16.68(b))"]],"source":"Nontrivial word maps on $\\mathrm{PSL}(2, \\mathbb{C})$ are surjective (Kourovka 16.68(b))"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#psl2-complex-word-maps-surjective-holds","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"can_prove","summary":"For every nontrivial element $w$ of the free group $F_2$ on two generators, the word map $\\mathrm{PSL}(2, \\mathbb{C}) \\times \\mathrm{PSL}(2, \\mathbb{C}) \\to \\mathrm{PSL}(2, \\mathbb{C})$, $(x, y) \\mapsto w(x, y)$, is surjective. That is, Mycielski's question has an affirmative answer for $\\mathrm{PSL}(2, \\mathbb{C})$.","title":"Nontrivial word maps on $\\mathrm{PSL}(2, \\mathbb{C})$ are surjective (Kourovka 16.68(b))"},{"id":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/so3-word-maps-surjective-refuted","kind":"node","lean":"MathlibExt.GroupTheory.WordMapsSurjectivityKourovka16Problem68Wanted.so3WordMapsSurjective_refuted","parent":3256,"reading_title":{"format":"inline-parts-v1","parts":[["text","Some nontrivial word map on "],["math","\\(\\mathrm{SO}(3, \\mathbb{R})\\)"],["text"," is not surjective (Kourovka 16.68(c))"]],"source":"Some nontrivial word map on $\\mathrm{SO}(3, \\mathbb{R})$ is not surjective (Kourovka 16.68(c))"},"result":true,"route":"../roadmap/GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups/index.html#so3-word-maps-surjective-refuted","scope":"GroupTheory/commutators-and-engel-conditions/word-maps-on-lie-groups","status":"can_prove","summary":"There is a nontrivial element $w$ of the free group $F_2$ on two generators whose word map $\\mathrm{SO}(3, \\mathbb{R}) \\times \\mathrm{SO}(3, \\mathbb{R}) \\to \\mathrm{SO}(3, \\mathbb{R})$, $(x, y) \\mapsto w(x, y)$, is not surjective. That is, Mycielski's question has a negative answer for $\\mathrm{SO}(3, \\mathbb{R})$.","title":"Some nontrivial word map on $\\mathrm{SO}(3, \\mathbb{R})$ is not surjective (Kourovka 16.68(c))"},{"data_url":"scopes/GroupTheory/automorphisms.json","id":"GroupTheory/automorphisms","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/automorphisms/index.html","scope":"GroupTheory/automorphisms","status":"can_prove","summary":"Automorphism and endomorphism groups of groups other than free groups: their orders and realizations, outer automorphism groups, class-preserving automorphisms, and automorphisms with few fixed points or twisted conjugacy classes.","title":"Automorphisms of groups"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups.json","id":"GroupTheory/automorphisms/automorphism-groups","kind":"scope","parent":3268,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/index.html","scope":"GroupTheory/automorphisms/automorphism-groups","status":"can_prove","summary":"The automorphism group as an object: orders of automorphism groups, realizing groups as (outer) automorphism groups, class-preserving and inner automorphisms, automorphism orbits, endomorphism counts.","title":"Automorphism groups"},{"id":"GroupTheory/automorphisms/automorphism-groups/has-finitely-many-automorphism-orbits","kind":"node","lean":"MetaMathlibExt.HasFinitelyManyAutomorphismOrbits","parent":3269,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/index.html#has-finitely-many-automorphism-orbits","scope":"GroupTheory/automorphisms/automorphism-groups","status":"fully_proved","summary":"A group $G$ has finitely many automorphism orbits if the action of its automorphism group $\\operatorname{Aut}(G)$ on $G$ has only finitely many orbits.","title":"Groups with finitely many automorphism orbits"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups.json","id":"GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups","kind":"scope","navigation":"module","parent":3269,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups/index.html","scope":"GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups","status":"can_prove","title":"Automorphism groups of p-groups"},{"id":"GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups/aut-iso-aut-of-proper-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismGroupProperSubgroupKourovka15Problem27Wanted.autIsoAutOfProperSubgroup","parent":3271,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups/index.html#aut-iso-aut-of-proper-subgroup","scope":"GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups","status":"fully_proved","summary":"The statement that for some prime $p$ there are a finite $p$-group $G$ of order greater than $2$ and a proper subgroup $H < G$ whose automorphism groups are isomorphic, $\\mathrm{Aut}(G) \\cong \\mathrm{Aut}(H)$.","title":"Berkovich's problem on automorphism groups of p-groups (Kourovka 15.27)"},{"id":"GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups/aut-iso-aut-of-proper-subgroup-holds","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismGroupProperSubgroupKourovka15Problem27Wanted.autIsoAutOfProperSubgroup_holds","parent":3271,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups/index.html#aut-iso-aut-of-proper-subgroup-holds","scope":"GroupTheory/automorphisms/automorphism-groups/automorphism-groups-of-p-groups","status":"can_prove","summary":"For some prime $p$ there are a finite $p$-group $G$ of order greater than $2$ and a proper subgroup $H < G$ with $\\mathrm{Aut}(G) \\cong \\mathrm{Aut}(H)$ as groups.","title":"A p-group and a proper subgroup with isomorphic automorphism groups (Kourovka 15.27)"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms.json","id":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms","kind":"scope","navigation":"module","parent":3269,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/index.html","scope":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms","status":"can_prove","title":"Class-preserving automorphisms"},{"id":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/class-preserving-automorphisms-inner","kind":"node","lean":"MathlibExt.GroupTheory.ClassPreservingAutomorphismsInnerKourovka16Problem62Wanted.classPreservingAutomorphismsInner","parent":3274,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/index.html#class-preserving-automorphisms-inner","scope":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms","status":"fully_proved","summary":"For every group $G$: if every automorphism of $G$ sends each element $g \\in G$ to a conjugate of $g$ (equivalently, maps every conjugacy class of $G$ into itself), then every automorphism of $G$ is inner, that is, of the form $x \\mapsto h x h^{-1}$ for some $h \\in G$.","title":"Claim of Kourovka Problem 16.62"},{"id":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/class-preserving-automorphisms-inner-ref","kind":"node","lean":"MathlibExt.GroupTheory.ClassPreservingAutomorphismsInnerKourovka16Problem62Wanted.classPreservingAutomorphismsInner_refuted","parent":3274,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/index.html#class-preserving-automorphisms-inner-ref","scope":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms","status":"can_prove","summary":"The claim of Kourovka Problem 16.62 is false: there is a group $G$ such that every automorphism of $G$ sends each element $g \\in G$ to a conjugate of $g$, but some automorphism of $G$ is not inner, that is, not of the form $x \\mapsto h x h^{-1}$ for any $h \\in G$.","title":"Groups whose automorphisms all preserve conjugacy classes can have outer automorphisms"},{"id":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/scottish-book-problem112-claim","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook112Wanted.ScottishBookProblem112Claim","parent":3274,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/index.html#scottish-book-problem112-claim","scope":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms","status":"fully_proved","summary":"The claim that for every group $G$, every automorphism $\\alpha$ of $G$ that sends each element to a conjugate of itself (for each $x \\in G$ there is $g \\in G$ with $\\alpha(x) = g x g^{-1}$) is inner, that is, there is a single $g \\in G$ with $\\alpha(x) = g x g^{-1}$ for all $x \\in G$.","title":"Claim of Scottish Book Problem 112"},{"id":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/scottish-book112-refuted","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook112Wanted.scottishBook112_refuted","parent":3274,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms/index.html#scottish-book112-refuted","scope":"GroupTheory/automorphisms/automorphism-groups/class-preserving-automorphisms","status":"can_prove","summary":"The claim of Scottish Book Problem 112 is false: there are a group $G$ and an automorphism $\\alpha$ of $G$ that sends every element to a conjugate of itself (for each $x \\in G$ there is $g \\in G$ with $\\alpha(x) = g x g^{-1}$) but is not inner (there is no $g \\in G$ with $\\alpha(x) = g x g^{-1}$ for all $x \\in G$).","title":"Class-preserving automorphisms need not be inner"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms.json","id":"GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms","kind":"scope","navigation":"module","parent":3269,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms/index.html","scope":"GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms","status":"can_prove","title":"Endomorphisms and partial isomorphisms"},{"id":"GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms/end-partial-iso-count-implies-abelian","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismPartialIsomorphismCountKourovka19Problem20Wanted.endPartialIsoCountImpliesAbelian","parent":3279,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms/index.html#end-partial-iso-count-implies-abelian","scope":"GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms","status":"fully_proved","summary":"The statement that every finite group $G$ with $|\\mathrm{End}(G)| = |\\mathrm{PIso}(G)|$ is abelian. Here $\\mathrm{End}(G)$ is the set of endomorphisms of $G$, and $\\mathrm{PIso}(G)$ is the set of partial isomorphisms of $G$: the triples $(H, K, f)$ of subgroups $H$, $K$ of $G$ and an isomorphism $f \\colon H \\to K$.","title":"Kourovka Problem 19.20 (Cameron)"},{"id":"GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms/end-partial-iso-count-implies-abelian-re","kind":"node","lean":"MathlibExt.GroupTheory.EndomorphismPartialIsomorphismCountKourovka19Problem20Wanted.endPartialIsoCountImpliesAbelian_refuted","parent":3279,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms/index.html#end-partial-iso-count-implies-abelian-re","scope":"GroupTheory/automorphisms/automorphism-groups/endomorphisms-and-partial-isomorphisms","status":"can_prove","summary":"There is a finite non-abelian group $G$ with $|\\mathrm{End}(G)| = |\\mathrm{PIso}(G)|$. Here $\\mathrm{End}(G)$ is the set of endomorphisms of $G$, and $\\mathrm{PIso}(G)$ is the set of partial isomorphisms of $G$: the triples $(H, K, f)$ of subgroups $H$, $K$ of $G$ and an isomorphism $f \\colon H \\to K$. This answers Kourovka Problem 19.20 (P. J. Cameron) in the negative.","title":"Equal endomorphism and partial isomorphism counts do not force commutativity"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations.json","id":"GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations","kind":"scope","navigation":"module","parent":3269,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations/index.html","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations","status":"can_prove","title":"Groups as automorphism groups of relations"},{"id":"GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations/is-automorphism-group-of","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook142Wanted.IsAutomorphismGroupOf","parent":3282,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations/index.html#is-automorphism-group-of","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations","status":"fully_proved","summary":"Let $G$ be a group, $Z$ a set and $X \\subseteq Z \\times Z$ a binary relation on $Z$. Then $G$ is the full automorphism group of $X$ when there is an injective group homomorphism $\\varphi$ from $G$ to the group of permutations of $Z$ whose image is exactly the set of permutations $f$ of $Z$ with $(x, y) \\in X \\iff (f(x), f(y)) \\in X$ for all $x, y \\in Z$.","title":"Full automorphism group of a binary relation"},{"id":"GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations/every-group-is-automorphism-group-of-bin","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook142Wanted.everyGroupIsAutomorphismGroupOfBinaryRelation","parent":3282,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations/index.html#every-group-is-automorphism-group-of-bin","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-automorphism-groups-of-relations","status":"can_prove","summary":"For every group $G$ there are a set $Z$ and a binary relation $X \\subseteq Z \\times Z$ such that $G$ is the full automorphism group of $X$: there is an injective group homomorphism $\\varphi$ from $G$ to the group of permutations of $Z$ whose image is exactly the set of permutations $f$ of $Z$ with $(x, y) \\in X \\iff (f(x), f(y)) \\in X$ for all $x, y \\in Z$.","title":"Birkhoff's theorem: every group is the automorphism group of a binary relation"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups.json","id":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups","kind":"scope","navigation":"module","parent":3269,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/index.html","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups","status":"can_prove","title":"Groups as outer automorphism groups"},{"id":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/every-finite-group-is-outer-automorphism","kind":"node","lean":"MathlibExt.GroupTheory.OuterAutomorphismGroupRealizationKourovka16Problem59Wanted.everyFiniteGroupIsOuterAutomorphismGroup","parent":3285,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/index.html#every-finite-group-is-outer-automorphism","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups","status":"fully_proved","summary":"The statement that for every finite group $K$ there are a finite group $G$ and a surjective homomorphism $\\mathrm{Aut}(G) \\to K$ whose kernel is the group $\\mathrm{Inn}(G)$ of inner automorphisms of $G$, that is, $K$ is isomorphic to $\\mathrm{Out}(G) = \\mathrm{Aut}(G) / \\mathrm{Inn}(G)$.","title":"Kourovka Problem 16.59 (finite groups as outer automorphism groups)"},{"id":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/every-finite-group-is-outer-automorphism-2","kind":"node","lean":"MathlibExt.GroupTheory.OuterAutomorphismGroupRealizationKourovka16Problem59Wanted.everyFiniteGroupIsOuterAutomorphismGroup_holds","parent":3285,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/index.html#every-finite-group-is-outer-automorphism-2","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups","status":"can_prove","summary":"For every finite group $K$ there are a finite group $G$ and a surjective homomorphism $\\mathrm{Aut}(G) \\to K$ whose kernel is the group $\\mathrm{Inn}(G)$ of inner automorphisms of $G$; equivalently, $\\mathrm{Out}(G) = \\mathrm{Aut}(G) / \\mathrm{Inn}(G)$ is isomorphic to $K$. That is, the answer to Kourovka Problem 16.59 is positive.","title":"Every finite group is the outer automorphism group of a finite group"},{"id":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/out-iso-to","kind":"node","lean":"MathlibExt.GroupTheory.OuterAutomorphismMetabelianKourovka11Problem26Wanted.OutIsoTo","parent":3285,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/index.html#out-iso-to","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups","status":"fully_proved","summary":"For groups $M$ and $G$, the statement that $\\operatorname{Out}(M) \\cong G$, expressed without forming the quotient: there is a surjective homomorphism $\\operatorname{Aut}(M) \\to G$ whose kernel is exactly the inner automorphism group $\\operatorname{Inn}(M)$, the image of the conjugation map $M \\to \\operatorname{Aut}(M)$.","title":"Outer automorphism group isomorphic to a given group"},{"id":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/group-not-outer-aut-of-metabelian","kind":"node","lean":"MathlibExt.GroupTheory.OuterAutomorphismMetabelianKourovka11Problem26Wanted.groupNotOuterAutOfMetabelian","parent":3285,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/index.html#group-not-outer-aut-of-metabelian","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups","status":"fully_proved","summary":"The statement (Kourovka Problem 11.26, G\u00f6bel) that there is a group $G$ such that for every metabelian group $M$ (one whose second derived subgroup $M''$ is trivial) with trivial centre, the outer automorphism group $\\operatorname{Out}(M)$ is not isomorphic to $G$.","title":"A group that is not an outer automorphism group of a centreless metabelian group"},{"id":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/group-not-outer-aut-of-metabelian-refute","kind":"node","lean":"MathlibExt.GroupTheory.OuterAutomorphismMetabelianKourovka11Problem26Wanted.groupNotOuterAutOfMetabelian_refuted","parent":3285,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups/index.html#group-not-outer-aut-of-metabelian-refute","scope":"GroupTheory/automorphisms/automorphism-groups/groups-as-outer-automorphism-groups","status":"can_prove","summary":"For every group $G$ there is a metabelian group $M$ (one whose second derived subgroup $M''$ is trivial) with trivial centre such that the outer automorphism group $\\operatorname{Out}(M) = \\operatorname{Aut}(M)/\\operatorname{Inn}(M)$ is isomorphic to $G$. This answers Kourovka Problem 11.26 (G\u00f6bel) in the negative (G\u00f6bel and Paras).","title":"Every group is the outer automorphism group of a centreless metabelian group"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups.json","id":"GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups","kind":"scope","navigation":"module","parent":3269,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups/index.html","scope":"GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups","status":"can_prove","title":"One-relator groups as automorphism groups"},{"id":"GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups/one-relator-group-is-automorphism-group","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorAutomorphismGroupKourovka16Problem75Wanted.oneRelatorGroupIsAutomorphismGroup","parent":3291,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups/index.html#one-relator-group-is-automorphism-group","scope":"GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups","status":"fully_proved","summary":"The proposition that there exist a natural number $n$ and an element $r$ of the free group on generators $x_0, \\dots, x_{n-1}$ such that the one-relator group $\\langle x_0, \\dots, x_{n-1} \\mid r \\rangle$ is non-abelian and is isomorphic to the automorphism group $\\operatorname{Aut}(H)$ of some group $H$. The problem is due to M. V. Neshchadim.","title":"Kourovka Problem 16.75: one-relator automorphism groups"},{"id":"GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups/one-relator-group-is-automorphism-group-2","kind":"node","lean":"MathlibExt.GroupTheory.OneRelatorAutomorphismGroupKourovka16Problem75Wanted.oneRelatorGroupIsAutomorphismGroup_holds","parent":3291,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups/index.html#one-relator-group-is-automorphism-group-2","scope":"GroupTheory/automorphisms/automorphism-groups/one-relator-groups-as-automorphism-groups","status":"can_prove","summary":"There exist a natural number $n$ and an element $r$ of the free group on generators $x_0, \\dots, x_{n-1}$ such that the one-relator group $\\langle x_0, \\dots, x_{n-1} \\mid r \\rangle$ is non-abelian and is isomorphic to the automorphism group $\\operatorname{Aut}(H)$ of some group $H$. This answers Kourovka Problem 16.75 in the affirmative.","title":"Some non-abelian one-relator group is an automorphism group"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups.json","id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","kind":"scope","navigation":"module","parent":3269,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"can_prove","title":"Orders of automorphism groups"},{"id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/totient-le-aut-order","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismGroupOrderTotientKourovka15Problem43Wanted.totientLeAutOrder","parent":3294,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html#totient-le-aut-order","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"fully_proved","summary":"The statement that every finite group $G$ satisfies $\\varphi(|G|) \\le |\\mathrm{Aut}(G)|$, where $\\varphi$ is Euler's totient function and $\\mathrm{Aut}(G)$ is the group of automorphisms of $G$.","title":"Deaconescu's totient lower bound conjecture (Kourovka 15.43(a))"},{"id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/aut-order-eq-totient-implies-cyclic","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismGroupOrderTotientKourovka15Problem43Wanted.autOrderEqTotientImpliesCyclic","parent":3294,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html#aut-order-eq-totient-implies-cyclic","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"fully_proved","summary":"The statement that every finite group $G$ with $|\\mathrm{Aut}(G)| = \\varphi(|G|)$ is cyclic, where $\\varphi$ is Euler's totient function and $\\mathrm{Aut}(G)$ is the group of automorphisms of $G$.","title":"Deaconescu's cyclicity conjecture (Kourovka 15.43(b))"},{"id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/totient-le-aut-order-refuted","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismGroupOrderTotientKourovka15Problem43Wanted.totientLeAutOrder_refuted","parent":3294,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html#totient-le-aut-order-refuted","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"can_prove","summary":"Not every finite group $G$ satisfies $\\varphi(|G|) \\le |\\mathrm{Aut}(G)|$: there is a finite group $G$ with $|\\mathrm{Aut}(G)| < \\varphi(|G|)$, where $\\varphi$ is Euler's totient function and $\\mathrm{Aut}(G)$ is the group of automorphisms of $G$.","title":"Deaconescu's totient lower bound conjecture fails (Kourovka 15.43(a))"},{"id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/aut-order-eq-totient-implies-cyclic-refu","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismGroupOrderTotientKourovka15Problem43Wanted.autOrderEqTotientImpliesCyclic_refuted","parent":3294,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html#aut-order-eq-totient-implies-cyclic-refu","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"can_prove","summary":"Not every finite group $G$ with $|\\mathrm{Aut}(G)| = \\varphi(|G|)$ is cyclic: there is a non-cyclic finite group $G$ with $|\\mathrm{Aut}(G)| = \\varphi(|G|)$, where $\\varphi$ is Euler's totient function and $\\mathrm{Aut}(G)$ is the group of automorphisms of $G$.","title":"Deaconescu's cyclicity conjecture fails (Kourovka 15.43(b))"},{"id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/aut-order-prime-power-bound","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismGroupPrimePowerKourovka18Problem82Wanted.autOrderPrimePowerBound","parent":3294,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html#aut-order-prime-power-bound","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"fully_proved","summary":"The statement that there is a function $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every $n \\in \\mathbb{N}$, every prime $p$ and every finite group $G$, if $p^{f(n)}$ divides $|G|$ then $p^n$ divides $|\\mathrm{Aut}(G)|$, where $\\mathrm{Aut}(G)$ is the group of automorphisms of $G$.","title":"Patne's prime-power divisibility problem (Kourovka 18.82)"},{"id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/aut-order-prime-power-bound-holds","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismGroupPrimePowerKourovka18Problem82Wanted.autOrderPrimePowerBound_holds","parent":3294,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html#aut-order-prime-power-bound-holds","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"can_prove","summary":"There is a function $f : \\mathbb{N} \\to \\mathbb{N}$, the same for all primes, such that for every $n \\in \\mathbb{N}$, every prime $p$ and every finite group $G$, if $p^{f(n)}$ divides $|G|$ then $p^n$ divides $|\\mathrm{Aut}(G)|$, where $\\mathrm{Aut}(G)$ is the group of automorphisms of $G$.","title":"Green's theorem on prime powers dividing the order of the automorphism group (Kourovka 18.82)"},{"id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/p-group-order-divides-aut-order","kind":"node","lean":"MathlibExt.GroupTheory.PGroupOrderDividesAutKourovka12Problem77Wanted.pGroupOrderDividesAutOrder","parent":3294,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility conjecture for orders of automorphism groups of "],["math","\\(p\\)"],["text","-groups (Kourovka 12.77)"]],"source":"Divisibility conjecture for orders of automorphism groups of $p$-groups (Kourovka 12.77)"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html#p-group-order-divides-aut-order","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"fully_proved","summary":"The statement that there is a prime $p$ such that for every finite non-cyclic $p$-group $P$ of order greater than $p^2$, the order of $P$ divides the order of its automorphism group $\\mathrm{Aut}(P)$.","title":"Divisibility conjecture for orders of automorphism groups of $p$-groups (Kourovka 12.77)"},{"id":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/p-group-order-divides-aut-order-refuted","kind":"node","lean":"MathlibExt.GroupTheory.PGroupOrderDividesAutKourovka12Problem77Wanted.pGroupOrderDividesAutOrder_refuted","parent":3294,"reading_title":{"format":"inline-parts-v1","parts":[["text","Orders of "],["math","\\(p\\)"],["text","-groups need not divide orders of their automorphism groups"]],"source":"Orders of $p$-groups need not divide orders of their automorphism groups"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups/index.html#p-group-order-divides-aut-order-refuted","scope":"GroupTheory/automorphisms/automorphism-groups/orders-of-automorphism-groups","status":"can_prove","summary":"For every prime $p$ there is a finite non-cyclic $p$-group $P$ of order greater than $p^2$ whose order does not divide the order of its automorphism group $\\mathrm{Aut}(P)$.","title":"Orders of $p$-groups need not divide orders of their automorphism groups"},{"data_url":"scopes/GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou.json","id":"GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou","kind":"scope","navigation":"module","parent":3269,"result":false,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou/index.html","scope":"GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou","status":"can_prove","title":"Outer automorphisms of poly-infinite-cyclic groups"},{"id":"GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou/has-normal-series-infinite-cyclic-factor","kind":"node","lean":"MathlibExt.GroupTheory.PolyInfiniteCyclicOuterAutomorphismKourovka13Problem23Wanted.HasNormalSeriesInfiniteCyclicFactors","parent":3303,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou/index.html#has-normal-series-infinite-cyclic-factor","scope":"GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou","status":"fully_proved","summary":"A group $G$ has a normal series with infinite cyclic factors when there are a natural number $n$ and subgroups $1 = H_0 \\le H_1 \\le \\cdots \\le H_n = G$, each normal in $G$, such that every factor $H_{i+1}/H_i$ is infinite cyclic.","title":"Normal series with infinite cyclic factors"},{"id":"GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou/poly-infinite-cyclic-has-outer-automorph","kind":"node","lean":"MathlibExt.GroupTheory.PolyInfiniteCyclicOuterAutomorphismKourovka13Problem23Wanted.polyInfiniteCyclicHasOuterAutomorphism","parent":3303,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou/index.html#poly-infinite-cyclic-has-outer-automorph","scope":"GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou","status":"fully_proved","summary":"The conjecture that every nontrivial group with a normal series with infinite cyclic factors has an automorphism that is not inner.","title":"Outer automorphisms of groups with an infinite cyclic normal series (Kourovka 13.23)"},{"id":"GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou/poly-infinite-cyclic-has-outer-automorph-2","kind":"node","lean":"MathlibExt.GroupTheory.PolyInfiniteCyclicOuterAutomorphismKourovka13Problem23Wanted.polyInfiniteCyclicHasOuterAutomorphism_refuted","parent":3303,"result":true,"route":"../roadmap/GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou/index.html#poly-infinite-cyclic-has-outer-automorph-2","scope":"GroupTheory/automorphisms/automorphism-groups/outer-automorphisms-of-poly-infinite-cyclic-grou","status":"can_prove","summary":"There is a nontrivial group $G$ with a normal series with infinite cyclic factors such that every automorphism of $G$ is inner. So the conjecture of Kourovka Problem 13.23 is false.","title":"Only inner automorphisms despite an infinite cyclic normal series (Kourovka 13.23 refuted)"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms","kind":"scope","parent":3268,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms","status":"can_prove","summary":"How an automorphism and its fixed points control a group: almost regular and fixed-point-free automorphisms, bounds on derived length and Fitting height, twisted conjugacy, e- and nil-automorphisms, squares of 2-automorphisms.","title":"Fixed points of automorphisms"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups","status":"can_prove","title":"Almost regular automorphisms of p-groups"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/bounded-index-subgroup-of-bounded-derive","kind":"node","lean":"MathlibExt.GroupTheory.AlmostRegularAutomorphismDerivedLengthKourovka14Problem96Wanted.boundedIndexSubgroupOfBoundedDerivedLength","parent":3308,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/index.html#bounded-index-subgroup-of-bounded-derive","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups","status":"fully_proved","summary":"The proposition that there are functions $f \\colon \\mathbb{N}^3 \\to \\mathbb{N}$ and $g \\colon \\mathbb{N} \\to \\mathbb{N}$ such that for every prime $p$, all natural numbers $n$ and $m$, every finite $p$-group $P$ and every automorphism $\\varphi$ of $P$ of order $p^n$ with exactly $p^m$ fixed points, $P$ has a subgroup of index at most $f(p, n, m)$ that is soluble of derived length at most $g(m)$.","title":"Bounded-index subgroups of bounded derived length (Kourovka 14.96)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/bounded-index-subgroup-of-bounded-derive-2","kind":"node","lean":"MathlibExt.GroupTheory.AlmostRegularAutomorphismDerivedLengthKourovka14Problem96Wanted.boundedIndexSubgroupOfBoundedDerivedLength_holds","parent":3308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Almost regular automorphisms of finite "],["math","\\(p\\)"],["text","-groups and derived length (Kourovka 14.96)"]],"source":"Almost regular automorphisms of finite $p$-groups and derived length (Kourovka 14.96)"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/index.html#bounded-index-subgroup-of-bounded-derive-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups","status":"can_prove","summary":"There are functions $f \\colon \\mathbb{N}^3 \\to \\mathbb{N}$ and $g \\colon \\mathbb{N} \\to \\mathbb{N}$ such that for every prime $p$, all natural numbers $n$ and $m$, every finite $p$-group $P$ and every automorphism $\\varphi$ of $P$ of order $p^n$ with exactly $p^m$ fixed points, $P$ has a subgroup of index at most $f(p, n, m)$ that is soluble of derived length at most $g(m)$. So Khukhro's question of Kourovka\u2026","title":"Almost regular automorphisms of finite $p$-groups and derived length (Kourovka 14.96)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/bounded-index-bounded-class-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismFewFixedPointsPGroupKourovka10Problem68Wanted.boundedIndexBoundedClassSubgroup","parent":3308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded index and bounded class under a "],["math","\\(p\\)"],["text","-automorphism (Kourovka 10.68)"]],"source":"Bounded index and bounded class under a $p$-automorphism (Kourovka 10.68)"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/index.html#bounded-index-bounded-class-subgroup","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups","status":"fully_proved","summary":"The proposition that there are functions $f(p, m)$ and $g(m)$ of natural numbers, with natural-number values, such that for every prime $p$, every $m \\in \\mathbb{N}$, every finite $p$-group $G$ and every automorphism $\\varphi$ of $G$ of order $p$ with exactly $p^m$ fixed points, $G$ has a subgroup $H$ of index at most $f(p, m)$ and nilpotency class at most $g(m)$, that is, $\\gamma_{g(m)+1}(H) = 1$, where\u2026","title":"Bounded index and bounded class under a $p$-automorphism (Kourovka 10.68)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/bounded-index-bounded-class-subgroup-hol","kind":"node","lean":"MathlibExt.GroupTheory.AutomorphismFewFixedPointsPGroupKourovka10Problem68Wanted.boundedIndexBoundedClassSubgroup_holds","parent":3308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Medvedev's theorem on automorphisms of order "],["math","\\(p\\)"],["text"," with "],["math","\\(p^m\\)"],["text"," fixed points"]],"source":"Medvedev's theorem on automorphisms of order $p$ with $p^m$ fixed points"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups/index.html#bounded-index-bounded-class-subgroup-hol","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/almost-regular-automorphisms-of-p-groups","status":"can_prove","summary":"There are functions $f(p, m)$ and $g(m)$ of natural numbers, with natural-number values, such that for every prime $p$, every $m \\in \\mathbb{N}$, every finite $p$-group $G$ and every automorphism $\\varphi$ of $G$ of order $p$ with exactly $p^m$ fixed points, $G$ has a subgroup $H$ of index at most $f(p, m)$ and nilpotency class at most $g(m)$, that is, $\\gamma_{g(m)+1}(H) = 1$, where $\\gamma_1(H) = H$ and\u2026","title":"Medvedev's theorem on automorphisms of order $p$ with $p^m$ fixed points"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point","status":"can_prove","title":"Automorphisms of order four with few fixed points"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point/order-four-automorphism-bounded-series","kind":"node","lean":"MathlibExt.GroupTheory.OrderFourAutomorphismNilpotentSeriesKourovka11Problem126Wanted.orderFourAutomorphismBoundedSeries","parent":3313,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point/index.html#order-four-automorphism-bounded-series","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point","status":"fully_proved","summary":"The statement that there are a natural number $h$ and a function $f : \\mathbb{N} \\to \\mathbb{N}$ such that, for every natural number $m$, every finite soluble group $G$ and every automorphism $\\varphi$ of $G$ of order $4$ with at most $m$ fixed points, there are normal subgroups $M \\le N$ of $G$ with $|G : N| \\le f(m)$, $[[N, N], N] \\le M$ (so $N/M$ is nilpotent of class at most $2$) and $\\gamma_{h+1}(M) = 1$ (so\u2026","title":"Kourovka Problem 11.126 (automorphisms of order four with few fixed points)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point/order-four-automorphism-bounded-series-h","kind":"node","lean":"MathlibExt.GroupTheory.OrderFourAutomorphismNilpotentSeriesKourovka11Problem126Wanted.orderFourAutomorphismBoundedSeries_holds","parent":3313,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point/index.html#order-four-automorphism-bounded-series-h","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/automorphisms-of-order-four-with-few-fixed-point","status":"can_prove","summary":"There are a natural number $h$ and a function $f : \\mathbb{N} \\to \\mathbb{N}$ such that, for every natural number $m$, every finite soluble group $G$ and every automorphism $\\varphi$ of $G$ of order $4$ with at most $m$ fixed points, there are normal subgroups $M \\le N$ of $G$ with $|G : N| \\le f(m)$, $[[N, N], N] \\le M$ (so $N/M$ is nilpotent of class at most $2$) and $\\gamma_{h+1}(M) = 1$ (so $M$ is nilpotent of\u2026","title":"Bounded normal series for soluble groups with an automorphism of order four"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism","status":"can_prove","title":"Derived length of p-groups with an automorphism"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism/sectional-rank-le","kind":"node","lean":"MathlibExt.GroupTheory.PGroupRankAutomorphismDerivedLengthKourovka13Problem56Wanted.SectionalRankLE","parent":3316,"reading_title":{"format":"inline-parts-v1","parts":[["text","Groups of sectional rank at most "],["math","\\(r\\)"]],"source":"Groups of sectional rank at most $r$"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism/index.html#sectional-rank-le","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism","status":"fully_proved","summary":"Let $G$ be a group and $r$ a natural number. The group $G$ has sectional rank at most $r$ when every subgroup of $G$ is generated by a set of at most $r$ elements.","title":"Groups of sectional rank at most $r$"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism/derived-length-bounded-by-rank-and-fixed","kind":"node","lean":"MathlibExt.GroupTheory.PGroupRankAutomorphismDerivedLengthKourovka13Problem56Wanted.derivedLengthBoundedByRankAndFixedPoints","parent":3316,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shalev and Khukhro's question on the derived length of "],["math","\\(p\\)"],["text","-groups with an automorphism"]],"source":"Shalev and Khukhro's question on the derived length of $p$-groups with an automorphism"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism/index.html#derived-length-bounded-by-rank-and-fixed","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism","status":"fully_proved","summary":"The statement that there is a function $f : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ such that for every prime $p$, every finite $p$-group $G$, all natural numbers $r$ and $m$ such that $G$ has sectional rank at most $r$ (every subgroup of $G$ is generated by at most $r$ elements), and every automorphism $\\varphi$ of $G$ with exactly $m$ fixed points, the derived length of $G$ is at most $f(r, m)$, that is, the\u2026","title":"Shalev and Khukhro's question on the derived length of $p$-groups with an automorphism"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism/derived-length-bounded-by-rank-and-fixed-2","kind":"node","lean":"MathlibExt.GroupTheory.PGroupRankAutomorphismDerivedLengthKourovka13Problem56Wanted.derivedLengthBoundedByRankAndFixedPoints_holds","parent":3316,"reading_title":{"format":"inline-parts-v1","parts":[["text","The derived length of a "],["math","\\(p\\)"],["text","-group is bounded by its sectional rank and the fixed points of an automorphism"]],"source":"The derived length of a $p$-group is bounded by its sectional rank and the fixed points of an automorphism"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism/index.html#derived-length-bounded-by-rank-and-fixed-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/derived-length-of-p-groups-with-an-automorphism","status":"can_prove","summary":"The derived length of a $p$-group is bounded by its sectional rank and the fixed points of an automorphism. There is a function $f : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ such that for every prime $p$, every finite $p$-group $G$, all natural numbers $r$ and $m$ such that $G$ has sectional rank at most $r$ (every subgroup of $G$ is generated by at most $r$ elements), and every automorphism $\\varphi$ of $G$\u2026","title":"The derived length of a $p$-group is bounded by its sectional rank and the fixed points of an automorphism"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms","status":"can_prove","title":"Fitting height bounds from automorphisms"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms/fitting-height-bounded-by-order-and-cent","kind":"node","lean":"MathlibExt.GroupTheory.FittingHeightAutomorphismCentralizerKourovka19Problem43Wanted.fittingHeightBoundedByOrderAndCentralizer","parent":3320,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms/index.html#fitting-height-bounded-by-order-and-cent","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms","status":"fully_proved","summary":"The statement that there is a function $f$ assigning a natural number $f(m, n)$ to every pair of natural numbers $m$, $n$ such that every finite soluble group $G$ with an automorphism $\\varphi$ has Fitting height at most $f(|\\varphi|, |C_G(\\varphi)|)$, where $|\\varphi|$ is the order of $\\varphi$ and $|C_G(\\varphi)|$ is the number of elements of $G$ fixed by $\\varphi$.","title":"Kourovka Problem 19.43: Fitting height, automorphism order and centralizer size"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms/fitting-height-bounded-by-order-and-cent-2","kind":"node","lean":"MathlibExt.GroupTheory.FittingHeightAutomorphismCentralizerKourovka19Problem43Wanted.fittingHeightBoundedByOrderAndCentralizer_holds","parent":3320,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms/index.html#fitting-height-bounded-by-order-and-cent-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/fitting-height-bounds-from-automorphisms","status":"can_prove","summary":"There is a function $f$ assigning a natural number $f(m, n)$ to every pair of natural numbers $m$, $n$ such that every finite soluble group $G$ with an automorphism $\\varphi$ has Fitting height at most $f(|\\varphi|, |C_G(\\varphi)|)$, where $|\\varphi|$ is the order of $\\varphi$ and $|C_G(\\varphi)|$ is the number of elements of $G$ fixed by $\\varphi$.","title":"Fitting height is bounded by the order of an automorphism and its centralizer"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups","status":"can_prove","title":"Groups acting freely on abelian groups"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups/free-action-embeds-in-skew-field","kind":"node","lean":"MathlibExt.GroupTheory.FreeActionSkewFieldEmbeddingKourovka16Problem54Wanted.freeActionEmbedsInSkewField","parent":3323,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups/index.html#free-action-embeds-in-skew-field","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups","status":"fully_proved","summary":"The statement, a question of V. D. Mazurov, that every group $G$ with a free action by automorphisms on a nontrivial abelian group $V$, meaning $g \\cdot v \\ne v$ for all $g \\ne 1$ in $G$ and all $v \\ne 0$ in $V$, has an injective homomorphism into the multiplicative group of some division ring.","title":"Kourovka Problem 16.54 (free actions and embeddings in skew-fields)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups/free-action-embeds-in-skew-field-refuted","kind":"node","lean":"MathlibExt.GroupTheory.FreeActionSkewFieldEmbeddingKourovka16Problem54Wanted.freeActionEmbedsInSkewField_refuted","parent":3323,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups/index.html#free-action-embeds-in-skew-field-refuted","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/groups-acting-freely-on-abelian-groups","status":"can_prove","summary":"There is a group $G$ with an action by automorphisms on a nontrivial abelian group $V$ such that $g \\cdot v \\ne v$ for all $g \\ne 1$ in $G$ and all $v \\ne 0$ in $V$, but with no injective homomorphism into the multiplicative group of any division ring. This answers Kourovka Problem 16.54 in the negative.","title":"Groups acting freely on abelian groups need not embed in skew-fields (Kourovka 16.54)"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms","status":"can_prove","title":"Nil-automorphisms"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/delta","kind":"node","lean":"MathlibExt.GroupTheory.EAutomorphismNilAutomorphismKourovka11Problem23Wanted.delta","parent":3326,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/index.html#delta","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms","status":"fully_proved","summary":"For an automorphism $\\varphi$ of a group $G$ and $x \\in G$, $\\delta_\\varphi(x) = x^{-1} \\varphi(x)$. This is the commutator $[x, \\varphi]$ in the Kourovka convention $[x, y] = x^{-1} y^{-1} x y$, with $\\varphi$ acting as $x \\mapsto \\varphi(x)$; the iterates $\\delta_\\varphi^n(x)$ are the iterated commutators $[x, {}_n \\varphi]$, with $[x, {}_0 \\varphi] = x$.","title":"The commutator of an element with an automorphism"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/is-nil-automorphism","kind":"node","lean":"MathlibExt.GroupTheory.EAutomorphismNilAutomorphismKourovka11Problem23Wanted.IsNilAutomorphism","parent":3326,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/index.html#is-nil-automorphism","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms","status":"fully_proved","summary":"An automorphism $\\varphi$ of a group $G$ is a nil-automorphism if for every $a \\in G$ there is $n \\ge 0$ with $\\delta_\\varphi^n(a) = 1$, where $\\delta_\\varphi(x) = x^{-1} \\varphi(x)$; that is, some iterated commutator $[a, {}_n \\varphi]$ is trivial.","title":"Nil-automorphisms"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/is-eautomorphism","kind":"node","lean":"MathlibExt.GroupTheory.EAutomorphismNilAutomorphismKourovka11Problem23Wanted.IsEAutomorphism","parent":3326,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(e\\)"],["text","-automorphisms"]],"source":"$e$-automorphisms"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/index.html#is-eautomorphism","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms","status":"fully_proved","summary":"An automorphism $\\varphi$ of a group $G$ is an $e$-automorphism if for any two subgroups $A$ and $B$ of $G$ with $\\varphi(A) = A$, $\\varphi(B) = B$ and $A \\not\\subseteq B$, some $a \\in A \\setminus B$ has $[a, \\varphi] = a^{-1} \\varphi(a) \\in B$.","title":"$e$-automorphisms"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/e-automorphism-is-nil","kind":"node","lean":"MathlibExt.GroupTheory.EAutomorphismNilAutomorphismKourovka11Problem23Wanted.eAutomorphismIsNil","parent":3326,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vilyatser's question on "],["math","\\(e\\)"],["text","-automorphisms (Kourovka 11.23)"]],"source":"Vilyatser's question on $e$-automorphisms (Kourovka 11.23)"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/index.html#e-automorphism-is-nil","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms","status":"fully_proved","summary":"The proposition that every $e$-automorphism $\\varphi$ of a group $G$ is a nil-automorphism of $G$.","title":"Vilyatser's question on $e$-automorphisms (Kourovka 11.23)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/e-automorphism-is-nil-refuted","kind":"node","lean":"MathlibExt.GroupTheory.EAutomorphismNilAutomorphismKourovka11Problem23Wanted.eAutomorphismIsNil_refuted","parent":3326,"reading_title":{"format":"inline-parts-v1","parts":[["text","An "],["math","\\(e\\)"],["text","-automorphism need not be a nil-automorphism"]],"source":"An $e$-automorphism need not be a nil-automorphism"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms/index.html#e-automorphism-is-nil-refuted","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/nil-automorphisms","status":"can_prove","summary":"Vilyatser's assertion in Kourovka Problem 11.23 is false: there are a group $G$ and an automorphism $\\varphi$ of $G$ that is an $e$-automorphism (for any subgroups $A$ and $B$ of $G$ with $\\varphi(A) = A$, $\\varphi(B) = B$ and $A \\not\\subseteq B$, some $a \\in A \\setminus B$ has $a^{-1} \\varphi(a) \\in B$) but not a nil-automorphism: writing $\\delta(x) = x^{-1} \\varphi(x)$, some $a \\in G$ has $\\delta^n(a) \\ne 1$ for\u2026","title":"An $e$-automorphism need not be a nil-automorphism"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi","status":"can_prove","title":"Prime-order automorphisms with centralizer of finite rank"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/rank-le","kind":"node","lean":"MathlibExt.GroupTheory.PrimeOrderAutomorphismRankNilpotentKourovka13Problem58Wanted.RankLE","parent":3332,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/index.html#rank-le","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi","status":"fully_proved","summary":"A group $G$ has rank at most $r$, for a natural number $r$, when every finitely generated subgroup of $G$ is generated by some set of at most $r$ elements.","title":"Groups of bounded rank"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/fixed-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.PrimeOrderAutomorphismRankNilpotentKourovka13Problem58Wanted.fixedSubgroup","parent":3332,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/index.html#fixed-subgroup","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi","status":"fully_proved","summary":"For an automorphism $\\varphi$ of a group $G$, the centralizer $C_G(\\varphi)$ is the subgroup $\\{x \\in G : \\varphi(x) = x\\}$ of fixed points of $\\varphi$.","title":"Fixed-point subgroup of an automorphism"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/nilpotent-by-bounded-rank-of-centralizer","kind":"node","lean":"MathlibExt.GroupTheory.PrimeOrderAutomorphismRankNilpotentKourovka13Problem58Wanted.nilpotentByBoundedRankOfCentralizerRank","parent":3332,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/index.html#nilpotent-by-bounded-rank-of-centralizer","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi","status":"fully_proved","summary":"The statement that there are functions $f \\colon \\mathbb{N} \\to \\mathbb{N}$ and $g \\colon \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ with the following property. Let $G$ be a nilpotent group in which every element has finite order, let $\\varphi$ be an automorphism of $G$ of prime order $p$, and let $r$ be a natural number such that the fixed-point subgroup $C_G(\\varphi)$ has rank at most $r$, that is, every\u2026","title":"Nilpotent-by-bounded-rank problem for automorphisms of prime order (Kourovka 13.58)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/nilpotent-by-bounded-rank-of-centralizer-2","kind":"node","lean":"MathlibExt.GroupTheory.PrimeOrderAutomorphismRankNilpotentKourovka13Problem58Wanted.nilpotentByBoundedRankOfCentralizerRank_holds","parent":3332,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi/index.html#nilpotent-by-bounded-rank-of-centralizer-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/prime-order-automorphisms-with-centralizer-of-fi","status":"can_prove","summary":"There are functions $f \\colon \\mathbb{N} \\to \\mathbb{N}$ and $g \\colon \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ with the following property. Let $G$ be a nilpotent group in which every element has finite order, let $\\varphi$ be an automorphism of $G$ of prime order $p$, and let $r$ be a natural number such that the fixed-point subgroup $C_G(\\varphi)$ has rank at most $r$, that is, every finitely generated\u2026","title":"Khukhro's theorem on automorphisms of prime order with centralizer of finite rank"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms","status":"can_prove","title":"Squares of 2-automorphisms"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms/two-automorphism-centralizing-sylow-squa","kind":"node","lean":"MathlibExt.GroupTheory.TwoAutomorphismSquareInnerKourovka14Problem1Wanted.twoAutomorphismCentralizingSylowSquareInner","parent":3337,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms/index.html#two-automorphism-centralizing-sylow-squa","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms","status":"fully_proved","summary":"The statement that for every finite group $G$ with no nontrivial normal subgroup of odd order and every automorphism $\\varphi$ of $G$ whose order is a power of $2$, if $\\varphi$ fixes every element of some Sylow $2$-subgroup of $G$, then $\\varphi^2$ is conjugation by some element of $G$, that is, $\\varphi^2$ is an inner automorphism.","title":"Kourovka Problem 14.1"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms/two-automorphism-centralizing-sylow-squa-2","kind":"node","lean":"MathlibExt.GroupTheory.TwoAutomorphismSquareInnerKourovka14Problem1Wanted.twoAutomorphismCentralizingSylowSquareInner_holds","parent":3337,"reading_title":{"format":"inline-parts-v1","parts":[["text","Glauberman's theorem on squares of "],["math","\\(2\\)"],["text","-automorphisms (Kourovka Problem 14.1)"]],"source":"Glauberman's theorem on squares of $2$-automorphisms (Kourovka Problem 14.1)"},"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms/index.html#two-automorphism-centralizing-sylow-squa-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/squares-of-2-automorphisms","status":"can_prove","summary":"Let $G$ be a finite group with no nontrivial normal subgroup of odd order, and let $\\varphi$ be an automorphism of $G$ whose order is a power of $2$. If $\\varphi$ fixes every element of some Sylow $2$-subgroup of $G$, then $\\varphi^2$ is an inner automorphism of $G$, that is, $\\varphi^2$ is conjugation by some $g \\in G$.","title":"Glauberman's theorem on squares of $2$-automorphisms (Kourovka Problem 14.1)"},{"data_url":"scopes/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes.json","id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","kind":"scope","navigation":"module","parent":3307,"result":false,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"can_prove","title":"Twisted conjugacy classes"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/twisted-unit-class","kind":"node","lean":"MathlibExt.GroupTheory.TrivialCentreTwistedClassKourovka18Problem15Wanted.twistedUnitClass","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#twisted-unit-class","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"fully_proved","summary":"For a group $G$ and an automorphism $\\varphi$ of $G$, the $\\varphi$-twisted conjugacy class of the identity is the set $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\} \\subseteq G$.","title":"Twisted conjugacy class of the identity (Kourovka 18.15)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/trivial-centre-has-inner-twisted-class-n","kind":"node","lean":"MathlibExt.GroupTheory.TrivialCentreTwistedClassKourovka18Problem15Wanted.trivialCentreHasInnerTwistedClassNotSubgroup","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#trivial-centre-has-inner-twisted-class-n","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"fully_proved","summary":"Every nontrivial group $G$ with trivial centre has an element $a \\in G$ such that, for the inner automorphism $\\varphi(g) = aga^{-1}$, the twisted conjugacy class of the identity $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\}$ is not a subgroup of $G$.","title":"Kourovka Problem 18.15: an inner twisted class that is not a subgroup"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/trivial-centre-has-inner-twisted-class-n-2","kind":"node","lean":"MathlibExt.GroupTheory.TrivialCentreTwistedClassKourovka18Problem15Wanted.trivialCentreHasInnerTwistedClassNotSubgroup_refuted","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#trivial-centre-has-inner-twisted-class-n-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"can_prove","summary":"The conjecture of Kourovka Problem 18.15 is false: there is a nontrivial group $G$ with trivial centre such that for every $a \\in G$, with $\\varphi$ the inner automorphism $\\varphi(g) = aga^{-1}$, the twisted conjugacy class of the identity $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\}$ is a subgroup of $G$.","title":"Negative answer to Kourovka Problem 18.15"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/twisted-unit-class-2","kind":"node","lean":"MathlibExt.GroupTheory.TwistedUnitClassSubgroupKourovka18Problem14Wanted.twistedUnitClass","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#twisted-unit-class-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"fully_proved","summary":"For a group $G$ and an automorphism $\\varphi$ of $G$, the $\\varphi$-twisted conjugacy class of the identity is the set $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\} \\subseteq G$.","title":"Twisted conjugacy class of the identity (Kourovka 18.14)"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/all-twisted-unit-classes-subgroups","kind":"node","lean":"MathlibExt.GroupTheory.TwistedUnitClassSubgroupKourovka18Problem14Wanted.AllTwistedUnitClassesSubgroups","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#all-twisted-unit-classes-subgroups","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"fully_proved","summary":"A group $G$ has this property when, for every automorphism $\\varphi$ of $G$, the twisted conjugacy class of the identity $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\}$ is a subgroup of $G$.","title":"Groups whose twisted classes of the identity are all subgroups"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/twisted-unit-classes-subgroups-implies-n","kind":"node","lean":"MathlibExt.GroupTheory.TwistedUnitClassSubgroupKourovka18Problem14Wanted.twistedUnitClassesSubgroupsImpliesNilpotent","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#twisted-unit-classes-subgroups-implies-n","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"fully_proved","summary":"Every group $G$ in which, for every automorphism $\\varphi$ of $G$, the twisted conjugacy class of the identity $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\}$ is a subgroup of $G$, is nilpotent.","title":"Kourovka Problem 18.14, nilpotency assertion"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/twisted-unit-classes-subgroups-fgimplies","kind":"node","lean":"MathlibExt.GroupTheory.TwistedUnitClassSubgroupKourovka18Problem14Wanted.twistedUnitClassesSubgroupsFGImpliesAbelian","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#twisted-unit-classes-subgroups-fgimplies","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"fully_proved","summary":"Every finitely generated group $G$ in which, for every automorphism $\\varphi$ of $G$, the twisted conjugacy class of the identity $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\}$ is a subgroup of $G$, is abelian.","title":"Kourovka Problem 18.14, abelian assertion for finitely generated groups"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/twisted-unit-classes-subgroups-implies-n-2","kind":"node","lean":"MathlibExt.GroupTheory.TwistedUnitClassSubgroupKourovka18Problem14Wanted.twistedUnitClassesSubgroupsImpliesNilpotent_refuted","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#twisted-unit-classes-subgroups-implies-n-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"can_prove","summary":"There is a group $G$ that is not nilpotent although, for every automorphism $\\varphi$ of $G$, the twisted conjugacy class of the identity $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\}$ is a subgroup of $G$.","title":"Negative answer to the nilpotency assertion of Kourovka Problem 18.14"},{"id":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/twisted-unit-classes-subgroups-fgimplies-2","kind":"node","lean":"MathlibExt.GroupTheory.TwistedUnitClassSubgroupKourovka18Problem14Wanted.twistedUnitClassesSubgroupsFGImpliesAbelian_refuted","parent":3340,"result":true,"route":"../roadmap/GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes/index.html#twisted-unit-classes-subgroups-fgimplies-2","scope":"GroupTheory/automorphisms/fixed-points-of-automorphisms/twisted-conjugacy-classes","status":"can_prove","summary":"There is a finitely generated group $G$ that is not abelian although, for every automorphism $\\varphi$ of $G$, the twisted conjugacy class of the identity $[e]_\\varphi = \\{g^{-1}\\varphi(g) : g \\in G\\}$ is a subgroup of $G$.","title":"Negative answer to the abelian assertion of Kourovka Problem 18.14"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups","status":"can_prove","summary":"Infinite groups defined by torsion and finiteness conditions: Burnside problems and free Burnside groups, Golod groups, Tarski monsters, locally finite simple groups, periodic Frobenius groups, minimal and weak chain conditions, and embeddings into simple periodic groups.","title":"Periodic and locally finite groups"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"can_prove","title":"Centralizer dimension of locally finite groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/is-locally-finite","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerDimensionSocleQuotientKourovka19Problem101Wanted.IsLocallyFinite","parent":3351,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html#is-locally-finite","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"fully_proved","summary":"A group $G$ is locally finite if every finitely generated subgroup of $G$ is finite.","title":"Locally finite groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/is-locally-soluble","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerDimensionSocleQuotientKourovka19Problem101Wanted.IsLocallySoluble","parent":3351,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html#is-locally-soluble","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is locally soluble if every finitely generated subgroup of $H$ is soluble.","title":"Locally soluble subgroups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/cdim-le","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerDimensionSocleQuotientKourovka19Problem101Wanted.CDimLE","parent":3351,"reading_title":{"format":"inline-parts-v1","parts":[["text","Centralizer dimension at most "],["math","\\(k\\)"]],"source":"Centralizer dimension at most $k$"},"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html#cdim-le","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"fully_proved","summary":"A group $G$ has $c$-dimension at most $k$ if every strictly decreasing chain $C_G(X_0) > C_G(X_1) > \\cdots > C_G(X_m)$ of centralizers of subsets $X_0, \\dots, X_m$ of $G$ has length $m \\le k$.","title":"Centralizer dimension at most $k$"},{"id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/is-minimal-normal","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerDimensionSocleQuotientKourovka19Problem101Wanted.IsMinimalNormal","parent":3351,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html#is-minimal-normal","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"fully_proved","summary":"A subgroup $M$ of a group $Q$ is a minimal normal subgroup if $M$ is normal in $Q$, $M \\ne \\{1\\}$, and every normal subgroup $N$ of $Q$ with $N \\le M$ is either $\\{1\\}$ or $M$.","title":"Minimal normal subgroups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/socle","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerDimensionSocleQuotientKourovka19Problem101Wanted.socle","parent":3351,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html#socle","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"fully_proved","summary":"The socle of a group $Q$ is the subgroup of $Q$ generated by all minimal normal subgroups of $Q$; it is the trivial subgroup when $Q$ has no minimal normal subgroup.","title":"Socle"},{"id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/socle-lift","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerDimensionSocleQuotientKourovka19Problem101Wanted.socleLift","parent":3351,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html#socle-lift","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"fully_proved","summary":"For a normal subgroup $R$ of a group $G$, the preimage in $G$ of the socle of $G/R$ under the quotient map $G \\to G/R$.","title":"Preimage of the socle of a quotient"},{"id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/centralizer-dimension-abelian-section-bo","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerDimensionSocleQuotientKourovka19Problem101Wanted.centralizerDimensionAbelianSectionBound","parent":3351,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html#centralizer-dimension-abelian-section-bo","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"fully_proved","summary":"The statement that there is a function $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every $k \\in \\mathbb{N}$ and every locally finite group $G$ of $c$-dimension at most $k$ the following holds: if $R$ is a normal locally soluble subgroup of $G$ containing every normal locally soluble subgroup of $G$ (the locally soluble radical) and $S$ is the preimage in $G$ of the socle of $G/R$, then there is a subgroup $A$ of\u2026","title":"Vasil'ev's abelian section bound (Kourovka 19.101)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/centralizer-dimension-abelian-section-bo-2","kind":"node","lean":"MathlibExt.GroupTheory.CentralizerDimensionSocleQuotientKourovka19Problem101Wanted.centralizerDimensionAbelianSectionBound_holds","parent":3351,"reading_title":{"format":"inline-parts-v1","parts":[["text","Buturlakin's theorem on locally finite groups of finite "],["math","\\(c\\)"],["text","-dimension"]],"source":"Buturlakin's theorem on locally finite groups of finite $c$-dimension"},"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups/index.html#centralizer-dimension-abelian-section-bo-2","scope":"GroupTheory/periodic-and-locally-finite-groups/centralizer-dimension-of-locally-finite-groups","status":"can_prove","summary":"There is a function $f : \\mathbb{N} \\to \\mathbb{N}$ such that the following holds for every $k \\in \\mathbb{N}$. Let $G$ be a locally finite group of $c$-dimension at most $k$, let $R$ be a normal locally soluble subgroup of $G$ that contains every normal locally soluble subgroup of $G$ (the locally soluble radical), and let $S$ be the preimage in $G$ of the socle of $G/R$. Then there is a subgroup $A$ of $G$ with\u2026","title":"Buturlakin's theorem on locally finite groups of finite $c$-dimension"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups","status":"can_prove","title":"Embeddings into simple periodic groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups/periodic-group-embeds-in-simple-periodic","kind":"node","lean":"MathlibExt.GroupTheory.PeriodicGroupSimpleEmbeddingKourovka11Problem106Wanted.periodicGroupEmbedsInSimplePeriodic","parent":3360,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups/index.html#periodic-group-embeds-in-simple-periodic","scope":"GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups","status":"fully_proved","summary":"The statement, posed by R. Phillips, that every periodic group embeds in a simple periodic group: for every group $G$ in which every element has finite order, there are a simple group $S$ in which every element has finite order and an injective homomorphism $G \\to S$.","title":"Kourovka Problem 11.106 (periodic groups in simple periodic groups)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups/periodic-group-embeds-in-simple-periodic-2","kind":"node","lean":"MathlibExt.GroupTheory.PeriodicGroupSimpleEmbeddingKourovka11Problem106Wanted.periodicGroupEmbedsInSimplePeriodic_holds","parent":3360,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups/index.html#periodic-group-embeds-in-simple-periodic-2","scope":"GroupTheory/periodic-and-locally-finite-groups/embeddings-into-simple-periodic-groups","status":"can_prove","summary":"For every group $G$ in which every element has finite order, there are a simple group $S$ in which every element has finite order and an injective homomorphism $G \\to S$. This answers Kourovka Problem 11.106 (R. Phillips) in the affirmative; the result is due to A. Yu. Olshanskii (1992).","title":"Every periodic group embeds in a simple periodic group"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups","status":"can_prove","title":"Free Burnside groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups/free-burnside-group","kind":"node","lean":"MathlibExt.GroupTheory.FreeBurnsideCyclicAutomorphismFixedPointKourovka17Problem70Wanted.FreeBurnsideGroup","parent":3363,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups/index.html#free-burnside-group","scope":"GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups","status":"fully_proved","summary":"For natural numbers $m$ and $n$, the free Burnside group $B(m, n)$ is the quotient of the free group $F_m$ on $m$ generators by the normal subgroup generated by all $n$-th powers $w^n$ with $w \\in F_m$.","title":"Free Burnside group"},{"id":"GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups/cyclic-automorphism-fixes-nontrivial","kind":"node","lean":"MathlibExt.GroupTheory.FreeBurnsideCyclicAutomorphismFixedPointKourovka17Problem70Wanted.cyclicAutomorphismFixesNontrivial","parent":3363,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups/index.html#cyclic-automorphism-fixes-nontrivial","scope":"GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups","status":"fully_proved","summary":"The assertion that for all primes $q$ and $p$ such that the free Burnside group $B(q, p)$ of rank $q$ and exponent $p$ is infinite, the automorphism of $B(q, p)$ induced by the cyclic permutation $x_i \\mapsto x_{i+1}$ (indices mod $q$) of its free generators $x_0, \\dots, x_{q-1}$ fixes some nontrivial element of $B(q, p)$.","title":"Mazurov's fixed-point question for free Burnside groups (Kourovka 17.70)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups/cyclic-automorphism-fixes-nontrivial-ref","kind":"node","lean":"MathlibExt.GroupTheory.FreeBurnsideCyclicAutomorphismFixedPointKourovka17Problem70Wanted.cyclicAutomorphismFixesNontrivial_refuted","parent":3363,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups/index.html#cyclic-automorphism-fixes-nontrivial-ref","scope":"GroupTheory/periodic-and-locally-finite-groups/free-burnside-groups","status":"can_prove","summary":"There exist primes $q$ and $p$ such that the free Burnside group $B(q, p)$ of rank $q$ and exponent $p$ is infinite and the automorphism of $B(q, p)$ induced by the cyclic permutation $x_i \\mapsto x_{i+1}$ (indices mod $q$) of its free generators $x_0, \\dots, x_{q-1}$ fixes no nontrivial element of $B(q, p)$.","title":"Mazurov's fixed-point question for free Burnside groups has a negative answer"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/golod-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups/golod-groups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/golod-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/golod-groups","status":"can_prove","title":"Golod groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/golod-groups/golod-group","kind":"node","lean":"MathlibExt.GroupTheory.GolodGroupQuotientResiduallyFiniteKourovka12Problem102Wanted.golodGroup","parent":3367,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/golod-groups/index.html#golod-group","scope":"GroupTheory/periodic-and-locally-finite-groups/golod-groups","status":"fully_proved","summary":"Let $k$ be a field, $N$ a non-unital associative $k$-algebra and $y_1, \\dots, y_r \\in N$. The Golod group of $y$ is the subgroup of the group of units of the unitization $k \\oplus N$ generated by the units that are equal to $1 + y_i$ for some $i$. When every $1 + y_i$ is invertible (for example when $N$ is nil), this is the subgroup $\\langle 1 + y_1, \\dots, 1 + y_r \\rangle$ of the adjoint group $1 + N$.","title":"Golod group (Kourovka 12.102)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/golod-groups/golod-proper-quotient-residually-finite","kind":"node","lean":"MathlibExt.GroupTheory.GolodGroupQuotientResiduallyFiniteKourovka12Problem102Wanted.golodProperQuotientResiduallyFinite","parent":3367,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/golod-groups/index.html#golod-proper-quotient-residually-finite","scope":"GroupTheory/periodic-and-locally-finite-groups/golod-groups","status":"fully_proved","summary":"This is the statement that the following holds for every field $k$ of prime characteristic $p$, every non-unital associative $k$-algebra $N$, every $r \\ge 2$ and all $y_1, \\dots, y_r \\in N$ such that $N$ is generated by $y_1, \\dots, y_r$ as a non-unital $k$-algebra, every element of $N$ is nilpotent, and for every natural number $n$ some product of $n$ elements of $N$ is nonzero: if $\\Gamma$ is the Golod group of\u2026","title":"Residual finiteness of proper quotients of Golod groups (Kourovka 12.102)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/golod-groups/golod-proper-quotient-residually-finite-2","kind":"node","lean":"MathlibExt.GroupTheory.GolodGroupQuotientResiduallyFiniteKourovka12Problem102Wanted.golodProperQuotientResiduallyFinite_refuted","parent":3367,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/golod-groups/index.html#golod-proper-quotient-residually-finite-2","scope":"GroupTheory/periodic-and-locally-finite-groups/golod-groups","status":"can_prove","summary":"The statement that every proper factor group of a Golod group is residually finite is false: there exist a field $k$ of prime characteristic $p$, an integer $r \\ge 2$, a non-unital associative $k$-algebra $N$ and elements $y_1, \\dots, y_r \\in N$ such that $N$ is generated by $y_1, \\dots, y_r$ as a non-unital $k$-algebra, every element of $N$ is nilpotent and for every natural number $n$ some product of $n$ elements\u2026","title":"Not every proper quotient of a Golod group is residually finite"},{"id":"GroupTheory/periodic-and-locally-finite-groups/golod-groups/golod-group-2","kind":"node","lean":"MathlibExt.GroupTheory.GolodGroupTrivialCentreKourovka11Problem101Wanted.golodGroup","parent":3367,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/golod-groups/index.html#golod-group-2","scope":"GroupTheory/periodic-and-locally-finite-groups/golod-groups","status":"fully_proved","summary":"Let $k$ be a field, $N$ a non-unital associative $k$-algebra and $y_1, \\dots, y_r \\in N$. The Golod group of $y$ is the subgroup of the group of units of the unitization $k \\oplus N$ generated by the units that are equal to $1 + y_i$ for some $i$. When every $1 + y_i$ is invertible (for example when $N$ is nil), this is the subgroup $\\langle 1 + y_1, \\dots, 1 + y_r \\rangle$ of the adjoint group $1 + N$.","title":"Golod group (Kourovka 11.101)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/golod-groups/golod-group-with-trivial-centre","kind":"node","lean":"MathlibExt.GroupTheory.GolodGroupTrivialCentreKourovka11Problem101Wanted.golodGroupWithTrivialCentre","parent":3367,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/golod-groups/index.html#golod-group-with-trivial-centre","scope":"GroupTheory/periodic-and-locally-finite-groups/golod-groups","status":"fully_proved","summary":"Kourovka 11.101 (A. V. Timofeenko), in the strong form proved: some Golod group has trivial centre (the problem asks for finite centre). A Golod group (9.76) is `\u27e81 + x\u2081 + I, \u2026, 1 + x_r + I\u27e9 \u2264 1 + F/I` for `r \u2265 2`, `F` the free associative algebra without constant term on `x\u2081, \u2026, x_r` over a field of characteristic `p > 0` and `F/I` a non-nilpotent nil-algebra; equivalently `N = F/I` is a non-unital algebra\u2026","title":"Existence of a Golod group with trivial centre"},{"id":"GroupTheory/periodic-and-locally-finite-groups/golod-groups/golod-group-with-trivial-centre-holds","kind":"node","lean":"MathlibExt.GroupTheory.GolodGroupTrivialCentreKourovka11Problem101Wanted.golodGroupWithTrivialCentre_holds","parent":3367,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/golod-groups/index.html#golod-group-with-trivial-centre-holds","scope":"GroupTheory/periodic-and-locally-finite-groups/golod-groups","status":"can_prove","summary":"There exist a field $k$ of prime characteristic $p$, an integer $r \\ge 2$, a non-unital associative $k$-algebra $N$ and elements $y_1, \\dots, y_r \\in N$ such that $N$ is generated by $y_1, \\dots, y_r$ as a non-unital $k$-algebra, every element of $N$ is nilpotent, for every natural number $n$ some product of $n$ elements of $N$ is nonzero, and the Golod group of $y$ has trivial centre. Since a trivial centre is\u2026","title":"There is a Golod group with trivial centre"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4.json","id":"GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4","status":"can_prove","title":"Groups of exponent 4"},{"id":"GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4/exponent-four-detected-on-bounded-subset","kind":"node","lean":"MathlibExt.GroupTheory.ExponentFourBoundedGenerationKourovka13Problem10Wanted.exponentFourDetectedOnBoundedSubsets","parent":3374,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4/index.html#exponent-four-detected-on-bounded-subset","scope":"GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4","status":"fully_proved","summary":"The proposition that there is a function $f : \\mathbb{N} \\to \\mathbb{N}$ with the following property: for every $k \\in \\mathbb{N}$, every group $G$ whose $k$-th derived subgroup $G^{(k)}$ is trivial (a soluble group of derived length at most $k$), and every generating set $A$ of $G$, if $x^4 = 1$ for every element $x$ of every subgroup generated by a finite subset of $A$ with at most $f(k)$ elements, then $x^4 = 1$\u2026","title":"Exponent 4 detected on boundedly generated subgroups (Kourovka 13.10)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4/exponent-four-detected-on-bounded-subset-2","kind":"node","lean":"MathlibExt.GroupTheory.ExponentFourBoundedGenerationKourovka13Problem10Wanted.exponentFourDetectedOnBoundedSubsets_refuted","parent":3374,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4/index.html#exponent-four-detected-on-bounded-subset-2","scope":"GroupTheory/periodic-and-locally-finite-groups/groups-of-exponent-4","status":"can_prove","summary":"There is no function $f : \\mathbb{N} \\to \\mathbb{N}$ with the following property: for every $k \\in \\mathbb{N}$, every group $G$ whose $k$-th derived subgroup $G^{(k)}$ is trivial, and every generating set $A$ of $G$, if $x^4 = 1$ for every element $x$ of every subgroup generated by a finite subset of $A$ with at most $f(k)$ elements, then $x^4 = 1$ for every $x \\in G$. This is the negative answer to Kourovka\u2026","title":"Exponent 4 is not detected on boundedly generated subgroups"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties.json","id":"GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties","status":"can_prove","title":"Higman embedding in Burnside varieties"},{"id":"GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties/burnside-presented","kind":"node","lean":"MathlibExt.GroupTheory.BurnsideHigmanEmbeddingKourovka12Problem32Wanted.BurnsidePresented","parent":3377,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties/index.html#burnside-presented","scope":"GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties","status":"fully_proved","summary":"For natural numbers $n$ and $m$ and a finite set $S$ of elements of the free group on $x_0, \\dots, x_{m-1}$, this is the group with generators $x_0, \\dots, x_{m-1}$ and relators $S$ together with $w^n$ for every element $w$ of the free group: the group finitely presented by $S$ in the Burnside variety $\\mathfrak{B}_n$ of groups of exponent $n$.","title":"Groups finitely presented in a Burnside variety"},{"id":"GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties/higman-embedding-burnside-variety","kind":"node","lean":"MathlibExt.GroupTheory.BurnsideHigmanEmbeddingKourovka12Problem32Wanted.higmanEmbeddingBurnsideVariety","parent":3377,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties/index.html#higman-embedding-burnside-variety","scope":"GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties","status":"fully_proved","summary":"There is $n_0$ such that for every odd $n \\ge n_0$ the following holds. Let $G$ be a group given by generators $x_0, x_1, \\dots$ and a recursively enumerable set of relators, and suppose that $g^n = 1$ for every $g \\in G$. Then for some $m$ and some finite set $S$ of elements of the free group on $x_0, \\dots, x_{m-1}$, there is an injective homomorphism from $G$ to the group finitely presented by $S$ in the\u2026","title":"Kourovka Problem 12.32: Higman embedding for Burnside varieties"},{"id":"GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties/higman-embedding-burnside-variety-holds","kind":"node","lean":"MathlibExt.GroupTheory.BurnsideHigmanEmbeddingKourovka12Problem32Wanted.higmanEmbeddingBurnsideVariety_holds","parent":3377,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties/index.html#higman-embedding-burnside-variety-holds","scope":"GroupTheory/periodic-and-locally-finite-groups/higman-embedding-in-burnside-varieties","status":"can_prove","summary":"There is $n_0$ such that for every odd $n \\ge n_0$, every group of exponent $n$ given by generators $x_0, x_1, \\dots$ and a recursively enumerable set of relators embeds in a group finitely presented in the Burnside variety $\\mathfrak{B}_n$: a group with generators $x_0, \\dots, x_{m-1}$ for some $m$ whose relators are a finite set $S$ together with $w^n$ for every word $w$.","title":"Higman embedding theorem for Burnside varieties"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups","status":"can_prove","title":"Infinite 2-group quotients of triangle groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups/wiegold-group","kind":"node","lean":"MathlibExt.GroupTheory.TriangleGroupTwoGroupQuotientsKourovka16Problem101Wanted.wiegoldGroup","parent":3381,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups/index.html#wiegold-group","scope":"GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups","status":"fully_proved","summary":"The $(2, 4, 8)$ triangle group is the group with presentation $\\langle x, y \\mid x^2 = y^4 = (xy)^8 = 1 \\rangle$.","title":"The (2, 4, 8) triangle group"},{"id":"GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups/uncountably-many-infinite-two-group-quot","kind":"node","lean":"MathlibExt.GroupTheory.TriangleGroupTwoGroupQuotientsKourovka16Problem101Wanted.uncountablyManyInfiniteTwoGroupQuotients","parent":3381,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups/index.html#uncountably-many-infinite-two-group-quot","scope":"GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups","status":"fully_proved","summary":"The statement that there are an uncountable index set $I$ and normal subgroups $N_i$ of $\\Gamma = \\langle x, y \\mid x^2 = y^4 = (xy)^8 = 1 \\rangle$, for $i \\in I$, such that every quotient $\\Gamma / N_i$ is an infinite $2$-group (a group in which every element has order a power of $2$), and $\\Gamma / N_i \\cong \\Gamma / N_j$ only when $i = j$.","title":"Wiegold's question on 2-group quotients of the (2, 4, 8) triangle group (Kourovka 16.101)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups/uncountably-many-infinite-two-group-quot-2","kind":"node","lean":"MathlibExt.GroupTheory.TriangleGroupTwoGroupQuotientsKourovka16Problem101Wanted.uncountablyManyInfiniteTwoGroupQuotients_holds","parent":3381,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups/index.html#uncountably-many-infinite-two-group-quot-2","scope":"GroupTheory/periodic-and-locally-finite-groups/infinite-2-group-quotients-of-triangle-groups","status":"can_prove","summary":"Let $\\Gamma = \\langle x, y \\mid x^2 = y^4 = (xy)^8 = 1 \\rangle$. There are an uncountable index set $I$ and normal subgroups $N_i$ of $\\Gamma$, for $i \\in I$, such that every quotient $\\Gamma / N_i$ is an infinite $2$-group (a group in which every element has order a power of $2$), and $\\Gamma / N_i \\cong \\Gamma / N_j$ only when $i = j$. So Wiegold's question has a positive answer.","title":"Uncountably many infinite 2-group quotients of the (2, 4, 8) triangle group (Kourovka 16.101)"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups","status":"can_prove","title":"Involutions in periodic groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups/periodic-group-generated-by-odd-involuti","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionClassOddProductsKourovka10Problem62Wanted.periodicGroupGeneratedByOddInvolutionClass","parent":3385,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups/index.html#periodic-group-generated-by-odd-involuti","scope":"GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups","status":"fully_proved","summary":"The proposition (Kourovka 10.62, A. I. Sozutov) that there is a periodic group $G$ that has no subgroup of index $2$ and contains an involution $x$ such that the conjugacy class $X$ of $x$ generates $G$ and the product $yz$ has odd order for all $y, z \\in X$.","title":"Kourovka Problem 10.62 on involution classes with products of odd order"},{"id":"GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups/periodic-group-generated-by-odd-involuti-2","kind":"node","lean":"MathlibExt.GroupTheory.InvolutionClassOddProductsKourovka10Problem62Wanted.periodicGroupGeneratedByOddInvolutionClass_holds","parent":3385,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups/index.html#periodic-group-generated-by-odd-involuti-2","scope":"GroupTheory/periodic-and-locally-finite-groups/involutions-in-periodic-groups","status":"can_prove","summary":"There is a periodic group $G$ that has no subgroup of index $2$ and contains an involution $x$ such that the conjugacy class $X$ of $x$ generates $G$ and the product $yz$ has odd order for all $y, z \\in X$.","title":"Positive answer to Kourovka Problem 10.62"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups","status":"can_prove","title":"Locally finite simple groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/is-locally-finite","kind":"node","lean":"MathlibExt.GroupTheory.LocallyFiniteSimpleAbsolutelySimpleKourovka11Problem108Wanted.IsLocallyFinite","parent":3388,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/index.html#is-locally-finite","scope":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups","status":"fully_proved","summary":"A group $G$ is locally finite when every subgroup of $G$ generated by finitely many elements is finite.","title":"Locally finite group"},{"id":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/is-composition-series","kind":"node","lean":"MathlibExt.GroupTheory.LocallyFiniteSimpleAbsolutelySimpleKourovka11Problem108Wanted.IsCompositionSeries","parent":3388,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/index.html#is-composition-series","scope":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups","status":"fully_proved","summary":"A set $S$ of subgroups of a group $G$ is a composition series of $G$ when it contains the trivial subgroup and $G$, it is totally ordered by inclusion, it contains the join and the intersection of every subset of $S$, and for every jump $A < B$ of $S$ (members $A \\subsetneq B$ of $S$ with no member of $S$ strictly between them) $A$ is normal in $B$ and the quotient $B / A$ is a simple group.","title":"Composition series in Kurosh's sense"},{"id":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/locally-finite-simple-is-absolutely-simp","kind":"node","lean":"MathlibExt.GroupTheory.LocallyFiniteSimpleAbsolutelySimpleKourovka11Problem108Wanted.locallyFiniteSimpleIsAbsolutelySimple","parent":3388,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/index.html#locally-finite-simple-is-absolutely-simp","scope":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups","status":"fully_proved","summary":"The statement, asked by R. Phillips, that every locally finite simple group $G$ is absolutely simple: every composition series of $G$ in Kurosh's sense consists of the trivial subgroup and $G$ alone.","title":"Absolute simplicity of locally finite simple groups (Kourovka 11.108)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/locally-finite-simple-is-absolutely-simp-2","kind":"node","lean":"MathlibExt.GroupTheory.LocallyFiniteSimpleAbsolutelySimpleKourovka11Problem108Wanted.locallyFiniteSimpleIsAbsolutelySimple_refuted","parent":3388,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups/index.html#locally-finite-simple-is-absolutely-simp-2","scope":"GroupTheory/periodic-and-locally-finite-groups/locally-finite-simple-groups","status":"can_prove","summary":"There is a locally finite simple group $G$ with a composition series, in Kurosh's sense, other than the series consisting of the trivial subgroup and $G$ alone. That is, the statement of Kourovka Problem 11.108, that every locally finite simple group is absolutely simple, is false.","title":"Locally finite simple groups need not be absolutely simple"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers.json","id":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers","status":"can_prove","title":"Minimal condition on centralizers"},{"id":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/is-locally-finite","kind":"node","lean":"MathlibExt.GroupTheory.TwoGroupMinimalConditionCentralizersKourovka11Problem103Wanted.IsLocallyFinite","parent":3393,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/index.html#is-locally-finite","scope":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers","status":"fully_proved","summary":"A group $G$ is locally finite when for every finite subset $S$ of $G$ the subgroup generated by $S$ is finite, that is, every finitely generated subgroup of $G$ is finite.","title":"Locally finite group"},{"id":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/minimal-condition-on-centralizers","kind":"node","lean":"MathlibExt.GroupTheory.TwoGroupMinimalConditionCentralizersKourovka11Problem103Wanted.MinimalConditionOnCentralizers","parent":3393,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/index.html#minimal-condition-on-centralizers","scope":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers","status":"fully_proved","summary":"For a subset $X$ of a group $G$, write $C_G(X)$ for the centralizer of $X$ in $G$. The group $G$ satisfies the minimal condition on centralizers when there is no sequence of subsets $X_0, X_1, X_2, \\ldots$ of $G$ whose centralizers form a strictly descending chain $C_G(X_0) > C_G(X_1) > C_G(X_2) > \\cdots$, where $H > K$ means that $K$ is a proper subgroup of $H$.","title":"Minimal condition on centralizers"},{"id":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/two-group-minimal-centralizers-locally-f","kind":"node","lean":"MathlibExt.GroupTheory.TwoGroupMinimalConditionCentralizersKourovka11Problem103Wanted.twoGroupMinimalCentralizersLocallyFinite","parent":3393,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wilson's question on "],["math","\\(2\\)"],["text","-groups with the minimal condition on centralizers (Kourovka problem 11.103)"]],"source":"Wilson's question on $2$-groups with the minimal condition on centralizers (Kourovka problem 11.103)"},"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/index.html#two-group-minimal-centralizers-locally-f","scope":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers","status":"fully_proved","summary":"Wilson's question on $2$-groups with the minimal condition on centralizers (Kourovka problem 11.103). The statement that every group $G$ in which every element has order a power of $2$ and which satisfies the minimal condition on centralizers is locally finite.","title":"Wilson's question on $2$-groups with the minimal condition on centralizers (Kourovka problem 11.103)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/two-group-minimal-centralizers-locally-f-2","kind":"node","lean":"MathlibExt.GroupTheory.TwoGroupMinimalConditionCentralizersKourovka11Problem103Wanted.twoGroupMinimalCentralizersLocallyFinite_holds","parent":3393,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every "],["math","\\(2\\)"],["text","-group with the minimal condition on centralizers is locally finite (Kourovka 11.103)"]],"source":"Every $2$-group with the minimal condition on centralizers is locally finite (Kourovka 11.103)"},"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers/index.html#two-group-minimal-centralizers-locally-f-2","scope":"GroupTheory/periodic-and-locally-finite-groups/minimal-condition-on-centralizers","status":"can_prove","summary":"Every group $G$ in which every element has order a power of $2$ and which satisfies the minimal condition on centralizers is locally finite.","title":"Every $2$-group with the minimal condition on centralizers is locally finite (Kourovka 11.103)"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind.json","id":"GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind","status":"can_prove","title":"Monsters of the first kind"},{"id":"GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind/is-monster-first-kind","kind":"node","lean":"MathlibExt.GroupTheory.MonsterTorsionFreeCentralExtensionKourovka14Problem104Wanted.IsMonsterFirstKind","parent":3398,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind/index.html#is-monster-first-kind","scope":"GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind","status":"fully_proved","summary":"A group $G$ is a *monster of the first kind* (V. P. Shunkov) when $G$ is infinite, $G$ has an element of order greater than $2$, and for every element $a$ of order greater than $2$ and every proper subgroup $H$ of $G$ there is $g \\in G \\setminus H$ such that $a$ and $g^{-1} a g$ generate $G$. Here an element of infinite order counts as having order greater than $2$.","title":"Monster of the first kind"},{"id":"GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind/monster-has-torsion-free-cyclic-central","kind":"node","lean":"MathlibExt.GroupTheory.MonsterTorsionFreeCentralExtensionKourovka14Problem104Wanted.monsterHasTorsionFreeCyclicCentralExtension","parent":3398,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind/index.html#monster-has-torsion-free-cyclic-central","scope":"GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind","status":"fully_proved","summary":"Kourovka Problem 14.104 (V. P. Shunkov) asks whether this proposition holds: for every monster of the first kind $M$ there are a torsion-free group $E$ and a cyclic subgroup $C$ contained in the center of $E$ such that $E / C \\cong M$.","title":"Shunkov's question on central extensions of monsters"},{"id":"GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind/monster-has-torsion-free-cyclic-central-2","kind":"node","lean":"MathlibExt.GroupTheory.MonsterTorsionFreeCentralExtensionKourovka14Problem104Wanted.monsterHasTorsionFreeCyclicCentralExtension_refuted","parent":3398,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind/index.html#monster-has-torsion-free-cyclic-central-2","scope":"GroupTheory/periodic-and-locally-finite-groups/monsters-of-the-first-kind","status":"can_prove","summary":"It is not the case that for every monster of the first kind $M$ there are a torsion-free group $E$ and a cyclic subgroup $C$ contained in the center of $E$ with $E / C \\cong M$. So Kourovka Problem 14.104 has a negative answer.","title":"Not every monster of the first kind is a torsion-free group modulo a central cyclic subgroup"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups.json","id":"GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups","status":"can_prove","title":"Periodic Frobenius groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups/is-frobenius-group","kind":"node","lean":"MathlibExt.GroupTheory.FrobeniusGroupShared.IsFrobeniusGroup","parent":3402,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups/index.html#is-frobenius-group","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups","status":"fully_proved","summary":"For subgroups $F$ and $H$ of a group $G$, $G$ is a Frobenius group with kernel $F$ and complement $H$ (Kourovka 6.53) if $G = F \\rtimes H$ (that is, $F$ is normal in $G$, $F \\cap H = 1$ and $FH = G$), $F \\ne 1$, $H \\ne 1$, $H \\cap gHg^{-1} = 1$ for every $g \\in G \\setminus H$, and the non-trivial elements of $F$ are exactly the elements of $G$ that lie in no conjugate $gHg^{-1}$ of $H$.","title":"Frobenius group with given kernel and complement"},{"id":"GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups/periodic-self-centralizing-abelian-norma","kind":"node","lean":"MathlibExt.GroupTheory.PeriodicSelfCentralizingAbelianFrobeniusKourovka20Problem96Wanted.periodicSelfCentralizingAbelianNormalFrobenius","parent":3402,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups/index.html#periodic-self-centralizing-abelian-norma","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups","status":"fully_proved","summary":"The statement, posed by A. I. Sozutov, that for every group $G$ in which every element has finite order and every normal abelian subgroup $A$ of $G$ that is neither trivial nor all of $G$ and such that for every $a \\in A$ with $a \\ne 1$ the centralizer $C_G(a)$ of $a$ in $G$ is contained in $A$, there are subgroups $F$ and $H$ of $G$ such that $G$ is a Frobenius group with kernel $F$ and complement $H$.","title":"Kourovka Problem 20.96 (periodic Frobenius groups)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups/periodic-self-centralizing-abelian-norma-2","kind":"node","lean":"MathlibExt.GroupTheory.PeriodicSelfCentralizingAbelianFrobeniusKourovka20Problem96Wanted.periodicSelfCentralizingAbelianNormalFrobenius_holds","parent":3402,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups/index.html#periodic-self-centralizing-abelian-norma-2","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-frobenius-groups","status":"can_prove","summary":"Let $G$ be a group in which every element has finite order, and let $A$ be a normal abelian subgroup of $G$ that is neither trivial nor all of $G$ and such that for every $a \\in A$ with $a \\ne 1$ the centralizer $C_G(a)$ of $a$ in $G$ is contained in $A$. Then there are subgroups $F$ and $H$ of $G$ such that $G$ is a Frobenius group with kernel $F$ and complement $H$. This answers Kourovka Problem 20.96 (A. I.\u2026","title":"Lytkina\u2013Mazurov theorem on periodic Frobenius groups"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions.json","id":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions","status":"can_prove","title":"Periodic groups with {3, 5}-involutions"},{"id":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/is-involution","kind":"node","lean":"MathlibExt.GroupTheory.ThreeFiveInvolutionsKourovka19Problem36Wanted.IsInvolution","parent":3406,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/index.html#is-involution","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions","status":"fully_proved","summary":"An element $x$ of a group $G$ is an involution when $x^2 = 1$ and $x \\ne 1$.","title":"Involution"},{"id":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/special-unitary-group-three-four","kind":"node","lean":"MathlibExt.GroupTheory.ThreeFiveInvolutionsKourovka19Problem36Wanted.specialUnitaryGroupThreeFour","parent":3406,"reading_title":{"format":"inline-parts-v1","parts":[["text","The special unitary group "],["math","\\(\\mathrm{SU}_3(4)\\)"]],"source":"The special unitary group $\\mathrm{SU}_3(4)$"},"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/index.html#special-unitary-group-three-four","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions","status":"fully_proved","summary":"Let $\\mathbb{F}_{16}$ be the field with $16$ elements and, for a matrix $A$ over $\\mathbb{F}_{16}$, let $\\bar{A}$ be the matrix obtained by applying the involutive field automorphism $x \\mapsto x^4$ of $\\mathbb{F}_{16}$ to each entry. Then $\\mathrm{SU}_3(4)$ is the subgroup of $\\mathrm{GL}_3(\\mathbb{F}_{16})$ generated by the invertible $3 \\times 3$ matrices $A$ over $\\mathbb{F}_{16}$ with $\\det A = 1$ and\u2026","title":"The special unitary group $\\mathrm{SU}_3(4)$"},{"id":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/psuthree-four","kind":"node","lean":"MathlibExt.GroupTheory.ThreeFiveInvolutionsKourovka19Problem36Wanted.PSUThreeFour","parent":3406,"reading_title":{"format":"inline-parts-v1","parts":[["text","The projective special unitary group "],["math","\\(\\mathrm{PSU}_3(4)\\)"]],"source":"The projective special unitary group $\\mathrm{PSU}_3(4)$"},"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/index.html#psuthree-four","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions","status":"fully_proved","summary":"It is the quotient $\\mathrm{PSU}_3(4) = \\mathrm{SU}_3(4) / Z(\\mathrm{SU}_3(4))$ of the special unitary group $\\mathrm{SU}_3(4)$ by its centre $Z(\\mathrm{SU}_3(4))$.","title":"The projective special unitary group $\\mathrm{PSU}_3(4)$"},{"id":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/three-five-involutions-classification","kind":"node","lean":"MathlibExt.GroupTheory.ThreeFiveInvolutionsKourovka19Problem36Wanted.threeFiveInvolutionsClassification","parent":3406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Jabara's conjecture on groups with "],["math","\\(\\{3, 5\\}\\)"],["text","-involutions"]],"source":"Jabara's conjecture on groups with $\\{3, 5\\}$-involutions"},"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/index.html#three-five-involutions-classification","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions","status":"fully_proved","summary":"Kourovka 19.36 (E. Jabara): let $G$ be a periodic group (every element has finite order) that is generated by its involutions (elements $x \\ne 1$ with $x^2 = 1$). Suppose that the order of $xy$ lies in $\\{1, 2, 3, 5\\}$ for all involutions $x, y$ of $G$, and that each of these four values is the order of $xy$ for some involutions $x, y$ of $G$. Then $G$ is isomorphic to the alternating group $A_5$ or to the\u2026","title":"Jabara's conjecture on groups with $\\{3, 5\\}$-involutions"},{"id":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/three-five-involutions-classification-ho","kind":"node","lean":"MathlibExt.GroupTheory.ThreeFiveInvolutionsKourovka19Problem36Wanted.threeFiveInvolutionsClassification_holds","parent":3406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Jabara's conjecture on groups with "],["math","\\(\\{3, 5\\}\\)"],["text","-involutions holds"]],"source":"Jabara's conjecture on groups with $\\{3, 5\\}$-involutions holds"},"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions/index.html#three-five-involutions-classification-ho","scope":"GroupTheory/periodic-and-locally-finite-groups/periodic-groups-with-3-5-involutions","status":"can_prove","summary":"Let $G$ be a periodic group (every element has finite order) that is generated by its involutions (elements $x \\ne 1$ with $x^2 = 1$). Suppose that the order of $xy$ lies in $\\{1, 2, 3, 5\\}$ for all involutions $x, y$ of $G$, and that each of these four values is the order of $xy$ for some involutions $x, y$ of $G$. Then $G$ is isomorphic to the alternating group $A_5$ or to the projective special unitary group\u2026","title":"Jabara's conjecture on groups with $\\{3, 5\\}$-involutions holds"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group.json","id":"GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group","status":"can_prove","title":"Products of abelian and locally quaternion groups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group/is-locally-quaternion","kind":"node","lean":"MathlibExt.GroupTheory.AbelianLocallyQuaternionProductKourovka18Problem95Wanted.IsLocallyQuaternion","parent":3412,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group/index.html#is-locally-quaternion","scope":"GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group","status":"fully_proved","summary":"A subgroup $B$ of a group $G$ is locally quaternion if there is a strictly increasing sequence $C_0 < C_1 < C_2 < \\cdots$ of subgroups of $G$, each isomorphic to a generalized quaternion group $Q_{2^{k+2}}$ of order $2^{k+2}$ for some $k \\ge 1$, whose supremum (for a chain, the union) is $B$. In particular $B$ is infinite.","title":"Locally quaternion subgroup"},{"id":"GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group/abelian-locally-quaternion-product-solub","kind":"node","lean":"MathlibExt.GroupTheory.AbelianLocallyQuaternionProductKourovka18Problem95Wanted.abelianLocallyQuaternionProductSoluble","parent":3412,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group/index.html#abelian-locally-quaternion-product-solub","scope":"GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group","status":"fully_proved","summary":"The proposition that every group $G$ with subgroups $A$ and $B$ such that $A$ is abelian, $B$ is locally quaternion, and $G = AB$ (every element of $G$ is a product $ab$ with $a \\in A$ and $b \\in B$) is soluble.","title":"Kourovka Problem 18.95 (Sozutov)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group/abelian-locally-quaternion-product-solub-2","kind":"node","lean":"MathlibExt.GroupTheory.AbelianLocallyQuaternionProductKourovka18Problem95Wanted.abelianLocallyQuaternionProductSoluble_holds","parent":3412,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group/index.html#abelian-locally-quaternion-product-solub-2","scope":"GroupTheory/periodic-and-locally-finite-groups/products-of-abelian-and-locally-quaternion-group","status":"can_prove","summary":"Let $G$ be a group with subgroups $A$ and $B$ such that $A$ is abelian, $B$ is locally quaternion, and $G = AB$, that is, every element of $G$ is a product $ab$ with $a \\in A$ and $b \\in B$. Then $G$ is soluble.","title":"Products of abelian and locally quaternion groups are soluble"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups.json","id":"GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups","status":"can_prove","title":"Products of periodic subgroups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups/subgroup-in-periodic-product-is-periodic","kind":"node","lean":"MathlibExt.GroupTheory.PeriodicSubgroupsInProductSetKourovka16Problem38Wanted.subgroupInPeriodicProductIsPeriodic","parent":3416,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups/index.html#subgroup-in-periodic-product-is-periodic","scope":"GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups","status":"fully_proved","summary":"The statement, posed by F. de Giovanni, that for every soluble group $G$ and all subgroups $A$ and $B$ of $G$ in which every element has finite order, every subgroup $H$ of $G$ contained in the set $AB$ of all products $ab$ with $a \\in A$ and $b \\in B$ has every element of finite order.","title":"Kourovka Problem 16.38 (subgroups inside a product of periodic subgroups)"},{"id":"GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups/subgroup-in-periodic-product-is-periodic-2","kind":"node","lean":"MathlibExt.GroupTheory.PeriodicSubgroupsInProductSetKourovka16Problem38Wanted.subgroupInPeriodicProductIsPeriodic_holds","parent":3416,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups/index.html#subgroup-in-periodic-product-is-periodic-2","scope":"GroupTheory/periodic-and-locally-finite-groups/products-of-periodic-subgroups","status":"can_prove","summary":"Let $G$ be a soluble group and let $A$ and $B$ be subgroups of $G$ in which every element has finite order. Then every subgroup $H$ of $G$ contained in the set $AB$ of all products $ab$ with $a \\in A$ and $b \\in B$ is periodic: every element of $H$ has finite order. This answers Kourovka Problem 16.38 (F. de Giovanni) in the affirmative; the proof is due to Kinyon and Urban (Proposition 3.8).","title":"Subgroups inside a product of two periodic subgroups of a soluble group are periodic"},{"data_url":"scopes/GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups.json","id":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups","kind":"scope","navigation":"module","parent":3350,"result":false,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/index.html","scope":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups","status":"can_prove","title":"Weak chain conditions on subgroups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/weak-maximal-condition","kind":"node","lean":"MathlibExt.GroupTheory.WeakChainConditionsKourovka10Problem14Wanted.WeakMaximalCondition","parent":3419,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/index.html#weak-maximal-condition","scope":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups","status":"fully_proved","summary":"A group $G$ satisfies the weak maximal condition on subgroups when there is no infinite ascending chain $H_0 \\le H_1 \\le \\cdots$ of subgroups of $G$ in which every index $|H_{i+1} : H_i|$ is infinite.","title":"Weak maximal condition on subgroups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/weak-minimal-condition","kind":"node","lean":"MathlibExt.GroupTheory.WeakChainConditionsKourovka10Problem14Wanted.WeakMinimalCondition","parent":3419,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/index.html#weak-minimal-condition","scope":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups","status":"fully_proved","summary":"The weak minimal condition on subgroups: there is no infinite descending chain `H\u2080 \u2265 H\u2081 \u2265 \u22ef` of subgroups with every index `|H\u1d62 : H\u1d62\u208a\u2081|` infinite.","title":"Weak minimal condition on subgroups"},{"id":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/minimal-implies-weak-maximal","kind":"node","lean":"MathlibExt.GroupTheory.WeakChainConditionsKourovka10Problem14Wanted.minimalImpliesWeakMaximal","parent":3419,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/index.html#minimal-implies-weak-maximal","scope":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups","status":"fully_proved","summary":"Kourovka 10.14(a) (D. I. Zaitsev): every group with the minimal condition on subgroups satisfies the weak maximal condition on subgroups.","title":"Minimal condition implies weak maximal condition"},{"id":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/maximal-implies-weak-minimal","kind":"node","lean":"MathlibExt.GroupTheory.WeakChainConditionsKourovka10Problem14Wanted.maximalImpliesWeakMinimal","parent":3419,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/index.html#maximal-implies-weak-minimal","scope":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups","status":"fully_proved","summary":"Kourovka 10.14(b): every group with the maximal condition on subgroups satisfies the weak minimal condition on subgroups.","title":"Maximal condition implies weak minimal condition"},{"id":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/minimal-implies-weak-maximal-refuted","kind":"node","lean":"MathlibExt.GroupTheory.WeakChainConditionsKourovka10Problem14Wanted.minimalImpliesWeakMaximal_refuted","parent":3419,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/index.html#minimal-implies-weak-maximal-refuted","scope":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups","status":"can_prove","summary":"There is a group $G$ satisfying the minimal condition on subgroups (there is no infinite strictly descending chain of subgroups of $G$) that has an infinite ascending chain $H_0 \\le H_1 \\le \\cdots$ of subgroups in which every index $|H_{i+1} : H_i|$ is infinite. This answers Kourovka Problem 10.14(a) negatively.","title":"A group with the minimal condition can fail the weak maximal condition"},{"id":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/maximal-implies-weak-minimal-refuted","kind":"node","lean":"MathlibExt.GroupTheory.WeakChainConditionsKourovka10Problem14Wanted.maximalImpliesWeakMinimal_refuted","parent":3419,"result":true,"route":"../roadmap/GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups/index.html#maximal-implies-weak-minimal-refuted","scope":"GroupTheory/periodic-and-locally-finite-groups/weak-chain-conditions-on-subgroups","status":"can_prove","summary":"There is a group $G$ satisfying the maximal condition on subgroups (there is no infinite strictly ascending chain of subgroups of $G$) that has an infinite descending chain $H_0 \\ge H_1 \\ge \\cdots$ of subgroups in which every index $|H_i : H_{i+1}|$ is infinite. This answers Kourovka Problem 10.14(b) negatively.","title":"A group with the maximal condition can fail the weak minimal condition"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups.json","id":"GroupTheory/profinite-and-topological-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/index.html","scope":"GroupTheory/profinite-and-topological-groups","status":"can_prove","summary":"Groups with topology or approximated by finite quotients: profinite and pro-p groups, residual finiteness and separability, profinite rigidity, Golod\u2013Shafarevich inequality, just-infinite groups, totally disconnected locally compact groups, group topologies, Lie groups and their dense subgroups.","title":"Profinite and topological groups"},{"id":"GroupTheory/profinite-and-topological-groups/semisimple-lie-group-two-generators-dens","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook90Wanted.semisimpleLieGroup_twoGeneratorsDense","parent":3426,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/index.html#semisimple-lie-group-two-generators-dens","scope":"GroupTheory/profinite-and-topological-groups","status":"can_prove","summary":"Every connected real analytic Lie group $G$ whose Lie algebra is semisimple contains two elements $a$ and $b$ such that the subgroup of $G$ generated by $a$ and $b$ is dense in $G$. In particular, for connected groups the number 4 in Problem 90 of the Scottish Book (four elements generating a dense subgroup) can be lowered to 2.","title":"Two elements generate a dense subgroup of a connected semisimple Lie group"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups.json","id":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/index.html","scope":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups","status":"can_prove","title":"Arcwise connectedness of metric groups"},{"id":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/is-hausdorff-connected","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook160Part2Wanted.IsHausdorffConnected","parent":3428,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/index.html#is-hausdorff-connected","scope":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups","status":"fully_proved","summary":"A topological space $G$ is connected in the sense of Hausdorff when it is not the union of two disjoint nonempty closed subsets.","title":"Connectedness in the sense of Hausdorff"},{"id":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/is-arcwise-connected","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook160Part2Wanted.IsArcwiseConnected","parent":3428,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/index.html#is-arcwise-connected","scope":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups","status":"fully_proved","summary":"A topological space $G$ is arcwise connected when for any two points $x, y \\in G$ there is a continuous map $\\gamma \\colon \\mathbb{R} \\to G$ with $\\gamma(0) = x$ and $\\gamma(1) = y$.","title":"Arcwise connected space"},{"id":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/scottish-book-problem160-part2-claim","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook160Part2Wanted.ScottishBookProblem160Part2Claim","parent":3428,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/index.html#scottish-book-problem160-part2-claim","scope":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups","status":"fully_proved","summary":"The claim that every metric group $G$ (a group with a metric for which multiplication and inversion are continuous) that is connected in the sense of Hausdorff is arcwise connected.","title":"Claim of Scottish Book Problem 160, part (2)"},{"id":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/scottish-book160-part2-refuted","kind":"node","lean":"MathlibExt.GroupTheory.ScottishBook160Part2Wanted.scottishBook160Part2_refuted","parent":3428,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups/index.html#scottish-book160-part2-refuted","scope":"GroupTheory/profinite-and-topological-groups/arcwise-connectedness-of-metric-groups","status":"can_prove","summary":"The claim of part (2) of Scottish Book Problem 160 is false: there is a metric group $G$ (a group with a metric for which multiplication and inversion are continuous) that is connected in the sense of Hausdorff but not arcwise connected, that is, there are points $x, y \\in G$ such that no continuous map $\\gamma \\colon \\mathbb{R} \\to G$ has $\\gamma(0) = x$ and $\\gamma(1) = y$.","title":"Connected metric groups need not be arcwise connected"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/conjugacy-separability.json","id":"GroupTheory/profinite-and-topological-groups/conjugacy-separability","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/conjugacy-separability/index.html","scope":"GroupTheory/profinite-and-topological-groups/conjugacy-separability","status":"can_prove","title":"Conjugacy separability"},{"id":"GroupTheory/profinite-and-topological-groups/conjugacy-separability/free-by-cyclic-conjugacy-separable","kind":"node","lean":"MathlibExt.GroupTheory.FreeByCyclicConjugacySeparableKourovka19Problem41Wanted.freeByCyclicConjugacySeparable","parent":3433,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/conjugacy-separability/index.html#free-by-cyclic-conjugacy-separable","scope":"GroupTheory/profinite-and-topological-groups/conjugacy-separability","status":"fully_proved","summary":"The assertion that for every natural number $n$ and every automorphism $\\varphi$ of the free group $F_n$ of rank $n$, the semidirect product $F_n \\rtimes_\\varphi \\mathbb{Z}$, in which $k \\in \\mathbb{Z}$ acts on $F_n$ by $\\varphi^k$, is conjugacy separable.","title":"Kapovich's conjugacy separability question (Kourovka 19.41)"},{"id":"GroupTheory/profinite-and-topological-groups/conjugacy-separability/free-by-cyclic-conjugacy-separable-holds","kind":"node","lean":"MathlibExt.GroupTheory.FreeByCyclicConjugacySeparableKourovka19Problem41Wanted.freeByCyclicConjugacySeparable_holds","parent":3433,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/conjugacy-separability/index.html#free-by-cyclic-conjugacy-separable-holds","scope":"GroupTheory/profinite-and-topological-groups/conjugacy-separability","status":"can_prove","summary":"For every natural number $n$ and every automorphism $\\varphi$ of the free group $F_n$ of rank $n$, the semidirect product $F_n \\rtimes_\\varphi \\mathbb{Z}$, in which $k \\in \\mathbb{Z}$ acts on $F_n$ by $\\varphi^k$, is conjugacy separable: any two of its elements whose images are conjugate in every quotient by a normal subgroup of finite index are conjugate.","title":"Finite-rank free-by-cyclic groups are conjugacy separable"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups.json","id":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/index.html","scope":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","status":"can_prove","title":"Finite images of minimax groups"},{"id":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/has-max-condition","kind":"node","lean":"MathlibExt.GroupTheory.MinimaxFiniteImagesKourovka12Problem90Wanted.HasMaxCondition","parent":3436,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/index.html#has-max-condition","scope":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","status":"fully_proved","summary":"A group $G$ satisfies the maximal condition on subgroups when every subgroup of $G$ is finitely generated.","title":"Maximal condition on subgroups"},{"id":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/has-min-condition","kind":"node","lean":"MathlibExt.GroupTheory.MinimaxFiniteImagesKourovka12Problem90Wanted.HasMinCondition","parent":3436,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/index.html#has-min-condition","scope":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","status":"fully_proved","summary":"A group $G$ satisfies the minimal condition on subgroups when the order on subgroups of $G$ by inclusion is well-founded: every nonempty set of subgroups of $G$ has a minimal element, equivalently there is no infinite strictly descending chain of subgroups of $G$.","title":"Minimal condition on subgroups"},{"id":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/is-minimax","kind":"node","lean":"MathlibExt.GroupTheory.MinimaxFiniteImagesKourovka12Problem90Wanted.IsMinimax","parent":3436,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/index.html#is-minimax","scope":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","status":"fully_proved","summary":"A group $G$ is a minimax group when it has a finite subnormal series $1 = H_0 \\le H_1 \\le \\dots \\le H_n = G$, with each $H_i$ normal in $H_{i+1}$, such that every factor $H_{i+1}/H_i$ satisfies the maximal condition on subgroups or the minimal condition on subgroups.","title":"Minimax groups"},{"id":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/same-finite-images","kind":"node","lean":"MathlibExt.GroupTheory.MinimaxFiniteImagesKourovka12Problem90Wanted.SameFiniteImages","parent":3436,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/index.html#same-finite-images","scope":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","status":"fully_proved","summary":"Two groups $G$ and $H$ have the same finite images when, for every finite group $Q$, there is a surjective homomorphism $G \\to Q$ exactly when there is a surjective homomorphism $H \\to Q$.","title":"Groups with the same finite images"},{"id":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/is-residually-finite","kind":"node","lean":"MathlibExt.GroupTheory.MinimaxFiniteImagesKourovka12Problem90Wanted.IsResiduallyFinite","parent":3436,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/index.html#is-residually-finite","scope":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","status":"fully_proved","summary":"A group $G$ is residually finite when for every element $g \\ne 1$ of $G$ there is a normal subgroup $N$ of finite index in $G$ with $g \\notin N$.","title":"Residually finite groups"},{"id":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/minimax-determined-by-finite-images","kind":"node","lean":"MathlibExt.GroupTheory.MinimaxFiniteImagesKourovka12Problem90Wanted.minimaxDeterminedByFiniteImages","parent":3436,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/index.html#minimax-determined-by-finite-images","scope":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","status":"fully_proved","summary":"The statement that for all groups $G$ and $H$, if $G$ is a finitely generated soluble minimax group and $H$ is a finitely generated residually finite group with the same finite images as $G$ (a finite group is a homomorphic image of $G$ exactly when it is a homomorphic image of $H$), then $H$ is a minimax group.","title":"Minimax groups and finite images (Kourovka 12.90)"},{"id":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/minimax-determined-by-finite-images-hold","kind":"node","lean":"MathlibExt.GroupTheory.MinimaxFiniteImagesKourovka12Problem90Wanted.minimaxDeterminedByFiniteImages_holds","parent":3436,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups/index.html#minimax-determined-by-finite-images-hold","scope":"GroupTheory/profinite-and-topological-groups/finite-images-of-minimax-groups","status":"can_prove","summary":"Let $G$ be a finitely generated soluble minimax group and let $H$ be a finitely generated residually finite group with the same finite images as $G$: a finite group is a homomorphic image of $G$ exactly when it is a homomorphic image of $H$. Then $H$ is a minimax group.","title":"Finite images determine minimaxity (Mann\u2013Segal)"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/free-pro-p-groups.json","id":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/free-pro-p-groups/index.html","scope":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups","status":"can_prove","title":"Free pro-p groups"},{"id":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups/is-pro-p","kind":"node","lean":"MathlibExt.GroupTheory.FreeProPFixedSubgroupKourovka12Problem70Wanted.IsProP","parent":3444,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pro-"],["math","\\(p\\)"],["text"," group"]],"source":"Pro-$p$ group"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/free-pro-p-groups/index.html#is-pro-p","scope":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups","status":"fully_proved","summary":"For a natural number $p$, a group $G$ with a topology is a pro-$p$ group when $G$ is compact, Hausdorff and totally disconnected, and every open normal subgroup of $G$ has index $p^k$ for some $k \\in \\mathbb{N}$.","title":"Pro-$p$ group"},{"id":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups/is-free-pro-pon","kind":"node","lean":"MathlibExt.GroupTheory.FreeProPFixedSubgroupKourovka12Problem70Wanted.IsFreeProPOn","parent":3444,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free pro-"],["math","\\(p\\)"],["text"," group on given generators"]],"source":"Free pro-$p$ group on given generators"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/free-pro-p-groups/index.html#is-free-pro-pon","scope":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups","status":"fully_proved","summary":"Let $p$ and $d$ be natural numbers, $F$ a topological group and $x_1, \\dots, x_d \\in F$. Then $F$ is a free pro-$p$ group on $x_1, \\dots, x_d$ when $F$ is a pro-$p$ group, the closure of the subgroup generated by $x_1, \\dots, x_d$ is all of $F$, and for every finite $p$-group $P$ and all $y_1, \\dots, y_d \\in P$ there is a group homomorphism $\\varphi \\colon F \\to P$ with open kernel such that $\\varphi(x_i) = y_i$\u2026","title":"Free pro-$p$ group on given generators"},{"id":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups/fixed-subgroup-finitely-generated","kind":"node","lean":"MathlibExt.GroupTheory.FreeProPFixedSubgroupKourovka12Problem70Wanted.fixedSubgroupFinitelyGenerated","parent":3444,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ribes's question on fixed subgroups of free pro-"],["math","\\(p\\)"],["text"," groups (Kourovka 12.70)"]],"source":"Ribes's question on fixed subgroups of free pro-$p$ groups (Kourovka 12.70)"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/free-pro-p-groups/index.html#fixed-subgroup-finitely-generated","scope":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups","status":"fully_proved","summary":"The statement that for every prime $p$, every free pro-$p$ group $F$ on finitely many generators $x_1, \\dots, x_d$ and every continuous automorphism $\\Delta \\ne 1$ of $F$ with $\\Delta^{p^k} = 1$ for some $k \\in \\mathbb{N}$, the fixed subgroup $\\mathrm{Fix}_F(\\Delta) = \\{z \\in F \\mid \\Delta(z) = z\\}$ has finite rank: there is a finite set $S \\subseteq \\mathrm{Fix}_F(\\Delta)$ such that the closure of the subgroup\u2026","title":"Ribes's question on fixed subgroups of free pro-$p$ groups (Kourovka 12.70)"},{"id":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups/fixed-subgroup-finitely-generated-holds","kind":"node","lean":"MathlibExt.GroupTheory.FreeProPFixedSubgroupKourovka12Problem70Wanted.fixedSubgroupFinitelyGenerated_holds","parent":3444,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/free-pro-p-groups/index.html#fixed-subgroup-finitely-generated-holds","scope":"GroupTheory/profinite-and-topological-groups/free-pro-p-groups","status":"can_prove","summary":"Resolved true: the Kourovka Archive answers 12.70 \"Yes, it is finite\" (C. Scheiderer, 1999); moreover, the fixed points of a finite `p`-group of automorphisms of a free pro-`p` group form a free factor (Herfort, Ribes and Zalesskii, 1999), which has finite rank when `F` does. Source: E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49\u2026","title":"Fixed subgroup of a free pro-p group is finitely generated"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality.json","id":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/index.html","scope":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality","status":"can_prove","title":"Golod\u2013Shafarevich inequality"},{"id":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/generator-rank","kind":"node","lean":"MathlibExt.GroupTheory.GolodShafarevichWanted.GroupTheory.generatorRank","parent":3449,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/index.html#generator-rank","scope":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality","status":"fully_proved","summary":"For a topological group $G$, the generator rank $d(G)$ is the least natural number $k$ such that some finite subset of $G$ with $k$ elements generates a dense subgroup of $G$, that is, a subgroup whose closure is all of $G$. If no finite subset of $G$ generates a dense subgroup, then $d(G)$ is defined to be $0$. For a finite discrete group this is the ordinary minimal number of generators.","title":"Generator rank"},{"id":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/trivial-zmodp-rep","kind":"node","lean":"MathlibExt.GroupTheory.GolodShafarevichWanted.GroupTheory.trivialZModpRep","parent":3449,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trivial representation on "],["math","\\(\\mathbb{Z}/p\\mathbb{Z}\\)"]],"source":"Trivial representation on $\\mathbb{Z}/p\\mathbb{Z}$"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/index.html#trivial-zmodp-rep","scope":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality","status":"fully_proved","summary":"For a natural number $p$ and a topological group $G$, this is $\\mathbb{Z}/p\\mathbb{Z}$, with its discrete topology, regarded as a continuous representation of $G$ over $\\mathbb{Z}/p\\mathbb{Z}$ on which every element of $G$ acts as the identity.","title":"Trivial representation on $\\mathbb{Z}/p\\mathbb{Z}$"},{"id":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/relation-rank","kind":"node","lean":"MathlibExt.GroupTheory.GolodShafarevichWanted.GroupTheory.relationRank","parent":3449,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/index.html#relation-rank","scope":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality","status":"fully_proved","summary":"For a prime $p$ and a topological group $G$, the relation rank $r(G)$ is the dimension over $\\mathbb{F}_p = \\mathbb{Z}/p\\mathbb{Z}$ of the continuous cohomology group $H^2(G, \\mathbb{F}_p)$, where $G$ acts trivially on $\\mathbb{F}_p$. If this space is infinite-dimensional, then $r(G)$ is defined to be $0$.","title":"Relation rank"},{"id":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/golod-shafarevich-inequality","kind":"node","lean":"MathlibExt.GroupTheory.GolodShafarevichWanted.GroupTheory.golod_shafarevich_inequality","parent":3449,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality/index.html#golod-shafarevich-inequality","scope":"GroupTheory/profinite-and-topological-groups/golod-shafarevich-inequality","status":"can_prove","summary":"Let $p$ be a prime and let $Q$ be a nontrivial finite $p$-group, viewed as a topological group with the discrete topology. Then the generator rank $d(Q)$ and the relation rank $r(Q)$, the dimension over $\\mathbb{F}_p$ of the continuous cohomology group $H^2(Q, \\mathbb{F}_p)$ with trivial coefficients, satisfy $d(Q)^2 < 4\\, r(Q)$.","title":"Golod\u2013Shafarevich inequality"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/group-topologies.json","id":"GroupTheory/profinite-and-topological-groups/group-topologies","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/group-topologies/index.html","scope":"GroupTheory/profinite-and-topological-groups/group-topologies","status":"can_prove","title":"Group topologies"},{"id":"GroupTheory/profinite-and-topological-groups/group-topologies/integer-group-topology-pow-sum-tendsto-z","kind":"node","lean":"MathlibExt.GroupTheory.IntegerGroupTopologyPowerSumKourovka15Problem79Wanted.integerGroupTopologyPowSumTendstoZero","parent":3454,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/group-topologies/index.html#integer-group-topology-pow-sum-tendsto-z","scope":"GroupTheory/profinite-and-topological-groups/group-topologies","status":"fully_proved","summary":"The statement that there is a topology on $\\mathbb{Z}$ that makes $(\\mathbb{Z}, +)$ a Hausdorff topological group and in which the sequence $(2^n + 3^n)_{n \\in \\mathbb{N}}$ converges to $0$.","title":"Kourovka Problem 15.79 (Protasov's question on the integers)"},{"id":"GroupTheory/profinite-and-topological-groups/group-topologies/integer-group-topology-pow-sum-tendsto-z-2","kind":"node","lean":"MathlibExt.GroupTheory.IntegerGroupTopologyPowerSumKourovka15Problem79Wanted.integerGroupTopologyPowSumTendstoZero_holds","parent":3454,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/group-topologies/index.html#integer-group-topology-pow-sum-tendsto-z-2","scope":"GroupTheory/profinite-and-topological-groups/group-topologies","status":"can_prove","summary":"There is a topology on $\\mathbb{Z}$ that makes $(\\mathbb{Z}, +)$ a Hausdorff topological group and in which the sequence $(2^n + 3^n)_{n \\in \\mathbb{N}}$ converges to $0$. That is, Kourovka Problem 15.79 has a positive answer.","title":"A Hausdorff group topology on the integers in which 2\u207f + 3\u207f tends to zero"},{"id":"GroupTheory/profinite-and-topological-groups/group-topologies/nilpotent-group-topology-lattice-not-mod","kind":"node","lean":"MathlibExt.GroupTheory.LatticeOfGroupTopologiesModularKourovka14Problem4Wanted.nilpotentGroupTopologyLatticeNotModular","parent":3454,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/group-topologies/index.html#nilpotent-group-topology-lattice-not-mod","scope":"GroupTheory/profinite-and-topological-groups/group-topologies","status":"fully_proved","summary":"This proposition states that some nilpotent group $G$ has a non-modular lattice $\\mathcal{L}(G)$ of all (not necessarily Hausdorff) group topologies.","title":"Kourovka Problem 14.4(a) (nilpotent groups with non-modular topology lattice)"},{"id":"GroupTheory/profinite-and-topological-groups/group-topologies/countable-nilpotent-nonabelian-topology","kind":"node","lean":"MathlibExt.GroupTheory.LatticeOfGroupTopologiesModularKourovka14Problem4Wanted.countableNilpotentNonabelianTopologyLatticeNotModular","parent":3454,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/group-topologies/index.html#countable-nilpotent-nonabelian-topology","scope":"GroupTheory/profinite-and-topological-groups/group-topologies","status":"fully_proved","summary":"This proposition states that for every countably infinite non-abelian nilpotent group $G$, the lattice $\\mathcal{L}(G)$ of all (not necessarily Hausdorff) group topologies on $G$ is not modular.","title":"Kourovka Problem 14.4(b) (countable nilpotent groups)"},{"id":"GroupTheory/profinite-and-topological-groups/group-topologies/nilpotent-group-topology-lattice-not-mod-2","kind":"node","lean":"MathlibExt.GroupTheory.LatticeOfGroupTopologiesModularKourovka14Problem4Wanted.nilpotentGroupTopologyLatticeNotModular_holds","parent":3454,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/group-topologies/index.html#nilpotent-group-topology-lattice-not-mod-2","scope":"GroupTheory/profinite-and-topological-groups/group-topologies","status":"can_prove","summary":"There is a nilpotent group $G$ whose lattice of all (not necessarily Hausdorff) group topologies is not modular.","title":"A nilpotent group with non-modular lattice of group topologies"},{"id":"GroupTheory/profinite-and-topological-groups/group-topologies/countable-nilpotent-nonabelian-topology-2","kind":"node","lean":"MathlibExt.GroupTheory.LatticeOfGroupTopologiesModularKourovka14Problem4Wanted.countableNilpotentNonabelianTopologyLatticeNotModular_refuted","parent":3454,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/group-topologies/index.html#countable-nilpotent-nonabelian-topology-2","scope":"GroupTheory/profinite-and-topological-groups/group-topologies","status":"can_prove","summary":"It is not the case that for every countably infinite non-abelian nilpotent group $G$ the lattice of all (not necessarily Hausdorff) group topologies on $G$ is not modular; that is, some countably infinite non-abelian nilpotent group has a modular lattice of group topologies.","title":"A countable nilpotent group with modular lattice of group topologies"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/nottingham-group.json","id":"GroupTheory/profinite-and-topological-groups/nottingham-group","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/nottingham-group/index.html","scope":"GroupTheory/profinite-and-topological-groups/nottingham-group","status":"can_prove","title":"Nottingham group"},{"id":"GroupTheory/profinite-and-topological-groups/nottingham-group/is-nottingham","kind":"node","lean":"MathlibExt.GroupTheory.NottinghamGroupFreeSubgroupKourovka12Problem24Wanted.IsNottingham","parent":3461,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/nottingham-group/index.html#is-nottingham","scope":"GroupTheory/profinite-and-topological-groups/nottingham-group","status":"fully_proved","summary":"Let $R$ be a commutative ring. A ring automorphism $\\varphi$ of the power series ring $R[[x]]$ satisfies the Nottingham condition when $\\varphi(x) - x$ lies in the ideal generated by $x^2$, that is, when $\\varphi(x) = x(1 + a_1 x + a_2 x^2 + \\cdots)$ for some $a_1, a_2, \\ldots \\in R$.","title":"Nottingham condition on automorphisms of power series"},{"id":"GroupTheory/profinite-and-topological-groups/nottingham-group/nottingham-contains-free-group","kind":"node","lean":"MathlibExt.GroupTheory.NottinghamGroupFreeSubgroupKourovka12Problem24Wanted.nottinghamContainsFreeGroup","parent":3461,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/nottingham-group/index.html#nottingham-contains-free-group","scope":"GroupTheory/profinite-and-topological-groups/nottingham-group","status":"fully_proved","summary":"For every prime $p$ there is an injective group homomorphism from the free group $F_2$ on two generators to the group of ring automorphisms of $(\\mathbb{Z}/p\\mathbb{Z})[[x]]$ whose image consists of automorphisms $\\varphi$ with $\\varphi(x) - x \\in (x^2)$, that is, of elements of the Nottingham group $N(\\mathbb{Z}/p\\mathbb{Z})$.","title":"Kourovka Problem 12.24 (free subgroups of the Nottingham group)"},{"id":"GroupTheory/profinite-and-topological-groups/nottingham-group/nottingham-contains-free-group-holds","kind":"node","lean":"MathlibExt.GroupTheory.NottinghamGroupFreeSubgroupKourovka12Problem24Wanted.nottinghamContainsFreeGroup_holds","parent":3461,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/nottingham-group/index.html#nottingham-contains-free-group-holds","scope":"GroupTheory/profinite-and-topological-groups/nottingham-group","status":"can_prove","summary":"For every prime $p$ there is an injective group homomorphism $f$ from the free group $F_2$ on two generators to the group of ring automorphisms of the power series ring $(\\mathbb{Z}/p\\mathbb{Z})[[x]]$ such that, for every $g \\in F_2$, the automorphism $\\varphi = f(g)$ satisfies $\\varphi(x) - x \\in (x^2)$; that is, the Nottingham group $N(\\mathbb{Z}/p\\mathbb{Z})$ contains a copy of $F_2$.","title":"The Nottingham group contains a free subgroup of rank 2"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/profinite-cn-groups.json","id":"GroupTheory/profinite-and-topological-groups/profinite-cn-groups","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-cn-groups/index.html","scope":"GroupTheory/profinite-and-topological-groups/profinite-cn-groups","status":"can_prove","title":"Profinite CN-groups"},{"id":"GroupTheory/profinite-and-topological-groups/profinite-cn-groups/is-pronilpotent","kind":"node","lean":"MathlibExt.GroupTheory.ProfiniteCNGroupNotProsolubleKourovka20Problem90Wanted.IsPronilpotent","parent":3465,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-cn-groups/index.html#is-pronilpotent","scope":"GroupTheory/profinite-and-topological-groups/profinite-cn-groups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ carrying a topology is pronilpotent when, for every open normal subgroup $N$ of $G$, the image of $H$ in the quotient $G/N$ is nilpotent.","title":"Pronilpotent subgroup"},{"id":"GroupTheory/profinite-and-topological-groups/profinite-cn-groups/exists-fginfinite-profinite-cngroup-not","kind":"node","lean":"MathlibExt.GroupTheory.ProfiniteCNGroupNotProsolubleKourovka20Problem90Wanted.existsFGInfiniteProfiniteCNGroupNotProsoluble","parent":3465,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-cn-groups/index.html#exists-fginfinite-profinite-cngroup-not","scope":"GroupTheory/profinite-and-topological-groups/profinite-cn-groups","status":"fully_proved","summary":"The statement that there is an infinite profinite group $G$, that is, a compact, Hausdorff, totally disconnected topological group, such that $G$ is topologically finitely generated (the closure of the subgroup generated by some finite set is $G$), the centralizer of every non-trivial element of $G$ is pronilpotent (so $G$ is a CN-group), and $G$ is not prosoluble: for some open normal subgroup $N$ of $G$, the\u2026","title":"Problem of a finitely generated non-prosoluble profinite CN-group (Kourovka 20.90)"},{"id":"GroupTheory/profinite-and-topological-groups/profinite-cn-groups/exists-fginfinite-profinite-cngroup-not-2","kind":"node","lean":"MathlibExt.GroupTheory.ProfiniteCNGroupNotProsolubleKourovka20Problem90Wanted.existsFGInfiniteProfiniteCNGroupNotProsoluble_holds","parent":3465,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-cn-groups/index.html#exists-fginfinite-profinite-cngroup-not-2","scope":"GroupTheory/profinite-and-topological-groups/profinite-cn-groups","status":"can_prove","summary":"There is an infinite profinite group $G$, that is, a compact, Hausdorff, totally disconnected topological group, such that $G$ is topologically finitely generated (the closure of the subgroup generated by some finite set is $G$), the centralizer of every non-trivial element of $G$ is pronilpotent, and $G$ is not prosoluble: for some open normal subgroup $N$ of $G$, the quotient $G/N$ is not soluble. This answers\u2026","title":"A finitely generated non-prosoluble profinite CN-group exists"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/profinite-rigidity.json","id":"GroupTheory/profinite-and-topological-groups/profinite-rigidity","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-rigidity/index.html","scope":"GroupTheory/profinite-and-topological-groups/profinite-rigidity","status":"can_prove","title":"Profinite rigidity"},{"id":"GroupTheory/profinite-and-topological-groups/profinite-rigidity/profinite-completion","kind":"node","lean":"MathlibExt.GroupTheory.AbsoluteProfiniteRigidityWanted.AbsoluteProfiniteRigidity.profiniteCompletion","parent":3469,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-rigidity/index.html#profinite-completion","scope":"GroupTheory/profinite-and-topological-groups/profinite-rigidity","status":"fully_proved","summary":"The profinite completion of a group $G$ is the inverse limit of the finite quotient groups $G/N$, where $N$ ranges over the normal subgroups of finite index in $G$ and the maps are the quotient maps $G/N \\to G/M$ for $N \\le M$. It is a profinite group.","title":"Profinite completion"},{"id":"GroupTheory/profinite-and-topological-groups/profinite-rigidity/profinitely-rigid","kind":"node","lean":"MathlibExt.GroupTheory.AbsoluteProfiniteRigidityWanted.AbsoluteProfiniteRigidity.ProfinitelyRigid","parent":3469,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-rigidity/index.html#profinitely-rigid","scope":"GroupTheory/profinite-and-topological-groups/profinite-rigidity","status":"fully_proved","summary":"A group $\\Gamma$ is *profinitely rigid* if it is finitely generated and residually finite, and $\\Gamma$ is isomorphic to every finitely generated residually finite group $\\Lambda$ whose profinite completion is isomorphic, as a group, to the profinite completion of $\\Gamma$.","title":"Profinitely rigid group"},{"id":"GroupTheory/profinite-and-topological-groups/profinite-rigidity/ring-of-integers","kind":"node","lean":"MathlibExt.GroupTheory.AbsoluteProfiniteRigidityWanted.AbsoluteProfiniteRigidity.RingOfIntegers","parent":3469,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ring of integers of a field obtained by adjoining a square root to "],["math","\\(\\mathbb{Q}\\)"]],"source":"Ring of integers of a field obtained by adjoining a square root to $\\mathbb{Q}$"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-rigidity/index.html#ring-of-integers","scope":"GroupTheory/profinite-and-topological-groups/profinite-rigidity","status":"fully_proved","summary":"For a natural number $d$, the ring of integers of the number field $\\mathbb{Q}(\\sqrt{-d})$, realized as the splitting field of $X^2 + d$ over $\\mathbb{Q}$.","title":"Ring of integers of a field obtained by adjoining a square root to $\\mathbb{Q}$"},{"id":"GroupTheory/profinite-and-topological-groups/profinite-rigidity/bianchi-group","kind":"node","lean":"MathlibExt.GroupTheory.AbsoluteProfiniteRigidityWanted.AbsoluteProfiniteRigidity.BianchiGroup","parent":3469,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-rigidity/index.html#bianchi-group","scope":"GroupTheory/profinite-and-topological-groups/profinite-rigidity","status":"fully_proved","summary":"For a natural number $d$, let $\\mathcal{O}_d$ be the ring of integers of $\\mathbb{Q}(\\sqrt{-d})$. The Bianchi group is $\\mathrm{PSL}(2, \\mathcal{O}_d)$, the projective special linear group of $2 \\times 2$ matrices over $\\mathcal{O}_d$.","title":"Bianchi group"},{"id":"GroupTheory/profinite-and-topological-groups/profinite-rigidity/theorem-7-1","kind":"node","lean":"MathlibExt.GroupTheory.AbsoluteProfiniteRigidityWanted.AbsoluteProfiniteRigidity.theorem_7_1","parent":3469,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/profinite-rigidity/index.html#theorem-7-1","scope":"GroupTheory/profinite-and-topological-groups/profinite-rigidity","status":"can_prove","summary":"Let $\\omega$ be a primitive cube root of unity, so that the ring $\\mathbb{Z}[\\omega]$ of Eisenstein integers is the ring of integers of $\\mathbb{Q}(\\sqrt{-3})$. The Bianchi group $\\mathrm{PSL}(2, \\mathbb{Z}[\\omega])$ is profinitely rigid: it is finitely generated and residually finite, and it is isomorphic to every finitely generated residually finite group whose profinite completion is isomorphic, as a group, to\u2026","title":"Profinite rigidity of the Bianchi group over the Eisenstein integers"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten.json","id":"GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten/index.html","scope":"GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten","status":"can_prove","title":"Residually finite p-groups and residual nilpotence"},{"id":"GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten/residually-finite-pimplies-residually-to","kind":"node","lean":"MathlibExt.GroupTheory.ResiduallyFinitePGroupTorsionFreeNilpotentKourovka14Problem52Wanted.residuallyFinitePImpliesResiduallyTorsionFreeNilpotent","parent":3475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kuz'min's question on residually finite "],["math","\\(p\\)"],["text","-groups (Kourovka 14.52)"]],"source":"Kuz'min's question on residually finite $p$-groups (Kourovka 14.52)"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten/index.html#residually-finite-pimplies-residually-to","scope":"GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten","status":"fully_proved","summary":"The statement that every finitely generated group $G$ that is residually a finite $p$-group for every prime $p$ is residually torsion-free nilpotent. Here $G$ is residually a finite $p$-group when every element $g \\ne 1$ lies outside some normal subgroup $N$ such that $G/N$ is a finite $p$-group, and residually torsion-free nilpotent when every element $g \\ne 1$ lies outside some normal subgroup $N$ such that $G/N$\u2026","title":"Kuz'min's question on residually finite $p$-groups (Kourovka 14.52)"},{"id":"GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten/residually-finite-pimplies-residually-to-2","kind":"node","lean":"MathlibExt.GroupTheory.ResiduallyFinitePGroupTorsionFreeNilpotentKourovka14Problem52Wanted.residuallyFinitePImpliesResiduallyTorsionFreeNilpotent_refuted","parent":3475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residual "],["math","\\(p\\)"],["text","-finiteness for all "],["math","\\(p\\)"],["text"," does not imply residual torsion-free nilpotence"]],"source":"Residual $p$-finiteness for all $p$ does not imply residual torsion-free nilpotence"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten/index.html#residually-finite-pimplies-residually-to-2","scope":"GroupTheory/profinite-and-topological-groups/residually-finite-p-groups-and-residual-nilpoten","status":"can_prove","summary":"It is not the case that every finitely generated group that is residually a finite $p$-group for every prime $p$ is residually torsion-free nilpotent. Here a group $G$ is residually a finite $p$-group when every element $g \\ne 1$ lies outside some normal subgroup $N$ such that $G/N$ is a finite $p$-group, and residually torsion-free nilpotent when every element $g \\ne 1$ lies outside some normal subgroup $N$ such\u2026","title":"Residual $p$-finiteness for all $p$ does not imply residual torsion-free nilpotence"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups.json","id":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/index.html","scope":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups","status":"can_prove","title":"Separability of isolated subgroups"},{"id":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/is-pprime-isolated","kind":"node","lean":"MathlibExt.GroupTheory.IsolatedSubgroupPSeparableKourovka15Problem60Wanted.IsPPrimeIsolated","parent":3478,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p'\\)"],["text","-isolated subgroup"]],"source":"$p'$-isolated subgroup"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/index.html#is-pprime-isolated","scope":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups","status":"fully_proved","summary":"Let $p$ be a natural number and let $H$ be a subgroup of a group $F$. Then $H$ is $p'$-isolated if, for every $x \\in F$ and every natural number $n > 0$ that is not divisible by $p$, $x^n \\in H$ implies $x \\in H$.","title":"$p'$-isolated subgroup"},{"id":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/is-pseparable","kind":"node","lean":"MathlibExt.GroupTheory.IsolatedSubgroupPSeparableKourovka15Problem60Wanted.IsPSeparable","parent":3478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Separability in the class of finite "],["math","\\(p\\)"],["text","-groups"]],"source":"Separability in the class of finite $p$-groups"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/index.html#is-pseparable","scope":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups","status":"fully_proved","summary":"Let $p$ be a natural number and let $H$ be a subgroup of a group $F$. Then $H$ is separable in the class of finite $p$-groups if for every $x \\in F$ with $x \\notin H$ there are a finite $p$-group $Q$ (a finite group in which every element $g$ satisfies $g^{p^k} = 1$ for some natural number $k$) and a group homomorphism $f \\colon F \\to Q$ with $f(x) \\notin f(H)$.","title":"Separability in the class of finite $p$-groups"},{"id":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/isolated-fgsubgroups-pseparable","kind":"node","lean":"MathlibExt.GroupTheory.IsolatedSubgroupPSeparableKourovka15Problem60Wanted.isolatedFGSubgroupsPSeparable","parent":3478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moldavanski\u012d's question on "],["math","\\(p\\)"],["text","-separability of isolated subgroups (Kourovka 15.60)"]],"source":"Moldavanski\u012d's question on $p$-separability of isolated subgroups (Kourovka 15.60)"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/index.html#isolated-fgsubgroups-pseparable","scope":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups","status":"fully_proved","summary":"The statement that there is a prime $p$ such that, for every set $\\alpha$, every finitely generated $p'$-isolated subgroup of the free group on $\\alpha$ is separable in the class of finite $p$-groups.","title":"Moldavanski\u012d's question on $p$-separability of isolated subgroups (Kourovka 15.60)"},{"id":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/isolated-fgsubgroups-pseparable-refuted","kind":"node","lean":"MathlibExt.GroupTheory.IsolatedSubgroupPSeparableKourovka15Problem60Wanted.isolatedFGSubgroupsPSeparable_refuted","parent":3478,"reading_title":{"format":"inline-parts-v1","parts":[["text","Isolated subgroups of free groups need not be "],["math","\\(p\\)"],["text","-separable"]],"source":"Isolated subgroups of free groups need not be $p$-separable"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups/index.html#isolated-fgsubgroups-pseparable-refuted","scope":"GroupTheory/profinite-and-topological-groups/separability-of-isolated-subgroups","status":"can_prove","summary":"For every prime $p$ there are a set $\\alpha$ and a finitely generated $p'$-isolated subgroup of the free group on $\\alpha$ that is not separable in the class of finite $p$-groups. That is, the statement of Kourovka Problem 15.60 fails for every prime $p$.","title":"Isolated subgroups of free groups need not be $p$-separable"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups.json","id":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/index.html","scope":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups","status":"can_prove","title":"Strong completeness of profinite groups"},{"id":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/is-strongly-complete","kind":"node","lean":"IsStronglyComplete","parent":3483,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/index.html#is-strongly-complete","scope":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups","status":"fully_proved","summary":"A group $G$ equipped with a topology is *strongly complete* when every subgroup of finite index in $G$ is open.","title":"Strongly complete group"},{"id":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/is-closed-to-subgroup-of-surjective-eta","kind":"node","lean":"ProfiniteGrp.ProfiniteCompletion.isClosed_toSubgroup_of_surjective_etaFn","parent":3483,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/index.html#is-closed-to-subgroup-of-surjective-eta","scope":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups","status":"fully_proved","summary":"Let $P$ be a profinite group and let $\\hat P$ be the profinite completion of $P$ as an abstract group, that is, the inverse limit of the finite quotients $P/N$ over all normal subgroups $N$ of finite index in $P$. If the canonical map $P \\to \\hat P$ is surjective, then every normal subgroup of finite index in $P$ is closed in $P$.","title":"Finite-index normal subgroups are closed when the completion map is onto"},{"id":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/is-open-to-subgroup-of-surjective-eta-fn","kind":"node","lean":"ProfiniteGrp.ProfiniteCompletion.isOpen_toSubgroup_of_surjective_etaFn","parent":3483,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/index.html#is-open-to-subgroup-of-surjective-eta-fn","scope":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups","status":"fully_proved","summary":"Let $P$ be a profinite group and let $\\hat P$ be the profinite completion of $P$ as an abstract group, that is, the inverse limit of the finite quotients $P/N$ over all normal subgroups $N$ of finite index in $P$. If the canonical map $P \\to \\hat P$ is surjective, then every normal subgroup of finite index in $P$ is open in $P$.","title":"Finite-index normal subgroups are open when the completion map is onto"},{"id":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/is-strongly-complete-iff-bijective-eta-f","kind":"node","lean":"ProfiniteGrp.ProfiniteCompletion.isStronglyComplete_iff_bijective_etaFn","parent":3483,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/index.html#is-strongly-complete-iff-bijective-eta-f","scope":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups","status":"fully_proved","summary":"Let $P$ be a profinite group and let $\\hat P$ be the profinite completion of $P$ as an abstract group, that is, the inverse limit of the finite quotients $P/N$ over all normal subgroups $N$ of finite index in $P$. Then $P$ is strongly complete if and only if the canonical map $P \\to \\hat P$ is bijective.","title":"Strongly complete profinite groups via the profinite completion"},{"id":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/is-topologically-finitely-generated","kind":"node","lean":"MathlibExt.GroupTheory.NikolovSegalWanted.GroupTheory.IsTopologicallyFinitelyGenerated","parent":3483,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/index.html#is-topologically-finitely-generated","scope":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups","status":"fully_proved","summary":"A topological group is topologically finitely generated if some finite set generates a dense abstract subgroup.","title":"Topologically finitely generated group"},{"id":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/nikolov-segal","kind":"node","lean":"MathlibExt.GroupTheory.NikolovSegalWanted.GroupTheory.nikolov_segal","parent":3483,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups/index.html#nikolov-segal","scope":"GroupTheory/profinite-and-topological-groups/strong-completeness-of-profinite-groups","status":"can_prove","summary":"Every finite-index subgroup of a topologically finitely generated profinite group is open.","title":"Nikolov\u2013Segal strong completeness theorem"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups.json","id":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/index.html","scope":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups","status":"can_prove","title":"Subgroup separability of Baumslag\u2013Solitar groups"},{"id":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/bs1","kind":"node","lean":"MathlibExt.GroupTheory.SolvableLERFBaumslagSolitarKourovka16Problem2Wanted.BS1","parent":3490,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/index.html#bs1","scope":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{BS}(1, n)$ is the group with presentation $\\langle a, b \\mid b a b^{-1} = a^n \\rangle$.","title":"The Baumslag\u2013Solitar group BS(1, n)"},{"id":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/is-separable-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.SolvableLERFBaumslagSolitarKourovka16Problem2Wanted.IsSeparableSubgroup","parent":3490,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/index.html#is-separable-subgroup","scope":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is separable when for every element $x \\in G$ with $x \\notin H$ there are a finite group $F$ and a group homomorphism $f \\colon G \\to F$ with $f(x) \\notin f(H)$.","title":"Separable subgroup"},{"id":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/is-locally-subgroup-separable","kind":"node","lean":"MathlibExt.GroupTheory.SolvableLERFBaumslagSolitarKourovka16Problem2Wanted.IsLocallySubgroupSeparable","parent":3490,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/index.html#is-locally-subgroup-separable","scope":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups","status":"fully_proved","summary":"A group $G$ is locally subgroup-separable, or LERF, when every finitely generated subgroup of $G$ is separable.","title":"Locally subgroup-separable (LERF) group"},{"id":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/solvable-lerfiff-no-baumslag-solitar","kind":"node","lean":"MathlibExt.GroupTheory.SolvableLERFBaumslagSolitarKourovka16Problem2Wanted.solvableLERFIffNoBaumslagSolitar","parent":3490,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/index.html#solvable-lerfiff-no-baumslag-solitar","scope":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups","status":"fully_proved","summary":"A finitely generated solvable group $G$ is locally subgroup-separable if and only if there is no $n > 1$ such that $G$ has a subgroup isomorphic to the Baumslag\u2013Solitar group $\\mathrm{BS}(1, n)$.","title":"Kourovka Problem 16.2 (R. C. Alperin)"},{"id":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/solvable-lerfiff-no-baumslag-solitar-ref","kind":"node","lean":"MathlibExt.GroupTheory.SolvableLERFBaumslagSolitarKourovka16Problem2Wanted.solvableLERFIffNoBaumslagSolitar_refuted","parent":3490,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups/index.html#solvable-lerfiff-no-baumslag-solitar-ref","scope":"GroupTheory/profinite-and-topological-groups/subgroup-separability-of-baumslag-solitar-groups","status":"can_prove","summary":"There is a finitely generated solvable group $G$ for which the following two conditions are not equivalent: $G$ is locally subgroup-separable; there is no $n > 1$ such that $G$ has a subgroup isomorphic to the Baumslag\u2013Solitar group $\\mathrm{BS}(1, n)$.","title":"Kourovka Problem 16.2 has a negative answer"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups.json","id":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/index.html","scope":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups","status":"can_prove","title":"Totally disconnected locally compact groups"},{"id":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/is-in-class-s","kind":"node","lean":"MathlibExt.GroupTheory.LocalIsomorphismClassesSimpleTdlcKourovka19Problem22Wanted.IsInClassS","parent":3496,"reading_title":{"format":"inline-parts-v1","parts":[["text","Caprace's class "],["math","\\(\\mathcal{S}\\)"]],"source":"Caprace's class $\\mathcal{S}$"},"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/index.html#is-in-class-s","scope":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups","status":"fully_proved","summary":"A group $G$ with a topology belongs to the class $\\mathcal{S}$ when it is a topological group that is Hausdorff, totally disconnected, locally compact and not discrete, is generated as a group by some compact subset, and is topologically simple: $G$ is nontrivial and its only closed normal subgroups are the trivial subgroup and $G$ itself.","title":"Caprace's class $\\mathcal{S}$"},{"id":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/locally-isomorphic","kind":"node","lean":"MathlibExt.GroupTheory.LocalIsomorphismClassesSimpleTdlcKourovka19Problem22Wanted.LocallyIsomorphic","parent":3496,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/index.html#locally-isomorphic","scope":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups","status":"fully_proved","summary":"Two topological groups are locally isomorphic: they contain open subgroups that are isomorphic as topological groups.","title":"Locally isomorphic topological groups"},{"id":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/uncountably-many-local-isomorphism-class","kind":"node","lean":"MathlibExt.GroupTheory.LocalIsomorphismClassesSimpleTdlcKourovka19Problem22Wanted.uncountablyManyLocalIsomorphismClasses","parent":3496,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/index.html#uncountably-many-local-isomorphism-class","scope":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups","status":"fully_proved","summary":"The statement that the groups in Caprace's class $\\mathcal{S}$ fall into uncountably many local isomorphism classes: there is a family $(G_i)_{i \\in I}$ of groups in $\\mathcal{S}$, indexed by an uncountable set $I$, such that $G_i$ and $G_j$ are not locally isomorphic whenever $i \\ne j$.","title":"Caprace's problem on local isomorphism classes (Kourovka 19.22)"},{"id":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/uncountably-many-local-isomorphism-class-2","kind":"node","lean":"MathlibExt.GroupTheory.LocalIsomorphismClassesSimpleTdlcKourovka19Problem22Wanted.uncountablyManyLocalIsomorphismClasses_holds","parent":3496,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups/index.html#uncountably-many-local-isomorphism-class-2","scope":"GroupTheory/profinite-and-topological-groups/totally-disconnected-locally-compact-groups","status":"can_prove","summary":"There is a family $(G_i)_{i \\in I}$ of groups in Caprace's class $\\mathcal{S}$ (non-discrete, compactly generated, topologically simple, totally disconnected, locally compact Hausdorff topological groups), indexed by an uncountable set $I$, such that $G_i$ and $G_j$ are not locally isomorphic whenever $i \\ne j$. This answers Kourovka Notebook Problem 19.22 (P.-E. Caprace) in the affirmative.","title":"Uncountably many local isomorphism classes of simple tdlc groups"},{"data_url":"scopes/GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated.json","id":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated","kind":"scope","navigation":"module","parent":3426,"result":false,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/index.html","scope":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated","status":"can_prove","title":"Upper composition factors of finitely generated groups"},{"id":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/is-composition-factor","kind":"node","lean":"MathlibExt.GroupTheory.UpperCompositionFactorsKourovka11Problem57Wanted.IsCompositionFactor","parent":3501,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/index.html#is-composition-factor","scope":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated","status":"fully_proved","summary":"For groups $T$ and $Q$, $T$ is a composition factor of $Q$ when some subnormal subgroup $H$ of $Q$ maps onto $T$, that is, there is a surjective group homomorphism $H \\to T$. Here $H$ is subnormal in $Q$ when there is a finite chain of subgroups $H = H_0 \\le H_1 \\le \\dots \\le H_k = Q$ in which each $H_j$ is normal in $H_{j+1}$.","title":"Composition factor of a group (via subnormal subgroups)"},{"id":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/is-upper-composition-factor","kind":"node","lean":"MathlibExt.GroupTheory.UpperCompositionFactorsKourovka11Problem57Wanted.IsUpperCompositionFactor","parent":3501,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/index.html#is-upper-composition-factor","scope":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated","status":"fully_proved","summary":"For groups $T$ and $G$, $T$ is an upper composition factor of $G$ when it is a composition factor of some finite quotient of $G$: there are a finite group $Q$ and a surjective group homomorphism $G \\to Q$ such that $T$ is a composition factor of $Q$, that is, $T$ is a homomorphic image of a subnormal subgroup of $Q$.","title":"Upper composition factors"},{"id":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/upper-composition-factors-any-set63","kind":"node","lean":"MathlibExt.GroupTheory.UpperCompositionFactorsKourovka11Problem57Wanted.upperCompositionFactorsAnySet63","parent":3501,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/index.html#upper-composition-factors-any-set63","scope":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated","status":"fully_proved","summary":"The statement that for every family $(T_i)_{i \\in I}$ of finite non-abelian simple groups there is a group $G$ generated by $63$ elements such that a finite simple group $Q$ is an upper composition factor of $G$ if and only if $Q$ is isomorphic to $T_i$ for some $i \\in I$.","title":"Kourovka Problem 11.57, 63-generator form (statement)"},{"id":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/nonabelian-upper-composition-factors-any","kind":"node","lean":"MathlibExt.GroupTheory.UpperCompositionFactorsKourovka11Problem57Wanted.nonabelianUpperCompositionFactorsAnySet3","parent":3501,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/index.html#nonabelian-upper-composition-factors-any","scope":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated","status":"fully_proved","summary":"The statement that for every family $(T_i)_{i \\in I}$ of finite non-abelian simple groups there is a group $G$ generated by $3$ elements such that a finite non-abelian simple group $Q$ is an upper composition factor of $G$ if and only if $Q$ is isomorphic to $T_i$ for some $i \\in I$. Abelian upper composition factors of $G$ are not restricted.","title":"Kourovka Problem 11.57, 3-generator form (statement)"},{"id":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/upper-composition-factors-any-set63-hold","kind":"node","lean":"MathlibExt.GroupTheory.UpperCompositionFactorsKourovka11Problem57Wanted.upperCompositionFactorsAnySet63_holds","parent":3501,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/index.html#upper-composition-factors-any-set63-hold","scope":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated","status":"can_prove","summary":"For every family $(T_i)_{i \\in I}$ of finite non-abelian simple groups there is a group $G$ generated by $63$ elements such that a finite simple group $Q$ is an upper composition factor of $G$ (a composition factor of some finite quotient of $G$) if and only if $Q$ is isomorphic to $T_i$ for some $i \\in I$. In particular, no cyclic group of prime order is an upper composition factor of $G$.","title":"Segal's theorem on upper composition factors, 63 generators (Kourovka Problem 11.57)"},{"id":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/nonabelian-upper-composition-factors-any-2","kind":"node","lean":"MathlibExt.GroupTheory.UpperCompositionFactorsKourovka11Problem57Wanted.nonabelianUpperCompositionFactorsAnySet3_holds","parent":3501,"result":true,"route":"../roadmap/GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated/index.html#nonabelian-upper-composition-factors-any-2","scope":"GroupTheory/profinite-and-topological-groups/upper-composition-factors-of-finitely-generated","status":"can_prove","summary":"For every family $(T_i)_{i \\in I}$ of finite non-abelian simple groups there is a group $G$ generated by $3$ elements such that a finite non-abelian simple group $Q$ is an upper composition factor of $G$ (a composition factor of some finite quotient of $G$) if and only if $Q$ is isomorphic to $T_i$ for some $i \\in I$. Abelian upper composition factors of $G$ are not restricted.","title":"Segal's theorem on upper composition factors, 3 generators (Kourovka Problem 11.57)"},{"data_url":"scopes/GroupTheory/linear-groups.json","id":"GroupTheory/linear-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/linear-groups/index.html","scope":"GroupTheory/linear-groups","status":"can_prove","summary":"Groups of matrices over fields and rings: SL2, PGL2 and SU(2) and their conjugacy and subgroups (Dickson's classification), Jordan, Malcev and Selberg theorems, transvections, eigenvalues of elements of prime order, and linearity of specific groups.","title":"Linear groups"},{"id":"GroupTheory/linear-groups/general-linear-group-exponent-eq-lcm-eq","kind":"node","lean":"MetaMathlibExt.generalLinearGroup_exponent_eq_lcm_eq_cyclotomicProduct","parent":3508,"result":true,"route":"../roadmap/GroupTheory/linear-groups/index.html#general-linear-group-exponent-eq-lcm-eq","scope":"GroupTheory/linear-groups","status":"fully_proved","summary":"Let $p$ be a prime and let $m \\ge 1$. The exponent of the group $\\mathrm{GL}_m(\\mathbb{F}_p)$ of invertible $m \\times m$ matrices over $\\mathbb{F}_p = \\mathbb{Z}/p\\mathbb{Z}$ is","title":"Exponent of the general linear group over a prime field"},{"id":"GroupTheory/linear-groups/jordan-linear-groups","kind":"node","lean":"MathlibExt.GroupTheory.JordanLinearGroupsWanted.jordan_linear_groups","parent":3508,"result":true,"route":"../roadmap/GroupTheory/linear-groups/index.html#jordan-linear-groups","scope":"GroupTheory/linear-groups","status":"fully_proved","summary":"For every natural number $n$ there is a natural number $J$ such that every finite subgroup $G$ of $\\mathrm{GL}_n(\\mathbb{C})$ contains an abelian subgroup $A$ that is normal in $G$ and has index $[G : A] \\le J$.","title":"Jordan's theorem on finite linear groups"},{"id":"GroupTheory/linear-groups/malcev-residually-finite","kind":"node","lean":"MathlibExt.GroupTheory.LinearGroupFinitenessWanted.malcev_residuallyFinite","parent":3508,"result":true,"route":"../roadmap/GroupTheory/linear-groups/index.html#malcev-residually-finite","scope":"GroupTheory/linear-groups","status":"fully_proved","summary":"Let $K$ be a field and $n$ a natural number. Every finitely generated subgroup of the general linear group $\\mathrm{GL}_n(K)$ is residually finite.","title":"Malcev's theorem on linear groups"},{"id":"GroupTheory/linear-groups/selberg-finite-index-torsion-free","kind":"node","lean":"MathlibExt.GroupTheory.LinearGroupFinitenessWanted.selberg_finiteIndex_torsionFree","parent":3508,"result":true,"route":"../roadmap/GroupTheory/linear-groups/index.html#selberg-finite-index-torsion-free","scope":"GroupTheory/linear-groups","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero and let $n$ be a natural number. Every finitely generated subgroup $G$ of $\\mathrm{GL}_n(K)$ has a subgroup $H$ of finite index in $G$ that is torsion-free: the identity is the only element of finite order in $H$.","title":"Selberg's lemma"},{"id":"GroupTheory/linear-groups/has-stable-series","kind":"node","lean":"MathlibExt.GroupTheory.StableSeriesShared.HasStableSeries","parent":3508,"result":true,"route":"../roadmap/GroupTheory/linear-groups/index.html#has-stable-series","scope":"GroupTheory/linear-groups","status":"fully_proved","summary":"Let $V$ be a vector space over a field $K$. A set $S$ of $K$-linear automorphisms of $V$ is stable when there are a natural number $m$ and subspaces $0 = W_0 \\le W_1 \\le \\cdots \\le W_m = V$ of $V$ such that $g(w) - w \\in W_i$ for every $i < m$, every $g \\in S$ and every $w \\in W_{i+1}$.","title":"Stable sets of linear automorphisms"},{"data_url":"scopes/GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl.json","id":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl","kind":"scope","navigation":"module","parent":3508,"result":false,"route":"../roadmap/GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/index.html","scope":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl","status":"fully_proved","title":"Conjugacy by eigenvalues in SL\u2082"},{"id":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/has-eigenvalues","kind":"node","lean":"SL2.HasEigenvalues","parent":3514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(\\mathrm{SL}_2(F)\\)"],["text"," with eigenvalues "],["math","\\(s\\)"],["text"," and "],["math","\\(s^{-1}\\)"]],"source":"Elements of $\\mathrm{SL}_2(F)$ with eigenvalues $s$ and $s^{-1}$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/index.html#has-eigenvalues","scope":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl","status":"fully_proved","summary":"Let $F$ be a field and let $s \\in F$. A matrix $M \\in \\mathrm{SL}_2(F)$ has eigenvalues $s$ and $s^{-1}$ if its characteristic polynomial is $(X - s)(X - s^{-1})$.","title":"Elements of $\\mathrm{SL}_2(F)$ with eigenvalues $s$ and $s^{-1}$"},{"id":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/is-conj-diag2-iff","kind":"node","lean":"SL2.isConj_diag2_iff","parent":3514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conjugacy to "],["math","\\(\\operatorname{diag}(s, s^{-1})\\)"],["text"," in terms of a conjugator"]],"source":"Conjugacy to $\\operatorname{diag}(s, s^{-1})$ in terms of a conjugator"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/index.html#is-conj-diag2-iff","scope":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl","status":"fully_proved","summary":"Let $F$ be a field, let $s \\in F$ with $s \\ne 0$, and let $M \\in \\mathrm{SL}_2(F)$. Then $M$ is conjugate in $\\mathrm{SL}_2(F)$ to the diagonal matrix $\\operatorname{diag}(s, s^{-1})$ if and only if there is $P \\in \\mathrm{SL}_2(F)$ such that the matrix $P M P^{-1}$ equals $\\operatorname{diag}(s, s^{-1})$.","title":"Conjugacy to $\\operatorname{diag}(s, s^{-1})$ in terms of a conjugator"},{"id":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/normal-subgroup-contains-all-conj-to-dia","kind":"node","lean":"SL2.normal_subgroup_contains_all_conj_to_diag","parent":3514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conjugates of "],["math","\\(\\operatorname{diag}(s, s^{-1})\\)"],["text"," lie in the same normal subgroups of "],["math","\\(\\mathrm{SL}_2(F)\\)"]],"source":"Conjugates of $\\operatorname{diag}(s, s^{-1})$ lie in the same normal subgroups of $\\mathrm{SL}_2(F)$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/index.html#normal-subgroup-contains-all-conj-to-dia","scope":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl","status":"fully_proved","summary":"Conjugates of $\\operatorname{diag}(s, s^{-1})$ lie in the same normal subgroups of $\\mathrm{SL}_2(F)$. Let $F$ be a field, let $N$ be a normal subgroup of $\\mathrm{SL}_2(F)$, and let $s \\in F$. Suppose $N$ contains a matrix $B$ for which some $P \\in \\mathrm{SL}_2(F)$ makes the matrix $P B P^{-1}$ equal to the diagonal matrix $\\operatorname{diag}(s, s^{-1})$. Then $N$ contains every $Q \\in \\mathrm{SL}_2(F)$ for\u2026","title":"Conjugates of $\\operatorname{diag}(s, s^{-1})$ lie in the same normal subgroups of $\\mathrm{SL}_2(F)$"},{"id":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/eigenvalues-implies-is-conj-diag2","kind":"node","lean":"SL2.eigenvalues_implies_isConj_diag2","parent":3514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Distinct eigenvalues give conjugacy to "],["math","\\(\\operatorname{diag}(s, s^{-1})\\)"],["text"," in "],["math","\\(\\mathrm{SL}_2(F)\\)"]],"source":"Distinct eigenvalues give conjugacy to $\\operatorname{diag}(s, s^{-1})$ in $\\mathrm{SL}_2(F)$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/index.html#eigenvalues-implies-is-conj-diag2","scope":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl","status":"fully_proved","summary":"Distinct eigenvalues give conjugacy to $\\operatorname{diag}(s, s^{-1})$ in $\\mathrm{SL}_2(F)$. Let $F$ be a field and let $s \\in F$ with $s \\ne s^{-1}$ and $s \\ne 0$. If $M \\in \\mathrm{SL}_2(F)$ has eigenvalues $s$ and $s^{-1}$, that is, characteristic polynomial $(X - s)(X - s^{-1})$, then $M$ is conjugate in $\\mathrm{SL}_2(F)$ to the diagonal matrix $\\operatorname{diag}(s, s^{-1})$.","title":"Distinct eigenvalues give conjugacy to $\\operatorname{diag}(s, s^{-1})$ in $\\mathrm{SL}_2(F)$"},{"id":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/eigenvalues-implies-conj-to-diag","kind":"node","lean":"SL2.eigenvalues_implies_conj_to_diag","parent":3514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonalizing elements of "],["math","\\(\\mathrm{SL}_2(F)\\)"],["text"," with distinct eigenvalues"]],"source":"Diagonalizing elements of $\\mathrm{SL}_2(F)$ with distinct eigenvalues"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/index.html#eigenvalues-implies-conj-to-diag","scope":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl","status":"fully_proved","summary":"Let $F$ be a field, let $s \\in F$ with $s \\ne s^{-1}$, and let $M \\in \\mathrm{SL}_2(F)$ have eigenvalues $s$ and $s^{-1}$, that is, characteristic polynomial $(X - s)(X - s^{-1})$. Then there is $P \\in \\mathrm{SL}_2(F)$ such that the matrix $P M P^{-1}$ equals the diagonal matrix $\\operatorname{diag}(s, s^{-1})$.","title":"Diagonalizing elements of $\\mathrm{SL}_2(F)$ with distinct eigenvalues"},{"id":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/normal-contains-all-with-eigenvalues","kind":"node","lean":"SL2.normal_contains_all_with_eigenvalues","parent":3514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Matrices with the same distinct eigenvalues lie in the same normal subgroups of "],["math","\\(\\mathrm{SL}_2(F)\\)"]],"source":"Matrices with the same distinct eigenvalues lie in the same normal subgroups of $\\mathrm{SL}_2(F)$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl/index.html#normal-contains-all-with-eigenvalues","scope":"GroupTheory/linear-groups/conjugacy-by-eigenvalues-in-sl","status":"fully_proved","summary":"Matrices with the same distinct eigenvalues lie in the same normal subgroups of $\\mathrm{SL}_2(F)$. Let $F$ be a field, let $N$ be a normal subgroup of $\\mathrm{SL}_2(F)$, and let $s \\in F$ with $s \\ne s^{-1}$. If $N$ contains a matrix with eigenvalues $s$ and $s^{-1}$, then $N$ contains every matrix in $\\mathrm{SL}_2(F)$ with eigenvalues $s$ and $s^{-1}$.","title":"Matrices with the same distinct eigenvalues lie in the same normal subgroups of $\\mathrm{SL}_2(F)$"},{"data_url":"scopes/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of.json","id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","kind":"scope","navigation":"module","parent":3508,"result":false,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","title":"Dickson's classification of finite subgroups of PGL\u2082"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/k","kind":"node","lean":"MetaMathlibExt.Dickson.K","parent":3521,"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#k","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"For a prime $p$, $K_p$ denotes a fixed algebraic closure of the finite field $\\mathbb{F}_p = \\mathbb{Z}/p\\mathbb{Z}$.","title":"Algebraic closure of the field with p elements"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/pgl","kind":"node","lean":"MetaMathlibExt.Dickson.PGL","parent":3521,"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#pgl","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"For a prime $p$, $\\mathrm{PGL}_2(K_p)$ is the quotient of the group $\\mathrm{GL}_2(K_p)$ of invertible $2 \\times 2$ matrices over the algebraic closure $K_p$ of $\\mathbb{F}_p$ by its center.","title":"Projective general linear group of degree two"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/psl","kind":"node","lean":"MetaMathlibExt.Dickson.PSL","parent":3521,"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#psl","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"For a prime $p$, $\\mathrm{PSL}_2(K_p)$ is the quotient of the group $\\mathrm{SL}_2(K_p)$ of $2 \\times 2$ matrices of determinant $1$ over the algebraic closure $K_p$ of $\\mathbb{F}_p$ by its center.","title":"Projective special linear group of degree two"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/classification-tame","kind":"node","lean":"MetaMathlibExt.Dickson.classification_tame","parent":3521,"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#classification-tame","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"Let $p$ be an odd prime and let $K$ be an algebraic closure of $\\mathbb{F}_p$. Every finite nontrivial subgroup $G$ of $\\mathrm{PGL}_2(K)$ whose order is not divisible by $p$ is cyclic, or isomorphic to the dihedral group of order $2n$ for some $n \\ge 2$, or isomorphic to one of the groups $A_4$, $S_4$, $A_5$.","title":"Dickson's classification, tame case"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/proj-gen-lin-group-map-injective","kind":"node","lean":"MetaMathlibExt.Dickson.projGenLinGroup_map_injective","parent":3521,"reading_title":{"format":"inline-parts-v1","parts":[["text","Injective ring maps induce injective maps of "],["math","\\(\\mathrm{PGL}_n\\)"]],"source":"Injective ring maps induce injective maps of $\\mathrm{PGL}_n$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#proj-gen-lin-group-map-injective","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"Let $f : R \\to S$ be an injective homomorphism of commutative rings and let $n$ be a finite index set. Then the homomorphism $\\mathrm{PGL}_n(R) \\to \\mathrm{PGL}_n(S)$ induced by applying $f$ to matrix entries is injective, where $\\mathrm{PGL}_n(R)$ denotes $\\mathrm{GL}_n(R)$ modulo its center.","title":"Injective ring maps induce injective maps of $\\mathrm{PGL}_n$"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/card-proj-gen-lin-group-fin-two","kind":"node","lean":"MetaMathlibExt.Dickson.card_projGenLinGroup_fin_two","parent":3521,"reading_title":{"format":"inline-parts-v1","parts":[["text","Order of "],["math","\\(\\mathrm{PGL}_2\\)"],["text"," over a finite field"]],"source":"Order of $\\mathrm{PGL}_2$ over a finite field"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#card-proj-gen-lin-group-fin-two","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"If $F$ is a finite field with $q$ elements, then the projective general linear group $\\mathrm{PGL}_2(F)$ has order $q(q^2 - 1)$.","title":"Order of $\\mathrm{PGL}_2$ over a finite field"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/card-projective-special-linear-group-fin","kind":"node","lean":"MetaMathlibExt.Dickson.card_projectiveSpecialLinearGroup_fin_two","parent":3521,"reading_title":{"format":"inline-parts-v1","parts":[["text","Order of "],["math","\\(\\mathrm{PSL}_2\\)"],["text"," over a finite field of odd characteristic"]],"source":"Order of $\\mathrm{PSL}_2$ over a finite field of odd characteristic"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#card-projective-special-linear-group-fin","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements whose characteristic $p$ is not $2$. Then the projective special linear group $\\mathrm{PSL}_2(F)$ has order $q(q^2 - 1)/2$.","title":"Order of $\\mathrm{PSL}_2$ over a finite field of odd characteristic"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/order-of-eq-char-of-char-dvd","kind":"node","lean":"MetaMathlibExt.Dickson.orderOf_eq_char_of_char_dvd","parent":3521,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(\\mathrm{PGL}_2\\)"],["text"," of order divisible by the characteristic"]],"source":"Elements of $\\mathrm{PGL}_2$ of order divisible by the characteristic"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#order-of-eq-char-of-char-dvd","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"Let $p$ be an odd prime and let $F$ be a field of characteristic $p$. If an element $g$ of $\\mathrm{PGL}_2(F)$ has finite order and $p$ divides the order of $g$, then the order of $g$ is $p$.","title":"Elements of $\\mathrm{PGL}_2$ of order divisible by the characteristic"},{"id":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/classification-wild","kind":"node","lean":"MetaMathlibExt.Dickson.classification_wild","parent":3521,"result":true,"route":"../roadmap/GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of/index.html#classification-wild","scope":"GroupTheory/linear-groups/dickson-s-classification-of-finite-subgroups-of","status":"fully_proved","summary":"Let $p$ be an odd prime, let $K$ be an algebraic closure of $\\mathbb{F}_p$, and let $G$ be a finite subgroup of $\\mathrm{PGL}_2(K)$ whose order is divisible by $p$. Then at least one of the following holds. (a) There are natural numbers $m \\ge 1$ and $t$ with $t$ coprime to $p$ and $t \\mid p^m - 1$, and an injective homomorphism $\\varphi$ from the cyclic group $\\mathbb{Z}/t\\mathbb{Z}$ to the automorphism group of\u2026","title":"Dickson's classification, wild case"},{"data_url":"scopes/GroupTheory/linear-groups/double-commutators-of-transvections.json","id":"GroupTheory/linear-groups/double-commutators-of-transvections","kind":"scope","navigation":"module","parent":3508,"result":false,"route":"../roadmap/GroupTheory/linear-groups/double-commutators-of-transvections/index.html","scope":"GroupTheory/linear-groups/double-commutators-of-transvections","status":"can_prove","title":"Double commutators of transvections"},{"id":"GroupTheory/linear-groups/double-commutators-of-transvections/is-elementary-transvection","kind":"node","lean":"MathlibExt.GroupTheory.TransvectionCommutatorUnipotentKourovka10Problem46Wanted.IsElementaryTransvection","parent":3531,"result":true,"route":"../roadmap/GroupTheory/linear-groups/double-commutators-of-transvections/index.html#is-elementary-transvection","scope":"GroupTheory/linear-groups/double-commutators-of-transvections","status":"fully_proved","summary":"For a ring $R$ and $n \\in \\mathbb{N}$, a matrix $\\tau \\in \\mathrm{GL}_n(R)$ is an elementary transvection when $\\tau = 1 + r e_{ij}$ for some indices $i \\ne j$ and some nonzero $r \\in R$, where $e_{ij}$ is the matrix unit with $1$ in position $(i, j)$ and $0$ elsewhere.","title":"Elementary transvection"},{"id":"GroupTheory/linear-groups/double-commutators-of-transvections/double-commutator-with-transvections-uni","kind":"node","lean":"MathlibExt.GroupTheory.TransvectionCommutatorUnipotentKourovka10Problem46Wanted.doubleCommutatorWithTransvectionsUnipotent","parent":3531,"result":true,"route":"../roadmap/GroupTheory/linear-groups/double-commutators-of-transvections/index.html#double-commutator-with-transvections-uni","scope":"GroupTheory/linear-groups/double-commutators-of-transvections","status":"fully_proved","summary":"The statement that for every nontrivial associative ring $R$ with identity, every $n \\ge 3$ and every $\\sigma \\in \\mathrm{GL}_n(R)$ there are elementary transvections $\\tau_1, \\tau_2 \\in \\mathrm{GL}_n(R)$ such that $[[\\sigma, \\tau_1], \\tau_2]$ is unipotent, that is, $[[\\sigma, \\tau_1], \\tau_2] - 1$ is a nilpotent matrix, where $[x, y] = x y x^{-1} y^{-1}$.","title":"Petechuk's double commutator question (Kourovka 10.46)"},{"id":"GroupTheory/linear-groups/double-commutators-of-transvections/double-commutator-with-transvections-uni-2","kind":"node","lean":"MathlibExt.GroupTheory.TransvectionCommutatorUnipotentKourovka10Problem46Wanted.doubleCommutatorWithTransvectionsUnipotent_refuted","parent":3531,"result":true,"route":"../roadmap/GroupTheory/linear-groups/double-commutators-of-transvections/index.html#double-commutator-with-transvections-uni-2","scope":"GroupTheory/linear-groups/double-commutators-of-transvections","status":"can_prove","summary":"There are a nontrivial associative ring $R$ with identity, an integer $n \\ge 3$ and a matrix $\\sigma \\in \\mathrm{GL}_n(R)$ such that for all elementary transvections $\\tau_1, \\tau_2 \\in \\mathrm{GL}_n(R)$ the matrix $[[\\sigma, \\tau_1], \\tau_2] - 1$ is not nilpotent, that is, $[[\\sigma, \\tau_1], \\tau_2]$ is not unipotent. Here $[x, y] = x y x^{-1} y^{-1}$, and an elementary transvection is a matrix $1 + r e_{ij}$\u2026","title":"Petechuk's double commutator question has a negative answer (Kourovka 10.46)"},{"data_url":"scopes/GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order.json","id":"GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order","kind":"scope","navigation":"module","parent":3508,"result":false,"route":"../roadmap/GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order/index.html","scope":"GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order","status":"can_prove","title":"Eigenvalues of elements of prime order"},{"id":"GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order/prime-order-element-many-eigenvalues","kind":"node","lean":"MathlibExt.GroupTheory.LinearGroupPrimeOrderEigenvaluesKourovka11Problem35Wanted.primeOrderElementManyEigenvalues","parent":3535,"result":true,"route":"../roadmap/GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order/index.html#prime-order-element-many-eigenvalues","scope":"GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order","status":"fully_proved","summary":"The statement that for every $n$, every finite subgroup $H$ of $\\mathrm{GL}_n(\\mathbb{C})$ and every $h \\in H$ of prime order $p$ that lies in no abelian normal subgroup of $H$, the element $h$ has at least $(p - 1)/2$ distinct eigenvalues, that is, $p - 1 \\le 2k$ where $k$ is the number of distinct roots of the characteristic polynomial of $h$.","title":"Zalesski\u012d's problem on eigenvalues of elements of prime order (Kourovka 11.35)"},{"id":"GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order/prime-order-element-many-eigenvalues-hol","kind":"node","lean":"MathlibExt.GroupTheory.LinearGroupPrimeOrderEigenvaluesKourovka11Problem35Wanted.primeOrderElementManyEigenvalues_holds","parent":3535,"result":true,"route":"../roadmap/GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order/index.html#prime-order-element-many-eigenvalues-hol","scope":"GroupTheory/linear-groups/eigenvalues-of-elements-of-prime-order","status":"can_prove","summary":"For every $n$, if $H$ is a finite subgroup of $\\mathrm{GL}_n(\\mathbb{C})$ and $h \\in H$ has prime order $p$ and lies in no abelian normal subgroup of $H$, then $h$ has at least $(p - 1)/2$ distinct eigenvalues, that is, $p - 1 \\le 2k$ where $k$ is the number of distinct roots of the characteristic polynomial of $h$. This answers Kourovka Notebook Problem 11.35 (A. E. Zalesski\u012d) in the affirmative.","title":"Robinson's theorem on eigenvalues of elements of prime order"},{"data_url":"scopes/GroupTheory/linear-groups/linear-groups-inside-division-rings.json","id":"GroupTheory/linear-groups/linear-groups-inside-division-rings","kind":"scope","navigation":"module","parent":3508,"result":false,"route":"../roadmap/GroupTheory/linear-groups/linear-groups-inside-division-rings/index.html","scope":"GroupTheory/linear-groups/linear-groups-inside-division-rings","status":"can_prove","title":"Linear groups inside division rings"},{"id":"GroupTheory/linear-groups/linear-groups-inside-division-rings/special-linear-rational-embeds-in-divisi","kind":"node","lean":"MathlibExt.GroupTheory.SpecialLinearRationalDivisionRingKourovka11Problem110Wanted.specialLinearRationalEmbedsInDivisionRing","parent":3538,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linear-groups-inside-division-rings/index.html#special-linear-rational-embeds-in-divisi","scope":"GroupTheory/linear-groups/linear-groups-inside-division-rings","status":"fully_proved","summary":"The group $\\mathrm{SL}(2, \\mathbb{Q})$ embeds in the multiplicative group of some division ring: there are a division ring $D$ and an injective group homomorphism $\\mathrm{SL}(2, \\mathbb{Q}) \\to D^\\times$.","title":"Kourovka Problem 11.110 (B. Hartley)"},{"id":"GroupTheory/linear-groups/linear-groups-inside-division-rings/special-linear-rational-embeds-in-divisi-2","kind":"node","lean":"MathlibExt.GroupTheory.SpecialLinearRationalDivisionRingKourovka11Problem110Wanted.specialLinearRationalEmbedsInDivisionRing_refuted","parent":3538,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linear-groups-inside-division-rings/index.html#special-linear-rational-embeds-in-divisi-2","scope":"GroupTheory/linear-groups/linear-groups-inside-division-rings","status":"can_prove","summary":"There is no division ring $D$ with an injective group homomorphism $\\mathrm{SL}(2, \\mathbb{Q}) \\to D^\\times$: the group $\\mathrm{SL}(2, \\mathbb{Q})$ does not embed in the multiplicative group of any division ring.","title":"SL(2, \u211a) does not embed in a division ring"},{"data_url":"scopes/GroupTheory/linear-groups/linearity-of-one-relator-groups.json","id":"GroupTheory/linear-groups/linearity-of-one-relator-groups","kind":"scope","navigation":"module","parent":3508,"result":false,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-one-relator-groups/index.html","scope":"GroupTheory/linear-groups/linearity-of-one-relator-groups","status":"can_prove","title":"Linearity of one-relator groups"},{"id":"GroupTheory/linear-groups/linearity-of-one-relator-groups/relator","kind":"node","lean":"MathlibExt.GroupTheory.UnitriangularOneRelatorLinearKourovka18Problem86Wanted.relator","parent":3541,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-one-relator-groups/index.html#relator","scope":"GroupTheory/linear-groups/linearity-of-one-relator-groups","status":"fully_proved","summary":"This is the element $[[a, b], b]$ of the free group on two generators $a$ and $b$, where $[x, y] = x^{-1} y^{-1} x y$; explicitly, $[[a, b], b] = c^{-1} b^{-1} c b$ with $c = a^{-1} b^{-1} a b$.","title":"The relator of Kourovka Problem 18.86"},{"id":"GroupTheory/linear-groups/linearity-of-one-relator-groups/one-relator-group","kind":"node","lean":"MathlibExt.GroupTheory.UnitriangularOneRelatorLinearKourovka18Problem86Wanted.OneRelatorGroup","parent":3541,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-one-relator-groups/index.html#one-relator-group","scope":"GroupTheory/linear-groups/linearity-of-one-relator-groups","status":"fully_proved","summary":"This is the group $G = \\langle a, b \\mid [[a, b], b] = 1 \\rangle$ with generators $a, b$ and the single defining relation $[[a, b], b] = 1$, where $[x, y] = x^{-1} y^{-1} x y$; that is, $G$ is the quotient of the free group on $a$ and $b$ by the normal subgroup generated by $[[a, b], b]$.","title":"The one-relator group of Kourovka Problem 18.86"},{"id":"GroupTheory/linear-groups/linearity-of-one-relator-groups/one-relator-group-is-linear","kind":"node","lean":"MathlibExt.GroupTheory.UnitriangularOneRelatorLinearKourovka18Problem86Wanted.oneRelatorGroupIsLinear","parent":3541,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-one-relator-groups/index.html#one-relator-group-is-linear","scope":"GroupTheory/linear-groups/linearity-of-one-relator-groups","status":"fully_proved","summary":"The statement that the group $G$ with generators $a, b$ and the single defining relation $[[a, b], b] = 1$, where $[x, y] = x^{-1} y^{-1} x y$, is linear: there are a field $K$, a natural number $n$ and an injective group homomorphism $G \\to \\mathrm{GL}_n(K)$.","title":"Kourovka Problem 18.86 (statement)"},{"id":"GroupTheory/linear-groups/linearity-of-one-relator-groups/one-relator-group-is-linear-holds","kind":"node","lean":"MathlibExt.GroupTheory.UnitriangularOneRelatorLinearKourovka18Problem86Wanted.oneRelatorGroupIsLinear_holds","parent":3541,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-one-relator-groups/index.html#one-relator-group-is-linear-holds","scope":"GroupTheory/linear-groups/linearity-of-one-relator-groups","status":"can_prove","summary":"Let $G$ be the group with generators $a, b$ and the single defining relation $[[a, b], b] = 1$, where $[x, y] = x^{-1} y^{-1} x y$. Then $G$ is linear: there are a field $K$, a natural number $n$ and an injective group homomorphism $G \\to \\mathrm{GL}_n(K)$.","title":"The one-relator group of Kourovka Problem 18.86 is linear"},{"data_url":"scopes/GroupTheory/linear-groups/linearity-of-sapir-s-group.json","id":"GroupTheory/linear-groups/linearity-of-sapir-s-group","kind":"scope","navigation":"module","parent":3508,"result":false,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-sapir-s-group/index.html","scope":"GroupTheory/linear-groups/linearity-of-sapir-s-group","status":"can_prove","title":"Linearity of Sapir's group"},{"id":"GroupTheory/linear-groups/linearity-of-sapir-s-group/sapir-group","kind":"node","lean":"MathlibExt.GroupTheory.MappingTorusLinearKourovka17Problem108Wanted.SapirGroup","parent":3546,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-sapir-s-group/index.html#sapir-group","scope":"GroupTheory/linear-groups/linearity-of-sapir-s-group","status":"fully_proved","summary":"Sapir's group is the group with presentation $\\langle a, b, t \\mid a^t = ab, b^t = ba \\rangle$, where $a^t = t^{-1} a t$: the quotient of the free group on $a$, $b$, $t$ by the normal subgroup generated by $t^{-1} a t (ab)^{-1}$ and $t^{-1} b t (ba)^{-1}$.","title":"Sapir's group"},{"id":"GroupTheory/linear-groups/linearity-of-sapir-s-group/sapir-group-integer-linear","kind":"node","lean":"MathlibExt.GroupTheory.MappingTorusLinearKourovka17Problem108Wanted.sapirGroupIntegerLinear","parent":3546,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-sapir-s-group/index.html#sapir-group-integer-linear","scope":"GroupTheory/linear-groups/linearity-of-sapir-s-group","status":"fully_proved","summary":"The statement that the group $\\langle a, b, t \\mid a^t = ab, b^t = ba \\rangle$, where $a^t = t^{-1} a t$, embeds in $\\mathrm{GL}_n(\\mathbb{Z})$ for some natural number $n$, that is, that it is linear over $\\mathbb{Z}$.","title":"Linearity of Sapir's group over the integers (Kourovka 17.108)"},{"id":"GroupTheory/linear-groups/linearity-of-sapir-s-group/sapir-group-integer-linear-holds","kind":"node","lean":"MathlibExt.GroupTheory.MappingTorusLinearKourovka17Problem108Wanted.sapirGroupIntegerLinear_holds","parent":3546,"result":true,"route":"../roadmap/GroupTheory/linear-groups/linearity-of-sapir-s-group/index.html#sapir-group-integer-linear-holds","scope":"GroupTheory/linear-groups/linearity-of-sapir-s-group","status":"can_prove","summary":"The group $\\langle a, b, t \\mid a^t = ab, b^t = ba \\rangle$, where $a^t = t^{-1} a t$, embeds in $\\mathrm{GL}_n(\\mathbb{Z})$ for some natural number $n$. That is, Kourovka Problem 17.108 has a positive answer.","title":"Sapir's group is linear over the integers"},{"data_url":"scopes/GroupTheory/linear-groups/the-special-unitary-group-su-2.json","id":"GroupTheory/linear-groups/the-special-unitary-group-su-2","kind":"scope","navigation":"module","parent":3508,"result":false,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","title":"The special unitary group SU(2)"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotation","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For a real number $\\theta$, the diagonal rotation through $\\theta$ is the $2 \\times 2$ complex diagonal matrix $\\operatorname{diag}(e^{i\\theta}, e^{-i\\theta})$.","title":"Diagonal rotation"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-eq-diagonal","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotation_eq_diagonal","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-eq-diagonal","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For every real number $\\theta$, the diagonal rotation through $\\theta$ is the diagonal matrix whose diagonal entries are $e^{i\\theta}$ and $e^{-i\\theta}$, in that order.","title":"Diagonal rotations as diagonal matrices"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/star-diag-rotation","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.star_diagRotation","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#star-diag-rotation","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For every real number $\\theta$, the conjugate transpose of the diagonal rotation through $\\theta$ is the diagonal rotation through $-\\theta$.","title":"Conjugate transpose of a diagonal rotation"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-mem-special-unitary-group","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotation_mem_specialUnitaryGroup","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonal rotations lie in "],["math","\\(\\mathrm{SU}(2)\\)"]],"source":"Diagonal rotations lie in $\\mathrm{SU}(2)$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-mem-special-unitary-group","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For every real number $\\theta$, the diagonal rotation through $\\theta$ is a $2 \\times 2$ complex matrix that is unitary and has determinant $1$, that is, it lies in $\\mathrm{SU}(2)$.","title":"Diagonal rotations lie in $\\mathrm{SU}(2)$"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-int-mul-two-pi","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotation_int_mul_two_pi","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonal rotation through a multiple of "],["math","\\(2\\pi\\)"]],"source":"Diagonal rotation through a multiple of $2\\pi$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-int-mul-two-pi","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For every integer $n$, the diagonal rotation through $2\\pi n$ is the $2 \\times 2$ identity matrix.","title":"Diagonal rotation through a multiple of $2\\pi$"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-add-int-mul-two-pi","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotation_add_int_mul_two_pi","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonal rotations are "],["math","\\(2\\pi\\)"],["text","-periodic"]],"source":"Diagonal rotations are $2\\pi$-periodic"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-add-int-mul-two-pi","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For every real number $\\theta$ and every integer $n$, the diagonal rotation through $\\theta + 2\\pi n$ equals the diagonal rotation through $\\theta$.","title":"Diagonal rotations are $2\\pi$-periodic"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/longitude-subgroup","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.longitudeSubgroup","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#longitude-subgroup","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The longitude subgroup of $\\mathrm{SU}(2)$ is the subgroup of those elements of $\\mathrm{SU}(2)$ that are the diagonal rotation through $\\theta$ for some real number $\\theta$.","title":"Longitude subgroup"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-to-longitude","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotationToLongitude","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-to-longitude","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For a real number $\\theta$, the diagonal rotation through $\\theta$, regarded as an element of the longitude subgroup of $\\mathrm{SU}(2)$.","title":"Diagonal rotation as a longitude element"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-to-longitude-mul","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotationToLongitude_mul","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-to-longitude-mul","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For real numbers $a$ and $b$, in the longitude subgroup the product of the diagonal rotations through $a$ and through $b$ is the diagonal rotation through $a + b$.","title":"Longitude elements multiply by adding angles"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-to-longitude-eq-of-diag-ro","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotationToLongitude_eq_of_diagRotation_eq","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-to-longitude-eq-of-diag-ro","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For real numbers $a$ and $b$, if the diagonal rotations through $a$ and through $b$ are equal as matrices, then the elements of the longitude subgroup given by the diagonal rotations through $a$ and through $b$ are equal.","title":"Equal rotation matrices give equal longitude elements"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-to-longitude-hom","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotationToLongitudeHom","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-to-longitude-hom","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The homomorphism from the additive group $\\mathbb{R}$ to the longitude subgroup, written additively, that sends a real number $\\theta$ to the diagonal rotation through $\\theta$, regarded as an element of the longitude subgroup.","title":"The angle homomorphism into the longitude subgroup"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/longitude-map","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.longitudeMap","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#longitude-map","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The longitude map is the group homomorphism from the circle group $\\mathbb{R}/2\\pi\\mathbb{Z}$, written multiplicatively, to the longitude subgroup of $\\mathrm{SU}(2)$ that sends the class of a real number $\\theta$ to the diagonal rotation through $\\theta$, that is, to $\\cos\\theta \\cdot I + \\sin\\theta \\cdot \\operatorname{diag}(i, -i) = \\operatorname{diag}(e^{i\\theta}, e^{-i\\theta})$, where $I$ is the $2 \\times 2$\u2026","title":"Longitude map"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/longitude-map-apply-of-add-mk","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.longitudeMap_apply_ofAdd_mk","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#longitude-map-apply-of-add-mk","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For every real number $\\theta$, the longitude map sends the class of $\\theta$ in $\\mathbb{R}/2\\pi\\mathbb{Z}$ to the diagonal rotation through $\\theta$, regarded as an element of the longitude subgroup.","title":"The longitude map on the class of an angle"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-rotation-eq-diag-rotation-iff","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.diagRotation_eq_diagRotation_iff","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-rotation-eq-diag-rotation-iff","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For real numbers $a$ and $b$, the diagonal rotations through $a$ and through $b$ are equal if and only if $a = b + 2\\pi n$ for some integer $n$.","title":"When two diagonal rotations are equal"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/exists-of-add-mk-eq","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.exists_ofAdd_mk_eq","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every point of "],["math","\\(\\mathbb{R}/2\\pi\\mathbb{Z}\\)"],["text"," comes from a real angle"]],"source":"Every point of $\\mathbb{R}/2\\pi\\mathbb{Z}$ comes from a real angle"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#exists-of-add-mk-eq","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"Every element $x$ of the circle group $\\mathbb{R}/2\\pi\\mathbb{Z}$, written multiplicatively, is the class of some real number $a$.","title":"Every point of $\\mathbb{R}/2\\pi\\mathbb{Z}$ comes from a real angle"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/longitude-map-injective","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.longitudeMap_injective","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#longitude-map-injective","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The longitude map from the circle group $\\mathbb{R}/2\\pi\\mathbb{Z}$, written multiplicatively, to the longitude subgroup of $\\mathrm{SU}(2)$ is injective.","title":"The longitude map is injective"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/longitude-map-surjective","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.longitudeMap_surjective","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#longitude-map-surjective","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The longitude map from the circle group $\\mathbb{R}/2\\pi\\mathbb{Z}$, written multiplicatively, to the longitude subgroup of $\\mathrm{SU}(2)$ is surjective: every element of the longitude subgroup is the image of some class.","title":"The longitude map is surjective"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/longitude-map-bijective","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.longitudeMap_bijective","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#longitude-map-bijective","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The longitude map from the circle group $\\mathbb{R}/2\\pi\\mathbb{Z}$, written multiplicatively, to the longitude subgroup of $\\mathrm{SU}(2)$ is bijective.","title":"The longitude map is bijective"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/longitude-mul-equiv","kind":"node","lean":"MathlibExt.GroupTheory.SU2Longitude.longitudeMulEquiv","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#longitude-mul-equiv","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The group isomorphism from the circle group $\\mathbb{R}/2\\pi\\mathbb{Z}$, written multiplicatively, onto the longitude subgroup of $\\mathrm{SU}(2)$ given by the longitude map, which sends the class of a real number $\\theta$ to the diagonal rotation through $\\theta$.","title":"Isomorphism of the circle group with the longitude subgroup"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/exists-eq-of-mem-special-unitary-group-f","kind":"node","lean":"Matrix.exists_eq_of_mem_specialUnitaryGroup_fin_two","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Matrix form of "],["math","\\(\\mathrm{SU}(2)\\)"]],"source":"Matrix form of $\\mathrm{SU}(2)$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#exists-eq-of-mem-special-unitary-group-f","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"Let $M$ be a complex $2 \\times 2$ matrix in the special unitary group $\\mathrm{SU}(2)$, that is, $M$ is unitary with determinant $1$. Then there are complex numbers $a, b$ with $|a|^2 + |b|^2 = 1$ such that the first row of $M$ is $(a, b)$ and the second row of $M$ is $(-\\bar b, \\bar a)$.","title":"Matrix form of $\\mathrm{SU}(2)$"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/trace-im-eq-zero-of-mem-special-unitary","kind":"node","lean":"Matrix.trace_im_eq_zero_of_mem_specialUnitaryGroup_fin_two","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","The trace of an "],["math","\\(\\mathrm{SU}(2)\\)"],["text"," matrix is real"]],"source":"The trace of an $\\mathrm{SU}(2)$ matrix is real"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#trace-im-eq-zero-of-mem-special-unitary","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For every complex $2 \\times 2$ matrix $M$ in the special unitary group $\\mathrm{SU}(2)$, the imaginary part of the trace of $M$ is $0$.","title":"The trace of an $\\mathrm{SU}(2)$ matrix is real"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/norm-trace-le-two-of-mem-special-unitary","kind":"node","lean":"Matrix.norm_trace_le_two_of_mem_specialUnitaryGroup_fin_two","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","The trace of an "],["math","\\(\\mathrm{SU}(2)\\)"],["text"," matrix has absolute value at most two"]],"source":"The trace of an $\\mathrm{SU}(2)$ matrix has absolute value at most two"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#norm-trace-le-two-of-mem-special-unitary","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For every complex $2 \\times 2$ matrix $M$ in the special unitary group $\\mathrm{SU}(2)$, the trace of $M$ satisfies $|\\operatorname{tr} M| \\le 2$.","title":"The trace of an $\\mathrm{SU}(2)$ matrix has absolute value at most two"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/mem-center-special-unitary-group-fin-two","kind":"node","lean":"Matrix.mem_center_specialUnitaryGroup_fin_two_iff","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","The center of "],["math","\\(\\mathrm{SU}(2)\\)"]],"source":"The center of $\\mathrm{SU}(2)$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#mem-center-special-unitary-group-fin-two","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"An element $M$ of the special unitary group $\\mathrm{SU}(2)$ of complex $2 \\times 2$ matrices lies in the center of $\\mathrm{SU}(2)$ if and only if $M$ is the identity matrix $I$ or $M = -I$.","title":"The center of $\\mathrm{SU}(2)$"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/mem-special-unitary-group-fin-two-iff","kind":"node","lean":"Matrix.mem_specialUnitaryGroup_fin_two_iff","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(\\mathrm{SU}(2)\\)"]],"source":"Characterization of $\\mathrm{SU}(2)$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#mem-special-unitary-group-fin-two-iff","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"A complex $2 \\times 2$ matrix $M$ lies in the special unitary group $\\mathrm{SU}(2)$, that is, $M$ is unitary with determinant $1$, if and only if there are complex numbers $\\alpha, \\beta$ with $|\\alpha|^2 + |\\beta|^2 = 1$ such that the first row of $M$ is $(\\alpha, \\beta)$ and the second row is $(-\\bar\\beta, \\bar\\alpha)$.","title":"Characterization of $\\mathrm{SU}(2)$"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/diag-i-mem","kind":"node","lean":"SU2.diag_I_mem","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(\\operatorname{diag}(i, -i)\\)"],["text"," is special unitary"]],"source":"The matrix $\\operatorname{diag}(i, -i)$ is special unitary"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#diag-i-mem","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The complex diagonal matrix $\\operatorname{diag}(i, -i)$, where $i$ is the imaginary unit, lies in the special unitary group $\\mathrm{SU}(2)$ of complex $2 \\times 2$ matrices.","title":"The matrix $\\operatorname{diag}(i, -i)$ is special unitary"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/antidiag-mem","kind":"node","lean":"SU2.antidiag_mem","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix with rows "],["math","\\((0, 1)\\)"],["text"," and "],["math","\\((-1, 0)\\)"],["text"," is special unitary"]],"source":"The matrix with rows $(0, 1)$ and $(-1, 0)$ is special unitary"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#antidiag-mem","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The complex $2 \\times 2$ matrix with rows $(0, 1)$ and $(-1, 0)$ lies in the special unitary group $\\mathrm{SU}(2)$ of complex $2 \\times 2$ matrices.","title":"The matrix with rows $(0, 1)$ and $(-1, 0)$ is special unitary"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/su2-center-ne-top","kind":"node","lean":"SU2.su2_center_ne_top","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","The center of "],["math","\\(\\mathrm{SU}(2)\\)"],["text"," is proper"]],"source":"The center of $\\mathrm{SU}(2)$ is proper"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#su2-center-ne-top","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"The center of the special unitary group $\\mathrm{SU}(2)$ of complex $2 \\times 2$ matrices is not all of $\\mathrm{SU}(2)$.","title":"The center of $\\mathrm{SU}(2)$ is proper"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/su2-conj-to-diag","kind":"node","lean":"SU2.su2_conj_to_diag","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonalization in "],["math","\\(\\mathrm{SU}(2)\\)"],["text"," up to conjugacy"]],"source":"Diagonalization in $\\mathrm{SU}(2)$ up to conjugacy"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#su2-conj-to-diag","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"Every element of $\\mathrm{SU}(2)$ is conjugate in $\\mathrm{SU}(2)$ to an element of $\\mathrm{SU}(2)$ that is the diagonal matrix $\\operatorname{diag}(z, \\bar z)$ for some complex number $z$ with $z \\bar z = 1$.","title":"Diagonalization in $\\mathrm{SU}(2)$ up to conjugacy"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/su2-trace-conj-inv","kind":"node","lean":"SU2.su2_trace_conj_inv","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trace is invariant under conjugation in "],["math","\\(\\mathrm{SU}(2)\\)"]],"source":"Trace is invariant under conjugation in $\\mathrm{SU}(2)$"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#su2-trace-conj-inv","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"For all $P$ and $A$ in $\\mathrm{SU}(2)$, the trace of $P A P^{-1}$ equals the trace of $A$.","title":"Trace is invariant under conjugation in $\\mathrm{SU}(2)$"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/su2-diag-conj-of-trace","kind":"node","lean":"SU2.su2_diag_conj_of_trace","parent":3550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonal elements of "],["math","\\(\\mathrm{SU}(2)\\)"],["text"," with equal trace are conjugate"]],"source":"Diagonal elements of $\\mathrm{SU}(2)$ with equal trace are conjugate"},"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#su2-diag-conj-of-trace","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"Let $z_1, z_2$ be complex numbers with $z_1 \\bar z_1 = 1$ and $z_2 \\bar z_2 = 1$, and let $D_1, D_2$ be elements of $\\mathrm{SU}(2)$ equal to the diagonal matrices $\\operatorname{diag}(z_1, \\bar z_1)$ and $\\operatorname{diag}(z_2, \\bar z_2)$. If $D_1$ and $D_2$ have the same trace, then they are conjugate in $\\mathrm{SU}(2)$.","title":"Diagonal elements of $\\mathrm{SU}(2)$ with equal trace are conjugate"},{"id":"GroupTheory/linear-groups/the-special-unitary-group-su-2/su2-conj-of-trace-eq","kind":"node","lean":"SU2.su2_conj_of_trace_eq","parent":3550,"result":true,"route":"../roadmap/GroupTheory/linear-groups/the-special-unitary-group-su-2/index.html#su2-conj-of-trace-eq","scope":"GroupTheory/linear-groups/the-special-unitary-group-su-2","status":"fully_proved","summary":"Two elements of `SU\u2082` with the same trace are conjugate in `SU\u2082`.","title":"Conjugacy in `SU\u2082` is determined by the trace"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups.json","id":"GroupTheory/combinatorics-in-groups","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/index.html","scope":"GroupTheory/combinatorics-in-groups","status":"can_prove","summary":"Additive and Ramsey-type combinatorics inside groups: approximate subgroups and product theorems, small doubling, partitions and colourings of groups, dense and coset-free partitions, subset factorizations, products of large normal subsets.","title":"Combinatorics in groups"},{"id":"GroupTheory/combinatorics-in-groups/product-theorem-special-linear","kind":"node","lean":"MetaMathlibExt.product_theorem_special_linear","parent":3582,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/index.html#product-theorem-special-linear","scope":"GroupTheory/combinatorics-in-groups","status":"can_prove","summary":"Let $n$ be a natural number. For every real $\\varepsilon$ with $0 < \\varepsilon < 1$ there is a real $\\delta > 0$ such that the following holds for every finite field $k$: if $A$ is a subset of $\\mathrm{SL}(n, k)$ with $A^{-1} = A$ and $1 \\in A$ that generates $\\mathrm{SL}(n, k)$, then $|A|^{1+\\delta} \\le |A^3|$ or $|\\mathrm{SL}(n, k)|^{1-\\varepsilon} \\le |A|$, where $A^3 = \\{abc : a, b, c \\in A\\}$. Thus $\\delta$\u2026","title":"Product theorem for special linear groups"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/approximate-subgroups.json","id":"GroupTheory/combinatorics-in-groups/approximate-subgroups","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/approximate-subgroups/index.html","scope":"GroupTheory/combinatorics-in-groups/approximate-subgroups","status":"can_prove","title":"Approximate subgroups"},{"id":"GroupTheory/combinatorics-in-groups/approximate-subgroups/breuillard-green-tao-approximate-subgrou","kind":"node","lean":"MetaMathlibExt.breuillard_green_tao_approximate_subgroup","parent":3584,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/approximate-subgroups/index.html#breuillard-green-tao-approximate-subgrou","scope":"GroupTheory/combinatorics-in-groups/approximate-subgroups","status":"can_prove","summary":"For every real number $K$ there is a natural number $C \\ge 1$ with the following property. Let $G$ be a group and let $A \\subseteq G$ be a $K$-approximate subgroup, that is, $1 \\in A$, $A^{-1} = A$, and $A^2$ is covered by at most $K$ left translates of $A$. Then there are a subgroup $H$ of $G$, a normal subgroup $N$ of $H$, a finite set $F \\subseteq G$ and a finite set $S \\subseteq H/N$ such that $H/N$ is\u2026","title":"Breuillard\u2013Green\u2013Tao structure theorem for approximate subgroups"},{"id":"GroupTheory/combinatorics-in-groups/approximate-subgroups/breuillard-green-tao-small-doubling","kind":"node","lean":"MetaMathlibExt.breuillard_green_tao_small_doubling","parent":3584,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/approximate-subgroups/index.html#breuillard-green-tao-small-doubling","scope":"GroupTheory/combinatorics-in-groups/approximate-subgroups","status":"can_prove","summary":"For every real number $K$ there is a natural number $C > 0$ with the following property. Let $G$ be a group and let $A$ be a nonempty finite subset of $G$ such that the product set $A A = \\{ab : a, b \\in A\\}$ satisfies $|A A| \\le K |A|$. Then there are a subgroup $N$ of $G$, a normal subgroup $D$ of $N$ and a finite subset $S$ of $N / D$ with at most $C$ elements such that $N / D$ is nilpotent of class at most $C$\u2026","title":"Breuillard\u2013Green\u2013Tao theorem (small doubling form)"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups.json","id":"GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups/index.html","scope":"GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups","status":"can_prove","title":"Coset-free partitions of groups"},{"id":"GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups/odd-exponent-coset-free-partition","kind":"node","lean":"MathlibExt.GroupTheory.OddExponentCosetFreePartitionKourovka13Problem46Wanted.oddExponentCosetFreePartition","parent":3587,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups/index.html#odd-exponent-coset-free-partition","scope":"GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups","status":"fully_proved","summary":"The proposition that every uncountable abelian group $G$ of finite odd exponent can be written as $G = A_1 \\cup A_2$ with $A_1$ and $A_2$ disjoint, so that for every infinite subgroup $H$ of $G$ and every $g \\in G$, the coset $gH$ is contained neither in $A_1$ nor in $A_2$. The problem is due to I. V. Protasov.","title":"Kourovka Problem 13.46: coset-free partitions"},{"id":"GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups/odd-exponent-coset-free-partition-holds","kind":"node","lean":"MathlibExt.GroupTheory.OddExponentCosetFreePartitionKourovka13Problem46Wanted.oddExponentCosetFreePartition_holds","parent":3587,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups/index.html#odd-exponent-coset-free-partition-holds","scope":"GroupTheory/combinatorics-in-groups/coset-free-partitions-of-groups","status":"can_prove","summary":"Every uncountable abelian group $G$ of finite odd exponent can be written as $G = A_1 \\cup A_2$ with $A_1$ and $A_2$ disjoint, so that for every infinite subgroup $H$ of $G$ and every $g \\in G$, the coset $gH$ is contained neither in $A_1$ nor in $A_2$. This answers Kourovka Problem 13.46 in the affirmative.","title":"Uncountable abelian groups of odd exponent have coset-free partitions"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/dense-partitions-of-groups.json","id":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/index.html","scope":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups","status":"can_prove","title":"Dense partitions of groups"},{"id":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/countable-absolutely-dense-partition","kind":"node","lean":"MathlibExt.GroupTheory.AbsolutelyDensePartitionKourovka13Problem47Wanted.countableAbsolutelyDensePartition","parent":3590,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/index.html#countable-absolutely-dense-partition","scope":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups","status":"fully_proved","summary":"The proposition that every countable abelian group $G$ with only finitely many elements of order $2$ can be partitioned into two disjoint sets $A_1$ and $A_2$ with $A_1 \\cup A_2 = G$ such that $A_1$ and $A_2$ are both dense in every non-discrete Hausdorff group topology on $G$.","title":"Protasov's question on absolutely dense partitions (Kourovka 13.47)"},{"id":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/countable-absolutely-dense-partition-hol","kind":"node","lean":"MathlibExt.GroupTheory.AbsolutelyDensePartitionKourovka13Problem47Wanted.countableAbsolutelyDensePartition_holds","parent":3590,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/index.html#countable-absolutely-dense-partition-hol","scope":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups","status":"can_prove","summary":"Every countable abelian group $G$ with only finitely many elements of order $2$ can be partitioned into two disjoint sets $A_1$ and $A_2$ with $A_1 \\cup A_2 = G$ such that $A_1$ and $A_2$ are both dense in every non-discrete Hausdorff group topology on $G$.","title":"Kourovka Problem 13.47 has a positive answer"},{"id":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/is-expansive-sequence","kind":"node","lean":"MathlibExt.GroupTheory.DensePartitionExpansiveSequenceKourovka15Problem80Wanted.IsExpansiveSequence","parent":3590,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/index.html#is-expansive-sequence","scope":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups","status":"fully_proved","summary":"In a topological space $G$, a sequence $(F_n)_{n \\in \\mathbb{N}}$ of finite subsets of $G$ is expansive if the sets $F_n$ are pairwise disjoint and for every nonempty open set $U \\subseteq G$ there is $m \\in \\mathbb{N}$ such that $F_n$ meets $U$ for all $n > m$.","title":"Expansive sequence"},{"id":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/countable-dense-partition-has-expansive","kind":"node","lean":"MathlibExt.GroupTheory.DensePartitionExpansiveSequenceKourovka15Problem80Wanted.countableDensePartitionHasExpansiveSequence","parent":3590,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/index.html#countable-dense-partition-has-expansive","scope":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups","status":"fully_proved","summary":"Every topological group $G$ admitting a sequence $(P_i)_{i \\in \\mathbb{N}}$ of pairwise disjoint dense subsets with union $G$ has an expansive sequence.","title":"Kourovka Problem 15.80: dense partitions and expansive sequences"},{"id":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/countable-dense-partition-has-expansive-2","kind":"node","lean":"MathlibExt.GroupTheory.DensePartitionExpansiveSequenceKourovka15Problem80Wanted.countableDensePartitionHasExpansiveSequence_refuted","parent":3590,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/dense-partitions-of-groups/index.html#countable-dense-partition-has-expansive-2","scope":"GroupTheory/combinatorics-in-groups/dense-partitions-of-groups","status":"can_prove","summary":"It is not true that every topological group $G$ admitting a sequence $(P_i)_{i \\in \\mathbb{N}}$ of pairwise disjoint dense subsets with union $G$ has an expansive sequence, that is, a sequence $(F_n)_{n \\in \\mathbb{N}}$ of pairwise disjoint finite subsets of $G$ such that every nonempty open set $U \\subseteq G$ meets $F_n$ for all $n$ larger than some $m$.","title":"Kourovka Problem 15.80 has a negative answer"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups.json","id":"GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups/index.html","scope":"GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups","status":"can_prove","title":"Direct products of subsets of groups"},{"id":"GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups/is-direct-product","kind":"node","lean":"MathlibExt.GroupTheory.DirectProductFactorCardinalityKourovka21Problem58Wanted.IsDirectProduct","parent":3596,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups/index.html#is-direct-product","scope":"GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups","status":"fully_proved","summary":"For subsets $A$ and $B$ of a group $G$, the product $AB = \\{ab \\mid a \\in A,\\ b \\in B\\}$ is direct when each element of $AB$ is $ab$ for unique $a \\in A$ and $b \\in B$: whenever $a_1 b_1 = a_2 b_2$ with $a_1, a_2 \\in A$ and $b_1, b_2 \\in B$, we have $a_1 = a_2$ and $b_1 = b_2$.","title":"Direct product of subsets"},{"id":"GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups/exists-infinite-group-stable-factors","kind":"node","lean":"MathlibExt.GroupTheory.DirectProductFactorCardinalityKourovka21Problem58Wanted.existsInfiniteGroupStableFactors","parent":3596,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups/index.html#exists-infinite-group-stable-factors","scope":"GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups","status":"fully_proved","summary":"The statement that there is an infinite group $G$ such that for every subset $A \\subseteq G$, any two subsets $B_1, B_2 \\subseteq G$ that are maximal under inclusion among the subsets $B$ for which the product $AB$ is direct have the same cardinality.","title":"Infinite groups with equicardinal maximal direct factors (Kourovka 21.58)"},{"id":"GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups/exists-infinite-group-stable-factors-ref","kind":"node","lean":"MathlibExt.GroupTheory.DirectProductFactorCardinalityKourovka21Problem58Wanted.existsInfiniteGroupStableFactors_refuted","parent":3596,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups/index.html#exists-infinite-group-stable-factors-ref","scope":"GroupTheory/combinatorics-in-groups/direct-products-of-subsets-of-groups","status":"can_prove","summary":"For every infinite group $G$ there are subsets $A, B_1, B_2 \\subseteq G$ such that $B_1$ and $B_2$ are both maximal under inclusion among the subsets $B$ for which the product $AB$ is direct, but $B_1$ and $B_2$ have different cardinalities. That is, the answer to Kourovka Problem 21.58 is negative.","title":"No infinite group has equicardinal maximal direct factors"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/disjoint-coset-systems.json","id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","title":"Disjoint coset systems"},{"id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems/is-gharmonic","kind":"node","lean":"IsGHarmonic","parent":3600,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html#is-gharmonic","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","summary":"Let $G$ be a group and $a = (a_1, \\dots, a_n)$ a tuple of natural numbers. The tuple $a$ is $G$-harmonic if every $a_i$ is positive and there exist subgroups $U_1, \\dots, U_n$ of $G$ and elements $g_1, \\dots, g_n \\in G$ such that $U_i$ has index $a_i$ in $G$ for every $i$ and the left cosets $g_1 U_1, \\dots, g_n U_n$ are pairwise disjoint.","title":"G-harmonic tuples"},{"id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems/intro","kind":"node","lean":"IsGHarmonic.intro","parent":3600,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html#intro","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","summary":"Let $G$ be a group and $a = (a_1, \\dots, a_n)$ a tuple of natural numbers with every $a_i$ positive. If $U_1, \\dots, U_n$ are subgroups of $G$ such that $U_i$ has index $a_i$ in $G$ for every $i$, and $g_1, \\dots, g_n \\in G$ are elements such that the left cosets $g_1 U_1, \\dots, g_n U_n$ are pairwise disjoint, then $a$ is $G$-harmonic.","title":"Constructing a G-harmonic tuple"},{"id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems/pos","kind":"node","lean":"IsGHarmonic.pos","parent":3600,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html#pos","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","summary":"Let $G$ be a group. If a tuple $a = (a_1, \\dots, a_n)$ of natural numbers is $G$-harmonic, then $a_i > 0$ for every $i$.","title":"Entries of a G-harmonic tuple are positive"},{"id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems/witnesses","kind":"node","lean":"IsGHarmonic.witnesses","parent":3600,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html#witnesses","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","summary":"Let $G$ be a group. If a tuple $a = (a_1, \\dots, a_n)$ of natural numbers is $G$-harmonic, then there exist subgroups $U_1, \\dots, U_n$ of $G$ and elements $g_1, \\dots, g_n \\in G$ such that $U_i$ has index $a_i$ in $G$ for every $i$ and the left cosets $g_1 U_1, \\dots, g_n U_n$ are pairwise disjoint.","title":"Witnesses for a G-harmonic tuple"},{"id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems/is-gharmonic-iff","kind":"node","lean":"isGHarmonic_iff","parent":3600,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html#is-gharmonic-iff","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","summary":"Let $G$ be a group and $a = (a_1, \\dots, a_n)$ a tuple of natural numbers. Then $a$ is $G$-harmonic if and only if every $a_i$ is positive and there are subgroups $U_1, \\dots, U_n$ of $G$ and elements $g_1, \\dots, g_n \\in G$ such that $U_i$ has index $a_i$ in $G$ for every $i$ and the left cosets $g_1 U_1, \\dots, g_n U_n$ are pairwise disjoint.","title":"Characterization of G-harmonic tuples"},{"id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems/is-gharmonic-empty","kind":"node","lean":"isGHarmonic_empty","parent":3600,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html#is-gharmonic-empty","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","summary":"For every group $G$, the empty tuple of natural numbers is (vacuously) $G$-harmonic.","title":"The empty tuple is G-harmonic"},{"id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems/is-gharmonic-singleton-one","kind":"node","lean":"isGHarmonic_singleton_one","parent":3600,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html#is-gharmonic-singleton-one","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","summary":"For every group $G$, the one-entry tuple $(1)$ is $G$-harmonic; the subgroup $G$ itself, of index $1$, with coset representative $1$ witnesses this.","title":"The tuple (1) is G-harmonic"},{"id":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems/not-is-gharmonic-of-zero","kind":"node","lean":"not_isGHarmonic_of_zero","parent":3600,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/disjoint-coset-systems/index.html#not-is-gharmonic-of-zero","scope":"GroupTheory/combinatorics-in-groups/disjoint-coset-systems","status":"fully_proved","summary":"Let $G$ be a group. If a tuple $a = (a_1, \\dots, a_n)$ of natural numbers has $a_i = 0$ for some $i$, then $a$ is not $G$-harmonic.","title":"Tuples with a zero entry are not G-harmonic"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets.json","id":"GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets/index.html","scope":"GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets","status":"can_prove","title":"Factorizations of groups into subsets"},{"id":"GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets/has-subset-factorization","kind":"node","lean":"MathlibExt.GroupTheory.SubsetFactorizationCounterexamplesKourovka19Problem35Wanted.HasSubsetFactorization","parent":3609,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets/index.html#has-subset-factorization","scope":"GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets","status":"fully_proved","summary":"Let $G$ be a group, $k$ a natural number and $a_0, \\dots, a_{k-1}$ natural numbers. The group $G$ has a subset factorization of type $(a_0, \\dots, a_{k-1})$ when there are finite subsets $A_0, \\dots, A_{k-1}$ of $G$ with $|A_i| = a_i$ for every $i$ such that $G = A_0 A_1 \\cdots A_{k-1}$, that is, every $g \\in G$ can be written as $g = x_0 x_1 \\cdots x_{k-1}$ with $x_i \\in A_i$ for every $i$.","title":"Factorization of a group into subsets of given sizes"},{"id":"GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets/subset-factorization-counterexample-ever","kind":"node","lean":"MathlibExt.GroupTheory.SubsetFactorizationCounterexamplesKourovka19Problem35Wanted.subsetFactorizationCounterexampleEveryLength","parent":3609,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets/index.html#subset-factorization-counterexample-ever","scope":"GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets","status":"fully_proved","summary":"The statement that for every natural number $k \\ge 3$ there are a finite group $G$ and natural numbers $a_0, \\dots, a_{k-1}$, each at least $2$, with $a_0 a_1 \\cdots a_{k-1} = |G|$, such that $G$ has no subset factorization of type $(a_0, \\dots, a_{k-1})$: there are no subsets $A_0, \\dots, A_{k-1}$ of $G$ with $|A_i| = a_i$ for every $i$ and $G = A_0 A_1 \\cdots A_{k-1}$.","title":"Kourovka Problem 19.35: counterexamples to subset factorization of every length"},{"id":"GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets/subset-factorization-counterexample-ever-2","kind":"node","lean":"MathlibExt.GroupTheory.SubsetFactorizationCounterexamplesKourovka19Problem35Wanted.subsetFactorizationCounterexampleEveryLength_holds","parent":3609,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets/index.html#subset-factorization-counterexample-ever-2","scope":"GroupTheory/combinatorics-in-groups/factorizations-of-groups-into-subsets","status":"can_prove","summary":"For every natural number $k \\ge 3$ there are a finite group $G$ and natural numbers $a_0, \\dots, a_{k-1}$, each at least $2$, with $a_0 a_1 \\cdots a_{k-1} = |G|$, such that there are no subsets $A_0, \\dots, A_{k-1}$ of $G$ with $|A_i| = a_i$ for every $i$ and $G = A_0 A_1 \\cdots A_{k-1}$, that is, with every element of $G$ a product $x_0 x_1 \\cdots x_{k-1}$ where $x_i \\in A_i$ for every $i$.","title":"Finite groups without subset factorizations exist for every length"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group.json","id":"GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group/index.html","scope":"GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group","status":"can_prove","title":"Monochromatic symmetric subsets of abelian groups"},{"id":"GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group/monochrome-symmetric-subset-exists","kind":"node","lean":"MathlibExt.GroupTheory.MonochromeSymmetricSubsetKourovka15Problem77Wanted.monochromeSymmetricSubsetExists","parent":3613,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group/index.html#monochrome-symmetric-subset-exists","scope":"GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group","status":"fully_proved","summary":"The statement, written additively, that for every infinite abelian group $G$, every cardinal $\\alpha < |G|$, every natural number $n$ and every colouring $\\chi \\colon G \\to \\{0, \\dots, n - 1\\}$, there are a set $A \\subseteq G$ of cardinality $\\alpha$ on which $\\chi$ is constant and an element $g \\in G$ such that $2g - a \\in A$ for every $a \\in A$; that is, $A$ is a monochrome set symmetric with respect to $g$.","title":"Monochrome symmetric subsets (Kourovka 15.77)"},{"id":"GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group/monochrome-symmetric-subset-exists-holds","kind":"node","lean":"MathlibExt.GroupTheory.MonochromeSymmetricSubsetKourovka15Problem77Wanted.monochromeSymmetricSubsetExists_holds","parent":3613,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group/index.html#monochrome-symmetric-subset-exists-holds","scope":"GroupTheory/combinatorics-in-groups/monochromatic-symmetric-subsets-of-abelian-group","status":"can_prove","summary":"Let $G$ be an infinite abelian group, written additively, let $\\alpha$ be a cardinal with $\\alpha < |G|$, and let $n$ be a natural number. For every colouring $\\chi \\colon G \\to \\{0, \\dots, n - 1\\}$ there are a set $A \\subseteq G$ of cardinality $\\alpha$ on which $\\chi$ is constant and an element $g \\in G$ such that $2g - a \\in A$ for every $a \\in A$.","title":"Monochromatic symmetric subsets in finite colourings (Jayadevan)"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups.json","id":"GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups/index.html","scope":"GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups","status":"can_prove","title":"Nilpotent pairs in finite subsets of groups"},{"id":"GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups/four-subset-nilpotent-pair-two-generated","kind":"node","lean":"MathlibExt.GroupTheory.FourSubsetNilpotentPairKourovka17Problem3Wanted.fourSubsetNilpotentPairTwoGeneratedNilpotent","parent":3616,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups/index.html#four-subset-nilpotent-pair-two-generated","scope":"GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups","status":"fully_proved","summary":"The proposition that for every group $G$ in which every $4$-element subset contains two distinct elements generating a nilpotent subgroup, every subgroup of $G$ generated by two elements is nilpotent.","title":"Kourovka Problem 17.3 (Abdollahi's question on nilpotent pairs)"},{"id":"GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups/four-subset-nilpotent-pair-two-generated-2","kind":"node","lean":"MathlibExt.GroupTheory.FourSubsetNilpotentPairKourovka17Problem3Wanted.fourSubsetNilpotentPairTwoGeneratedNilpotent_refuted","parent":3616,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups/index.html#four-subset-nilpotent-pair-two-generated-2","scope":"GroupTheory/combinatorics-in-groups/nilpotent-pairs-in-finite-subsets-of-groups","status":"can_prove","summary":"There is a group $G$ in which every $4$-element subset contains two distinct elements generating a nilpotent subgroup, but some subgroup of $G$ generated by two elements is not nilpotent.","title":"Kourovka Problem 17.3 has a negative answer"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset.json","id":"GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset/index.html","scope":"GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset","status":"can_prove","title":"Partitions meeting every infinite coset"},{"id":"GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset/three-partition-meets-infinite-cosets","kind":"node","lean":"MathlibExt.GroupTheory.PartitionMeetingInfiniteCosetsKourovka16Problem43Wanted.threePartitionMeetsInfiniteCosets","parent":3619,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kourovka 16.43: partitioning "],["math","\\(\\bigoplus_{\\omega_1} \\mathbb{Z}/3\\mathbb{Z}\\)"],["text"," into three parts meeting every infinite coset"]],"source":"Kourovka 16.43: partitioning $\\bigoplus_{\\omega_1} \\mathbb{Z}/3\\mathbb{Z}$ into three parts meeting every infinite coset"},"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset/index.html#three-partition-meets-infinite-cosets","scope":"GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset","status":"fully_proved","summary":"Kourovka 16.43: partitioning $\\bigoplus_{\\omega_1} \\mathbb{Z}/3\\mathbb{Z}$ into three parts meeting every infinite coset. The proposition that for every set $I$ of cardinality $\\aleph_1$, the group $A = \\bigoplus_{i \\in I} \\mathbb{Z}/3\\mathbb{Z}$ of finitely supported functions from $I$ to $\\mathbb{Z}/3\\mathbb{Z}$ can be partitioned into three subsets $A_0, A_1, A_2$ such that for every infinite subgroup $H$ of $A$\u2026","title":"Kourovka 16.43: partitioning $\\bigoplus_{\\omega_1} \\mathbb{Z}/3\\mathbb{Z}$ into three parts meeting every infinite coset"},{"id":"GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset/three-partition-meets-infinite-cosets-ho","kind":"node","lean":"MathlibExt.GroupTheory.PartitionMeetingInfiniteCosetsKourovka16Problem43Wanted.threePartitionMeetsInfiniteCosets_holds","parent":3619,"reading_title":{"format":"inline-parts-v1","parts":[["text","A partition of "],["math","\\(\\bigoplus_{\\omega_1} \\mathbb{Z}/3\\mathbb{Z}\\)"],["text"," into three parts meeting every infinite coset"]],"source":"A partition of $\\bigoplus_{\\omega_1} \\mathbb{Z}/3\\mathbb{Z}$ into three parts meeting every infinite coset"},"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset/index.html#three-partition-meets-infinite-cosets-ho","scope":"GroupTheory/combinatorics-in-groups/partitions-meeting-every-infinite-coset","status":"can_prove","summary":"A partition of $\\bigoplus_{\\omega_1} \\mathbb{Z}/3\\mathbb{Z}$ into three parts meeting every infinite coset. Let $I$ be a set of cardinality $\\aleph_1$ and let $A = \\bigoplus_{i \\in I} \\mathbb{Z}/3\\mathbb{Z}$ be the group of finitely supported functions from $I$ to $\\mathbb{Z}/3\\mathbb{Z}$. Then $A$ can be partitioned into three subsets $A_0, A_1, A_2$ such that for every infinite subgroup $H$ of $A$ and every\u2026","title":"A partition of $\\bigoplus_{\\omega_1} \\mathbb{Z}/3\\mathbb{Z}$ into three parts meeting every infinite coset"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality.json","id":"GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality/index.html","scope":"GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality","status":"can_prove","title":"Partitions of groups of singular cardinality"},{"id":"GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality/singular-cardinality-partition","kind":"node","lean":"MathlibExt.GroupTheory.SingularCardinalityPartitionKourovka13Problem45Wanted.singularCardinalityPartition","parent":3622,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality/index.html#singular-cardinality-partition","scope":"GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality","status":"fully_proved","summary":"The statement that every infinite group $G$ whose cardinality $\\mathfrak{m} = |G|$ is singular (not regular) can be split into two disjoint sets $A_1$ and $A_2$ with $A_1 \\cup A_2 = G$ such that $A_1 F \\ne G$ and $A_2 F \\ne G$ for every subset $F \\subseteq G$ with $|F| < \\mathfrak{m}$, where $A F = \\{a f : a \\in A, f \\in F\\}$.","title":"Partitions of groups of singular cardinality (Kourovka 13.45)"},{"id":"GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality/singular-cardinality-partition-refuted","kind":"node","lean":"MathlibExt.GroupTheory.SingularCardinalityPartitionKourovka13Problem45Wanted.singularCardinalityPartition_refuted","parent":3622,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality/index.html#singular-cardinality-partition-refuted","scope":"GroupTheory/combinatorics-in-groups/partitions-of-groups-of-singular-cardinality","status":"can_prove","summary":"There is an infinite group $G$ whose cardinality $\\mathfrak{m} = |G|$ is singular (not regular) such that for all disjoint sets $A_1$ and $A_2$ with $A_1 \\cup A_2 = G$ there is a subset $F \\subseteq G$ with $|F| < \\mathfrak{m}$ and $A_1 F = G$ or $A_2 F = G$, where $A F = \\{a f : a \\in A, f \\in F\\}$. That is, the statement of Kourovka Problem 13.45 fails.","title":"Some group of singular cardinality has no such partition"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin.json","id":"GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin/index.html","scope":"GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin","status":"can_prove","title":"Product growth in special linear groups over finite fields"},{"id":"GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin/aimcombinatorics0241-prod-set","kind":"node","lean":"MathlibExt.Combinatorics.SL2TwoStepProductGrowthWanted.AIMCombinatorics0241_prodSet","parent":3625,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin/index.html#aimcombinatorics0241-prod-set","scope":"GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin","status":"fully_proved","summary":"For a finite subset $A$ of a type with a multiplication, the product set $A \\cdot A$ is the finite set $\\{ab : a, b \\in A\\}$ of all products of two elements of $A$.","title":"Product set of a finite set"},{"id":"GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.SL2TwoStepProductGrowthWanted.conjecture","parent":3625,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-step product growth in "],["math","\\(\\mathrm{SL}_2(\\mathbb{F}_p)\\)"],["text"," (AIM Problem 3.4)"]],"source":"Two-step product growth in $\\mathrm{SL}_2(\\mathbb{F}_p)$ (AIM Problem 3.4)"},"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin/index.html#conjecture","scope":"GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin","status":"fully_proved","summary":"The statement that for every real $C \\ge 1$ there are a real $\\delta > 0$ and a natural number $P_0$ such that for every prime $p \\ge P_0$ and every finite subset $A$ of $\\mathrm{SL}_2(\\mathbb{F}_p)$, where $\\mathbb{F}_p = \\mathbb{Z}/p\\mathbb{Z}$, with $C^{-1} p^{5/2} \\le |A| \\le C p^{5/2}$, the product set $A \\cdot A = \\{ab : a, b \\in A\\}$ satisfies $|A \\cdot A| > p^{5/2 + \\delta}$.","title":"Two-step product growth in $\\mathrm{SL}_2(\\mathbb{F}_p)$ (AIM Problem 3.4)"},{"id":"GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.SL2TwoStepProductGrowthWanted.conjecture_holds","parent":3625,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sets of size about "],["math","\\(p^{5/2}\\)"],["text"," in "],["math","\\(\\mathrm{SL}_2(\\mathbb{F}_p)\\)"],["text"," grow under multiplication"]],"source":"Sets of size about $p^{5/2}$ in $\\mathrm{SL}_2(\\mathbb{F}_p)$ grow under multiplication"},"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin/index.html#conjecture-holds","scope":"GroupTheory/combinatorics-in-groups/product-growth-in-special-linear-groups-over-fin","status":"can_prove","summary":"For every real $C \\ge 1$ there are a real $\\delta > 0$ and a natural number $P_0$ such that for every prime $p \\ge P_0$ and every finite subset $A$ of $\\mathrm{SL}_2(\\mathbb{F}_p)$, where $\\mathbb{F}_p = \\mathbb{Z}/p\\mathbb{Z}$, with $C^{-1} p^{5/2} \\le |A| \\le C p^{5/2}$, the product set $A \\cdot A = \\{ab : a, b \\in A\\}$ satisfies $|A \\cdot A| > p^{5/2 + \\delta}$.","title":"Sets of size about $p^{5/2}$ in $\\mathrm{SL}_2(\\mathbb{F}_p)$ grow under multiplication"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets.json","id":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/index.html","scope":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets","status":"can_prove","title":"Products of large normal subsets"},{"id":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/has-large-inverse-closed-normal-subset-w","kind":"node","lean":"MathlibExt.GroupTheory.LargeNormalSubsetSquareKourovka20Problem74Wanted.HasLargeInverseClosedNormalSubsetWithSmallSquare","parent":3629,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/index.html#has-large-inverse-closed-normal-subset-w","scope":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets","status":"fully_proved","summary":"A group $G$ has this property when it has a subset $S$ such that $g s g^{-1} \\in S$ and $s^{-1} \\in S$ for all $g \\in G$ and $s \\in S$, $S$ has more than $|G| / \\log_2 |G|$ elements, and $S^2 = \\{st : s, t \\in S\\}$ is not all of $G$.","title":"Large inverse-closed normal subsets with small square"},{"id":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/finitely-many-simple-groups-with-large-n","kind":"node","lean":"MathlibExt.GroupTheory.LargeNormalSubsetSquareKourovka20Problem74Wanted.finitelyManySimpleGroupsWithLargeNormalSubset","parent":3629,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/index.html#finitely-many-simple-groups-with-large-n","scope":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets","status":"fully_proved","summary":"This proposition states that there is a natural number $N$ such that every nonabelian finite simple group $G$ that has a subset $S$ closed under conjugation and under inversion with $|S| > |G| / \\log_2 |G|$ and $S^2 \\ne G$, where $S^2 = \\{st : s, t \\in S\\}$, has order $|G| \\le N$. Since there are only finitely many groups of each order, this says that only finitely many such groups exist up to isomorphism.","title":"Pyber's conjecture for finite simple groups (Kourovka 20.74(a))"},{"id":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/finitely-many-psl2-with-large-normal-sub","kind":"node","lean":"MathlibExt.GroupTheory.LargeNormalSubsetSquareKourovka20Problem74Wanted.finitelyManyPSL2WithLargeNormalSubset","parent":3629,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pyber's conjecture for "],["math","\\(\\mathrm{PSL}(2, p)\\)"],["text"," (Kourovka 20.74(b))"]],"source":"Pyber's conjecture for $\\mathrm{PSL}(2, p)$ (Kourovka 20.74(b))"},"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/index.html#finitely-many-psl2-with-large-normal-sub","scope":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets","status":"fully_proved","summary":"This proposition states that there is a natural number $N$ such that $p \\le N$ for every prime $p$ for which the group $G = \\mathrm{PSL}(2, p)$, the quotient of $\\mathrm{SL}(2, \\mathbb{Z}/p\\mathbb{Z})$ by its center, has a subset $S$ closed under conjugation and under inversion with $|S| > |G| / \\log_2 |G|$ and $S^2 \\ne G$, where $S^2 = \\{st : s, t \\in S\\}$. In other words, only finitely many groups\u2026","title":"Pyber's conjecture for $\\mathrm{PSL}(2, p)$ (Kourovka 20.74(b))"},{"id":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/finitely-many-simple-groups-with-large-n-2","kind":"node","lean":"MathlibExt.GroupTheory.LargeNormalSubsetSquareKourovka20Problem74Wanted.finitelyManySimpleGroupsWithLargeNormalSubset_refuted","parent":3629,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/index.html#finitely-many-simple-groups-with-large-n-2","scope":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets","status":"can_prove","summary":"There is no natural number $N$ such that every nonabelian finite simple group $G$ that has a subset $S$ with $g s g^{-1} \\in S$ and $s^{-1} \\in S$ for all $g \\in G$ and $s \\in S$, with $|S| > |G| / \\log_2 |G|$ and with $S^2 \\ne G$, where $S^2 = \\{st : s, t \\in S\\}$, has order $|G| \\le N$.","title":"Pyber's conjecture for finite simple groups is false"},{"id":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/finitely-many-psl2-with-large-normal-sub-2","kind":"node","lean":"MathlibExt.GroupTheory.LargeNormalSubsetSquareKourovka20Problem74Wanted.finitelyManyPSL2WithLargeNormalSubset_holds","parent":3629,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pyber's conjecture for "],["math","\\(\\mathrm{PSL}(2, p)\\)"],["text"," holds"]],"source":"Pyber's conjecture for $\\mathrm{PSL}(2, p)$ holds"},"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets/index.html#finitely-many-psl2-with-large-normal-sub-2","scope":"GroupTheory/combinatorics-in-groups/products-of-large-normal-subsets","status":"can_prove","summary":"There is a natural number $N$ such that for every prime $p$ the following holds. Let $G = \\mathrm{PSL}(2, p)$ be the quotient of $\\mathrm{SL}(2, \\mathbb{Z}/p\\mathbb{Z})$ by its center. If $G$ has a subset $S$ with $g s g^{-1} \\in S$ and $s^{-1} \\in S$ for all $g \\in G$ and $s \\in S$, with more than $|G| / \\log_2 |G|$ elements, and with $S^2 = \\{st : s, t \\in S\\}$ different from $G$, then $p \\le N$. In other words,\u2026","title":"Pyber's conjecture for $\\mathrm{PSL}(2, p)$ holds"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups.json","id":"GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups/index.html","scope":"GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups","status":"can_prove","title":"Products of permuted sequences in groups"},{"id":"GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups/permuted-products-take-every-count","kind":"node","lean":"MathlibExt.GroupTheory.PermutedProductsCountKourovka18Problem50Wanted.permutedProductsTakeEveryCount","parent":3635,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups/index.html#permuted-products-take-every-count","scope":"GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups","status":"fully_proved","summary":"The assertion that for all natural numbers $n$ and $k$ with $1 \\le k \\le n!$ there are a group $G$ and pairwise distinct elements $g_1, \\dots, g_n \\in G$ such that the set $\\lbrace g_{\\sigma(1)} g_{\\sigma(2)} \\cdots g_{\\sigma(n)} : \\sigma \\in S_n \\rbrace$ of products obtained by permuting the factors has exactly $k$ elements.","title":"Kohl's permuted-products statement (Kourovka 18.50)"},{"id":"GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups/permuted-products-take-every-count-holds","kind":"node","lean":"MathlibExt.GroupTheory.PermutedProductsCountKourovka18Problem50Wanted.permutedProductsTakeEveryCount_holds","parent":3635,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups/index.html#permuted-products-take-every-count-holds","scope":"GroupTheory/combinatorics-in-groups/products-of-permuted-sequences-in-groups","status":"can_prove","summary":"For all natural numbers $n$ and $k$ with $1 \\le k \\le n!$ there are a group $G$ and pairwise distinct elements $g_1, \\dots, g_n \\in G$ such that the set $\\lbrace g_{\\sigma(1)} g_{\\sigma(2)} \\cdots g_{\\sigma(n)} : \\sigma \\in S_n \\rbrace$ of products obtained by permuting the factors has exactly $k$ elements.","title":"Every number of distinct permuted products occurs (Kourovka 18.50)"},{"data_url":"scopes/GroupTheory/combinatorics-in-groups/separated-subsets-of-groups.json","id":"GroupTheory/combinatorics-in-groups/separated-subsets-of-groups","kind":"scope","navigation":"module","parent":3582,"result":false,"route":"../roadmap/GroupTheory/combinatorics-in-groups/separated-subsets-of-groups/index.html","scope":"GroupTheory/combinatorics-in-groups/separated-subsets-of-groups","status":"can_prove","title":"Separated subsets of groups"},{"id":"GroupTheory/combinatorics-in-groups/separated-subsets-of-groups/are-separated","kind":"node","lean":"MathlibExt.GroupTheory.SmallSubsetsSeparatedKourovka17Problem102Wanted.AreSeparated","parent":3638,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/separated-subsets-of-groups/index.html#are-separated","scope":"GroupTheory/combinatorics-in-groups/separated-subsets-of-groups","status":"fully_proved","summary":"Two subsets $A$ and $B$ of a group $G$ are separated, in the sense of I. V. Protasov, when there is an infinite subset $X \\subseteq G$ with $1 \\in X$, $X^{-1} = X$ and $XAX \\cap B = \\emptyset$, where $XAX = \\{x a y : x, y \\in X,\\ a \\in A\\}$.","title":"Separated subsets of a group"},{"id":"GroupTheory/combinatorics-in-groups/separated-subsets-of-groups/small-disjoint-subsets-separated","kind":"node","lean":"MathlibExt.GroupTheory.SmallSubsetsSeparatedKourovka17Problem102Wanted.smallDisjointSubsetsSeparated","parent":3638,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/separated-subsets-of-groups/index.html#small-disjoint-subsets-separated","scope":"GroupTheory/combinatorics-in-groups/separated-subsets-of-groups","status":"fully_proved","summary":"The statement, posed by I. V. Protasov, that for every infinite group $G$ and all disjoint subsets $A, B \\subseteq G$ with $|A| < |G|$ and $|B| < |G|$, the sets $A$ and $B$ are separated.","title":"Separation of small disjoint subsets of infinite groups (Kourovka 17.102)"},{"id":"GroupTheory/combinatorics-in-groups/separated-subsets-of-groups/small-disjoint-subsets-separated-refuted","kind":"node","lean":"MathlibExt.GroupTheory.SmallSubsetsSeparatedKourovka17Problem102Wanted.smallDisjointSubsetsSeparated_refuted","parent":3638,"result":true,"route":"../roadmap/GroupTheory/combinatorics-in-groups/separated-subsets-of-groups/index.html#small-disjoint-subsets-separated-refuted","scope":"GroupTheory/combinatorics-in-groups/separated-subsets-of-groups","status":"can_prove","summary":"There exist an infinite group $G$ and disjoint subsets $A, B \\subseteq G$ with $|A| < |G|$ and $|B| < |G|$ such that $A$ and $B$ are not separated. That is, the statement of Kourovka Problem 17.102 is false.","title":"Small disjoint subsets of an infinite group need not be separated"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology.json","id":"GroupTheory/extensions-and-cohomology","kind":"scope","parent":1661,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/index.html","scope":"GroupTheory/extensions-and-cohomology","status":"can_prove","summary":"Extensions and complements (Schur\u2013Zassenhaus), splitting of extensions, homology and first cohomology of groups, cohomological dimension, and integrability of groups as derived subgroups.","title":"Group extensions and cohomology"},{"id":"GroupTheory/extensions-and-cohomology/schur-zassenhaus-conjugate-complements","kind":"node","lean":"MathlibExt.GroupTheory.SchurZassenhausWanted.schur_zassenhaus_conjugate_complements","parent":3642,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/index.html#schur-zassenhaus-conjugate-complements","scope":"GroupTheory/extensions-and-cohomology","status":"fully_proved","summary":"Let $G$ be a finite group and $N$ a normal subgroup of $G$ whose order is coprime to its index, and suppose $N$ is solvable. If $K_1$ and $K_2$ are complements of $N$ in $G$, then there is an element $g \\in N$ with $g K_1 g^{-1} = K_2$.","title":"Schur\u2013Zassenhaus theorem: conjugacy of complements"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g.json","id":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g","kind":"scope","navigation":"module","parent":3642,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/index.html","scope":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g","status":"can_prove","title":"Cohomological dimension of just-non-polycyclic groups"},{"id":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/base-relators","kind":"node","lean":"MathlibExt.GroupTheory.CohomologicalDimensionJustNonPolycyclicKourovka12Problem41Wanted.baseRelators","parent":3644,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/index.html#base-relators","scope":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g","status":"fully_proved","summary":"Let $F$ be the free group on $x, y$, viewed inside the free group on $x, y, t$. The base relators are the elements $[[a, b], [c, d]]$ and $[a, b]^2$ for all $a, b, c, d \\in F$.","title":"Base relators of the Kropholler group"},{"id":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/twist-relators","kind":"node","lean":"MathlibExt.GroupTheory.CohomologicalDimensionJustNonPolycyclicKourovka12Problem41Wanted.twistRelators","parent":3644,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/index.html#twist-relators","scope":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g","status":"fully_proved","summary":"In the free group on $x, y, t$, the twist relators are the two elements $t x t^{-1} y^{-1}$ and $t y t^{-1} (x y)^{-1}$; they encode the relations $t x t^{-1} = y$ and $t y t^{-1} = x y$, so conjugation by $t$ acts as the automorphism $\\varphi \\colon x \\mapsto y,\\ y \\mapsto x y$.","title":"Twist relators of the Kropholler group"},{"id":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/kropholler-group","kind":"node","lean":"MathlibExt.GroupTheory.CohomologicalDimensionJustNonPolycyclicKourovka12Problem41Wanted.KrophollerGroup","parent":3644,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/index.html#kropholler-group","scope":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g","status":"fully_proved","summary":"The Kropholler group $G$ is the group with generators $x, y, t$ whose relators are the base relators, namely $[[a, b], [c, d]]$ and $[a, b]^2$ for all $a, b, c, d$ in the free group $F$ on $x, y$, together with the twist relators $t x t^{-1} y^{-1}$ and $t y t^{-1} (x y)^{-1}$.","title":"The Kropholler group"},{"id":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/rational-cohomological-dimension-eq-four","kind":"node","lean":"MathlibExt.GroupTheory.CohomologicalDimensionJustNonPolycyclicKourovka12Problem41Wanted.rationalCohomologicalDimensionEqFour","parent":3644,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/index.html#rational-cohomological-dimension-eq-four","scope":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g","status":"fully_proved","summary":"Let $G$ be the Kropholler group. This is the proposition that $H^4(G, M) \\neq 0$ for some $\\mathbb{Q}G$-module $M$, and $H^n(G, M) = 0$ for every $\\mathbb{Q}G$-module $M$ and every integer $n > 4$; that is, the cohomological dimension of $G$ over $\\mathbb{Q}$ is $4$.","title":"Rational cohomological dimension 4 of the Kropholler group (Kourovka 12.41)"},{"id":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/rational-cohomological-dimension-eq-four-2","kind":"node","lean":"MathlibExt.GroupTheory.CohomologicalDimensionJustNonPolycyclicKourovka12Problem41Wanted.rationalCohomologicalDimensionEqFour_holds","parent":3644,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g/index.html#rational-cohomological-dimension-eq-four-2","scope":"GroupTheory/extensions-and-cohomology/cohomological-dimension-of-just-non-polycyclic-g","status":"can_prove","summary":"Let $G$ be the Kropholler group. Then $H^4(G, M) \\neq 0$ for some $\\mathbb{Q}G$-module $M$, and $H^n(G, M) = 0$ for every $\\mathbb{Q}G$-module $M$ and every integer $n > 4$. This resolves Kourovka Notebook Problem 12.41 by P. H. Kropholler, which asks whether this cohomological dimension is 3 or 4.","title":"The Kropholler group has rational cohomological dimension 4"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups.json","id":"GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups","kind":"scope","navigation":"module","parent":3642,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups/index.html","scope":"GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups","status":"can_prove","title":"Dimension of first cohomology groups"},{"id":"GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups/is-absolutely-irreducible","kind":"node","lean":"MathlibExt.GroupTheory.FirstCohomologyDimensionKourovka16Problem55Wanted.IsAbsolutelyIrreducible","parent":3650,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups/index.html#is-absolutely-irreducible","scope":"GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups","status":"fully_proved","summary":"A representation $\\rho$ of a group $G$ on a vector space $V$ over a field $k$ is absolutely irreducible when $V \\neq 0$, the only $G$-invariant subspaces of $V$ are $0$ and $V$, and every $k$-linear endomorphism of $V$ commuting with $\\rho(g)$ for all $g \\in G$ is a scalar multiple of the identity.","title":"Absolutely irreducible representations"},{"id":"GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups/first-cohomology-dim-at-most-two","kind":"node","lean":"MathlibExt.GroupTheory.FirstCohomologyDimensionKourovka16Problem55Wanted.firstCohomologyDimAtMostTwo","parent":3650,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups/index.html#first-cohomology-dim-at-most-two","scope":"GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups","status":"fully_proved","summary":"The proposition that for every field $k$, every finite group $G$ and every faithful absolutely irreducible representation $\\rho$ of $G$ on a finite-dimensional $k$-vector space $V$, the first cohomology group $H^1(G, V)$ has dimension at most $2$ over $k$.","title":"Mazurov's question on first cohomology (Kourovka 16.55)"},{"id":"GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups/first-cohomology-dim-at-most-two-refuted","kind":"node","lean":"MathlibExt.GroupTheory.FirstCohomologyDimensionKourovka16Problem55Wanted.firstCohomologyDimAtMostTwo_refuted","parent":3650,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups/index.html#first-cohomology-dim-at-most-two-refuted","scope":"GroupTheory/extensions-and-cohomology/dimension-of-first-cohomology-groups","status":"can_prove","summary":"There exist a field $k$, a finite group $G$ and a faithful absolutely irreducible representation of $G$ on a finite-dimensional $k$-vector space $V$ such that the first cohomology group $H^1(G, V)$ has dimension greater than $2$ over $k$.","title":"Mazurov's question on first cohomology (Kourovka 16.55) has a negative answer"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r.json","id":"GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r","kind":"scope","navigation":"module","parent":3642,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r/index.html","scope":"GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r","status":"can_prove","title":"Extensions of the additive group of a division ring"},{"id":"GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r/division-ring-multiplication-extensions","kind":"node","lean":"MathlibExt.GroupTheory.DivisionRingExtensionSplitKourovka18Problem76Wanted.divisionRingMultiplicationExtensionsSplit","parent":3654,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r/index.html#division-ring-multiplication-extensions","scope":"GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r","status":"fully_proved","summary":"The statement that for every division ring $A$, every subgroup $G$ of its unit group $A^\\times$ and every group $E$, if $i$ is an injective homomorphism from the additive group of $A$ to $E$ and $\\pi : E \\to G$ is a surjective homomorphism whose kernel is the image of $i$, such that $e \\, i(a) \\, e^{-1} = i(\\pi(e) a)$ for all $e \\in E$ and $a \\in A$ (so $G$ acts on the kernel by left multiplication), then the\u2026","title":"Palyutin's splitting problem for division rings (Kourovka 18.76)"},{"id":"GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r/division-ring-multiplication-extensions-2","kind":"node","lean":"MathlibExt.GroupTheory.DivisionRingExtensionSplitKourovka18Problem76Wanted.divisionRingMultiplicationExtensionsSplit_refuted","parent":3654,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r/index.html#division-ring-multiplication-extensions-2","scope":"GroupTheory/extensions-and-cohomology/extensions-of-the-additive-group-of-a-division-r","status":"can_prove","summary":"There exist a division ring $A$, a subgroup $G$ of its unit group $A^\\times$, a group $E$, an injective homomorphism $i$ from the additive group of $A$ to $E$ and a surjective homomorphism $\\pi : E \\to G$ whose kernel is the image of $i$, with $e \\, i(a) \\, e^{-1} = i(\\pi(e) a)$ for all $e \\in E$ and $a \\in A$, such that no homomorphism $s : G \\to E$ satisfies $\\pi(s(g)) = g$ for all $g \\in G$. That is, the\u2026","title":"Extensions of a division ring's additive group by units need not split (Kourovka 18.76)"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups.json","id":"GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups","kind":"scope","navigation":"module","parent":3642,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups/index.html","scope":"GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups","status":"can_prove","title":"Homology of finitely generated metabelian groups"},{"id":"GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups/metabelian-homology-finite-rank","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianHomologyFiniteRankKourovka10Problem37Wanted.metabelianHomologyFiniteRank","parent":3657,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups/index.html#metabelian-homology-finite-rank","scope":"GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups","status":"fully_proved","summary":"The statement that every finitely generated metabelian group $G$ (that is, one whose second derived subgroup $G''$ is trivial) all of whose integral homology groups $H_n(G, \\mathbb{Z})$ with trivial coefficients, $n \\ge 0$, are finitely generated has finite Pr\u00fcfer rank: there is a natural number $r$ such that every finitely generated subgroup of $G$ is generated by at most $r$ elements.","title":"Kourovka Problem 10.37 (statement)"},{"id":"GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups/metabelian-homology-finite-rank-holds","kind":"node","lean":"MathlibExt.GroupTheory.MetabelianHomologyFiniteRankKourovka10Problem37Wanted.metabelianHomologyFiniteRank_holds","parent":3657,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups/index.html#metabelian-homology-finite-rank-holds","scope":"GroupTheory/extensions-and-cohomology/homology-of-finitely-generated-metabelian-groups","status":"can_prove","summary":"Finitely generated metabelian groups with finitely generated integral homology have finite rank. Let $G$ be a finitely generated metabelian group, that is, its second derived subgroup $G''$ is trivial, such that for every $n \\ge 0$ the integral homology group $H_n(G, \\mathbb{Z})$ with trivial coefficients is finitely generated. Then $G$ has finite Pr\u00fcfer rank: there is a natural number $r$ such that every finitely\u2026","title":"Finitely generated metabelian groups with finitely generated integral homology have finite rank"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology/integrable-groups.json","id":"GroupTheory/extensions-and-cohomology/integrable-groups","kind":"scope","navigation":"module","parent":3642,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/integrable-groups/index.html","scope":"GroupTheory/extensions-and-cohomology/integrable-groups","status":"can_prove","title":"Integrable groups"},{"id":"GroupTheory/extensions-and-cohomology/integrable-groups/is-integrable-group-table","kind":"node","lean":"MathlibExt.GroupTheory.IntegrableFiniteGroupDecidableKourovka21Problem32Wanted.IsIntegrableGroupTable","parent":3660,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/integrable-groups/index.html#is-integrable-group-table","scope":"GroupTheory/extensions-and-cohomology/integrable-groups","status":"fully_proved","summary":"A finite list $t = (t_0, \\dots, t_{n-1})$ of finite lists of natural numbers is an integrable group table when there is a group structure on $\\{0, \\dots, n-1\\}$ such that, for all $i, j < n$, the product of $i$ and $j$ is the entry of $t_i$ at index $j$, counting from $0$ (an entry missing from a short row is read as $0$), and some finite group $H$ has derived subgroup $[H, H]$ isomorphic to this group. Lists that\u2026","title":"Integrable group tables"},{"id":"GroupTheory/extensions-and-cohomology/integrable-groups/integrability-decidable","kind":"node","lean":"MathlibExt.GroupTheory.IntegrableFiniteGroupDecidableKourovka21Problem32Wanted.integrabilityDecidable","parent":3660,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/integrable-groups/index.html#integrability-decidable","scope":"GroupTheory/extensions-and-cohomology/integrable-groups","status":"fully_proved","summary":"The statement that integrable group tables form a decidable set: some computable function decides, for every finite list $t = (t_0, \\dots, t_{n-1})$ of finite lists of natural numbers, whether $t$ is an integrable group table, that is, whether there is a group structure on $\\{0, \\dots, n-1\\}$ in which, for all $i, j < n$, the product of $i$ and $j$ is the entry of $t_i$ at index $j$, counting from $0$ (an entry\u2026","title":"Kourovka Problem 21.32 (decidability of integrability)"},{"id":"GroupTheory/extensions-and-cohomology/integrable-groups/integrability-decidable-holds","kind":"node","lean":"MathlibExt.GroupTheory.IntegrableFiniteGroupDecidableKourovka21Problem32Wanted.integrabilityDecidable_holds","parent":3660,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/integrable-groups/index.html#integrability-decidable-holds","scope":"GroupTheory/extensions-and-cohomology/integrable-groups","status":"can_prove","summary":"Integrable group tables form a decidable set: some computable function decides, for every finite list $t = (t_0, \\dots, t_{n-1})$ of finite lists of natural numbers, whether there is a group structure on $\\{0, \\dots, n-1\\}$ in which, for all $i, j < n$, the product of $i$ and $j$ is the entry of $t_i$ at index $j$, counting from $0$ (an entry missing from a short row is read as $0$), and which is isomorphic to the\u2026","title":"Integrability of finite groups is decidable"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions.json","id":"GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions","kind":"scope","navigation":"module","parent":3642,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions/index.html","scope":"GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions","status":"can_prove","title":"Ranks in elementary abelian extensions"},{"id":"GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions/is-elementary-abelian","kind":"node","lean":"MathlibExt.GroupTheory.CentreDerivedRankExtensionKourovka21Problem150Wanted.IsElementaryAbelian","parent":3664,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions/index.html#is-elementary-abelian","scope":"GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions","status":"fully_proved","summary":"A subgroup $H$ of a group $G$ is elementary abelian if any two of its elements commute and there is a prime $p$ with $x^p = 1$ for all $x \\in H$.","title":"Elementary abelian subgroups"},{"id":"GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions/centre-derived-rank-le-complement-rank","kind":"node","lean":"MathlibExt.GroupTheory.CentreDerivedRankExtensionKourovka21Problem150Wanted.centreDerivedRankLeComplementRank","parent":3664,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions/index.html#centre-derived-rank-le-complement-rank","scope":"GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions","status":"fully_proved","summary":"Call a subgroup elementary abelian if it is abelian and there is a prime $p$ with $x^p = 1$ for all its elements $x$. This is the proposition that for every finite group $G$, all elementary abelian subgroups $A$ and $B$ of $G$ (possibly for different primes) with $A$ normal, $A \\cap B = 1$ and $\\langle A, B \\rangle = G$ (so $B$ is a complement to $A$ and $B \\cong G/A$), and every $a \\in A$ whose centralizer\u2026","title":"Zenkov's rank bound in elementary abelian extensions (Kourovka 21.150)"},{"id":"GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions/centre-derived-rank-le-complement-rank-r","kind":"node","lean":"MathlibExt.GroupTheory.CentreDerivedRankExtensionKourovka21Problem150Wanted.centreDerivedRankLeComplementRank_refuted","parent":3664,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions/index.html#centre-derived-rank-le-complement-rank-r","scope":"GroupTheory/extensions-and-cohomology/ranks-in-elementary-abelian-extensions","status":"can_prove","summary":"Call a subgroup elementary abelian if it is abelian and there is a prime $p$ with $x^p = 1$ for all its elements $x$. There are a finite group $G$, elementary abelian subgroups $A$ and $B$ of $G$ with $A$ normal, $A \\cap B = 1$ and $\\langle A, B \\rangle = G$, and an element $a \\in A$ whose centralizer $C_B(a)$ in $B$ is trivial, such that, for $H = \\langle a, B \\rangle$, with centre $Z(H)$ and derived subgroup\u2026","title":"Zenkov's rank bound fails (Kourovka 21.150)"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology/self-extensions-of-groups.json","id":"GroupTheory/extensions-and-cohomology/self-extensions-of-groups","kind":"scope","navigation":"module","parent":3642,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/self-extensions-of-groups/index.html","scope":"GroupTheory/extensions-and-cohomology/self-extensions-of-groups","status":"can_prove","title":"Self-extensions of groups"},{"id":"GroupTheory/extensions-and-cohomology/self-extensions-of-groups/is-hall-group","kind":"node","lean":"MathlibExt.GroupTheory.HallGroupSelfExtensionKourovka12Problem44Wanted.IsHallGroup","parent":3668,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/self-extensions-of-groups/index.html#is-hall-group","scope":"GroupTheory/extensions-and-cohomology/self-extensions-of-groups","status":"fully_proved","summary":"A group $G$ is a Hall group (in the sense of G\u00f6bel and Shelah) when it is isomorphic to every extension of itself by itself: for every group $H$ and every normal subgroup $N$ of $H$ with $N \\cong G$ and $H/N \\cong G$, also $H \\cong G$.","title":"Hall group"},{"id":"GroupTheory/extensions-and-cohomology/self-extensions-of-groups/exists-group-iso-to-self-extensions","kind":"node","lean":"MathlibExt.GroupTheory.HallGroupSelfExtensionKourovka12Problem44Wanted.existsGroupIsoToSelfExtensions","parent":3668,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/self-extensions-of-groups/index.html#exists-group-iso-to-self-extensions","scope":"GroupTheory/extensions-and-cohomology/self-extensions-of-groups","status":"fully_proved","summary":"There is a nontrivial group $G$ that is a Hall group, that is, isomorphic to every extension of itself by itself: every group $H$ with a normal subgroup $N \\cong G$ such that $H/N \\cong G$ is isomorphic to $G$.","title":"Hall's problem on self-extensions (Kourovka Problem 12.44)"},{"id":"GroupTheory/extensions-and-cohomology/self-extensions-of-groups/exists-group-iso-to-self-extensions-hold","kind":"node","lean":"MathlibExt.GroupTheory.HallGroupSelfExtensionKourovka12Problem44Wanted.existsGroupIsoToSelfExtensions_holds","parent":3668,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/self-extensions-of-groups/index.html#exists-group-iso-to-self-extensions-hold","scope":"GroupTheory/extensions-and-cohomology/self-extensions-of-groups","status":"can_prove","summary":"There is a nontrivial group $G$ that is isomorphic to every extension of itself by itself: every group $H$ with a normal subgroup $N \\cong G$ such that $H/N \\cong G$ is isomorphic to $G$.","title":"Existence of Hall groups (G\u00f6bel\u2013Shelah; Kourovka Problem 12.44)"},{"data_url":"scopes/GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup.json","id":"GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup","kind":"scope","navigation":"module","parent":3642,"result":false,"route":"../roadmap/GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup/index.html","scope":"GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup","status":"can_prove","title":"Surjections injective on no nontrivial subgroup"},{"id":"GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup/surjection-noninjective-on-subgroups","kind":"node","lean":"MathlibExt.GroupTheory.SurjectionNoninjectiveOnSubgroupsKourovka17Problem50Wanted.surjectionNoninjectiveOnSubgroups","parent":3672,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup/index.html#surjection-noninjective-on-subgroups","scope":"GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup","status":"fully_proved","summary":"The statement that for every finite group $G$ there are a finite group $F$ and a surjective homomorphism $f \\colon F \\to G$ such that for every nontrivial subgroup $H$ of $F$ the restriction of $f$ to $H$ is not injective.","title":"Kourovka Problem 17.50: surjections injective on no nontrivial subgroup"},{"id":"GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup/surjection-noninjective-on-subgroups-hol","kind":"node","lean":"MathlibExt.GroupTheory.SurjectionNoninjectiveOnSubgroupsKourovka17Problem50Wanted.surjectionNoninjectiveOnSubgroups_holds","parent":3672,"result":true,"route":"../roadmap/GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup/index.html#surjection-noninjective-on-subgroups-hol","scope":"GroupTheory/extensions-and-cohomology/surjections-injective-on-no-nontrivial-subgroup","status":"can_prove","summary":"For every finite group $G$ there are a finite group $F$ and a surjective homomorphism $f \\colon F \\to G$ such that for every nontrivial subgroup $H$ of $F$ the restriction of $f$ to $H$ is not injective.","title":"Burichenko's theorem on surjections injective on no nontrivial subgroup"},{"area":"Algebra & Number Theory","data_url":"chapters/RingTheory.json","id":"RingTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/RingTheory/index.html","scope":"RingTheory","status":"can_prove","summary":"Ring theory studies addition and multiplication through ideals, modules, localization, dimension, and homological structure.","title":"Ring Theory"},{"data_url":"scopes/RingTheory/ramification.json","id":"RingTheory/ramification","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/ramification/index.html","scope":"RingTheory/ramification","status":"fully_proved","summary":"How primes split, ramify and stay inert in finite extensions of Dedekind domains and discrete valuation rings, measured by the different, the discriminant, ramification indices, inertia degrees and Frobenius elements.","title":"Ramification theory"},{"data_url":"scopes/RingTheory/ramification/decomposition-and-inertia-groups.json","id":"RingTheory/ramification/decomposition-and-inertia-groups","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","title":"Decomposition and inertia groups"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/le-of-card-subgroup-of-eq","kind":"node","lean":"Subgroup.le_of_card_subgroupOf_eq","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#le-of-card-subgroup-of-eq","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $H$ and $K$ be subgroups of a group $G$ with $H$ finite. If $H \\cap K$ has as many elements as $H$, then $H \\le K$.","title":"Equal cardinality of an intersection forces containment"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/card-inertia-tower-formula","kind":"node","lean":"Ideal.card_inertia_tower_formula","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#card-inertia-tower-formula","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $L/K$ be a finite Galois extension with Galois group $G$. Let $A$ and $B$ be Dedekind domains with $B$ a finite torsion-free $A$-algebra, and let $G$ act on $B$ by ring automorphisms as a Galois group of $B$ over $A$: faithfully, fixing the image of $A$, and with every element of $B$ fixed by $G$ in the image of $A$. Let $E$ be an intermediate field of $L/K$ and $H \\le G$ the subgroup fixing $E$ pointwise. Let\u2026","title":"Tower formula for inertia groups"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/ramification-idx-mul-card-inertia-subgro","kind":"node","lean":"Ideal.ramificationIdx_mul_card_inertia_subgroupOf","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#ramification-idx-mul-card-inertia-subgro","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $L/K$ be a finite Galois extension with Galois group $G$. Let $A$ and $B$ be Dedekind domains with $B$ a finite torsion-free $A$-algebra, and let $G$ act on $B$ by ring automorphisms as a Galois group of $B$ over $A$: faithfully, fixing the image of $A$, and with every element of $B$ fixed by $G$ in the image of $A$. Let $E$ be an intermediate field of $L/K$ and $H \\le G$ the subgroup fixing $E$ pointwise. Let\u2026","title":"Tower formula for inertia groups, intersected with the fixing group"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/ramification-idx-eq-one-iff-inertia-le-f","kind":"node","lean":"Ideal.ramificationIdx_eq_one_iff_inertia_le_fixingSubgroup","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#ramification-idx-eq-one-iff-inertia-le-f","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $L/K$ be a finite Galois extension with Galois group $G$. Let $A$ and $B$ be Dedekind domains with $B$ a finite torsion-free $A$-algebra, and let $G$ act on $B$ by ring automorphisms as a Galois group of $B$ over $A$: faithfully, fixing the image of $A$, and with every element of $B$ fixed by $G$ in the image of $A$. Let $E$ be an intermediate field of $L/K$ and $H \\le G$ the subgroup fixing $E$ pointwise. Let\u2026","title":"Ramification index one and the inertia group"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/ramification-idx-eq-one-iff-le-inertia-f","kind":"node","lean":"Ideal.ramificationIdx_eq_one_iff_le_inertiaField","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#ramification-idx-eq-one-iff-le-inertia-f","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $L/K$ be a finite Galois extension with Galois group $G$. Let $A$ and $B$ be Dedekind domains with $B$ a finite torsion-free $A$-algebra, and let $G$ act on $B$ by ring automorphisms as a Galois group of $B$ over $A$: faithfully, fixing the image of $A$, and with every element of $B$ fixed by $G$ in the image of $A$. Let $E$ be an intermediate field of $L/K$ and $H \\le G$ the subgroup fixing $E$ pointwise. Let\u2026","title":"Ramification index one and the inertia field"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/ramification-idx-mul-inertia-deg-mul-car","kind":"node","lean":"Ideal.ramificationIdx_mul_inertiaDeg_mul_card_stabilizer_subgroupOf","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#ramification-idx-mul-inertia-deg-mul-car","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $L/K$ be a finite Galois extension with Galois group $G$. Let $A$ and $B$ be Dedekind domains with $B$ a finite torsion-free $A$-algebra, and let $G$ act on $B$ by ring automorphisms as a Galois group of $B$ over $A$: faithfully, fixing the image of $A$, and with every element of $B$ fixed by $G$ in the image of $A$. Let $E$ be an intermediate field of $L/K$ and $H \\le G$ the subgroup fixing $E$ pointwise. Let\u2026","title":"Tower formula for decomposition groups"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/ramification-idx-inertia-deg-eq-one-iff","kind":"node","lean":"Ideal.ramificationIdx_inertiaDeg_eq_one_iff_stabilizer_le_fixingSubgroup","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#ramification-idx-inertia-deg-eq-one-iff","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $L/K$ be a finite Galois extension with Galois group $G$. Let $A$ and $B$ be Dedekind domains with $B$ a finite torsion-free $A$-algebra, and let $G$ act on $B$ by ring automorphisms as a Galois group of $B$ over $A$: faithfully, fixing the image of $A$, and with every element of $B$ fixed by $G$ in the image of $A$. Let $E$ be an intermediate field of $L/K$ and $H \\le G$ the subgroup fixing $E$ pointwise. Let\u2026","title":"Ramification index and inertia degree one and the decomposition group"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/ramification-idx-inertia-deg-eq-one-iff-2","kind":"node","lean":"Ideal.ramificationIdx_inertiaDeg_eq_one_iff_le_decompositionField","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#ramification-idx-inertia-deg-eq-one-iff-2","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $L/K$ be a finite Galois extension with Galois group $G$. Let $A$ and $B$ be Dedekind domains with $B$ a finite torsion-free $A$-algebra, and let $G$ act on $B$ by ring automorphisms as a Galois group of $B$ over $A$: faithfully, fixing the image of $A$, and with every element of $B$ fixed by $G$ in the image of $A$. Let $E$ be an intermediate field of $L/K$ and $H \\le G$ the subgroup fixing $E$ pointwise. Let\u2026","title":"Ramification index and inertia degree one and the decomposition field"},{"id":"RingTheory/ramification/decomposition-and-inertia-groups/ramification-idx-inertia-deg-eq-one-iff-3","kind":"node","lean":"Ideal.ramificationIdx_inertiaDeg_eq_one_iff_le_inertiaField_decompositionField","parent":3677,"result":true,"route":"../roadmap/RingTheory/ramification/decomposition-and-inertia-groups/index.html#ramification-idx-inertia-deg-eq-one-iff-3","scope":"RingTheory/ramification/decomposition-and-inertia-groups","status":"fully_proved","summary":"Let $L/K$ be a finite Galois extension with Galois group $G$. Let $A$ and $B$ be Dedekind domains with $B$ a finite torsion-free $A$-algebra, and let $G$ act on $B$ by ring automorphisms as a Galois group of $B$ over $A$: faithfully, fixing the image of $A$, and with every element of $B$ fixed by $G$ in the image of $A$. Let $E$ be an intermediate field of $L/K$ and $H \\le G$ the subgroup fixing $E$ pointwise. Let\u2026","title":"Fields below the inertia and decomposition fields"},{"data_url":"scopes/RingTheory/ramification/different-and-discriminant-ideals.json","id":"RingTheory/ramification/different-and-discriminant-ideals","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","title":"Different and discriminant ideals"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/le-different-ideal-iff-mul-dual-le-one","kind":"node","lean":"Ideal.le_differentIdeal_iff_mul_dual_le_one","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#le-different-ideal-iff-mul-dual-le-one","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be an integrally closed domain with fraction field $K$, let $L/K$ be a finite separable extension, and let $B$ be the integral closure of $A$ in $L$; assume that $B$ is a Dedekind domain with fraction field $L$ and is torsion-free as an $A$-module. Write $B^\\vee = \\{x \\in L : \\operatorname{Tr}_{L/K}(xB) \\subseteq A\\}$ for the trace dual of $B$ and $\\mathfrak{D}_{B/A}$ for the different ideal of $B$ over\u2026","title":"Ideals below the different"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/extended-hom-dual-one-eq-dual-one-of-fie","kind":"node","lean":"FractionalIdeal.extendedHom_dual_one_eq_dual_one_ofField","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#extended-hom-dual-one-eq-dual-one-of-fie","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $L/K$ be a finite separable extension, and let $B$ be the integral closure of $A$ in $L$, a Dedekind domain with fraction field $L$. Let $S \\subseteq A$ be a multiplicative set of nonzerodivisors, let $A_S \\subseteq K$ be the localization of $A$ at $S$ inside $K$, and let $B_S$ be a localization of $B$ at the image of $S$ that is also an $A_S$-algebra, with\u2026","title":"The trace dual commutes with localization"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/map-different-ideal-eq-localization-of-f","kind":"node","lean":"Ideal.map_differentIdeal_eq_localization_ofField","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#map-different-ideal-eq-localization-of-f","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $L/K$ be a finite separable extension, and let $B$ be the integral closure of $A$ in $L$, a Dedekind domain with fraction field $L$ that is torsion-free as an $A$-module. Let $S \\subseteq A$ be a multiplicative set of nonzerodivisors, let $A_S \\subseteq K$ be the localization of $A$ at $S$ inside $K$, and let $B_S$ be a localization of $B$ at the image of\u2026","title":"The different commutes with localization"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/discriminant-ideal","kind":"node","lean":"Ideal.discriminantIdeal","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#discriminant-ideal","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ and $B$ be Dedekind domains, with $B$ an $A$-algebra that is finite and torsion-free as an $A$-module. The discriminant ideal $\\mathfrak{d}_{B/A}$ is the ideal of $A$ generated by the norms $N_{B/A}(x)$ of the elements $x$ of the different ideal $\\mathfrak{D}_{B/A}$ of $B$ over $A$.","title":"Discriminant ideal of a Dedekind extension"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/discriminant-ideal-eq-span-norm","kind":"node","lean":"Ideal.discriminantIdeal_eq_spanNorm","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#discriminant-ideal-eq-span-norm","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ and $B$ be Dedekind domains, with $B$ an $A$-algebra that is finite and torsion-free as an $A$-module. Then the discriminant ideal $\\mathfrak{d}_{B/A}$ is the ideal of $A$ generated by the norms $N_{B/A}(x)$ of the elements $x$ of the different ideal $\\mathfrak{D}_{B/A}$ of $B$ over $A$; this holds by definition.","title":"The discriminant ideal is the norm of the different"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/discriminant-ideal-tower","kind":"node","lean":"Ideal.discriminantIdeal_tower","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#discriminant-ideal-tower","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$, $B$, $C$ be Dedekind domains with algebra maps $A \\to B \\to C$ whose composite is the algebra map $A \\to C$, such that $B$ is finite and torsion-free over $A$, $C$ is finite and torsion-free over $A$ and over $B$, and the fraction field of $C$ is separable over the fraction field of $A$. Write $\\mathfrak{d}_{B/A}$, $\\mathfrak{d}_{C/A}$ and $\\mathfrak{d}_{C/B}$ for the discriminant ideals of $B$ over $A$,\u2026","title":"Tower formula for discriminant ideals"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/map-span-norm-different-ideal-eq-localiz","kind":"node","lean":"Ideal.map_spanNorm_differentIdeal_eq_localization_ofField","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#map-span-norm-different-ideal-eq-localiz","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $L/K$ be a finite separable extension, and let $B$ be the integral closure of $A$ in $L$, a Dedekind domain with fraction field $L$ that is finite and torsion-free as an $A$-module. Let $S \\subseteq A$ be a multiplicative set of nonzerodivisors, let $A_S \\subseteq K$ be the localization of $A$ at $S$ inside $K$, and let $B_S$ be a localization of $B$ at the\u2026","title":"The norm of the different commutes with localization"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/map-discriminant-ideal-eq-localization-o","kind":"node","lean":"Ideal.map_discriminantIdeal_eq_localization_ofField","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#map-discriminant-ideal-eq-localization-o","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $L/K$ be a finite separable extension, and let $B$ be the integral closure of $A$ in $L$, a Dedekind domain with fraction field $L$ that is finite and torsion-free as an $A$-module. Let $S \\subseteq A$ be a multiplicative set of nonzerodivisors, let $A_S \\subseteq K$ be the localization of $A$ at $S$ inside $K$, and let $B_S$ be a localization of $B$ at the\u2026","title":"The discriminant commutes with localization"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/element-different","kind":"node","lean":"elementDifferent","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#element-different","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be a commutative ring, $B$ a commutative $A$-algebra, $L$ a field that is a $B$-algebra, and $K$ a field with $L$ a $K$-algebra. For $\\alpha \\in B$, let $f \\in A[X]$ be the minimal polynomial of $\\alpha$ over $A$ (taken to be $0$ when $\\alpha$ is not integral over $A$). The element different $\\delta_{B/A}(\\alpha) \\in B$ is $f'(\\alpha)$ when the image of $\\alpha$ in $L$ generates $L$ as a $K$-algebra, and\u2026","title":"Element different"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/element-different-of-adjoin-eq-top","kind":"node","lean":"elementDifferent_of_adjoin_eq_top","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#element-different-of-adjoin-eq-top","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be a commutative ring, $B$ a commutative $A$-algebra, $L$ a field that is a $B$-algebra, and $K$ a field with $L$ a $K$-algebra. Let $\\alpha \\in B$ and let $f$ be the minimal polynomial of $\\alpha$ over $A$ (taken to be $0$ when $\\alpha$ is not integral over $A$). If the image of $\\alpha$ in $L$ generates $L$ as a $K$-algebra, then the element different $\\delta_{B/A}(\\alpha)$ equals $f'(\\alpha)$.","title":"Element different of a generator"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/element-different-of-adjoin-ne-top","kind":"node","lean":"elementDifferent_of_adjoin_ne_top","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#element-different-of-adjoin-ne-top","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be a commutative ring, $B$ a commutative $A$-algebra, $L$ a field that is a $B$-algebra, and $K$ a field with $L$ a $K$-algebra. If $\\alpha \\in B$ and the image of $\\alpha$ in $L$ does not generate $L$ as a $K$-algebra, then the element different $\\delta_{B/A}(\\alpha)$ is $0$.","title":"Element different of a non-generator"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/element-different-eq-zero-iff","kind":"node","lean":"elementDifferent_eq_zero_iff","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#element-different-eq-zero-iff","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be a commutative ring, $B$ a commutative $A$-algebra, $L$ a field that is a $B$-algebra, and $K$ a field with $L$ a $K$-algebra. Let $\\alpha \\in B$ and let $f$ be the minimal polynomial of $\\alpha$ over $A$ (taken to be $0$ when $\\alpha$ is not integral over $A$). Then the element different $\\delta_{B/A}(\\alpha)$ is $0$ if and only if the image of $\\alpha$ in $L$ does not generate $L$ as a $K$-algebra or\u2026","title":"Vanishing of the element different"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/element-different-mem-different-ideal","kind":"node","lean":"elementDifferent_mem_differentIdeal","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#element-different-mem-different-ideal","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be an integrally closed domain with fraction field $K$, let $L$ be a finite separable extension of $K$, and let $B$ be the integral closure of $A$ in $L$, assumed to be a Dedekind domain with fraction field $L$ that is torsion-free as an $A$-module, with compatible algebra maps $A \\to K \\to L$ and $A \\to B \\to L$. If $\\alpha \\in B$ and the image of $\\alpha$ in $L$ generates $L$ as a $K$-algebra, then the\u2026","title":"The element different lies in the different"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/different-ideal-eq-span-derivative-of-ad","kind":"node","lean":"differentIdeal_eq_span_derivative_of_adjoin_eq_top","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#different-ideal-eq-span-derivative-of-ad","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ be an integrally closed domain with fraction field $K$, let $L$ be a finite separable extension of $K$, and let $B$ be the integral closure of $A$ in $L$, assumed to be a Dedekind domain with fraction field $L$ that is torsion-free as an $A$-module, with compatible algebra maps $A \\to K \\to L$ and $A \\to B \\to L$. If $\\alpha \\in B$ generates $B$ as an $A$-algebra, that is $B = A[\\alpha]$, and $f$ is the\u2026","title":"The different of a monogenic extension"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/finite-ramified-primes","kind":"node","lean":"Ideal.finite_ramified_primes","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#finite-ramified-primes","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ and $B$ be Dedekind domains, where $B$ is an $A$-algebra that is finitely generated and torsion-free as an $A$-module and the field of fractions of $B$ is separable over the field of fractions of $A$. Then only finitely many nonzero prime ideals $\\mathfrak P$ of $B$ are ramified over $A$. Here $\\mathfrak P$ is unramified over $A$ when the localization $B_{\\mathfrak P}$ is formally unramified over $A$.","title":"Only finitely many primes of the extension ramify"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/finite-ramified-primes-below","kind":"node","lean":"Ideal.finite_ramified_primes_below","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#finite-ramified-primes-below","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Let $A$ and $B$ be Dedekind domains, where $B$ is an $A$-algebra that is finitely generated and torsion-free as an $A$-module and the field of fractions of $B$ is separable over the field of fractions of $A$. Then only finitely many nonzero prime ideals $\\mathfrak p$ of $A$ are ramified in $B$. Here $\\mathfrak p$ is unramified in $B$ when every prime ideal $\\mathfrak P$ of $B$ lying over $\\mathfrak p$ is unramified\u2026","title":"Only finitely many primes of the base ramify"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/is-unramified-in-iff-not-dvd-discriminan","kind":"node","lean":"Ideal.isUnramifiedIn_iff_not_dvd_discriminantIdeal","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#is-unramified-in-iff-not-dvd-discriminan","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Theorem 12.19, discriminant half, in Mathlib's standard predicate: `\ud835\udd2d` is unramified in `B` exactly when it does not divide the discriminant ideal.","title":"Unramified primes are those not dividing the discriminant"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/not-dvd-discriminant-ideal-of-is-unramif","kind":"node","lean":"Ideal.not_dvd_discriminantIdeal_of_isUnramifiedAt","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#not-dvd-discriminant-ideal-of-is-unramif","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Height-one compatibility: derive nondivisibility from the standard unramified-in criterion after converting the original height-one premise.","title":"Unramified at a height-one prime implies not dividing discriminant"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/is-unramified-at-of-not-dvd-discriminant","kind":"node","lean":"Ideal.isUnramifiedAt_of_not_dvd_discriminantIdeal","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#is-unramified-at-of-not-dvd-discriminant","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"Height-one compatibility: evaluate the standard unramified-in conclusion at the particular height-one prime supplied by an original client.","title":"Not dividing the discriminant implies unramified at a prime"},{"id":"RingTheory/ramification/different-and-discriminant-ideals/is-unramified-at-iff-not-dvd-discriminan","kind":"node","lean":"Ideal.isUnramifiedAt_iff_not_dvd_discriminantIdeal","parent":3687,"result":true,"route":"../roadmap/RingTheory/ramification/different-and-discriminant-ideals/index.html#is-unramified-at-iff-not-dvd-discriminan","scope":"RingTheory/ramification/different-and-discriminant-ideals","status":"fully_proved","summary":"The original height-one formulation, derived from the standard primary criterion.","title":"Height-one form of the discriminant ramification criterion"},{"data_url":"scopes/RingTheory/ramification/extensions-of-valuations-above-a-prime.json","id":"RingTheory/ramification/extensions-of-valuations-above-a-prime","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/extensions-of-valuations-above-a-prime/index.html","scope":"RingTheory/ramification/extensions-of-valuations-above-a-prime","status":"fully_proved","title":"Extensions of valuations above a prime"},{"id":"RingTheory/ramification/extensions-of-valuations-above-a-prime/extends-valuation","kind":"node","lean":"ValuationBijection.extendsValuation","parent":3708,"result":true,"route":"../roadmap/RingTheory/ramification/extensions-of-valuations-above-a-prime/index.html#extends-valuation","scope":"RingTheory/ramification/extensions-of-valuations-above-a-prime","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $L$ be a field extension of $K$, and let $\\mathfrak{p}$ be a height-one prime of $A$. A valuation $w$ on $L$ with values in the multiplicative group $\\mathbb{Z}$ with a zero adjoined extends the $\\mathfrak{p}$-adic valuation of $K$ when its restriction to $K$ is equivalent to the $\\mathfrak{p}$-adic valuation of $K$.","title":"Valuations extending a prime-adic valuation"},{"id":"RingTheory/ramification/extensions-of-valuations-above-a-prime/primes-above","kind":"node","lean":"ValuationBijection.primesAbove","parent":3708,"result":true,"route":"../roadmap/RingTheory/ramification/extensions-of-valuations-above-a-prime/index.html#primes-above","scope":"RingTheory/ramification/extensions-of-valuations-above-a-prime","status":"fully_proved","summary":"Let $A$ and $B$ be Dedekind domains with $B$ an $A$-algebra, and let $\\mathfrak{p}$ be a height-one prime of $A$. The primes above $\\mathfrak{p}$ form the set of height-one primes $\\mathfrak{q}$ of $B$ lying over $\\mathfrak{p}$, that is, whose preimage in $A$ is $\\mathfrak{p}$.","title":"Height-one primes above a prime"},{"id":"RingTheory/ramification/extensions-of-valuations-above-a-prime/mem-of-lies-over","kind":"node","lean":"ValuationBijection.mem_of_liesOver","parent":3708,"result":true,"route":"../roadmap/RingTheory/ramification/extensions-of-valuations-above-a-prime/index.html#mem-of-lies-over","scope":"RingTheory/ramification/extensions-of-valuations-above-a-prime","status":"fully_proved","summary":"Let $A$ and $B$ be Dedekind domains with $B$ an $A$-algebra, let $\\mathfrak{p}$ be a height-one prime of $A$, and let $\\mathfrak{q}$ be a height-one prime of $B$ lying over $\\mathfrak{p}$. Then for every $a \\in A$, $a \\in \\mathfrak{p}$ if and only if the image of $a$ in $B$ lies in $\\mathfrak{q}$.","title":"Membership in a prime is detected by a prime lying over it"},{"id":"RingTheory/ramification/extensions-of-valuations-above-a-prime/valuation-extends-of-lies-over","kind":"node","lean":"ValuationBijection.valuation_extends_of_liesOver","parent":3708,"result":true,"route":"../roadmap/RingTheory/ramification/extensions-of-valuations-above-a-prime/index.html#valuation-extends-of-lies-over","scope":"RingTheory/ramification/extensions-of-valuations-above-a-prime","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $L$ be a finite separable extension of $K$, and let $B$ be a Dedekind domain that is the integral closure of $A$ in $L$ and has fraction field $L$. If $\\mathfrak{q}$ is a height-one prime of $B$ lying over a height-one prime $\\mathfrak{p}$ of $A$, then the $\\mathfrak{q}$-adic valuation of $L$ extends the $\\mathfrak{p}$-adic valuation of $K$: its restriction\u2026","title":"The adic valuation of a prime lying over extends the adic valuation below"},{"id":"RingTheory/ramification/extensions-of-valuations-above-a-prime/dvr-overring-equiv","kind":"node","lean":"ValuationBijection.dvr_overring_equiv","parent":3708,"result":true,"route":"../roadmap/RingTheory/ramification/extensions-of-valuations-above-a-prime/index.html#dvr-overring-equiv","scope":"RingTheory/ramification/extensions-of-valuations-above-a-prime","status":"fully_proved","summary":"Let $B$ be a Dedekind domain with fraction field $L$, let $\\mathfrak{q}$ be a height-one prime of $B$, and let $w$ be a valuation on $L$ with values in the multiplicative group $\\mathbb{Z}$ with a zero adjoined. Suppose that $w(b) \\le 1$ for every $b \\in B$, and that for every $b \\in B$ we have $b \\in \\mathfrak{q}$ if and only if $w(b) < 1$. Then the $\\mathfrak{q}$-adic valuation of $L$ is equivalent to $w$.","title":"A valuation centered at a height-one prime is equivalent to its adic valuation"},{"id":"RingTheory/ramification/extensions-of-valuations-above-a-prime/valuation-surjective-of-extends","kind":"node","lean":"ValuationBijection.valuation_surjective_of_extends","parent":3708,"result":true,"route":"../roadmap/RingTheory/ramification/extensions-of-valuations-above-a-prime/index.html#valuation-surjective-of-extends","scope":"RingTheory/ramification/extensions-of-valuations-above-a-prime","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $L$ be a finite separable extension of $K$, and let $B$ be a Dedekind domain that is the integral closure of $A$ in $L$ and has fraction field $L$. Let $\\mathfrak{p}$ be a height-one prime of $A$, and let $w$ be a valuation on $L$ with values in the multiplicative group $\\mathbb{Z}$ with a zero adjoined whose restriction to $K$ is equivalent to the\u2026","title":"Each extension of a prime-adic valuation comes from some prime above"},{"id":"RingTheory/ramification/extensions-of-valuations-above-a-prime/extending-valuation-bijection-primes-abo","kind":"node","lean":"ValuationBijection.extending_valuation_bijection_primes_above","parent":3708,"result":true,"route":"../roadmap/RingTheory/ramification/extensions-of-valuations-above-a-prime/index.html#extending-valuation-bijection-primes-abo","scope":"RingTheory/ramification/extensions-of-valuations-above-a-prime","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $L$ be a finite separable extension of $K$, and let $B$ be a Dedekind domain that is the integral closure of $A$ in $L$ and has fraction field $L$. Let $\\mathfrak{p}$ be a height-one prime of $A$, and consider valuations on $L$ with values in the multiplicative group $\\mathbb{Z}$ with a zero adjoined. Then (1) if $\\mathfrak{q}_1$ and $\\mathfrak{q}_2$ are\u2026","title":"Each extension of a prime-adic valuation comes from exactly one prime above"},{"data_url":"scopes/RingTheory/ramification/frobenius-elements.json","id":"RingTheory/ramification/frobenius-elements","kind":"scope","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/frobenius-elements/index.html","scope":"RingTheory/ramification/frobenius-elements","status":"fully_proved","title":"Frobenius elements"},{"id":"RingTheory/ramification/frobenius-elements/arith-frob-at-eq-of-under-eq","kind":"node","lean":"arithFrobAt_eq_of_under_eq","parent":3716,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/index.html#arith-frob-at-eq-of-under-eq","scope":"RingTheory/ramification/frobenius-elements","status":"fully_proved","summary":"Let $S$ be a commutative algebra over a commutative ring $R$, and let a finite commutative group $G$ act on $S$ by $R$-algebra automorphisms such that every element of $S$ fixed by $G$ is the image of an element of $R$. For a prime $Q$ of $S$ with $S/Q$ finite, let $\\mathrm{Frob}_Q \\in G$ be the chosen arithmetic Frobenius element at $Q$: an element $\\sigma \\in G$ with $\\sigma(x) \\equiv x^q \\pmod Q$ for all\u2026","title":"In an abelian group the Frobenius element depends only on the prime below"},{"id":"RingTheory/ramification/frobenius-elements/restrict-normal-hom-eq-of-is-arith-frob","kind":"node","lean":"restrictNormalHom_eq_of_isArithFrobAt","parent":3716,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/index.html#restrict-normal-hom-eq-of-is-arith-frob","scope":"RingTheory/ramification/frobenius-elements","status":"fully_proved","summary":"Let $K \\subseteq L \\subseteq M$ be a tower of number fields with $L/K$ and $M/K$ Galois, and let $\\mathcal{O}_K$, $\\mathcal{O}_L$ and $\\mathcal{O}_M$ be their rings of integers. Let $Q_L$ be a prime of $\\mathcal{O}_L$ at which $\\mathcal{O}_L$ is unramified over $\\mathcal{O}_K$, let $Q_M$ be a prime of $\\mathcal{O}_M$ lying over $Q_L$ with $\\mathcal{O}_M/Q_M$ finite, and let $q$ be the number of elements of\u2026","title":"Restricting a Frobenius element to an intermediate field"},{"data_url":"scopes/RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a.json","id":"RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a","kind":"scope","navigation":"module","parent":3716,"result":false,"route":"../roadmap/RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a/index.html","scope":"RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a","status":"fully_proved","title":"Conjugacy class of the arithmetic Frobenius at an unramified prime"},{"id":"RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a/arith-frob-conj-class","kind":"node","lean":"arithFrobConjClass","parent":3719,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a/index.html#arith-frob-conj-class","scope":"RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a","status":"fully_proved","summary":"Let $S$ be a commutative algebra over a commutative ring $R$, and let a finite group $G$ act on $S$ by $R$-algebra automorphisms such that every element of $S$ fixed by $G$ is the image of an element of $R$. Let $Q$ be a prime of $S$ with $S/Q$ finite at which $S$ is unramified over $R$ (the localization $S_Q$ is formally unramified over $R$). The Frobenius conjugacy class at $Q$ is the conjugacy class in $G$ of\u2026","title":"Frobenius conjugacy class"},{"id":"RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a/arith-frob-conj-class-mem-carrier","kind":"node","lean":"arithFrobConjClass_mem_carrier","parent":3719,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a/index.html#arith-frob-conj-class-mem-carrier","scope":"RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a","status":"fully_proved","summary":"Let a finite group $G$ act on a commutative algebra $S$ over a commutative ring $R$ by $R$-algebra automorphisms such that every element of $S$ fixed by $G$ is the image of an element of $R$. For every prime $Q$ of $S$ with $S/Q$ finite at which $S$ is unramified over $R$, the chosen arithmetic Frobenius element $\\mathrm{Frob}_Q$ at $Q$ belongs to the Frobenius conjugacy class at $Q$.","title":"The Frobenius element lies in the Frobenius conjugacy class"},{"id":"RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a/arith-frob-conj-class-eq-of-under-eq","kind":"node","lean":"arithFrobConjClass_eq_of_under_eq","parent":3719,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a/index.html#arith-frob-conj-class-eq-of-under-eq","scope":"RingTheory/ramification/frobenius-elements/conjugacy-class-of-the-arithmetic-frobenius-at-a","status":"fully_proved","summary":"Let a finite group $G$ act on a commutative algebra $S$ over a commutative ring $R$ by $R$-algebra automorphisms such that every element of $S$ fixed by $G$ is the image of an element of $R$. Let $Q$ and $Q'$ be primes of $S$ with $S/Q$ and $S/Q'$ finite at which $S$ is unramified over $R$. If $Q$ and $Q'$ have the same preimage in $R$, then the Frobenius conjugacy classes at $Q$ and $Q'$ are equal.","title":"The Frobenius conjugacy class depends only on the prime below"},{"data_url":"scopes/RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting.json","id":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting","kind":"scope","navigation":"module","parent":3716,"result":false,"route":"../roadmap/RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/index.html","scope":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting","status":"fully_proved","title":"Frobenius elements and complete splitting"},{"id":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/stabilizer-eq-bot-iff-ncard-primes-over","kind":"node","lean":"Ideal.stabilizer_eq_bot_iff_ncard_primesOver_eq_card","parent":3723,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/index.html#stabilizer-eq-bot-iff-ncard-primes-over","scope":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting","status":"fully_proved","summary":"Let $B$ be a commutative algebra over a commutative ring $A$, and let a finite group $G$ act on $B$ by $A$-algebra automorphisms such that every element of $B$ fixed by $G$ is the image of an element of $A$. Let $p$ be an ideal of $A$ and $P$ a prime of $B$ lying over $p$. Then the stabilizer of $P$ in $G$ (the decomposition group of $P$) is trivial if and only if the number of primes of $B$ lying over $p$ equals\u2026","title":"Trivial decomposition group and the number of primes above"},{"id":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/arith-frob-at-eq-one-iff-ncard-primes-ov","kind":"node","lean":"arithFrobAt_eq_one_iff_ncard_primesOver_eq_card_of_generates","parent":3723,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/index.html#arith-frob-at-eq-one-iff-ncard-primes-ov","scope":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting","status":"fully_proved","summary":"Let $B$ be a commutative algebra over a commutative ring $A$, and let a finite group $G$ act on $B$ by $A$-algebra automorphisms such that every element of $B$ fixed by $G$ is the image of an element of $A$. Let $p$ be an ideal of $A$ and $P$ a prime of $B$ lying over $p$ with $B/P$ finite, and let $\\mathrm{Frob}_P \\in G$ be the chosen arithmetic Frobenius element at $P$, an element $\\sigma \\in G$ with\u2026","title":"Trivial Frobenius and the number of primes above, assuming the Frobenius generates"},{"id":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/arith-frob-at-generates-stabilizer-of-is","kind":"node","lean":"arithFrobAt_generates_stabilizer_of_isUnramifiedAt","parent":3723,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/index.html#arith-frob-at-generates-stabilizer-of-is","scope":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting","status":"fully_proved","summary":"Let $A$ and $B$ be integral domains with $B$ a finite flat $A$-algebra, and let $G$ be a finite group acting faithfully on $B$ by $A$-algebra automorphisms such that every element of $B$ fixed by $G$ is the image of an element of $A$. Let $p$ be a prime of $A$ and $P$ a prime of $B$ lying over $p$, with $B/P$ finite and $B$ unramified over $A$ at $P$ (the localization $B_P$ is formally unramified over $A$). Let\u2026","title":"The Frobenius element generates the decomposition group at an unramified prime"},{"id":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/arith-frob-at-eq-one-iff-ncard-primes-ov-2","kind":"node","lean":"arithFrobAt_eq_one_iff_ncard_primesOver_eq_card","parent":3723,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting/index.html#arith-frob-at-eq-one-iff-ncard-primes-ov-2","scope":"RingTheory/ramification/frobenius-elements/frobenius-elements-and-complete-splitting","status":"fully_proved","summary":"Let $A$ and $B$ be integral domains with $B$ a finite flat $A$-algebra, and let $G$ be a finite group acting faithfully on $B$ by $A$-algebra automorphisms such that every element of $B$ fixed by $G$ is the image of an element of $A$. Let $p$ be a prime of $A$ and $P$ a prime of $B$ lying over $p$, with $B/P$ finite and $B$ unramified over $A$ at $P$ (the localization $B_P$ is formally unramified over $A$). Let\u2026","title":"The Frobenius element is trivial iff the prime splits completely"},{"data_url":"scopes/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions.json","id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","kind":"scope","navigation":"module","parent":3716,"result":false,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","title":"Arithmetic Frobenius in towers and restrictions"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/mem-inertia-of-forall-smul-sub-pow-mem","kind":"node","lean":"Ideal.mem_inertia_of_forall_smul_sub_pow_mem","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#mem-inertia-of-forall-smul-sub-pow-mem","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"Two elements satisfying the same powered congruence differ by an element of the inertia subgroup of `Q`.","title":"Elements with the same powered congruence differ by inertia"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/pow-apply-sub-pow-mem","kind":"node","lean":"AlgHom.IsArithFrobAt.pow_apply_sub_pow_mem","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#pow-apply-sub-pow-mem","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"Powers of an arithmetic Frobenius map satisfy the powered congruence: if `\u03c6 x - x ^ q \u2208 Q` for all `x`, then `(\u03c6 ^ f) x - x ^ q ^ f \u2208 Q`.","title":"Powers of a Frobenius map satisfy the powered congruence"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/pow-smul-sub-pow-mem","kind":"node","lean":"IsArithFrobAt.pow_smul_sub_pow_mem","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#pow-smul-sub-pow-mem","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"Powers of an arithmetic Frobenius element satisfy the powered congruence: if `\u03c3 \u2022 x - x ^ q \u2208 Q` for all `x`, with `q = Nat.card (R \u29f8 Q.under R)`, then `(\u03c3 ^ f) \u2022 x - x ^ q ^ f \u2208 Q`.","title":"Powers of a Frobenius element satisfy the powered congruence"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/comap","kind":"node","lean":"IsArithFrobAt.comap","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#comap","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"A Frobenius congruence restricts along a compatible pair: if `\u03b9 : C \u2192\u2090[R] B` and `\u03c1 : G \u2192* H` satisfy `\u03b9 (\u03c1 g \u2022 c) = g \u2022 \u03b9 c`, and `\u03c3` is a Frobenius at `Q`, then `\u03c1 \u03c3` is a Frobenius at `Q.comap \u03b9`.","title":"Restriction of a Frobenius element along a compatible pair"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/pow-eq-of-forall-smul-sub-pow-mem-of-ine","kind":"node","lean":"IsArithFrobAt.pow_eq_of_forall_smul_sub_pow_mem_of_inertia_eq_bot","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#pow-eq-of-forall-smul-sub-pow-mem-of-ine","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"If `\u03c3` is a Frobenius at `Q` and `\u03c4` satisfies the powered congruence with exponent `q ^ f`, then `\u03c3 ^ f = \u03c4` when the inertia subgroup of `Q` is trivial. This is a uniqueness helper for an already supplied powered congruence. The actual tower theorem below derives that congruence from the Frobenius premise over the intermediate ring and its residue cardinality.","title":"Uniqueness of Frobenius powers with trivial inertia"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/eq-comap-of-forall-smul-sub-pow-mem-of-i","kind":"node","lean":"IsArithFrobAt.eq_comap_of_forall_smul_sub_pow_mem_of_inertia_eq_bot","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#eq-comap-of-forall-smul-sub-pow-mem-of-i","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"If `\u03c3` is a Frobenius at `Q` and `\u03c4` satisfies the Frobenius congruence at `Q.comap \u03b9`, then `\u03c1 \u03c3 = \u03c4` when the inertia subgroup of `Q.comap \u03b9` is trivial. This is the equality form of the restriction statement of Proposition 7.22 under its unramified hypothesis.","title":"Restriction of Frobenius equals Frobenius with trivial inertia"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/nat-card-quotient-under-eq-pow-inertia-d","kind":"node","lean":"Ideal.natCard_quotient_under_eq_pow_inertiaDeg","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#nat-card-quotient-under-eq-pow-inertia-d","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"In a finite ring tower `R \u2192 T \u2192 B`, the intermediate prime residue quotient has cardinality equal to the base residue cardinality raised to its canonical inertia degree. The cardinality theorem is Mathlib's `cardQuot_pow_inertiaDeg`.","title":"Intermediate residue cardinality as a power of the base residue cardinality"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/tower-eq-of-inertia-eq-bot","kind":"node","lean":"IsArithFrobAt.tower_eq_of_inertia_eq_bot","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#tower-eq-of-inertia-eq-bot","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"In a finite ring tower `R \u2192 T \u2192 B`, the Frobenius over `T`, embedded into an action over `R`, equals the Frobenius over `R` raised to the intermediate residue degree when the inertia subgroup is trivial. Only `H` must fix `T`.","title":"Frobenius in a tower is a power of the base Frobenius"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/tower-eq-of-inertia-eq-bot-alg-equiv","kind":"node","lean":"IsArithFrobAt.tower_eq_of_inertia_eq_bot_algEquiv","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#tower-eq-of-inertia-eq-bot-alg-equiv","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"For algebra automorphisms in a finite ring tower, the inclusion of the intermediate Frobenius is the canonical `AlgEquiv.restrictScalars`.","title":"Frobenius tower equality for algebra automorphisms"},{"id":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/tower-eq-of-inertia-eq-bot-fixing-subgro","kind":"node","lean":"IsArithFrobAt.tower_eq_of_inertia_eq_bot_fixingSubgroup","parent":3728,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions/index.html#tower-eq-of-inertia-eq-bot-fixing-subgro","scope":"RingTheory/ramification/frobenius-elements/arithmetic-frobenius-in-towers-and-restrictions","status":"fully_proved","summary":"The tower equality for the source's intermediate field `E \u2282 L`. Here `K` is the fraction field of the base ring `R`, `T` is the integral closure of `R` in `E`, and `B` is its integral closure in `L`. In the number-field application these are the three rings of integers. The intermediate action is `E.fixingSubgroup`; the whole `Gal(L/K)` need not fix `T`.","title":"Frobenius tower equality for an intermediate field"},{"data_url":"scopes/RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia.json","id":"RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia","kind":"scope","navigation":"module","parent":3716,"result":false,"route":"../roadmap/RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia/index.html","scope":"RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia","status":"fully_proved","title":"Frobenius elements under trivial inertia"},{"id":"RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia/eq-of-inertia-eq-bot","kind":"node","lean":"IsArithFrobAt.eq_of_inertia_eq_bot","parent":3739,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia/index.html#eq-of-inertia-eq-bot","scope":"RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, let a group $G$ act on $S$ by $R$-algebra automorphisms, and let $Q$ be an ideal of $S$. Write $P$ for the preimage of $Q$ in $R$ and $q$ for the number of elements of $R/P$ (with $q = 0$ when $R/P$ is infinite). Call $\\sigma \\in G$ an arithmetic Frobenius element at $Q$ if $\\sigma x - x^q \\in Q$ for all $x \\in S$, and call the inertia subgroup of $Q$\u2026","title":"Frobenius elements coincide when inertia is trivial"},{"id":"RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia/exists-unique-of-inertia-eq-bot","kind":"node","lean":"IsArithFrobAt.existsUnique_of_inertia_eq_bot","parent":3739,"result":true,"route":"../roadmap/RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia/index.html#exists-unique-of-inertia-eq-bot","scope":"RingTheory/ramification/frobenius-elements/frobenius-elements-under-trivial-inertia","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, and let a finite group $G$ act on $S$ by $R$-algebra automorphisms such that every element of $S$ fixed by $G$ is the image of an element of $R$. Let $Q$ be a prime ideal of $S$ with $S/Q$ finite, write $P$ for the preimage of $Q$ in $R$ and $q$ for the number of elements of $R/P$. Call $\\sigma \\in G$ an arithmetic Frobenius element at $Q$ if\u2026","title":"Existence and uniqueness of the Frobenius element under trivial inertia"},{"data_url":"scopes/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings.json","id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","title":"Monogenic extensions of discrete valuation rings"},{"id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/of-is-unit-aeval-derivative-of-adjoin-eq","kind":"node","lean":"Algebra.FormallyUnramified.of_isUnit_aeval_derivative_of_adjoin_eq_top","parent":3742,"result":true,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html#of-is-unit-aeval-derivative-of-adjoin-eq","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings, let $S$ be an $R$-algebra, and let $x \\in S$ generate $S$ as an $R$-algebra, that is $R[x] = S$. If some polynomial $P \\in R[X]$ satisfies $P(x) = 0$ and $P'(x)$ is a unit of $S$, then $S$ is formally unramified over $R$.","title":"Algebras generated by a root with invertible derivative are formally unramified"},{"id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/etale-iff-exists-adjoin-eq-top-and-separ","kind":"node","lean":"Algebra.etale_iff_exists_adjoin_eq_top_and_separable_map_minpoly","parent":3742,"result":true,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html#etale-iff-exists-adjoin-eq-top-and-separ","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","summary":"Let $R$ and $S$ be local rings and let $S$ be an $R$-algebra whose structure map $R \\to S$ is injective, which is finitely generated and flat as an $R$-module and finitely presented as an $R$-algebra. Let $k$ be the residue field of $R$. Then $S$ is \u00e9tale over $R$ if and only if there is an element $x \\in S$ with $R[x] = S$ such that the image in $k[X]$ of the minimal polynomial of $x$ over $R$ (a monic polynomial\u2026","title":"\u00c9tale finite local algebras are generated by one element with separable reduction"},{"id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/exists-generator-with-residue-minpoly-un","kind":"node","lean":"IsDiscreteValuationRing.exists_generator_with_residue_minpoly_uniformizer","parent":3742,"result":true,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html#exists-generator-with-residue-minpoly-un","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$, $\\mathfrak m_B$ and residue fields $\\kappa_A$, $\\kappa_B$, and let $B$ be an $A$-algebra that is finitely generated as an $A$-module and whose structure map is a local homomorphism. If $\\kappa_B / \\kappa_A$ is separable, then there are $\\alpha \\in B$ and a polynomial $F \\in A[X]$ such that, writing $\\bar\\alpha$ for the residue of\u2026","title":"Monogenic generator with a uniformizing residue polynomial"},{"id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/exists-adjoin-eq-top-of-separable-residu","kind":"node","lean":"IsDiscreteValuationRing.exists_adjoin_eq_top_of_separable_residueField","parent":3742,"result":true,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html#exists-adjoin-eq-top-of-separable-residu","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with residue fields $\\kappa_A$, $\\kappa_B$, and let $B$ be an $A$-algebra that is finitely generated as an $A$-module and whose structure map is a local homomorphism. If $\\kappa_B / \\kappa_A$ is separable, then there is $\\alpha \\in B$ with $A[\\alpha] = B$.","title":"Monogenicity of extensions with separable residue field extension"},{"id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/adjoin-eq-top-of-lift-of-map-maximal-ide","kind":"node","lean":"IsDiscreteValuationRing.adjoin_eq_top_of_lift_of_map_maximalIdeal","parent":3742,"result":true,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html#adjoin-eq-top-of-lift-of-map-maximal-ide","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$, $\\mathfrak m_B$ and residue fields $\\kappa_A$, $\\kappa_B$, and let $B$ be an $A$-algebra that is finitely generated as an $A$-module and whose structure map is a local homomorphism. Suppose $\\kappa_B / \\kappa_A$ is separable and the ideal $\\mathfrak m_A B$ generated by the image of $\\mathfrak m_A$ equals $\\mathfrak m_B$. If\u2026","title":"Lifts of residue generators generate unramified extensions"},{"id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/etale-of-is-separable-residue-field-of-f","kind":"node","lean":"IsDiscreteValuationRing.etale_of_isSeparable_residueField_of_finrank_eq","parent":3742,"result":true,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html#etale-of-is-separable-residue-field-of-f","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with residue fields $\\kappa_A$, $\\kappa_B$, and let $B$ be an $A$-algebra that is finitely generated as an $A$-module and whose structure map is a local homomorphism. If $\\kappa_B / \\kappa_A$ is separable and the rank of $B$ as an $A$-module equals the degree $[\\kappa_B : \\kappa_A]$, then $B$ is \u00e9tale over $A$.","title":"\u00c9tale criterion for extensions of discrete valuation rings"},{"id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/unramified-iff-exists-monogenic-separabl","kind":"node","lean":"IsDiscreteValuationRing.unramified_iff_exists_monogenic_separable_reduced_minpoly","parent":3742,"result":true,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html#unramified-iff-exists-monogenic-separabl","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings, with $A$ complete with respect to its maximal ideal, and let $B$ be an $A$-algebra via a local homomorphism $A \\to B$, such that $B$ is finitely generated and torsion-free as an $A$-module. Let $k_A$ and $k_B$ be the residue fields of $A$ and $B$. Then the following are equivalent: (a) $k_B$ is separable over $k_A$ and the rank of $B$ as an $A$-module equals\u2026","title":"Characterization of unramified extensions of a complete DVR"},{"id":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/unramified-of-primitive-root-of-unity-of","kind":"node","lean":"IsDiscreteValuationRing.unramified_of_primitive_root_of_unity_of_coprime","parent":3742,"result":true,"route":"../roadmap/RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings/index.html#unramified-of-primitive-root-of-unity-of","scope":"RingTheory/ramification/monogenic-extensions-of-discrete-valuation-rings","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings, with $A$ complete with respect to its maximal ideal, and let $B$ be an $A$-algebra via a local homomorphism $A \\to B$, such that $B$ is finitely generated and torsion-free as an $A$-module. Let $k_A$ and $k_B$ be the residue fields of $A$ and $B$. Let $n$ be a natural number coprime to the characteristic of $k_A$, and let $\\zeta \\in B$ be a primitive $n$-th root of unity\u2026","title":"Adjoining a root of unity of order coprime to the residue characteristic is unramified"},{"data_url":"scopes/RingTheory/ramification/ramification-index-and-inertia-degree.json","id":"RingTheory/ramification/ramification-index-and-inertia-degree","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","title":"Ramification index and inertia degree"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/ramification-idx-mem-icc","kind":"node","lean":"Ideal.ramificationIdx_mem_Icc","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#ramification-idx-mem-icc","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $R$ and $S$ be Dedekind domains, with $S$ an $R$-algebra that is finitely generated and torsion-free as an $R$-module, and let $n$ be the rank of $S$ as an $R$-module. If $\\mathfrak{p}$ is a maximal ideal of $R$ and $\\mathfrak{q}$ is a prime ideal of $S$ lying over $\\mathfrak{p}$, then the ramification index $e_{\\mathfrak{q}}$ of $\\mathfrak{q}$ over $\\mathfrak{p}$ satisfies $1 \\le e_{\\mathfrak{q}} \\le n$.","title":"Bounds on the ramification index"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/inertia-deg-mem-icc","kind":"node","lean":"Ideal.inertiaDeg_mem_Icc","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#inertia-deg-mem-icc","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $R$ and $S$ be Dedekind domains, with $S$ an $R$-algebra that is finitely generated and torsion-free as an $R$-module, and let $n$ be the rank of $S$ as an $R$-module. If $\\mathfrak{p}$ is a maximal ideal of $R$ and $\\mathfrak{q}$ is a prime ideal of $S$ lying over $\\mathfrak{p}$, then the inertia degree $f_{\\mathfrak{q}}$ of $\\mathfrak{q}$ over $\\mathfrak{p}$ satisfies $1 \\le f_{\\mathfrak{q}} \\le n$.","title":"Bounds on the inertia degree"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/ncard-primes-over-mem-icc","kind":"node","lean":"Ideal.ncard_primesOver_mem_Icc","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#ncard-primes-over-mem-icc","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $R$ and $S$ be Dedekind domains, with $S$ an $R$-algebra that is finitely generated and torsion-free as an $R$-module, and let $n$ be the rank of $S$ as an $R$-module. If $\\mathfrak{p}$ is a maximal ideal of $R$, then the number $g_{\\mathfrak{p}}$ of prime ideals of $S$ lying over $\\mathfrak{p}$ satisfies $1 \\le g_{\\mathfrak{p}} \\le n$.","title":"Bounds on the number of primes above a prime"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/ramification-inertia-card-mem-icc","kind":"node","lean":"Ideal.ramification_inertia_card_mem_Icc","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#ramification-inertia-card-mem-icc","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $R$ and $S$ be Dedekind domains, with $S$ an $R$-algebra that is finitely generated and torsion-free as an $R$-module, and let $n$ be the rank of $S$ as an $R$-module. If $\\mathfrak{p}$ is a maximal ideal of $R$ and $\\mathfrak{q}$ is a prime ideal of $S$ lying over $\\mathfrak{p}$, then the ramification index $e_{\\mathfrak{q}}$ and the inertia degree $f_{\\mathfrak{q}}$ of $\\mathfrak{q}$ over $\\mathfrak{p}$ and\u2026","title":"Bounds on ramification, inertia and splitting"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/ramification-inertia-card-mem-icc-of-is","kind":"node","lean":"Ideal.ramification_inertia_card_mem_Icc_of_isFractionRing","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#ramification-inertia-card-mem-icc-of-is","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $R$ and $S$ be Dedekind domains, with $S$ an $R$-algebra that is finitely generated and torsion-free as an $R$-module. Let $K$ be the fraction field of $R$ and $L$ the fraction field of $S$, with $L$ a $K$-algebra compatible with the maps $R \\to K \\to L$ and $R \\to S \\to L$, and let $n = [L : K]$. If $\\mathfrak{p}$ is a maximal ideal of $R$ and $\\mathfrak{q}$ is a prime ideal of $S$ lying over $\\mathfrak{p}$,\u2026","title":"Bounds on ramification, inertia and splitting, fraction-field form"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/map-maximal-ideal-eq-pow-ramification-id","kind":"node","lean":"IsDiscreteValuationRing.map_maximalIdeal_eq_pow_ramificationIdx","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#map-maximal-ideal-eq-pow-ramification-id","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$, $\\mathfrak m_B$, and let $B$ be an $A$-algebra that is torsion-free as an $A$-module and whose structure map is a local homomorphism. Let $e$ be the ramification index of $\\mathfrak m_B$ over $A$. Then the ideal $\\mathfrak m_A B$ of $B$ generated by the image of $\\mathfrak m_A$ equals $\\mathfrak m_B^e$.","title":"The extended maximal ideal is a power of the maximal ideal"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/algebra-map-mem-maximal-ideal-pow-ramifi","kind":"node","lean":"IsDiscreteValuationRing.algebraMap_mem_maximalIdeal_pow_ramificationIdx","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#algebra-map-mem-maximal-ideal-pow-ramifi","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$, $\\mathfrak m_B$, and let $B$ be an $A$-algebra that is torsion-free as an $A$-module and whose structure map is a local homomorphism. Let $e$ be the ramification index of $\\mathfrak m_B$ over $A$. Then for every $x \\in \\mathfrak m_A$, the image of $x$ in $B$ lies in $\\mathfrak m_B^e$.","title":"Images of the base maximal ideal lie in the ramification power"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/rel-norm-eq-pow-inertia-deg-of-is-separa","kind":"node","lean":"Ideal.relNorm_eq_pow_inertiaDeg_of_isSeparable","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#rel-norm-eq-pow-inertia-deg-of-is-separa","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Theorem 6.10 of ATLAS `NumberTheoryI`, in the separable (rather than Galois or perfect-field) form: the relative norm of a maximal ideal lying over a maximal ideal is the base prime raised to the inertia degree.","title":"Relative norm is the base prime to the inertia degree, separable case"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/emultiplicity-maximal-ideal-span-eq-add","kind":"node","lean":"IsDiscreteValuationRing.emultiplicity_maximalIdeal_span_eq_addVal","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#emultiplicity-maximal-ideal-span-eq-add","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and normalized additive valuation $v \\colon R \\to \\mathbb{N} \\cup \\{\\infty\\}$. For every $x \\in R$, the multiplicity of $\\mathfrak{m}$ in the principal ideal $(x)$, that is, the supremum in $\\mathbb{N} \\cup \\{\\infty\\}$ of the $n \\in \\mathbb{N}$ such that $\\mathfrak{m}^n$ divides $(x)$ as ideals, equals $v(x)$.","title":"Multiplicity of the maximal ideal equals the additive valuation"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/add-val-algebra-map-eq-ramification-idx","kind":"node","lean":"IsDiscreteValuationRing.addVal_algebraMap_eq_ramificationIdx_mul","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#add-val-algebra-map-eq-ramification-idx","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings, and let $B$ be an $A$-algebra that is torsion-free as an $A$-module and whose structure map $\\varphi \\colon A \\to B$ is a local homomorphism. Let $v_A$ and $v_B$ be the normalized additive valuations of $A$ and $B$, with values in $\\mathbb{N} \\cup \\{\\infty\\}$, and let $e$ be the ramification index of the maximal ideal of $B$ over $A$. Then\u2026","title":"Additive valuations scale by the ramification index"},{"id":"RingTheory/ramification/ramification-index-and-inertia-degree/is-unit-nat-cast-iff-ring-char-not-dvd","kind":"node","lean":"IsLocalRing.isUnit_natCast_iff_ringChar_not_dvd","parent":3751,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-index-and-inertia-degree/index.html#is-unit-nat-cast-iff-ring-char-not-dvd","scope":"RingTheory/ramification/ramification-index-and-inertia-degree","status":"fully_proved","summary":"Let $R$ be a commutative local ring and let $p$ be the characteristic of its residue field. For every natural number $n$, the image of $n$ in $R$ is a unit if and only if $p$ does not divide $n$.","title":"Natural numbers that are units in a local ring"},{"data_url":"scopes/RingTheory/ramification/ramification-of-cyclotomic-extensions.json","id":"RingTheory/ramification/ramification-of-cyclotomic-extensions","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/ramification-of-cyclotomic-extensions/index.html","scope":"RingTheory/ramification/ramification-of-cyclotomic-extensions","status":"fully_proved","title":"Ramification of cyclotomic extensions"},{"id":"RingTheory/ramification/ramification-of-cyclotomic-extensions/pth-root-unity-eq-one-of-no-prim-root","kind":"node","lean":"pth_root_unity_eq_one_of_no_prim_root","parent":3763,"reading_title":{"format":"inline-parts-v1","parts":[["text","Without primitive "],["math","\\(p\\)"],["text","-th roots of unity, every "],["math","\\(p\\)"],["text","-th root of unity is trivial"]],"source":"Without primitive $p$-th roots of unity, every $p$-th root of unity is trivial"},"result":true,"route":"../roadmap/RingTheory/ramification/ramification-of-cyclotomic-extensions/index.html#pth-root-unity-eq-one-of-no-prim-root","scope":"RingTheory/ramification/ramification-of-cyclotomic-extensions","status":"fully_proved","summary":"Let $p$ be a prime and let $L$ be a field containing no primitive $p$-th root of unity. Then every $\\eta \\in L$ with $\\eta^p = 1$ equals $1$.","title":"Without primitive $p$-th roots of unity, every $p$-th root of unity is trivial"},{"id":"RingTheory/ramification/ramification-of-cyclotomic-extensions/no-prim-root-in-unramified-ext-of-prime","kind":"node","lean":"no_prim_root_in_unramified_ext_of_prime_spans_maximal_ideal","parent":3763,"result":true,"route":"../roadmap/RingTheory/ramification/ramification-of-cyclotomic-extensions/index.html#no-prim-root-in-unramified-ext-of-prime","scope":"RingTheory/ramification/ramification-of-cyclotomic-extensions","status":"fully_proved","summary":"For a complete DVR base whose maximal ideal is generated by `p`, a primitive `p`-th root cannot appear in an index-one DVR extension: if the residue characteristic is `p`, `(p)` is the maximal ideal, the ramification index is `1`, and the base field has no primitive `p`-th root, then neither does the extension.","title":"No primitive p-th root in an unramified DVR extension"},{"id":"RingTheory/ramification/ramification-of-cyclotomic-extensions/no-prim-pth-root-in-unramified-ext-of-pr","kind":"node","lean":"no_prim_pth_root_in_unramified_ext_of_prime_spans_maximal_ideal","parent":3763,"reading_title":{"format":"inline-parts-v1","parts":[["text","Over a complete base with maximal ideal "],["math","\\((p)\\)"],["text"," and no primitive "],["math","\\(p\\)"],["text","-th root of unity, an index-one extension has no nontrivial "],["math","\\(p\\)"],["text","-th roots of unity"]],"source":"Over a complete base with maximal ideal $(p)$ and no primitive $p$-th root of unity, an index-one extension has no nontrivial $p$-th roots of unity"},"result":true,"route":"../roadmap/RingTheory/ramification/ramification-of-cyclotomic-extensions/index.html#no-prim-pth-root-in-unramified-ext-of-pr","scope":"RingTheory/ramification/ramification-of-cyclotomic-extensions","status":"fully_proved","summary":"Over a complete base with maximal ideal $(p)$ and no primitive $p$-th root of unity, an index-one extension has no nontrivial $p$-th roots of unity. Let $A$ be a discrete valuation ring, complete with respect to its maximal ideal $\\mathfrak m_A$, with fraction field $K$ and finite residue field of characteristic a prime $p$, and suppose $\\mathfrak m_A = (p)$. Let $L$ be a finite extension of $K$, and let $B$ be a\u2026","title":"Over a complete base with maximal ideal $(p)$ and no primitive $p$-th root of unity, an index-one extension has no nontrivial $p$-th roots of unity"},{"id":"RingTheory/ramification/ramification-of-cyclotomic-extensions/unramified-pth-root-in-base-of-prime-spa","kind":"node","lean":"unramified_pth_root_in_base_of_prime_spans_maximal_ideal","parent":3763,"reading_title":{"format":"inline-parts-v1","parts":[["text","Over a complete base with maximal ideal "],["math","\\((p)\\)"],["text",", the "],["math","\\(p\\)"],["text","-th roots of unity in an index-one extension come from the base"]],"source":"Over a complete base with maximal ideal $(p)$, the $p$-th roots of unity in an index-one extension come from the base"},"result":true,"route":"../roadmap/RingTheory/ramification/ramification-of-cyclotomic-extensions/index.html#unramified-pth-root-in-base-of-prime-spa","scope":"RingTheory/ramification/ramification-of-cyclotomic-extensions","status":"fully_proved","summary":"Over a complete base with maximal ideal $(p)$, the $p$-th roots of unity in an index-one extension come from the base. Let $A$ be a discrete valuation ring, complete with respect to its maximal ideal $\\mathfrak m_A$, with fraction field $K$ and finite residue field of characteristic a prime $p$, and suppose $\\mathfrak m_A = (p)$. Let $L$ be a finite extension of $K$, and let $B$ be a discrete valuation ring with\u2026","title":"Over a complete base with maximal ideal $(p)$, the $p$-th roots of unity in an index-one extension come from the base"},{"id":"RingTheory/ramification/ramification-of-cyclotomic-extensions/dvr-pth-root-ramified-of-prime-spans-max","kind":"node","lean":"dvr_pth_root_ramified_of_prime_spans_maximal_ideal","parent":3763,"reading_title":{"format":"inline-parts-v1","parts":[["text","Over a complete base with maximal ideal "],["math","\\((p)\\)"],["text"," and no nontrivial "],["math","\\(p\\)"],["text","-th roots of unity, cyclotomic extensions of order divisible by "],["math","\\(p\\)"],["text"," ramify"]],"source":"Over a complete base with maximal ideal $(p)$ and no nontrivial $p$-th roots of unity, cyclotomic extensions of order divisible by $p$ ramify"},"result":true,"route":"../roadmap/RingTheory/ramification/ramification-of-cyclotomic-extensions/index.html#dvr-pth-root-ramified-of-prime-spans-max","scope":"RingTheory/ramification/ramification-of-cyclotomic-extensions","status":"fully_proved","summary":"Over a complete base with maximal ideal $(p)$ and no nontrivial $p$-th roots of unity, cyclotomic extensions of order divisible by $p$ ramify. Let $A$ be a discrete valuation ring, complete with respect to its maximal ideal $\\mathfrak m_A$, with fraction field $K$ and finite residue field of characteristic a prime $p$, and suppose $\\mathfrak m_A = (p)$. Suppose $1$ is the only $p$-th root of unity in $K$. Let\u2026","title":"Over a complete base with maximal ideal $(p)$ and no nontrivial $p$-th roots of unity, cyclotomic extensions of order divisible by $p$ ramify"},{"data_url":"scopes/RingTheory/ramification/tame-and-wild-ramification.json","id":"RingTheory/ramification/tame-and-wild-ramification","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","title":"Tame and wild ramification"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-residue-separable-at","kind":"node","lean":"Algebra.IsResidueSeparableAt","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-residue-separable-at","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, let $q$ be a prime ideal of $S$, and let $p$ be its preimage in $R$. Then $q$ is residue separable over $R$ when the residue field $\\kappa(q)$ is separable over the residue field $\\kappa(p)$, for the algebra structure induced by the canonical local homomorphism $R_p \\to S_q$ of localizations.","title":"Residue separability at a prime"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-tamely-ramified-at","kind":"node","lean":"Algebra.IsTamelyRamifiedAt","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-tamely-ramified-at","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, and let $q$ be a prime ideal of $S$. Then $q$ is *tamely ramified* over $R$ when $q$ is residue separable over $R$ and the characteristic of the residue field $\\kappa(q)$ does not divide the ramification index of $q$ over $R$.","title":"Tamely ramified prime"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-wildly-ramified-at","kind":"node","lean":"Algebra.IsWildlyRamifiedAt","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-wildly-ramified-at","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ be a commutative ring, $S$ a commutative $R$-algebra and $q$ a prime ideal of $S$. Then $q$ is *wildly ramified* over $R$ if it is not tamely ramified over $R$.","title":"Wildly ramified prime"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-residue-separable-at-iff","kind":"node","lean":"Algebra.isResidueSeparableAt_iff","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-residue-separable-at-iff","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, let $q$ be a prime ideal of $S$, and let $p$ be its preimage in $R$. Then $q$ is residue separable over $R$ if and only if the residue field $\\kappa(q)$ is separable over the residue field $\\kappa(p)$, for the algebra structure induced by the canonical local homomorphism $R_p \\to S_q$ of localizations.","title":"Residue separability, unfolded"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-tamely-ramified-at-iff","kind":"node","lean":"Algebra.isTamelyRamifiedAt_iff","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-tamely-ramified-at-iff","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, and let $q$ be a prime ideal of $S$. Then $q$ is tamely ramified over $R$ if and only if $q$ is residue separable over $R$ and the characteristic of the residue field $\\kappa(q)$ does not divide the ramification index of $q$ over $R$.","title":"Characterization of tame ramification"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-wildly-ramified-at-iff","kind":"node","lean":"Algebra.isWildlyRamifiedAt_iff","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-wildly-ramified-at-iff","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ be a commutative ring, $S$ a commutative $R$-algebra and $q$ a prime ideal of $S$. Then $q$ is wildly ramified over $R$ if and only if $q$ is not tamely ramified over $R$.","title":"Wild ramification is the failure of tame ramification"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-tamely-ramified-at-mk","kind":"node","lean":"Algebra.isTamelyRamifiedAt_mk","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-tamely-ramified-at-mk","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, and let $q$ be a prime ideal of $S$. If $q$ is residue separable over $R$ and the characteristic of the residue field $\\kappa(q)$ does not divide the ramification index of $q$ over $R$, then $q$ is tamely ramified over $R$.","title":"Criterion for tame ramification"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-separable","kind":"node","lean":"Algebra.IsTamelyRamifiedAt.isSeparable","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-separable","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, and let $q$ be a prime ideal of $S$. If $q$ is tamely ramified over $R$, then $q$ is residue separable over $R$.","title":"Tame primes are residue separable"},{"id":"RingTheory/ramification/tame-and-wild-ramification/not-dvd","kind":"node","lean":"Algebra.IsTamelyRamifiedAt.not_dvd","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#not-dvd","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ be a commutative ring, $S$ a commutative $R$-algebra and $q$ a prime ideal of $S$ with residue field $\\kappa(q)$. If $q$ is tamely ramified over $R$, then the characteristic of $\\kappa(q)$ does not divide the ramification index of $q$ over $R$.","title":"The residue characteristic of a tame prime does not divide its ramification index"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-wildly-ramified-at-iff-char-dvd","kind":"node","lean":"Algebra.isWildlyRamifiedAt_iff_char_dvd","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-wildly-ramified-at-iff-char-dvd","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra, and let $q$ be a prime ideal of $S$ that is residue separable over $R$. Then $q$ is wildly ramified over $R$ if and only if the characteristic of the residue field $\\kappa(q)$ divides the ramification index of $q$ over $R$.","title":"Wild ramification under residue separability"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-wildly-ramified-at-of-char-dvd","kind":"node","lean":"Algebra.isWildlyRamifiedAt_of_char_dvd","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-wildly-ramified-at-of-char-dvd","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ be a commutative ring, $S$ a commutative $R$-algebra and $q$ a prime ideal of $S$ with residue field $\\kappa(q)$. If the characteristic of $\\kappa(q)$ divides the ramification index of $q$ over $R$, then $q$ is wildly ramified over $R$.","title":"Divisibility of the ramification index forces wild ramification"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-wildly-ramified-at-of-not-is-residue","kind":"node","lean":"Algebra.isWildlyRamifiedAt_of_not_isResidueSeparableAt","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-wildly-ramified-at-of-not-is-residue","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ be a commutative ring, $S$ a commutative $R$-algebra, $q$ a prime ideal of $S$ and $p$ the preimage of $q$ in $R$. If the residue field extension $\\kappa(p) \\to \\kappa(q)$ is not separable, then $q$ is wildly ramified over $R$.","title":"An inseparable residue field extension forces wild ramification"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-residue-separable-at-of-is-unramified","kind":"node","lean":"Algebra.isResidueSeparableAt_of_isUnramifiedAt","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-residue-separable-at-of-is-unramified","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra that is essentially of finite type, and let $q$ be a prime ideal of $S$ at which $S$ is unramified over $R$, that is, the localization $S_q$ is formally unramified over $R$. Then $q$ is residue separable over $R$.","title":"Unramified primes are residue separable"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-tamely-ramified-at-of-is-unramified-a","kind":"node","lean":"Algebra.isTamelyRamifiedAt_of_isUnramifiedAt","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-tamely-ramified-at-of-is-unramified-a","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra that is essentially of finite type, and let $q$ be a prime ideal of $S$ at which $S$ is unramified over $R$, that is, the localization $S_q$ is formally unramified over $R$. Then $q$ is tamely ramified over $R$.","title":"Unramified primes are tamely ramified"},{"id":"RingTheory/ramification/tame-and-wild-ramification/is-tamely-ramified-at-iff-is-separable-o","kind":"node","lean":"Algebra.isTamelyRamifiedAt_iff_isSeparable_of_charZero","parent":3769,"result":true,"route":"../roadmap/RingTheory/ramification/tame-and-wild-ramification/index.html#is-tamely-ramified-at-iff-is-separable-o","scope":"RingTheory/ramification/tame-and-wild-ramification","status":"fully_proved","summary":"Let $R$ and $S$ be commutative rings with $S$ an $R$-algebra that is finite as an $R$-module, and let $q$ be a prime ideal of $S$ whose residue field $\\kappa(q)$ has characteristic $0$. Then $q$ is tamely ramified over $R$ if and only if $q$ is residue separable over $R$.","title":"Tameness in residue characteristic zero"},{"data_url":"scopes/RingTheory/ramification/tame-ramification-and-the-different.json","id":"RingTheory/ramification/tame-ramification-and-the-different","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","title":"Tame ramification and the different"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/maximal-ideal-graded-piece","kind":"node","lean":"IsDiscreteValuationRing.MaximalIdealGradedPiece","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#maximal-ideal-graded-piece","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$. For $i \\in \\mathbb{N}$, the $i$-th graded piece is the $R$-module $\\mathfrak{m}^i/\\mathfrak{m}^{i+1}$, realized as the quotient of the ideal $\\mathfrak{m}^i$, viewed as an $R$-module, by its submodule $\\mathfrak{m} \\cdot \\mathfrak{m}^i$.","title":"Graded pieces of the maximal-ideal filtration"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-linear-equiv-maximal-ideal","kind":"node","lean":"IsDiscreteValuationRing.residueFieldLinearEquivMaximalIdealGradedPiece","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-linear-equiv-maximal-ideal","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and residue field $\\kappa = R/\\mathfrak{m}$, and let $\\varpi$ be a generator of $\\mathfrak{m}$ chosen once and for all. For $i \\in \\mathbb{N}$, the map $\\kappa \\to \\mathfrak{m}^i/\\mathfrak{m}^{i+1}$ sending the class of $r \\in R$ to the class of $r \\varpi^i$ is a $\\kappa$-linear isomorphism.","title":"Graded pieces of a DVR as copies of the residue field"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/finrank-maximal-ideal-graded-piece","kind":"node","lean":"IsDiscreteValuationRing.finrank_maximalIdealGradedPiece","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#finrank-maximal-ideal-graded-piece","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and residue field $\\kappa = R/\\mathfrak{m}$. For every $i \\in \\mathbb{N}$, the graded piece $\\mathfrak{m}^i/\\mathfrak{m}^{i+1}$ is a vector space of dimension one over $\\kappa$.","title":"Graded pieces of a DVR are one-dimensional"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-mul-maximal-ideal-graded-p","kind":"node","lean":"IsDiscreteValuationRing.residueFieldMulMaximalIdealGradedPiece","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-mul-maximal-ideal-graded-p","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and residue field $\\kappa = R/\\mathfrak{m}$. For $a \\in \\kappa$ and $i \\in \\mathbb{N}$, this is the $\\kappa$-linear endomorphism $x \\mapsto a x$ of the graded piece $\\mathfrak{m}^i/\\mathfrak{m}^{i+1}$.","title":"Multiplication by a residue class on a graded piece"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-mul-maximal-ideal-graded-p-2","kind":"node","lean":"IsDiscreteValuationRing.residueFieldMulMaximalIdealGradedPiece_residue_mk","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-mul-maximal-ideal-graded-p-2","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and residue field $\\kappa = R/\\mathfrak{m}$. For $x \\in R$, $i \\in \\mathbb{N}$ and $y \\in \\mathfrak{m}^i$, multiplication by the residue class $\\bar{x} \\in \\kappa$ on $\\mathfrak{m}^i/\\mathfrak{m}^{i+1}$ sends the class of $y$ to the class of $x y$.","title":"Multiplication by a residue class on representatives"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-linear-equiv-maximal-ideal-2","kind":"node","lean":"IsDiscreteValuationRing.residueFieldLinearEquivMaximalIdealGradedPiece_conj_lmul","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-linear-equiv-maximal-ideal-2","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and residue field $\\kappa = R/\\mathfrak{m}$, let $i \\in \\mathbb{N}$, and let $e : \\kappa \\to \\mathfrak{m}^i/\\mathfrak{m}^{i+1}$ be the $\\kappa$-linear isomorphism of the residue field with the graded piece. Let $K$ be a field, with a $K$-algebra structure on $\\kappa$ and a $K$-module structure on $\\mathfrak{m}^i/\\mathfrak{m}^{i+1}$ such that\u2026","title":"The residue-field isomorphism intertwines multiplication"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/trace-residue-field-mul-maximal-ideal-gr","kind":"node","lean":"IsDiscreteValuationRing.trace_residueFieldMulMaximalIdealGradedPiece","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#trace-residue-field-mul-maximal-ideal-gr","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and residue field $\\kappa = R/\\mathfrak{m}$, let $i \\in \\mathbb{N}$, and let $K$ be a field, with a $K$-algebra structure on $\\kappa$ and a $K$-module structure on $\\mathfrak{m}^i/\\mathfrak{m}^{i+1}$ such that $(c b) x = c (b x)$ for all $c \\in K$, $b \\in \\kappa$ and $x \\in \\mathfrak{m}^i/\\mathfrak{m}^{i+1}$. Then for every $a \\in \\kappa$, the\u2026","title":"Trace of multiplication on a graded piece"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-maximal-ideal-pow-quotient","kind":"node","lean":"ResidueFieldMaximalIdealPowQuotient","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-maximal-ideal-pow-quotient","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, let $B$ be an $A$-algebra, and write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$. For a natural number $n$, this is the quotient ring $B / (\\mathfrak m_A B + \\mathfrak m_B^n)$.","title":"Quotient by the extended maximal ideal plus a maximal-ideal power"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/maximal-ideal-pow-quotient-ideal-eq","kind":"node","lean":"maximalIdealPowQuotientIdeal_eq","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#maximal-ideal-pow-quotient-ideal-eq","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, let $B$ be an $A$-algebra, and write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$. For every natural number $n$, if $\\mathfrak m_A B \\subseteq \\mathfrak m_B^n$, then $\\mathfrak m_A B + \\mathfrak m_B^n = \\mathfrak m_B^n$.","title":"Collapse of the quotient-defining ideal"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/maximal-ideal-pow-quotient-ideal-le","kind":"node","lean":"maximalIdealPowQuotientIdeal_le","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#maximal-ideal-pow-quotient-ideal-le","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, let $B$ be an $A$-algebra, and write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$. For every natural number $n$, the ideal $\\mathfrak m_A B + \\mathfrak m_B^{n+1}$ is contained in $\\mathfrak m_A B + \\mathfrak m_B^n$.","title":"The quotient-defining ideals decrease"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/factor-residue-field-maximal-ideal-pow-q","kind":"node","lean":"factorResidueFieldMaximalIdealPowQuotient","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#factor-residue-field-maximal-ideal-pow-q","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$ and let $\\kappa_A = A/\\mathfrak m_A$ be the residue field of $A$. Let $B$ be an $A$-algebra, write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$, and for each $m$ let $Q_m = B/(\\mathfrak m_A B + \\mathfrak m_B^m)$, a $\\kappa_A$-algebra in which the class of $r \\in A$ acts as\u2026","title":"Transition map between maximal-ideal-power quotients"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/factor-residue-field-maximal-ideal-pow-q-2","kind":"node","lean":"factorResidueFieldMaximalIdealPowQuotient_surjective","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#factor-residue-field-maximal-ideal-pow-q-2","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, let $B$ be an $A$-algebra, write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$, and for each $m$ let $Q_m = B/(\\mathfrak m_A B + \\mathfrak m_B^m)$. For every natural number $n$, the transition homomorphism $Q_{n+1} \\to Q_n$ that sends the class of each $b \\in B$ to its class in\u2026","title":"Surjectivity of the transition maps between maximal-ideal-power quotients"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/factor-residue-field-maximal-ideal-pow-q-3","kind":"node","lean":"factorResidueFieldMaximalIdealPowQuotient_mk","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#factor-residue-field-maximal-ideal-pow-q-3","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, let $B$ be an $A$-algebra, write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$, and for each $m$ let $Q_m = B/(\\mathfrak m_A B + \\mathfrak m_B^m)$. For every natural number $n$ and every $b \\in B$, the transition homomorphism $Q_{n+1} \\to Q_n$ maps the class of $b$ in $Q_{n+1}$ to\u2026","title":"The transition map on residue classes"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-maximal-ideal-pow-quotient-2","kind":"node","lean":"residueFieldMaximalIdealPowQuotientEquivPow","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-maximal-ideal-pow-quotient-2","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, let $B$ be an $A$-algebra, and write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$. For a natural number $n$ with $\\mathfrak m_A B \\subseteq \\mathfrak m_B^n$, this is the ring isomorphism $B/(\\mathfrak m_A B + \\mathfrak m_B^n) \\cong B/\\mathfrak m_B^n$ sending the class of each\u2026","title":"Ring isomorphism with the quotient by a power of the maximal ideal"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-maximal-ideal-pow-quotient-3","kind":"node","lean":"residueFieldMaximalIdealPowQuotientEquivPow_mk","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-maximal-ideal-pow-quotient-3","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, let $B$ be an $A$-algebra, and write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$. For every natural number $n$ with $\\mathfrak m_A B \\subseteq \\mathfrak m_B^n$ and every $b \\in B$, the ring isomorphism $B/(\\mathfrak m_A B + \\mathfrak m_B^n) \\cong B/\\mathfrak m_B^n$ induced by the\u2026","title":"The power-quotient isomorphism on residue classes"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-linear-equiv-maximal-ideal-3","kind":"node","lean":"residueFieldLinearEquivMaximalIdealGradedPieceKerFactor","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-linear-equiv-maximal-ideal-3","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$ and residue fields $\\kappa_A = A/\\mathfrak m_A$ and $\\kappa_B = B/\\mathfrak m_B$, let $B$ be an $A$-algebra whose algebra map $A \\to B$ is a local homomorphism, so that it induces a field homomorphism $\\kappa_A \\to \\kappa_B$, and write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$.\u2026","title":"Graded pieces as kernels of the transition maps"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-field-linear-equiv-maximal-ideal-4","kind":"node","lean":"residueFieldLinearEquivMaximalIdealGradedPieceKerFactor_mk","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-field-linear-equiv-maximal-ideal-4","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$ and residue fields $\\kappa_A = A/\\mathfrak m_A$ and $\\kappa_B = B/\\mathfrak m_B$, let $B$ be an $A$-algebra whose algebra map $A \\to B$ is a local homomorphism, so that it induces a field homomorphism $\\kappa_A \\to \\kappa_B$, and write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$.\u2026","title":"The kernel identification on representatives"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/trace-residue-field-maximal-ideal-pow-qu","kind":"node","lean":"trace_residueFieldMaximalIdealPowQuotient_succ","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#trace-residue-field-maximal-ideal-pow-qu","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$, $\\mathfrak m_B$ and residue fields $\\kappa_A$, $\\kappa_B$, and let $B$ be an $A$-algebra whose structure map is a local homomorphism. Write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$, and for each $m$ let $Q_m = B / (\\mathfrak m_A B + \\mathfrak m_B^m)$, a $\\kappa_A$-algebra. Let $n$ be a natural\u2026","title":"One-step trace recurrence for the maximal-ideal-power quotients"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/trace-residue-field-maximal-ideal-pow-qu-2","kind":"node","lean":"trace_residueFieldMaximalIdealPowQuotient","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#trace-residue-field-maximal-ideal-pow-qu-2","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$, $\\mathfrak m_B$ and residue fields $\\kappa_A$, $\\kappa_B$, and let $B$ be an $A$-algebra whose structure map is a local homomorphism. Write $\\mathfrak m_A B$ for the ideal of $B$ generated by the image of $\\mathfrak m_A$. Let $n$ be a natural number with $\\mathfrak m_A B \\subseteq \\mathfrak m_B^n$, and suppose the $\\kappa_A$-algebra\u2026","title":"Trace on the maximal-ideal-power quotients"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/residue-int-trace-eq-nsmul-trace-residue","kind":"node","lean":"residue_intTrace_eq_nsmul_trace_residue_of_map_maximalIdeal_eq_pow","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#residue-int-trace-eq-nsmul-trace-residue","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$, $\\mathfrak m_B$ and residue fields $\\kappa_A$, $\\kappa_B$, and let $B$ be an $A$-algebra that is finitely generated and torsion-free as an $A$-module. Let $e$ be a natural number such that the ideal $\\mathfrak m_A B$ generated by the image of $\\mathfrak m_A$ equals $\\mathfrak m_B^e$. Then for every $b \\in B$, writing $\\bar b$ for the\u2026","title":"Residue of the integral trace"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/int-trace-surjective-of-is-unit-nat-cast","kind":"node","lean":"intTrace_surjective_of_isUnit_natCast_of_map_maximalIdeal_eq_pow","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#int-trace-surjective-of-is-unit-nat-cast","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak m_A$, $\\mathfrak m_B$ and residue fields $\\kappa_A$, $\\kappa_B$, and let $B$ be an $A$-algebra that is finitely generated and torsion-free as an $A$-module, with $\\kappa_B / \\kappa_A$ separable. Let $e$ be a natural number that is a unit in $B$ and such that the ideal $\\mathfrak m_A B$ generated by the image of $\\mathfrak m_A$ equals\u2026","title":"Tame extensions have surjective integral trace"},{"id":"RingTheory/ramification/tame-ramification-and-the-different/different-ideal-eq-maximal-ideal-pow-sub","kind":"node","lean":"differentIdeal_eq_maximalIdeal_pow_sub_one_of_isUnit_natCast","parent":3785,"result":true,"route":"../roadmap/RingTheory/ramification/tame-ramification-and-the-different/index.html#different-ideal-eq-maximal-ideal-pow-sub","scope":"RingTheory/ramification/tame-ramification-and-the-different","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with maximal ideals $\\mathfrak{m}_A$ and $\\mathfrak{m}_B$, and let $B$ be an $A$-algebra that is finite and torsion-free as an $A$-module, such that the extension of fraction fields and the extension of residue fields are both separable. Let $e$ be a natural number whose image in $B$ is a unit, and suppose that the ideal of $B$ generated by the image of $\\mathfrak{m}_A$\u2026","title":"Different ideal of a tamely ramified extension of DVRs"},{"data_url":"scopes/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn.json","id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","kind":"scope","navigation":"module","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","title":"Totally ramified extensions and Eisenstein polynomials"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/ramification-idx-eq-finrank-of-adjoin-un","kind":"node","lean":"IsDiscreteValuationRing.ramificationIdx_eq_finrank_of_adjoin_uniformizer","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#ramification-idx-eq-finrank-of-adjoin-un","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A \\subseteq B$ be an extension of discrete valuation rings with $B$ finite over $A$ and the inclusion a local homomorphism, and let $K \\subseteq L$ be their fraction fields, with the maps $A \\to B \\to L$ and $A \\to K \\to L$ agreeing. If a uniformizer $\\pi$ of $B$ satisfies $B = A[\\pi]$, then the ramification index of the maximal ideal of $B$ over $A$ equals $[L : K]$.","title":"A generating uniformizer gives total ramification"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/ramification-idx-eq-finrank-of-eisenstei","kind":"node","lean":"IsDiscreteValuationRing.ramificationIdx_eq_finrank_of_eisenstein_adjoin","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#ramification-idx-eq-finrank-of-eisenstei","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A \\subseteq B$ be an extension of discrete valuation rings with $B$ finite over $A$ and the inclusion a local homomorphism, and let $K \\subseteq L$ be their fraction fields, with the maps $A \\to B \\to L$ and $A \\to K \\to L$ agreeing. If a uniformizer $\\pi$ of $B$ satisfies $B = A[\\pi]$ and its minimal polynomial over $A$ is Eisenstein at the maximal ideal of $A$, then the ramification index of the maximal\u2026","title":"An Eisenstein generating uniformizer gives total ramification"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/eisenstein-adjoin-of-ramification-idx-eq","kind":"node","lean":"IsDiscreteValuationRing.eisenstein_adjoin_of_ramificationIdx_eq_finrank","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#eisenstein-adjoin-of-ramification-idx-eq","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A \\subseteq B$ be an extension of discrete valuation rings with $B$ finite over $A$ and the inclusion a local homomorphism, and let $K \\subseteq L$ be their fraction fields, with $L$ finite over $K$ and the maps $A \\to B \\to L$ and $A \\to K \\to L$ agreeing. If the ramification index of the maximal ideal of $B$ over $A$ equals $[L : K]$, then every uniformizer $\\pi$ of $B$ satisfies $B = A[\\pi]$, and the\u2026","title":"Totally ramified extensions are generated by an Eisenstein uniformizer"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/ramification-idx-eq-finrank-iff-eisenste","kind":"node","lean":"IsDiscreteValuationRing.ramificationIdx_eq_finrank_iff_eisenstein_adjoin","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#ramification-idx-eq-finrank-iff-eisenste","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A \\subseteq B$ be an extension of discrete valuation rings with $B$ finite over $A$ and the inclusion a local homomorphism, and let $K \\subseteq L$ be their fraction fields, with $L$ finite over $K$ and the maps $A \\to B \\to L$ and $A \\to K \\to L$ agreeing. For a uniformizer $\\pi$ of $B$, the ramification index of the maximal ideal of $B$ over $A$ equals $[L : K]$ if and only if $B = A[\\pi]$ and the minimal\u2026","title":"Eisenstein criterion for total ramification of discrete valuation rings"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/algebra-map-coeff-zero-mem-span-root","kind":"node","lean":"AdjoinRoot.algebraMap_coeff_zero_mem_span_root","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#algebra-map-coeff-zero-mem-span-root","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a commutative ring, let $f \\in A[X]$ be a polynomial, and let $\\alpha$ be the image of $X$ in $A[X]/(f)$. Then the image in $A[X]/(f)$ of the constant coefficient $f(0)$ of $f$ lies in the principal ideal $(\\alpha)$.","title":"The constant coefficient lies in the ideal generated by the adjoined root"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/maximal-ideal-eq-span-coeff-zero","kind":"node","lean":"Polynomial.IsEisensteinAt.maximalIdeal_eq_span_coeff_zero","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#maximal-ideal-eq-span-coeff-zero","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak m$, and let $f \\in A[X]$ be a polynomial of positive degree that is Eisenstein with respect to $\\mathfrak m$. Then $\\mathfrak m$ is the principal ideal generated by the constant coefficient $f(0)$ of $f$.","title":"The constant term of an Eisenstein polynomial generates the maximal ideal of a DVR"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/map-maximal-ideal-le-span-root","kind":"node","lean":"AdjoinRoot.map_maximalIdeal_le_span_root","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#map-maximal-ideal-le-span-root","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak m$, let $f \\in A[X]$ be a polynomial of positive degree that is Eisenstein with respect to $\\mathfrak m$, and let $\\alpha$ be the image of $X$ in $A[X]/(f)$. Then the ideal of $A[X]/(f)$ generated by the image of $\\mathfrak m$ is contained in the principal ideal $(\\alpha)$.","title":"The maximal ideal of a DVR maps into the ideal of an Eisenstein root"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/root-ne-zero","kind":"node","lean":"AdjoinRoot.root_ne_zero","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#root-ne-zero","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a local integral domain with maximal ideal $\\mathfrak m$, let $f \\in A[X]$ be a monic polynomial of positive degree that is Eisenstein with respect to $\\mathfrak m$, and let $\\alpha$ be the image of $X$ in $A[X]/(f)$. Then $\\alpha \\neq 0$.","title":"The adjoined root of an Eisenstein polynomial is nonzero"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/is-maximal-span-root","kind":"node","lean":"AdjoinRoot.isMaximal_span_root","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#is-maximal-span-root","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak m$, let $f \\in A[X]$ be a monic polynomial of positive degree that is Eisenstein with respect to $\\mathfrak m$, and let $\\alpha$ be the image of $X$ in $A[X]/(f)$. Then the principal ideal $(\\alpha)$ of $A[X]/(f)$ is a maximal ideal.","title":"The adjoined root of an Eisenstein polynomial generates a maximal ideal"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/root-mem-of-is-maximal","kind":"node","lean":"AdjoinRoot.root_mem_of_isMaximal","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#root-mem-of-is-maximal","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak m$, let $f \\in A[X]$ be a monic polynomial that is Eisenstein with respect to $\\mathfrak m$, and let $\\alpha$ be the image of $X$ in $A[X]/(f)$. Then $\\alpha$ lies in every maximal ideal of $A[X]/(f)$.","title":"Every maximal ideal of an Eisenstein extension of a DVR contains the root"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/is-local-ring-of-is-eisenstein-at","kind":"node","lean":"AdjoinRoot.isLocalRing_of_isEisensteinAt","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#is-local-ring-of-is-eisenstein-at","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak m$, and let $f \\in A[X]$ be a monic polynomial of positive degree that is Eisenstein with respect to $\\mathfrak m$. Then $A[X]/(f)$ is a local ring.","title":"An Eisenstein extension of a DVR is local"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/is-local-ring-maximal-ideal-eq-span-root","kind":"node","lean":"AdjoinRoot.isLocalRing_maximalIdeal_eq_span_root","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#is-local-ring-maximal-ideal-eq-span-root","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak m$, let $f \\in A[X]$ be a monic polynomial of positive degree that is Eisenstein with respect to $\\mathfrak m$, and let $\\alpha$ be the image of $X$ in $A[X]/(f)$. If $A[X]/(f)$ is a local ring, then its maximal ideal is the principal ideal $(\\alpha)$.","title":"The maximal ideal of an Eisenstein extension of a DVR is generated by the root"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/is-discrete-valuation-ring-of-is-eisenst","kind":"node","lean":"AdjoinRoot.isDiscreteValuationRing_of_isEisensteinAt","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#is-discrete-valuation-ring-of-is-eisenst","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak m$, and let $f \\in A[X]$ be a monic polynomial of positive degree that is Eisenstein with respect to $\\mathfrak m$. Then $A[X]/(f)$ is a discrete valuation ring.","title":"Adjoining a root of an Eisenstein polynomial to a DVR gives a DVR"},{"id":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/irreducible-root-of-is-eisenstein-at","kind":"node","lean":"AdjoinRoot.irreducible_root_of_isEisensteinAt","parent":3808,"result":true,"route":"../roadmap/RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn/index.html#irreducible-root-of-is-eisenstein-at","scope":"RingTheory/ramification/totally-ramified-extensions-and-eisenstein-polyn","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak m$, let $f \\in A[X]$ be a monic polynomial of positive degree that is Eisenstein with respect to $\\mathfrak m$, and let $\\alpha$ be the image of $X$ in $A[X]/(f)$. Then $\\alpha$ is an irreducible element of $A[X]/(f)$.","title":"The adjoined root of an Eisenstein polynomial over a DVR is irreducible"},{"data_url":"scopes/RingTheory/ramification/unramified-extensions-and-residue-field-extensio.json","id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio","kind":"scope","parent":3676,"result":false,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/index.html","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio","status":"fully_proved","title":"Unramified extensions and residue field extensions"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-partial-equivalence-packag","kind":"node","lean":"DVRResidue.residueField_partial_equivalence_package_of_isAdicComplete","parent":3823,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/index.html#residue-field-partial-equivalence-packag","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring, and write $\\kappa(R)$ for the residue field of a local ring $R$. Let $B_1$ and $B_2$ be discrete valuation rings that are finite $A$-algebras with local structure maps, complete and separated for their maximal-ideal-adic topologies, such that for $i = 1, 2$ the field $\\kappa(B_i)$ is separable over $\\kappa(A)$ and $\\operatorname{rank}_A B_i$ equals\u2026","title":"Partial residue-field equivalence for complete finite DVR extensions"},{"data_url":"scopes/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d.json","id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","kind":"scope","navigation":"module","parent":3823,"result":false,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","title":"Lifting irreducible residue polynomials over a DVR"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/exists-monic-lift-adjoin-root-domain","kind":"node","lean":"DVRResidue.exists_monic_lift_adjoinRoot_domain","parent":3825,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#exists-monic-lift-adjoin-root-domain","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with residue field $k$, and let $\\bar g \\in k[X]$ be monic and irreducible. Then there is a monic polynomial $g \\in A[X]$ whose reduction modulo the maximal ideal of $A$ is $\\bar g$, such that $A[X]/(g)$ is an integral domain and is finitely generated as an $A$-module.","title":"Monic lifts of irreducible residue polynomials"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/adjoin-root-quotient-equiv-of-map-eq","kind":"node","lean":"DVRResidue.adjoinRootQuotientEquivOfMapEq","parent":3825,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reducing "],["math","\\(R(X)/(g)\\)"],["text"," modulo the maximal ideal of "],["math","\\(R\\)"]],"source":"Reducing $R(X)/(g)$ modulo the maximal ideal of $R$"},"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#adjoin-root-quotient-equiv-of-map-eq","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $k$, let $g \\in R[X]$, and let $\\bar g \\in k[X]$ be a polynomial equal to the reduction of $g$ modulo $\\mathfrak m$. This is the canonical ring isomorphism from the quotient of $R[X]/(g)$ by the ideal generated by the image of $\\mathfrak m$ onto $k[X]/(\\bar g)$; it sends the class of the image of $p \\in R[X]$ to the class of the reduction\u2026","title":"Reducing $R(X)/(g)$ modulo the maximal ideal of $R$"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/adjoin-root-map-maximal-ideal-is-maximal","kind":"node","lean":"DVRResidue.adjoinRoot_map_maximalIdeal_isMaximal","parent":3825,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#adjoin-root-map-maximal-ideal-is-maximal","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $k$, and let $g \\in A[X]$ be a polynomial whose reduction $\\bar g \\in k[X]$ modulo $\\mathfrak m$ is irreducible. Then the ideal of $A[X]/(g)$ generated by the image of $\\mathfrak m$ under the canonical map $A \\to A[X]/(g)$ is maximal.","title":"The extended maximal ideal is maximal for irreducible reduction"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/adjoin-root-is-local-ring-of-monic-of-ir","kind":"node","lean":"DVRResidue.adjoinRoot_isLocalRing_of_monic_of_irreducible_map","parent":3825,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#adjoin-root-is-local-ring-of-monic-of-ir","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a local ring with residue field $k$, and let $g \\in A[X]$ be a monic polynomial whose reduction $\\bar g \\in k[X]$ modulo the maximal ideal of $A$ is irreducible. Then $A[X]/(g)$ is a local ring.","title":"Monic polynomials with irreducible reduction give local rings"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/maximal-ideal-adjoin-root-eq-map","kind":"node","lean":"DVRResidue.maximalIdeal_adjoinRoot_eq_map","parent":3825,"reading_title":{"format":"inline-parts-v1","parts":[["text","The maximal ideal of "],["math","\\(A(X)/(g)\\)"],["text"," is extended from "],["math","\\(A\\)"]],"source":"The maximal ideal of $A(X)/(g)$ is extended from $A$"},"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#maximal-ideal-adjoin-root-eq-map","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $k$, and let $g \\in A[X]$ be a polynomial whose reduction $\\bar g \\in k[X]$ modulo $\\mathfrak m$ is irreducible. If $A[X]/(g)$ is a local ring, then its maximal ideal is the ideal generated by the image of $\\mathfrak m$ under the canonical map $A \\to A[X]/(g)$.","title":"The maximal ideal of $A(X)/(g)$ is extended from $A$"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/residue-field-equiv-adjoin-root-of-map-e","kind":"node","lean":"DVRResidue.residueFieldEquivAdjoinRootOfMapEq","parent":3825,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residue field of "],["math","\\(A(X)/(g)\\)"]],"source":"Residue field of $A(X)/(g)$"},"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#residue-field-equiv-adjoin-root-of-map-e","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a local ring with residue field $\\kappa(A)$, let $g \\in A[X]$ be such that $A[X]/(g)$ is a local ring, and let $\\bar g \\in \\kappa(A)[X]$ be the reduction of $g$ modulo the maximal ideal of $A$, assumed irreducible. This is the ring isomorphism from the residue field of $A[X]/(g)$ onto $\\kappa(A)[X]/(\\bar g)$ that sends the residue class of the image of each $p \\in A[X]$ to the class of $\\bar p$, the\u2026","title":"Residue field of $A(X)/(g)$"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/residue-field-equiv-adjoin-root-of-map-e-2","kind":"node","lean":"DVRResidue.residueFieldEquivAdjoinRootOfMapEq_residue_mk","parent":3825,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residue field of "],["math","\\(A(X)/(g)\\)"],["text"," on representatives"]],"source":"Residue field of $A(X)/(g)$ on representatives"},"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#residue-field-equiv-adjoin-root-of-map-e-2","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a local ring with residue field $\\kappa(A)$, let $g \\in A[X]$ be such that $A[X]/(g)$ is a local ring, and let $\\bar g \\in \\kappa(A)[X]$ be the reduction of $g$ modulo the maximal ideal of $A$, assumed irreducible. Then for every $p \\in A[X]$, the ring isomorphism from the residue field of $A[X]/(g)$ onto $\\kappa(A)[X]/(\\bar g)$ sends the residue class of the image of $p$ in $A[X]/(g)$ to the class of\u2026","title":"Residue field of $A(X)/(g)$ on representatives"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/adjoin-root-is-local-hom-of-irreducible","kind":"node","lean":"DVRResidue.adjoinRoot_isLocalHom_of_irreducible_map","parent":3825,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#adjoin-root-is-local-hom-of-irreducible","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a local ring with residue field $k$, and let $g \\in A[X]$ be a polynomial whose reduction $\\bar g \\in k[X]$ modulo the maximal ideal of $A$ is irreducible. If $A[X]/(g)$ is a local ring, then the canonical map $A \\to A[X]/(g)$ is a local ring homomorphism: an element of $A$ whose image in $A[X]/(g)$ is a unit is itself a unit.","title":"Local base map for polynomials with irreducible reduction"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/residue-field-alg-equiv-adjoin-root-of-m","kind":"node","lean":"DVRResidue.residueFieldAlgEquivAdjoinRootOfMapEq","parent":3825,"reading_title":{"format":"inline-parts-v1","parts":[["text","The residue field of "],["math","\\(A(X)/(g)\\)"],["text"," as a "],["math","\\(k\\)"],["text","-algebra"]],"source":"The residue field of $A(X)/(g)$ as a $k$-algebra"},"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#residue-field-alg-equiv-adjoin-root-of-m","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $k$, and let $g \\in A[X]$ be a polynomial such that $A[X]/(g)$ is a local ring, the canonical map $A \\to A[X]/(g)$ is a local homomorphism, and the reduction $\\bar g \\in k[X]$ of $g$ modulo $\\mathfrak m$ is irreducible. This is the isomorphism of $k$-algebras from the residue field of $A[X]/(g)$ (a $k$-algebra via $A \\to A[X]/(g)$) onto\u2026","title":"The residue field of $A(X)/(g)$ as a $k$-algebra"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/residue-field-alg-equiv-adjoin-root-of-m-2","kind":"node","lean":"DVRResidue.residueFieldAlgEquivAdjoinRootOfMapEq_residue_mk","parent":3825,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#residue-field-alg-equiv-adjoin-root-of-m-2","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a local ring with maximal ideal $\\mathfrak m$ and residue field $k$, let $g \\in A[X]$ be a polynomial such that $A[X]/(g)$ is a local ring, the canonical map $A \\to A[X]/(g)$ is a local homomorphism, and the reduction $\\bar g \\in k[X]$ of $g$ modulo $\\mathfrak m$ is irreducible. Then for every $p \\in A[X]$, the canonical $k$-algebra isomorphism from the residue field of $A[X]/(g)$ onto $k[X]/(\\bar g)$\u2026","title":"The residue-field algebra isomorphism on polynomial classes"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/adjoin-root-is-discrete-valuation-ring-o","kind":"node","lean":"DVRResidue.adjoinRoot_isDiscreteValuationRing_of_monic_of_irreducible_map","parent":3825,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#adjoin-root-is-discrete-valuation-ring-o","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with residue field $k$, and let $g \\in A[X]$ be a monic polynomial whose reduction $\\bar g \\in k[X]$ modulo the maximal ideal of $A$ is irreducible. If $A[X]/(g)$ is an integral domain and a local ring, then $A[X]/(g)$ is a discrete valuation ring.","title":"Monic lifts of irreducible residue polynomials give DVRs"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/exists-finite-dvr-extension-residue-fiel","kind":"node","lean":"DVRResidue.exists_finite_dvr_extension_residueField_algEquiv","parent":3825,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d/index.html#exists-finite-dvr-extension-residue-fiel","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/lifting-irreducible-residue-polynomials-over-a-d","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with residue field $k$, and let $L$ be a finite separable field extension of $k$. Then there is a discrete valuation ring $B$ with an $A$-algebra structure such that $B$ is finitely generated as an $A$-module and the structure map $A \\to B$ is a local homomorphism, together with an isomorphism of $k$-algebras between the residue field of $B$ (a $k$-algebra via $A \\to B$) and $L$.","title":"Finite separable residue extensions lift to finite DVR extensions"},{"data_url":"scopes/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for.json","id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for","kind":"scope","navigation":"module","parent":3823,"result":false,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for/index.html","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for","status":"fully_proved","title":"Residue-field functoriality and faithfulness for DVR extensions"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for/is-local-hom-of-is-discrete-valuation-ri","kind":"node","lean":"DVRResidue.AlgHom.isLocalHom_of_isDiscreteValuationRing","parent":3838,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for/index.html#is-local-hom-of-is-discrete-valuation-ri","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for","status":"fully_proved","summary":"Let $A$, $B$, $C$ be discrete valuation rings, with $B$ and $C$ algebras over $A$, where $C$ is finite as an $A$-module and the structure map $A \\to C$ is a local homomorphism. Then every $A$-algebra homomorphism $B \\to C$ is a local homomorphism.","title":"Algebra maps from a DVR into a finite local DVR extension are local"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for/residue-field-map-alg-hom","kind":"node","lean":"DVRResidue.residueFieldMapAlgHom","parent":3838,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for/index.html#residue-field-map-alg-hom","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for","status":"fully_proved","summary":"Let $A$, $B$, $C$ be discrete valuation rings, with $B$ and $C$ algebras over $A$ whose structure maps are local homomorphisms, where $C$ is finite as an $A$-module. Write $\\kappa(R)$ for the residue field of a local ring $R$. An $A$-algebra homomorphism $f \\colon B \\to C$ induces the $\\kappa(A)$-algebra homomorphism $\\kappa(B) \\to \\kappa(C)$ that sends the residue class of $b \\in B$ to the residue class of $f(b)$.","title":"Induced map on residue fields of DVR extensions"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for/residue-field-map-alg-hom-injective","kind":"node","lean":"DVRResidue.residueFieldMapAlgHom_injective","parent":3838,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for/index.html#residue-field-map-alg-hom-injective","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/residue-field-functoriality-and-faithfulness-for","status":"fully_proved","summary":"Let $A$, $B$, $C$ be discrete valuation rings, with $B$ and $C$ algebras over $A$ whose structure maps are local homomorphisms, where $C$ is finite as an $A$-module and $B$ is formally unramified over $A$. Write $\\kappa(R)$ for the residue field of a local ring $R$. Then two $A$-algebra homomorphisms $B \\to C$ that induce the same $\\kappa(A)$-algebra homomorphism $\\kappa(B) \\to \\kappa(C)$ are equal.","title":"Residue-field maps determine algebra maps out of a formally unramified DVR extension"},{"data_url":"scopes/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex.json","id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex","kind":"scope","navigation":"module","parent":3823,"result":false,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex/index.html","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex","status":"fully_proved","title":"Fullness of the residue-field functor for DVR extensions"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex/exists-root-lift-of-adjoin-eq-top","kind":"node","lean":"DVRResidue.exists_root_lift_of_adjoin_eq_top","parent":3842,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex/index.html#exists-root-lift-of-adjoin-eq-top","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings with $B$ a finite \u00e9tale $A$-algebra whose structure map is local, and let $C$ be a Henselian local ring that is an $A$-algebra whose structure map is local. Write $k_A$, $k_B$, $k_C$ for the residue fields. Let $\\beta \\in B$ generate $B$ as an $A$-algebra, and let $\\bar\\beta \\in k_B$ be its residue class. Then for every $k_A$-algebra homomorphism $f : k_B \\to k_C$ there\u2026","title":"Lifting residue-field images to roots of a minimal polynomial"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex/residue-field-map-alg-hom-surjective","kind":"node","lean":"DVRResidue.residueFieldMapAlgHom_surjective","parent":3842,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex/index.html#residue-field-map-alg-hom-surjective","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/fullness-of-the-residue-field-functor-for-dvr-ex","status":"fully_proved","summary":"Let $A$, $B$, $C$ be discrete valuation rings, with $B$ and $C$ finite $A$-algebras whose structure maps are local, $B$ \u00e9tale over $A$ and $C$ Henselian. Write $k_A$, $k_B$, $k_C$ for the residue fields. Then every $k_A$-algebra homomorphism $k_B \\to k_C$ is induced by an $A$-algebra homomorphism: the map sending an $A$-algebra homomorphism $\\varphi : B \\to C$ to the induced homomorphism of residue fields\u2026","title":"Lifting residue-field homomorphisms of finite \u00e9tale DVR extensions"},{"data_url":"scopes/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local.json","id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local","kind":"scope","navigation":"module","parent":3823,"result":false,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local/index.html","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local","status":"fully_proved","title":"Bijections on residue-field morphisms for local DVR extensions"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local/residue-field-map-alg-hom-bijective","kind":"node","lean":"DVRResidue.residueFieldMapAlgHom_bijective","parent":3845,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local/index.html#residue-field-map-alg-hom-bijective","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local","status":"fully_proved","summary":"Let $A$, $B$, $C$ be discrete valuation rings, with $B$ and $C$ finite $A$-algebras whose structure maps are local homomorphisms, $B$ \u00e9tale over $A$, and $C$ a Henselian local ring. Write $\\kappa(R)$ for the residue field of a local ring $R$. Then sending an $A$-algebra homomorphism $B \\to C$ to the $\\kappa(A)$-algebra homomorphism $\\kappa(B) \\to \\kappa(C)$ it induces is a bijection.","title":"Algebra maps from an \u00e9tale DVR extension into a Henselian one match residue-field maps"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local/exists-alg-equiv-lift-residue-field","kind":"node","lean":"DVRResidue.exists_algEquiv_lift_residueField","parent":3845,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local/index.html#exists-alg-equiv-lift-residue-field","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/bijections-on-residue-field-morphisms-for-local","status":"fully_proved","summary":"Let $A$, $B$, $C$ be discrete valuation rings, with $B$ and $C$ finite $A$-algebras whose structure maps are local homomorphisms, both \u00e9tale over $A$ and both Henselian local rings. Write $\\kappa(R)$ for the residue field of a local ring $R$. Then every $\\kappa(A)$-algebra isomorphism $e \\colon \\kappa(B) \\to \\kappa(C)$ lifts to an $A$-algebra isomorphism $e' \\colon B \\to C$: the map that $e'$ induces on residue\u2026","title":"Lifting residue-field isomorphisms between Henselian \u00e9tale DVR extensions"},{"data_url":"scopes/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom.json","id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom","kind":"scope","navigation":"module","parent":3823,"result":false,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom/index.html","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom","status":"fully_proved","title":"Complete-DVR wrappers for the residue-field Hom bijection"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom/residue-field-map-alg-hom-bijective-of-i","kind":"node","lean":"DVRResidue.residueFieldMapAlgHom_bijective_of_isAdicComplete","parent":3848,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom/index.html#residue-field-map-alg-hom-bijective-of-i","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom","status":"fully_proved","summary":"Let $A$, $B$, $C$ be discrete valuation rings, with $B$ and $C$ finite $A$-algebras whose structure maps are local homomorphisms, $B$ \u00e9tale over $A$, and $C$ complete and separated for its maximal-ideal-adic topology. Write $\\kappa(R)$ for the residue field of a local ring $R$. Then sending an $A$-algebra homomorphism $B \\to C$ to the $\\kappa(A)$-algebra homomorphism $\\kappa(B) \\to \\kappa(C)$ it induces is a\u2026","title":"Algebra maps from an \u00e9tale DVR extension into a complete one match residue-field maps"},{"id":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom/exists-alg-equiv-lift-residue-field-of-i","kind":"node","lean":"DVRResidue.exists_algEquiv_lift_residueField_of_isAdicComplete","parent":3848,"result":true,"route":"../roadmap/RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom/index.html#exists-alg-equiv-lift-residue-field-of-i","scope":"RingTheory/ramification/unramified-extensions-and-residue-field-extensio/complete-dvr-wrappers-for-the-residue-field-hom","status":"fully_proved","summary":"Let $A$, $B$, $C$ be discrete valuation rings, with $B$ and $C$ finite $A$-algebras whose structure maps are local homomorphisms, both \u00e9tale over $A$, and both complete and separated for their maximal-ideal-adic topologies. Write $\\kappa(R)$ for the residue field of a local ring $R$. Then every $\\kappa(A)$-algebra isomorphism $e \\colon \\kappa(B) \\to \\kappa(C)$ lifts to an $A$-algebra isomorphism\u2026","title":"Lifting residue-field isomorphisms between complete \u00e9tale DVR extensions"},{"data_url":"scopes/RingTheory/dedekind-domains.json","id":"RingTheory/dedekind-domains","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/index.html","scope":"RingTheory/dedekind-domains","status":"fully_proved","summary":"Dedekind domains and their characterizations, unique factorization of fractional ideals into primes, the ideal class group, invertible fractional ideals and the conductor of an order.","title":"Dedekind domains and ideal theory"},{"data_url":"scopes/RingTheory/dedekind-domains/bhargava-p-orderings.json","id":"RingTheory/dedekind-domains/bhargava-p-orderings","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","title":"Bhargava p-orderings"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/p-ordering-valuation","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.pOrderingValuation","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#p-ordering-valuation","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a commutative ring, $\\mathfrak p$ a nonzero prime ideal of $R$ and $(a_k)_{k \\ge 0}$ a sequence in $R$. For $n \\in \\mathbb N$ and $x \\in R$, the stage-$n$ prefix-product valuation of $x$ is the multiplicity of $\\mathfrak p$ in the principal ideal generated by $\\prod_{k < n} (x - a_k)$: the largest $m \\in \\mathbb N$ such that $\\mathfrak p^m$ divides this ideal, or $\\infty$ if every power of $\\mathfrak p$\u2026","title":"Prefix-product valuation"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/is-pordering","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.IsPOrdering","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#is-pordering","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a nonzero prime ideal of $R$, $S \\subseteq R$ and $(a_n)_{n \\ge 0}$ a sequence in $R$. Write $v_n(x)$ for the stage-$n$ prefix-product valuation of $x \\in R$ with respect to $\\mathfrak p$ and $(a_n)$, the multiplicity of $\\mathfrak p$ in the ideal generated by $\\prod_{k < n} (x - a_k)$. Then $(a_n)$ is a $\\mathfrak p$-ordering of $S$ when $S$ is infinite, $a_n \\in S$ for\u2026","title":"Bhargava p-ordering"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/is-pordering-unfold","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.isPOrdering_unfold","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#is-pordering-unfold","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a nonzero prime ideal of $R$, $S \\subseteq R$ and $(a_n)_{n \\ge 0}$ a sequence in $R$, and write $v_n$ for the stage-$n$ prefix-product valuation with respect to $\\mathfrak p$ and $(a_n)$. Then $(a_n)$ is a $\\mathfrak p$-ordering of $S$ if and only if $S$ is infinite, $a_n \\in S$ for every $n$, and $v_n(a_n) \\le v_n(x)$ for every $n$ and every $x \\in S$.","title":"Unfolding the definition of a p-ordering"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/infinite","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.IsPOrdering.infinite","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#infinite","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a nonzero prime ideal of $R$ and $S \\subseteq R$. If some sequence in $R$ is a $\\mathfrak p$-ordering of $S$, then $S$ is infinite.","title":"A set with a p-ordering is infinite"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/mem","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.IsPOrdering.mem","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#mem","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a nonzero prime ideal of $R$, $S \\subseteq R$ and $(a_n)_{n \\ge 0}$ a $\\mathfrak p$-ordering of $S$. Then $a_n \\in S$ for every $n$.","title":"Terms of a p-ordering lie in the set"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/min-le","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.IsPOrdering.min_le","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#min-le","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a nonzero prime ideal of $R$, $S \\subseteq R$ and $(a_n)_{n \\ge 0}$ a $\\mathfrak p$-ordering of $S$, and write $v_n$ for the stage-$n$ prefix-product valuation with respect to $\\mathfrak p$ and $(a_n)$. Then $v_n(a_n) \\le v_n(x)$ for every $n$ and every $x \\in S$.","title":"Minimality of a p-ordering"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/is-least","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.IsPOrdering.isLeast","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#is-least","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a nonzero prime ideal of $R$, $S \\subseteq R$ and $(a_n)_{n \\ge 0}$ a $\\mathfrak p$-ordering of $S$, and write $v_n$ for the stage-$n$ prefix-product valuation with respect to $\\mathfrak p$ and $(a_n)$. Then for every $n$, $v_n(a_n)$ is the least element of $\\{v_n(x) : x \\in S\\}$.","title":"Each term of a p-ordering attains the least valuation"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/intro","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.IsPOrdering.intro","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#intro","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a nonzero prime ideal of $R$, $S \\subseteq R$ and $(a_n)_{n \\ge 0}$ a sequence in $R$, and write $v_n$ for the stage-$n$ prefix-product valuation with respect to $\\mathfrak p$ and $(a_n)$. If $S$ is infinite, $a_n \\in S$ for every $n$, and $v_n(a_n) \\le v_n(x)$ for every $n$ and every $x \\in S$, then $(a_n)$ is a $\\mathfrak p$-ordering of $S$.","title":"Building a p-ordering"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/p-ordering-valuation-zero","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.pOrderingValuation_zero","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#p-ordering-valuation-zero","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a commutative ring, $\\mathfrak p$ a nonzero prime ideal of $R$ and $(a_n)_{n \\ge 0}$ a sequence in $R$. For every $x \\in R$, the stage-$0$ prefix-product valuation of $x$ with respect to $\\mathfrak p$ and $(a_n)$ is $0$: the product is empty, so the ideal is all of $R$.","title":"The stage-zero valuation vanishes"},{"id":"RingTheory/dedekind-domains/bhargava-p-orderings/is-pordering-iff-pos","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.isPOrdering_iff_pos","parent":3852,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/bhargava-p-orderings/index.html#is-pordering-iff-pos","scope":"RingTheory/dedekind-domains/bhargava-p-orderings","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a nonzero prime ideal of $R$, $S \\subseteq R$ and $(a_n)_{n \\ge 0}$ a sequence in $R$, and write $v_n$ for the stage-$n$ prefix-product valuation with respect to $\\mathfrak p$ and $(a_n)$. Then $(a_n)$ is a $\\mathfrak p$-ordering of $S$ if and only if $S$ is infinite, $a_n \\in S$ for every $n$, and $v_n(a_n) \\le v_n(x)$ for every $n \\ge 1$ and every $x \\in S$.","title":"The stage-zero condition of a p-ordering is automatic"},{"data_url":"scopes/RingTheory/dedekind-domains/characterizations-of-dedekind-domains.json","id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","title":"Characterizations of Dedekind domains"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-dedekind-domain-iff-contain-is-divide","kind":"node","lean":"isDedekindDomain_iff_contain_isDivide","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-dedekind-domain-iff-contain-is-divide","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"An integral domain $A$ is a Dedekind domain if and only if $A$ is Noetherian and, for all ideals $I$ and $J$ of $A$, $I$ divides $J$ if and only if $J \\subseteq I$.","title":"Dedekind domains as the Noetherian domains where to contain is to divide"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-dedekind-domain-iff-principal-product","kind":"node","lean":"isDedekindDomain_iff_principalProduct","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-dedekind-domain-iff-principal-product","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"An integral domain $A$ is a Dedekind domain if and only if for every nonzero ideal $I$ of $A$ there is an ideal $J$ of $A$ such that $IJ$ is a nonzero principal ideal.","title":"Dedekind domains via principal products"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-dedekind-domain-tfae-contain-is-divid","kind":"node","lean":"isDedekindDomain_tfae_contain_isDivide_principalProduct","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-dedekind-domain-tfae-contain-is-divid","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"For an integral domain $A$, the following are equivalent: (1) $A$ is a Dedekind domain; (2) $A$ is Noetherian and, for all ideals $I$ and $J$ of $A$, $I$ divides $J$ if and only if $J \\subseteq I$; (3) for every nonzero ideal $I$ of $A$ there is an ideal $J$ of $A$ such that $IJ$ is a nonzero principal ideal.","title":"Equivalent characterizations of Dedekind domains"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/integrally-closed-dimension-leone-iff-is","kind":"node","lean":"integrallyClosed_dimensionLEOne_iff_isDedekindDomain","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#integrally-closed-dimension-leone-iff-is","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"Let $A$ be a Noetherian integral domain. Then $A$ is integrally closed and has Krull dimension at most one if and only if $A$ is a Dedekind domain.","title":"Dedekind domains are the integrally closed Noetherian domains of dimension at most one"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-discrete-valuation-ring-localization","kind":"node","lean":"isDiscreteValuationRing_localization_atPrime_iff_integrallyClosed_dimensionLEOne","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-discrete-valuation-ring-localization","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"Let $A$ be a Noetherian integral domain. Then the localizations $A_P$ at all nonzero prime ideals $P$ of $A$ are discrete valuation rings if and only if $A$ is integrally closed and has Krull dimension at most one.","title":"DVR localizations in a Noetherian domain"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-principal-ideal-ring-quotient","kind":"node","lean":"IsDedekindDomain.isPrincipalIdealRing_quotient","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-principal-ideal-ring-quotient","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"Every quotient of a Dedekind domain by a nonzero ideal is a principal ideal ring.","title":"Quotients of Dedekind domains by nonzero ideals are principal ideal rings"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-noetherian-ring-of-forall-is-principa","kind":"node","lean":"isNoetherianRing_of_forall_isPrincipalIdealRing_quotient","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-noetherian-ring-of-forall-is-principa","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"Let $A$ be an integral domain such that for every nonzero ideal $I$ of $A$ the quotient ring $A/I$ is a principal ideal ring. Then $A$ is a Noetherian ring.","title":"Domains with principal quotients are Noetherian"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-discrete-valuation-ring-localization-2","kind":"node","lean":"isDiscreteValuationRing_localization_atPrime_of_forall_isPrincipalIdealRing_quotient","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-discrete-valuation-ring-localization-2","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"Let $A$ be an integral domain such that for every nonzero ideal $I$ of $A$ the quotient ring $A/I$ is a principal ideal ring. Then for every nonzero prime ideal $P$ of $A$, the localization $A_P$ is a discrete valuation ring.","title":"Principal quotients give discrete valuation rings at nonzero primes"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-dedekind-domain-of-forall-is-principa","kind":"node","lean":"isDedekindDomain_of_forall_isPrincipalIdealRing_quotient","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-dedekind-domain-of-forall-is-principa","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"Let $A$ be an integral domain such that for every nonzero ideal $I$ of $A$ the quotient ring $A/I$ is a principal ideal ring. Then $A$ is a Dedekind domain.","title":"Dedekind domains from principal quotients"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-principal-ideal-ring-quotient-of-fora","kind":"node","lean":"isPrincipalIdealRing_quotient_of_forall_exists_span_pair","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-principal-ideal-ring-quotient-of-fora","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"Let $A$ be an integral domain such that for every nonzero ideal $I$ of $A$ and every nonzero $a \\in I$ there is $b \\in A$ with $I = (a, b)$. Then for every nonzero ideal $I$ of $A$, the quotient ring $A/I$ is a principal ideal ring.","title":"Two generators with one prescribed give principal quotients"},{"id":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains/is-dedekind-domain-of-forall-exists-span","kind":"node","lean":"isDedekindDomain_of_forall_exists_span_pair","parent":3863,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/characterizations-of-dedekind-domains/index.html#is-dedekind-domain-of-forall-exists-span","scope":"RingTheory/dedekind-domains/characterizations-of-dedekind-domains","status":"fully_proved","summary":"Let $A$ be an integral domain such that for every nonzero ideal $I$ of $A$ and every nonzero $a \\in I$ there is $b \\in A$ with $I = (a, b)$. Then $A$ is a Dedekind domain.","title":"Dedekind domains from two generators with one prescribed"},{"data_url":"scopes/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes.json","id":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/index.html","scope":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","status":"fully_proved","title":"Completions of Dedekind domains at primes"},{"id":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/map-as-ideal-eq-maximal-ideal","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.map_asIdeal_eq_maximalIdeal","parent":3875,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/index.html#map-as-ideal-eq-maximal-ideal","scope":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $v$ be a height-one prime of $A$, and let $\\mathcal{O}_v$ be the valuation ring of the $v$-adic completion of $K$. Then the ideal of $\\mathcal{O}_v$ generated by the image of $v$ under the canonical map $A \\to \\mathcal{O}_v$ is the maximal ideal of $\\mathcal{O}_v$.","title":"Extension of a prime to the completed valuation ring"},{"id":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/completion-residue-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionResidueMap","parent":3875,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/index.html#completion-residue-map","scope":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $\\mathfrak{p}$ be a height-one prime of $A$, and let $\\mathcal{O}_\\mathfrak{p}$ be the valuation ring of the $\\mathfrak{p}$-adic completion of $K$, with maximal ideal $\\mathfrak{m}_\\mathfrak{p}$. The residue map is the ring homomorphism $A \\to \\mathcal{O}_\\mathfrak{p}/\\mathfrak{m}_\\mathfrak{p}$ obtained by composing the canonical map\u2026","title":"Residue map to the completed residue field"},{"id":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/ker-completion-residue-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.ker_completionResidueMap","parent":3875,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/index.html#ker-completion-residue-map","scope":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $\\mathfrak{p}$ be a height-one prime of $A$, and let $\\mathcal{O}_\\mathfrak{p}$ be the valuation ring of the $\\mathfrak{p}$-adic completion of $K$, with maximal ideal $\\mathfrak{m}_\\mathfrak{p}$. The kernel of the residue map $A \\to \\mathcal{O}_\\mathfrak{p}/\\mathfrak{m}_\\mathfrak{p}$, the canonical map $A \\to \\mathcal{O}_\\mathfrak{p}$ followed by the\u2026","title":"Kernel of the completed residue map"},{"id":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/surjective-completion-residue-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.surjective_completionResidueMap","parent":3875,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/index.html#surjective-completion-residue-map","scope":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $\\mathfrak{p}$ be a height-one prime of $A$, and let $\\mathcal{O}_\\mathfrak{p}$ be the valuation ring of the $\\mathfrak{p}$-adic completion of $K$, with maximal ideal $\\mathfrak{m}_\\mathfrak{p}$. The residue map $A \\to \\mathcal{O}_\\mathfrak{p}/\\mathfrak{m}_\\mathfrak{p}$, the canonical map $A \\to \\mathcal{O}_\\mathfrak{p}$ followed by the quotient map, is\u2026","title":"Surjectivity of the completed residue map"},{"id":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/completion-residue-field-equiv","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionResidueFieldEquiv","parent":3875,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/index.html#completion-residue-field-equiv","scope":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $\\mathfrak{p}$ be a height-one prime of $A$, and let $\\mathcal{O}_\\mathfrak{p}$ be the valuation ring of the $\\mathfrak{p}$-adic completion of $K$, with maximal ideal $\\mathfrak{m}_\\mathfrak{p}$. This is the ring isomorphism $\\mathcal{O}_\\mathfrak{p}/\\mathfrak{m}_\\mathfrak{p} \\cong A/\\mathfrak{p}$ that sends the residue class of the image of each $a \\in A$\u2026","title":"Residue field of the completion"},{"id":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/valuation-subring-at-prime-eq-comap-adic","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.valuationSubringAtPrime_eq_comap_adicCompletionIntegers","parent":3875,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/index.html#valuation-subring-at-prime-eq-comap-adic","scope":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, let $v$ be a height-one prime of $R$, and let $\\mathcal{O}_v$ be the valuation ring of the $v$-adic completion $K_v$ of $K$. Then the localization $R_v$ of $R$ at $v$, viewed as a subring of $K$, is the preimage of $\\mathcal{O}_v$ under the canonical embedding $K \\to K_v$.","title":"Localization at a prime as a preimage of the completed valuation ring"},{"id":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/finite-residue-field-adic-completion-int","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.finite_residueField_adicCompletionIntegers","parent":3875,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes/index.html#finite-residue-field-adic-completion-int","scope":"RingTheory/dedekind-domains/completions-of-dedekind-domains-at-primes","status":"fully_proved","summary":"The residue field of the adic completion integers at a place with finite residue field is finite.","title":"Finite residue field of adic completion integers"},{"data_url":"scopes/RingTheory/dedekind-domains/conductor-ideal.json","id":"RingTheory/dedekind-domains/conductor-ideal","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","title":"Conductor ideal"},{"id":"RingTheory/dedekind-domains/conductor-ideal/conductor-ideal","kind":"node","lean":"Subalgebra.conductorIdeal","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#conductor-ideal","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $S$ a commutative $R$-algebra and $A$ an $R$-subalgebra of $S$. The conductor ideal of $A$ is the ideal of $S$ consisting of all $s \\in S$ such that $st \\in A$ for every $t \\in S$; it is the largest ideal of $S$ contained in $A$.","title":"Conductor ideal of a subalgebra"},{"id":"RingTheory/dedekind-domains/conductor-ideal/mem-conductor-ideal","kind":"node","lean":"Subalgebra.mem_conductorIdeal","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#mem-conductor-ideal","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ and $S$ be commutative semirings with $S$ an $R$-algebra, and let $A$ be an $R$-subalgebra of $S$. An element $s \\in S$ lies in the conductor ideal of $A$ if and only if $st \\in A$ for every $t \\in S$.","title":"Membership in the conductor ideal"},{"id":"RingTheory/dedekind-domains/conductor-ideal/le-conductor-ideal-iff","kind":"node","lean":"Subalgebra.le_conductorIdeal_iff","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#le-conductor-ideal-iff","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ and $S$ be commutative semirings with $S$ an $R$-algebra, let $A$ be an $R$-subalgebra of $S$, and let $I$ be an ideal of $S$. Then $I$ is contained in the conductor ideal of $A$ if and only if $I$ is contained in $A$.","title":"Universal property of the conductor ideal"},{"id":"RingTheory/dedekind-domains/conductor-ideal/conductor-ideal-subset","kind":"node","lean":"Subalgebra.conductorIdeal_subset","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#conductor-ideal-subset","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $S$ a commutative $R$-algebra and $A$ an $R$-subalgebra of $S$. Then the conductor ideal of $A$ is contained in $A$.","title":"The conductor ideal lies in the subalgebra"},{"id":"RingTheory/dedekind-domains/conductor-ideal/conductor-ideal-mono","kind":"node","lean":"Subalgebra.conductorIdeal_mono","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#conductor-ideal-mono","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $S$ a commutative $R$-algebra and $A \\subseteq B$ two $R$-subalgebras of $S$. Then the conductor ideal of $A$ is contained in the conductor ideal of $B$.","title":"Monotonicity of the conductor ideal"},{"id":"RingTheory/dedekind-domains/conductor-ideal/conductor-ideal-top","kind":"node","lean":"Subalgebra.conductorIdeal_top","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#conductor-ideal-top","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a commutative semiring and $S$ a commutative $R$-algebra. The conductor ideal of $S$, viewed as an $R$-subalgebra of itself, is the unit ideal $S$.","title":"Conductor ideal of the whole algebra"},{"id":"RingTheory/dedekind-domains/conductor-ideal/conductor-ideal-eq-top-iff","kind":"node","lean":"Subalgebra.conductorIdeal_eq_top_iff","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#conductor-ideal-eq-top-iff","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $S$ a commutative $R$-algebra and $A$ an $R$-subalgebra of $S$. The conductor ideal of $A$ is all of $S$ if and only if $A = S$.","title":"The conductor ideal is the unit ideal only for the whole algebra"},{"id":"RingTheory/dedekind-domains/conductor-ideal/conductor-ideal-comap-le","kind":"node","lean":"Subalgebra.conductorIdeal_comap_le","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#conductor-ideal-comap-le","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $S$ and $T$ commutative $R$-algebras, $f \\colon S \\to T$ a homomorphism of $R$-algebras and $B$ an $R$-subalgebra of $T$. Then the preimage under $f$ of the conductor ideal of $B$ is contained in the conductor ideal of the subalgebra $f^{-1}(B)$ of $S$.","title":"Preimage of a conductor ideal"},{"id":"RingTheory/dedekind-domains/conductor-ideal/adjoin-conductor-ideal","kind":"node","lean":"Subalgebra.adjoin_conductorIdeal","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#adjoin-conductor-ideal","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a commutative ring, $S$ a commutative $R$-algebra and $x \\in S$. The conductor ideal of the subalgebra $R[x]$ of $S$ generated by $x$ equals the conductor of $x$, the ideal of all $a \\in S$ such that $ab \\in R[x]$ for every $b \\in S$.","title":"Conductor of a monogenic subalgebra"},{"id":"RingTheory/dedekind-domains/conductor-ideal/conductor-ideal-bot-ne-bot-of-finite","kind":"node","lean":"Subalgebra.conductorIdeal_bot_ne_bot_of_finite","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#conductor-ideal-bot-ne-bot-of-finite","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be an integral domain with fraction field $K$, and let $T$ be an $R$-subalgebra of $K$ that is finitely generated as an $R$-module. Then the conductor ideal of the image of $R$ in $T$ (the smallest $R$-subalgebra of $T$) is nonzero.","title":"A finite overring has nonzero conductor"},{"id":"RingTheory/dedekind-domains/conductor-ideal/finite-of-conductor-ideal-bot-ne-bot","kind":"node","lean":"Subalgebra.finite_of_conductorIdeal_bot_ne_bot","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#finite-of-conductor-ideal-bot-ne-bot","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a Noetherian commutative ring, $K$ a field that is an $R$-algebra, and $T$ an $R$-subalgebra of $K$. If the conductor ideal of the image of $R$ in $T$ (the smallest $R$-subalgebra of $T$) is nonzero, then $T$ is finitely generated as an $R$-module.","title":"A subalgebra of a field with nonzero conductor is finite"},{"id":"RingTheory/dedekind-domains/conductor-ideal/bot-conductor-ideal-ne-bot-iff","kind":"node","lean":"Subalgebra.bot_conductorIdeal_ne_bot_iff","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#bot-conductor-ideal-ne-bot-iff","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a Noetherian integral domain with fraction field $K$, and let $T$ be an $R$-subalgebra of $K$. Then the conductor ideal of the image of $R$ in $T$ (the smallest $R$-subalgebra of $T$) is nonzero if and only if $T$ is finitely generated as an $R$-module.","title":"Finiteness of an overring via its conductor"},{"id":"RingTheory/dedekind-domains/conductor-ideal/integral-closure-bot-conductor-ideal-ne","kind":"node","lean":"Subalgebra.integralClosure_bot_conductorIdeal_ne_bot_iff","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#integral-closure-bot-conductor-ideal-ne","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $R$ be a Noetherian integral domain with fraction field $K$, and let $\\overline{R}$ be the integral closure of $R$ in $K$, viewed as an $R$-algebra. The conductor ideal of the smallest $R$-subalgebra of $\\overline{R}$ (the image of $R$) is a nonzero ideal of $\\overline{R}$ if and only if $\\overline{R}$ is finitely generated as an $R$-module.","title":"Finiteness of the integral closure via the conductor"},{"id":"RingTheory/dedekind-domains/conductor-ideal/quotient-coprime-residue-equiv","kind":"node","lean":"Ideal.quotient_coprime_residue_equiv","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#quotient-coprime-residue-equiv","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $B$ be an algebra over $O$, where $O$ and $B$ are commutative rings, with structure map $\\varphi : O \\to B$. Let $\\mathfrak q$ be a maximal ideal of $B$ and let $\\mathfrak c$ be an ideal of $B$ with $\\mathfrak c \\not\\subseteq \\mathfrak q$ such that for every $x \\in \\mathfrak c$ and every $b \\in B$ there is $o \\in O$ with $\\varphi(o) = xb$. This is the ring isomorphism\u2026","title":"Residue-ring isomorphism away from a denominator-clearing ideal"},{"id":"RingTheory/dedekind-domains/conductor-ideal/quotient-coprime-residue-equiv-apply-mk","kind":"node","lean":"Ideal.quotient_coprime_residue_equiv_apply_mk","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#quotient-coprime-residue-equiv-apply-mk","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $B$ be an algebra over $O$, where $O$ and $B$ are commutative rings, with structure map $\\varphi : O \\to B$. Let $\\mathfrak q$ be a maximal ideal of $B$ and let $\\mathfrak c$ be an ideal of $B$ with $\\mathfrak c \\not\\subseteq \\mathfrak q$ such that for every $x \\in \\mathfrak c$ and every $b \\in B$ there is $o \\in O$ with $\\varphi(o) = xb$. Then for every $o \\in O$, the residue-ring isomorphism\u2026","title":"The residue-ring isomorphism on residue classes"},{"id":"RingTheory/dedekind-domains/conductor-ideal/nat-card-quotient-coprime-residue-eq","kind":"node","lean":"Ideal.nat_card_quotient_coprime_residue_eq","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#nat-card-quotient-coprime-residue-eq","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $B$ be an algebra over $O$, where $O$ and $B$ are commutative rings, with structure map $\\varphi : O \\to B$. Let $\\mathfrak q$ be a maximal ideal of $B$ and let $\\mathfrak c$ be an ideal of $B$ with $\\mathfrak c \\not\\subseteq \\mathfrak q$ such that for every $x \\in \\mathfrak c$ and every $b \\in B$ there is $o \\in O$ with $\\varphi(o) = xb$. Then $O / \\varphi^{-1}(\\mathfrak q)$ and $B / \\mathfrak q$ have the same\u2026","title":"Equal cardinality of residue rings away from a denominator-clearing ideal"},{"id":"RingTheory/dedekind-domains/conductor-ideal/card-quot-coprime-residue-eq","kind":"node","lean":"Ideal.cardQuot_coprime_residue_eq","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#card-quot-coprime-residue-eq","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $B$ be an algebra over $O$, where $O$ and $B$ are commutative rings, with structure map $\\varphi : O \\to B$. Let $\\mathfrak q$ be a maximal ideal of $B$ and let $\\mathfrak c$ be an ideal of $B$ with $\\mathfrak c \\not\\subseteq \\mathfrak q$ such that for every $x \\in \\mathfrak c$ and every $b \\in B$ there is $o \\in O$ with $\\varphi(o) = xb$. Then the index of $\\varphi^{-1}(\\mathfrak q)$ in $O$ equals the index of\u2026","title":"Equal ideal indices away from a denominator-clearing ideal"},{"id":"RingTheory/dedekind-domains/conductor-ideal/is-prime-map-of-coprime-clear-denom","kind":"node","lean":"Ideal.isPrime_map_of_coprime_clearDenom","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#is-prime-map-of-coprime-clear-denom","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $O$ and $B$ be commutative rings, let $B$ be an $O$-algebra whose structure map $f \\colon O \\to B$ is injective, let $\\mathfrak p$ be a prime ideal of $O$, and let $\\mathfrak c$ be an ideal of $O$ with $\\mathfrak c + \\mathfrak p = O$ such that $f(x)\\,b$ lies in the image of $f$ for every $x \\in \\mathfrak c$ and every $b \\in B$. Then the ideal $\\mathfrak p B$ of $B$ generated by $f(\\mathfrak p)$ is prime.","title":"Prime ideals stay prime under denominator clearing"},{"id":"RingTheory/dedekind-domains/conductor-ideal/is-maximal-map-of-coprime-clear-denom","kind":"node","lean":"Ideal.isMaximal_map_of_coprime_clearDenom","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#is-maximal-map-of-coprime-clear-denom","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Let $O$ and $B$ be commutative rings, let $B$ be an $O$-algebra whose structure map $f \\colon O \\to B$ is injective, let $\\mathfrak p$ be a maximal ideal of $O$, and let $\\mathfrak c$ be an ideal of $O$ with $\\mathfrak c + \\mathfrak p = O$ such that $f(x)\\,b$ lies in the image of $f$ for every $x \\in \\mathfrak c$ and every $b \\in B$. Then the ideal $\\mathfrak p B$ of $B$ generated by $f(\\mathfrak p)$ is maximal.","title":"Maximal ideals stay maximal under denominator clearing"},{"id":"RingTheory/dedekind-domains/conductor-ideal/is-maximal-map-of-dimension-leone","kind":"node","lean":"Ideal.isMaximal_map_of_dimensionLEOne","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#is-maximal-map-of-dimension-leone","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"In dimension at most one, nonzero primes extend to maximal ideals under denominator clearing. Let $O$ and $B$ be commutative rings such that every nonzero prime ideal of $O$ is maximal, let $B$ be an $O$-algebra whose structure map $f \\colon O \\to B$ is injective, let $\\mathfrak p$ be a nonzero prime ideal of $O$, and let $\\mathfrak c$ be an ideal of $O$ with $\\mathfrak c + \\mathfrak p = O$ such that $f(x)\\,b$ lies\u2026","title":"In dimension at most one, nonzero primes extend to maximal ideals under denominator clearing"},{"id":"RingTheory/dedekind-domains/conductor-ideal/comap-map-eq-of-not-le-conductor","kind":"node","lean":"Ideal.comap_map_eq_of_not_le_conductor","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#comap-map-eq-of-not-le-conductor","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Contraction after extension is the identity on primes not containing `\ud835\udd20`.","title":"Contraction after extension fixes primes away from the conductor"},{"id":"RingTheory/dedekind-domains/conductor-ideal/comap-ne-bot-of-ne-bot","kind":"node","lean":"Ideal.comap_ne_bot_of_ne_bot","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#comap-ne-bot-of-ne-bot","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Contraction of a nonzero ideal is nonzero when `\ud835\udd20` is nonzero: a nonzero element of `\ud835\udd20` clears a nonzero element of `\ud835\udd2e` back to a nonzero element of the contraction.","title":"Contraction of a nonzero ideal is nonzero"},{"id":"RingTheory/dedekind-domains/conductor-ideal/map-comap-eq-of-not-le-map-conductor","kind":"node","lean":"Ideal.map_comap_eq_of_not_le_map_conductor","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#map-comap-eq-of-not-le-map-conductor","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Extension after contraction is the identity on primes of `B` not containing `\ud835\udd20.map f`, using that `B` has Krull dimension at most one.","title":"Extension after contraction fixes primes away from the conductor"},{"id":"RingTheory/dedekind-domains/conductor-ideal/prime-equiv-of-not-le-conductor","kind":"node","lean":"Ideal.primeEquivOfNotLeConductor","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#prime-equiv-of-not-le-conductor","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Contraction and extension give an equivalence between all primes of `B` not containing `\ud835\udd20.map (algebraMap O B)` and all primes of `O` not containing `\ud835\udd20`, including the zero prime on both sides. This restores the full spectra in ATLAS `NumberTheoryI` Corollary 6.27.","title":"Bijection of primes away from the conductor"},{"id":"RingTheory/dedekind-domains/conductor-ideal/prime-equiv-of-coprime-conductor","kind":"node","lean":"Ideal.primeEquivOfCoprimeConductor","parent":3883,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/conductor-ideal/index.html#prime-equiv-of-coprime-conductor","scope":"RingTheory/dedekind-domains/conductor-ideal","status":"fully_proved","summary":"Contraction and extension give an equivalence between the nonzero primes of `B` not containing `\ud835\udd20.map (algebraMap O B)` and the nonzero primes of `O` not containing `\ud835\udd20`. This is the restriction of `Ideal.primeEquivOfNotLeConductor` to nonzero prime ideals.","title":"Bijection of nonzero primes away from the conductor"},{"data_url":"scopes/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal.json","id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","title":"Fractional ideals prime to an ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/non-zero-divisors-le-comap-algebra-map-a","kind":"node","lean":"FractionalIdeal.nonZeroDivisors_le_comap_algebraMap_atPrime","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#non-zero-divisors-le-comap-algebra-map-a","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be a commutative ring and $\\mathfrak p$ a prime ideal of $A$. Every non-zero-divisor of $A$ maps to a non-zero-divisor of the localization $A_{\\mathfrak p}$.","title":"Non-zero-divisors stay non-zero-divisors in a localization at a prime"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/is-prime-to","kind":"node","lean":"FractionalIdeal.IsPrimeTo","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#is-prime-to","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $I$ be a fractional ideal of $A$ in $K$ and let $J$ be an ideal of $A$. Then $I$ is prime to $J$ if for every prime ideal $\\mathfrak p$ of $A$ with $J \\subseteq \\mathfrak p$, the extension $I A_{\\mathfrak p}$ of $I$ to the localization $A_{\\mathfrak p}$, a fractional ideal of $A_{\\mathfrak p}$ in $K$, is the unit fractional ideal $A_{\\mathfrak p}$.","title":"Fractional ideals prime to an ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/is-prime-to-one","kind":"node","lean":"FractionalIdeal.isPrimeTo_one","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#is-prime-to-one","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$. For every ideal $J$ of $A$, the unit fractional ideal $A$ of $A$ in $K$ is prime to $J$.","title":"The unit fractional ideal is prime to every ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/mul","kind":"node","lean":"FractionalIdeal.IsPrimeTo.mul","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#mul","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$ and let $J$ be an ideal of $A$. If fractional ideals $I$ and $I'$ of $A$ in $K$ are both prime to $J$, then their product $I I'$ is prime to $J$.","title":"Products of fractional ideals prime to an ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/inv-unit","kind":"node","lean":"FractionalIdeal.IsPrimeTo.inv_unit","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#inv-unit","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$ and let $J$ be an ideal of $A$. If an invertible fractional ideal $I$ of $A$ in $K$ is prime to $J$, then its inverse $I^{-1}$ is prime to $J$.","title":"Inverses of invertible fractional ideals prime to an ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/units-prime-to","kind":"node","lean":"FractionalIdeal.unitsPrimeTo","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#units-prime-to","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$ and let $J$ be an ideal of $A$. The invertible fractional ideals of $A$ in $K$ that are prime to $J$ form a subgroup of the group of invertible fractional ideals of $A$ in $K$.","title":"Invertible fractional ideals prime to an ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/mem-units-prime-to-iff","kind":"node","lean":"FractionalIdeal.mem_unitsPrimeTo_iff","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#mem-units-prime-to-iff","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$ and let $J$ be an ideal of $A$. An invertible fractional ideal $I$ of $A$ in $K$ lies in the subgroup of invertible fractional ideals prime to $J$ if and only if $I$ is prime to $J$.","title":"Membership in the subgroup of units prime to an ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/mono","kind":"node","lean":"FractionalIdeal.IsPrimeTo.mono","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#mono","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $I$ be a fractional ideal of $A$ in $K$ and let $J \\subseteq J'$ be ideals of $A$. If $I$ is prime to $J$, then $I$ is prime to $J'$.","title":"Prime to an ideal implies prime to every larger ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/units-prime-to-mono","kind":"node","lean":"FractionalIdeal.unitsPrimeTo_mono","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#units-prime-to-mono","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$ and let $J \\subseteq J'$ be ideals of $A$. Then the subgroup of invertible fractional ideals of $A$ in $K$ prime to $J$ is contained in the subgroup of invertible fractional ideals of $A$ in $K$ prime to $J'$.","title":"Monotonicity of the subgroups of units prime to an ideal"},{"id":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/units-prime-to-top","kind":"node","lean":"FractionalIdeal.unitsPrimeTo_top","parent":3909,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal/index.html#units-prime-to-top","scope":"RingTheory/dedekind-domains/fractional-ideals-prime-to-an-ideal","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$. The subgroup of invertible fractional ideals of $A$ in $K$ prime to the unit ideal $A$ is the whole group of invertible fractional ideals of $A$ in $K$.","title":"Every invertible fractional ideal is prime to the unit ideal"},{"data_url":"scopes/RingTheory/dedekind-domains/ideal-class-group.json","id":"RingTheory/dedekind-domains/ideal-class-group","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/ideal-class-group/index.html","scope":"RingTheory/dedekind-domains/ideal-class-group","status":"fully_proved","title":"Ideal class group"},{"id":"RingTheory/dedekind-domains/ideal-class-group/exists-is-coprime-mul-is-principal","kind":"node","lean":"Ideal.exists_isCoprime_mul_isPrincipal","parent":3920,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ideal-class-group/index.html#exists-is-coprime-mul-is-principal","scope":"RingTheory/dedekind-domains/ideal-class-group","status":"fully_proved","summary":"Let $A$ be a Dedekind domain and let $I$ and $\\mathfrak{a}$ be nonzero ideals of $A$. Then there is an ideal $J$ of $A$ with $J + \\mathfrak{a} = A$ such that the product $IJ$ is a principal ideal.","title":"Making an ideal principal with a factor coprime to a given ideal"},{"id":"RingTheory/dedekind-domains/ideal-class-group/exists-mk0-eq-and-is-coprime","kind":"node","lean":"ClassGroup.exists_mk0_eq_and_isCoprime","parent":3920,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ideal-class-group/index.html#exists-mk0-eq-and-is-coprime","scope":"RingTheory/dedekind-domains/ideal-class-group","status":"fully_proved","summary":"Let $A$ be a Dedekind domain and let $\\mathfrak{a}$ be a nonzero ideal of $A$. Then every class $c$ in the ideal class group of $A$ is the class of some nonzero ideal $K$ of $A$ with $K + \\mathfrak{a} = A$.","title":"Ideal classes have representatives coprime to a given ideal"},{"id":"RingTheory/dedekind-domains/ideal-class-group/ideals-similar","kind":"node","lean":"IdealFactorization.IdealsSimilar","parent":3920,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ideal-class-group/index.html#ideals-similar","scope":"RingTheory/dedekind-domains/ideal-class-group","status":"fully_proved","summary":"Let $R$ be a Dedekind domain. Two nonzero ideals $I$ and $J$ of $R$ are similar when their classes in the ideal class group of $R$ are equal.","title":"Similar ideals"},{"id":"RingTheory/dedekind-domains/ideal-class-group/unique-factorization-monoid-iff-is-princ","kind":"node","lean":"IsDedekindDomain.uniqueFactorizationMonoid_iff_isPrincipalIdealRing","parent":3920,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ideal-class-group/index.html#unique-factorization-monoid-iff-is-princ","scope":"RingTheory/dedekind-domains/ideal-class-group","status":"fully_proved","summary":"A Dedekind domain $R$ is a unique factorization domain if and only if it is a principal ideal domain.","title":"Dedekind domains: UFD iff PID"},{"id":"RingTheory/dedekind-domains/ideal-class-group/is-principal-ideal-ring-iff-subsingleton","kind":"node","lean":"IsDedekindDomain.isPrincipalIdealRing_iff_subsingleton_classGroup","parent":3920,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ideal-class-group/index.html#is-principal-ideal-ring-iff-subsingleton","scope":"RingTheory/dedekind-domains/ideal-class-group","status":"fully_proved","summary":"A Dedekind domain $R$ is a principal ideal domain if and only if its ideal class group is trivial.","title":"Dedekind domains: PID iff trivial class group"},{"id":"RingTheory/dedekind-domains/ideal-class-group/unique-factorization-monoid-iff-subsingl","kind":"node","lean":"IsDedekindDomain.uniqueFactorizationMonoid_iff_subsingleton_classGroup","parent":3920,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ideal-class-group/index.html#unique-factorization-monoid-iff-subsingl","scope":"RingTheory/dedekind-domains/ideal-class-group","status":"fully_proved","summary":"A Dedekind domain $R$ is a unique factorization domain if and only if its ideal class group is trivial.","title":"Dedekind domains: UFD iff trivial class group"},{"data_url":"scopes/RingTheory/dedekind-domains/invertible-fractional-ideals.json","id":"RingTheory/dedekind-domains/invertible-fractional-ideals","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","title":"Invertible fractional ideals"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/dual-submodule-mul-eq-div","kind":"node","lean":"dualSubmodule_mul_eq_div","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#dual-submodule-mul-eq-div","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be a commutative ring and $K$ a field that is an $A$-algebra, and write $1$ for the $A$-submodule $A \\cdot 1$ of $K$. For every $A$-submodule $M$ of $K$, the dual submodule of $M$ with respect to the multiplication form $(x, y) \\mapsto xy$ on $K$ equals the quotient $1 / M$, the set of $x \\in K$ with $xy \\in A \\cdot 1$ for all $y \\in M$.","title":"The multiplication dual is a colon quotient"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/exists-mul-rep","kind":"node","lean":"Submodule.exists_mul_rep","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#exists-mul-rep","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$ and let $M$ be a nonzero $A$-submodule of $K$. For every $A$-linear map $f \\colon M \\to A$ there is $x \\in K$ such that the image of $f(m)$ in $K$ equals $xm$ for every $m \\in M$.","title":"Functionals on a submodule of the fraction field are multiplications"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/mul-nondegenerate","kind":"node","lean":"LinearMap.BilinForm.mul_nondegenerate","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#mul-nondegenerate","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"For every field $K$, the $K$-bilinear form $(x, y) \\mapsto xy$ on $K$ is nondegenerate: if $xy = 0$ for all $y \\in K$ then $x = 0$, and if $xy = 0$ for all $x \\in K$ then $y = 0$.","title":"Multiplication is a nondegenerate bilinear form"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/span-coe-eq-top","kind":"node","lean":"Submodule.span_coe_eq_top","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#span-coe-eq-top","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be a commutative ring and $K$ a field that is an $A$-algebra. Then every nonzero $A$-submodule $M$ of $K$ spans $K$ as a $K$-vector space.","title":"A nonzero submodule spans the field"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/mul-dual-submodule-to-dual-surjective","kind":"node","lean":"Submodule.mul_dualSubmoduleToDual_surjective","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#mul-dual-submodule-to-dual-surjective","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, identify $A$ with its image in $K$, and let $M$ be a nonzero $A$-submodule of $K$. Let $M^\\vee = \\{x \\in K : xy \\in A \\text{ for all } y \\in M\\}$ be the dual submodule of $M$ for the multiplication form $(x, y) \\mapsto xy$ on $K$. Then the $A$-linear map $M^\\vee \\to \\operatorname{Hom}_A(M, A)$ sending $x$ to the functional $y \\mapsto xy$ on $M$ is surjective.","title":"Every functional comes from the multiplication dual"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/dual-equiv-mul-dual-submodule","kind":"node","lean":"Submodule.dualEquivMulDualSubmodule","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#dual-equiv-mul-dual-submodule","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, identify $A$ with its image in $K$, and let $M$ be a nonzero $A$-submodule of $K$. Let $M^\\vee = \\{x \\in K : xy \\in A \\text{ for all } y \\in M\\}$ be the dual submodule of $M$ for the multiplication form $(x, y) \\mapsto xy$ on $K$. This is the $A$-linear isomorphism $\\operatorname{Hom}_A(M, A) \\to M^\\vee$ inverse to the map sending $x \\in M^\\vee$ to the\u2026","title":"Dual of a nonzero submodule of the fraction field"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/dual-equiv-mul-dual-submodule-spec","kind":"node","lean":"Submodule.dualEquivMulDualSubmodule_spec","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#dual-equiv-mul-dual-submodule-spec","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $M$ be a nonzero $A$-submodule of $K$, and let $\\varphi \\colon \\operatorname{Hom}_A(M, A) \\to M^\\vee$ be the isomorphism onto the dual submodule $M^\\vee$ of $M$ for the multiplication form $(x, y) \\mapsto xy$ on $K$. Then for every $f \\in \\operatorname{Hom}_A(M, A)$ and every $m \\in M$, the image of $f(m)$ in $K$ equals $\\varphi(f) \\, m$.","title":"Formula for the isomorphism onto the multiplication dual"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/coe-inv-eq-mul-dual-submodule","kind":"node","lean":"FractionalIdeal.coe_inv_eq_mul_dualSubmodule","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#coe-inv-eq-mul-dual-submodule","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, and let $I$ be a nonzero fractional ideal of $A$ in $K$. Then $I^{-1}$, viewed as an $A$-submodule of $K$, equals the dual submodule of $I$ with respect to the multiplication form $(x, y) \\mapsto xy$ on $K$, namely the set $\\{x \\in K : xy \\in A \\text{ for all } y \\in I\\}$, where $A$ is identified with its image in $K$.","title":"The inverse of a nonzero fractional ideal is its multiplication dual"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/dual-equiv-inv","kind":"node","lean":"FractionalIdeal.dualEquivInv","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#dual-equiv-inv","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$ and let $I$ be a nonzero fractional ideal of $A$ in $K$. This is the $A$-linear isomorphism $\\operatorname{Hom}_A(I, A) \\to I^{-1}$, with $I^{-1}$ viewed as an $A$-submodule of $K$, sending $f$ to the unique $x \\in I^{-1}$ such that $f(y) = xy$ in $K$ for all $y \\in I$ (where $A$ is identified with its image in $K$).","title":"Dual of a nonzero fractional ideal is its inverse"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/double-dual-equiv-of-is-unit","kind":"node","lean":"FractionalIdeal.doubleDualEquivOfIsUnit","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#double-dual-equiv-of-is-unit","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$ and let $I$ be an invertible fractional ideal of $A$ in $K$. This is the $A$-linear isomorphism $\\operatorname{Hom}_A(\\operatorname{Hom}_A(I, A), A) \\to I$ inverse to the canonical evaluation map $I \\to \\operatorname{Hom}_A(\\operatorname{Hom}_A(I, A), A)$, $x \\mapsto (f \\mapsto f(x))$.","title":"Double dual of an invertible fractional ideal"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/localized-div-of-fg","kind":"node","lean":"Submodule.localized'_div_of_fg","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#localized-div-of-fg","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $R$, $S$ and $K$ be commutative rings with algebra maps $R \\to S \\to K$ forming a scalar tower, let $p$ be a submonoid of $R$ such that $S$ is the localization of $R$ at $p$, and suppose the identity map of $K$ is a localization of the $R$-module $K$ at $p$. For an $R$-submodule $M$ of $K$ write $M_p = \\{x/s : x \\in M,\\ s \\in p\\}$ for its localization, an $S$-submodule of $K$, where $x/s$ is the element\u2026","title":"Localization commutes with division by a finitely generated submodule"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/coe-extended-hom-eq-localized","kind":"node","lean":"FractionalIdeal.coe_extendedHom_eq_localized'","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#coe-extended-hom-eq-localized","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ and $B$ be integral domains with $B$ a torsion-free $A$-algebra, and let $K$ be a field that is a fraction field of both $A$ and $B$, compatibly with $A \\to B \\to K$. Let $S$ be a submonoid of $A$ such that $B$ is the localization of $A$ at $S$ and the identity map of $K$ is a localization of the $A$-module $K$ at $S$. Then for every fractional ideal $I$ of $A$ in $K$, the $B$-submodule of $K$ underlying\u2026","title":"Extension to a localization is localization"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/extended-hom-div-of-fg","kind":"node","lean":"FractionalIdeal.extendedHom_div_of_fg","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#extended-hom-div-of-fg","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ and $B$ be integral domains with $B$ a torsion-free $A$-algebra, and let $K$ be a field that is a fraction field of both $A$ and $B$, compatibly with $A \\to B \\to K$. Let $S$ be a submonoid of $A$ such that $B$ is the localization of $A$ at $S$. Let $I$ and $J$ be fractional ideals of $A$ in $K$ with $J \\ne 0$ and $J$ finitely generated as an $A$-module, and write $IB$ for the extension of $I$ to a\u2026","title":"Extension to a localization commutes with division by a finitely generated ideal"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/extended-hom-div","kind":"node","lean":"FractionalIdeal.extendedHom_div","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#extended-hom-div","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be a Noetherian integral domain, $B$ an integral domain that is a torsion-free $A$-algebra, and $K$ a field that is a fraction field of both $A$ and $B$, compatibly with $A \\to B \\to K$. Let $S$ be a submonoid of $A$ such that $B$ is the localization of $A$ at $S$. Let $I$ and $J$ be fractional ideals of $A$ in $K$ with $J \\ne 0$, and write $IB$ for the extension of $I$ to a fractional ideal of $B$. Then\u2026","title":"Over a Noetherian domain, extension to a localization commutes with division"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/fractional-ideal-is-unit-iff-is-principa","kind":"node","lean":"fractionalIdeal_isUnit_iff_isPrincipal","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#fractional-ideal-is-unit-iff-is-principa","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be a Noetherian local integral domain with fraction field $K$, and let $I$ be a nonzero fractional ideal of $A$ in $K$. Then $I$ is invertible if and only if $I$ is principal.","title":"Invertible fractional ideals of a Noetherian local domain are principal"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/localize-at-prime","kind":"node","lean":"FractionalIdeal.localizeAtPrime","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#localize-at-prime","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $\\mathfrak p$ be a prime ideal of $A$, and write $A_{\\mathfrak p}$ for the localization of $A$ at $\\mathfrak p$. The localization $I_{\\mathfrak p}$ of a fractional ideal $I$ of $A$ in $K$ is the fractional ideal of $A_{\\mathfrak p}$ in $K$ obtained by extending $I$ along $A \\to A_{\\mathfrak p}$, that is, the $A_{\\mathfrak p}$-submodule of $K$ generated by\u2026","title":"Localization of a fractional ideal at a prime"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/is-locally-principal","kind":"node","lean":"FractionalIdeal.IsLocallyPrincipal","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#is-locally-principal","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$. For a maximal ideal $\\mathfrak m$ of $A$, write $A_{\\mathfrak m}$ for the localization of $A$ at $\\mathfrak m$ and $I_{\\mathfrak m}$ for the localization of a fractional ideal $I$ of $A$ in $K$, the $A_{\\mathfrak m}$-submodule of $K$ generated by $I$. A fractional ideal $I$ of $A$ in $K$ is locally principal if for every maximal ideal $\\mathfrak m$ of $A$ the\u2026","title":"Locally principal fractional ideal"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/localize-at-prime-inv","kind":"node","lean":"FractionalIdeal.localizeAtPrime_inv","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#localize-at-prime-inv","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Localization commutes with inversion of fractional ideals.","title":"Localization commutes with inverting fractional ideals"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/inv-localize-at-prime-le","kind":"node","lean":"FractionalIdeal.inv_localizeAtPrime_le","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#inv-localize-at-prime-le","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"The inverse/localization inequality follows from equality.","title":"Inverse of a localized fractional ideal is bounded by localization"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/is-unit-iff-is-unit-localize-at-prime","kind":"node","lean":"FractionalIdeal.isUnit_iff_isUnit_localizeAtPrime","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#is-unit-iff-is-unit-localize-at-prime","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Theorem 3.2 (maximal-ideal form). A fractional ideal of a Noetherian domain is invertible iff its localization at every maximal ideal is invertible.","title":"Theorem 3.2"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/is-unit-iff-is-unit-localize-at-prime-al","kind":"node","lean":"FractionalIdeal.isUnit_iff_isUnit_localizeAtPrime_all_primes","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#is-unit-iff-is-unit-localize-at-prime-al","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"Theorem 3.2 (prime-ideal form). A fractional ideal of a Noetherian domain is invertible iff its localization at every prime ideal is invertible.","title":"Theorem 3.2"},{"id":"RingTheory/dedekind-domains/invertible-fractional-ideals/is-unit-iff-is-locally-principal","kind":"node","lean":"FractionalIdeal.isUnit_iff_isLocallyPrincipal","parent":3927,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/invertible-fractional-ideals/index.html#is-unit-iff-is-locally-principal","scope":"RingTheory/dedekind-domains/invertible-fractional-ideals","status":"fully_proved","summary":"A nonzero fractional ideal of a Noetherian domain is invertible iff it is locally principal.","title":"Corollary 3.5"},{"data_url":"scopes/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals.json","id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","title":"Prime factorization of fractional ideals"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/units-support-finset","kind":"node","lean":"FractionalIdeal.unitsSupportFinset","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#units-support-finset","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. For a unit fractional ideal $I$ of $R$ in $K$, this is the finite set of height-one primes $v$ of $R$ at which the $v$-adic valuation of $I$ is nonzero.","title":"Support of the prime exponents of a fractional ideal"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/units-to-finsupp","kind":"node","lean":"FractionalIdeal.unitsToFinsupp","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#units-to-finsupp","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, and let $I$ be a unit of the monoid of fractional ideals of $R$ in $K$. The prime exponents of $I$ form the finitely supported function from the height-one primes of $R$ to $\\mathbb{Z}$ that sends each height-one prime $v$ to the exponent $\\operatorname{ord}_v(I)$ of $v$ in the factorization $I = \\prod_v v^{\\operatorname{ord}_v(I)}$; its support is the finite\u2026","title":"Prime exponents of an invertible fractional ideal"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/finsupp-prod-ne-zero","kind":"node","lean":"FractionalIdeal.finsuppProd_ne_zero","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#finsupp-prod-ne-zero","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. For every finitely supported function $e$ from the height-one primes of $R$ to $\\mathbb{Z}$, the fractional ideal $\\prod_v v^{e(v)}$ of $R$ in $K$, the product over the primes $v$ with $e(v) \\neq 0$, is nonzero.","title":"Products of prime powers are nonzero"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/finsupp-to-units","kind":"node","lean":"FractionalIdeal.finsuppToUnits","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#finsupp-to-units","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. For a finitely supported function $e$ from the height-one primes of $R$ to $\\mathbb{Z}$, this is the fractional ideal $\\prod_v v^{e(v)}$ of $R$ in $K$, the product over the primes $v$ with $e(v) \\neq 0$, regarded as a unit of the monoid of fractional ideals (it is nonzero, hence invertible).","title":"Fractional ideal with prescribed prime exponents"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/count-finsupp-to-units","kind":"node","lean":"FractionalIdeal.count_finsuppToUnits","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#count-finsupp-to-units","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. For every finitely supported function $e$ from the height-one primes of $R$ to $\\mathbb{Z}$ and every height-one prime $v$ of $R$, the $v$-adic valuation of the fractional ideal $\\prod_w w^{e(w)}$, the product over the primes $w$ with $e(w) \\neq 0$, equals $e(v)$.","title":"Valuations of a product of prime powers"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/to-units-left-inv","kind":"node","lean":"FractionalIdeal.toUnits_left_inv","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#to-units-left-inv","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, and for a unit fractional ideal $I$ of $R$ in $K$ and a height-one prime $v$ of $R$ write $\\operatorname{val}_v(I)$ for the $v$-adic valuation of $I$. Every unit fractional ideal $I$ of $R$ in $K$ equals $\\prod_v v^{\\operatorname{val}_v(I)}$, the product over the height-one primes $v$ of $R$ with $\\operatorname{val}_v(I) \\neq 0$.","title":"Reconstructing a fractional ideal from its prime exponents"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/to-units-right-inv","kind":"node","lean":"FractionalIdeal.toUnits_right_inv","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#to-units-right-inv","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, and let $e$ be a finitely supported function from the height-one primes of $R$ to $\\mathbb{Z}$. The prime-exponent function of the fractional ideal $\\prod_w w^{e(w)}$, the product over the primes $w$ with $e(w) \\neq 0$, is $e$: its $v$-adic valuation is $e(v)$ for every height-one prime $v$ of $R$.","title":"Recovering prime exponents from a product of prime powers"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/units-to-finsupp-mul","kind":"node","lean":"FractionalIdeal.unitsToFinsupp_mul","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#units-to-finsupp-mul","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, and for a unit fractional ideal $I$ of $R$ in $K$ and a height-one prime $v$ of $R$ write $\\operatorname{val}_v(I)$ for the $v$-adic valuation of $I$. For all unit fractional ideals $I$ and $J$ of $R$ in $K$ and every height-one prime $v$ of $R$, $\\operatorname{val}_v(IJ) = \\operatorname{val}_v(I) + \\operatorname{val}_v(J)$.","title":"Prime exponents are additive"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/factorization-mul-equiv","kind":"node","lean":"FractionalIdeal.factorizationMulEquiv","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#factorization-mul-equiv","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, and for a unit fractional ideal $I$ of $R$ in $K$ and a height-one prime $v$ of $R$ write $\\operatorname{val}_v(I)$ for the $v$-adic valuation of $I$. The factorization isomorphism is the group isomorphism from the unit group of the fractional ideals of $R$ in $K$ to the additive group of finitely supported functions from the height-one primes of $R$ to\u2026","title":"Prime factorization isomorphism for fractional ideals"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/factorization-mul-equiv-prime-self","kind":"node","lean":"FractionalIdeal.factorizationMulEquiv_prime_self","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#factorization-mul-equiv-prime-self","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$ and let $w$ be a height-one prime of $R$. The factorization isomorphism sends the fractional ideal $w$, as a unit fractional ideal of $R$ in $K$, to the function on height-one primes of $R$ that is $1$ at $w$ and $0$ elsewhere.","title":"Exponent function of a prime ideal"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/symm-single-zpow","kind":"node","lean":"FractionalIdeal.symm_single_zpow","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#symm-single-zpow","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, let $w$ be a height-one prime of $R$ and let $n \\in \\mathbb{Z}$. The inverse of the factorization isomorphism sends the function on height-one primes of $R$ that is $n$ at $w$ and $0$ elsewhere to the fractional ideal $w^n$.","title":"Inverse factorization of a single prime power"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/finite-set-of-is-prime-and-mem","kind":"node","lean":"Ideal.finite_setOf_isPrime_and_mem","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#finite-set-of-is-prime-and-mem","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"In a Dedekind domain $A$, every nonzero element $a$ lies in only finitely many prime ideals of $A$.","title":"Finitely many primes contain a nonzero element"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/finite-set-of-is-prime-and-le","kind":"node","lean":"Ideal.finite_setOf_isPrime_and_le","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#finite-set-of-is-prime-and-le","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"In a Dedekind domain $A$, every nonzero ideal $I$ is contained in only finitely many prime ideals of $A$.","title":"Finitely many primes contain a nonzero ideal"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/count-normalized-factors-eq-zero-iff","kind":"node","lean":"Ideal.count_normalizedFactors_eq_zero_iff","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#count-normalized-factors-eq-zero-iff","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $A$ be a Dedekind domain, let $\\mathfrak p$ be a prime ideal of $A$ and let $I$ be a nonzero ideal of $A$. Then the multiplicity of $\\mathfrak p$ in the factorization of $I$ into prime ideals is zero if and only if $I \\not\\subseteq \\mathfrak p$.","title":"Vanishing multiplicity of a prime factor"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/count-eq-zero-for-all-but-finitely-many","kind":"node","lean":"FractionalIdeal.count_eq_zero_for_all_but_finitely_many","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#count-eq-zero-for-all-but-finitely-many","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with field of fractions $K$, and let $I$ be a fractional ideal of $A$ in $K$. For a nonzero prime ideal $\\mathfrak p$ of $A$, let $v_{\\mathfrak p}(I) \\in \\mathbb Z$ be the exponent of $\\mathfrak p$ in the factorization of $I$ into prime ideals, with $v_{\\mathfrak p}(0) = 0$. Then $v_{\\mathfrak p}(I) \\neq 0$ for only finitely many nonzero prime ideals $\\mathfrak p$ of $A$.","title":"Fractional ideals have finite support"},{"id":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/emultiplicity-maximal-ideal-map-eq","kind":"node","lean":"IsLocalization.AtPrime.emultiplicity_maximalIdeal_map_eq","parent":3950,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals/index.html#emultiplicity-maximal-ideal-map-eq","scope":"RingTheory/dedekind-domains/prime-factorization-of-fractional-ideals","status":"fully_proved","summary":"Let $R$ be a Dedekind domain, $\\mathfrak p$ a maximal ideal of $R$, and $R_{\\mathfrak p}$ a local ring that is a localization of $R$ at $\\mathfrak p$, with maximal ideal $\\mathfrak m$. Then for every ideal $I$ of $R$, the multiplicity of $\\mathfrak m$ in the extended ideal $I R_{\\mathfrak p}$ equals the multiplicity of $\\mathfrak p$ in $I$. Here the multiplicity of an ideal $J$ in an ideal $K$ is the largest $n$\u2026","title":"Ideal multiplicity is unchanged by localization at a maximal ideal"},{"data_url":"scopes/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals.json","id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","kind":"scope","navigation":"module","parent":3851,"result":false,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","title":"Ring automorphisms acting on ideals"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/ring-equiv-of-ring-equiv-hom","kind":"node","lean":"FractionalIdeal.ringEquivOfRingEquivHom","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#ring-equiv-of-ring-equiv-hom","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let $R$ be a commutative domain and $K$ a field of fractions of $R$. For a ring automorphism $\\sigma$ of $R$, let $\\tilde\\sigma$ be the automorphism of $K$ extending $\\sigma$. Sending $\\sigma$ to the map $I \\mapsto \\tilde\\sigma(I)$ on fractional ideals of $R$ in $K$ defines a group homomorphism from the group of ring automorphisms of $R$ to the group of ring automorphisms of the semiring of fractional ideals of $R$\u2026","title":"Ring automorphisms act on fractional ideals"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/units-mul-distrib-mul-action","kind":"node","lean":"FractionalIdeal.unitsMulDistribMulAction","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#units-mul-distrib-mul-action","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let a group $G$ act on a commutative domain $R$ by ring automorphisms, let $K$ be a field of fractions of $R$, and let $G$ act on fractional ideals of $R$ in $K$ by $g \\cdot I = \\tilde\\sigma_g(I)$, where $\\tilde\\sigma_g$ is the automorphism of $K$ extending $r \\mapsto g \\cdot r$. This induces an action of $G$ on the group of invertible fractional ideals: $g$ sends an invertible fractional ideal $I$ to $g \\cdot I$,\u2026","title":"Action on invertible fractional ideals"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/units-smul-val","kind":"node","lean":"FractionalIdeal.units_smul_val","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#units-smul-val","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let a group $G$ act on a commutative domain $R$ by ring automorphisms, let $K$ be a field of fractions of $R$, and let $G$ act on fractional ideals of $R$ in $K$ by $g \\cdot I = \\tilde\\sigma_g(I)$, where $\\tilde\\sigma_g$ is the automorphism of $K$ extending $r \\mapsto g \\cdot r$. For every $g \\in G$ and every invertible fractional ideal $I$, the translate of $I$ under the induced action on invertible fractional\u2026","title":"Translating an invertible fractional ideal"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/smul-def","kind":"node","lean":"FractionalIdeal.smul_def","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#smul-def","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let a group $G$ act on a commutative domain $R$ by ring automorphisms and let $K$ be a field of fractions of $R$. For $g \\in G$ let $\\tilde\\sigma_g$ be the automorphism of $K$ extending the automorphism $r \\mapsto g \\cdot r$ of $R$. Then for every $g \\in G$ and every fractional ideal $I$ of $R$ in $K$, the induced action satisfies $g \\cdot I = \\tilde\\sigma_g(I)$, the image of $I$ under $\\tilde\\sigma_g$.","title":"The action on fractional ideals is by images"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/mem-smul-iff-exists","kind":"node","lean":"FractionalIdeal.mem_smul_iff_exists","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#mem-smul-iff-exists","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let a group $G$ act on a commutative domain $R$ by ring automorphisms, let $K$ be a field of fractions of $R$, and let $G$ act on fractional ideals of $R$ in $K$ by $g \\cdot I = \\tilde\\sigma_g(I)$, where $\\tilde\\sigma_g$ is the automorphism of $K$ extending $r \\mapsto g \\cdot r$. For every $g \\in G$, every fractional ideal $I$ and every $x \\in K$, we have $x \\in g \\cdot I$ if and only if $\\tilde\\sigma_g(y) = x$ for\u2026","title":"Elements of a translated fractional ideal are images"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/mem-smul-iff","kind":"node","lean":"FractionalIdeal.mem_smul_iff","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#mem-smul-iff","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let a group $G$ act on a commutative domain $R$ by ring automorphisms, let $K$ be a field of fractions of $R$, and let $G$ act on fractional ideals of $R$ in $K$ by $g \\cdot I = \\tilde\\sigma_g(I)$, where $\\tilde\\sigma_g$ is the automorphism of $K$ extending $r \\mapsto g \\cdot r$. For every $g \\in G$, every fractional ideal $I$ and every $x \\in K$, we have $x \\in g \\cdot I$ if and only if\u2026","title":"Membership in a translated fractional ideal"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/smul-coe-ideal","kind":"node","lean":"FractionalIdeal.smul_coeIdeal","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#smul-coe-ideal","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let a group $G$ act on a commutative domain $R$ by ring automorphisms, let $K$ be a field of fractions of $R$, and let $G$ act on fractional ideals of $R$ in $K$ by $g \\cdot J = \\tilde\\sigma_g(J)$, where $\\tilde\\sigma_g$ is the automorphism of $K$ extending $r \\mapsto g \\cdot r$. For every $g \\in G$ and every ideal $I$ of $R$, the translate $g \\cdot I$ of $I$ viewed as a fractional ideal is the ideal\u2026","title":"Translating an integral ideal"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/smul-as-ideal","kind":"node","lean":"PrimeSpectrum.smul_asIdeal","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#smul-as-ideal","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let a group $G$ act on a commutative semiring $R$ by semiring automorphisms, and let $G$ act on the prime spectrum of $R$ by the induced action. For every $g \\in G$ and every point $\\mathfrak p$ of the prime spectrum of $R$, the prime ideal underlying the translated point $g \\cdot \\mathfrak p$ is the image of the ideal $\\mathfrak p$ under the automorphism of $R$ given by $g$.","title":"Translating a prime ideal by a group element"},{"id":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/smul-as-ideal-coe","kind":"node","lean":"PrimeSpectrum.smul_asIdeal_coe","parent":3967,"result":true,"route":"../roadmap/RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals/index.html#smul-as-ideal-coe","scope":"RingTheory/dedekind-domains/ring-automorphisms-acting-on-ideals","status":"fully_proved","summary":"Let a group $G$ act on an integral domain $R$ by ring automorphisms, let $K$ be a fraction field of $R$, and let $G$ act on the fractional ideals of $R$ in $K$ through the extension of each automorphism of $R$ to $K$. For $g \\in G$ and a prime ideal $\\mathfrak p$ of $R$, write $g \\cdot \\mathfrak p = \\{g \\cdot x : x \\in \\mathfrak p\\}$ for the image of $\\mathfrak p$ under $g$, again a prime ideal of $R$. Then for\u2026","title":"Viewing prime ideals as fractional ideals commutes with the group action"},{"data_url":"scopes/RingTheory/formal-power-series.json","id":"RingTheory/formal-power-series","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/formal-power-series/index.html","scope":"RingTheory/formal-power-series","status":"fully_proved","summary":"Rings of formal power series in one or several variables and Hahn series, with formal exponentials, derivatives, coefficient operators and Lagrange inversion.","title":"Formal power series"},{"id":"RingTheory/formal-power-series/taylor-formula","kind":"node","lean":"PowerSeries.taylorFormula","parent":3977,"result":true,"route":"../roadmap/RingTheory/formal-power-series/index.html#taylor-formula","scope":"RingTheory/formal-power-series","status":"fully_proved","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra and $f$ a formal power series over $R$, with $j$-th formal derivative $f^{(j)}$, and write $[X^i]h$ for the coefficient of $X^i$ in a power series $h$. In the ring of formal power series over $R$ in two variables $t$ and $u$,","title":"Taylor's formula for formal power series"},{"data_url":"scopes/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series.json","id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","kind":"scope","navigation":"module","parent":3977,"result":false,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","title":"Exponential of multivariate formal power series"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/exp-trunc","kind":"node","lean":"MvPowerSeries.expTrunc","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#exp-trunc","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The finite sum `\u2211 n < N, w\u207f/n!`, defined without a constant coefficient restriction.","title":"Truncated exponential of a multivariate power series"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/exp-of-constant-coeff-zero","kind":"node","lean":"MvPowerSeries.expOfConstantCoeffZero","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#exp-of-constant-coeff-zero","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The formal exponential of a series with zero constant coefficient, obtained by substituting it into Mathlib's canonical exponential power series.","title":"Exponential of a power series with zero constant coefficient"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/exp-of-constant-coeff-zero-eq-subst","kind":"node","lean":"MvPowerSeries.expOfConstantCoeffZero_eq_subst","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#exp-of-constant-coeff-zero-eq-subst","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The exponential API is the canonical univariate exponential substituted at `w`.","title":"Multivariate exponential as substitution into exp"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/coeff-exp-of-constant-coeff-zero","kind":"node","lean":"MvPowerSeries.coeff_expOfConstantCoeffZero","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#coeff-exp-of-constant-coeff-zero","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"A coefficient of the exponential is computed by any truncation extending past its degree. This is the exact finite-sum formula used by the source's degreewise constructor.","title":"Coefficients of the exponential via truncation"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/coeff-exp-of-constant-coeff-zero-degree","kind":"node","lean":"MvPowerSeries.coeff_expOfConstantCoeffZero_degree","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#coeff-exp-of-constant-coeff-zero-degree","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"In particular, the coefficient of degree `d.degree` uses exactly the terms `n \u2264 d.degree`.","title":"Coefficient of the exponential in a given degree"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/constant-coeff-exp-of-constant-coeff-zer","kind":"node","lean":"MvPowerSeries.constantCoeff_expOfConstantCoeffZero","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#constant-coeff-exp-of-constant-coeff-zer","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The constant coefficient of the formal exponential is one.","title":"Constant coefficient of the formal exponential is one"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/exp-of-constant-coeff-zero-zero","kind":"node","lean":"MvPowerSeries.expOfConstantCoeffZero_zero","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#exp-of-constant-coeff-zero-zero","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The exponential of the zero series is one.","title":"Exponential of the zero series is one"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/map-exp-of-constant-coeff-zero","kind":"node","lean":"MvPowerSeries.map_expOfConstantCoeffZero","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#map-exp-of-constant-coeff-zero","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"Applying a coefficient ring homomorphism commutes with the formal exponential.","title":"Coefficient ring maps commute with the formal exponential"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/exp-of-constant-coeff-zero-x","kind":"node","lean":"MvPowerSeries.expOfConstantCoeffZero_X","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#exp-of-constant-coeff-zero-x","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"Substituting a variable recovers the canonical exponential in that variable.","title":"Exponential of a variable"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/exp-of-constant-coeff-zero-add","kind":"node","lean":"MvPowerSeries.expOfConstantCoeffZero_add","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#exp-of-constant-coeff-zero-add","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The formal exponential carries sums of series with zero constant coefficient to products.","title":"Formal exponential turns sums into products"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/pderiv-exp-trunc","kind":"node","lean":"MvPowerSeries.pderiv_expTrunc","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#pderiv-exp-trunc","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The partial derivative of the finite exponential sum containing the terms `w ^ n / n!` for `n \u2264 N` is the derivative of its exponent times the sum containing those terms for `n < N`.","title":"Partial derivative of the truncated exponential"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/pderiv-exp-of-constant-coeff-zero","kind":"node","lean":"MvPowerSeries.pderiv_expOfConstantCoeffZero","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#pderiv-exp-of-constant-coeff-zero","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The partial derivative of the canonical exponential is the derivative of its exponent times the exponential.","title":"Partial derivative of the formal exponential"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/map-exp-trunc","kind":"node","lean":"MvPowerSeries.map_expTrunc","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#map-exp-trunc","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"A ring homomorphism between multivariate power-series rings commutes with finite exponential truncations.","title":"Ring maps commute with truncated exponentials"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/hom-exp-of-constant-coeff-zero","kind":"node","lean":"MvPowerSeries.hom_expOfConstantCoeffZero","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#hom-exp-of-constant-coeff-zero","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"A ring endomorphism preserving the total-degree coefficient filtration commutes with the formal exponential. The filtration premise itself implies that the image of a zero-constant series again has zero constant coefficient.","title":"Filtration-preserving endomorphisms commute with the formal exponential"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/coeff-rescale-map-of-weight","kind":"node","lean":"MvPowerSeries.coeff_rescale_map_of_weight","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#coeff-rescale-map-of-weight","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"Mapping rational coefficients and rescaling variable `i` by `\u03b5 ^ wt (Finsupp.single i 1)` multiplies coefficient `d` by `\u03b5 ^ wt d`.","title":"Coefficients of a weighted rescaling of a rational power series"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/eq-rescale-map-of-weight-iff","kind":"node","lean":"MvPowerSeries.eq_rescale_map_of_weight_iff","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#eq-rescale-map-of-weight-iff","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"A series has the weighted coefficients of `q` exactly when it is the canonical coefficient map of `q` followed by the corresponding rescaling.","title":"Characterization of weighted rescalings of a power series"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/exp-trunc-rescale-map-of-weight","kind":"node","lean":"MvPowerSeries.expTrunc_rescale_map_of_weight","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#exp-trunc-rescale-map-of-weight","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"Weighted coefficient mapping and rescaling commutes with every finite exponential truncation, without a constant coefficient restriction.","title":"Weighted rescaling commutes with truncated exponentials"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/coeff-exp-trunc-rescale-map-of-weight","kind":"node","lean":"MvPowerSeries.coeff_expTrunc_rescale_map_of_weight","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#coeff-exp-trunc-rescale-map-of-weight","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"The coefficient form of weighted finite exponential commutation. It preserves the source's degreewise expression for arbitrary input series.","title":"Coefficients of weighted rescaling of truncated exponentials"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/exp-of-constant-coeff-zero-rescale-map-o","kind":"node","lean":"MvPowerSeries.expOfConstantCoeffZero_rescale_map_of_weight","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#exp-of-constant-coeff-zero-rescale-map-o","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"On the zero-constant domain, weighted coefficient mapping and rescaling commutes with the lawful full formal exponential.","title":"Weighted rescaling commutes with the formal exponential"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/constant-coeff-eq-zero-of-weight","kind":"node","lean":"MvPowerSeries.constantCoeff_eq_zero_of_weight","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#constant-coeff-eq-zero-of-weight","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"Weighted coefficient equality transfers a zero constant coefficient from the rational source series to the target series.","title":"Weighted coefficient equality transfers zero constant coefficient"},{"id":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/coeff-exp-of-constant-coeff-zero-of-weig","kind":"node","lean":"MvPowerSeries.coeff_expOfConstantCoeffZero_of_weight","parent":3979,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series/index.html#coeff-exp-of-constant-coeff-zero-of-weig","scope":"RingTheory/formal-power-series/exponential-of-multivariate-formal-power-series","status":"fully_proved","summary":"If `w` has the coefficients of `q` scaled by an additive weight, then the same coefficient relation holds for their lawful formal exponentials. The coefficient hypothesis and `q`'s zero constant imply the required premise for `w`; no separate zero-constant assumption on `w` is needed.","title":"Weighted coefficient relation passes to formal exponentials"},{"data_url":"scopes/RingTheory/formal-power-series/exponentials-of-coefficient-operators.json","id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","kind":"scope","navigation":"module","parent":3977,"result":false,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","title":"Exponentials of coefficient operators"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/coefficient-operator","kind":"node","lean":"MvPowerSeries.coefficientOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#coefficient-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The operator `\u2211 i, X i * O i`, defined by finite coefficient sums.","title":"Coefficient operator on multivariate power series"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/coeff-coefficient-operator","kind":"node","lean":"MvPowerSeries.coeff_coefficientOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#coeff-coefficient-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The coefficient of the formal operator is a sum over the exponent support.","title":"Coefficient formula for a coefficient operator"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/add-one-le-order-coefficient-operator","kind":"node","lean":"MvPowerSeries.add_one_le_order_coefficientOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#add-one-le-order-coefficient-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"A coefficient operator raises every natural lower bound on order by one.","title":"Coefficient operators raise the order by one"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/add-le-order-iterate-coefficient-operato","kind":"node","lean":"MvPowerSeries.add_le_order_iterate_coefficientOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#add-le-order-iterate-coefficient-operato","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"Iteration raises every natural lower bound on order by the number of steps.","title":"Iterated coefficient operators raise the order"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/coeff-iterate-coefficient-operator-eq-ze","kind":"node","lean":"MvPowerSeries.coeff_iterate_coefficientOperator_eq_zero","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#coeff-iterate-coefficient-operator-eq-ze","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The `n`th iterate has no coefficient in total degree less than `n`.","title":"Low-degree coefficients of an iterated coefficient operator vanish"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/coeff-coefficient-operator-eq-sum","kind":"node","lean":"MvPowerSeries.coeff_coefficientOperator_eq_sum","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#coeff-coefficient-operator-eq-sum","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"At any exponent, sum over any finite set containing its support.","title":"Coefficient of a coefficient operator over any finite superset"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/pderiv-comp-left","kind":"node","lean":"MvPowerSeries.pderiv_compLeft","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#pderiv-comp-left","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"Partial differentiation commutes with canonical coefficient lifting.","title":"Partial derivatives commute with coefficient lifting"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/pderiv-coefficient-operator","kind":"node","lean":"MvPowerSeries.pderiv_coefficientOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#pderiv-coefficient-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The operator derivative is the operator of the derivative plus the lifted `O i`.","title":"Partial derivative of a coefficient operator"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/comp-left-coefficient-operator","kind":"node","lean":"MvPowerSeries.compLeft_coefficientOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#comp-left-coefficient-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"Lifted coefficient maps commute with the operator under the source pairwise premise.","title":"Lifted coefficient maps commute with the coefficient operator"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/coefficient-operator-mul","kind":"node","lean":"MvPowerSeries.coefficientOperator_mul","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#coefficient-operator-mul","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"A family of coefficient derivations gives the source product rule, without commutation.","title":"Product rule for coefficient operators"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-partial","kind":"node","lean":"MvPowerSeries.expOperatorPartial","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-partial","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The first `N` operator iterates, `\u2211 n < N, \ud835\udcaa\u207f / n!`, acting on the full input series.","title":"Partial sums of the exponential of a coefficient operator"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-partial-apply","kind":"node","lean":"MvPowerSeries.expOperatorPartial_apply","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-partial-apply","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The finite exponential sum agrees with the source's function-iterate formula.","title":"Partial operator exponential as a sum of function iterates"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator","kind":"node","lean":"MvPowerSeries.expOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The formal exponential: coefficient `d` uses operator iterates `n \u2264 d.degree`.","title":"Formal exponential of a coefficient operator"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/coeff-exp-operator-def","kind":"node","lean":"MvPowerSeries.coeff_expOperator_def","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#coeff-exp-operator-def","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The defining iteration bound is exactly the source constructor's bound.","title":"Defining coefficient formula of the operator exponential"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/coeff-exp-operator","kind":"node","lean":"MvPowerSeries.coeff_expOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#coeff-exp-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"For `d.degree < N`, the first `N` operator iterates determine coefficient `d`.","title":"Coefficients of the operator exponential via partial sums"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/constant-coeff-exp-operator","kind":"node","lean":"MvPowerSeries.constantCoeff_expOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#constant-coeff-exp-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The formal exponential preserves every constant coefficient, including nonzero ones.","title":"The operator exponential preserves the constant coefficient"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-of-sq-zero","kind":"node","lean":"MvPowerSeries.expOperator_of_sq_zero","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-of-sq-zero","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"If the second operator iterate vanishes on `f`, its exponential is `f + \ud835\udcaa f`.","title":"Operator exponential when the square of the operator vanishes"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-of-zero","kind":"node","lean":"MvPowerSeries.expOperator_of_zero","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-of-zero","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The exponential fixes an input annihilated by the coefficient operator.","title":"The operator exponential fixes series annihilated by the operator"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-zero-family","kind":"node","lean":"MvPowerSeries.expOperator_zero_family","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-zero-family","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The exponential of the zero coefficient-map family is the identity linear map.","title":"Exponential of the zero coefficient-operator family is the identity"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/comp-left-iterate-coefficient-operator","kind":"node","lean":"MvPowerSeries.compLeft_iterate_coefficientOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#comp-left-iterate-coefficient-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"A lifted coefficient map commutes with every coefficient-operator iterate when the coefficient maps commute pairwise.","title":"Lifted coefficient maps commute with coefficient-operator iterates"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/pderiv-iterate-coefficient-operator","kind":"node","lean":"MvPowerSeries.pderiv_iterate_coefficientOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#pderiv-iterate-coefficient-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The partial derivative of the `(n + 1)`st coefficient-operator iterate is the iterate of the partial derivative plus `(n + 1)` copies of the lifted coefficient map.","title":"Partial derivative of a coefficient-operator iterate"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/pderiv-exp-operator-partial","kind":"node","lean":"MvPowerSeries.pderiv_expOperatorPartial","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#pderiv-exp-operator-partial","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The partial derivative of the first `N + 1` exponential terms is the corresponding partial sum on the derivative plus the first `N` terms on the lifted coefficient map.","title":"Partial derivative of partial sums of the operator exponential"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/pderiv-exp-operator","kind":"node","lean":"MvPowerSeries.pderiv_expOperator","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#pderiv-exp-operator","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The partial derivative of a formal coefficient-operator exponential is the exponential of the input derivative plus the exponential of the lifted coefficient map.","title":"Partial derivative of a coefficient-operator exponential"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/pderiv-exp-operator-derivation","kind":"node","lean":"MvPowerSeries.pderiv_expOperator_derivation","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#pderiv-exp-operator-derivation","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The exponential gradient law specialized to a pairwise commuting family of coefficient derivations, using their canonical rational linear maps.","title":"Gradient law for the exponential of commuting coefficient derivations"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-mul","kind":"node","lean":"MvPowerSeries.expOperator_mul","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-mul","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The formal exponential of a family of coefficient derivations preserves products. No pairwise commutation or finiteness hypothesis on the family is needed.","title":"The exponential of coefficient derivations is multiplicative"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-one","kind":"node","lean":"MvPowerSeries.expOperator_one","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-one","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The formal exponential of a family of coefficient derivations preserves one.","title":"The exponential of coefficient derivations preserves one"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-hom","kind":"node","lean":"MvPowerSeries.expOperatorHom","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-hom","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The formal exponential of coefficient derivations as a ring endomorphism.","title":"Exponential of coefficient derivations as a ring endomorphism"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-hom-apply","kind":"node","lean":"MvPowerSeries.expOperatorHom_apply","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-hom-apply","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"Applying the operator-exponential ring homomorphism is the existing formal operator exponential.","title":"Evaluation of the operator-exponential ring homomorphism"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/coeff-exp-operator-congr","kind":"node","lean":"MvPowerSeries.coeff_expOperator_congr","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#coeff-exp-operator-congr","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"A coefficient of the operator exponential depends only on input coefficients of no greater total degree.","title":"Operator exponential coefficients depend only on lower-degree inputs"},{"id":"RingTheory/formal-power-series/exponentials-of-coefficient-operators/exp-operator-exp-of-constant-coeff-zero","kind":"node","lean":"MvPowerSeries.expOperator_expOfConstantCoeffZero","parent":4001,"result":true,"route":"../roadmap/RingTheory/formal-power-series/exponentials-of-coefficient-operators/index.html#exp-operator-exp-of-constant-coeff-zero","scope":"RingTheory/formal-power-series/exponentials-of-coefficient-operators","status":"fully_proved","summary":"The operator exponential commutes with the lawful ordinary formal exponential.","title":"The operator exponential commutes with the formal exponential"},{"data_url":"scopes/RingTheory/formal-power-series/hahn-series.json","id":"RingTheory/formal-power-series/hahn-series","kind":"scope","parent":3977,"result":false,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/index.html","scope":"RingTheory/formal-power-series/hahn-series","status":"fully_proved","title":"Hahn series"},{"data_url":"scopes/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series.json","id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","kind":"scope","navigation":"module","parent":4032,"result":false,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","title":"Coefficientwise derivations of Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/coefficient-linear-map","kind":"node","lean":"HahnSeries.coefficientLinearMap","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#coefficient-linear-map","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"Apply a coefficient derivation to a Hahn series as an `R`-linear operator.","title":"Coefficientwise linear operator on Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/coeff-coefficient-linear-map","kind":"node","lean":"HahnSeries.coeff_coefficientLinearMap","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#coeff-coefficient-linear-map","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"The coefficient formula for the linear operator underlying the lifted derivation.","title":"Coefficient formula for the coefficientwise linear operator"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/support-coefficient-linear-map","kind":"node","lean":"HahnSeries.support_coefficientLinearMap","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#support-coefficient-linear-map","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"Acting on coefficients cannot introduce new exponents into the support.","title":"Coefficientwise linear operator does not enlarge the support"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/coefficient-derivation","kind":"node","lean":"HahnSeries.coefficientDerivation","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#coefficient-derivation","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"Extend a derivation of the coefficient algebra by applying it to each coefficient.","title":"Coefficientwise derivation of Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/coeff-coefficient-derivation","kind":"node","lean":"HahnSeries.coeff_coefficientDerivation","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#coeff-coefficient-derivation","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"The coefficient of the lifted derivation is the derivative of that coefficient.","title":"Coefficients of the coefficientwise derivation"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/coefficient-derivation-zero","kind":"node","lean":"HahnSeries.coefficientDerivation_zero","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#coefficient-derivation-zero","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"The lift of the zero derivation is zero.","title":"Coefficientwise lift of the zero derivation"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/coefficient-derivation-single","kind":"node","lean":"HahnSeries.coefficientDerivation_single","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#coefficient-derivation-single","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"A one-term series keeps its exponent and differentiates its coefficient.","title":"Coefficientwise derivation of a single-term series"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/coefficient-derivation-c","kind":"node","lean":"HahnSeries.coefficientDerivation_C","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#coefficient-derivation-c","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"On a constant series, the lift acts on its constant coefficient.","title":"Coefficientwise derivation of a constant series"},{"id":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/coefficient-derivation-commute","kind":"node","lean":"HahnSeries.coefficientDerivation_commute","parent":4033,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series/index.html#coefficient-derivation-commute","scope":"RingTheory/formal-power-series/hahn-series/coefficientwise-derivations-of-hahn-series","status":"fully_proved","summary":"Commuting coefficient derivations lift to commuting derivations of Hahn series.","title":"Commuting derivations lift to commuting coefficientwise derivations"},{"data_url":"scopes/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series.json","id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","kind":"scope","navigation":"module","parent":4032,"result":false,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","title":"Diagonal Laurent-coefficient Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/diagonal-laurent-series","kind":"node","lean":"HahnSeries.diagonalLaurentSeries","parent":4043,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html#diagonal-laurent-series","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","summary":"The Hahn series with coefficient `c y` at inner exponent `d - y` and outer exponent `y` for every `y \u2265 j\u2080`.","title":"Diagonal Laurent-coefficient Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/coeff-diagonal-laurent-series","kind":"node","lean":"HahnSeries.coeff_diagonalLaurentSeries","parent":4043,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html#coeff-diagonal-laurent-series","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","summary":"The outer coefficient of a diagonal Laurent series.","title":"Outer coefficient of a diagonal Laurent series"},{"id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/coeff-coeff-diagonal-laurent-series","kind":"node","lean":"HahnSeries.coeff_coeff_diagonalLaurentSeries","parent":4043,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html#coeff-coeff-diagonal-laurent-series","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","summary":"Both coefficients of a diagonal Laurent series, exposing its cutoff and diagonal equation.","title":"Double coefficients of a diagonal Laurent series"},{"id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/coeff-coeff-variable-difference-mul","kind":"node","lean":"HahnSeries.coeff_coeff_variableDifference_mul","parent":4043,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html#coeff-coeff-variable-difference-mul","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","summary":"Multiplication by the difference of the two variables takes a difference of adjacent diagonal coefficients.","title":"Multiplying by the variable difference on diagonal coefficients"},{"id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/variable-difference-unit","kind":"node","lean":"HahnSeries.variableDifferenceUnit","parent":4043,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html#variable-difference-unit","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","summary":"The formal difference of the inner and outer variables, bundled with its diagonal-series inverse.","title":"Variable difference as a unit of Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/coe-variable-difference-unit","kind":"node","lean":"HahnSeries.coe_variableDifferenceUnit","parent":4043,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html#coe-variable-difference-unit","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","summary":"The value of `variableDifferenceUnit` is the difference of the two formal variables.","title":"Value of the variable difference unit"},{"id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/coe-inv-variable-difference-unit","kind":"node","lean":"HahnSeries.coe_inv_variableDifferenceUnit","parent":4043,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html#coe-inv-variable-difference-unit","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","summary":"The inverse of the formal variable difference is its infinite diagonal Laurent series.","title":"Inverse of the variable difference as a diagonal series"},{"id":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/coe-inv-variable-difference-unit-pow-suc","kind":"node","lean":"HahnSeries.coe_inv_variableDifferenceUnit_pow_succ","parent":4043,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series/index.html#coe-inv-variable-difference-unit-pow-suc","scope":"RingTheory/formal-power-series/hahn-series/diagonal-laurent-coefficient-hahn-series","status":"fully_proved","summary":"The `(N + 1)`-st power of the inverse variable difference has negative-binomial coefficients on one diagonal.","title":"Powers of the inverse variable difference have binomial coefficients"},{"data_url":"scopes/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series.json","id":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","kind":"scope","navigation":"module","parent":4032,"result":false,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/index.html","scope":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","status":"fully_proved","title":"Exponent-indexed coefficient maps on Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/graded-coefficient-map","kind":"node","lean":"HahnSeries.gradedCoefficientMap","parent":4052,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/index.html#graded-coefficient-map","scope":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","status":"fully_proved","summary":"Apply the map indexed by each exponent to the coefficient at that exponent.","title":"Exponent-indexed coefficient map on Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/coeff-graded-coefficient-map","kind":"node","lean":"HahnSeries.coeff_gradedCoefficientMap","parent":4052,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/index.html#coeff-graded-coefficient-map","scope":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","status":"fully_proved","summary":"The coefficient formula for an exponent-indexed coefficient map.","title":"Coefficient formula for an exponent-indexed coefficient map"},{"id":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/support-graded-coefficient-map","kind":"node","lean":"HahnSeries.support_gradedCoefficientMap","parent":4052,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/index.html#support-graded-coefficient-map","scope":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","status":"fully_proved","summary":"Acting coefficientwise cannot introduce new exponents into the support.","title":"Exponent-indexed coefficient maps do not enlarge the support"},{"id":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/graded-coefficient-map-zero-family","kind":"node","lean":"HahnSeries.gradedCoefficientMap_zero_family","parent":4052,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/index.html#graded-coefficient-map-zero-family","scope":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","status":"fully_proved","summary":"The family consisting of zero maps induces the zero additive map.","title":"Exponent-indexed coefficient map of the zero family"},{"id":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/graded-coefficient-map-single","kind":"node","lean":"HahnSeries.gradedCoefficientMap_single","parent":4052,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/index.html#graded-coefficient-map-single","scope":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","status":"fully_proved","summary":"A one-term series keeps its exponent and applies the map indexed by that exponent.","title":"Exponent-indexed coefficient map on a single-term series"},{"id":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/graded-coefficient-map-c","kind":"node","lean":"HahnSeries.gradedCoefficientMap_C","parent":4052,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/index.html#graded-coefficient-map-c","scope":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","status":"fully_proved","summary":"On a constant series, the map indexed by zero acts on the constant coefficient.","title":"Exponent-indexed coefficient map on a constant series"},{"id":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/graded-coefficient-map-mul","kind":"node","lean":"HahnSeries.gradedCoefficientMap_mul","parent":4052,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series/index.html#graded-coefficient-map-mul","scope":"RingTheory/formal-power-series/hahn-series/exponent-indexed-coefficient-maps-on-hahn-series","status":"fully_proved","summary":"An exponent-indexed coefficient map satisfying the graded Leibniz identity obeys the product rule on Hahn series.","title":"Product rule for graded Leibniz coefficient maps"},{"data_url":"scopes/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series.json","id":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","kind":"scope","navigation":"module","parent":4032,"result":false,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/index.html","scope":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","status":"fully_proved","title":"Finite monoid algebras as Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/of-add-monoid-algebra","kind":"node","lean":"HahnSeries.ofAddMonoidAlgebra","parent":4060,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/index.html#of-add-monoid-algebra","scope":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","status":"fully_proved","summary":"Embed a finite additive monoid algebra into Hahn series.","title":"Embedding of a finite additive monoid algebra into Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/coeff-of-add-monoid-algebra","kind":"node","lean":"HahnSeries.coeff_ofAddMonoidAlgebra","parent":4060,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/index.html#coeff-of-add-monoid-algebra","scope":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","status":"fully_proved","summary":"The finite Hahn embedding preserves every coefficient.","title":"Monoid algebra embedding into Hahn series preserves coefficients"},{"id":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/of-add-monoid-algebra-apply","kind":"node","lean":"HahnSeries.ofAddMonoidAlgebra_apply","parent":4060,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/index.html#of-add-monoid-algebra-apply","scope":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","status":"fully_proved","summary":"The finite Hahn embedding is Mathlib's `ofFinsupp`, bundled as a ring homomorphism.","title":"Monoid algebra embedding into Hahn series as a ring map"},{"id":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/of-add-monoid-algebra-injective","kind":"node","lean":"HahnSeries.ofAddMonoidAlgebra_injective","parent":4060,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/index.html#of-add-monoid-algebra-injective","scope":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","status":"fully_proved","summary":"The finite Hahn embedding is injective.","title":"Injectivity of the monoid algebra embedding into Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/of-bivariate-add-monoid-algebra","kind":"node","lean":"HahnSeries.ofBivariateAddMonoidAlgebra","parent":4060,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/index.html#of-bivariate-add-monoid-algebra","scope":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","status":"fully_proved","summary":"Embed a bivariate finite monoid algebra into Hahn series, with the second coordinate outer.","title":"Embedding of a bivariate monoid algebra into Hahn series"},{"id":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/coeff-coeff-of-bivariate-add-monoid-alge","kind":"node","lean":"HahnSeries.coeff_coeff_ofBivariateAddMonoidAlgebra","parent":4060,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/index.html#coeff-coeff-of-bivariate-add-monoid-alge","scope":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","status":"fully_proved","summary":"The two coefficient projections of the bivariate embedding recover the original coefficient.","title":"Coefficients of the bivariate monoid algebra embedding"},{"id":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/of-bivariate-add-monoid-algebra-injectiv","kind":"node","lean":"HahnSeries.ofBivariateAddMonoidAlgebra_injective","parent":4060,"result":true,"route":"../roadmap/RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series/index.html#of-bivariate-add-monoid-algebra-injectiv","scope":"RingTheory/formal-power-series/hahn-series/finite-monoid-algebras-as-hahn-series","status":"fully_proved","summary":"The bivariate finite Hahn embedding is injective.","title":"Injectivity of the bivariate monoid algebra embedding"},{"data_url":"scopes/RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings.json","id":"RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings","kind":"scope","navigation":"module","parent":3977,"result":false,"route":"../roadmap/RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings/index.html","scope":"RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings","status":"fully_proved","title":"Ideals of multivariate power series rings"},{"id":"RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings/mem-span-c-sup-of-list-take-iff","kind":"node","lean":"MvPowerSeries.mem_span_C_sup_ofList_take_iff","parent":4068,"result":true,"route":"../roadmap/RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings/index.html#mem-span-c-sup-of-list-take-iff","scope":"RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings","status":"fully_proved","summary":"A power series lies in the span of `C \u03d6` and the first `k` variables exactly when every coefficient of a monomial supported on the last `n - k` variables is divisible by `\u03d6`.","title":"Membership in the span of a constant and initial variables"},{"id":"RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings/span-range-x-eq-ker-constant-coeff","kind":"node","lean":"MvPowerSeries.span_range_X_eq_ker_constantCoeff","parent":4068,"result":true,"route":"../roadmap/RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings/index.html#span-range-x-eq-ker-constant-coeff","scope":"RingTheory/formal-power-series/ideals-of-multivariate-power-series-rings","status":"fully_proved","summary":"The kernel of the constant coefficient map on multivariate power series is the span of the variables.","title":"The variables span the kernel of the constant coefficient map"},{"data_url":"scopes/RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series.json","id":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series","kind":"scope","navigation":"module","parent":3977,"result":false,"route":"../roadmap/RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/index.html","scope":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series","status":"fully_proved","title":"Kaluza's theorem on reciprocal power series"},{"id":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/has-positive-log-convex-coefficients","kind":"node","lean":"MathlibExt.Algebra.PowerSeries.KaluzaCoefficientBoundsWanted.HasPositiveLogConvexCoefficients","parent":4071,"result":true,"route":"../roadmap/RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/index.html#has-positive-log-convex-coefficients","scope":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series","status":"fully_proved","summary":"A real formal power series $f = \\sum_n a_n X^n$ has positive log-convex coefficients if $a_n > 0$ for every natural number $n$ and $a_{n+1}^2 \\le a_n a_{n+2}$ for every natural number $n$. This is the hypothesis of Kaluza's bounds for the coefficients of the reciprocal series.","title":"Positive log-convex coefficients"},{"id":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/coeff-pos","kind":"node","lean":"MathlibExt.Algebra.PowerSeries.KaluzaCoefficientBoundsWanted.HasPositiveLogConvexCoefficients.coeff_pos","parent":4071,"result":true,"route":"../roadmap/RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/index.html#coeff-pos","scope":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series","status":"fully_proved","summary":"If a real formal power series $f$ has positive log-convex coefficients, then the coefficient of $X^n$ in $f$ is strictly positive for every natural number $n$.","title":"Coefficients of a positive log-convex series are positive"},{"id":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/sq-coeff-succ-le","kind":"node","lean":"MathlibExt.Algebra.PowerSeries.KaluzaCoefficientBoundsWanted.HasPositiveLogConvexCoefficients.sq_coeff_succ_le","parent":4071,"result":true,"route":"../roadmap/RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/index.html#sq-coeff-succ-le","scope":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series","status":"fully_proved","summary":"If a real formal power series $f = \\sum_n a_n X^n$ has positive log-convex coefficients, then $a_{n+1}^2 \\le a_n a_{n+2}$ for every natural number $n$.","title":"Log-convexity of the coefficients"},{"id":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/kaluza-coeff-inv-bounds","kind":"node","lean":"MathlibExt.Algebra.PowerSeries.KaluzaCoefficientBoundsWanted.kaluza_coeff_inv_bounds","parent":4071,"result":true,"route":"../roadmap/RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series/index.html#kaluza-coeff-inv-bounds","scope":"RingTheory/formal-power-series/kaluza-s-theorem-on-reciprocal-power-series","status":"fully_proved","summary":"Let $f = \\sum_n a_n X^n$ be a real formal power series with positive log-convex coefficients, and let $f^{-1} = \\sum_n b_n X^n$ be its reciprocal in the ring of formal power series. Then $b_0 = a_0^{-1}$, $b_0 > 0$, and $-a_n / a_0^2 \\le b_n \\le 0$ for every $n \\ge 1$.","title":"Kaluza's bounds for the coefficients of a reciprocal power series"},{"data_url":"scopes/RingTheory/formal-power-series/lagrange-inversion-theorem.json","id":"RingTheory/formal-power-series/lagrange-inversion-theorem","kind":"scope","navigation":"module","parent":3977,"result":false,"route":"../roadmap/RingTheory/formal-power-series/lagrange-inversion-theorem/index.html","scope":"RingTheory/formal-power-series/lagrange-inversion-theorem","status":"fully_proved","title":"Lagrange inversion theorem"},{"id":"RingTheory/formal-power-series/lagrange-inversion-theorem/coeff-mul-pow-eq-coeff-of-eq-x-mul-subst","kind":"node","lean":"MetaMathlibExt.LagrangeInversion.coeff_mul_pow_eq_coeff_of_eq_X_mul_subst","parent":4076,"result":true,"route":"../roadmap/RingTheory/formal-power-series/lagrange-inversion-theorem/index.html#coeff-mul-pow-eq-coeff-of-eq-x-mul-subst","scope":"RingTheory/formal-power-series/lagrange-inversion-theorem","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $F$, $\\varphi$, $w$ and $G$ be formal power series over $R$, and write $[X^n]H$ for the coefficient of $X^n$ in a power series $H$. Write $\\varphi'$ for the formal derivative of $\\varphi$, and $\\varphi(w)$, $\\varphi'(w)$, $F(w)$ for the substitution of $w$ into $\\varphi$, $\\varphi'$, $F$. If $w = X\\,\\varphi(w)$ and $(1 - X\\,\\varphi'(w))\\,G = F(w)$, then $X^n = [X^n]G$ for every\u2026","title":"Hermite's diagonal form of Lagrange inversion without side conditions"},{"id":"RingTheory/formal-power-series/lagrange-inversion-theorem/hermite-lagrange-diagonal-formula","kind":"node","lean":"MetaMathlibExt.LagrangeInversion.hermite_lagrange_diagonal_formula","parent":4076,"result":true,"route":"../roadmap/RingTheory/formal-power-series/lagrange-inversion-theorem/index.html#hermite-lagrange-diagonal-formula","scope":"RingTheory/formal-power-series/lagrange-inversion-theorem","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $F$, $\\varphi$, $w$ and $G$ be formal power series over $R$, and write $[X^n]H$ for the coefficient of $X^n$ in a power series $H$. Write $\\varphi'$ for the formal derivative of $\\varphi$, and $\\varphi(w)$, $\\varphi'(w)$, $F(w)$ for the substitution of $w$ into $\\varphi$, $\\varphi'$, $F$. Suppose that $w$ has constant term $0$, that $\\varphi$ has nonzero constant term, that\u2026","title":"Hermite's diagonal form of Lagrange inversion"},{"id":"RingTheory/formal-power-series/lagrange-inversion-theorem/hermite-lagrange-inversion","kind":"node","lean":"MetaMathlibExt.hermite_lagrange_inversion","parent":4076,"result":true,"route":"../roadmap/RingTheory/formal-power-series/lagrange-inversion-theorem/index.html#hermite-lagrange-inversion","scope":"RingTheory/formal-power-series/lagrange-inversion-theorem","status":"fully_proved","summary":"Let $\\varphi$, $g$, $F$ and $\\psi$ be formal power series over $\\mathbb{C}$, write $F'$ for the formal derivative of $F$, and write $h(k)$ for the substitution of a power series $k$ into a power series $h$. Suppose that $\\varphi$ has nonzero constant term, that $F\\varphi = X$, and that $\\psi$ is a two-sided compositional inverse of $F$, that is, $F(\\psi) = X$ and $\\psi(F) = X$. Let $H = \\sum_{n \\ge 0} c_n X^n$,\u2026","title":"Hermite's form of Lagrange inversion as a series identity"},{"id":"RingTheory/formal-power-series/lagrange-inversion-theorem/fps-lagrange-burmann-coeff","kind":"node","lean":"fps_lagrange_burmann_coeff'","parent":4076,"result":true,"route":"../roadmap/RingTheory/formal-power-series/lagrange-inversion-theorem/index.html#fps-lagrange-burmann-coeff","scope":"RingTheory/formal-power-series/lagrange-inversion-theorem","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $A$, $\\varphi$ and $g$ be formal power series over $R$ with $A = X\\,\\varphi(A)$, where $\\varphi(A)$ and $g(A)$ denote the substitution of $A$ into $\\varphi$ and $g$. Write $[X^m]H$ for the coefficient of $X^m$ in a power series $H$ and $g'$ for the formal derivative of $g$. Then for every natural number $n \\ge 1$,","title":"Lagrange\u2013B\u00fcrmann formula from the fixed-point equation alone"},{"id":"RingTheory/formal-power-series/lagrange-inversion-theorem/fps-lagrange-burmann-coeff-2","kind":"node","lean":"fps_lagrange_burmann_coeff","parent":4076,"result":true,"route":"../roadmap/RingTheory/formal-power-series/lagrange-inversion-theorem/index.html#fps-lagrange-burmann-coeff-2","scope":"RingTheory/formal-power-series/lagrange-inversion-theorem","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $A$, $\\varphi$ and $g$ be formal power series over $R$ such that the constant term of $A$ is nilpotent, $\\varphi$ has nonzero constant term, and $A = X\\,\\varphi(A)$, where $\\varphi(A)$ and $g(A)$ denote the substitution of $A$ into $\\varphi$ and $g$. Write $[X^m]H$ for the coefficient of $X^m$ in a power series $H$ and $g'$ for the formal derivative of $g$. Then for every\u2026","title":"Lagrange\u2013B\u00fcrmann formula, coefficient form"},{"data_url":"scopes/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series.json","id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","kind":"scope","navigation":"module","parent":3977,"result":false,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","title":"Partial derivatives of multivariate power series"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/iterated-pderiv","kind":"node","lean":"MvPowerSeries.iteratedPDeriv","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#iterated-pderiv","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"Partial derivatives in list order, with the first entry applied first, as a linear map.","title":"Iterated partial derivative of a multivariate power series"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/iterated-pderiv-append","kind":"node","lean":"MvPowerSeries.iteratedPDeriv_append","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#iterated-pderiv-append","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"Concatenating lists composes the second iterated derivative after the first.","title":"Iterated partial derivatives along concatenated lists compose"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/pderiv-pderiv","kind":"node","lean":"MvPowerSeries.pderiv_pderiv","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#pderiv-pderiv","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"Two formal partial derivatives commute.","title":"Formal partial derivatives commute"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/pderiv-iterated-pderiv","kind":"node","lean":"MvPowerSeries.pderiv_iteratedPDeriv","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#pderiv-iterated-pderiv","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"A partial derivative commutes with any list of partial derivatives.","title":"A partial derivative commutes with iterated partial derivatives"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/to-finsupp-coe-list-eq-sum","kind":"node","lean":"Multiset.toFinsupp_coe_list_eq_sum","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#to-finsupp-coe-list-eq-sum","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"List variable multiplicities equal the sum of singleton multi-indices.","title":"Variable multiplicities of a list as a sum of singleton multi-indices"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/factorial-smul-coeff-iterated-pderiv","kind":"node","lean":"MvPowerSeries.factorial_smul_coeff_iteratedPDeriv","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#factorial-smul-coeff-iterated-pderiv","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"The factorial-weighted coefficient after iterated differentiation. No factorial is cancelled, so this identity also holds in positive characteristic.","title":"Factorial-weighted coefficient of an iterated partial derivative"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/factorial-smul-coeff-eq-constant-coeff-i","kind":"node","lean":"MvPowerSeries.factorial_smul_coeff_eq_constantCoeff_iteratedPDeriv","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#factorial-smul-coeff-eq-constant-coeff-i","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"A monomial's factorial times its coefficient is the constant coefficient of the corresponding iterated partial derivative.","title":"Factorial times a coefficient is a constant term of an iterated derivative"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/factorial-smul-coeff-iterated-pderiv-rat","kind":"node","lean":"MvPowerSeries.factorial_smul_coeff_iteratedPDeriv_rat","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#factorial-smul-coeff-iterated-pderiv-rat","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"The source rational-factorial scalar equation, obtained by casting the natural factorial product and its scalar action.","title":"Rational factorial-weighted coefficient of an iterated derivative"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/factorial-smul-coeff-eq-constant-coeff-i-2","kind":"node","lean":"MvPowerSeries.factorial_smul_coeff_eq_constantCoeff_iteratedPDeriv_rat","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#factorial-smul-coeff-eq-constant-coeff-i-2","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"The source rational-factorial constant coefficient equation.","title":"Rational factorial coefficient as a constant term of an iterated derivative"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/linear-form","kind":"node","lean":"MvPowerSeries.linearForm","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#linear-form","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"The formal linear form with variable coefficient family `b : \u03c3 \u2192 R`.","title":"Formal linear form in multivariate power series"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/coeff-linear-form","kind":"node","lean":"MvPowerSeries.coeff_linearForm","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#coeff-linear-form","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"The coefficient formula defining a formal linear form.","title":"Coefficient formula for a formal linear form"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/coeff-linear-form-single","kind":"node","lean":"MvPowerSeries.coeff_linearForm_single","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#coeff-linear-form-single","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"The coefficient of variable `i` is the prescribed coefficient `b i`.","title":"Coefficient of a single variable in a formal linear form"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/constant-coeff-linear-form","kind":"node","lean":"MvPowerSeries.constantCoeff_linearForm","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#constant-coeff-linear-form","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"A formal linear form has zero constant coefficient.","title":"A formal linear form has zero constant coefficient"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/pderiv-linear-form","kind":"node","lean":"MvPowerSeries.pderiv_linearForm","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#pderiv-linear-form","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"The partial derivative of a formal linear form is its corresponding constant coefficient.","title":"Partial derivative of a formal linear form"},{"id":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/map-linear-form","kind":"node","lean":"MvPowerSeries.map_linearForm","parent":4082,"result":true,"route":"../roadmap/RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series/index.html#map-linear-form","scope":"RingTheory/formal-power-series/partial-derivatives-of-multivariate-power-series","status":"fully_proved","summary":"Mapping the coefficient ring commutes with forming a formal linear form.","title":"Coefficient ring maps commute with formal linear forms"},{"data_url":"scopes/RingTheory/valuation-rings.json","id":"RingTheory/valuation-rings","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/valuation-rings/index.html","scope":"RingTheory/valuation-rings","status":"fully_proved","summary":"Discrete valuation rings and general valuation rings, their completions and finite extensions, residue fields, Hensel's lemma and the Krull\u2013Akizuki theorem.","title":"Valuation rings and Henselian rings"},{"id":"RingTheory/valuation-rings/is-principal-ideal-ring-of-finite-dimens","kind":"node","lean":"ValuationSubring.isPrincipalIdealRing_of_finiteDimensional_adjoin","parent":4098,"result":true,"route":"../roadmap/RingTheory/valuation-rings/index.html#is-principal-ideal-ring-of-finite-dimens","scope":"RingTheory/valuation-rings","status":"fully_proved","summary":"A valuation subring of a function field in one variable over `K`, containing `K` and different from the whole field, is a principal ideal ring.","title":"Valuation rings of one-variable function fields are PIDs"},{"data_url":"scopes/RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien.json","id":"RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien","kind":"scope","navigation":"module","parent":4098,"result":false,"route":"../roadmap/RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien/index.html","scope":"RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien","status":"fully_proved","title":"Discrete valuation rings as power series quotients"},{"id":"RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien/is-discrete-valuation-ring-quotient-span","kind":"node","lean":"PowerSeries.isDiscreteValuationRing_quotient_span_X_sub_C_pow_of_irreducible","parent":4100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power series over a DVR modulo "],["math","\\(X - \\varpi^e\\)"],["text"," form a DVR"]],"source":"Power series over a DVR modulo $X - \\varpi^e$ form a DVR"},"result":true,"route":"../roadmap/RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien/index.html#is-discrete-valuation-ring-quotient-span","scope":"RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien","status":"fully_proved","summary":"Modulo `X - \u03d6 ^ e`, a power series ring over a discrete valuation ring is a discrete valuation ring, with uniformizer the image of `\u03d6`.","title":"Power series over a DVR modulo $X - \\varpi^e$ form a DVR"},{"id":"RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien/quotient-span-x-sub-c-pow-of-irreducible","kind":"node","lean":"PowerSeries.quotient_span_X_sub_C_pow_of_irreducible","parent":4100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power series over a DVR modulo "],["math","\\(X - \\varpi^e\\)"],["text"," form a local PID"]],"source":"Power series over a DVR modulo $X - \\varpi^e$ form a local PID"},"result":true,"route":"../roadmap/RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien/index.html#quotient-span-x-sub-c-pow-of-irreducible","scope":"RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien","status":"fully_proved","summary":"Modulo `X - \u03d6 ^ e`, a power series ring over a discrete valuation ring is a principal ideal ring and a local ring, and the image of `\u03d6` is irreducible.","title":"Power series over a DVR modulo $X - \\varpi^e$ form a local PID"},{"id":"RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien/is-adic-complete-quotient-span-x-sub-c-o","kind":"node","lean":"PowerSeries.isAdicComplete_quotient_span_X_sub_C_of_irreducible","parent":4100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power series over a DVR modulo "],["math","\\(X - \\varpi\\)"],["text"," form a complete DVR"]],"source":"Power series over a DVR modulo $X - \\varpi$ form a complete DVR"},"result":true,"route":"../roadmap/RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien/index.html#is-adic-complete-quotient-span-x-sub-c-o","scope":"RingTheory/valuation-rings/discrete-valuation-rings-as-power-series-quotien","status":"fully_proved","summary":"Modulo `X - \u03d6`, a power series ring over a discrete valuation ring is a discrete valuation ring, adically complete at its maximal ideal, with uniformizer the image of `\u03d6`.","title":"Power series over a DVR modulo $X - \\varpi$ form a complete DVR"},{"data_url":"scopes/RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes.json","id":"RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes","kind":"scope","navigation":"module","parent":4098,"result":false,"route":"../roadmap/RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes/index.html","scope":"RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes","status":"fully_proved","title":"Discrete valuation rings at height-one primes"},{"id":"RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes/is-discrete-valuation-ring-of-height-eq","kind":"node","lean":"Localization.AtPrime.isDiscreteValuationRing_of_height_eq_one","parent":4104,"result":true,"route":"../roadmap/RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes/index.html#is-discrete-valuation-ring-of-height-eq","scope":"RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes","status":"fully_proved","summary":"In a Noetherian integrally closed domain, the localization at a height-one prime ideal is a discrete valuation ring.","title":"Localization at a height-one prime is a DVR"},{"id":"RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes/exists-valuation-subring-mem-iff-of-heig","kind":"node","lean":"IsIntegrallyClosed.exists_valuationSubring_mem_iff_of_height_eq_one","parent":4104,"result":true,"route":"../roadmap/RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes/index.html#exists-valuation-subring-mem-iff-of-heig","scope":"RingTheory/valuation-rings/discrete-valuation-rings-at-height-one-primes","status":"fully_proved","summary":"At a height-one prime of a Noetherian integrally closed domain, there is a principal ideal valuation subring of the fraction field whose membership is the usual numerator-denominator criterion.","title":"Valuation subring at a height-one prime of a normal domain"},{"data_url":"scopes/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma.json","id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","kind":"scope","navigation":"module","parent":4098,"result":false,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","title":"Henselian rings and Hensel's lemma"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/exists-monic-coprime-factorization-of-is","kind":"node","lean":"Polynomial.exists_monic_coprime_factorization_of_isAdicComplete","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#exists-monic-coprime-factorization-of-is","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Let $A$ be a commutative ring and $I$ an ideal of $A$ such that $A$ is $I$-adically complete, and let $f \\in A[X]$ be monic. Suppose the reduction of $f$ modulo $I$ equals $\\bar g \\bar h$, where $\\bar g, \\bar h \\in (A/I)[X]$ are monic and coprime (that is, $a \\bar g + b \\bar h = 1$ for some $a, b \\in (A/I)[X]$). Then there exist monic $g, h \\in A[X]$ with $f = gh$ whose reductions modulo $I$ are $\\bar g$ and\u2026","title":"Hensel lifting of monic coprime factorizations"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/exists-monic-coprime-factorization-of-is-2","kind":"node","lean":"Polynomial.exists_monic_coprime_factorization_of_isAdicComplete_natDegree","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#exists-monic-coprime-factorization-of-is-2","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Let $A$ be a commutative ring and $I$ an ideal of $A$ such that $A$ is $I$-adically complete and $A/I$ is not the zero ring, and let $f \\in A[X]$ be monic. Suppose the reduction of $f$ modulo $I$ equals $\\bar g \\bar h$, where $\\bar g, \\bar h \\in (A/I)[X]$ are monic and coprime (that is, $a \\bar g + b \\bar h = 1$ for some $a, b \\in (A/I)[X]$). Then there exist monic $g, h \\in A[X]$ with $f = gh$ whose reductions\u2026","title":"Hensel lifting of monic coprime factorizations, with degrees"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/is-coprime-of-is-coprime-map","kind":"node","lean":"Polynomial.Monic.isCoprime_of_isCoprime_map","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#is-coprime-of-is-coprime-map","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"A local ring homomorphism reflects coprimality of polynomials when the first polynomial is monic. Neither ring needs to be local.","title":"Local homomorphisms reflect coprimality with a monic polynomial"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/exists-unique-monic-map-eq-dvd-of-is-cop","kind":"node","lean":"Polynomial.existsUnique_monic_map_eq_dvd_of_isCoprime_of_ker_pow_eq_bot","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#exists-unique-monic-map-eq-dvd-of-is-cop","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"A monic factor of a coprime factorization lifts uniquely through a surjective homomorphism with nilpotent kernel. The original polynomial and the complementary reduced factor are not required to be monic.","title":"Unique lift of a monic factor through a nilpotent-kernel surjection"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/exists-monic-mul-eq-and-map-eq-of-is-cop","kind":"node","lean":"Polynomial.exists_monic_mul_eq_and_map_eq_of_isCoprime_of_isAdicComplete","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#exists-monic-mul-eq-and-map-eq-of-is-cop","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Coprime monic factors lift over a ring complete at the kernel of a surjective homomorphism. The lifts are coprime, and the first is the unique monic divisor of the original polynomial with its specified reduction.","title":"Hensel lifting of coprime monic factors over a complete ring"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/exists-is-root-sub-mem-maximal-ideal-of","kind":"node","lean":"Polynomial.exists_isRoot_sub_mem_maximalIdeal_of_isUnit_derivative","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#exists-is-root-sub-mem-maximal-ideal-of","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ that is complete and Hausdorff for the $\\mathfrak{m}$-adic topology. Let $f \\in A[X]$ have derivative $f'$, and let $a_0 \\in A$ satisfy $f(a_0) \\in \\mathfrak{m}$ with $f'(a_0)$ a unit of $A$. Then $f$ has a root $a \\in A$ with $a - a_0 \\in \\mathfrak{m}$.","title":"Hensel's lemma for complete discrete valuation rings"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/add-val-sub-le-add-val-eval-sub","kind":"node","lean":"Polynomial.addVal_sub_le_addVal_eval_sub","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#add-val-sub-le-add-val-eval-sub","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with normalized additive valuation $v \\colon A \\to \\mathbb{N} \\cup \\{\\infty\\}$. For every polynomial $p \\in A[X]$ and all $a, a_0 \\in A$, we have $v(a - a_0) \\le v(p(a) - p(a_0))$.","title":"Polynomials do not decrease the valuation of differences"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/add-val-derivative-eval-eq-of-lt-add-val","kind":"node","lean":"Polynomial.addVal_derivative_eval_eq_of_lt_addVal_sub","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#add-val-derivative-eval-eq-of-lt-add-val","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"A root sufficiently close to the approximation has the same derivative valuation.","title":"Nearby roots share the derivative valuation"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/eq-of-is-root-of-add-val-derivative-lt-a","kind":"node","lean":"Polynomial.eq_of_isRoot_of_addVal_derivative_lt_addVal_sub","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#eq-of-is-root-of-add-val-derivative-lt-a","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with normalized additive valuation $v \\colon A \\to \\mathbb{N} \\cup \\{\\infty\\}$, let $f \\in A[X]$ have derivative $f'$, and let $a_0 \\in A$. If $a, a' \\in A$ are roots of $f$ with $v(f'(a_0)) < v(a - a_0)$ and $v(f'(a_0)) < v(a' - a_0)$, then $a = a'$.","title":"Uniqueness of roots near a point"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/add-val-lt-add-val-add-of-mem-maximal-id","kind":"node","lean":"IsDiscreteValuationRing.addVal_lt_addVal_add_of_mem_maximalIdeal","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#add-val-lt-add-val-add-of-mem-maximal-id","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ and normalized additive valuation $v \\colon A \\to \\mathbb{N} \\cup \\{\\infty\\}$. If $d \\in A$ is nonzero and $q \\in \\mathfrak{m}$, then $v(d) < v(d) + v(q)$.","title":"Elements of the maximal ideal have positive valuation"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/exists-is-root-of-two-mul-add-val-deriva","kind":"node","lean":"Polynomial.exists_isRoot_of_two_mul_addVal_derivative_lt_addVal","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#exists-is-root-of-two-mul-add-val-deriva","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring with maximal ideal $\\mathfrak{m}$ that is complete and Hausdorff for the $\\mathfrak{m}$-adic topology, and let $v \\colon A \\to \\mathbb{N} \\cup \\{\\infty\\}$ be its normalized additive valuation. Let $f \\in A[X]$ have derivative $f'$, and let $a_0 \\in A$ satisfy $2 v(f'(a_0)) < v(f(a_0))$. Then $f$ has a root $a \\in A$ such that $a - a_0 \\in \\mathfrak{m}$,\u2026","title":"Hensel's lemma with a derivative bound"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/exists-alg-hom-lift-of-etale","kind":"node","lean":"HenselianLocalRing.exists_algHom_lift_of_etale","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#exists-alg-hom-lift-of-etale","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Over a Henselian local ring, an \u00e9tale algebra has a section lifting any residue-field point.","title":"Sections of \u00e9tale algebras over Henselian local rings"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/of-is-adic-complete-maximal-ideal","kind":"node","lean":"HenselianLocalRing.of_isAdicComplete_maximalIdeal","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#of-is-adic-complete-maximal-ideal","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$. If $R$ is Hausdorff and complete for the $\\mathfrak m$-adic topology, then $R$ is a Henselian local ring.","title":"Complete local rings are Henselian"},{"id":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/henselian-local-ring-of-is-alg-closed","kind":"node","lean":"ValuationSubring.henselianLocalRing_of_isAlgClosed","parent":4107,"result":true,"route":"../roadmap/RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma/index.html#henselian-local-ring-of-is-alg-closed","scope":"RingTheory/valuation-rings/henselian-rings-and-hensel-s-lemma","status":"fully_proved","summary":"A valuation subring of an algebraically closed field is Henselian.","title":"Valuation subrings of algebraically closed fields are Henselian"},{"data_url":"scopes/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation.json","id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","kind":"scope","navigation":"module","parent":4098,"result":false,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","title":"Finite extensions of complete discrete valuation rings"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/is-adic-complete-of-finite-local-extensi","kind":"node","lean":"IsDiscreteValuationRing.isAdicComplete_of_finite_local_extension","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#is-adic-complete-of-finite-local-extensi","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ and $B$ be discrete valuation rings, and let $B$ be an $A$-algebra that is finitely generated as an $A$-module and whose structure map $A \\to B$ is injective. If $A$ is complete and Hausdorff for the adic topology of its maximal ideal, then $B$ is complete and Hausdorff for the adic topology of its maximal ideal.","title":"Completeness passes to finite extensions of discrete valuation rings"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/comap-eq-base-maximal-of-is-adic-complet","kind":"node","lean":"Ideal.comap_eq_base_maximal_of_isAdicComplete_of_isIntegral","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#comap-eq-base-maximal-of-is-adic-complet","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings with $B$ an $A$-algebra that is integral over $A$, and let $\\mathfrak p$ be a maximal ideal of $A$ such that $A$ is $\\mathfrak p$-adically complete. Then every maximal ideal $Q$ of $B$ has preimage $\\mathfrak p$ under the structure map $A \\to B$.","title":"Maximal ideals contract to the maximal ideal of a complete base"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/is-local-ring-of-is-integral-of-is-adic","kind":"node","lean":"isLocalRing_of_isIntegral_of_isAdicComplete_maximal","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#is-local-ring-of-is-integral-of-is-adic","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ be an integral domain and $\\mathfrak p$ a maximal ideal of $A$ such that $A$ is $\\mathfrak p$-adically complete. If $B$ is an integral domain and an $A$-algebra that is integral over $A$, then $B$ is a local ring.","title":"Domains integral over a complete local domain are local"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/is-local-ring-of-is-adic-complete","kind":"node","lean":"IsIntegralClosure.isLocalRing_of_isAdicComplete","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#is-local-ring-of-is-adic-complete","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring that is complete with respect to its maximal ideal, $K$ its fraction field and $L$ a finite separable extension of $K$. Then the integral closure of $A$ in $L$ is a local ring.","title":"The integral closure of a complete DVR is local"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/is-discrete-valuation-ring-of-is-adic-co","kind":"node","lean":"IsIntegralClosure.isDiscreteValuationRing_of_isAdicComplete","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#is-discrete-valuation-ring-of-is-adic-co","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring that is complete with respect to its maximal ideal, $K$ its fraction field and $L$ a finite separable extension of $K$. Then the integral closure of $A$ in $L$ is a discrete valuation ring.","title":"The integral closure of a complete DVR is a DVR"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/is-adic-complete-of-is-adic-complete","kind":"node","lean":"IsIntegralClosure.isAdicComplete_of_isAdicComplete","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#is-adic-complete-of-is-adic-complete","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring that is complete with respect to its maximal ideal, $K$ its fraction field, $L$ a finite separable extension of $K$, and $B$ the integral closure of $A$ in $L$. If $B$ is local (as it always is here), then $B$ is complete with respect to its maximal ideal.","title":"The integral closure of a complete DVR is complete"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/exists-unique-primes-over-maximal-ideal","kind":"node","lean":"IsIntegralClosure.existsUnique_primesOver_maximalIdeal_of_isAdicComplete","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#exists-unique-primes-over-maximal-ideal","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring that is complete with respect to its maximal ideal $\\mathfrak m$, $K$ its fraction field, $L$ a finite separable extension of $K$, and $B$ the integral closure of $A$ in $L$. Then there is exactly one prime ideal of $B$ whose preimage under $A \\to B$ is $\\mathfrak m$.","title":"Exactly one prime lies over the maximal ideal of a complete DVR"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/integer-is-integral-closure-of-is-adic-c","kind":"node","lean":"Valuation.integer_isIntegralClosure_of_isAdicComplete","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#integer-is-integral-closure-of-is-adic-c","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring that is complete with respect to its maximal ideal, $K$ its fraction field, and $L$ a finite separable extension of $K$. Let $v$ be a valuation on $L$ with values in a linearly ordered commutative group with zero, with valuation ring $\\mathcal O_v = \\{x \\in L : v(x) \\le 1\\}$. Suppose $\\mathcal O_v$ is an $A$-algebra such that the composite $A \\to \\mathcal O_v \\subseteq L$ is the\u2026","title":"Valuation rings dominating a complete DVR are integral closures"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/module-finite-integer-of-is-adic-complet","kind":"node","lean":"Valuation.module_finite_integer_of_isAdicComplete","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#module-finite-integer-of-is-adic-complet","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"Let $A$ be a discrete valuation ring that is complete with respect to its maximal ideal, $K$ its fraction field, and $L$ a finite separable extension of $K$. Let $v$ be a valuation on $L$ with values in a linearly ordered commutative group with zero, with valuation ring $\\mathcal O_v = \\{x \\in L : v(x) \\le 1\\}$. Suppose $\\mathcal O_v$ is an $A$-algebra such that the composite $A \\to \\mathcal O_v \\subseteq L$ is the\u2026","title":"Valuation rings dominating a complete DVR are finite over it"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/finite-and-is-discrete-valuation-ring-an","kind":"node","lean":"integralClosure.finite_and_isDiscreteValuationRing_and_isAdicComplete_maximalIdeal","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#finite-and-is-discrete-valuation-ring-an","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"The integral closure of a complete discrete valuation ring in a finite separable extension of its fraction field is finite over the base, is a discrete valuation ring, and is complete for its maximal ideal.","title":"Integral closure of a complete DVR in a separable extension"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/of-is-domain-of-module-finite-of-is-adic","kind":"node","lean":"IsLocalRing.of_isDomain_of_module_finite_of_isAdicComplete","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#of-is-domain-of-module-finite-of-is-adic","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"A finite algebra over an adically complete Noetherian local ring that is a domain is itself local.","title":"Finite domains over adically complete local rings are local"},{"id":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/exists-fin-points-dvr-i-inf-ker-eq-bot","kind":"node","lean":"IsLocalRing.exists_fin_points_dvr_iInf_ker_eq_bot","parent":4122,"result":true,"route":"../roadmap/RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation/index.html#exists-fin-points-dvr-i-inf-ker-eq-bot","scope":"RingTheory/valuation-rings/finite-extensions-of-complete-discrete-valuation","status":"fully_proved","summary":"A finite torsion-free local reduced algebra over an adically complete DVR with finite residue field and characteristic zero admits finitely many adically complete DVR points whose kernels intersect to `\u22a5`, with a common denominator clearing all denominators.","title":"Finitely many DVR points with trivial common kernel"},{"data_url":"scopes/RingTheory/valuation-rings/krull-akizuki-theorem.json","id":"RingTheory/valuation-rings/krull-akizuki-theorem","kind":"scope","navigation":"module","parent":4098,"result":false,"route":"../roadmap/RingTheory/valuation-rings/krull-akizuki-theorem/index.html","scope":"RingTheory/valuation-rings/krull-akizuki-theorem","status":"fully_proved","title":"Krull\u2013Akizuki theorem"},{"id":"RingTheory/valuation-rings/krull-akizuki-theorem/is-noetherian-ring-and-dimension-leone-o","kind":"node","lean":"Subalgebra.isNoetherianRing_and_dimensionLEOne_of_isFractionRing_of_finite","parent":4135,"result":true,"route":"../roadmap/RingTheory/valuation-rings/krull-akizuki-theorem/index.html#is-noetherian-ring-and-dimension-leone-o","scope":"RingTheory/valuation-rings/krull-akizuki-theorem","status":"fully_proved","summary":"The Krull\u2013Akizuki theorem: a subalgebra of a finite extension of the fraction field of a Noetherian domain of Krull dimension at most one is Noetherian of Krull dimension at most one, and every nonzero quotient has finite length over the base.","title":"Krull\u2013Akizuki theorem"},{"id":"RingTheory/valuation-rings/krull-akizuki-theorem/is-discrete-valuation-ring-of-algebra-ma","kind":"node","lean":"ValuationSubring.isDiscreteValuationRing_of_algebraMap_mem_of_finite","parent":4135,"result":true,"route":"../roadmap/RingTheory/valuation-rings/krull-akizuki-theorem/index.html#is-discrete-valuation-ring-of-algebra-ma","scope":"RingTheory/valuation-rings/krull-akizuki-theorem","status":"fully_proved","summary":"A valuation subring of a finite extension of the fraction field of a Noetherian domain of Krull dimension at most one, containing the base ring and different from the whole field, is a discrete valuation ring.","title":"Valuation subrings of finite extensions of dimension-one domains are DVRs"},{"data_url":"scopes/RingTheory/valuation-rings/residue-fields-of-valuation-rings.json","id":"RingTheory/valuation-rings/residue-fields-of-valuation-rings","kind":"scope","navigation":"module","parent":4098,"result":false,"route":"../roadmap/RingTheory/valuation-rings/residue-fields-of-valuation-rings/index.html","scope":"RingTheory/valuation-rings/residue-fields-of-valuation-rings","status":"fully_proved","title":"Residue fields of valuation rings"},{"id":"RingTheory/valuation-rings/residue-fields-of-valuation-rings/exists-root-mem-of-monic","kind":"node","lean":"ValuationSubring.exists_root_mem_of_monic","parent":4138,"result":true,"route":"../roadmap/RingTheory/valuation-rings/residue-fields-of-valuation-rings/index.html#exists-root-mem-of-monic","scope":"RingTheory/valuation-rings/residue-fields-of-valuation-rings","status":"fully_proved","summary":"A monic polynomial over a valuation subring of an algebraically closed field, of nonzero degree, has a root in the subring: take a root in the field, it is integral over the subring, and valuation subrings are integrally closed.","title":"Monic polynomials have roots in valuation subrings of closed fields"},{"id":"RingTheory/valuation-rings/residue-fields-of-valuation-rings/is-alg-closed-residue-field","kind":"node","lean":"ValuationSubring.isAlgClosed_residueField","parent":4138,"result":true,"route":"../roadmap/RingTheory/valuation-rings/residue-fields-of-valuation-rings/index.html#is-alg-closed-residue-field","scope":"RingTheory/valuation-rings/residue-fields-of-valuation-rings","status":"fully_proved","summary":"The residue field of a valuation subring of an algebraically closed field is algebraically closed: lift a monic irreducible polynomial to the valuation subring, take a root there, and reduce it back.","title":"Residue field of a valuation subring of a closed field is closed"},{"id":"RingTheory/valuation-rings/residue-fields-of-valuation-rings/algebra-is-algebraic-zmod-residue-field","kind":"node","lean":"ValuationSubring.algebra_isAlgebraic_zmod_residueField_of_isAlgebraic_rat","parent":4138,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residue fields in algebraic extensions of "],["math","\\(\\mathbb{Q}\\)"],["text"," are algebraic"]],"source":"Residue fields in algebraic extensions of $\\mathbb{Q}$ are algebraic"},"result":true,"route":"../roadmap/RingTheory/valuation-rings/residue-fields-of-valuation-rings/index.html#algebra-is-algebraic-zmod-residue-field","scope":"RingTheory/valuation-rings/residue-fields-of-valuation-rings","status":"fully_proved","summary":"The residue field of a valuation subring of an algebraic extension of `\u211a` is algebraic over `\ud835\udd3d_p`.","title":"Residue fields in algebraic extensions of $\\mathbb{Q}$ are algebraic"},{"id":"RingTheory/valuation-rings/residue-fields-of-valuation-rings/residue-inj-on-pow-eq-one-and-exists-res","kind":"node","lean":"ValuationSubring.residue_injOn_pow_eq_one_and_exists_residue_eq_of_isAlgClosed","parent":4138,"result":true,"route":"../roadmap/RingTheory/valuation-rings/residue-fields-of-valuation-rings/index.html#residue-inj-on-pow-eq-one-and-exists-res","scope":"RingTheory/valuation-rings/residue-fields-of-valuation-rings","status":"fully_proved","summary":"The residue map is injective on `m`-th roots of unity, and every `m`-th root of unity in the residue field lifts, when `m` is nonzero in the residue field.","title":"Roots of unity under the residue map of a valuation subring"},{"data_url":"scopes/RingTheory/local-rings.json","id":"RingTheory/local-rings","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/local-rings/index.html","scope":"RingTheory/local-rings","status":"can_prove","summary":"Local rings and their residue fields, Krull dimension, adic completion, regular local rings and length of modules.","title":"Local rings, dimension and completion"},{"id":"RingTheory/local-rings/is-domain-and-is-integrally-closed-of-fl","kind":"node","lean":"IsIntegrallyClosed.isDomain_and_isIntegrallyClosed_of_flat_of_isLocalHom","parent":4143,"result":true,"route":"../roadmap/RingTheory/local-rings/index.html#is-domain-and-is-integrally-closed-of-fl","scope":"RingTheory/local-rings","status":"fully_proved","summary":"A flat local homomorphism from a local ring to a domain and integrally closed local ring forces the source to be a domain and integrally closed. The flatness is stated with the `Flat` class of Mathlib 4.35, where the FLT source used the older `RingHom.Flat` spelling.","title":"Flat local maps reflect integrally closed domains"},{"id":"RingTheory/local-rings/length-quotient-le-of-ker-le","kind":"node","lean":"Module.length_quotient_le_of_ker_le","parent":4143,"result":true,"route":"../roadmap/RingTheory/local-rings/index.html#length-quotient-le-of-ker-le","scope":"RingTheory/local-rings","status":"fully_proved","summary":"If `ker f \u2264 K`, the length of `M \u29f8 K` is at most the length of `N`.","title":"Length bound for a quotient containing a kernel"},{"data_url":"scopes/RingTheory/local-rings/adic-completion.json","id":"RingTheory/local-rings/adic-completion","kind":"scope","parent":4143,"result":false,"route":"../roadmap/RingTheory/local-rings/adic-completion/index.html","scope":"RingTheory/local-rings/adic-completion","status":"fully_proved","title":"Adic completion"},{"id":"RingTheory/local-rings/adic-completion/of-module-finite-free-span-nat-cast","kind":"node","lean":"IsAdicComplete.of_module_finite_free_span_natCast","parent":4146,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/index.html#of-module-finite-free-span-nat-cast","scope":"RingTheory/local-rings/adic-completion","status":"fully_proved","summary":"If `\ud835\udcde` is adically complete at `(p)` and `R` is finite free over `\ud835\udcde`, then `R` is adically complete at `(p)`.","title":"Finite free modules over adically complete rings are complete"},{"id":"RingTheory/local-rings/adic-completion/two-le-ring-krull-dim-adic-completion-of","kind":"node","lean":"IsLocalRing.two_le_ringKrullDim_adicCompletion_of_two_le","parent":4146,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/index.html#two-le-ring-krull-dim-adic-completion-of","scope":"RingTheory/local-rings/adic-completion","status":"fully_proved","summary":"A Noetherian local ring of Krull dimension at least two has an adic completion of Krull dimension at least two.","title":"Adic completion preserves Krull dimension at least two"},{"id":"RingTheory/local-rings/adic-completion/is-adic-complete-of-module-finite","kind":"node","lean":"IsLocalRing.isAdicComplete_of_module_finite","parent":4146,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/index.html#is-adic-complete-of-module-finite","scope":"RingTheory/local-rings/adic-completion","status":"fully_proved","summary":"A finite local extension of an adically complete local ring is adically complete at its maximal ideal.","title":"Finite local extensions of adically complete rings are complete"},{"id":"RingTheory/local-rings/adic-completion/of-is-nilpotent","kind":"node","lean":"IsAdicComplete.of_isNilpotent","parent":4146,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/index.html#of-is-nilpotent","scope":"RingTheory/local-rings/adic-completion","status":"fully_proved","summary":"A ring is adically complete at a nilpotent ideal.","title":"Adic completeness at a nilpotent ideal"},{"id":"RingTheory/local-rings/adic-completion/is-noetherian-ring-of-is-noetherian-ring","kind":"node","lean":"AdicCompletion.isNoetherianRing_of_isNoetherianRing","parent":4146,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/index.html#is-noetherian-ring-of-is-noetherian-ring","scope":"RingTheory/local-rings/adic-completion","status":"fully_proved","summary":"The adic completion of a Noetherian ring is Noetherian.","title":"Adic completion of a Noetherian ring is Noetherian"},{"id":"RingTheory/local-rings/adic-completion/is-adic-complete-maximal-ideal","kind":"node","lean":"MvPowerSeries.isAdicComplete_maximalIdeal","parent":4146,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/index.html#is-adic-complete-maximal-ideal","scope":"RingTheory/local-rings/adic-completion","status":"fully_proved","summary":"If `R` is adically complete at its maximal ideal, then `MvPowerSeries \u03c3 R` is adically complete at its maximal ideal.","title":"Adic completeness of multivariate power series"},{"data_url":"scopes/RingTheory/local-rings/adic-completion/functoriality-of-adic-completions.json","id":"RingTheory/local-rings/adic-completion/functoriality-of-adic-completions","kind":"scope","navigation":"module","parent":4146,"result":false,"route":"../roadmap/RingTheory/local-rings/adic-completion/functoriality-of-adic-completions/index.html","scope":"RingTheory/local-rings/adic-completion/functoriality-of-adic-completions","status":"fully_proved","title":"Functoriality of adic completions"},{"id":"RingTheory/local-rings/adic-completion/functoriality-of-adic-completions/exists-ring-hom-comp-algebra-map-eq-of-f","kind":"node","lean":"AdicCompletion.exists_ringHom_comp_algebraMap_eq_of_forall_exists_pow_le_comap","parent":4153,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/functoriality-of-adic-completions/index.html#exists-ring-hom-comp-algebra-map-eq-of-f","scope":"RingTheory/local-rings/adic-completion/functoriality-of-adic-completions","status":"fully_proved","summary":"A ring homomorphism compatible with the adic topologies lifts to a map from the adic completion into an adically complete target.","title":"Lifting compatible ring maps to adic completions"},{"id":"RingTheory/local-rings/adic-completion/functoriality-of-adic-completions/exists-ring-equiv-map-of-ring-equiv","kind":"node","lean":"AdicCompletion.exists_ringEquiv_map_of_ringEquiv","parent":4153,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/functoriality-of-adic-completions/index.html#exists-ring-equiv-map-of-ring-equiv","scope":"RingTheory/local-rings/adic-completion/functoriality-of-adic-completions","status":"fully_proved","summary":"A ring equivalence induces an equivalence of adic completions.","title":"Ring equivalences induce equivalences of adic completions"},{"id":"RingTheory/local-rings/adic-completion/functoriality-of-adic-completions/exists-ring-equiv-adic-completion-maxima","kind":"node","lean":"IsLocalRing.exists_ringEquiv_adicCompletion_maximalIdeal_comp_algebraMap_of_ringEquiv","parent":4153,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/functoriality-of-adic-completions/index.html#exists-ring-equiv-adic-completion-maxima","scope":"RingTheory/local-rings/adic-completion/functoriality-of-adic-completions","status":"fully_proved","summary":"A ring equivalence between local rings induces an equivalence of their maximal-ideal adic completions.","title":"Ring equivalences of local rings induce equivalent completions"},{"data_url":"scopes/RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio.json","id":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio","kind":"scope","navigation":"module","parent":4146,"result":false,"route":"../roadmap/RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/index.html","scope":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio","status":"fully_proved","title":"Kernels of evaluation maps out of adic completions"},{"id":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/faithfully-flat-adic-completion-maximal","kind":"node","lean":"IsLocalRing.faithfullyFlat_adicCompletion_maximalIdeal","parent":4157,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/index.html#faithfully-flat-adic-completion-maximal","scope":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio","status":"fully_proved","summary":"The adic completion of a Noetherian local ring at its maximal ideal is faithfully flat.","title":"Completion of a Noetherian local ring is faithfully flat"},{"id":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/comap-map-adic-completion-eq","kind":"node","lean":"IsLocalRing.comap_map_adicCompletion_eq","parent":4157,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/index.html#comap-map-adic-completion-eq","scope":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio","status":"fully_proved","summary":"Ideals contract back to themselves along the adic completion map of a Noetherian local ring.","title":"Ideals contract to themselves along local adic completion"},{"id":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/quotient-maximal-ideal-pow-alg-hom","kind":"node","lean":"AdicCompletion.quotientMaximalIdealPowAlgHom","parent":4157,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/index.html#quotient-maximal-ideal-pow-alg-hom","scope":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio","status":"fully_proved","summary":"The finite quotient of the completion by a power of its maximal ideal identifies with the corresponding quotient of the ring.","title":"Algebra map between maximal ideal power quotients of a ring and its completion"},{"id":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/quotient-maximal-ideal-pow-alg-equiv","kind":"node","lean":"AdicCompletion.quotientMaximalIdealPowAlgEquiv","parent":4157,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/index.html#quotient-maximal-ideal-pow-alg-equiv","scope":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio","status":"fully_proved","summary":"The finite quotient of the completion by a power of its maximal ideal identifies with the corresponding quotient of the ring.","title":"Isomorphism of maximal ideal power quotients of a ring and its completion"},{"id":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/quotient-maximal-ideal-pow-alg-equiv-of","kind":"node","lean":"AdicCompletion.quotientMaximalIdealPowAlgEquivOfAlgEquiv","parent":4157,"result":true,"route":"../roadmap/RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio/index.html#quotient-maximal-ideal-pow-alg-equiv-of","scope":"RingTheory/local-rings/adic-completion/kernels-of-evaluation-maps-out-of-adic-completio","status":"fully_proved","summary":"The finite quotients of `B` by powers of its maximal ideal agree with those of `A`, transported along an algebra isomorphism with the completion of `A`.","title":"Maximal ideal power quotients transported along an isomorphism"},{"data_url":"scopes/RingTheory/local-rings/krull-dimension.json","id":"RingTheory/local-rings/krull-dimension","kind":"scope","navigation":"module","parent":4143,"result":false,"route":"../roadmap/RingTheory/local-rings/krull-dimension/index.html","scope":"RingTheory/local-rings/krull-dimension","status":"fully_proved","title":"Krull dimension"},{"id":"RingTheory/local-rings/krull-dimension/ring-krull-dim-le-of-integral","kind":"node","lean":"ringKrullDim_le_of_integral","parent":4163,"result":true,"route":"../roadmap/RingTheory/local-rings/krull-dimension/index.html#ring-krull-dim-le-of-integral","scope":"RingTheory/local-rings/krull-dimension","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and let $B$ be an $A$-algebra that is integral over $A$. Then the Krull dimension of $B$ is at most the Krull dimension of $A$.","title":"Krull dimension does not increase along integral algebras"},{"id":"RingTheory/local-rings/krull-dimension/exists-lt-series-of-integral-of-injectiv","kind":"node","lean":"exists_ltSeries_of_integral_of_injective","parent":4163,"result":true,"route":"../roadmap/RingTheory/local-rings/krull-dimension/index.html#exists-lt-series-of-integral-of-injectiv","scope":"RingTheory/local-rings/krull-dimension","status":"fully_proved","summary":"Every finite chain of prime ideals lifts along an injective integral algebra map.","title":"Prime chains lift along injective integral maps"},{"id":"RingTheory/local-rings/krull-dimension/ring-krull-dim-ge-of-integral-of-injecti","kind":"node","lean":"ringKrullDim_ge_of_integral_of_injective","parent":4163,"result":true,"route":"../roadmap/RingTheory/local-rings/krull-dimension/index.html#ring-krull-dim-ge-of-integral-of-injecti","scope":"RingTheory/local-rings/krull-dimension","status":"fully_proved","summary":"Krull dimension cannot decrease along an injective integral algebra map.","title":"Krull dimension does not decrease along injective integral maps"},{"id":"RingTheory/local-rings/krull-dimension/ring-krull-dim-eq-of-integral-of-injecti","kind":"node","lean":"ringKrullDim_eq_of_integral_of_injective","parent":4163,"result":true,"route":"../roadmap/RingTheory/local-rings/krull-dimension/index.html#ring-krull-dim-eq-of-integral-of-injecti","scope":"RingTheory/local-rings/krull-dimension","status":"fully_proved","summary":"An injective integral algebra map preserves Krull dimension.","title":"Injective integral maps preserve Krull dimension"},{"id":"RingTheory/local-rings/krull-dimension/eq-bot-of-lt-of-ne-maximal-ideal-of-ring","kind":"node","lean":"IsLocalRing.eq_bot_of_lt_of_ne_maximalIdeal_of_ringKrullDim_le_two","parent":4163,"result":true,"route":"../roadmap/RingTheory/local-rings/krull-dimension/index.html#eq-bot-of-lt-of-ne-maximal-ideal-of-ring","scope":"RingTheory/local-rings/krull-dimension","status":"fully_proved","summary":"In a Noetherian local domain whose adic completion has Krull dimension at most `2`, any prime strictly below a non-maximal prime is `\u22a5`.","title":"Short prime chains when the completion has dimension at most two"},{"data_url":"scopes/RingTheory/local-rings/local-rings-and-residue-fields.json","id":"RingTheory/local-rings/local-rings-and-residue-fields","kind":"scope","navigation":"module","parent":4143,"result":false,"route":"../roadmap/RingTheory/local-rings/local-rings-and-residue-fields/index.html","scope":"RingTheory/local-rings/local-rings-and-residue-fields","status":"fully_proved","title":"Local rings and residue fields"},{"id":"RingTheory/local-rings/local-rings-and-residue-fields/char-p-residue-field-of-nat-cast-mem-max","kind":"node","lean":"IsLocalRing.charP_residueField_of_natCast_mem_maximalIdeal","parent":4169,"result":true,"route":"../roadmap/RingTheory/local-rings/local-rings-and-residue-fields/index.html#char-p-residue-field-of-nat-cast-mem-max","scope":"RingTheory/local-rings/local-rings-and-residue-fields","status":"fully_proved","summary":"If `(p : A)` lies in the maximal ideal of a local ring `A`, the residue field has characteristic `p`: `p` maps to zero in the residue field.","title":"Characteristic of the residue field of a local ring"},{"id":"RingTheory/local-rings/local-rings-and-residue-fields/quotient-of-ne-top","kind":"node","lean":"IsLocalRing.quotient_of_ne_top","parent":4169,"result":true,"route":"../roadmap/RingTheory/local-rings/local-rings-and-residue-fields/index.html#quotient-of-ne-top","scope":"RingTheory/local-rings/local-rings-and-residue-fields","status":"fully_proved","summary":"A quotient of a local ring by a proper ideal is local. Mathlib uses this fact inline (via `IsLocalRing.of_surjective'`), but does not package it as a standalone lemma.","title":"Quotient of a local ring by a proper ideal is local"},{"data_url":"scopes/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r.json","id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","kind":"scope","navigation":"module","parent":4143,"result":false,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","title":"Maximal ideals in integral extensions of local rings"},{"id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/comap-map-ring-hom-bot-eq-map-c-ker","kind":"node","lean":"Polynomial.comap_mapRingHom_bot_eq_map_C_ker","parent":4172,"result":true,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html#comap-map-ring-hom-bot-eq-map-c-ker","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","summary":"Let $A$ be a commutative ring and $I$ an ideal of $A$. The kernel of the ring homomorphism $A[X] \\to (A/I)[X]$ that reduces every coefficient modulo $I$ is the ideal of $A[X]$ generated by the constant polynomials $a$ with $a \\in I$.","title":"Kernel of reducing polynomial coefficients modulo an ideal"},{"id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/exists-irreducible-factor-of-is-maximal","kind":"node","lean":"AdjoinRoot.exists_irreducible_factor_of_is_maximal","parent":4172,"result":true,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html#exists-irreducible-factor-of-is-maximal","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","summary":"Maximal ideals of a monogenic algebra over a local ring come from residual irreducible factors. Let $A$ be a commutative local ring with maximal ideal $\\mathfrak{m}$, and for $f \\in A[X]$ write $\\bar{f}$ for its image in $(A/\\mathfrak{m})[X]$. Let $g \\in A[X]$ and suppose that $A[X]/(g)$ is integral over $A$. Then for every maximal ideal $M$ of $A[X]/(g)$ there is a polynomial $q \\in A[X]$ such that $\\bar{q}$ is\u2026","title":"Maximal ideals of a monogenic algebra over a local ring come from residual irreducible factors"},{"id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/is-maximal-of-irreducible-factor","kind":"node","lean":"AdjoinRoot.is_maximal_of_irreducible_factor","parent":4172,"result":true,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html#is-maximal-of-irreducible-factor","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","summary":"Let $A$ be a commutative local ring with maximal ideal $\\mathfrak{m}$, and for $f \\in A[X]$ write $\\bar{f}$ for its image in $(A/\\mathfrak{m})[X]$. Let $g, q \\in A[X]$ be such that $\\bar{q}$ is irreducible in $(A/\\mathfrak{m})[X]$ and $\\bar{q}$ divides $\\bar{g}$. Then the ideal of $A[X]/(g)$ generated by the image of $\\mathfrak{m}$ and the class of $q$ is maximal.","title":"Residual irreducible factors give maximal ideals of a monogenic algebra"},{"id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/is-maximal-iff-irreducible-factor","kind":"node","lean":"AdjoinRoot.is_maximal_iff_irreducible_factor","parent":4172,"result":true,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html#is-maximal-iff-irreducible-factor","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","summary":"Let $A$ be a commutative local ring with maximal ideal $\\mathfrak{m}$, and for $f \\in A[X]$ write $\\bar{f}$ for its image in $(A/\\mathfrak{m})[X]$. Let $g \\in A[X]$ and suppose that $A[X]/(g)$ is integral over $A$. Then an ideal $M$ of $A[X]/(g)$ is maximal if and only if there is a polynomial $q \\in A[X]$ such that $\\bar{q}$ is irreducible in $(A/\\mathfrak{m})[X]$, $\\bar{q}$ divides $\\bar{g}$, and $M$ is the ideal\u2026","title":"Maximal ideals of a monogenic algebra over a local ring"},{"id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/is-maximal-iff-irreducible-factor-of-mon","kind":"node","lean":"AdjoinRoot.is_maximal_iff_irreducible_factor_of_monic","parent":4172,"result":true,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html#is-maximal-iff-irreducible-factor-of-mon","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","summary":"Let $A$ be a commutative local ring with maximal ideal $\\mathfrak{m}$, and for $f \\in A[X]$ write $\\bar{f}$ for its image in $(A/\\mathfrak{m})[X]$. Let $g \\in A[X]$ be monic. Then an ideal $M$ of $A[X]/(g)$ is maximal if and only if there is a polynomial $q \\in A[X]$ such that $\\bar{q}$ is irreducible in $(A/\\mathfrak{m})[X]$, $\\bar{q}$ divides $\\bar{g}$, and $M$ is the ideal of $A[X]/(g)$ generated by the image of\u2026","title":"Maximal ideals of a monogenic algebra over a local ring, monic case"},{"id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/map-maximal-ideal-le","kind":"node","lean":"Ideal.IsMaximal.map_maximalIdeal_le","parent":4172,"result":true,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html#map-maximal-ideal-le","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","summary":"Let $A$ be a commutative local ring with maximal ideal $\\mathfrak{m}$, and let $B$ be a commutative $A$-algebra that is integral over $A$. Then every maximal ideal $M$ of $B$ contains the ideal of $B$ generated by the image of $\\mathfrak{m}$.","title":"Maximal ideals of an integral algebra over a local ring contain the base maximal ideal"},{"id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/is-maximal-map-maximal-ideal-le","kind":"node","lean":"AdjoinRoot.isMaximal_map_maximalIdeal_le","parent":4172,"result":true,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html#is-maximal-map-maximal-ideal-le","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","summary":"Maximal ideals of an integral monogenic algebra over a local ring contain the base maximal ideal. Let $A$ be a commutative local ring with maximal ideal $\\mathfrak{m}$, let $g \\in A[X]$, and suppose that $A[X]/(g)$ is integral over $A$. Then every maximal ideal $M$ of $A[X]/(g)$ contains the ideal of $A[X]/(g)$ generated by the image of $\\mathfrak{m}$.","title":"Maximal ideals of an integral monogenic algebra over a local ring contain the base maximal ideal"},{"id":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/is-maximal-map-maximal-ideal-le-of-monic","kind":"node","lean":"AdjoinRoot.isMaximal_map_maximalIdeal_le_of_monic","parent":4172,"result":true,"route":"../roadmap/RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r/index.html#is-maximal-map-maximal-ideal-le-of-monic","scope":"RingTheory/local-rings/maximal-ideals-in-integral-extensions-of-local-r","status":"fully_proved","summary":"Maximal ideals of a monogenic algebra over a local ring contain the base maximal ideal, monic case. Let $A$ be a commutative local ring with maximal ideal $\\mathfrak{m}$ and let $g \\in A[X]$ be monic. Then every maximal ideal $M$ of $A[X]/(g)$ contains the ideal of $A[X]/(g)$ generated by the image of $\\mathfrak{m}$.","title":"Maximal ideals of a monogenic algebra over a local ring contain the base maximal ideal, monic case"},{"data_url":"scopes/RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien.json","id":"RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien","kind":"scope","navigation":"module","parent":4143,"result":false,"route":"../roadmap/RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien/index.html","scope":"RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien","status":"can_prove","title":"Reconstructing complete local rings from quotients"},{"id":"RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien/conjecture","kind":"node","lean":"MathlibExt.Algebra.CompleteLocalRingReconstructionWanted.conjecture","parent":4181,"result":true,"route":"../roadmap/RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien/index.html#conjecture","scope":"RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien","status":"fully_proved","summary":"The statement that any two commutative Noetherian local rings $A$ and $B$, with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, that are complete and Hausdorff for the $\\mathfrak m_A$-adic and $\\mathfrak m_B$-adic topologies are isomorphic as rings whenever $A/\\mathfrak m_A^r$ and $B/\\mathfrak m_B^r$ are isomorphic as rings for every natural number $r$.","title":"Complete local ring reconstruction conjecture"},{"id":"RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien/conjecture-refuted","kind":"node","lean":"MathlibExt.Algebra.CompleteLocalRingReconstructionWanted.conjecture_refuted","parent":4181,"result":true,"route":"../roadmap/RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien/index.html#conjecture-refuted","scope":"RingTheory/local-rings/reconstructing-complete-local-rings-from-quotien","status":"can_prove","summary":"There exist commutative Noetherian local rings $A$ and $B$, with maximal ideals $\\mathfrak m_A$ and $\\mathfrak m_B$, that are complete and Hausdorff for the $\\mathfrak m_A$-adic and $\\mathfrak m_B$-adic topologies, such that $A/\\mathfrak m_A^r$ and $B/\\mathfrak m_B^r$ are isomorphic as rings for every natural number $r$, but $A$ and $B$ are not isomorphic as rings.","title":"The complete local ring reconstruction conjecture is false"},{"data_url":"scopes/RingTheory/local-rings/regular-local-rings.json","id":"RingTheory/local-rings/regular-local-rings","kind":"scope","parent":4143,"result":false,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/index.html","scope":"RingTheory/local-rings/regular-local-rings","status":"fully_proved","title":"Regular local rings"},{"id":"RingTheory/local-rings/regular-local-rings/quotient-span-singleton-of-not-mem-sq-of","kind":"node","lean":"IsRegularLocalRing.quotient_span_singleton_of_notMem_sq_of_forall_minimalPrimes_notMem","parent":4184,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/index.html#quotient-span-singleton-of-not-mem-sq-of","scope":"RingTheory/local-rings/regular-local-rings","status":"fully_proved","summary":"Quotienting a regular local ring by a parameter outside the square of the maximal ideal and outside every minimal prime stays regular, with the embedding dimension dropping by exactly one.","title":"Quotient of a regular local ring by a regular parameter is regular"},{"id":"RingTheory/local-rings/regular-local-rings/unique-factorization-monoid-of-is-princi","kind":"node","lean":"IsLocalRing.uniqueFactorizationMonoid_of_isPrincipalIdealRing_quotient","parent":4184,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/index.html#unique-factorization-monoid-of-is-princi","scope":"RingTheory/local-rings/regular-local-rings","status":"fully_proved","summary":"A Noetherian local domain is a unique factorization monoid when quotienting by a principal prime ideal gives a principal ideal ring.","title":"Unique factorization from a principal ideal ring quotient"},{"id":"RingTheory/local-rings/regular-local-rings/mv-power-series-fin","kind":"node","lean":"IsRegularLocalRing.mvPowerSeries_fin","parent":4184,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/index.html#mv-power-series-fin","scope":"RingTheory/local-rings/regular-local-rings","status":"fully_proved","summary":"Multivariable power series over a regular local ring form a regular local ring. The discrete-valuation-ring case follows by typeclass inference, since a DVR is a regular local ring.","title":"Multivariable power series over a regular local ring are regular local"},{"id":"RingTheory/local-rings/regular-local-rings/is-domain-and-is-integrally-closed-of-ri","kind":"node","lean":"IsRegularLocalRing.isDomain_and_isIntegrallyClosed_of_ringKrullDim_le_one","parent":4184,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/index.html#is-domain-and-is-integrally-closed-of-ri","scope":"RingTheory/local-rings/regular-local-rings","status":"fully_proved","summary":"A regular local ring of Krull dimension at most one is a domain and integrally closed.","title":"Regular local rings of dimension at most one are normal domains"},{"id":"RingTheory/local-rings/regular-local-rings/adjoin-root-of-monic-of-coeff-mem-maxima","kind":"node","lean":"IsRegularLocalRing.adjoinRoot_of_monic_of_coeff_mem_maximalIdeal_of_coeff_zero_not_mem_sq","parent":4184,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/index.html#adjoin-root-of-monic-of-coeff-mem-maxima","scope":"RingTheory/local-rings/regular-local-rings","status":"fully_proved","summary":"Adjoining a root of a monic polynomial whose non-leading coefficients lie in the maximal ideal and whose constant coefficient lies outside the square of the maximal ideal preserves regularity and Krull dimension.","title":"Adjoining a root of an Eisenstein-like polynomial preserves regularity"},{"id":"RingTheory/local-rings/regular-local-rings/unique-factorization-monoid-of-ring-krul","kind":"node","lean":"IsRegularLocalRing.uniqueFactorizationMonoid_of_ringKrullDim_le_two","parent":4184,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/index.html#unique-factorization-monoid-of-ring-krul","scope":"RingTheory/local-rings/regular-local-rings","status":"fully_proved","summary":"A regular local ring of Krull dimension at most two is a unique factorization monoid.","title":"Regular local rings of dimension at most two are UFDs"},{"data_url":"scopes/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring.json","id":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","kind":"scope","navigation":"module","parent":4184,"result":false,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/index.html","scope":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","status":"fully_proved","title":"Power series over a regular local ring"},{"id":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/is-smul-regular-x","kind":"node","lean":"PowerSeriesRegularAux.isSMulRegular_X","parent":4191,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(X\\)"],["text"," is a regular element of the power series ring"]],"source":"$X$ is a regular element of the power series ring"},"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/index.html#is-smul-regular-x","scope":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","status":"fully_proved","summary":"`X` is a regular element of the power series ring.","title":"$X$ is a regular element of the power series ring"},{"id":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/ker-constant-coeff","kind":"node","lean":"PowerSeriesRegularAux.ker_constantCoeff","parent":4191,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/index.html#ker-constant-coeff","scope":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","status":"fully_proved","summary":"The kernel of the constant coefficient map is the span of `X`.","title":"Kernel of the constant coefficient map of power series"},{"id":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/quot-span-xequiv","kind":"node","lean":"PowerSeriesRegularAux.quotSpanXEquiv","parent":4191,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power series modulo "],["math","\\(X\\)"],["text"," recover the coefficient ring"]],"source":"Power series modulo $X$ recover the coefficient ring"},"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/index.html#quot-span-xequiv","scope":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","status":"fully_proved","summary":"Quotienting `R\u27e6X\u27e7` by the span of `X` recovers `R`.","title":"Power series modulo $X$ recover the coefficient ring"},{"id":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/ring-krull-dim-power-series","kind":"node","lean":"PowerSeries.ringKrullDim_powerSeries","parent":4191,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/index.html#ring-krull-dim-power-series","scope":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","status":"fully_proved","summary":"The Krull dimension of `R\u27e6X\u27e7` is one more than that of `R`.","title":"Krull dimension of a power series ring"},{"id":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/maximal-ideal-eq-span-image-union-x","kind":"node","lean":"PowerSeries.maximalIdeal_eq_span_image_union_X","parent":4191,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/index.html#maximal-ideal-eq-span-image-union-x","scope":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","status":"fully_proved","summary":"The maximal ideal of `R\u27e6X\u27e7` is generated by the image of a generating set of the maximal ideal of `R` together with `X`.","title":"Generators of the maximal ideal of a power series ring"},{"id":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/span-finrank-maximal-ideal-le-succ","kind":"node","lean":"PowerSeries.spanFinrank_maximalIdeal_le_succ","parent":4191,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/index.html#span-finrank-maximal-ideal-le-succ","scope":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","status":"fully_proved","summary":"The embedding dimension of `R\u27e6X\u27e7` is at most one more than that of `R`.","title":"Embedding dimension of a power series ring"},{"id":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/is-regular-local-ring-power-series","kind":"node","lean":"PowerSeries.isRegularLocalRing_powerSeries","parent":4191,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring/index.html#is-regular-local-ring-power-series","scope":"RingTheory/local-rings/regular-local-rings/power-series-over-a-regular-local-ring","status":"fully_proved","summary":"Power series over a regular local ring are regular local.","title":"Power series over a regular local ring are regular local"},{"data_url":"scopes/RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains.json","id":"RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains","kind":"scope","navigation":"module","parent":4184,"result":false,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains/index.html","scope":"RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains","status":"fully_proved","title":"Regular local rings are domains"},{"id":"RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains/exists-mem-maximal-ideal-not-mem-sq-fora","kind":"node","lean":"IsLocalRing.exists_mem_maximalIdeal_notMem_sq_forall_minimalPrimes_notMem","parent":4199,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains/index.html#exists-mem-maximal-ideal-not-mem-sq-fora","scope":"RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains","status":"fully_proved","summary":"In a Noetherian local ring of positive Krull dimension, there is a parameter in the maximal ideal, outside its square and outside every minimal prime.","title":"Existence of a parameter avoiding minimal primes and the square"},{"id":"RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains/is-domain","kind":"node","lean":"IsRegularLocalRing.isDomain","parent":4199,"result":true,"route":"../roadmap/RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains/index.html#is-domain","scope":"RingTheory/local-rings/regular-local-rings/regular-local-rings-are-domains","status":"fully_proved","summary":"Every regular local ring is a domain.","title":"Regular local rings are domains"},{"data_url":"scopes/RingTheory/ideals-and-constructions.json","id":"RingTheory/ideals-and-constructions","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/ideals-and-constructions/index.html","scope":"RingTheory/ideals-and-constructions","status":"can_prove","summary":"Ideals and submodules under localization, Hilbert's Nullstellensatz, zero divisors, products of rings and fields, and ring constructions such as Witt vectors.","title":"Ideals, localization and ring constructions"},{"data_url":"scopes/RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien.json","id":"RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien","kind":"scope","navigation":"module","parent":4202,"result":false,"route":"../roadmap/RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien/index.html","scope":"RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien","status":"fully_proved","title":"Chinese remainder theorem for polynomial quotients"},{"id":"RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien/quotient-inf-alg-equiv-pi-quotient","kind":"node","lean":"Ideal.quotientInfAlgEquivPiQuotient","parent":4203,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien/index.html#quotient-inf-alg-equiv-pi-quotient","scope":"RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien","status":"fully_proved","summary":"Let $K$ be a commutative semiring, let $R$ be a commutative ring that is a $K$-algebra, and let $(I_i)_{i \\in \\iota}$ be a family of ideals of $R$ indexed by a finite type $\\iota$ that are pairwise coprime, that is, $I_i + I_j = R$ for $i \\ne j$. Then the map sending the class of $r \\in R$ to $(r \\bmod I_i)_{i \\in \\iota}$ is an isomorphism of $K$-algebras from $R / \\bigcap_i I_i$ to $\\prod_i R / I_i$.","title":"Chinese remainder theorem as an algebra isomorphism"},{"id":"RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien/tensor-product-alg-equiv-pi","kind":"node","lean":"AdjoinRoot.tensorProductAlgEquivPi","parent":4203,"reading_title":{"format":"inline-parts-v1","parts":[["text","Base change of "],["math","\\(K(X)/(f)\\)"],["text"," along a factorization of "],["math","\\(f\\)"],["text"," over "],["math","\\(K'\\)"]],"source":"Base change of $K(X)/(f)$ along a factorization of $f$ over $K'$"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien/index.html#tensor-product-alg-equiv-pi","scope":"RingTheory/ideals-and-constructions/chinese-remainder-theorem-for-polynomial-quotien","status":"fully_proved","summary":"Let $K$ and $K'$ be fields with $K'$ a $K$-algebra, let $f \\in K[X]$, and let $(g_i)_{i \\in \\iota}$ be a family of irreducible polynomials in $K'[X]$, indexed by a finite type $\\iota$, that are pairwise non-associated and whose product $\\prod_i g_i$ is associated (equal up to a unit) to the image of $f$ in $K'[X]$. Then there is an isomorphism of $K'$-algebras","title":"Base change of $K(X)/(f)$ along a factorization of $f$ over $K'$"},{"data_url":"scopes/RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz.json","id":"RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz","kind":"scope","navigation":"module","parent":4202,"result":false,"route":"../roadmap/RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz/index.html","scope":"RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz","status":"fully_proved","title":"Hilbert's Nullstellensatz"},{"id":"RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz/vanishing-ideal-singleton-eq-ker-aeval","kind":"node","lean":"HilbertNullstellensatz.vanishingIdeal_singleton_eq_ker_aeval","parent":4206,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz/index.html#vanishing-ideal-singleton-eq-ker-aeval","scope":"RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz","status":"fully_proved","summary":"For a point $a \\in \\mathbb{C}^n$, the ideal of polynomials in $\\mathbb{C}[x_1, \\ldots, x_n]$ that vanish on $\\{a\\}$ equals the kernel of the evaluation homomorphism $\\mathbb{C}[x_1, \\ldots, x_n] \\to \\mathbb{C}$, $f \\mapsto f(a)$.","title":"Vanishing ideal of a point"},{"id":"RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz/is-maximal-iff-eq-ker-aeval","kind":"node","lean":"HilbertNullstellensatz.isMaximal_iff_eq_ker_aeval","parent":4206,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz/index.html#is-maximal-iff-eq-ker-aeval","scope":"RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz","status":"fully_proved","summary":"Let $n$ be a natural number. An ideal $I$ of the polynomial ring $\\mathbb{C}[x_1, \\ldots, x_n]$ is maximal if and only if there is a point $a \\in \\mathbb{C}^n$ such that $I$ is the kernel of the evaluation homomorphism $\\mathbb{C}[x_1, \\ldots, x_n] \\to \\mathbb{C}$, $f \\mapsto f(a)$.","title":"Weak Nullstellensatz, kernel form"},{"id":"RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz/is-maximal-quotient-iff-exists-common-ze","kind":"node","lean":"HilbertNullstellensatz.isMaximal_quotient_iff_exists_common_zero","parent":4206,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz/index.html#is-maximal-quotient-iff-exists-common-ze","scope":"RingTheory/ideals-and-constructions/hilbert-s-nullstellensatz","status":"fully_proved","summary":"Let $P_1, \\ldots, P_m$ be polynomials in $\\mathbb{C}[x_1, \\ldots, x_n]$ and let $A = \\mathbb{C}[x_1, \\ldots, x_n]/(P_1, \\ldots, P_m)$. An ideal $J$ of $A$ is maximal if and only if there is a common zero $a \\in \\mathbb{C}^n$ of $P_1, \\ldots, P_m$ such that $J$ is the image in $A$ of the ideal of polynomials vanishing at $a$.","title":"Weak Nullstellensatz, quotient form"},{"data_url":"scopes/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules.json","id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","kind":"scope","navigation":"module","parent":4202,"result":false,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","title":"Localization of ideals and submodules"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/localized-at","kind":"node","lean":"Submodule.localizedAt","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#localized-at","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring, $V$ an $A$-module, $M$ a submodule of $V$ and $S$ a submonoid of $A$. The saturation of $M$ at $S$ is the submodule of $V$ consisting of the elements $v \\in V$ such that $s v \\in M$ for some $s \\in S$.","title":"Saturation of a submodule at a submonoid"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/mem-localized-at","kind":"node","lean":"Submodule.mem_localizedAt","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#mem-localized-at","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring, $V$ an $A$-module, $M$ a submodule of $V$ and $S$ a submonoid of $A$. An element $v \\in V$ lies in the saturation of $M$ at $S$ if and only if $s v \\in M$ for some $s \\in S$.","title":"Membership in the saturation"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/mem-of-forall-mem-localized-at-maximal","kind":"node","lean":"Submodule.mem_of_forall_mem_localizedAt_maximal","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#mem-of-forall-mem-localized-at-maximal","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring, $V$ an $A$-module, $M$ a submodule of $V$ and $v \\in V$. Suppose that for every maximal ideal $\\mathfrak m$ of $A$, $v$ lies in the saturation of $M$ at $A \\setminus \\mathfrak m$, that is, $s v \\in M$ for some $s \\in A \\setminus \\mathfrak m$. Then $v \\in M$.","title":"Membership is detected at the maximal ideals"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/mem-of-forall-mem-localized-at-prime","kind":"node","lean":"Submodule.mem_of_forall_mem_localizedAt_prime","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#mem-of-forall-mem-localized-at-prime","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring, $V$ an $A$-module, $M$ a submodule of $V$ and $v \\in V$. Suppose that for every prime ideal $\\mathfrak p$ of $A$, $v$ lies in the saturation of $M$ at $A \\setminus \\mathfrak p$, that is, $s v \\in M$ for some $s \\in A \\setminus \\mathfrak p$. Then $v \\in M$.","title":"Membership is detected at the prime ideals"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/eq-i-inf-localized-at-maximal-2","kind":"node","lean":"Submodule.eq_iInf_localizedAt_maximal","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#eq-i-inf-localized-at-maximal-2","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring, $V$ an $A$-module and $M$ a submodule of $V$. For a maximal ideal $\\mathfrak m$ of $A$, let $M_{\\mathfrak m}$ be the saturation of $M$ at $A \\setminus \\mathfrak m$, that is, the set of $v \\in V$ with $s v \\in M$ for some $s \\in A \\setminus \\mathfrak m$. Then $M$ is the intersection of the submodules $M_{\\mathfrak m}$ over all maximal ideals $\\mathfrak m$ of $A$.","title":"A submodule is the intersection of its saturations at the maximal ideals"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/eq-i-inf-localized-at-prime-2","kind":"node","lean":"Submodule.eq_iInf_localizedAt_prime","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#eq-i-inf-localized-at-prime-2","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring, $V$ an $A$-module and $M$ a submodule of $V$. For a prime ideal $\\mathfrak p$ of $A$, let $M_{\\mathfrak p}$ be the saturation of $M$ at $A \\setminus \\mathfrak p$, that is, the set of $v \\in V$ with $s v \\in M$ for some $s \\in A \\setminus \\mathfrak p$. Then $M$ is the intersection of the submodules $M_{\\mathfrak p}$ over all prime ideals $\\mathfrak p$ of $A$.","title":"A submodule is the intersection of its saturations at the prime ideals"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/eq-i-inf-localized-at-maximal-eq-prime","kind":"node","lean":"Submodule.eq_iInf_localizedAt_maximal_eq_prime","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#eq-i-inf-localized-at-maximal-eq-prime","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring, $V$ an $A$-module and $M$ a submodule of $V$. For a prime ideal $\\mathfrak p$ of $A$, let $M_{\\mathfrak p}$ be the saturation of $M$ at $A \\setminus \\mathfrak p$, that is, the set of $v \\in V$ with $s v \\in M$ for some $s \\in A \\setminus \\mathfrak p$. Then the intersection of the $M_{\\mathfrak m}$ over all maximal ideals $\\mathfrak m$ of $A$ equals the intersection of the\u2026","title":"Intersections of saturations over maximal and over prime ideals agree"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/eq-i-inf-comap-map-localization-maximal","kind":"node","lean":"Ideal.eq_iInf_comap_map_localization_maximal","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#eq-i-inf-comap-map-localization-maximal","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring and $I$ an ideal of $A$. For a maximal ideal $\\mathfrak{m}$ of $A$, let $\\varphi_{\\mathfrak{m}} \\colon A \\to A_{\\mathfrak{m}}$ be the canonical map to the localization of $A$ at $\\mathfrak{m}$. Then $I = \\bigcap_{\\mathfrak{m}} \\varphi_{\\mathfrak{m}}^{-1}(I A_{\\mathfrak{m}})$, the intersection over all maximal ideals $\\mathfrak{m}$ of $A$: an ideal is the intersection of the\u2026","title":"An ideal is the intersection of its contracted localizations at maximal ideals"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/eq-i-inf-comap-map-localization-prime","kind":"node","lean":"Ideal.eq_iInf_comap_map_localization_prime","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#eq-i-inf-comap-map-localization-prime","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring and $I$ an ideal of $A$. For a prime ideal $\\mathfrak{p}$ of $A$, let $\\varphi_{\\mathfrak{p}} \\colon A \\to A_{\\mathfrak{p}}$ be the canonical map to the localization of $A$ at $\\mathfrak{p}$. Then $I = \\bigcap_{\\mathfrak{p}} \\varphi_{\\mathfrak{p}}^{-1}(I A_{\\mathfrak{p}})$, the intersection over all prime ideals $\\mathfrak{p}$ of $A$: an ideal is the intersection of the contractions to\u2026","title":"An ideal is the intersection of its contracted localizations at prime ideals"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/i-inf-comap-map-localization-maximal-eq","kind":"node","lean":"Ideal.iInf_comap_map_localization_maximal_eq_prime","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#i-inf-comap-map-localization-maximal-eq","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"Let $A$ be a commutative ring and $I$ an ideal of $A$. For a prime ideal $\\mathfrak{p}$ of $A$, let $\\varphi_{\\mathfrak{p}} \\colon A \\to A_{\\mathfrak{p}}$ be the canonical map to the localization of $A$ at $\\mathfrak{p}$. Then $\\bigcap_{\\mathfrak{m}} \\varphi_{\\mathfrak{m}}^{-1}(I A_{\\mathfrak{m}}) = \\bigcap_{\\mathfrak{p}} \\varphi_{\\mathfrak{p}}^{-1}(I A_{\\mathfrak{p}})$, where the left intersection runs over all\u2026","title":"Contracted localizations of an ideal: the maximal and prime intersections agree"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/eq-i-inf-localized-at-maximal","kind":"node","lean":"Ideal.eq_iInf_localizedAt_maximal","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#eq-i-inf-localized-at-maximal","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"The image of an ideal in the fraction field is the intersection of its localizations at maximal ideals. Let $A$ be an integral domain with fraction field $K$, let $I$ be an ideal of $A$, and let $M \\subseteq K$ be the image of $I$ under the canonical map $A \\to K$, an $A$-submodule of $K$. For a maximal ideal $\\mathfrak{m}$ of $A$, let $M_{\\mathfrak{m}}$ be the localization of $M$ at $\\mathfrak{m}$: the set of\u2026","title":"The image of an ideal in the fraction field is the intersection of its localizations at maximal ideals"},{"id":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/eq-i-inf-localized-at-prime","kind":"node","lean":"Ideal.eq_iInf_localizedAt_prime","parent":4210,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules/index.html#eq-i-inf-localized-at-prime","scope":"RingTheory/ideals-and-constructions/localization-of-ideals-and-submodules","status":"fully_proved","summary":"The image of an ideal in the fraction field is the intersection of its localizations at prime ideals. Let $A$ be an integral domain with fraction field $K$, let $I$ be an ideal of $A$, and let $M \\subseteq K$ be the image of $I$ under the canonical map $A \\to K$, an $A$-submodule of $K$. For a prime ideal $\\mathfrak{p}$ of $A$, let $M_{\\mathfrak{p}}$ be the localization of $M$ at $\\mathfrak{p}$: the set of\u2026","title":"The image of an ideal in the fraction field is the intersection of its localizations at prime ideals"},{"data_url":"scopes/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions.json","id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","kind":"scope","navigation":"module","parent":4202,"result":false,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","title":"Locally integer polynomial functions"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-subring","kind":"node","lean":"MetaMathlibExt.lipSubring","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ring "],["math","\\(\\mathrm{LIP}(U)\\)"],["text"," of locally integer polynomial functions"]],"source":"The ring $\\mathrm{LIP}(U)$ of locally integer polynomial functions"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-subring","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"fully_proved","summary":"For $U \\subseteq \\mathbb{Z}$, $\\mathrm{LIP}(U)$ is the subring of the ring of all functions $U \\to \\mathbb{Z}$ (with pointwise operations) consisting of the functions $f$ such that for every finite subset $X \\subseteq U$ there is a polynomial $p \\in \\mathbb{Z}[x]$ with $p(x) = f(x)$ for all $x \\in X$.","title":"The ring $\\mathrm{LIP}(U)$ of locally integer polynomial functions"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/mem-lip-subring-iff","kind":"node","lean":"MetaMathlibExt.mem_lipSubring_iff","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in "],["math","\\(\\mathrm{LIP}(U)\\)"]],"source":"Membership in $\\mathrm{LIP}(U)$"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#mem-lip-subring-iff","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"fully_proved","summary":"For $U \\subseteq \\mathbb{Z}$ and $f : U \\to \\mathbb{Z}$, $f$ lies in $\\mathrm{LIP}(U)$ if and only if for every finite subset $X \\subseteq U$ there is a polynomial $p \\in \\mathbb{Z}[x]$ with $p(x) = f(x)$ for all $x \\in X$.","title":"Membership in $\\mathrm{LIP}(U)$"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/triangular-not-mem-lip-subring-univ","kind":"node","lean":"MetaMathlibExt.triangular_not_mem_lipSubring_univ","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#triangular-not-mem-lip-subring-univ","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"fully_proved","summary":"The function $x \\mapsto x(x+1)/2$ on $\\mathbb{Z}$ (an integer, since $x(x+1)$ is even) does not belong to $\\mathrm{LIP}(\\mathbb{Z})$: no integer polynomial $p$ takes the values $0, 0, 1$ at $-1, 0, 1$, since $1 - (-1)$ must divide $p(1) - p(-1)$.","title":"The triangular-number function is not LIP on the integers"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/polynomial-to-lip","kind":"node","lean":"MetaMathlibExt.polynomialToLIP","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["text","The evaluation map from "],["math","\\(\\mathbb{Z}(x)\\)"],["text"," to "],["math","\\(\\mathrm{LIP}(U)\\)"]],"source":"The evaluation map from $\\mathbb{Z}(x)$ to $\\mathrm{LIP}(U)$"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#polynomial-to-lip","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"fully_proved","summary":"For $U \\subseteq \\mathbb{Z}$, the evaluation map is the ring homomorphism from $\\mathbb{Z}[x]$ to the ring $\\mathrm{LIP}(U)$ of locally integer polynomial functions on $U$ that sends each integer polynomial $p$ to the function $U \\to \\mathbb{Z}$, $u \\mapsto p(u)$.","title":"The evaluation map from $\\mathbb{Z}(x)$ to $\\mathrm{LIP}(U)$"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/polynomial-to-lip-apply","kind":"node","lean":"MetaMathlibExt.polynomialToLIP_apply","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#polynomial-to-lip-apply","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"fully_proved","summary":"For $U \\subseteq \\mathbb{Z}$, an integer polynomial $p \\in \\mathbb{Z}[x]$ and $u \\in U$, the image of $p$ under the evaluation map $\\mathbb{Z}[x] \\to \\mathrm{LIP}(U)$ takes the value $p(u)$ at $u$.","title":"Values of the evaluation map into LIP(U)"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/interpolation-poly","kind":"node","lean":"MetaMathlibExt.interpolationPoly","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#interpolation-poly","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"fully_proved","summary":"Let $X \\subseteq \\mathbb{Z}$ be finite and $f : \\mathbb{Z} \\to \\mathbb{Z}$. The interpolation polynomial $f_X \\in \\mathbb{Q}[x]$ of $f$ on $X$ is the Lagrange interpolation polynomial: the unique polynomial with rational coefficients of degree less than $|X|$ such that $f_X(x) = f(x)$ for every $x \\in X$.","title":"Interpolation polynomial"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/eval-interpolation-poly","kind":"node","lean":"MetaMathlibExt.eval_interpolationPoly","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#eval-interpolation-poly","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"fully_proved","summary":"Let $X \\subseteq \\mathbb{Z}$ be finite and $f : \\mathbb{Z} \\to \\mathbb{Z}$. The interpolation polynomial $f_X \\in \\mathbb{Q}[x]$ of $f$ on $X$ satisfies $f_X(x) = f(x)$ for every $x \\in X$.","title":"The interpolation polynomial agrees with the function on its nodes"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/degree-interpolation-poly-lt","kind":"node","lean":"MetaMathlibExt.degree_interpolationPoly_lt","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#degree-interpolation-poly-lt","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"fully_proved","summary":"Let $X \\subseteq \\mathbb{Z}$ be finite and $f : \\mathbb{Z} \\to \\mathbb{Z}$. The interpolation polynomial $f_X \\in \\mathbb{Q}[x]$ of $f$ on $X$ has degree less than $|X|$, where the zero polynomial has degree $-\\infty$.","title":"Degree bound for the interpolation polynomial"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/polynomial-to-lip-injective","kind":"node","lean":"MetaMathlibExt.polynomialToLIP_injective","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#polynomial-to-lip-injective","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"For every infinite set $U \\subseteq \\mathbb{Z}$, the evaluation map $\\mathbb{Z}[x] \\to \\mathrm{LIP}(U)$, sending $p$ to the function $u \\mapsto p(u)$, is injective: distinct integer polynomials give distinct functions on $U$.","title":"Integer polynomials embed in LIP(U)"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/mem-lifts-of-degree-lt-of-eval-eq","kind":"node","lean":"MetaMathlibExt.mem_lifts_of_degree_lt_of_eval_eq","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#mem-lifts-of-degree-lt-of-eval-eq","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $X \\subseteq \\mathbb{Z}$ be finite, let $q \\in \\mathbb{Q}[x]$ have degree less than $|X|$, and let $p \\in \\mathbb{Z}[x]$ satisfy $q(x) = p(x)$ for all $x \\in X$. Then $q$ has integer coefficients, that is, $q$ is the image in $\\mathbb{Q}[x]$ of a polynomial in $\\mathbb{Z}[x]$.","title":"Integrality of low-degree interpolants"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-eq-zero-of-infinite-zeros","kind":"node","lean":"MetaMathlibExt.lip_eq_zero_of_infinite_zeros","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-eq-zero-of-infinite-zeros","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{Z}$ and let $f : U \\to \\mathbb{Z}$ lie in the ring $\\mathrm{LIP}(U)$ of locally integer polynomial functions on $U$. If $f(x) = 0$ for infinitely many $x \\in U$, then $f(x) = 0$ for every $x \\in U$.","title":"A locally integer polynomial function with infinitely many zeros is zero"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-restrict-injective","kind":"node","lean":"MetaMathlibExt.lip_restrict_injective","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-restrict-injective","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq V \\subseteq \\mathbb{Z}$ with $U$ infinite, and let $f, g : V \\to \\mathbb{Z}$ lie in the ring $\\mathrm{LIP}(V)$ of locally integer polynomial functions on $V$. If $f(x) = g(x)$ for every $x \\in U$, then $f = g$.","title":"Locally integer polynomial functions agreeing on an infinite subset are equal"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-subring-is-domain","kind":"node","lean":"MetaMathlibExt.lipSubring_isDomain","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-subring-is-domain","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"For every infinite $U \\subseteq \\mathbb{Z}$, the ring $\\mathrm{LIP}(U)$ of locally integer polynomial functions $U \\to \\mathbb{Z}$ is an integral domain.","title":"Locally integer polynomial functions on an infinite set form a domain"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-subring-units-eq","kind":"node","lean":"MetaMathlibExt.lipSubring_units_eq","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["text","Units of "],["math","\\(\\mathrm{LIP}(U)\\)"]],"source":"Units of $\\mathrm{LIP}(U)$"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-subring-units-eq","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"For every infinite set $U \\subseteq \\mathbb{Z}$, the only units of the ring $\\mathrm{LIP}(U)$ of locally integer polynomial functions on $U$ are the constant functions $1$ and $-1$.","title":"Units of $\\mathrm{LIP}(U)$"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/exists-factorial-div-two-pow-le-abs-eval","kind":"node","lean":"MetaMathlibExt.exists_factorial_div_two_pow_le_abs_eval","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#exists-factorial-div-two-pow-le-abs-eval","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x]$ have degree $d$. Then there is an integer $x$ with $0 \\le x \\le d$ such that $|f(x)| \\ge d!/2^d$.","title":"Integer polynomials take a large value at a small nonnegative integer"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/interpolation-poly-exchange","kind":"node","lean":"MetaMathlibExt.interpolationPoly_exchange","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#interpolation-poly-exchange","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $f : \\mathbb{Z} \\to \\mathbb{Z}$, and for a finite set $Y \\subseteq \\mathbb{Z}$ write $f_Y \\in \\mathbb{Q}[x]$ for the interpolation polynomial of $f$ on $Y$. Let $X \\subseteq \\mathbb{Z}$ be finite and $a, b \\in \\mathbb{Z} \\setminus X$, and let $c$ be the coefficient of $x^{|X|+1}$ in $f_{X \\cup \\{a, b\\}}$. Then","title":"Exchange formula for interpolation polynomials"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/circuit-coeff-den","kind":"node","lean":"MetaMathlibExt.circuit_coeff_den","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#circuit-coeff-den","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $f : \\mathbb{Z} \\to \\mathbb{Z}$, and for a finite set $Y \\subseteq \\mathbb{Z}$ write $f_Y \\in \\mathbb{Q}[x]$ for the interpolation polynomial of $f$ on $Y$. Let $X \\subseteq \\mathbb{Z}$ be a finite circuit for $f$: $f_X$ does not have integer coefficients, but $f_{X \\setminus \\{a\\}}$ has integer coefficients for every $a \\in X$. Let $c$ be the coefficient of $x^{|X|-1}$ in $f_X$ and let $q \\ge 1$ be the\u2026","title":"Denominator of the top coefficient on a circuit"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/nat-progression-inter","kind":"node","lean":"MetaMathlibExt.natProgression_inter","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#nat-progression-inter","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $a_1, a_2, d_1, d_2$ be natural numbers with $d_1, d_2 > 0$, where $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$, and let $P_i = \\{a_i + d_i n : n \\in \\mathbb{N}\\}$ for $i = 1, 2$. Then $P_1 \\cap P_2$ is nonempty if and only if $\\gcd(d_1, d_2)$ divides the integer $a_1 - a_2$, and if $P_1 \\cap P_2$ is nonempty, then $P_1 \\cap P_2 = \\{a_3 + \\operatorname{lcm}(d_1, d_2)\\, n : n \\in \\mathbb{N}\\}$ for some $a_3 \\in \\mathbb{N}$.","title":"Intersection of two arithmetic progressions of natural numbers"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/nat-progression-inter-infinite","kind":"node","lean":"MetaMathlibExt.natProgression_inter_infinite","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#nat-progression-inter-infinite","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $a_1, a_2, d_1, d_2, a, b$ be natural numbers with $d_1, d_2 > 0$, where $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$, and let $P_i = \\{a_i + d_i n : n \\in \\mathbb{N}\\}$ for $i = 1, 2$. If $a \\in P_1$, $b \\in P_2$, $P_1 \\cap P_2$ is nonempty and $a \\neq b$, then $P_1 \\cap P_2$ contains infinitely many elements of the progression $\\{\\min(a, b) + |a - b|\\, n : n \\in \\mathbb{N}\\}$.","title":"Two intersecting progressions share infinitely many terms with a third"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-subring-newton-series","kind":"node","lean":"MetaMathlibExt.lipSubring_newton_series","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-subring-newton-series","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{Z}$, let $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$, and let $\\sigma : \\mathbb{N} \\to U$ be a bijection. For a sequence of integers $a = (a_k)_{k \\in \\mathbb{N}}$ define $f_a : U \\to \\mathbb{Z}$ by","title":"Newton series of locally integer polynomial functions"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/irreducible-of-infinite-prime-values","kind":"node","lean":"MetaMathlibExt.irreducible_of_infinite_prime_values","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#irreducible-of-infinite-prime-values","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{Z}$, and write $\\mathrm{LIP}(U)$ for the ring of locally integer polynomial functions on $U$: the functions $U \\to \\mathbb{Z}$ whose restriction to every finite subset of $U$ agrees with a polynomial with integer coefficients. Let $f \\in \\mathrm{LIP}(U)$. If $|f(x)|$ is a prime number for infinitely many $x \\in U$, then $f$ is irreducible in the ring $\\mathrm{LIP}(U)$.","title":"LIP functions with infinitely many prime values are irreducible"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/mem-range-polynomial-to-lip-of-bound","kind":"node","lean":"MetaMathlibExt.mem_range_polynomialToLIP_of_bound","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["text","A LIP function bounded by "],["math","\\(C|x|^d\\)"],["text"," infinitely often is a polynomial"]],"source":"A LIP function bounded by $C|x|^d$ infinitely often is a polynomial"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#mem-range-polynomial-to-lip-of-bound","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{Z}$, let $C$ be a real number and $d \\ge 0$ an integer, and let $f \\in \\mathrm{LIP}(U)$. If $|f(x)| \\le C|x|^d$ for infinitely many $x \\in U$, then there is a polynomial $p \\in \\mathbb{Z}[x]$ with $f(x) = p(x)$ for all $x \\in U$.","title":"A LIP function bounded by $C|x|^d$ infinitely often is a polynomial"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/prime-polynomial-to-lip","kind":"node","lean":"MetaMathlibExt.prime_polynomialToLIP","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#prime-polynomial-to-lip","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{Z}$ be infinite and let $p$ be a prime element of $\\mathbb{Z}[x]$ (a prime of $\\mathbb{Z}$, or an irreducible integer polynomial of content $1$). Then the function $u \\mapsto p(u)$, the image of $p$ under the evaluation map $\\mathbb{Z}[x] \\to \\mathrm{LIP}(U)$, is a prime element of $\\mathrm{LIP}(U)$.","title":"Prime integer polynomials stay prime in LIP(U)"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/mem-range-polynomial-to-lip-of-is-algebr","kind":"node","lean":"MetaMathlibExt.mem_range_polynomialToLIP_of_isAlgebraic","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathbb{Z}(x)\\)"],["text"," is algebraically closed in "],["math","\\(\\mathrm{LIP}(U)\\)"]],"source":"$\\mathbb{Z}(x)$ is algebraically closed in $\\mathrm{LIP}(U)$"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#mem-range-polynomial-to-lip-of-is-algebr","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{Z}$ be infinite, and map each $p \\in \\mathbb{Z}[x]$ to the function $u \\mapsto p(u)$ in $\\mathrm{LIP}(U)$. Let $f \\in \\mathrm{LIP}(U)$ and let $Q$ be a nonzero polynomial with coefficients in $\\mathbb{Z}[x]$ such that $Q(f) = 0$ in $\\mathrm{LIP}(U)$, where the coefficients of $Q$ are mapped into $\\mathrm{LIP}(U)$ in this way. Then there is $p \\in \\mathbb{Z}[x]$ with $f(u) = p(u)$ for all\u2026","title":"$\\mathbb{Z}(x)$ is algebraically closed in $\\mathrm{LIP}(U)$"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/exists-strict-mono-principal-ideals","kind":"node","lean":"MetaMathlibExt.exists_strictMono_principal_ideals","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#exists-strict-mono-principal-ideals","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{Z}$ be infinite, and write $\\mathrm{LIP}(U)$ for the ring of locally integer polynomial functions on $U$: the functions $U \\to \\mathbb{Z}$ whose restriction to every finite subset of $U$ agrees with a polynomial with integer coefficients. Then there are $g_0, g_1, g_2, \\ldots \\in \\mathrm{LIP}(U)$ such that the principal ideals of $\\mathrm{LIP}(U)$ they generate form a strictly increasing\u2026","title":"Infinite ascending chains of principal ideals in LIP rings"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-univ-comp","kind":"node","lean":"MetaMathlibExt.lip_univ_comp","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{LIP}(\\mathbb{Z})\\)"],["text"," is closed under composition"]],"source":"$\\mathrm{LIP}(\\mathbb{Z})$ is closed under composition"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-univ-comp","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"If $f$ and $g$ lie in $\\mathrm{LIP}(\\mathbb{Z})$, the ring of locally integer polynomial functions $\\mathbb{Z} \\to \\mathbb{Z}$, then so does the composite $x \\mapsto f(g(x))$.","title":"$\\mathrm{LIP}(\\mathbb{Z})$ is closed under composition"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-univ-sub-shift","kind":"node","lean":"MetaMathlibExt.lip_univ_sub_shift","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{LIP}(\\mathbb{Z})\\)"],["text"," is closed under the discrete derivative"]],"source":"$\\mathrm{LIP}(\\mathbb{Z})$ is closed under the discrete derivative"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-univ-sub-shift","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"If $f$ lies in $\\mathrm{LIP}(\\mathbb{Z})$, the ring of locally integer polynomial functions $\\mathbb{Z} \\to \\mathbb{Z}$, then so does the function $x \\mapsto f(x) - f(x - 1)$.","title":"$\\mathrm{LIP}(\\mathbb{Z})$ is closed under the discrete derivative"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/exists-lip-univ-not-polynomial-lt-rpow","kind":"node","lean":"MetaMathlibExt.exists_lip_univ_not_polynomial_lt_rpow","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#exists-lip-univ-not-polynomial-lt-rpow","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $\\tau : \\mathbb{N} \\to \\mathbb{R}$ with $\\tau(n) \\to +\\infty$ as $n \\to \\infty$. Write $\\mathrm{LIP}(\\mathbb{Z})$ for the ring of locally integer polynomial functions on $\\mathbb{Z}$: the functions $\\mathbb{Z} \\to \\mathbb{Z}$ whose restriction to every finite subset of $\\mathbb{Z}$ agrees with a polynomial with integer coefficients. Then there is $f \\in \\mathrm{LIP}(\\mathbb{Z})$ that is not of the form\u2026","title":"Non-polynomial LIP functions on the integers that are small on an infinite set"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/mem-range-polynomial-to-lip-of-factorial","kind":"node","lean":"MetaMathlibExt.mem_range_polynomialToLIP_of_factorial_bound","parent":4223,"reading_title":{"format":"inline-parts-v1","parts":[["text","LIP functions on "],["math","\\(\\mathbb{Z}\\)"],["text"," below the factorial bound are polynomials"]],"source":"LIP functions on $\\mathbb{Z}$ below the factorial bound are polynomials"},"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#mem-range-polynomial-to-lip-of-factorial","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $f$ lie in $\\mathrm{LIP}(\\mathbb{Z})$, the ring of locally integer polynomial functions $\\mathbb{Z} \\to \\mathbb{Z}$. If $|f(x)| < (2|x| - 1)!/2^{2|x| - 1}$ for all but finitely many $x \\in \\mathbb{Z}$, then there is a polynomial $p \\in \\mathbb{Z}[x]$ with $f(x) = p(x)$ for all $x \\in \\mathbb{Z}$.","title":"LIP functions on $\\mathbb{Z}$ below the factorial bound are polynomials"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-continuation-two-points","kind":"node","lean":"MetaMathlibExt.lip_continuation_two_points","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-continuation-two-points","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $f : \\mathbb{Z} \\to \\mathbb{Z}$, and for a finite set $Y \\subseteq \\mathbb{Z}$ write $f_Y \\in \\mathbb{Q}[x]$ for the interpolation polynomial of $f$ on $Y$. For $S \\subseteq \\mathbb{Z}$ write $\\mathrm{LIP}(S)$ for the ring of locally integer polynomial functions on $S$: the functions $S \\to \\mathbb{Z}$ whose restriction to every finite subset of $S$ agrees with a polynomial with integer coefficients. Let\u2026","title":"LIP continuation at two points"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-i-union-iff","kind":"node","lean":"MetaMathlibExt.lip_iUnion_iff","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-i-union-iff","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"For $S \\subseteq \\mathbb{Z}$, write $\\mathrm{LIP}(S)$ for the ring of locally integer polynomial functions $S \\to \\mathbb{Z}$. Let $(V_i)_{i \\in I}$ be a family of subsets of $\\mathbb{Z}$ with union $U = \\bigcup_{i \\in I} V_i$. Suppose that for all $i, j \\in I$, not necessarily distinct, and all $a_1 \\in V_i$ and $a_2 \\in V_j$ with $a_1 \\ne a_2$, the set $V_i \\cap V_j$ contains infinitely many elements of\u2026","title":"Locally integer polynomial functions on a union of sets"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-sheaf-cofinite-nat","kind":"node","lean":"MetaMathlibExt.lip_sheaf_cofinite_nat","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-sheaf-cofinite-nat","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $\\mathbb{N}^+ = \\{n \\in \\mathbb{Z} : n > 0\\}$, let $U \\subseteq \\mathbb{N}^+$ with $\\mathbb{N}^+ \\setminus U$ finite, let $f : \\mathbb{Z} \\to \\mathbb{Z}$, and for $S \\subseteq \\mathbb{Z}$ write $\\mathrm{LIP}(S)$ for the ring of locally integer polynomial (LIP) functions $S \\to \\mathbb{Z}$. Then the restriction of $f$ to $U$ lies in $\\mathrm{LIP}(U)$ if and only if every $a \\in U$ lies in some $W \\subseteq U$\u2026","title":"Locally integer polynomial functions on cofinite sets of positive integers"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-sheaf-cofinite-int","kind":"node","lean":"MetaMathlibExt.lip_sheaf_cofinite_int","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-sheaf-cofinite-int","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{Z}$ have finite complement, let $f : \\mathbb{Z} \\to \\mathbb{Z}$, and for $S \\subseteq \\mathbb{Z}$ write $\\mathrm{LIP}(S)$ for the ring of locally integer polynomial (LIP) functions $S \\to \\mathbb{Z}$. Then the restriction of $f$ to $U$ lies in $\\mathrm{LIP}(U)$ if and only if every $a \\in U$ lies in some $W \\subseteq U$ with finite complement in $\\mathbb{Z}$ such that the restriction of $f$\u2026","title":"Locally integer polynomial functions on cofinite subsets of the integers"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-sheaf-density-one-nat","kind":"node","lean":"MetaMathlibExt.lip_sheaf_densityOne_nat","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-sheaf-density-one-nat","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Say that a set $S$ of positive integers has natural density one if $\\#(S \\cap [1, x]) / x \\to 1$ as the natural number $x$ tends to infinity, and write $\\mathrm{LIP}(S)$ for the ring of locally integer polynomial (LIP) functions $S \\to \\mathbb{Z}$. Let $U$ be a set of positive integers of natural density one and let $f : \\mathbb{Z} \\to \\mathbb{Z}$. Then the restriction of $f$ to $U$ lies in $\\mathrm{LIP}(U)$ if and\u2026","title":"Locally integer polynomial functions on density one sets of positive integers"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-sheaf-density-one-int","kind":"node","lean":"MetaMathlibExt.lip_sheaf_densityOne_int","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-sheaf-density-one-int","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Say that $S \\subseteq \\mathbb{Z}$ has density one if $\\#(S \\cap [-x, x]) / (2x + 1) \\to 1$ as the natural number $x$ tends to infinity, and write $\\mathrm{LIP}(S)$ for the ring of locally integer polynomial (LIP) functions $S \\to \\mathbb{Z}$. Let $U \\subseteq \\mathbb{Z}$ have density one and let $f : \\mathbb{Z} \\to \\mathbb{Z}$. Then the restriction of $f$ to $U$ lies in $\\mathrm{LIP}(U)$ if and only if every\u2026","title":"Locally integer polynomial functions on density one subsets of the integers"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-sheaf-cofinite-primes","kind":"node","lean":"MetaMathlibExt.lip_sheaf_cofinite_primes","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-sheaf-cofinite-primes","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $P$ be the set of positive prime numbers in $\\mathbb{Z}$, let $U \\subseteq P$ with $P \\setminus U$ finite, let $f : \\mathbb{Z} \\to \\mathbb{Z}$, and for $S \\subseteq \\mathbb{Z}$ write $\\mathrm{LIP}(S)$ for the ring of locally integer polynomial (LIP) functions $S \\to \\mathbb{Z}$. Then the restriction of $f$ to $U$ lies in $\\mathrm{LIP}(U)$ if and only if every $a \\in U$ lies in some $W \\subseteq U$ with\u2026","title":"Locally integer polynomial functions on cofinite sets of primes"},{"id":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/lip-union-nat-progression","kind":"node","lean":"MetaMathlibExt.lip_union_natProgression","parent":4223,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/locally-integer-polynomial-functions/index.html#lip-union-nat-progression","scope":"RingTheory/ideals-and-constructions/locally-integer-polynomial-functions","status":"can_prove","summary":"Let $a_1, a_2 \\ge 0$ and $d_1, d_2 \\ge 1$ be integers, let $U_i = \\{a_i + d_i n : n \\in \\mathbb{Z}, n \\ge 0\\}$ for $i = 1, 2$, and suppose $U_1 \\cap U_2$ is nonempty. If $f : \\mathbb{Z} \\to \\mathbb{Z}$ restricts to an element of $\\mathrm{LIP}(U_1)$ on $U_1$ and to an element of $\\mathrm{LIP}(U_2)$ on $U_2$, then the restriction of $f$ to $U_1 \\cup U_2$ lies in $\\mathrm{LIP}(U_1 \\cup U_2)$.","title":"Gluing LIP functions on two intersecting progressions"},{"data_url":"scopes/RingTheory/ideals-and-constructions/products-of-rings-and-fields.json","id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","kind":"scope","navigation":"module","parent":4202,"result":false,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","title":"Products of rings and fields"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/restrict","kind":"node","lean":"PiFieldQuotient.restrict","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#restrict","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field and $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras indexed by a type $\\iota$. For a finite subset $S \\subseteq \\iota$, the coordinate restriction is the $K$-algebra homomorphism $\\prod_{i \\in \\iota} F_i \\to \\prod_{i \\in S} F_i$ sending $(x_i)_{i \\in \\iota}$ to $(x_i)_{i \\in S}$.","title":"Coordinate restriction"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/restrict-apply","kind":"node","lean":"PiFieldQuotient.restrict_apply","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#restrict-apply","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras indexed by a type $\\iota$, and $S \\subseteq \\iota$ a finite subset. For $x \\in \\prod_{i \\in \\iota} F_i$ and $s \\in S$, the $s$-th coordinate of the coordinate restriction of $x$ to $S$ is $x_s$.","title":"Coordinates of a restriction"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/restrict-surjective","kind":"node","lean":"PiFieldQuotient.restrict_surjective","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#restrict-surjective","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras indexed by a type $\\iota$, and $S \\subseteq \\iota$ a finite subset. The coordinate restriction $\\prod_{i \\in \\iota} F_i \\to \\prod_{i \\in S} F_i$ is surjective.","title":"Coordinate restriction is surjective"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/mem-ker-restrict-iff","kind":"node","lean":"PiFieldQuotient.mem_ker_restrict_iff","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#mem-ker-restrict-iff","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras indexed by a type $\\iota$, and $S \\subseteq \\iota$ a finite subset. An element $x \\in \\prod_{i \\in \\iota} F_i$ is sent to $0$ by the coordinate restriction $\\prod_{i \\in \\iota} F_i \\to \\prod_{i \\in S} F_i$ if and only if $x_i = 0$ for every $i \\in S$.","title":"Kernel of coordinate restriction"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/selected","kind":"node","lean":"PiFieldQuotient.selected","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#selected","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, $\\iota$ a finite type, $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras, $B$ a commutative $K$-algebra and $\\varphi : \\prod_{i \\in \\iota} F_i \\to B$ a $K$-algebra homomorphism. The selected indices of $\\varphi$ form the finite set of those $i \\in \\iota$ for which the ideal of $F_i$ generated by the image of $\\ker \\varphi$ under the $i$-th projection\u2026","title":"Selected indices of a homomorphism"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/mem-selected","kind":"node","lean":"PiFieldQuotient.mem_selected","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#mem-selected","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, $\\iota$ a finite type, $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras, $B$ a commutative $K$-algebra and $\\varphi : \\prod_{i \\in \\iota} F_i \\to B$ a $K$-algebra homomorphism. An index $i \\in \\iota$ belongs to the selected indices of $\\varphi$ if and only if the ideal of $F_i$ generated by the image of $\\ker \\varphi$ under the $i$-th projection $\\prod_{j \\in \\iota} F_j \\to F_i$ is\u2026","title":"Membership in the selected indices"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/mem-ker-iff","kind":"node","lean":"PiFieldQuotient.mem_ker_iff","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#mem-ker-iff","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, $\\iota$ a finite type, $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras, $B$ a commutative $K$-algebra and $\\varphi : \\prod_{i \\in \\iota} F_i \\to B$ a $K$-algebra homomorphism. For $x \\in \\prod_{i \\in \\iota} F_i$, $\\varphi(x) = 0$ if and only if for every $i \\in \\iota$ the coordinate $x_i$ lies in the ideal of $F_i$ generated by the image of $\\ker \\varphi$ under the $i$-th\u2026","title":"Kernel membership in a finite product of fields is componentwise"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/exists-alg-equiv-restrict","kind":"node","lean":"PiFieldQuotient.exists_algEquiv_restrict","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#exists-alg-equiv-restrict","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, $\\iota$ a finite type, $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras, and $B$ a commutative $K$-algebra. For every surjective $K$-algebra homomorphism $\\varphi : \\prod_{i \\in \\iota} F_i \\to B$ there are a finite subset $S \\subseteq \\iota$ and a $K$-algebra isomorphism $e : B \\to \\prod_{i \\in S} F_i$ such that $e \\circ \\varphi$ is the coordinate restriction\u2026","title":"Surjections from a finite product of fields are coordinate restrictions up to isomorphism"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/exists-alg-equiv","kind":"node","lean":"PiFieldQuotient.exists_algEquiv","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#exists-alg-equiv","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, $\\iota$ a finite type, $(F_i)_{i \\in \\iota}$ a family of fields that are $K$-algebras, and $B$ a commutative $K$-algebra. If $\\varphi : \\prod_{i \\in \\iota} F_i \\to B$ is a surjective $K$-algebra homomorphism, then there is a finite subset $S \\subseteq \\iota$ such that $B$ is isomorphic as a $K$-algebra to $\\prod_{i \\in S} F_i$.","title":"A surjective image of a finite product of fields is isomorphic to a subproduct"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/exists-index-equiv-alg-equiv","kind":"node","lean":"PiFieldQuotient.exists_indexEquiv_algEquiv","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#exists-index-equiv-alg-equiv","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"Let $K$ be a field, and let $(F_i)_{i \\in I}$ and $(G_j)_{j \\in J}$ be families of fields that are $K$-algebras, indexed by finite types $I$ and $J$. If $e : \\prod_{i \\in I} F_i \\to \\prod_{j \\in J} G_j$ is an isomorphism of $K$-algebras, then there is a bijection $\\sigma : I \\to J$ such that $F_i$ is isomorphic to $G_{\\sigma(i)}$ as a $K$-algebra for every $i \\in I$.","title":"Uniqueness of a finite product of fields up to permutation"},{"id":"RingTheory/ideals-and-constructions/products-of-rings-and-fields/ring-product-iff-nontrivial-idempotent","kind":"node","lean":"ProductRings.ring_product_iff_nontrivial_idempotent","parent":4261,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/products-of-rings-and-fields/index.html#ring-product-iff-nontrivial-idempotent","scope":"RingTheory/ideals-and-constructions/products-of-rings-and-fields","status":"fully_proved","summary":"A commutative ring $Q$ is isomorphic to a product $R \\times S$ of two nontrivial rings $R$ and $S$ if and only if $Q$ contains an idempotent $e$ with $e \\neq 0$ and $e \\neq 1$.","title":"Product decompositions and idempotents"},{"data_url":"scopes/RingTheory/ideals-and-constructions/witt-vectors.json","id":"RingTheory/ideals-and-constructions/witt-vectors","kind":"scope","navigation":"module","parent":4202,"result":false,"route":"../roadmap/RingTheory/ideals-and-constructions/witt-vectors/index.html","scope":"RingTheory/ideals-and-constructions/witt-vectors","status":"fully_proved","title":"Witt vectors"},{"id":"RingTheory/ideals-and-constructions/witt-vectors/coeff-eq-coeff-of-forall-ghost-component","kind":"node","lean":"WittVector.coeff_eq_coeff_of_forall_ghostComponent_eq","parent":4273,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/witt-vectors/index.html#coeff-eq-coeff-of-forall-ghost-component","scope":"RingTheory/ideals-and-constructions/witt-vectors","status":"fully_proved","summary":"Ghost components determine Witt vector coefficients when `p` is a non-zero-divisor.","title":"Ghost components determine Witt vector coefficients"},{"id":"RingTheory/ideals-and-constructions/witt-vectors/exists-forall-ghost-component-eq-of-sub","kind":"node","lean":"WittVector.exists_forall_ghostComponent_eq_of_sub_frobeniusLift_mem","parent":4273,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/witt-vectors/index.html#exists-forall-ghost-component-eq-of-sub","scope":"RingTheory/ideals-and-constructions/witt-vectors","status":"fully_proved","summary":"Compatible ghost data with a Frobenius lift is realized by a Witt vector.","title":"Ghost data compatible with a Frobenius lift come from a Witt vector"},{"id":"RingTheory/ideals-and-constructions/witt-vectors/ring-hom-map-frobenius-of-finite","kind":"node","lean":"WittVector.ringHom_map_frobenius_of_finite","parent":4273,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/witt-vectors/index.html#ring-hom-map-frobenius-of-finite","scope":"RingTheory/ideals-and-constructions/witt-vectors","status":"fully_proved","summary":"Ring homomorphisms out of Witt vectors over a finite field commute with Frobenius.","title":"Homomorphisms from Witt vectors over finite fields commute with Frobenius"},{"id":"RingTheory/ideals-and-constructions/witt-vectors/coeff-mem-span-of-ghost-component-mem-sp","kind":"node","lean":"WittVector.coeff_mem_span_of_ghostComponent_mem_span_pow_of_isReduced","parent":4273,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/witt-vectors/index.html#coeff-mem-span-of-ghost-component-mem-sp","scope":"RingTheory/ideals-and-constructions/witt-vectors","status":"fully_proved","summary":"If a ghost component of a Witt vector is highly `p`-divisible and the reduction mod `p` is reduced, all lower coefficients lie in `(p)`.","title":"Divisibility of Witt coefficients from divisible ghost components"},{"data_url":"scopes/RingTheory/ideals-and-constructions/zero-divisor-graph.json","id":"RingTheory/ideals-and-constructions/zero-divisor-graph","kind":"scope","navigation":"module","parent":4202,"result":false,"route":"../roadmap/RingTheory/ideals-and-constructions/zero-divisor-graph/index.html","scope":"RingTheory/ideals-and-constructions/zero-divisor-graph","status":"fully_proved","title":"Zero-divisor graph"},{"id":"RingTheory/ideals-and-constructions/zero-divisor-graph/vertex","kind":"node","lean":"ZeroDivisorGraph.Vertex","parent":4278,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/zero-divisor-graph/index.html#vertex","scope":"RingTheory/ideals-and-constructions/zero-divisor-graph","status":"fully_proved","summary":"For a commutative ring $R$, this is the set of elements $r \\in R$ with $r \\neq 0$ such that $sr = 0$ for some nonzero $s \\in R$, that is, the nonzero zero divisors of $R$. They are the vertices of the zero-divisor graph of $R$.","title":"Nonzero zero divisors"},{"id":"RingTheory/ideals-and-constructions/zero-divisor-graph/zero-divisor-graph","kind":"node","lean":"ZeroDivisorGraph.zeroDivisorGraph","parent":4278,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/zero-divisor-graph/index.html#zero-divisor-graph","scope":"RingTheory/ideals-and-constructions/zero-divisor-graph","status":"fully_proved","summary":"For a commutative ring $R$, the zero-divisor graph of $R$ is the simple graph whose vertices are the nonzero zero divisors of $R$, in which two distinct vertices $v$ and $w$ are adjacent exactly when $vw = 0$.","title":"The zero-divisor graph"},{"id":"RingTheory/ideals-and-constructions/zero-divisor-graph/exists-vertex-iff","kind":"node","lean":"ZeroDivisorGraph.exists_vertex_iff","parent":4278,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/zero-divisor-graph/index.html#exists-vertex-iff","scope":"RingTheory/ideals-and-constructions/zero-divisor-graph","status":"fully_proved","summary":"Let $R$ be a commutative ring. An element $r \\in R$ is a vertex of the zero-divisor graph of $R$ if and only if $r \\neq 0$ and $r$ is a zero divisor, that is, $sr = 0$ for some nonzero $s \\in R$.","title":"Vertices are the nonzero zero divisors"},{"id":"RingTheory/ideals-and-constructions/zero-divisor-graph/adj-iff","kind":"node","lean":"ZeroDivisorGraph.adj_iff","parent":4278,"result":true,"route":"../roadmap/RingTheory/ideals-and-constructions/zero-divisor-graph/index.html#adj-iff","scope":"RingTheory/ideals-and-constructions/zero-divisor-graph","status":"fully_proved","summary":"Let $R$ be a commutative ring. Two vertices $v$ and $w$ of the zero-divisor graph of $R$ are adjacent if and only if $v \\neq w$ and $vw = 0$.","title":"Adjacency in the zero-divisor graph"},{"data_url":"scopes/RingTheory/algebras-and-differentials.json","id":"RingTheory/algebras-and-differentials","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/index.html","scope":"RingTheory/algebras-and-differentials","status":"fully_proved","summary":"Algebras over commutative rings: \u00e9tale algebras and the trace form, norm and trace maps, K\u00e4hler differentials and derivations.","title":"\u00c9tale algebras, norms and differentials"},{"id":"RingTheory/algebras-and-differentials/is-pushout-of-forall-exists-unique-alg-h","kind":"node","lean":"Algebra.isPushout_of_forall_existsUnique_algHom_comp_eq","parent":4283,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/index.html#is-pushout-of-forall-exists-unique-alg-h","scope":"RingTheory/algebras-and-differentials","status":"fully_proved","summary":"A square of algebras is a pushout when it satisfies the universal property for algebra homomorphisms: for every `R'`-algebra `T` and every `g : S \u2192+* T` compatible with the maps from `R`, there is a unique `R'`-algebra map `S' \u2192\u2090[R'] T` extending `g`. The rings may lie in independent universes; it suffices to quantify `T` in the maximum of the universes of `S`, `R'`, and `S'`.","title":"Universal-property criterion for a pushout square of algebras"},{"data_url":"scopes/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos.json","id":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","kind":"scope","navigation":"module","parent":4283,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/index.html","scope":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","status":"fully_proved","title":"Algebra homomorphisms into an algebraically closed field"},{"id":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/eq-zero-of-forall-alg-hom-apply-eq-zero","kind":"node","lean":"Algebra.eq_zero_of_forall_algHom_apply_eq_zero_of_isReduced_tensorProduct","parent":4285,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/index.html#eq-zero-of-forall-alg-hom-apply-eq-zero","scope":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","status":"fully_proved","summary":"An element of a torsion-free algebra that vanishes under every algebra homomorphism into an algebraically closed field is zero, provided the tensor product with the fraction field is reduced and finite.","title":"Geometric points detect elements when the generic fiber is reduced"},{"id":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/nat-card-alg-hom-eq-finrank-of-is-alg-cl","kind":"node","lean":"Algebra.Etale.natCard_algHom_eq_finrank_of_isAlgClosed","parent":4285,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/index.html#nat-card-alg-hom-eq-finrank-of-is-alg-cl","scope":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","status":"fully_proved","summary":"Over an algebraically closed field, the number of algebra homomorphisms from an \u00e9tale algebra to the base field equals its rank.","title":"Counting algebra homomorphisms from an \u00e9tale algebra"},{"id":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/nat-card-eq-finrank-of-is-reduced-of-is","kind":"node","lean":"AlgHom.natCard_eq_finrank_of_isReduced_of_isAlgClosed","parent":4285,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/index.html#nat-card-eq-finrank-of-is-reduced-of-is","scope":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","status":"fully_proved","summary":"A finite reduced algebra over an algebraically closed field has `finrank` many algebra homomorphisms to the field.","title":"Counting algebra homomorphisms of a finite reduced algebra"},{"id":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/eq-zero-of-forall-lift-apply-eq-zero","kind":"node","lean":"Algebra.TensorProduct.eq_zero_of_forall_lift_apply_eq_zero","parent":4285,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/index.html#eq-zero-of-forall-lift-apply-eq-zero","scope":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","status":"fully_proved","summary":"If the scalar-extended evaluation map is injective, a tensor vanishing at all pairs of selected algebraic points is zero. The algebra and the point index may be infinite; no closedness or perfection assumption is needed.","title":"A tensor vanishing at all pairs of algebraic points is zero"},{"id":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/exists-alg-hom-adjoin-of-linear-relation","kind":"node","lean":"Algebra.exists_algHom_adjoin_of_linearRelations","parent":4285,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/index.html#exists-alg-hom-adjoin-of-linear-relation","scope":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","status":"fully_proved","summary":"A unital, multiplicatively consistent assignment on generators extends to an algebra homomorphism when it preserves every finite linear relation.","title":"Algebra maps from relation-preserving generator assignments"},{"id":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/exists-alg-hom-adjoin-of-prime-subfield","kind":"node","lean":"Algebra.exists_algHom_adjoin_of_primeSubfield","parent":4285,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/index.html#exists-alg-hom-adjoin-of-prime-subfield","scope":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","status":"fully_proved","summary":"A ring character extends to the algebra generated by another ring image when its kernel contains the first kernel and a separating family of linear forms takes prime-subfield values on that image.","title":"Extending a ring character via prime-subfield-valued linear forms"},{"id":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/exists-comp-algebra-map-eq-and-ker-eq-of","kind":"node","lean":"RingHom.exists_comp_algebraMap_eq_and_ker_eq_of_isIntegral_of_isAlgClosed","parent":4285,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos/index.html#exists-comp-algebra-map-eq-and-ker-eq-of","scope":"RingTheory/algebras-and-differentials/algebra-homomorphisms-into-an-algebraically-clos","status":"fully_proved","summary":"A homomorphism from `R` to an algebraically closed field extends to an integral extension `S` with any prescribed prime kernel lying over `ker \u03c7`.","title":"Lifting maps to algebraically closed fields along integral extensions"},{"data_url":"scopes/RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule.json","id":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule","kind":"scope","parent":4283,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/index.html","scope":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule","status":"fully_proved","title":"Iterated derivations and the Leibniz rule"},{"id":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterate-pow-of-apply-eq-smul-sq","kind":"node","lean":"Derivation.iterate_pow_of_apply_eq_smul_sq","parent":4293,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterated derivation of a power when "],["math","\\(D\\varepsilon = r\\varepsilon^2\\)"]],"source":"Iterated derivation of a power when $D\\varepsilon = r\\varepsilon^2$"},"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/index.html#iterate-pow-of-apply-eq-smul-sq","scope":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule","status":"fully_proved","summary":"If `D epsilon = r \u2022 epsilon\u00b2`, then the `n`th iterate of `D` on `epsilon^m` has coefficient `r^n` times the ascending factorial of `m`. The coefficient is not cancelled, so the formula remains valid in positive characteristic.","title":"Iterated derivation of a power when $D\\varepsilon = r\\varepsilon^2$"},{"data_url":"scopes/RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule.json","id":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule","kind":"scope","navigation":"module","parent":4293,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule/index.html","scope":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule","status":"fully_proved","title":"Iterated Leibniz rule"},{"id":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule/iterate-mul","kind":"node","lean":"Derivation.iterate_mul","parent":4295,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule/index.html#iterate-mul","scope":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule","status":"fully_proved","summary":"Leibniz's rule for an iterated derivation. The natural weights are not cancelled, so the identity remains valid in positive characteristic.","title":"Leibniz rule for iterated derivations"},{"id":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule/iterate-mul-div","kind":"node","lean":"Derivation.iterate_mul_div","parent":4295,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule/index.html#iterate-mul-div","scope":"RingTheory/algebras-and-differentials/iterated-derivations-and-the-leibniz-rule/iterated-leibniz-rule","status":"fully_proved","summary":"The factorial-normalized iterated Leibniz rule for a rational derivation.","title":"Factorial-normalized iterated Leibniz rule"},{"data_url":"scopes/RingTheory/algebras-and-differentials/k-hler-differentials.json","id":"RingTheory/algebras-and-differentials/k-hler-differentials","kind":"scope","navigation":"module","parent":4283,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/k-hler-differentials/index.html","scope":"RingTheory/algebras-and-differentials/k-hler-differentials","status":"fully_proved","title":"K\u00e4hler differentials"},{"id":"RingTheory/algebras-and-differentials/k-hler-differentials/d-ne-zero-of-transcendental","kind":"node","lean":"KaehlerDifferential.D_ne_zero_of_transcendental","parent":4298,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/k-hler-differentials/index.html#d-ne-zero-of-transcendental","scope":"RingTheory/algebras-and-differentials/k-hler-differentials","status":"fully_proved","summary":"The differential of a transcendental separable element is nonzero.","title":"Differential of a transcendental element is nonzero"},{"id":"RingTheory/algebras-and-differentials/k-hler-differentials/span-d-eq-top-of-transcendental","kind":"node","lean":"KaehlerDifferential.span_D_eq_top_of_transcendental","parent":4298,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/k-hler-differentials/index.html#span-d-eq-top-of-transcendental","scope":"RingTheory/algebras-and-differentials/k-hler-differentials","status":"fully_proved","summary":"The differential of a transcendental separable element spans the K\u00e4hler differentials.","title":"Differential of a transcendental element spans the differentials"},{"id":"RingTheory/algebras-and-differentials/k-hler-differentials/exists-unique-smul-d-of-transcendental","kind":"node","lean":"KaehlerDifferential.exists_unique_smul_D_of_transcendental","parent":4298,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/k-hler-differentials/index.html#exists-unique-smul-d-of-transcendental","scope":"RingTheory/algebras-and-differentials/k-hler-differentials","status":"fully_proved","summary":"Every K\u00e4hler differential is uniquely a scalar multiple of the differential of a transcendental separable element.","title":"K\u00e4hler differentials are unique multiples of dt for transcendental t"},{"data_url":"scopes/RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras.json","id":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras","kind":"scope","navigation":"module","parent":4283,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/index.html","scope":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras","status":"fully_proved","title":"Localizations of smooth and \u00e9tale algebras"},{"id":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/is-domain-and-is-integrally-closed-of-is","kind":"node","lean":"Algebra.Etale.isDomain_and_isIntegrallyClosed_of_isLocalization_atPrime","parent":4302,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/index.html#is-domain-and-is-integrally-closed-of-is","scope":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras","status":"fully_proved","summary":"The localization of an \u00e9tale algebra at a prime is a domain and integrally closed.","title":"Localizations of \u00e9tale algebras are integrally closed domains"},{"id":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/is-domain-and-is-integrally-closed-of-is-2","kind":"node","lean":"Algebra.Smooth.isDomain_and_isIntegrallyClosed_of_isLocalization_atPrime","parent":4302,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/index.html#is-domain-and-is-integrally-closed-of-is-2","scope":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras","status":"fully_proved","summary":"The localization of a smooth algebra at a prime is a domain and integrally closed.","title":"Localizations of smooth algebras are integrally closed domains"},{"id":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/is-domain-of-maximal-ideal-eq-span","kind":"node","lean":"IsLocalRing.isDomain_of_maximalIdeal_eq_span","parent":4302,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/index.html#is-domain-of-maximal-ideal-eq-span","scope":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras","status":"fully_proved","summary":"A Noetherian local ring whose maximal ideal is generated by a single regular element is a domain.","title":"Local rings with principal regular maximal ideal are domains"},{"id":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/is-discrete-valuation-ring-localization","kind":"node","lean":"Algebra.IsStandardSmoothOfRelativeDimension.isDiscreteValuationRing_localization_atPrime","parent":4302,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras/index.html#is-discrete-valuation-ring-localization","scope":"RingTheory/algebras-and-differentials/localizations-of-smooth-and-tale-algebras","status":"fully_proved","summary":"The localization at a maximal ideal of a standard smooth algebra of relative dimension `1` over a field is a discrete valuation ring.","title":"Smooth curves have discrete valuation rings as local rings"},{"data_url":"scopes/RingTheory/algebras-and-differentials/norm-and-trace-of-algebras.json","id":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras","kind":"scope","navigation":"module","parent":4283,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/index.html","scope":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras","status":"fully_proved","title":"Norm and trace of algebras"},{"id":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/norm-base-change","kind":"node","lean":"Algebra.norm_baseChange","parent":4307,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/index.html#norm-base-change","scope":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras","status":"fully_proved","summary":"Let $A$, $A'$ and $B$ be commutative rings, where $A'$ and $B$ are $A$-algebras and $B$ is a free $A$-module of finite rank. Then for every $b \\in B$, the image of the norm $N_{B/A}(b)$ under the structure map $A \\to A'$ equals the norm $N_{(A' \\otimes_A B)/A'}(1 \\otimes b)$ of $1 \\otimes b$ in the $A'$-algebra $A' \\otimes_A B$.","title":"The norm commutes with base change"},{"id":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/trace-base-change","kind":"node","lean":"Algebra.trace_baseChange","parent":4307,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/index.html#trace-base-change","scope":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras","status":"fully_proved","summary":"Let $A$, $A'$ and $B$ be commutative rings, where $A'$ and $B$ are $A$-algebras and $B$ is a free $A$-module of finite rank. Then for every $b \\in B$, the image of the trace $T_{B/A}(b)$ under the structure map $A \\to A'$ equals the trace $T_{(A' \\otimes_A B)/A'}(1 \\otimes b)$ of $1 \\otimes b$ in the $A'$-algebra $A' \\otimes_A B$.","title":"The trace commutes with base change"},{"id":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/norm-prod","kind":"node","lean":"Algebra.norm_prod","parent":4307,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/index.html#norm-prod","scope":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras","status":"fully_proved","summary":"The norm of a product algebra is the product of the norms of the components: left multiplication by `(x\u2081, x\u2082)` is the product map of the left multiplications, and the determinant of a product map is the product of the determinants.","title":"Norm in a product algebra is the product of norms"},{"id":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/norm-of-subsingleton","kind":"node","lean":"Algebra.norm_of_subsingleton","parent":4307,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/norm-and-trace-of-algebras/index.html#norm-of-subsingleton","scope":"RingTheory/algebras-and-differentials/norm-and-trace-of-algebras","status":"fully_proved","summary":"The norm on a subsingleton algebra is `1`: left multiplication by any element is the identity map on a subsingleton module, hence has determinant `1`.","title":"Norm on a subsingleton algebra is one"},{"data_url":"scopes/RingTheory/algebras-and-differentials/point-derivations.json","id":"RingTheory/algebras-and-differentials/point-derivations","kind":"scope","navigation":"module","parent":4283,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/point-derivations/index.html","scope":"RingTheory/algebras-and-differentials/point-derivations","status":"fully_proved","title":"Point derivations"},{"id":"RingTheory/algebras-and-differentials/point-derivations/point-derivations","kind":"node","lean":"Algebra.PointDerivations","parent":4312,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/point-derivations/index.html#point-derivations","scope":"RingTheory/algebras-and-differentials/point-derivations","status":"fully_proved","summary":"Point derivations at a `k`-point `ev` of a `k`-algebra `A` with values in a `k`-module `M`.","title":"Point derivations of an algebra"},{"id":"RingTheory/algebras-and-differentials/point-derivations/map","kind":"node","lean":"Algebra.PointDerivations.map","parent":4312,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/point-derivations/index.html#map","scope":"RingTheory/algebras-and-differentials/point-derivations","status":"fully_proved","summary":"Functoriality of point derivations in the coefficient module.","title":"Functoriality of point derivations in the coefficient module"},{"data_url":"scopes/RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains.json","id":"RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains","kind":"scope","navigation":"module","parent":4283,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains/index.html","scope":"RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains","status":"fully_proved","title":"Tensor products of reduced algebras and domains"},{"id":"RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains/is-reduced-tensor-product-of-perfect-fie","kind":"node","lean":"Algebra.isReduced_tensorProduct_of_perfectField","parent":4315,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains/index.html#is-reduced-tensor-product-of-perfect-fie","scope":"RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains","status":"fully_proved","summary":"Tensoring a reduced algebra with a field extension over a perfect field preserves reducedness.","title":"Reduced algebras stay reduced under base change over a perfect field"},{"id":"RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains/is-domain-of-is-alg-closed-of-finite-typ","kind":"node","lean":"Algebra.TensorProduct.isDomain_of_isAlgClosed_of_finiteType","parent":4315,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains/index.html#is-domain-of-is-alg-closed-of-finite-typ","scope":"RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains","status":"fully_proved","summary":"Over an algebraically closed field, the tensor product of a finite-type domain with a domain is a domain.","title":"Tensor products of domains over an algebraically closed field"},{"id":"RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains/is-domain-of-is-alg-closed-of-field","kind":"node","lean":"Algebra.TensorProduct.isDomain_of_isAlgClosed_of_field","parent":4315,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains/index.html#is-domain-of-is-alg-closed-of-field","scope":"RingTheory/algebras-and-differentials/tensor-products-of-reduced-algebras-and-domains","status":"fully_proved","summary":"Over an algebraically closed field, the tensor product of a field with a domain is a domain.","title":"Field-with-domain tensor products over an algebraically closed field"},{"data_url":"scopes/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form.json","id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","kind":"scope","navigation":"module","parent":4283,"result":false,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","title":"\u00c9tale algebras and the trace form"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/of-is-semisimple-ring-tensor-product","kind":"node","lean":"Algebra.Etale.of_isSemisimpleRing_tensorProduct","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#of-is-semisimple-ring-tensor-product","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $K$ be a field, $\\Omega$ an algebraically closed field extension of $K$, and $A$ a commutative $K$-algebra that is finite-dimensional over $K$. If the ring $\\Omega \\otimes_K A$ is semisimple, then $A$ is \u00e9tale over $K$.","title":"\u00c9tale criterion via semisimple base change"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/trace-pi","kind":"node","lean":"Algebra.trace_pi","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#trace-pi","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $K$ be a field, and for a finite-dimensional commutative $K$-algebra $A$ and $z \\in A$ write $\\operatorname{Tr}_{A/K}(z)$ for the trace of the $K$-linear map given by multiplication by $z$. Let $(A_i)_{i \\in I}$ be a family of finite-dimensional commutative $K$-algebras indexed by a finite set $I$, and let $P = \\prod_{i \\in I} A_i$. Then for every $x = (x_i)_{i \\in I}$ in $P$,\u2026","title":"Trace on a finite product of algebras"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/trace-form-pi","kind":"node","lean":"Algebra.traceForm_pi","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#trace-form-pi","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $K$ be a field, and for a finite-dimensional commutative $K$-algebra $A$ and $z \\in A$ write $\\operatorname{Tr}_{A/K}(z)$ for the trace of the $K$-linear map given by multiplication by $z$. Let $(A_i)_{i \\in I}$ be a family of finite-dimensional commutative $K$-algebras indexed by a finite set $I$, and let $P = \\prod_{i \\in I} A_i$. Then for all $x = (x_i)_{i \\in I}$ and $y = (y_i)_{i \\in I}$ in $P$,\u2026","title":"Trace form of a finite product of algebras"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/trace-form-apply-of-alg-equiv","kind":"node","lean":"Algebra.traceForm_apply_of_algEquiv","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#trace-form-apply-of-alg-equiv","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"For a commutative algebra $A$ over a field $K$ and $z \\in A$, write $\\operatorname{Tr}_{A/K}(z)$ for the trace of the $K$-linear map given by multiplication by $z$, taken to be $0$ when $A$ is infinite-dimensional over $K$. Let $R$ and $S$ be commutative algebras over $K$ and let $e : R \\to S$ be an isomorphism of $K$-algebras. Then for all $x, y \\in R$ we have\u2026","title":"The trace form is invariant under algebra isomorphisms"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/trace-form-apply-eq-zero-of-is-nilpotent","kind":"node","lean":"Algebra.traceForm_apply_eq_zero_of_isNilpotent","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#trace-form-apply-eq-zero-of-is-nilpotent","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $R$ be a commutative algebra over a field $K$, and for $z \\in R$ write $\\operatorname{Tr}_{R/K}(z)$ for the trace of the $K$-linear map given by multiplication by $z$, taken to be $0$ when $R$ is infinite-dimensional over $K$. If $x, y \\in R$ and $x$ is nilpotent, then the trace form of $R$ over $K$ vanishes at $(x, y)$, that is, $\\operatorname{Tr}_{R/K}(xy) = 0$.","title":"Nilpotent elements pair to zero under the trace form"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/is-reduced-of-trace-form-nondegenerate","kind":"node","lean":"Algebra.isReduced_of_traceForm_nondegenerate","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#is-reduced-of-trace-form-nondegenerate","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $R$ be a commutative algebra over a field $K$. For $z \\in R$ write $\\operatorname{Tr}_{R/K}(z)$ for the trace of the $K$-linear map given by multiplication by $z$, taken to be $0$ when $R$ is infinite-dimensional over $K$. If the trace form $(x, y) \\mapsto \\operatorname{Tr}_{R/K}(xy)$ of $R$ over $K$ is nondegenerate, then $R$ is reduced, that is, its only nilpotent element is $0$.","title":"A nondegenerate trace form forces the algebra to be reduced"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/trace-form-nondegenerate-iff-etale","kind":"node","lean":"Algebra.traceForm_nondegenerate_iff_etale","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#trace-form-nondegenerate-iff-etale","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $R$ be a finite-dimensional commutative algebra over a field $K$, and for $z \\in R$ write $\\operatorname{Tr}_{R/K}(z)$ for the trace of the $K$-linear map given by multiplication by $z$. Then the trace form $(x, y) \\mapsto \\operatorname{Tr}_{R/K}(xy)$ of $R$ over $K$ is nondegenerate if and only if $R$ is \u00e9tale over $K$.","title":"Trace form criterion for \u00e9tale algebras"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/discr-ne-zero-iff-trace-form-nondegenera","kind":"node","lean":"Algebra.discr_ne_zero_iff_traceForm_nondegenerate","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#discr-ne-zero-iff-trace-form-nondegenera","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $R$ be a commutative algebra over a field $K$ and let $b$ be a $K$-basis of $R$ indexed by a finite set. For $z \\in R$ write $\\operatorname{Tr}_{R/K}(z)$ for the trace of the $K$-linear map given by multiplication by $z$. Then the discriminant of $b$ over $K$ is nonzero if and only if the trace form $(x, y) \\mapsto \\operatorname{Tr}_{R/K}(xy)$ of $R$ over $K$ is nondegenerate.","title":"Discriminant and nondegeneracy of the trace form"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/discr-ne-zero-iff-etale","kind":"node","lean":"Algebra.discr_ne_zero_iff_etale","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#discr-ne-zero-iff-etale","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $R$ be a commutative algebra over a field $K$ and let $b$ be a $K$-basis of $R$ indexed by a finite set. Then the discriminant of $b$ over $K$ is nonzero if and only if $R$ is \u00e9tale over $K$.","title":"\u00c9tale algebras are detected by the discriminant"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/discr-ne-zero-of-etale","kind":"node","lean":"Algebra.discr_ne_zero_of_etale","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#discr-ne-zero-of-etale","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $R$ be a commutative algebra over a field $K$ that is \u00e9tale over $K$. Then for every $K$-basis $b$ of $R$ indexed by a finite set, the discriminant of $b$ over $K$ is nonzero.","title":"\u00c9tale algebras have nonzero discriminant"},{"id":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/etale-of-discr-ne-zero","kind":"node","lean":"Algebra.etale_of_discr_ne_zero","parent":4319,"result":true,"route":"../roadmap/RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form/index.html#etale-of-discr-ne-zero","scope":"RingTheory/algebras-and-differentials/tale-algebras-and-the-trace-form","status":"fully_proved","summary":"Let $R$ be a commutative algebra over a field $K$ and let $b$ be a $K$-basis of $R$ indexed by a finite set. If the discriminant of $b$ over $K$ is nonzero, then $R$ is \u00e9tale over $K$.","title":"A nonzero discriminant forces \u00e9taleness"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions.json","id":"RingTheory/syzygies-and-resolutions","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/index.html","scope":"RingTheory/syzygies-and-resolutions","status":"fully_proved","summary":"Free resolutions of modules over polynomial rings, Hilbert's syzygy theorem, and graded Betti numbers and Poincar\u00e9 series of multigraded modules.","title":"Syzygies and free resolutions"},{"id":"RingTheory/syzygies-and-resolutions/hilbert-syzygy","kind":"node","lean":"MathlibExt.RingTheory.Polynomial.HilbertSyzygyWanted.hilbert_syzygy","parent":4331,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/index.html#hilbert-syzygy","scope":"RingTheory/syzygies-and-resolutions","status":"fully_proved","summary":"Let $k$ be a field, let $n$ be a natural number, and let $M$ be a finitely generated module over the polynomial ring $k[X_1, \\dots, X_n]$ in $n$ variables. Then $M$ has projective dimension at most $n$ in the category of $k[X_1, \\dots, X_n]$-modules.","title":"Hilbert's syzygy theorem"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series.json","id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","kind":"scope","parent":4331,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","title":"Multigraded Betti numbers and Poincar\u00e9 series"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/exists-finset-homogeneous-generators","kind":"node","lean":"MetaMathlibExt.MultigradedPolynomialModule.exists_finset_homogeneous_generators","parent":4333,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html#exists-finset-homogeneous-generators","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be a multigraded polynomial module over $R$, with degree-$\\alpha$ components $M_\\alpha$ for $\\alpha \\in \\mathbb{Z}^n$, that is finitely generated as an $R$-module. Then there is a finite set $s \\subseteq M$ of homogeneous elements, each lying in $M_\\alpha$ for some $\\alpha \\in \\mathbb{Z}^n$, that generates $M$ as an $R$-module.","title":"Finite homogeneous generating sets"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/base-changed-degree-piece-finite","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution.baseChangedDegreePiece_finite","parent":4333,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html#base-changed-degree-piece-finite","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded polynomial module over $R$, and let $F$ be a multigraded free resolution of $M$ whose term $F_i$ has the homogeneous basis $(e_j)_{j \\in \\iota_i}$. Write $K \\otimes_R F$ for the chain complex of $K$-vector spaces obtained from $F$ by extending scalars along the evaluation $R \\to K$ at $x = 0$. If $\\iota_i$ is finite, then for every\u2026","title":"Finite bases give finite-dimensional multidegree pieces"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/base-changed-degree-decomposition","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution.baseChangedDegreeDecomposition","parent":4333,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html#base-changed-degree-decomposition","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded polynomial module over $R$, and let $F$ be a multigraded free resolution of $M$ whose term $F_i$ has the homogeneous basis $(e_j)_{j \\in \\iota_i}$. Write $K \\otimes_R F$ for the chain complex of $K$-vector spaces obtained from $F$ by extending scalars along the evaluation $R \\to K$ at $x = 0$, and for $\\alpha \\in \\mathbb{Z}^n$ call the\u2026","title":"Multidegree decomposition of the base-changed resolution"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-number-eq-of-base-chan","kind":"node","lean":"MetaMathlibExt.multigradedBettiNumber_eq_of_baseChange","parent":4333,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html#multigraded-betti-number-eq-of-base-chan","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be a $\\mathbb{Z}^n$-graded $R$-module with a fixed multigraded free resolution $P_\\bullet \\to M$. Any two choices of multigraded base change data for $P_\\bullet$ define the same multigraded Betti numbers, in every multidegree $\\alpha \\in \\mathbb{Z}^n$ and every homological index $i \\in \\mathbb{N}$.","title":"Betti numbers do not depend on the choice of base change data"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/exists-finite-multigraded-free-resolutio","kind":"node","lean":"MetaMathlibExt.exists_finite_multigraded_free_resolution","parent":4333,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html#exists-finite-multigraded-free-resolutio","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be a multigraded $R$-module that is finitely generated over $R$. Then there are a projective resolution $P_\\bullet \\to M$ over $R$, index types $\\iota_i$ for $i \\in \\mathbb{N}$, a multigraded free resolution structure on $P_\\bullet$ for the given grading of $M$ in which $P_i$ has basis indexed by $\\iota_i$, and a number $b \\in \\mathbb{N}$ such that every\u2026","title":"Existence of finite multigraded free resolutions"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincare-coefficients-has-fi","kind":"node","lean":"MetaMathlibExt.multigradedPoincareCoefficients_hasFiniteSupport_of_finiteResolution","parent":4333,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html#multigraded-poincare-coefficients-has-fi","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded $R$-module, and let $D$ be base-change data for a multigraded free resolution $F_\\bullet$ of $M$ in which $F_i$ has a homogeneous basis indexed by $\\iota_i$. Suppose every $\\iota_i$ is finite and, for some $b \\in \\mathbb{N}$, $\\iota_i$ is empty for every $i > b$. Then the multigraded Poincar\u00e9 coefficient family of $D$ is nonzero at only\u2026","title":"Finite support of the multigraded Poincar\u00e9 coefficients"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/base-change-mem-degree-span","kind":"node","lean":"MetaMathlibExt.MultigradedFreeModule.baseChange_mem_degreeSpan","parent":4333,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html#base-change-mem-degree-span","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","summary":"Let $K$ be a field, $R = K[x_1, \\dots, x_n]$, and let $F$ be a multigraded free $R$-module with basis $(b_j)_{j \\in \\iota}$, where each $b_j$ is homogeneous of degree $d_j \\in \\mathbb{Z}^n$. Make $K$ an $R$-algebra through the zero evaluation $R \\to K$, which sends every variable to $0$. If $x \\in F$ is homogeneous of degree $\\alpha \\in \\mathbb{Z}^n$, then $1 \\otimes x \\in K \\otimes_R F$ lies in the $K$-linear span\u2026","title":"Base change of a homogeneous element along zero evaluation"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/base-changed-differential-preserves-degr","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution.baseChangedDifferential_preservesDegree","parent":4333,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/index.html#base-changed-differential-preserves-degr","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded polynomial module over $R$, and let $F$ be a multigraded free resolution of $M$ whose term $F_i$ has the homogeneous basis $(e_j)_{j \\in \\iota_i}$. Write $K \\otimes_R F$ for the chain complex of $K$-vector spaces obtained from $F$ by extending scalars along the evaluation $R \\to K$ at $x = 0$, and for $\\alpha \\in \\mathbb{Z}^n$ call the\u2026","title":"The base-changed differential preserves multidegree"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules.json","id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules","kind":"scope","navigation":"module","parent":4333,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules/index.html","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules","status":"fully_proved","title":"Multigraded polynomial modules"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules/nat-degree-to-int","kind":"node","lean":"MetaMathlibExt.natDegreeToInt","parent":4342,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules/index.html#nat-degree-to-int","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules","status":"fully_proved","summary":"For an exponent vector $a = (a_1, \\dots, a_n) \\in \\mathbb{N}^n$ of a monomial, this is the vector $(a_1, \\dots, a_n) \\in \\mathbb{Z}^n$ obtained by viewing each entry $a_i$ as an integer.","title":"Exponent vectors as integer vectors"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules/multigraded-polynomial-module","kind":"node","lean":"MetaMathlibExt.MultigradedPolynomialModule","parent":4342,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules/index.html#multigraded-polynomial-module","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-polynomial-modules","status":"fully_proved","summary":"Let $K$ be a field, $n \\in \\mathbb{N}$ and $R = K[x_1, \\dots, x_n]$, and let $M$ be an $R$-module that is also a $K$-vector space. A $\\mathbb{Z}^n$-grading of $M$ compatible with monomial multiplication consists of $K$-subspaces $M_\\alpha \\subseteq M$, one for each $\\alpha \\in \\mathbb{Z}^n$, such that $M$ is the direct sum of the $M_\\alpha$, the action of each $k \\in K$ on $M$ agrees with multiplication by the\u2026","title":"Multigraded modules over a polynomial ring"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut.json","id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut","kind":"scope","navigation":"module","parent":4333,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut/index.html","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut","status":"fully_proved","title":"Multigraded free modules and graded free resolutions"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut/is-degree-zero","kind":"node","lean":"MetaMathlibExt.IsDegreeZero","parent":4345,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut/index.html#is-degree-zero","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ and $N$ be multigraded $R$-modules with homogeneous pieces $M_\\alpha$ and $N_\\alpha$ for $\\alpha \\in \\mathbb{Z}^n$. An $R$-linear map $f : M \\to N$ is degree-zero when $f(M_\\alpha) \\subseteq N_\\alpha$ for every $\\alpha \\in \\mathbb{Z}^n$.","title":"Degree-zero homomorphisms of multigraded modules"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut/multigraded-free-module","kind":"node","lean":"MetaMathlibExt.MultigradedFreeModule","parent":4345,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut/index.html#multigraded-free-module","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $\\iota$ be an index type. A multigraded free module structure on an $R$-module $F$ consists of a multigraded $R$-module structure on $F$, with homogeneous pieces $F_\\alpha$ for $\\alpha \\in \\mathbb{Z}^n$, together with an $R$-basis $(b_j)_{j \\in \\iota}$ of $F$ and a degree $d_j \\in \\mathbb{Z}^n$ for each $j \\in \\iota$ such that $b_j \\in F_{d_j}$ for every\u2026","title":"Multigraded free module"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut/multigraded-free-resolution","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution","parent":4345,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut/index.html#multigraded-free-resolution","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-free-modules-and-graded-free-resolut","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded $R$-module, let $\\cdots \\to P_1 \\to P_0 \\to M$ be a projective resolution of $M$ over $R$, and for each $i \\in \\mathbb{N}$ let $\\iota_i$ be an index type. A multigraded free resolution structure on this resolution consists of a multigraded free module structure on each $P_i$ with basis indexed by $\\iota_i$, such that every differential\u2026","title":"Multigraded free resolution"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba.json","id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba","kind":"scope","navigation":"module","parent":4333,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/index.html","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba","status":"fully_proved","title":"Multigraded Betti numbers after residue-field base change"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/zero-eval","kind":"node","lean":"MetaMathlibExt.zeroEval","parent":4349,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/index.html#zero-eval","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba","status":"fully_proved","summary":"For a field $K$ and $n \\in \\mathbb{N}$, the augmentation $K[x_1, \\dots, x_n] \\to K$ is the ring homomorphism sending each polynomial $p$ to its value $p(0, \\dots, 0)$: it is the identity on constants and sends every variable to $0$.","title":"Augmentation of the polynomial ring"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/base-changed-complex","kind":"node","lean":"MetaMathlibExt.baseChangedComplex","parent":4349,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/index.html#base-changed-complex","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be an $R$-module. Given a projective resolution $P_\\bullet \\to M$ over $R$, its base change is the chain complex of $K$-vector spaces $K \\otimes_R P_\\bullet$, indexed by $\\mathbb{N}$ with differentials lowering the index, obtained by extending scalars termwise along the augmentation $R \\to K$ that sends every variable to $0$: its $i$th term is\u2026","title":"Base change of a resolution to the residue field"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/base-changed-degree-piece","kind":"node","lean":"MetaMathlibExt.baseChangedDegreePiece","parent":4349,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/index.html#base-changed-degree-piece","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be a $\\mathbb{Z}^n$-graded $R$-module with a multigraded free resolution $P_\\bullet \\to M$, in which each $P_i$ has an $R$-basis $(e_j)_{j \\in J_i}$ with $e_j$ homogeneous of degree $\\deg e_j \\in \\mathbb{Z}^n$. For $i \\in \\mathbb{N}$ and $\\alpha \\in \\mathbb{Z}^n$, the degree-$\\alpha$ piece of the $i$th term $K \\otimes_R P_i$ of the base-changed complex\u2026","title":"Degree pieces of the base-changed resolution"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/multigraded-base-change","kind":"node","lean":"MetaMathlibExt.MultigradedBaseChange","parent":4349,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/index.html#multigraded-base-change","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be a $\\mathbb{Z}^n$-graded $R$-module with a multigraded free resolution $P_\\bullet \\to M$. Write $K \\otimes_R P_\\bullet$ for its base change along the augmentation $R \\to K$ and $(K \\otimes_R P_i)_\\beta$ for the degree-$\\beta$ piece of its $i$th term. Multigraded base change data for $P_\\bullet$ consist of: for each $i \\in \\mathbb{N}$, a decomposition\u2026","title":"Multigraded base change data"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/multigraded-betti-number","kind":"node","lean":"MetaMathlibExt.multigradedBettiNumber","parent":4349,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba/index.html#multigraded-betti-number","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-betti-numbers-after-residue-field-ba","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be a $\\mathbb{Z}^n$-graded $R$-module with a multigraded free resolution $P_\\bullet \\to M$. Fix multigraded base change data for $P_\\bullet$; for each $\\alpha \\in \\mathbb{Z}^n$ they supply a chain complex $C^\\alpha$ of $K$-vector spaces embedded in the base-changed complex $K \\otimes_R P_\\bullet$ with image its degree-$\\alpha$ pieces. For\u2026","title":"Multigraded Betti number"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade.json","id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade","kind":"scope","navigation":"module","parent":4333,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/index.html","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade","status":"fully_proved","title":"Degree components of the base-changed multigraded resolution"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/base-changed-degree-complex","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution.baseChangedDegreeComplex","parent":4355,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/index.html#base-changed-degree-complex","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded polynomial module over $R$, and let $F$ be a multigraded free resolution of $M$ whose term $F_i$ has the homogeneous basis $(e_j)_{j \\in \\iota_i}$. Write $K \\otimes_R F$ for the chain complex of $K$-vector spaces obtained from $F$ by extending scalars along the evaluation $R \\to K$ at $x = 0$, and for $\\alpha \\in \\mathbb{Z}^n$ call the\u2026","title":"Multidegree component complex of the base-changed resolution"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/base-changed-degree-inclusion","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution.baseChangedDegreeInclusion","parent":4355,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/index.html#base-changed-degree-inclusion","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded polynomial module over $R$, and let $F$ be a multigraded free resolution of $M$ whose term $F_i$ has the homogeneous basis $(e_j)_{j \\in \\iota_i}$. Write $K \\otimes_R F$ for the chain complex of $K$-vector spaces obtained from $F$ by extending scalars along the evaluation $R \\to K$ at $x = 0$, and for $\\alpha \\in \\mathbb{Z}^n$ call the\u2026","title":"Inclusion of a multidegree component complex"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/injectivity-of-the-base-changed-degree-i","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution.baseChangedDegreeInclusion_injective","parent":4355,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/index.html#injectivity-of-the-base-changed-degree-i","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade","status":"fully_proved","summary":"Each component of the degree inclusion is injective. Source: Benjamin Braun and Brian Davis, \"Antichain Simplices,\" Journal of Integer Sequences 23 (2020), lines 864\u2013867, `https://cs.uwaterloo.ca/journals/JIS/VOL23/Braun/braun4.tex`.","title":"Injectivity of the base-changed degree inclusion"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/range-of-the-base-changed-degree-inclusi","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution.baseChangedDegreeInclusion_range","parent":4355,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/index.html#range-of-the-base-changed-degree-inclusi","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade","status":"fully_proved","summary":"Each component of the degree inclusion has range exactly the degree piece `baseChangedDegreePiece`. Source: Benjamin Braun and Brian Davis, \"Antichain Simplices,\" Journal of Integer Sequences 23 (2020), lines 864\u2013867, `https://cs.uwaterloo.ca/journals/JIS/VOL23/Braun/braun4.tex`.","title":"Range of the base-changed degree inclusion"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/multigraded-base-change-nonempty","kind":"node","lean":"MetaMathlibExt.MultigradedFreeResolution.multigradedBaseChange_nonempty","parent":4355,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade/index.html#multigraded-base-change-nonempty","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/degree-components-of-the-base-changed-multigrade","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded polynomial module over $R$, and let $F$ be a multigraded free resolution of $M$ whose term $F_i$ has the homogeneous basis $(e_j)_{j \\in \\iota_i}$, where every index set $\\iota_i$ is finite. Write $K \\otimes_R F$ for the chain complex of $K$-vector spaces obtained from $F$ by extending scalars along the evaluation $R \\to K$ at $x = 0$, and\u2026","title":"Existence of multigraded base-change data"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g.json","id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g","kind":"scope","navigation":"module","parent":4333,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g/index.html","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g","status":"fully_proved","title":"Invariance of multigraded Betti numbers across graded free resolutions"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g/multigraded-resolution-comparison","kind":"node","lean":"MetaMathlibExt.MultigradedResolutionComparison","parent":4361,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g/index.html#multigraded-resolution-comparison","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be a $\\mathbb{Z}^n$-graded $R$-module with two multigraded free resolutions, each equipped with multigraded base change data, whose degree-$\\alpha$ component complexes are $C^\\alpha$ and $D^\\alpha$ for $\\alpha \\in \\mathbb{Z}^n$. A comparison between the two base change data is a choice, for every $\\alpha \\in \\mathbb{Z}^n$, of a chain-homotopy equivalence\u2026","title":"Multigraded resolution comparison"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g/multigraded-betti-number-eq-of-resolutio","kind":"node","lean":"MetaMathlibExt.multigradedBettiNumber_eq_of_resolutionComparison","parent":4361,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g/index.html#multigraded-betti-number-eq-of-resolutio","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/invariance-of-multigraded-betti-numbers-across-g","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, and let $M$ be a $\\mathbb{Z}^n$-graded $R$-module with two multigraded free resolutions, each equipped with multigraded base change data, whose degree-$\\alpha$ component complexes are $C^\\alpha$ and $D^\\alpha$ for $\\alpha \\in \\mathbb{Z}^n$. If the two are related by a multigraded resolution comparison, that is, for every $\\alpha$ a chain-homotopy equivalence between\u2026","title":"Betti numbers agree across comparable resolutions"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family.json","id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family","kind":"scope","navigation":"module","parent":4333,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family/index.html","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family","status":"fully_proved","title":"Multigraded Poincar\u00e9-series coefficient family"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family/multigraded-poincare-coefficients","kind":"node","lean":"MetaMathlibExt.multigradedPoincareCoefficients","parent":4364,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family/index.html#multigraded-poincare-coefficients","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded $R$-module, and let $D$ be base-change data for a multigraded free resolution of $M$. The multigraded Poincar\u00e9 coefficient family of $D$ is the function $\\mathbb{N} \\times \\mathbb{Z}^n \\to \\mathbb{N}$ sending $(i, \\alpha)$ to the multigraded Betti number $\\beta_{i,\\alpha}(D)$ of $D$ in homological degree $i$ and multidegree $\\alpha$.","title":"Multigraded Poincar\u00e9 coefficients"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family/multigraded-poincare-coefficients-apply","kind":"node","lean":"MetaMathlibExt.multigradedPoincareCoefficients_apply","parent":4364,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family/index.html#multigraded-poincare-coefficients-apply","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded $R$-module, and let $D$ be base-change data for a multigraded free resolution of $M$. For every $i \\in \\mathbb{N}$ and $\\alpha \\in \\mathbb{Z}^n$, the multigraded Poincar\u00e9 coefficient family of $D$ takes the value $\\beta_{i,\\alpha}(D)$ at $(i, \\alpha)$, where $\\beta_{i,\\alpha}(D)$ is the multigraded Betti number of $D$ in homological degree\u2026","title":"Multigraded Poincar\u00e9 coefficients are Betti numbers"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family/multigraded-poincare-coefficients-eq-of","kind":"node","lean":"MetaMathlibExt.multigradedPoincareCoefficients_eq_of_resolutionComparison","parent":4364,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family/index.html#multigraded-poincare-coefficients-eq-of","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-series-coefficient-family","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded $R$-module, and let $D$ and $D'$ be base-change data for two multigraded free resolutions of $M$. Given a multigraded resolution comparison between $D$ and $D'$, that is, for every $\\alpha \\in \\mathbb{Z}^n$ a chain-homotopy equivalence between the degree-$\\alpha$ component complexes of $D$ and $D'$, the multigraded Poincar\u00e9 coefficient\u2026","title":"Resolution comparisons preserve multigraded Poincar\u00e9 coefficients"},{"data_url":"scopes/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial.json","id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial","kind":"scope","navigation":"module","parent":4333,"result":false,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial/index.html","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial","status":"fully_proved","title":"Multigraded Poincar\u00e9 polynomial"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial/multigraded-poincare-polynomial","kind":"node","lean":"MetaMathlibExt.multigradedPoincarePolynomial","parent":4368,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial/index.html#multigraded-poincare-polynomial","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded $R$-module, and let $D$ be base-change data for a multigraded free resolution $F_\\bullet$ of $M$ in which $F_i$ has a homogeneous basis indexed by a finite set $\\iota_i$, with $\\iota_i$ empty for every $i > b$ for a fixed $b \\in \\mathbb{N}$. The multigraded Poincar\u00e9 polynomial of $D$ is the element of the monoid algebra\u2026","title":"Multigraded Poincar\u00e9 polynomial"},{"id":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial/multigraded-poincare-polynomial-coeff","kind":"node","lean":"MetaMathlibExt.multigradedPoincarePolynomial_coeff","parent":4368,"result":true,"route":"../roadmap/RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial/index.html#multigraded-poincare-polynomial-coeff","scope":"RingTheory/syzygies-and-resolutions/multigraded-betti-numbers-and-poincar-series/multigraded-poincar-polynomial","status":"fully_proved","summary":"Let $K$ be a field, let $R = K[x_1, \\dots, x_n]$, let $M$ be a multigraded $R$-module, and let $D$ be base-change data for a multigraded free resolution $F_\\bullet$ of $M$ in which $F_i$ has a homogeneous basis indexed by a finite set $\\iota_i$, with $\\iota_i$ empty for every $i > b$ for a fixed $b \\in \\mathbb{N}$. For every $i \\in \\mathbb{N}$ and $\\alpha \\in \\mathbb{Z}^n$, the coefficient at $(i, \\alpha)$ of the\u2026","title":"Coefficients of the multigraded Poincar\u00e9 polynomial"},{"data_url":"scopes/RingTheory/modules.json","id":"RingTheory/modules","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/modules/index.html","scope":"RingTheory/modules","status":"can_prove","summary":"Projective, free and finitely generated modules over commutative rings and module-finite extensions: Kaplansky's theorem, the Quillen\u2013Suslin theorem, descent of Noetherianness (Eakin\u2013Nagata) and the direct summand theorem.","title":"Modules over commutative rings"},{"id":"RingTheory/modules/eakin-nagata","kind":"node","lean":"MathlibExt.Algebra.CommutativeAlgebra.EakinNagataWanted.eakin_nagata","parent":4371,"result":true,"route":"../roadmap/RingTheory/modules/index.html#eakin-nagata","scope":"RingTheory/modules","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and let $B$ be an $A$-algebra that is finitely generated as an $A$-module. If the structure map $A \\to B$ is injective and $B$ is Noetherian, then $A$ is Noetherian.","title":"Eakin\u2013Nagata theorem"},{"id":"RingTheory/modules/bijective-algebra-map-of-faithfully-flat","kind":"node","lean":"IsLocalization.bijective_algebraMap_of_faithfullyFlat","parent":4371,"result":true,"route":"../roadmap/RingTheory/modules/index.html#bijective-algebra-map-of-faithfully-flat","scope":"RingTheory/modules","status":"fully_proved","summary":"A faithfully flat `R`-algebra `S` which embeds compatibly in a localization `K` of `R` has bijective structure map. No restriction on the denominators or domain assumptions on the rings are required.","title":"Faithfully flat algebras inside a localization equal the base ring"},{"id":"RingTheory/modules/of-projective-of-forall-finrank-eq-one","kind":"node","lean":"Module.Invertible.of_projective_of_forall_finrank_eq_one","parent":4371,"result":true,"route":"../roadmap/RingTheory/modules/index.html#of-projective-of-forall-finrank-eq-one","scope":"RingTheory/modules","status":"fully_proved","summary":"A finite projective module whose base change to every field has rank one is invertible.","title":"Finite projective modules of fiberwise rank one are invertible"},{"id":"RingTheory/modules/kaplansky-projective-local-free","kind":"node","lean":"MathlibExt.Algebra.Module.KaplanskyProjectiveWanted.kaplansky_projective_local_free","parent":4371,"result":true,"route":"../roadmap/RingTheory/modules/index.html#kaplansky-projective-local-free","scope":"RingTheory/modules","status":"fully_proved","summary":"Every projective module over a commutative local ring is free.","title":"Kaplansky's theorem on projective modules over local rings"},{"id":"RingTheory/modules/quillen-suslin","kind":"node","lean":"MathlibExt.Algebra.Module.QuillenSuslinWanted.quillen_suslin","parent":4371,"result":true,"route":"../roadmap/RingTheory/modules/index.html#quillen-suslin","scope":"RingTheory/modules","status":"fully_proved","summary":"Let $k$ be a field and let $n$ be a natural number. Every finitely generated projective module over the polynomial ring $k[x_1, \\dots, x_n]$ in $n$ variables is free.","title":"Quillen\u2013Suslin theorem"},{"data_url":"scopes/RingTheory/modules/direct-summand-theorem.json","id":"RingTheory/modules/direct-summand-theorem","kind":"scope","navigation":"module","parent":4371,"result":false,"route":"../roadmap/RingTheory/modules/direct-summand-theorem/index.html","scope":"RingTheory/modules/direct-summand-theorem","status":"can_prove","title":"Direct summand theorem"},{"id":"RingTheory/modules/direct-summand-theorem/single-functor-plus-iso-comp-qh","kind":"node","lean":"MathlibExt.AlgebraicGeometry.DirectSummandWanted.AlgebraicGeometry.DirectSummand.singleFunctorPlusIsoCompQh","parent":4377,"result":true,"route":"../roadmap/RingTheory/modules/direct-summand-theorem/index.html#single-functor-plus-iso-comp-qh","scope":"RingTheory/modules/direct-summand-theorem","status":"fully_proved","summary":"For an abelian category $C$ with a derived category and an integer $n$, the functor sending an object of $C$ to the complex concentrated in degree $n$, viewed in the bounded-below derived category $D^+(C)$, is naturally isomorphic to the corresponding single-complex functor into the bounded-below homotopy category $K^+(C)$ followed by the localization functor $K^+(C) \\to D^+(C)$.","title":"Single complexes in the bounded-below derived category"},{"id":"RingTheory/modules/direct-summand-theorem/derived-direct-summand","kind":"node","lean":"MathlibExt.AlgebraicGeometry.DirectSummandWanted.AlgebraicGeometry.DirectSummand.derived_direct_summand","parent":4377,"result":true,"route":"../roadmap/RingTheory/modules/direct-summand-theorem/index.html#derived-direct-summand","scope":"RingTheory/modules/direct-summand-theorem","status":"can_prove","summary":"Let $A$ be a regular commutative ring, let $X$ be a scheme and let $f \\colon X \\to \\operatorname{Spec} A$ be a proper morphism that is surjective on underlying points. Let $\\varphi \\colon A \\to \\Gamma(X, \\mathcal{O}_X)$ be the ring map corresponding to $f$, and let $R\\Gamma$ be the right derived functor, between bounded-below derived categories, of global sections of sheaves of $\\mathcal{O}_X$-modules followed by\u2026","title":"Derived direct summand theorem"},{"id":"RingTheory/modules/direct-summand-theorem/direct-summand","kind":"node","lean":"MathlibExt.AlgebraicGeometry.DirectSummandWanted.AlgebraicGeometry.DirectSummand.direct_summand","parent":4377,"result":true,"route":"../roadmap/RingTheory/modules/direct-summand-theorem/index.html#direct-summand","scope":"RingTheory/modules/direct-summand-theorem","status":"can_prove","summary":"Let $A$ be a regular commutative ring and let $B$ be a commutative $A$-algebra that is finitely generated as an $A$-module and on which $A$ acts faithfully (equivalently, the structure map $A \\to B$ is injective). Then there is an $A$-linear map $\\pi \\colon B \\to A$ with $\\pi(a \\cdot 1) = a$ for every $a \\in A$; that is, $A$ is a direct summand of $B$ as an $A$-module.","title":"Direct summand theorem"},{"data_url":"scopes/RingTheory/invariant-theory.json","id":"RingTheory/invariant-theory","kind":"scope","parent":3675,"result":false,"route":"../roadmap/RingTheory/invariant-theory/index.html","scope":"RingTheory/invariant-theory","status":"can_prove","summary":"Rings of invariants of finite groups acting on commutative rings: finiteness of invariant rings and Hilbert\u2013Poincar\u00e9 series of modular invariant rings of polynomial algebras.","title":"Invariant theory"},{"id":"RingTheory/invariant-theory/module-finite-and-finite-type-of-finite","kind":"node","lean":"Algebra.IsInvariant.moduleFinite_and_finiteType_of_finiteType","parent":4381,"result":true,"route":"../roadmap/RingTheory/invariant-theory/index.html#module-finite-and-finite-type-of-finite","scope":"RingTheory/invariant-theory","status":"fully_proved","summary":"Over a Noetherian ring `R`, suppose `B` is finite type over `R` and the faithful image of `A` in `B` contains every fixed element of a finite group acting on `B` by semiring automorphisms. Then `B` is finite over `A` and `A` is finite type over `R`.","title":"Finiteness over a subring containing the invariants"},{"data_url":"scopes/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings.json","id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","kind":"scope","navigation":"module","parent":4381,"result":false,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"can_prove","title":"Poincar\u00e9 series of modular invariant rings"},{"id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/lin-subst","kind":"node","lean":"MathlibExt.GroupTheory.InvariantRingPoincareSeriesKourovka12Problem36Wanted.linSubst","parent":4383,"result":true,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html#lin-subst","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"fully_proved","summary":"For natural numbers $p$ and $n$ and an $n \\times n$ matrix $g$ over $\\mathbb{Z}/p\\mathbb{Z}$, the linear substitution by $g$ is the $\\mathbb{Z}/p\\mathbb{Z}$-algebra endomorphism of the polynomial ring $(\\mathbb{Z}/p\\mathbb{Z})[x_0, \\dots, x_{n-1}]$ that sends each variable $x_i$ to $\\sum_j g_{ij} x_j$. Viewing polynomials as functions on $V = (\\mathbb{Z}/p\\mathbb{Z})^n$, it sends $f$ to the function\u2026","title":"Linear substitution by a matrix"},{"id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/invariant-component","kind":"node","lean":"MathlibExt.GroupTheory.InvariantRingPoincareSeriesKourovka12Problem36Wanted.invariantComponent","parent":4383,"result":true,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html#invariant-component","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"fully_proved","summary":"For natural numbers $p$ and $n$, a subgroup $G$ of $\\mathrm{GL}_n(\\mathbb{Z}/p\\mathbb{Z})$ and $m \\in \\mathbb{N}$, the degree-$m$ component $T_m$ of the invariant ring $T$ of $G$ is the $\\mathbb{Z}/p\\mathbb{Z}$-submodule of the polynomial ring $(\\mathbb{Z}/p\\mathbb{Z})[x_0, \\dots, x_{n-1}]$ consisting of the homogeneous polynomials $f$ of degree $m$ that are fixed by the linear substitution\u2026","title":"Homogeneous component of the invariant ring"},{"id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/poincare-series","kind":"node","lean":"MathlibExt.GroupTheory.InvariantRingPoincareSeriesKourovka12Problem36Wanted.poincareSeries","parent":4383,"result":true,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html#poincare-series","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"fully_proved","summary":"For natural numbers $p$ and $n$ and a subgroup $G$ of $\\mathrm{GL}_n(\\mathbb{Z}/p\\mathbb{Z})$, the Poincar\u00e9 series of the invariant ring $T$ of $G$ is the formal power series $P(t) = \\sum_{m \\ge 0} b_m t^m$ over $\\mathbb{Q}$, where $b_m$ is the rank over $\\mathbb{Z}/p\\mathbb{Z}$ (the dimension, when $p$ is prime) of the degree-$m$ homogeneous component $T_m$ of $T$.","title":"Poincar\u00e9 series of the invariant ring"},{"id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/pointwise-stabilizer","kind":"node","lean":"MathlibExt.GroupTheory.InvariantRingPoincareSeriesKourovka12Problem36Wanted.pointwiseStabilizer","parent":4383,"result":true,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html#pointwise-stabilizer","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"fully_proved","summary":"For natural numbers $p$ and $n$, a subgroup $G$ of $\\mathrm{GL}_n(\\mathbb{Z}/p\\mathbb{Z})$ and a submodule $W$ of $V = (\\mathbb{Z}/p\\mathbb{Z})^n$, the pointwise stabilizer $G_W$ of $W$ in $G$ is the set of all $g \\in G$ with $g w = w$ for every $w \\in W$, where $g$ acts on column vectors by matrix multiplication.","title":"Pointwise stabilizer"},{"id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/hyperplane-term","kind":"node","lean":"MathlibExt.GroupTheory.InvariantRingPoincareSeriesKourovka12Problem36Wanted.hyperplaneTerm","parent":4383,"result":true,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html#hyperplane-term","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"fully_proved","summary":"For natural numbers $p$ and $n$, a subgroup $G$ of $\\mathrm{GL}_n(\\mathbb{Z}/p\\mathbb{Z})$ and a submodule $W$ of $V = (\\mathbb{Z}/p\\mathbb{Z})^n$, let $N$ be the number of elements of the pointwise stabilizer $G_W$ of $W$ in $G$ (taken to be $0$ if $G_W$ is infinite), let $\\alpha_W$ be the exponent of $p$ in the prime factorization of $N$ (which is $0$ unless $p$ is prime and $N \\ne 0$), and let\u2026","title":"Hyperplane term"},{"id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/hyperplane-sum","kind":"node","lean":"MathlibExt.GroupTheory.InvariantRingPoincareSeriesKourovka12Problem36Wanted.hyperplaneSum","parent":4383,"result":true,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html#hyperplane-sum","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"fully_proved","summary":"For natural numbers $p$ and $n$ and a subgroup $G$ of $\\mathrm{GL}_n(\\mathbb{Z}/p\\mathbb{Z})$, the hyperplane sum of $G$ is the rational number $r = \\sum_W ((p - 1) \\alpha_W + s_W - 1)$, where $W$ runs over the maximal proper submodules of $V = (\\mathbb{Z}/p\\mathbb{Z})^n$ (the hyperplanes of $V$ when $p$ is prime) and $(p - 1) \\alpha_W + s_W - 1$ is the hyperplane term of $W$: here $N_W$ is the number of elements\u2026","title":"Hyperplane sum"},{"id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/poincare-series-second-coefficient","kind":"node","lean":"MathlibExt.GroupTheory.InvariantRingPoincareSeriesKourovka12Problem36Wanted.poincareSeriesSecondCoefficient","parent":4383,"result":true,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html#poincare-series-second-coefficient","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"fully_proved","summary":"The statement that for every prime $p$, every $n \\in \\mathbb{N}$ and every subgroup $G$ of $\\mathrm{GL}_n(\\mathbb{F}_p)$, with $P(t)$ the Poincar\u00e9 series of the invariant ring of $G$ and $r$ the hyperplane sum of $G$, there are polynomials $A, B \\in \\mathbb{Q}[t]$ with $B(1) \\ne 0$ and $B(t) (1 - t)^n P(t) = A(t)$ in $\\mathbb{Q}[[t]]$ such that","title":"Carlisle\u2013Kropholler conjecture on the second Laurent coefficient (Kourovka 12.36)"},{"id":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/poincare-series-second-coefficient-holds","kind":"node","lean":"MathlibExt.GroupTheory.InvariantRingPoincareSeriesKourovka12Problem36Wanted.poincareSeriesSecondCoefficient_holds","parent":4383,"result":true,"route":"../roadmap/RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings/index.html#poincare-series-second-coefficient-holds","scope":"RingTheory/invariant-theory/poincar-series-of-modular-invariant-rings","status":"can_prove","summary":"For every prime $p$, every $n \\in \\mathbb{N}$ and every subgroup $G$ of $\\mathrm{GL}_n(\\mathbb{F}_p)$, with $P(t)$ the Poincar\u00e9 series of the invariant ring of $G$ and $r$ the hyperplane sum of $G$, there are polynomials $A, B \\in \\mathbb{Q}[t]$ with $B(1) \\ne 0$ and $B(t) (1 - t)^n P(t) = A(t)$ in $\\mathbb{Q}[[t]]$ such that","title":"Benson\u2013Crawley-Boevey hyperplane formula (Kourovka 12.36)"},{"area":"Algebra & Number Theory","data_url":"chapters/FieldTheory.json","id":"FieldTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/FieldTheory/index.html","scope":"FieldTheory","status":"can_prove","summary":"Field theory studies extensions of fields, algebraic elements, automorphisms, and Galois-theoretic structure.","title":"Field Theory"},{"data_url":"scopes/FieldTheory/galois-theory.json","id":"FieldTheory/galois-theory","kind":"scope","parent":4392,"result":false,"route":"../roadmap/FieldTheory/galois-theory/index.html","scope":"FieldTheory/galois-theory","status":"can_prove","summary":"Galois theory relates a field extension to its group of automorphisms, covering the Galois correspondence, infinite Galois groups, Hilbert's theorem 90, solvability by radicals and essential dimension.","title":"Galois theory"},{"id":"FieldTheory/galois-theory/is-galois-and-order-of-eq-finrank-of-fin","kind":"node","lean":"AlgEquiv.isGalois_and_orderOf_eq_finrank_of_finrank_prime_of_ne_one","parent":4393,"result":true,"route":"../roadmap/FieldTheory/galois-theory/index.html#is-galois-and-order-of-eq-finrank-of-fin","scope":"FieldTheory/galois-theory","status":"fully_proved","summary":"In an extension of prime degree, a nontrivial automorphism witnesses that the extension is Galois, has order equal to the degree, and generates the automorphism group.","title":"Extensions of prime degree with a nontrivial automorphism are Galois"},{"id":"FieldTheory/galois-theory/is-modular-lattice-intermediate-field","kind":"node","lean":"IsAbelianGalois.isModularLattice_intermediateField","parent":4393,"result":true,"route":"../roadmap/FieldTheory/galois-theory/index.html#is-modular-lattice-intermediate-field","scope":"FieldTheory/galois-theory","status":"fully_proved","summary":"If $L/K$ is a finite-dimensional abelian Galois extension, then the lattice of intermediate fields of $L/K$ is modular.","title":"Intermediate fields of an abelian extension form a modular lattice"},{"id":"FieldTheory/galois-theory/algebra-map-norm-eq-prod-map-alg-equiv","kind":"node","lean":"Algebra.TensorProduct.algebraMap_norm_eq_prod_map_algEquiv","parent":4393,"result":true,"route":"../roadmap/FieldTheory/galois-theory/index.html#algebra-map-norm-eq-prod-map-alg-equiv","scope":"FieldTheory/galois-theory","status":"fully_proved","summary":"For a Galois extension `M / F` and any `F`-algebra `R`, the norm of an element of `R \u2297[F] M` over `R` is the product of its Galois conjugates.","title":"Norm in a tensor product with a Galois extension"},{"id":"FieldTheory/galois-theory/topological-closure-eq-fixing-subgroup-f","kind":"node","lean":"InfiniteGalois.topologicalClosure_eq_fixingSubgroup_fixedField","parent":4393,"result":true,"route":"../roadmap/FieldTheory/galois-theory/index.html#topological-closure-eq-fixing-subgroup-f","scope":"FieldTheory/galois-theory","status":"fully_proved","summary":"Let $K/k$ be a Galois extension, not necessarily finite, and give $\\operatorname{Gal}(K/k)$ the Krull topology. For every subgroup $H$ of $\\operatorname{Gal}(K/k)$ with fixed field $K^H$, the closure of $H$ equals the subgroup of $\\operatorname{Gal}(K/k)$ consisting of the automorphisms that fix $K^H$ pointwise.","title":"The closure of a Galois subgroup is the fixing group of its fixed field"},{"data_url":"scopes/FieldTheory/galois-theory/additive-hilbert-s-theorem-90.json","id":"FieldTheory/galois-theory/additive-hilbert-s-theorem-90","kind":"scope","navigation":"module","parent":4393,"result":false,"route":"../roadmap/FieldTheory/galois-theory/additive-hilbert-s-theorem-90/index.html","scope":"FieldTheory/galois-theory/additive-hilbert-s-theorem-90","status":"fully_proved","title":"Additive Hilbert's theorem 90"},{"id":"FieldTheory/galois-theory/additive-hilbert-s-theorem-90/ker-trace-eq-range-alg-equiv-sub-id","kind":"node","lean":"Algebra.ker_trace_eq_range_algEquiv_sub_id","parent":4398,"result":true,"route":"../roadmap/FieldTheory/galois-theory/additive-hilbert-s-theorem-90/index.html#ker-trace-eq-range-alg-equiv-sub-id","scope":"FieldTheory/galois-theory/additive-hilbert-s-theorem-90","status":"fully_proved","summary":"Let $L/K$ be a finite-dimensional Galois extension and let $\\sigma$ be a $K$-algebra automorphism of $L$ that generates its Galois group, that is, every $K$-algebra automorphism of $L$ is an integer power of $\\sigma$. Then the kernel of the trace $\\operatorname{Tr}_{L/K} \\colon L \\to K$ equals the image of the $K$-linear map $\\sigma - 1 \\colon L \\to L$, $y \\mapsto \\sigma(y) - y$.","title":"Additive Hilbert 90 for cyclic extensions"},{"id":"FieldTheory/galois-theory/additive-hilbert-s-theorem-90/exists-alg-equiv-sub-eq-of-trace-eq-zero","kind":"node","lean":"Algebra.exists_algEquiv_sub_eq_of_trace_eq_zero","parent":4398,"result":true,"route":"../roadmap/FieldTheory/galois-theory/additive-hilbert-s-theorem-90/index.html#exists-alg-equiv-sub-eq-of-trace-eq-zero","scope":"FieldTheory/galois-theory/additive-hilbert-s-theorem-90","status":"fully_proved","summary":"Let $L/K$ be a finite-dimensional Galois extension and let $\\sigma$ be a $K$-algebra automorphism of $L$ that generates its Galois group, that is, every $K$-algebra automorphism of $L$ is an integer power of $\\sigma$. If $x \\in L$ satisfies $\\operatorname{Tr}_{L/K}(x) = 0$, then there is $y \\in L$ with $\\sigma(y) - y = x$.","title":"Additive Hilbert 90, pointwise form"},{"data_url":"scopes/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups.json","id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","kind":"scope","navigation":"module","parent":4393,"result":false,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"can_prove","title":"Essential dimension of symmetric groups"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/roots-field","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.rootsField","parent":4401,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function field "],["math","\\(\\mathbb{C}(x_1, \\dots, x_n)\\)"]],"source":"The function field $\\mathbb{C}(x_1, \\dots, x_n)$"},"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#roots-field","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the function field $\\mathbb{C}(x_1, \\dots, x_n)$ is the fraction field of the polynomial ring $\\mathbb{C}[x_1, \\dots, x_n]$ in $n$ indeterminates.","title":"The function field $\\mathbb{C}(x_1, \\dots, x_n)$"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/root","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.root","parent":4401,"reading_title":{"format":"inline-parts-v1","parts":[["text","The roots "],["math","\\(x_i\\)"]],"source":"The roots $x_i$"},"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#root","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and $i \\in \\{1, \\dots, n\\}$, $x_i$ is the $i$-th indeterminate of the polynomial ring $\\mathbb{C}[x_1, \\dots, x_n]$, viewed as an element of its fraction field, the function field $\\mathbb{C}(x_1, \\dots, x_n)$.","title":"The roots $x_i$"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/coeff","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.coeff","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#coeff","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For $0 \\le i < n$, the coefficient $a_{i+1}$ is the $(i+1)$-st elementary symmetric function of $x_1, \\dots, x_n$, as an element of the function field $\\mathbb{C}(x_1, \\dots, x_n)$.","title":"The coefficients of the general polynomial"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/general-base-field","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.generalBaseField","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#general-base-field","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For a natural number $n$, let $\\mathbb{C}(x_1, \\dots, x_n)$ be the field of rational functions in $n$ indeterminates and, for $1 \\le k \\le n$, let $a_k$ be the $k$-th elementary symmetric function of $x_1, \\dots, x_n$. The coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$ is the intermediate field of $\\mathbb{C} \\subseteq \\mathbb{C}(x_1, \\dots, x_n)$ generated over $\\mathbb{C}$ by $a_1, \\dots, a_n$.","title":"Coefficient field of the general polynomial"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/coeff-elem","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.coeffElem","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#coeff-elem","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For $0 \\le i < n$, the coefficient $a_{i+1}$, the $(i+1)$-st elementary symmetric function of $x_1, \\dots, x_n$, regarded as an element of the coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$.","title":"Coefficients in the coefficient field"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/root-injective","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.root_injective","parent":4401,"reading_title":{"format":"inline-parts-v1","parts":[["text","The roots "],["math","\\(x_i\\)"],["text"," are pairwise distinct"]],"source":"The roots $x_i$ are pairwise distinct"},"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#root-injective","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the elements $x_1, \\dots, x_n$ of the function field $\\mathbb{C}(x_1, \\dots, x_n)$ are pairwise distinct: $x_i = x_j$ implies $i = j$.","title":"The roots $x_i$ are pairwise distinct"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/algebraic-independent-esymm","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.algebraicIndependent_esymm","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#algebraic-independent-esymm","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every natural number $n$, the elementary symmetric polynomials $e_1, \\dots, e_n \\in \\mathbb{C}[X_1, \\dots, X_n]$ are algebraically independent over $\\mathbb{C}$.","title":"Algebraic independence of the elementary symmetric polynomials"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/algebraic-independent-map","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.algebraicIndependent_map","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#algebraic-independent-map","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"Let $A$ and $B$ be commutative $\\mathbb{C}$-algebras, let $f : A \\to B$ be an injective $\\mathbb{C}$-algebra homomorphism, and let $(x_i)_{i \\in \\iota}$ be a family in $A$ that is algebraically independent over $\\mathbb{C}$. Then the family $(f(x_i))_{i \\in \\iota}$ is algebraically independent over $\\mathbb{C}$.","title":"Injective algebra maps preserve algebraic independence"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/algebraic-independent-coeff","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.algebraicIndependent_coeff","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#algebraic-independent-coeff","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every natural number $n$, the coefficients $a_1, \\dots, a_n$ of the general polynomial, the elementary symmetric functions of $x_1, \\dots, x_n$, are algebraically independent over $\\mathbb{C}$ in the function field $\\mathbb{C}(x_1, \\dots, x_n)$.","title":"The coefficients are algebraically independent"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/algebraic-independent-root","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.algebraicIndependent_root","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#algebraic-independent-root","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every natural number $n$, the roots $x_1, \\dots, x_n$ are algebraically independent over $\\mathbb{C}$ in the function field $\\mathbb{C}(x_1, \\dots, x_n)$.","title":"The roots are algebraically independent"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/algebra-map-mem-adjoin-roots","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.algebraMap_mem_adjoin_roots","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#algebra-map-mem-adjoin-roots","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every natural number $n$ and every polynomial $p \\in \\mathbb{C}[X_1, \\dots, X_n]$, the image of $p$ in the function field $\\mathbb{C}(x_1, \\dots, x_n)$ lies in the $\\mathbb{C}$-subalgebra generated by the roots $x_1, \\dots, x_n$.","title":"Polynomials in the roots"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/is-algebraic-adjoin-roots","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.isAlgebraic_adjoin_roots","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#is-algebraic-adjoin-roots","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For a natural number $n$, every element of the field $\\mathbb{C}(x_1, \\dots, x_n)$ of rational functions in $n$ indeterminates is algebraic over the $\\mathbb{C}$-subalgebra generated by $x_1, \\dots, x_n$: it is a ratio of two polynomials in $x_1, \\dots, x_n$.","title":"Rational functions are algebraic over the polynomial subalgebra"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/trdeg-roots-field","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.trdeg_rootsField","parent":4401,"reading_title":{"format":"inline-parts-v1","parts":[["text","Transcendence degree of "],["math","\\(\\mathbb{C}(x_1, \\dots, x_n)\\)"]],"source":"Transcendence degree of $\\mathbb{C}(x_1, \\dots, x_n)$"},"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#trdeg-roots-field","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the function field $\\mathbb{C}(x_1, \\dots, x_n)$ has transcendence degree $n$ over $\\mathbb{C}$, as a cardinal.","title":"Transcendence degree of $\\mathbb{C}(x_1, \\dots, x_n)$"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/trans-deg-nat","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.transDegNat","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#trans-deg-nat","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For a field $F$ that is a $\\mathbb{C}$-algebra, the numeric transcendence degree of $F$ is its transcendence degree over $\\mathbb{C}$ as a natural number when that degree is finite, and $0$ when it is infinite.","title":"Numeric transcendence degree"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/algebra-map-esymm","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.algebraMap_esymm","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#algebra-map-esymm","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the image of the $k$-th elementary symmetric polynomial $e_k \\in \\mathbb{C}[X_1, \\dots, X_n]$ in the function field $\\mathbb{C}(x_1, \\dots, x_n)$ is the $k$-th elementary symmetric function of the multiset of roots $\\{x_1, \\dots, x_n\\}$, that is, the sum of the products $x_{i_1} \\cdots x_{i_k}$ over all $i_1 < \\dots < i_k$.","title":"Elementary symmetric polynomials evaluated at the roots"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/coeff-nat","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.coeffNat","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#coeff-nat","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"Let $n$ be a natural number, for $1 \\le k \\le n$ let $a_k$ be the $k$-th elementary symmetric function of $x_1, \\dots, x_n$ in the function field $\\mathbb{C}(x_1, \\dots, x_n)$, and let $\\mathbb{C}(a_1, \\dots, a_n)$ be the coefficient field. For each natural number $k$, the element $c_k$ of $\\mathbb{C}(a_1, \\dots, a_n)$ is defined by $c_0 = 1$, $c_k = a_k$ for $1 \\le k \\le n$, and $c_k = 0$ for $k > n$.","title":"Coefficients indexed by degree"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/general-polynomial","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.generalPolynomial","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#general-polynomial","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For a natural number $n$, let $x_1, \\dots, x_n$ be the indeterminates of the rational function field $\\mathbb{C}(x_1, \\dots, x_n)$, let $a_0 = 1$ and, for $1 \\le k \\le n$, let $a_k$ be the $k$-th elementary symmetric function of $x_1, \\dots, x_n$. The general polynomial of degree $n$ is the polynomial $\\sum_{j=0}^{n} (-1)^j a_j X^{n-j} = X^n - a_1 X^{n-1} + \\cdots + (-1)^n a_n$ with coefficients in the coefficient\u2026","title":"General polynomial"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/is-coprime-x-sub-c-of-ne","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.isCoprime_X_sub_C_of_ne","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#is-coprime-x-sub-c-of-ne","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For a natural number $n$ and distinct elements $a \\ne b$ of the field $\\mathbb{C}(x_1, \\dots, x_n)$ of rational functions in $n$ indeterminates, the polynomials $X - a$ and $X - b$ are coprime in $\\mathbb{C}(x_1, \\dots, x_n)[X]$: there are polynomials $u$ and $v$ with $u (X - a) + v (X - b) = 1$.","title":"Distinct linear factors are coprime"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/separable-prod-roots","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.separable_prod_roots","parent":4401,"reading_title":{"format":"inline-parts-v1","parts":[["text","The product of the linear factors "],["math","\\(X - x_i\\)"],["text"," is separable"]],"source":"The product of the linear factors $X - x_i$ is separable"},"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#separable-prod-roots","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the polynomial $\\prod_{i=1}^{n} (X - x_i)$ over the function field $\\mathbb{C}(x_1, \\dots, x_n)$ is separable.","title":"The product of the linear factors $X - x_i$ is separable"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/adjoin-roots-eq-top","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.adjoin_roots_eq_top","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#adjoin-roots-eq-top","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every natural number $n$, the intermediate field of $\\mathbb{C}(x_1, \\dots, x_n)$ generated over $\\mathbb{C}$ by the roots $x_1, \\dots, x_n$ is the whole function field $\\mathbb{C}(x_1, \\dots, x_n)$.","title":"The roots generate the function field"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/root-mem-general-polynomial-roots","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.root_mem_generalPolynomial_roots","parent":4401,"reading_title":{"format":"inline-parts-v1","parts":[["text","Each "],["math","\\(x_i\\)"],["text"," is a root of the general polynomial"]],"source":"Each $x_i$ is a root of the general polynomial"},"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#root-mem-general-polynomial-roots","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, let $E_n = \\mathbb{C}(x_1, \\dots, x_n)$, let $a_k$ be the $k$-th elementary symmetric function of $x_1, \\dots, x_n$, and let $f_n = X^n - a_1 X^{n-1} + \\cdots + (-1)^n a_n$ be the general polynomial, with coefficients in $F_n = \\mathbb{C}(a_1, \\dots, a_n) \\subseteq E_n$. Then $f_n(x_i) = 0$ in $E_n$ for every $i \\in \\{1, \\dots, n\\}$.","title":"Each $x_i$ is a root of the general polynomial"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/trans-deg-nat-general-base-field","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.transDegNat_generalBaseField","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#trans-deg-nat-general-base-field","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, let $a_k$ be the $k$-th elementary symmetric function of $x_1, \\dots, x_n$ in $\\mathbb{C}(x_1, \\dots, x_n)$. The coefficient field $F_n = \\mathbb{C}(a_1, \\dots, a_n)$ of the general polynomial has numeric transcendence degree $n$ over $\\mathbb{C}$.","title":"Transcendence degree of the coefficient field"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/is-descent","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.IsDescent","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#is-descent","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For a natural number $n$, let $E = \\mathbb{C}(x_1, \\dots, x_n)$ be the field of rational functions in $n$ indeterminates and let $F = \\mathbb{C}(a_1, \\dots, a_n) \\subseteq E$ be the coefficient field of the general polynomial, where $a_k$ is the $k$-th elementary symmetric function of $x_1, \\dots, x_n$. A pair $(F_0, E_0)$ of intermediate fields of $\\mathbb{C} \\subseteq E$ is a descent when $F_0 \\subseteq F$,\u2026","title":"Descent of the general polynomial extension"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/restrict-scalars-adjoin-general-polynomi","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.restrictScalars_adjoin_generalPolynomial","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#restrict-scalars-adjoin-general-polynomi","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, let $E_n = \\mathbb{C}(x_1, \\dots, x_n)$, let $a_k$ be the $k$-th elementary symmetric function of $x_1, \\dots, x_n$, let $F_n = \\mathbb{C}(a_1, \\dots, a_n) \\subseteq E_n$ be the coefficient field, and let $f_n = X^n - a_1 X^{n-1} + \\cdots + (-1)^n a_n \\in F_n[X]$ be the general polynomial. The subfield of $E_n$ generated over $F_n$ by all roots of $f_n$ in $E_n$ is $E_n$ itself.","title":"The roots of the general polynomial generate the function field"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/is-descent-general-base-field","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.isDescent_generalBaseField","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#is-descent-general-base-field","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every natural number $n$, the pair $(F, E)$ is a descent of the general polynomial extension, where $E = \\mathbb{C}(x_1, \\dots, x_n)$ is the field of rational functions in $n$ indeterminates and $F = \\mathbb{C}(a_1, \\dots, a_n)$ is the coefficient field, generated over $\\mathbb{C}$ by the elementary symmetric functions $a_1, \\dots, a_n$ of $x_1, \\dots, x_n$.","title":"The trivial descent"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ed-sn","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.edSn","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ed-sn","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For a natural number $n$, let $\\mathbb{C}(x_1, \\dots, x_n)$ be the field of rational functions in $n$ indeterminates. The essential dimension of $S_n$ over $\\mathbb{C}$, in the general-polynomial parameter sense, is the least natural number $d$ such that some intermediate field $F_0$ of $\\mathbb{C} \\subseteq \\mathbb{C}(x_1, \\dots, x_n)$ of transcendence degree at most $d$ over $\\mathbb{C}$ admits an intermediate\u2026","title":"Essential dimension of the symmetric group"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ed-sn-le","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.edSn_le","parent":4401,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trivial upper bound for the essential dimension of "],["math","\\(S_n\\)"]],"source":"Trivial upper bound for the essential dimension of $S_n$"},"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ed-sn-le","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"For every natural number $n$, the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$, in the general-polynomial parameter sense, is at most $n$.","title":"Trivial upper bound for the essential dimension of $S_n$"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ess-dim-ratio","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.essDimRatio","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ess-dim-ratio","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"The catalogue constant $C_{79}$ is the limit superior, as $n \\to \\infty$, of $\\operatorname{ed}_{\\mathbb{C}}(S_n) / n$, where $\\operatorname{ed}_{\\mathbb{C}}(S_n)$ is the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$ in the general-polynomial parameter sense.","title":"Asymptotic essential-dimension ratio of the symmetric groups"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ess-dim-ratio-le-one","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetric.essDimRatio_le_one","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ess-dim-ratio-le-one","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"fully_proved","summary":"The constant $C_{79}$, the limit superior as $n \\to \\infty$ of $\\operatorname{ed}_{\\mathbb{C}}(S_n) / n$ with the essential dimension of $S_n$ over $\\mathbb{C}$ taken in the general-polynomial parameter sense, satisfies $C_{79} \\le 1$.","title":"The essential-dimension ratio of the symmetric groups is at most one"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/buhler-reichstein1997-upper-bound-wanted","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetricWanted.buhlerReichstein1997UpperBound_wanted","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#buhler-reichstein1997-upper-bound-wanted","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"can_prove","summary":"Write $\\operatorname{ed}(S_n)$ for the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$ in the general-polynomial sense: the least transcendence degree over $\\mathbb{C}$ of a subfield $F_0$ of the coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$ of the general polynomial of degree $n$ over which that polynomial descends up to Tschirnhaus transformation. Then $\\operatorname{ed}(S_n) \\le n - 3$ for\u2026","title":"Upper bound for the essential dimension of symmetric groups"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/buhler-reichstein-duncan-lower-bound-wan","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetricWanted.buhlerReichsteinDuncanLowerBound_wanted","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#buhler-reichstein-duncan-lower-bound-wan","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"can_prove","summary":"Write $\\operatorname{ed}(S_n)$ for the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$ in the general-polynomial sense: the least transcendence degree over $\\mathbb{C}$ of a subfield $F_0$ of the coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$ of the general polynomial of degree $n$ over which that polynomial descends up to Tschirnhaus transformation. Then\u2026","title":"Lower bound for the essential dimension of symmetric groups"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ess-dim-ratio-lower-bound-wanted","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetricWanted.essDimRatioLowerBound_wanted","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ess-dim-ratio-lower-bound-wanted","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"can_prove","summary":"Write $\\operatorname{ed}(S_n)$ for the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$ in the general-polynomial sense: the least transcendence degree over $\\mathbb{C}$ of a subfield $F_0$ of the coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$ of the general polynomial of degree $n$ over which that polynomial descends up to Tschirnhaus transformation. Then\u2026","title":"Lower bound for the asymptotic essential-dimension ratio"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ed-sn-five-wanted","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetricWanted.edSn_five_wanted","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ed-sn-five-wanted","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"can_prove","summary":"Write $\\operatorname{ed}(S_n)$ for the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$ in the general-polynomial sense: the least transcendence degree over $\\mathbb{C}$ of a subfield $F_0$ of the coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$ of the general polynomial of degree $n$ over which that polynomial descends up to Tschirnhaus transformation. Then $\\operatorname{ed}(S_5) = 2$.","title":"Essential dimension of the symmetric group on five letters"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ed-sn-four-wanted","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetricWanted.edSn_four_wanted","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ed-sn-four-wanted","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"can_prove","summary":"Write $\\operatorname{ed}(S_n)$ for the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$ in the general-polynomial sense: the least transcendence degree over $\\mathbb{C}$ of a subfield $F_0$ of the coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$ of the general polynomial of degree $n$ over which that polynomial descends up to Tschirnhaus transformation. Then $\\operatorname{ed}(S_4) = 2$.","title":"Essential dimension of the symmetric group on four letters"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ed-sn-six-wanted","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetricWanted.edSn_six_wanted","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ed-sn-six-wanted","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"can_prove","summary":"Write $\\operatorname{ed}(S_n)$ for the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$ in the general-polynomial sense: the least transcendence degree over $\\mathbb{C}$ of a subfield $F_0$ of the coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$ of the general polynomial of degree $n$ over which that polynomial descends up to Tschirnhaus transformation. Then $\\operatorname{ed}(S_6) = 3$.","title":"Essential dimension of the symmetric group on six letters"},{"id":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/ed-sn-seven-wanted","kind":"node","lean":"MathlibExt.FieldTheory.EssentialDimensionSymmetricWanted.edSn_seven_wanted","parent":4401,"result":true,"route":"../roadmap/FieldTheory/galois-theory/essential-dimension-of-symmetric-groups/index.html#ed-sn-seven-wanted","scope":"FieldTheory/galois-theory/essential-dimension-of-symmetric-groups","status":"can_prove","summary":"Write $\\operatorname{ed}(S_n)$ for the essential dimension of the symmetric group $S_n$ over $\\mathbb{C}$ in the general-polynomial sense: the least transcendence degree over $\\mathbb{C}$ of a subfield $F_0$ of the coefficient field $\\mathbb{C}(a_1, \\dots, a_n)$ of the general polynomial of degree $n$ over which that polynomial descends up to Tschirnhaus transformation. Then $\\operatorname{ed}(S_7) = 4$.","title":"Essential dimension of the symmetric group on seven letters"},{"data_url":"scopes/FieldTheory/galois-theory/galois-extensions-of-the-real-numbers.json","id":"FieldTheory/galois-theory/galois-extensions-of-the-real-numbers","kind":"scope","navigation":"module","parent":4393,"result":false,"route":"../roadmap/FieldTheory/galois-theory/galois-extensions-of-the-real-numbers/index.html","scope":"FieldTheory/galois-theory/galois-extensions-of-the-real-numbers","status":"fully_proved","title":"Galois extensions of the real numbers"},{"id":"FieldTheory/galois-theory/galois-extensions-of-the-real-numbers/finrank-eq-one-or-two","kind":"node","lean":"RealGalois.finrank_eq_one_or_two","parent":4438,"result":true,"route":"../roadmap/FieldTheory/galois-theory/galois-extensions-of-the-real-numbers/index.html#finrank-eq-one-or-two","scope":"FieldTheory/galois-theory/galois-extensions-of-the-real-numbers","status":"fully_proved","summary":"If $E$ is a field that is a finite-dimensional algebra over $\\mathbb{R}$, then the dimension of $E$ as a real vector space is $1$ or $2$.","title":"Degree of a finite extension of the reals"},{"id":"FieldTheory/galois-theory/galois-extensions-of-the-real-numbers/gal-card-eq-one-or-two","kind":"node","lean":"RealGalois.gal_card_eq_one_or_two","parent":4438,"result":true,"route":"../roadmap/FieldTheory/galois-theory/galois-extensions-of-the-real-numbers/index.html#gal-card-eq-one-or-two","scope":"FieldTheory/galois-theory/galois-extensions-of-the-real-numbers","status":"fully_proved","summary":"If $E$ is a finite Galois extension of $\\mathbb{R}$, then the Galois group $\\mathrm{Gal}(E/\\mathbb{R})$ has order $1$ or $2$.","title":"Galois groups over the reals have order 1 or 2"},{"id":"FieldTheory/galois-theory/galois-extensions-of-the-real-numbers/gal-card-pow-two","kind":"node","lean":"RealGalois.gal_card_pow_two","parent":4438,"result":true,"route":"../roadmap/FieldTheory/galois-theory/galois-extensions-of-the-real-numbers/index.html#gal-card-pow-two","scope":"FieldTheory/galois-theory/galois-extensions-of-the-real-numbers","status":"fully_proved","summary":"If $E$ is a finite Galois extension of $\\mathbb{R}$, then the Galois group $\\mathrm{Gal}(E/\\mathbb{R})$ has order $2^n$ for some natural number $n$.","title":"Galois groups over the reals have order a power of 2"},{"data_url":"scopes/FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons.json","id":"FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons","kind":"scope","navigation":"module","parent":4393,"result":false,"route":"../roadmap/FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons/index.html","scope":"FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons","status":"can_prove","title":"Gauss\u2013Wantzel theorem on constructible polygons"},{"id":"FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons/is-constructible","kind":"node","lean":"MathlibExt.NumberTheory.GaussWantzelWanted.NumberTheory.GaussWantzel.IsConstructible","parent":4442,"result":true,"route":"../roadmap/FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons/index.html#is-constructible","scope":"FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons","status":"fully_proved","summary":"The constructible real numbers form the smallest set of real numbers that contains every rational number and is closed under addition, negation, multiplication, inverses of nonzero elements and square roots of nonnegative elements; equivalently, the smallest subfield of $\\mathbb{R}$ closed under square roots.","title":"Constructible real numbers"},{"id":"FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons/gauss-wantzel-number","kind":"node","lean":"MathlibExt.NumberTheory.GaussWantzelWanted.NumberTheory.GaussWantzel.GaussWantzelNumber","parent":4442,"result":true,"route":"../roadmap/FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons/index.html#gauss-wantzel-number","scope":"FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons","status":"fully_proved","summary":"A natural number $n$ is a Gauss\u2013Wantzel number if $n > 0$, every prime factor of $n$ is either $2$ or a Fermat prime, and $p^2$ does not divide $n$ for every odd prime factor $p$ of $n$.","title":"Gauss\u2013Wantzel number"},{"id":"FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons/gauss-wantzel-constructible-polygon","kind":"node","lean":"MathlibExt.NumberTheory.GaussWantzelWanted.NumberTheory.GaussWantzel.gauss_wantzel_constructible_polygon","parent":4442,"result":true,"route":"../roadmap/FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons/index.html#gauss-wantzel-constructible-polygon","scope":"FieldTheory/galois-theory/gauss-wantzel-theorem-on-constructible-polygons","status":"can_prove","summary":"For every natural number $n \\ge 3$, the real number $\\cos(2\\pi/n)$ is constructible (it lies in the smallest subfield of $\\mathbb{R}$ closed under square roots) if and only if $n$ is a Gauss\u2013Wantzel number, that is, every prime factor of $n$ is $2$ or a Fermat prime and every odd prime factor of $n$ divides $n$ only once.","title":"Gauss\u2013Wantzel theorem"},{"data_url":"scopes/FieldTheory/galois-theory/solvability-by-radicals.json","id":"FieldTheory/galois-theory/solvability-by-radicals","kind":"scope","navigation":"module","parent":4393,"result":false,"route":"../roadmap/FieldTheory/galois-theory/solvability-by-radicals/index.html","scope":"FieldTheory/galois-theory/solvability-by-radicals","status":"can_prove","title":"Solvability by radicals"},{"id":"FieldTheory/galois-theory/solvability-by-radicals/not-solvable-by-rad-of-gal-iso-perm","kind":"node","lean":"GaloisTheory.not_solvableByRad_of_gal_iso_perm","parent":4446,"result":true,"route":"../roadmap/FieldTheory/galois-theory/solvability-by-radicals/index.html#not-solvable-by-rad-of-gal-iso-perm","scope":"FieldTheory/galois-theory/solvability-by-radicals","status":"fully_proved","summary":"Let $F$ be a field, let $E$ be a field extension of $F$, and let $n \\ge 5$. If $p \\in F[X]$ is irreducible and the Galois group of $p$ over $F$ is isomorphic to the symmetric group $S_n$, then no root of $p$ in $E$ is solvable by radicals over $F$.","title":"Abel\u2013Ruffini, contrapositive form"},{"id":"FieldTheory/galois-theory/solvability-by-radicals/not-solvable-by-rad-of-gal-iso-s5","kind":"node","lean":"GaloisTheory.not_solvableByRad_of_gal_iso_S5","parent":4446,"reading_title":{"format":"inline-parts-v1","parts":[["text","Abel\u2013Ruffini for "],["math","\\(S_5\\)"]],"source":"Abel\u2013Ruffini for $S_5$"},"result":true,"route":"../roadmap/FieldTheory/galois-theory/solvability-by-radicals/index.html#not-solvable-by-rad-of-gal-iso-s5","scope":"FieldTheory/galois-theory/solvability-by-radicals","status":"fully_proved","summary":"Let $F$ be a field and let $E$ be a field extension of $F$. If $p \\in F[X]$ is irreducible and the Galois group of $p$ over $F$ is isomorphic to the symmetric group $S_5$, then no root of $p$ in $E$ is solvable by radicals over $F$.","title":"Abel\u2013Ruffini for $S_5$"},{"id":"FieldTheory/galois-theory/solvability-by-radicals/abel-ruffini","kind":"node","lean":"MathlibExt.Algebra.AbelRuffiniWanted.abel_ruffini","parent":4446,"result":true,"route":"../roadmap/FieldTheory/galois-theory/solvability-by-radicals/index.html#abel-ruffini","scope":"FieldTheory/galois-theory/solvability-by-radicals","status":"can_prove","summary":"Let $n \\ge 1$ be a natural number. Every complex root of every polynomial with rational coefficients of degree exactly $n$ is expressible by radicals over $\\mathbb{Q}$ if and only if $n \\le 4$. Here a complex number is expressible by radicals over $\\mathbb{Q}$ when it lies in the smallest subfield $S$ of $\\mathbb{C}$ such that $z \\in S$ whenever $z \\in \\mathbb{C}$ and $z^k \\in S$ for some integer $k \\ge 1$.","title":"Abel\u2013Ruffini theorem"},{"id":"FieldTheory/galois-theory/solvability-by-radicals/solvable-iff-solvable-by-rad","kind":"node","lean":"MathlibExt.Algebra.SolvableByRadicalsWanted.Algebra.solvable_iff_solvableByRad","parent":4446,"result":true,"route":"../roadmap/FieldTheory/galois-theory/solvability-by-radicals/index.html#solvable-iff-solvable-by-rad","scope":"FieldTheory/galois-theory/solvability-by-radicals","status":"can_prove","summary":"Let $F$ be a field of characteristic zero and let $p \\in F[X]$ be a nonzero polynomial. Every root of $p$ in an algebraic closure $\\overline{F}$ of $F$ is solvable by radicals over $F$ (obtainable from $F$ by field operations and $n$-th roots inside $\\overline{F}$) if and only if the Galois group of $p$ over $F$ is solvable.","title":"Solvable extensions \u2194 solvable groups"},{"data_url":"scopes/FieldTheory/finite-fields.json","id":"FieldTheory/finite-fields","kind":"scope","parent":4392,"result":false,"route":"../roadmap/FieldTheory/finite-fields/index.html","scope":"FieldTheory/finite-fields","status":"can_prove","summary":"Finite fields and polynomials over them: irreducible and primitive polynomials, permutation polynomials, and the arithmetic of squares.","title":"Finite fields"},{"id":"FieldTheory/finite-fields/exists-sq-ne-zero-one-neg-one","kind":"node","lean":"exists_sq_ne_zero_one_neg_one","parent":4451,"reading_title":{"format":"inline-parts-v1","parts":[["text","A square avoiding "],["math","\\(0\\)"],["text",", "],["math","\\(1\\)"],["text"," and "],["math","\\(-1\\)"],["text"," in a finite field"]],"source":"A square avoiding $0$, $1$ and $-1$ in a finite field"},"result":true,"route":"../roadmap/FieldTheory/finite-fields/index.html#exists-sq-ne-zero-one-neg-one","scope":"FieldTheory/finite-fields","status":"fully_proved","summary":"If $F$ is a finite field with more than five elements, then there is an element $r \\in F$ with $r^2 \\neq 0$, $r^2 \\neq 1$ and $r^2 \\neq -1$.","title":"A square avoiding $0$, $1$ and $-1$ in a finite field"},{"id":"FieldTheory/finite-fields/aeval-pow-card-eq-pow-card","kind":"node","lean":"Polynomial.aeval_pow_card_eq_pow_card","parent":4451,"result":true,"route":"../roadmap/FieldTheory/finite-fields/index.html#aeval-pow-card-eq-pow-card","scope":"FieldTheory/finite-fields","status":"fully_proved","summary":"Evaluating `p` at `x ^ q` equals the `q`-th power of evaluating `p` at `x`, where `q` is the cardinality of the finite field `F`.","title":"Polynomial evaluation commutes with the Frobenius power"},{"data_url":"scopes/FieldTheory/finite-fields/permutation-polynomials.json","id":"FieldTheory/finite-fields/permutation-polynomials","kind":"scope","navigation":"module","parent":4451,"result":false,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","title":"Permutation polynomials"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-permutation-polynomial","kind":"node","lean":"Polynomial.IsPermutationPolynomial","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-permutation-polynomial","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"fully_proved","summary":"A polynomial $f$ over a finite field $F$ is a permutation polynomial when its evaluation map $F \\to F$, $x \\mapsto f(x)$, is bijective.","title":"Permutation polynomial over a finite field"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-kcomplete","kind":"node","lean":"Polynomial.IsKComplete","parent":4454,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial complete to level "],["math","\\(k\\)"]],"source":"Polynomial complete to level $k$"},"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-kcomplete","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field, $f \\in F[X]$ and $k$ a natural number. The polynomial $f$ is complete to level $k$ ($k$-complete) when $f + iX$ is a permutation polynomial for every natural number $i \\le k$, where $i$ is viewed as an element of $F$.","title":"Polynomial complete to level $k$"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-kcomplete-iff","kind":"node","lean":"Polynomial.isKComplete_iff","parent":4454,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unfolded form of "],["math","\\(k\\)"],["text","-completeness"]],"source":"Unfolded form of $k$-completeness"},"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-kcomplete-iff","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field, $f \\in F[X]$ and $k$ a natural number. Then $f$ is complete to level $k$ if and only if for every natural number $i \\le k$ the map $F \\to F$, $x \\mapsto (f + iX)(x)$, is bijective, where $i$ is viewed as an element of $F$.","title":"Unfolded form of $k$-completeness"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-kcomplete-mono","kind":"node","lean":"Polynomial.isKComplete_mono","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-kcomplete-mono","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field, $f \\in F[X]$ and let $j \\le k$ be natural numbers. If $f$ is complete to level $k$, then $f$ is complete to level $j$.","title":"Completeness at a lower level"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-kcomplete-zero","kind":"node","lean":"Polynomial.isKComplete_zero","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-kcomplete-zero","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field and $f \\in F[X]$. Then $f$ is complete to level $0$ if and only if $f$ is a permutation polynomial.","title":"Completeness at level zero"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-kcomplete-succ-iff","kind":"node","lean":"Polynomial.isKComplete_succ_iff","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-kcomplete-succ-iff","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field, $f \\in F[X]$ and $k$ a natural number. Then $f$ is complete to level $k + 1$ if and only if $f$ is complete to level $k$ and $f + (k + 1)X$ is a permutation polynomial, where $k + 1$ is viewed as an element of $F$.","title":"Completeness at the next level"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-kcomplete-self","kind":"node","lean":"Polynomial.isKComplete_self","parent":4454,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(k\\)"],["text","-complete polynomial is a permutation polynomial"]],"source":"A $k$-complete polynomial is a permutation polynomial"},"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-kcomplete-self","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field, $f \\in F[X]$ and $k$ a natural number. If $f$ is complete to level $k$, then $f$ is a permutation polynomial.","title":"A $k$-complete polynomial is a permutation polynomial"},{"id":"FieldTheory/finite-fields/permutation-polynomials/hermite-dickson-permutation-criterion","kind":"node","lean":"MathlibExt.FieldTheory.HermiteDickson.hermite_dickson_permutation_criterion","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#hermite-dickson-permutation-criterion","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"fully_proved","summary":"Let $\\mathbb{F}_q$ be a finite field with $q$ elements and characteristic $p$, and let $f \\in \\mathbb{F}_q[X]$. Then $f$ is a permutation polynomial of $\\mathbb{F}_q$, that is, $x \\mapsto f(x)$ is a bijection of $\\mathbb{F}_q$, if and only if $f$ has exactly one root in $\\mathbb{F}_q$ and, for every integer $t$ with $1 \\le t \\le q - 2$ and $p \\nmid t$, the remainder of $f^t$ upon division by $X^q - X$ has degree at\u2026","title":"Hermite\u2013Dickson criterion"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-permutation-polynomial-x-pow-iff-copr","kind":"node","lean":"MetaMathlibExt.isPermutationPolynomial_X_pow_iff_coprime","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-permutation-polynomial-x-pow-iff-copr","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $K$ be a finite field with $q$ elements and let $d \\ge 1$ be an integer. Then $X^d$ is a permutation polynomial of $K$ if and only if $\\gcd(d, q - 1) = 1$.","title":"Monomial permutation polynomials of a finite field"},{"id":"FieldTheory/finite-fields/permutation-polynomials/injective-iff-sum-pow-eq","kind":"node","lean":"MetaMathlibExt.injective_iff_sum_pow_eq","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#injective-iff-sum-pow-eq","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $K$ be a finite field with $q$ elements and let $a_1, \\dots, a_q \\in K$. Then $a_1, \\dots, a_q$ are pairwise distinct if and only if $\\sum_{i=1}^{q} a_i^t = 0$ for every $t = 0, 1, \\dots, q - 2$ and $\\sum_{i=1}^{q} a_i^{q-1} = -1$, where $0^0 = 1$.","title":"Power-sum criterion for distinct elements of a finite field"},{"id":"FieldTheory/finite-fields/permutation-polynomials/sum-neg-one-pow-choose-mul-pow","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_mul_pow","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#sum-neg-one-pow-choose-mul-pow","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $x, y, k \\in \\mathbb{C}$ and let $n, m$ be positive integers. If $n = m$, then","title":"Alternating binomial sums of powers"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-permutation-polynomial-sum-pow-mul-x","kind":"node","lean":"MetaMathlibExt.isPermutationPolynomial_sum_pow_mul_X_pow_add","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-permutation-polynomial-sum-pow-mul-x","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $p$ be an odd prime and let $k, t, a$ be positive integers with $\\gcd(a + 1, p - 1) = 1$ and $k \\equiv 1 \\pmod{p(p-1)}$. Suppose that $p - 1 \\nmid t$ or $p^2 \\mid k - 1$. Then the polynomial $\\sum_{n=1}^{k} n^t X^{n+a}$, with coefficients reduced modulo $p$, is a permutation polynomial of $\\mathbb{F}_p$.","title":"Shifted power-sum permutation polynomials of a prime field"},{"id":"FieldTheory/finite-fields/permutation-polynomials/is-permutation-polynomial-sum-mul-pow-mu","kind":"node","lean":"MetaMathlibExt.isPermutationPolynomial_sum_mul_pow_mul_X_pow_mul","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#is-permutation-polynomial-sum-mul-pow-mu","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $p$ be an odd prime and let $k, t, a$ be positive integers with $p \\nmid a$, $\\gcd(a, p - 1) = 1$ and $k \\equiv 1 \\pmod{p(p-1)}$. Suppose that $p - 1 \\nmid t$ or $p^2 \\mid k - 1$. Then the polynomial $\\sum_{n=1}^{k} (an)^t X^{an}$, with coefficients reduced modulo $p$, is a permutation polynomial of $\\mathbb{F}_p$.","title":"Dilated power-sum permutation polynomials of a prime field"},{"id":"FieldTheory/finite-fields/permutation-polynomials/one-sub-x-pow-mul-sum-pow-mul-x-pow","kind":"node","lean":"MetaMathlibExt.one_sub_X_pow_mul_sum_pow_mul_X_pow","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#one-sub-x-pow-mul-sum-pow-mul-x-pow","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"For positive integers $t$ and $k$, the following identity holds in $\\mathbb{Z}[X]$:","title":"Rational form of the power-weighted geometric sum"},{"id":"FieldTheory/finite-fields/permutation-polynomials/prime-dvd-sum-compositions-prod-pow","kind":"node","lean":"MetaMathlibExt.prime_dvd_sum_compositions_prod_pow","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#prime-dvd-sum-compositions-prod-pow","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $p$ be an odd prime and let $k, s, t$ be positive integers with $s \\le p - 2$ and $k \\equiv 1 \\pmod{p(p-1)}$, such that $p^2 \\mid k - 1$ whenever $p - 1 \\mid t$. Let $\\mu = \\lfloor ks/(p - 1) \\rfloor$. Then","title":"Vishwakarma\u2013Singh congruence for ordered partitions"},{"id":"FieldTheory/finite-fields/permutation-polynomials/prime-dvd-sum-compositions-prod","kind":"node","lean":"MetaMathlibExt.prime_dvd_sum_compositions_prod","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#prime-dvd-sum-compositions-prod","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $p$ be an odd prime and let $k, s$ be positive integers with $s \\le p - 2$ and $k \\equiv 1 \\pmod{p(p-1)}$. Let $\\mu = \\lfloor ks/(p - 1) \\rfloor$. Then","title":"Vishwakarma\u2013Singh congruence for ordered partitions, linear case"},{"id":"FieldTheory/finite-fields/permutation-polynomials/prime-dvd-sum-compositions-mul-prod-pow","kind":"node","lean":"MetaMathlibExt.prime_dvd_sum_compositions_mul_prod_pow","parent":4454,"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#prime-dvd-sum-compositions-mul-prod-pow","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $p$ be an odd prime and let $a, k, s, t$ be positive integers with $\\gcd(a, p - 1) = 1$, $s \\le p - 2$ and $k \\equiv 1 \\pmod{p(p-1)}$, such that $p^2 \\mid k - 1$ whenever $p - 1 \\mid t$. Let $\\mu = \\lfloor aks/(p - 1) \\rfloor$. Then","title":"Vishwakarma\u2013Singh congruence for ordered partitions into multiples"},{"id":"FieldTheory/finite-fields/permutation-polynomials/sum-zmod-pow-eq-ite","kind":"node","lean":"MathlibExt.Algebra.PermutationPolynomialsWanted.sum_zmod_pow_eq_ite","parent":4454,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power sums over "],["math","\\(\\mathbb{Z}/p\\mathbb{Z}\\)"]],"source":"Power sums over $\\mathbb{Z}/p\\mathbb{Z}$"},"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#sum-zmod-pow-eq-ite","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $p$ be a prime and $t$ a positive integer. Then $\\sum_{a \\in \\mathbb{Z}/p\\mathbb{Z}} a^t$ equals $-1$ if $p - 1$ divides $t$, and $0$ otherwise.","title":"Power sums over $\\mathbb{Z}/p\\mathbb{Z}$"},{"id":"FieldTheory/finite-fields/permutation-polynomials/bijective-sum-pow-mul-pow-zmod","kind":"node","lean":"MathlibExt.Algebra.PermutationPolynomialsWanted.bijective_sum_pow_mul_pow_zmod","parent":4454,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutation polynomials "],["math","\\(\\sum_{n=1}^k n^t X^n\\)"],["text"," over "],["math","\\(\\mathbb{F}_p\\)"]],"source":"Permutation polynomials $\\sum_{n=1}^k n^t X^n$ over $\\mathbb{F}_p$"},"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#bijective-sum-pow-mul-pow-zmod","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $p$ be an odd prime and let $k$ and $t$ be positive integers with $k \\equiv 1 \\pmod{p(p-1)}$. Suppose that $k \\equiv 1 \\pmod{p^2}$ or that $p - 1$ does not divide $t$. Then the map $c \\mapsto \\sum_{n=1}^{k} n^t c^n$ is a bijection from $\\mathbb{Z}/p\\mathbb{Z}$ to itself.","title":"Permutation polynomials $\\sum_{n=1}^k n^t X^n$ over $\\mathbb{F}_p$"},{"id":"FieldTheory/finite-fields/permutation-polynomials/finite-field-sum-eval-mul-pow-eq-id-iff","kind":"node","lean":"MathlibExt.Algebra.PermutationPolynomialsWanted.finiteField_sum_eval_mul_pow_eq_id_iff","parent":4454,"reading_title":{"format":"inline-parts-v1","parts":[["text","When "],["math","\\(\\sum_{n=1}^k h(n) X^n\\)"],["text"," induces the identity on "],["math","\\(\\mathbb{F}_q\\)"]],"source":"When $\\sum_{n=1}^k h(n) X^n$ induces the identity on $\\mathbb{F}_q$"},"result":true,"route":"../roadmap/FieldTheory/finite-fields/permutation-polynomials/index.html#finite-field-sum-eval-mul-pow-eq-id-iff","scope":"FieldTheory/finite-fields/permutation-polynomials","status":"can_prove","summary":"Let $p$ be a prime, $\\ell \\ge 1$ and $q = p^\\ell > 2$, and let $K$ be a finite field of characteristic $p$ with $q$ elements. Let $k$ be a positive integer with $k \\equiv 1 \\pmod{p(q-1)}$ and let $h \\in K[X]$, with degree $d$ (taken to be $0$ when $h = 0$) and coefficient $h_i$ of $X^i$. Then $\\sum_{n=1}^{k} h(n)\\,c^n = c$ for every $c \\in K$ if and only if $h(1) = 1$ and either $k \\equiv 1 \\pmod{p^2}$ or\u2026","title":"When $\\sum_{n=1}^k h(n) X^n$ induces the identity on $\\mathbb{F}_q$"},{"data_url":"scopes/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields.json","id":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","kind":"scope","navigation":"module","parent":4451,"result":false,"route":"../roadmap/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/index.html","scope":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","status":"fully_proved","title":"Primitive polynomials over finite fields"},{"id":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/is-primitive-polynomial","kind":"node","lean":"Polynomial.IsPrimitivePolynomial","parent":4475,"result":true,"route":"../roadmap/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/index.html#is-primitive-polynomial","scope":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $f \\in F[X]$, and let $n$ be the degree of $f$ (with $n = 0$ if $f = 0$). Then $f$ is primitive if it is monic, it is irreducible, and the class of $X$ in the quotient ring $F[X]/(f)$ has multiplicative order $q^n - 1$.","title":"Primitive polynomial over a finite field"},{"id":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/monic","kind":"node","lean":"Polynomial.IsPrimitivePolynomial.monic","parent":4475,"result":true,"route":"../roadmap/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/index.html#monic","scope":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","status":"fully_proved","summary":"Every primitive polynomial over a finite field is monic.","title":"Primitive polynomials are monic"},{"id":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/irreducible","kind":"node","lean":"Polynomial.IsPrimitivePolynomial.irreducible","parent":4475,"result":true,"route":"../roadmap/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/index.html#irreducible","scope":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","status":"fully_proved","summary":"Every primitive polynomial over a finite field is irreducible.","title":"Primitive polynomials are irreducible"},{"id":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/root-order-of","kind":"node","lean":"Polynomial.IsPrimitivePolynomial.root_orderOf","parent":4475,"result":true,"route":"../roadmap/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/index.html#root-order-of","scope":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $f \\in F[X]$ be a primitive polynomial of degree $n$. Then the class of $X$ in the quotient ring $F[X]/(f)$ has multiplicative order $q^n - 1$.","title":"Order of the root of a primitive polynomial"},{"id":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/iff","kind":"node","lean":"Polynomial.IsPrimitivePolynomial.iff","parent":4475,"result":true,"route":"../roadmap/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/index.html#iff","scope":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $f \\in F[X]$, and let $n$ be the degree of $f$ (with $n = 0$ if $f = 0$). Then $f$ is primitive if and only if $f$ is monic, $f$ is irreducible, and the class of $X$ in the quotient ring $F[X]/(f)$ has multiplicative order $q^n - 1$.","title":"Defining characterization of primitive polynomials"},{"id":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/iff-is-primitive-root","kind":"node","lean":"Polynomial.IsPrimitivePolynomial.iff_isPrimitiveRoot","parent":4475,"result":true,"route":"../roadmap/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/index.html#iff-is-primitive-root","scope":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $f \\in F[X]$, and let $n$ be the degree of $f$ (with $n = 0$ if $f = 0$). Let $\\alpha$ be the class of $X$ in the quotient ring $F[X]/(f)$. Then $f$ is primitive if and only if $f$ is monic, $f$ is irreducible, and $\\alpha$ is a primitive $(q^n - 1)$-th root of unity, that is, $\\alpha^{q^n - 1} = 1$ and $q^n - 1$ divides every natural number $l$ with $\\alpha^l = 1$.","title":"Primitive polynomials via primitive roots of unity"},{"id":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/zmod-iff","kind":"node","lean":"Polynomial.IsPrimitivePolynomial.zmod_iff","parent":4475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primitive polynomials over "],["math","\\(\\mathbb{Z}/p\\mathbb{Z}\\)"]],"source":"Primitive polynomials over $\\mathbb{Z}/p\\mathbb{Z}$"},"result":true,"route":"../roadmap/FieldTheory/finite-fields/primitive-polynomials-over-finite-fields/index.html#zmod-iff","scope":"FieldTheory/finite-fields/primitive-polynomials-over-finite-fields","status":"fully_proved","summary":"Let $p$ be a prime, let $f$ be a polynomial over $\\mathbb{Z}/p\\mathbb{Z}$, and let $n$ be the degree of $f$ (with $n = 0$ if $f = 0$). Then $f$ is primitive if and only if $f$ is monic, $f$ is irreducible, and the class of $X$ in the quotient ring $(\\mathbb{Z}/p\\mathbb{Z})[X]/(f)$ has multiplicative order $p^n - 1$.","title":"Primitive polynomials over $\\mathbb{Z}/p\\mathbb{Z}$"},{"data_url":"scopes/FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2.json","id":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2","kind":"scope","navigation":"module","parent":4451,"result":false,"route":"../roadmap/FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/index.html","scope":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2","status":"fully_proved","title":"Searching for irreducible polynomials over GF(2)"},{"id":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/exists-monic-irreducible-zmod-two","kind":"node","lean":"Polynomial.exists_monic_irreducible_zmod_two","parent":4483,"result":true,"route":"../roadmap/FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/index.html#exists-monic-irreducible-zmod-two","scope":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2","status":"fully_proved","summary":"Over `ZMod 2`, there is a monic irreducible polynomial of every positive degree.","title":"Monic irreducible binary polynomials exist in every degree"},{"id":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/passes-inverse-test","kind":"node","lean":"BitVec.passesInverseTest","parent":4483,"result":true,"route":"../roadmap/FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/index.html#passes-inverse-test","scope":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2","status":"fully_proved","summary":"Test exhaustively whether every nonzero word has a multiplicative inverse modulo `g.monicModulus`.","title":"Exhaustive inverse test for a binary polynomial modulus"},{"id":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/passes-inverse-test-iff-irreducible","kind":"node","lean":"BitVec.passesInverseTest_iff_irreducible","parent":4483,"result":true,"route":"../roadmap/FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/index.html#passes-inverse-test-iff-irreducible","scope":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2","status":"fully_proved","summary":"At positive width, the exhaustive inverse test succeeds exactly for irreducible monic moduli.","title":"Inverse test succeeds exactly for irreducible moduli"},{"id":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/find-irreducible-modulus","kind":"node","lean":"BitVec.findIrreducibleModulus","parent":4483,"result":true,"route":"../roadmap/FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/index.html#find-irreducible-modulus","scope":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2","status":"fully_proved","summary":"Search the lower coefficient words in ascending order for the first word passing the exhaustive inverse test. Width zero is deliberately reported as failure.","title":"Exhaustive search for an irreducible binary modulus"},{"id":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/find-irreducible-modulus-spec","kind":"node","lean":"BitVec.findIrreducibleModulus_spec","parent":4483,"result":true,"route":"../roadmap/FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/index.html#find-irreducible-modulus-spec","scope":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2","status":"fully_proved","summary":"At every positive width, exhaustive search returns an irreducible binary modulus.","title":"Exhaustive search returns an irreducible binary modulus"},{"id":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/find-irreducible-modulus-minimal","kind":"node","lean":"BitVec.findIrreducibleModulus_minimal","parent":4483,"result":true,"route":"../roadmap/FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2/index.html#find-irreducible-modulus-minimal","scope":"FieldTheory/finite-fields/searching-for-irreducible-polynomials-over-gf-2","status":"fully_proved","summary":"The returned modulus is no larger than any lower coefficient word whose monic modulus is irreducible.","title":"Minimality of the binary modulus found by exhaustive search"},{"data_url":"scopes/FieldTheory/extensions-and-closure.json","id":"FieldTheory/extensions-and-closure","kind":"scope","parent":4392,"result":false,"route":"../roadmap/FieldTheory/extensions-and-closure/index.html","scope":"FieldTheory/extensions-and-closure","status":"fully_proved","summary":"Algebraic field extensions and closures: minimal polynomials, separable and purely inseparable extensions, base change to separably closed fields, algebraic closures of series fields and quasi-algebraically closed fields.","title":"Field extensions and algebraic closure"},{"id":"FieldTheory/extensions-and-closure/finite-dimensional-adjoin-range-of-finit","kind":"node","lean":"RingHom.finiteDimensional_adjoin_range_of_finite_of_forall_mem_range","parent":4490,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/index.html#finite-dimensional-adjoin-range-of-finit","scope":"FieldTheory/extensions-and-closure","status":"fully_proved","summary":"If `S` is finite over `R` and `\u03c7' : S \u2192+* K` sends `R` into the image of `E` in `K`, then the field generated over `E` by the range of `\u03c7'` is finite-dimensional over `E`.","title":"Field generated by the image of a finite algebra is finite-dimensional"},{"id":"FieldTheory/extensions-and-closure/is-separable-iff-exists-monic-irreducibl","kind":"node","lean":"Algebra.isSeparable_iff_exists_monic_irreducible_separable","parent":4490,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/index.html#is-separable-iff-exists-monic-irreducibl","scope":"FieldTheory/extensions-and-closure","status":"fully_proved","summary":"Let $L/K$ be a finite field extension. Then $L/K$ is separable if and only if there is a monic, irreducible, separable polynomial $f \\in K[X]$ such that $L$ is isomorphic as a $K$-algebra to $K[X]/(f)$.","title":"Finite separable extensions are quotients by separable irreducible polynomials"},{"id":"FieldTheory/extensions-and-closure/exists-minpoly-x-pow-sub-c-of-finrank-eq","kind":"node","lean":"IsPurelyInseparable.exists_minpoly_X_pow_sub_C_of_finrank_eq_prime","parent":4490,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/index.html#exists-minpoly-x-pow-sub-c-of-finrank-eq","scope":"FieldTheory/extensions-and-closure","status":"fully_proved","summary":"Let $p$ be a prime, let $K$ be a field of characteristic $p$, and let $L/K$ be a finite purely inseparable field extension of degree $[L : K] = p$. Then there are $\\alpha \\in L$ and $a \\in K$ such that the minimal polynomial of $\\alpha$ over $K$ is $X^p - a$, $a$ is not a $p$-th power in $K$ (that is, $b^p \\neq a$ for every $b \\in K$), and $L = K(\\alpha)$.","title":"Purely inseparable extensions of prime degree"},{"id":"FieldTheory/extensions-and-closure/tsen","kind":"node","lean":"MathlibExt.FieldTheory.TsenWanted.tsen","parent":4490,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/index.html#tsen","scope":"FieldTheory/extensions-and-closure","status":"fully_proved","summary":"Let $k$ be an algebraically closed field and $k(X)$ the field of rational functions over $k$. For all natural numbers $n$ and $d$ with $0 < d < n$, every homogeneous polynomial $f$ of degree $d$ in $n$ variables over $k(X)$ has a nontrivial zero: there is $x = (x_1, \\dots, x_n) \\in k(X)^n$ with $x_i \\neq 0$ for some $i$ and $f(x) = 0$. In other words, $k(X)$ is a $C_1$ field.","title":"Tsen's theorem"},{"data_url":"scopes/FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality.json","id":"FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality","kind":"scope","navigation":"module","parent":4490,"result":false,"route":"../roadmap/FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality/index.html","scope":"FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality","status":"fully_proved","title":"Minimal polynomials and integrality"},{"id":"FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality/is-integral-iff-mem-lifts","kind":"node","lean":"minpoly.isIntegral_iff_mem_lifts","parent":4495,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality/index.html#is-integral-iff-mem-lifts","scope":"FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality","status":"fully_proved","summary":"Let $R$ be an integrally closed domain with fraction field $K$, and let $L$ be a field extension of $K$ that is also an $R$-algebra, with the $R$-algebra structure on $L$ the composite of $R \\to K$ and $K \\to L$. An element $x \\in L$ that is algebraic over $K$ is integral over $R$ if and only if the minimal polynomial of $x$ over $K$ is the image of a polynomial in $R[X]$ under the map $R[X] \\to K[X]$ that applies\u2026","title":"Integrality via a lift of the minimal polynomial"},{"id":"FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality/is-integral-iff-coeff-mem-range","kind":"node","lean":"minpoly.isIntegral_iff_coeff_mem_range","parent":4495,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality/index.html#is-integral-iff-coeff-mem-range","scope":"FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality","status":"fully_proved","summary":"Let $R$ be an integrally closed domain with fraction field $K$, and let $L$ be a field extension of $K$ that is also an $R$-algebra, with the $R$-algebra structure on $L$ the composite of $R \\to K$ and $K \\to L$. An element $x \\in L$ that is algebraic over $K$ is integral over $R$ if and only if every coefficient of the minimal polynomial of $x$ over $K$ lies in the image of $R$ in $K$.","title":"Integrality via the coefficients of the minimal polynomial"},{"id":"FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality/minpoly-eq-map-iff-is-integral","kind":"node","lean":"IsIntegrallyClosed.minpoly_eq_map_iff_isIntegral","parent":4495,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality/index.html#minpoly-eq-map-iff-is-integral","scope":"FieldTheory/extensions-and-closure/minimal-polynomials-and-integrality","status":"fully_proved","summary":"Let $A$ be an integrally closed domain with fraction field $K$, and let $L$ be a field extension of $K$ that is also an $A$-algebra, with the $A$-algebra structure on $L$ the composite of $A \\to K$ and $K \\to L$. An element $\\alpha \\in L$ that is integral over $K$ is integral over $A$ if and only if the minimal polynomial of $\\alpha$ over $K$ equals the image of the minimal polynomial of $\\alpha$ over $A$ under the\u2026","title":"Integrality via equality of minimal polynomials"},{"data_url":"scopes/FieldTheory/extensions-and-closure/puiseux-s-theorem.json","id":"FieldTheory/extensions-and-closure/puiseux-s-theorem","kind":"scope","navigation":"module","parent":4490,"result":false,"route":"../roadmap/FieldTheory/extensions-and-closure/puiseux-s-theorem/index.html","scope":"FieldTheory/extensions-and-closure/puiseux-s-theorem","status":"fully_proved","title":"Puiseux's theorem"},{"id":"FieldTheory/extensions-and-closure/puiseux-s-theorem/is-alg-closed-rat","kind":"node","lean":"HahnSeries.isAlgClosed_rat","parent":4499,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/puiseux-s-theorem/index.html#is-alg-closed-rat","scope":"FieldTheory/extensions-and-closure/puiseux-s-theorem","status":"fully_proved","summary":"Let $K$ be an algebraically closed field of characteristic zero. Then the field of Hahn series over $K$ with exponents in $\\mathbb{Q}$ (formal sums $\\sum_r a_r t^r$ with $a_r \\in K$ whose support $\\{r \\in \\mathbb{Q} : a_r \\neq 0\\}$ is well-ordered) is algebraically closed.","title":"Algebraic closedness of rational Hahn series"},{"id":"FieldTheory/extensions-and-closure/puiseux-s-theorem/puiseux-theorem","kind":"node","lean":"MetaMathlibExt.puiseux_theorem","parent":4499,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/puiseux-s-theorem/index.html#puiseux-theorem","scope":"FieldTheory/extensions-and-closure/puiseux-s-theorem","status":"fully_proved","summary":"Let $K$ be an algebraically closed field of characteristic zero, let $K((t))$ be the field of formal Laurent series over $K$, and let $K((t^{\\mathbb{Q}}))$ be the field of Hahn series over $K$ with exponents in $\\mathbb{Q}$ (formal sums $\\sum_r a_r t^r$ with $a_r \\in K$ whose support $\\{r \\in \\mathbb{Q} : a_r \\neq 0\\}$ is well-ordered). Then: (i) $K((t^{\\mathbb{Q}}))$ is algebraically closed; (ii) there are an\u2026","title":"Puiseux's theorem"},{"data_url":"scopes/FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials.json","id":"FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials","kind":"scope","navigation":"module","parent":4490,"result":false,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials/index.html","scope":"FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials","status":"fully_proved","title":"Separable contraction of irreducible polynomials"},{"id":"FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials/exists-irreducible-separable-expand-eq","kind":"node","lean":"Polynomial.exists_irreducible_separable_expand_eq","parent":4502,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials/index.html#exists-irreducible-separable-expand-eq","scope":"FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials","status":"fully_proved","summary":"Let $K$ be a field of characteristic $p$ (with $p = 0$ in characteristic zero) and let $f \\in K[x]$ be irreducible. Then there exist $n \\in \\mathbb{N}$ and an irreducible separable polynomial $g \\in K[x]$ such that $f(x) = g(x^{p^n})$, where $0^0 = 1$.","title":"Irreducible polynomials are expansions of separable irreducibles"},{"id":"FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials/unique-irreducible-separable-expand-eq","kind":"node","lean":"Polynomial.unique_irreducible_separable_expand_eq","parent":4502,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials/index.html#unique-irreducible-separable-expand-eq","scope":"FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials","status":"fully_proved","summary":"Let $K$ be a field of characteristic $p$ (with $p = 0$ in characteristic zero) and let $f \\in K[x]$ be irreducible. If $n_1, n_2 \\in \\mathbb{N}$ and irreducible separable polynomials $g_1, g_2 \\in K[x]$ satisfy $f(x) = g_1(x^{p^{n_1}})$ and $f(x) = g_2(x^{p^{n_2}})$, where $0^0 = 1$, then $n_1 = n_2$ and $g_1 = g_2$.","title":"The separable irreducible underlying an irreducible polynomial is unique"},{"id":"FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials/exists-unique-irreducible-separable-expa","kind":"node","lean":"Polynomial.exists_unique_irreducible_separable_expand_eq","parent":4502,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials/index.html#exists-unique-irreducible-separable-expa","scope":"FieldTheory/extensions-and-closure/separable-contraction-of-irreducible-polynomials","status":"fully_proved","summary":"Let $K$ be a field of characteristic $p$ (with $p = 0$ in characteristic zero, and $0^0 = 1$) and let $f \\in K[x]$ be irreducible. Then there exist $n \\in \\mathbb{N}$ and an irreducible separable polynomial $g \\in K[x]$ with $f(x) = g(x^{p^n})$, and for every $n' \\in \\mathbb{N}$ and every irreducible separable polynomial $g' \\in K[x]$ with $f(x) = g'(x^{p^{n'}})$ we have $n' = n$ and $g' = g$.","title":"Irreducible polynomials are uniquely expansions of separable irreducibles"},{"data_url":"scopes/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field.json","id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","kind":"scope","navigation":"module","parent":4490,"result":false,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","title":"Separable extensions over separably closed fields"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/alg-hom-card-eq-finrank-of-is-sep-closed","kind":"node","lean":"Algebra.TensorProduct.algHom_card_eq_finrank_of_isSepClosed","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#alg-hom-card-eq-finrank-of-is-sep-closed","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension and let $\\Omega$ be a separably closed field that is a $K$-algebra. Then the number of $K$-algebra homomorphisms $L \\to \\Omega$ equals the degree $[L : K]$.","title":"Number of embeddings into a separably closed field"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/sep-closed-eval-alg-hom","kind":"node","lean":"Algebra.TensorProduct.sepClosedEvalAlgHom","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#sep-closed-eval-alg-hom","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension, let $\\Omega$ be a separably closed field that is a $K$-algebra, and write $\\mathrm{Hom}_K(L, \\Omega)$ for the set of $K$-algebra homomorphisms $L \\to \\Omega$. The evaluation map is the $K$-algebra homomorphism from $L$ to the algebra of functions $\\mathrm{Hom}_K(L, \\Omega) \\to \\Omega$ sending $l \\in L$ to the function $\\sigma \\mapsto \\sigma(l)$.","title":"Evaluation of embeddings at an element"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/sep-closed-tensor-hom","kind":"node","lean":"Algebra.TensorProduct.sepClosedTensorHom","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#sep-closed-tensor-hom","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L / K$ be a finite separable field extension and let $\\Omega$ be a separably closed field extension of $K$. This is the homomorphism of $\\Omega$-algebras from $\\Omega \\otimes_K L$ to the algebra of functions from the set of $K$-algebra homomorphisms $L \\to \\Omega$ to $\\Omega$, which sends each pure tensor $\\omega \\otimes l$ to the function $\\sigma \\mapsto \\omega \\, \\sigma(l)$.","title":"Evaluation map on the base change to a separably closed field"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/alg-hom-linear-independent","kind":"node","lean":"Algebra.TensorProduct.algHom_linearIndependent","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#alg-hom-linear-independent","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $K$, $L$ and $\\Omega$ be fields such that $L$ and $\\Omega$ are $K$-algebras. Then the $K$-algebra homomorphisms $L \\to \\Omega$, viewed as functions $L \\to \\Omega$, are linearly independent over $\\Omega$.","title":"Dedekind independence of embeddings"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/column-functions-linear-independent","kind":"node","lean":"Algebra.TensorProduct.column_functions_linearIndependent","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#column-functions-linear-independent","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension of degree $n$, let $\\Omega$ be a separably closed field that is a $K$-algebra, and write $\\mathrm{Hom}_K(L, \\Omega)$ for the set of $K$-algebra homomorphisms $L \\to \\Omega$. Let $b_1, \\dots, b_n$ be a fixed basis of $L$ over $K$ (one particular basis chosen once and for all, not an arbitrary one). Then the $n$ functions $\\mathrm{Hom}_K(L, \\Omega) \\to \\Omega$ given by\u2026","title":"Embeddings evaluated on a basis"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/sep-closed-tensor-hom-injective","kind":"node","lean":"Algebra.TensorProduct.sepClosedTensorHom_injective","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#sep-closed-tensor-hom-injective","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension, let $\\Omega$ be a separably closed field that is a $K$-algebra, and write $\\mathrm{Hom}_K(L, \\Omega)$ for the set of $K$-algebra homomorphisms $L \\to \\Omega$. Then the evaluation homomorphism of $\\Omega$-algebras from $\\Omega \\otimes_K L$ to the algebra of functions $\\mathrm{Hom}_K(L, \\Omega) \\to \\Omega$, which sends $\\omega \\otimes l$ to the function\u2026","title":"Injectivity of the evaluation homomorphism"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/finrank-tensor-eq-finrank-pi-of-is-sep-c","kind":"node","lean":"Algebra.TensorProduct.finrank_tensor_eq_finrank_pi_of_isSepClosed","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#finrank-tensor-eq-finrank-pi-of-is-sep-c","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension, let $\\Omega$ be a separably closed field that is a $K$-algebra, and write $\\mathrm{Hom}_K(L, \\Omega)$ for the set of $K$-algebra homomorphisms $L \\to \\Omega$. Then the base change $\\Omega \\otimes_K L$ and the space of functions $\\mathrm{Hom}_K(L, \\Omega) \\to \\Omega$ have the same dimension as vector spaces over $\\Omega$.","title":"Dimensions of the two sides of the splitting"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/sep-closed-tensor-hom-surjective","kind":"node","lean":"Algebra.TensorProduct.sepClosedTensorHom_surjective","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#sep-closed-tensor-hom-surjective","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension, let $\\Omega$ be a separably closed field that is a $K$-algebra, and write $\\mathrm{Hom}_K(L, \\Omega)$ for the set of $K$-algebra homomorphisms $L \\to \\Omega$. Then the evaluation homomorphism of $\\Omega$-algebras from $\\Omega \\otimes_K L$ to the algebra of functions $\\mathrm{Hom}_K(L, \\Omega) \\to \\Omega$, which sends $\\omega \\otimes l$ to the function\u2026","title":"Surjectivity of the evaluation homomorphism"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/sep-closed-tensor-hom-bijective","kind":"node","lean":"Algebra.TensorProduct.sepClosedTensorHom_bijective","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#sep-closed-tensor-hom-bijective","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension, let $\\Omega$ be a separably closed field that is a $K$-algebra, and write $\\mathrm{Hom}_K(L, \\Omega)$ for the set of $K$-algebra homomorphisms $L \\to \\Omega$. Then the evaluation homomorphism of $\\Omega$-algebras from $\\Omega \\otimes_K L$ to the algebra of functions $\\mathrm{Hom}_K(L, \\Omega) \\to \\Omega$, which sends $\\omega \\otimes l$ to the function\u2026","title":"Bijectivity of the evaluation homomorphism"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/sep-closed-alg-equiv","kind":"node","lean":"Algebra.TensorProduct.sepClosedAlgEquiv","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#sep-closed-alg-equiv","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension, let $\\Omega$ be a separably closed field that is a $K$-algebra, and write $\\mathrm{Hom}_K(L, \\Omega)$ for the set of $K$-algebra homomorphisms $L \\to \\Omega$. The splitting isomorphism is the isomorphism of $\\Omega$-algebras from $\\Omega \\otimes_K L$ onto the algebra $\\Omega^{\\mathrm{Hom}_K(L, \\Omega)}$ of functions $\\mathrm{Hom}_K(L, \\Omega) \\to \\Omega$ given by the\u2026","title":"Splitting of a separable extension over a separably closed field"},{"id":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/sep-closed-alg-equiv-apply","kind":"node","lean":"Algebra.TensorProduct.sepClosedAlgEquiv_apply","parent":4506,"result":true,"route":"../roadmap/FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field/index.html#sep-closed-alg-equiv-apply","scope":"FieldTheory/extensions-and-closure/separable-extensions-over-separably-closed-field","status":"fully_proved","summary":"Let $L/K$ be a finite separable field extension, let $\\Omega$ be a separably closed field that is a $K$-algebra, and write $\\mathrm{Hom}_K(L, \\Omega)$ for the set of $K$-algebra homomorphisms $L \\to \\Omega$. For all $\\omega \\in \\Omega$ and $l \\in L$, the splitting isomorphism from $\\Omega \\otimes_K L$ onto the algebra $\\Omega^{\\mathrm{Hom}_K(L, \\Omega)}$ of functions $\\mathrm{Hom}_K(L, \\Omega) \\to \\Omega$ sends\u2026","title":"The splitting isomorphism on pure tensors"},{"area":"Algebra & Number Theory","data_url":"chapters/LinearAlgebra.json","id":"LinearAlgebra","kind":"scope","parent":0,"result":false,"route":"../roadmap/LinearAlgebra/index.html","scope":"LinearAlgebra","status":"can_prove","summary":"Linear algebra studies vector spaces, linear maps, matrices, duality, dimension, and spectral structure.","title":"Linear Algebra"},{"data_url":"scopes/LinearAlgebra/special-matrices.json","id":"LinearAlgebra/special-matrices","kind":"scope","parent":4518,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/index.html","scope":"LinearAlgebra/special-matrices","status":"can_prove","summary":"Classes of matrices defined by their structure or entries: Toeplitz, Hankel and circulant matrices, Hadamard and sign matrices, combinatorial matrices, nonnegative, doubly stochastic and totally positive matrices.","title":"Special matrices"},{"id":"LinearAlgebra/special-matrices/hadamard-zero-one-determinant-bound","kind":"node","lean":"MetaMathlibExt.hadamard_zero_one_determinant_bound","parent":4519,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/index.html#hadamard-zero-one-determinant-bound","scope":"LinearAlgebra/special-matrices","status":"fully_proved","summary":"Let $n \\ge 1$ and let $B$ be an $n \\times n$ real matrix each of whose entries is $0$ or $1$. Then","title":"Hadamard bound for zero-one determinants"},{"id":"LinearAlgebra/special-matrices/van-der-waerden-permanent","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.VanDerWaerdenWanted.van_der_waerden_permanent","parent":4519,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/index.html#van-der-waerden-permanent","scope":"LinearAlgebra/special-matrices","status":"fully_proved","summary":"For every integer $n \\ge 1$ and every doubly stochastic real $n \\times n$ matrix $A$ (nonnegative entries, every row and every column summing to $1$), the permanent of $A$ is at least $n!/n^n$.","title":"The van der Waerden permanent bound"},{"data_url":"scopes/LinearAlgebra/special-matrices/circulant-matrices.json","id":"LinearAlgebra/special-matrices/circulant-matrices","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","title":"Circulant matrices"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/zeta","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.zeta","parent":4522,"reading_title":{"format":"inline-parts-v1","parts":[["text","Standard primitive n-th root of unity "],["math","\\(e^{2\\pi i/n}\\)"]],"source":"Standard primitive n-th root of unity $e^{2\\pi i/n}$"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#zeta","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The exponential `exp (2 * \u03c0 * I / n)`, a primitive `n`-th root for nonzero `n`. This definition is total: at `n = 0`, it is `exp 0 = 1`.","title":"Standard primitive n-th root of unity $e^{2\\pi i/n}$"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/zeta-eq-std-add-char","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.zeta_eq_stdAddChar","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#zeta-eq-std-add-char","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"For nonzero `n`, `zeta n` is the canonical standard character at `1`.","title":"Standard root of unity as the standard additive character"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/zeta-is-primitive-root","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.zeta_isPrimitiveRoot","parent":4522,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(e^{2\\pi i/n}\\)"],["text"," is a primitive n-th root of unity"]],"source":"$e^{2\\pi i/n}$ is a primitive n-th root of unity"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#zeta-is-primitive-root","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"For nonzero `n`, the exponential is a primitive `n`-th root of unity.","title":"$e^{2\\pi i/n}$ is a primitive n-th root of unity"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-vec","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierVec","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-vec","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The `j`-th Fourier vector on `ZMod n`, using the canonical positive character.","title":"Fourier vector on the integers modulo n"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-vec-apply","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierVec_apply","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-vec-apply","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The character definition of a Fourier vector agrees with its exponential formula.","title":"Exponential formula for Fourier vectors"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/circulant-eigenvalue","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.circulantEigenvalue","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#circulant-eigenvalue","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The `j`-th Fourier eigenvalue of a circulant matrix is its canonical DFT.","title":"Fourier eigenvalue of a circulant matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/circulant-eigenvalue-apply","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.circulantEigenvalue_apply","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#circulant-eigenvalue-apply","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The canonical DFT definition agrees with the original negative-exponent eigenvalue sum.","title":"Exponential-sum formula for circulant eigenvalues"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/circulant-mul-vec-fourier-vec","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.circulant_mulVec_fourierVec","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#circulant-mul-vec-fourier-vec","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"Every Fourier vector is an eigenvector of every circulant matrix.","title":"Fourier vectors are eigenvectors of circulant matrices"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-matrix","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierMatrix","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-matrix","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The Fourier matrix, whose `j`-th column is the standard character `fourierVec n j`.","title":"Fourier matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-matrix-apply","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierMatrix_apply","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-matrix-apply","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The Fourier matrix agrees entrywise with its original exponential formula.","title":"Entrywise exponential formula for the Fourier matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-matrix-inv","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierMatrixInv","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-matrix-inv","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The inverse Fourier matrix, `(1 / n)` times the negative standard character.","title":"Inverse Fourier matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-matrix-inv-apply","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierMatrixInv_apply","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-matrix-inv-apply","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The inverse Fourier matrix agrees entrywise with its original exponential formula.","title":"Entrywise exponential formula for the inverse Fourier matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-matrix-mul-vec-eq-inv-dft","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierMatrix_mulVec_eq_invDFT","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-matrix-mul-vec-eq-inv-dft","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"Fourier matrix multiplication is the canonical inverse DFT multiplied by `n`.","title":"Fourier matrix acts as n times the inverse DFT"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-matrix-inv-mul-vec-eq-dft","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierMatrixInv_mulVec_eq_dft","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-matrix-inv-mul-vec-eq-dft","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"Inverse Fourier matrix multiplication is the canonical DFT divided by `n`.","title":"Inverse Fourier matrix acts as the DFT divided by n"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/fourier-matrix-mul-inv","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.fourierMatrix_mul_inv","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#fourier-matrix-mul-inv","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The Fourier matrix is invertible, with inverse `fourierMatrixInv`.","title":"Fourier matrix times its inverse is the identity"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/circulant-mul-fourier-matrix","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.circulant_mul_fourierMatrix","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#circulant-mul-fourier-matrix","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"Circulant matrices are diagonalized by the Fourier matrix.","title":"Circulant matrices are diagonalized by the Fourier matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/det-circulant","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.det_circulant","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#det-circulant","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The determinant of a circulant matrix is the product of its Fourier eigenvalues.","title":"Determinant of a circulant matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/circulant-eigenvalue-zero","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.circulantEigenvalue_zero","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#circulant-eigenvalue-zero","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The Fourier eigenvalue at `j = 0` is the sum of the entries of `v`.","title":"Zeroth circulant eigenvalue is the sum of entries"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/sum-circulant-eigenvalue","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.CirculantSpectrum.sum_circulantEigenvalue","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#sum-circulant-eigenvalue","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The Fourier eigenvalues sum to `n * v 0`.","title":"Sum of the eigenvalues of a circulant matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/ons-block-circulant","kind":"node","lean":"StatMech.Onsager.ons_blockCirculant","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#ons-block-circulant","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The block-circulant matrix on `\u2124_n \u00d7 Fin d` with block `v (p - q)` in position `(p, q)`.","title":"Block-circulant matrix over \u2124_n"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/ons-block-symbol","kind":"node","lean":"StatMech.Onsager.ons_blockSymbol","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#ons-block-symbol","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The Fourier symbol `\u2211_k \u03c9 ^ (k j) \u2022 v k` of a block-circulant matrix at frequency `j`.","title":"Fourier symbol of a block-circulant matrix"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/ons-block-circulant2-d","kind":"node","lean":"StatMech.Onsager.ons_blockCirculant2D","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#ons-block-circulant2-d","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The block-circulant matrix on `(\u2124_L \u00d7 \u2124_L) \u00d7 Fin d` with block `v (p - q)` in position `(p, q)`.","title":"Two-dimensional block-circulant matrix on the torus"},{"id":"LinearAlgebra/special-matrices/circulant-matrices/ons-block-symbol2-d","kind":"node","lean":"StatMech.Onsager.ons_blockSymbol2D","parent":4522,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/circulant-matrices/index.html#ons-block-symbol2-d","scope":"LinearAlgebra/special-matrices/circulant-matrices","status":"fully_proved","summary":"The Fourier symbol `\u2211_k \u03c9 ^ (k\u2081 j\u2081 + k\u2082 j\u2082) \u2022 v k` at frequency `j`.","title":"Fourier symbol of a two-dimensional block-circulant matrix"},{"data_url":"scopes/LinearAlgebra/special-matrices/hadamard-matrices.json","id":"LinearAlgebra/special-matrices/hadamard-matrices","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/hadamard-matrices/index.html","scope":"LinearAlgebra/special-matrices/hadamard-matrices","status":"fully_proved","title":"Hadamard matrices"},{"id":"LinearAlgebra/special-matrices/hadamard-matrices/is-hadamard-matrix","kind":"node","lean":"MetaMathlibExt.isHadamardMatrix","parent":4546,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hadamard-matrices/index.html#is-hadamard-matrix","scope":"LinearAlgebra/special-matrices/hadamard-matrices","status":"fully_proved","summary":"For a natural number $m$, an $m \\times m$ matrix $H$ with integer entries is a Hadamard matrix when every entry of $H$ is $1$ or $-1$ and $H^{\\mathsf T} H = m I$, where $I$ is the $m \\times m$ identity matrix.","title":"Hadamard matrix"},{"id":"LinearAlgebra/special-matrices/hadamard-matrices/exists-hadamard-matrix-with-circulant-co","kind":"node","lean":"MetaMathlibExt.exists_hadamard_matrix_with_circulant_core","parent":4546,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hadamard-matrices/index.html#exists-hadamard-matrix-with-circulant-co","scope":"LinearAlgebra/special-matrices/hadamard-matrices","status":"fully_proved","summary":"Let $n$ be a natural number such that either $n$ is a prime with $n \\equiv 3 \\pmod 4$, or $n = p(p + 2)$ for some natural number $p$ with $p$ and $p + 2$ both prime. Then there is an $(n + 1) \\times (n + 1)$ Hadamard matrix $H$ with integer entries, rows and columns indexed by $0, 1, \\dots, n$ (so every entry of $H$ is $1$ or $-1$ and $H^{\\mathsf T} H = (n + 1) I$), that has a circulant core: there is a function\u2026","title":"Hadamard matrices with circulant core: Paley and twin-prime cases"},{"id":"LinearAlgebra/special-matrices/hadamard-matrices/is-hadamard-matrix-iff-is-hadamard","kind":"node","lean":"MetaMathlibExt.isHadamardMatrix_iff_isHadamard","parent":4546,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hadamard-matrices/index.html#is-hadamard-matrix-iff-is-hadamard","scope":"LinearAlgebra/special-matrices/hadamard-matrices","status":"fully_proved","summary":"Let $m$ be a natural number and let $H$ be an $m \\times m$ integer matrix. Then every entry of $H$ is $1$ or $-1$ and $H^{\\mathsf T} H = m I$ if and only if every entry of $H$ is a unitary element of $\\mathbb{Z}$ (that is, $1$ or $-1$), $H H^{\\mathsf T} = m I$ and $H^{\\mathsf T} H = m I$. The right-hand side is the general notion of a Hadamard matrix, with entries unitary and $H H^{*} = H^{*} H = m I$; over\u2026","title":"One-sided orthogonality characterizes Hadamard matrices"},{"id":"LinearAlgebra/special-matrices/hadamard-matrices/has-circulant-core-iff-circulant","kind":"node","lean":"MetaMathlibExt.hasCirculantCore_iff_circulant","parent":4546,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hadamard-matrices/index.html#has-circulant-core-iff-circulant","scope":"LinearAlgebra/special-matrices/hadamard-matrices","status":"fully_proved","summary":"Let $n$ be a natural number and let $H$ be an $(n + 1) \\times (n + 1)$ integer matrix with rows and columns indexed by $0, 1, \\dots, n$. Then $H$ has a circulant core, meaning that there is a function $v \\colon \\{0, \\dots, n - 1\\} \\to \\mathbb{Z}$ with $H_{i+1, j+1} = v(i - j)$ for all $i, j \\in \\{0, \\dots, n - 1\\}$, if and only if there is a function $v \\colon \\{0, \\dots, n - 1\\} \\to \\mathbb{Z}$ such that the\u2026","title":"A circulant core is a circulant submatrix"},{"id":"LinearAlgebra/special-matrices/hadamard-matrices/exists-is-hadamard-with-circulant-core","kind":"node","lean":"MetaMathlibExt.exists_isHadamard_with_circulant_core","parent":4546,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hadamard-matrices/index.html#exists-is-hadamard-with-circulant-core","scope":"LinearAlgebra/special-matrices/hadamard-matrices","status":"fully_proved","summary":"Let $n$ be a natural number such that either $n$ is a prime with $n \\equiv 3 \\pmod 4$, or $n = p(p + 2)$ for some natural number $p$ with $p$ and $p + 2$ both prime. Then there is an $(n + 1) \\times (n + 1)$ integer matrix $H$, with rows and columns indexed by $0, 1, \\dots, n$, such that every entry of $H$ is $1$ or $-1$, $H H^{\\mathsf T} = (n + 1) I$ and $H^{\\mathsf T} H = (n + 1) I$, and there is a function\u2026","title":"Circulant-core Hadamard matrices with orthogonal rows and columns"},{"data_url":"scopes/LinearAlgebra/special-matrices/hankel-matrices.json","id":"LinearAlgebra/special-matrices/hankel-matrices","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/hankel-matrices/index.html","scope":"LinearAlgebra/special-matrices/hankel-matrices","status":"fully_proved","title":"Hankel matrices"},{"id":"LinearAlgebra/special-matrices/hankel-matrices/hankel","kind":"node","lean":"Matrix.hankel","parent":4552,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hankel-matrices/index.html#hankel","scope":"LinearAlgebra/special-matrices/hankel-matrices","status":"fully_proved","summary":"For a sequence $s \\colon \\mathbb{N} \\to \\alpha$ with values in a set $\\alpha$, the infinite Hankel matrix generated by $s$ has rows and columns indexed by $\\mathbb{N}$ and $(i, j)$ entry $s_{i+j}$.","title":"Infinite Hankel matrix"},{"id":"LinearAlgebra/special-matrices/hankel-matrices/hankel-fin","kind":"node","lean":"Matrix.hankelFin","parent":4552,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hankel-matrices/index.html#hankel-fin","scope":"LinearAlgebra/special-matrices/hankel-matrices","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and a sequence $s \\colon \\mathbb{N} \\to \\alpha$ with values in a set $\\alpha$, the $n \\times n$ Hankel matrix generated by $s$ has rows and columns indexed by $\\{0, \\dots, n-1\\}$ and $(i, j)$ entry $s_{i+j}$, where $i + j$ is the sum in $\\mathbb{N}$.","title":"Finite Hankel matrix"},{"data_url":"scopes/LinearAlgebra/special-matrices/hurwitz-transform-of-sequences.json","id":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/index.html","scope":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences","status":"fully_proved","title":"Hurwitz transform of sequences"},{"id":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/hurwitz-entry","kind":"node","lean":"MetaMathlibExt.hurwitzEntry","parent":4555,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/index.html#hurwitz-entry","scope":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences","status":"fully_proved","summary":"For sequences $a, b \\colon \\mathbb{N} \\to R$ in a commutative ring $R$ and indices $i, j \\in \\mathbb{N}$ (counted from zero), the Hurwitz entry $h_{ij}$ is $0$ if $i \\ge 2j + 2$; otherwise it is $a_{j - i/2}$ if $i$ is even and $b_{j - (i-1)/2}$ if $i$ is odd.","title":"Entries of the Hurwitz matrix"},{"id":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/hurwitz-matrix","kind":"node","lean":"MetaMathlibExt.hurwitzMatrix","parent":4555,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/index.html#hurwitz-matrix","scope":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences","status":"fully_proved","summary":"For sequences $a, b \\colon \\mathbb{N} \\to R$ in a commutative ring $R$ and $n \\in \\mathbb{N}$, the Hurwitz matrix of order $n$ is the $(n+1) \\times (n+1)$ matrix whose entry in row $i$ and column $j$, for $0 \\le i, j \\le n$, is the Hurwitz entry $h_{ij}$ of $a$ and $b$.","title":"Hurwitz matrix"},{"id":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/hurwitz-transform","kind":"node","lean":"MetaMathlibExt.hurwitzTransform","parent":4555,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/index.html#hurwitz-transform","scope":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences","status":"fully_proved","summary":"For sequences $a, b \\colon \\mathbb{N} \\to R$ in a commutative ring $R$, the Hurwitz transform of $a$ and $b$ is the sequence whose $n$-th term, for $n \\in \\mathbb{N}$, is the determinant of the Hurwitz matrix of $a$ and $b$ of order $n$.","title":"Hurwitz transform"},{"id":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/hurwitz-single","kind":"node","lean":"MetaMathlibExt.hurwitzSingle","parent":4555,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/hurwitz-transform-of-sequences/index.html#hurwitz-single","scope":"LinearAlgebra/special-matrices/hurwitz-transform-of-sequences","status":"fully_proved","summary":"For a sequence $a \\colon \\mathbb{N} \\to R$ in a commutative ring $R$, its Hurwitz transform is the Hurwitz transform of $a$ and the sequence $(0^k)_{k \\in \\mathbb{N}}$, which is $1$ at $k = 0$ and $0$ for $k \\ge 1$.","title":"Hurwitz transform of a single sequence"},{"data_url":"scopes/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability.json","id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","title":"M-matrices and Hurwitz stability"},{"id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/is-hurwitz","kind":"node","lean":"Matrix.IsHurwitz","parent":4560,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html#is-hurwitz","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","summary":"A real square matrix is Hurwitz if every complex eigenvalue has negative real part.","title":"Hurwitz matrix"},{"id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/is-zmatrix","kind":"node","lean":"Matrix.IsZMatrix","parent":4560,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html#is-zmatrix","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","summary":"A Z-matrix has nonpositive off-diagonal entries.","title":"Z-matrix"},{"id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/is-nonsingular-mmatrix","kind":"node","lean":"Matrix.IsNonsingularMMatrix","parent":4560,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html#is-nonsingular-mmatrix","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","summary":"A nonsingular M-matrix is a Z-matrix all of whose complex eigenvalues have positive real part (Berman\u2013Plemmons, Ch. 6, Theorem 2.3, condition (G20)).","title":"Nonsingular M-matrix"},{"id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/is-unit-one-sub-of-spectral-radius-lt-on","kind":"node","lean":"Matrix.isUnit_one_sub_of_spectralRadius_lt_one","parent":4560,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html#is-unit-one-sub-of-spectral-radius-lt-on","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","summary":"If the spectral radius of `K` is `< 1`, then `1` is not an eigenvalue of `K`, so `1 - K` is invertible.","title":"Invertibility of 1 \u2212 K when the spectral radius is below one"},{"id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/inv-one-sub-nonneg","kind":"node","lean":"Matrix.inv_one_sub_nonneg","parent":4560,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html#inv-one-sub-nonneg","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","summary":"If `K` is entrywise nonnegative with spectral radius `< 1`, then `(1 - K)\u207b\u00b9` is entrywise nonnegative (the proof identifies it with the limit of the partial sums `\u2211_{n<N} K\u207f`).","title":"Neumann series"},{"id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/le-spectral-radius-of-smul-le-mul-vec","kind":"node","lean":"Matrix.le_spectralRadius_of_smul_le_mulVec","parent":4560,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html#le-spectral-radius-of-smul-le-mul-vec","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","summary":"If `K \u2265 0` entrywise, `z \u2265 0`, `z \u2260 0` and `r \u2022 z \u2264 K *\u1d65 z` with `r \u2265 0`, then `r` is at most the spectral radius of `K`.","title":"Collatz\u2013Wielandt lower bound"},{"id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/is-unit","kind":"node","lean":"Matrix.IsNonsingularMMatrix.isUnit","parent":4560,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html#is-unit","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","summary":"A nonsingular M-matrix is invertible.","title":"Nonsingular M-matrices are invertible"},{"id":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/inv-nonneg","kind":"node","lean":"Matrix.IsNonsingularMMatrix.inv_nonneg","parent":4560,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability/index.html#inv-nonneg","scope":"LinearAlgebra/special-matrices/m-matrices-and-hurwitz-stability","status":"fully_proved","summary":"The inverse of a nonsingular M-matrix is entrywise nonnegative (Berman\u2013Plemmons, Ch. 6, Theorem 2.3, (G20) \u21d2 (N38)).","title":"Inverse positivity of M-matrices"},{"data_url":"scopes/LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form.json","id":"LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form/index.html","scope":"LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form","status":"fully_proved","title":"Matrix scaling to doubly stochastic form"},{"id":"LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form/exists-symmetric-balancing-of-has-total","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.exists_symmetric_balancing_of_hasTotalSupport","parent":4569,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form/index.html#exists-symmetric-balancing-of-has-total","scope":"LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form","status":"fully_proved","summary":"A symmetric nonnegative matrix with total support has a positive vector `d` with `diag(d) A diag(d)` doubly stochastic.","title":"Csima\u2013Datta: sufficiency of total support"},{"id":"LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form/exists-symmetric-balancing-iff-has-total","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.exists_symmetric_balancing_iff_hasTotalSupport","parent":4569,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form/index.html#exists-symmetric-balancing-iff-has-total","scope":"LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form","status":"fully_proved","summary":"For `0 < n`, a symmetric nonnegative matrix has a positive vector `d` with `diag(d) A diag(d)` doubly stochastic if and only if it has total support.","title":"Csima\u2013Datta theorem"},{"id":"LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form/sinkhorn-scaling","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.SinkhornWanted.sinkhorn_scaling","parent":4569,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form/index.html#sinkhorn-scaling","scope":"LinearAlgebra/special-matrices/matrix-scaling-to-doubly-stochastic-form","status":"fully_proved","summary":"Let $n \\ge 1$ and let $A$ be a real $n \\times n$ matrix all of whose entries are strictly positive. Then there are vectors $r, c \\in \\mathbb{R}^n$ with strictly positive entries such that the matrix with entries $r_i A_{ij} c_j$ is doubly stochastic, that is, its entries are nonnegative and each of its rows and each of its columns sums to $1$.","title":"Sinkhorn's scaling theorem"},{"data_url":"scopes/LinearAlgebra/special-matrices/pascal-like-matrices.json","id":"LinearAlgebra/special-matrices/pascal-like-matrices","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/pascal-like-matrices/index.html","scope":"LinearAlgebra/special-matrices/pascal-like-matrices","status":"fully_proved","title":"Pascal-like matrices"},{"id":"LinearAlgebra/special-matrices/pascal-like-matrices/is-nu-lambda-pascal-matrix","kind":"node","lean":"MetaMathlibExt.IsNuLambdaPascalMatrix","parent":4573,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/pascal-like-matrices/index.html#is-nu-lambda-pascal-matrix","scope":"LinearAlgebra/special-matrices/pascal-like-matrices","status":"fully_proved","summary":"Let $N$ and $n$ be natural numbers with $n > 0$, let $\\nu, \\lambda \\in \\mathbb{C}$ and $\\beta_0, \\dots, \\beta_{N-1} \\in \\mathbb{C}$. An array $S = (S_{i,j})_{i,j \\ge 0}$ of complex numbers, indexed from $0$, is a $(\\nu, \\lambda)$-Pascal matrix of size $2n$ with coefficients $\\beta$ when the following hold: $S_{i,j} = 0$ whenever $i \\ge 2n$ or $j \\ge 2n$; $S_{i,j} = -S_{j,i}$ for all $i, j$; $S_{2n-2,\\,2n-1} = 1$;\u2026","title":"Nu-lambda Pascal matrix with coefficients"},{"id":"LinearAlgebra/special-matrices/pascal-like-matrices/finite-support","kind":"node","lean":"MetaMathlibExt.IsNuLambdaPascalMatrix.finite_support","parent":4573,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/pascal-like-matrices/index.html#finite-support","scope":"LinearAlgebra/special-matrices/pascal-like-matrices","status":"fully_proved","summary":"Let $N$ and $n$ be natural numbers with $n > 0$, let $\\nu, \\lambda \\in \\mathbb{C}$ and $\\beta_0, \\dots, \\beta_{N-1} \\in \\mathbb{C}$, and let $S = (S_{i,j})_{i,j \\ge 0}$ be a $(\\nu, \\lambda)$-Pascal matrix of size $2n$ with coefficients $\\beta$. Then $S_{i,j} = 0$ for all $i, j$ with $i \\ge 2n$ or $j \\ge 2n$.","title":"Finite support of a nu-lambda Pascal matrix"},{"id":"LinearAlgebra/special-matrices/pascal-like-matrices/skew","kind":"node","lean":"MetaMathlibExt.IsNuLambdaPascalMatrix.skew","parent":4573,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/pascal-like-matrices/index.html#skew","scope":"LinearAlgebra/special-matrices/pascal-like-matrices","status":"fully_proved","summary":"Let $N$ and $n$ be natural numbers with $n > 0$, let $\\nu, \\lambda \\in \\mathbb{C}$ and $\\beta_0, \\dots, \\beta_{N-1} \\in \\mathbb{C}$, and let $S = (S_{i,j})_{i,j \\ge 0}$ be a $(\\nu, \\lambda)$-Pascal matrix of size $2n$ with coefficients $\\beta$. Then $S_{i,j} = -S_{j,i}$ for all $i, j$.","title":"Nu-lambda Pascal matrices are skew-symmetric"},{"id":"LinearAlgebra/special-matrices/pascal-like-matrices/corner","kind":"node","lean":"MetaMathlibExt.IsNuLambdaPascalMatrix.corner","parent":4573,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/pascal-like-matrices/index.html#corner","scope":"LinearAlgebra/special-matrices/pascal-like-matrices","status":"fully_proved","summary":"Let $N$ and $n$ be natural numbers with $n > 0$, let $\\nu, \\lambda \\in \\mathbb{C}$ and $\\beta_0, \\dots, \\beta_{N-1} \\in \\mathbb{C}$, and let $S = (S_{i,j})_{i,j \\ge 0}$ be a $(\\nu, \\lambda)$-Pascal matrix of size $2n$ with coefficients $\\beta$. Then $S_{2n-2,\\,2n-1} = 1$.","title":"Corner entry of a nu-lambda Pascal matrix"},{"id":"LinearAlgebra/special-matrices/pascal-like-matrices/last-column","kind":"node","lean":"MetaMathlibExt.IsNuLambdaPascalMatrix.last_column","parent":4573,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/pascal-like-matrices/index.html#last-column","scope":"LinearAlgebra/special-matrices/pascal-like-matrices","status":"fully_proved","summary":"Let $N$ and $n$ be natural numbers with $n > 0$, let $\\nu, \\lambda \\in \\mathbb{C}$ and $\\beta_0, \\dots, \\beta_{N-1} \\in \\mathbb{C}$, and let $S = (S_{i,j})_{i,j \\ge 0}$ be a $(\\nu, \\lambda)$-Pascal matrix of size $2n$ with coefficients $\\beta$. Then for every $j < 2n - 2$, $S_{j,\\,2n-1} = \\sum_{t=0}^{N-1} \\beta_t S_{j+t+1,\\,2n-1}$.","title":"Last-column recurrence of a nu-lambda Pascal matrix"},{"id":"LinearAlgebra/special-matrices/pascal-like-matrices/interior","kind":"node","lean":"MetaMathlibExt.IsNuLambdaPascalMatrix.interior","parent":4573,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/pascal-like-matrices/index.html#interior","scope":"LinearAlgebra/special-matrices/pascal-like-matrices","status":"fully_proved","summary":"Let $N$ and $n$ be natural numbers with $n > 0$, let $\\nu, \\lambda \\in \\mathbb{C}$ and $\\beta_0, \\dots, \\beta_{N-1} \\in \\mathbb{C}$, and let $S = (S_{i,j})_{i,j \\ge 0}$ be a $(\\nu, \\lambda)$-Pascal matrix of size $2n$ with coefficients $\\beta$. Then for all $i, j < 2n - 1$, $S_{i,j} = \\lambda S_{i+1,j+1} + \\nu (S_{i+1,j} + S_{i,j+1})$.","title":"Interior recurrence of a nu-lambda Pascal matrix"},{"data_url":"scopes/LinearAlgebra/special-matrices/perron-frobenius-theorem.json","id":"LinearAlgebra/special-matrices/perron-frobenius-theorem","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/perron-frobenius-theorem/index.html","scope":"LinearAlgebra/special-matrices/perron-frobenius-theorem","status":"fully_proved","title":"Perron\u2013Frobenius theorem"},{"id":"LinearAlgebra/special-matrices/perron-frobenius-theorem/to-complex-matrix","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PerronFrobeniusWanted.toComplexMatrix","parent":4580,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/perron-frobenius-theorem/index.html#to-complex-matrix","scope":"LinearAlgebra/special-matrices/perron-frobenius-theorem","status":"fully_proved","summary":"For a real square matrix $A$ with rows and columns indexed by a nonempty finite set $S$, its complexification $A_{\\mathbb{C}}$ is the complex matrix with entries $(A_{\\mathbb{C}})_{ij} = A_{ij}$ for $i, j \\in S$, each real entry viewed as a complex number.","title":"Complexification of a real matrix"},{"id":"LinearAlgebra/special-matrices/perron-frobenius-theorem/perron-frobenius","kind":"node","lean":"Matrix.IsIrreducible.perron_frobenius","parent":4580,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/perron-frobenius-theorem/index.html#perron-frobenius","scope":"LinearAlgebra/special-matrices/perron-frobenius-theorem","status":"fully_proved","summary":"Let $A$ be a real square matrix with rows and columns indexed by a nonempty finite set $S$, and suppose $A$ is irreducible: its entries are nonnegative, and for all $i, j \\in S$ there is a path $i = i_0, i_1, \\dots, i_k = j$ of length $k \\ge 1$ with $A_{i_{t-1} i_t} > 0$ for every $t$. Then there are a real number $r > 0$ and a vector $v \\in \\mathbb{R}^S$ with $v_i > 0$ for every $i \\in S$ such that $Av = rv$, and\u2026","title":"Perron\u2013Frobenius theorem for irreducible nonnegative matrices"},{"id":"LinearAlgebra/special-matrices/perron-frobenius-theorem/perron-frobenius-2","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PerronFrobeniusWanted.perron_frobenius","parent":4580,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/perron-frobenius-theorem/index.html#perron-frobenius-2","scope":"LinearAlgebra/special-matrices/perron-frobenius-theorem","status":"fully_proved","summary":"Let $A$ be a nonzero real square matrix with rows and columns indexed by a nonempty finite set $S$, and suppose $A$ is irreducible: its entries are nonnegative, and for all $i, j \\in S$ there is a path $i = i_0, i_1, \\dots, i_k = j$ of length $k \\ge 1$ with $A_{i_{t-1} i_t} > 0$ for every $t$. Then there are a real number $r > 0$ and a vector $v \\in \\mathbb{R}^S$ with $v_i > 0$ for every $i \\in S$ such that\u2026","title":"Perron\u2013Frobenius theorem for nonzero irreducible nonnegative matrices"},{"data_url":"scopes/LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix.json","id":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/index.html","scope":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix","status":"fully_proved","title":"Principal minors of the Fourier matrix"},{"id":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/root-data","kind":"node","lean":"Fourier.RootData","parent":4584,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/index.html#root-data","scope":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix","status":"fully_proved","summary":"For a natural number $M$ and a commutative ring $R$, root data of order $M$ in $R$ consists of an element $\\zeta \\in R$ that is a primitive $M$-th root of unity, together with the condition $M \\ge 2$.","title":"Primitive root of unity data"},{"id":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/principal-minor-matrix","kind":"node","lean":"Fourier.principalMinorMatrix","parent":4584,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/index.html#principal-minor-matrix","scope":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix","status":"fully_proved","summary":"Let $M \\ge 2$, let $R$ be a commutative ring, let $\\zeta \\in R$ be a primitive $M$-th root of unity, and let $S$ be a finite subset of $\\mathbb{Z}/M\\mathbb{Z}$. The principal Fourier minor indexed by $S$ is the $S \\times S$ matrix over $R$ whose $(i, j)$ entry is $\\zeta^{n}$, where $n \\in \\{0, \\dots, M-1\\}$ is the representative of the product $ij$ in $\\mathbb{Z}/M\\mathbb{Z}$.","title":"Principal Fourier minor"},{"id":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/principal-minor-det","kind":"node","lean":"Fourier.principalMinorDet","parent":4584,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/index.html#principal-minor-det","scope":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix","status":"fully_proved","summary":"Let $M \\ge 2$, let $R$ be a commutative ring, let $\\zeta \\in R$ be a primitive $M$-th root of unity, and let $S$ be a finite subset of $\\mathbb{Z}/M\\mathbb{Z}$. The principal Fourier minor determinant of $S$ is the determinant in $R$ of the principal Fourier minor indexed by $S$.","title":"Determinant of a principal Fourier minor"},{"id":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/principal-minor-matrix-apply","kind":"node","lean":"Fourier.principalMinorMatrix_apply","parent":4584,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/index.html#principal-minor-matrix-apply","scope":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix","status":"fully_proved","summary":"Let $M \\ge 2$, let $R$ be a commutative ring, let $\\zeta \\in R$ be a primitive $M$-th root of unity, and let $S$ be a finite subset of $\\mathbb{Z}/M\\mathbb{Z}$. For all $i, j \\in S$, the $(i, j)$ entry of the principal Fourier minor indexed by $S$ equals $\\zeta^{n}$, where $n \\in \\{0, \\dots, M-1\\}$ is the representative of $ij$ in $\\mathbb{Z}/M\\mathbb{Z}$.","title":"Entries of a principal Fourier minor"},{"id":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/principal-minor-det-empty","kind":"node","lean":"Fourier.principalMinorDet_empty","parent":4584,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/index.html#principal-minor-det-empty","scope":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix","status":"fully_proved","summary":"Let $M \\ge 2$, let $R$ be a commutative ring and let $\\zeta \\in R$ be a primitive $M$-th root of unity. The determinant of the principal Fourier minor indexed by the empty subset of $\\mathbb{Z}/M\\mathbb{Z}$ equals $1$.","title":"Determinant of the empty principal Fourier minor"},{"id":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/principal-minor-det-singleton","kind":"node","lean":"Fourier.principalMinorDet_singleton","parent":4584,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix/index.html#principal-minor-det-singleton","scope":"LinearAlgebra/special-matrices/principal-minors-of-the-fourier-matrix","status":"fully_proved","summary":"Let $M \\ge 2$, let $R$ be a commutative ring, let $\\zeta \\in R$ be a primitive $M$-th root of unity, and let $a \\in \\mathbb{Z}/M\\mathbb{Z}$. The determinant of the principal Fourier minor indexed by $\\{a\\}$ equals $\\zeta^{n}$, where $n \\in \\{0, \\dots, M-1\\}$ is the representative of $a^2$ in $\\mathbb{Z}/M\\mathbb{Z}$.","title":"Determinant of a singleton principal Fourier minor"},{"data_url":"scopes/LinearAlgebra/special-matrices/toeplitz-matrices.json","id":"LinearAlgebra/special-matrices/toeplitz-matrices","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","title":"Toeplitz matrices"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/lower-toeplitz","kind":"node","lean":"Matrix.lowerToeplitz","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#lower-toeplitz","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"Let $R$ be a set with a zero element $0$ and let $a = (a_0, a_1, a_2, \\dots)$ be a sequence in $R$. The lower-triangular Toeplitz matrix of $a$ is the matrix with rows and columns indexed by $\\mathbb{N}$ whose $(i, j)$ entry is $a_{i-j}$ when $j \\le i$ and $0$ when $j > i$. In particular it is constant along each diagonal and vanishes strictly above the main diagonal.","title":"Lower-triangular Toeplitz matrix"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/lower-toeplitz-apply-of-le","kind":"node","lean":"Matrix.lowerToeplitz_apply_of_le","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#lower-toeplitz-apply-of-le","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"Let $R$ be a set with a zero element and $a = (a_0, a_1, \\dots)$ a sequence in $R$. For natural numbers $j \\le i$, the $(i, j)$ entry of the lower-triangular Toeplitz matrix of $a$ is $a_{i-j}$.","title":"Entries on and below the diagonal of a lower-triangular Toeplitz matrix"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/lower-toeplitz-apply-of-gt","kind":"node","lean":"Matrix.lowerToeplitz_apply_of_gt","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#lower-toeplitz-apply-of-gt","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"Let $R$ be a set with a zero element and $a = (a_0, a_1, \\dots)$ a sequence in $R$. For natural numbers $i < j$, the $(i, j)$ entry of the lower-triangular Toeplitz matrix of $a$ is $0$.","title":"Entries above the diagonal of a lower-triangular Toeplitz matrix"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/lower-toeplitz-apply-diag","kind":"node","lean":"Matrix.lowerToeplitz_apply_diag","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#lower-toeplitz-apply-diag","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"Let $R$ be a set with a zero element and $a = (a_0, a_1, \\dots)$ a sequence in $R$. For every natural number $i$, the $(i, i)$ entry of the lower-triangular Toeplitz matrix of $a$ is $a_0$.","title":"Diagonal entries of a lower-triangular Toeplitz matrix"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/is-toeplitz-matrix","kind":"node","lean":"MetaMathlibExt.IsToeplitzMatrix","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#is-toeplitz-matrix","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"A real $(n+1) \\times (n+1)$ matrix $A = (a_{ij})$, with rows and columns indexed by $\\{0, \\dots, n\\}$, is Toeplitz if there is a function $c : \\mathbb{Z} \\to \\mathbb{R}$ such that $a_{ij} = c(i - j)$ for all $i, j$. Equivalently, $A$ is constant along each of its diagonals.","title":"Toeplitz matrix"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/is-symmetric-toeplitz-matrix","kind":"node","lean":"MetaMathlibExt.IsSymmetricToeplitzMatrix","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#is-symmetric-toeplitz-matrix","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"A real $(n+1) \\times (n+1)$ matrix $A = (a_{ij})$, with rows and columns indexed by $\\{0, \\dots, n\\}$, is symmetric Toeplitz if there is a function $c : \\mathbb{Z} \\to \\mathbb{R}$ such that $c(-k) = c(k)$ for every integer $k$ and $a_{ij} = c(i - j)$ for all $i, j$.","title":"Symmetric Toeplitz matrix"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/is-toeplitz-matrix-of-witness","kind":"node","lean":"MetaMathlibExt.isToeplitzMatrix_of_witness","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#is-toeplitz-matrix-of-witness","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"Let $A = (a_{ij})$ be a real $(n+1) \\times (n+1)$ matrix with rows and columns indexed by $\\{0, \\dots, n\\}$. If $c : \\mathbb{Z} \\to \\mathbb{R}$ satisfies $a_{ij} = c(i - j)$ for all $i, j$, then $A$ is a Toeplitz matrix.","title":"Toeplitz matrices from coefficients"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/is-symmetric-toeplitz-matrix-of-witness","kind":"node","lean":"MetaMathlibExt.isSymmetricToeplitzMatrix_of_witness","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#is-symmetric-toeplitz-matrix-of-witness","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"Let $A = (a_{ij})$ be a real $(n+1) \\times (n+1)$ matrix with rows and columns indexed by $\\{0, \\dots, n\\}$. If $c : \\mathbb{Z} \\to \\mathbb{R}$ satisfies $c(-k) = c(k)$ for every integer $k$ and $a_{ij} = c(i - j)$ for all $i, j$, then $A$ is a symmetric Toeplitz matrix.","title":"Symmetric Toeplitz matrices from even coefficients"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/is-toeplitz","kind":"node","lean":"MetaMathlibExt.IsSymmetricToeplitzMatrix.isToeplitz","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#is-toeplitz","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"Every real $(n+1) \\times (n+1)$ symmetric Toeplitz matrix is a Toeplitz matrix.","title":"Symmetric Toeplitz matrices are Toeplitz"},{"id":"LinearAlgebra/special-matrices/toeplitz-matrices/entry-comm","kind":"node","lean":"MetaMathlibExt.IsSymmetricToeplitzMatrix.entry_comm","parent":4591,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-matrices/index.html#entry-comm","scope":"LinearAlgebra/special-matrices/toeplitz-matrices","status":"fully_proved","summary":"If $A = (a_{ij})$ is a real $(n+1) \\times (n+1)$ symmetric Toeplitz matrix, with rows and columns indexed by $\\{0, \\dots, n\\}$, then $a_{ij} = a_{ji}$ for all $i, j$.","title":"Symmetric Toeplitz matrices are symmetric"},{"data_url":"scopes/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants.json","id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","title":"Toeplitz\u2013Hessenberg determinants"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/det-toeplitz-hessenberg-eq-sum-compositi","kind":"node","lean":"MetaMathlibExt.Matrix.det_toeplitzHessenberg_eq_sum_compositions_general","parent":4602,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#det-toeplitz-hessenberg-eq-sum-compositi","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $n \\ge 0$, and let $a_0, a_1, a_2, \\dots$ be any elements of $R$. Let $A_n$ be the $n \\times n$ matrix whose entry in row $i$ and column $j$ ($0 \\le i, j < n$) is $a_{i+1-j}$ if $j \\le i + 1$ and $0$ otherwise; for $n = 0$ it is the empty matrix, whose determinant is $1$. Then","title":"Trudi's formula for Toeplitz\u2013Hessenberg determinants, general case"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/det-toeplitz-hessenberg-eq-sum-compositi-2","kind":"node","lean":"MetaMathlibExt.Matrix.det_toeplitzHessenberg_eq_sum_compositions","parent":4602,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#det-toeplitz-hessenberg-eq-sum-compositi-2","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $n \\ge 1$, and let $a_0, a_1, a_2, \\dots$ be elements of $R$ with $a_0 \\ne 0$. Let $A_n$ be the $n \\times n$ matrix whose entry in row $i$ and column $j$ ($0 \\le i, j < n$) is $a_{i+1-j}$ if $j \\le i + 1$ and $0$ otherwise. Then","title":"Trudi's formula for Toeplitz\u2013Hessenberg determinants"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/toeplitz-hessenberg-det","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet","parent":4602,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#toeplitz-hessenberg-det","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $a_0, a_1, a_2, \\dots$ be elements of $R$ and let $n \\ge 0$. The Toeplitz\u2013Hessenberg determinant $\\det A_n$ is the determinant of the $n \\times n$ matrix $A_n = A_n(a_0; a_1, \\dots, a_n)$ whose entry in row $i$ and column $j$ ($0 \\le i, j < n$) is $a_{i+1-j}$ if $j \\le i + 1$ and $0$ otherwise. Thus $a_0$ fills the superdiagonal, $a_1$ the diagonal, and every entry above the\u2026","title":"Toeplitz\u2013Hessenberg determinant"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/toeplitz-hessenberg-det-one","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_one","parent":4602,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(1 \\times 1\\)"],["text"," Toeplitz\u2013Hessenberg determinant"]],"source":"The $1 \\times 1$ Toeplitz\u2013Hessenberg determinant"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#toeplitz-hessenberg-det-one","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"For every commutative ring $R$ and every sequence $a_0, a_1, a_2, \\dots$ in $R$, the Toeplitz\u2013Hessenberg determinant of size $1$ built from $a$ is $\\det A_1 = a_1$.","title":"The $1 \\times 1$ Toeplitz\u2013Hessenberg determinant"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/toeplitz-hessenberg-det-two","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_two","parent":4602,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(2 \\times 2\\)"],["text"," Toeplitz\u2013Hessenberg determinant"]],"source":"The $2 \\times 2$ Toeplitz\u2013Hessenberg determinant"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#toeplitz-hessenberg-det-two","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"For every commutative ring $R$ and every sequence $a_0, a_1, a_2, \\dots$ in $R$, the Toeplitz\u2013Hessenberg determinant of size $2$ built from $a$ is $\\det A_2 = a_1^2 - a_0 a_2$.","title":"The $2 \\times 2$ Toeplitz\u2013Hessenberg determinant"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/toeplitz-hessenberg-det-eq-sum-compositi","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_eq_sum_compositions","parent":4602,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#toeplitz-hessenberg-det-eq-sum-compositi","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $a_0, a_1, a_2, \\dots$ be elements of $R$ and let $n \\ge 0$. The Toeplitz\u2013Hessenberg determinant $\\det A_n$ of size $n$ built from $a$ satisfies","title":"Trudi's formula in composition form"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/trudi-partition-sum","kind":"node","lean":"MetaMathlibExt.trudiPartitionSum","parent":4602,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#trudi-partition-sum","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $s \\in R$, let $n$ be a natural number, and let $a : \\mathbb{N} \\to R$ (only $a_1, \\dots, a_n$ enter). Trudi's partition sum is","title":"Trudi's partition sum"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/trudi-partition-sum-eq-sum-compositions","kind":"node","lean":"MetaMathlibExt.trudiPartitionSum_eq_sum_compositions","parent":4602,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#trudi-partition-sum-eq-sum-compositions","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $s \\in R$, let $a_0, a_1, a_2, \\dots$ be elements of $R$ and let $n \\ge 0$. Trudi's partition sum","title":"Trudi's partition sum as a sum over compositions"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/toeplitz-hessenberg-det-eq-neg-one-pow-m","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_eq_neg_one_pow_mul_trudiPartitionSum","parent":4602,"reading_title":{"format":"inline-parts-v1","parts":[["text","Brioschi's formula: the case "],["math","\\(a_0 = 1\\)"],["text"," of Trudi's formula"]],"source":"Brioschi's formula: the case $a_0 = 1$ of Trudi's formula"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#toeplitz-hessenberg-det-eq-neg-one-pow-m","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $a_0, a_1, a_2, \\dots$ be elements of $R$ with $a_0 = 1$. For every $n \\ge 0$, the Toeplitz\u2013Hessenberg determinant $\\det A_n$ of size $n$ built from $a$ satisfies","title":"Brioschi's formula: the case $a_0 = 1$ of Trudi's formula"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/toeplitz-hessenberg-det-eq-trudi-partiti","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_eq_trudiPartitionSum_one","parent":4602,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trudi's formula when "],["math","\\(a_0 = -1\\)"]],"source":"Trudi's formula when $a_0 = -1$"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#toeplitz-hessenberg-det-eq-trudi-partiti","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $a_0, a_1, a_2, \\dots$ be elements of $R$ with $a_0 = -1$. For every $n \\ge 0$, the Toeplitz\u2013Hessenberg determinant $\\det A_n$ of size $n$ built from $a$ satisfies","title":"Trudi's formula when $a_0 = -1$"},{"id":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/trudi-partition-sum-examples","kind":"node","lean":"MetaMathlibExt.trudiPartitionSum_examples","parent":4602,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants/index.html#trudi-partition-sum-examples","scope":"LinearAlgebra/special-matrices/toeplitz-hessenberg-determinants","status":"fully_proved","summary":"For an integer $s$, integers $a_1, a_2, \\dots$ and $n \\ge 0$, write $T_n(s; a) = \\sum_\\alpha s^{|\\alpha|} m_n(\\alpha) \\, a_1^{\\alpha_1} \\cdots a_n^{\\alpha_n}$ for Trudi's partition sum, where $\\alpha = (\\alpha_1, \\dots, \\alpha_n)$ runs over the tuples of nonnegative integers with $\\alpha_1 + 2\\alpha_2 + \\dots + n\\alpha_n = n$, with $|\\alpha| = \\alpha_1 + \\dots + \\alpha_n$ and\u2026","title":"Two values of Trudi's partition sum"},{"data_url":"scopes/LinearAlgebra/special-matrices/total-positivity.json","id":"LinearAlgebra/special-matrices/total-positivity","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","title":"Total positivity"},{"id":"LinearAlgebra/special-matrices/total-positivity/has-nonnegative-minors","kind":"node","lean":"MetaMathlibExt.Matrix.HasNonnegativeMinors","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#has-nonnegative-minors","scope":"LinearAlgebra/special-matrices/total-positivity","status":"fully_proved","summary":"Let $\\iota$ and $\\kappa$ be linearly ordered types and let $M$ be a real matrix with rows indexed by $\\iota$ and columns indexed by $\\kappa$. Then $M$ has nonnegative minors when, for every $k \\in \\mathbb{N}$ and all strictly increasing maps $r \\colon \\{0, \\dots, k-1\\} \\to \\iota$ and $c \\colon \\{0, \\dots, k-1\\} \\to \\kappa$, the $k \\times k$ submatrix $(M_{r(a), c(b)})_{a, b < k}$ has nonnegative determinant. The\u2026","title":"Matrices with nonnegative minors"},{"id":"LinearAlgebra/special-matrices/total-positivity/has-positive-minors","kind":"node","lean":"MetaMathlibExt.Matrix.HasPositiveMinors","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#has-positive-minors","scope":"LinearAlgebra/special-matrices/total-positivity","status":"fully_proved","summary":"A real matrix $M$ whose rows and columns are indexed by linearly ordered sets $R$ and $C$ has positive minors (is totally positive) when, for every natural number $k$ and all strictly increasing maps $r : \\{1, \\dots, k\\} \\to R$ and $c : \\{1, \\dots, k\\} \\to C$, the $k \\times k$ submatrix $(M_{r(a), c(b)})_{1 \\le a, b \\le k}$ has positive determinant. The case $k = 0$ is the empty minor, whose determinant is $1$.","title":"Matrices with positive minors"},{"id":"LinearAlgebra/special-matrices/total-positivity/entry-pos","kind":"node","lean":"MetaMathlibExt.Matrix.HasPositiveMinors.entry_pos","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#entry-pos","scope":"LinearAlgebra/special-matrices/total-positivity","status":"fully_proved","summary":"Let $M$ be a real matrix whose rows and columns are indexed by linearly ordered sets $R$ and $C$. If $M$ has positive minors, then $M_{ij} > 0$ for all $i \\in R$ and $j \\in C$; these entries are the $1 \\times 1$ minors of $M$.","title":"Entries of a matrix with positive minors are positive"},{"id":"LinearAlgebra/special-matrices/total-positivity/has-positive-minors-iff-initial-minors-p","kind":"node","lean":"MetaMathlibExt.hasPositiveMinors_iff_initial_minors_pos","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#has-positive-minors-iff-initial-minors-p","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $A$ be a real $n \\times n$ matrix whose rows and columns are indexed by $0, \\dots, n - 1$. Then $A$ is totally positive (has positive minors) if and only if all its initial minors are positive: for all natural numbers $k$, $i_0$ and $j_0$ with $i_0 + k \\le n$, $j_0 + k \\le n$ and $i_0 = 0$ or $j_0 = 0$, the $k \\times k$ submatrix of $A$ with rows $i_0, \\dots, i_0 + k - 1$ and columns $j_0, \\dots, j_0 + k - 1$\u2026","title":"Gasca\u2013Pe\u00f1a criterion for total positivity"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-pfibonacci","kind":"node","lean":"MetaMathlibExt.generalizedPFibonacci","parent":4614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized "],["math","\\(p\\)"],["text","-Fibonacci sequence"]],"source":"Generalized $p$-Fibonacci sequence"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-pfibonacci","scope":"LinearAlgebra/special-matrices/total-positivity","status":"fully_proved","summary":"For integers $p$, $a$ and $b$, the generalized Fibonacci sequence $(W_{p,n})_{n \\ge 0}$ of integers is defined by $W_{p,0} = a$, $W_{p,1} = b$ and $W_{p,n} = p W_{p,n-1} + W_{p,n-2}$ for $n \\ge 2$.","title":"Generalized $p$-Fibonacci sequence"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-pfibonacci-zero-one","kind":"node","lean":"MetaMathlibExt.generalizedPFibonacci_zero_one","parent":4614,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-Fibonacci numbers as a Lucas sequence"]],"source":"$p$-Fibonacci numbers as a Lucas sequence"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-pfibonacci-zero-one","scope":"LinearAlgebra/special-matrices/total-positivity","status":"fully_proved","summary":"For every integer $p$, the generalized Fibonacci sequence $W_{p,n}$ with initial values $a = 0$ and $b = 1$ (the $p$-Fibonacci numbers $F_{p,n}$) is the Lucas sequence $U(p, -1)$: $W_{p,n} = U_n(p, -1)$ for every natural number $n$.","title":"$p$-Fibonacci numbers as a Lucas sequence"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-pfibonacci-two-p","kind":"node","lean":"MetaMathlibExt.generalizedPFibonacci_two_p","parent":4614,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-Lucas numbers as a companion Lucas sequence"]],"source":"$p$-Lucas numbers as a companion Lucas sequence"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-pfibonacci-two-p","scope":"LinearAlgebra/special-matrices/total-positivity","status":"fully_proved","summary":"For every integer $p$, the generalized Fibonacci sequence $W_{p,n}$ with initial values $a = 2$ and $b = p$ (the $p$-Lucas numbers $L_{p,n}$) is the companion Lucas sequence $V(p, -1)$: $W_{p,n} = V_n(p, -1)$ for every natural number $n$.","title":"$p$-Lucas numbers as a companion Lucas sequence"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci","scope":"LinearAlgebra/special-matrices/total-positivity","status":"fully_proved","summary":"Let $p, a, b$ be integers and let $W_{p,n}$ denote the generalized Fibonacci sequence, with $W_{p,0} = a$, $W_{p,1} = b$ and $W_{p,n+2} = p W_{p,n+1} + W_{p,n}$. The generalized hyper-Fibonacci numbers $W^{(r)}_{p,n}$, for natural numbers $r$ and $n$, are the integers defined by $W^{(0)}_{p,n} = W_{p,n}$ and, for $r \\ge 1$, by $W^{(r)}_{p,n} = \\sum_{k=0}^{n} p^{n-k} W^{(r-1)}_{p,k}$.","title":"Generalized hyper-Fibonacci numbers"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-zero-one","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_zero_one","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-zero-one","scope":"LinearAlgebra/special-matrices/total-positivity","status":"fully_proved","summary":"Let $p, a, b$ be integers and let $W^{(r)}_{p,n}$ denote the generalized hyper-Fibonacci numbers. For every natural number $r$, $W^{(r)}_{p,0} = a$ and $W^{(r)}_{p,1} = p \\cdot a \\cdot r + b$.","title":"Initial values of generalized hyper-Fibonacci numbers"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-succ-succ","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_succ_succ","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-succ-succ","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p, a, b$ be integers and let $W^{(r)}_{p,n}$ denote the generalized hyper-Fibonacci numbers. For all natural numbers $r \\ge 1$ and $n \\ge 1$, $W^{(r)}_{p,n} = p W^{(r)}_{p,n-1} + W^{(r-1)}_{p,n}$.","title":"Recurrence for generalized hyper-Fibonacci numbers"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-sub-eq-sub","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_sub_eq_sub","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-sub-eq-sub","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p, a, b$ be integers with $p \\ne 0$, let $W_{p,n}$ denote the generalized Fibonacci sequence and $W^{(r)}_{p,n}$ the generalized hyper-Fibonacci numbers. For all natural numbers $r$ and $n$ with $1 \\le r \\le n$, the following identity of rational numbers holds:","title":"A binomial-sum formula for generalized hyper-Fibonacci numbers"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-hankel-det","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_hankel_det","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-hankel-det","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p, a, b$ be integers, let $W^{(r)}_{p,n}$ denote the generalized hyper-Fibonacci numbers and put $\\Delta = a^2 + pab - b^2$. For all natural numbers $r$ and $n$, the $(r+2) \\times (r+2)$ Hankel matrix $H_{r,n} = (W^{(r)}_{p,n+i+j})_{0 \\le i, j \\le r+1}$ has determinant $\\det H_{r,n} = (-1)^{n+1+\\lfloor (r+3)/2 \\rfloor} p^{nr+r^2} b^r \\Delta$.","title":"Hankel determinants of generalized hyper-Fibonacci numbers"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-toeplitz-det","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_toeplitz_det_eventually_pos","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-toeplitz-det","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p, a, b$ be integers with $p \\ge 1$, $a \\ge 0$ and $b > 0$, and let $W^{(r)}_{p,n}$ denote the generalized hyper-Fibonacci numbers. For every natural number $m \\ge 1$ there is a natural number $N$ such that for every $n \\ge N$ the $m \\times m$ Toeplitz matrix with entry $W^{(m-1)}_{p,n+i-j}$ in row $i$ and column $j$ ($0 \\le i, j < m$) has positive determinant.","title":"Eventual positivity of hyper-Fibonacci Toeplitz determinants"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-toeplitz-det-2","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_toeplitz_det_eventually_pos_of_le","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-toeplitz-det-2","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p, a, b$ be integers with $p \\ge 1$, $a \\ge 0$ and $b > 0$, and let $W^{(r)}_{p,n}$ denote the generalized hyper-Fibonacci numbers. For natural numbers $r$, $m$ and $n$ write $T^{(r)}_{m,n}$ for the $m \\times m$ Toeplitz matrix with entry $W^{(r)}_{p,n+i-j}$ in row $i$ and column $j$ ($0 \\le i, j < m$). If $r \\ge 1$ and $1 \\le m \\le r + 1$, then there is a natural number $N$ such that $\\det T^{(r)}_{m,n} > 0$\u2026","title":"Eventual positivity of lower-order hyper-Fibonacci Toeplitz determinants"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-eventually-l","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_eventually_logConcave","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-eventually-l","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p, a, b$ be integers with $p \\ge 1$, $a \\ge 0$ and $b > 0$, let $W^{(r)}_{p,n}$ denote the generalized hyper-Fibonacci numbers, and let $r \\ge 1$ be a natural number. Then there is a natural number $N$ such that for every $n \\ge N$ we have $W^{(r)}_{p,n} > 0$ and $W^{(r)}_{p,n} W^{(r)}_{p,n+2} \\le (W^{(r)}_{p,n+1})^2$.","title":"Eventual log-concavity of generalized hyper-Fibonacci numbers"},{"id":"LinearAlgebra/special-matrices/total-positivity/hyper-pfibonacci-hyper-plucas-eventually","kind":"node","lean":"MetaMathlibExt.hyperPFibonacci_hyperPLucas_eventually_logConcave","parent":4614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual log-concavity of hyper "],["math","\\(p\\)"],["text","-Fibonacci and hyper "],["math","\\(p\\)"],["text","-Lucas numbers"]],"source":"Eventual log-concavity of hyper $p$-Fibonacci and hyper $p$-Lucas numbers"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#hyper-pfibonacci-hyper-plucas-eventually","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p \\ge 1$ be an integer and $r \\ge 1$ a natural number, and let $F_{p,n}^{(r)}$ and $L_{p,n}^{(r)}$ be the generalized hyper-Fibonacci numbers $W_{p,n}^{(r)}$ with initial values $(a, b) = (0, 1)$ and $(a, b) = (2, p)$ respectively (the hyper $p$-Fibonacci and hyper $p$-Lucas numbers). Both sequences are eventually log-concave: there is $N$ such that for all $n \\ge N$, $F_{p,n}^{(r)} > 0$ and\u2026","title":"Eventual log-concavity of hyper $p$-Fibonacci and hyper $p$-Lucas numbers"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-toeplitz-has","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_toeplitz_hasPositiveMinors","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-toeplitz-has","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p, a, b$ be integers with $p \\ge 1$, $a \\ge 0$ and $b > 0$, put $\\Delta = a^2 + pab - b^2$, let $W^{(r)}_{p,n}$ denote the generalized hyper-Fibonacci numbers, and let $r \\ge 1$ be a natural number. For a natural number $s$ write $T_s$ for the real $(r+2) \\times (r+2)$ Toeplitz matrix with entry $W^{(r)}_{p,s+i-j}$ in row $i$ and column $j$ ($0 \\le i, j \\le r+1$). If $\\Delta > 0$, there is a natural number $N$\u2026","title":"Eventual total positivity of hyper-Fibonacci Toeplitz matrices"},{"id":"LinearAlgebra/special-matrices/total-positivity/generalized-hyper-fibonacci-toeplitz-pos","kind":"node","lean":"MetaMathlibExt.generalizedHyperFibonacci_toeplitz_positiveMinors_le","parent":4614,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#generalized-hyper-fibonacci-toeplitz-pos","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p, a, b$ be integers with $p \\ge 1$, $a \\ge 0$ and $b > 0$, let $W^{(r)}_{p,n}$ denote the generalized hyper-Fibonacci numbers, and let $r \\ge 1$ be a natural number. For a natural number $n$ let $T_n$ be the real $(r+2) \\times (r+2)$ Toeplitz matrix with entry $W^{(r)}_{p,n+i-j}$ in row $i$ and column $j$ ($0 \\le i, j \\le r+1$). Then there is a natural number $N$ such that for every $n \\ge N$, every natural\u2026","title":"Eventually positive bounded-order minors of hyper-Fibonacci Toeplitz matrices"},{"id":"LinearAlgebra/special-matrices/total-positivity/hyper-pfibonacci-toeplitz-has-positive-m","kind":"node","lean":"MetaMathlibExt.hyperPFibonacci_toeplitz_hasPositiveMinors","parent":4614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total positivity of hyper "],["math","\\(p\\)"],["text","-Fibonacci Toeplitz matrices"]],"source":"Total positivity of hyper $p$-Fibonacci Toeplitz matrices"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#hyper-pfibonacci-toeplitz-has-positive-m","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p \\ge 1$ be an integer and $r \\ge 1$ a natural number, and let $F_{p,n}^{(r)}$ be the hyper $p$-Fibonacci numbers, the generalized hyper-Fibonacci numbers $W_{p,n}^{(r)}$ with initial values $a = 0$ and $b = 1$. Consider real $(r + 2) \\times (r + 2)$ matrices with rows and columns indexed by $0, \\dots, r + 1$. For all sufficiently large $n$, the Toeplitz matrix with entry $F_{p,2n+1+i-j}^{(r)}$ in row $i$ and\u2026","title":"Total positivity of hyper $p$-Fibonacci Toeplitz matrices"},{"id":"LinearAlgebra/special-matrices/total-positivity/hyper-plucas-toeplitz-has-positive-minor","kind":"node","lean":"MetaMathlibExt.hyperPLucas_toeplitz_hasPositiveMinors","parent":4614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total positivity of hyper "],["math","\\(p\\)"],["text","-Lucas Toeplitz matrices"]],"source":"Total positivity of hyper $p$-Lucas Toeplitz matrices"},"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/total-positivity/index.html#hyper-plucas-toeplitz-has-positive-minor","scope":"LinearAlgebra/special-matrices/total-positivity","status":"can_prove","summary":"Let $p \\ge 1$ be an integer and $r \\ge 1$ a natural number, and let $L_{p,n}^{(r)}$ be the hyper $p$-Lucas numbers, the generalized hyper-Fibonacci numbers $W_{p,n}^{(r)}$ with initial values $a = 2$ and $b = p$. Consider real $(r + 2) \\times (r + 2)$ matrices with rows and columns indexed by $0, \\dots, r + 1$. For all sufficiently large $n$, the Toeplitz matrix with entry $L_{p,2n+i-j}^{(r)}$ in row $i$ and column\u2026","title":"Total positivity of hyper $p$-Lucas Toeplitz matrices"},{"data_url":"scopes/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers.json","id":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","kind":"scope","navigation":"module","parent":4519,"result":false,"route":"../roadmap/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/index.html","scope":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","status":"fully_proved","title":"Zagier's matrix and Bernoulli numbers"},{"id":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/zagier-matrix","kind":"node","lean":"MetaMathlibExt.zagierMatrix","parent":4635,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/index.html#zagier-matrix","scope":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","status":"fully_proved","summary":"For a natural number $K$, Zagier's matrix $A_K$ is the rational matrix with rows and columns indexed by $1 \\le i, j \\le K - 1$ (empty when $K \\le 1$) whose $(i, j)$ entry is","title":"Zagier's matrix"},{"id":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/zagier-inverse-p","kind":"node","lean":"MetaMathlibExt.zagierInverseP","parent":4635,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/index.html#zagier-inverse-p","scope":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","status":"fully_proved","summary":"For a natural number $K$, $P_K$ is the rational matrix with rows and columns indexed by $1 \\le i, j \\le K - 1$ (empty when $K \\le 1$) whose $(i, j)$ entry is","title":"First Bernoulli formula for the inverse of Zagier's matrix"},{"id":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/zagier-inverse-q","kind":"node","lean":"MetaMathlibExt.zagierInverseQ","parent":4635,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/index.html#zagier-inverse-q","scope":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","status":"fully_proved","summary":"For a natural number $K$, $Q_K$ is the rational matrix with rows and columns indexed by $1 \\le i, j \\le K - 1$ (empty when $K \\le 1$) whose $(i, j)$ entry is","title":"Second Bernoulli formula for the inverse of Zagier's matrix"},{"id":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/zagier-inverse-p-eq-zagier-inverse-q","kind":"node","lean":"MetaMathlibExt.zagierInverseP_eq_zagierInverseQ","parent":4635,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/index.html#zagier-inverse-p-eq-zagier-inverse-q","scope":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","status":"fully_proved","summary":"For every natural number $K$, the matrices $P_K$ and $Q_K$ given by the first and second Bernoulli formulas for the inverse of Zagier's matrix are equal: $P_K = Q_K$.","title":"The two Bernoulli formulas for the inverse of Zagier's matrix agree"},{"id":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/zagier-inverse-p-mul-zagier-matrix","kind":"node","lean":"MetaMathlibExt.zagierInverseP_mul_zagierMatrix","parent":4635,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/index.html#zagier-inverse-p-mul-zagier-matrix","scope":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","status":"fully_proved","summary":"For every natural number $K$, the matrix $P_K$ given by the first Bernoulli formula is a left inverse of Zagier's matrix $A_K$: $P_K A_K = I$, where $I$ is the $(K - 1) \\times (K - 1)$ identity matrix.","title":"Left inverse of Zagier's matrix"},{"id":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/zagier-matrix-mul-zagier-inverse-p","kind":"node","lean":"MetaMathlibExt.zagierMatrix_mul_zagierInverseP","parent":4635,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/index.html#zagier-matrix-mul-zagier-inverse-p","scope":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","status":"fully_proved","summary":"For every natural number $K$, the matrix $P_K$ given by the first Bernoulli formula is a right inverse of Zagier's matrix $A_K$: $A_K P_K = I$, where $I$ is the $(K - 1) \\times (K - 1)$ identity matrix.","title":"Right inverse of Zagier's matrix"},{"id":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/zagier-matrix-inverse","kind":"node","lean":"MetaMathlibExt.zagier_matrix_inverse","parent":4635,"result":true,"route":"../roadmap/LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers/index.html#zagier-matrix-inverse","scope":"LinearAlgebra/special-matrices/zagier-s-matrix-and-bernoulli-numbers","status":"fully_proved","summary":"Let $K$ be a natural number, let $B_n$ be the $n$-th Bernoulli number with $B_1 = -\\frac{1}{2}$, and let $\\binom{m}{k} = 0$ for $k > m$. Let $A$, $P$ and $Q$ be the $(K - 1) \\times (K - 1)$ rational matrices, with rows and columns indexed by $1 \\le i, j \\le K - 1$, whose entries are","title":"Zagier's matrix conjecture"},{"data_url":"scopes/LinearAlgebra/matrix-decompositions.json","id":"LinearAlgebra/matrix-decompositions","kind":"scope","parent":4518,"result":false,"route":"../roadmap/LinearAlgebra/matrix-decompositions/index.html","scope":"LinearAlgebra/matrix-decompositions","status":"can_prove","summary":"Factorizations of matrices (singular value, polar, Schur and LU decompositions), Gaussian elimination and the solvability of linear systems, iterative methods and condition numbers.","title":"Matrix decompositions and linear systems"},{"id":"LinearAlgebra/matrix-decompositions/jacobi-iteration-converges","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.JacobiIterationWanted.jacobi_iteration_converges","parent":4643,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/index.html#jacobi-iteration-converges","scope":"LinearAlgebra/matrix-decompositions","status":"can_prove","summary":"Let $A$ be a real square matrix with rows and columns indexed by a finite type, and let $b$ be a real vector. Suppose $A$ is strictly diagonally dominant: $\\sum_{j \\ne i} |A_{ij}| < |A_{ii}|$ for every $i$. Define the Jacobi sweep $J$ by $J(x)_i = \\bigl(b_i - \\sum_{j \\ne i} A_{ij} x_j\\bigr) / A_{ii}$. Then there is a vector $x^*$ with $A x^* = b$ such that, for every starting vector $x_0$, the iterates $J^k(x_0)$\u2026","title":"Convergence of the Jacobi method for strictly diagonally dominant matrices"},{"id":"LinearAlgebra/matrix-decompositions/rouche-capelli","kind":"node","lean":"MetaMathlibExt.rouche_capelli","parent":4643,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/index.html#rouche-capelli","scope":"LinearAlgebra/matrix-decompositions","status":"fully_proved","summary":"Let $K$ be a field, let $m$ be a set and $n$ a finite set, let $A$ be an $m \\times n$ matrix over $K$, and let $b \\in K^m$. Then the linear system $Ax = b$ has a solution $x \\in K^n$ if and only if the rank of $A$ equals the rank of the augmented matrix $(A \\mid b)$, whose columns are the columns of $A$ followed by the single column $b$.","title":"Rouch\u00e9\u2013Capelli theorem"},{"id":"LinearAlgebra/matrix-decompositions/exists-unitary-group-is-upper-triangular","kind":"node","lean":"Matrix.exists_unitaryGroup_isUpperTriangular","parent":4643,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/index.html#exists-unitary-group-is-upper-triangular","scope":"LinearAlgebra/matrix-decompositions","status":"fully_proved","summary":"Let $A$ be a square complex matrix whose rows and columns are indexed by a finite linearly ordered type $n$. Then there are a unitary matrix $U$, with conjugate transpose $U^*$, and a matrix $T$ that is upper triangular with respect to the order on $n$ (that is, $T_{ij} = 0$ whenever $j < i$) such that $A = U T U^*$.","title":"Schur decomposition of a complex square matrix"},{"data_url":"scopes/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices.json","id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","kind":"scope","navigation":"module","parent":4643,"result":false,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","title":"Condition numbers of sign matrices"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/sign-matrix","kind":"node","lean":"Matrix.SignConditionNumber.SignMatrix","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#sign-matrix","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"For a natural number $n$, a sign matrix of order $n$ is encoded as an $n \\times n$ matrix with entries in the Booleans $\\{\\mathrm{true}, \\mathrm{false}\\}$, whose rows and columns are indexed by $\\{0, \\dots, n - 1\\}$.","title":"Boolean encoding of sign matrices"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/realize","kind":"node","lean":"Matrix.SignConditionNumber.SignMatrix.realize","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#realize","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"The realization of a Boolean $n \\times n$ matrix $A$ is the real $n \\times n$ matrix whose $(i, j)$ entry is $-1$ if $A_{ij}$ is true and $1$ if $A_{ij}$ is false.","title":"Real sign matrix of a Boolean matrix"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/realize-entries","kind":"node","lean":"Matrix.SignConditionNumber.SignMatrix.realize_entries","parent":4647,"reading_title":{"format":"inline-parts-v1","parts":[["text","Realized sign matrices have entries "],["math","\\(\\pm 1\\)"]],"source":"Realized sign matrices have entries $\\pm 1$"},"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#realize-entries","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"For every Boolean $n \\times n$ matrix $A$ and all indices $i, j$, the $(i, j)$ entry of the real realization of $A$ is $1$ or $-1$.","title":"Realized sign matrices have entries $\\pm 1$"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/exists-realize-iff","kind":"node","lean":"Matrix.SignConditionNumber.exists_realize_iff","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#exists-realize-iff","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"Send each $n \\times n$ Boolean matrix to the real matrix whose entry is $-1$ where the Boolean entry is true and $1$ where it is false. A real $n \\times n$ matrix $M$ arises in this way if and only if every entry of $M$ is $1$ or $-1$.","title":"Boolean matrices realize exactly the sign matrices"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/condition-number","kind":"node","lean":"Matrix.SignConditionNumber.conditionNumber","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#condition-number","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"For a real $n \\times n$ matrix $A$, let $\\sigma_0 \\ge \\sigma_1 \\ge \\dots$ be the singular values of $A$ acting on Euclidean space $\\mathbb{R}^n$, indexed from $0$ and extended by $\\sigma_i = 0$ for $i \\ge n$. The condition number of $A$, an element of $[0, \\infty]$, is $\\infty$ if $\\sigma_{n-1} = 0$ and $\\sigma_0 / \\sigma_{n-1}$ otherwise, where $n - 1$ is read as $0$ when $n = 0$.","title":"Condition number of a square real matrix"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/minimum-condition-number","kind":"node","lean":"Matrix.SignConditionNumber.minimumConditionNumber","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#minimum-condition-number","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"For a positive integer $n$, the minimum condition number of order $n$ is the least condition number of a real $n \\times n$ sign matrix, that is, a matrix all of whose entries are $1$ or $-1$; the minimum is taken over the finitely many such matrices.","title":"Minimum condition number of sign matrices"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/condition-number-eq-top-of-smallest-eq-z","kind":"node","lean":"Matrix.SignConditionNumber.conditionNumber_eq_top_of_smallest_eq_zero","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#condition-number-eq-top-of-smallest-eq-z","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"Let $A$ be a real $n \\times n$ matrix with singular values $\\sigma_0 \\ge \\sigma_1 \\ge \\dots$ on Euclidean space $\\mathbb{R}^n$, indexed from $0$ and extended by zeros, where $n - 1$ is read as $0$ when $n = 0$. If $\\sigma_{n-1} = 0$, then the condition number of $A$ is $\\infty$.","title":"Infinite condition number from a vanishing singular value"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/condition-number-eq-ratio-of-smallest-ne","kind":"node","lean":"Matrix.SignConditionNumber.conditionNumber_eq_ratio_of_smallest_ne_zero","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#condition-number-eq-ratio-of-smallest-ne","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"Let $A$ be a real $n \\times n$ matrix with singular values $\\sigma_0 \\ge \\sigma_1 \\ge \\dots$ on Euclidean space $\\mathbb{R}^n$, indexed from $0$ and extended by zeros, where $n - 1$ is read as $0$ when $n = 0$. If $\\sigma_{n-1} \\ne 0$, then the condition number of $A$ equals $\\sigma_0 / \\sigma_{n-1}$.","title":"Condition number as a singular-value ratio"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/condition-number-zero","kind":"node","lean":"Matrix.SignConditionNumber.conditionNumber_zero","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#condition-number-zero","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"For every natural number $n$, the condition number of the real $n \\times n$ zero matrix is $\\infty$; this includes the $0 \\times 0$ matrix.","title":"The zero matrix has infinite condition number"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/minimum-condition-number-attained","kind":"node","lean":"Matrix.SignConditionNumber.minimumConditionNumber_attained","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#minimum-condition-number-attained","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"For every positive integer $n$, there is an $n \\times n$ Boolean matrix whose realization as a real sign matrix ($-1$ for true, $1$ for false) has condition number equal to the minimum condition number of order $n$.","title":"The minimum condition number is attained"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/one-le-condition-number","kind":"node","lean":"Matrix.SignConditionNumber.one_le_conditionNumber","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#one-le-condition-number","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"For every natural number $n$ and every real $n \\times n$ matrix $A$, the condition number of $A$ is at least $1$.","title":"Condition numbers are at least one"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/one-le-minimum-condition-number","kind":"node","lean":"Matrix.SignConditionNumber.one_le_minimumConditionNumber","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#one-le-minimum-condition-number","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"For every positive integer $n$, the minimum condition number of the real $n \\times n$ sign matrices is at least $1$.","title":"The minimum condition number is at least one"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/condition-number-one-order-one","kind":"node","lean":"Matrix.SignConditionNumber.conditionNumber_one_order_one","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#condition-number-one-order-one","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"The condition number of the $1 \\times 1$ real identity matrix is $1$.","title":"Condition number of the order-one identity"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/minimum-condition-number-one","kind":"node","lean":"Matrix.SignConditionNumber.minimumConditionNumber_one","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#minimum-condition-number-one","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"The minimum condition number of order $1$ is $1$: the least condition number of a $1 \\times 1$ real sign matrix is $1$.","title":"Minimum condition number of order one"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/is-subpolynomial","kind":"node","lean":"Matrix.SignConditionNumber.IsSubpolynomial","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#is-subpolynomial","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"A sequence $f \\colon \\mathbb{N} \\to [0, \\infty)$ is subpolynomial when for every real $\\varepsilon > 0$ we have $f(n) = o(n^{\\varepsilon})$ as $n \\to \\infty$.","title":"Subpolynomial sequences"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/is-admissible-exponent","kind":"node","lean":"Matrix.SignConditionNumber.IsAdmissibleExponent","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#is-admissible-exponent","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"A real number $\\alpha$ is an admissible exponent when there is a subpolynomial sequence $f \\colon \\mathbb{N} \\to [0, \\infty)$ such that for every positive integer $n$ the minimum condition number $\\kappa_n$ of the real $n \\times n$ sign matrices satisfies $\\kappa_n = 1 + f(n) / n^{\\alpha}$, as an equality in $[0, \\infty]$.","title":"Admissible decay exponents"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/decay-exponent","kind":"node","lean":"Matrix.SignConditionNumber.decayExponent","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#decay-exponent","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"The decay exponent is the supremum, taken in the extended reals $[-\\infty, \\infty]$, of the set of admissible exponents $\\alpha \\in \\mathbb{R}$.","title":"Decay exponent of minimum sign-matrix condition numbers"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/hadamard-hypothesis","kind":"node","lean":"Matrix.SignConditionNumber.HadamardHypothesis","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#hadamard-hypothesis","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"The Hadamard hypothesis is the proposition that for every natural number $n$ divisible by $4$ there is an $n \\times n$ Hadamard matrix with integer entries, that is, an integer matrix $H$ with all entries $\\pm 1$ such that $H H^{\\mathsf{T}} = H^{\\mathsf{T}} H = n I$.","title":"Hadamard hypothesis"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/is-symmetric-conference","kind":"node","lean":"Matrix.SignConditionNumber.IsSymmetricConference","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#is-symmetric-conference","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"A real $n \\times n$ matrix $C$ is a symmetric conference matrix when $n \\ge 2$, $C^{\\mathsf{T}} = C$, every diagonal entry of $C$ is $0$, every off-diagonal entry of $C$ is $1$ or $-1$, and $C C^{\\mathsf{T}} = (n - 1) I$, where $I$ is the identity matrix.","title":"Symmetric conference matrices"},{"id":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/exists-realize-add-one","kind":"node","lean":"Matrix.SignConditionNumber.IsSymmetricConference.exists_realize_add_one","parent":4647,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices/index.html#exists-realize-add-one","scope":"LinearAlgebra/matrix-decompositions/condition-numbers-of-sign-matrices","status":"fully_proved","summary":"If $C$ is a real $n \\times n$ symmetric conference matrix and $I$ is the $n \\times n$ identity matrix, then there is a Boolean $n \\times n$ matrix whose realization as a real sign matrix equals $C + I$.","title":"Adding the identity to a symmetric conference matrix gives a sign matrix"},{"data_url":"scopes/LinearAlgebra/matrix-decompositions/conjugate-gradient-method.json","id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","kind":"scope","navigation":"module","parent":4643,"result":false,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","title":"Conjugate gradient method"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/energy-norm","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.energyNorm","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#energy-norm","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"The energy norm `\u2016v\u2016_A = \u221a(v\u1d40 A v)`; a norm when `A` is positive definite.","title":"Energy norm induced by a matrix"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/krylov","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.krylov","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#krylov","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"The Krylov space `\ud835\udca6_k(A, r) = span {r, A r, \u2026, A^{k-1} r}`.","title":"Krylov subspace"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/state","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.State","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#state","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"The state of the conjugate gradient method: iterate `x`, residual `r = b - A x` and search direction `d`.","title":"State of the conjugate gradient method"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/step","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.step","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#step","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"One Hestenes\u2013Stiefel conjugate gradient step.","title":"One Hestenes\u2013Stiefel conjugate gradient step"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/init","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.init","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#init","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"The initial state `x\u2080`, `r\u2080 = d\u2080 = b - A x\u2080`.","title":"Initial conjugate gradient state"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/iterate","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.iterate","parent":4668,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conjugate gradient state after "],["math","\\(k\\)"],["text"," steps"]],"source":"Conjugate gradient state after $k$ steps"},"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#iterate","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"The `k`-th conjugate gradient state from `x\u2080` for the system `A x = b`.","title":"Conjugate gradient state after $k$ steps"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/iterate-x-sub-mem-krylov","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.iterate_x_sub_mem_krylov","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#iterate-x-sub-mem-krylov","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"`x_k - x\u2080 \u2208 \ud835\udca6_k(A, r\u2080)`.","title":"CG stays in the Krylov space"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/energy-norm-iterate-le","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.energyNorm_iterate_le","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#energy-norm-iterate-le","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"For the solution `x\u22c6` of `A x\u22c6 = b`, `\u2016x_k - x\u22c6\u2016_A \u2264 \u2016x\u2080 + z - x\u22c6\u2016_A` for every `z \u2208 \ud835\udca6_k(A, r\u2080)`.","title":"CG minimizes the energy norm of the error over the Krylov space"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/energy-norm-le-of-is-min-krylov","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.energyNorm_le_of_isMin_krylov","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#energy-norm-le-of-is-min-krylov","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"If the eigenvalues of `A` lie in `[a, c]` with `0 < a`, and `x` minimizes `\u2016\u00b7 - x\u22c6\u2016_A` over `x\u2080 + \ud835\udca6_k(A, A (x\u22c6 - x\u2080))`, then `\u2016x - x\u22c6\u2016_A \u2264 2 ((\u221a\u03ba - 1) / (\u221a\u03ba + 1))^k \u2016x\u2080 - x\u22c6\u2016_A` with `\u03ba = c / a`.","title":"The Chebyshev bound for Krylov energy minimizers"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/energy-norm-iterate-sub-le","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.energyNorm_iterate_sub_le","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#energy-norm-iterate-sub-le","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"If the eigenvalues of `A` lie in `[a, c]` with `0 < a` and `A x\u22c6 = b`, the CG iterates satisfy `\u2016x_k - x\u22c6\u2016_A \u2264 2 ((\u221a\u03ba - 1) / (\u221a\u03ba + 1))^k \u2016x\u2080 - x\u22c6\u2016_A` with `\u03ba = c / a`.","title":"The conjugate gradient error bound"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/cond-num","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.condNum","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#cond-num","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"The spectral condition number `\u03bb_max / \u03bb_min` of a positive definite matrix. For `n = 0` it is `0 / 0 = 0` (no eigenvalues), a junk value; the bound below is stated for `0 < n`.","title":"Spectral condition number of a positive definite matrix"},{"id":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method/energy-norm-iterate-sub-le-cond-num","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ConjugateGradient.energyNorm_iterate_sub_le_condNum","parent":4668,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/conjugate-gradient-method/index.html#energy-norm-iterate-sub-le-cond-num","scope":"LinearAlgebra/matrix-decompositions/conjugate-gradient-method","status":"fully_proved","summary":"The conjugate gradient error bound in terms of the condition number `\u03ba` of `A`: `\u2016x_k - x\u22c6\u2016_A \u2264 2 ((\u221a\u03ba - 1) / (\u221a\u03ba + 1))^k \u2016x\u2080 - x\u22c6\u2016_A`.","title":"The conjugate gradient error bound in terms of the condition number"},{"data_url":"scopes/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form.json","id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","kind":"scope","navigation":"module","parent":4643,"result":false,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","title":"Gaussian elimination and row echelon form"},{"id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/exists-unit-lower-triangular-mul-submatr","kind":"node","lean":"Matrix.exists_unitLowerTriangular_mul_submatrix_isRowEchelon","parent":4681,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html#exists-unit-lower-triangular-mul-submatr","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","summary":"Let $K$ be a field and let $A$ be a matrix over $K$ whose rows and columns are indexed by finite linearly ordered sets. Then there are a permutation $\\sigma$ of the row indices and a square matrix $L$ over $K$ indexed by the row indices, lower triangular with every diagonal entry equal to $1$, such that the matrix $L A_\\sigma$ is in row echelon form. Here $A_\\sigma$ is the matrix with entries\u2026","title":"Row echelon form via a row permutation and a unit lower-triangular matrix"},{"id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/gauss-pivot-elim-step","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.GaussPivotWanted.gauss_pivot_elim_step","parent":4681,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html#gauss-pivot-elim-step","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","summary":"Let $K$ be a field and let $A$ be a matrix over $K$ whose rows are indexed by a finite linearly ordered set and whose columns are indexed by any set. Let $r$ and $i_0$ be row indices and $j$ a column index with $A_{i_0 j} \\ne 0$, and let $A'$ be the matrix obtained from $A$ by exchanging rows $r$ and $i_0$. Then there is a square matrix $P$ over $K$ indexed by the row indices, lower triangular with every diagonal\u2026","title":"One pivot step of Gaussian elimination"},{"id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/exists-gauss-elimination-op-list","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.GaussPivotWanted.exists_gauss_elimination_opList","parent":4681,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html#exists-gauss-elimination-op-list","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","summary":"Let $K$ be a field and let $A$ be a matrix over $K$ whose rows and columns are indexed by finite linearly ordered sets. Then there are a permutation $\\sigma$ of the row indices and a finite, possibly empty, list $M_1, \\dots, M_\\ell$ of square matrices over $K$ indexed by the row indices, each lower triangular with every diagonal entry equal to $1$, such that the matrix $M_1 M_2 \\cdots M_\\ell A_\\sigma$ is in row\u2026","title":"Gaussian elimination as a product of row operations"},{"id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/col-eq-single-of-is-leading-entry","kind":"node","lean":"Matrix.IsReducedRowEchelon.col_eq_single_of_isLeadingEntry","parent":4681,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html#col-eq-single-of-is-leading-entry","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","summary":"Let $R$ be a type with elements $0$ and $1$, and let $A$ be a matrix over $R$ in reduced row echelon form whose rows are indexed by a linearly ordered set and whose columns are indexed by a set with a relation $<$. Suppose that the leading entry of row $i$ of $A$ is in column $c$, that is, $A_{ic} \\neq 0$ and $A_{ij} = 0$ for every column $j < c$. Then column $c$ of $A$ is the standard basis vector $e_i$:\u2026","title":"Pivot columns of a reduced row echelon matrix"},{"id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/submatrix-order-iso","kind":"node","lean":"Matrix.IsReducedRowEchelon.submatrix_orderIso","parent":4681,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html#submatrix-order-iso","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","summary":"Let $R$ be a type with elements $0$ and $1$, and let $A$ be a matrix over $R$ in reduced row echelon form whose rows are indexed by a linearly ordered set and whose columns are indexed by a linearly ordered set $N$. Let $e : O \\to N$ be an order isomorphism from a linearly ordered set $O$. Then the matrix $B$ with columns indexed by $O$ and entries $B_{ik} = A_{i\\,e(k)}$ is in reduced row echelon form.","title":"Order-preserving reindexing of columns preserves reduced row echelon form"},{"id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/exists-is-reduced-row-echelon-row-equiva","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ReducedRowEchelonWanted.exists_isReducedRowEchelon_rowEquivalent","parent":4681,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html#exists-is-reduced-row-echelon-row-equiva","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","summary":"Let $K$ be a field, let $m$ and $n$ be finite linearly ordered sets, and let $A$ be an $m \\times n$ matrix over $K$. Then there are an $m \\times n$ matrix $R$ over $K$ in reduced row echelon form and an invertible $m \\times m$ matrix $P$ over $K$ with $R = PA$. In other words, $A$ is row-equivalent to a matrix in reduced row echelon form.","title":"Existence of the reduced row echelon form"},{"id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/is-reduced-row-echelon-unique-of-row-equ","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ReducedRowEchelonWanted.isReducedRowEchelon_unique_of_rowEquivalent","parent":4681,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html#is-reduced-row-echelon-unique-of-row-equ","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","summary":"Let $K$ be a field, let $m$ and $n$ be finite linearly ordered sets, and let $R_1$ and $R_2$ be $m \\times n$ matrices over $K$ in reduced row echelon form. If $R_2 = P R_1$ for some invertible $m \\times m$ matrix $P$ over $K$, then $R_1 = R_2$. In other words, a row-equivalence class contains at most one matrix in reduced row echelon form.","title":"Uniqueness of the reduced row echelon form"},{"id":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/ker-mul-vec-lin-mul-left-is-unit","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ReducedRowEchelonWanted.ker_mulVecLin_mul_left_isUnit","parent":4681,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form/index.html#ker-mul-vec-lin-mul-left-is-unit","scope":"LinearAlgebra/matrix-decompositions/gaussian-elimination-and-row-echelon-form","status":"fully_proved","summary":"Let $K$ be a field, let $m$ and $n$ be finite sets, let $A$ be an $m \\times n$ matrix over $K$, and let $P$ be an invertible $m \\times m$ matrix over $K$. Then the linear maps $x \\mapsto PAx$ and $x \\mapsto Ax$ on $K^n$ have the same kernel; that is, the homogeneous systems $PAx = 0$ and $Ax = 0$ have the same solutions.","title":"Invertible row operations preserve the null space"},{"data_url":"scopes/LinearAlgebra/matrix-decompositions/lu-decomposition.json","id":"LinearAlgebra/matrix-decompositions/lu-decomposition","kind":"scope","navigation":"module","parent":4643,"result":false,"route":"../roadmap/LinearAlgebra/matrix-decompositions/lu-decomposition/index.html","scope":"LinearAlgebra/matrix-decompositions/lu-decomposition","status":"fully_proved","title":"LU decomposition"},{"id":"LinearAlgebra/matrix-decompositions/lu-decomposition/exists-lu-decomposition-of-minors-ne-zer","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.LUDecompositionWanted.exists_lu_decomposition_of_minors_ne_zero","parent":4690,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/lu-decomposition/index.html#exists-lu-decomposition-of-minors-ne-zer","scope":"LinearAlgebra/matrix-decompositions/lu-decomposition","status":"fully_proved","summary":"Let $K$ be a field and $A$ an $n \\times n$ matrix over $K$ whose leading principal minors are all nonzero: for every $k \\le n$, the determinant of the upper-left $k \\times k$ block of $A$ is nonzero. Then $A = LU$ for some $n \\times n$ matrices $L$ and $U$ over $K$ such that $L$ is lower triangular with every diagonal entry equal to $1$ and $U$ is upper triangular.","title":"LU decomposition without pivoting"},{"id":"LinearAlgebra/matrix-decompositions/lu-decomposition/lu-decomposition-unique","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.LUDecompositionWanted.lu_decomposition_unique","parent":4690,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/lu-decomposition/index.html#lu-decomposition-unique","scope":"LinearAlgebra/matrix-decompositions/lu-decomposition","status":"fully_proved","summary":"Let $K$ be a field and $A$ an invertible $n \\times n$ matrix over $K$. Suppose $A = L_1 U_1 = L_2 U_2$, where $L_1$ and $L_2$ are lower triangular $n \\times n$ matrices over $K$ with every diagonal entry equal to $1$, and $U_1$ and $U_2$ are upper triangular $n \\times n$ matrices over $K$. Then $L_1 = L_2$ and $U_1 = U_2$.","title":"Uniqueness of the LU decomposition"},{"data_url":"scopes/LinearAlgebra/matrix-decompositions/singular-value-decomposition.json","id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","kind":"scope","navigation":"module","parent":4643,"result":false,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","title":"Singular value decomposition"},{"id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition/exists-unitary-group-mul-eq-rectangular","kind":"node","lean":"Matrix.exists_unitaryGroup_mul_eq_rectangularDiagonal","parent":4693,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html#exists-unitary-group-mul-eq-rectangular","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","summary":"Every rectangular complex matrix becomes rectangular diagonal after a unitary change of basis on the right, with a further unitary change of basis on the codomain (S. Axler, Linear Algebra Done Right, 4th ed., Section 7E).","title":"Unitary reduction of a complex matrix to rectangular diagonal form"},{"id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition/exists-unitary-group-rectangular-diagona","kind":"node","lean":"Matrix.exists_unitaryGroup_rectangularDiagonal_antitone","parent":4693,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html#exists-unitary-group-rectangular-diagona","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","summary":"This is the rectangular complex SVD with singular values in nonincreasing order.","title":"Complex SVD with nonincreasing singular values"},{"id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition/sum-sq-singular-values-eq-sum-sq-norm-of","kind":"node","lean":"Matrix.sum_sq_singularValues_eq_sum_sq_norm_of_svd","parent":4693,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html#sum-sq-singular-values-eq-sum-sq-norm-of","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","summary":"Frobenius-norm identity for an SVD: the sum of the squared singular values is the sum of the squared norms of the entries.","title":"Squared Frobenius norm as the sum of squared singular values"},{"id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition/complex-singular-value-decomposition","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.ComplexSVDWanted.complex_singular_value_decomposition","parent":4693,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html#complex-singular-value-decomposition","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","summary":"Rectangular complex singular value decomposition: every complex `m \u00d7 n` matrix factors as `U * \u03a3 * V\u1d34` with `U` and `V` unitary and `\u03a3` rectangular diagonal with nonnegative entries; the singular values beyond the first `min m n` (indices `j` with `m \u2264 j.val`) are zero.","title":"Singular value decomposition of a complex matrix"},{"id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition/exists-unitary-group-diagonal-nonneg","kind":"node","lean":"Matrix.exists_unitaryGroup_diagonal_nonneg","parent":4693,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html#exists-unitary-group-diagonal-nonneg","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","summary":"Let $A$ be a square complex matrix whose rows and columns are indexed by a finite type $n$. Then there are unitary matrices $U$ and $V$ and nonnegative real numbers $\\sigma_i$ for $i \\in n$ such that, writing $D$ for the diagonal matrix with diagonal entries $\\sigma_i$ and $V^*$ for the conjugate transpose of $V$, we have $A = U D V^*$.","title":"Singular value decomposition of a complex square matrix"},{"id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition/exists-orthogonal-group-mul-eq-rectangul","kind":"node","lean":"Matrix.exists_orthogonalGroup_mul_eq_rectangularDiagonal","parent":4693,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html#exists-orthogonal-group-mul-eq-rectangul","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","summary":"Let $A$ be a real $m \\times n$ matrix. Then there are orthogonal matrices $U$ of size $m \\times m$ and $V$ of size $n \\times n$ and real numbers $\\sigma_1 \\ge \\sigma_2 \\ge \\dots \\ge \\sigma_n \\ge 0$ with $\\sigma_j = 0$ for every $j > m$, such that $AV = U\\Sigma$, where $\\Sigma$ is the $m \\times n$ matrix with $\\Sigma_{jj} = \\sigma_j$ for $j \\le \\min(m, n)$ and all other entries zero.","title":"Orthogonal reduction of a real matrix to rectangular diagonal form"},{"id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition/exists-orthogonal-group-rectangular-diag","kind":"node","lean":"Matrix.exists_orthogonalGroup_rectangularDiagonal_antitone","parent":4693,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html#exists-orthogonal-group-rectangular-diag","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","summary":"Let $A$ be a real $m \\times n$ matrix. Then there are orthogonal matrices $U$ of size $m \\times m$ and $V$ of size $n \\times n$ and real numbers $\\sigma_1 \\ge \\sigma_2 \\ge \\dots \\ge \\sigma_n \\ge 0$ with $\\sigma_j = 0$ for every $j > m$, such that $A = U \\Sigma V^{\\mathsf{T}}$, where $\\Sigma$ is the $m \\times n$ matrix with $\\Sigma_{jj} = \\sigma_j$ for $j \\le \\min(m, n)$ and all other entries zero.","title":"Real singular value decomposition with nonincreasing singular values"},{"id":"LinearAlgebra/matrix-decompositions/singular-value-decomposition/real-singular-value-decomposition","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.RealSVDWanted.real_singular_value_decomposition","parent":4693,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/singular-value-decomposition/index.html#real-singular-value-decomposition","scope":"LinearAlgebra/matrix-decompositions/singular-value-decomposition","status":"fully_proved","summary":"Let $A$ be a real $m \\times n$ matrix. Then there are orthogonal matrices $U$ of size $m \\times m$ and $V$ of size $n \\times n$ and real numbers $\\sigma_1, \\dots, \\sigma_n \\ge 0$ with $\\sigma_j = 0$ for every $j > m$, such that $A = U \\Sigma V^{\\mathsf{T}}$, where $\\Sigma$ is the $m \\times n$ matrix with $\\Sigma_{jj} = \\sigma_j$ for $j \\le \\min(m, n)$ and all other entries zero.","title":"Real singular value decomposition"},{"data_url":"scopes/LinearAlgebra/matrix-decompositions/polar-decomposition.json","id":"LinearAlgebra/matrix-decompositions/polar-decomposition","kind":"scope","navigation":"module","parent":4643,"result":false,"route":"../roadmap/LinearAlgebra/matrix-decompositions/polar-decomposition/index.html","scope":"LinearAlgebra/matrix-decompositions/polar-decomposition","status":"fully_proved","title":"Polar decomposition"},{"id":"LinearAlgebra/matrix-decompositions/polar-decomposition/exists-unitary-group-pos-semidef","kind":"node","lean":"Matrix.exists_unitaryGroup_posSemidef","parent":4702,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/polar-decomposition/index.html#exists-unitary-group-pos-semidef","scope":"LinearAlgebra/matrix-decompositions/polar-decomposition","status":"fully_proved","summary":"Let $A$ be a square complex matrix indexed by a finite type, and let $A^*$ be its conjugate transpose. Then there are a unitary matrix $U$ and a positive semidefinite matrix $P$ such that $P^2 = A^* A$ and $A = UP$.","title":"Polar decomposition of a complex square matrix"},{"id":"LinearAlgebra/matrix-decompositions/polar-decomposition/exists-unique-unitary-group-pos-def-of-i","kind":"node","lean":"Matrix.exists_unique_unitaryGroup_posDef_of_isUnit_det","parent":4702,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/polar-decomposition/index.html#exists-unique-unitary-group-pos-def-of-i","scope":"LinearAlgebra/matrix-decompositions/polar-decomposition","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $A$ be a square matrix over $\\mathbb{K}$, indexed by a finite type, with $\\det A \\neq 0$. Then there are a unitary matrix $U$ and a positive-definite matrix $P$ with $A = UP$, and they are unique: if $U'$ is unitary, $P'$ is positive definite and $A = U'P'$, then $U' = U$ and $P' = P$.","title":"Uniqueness of the polar decomposition of an invertible matrix"},{"id":"LinearAlgebra/matrix-decompositions/polar-decomposition/polar-decomposition-gl-real","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PolarDecompositionGLWanted.polar_decomposition_gl_real","parent":4702,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/polar-decomposition/index.html#polar-decomposition-gl-real","scope":"LinearAlgebra/matrix-decompositions/polar-decomposition","status":"fully_proved","summary":"Let $A$ be a real square matrix with rows and columns indexed by a finite set, and suppose $\\det A \\ne 0$. Then there are an orthogonal matrix $O$ and a positive-definite symmetric matrix $S$ with $A = OS$, and they are unique: if $O'$ is orthogonal, $S'$ is positive-definite symmetric and $A = O'S'$, then $O' = O$ and $S' = S$.","title":"Polar decomposition of invertible real matrices"},{"id":"LinearAlgebra/matrix-decompositions/polar-decomposition/polar-decomposition-gl-complex","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PolarDecompositionGLWanted.polar_decomposition_gl_complex","parent":4702,"result":true,"route":"../roadmap/LinearAlgebra/matrix-decompositions/polar-decomposition/index.html#polar-decomposition-gl-complex","scope":"LinearAlgebra/matrix-decompositions/polar-decomposition","status":"fully_proved","summary":"Let $A$ be a complex square matrix with rows and columns indexed by a finite set, and suppose $\\det A \\ne 0$. Then there are a unitary matrix $U$ and a positive-definite Hermitian matrix $P$ with $A = UP$, and they are unique: if $U'$ is unitary, $P'$ is positive-definite Hermitian and $A = U'P'$, then $U' = U$ and $P' = P$.","title":"Polar decomposition of invertible complex matrices"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms","kind":"scope","parent":4518,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms","status":"can_prove","summary":"Eigenvalues, eigenspaces and characteristic polynomials of matrices and endomorphisms, and the normal forms they lead to: diagonalization, triangularization, Jordan and Smith normal forms.","title":"Eigenvalues and canonical forms"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/inverse-eigenvalues","kind":"node","lean":"MetaMathlibExt.inverse_eigenvalues","parent":4707,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/index.html#inverse-eigenvalues","scope":"LinearAlgebra/eigenvalues-and-canonical-forms","status":"can_prove","summary":"Let $M$ be a square matrix over a field $K$, with rows and columns indexed by a finite type, whose determinant is a unit of $K$ (so $M$ is invertible), and let $\\mu \\in K$ be nonzero. Then a vector $x$ is an eigenvector of $M$ with eigenvalue $\\mu$ (that is, $x \\ne 0$ and $Mx = \\mu x$) if and only if $x$ is an eigenvector of $M^{-1}$ with eigenvalue $\\mu^{-1}$.","title":"Eigenvalues of the inverse matrix"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/exists-unitary-diagonal-of-is-star-norma","kind":"node","lean":"Matrix.exists_unitary_diagonal_of_isStarNormal","parent":4707,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/index.html#exists-unitary-diagonal-of-is-star-norma","scope":"LinearAlgebra/eigenvalues-and-canonical-forms","status":"fully_proved","summary":"Let $M$ be a complex square matrix whose rows and columns are indexed by a finite set $n$, and write $M^*$ for its conjugate transpose. If $M$ is normal, that is, $M^* M = M M^*$, then there exist a unitary matrix $P$ indexed by $n$ (so $P^* P = P P^* = I$) and a family of complex numbers $(d_i)_{i \\in n}$ such that $P^* M P$ is the diagonal matrix with diagonal entries $d_i$.","title":"Spectral theorem for normal matrices"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization","status":"fully_proved","title":"Diagonalization"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization/exists-eigenbasis-of-is-semisimple-of-sp","kind":"node","lean":"MathlibExt.LinearAlgebra.DiagonalizationWanted.exists_eigenbasis_of_isSemisimple_of_splits","parent":4710,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization/index.html#exists-eigenbasis-of-is-semisimple-of-sp","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization","status":"fully_proved","summary":"Let $V$ be a finite-dimensional vector space over a field $K$ and let $f$ be a semisimple endomorphism of $V$ (every $f$-invariant subspace has an $f$-invariant complement) whose characteristic polynomial splits over $K$. Then for every finite type $\\iota$ with exactly $\\dim_K V$ elements there are a basis $(b_i)_{i \\in \\iota}$ of $V$ and scalars $(\\mu_i)_{i \\in \\iota}$ in $K$ such that each $b_i$ is an eigenvector\u2026","title":"Eigenbasis of a semisimple endomorphism with split characteristic polynomial"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization/exists-diagonal-similar-of-splits-of-squ","kind":"node","lean":"MathlibExt.LinearAlgebra.DiagonalizationWanted.exists_diagonal_similar_of_splits_of_squarefree_minpoly","parent":4710,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization/index.html#exists-diagonal-similar-of-splits-of-squ","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/diagonalization","status":"fully_proved","summary":"Let $K$ be a field and let $A$ be a square matrix over $K$ with rows and columns indexed by a finite type $n$. If the characteristic polynomial of $A$ splits over $K$ and the minimal polynomial of $A$ (that is, of the linear map $x \\mapsto Ax$ on $K^n$) is squarefree, then there are matrices $P$ and $D$ over $K$ indexed by $n$, with $D$ diagonal and $P$ invertible, such that $A = P D P^{-1}$.","title":"Diagonalizability from a split characteristic polynomial and a squarefree minimal polynomial"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri","status":"fully_proved","title":"Eigenvalues and spectral norm of Hermitian matrices"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri/charpoly-is-real-rooted","kind":"node","lean":"Matrix.IsHermitian.charpoly_isRealRooted","parent":4713,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri/index.html#charpoly-is-real-rooted","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri","status":"fully_proved","summary":"The characteristic polynomial of a Hermitian complex matrix is real-rooted.","title":"Hermitian matrices have real-rooted characteristic polynomials"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri/l2-op-norm-eq-max-real-root-charpoly","kind":"node","lean":"Matrix.PosSemidef.l2_opNorm_eq_maxRealRoot_charpoly","parent":4713,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri/index.html#l2-op-norm-eq-max-real-root-charpoly","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-and-spectral-norm-of-hermitian-matri","status":"fully_proved","summary":"Let $A$ be a positive semidefinite complex square matrix indexed by a nonempty finite set. Then the operator norm of $A$ with respect to the Euclidean ($\\ell^2$) norm equals the largest real part of a root of the characteristic polynomial of $A$. Since these roots are real, this is its largest root.","title":"Spectral norm of a positive semidefinite matrix"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","title":"Eigenvalues of the products AB and BA"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/maps-to-eigenspace-comp","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.mapsTo_eigenspace_comp","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#maps-to-eigenspace-comp","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"`g` sends the `\u03bc`-eigenspace of `f \u2218\u2097 g` into the `\u03bc`-eigenspace of `g \u2218\u2097 f`: if `(f \u2218\u2097 g) x = \u03bc \u2022 x` then `(g \u2218\u2097 f) (g x) = \u03bc \u2022 (g x)`.","title":"Eigenspace of fg maps into the eigenspace of gf"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/injective-restrict-eigenspace-comp","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.injective_restrict_eigenspace_comp","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#injective-restrict-eigenspace-comp","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The map `g` restricted to the eigenspaces is injective when `\u03bc \u2260 0`: if `g x = 0` for an eigenvector `x` of `f \u2218\u2097 g`, then `(f \u2218\u2097 g) x = 0 = \u03bc \u2022 x`, so `x = 0`.","title":"Injectivity of g between eigenspaces of fg and gf"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/surjective-restrict-eigenspace-comp","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.surjective_restrict_eigenspace_comp","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#surjective-restrict-eigenspace-comp","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The map `g` restricted to the eigenspaces is surjective when `\u03bc \u2260 0`: an eigenvector `y` of `g \u2218\u2097 f` is the image of `\u03bc\u207b\u00b9 \u2022 f y`.","title":"Surjectivity of g between eigenspaces of fg and gf"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/eigenspace-comp-equiv","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.eigenspaceCompEquiv","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#eigenspace-comp-equiv","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"For a nonzero eigenvalue, the eigenspaces of `f \u2218\u2097 g` and of `g \u2218\u2097 f` are linearly equivalent, via `g` in one direction.","title":"Linear equivalence of eigenspaces of fg and gf"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finrank-eigenspace-comp-eq","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.finrank_eigenspace_comp_eq","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finrank-eigenspace-comp-eq","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The geometric multiplicities of a nonzero eigenvalue of `f \u2218\u2097 g` and of `g \u2218\u2097 f` agree.","title":"Geometric multiplicities of fg and gf agree"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/has-eigenvector-comp","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.hasEigenvector_comp","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#has-eigenvector-comp","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"An eigenvector of `f \u2218\u2097 g` for a nonzero eigenvalue is sent by `g` to an eigenvector of `g \u2218\u2097 f` for the same eigenvalue.","title":"Eigenvectors of fg map to eigenvectors of gf"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/has-eigenvalue-comp-iff","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.hasEigenvalue_comp_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#has-eigenvalue-comp-iff","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"A nonzero scalar is an eigenvalue of `f \u2218\u2097 g` iff it is an eigenvalue of `g \u2218\u2097 f`.","title":"Nonzero eigenvalues of fg and gf coincide"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/maps-to-gen-eigenspace-comp","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.mapsTo_genEigenspace_comp","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#maps-to-gen-eigenspace-comp","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"`g` sends the generalized `\u03bc`-eigenspace of `f \u2218\u2097 g` into the generalized `\u03bc`-eigenspace of `g \u2218\u2097 f`, for every exponent. This is the intertwining `(g \u2218\u2097 f) \u2218\u2097 g = g \u2218\u2097 (f \u2218\u2097 g)` applied to Mathlib's `Module.End.mapsTo_genEigenspace_of_comp`.","title":"Generalized eigenspace of fg maps into that of gf"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/eq-zero-of-apply-eq-zero-of-mem-gen-eige","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.eq_zero_of_apply_eq_zero_of_mem_genEigenspace","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#eq-zero-of-apply-eq-zero-of-mem-gen-eige","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"If `g x = 0` and `x` lies in the generalized `\u03bc`-eigenspace of `f \u2218\u2097 g` with `\u03bc \u2260 0`, then `x = 0`. Indeed, on the kernel of `g` the map `f \u2218\u2097 g` vanishes, so `(f \u2218\u2097 g - \u03bc \u2022 1) ^ l` acts as `(-\u03bc) ^ l \u2022 1` there.","title":"Kernel of g meets a generalized eigenspace of fg trivially"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/gen-eigenspace-comp-map","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.genEigenspaceCompMap","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#gen-eigenspace-comp-map","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The restriction of `g` to the generalized `\u03bc`-eigenspaces.","title":"Restriction of g to generalized eigenspaces of fg and gf"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/injective-gen-eigenspace-comp-map","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.injective_genEigenspaceCompMap","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#injective-gen-eigenspace-comp-map","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The restriction of `g` to a generalized `\u03bc`-eigenspace with `\u03bc \u2260 0` is injective.","title":"Injectivity of g on generalized eigenspaces of fg"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finite-gen-eigenspace-comp-iff","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.finite_genEigenspace_comp_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finite-gen-eigenspace-comp-iff","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"For a nonzero scalar, the generalized eigenspaces of `f \u2218\u2097 g` and of `g \u2218\u2097 f` are finite-dimensional together, for every exponent `k : \u2115\u221e`.","title":"Generalized eigenspaces of fg and gf are finite-dimensional together"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finrank-gen-eigenspace-comp-eq","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.finrank_genEigenspace_comp_eq","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finrank-gen-eigenspace-comp-eq","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The generalized `\u03bc`-eigenspaces of `f \u2218\u2097 g` and of `g \u2218\u2097 f` have equal `Module.finrank`, for every exponent `k : \u2115\u221e` and every nonzero `\u03bc`. They are finite-dimensional together; if both are infinite-dimensional, both finranks are zero. In finite dimension only the stabilized generalized eigenspace (for example at exponent `\u22a4`) measures algebraic multiplicity.","title":"Generalized eigenspaces of fg and gf have equal dimension"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/eigenspace-mul-equiv","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.eigenspaceMulEquiv","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#eigenspace-mul-equiv","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The `*` form of `eigenspaceCompEquiv`, for endomorphisms of one space.","title":"Eigenspaces of fg and gf are equivalent, for endomorphisms"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finrank-eigenspace-mul-eq-2","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.finrank_eigenspace_mul_eq","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finrank-eigenspace-mul-eq-2","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The `*` form of `finrank_eigenspace_comp_eq`.","title":"Geometric multiplicities of fg and gf agree, for endomorphisms"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/has-eigenvector-mul","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.hasEigenvector_mul","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#has-eigenvector-mul","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The `*` form of `hasEigenvector_comp`.","title":"Eigenvectors of fg map to eigenvectors of gf, for endomorphisms"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/has-eigenvalue-mul-iff-2","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.hasEigenvalue_mul_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#has-eigenvalue-mul-iff-2","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The `*` form of `hasEigenvalue_comp_iff`.","title":"Nonzero eigenvalues of fg and gf coincide, for endomorphisms"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finrank-gen-eigenspace-mul-eq-2","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.finrank_genEigenspace_mul_eq","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finrank-gen-eigenspace-mul-eq-2","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The `*` form of `finrank_genEigenspace_comp_eq`.","title":"Generalized eigenspace dimensions agree, for endomorphisms"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finite-gen-eigenspace-mul-iff-2","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.End.finite_genEigenspace_mul_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finite-gen-eigenspace-mul-iff-2","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The `*` form of `finite_genEigenspace_comp_iff`.","title":"Generalized eigenspaces finite-dimensional together, for endomorphisms"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/has-eigenvalue-mul-iff-3","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.Matrix.hasEigenvalue_mul_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#has-eigenvalue-mul-iff-3","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"A nonzero scalar is an eigenvalue of the matrix product `A * B` iff it is an eigenvalue of `B * A`.","title":"Nonzero eigenvalues of matrix products AB and BA coincide"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finrank-eigenspace-mul-eq-3","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.Matrix.finrank_eigenspace_mul_eq","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finrank-eigenspace-mul-eq-3","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The geometric multiplicities of a nonzero eigenvalue of `A * B` and of `B * A` agree.","title":"Geometric multiplicities of matrix products AB and BA agree"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finrank-gen-eigenspace-mul-eq-3","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.Matrix.finrank_genEigenspace_mul_eq","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finrank-gen-eigenspace-mul-eq-3","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The generalized `\u03bc`-eigenspaces of `A * B` and of `B * A` have equal `Module.finrank`, for every exponent `k : \u2115\u221e`.","title":"Generalized eigenspaces of AB and BA have equal dimension"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finite-gen-eigenspace-mul-iff-3","kind":"node","lean":"MathlibExt.LinearAlgebra.Eigenspace.MulComm.Matrix.finite_genEigenspace_mul_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finite-gen-eigenspace-mul-iff-3","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The generalized eigenspaces of rectangular products in opposite orders are finite-dimensional together for a nonzero scalar and every exponent.","title":"Generalized eigenspaces of AB and BA are finite-dimensional together"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/has-eigenvalue-mul-iff","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.EigenvalueMulComm.ContinuousLinearMap.hasEigenvalue_mul_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#has-eigenvalue-mul-iff","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"A nonzero scalar is an eigenvalue of `S * T` iff it is an eigenvalue of `T * S`, for bounded operators.","title":"Nonzero eigenvalues of ST and TS coincide"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finrank-eigenspace-mul-eq","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.EigenvalueMulComm.ContinuousLinearMap.finrank_eigenspace_mul_eq","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finrank-eigenspace-mul-eq","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The geometric multiplicities of a nonzero eigenvalue of `S * T` and of `T * S` agree, for bounded operators.","title":"Equal geometric multiplicities of nonzero eigenvalues of ST and TS"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finrank-gen-eigenspace-mul-eq","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.EigenvalueMulComm.ContinuousLinearMap.finrank_genEigenspace_mul_eq","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finrank-gen-eigenspace-mul-eq","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"The generalized `\u03bc`-eigenspaces of `S * T` and of `T * S` have equal `Module.finrank`, for every exponent `k : \u2115\u221e`, for bounded operators.","title":"Equal dimensions of generalized eigenspaces of ST and TS"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/finite-gen-eigenspace-mul-iff","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.EigenvalueMulComm.ContinuousLinearMap.finite_genEigenspace_mul_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#finite-gen-eigenspace-mul-iff","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"Generalized eigenspaces of reversed bounded-operator products are finite-dimensional together for a nonzero scalar and every exponent.","title":"Finite-dimensionality of generalized eigenspaces of ST and TS"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/has-eigenvalue-adjoint-mul-self-iff","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.EigenvalueMulComm.ContinuousLinearMap.hasEigenvalue_adjoint_mul_self_iff","parent":4716,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba/index.html#has-eigenvalue-adjoint-mul-self-iff","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/eigenvalues-of-the-products-ab-and-ba","status":"fully_proved","summary":"A nonzero scalar is an eigenvalue of `T\u2020 * T` iff it is an eigenvalue of `T * T\u2020`, on a Hilbert space.","title":"Nonzero eigenvalues of an operator times its adjoint, in either order"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices","status":"can_prove","title":"Generalized eigenvalues of max and min matrices"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices/det-max-matrix-sub-x-smul-min-matrix","kind":"node","lean":"MetaMathlibExt.det_maxMatrix_sub_X_smul_minMatrix","parent":4745,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices/index.html#det-max-matrix-sub-x-smul-min-matrix","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices","status":"can_prove","summary":"Let $n > 2$ and let $0 < s_1 < s_2 < \\dots < s_n$ be real numbers. Let $M$ be the $n \\times n$ matrix with entries $M_{ij} = \\max(s_i, s_j)$, let $N$ be the $n \\times n$ matrix with entries $N_{ij} = \\min(s_i, s_j)$, and put $\\rho = \\sqrt{s_n / s_1}$. Then, as polynomials in $X$ with real coefficients,","title":"Generalized eigenvalues of a MAX matrix to a MIN matrix"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices/generalized-eigenvalue-interlacing","kind":"node","lean":"MetaMathlibExt.generalizedEigenvalue_interlacing","parent":4745,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices/index.html#generalized-eigenvalue-interlacing","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/generalized-eigenvalues-of-max-and-min-matrices","status":"can_prove","summary":"Let $n \\ge 1$, let $A$ be a Hermitian and $B$ a positive definite complex $(n+1) \\times (n+1)$ matrix, and let $A'$ and $B'$ be their leading principal $n \\times n$ submatrices (rows and columns $1, \\dots, n$). Let $\\lambda_1 \\ge \\dots \\ge \\lambda_{n+1}$ be real numbers such that the roots of the polynomial $\\det(A - XB)$, counted with multiplicity, are exactly $\\lambda_1, \\dots, \\lambda_{n+1}$, and let\u2026","title":"Cauchy interlacing for generalized eigenvalues"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","status":"fully_proved","title":"Jordan normal form"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/jordan-block","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.JordanNormalFormWanted.jordanBlock","parent":4748,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/index.html#jordan-block","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","status":"fully_proved","summary":"For a natural number $s$ and a scalar $\\mu$ in a field $K$, the Jordan block of size $s$ with eigenvalue $\\mu$ is the $s \\times s$ matrix over $K$, with rows and columns indexed by $\\{0, \\dots, s-1\\}$, whose $(i, j)$ entry is $1$ if $j = i + 1$, $\\mu$ if $i = j$, and $0$ otherwise: it has $\\mu$ on the diagonal, $1$ on the first superdiagonal and $0$ elsewhere.","title":"Jordan block"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/jordan-block-apply","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.JordanNormalFormWanted.jordanBlock_apply","parent":4748,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/index.html#jordan-block-apply","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","status":"fully_proved","summary":"For a natural number $s$, a scalar $\\mu$ in a field $K$ and indices $i, j \\in \\{0, \\dots, s-1\\}$, the $(i, j)$ entry of the Jordan block of size $s$ with eigenvalue $\\mu$ is $1$ if $j = i + 1$, is $\\mu$ if $i = j$, and is $0$ otherwise.","title":"Entries of a Jordan block"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/to-matrix-eq-jordan-block","kind":"node","lean":"LinearMap.toMatrix_eq_jordanBlock","parent":4748,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/index.html#to-matrix-eq-jordan-block","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","status":"fully_proved","summary":"Let $V$ be a vector space over a field $K$, let $f$ be a linear endomorphism of $V$, let $\\mu \\in K$, and let $b_0, \\dots, b_d$ be a basis of $V$ such that $f(b_0) = \\mu b_0$ and $f(b_{j+1}) = \\mu b_{j+1} + b_j$ for every $j < d$. Then the matrix of $f$ with respect to the basis $(b_0, \\dots, b_d)$ is the Jordan block of size $d + 1$ with eigenvalue $\\mu$.","title":"Matrix of an endomorphism on a Jordan chain"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/to-matrix-eq-block-diagonal-jordan-block","kind":"node","lean":"LinearMap.toMatrix_eq_blockDiagonal_jordanBlock","parent":4748,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/index.html#to-matrix-eq-block-diagonal-jordan-block","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","status":"fully_proved","summary":"Let $V$ be a vector space over a field $K$, let $f$ be a linear endomorphism of $V$, and let $\\iota$ be a finite set. For each $i \\in \\iota$ let $d_i \\in \\mathbb{N}$ and $\\mu_i \\in K$, and let $(b_{i,r})$, indexed by $i \\in \\iota$ and $0 \\le r \\le d_i$, be a basis of $V$ such that $f(b_{i,0}) = \\mu_i b_{i,0}$ and $f(b_{i,j+1}) = \\mu_i b_{i,j+1} + b_{i,j}$ for every $i \\in \\iota$ and every $j < d_i$. Then the matrix\u2026","title":"Matrix of an endomorphism on a basis of Jordan chains"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/exists-jordan-basis-of-is-nilpotent","kind":"node","lean":"Module.End.exists_jordanBasis_of_isNilpotent","parent":4748,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/index.html#exists-jordan-basis-of-is-nilpotent","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","status":"fully_proved","summary":"Let $V$ be a finite-dimensional vector space over a field $K$ and let $f$ be a nilpotent linear endomorphism of $V$. Then there are $k \\in \\mathbb{N}$, natural numbers $d_0, \\dots, d_{k-1}$ and a basis $(b_{i,r})$ of $V$, indexed by $i < k$ and $0 \\le r \\le d_i$, such that $f(b_{i,0}) = 0$ and $f(b_{i,j+1}) = b_{i,j}$ for every $i < k$ and every $j < d_i$, and such that the matrix of $f$ with respect to $b$ is the\u2026","title":"Jordan chain basis for a nilpotent endomorphism"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/exists-jordan-basis-of-charpoly-splits","kind":"node","lean":"Module.End.exists_jordanBasis_of_charpoly_splits","parent":4748,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/index.html#exists-jordan-basis-of-charpoly-splits","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","status":"fully_proved","summary":"Let $V$ be a finite-dimensional vector space over a field $K$ and let $f$ be a linear endomorphism of $V$ whose characteristic polynomial splits into linear factors over $K$. Then there are $k \\in \\mathbb{N}$, natural numbers $d_0, \\dots, d_{k-1}$, scalars $\\mu_0, \\dots, \\mu_{k-1} \\in K$ and a basis $(b_{i,r})$ of $V$, indexed by $i < k$ and $0 \\le r \\le d_i$, such that $f(b_{i,0}) = \\mu_i b_{i,0}$ and\u2026","title":"Jordan chain basis for an endomorphism with split characteristic polynomial"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/exists-jordan-normal-form-of-splits","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.JordanNormalFormWanted.exists_jordan_normal_form_of_splits","parent":4748,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form/index.html#exists-jordan-normal-form-of-splits","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/jordan-normal-form","status":"fully_proved","summary":"Let $A$ be a square matrix over a field $K$, with rows and columns indexed by a finite set $N$, whose characteristic polynomial splits into linear factors over $K$. Then $A$ is similar to a block-diagonal matrix of Jordan blocks: there are $k \\in \\mathbb{N}$, positive integers $s_0, \\dots, s_{k-1}$, scalars $\\mu_0, \\dots, \\mu_{k-1} \\in K$, an invertible matrix $P$ with rows and columns indexed by $N$, and a\u2026","title":"Jordan normal form"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials","status":"fully_proved","title":"Kernels lemma for coprime polynomials"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/i-sup-ker-aeval-eq-ker-aeval-prod-of-pai","kind":"node","lean":"MathlibExt.LinearAlgebra.KernelsLemmaWanted.iSup_ker_aeval_eq_ker_aeval_prod_of_pairwise_isCoprime","parent":4756,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/index.html#i-sup-ker-aeval-eq-ker-aeval-prod-of-pai","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, $M$ an $R$-module, $f$ an $R$-linear endomorphism of $M$, and $(p_i)_{i \\in \\iota}$ a family of polynomials in $R[X]$ that are pairwise coprime, meaning that for all $i \\ne j$ there are $a, b \\in R[X]$ with $a p_i + b p_j = 1$. Then for every finite set $s \\subseteq \\iota$, the sum of the submodules $\\ker p_i(f)$ over $i \\in s$ equals $\\ker \\bigl(\\prod_{i \\in s} p_i\\bigr)(f)$.","title":"Kernel of a product of pairwise coprime polynomials"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/sup-indep-ker-aeval-of-pairwise-is-copri","kind":"node","lean":"MathlibExt.LinearAlgebra.KernelsLemmaWanted.supIndep_ker_aeval_of_pairwise_isCoprime","parent":4756,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/index.html#sup-indep-ker-aeval-of-pairwise-is-copri","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, $M$ an $R$-module, $f$ an $R$-linear endomorphism of $M$, and $(p_i)_{i \\in \\iota}$ a family of polynomials in $R[X]$ that are pairwise coprime, meaning that for all $i \\ne j$ there are $a, b \\in R[X]$ with $a p_i + b p_j = 1$. Then for every finite set $s \\subseteq \\iota$, every $i \\in s$ and every $t \\subseteq s$ with $i \\notin t$, the submodule $\\ker p_i(f)$ meets the sum of the\u2026","title":"Kernels of pairwise coprime polynomials are independent over a finite set"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/i-sup-indep-ker-aeval-and-i-sup-eq-of-pa","kind":"node","lean":"MathlibExt.LinearAlgebra.KernelsLemmaWanted.iSupIndep_ker_aeval_and_iSup_eq_of_pairwise_isCoprime","parent":4756,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/index.html#i-sup-indep-ker-aeval-and-i-sup-eq-of-pa","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, $M$ an $R$-module, $f$ an $R$-linear endomorphism of $M$, and $(p_i)_{i \\in \\iota}$ a family of polynomials in $R[X]$, indexed by a finite type $\\iota$, that are pairwise coprime, meaning that for all $i \\ne j$ there are $a, b \\in R[X]$ with $a p_i + b p_j = 1$. Then the submodules $\\ker p_i(f)$ are independent (each meets the sum of the others only in $0$), and their sum is\u2026","title":"Independence and sum of kernels of pairwise coprime polynomials"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/kernels-lemma-pairwise-is-coprime","kind":"node","lean":"MathlibExt.LinearAlgebra.KernelsLemmaWanted.kernels_lemma_pairwise_isCoprime","parent":4756,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials/index.html#kernels-lemma-pairwise-is-coprime","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/kernels-lemma-for-coprime-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring, $M$ an $R$-module, $f$ an $R$-linear endomorphism of $M$, and $(p_i)_{i \\in \\iota}$ a family of polynomials in $R[X]$, indexed by a finite type $\\iota$, that are pairwise coprime, meaning that for all $i \\ne j$ there are $a, b \\in R[X]$ with $a p_i + b p_j = 1$. Then the submodules $\\ker p_i(f)$ are independent (each meets the sum of the others only in $0$), and their sum is\u2026","title":"Kernels lemma"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers","status":"can_prove","title":"Newton's identities for traces of matrix powers"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/trace-pow-two","kind":"node","lean":"MathlibExt.LinearAlgebra.PowerSumMatrixWanted.trace_pow_two","parent":4761,"reading_title":{"format":"inline-parts-v1","parts":[["text","Traces of powers of a "],["math","\\(2 \\times 2\\)"],["text"," matrix"]],"source":"Traces of powers of a $2 \\times 2$ matrix"},"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/index.html#trace-pow-two","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers","status":"can_prove","summary":"Let $M$ be a real $2 \\times 2$ matrix and let $m \\ge 1$ be an integer. Then","title":"Traces of powers of a $2 \\times 2$ matrix"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/integral-pow-one-sub-pow","kind":"node","lean":"MathlibExt.LinearAlgebra.PowerSumMatrixWanted.integral_pow_one_sub_pow","parent":4761,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(x^r (1 - x)^r\\)"],["text"," over "],["math","\\((0, 1)\\)"]],"source":"Integral of $x^r (1 - x)^r$ over $(0, 1)$"},"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/index.html#integral-pow-one-sub-pow","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers","status":"can_prove","summary":"For every natural number $r$ (including $r = 0$),","title":"Integral of $x^r (1 - x)^r$ over $(0, 1)$"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/charpoly-descending-coeff","kind":"node","lean":"MathlibExt.LinearAlgebra.TraceNewtonWanted.LinearAlgebra.TraceNewton.charpolyDescendingCoeff","parent":4761,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/index.html#charpoly-descending-coeff","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers","status":"fully_proved","summary":"Let $A$ be a square matrix over a commutative ring $R$ whose rows and columns are indexed by a finite set of size $N$, and write its characteristic polynomial as $\\chi_A(X) = \\det(X I - A) = X^N + c_1 X^{N-1} + \\dots + c_N$. For a natural number $k$, the descending coefficient $c_k$ of $A$ is the coefficient of $X^{N-k}$ in $\\chi_A$ when $k \\le N$, and $0$ when $k > N$.","title":"Characteristic-polynomial coefficients in descending order"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/trace-cayley-hamilton-newton","kind":"node","lean":"MathlibExt.LinearAlgebra.TraceNewtonWanted.LinearAlgebra.TraceNewton.trace_cayley_hamilton_newton","parent":4761,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers/index.html#trace-cayley-hamilton-newton","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/newton-s-identities-for-traces-of-matrix-powers","status":"can_prove","summary":"Let $A$ be a square matrix over a commutative ring $R$ whose rows and columns are indexed by a finite set of size $N$. For each natural number $k$, let $c_k$ be the coefficient of $X^{N-k}$ in the characteristic polynomial $\\chi_A(X) = \\det(X I - A)$ when $k \\le N$, and $c_k = 0$ when $k > N$, so that $\\chi_A(X) = X^N + c_1 X^{N-1} + \\dots + c_N$. Then for every natural number $k \\ge 1$,","title":"Newton's trace identities for the characteristic polynomial"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form","status":"fully_proved","title":"Smith normal form"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form/exists-smith-normal-form-dvd-chain","kind":"node","lean":"Submodule.exists_smith_normal_form_dvd_chain","parent":4766,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form/index.html#exists-smith-normal-form-dvd-chain","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form","status":"fully_proved","summary":"Let $R$ be a principal ideal domain, let $M$ be a finitely generated free $R$-module and let $N$ be a submodule of $M$. Then there are natural numbers $n \\le m$, a basis $e_0, \\dots, e_{m-1}$ of $M$, a basis $f_0, \\dots, f_{n-1}$ of $N$ and nonzero elements $a_0, \\dots, a_{n-1}$ of $R$ such that $f_i = a_i e_i$ in $M$ for every $i < n$ and $a_i \\mid a_{i+1}$ whenever $i + 1 < n$.","title":"Smith normal form with a divisibility chain"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form/smith-normal-form","kind":"node","lean":"MathlibExt.LinearAlgebra.SmithNormalFormWanted.smithNormalForm","parent":4766,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form/index.html#smith-normal-form","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/smith-normal-form","status":"fully_proved","summary":"Let $R$ be a principal ideal domain, let $M$ be a finitely generated free $R$-module and let $N$ be a submodule of $M$. Then there are natural numbers $n \\le m$, a basis $e_0, \\dots, e_{m-1}$ of $M$, a basis $f_0, \\dots, f_{n-1}$ of $N$ and nonzero elements $a_0, \\dots, a_{n-1}$ of $R$ such that $f_i = a_i e_i$ in $M$ for every $i < n$ and $a_i \\mid a_{i+1}$ whenever $i + 1 < n$.","title":"Smith normal form"},{"data_url":"scopes/LinearAlgebra/eigenvalues-and-canonical-forms/triangularization.json","id":"LinearAlgebra/eigenvalues-and-canonical-forms/triangularization","kind":"scope","navigation":"module","parent":4707,"result":false,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/triangularization/index.html","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/triangularization","status":"fully_proved","title":"Triangularization"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/triangularization/exists-basis-upper-triangular-to-matrix","kind":"node","lean":"MathlibExt.LinearAlgebra.TriangularizationWanted.exists_basis_upperTriangular_toMatrix","parent":4769,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/triangularization/index.html#exists-basis-upper-triangular-to-matrix","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/triangularization","status":"fully_proved","summary":"Let $V$ be a finite-dimensional vector space over a field $K$, let $f$ be an endomorphism of $V$ whose characteristic polynomial splits over $K$, and let $\\iota$ be a finite linearly ordered set with exactly $\\dim_K V$ elements. Then there is a basis $(b_i)_{i \\in \\iota}$ of $V$ in which the matrix of $f$ is upper triangular: its $(i, j)$ entry is $0$ whenever $j < i$.","title":"Triangularization of an endomorphism with split characteristic polynomial"},{"id":"LinearAlgebra/eigenvalues-and-canonical-forms/triangularization/exists-upper-triangular-similar-of-split","kind":"node","lean":"MathlibExt.LinearAlgebra.TriangularizationWanted.exists_upperTriangular_similar_of_splits","parent":4769,"result":true,"route":"../roadmap/LinearAlgebra/eigenvalues-and-canonical-forms/triangularization/index.html#exists-upper-triangular-similar-of-split","scope":"LinearAlgebra/eigenvalues-and-canonical-forms/triangularization","status":"fully_proved","summary":"Let $K$ be a field, let $n$ be a finite linearly ordered index set, and let $A$ be a square matrix over $K$ with rows and columns indexed by $n$ whose characteristic polynomial splits over $K$. Then there are an invertible matrix $P$ and a matrix $T$, both over $K$ and indexed by $n$, such that $T$ is upper triangular ($T_{ij} = 0$ whenever $j < i$) and $A = P T P^{-1}$.","title":"Triangularization of a matrix with split characteristic polynomial"},{"data_url":"scopes/LinearAlgebra/positive-definite-matrices.json","id":"LinearAlgebra/positive-definite-matrices","kind":"scope","parent":4518,"result":false,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/index.html","scope":"LinearAlgebra/positive-definite-matrices","status":"can_prove","summary":"Positive definite and semidefinite matrices, criteria for definiteness, and inequalities for their determinants, traces and eigenvalues.","title":"Positive definite matrices and matrix inequalities"},{"id":"LinearAlgebra/positive-definite-matrices/hoffman-wielandt-hermitian","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HoffmanWielandtWanted.hoffman_wielandt_hermitian","parent":4772,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/index.html#hoffman-wielandt-hermitian","scope":"LinearAlgebra/positive-definite-matrices","status":"fully_proved","summary":"Let $A$ and $B$ be Hermitian $n \\times n$ complex matrices with eigenvalues $\\lambda_1 \\ge \\dots \\ge \\lambda_n$ and $\\mu_1 \\ge \\dots \\ge \\mu_n$ respectively, counted with multiplicity. Then $\\sum_{i=1}^n (\\lambda_i - \\mu_i)^2 \\le \\sum_{i=1}^n \\sum_{j=1}^n |A_{ij} - B_{ij}|^2$, that is, the left side is at most the squared Frobenius norm of $A - B$.","title":"Hoffman\u2013Wielandt inequality for Hermitian matrices"},{"id":"LinearAlgebra/positive-definite-matrices/log-det-mul-le-trace","kind":"node","lean":"Matrix.PosDef.log_det_mul_le_trace","parent":4772,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/index.html#log-det-mul-le-trace","scope":"LinearAlgebra/positive-definite-matrices","status":"fully_proved","summary":"Let $A$ and $B$ be positive definite real square matrices indexed by a finite set with $n$ elements. Then $\\log(\\det A \\cdot \\det B) \\le \\operatorname{tr}(AB) - n$.","title":"Log-determinant trace inequality"},{"data_url":"scopes/LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for.json","id":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for","kind":"scope","navigation":"module","parent":4772,"result":false,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/index.html","scope":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for","status":"fully_proved","title":"Hadamard's inequality and determinant bounds for positive semidefinite matrices"},{"id":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/hadamard-determinant-inequality","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.HadamardDeterminant.hadamard_determinant_inequality","parent":4775,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/index.html#hadamard-determinant-inequality","scope":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for","status":"fully_proved","summary":"For every natural number $n$ and every $n \\times n$ complex matrix $A = (a_{ij})$,","title":"Hadamard's determinant inequality"},{"id":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/det-re-le-prod-diag-re","kind":"node","lean":"Matrix.PosSemidef.det_re_le_prod_diag_re","parent":4775,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/index.html#det-re-le-prod-diag-re","scope":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for","status":"fully_proved","summary":"Let $A$ be a positive semidefinite complex $n \\times n$ matrix. Then the real part of the determinant of $A$ is at most the product of the real parts of its diagonal entries: $\\operatorname{Re} \\det A \\le \\prod_{i=1}^{n} \\operatorname{Re} A_{ii}$.","title":"Hadamard's inequality for positive semidefinite matrices"},{"id":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/geom-mean-det-le-trace-div","kind":"node","lean":"Matrix.PosSemidef.geomMean_det_le_trace_div","parent":4775,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/index.html#geom-mean-det-le-trace-div","scope":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for","status":"fully_proved","summary":"Let $n \\ge 1$ and let $A$ be a positive semidefinite complex $n \\times n$ matrix. Then $(\\operatorname{Re} \\det A)^{1/n} \\le \\operatorname{Re}(\\operatorname{tr} A)/n$.","title":"AM\u2013GM inequality for the determinant of a positive semidefinite matrix"},{"id":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/det-re-le-pow-trace-div","kind":"node","lean":"Matrix.PosSemidef.det_re_le_pow_trace_div","parent":4775,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/index.html#det-re-le-pow-trace-div","scope":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for","status":"fully_proved","summary":"Let $n \\ge 1$ and let $A$ be a positive semidefinite complex $n \\times n$ matrix. Then $\\operatorname{Re} \\det A \\le (\\operatorname{Re}(\\operatorname{tr} A)/n)^n$.","title":"AM\u2013GM inequality for the determinant of a positive semidefinite matrix, power form"},{"id":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/amgm-eq-iff","kind":"node","lean":"Matrix.PosSemidef.amgm_eq_iff","parent":4775,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/index.html#amgm-eq-iff","scope":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for","status":"fully_proved","summary":"Let $n \\ge 1$ and let $A$ be a positive semidefinite complex $n \\times n$ matrix. Then $(\\operatorname{Re} \\det A)^{1/n} = \\operatorname{Re}(\\operatorname{tr} A)/n$ if and only if $A = cI$ for some real number $c \\ge 0$, where $I$ is the $n \\times n$ identity matrix.","title":"Equality in the AM\u2013GM bound for the determinant"},{"id":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/trace-mul-nonneg-re","kind":"node","lean":"Matrix.PosSemidef.trace_mul_nonneg_re","parent":4775,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for/index.html#trace-mul-nonneg-re","scope":"LinearAlgebra/positive-definite-matrices/hadamard-s-inequality-and-determinant-bounds-for","status":"fully_proved","summary":"Let $A$ and $B$ be positive semidefinite complex $n \\times n$ matrices. Then the real part of the trace of $AB$ is nonnegative: $\\operatorname{Re} \\operatorname{tr}(AB) \\ge 0$.","title":"The trace of a product of positive semidefinite matrices has nonnegative real part"},{"data_url":"scopes/LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality.json","id":"LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality","kind":"scope","navigation":"module","parent":4772,"result":false,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality/index.html","scope":"LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality","status":"can_prove","title":"Lidskii's eigenvalue inequality"},{"id":"LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality/lidskii-inequality","kind":"node","lean":"MathlibExt.LinearAlgebra.LidskiiInequalityWanted.LinearAlgebra.lidskii_inequality","parent":4782,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality/index.html#lidskii-inequality","scope":"LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality","status":"can_prove","summary":"Let $A$ and $B$ be Hermitian complex square matrices indexed by a finite set with $N$ elements. List the eigenvalues of $A$, of $B$ and of $B - A$, each with multiplicity and in decreasing order, as $\\alpha_1 \\ge \\dots \\ge \\alpha_N$, $\\beta_1 \\ge \\dots \\ge \\beta_N$ and $\\gamma_1 \\ge \\dots \\ge \\gamma_N$. Then for every real $p \\ge 1$,","title":"Lidskii's inequality"},{"id":"LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality/lidskii-last","kind":"node","lean":"MathlibExt.LinearAlgebra.LidskiiWanted.LinearAlgebra.lidskii_last","parent":4782,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality/index.html#lidskii-last","scope":"LinearAlgebra/positive-definite-matrices/lidskii-s-eigenvalue-inequality","status":"can_prove","summary":"Let $A$ and $B$ be Hermitian complex square matrices indexed by a finite set with $N$ elements, and list the eigenvalues of $A$ and of $B$, each with multiplicity and in decreasing order, as $\\alpha_1 \\ge \\dots \\ge \\alpha_N$ and $\\beta_1 \\ge \\dots \\ge \\beta_N$. Then","title":"Lidskii\u2013Last theorem"},{"data_url":"scopes/LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality.json","id":"LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality","kind":"scope","navigation":"module","parent":4772,"result":false,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality/index.html","scope":"LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality","status":"fully_proved","title":"Minkowski's determinant inequality"},{"id":"LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality/minkowski-det","kind":"node","lean":"Matrix.PosSemidef.minkowski_det","parent":4785,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality/index.html#minkowski-det","scope":"LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality","status":"fully_proved","summary":"For positive semidefinite `A` and `B`, `(det (A + B)) ^ (1 / n) \u2265 (det A) ^ (1 / n) + (det B) ^ (1 / n)`.","title":"Minkowski's determinant inequality"},{"id":"LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality/det-le-det-add","kind":"node","lean":"Matrix.PosSemidef.det_le_det_add","parent":4785,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality/index.html#det-le-det-add","scope":"LinearAlgebra/positive-definite-matrices/minkowski-s-determinant-inequality","status":"fully_proved","summary":"`det B \u2264 det (A + B)` for `A` and `B` positive semidefinite.","title":"Monotonicity of the determinant on the positive semidefinite cone"},{"data_url":"scopes/LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality.json","id":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality","kind":"scope","navigation":"module","parent":4772,"result":false,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/index.html","scope":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality","status":"fully_proved","title":"Fischer's determinant inequality"},{"id":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/one-le-det-of-pos-semidef-sub-one","kind":"node","lean":"Matrix.one_le_det_of_posSemidef_sub_one","parent":4788,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/index.html#one-le-det-of-pos-semidef-sub-one","scope":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality","status":"fully_proved","summary":"If `W - 1` is positive semidefinite, then `1 \u2264 det W`, since every eigenvalue of `W` is at least `1`.","title":"Determinant at least one when W minus identity is positive semidefinite"},{"id":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/det-le-det-of-pos-semidef-le","kind":"node","lean":"Matrix.det_le_det_of_posSemidef_le","parent":4788,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/index.html#det-le-det-of-pos-semidef-le","scope":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality","status":"fully_proved","summary":"Monotonicity of the determinant on the positive semidefinite cone: if `S` and `D - S` are positive semidefinite, then `det S \u2264 det D`.","title":"Monotonicity of the determinant on the positive semidefinite cone"},{"id":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/fischer-det-from-blocks-le","kind":"node","lean":"Matrix.fischer_det_fromBlocks_le","parent":4788,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/index.html#fischer-det-from-blocks-le","scope":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality","status":"fully_proved","summary":"For a positive semidefinite block matrix `[[A, B], [B\u1d34, D]]` with real entries, `det [[A, B], [B\u1d34, D]] \u2264 det A * det D`.","title":"Fischer's determinant inequality"},{"id":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/fischer-det-from-blocks-eq-iff","kind":"node","lean":"Matrix.fischer_det_fromBlocks_eq_iff","parent":4788,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality/index.html#fischer-det-from-blocks-eq-iff","scope":"LinearAlgebra/positive-definite-matrices/fischer-s-determinant-inequality","status":"fully_proved","summary":"For `A` and `D` positive definite and `[[A, B], [B\u1d34, D]]` positive semidefinite, `det [[A, B], [B\u1d34, D]] = det A * det D` iff `B = 0`.","title":"Equality case of Fischer's inequality"},{"data_url":"scopes/LinearAlgebra/positive-definite-matrices/sylvester-s-criterion.json","id":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion","kind":"scope","navigation":"module","parent":4772,"result":false,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/index.html","scope":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion","status":"fully_proved","title":"Sylvester's criterion"},{"id":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/leading-principal-submatrix","kind":"node","lean":"Matrix.leadingPrincipalSubmatrix","parent":4793,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/index.html#leading-principal-submatrix","scope":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion","status":"fully_proved","summary":"The leading `k \u00d7 k` principal submatrix of `A`: the submatrix on the first `k` rows and columns.","title":"Leading principal submatrix"},{"id":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/leading-principal-submatrix-self","kind":"node","lean":"Matrix.leadingPrincipalSubmatrix_self","parent":4793,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/index.html#leading-principal-submatrix-self","scope":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion","status":"fully_proved","summary":"The full leading principal submatrix is the matrix itself.","title":"Full leading principal submatrix is the matrix itself"},{"id":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/from-blocks","kind":"node","lean":"Matrix.PosDef.fromBlocks","parent":4793,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/index.html#from-blocks","scope":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion","status":"fully_proved","summary":"A block matrix with a positive definite top-left block and a positive definite Schur complement is positive definite. This is the one-step induction step of Sylvester's criterion, stated for a `1 \u00d7 1` bottom-right block.","title":"Positive definiteness of a block matrix via the Schur complement"},{"id":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/pos-def-iff-leading-principal-minors-pos","kind":"node","lean":"Matrix.posDef_iff_leadingPrincipalMinors_pos","parent":4793,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/sylvester-s-criterion/index.html#pos-def-iff-leading-principal-minors-pos","scope":"LinearAlgebra/positive-definite-matrices/sylvester-s-criterion","status":"fully_proved","summary":"A Hermitian real matrix is positive definite if and only if all of its leading principal minors are positive. The Hermitian hypothesis is essential: `!![1, 2; 0, 1]` has leading minors `1` and `1` but is not Hermitian, hence not positive definite.","title":"Sylvester's criterion"},{"data_url":"scopes/LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality.json","id":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality","kind":"scope","navigation":"module","parent":4772,"result":false,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/index.html","scope":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality","status":"fully_proved","title":"Von Neumann's trace inequality"},{"id":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/inner-to-euclidean-lin-self-eq-sum","kind":"node","lean":"MetaMathlibExt.inner_toEuclideanLin_self_eq_sum","parent":4798,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/index.html#inner-to-euclidean-lin-self-eq-sum","scope":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality","status":"fully_proved","summary":"Let $B$ be a real symmetric matrix with rows and columns indexed by a finite set $I$, and let $(v_j)_{j \\in I}$ be the orthonormal basis of $\\mathbb{R}^I$ of eigenvectors of $B$ chosen by the spectral theorem, with $B v_j = \\beta_j v_j$. Then for every $x \\in \\mathbb{R}^I$, $\\langle x, Bx \\rangle = \\sum_j \\beta_j \\langle v_j, x \\rangle^2$.","title":"The quadratic form of a symmetric matrix in an orthonormal eigenbasis"},{"id":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/trace-mul-eq-sum-eigenvalues","kind":"node","lean":"MetaMathlibExt.trace_mul_eq_sum_eigenvalues","parent":4798,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/index.html#trace-mul-eq-sum-eigenvalues","scope":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality","status":"fully_proved","summary":"Let $A$ and $B$ be real symmetric matrices with rows and columns indexed by a finite set $I$. Let $(u_i)_{i \\in I}$ and $(v_j)_{j \\in I}$ be the orthonormal bases of $\\mathbb{R}^I$ of eigenvectors of $A$ and $B$ chosen by the spectral theorem, with $A u_i = \\alpha_i u_i$ and $B v_j = \\beta_j v_j$. Then $\\operatorname{tr}(AB) = \\sum_i \\sum_j \\alpha_i \\beta_j \\langle v_j, u_i \\rangle^2$.","title":"The trace of a product of symmetric matrices in their eigenbases"},{"id":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/overlap-mem-doubly-stochastic","kind":"node","lean":"MetaMathlibExt.overlap_mem_doublyStochastic","parent":4798,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/index.html#overlap-mem-doubly-stochastic","scope":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality","status":"fully_proved","summary":"Let $I$ be a finite set and let $(u_i)_{i \\in I}$ and $(v_j)_{j \\in I}$ be orthonormal bases of $\\mathbb{R}^I$ with its standard inner product. Then the $I \\times I$ matrix with entries $P_{ij} = \\langle v_j, u_i \\rangle^2$ is doubly stochastic: its entries are nonnegative and each of its rows and columns sums to $1$.","title":"The overlap matrix of two orthonormal bases is doubly stochastic"},{"id":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/exists-perm-sum-le-of-mem-doubly-stochas","kind":"node","lean":"MetaMathlibExt.exists_perm_sum_le_of_mem_doublyStochastic","parent":4798,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/index.html#exists-perm-sum-le-of-mem-doubly-stochas","scope":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality","status":"fully_proved","summary":"Let $I$ be a finite set, let $P$ be a doubly stochastic real $I \\times I$ matrix (its entries are nonnegative and each of its rows and columns sums to $1$), and let $c_{ij} \\in \\mathbb{R}$ for $i, j \\in I$. Then there is a permutation $\\sigma$ of $I$ with $\\sum_i c_{i \\sigma(i)} \\le \\sum_i \\sum_j c_{ij} P_{ij}$. That is, the linear functional $M \\mapsto \\sum_{i, j} c_{ij} M_{ij}$ takes a value at some permutation\u2026","title":"Linear functionals on doubly stochastic matrices are bounded below at a permutation"},{"id":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/exists-perm-sum-mul-eigenvalues-le-trace","kind":"node","lean":"MetaMathlibExt.exists_perm_sum_mul_eigenvalues_le_trace_mul","parent":4798,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/index.html#exists-perm-sum-mul-eigenvalues-le-trace","scope":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality","status":"fully_proved","summary":"Let $A$ and $B$ be real symmetric matrices with rows and columns indexed by a finite set $I$, and let $(\\alpha_i)_{i \\in I}$ and $(\\beta_i)_{i \\in I}$ be their eigenvalues, listed with multiplicity. Then there is a permutation $\\sigma$ of $I$ with $\\sum_i \\alpha_i \\beta_{\\sigma(i)} \\le \\operatorname{tr}(AB)$.","title":"A pairing of eigenvalues bounds the trace of a product from below"},{"id":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/exists-perm-trace-mul-le-sum-mul-eigenva","kind":"node","lean":"MetaMathlibExt.exists_perm_trace_mul_le_sum_mul_eigenvalues","parent":4798,"result":true,"route":"../roadmap/LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality/index.html#exists-perm-trace-mul-le-sum-mul-eigenva","scope":"LinearAlgebra/positive-definite-matrices/von-neumann-s-trace-inequality","status":"fully_proved","summary":"Let $A$ and $B$ be real symmetric matrices with rows and columns indexed by a finite set $I$, and let $(\\alpha_i)_{i \\in I}$ and $(\\beta_i)_{i \\in I}$ be their eigenvalues, listed with multiplicity. Then there is a permutation $\\sigma$ of $I$ with $\\operatorname{tr}(AB) \\le \\sum_i \\alpha_i \\beta_{\\sigma(i)}$.","title":"A pairing of eigenvalues bounds the trace of a product from above"},{"data_url":"scopes/LinearAlgebra/linear-maps-and-duality.json","id":"LinearAlgebra/linear-maps-and-duality","kind":"scope","parent":4518,"result":false,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/index.html","scope":"LinearAlgebra/linear-maps-and-duality","status":"fully_proved","summary":"Vector spaces and modules, linear maps and their invariants such as trace, dual spaces, dual bases and perfect pairings, and exactness of sequences of linear maps.","title":"Linear maps and duality"},{"id":"LinearAlgebra/linear-maps-and-duality/affine-evaluation","kind":"node","lean":"LinearEquiv.affineEvaluation","parent":4805,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/index.html#affine-evaluation","scope":"LinearAlgebra/linear-maps-and-duality","status":"fully_proved","summary":"Evaluation of the affine function `x \u21a6 slope * x + intercept` at two distinct points.","title":"Evaluation of an affine function at two points"},{"id":"LinearAlgebra/linear-maps-and-duality/exact-and-surjective-iff-forall-lcomp","kind":"node","lean":"LinearMap.exact_and_surjective_iff_forall_lcomp","parent":4805,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/index.html#exact-and-surjective-iff-forall-lcomp","scope":"LinearAlgebra/linear-maps-and-duality","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $M$, $N$, $P$ be $R$-modules, and let $\\varphi : M \\to N$ and $\\psi : N \\to P$ be $R$-linear maps. Then the image of $\\varphi$ equals the kernel of $\\psi$ and $\\psi$ is surjective if and only if, for every $R$-module $D$ in the same universe as $N$ and $P$, precomposition with $\\psi$, $g \\mapsto g \\circ \\psi$, is an injective map\u2026","title":"Hom detects exact sequences ending in a surjection"},{"data_url":"scopes/LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings.json","id":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings","kind":"scope","navigation":"module","parent":4805,"result":false,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/index.html","scope":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings","status":"fully_proved","title":"Dual bases of perfect pairings"},{"id":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/dual-basis","kind":"node","lean":"LinearMap.IsPerfPair.dualBasis","parent":4808,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/index.html#dual-basis","scope":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings","status":"fully_proved","summary":"Let $A$ be a commutative ring, let $M$ and $N$ be $A$-modules, and let $B : M \\times N \\to A$ be a perfect pairing, that is, an $A$-bilinear map such that $x \\mapsto B(x, \\cdot)$ is a linear isomorphism $M \\to N^*$ and $y \\mapsto B(\\cdot, y)$ is a linear isomorphism $N \\to M^*$. Let $b = (b_i)_{i \\in \\iota}$ be a basis of $N$ indexed by a finite type $\\iota$, and let $(b_i^*)_{i \\in \\iota}$ be its dual basis of\u2026","title":"Dual basis with respect to a perfect pairing"},{"id":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/dual-basis-apply","kind":"node","lean":"LinearMap.IsPerfPair.dualBasis_apply","parent":4808,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/index.html#dual-basis-apply","scope":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings","status":"fully_proved","summary":"Let $A$ be a commutative ring, let $M$ and $N$ be $A$-modules, let $B : M \\times N \\to A$ be a perfect pairing, and let $b = (b_i)_{i \\in \\iota}$ be a basis of $N$ indexed by a finite type $\\iota$. Write $(b_i^\\vee)_{i \\in \\iota}$ for the dual basis of $b$ with respect to $B$, a basis of $M$. Then for all $i, j \\in \\iota$, the value $B(b_i^\\vee, b_j)$ equals $1$ if $j = i$ and $0$ otherwise.","title":"Evaluation of the dual basis"},{"id":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/eq-dual-basis-of","kind":"node","lean":"LinearMap.IsPerfPair.eq_dualBasis_of","parent":4808,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/index.html#eq-dual-basis-of","scope":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings","status":"fully_proved","summary":"Let $A$ be a commutative ring, let $M$ and $N$ be $A$-modules, let $B : M \\times N \\to A$ be a perfect pairing, and let $b = (b_i)_{i \\in \\iota}$ be a basis of $N$ indexed by a finite type $\\iota$. If $b' = (b'_i)_{i \\in \\iota}$ is a basis of $M$ such that for all $i, j \\in \\iota$ the value $B(b'_i, b_j)$ equals $1$ if $j = i$ and $0$ otherwise, then $b'$ is the dual basis of $b$ with respect to $B$.","title":"Uniqueness of the dual basis"},{"id":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/exists-dual-basis","kind":"node","lean":"LinearMap.IsPerfPair.exists_dualBasis","parent":4808,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/index.html#exists-dual-basis","scope":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings","status":"fully_proved","summary":"Let $A$ be a commutative ring, let $M$ and $N$ be $A$-modules, let $B : M \\times N \\to A$ be a perfect pairing, and let $b = (b_i)_{i \\in \\iota}$ be a basis of $N$ indexed by a finite type $\\iota$. Then there is a basis $(b'_i)_{i \\in \\iota}$ of $M$ such that for all $i, j \\in \\iota$ the value $B(b'_i, b_j)$ equals $1$ if $j = i$ and $0$ otherwise.","title":"Existence of a dual basis for a perfect pairing"},{"id":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/exists-unique-dual-basis","kind":"node","lean":"LinearMap.IsPerfPair.existsUnique_dualBasis","parent":4808,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings/index.html#exists-unique-dual-basis","scope":"LinearAlgebra/linear-maps-and-duality/dual-bases-of-perfect-pairings","status":"fully_proved","summary":"Let $A$ be a commutative ring, let $M$ and $N$ be $A$-modules, let $B : M \\times N \\to A$ be a perfect pairing, and let $b = (b_i)_{i \\in \\iota}$ be a basis of $N$ indexed by a finite type $\\iota$. Then there is exactly one basis $(b'_i)_{i \\in \\iota}$ of $M$ such that for all $i, j \\in \\iota$ the value $B(b'_i, b_j)$ equals $1$ if $j = i$ and $0$ otherwise.","title":"Existence and uniqueness of the dual basis for a perfect pairing"},{"data_url":"scopes/LinearAlgebra/linear-maps-and-duality/partial-trace.json","id":"LinearAlgebra/linear-maps-and-duality/partial-trace","kind":"scope","navigation":"module","parent":4805,"result":false,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","title":"Partial trace"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/partial-trace-left","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.partialTraceLeft","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#partial-trace-left","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace over the first factor: `\u2211 i, A (i, j) (i, j')`.","title":"Partial trace over the first factor"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/partial-trace-right","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.partialTraceRight","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#partial-trace-right","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace over the second factor: `\u2211 j, A (i, j) (i', j)`.","title":"Partial trace over the second factor"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/trace-partial-trace-left-of-trace-eq-one","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.trace_partialTraceLeft_of_trace_eq_one","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#trace-partial-trace-left-of-trace-eq-one","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace of a density matrix has trace `1`.","title":"Left partial trace of a density matrix has trace one"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/trace-partial-trace-right-of-trace-eq-on","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.trace_partialTraceRight_of_trace_eq_one","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#trace-partial-trace-right-of-trace-eq-on","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace of a density matrix has trace `1`.","title":"Right partial trace of a density matrix has trace one"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/partial-trace-left-vec-mul-vec-star","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.partialTraceLeft_vecMulVec_star","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#partial-trace-left-vec-mul-vec-star","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace of a rank-one positive semidefinite matrix is the sum of the rank-one positive semidefinite matrices of its slices.","title":"Left partial trace of a rank-one matrix"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/partial-trace-right-vec-mul-vec-star","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.partialTraceRight_vecMulVec_star","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#partial-trace-right-vec-mul-vec-star","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace of a rank-one positive semidefinite matrix is the sum of the rank-one positive semidefinite matrices of its slices.","title":"Right partial trace of a rank-one matrix"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/pos-semidef-partial-trace-left","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.posSemidef_partialTraceLeft","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#pos-semidef-partial-trace-left","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace over the first factor preserves positive semidefiniteness.","title":"Left partial trace preserves positive semidefiniteness"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/pos-semidef-partial-trace-right","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.posSemidef_partialTraceRight","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#pos-semidef-partial-trace-right","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace over the second factor preserves positive semidefiniteness.","title":"Right partial trace preserves positive semidefiniteness"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/trace-partial-trace-left-mul","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.trace_partialTraceLeft_mul","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#trace-partial-trace-left-mul","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace over the first factor is adjoint to amplification by the identity, in the trace pairing.","title":"Left partial trace is adjoint to identity amplification"},{"id":"LinearAlgebra/linear-maps-and-duality/partial-trace/trace-partial-trace-right-mul","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.PartialTrace.trace_partialTraceRight_mul","parent":4814,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/partial-trace/index.html#trace-partial-trace-right-mul","scope":"LinearAlgebra/linear-maps-and-duality/partial-trace","status":"fully_proved","summary":"The partial trace over the second factor is adjoint to amplification by the identity, in the trace pairing.","title":"Right partial trace is adjoint to identity amplification"},{"data_url":"scopes/LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps.json","id":"LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps","kind":"scope","navigation":"module","parent":4805,"result":false,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps/index.html","scope":"LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps","status":"fully_proved","title":"Similarity invariants of linear maps"},{"id":"LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps/charpoly-of-finrank-eq-two","kind":"node","lean":"LinearMap.charpoly_of_finrank_eq_two","parent":4825,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps/index.html#charpoly-of-finrank-eq-two","scope":"LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps","status":"fully_proved","summary":"The characteristic polynomial of an endomorphism of a free rank-two module is `X\u00b2 - tr f \u00b7 X + det f`, the familiar quadratic formula.","title":"Characteristic polynomial of a rank-two endomorphism"},{"id":"LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps/trace-eq-and-det-eq-of-semiconj","kind":"node","lean":"LinearMap.trace_eq_and_det_eq_of_semiconj","parent":4825,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps/index.html#trace-eq-and-det-eq-of-semiconj","scope":"LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps","status":"fully_proved","summary":"Trace and determinant are preserved under semiconjugacy: if `e` intertwines `f` and `g`, then `g` is the conjugate of `f` by `e`, so both invariants agree.","title":"Trace and determinant are invariant under semiconjugacy"},{"id":"LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps/finrank-ker-dual-map-eq-finrank-ker","kind":"node","lean":"LinearMap.finrank_ker_dualMap_eq_finrank_ker","parent":4825,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps/index.html#finrank-ker-dual-map-eq-finrank-ker","scope":"LinearAlgebra/linear-maps-and-duality/similarity-invariants-of-linear-maps","status":"fully_proved","summary":"The kernel of the dual map has the same finite dimension as the kernel: both are complementary to the range, by rank-nullity and the dual-annihilator dimension formula.","title":"Kernel of the dual map has the same dimension as the kernel"},{"data_url":"scopes/LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces.json","id":"LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces","kind":"scope","navigation":"module","parent":4805,"result":false,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces/index.html","scope":"LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces","status":"fully_proved","title":"Trace on invariant subspaces"},{"id":"LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces/trace-eq-trace-restrict-add-trace-map-q","kind":"node","lean":"LinearMap.trace_eq_trace_restrict_add_trace_mapQ","parent":4829,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces/index.html#trace-eq-trace-restrict-add-trace-map-q","scope":"LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces","status":"fully_proved","summary":"Let $V$ be a finite-dimensional vector space over a field $K$, let $e$ be a linear endomorphism of $V$, and let $W$ be a subspace of $V$ with $e(W) \\subseteq W$. Then the trace of $e$ equals the trace of the restriction $e|_W$ of $e$ to $W$ plus the trace of the endomorphism $\\bar e$ of the quotient $V/W$ induced by $e$: $\\operatorname{tr}(e) = \\operatorname{tr}(e|_W) + \\operatorname{tr}(\\bar e)$.","title":"Trace is additive over an invariant subspace"},{"id":"LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces/trace-eq-trace-restrict-ker-add-of-surje","kind":"node","lean":"LinearMap.trace_eq_trace_restrict_ker_add_of_surjective","parent":4829,"result":true,"route":"../roadmap/LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces/index.html#trace-eq-trace-restrict-ker-add-of-surje","scope":"LinearAlgebra/linear-maps-and-duality/trace-on-invariant-subspaces","status":"fully_proved","summary":"Let $V$ be a finite-dimensional vector space over a field $K$, let $W$ be a vector space over $K$, and let $q : V \\to W$ be a surjective linear map. If $f$ is a linear endomorphism of $V$ and $g$ a linear endomorphism of $W$ with $q \\circ f = g \\circ q$, then $f$ maps $\\ker q$ into itself, and the trace of $f$ equals the trace of the restriction $f|_{\\ker q}$ of $f$ to $\\ker q$ plus the trace of $g$.","title":"Trace splits along a surjective intertwining map"},{"data_url":"scopes/LinearAlgebra/determinants-and-matrix-identities.json","id":"LinearAlgebra/determinants-and-matrix-identities","kind":"scope","parent":4518,"result":false,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/index.html","scope":"LinearAlgebra/determinants-and-matrix-identities","status":"can_prove","summary":"Determinant expansions and identities, determinants of special products and exponentials, derivatives of determinants, and algebraic identities between matrices.","title":"Determinants and matrix identities"},{"id":"LinearAlgebra/determinants-and-matrix-identities/carroll-identity","kind":"node","lean":"MetaMathlibExt.CarrollIdentity.carroll_identity","parent":4832,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/index.html#carroll-identity","scope":"LinearAlgebra/determinants-and-matrix-identities","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $n$ be a natural number, and let $C$ be an $(n + 2) \\times (n + 2)$ matrix over $R$. For indices $a$ and $b$, write $C^{a}_{b}$ for the $(n + 1) \\times (n + 1)$ matrix obtained from $C$ by deleting row $a$ and column $b$. Let $i \\ne j$ be distinct indices and let $C^{ij}_{ij}$ be the $n \\times n$ matrix obtained from $C$ by deleting rows $i$ and $j$ and columns $i$ and $j$. Then","title":"Carroll (Desnanot\u2013Jacobi) identity"},{"id":"LinearAlgebra/determinants-and-matrix-identities/jacobi-det-derivative","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.JacobiDetDerivativeWanted.jacobi_det_derivative","parent":4832,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/index.html#jacobi-det-derivative","scope":"LinearAlgebra/determinants-and-matrix-identities","status":"can_prove","summary":"Let $n$ be a finite type, and let $A$ be a curve $\\mathbb{R} \\to \\mathbb{R}^{n \\times n}$ of square real matrices with rows and columns indexed by $n$. Suppose $A$ is differentiable at $t \\in \\mathbb{R}$. Then the derivative of $s \\mapsto \\det A(s)$ at $t$ is $\\operatorname{tr}\\bigl(\\operatorname{adj}(A(t))\\, A'(t)\\bigr)$, where $\\operatorname{adj}$ denotes the adjugate matrix and $A'(t)$ the derivative of $A$ at\u2026","title":"Jacobi's formula"},{"id":"LinearAlgebra/determinants-and-matrix-identities/is-symm-prod-iff","kind":"node","lean":"Matrix.isSymm_prod_iff","parent":4832,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/index.html#is-symm-prod-iff","scope":"LinearAlgebra/determinants-and-matrix-identities","status":"fully_proved","summary":"A matrix on a product is symmetric exactly when its restrictions to each row and column are symmetric and both diagonals of every ordered rectangle agree in opposite directions.","title":"Symmetry criterion for matrices indexed by a product"},{"id":"LinearAlgebra/determinants-and-matrix-identities/symplectic-matrix-det","kind":"node","lean":"MathlibExt.LinearAlgebra.SymplecticDetWanted.LinearAlgebra.symplectic_matrix_det","parent":4832,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/index.html#symplectic-matrix-det","scope":"LinearAlgebra/determinants-and-matrix-identities","status":"can_prove","summary":"Let $R$ be a commutative ring, $l$ a finite set, $I$ the identity matrix indexed by $l$, and $J = \\begin{pmatrix} 0 & -I \\\\ I & 0 \\end{pmatrix}$ the matrix over $R$ indexed by the disjoint union $l \\sqcup l$. Every square matrix $A$ over $R$ indexed by $l \\sqcup l$ with $A J A^{\\mathsf{T}} = J$ satisfies $\\det A = 1$.","title":"Symplectic matrices have determinant one"},{"data_url":"scopes/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix.json","id":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","kind":"scope","navigation":"module","parent":4832,"result":false,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/index.html","scope":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","status":"fully_proved","title":"Decomposition of the bit-swap permutation matrix"},{"id":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/bit-singlet","kind":"node","lean":"Matrix.bitSinglet","parent":4837,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/index.html#bit-singlet","scope":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","status":"fully_proved","summary":"The unnormalized antisymmetric two-bit vector, in order `00, 01, 10, 11`.","title":"Antisymmetric two-bit singlet vector"},{"id":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/bit-swap-eq-one-sub-vec-mul-vec","kind":"node","lean":"Matrix.bitSwap_eq_one_sub_vecMulVec","parent":4837,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/index.html#bit-swap-eq-one-sub-vec-mul-vec","scope":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","status":"fully_proved","summary":"The one-bit swap matrix is the identity minus the singlet outer product.","title":"One-bit swap matrix as identity minus singlet outer product"},{"id":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/bit-swap-piece","kind":"node","lean":"Matrix.bitSwapPiece","parent":4837,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/index.html#bit-swap-piece","scope":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","status":"fully_proved","summary":"The part of an `n`-bit swap with exactly `i` antisymmetric factors.","title":"Piece of an n-bit swap with i antisymmetric factors"},{"id":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/sum-bit-swap-piece","kind":"node","lean":"Matrix.sum_bitSwapPiece","parent":4837,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/index.html#sum-bit-swap-piece","scope":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","status":"fully_proved","summary":"The `n + 1` grouped pieces sum to the permutation matrix swapping the two bit strings.","title":"Bit-swap pieces sum to the swap permutation matrix"},{"id":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/bit-swap-piece-eq-zero-of-lt","kind":"node","lean":"Matrix.bitSwapPiece_eq_zero_of_lt","parent":4837,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/index.html#bit-swap-piece-eq-zero-of-lt","scope":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","status":"fully_proved","summary":"There is no piece containing more antisymmetric factors than bit positions.","title":"Bit-swap pieces vanish beyond the number of bits"},{"id":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/abs-bit-swap-piece-le-choose","kind":"node","lean":"Matrix.abs_bitSwapPiece_le_choose","parent":4837,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/index.html#abs-bit-swap-piece-le-choose","scope":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","status":"fully_proved","summary":"Each entry of the `i`th piece is bounded by the number of `i`-subsets of the bit positions.","title":"Binomial entry bound for bit-swap pieces"},{"id":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/abs-bit-swap-piece-le-two-pow","kind":"node","lean":"Matrix.abs_bitSwapPiece_le_two_pow","parent":4837,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix/index.html#abs-bit-swap-piece-le-two-pow","scope":"LinearAlgebra/determinants-and-matrix-identities/decomposition-of-the-bit-swap-permutation-matrix","status":"fully_proved","summary":"The uniform entry bound used in the source: every piece has absolute value at most `2 ^ n`.","title":"Uniform power-of-two entry bound for bit-swap pieces"},{"data_url":"scopes/LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant.json","id":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant","kind":"scope","navigation":"module","parent":4832,"result":false,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/index.html","scope":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant","status":"fully_proved","title":"First-order expansion of the determinant"},{"id":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/det-add-smul-vec-mul-vec","kind":"node","lean":"Matrix.det_add_smul_vecMulVec","parent":4845,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/index.html#det-add-smul-vec-mul-vec","scope":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant","status":"fully_proved","summary":"Let $K$ be a field, let $B$ be a square matrix over $K$ indexed by a finite type $n$, and let $u, v \\in K^n$. Then for every $t \\in K$, $\\det(B + t\\,uv^{\\mathsf T}) = \\det B + t\\,(\\det(B + uv^{\\mathsf T}) - \\det B)$.","title":"The determinant is affine along a rank-one direction"},{"id":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/det-linear","kind":"node","lean":"Matrix.detLinear","parent":4845,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/index.html#det-linear","scope":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $A$ and $B$ be square matrices over $R$ indexed by a finite type $n$. The first-order term of the determinant of $B$ in the direction $A$ is $L_B(A) = \\sum_{i \\in n} \\det B^{(i)}$, where $B^{(i)}$ is the matrix $B$ with its $i$-th row replaced by the $i$-th row of $A$.","title":"First-order term of the determinant"},{"id":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/coeff-one-det-add-x-smul","kind":"node","lean":"Matrix.coeff_one_det_add_X_smul","parent":4845,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient of "],["math","\\(X\\)"],["text"," in "],["math","\\(\\det(B + XA)\\)"]],"source":"The coefficient of $X$ in $\\det(B + XA)$"},"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/index.html#coeff-one-det-add-x-smul","scope":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $A$ and $B$ be square matrices over $R$ indexed by a finite type $n$. Regard $B + XA$ as a matrix over the polynomial ring $R[X]$. Then the coefficient of $X$ in the polynomial $\\det(B + XA)$ equals the first-order term of the determinant $L_B(A) = \\sum_{i \\in n} \\det B^{(i)}$, where $B^{(i)}$ is the matrix $B$ with its $i$-th row replaced by the $i$-th row of $A$.","title":"The coefficient of $X$ in $\\det(B + XA)$"},{"id":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/det-linear-add","kind":"node","lean":"Matrix.detLinear_add","parent":4845,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/index.html#det-linear-add","scope":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $A$, $B$, $C$ be square matrices over $R$ indexed by a finite type $n$. Write $L_B(A) = \\sum_{i \\in n} \\det B^{(i)}_A$, where $B^{(i)}_A$ is the matrix $B$ with its $i$-th row replaced by the $i$-th row of $A$. Then $L_B(A + C) = L_B(A) + L_B(C)$.","title":"Additivity of the first-order term of the determinant"},{"id":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/det-linear-smul","kind":"node","lean":"Matrix.detLinear_smul","parent":4845,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant/index.html#det-linear-smul","scope":"LinearAlgebra/determinants-and-matrix-identities/first-order-expansion-of-the-determinant","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $a \\in R$, and let $A$ and $B$ be square matrices over $R$ indexed by a finite type $n$. Write $L_B(A) = \\sum_{i \\in n} \\det B^{(i)}_A$, where $B^{(i)}_A$ is the matrix $B$ with its $i$-th row replaced by the $i$-th row of $A$. Then $L_B(aA) = a \\, L_B(A)$.","title":"Homogeneity of the first-order term of the determinant"},{"data_url":"scopes/LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices.json","id":"LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices","kind":"scope","navigation":"module","parent":4832,"result":false,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices/index.html","scope":"LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices","status":"fully_proved","title":"Generalized factorial matrices"},{"id":"LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices/pan-vandermonde-general","kind":"node","lean":"MetaMathlibExt.pan_vandermonde_general","parent":4851,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices/index.html#pan-vandermonde-general","scope":"LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices","status":"fully_proved","summary":"Let $K$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $p$ be a natural number, let $x_0, \\dots, x_{p-1} \\in K$ be pairwise distinct, and let $\\alpha, \\beta, \\gamma \\in \\mathbb{R}$. For $y \\in K$, $a \\in \\mathbb{R}$ and $n \\in \\mathbb{N}$ write $(y \\mid a)_n = \\prod_{j=0}^{n-1} (y - ja)$. Let numbers $S(a, b, c; n, k) \\in K$, for $a, b, c \\in \\mathbb{R}$ and $n, k \\in \\mathbb{N}$, satisfy $S(a, b, c; n, k) = 0$ whenever\u2026","title":"Generalized factorial matrices: decomposition, determinant and inverse"},{"id":"LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices/pan-vandermonde","kind":"node","lean":"MetaMathlibExt.pan_vandermonde","parent":4851,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices/index.html#pan-vandermonde","scope":"LinearAlgebra/determinants-and-matrix-identities/generalized-factorial-matrices","status":"fully_proved","summary":"Let $K$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $p$ be a positive integer, let $x_0, \\dots, x_{p-1} \\in K$ be pairwise distinct, and let $\\alpha, \\beta, \\gamma \\in \\mathbb{R}$. For $y \\in K$, $a \\in \\mathbb{R}$ and $n \\in \\mathbb{N}$ write $(y \\mid a)_n = \\prod_{j=0}^{n-1} (y - ja)$. Let numbers $S(a, b, c; n, k) \\in K$, for $a, b, c \\in \\mathbb{R}$ and $n, k \\in \\mathbb{N}$, satisfy $S(a, b, c; n, k) = 0$ whenever\u2026","title":"Pan's theorem on generalized factorial matrices"},{"data_url":"scopes/LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma.json","id":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma","kind":"scope","navigation":"module","parent":4832,"result":false,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/index.html","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma","status":"fully_proved","title":"Matrix determinant lemma"},{"id":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/det-add-vec-mul-vec","kind":"node","lean":"Matrix.det_add_vecMulVec","parent":4854,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/index.html#det-add-vec-mul-vec","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma","status":"fully_proved","summary":"The matrix determinant lemma, without any invertibility hypothesis: the determinant of a rank-one update of `A` is `det A` plus the adjugate pairing `v \u2b1d\u1d65 (adjugate A *\u1d65 u)`. This is the general version left as a TODO in Mathlib's `Matrix.det_add_replicateCol_mul_replicateRow`.","title":"Matrix determinant lemma without invertibility"},{"id":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/det-add-replicate-col-mul-replicate-row","kind":"node","lean":"Matrix.det_add_replicateCol_mul_replicateRow_general","parent":4854,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/index.html#det-add-replicate-col-mul-replicate-row","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma","status":"fully_proved","summary":"The matrix determinant lemma in the `replicateCol` / `replicateRow` form stated in Mathlib's TODO for `Matrix.det_add_replicateCol_mul_replicateRow`. The `_general` suffix avoids clashing with Mathlib's invertible version of the same name.","title":"Matrix determinant lemma in column-times-row form"},{"id":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/is-unit-add-vec-mul-vec-iff","kind":"node","lean":"Matrix.isUnit_add_vecMulVec_iff","parent":4854,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/index.html#is-unit-add-vec-mul-vec-iff","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma","status":"fully_proved","summary":"A rank-one update of `A` is a unit exactly when the adjugate correction term is a unit; no invertibility hypothesis on `A` is needed.","title":"Invertibility criterion for a rank-one update"},{"id":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/det-add-vec-mul-vec-of-is-unit-det","kind":"node","lean":"Matrix.det_add_vecMulVec_of_isUnit_det","parent":4854,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma/index.html#det-add-vec-mul-vec-of-is-unit-det","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-determinant-lemma","status":"fully_proved","summary":"The matrix determinant lemma in multiplicative form, when `A.det` is a unit: this recovers Mathlib's `Matrix.det_add_replicateCol_mul_replicateRow` from the general version.","title":"Multiplicative matrix determinant lemma for invertible matrices"},{"data_url":"scopes/LinearAlgebra/determinants-and-matrix-identities/matrix-exponential.json","id":"LinearAlgebra/determinants-and-matrix-identities/matrix-exponential","kind":"scope","navigation":"module","parent":4832,"result":false,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-exponential/index.html","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-exponential","status":"fully_proved","title":"Matrix exponential"},{"id":"LinearAlgebra/determinants-and-matrix-identities/matrix-exponential/exp-mul-eq-mul-exp-of-mul-eq","kind":"node","lean":"Matrix.exp_mul_eq_mul_exp_of_mul_eq","parent":4859,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-exponential/index.html#exp-mul-eq-mul-exp-of-mul-eq","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-exponential","status":"fully_proved","summary":"If `A * X = X * B` for a rectangular matrix `X`, then `exp A * X = X * exp B`.","title":"Intertwining passes to the exponential"},{"id":"LinearAlgebra/determinants-and-matrix-identities/matrix-exponential/matrix-det-exp-eq-exp-trace","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.MatrixExpDetTraceWanted.matrix_det_exp_eq_exp_trace","parent":4859,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-exponential/index.html#matrix-det-exp-eq-exp-trace","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-exponential","status":"fully_proved","summary":"For every complex square matrix $A$ indexed by a finite set, $\\det(e^A) = e^{\\operatorname{tr} A}$, where $e^A$ is the matrix exponential. This identity is sometimes called Jacobi's formula.","title":"Determinant of the matrix exponential"},{"id":"LinearAlgebra/determinants-and-matrix-identities/matrix-exponential/matrix-exp-eq-sum-of-is-nilpotent","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.MatrixExpDetTraceWanted.matrix_exp_eq_sum_of_isNilpotent","parent":4859,"result":true,"route":"../roadmap/LinearAlgebra/determinants-and-matrix-identities/matrix-exponential/index.html#matrix-exp-eq-sum-of-is-nilpotent","scope":"LinearAlgebra/determinants-and-matrix-identities/matrix-exponential","status":"fully_proved","summary":"If $A$ is a nilpotent complex square matrix indexed by a finite set, then there is a natural number $m$ such that $e^A = \\sum_{k=0}^{m-1} \\frac{1}{k!} A^k$, where $e^A$ is the matrix exponential.","title":"Exponential of a nilpotent matrix"},{"data_url":"scopes/LinearAlgebra/bilinear-and-quadratic-forms.json","id":"LinearAlgebra/bilinear-and-quadratic-forms","kind":"scope","parent":4518,"result":false,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/index.html","scope":"LinearAlgebra/bilinear-and-quadratic-forms","status":"fully_proved","summary":"Bilinear and quadratic forms on vector spaces and modules: rank, orthogonal sums, inertia and congruence, and lattices with their duals and neighbours.","title":"Bilinear and quadratic forms"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/sylvester-law-of-inertia","kind":"node","lean":"MetaMathlibExt.sylvester_law_of_inertia","parent":4863,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/index.html#sylvester-law-of-inertia","scope":"LinearAlgebra/bilinear-and-quadratic-forms","status":"fully_proved","summary":"Let $A$ and $B$ be real symmetric square matrices with rows and columns indexed by a finite set $n$, and suppose $B = P^{\\mathsf T} A P$ for some invertible real matrix $P$ indexed by $n$. Then $A$ and $B$ have the same number of positive eigenvalues, the same number of negative eigenvalues and the same number of zero eigenvalues, each counted with multiplicity.","title":"Sylvester's law of inertia"},{"data_url":"scopes/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices.json","id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","kind":"scope","navigation":"module","parent":4863,"result":false,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","title":"Dual lattices"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/is-localized-module-subtype-of-span-eq-t","kind":"node","lean":"Submodule.isLocalizedModule_subtype_of_span_eq_top","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#is-localized-module-subtype-of-span-eq-t","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be a commutative ring with fraction field $K$, let $V$ be a $K$-vector space, viewed as an $A$-module through $A \\to K$, and let $M$ be an $A$-submodule of $V$ that spans $V$ over $K$. Then $V$, together with the inclusion $M \\hookrightarrow V$, is the localization of the $A$-module $M$ at the non-zero-divisors of $A$.","title":"A spanning submodule localizes to the whole space"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-to-dual-surjective","kind":"node","lean":"LinearMap.BilinForm.dualSubmoduleToDual_surjective","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-to-dual-surjective","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $V$ be a finite-dimensional $K$-vector space, viewed as an $A$-module through $A \\subseteq K$, let $B$ be a nondegenerate bilinear form on $V$, and let $M$ be an $A$-submodule of $V$ that spans $V$ over $K$. Then the map from $M^\\vee = \\{x \\in V : B(x, y) \\in A \\text{ for all } y \\in M\\}$ to $\\operatorname{Hom}_A(M, A)$ sending $x$ to the functional\u2026","title":"Surjectivity of the dual pairing"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-to-dual-bijective","kind":"node","lean":"LinearMap.BilinForm.dualSubmoduleToDual_bijective","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-to-dual-bijective","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $V$ be a finite-dimensional $K$-vector space, viewed as an $A$-module through $A \\subseteq K$, let $B$ be a nondegenerate bilinear form on $V$, and let $M$ be an $A$-submodule of $V$ that spans $V$ over $K$. Then the map from $M^\\vee = \\{x \\in V : B(x, y) \\in A \\text{ for all } y \\in M\\}$ to $\\operatorname{Hom}_A(M, A)$ sending $x$ to the functional\u2026","title":"Bijectivity of the dual pairing"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-fg","kind":"node","lean":"LinearMap.BilinForm.dualSubmodule_fg","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-fg","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be a Noetherian integral domain with fraction field $K$, let $V$ be a $K$-vector space, viewed as an $A$-module through $A \\subseteq K$, and let $B$ be a nondegenerate bilinear form on $V$. Let $M$ be a finitely generated $A$-submodule of $V$ that spans $V$ over $K$. Then the dual submodule $M^\\vee = \\{x \\in V : B(x, y) \\in A \\text{ for all } y \\in M\\}$ is a finitely generated $A$-submodule of $V$.","title":"The dual of a lattice is finitely generated"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-span-eq-top","kind":"node","lean":"LinearMap.BilinForm.dualSubmodule_span_eq_top","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-span-eq-top","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be a commutative ring with fraction field $K$, let $V$ be a $K$-vector space, viewed as an $A$-module through $A \\to K$, and let $B$ be a bilinear form on $V$. If $M$ is a finitely generated $A$-submodule of $V$, then its dual submodule $M^\\vee = \\{x \\in V : B(x, y) \\in A \\text{ for all } y \\in M\\}$ spans $V$ over $K$.","title":"The dual of a finitely generated submodule spans"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-is-lattice","kind":"node","lean":"LinearMap.BilinForm.dualSubmodule_isLattice","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-is-lattice","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be a Noetherian integral domain with fraction field $K$, let $V$ be a $K$-vector space, viewed as an $A$-module through $A \\subseteq K$, and let $B$ be a nondegenerate bilinear form on $V$. If $M$ is an $A$-lattice in $V$, that is, a finitely generated $A$-submodule that spans $V$ over $K$, then so is its dual submodule $M^\\vee = \\{x \\in V : B(x, y) \\in A \\text{ for all } y \\in M\\}$.","title":"The dual of a lattice is a lattice"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-is-lattice-and-dual-submo","kind":"node","lean":"LinearMap.BilinForm.dualSubmodule_isLattice_and_dualSubmoduleToDual_bijective","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-is-lattice-and-dual-submo","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be a Noetherian integral domain with fraction field $K$, let $V$ be a finite-dimensional $K$-vector space, viewed as an $A$-module through $A \\subseteq K$, and let $B$ be a nondegenerate bilinear form on $V$. Let $M$ be an $A$-lattice in $V$, that is, a finitely generated $A$-submodule that spans $V$ over $K$, and let $M^\\vee = \\{x \\in V : B(x, y) \\in A \\text{ for all } y \\in M\\}$. Then $M^\\vee$ is an\u2026","title":"The dual lattice and the dual module"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-linear-equiv","kind":"node","lean":"LinearMap.BilinForm.dualSubmoduleLinearEquiv","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-linear-equiv","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $V$ be a finite-dimensional $K$-vector space, viewed as an $A$-module through $A \\subseteq K$, let $B$ be a nondegenerate bilinear form on $V$, and let $M$ be an $A$-submodule of $V$ that spans $V$ over $K$. Write $M^\\vee = \\{x \\in V : B(x, y) \\in A \\text{ for all } y \\in M\\}$. This is the isomorphism of $A$-modules $M^\\vee \\cong \\operatorname{Hom}_A(M, A)$\u2026","title":"The dual submodule is the dual module"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-flip-dual-submodule-of-is","kind":"node","lean":"LinearMap.BilinForm.dualSubmodule_flip_dualSubmodule_of_isLattice","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-flip-dual-submodule-of-is","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $V$ be a finite-dimensional $K$-vector space, viewed as an $A$-module through $A \\subseteq K$, and let $B$ be a nondegenerate bilinear form on $V$. Let $M$ be an $A$-lattice in $V$, that is, a finitely generated $A$-submodule that spans $V$ over $K$, and suppose that $M$ is reflexive as an $A$-module. Let\u2026","title":"Double dual of a reflexive lattice"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-dual-submodule-of-is-latt","kind":"node","lean":"LinearMap.BilinForm.dualSubmodule_dualSubmodule_of_isLattice","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-dual-submodule-of-is-latt","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be a Dedekind domain with fraction field $K$, let $V$ be a finite-dimensional $K$-vector space, viewed as an $A$-module through $A \\subseteq K$, and let $B$ be a nondegenerate symmetric bilinear form on $V$. For an $A$-submodule $N$ of $V$ write $N^\\vee = \\{x \\in V : B(x, y) \\in A \\text{ for all } y \\in N\\}$. If $M$ is an $A$-lattice in $V$, that is, a finitely generated $A$-submodule that spans $V$ over\u2026","title":"Double dual of a lattice over a Dedekind domain"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/dual-submodule-span-localization","kind":"node","lean":"LinearMap.BilinForm.dualSubmodule_span_localization","parent":4865,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices/index.html#dual-submodule-span-localization","scope":"LinearAlgebra/bilinear-and-quadratic-forms/dual-lattices","status":"fully_proved","summary":"Let $A$ be an integral domain with fraction field $K$, let $V$ be a $K$-vector space that is also an $A$-module compatibly with $A \\to K$, and let $B$ be a $K$-bilinear form on $V$. For a subring $R \\subseteq K$ and an $R$-submodule $N$ of $V$, write $N^\\vee_R = \\{x \\in V : B(x, y) \\in R \\text{ for all } y \\in N\\}$ for the dual of $N$ over $R$. Let $S$ be a submonoid of $A$ consisting of non-zero-divisors, and let\u2026","title":"Dual submodules commute with localization"},{"data_url":"scopes/LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors.json","id":"LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors","kind":"scope","navigation":"module","parent":4863,"result":false,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors/index.html","scope":"LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors","status":"fully_proved","title":"Kneser p-neighbors"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors/are-pneighbors","kind":"node","lean":"Submodule.ArePNeighbors","parent":4877,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-neighbors of submodules"]],"source":"$p$-neighbors of submodules"},"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors/index.html#are-pneighbors","scope":"LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors","status":"fully_proved","summary":"Let $R$ be a commutative ring, $M$ an $R$-module and $p$ a natural number. Two submodules $L_1$ and $L_2$ of $M$ are $p$-neighbors when $p$ is prime and the intersection $L_1 \\cap L_2$ has index $p$ in each of them, that is, the quotients $L_1 / (L_1 \\cap L_2)$ and $L_2 / (L_1 \\cap L_2)$ both have exactly $p$ elements.","title":"$p$-neighbors of submodules"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors/are-pneighbors-comm","kind":"node","lean":"Submodule.arePNeighbors_comm","parent":4877,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-neighbor relation is symmetric"]],"source":"The $p$-neighbor relation is symmetric"},"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors/index.html#are-pneighbors-comm","scope":"LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors","status":"fully_proved","summary":"For every natural number $p$ and all submodules $L_1$, $L_2$ of a module $M$ over a commutative ring $R$, $L_1$ and $L_2$ are $p$-neighbors if and only if $L_2$ and $L_1$ are $p$-neighbors.","title":"The $p$-neighbor relation is symmetric"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors/not-are-pneighbors-one","kind":"node","lean":"Submodule.not_arePNeighbors_one","parent":4877,"reading_title":{"format":"inline-parts-v1","parts":[["text","No submodules are "],["math","\\(1\\)"],["text","-neighbors"]],"source":"No submodules are $1$-neighbors"},"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors/index.html#not-are-pneighbors-one","scope":"LinearAlgebra/bilinear-and-quadratic-forms/kneser-p-neighbors","status":"fully_proved","summary":"For any submodules $L_1$ and $L_2$ of a module $M$ over a commutative ring $R$, $L_1$ and $L_2$ are not $1$-neighbors, since $1$ is not prime.","title":"No submodules are $1$-neighbors"},{"data_url":"scopes/LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms.json","id":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms","kind":"scope","navigation":"module","parent":4863,"result":false,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/index.html","scope":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms","status":"fully_proved","title":"Orthogonal product of bilinear forms"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/prod","kind":"node","lean":"LinearMap.BilinForm.prod","parent":4881,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/index.html#prod","scope":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms","status":"fully_proved","summary":"Let $B_1$ and $B_2$ be bilinear forms on modules $M_1$ and $M_2$ over a commutative semiring $R$. Their orthogonal product $B_1 \\oplus B_2$ is the bilinear form on $M_1 \\times M_2$ given by $(B_1 \\oplus B_2)((x_1, x_2), (y_1, y_2)) = B_1(x_1, y_1) + B_2(x_2, y_2)$.","title":"Orthogonal product of bilinear forms"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/prod-apply","kind":"node","lean":"LinearMap.BilinForm.prod_apply","parent":4881,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/index.html#prod-apply","scope":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms","status":"fully_proved","summary":"Let $B_1$ and $B_2$ be bilinear forms on modules $M_1$ and $M_2$ over a commutative semiring $R$, with orthogonal product $B_1 \\oplus B_2$ on $M_1 \\times M_2$. For all $x = (x_1, x_2)$ and $y = (y_1, y_2)$ in $M_1 \\times M_2$, $(B_1 \\oplus B_2)(x, y) = B_1(x_1, y_1) + B_2(x_2, y_2)$.","title":"Evaluation of an orthogonal product"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/nondegenerate-prod-iff","kind":"node","lean":"LinearMap.BilinForm.nondegenerate_prod_iff","parent":4881,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/index.html#nondegenerate-prod-iff","scope":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms","status":"fully_proved","summary":"Let $B_1$ and $B_2$ be bilinear forms on modules $M_1$ and $M_2$ over a commutative semiring $R$. Their orthogonal product $B_1 \\oplus B_2$ on $M_1 \\times M_2$ is nondegenerate (both left- and right-separating) if and only if $B_1$ and $B_2$ are both nondegenerate.","title":"Nondegeneracy of an orthogonal product"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/dual-submodule-prod","kind":"node","lean":"LinearMap.BilinForm.dualSubmodule_prod","parent":4881,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms/index.html#dual-submodule-prod","scope":"LinearAlgebra/bilinear-and-quadratic-forms/orthogonal-product-of-bilinear-forms","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $S$ be a field that is an $R$-algebra, and let $M_1$, $M_2$ be vector spaces over $S$ that are also $R$-modules compatibly with $R \\to S$. For a bilinear form $B$ over $S$ on such a space $M$ and an $R$-submodule $N$ of $M$, the dual of $N$ is $N^\\vee = \\{x \\in M : B(x, y) \\in R \\text{ for all } y \\in N\\}$, where $R$ stands for its image in $S$. Let $B_1$, $B_2$ be bilinear forms\u2026","title":"Dual submodule of an orthogonal product"},{"data_url":"scopes/LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form.json","id":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form","kind":"scope","navigation":"module","parent":4863,"result":false,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/index.html","scope":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form","status":"fully_proved","title":"Rank of a bilinear form"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/rank-to-matrix-eq-rank-to-matrix","kind":"node","lean":"LinearMap.BilinForm.rank_toMatrix_eq_rank_toMatrix","parent":4886,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/index.html#rank-to-matrix-eq-rank-to-matrix","scope":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form","status":"fully_proved","summary":"Let $V$ be a vector space over a field $K$, let $B$ be a bilinear form on $V$, and let $b = (b_i)_{i \\in n_1}$ and $c = (c_j)_{j \\in n_2}$ be bases of $V$ indexed by finite types $n_1$ and $n_2$. Then the Gram matrices $(B(b_i, b_j))_{i, j \\in n_1}$ and $(B(c_i, c_j))_{i, j \\in n_2}$ have the same rank.","title":"The Gram matrices of a bilinear form in two bases have the same rank"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/bilin-form-rank","kind":"node","lean":"MathlibExt.LinearAlgebra.BilinearFormRankWanted.bilinFormRank","parent":4886,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/index.html#bilin-form-rank","scope":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form","status":"fully_proved","summary":"Let $V$ be a vector space over a field $K$, let $B$ be a bilinear form on $V$, and let $b = (b_i)_{i \\in n}$ be a basis of $V$ indexed by a finite type $n$. The rank of $B$ with respect to $b$ is the rank of the Gram matrix $(B(b_i, b_j))_{i, j \\in n}$ of $B$ in $b$.","title":"Rank of a bilinear form"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/bilin-form-rank-eq-matrix-rank","kind":"node","lean":"MathlibExt.LinearAlgebra.BilinearFormRankWanted.bilinFormRank_eq_matrix_rank","parent":4886,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/index.html#bilin-form-rank-eq-matrix-rank","scope":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form","status":"fully_proved","summary":"Let $V$ be a vector space over a field $K$, let $B$ be a bilinear form on $V$, and let $b = (b_i)_{i \\in n}$ be a basis of $V$ indexed by a finite type $n$. Then the rank of $B$ with respect to $b$ equals the rank of the Gram matrix $(B(b_i, b_j))_{i, j \\in n}$. This holds by definition.","title":"Rank of a bilinear form via its Gram matrix"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/bilin-form-rank-basis-independent","kind":"node","lean":"MathlibExt.LinearAlgebra.BilinearFormRankWanted.bilinFormRank_basis_independent","parent":4886,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form/index.html#bilin-form-rank-basis-independent","scope":"LinearAlgebra/bilinear-and-quadratic-forms/rank-of-a-bilinear-form","status":"fully_proved","summary":"Let $V$ be a vector space over a field $K$, let $B$ be a bilinear form on $V$, and let $b = (b_i)_{i \\in n_1}$ and $c = (c_j)_{j \\in n_2}$ be bases of $V$ indexed by finite types $n_1$ and $n_2$. Then the rank of $B$ with respect to $b$ equals the rank of the Gram matrix $(B(c_i, c_j))_{i, j \\in n_2}$ of $B$ in $c$.","title":"Basis independence of the rank of a bilinear form"},{"data_url":"scopes/LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence.json","id":"LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence","kind":"scope","navigation":"module","parent":4863,"result":false,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence/index.html","scope":"LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence","status":"fully_proved","title":"Simultaneous diagonalization by congruence"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence/exists-is-unit-transpose-mul-mul-eq-one","kind":"node","lean":"Matrix.PosDef.exists_isUnit_transpose_mul_mul_eq_one","parent":4891,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence/index.html#exists-is-unit-transpose-mul-mul-eq-one","scope":"LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence","status":"fully_proved","summary":"Let $P$ be a real square matrix with rows and columns indexed by a finite set, and suppose $P$ is positive definite: $P$ is symmetric and $x^{\\mathsf{T}} P x > 0$ for every nonzero real vector $x$. Then there is an invertible real matrix $Q$ of the same size such that $Q^{\\mathsf{T}} P Q = I$, where $I$ is the identity matrix.","title":"Positive-definite matrices are congruent to the identity"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence/simultaneous-diagonalization-pos-def","kind":"node","lean":"MathlibExt.LinearAlgebra.Matrix.SimultaneousDiagonalizationWanted.simultaneous_diagonalization_posDef","parent":4891,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence/index.html#simultaneous-diagonalization-pos-def","scope":"LinearAlgebra/bilinear-and-quadratic-forms/simultaneous-diagonalization-by-congruence","status":"fully_proved","summary":"Let $S$ and $P$ be real square matrices with rows and columns indexed by a finite set $n$, where $S$ is symmetric and $P$ is positive definite. Then there is an invertible real matrix $C$, indexed by $n$, such that $C^{\\mathsf T} S C$ is diagonal and $C^{\\mathsf T} P C$ is the identity matrix.","title":"Simultaneous diagonalization by congruence"},{"data_url":"scopes/LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms.json","id":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms","kind":"scope","navigation":"module","parent":4863,"result":false,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/index.html","scope":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms","status":"fully_proved","title":"Systems of quadratic forms"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/system-orthogonal-sum","kind":"node","lean":"QuadraticForm.systemOrthogonalSum","parent":4894,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/index.html#system-orthogonal-sum","scope":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $M$ and $N$ be $R$-modules, and let $(Q_i)_{i \\in \\iota}$ and $(Q'_i)_{i \\in \\iota}$ be families of quadratic forms on $M$ and on $N$ indexed by the same type $\\iota$. Their pointwise orthogonal sum is the family $(Q_i \\oplus Q'_i)_{i \\in \\iota}$ of quadratic forms on $M \\times N$, where $Q_i \\oplus Q'_i$ is the orthogonal sum given by\u2026","title":"Orthogonal sum of systems of quadratic forms"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/system-orthogonal-sum-apply","kind":"node","lean":"QuadraticForm.systemOrthogonalSum_apply","parent":4894,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/index.html#system-orthogonal-sum-apply","scope":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $M$ and $N$ be $R$-modules, and let $(Q_i)_{i \\in \\iota}$ and $(Q'_i)_{i \\in \\iota}$ be families of quadratic forms on $M$ and on $N$ indexed by the same type $\\iota$. For every index $i$ and every $(x, y) \\in M \\times N$, the $i$-th form of their pointwise orthogonal sum takes the value $Q_i(x) + Q'_i(y)$ at $(x, y)$.","title":"Evaluating an orthogonal sum of systems of quadratic forms"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/system-orthogonal-sum-inl","kind":"node","lean":"QuadraticForm.systemOrthogonalSum_inl","parent":4894,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/index.html#system-orthogonal-sum-inl","scope":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $M$ and $N$ be $R$-modules, and let $(Q_i)_{i \\in \\iota}$ and $(Q'_i)_{i \\in \\iota}$ be families of quadratic forms on $M$ and on $N$ indexed by the same type $\\iota$. For every index $i$ and every $x \\in M$, the $i$-th form of their pointwise orthogonal sum takes the value $Q_i(x)$ at $(x, 0) \\in M \\times N$.","title":"Orthogonal sum of systems on the left summand"},{"id":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/system-orthogonal-sum-inr","kind":"node","lean":"QuadraticForm.systemOrthogonalSum_inr","parent":4894,"result":true,"route":"../roadmap/LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms/index.html#system-orthogonal-sum-inr","scope":"LinearAlgebra/bilinear-and-quadratic-forms/systems-of-quadratic-forms","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $M$ and $N$ be $R$-modules, and let $(Q_i)_{i \\in \\iota}$ and $(Q'_i)_{i \\in \\iota}$ be families of quadratic forms on $M$ and on $N$ indexed by the same type $\\iota$. For every index $i$ and every $y \\in N$, the $i$-th form of their pointwise orthogonal sum takes the value $Q'_i(y)$ at $(0, y) \\in M \\times N$.","title":"Orthogonal sum of systems on the right summand"},{"area":"Algebra & Number Theory","data_url":"chapters/RepresentationTheory.json","id":"RepresentationTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/RepresentationTheory/index.html","scope":"RepresentationTheory","status":"can_prove","summary":"Representation theory studies symmetry by realizing groups and algebras as transformations of linear spaces.","title":"Representation Theory"},{"data_url":"scopes/RepresentationTheory/finite-group-representations.json","id":"RepresentationTheory/finite-group-representations","kind":"scope","parent":4899,"result":false,"route":"../roadmap/RepresentationTheory/finite-group-representations/index.html","scope":"RepresentationTheory/finite-group-representations","status":"fully_proved","summary":"Ordinary representation theory of finite groups over algebraically closed fields of characteristic zero: characters, irreducibility, Schur's lemma, multiplicities and the Wedderburn decomposition of the group algebra.","title":"Representations of finite groups"},{"id":"RepresentationTheory/finite-group-representations/irred-dim-dvd-card","kind":"node","lean":"CharacterTheory.irred_dim_dvd_card","parent":4900,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/index.html#irred-dim-dvd-card","scope":"RepresentationTheory/finite-group-representations","status":"fully_proved","summary":"If `\u03c1` is an irreducible representation of a finite group `G` over an algebraically closed field `k` of characteristic zero, then `dim_k \u03c1 \u2223 |G|`. In particular, over `\u2102`, every irreducible degree divides the order of the group.","title":"Dimension of an irreducible representation divides the group order"},{"id":"RepresentationTheory/finite-group-representations/maschke-decomposition","kind":"node","lean":"MathlibExt.GroupTheory.MaschkeDecomposition.maschke_decomposition","parent":4900,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/index.html#maschke-decomposition","scope":"RepresentationTheory/finite-group-representations","status":"fully_proved","summary":"Every finite-dimensional complex representation $\\sigma$ of a finite group is equivalent to the direct sum of a finite family of irreducible subrepresentations of $\\sigma$.","title":"Maschke's theorem"},{"data_url":"scopes/RepresentationTheory/finite-group-representations/characters-of-representations.json","id":"RepresentationTheory/finite-group-representations/characters-of-representations","kind":"scope","navigation":"module","parent":4900,"result":false,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","title":"Characters of representations"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/char-inv-eq-conj","kind":"node","lean":"CharacterTheory.char_inv_eq_conj","parent":4903,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#char-inv-eq-conj","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, with character $\\chi_\\rho$. For every $g \\in G$, $\\chi_\\rho(g^{-1}) = \\overline{\\chi_\\rho(g)}$.","title":"Character values at inverses are complex conjugates"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/char-dual-eq-conj","kind":"node","lean":"CharacterTheory.char_dual_eq_conj","parent":4903,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#char-dual-eq-conj","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and let $\\rho^*$ be its dual representation. For every $g \\in G$, the character of $\\rho^*$ at $g$ is the complex conjugate of the character of $\\rho$ at $g$: $\\chi_{\\rho^*}(g) = \\overline{\\chi_\\rho(g)}$.","title":"Character of the dual representation"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/class-inner-product","kind":"node","lean":"CharacterTheory.classInnerProduct","parent":4903,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#class-inner-product","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"Let $G$ be a finite group and $k$ a field in which $|G|$ is invertible. For functions $f_1, f_2 \\colon G \\to k$, their class inner product is $\\langle f_1, f_2 \\rangle = |G|^{-1} \\sum_{g \\in G} f_1(g) f_2(g^{-1})$.","title":"Class inner product"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/irreducible-iff-class-inner-product-self","kind":"node","lean":"CharacterTheory.irreducible_iff_classInnerProduct_self_eq_one","parent":4903,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#irreducible-iff-class-inner-product-self","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"A finite-dimensional representation `\u03c3` of a finite group over an algebraically closed field of characteristic zero is irreducible if and only if the self-pairing of its character is `1`.","title":"Character criterion for irreducibility"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/has-eigenvalue-pow-order-of-eq-one","kind":"node","lean":"CharacterTheory.hasEigenvalue_pow_orderOf_eq_one","parent":4903,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eigenvalues of "],["math","\\(\\rho(g)\\)"],["text"," are roots of unity of order dividing that of "],["math","\\(g\\)"]],"source":"Eigenvalues of $\\rho(g)$ are roots of unity of order dividing that of $g$"},"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#has-eigenvalue-pow-order-of-eq-one","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, let $g \\in G$, and let $\\operatorname{ord}(g)$ be the order of $g$. Every eigenvalue $\\mu$ of $\\rho(g)$ satisfies $\\mu^{\\operatorname{ord}(g)} = 1$.","title":"Eigenvalues of $\\rho(g)$ are roots of unity of order dividing that of $g$"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/has-eigenvalue-pow-card-eq-one","kind":"node","lean":"CharacterTheory.hasEigenvalue_pow_card_eq_one","parent":4903,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eigenvalues of "],["math","\\(\\rho(g)\\)"],["text"," are "],["math","\\(|G|\\)"],["text","-th roots of unity"]],"source":"Eigenvalues of $\\rho(g)$ are $|G|$-th roots of unity"},"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#has-eigenvalue-pow-card-eq-one","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and let $g \\in G$. Every eigenvalue $\\mu$ of $\\rho(g)$ satisfies $\\mu^{|G|} = 1$.","title":"Eigenvalues of $\\rho(g)$ are $|G|$-th roots of unity"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/char-charpoly-roots-pow-eq-one","kind":"node","lean":"CharacterTheory.char_charpoly_roots_pow_eq_one","parent":4903,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characteristic roots of "],["math","\\(\\rho(g)\\)"],["text"," are "],["math","\\(|G|\\)"],["text","-th roots of unity"]],"source":"Characteristic roots of $\\rho(g)$ are $|G|$-th roots of unity"},"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#char-charpoly-roots-pow-eq-one","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and let $g \\in G$. Every root $\\mu$ of the characteristic polynomial of the matrix of $\\rho(g)$ in a fixed basis of $V$ satisfies $\\mu^{|G|} = 1$.","title":"Characteristic roots of $\\rho(g)$ are $|G|$-th roots of unity"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/character-self-pairing-eq-sum-sq","kind":"node","lean":"CharacterTheory.character_selfPairing_eq_sum_sq","parent":4903,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#character-self-pairing-eq-sum-sq","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"If `\u03c7` is a combination of the characters of pairwise non-equivalent irreducible representations with coefficients `n i`, then the self-pairing of `\u03c7` is the sum of the squares of the coefficients. In particular, for a single irreducible `\u03c1`, taking `\u03c7 = \u03c7_\u03c1` gives self-pairing `1`, and for the regular character the formula recovers `\u2211 (dim \u03c1_i) ^ 2 = |G|` once the regular character is expanded in irreducibles.","title":"Parseval / sum-of-squares for characters"},{"id":"RepresentationTheory/finite-group-representations/characters-of-representations/wedderburn-sum-sq-eq-card","kind":"node","lean":"CharacterTheory.wedderburn_sum_sq_eq_card","parent":4903,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/characters-of-representations/index.html#wedderburn-sum-sq-eq-card","scope":"RepresentationTheory/finite-group-representations/characters-of-representations","status":"fully_proved","summary":"If `|G|` is nonzero in an algebraically closed field, the group algebra of a finite group is a product of matrix algebras with nonzero block sizes whose squares sum to `|G|`.","title":"Sum of squares of the Wedderburn block sizes"},{"data_url":"scopes/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups.json","id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","kind":"scope","navigation":"module","parent":4900,"result":false,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","title":"Representations of finite abelian groups"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/is-irreducible-of-simple","kind":"node","lean":"FDRep.isIrreducible_of_simple","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#is-irreducible-of-simple","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be a field and let $G$ be a monoid. If a finite-dimensional representation $V$ of $G$ over $k$ is simple as an object of the category of finite-dimensional representations of $G$ over $k$, then $V$ is irreducible as a representation: $V \\ne 0$, and the only subrepresentations of $V$ are $0$ and $V$.","title":"Simple finite-dimensional representations of a monoid are irreducible"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/finrank-eq-one-of-is-mul-commutative","kind":"node","lean":"FDRep.finrank_eq_one_of_isMulCommutative","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#finrank-eq-one-of-is-mul-commutative","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be an algebraically closed field and let $G$ be a commutative monoid. Every finite-dimensional representation $V$ of $G$ over $k$ that is simple as an object of the category of finite-dimensional representations of $G$ over $k$ has dimension $1$ over $k$.","title":"Simple finite-dimensional representations of commutative monoids are one-dimensional"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/rho-eq-character-smul-of-simple","kind":"node","lean":"FDRep.rho_eq_character_smul_of_simple","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#rho-eq-character-smul-of-simple","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be an algebraically closed field, $G$ a monoid with commutative multiplication, and $V$ a simple finite-dimensional representation of $G$ over $k$ (a simple object of the category of finite-dimensional representations), with action $\\rho$ and character $\\chi_V(g) = \\operatorname{tr} \\rho(g)$. Then for every $g \\in G$ the map $\\rho(g)$ is scalar multiplication by $\\chi_V(g)$, that is,\u2026","title":"Simple representations of commutative monoids act by scalars"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/character-mul-of-simple","kind":"node","lean":"FDRep.character_mul_of_simple","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#character-mul-of-simple","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be an algebraically closed field, let $G$ be a commutative monoid, and let $V$ be a finite-dimensional representation of $G$ over $k$ that is simple as an object of the category of finite-dimensional representations of $G$ over $k$. Let $\\chi_V : G \\to k$ be the character of $V$, where $\\chi_V(g)$ is the trace of the action of $g$ on $V$. Then $\\chi_V(gh) = \\chi_V(g)\\,\\chi_V(h)$ for all $g, h \\in G$.","title":"Characters of simple representations of commutative monoids are multiplicative"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/character-monoid-hom","kind":"node","lean":"FDRep.characterMonoidHom","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#character-monoid-hom","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be an algebraically closed field, let $G$ be a commutative monoid, and let $V$ be a finite-dimensional representation of $G$ over $k$ that is simple as an object of the category of finite-dimensional representations of $G$ over $k$. The character $\\chi_V : G \\to k$ of $V$, where $\\chi_V(g)$ is the trace of the action of $g$ on $V$, satisfies $\\chi_V(1) = 1$ and $\\chi_V(gh) = \\chi_V(g)\\,\\chi_V(h)$ for all\u2026","title":"The character of a simple representation as a monoid homomorphism"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/coe-character-monoid-hom","kind":"node","lean":"FDRep.coe_characterMonoidHom","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#coe-character-monoid-hom","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be an algebraically closed field, let $G$ be a commutative monoid, and let $V$ be a finite-dimensional representation of $G$ over $k$ that is simple as an object of the category of finite-dimensional representations of $G$ over $k$. The character of $V$, regarded as a monoid homomorphism from $G$ to $k$, has the character $\\chi_V : G \\to k$ of $V$ as its underlying function, where $\\chi_V(g)$ is the trace\u2026","title":"The character homomorphism has the character as its underlying function"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/of-monoid-hom","kind":"node","lean":"FDRep.ofMonoidHom","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#of-monoid-hom","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be a field, let $G$ be a monoid, and let $\\varphi : G \\to k$ be a monoid homomorphism. This is the finite-dimensional representation $k_\\varphi$ of $G$ on the one-dimensional space $k$ in which each $g \\in G$ acts by multiplication by $\\varphi(g)$, that is, by the linear map $x \\mapsto \\varphi(g)\\,x$.","title":"The one-dimensional representation of a homomorphism"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/of-monoid-hom-2","kind":"node","lean":"FDRep.ofMonoidHom_\u03c1","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#of-monoid-hom-2","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be a field, let $G$ be a monoid, let $\\varphi : G \\to k$ be a monoid homomorphism, and let $k_\\varphi$ be the one-dimensional representation of $G$ on $k$ defined by $\\varphi$. For every $g \\in G$, the action of $g$ on $k_\\varphi$ is $\\varphi(g)$ times the identity map of $k$.","title":"The action in the one-dimensional representation of a homomorphism"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/of-monoid-hom-finrank","kind":"node","lean":"FDRep.ofMonoidHom_finrank","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#of-monoid-hom-finrank","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be a field, let $G$ be a monoid, let $\\varphi : G \\to k$ be a monoid homomorphism, and let $k_\\varphi$ be the representation of $G$ on $k$ in which each $g \\in G$ acts by multiplication by $\\varphi(g)$. Then $k_\\varphi$ has dimension $1$ over $k$.","title":"The representation of a homomorphism is one-dimensional"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/of-monoid-hom-character","kind":"node","lean":"FDRep.ofMonoidHom_character","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#of-monoid-hom-character","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be a field, let $G$ be a monoid, let $\\varphi : G \\to k$ be a monoid homomorphism, and let $k_\\varphi$ be the one-dimensional representation of $G$ on $k$ in which each $g \\in G$ acts by multiplication by $\\varphi(g)$. The character of $k_\\varphi$ is $\\varphi$: the trace of the action of each $g \\in G$ on $k_\\varphi$ is $\\varphi(g)$.","title":"The character of the representation of a homomorphism is the homomorphism"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/of-monoid-hom-simple","kind":"node","lean":"FDRep.ofMonoidHom_simple","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#of-monoid-hom-simple","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $k$ be a field, let $G$ be a monoid, let $\\varphi : G \\to k$ be a monoid homomorphism, and let $k_\\varphi$ be the one-dimensional representation of $G$ on $k$ in which each $g \\in G$ acts by multiplication by $\\varphi(g)$. Then $k_\\varphi$ is a simple object of the category of finite-dimensional representations of $G$ over $k$.","title":"The representation of a homomorphism is simple"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/abelian-simple-char-is-hom","kind":"node","lean":"MathlibExt.GroupTheory.AbelianRepresentationWanted.abelian_simple_char_is_hom","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#abelian-simple-char-is-hom","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group and let $V$ be a simple finite-dimensional complex representation of $G$. Then there is a monoid homomorphism $\\varphi : G \\to \\mathbb{C}$ such that the character of $V$ equals $\\varphi$.","title":"Characters of simple representations of a finite abelian group are homomorphisms"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/abelian-hom-is-simple-char","kind":"node","lean":"MathlibExt.GroupTheory.AbelianRepresentationWanted.abelian_hom_is_simple_char","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#abelian-hom-is-simple-char","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group and let $\\varphi : G \\to \\mathbb{C}$ be a monoid homomorphism. Then there is a simple finite-dimensional complex representation $V$ of $G$ whose character equals $\\varphi$.","title":"Every homomorphism of a finite abelian group to the complex numbers is a simple character"},{"id":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/abelian-simple-iso-of-char-eq","kind":"node","lean":"MathlibExt.GroupTheory.AbelianRepresentationWanted.abelian_simple_iso_of_char_eq","parent":4913,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups/index.html#abelian-simple-iso-of-char-eq","scope":"RepresentationTheory/finite-group-representations/representations-of-finite-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group and let $V$ and $W$ be simple finite-dimensional complex representations of $G$, with characters $\\chi_V$ and $\\chi_W$. If $\\chi_V = \\chi_W$, then $V$ and $W$ are isomorphic.","title":"Characters determine simple representations of a finite abelian group"},{"data_url":"scopes/RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities.json","id":"RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities","kind":"scope","navigation":"module","parent":4900,"result":false,"route":"../roadmap/RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities/index.html","scope":"RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities","status":"fully_proved","title":"Schur's lemma and multiplicities"},{"id":"RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities/finrank-intertwining-map-eq-one-or-zero","kind":"node","lean":"CharacterTheory.finrank_intertwiningMap_eq_one_or_zero","parent":4928,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities/index.html#finrank-intertwining-map-eq-one-or-zero","scope":"RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities","status":"fully_proved","summary":"Let $G$ be a finite group, $k$ an algebraically closed field, and $\\rho$, $\\sigma$ irreducible representations of $G$ on finite-dimensional $k$-vector spaces. Then the space $\\operatorname{Hom}_G(\\rho, \\sigma)$ of intertwining maps from $\\rho$ to $\\sigma$ has dimension $1$ over $k$ if $\\sigma$ and $\\rho$ are equivalent, and dimension $0$ otherwise.","title":"Schur's lemma, dimension form"},{"id":"RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities/multiplicity-eq-finrank-intertwining-map","kind":"node","lean":"CharacterTheory.multiplicity_eq_finrank_intertwiningMap","parent":4928,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities/index.html#multiplicity-eq-finrank-intertwining-map","scope":"RepresentationTheory/finite-group-representations/schur-s-lemma-and-multiplicities","status":"fully_proved","summary":"Let $G$ be a finite group, $k$ an algebraically closed field, and $(\\rho_i)_{i \\in I}$ a family, indexed by a finite set $I$, of irreducible representations of $G$ on finite-dimensional $k$-vector spaces that are pairwise non-equivalent: $\\rho_i$ and $\\rho_j$ are not equivalent whenever $i \\ne j$. Let $\\sigma$ be a representation of $G$ on a finite-dimensional $k$-vector space, and let $(d_i)_{i \\in I}$ be natural\u2026","title":"Multiplicity equals the dimension of the intertwining space"},{"data_url":"scopes/RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra.json","id":"RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra","kind":"scope","navigation":"module","parent":4900,"result":false,"route":"../roadmap/RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra/index.html","scope":"RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra","status":"fully_proved","title":"Wedderburn decomposition of the group algebra"},{"id":"RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra/wedderburn-alg-hom","kind":"node","lean":"CharacterTheory.wedderburnAlgHom","parent":4931,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra/index.html#wedderburn-alg-hom","scope":"RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra","status":"fully_proved","summary":"Let $k$ be an algebraically closed field, $G$ a finite group whose order is nonzero in $k$, and $e \\colon k[G] \\to \\prod_{j < n} M_{d_j}(k)$ an isomorphism of $k$-algebras from the group algebra onto a product of full matrix algebras. For $i < n$, this is the $k$-algebra homomorphism $k[G] \\to \\operatorname{End}_k(k^{d_i})$ that sends $a \\in k[G]$ to the linear map $v \\mapsto e(a)_i \\, v$, where $e(a)_i$ is the\u2026","title":"Algebra map of a Wedderburn block"},{"id":"RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra/wedderburn-irrep","kind":"node","lean":"CharacterTheory.wedderburnIrrep","parent":4931,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra/index.html#wedderburn-irrep","scope":"RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra","status":"fully_proved","summary":"Let $k$ be an algebraically closed field, $G$ a finite group whose order is nonzero in $k$, and $e \\colon k[G] \\to \\prod_{j < n} M_{d_j}(k)$ an isomorphism of $k$-algebras from the group algebra $k[G]$ onto a product of full matrix algebras $M_{d_j}(k)$ of $d_j \\times d_j$ matrices over $k$. For $i < n$, this is the representation of $G$ on $k^{d_i}$ in which $g \\in G$ acts by $v \\mapsto e(g)_i \\, v$, where $g$ is\u2026","title":"Representation of a Wedderburn block"},{"id":"RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra/wedderburn-irrep-is-irreducible","kind":"node","lean":"CharacterTheory.wedderburnIrrep_isIrreducible","parent":4931,"result":true,"route":"../roadmap/RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra/index.html#wedderburn-irrep-is-irreducible","scope":"RepresentationTheory/finite-group-representations/wedderburn-decomposition-of-the-group-algebra","status":"fully_proved","summary":"Let $k$ be an algebraically closed field, $G$ a finite group whose order is nonzero in $k$, and $e \\colon k[G] \\to \\prod_{j < n} M_{d_j}(k)$ an isomorphism of $k$-algebras from the group algebra $k[G]$ onto a product of full matrix algebras $M_{d_j}(k)$ of $d_j \\times d_j$ matrices over $k$. For every $i < n$ with $d_i \\ne 0$, the representation of $G$ on $k^{d_i}$ in which $g \\in G$ acts by\u2026","title":"Wedderburn blocks are irreducible"},{"data_url":"scopes/RepresentationTheory/group-cohomology.json","id":"RepresentationTheory/group-cohomology","kind":"scope","parent":4899,"result":false,"route":"../roadmap/RepresentationTheory/group-cohomology/index.html","scope":"RepresentationTheory/group-cohomology","status":"fully_proved","summary":"Homology, cohomology and Tate cohomology of groups with coefficients in modules, with their functoriality, additivity, periodicity and the Herbrand quotient.","title":"Group cohomology"},{"data_url":"scopes/RepresentationTheory/group-cohomology/group-homology-and-cohomology.json","id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","kind":"scope","navigation":"module","parent":4935,"result":false,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","title":"Group homology and cohomology"},{"id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology/nonempty-linear-equiv-of-iso-res-mul-equ","kind":"node","lean":"groupCohomology.nonempty_linearEquiv_of_iso_res_mulEquiv","parent":4936,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html#nonempty-linear-equiv-of-iso-res-mul-equ","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","summary":"Group cohomology is invariant under group isomorphism.","title":"Invariance of group cohomology under group isomorphism"},{"id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology/biprod-iso","kind":"node","lean":"groupHomology.biprodIso","parent":4936,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html#biprod-iso","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","summary":"Let $k$ be a commutative ring, $G$ a group, $A$ and $B$ representations of $G$ over $k$, and $n \\in \\mathbb{N}$, and write $H_n(G, M)$ for the $n$th group homology of a representation $M$. This is the isomorphism $H_n(G, A \\oplus B) \\cong H_n(G, A) \\oplus H_n(G, B)$ obtained by applying the $n$th group homology functor, which is additive, to the biproduct $A \\oplus B$.","title":"Group homology of a direct sum"},{"id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology/biprod-iso-hom","kind":"node","lean":"groupHomology.biprodIso_hom","parent":4936,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html#biprod-iso-hom","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","summary":"Let $k$ be a commutative ring, $G$ a group, $A$ and $B$ representations of $G$ over $k$, and $n \\in \\mathbb{N}$, and write $H_n(G, M)$ for the $n$th group homology of a representation $M$. The forward map of the isomorphism $H_n(G, A \\oplus B) \\cong H_n(G, A) \\oplus H_n(G, B)$ is the map into the biproduct whose components are the maps $H_n(G, A \\oplus B) \\to H_n(G, A)$ and $H_n(G, A \\oplus B) \\to H_n(G, B)$\u2026","title":"Forward map of the direct-sum isomorphism in group homology"},{"id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology/biprod-iso-inv","kind":"node","lean":"groupHomology.biprodIso_inv","parent":4936,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html#biprod-iso-inv","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","summary":"Let $k$ be a commutative ring, $G$ a group, $A$ and $B$ representations of $G$ over $k$, and $n \\in \\mathbb{N}$, and write $H_n(G, M)$ for the $n$th group homology of a representation $M$. The inverse map of the isomorphism $H_n(G, A \\oplus B) \\cong H_n(G, A) \\oplus H_n(G, B)$ is the map out of the biproduct whose components are the maps $H_n(G, A) \\to H_n(G, A \\oplus B)$ and $H_n(G, B) \\to H_n(G, A \\oplus B)$\u2026","title":"Inverse map of the direct-sum isomorphism in group homology"},{"id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology/biprod-iso-hom-fst","kind":"node","lean":"groupHomology.biprodIso_hom_fst","parent":4936,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html#biprod-iso-hom-fst","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","summary":"Let $k$ be a commutative ring, $G$ a group, $A$ and $B$ representations of $G$ over $k$, and $n \\in \\mathbb{N}$, and write $H_n(G, M)$ for the $n$th group homology of a representation $M$. The isomorphism $H_n(G, A \\oplus B) \\cong H_n(G, A) \\oplus H_n(G, B)$ followed by the projection onto $H_n(G, A)$ equals the map $H_n(G, A \\oplus B) \\to H_n(G, A)$ induced by the projection $A \\oplus B \\to A$.","title":"Direct-sum isomorphism in group homology and the first projection"},{"id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology/biprod-iso-hom-snd","kind":"node","lean":"groupHomology.biprodIso_hom_snd","parent":4936,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html#biprod-iso-hom-snd","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","summary":"Let $k$ be a commutative ring, $G$ a group, $A$ and $B$ representations of $G$ over $k$, and $n \\in \\mathbb{N}$, and write $H_n(G, M)$ for the $n$th group homology of a representation $M$. The isomorphism $H_n(G, A \\oplus B) \\cong H_n(G, A) \\oplus H_n(G, B)$ followed by the projection onto $H_n(G, B)$ equals the map $H_n(G, A \\oplus B) \\to H_n(G, B)$ induced by the projection $A \\oplus B \\to B$.","title":"Direct-sum isomorphism in group homology and the second projection"},{"id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology/biprod-iso-inv-inl","kind":"node","lean":"groupHomology.biprodIso_inv_inl","parent":4936,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html#biprod-iso-inv-inl","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","summary":"Let $k$ be a commutative ring, $G$ a group, $A$ and $B$ representations of $G$ over $k$, and $n \\in \\mathbb{N}$, and write $H_n(G, M)$ for the $n$th group homology of a representation $M$. The inclusion of $H_n(G, A)$ into $H_n(G, A) \\oplus H_n(G, B)$ followed by the inverse of the isomorphism $H_n(G, A \\oplus B) \\cong H_n(G, A) \\oplus H_n(G, B)$ equals the map $H_n(G, A) \\to H_n(G, A \\oplus B)$ induced by the\u2026","title":"Direct-sum isomorphism in group homology and the first inclusion"},{"id":"RepresentationTheory/group-cohomology/group-homology-and-cohomology/biprod-iso-inv-inr","kind":"node","lean":"groupHomology.biprodIso_inv_inr","parent":4936,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/group-homology-and-cohomology/index.html#biprod-iso-inv-inr","scope":"RepresentationTheory/group-cohomology/group-homology-and-cohomology","status":"fully_proved","summary":"Let $k$ be a commutative ring, $G$ a group, $A$ and $B$ representations of $G$ over $k$, and $n \\in \\mathbb{N}$, and write $H_n(G, M)$ for the $n$th group homology of a representation $M$. The inclusion of $H_n(G, B)$ into $H_n(G, A) \\oplus H_n(G, B)$ followed by the inverse of the isomorphism $H_n(G, A \\oplus B) \\cong H_n(G, A) \\oplus H_n(G, B)$ equals the map $H_n(G, B) \\to H_n(G, A \\oplus B)$ induced by the\u2026","title":"Direct-sum isomorphism in group homology and the second inclusion"},{"data_url":"scopes/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient.json","id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","kind":"scope","navigation":"module","parent":4935,"result":false,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","title":"Tate cohomology and the Herbrand quotient"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/nonempty-periodicity-iso","kind":"node","lean":"TateCohomology.nonempty_periodicityIso","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#nonempty-periodicity-iso","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $R$ be a commutative ring, $G$ a finite cyclic group and $A$ a representation of $G$ over $R$. Then for every integer $n$, the Tate cohomology groups $\\hat{H}^n(G, A)$ and $\\hat{H}^{n+2}(G, A)$ are isomorphic as $R$-modules.","title":"Periodicity of Tate cohomology for finite cyclic groups"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/tate-cohomology-card","kind":"node","lean":"TateCohomology.tateCohomologyCard","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#tate-cohomology-card","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring, $M$ an $R$-linear representation of $G$ and $n$ an integer. This is the cardinality of the Tate cohomology group $\\hat H^n(G, M)$, as a cardinal number (possibly infinite).","title":"Cardinality of Tate cohomology"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/tate-homology-card","kind":"node","lean":"TateCohomology.tateHomologyCard","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#tate-homology-card","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$ an $R$-linear representation of $G$. For an integer $n$, the cardinality $h_n(M)$ of the degree-$n$ Tate homology of $M$ is defined as the cardinal number of the Tate cohomology group $\\hat H^{-n-1}(G, M)$, following the convention $\\hat H_n = \\hat H^{-n-1}$.","title":"Cardinality of Tate homology"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/tate-homology-card-zero","kind":"node","lean":"TateCohomology.tateHomologyCard_zero","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#tate-homology-card-zero","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$ an $R$-linear representation of $G$. The cardinality of the degree-$0$ Tate homology of $M$ equals the cardinality of the Tate cohomology group $\\hat H^{-1}(G, M)$.","title":"Cardinality of Tate homology in degree zero"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/tate-cohomology-card-eq-nat-cast","kind":"node","lean":"TateCohomology.tateCohomologyCard_eq_natCast","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#tate-cohomology-card-eq-nat-cast","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring, $M$ an $R$-linear representation of $G$ and $n$ an integer such that the Tate cohomology group $\\hat H^n(G, M)$ is finite. Then the cardinality of $\\hat H^n(G, M)$ equals its number of elements, a natural number viewed as a cardinal number.","title":"Cardinality of finite Tate cohomology as a natural number"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/herbrand-finite","kind":"node","lean":"TateCohomology.HerbrandFinite","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#herbrand-finite","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$ an $R$-linear representation of $G$. The representation $M$ satisfies the Herbrand finiteness hypothesis when its Tate cohomology groups $\\hat H^0(G, M)$ and $\\hat H^{-1}(G, M)$ are both finite.","title":"Finiteness hypothesis for the Herbrand quotient"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/herbrand-quotient","kind":"node","lean":"TateCohomology.herbrandQuotient","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#herbrand-quotient","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$ an $R$-linear representation of $G$ whose Tate cohomology groups $\\hat H^0(G, M)$ and $\\hat H^{-1}(G, M)$ are finite. The Herbrand quotient of $M$ is the rational number $h(M) = |\\hat H^0(G, M)| / |\\hat H^{-1}(G, M)|$, where $|X|$ is the number of elements of $X$.","title":"Herbrand quotient"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/herbrand-quotient-eq","kind":"node","lean":"TateCohomology.herbrandQuotient_eq","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#herbrand-quotient-eq","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$ an $R$-linear representation of $G$ whose Tate cohomology groups $\\hat H^0(G, M)$ and $\\hat H^{-1}(G, M)$ are finite. Then the Herbrand quotient of $M$ equals the rational number $|\\hat H^0(G, M)| / |\\hat H^{-1}(G, M)|$, where $|X|$ is the number of elements of $X$.","title":"Herbrand quotient as a ratio of orders"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/herbrand-quotient-den-pos","kind":"node","lean":"TateCohomology.herbrandQuotient_den_pos","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#herbrand-quotient-den-pos","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$ an $R$-linear representation of $G$ whose Tate cohomology groups $\\hat H^0(G, M)$ and $\\hat H^{-1}(G, M)$ are finite. Then the number of elements of $\\hat H^{-1}(G, M)$ is positive.","title":"The denominator of the Herbrand quotient is positive"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/herbrand-quotient-den-ne-zero","kind":"node","lean":"TateCohomology.herbrandQuotient_den_ne_zero","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#herbrand-quotient-den-ne-zero","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$ an $R$-linear representation of $G$ whose Tate cohomology groups $\\hat H^0(G, M)$ and $\\hat H^{-1}(G, M)$ are finite. Then the number of elements of $\\hat H^{-1}(G, M)$ is nonzero.","title":"The denominator of the Herbrand quotient is nonzero"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/herbrand-quotient-pos","kind":"node","lean":"TateCohomology.herbrandQuotient_pos","parent":4945,"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#herbrand-quotient-pos","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$ an $R$-linear representation of $G$ whose Tate cohomology groups $\\hat H^0(G, M)$ and $\\hat H^{-1}(G, M)$ are finite. Then the Herbrand quotient of $M$ is strictly positive.","title":"Positivity of the Herbrand quotient"},{"id":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/herbrand-quotient-congr","kind":"node","lean":"TateCohomology.herbrandQuotient_congr","parent":4945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Herbrand quotients agree when Tate cohomology in degrees "],["math","\\(0\\)"],["text"," and "],["math","\\(-1\\)"],["text"," is in bijection"]],"source":"Herbrand quotients agree when Tate cohomology in degrees $0$ and $-1$ is in bijection"},"result":true,"route":"../roadmap/RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient/index.html#herbrand-quotient-congr","scope":"RepresentationTheory/group-cohomology/tate-cohomology-and-the-herbrand-quotient","status":"fully_proved","summary":"Let $G$ be a finite group, $R$ a commutative ring and $M$, $N$ two $R$-linear representations of $G$ whose Tate cohomology groups $\\hat H^0(G, M)$, $\\hat H^{-1}(G, M)$, $\\hat H^0(G, N)$ and $\\hat H^{-1}(G, N)$ are finite. If there are bijections between the underlying sets of $\\hat H^0(G, M)$ and $\\hat H^0(G, N)$ and between those of $\\hat H^{-1}(G, M)$ and $\\hat H^{-1}(G, N)$, then the Herbrand quotients of $M$\u2026","title":"Herbrand quotients agree when Tate cohomology in degrees $0$ and $-1$ is in bijection"},{"data_url":"scopes/RepresentationTheory/lie-and-topological-representations.json","id":"RepresentationTheory/lie-and-topological-representations","kind":"scope","parent":4899,"result":false,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/index.html","scope":"RepresentationTheory/lie-and-topological-representations","status":"can_prove","summary":"Representations of infinite groups and Lie algebras: continuous and unitary representations of compact and topological groups (Peter\u2013Weyl theorem, complete reducibility), continuous representations over topological rings, representations of Lie algebras and of general linear groups (Schur\u2013Weyl duality), and groups characterized by their irreducible representations.","title":"Representations of topological and Lie groups"},{"id":"RepresentationTheory/lie-and-topological-representations/is-continuous-semilinear-representation","kind":"node","lean":"IsContinuousSemilinearRepresentation","parent":4958,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/index.html#is-continuous-semilinear-representation","scope":"RepresentationTheory/lie-and-topological-representations","status":"fully_proved","summary":"Let $G$ be a topological group, $R$ a topological ring and $M$ a topological abelian group that is a finite free $R$-module. Suppose that $G$ acts on $R$ by ring automorphisms and on $M$ by additive automorphisms, and that the actions of $G$ on $R$, of $G$ on $M$ and of $R$ on $M$ are jointly continuous. This is a continuous semilinear representation of $G$ on $M$ when $g \\cdot (r x) = (g \\cdot r)(g \\cdot x)$ for\u2026","title":"Continuous semilinear representation"},{"data_url":"scopes/RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea.json","id":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea","kind":"scope","navigation":"module","parent":4958,"result":false,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/index.html","scope":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea","status":"can_prove","title":"Groups with one irreducible representation in each dimension"},{"id":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/is-irreducible-rep","kind":"node","lean":"MathlibExt.GroupTheory.UniqueIrreducibleRepresentationsSU2Kourovka16Problem58Wanted.IsIrreducibleRep","parent":4960,"reading_title":{"format":"inline-parts-v1","parts":[["text","Irreducible representations into "],["math","\\(\\mathrm{GL}_n(\\mathbb{C})\\)"]],"source":"Irreducible representations into $\\mathrm{GL}_n(\\mathbb{C})$"},"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/index.html#is-irreducible-rep","scope":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea","status":"fully_proved","summary":"Let $G$ be a group and $\\rho \\colon G \\to \\mathrm{GL}_n(\\mathbb{C})$ a homomorphism. Then $\\rho$ is irreducible when $n \\ge 1$ and the only subspaces $W$ of $\\mathbb{C}^n$ with $\\rho(g)v \\in W$ for all $g \\in G$ and $v \\in W$ are $0$ and $\\mathbb{C}^n$.","title":"Irreducible representations into $\\mathrm{GL}_n(\\mathbb{C})$"},{"id":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/is-equivalent-rep","kind":"node","lean":"MathlibExt.GroupTheory.UniqueIrreducibleRepresentationsSU2Kourovka16Problem58Wanted.IsEquivalentRep","parent":4960,"reading_title":{"format":"inline-parts-v1","parts":[["text","Equivalent representations into "],["math","\\(\\mathrm{GL}_n(\\mathbb{C})\\)"]],"source":"Equivalent representations into $\\mathrm{GL}_n(\\mathbb{C})$"},"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/index.html#is-equivalent-rep","scope":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea","status":"fully_proved","summary":"Let $G$ be a group. Two homomorphisms $\\rho, \\sigma \\colon G \\to \\mathrm{GL}_n(\\mathbb{C})$ are equivalent when there is a matrix $P \\in \\mathrm{GL}_n(\\mathbb{C})$ with $\\sigma(g) = P \\rho(g) P^{-1}$ for all $g \\in G$.","title":"Equivalent representations into $\\mathrm{GL}_n(\\mathbb{C})$"},{"id":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/has-unique-irreducible-rep-of-each-dim","kind":"node","lean":"MathlibExt.GroupTheory.UniqueIrreducibleRepresentationsSU2Kourovka16Problem58Wanted.HasUniqueIrreducibleRepOfEachDim","parent":4960,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/index.html#has-unique-irreducible-rep-of-each-dim","scope":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea","status":"fully_proved","summary":"A group $G$ has exactly one irreducible complex representation of each dimension when for every $n \\ge 1$ there is an irreducible homomorphism $G \\to \\mathrm{GL}_n(\\mathbb{C})$ and any two irreducible homomorphisms $G \\to \\mathrm{GL}_n(\\mathbb{C})$ are equivalent. Representations are arbitrary homomorphisms of the abstract group $G$; no continuity is required.","title":"Groups with exactly one irreducible representation of each dimension"},{"id":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/su2-only-group-with-unique-irreducible-r","kind":"node","lean":"MathlibExt.GroupTheory.UniqueIrreducibleRepresentationsSU2Kourovka16Problem58Wanted.su2OnlyGroupWithUniqueIrreducibleReps","parent":4960,"reading_title":{"format":"inline-parts-v1","parts":[["text","McKay's question on "],["math","\\(\\mathrm{SU}_2(\\mathbb{C})\\)"],["text"," (Kourovka 16.58)"]],"source":"McKay's question on $\\mathrm{SU}_2(\\mathbb{C})$ (Kourovka 16.58)"},"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/index.html#su2-only-group-with-unique-irreducible-r","scope":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea","status":"fully_proved","summary":"A group $G$ has exactly one irreducible complex representation of each dimension $n = 1, 2, \\dots$, up to equivalence, if and only if $G$ is isomorphic to $\\mathrm{SU}_2(\\mathbb{C})$. Here a representation of dimension $n$ is any homomorphism $G \\to \\mathrm{GL}_n(\\mathbb{C})$ of abstract groups, with no continuity required.","title":"McKay's question on $\\mathrm{SU}_2(\\mathbb{C})$ (Kourovka 16.58)"},{"id":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/su2-only-group-with-unique-irreducible-r-2","kind":"node","lean":"MathlibExt.GroupTheory.UniqueIrreducibleRepresentationsSU2Kourovka16Problem58Wanted.su2OnlyGroupWithUniqueIrreducibleReps_refuted","parent":4960,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea/index.html#su2-only-group-with-unique-irreducible-r-2","scope":"RepresentationTheory/lie-and-topological-representations/groups-with-one-irreducible-representation-in-ea","status":"can_prove","summary":"It is not the case that a group has exactly one irreducible complex representation of each dimension $n \\ge 1$, up to equivalence, if and only if it is isomorphic to $\\mathrm{SU}_2(\\mathbb{C})$. Here a representation of dimension $n$ is any homomorphism to $\\mathrm{GL}_n(\\mathbb{C})$ of abstract groups, with no continuity required.","title":"McKay's question 16.58 has a negative answer"},{"data_url":"scopes/RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups.json","id":"RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups","kind":"scope","navigation":"module","parent":4958,"result":false,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups/index.html","scope":"RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups","status":"can_prove","title":"Representations of compact groups"},{"id":"RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups/peter-weyl-point-separation","kind":"node","lean":"MathlibExt.Topology.Algebra.PeterWeylWanted.peter_weyl_point_separation","parent":4966,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups/index.html#peter-weyl-point-separation","scope":"RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups","status":"fully_proved","summary":"Let $G$ be a compact Hausdorff topological group and let $x \\ne y$ be elements of $G$. Then there are an integer $n > 0$ and a continuous map $\\rho$ from $G$ to the complex $n \\times n$ matrices such that $\\rho(1) = 1$, $\\rho(gh) = \\rho(g)\\rho(h)$ for all $g, h \\in G$, the matrix $\\rho(g)$ is unitary for every $g \\in G$, and $\\rho(x) \\ne \\rho(y)$.","title":"Peter\u2013Weyl theorem: unitary representations separate points"},{"id":"RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups/compact-group-semisimple","kind":"node","lean":"MathlibExt.GroupTheory.CompactGroupSemisimpleWanted.RepresentationTheory.CompactGroupSemisimple.compact_group_semisimple","parent":4966,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups/index.html#compact-group-semisimple","scope":"RepresentationTheory/lie-and-topological-representations/representations-of-compact-groups","status":"can_prove","summary":"Let $G$ be a compact topological group and let $\\rho$ be a representation of $G$ on a finite-dimensional real normed space $V$ such that the action map $G \\times V \\to V$, $(g, v) \\mapsto \\rho(g)(v)$, is continuous. Then $\\rho$ is semisimple: every subrepresentation of $\\rho$ has a complementary subrepresentation.","title":"Representations of compact groups are semisimple"},{"data_url":"scopes/RepresentationTheory/lie-and-topological-representations/schur-weyl-duality.json","id":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality","kind":"scope","navigation":"module","parent":4958,"result":false,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/index.html","scope":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality","status":"can_prove","title":"Schur\u2013Weyl duality"},{"id":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/sym-action","kind":"node","lean":"MathlibExt.GroupTheory.SchurWeylWanted.RepresentationTheory.symAction","parent":4969,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/index.html#sym-action","scope":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $M$ an $R$-module and $k$ a natural number. The symmetric group $S_k$ of permutations of $\\{1, \\dots, k\\}$ acts on the $k$-th tensor power $M^{\\otimes k}$ by permuting tensor factors: this is the monoid homomorphism from $S_k$ to the $R$-linear endomorphisms of $M^{\\otimes k}$ that sends $\\sigma$ to the map\u2026","title":"Permutation action of the symmetric group on tensor powers"},{"id":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/gl-action","kind":"node","lean":"MathlibExt.GroupTheory.SchurWeylWanted.RepresentationTheory.glAction","parent":4969,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/index.html#gl-action","scope":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $M$ an $R$-module and $k$ a natural number. Write $\\mathrm{GL}(M)$ for the group of invertible $R$-linear endomorphisms of $M$. The diagonal action of $\\mathrm{GL}(M)$ on the $k$-th tensor power $M^{\\otimes k}$ is the monoid homomorphism from $\\mathrm{GL}(M)$ to the $R$-linear endomorphisms of $M^{\\otimes k}$ that sends $g$ to $g^{\\otimes k}$, so that\u2026","title":"Diagonal action of the general linear group on tensor powers"},{"id":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/sym-action-range-eq-centralizer-gl-actio","kind":"node","lean":"MathlibExt.GroupTheory.SchurWeylWanted.RepresentationTheory.symAction_range_eq_centralizer_glAction","parent":4969,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/index.html#sym-action-range-eq-centralizer-gl-actio","scope":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality","status":"can_prove","summary":"Let $R$ be a field, $M$ a finite-dimensional vector space over $R$, and $k$ a natural number such that $k!$ is invertible in $R$. In the algebra of $R$-linear endomorphisms of the tensor power $M^{\\otimes k}$, the subalgebra generated by the image of the action of the symmetric group $S_k$ by permuting tensor factors equals the centralizer of the image of the diagonal action $g \\mapsto g^{\\otimes k}$ of\u2026","title":"Schur\u2013Weyl duality: the symmetric group side"},{"id":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/gl-action-range-eq-centralizer-sym-actio","kind":"node","lean":"MathlibExt.GroupTheory.SchurWeylWanted.RepresentationTheory.glAction_range_eq_centralizer_symAction","parent":4969,"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/schur-weyl-duality/index.html#gl-action-range-eq-centralizer-sym-actio","scope":"RepresentationTheory/lie-and-topological-representations/schur-weyl-duality","status":"can_prove","summary":"Let $R$ be a field, $M$ a finite-dimensional vector space over $R$, and $k$ a natural number such that $k!$ is invertible in $R$. In the algebra of $R$-linear endomorphisms of the tensor power $M^{\\otimes k}$, the subalgebra generated by the image of the diagonal action $g \\mapsto g^{\\otimes k}$ of $\\mathrm{GL}(M)$ equals the centralizer of the image of the action of the symmetric group $S_k$ by permuting tensor\u2026","title":"Schur\u2013Weyl duality: the general linear side"},{"data_url":"scopes/RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres.json","id":"RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres","kind":"scope","navigation":"module","parent":4958,"result":false,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres/index.html","scope":"RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres","status":"can_prove","title":"Tensor squares of exceptional Lie algebra representations"},{"id":"RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres/g2-irrep-tensor-square-decomp","kind":"node","lean":"MathlibExt.GroupTheory.ExceptionalLieTensorSquareWanted.RepresentationTheory.g2_irrep_tensor_square_decomp","parent":4974,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tensor square of an irreducible "],["math","\\(\\mathfrak{g}_2\\)"],["text","-representation of dimension "],["math","\\(64\\)"]],"source":"Tensor square of an irreducible $\\mathfrak{g}_2$-representation of dimension $64$"},"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres/index.html#g2-irrep-tensor-square-decomp","scope":"RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres","status":"can_prove","summary":"There exist a complex vector space $V$ and a representation of the exceptional Lie algebra $\\mathfrak{g}_2$ over $\\mathbb{C}$ on $V$ such that $V$ has dimension $64$ over $\\mathbb{C}$, $V$ is irreducible, and the tensor square $V \\otimes_{\\mathbb{C}} V$, with the representation $x \\cdot (v \\otimes w) = (x \\cdot v) \\otimes w + v \\otimes (x \\cdot w)$ viewed as a module over the universal enveloping algebra of\u2026","title":"Tensor square of an irreducible $\\mathfrak{g}_2$-representation of dimension $64$"},{"id":"RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres/e8-irrep-tensor-square-decomp","kind":"node","lean":"MathlibExt.GroupTheory.ExceptionalLieTensorSquareWanted.RepresentationTheory.e8_irrep_tensor_square_decomp","parent":4974,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tensor square of an irreducible "],["math","\\(\\mathfrak{e}_8\\)"],["text","-representation of dimension "],["math","\\(779247\\)"]],"source":"Tensor square of an irreducible $\\mathfrak{e}_8$-representation of dimension $779247$"},"result":true,"route":"../roadmap/RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres/index.html#e8-irrep-tensor-square-decomp","scope":"RepresentationTheory/lie-and-topological-representations/tensor-squares-of-exceptional-lie-algebra-repres","status":"can_prove","summary":"There exist a complex vector space $V$ and a representation of the exceptional Lie algebra $\\mathfrak{e}_8$ over $\\mathbb{C}$ on $V$ such that $V$ has dimension $779247$ over $\\mathbb{C}$, $V$ is irreducible, and the tensor square $V \\otimes_{\\mathbb{C}} V$, with the representation $x \\cdot (v \\otimes w) = (x \\cdot v) \\otimes w + v \\otimes (x \\cdot w)$ viewed as a module over the universal enveloping algebra of\u2026","title":"Tensor square of an irreducible $\\mathfrak{e}_8$-representation of dimension $779247$"},{"area":"Algebra & Number Theory","data_url":"chapters/NumberTheory.json","id":"NumberTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/NumberTheory/index.html","scope":"NumberTheory","status":"can_state","summary":"Number theory studies integers, primes, Diophantine equations, arithmetic functions, and their analytic and algebraic structure.","title":"Number Theory"},{"data_url":"scopes/NumberTheory/congruences.json","id":"NumberTheory/congruences","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/congruences/index.html","scope":"NumberTheory/congruences","status":"can_prove","summary":"Modular arithmetic: Wilson, Wolstenholme and Fermat-quotient congruences, lifting the exponent, primitive roots, Legendre/Jacobi symbols, reciprocity, covering systems.","title":"Congruences and power residues"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic.json","id":"NumberTheory/congruences/congruences-modular-arithmetic","kind":"scope","parent":4978,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic","status":"can_prove","summary":"Congruences for factorials, quotients and power sums (Wilson, Wolstenholme, Fermat quotients), lifting the exponent, multiplicative order, primitive roots, residue systems.","title":"Congruences and modular arithmetic"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/power-sum-vanishing-mod-prime","kind":"node","lean":"MetaMathlibExt.powerSum_vanishing_mod_prime","parent":4979,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/index.html#power-sum-vanishing-mod-prime","scope":"NumberTheory/congruences/congruences-modular-arithmetic","status":"fully_proved","summary":"Let $p$ be a prime and let $e \\ge 0$ be an integer such that $p - 1$ does not divide $e$ whenever $e \\ge 1$. Then $p$ divides $\\sum_{t=1}^{p} t^e$.","title":"Power sums vanish modulo a prime"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","title":"Arithmetic progressions in reduced residue systems"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/reduced-residue-window","kind":"node","lean":"MetaMathlibExt.reducedResidueWindow","parent":4981,"reading_title":{"format":"inline-parts-v1","parts":[["text","The reduced residue system "],["math","\\(R_m(n)\\)"]],"source":"The reduced residue system $R_m(n)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#reduced-residue-window","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"fully_proved","summary":"For an integer $m$ and a natural number $n$, the reduced residue system $R_m(n)$ modulo $n$ is the finite set $R_m(n) = \\{a \\in \\mathbb{Z} : m + 1 \\le a \\le m + n,\\ \\gcd(a, n) = 1\\}$.","title":"The reduced residue system $R_m(n)$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/longest-aplength-with-diff","kind":"node","lean":"MetaMathlibExt.longestAPLengthWithDiff","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#longest-aplength-with-diff","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"fully_proved","summary":"For an integer $m$ and natural numbers $n$ and $d$, let $R_m(n)$ be the set of integers $a$ with $m + 1 \\le a \\le m + n$ and $\\gcd(a, n) = 1$. Then $f_m(d, n)$ is the supremum of the natural numbers $s$ for which some integer $b$ has $b + i d \\in R_m(n)$ for every natural number $i < s$. For $d > 0$ every such $s$ is at most $n$, so $f_m(d, n)$ is the length of the longest arithmetic progression with common\u2026","title":"Longest progression with a given common difference"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/longest-aplength","kind":"node","lean":"MetaMathlibExt.longestAPLength","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#longest-aplength","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"fully_proved","summary":"For an integer $m$ and a natural number $n$, let $R_m(n)$ be the reduced residue system of the integers $a$ with $m + 1 \\le a \\le m + n$ and $\\gcd(a, n) = 1$. Then $f_m(n)$ is the largest natural number $s$ for which there are an integer $b$ and a positive integer $d$ such that $b, b + d, \\dots, b + (s - 1)d$ all lie in $R_m(n)$. So it is the length of the longest arithmetic progression with positive common\u2026","title":"Length of the longest arithmetic progression in a reduced residue system"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength","kind":"node","lean":"MetaMathlibExt.absLeastAPLength","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"fully_proved","summary":"For a natural number $n$, let $R(n)$ be the set of integers $a$ with $1 - \\lceil n/2 \\rceil \\le a \\le n - \\lceil n/2 \\rceil$ and $\\gcd(a, n) = 1$, the system of absolute least reduced residues modulo $n$. Then $g(n)$ is the largest $s \\ge 0$ for which there are an integer $b$ and a positive integer $d$ with $b + i d \\in R(n)$ for every $0 \\le i < s$: the length of the longest arithmetic progression with positive\u2026","title":"Longest progression in the absolute least reduced residues"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/ap-length-le","kind":"node","lean":"MetaMathlibExt.apLength_le","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#ap-length-le","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"fully_proved","summary":"Let $m$ be an integer and $n$ a natural number, and let $R_m(n)$ be the set of integers $a$ with $m + 1 \\le a \\le m + n$ and $\\gcd(a, n) = 1$. If $b$ is an integer, $d$ a positive integer and $s$ a natural number such that $b + i d \\in R_m(n)$ for every $0 \\le i < s$, then $s \\le n$.","title":"Length bound for progressions in a reduced residue window"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/longest-aplength-le","kind":"node","lean":"MetaMathlibExt.longestAPLength_le","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#longest-aplength-le","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"fully_proved","summary":"For every integer $m$ and natural number $n$, $f_m(n) \\le n$, where $f_m(n)$ is the length of the longest arithmetic progression with positive common difference contained in the set $R_m(n)$ of integers $a$ with $m + 1 \\le a \\le m + n$ and $\\gcd(a, n) = 1$.","title":"The longest progression length is at most the modulus"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/mem-reduced-residue-window-neg-three-fiv","kind":"node","lean":"MetaMathlibExt.mem_reducedResidueWindow_neg_three_five","parent":4981,"reading_title":{"format":"inline-parts-v1","parts":[["text","The reduced residue system "],["math","\\(R_{-3}(5)\\)"]],"source":"The reduced residue system $R_{-3}(5)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#mem-reduced-residue-window-neg-three-fiv","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"fully_proved","summary":"The reduced residue system $R_{-3}(5)$ of the integers $a$ with $-2 \\le a \\le 2$ and $\\gcd(a, 5) = 1$ is $\\{-2, -1, 1, 2\\}$: an integer $a$ lies in $R_{-3}(5)$ if and only if $a$ is one of $-2$, $-1$, $1$, $2$.","title":"The reduced residue system $R_{-3}(5)$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength-five","kind":"node","lean":"MetaMathlibExt.absLeastAPLength_five","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength-five","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"fully_proved","summary":"The length $g(5)$ of the longest arithmetic progression in the absolute least reduced residue system modulo $5$ is $2$.","title":"Longest progressions of absolute least residues modulo 5"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/longest-aplength-with-diff-le-prime-sub","kind":"node","lean":"MetaMathlibExt.longestAPLengthWithDiff_le_prime_sub_one","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#longest-aplength-with-diff-le-prime-sub","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $m$ be an integer, $n > 1$ and $d > 0$ natural numbers, and $p$ a prime that divides $n$ but not $d$. Then $f_m(d, n) \\le p - 1$, where $f_m(d, n)$ is the length of the longest arithmetic progression with common difference $d$ contained in the set $R_m(n)$ of integers $a$ with $m + 1 \\le a \\le m + n$ and $\\gcd(a, n) = 1$.","title":"Prime bound for progressions with a fixed difference"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/longest-aplength-with-diff-le-gpf-sub-on","kind":"node","lean":"MetaMathlibExt.longestAPLengthWithDiff_le_gpf_sub_one_of_squarefree","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#longest-aplength-with-diff-le-gpf-sub-on","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $m$ be an integer, $n > 1$ a squarefree natural number with largest prime factor $\\operatorname{gpf}(n)$, and $d > 0$ a natural number. Then $f_m(d, n) \\le \\operatorname{gpf}(n) - 1$, where $f_m(d, n)$ is the length of the longest arithmetic progression with common difference $d$ contained in the set $R_m(n)$ of integers $a$ with $m + 1 \\le a \\le m + n$ and $\\gcd(a, n) = 1$.","title":"Squarefree bound for progressions with a fixed difference"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength-prime-of-mod-four-eq","kind":"node","lean":"MetaMathlibExt.absLeastAPLength_prime_of_mod_four_eq_one","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength-prime-of-mod-four-eq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"For a prime $p \\equiv 1 \\pmod 4$, the length $g(p)$ of the longest arithmetic progression in the absolute least reduced residue system modulo $p$ is $(p - 1)/2$.","title":"Longest progressions of absolute least residues: primes congruent to 1 mod 4"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength-prime-of-mod-four-eq-2","kind":"node","lean":"MetaMathlibExt.absLeastAPLength_prime_of_mod_four_eq_three","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength-prime-of-mod-four-eq-2","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"For a prime $p \\equiv 3 \\pmod 4$, the length $g(p)$ of the longest arithmetic progression in the absolute least reduced residue system modulo $p$ is $(p + 1)/2$.","title":"Longest progressions of absolute least residues: primes congruent to 3 mod 4"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength-prime-pow","kind":"node","lean":"MetaMathlibExt.absLeastAPLength_prime_pow","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength-prime-pow","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $p$ be a prime and $r \\ge 1$, with $r \\ge 2$ when $p$ is odd. Then the length $g(p^r)$ of the longest arithmetic progression in the absolute least reduced residue system modulo $p^r$ is $p^{r-1}$.","title":"Longest progressions of absolute least residues: prime powers"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength-odd-squarefree","kind":"node","lean":"MetaMathlibExt.absLeastAPLength_odd_squarefree","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength-odd-squarefree","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $n > 1$ be an odd squarefree natural number that is not prime, let $P$ be its largest prime factor and $d = n / P$, and let $g(n)$ be the length of the longest arithmetic progression in the absolute least reduced residue system modulo $n$. Then $g(n) = \\lfloor P - (P + 1)/(2d) \\rfloor$.","title":"Longest progressions of absolute least residues: odd squarefree composite moduli"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength-two-mul-prime","kind":"node","lean":"MetaMathlibExt.absLeastAPLength_two_mul_prime","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength-two-mul-prime","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"For an odd prime $p$, the length $g(2p)$ of the longest arithmetic progression in the absolute least reduced residue system modulo $2p$ is $p - 1$.","title":"Longest progressions of absolute least residues: twice an odd prime"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength-of-not-squarefree-of","kind":"node","lean":"MetaMathlibExt.absLeastAPLength_of_not_squarefree_of_not_isPrimePow","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength-of-not-squarefree-of","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $n > 1$ be a natural number that is neither squarefree nor a prime power, let $\\operatorname{rad}(n)$ be the product of the distinct primes dividing $n$ and $P$ the largest prime factor of $n$, and let $g(n)$ be the length of the longest arithmetic progression in the absolute least reduced residue system modulo $n$. Then $g(n) = \\max(n / \\operatorname{rad}(n), P - 1)$.","title":"Longest progressions of absolute least residues: neither squarefree nor prime power"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/abs-least-aplength-even-squarefree-bound","kind":"node","lean":"MetaMathlibExt.absLeastAPLength_even_squarefree_bounds","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#abs-least-aplength-even-squarefree-bound","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $n$ be an even squarefree natural number with at least three prime factors, let $P$ be its largest prime factor and $d = n / P$, and let $g(n)$ be the length of the longest arithmetic progression in the absolute least reduced residue system modulo $n$. Then $\\lfloor P - 2P/d \\rfloor \\le g(n) \\le P - 1$, and if $d \\ge 2P$ then $g(n) = P - 1$.","title":"Longest progressions of absolute least residues: even squarefree, three or more primes"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/is-greatest-longest-aplength","kind":"node","lean":"MetaMathlibExt.isGreatest_longestAPLength","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#is-greatest-longest-aplength","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $n > 1$ be a natural number, $\\operatorname{rad}(n)$ the product of its distinct prime factors and $\\operatorname{gpf}(n)$ its largest prime factor. For an integer $m$ let $f_m(n)$ be the length of the longest arithmetic progression with positive common difference contained in the set $R_m(n)$ of integers $a$ with $m + 1 \\le a \\le m + n$ and $\\gcd(a, n) = 1$. Then the largest value of $f_m(n)$ over all integers\u2026","title":"Maximum longest progression length over all reduced residue windows"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/is-least-longest-aplength-prime-of-mod-f","kind":"node","lean":"MetaMathlibExt.isLeast_longestAPLength_prime_of_mod_four_eq_one","parent":4981,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimum longest progression length modulo a prime "],["math","\\(p \\equiv 1 \\pmod 4\\)"]],"source":"Minimum longest progression length modulo a prime $p \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#is-least-longest-aplength-prime-of-mod-f","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$. For an integer $m$ let $f_m(p)$ be the length of the longest arithmetic progression with positive common difference contained in the set $R_m(p)$ of integers $a$ with $m + 1 \\le a \\le m + p$ and $\\gcd(a, p) = 1$. Then the least value of $f_m(p)$ over all integers $m$ is $(p - 1) / 2$.","title":"Minimum longest progression length modulo a prime $p \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/is-least-longest-aplength-prime-of-mod-f-2","kind":"node","lean":"MetaMathlibExt.isLeast_longestAPLength_prime_of_mod_four_eq_three","parent":4981,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimum longest progression length modulo a prime "],["math","\\(p \\equiv 3 \\pmod 4\\)"]],"source":"Minimum longest progression length modulo a prime $p \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#is-least-longest-aplength-prime-of-mod-f-2","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 4$. For an integer $m$ let $f_m(p)$ be the length of the longest arithmetic progression with positive common difference contained in the set $R_m(p)$ of integers $a$ with $m + 1 \\le a \\le m + p$ and $\\gcd(a, p) = 1$. Then the least value of $f_m(p)$ over all integers $m$ is $(p + 1) / 2$.","title":"Minimum longest progression length modulo a prime $p \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/is-least-longest-aplength-prime-pow","kind":"node","lean":"MetaMathlibExt.isLeast_longestAPLength_prime_pow","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#is-least-longest-aplength-prime-pow","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $p$ be a prime and $r > 1$ a natural number. For an integer $m$ let $f_m(p^r)$ be the length of the longest arithmetic progression with positive common difference contained in the set $R_m(p^r)$ of integers $a$ with $m + 1 \\le a \\le m + p^r$ and $\\gcd(a, p^r) = 1$. Then the least value of $f_m(p^r)$ over all integers $m$ is $p^{r-1}$.","title":"Minimum longest progression length modulo a prime power"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/longest-aplength-min-of-not-squarefree-o","kind":"node","lean":"MetaMathlibExt.longestAPLength_min_of_not_squarefree_of_not_isPrimePow","parent":4981,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimum longest progression length when "],["math","\\(n\\)"],["text"," is neither squarefree nor a prime power"]],"source":"Minimum longest progression length when $n$ is neither squarefree nor a prime power"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#longest-aplength-min-of-not-squarefree-o","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $n > 1$ be a natural number that is neither squarefree nor a prime power, $\\operatorname{rad}(n)$ the product of its distinct prime factors and $\\operatorname{gpf}(n)$ its largest prime factor. For an integer $m$ let $f_m(n)$ be the length of the longest arithmetic progression with positive common difference contained in the set $R_m(n)$ of integers $a$ with $m + 1 \\le a \\le m + n$ and $\\gcd(a, n) = 1$. Then\u2026","title":"Minimum longest progression length when $n$ is neither squarefree nor a prime power"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/average-longest-aplength-of-not-squarefr","kind":"node","lean":"MetaMathlibExt.average_longestAPLength_of_not_squarefree","parent":4981,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste/index.html#average-longest-aplength-of-not-squarefr","scope":"NumberTheory/congruences/congruences-modular-arithmetic/arithmetic-progressions-in-reduced-residue-syste","status":"can_prove","summary":"Let $n > 1$ be a natural number that is not squarefree, $\\operatorname{rad}(n)$ the product of its distinct prime factors and $\\operatorname{gpf}(n)$ its largest prime factor, and suppose that $\\operatorname{gpf}(n) - 1 \\le n / \\operatorname{rad}(n)$. For an integer $m$ let $f_m(n)$ be the length of the longest arithmetic progression with positive common difference contained in the set $R_m(n)$ of integers $a$ with\u2026","title":"Average longest progression length for non-squarefree moduli"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers","status":"fully_proved","title":"Euler\u2013Fermat congruences modulo prime powers"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers/euler-fermat-prime-power-congruence-gene","kind":"node","lean":"Nat.euler_fermat_prime_power_congruence_general","parent":5005,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers/index.html#euler-fermat-prime-power-congruence-gene","scope":"NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers","status":"fully_proved","summary":"For every natural number $a$, every prime $p$ and every natural number $r \\ge 1$, we have $a^{p^r} \\equiv a^{p^{r-1}} \\pmod{p^r}$.","title":"Euler\u2013Fermat congruence modulo prime powers"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers/euler-fermat-prime-power-congruence","kind":"node","lean":"Nat.euler_fermat_prime_power_congruence","parent":5005,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers/index.html#euler-fermat-prime-power-congruence","scope":"NumberTheory/congruences/congruences-modular-arithmetic/euler-fermat-congruences-modulo-prime-powers","status":"fully_proved","summary":"For every positive natural number $a$, every prime $p$ and every natural number $r \\ge 1$, we have $a^{p^r} \\equiv a^{p^{r-1}} \\pmod{p^r}$.","title":"Euler\u2013Fermat congruence modulo prime powers, for positive bases"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients","status":"fully_proved","title":"Fermat quotients"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients/fermat-quotient","kind":"node","lean":"Int.fermatQuotient","parent":5008,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients/index.html#fermat-quotient","scope":"NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients","status":"fully_proved","summary":"For a prime $p$ and an integer $a$ coprime to $p$, the Fermat quotient is the integer $q_p(a) = (a^{p-1} - 1)/p$, where $/$ is integer division rounding down. The division is exact, since $p$ divides $a^{p-1} - 1$ by Fermat's little theorem.","title":"Fermat quotient"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients/fermat-quotient-eq","kind":"node","lean":"Int.fermatQuotient_eq","parent":5008,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients/index.html#fermat-quotient-eq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients","status":"fully_proved","summary":"For a prime $p$ and an integer $a$ coprime to $p$, the Fermat quotient satisfies $q_p(a) = (a^{p-1} - 1)/p$, where $/$ is integer division rounding down.","title":"Formula for the Fermat quotient"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients/nat-cast-mul-fermat-quotient","kind":"node","lean":"Int.natCast_mul_fermatQuotient","parent":5008,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients/index.html#nat-cast-mul-fermat-quotient","scope":"NumberTheory/congruences/congruences-modular-arithmetic/fermat-quotients","status":"fully_proved","summary":"For a prime $p$ and an integer $a$ coprime to $p$, the Fermat quotient $q_p(a)$ satisfies $p \\, q_p(a) = a^{p-1} - 1$.","title":"Fermat quotient times the prime"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","title":"Congruences for Delannoy numbers"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/harmonic-mod","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.harmonicMod","parent":5012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Harmonic numbers modulo "],["math","\\(m\\)"]],"source":"Harmonic numbers modulo $m$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#harmonic-mod","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"fully_proved","summary":"For natural numbers $m$ and $k$, the harmonic number $H_k$ modulo $m$ is the residue $\\sum_{i=1}^{k} i^{-1}$ in $\\mathbb{Z}/m\\mathbb{Z}$, where $i^{-1}$ is the inverse of $i$ modulo $m$ when $i$ is coprime to $m$; for other $i$ it is the conventional value that Mathlib assigns to the inverse of a non-unit.","title":"Harmonic numbers modulo $m$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/harmonic2-mod","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.harmonic2Mod","parent":5012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Harmonic numbers of order two modulo "],["math","\\(m\\)"]],"source":"Harmonic numbers of order two modulo $m$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#harmonic2-mod","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"fully_proved","summary":"For natural numbers $m$ and $k$, the harmonic number $H_{k,2}$ of order two modulo $m$ is the residue $\\sum_{i=1}^{k} (i^2)^{-1}$ in $\\mathbb{Z}/m\\mathbb{Z}$, where $(i^2)^{-1}$ is the inverse of $i^2$ modulo $m$ when $i$ is coprime to $m$; for other $i$ it is the conventional value that Mathlib assigns to the inverse of a non-unit.","title":"Harmonic numbers of order two modulo $m$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/delannoy-two-mul","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.delannoy_two_mul","parent":5012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Delannoy triangle entries "],["math","\\(T(2p, p; 1)\\)"],["text"," and "],["math","\\(T(2p-1, p-1; 1)\\)"],["text"," modulo "],["math","\\(p^3\\)"]],"source":"Delannoy triangle entries $T(2p, p; 1)$ and $T(2p-1, p-1; 1)$ modulo $p^3$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#delannoy-two-mul","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","summary":"Let $p \\ge 5$ be a prime, let $q_p(2) = (2^{p-1} - 1)/p$ be the Fermat quotient of $2$, and let $T(n, k; 1)$ denote the entry of the ordinary Riordan Pascal-like triangle with $r = 1$ (the Delannoy triangle). Then","title":"Delannoy triangle entries $T(2p, p; 1)$ and $T(2p-1, p-1; 1)$ modulo $p^3$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/delannoy-mul-sub-one","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.delannoy_mul_sub_one","parent":5012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Delannoy triangle entry "],["math","\\(T(np-1, p-1; 1)\\)"],["text"," modulo "],["math","\\(p^3\\)"]],"source":"Delannoy triangle entry $T(np-1, p-1; 1)$ modulo $p^3$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#delannoy-mul-sub-one","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","summary":"Let $p \\ge 5$ be a prime and $n \\ge 1$ a natural number, let $q_p(2) = (2^{p-1} - 1)/p$ be the Fermat quotient of $2$, and let $T(a, b; 1)$ denote the entry of the ordinary Riordan Pascal-like triangle with $r = 1$ (the Delannoy triangle). Then","title":"Delannoy triangle entry $T(np-1, p-1; 1)$ modulo $p^3$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/choose-mul-sub-one-mod","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.choose_mul_sub_one_mod","parent":5012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Expansion of "],["math","\\(\\binom{sp-1}{k}\\)"],["text"," modulo "],["math","\\(p^3\\)"]],"source":"Expansion of $\\binom{sp-1}{k}$ modulo $p^3$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#choose-mul-sub-one-mod","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","summary":"Let $p$ be an odd prime, $s$ an integer and $k$ a natural number with $1 \\le k \\le p - 1$, and let $H_k = \\sum_{i=1}^{k} i^{-1}$ and $H_{k,2} = \\sum_{i=1}^{k} (i^2)^{-1}$ be the harmonic numbers of orders one and two in $\\mathbb{Z}/p^3\\mathbb{Z}$. Then in $\\mathbb{Z}/p^3\\mathbb{Z}$","title":"Expansion of $\\binom{sp-1}{k}$ modulo $p^3$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/sum-two-pow-div","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.sum_two_pow_div","parent":5012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of "],["math","\\(2^k/k\\)"],["text"," and "],["math","\\(2^k/k^2\\)"],["text"," over "],["math","\\(1 \\le k \\le p - 1\\)"]],"source":"Sums of $2^k/k$ and $2^k/k^2$ over $1 \\le k \\le p - 1$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#sum-two-pow-div","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","summary":"Let $p > 3$ be a prime and let $q_p(2) = (2^{p-1} - 1)/p$ be the Fermat quotient of $2$. Then","title":"Sums of $2^k/k$ and $2^k/k^2$ over $1 \\le k \\le p - 1$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/sum-harmonic-two-pow-div","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.sum_harmonic_two_pow_div","parent":5012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(H_k 2^k / k\\)"],["text"," over "],["math","\\(1 \\le k \\le p - 1\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Sum of $H_k 2^k / k$ over $1 \\le k \\le p - 1$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#sum-harmonic-two-pow-div","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","summary":"Let $p \\ge 5$ be a prime, let $q_p(2) = (2^{p-1} - 1)/p$ be the Fermat quotient of $2$, and let $H_k = \\sum_{i=1}^{k} 1/i$ be the harmonic number modulo $p$, where $1/i$ is the inverse of $i$ modulo $p$. Then","title":"Sum of $H_k 2^k / k$ over $1 \\le k \\le p - 1$ modulo $p$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/sum-delannoy","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.sum_delannoy","parent":5012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of the Delannoy triangle entries in rows "],["math","\\(0\\)"],["text"," to "],["math","\\(p - 1\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Sum of the Delannoy triangle entries in rows $0$ to $p - 1$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#sum-delannoy","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","summary":"Let $p$ be an odd prime and let $T(n, k; 1)$ denote the entry of the ordinary Riordan Pascal-like triangle with $r = 1$ (the Delannoy triangle). Then","title":"Sum of the Delannoy triangle entries in rows $0$ to $p - 1$ modulo $p$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/choose-add-choose-mod","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.choose_add_choose_mod","parent":5012,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#choose-add-choose-mod","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","summary":"For a prime $p$ and natural numbers $j$ and $k$ with $j \\le k \\le p - 2$,","title":"Reflection congruence for binomial coefficients modulo a prime"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/sum-choose-choose-binomial-power-seq","kind":"node","lean":"MathlibExt.NumberTheory.DelannoyTriangleWanted.sum_choose_choose_binomialPowerSeq","parent":5012,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers/index.html#sum-choose-choose-binomial-power-seq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/congruences-for-delannoy-numbers","status":"can_prove","summary":"For every natural number $k$,","title":"A binomial sum through central Delannoy numbers"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma","status":"fully_proved","title":"Lifting the exponent lemma"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma/padic-val-int-pow-sub-pow","kind":"node","lean":"MetaMathlibExt.padicValInt_pow_sub_pow","parent":5023,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma/index.html#padic-val-int-pow-sub-pow","scope":"NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma","status":"fully_proved","summary":"Let $p$ be an odd prime, let $a \\ne b$ be integers such that $p$ divides $a - b$ but does not divide $ab$, and let $n \\ge 1$ be a natural number. Then","title":"Lifting the exponent for odd primes"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma/pow-pow-mod-eq-one-of-mod-eq-one","kind":"node","lean":"MetaMathlibExt.pow_pow_modEq_one_of_modEq_one","parent":5023,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma/index.html#pow-pow-mod-eq-one-of-mod-eq-one","scope":"NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma","status":"fully_proved","summary":"Let $p$, $n$ and $a$ be natural numbers. If $a \\equiv 1 \\pmod p$, then $a^{p^n} \\equiv 1 \\pmod{p^{n+1}}$.","title":"Lifting a congruence to one along powers of the modulus"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma/prime-power-lifting-mod-eq-one","kind":"node","lean":"MetaMathlibExt.prime_power_lifting_modEq_one","parent":5023,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma/index.html#prime-power-lifting-mod-eq-one","scope":"NumberTheory/congruences/congruences-modular-arithmetic/lifting-the-exponent-lemma","status":"fully_proved","summary":"Let $p$ be a prime and let $n$ and $a$ be positive integers. If $a \\equiv 1 \\pmod p$, then $a^{p^n} \\equiv 1 \\pmod{p^{n+1}}$.","title":"Prime-power lifting of a congruence to one"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","title":"Modular inverses from the Euclidean algorithm"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/is-least-abs-inverse","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.IsLeastAbsInverse","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#is-least-abs-inverse","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"fully_proved","summary":"For integers $x$ and $a$ and a natural number $n$, $x$ is the least absolute inverse of $a$ modulo $n$ when $xa \\equiv 1 \\pmod{n}$ and $-n < 2x \\le n$, that is, when $x$ is an inverse of $a$ modulo $n$ in the half-open interval $-n/2 < x \\le n/2$.","title":"Least absolute inverse"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/is-euclid-switch-seq","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.IsEuclidSwitchSeq","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#is-euclid-switch-seq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"fully_proved","summary":"A natural number $N$, a sequence $r_0, r_1, \\dots$ of natural numbers and a sequence $s_0, s_1, \\dots$ of integers form Euclidean switching data when $N \\ge 1$; $r_0 > r_1 > \\cdots > r_{N+1} = 1$, with $r_0, \\dots, r_{N+1}$ all positive; for each $i = 1, \\dots, N$ there is a positive integer $q_i$ with $r_{i-1} = q_i r_i + r_{i+1}$; $s_{N+1} = 1$; and $s_{i+1} r_{i-1} + s_i r_i = 1$ for each $i = 1, \\dots, N$.","title":"Euclidean switching data"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/bezout-coeffs-are-inverses","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.bezout_coeffs_are_inverses","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#bezout-coeffs-are-inverses","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $m$ and $n$ be coprime positive integers, and let $x$ and $y$ be integers with $xm + yn = 1$. Then $xm \\equiv 1 \\pmod{n}$ and $yn \\equiv 1 \\pmod{m}$; that is, $x$ is an inverse of $m$ modulo $n$ and $y$ is an inverse of $n$ modulo $m$.","title":"B\u00e9zout coefficients are modular inverses"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/exists-eq-mul-add-iff-switching","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.exists_eq_mul_add_iff_switching","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#exists-eq-mul-add-iff-switching","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $m$ and $n$ be positive integers with $\\gcd(m, n) = 1$, and let $t$ be an integer with $\\gcd(|t|, n) = 1$. Then $m = kn + t$ for some integer $k$ if and only if there are integers $x$ and $y$ with $xt \\equiv 1 \\pmod{n}$, $yn \\equiv 1 \\pmod{m}$ and $xm + yn = 1$.","title":"Modular inverse switching"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/is-least-abs-inverse-succ-imp","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.isLeastAbsInverse_succ_imp","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#is-least-abs-inverse-succ-imp","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $r_0, r_1, \\ldots$ be natural numbers, $s_0, s_1, \\ldots$ integers and $N \\ge 1$ such that $(r, s, N)$ is Euclidean switching data: $r_0 > r_1 > \\cdots > r_{N+1} = 1$ are the successive remainders of the Euclidean algorithm, $s_{N+1} = 1$, and $s_{i+1} r_{i-1} + s_i r_i = 1$ for $1 \\le i \\le N$. If $1 \\le j \\le N$ and $s_{j+1}$ is the least absolute inverse of $r_{j+1}$ modulo $r_j$, then $s_j$ is the least\u2026","title":"Least absolute inverses propagate down the Euclidean sequence"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/is-least-abs-inverse-all","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.isLeastAbsInverse_all","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#is-least-abs-inverse-all","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $(r_i)$, $(s_i)$ and $N$ be Euclidean switching data. Then for every $i$ with $1 \\le i \\le N + 1$, $s_i$ is the least absolute inverse of $r_i$ modulo $r_{i-1}$.","title":"Every switching inverse is a least absolute inverse"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/euclid-bezout-coeffs-least-abs-inverse","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.euclid_bezout_coeffs_leastAbsInverse","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#euclid-bezout-coeffs-least-abs-inverse","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $m > n > 1$ be integers, write $g = \\gcd(m, n)$, and let $(r_i)$, $(s_i)$ and $N$ be Euclidean switching data with $r_0 = m/g$ and $r_1 = n/g$. Then $s_2 m + s_1 n = g$, $s_2$ is the least absolute inverse of $m/g$ modulo $n/g$, and $s_1$ is the least absolute inverse of $n/g$ modulo $m/g$.","title":"The Euclidean B\u00e9zout coefficients are least absolute inverses"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/euclid-bezout-coeffs-abs-le","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.euclid_bezout_coeffs_abs_le","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#euclid-bezout-coeffs-abs-le","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $m > n > 1$ be integers, write $g = \\gcd(m, n)$, and let $(r_i)$, $(s_i)$ and $N$ be Euclidean switching data with $r_0 = m/g$ and $r_1 = n/g$. Then for all integers $x$ and $y$ with $xm + yn = g$, we have $|s_2| \\le |x|$ and $|s_1| \\le |y|$.","title":"The Euclidean B\u00e9zout coefficients are smallest in absolute value"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/euclid-bezout-coeffs-sq-add-le","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.euclid_bezout_coeffs_sq_add_le","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#euclid-bezout-coeffs-sq-add-le","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $m > n > 1$ be integers, write $g = \\gcd(m, n)$, and let $(r_i)$, $(s_i)$ and $N$ be Euclidean switching data with $r_0 = m/g$ and $r_1 = n/g$. Then for all integers $x$ and $y$ with $xm + yn = g$, we have $s_2^2 + s_1^2 \\le x^2 + y^2$.","title":"Rankin's minimality of the Euclidean B\u00e9zout pair"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/abs-eq-halves-iff-sub-eq-two-of-odd","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.abs_eq_halves_iff_sub_eq_two_of_odd","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#abs-eq-halves-iff-sub-eq-two-of-odd","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $(r_i)$, $(s_i)$ and $N$ be Euclidean switching data with $r_0$ and $r_1$ both odd. Then $2|s_1| = r_0 - 1$ and $2|s_2| = r_1 - 1$ hold simultaneously if and only if $r_0 = r_1 + 2$.","title":"Both first inverses extremal, odd\u2013odd case"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/abs-eq-halves-iff-eq-two-of-odd-even","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.abs_eq_halves_iff_eq_two_of_odd_even","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#abs-eq-halves-iff-eq-two-of-odd-even","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $(r_i)$, $(s_i)$ and $N$ be Euclidean switching data with $r_0$ odd and $r_1$ even. Then $2|s_1| = r_0 - 1$ and $2|s_2| = r_1$ hold simultaneously if and only if $r_1 = 2$.","title":"Both first inverses extremal, odd\u2013even case"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/abs-lt-of-even-odd","kind":"node","lean":"MathlibExt.NumberTheory.ModInverseSwitchingWanted.abs_lt_of_even_odd","parent":5027,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm/index.html#abs-lt-of-even-odd","scope":"NumberTheory/congruences/congruences-modular-arithmetic/modular-inverses-from-the-euclidean-algorithm","status":"can_prove","summary":"Let $(r_i)$, $(s_i)$ and $N$ be Euclidean switching data with $r_0$ even and $r_1$ odd. Then $2|s_1| < r_0$ or $2|s_2| < r_1 - 1$; that is, $|s_1| < r_0/2$ or $|s_2| < (r_1 - 1)/2$.","title":"First inverses not both extremal, even\u2013odd case"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","title":"Primitive roots modulo prime powers"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/mul-ord-mod","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.mulOrdMod","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicative order modulo "],["math","\\(n\\)"]],"source":"Multiplicative order modulo $n$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#mul-ord-mod","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"fully_proved","summary":"For natural numbers $n$ and $a$, $\\operatorname{ord}_n(a)$ is the multiplicative order of the residue class of $a$ in the unit group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ when $a$ is coprime to $n$, and $0$ otherwise.","title":"Multiplicative order modulo $n$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/unit-reduce-sq","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.unitReduceSq","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reduction of units modulo "],["math","\\(p^2\\)"],["text"," to units modulo "],["math","\\(p\\)"]],"source":"Reduction of units modulo $p^2$ to units modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#unit-reduce-sq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"fully_proved","summary":"For a natural number $p$, this is the group homomorphism $(\\mathbb{Z}/p^2\\mathbb{Z})^\\times \\to (\\mathbb{Z}/p\\mathbb{Z})^\\times$ induced on units by the reduction ring homomorphism $\\mathbb{Z}/p^2\\mathbb{Z} \\to \\mathbb{Z}/p\\mathbb{Z}$, which sends the class of $x$ modulo $p^2$ to the class of $x$ modulo $p$.","title":"Reduction of units modulo $p^2$ to units modulo $p$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/lift-fiber","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.liftFiber","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibers of reduction modulo "],["math","\\(p\\)"],["text"," on units modulo "],["math","\\(p^2\\)"]],"source":"Fibers of reduction modulo $p$ on units modulo $p^2$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#lift-fiber","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"fully_proved","summary":"For a natural number $p$ and a unit $a$ of $\\mathbb{Z}/p^2\\mathbb{Z}$, the fiber $P_a$ is the set of units $x$ of $\\mathbb{Z}/p^2\\mathbb{Z}$ with $x \\equiv a \\pmod p$, that is, the units whose image under the reduction map $(\\mathbb{Z}/p^2\\mathbb{Z})^\\times \\to (\\mathbb{Z}/p\\mathbb{Z})^\\times$ equals the image of $a$.","title":"Fibers of reduction modulo $p$ on units modulo $p^2$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/order-add-mul-prime-eq-or-mul","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.order_add_mul_prime_eq_or_mul","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","The order of "],["math","\\(a + kp\\)"],["text"," modulo "],["math","\\(p^2\\)"]],"source":"The order of $a + kp$ modulo $p^2$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#order-add-mul-prime-eq-or-mul","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime and let $a$ and $k$ be natural numbers with $a$ coprime to $p$. Writing $\\operatorname{ord}_n(b)$ for the multiplicative order of $b$ modulo $n$, the order $\\operatorname{ord}_{p^2}(a + kp)$ is either $\\operatorname{ord}_p(a)$ or $p \\cdot \\operatorname{ord}_p(a)$.","title":"The order of $a + kp$ modulo $p^2$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/exists-unique-order-add-mul-prime-eq","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.existsUnique_order_add_mul_prime_eq","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unique order-preserving lift modulo "],["math","\\(p^2\\)"]],"source":"Unique order-preserving lift modulo $p^2$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#exists-unique-order-add-mul-prime-eq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime and let $a$ be a natural number coprime to $p$. Writing $\\operatorname{ord}_n(b)$ for the multiplicative order of $b$ modulo $n$, there is exactly one $m \\in \\{0, 1, \\dots, p - 1\\}$ such that $\\operatorname{ord}_{p^2}(a + mp) = \\operatorname{ord}_p(a)$.","title":"Unique order-preserving lift modulo $p^2$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/exists-unique-mem-lift-fiber-order-eq","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.existsUnique_mem_liftFiber_order_eq","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unique order-preserving element of a fiber modulo "],["math","\\(p^2\\)"]],"source":"Unique order-preserving element of a fiber modulo $p^2$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#exists-unique-mem-lift-fiber-order-eq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime, let $x_0$ be a unit modulo $p^2$, and let $P_{x_0}$ be the set of units $x$ modulo $p^2$ with $x \\equiv x_0 \\pmod p$. There is exactly one $x \\in P_{x_0}$ whose multiplicative order modulo $p^2$ equals the multiplicative order of its reduction modulo $p$.","title":"Unique order-preserving element of a fiber modulo $p^2$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/exists-primitive-root-add-mul-prime-pow","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.exists_primitiveRoot_add_mul_prime_pow","parent":5040,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#exists-primitive-root-add-mul-prime-pow","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime and let $g$ be a natural number that is a primitive root modulo $p$, where a primitive root modulo $n$ is a residue class modulo $n$ of multiplicative order $\\varphi(n)$. Then there is a natural number $k$ such that $g + kp$ is a primitive root modulo $p^\\ell$ for every $\\ell \\ge 2$.","title":"Gauss's theorem on primitive roots modulo prime powers"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/exists-unique-not-primitive-root-add-mul","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.existsUnique_not_primitiveRoot_add_mul_prime","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exactly one lift of a primitive root fails modulo "],["math","\\(p^2\\)"]],"source":"Exactly one lift of a primitive root fails modulo $p^2$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#exists-unique-not-primitive-root-add-mul","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime and let $g$ be a natural number that is a primitive root modulo $p$, where a primitive root modulo $n$ is a residue class modulo $n$ of multiplicative order $\\varphi(n)$. Then there is exactly one $m \\in \\{0, 1, \\dots, p - 1\\}$ such that $g + mp$ is not a primitive root modulo $p^2$, and $g + kp$ is a primitive root modulo $p^2$ for every other $k \\in \\{0, 1, \\dots, p - 1\\}$.","title":"Exactly one lift of a primitive root fails modulo $p^2$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/exists-unique-mem-lift-fiber-order-eq-su","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.existsUnique_mem_liftFiber_order_eq_sub_one","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","The lift of order "],["math","\\(p - 1\\)"],["text"," of a primitive root"]],"source":"The lift of order $p - 1$ of a primitive root"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#exists-unique-mem-lift-fiber-order-eq-su","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime and let $g$ be a natural number that is a primitive root modulo $p$, that is, a residue class modulo $p$ of multiplicative order $\\varphi(p) = p - 1$. Let $P_g$ be the set of units $x$ modulo $p^2$ with $x \\equiv g \\pmod p$. Then exactly one $x \\in P_g$ has multiplicative order $p - 1$ modulo $p^2$.","title":"The lift of order $p - 1$ of a primitive root"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/primitive-root-pow-add-mul-prime-pow","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.primitiveRoot_pow_add_mul_prime_pow","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primitive roots modulo "],["math","\\(p^\\ell\\)"],["text"," of the form "],["math","\\(g^p + kp\\)"]],"source":"Primitive roots modulo $p^\\ell$ of the form $g^p + kp$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#primitive-root-pow-add-mul-prime-pow","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime and let $g$ be a natural number that is a primitive root modulo $p$. If $k$ is a natural number not divisible by $p$, then $g^p + kp$ is a primitive root modulo $p^\\ell$ for every natural number $\\ell > 1$.","title":"Primitive roots modulo $p^\\ell$ of the form $g^p + kp$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/monoid-hom-image-lift-fiber-subset","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.monoidHom_image_liftFiber_subset","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Endomorphisms of the units modulo "],["math","\\(p^2\\)"],["text"," respect the fibers"]],"source":"Endomorphisms of the units modulo $p^2$ respect the fibers"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#monoid-hom-image-lift-fiber-subset","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime, let $f$ be a group endomorphism of $(\\mathbb{Z}/p^2\\mathbb{Z})^\\times$, and let $a$ be a unit modulo $p^2$. Writing $P_b$ for the set of units $x$ modulo $p^2$ with $x \\equiv b \\pmod p$, the image of $P_a$ under $f$ is contained in $P_{f(a)}$.","title":"Endomorphisms of the units modulo $p^2$ respect the fibers"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/pow-injective-on-lift-fiber-and-order-eq","kind":"node","lean":"MathlibExt.NumberTheory.GaussPrimitiveRootsWanted.pow_injectiveOn_liftFiber_and_order_eq","parent":5040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power maps on the fibers modulo "],["math","\\(p^2\\)"]],"source":"Power maps on the fibers modulo $p^2$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers/index.html#pow-injective-on-lift-fiber-and-order-eq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots-modulo-prime-powers","status":"can_prove","summary":"Let $p$ be an odd prime, let $a$ be a unit modulo $p^2$, and let $n$ be a natural number coprime to $p$ and to the multiplicative order of $a$. Writing $P_b$ for the set of units $x$ modulo $p^2$ with $x \\equiv b \\pmod p$, the map $x \\mapsto x^n$ is injective on $P_a$, maps $P_a$ onto $P_{a^n}$, and preserves the multiplicative order of every element of $P_a$.","title":"Power maps on the fibers modulo $p^2$"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots","status":"can_prove","title":"Primitive \u03bb-roots"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/lambda-delta","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveLambdaRootsWanted.lambdaDelta","parent":5053,"reading_title":{"format":"inline-parts-v1","parts":[["text","The count "],["math","\\(\\Delta_q(n)\\)"],["text"," for primitive \u03bb-roots"]],"source":"The count $\\Delta_q(n)$ for primitive \u03bb-roots"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/index.html#lambda-delta","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots","status":"fully_proved","summary":"Let $\\lambda$ be the Carmichael function and, for natural numbers $p$ and $m$, let $v_p(m)$ be the exponent of $p$ in the prime factorization of $m$ (taken to be $0$ when $p$ is not prime or $m = 0$). For natural numbers $q$ and $n$, the natural number $\\Delta_q(n)$ is defined as follows. If $q = 2$, $v_2(n) = 3$ and $v_2(\\lambda(n)) = 1$, then $\\Delta_q(n)$ is one more than the number of distinct prime factors of\u2026","title":"The count $\\Delta_q(n)$ for primitive \u03bb-roots"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/lambda-h","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveLambdaRootsWanted.lambdaH","parent":5053,"reading_title":{"format":"inline-parts-v1","parts":[["text","The local factor "],["math","\\(h_q(n)\\)"],["text"," for primitive \u03bb-roots"]],"source":"The local factor $h_q(n)$ for primitive \u03bb-roots"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/index.html#lambda-h","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots","status":"fully_proved","summary":"Let $\\Delta_q(n)$ be the count for primitive \u03bb-roots. For natural numbers $q$ and $n$, the rational number $h_q(n)$ equals $1$ if $\\Delta_q(n) = 0$, and $1 - 1/q^{\\Delta_q(n)}$ otherwise.","title":"The local factor $h_q(n)$ for primitive \u03bb-roots"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/primitive-lambda-root-count","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveLambdaRootsWanted.primitiveLambdaRootCount","parent":5053,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/index.html#primitive-lambda-root-count","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots","status":"fully_proved","summary":"For a natural number $n$, $R(n)$ is the number of units $a$ of $\\mathbb{Z}/n\\mathbb{Z}$ whose multiplicative order equals $\\lambda(n)$, where $\\lambda$ is the Carmichael function.","title":"Number of primitive \u03bb-roots"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/carmichael-mul-eq-lcm","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveLambdaRootsWanted.carmichael_mul_eq_lcm","parent":5053,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/index.html#carmichael-mul-eq-lcm","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots","status":"can_prove","summary":"For coprime natural numbers $m, n > 1$, the Carmichael function $\\lambda$ satisfies $\\lambda(mn) = \\operatorname{lcm}(\\lambda(m), \\lambda(n))$.","title":"Carmichael's function on coprime products"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/lambda-h-mul-ge-and-gt-iff","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveLambdaRootsWanted.lambdaH_mul_ge_and_gt_iff","parent":5053,"reading_title":{"format":"inline-parts-v1","parts":[["text","Huang\u2013Li inequality for the local factors "],["math","\\(h_q\\)"]],"source":"Huang\u2013Li inequality for the local factors $h_q$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/index.html#lambda-h-mul-ge-and-gt-iff","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots","status":"can_prove","summary":"Let $\\lambda$ be the Carmichael function, and let $\\Delta_q$ and $h_q$ be the count and the local factor for primitive \u03bb-roots. Let $m, n > 1$ be coprime natural numbers and let $q$ be a prime dividing $\\lambda(mn)$. Then $h_q(m)\\,h_q(n) \\le h_q(mn)$. Moreover, $h_q(m)\\,h_q(n) < h_q(mn)$ if and only if at least one of the following holds: $0 < \\Delta_q(m) < \\Delta_q(mn)$; $0 < \\Delta_q(n) < \\Delta_q(mn)$; or\u2026","title":"Huang\u2013Li inequality for the local factors $h_q$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/primitive-lambda-root-count-mul-ge-and-e","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveLambdaRootsWanted.primitiveLambdaRootCount_mul_ge_and_eq_iff","parent":5053,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots/index.html#primitive-lambda-root-count-mul-ge-and-e","scope":"NumberTheory/congruences/congruences-modular-arithmetic/primitive-roots","status":"can_prove","summary":"Let $R(n)$ be the number of primitive \u03bb-roots modulo $n$. For coprime natural numbers $m, n > 1$, we have $R(m)\\,R(n) \\le R(mn)$, and $R(mn) = R(m)\\,R(n)$ if and only if $m = 2$ and $n$ is odd, or $n = 2$ and $m$ is odd.","title":"Supermultiplicativity of the number of primitive \u03bb-roots"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences","status":"can_prove","title":"Small solutions of diagonal congruences"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/product-add-add-eq-univ-of-card-mul-gt","kind":"node","lean":"MetaMathlibExt.product_add_add_eq_univ_of_card_mul_gt","parent":5060,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/index.html#product-add-add-eq-univ-of-card-mul-gt","scope":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences","status":"can_prove","summary":"Let $p$ be a prime and let $A_1, A_2, A_3, D$ be subsets of $\\mathbb{Z}/p\\mathbb{Z}$ with $|A_1| |A_2| |A_3| |D| > p^3$. Then every residue class modulo $p$ can be written as $a_1 d + a_2 + a_3$ with $a_1 \\in A_1$, $d \\in D$, $a_2 \\in A_2$ and $a_3 \\in A_3$; that is, $A_1 D + A_2 + A_3 = \\mathbb{Z}/p\\mathbb{Z}$.","title":"Sum-product sets cover the integers modulo a prime"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/diagonal-congruence-solvable-of-card-eq","kind":"node","lean":"MetaMathlibExt.diagonal_congruence_solvable_of_card_eq_exponent","parent":5060,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonal congruences in "],["math","\\(k\\)"],["text"," variables are solvable"]],"source":"Diagonal congruences in $k$ variables are solvable"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/index.html#diagonal-congruence-solvable-of-card-eq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences","status":"can_prove","summary":"Let $p$ be a prime and $k \\ge 1$. For all integers $a_1, \\dots, a_k$ not divisible by $p$ and every integer $c$, there are integers $x_1, \\dots, x_k$ with $a_1 x_1^k + \\dots + a_k x_k^k \\equiv c \\pmod p$.","title":"Diagonal congruences in $k$ variables are solvable"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/smooth-diagonal-solution-count-asymp","kind":"node","lean":"MetaMathlibExt.smooth_diagonal_solution_count_asymp","parent":5060,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/index.html#smooth-diagonal-solution-count-asymp","scope":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences","status":"can_prove","summary":"For a natural number $N$ and real $\\eta > 0$, let $A(N, N^\\eta)$ be the set of integers $1 \\le x \\le N$ all of whose prime factors are at most $N^\\eta$. There is a real-valued function $g$ on the natural numbers with $g(k) = O(\\log \\log k / \\log k)$ as $k \\to \\infty$ such that for every $k \\ge 2$ there is $\\eta_0 > 0$ with the following property. Let $c_1, \\dots, c_s$ be nonzero integers such that\u2026","title":"Smooth solutions of diagonal equations (Chow\u2013Lindqvist\u2013Prendiville)"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/card-smooth-diagonal-values-ge","kind":"node","lean":"MetaMathlibExt.card_smooth_diagonal_values_ge","parent":5060,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/index.html#card-smooth-diagonal-values-ge","scope":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences","status":"can_prove","summary":"For a natural number $B$ and real $\\eta > 0$, let $A(B, B^\\eta)$ be the set of integers $1 \\le x \\le B$ all of whose prime factors are at most $B^\\eta$. There is a real-valued function $g$ on the natural numbers with $g(k) = O(\\log \\log k / \\log k)$ as $k \\to \\infty$ such that for every $k \\ge 2$ there is $\\eta_0 > 0$ with the following property. Let $0 < \\eta \\le \\eta_0$ and let $n \\ge 1$ satisfy $2n \\ge 5$ if\u2026","title":"Many values of a smooth diagonal form modulo a prime"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/exists-small-solution-diagonal-congruenc","kind":"node","lean":"MetaMathlibExt.exists_small_solution_diagonal_congruence","parent":5060,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences/index.html#exists-small-solution-diagonal-congruenc","scope":"NumberTheory/congruences/congruences-modular-arithmetic/small-solutions-of-diagonal-congruences","status":"can_prove","summary":"There is a real-valued function $g$ on the natural numbers with $g(k) = O(\\log \\log k / \\log k)$ as $k \\to \\infty$ such that for every $k \\ge 2$ there is a constant $C > 0$ with the following property. Let $n$ satisfy $n \\ge 9$ if $k = 2$, $n \\ge 12$ if $k = 3$, and $n \\ge \\frac{3}{2} k (\\log k + \\log \\log k + 2 + g(k))$ if $k \\ge 4$. Then for every prime $p$, every integer $c$ and all integers $a_1, \\dots, a_n$\u2026","title":"Small solutions of diagonal congruences"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma","status":"fully_proved","title":"Thue's lemma"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma/thue-lemma-general","kind":"node","lean":"MetaMathlibExt.thue_lemma_general","parent":5066,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma/index.html#thue-lemma-general","scope":"NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma","status":"fully_proved","summary":"Let $a$ and $n$ be integers with $n > 1$ and $\\gcd(a, n) = 1$. Then there exist nonzero integers $x$ and $y$ with $|x| < \\sqrt{n}$, $|y| < \\sqrt{n}$ and $x \\equiv a y \\pmod{n}$. Unlike the source form of Thue's lemma, no lower bound on $a$ is assumed.","title":"Thue's lemma, general form"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma/thue-lemma","kind":"node","lean":"MetaMathlibExt.thue_lemma","parent":5066,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma/index.html#thue-lemma","scope":"NumberTheory/congruences/congruences-modular-arithmetic/thue-s-lemma","status":"fully_proved","summary":"Let $a$ and $n$ be integers with $a > 1$, $n > 1$ and $\\gcd(a, n) = 1$. Then there exist nonzero integers $x$ and $y$ with $|x| < \\sqrt{n}$, $|y| < \\sqrt{n}$ and $x \\equiv a y \\pmod{n}$.","title":"Thue's lemma"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","title":"Wilson quotients"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/wilson-quotient","kind":"node","lean":"Int.wilsonQuotient","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#wilson-quotient","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"fully_proved","summary":"For a natural number $p$, the Wilson quotient is the integer $W(p) = \\lfloor ((p-1)! + 1) / p \\rfloor$ for $p \\ge 1$, and $W(0) = 0$.","title":"Wilson quotient"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/mul-wilson-quotient","kind":"node","lean":"Int.mul_wilsonQuotient","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#mul-wilson-quotient","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"fully_proved","summary":"For every prime $p$, the Wilson quotient $W(p)$ of $p$ satisfies $p \\, W(p) = (p-1)! + 1$ in $\\mathbb{Z}$.","title":"Exactness of the Wilson quotient"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/factorial-eq-mul-wilson-quotient-sub-one","kind":"node","lean":"Int.factorial_eq_mul_wilsonQuotient_sub_one","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#factorial-eq-mul-wilson-quotient-sub-one","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"fully_proved","summary":"For every prime $p$, the Wilson quotient $W(p)$ of $p$ satisfies $(p-1)! = p \\, W(p) - 1$ in $\\mathbb{Z}$.","title":"Factorial form of the Wilson quotient identity"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/wilson-mod-prime-sq","kind":"node","lean":"MathlibExt.NumberTheory.WilsonFaulhaberWanted.wilson_mod_prime_sq","parent":5069,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wilson's theorem modulo "],["math","\\(p^2\\)"]],"source":"Wilson's theorem modulo $p^2$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#wilson-mod-prime-sq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","summary":"Let $p$ be an odd prime and, for $1 \\le k \\le p - 1$, let $q_p(k)$ be the Fermat quotient of $k$, the integer with $k^{p-1} = 1 + p\\, q_p(k)$. Then","title":"Wilson's theorem modulo $p^2$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/wilson-quotient-m","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.wilsonQuotientM","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#wilson-quotient-m","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $M_k(n)$ is the rational number","title":"Morozov's Wilson quotients"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/wilson-quotient-mplus","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.wilsonQuotientMPlus","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#wilson-quotient-mplus","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $M_k^+(n)$ is the rational number","title":"Morozov's sign-adjusted Wilson quotients"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/wilson-quotient-z","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.wilsonQuotientZ","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#wilson-quotient-z","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"fully_proved","summary":"For a natural number $n$, $Z(n)$ is the rational number $Z(n) = \\sum_{k=0}^{n-1} M_k(n)$, where $M_k(n)$ are Morozov's Wilson quotients.","title":"Morozov's Wilson quotient sum"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/wilson-quotient-zplus","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.wilsonQuotientZPlus","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#wilson-quotient-zplus","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"fully_proved","summary":"For a natural number $n$, $Z^+(n)$ is the rational number $\\sum_{k=0}^{n-1} M_k^+(n)$, where $M_k^+(n) = ((-1)^k + k!\\,(n-k-1)!)/n$ is Morozov's sign-adjusted quotient.","title":"Morozov's sign-adjusted Wilson quotient sum"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/neg-one-pow-mul-factorial-mul-factorial","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.neg_one_pow_mul_factorial_mul_factorial_modEq_neg_one_iff","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#neg-one-pow-mul-factorial-mul-factorial","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","summary":"Let $n \\ge 1$ and $k < n$ be natural numbers. Then $(-1)^k \\, k! \\, (n - k - 1)! \\equiv -1 \\pmod{n}$ if and only if $n = 1$ or $n$ is prime.","title":"A generalized Wilson criterion for primality"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/exists-int-eq-wilson-quotient-z","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.exists_int_eq_wilsonQuotientZ","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#exists-int-eq-wilson-quotient-z","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","summary":"For every natural number $n \\ge 1$, Morozov's Wilson quotient sum $Z(n)$ is an integer.","title":"The Wilson quotient sum is an integer"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/exists-nat-eq-wilson-quotient-zplus-iff","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.exists_nat_eq_wilsonQuotientZPlus_iff","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#exists-nat-eq-wilson-quotient-zplus-iff","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","summary":"For every natural number $n \\ge 1$, Morozov's sign-adjusted Wilson quotient sum $Z^+(n)$ is a natural number if and only if $n$ is even, $n = 1$, or $n$ is prime.","title":"When the sign-adjusted Wilson quotient sum is a natural number"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/exists-int-eq-wilson-quotient-z-and-mod","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.exists_int_eq_wilsonQuotientZ_and_modEq","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#exists-int-eq-wilson-quotient-z-and-mod","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","summary":"Let $n \\ge 1$ be a natural number. Morozov's Wilson quotient sum $Z(n)$ is an integer; moreover $Z(n) \\equiv 1 \\pmod{n}$ if $n$ is neither $1$ nor prime, and $Z(n) \\equiv -1 \\pmod{n}$ if $n = 1$ or $n$ is prime.","title":"The Wilson quotient sum Z(n) modulo n"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/exists-int-eq-wilson-quotient-z-two-mul","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.exists_int_eq_wilsonQuotientZ_two_mul_sub_one_modEq_zero_iff","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#exists-int-eq-wilson-quotient-z-two-mul","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","summary":"For every natural number $n \\ge 1$, Morozov's Wilson quotient sum $Z(2n - 1)$ is an integer, and it is divisible by $n$ if and only if $n = 1$ or $n$ is prime.","title":"A primality criterion from the Wilson quotient sum Z(2n \u2212 1)"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/exists-int-eq-wilson-quotient-zplus-two","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.exists_int_eq_wilsonQuotientZPlus_two_mul_and_modEq","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#exists-int-eq-wilson-quotient-zplus-two","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","summary":"Let $n \\ge 1$ be a natural number. Morozov's sign-adjusted Wilson quotient sum $Z^+(2n)$ is an integer; it satisfies $Z^+(2n) \\equiv 0 \\pmod{2n}$ if $n = 2$ or $n$ is neither $1$ nor prime, and $Z^+(2n) \\equiv n + 1 \\pmod{2n}$ if $n$ is an odd prime.","title":"The sign-adjusted Wilson quotient sum Z\u207a(2n) modulo 2n"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/prime-dvd-num-wilson-quotient-zplus-sub","kind":"node","lean":"MathlibExt.NumberTheory.WilsonQuotientsWanted.prime_dvd_num_wilsonQuotientZPlus_sub_bernoulli_alternating","parent":5069,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients/index.html#prime-dvd-num-wilson-quotient-zplus-sub","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wilson-quotients","status":"can_prove","summary":"Let $p$ be an odd prime, let $B_m$ denote the Bernoulli numbers, and let $H_k = \\sum_{j=1}^{k} 1/j$ denote the harmonic numbers, with $H_0 = 0$. Then $p$ divides the numerator, in lowest terms, of the rational number","title":"A Bernoulli\u2013harmonic congruence for the sign-adjusted Wilson quotient sum"},{"data_url":"scopes/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem.json","id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","kind":"scope","navigation":"module","parent":4979,"result":false,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","title":"Wolstenholme's theorem"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/carlitz-sum-inv-block-ico-eq-zero-of-fiv","kind":"node","lean":"ZMod.carlitz_sum_inv_block_Ico_eq_zero_of_five_le","parent":5085,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#carlitz-sum-inv-block-ico-eq-zero-of-fiv","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"fully_proved","summary":"Let $p \\ge 5$ be a prime and let $k \\in \\mathbb{N}$. Then in $\\mathbb{Z}/p^2\\mathbb{Z}$","title":"Carlitz's block harmonic congruence"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/wolstenholme","kind":"node","lean":"MathlibExt.NumberTheory.Wolstenholme.wolstenholme","parent":5085,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#wolstenholme","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"fully_proved","summary":"If $p \\ge 5$ is prime, then $\\sum_{k=1}^{p-1} k^{-1} = 0$ in $\\mathbb{Z}/p^2\\mathbb{Z}$, where $k^{-1}$ is the inverse of $k$ modulo $p^2$.","title":"Wolstenholme's theorem"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/wolstenholme-harmonic-power-sum","kind":"node","lean":"MetaMathlibExt.wolstenholme_harmonic_power_sum","parent":5085,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#wolstenholme-harmonic-power-sum","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"fully_proved","summary":"Let $\\nu \\ge 1$ be odd and let $p$ be a prime with $p \\ge \\nu + 4$. Then in $\\mathbb{Z}/p^2\\mathbb{Z}$","title":"Wolstenholme's congruence for odd powers"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/coeff-add-one-pow-mul-x-pow-add-one-pow","kind":"node","lean":"MetaMathlibExt.coeff_add_one_pow_mul_X_pow_add_one_pow","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of "],["math","\\((x + 1)^{jp} (x^p + 1)^{n-j}\\)"]],"source":"Coefficients of $(x + 1)^{jp} (x^p + 1)^{n-j}$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#coeff-add-one-pow-mul-x-pow-add-one-pow","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $n$, $j$ and $k$ be natural numbers and $p$ a positive integer, and let $d = \\max(n - j, 0)$. Then the coefficient of $x^{kp}$ in the polynomial $(x + 1)^{jp} (x^p + 1)^{d}$ is","title":"Coefficients of $(x + 1)^{jp} (x^p + 1)^{n-j}$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-neg-mul-mul-eq","kind":"node","lean":"MetaMathlibExt.ringChoose_neg_mul_mul_eq","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial coefficients with upper index "],["math","\\(-np\\)"]],"source":"Binomial coefficients with upper index $-np$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-neg-mul-mul-eq","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p$, $n$ and $k$ be natural numbers with $p \\ge 1$ and $k \\ge 1$. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$. Then","title":"Binomial coefficients with upper index $-np$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/choose-mul-prime-prime-harmonic-congr","kind":"node","lean":"MetaMathlibExt.choose_mul_prime_prime_harmonic_congr","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences for "],["math","\\(\\binom{tp}{p}\\)"],["text"," modulo "],["math","\\(p^5\\)"],["text",", "],["math","\\(p^6\\)"],["text"," and "],["math","\\(p^7\\)"]],"source":"Congruences for $\\binom{tp}{p}$ modulo $p^5$, $p^6$ and $p^7$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#choose-mul-prime-prime-harmonic-congr","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p \\ge 11$ be a prime and $t$ a natural number. Write $H_{p-1}^{(r)} = \\sum_{j=1}^{p-1} 1/j^r$ for the harmonic number of order $r$ and $H_{p-1}(1, 1) = \\sum_{1 \\le i < j \\le p-1} 1/(ij)$. For rational numbers $x$, $y$ and a natural number $s$, write $x \\equiv y \\pmod{p^s}$ when $x = y$ or $v_p(x - y) \\ge s$, where $v_p$ is the $p$-adic valuation on $\\mathbb{Q}$. Then","title":"Congruences for $\\binom{tp}{p}$ modulo $p^5$, $p^6$ and $p^7$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-one-mul-congr","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_one_mul_congr","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo "],["math","\\(p^7\\)"],["text"," for "],["math","\\(\\binom{\\alpha p - 1}{p - 1} \\binom{(\\alpha + 1)p - 1}{p - 1}\\)"],["text"," and "],["math","\\(\\binom{2p - 1}{p - 1}^{-1}\\)"]],"source":"Congruences modulo $p^7$ for $\\binom{\\alpha p - 1}{p - 1} \\binom{(\\alpha + 1)p - 1}{p - 1}$ and $\\binom{2p - 1}{p - 1}^{-1}$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-one-mul-congr","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Congruences modulo $p^7$ for $\\binom{\\alpha p - 1}{p - 1} \\binom{(\\alpha + 1)p - 1}{p - 1}$ and $\\binom{2p - 1}{p - 1}^{-1}$. Let $p \\ge 11$ be a prime and let $\\alpha$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, let $H_{p-1}^{(r)} = \\sum_{j=1}^{p-1} 1/j^r$ be the harmonic number of order $r$, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$. Put","title":"Congruences modulo $p^7$ for $\\binom{\\alpha p - 1}{p - 1} \\binom{(\\alpha + 1)p - 1}{p - 1}$ and $\\binom{2p - 1}{p - 1}^{-1}$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/norm-choose-nsmul-coeff-le","kind":"node","lean":"MetaMathlibExt.norm_choose_nsmul_coeff_le","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficient bound for the divided derivatives "],["math","\\(D^n/n!\\)"]],"source":"Coefficient bound for the divided derivatives $D^n/n!$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#norm-choose-nsmul-coeff-le","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p$ be a prime and $B$ a normed vector space over $\\mathbb{Q}_p$ whose norm is ultrametric. Let $(a_k)_{k \\ge 0}$ be a sequence in $B$ and $C$ a real number with $\\|a_k\\| \\le C$ for every $k$. Then for all natural numbers $n$ and $k$,","title":"Coefficient bound for the divided derivatives $D^n/n!$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/norm-sub-taylor-le-of-norm-le","kind":"node","lean":"MetaMathlibExt.norm_sub_taylor_le_of_norm_le","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Higher "],["math","\\(p\\)"],["text","-adic mean value theorem"]],"source":"Higher $p$-adic mean value theorem"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#norm-sub-taylor-le-of-norm-le","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p$ be a prime, let $|\\cdot|$ be the $p$-adic absolute value on $\\mathbb{Q}_p$, and let $B$ be a normed vector space over $\\mathbb{Q}_p$ whose norm is ultrametric. Let $N$ and $m$ be natural numbers with $1 \\le m \\le p$, let $f(t) = \\sum_{j\\lt N} a_j t^j$ be a polynomial with coefficients $a_j \\in B$, and let $t, h \\in \\mathbb{Q}_p$ with $|t| \\le 1$. Suppose that $|h| \\le p^{-1/(p-m)}$ if $m < p$, that\u2026","title":"Higher $p$-adic mean value theorem"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/norm-sub-taylor-le-zpow-valuation","kind":"node","lean":"MetaMathlibExt.norm_sub_taylor_le_zpow_valuation","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruence form of the higher "],["math","\\(p\\)"],["text","-adic mean value theorem"]],"source":"Congruence form of the higher $p$-adic mean value theorem"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#norm-sub-taylor-le-zpow-valuation","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p$ be a prime, let $|\\cdot|$ be the $p$-adic absolute value on $\\mathbb{Q}_p$, and let $B$ be a normed vector space over $\\mathbb{Q}_p$ whose norm is ultrametric. Let $N$ and $m$ be natural numbers with $1 \\le m \\le p$, let $f(t) = \\sum_{j\\lt N} a_j t^j$ with $a_j \\in B$, let $t, h \\in \\mathbb{Q}_p$ with $|t| \\le 1$, and let $w$ be a real number with $\\|j(j-1)\\cdots(j-m+1) \\, a_j\\| \\le p^{-w}$ for every\u2026","title":"Congruence form of the higher $p$-adic mean value theorem"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/alternating-choose-sum-identities","kind":"node","lean":"MetaMathlibExt.alternating_choose_sum_identities","parent":5085,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#alternating-choose-sum-identities","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$ for the generalized binomial coefficient. Let $m$, $n$ and $k$ be natural numbers. Then for every natural number $j \\le m$,","title":"Alternating binomial sum identities"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-sum-d","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_sum_d","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The general congruence for "],["math","\\(\\binom{np}{kp}\\)"]],"source":"The general congruence for $\\binom{np}{kp}$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-sum-d","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p$ be a prime, let $m$ and $k$ be natural numbers with $m + 1 < p$, and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$. For natural numbers $j$ and $l$ put","title":"The general congruence for $\\binom{np}{kp}$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-cbar","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_cbar","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruence for "],["math","\\(\\binom{np}{kp}\\)"],["text"," with the coefficients "],["math","\\(\\bar C_{ji}\\)"]],"source":"Congruence for $\\binom{np}{kp}$ with the coefficients $\\bar C_{ji}$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-cbar","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p$ be a prime, let $m$ and $k$ be natural numbers with $m + 1 < p$, and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and set $\\binom{x}{r} = 0$ for negative $r$. Let","title":"Congruence for $\\binom{np}{kp}$ with the coefficients $\\bar C_{ji}$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-cprime","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_cprime","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence for "],["math","\\(\\binom{np}{kp}\\)"],["text"," with symmetrized coefficients"]],"source":"The congruence for $\\binom{np}{kp}$ with symmetrized coefficients"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-cprime","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p$ be a prime, let $m$ and $k$ be natural numbers with $m + 1 < p$, and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, for a rational number $x$ write $(x)_r = x(x-1)\\cdots(x-r+1)$ for the falling factorial, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$. Put","title":"The congruence for $\\binom{np}{kp}$ with symmetrized coefficients"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/exists-polynomial-part-eq-mul-half-mul","kind":"node","lean":"MetaMathlibExt.exists_polynomial_part_eq_mul_half_mul","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility of the polynomial parts by "],["math","\\(k(n-k)/2\\)"]],"source":"Divisibility of the polynomial parts by $k(n-k)/2$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#exists-polynomial-part-eq-mul-half-mul","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $m$ be a natural number and $n$ an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and for a rational number $x$ write $(x)_r = x(x-1)\\cdots(x-r+1)$ for the falling factorial. Then there is a polynomial $Q \\in \\mathbb{Q}[X]$ such that for every natural number $k$","title":"Divisibility of the polynomial parts by $k(n-k)/2$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-pow-thre","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_pow_three","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence for "],["math","\\(\\binom{np}{kp}\\)"],["text"," modulo "],["math","\\(p^3\\)"]],"source":"The congruence for $\\binom{np}{kp}$ modulo $p^3$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-pow-thre","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p \\ge 5$ be a prime, let $k$ be a natural number and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$. Let","title":"The congruence for $\\binom{np}{kp}$ modulo $p^3$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/choose-two-mul-prime-congr-iff-pow-three","kind":"node","lean":"MetaMathlibExt.choose_two_mul_prime_congr_iff_pow_three","parent":5085,"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#choose-two-mul-prime-congr-iff-pow-three","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p \\ge 5$ be a prime and let $n$ and $k$ be natural numbers. Then the congruences $\\binom{2p}{p} \\equiv 2$, $\\binom{np}{p} \\equiv n$ and $\\binom{np}{kp} \\equiv \\binom{n}{k}$ modulo $p^3$ are equivalent.","title":"Equivalent forms of Wolstenholme's congruence"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-pow-four","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_pow_four","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence for "],["math","\\(\\binom{np}{kp}\\)"],["text"," modulo "],["math","\\(p^4\\)"],["text"," via "],["math","\\(\\binom{2p}{p}\\)"],["text"," and "],["math","\\(\\binom{3p}{p}\\)"]],"source":"The congruence for $\\binom{np}{kp}$ modulo $p^4$ via $\\binom{2p}{p}$ and $\\binom{3p}{p}$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-pow-four","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p \\ge 7$ be a prime, let $k$ be a natural number and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$. Let","title":"The congruence for $\\binom{np}{kp}$ modulo $p^4$ via $\\binom{2p}{p}$ and $\\binom{3p}{p}$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-pow-four-2","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_pow_four_two_mul","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence for "],["math","\\(\\binom{np}{kp}\\)"],["text"," modulo "],["math","\\(p^4\\)"],["text"," via "],["math","\\(\\binom{2p}{p}\\)"],["text"," alone"]],"source":"The congruence for $\\binom{np}{kp}$ modulo $p^4$ via $\\binom{2p}{p}$ alone"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-pow-four-2","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p \\ge 7$ be a prime, let $k$ be a natural number and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$, with the convention $v_p(0) = 0$. Let","title":"The congruence for $\\binom{np}{kp}$ modulo $p^4$ via $\\binom{2p}{p}$ alone"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-pow-five","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_pow_five","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence for "],["math","\\(\\binom{np}{kp}\\)"],["text"," modulo "],["math","\\(p^5\\)"]],"source":"The congruence for $\\binom{np}{kp}$ modulo $p^5$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-pow-five","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p \\ge 11$ be a prime, let $k$ be a natural number and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$. Put $C_{00} = -(k^2 - kn + 2n^2 - 15n + 35)/6$, $C_{21} = (n-4)(n-3)/2$, $C_{31} = (n-2)(4-n)/3$, $C_{41} = (2k^2 - 2kn + n^2 - 3n + 4)/12$ and $C_{42} = (n-k-1)(k-1)/12$, and let","title":"The congruence for $\\binom{np}{kp}$ modulo $p^5$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-pow-six","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_pow_six","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence for "],["math","\\(\\binom{np}{kp}\\)"],["text"," modulo "],["math","\\(p^6\\)"]],"source":"The congruence for $\\binom{np}{kp}$ modulo $p^6$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-pow-six","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p \\ge 11$ be a prime, let $k$ be a natural number and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$. Put $C_{00} = (n-6)(n^2 - nk - 6n + k^2 + 15)/12$, $C_{21} = (5-n)(4-n)(3-n)/6$, $C_{31} = (n-2)(n-5)(n-4)/6$, $C_{41} = (2k^2 - 2kn + n^2 - 3n + 4)(5-n)/12$,\u2026","title":"The congruence for $\\binom{np}{kp}$ modulo $p^6$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/ring-choose-mul-prime-sub-congr-pow-seve","kind":"node","lean":"MetaMathlibExt.ringChoose_mul_prime_sub_congr_pow_seven","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence for "],["math","\\(\\binom{np}{kp}\\)"],["text"," modulo "],["math","\\(p^7\\)"]],"source":"The congruence for $\\binom{np}{kp}$ modulo $p^7$"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#ring-choose-mul-prime-sub-congr-pow-seve","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p \\ge 11$ be a prime, let $k$ be a natural number and let $n$ be an integer. For an integer $x$ and a natural number $r$ write $\\binom{x}{r} = x(x-1)\\cdots(x-r+1)/r!$, and let $v_p$ be the $p$-adic valuation on $\\mathbb{Q}$. Put","title":"The congruence for $\\binom{np}{kp}$ modulo $p^7$"},{"id":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/choose-mul-prime-eq-prod-div-prod","kind":"node","lean":"MetaMathlibExt.choose_mul_prime_eq_prod_div_prod","parent":5085,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\binom{np}{mp}\\)"],["text"," as a quotient of products"]],"source":"$\\binom{np}{mp}$ as a quotient of products"},"result":true,"route":"../roadmap/NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem/index.html#choose-mul-prime-eq-prod-div-prod","scope":"NumberTheory/congruences/congruences-modular-arithmetic/wolstenholme-s-theorem","status":"can_prove","summary":"Let $p$ be a prime and let $m$ and $n$ be natural numbers with $1 \\le m \\le n$. Then","title":"$\\binom{np}{mp}$ as a quotient of products"},{"data_url":"scopes/NumberTheory/congruences/power-residues-reciprocity.json","id":"NumberTheory/congruences/power-residues-reciprocity","kind":"scope","parent":4978,"result":false,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/index.html","scope":"NumberTheory/congruences/power-residues-reciprocity","status":"fully_proved","summary":"Quadratic and cubic residues, Legendre, Jacobi and Kronecker symbols, quadratic reciprocity and its extensions, Legendre-symbol sums.","title":"Power residues and reciprocity"},{"id":"NumberTheory/congruences/power-residues-reciprocity/sum-linear-eq-zero","kind":"node","lean":"legendreSym.sum_linear_eq_zero","parent":5109,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/index.html#sum-linear-eq-zero","scope":"NumberTheory/congruences/power-residues-reciprocity","status":"fully_proved","summary":"Let $p$ be an odd prime and let $b$ and $c$ be integers with $p \\nmid b$. Then the Legendre symbols $\\left(\\frac{b\\ell + c}{p}\\right)$ for $\\ell = 0, 1, \\dots, p - 1$ sum to $0$.","title":"Linear Legendre-symbol sums vanish"},{"data_url":"scopes/NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol.json","id":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol","kind":"scope","navigation":"module","parent":5109,"result":false,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/index.html","scope":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol","status":"fully_proved","title":"Cubic residue symbol"},{"id":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/prime-ideal-cubic-symbol-factor-raw","kind":"node","lean":"MetaMathlibExt.primeIdealCubicSymbolFactorRaw","parent":5111,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/index.html#prime-ideal-cubic-symbol-factor-raw","scope":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol","status":"fully_proved","summary":"Let $K$ be a number field that is the third cyclotomic extension of $\\mathbb{Q}$, with ring of integers $\\mathcal{O}_K$, let $\\zeta \\in K$ be a primitive cube root of unity and let $\\omega$ be $\\zeta$ regarded as an element of $\\mathcal{O}_K$. For an ideal $P$ of $\\mathcal{O}_K$ and $a \\in \\mathcal{O}_K$: if $P$ is a nonzero prime ideal whose absolute norm $N(P)$ is coprime to $3$, the value is $0$ when $a \\in P$,\u2026","title":"Totalized prime-ideal cubic symbol"},{"id":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/cubic-residue-symbol-raw","kind":"node","lean":"MetaMathlibExt.cubicResidueSymbolRaw","parent":5111,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/index.html#cubic-residue-symbol-raw","scope":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol","status":"fully_proved","summary":"Let $K$ be a number field that is the third cyclotomic extension of $\\mathbb{Q}$, with ring of integers $\\mathcal{O}_K$, and let $\\zeta \\in K$ be a primitive cube root of unity. For $a, n \\in \\mathcal{O}_K$, the value is $0$ if $n = 0$; otherwise it is the product, over the prime ideals $P$ in the factorization of the principal ideal $(n)$ counted with multiplicity, of the totalized prime-ideal cubic symbol of $a$\u2026","title":"Totalized cubic residue symbol"},{"id":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/prime-ideal-cubic-symbol","kind":"node","lean":"MetaMathlibExt.primeIdealCubicSymbol","parent":5111,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/index.html#prime-ideal-cubic-symbol","scope":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol","status":"fully_proved","summary":"Let $K$ be a number field that is the third cyclotomic extension of $\\mathbb{Q}$, with ring of integers $\\mathcal{O}_K$, let $\\zeta \\in K$ be a primitive cube root of unity and let $\\omega$ be $\\zeta$ regarded as an element of $\\mathcal{O}_K$. For a nonzero prime ideal $P$ of $\\mathcal{O}_K$ whose absolute norm $N(P)$ is coprime to $3$ and for $a \\in \\mathcal{O}_K$, the cubic residue symbol $(a/P)_3$ is $0$ if\u2026","title":"Cubic residue symbol at a prime ideal"},{"id":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/is-cubic-residue-denominator","kind":"node","lean":"MetaMathlibExt.IsCubicResidueDenominator","parent":5111,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/index.html#is-cubic-residue-denominator","scope":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol","status":"fully_proved","summary":"Let $K$ be a number field that is the third cyclotomic extension of $\\mathbb{Q}$, with ring of integers $\\mathcal{O}_K$. An element $n \\in \\mathcal{O}_K$ is a cubic residue denominator when $n \\neq 0$ and the absolute norm of the principal ideal $(n)$ is coprime to $3$.","title":"Cubic residue denominators"},{"id":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/cubic-residue-symbol","kind":"node","lean":"MetaMathlibExt.cubicResidueSymbol","parent":5111,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol/index.html#cubic-residue-symbol","scope":"NumberTheory/congruences/power-residues-reciprocity/cubic-residue-symbol","status":"fully_proved","summary":"Let $K$ be a number field that is the third cyclotomic extension of $\\mathbb{Q}$, with ring of integers $\\mathcal{O}_K$, let $\\zeta \\in K$ be a primitive cube root of unity and let $\\omega$ be $\\zeta$ regarded as an element of $\\mathcal{O}_K$. For $a \\in \\mathcal{O}_K$ and a cubic residue denominator $n$ (that is, $n \\neq 0$ and the absolute norm of the principal ideal $(n)$ is coprime to $3$), the cubic residue\u2026","title":"Cubic residue symbol modulo an element"},{"data_url":"scopes/NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma.json","id":"NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma","kind":"scope","navigation":"module","parent":5109,"result":false,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma/index.html","scope":"NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma","status":"fully_proved","title":"Eisenstein's lemma"},{"id":"NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma/eisenstein-lemma-jacobi-symbol-general","kind":"node","lean":"MetaMathlibExt.eisenstein_lemma_jacobi_symbol_general","parent":5117,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma/index.html#eisenstein-lemma-jacobi-symbol-general","scope":"NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma","status":"fully_proved","summary":"Let $a$ and $b$ be odd natural numbers that are coprime. Then the Jacobi symbol $\\left(\\frac{a}{b}\\right)$ equals $(-1)^{S}$, where $S = \\sum_{i=1}^{(b-1)/2} \\lfloor ia/b \\rfloor$.","title":"Eisenstein's lemma for Jacobi symbols (odd coprime form)"},{"id":"NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma/eisenstein-lemma-jacobi-symbol","kind":"node","lean":"MetaMathlibExt.eisenstein_lemma_jacobi_symbol","parent":5117,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma/index.html#eisenstein-lemma-jacobi-symbol","scope":"NumberTheory/congruences/power-residues-reciprocity/eisenstein-s-lemma","status":"fully_proved","summary":"Let $a$ and $b$ be positive odd natural numbers that are coprime. Then the Jacobi symbol $\\left(\\frac{a}{b}\\right)$ equals $(-1)^{S}$, where $S = \\sum_{i=1}^{(b-1)/2} \\lfloor ia/b \\rfloor$.","title":"Eisenstein's lemma for Jacobi symbols"},{"data_url":"scopes/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters.json","id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","kind":"scope","navigation":"module","parent":5109,"result":false,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","title":"Kronecker symbol and quadratic characters"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/is-fundamental-discriminant","kind":"node","lean":"MetaMathlibExt.IsFundamentalDiscriminant","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#is-fundamental-discriminant","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"An integer $D$ is a fundamental discriminant if either $D$ is squarefree and $D \\equiv 1 \\pmod 4$, or $D = 4m$ for some squarefree integer $m$ with $m \\equiv 2 \\pmod 4$ or $m \\equiv 3 \\pmod 4$.","title":"Fundamental discriminant"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/kronecker-at-two","kind":"node","lean":"MetaMathlibExt.kroneckerAtTwo","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#kronecker-at-two","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"For an integer $D$, the Kronecker symbol of $D$ at $2$ is $0$ if $D$ is even, $1$ if $D \\equiv \\pm 1 \\pmod 8$, and $-1$ if $D \\equiv \\pm 3 \\pmod 8$.","title":"Kronecker symbol at two"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/kronecker-at-prime","kind":"node","lean":"MetaMathlibExt.kroneckerAtPrime","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#kronecker-at-prime","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"For an integer $D$ and a natural number $p$, the Kronecker symbol of $D$ at $p$ is the Kronecker symbol of $D$ at two if $p = 2$, the Legendre symbol $\\left(\\frac{D}{p}\\right)$ if $p$ is an odd prime, and $0$ if $p$ is not prime.","title":"Kronecker symbol at a prime"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/kronecker-sym","kind":"node","lean":"MetaMathlibExt.kroneckerSym","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#kronecker-sym","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"For an integer $D$ and a natural number $n$, the Kronecker symbol $\\left(\\frac{D}{n}\\right)$ is defined as follows. If $n = 0$, it is $1$ when $D = \\pm 1$ and $0$ otherwise. If $n \\ge 1$ and $n = p_1 p_2 \\cdots p_k$ is the factorization of $n$ into primes, listed with multiplicity, it is the product of the Kronecker symbols of $D$ at the primes $p_1, \\ldots, p_k$ (the empty product $1$ when $n = 1$).","title":"Kronecker symbol"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/kronecker-sym-prime","kind":"node","lean":"MetaMathlibExt.kroneckerSym_prime","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#kronecker-sym-prime","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"For every integer $D$ and every prime $p$, the Kronecker symbol $\\left(\\frac{D}{p}\\right)$ equals the Kronecker symbol of $D$ at the prime $p$.","title":"Kronecker symbol with a prime denominator"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/kronecker-sym-eq-jacobi-sym-of-prime-ne","kind":"node","lean":"MetaMathlibExt.kroneckerSym_eq_jacobiSym_of_prime_ne_two","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#kronecker-sym-eq-jacobi-sym-of-prime-ne","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"For every integer $D$ and every prime $p \\ne 2$, the Kronecker symbol $\\left(\\frac{D}{p}\\right)$ equals the Jacobi symbol $\\left(\\frac{D}{p}\\right)$.","title":"Kronecker and Jacobi symbols agree at odd primes"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/quadratic-dirichlet-character","kind":"node","lean":"MetaMathlibExt.quadraticDirichletCharacter","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#quadratic-dirichlet-character","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"Let $D$ be a fundamental discriminant. Then there is a unique integer-valued Dirichlet character $\\chi$ modulo $|D|$ such that $\\chi(n \\bmod |D|) = \\left(\\frac{D}{n}\\right)$ for every natural number $n$, where $\\left(\\frac{D}{n}\\right)$ is the Kronecker symbol.","title":"Quadratic Dirichlet character of a fundamental discriminant"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/jacobi-sym-nat-abs-eq-kronecker-sym-of-p","kind":"node","lean":"MetaMathlibExt.jacobiSym_natAbs_eq_kroneckerSym_of_prime_ne_two_of_one_mod_four_pos","parent":5120,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quadratic reciprocity for the Kronecker symbol, positive "],["math","\\(D \\equiv 1 \\pmod 4\\)"]],"source":"Quadratic reciprocity for the Kronecker symbol, positive $D \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#jacobi-sym-nat-abs-eq-kronecker-sym-of-p","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"Let $D$ be a positive integer with $D \\equiv 1 \\pmod 4$ and let $p$ be an odd prime. Then the Jacobi symbol $\\left(\\frac{p}{|D|}\\right)$ equals the Kronecker symbol $\\left(\\frac{D}{p}\\right)$.","title":"Quadratic reciprocity for the Kronecker symbol, positive $D \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/jacobi-sym-nat-abs-eq-kronecker-sym-of-p-2","kind":"node","lean":"MetaMathlibExt.jacobiSym_natAbs_eq_kroneckerSym_of_prime_ne_two_of_one_mod_four_neg","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#jacobi-sym-nat-abs-eq-kronecker-sym-of-p-2","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"Let $D$ be a negative integer with $D \\equiv 1 \\pmod 4$ and let $p$ be an odd prime. Then the Jacobi symbol $\\left(\\frac{p}{|D|}\\right)$ equals the Kronecker symbol $\\left(\\frac{D}{p}\\right)$.","title":"Kronecker symbol at odd primes for negative integers congruent to 1 mod 4"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/jacobi-sym-nat-abs-eq-kronecker-sym-of-p-3","kind":"node","lean":"MetaMathlibExt.jacobiSym_natAbs_eq_kroneckerSym_of_prime_ne_two_of_one_mod_four","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#jacobi-sym-nat-abs-eq-kronecker-sym-of-p-3","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"Let $D$ be an integer with $D \\equiv 1 \\pmod 4$ and let $p$ be an odd prime. Then the Jacobi symbol $\\left(\\frac{p}{|D|}\\right)$ equals the Kronecker symbol $\\left(\\frac{D}{p}\\right)$.","title":"Kronecker symbol at odd primes for integers congruent to 1 mod 4"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/chi4-mul-jacobi-sym-nat-abs-eq-kronecker","kind":"node","lean":"MetaMathlibExt.chi4_mul_jacobiSym_natAbs_eq_kroneckerSym_of_odd_of_three_mod_four","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#chi4-mul-jacobi-sym-nat-abs-eq-kronecker","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"Let $\\chi_4$ be the nontrivial Dirichlet character modulo $4$, so $\\chi_4(n) = 1$ for $n \\equiv 1 \\pmod 4$ and $\\chi_4(n) = -1$ for $n \\equiv 3 \\pmod 4$. Let $D$ be an integer with $D \\equiv 3 \\pmod 4$ and let $p$ be an odd natural number. Then $\\chi_4(p) \\cdot \\left(\\frac{p}{|D|}\\right) = \\left(\\frac{D}{p}\\right)$, where $\\left(\\frac{p}{|D|}\\right)$ is the Jacobi symbol and $\\left(\\frac{D}{p}\\right)$ is the\u2026","title":"Kronecker symbol at odd numbers for integers congruent to 3 mod 4"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/jacobi-sym-four-mul-of-odd","kind":"node","lean":"MetaMathlibExt.jacobiSym_four_mul_of_odd","parent":5120,"reading_title":{"format":"inline-parts-v1","parts":[["text","Jacobi symbol of "],["math","\\(4m\\)"],["text"," at an odd denominator"]],"source":"Jacobi symbol of $4m$ at an odd denominator"},"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#jacobi-sym-four-mul-of-odd","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"For every integer $m$ and every odd natural number $p$, the Jacobi symbol $\\left(\\frac{4m}{p}\\right)$ equals $\\left(\\frac{m}{p}\\right)$.","title":"Jacobi symbol of $4m$ at an odd denominator"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/jacobi-sym-two-mul-of-odd","kind":"node","lean":"MetaMathlibExt.jacobiSym_two_mul_of_odd","parent":5120,"reading_title":{"format":"inline-parts-v1","parts":[["text","Jacobi symbol of "],["math","\\(2m\\)"],["text"," at an odd denominator"]],"source":"Jacobi symbol of $2m$ at an odd denominator"},"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#jacobi-sym-two-mul-of-odd","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"For every integer $m$ and every odd natural number $p$, the Jacobi symbol $\\left(\\frac{2m}{p}\\right)$ equals $\\chi_8(p) \\left(\\frac{m}{p}\\right)$, where $\\chi_8$ is the Dirichlet character modulo $8$ with $\\chi_8(n) = 1$ for $n \\equiv \\pm 1 \\pmod 8$, $\\chi_8(n) = -1$ for $n \\equiv \\pm 3 \\pmod 8$, and $\\chi_8(n) = 0$ for even $n$.","title":"Jacobi symbol of $2m$ at an odd denominator"},{"id":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/kronecker-jacobi-bridge-for-4m-with-m-3","kind":"node","lean":"MetaMathlibExt.chi4_mul_jacobiSym_natAbs_eq_kroneckerSym_of_odd_of_four_mul_three_mod_four","parent":5120,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters/index.html#kronecker-jacobi-bridge-for-4m-with-m-3","scope":"NumberTheory/congruences/power-residues-reciprocity/kronecker-symbol-and-quadratic-characters","status":"fully_proved","summary":"Even `m % 4 = 3` subcase for odd `p`: reduce `4 * m` to `m`, identify `kroneckerSym` with `jacobiSym` at odd denominators, then apply the `D % 4 = 3` bridge to `m` itself. Covers prime and composite odd denominators alike.","title":"Kronecker\u2013Jacobi bridge for 4m with m \u2261 3 mod 4"},{"data_url":"scopes/NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two.json","id":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two","kind":"scope","navigation":"module","parent":5109,"result":false,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/index.html","scope":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two","status":"fully_proved","title":"Squares modulo powers of two"},{"id":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/sq-mod-eight","kind":"node","lean":"Nat.sq_mod_eight","parent":5135,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/index.html#sq-mod-eight","scope":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two","status":"fully_proved","summary":"For every natural number $r$, the remainder of $r^2$ on division by $8$ is $0$, $1$ or $4$.","title":"Squares of natural numbers modulo 8"},{"id":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/not-is-square-of-mod-eight-eq-three","kind":"node","lean":"Nat.not_isSquare_of_mod_eight_eq_three","parent":5135,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/index.html#not-is-square-of-mod-eight-eq-three","scope":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two","status":"fully_proved","summary":"If a natural number $n$ satisfies $n \\equiv 3 \\pmod 8$, then $n$ is not the square of a natural number.","title":"Numbers congruent to 3 mod 8 are not squares"},{"id":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/not-is-square-of-mod-eight-eq-six","kind":"node","lean":"Nat.not_isSquare_of_mod_eight_eq_six","parent":5135,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/index.html#not-is-square-of-mod-eight-eq-six","scope":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two","status":"fully_proved","summary":"If a natural number $n$ satisfies $n \\equiv 6 \\pmod 8$, then $n$ is not the square of a natural number.","title":"Numbers congruent to 6 mod 8 are not squares"},{"id":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/not-is-square-sub-sq-of-mod-eight-eq-six","kind":"node","lean":"Nat.not_isSquare_sub_sq_of_mod_eight_eq_six","parent":5135,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/index.html#not-is-square-sub-sq-of-mod-eight-eq-six","scope":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two","status":"fully_proved","summary":"Let $N$ and $k$ be natural numbers with $N \\equiv 6 \\pmod 8$ and $k^2 < N$. Then $N - k^2$ is not the square of a natural number.","title":"A number that is 6 mod 8 minus a smaller square is not a square"},{"id":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/sq-mod-sixteen-of-odd","kind":"node","lean":"Nat.sq_mod_sixteen_of_odd","parent":5135,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/index.html#sq-mod-sixteen-of-odd","scope":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two","status":"fully_proved","summary":"For every odd natural number $k$, either $k^2 \\equiv 1 \\pmod{16}$ and $k \\equiv 1$ or $7 \\pmod 8$, or $k^2 \\equiv 9 \\pmod{16}$ and $k \\equiv 3$ or $5 \\pmod 8$.","title":"Odd squares modulo 16"},{"id":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/sq-emod-eight","kind":"node","lean":"Int.sq_emod_eight","parent":5135,"result":true,"route":"../roadmap/NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two/index.html#sq-emod-eight","scope":"NumberTheory/congruences/power-residues-reciprocity/squares-modulo-powers-of-two","status":"fully_proved","summary":"For every integer $x$, the remainder of $x^2$ on division by $8$, taken in $\\{0, 1, \\dots, 7\\}$, is $0$, $1$ or $4$.","title":"Squares of integers modulo 8"},{"data_url":"scopes/NumberTheory/congruences/covering-systems.json","id":"NumberTheory/congruences/covering-systems","kind":"scope","parent":4978,"result":false,"route":"../roadmap/NumberTheory/congruences/covering-systems/index.html","scope":"NumberTheory/congruences/covering-systems","status":"can_prove","summary":"Covering and exact covering systems of residue classes, minimum moduli, Erd\u0151s\u2013Graham and Fraenkel-type results.","title":"Covering systems"},{"id":"NumberTheory/congruences/covering-systems/covering-system","kind":"node","lean":"CoveringSystem","parent":5142,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/index.html#covering-system","scope":"NumberTheory/congruences/covering-systems","status":"fully_proved","summary":"A covering system of a commutative semiring $R$ is a finite family of cosets $r_i + I_i = \\{r_i + a : a \\in I_i\\}$, where each $I_i$ is an ideal of $R$ different from both $\\{0\\}$ and $R$, whose union is all of $R$. In the case $R = \\mathbb{Z}$, this corresponds to a choice of finitely many congruences $x \\equiv r_i \\pmod{n_i}$ with moduli $n_i \\ge 2$ such that every integer satisfies at least one of them.","title":"Covering system"},{"id":"NumberTheory/congruences/covering-systems/strict-covering-system","kind":"node","lean":"StrictCoveringSystem","parent":5142,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/index.html#strict-covering-system","scope":"NumberTheory/congruences/covering-systems","status":"fully_proved","summary":"A covering system of a commutative semiring $R$, given by cosets $r_i + I_i$, is strict if the ideals $I_i$ are pairwise distinct. In the case $R = \\mathbb{Z}$, this means that the congruences defining it have pairwise distinct moduli.","title":"Strict covering system"},{"data_url":"scopes/NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste.json","id":"NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste","kind":"scope","navigation":"module","parent":5142,"result":false,"route":"../roadmap/NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste/index.html","scope":"NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste","status":"can_prove","title":"A question of Erd\u0151s and Graham on covering systems"},{"id":"NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste/not-covering-of-nonintersecting","kind":"node","lean":"MathlibExt.NumberTheory.ErdosGrahamCoveringWanted.not_covering_of_nonintersecting","parent":5145,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste/index.html#not-covering-of-nonintersecting","scope":"NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste","status":"can_prove","summary":"Let $n$ be a natural number and let $D$ be the finite set of divisors of $n$ greater than $1$, taken empty when $n = 0$. Call $n$ non-intersecting if there are integers $a_d$, one for each natural number $d$, such that for all distinct natural numbers $d, d' > 1$ dividing $n$ (every $d$ divides $0$), if the classes $a_d \\pmod d$ and $a_{d'} \\pmod{d'}$ have an integer in common, then $d$ and $d'$ are coprime. If $n$\u2026","title":"Non-intersecting integers admit no covering system of divisor form"},{"id":"NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste/one-le-sum-inv-moduli","kind":"node","lean":"MathlibExt.NumberTheory.ErdosGrahamCoveringWanted.one_le_sum_inv_moduli","parent":5145,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste/index.html#one-le-sum-inv-moduli","scope":"NumberTheory/congruences/covering-systems/a-question-of-erd-s-and-graham-on-covering-syste","status":"can_prove","summary":"Let $d_1, \\dots, d_t$ be positive integers and $a_1, \\dots, a_t$ integers such that every integer $z$ satisfies $z \\equiv a_i \\pmod{d_i}$ for some $i$. Then $\\sum_{i=1}^t 1/d_i \\ge 1$.","title":"Covering systems have reciprocal-moduli sum at least one"},{"data_url":"scopes/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs.json","id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","kind":"scope","navigation":"module","parent":5142,"result":false,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","title":"Exact covering system digraphs"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-adj","kind":"node","lean":"MetaMathlibExt.ecsdAdj","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-adj","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"fully_proved","summary":"Let $r \\in \\mathbb{N}$ and let $d_1, \\dots, d_r$ and $a_1, \\dots, a_r$ be integers. The digraph $G(d_1 n + a_1, \\dots, d_r n + a_r)$ has vertex set $\\mathbb{Z}$ and an edge $x \\to y$ iff $y = d_i x + a_i$ for some $i \\in \\{1, \\dots, r\\}$.","title":"Edge relation of an exact covering system digraph"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-adj-two","kind":"node","lean":"MetaMathlibExt.ecsdAdj_two","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-adj-two","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"fully_proved","summary":"For all integers $d_1, d_2, a_1, a_2, x, y$, the digraph $G(d_1 n + a_1, d_2 n + a_2)$ has an edge $x \\to y$ iff $y = d_1 x + a_1$ or $y = d_2 x + a_2$.","title":"Out-neighbours in a two-rule digraph"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/is-kdescendant","kind":"node","lean":"MetaMathlibExt.IsKDescendant","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-descendants in a digraph on the integers"]],"source":"$k$-descendants in a digraph on the integers"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#is-kdescendant","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"fully_proved","summary":"Let $R$ be a digraph on $\\mathbb{Z}$, let $k \\in \\mathbb{N}$ and let $m, n \\in \\mathbb{Z}$. Then $n$ is a $k$-descendant of $m$ in $R$ when there is a directed walk of exactly $k$ edges from $m$ to $n$, that is, integers $m = x_0, x_1, \\dots, x_k = n$ such that $R$ has an edge $x_{i-1} \\to x_i$ for every $1 \\le i \\le k$.","title":"$k$-descendants in a digraph on the integers"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/is-kdescendant-zero","kind":"node","lean":"MetaMathlibExt.isKDescendant_zero","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","The only "],["math","\\(0\\)"],["text","-descendant of a vertex is itself"]],"source":"The only $0$-descendant of a vertex is itself"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#is-kdescendant-zero","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"fully_proved","summary":"Let $R$ be a digraph on $\\mathbb{Z}$ and let $m, n \\in \\mathbb{Z}$. Then $n$ is a $0$-descendant of $m$ in $R$ if and only if $n = m$.","title":"The only $0$-descendant of a vertex is itself"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/is-kdescendant-one","kind":"node","lean":"MetaMathlibExt.isKDescendant_one","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(1\\)"],["text","-descendants are the out-neighbours"]],"source":"The $1$-descendants are the out-neighbours"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#is-kdescendant-one","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"fully_proved","summary":"Let $R$ be a digraph on $\\mathbb{Z}$ and let $m, n \\in \\mathbb{Z}$. Then $n$ is a $1$-descendant of $m$ in $R$ if and only if $R$ has an edge $m \\to n$.","title":"The $1$-descendants are the out-neighbours"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/is-digraph-cycle","kind":"node","lean":"MetaMathlibExt.IsDigraphCycle","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#is-digraph-cycle","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"fully_proved","summary":"Let $R$ be a digraph on $\\mathbb{Z}$, given by its edge relation. A list $c = (c_1, \\dots, c_t)$ of integers is a cycle of $R$ when it is nonempty, its entries are pairwise distinct, and $R$ has the edges $c_1 \\to c_2, \\dots, c_{t-1} \\to c_t$ and $c_t \\to c_1$ (for $t = 1$ this means a loop $c_1 \\to c_1$).","title":"Cycles of a digraph on the integers"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/is-digraph-cycle-singleton","kind":"node","lean":"MetaMathlibExt.isDigraphCycle_singleton","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#is-digraph-cycle-singleton","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"fully_proved","summary":"Let $R$ be a digraph on $\\mathbb{Z}$ and let $x \\in \\mathbb{Z}$. The one-element list $(x)$ is a cycle of $R$ if and only if $R$ has a loop $x \\to x$.","title":"One-vertex cycles of a digraph on the integers"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/is-digraph-cycle-pair","kind":"node","lean":"MetaMathlibExt.isDigraphCycle_pair","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#is-digraph-cycle-pair","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"fully_proved","summary":"Let $R$ be a digraph on $\\mathbb{Z}$ and let $x, y \\in \\mathbb{Z}$. The list $(x, y)$ is a cycle of $R$ if and only if $x \\ne y$ and $R$ has both edges $x \\to y$ and $y \\to x$.","title":"Two-vertex cycles of a digraph on the integers"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-adj-shift-rel-iso","kind":"node","lean":"MetaMathlibExt.ecsdAdj_shift_relIso","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-adj-shift-rel-iso","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $r \\in \\mathbb{N}$, let $d_1, \\dots, d_r$, $a_1, \\dots, a_r$ and $k$ be integers, and put $a_i' = a_i - (d_i - 1)k$. Then the map $x \\mapsto x + k$ is a digraph isomorphism from $G(d_1 n + a_1, \\dots, d_r n + a_r)$ onto $G(d_1 n + a_1', \\dots, d_r n + a_r')$.","title":"Translation isomorphism of exact covering system digraphs"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-adj-neg-shift-rel-iso","kind":"node","lean":"MetaMathlibExt.ecsdAdj_neg_shift_relIso","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-adj-neg-shift-rel-iso","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $r \\in \\mathbb{N}$, let $d_1, \\dots, d_r$, $a_1, \\dots, a_r$ and $k$ be integers, and put $a_i' = -a_i + (d_i - 1)k$. Then the map $x \\mapsto -(x + k)$ is a digraph isomorphism from $G(d_1 n + a_1, \\dots, d_r n + a_r)$ onto $G(d_1 n + a_1', \\dots, d_r n + a_r')$.","title":"Reflection isomorphism of exact covering system digraphs"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-type-one-rel-iso","kind":"node","lean":"MetaMathlibExt.ecsd_type_one_relIso","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normal form of the ECSDs "],["math","\\(G(2n + a_1, 2n + a_2)\\)"]],"source":"Normal form of the ECSDs $G(2n + a_1, 2n + a_2)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-type-one-rel-iso","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"For all integers $a_1, a_2$ with $a_1 - a_2$ odd, the exact covering system digraph $G(2n + a_1, 2n + a_2)$, with an edge $m \\to 2m + a_1$ and an edge $m \\to 2m + a_2$ for every integer $m$, is isomorphic as a digraph to $G(2n, 2n - a)$ for some odd natural number $a$.","title":"Normal form of the ECSDs $G(2n + a_1, 2n + a_2)$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-type-two-rel-iso","kind":"node","lean":"MetaMathlibExt.ecsd_type_two_relIso","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-type-two-rel-iso","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a_1, a_2 \\in \\mathbb{Z}$ with $a_1 - a_2$ odd, and write $G(2n + a_1, -2n + a_2)$ for the exact covering system digraph on $\\mathbb{Z}$ with an edge $m \\to n$ exactly when $n = 2m + a_1$ or $n = -2m + a_2$. Then there is an odd natural number $a$ such that $G(2n + a_1, -2n + a_2)$ is isomorphic as a digraph to $G(2n, -2n + a)$, the digraph on $\\mathbb{Z}$ with an edge $m \\to n$ exactly when $n = 2m$ or\u2026","title":"Normal form of Type 2 exact covering system digraphs"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-type-three-rel-iso","kind":"node","lean":"MetaMathlibExt.ecsd_type_three_relIso","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normal forms of the ECSDs "],["math","\\(G(-2n + a_1, -2n + a_2)\\)"]],"source":"Normal forms of the ECSDs $G(-2n + a_1, -2n + a_2)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-type-three-rel-iso","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"For all integers $a_1, a_2$ with $a_1 - a_2$ odd, the exact covering system digraph $G(-2n + a_1, -2n + a_2)$, with an edge $m \\to -2m + a_1$ and an edge $m \\to -2m + a_2$ for every integer $m$, is isomorphic as a digraph to one of the following: $G(-2n, -2n + a)$ for an odd natural number $a$ (Type 3.1); $G(-2n + 1, -2n + a)$ for an even natural number $a$ with $a \\equiv 1 \\pmod 3$ (Type 3.2); or\u2026","title":"Normal forms of the ECSDs $G(-2n + a_1, -2n + a_2)$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-two-two-sub-cycles","kind":"node","lean":"MetaMathlibExt.ecsd_two_two_sub_cycles","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycles of the digraph "],["math","\\(G(2n, 2n - a)\\)"]],"source":"Cycles of the digraph $G(2n, 2n - a)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-two-two-sub-cycles","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"For an integer $b$ write $G_b$ for the digraph $G(2n, 2n - b)$ on $\\mathbb{Z}$, with an edge $m \\to 2m$ and an edge $m \\to 2m - b$ for every integer $m$. Let $a$ be an odd natural number, write $x \\bmod a$ for the residue of $x$ in $\\{0, \\dots, a - 1\\}$, and write $\\operatorname{ord}(2, q)$ for the multiplicative order of $2$ modulo $q$. Then (i) $G_a$ has a loop at $m$ if and only if $m = 0$ or $m = a$; (ii) $G_a$\u2026","title":"Cycles of the digraph $G(2n, 2n - a)$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-two-neg-two-descendants","kind":"node","lean":"MetaMathlibExt.ecsd_two_neg_two_descendants","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Descendants in the digraph "],["math","\\(G(2n, -2n + a)\\)"]],"source":"Descendants in the digraph $G(2n, -2n + a)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-two-neg-two-descendants","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a$ and $m$ be integers and let $k \\ge 1$. In the digraph $G(2n, -2n + a)$ on $\\mathbb{Z}$, with an edge $x \\to 2x$ and an edge $x \\to -2x + a$ for every integer $x$, the $k$-descendants of $m$ (the ends of directed walks of exactly $k$ edges from $m$) are exactly the integers $2^k m$, the integers $2^k m - ba$ and $-(2^k m - ba)$ for natural numbers $b$ with $1 \\le b < 2^{k-1}$, and $-(2^k m - 2^{k-1} a)$.","title":"Descendants in the digraph $G(2n, -2n + a)$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-two-neg-two-cycles","kind":"node","lean":"MetaMathlibExt.ecsd_two_neg_two_cycles","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycles of the digraph "],["math","\\(G(2n, -2n + a)\\)"]],"source":"Cycles of the digraph $G(2n, -2n + a)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-two-neg-two-cycles","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"For an integer $b$ write $G_b$ for the digraph $G(2n, -2n + b)$ on $\\mathbb{Z}$, with an edge $m \\to 2m$ and an edge $m \\to -2m + b$ for every integer $m$. Let $a$ be an odd natural number. Then (i) $G_a$ has a loop at $m$ if and only if $m = 0$ or $3m = a$; (ii) $G_a$ has a directed walk of positive length from $m$ back to $m$ if and only if $0 \\le m < a/2$; (iii) for all natural numbers $t$ and $a'$ with\u2026","title":"Cycles of the digraph $G(2n, -2n + a)$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-neg-two-descendant-mod-eq","kind":"node","lean":"MetaMathlibExt.ecsd_neg_two_descendant_modEq","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-neg-two-descendant-mod-eq","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a_1, a_2, x, y$ be integers and $k \\in \\mathbb{N}$. If $y$ is a $k$-descendant of $x$ in $G(-2n + a_1, -2n + a_2)$, that is, the end of a directed walk of exactly $k$ edges from $x$, then","title":"Descendants modulo the difference of offsets"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-neg-two-cyclic-residues","kind":"node","lean":"MetaMathlibExt.ecsd_neg_two_cyclic_residues","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-neg-two-cyclic-residues","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a_1, a_2$ be integers with $a_1 - a_2$ odd. Then every residue class modulo $|a_1 - a_2|$ contains a vertex that lies on a directed cycle of $G(-2n + a_1, -2n + a_2)$.","title":"Cyclic vertices in every residue class"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-neg-two-neg-two-cycles","kind":"node","lean":"MetaMathlibExt.ecsd_neg_two_neg_two_cycles","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycles of "],["math","\\(G(-2n, -2n + a)\\)"]],"source":"Cycles of $G(-2n, -2n + a)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-neg-two-neg-two-cycles","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a$ be an odd natural number and $G = G(-2n, -2n + a)$. A cycle of $G$ is a list $(c_1, \\dots, c_t)$, $t \\ge 1$, of distinct integers with edges $c_1 \\to c_2, \\dots, c_{t-1} \\to c_t$ and $c_t \\to c_1$; a vertex is cyclic if it lies on a cycle. For an integer $x$ let $\\hat{x}$ be the representative of $x$ modulo $a$ in the interval $(-a/3, 2a/3]$, and let $\\operatorname{ord}_q(-2)$ be the multiplicative order of\u2026","title":"Cycles of $G(-2n, -2n + a)$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-type-three-two-cyclic-iff","kind":"node","lean":"MetaMathlibExt.ecsd_type_three_two_cyclic_iff","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cyclic vertices of "],["math","\\(G(-2n + 1, -2n + a)\\)"],["text"," when "],["math","\\(a \\equiv 1 \\pmod 3\\)"]],"source":"Cyclic vertices of $G(-2n + 1, -2n + a)$ when $a \\equiv 1 \\pmod 3$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-type-three-two-cyclic-iff","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a$ be an even natural number with $a \\equiv 1 \\pmod 3$ and let $m$ be an integer. The digraph $G(-2n + 1, -2n + a)$ on $\\mathbb{Z}$, with an edge $x \\to -2x + 1$ and an edge $x \\to -2x + a$ for every integer $x$, has a directed walk of positive length from $m$ back to $m$ if and only if $-(a - 1)/3 < m < 2a/3$.","title":"Cyclic vertices of $G(-2n + 1, -2n + a)$ when $a \\equiv 1 \\pmod 3$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-type-three-three-cyclic-bounds","kind":"node","lean":"MetaMathlibExt.ecsd_type_three_three_cyclic_bounds","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds on the cyclic vertices of "],["math","\\(G(-2n + 1, -2n + a)\\)"],["text"," when "],["math","\\(a \\equiv 2 \\pmod 3\\)"]],"source":"Bounds on the cyclic vertices of $G(-2n + 1, -2n + a)$ when $a \\equiv 2 \\pmod 3$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-type-three-three-cyclic-bounds","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a$ be an even integer with $a \\equiv 2 \\pmod 3$, and let $m$ be an integer such that the digraph $G(-2n + 1, -2n + a)$ on $\\mathbb{Z}$, with an edge $x \\to -2x + 1$ and an edge $x \\to -2x + a$ for every integer $x$, has a directed walk of positive length from $m$ back to $m$. If $a \\ge 2$, then $-(a - 1)/3 < m < 2a/3$; if $a \\le -2$, then $-2(|a| + 1)/3 < m < (|a| + 3)/3$.","title":"Bounds on the cyclic vertices of $G(-2n + 1, -2n + a)$ when $a \\equiv 2 \\pmod 3$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-type-three-three-cyclic","kind":"node","lean":"MetaMathlibExt.ecsd_type_three_three_cyclic","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cyclic vertices of "],["math","\\(G(-2n + 1, -2n + a)\\)"],["text"," and their residues when "],["math","\\(a \\equiv 2 \\pmod 3\\)"]],"source":"Cyclic vertices of $G(-2n + 1, -2n + a)$ and their residues when $a \\equiv 2 \\pmod 3$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-type-three-three-cyclic","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a$ be an even integer with $a \\equiv 2 \\pmod 3$, and call an integer $m$ cyclic if the digraph $G(-2n + 1, -2n + a)$ on $\\mathbb{Z}$, with an edge $x \\to -2x + 1$ and an edge $x \\to -2x + a$ for every integer $x$, has a directed walk of positive length from $m$ back to $m$. Then: if $a \\ge 2$, every integer $m$ with $-(a - 1)/3 < m < 2a/3$ is cyclic; if $a \\le -2$, every integer $m$ with\u2026","title":"Cyclic vertices of $G(-2n + 1, -2n + a)$ and their residues when $a \\equiv 2 \\pmod 3$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-digit-representation-iff","kind":"node","lean":"MetaMathlibExt.ecsd_digit_representation_iff","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-digit-representation-iff","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $d \\ge 2$ be a natural number, let $D$ be an integer with $|D| = d$, and let $a_1, \\dots, a_d$ be integers such that for every integer $n$ there is exactly one $i$ with $D \\mid n - a_i$. Let $G = G(Dn + a_1, \\dots, Dn + a_d)$. Then every integer $z$ can be written as $z = \\sum_{j=0}^{k} b_j D^j$ with $k \\in \\mathbb{N}$ and digits $b_0, \\dots, b_k \\in \\{a_1, \\dots, a_d\\}$ if and only if the underlying undirected\u2026","title":"Digit representations and connected digraphs"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-one-component-iff","kind":"node","lean":"MetaMathlibExt.ecsd_one_component_iff","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Connectivity of "],["math","\\(G(-2n + 1, -2n + a)\\)"]],"source":"Connectivity of $G(-2n + 1, -2n + a)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-one-component-iff","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"For an even integer $a$, the underlying undirected graph of $G(-2n + 1, -2n + a)$ is connected iff $a = 1 + 3^m$ or $a = 1 - 3^m$ for some integer $m \\ge 0$.","title":"Connectivity of $G(-2n + 1, -2n + a)$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-path-length-to-a-sub-one","kind":"node","lean":"MetaMathlibExt.ecsd_path_length_to_a_sub_one","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shortest walks to "],["math","\\(\\pm(a - 1)\\)"],["text"," in "],["math","\\(G(-2n + 1, -2n + a)\\)"]],"source":"Shortest walks to $\\pm(a - 1)$ in $G(-2n + 1, -2n + a)$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-path-length-to-a-sub-one","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $m \\ge 1$ be an integer and $a = 1 + 3^m$ or $a = 1 - 3^m$. Then in $G(-2n + 1, -2n + a)$ the least $k \\in \\mathbb{N}$ such that $a - 1$ is a $k$-descendant of $0$ (the end of a directed walk of exactly $k$ edges from $0$) is $3^m$, and the least $k$ such that $-(a - 1)$ is a $k$-descendant of $0$ is also $3^m$.","title":"Shortest walks to $\\pm(a - 1)$ in $G(-2n + 1, -2n + a)$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-single-component-zero-cyclic-iff","kind":"node","lean":"MetaMathlibExt.ecsd_single_component_zero_cyclic_iff","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Degree-2 ECSDs with one component and "],["math","\\(0\\)"],["text"," on a cycle"]],"source":"Degree-2 ECSDs with one component and $0$ on a cycle"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-single-component-zero-cyclic-iff","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $d_1, d_2 \\in \\{2, -2\\}$ and let $a_1, a_2$ be integers with $a_1 - a_2$ odd, excluding the case $d_1 = d_2 = -2$ with $|a_1 - a_2| = 1$. Consider the exact covering system digraph $G(d_1 n + a_1, d_2 n + a_2)$, with an edge $m \\to d_i m + a_i$ for every integer $m$ and $i = 1, 2$. It has a single component (any two integers are joined by a path in its underlying undirected graph) and a directed walk of\u2026","title":"Degree-2 ECSDs with one component and $0$ on a cycle"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/negabinary-digits-one-a-complete-iff","kind":"node","lean":"MetaMathlibExt.negabinary_digits_one_a_complete_iff","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Negabinary representations with digits "],["math","\\(1\\)"],["text"," and "],["math","\\(a\\)"]],"source":"Negabinary representations with digits $1$ and $a$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#negabinary-digits-one-a-complete-iff","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a \\in \\mathbb{Z}$. Every integer $z$ can be written as $z = \\sum_{j=0}^{k} b_j (-2)^j$ for some $k \\in \\mathbb{N}$ and digits $b_0, \\dots, b_k \\in \\{1, a\\}$ if and only if $a = 1 + 3^m$ or $a = 1 - 3^m$ for some integer $m \\ge 0$.","title":"Negabinary representations with digits $1$ and $a$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/negabinary-two-digits-complete-iff","kind":"node","lean":"MetaMathlibExt.negabinary_two_digits_complete_iff","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#negabinary-two-digits-complete-iff","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $a_1, a_2 \\in \\mathbb{Z}$ with $|a_1 - a_2| \\ne 1$. Every integer $z$ can be written as $z = \\sum_{j=0}^{k} b_j (-2)^j$ for some $k \\in \\mathbb{N}$ and digits $b_0, \\dots, b_k \\in \\{a_1, a_2\\}$ if and only if $|a_1 - a_2| = 3^m$ for some integer $m \\ge 0$, neither $a_1$ nor $a_2$ is divisible by $3$, and $\\max\\{|a_1|, |a_2|\\} \\le 2|a_1 - a_2| - 1$.","title":"Negabinary representations with two digits"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/three-mul-dvd-two-pow-add-one-cube","kind":"node","lean":"MetaMathlibExt.three_mul_dvd_two_pow_add_one_cube","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility of "],["math","\\(2^{3n} + 1\\)"],["text"," from "],["math","\\(2^n + 1\\)"]],"source":"Divisibility of $2^{3n} + 1$ from $2^n + 1$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#three-mul-dvd-two-pow-add-one-cube","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"For every natural number $n$, if $3n$ divides $2^n + 1$, then $9n$ divides $2^{3n} + 1$.","title":"Divisibility of $2^{3n} + 1$ from $2^n + 1$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/three-mul-dvd-one-sub-neg-two-pow-cube","kind":"node","lean":"MetaMathlibExt.three_mul_dvd_one_sub_neg_two_pow_cube","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility of "],["math","\\(1 - (-2)^{3k}\\)"],["text"," from "],["math","\\(1 - (-2)^k\\)"]],"source":"Divisibility of $1 - (-2)^{3k}$ from $1 - (-2)^k$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#three-mul-dvd-one-sub-neg-two-pow-cube","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"For every integer $n$ and every natural number $k$, if $3n$ divides $1 - (-2)^k$, then $9n$ divides $1 - (-2)^{3k}$.","title":"Divisibility of $1 - (-2)^{3k}$ from $1 - (-2)^k$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/three-pow-not-dvd-two-pow-add-one-cube","kind":"node","lean":"MetaMathlibExt.three_pow_not_dvd_two_pow_add_one_cube","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounding the power of "],["math","\\(3\\)"],["text"," dividing "],["math","\\(2^{3k} + 1\\)"]],"source":"Bounding the power of $3$ dividing $2^{3k} + 1$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#three-pow-not-dvd-two-pow-add-one-cube","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"For all natural numbers $m$ and $k$, if $3^{m+1}$ does not divide $2^k + 1$, then $3^{m+2}$ does not divide $2^{3k} + 1$.","title":"Bounding the power of $3$ dividing $2^{3k} + 1$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/nine-dvd-of-three-mul-dvd-two-pow-add-on","kind":"node","lean":"MetaMathlibExt.nine_dvd_of_three_mul_dvd_two_pow_add_one","parent":5148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nine divides "],["math","\\(n\\)"],["text"," when "],["math","\\(n > 3\\)"],["text"," and "],["math","\\(3n\\)"],["text"," divides "],["math","\\(2^n + 1\\)"]],"source":"Nine divides $n$ when $n > 3$ and $3n$ divides $2^n + 1$"},"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#nine-dvd-of-three-mul-dvd-two-pow-add-on","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $n > 3$ be a natural number. If $3n$ divides $2^n + 1$, then $9$ divides $n$.","title":"Nine divides $n$ when $n > 3$ and $3n$ divides $2^n + 1$"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/ecsd-single-component-rel-iso","kind":"node","lean":"MetaMathlibExt.ecsd_single_component_relIso","parent":5148,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-system-digraphs/index.html#ecsd-single-component-rel-iso","scope":"NumberTheory/congruences/covering-systems/exact-covering-system-digraphs","status":"can_prove","summary":"Let $d_1, d_2 \\in \\{2, -2\\}$ and let $a_1, a_2$ be integers with $a_1 - a_2$ odd. If the exact covering system digraph $G(d_1 n + a_1, d_2 n + a_2)$, with an edge $m \\to d_i m + a_i$ for every integer $m$ and $i = 1, 2$, has a single component, that is, any two integers are joined by a path in its underlying undirected graph, then it is isomorphic as a digraph to $G(2n, -2n + 1)$ or to $G(-2n + 1, -2n + a)$ for\u2026","title":"Single-component degree-2 ECSDs up to isomorphism"},{"data_url":"scopes/NumberTheory/congruences/covering-systems/exact-covering-systems.json","id":"NumberTheory/congruences/covering-systems/exact-covering-systems","kind":"scope","navigation":"module","parent":5142,"result":false,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-systems/index.html","scope":"NumberTheory/congruences/covering-systems/exact-covering-systems","status":"fully_proved","title":"Exact covering systems"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-systems/indexed-cover-multiplicity","kind":"node","lean":"Int.indexedCoverMultiplicity","parent":5182,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-systems/index.html#indexed-cover-multiplicity","scope":"NumberTheory/congruences/covering-systems/exact-covering-systems","status":"fully_proved","summary":"Given $k \\in \\mathbb{N}$, moduli $m_0, \\dots, m_{k-1} \\in \\mathbb{N}$ and residues $r_0, \\dots, r_{k-1} \\in \\mathbb{Z}$, the multiplicity of an integer $z$ is the number of indices $i < k$ with $z \\equiv r_i \\pmod{m_i}$.","title":"Multiplicity of an indexed system of residue classes"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-systems/is-exact-cover-of","kind":"node","lean":"Int.IsExactCoverOf","parent":5182,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-systems/index.html#is-exact-cover-of","scope":"NumberTheory/congruences/covering-systems/exact-covering-systems","status":"fully_proved","summary":"Let $k \\in \\mathbb{N}$, let $m_0, \\dots, m_{k-1} \\in \\mathbb{N}$ be moduli and $r_0, \\dots, r_{k-1} \\in \\mathbb{Z}$ residues, and let $M \\in \\mathbb{N}$ and $T \\subseteq \\mathbb{Z}$. The system of residue classes $r_i \\bmod m_i$ is an exact $M$-cover of $T$ when $k \\ge 2$, every $m_i$ is positive, every $z \\in T$ satisfies $z \\equiv r_i \\pmod{m_i}$ for exactly $M$ indices $i < k$, and every integer $z \\notin T$\u2026","title":"Exact covers by residue classes"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-systems/fraenkel-exact-m-cover-bernoulli-recurre","kind":"node","lean":"MetaMathlibExt.fraenkel_exact_M_cover_bernoulli_recurrence_general","parent":5182,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-systems/index.html#fraenkel-exact-m-cover-bernoulli-recurre","scope":"NumberTheory/congruences/covering-systems/exact-covering-systems","status":"fully_proved","summary":"Let $q$, $M$ and $m \\ge 1$ be natural numbers, and for $i = 1, \\dots, q$ let $n_i$ be a positive integer and $a_i$ an integer with $0 \\le a_i < n_i$. Suppose the residue classes $a_i \\bmod n_i$, counted with repetition, form an exact $M$-cover of $\\mathbb{Z}$: every integer $k$ satisfies $k \\equiv a_i \\pmod{n_i}$ for exactly $M$ indices $i$. Then","title":"Fraenkel's Bernoulli recurrence for exact covers by any number of classes"},{"id":"NumberTheory/congruences/covering-systems/exact-covering-systems/fraenkel-exact-m-cover-bernoulli-recurre-2","kind":"node","lean":"MetaMathlibExt.fraenkel_exact_M_cover_bernoulli_recurrence","parent":5182,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/exact-covering-systems/index.html#fraenkel-exact-m-cover-bernoulli-recurre-2","scope":"NumberTheory/congruences/covering-systems/exact-covering-systems","status":"fully_proved","summary":"Let $q \\ge 2$, $m \\ge 1$ and $M$ be natural numbers, and for $i = 1, \\dots, q$ let $n_i$ be a positive integer and $a_i$ an integer with $0 \\le a_i < n_i$. Suppose the residue classes $a_i \\bmod n_i$, counted with repetition, form an exact $M$-cover of $\\mathbb{Z}$: every integer $k$ satisfies $k \\equiv a_i \\pmod{n_i}$ for exactly $M$ indices $i$. Then","title":"Fraenkel's Bernoulli recurrence for exact covers"},{"data_url":"scopes/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems.json","id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","kind":"scope","navigation":"module","parent":5142,"result":false,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","title":"Minimum modulus of covering systems"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/exists-modulus-ge-108-of-distinct-coveri","kind":"node","lean":"MetaMathlibExt.exists_modulus_ge_108_of_distinct_covering_min_five","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#exists-modulus-ge-108-of-distinct-coveri","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $C$ be a distinct covering system: a finite set of congruences $x \\equiv r \\pmod m$ with $r \\in \\mathbb{Z}$ and $m \\in \\mathbb{N}$, no two with the same modulus, such that every integer satisfies at least one of them. If every modulus of $C$ is at least $5$ and some modulus equals $5$, then some modulus of $C$ is at least $108$.","title":"Krukenberg's conjecture for minimum modulus five"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/exists-distinct-covering-min-six-max-168","kind":"node","lean":"MetaMathlibExt.exists_distinct_covering_min_six_max_168","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#exists-distinct-covering-min-six-max-168","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"There is a distinct covering system, that is, a finite set of congruences $x \\equiv r \\pmod m$ with $r \\in \\mathbb{Z}$, $m \\in \\mathbb{N}$ and no two with the same modulus, such that every integer satisfies at least one of them, whose moduli all lie between $6$ and $168$ and include both $6$ and $168$.","title":"A distinct covering system with moduli from 6 to 168"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/lcm-moduli-ge-1440-of-distinct-covering","kind":"node","lean":"MetaMathlibExt.lcm_moduli_ge_1440_of_distinct_covering_min_five","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#lcm-moduli-ge-1440-of-distinct-covering","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $C$ be a distinct covering system: a finite set of congruences $x \\equiv r \\pmod m$ with $r \\in \\mathbb{Z}$ and $m \\in \\mathbb{N}$, no two with the same modulus, such that every integer satisfies at least one of them. If every modulus of $C$ is at least $5$ and some modulus equals $5$, then the least common multiple of the moduli of $C$ is at least $1440$.","title":"Least common multiple of a covering system with minimum modulus five"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/lcm-moduli-ge-5040-of-distinct-covering","kind":"node","lean":"MetaMathlibExt.lcm_moduli_ge_5040_of_distinct_covering_min_six","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#lcm-moduli-ge-5040-of-distinct-covering","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $C$ be a distinct covering system: a finite set of congruences $x \\equiv r \\pmod m$ with $r \\in \\mathbb{Z}$ and $m \\in \\mathbb{N}$, no two with the same modulus, such that every integer satisfies at least one of them. If every modulus of $C$ is at least $6$ and some modulus equals $6$, then the least common multiple of the moduli of $C$ is at least $5040$.","title":"Least common multiple of a covering system with minimum modulus six"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/exists-distinct-covering-min-seven-lcm-1","kind":"node","lean":"MetaMathlibExt.exists_distinct_covering_min_seven_lcm_15120","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#exists-distinct-covering-min-seven-lcm-1","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"There is a distinct covering system, that is, a finite set of congruences $x \\equiv r \\pmod m$ with $r \\in \\mathbb{Z}$, $m \\in \\mathbb{N}$ and no two with the same modulus, such that every integer satisfies at least one of them, whose moduli are all at least $7$, include $7$, and have least common multiple $15120$.","title":"A distinct covering system with minimum modulus 7 and moduli of lcm 15120"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/is-covering-system-filter-not-dvd-of-lt","kind":"node","lean":"MetaMathlibExt.isCoveringSystem_filter_not_dvd_of_lt","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#is-covering-system-filter-not-dvd-of-lt","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $C$ be a distinct covering system: a finite set of congruences $x \\equiv r \\pmod m$ with $r \\in \\mathbb{Z}$ and $m \\in \\mathbb{N}$ (where $x \\equiv r \\pmod 0$ means $x = r$), no two with the same modulus, such that every integer satisfies at least one of them. Let $B \\in \\mathbb{N}$ be at least every modulus of $C$, let $p$ be a prime and $a \\ge 1$. If $B < p^a (p + 1)$, then the congruences of $C$ whose\u2026","title":"Discarding the classes whose modulus is divisible by a large prime power"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/exists-is-covering-system-merge-prime-po","kind":"node","lean":"MetaMathlibExt.exists_isCoveringSystem_merge_prime_power","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#exists-is-covering-system-merge-prime-po","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $C$ be a covering system with positive moduli: a finite set of congruences $x \\equiv r \\pmod m$ with $r \\in \\mathbb{Z}$ and $m \\ge 1$ such that every integer satisfies at least one of them. Let $p$ be a prime and $a \\ge 1$, and suppose exactly $p$ congruences of $C$ have modulus divisible by $p^a$, with moduli $p^a m_1, \\dots, p^a m_p$. Then for some $r \\in \\mathbb{Z}$, replacing these $p$ congruences by the\u2026","title":"Merging the p classes whose modulus is divisible by a prime power"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/is-covering-system-translate","kind":"node","lean":"MetaMathlibExt.isCoveringSystem_translate","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#is-covering-system-translate","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $C$ be a covering system: a finite set of congruences $x \\equiv a_i \\pmod{m_i}$ with $a_i \\in \\mathbb{Z}$ and $m_i \\in \\mathbb{N}$ (where $x \\equiv a_i \\pmod 0$ means $x = a_i$) such that every integer satisfies at least one of them. Then for every $t \\in \\mathbb{Z}$ the congruences $x \\equiv a_i + t \\pmod{m_i}$ also form a covering system.","title":"Translating a covering system"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/is-covering-system-swap-residues-mod-pri","kind":"node","lean":"MetaMathlibExt.isCoveringSystem_swap_residues_mod_prime","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#is-covering-system-swap-residues-mod-pri","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $C$ be a covering system with positive moduli: a finite set of congruences $x \\equiv a \\pmod m$ with $a \\in \\mathbb{Z}$ and $m \\ge 1$ such that every integer satisfies at least one of them. Let $L$ be the least common multiple of its moduli, $p$ a prime dividing $L$, and $a_1 < a_2 \\le p - 1$ natural numbers. For a prime $\\ell$ write $\\nu_\\ell(L)$ for the exponent of $\\ell$ in $L$, and let $C_p(\\alpha)$ be the\u2026","title":"Swapping two residues modulo a prime in a covering system"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/one-lt-sum-inv-moduli-of-distinct-coveri","kind":"node","lean":"MetaMathlibExt.one_lt_sum_inv_moduli_of_distinct_covering","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#one-lt-sum-inv-moduli-of-distinct-coveri","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $a_1 \\bmod m_1, \\dots, a_k \\bmod m_k$ be residue classes that cover $\\mathbb{Z}$, that is, every integer lies in at least one of them. If the moduli $m_1, \\dots, m_k$ are pairwise distinct and all at least $2$, then $\\sum_{i=1}^{k} 1/m_i > 1$.","title":"Mirsky\u2013Newman bound for distinct covering systems"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/exists-covering-divisors-replace-largest","kind":"node","lean":"MetaMathlibExt.exists_covering_divisors_replace_largest_prime","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#exists-covering-divisors-replace-largest","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"Let $m, L \\in \\mathbb{N}$, let $p \\ge m$ be a prime with $p \\nmid L$, let $q > p$ be the largest prime dividing $L$, and write $\\nu_q(L)$ for the exponent of $q$ in $L$. Suppose there are residues $r_d \\in \\mathbb{Z}$, one for each divisor $d \\ge m$ of $L$, such that every integer $x$ satisfies $x \\equiv r_d \\pmod d$ for some such $d$. Then the same holds with $L$ replaced by $p^{\\nu_q(L)} L / q^{\\nu_q(L)}$: there\u2026","title":"Replacing the largest prime factor of the modulus set"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.LargeModulusCoveringSystemsWanted.conjecture","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#conjecture","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"fully_proved","summary":"The statement that for every natural number $N$ there are finitely many residue classes $a_1 \\bmod m_1, \\dots, a_k \\bmod m_k$ that cover $\\mathbb{Z}$ (every integer lies in at least one of them) whose moduli are pairwise distinct and all at least $N$.","title":"Covering systems with arbitrarily large distinct moduli (OPG-493)"},{"id":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.LargeModulusCoveringSystemsWanted.conjecture_refuted","parent":5187,"result":true,"route":"../roadmap/NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems/index.html#conjecture-refuted","scope":"NumberTheory/congruences/covering-systems/minimum-modulus-of-covering-systems","status":"can_prove","summary":"There is a natural number $N$ such that no finite set of residue classes covering $\\mathbb{Z}$ has pairwise distinct moduli that are all at least $N$. Hence the conjecture of Open Garden problem OPG-493 is false.","title":"Hough's minimum modulus theorem"},{"data_url":"scopes/NumberTheory/algebraic-number-theory.json","id":"NumberTheory/algebraic-number-theory","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/index.html","scope":"NumberTheory/algebraic-number-theory","status":"can_state","summary":"Number fields and rings of integers: quadratic and cyclotomic fields, discriminants, ramification, ideals, adeles and ideles, class field theory.","title":"Algebraic number theory"},{"id":"NumberTheory/algebraic-number-theory/shanks-cubic-polynomial","kind":"node","lean":"MetaMathlibExt.shanksCubicPolynomial","parent":5201,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/index.html#shanks-cubic-polynomial","scope":"NumberTheory/algebraic-number-theory","status":"fully_proved","summary":"For an element $h$ of a ring $R$, the Shanks cubic polynomial is","title":"Shanks cubic polynomial"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/quadratic-fields.json","id":"NumberTheory/algebraic-number-theory/quadratic-fields","kind":"scope","parent":5201,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/index.html","scope":"NumberTheory/algebraic-number-theory/quadratic-fields","status":"can_prove","summary":"Gaussian, Eisenstein and \u2124[\u221ad] integers, prime factorization there, imaginary quadratic discriminants, class numbers, Stark\u2013Heegner.","title":"Quadratic fields and integer rings"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/stark-heegner-theorem","kind":"node","lean":"MetaMathlibExt.stark_heegner_theorem","parent":5203,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/index.html#stark-heegner-theorem","scope":"NumberTheory/algebraic-number-theory/quadratic-fields","status":"can_prove","summary":"Let $d$ be a positive squarefree integer and let $K$ be a number field of degree $2$ over $\\mathbb{Q}$ that contains an element $x$ with $x^2 = -d$, so that $K = \\mathbb{Q}(\\sqrt{-d})$. Then $K$ has class number $1$ if and only if $d \\in \\{1, 2, 3, 7, 11, 19, 43, 67, 163\\}$.","title":"Stark\u2013Heegner theorem"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/golden-ratio-integer-ring-units","kind":"node","lean":"MetaMathlibExt.golden_ratio_integer_ring_units","parent":5203,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/index.html#golden-ratio-integer-ring-units","scope":"NumberTheory/algebraic-number-theory/quadratic-fields","status":"fully_proved","summary":"Let $\\alpha = (1 + \\sqrt{5})/2$ and let $\\mathbb{Z}[\\alpha]$ be the subring of $\\mathbb{R}$ generated by $\\alpha$. A real number is a unit of $\\mathbb{Z}[\\alpha]$ if and only if it equals $\\alpha^n$ or $-\\alpha^n$ for some integer $n$.","title":"Units of the ring generated by the golden ratio"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic.json","id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","kind":"scope","navigation":"module","parent":5203,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","title":"Class number congruences for imaginary quadratic fields"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/class-number-sqrt-neg-prime-congr-bernou","kind":"node","lean":"MetaMathlibExt.classNumber_sqrt_neg_prime_congr_bernoulli_euler","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Class number congruences for "],["math","\\(\\mathbb{Q}(\\sqrt{-p})\\)"]],"source":"Class number congruences for $\\mathbb{Q}(\\sqrt{-p})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#class-number-sqrt-neg-prime-congr-bernou","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $p > 3$ be a prime and let $K$ be a number field of degree $2$ over $\\mathbb{Q}$ that contains a square root of $-p$, with class number $h$. Let $B_n$ be the Bernoulli numbers, $t/(e^t - 1) = \\sum_n B_n t^n/n!$, and $E_n$ the Euler numbers, $2/(e^t + e^{-t}) = \\sum_n E_n t^n/n!$ (so $E_2 = -1$). If $p \\equiv 3 \\pmod 4$, then $p$ divides the numerator, in lowest terms, of the rational number $h + 2B_{(p+1)/2}$.\u2026","title":"Class number congruences for $\\mathbb{Q}(\\sqrt{-p})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/eisenstein-real-period-eq-beta-gamma","kind":"node","lean":"MetaMathlibExt.eisensteinRealPeriod_eq_beta_gamma","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#eisenstein-real-period-eq-beta-gamma","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"The real number $\\varpi_1 = \\int_0^1 (1 - t^3)^{-2/3}\\,dt$ satisfies $\\varpi_1 = \\frac13 B(\\frac13, \\frac13) = \\frac{\\Gamma(1/3)^3}{2\\pi\\sqrt3} = \\frac{2^{1/3}}{3\\sqrt3} B(\\frac16, \\frac12)$, where $B$ is the Beta function.","title":"Beta and Gamma values of the real Eisenstein period"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/weierstrass-p-eisenstein-lattice","kind":"node","lean":"MetaMathlibExt.weierstrassP_eisenstein_lattice","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#weierstrass-p-eisenstein-lattice","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\rho = e^{2\\pi i/3}$, let $\\varpi_1 = \\int_0^1 (1 - t^3)^{-2/3}\\,dt$, and let $\\wp$ be the Weierstrass function of the lattice $\\Lambda = \\varpi_1 \\mathbb{Z} + \\varpi_1 \\rho \\mathbb{Z}$, with derivative $\\wp'$. Then for every $u \\in \\mathbb{C} \\setminus \\Lambda$: (a) $\\wp(\\rho u) = \\rho\\,\\wp(u)$, $\\wp(-u) = \\wp(u)$, $\\wp'(\\rho u) = \\wp'(u)$ and $\\wp'(-u) = -\\wp'(u)$; (b) $\\wp(\\bar u) = \\overline{\\wp(u)}$ and\u2026","title":"The Weierstrass function of the Eisenstein lattice"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/sl-add-formula","kind":"node","lean":"MetaMathlibExt.sl_add_formula","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#sl-add-formula","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\mathrm{Sl}, \\mathrm{Cl}$ be formal power series in $\\mathbb{Q}[[u]]$ with $\\mathrm{Sl}' = \\mathrm{Cl}^2$, $\\mathrm{Cl}' = -\\mathrm{Sl}^2$, $\\mathrm{Sl}(0) = 0$ and $\\mathrm{Cl}(0) = 1$. In the ring $\\mathbb{Q}[[u, v]]$ of formal power series in two variables,","title":"Addition formula for the Eisenstein function Sl"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/sl-rescale-int-expansion","kind":"node","lean":"MetaMathlibExt.sl_rescale_int_expansion","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#sl-rescale-int-expansion","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\mathrm{Sl}, \\mathrm{Cl}$ be formal power series in $\\mathbb{Q}[[u]]$ with $\\mathrm{Sl}' = \\mathrm{Cl}^2$, $\\mathrm{Cl}' = -\\mathrm{Sl}^2$, $\\mathrm{Sl}(0) = 0$ and $\\mathrm{Cl}(0) = 1$, and let $r$ be an integer. Then there is a power series $g \\in \\mathbb{Q}[[x]]$, with every coefficient in $\\mathbb{Z}[1/3]$, such that $\\mathrm{Sl}(ru) = r\\,\\mathrm{Sl}(u) + \\mathrm{Sl}(u)^4\\,g(\\mathrm{Sl}(u)^3)$. If moreover\u2026","title":"Sl(ru) as a power series in Sl(u)"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/arc-sl-subst-sl","kind":"node","lean":"MetaMathlibExt.arcSl_subst_sl","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power series inverse of "],["math","\\(\\mathrm{Sl}\\)"]],"source":"Power series inverse of $\\mathrm{Sl}$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#arc-sl-subst-sl","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $S, C \\in \\mathbb{Q}[[u]]$ be formal power series with $S' = C^2$, $C' = -S^2$, $S(0) = 0$ and $C(0) = 1$. Then","title":"Power series inverse of $\\mathrm{Sl}$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/factorial-mul-coeff-sl-is-int","kind":"node","lean":"MetaMathlibExt.factorial_mul_coeff_sl_isInt","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#factorial-mul-coeff-sl-is-int","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\mathrm{Sl}, \\mathrm{Cl}$ be formal power series in $\\mathbb{Q}[[u]]$ with $\\mathrm{Sl}' = \\mathrm{Cl}^2$, $\\mathrm{Cl}' = -\\mathrm{Sl}^2$, $\\mathrm{Sl}(0) = 0$ and $\\mathrm{Cl}(0) = 1$, and let $C_n$ be the coefficient of $u^n$ in $\\mathrm{Sl}$. Then $(3m + 1)!\\,C_{3m+1}$ is an integer for every $m \\ge 0$.","title":"The Hurwitz-normalized coefficients of Sl are integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/two-mul-factorial-coeff-inv-sl-mem-int-i","kind":"node","lean":"MetaMathlibExt.two_mul_factorial_coeff_inv_sl_mem_int_inv_three","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#two-mul-factorial-coeff-inv-sl-mem-int-i","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\mathrm{Sl}, \\mathrm{Cl}$ be formal power series in $\\mathbb{Q}[[u]]$ with $\\mathrm{Sl}' = \\mathrm{Cl}^2$, $\\mathrm{Cl}' = -\\mathrm{Sl}^2$, $\\mathrm{Sl}(0) = 0$ and $\\mathrm{Cl}(0) = 1$. Write $\\mathrm{Sl}(u) = u\\,S_0(u)$ with $S_0 \\in \\mathbb{Q}[[u]]$, and let $D_n$ be the coefficient of $u^{n+1}$ in $S_0^{-1}$, so that $1/\\mathrm{Sl}(u) = 1/u + \\sum_{n \\ge 0} D_n u^n$. Then for every $m \\ge 0$ the number\u2026","title":"The Hurwitz-normalized coefficients of 1/Sl"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/sl-hurwitz-numbers-den-coprime","kind":"node","lean":"MetaMathlibExt.sl_hurwitz_numbers_den_coprime","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#sl-hurwitz-numbers-den-coprime","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\mathrm{Sl}, \\mathrm{Cl}$ be formal power series in $\\mathbb{Q}[[u]]$ with $\\mathrm{Sl}' = \\mathrm{Cl}^2$, $\\mathrm{Cl}' = -\\mathrm{Sl}^2$, $\\mathrm{Sl}(0) = 0$ and $\\mathrm{Cl}(0) = 1$. Let $C_n$ be the coefficient of $u^n$ in $\\mathrm{Sl}$; write $\\mathrm{Sl}(u) = u\\,S_0(u)$ with $S_0 \\in \\mathbb{Q}[[u]]$ and let $D_n$ be the coefficient of $u^{n+1}$ in $S_0^{-1}$, so that\u2026","title":"Denominators of the Hurwitz-type numbers of Sl"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/eisenstein-phi-psi-properties","kind":"node","lean":"MetaMathlibExt.eisensteinPhiPsi_properties","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Identities for the Eisenstein elliptic functions "],["math","\\(\\varphi\\)"],["text"," and "],["math","\\(\\psi\\)"]],"source":"Identities for the Eisenstein elliptic functions $\\varphi$ and $\\psi$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#eisenstein-phi-psi-properties","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\rho = e^{2\\pi i/3}$, let $\\varpi_1 = \\int_0^1 (1 - t^3)^{-2/3}\\,dt$, and let $\\wp$ be the Weierstrass function of the lattice spanned by $\\varpi_1$ and $\\varpi_1\\rho$. Put $\\varphi(u) = 6\\wp(\\varpi_1 u)/(9 + \\wp'(\\varpi_1 u))$ and $\\psi(u) = (\\wp'(\\varpi_1 u) - 9)/(\\wp'(\\varpi_1 u) + 9)$. For every $u \\in \\mathbb{C}$ with $\\varpi_1 u$ off the lattice and $\\wp'(\\varpi_1 u) \\ne -9$:\u2026","title":"Identities for the Eisenstein elliptic functions $\\varphi$ and $\\psi$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/eisenstein-sl-inverse-arc-sl","kind":"node","lean":"MetaMathlibExt.eisensteinSl_inverse_arcSl","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{Sl}\\)"],["text"," and "],["math","\\(\\mathrm{ArcSl}\\)"],["text"," are inverse on the real segment"]],"source":"$\\mathrm{Sl}$ and $\\mathrm{ArcSl}$ are inverse on the real segment"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#eisenstein-sl-inverse-arc-sl","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\rho = e^{2\\pi i/3}$, let $\\varpi_1 = \\int_0^1 (1 - t^3)^{-2/3}\\,dt$, let $\\wp$ be the Weierstrass function of the lattice spanned by $\\varpi_1$ and $\\varpi_1\\rho$, and put $\\varphi(u) = 6\\wp(\\varpi_1 u)/(9 + \\wp'(\\varpi_1 u))$. For real $x$ put $\\mathrm{Sl}(x) = \\varphi(x/(-3\\varpi_1))$, and for real $t$ put $\\mathrm{ArcSl}(t) = \\int_0^t (1 - s^3)^{-2/3}\\,ds$, so that $\\varpi_1 = \\mathrm{ArcSl}(1)$. Then: for\u2026","title":"$\\mathrm{Sl}$ and $\\mathrm{ArcSl}$ are inverse on the real segment"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/jacobi-sum-quadratic-char-cubic-residue","kind":"node","lean":"MetaMathlibExt.jacobiSum_quadraticChar_cubicResidueChar","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#jacobi-sum-quadratic-char-cubic-residue","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $K$ be a number field generated over $\\mathbb{Q}$ by a primitive cube root of unity, with ring of integers $\\mathcal{O}_K$. Let $\\mu \\in \\mathcal{O}_K$ with $\\mu \\equiv 1 \\pmod 3$, and let $p$ be a prime such that the norm of $\\mu$ from $\\mathcal{O}_K$ to $\\mathbb{Z}$ is $p$. Let $\\chi : \\mathbb{Z}/p\\mathbb{Z} \\to \\mathcal{O}_K$ be a multiplicative character with $\\chi(t) \\equiv t^{(p-1)/3} \\pmod{\\mu}$ for\u2026","title":"Jacobi sums of the quadratic and the cubic residue character"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/jacobi-sum-quadratic-char-cubic-char-car","kind":"node","lean":"MetaMathlibExt.jacobiSum_quadraticChar_cubicChar_card_sq","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Jacobi sum of the quadratic and a cubic character of a field with "],["math","\\(q^2\\)"],["text"," elements"]],"source":"The Jacobi sum of the quadratic and a cubic character of a field with $q^2$ elements"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#jacobi-sum-quadratic-char-cubic-char-car","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $q$ be an odd prime with $q \\equiv 2 \\pmod 3$, let $F$ be a finite field with $q^2$ elements, let $\\phi$ be the quadratic character of $F$, and let $\\chi : F \\to \\mathbb{C}$ be a multiplicative character of order $3$, that is, $\\chi^3 = 1$ and $\\chi \\ne 1$ (characters take the value $0$ at $0$). Then the Jacobi sum $J(\\phi, \\chi) = \\sum_{t \\in F} \\phi(t)\\,\\chi(1 - t)$ equals $q$.","title":"The Jacobi sum of the quadratic and a cubic character of a field with $q^2$ elements"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/cubic-residue-symbol-reciprocity","kind":"node","lean":"MetaMathlibExt.cubicResidueSymbol_reciprocity","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#cubic-residue-symbol-reciprocity","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $K$ be the cyclotomic field generated by a primitive cube root of unity, with ring of integers $\\mathcal{O}_K$, and let $\\zeta \\in \\mathcal{O}_K$ be a primitive cube root of unity. Let $\\lambda_1, \\lambda_2 \\in \\mathcal{O}_K$ be primes with $\\lambda_1 \\equiv \\lambda_2 \\equiv 1 \\pmod 3$, and write $N$ for the absolute value of the norm from $\\mathcal{O}_K$ to $\\mathbb{Z}$. If\u2026","title":"Cubic reciprocity"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/eisenstein-twist-card-points","kind":"node","lean":"MetaMathlibExt.eisensteinTwist_card_points","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Point counts of "],["math","\\(y^2 + \\lambda y = x^3\\)"],["text"," modulo primes of "],["math","\\(\\mathbb{Z}(\\zeta_3)\\)"]],"source":"Point counts of $y^2 + \\lambda y = x^3$ modulo primes of $\\mathbb{Z}(\\zeta_3)$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#eisenstein-twist-card-points","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $K$ be the cyclotomic field generated by a primitive cube root of unity, with ring of integers $\\mathcal{O}_K$, let $\\zeta \\in \\mathcal{O}_K$ be a primitive cube root of unity, and write $N$ and $\\mathrm{Tr}$ for the norm and trace from $\\mathcal{O}_K$ to $\\mathbb{Z}$. Let $\\lambda \\in \\mathcal{O}_K$ satisfy $\\lambda \\equiv 1 \\pmod 3$ with $N\\lambda = \\ell$ a prime. For a principal ideal $I$ of $\\mathcal{O}_K$,\u2026","title":"Point counts of $y^2 + \\lambda y = x^3$ modulo primes of $\\mathbb{Z}(\\zeta_3)$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/eisenstein-twist-torsion","kind":"node","lean":"MetaMathlibExt.eisensteinTwist_torsion","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Torsion of "],["math","\\(y^2 = x^3 + \\lambda^2/4\\)"],["text"," over "],["math","\\(\\mathbb{Q}(\\zeta_3)\\)"]],"source":"Torsion of $y^2 = x^3 + \\lambda^2/4$ over $\\mathbb{Q}(\\zeta_3)$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#eisenstein-twist-torsion","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $K$ be the cyclotomic field generated by a primitive cube root of unity, with ring of integers $\\mathcal{O}_K$, and let $\\lambda \\in \\mathcal{O}_K$ satisfy $\\lambda \\equiv 1 \\pmod 3$, with norm from $\\mathcal{O}_K$ to $\\mathbb{Z}$ a prime number. Let $E$ be the Weierstrass curve $y^2 = x^3 + \\lambda^2/4$ over $K$, with point at infinity $O$. Then a point $P \\in E(K)$ has finite order if and only if $3P = O$;\u2026","title":"Torsion of $y^2 = x^3 + \\lambda^2/4$ over $\\mathbb{Q}(\\zeta_3)$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/eisenstein-lambda-tilde-sq-congr","kind":"node","lean":"MetaMathlibExt.eisensteinLambdaTilde_sq_congr","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruence for the square of "],["math","\\(\\tilde\\lambda\\)"]],"source":"Congruence for the square of $\\tilde\\lambda$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#eisenstein-lambda-tilde-sq-congr","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\rho = e^{2\\pi i/3}$, let $\\varpi_1 = \\int_0^1 (1 - t^3)^{-2/3}\\,dt$, let $\\wp$ be the Weierstrass function of the lattice spanned by $\\varpi_1$ and $\\varpi_1\\rho$, and put $\\varphi(u) = 6\\wp(\\varpi_1 u)/(9 + \\wp'(\\varpi_1 u))$. Let $a, b$ be integers with $a \\equiv 1$ and $b \\equiv 0 \\pmod 3$ such that $\\ell = a^2 - ab + b^2$ is prime, and put $\\lambda = a + b\\rho$. Let $r_0 \\in \\mathbb{Z}/\\ell$ satisfy\u2026","title":"Congruence for the square of $\\tilde\\lambda$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/elliptic-gauss-sum-mem-integers-eq-sum-k","kind":"node","lean":"MetaMathlibExt.ellipticGaussSum_mem_integers_eq_sum_ker","parent":5206,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#elliptic-gauss-sum-mem-integers-eq-sum-k","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\rho = e^{2\\pi i/3}$, let $\\varpi_1 = \\int_0^1 (1 - t^3)^{-2/3}\\,dt$, let $\\wp$ be the Weierstrass function of the lattice $\\varpi_1 \\mathbb{Z} + \\varpi_1 \\rho \\mathbb{Z}$, and let $\\varphi(u) = 6\\wp(\\varpi_1 u)/(9 + \\wp'(\\varpi_1 u))$. Let $a, b$ be integers with $a \\equiv 1 \\pmod 3$ and $b \\equiv 0 \\pmod 3$ such that $\\ell = a^2 - ab + b^2$ is a prime with $\\ell \\equiv 7 \\pmod 9$ or $\\ell \\equiv 4 \\pmod 9$,\u2026","title":"Integrality of the elliptic Gauss sum"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/elliptic-gauss-sum-coeff-norm-sq-congr","kind":"node","lean":"MetaMathlibExt.ellipticGaussSum_coeff_normSq_congr","parent":5206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences for "],["math","\\(|\\alpha_\\lambda|^2\\)"]],"source":"Congruences for $|\\alpha_\\lambda|^2$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic/index.html#elliptic-gauss-sum-coeff-norm-sq-congr","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/class-number-congruences-for-imaginary-quadratic","status":"can_prove","summary":"Let $\\rho = e^{2\\pi i/3}$, let $\\varpi_1 = \\int_0^1 (1 - t^3)^{-2/3}\\,dt$, let $\\wp$ be the Weierstrass function of the lattice spanned by $\\varpi_1$ and $\\varpi_1\\rho$, and put $\\varphi(u) = 6\\wp(\\varpi_1 u)/(9 + \\wp'(\\varpi_1 u))$. Put $\\mathrm{Sl}(u) = \\varphi(u/(-3\\varpi_1))$. Let $a, b$ be integers with $a \\equiv 1$ and $b \\equiv 0 \\pmod 3$ such that $\\ell = a^2 - ab + b^2$ is prime, and put\u2026","title":"Congruences for $|\\alpha_\\lambda|^2$"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants.json","id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","kind":"scope","navigation":"module","parent":5203,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","title":"Imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-imaginary-quadratic-discriminant","kind":"node","lean":"ImaginaryQuadraticDiscriminant.IsImaginaryQuadraticDiscriminant","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-imaginary-quadratic-discriminant","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"An integer $D$ is an imaginary quadratic discriminant if $D < 0$ and either $D \\equiv 0 \\pmod 4$ or $D \\equiv 1 \\pmod 4$.","title":"Imaginary quadratic discriminant"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-fundamental-imaginary-quadratic-discr","kind":"node","lean":"ImaginaryQuadraticDiscriminant.IsFundamentalImaginaryQuadraticDiscriminant","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-fundamental-imaginary-quadratic-discr","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"An integer $D$ is a fundamental imaginary quadratic discriminant if $D < 0$ and $D$ is a fundamental discriminant.","title":"Fundamental imaginary quadratic discriminant"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/imaginary-quadratic-discriminants","kind":"node","lean":"ImaginaryQuadraticDiscriminant.imaginaryQuadraticDiscriminants","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#imaginary-quadratic-discriminants","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"The set of imaginary quadratic discriminants is the set of integers $D$ such that $D < 0$ and $D$ is congruent to $0$ or $1$ modulo $4$.","title":"Imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/fundamental-imaginary-quadratic-discrimi","kind":"node","lean":"ImaginaryQuadraticDiscriminant.fundamentalImaginaryQuadraticDiscriminants","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#fundamental-imaginary-quadratic-discrimi","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"The set of fundamental imaginary quadratic discriminants is the set of integers $D$ with $D < 0$ that are fundamental discriminants.","title":"Fundamental imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/imaginary-quadratic-discriminant-decompo","kind":"node","lean":"ImaginaryQuadraticDiscriminant.ImaginaryQuadraticDiscriminantDecomposition","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#imaginary-quadratic-discriminant-decompo","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"For an integer $D$, a decomposition of $D$ consists of integers $u$ and $D_K$ such that $u \\ge 1$, $D_K$ is a fundamental imaginary quadratic discriminant, and $D = u^2 D_K$. The integer $u$ is called the conductor.","title":"Conductor decomposition of an imaginary quadratic discriminant"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-imaginary-quadratic-discriminant-iff","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isImaginaryQuadraticDiscriminant_iff","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-imaginary-quadratic-discriminant-iff","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"An integer $D$ is an imaginary quadratic discriminant if and only if $D < 0$ and $D$ is congruent to $0$ or $1$ modulo $4$.","title":"Characterization of imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-imaginary-quadratic-discriminant-mk","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isImaginaryQuadraticDiscriminant_mk","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-imaginary-quadratic-discriminant-mk","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"If an integer $D$ satisfies $D < 0$ and either $D \\equiv 0 \\pmod 4$ or $D \\equiv 1 \\pmod 4$, then $D$ is an imaginary quadratic discriminant.","title":"Criterion for an imaginary quadratic discriminant"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-imaginary-quadratic-discriminant-neg","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isImaginaryQuadraticDiscriminant_neg","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-imaginary-quadratic-discriminant-neg","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"Every imaginary quadratic discriminant $D$ satisfies $D < 0$.","title":"Imaginary quadratic discriminants are negative"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-imaginary-quadratic-discriminant-mod","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isImaginaryQuadraticDiscriminant_mod","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-imaginary-quadratic-discriminant-mod","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"Every imaginary quadratic discriminant $D$ satisfies $D \\equiv 0 \\pmod 4$ or $D \\equiv 1 \\pmod 4$.","title":"Imaginary quadratic discriminants are congruent to 0 or 1 mod 4"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-imaginary-quadratic-discriminant-of-m","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isImaginaryQuadraticDiscriminant_of_mod_zero","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-imaginary-quadratic-discriminant-of-m","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"If an integer $D$ satisfies $D < 0$ and $D \\equiv 0 \\pmod 4$, then $D$ is an imaginary quadratic discriminant.","title":"Negative multiples of 4 are imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-imaginary-quadratic-discriminant-of-m-2","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isImaginaryQuadraticDiscriminant_of_mod_one","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-imaginary-quadratic-discriminant-of-m-2","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"If an integer $D$ satisfies $D < 0$ and $D \\equiv 1 \\pmod 4$, then $D$ is an imaginary quadratic discriminant.","title":"Negative integers congruent to 1 mod 4 are imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-fundamental-imaginary-quadratic-discr-2","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isFundamentalImaginaryQuadraticDiscriminant_iff","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-fundamental-imaginary-quadratic-discr-2","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"An integer $D$ is a fundamental imaginary quadratic discriminant if and only if $D < 0$ and $D$ is a fundamental discriminant.","title":"Characterization of fundamental imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-fundamental-imaginary-quadratic-discr-3","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isFundamentalImaginaryQuadraticDiscriminant_mk","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-fundamental-imaginary-quadratic-discr-3","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"If an integer $D$ satisfies $D < 0$ and is a fundamental discriminant, then $D$ is a fundamental imaginary quadratic discriminant.","title":"Negative fundamental discriminants are fundamental imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-fundamental-imaginary-quadratic-discr-4","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isFundamentalImaginaryQuadraticDiscriminant_isFundamentalDiscriminant","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-fundamental-imaginary-quadratic-discr-4","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"Every fundamental imaginary quadratic discriminant $D$ is a fundamental discriminant.","title":"Fundamental imaginary quadratic discriminants are fundamental discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-fundamental-imaginary-quadratic-discr-5","kind":"node","lean":"ImaginaryQuadraticDiscriminant.isFundamentalImaginaryQuadraticDiscriminant_isImaginary","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-fundamental-imaginary-quadratic-discr-5","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"Every fundamental imaginary quadratic discriminant is an imaginary quadratic discriminant.","title":"Fundamental imaginary quadratic discriminants are imaginary quadratic discriminants"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/is-fundamental","kind":"node","lean":"ImaginaryQuadraticDiscriminant.ImaginaryQuadraticDiscriminantDecomposition.isFundamental","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#is-fundamental","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"For every conductor decomposition of an integer $D$, consisting of an integer $u \\ge 1$ and a fundamental imaginary quadratic discriminant $D_K$ with $D = u^2 D_K$, the integer $D_K$ is a fundamental imaginary quadratic discriminant.","title":"The fundamental part of a conductor decomposition"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/u-ge-one","kind":"node","lean":"ImaginaryQuadraticDiscriminant.ImaginaryQuadraticDiscriminantDecomposition.u_ge_one","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#u-ge-one","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"For every decomposition of an integer $D$ into a conductor $u$ and a fundamental imaginary quadratic discriminant $D_K$, we have $u \\ge 1$.","title":"The conductor of a discriminant decomposition is at least one"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/prop-eq","kind":"node","lean":"ImaginaryQuadraticDiscriminant.ImaginaryQuadraticDiscriminantDecomposition.prop_eq","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#prop-eq","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"For every decomposition of an integer $D$ into a conductor $u$ and a fundamental imaginary quadratic discriminant $D_K$, we have $D = u^2 D_K$.","title":"The equation of a discriminant decomposition"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/of-fundamental","kind":"node","lean":"ImaginaryQuadraticDiscriminant.ImaginaryQuadraticDiscriminantDecomposition.ofFundamental","parent":5226,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants/index.html#of-fundamental","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/imaginary-quadratic-discriminants","status":"fully_proved","summary":"Let $D$ be an integer. If $u \\ge 1$ is an integer, $D_K$ is a fundamental imaginary quadratic discriminant and $D = u^2 D_K$, then $u$ and $D_K$ form a decomposition of $D$ into the conductor $u$ and the fundamental imaginary quadratic discriminant $D_K$.","title":"Decomposition from a conductor and a fundamental discriminant"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings.json","id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","kind":"scope","navigation":"module","parent":5203,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","title":"Norms in imaginary quadratic integer rings"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/prime-of-nat-abs-norm-prime","kind":"node","lean":"Zsqrtd.prime_of_natAbs_norm_prime","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(\\mathbb{Z}(\\sqrt{d})\\)"],["text"," of prime norm are prime"]],"source":"Elements of $\\mathbb{Z}(\\sqrt{d})$ of prime norm are prime"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#prime-of-nat-abs-norm-prime","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d$ be an integer such that the ring $\\mathbb{Z}[\\sqrt{d}]$ of formal expressions $a + b\\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ and $\\sqrt{d}^2 = d$ is a unique factorization domain, with norm $N(a + b\\sqrt{d}) = a^2 - db^2$. If $\\pi \\in \\mathbb{Z}[\\sqrt{d}]$ and $|N(\\pi)|$ is a prime number, then $\\pi$ is prime in $\\mathbb{Z}[\\sqrt{d}]$.","title":"Elements of $\\mathbb{Z}(\\sqrt{d})$ of prime norm are prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/nat-abs-norm-dvd-of-dvd","kind":"node","lean":"Zsqrtd.natAbs_norm_dvd_of_dvd","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","The norm respects divisibility in "],["math","\\(\\mathbb{Z}(\\sqrt{d})\\)"]],"source":"The norm respects divisibility in $\\mathbb{Z}(\\sqrt{d})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#nat-abs-norm-dvd-of-dvd","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d$ be an integer and let $N(a + b\\sqrt{d}) = a^2 - db^2$ be the norm on the ring $\\mathbb{Z}[\\sqrt{d}]$ of formal expressions $a + b\\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ and $\\sqrt{d}^2 = d$. If $x$ divides $y$ in $\\mathbb{Z}[\\sqrt{d}]$, then $|N(x)|$ divides $|N(y)|$.","title":"The norm respects divisibility in $\\mathbb{Z}(\\sqrt{d})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/eq-zero-of-norm-nat-abs-eq-zero","kind":"node","lean":"Zsqrtd.eq_zero_of_norm_natAbs_eq_zero","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Only zero has norm zero in "],["math","\\(\\mathbb{Z}(\\sqrt d)\\)"],["text"," for "],["math","\\(d < 0\\)"]],"source":"Only zero has norm zero in $\\mathbb{Z}(\\sqrt d)$ for $d < 0$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#eq-zero-of-norm-nat-abs-eq-zero","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm on the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$). If $\\alpha \\in \\mathbb{Z}[\\sqrt d]$ satisfies $|N(\\alpha)| = 0$, then $\\alpha = 0$.","title":"Only zero has norm zero in $\\mathbb{Z}(\\sqrt d)$ for $d < 0$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/dvd-of-coprime-norms","kind":"node","lean":"Zsqrtd.dvd_of_coprime_norms","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#dvd-of-coprime-norms","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is a unique factorization domain, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Let $u$ and $v$ be coprime natural numbers with $u \\ne 0$. If $\\beta, \\beta_0, \\gamma \\in \\mathbb{Z}[\\sqrt d]$ satisfy $|N(\\beta)| \\mid u$, $|N(\\gamma)| = v$ and $\\beta \\mid \\beta_0\\gamma$, then $\\beta \\mid \\beta_0$.","title":"Cancelling a factor of coprime norm"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/prime-pow-norm-of-prime","kind":"node","lean":"Zsqrtd.prime_pow_norm_of_prime","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes of "],["math","\\(\\mathbb{Z}(\\sqrt{d})\\)"],["text"," have prime-power norm"]],"source":"Primes of $\\mathbb{Z}(\\sqrt{d})$ have prime-power norm"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#prime-pow-norm-of-prime","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer and let $N(a + b\\sqrt{d}) = a^2 - db^2$ be the norm on the ring $\\mathbb{Z}[\\sqrt{d}]$ of numbers $a + b\\sqrt{d}$ with $a, b \\in \\mathbb{Z}$. For every prime $\\varpi$ of $\\mathbb{Z}[\\sqrt{d}]$ there are a prime number $\\ell$ and a natural number $s \\ne 0$ such that $|N(\\varpi)| = \\ell^s$ and $\\varpi$ divides $\\ell$ in $\\mathbb{Z}[\\sqrt{d}]$.","title":"Primes of $\\mathbb{Z}(\\sqrt{d})$ have prime-power norm"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/exists-factor-of-coprime","kind":"node","lean":"Zsqrtd.exists_factor_of_coprime","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#exists-factor-of-coprime","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is a unique factorization domain, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Let $u$ and $v$ be coprime nonzero natural numbers. Every $\\alpha \\in \\mathbb{Z}[\\sqrt d]$ with $|N(\\alpha)| = uv$ factors as $\\alpha = \\beta\\gamma$ with $|N(\\beta)| = u$ and $|N(\\gamma)| = v$.","title":"Splitting an element along coprime norms"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/card-repr-of-shape","kind":"node","lean":"Zsqrtd.card_repr_of_shape","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#card-repr-of-shape","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of formal sums $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is an integral domain with exactly $k$ units, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Assume that for each $m \\in \\mathbb{N}$ the set of $\\alpha \\in \\mathbb{Z}[\\sqrt d]$ with $|N(\\alpha)| = m$ is finite. Let $n \\in \\mathbb{N}$ and let $s \\ne 0$ satisfy $|N(s)| = n$. If every $\\alpha$ with\u2026","title":"Counting a norm class generated by one element"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/card-mul-of-coprime","kind":"node","lean":"Zsqrtd.card_mul_of_coprime","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#card-mul-of-coprime","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is a unique factorization domain with exactly $k$ units, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Assume that for each $n \\in \\mathbb{N}$ the set of $\\alpha \\in \\mathbb{Z}[\\sqrt d]$ with $|N(\\alpha)| = n$ is finite, and let $r(n)$ be its size. If $u$ and $v$ are coprime nonzero natural\u2026","title":"Norm counts are multiplicative up to units"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/norm-pow","kind":"node","lean":"Zsqrtd.norm_pow","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","The norm of a power in "],["math","\\(\\mathbb{Z}(\\sqrt{d})\\)"]],"source":"The norm of a power in $\\mathbb{Z}(\\sqrt{d})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#norm-pow","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d$ be an integer and let $N(a + b\\sqrt{d}) = a^2 - db^2$ be the norm on the ring $\\mathbb{Z}[\\sqrt{d}]$ of formal expressions $a + b\\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ and $\\sqrt{d}^2 = d$. Then $N(x^n) = N(x)^n$ for every $x \\in \\mathbb{Z}[\\sqrt{d}]$ and every natural number $n$.","title":"The norm of a power in $\\mathbb{Z}(\\sqrt{d})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/exists-unit-mul-pow-of-norm-two-pow","kind":"node","lean":"Zsqrtd.exists_unit_mul_pow_of_norm_two_pow","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#exists-unit-mul-pow-of-norm-two-pow","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is a unique factorization domain, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Suppose $2 = \\varepsilon\\rho^2$ for a unit $\\varepsilon$ and a prime $\\rho$ with $|N(\\rho)| = 2$. Then for every $e \\in \\mathbb{N}$, every $\\alpha$ with $|N(\\alpha)| = 2^e$ equals $u\\rho^e$ for some unit $u$.","title":"Elements whose norm is a power of a ramified two"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/card-norm-two-pow","kind":"node","lean":"Zsqrtd.card_norm_two_pow","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#card-norm-two-pow","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is a unique factorization domain with exactly $k$ units, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Assume that for each $n \\in \\mathbb{N}$ the set of $\\alpha \\in \\mathbb{Z}[\\sqrt d]$ with $|N(\\alpha)| = n$ is finite, and let $r(n)$ be its size. If $2 = \\varepsilon\\rho^2$ for a unit\u2026","title":"Norm counts at powers of a ramified two"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/inert-peel","kind":"node","lean":"Zsqrtd.inert_peel","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peeling an inert prime off an element of norm "],["math","\\(q^f\\)"]],"source":"Peeling an inert prime off an element of norm $q^f$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#inert-peel","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt{d}]$ of numbers $a + b\\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ is a unique factorization domain, with norm $N(a + b\\sqrt{d}) = a^2 - db^2$. Let $q$ be a natural number that is prime in $\\mathbb{Z}[\\sqrt{d}]$, let $f$ be a natural number, and let $\\alpha \\in \\mathbb{Z}[\\sqrt{d}]$ be nonzero and not a unit with $|N(\\alpha)| = q^f$. Then $\\alpha = wq\\beta$ for\u2026","title":"Peeling an inert prime off an element of norm $q^f$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/inert-even","kind":"node","lean":"Zsqrtd.inert_even","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of norm "],["math","\\(q^{2n}\\)"],["text"," over an inert prime"]],"source":"Elements of norm $q^{2n}$ over an inert prime"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#inert-even","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt{d}]$ of numbers $a + b\\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ is a unique factorization domain, with norm $N(a + b\\sqrt{d}) = a^2 - db^2$. Let $q$ be a prime number that is also prime in $\\mathbb{Z}[\\sqrt{d}]$. Then for every natural number $n$, every $\\alpha \\in \\mathbb{Z}[\\sqrt{d}]$ with $|N(\\alpha)| = q^{2n}$ is of the form $\\alpha = uq^n$ for some unit\u2026","title":"Elements of norm $q^{2n}$ over an inert prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/inert-odd","kind":"node","lean":"Zsqrtd.inert_odd","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","No element has norm "],["math","\\(q^{2n+1}\\)"],["text"," over an inert prime"]],"source":"No element has norm $q^{2n+1}$ over an inert prime"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#inert-odd","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt{d}]$ of numbers $a + b\\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ is a unique factorization domain, with norm $N(a + b\\sqrt{d}) = a^2 - db^2$. Let $q$ be a prime number that is also prime in $\\mathbb{Z}[\\sqrt{d}]$. Then for every natural number $n$, no $\\alpha \\in \\mathbb{Z}[\\sqrt{d}]$ satisfies $|N(\\alpha)| = q^{2n+1}$.","title":"No element has norm $q^{2n+1}$ over an inert prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/card-norm-inert-pow","kind":"node","lean":"Zsqrtd.card_norm_inert_pow","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#card-norm-inert-pow","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is a unique factorization domain with exactly $k$ units, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Assume that for each $n \\in \\mathbb{N}$ the set of $\\alpha \\in \\mathbb{Z}[\\sqrt d]$ with $|N(\\alpha)| = n$ is finite, and let $r(n)$ be its size. If $q$ is a prime number that is also a prime\u2026","title":"Norm counts at an inert prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/exists-unit-mul-split-pow","kind":"node","lean":"Zsqrtd.exists_unit_mul_split_pow","parent":5246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of norm "],["math","\\(p^e\\)"],["text"," over a split prime"]],"source":"Elements of norm $p^e$ over a split prime"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#exists-unit-mul-split-pow","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt{d}]$ of numbers $a + b\\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ is a unique factorization domain, with norm $N(a + b\\sqrt{d}) = a^2 - db^2$ and conjugation $\\overline{a + b\\sqrt{d}} = a - b\\sqrt{d}$. Let $p$ be a prime number and let $\\pi$ be a prime of $\\mathbb{Z}[\\sqrt{d}]$ with $N(\\pi) = p$ and $p = \\pi\\bar{\\pi}$ such that $\\bar{\\pi}$ is also prime. Then\u2026","title":"Elements of norm $p^e$ over a split prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/card-norm-split-pow","kind":"node","lean":"Zsqrtd.card_norm_split_pow","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#card-norm-split-pow","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is a unique factorization domain with exactly $k$ units, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Assume that for each $n \\in \\mathbb{N}$ the set of $\\alpha \\in \\mathbb{Z}[\\sqrt d]$ with $|N(\\alpha)| = n$ is finite, and let $r(n)$ be its size. Let $p$ be a prime number and\u2026","title":"Norm counts at a split prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/card-norm-eq-mul-of-prime-pow","kind":"node","lean":"Zsqrtd.card_norm_eq_mul_of_prime_pow","parent":5246,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings/index.html#card-norm-eq-mul-of-prime-pow","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/norms-in-imaginary-quadratic-integer-rings","status":"fully_proved","summary":"Let $d < 0$ be an integer such that the ring $\\mathbb{Z}[\\sqrt d]$ of elements $a + b\\sqrt d$ ($a, b \\in \\mathbb{Z}$) is a unique factorization domain with exactly $k$ units, and let $N(a + b\\sqrt d) = a^2 - db^2$ be the norm. Assume that for each $n \\in \\mathbb{N}$ the set of $\\alpha \\in \\mathbb{Z}[\\sqrt d]$ with $|N(\\alpha)| = n$ is finite, and let $r(n)$ be its size. Let $D : \\mathbb{N} \\to \\mathbb{Z}$ satisfy\u2026","title":"Norm counts determined by prime powers"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes.json","id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","kind":"scope","navigation":"module","parent":5203,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","title":"Gaussian primes"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-is-unit-iff","kind":"node","lean":"GaussianInt.gaussianInt_isUnit_iff","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-is-unit-iff","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"A Gaussian integer $x \\in \\mathbb{Z}[i]$ is a unit if and only if $x$ is one of $1$, $-1$, $i$ and $-i$.","title":"Units of the Gaussian integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-prime-of-nat-abs-norm-prime","kind":"node","lean":"GaussianInt.gaussianInt_prime_of_natAbs_norm_prime","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-prime-of-nat-abs-norm-prime","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Let $\\pi = a + bi \\in \\mathbb{Z}[i]$ and let $N(\\pi) = a^2 + b^2$ be its norm. If the absolute value of $N(\\pi)$ is a prime number, then $\\pi$ is a prime element of $\\mathbb{Z}[i]$.","title":"Gaussian integers of prime norm are prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-prime-one-add-i","kind":"node","lean":"GaussianInt.gaussianInt_prime_one_add_I","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-prime-one-add-i","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"The Gaussian integer $1 + i$ is a prime element of $\\mathbb{Z}[i]$; its norm is $2$.","title":"1 + i is a Gaussian prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-two-eq","kind":"node","lean":"GaussianInt.gaussianInt_two_eq","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-two-eq","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"In $\\mathbb{Z}[i]$ we have $2 = -i \\cdot (1 + i)^2$, where $-i$ is a unit with inverse $i$.","title":"Factorization of 2 in the Gaussian integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/unit-val","kind":"node","lean":"GaussianInt.unitVal","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#unit-val","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"The map $u$ on $\\{0, 1, 2, 3\\}$ with $u_0 = 1$, $u_1 = -1$, $u_2 = i$ and $u_3 = -i$ in $\\mathbb{Z}[i]$, listing the four units of $\\mathbb{Z}[i]$.","title":"Enumeration of the Gaussian units"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/unit-val-inj","kind":"node","lean":"GaussianInt.unitVal_inj","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#unit-val-inj","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Let $u_0 = 1$, $u_1 = -1$, $u_2 = i$ and $u_3 = -i$ in $\\mathbb{Z}[i]$. For $j, k \\in \\{0, 1, 2, 3\\}$, if $u_j = u_k$ then $j = k$.","title":"The four Gaussian units are distinct"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/unit-val-complete","kind":"node","lean":"GaussianInt.unitVal_complete","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#unit-val-complete","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Let $u_0 = 1$, $u_1 = -1$, $u_2 = i$ and $u_3 = -i$. Every unit $u$ of $\\mathbb{Z}[i]$ equals $u_j$ for some $j \\in \\{0, 1, 2, 3\\}$.","title":"The enumeration of the Gaussian units is complete"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/unit-val-inv","kind":"node","lean":"GaussianInt.unitValInv","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#unit-val-inv","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"The map $v$ on $\\{0, 1, 2, 3\\}$ with $v_0 = 1$, $v_1 = -1$, $v_2 = -i$ and $v_3 = i$ in $\\mathbb{Z}[i]$, listing the inverses of the units $1$, $-1$, $i$ and $-i$ in that order.","title":"Inverses of the Gaussian units"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/unit-val-mul-unit-val-inv","kind":"node","lean":"GaussianInt.unitVal_mul_unitValInv","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#unit-val-mul-unit-val-inv","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Let $u_0 = 1$, $u_1 = -1$, $u_2 = i$, $u_3 = -i$ and $v_0 = 1$, $v_1 = -1$, $v_2 = -i$, $v_3 = i$ in $\\mathbb{Z}[i]$. Then $u_j v_j = 1$ for every $j \\in \\{0, 1, 2, 3\\}$.","title":"Each Gaussian unit times its listed inverse is one"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-split-prime-sq-add-sq","kind":"node","lean":"GaussianInt.gaussianInt_split_prime_sq_add_sq","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-split-prime-sq-add-sq","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$. Then there are natural numbers $a$ and $b$ with $a^2 + b^2 = p$ such that both $a + bi$ and $a - bi$ are prime elements of the Gaussian integers $\\mathbb{Z}[i]$.","title":"Explicit Gaussian prime factors of a prime congruent to 1 mod 4"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-prime-associated","kind":"node","lean":"GaussianInt.gaussianInt_prime_associated","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-prime-associated","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Every prime element $z$ of the Gaussian integers $\\mathbb{Z}[i]$ is associated to one of the following: a rational prime $p$ with $p \\equiv 3 \\pmod 4$; one of $a + bi$ and $a - bi$, where $a$ and $b$ are natural numbers with $a^2 + b^2 = p$ for some rational prime $p \\equiv 1 \\pmod 4$; or $1 + i$.","title":"Classification of the Gaussian primes"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-split-prime","kind":"node","lean":"GaussianInt.gaussianInt_split_prime","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-split-prime","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$. Then there is a prime element $\\pi = a + bi$ of $\\mathbb{Z}[i]$ whose norm $N(\\pi) = a^2 + b^2$ equals $p$.","title":"Primes congruent to 1 mod 4 split in the Gaussian integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-split-conj","kind":"node","lean":"GaussianInt.gaussianInt_split_conj","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-split-conj","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Let $p$ be a prime, write $N(a + bi) = a^2 + b^2$ for the norm on $\\mathbb{Z}[i]$, and let $\\pi \\in \\mathbb{Z}[i]$ satisfy $N(\\pi) = p$. Then $p = \\pi \\bar{\\pi}$ in $\\mathbb{Z}[i]$, where $\\bar{\\pi}$ is the complex conjugate of $\\pi$, and $\\bar{\\pi}$ is a prime element of $\\mathbb{Z}[i]$.","title":"Factoring a prime as a Gaussian integer times its conjugate"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-split-nonassociated","kind":"node","lean":"GaussianInt.gaussianInt_split_nonassociated","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-split-nonassociated","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Write $N(a + bi) = a^2 + b^2$ for the norm on $\\mathbb{Z}[i]$. Let $p$ be a prime with $p \\equiv 1 \\pmod 4$ and let $\\pi \\in \\mathbb{Z}[i]$ satisfy $N(\\pi) = p$. Then the complex conjugate $\\bar{\\pi}$ is not associated to $\\pi$: there is no unit $u$ of $\\mathbb{Z}[i]$ with $\\bar{\\pi} u = \\pi$.","title":"Gaussian integers of split prime norm are not associate to their conjugates"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/gaussian-int-card-repr-prime-pow","kind":"node","lean":"GaussianInt.gaussianInt_card_repr_prime_pow","parent":5265,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes/index.html#gaussian-int-card-repr-prime-pow","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/gaussian-primes","status":"fully_proved","summary":"Write $N(a + bi) = a^2 + b^2$ for the norm on $\\mathbb{Z}[i]$. Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, let $e$ be a natural number, and suppose $\\pi \\in \\mathbb{Z}[i]$ is a prime element with $N(\\pi) = p$. Then exactly $4(e + 1)$ Gaussian integers $\\alpha$ satisfy $|N(\\alpha)| = p^e$.","title":"Number of Gaussian integers of norm a power of a split prime"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems.json","id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","kind":"scope","navigation":"module","parent":5203,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","title":"Schatunowsky-type theorems"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/gaussian-int-exists-prime-norm-between","kind":"node","lean":"MetaMathlibExt.gaussianInt_exists_prime_norm_between","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#gaussian-int-exists-prime-norm-between","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $N(a + bi) = a^2 + b^2$ be the norm on the Gaussian integers $\\mathbb{Z}[i]$. For every prime element $\\pi$ of $\\mathbb{Z}[i]$ with $N(\\pi) \\ne 2$ (that is, $\\pi$ is not an associate of $1 + i$) there is a prime element $\\rho$ of $\\mathbb{Z}[i]$ with $N(\\pi) < N(\\rho) \\le 2N(\\pi)$.","title":"Bertrand's postulate for Gaussian integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/eisenstein-exists-prime-norm-between","kind":"node","lean":"MetaMathlibExt.eisenstein_exists_prime_norm_between","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#eisenstein-exists-prime-norm-between","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $\\mathcal{O}$ be the ring of integers of the cyclotomic field $\\mathbb{Q}(\\zeta_3)$, and let $N$ be the field norm from $\\mathbb{Q}(\\zeta_3)$ to $\\mathbb{Q}$. For every prime element $\\pi$ of $\\mathcal{O}$ there is a prime element $\\rho$ of $\\mathcal{O}$ with $N(\\pi) < N(\\rho) \\le 2N(\\pi)$.","title":"Bertrand's postulate for Eisenstein integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/nat-schatunowsky","kind":"node","lean":"MetaMathlibExt.nat_schatunowsky","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#nat-schatunowsky","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"The natural numbers $a \\ge 2$ such that every natural number $b$ with $1 < b < a$ and $\\gcd(a, b) = 1$ is prime are exactly $2, 3, 4, 6, 8, 12, 18, 24$ and $30$.","title":"Schatunowsky's theorem"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/nat-iwata-finite","kind":"node","lean":"MetaMathlibExt.nat_iwata_finite","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#nat-iwata-finite","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"For every natural number $\\ell$, there are only finitely many natural numbers $a$ such that every natural number $b$ with $1 < b < a$ and $\\gcd(a, b) = 1$ satisfies $\\Omega(b) \\le \\ell$, where $\\Omega(b)$ is the number of prime factors of $b$ counted with multiplicity.","title":"Iwata's generalization of Schatunowsky's theorem"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/gaussian-int-schatunowsky-max-norm-five","kind":"node","lean":"MetaMathlibExt.gaussianInt_schatunowsky_max_norm_five_add_five_i","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#gaussian-int-schatunowsky-max-norm-five","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $N(a + bi) = a^2 + b^2$ be the norm on the Gaussian integers $\\mathbb{Z}[i]$, and let $S$ be the set of $\\alpha \\in \\mathbb{Z}[i]$ such that every $\\beta \\in \\mathbb{Z}[i]$ coprime to $\\alpha$ with $1 < N(\\beta) < N(\\alpha)$ is a Gaussian prime. Then $5 + 5i \\in S$, every element of $S$ has norm at most $50$, and every element of $S$ of norm $50$ is an associate of $5 + 5i$.","title":"Dalezman's Schatunowsky theorem for the Gaussian integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/quad-prime-norm","kind":"node","lean":"MetaMathlibExt.quadPrimeNorm","parent":5281,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norm of the "],["math","\\(n\\)"],["text","-th prime"]],"source":"Norm of the $n$-th prime"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#quad-prime-norm","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$. For $n \\ge 1$, $N(\\pi_n)$ is the $n$-th smallest natural number that is the absolute norm of some nonzero prime ideal of $\\mathcal{O}_K$, each such norm counted once however many prime ideals have it; and $N(\\pi_0) = 1$.","title":"Norm of the $n$-th prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/quad-prime-norm-zero","kind":"node","lean":"MetaMathlibExt.quadPrimeNorm_zero","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#quad-prime-norm-zero","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"fully_proved","summary":"For every number field $K$, the zeroth prime norm $N(\\pi_0)$ of $K$ equals $1$.","title":"The zeroth prime norm is one"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/quad-primorial-norm","kind":"node","lean":"MetaMathlibExt.quadPrimorialNorm","parent":5281,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norm of the "],["math","\\(n\\)"],["text","-th primorial"]],"source":"Norm of the $n$-th primorial"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#quad-primorial-norm","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$, and let $n$ be a natural number. The primorial norm $N(\\pi_n\\#)$ is the product of the absolute norms of the nonzero prime ideals of $\\mathcal{O}_K$ whose norm is at most $N(\\pi_n)$, where $N(\\pi_n)$ is the $n$-th smallest norm of a nonzero prime ideal of $\\mathcal{O}_K$ for $n \\ge 1$ and $N(\\pi_0) = 1$.","title":"Norm of the $n$-th primorial"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/quad-primorial-norm-zero","kind":"node","lean":"MetaMathlibExt.quadPrimorialNorm_zero","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#quad-primorial-norm-zero","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"fully_proved","summary":"For every number field $K$ with ring of integers $\\mathcal{O}_K$, the primorial norm $N(\\pi_0\\#)$ of $K$ equals $1$: no nonzero prime ideal of $\\mathcal{O}_K$ has norm at most $1$.","title":"The zeroth primorial norm is one"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/quad-schatunowsky-set","kind":"node","lean":"MetaMathlibExt.quadSchatunowskySet","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#quad-schatunowsky-set","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ and let $\\ell$ be a natural number. For $\\alpha \\in \\mathcal{O}_K$ write $N(\\alpha)$ for the absolute norm of the principal ideal $(\\alpha)$ and $\\Omega(\\alpha)$ for the number of prime ideal factors of $(\\alpha)$ counted with multiplicity. The Schatunowsky set $Q_\\ell(K)$ is the set of $\\alpha \\in \\mathcal{O}_K$ with $N(\\alpha) > 1$ such that every\u2026","title":"Schatunowsky set of a number field"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/one-not-mem-quad-schatunowsky-set","kind":"node","lean":"MetaMathlibExt.one_not_mem_quadSchatunowskySet","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#one-not-mem-quad-schatunowsky-set","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"fully_proved","summary":"For every number field $K$ and every natural number $\\ell$, the element $1$ of the ring of integers $\\mathcal{O}_K$ does not lie in the Schatunowsky set $Q_\\ell(K)$, because the ideal it generates has norm $1$.","title":"One is not in the Schatunowsky set"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/imag-quadratic-units-is-cyclic-card","kind":"node","lean":"MetaMathlibExt.imagQuadratic_units_isCyclic_card","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#imag-quadratic-units-is-cyclic-card","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be an imaginary quadratic field with discriminant $d_K$. The unit group of its ring of integers $\\mathcal{O}_K$ is cyclic, of order $4$ if $d_K = -4$, of order $6$ if $d_K = -3$, and of order $2$ otherwise.","title":"Unit groups of imaginary quadratic fields"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/tendsto-quad-prime-norm-succ-div","kind":"node","lean":"MetaMathlibExt.tendsto_quadPrimeNorm_succ_div","parent":5281,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ratios of consecutive prime ideal norms in an imaginary quadratic field tend to "],["math","\\(1\\)"]],"source":"Ratios of consecutive prime ideal norms in an imaginary quadratic field tend to $1$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#tendsto-quad-prime-norm-succ-div","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be an imaginary quadratic field, that is, a totally complex number field of degree $2$ over $\\mathbb{Q}$, with ring of integers $\\mathcal{O}_K$. For $m \\ge 1$ let $r_m$ be the least natural number $x$ such that at least $m$ nonzero prime ideals of $\\mathcal{O}_K$ have absolute norm at most $x$; this is the $m$-th smallest norm of a nonzero prime ideal, counted with multiplicity. Let $N(\\pi_m)$ be the $m$-th\u2026","title":"Ratios of consecutive prime ideal norms in an imaginary quadratic field tend to $1$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/exists-quad-prime-norm-pow-lt-quad-primo","kind":"node","lean":"MetaMathlibExt.exists_quadPrimeNorm_pow_lt_quadPrimorialNorm","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#exists-quad-prime-norm-pow-lt-quad-primo","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be an imaginary quadratic field. Put $N(\\pi_0) = 1$ and, for $n \\ge 1$, let $N(\\pi_n)$ be the $n$-th smallest number that is the norm of a nonzero prime ideal of $\\mathcal{O}_K$; let $N(\\pi_n\\#)$ be the product of the norms of the nonzero prime ideals of $\\mathcal{O}_K$ of norm at most $N(\\pi_n)$. For every natural number $\\ell$ there is $n_0$ such that $N(\\pi_{n+1})^{\\ell+1} < N(\\pi_n\\#)$ for all\u2026","title":"Prime norm powers eventually fall below the quadratic primorial"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/mem-prime-of-mem-quad-schatunowsky-set","kind":"node","lean":"MetaMathlibExt.mem_prime_of_mem_quadSchatunowskySet","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#mem-prime-of-mem-quad-schatunowsky-set","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be an imaginary quadratic field of class number $1$. For $\\alpha \\in \\mathcal{O}_K$ let $N(\\alpha)$ be the absolute norm of the ideal $(\\alpha)$. Put $N(\\pi_0) = 1$ and, for $n \\ge 1$, let $N(\\pi_n)$ be the $n$-th smallest number that is the norm of a nonzero prime ideal of $\\mathcal{O}_K$. For a natural number $\\ell$, let $Q_\\ell$ be the set of $\\alpha \\in \\mathcal{O}_K$ with $N(\\alpha) > 1$ such that\u2026","title":"Large elements of a Schatunowsky set lie in all small prime ideals"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/quad-schatunowsky-set-norm-bound-and-fin","kind":"node","lean":"MetaMathlibExt.quadSchatunowskySet_norm_bound_and_finite","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#quad-schatunowsky-set-norm-bound-and-fin","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be an imaginary quadratic field (a totally complex number field of degree $2$ over $\\mathbb{Q}$) with class number $1$ and ring of integers $\\mathcal{O}_K$, and let $\\ell$ and $n_0$ be natural numbers such that $N(\\pi_{n+1})^{\\ell+1} < N(\\pi_n\\#)$ for every $n \\ge n_0$, where $N(\\pi_n)$ and $N(\\pi_n\\#)$ are the prime and primorial norms of $K$. Then no $\\alpha \\in \\mathcal{O}_K$ whose principal ideal\u2026","title":"Norm bound and finiteness of the Schatunowsky set"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/quad-schatunowsky-set-eq-i-union","kind":"node","lean":"MetaMathlibExt.quadSchatunowskySet_eq_iUnion","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#quad-schatunowsky-set-eq-i-union","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be an imaginary quadratic field (a totally complex number field of degree $2$ over $\\mathbb{Q}$) with class number $1$ and ring of integers $\\mathcal{O}_K$, and let $\\ell$ and $n_0$ be natural numbers such that $N(\\pi_{n+1})^{\\ell+1} < N(\\pi_n\\#)$ for every $n \\ge n_0$, where $N(\\pi_n)$ and $N(\\pi_n\\#)$ are the prime and primorial norms of $K$. Write $N(\\alpha)$ for the absolute norm of the principal ideal\u2026","title":"Decomposition of the Schatunowsky set"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/gaussian-quad-prime-norm-pow-lt-quad-pri","kind":"node","lean":"MetaMathlibExt.gaussian_quadPrimeNorm_pow_lt_quadPrimorialNorm","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#gaussian-quad-prime-norm-pow-lt-quad-pri","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be a number field of discriminant $-4$ (so $K \\cong \\mathbb{Q}(i)$). Put $N(\\pi_0) = 1$ and, for $n \\ge 1$, let $N(\\pi_n)$ be the $n$-th smallest number that is the norm of a nonzero prime ideal of $\\mathcal{O}_K$; let $N(\\pi_n\\#)$ be the product of the norms of the nonzero prime ideals of $\\mathcal{O}_K$ of norm at most $N(\\pi_n)$. For each pair $(\\ell, n_\\ell)$ among $(1, 3)$, $(2, 4)$, $(3, 5)$,\u2026","title":"Explicit primorial thresholds for the Gaussian integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/eisenstein-quad-prime-norm-pow-lt-quad-p","kind":"node","lean":"MetaMathlibExt.eisenstein_quadPrimeNorm_pow_lt_quadPrimorialNorm","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#eisenstein-quad-prime-norm-pow-lt-quad-p","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be a number field of discriminant $-3$ (so $K \\cong \\mathbb{Q}(\\sqrt{-3})$). Put $N(\\pi_0) = 1$ and, for $n \\ge 1$, let $N(\\pi_n)$ be the $n$-th smallest number that is the norm of a nonzero prime ideal of $\\mathcal{O}_K$; let $N(\\pi_n\\#)$ be the product of the norms of the nonzero prime ideals of $\\mathcal{O}_K$ of norm at most $N(\\pi_n)$. For each pair $(\\ell, n_\\ell)$ among $(1, 3)$, $(2, 4)$, $(3, 5)$,\u2026","title":"Explicit primorial thresholds for the Eisenstein integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/gaussian-quad-schatunowsky-set-eq","kind":"node","lean":"MetaMathlibExt.gaussian_quadSchatunowskySet_eq","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#gaussian-quad-schatunowsky-set-eq","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be a number field of discriminant $-4$ (so $K \\cong \\mathbb{Q}(i)$). For $\\alpha \\in \\mathcal{O}_K$ let $N(\\alpha)$ be the absolute norm of the ideal $(\\alpha)$, and let $Q_\\ell$ be the set of $\\alpha \\in \\mathcal{O}_K$ with $N(\\alpha) > 1$ such that every $\\beta \\in \\mathcal{O}_K$ coprime to $\\alpha$ with $1 < N(\\beta) < N(\\alpha)$ has at most $\\ell$ prime ideal factors, counted with multiplicity. Below,\u2026","title":"The first three Schatunowsky sets of the Gaussian integers"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/eisenstein-quad-schatunowsky-set-eq","kind":"node","lean":"MetaMathlibExt.eisenstein_quadSchatunowskySet_eq","parent":5281,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems/index.html#eisenstein-quad-schatunowsky-set-eq","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/schatunowsky-type-theorems","status":"can_prove","summary":"Let $K$ be a number field of discriminant $-3$ (so $K \\cong \\mathbb{Q}(\\sqrt{-3})$). For $\\alpha \\in \\mathcal{O}_K$ let $N(\\alpha)$ be the absolute norm of the ideal $(\\alpha)$, and let $Q_\\ell$ be the set of $\\alpha \\in \\mathcal{O}_K$ with $N(\\alpha) > 1$ such that every $\\beta \\in \\mathcal{O}_K$ coprime to $\\alpha$ with $1 < N(\\beta) < N(\\alpha)$ has at most $\\ell$ prime ideal factors, counted with multiplicity.\u2026","title":"The first three Schatunowsky sets of the Eisenstein integers"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2.json","id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","kind":"scope","navigation":"module","parent":5203,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","title":"The ring \u2124[\u221a\u22122]"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/norm-nonneg","kind":"node","lean":"Zsqrtd.NegTwo.norm_nonneg'","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","The norm on "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," is nonnegative"]],"source":"The norm on $\\mathbb{Z}(\\sqrt{-2})$ is nonnegative"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#norm-nonneg","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"For every $x = a + b\\sqrt{-2}$ in $\\mathbb{Z}[\\sqrt{-2}]$, the norm $N(x) = a^2 + 2b^2$ satisfies $N(x) \\ge 0$.","title":"The norm on $\\mathbb{Z}(\\sqrt{-2})$ is nonnegative"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/norm-eq-zero","kind":"node","lean":"Zsqrtd.NegTwo.norm_eq_zero'","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","The norm on "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," vanishes only at zero"]],"source":"The norm on $\\mathbb{Z}(\\sqrt{-2})$ vanishes only at zero"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#norm-eq-zero","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"For $x = a + b\\sqrt{-2}$ in $\\mathbb{Z}[\\sqrt{-2}]$ with norm $N(x) = a^2 + 2b^2$, we have $N(x) = 0$ if and only if $x = 0$.","title":"The norm on $\\mathbb{Z}(\\sqrt{-2})$ vanishes only at zero"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/norm-pos","kind":"node","lean":"Zsqrtd.NegTwo.norm_pos'","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonzero elements of "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," have positive norm"]],"source":"Nonzero elements of $\\mathbb{Z}(\\sqrt{-2})$ have positive norm"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#norm-pos","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"For $x = a + b\\sqrt{-2}$ in $\\mathbb{Z}[\\sqrt{-2}]$ with norm $N(x) = a^2 + 2b^2$, we have $N(x) > 0$ if and only if $x \\ne 0$.","title":"Nonzero elements of $\\mathbb{Z}(\\sqrt{-2})$ have positive norm"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/abs-nat-cast-norm","kind":"node","lean":"Zsqrtd.NegTwo.abs_natCast_norm'","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","The norm on "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," equals its absolute value"]],"source":"The norm on $\\mathbb{Z}(\\sqrt{-2})$ equals its absolute value"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#abs-nat-cast-norm","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"For every $x = a + b\\sqrt{-2}$ in $\\mathbb{Z}[\\sqrt{-2}]$, the norm $N(x) = a^2 + 2b^2$ satisfies $|N(x)| = N(x)$, where $|N(x)|$ is taken as a natural number and read back as an integer.","title":"The norm on $\\mathbb{Z}(\\sqrt{-2})$ equals its absolute value"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/div-def","kind":"node","lean":"Zsqrtd.NegTwo.div_def'","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rounding division on "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"Rounding division on $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#div-def","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $x, y \\in \\mathbb{Z}[\\sqrt{-2}]$ with $y = c + d\\sqrt{-2}$, let $\\bar{y} = c - d\\sqrt{-2}$ be its conjugate and $N(y) = c^2 + 2d^2$ its norm, and write $x\\bar{y} = u + v\\sqrt{-2}$ with $u, v \\in \\mathbb{Z}$. The quotient is $x / y = \\operatorname{round}(u / N(y)) + \\operatorname{round}(v / N(y))\\sqrt{-2}$, where $u / N(y)$ and $v / N(y)$ are computed in $\\mathbb{Q}$ (division by $0$ gives $0$) and\u2026","title":"Rounding division on $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/mod-def","kind":"node","lean":"Zsqrtd.NegTwo.mod_def'","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Remainder of the rounding division on "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"Remainder of the rounding division on $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#mod-def","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"For all $x, y$ in $\\mathbb{Z}[\\sqrt{-2}]$, the remainder is $x \\bmod y = x - y\\,(x / y)$, where $x / y$ is the rounding division of $\\mathbb{Z}[\\sqrt{-2}]$.","title":"Remainder of the rounding division on $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/round-error","kind":"node","lean":"Zsqrtd.NegTwo.round_error","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#round-error","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $a$ and $N$ be integers with $N > 0$, and let $q$ be the integer nearest to the rational number $a/N$ (halves rounded up). Then $|2(a - qN)| \\le N$, that is, $a$ differs from $qN$ by at most $N/2$.","title":"Error of rounded integer division"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/norm-key","kind":"node","lean":"Zsqrtd.NegTwo.norm_key","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#norm-key","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $x, y, q \\in \\mathbb{Z}[\\sqrt{-2}]$, write $N(a + b\\sqrt{-2}) = a^2 + 2b^2$ for the norm and $\\bar{y}$ for the conjugate of $y$, and write $x\\bar{y} = u + v\\sqrt{-2}$ and $q = e + f\\sqrt{-2}$ with $u, v, e, f \\in \\mathbb{Z}$. Then $N(y)\\,N(x - yq) = (u - N(y)\\,e)^2 + 2\\,(v - N(y)\\,f)^2$.","title":"Norm identity for the remainder"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/norm-mod-lt","kind":"node","lean":"Zsqrtd.NegTwo.norm_mod_lt'","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","The remainder is smaller than the divisor in "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"The remainder is smaller than the divisor in $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#norm-mod-lt","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $x, y$ be elements of $\\mathbb{Z}[\\sqrt{-2}]$ with $y \\ne 0$, and let $N(a + b\\sqrt{-2}) = a^2 + 2b^2$ be the norm. The remainder $x \\bmod y = x - y\\,(x / y)$ of the rounding division of $\\mathbb{Z}[\\sqrt{-2}]$ satisfies $N(x \\bmod y) < N(y)$.","title":"The remainder is smaller than the divisor in $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/nat-abs-norm-mod-lt","kind":"node","lean":"Zsqrtd.NegTwo.natAbs_norm_mod_lt'","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#nat-abs-norm-mod-lt","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $x, y$ be elements of $\\mathbb{Z}[\\sqrt{-2}]$ with $y \\ne 0$, and let $N(a + b\\sqrt{-2}) = a^2 + 2b^2$ be the norm. The remainder $x \\bmod y = x - y\\,(x / y)$ of the rounding division of $\\mathbb{Z}[\\sqrt{-2}]$ satisfies $|N(x \\bmod y)| < |N(y)|$ as natural numbers.","title":"The remainder is smaller than the divisor, in absolute norm"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/norm-le-norm-mul-left","kind":"node","lean":"Zsqrtd.NegTwo.norm_le_norm_mul_left'","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplying by a nonzero element does not decrease the norm in "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"Multiplying by a nonzero element does not decrease the norm in $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#norm-le-norm-mul-left","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $N(a + b\\sqrt{-2}) = a^2 + 2b^2$ be the norm on $\\mathbb{Z}[\\sqrt{-2}]$. For $x, y \\in \\mathbb{Z}[\\sqrt{-2}]$ with $y \\ne 0$, we have $|N(x)| \\le |N(xy)|$ as natural numbers.","title":"Multiplying by a nonzero element does not decrease the norm in $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/zsqrtd-neg-two-is-unit-iff","kind":"node","lean":"Zsqrtd.NegTwo.zsqrtd_neg_two_isUnit_iff","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#zsqrtd-neg-two-is-unit-iff","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"An element $x$ of $\\mathbb{Z}[\\sqrt{-2}]$ is a unit if and only if $x = 1$ or $x = -1$.","title":"Units of \u2124(\u221a\u22122)"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/zsqrtd-neg-two-prime-of-nat-abs-norm-pri","kind":"node","lean":"Zsqrtd.NegTwo.zsqrtd_neg_two_prime_of_natAbs_norm_prime","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#zsqrtd-neg-two-prime-of-nat-abs-norm-pri","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"For $\\pi = a + b\\sqrt{-2}$ in $\\mathbb{Z}[\\sqrt{-2}]$ let $N(\\pi) = a^2 + 2b^2$ be its norm. If $|N(\\pi)|$ is a prime number, then $\\pi$ is a prime element of $\\mathbb{Z}[\\sqrt{-2}]$.","title":"Elements of \u2124(\u221a\u22122) of prime norm are prime"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/zsqrtd-neg-two-prime-sqrtd","kind":"node","lean":"Zsqrtd.NegTwo.zsqrtd_neg_two_prime_sqrtd","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#zsqrtd-neg-two-prime-sqrtd","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"The element $\\sqrt{-2}$ is a prime element of $\\mathbb{Z}[\\sqrt{-2}]$.","title":"\u221a\u22122 is prime in \u2124(\u221a\u22122)"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/zsqrtd-neg-two-two-eq","kind":"node","lean":"Zsqrtd.NegTwo.zsqrtd_neg_two_two_eq","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text"," is minus the square of "],["math","\\(\\sqrt{-2}\\)"]],"source":"$2$ is minus the square of $\\sqrt{-2}$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#zsqrtd-neg-two-two-eq","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"In $\\mathbb{Z}[\\sqrt{-2}]$ one has $2 = -(\\sqrt{-2})^2$.","title":"$2$ is minus the square of $\\sqrt{-2}$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/zsqrtd-neg-two-mod8-obstruction","kind":"node","lean":"Zsqrtd.NegTwo.zsqrtd_neg_two_mod8_obstruction","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#zsqrtd-neg-two-mod8-obstruction","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"For all $x, y \\in \\mathbb{Z}/8\\mathbb{Z}$, $x^2 + 2y^2 \\ne 5$ and $x^2 + 2y^2 \\ne 7$.","title":"x\u00b2 + 2y\u00b2 is never 5 or 7 modulo 8"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/zsqrtd-neg-two-inert","kind":"node","lean":"Zsqrtd.NegTwo.zsqrtd_neg_two_inert","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#zsqrtd-neg-two-inert","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"If $p$ is a prime number with $p \\equiv 5 \\pmod 8$ or $p \\equiv 7 \\pmod 8$, then $p$ is a prime element of $\\mathbb{Z}[\\sqrt{-2}]$.","title":"Primes congruent to 5 or 7 mod 8 are inert in \u2124(\u221a\u22122)"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/zsqrtd-neg-two-split","kind":"node","lean":"Zsqrtd.NegTwo.zsqrtd_neg_two_split","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes "],["math","\\(p \\equiv 1, 3 \\pmod 8\\)"],["text"," are norms of primes of "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"Primes $p \\equiv 1, 3 \\pmod 8$ are norms of primes of $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#zsqrtd-neg-two-split","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $p$ be a prime number with $p \\equiv 1 \\pmod 8$ or $p \\equiv 3 \\pmod 8$. Then there is a prime element $\\pi = a + b\\sqrt{-2}$ of $\\mathbb{Z}[\\sqrt{-2}]$ whose norm $a^2 + 2b^2$ equals $p$.","title":"Primes $p \\equiv 1, 3 \\pmod 8$ are norms of primes of $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/zsqrtd-neg-two-split-conj","kind":"node","lean":"Zsqrtd.NegTwo.zsqrtd_neg_two_split_conj","parent":5303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conjugate factorization of an odd prime in "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"Conjugate factorization of an odd prime in $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#zsqrtd-neg-two-split-conj","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $p$ be an odd prime number and let $\\pi = a + b\\sqrt{-2}$ be an element of $\\mathbb{Z}[\\sqrt{-2}]$ whose norm $a^2 + 2b^2$ equals $p$. Write $\\bar\\pi = a - b\\sqrt{-2}$ for the conjugate of $\\pi$. Then $p = \\pi\\bar\\pi$, $\\bar\\pi$ is prime in $\\mathbb{Z}[\\sqrt{-2}]$, and $\\bar\\pi$ is not associated to $\\pi$.","title":"Conjugate factorization of an odd prime in $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/unit-val","kind":"node","lean":"Zsqrtd.NegTwo.unitVal","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#unit-val","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"The map $\\{0, 1\\} \\to \\mathbb{Z}[\\sqrt{-2}]$ sending $0$ to $1$ and $1$ to $-1$, which lists the two units of $\\mathbb{Z}[\\sqrt{-2}]$.","title":"Enumeration of the units of \u2124(\u221a\u22122)"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/unit-val-inj","kind":"node","lean":"Zsqrtd.NegTwo.unitVal_inj","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#unit-val-inj","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $\\varepsilon_0 = 1$ and $\\varepsilon_1 = -1$ in $\\mathbb{Z}[\\sqrt{-2}]$. For $i, j \\in \\{0, 1\\}$, if $\\varepsilon_i = \\varepsilon_j$ then $i = j$.","title":"The enumeration of the units of \u2124(\u221a\u22122) is injective"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/unit-val-complete","kind":"node","lean":"Zsqrtd.NegTwo.unitVal_complete","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#unit-val-complete","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $\\varepsilon_0 = 1$ and $\\varepsilon_1 = -1$ be the enumeration of the units of $\\mathbb{Z}[\\sqrt{-2}]$. Every unit $u$ of $\\mathbb{Z}[\\sqrt{-2}]$ equals $\\varepsilon_i$ for some $i \\in \\{0, 1\\}$.","title":"Every unit of \u2124(\u221a\u22122) is enumerated"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/unit-val-inv","kind":"node","lean":"Zsqrtd.NegTwo.unitValInv","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#unit-val-inv","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"The map $\\{0, 1\\} \\to \\mathbb{Z}[\\sqrt{-2}]$ sending $0$ to $1$ and $1$ to $-1$. Since $1$ and $-1$ are their own inverses, its value at $i$ is the inverse of the $i$-th enumerated unit.","title":"Inverses of the enumerated units of \u2124(\u221a\u22122)"},{"id":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/unit-val-mul-unit-val-inv","kind":"node","lean":"Zsqrtd.NegTwo.unitVal_mul_unitValInv","parent":5303,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2/index.html#unit-val-mul-unit-val-inv","scope":"NumberTheory/algebraic-number-theory/quadratic-fields/the-ring-2","status":"fully_proved","summary":"Let $\\varepsilon_0 = 1$, $\\varepsilon_1 = -1$ be the enumerated units of $\\mathbb{Z}[\\sqrt{-2}]$ and $\\eta_0 = 1$, $\\eta_1 = -1$ their listed inverses. Then $\\varepsilon_i \\eta_i = 1$ for each $i \\in \\{0, 1\\}$.","title":"Each enumerated unit of \u2124(\u221a\u22122) times its listed inverse is one"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/cyclotomic-fields.json","id":"NumberTheory/algebraic-number-theory/cyclotomic-fields","kind":"scope","parent":5201,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/index.html","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields","status":"can_state","summary":"Cyclotomic fields and polynomials, real cyclotomic polynomials, 2cos(2\u03c0k/n), cyclotomic units, ramification and inertness in cyclotomic fields.","title":"Cyclotomic fields"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/exists-ring-hom-zeta-eq-of-is-cyclotomic","kind":"node","lean":"IsPrimitiveRoot.exists_ringHom_zeta_eq_of_isCyclotomicExtension","parent":5328,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/index.html#exists-ring-hom-zeta-eq-of-is-cyclotomic","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields","status":"fully_proved","summary":"A cyclotomic extension `K` of `\u211a` embeds into any characteristic-zero field `L` containing a primitive `n`-th root of unity, sending the distinguished root `\u03b6` to it.","title":"Embedding a cyclotomic extension of the rationals"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials.json","id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials","kind":"scope","navigation":"module","parent":5328,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/index.html","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials","status":"fully_proved","title":"Homogenized cyclotomic polynomials"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/homogenized-cyclotomic","kind":"node","lean":"MetaMathlibExt.homogenizedCyclotomic","parent":5330,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/index.html#homogenized-cyclotomic","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials","status":"fully_proved","summary":"For a natural number $n$ and integers $a, b$, let $c_0, \\dots, c_{\\varphi(n)}$ be the integer coefficients of the $n$-th cyclotomic polynomial, whose degree is $\\varphi(n)$, where $\\varphi$ is Euler's totient function. The homogenized cyclotomic value is the integer","title":"Homogenized cyclotomic value"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/homogenized-cyclotomic-one","kind":"node","lean":"MetaMathlibExt.homogenizedCyclotomic_one","parent":5330,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/index.html#homogenized-cyclotomic-one","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials","status":"fully_proved","summary":"For all integers $a$ and $b$, the homogenized cyclotomic value $\\Phi_1(a, b)$ equals $a - b$.","title":"Homogenized cyclotomic value of index one"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/homogenized-cyclotomic-two","kind":"node","lean":"MetaMathlibExt.homogenizedCyclotomic_two","parent":5330,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/index.html#homogenized-cyclotomic-two","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials","status":"fully_proved","summary":"For all integers $a$ and $b$, the homogenized cyclotomic value $\\Phi_2(a, b)$ equals $a + b$.","title":"Homogenized cyclotomic value of index two"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/homogenized-cyclotomic-one-right","kind":"node","lean":"MetaMathlibExt.homogenizedCyclotomic_one_right","parent":5330,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/index.html#homogenized-cyclotomic-one-right","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials","status":"fully_proved","summary":"For every natural number $n$ and every integer $a$, the homogenized cyclotomic value $\\Phi_n(a, 1)$ equals $\\Phi_n(a)$, the value at $a$ of the $n$-th cyclotomic polynomial with integer coefficients.","title":"Homogenized cyclotomic value with second argument one"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/homogenized-cyclotomic-cast","kind":"node","lean":"MetaMathlibExt.homogenizedCyclotomic_cast","parent":5330,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials/index.html#homogenized-cyclotomic-cast","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/homogenized-cyclotomic-polynomials","status":"fully_proved","summary":"Let $K$ be a field, $n$ a natural number and $a, b$ integers such that the image of $b$ in $K$ is nonzero. Then the image in $K$ of the homogenized cyclotomic value $\\Phi_n(a, b)$ equals $b^{\\varphi(n)} \\Phi_n(a/b)$, where $\\Phi_n$ is the $n$-th cyclotomic polynomial over $K$, $\\varphi$ is Euler's totient function, and $b$ and $a/b$ are computed in $K$.","title":"Homogenized cyclotomic value in a field"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products.json","id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products","kind":"scope","navigation":"module","parent":5328,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products/index.html","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products","status":"can_prove","title":"Infinite cyclotomic products"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products/tendsto-prod-cyclotomic-hat-inv-root-one","kind":"node","lean":"MetaMathlibExt.tendsto_prod_cyclotomicHat_inv_root_one","parent":5336,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products/index.html#tendsto-prod-cyclotomic-hat-inv-root-one","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products","status":"can_prove","summary":"For $n \\ge 1$ and complex $z$, let $\\hat\\Phi_n(z) = \\prod_\\omega (1 - z\\omega)$, the product over the primitive $n$-th roots of unity $\\omega$; thus $\\hat\\Phi_1(z) = 1 - z$ and $\\hat\\Phi_n = \\Phi_n$ for $n \\ge 2$, where $\\Phi_n$ is the $n$-th cyclotomic polynomial. For every complex $z$ with $|z| < 1$, the partial products $\\prod_{n=1}^{N} \\hat\\Phi_n(z)^{-1/n}$ tend to $1$ as $N \\to \\infty$, where\u2026","title":"Constancy of an infinite cyclotomic product"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products/exists-tendsto-prod-one-sub-pow-and-cycl","kind":"node","lean":"MetaMathlibExt.exists_tendsto_prod_one_sub_pow_and_cyclotomicHat_zeta","parent":5336,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products/index.html#exists-tendsto-prod-one-sub-pow-and-cycl","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/infinite-cyclotomic-products","status":"can_prove","summary":"For $n \\ge 1$ and complex $z$, let $\\hat\\Phi_n(z) = \\prod_\\omega (1 - z\\omega)$, the product over the primitive $n$-th roots of unity $\\omega$. Let $z$ be a complex number with $|z| < 1$, let $s > 1$ be real, and let $\\zeta$ be the Riemann zeta function. Then","title":"A zeta-twisted cyclotomic product identity"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values.json","id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","kind":"scope","navigation":"module","parent":5328,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","title":"Prime divisor bounds for cyclotomic values"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/prime-eq-of-dvd-sub-one-of-dvd-cyclotomi","kind":"node","lean":"MetaMathlibExt.prime_eq_of_dvd_sub_one_of_dvd_cyclotomic_eval","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","A common prime divisor of "],["math","\\(x - 1\\)"],["text"," and "],["math","\\(\\Phi_p(x)\\)"],["text"," is "],["math","\\(p\\)"]],"source":"A common prime divisor of $x - 1$ and $\\Phi_p(x)$ is $p$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#prime-eq-of-dvd-sub-one-of-dvd-cyclotomi","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $\\rho$ and $p$ be primes, let $x$ be a positive integer, and let $\\Phi_p$ be the $p$-th cyclotomic polynomial with integer coefficients. If $\\rho \\mid x - 1$ and $\\rho \\mid \\Phi_p(x)$, then $\\rho = p$.","title":"A common prime divisor of $x - 1$ and $\\Phi_p(x)$ is $p$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/prime-dvd-pow-self-sub-one-iff","kind":"node","lean":"MetaMathlibExt.prime_dvd_pow_self_sub_one_iff","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility of "],["math","\\(x^p - 1\\)"],["text"," by "],["math","\\(p\\)"]],"source":"Divisibility of $x^p - 1$ by $p$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#prime-dvd-pow-self-sub-one-iff","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $p$ be a prime, let $x$ be a positive integer, and let $\\Phi_p$ be the $p$-th cyclotomic polynomial with integer coefficients. Then $p \\mid x^p - 1$ if and only if $p \\mid x - 1$ and $p \\mid \\Phi_p(x)$.","title":"Divisibility of $x^p - 1$ by $p$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/prime-eq-of-dvd-sub-one-of-dvd-cyclotomi-2","kind":"node","lean":"MetaMathlibExt.prime_eq_of_dvd_sub_one_of_dvd_cyclotomic_eval_pow","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","A common prime divisor of "],["math","\\(y - 1\\)"],["text"," and "],["math","\\(\\Phi_p(y^{p^{a-1}})\\)"],["text"," is "],["math","\\(p\\)"]],"source":"A common prime divisor of $y - 1$ and $\\Phi_p(y^{p^{a-1}})$ is $p$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#prime-eq-of-dvd-sub-one-of-dvd-cyclotomi-2","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $\\rho$ and $p$ be primes, let $a \\ge 1$ be a natural number, let $y$ be a positive integer, and let $\\Phi_p$ be the $p$-th cyclotomic polynomial with integer coefficients. If $\\rho \\mid y - 1$ and $\\rho \\mid \\Phi_p(y^{p^{a-1}})$, then $\\rho = p$.","title":"A common prime divisor of $y - 1$ and $\\Phi_p(y^{p^{a-1}})$ is $p$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/prime-dvd-pow-sub-one-iff-cyclotomic","kind":"node","lean":"MetaMathlibExt.prime_dvd_pow_sub_one_iff_cyclotomic","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility of "],["math","\\(x^{rp^a} - 1\\)"],["text"," by "],["math","\\(p\\)"]],"source":"Divisibility of $x^{rp^a} - 1$ by $p$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#prime-dvd-pow-sub-one-iff-cyclotomic","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $p$ be a prime, let $r$ and $a$ be natural numbers with $r$ coprime to $p$, let $x$ be a positive integer, and let $\\Phi_p$ be the $p$-th cyclotomic polynomial with integer coefficients. Then $p \\mid x^{rp^a} - 1$ if and only if $p \\mid x^r - 1$ and $p \\mid \\Phi_p(x^{rp^{i-1}})$ for every $i \\in \\{1, \\dots, a\\}$.","title":"Divisibility of $x^{rp^a} - 1$ by $p$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/prime-mod-eq-one-of-dvd-cyclotomic-eval","kind":"node","lean":"MetaMathlibExt.prime_modEq_one_of_dvd_cyclotomic_eval","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime divisors of "],["math","\\(\\Phi_p(x)\\)"],["text"," other than "],["math","\\(p\\)"],["text"," are "],["math","\\(1\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Prime divisors of $\\Phi_p(x)$ other than $p$ are $1$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#prime-mod-eq-one-of-dvd-cyclotomic-eval","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $\\rho$ and $p$ be distinct primes, let $x$ be a positive integer, and let $\\Phi_p$ be the $p$-th cyclotomic polynomial with integer coefficients. If $\\rho \\mid \\Phi_p(x)$, then $\\rho \\equiv 1 \\pmod p$.","title":"Prime divisors of $\\Phi_p(x)$ other than $p$ are $1$ modulo $p$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/hybrid-bound-prime-divisors","kind":"node","lean":"MetaMathlibExt.hybrid_bound_prime_divisors","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hybrid bound for prime divisors of "],["math","\\(x^n - 1\\)"]],"source":"Hybrid bound for prime divisors of $x^n - 1$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#hybrid-bound-prime-divisors","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"The hybrid bound. Let `n = \u220f_{i=1}^m p_i^{a_i}` with distinct primes `p_i` (in any order), `x > 1`, and `\u03c9_n = \u03c9(x^n - 1)`. With `p_0 := 1`, let `\u03a0^j = {\u03c1 prime : \u03c1 \u2261 1 (mod p_j), \u03c1 \u2262 1 (mod p_s) for j < s \u2264 m}` and `r_{i,j}` its `i`-th element (`Nat.nth`, `0`-indexed here). The source's `x \u2265 min_{k_0 + \u22ef + k_m = \u03c9_n} max_{0 \u2264 t \u2264 m} (1 + \u220f_{j \u2264 t} \u220f_{i \u2264 k_j} r_{i,j})^{T(t)}` with `T(t) = \u220f_{s=1}^t p_s^{-a_s}` is\u2026","title":"Hybrid bound for prime divisors of $x^n - 1$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/prime-hybrid-bound","kind":"node","lean":"MetaMathlibExt.prime_hybrid_bound","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hybrid bound for prime divisors of "],["math","\\(x^p - 1\\)"]],"source":"Hybrid bound for prime divisors of $x^p - 1$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#prime-hybrid-bound","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"The `p`-hybrid bound (Theorem 1 for prime `n = p`). For a prime `p`, `x > 1` and `\u03c9_p = \u03c9(x^p - 1)`, let `r_{i,0}` and `r_{j,1}` enumerate the primes `\u2262 1` and `\u2261 1 (mod p)` (`Nat.nth`, `0`-indexed here). The source's `x \u2265 min_{0 \u2264 k < \u03c9_p} max(1 + \u220f_{i \u2264 k} r_{i,0}, (1 + \u220f_{i \u2264 k} r_{i,0} \u220f_{j \u2264 \u03c9_p - k} r_{j,1})^{1/p})` is stated equivalently as the existence of `k < \u03c9_p` meeting both bounds, the second after\u2026","title":"Hybrid bound for prime divisors of $x^p - 1$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/two-pow-ramified-bound","kind":"node","lean":"MetaMathlibExt.two_pow_ramified_bound","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ramified bound for "],["math","\\(n = 2^a\\)"]],"source":"The ramified bound for $n = 2^a$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#two-pow-ramified-bound","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $a \\ge 1$, let $n = 2^a$, let $x > 1$ be a natural number, let $\\omega_n$ be the number of distinct prime factors of $x^n - 1$, and let $s_1 = 2 < s_2 = 3 < s_3 = 5 < \\cdots$ be the primes in increasing order. Then","title":"The ramified bound for $n = 2^a$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/two-pow-trivial-ramified-bound","kind":"node","lean":"MetaMathlibExt.two_pow_trivial_ramified_bound","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","A weaker ramified bound for "],["math","\\(n = 2^a\\)"]],"source":"A weaker ramified bound for $n = 2^a$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#two-pow-trivial-ramified-bound","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $a \\ge 1$, let $n = 2^a$, let $x > 1$ be a natural number, let $\\omega_n$ be the number of distinct prime factors of $x^n - 1$, and let $s_1 = 2 < s_2 = 3 < s_3 = 5 < \\cdots$ be the primes in increasing order. Then","title":"A weaker ramified bound for $n = 2^a$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/lt-sixteen-iff-prod-first-primes-lt","kind":"node","lean":"MetaMathlibExt.lt_sixteen_iff_prod_first_primes_lt","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#lt-sixteen-iff-prod-first-primes-lt","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"For every positive integer $w$, we have $w < 16$ if and only if the product $p_1 p_2 \\cdots p_w$ of the first $w$ primes is less than $16^w$.","title":"Primorials below powers of sixteen"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/ramified-bound-prime-divisors","kind":"node","lean":"MetaMathlibExt.ramified_bound_prime_divisors","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramified bound on the number of prime divisors of "],["math","\\(x^n - 1\\)"]],"source":"Ramified bound on the number of prime divisors of $x^n - 1$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#ramified-bound-prime-divisors","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"The ramified bound. Let `n = \u220f_{i=1}^m p_i^{a_i}` with primes `p_1 < \u22ef < p_m` (the ordering the source fixes at the start of its Section 5), `x > 1`, `\u03c9_n = \u03c9(x^n - 1)`, and, with `p_0 := 1`, let `r_{i,j}` be the `i`-th element (`Nat.nth`, `0`-indexed here) of `\u03a0^j = {\u03c1 prime : \u03c1 \u2261 1 (mod p_j), \u03c1 \u2262 1 (mod p_s) for j < s \u2264 m}`. Fix a prime factor `p_\u03b1` of `n` and write `p_\u03b1 = r_{b,\u03c4}` with `\u03c4 \u2264 m`. As in\u2026","title":"Ramified bound on the number of prime divisors of $x^n - 1$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/prime-ramified-bound","kind":"node","lean":"MetaMathlibExt.prime_ramified_bound","parent":5339,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramified bound on prime divisors of "],["math","\\(x^p - 1\\)"],["text"," for prime "],["math","\\(p\\)"]],"source":"Ramified bound on prime divisors of $x^p - 1$ for prime $p$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#prime-ramified-bound","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"The `p`-ramified bound (Theorem 2 for prime `n = p`). For a prime `p`, `x > 1` and `\u03c9_p = \u03c9(x^p - 1)`, let `r_{i,0}` and `r_{j,1}` enumerate the primes `\u2262 1` and `\u2261 1 (mod p)` (`Nat.nth`, `0`-indexed here), `p = r_{b,0}`, `R^0_k = \u220f_{i < k} r_{i,0}` and `R^1_k = \u220f_{j < \u03c9_p - k} r_{j,1}`. The source's minima over `k \u2208 [0, k') \u222a [k', \u03c9_p)` (with `k' = b + 1`) are stated as the existence of `k < \u03c9_p` meeting both\u2026","title":"Ramified bound on prime divisors of $x^p - 1$ for prime $p$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/x-pow-sub-one-eq-mul-prod-cyclotomic-com","kind":"node","lean":"MetaMathlibExt.X_pow_sub_one_eq_mul_prod_cyclotomic_comp","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#x-pow-sub-one-eq-mul-prod-cyclotomic-com","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $x$ be the indeterminate of the polynomial ring $\\mathbb{Z}[x]$. Let $m$ be a natural number, let $p_1, \\dots, p_m$ be primes, not necessarily distinct, let $a_1, \\dots, a_m$ be natural numbers, and put $n = \\prod_{i=1}^m p_i^{a_i}$. For $1 \\le i \\le m$ let $x_i = x^{N_i}$ with $N_i = \\prod_{1 \\le r < i} p_r^{a_r}$. Then","title":"Factorization into composed cyclotomic polynomials"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/mem-line-problem-of-prime-sieve","kind":"node","lean":"MetaMathlibExt.mem_lineProblem_of_prime_sieve","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#mem-line-problem-of-prime-sieve","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $q$ and $n$ be natural numbers, let $\\omega$ be the number of distinct prime factors of $q^n - 1$, and let $S$ be a set of $s$ prime factors of $q^n - 1$ with $\\delta = 1 - \\sum_{\\rho \\in S} 1/\\rho > 0$. Suppose that $q > (n - 1)^2 \\, 2^{2(\\omega - s)} \\left(\\frac{s - 1}{\\delta} + 2\\right)^2$. Then for all finite fields $K$ and $L$ with $L$ an extension of $K$, $|K| = q$ and $[L : K] = n$, and all nonzero\u2026","title":"Cohen's prime sieve for the line problem"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/mem-line-problem-of-modified-prime-sieve","kind":"node","lean":"MetaMathlibExt.mem_lineProblem_of_modified_prime_sieve","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#mem-line-problem-of-modified-prime-sieve","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $q$ and $n$ be natural numbers such that $q^n - 1$ has $\\omega \\ge 2$ distinct prime factors. Let $S$ be a set of $s$ prime factors of $q^n - 1$, let $l$ be a prime factor of $q^n - 1$ not in $S$, let $k$ be the product of the prime factors of $q^n - 1$ that are neither $l$ nor in $S$, and put $\\delta = 1 - \\sum_{\\rho \\in S} 1/\\rho$. Suppose that $\\delta > 1/(l\\varphi(k))$, where $\\varphi$ is Euler's totient\u2026","title":"Cohen's modified prime sieve for the line problem"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/mem-line-problem-of-general-prime-sieve","kind":"node","lean":"MetaMathlibExt.mem_lineProblem_of_general_prime_sieve","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#mem-line-problem-of-general-prime-sieve","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $q$ and $n$ be natural numbers, let $\\omega$ be the number of distinct prime factors of $q^n - 1$, and let $S$ and $T$ be disjoint sets of prime factors of $q^n - 1$, with $s = |S|$ and $r = |T|$. Let $k$ be the product of the prime factors of $q^n - 1$ that lie in neither $S$ nor $T$, and put $\\delta = 1 - \\sum_{\\rho \\in S} 1/\\rho$, $\\varepsilon = \\sum_{l \\in T} 1/l$ and $\\mu = \\varphi(k)/k$, where $\\varphi$\u2026","title":"General prime sieve for the line problem"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/mem-line-problem-five","kind":"node","lean":"MetaMathlibExt.mem_lineProblem_five","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#mem-line-problem-five","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $E_5$ be the set of $647$ prime powers listed in the appendix of arXiv:2412.00010, the smallest being $2$ and the largest $62791$. Let $q$ be a natural number not in $E_5$. Then for all finite fields $K$ and $L$ with $L$ an extension of $K$, $|K| = q$ and $[L : K] = 5$, and all nonzero $\\gamma_1, \\gamma_2 \\in L$ with $K(\\gamma_1/\\gamma_2) = L$, there is $a \\in K$ such that $a\\gamma_2 + \\gamma_1$ generates the\u2026","title":"The line problem for quintic extensions"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/mem-primitive-sum-of-sixteen-pow-lt","kind":"node","lean":"MetaMathlibExt.mem_primitiveSum_of_sixteen_pow_lt","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#mem-primitive-sum-of-sixteen-pow-lt","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $q$ and $n$ be coprime natural numbers with $q^n > 2$, and let $\\omega$ be the number of distinct prime factors of $q^n - 1$. If $16^\\omega < q^n$, that is $q^{n/2} > 2^{2\\omega}$, then every finite field $L$ with $q^n$ elements contains an element $\\alpha$ such that both $\\alpha$ and $\\alpha + \\alpha^{-1}$ generate the multiplicative group $L^\\times$.","title":"Liao\u2013Li\u2013Pu criterion for primitive element sums"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/mem-primitive-sum-one-iff","kind":"node","lean":"MetaMathlibExt.mem_primitiveSum_one_iff","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#mem-primitive-sum-one-iff","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"For every natural number $q$, we have $q \\notin \\{2, 3, 4, 5, 7, 9, 13, 25, 121\\}$ if and only if every finite field $L$ with $q$ elements contains an element $\\alpha$ such that both $\\alpha$ and $\\alpha + \\alpha^{-1}$ generate the multiplicative group $L^\\times$.","title":"Finite fields with a primitive element sum"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/mem-primitive-sum-of-not-mem-exceptional","kind":"node","lean":"MetaMathlibExt.mem_primitiveSum_of_not_mem_exceptional","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#mem-primitive-sum-of-not-mem-exceptional","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $q$ and $n > 1$ be coprime natural numbers, and let $\\omega$ be the number of distinct prime factors of $q^n - 1$. Suppose that the pair $(n, \\omega)$ is none of the following $33$ pairs: $(2, w)$ for $1 \\le w \\le 11$; $(3, w)$ for $2 \\le w \\le 7$; $(4, w)$ for $w \\in \\{2, 3, 4, 5, 6, 7, 9\\}$; $(5, 3)$ and $(5, 5)$; $(6, w)$ for $3 \\le w \\le 6$; $(8, 4)$ and $(8, 6)$; and $(12, 6)$. Then every finite field $L$\u2026","title":"Primitive element sums away from exceptional pairs"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/card-prime-factors-le-fifteen-of-not-mem","kind":"node","lean":"MetaMathlibExt.card_primeFactors_le_fifteen_of_not_mem_primitiveSum","parent":5339,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values/index.html#card-prime-factors-le-fifteen-of-not-mem","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/prime-divisor-bounds-for-cyclotomic-values","status":"can_prove","summary":"Let $q$ and $n$ be natural numbers with $\\gcd(q, n) = 1$. If it is not the case that every finite field $L$ with $q^n$ elements contains an element $\\alpha$ such that both $\\alpha$ and $\\alpha + \\alpha^{-1}$ generate the multiplicative group $L^\\times$, then $q^n - 1$ has at most $15$ distinct prime factors.","title":"Fields without primitive element sums have few prime factors"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields.json","id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields","kind":"scope","navigation":"module","parent":5328,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/index.html","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields","status":"fully_proved","title":"Ramification and inertia in cyclotomic fields"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/is-field-quotient-span-nat-cast-iff-orde","kind":"node","lean":"IsCyclotomicExtension.Rat.isField_quotient_span_natCast_iff_orderOf_eq","parent":5361,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/index.html#is-field-quotient-span-nat-cast-iff-orde","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields","status":"fully_proved","summary":"Let $\\ell$ and $p$ be distinct prime numbers, and let $K$ be a number field that is an $\\ell$-th cyclotomic extension of $\\mathbb{Q}$, with ring of integers $\\mathcal{O}_K$. Then the quotient ring $\\mathcal{O}_K / p\\mathcal{O}_K$ is a field (that is, $p$ is inert in $K$) if and only if the multiplicative order of $p$ modulo $\\ell$ is $\\ell - 1$.","title":"Inert primes in prime cyclotomic fields"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/is-unramified-in-of-not-dvd","kind":"node","lean":"IsCyclotomicExtension.Rat.isUnramifiedIn_of_not_dvd","parent":5361,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cyclotomic fields are unramified at primes not dividing "],["math","\\(m\\)"]],"source":"Cyclotomic fields are unramified at primes not dividing $m$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/index.html#is-unramified-in-of-not-dvd","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields","status":"fully_proved","summary":"Let $m \\ge 1$ and let $p$ be a prime number with $p \\nmid m$. If $K$ is a number field that is an $m$-th cyclotomic extension of $\\mathbb{Q}$ (that is, $K = \\mathbb{Q}(\\zeta_m)$ for a primitive $m$-th root of unity $\\zeta_m$), then $p$ is unramified in $K$: every prime ideal of the ring of integers $\\mathcal{O}_K$ lying over $p\\mathbb{Z}$ is unramified over $\\mathbb{Z}$.","title":"Cyclotomic fields are unramified at primes not dividing $m$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/is-unramified-in-div-max-pow","kind":"node","lean":"IsCyclotomicExtension.Rat.isUnramifiedIn_divMaxPow","parent":5361,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-free part of "],["math","\\(m\\)"],["text"," gives a cyclotomic field unramified at "],["math","\\(p\\)"]],"source":"The $p$-free part of $m$ gives a cyclotomic field unramified at $p$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/index.html#is-unramified-in-div-max-pow","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields","status":"fully_proved","summary":"Let $m \\ge 1$, let $p$ be a prime number, and let $m'$ be $m$ divided by the largest power of $p$ dividing it, so that $m = p^a m'$ for some $a \\ge 0$ with $p \\nmid m'$. Then $p$ is unramified in the $m'$-th cyclotomic field $\\mathbb{Q}(\\zeta_{m'})$, where $\\zeta_{m'}$ is a primitive $m'$-th root of unity: every prime ideal of its ring of integers lying over $p\\mathbb{Z}$ is unramified over $\\mathbb{Z}$.","title":"The $p$-free part of $m$ gives a cyclotomic field unramified at $p$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/eq-bot-of-is-unramified-in-prime-pow","kind":"node","lean":"IsCyclotomicExtension.Rat.eq_bot_of_isUnramifiedIn_prime_pow","parent":5361,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subfields of a "],["math","\\(p\\)"],["text","-power cyclotomic field unramified at "],["math","\\(p\\)"],["text"," are trivial"]],"source":"Subfields of a $p$-power cyclotomic field unramified at $p$ are trivial"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields/index.html#eq-bot-of-is-unramified-in-prime-pow","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/ramification-and-inertia-in-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a prime number, $e \\ge 0$ an integer and $L$ a number field. Let $E_1$ and $F$ be subfields of $L$ such that $E_1$ is a $p^e$-th cyclotomic extension of $\\mathbb{Q}$ and $F \\subseteq E_1$. If $p$ is unramified in $F$, that is, every prime ideal of the ring of integers $\\mathcal{O}_F$ lying over $p\\mathbb{Z}$ is unramified over $\\mathbb{Z}$, then $F = \\mathbb{Q}$.","title":"Subfields of a $p$-power cyclotomic field unramified at $p$ are trivial"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n.json","id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","kind":"scope","navigation":"module","parent":5328,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"can_state","title":"Real cyclotomic polynomials and 2cos(2\u03c0k/n)"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/nat-degree-real-cyclotomic-polynomial-2","kind":"node","parent":5366,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html#nat-degree-real-cyclotomic-polynomial-2","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"can_state","summary":"For `2 < n`, `n.totient` is even and the `natDegree` of the real cyclotomic polynomial `realCyclotomicPolynomial n` equals `n.totient / 2`. The `Even` conjunct makes the source's exact half machine-visible under natural-number division.","title":"Degree of the real cyclotomic polynomial"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/real-cyclotomic-polynomial","kind":"node","lean":"MetaMathlibExt.realCyclotomicPolynomial","parent":5366,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html#real-cyclotomic-polynomial","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"fully_proved","summary":"For a natural number $n$, the real cyclotomic polynomial $\\psi_n \\in \\mathbb{Z}[X]$ is the minimal polynomial over $\\mathbb{Z}$ of the complex number $\\zeta + \\zeta^{-1}$, where $\\zeta = e^{2\\pi i/n}$; for $n \\ge 1$ this number is $2\\cos(2\\pi/n)$.","title":"Real cyclotomic polynomial"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/real-cyclotomic-polynomial-eq-minpoly-tw","kind":"node","lean":"MetaMathlibExt.realCyclotomicPolynomial_eq_minpoly_two_cos","parent":5366,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimal polynomial of "],["math","\\(2\\cos(2\\pi k/n)\\)"],["text"," for "],["math","\\(k\\)"],["text"," coprime to "],["math","\\(n\\)"]],"source":"Minimal polynomial of $2\\cos(2\\pi k/n)$ for $k$ coprime to $n$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html#real-cyclotomic-polynomial-eq-minpoly-tw","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $n > 2$ and $\\gcd(k, n) = 1$. Then the real cyclotomic polynomial $\\psi_n$ equals the minimal polynomial over $\\mathbb{Z}$ of the complex number $2\\cos(2\\pi k/n)$.","title":"Minimal polynomial of $2\\cos(2\\pi k/n)$ for $k$ coprime to $n$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/real-cyclotomic-polynomial-eq-minpoly-tw-2","kind":"node","lean":"MetaMathlibExt.realCyclotomicPolynomial_eq_minpoly_two_cos_of_coprime","parent":5366,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimal polynomial of "],["math","\\(2\\cos(2\\pi k/n)\\)"],["text"," for "],["math","\\(1 \\le k \\le n\\)"],["text"," coprime to "],["math","\\(n\\)"]],"source":"Minimal polynomial of $2\\cos(2\\pi k/n)$ for $1 \\le k \\le n$ coprime to $n$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html#real-cyclotomic-polynomial-eq-minpoly-tw-2","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $n > 2$, $1 \\le k \\le n$ and $\\gcd(k, n) = 1$. Then the real cyclotomic polynomial $\\psi_n$ equals the minimal polynomial over $\\mathbb{Z}$ of the complex number $2\\cos(2\\pi k/n)$.","title":"Minimal polynomial of $2\\cos(2\\pi k/n)$ for $1 \\le k \\le n$ coprime to $n$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/nat-degree-real-cyclotomic-polynomial","kind":"node","parent":5366,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html#nat-degree-real-cyclotomic-polynomial","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"can_state","summary":"For `2 < n`, `n.totient` is even and the `natDegree` of the real cyclotomic polynomial `realCyclotomicPolynomial n` equals `n.totient / 2`. The `Even` conjunct makes the source's exact half machine-visible under natural-number division.","title":"Real cyclotomic polynomial has degree half the totient"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/lehmer-cyclotomic-identity","kind":"node","lean":"MetaMathlibExt.lehmer_cyclotomic_identity","parent":5366,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html#lehmer-cyclotomic-identity","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"fully_proved","summary":"Let $n > 2$ be a natural number, let $\\psi_n$ be the real cyclotomic polynomial, let $\\Phi_n \\in \\mathbb{Z}[X]$ be the $n$-th cyclotomic polynomial, and let $\\varphi$ be Euler's totient function. Then in the ring $\\mathbb{Z}[x, x^{-1}]$ of Laurent polynomials over $\\mathbb{Z}$, $x^{\\varphi(n)/2} \\, \\psi_n(x + x^{-1}) = \\Phi_n(x)$, that is, $\\psi_n(x + x^{-1}) = x^{-\\varphi(n)/2} \\, \\Phi_n(x)$.","title":"Lehmer's identity for the real cyclotomic polynomial"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/real-cyclotomic-polynomial-eq-minpoly-tr","kind":"node","lean":"MetaMathlibExt.realCyclotomicPolynomial_eq_minpoly_trace","parent":5366,"reading_title":{"format":"inline-parts-v1","parts":[["text","The real cyclotomic polynomial is the minimal polynomial of "],["math","\\(\\zeta + \\zeta^{-1}\\)"]],"source":"The real cyclotomic polynomial is the minimal polynomial of $\\zeta + \\zeta^{-1}$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html#real-cyclotomic-polynomial-eq-minpoly-tr","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero, let $n > 2$ be a natural number, and let $\\zeta \\in K$ be a primitive $n$-th root of unity. Then the real cyclotomic polynomial $\\psi_n$ equals the minimal polynomial of $\\zeta + \\zeta^{-1}$ over $\\mathbb{Z}$.","title":"The real cyclotomic polynomial is the minimal polynomial of $\\zeta + \\zeta^{-1}$"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/finrank-adjoin-two-cos","kind":"node","lean":"MetaMathlibExt.finrank_adjoin_two_cos","parent":5366,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n/index.html#finrank-adjoin-two-cos","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/real-cyclotomic-polynomials-and-2cos-2-k-n","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $n > 2$ and $\\gcd(k, n) = 1$. Then the $\\mathbb{Q}$-subalgebra of $\\mathbb{R}$ generated by the real number $2\\cos(2\\pi k/n)$ has dimension $\\varphi(n)/2$ as a vector space over $\\mathbb{Q}$, where $\\varphi$ is Euler's totient function.","title":"Lehmer's theorem on the degree of 2cos(2\u03c0k/n)"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields.json","id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields","kind":"scope","navigation":"module","parent":5328,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/index.html","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields","status":"fully_proved","title":"Subfields of cyclotomic fields"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/exists-unique-intermediate-field-finrank","kind":"node","lean":"CyclotomicField.existsUnique_intermediateField_finrank","parent":5375,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unique subfield of each degree in the "],["math","\\(p\\)"],["text","-th cyclotomic field"]],"source":"Unique subfield of each degree in the $p$-th cyclotomic field"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/index.html#exists-unique-intermediate-field-finrank","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a prime and let $s$ be a natural number dividing $p - 1$. Then the $p$-th cyclotomic field $\\mathbb{Q}(\\zeta_p)$ has exactly one intermediate field $K$ over $\\mathbb{Q}$ with $[K : \\mathbb{Q}] = s$.","title":"Unique subfield of each degree in the $p$-th cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/intermediate-field-of-finrank","kind":"node","lean":"CyclotomicField.intermediateFieldOfFinrank","parent":5375,"reading_title":{"format":"inline-parts-v1","parts":[["text","The degree-"],["math","\\(s\\)"],["text"," subfield of the "],["math","\\(p\\)"],["text","-th cyclotomic field"]],"source":"The degree-$s$ subfield of the $p$-th cyclotomic field"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/index.html#intermediate-field-of-finrank","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields","status":"fully_proved","summary":"For a prime $p$ and a natural number $s$ dividing $p - 1$, this is the unique intermediate field $K$ of the $p$-th cyclotomic field $\\mathbb{Q}(\\zeta_p)$ over $\\mathbb{Q}$ with $[K : \\mathbb{Q}] = s$.","title":"The degree-$s$ subfield of the $p$-th cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/eq-intermediate-field-of-finrank-iff","kind":"node","lean":"CyclotomicField.eq_intermediateFieldOfFinrank_iff","parent":5375,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of the degree-"],["math","\\(s\\)"],["text"," subfield of the "],["math","\\(p\\)"],["text","-th cyclotomic field"]],"source":"Characterization of the degree-$s$ subfield of the $p$-th cyclotomic field"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/index.html#eq-intermediate-field-of-finrank-iff","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a prime, let $s$ be a natural number dividing $p - 1$, and let $K$ be an intermediate field of the $p$-th cyclotomic field $\\mathbb{Q}(\\zeta_p)$ over $\\mathbb{Q}$. Then $K$ is the degree-$s$ subfield of $\\mathbb{Q}(\\zeta_p)$ if and only if $[K : \\mathbb{Q}] = s$.","title":"Characterization of the degree-$s$ subfield of the $p$-th cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/sup-eq-top-of-lcm-eq","kind":"node","lean":"CyclotomicField.sup_eq_top_of_lcm_eq","parent":5375,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two cyclotomic subfields whose levels have lcm "],["math","\\(m\\)"],["text"," generate the "],["math","\\(m\\)"],["text","-th cyclotomic field"]],"source":"Two cyclotomic subfields whose levels have lcm $m$ generate the $m$-th cyclotomic field"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/index.html#sup-eq-top-of-lcm-eq","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields","status":"fully_proved","summary":"Let $n_0$, $n_1$ and $m$ be natural numbers with $n_0, n_1 \\neq 0$ and $\\operatorname{lcm}(n_0, n_1) = m$. Let $E_0$ and $E_1$ be intermediate fields of the $m$-th cyclotomic field $\\mathbb{Q}(\\zeta_m)$ over $\\mathbb{Q}$ such that $E_0$ is generated over $\\mathbb{Q}$ by a primitive $n_0$-th root of unity and $E_1$ by a primitive $n_1$-th root of unity. Then the compositum of $E_0$ and $E_1$ is all of\u2026","title":"Two cyclotomic subfields whose levels have lcm $m$ generate the $m$-th cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/inf-ne-bot-of-lt-of-lcm-eq","kind":"node","lean":"CyclotomicField.inf_ne_bot_of_lt_of_lcm_eq","parent":5375,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields/index.html#inf-ne-bot-of-lt-of-lcm-eq","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/subfields-of-cyclotomic-fields","status":"fully_proved","summary":"Let $n_0$, $n_1$ and $m$ be natural numbers with $n_0, n_1 \\neq 0$ and $\\operatorname{lcm}(n_0, n_1) = m$. Let $E_0$ and $E_1$ be intermediate fields of the $m$-th cyclotomic field $\\mathbb{Q}(\\zeta_m)$ over $\\mathbb{Q}$ such that $E_0$ is generated over $\\mathbb{Q}$ by a primitive $n_0$-th root of unity and $E_1$ by a primitive $n_1$-th root of unity. If $F$ is an intermediate field of $\\mathbb{Q}(\\zeta_m)$ with\u2026","title":"A field strictly above one cyclotomic subfield meets the other"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields.json","id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","kind":"scope","navigation":"module","parent":5328,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","title":"Trace metric on cyclotomic fields"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-norm-sq","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceNormSq","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-norm-sq","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a positive integer, let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$, and let $\\omega$ be the chosen primitive $p$-th root of unity in $K$. The squared trace norm of $\\alpha \\in K$ is the rational number","title":"Squared trace norm on a cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-norm","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceNorm","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-norm","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a positive integer, let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$, and let $\\omega$ be the chosen primitive $p$-th root of unity in $K$. The trace norm of $\\alpha \\in K$ is the real number","title":"Trace norm on a cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-dist","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceDist","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-dist","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a positive integer and let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$. The trace distance between $\\alpha, \\beta \\in K$ is the real number $d(\\alpha, \\beta) = \\lVert \\alpha - \\beta \\rVert$, where $\\lVert \\cdot \\rVert$ is the trace norm on $K$.","title":"Trace metric on a cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-lattice-point","kind":"node","lean":"MetaMathlibExt.cyclotomicLatticePoint","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-lattice-point","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a positive integer, let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$, and let $\\omega$ be the chosen primitive $p$-th root of unity in $K$. For integers $a_1, \\dots, a_{p-1}$, the lattice point with coefficients $a_1, \\dots, a_{p-1}$ is the element $a_1 \\omega + a_2 \\omega^2 + \\cdots + a_{p-1} \\omega^{p-1}$ of $K$.","title":"Lattice points of a cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-box","kind":"node","lean":"MetaMathlibExt.cyclotomicBox","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-box","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"For natural numbers $p$ and $N$, the box $B(p, N)$ is the finite set of integer vectors $a = (a_1, \\dots, a_{p-1}) \\in \\mathbb{Z}^{p-1}$ with $-N \\le a_i \\le N$ for every $i$.","title":"The cyclotomic box"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-dist-moment","kind":"node","lean":"MetaMathlibExt.cyclotomicDistMoment","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-dist-moment","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p \\ge 1$, $N$ and $k$ be natural numbers, and let $K$ be a field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity, with fixed primitive $p$-th root of unity $\\omega \\in K$. Let $d$ be the trace metric on $K$, given by $d(\\alpha, \\beta) \\ge 0$ and $d(\\alpha, \\beta)^2 = \\sum_{j=1}^{p-1} \\mathrm{Tr}_{K/\\mathbb{Q}}((\\alpha - \\beta) \\omega^j)^2$. Let $B(p, N)$ be the set of integer vectors\u2026","title":"Moments of the trace distance on the cyclotomic box"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-norm-sq-zero","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceNormSq_zero","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-norm-sq-zero","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a positive integer and let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$. The squared trace norm of $0 \\in K$ is $0$, that is, $\\lVert 0 \\rVert^2 = 0$.","title":"Squared trace norm of zero"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-norm-sq-neg","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceNormSq_neg","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-norm-sq-neg","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a positive integer and let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$. For every $\\alpha \\in K$, the squared trace norm satisfies $\\lVert -\\alpha \\rVert^2 = \\lVert \\alpha \\rVert^2$.","title":"The squared trace norm is invariant under negation"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-dist-self","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceDist_self","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-dist-self","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a positive integer and let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$. For every $\\alpha \\in K$, the trace distance satisfies $d(\\alpha, \\alpha) = 0$.","title":"Trace distance from a point to itself"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-dist-comm","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceDist_comm","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-dist-comm","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a positive integer and let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$. For all $\\alpha, \\beta \\in K$, the trace distance satisfies $d(\\alpha, \\beta) = d(\\beta, \\alpha)$.","title":"Symmetry of the trace metric"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-lattice-point-zero","kind":"node","lean":"MetaMathlibExt.cyclotomicLatticePoint_zero","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-lattice-point-zero","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p \\ge 1$ be a natural number and let $K$ be a field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity, with fixed primitive $p$-th root of unity $\\omega \\in K$. For an integer vector $a = (a_1, \\dots, a_{p-1})$ write $\\alpha_a = a_1 \\omega + \\dots + a_{p-1} \\omega^{p-1}$. Then $\\alpha_0 = 0$.","title":"The zero coefficient vector gives the zero lattice point"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-box-zero","kind":"node","lean":"MetaMathlibExt.cyclotomicBox_zero","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-box-zero","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p$ be a natural number and, for a natural number $N$, let $B(p, N)$ be the set of integer vectors $a = (a_1, \\dots, a_{p-1})$ with $-N \\le a_i \\le N$ for all $i$. Then $B(p, 0) = \\{0\\}$: the zero vector is the only element of the box of radius $0$.","title":"The cyclotomic box of radius zero"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/zero-mem-cyclotomic-box","kind":"node","lean":"MetaMathlibExt.zero_mem_cyclotomicBox","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#zero-mem-cyclotomic-box","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"For all natural numbers $p$ and $N$, the zero vector lies in the box $B(p, N)$, the set of integer vectors $(a_1, \\dots, a_{p-1})$ with $|a_i| \\le N$ for all $i$.","title":"The zero vector lies in every box"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-dist-moment-zero","kind":"node","lean":"MetaMathlibExt.cyclotomicDistMoment_zero","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-dist-moment-zero","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"fully_proved","summary":"Let $p \\ge 1$ and $N$ be natural numbers, and let $K$ be a field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity, with fixed primitive $p$-th root of unity $\\omega \\in K$. Let $d$ be the trace metric on $K$, given by $d(\\alpha, \\beta) \\ge 0$ and $d(\\alpha, \\beta)^2 = \\sum_{j=1}^{p-1} \\mathrm{Tr}_{K/\\mathbb{Q}}((\\alpha - \\beta) \\omega^j)^2$. Let $B(p, N)$ be the set of integer vectors\u2026","title":"The zeroth moment of the trace distance is one"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-dist-alg-equiv","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceDist_algEquiv","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-dist-alg-equiv","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be a prime and let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$. For every $\\mathbb{Q}$-algebra automorphism $\\sigma$ of $K$ and all $\\alpha, \\beta \\in K$, the trace distance satisfies $d(\\sigma \\alpha, \\sigma \\beta) = d(\\alpha, \\beta)$.","title":"Galois invariance of the trace metric"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-norm-is-norm","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceNorm_isNorm","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-norm-is-norm","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be a prime and let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$, viewed as a $\\mathbb{Q}$-vector space. The trace norm $\\lVert \\cdot \\rVert$ on $K$ is a norm: for all $\\alpha, \\beta \\in K$ and $q \\in \\mathbb{Q}$, we have $\\lVert \\alpha \\rVert = 0$ if and only if $\\alpha = 0$, $\\lVert \\alpha + \\beta \\rVert \\le \\lVert \\alpha \\rVert + \\lVert \\beta \\rVert$, and\u2026","title":"The trace norm is a norm"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/adjoin-le-adjoin-of-cyclotomic-trace-dis","kind":"node","lean":"MetaMathlibExt.adjoin_le_adjoin_of_cyclotomicTraceDist_lt","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#adjoin-le-adjoin-of-cyclotomic-trace-dis","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be a prime and let $K$ be a field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity, with fixed primitive $p$-th root of unity $\\omega \\in K$. Write $d(\\alpha, \\beta) = \\lVert \\alpha - \\beta \\rVert$ for the trace metric on $K$, where $\\lVert \\gamma \\rVert \\ge 0$ and $\\lVert \\gamma \\rVert^2 = \\sum_{j=1}^{p-1} \\operatorname{Tr}_{K/\\mathbb{Q}}(\\gamma \\omega^j)^2$. Let $\\alpha, \\beta \\in K$ and\u2026","title":"Krasner-type lemma for the cyclotomic trace metric"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/eventually-adjoin-add-nsmul-eq","kind":"node","lean":"MetaMathlibExt.eventually_adjoin_add_nsmul_eq","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#eventually-adjoin-add-nsmul-eq","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be a prime and let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$. For all $\\alpha, \\beta \\in K$ there is a natural number $n_0$ such that, for every natural number $n \\ge n_0$, the subfields $\\mathbb{Q}(\\alpha + n\\beta)$ and $\\mathbb{Q}(\\alpha, \\beta)$ of $K$ are equal.","title":"Primitive elements \u03b1 + n\u03b2 in a prime cyclotomic field"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-trace-norm-sq-eq-and-ge","kind":"node","lean":"MetaMathlibExt.cyclotomicTraceNormSq_eq_and_ge","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-trace-norm-sq-eq-and-ge","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be a prime, let $K$ be a $p$-th cyclotomic extension of $\\mathbb{Q}$, let $\\omega$ be the chosen primitive $p$-th root of unity in $K$, and let $\\mathrm{Tr}$ be the trace from $K$ to $\\mathbb{Q}$. For rational numbers $a_1, \\dots, a_{p-1}$, put $\\alpha = a_1 \\omega + a_2 \\omega^2 + \\cdots + a_{p-1} \\omega^{p-1}$ and $\\lVert \\alpha \\rVert_E^2 = a_1^2 + a_2^2 + \\cdots + a_{p-1}^2$. Then the squared trace norm\u2026","title":"Squared trace norm in coordinates"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-box-diam","kind":"node","lean":"MetaMathlibExt.cyclotomicBox_diam","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-box-diam","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be an odd prime, let $N$ be a natural number, and let $K$ be a field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity, with fixed primitive $p$-th root of unity $\\omega \\in K$. Let $d(\\alpha, \\beta) = \\lVert \\alpha - \\beta \\rVert$ be the trace metric on $K$, where $\\lVert \\gamma \\rVert \\ge 0$ and $\\lVert \\gamma \\rVert^2 = \\sum_{j=1}^{p-1} \\operatorname{Tr}_{K/\\mathbb{Q}}(\\gamma \\omega^j)^2$.\u2026","title":"Diameter of the cyclotomic box"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/symmetric-power-sums","kind":"node","lean":"MetaMathlibExt.symmetric_power_sums","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#symmetric-power-sums","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"For natural numbers $N$ and $r$, let $S_r(N) = \\sum_{a=-N}^{N} a^r$, the sum over the integers $a$ with $-N \\le a \\le N$. Then $S_r(N) = 0$ for every odd $r$, and","title":"Power sums over a symmetric interval"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-dist-moment-two","kind":"node","lean":"MetaMathlibExt.cyclotomicDistMoment_two","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-dist-moment-two","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be a prime, let $N$ be a natural number, and let $K$ be a field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity, with fixed primitive $p$-th root of unity $\\omega \\in K$. Let $M_2(p, N)$ be the second moment of the trace metric $d$ of $K$ over the box $B(p, N)$: the average of $d(\\alpha_a, \\alpha_b)^2$ over all pairs $(a, b)$ of integer vectors in $\\mathbb{Z}^{p-1}$ whose coordinates lie in\u2026","title":"Second moment of the trace distance on the cyclotomic box"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/sum-sq-sub-mul-sq-sub-box","kind":"node","lean":"MetaMathlibExt.sum_sq_sub_mul_sq_sub_box","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#sum-sq-sub-mul-sq-sub-box","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be a prime and $N$ a natural number, and let $B(p, N)$ be the set of integer vectors $a = (a_1, \\dots, a_{p-1})$ with $|a_i| \\le N$ for all $i$. Then","title":"Fourth-power sum of coordinate differences over a box"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-dist-moment-four","kind":"node","lean":"MetaMathlibExt.cyclotomicDistMoment_four","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-dist-moment-four","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"Let $p$ be a prime, let $N$ be a natural number, and let $K$ be a field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity, with fixed primitive $p$-th root of unity $\\omega \\in K$. Let $M_4(p, N)$ be the fourth moment of the trace metric $d$ of $K$ over the box $B(p, N)$: the average of $d(\\alpha_a, \\alpha_b)^4$ over all pairs $(a, b)$ of integer vectors in $\\mathbb{Z}^{p-1}$ whose coordinates lie in\u2026","title":"Fourth moment of the trace distance on the cyclotomic box"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/exists-cyclotomic-dist-variance-le","kind":"node","lean":"MetaMathlibExt.exists_cyclotomicDist_variance_le","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#exists-cyclotomic-dist-variance-le","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"There is a real number $C$ such that the following holds for every prime $p$, every integer $N \\ge 1$ and every $p$-th cyclotomic extension $K$ of $\\mathbb{Q}$. Let $\\omega$ be the chosen primitive $p$-th root of unity in $K$, let $d$ be the trace distance on $K$, and let $B(p, N)$ be the set of integer vectors $a = (a_1, \\dots, a_{p-1})$ with $|a_i| \\le N$ for all $i$, where $a$ stands for the point\u2026","title":"Variance bound for squared trace distances in the box"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/cyclotomic-box-almost-equidistant","kind":"node","lean":"MetaMathlibExt.cyclotomicBox_almost_equidistant","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#cyclotomic-box-almost-equidistant","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"For every real $\\varepsilon > 0$ there is a natural number $A$ with the following property. Let $p$ be a prime and $N$ a natural number with $p > A$ and $N > A$, and let $K$ be a field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity, with fixed primitive $p$-th root of unity $\\omega \\in K$. Let $d(\\alpha, \\beta) = \\lVert \\alpha - \\beta \\rVert$ be the trace metric on $K$, where\u2026","title":"Almost all pairs in the cyclotomic box are nearly equidistant"},{"id":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/exists-cyclotomic-box-deviation-le","kind":"node","lean":"MetaMathlibExt.exists_cyclotomicBox_deviation_le","parent":5381,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields/index.html#exists-cyclotomic-box-deviation-le","scope":"NumberTheory/algebraic-number-theory/cyclotomic-fields/trace-metric-on-cyclotomic-fields","status":"can_prove","summary":"There is a real number $C$ such that the following holds for every real $\\varepsilon > 0$, every prime $p$, every integer $N \\ge 1$ and every $p$-th cyclotomic extension $K$ of $\\mathbb{Q}$. Let $\\omega$ be the chosen primitive $p$-th root of unity in $K$, let $d$ be the trace distance on $K$, and let $B(p, N)$ be the set of integer vectors $a = (a_1, \\dots, a_{p-1})$ with $|a_i| \\le N$ for all $i$, where $a$\u2026","title":"Concentration of trace distances in the box, quantitative form"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification","kind":"scope","parent":5201,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification","status":"can_prove","summary":"Discriminants and their bounds, ramification and inertia, orders, units, ideal counting, local degrees under completion, field enumeration and towers.","title":"Discriminants, ramification and ideals"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/exists-totally-real-discr-le","kind":"node","lean":"MathlibExt.NumberTheory.MartinetTotallyRealTowersWanted.NumberTheory.exists_totallyReal_discr_le","parent":5409,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/index.html#exists-totally-real-discr-le","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification","status":"can_prove","summary":"There is a real constant $C > 0$ such that for every $N \\in \\mathbb{N}$ there exist a natural number $d \\ge N$ and a totally real number field $K$ of degree $[K : \\mathbb{Q}] = d$ whose discriminant $\\Delta_K$ satisfies $|\\Delta_K| \\le C^d$.","title":"Martinet's totally real towers"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/odlyzko-root-discr-ge","kind":"node","lean":"MetaMathlibExt.odlyzko_root_discr_ge","parent":5409,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/index.html#odlyzko-root-discr-ge","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification","status":"can_prove","summary":"Let $K$ be a totally complex number field of degree $n = [K : \\mathbb{Q}] \\ge 18$ and discriminant $d_K$. Then $|d_K| \\ge 8.25^n$; equivalently, the root discriminant $|d_K|^{1/n}$ of $K$ is at least $8.25$.","title":"Odlyzko's root discriminant bound in degree at least 18"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/card-inertia-eq-ramification-idx-in-mul","kind":"node","lean":"Ideal.card_inertia_eq_ramificationIdxIn_mul_finInsepDegree","parent":5409,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/index.html#card-inertia-eq-ramification-idx-in-mul","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification","status":"fully_proved","summary":"Let $A$ and $B$ be integral domains, with $B$ an $A$-algebra that is finite and flat as an $A$-module, and let $G$ be a finite group acting faithfully on $B$ by $A$-algebra automorphisms such that every $G$-invariant element of $B$ comes from $A$. Let $\\mathfrak p$ be a maximal ideal of $A$ and $\\mathfrak P$ a maximal ideal of $B$ lying over $\\mathfrak p$. Then the inertia group of $\\mathfrak P$, consisting of the\u2026","title":"Order of the inertia group"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/finite-index-closure-range-of-lt-one-off","kind":"node","lean":"NumberField.Units.finiteIndex_closure_range_of_lt_one_off_diagonal","parent":5409,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/index.html#finite-index-closure-range-of-lt-one-off","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification","status":"fully_proved","summary":"Let $F$ be a number field and let $(u_v)$ be a family of units of the ring of integers of $F$, indexed by the infinite places $v$ of $F$. Write $w(x)$ for the absolute value of $x$ at an infinite place $w$. If $w(u_v) < 1$ for all infinite places $v$ and $w$ with $w \\ne v$, then the subgroup of the unit group of the ring of integers generated by the $u_v$ has finite index.","title":"Units small away from the diagonal generate a subgroup of finite index"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"can_prove","title":"Class field towers"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/is-everywhere-unramified","kind":"node","lean":"MathlibExt.NumberTheory.ShafarevichRelationRankWanted.NumberTheory.IsEverywhereUnramified","parent":5414,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html#is-everywhere-unramified","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"fully_proved","summary":"Let $M/F$ be an extension of number fields, with rings of integers $\\mathcal{O}_F$ and $\\mathcal{O}_M$. Then $M/F$ is everywhere unramified when every prime ideal $\\mathfrak{P}$ of $\\mathcal{O}_M$ lying over a nonzero prime ideal $\\mathfrak{p}$ of $\\mathcal{O}_F$ has ramification index $1$ over $\\mathcal{O}_F$, and every infinite place of $M$ lying over a real place of $F$ is real.","title":"Everywhere unramified extension"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/maximal-unramified-pro-pf","kind":"node","lean":"MathlibExt.NumberTheory.ShafarevichRelationRankWanted.NumberTheory.maximalUnramifiedProPF","parent":5414,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal unramified pro-"],["math","\\(p\\)"],["text"," extension"]],"source":"Maximal unramified pro-$p$ extension"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html#maximal-unramified-pro-pf","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"fully_proved","summary":"Let $F$ be a number field, $\\overline{F}$ an algebraic closure of $F$, and $p$ a prime. The maximal unramified pro-$p$ extension $F^{\\mathrm{un},p}$ of $F$ is the compositum in $\\overline{F}$ of all intermediate fields $F \\subseteq M \\subseteq \\overline{F}$ such that $M/F$ is finite, Galois and everywhere unramified, and its Galois group $\\mathrm{Gal}(M/F)$ is a $p$-group (each such $M$ is a number field).","title":"Maximal unramified pro-$p$ extension"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/max-unramified-pro-pgalois-group","kind":"node","lean":"MathlibExt.NumberTheory.ShafarevichRelationRankWanted.NumberTheory.MaxUnramifiedProPGaloisGroup","parent":5414,"reading_title":{"format":"inline-parts-v1","parts":[["text","Galois group of the maximal unramified pro-"],["math","\\(p\\)"],["text"," extension"]],"source":"Galois group of the maximal unramified pro-$p$ extension"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html#max-unramified-pro-pgalois-group","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"fully_proved","summary":"For a number field $F$ and a prime $p$, this is the group $\\mathrm{Gal}(F^{\\mathrm{un},p}/F)$ of field automorphisms of the maximal unramified pro-$p$ extension $F^{\\mathrm{un},p}$ of $F$ that fix $F$, under composition and with the Krull topology.","title":"Galois group of the maximal unramified pro-$p$ extension"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/generator-rank","kind":"node","lean":"MathlibExt.NumberTheory.ShafarevichRelationRankWanted.NumberTheory.generatorRank","parent":5414,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html#generator-rank","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"fully_proved","summary":"For a topological group $G$, the generator rank $d(G)$ is the least natural number $k$ such that some finite subset $S \\subseteq G$ with $k$ elements generates a dense subgroup of $G$. If no finite subset of $G$ generates a dense subgroup, then $d(G) = 0$ by convention.","title":"Generator rank of a topological group"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/trivial-zmodp-rep","kind":"node","lean":"MathlibExt.NumberTheory.ShafarevichRelationRankWanted.NumberTheory.trivialZModpRep","parent":5414,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trivial representation on "],["math","\\(\\mathbb{Z}/p\\mathbb{Z}\\)"]],"source":"Trivial representation on $\\mathbb{Z}/p\\mathbb{Z}$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html#trivial-zmodp-rep","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"fully_proved","summary":"For a natural number $p$ and a topological group $G$, this is $\\mathbb{Z}/p\\mathbb{Z}$ with the discrete topology, regarded as a continuous representation of $G$ over $\\mathbb{Z}/p\\mathbb{Z}$ on which every element of $G$ acts as the identity.","title":"Trivial representation on $\\mathbb{Z}/p\\mathbb{Z}$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/relation-rank","kind":"node","lean":"MathlibExt.NumberTheory.ShafarevichRelationRankWanted.NumberTheory.relationRank","parent":5414,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html#relation-rank","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"fully_proved","summary":"For a prime $p$ and a topological group $G$, the relation rank $r(G)$ is the dimension over $\\mathbb{F}_p$ of the continuous cohomology group $H^2(G, \\mathbb{F}_p)$ of degree $2$, with coefficients in the trivial representation of $G$ on $\\mathbb{F}_p$. If this space is infinite-dimensional, then $r(G) = 0$ by convention.","title":"Relation rank of a topological group"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/h2-finite","kind":"node","lean":"MathlibExt.NumberTheory.ShafarevichRelationRankWanted.NumberTheory.H2Finite","parent":5414,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite-dimensional second cohomology mod "],["math","\\(p\\)"]],"source":"Finite-dimensional second cohomology mod $p$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html#h2-finite","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"fully_proved","summary":"For a prime $p$ and a topological group $G$, this is the statement that the continuous cohomology group $H^2(G, \\mathbb{F}_p)$ of degree $2$, with coefficients in the trivial representation of $G$ on $\\mathbb{F}_p$, is finite-dimensional over $\\mathbb{F}_p$.","title":"Finite-dimensional second cohomology mod $p$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/shafarevich-relation-rank-bound","kind":"node","lean":"MathlibExt.NumberTheory.ShafarevichRelationRankWanted.NumberTheory.shafarevich_relation_rank_bound","parent":5414,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers/index.html#shafarevich-relation-rank-bound","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/class-field-towers","status":"can_prove","summary":"Let $F$ be a number field, let $p$ be an odd prime, and let $G = \\mathrm{Gal}(F^{\\mathrm{un},p}/F)$ be the Galois group of the maximal unramified pro-$p$ extension of $F$. Write $d(G)$ for the generator rank and $r(G)$ for the relation rank of $G$, $r_1$ and $r_2$ for the numbers of real and complex places of $F$, and $\\delta$ for $1$ if $F$ contains a primitive $p$-th root of unity and $0$ otherwise. Then the\u2026","title":"Shafarevich's relation-rank bound"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places","kind":"scope","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places","status":"fully_proved","title":"Completions of number fields at finite places"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finrank-adic-completion-le","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.finrank_adicCompletion_le","parent":5423,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/index.html#finrank-adic-completion-le","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ (a finite place of $K$) and let $w$ be a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Regard the $w$-adic completion $L_w$ as an algebra over the $v$-adic completion $K_v$ via the canonical map $K_v \\to L_w$. Then the dimension of $L_w$ over $K_v$ is at most $[L : K]$.","title":"Local degree is at most the global degree"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/comap-maximal-ideal-adic-completion-inte","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.comap_maximalIdeal_adicCompletionIntegersMap","parent":5423,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/index.html#comap-maximal-ideal-adic-completion-inte","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Let $\\mathcal{O}_{K_v}$ and $\\mathcal{O}_{L_w}$ be the rings of integers of the completions $K_v$ and $L_w$. Then the preimage of the maximal ideal of $\\mathcal{O}_{L_w}$ under the canonical map $\\mathcal{O}_{K_v} \\to \\mathcal{O}_{L_w}$ is the maximal ideal\u2026","title":"Maximal ideals of completed integer rings lie over each other"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/inertia-deg-adic-completion-integers-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.inertiaDeg_adicCompletionIntegersMap","parent":5423,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/index.html#inertia-deg-adic-completion-integers-map","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Let $\\mathcal{O}_{K_v}$ and $\\mathcal{O}_{L_w}$ be the rings of integers of the completions $K_v$ and $L_w$, with $\\mathcal{O}_{L_w}$ an algebra over $\\mathcal{O}_{K_v}$ via the canonical map. Then the inertia degree of the maximal ideal of\u2026","title":"Completion preserves the inertia degree"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finrank-adic-completion-eq-ramification","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.finrank_adicCompletion_eq_ramificationIdx_mul_inertiaDeg","parent":5423,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/index.html#finrank-adic-completion-eq-ramification","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ (a finite place of $K$) and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Regard the completion $L_w$ as an algebra over $K_v$ via the canonical map $K_v \\to L_w$. Then the dimension of $L_w$ over $K_v$ equals $e(w \\mid v) \\, f(w \\mid v)$, the ramification index of $w$ over $v$ times the inertia degree of\u2026","title":"The local degree is the product of ramification index and inertia degree"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/adic-completion-integers-map-algebra-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegersMap_algebraMap","parent":5423,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/index.html#adic-completion-integers-map-algebra-map","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Let $\\mathcal{O}_{K_v}$ and $\\mathcal{O}_{L_w}$ be the rings of integers of the completions $K_v$ and $L_w$. For every $a \\in \\mathcal{O}_K$, the canonical map $\\mathcal{O}_{K_v} \\to \\mathcal{O}_{L_w}$ sends the image of $a$ in $\\mathcal{O}_{K_v}$ to the\u2026","title":"The completed integer map is compatible with the rings of integers"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place","kind":"scope","navigation":"module","parent":5423,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place","status":"fully_proved","title":"Valuation formula for maps between finite-place completions"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place/valuation-adic-completion-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.valuation_adicCompletionMap","parent":5429,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place/index.html#valuation-adic-completion-map","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$, let $w$ be a finite place of $L$ lying over $v$, and let $e$ be the ramification index of $w$ over $v$. Let $\\iota : K_v \\to L_w$ be the continuous ring homomorphism between the completions that extends $K \\to L$. Then for every $x \\in K_v$, the valuation of $\\iota(x)$\u2026","title":"Valuation of the image under the completion map"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place/adic-completion-map-mem-integers","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletionMap_mem_integers","parent":5429,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place/index.html#adic-completion-map-mem-integers","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$, and let $w$ be a finite place of $L$ lying over $v$. The continuous ring homomorphism $K_v \\to L_w$ between the completions that extends $K \\to L$ maps every element of the ring of integers of $K_v$ into the ring of integers of $L_w$.","title":"The completion map preserves integrality"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place/adic-completion-integers-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegersMap","parent":5429,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place/index.html#adic-completion-integers-map","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/valuation-formula-for-maps-between-finite-place","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$, and let $w$ be a finite place of $L$ lying over $v$. Let $K_v$ and $L_w$ be the completions, with rings of integers $\\mathcal{O}_v$ and $\\mathcal{O}_w$. The continuous ring homomorphism $K_v \\to L_w$ extending $K \\to L$ restricts to a ring homomorphism\u2026","title":"Restriction of the completion map to the rings of integers"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion","kind":"scope","navigation":"module","parent":5423,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion","status":"fully_proved","title":"Ramification index under finite-place completion"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion/adic-completion-integers-map-injective","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegersMap_injective","parent":5433,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion/index.html#adic-completion-integers-map-injective","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Then the canonical map $\\mathcal{O}_{K_v} \\to \\mathcal{O}_{L_w}$ from the ring of integers of the completion $K_v$ to the ring of integers of the completion $L_w$ is injective.","title":"The map of completed integer rings is injective"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion/ramification-idx-adic-completion-integer","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.ramificationIdx_adicCompletionIntegersMap","parent":5433,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion/index.html#ramification-idx-adic-completion-integer","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/ramification-index-under-finite-place-completion","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Let $\\mathcal{O}_{K_v}$ and $\\mathcal{O}_{L_w}$ be the rings of integers of the completions $K_v$ and $L_w$, with $\\mathcal{O}_{L_w}$ an algebra over $\\mathcal{O}_{K_v}$ via the canonical map. Then the ramification index of the maximal ideal of\u2026","title":"Completion preserves the ramification index"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square","kind":"scope","navigation":"module","parent":5423,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square","status":"fully_proved","title":"Explicit residue-field naturality square"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square/ring-of-integers-residue-quotient-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.ringOfIntegersResidueQuotientMap","parent":5436,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square/index.html#ring-of-integers-residue-quotient-map","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. This is the ring homomorphism $\\mathcal{O}_K/v \\to \\mathcal{O}_L/w$ induced by the canonical map $\\mathcal{O}_K \\to \\mathcal{O}_L$.","title":"Residue map of rings of integers"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square/adic-completion-residue-quotient-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.adicCompletionResidueQuotientMap","parent":5436,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square/index.html#adic-completion-residue-quotient-map","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Let $\\mathcal{O}_{K_v}$ and $\\mathcal{O}_{L_w}$ be the rings of integers of the completions $K_v$ and $L_w$, with maximal ideals $\\mathfrak{m}_v$ and $\\mathfrak{m}_w$. This is the ring homomorphism\u2026","title":"Residue map of completed integer rings"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square/completion-residue-field-equiv-naturalit","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionResidueFieldEquiv_naturality","parent":5436,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square/index.html#completion-residue-field-equiv-naturalit","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/explicit-residue-field-naturality-square","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a nonzero prime ideal of $\\mathcal{O}_K$ and $w$ a nonzero prime ideal of $\\mathcal{O}_L$ lying over $v$. Let $\\mathcal{O}_{K_v}$ and $\\mathcal{O}_{L_w}$ be the rings of integers of the completions $K_v$ and $L_w$, with maximal ideals $\\mathfrak{m}_v$ and $\\mathfrak{m}_w$. Then the two maps $\\mathcal{O}_{K_v}/\\mathfrak{m}_v \\to \\mathcal{O}_L/w$ agree: the\u2026","title":"Naturality of the residue field identifications"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a","kind":"scope","navigation":"module","parent":5423,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a","status":"fully_proved","title":"Places of a number field extension lying over a fixed finite place"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a/places-over","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.placesOver","parent":5440,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a/index.html#places-over","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields and let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$. The finite places of $L$ lying over $v$ form the type of height-one primes $w$ of $\\mathcal{O}_L$ that lie over $v$, that is, whose preimage in $\\mathcal{O}_K$ is $v$.","title":"Finite places lying over a finite place"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a/places-over-equiv-primes-over","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.placesOverEquivPrimesOver","parent":5440,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a/index.html#places-over-equiv-primes-over","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/places-of-a-number-field-extension-lying-over-a","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields and let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$. Sending a finite place $w$ of $L$ lying over $v$ to its prime ideal of $\\mathcal{O}_L$ is a bijection from the finite places of $L$ lying over $v$ onto the set of prime ideals of $\\mathcal{O}_L$ lying over $v$.","title":"Places over a finite place correspond to primes over it"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps","kind":"scope","navigation":"module","parent":5423,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps","status":"fully_proved","title":"Finite-place completion tensor product maps"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps/completion-tensor-product-component","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionTensorProductComponent","parent":5443,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps/index.html#completion-tensor-product-component","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$, let $K_v$ be the completion of $K$ at $v$, and let $w$ be a finite place of $L$ lying over $v$, with completion $L_w$. Let $\\iota_w : K_v \\to L_w$ be the continuous ring homomorphism extending $K \\to L$. This is the $K_v$-algebra homomorphism $K_v \\otimes_K L \\to L_w$\u2026","title":"Component at a place of the completion tensor product map"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps/completion-tensor-product-map","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionTensorProductMap","parent":5443,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps/index.html#completion-tensor-product-map","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/finite-place-completion-tensor-product-maps","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$, and let $K_v$ be the completion of $K$ at $v$. For each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$ and $\\iota_w : K_v \\to L_w$ the continuous ring homomorphism extending $K \\to L$. This is the $K_v$-algebra homomorphism\u2026","title":"The completion tensor product map"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","kind":"scope","navigation":"module","parent":5423,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","title":"The finite-place completion tensor product"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/sum-finrank-adic-completion-places-over","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.sum_finrank_adicCompletion_placesOver","parent":5446,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html#sum-finrank-adic-completion-places-over","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, $v$ a finite place of $K$ and $K_v$ the completion of $K$ at $v$; for each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$, a $K_v$-algebra via the canonical map $K_v \\to L_w$. Then $\\sum_{w \\mid v} [L_w : K_v] = [L : K]$.","title":"Local degrees sum to the global degree"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/finrank-completion-tensor-product-eq-pi","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.finrank_completionTensorProduct_eq_pi","parent":5446,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html#finrank-completion-tensor-product-eq-pi","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, $v$ a finite place of $K$ and $K_v$ the completion of $K$ at $v$; for each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$, a $K_v$-algebra via the canonical map $K_v \\to L_w$. Then $K_v \\otimes_K L$ and $\\prod_{w \\mid v} L_w$ have the same dimension as $K_v$-vector spaces.","title":"Dimension count for the base change to a finite completion"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/completion-tensor-product-map-injective","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionTensorProductMap_injective","parent":5446,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html#completion-tensor-product-map-injective","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, $v$ a finite place of $K$ and $K_v$ the completion of $K$ at $v$; for each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$, a $K_v$-algebra via the canonical map $K_v \\to L_w$. The $K_v$-algebra map $K_v \\otimes_K L \\to \\prod_{w \\mid v} L_w$ sending $x \\otimes y$ to $(x y)_w$, where $x \\in K_v$ and $y \\in L$ are mapped into each $L_w$, is\u2026","title":"The canonical map to the product of completions is injective"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/completion-tensor-product-map-bijective","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionTensorProductMap_bijective","parent":5446,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html#completion-tensor-product-map-bijective","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, $v$ a finite place of $K$ and $K_v$ the completion of $K$ at $v$; for each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$, a $K_v$-algebra via the canonical map $K_v \\to L_w$. The $K_v$-algebra map $K_v \\otimes_K L \\to \\prod_{w \\mid v} L_w$ sending $x \\otimes y$ to $(x y)_w$, where $x \\in K_v$ and $y \\in L$ are mapped into each $L_w$, is\u2026","title":"The canonical map to the product of completions is bijective"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/completion-tensor-product-alg-equiv","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionTensorProductAlgEquiv","parent":5446,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html#completion-tensor-product-alg-equiv","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, $v$ a finite place of $K$ and $K_v$ the completion of $K$ at $v$; for each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$, a $K_v$-algebra via the canonical map $K_v \\to L_w$. This is the isomorphism of $K_v$-algebras $K_v \\otimes_K L \\cong \\prod_{w \\mid v} L_w$ sending $x \\otimes y$ to $(x y)_w$, where $x \\in K_v$ and $y \\in L$ are\u2026","title":"Base change to a finite completion as a product of completions"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/completion-tensor-product-right-alg-equi","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionTensorProductRightAlgEquiv","parent":5446,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html#completion-tensor-product-right-alg-equi","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, $v$ a finite place of $K$ and $K_v$ the completion of $K$ at $v$; for each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$, a $K_v$-algebra via the canonical map $K_v \\to L_w$, and make $L \\otimes_K K_v$ a $K_v$-algebra through its second factor. This is the isomorphism of $K_v$-algebras $L \\otimes_K K_v \\cong \\prod_{w \\mid v} L_w$\u2026","title":"Tensor product decomposition at a finite place"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/completion-tensor-product-right-alg-equi-2","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionTensorProductRightAlgEquiv_tmul_one","parent":5446,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html#completion-tensor-product-right-alg-equi-2","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, $v$ a finite place of $K$ and $K_v$ the completion of $K$ at $v$; for each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$, a $K_v$-algebra via the canonical map $K_v \\to L_w$, and make $L \\otimes_K K_v$ a $K_v$-algebra through its second factor. Consider the isomorphism of $K_v$-algebras $L \\otimes_K K_v \\cong \\prod_{w \\mid v} L_w$\u2026","title":"Coordinate law for the tensor product decomposition"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/completion-tensor-product-right-alg-equi-3","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.completionTensorProductRightAlgEquiv_K","parent":5446,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product/index.html#completion-tensor-product-right-alg-equi-3","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/the-finite-place-completion-tensor-product","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, $v$ a finite place of $K$ and $K_v$ the completion of $K$ at $v$; for each finite place $w$ of $L$ lying over $v$, let $L_w$ be the completion of $L$ at $w$. There exists an isomorphism of $K$-algebras $L \\otimes_K K_v \\cong \\prod_{w \\mid v} L_w$.","title":"Tensor product decomposition at a finite place over the base field"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over","kind":"scope","navigation":"module","parent":5423,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over","status":"fully_proved","title":"Global norm and trace as products and sums over finite-place completions"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over/algebra-map-norm-eq-prod-adic-completion","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.algebraMap_norm_eq_prod_adicCompletion","parent":5455,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over/index.html#algebra-map-norm-eq-prod-adic-completion","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$, and let $K_v$ be the completion of $K$ at $v$. For each of the finitely many finite places $w$ of $L$ lying over $v$, regard the completion $L_w$ as a $K_v$-algebra through the continuous ring homomorphism $K_v \\to L_w$ extending $K \\to L$. Then for every $\\alpha \\in L$,\u2026","title":"Global norm as a product of local norms"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over/algebra-map-trace-eq-sum-adic-completion","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.algebraMap_trace_eq_sum_adicCompletion","parent":5455,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over/index.html#algebra-map-trace-eq-sum-adic-completion","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/completions-of-number-fields-at-finite-places/global-norm-and-trace-as-products-and-sums-over","status":"fully_proved","summary":"Let $L/K$ be an extension of number fields, let $v$ be a finite place of $K$, given as a height-one prime of the ring of integers $\\mathcal{O}_K$, and let $K_v$ be the completion of $K$ at $v$. For each of the finitely many finite places $w$ of $L$ lying over $v$, regard the completion $L_w$ as a $K_v$-algebra through the continuous ring homomorphism $K_v \\to L_w$ extending $K \\to L$. Then for every $\\alpha \\in L$,\u2026","title":"Global trace as a sum of local traces"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm","status":"fully_proved","title":"Counting ideals of bounded norm"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm/card-nonzero-abs-norm-le","kind":"node","lean":"NumberField.Ideal.card_nonzero_absNorm_le","parent":5458,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm/index.html#card-nonzero-abs-norm-le","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm","status":"fully_proved","summary":"Let $K$ be a number field of degree $n = [K : \\mathbb{Q}]$ with ring of integers $\\mathcal{O}_K$, let $M$ be a natural number, and let $k$ be the largest natural number with $2^k \\le M$, with $k = 0$ if $M = 0$. Then the number of nonzero ideals $I$ of $\\mathcal{O}_K$ whose absolute norm $N(I) = |\\mathcal{O}_K / I|$ satisfies $N(I) \\le M$ is at most $(nM)^k$.","title":"Bound on the number of ideals of bounded norm, integer threshold"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm/card-nonzero-abs-norm-le-real","kind":"node","lean":"NumberField.Ideal.card_nonzero_absNorm_le_real","parent":5458,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm/index.html#card-nonzero-abs-norm-le-real","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm","status":"fully_proved","summary":"Let $K$ be a number field of degree $n = [K : \\mathbb{Q}]$ with ring of integers $\\mathcal{O}_K$, let $M$ be any real number, and let $m = \\max(0, \\lfloor M \\rfloor)$. Let $k$ be the largest natural number with $2^k \\le m$, and $k = 0$ if $m = 0$. Then the number of nonzero ideals $I$ of $\\mathcal{O}_K$ whose absolute norm $N(I) = |\\mathcal{O}_K / I|$ satisfies $N(I) \\le M$ is at most $(nm)^k$.","title":"Bound on the number of ideals of bounded norm, floor form"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm/card-nonzero-abs-norm-le-real-rpow","kind":"node","lean":"NumberField.Ideal.card_nonzero_absNorm_le_real_rpow","parent":5458,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm/index.html#card-nonzero-abs-norm-le-real-rpow","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/counting-ideals-of-bounded-norm","status":"fully_proved","summary":"Let $K$ be a number field of degree $n = [K : \\mathbb{Q}]$ with ring of integers $\\mathcal{O}_K$, and let $M > 0$ be a real number. The number of nonzero ideals $I$ of $\\mathcal{O}_K$ whose absolute norm $N(I) = |\\mathcal{O}_K / I|$ satisfies $N(I) \\le M$ is at most $(nM)^{\\log_2 M}$, where the exponent $\\log_2 M$ is a real number.","title":"Bound on the number of ideals of bounded norm"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","title":"Discriminants of number fields"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/abs-norm-discriminant-ideal","kind":"node","lean":"Ideal.absNorm_discriminantIdeal","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#abs-norm-discriminant-ideal","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ and discriminant $d_K$. The absolute norm of the discriminant ideal of $\\mathcal{O}_K$ over $\\mathbb{Z}$ equals $|d_K|$.","title":"Absolute norm of the discriminant ideal of a number field"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/pow-div-factorial-sq-mul-pi-div-four-pow","kind":"node","lean":"NumberField.pow_div_factorial_sq_mul_pi_div_four_pow_gt","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#pow-div-factorial-sq-mul-pi-div-four-pow","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $n$ and $s$ be natural numbers with $n \\ne 0$ and $2s \\le n$, and let $e$ be Euler's number. Then","title":"Comparison of the Minkowski and exponential discriminant bounds"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/middle-term-le-abs-discr","kind":"node","lean":"NumberField.middle_term_le_abs_discr","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#middle-term-le-abs-discr","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $K$ be a number field of degree $n = [K : \\mathbb{Q}]$ with $s$ complex places and discriminant $d_K$. Then","title":"Minkowski's lower bound for the discriminant"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/abs-discr-gt-exp-term","kind":"node","lean":"NumberField.abs_discr_gt_exp_term","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#abs-discr-gt-exp-term","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $K$ be a number field of degree $n = [K : \\mathbb{Q}]$ and discriminant $d_K$, and let $e$ be Euler's number. Then","title":"Exponential lower bound for the discriminant"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant-set","kind":"node","lean":"latticeDiscriminantSet","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant-set","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. For an $A$-submodule $M$ of $L$, this is the set of all discriminants $\\operatorname{disc}_{L/K}(b_1, \\dots, b_n) = \\det(\\operatorname{Tr}_{L/K}(b_i b_j))$ in $K$ of $n$-tuples $(b_1, \\dots, b_n)$ of elements of $M$.","title":"Discriminants of tuples from a submodule"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant","kind":"node","lean":"latticeDiscriminant","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. The discriminant $D(M)$ of an $A$-submodule $M$ of $L$ is the $A$-submodule of $K$ generated by the discriminants $\\operatorname{disc}_{L/K}(b_1, \\dots, b_n) = \\det(\\operatorname{Tr}_{L/K}(b_i b_j))$ of all $n$-tuples $(b_1, \\dots, b_n)$ of\u2026","title":"Discriminant of a lattice"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant-le-span-singleton","kind":"node","lean":"latticeDiscriminant_le_span_singleton","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant-le-span-singleton","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. For an $A$-submodule $M$ of $L$, let $D(M)$ be its discriminant, the $A$-submodule of $K$ generated by the discriminants over $K$ of all $n$-tuples of elements of $M$. If $b_1, \\dots, b_n$ is a $K$-basis of $L$ and $M$ is contained in the\u2026","title":"The discriminant of a lattice inside a basis lattice"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant-is-fractional","kind":"node","lean":"latticeDiscriminant_isFractional","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant-is-fractional","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. For an $A$-submodule $M$ of $L$, let $D(M)$ be its discriminant, the $A$-submodule of $K$ generated by the discriminants over $K$ of all $n$-tuples of elements of $M$. Assume $K$ is the fraction field of $A$. If $M$ is contained in the\u2026","title":"The discriminant of a lattice inside a basis lattice is fractional"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant-ne-bot-of-basis","kind":"node","lean":"latticeDiscriminant_ne_bot_of_basis","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant-ne-bot-of-basis","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. For an $A$-submodule $M$ of $L$, let $D(M)$ be its discriminant, the $A$-submodule of $K$ generated by the discriminants over $K$ of all $n$-tuples of elements of $M$. Assume $L/K$ is separable. If $M$ contains a $K$-basis $b_1, \\dots, b_n$\u2026","title":"The discriminant of a lattice containing a basis is nonzero"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant-is-fractional-and-n","kind":"node","lean":"latticeDiscriminant_isFractional_and_ne_bot","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant-is-fractional-and-n","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. For an $A$-submodule $M$ of $L$, let $D(M)$ be its discriminant, the $A$-submodule of $K$ generated by the discriminants over $K$ of all $n$-tuples of elements of $M$. Assume $K$ is the fraction field of $A$ and $L/K$ is separable. If $M$\u2026","title":"The discriminant of a sandwiched lattice is a nonzero fractional ideal"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant-eq-span-singleton","kind":"node","lean":"latticeDiscriminant_eq_span_singleton","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant-eq-span-singleton","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. For an $A$-submodule $M$ of $L$, let $D(M)$ be its discriminant, the $A$-submodule of $K$ generated by the discriminants over $K$ of all $n$-tuples of elements of $M$. If $b_1, \\dots, b_n$ is a $K$-basis of $L$, every $b_i$ lies in $M$ and\u2026","title":"The discriminant of a lattice spanned by a basis"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant-eq-imp-eq","kind":"node","lean":"latticeDiscriminant_eq_imp_eq","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant-eq-imp-eq","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. For an $A$-submodule $M$ of $L$, let $D(M)$ be its discriminant, the $A$-submodule of $K$ generated by the discriminants over $K$ of all $n$-tuples of elements of $M$. Assume $L/K$ is separable and the map $A \\to K$ is injective. Let\u2026","title":"Equal discriminants force equal lattices"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/lattice-discriminant-eq-imp-eq-of-is-fra","kind":"node","lean":"latticeDiscriminant_eq_imp_eq_of_isFractionRing","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#lattice-discriminant-eq-imp-eq-of-is-fra","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a finite extension of fields, with $A$-algebra structures on $K$ and $L$ forming a scalar tower $A \\to K \\to L$, and let $n = [L : K]$. For an $A$-submodule $M$ of $L$, let $D(M)$ be its discriminant, the $A$-submodule of $K$ generated by the discriminants over $K$ of all $n$-tuples of elements of $M$. Assume $K$ is the fraction field of $A$ and $L/K$ is separable. Let\u2026","title":"Equal discriminants force equal lattices (fraction field version)"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/image-of-b","kind":"node","lean":"imageOfB","parent":5462,"reading_title":{"format":"inline-parts-v1","parts":[["text","Image of "],["math","\\(B\\)"],["text"," in "],["math","\\(L\\)"]],"source":"Image of $B$ in $L$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#image-of-b","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and $L$ a field, with algebra maps $A \\to B$, $B \\to L$ and $A \\to L$ forming a scalar tower. The image of $B$ in $L$ is the range of the algebra map $B \\to L$, regarded as an $A$-submodule of $L$.","title":"Image of $B$ in $L$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/mem-image-of-b","kind":"node","lean":"mem_imageOfB","parent":5462,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in the image of "],["math","\\(B\\)"]],"source":"Membership in the image of $B$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#mem-image-of-b","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and $L$ a field, with algebra maps $A \\to B$, $B \\to L$ and $A \\to L$ forming a scalar tower. An element $x \\in L$ lies in the image of $B$ in $L$ if and only if $x$ is the image of some $b \\in B$ under the algebra map $B \\to L$.","title":"Membership in the image of $B$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/algebra-map-mem-image-of-b","kind":"node","lean":"algebraMap_mem_imageOfB","parent":5462,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(B\\)"],["text"," lie in the image of "],["math","\\(B\\)"]],"source":"Elements of $B$ lie in the image of $B$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#algebra-map-mem-image-of-b","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and $L$ a field, with algebra maps $A \\to B$, $B \\to L$ and $A \\to L$ forming a scalar tower. For every $b \\in B$, the image of $b$ under the algebra map $B \\to L$ lies in the image of $B$ in $L$, regarded as an $A$-submodule of $L$.","title":"Elements of $B$ lie in the image of $B$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/extension-discriminant","kind":"node","lean":"extensionDiscriminant","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#extension-discriminant","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and $K$ and $L$ fields with $L$ finite-dimensional over $K$, with algebra maps forming scalar towers $A \\to K \\to L$ and $A \\to B \\to L$, and let $n = [L : K]$. The extension discriminant is the $A$-submodule of $K$ spanned by the discriminants $\\operatorname{disc}_{L/K}(b_1, \\dots, b_n)$ of all $n$-tuples of elements of $L$ lying in the image of $B$ in $L$.","title":"Extension discriminant"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/extension-discriminant-eq","kind":"node","lean":"extensionDiscriminant_eq","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#extension-discriminant-eq","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and $K$ and $L$ fields with $L$ finite-dimensional over $K$, with algebra maps forming scalar towers $A \\to K \\to L$ and $A \\to B \\to L$. The extension discriminant of the tower equals the lattice discriminant over $A$ and $K$ of the image of $B$ in $L$, regarded as an $A$-submodule of $L$.","title":"The extension discriminant is a lattice discriminant"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/discr-mem-extension-discriminant","kind":"node","lean":"discr_mem_extensionDiscriminant","parent":5462,"reading_title":{"format":"inline-parts-v1","parts":[["text","Discriminants of tuples from "],["math","\\(B\\)"],["text"," lie in the extension discriminant"]],"source":"Discriminants of tuples from $B$ lie in the extension discriminant"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#discr-mem-extension-discriminant","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and $K$ and $L$ fields with $L$ finite-dimensional over $K$, with algebra maps forming scalar towers $A \\to K \\to L$ and $A \\to B \\to L$, and let $n = [L : K]$. If $b_1, \\dots, b_n \\in L$ all lie in the image of $B$ in $L$, then their discriminant $\\operatorname{disc}_{L/K}(b_1, \\dots, b_n)$ lies in the extension discriminant of the tower, an $A$-submodule of $K$.","title":"Discriminants of tuples from $B$ lie in the extension discriminant"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/extension-discriminant-localization","kind":"node","lean":"extensionDiscriminant_localization","parent":5462,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields/index.html#extension-discriminant-localization","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-number-fields","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings and $K$ and $L$ fields, with ring maps $A \\to K \\to L$ and $A \\to B \\to L$ whose composites $A \\to L$ agree, and with $L$ finite-dimensional over $K$ of degree $n$. Write $D_{B/A}$ for the $A$-submodule of $K$ generated by the discriminants $\\operatorname{disc}_{L/K}(b_1, \\dots, b_n)$ of all $n$-tuples of elements of the image of $B$ in $L$. Let $S$ be a submonoid of $A$, let\u2026","title":"Localization of the extension discriminant"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","title":"Discriminants of truncated logarithmic polynomials"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-polynomial","kind":"node","lean":"MetaMathlibExt.truncatedLogPolynomial","parent":5483,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-polynomial","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"For a natural number $n$, the truncated logarithmic polynomial is $F_n(x) = 1 + x + \\frac{x^2}{2} + \\dots + \\frac{x^n}{n} \\in \\mathbb{Q}[x]$.","title":"Truncated logarithmic polynomial"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-lcm","kind":"node","lean":"MetaMathlibExt.truncatedLogLcm","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Least common multiple of "],["math","\\(1, \\dots, n\\)"]],"source":"Least common multiple of $1, \\dots, n$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-lcm","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"For a natural number $n$, $L_n$ is the least common multiple $\\operatorname{lcm}(1, 2, \\dots, n)$ of the integers from $1$ to $n$; in particular $L_0 = 1$.","title":"Least common multiple of $1, \\dots, n$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-lcm-four","kind":"node","lean":"MetaMathlibExt.truncatedLogLcm_four","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Least common multiple of "],["math","\\(1, 2, 3, 4\\)"]],"source":"Least common multiple of $1, 2, 3, 4$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-lcm-four","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"The least common multiple of the integers from $1$ to $4$ is $L_4 = \\operatorname{lcm}(1, 2, 3, 4) = 12$.","title":"Least common multiple of $1, 2, 3, 4$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/scaled-truncated-log-polynomial","kind":"node","lean":"MetaMathlibExt.scaledTruncatedLogPolynomial","parent":5483,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#scaled-truncated-log-polynomial","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"For a natural number $n$, let $L_n = \\operatorname{lcm}(1, 2, \\dots, n)$. The scaled truncated logarithmic polynomial is the integer polynomial $\\tilde F_n(x) = L_n + \\sum_{j=1}^{n} (L_n / j)\\, x^j$, where each quotient $L_n / j$ is an integer.","title":"Scaled truncated logarithmic polynomial"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/map-scaled-truncated-log-polynomial","kind":"node","lean":"MetaMathlibExt.map_scaledTruncatedLogPolynomial","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","The scaled truncated logarithmic polynomial is "],["math","\\(L_n F_n\\)"]],"source":"The scaled truncated logarithmic polynomial is $L_n F_n$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#map-scaled-truncated-log-polynomial","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"For a natural number $n$, let $L_n = \\operatorname{lcm}(1, 2, \\dots, n)$, let $F_n(x) = 1 + x + x^2/2 + \\cdots + x^n/n$ be the truncated logarithmic polynomial and let $\\tilde F_n \\in \\mathbb{Z}[x]$ be the scaled truncated logarithmic polynomial. Mapping the coefficients of $\\tilde F_n$ from $\\mathbb{Z}$ to $\\mathbb{Q}$ gives the polynomial $L_n F_n$ in $\\mathbb{Q}[x]$.","title":"The scaled truncated logarithmic polynomial is $L_n F_n$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-resultant-product","kind":"node","lean":"MetaMathlibExt.truncatedLogResultantProduct","parent":5483,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-resultant-product","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"For a natural number $n$, let $F_n(x) = 1 + x + \\frac{x^2}{2} + \\dots + \\frac{x^n}{n}$, let $L_n = \\operatorname{lcm}(1, \\dots, n)$, and let $\\tilde F_n = L_n F_n \\in \\mathbb{Z}[x]$. The integer $\\mathcal{P}_n$ is the resultant $\\operatorname{Res}(1 + x + \\dots + x^{n-1}, \\tilde F_n)$. For $n \\ge 1$ the polynomial $1 + x + \\dots + x^{n-1} = F_n'$ is monic, so $\\mathcal{P}_n = \\prod_{\\theta} \\tilde F_n(\\theta)$, the\u2026","title":"Resultant product for truncated logarithmic polynomials"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-aux-product","kind":"node","lean":"MetaMathlibExt.truncatedLogAuxProduct","parent":5483,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-aux-product","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"For a natural number $m$, let $f_m(x) = \\sum_{k=0}^{m-1} x^k$ and $g_m(x) = c_m + \\sum_{j=1}^{m-1} \\frac{x^j}{j}$ in $\\mathbb{Q}[x]$, where $c_m = \\frac{1}{m}$ for $m \\ge 1$ and $c_0 = 0$. The rational number $X(m)$ is the resultant $\\operatorname{Res}(f_m, g_m)$. For $m \\ge 1$, since $f_m$ is monic with roots $\\omega^k$ for $1 \\le k \\le m-1$, where $\\omega$ is a primitive $m$-th root of unity, this is\u2026","title":"Auxiliary product for truncated logarithmic discriminants"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-aux-product-one","kind":"node","lean":"MetaMathlibExt.truncatedLogAuxProduct_one","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Auxiliary product at "],["math","\\(m = 1\\)"]],"source":"Auxiliary product at $m = 1$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-aux-product-one","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"The auxiliary product $X(m)$ takes the value $X(1) = 1$ at $m = 1$; it is an empty product.","title":"Auxiliary product at $m = 1$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-resultant-product-one","kind":"node","lean":"MetaMathlibExt.truncatedLogResultantProduct_one","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Resultant product at "],["math","\\(n = 1\\)"]],"source":"Resultant product at $n = 1$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-resultant-product-one","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"The resultant product satisfies $\\mathcal{P}_1 = 1$; it is an empty product.","title":"Resultant product at $n = 1$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-exceptional-primes","kind":"node","lean":"MetaMathlibExt.truncatedLogExceptionalPrimes","parent":5483,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-exceptional-primes","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"fully_proved","summary":"For a natural number $m$, the exceptional set $E_m$ is the set of primes $\\ell$ with $m\\ell \\equiv 1 \\pmod 4$ such that the $\\ell$-adic valuation of the auxiliary product $X(m)$ is positive or the $\\ell$-adic valuation of the harmonic number $Y(m) = H_m = 1 + \\frac{1}{2} + \\dots + \\frac{1}{m}$ is positive.","title":"Exceptional primes for truncated logarithmic discriminants"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/truncated-log-resultant-product-pos","kind":"node","lean":"MetaMathlibExt.truncatedLogResultantProduct_pos","parent":5483,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#truncated-log-resultant-product-pos","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"For every natural number $n \\ge 1$, the resultant product $\\mathcal{P}_n$ is a positive integer.","title":"Positivity of the resultant product"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/not-is-square-discr-truncated-log-of-mod","kind":"node","lean":"MetaMathlibExt.not_isSquare_discr_truncatedLog_of_mod_four","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonsquare discriminants of "],["math","\\(F_n\\)"],["text"," for positive "],["math","\\(n \\equiv 0, 2, 3 \\pmod 4\\)"]],"source":"Nonsquare discriminants of $F_n$ for positive $n \\equiv 0, 2, 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#not-is-square-discr-truncated-log-of-mod","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"Let $F_n(x) = 1 + x + x^2/2 + \\cdots + x^n/n$ be the truncated logarithmic polynomial and let $n \\ge 1$ be a natural number with $n \\equiv 0$, $2$ or $3 \\pmod 4$. Then the discriminant of $F_n$ is not a square in $\\mathbb{Q}$.","title":"Nonsquare discriminants of $F_n$ for positive $n \\equiv 0, 2, 3 \\pmod 4$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/not-is-square-discr-truncated-log","kind":"node","lean":"MetaMathlibExt.not_isSquare_discr_truncatedLog","parent":5483,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#not-is-square-discr-truncated-log","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"Let $F_n(x) = 1 + x + x^2/2 + \\cdots + x^n/n$ be the truncated logarithmic polynomial and let $n \\ge 1$. If $n \\equiv 0$, $2$ or $3 \\pmod 4$, or if $n \\equiv 1 \\pmod 4$ and $n = p^e$ for a prime $p$ and an odd $e$, then the discriminant of $F_n$ is not a square in $\\mathbb{Q}$.","title":"Nonsquare discriminants of truncated logarithmic polynomials"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/not-dvd-truncated-log-resultant-product","kind":"node","lean":"MetaMathlibExt.not_dvd_truncatedLogResultantProduct_of_even_power","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","An odd prime "],["math","\\(p\\)"],["text"," does not divide "],["math","\\(\\mathcal{P}_{p^{2e}}\\)"]],"source":"An odd prime $p$ does not divide $\\mathcal{P}_{p^{2e}}$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#not-dvd-truncated-log-resultant-product","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"Let $p$ be an odd prime, let $e \\ge 1$, and let $\\mathcal{P}_n$ be the truncated logarithmic resultant product. Then $p$ does not divide $\\mathcal{P}_{p^{2e}}$.","title":"An odd prime $p$ does not divide $\\mathcal{P}_{p^{2e}}$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/finite-primes-is-square-discr-truncated","kind":"node","lean":"MetaMathlibExt.finite_primes_isSquare_discr_truncatedLog","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finiteness of square discriminants of "],["math","\\(F_{mq}\\)"],["text"," with "],["math","\\(mq \\equiv 1 \\pmod 4\\)"]],"source":"Finiteness of square discriminants of $F_{mq}$ with $mq \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#finite-primes-is-square-discr-truncated","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"Let $F_n(x) = 1 + x + x^2/2 + \\cdots + x^n/n$ be the truncated logarithmic polynomial and let $m \\ge 1$ be a natural number. Then there are only finitely many primes $q$ with $mq \\equiv 1 \\pmod 4$ such that the discriminant of $F_{mq}$ is a square in $\\mathbb{Q}$.","title":"Finiteness of square discriminants of $F_{mq}$ with $mq \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/not-dvd-truncated-log-resultant-product-2","kind":"node","lean":"MetaMathlibExt.not_dvd_truncatedLogResultantProduct_of_prime_power","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prime "],["math","\\(p \\equiv 1 \\pmod 4\\)"],["text"," does not divide "],["math","\\(\\mathcal{P}_{p^e}\\)"]],"source":"A prime $p \\equiv 1 \\pmod 4$ does not divide $\\mathcal{P}_{p^e}$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#not-dvd-truncated-log-resultant-product-2","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, let $e \\ge 1$, and let $\\mathcal{P}_n$ be the truncated logarithmic resultant product. Then $p$ does not divide $\\mathcal{P}_{p^e}$.","title":"A prime $p \\equiv 1 \\pmod 4$ does not divide $\\mathcal{P}_{p^e}$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/discr-truncated-log-prime-power","kind":"node","lean":"MetaMathlibExt.discr_truncatedLog_prime_power","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Discriminants of "],["math","\\(F_{p^e}\\)"],["text"," for primes "],["math","\\(p \\equiv 1 \\pmod 4\\)"]],"source":"Discriminants of $F_{p^e}$ for primes $p \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#discr-truncated-log-prime-power","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, let $e \\ge 1$, let $F_n(x) = 1 + x + x^2/2 + \\cdots + x^n/n$ be the truncated logarithmic polynomial and let $\\mathcal{P}_n$ be the truncated logarithmic resultant product. If $e$ is odd, the discriminant of $F_{p^e}$ is not a square in $\\mathbb{Q}$. If $e$ is even, the discriminant of $F_{p^e}$ is not a square in $\\mathbb{Q}$ if and only if the integer\u2026","title":"Discriminants of $F_{p^e}$ for primes $p \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/not-is-square-discr-truncated-log-mul-pr","kind":"node","lean":"MetaMathlibExt.not_isSquare_discr_truncatedLog_mul_prime","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonsquare discriminants of "],["math","\\(F_{mq}\\)"],["text"," for non-exceptional primes "],["math","\\(q > m\\)"]],"source":"Nonsquare discriminants of $F_{mq}$ for non-exceptional primes $q > m$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#not-is-square-discr-truncated-log-mul-pr","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"Let $m$ be a natural number, let $E_m$ be the set of exceptional primes for truncated logarithmic discriminants, and let $F_n(x) = 1 + x + x^2/2 + \\cdots + x^n/n$ be the truncated logarithmic polynomial. If $q$ is a prime with $q \\notin E_m$, $q > m$ and $mq \\equiv 1 \\pmod 4$, then the discriminant of $F_{mq}$ is not a square in $\\mathbb{Q}$.","title":"Nonsquare discriminants of $F_{mq}$ for non-exceptional primes $q > m$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/not-is-square-discr-truncated-log-three","kind":"node","lean":"MetaMathlibExt.not_isSquare_discr_truncatedLog_three_five_seven_mul","parent":5483,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonsquare discriminants of "],["math","\\(F_{3q}\\)"],["text",", "],["math","\\(F_{5q}\\)"],["text"," and "],["math","\\(F_{7q}\\)"]],"source":"Nonsquare discriminants of $F_{3q}$, $F_{5q}$ and $F_{7q}$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia/index.html#not-is-square-discr-truncated-log-three","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/discriminants-of-truncated-logarithmic-polynomia","status":"can_prove","summary":"Let $F_n(x) = 1 + x + x^2/2 + \\cdots + x^n/n$ be the truncated logarithmic polynomial. For every prime $q$, if $n = 3q$, $n = 5q$ or $n = 7q$, then the discriminant of $F_n$ is not a square in $\\mathbb{Q}$.","title":"Nonsquare discriminants of $F_{3q}$, $F_{5q}$ and $F_{7q}$"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"can_prove","title":"Enumerating number fields"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/number-field-of-degree","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.NumberFieldOfDegree","parent":5503,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number fields of degree "],["math","\\(n\\)"]],"source":"Number fields of degree $n$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#number-field-of-degree","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"fully_proved","summary":"For a natural number $n$, the type of number fields of degree $n$ over $\\mathbb{Q}$: an element is a type $K$ with a field structure making $K$ a number field (a field of characteristic zero that is finite-dimensional over $\\mathbb{Q}$) such that $\\dim_{\\mathbb{Q}} K = n$.","title":"Number fields of degree $n$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/quotient-ring","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.quotientRing","parent":5503,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quotient of a polynomial ring by "],["math","\\(r\\)"],["text"," polynomials"]],"source":"Quotient of a polynomial ring by $r$ polynomials"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#quotient-ring","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"fully_proved","summary":"For a natural number $r$ and polynomials $E_1, \\dots, E_r \\in \\mathbb{Q}[x_1, \\dots, x_r]$, the quotient ring $\\mathbb{Q}[x_1, \\dots, x_r]/(E_1, \\dots, E_r)$ of the polynomial ring by the ideal generated by $E_1, \\dots, E_r$.","title":"Quotient of a polynomial ring by $r$ polynomials"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/jacobian","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.jacobian","parent":5503,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#jacobian","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"fully_proved","summary":"For polynomials $E_1, \\dots, E_r \\in \\mathbb{Q}[x_1, \\dots, x_r]$, the image in the quotient ring $\\mathbb{Q}[x_1, \\dots, x_r]/(E_1, \\dots, E_r)$ of the Jacobian determinant $\\det(\\partial E_i / \\partial x_j)_{1 \\le i, j \\le r}$.","title":"Jacobian determinant in the quotient ring"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/nonsingular-open","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.nonsingularOpen","parent":5503,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#nonsingular-open","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"fully_proved","summary":"For polynomials $E_1, \\dots, E_r \\in \\mathbb{Q}[x_1, \\dots, x_r]$, the open subscheme of $\\operatorname{Spec} \\mathbb{Q}[x_1, \\dots, x_r]/(E_1, \\dots, E_r)$ where the Jacobian determinant $\\det(\\partial E_i / \\partial x_j)$ does not vanish, that is, the basic open set of the image of this determinant in the quotient ring.","title":"The open subscheme where the Jacobian does not vanish"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/theorem-1","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.theorem_1","parent":5503,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#theorem-1","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"can_prove","summary":"There is a positive integer $Q$ such that for every natural number $n \\ge Q$ and every number field $K$ of degree $n$ over $\\mathbb{Q}$, with discriminant $d_K$ and root discriminant $\\delta_K = |d_K|^{1/n}$, there are natural numbers $r$ and $d$ with $r \\le Q \\log n$, $d \\le Q \\log n$ and $\\binom{d + r}{r} \\le Q n \\log n$, and polynomials $E_1, \\dots, E_r \\in \\mathbb{Z}[x_1, \\dots, x_r]$ of total degree at most\u2026","title":"Couveignes: number fields have small models"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/number-field-of-bounded-discriminant","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.NumberFieldOfBoundedDiscriminant","parent":5503,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number fields of degree "],["math","\\(n\\)"],["text"," with discriminant bounded by "],["math","\\(H\\)"]],"source":"Number fields of degree $n$ with discriminant bounded by $H$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#number-field-of-bounded-discriminant","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"fully_proved","summary":"For natural numbers $n$ and $H$, the type of number fields $K$ of degree $n$ over $\\mathbb{Q}$ whose discriminant $d_K$ satisfies $|d_K| \\le H$.","title":"Number fields of degree $n$ with discriminant bounded by $H$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/iso","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.NumberFieldOfBoundedDiscriminant.iso","parent":5503,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#iso","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"fully_proved","summary":"For natural numbers $n$ and $H$, two number fields $K$ and $L$ of degree $n$ over $\\mathbb{Q}$, each with discriminant of absolute value at most $H$, are related when there exists a ring isomorphism from $K$ to $L$.","title":"Isomorphism of number fields with bounded discriminant"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/number-field-of-bounded-discriminant-up","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.NumberFieldOfBoundedDiscriminantUpToIsomorphism","parent":5503,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#number-field-of-bounded-discriminant-up","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"fully_proved","summary":"For natural numbers $n$ and $H$, the quotient of the type of number fields of degree $n$ over $\\mathbb{Q}$ whose discriminant has absolute value at most $H$ by the relation of being isomorphic as rings.","title":"Isomorphism classes of number fields with bounded discriminant"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/theorem-2","kind":"node","lean":"MathlibExt.NumberTheory.EnumeratingNumberFieldsWanted.EnumeratingNumberFields.theorem_2","parent":5503,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields/index.html#theorem-2","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/enumerating-number-fields","status":"can_prove","summary":"There is a positive integer $Q$ such that for all natural numbers $n \\ge Q$ and $H \\ge 1$, there are only finitely many isomorphism classes of number fields of degree $n$ over $\\mathbb{Q}$ whose discriminant has absolute value at most $H$, and their number is at most $n^{Q n (\\log n)^3} H^{Q (\\log n)^3}$.","title":"Couveignes: counting number fields with bounded discriminant"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension","status":"fully_proved","title":"Maximal unramified subextension"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/is-finite-unramified","kind":"node","lean":"IntermediateField.IsFiniteUnramified","parent":5513,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/index.html#is-finite-unramified","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a field extension such that the map $A \\to L$ factors as $A \\to K \\to L$. An intermediate field $E$ of $L/K$ is finite unramified relative to $A$ if $E$ is finite-dimensional over $K$ and the integral closure of $A$ in $E$ is formally unramified over $A$.","title":"Finite unramified intermediate fields"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/maximal-unramified-subextension","kind":"node","lean":"IntermediateField.maximalUnramifiedSubextension","parent":5513,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/index.html#maximal-unramified-subextension","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a field extension such that the map $A \\to L$ factors as $A \\to K \\to L$. The maximal unramified subextension of $L/K$ relative to $A$ is the supremum, in the lattice of intermediate fields of $L/K$, of all intermediate fields of $L/K$ that are finite unramified relative to $A$.","title":"Maximal unramified subextension"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/le-maximal-unramified-subextension","kind":"node","lean":"IntermediateField.le_maximalUnramifiedSubextension","parent":5513,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/index.html#le-maximal-unramified-subextension","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a field extension such that the map $A \\to L$ factors as $A \\to K \\to L$. Every intermediate field $E$ of $L/K$ that is finite unramified relative to $A$ is contained in the maximal unramified subextension of $L/K$ relative to $A$.","title":"Finite unramified subfields lie in the maximal unramified subextension"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/maximal-unramified-subextension-le-iff","kind":"node","lean":"IntermediateField.maximalUnramifiedSubextension_le_iff","parent":5513,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/index.html#maximal-unramified-subextension-le-iff","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a field extension such that the map $A \\to L$ factors as $A \\to K \\to L$. An intermediate field $F$ of $L/K$ contains the maximal unramified subextension of $L/K$ relative to $A$ if and only if $F$ contains every intermediate field of $L/K$ that is finite unramified relative to $A$.","title":"Universal property of the maximal unramified subextension"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/map","kind":"node","lean":"IntermediateField.IsFiniteUnramified.map","parent":5513,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/index.html#map","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a field extension such that the map $A \\to L$ factors as $A \\to K \\to L$. If an intermediate field $E$ of $L/K$ is finite unramified relative to $A$, then for every $K$-algebra automorphism $\\sigma$ of $L$ the image $\\sigma(E)$ is finite unramified relative to $A$.","title":"Automorphisms preserve finite unramified subfields"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/map-maximal-unramified-subextension","kind":"node","lean":"IntermediateField.map_maximalUnramifiedSubextension","parent":5513,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension/index.html#map-maximal-unramified-subextension","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/maximal-unramified-subextension","status":"fully_proved","summary":"Let $A$ be a commutative ring and $L/K$ a field extension such that the map $A \\to L$ factors as $A \\to K \\to L$, and let $M$ be the maximal unramified subextension of $L/K$ relative to $A$. For every $K$-algebra automorphism $\\sigma$ of $L$, $\\sigma(M) = M$.","title":"The maximal unramified subextension is stable under automorphisms"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields","status":"fully_proved","title":"Orders in number fields"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/le-integral-closure","kind":"node","lean":"NumberField.IsOrder.le_integralClosure","parent":5520,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/index.html#le-integral-closure","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields","status":"fully_proved","summary":"Let $K$ be a number field and $O$ an order in $K$, that is, a subring of $K$ contained in the ring of integers $\\mathcal{O}_K$ (the integral closure of $\\mathbb{Z}$ in $K$) whose field of fractions is $K$. Then $O \\subseteq \\mathcal{O}_K$.","title":"Orders lie in the ring of integers"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/is-fraction-ring","kind":"node","lean":"NumberField.IsOrder.isFractionRing","parent":5520,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/index.html#is-fraction-ring","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields","status":"fully_proved","summary":"Let $K$ be a number field and $O$ an order in $K$, that is, a subring of $K$ contained in the ring of integers $\\mathcal{O}_K$ (the integral closure of $\\mathbb{Z}$ in $K$) whose field of fractions is $K$. Then $K$, with the inclusion $O \\subseteq K$, is a field of fractions of $O$.","title":"The fraction field of an order"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/exists-div-eq","kind":"node","lean":"NumberField.IsOrder.exists_div_eq","parent":5520,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/index.html#exists-div-eq","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields","status":"fully_proved","summary":"Let $K$ be a number field and $O$ an order in $K$, that is, a subring of $K$ contained in the ring of integers $\\mathcal{O}_K$ (the integral closure of $\\mathbb{Z}$ in $K$) whose field of fractions is $K$. Then for every $x \\in K$ there are $a, b \\in O$ with $b \\neq 0$ and $x = a / b$.","title":"Elements of a number field are quotients of elements of an order"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/is-order-integral-closure","kind":"node","lean":"NumberField.isOrder_integralClosure","parent":5520,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields/index.html#is-order-integral-closure","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/orders-in-number-fields","status":"fully_proved","summary":"For every number field $K$, the ring of integers $\\mathcal{O}_K$, the integral closure of $\\mathbb{Z}$ in $K$, is an order in $K$, that is, a subring of $K$ contained in $\\mathcal{O}_K$ whose field of fractions is $K$.","title":"The ring of integers is an order"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","title":"Ramification in extensions of number fields"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/finite-ramified-primes","kind":"node","lean":"NumberField.finite_ramified_primes","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#finite-ramified-primes","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Only finitely many height-one primes of a number field ramify in a number-field extension.","title":"Only finitely many primes ramify in a number field extension"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/is-unramified-in-mono","kind":"node","lean":"NumberField.isUnramifiedIn_mono","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#is-unramified-in-mono","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E \\subseteq F$ be intermediate fields of $L/\\mathbb{Q}$, and let $\\mathfrak{p}$ be an ideal of $\\mathbb{Z}$. If $\\mathfrak{p}$ is unramified in $\\mathcal{O}_F$, that is, every prime ideal of $\\mathcal{O}_F$ lying over $\\mathfrak{p}$ is unramified over $\\mathbb{Z}$, then $\\mathfrak{p}$ is unramified in $\\mathcal{O}_E$.","title":"Unramifiedness descends to smaller intermediate fields"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/etale-localization","kind":"node","lean":"NumberField.IsUnramifiedIn.etale_localization","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#etale-localization","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$, and let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_K$, that is, every prime ideal of $\\mathcal{O}_K$ lying over $p$ is unramified over $\\mathbb{Z}$. Let $S$ be the localization of $\\mathcal{O}_K$ at the image of the multiplicative set $\\mathbb{Z} \\setminus p$. Then $S$ is an \u00e9tale algebra over the local ring\u2026","title":"Localizing the ring of integers at an unramified prime gives an \u00e9tale algebra"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/tensor-product-localization-equiv-integr","kind":"node","lean":"NumberField.IsUnramifiedIn.tensorProductLocalizationEquivIntegralClosure","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#tensor-product-localization-equiv-integr","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $E$ and $F$ be number fields and let $\\mathfrak p$ be a prime ideal of $\\mathbb Z$ that is unramified in the ring of integers $\\mathcal O_F$. Write $R = \\mathbb Z_{\\mathfrak p}$ for the localization of $\\mathbb Z$ at $\\mathfrak p$, and for $K = E, F$ write $S_K$ for the localization of $\\mathcal O_K$ at the image of $\\mathbb Z \\setminus \\mathfrak p$. Let $K_E$ be the fraction field of $\\mathcal O_E$ and let\u2026","title":"Normalization commutes with base change along an unramified prime"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/semilocalization-to-at-prime","kind":"node","lean":"Localization.semilocalizationToAtPrime","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#semilocalization-to-at-prime","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $R \\to S \\to T$ be homomorphisms of commutative rings whose algebra structures form a scalar tower, let $\\mathfrak p$ be a prime ideal of $R$, and let $\\mathfrak Q$ be a prime ideal of $T$ lying over $\\mathfrak p$. Write $S_{\\mathfrak p}$ for the localization of $S$ at the image of $R \\setminus \\mathfrak p$. This is the homomorphism of $S$-algebras $S_{\\mathfrak p} \\to T_{\\mathfrak Q}$ extending the composite\u2026","title":"Map from the localization away from a prime to the local ring at a prime above it"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/semilocalization-to-at-prime-mk-eq-iff","kind":"node","lean":"Localization.semilocalizationToAtPrime_mk'_eq_iff","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#semilocalization-to-at-prime-mk-eq-iff","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $R \\to S \\to T$ be homomorphisms of commutative rings whose algebra structures form a scalar tower, let $\\mathfrak p$ be a prime ideal of $R$, and let $\\mathfrak Q$ be a prime ideal of $T$ lying over $\\mathfrak p$. Write $S_{\\mathfrak p}$ for the localization of $S$ at the image of $R \\setminus \\mathfrak p$, and let $\\varphi : S_{\\mathfrak p} \\to T_{\\mathfrak Q}$ be the $S$-algebra homomorphism extending the\u2026","title":"Values of the localization-to-local-ring map on fractions"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-to-compositum-at-prime","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorToCompositumAtPrime","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-to-compositum-at-prime","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $E$ and $F$ be intermediate fields of a number field $L$ over $\\mathbb Q$, let $\\mathfrak p$ be a prime ideal of $\\mathbb Z$ that is unramified in $\\mathcal O_F$, and let $\\mathfrak Q$ be a prime ideal of the ring of integers $\\mathcal O_{EF}$ of the compositum $EF$ of $E$ and $F$ in $L$ lying over $\\mathfrak p$. Write $R = \\mathbb Z_{\\mathfrak p}$ for the localization of $\\mathbb Z$ at $\\mathfrak p$; for\u2026","title":"Map from the normalized tensor algebra to the local ring of the compositum"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-to-compositum-at-prime-2","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorToCompositumAtPrime_equiv_tmul","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-to-compositum-at-prime-2","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E$ and $F$ be intermediate fields of $L/\\mathbb{Q}$ with compositum $EF$, let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_F$, and let $Q$ be a prime ideal of $\\mathcal{O}_{EF}$ lying over $p$. Write $R = \\mathbb{Z}_{(p)}$, let $S_F$ and $S_E$ be the localizations of $\\mathcal{O}_F$ and $\\mathcal{O}_E$ at the image of $\\mathbb{Z} \\setminus p$, let $C$ be\u2026","title":"The normalized tensor map on simple tensors"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-to-compositum-at-prime-3","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorToCompositumAtPrime_ker_isPrime","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-to-compositum-at-prime-3","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E$ and $F$ be intermediate fields of $L/\\mathbb{Q}$ with compositum $EF$, let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_F$, and let $Q$ be a prime ideal of $\\mathcal{O}_{EF}$ lying over $p$. Write $R = \\mathbb{Z}_{(p)}$, let $S_F$ be the localization of $\\mathcal{O}_F$ at the image of $\\mathbb{Z} \\setminus p$, let $C$ be the integral closure of $S_F$ in\u2026","title":"The kernel of the normalized tensor map is prime"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-compositum-prime","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorCompositumPrime","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-compositum-prime","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E$ and $F$ be intermediate fields of $L/\\mathbb{Q}$ with compositum $EF$, let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_F$, and let $Q$ be a prime ideal of $\\mathcal{O}_{EF}$ lying over $p$. Write $R = \\mathbb{Z}_{(p)}$, let $S_F$ be the localization of $\\mathcal{O}_F$ at the image of $\\mathbb{Z} \\setminus p$, let $C$ be the integral closure of $S_F$ in\u2026","title":"The contracted maximal prime of the normalized tensor algebra"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-compositum-prime-is-pr","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorCompositumPrime_isPrime","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-compositum-prime-is-pr","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E$ and $F$ be intermediate fields of $L/\\mathbb{Q}$ with compositum $EF$, let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_F$, and let $Q$ be a prime ideal of $\\mathcal{O}_{EF}$ lying over $p$. Then the contracted maximal prime of the normalized tensor algebra at $Q$ is a prime ideal.","title":"The contracted maximal prime is prime"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-compositum-prime-mem-i","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorCompositumPrime_mem_iff","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-compositum-prime-mem-i","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E$ and $F$ be intermediate fields of $L/\\mathbb{Q}$ with compositum $EF$, let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_F$, and let $Q$ be a prime ideal of $\\mathcal{O}_{EF}$ lying over $p$. Write $R = \\mathbb{Z}_{(p)}$, let $S_F$ be the localization of $\\mathcal{O}_F$ at the image of $\\mathbb{Z} \\setminus p$, let $C$ be the integral closure of $S_F$ in\u2026","title":"Membership in the contracted maximal prime"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-to-compositum-at-prime-4","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorToCompositumAtPrime_ker_le","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-to-compositum-at-prime-4","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $E$ and $F$ be intermediate fields of a number field $L$ over $\\mathbb Q$, let $\\mathfrak p$ be a prime ideal of $\\mathbb Z$ that is unramified in $\\mathcal O_F$, and let $\\mathfrak Q$ be a prime ideal of the ring of integers $\\mathcal O_{EF}$ of the compositum $EF$ of $E$ and $F$ in $L$ lying over $\\mathfrak p$. Write $R = \\mathbb Z_{\\mathfrak p}$ for the localization of $\\mathbb Z$ at $\\mathfrak p$; for\u2026","title":"The kernel lies in the contracted maximal ideal"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-localization-to-compos","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorLocalizationToCompositumAtPrime","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-localization-to-compos","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E$ and $F$ be intermediate fields of $L/\\mathbb{Q}$ with compositum $EF$, let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_F$, and let $Q$ be a prime ideal of $\\mathcal{O}_{EF}$ lying over $p$. Write $R = \\mathbb{Z}_{(p)}$, let $S_F$ be the localization of $\\mathcal{O}_F$ at the image of $\\mathbb{Z} \\setminus p$, let $C$ be the integral closure of $S_F$ in\u2026","title":"The localization map to the compositum at a prime"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-localization-to-compos-2","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorLocalizationToCompositumAtPrime_algebraMap","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-localization-to-compos-2","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E$ and $F$ be intermediate fields of $L/\\mathbb{Q}$ with compositum $EF$, let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_F$, and let $Q$ be a prime ideal of $\\mathcal{O}_{EF}$ lying over $p$. Write $R = \\mathbb{Z}_{(p)}$, let $S_F$ be the localization of $\\mathcal{O}_F$ at the image of $\\mathbb{Z} \\setminus p$, let $C$ be the integral closure of $S_F$ in\u2026","title":"The localization map restricts to the normalized tensor map"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/normalized-tensor-localization-to-compos-3","kind":"node","lean":"NumberField.IsUnramifiedIn.normalizedTensorLocalizationToCompositumAtPrime_mk'_spec","parent":5525,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields/index.html#normalized-tensor-localization-to-compos-3","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/ramification-in-extensions-of-number-fields","status":"fully_proved","summary":"Let $L$ be a number field, let $E$ and $F$ be intermediate fields of $L/\\mathbb{Q}$ with compositum $EF$, let $p$ be a prime ideal of $\\mathbb{Z}$ that is unramified in $\\mathcal{O}_F$, and let $Q$ be a prime ideal of $\\mathcal{O}_{EF}$ lying over $p$. Write $R = \\mathbb{Z}_{(p)}$, let $S_F$ be the localization of $\\mathcal{O}_F$ at the image of $\\mathbb{Z} \\setminus p$, let $C$ be the integral closure of $S_F$ in\u2026","title":"Value of the localization map on fractions"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras","status":"fully_proved","title":"Residue maps of singly generated algebras"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras/quotient-algebra-map-surjective-of-adjoi","kind":"node","lean":"Ideal.quotient_algebraMap_surjective_of_adjoin_singleton_eq_top","parent":5542,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras/index.html#quotient-algebra-map-surjective-of-adjoi","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings, $B$ an $A$-algebra, $\\mathfrak q$ an ideal of $B$ and $\\mathfrak p$ its preimage under $A \\to B$. If $B$ is generated as an $A$-algebra by a single element $\\pi \\in \\mathfrak q$, then the canonical map $A/\\mathfrak p \\to B/\\mathfrak q$ is surjective.","title":"Surjectivity of the residue map for a singly generated algebra"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras/inertia-deg-eq-one-of-adjoin-singleton-e","kind":"node","lean":"Ideal.inertiaDeg_eq_one_of_adjoin_singleton_eq_top","parent":5542,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras/index.html#inertia-deg-eq-one-of-adjoin-singleton-e","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/residue-maps-of-singly-generated-algebras","status":"fully_proved","summary":"Let $A$ and $B$ be commutative rings, $B$ an $A$-algebra, and $\\mathfrak q$ a maximal ideal of $B$ whose preimage $\\mathfrak p$ under $A \\to B$ is maximal. If $B$ is generated as an $A$-algebra by a single element $\\pi \\in \\mathfrak q$, then the inertia degree of $\\mathfrak q$ over $\\mathfrak p$ is $1$.","title":"Inertia degree one for a singly generated algebra"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups.json","id":"NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups","kind":"scope","navigation":"module","parent":5409,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups/index.html","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups","status":"can_prove","title":"Van der Waerden's conjecture on Galois groups"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups/e","kind":"node","lean":"MathlibExt.NumberTheory.VanDerWaerdenConjectureWanted.vanDerWaerdenConjecture.E","parent":5545,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monic integer polynomials whose Galois group is not "],["math","\\(S_n\\)"]],"source":"Monic integer polynomials whose Galois group is not $S_n$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups/index.html#e","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups","status":"fully_proved","summary":"For natural numbers $n$ and $H$, $E_n(H)$ is the number of monic polynomials $f(x) = x^n + a_1 x^{n-1} + \\dots + a_n$ of degree $n$ with integer coefficients and $|a_i| \\le H$ for $i = 1, \\dots, n$ whose Galois group over $\\mathbb{Q}$ has order less than $n!$, that is, is not the full symmetric group $S_n$.","title":"Monic integer polynomials whose Galois group is not $S_n$"},{"id":"NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups/theorem-1","kind":"node","lean":"MathlibExt.NumberTheory.VanDerWaerdenConjectureWanted.vanDerWaerdenConjecture.theorem_1","parent":5545,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups/index.html#theorem-1","scope":"NumberTheory/algebraic-number-theory/discriminants-ramification/van-der-waerden-s-conjecture-on-galois-groups","status":"can_prove","summary":"For every natural number $n \\ge 3$ there is a constant $C$ such that $E_n(H) \\le C H^{n-1}$ for all sufficiently large natural numbers $H$, where $E_n(H)$ is the number of monic polynomials of degree $n$ with integer coefficients, all non-leading coefficients of absolute value at most $H$, whose Galois group over $\\mathbb{Q}$ has order less than $n!$.","title":"Van der Waerden's conjecture (Bhargava)"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory","kind":"scope","parent":5201,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory","status":"can_prove","summary":"Places, adeles and ideles, Arakelov divisors, ray class groups, Artin map, Hilbert class fields, Neukirch\u2013Uchida-type Galois characterization.","title":"Adeles, ideles and class field theory"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/neukirch-uchida","kind":"node","lean":"MathlibExt.NumberTheory.NeukirchUchidaWanted.NumberTheory.neukirch_uchida","parent":5548,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/index.html#neukirch-uchida","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory","status":"can_prove","summary":"Let $K_1$ and $K_2$ be number fields with separable closures $\\overline{K_1}$ and $\\overline{K_2}$, and let $\\varphi \\colon \\mathrm{Gal}(\\overline{K_1}/K_1) \\to \\mathrm{Gal}(\\overline{K_2}/K_2)$ be a group isomorphism that is a homeomorphism for the Krull topologies. Then there is a unique field isomorphism $\\sigma \\colon \\overline{K_2} \\to \\overline{K_1}$ that maps the image of $K_2$ in $\\overline{K_2}$ onto the\u2026","title":"Neukirch\u2013Uchida theorem"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","title":"Adeles and ideles"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/finite-residue-field","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.finite_residueField","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#finite-residue-field","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field and $v$ a nonzero prime ideal of its ring of integers $\\mathcal{O}_K$. The residue field of the ring of integers $\\mathcal{O}_v$ of the $v$-adic completion $K_v$ of $K$ is finite.","title":"Finite residue field of a completed number ring"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/is-compact-adic-completion-integers","kind":"node","lean":"IsDedekindDomain.HeightOneSpectrum.isCompact_adicCompletionIntegers","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#is-compact-adic-completion-integers","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field and $v$ a nonzero prime ideal of its ring of integers $\\mathcal{O}_K$. The ring of integers $\\mathcal{O}_v$ of the $v$-adic completion $K_v$ of $K$ is a compact subset of $K_v$.","title":"Compactness of the completed ring of integers"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/local-integral-unit-subgroup","kind":"node","lean":"NumberField.localIntegralUnitSubgroup","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#local-integral-unit-subgroup","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, let $v$ be a height-one prime of $R$, and let $\\mathcal{O}_v$ be the ring of integers of the $v$-adic completion $K_v$ of $K$. The local integral unit subgroup is the subgroup $\\mathcal{O}_v^\\times$ of $K_v^\\times$, consisting of the units $u$ of $K_v$ with $u \\in \\mathcal{O}_v$ and $u^{-1} \\in \\mathcal{O}_v$.","title":"Local integral unit subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/finite-idele-group","kind":"node","lean":"NumberField.FiniteIdeleGroup","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#finite-idele-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. For each height-one prime $v$ of $R$, let $K_v$ be the $v$-adic completion of $K$ and $\\mathcal{O}_v$ its ring of integers. The finite idele group is the restricted product of the unit groups $K_v^\\times$ over all height-one primes $v$ of $R$ with respect to the subgroups $\\mathcal{O}_v^\\times$: the families $(x_v)_v$ with $x_v \\in K_v^\\times$ for every $v$ and\u2026","title":"Finite idele group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/is-open-local-integral-unit-subgroup","kind":"node","lean":"NumberField.FiniteIdeleGroup.isOpen_localIntegralUnitSubgroup","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#is-open-local-integral-unit-subgroup","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, let $v$ be a height-one prime of $R$, and let $\\mathcal{O}_v$ be the ring of integers of the $v$-adic completion $K_v$ of $K$. Then $\\mathcal{O}_v^\\times$ is an open subset of the unit group $K_v^\\times$.","title":"The local integral units are open"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/infinite-idele-group","kind":"node","lean":"NumberField.InfiniteIdeleGroup","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#infinite-idele-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"For a field $K$, the infinite idele group is the product $\\prod_v K_v^\\times$ of the unit groups of the completions $K_v$ of $K$ over all infinite places $v$ of $K$, with componentwise multiplication and the product topology.","title":"Infinite idele group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/topological-idele-group","kind":"node","lean":"NumberField.TopologicalIdeleGroup","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#topological-idele-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. The topological idele group is the product $\\mathbb{I}_{K,\\infty} \\times \\mathbb{I}_{K,f}$ of the infinite idele group $\\mathbb{I}_{K,\\infty} = \\prod_w K_w^\\times$, the product of the unit groups of the completions of $K$ at its infinite places $w$, and the finite idele group $\\mathbb{I}_{K,f}$, the restricted product of the unit groups $K_v^\\times$ of the\u2026","title":"Topological idele group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/infinite-principal-embedding","kind":"node","lean":"NumberField.TopologicalIdeleGroup.infinitePrincipalEmbedding","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#infinite-principal-embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"For a field $K$, this is the group homomorphism from $K^\\times$ to the infinite idele group of $K$ that sends $x$ to the family of its images in the completions $K_v$ of $K$, one for each infinite place $v$ of $K$.","title":"Diagonal map into the infinite ideles"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/finite-principal-embedding","kind":"node","lean":"NumberField.TopologicalIdeleGroup.finitePrincipalEmbedding","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#finite-principal-embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. This is the group homomorphism from $K^\\times$ to the finite idele group of $R$ and $K$ that sends $x$ to the family of its images in the $v$-adic completions $K_v$ of $K$, one for each height-one prime $v$ of $R$.","title":"Diagonal map into the finite ideles"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/principal-embedding","kind":"node","lean":"NumberField.TopologicalIdeleGroup.principalEmbedding","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#principal-embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. This is the group homomorphism from $K^\\times$ to the topological idele group of $R$ and $K$ whose infinite component sends $x$ to its images in the completions of $K$ at all infinite places of $K$, and whose finite component sends $x$ to its images in the $v$-adic completions $K_v$ of $K$ at all height-one primes $v$ of $R$.","title":"Diagonal map into the ideles"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/principal-ideles","kind":"node","lean":"NumberField.TopologicalIdeleGroup.principalIdeles","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#principal-ideles","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. The subgroup of principal ideles is the image of the diagonal homomorphism from $K^\\times$ to the topological idele group of $R$ and $K$, which sends $x$ to its images in the completions of $K$ at all infinite places of $K$ and at all height-one primes of $R$.","title":"Principal ideles"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/units-equiv","kind":"node","lean":"NumberField.TopologicalIdeleGroup.unitsEquiv","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#units-equiv","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$, and let $\\mathbb{A}_K$ be the adele ring of $K$ built from the infinite places of $K$ and the height-one primes of $R$. For an infinite place $w$ of $K$ write $K_w$ for the completion of $K$ at $w$, and for a height-one prime $v$ of $R$ write $K_v$ for the $v$-adic completion of $K$. This is the group isomorphism from the unit group $\\mathbb{A}_K^\\times$ to the\u2026","title":"Ideles as units of the adele ring"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/principal-embedding-injective","kind":"node","lean":"NumberField.TopologicalIdeleGroup.principalEmbedding_injective","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#principal-embedding-injective","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field and let $R$ be a Dedekind domain with fraction field $K$. The diagonal homomorphism from $K^\\times$ to the topological idele group of $R$ and $K$, which sends $x$ to its images in the completions of $K$ at all infinite places of $K$ and at all height-one primes of $R$, is injective.","title":"The diagonal map into the ideles is injective"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/topological-idele-class-group","kind":"node","lean":"NumberField.TopologicalIdeleClassGroup","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#topological-idele-class-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $R$ be a Dedekind domain with fraction field $K$. The topological idele class group is the quotient group of the topological idele group of $R$ and $K$ by its subgroup of principal ideles, the image of the diagonal homomorphism from $K^\\times$.","title":"Topological idele class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/is-compact-local-integral-unit-subgroup","kind":"node","lean":"NumberField.FiniteIdeleGroup.isCompact_localIntegralUnitSubgroup","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#is-compact-local-integral-unit-subgroup","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field, let $v$ be a height-one prime of its ring of integers $\\mathcal{O}_K$, and let $\\mathcal{O}_v$ be the ring of integers of the $v$-adic completion $K_v$ of $K$. Then the local integral unit subgroup $\\mathcal{O}_v^\\times$ is a compact subset of the unit group $K_v^\\times$.","title":"Compactness of the local integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/adelic-norm","kind":"node","lean":"NumberField.TopologicalIdeleGroup.adelicNorm","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#adelic-norm","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$. For each infinite place $w$ of $K$ let $K_w$ be the completion of $K$ at $w$, with the absolute value $\\|\\cdot\\|_w$ extending $w$, and let $m_w$ be $1$ if $w$ is real and $2$ if $w$ is complex. For each height-one prime $v$ of $\\mathcal{O}_K$ let $K_v$ be the $v$-adic completion of $K$, with the normalized absolute value $\\|\\cdot\\|_v$. The adelic norm\u2026","title":"Adelic norm"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/adelic-norm-apply","kind":"node","lean":"NumberField.TopologicalIdeleGroup.adelicNorm_apply","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#adelic-norm-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field. For every idele $a$ of $K$ (with $R = \\mathcal{O}_K$), the adelic norm of $a$ equals $\\bigl(\\prod_w \\|a_w\\|_w^{m_w}\\bigr) \\cdot \\prod_v \\|a_v\\|_v$, where $w$ runs over the infinite places of $K$, $\\|\\cdot\\|_w$ is the absolute value on the completion $K_w$ extending $w$, $m_w$ is $1$ for a real place and $2$ for a complex place, $v$ runs over the height-one primes of $\\mathcal{O}_K$,\u2026","title":"Formula for the adelic norm"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/norm-one-ideles","kind":"node","lean":"NumberField.TopologicalIdeleGroup.normOneIdeles","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#norm-one-ideles","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field. The norm-one ideles of $K$ form the kernel of the adelic norm, the subgroup of the topological idele group of $K$ (with $R = \\mathcal{O}_K$) consisting of the ideles whose adelic norm is $1$.","title":"Norm-one ideles"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/mem-norm-one-ideles","kind":"node","lean":"NumberField.TopologicalIdeleGroup.mem_normOneIdeles","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#mem-norm-one-ideles","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field. An idele $a$ of $K$ (with $R = \\mathcal{O}_K$) lies in the subgroup of norm-one ideles if and only if its adelic norm is $1$.","title":"Membership in the norm-one ideles"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/adelic-norm-principal-embedding","kind":"node","lean":"NumberField.TopologicalIdeleGroup.adelicNorm_principalEmbedding","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#adelic-norm-principal-embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field. For every $x \\in K^\\times$, the principal idele of $x$ (its image under the diagonal embedding of $K^\\times$ into the topological idele group, with $R = \\mathcal{O}_K$) has adelic norm $1$.","title":"Product formula for principal ideles"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/principal-ideles-le-norm-one-ideles","kind":"node","lean":"NumberField.TopologicalIdeleGroup.principalIdeles_le_normOneIdeles","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#principal-ideles-le-norm-one-ideles","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field. The subgroup of principal ideles of $K$ (with $R = \\mathcal{O}_K$) is contained in the subgroup of norm-one ideles; that is, every principal idele has adelic norm $1$.","title":"Principal ideles have norm one"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/norm-one-principal-ideles","kind":"node","lean":"NumberField.TopologicalIdeleGroup.normOnePrincipalIdeles","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#norm-one-principal-ideles","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"For a number field $K$, this is the subgroup of the group $I_K^1$ of ideles of $K$ with adelic norm $1$ consisting of the principal ideles, that is, of the elements of $I_K^1$ that are images of elements of $K^\\times$ under the diagonal embedding.","title":"Principal ideles of norm one"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/norm-one-idele-class-group","kind":"node","lean":"NumberField.TopologicalIdeleGroup.NormOneIdeleClassGroup","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#norm-one-idele-class-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"For a number field $K$, the norm-one idele class group $C_K^1 = I_K^1 / K^\\times$ is the quotient of the group $I_K^1$ of ideles of $K$ with adelic norm $1$ by its subgroup of principal ideles, the images of the elements of $K^\\times$ under the diagonal embedding.","title":"Norm-one idele class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/norm-one-mk-principal-embedding","kind":"node","lean":"NumberField.TopologicalIdeleGroup.normOne_mk_principalEmbedding","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#norm-one-mk-principal-embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"Let $K$ be a number field. For every $x \\in K^\\times$, the principal idele of $x$, viewed as an element of the group $I_K^1$ of ideles of $K$ with adelic norm $1$, has trivial class in the norm-one idele class group $C_K^1 = I_K^1 / K^\\times$.","title":"Principal ideles are trivial in the norm-one idele class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/principal-ideles-discrete","kind":"node","lean":"NumberField.TopologicalIdeleGroup.principalIdeles_discrete","parent":5550,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles/index.html#principal-ideles-discrete","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/adeles-and-ideles","status":"fully_proved","summary":"For a number field $K$, the subgroup of principal ideles is discrete in the idele group of $K$, where the idele group carries the product of the product topology at the infinite places and the restricted-product topology at the finite places.","title":"The principal ideles are discrete"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","title":"Arakelov divisors"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/abs-norm-units-ne-zero","kind":"node","lean":"FractionalIdeal.absNorm_units_ne_zero","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#abs-norm-units-ne-zero","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $R$ be a Dedekind domain that is a free $\\mathbb{Z}$-module of finite rank, and let $F$ be its field of fractions. For every invertible fractional ideal $I$ of $R$ in $F$, the absolute norm of $I$, a rational number, is nonzero.","title":"Invertible fractional ideals have nonzero absolute norm"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/abs-norm-units-pos","kind":"node","lean":"FractionalIdeal.absNorm_units_pos","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#abs-norm-units-pos","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $R$ be a Dedekind domain that is a free $\\mathbb{Z}$-module of finite rank, and let $F$ be its field of fractions. For every invertible fractional ideal $I$ of $R$ in $F$, the absolute norm of $I$, a rational number, is positive.","title":"Invertible fractional ideals have positive absolute norm"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/arakelov-divisor","kind":"node","lean":"NumberField.ArakelovDivisor","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#arakelov-divisor","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$. An Arakelov divisor on $K$ is a pair $(I, r)$, where $I$ is an invertible fractional ideal of $\\mathcal{O}_K$ in $K$ and $r$ assigns a positive real number $r_w$ to each infinite place $w$ of $K$.","title":"Arakelov divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/archimedean-factor","kind":"node","lean":"NumberField.ArakelovDivisor.archimedeanFactor","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#archimedean-factor","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $D = (I, r)$ be an Arakelov divisor on a number field $K$. Its archimedean factor is the positive real number $\\prod_w r_w^{m_w}$, where the product runs over all infinite places $w$ of $K$ and $m_w$ is $1$ if $w$ is real and $2$ if $w$ is complex.","title":"Archimedean factor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/archimedean-factor-coe","kind":"node","lean":"NumberField.ArakelovDivisor.archimedeanFactor_coe","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#archimedean-factor-coe","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $D = (I, r)$ be an Arakelov divisor on a number field $K$, and for each infinite place $w$ of $K$ let $m_w$ be $1$ if $w$ is real and $2$ if $w$ is complex. Viewed as a real number, the archimedean factor of $D$ equals the real product $\\prod_w r_w^{m_w}$ over all infinite places $w$ of $K$.","title":"Archimedean factor as a real number"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/ideal-norm","kind":"node","lean":"NumberField.ArakelovDivisor.idealNorm","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#ideal-norm","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For an Arakelov divisor $D$ of a number field $K$ with invertible fractional ideal $I$, the ideal norm of $D$ is the absolute norm $N(I)$ of $I$, regarded as a positive real number; it is positive because $I$ is invertible.","title":"Ideal norm of an Arakelov divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/ideal-norm-coe","kind":"node","lean":"NumberField.ArakelovDivisor.idealNorm_coe","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#ideal-norm-coe","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For every Arakelov divisor $D$ of a number field $K$ with fractional ideal $I$, the ideal norm of $D$, viewed as a real number, equals the absolute norm $N(I) \\in \\mathbb{Q}$ of $I$, viewed as a real number.","title":"Real value of the ideal norm"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/arch-hom","kind":"node","lean":"NumberField.ArakelovDivisor.archHom","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#arch-hom","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a number field $K$, the map sending an Arakelov divisor on $K$ to its archimedean factor is a monoid homomorphism from the Arakelov divisors on $K$, multiplied componentwise, to the positive real numbers under multiplication.","title":"Archimedean factor homomorphism"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/ideal-hom","kind":"node","lean":"NumberField.ArakelovDivisor.idealHom","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#ideal-hom","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a number field $K$, the map sending an Arakelov divisor $(I, r)$ on $K$ to the absolute norm of $I$, viewed as a positive real number, is a monoid homomorphism from the Arakelov divisors on $K$, multiplied componentwise, to the positive real numbers under multiplication.","title":"Ideal norm homomorphism"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/size-hom","kind":"node","lean":"NumberField.ArakelovDivisor.sizeHom","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#size-hom","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a number field $K$, the size homomorphism sends an Arakelov divisor $D = (I, (r_w)_w)$, with $I$ an invertible fractional ideal of the ring of integers of $K$ and $r_w > 0$ for each infinite place $w$, to the positive real number $\\prod_w r_w^{m_w} / N(I)$, where $w$ runs over the infinite places of $K$, $m_w$ is $1$ for a real place and $2$ for a complex place, and $N(I)$ is the absolute norm of $I$. It is a\u2026","title":"Size of an Arakelov divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/size-hom-apply","kind":"node","lean":"NumberField.ArakelovDivisor.sizeHom_apply","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#size-hom-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For every Arakelov divisor $D$ of a number field $K$, the size of $D$ equals the archimedean-factor homomorphism evaluated at $D$ divided by the ideal-norm homomorphism evaluated at $D$, the quotient taken in the group of positive reals.","title":"Size as a quotient of homomorphisms"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/size-hom-coe","kind":"node","lean":"NumberField.ArakelovDivisor.sizeHom_coe","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#size-hom-coe","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $D = (I, (r_w)_w)$ be an Arakelov divisor of a number field $K$, where $I$ is an invertible fractional ideal of the ring of integers of $K$ and $r_w > 0$ for each infinite place $w$ of $K$. For each $w$ let $m_w$ be $1$ if $w$ is real and $2$ if $w$ is complex, and let $N(I)$ be the absolute norm of $I$. Then the size of $D$, as a real number, is","title":"Explicit formula for the size of an Arakelov divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/section-set","kind":"node","lean":"NumberField.ArakelovDivisor.sectionSet","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#section-set","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $D$ be an Arakelov divisor of a number field $K$, with invertible fractional ideal $I$ and positive real number $r_w$ at each infinite place $w$ of $K$. The section set of $D$ is the set of $x \\in I$ such that $w(x) \\le r_w$ for every infinite place $w$ of $K$.","title":"Section set of an Arakelov divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/mem-section-set","kind":"node","lean":"NumberField.ArakelovDivisor.mem_sectionSet","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#mem-section-set","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $D$ be an Arakelov divisor of a number field $K$, with invertible fractional ideal $I$ and positive real number $r_w$ at each infinite place $w$ of $K$. An element $x \\in K$ lies in the section set of $D$ if and only if $x \\in I$ and $w(x) \\le r_w$ for every infinite place $w$ of $K$.","title":"Membership in the section set"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/section-set-finite","kind":"node","lean":"NumberField.ArakelovDivisor.sectionSet_finite","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#section-set-finite","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For every Arakelov divisor $D$ of a number field $K$, the section set of $D$ is finite.","title":"Finiteness of section sets"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/principal-arch","kind":"node","lean":"NumberField.principalArch","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#principal-arch","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a nonzero element $x$ of a number field $K$, this is the family that assigns to each infinite place $w$ of $K$ the positive real number $w(x)$.","title":"Archimedean part of a principal divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/principal-divisor","kind":"node","lean":"NumberField.principalDivisor","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#principal-divisor","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a nonzero element $x$ of a number field $K$, the principal Arakelov divisor of $x$ consists of the principal fractional ideal $x\\mathcal{O}_K$ together with the positive real number $w(x)$ at each infinite place $w$ of $K$.","title":"Principal Arakelov divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/coe-principal-divisor-snd","kind":"node","lean":"NumberField.coe_principalDivisor_snd","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#coe-principal-divisor-snd","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $K$ be a number field, $x \\in K$ nonzero and $w$ an infinite place of $K$. The positive real number that the principal Arakelov divisor of $x$ assigns to $w$ is $w(x)$, the absolute value of $x$ at $w$.","title":"Infinite component of a principal divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/coe-principal-divisor-fst","kind":"node","lean":"NumberField.coe_principalDivisor_fst","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#coe-principal-divisor-fst","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$. For every nonzero $x \\in K$, the fractional ideal component of the principal Arakelov divisor of $x$ is the principal fractional ideal $x\\mathcal{O}_K$.","title":"Finite component of a principal divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/principal-hom","kind":"node","lean":"NumberField.principalHom","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#principal-hom","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a number field $K$, the map sending a nonzero element $x \\in K$ to its principal Arakelov divisor is a group homomorphism from the multiplicative group $K^\\times$ to the group of Arakelov divisors of $K$.","title":"Principal divisor homomorphism"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/principal-subgroup","kind":"node","lean":"NumberField.principalSubgroup","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#principal-subgroup","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a number field $K$, the principal subgroup is the image of the principal divisor homomorphism, that is, the subgroup of the group of Arakelov divisors of $K$ consisting of the principal Arakelov divisors of the nonzero elements of $K$.","title":"Subgroup of principal Arakelov divisors"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/degree-zero","kind":"node","lean":"NumberField.degreeZero","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#degree-zero","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a number field $K$, the degree-zero subgroup of the group of Arakelov divisors on $K$ is the kernel of the size homomorphism, that is, the subgroup of Arakelov divisors of size $1$.","title":"Degree-zero Arakelov divisors"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/size-principal","kind":"node","lean":"NumberField.size_principal","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#size-principal","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$. For every nonzero $x \\in K$, the principal Arakelov divisor of $x$, formed by the principal fractional ideal $x\\mathcal{O}_K$ and the values $w(x)$ at the infinite places $w$ of $K$, has size $1$.","title":"Principal Arakelov divisors have size one"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/principal-subgroup-le-degree-zero","kind":"node","lean":"NumberField.principalSubgroup_le_degreeZero","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#principal-subgroup-le-degree-zero","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"For a number field $K$, the subgroup of principal Arakelov divisors of $K$ is contained in the degree-zero subgroup.","title":"Principal Arakelov divisors have degree zero"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/exists-ne-zero-mem-section-set-of-size-g","kind":"node","lean":"NumberField.ArakelovDivisor.exists_ne_zero_mem_sectionSet_of_size_gt","parent":5576,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors/index.html#exists-ne-zero-mem-section-set-of-size-g","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/arakelov-divisors","status":"fully_proved","summary":"Let $K$ be a number field with $s$ complex places and discriminant $d_K$, and let $D$ be an Arakelov divisor on $K$. If the size of $D$ is strictly greater than $(2/\\pi)^s \\sqrt{|d_K|}$, then the section set of $D$ contains a nonzero element of $K$.","title":"Large Arakelov divisors have a nonzero section"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field","status":"fully_proved","title":"Hilbert class field"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field/is-hilbert-class-field","kind":"node","lean":"NumberField.IsHilbertClassField","parent":5602,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field/index.html#is-hilbert-class-field","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field","status":"fully_proved","summary":"Let $K$ and $L$ be number fields with $L$ an extension of $K$. Then $L$ is a Hilbert class field of $K$ if $L/K$ is a finite-dimensional abelian Galois extension, every nonzero prime ideal of the ring of integers $\\mathcal{O}_K$ is unramified in $\\mathcal{O}_L$, every infinite place of $L$ is unramified over $K$, and $L$ is maximal with these properties: every number field $M$ that is a finite-dimensional abelian\u2026","title":"Hilbert class field of a number field"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field/apply-maximal","kind":"node","lean":"NumberField.IsHilbertClassField.apply_maximal","parent":5602,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field/index.html#apply-maximal","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/hilbert-class-field","status":"fully_proved","summary":"Let $K$ and $L$ be number fields with $L$ an extension of $K$ that is a Hilbert class field of $K$. If $M$ is a number field that is a finite-dimensional abelian Galois extension of $K$, such that every nonzero prime ideal of the ring of integers $\\mathcal{O}_K$ is unramified in $\\mathcal{O}_M$ and every infinite place of $M$ is unramified over $K$, then there is a $K$-algebra homomorphism $M \\to L$.","title":"Abelian unramified extensions embed in the Hilbert class field"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","title":"Places and completions of the rationals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/place","kind":"node","lean":"Rat.Place","parent":5605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Places of "],["math","\\(\\mathbb{Q}\\)"]],"source":"Places of $\\mathbb{Q}$"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#place","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"The set $M_{\\mathbb{Q}}$ of places of $\\mathbb{Q}$, which indexes the inequivalent normalised absolute values on $\\mathbb{Q}$. It consists of one infinite place, corresponding to the ordinary archimedean absolute value, and one finite place for each prime number $p$, corresponding to the $p$-adic absolute value.","title":"Places of $\\mathbb{Q}$"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/absolute-value","kind":"node","lean":"Rat.Place.absoluteValue","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#absolute-value","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"For a place $v$ of $\\mathbb{Q}$, the absolute value $|\\cdot|_v : \\mathbb{Q} \\to \\mathbb{R}$ sends $q \\in \\mathbb{Q}$ to the ordinary absolute value $|q|$ when $v$ is the infinite place. When $v$ is the finite place at a prime $p$, it sends $q$ to the normalised $p$-adic absolute value $|q|_p$, where $|0|_p = 0$ and $|q|_p = p^{-v_p(q)}$ for $q \\neq 0$, with $v_p(q)$ the $p$-adic valuation of $q$.","title":"Absolute value at a rational place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/completed-field-bundle","kind":"node","lean":"Rat.Place.CompletedFieldBundle","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#completed-field-bundle","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"A complete normed field bundled with its structure: a type $K$ together with a normed field structure on $K$ for which $K$ is complete with respect to the metric induced by the norm. It is used to package the completed fields $\\mathbb{Q}_v$ of $\\mathbb{Q}$ at its places $v$.","title":"Bundled complete normed field"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/bundle","kind":"node","lean":"Rat.Place.bundle","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#bundle","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"To each place $v$ of $\\mathbb{Q}$ this assigns a complete normed field: the real numbers $\\mathbb{R}$ with the usual absolute value when $v$ is the infinite place, and, when $v$ is the finite place at a prime $p$, the field $\\mathbb{Q}_p$ of $p$-adic numbers with the $p$-adic norm.","title":"Bundled completed field at a rational place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/completed-field","kind":"node","lean":"Rat.Place.CompletedField","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#completed-field","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"For a place $v$ of $\\mathbb{Q}$, the completed field $\\mathbb{Q}_v$ is the field $\\mathbb{R}$ of real numbers when $v$ is the infinite place and, when $v$ is the finite place at a prime $p$, the field $\\mathbb{Q}_p$ of $p$-adic numbers.","title":"Completed field at a rational place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/embedding","kind":"node","lean":"Rat.Place.embedding","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"For a place $v$ of $\\mathbb{Q}$ with completed field $\\mathbb{Q}_v$, the canonical ring homomorphism $\\mathbb{Q} \\to \\mathbb{Q}_v$ sends each rational number to itself, viewed as a real number when $v$ is the infinite place and as a $p$-adic number when $v$ is the finite place at a prime $p$.","title":"Embedding of the rationals into a completion"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/absolute-value-infinite-apply","kind":"node","lean":"Rat.Place.absoluteValue_infinite_apply","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#absolute-value-infinite-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"For every $q \\in \\mathbb{Q}$, the absolute value attached to the infinite place of $\\mathbb{Q}$, evaluated at $q$, equals the ordinary absolute value $|q|$ of $q$ viewed as a real number.","title":"Absolute value at the infinite rational place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/absolute-value-finite-apply","kind":"node","lean":"Rat.Place.absoluteValue_finite_apply","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#absolute-value-finite-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"For every prime $p$ and every $q \\in \\mathbb{Q}$, the absolute value attached to the finite place of $\\mathbb{Q}$ at $p$, evaluated at $q$, equals the $p$-adic absolute value $|q|_p$ of $q$. Here $|0|_p = 0$, and $|q|_p = p^{-v_p(q)}$ for $q \\neq 0$, where $v_p(q)$ is the $p$-adic valuation of $q$.","title":"Absolute value at a finite rational place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/embedding-norm-eq-absolute-value","kind":"node","lean":"Rat.Place.embedding_norm_eq_absoluteValue","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#embedding-norm-eq-absolute-value","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"For every place $v$ of $\\mathbb{Q}$ and every $q \\in \\mathbb{Q}$, the norm in the completed field $\\mathbb{Q}_v$ of the image of $q$ under the canonical embedding $\\mathbb{Q} \\to \\mathbb{Q}_v$ equals $|q|_v$, the absolute value attached to $v$ evaluated at $q$.","title":"Norm of a rational in a completion"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/dense-range-embedding","kind":"node","lean":"Rat.Place.denseRange_embedding","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#dense-range-embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"For every place $v$ of $\\mathbb{Q}$, with completed field $\\mathbb{Q}_v$, the image of $\\mathbb{Q}$ under the canonical embedding $\\mathbb{Q} \\to \\mathbb{Q}_v$ is dense in $\\mathbb{Q}_v$.","title":"The rationals are dense in each completion"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/exists-place-equiv","kind":"node","lean":"Rat.Place.exists_place_equiv","parent":5605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every nontrivial absolute value on "],["math","\\(\\mathbb{Q}\\)"],["text"," is equivalent to that of a place"]],"source":"Every nontrivial absolute value on $\\mathbb{Q}$ is equivalent to that of a place"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#exists-place-equiv","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"Let $f$ be a real-valued absolute value on $\\mathbb{Q}$ that is nontrivial, that is, $f(x) \\ne 1$ for some $x \\ne 0$. Then there is a place $v$ of $\\mathbb{Q}$ such that $f$ is equivalent to the absolute value $|\\cdot|_v$ attached to $v$ (the ordinary absolute value at the infinite place, the $p$-adic absolute value at the prime $p$).","title":"Every nontrivial absolute value on $\\mathbb{Q}$ is equivalent to that of a place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/exists-unique-place-equiv","kind":"node","lean":"Rat.Place.exists_unique_place_equiv","parent":5605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ostrowski's theorem for "],["math","\\(\\mathbb{Q}\\)"],["text",", indexed by places"]],"source":"Ostrowski's theorem for $\\mathbb{Q}$, indexed by places"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#exists-unique-place-equiv","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"Let $f$ be a real-valued absolute value on $\\mathbb{Q}$ that is nontrivial, that is, $f(x) \\ne 1$ for some $x \\ne 0$. Then there is exactly one place $v$ of $\\mathbb{Q}$ such that $f$ is equivalent to the absolute value $|\\cdot|_v$ attached to $v$ (the ordinary absolute value at the infinite place, the $p$-adic absolute value at the prime $p$).","title":"Ostrowski's theorem for $\\mathbb{Q}$, indexed by places"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/with-abs-embedding","kind":"node","lean":"Rat.Place.withAbsEmbedding","parent":5605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Embedding of "],["math","\\(\\mathbb{Q}\\)"],["text",", normed by a place, into its completion"]],"source":"Embedding of $\\mathbb{Q}$, normed by a place, into its completion"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#with-abs-embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"For a place $v$ of $\\mathbb{Q}$, the ring homomorphism from $\\mathbb{Q}$, equipped with the absolute value $|\\cdot|_v$ attached to $v$ as its norm, into the completed field $\\mathbb{Q}_v$ given by the canonical embedding of $\\mathbb{Q}$ ($\\mathbb{Q}_v = \\mathbb{R}$ at the infinite place and $\\mathbb{Q}_v = \\mathbb{Q}_p$ at a prime $p$).","title":"Embedding of $\\mathbb{Q}$, normed by a place, into its completion"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/with-abs-embedding-isometry","kind":"node","lean":"Rat.Place.withAbsEmbedding_isometry","parent":5605,"reading_title":{"format":"inline-parts-v1","parts":[["text","The embedding of "],["math","\\(\\mathbb{Q}\\)"],["text"," into its completion at a place is an isometry"]],"source":"The embedding of $\\mathbb{Q}$ into its completion at a place is an isometry"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#with-abs-embedding-isometry","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"Let $v$ be a place of $\\mathbb{Q}$, let $|\\cdot|_v$ be the absolute value attached to $v$, and let $\\mathbb{Q}_v$ be the completed field at $v$. The canonical embedding of $\\mathbb{Q}$, equipped with the metric $d(x, y) = |x - y|_v$, into $\\mathbb{Q}_v$ is an isometry: for all $x, y \\in \\mathbb{Q}$, the distance between the images of $x$ and $y$ in $\\mathbb{Q}_v$ equals $|x - y|_v$.","title":"The embedding of $\\mathbb{Q}$ into its completion at a place is an isometry"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/dense-range-with-abs-embedding","kind":"node","lean":"Rat.Place.denseRange_withAbsEmbedding","parent":5605,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals/index.html#dense-range-with-abs-embedding","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-and-completions-of-the-rationals","status":"fully_proved","summary":"Let $v$ be a place of $\\mathbb{Q}$, let $|\\cdot|_v$ be the absolute value attached to $v$, let $\\mathbb{Q}_v$ be the completed field at $v$, and write $(\\mathbb{Q}, |\\cdot|_v)$ for the field $\\mathbb{Q}$ equipped with $|\\cdot|_v$. The image of the canonical ring homomorphism $(\\mathbb{Q}, |\\cdot|_v) \\to \\mathbb{Q}_v$, which sends each rational number to itself, is dense in $\\mathbb{Q}_v$.","title":"Density of the rationals with a place's absolute value in the completion"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","title":"Places of a number field"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/normalized-absolute-value","kind":"node","lean":"NumberField.FinitePlace.normalizedAbsoluteValue","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#normalized-absolute-value","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"Let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$ and let $v$ be a finite place of $K$, with absolute value $|\\cdot|_v$. The degree-normalized absolute value at $v$ is the absolute value $x \\mapsto |x|_v^{1/d}$ on $K$.","title":"Degree-normalized absolute value at a finite place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/normalized-absolute-value-apply","kind":"node","lean":"NumberField.FinitePlace.normalizedAbsoluteValue_apply","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#normalized-absolute-value-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"Let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$ and let $v$ be a finite place of $K$, with absolute value $|\\cdot|_v$. For every $x \\in K$, the degree-normalized absolute value of $x$ at $v$ equals $|x|_v^{1/d}$.","title":"Value of the degree-normalized absolute value at a finite place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/normalized-absolute-value-apply-eq-adic","kind":"node","lean":"NumberField.FinitePlace.normalizedAbsoluteValue_apply_eq_adicAbv","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#normalized-absolute-value-apply-eq-adic","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"Let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$, let $v$ be a finite place of $K$, and let $\\mathfrak{p}$ be the nonzero prime ideal of the ring of integers of $K$ that defines $v$. For every $x \\in K$, the degree-normalized absolute value of $x$ at $v$ equals $|x|_{\\mathfrak{p}}^{1/d}$, where $|x|_{\\mathfrak{p}} = N(\\mathfrak{p})^{-v_{\\mathfrak{p}}(x)}$ for $x \\neq 0$ and $|0|_{\\mathfrak{p}} = 0$, with\u2026","title":"Degree-normalized finite absolute value through the prime ideal"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/normalized-absolute-value-2","kind":"node","lean":"NumberField.InfinitePlace.normalizedAbsoluteValue","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#normalized-absolute-value-2","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"Let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$ and let $v$ be an infinite place of $K$, with absolute value $|\\cdot|_v$ and multiplicity $m_v$, equal to $1$ if $v$ is real and $2$ if $v$ is complex. The degree-normalized absolute value at $v$ is the absolute value $x \\mapsto |x|_v^{m_v/d}$ on $K$.","title":"Degree-normalized absolute value at an infinite place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/normalized-absolute-value-apply-2","kind":"node","lean":"NumberField.InfinitePlace.normalizedAbsoluteValue_apply","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#normalized-absolute-value-apply-2","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"Let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$ and let $v$ be an infinite place of $K$, with absolute value $|\\cdot|_v$ and multiplicity $m_v$, equal to $1$ if $v$ is real and $2$ if $v$ is complex. For every $x \\in K$, the degree-normalized absolute value of $x$ at $v$ equals $|x|_v^{m_v/d}$.","title":"Value of the degree-normalized absolute value at an infinite place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/normalized-absolute-value-apply-of-is-re","kind":"node","lean":"NumberField.InfinitePlace.normalizedAbsoluteValue_apply_of_isReal","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#normalized-absolute-value-apply-of-is-re","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"Let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$ and let $v$ be a real infinite place of $K$, with absolute value $|\\cdot|_v$. For every $x \\in K$, the degree-normalized absolute value of $x$ at $v$ equals $|x|_v^{1/d}$.","title":"Degree-normalized absolute value at a real place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/normalized-absolute-value-apply-of-is-co","kind":"node","lean":"NumberField.InfinitePlace.normalizedAbsoluteValue_apply_of_isComplex","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#normalized-absolute-value-apply-of-is-co","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"Let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$ and let $v$ be a complex infinite place of $K$, with absolute value $|\\cdot|_v$. For every $x \\in K$, the degree-normalized absolute value of $x$ at $v$ equals $|x|_v^{2/d}$.","title":"Degree-normalized absolute value at a complex place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/place","kind":"node","lean":"NumberField.Place","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#place","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"For a number field $K$, the places of $K$ form the disjoint union of the infinite places of $K$ (the absolute values coming from complex embeddings of $K$) and the finite places of $K$ (the absolute values coming from nonzero prime ideals of the ring of integers $\\mathcal{O}_K$).","title":"Places of a number field"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/infinite-place","kind":"node","lean":"NumberField.Place.infinitePlace","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#infinite-place","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"For a number field $K$, each infinite place $v$ of $K$ is regarded as a place of $K$ via the inclusion of the infinite places into the disjoint union of the infinite and finite places of $K$.","title":"Infinite places as places"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/finite-place","kind":"node","lean":"NumberField.Place.finitePlace","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#finite-place","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"For a number field $K$, each finite place $v$ of $K$ is regarded as a place of $K$ via the inclusion of the finite places into the disjoint union of the infinite and finite places of $K$.","title":"Finite places as places"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/residue-norm","kind":"node","lean":"NumberField.FinitePlace.residueNorm","parent":5621,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field/index.html#residue-norm","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/places-of-a-number-field","status":"fully_proved","summary":"Let $v$ be a finite place of a number field $K$ and $\\mathfrak{p}_v$ the corresponding nonzero prime ideal of the ring of integers $\\mathcal{O}_K$. The residue norm $N(v)$ is the absolute norm $|\\mathcal{O}_K / \\mathfrak{p}_v|$ of $\\mathfrak{p}_v$, the cardinality of the residue field at $v$, regarded as a real number.","title":"Residue norm of a finite place"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","title":"Ray class groups"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/infinite-finite-part","kind":"node","lean":"NumberField.Modulus.infinite_finitePart","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#infinite-finite-part","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"For a real place $v$ of a number field $K$, let $\\mathfrak m_v$ be the modulus of $K$ with finite part the unit ideal $\\mathcal O_K$ and infinite part $\\{v\\}$. The finite part of $\\mathfrak m_v$ is the unit ideal $\\mathcal O_K$.","title":"Finite part of the modulus of a real place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/infinite-infinite-part","kind":"node","lean":"NumberField.Modulus.infinite_infinitePart","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#infinite-infinite-part","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"For a real place $v$ of a number field $K$, let $\\mathfrak m_v$ be the modulus of $K$ with finite part the unit ideal $\\mathcal O_K$ and infinite part $\\{v\\}$. The infinite part of $\\mathfrak m_v$ is the singleton $\\{v\\}$.","title":"Infinite part of the modulus of a real place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/finite-supported","kind":"node","lean":"NumberField.Modulus.finiteSupported","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#finite-supported","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, with finite part the nonzero ideal $\\mathfrak m_0$ of $\\mathcal O_K$. A nonzero prime ideal $\\mathfrak p$ of $\\mathcal O_K$ lies in the finite support of $\\mathfrak m$ when $\\mathfrak m_0 \\subseteq \\mathfrak p$, that is, when $\\mathfrak p$ divides $\\mathfrak m_0$.","title":"Primes dividing a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/finite-exponent","kind":"node","lean":"NumberField.Modulus.finiteExponent","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#finite-exponent","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, with finite part the nonzero ideal $\\mathfrak m_0$ of $\\mathcal O_K$. For a nonzero prime ideal $\\mathfrak p$ of $\\mathcal O_K$, the finite exponent of $\\mathfrak m$ at $\\mathfrak p$ is the integer $v_{\\mathfrak p}(\\mathfrak m_0)$, the exponent of $\\mathfrak p$ in the prime factorization of $\\mathfrak m_0$.","title":"Exponent of a modulus at a prime"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/coprime-fractional-ideals","kind":"node","lean":"NumberField.Modulus.coprimeFractionalIdeals","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#coprime-fractional-ideals","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, with finite part the nonzero ideal $\\mathfrak m_0$ of $\\mathcal O_K$. For an invertible fractional ideal $I$ of $\\mathcal O_K$ and a nonzero prime ideal $\\mathfrak p$, write $v_{\\mathfrak p}(I)$ for the exponent of $\\mathfrak p$ in the factorization of $I$. The fractional ideals coprime to $\\mathfrak m$ form the subgroup of the group of invertible fractional\u2026","title":"Fractional ideals coprime to a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-elements","kind":"node","lean":"NumberField.Modulus.rayElements","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-elements","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, with finite part the nonzero ideal $\\mathfrak m_0$ of $\\mathcal O_K$. The elements coprime to $\\mathfrak m$ form the subgroup of $K^\\times$ consisting of those $x$ whose principal fractional ideal $(x)$ is coprime to $\\mathfrak m$, that is, has exponent $0$ at every nonzero prime ideal $\\mathfrak p$ of $\\mathcal O_K$ containing $\\mathfrak m_0$.","title":"Elements coprime to a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/congruent-one-at-finite-part","kind":"node","lean":"NumberField.Modulus.CongruentOneAtFinitePart","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#congruent-one-at-finite-part","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, with finite part the nonzero ideal $\\mathfrak m_0$ of $\\mathcal O_K$. For a nonzero prime ideal $\\mathfrak p$ of $\\mathcal O_K$, write $v_{\\mathfrak p}(I)$ for the exponent of $\\mathfrak p$ in the factorization of a nonzero fractional ideal $I$ of $\\mathcal O_K$, and for $y \\in K$ write $v_{\\mathfrak p}(y) = v_{\\mathfrak p}((y))$ (with $v_{\\mathfrak p}(0) = 0$).\u2026","title":"Multiplicative congruence to one at the finite part"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/positive-at-infinite-part","kind":"node","lean":"NumberField.Modulus.PositiveAtInfinitePart","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#positive-at-infinite-part","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, with infinite part the finite set $\\mathfrak m_\\infty$ of real places of $K$. An element $x \\in K^\\times$ is positive at the infinite part of $\\mathfrak m$ when $\\sigma_w(x) > 0$ for every real place $w \\in \\mathfrak m_\\infty$, where $\\sigma_w \\colon K \\to \\mathbb R$ is the real embedding corresponding to $w$.","title":"Positivity at the infinite part of a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/is-ray-element-one","kind":"node","lean":"NumberField.Modulus.IsRayElementOne","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#is-ray-element-one","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, with finite part the nonzero ideal $\\mathfrak m_0$ of $\\mathcal O_K$ and infinite part the finite set $\\mathfrak m_\\infty$ of real places of $K$. For a nonzero prime ideal $\\mathfrak p$ of $\\mathcal O_K$, write $v_{\\mathfrak p}(I)$ for the exponent of $\\mathfrak p$ in the factorization of a nonzero fractional ideal $I$ of $\\mathcal O_K$, and for $y \\in K$ write\u2026","title":"Elements congruent to one modulo a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-generators","kind":"node","lean":"NumberField.Modulus.rayGenerators","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-generators","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. The ray generators of $\\mathfrak m$ are the fractional ideals coprime to $\\mathfrak m$ of the form $(x)$, the principal fractional ideal of an element $x \\in K^\\times$ that is congruent to $1$ modulo $\\mathfrak m$.","title":"Generators of the ray group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-group","kind":"node","lean":"NumberField.Modulus.rayGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. The ray group of $\\mathfrak m$ is the subgroup of the group of fractional ideals coprime to $\\mathfrak m$ generated by the principal ideals $(x)$ of the elements $x \\in K^\\times$ congruent to $1$ modulo $\\mathfrak m$.","title":"The ray group of a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/count-span-singleton-add-ge-of-ne-zero","kind":"node","lean":"NumberField.Modulus.count_spanSingleton_add_ge_of_ne_zero","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#count-span-singleton-add-ge-of-ne-zero","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak p$ a nonzero prime ideal of $\\mathcal O_K$. For $x \\in K$ write $c_{\\mathfrak p}(x)$ for the exponent of $\\mathfrak p$ in the principal fractional ideal $x \\mathcal O_K$, with the convention $c_{\\mathfrak p}(0) = 0$. If $x, y \\in K$ satisfy $x + y \\neq 0$ and $n \\in \\mathbb Z$ satisfies $n \\le c_{\\mathfrak p}(x)$ and $n \\le c_{\\mathfrak p}(y)$, then\u2026","title":"Lower bound for the exponent of a sum"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-element-count-span-singleton-eq-zero","kind":"node","lean":"NumberField.Modulus.rayElement_count_spanSingleton_eq_zero","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-element-count-span-singleton-eq-zero","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ and let $x \\in K^\\times$ be a ray element of $m$. Then for every finite prime $\\mathfrak p$ of $K$ dividing the finite part of $m$, the exponent of $\\mathfrak p$ in the principal fractional ideal $x \\mathcal O_K$ is $0$.","title":"Ray elements have order zero at the primes dividing the modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-element-valuation-eq-one","kind":"node","lean":"NumberField.Modulus.rayElement_valuation_eq_one","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-element-valuation-eq-one","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ and let $x \\in K^\\times$ be a ray element of $m$. Then for every finite prime $\\mathfrak p$ of $K$ dividing the finite part of $m$, the multiplicative $\\mathfrak p$-adic valuation of $x$ equals $1$.","title":"Ray elements have valuation one at the primes dividing the modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/mem-ray-elements-of-valuation-eq-one","kind":"node","lean":"NumberField.Modulus.mem_rayElements_of_valuation_eq_one","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#mem-ray-elements-of-valuation-eq-one","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, and $a \\in K^\\times$. If the multiplicative $\\mathfrak p$-adic valuation of $a$ equals $1$ at every nonzero prime ideal $\\mathfrak p$ of $\\mathcal O_K$ containing $\\mathfrak m_0$, then $a$ is coprime to $\\mathfrak m$, that is, a ray element of $\\mathfrak m$: its principal fractional ideal $(a)$ has exponent $0$ at every such\u2026","title":"Elements of valuation one at the primes dividing a modulus are ray elements"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/mul","kind":"node","lean":"NumberField.Modulus.CongruentOneAtFinitePart.mul","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#mul","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ and let $x, y \\in K^\\times$ both be congruent to $1$ at the finite part of $m$. If moreover $x$ is a ray element of $m$, then $xy$ is congruent to $1$ at the finite part of $m$.","title":"Product of two elements congruent to one at the finite part"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/inv","kind":"node","lean":"NumberField.Modulus.CongruentOneAtFinitePart.inv","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#inv","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$. If a ray element $x$ of $\\mathfrak m$ is congruent to $1$ at the finite part of $\\mathfrak m$, then so is $x^{-1}$.","title":"Inverses preserve congruence to one at the finite part"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/finite-ray-one-elements","kind":"node","lean":"NumberField.Modulus.finiteRayOneElements","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#finite-ray-one-elements","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$. The finite ray-one subgroup of $\\mathfrak m$ is the subgroup of the group of ray elements of $\\mathfrak m$ consisting of the $x$ that are congruent to $1$ at the finite part of $\\mathfrak m$: for every nonzero prime $\\mathfrak p$ of $\\mathcal O_K$ containing $\\mathfrak m_0$, either $x = 1$ or the exponent of $\\mathfrak p$\u2026","title":"The finite ray-one subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/exists-is-unit-mod-finite-part-div-eq","kind":"node","lean":"NumberField.Modulus.exists_isUnit_mod_finitePart_div_eq","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#exists-is-unit-mod-finite-part-div-eq","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, a nonzero ideal of $\\mathcal O_K$. Every ray element $x$ of $\\mathfrak m$ can be written as $x = a / b$ in $K$ with $a, b \\in \\mathcal O_K$ whose residue classes are units of $\\mathcal O_K / \\mathfrak m_0$.","title":"Coprime representation of ray elements"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/quotient-units-div-eq-of-div-eq","kind":"node","lean":"NumberField.Modulus.quotientUnits_div_eq_of_div_eq","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#quotient-units-div-eq-of-div-eq","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ with finite part $\\mathfrak m_0$, an ideal of $\\mathcal O_K$, and write $\\bar x$ for the image of $x \\in \\mathcal O_K$ in $\\mathcal O_K / \\mathfrak m_0$. Let $a, b, c, d \\in \\mathcal O_K$ be such that $\\bar a$, $\\bar b$, $\\bar c$ and $\\bar d$ are units. If $a / b = c / d$ in $K$, then $\\bar a \\bar b^{-1} = \\bar c \\bar d^{-1}$ in the unit group\u2026","title":"Equal fractions give equal residues modulo the finite part"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/finite-residue-map","kind":"node","lean":"NumberField.Modulus.finiteResidueMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#finite-residue-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, a nonzero ideal of $\\mathcal O_K$. The finite residue map of $\\mathfrak m$ is the group homomorphism from the ray elements of $\\mathfrak m$ to $(\\mathcal O_K / \\mathfrak m_0)^\\times$ sending $x$ to $\\bar a \\bar b^{-1}$, where $x = a / b$ with $a, b \\in \\mathcal O_K$ whose residue classes $\\bar a, \\bar b$ are units of\u2026","title":"Finite residue map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/finite-residue-map-apply-of-div-eq","kind":"node","lean":"NumberField.Modulus.finiteResidueMap_apply_of_div_eq","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#finite-residue-map-apply-of-div-eq","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$. Let $x$ be a ray element of $\\mathfrak m$, and let $a, b \\in \\mathcal O_K$ have residue classes $\\bar a, \\bar b$ that are units of $\\mathcal O_K / \\mathfrak m_0$ and satisfy $a / b = x$ in $K$. Then the finite residue map of $\\mathfrak m$ sends $x$ to $\\bar a \\bar b^{-1}$ in $(\\mathcal O_K / \\mathfrak m_0)^\\times$.","title":"Computing the finite residue map from a representation"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/finite-residue-map-eq-one-iff","kind":"node","lean":"NumberField.Modulus.finiteResidueMap_eq_one_iff","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#finite-residue-map-eq-one-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$. A ray element $x$ of $\\mathfrak m$ is sent to $1$ by the finite residue map of $\\mathfrak m$ if and only if $x$ is congruent to $1$ at the finite part of $\\mathfrak m$.","title":"Kernel criterion for the finite residue map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/finite-residue-map-ker","kind":"node","lean":"NumberField.Modulus.finiteResidueMap_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#finite-residue-map-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$. The kernel of the finite residue map of $\\mathfrak m$ is the finite ray-one subgroup of $\\mathfrak m$.","title":"Kernel of the finite residue map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/finite-residue-map-eq-one-of-valuation-l","kind":"node","lean":"NumberField.Modulus.finiteResidueMap_eq_one_of_valuation_le","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#finite-residue-map-eq-one-of-valuation-l","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, and $z$ a ray element of $\\mathfrak m$. Suppose that at every nonzero prime ideal $\\mathfrak p$ of $\\mathcal O_K$ containing $\\mathfrak m_0$, the multiplicative $\\mathfrak p$-adic valuation of $z - 1$ is at most $\\exp(-e_{\\mathfrak p})$, where $e_{\\mathfrak p}$ is the exponent of $\\mathfrak p$ in $\\mathfrak m_0$;\u2026","title":"Valuation criterion for the finite residue map to be trivial"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-group","kind":"node","lean":"NumberField.Modulus.RayClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"The ray class group of a modulus $\\mathfrak m$ of a number field $K$ is the quotient of the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$ by the ray group of $\\mathfrak m$.","title":"Ray class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-map","kind":"node","lean":"NumberField.Modulus.rayClassMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"For a modulus $\\mathfrak m$ of a number field $K$, the ray class map is the canonical quotient homomorphism from the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$ onto the ray class group of $\\mathfrak m$, sending each such ideal to its class modulo the ray group of $\\mathfrak m$.","title":"Ray class map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/sign-at-real-place","kind":"node","lean":"NumberField.Modulus.signAtRealPlace","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#sign-at-real-place","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"For a real place $w$ of a number field $K$, let $\\sigma_w \\colon K \\to \\mathbb{R}$ be the corresponding real embedding. The sign at $w$ is the group homomorphism from $K^\\times$ to $\\mathbb{Z}/2\\mathbb{Z}$, written multiplicatively, that sends $x$ to the class of $0$ (the identity) if $\\sigma_w(x) > 0$ and to the class of $1$ otherwise.","title":"Sign of a unit at a real place"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/sign-at-real-place-apply-eq-one-iff","kind":"node","lean":"NumberField.Modulus.signAtRealPlace_apply_eq_one_iff","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#sign-at-real-place-apply-eq-one-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $w$ be a real place of a number field $K$ with real embedding $\\sigma_w \\colon K \\to \\mathbb{R}$. For every $x \\in K^\\times$, the sign of $x$ at $w$, an element of $\\mathbb{Z}/2\\mathbb{Z}$ written multiplicatively, is the identity if and only if $\\sigma_w(x) > 0$.","title":"Trivial sign at a real place means positivity"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/infinite-sign-map","kind":"node","lean":"NumberField.Modulus.infiniteSignMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#infinite-sign-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ and let $m_\\infty$ be its finite set of real places. The infinite sign map of $m$ is the group homomorphism from the ray elements of $m$ to the group of functions $m_\\infty \\to \\mathbb{Z}/2\\mathbb{Z}$, with $\\mathbb{Z}/2\\mathbb{Z}$ written multiplicatively and the pointwise product, that sends $x$ to the function whose value at $w \\in m_\\infty$ is the sign of $x$ at the\u2026","title":"Infinite sign map of a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/infinite-sign-map-apply","kind":"node","lean":"NumberField.Modulus.infiniteSignMap_apply","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#infinite-sign-map-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ with finite set of real places $m_\\infty$. For every ray element $x$ of $m$ and every $w \\in m_\\infty$, the value at $w$ of the infinite sign map of $m$ applied to $x$ is the sign of $x$ at the real place $w$.","title":"Coordinates of the infinite sign map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/infinite-sign-map-eq-one-iff","kind":"node","lean":"NumberField.Modulus.infiniteSignMap_eq_one_iff","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#infinite-sign-map-eq-one-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ with finite set of real places $m_\\infty$, and for a real place $w$ let $\\sigma_w$ be the corresponding real embedding of $K$. For every ray element $x$ of $m$, the infinite sign map of $m$ sends $x$ to the identity if and only if $x$ is positive at the infinite part of $m$, that is, $\\sigma_w(x) > 0$ for every $w \\in m_\\infty$.","title":"Trivial infinite sign means positivity at the modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/positive-ray-elements","kind":"node","lean":"NumberField.Modulus.positiveRayElements","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#positive-ray-elements","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ with finite set of real places $m_\\infty$, and for a real place $w$ let $\\sigma_w$ be the real embedding of $K$ at $w$. The positive ray elements of $m$ form the subgroup of the ray elements of $m$ defined as the kernel of the infinite sign map of $m$; equivalently, they are the ray elements $x$ with $\\sigma_w(x) > 0$ for every $w \\in m_\\infty$.","title":"Ray elements positive at the infinite part"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/mem-positive-ray-elements-iff","kind":"node","lean":"NumberField.Modulus.mem_positiveRayElements_iff","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#mem-positive-ray-elements-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ with finite set of real places $m_\\infty$, and for a real place $w$ let $\\sigma_w$ be the real embedding of $K$ at $w$. A ray element $x$ of $m$ is a positive ray element of $m$ if and only if $x$ is positive at the infinite part of $m$, that is, $\\sigma_w(x) > 0$ for every $w \\in m_\\infty$.","title":"Membership in the positive ray elements"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/exists-ne-zero-with-target-and-sign","kind":"node","lean":"NumberField.Modulus.exists_ne_zero_with_target_and_sign","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#exists-ne-zero-with-target-and-sign","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$, with finite part $\\mathfrak m_0$ (a nonzero ideal of $\\mathcal O_K$) and infinite part $\\mathfrak m_\\infty$ (a finite set of real places of $K$). For a nonzero prime $\\mathfrak p$ of $\\mathcal O_K$ write $v_{\\mathfrak p}$ for the $\\mathfrak p$-adic valuation on $K$ (with $v_{\\mathfrak p}(0) = \\infty$) and $e_{\\mathfrak p}$ for the exponent of\u2026","title":"Weak approximation with a finite target and prescribed signs"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/exists-ne-zero-with-valuation-one-and-si","kind":"node","lean":"NumberField.Modulus.exists_ne_zero_with_valuation_one_and_sign","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#exists-ne-zero-with-valuation-one-and-si","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$, with finite part $\\mathfrak m_0$ (a nonzero ideal of $\\mathcal O_K$) and infinite part $\\mathfrak m_\\infty$ (a finite set of real places of $K$); for a real place $w$ write $\\sigma_w : K \\to \\mathbb R$ for its real embedding. For every set $P$ of places in $\\mathfrak m_\\infty$ there is a nonzero $x \\in K$ whose $\\mathfrak p$-adic valuation is $0$ for\u2026","title":"Weak approximation with prescribed signs at a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-target","kind":"node","lean":"NumberField.Modulus.ResidueSignTarget","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-target","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$, with finite part the nonzero ideal $\\mathfrak{m}_0$ of the ring of integers $\\mathcal{O}_K$ and infinite part the finite set $S$ of real places of $K$. Let $G$ denote $\\mathbb{Z}/2\\mathbb{Z}$ written multiplicatively, let $G^S$ denote the group of functions from $S$ to $G$ (one sign for each place in $S$), and let $(\\mathcal{O}_K/\\mathfrak{m}_0)^\\times$ denote the unit\u2026","title":"Residue-sign target"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-map","kind":"node","lean":"NumberField.Modulus.residueSignMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. The residue-sign map of $m$ is the group homomorphism from the ray elements of $m$ to the residue-sign target of $m$ that sends a ray element $x$ to the pair formed by the image of $x$ under the infinite sign map of $m$ and the image of $x$ under the finite residue map of $m$.","title":"Residue-sign map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-map-apply","kind":"node","lean":"NumberField.Modulus.residueSignMap_apply","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-map-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. For every ray element $x$ of $m$, the image of $x$ under the residue-sign map of $m$ is the pair whose first component is the image of $x$ under the infinite sign map of $m$ and whose second component is the image of $x$ under the finite residue map of $m$.","title":"Components of the residue-sign map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-one-elements","kind":"node","lean":"NumberField.Modulus.rayOneElements","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-one-elements","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. The ray-one subgroup of $m$ is the subgroup of the ray elements of $m$ consisting of the ray elements $x$ that lie in the finite ray-one subgroup of $m$ (those congruent to $1$ at the finite part of $m$) and in the subgroup of ray elements positive at every real place in the infinite part of $m$.","title":"Ray-one subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/mem-ray-one-elements-iff","kind":"node","lean":"NumberField.Modulus.mem_rayOneElements_iff","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#mem-ray-one-elements-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. A ray element $x$ of $m$ lies in the ray-one subgroup of $m$ if and only if $x$ is congruent to $1$ at the finite part of $m$ and $x$ is positive at every real place in the infinite part of $m$.","title":"Membership in the ray-one subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-map-surjective","kind":"node","lean":"NumberField.Modulus.residueSignMap_surjective","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-map-surjective","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. Every element of the residue-sign target of $m$ is the image of some ray element of $m$ under the residue-sign map of $m$.","title":"Surjectivity of the residue-sign map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-map-ker","kind":"node","lean":"NumberField.Modulus.residueSignMap_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-map-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. The kernel of the residue-sign map of $m$ is the ray-one subgroup of $m$.","title":"Kernel of the residue-sign map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-map-eq-one-iff","kind":"node","lean":"NumberField.Modulus.residueSignMap_eq_one_iff","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-map-eq-one-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. A ray element $x$ of $m$ is sent to $1$ by the residue-sign map of $m$ if and only if $x$ is a ray element congruent to $1$ modulo $m$: $x$ is a ray element of $m$, $x$ is congruent to $1$ at the finite part of $m$, and $x$ is positive at every real place in the infinite part of $m$.","title":"Kernel criterion for the residue-sign map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-elements-quotient-equiv-residue-sign","kind":"node","lean":"NumberField.Modulus.rayElementsQuotientEquivResidueSign","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-elements-quotient-equiv-residue-sign","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. The residue-sign isomorphism of $m$ is the group isomorphism from the quotient of the ray elements of $m$ by the ray-one subgroup of $m$ onto the residue-sign target of $m$ induced by the residue-sign map, which is surjective with kernel the ray-one subgroup; it sends the class of a ray element $x$ to the image of $x$ under the residue-sign map.","title":"The residue-sign isomorphism"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-elements-quotient-equiv-residue-sign-2","kind":"node","lean":"NumberField.Modulus.rayElementsQuotientEquivResidueSign_mk","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-elements-quotient-equiv-residue-sign-2","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. For every ray element $x$ of $m$, the residue-sign isomorphism of $m$ sends the class of $x$ in the quotient of the ray elements of $m$ by the ray-one subgroup to the image of $x$ under the residue-sign map of $m$.","title":"The residue-sign isomorphism on classes"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-field-units","kind":"node","lean":"NumberField.Modulus.integralUnitsToFieldUnits","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-field-units","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a field and $\\mathcal{O}_K$ its ring of integers, the integral closure of $\\mathbb{Z}$ in $K$. This is the group homomorphism $\\mathcal{O}_K^\\times \\to K^\\times$ induced by the inclusion $\\mathcal{O}_K \\subseteq K$: it sends a unit of $\\mathcal{O}_K$ to the same element regarded as a unit of $K$.","title":"Embedding of integral units into field units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-field-units-injective","kind":"node","lean":"NumberField.Modulus.integralUnitsToFieldUnits_injective","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-field-units-injective","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a field and $\\mathcal{O}_K$ its ring of integers, the integral closure of $\\mathbb{Z}$ in $K$. The group homomorphism $\\mathcal{O}_K^\\times \\to K^\\times$ induced by the inclusion $\\mathcal{O}_K \\subseteq K$ is injective.","title":"Integral units embed injectively into field units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-ray-elements","kind":"node","lean":"NumberField.Modulus.integralUnitsToRayElements","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-ray-elements","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$, let $\\mathfrak{m}$ be a modulus of $K$, and let $K_{\\mathfrak{m}} \\subseteq K^\\times$ be the group of ray elements for $\\mathfrak{m}$. Every unit of $\\mathcal{O}_K$ has $\\mathfrak{p}$-adic valuation $0$ at every prime $\\mathfrak{p}$ of $\\mathcal{O}_K$, so it lies in $K_{\\mathfrak{m}}$. This is the group homomorphism\u2026","title":"Integral units as ray elements"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-ray-elements-coe","kind":"node","lean":"NumberField.Modulus.integralUnitsToRayElements_coe","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-ray-elements-coe","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$, let $\\mathfrak{m}$ be a modulus of $K$, and let $u \\in \\mathcal{O}_K^\\times$. The image of $u$ in the group of ray elements for $\\mathfrak{m}$, regarded as an element of $K^\\times$, equals the image of $u$ under the embedding $\\mathcal{O}_K^\\times \\to K^\\times$.","title":"Ray-element image of an integral unit as a field unit"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-ray-elements-injective","kind":"node","lean":"NumberField.Modulus.integralUnitsToRayElements_injective","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-ray-elements-injective","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ and let $\\mathfrak{m}$ be a modulus of $K$. The homomorphism from $\\mathcal{O}_K^\\times$ to the group of ray elements for $\\mathfrak{m}$ that sends a unit to its image in $K^\\times$ is injective.","title":"Integral units embed injectively into the ray elements"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-ray-one-units","kind":"node","lean":"NumberField.Modulus.integralRayOneUnits","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-ray-one-units","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$, let $\\mathfrak{m}$ be a modulus of $K$ with finite part $\\mathfrak{m}_0$, and send each unit of $\\mathcal{O}_K$ to its image in the group of ray elements for $\\mathfrak{m}$. For a nonzero prime $\\mathfrak{p}$ of $\\mathcal{O}_K$, write $v_{\\mathfrak{p}}$ for the $\\mathfrak{p}$-adic valuation of nonzero elements and nonzero ideals. The ray-one integral\u2026","title":"Ray-one integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/mem-integral-ray-one-units-iff","kind":"node","lean":"NumberField.Modulus.mem_integralRayOneUnits_iff","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#mem-integral-ray-one-units-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ with finite part $\\mathfrak m_0$ and set of real places $\\mathfrak m_\\infty$, and for a nonzero prime $\\mathfrak p$ of the ring of integers $\\mathcal O_K$ write $v_{\\mathfrak p}$ for the exponent of $\\mathfrak p$ in a fractional ideal of $K$, applied to an element through its principal fractional ideal. A unit $u$ of $\\mathcal O_K$ lies in the subgroup of ray-one\u2026","title":"Membership in the ray-one integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-ray-one-units-inclusion","kind":"node","lean":"NumberField.Modulus.integralRayOneUnitsInclusion","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-ray-one-units-inclusion","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ and let $\\mathfrak{m}$ be a modulus of $K$. The inclusion of the ray-one integral units for $\\mathfrak{m}$ is the group homomorphism from the subgroup of ray-one integral units for $\\mathfrak{m}$ to $\\mathcal{O}_K^\\times$ that sends each such unit to itself.","title":"Inclusion of the ray-one integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-ray-quotient","kind":"node","lean":"NumberField.Modulus.integralUnitsToRayQuotient","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-ray-quotient","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $K_{\\mathfrak m} \\le K^\\times$ be the group of ray elements (elements whose principal fractional ideal is coprime to the finite part of $\\mathfrak m$) and let $K_{\\mathfrak m,1} \\le K_{\\mathfrak m}$ be the ray-one subgroup. This is the group homomorphism $\\mathcal O_K^\\times \\to K_{\\mathfrak m}/K_{\\mathfrak m,1}$ sending a unit $u$ of the ring of integers\u2026","title":"Map from integral units to the ray quotient"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-ray-quotient-apply","kind":"node","lean":"NumberField.Modulus.integralUnitsToRayQuotient_apply","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-ray-quotient-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $K_{\\mathfrak m} \\le K^\\times$ be the group of ray elements (elements whose principal fractional ideal is coprime to the finite part of $\\mathfrak m$) and let $K_{\\mathfrak m,1} \\le K_{\\mathfrak m}$ be the ray-one subgroup. For every unit $u$ of the ring of integers $\\mathcal O_K$, the map $\\mathcal O_K^\\times \\to K_{\\mathfrak m}/K_{\\mathfrak m,1}$ sends $u$\u2026","title":"Value of the ray-quotient map on integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-residue-sign","kind":"node","lean":"NumberField.Modulus.integralUnitsToResidueSign","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-residue-sign","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ with finite part $\\mathfrak m_0$ and set of real places $\\mathfrak m_\\infty$, let $K_{\\mathfrak m} \\le K^\\times$ be the group of ray elements and let $K_{\\mathfrak m,1} \\le K_{\\mathfrak m}$ be the ray-one subgroup. The residue-sign map on integral units is the group homomorphism from $\\mathcal O_K^\\times$ to\u2026","title":"Residue-sign map on integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-residue-sign-apply","kind":"node","lean":"NumberField.Modulus.integralUnitsToResidueSign_apply","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-residue-sign-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ and let $K_{\\mathfrak m} \\le K^\\times$ be the group of ray elements. For every unit $u$ of the ring of integers $\\mathcal O_K$, the residue-sign map on integral units sends $u$ to the image of $u$, viewed as an element of $K_{\\mathfrak m}$, under the residue-sign homomorphism on $K_{\\mathfrak m}$.","title":"Value of the residue-sign map on integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-ray-quotient-ker","kind":"node","lean":"NumberField.Modulus.integralUnitsToRayQuotient_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-ray-quotient-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $K_{\\mathfrak m} \\le K^\\times$ be the group of ray elements and $K_{\\mathfrak m,1} \\le K_{\\mathfrak m}$ the ray-one subgroup. The kernel of the homomorphism $\\mathcal O_K^\\times \\to K_{\\mathfrak m}/K_{\\mathfrak m,1}$ sending a unit of the ring of integers $\\mathcal O_K$ to its coset is the subgroup of ray-one integral units, the units of $\\mathcal O_K$ that\u2026","title":"Kernel of the ray-quotient map on integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-residue-sign-ker","kind":"node","lean":"NumberField.Modulus.integralUnitsToResidueSign_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-residue-sign-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. The kernel of the residue-sign map on the unit group $\\mathcal O_K^\\times$ of the ring of integers is the subgroup of ray-one integral units, the units of $\\mathcal O_K$ whose image in the group of ray elements modulo $\\mathfrak m$ lies in its ray-one subgroup.","title":"Kernel of the residue-sign map on integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-ray-one-units-inclusion-injecti","kind":"node","lean":"NumberField.Modulus.integralRayOneUnitsInclusion_injective","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-ray-one-units-inclusion-injecti","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ and let $\\mathfrak{m}$ be a modulus of $K$. The inclusion homomorphism from the subgroup of ray-one integral units for $\\mathfrak{m}$ into the unit group $\\mathcal{O}_K^\\times$ is injective.","title":"The inclusion of the ray-one integral units is injective"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-mul-exact-quotient","kind":"node","lean":"NumberField.Modulus.integralUnits_mulExact_quotient","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-mul-exact-quotient","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ and let $\\mathfrak{m}$ be a modulus of $K$. Write $K_{\\mathfrak{m}}$ for the group of ray elements for $\\mathfrak{m}$, $K_{\\mathfrak{m},1}$ for its ray-one subgroup, and $U_{\\mathfrak{m},1} \\subseteq \\mathcal{O}_K^\\times$ for the subgroup of ray-one integral units. Consider the sequence","title":"Exactness of the ray sequence at the integral units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-mul-exact-residue-sign","kind":"node","lean":"NumberField.Modulus.integralUnits_mulExact_residueSign","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-mul-exact-residue-sign","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ and let $\\mathfrak{m}$ be a modulus of $K$ with finite part $\\mathfrak{m}_0$. Let $T_{\\mathfrak{m}}$ be the residue-sign target of $\\mathfrak{m}$: the product of a copy of $\\mathbb{Z}/2\\mathbb{Z}$, written multiplicatively, for each real place in the infinite part of $\\mathfrak{m}$, with $(\\mathcal{O}_K/\\mathfrak{m}_0)^\\times$. Let\u2026","title":"Exactness at the integral units, residue-sign form"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/principal-ray-ideal-map","kind":"node","lean":"NumberField.Modulus.principalRayIdealMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#principal-ray-ideal-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ with ring of integers $\\mathcal O_K$. The principal ray ideal map is the group homomorphism from the group of ray elements for $\\mathfrak m$ (the $x \\in K^\\times$ whose principal ideal is coprime to $\\mathfrak m$) to the group of fractional ideals coprime to $\\mathfrak m$ that sends $x$ to the principal fractional ideal $x\\mathcal O_K$.","title":"Principal ray ideal map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/principal-ray-ideal-map-apply","kind":"node","lean":"NumberField.Modulus.principalRayIdealMap_apply","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#principal-ray-ideal-map-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ with ring of integers $\\mathcal O_K$. For every ray element $x \\in K^\\times$ for $\\mathfrak m$, the image of $x$ under the principal ray ideal map, viewed as an invertible fractional ideal of $K$, is the principal fractional ideal $x\\mathcal O_K$.","title":"Value of the principal ray ideal map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/to-principal-ideal-eq-iff-exists-integra","kind":"node","lean":"NumberField.Modulus.toPrincipalIdeal_eq_iff_exists_integralUnit_mul","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#to-principal-ideal-eq-iff-exists-integra","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal O_K$. For all $x, y \\in K^\\times$, the principal fractional ideals $x\\mathcal O_K$ and $y\\mathcal O_K$ are equal if and only if $x = uy$ for some unit $u \\in \\mathcal O_K^\\times$.","title":"Principal ideals agree exactly up to an integral unit"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-group-eq-map-ray-one-elements","kind":"node","lean":"NumberField.Modulus.rayGroup_eq_map_rayOneElements","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-group-eq-map-ray-one-elements","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. The ray group of $\\mathfrak m$ equals the image under the principal ray ideal map of the subgroup of ray-one elements, that is, of the ray elements for $\\mathfrak m$ that are congruent to $1$ modulo the finite part of $\\mathfrak m$ and positive at every real place in its infinite part.","title":"The ray group is the image of the ray-one elements"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-element-class-map","kind":"node","lean":"NumberField.Modulus.rayElementClassMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-element-class-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"For a modulus $\\mathfrak m$ of a number field $K$, the ray-element class map is the group homomorphism from the group of ray elements of $\\mathfrak m$ to the ray class group $\\mathrm{Cl}_{\\mathfrak m}$ that sends $x$ to the ray class of its principal fractional ideal $(x)$, which is coprime to $\\mathfrak m$; that is, the principal-ideal map followed by the projection onto the ray class group.","title":"Ray-element class map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-element-class-map-apply","kind":"node","lean":"NumberField.Modulus.rayElementClassMap_apply","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-element-class-map-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ and let $\\mathrm{Cl}_{\\mathfrak m}$ be its ray class group. For every ray element $x$ of $\\mathfrak m$, the ray-element class map sends $x$ to the ray class in $\\mathrm{Cl}_{\\mathfrak m}$ of the principal fractional ideal $(x)$, viewed as a fractional ideal coprime to $\\mathfrak m$.","title":"Value of the ray-element class map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-one-elements-le-ray-element-class-ma","kind":"node","lean":"NumberField.Modulus.rayOneElements_le_rayElementClassMap_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-one-elements-le-ray-element-class-ma","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. Every ray-one element of $\\mathfrak m$ lies in the kernel of the ray-element class map; that is, its principal fractional ideal has trivial class in the ray class group $\\mathrm{Cl}_{\\mathfrak m}$.","title":"Ray-one elements have trivial ray class"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-quotient-to-ray-class-group","kind":"node","lean":"NumberField.Modulus.rayQuotientToRayClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-quotient-to-ray-class-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"For a modulus $\\mathfrak m$ of a number field $K$, the quotient-to-ray-class map is the group homomorphism from the quotient of the group of ray elements of $\\mathfrak m$ by the subgroup of ray-one elements to the ray class group $\\mathrm{Cl}_{\\mathfrak m}$ induced by the ray-element class map: the class of a ray element $x$ goes to the ray class of its principal fractional ideal $(x)$. It is well defined because\u2026","title":"Quotient-to-ray-class map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-quotient-to-ray-class-group-mk","kind":"node","lean":"NumberField.Modulus.rayQuotientToRayClassGroup_mk","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-quotient-to-ray-class-group-mk","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. For every ray element $x$ of $\\mathfrak m$, the quotient-to-ray-class map sends the class of $x$ modulo the ray-one elements to the image of $x$ under the ray-element class map.","title":"Quotient-to-ray-class map on representatives"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/principal-ray-ideal-map-integral-units-t","kind":"node","lean":"NumberField.Modulus.principalRayIdealMap_integralUnitsToRayElements","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#principal-ray-ideal-map-integral-units-t","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. For every unit $u \\in \\mathcal O_K^\\times$, viewed as a ray element of $\\mathfrak m$, the principal fractional ideal $(u)$, as an element of the group of fractional ideals coprime to $\\mathfrak m$, is the identity element $\\mathcal O_K$.","title":"Integral units generate the unit ideal"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-quotient-to-ray-class-group-ker","kind":"node","lean":"NumberField.Modulus.rayQuotientToRayClassGroup_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-quotient-to-ray-class-group-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. The kernel of the quotient-to-ray-class map, from the quotient of the ray elements of $\\mathfrak m$ by the ray-one elements to the ray class group $\\mathrm{Cl}_{\\mathfrak m}$, equals the image of the map sending each unit $u \\in \\mathcal O_K^\\times$ to the class of $u$ in that quotient.","title":"Exactness at the ray-element quotient (kernel form)"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-ray-quotient-mul-exact","kind":"node","lean":"NumberField.Modulus.integralUnitsToRayQuotient_mulExact_rayQuotientToRayClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-ray-quotient-mul-exact","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $K_{\\mathfrak m}$ be the group of ray elements (elements of $K^\\times$ whose principal ideal is coprime to the finite part of $\\mathfrak m$), let $K_{\\mathfrak m,1} \\le K_{\\mathfrak m}$ be the subgroup of ray-one elements, and let $\\mathrm{Cl}_{\\mathfrak m}$ be the ray class group of $\\mathfrak m$. In the sequence\u2026","title":"Exactness at the ray-element quotient"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-to-ray-class-group","kind":"node","lean":"NumberField.Modulus.residueSignToRayClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-to-ray-class-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ with finite part $\\mathfrak m_0$, let $K_{\\mathfrak m}$ be the group of elements of $K^\\times$ whose principal ideal is coprime to $\\mathfrak m_0$, and let $K_{\\mathfrak m,1}$ be its subgroup of elements congruent to $1$ modulo $\\mathfrak m_0$ and positive at every real place of $\\mathfrak m$. Let $T_{\\mathfrak m}$ be the residue-sign target of $\\mathfrak m$ and\u2026","title":"Ray class map on residues and signs"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-to-ray-class-group-equiv","kind":"node","lean":"NumberField.Modulus.residueSignToRayClassGroup_equiv","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-to-ray-class-group-equiv","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ with finite part $\\mathfrak m_0$, let $K_{\\mathfrak m}$ be the group of elements of $K^\\times$ whose principal ideal is coprime to $\\mathfrak m_0$, and let $K_{\\mathfrak m,1}$ be its subgroup of elements congruent to $1$ modulo $\\mathfrak m_0$ and positive at every real place of $\\mathfrak m$. Let $T_{\\mathfrak m}$ be the residue-sign target of $\\mathfrak m$, let\u2026","title":"Residue-sign-to-ray-class map through the quotient isomorphism"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-to-ray-class-group-residue","kind":"node","lean":"NumberField.Modulus.residueSignToRayClassGroup_residueSignMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-to-ray-class-group-residue","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ and let $x$ be a ray element of $\\mathfrak m$, that is, $x \\in K^\\times$ with principal ideal coprime to the finite part of $\\mathfrak m$. Then the residue-sign-to-ray-class map sends the residue-sign value of $x$ (its signs at the real places of $\\mathfrak m$ together with its residue modulo the finite part) to the ray class of the principal ideal $(x)$.","title":"Ray class map on residue-sign values"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-to-ray-class-group-comp-res","kind":"node","lean":"NumberField.Modulus.residueSignToRayClassGroup_comp_residueSignMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-to-ray-class-group-comp-res","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. The residue-sign map from the ray elements of $\\mathfrak m$ to the residue-sign target, followed by the transported residue-sign-to-ray-class map, equals the ray-element class map.","title":"The residue-sign transport square commutes"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-to-ray-class-group-ker","kind":"node","lean":"NumberField.Modulus.residueSignToRayClassGroup_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-to-ray-class-group-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. The kernel of the residue-sign-to-ray-class map from the residue-sign target of $\\mathfrak m$ to the ray class group of $\\mathfrak m$ equals the image of the homomorphism $\\mathcal O_K^\\times \\to$ (residue-sign target) that sends a unit $u$ to its residue-sign value, namely its signs at the real places of $\\mathfrak m$ together with its residue modulo the finite\u2026","title":"Kernel of the ray class map on residue-sign values"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/integral-units-to-residue-sign-mul-exact","kind":"node","lean":"NumberField.Modulus.integralUnitsToResidueSign_mulExact_residueSignToRayClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#integral-units-to-residue-sign-mul-exact","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ with finite part $\\mathfrak m_0$ and set of real places $\\mathfrak m_\\infty$, let $\\mathrm{Cl}_{\\mathfrak m}$ be its ray class group, and let $R_{\\mathfrak m} = \\{\\pm 1\\}^{\\mathfrak m_\\infty} \\times (\\mathcal O_K/\\mathfrak m_0)^\\times$. Consider the maps $\\mathcal O_K^\\times \\to R_{\\mathfrak m} \\to \\mathrm{Cl}_{\\mathfrak m}$. The first sends a unit $u$ to its\u2026","title":"Exactness at the residue-sign group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/coprime-ideal-class-map","kind":"node","lean":"NumberField.Modulus.coprimeIdealClassMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#coprime-ideal-class-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$, and $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to the finite part of $\\mathfrak m$. The coprime ideal-class map is the group homomorphism $I_{\\mathfrak m} \\to \\mathrm{Cl}(\\mathcal O_K)$ that sends $I$ to its ideal class $[I]$: the inclusion of $I_{\\mathfrak m}$ into the group of nonzero fractional ideals followed by\u2026","title":"Ideal-class map on coprime fractional ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/coprime-ideal-class-map-apply","kind":"node","lean":"NumberField.Modulus.coprimeIdealClassMap_apply","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#coprime-ideal-class-map-apply","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$, and $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to the finite part of $\\mathfrak m$. For every $I \\in I_{\\mathfrak m}$, the coprime ideal-class map sends $I$ to the class $[I]$ in the class group $\\mathrm{Cl}(\\mathcal O_K)$ of $I$ viewed as a nonzero fractional ideal.","title":"Value of the ideal-class map on a coprime ideal"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-group-le-coprime-ideal-class-map-ker","kind":"node","lean":"NumberField.Modulus.rayGroup_le_coprimeIdealClassMap_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-group-le-coprime-ideal-class-map-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$, $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to the finite part of $\\mathfrak m$, and $P_{\\mathfrak m} \\subseteq I_{\\mathfrak m}$ the ray group. Then $P_{\\mathfrak m}$ is contained in the kernel of the coprime ideal-class map $I_{\\mathfrak m} \\to \\mathrm{Cl}(\\mathcal O_K)$, $I \\mapsto [I]$: every ideal in\u2026","title":"The ray group lies in the kernel of the ideal-class map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-to-class-group","kind":"node","lean":"NumberField.Modulus.rayClassToClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-to-class-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$, $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to the finite part of $\\mathfrak m$, $P_{\\mathfrak m}$ the ray group, and $\\mathrm{Cl}_{\\mathfrak m} = I_{\\mathfrak m}/P_{\\mathfrak m}$ the ray class group. The projection $\\mathrm{Cl}_{\\mathfrak m} \\to \\mathrm{Cl}(\\mathcal O_K)$ is the group homomorphism induced by the\u2026","title":"Projection from the ray class group to the class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-to-class-group-ray-class-map","kind":"node","lean":"NumberField.Modulus.rayClassToClassGroup_rayClassMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-to-class-group-ray-class-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$, $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to the finite part of $\\mathfrak m$, and $\\mathrm{Cl}_{\\mathfrak m}$ the ray class group. For every $I \\in I_{\\mathfrak m}$, the projection $\\mathrm{Cl}_{\\mathfrak m} \\to \\mathrm{Cl}(\\mathcal O_K)$ sends the ray class of $I$ to $[I]$, the value of the coprime ideal-class map\u2026","title":"Projection to the class group on ray class representatives"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-to-class-group-comp-ray-class","kind":"node","lean":"NumberField.Modulus.rayClassToClassGroup_comp_rayClassMap","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-to-class-group-comp-ray-class","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$, $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to the finite part of $\\mathfrak m$, $P_{\\mathfrak m} \\subseteq I_{\\mathfrak m}$ the ray group, and $\\mathrm{Cl}_{\\mathfrak m} = I_{\\mathfrak m}/P_{\\mathfrak m}$ the ray class group. As group homomorphisms, the quotient map $I_{\\mathfrak m} \\to \\mathrm{Cl}_{\\mathfrak m}$\u2026","title":"The ideal-class map factors through the ray class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-to-class-group-ray-quotient-to","kind":"node","lean":"NumberField.Modulus.rayClassToClassGroup_rayQuotientToRayClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-to-class-group-ray-quotient-to","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to $\\mathfrak m_0$, $\\mathrm{Cl}_{\\mathfrak m}$ the ray class group, $K_{\\mathfrak m}$ the group of $x \\in K^\\times$ with $x\\mathcal O_K \\in I_{\\mathfrak m}$, and $K_{\\mathfrak m,1} \\subseteq K_{\\mathfrak m}$ the subgroup of those $x$ with\u2026","title":"Projecting the image of a ray element to the class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-to-class-group-residue-sign-to","kind":"node","lean":"NumberField.Modulus.rayClassToClassGroup_residueSignToRayClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-to-class-group-residue-sign-to","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to $\\mathfrak m_0$, $\\mathrm{Cl}_{\\mathfrak m}$ the ray class group, and $K_{\\mathfrak m}$ the group of $x \\in K^\\times$ with $x\\mathcal O_K \\in I_{\\mathfrak m}$. Let $T_{\\mathfrak m}$ be the product of one group of order $2$ for each real\u2026","title":"Projecting the image of a residue-sign value to the class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-to-class-group-ker","kind":"node","lean":"NumberField.Modulus.rayClassToClassGroup_ker","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-to-class-group-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to $\\mathfrak m_0$, $\\mathrm{Cl}_{\\mathfrak m}$ the ray class group, $K_{\\mathfrak m}$ the group of $x \\in K^\\times$ with $x\\mathcal O_K \\in I_{\\mathfrak m}$, and $K_{\\mathfrak m,1} \\subseteq K_{\\mathfrak m}$ the subgroup of those $x$ with\u2026","title":"Kernel of the projection from the ray class group to the class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-quotient-to-ray-class-group-mul-exac","kind":"node","lean":"NumberField.Modulus.rayQuotientToRayClassGroup_mulExact_rayClassToClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-quotient-to-ray-class-group-mul-exac","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field, $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, $I_{\\mathfrak m}$ the group of nonzero fractional ideals of $\\mathcal O_K$ coprime to $\\mathfrak m_0$, $\\mathrm{Cl}_{\\mathfrak m}$ the ray class group, $K_{\\mathfrak m}$ the group of $x \\in K^\\times$ with $x\\mathcal O_K \\in I_{\\mathfrak m}$, and $K_{\\mathfrak m,1} \\subseteq K_{\\mathfrak m}$ the subgroup of those $x$ with\u2026","title":"Exactness at the ray class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/residue-sign-to-ray-class-group-mul-exac","kind":"node","lean":"NumberField.Modulus.residueSignToRayClassGroup_mulExact_rayClassToClassGroup","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#residue-sign-to-ray-class-group-mul-exac","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, with finite part the nonzero ideal $\\mathfrak m_0$ of $\\mathcal O_K$ and infinite part the finite set $\\mathfrak m_\\infty$ of real places of $K$. Let $\\varphi$ be the homomorphism from the residue-sign group $\\{\\pm 1\\}^{\\mathfrak m_\\infty} \\times (\\mathcal O_K/\\mathfrak m_0)^\\times$ to the ray class group $\\mathrm{Cl}_{\\mathfrak m}(K)$ that sends the signs at\u2026","title":"Exactness at the ray class group, residue-sign form"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/fractional-ideal-mk0-mem-coprime-fractio","kind":"node","lean":"NumberField.Modulus.fractionalIdealMk0_mem_coprimeFractionalIdeals","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#fractional-ideal-mk0-mem-coprime-fractio","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal O_K$, let $\\mathfrak m$ be a modulus of $K$ with finite part $\\mathfrak m_0$, and let $I$ be a nonzero ideal of $\\mathcal O_K$ with $I + \\mathfrak m_0 = \\mathcal O_K$. Then $I$, viewed as an invertible fractional ideal of $K$, belongs to the group of fractional ideals coprime to $\\mathfrak m_0$: its exponent at every prime dividing $\\mathfrak m_0$ is $0$.","title":"Nonzero integral ideals coprime to the modulus are coprime fractional ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/coprime-ideal-class-map-surjective","kind":"node","lean":"NumberField.Modulus.coprimeIdealClassMap_surjective","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#coprime-ideal-class-map-surjective","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$. The homomorphism from the group of fractional ideals of $K$ coprime to $\\mathfrak m_0$ (those whose exponent at every prime dividing $\\mathfrak m_0$ is $0$) to the class group of $\\mathcal O_K$, sending each such fractional ideal to its ideal class, is surjective.","title":"Every ideal class contains a fractional ideal coprime to the modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-to-class-group-surjective","kind":"node","lean":"NumberField.Modulus.rayClassToClassGroup_surjective","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-to-class-group-surjective","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$. The homomorphism from the ray class group of $\\mathfrak m$ to the class group of $\\mathcal O_K$, sending the ray class of a fractional ideal coprime to the finite part of $\\mathfrak m$ to its ideal class, is surjective.","title":"The ray class group surjects onto the class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-six-term-exact","kind":"node","lean":"NumberField.Modulus.rayClass_six_term_exact","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-six-term-exact","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal O_K$, and let $\\mathfrak m$ be a modulus of $K$ with finite part $\\mathfrak m_0$ (a nonzero ideal of $\\mathcal O_K$) and infinite part $\\mathfrak m_\\infty$ (a finite set of real places). Let $K_{\\mathfrak m}$ be the group of $x \\in K^\\times$ whose principal fractional ideal is coprime to $\\mathfrak m_0$, and let $K_{\\mathfrak m,1} \\subseteq K_{\\mathfrak m}$\u2026","title":"The ray class exact sequence"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/ray-class-residue-sign-six-term-exact","kind":"node","lean":"NumberField.Modulus.rayClass_residueSign_six_term_exact","parent":5633,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups/index.html#ray-class-residue-sign-six-term-exact","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-groups","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal O_K$, and let $\\mathfrak m$ be a modulus of $K$ with finite part $\\mathfrak m_0$ (a nonzero ideal of $\\mathcal O_K$) and infinite part $\\mathfrak m_\\infty$ (a finite set of real places). Let $K_{\\mathfrak m}$ be the group of $x \\in K^\\times$ whose principal fractional ideal is coprime to $\\mathfrak m_0$, and let $K_{\\mathfrak m,1} \\subseteq K_{\\mathfrak m}$\u2026","title":"The ray class exact sequence, sign and residue form"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","title":"Artin map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/prime-unit-fractional-ideal","kind":"node","lean":"NumberField.primeUnitFractionalIdeal","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#prime-unit-fractional-ideal","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"For a number field $K$ with ring of integers $\\mathcal O_K$ and a height-one prime $\\mathfrak p$ of $\\mathcal O_K$, this is $\\mathfrak p$ regarded as a fractional ideal of $K$; it is nonzero, hence a unit in the group of nonzero fractional ideals of $K$.","title":"A prime as an invertible fractional ideal"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/factorization-mul-equiv-prime-unit-fract","kind":"node","lean":"FractionalIdeal.factorizationMulEquiv_primeUnitFractionalIdeal","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#factorization-mul-equiv-prime-unit-fract","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal O_K$ and let $\\mathfrak p$ be a height-one prime of $\\mathcal O_K$. The factorization isomorphism sends $\\mathfrak p$, regarded as an invertible fractional ideal of $K$, to the function on height-one primes of $\\mathcal O_K$ that is $1$ at $\\mathfrak p$ and $0$ elsewhere.","title":"Factorization of a prime of a number field"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/mem-coprime-fractional-ideals-prime-unit","kind":"node","lean":"NumberField.Modulus.mem_coprimeFractionalIdeals_primeUnit","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#mem-coprime-fractional-ideals-prime-unit","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ and let $\\mathfrak p$ be a height-one prime of the ring of integers $\\mathcal O_K$ that does not contain the finite part of $\\mathfrak m$. Then $\\mathfrak p$, viewed as an invertible fractional ideal of $K$, lies in the group of fractional ideals coprime to $\\mathfrak m$: its exponent at every height-one prime containing the finite part of $\\mathfrak m$ is zero.","title":"Primes not dividing the modulus are coprime to it"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/prime-coprime-unit","kind":"node","lean":"NumberField.Modulus.primeCoprimeUnit","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#prime-coprime-unit","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$ and let $\\mathfrak p$ be a height-one prime of the ring of integers $\\mathcal O_K$ that does not contain the finite part of $\\mathfrak m$. This is $\\mathfrak p$, viewed as an invertible fractional ideal of $K$, regarded as an element of the group of fractional ideals coprime to $\\mathfrak m$.","title":"The prime as a coprime fractional ideal"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/coprime-primes","kind":"node","lean":"NumberField.Modulus.CoprimePrimes","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#coprime-primes","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$. The primes coprime to $\\mathfrak m$ are the nonzero prime ideals $\\mathfrak p$ of the ring of integers $\\mathcal O_K$ that do not contain the finite part of $\\mathfrak m$, that is, that do not divide it. They index the generators of a free group that maps to the group of fractional ideals coprime to the finite part of $\\mathfrak m$.","title":"Primes coprime to a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/coprime-prime-units","kind":"node","lean":"NumberField.Modulus.coprimePrimeUnits","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#coprime-prime-units","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, and let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. The prime coprime units of $\\mathfrak m$ are the elements of $I_K^{\\mathfrak m}$ of the form $\\mathfrak p$, where $\\mathfrak p$ is a nonzero prime ideal of the ring of integers $\\mathcal O_K$ that does not divide the finite part of\u2026","title":"Prime coprime units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/mem-coprime-prime-units","kind":"node","lean":"NumberField.Modulus.mem_coprimePrimeUnits","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#mem-coprime-prime-units","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, and let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. An element $I$ of $I_K^{\\mathfrak m}$ is a prime coprime unit of $\\mathfrak m$ if and only if $I = \\mathfrak p$ for some nonzero prime ideal $\\mathfrak p$ of the ring of integers $\\mathcal O_K$ that does not divide the finite part of\u2026","title":"Membership in the prime coprime units"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/coprime-prime-units-eq-range","kind":"node","lean":"NumberField.Modulus.coprimePrimeUnits_eq_range","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#coprime-prime-units-eq-range","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$, and let $P$ be the set of nonzero prime ideals of the ring of integers $\\mathcal O_K$ that do not divide the finite part of $\\mathfrak m$. Then the set of prime coprime units of $\\mathfrak m$ equals the range of the map\u2026","title":"The prime coprime units are the image of the coprime primes"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/closure-coprime-prime-units-eq-top","kind":"node","lean":"NumberField.Modulus.closure_coprimePrimeUnits_eq_top","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#closure-coprime-prime-units-eq-top","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, and let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. Then $I_K^{\\mathfrak m}$ is generated, as a group, by the nonzero prime ideals $\\mathfrak p$ of the ring of integers $\\mathcal O_K$ that do not divide the finite part of $\\mathfrak m$.","title":"The coprime primes generate the coprime ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/free-group-to-coprime","kind":"node","lean":"NumberField.Modulus.freeGroupToCoprime","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#free-group-to-coprime","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $P$ be the set of nonzero prime ideals of the ring of integers $\\mathcal O_K$ that do not divide the finite part of $\\mathfrak m$, and let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. This is the group homomorphism $F(P) \\to I_K^{\\mathfrak m}$ from the free group on $P$ that sends each\u2026","title":"The free group on the coprime primes maps to the coprime ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/free-group-to-coprime-of","kind":"node","lean":"NumberField.Modulus.freeGroupToCoprime_of","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#free-group-to-coprime-of","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $P$ be the set of nonzero prime ideals of the ring of integers $\\mathcal O_K$ that do not divide the finite part of $\\mathfrak m$, and let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. For every $\\mathfrak p \\in P$, the homomorphism $F(P) \\to I_K^{\\mathfrak m}$ from the free group on $P$\u2026","title":"The free-group map on generators"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/free-group-to-coprime-surjective","kind":"node","lean":"NumberField.Modulus.freeGroupToCoprime_surjective","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#free-group-to-coprime-surjective","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $P$ be the set of nonzero prime ideals of the ring of integers $\\mathcal O_K$ that do not divide the finite part of $\\mathfrak m$, and let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. The homomorphism $F(P) \\to I_K^{\\mathfrak m}$ from the free group on $P$ sending each generator\u2026","title":"The free group on the coprime primes surjects onto the coprime ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/free-group-to-coprime-ker-le-commutator","kind":"node","lean":"NumberField.Modulus.freeGroupToCoprime_ker_le_commutator","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#free-group-to-coprime-ker-le-commutator","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $P$ be the set of nonzero prime ideals of the ring of integers $\\mathcal O_K$ that do not divide the finite part of $\\mathfrak m$, and let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. The kernel of the homomorphism $F(P) \\to I_K^{\\mathfrak m}$ from the free group on $P$ sending each\u2026","title":"The kernel of the free-group map lies in the commutator subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/free-group-to-coprime-ker","kind":"node","lean":"NumberField.Modulus.freeGroupToCoprime_ker","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#free-group-to-coprime-ker","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $\\mathfrak m$ be a modulus of a number field $K$, let $P$ be the set of nonzero prime ideals of the ring of integers $\\mathcal O_K$ that do not divide the finite part of $\\mathfrak m$, and let $I_K^{\\mathfrak m}$ be the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. The kernel of the homomorphism $F(P) \\to I_K^{\\mathfrak m}$ from the free group on $P$ sending each\u2026","title":"The kernel of the free-group map is the commutator subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/is-unramified-outside","kind":"node","lean":"NumberField.Modulus.IsUnramifiedOutside","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#is-unramified-outside","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. An extension of number fields $L / K$ is unramified outside $m$ when every finite prime $\\mathfrak p$ of $K$ not dividing the finite part of $m$ is unramified in $\\mathcal O_L$, that is, every prime of $\\mathcal O_L$ lying over $\\mathfrak p$ is unramified over $\\mathcal O_K$.","title":"Extensions unramified outside a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/at-coprime-prime","kind":"node","lean":"NumberField.Modulus.IsUnramifiedOutside.at_coprimePrime","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#at-coprime-prime","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ and let $L / K$ be an extension of number fields that is unramified outside $m$. Then every finite prime $\\mathfrak p$ of $K$ not dividing the finite part of $m$ is unramified in $\\mathcal O_L$.","title":"Unramified outside a modulus implies unramified at coprime primes"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/frobenius-at-coprime-prime","kind":"node","lean":"NumberField.Modulus.frobeniusAtCoprimePrime","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#frobenius-at-coprime-prime","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$, let $L / K$ be a Galois extension of number fields that is unramified outside $m$, and let $\\mathfrak p$ be a finite prime of $K$ not dividing the finite part of $m$. Fix a prime $\\mathfrak P$ of $\\mathcal O_L$ lying over $\\mathfrak p$. The chosen Frobenius element at $\\mathfrak p$ is a fixed element $\\sigma \\in \\mathrm{Gal}(L/K)$ that is an arithmetic Frobenius at\u2026","title":"Chosen Frobenius element at a coprime prime"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/frobenius-at-coprime-prime-spec","kind":"node","lean":"NumberField.Modulus.frobeniusAtCoprimePrime_spec","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#frobenius-at-coprime-prime-spec","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$, let $L / K$ be a Galois extension of number fields that is unramified outside $m$, and let $\\mathfrak p$ be a finite prime of $K$ not dividing the finite part of $m$, with chosen Frobenius element $\\sigma \\in \\mathrm{Gal}(L/K)$. Then there is a prime $\\mathfrak P$ of $\\mathcal O_L$ lying over $\\mathfrak p$ such that $\\mathcal O_L / \\mathfrak P$ is finite, $\\mathfrak P$ is\u2026","title":"The chosen Frobenius is an arithmetic Frobenius at an unramified prime"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/free-group-to-gal","kind":"node","lean":"NumberField.Modulus.freeGroupToGal","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#free-group-to-gal","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ and let $L / K$ be a Galois extension of number fields that is unramified outside $m$. The Frobenius homomorphism is the group homomorphism from the free group on the finite primes of $K$ not dividing the finite part of $m$ to $\\mathrm{Gal}(L/K)$ that sends each such prime $\\mathfrak p$ to the chosen Frobenius element at $\\mathfrak p$.","title":"Frobenius homomorphism on the free group of coprime primes"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/free-group-to-gal-of","kind":"node","lean":"NumberField.Modulus.freeGroupToGal_of","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#free-group-to-gal-of","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ and let $L / K$ be a Galois extension of number fields that is unramified outside $m$. For every finite prime $\\mathfrak p$ of $K$ not dividing the finite part of $m$, the Frobenius homomorphism from the free group on these primes to $\\mathrm{Gal}(L/K)$ sends the generator $\\mathfrak p$ to the chosen Frobenius element at $\\mathfrak p$.","title":"The Frobenius homomorphism on generators"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/free-group-to-coprime-ker-le-free-group","kind":"node","lean":"NumberField.Modulus.freeGroupToCoprime_ker_le_freeGroupToGal_ker","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#free-group-to-coprime-ker-le-free-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $L/K$ be an abelian Galois extension of number fields and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, such that every prime of $\\mathcal O_K$ not dividing $\\mathfrak m_0$ is unramified in $L$. Let $F$ be the free group on the primes $\\mathfrak p$ of $\\mathcal O_K$ not dividing $\\mathfrak m_0$. Then the kernel of the homomorphism from $F$ to the group of fractional ideals of $K$ coprime to\u2026","title":"Kernel containment for the free-group Frobenius map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/artin-map","kind":"node","lean":"NumberField.Modulus.artinMap","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#artin-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $L/K$ be an abelian Galois extension of number fields and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, such that every prime of $\\mathcal O_K$ not dividing $\\mathfrak m_0$ is unramified in $L$. Let $I^{\\mathfrak m}$ be the group of fractional ideals of $K$ coprime to $\\mathfrak m_0$ and $F$ the free group on the primes $\\mathfrak p$ of $\\mathcal O_K$ not dividing $\\mathfrak m_0$. The\u2026","title":"Artin map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/artin-map-comp-free-group-to-coprime","kind":"node","lean":"NumberField.Modulus.artinMap_comp_freeGroupToCoprime","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#artin-map-comp-free-group-to-coprime","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $L/K$ be an abelian Galois extension of number fields and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, such that every prime of $\\mathcal O_K$ not dividing $\\mathfrak m_0$ is unramified in $L$. Let $F$ be the free group on the primes $\\mathfrak p$ of $\\mathcal O_K$ not dividing $\\mathfrak m_0$, and let $I^{\\mathfrak m}$ be the group of fractional ideals of $K$ coprime to $\\mathfrak m_0$.\u2026","title":"The Artin map factors the free-group Frobenius map"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/artin-map-prime-coprime-unit","kind":"node","lean":"NumberField.Modulus.artinMap_primeCoprimeUnit","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#artin-map-prime-coprime-unit","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $L/K$ be an abelian Galois extension of number fields and $\\mathfrak m$ a modulus of $K$ with finite part $\\mathfrak m_0$, such that every prime of $\\mathcal O_K$ not dividing $\\mathfrak m_0$ is unramified in $L$. For every prime $\\mathfrak p$ of $\\mathcal O_K$ not dividing $\\mathfrak m_0$, the Artin map sends $\\mathfrak p$, viewed as a fractional ideal coprime to $\\mathfrak m_0$, to the arithmetic Frobenius in\u2026","title":"The Artin map at a prime"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/restrict-normal-hom-frobenius-at-coprime","kind":"node","lean":"NumberField.Modulus.restrictNormalHom_frobeniusAtCoprimePrime","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#restrict-normal-hom-frobenius-at-coprime","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $K \\subseteq L \\subseteq M$ be a tower of number fields with $L/K$ abelian Galois and $M/K$ Galois, and let $\\mathfrak m$ be a modulus of $K$ such that $L$ and $M$ are both unramified outside $\\mathfrak m$. For every finite prime $\\mathfrak p$ of $K$ not dividing the finite part of $\\mathfrak m$, the restriction to $L$ of the chosen Frobenius element of $\\mathfrak p$ in $\\mathrm{Gal}(M/K)$ equals the chosen\u2026","title":"Restriction of Frobenius elements in a tower"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/restrict-normal-hom-artin-map-prime-copr","kind":"node","lean":"NumberField.Modulus.restrictNormalHom_artinMap_primeCoprimeUnit","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#restrict-normal-hom-artin-map-prime-copr","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $K \\subseteq L \\subseteq M$ be a tower of number fields with $L/K$ and $M/K$ abelian Galois, and let $\\mathfrak m$ be a modulus of $K$ with finite part $\\mathfrak m_0$ such that every prime of $\\mathcal O_K$ not dividing $\\mathfrak m_0$ is unramified in $L$ and in $M$. For every prime $\\mathfrak p$ of $\\mathcal O_K$ not dividing $\\mathfrak m_0$, restricting to $L$ the image of $\\mathfrak p$ under the Artin map\u2026","title":"Restriction of the Artin map at a prime"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/restrict-normal-hom-comp-artin-map","kind":"node","lean":"NumberField.Modulus.restrictNormalHom_comp_artinMap","parent":5731,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map/index.html#restrict-normal-hom-comp-artin-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/artin-map","status":"fully_proved","summary":"Let $K \\subseteq L \\subseteq M$ be a tower of number fields with $L/K$ and $M/K$ abelian Galois, and let $\\mathfrak m$ be a modulus of $K$ with finite part $\\mathfrak m_0$ such that every prime of $\\mathcal O_K$ not dividing $\\mathfrak m_0$ is unramified in $L$ and in $M$. Then the Artin map of $M/K$ followed by restriction $\\mathrm{Gal}(M/K) \\to \\mathrm{Gal}(L/K)$ equals the Artin map of $L/K$, as homomorphisms\u2026","title":"The Artin map commutes with restriction"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","title":"Congruence subgroups for a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/congruence-subgroup","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroup","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#congruence-subgroup","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $K$ be a number field and $\\mathfrak m$ a modulus of $K$. A congruence subgroup for $\\mathfrak m$ is a subgroup $H$ of the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$ such that $H$ contains the ray group of $\\mathfrak m$.","title":"Congruence subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/image","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroup.image","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#image","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $H$ be a congruence subgroup for a modulus $\\mathfrak m$ of a number field $K$, and let $\\pi$ be the canonical projection from the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$ onto the ray class group of $\\mathfrak m$. The image of $H$ is the subgroup $\\pi(H)$ of the ray class group.","title":"Image of a congruence subgroup in the ray class group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/mem-image-iff","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroup.mem_image_iff","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#mem-image-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $H$ be a congruence subgroup for a modulus $\\mathfrak m$ of a number field $K$, and let $\\pi$ be the canonical projection from the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$ onto the ray class group of $\\mathfrak m$. An element $x$ of the ray class group lies in the image $\\pi(H)$ if and only if $\\pi(y) = x$ for some $y \\in H$.","title":"Membership in the image of a congruence subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/to-ambient-subgroup","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroup.toAmbientSubgroup","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#to-ambient-subgroup","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"For a congruence subgroup $H$ for a modulus $\\mathfrak m$ of a number field $K$, this is the image of $H$ under the inclusion of the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$ into the group of all invertible fractional ideals of $K$.","title":"Congruence subgroup as a subgroup of all invertible fractional ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/smallest","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroup.smallest","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#smallest","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"For a modulus $\\mathfrak m$ of a number field $K$, the smallest congruence subgroup for $\\mathfrak m$ is the ray group of $\\mathfrak m$ itself, a subgroup of the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$.","title":"Smallest congruence subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/largest","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroup.largest","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#largest","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"For a modulus $\\mathfrak m$ of a number field $K$, the largest congruence subgroup for $\\mathfrak m$ is the whole group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$, which contains the ray group of $\\mathfrak m$.","title":"Largest congruence subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/congruence-subgroup-pair","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroupPair","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#congruence-subgroup-pair","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $K$ be a number field. A congruence subgroup pair for $K$ consists of a modulus $\\mathfrak m$ of $K$ together with a congruence subgroup for $\\mathfrak m$, that is, a subgroup of the group of invertible fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$ that contains the ray group of $\\mathfrak m$.","title":"Congruence subgroup pair"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/is-equivalent","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroupPair.IsEquivalent","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#is-equivalent","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $K$ be a number field. For a modulus $\\mathfrak m$ of $K$, write $I^{\\mathfrak m}$ for the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. Two pairs $(\\mathfrak m, H)$ and $(\\mathfrak n, H')$, each consisting of a modulus of $K$ and a congruence subgroup for it, are equivalent when $I^{\\mathfrak m} \\cap H' = I^{\\mathfrak n} \\cap H$, where both congruence subgroups are\u2026","title":"Equivalence of congruence subgroup pairs"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/is-equivalent-iff","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroupPair.isEquivalent_iff","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#is-equivalent-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $K$ be a number field, and for a modulus $\\mathfrak m$ of $K$ write $I^{\\mathfrak m}$ for the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$. Two pairs $(\\mathfrak m, H)$ and $(\\mathfrak n, H')$, each consisting of a modulus of $K$ and a congruence subgroup for it, are equivalent if and only if $I^{\\mathfrak m} \\cap H' = I^{\\mathfrak n} \\cap H$ in the group of all nonzero\u2026","title":"Criterion for equivalence of congruence subgroup pairs"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/refl","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroupPair.IsEquivalent.refl","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#refl","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $K$ be a number field. Every pair consisting of a modulus of $K$ and a congruence subgroup for it is equivalent to itself.","title":"Equivalence of congruence subgroup pairs is reflexive"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/symm","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroupPair.IsEquivalent.symm","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#symm","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $K$ be a number field. If a pair $P$, consisting of a modulus of $K$ and a congruence subgroup for it, is equivalent to such a pair $Q$, then $Q$ is equivalent to $P$.","title":"Equivalence of congruence subgroup pairs is symmetric"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/ray-group-ambient","kind":"node","lean":"NumberField.Modulus.rayGroupAmbient","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#ray-group-ambient","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"For a modulus $\\mathfrak m$ of a number field $K$, the ambient ray group of $\\mathfrak m$ is the image of the ray group of $\\mathfrak m$ under the inclusion of the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m$ into the group of all nonzero fractional ideals of $K$.","title":"Ray group among all fractional ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/ray-group-ambient-le-coprime","kind":"node","lean":"NumberField.Modulus.rayGroupAmbient_le_coprime","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#ray-group-ambient-le-coprime","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"For every modulus $\\mathfrak m$ of a number field $K$, the ray group of $\\mathfrak m$, viewed as a subgroup of the group of nonzero fractional ideals of $K$, is contained in the group of nonzero fractional ideals coprime to the finite part of $\\mathfrak m$.","title":"The ray group consists of coprime ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/ray-group-ambient-le-to-ambient-subgroup","kind":"node","lean":"NumberField.Modulus.CongruenceSubgroup.rayGroupAmbient_le_toAmbientSubgroup","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#ray-group-ambient-le-to-ambient-subgroup","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $C$ be a congruence subgroup for a modulus $\\mathfrak m$ of a number field $K$. Viewed as subgroups of the group of nonzero fractional ideals of $K$, the ray group of $\\mathfrak m$ is contained in $C$.","title":"A congruence subgroup contains the ray group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/coprime-fractional-ideals-le-of-dvd","kind":"node","lean":"NumberField.Modulus.coprimeFractionalIdeals_le_of_dvd","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#coprime-fractional-ideals-le-of-dvd","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $\\mathfrak m_1$ and $\\mathfrak m_2$ be moduli of a number field $K$ with $\\mathfrak m_2 \\mid \\mathfrak m_1$. Then every nonzero fractional ideal of $K$ coprime to the finite part of $\\mathfrak m_1$ is coprime to the finite part of $\\mathfrak m_2$.","title":"Coprimality passes to divisors of a modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/change-modulus","kind":"node","lean":"NumberField.Modulus.changeModulus","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#change-modulus","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $C_1$ be a congruence subgroup for a modulus $\\mathfrak m_1$ of a number field $K$, viewed as a subgroup of the group of nonzero fractional ideals of $K$, and let $\\mathfrak m_2$ be any modulus of $K$. Write $I^{\\mathfrak m_2}$ for the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m_2$ and $P_{\\mathfrak m_2} \\subseteq I^{\\mathfrak m_2}$ for the ray group of $\\mathfrak m_2$.\u2026","title":"Change of modulus of a congruence subgroup"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/change-modulus-to-ambient-subgroup","kind":"node","lean":"NumberField.Modulus.changeModulus_toAmbientSubgroup","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#change-modulus-to-ambient-subgroup","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $\\mathfrak m_1$ and $\\mathfrak m_2$ be moduli of a number field $K$ and let $C_1$ be a congruence subgroup for $\\mathfrak m_1$. Write $I^{\\mathfrak m_2}$ for the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m_2$ and $P_{\\mathfrak m_2}$ for the ray group of $\\mathfrak m_2$. Viewing all of these as subgroups of the group of nonzero fractional ideals of $K$, the change of\u2026","title":"Ambient subgroup of a change of modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/change-modulus-to-ambient-subgroup-of-dv","kind":"node","lean":"NumberField.Modulus.changeModulus_toAmbientSubgroup_of_dvd","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#change-modulus-to-ambient-subgroup-of-dv","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $\\mathfrak m_1$ and $\\mathfrak m_2$ be moduli of a number field $K$ with $\\mathfrak m_2 \\mid \\mathfrak m_1$, and let $C_1$ be a congruence subgroup for $\\mathfrak m_1$. Viewed as subgroups of the group of nonzero fractional ideals of $K$, the change of modulus of $C_1$ to $\\mathfrak m_2$ equals the subgroup generated by $C_1$ and the ray group $P_{\\mathfrak m_2}$ of $\\mathfrak m_2$.","title":"Ambient subgroup of a change of modulus to a divisor"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/change-modulus-is-equivalent","kind":"node","lean":"NumberField.Modulus.changeModulus_isEquivalent","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#change-modulus-is-equivalent","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $\\mathfrak m_1$ and $\\mathfrak m_2$ be moduli of a number field $K$ with $\\mathfrak m_2 \\mid \\mathfrak m_1$, and let $C_1$ be a congruence subgroup for $\\mathfrak m_1$. Write $I^{\\mathfrak m_1}$ for the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m_1$ and $P_{\\mathfrak m_2}$ for the ray group of $\\mathfrak m_2$, and view all of these as subgroups of the group of nonzero\u2026","title":"The change of modulus is equivalent to the original pair"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/exists-equivalent-iff-change-modulus","kind":"node","lean":"NumberField.Modulus.exists_equivalent_iff_changeModulus","parent":5759,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus/index.html#exists-equivalent-iff-change-modulus","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/congruence-subgroups-for-a-modulus","status":"fully_proved","summary":"Let $\\mathfrak m_1$ and $\\mathfrak m_2$ be moduli of a number field $K$ with $\\mathfrak m_2 \\mid \\mathfrak m_1$, and let $C_1$ be a congruence subgroup for $\\mathfrak m_1$. Write $I^{\\mathfrak m_1}$ for the group of nonzero fractional ideals of $K$ coprime to the finite part of $\\mathfrak m_1$ and $P_{\\mathfrak m_2}$ for the ray group of $\\mathfrak m_2$, and view all of these as subgroups of the group of nonzero\u2026","title":"Criterion for an equivalent congruence subgroup at a divisor of the modulus"},{"data_url":"scopes/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters.json","id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","kind":"scope","navigation":"module","parent":5548,"result":false,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","title":"Ray class characters"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/ray-class-character","kind":"node","lean":"NumberField.Modulus.RayClassCharacter","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#ray-class-character","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"A ray class character of a modulus $m$ of a number field $K$ is a group homomorphism from the ray class group of $m$ to the unit circle $\\{z \\in \\mathbb C : |z| = 1\\}$.","title":"Ray class character"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/trivial-character","kind":"node","lean":"NumberField.Modulus.trivialCharacter","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#trivial-character","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"The trivial ray class character of a modulus $m$ of a number field $K$ is the ray class character sending every ray class of $m$ to $1$.","title":"Trivial character"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/pullback","kind":"node","lean":"NumberField.Modulus.pullback","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#pullback","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"For a ray class character $\\chi$ of a modulus $m$ of a number field $K$, its pullback is the group homomorphism from the group of nonzero fractional ideals of $K$ coprime to the finite part of $m$ to the unit circle sending $\\mathfrak a$ to $\\chi([\\mathfrak a])$, where $[\\mathfrak a]$ is the ray class of $\\mathfrak a$.","title":"Pullback of a ray class character to ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/pullback-eq-one-of-mem-ray-group","kind":"node","lean":"NumberField.Modulus.pullback_eq_one_of_mem_rayGroup","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#pullback-eq-one-of-mem-ray-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $\\chi$ be a ray class character of a modulus $m$ of a number field $K$. If a nonzero fractional ideal $\\mathfrak a$ of $K$ coprime to the finite part of $m$ lies in the ray group of $m$, then the pullback of $\\chi$ to the fractional ideals coprime to the finite part of $m$ takes the value $1$ at $\\mathfrak a$.","title":"Pulled-back ray class characters are trivial on the ray group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/eval-complex","kind":"node","lean":"NumberField.Modulus.evalComplex","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#eval-complex","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"For a ray class character $\\chi$ of a modulus $m$ and a ray class $x$ of $m$, the complex value of $\\chi$ at $x$ is $\\chi(x)$, a point of the unit circle, regarded as a complex number.","title":"Complex value of a ray class character"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/eval-complex-ray-group","kind":"node","lean":"NumberField.Modulus.evalComplex_rayGroup","parent":5780,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ray class characters take the complex value "],["math","\\(1\\)"],["text"," on the ray group"]],"source":"Ray class characters take the complex value $1$ on the ray group"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#eval-complex-ray-group","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$ and $\\chi$ a ray class character of $m$. If a nonzero fractional ideal $\\mathfrak a$ of $K$ coprime to the finite part of $m$ lies in the ray group of $m$, then the value of $\\chi$ at the ray class of $\\mathfrak a$, regarded as a complex number, is $1$.","title":"Ray class characters take the complex value $1$ on the ray group"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/extend-by-zero","kind":"node","lean":"NumberField.Modulus.extendByZero","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#extend-by-zero","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $\\chi$ be a ray class character of a modulus $m$ of a number field $K$. Its extension by zero assigns to a nonzero fractional ideal $\\mathfrak a$ of $K$ the complex number $\\chi([\\mathfrak a])$ if $\\mathfrak a$ is coprime to the finite part of $m$, where $[\\mathfrak a]$ is the ray class of $\\mathfrak a$, and the value $0$ otherwise.","title":"Extension by zero of a ray class character"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/finite-exponent-le-of-dvd","kind":"node","lean":"NumberField.Modulus.finiteExponent_le_of_dvd","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#finite-exponent-le-of-dvd","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m_1$ and $m_2$ be moduli of a number field $K$ with $m_1 \\mid m_2$, that is, the finite part of $m_2$ is contained in the finite part of $m_1$ and every real place of $m_1$ is a real place of $m_2$. Then for every finite prime $v$ of $K$ (a height-one prime of the ring of integers $\\mathcal{O}_K$), the exponent of $v$ in the finite part of $m_1$ is at most the exponent of $v$ in the finite part of $m_2$.","title":"Exponents of a modulus grow under divisibility"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/coprime-inclusion","kind":"node","lean":"NumberField.Modulus.coprimeInclusion","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#coprime-inclusion","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m_1$ and $m_2$ be moduli of a number field $K$ with $m_1 \\mid m_2$, that is, the finite part of $m_2$ is contained in the finite part of $m_1$ and every real place of $m_1$ is a real place of $m_2$. Then every fractional ideal of $K$ coprime to $m_2$ is coprime to $m_1$, and this map is the inclusion homomorphism from the group of fractional ideals coprime to $m_2$ into the group of fractional ideals coprime\u2026","title":"Inclusion of coprime ideal groups under divisibility of moduli"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/of-dvd","kind":"node","lean":"NumberField.Modulus.IsRayElementOne.of_dvd","parent":5780,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ray congruence to "],["math","\\(1\\)"],["text"," passes to divisors of the modulus"]],"source":"Ray congruence to $1$ passes to divisors of the modulus"},"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#of-dvd","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m_1$ and $m_2$ be moduli of a number field $K$ with $m_1 \\mid m_2$, that is, the finite part of $m_2$ is contained in the finite part of $m_1$ and every real place of $m_1$ is a real place of $m_2$. If a nonzero element $x$ of $K$ is congruent to $1$ modulo $m_2$ in the ray sense (its principal ideal is coprime to $m_2$, for each prime $v$ dividing the finite part of $m_2$ either $x = 1$ or the exponent of $v$\u2026","title":"Ray congruence to $1$ passes to divisors of the modulus"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/map-of-dvd","kind":"node","lean":"NumberField.Modulus.RayClassGroup.mapOfDvd","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#map-of-dvd","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m_1$ and $m_2$ be moduli of a number field $K$ with $m_1 \\mid m_2$, that is, the finite part of $m_2$ is contained in the finite part of $m_1$ and every real place of $m_1$ is a real place of $m_2$. The inclusion of the group of fractional ideals coprime to $m_2$ into the group of fractional ideals coprime to $m_1$ maps the ray group modulo $m_2$ into the ray group modulo $m_1$, so it induces a group\u2026","title":"Level-change map on ray class groups"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/map-of-dvd-ray-class-map","kind":"node","lean":"NumberField.Modulus.RayClassGroup.mapOfDvd_rayClassMap","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#map-of-dvd-ray-class-map","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m_1$ and $m_2$ be moduli of a number field $K$ with $m_1 \\mid m_2$, that is, the finite part of $m_2$ is contained in the finite part of $m_1$ and every real place of $m_1$ is a real place of $m_2$. For every fractional ideal $\\mathfrak{a}$ of $K$ coprime to $m_2$, the level-change homomorphism from the ray class group modulo $m_2$ to the ray class group modulo $m_1$ sends the ray class of $\\mathfrak{a}$\u2026","title":"The level-change map on ray classes"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/is-induced-by","kind":"node","lean":"NumberField.Modulus.RayClassCharacter.IsInducedBy","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#is-induced-by","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m_1$ and $m_2$ be moduli of a number field $K$ with $m_1 \\mid m_2$, that is, the finite part of $m_2$ is contained in the finite part of $m_1$ and every real place of $m_1$ is a real place of $m_2$. A ray class character $\\chi_2$ modulo $m_2$ is induced by a ray class character $\\chi_1$ modulo $m_1$ when $\\chi_2 = \\chi_1 \\circ \\pi$, where $\\pi$ is the level-change homomorphism from the ray class group modulo\u2026","title":"Induced ray class character"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/is-induced-by-iff","kind":"node","lean":"NumberField.Modulus.RayClassCharacter.isInducedBy_iff","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#is-induced-by-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m_1$ and $m_2$ be moduli of a number field $K$ with $m_1 \\mid m_2$, that is, the finite part of $m_2$ is contained in the finite part of $m_1$ and every real place of $m_1$ is a real place of $m_2$. Let $\\chi_1$ be a ray class character modulo $m_1$ and $\\chi_2$ a ray class character modulo $m_2$. Then $\\chi_2$ is induced by $\\chi_1$ if and only if, for every fractional ideal $\\mathfrak{a}$ of $K$ coprime to\u2026","title":"Induced characters agree on coprime ideals"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/is-primitive","kind":"node","lean":"NumberField.Modulus.RayClassCharacter.IsPrimitive","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#is-primitive","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. A ray class character $\\chi$ modulo $m$ is primitive when, for every modulus $m_0$ of $K$ with $m_0 \\mid m$ (the finite part of $m$ is contained in the finite part of $m_0$ and every real place of $m_0$ is a real place of $m$) and every ray class character $\\chi_0$ modulo $m_0$ by which $\\chi$ is induced, $m_0 = m$.","title":"Primitive ray class character"},{"id":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/is-primitive-iff","kind":"node","lean":"NumberField.Modulus.RayClassCharacter.isPrimitive_iff","parent":5780,"result":true,"route":"../roadmap/NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters/index.html#is-primitive-iff","scope":"NumberTheory/algebraic-number-theory/adeles-class-field-theory/ray-class-characters","status":"fully_proved","summary":"Let $m$ be a modulus of a number field $K$. A ray class character $\\chi$ modulo $m$ is primitive if and only if, for every modulus $m_0$ of $K$ with $m_0 \\mid m$ (the finite part of $m$ is contained in the finite part of $m_0$ and every real place of $m_0$ is a real place of $m$) and every ray class character $\\chi_0$ modulo $m_0$ with $\\chi = \\chi_0 \\circ \\pi$, where $\\pi$ is the level-change homomorphism from the\u2026","title":"Primitivity in terms of the level-change map"},{"data_url":"scopes/NumberTheory/p-adic-numbers.json","id":"NumberTheory/p-adic-numbers","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/index.html","scope":"NumberTheory/p-adic-numbers","status":"can_prove","summary":"p-adic integers and expansions, valuations of integer sequences, Hensel lifting, Strassmann's theorem, Hilbert symbols, p-adic gamma, local fields.","title":"p-adic numbers and local fields"},{"id":"NumberTheory/p-adic-numbers/exists-injective-ring-hom-closure-padic","kind":"node","lean":"Subring.exists_injective_ringHom_closure_padicInt","parent":5797,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/index.html#exists-injective-ring-hom-closure-padic","scope":"NumberTheory/p-adic-numbers","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero and $S$ a finite subset of $K$. Then there are a prime $p \\ge 3$ and an injective ring homomorphism from the subring of $K$ generated by $S$ into the ring $\\mathbb{Z}_p$ of $p$-adic integers.","title":"Cassels' embedding theorem"},{"id":"NumberTheory/p-adic-numbers/is-integral-padic-int-iff-norm-le-one","kind":"node","lean":"PadicAlgCl.isIntegral_padicInt_iff_norm_le_one","parent":5797,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/index.html#is-integral-padic-int-iff-norm-le-one","scope":"NumberTheory/p-adic-numbers","status":"fully_proved","summary":"An element of the algebraic closure of `\u211a_[p]` is integral over `\u2124_[p]` if and only if its norm is at most one.","title":"p-adic integrality is equivalent to norm at most one"},{"id":"NumberTheory/p-adic-numbers/strassmann","kind":"node","lean":"Padic.strassmann","parent":5797,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/index.html#strassmann","scope":"NumberTheory/p-adic-numbers","status":"fully_proved","summary":"Let $p$ be a prime and let $(b_m)_{m \\ge 0}$ be a sequence in $\\mathbb{Q}_p$, not identically zero, with $b_m \\to 0$. Then the power series $\\sum_{m \\ge 0} b_m y^m$ has only finitely many zeros $y \\in \\mathbb{Q}_p$ with $|y|_p \\le 1$.","title":"Strassmann's theorem"},{"data_url":"scopes/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two.json","id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","title":"2-adic odd part of factorials of powers of two"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-part","kind":"node","lean":"MetaMathlibExt.oddPart","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-part","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For a natural number $n \\ge 1$, the odd part $\\operatorname{od}(n)$ is $n / 2^{\\nu_2(n)}$, where $2^{\\nu_2(n)}$ is the largest power of $2$ dividing $n$; it is the odd factor of $n$ left after removing that power of $2$. For $n = 0$ the definition gives $\\operatorname{od}(0) = 0$.","title":"Odd part"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-part-eq","kind":"node","lean":"MetaMathlibExt.oddPart_eq","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-part-eq","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For every natural number $n$, $\\mathrm{od}(n) = n / 2^{\\nu_2(n)}$, where $\\mathrm{od}(n)$ is the odd part of $n$ and $\\nu_2(n)$ is the exponent of $2$ in the prime factorization of $n$, with $\\nu_2(0) = 0$.","title":"Odd part as a quotient by a power of two"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-range-product","kind":"node","lean":"MetaMathlibExt.oddRangeProduct","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-range-product","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For natural numbers $l$ and $m$, the odd-range product $\\operatorname{odpr}(l, m)$ is the product of the odd natural numbers $j$ with $l \\le j \\le m$; it equals $1$ when there are none.","title":"Odd-range product"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-range-product-eq","kind":"node","lean":"MetaMathlibExt.oddRangeProduct_eq","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-range-product-eq","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For all natural numbers $l$ and $m$, the odd-range product $\\operatorname{odpr}(l, m)$ equals the product $\\prod j$ over the odd natural numbers $j$ in the closed interval $[l, m]$.","title":"Odd-range product as a filtered product"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-product-block","kind":"node","lean":"MetaMathlibExt.oddProductBlock","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-product-block","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For a natural number $m$, the odd product block $h(m)$ is the product of the odd integers $j$ with $2^{m-1}+1 \\le j \\le 2^m-1$. In particular $h(0) = h(1) = 1$, as empty products.","title":"Odd product block"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-product-block-eq","kind":"node","lean":"MetaMathlibExt.oddProductBlock_eq","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-product-block-eq","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For every natural number $m$, the odd product block $h(m)$ equals $\\operatorname{odpr}(2^{m-1}+1, 2^m-1)$, the product of the odd integers $j$ with $2^{m-1}+1 \\le j \\le 2^m-1$.","title":"Odd product block as an odd-range product"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/unstable-bits","kind":"node","lean":"MetaMathlibExt.unstableBits","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#unstable-bits","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"Write $\\operatorname{od}(n)$ for the odd part of a natural number $n$. For natural numbers $e$ and $d$, the unstable bits $\\operatorname{uns}(e,d)$ are the natural number $\\lfloor \\operatorname{od}((2^e)!) / 2^{e+1} \\rfloor \\bmod 2^d$, whose binary digits are the $d$ bits of $\\operatorname{od}((2^e)!)$ at positions $e+1, \\dots, e+d$ (position $0$ being the least significant bit).","title":"Unstable bits"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/unstable-bits-eq","kind":"node","lean":"MetaMathlibExt.unstableBits_eq","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#unstable-bits-eq","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"Write $\\operatorname{od}(n)$ for the odd part of a natural number $n$. For all natural numbers $e$ and $d$, the unstable bits are $\\operatorname{uns}(e,d) = \\lfloor \\operatorname{od}((2^e)!) / 2^{e+1} \\rfloor \\bmod 2^d$.","title":"Formula for the unstable bits"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/stable-bits","kind":"node","lean":"MetaMathlibExt.stableBits","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#stable-bits","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For natural numbers $s$ and $d$, let $\\operatorname{stab}(s, d)$ be the number whose binary digits are the digits of $\\operatorname{od}((2^{s+d-1})!)$ in positions $s, \\dots, s + d - 1$ (position $0$ being the units digit), where $\\operatorname{od}(m)$ is the odd part of $m$ and $s + d - 1$ is truncated subtraction. That is,","title":"Stable bits of the odd part of a factorial"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/stable-bits-eq","kind":"node","lean":"MetaMathlibExt.stableBits_eq","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#stable-bits-eq","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"Write $\\operatorname{od}(n)$ for the odd part of a natural number $n$. For all natural numbers $s$ and $d$, the stable bits are $\\operatorname{stab}(s, d) = \\lfloor \\operatorname{od}((2^{s+d-1})!) / 2^s \\rfloor \\bmod 2^d$, where $s + d - 1$ is read as $0$ when $s = d = 0$.","title":"Formula for the stable bits"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/unstable-bits-four-two","kind":"node","lean":"MetaMathlibExt.unstableBits_four_two","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","The unstable bits for "],["math","\\(e = 4\\)"],["text"," and "],["math","\\(d = 2\\)"]],"source":"The unstable bits for $e = 4$ and $d = 2$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#unstable-bits-four-two","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"The unstable bits $\\operatorname{uns}(4,2)$ equal $3$; that is, $\\lfloor \\operatorname{od}(16!) / 2^5 \\rfloor \\bmod 2^2 = 3$, where $\\operatorname{od}(16!)$ is the odd part of $16!$.","title":"The unstable bits for $e = 4$ and $d = 2$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/stable-bits-five-two","kind":"node","lean":"MetaMathlibExt.stableBits_five_two","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","The stable bits "],["math","\\(\\operatorname{stab}(5, 2)\\)"],["text"," vanish"]],"source":"The stable bits $\\operatorname{stab}(5, 2)$ vanish"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#stable-bits-five-two","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"Writing $\\operatorname{od}(m)$ for the odd part of $m$, the binary digits of $\\operatorname{od}(64!)$ in positions $5$ and $6$ (the units digit being position $0$) are both $0$; that is, $\\operatorname{stab}(5, 2) = \\lfloor \\operatorname{od}(64!) / 2^5 \\rfloor \\bmod 4 = 0$.","title":"The stable bits $\\operatorname{stab}(5, 2)$ vanish"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/dyadic-upper-half","kind":"node","lean":"MetaMathlibExt.dyadicUpperHalf","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#dyadic-upper-half","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For a natural number $e$, $I_e$ is the finite set of natural numbers $i$ with $2^{e-1} < i \\le 2^e$, where $e - 1$ is truncated subtraction, so that $I_0$ is empty.","title":"Dyadic upper half interval"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/mem-dyadic-upper-half","kind":"node","lean":"MetaMathlibExt.mem_dyadicUpperHalf","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#mem-dyadic-upper-half","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For natural numbers $e$ and $i$, $i$ belongs to the dyadic upper half $I_e$ if and only if $2^{e-1} < i$ and $i \\le 2^e$, where $e - 1$ is truncated subtraction.","title":"Membership in the dyadic upper half"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-below-two-pow","kind":"node","lean":"MetaMathlibExt.oddBelowTwoPow","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-below-two-pow","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For a natural number $e$, $S_e$ is the finite set of odd natural numbers $i$ with $1 \\le i < 2^e$.","title":"Odd numbers below a power of two"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/mem-odd-set","kind":"node","lean":"MetaMathlibExt.mem_oddSet","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in the odd numbers below "],["math","\\(2^e\\)"]],"source":"Membership in the odd numbers below $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#mem-odd-set","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For natural numbers $e$ and $i$, $i$ belongs to the set $S_e$ of odd positive integers below $2^e$ if and only if $1 \\le i$, $i < 2^e$ and $i$ is odd.","title":"Membership in the odd numbers below $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/integer-multiset","kind":"node","lean":"MetaMathlibExt.integerMultiset","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#integer-multiset","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For a finite set $s$ of natural numbers, its integer multiset is the multiset of integers obtained by sending each element $n \\in s$ to $n$ viewed as an integer, so each element of $s$ occurs exactly once.","title":"Integer multiset of a finite set of naturals"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/integer-multiset-card","kind":"node","lean":"MetaMathlibExt.integerMultiset_card","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#integer-multiset-card","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For every finite set $s$ of natural numbers, the integer multiset of $s$ has the same number of elements, counted with multiplicity, as $s$.","title":"Cardinality of the integer multiset"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/reverse-esymm","kind":"node","lean":"MetaMathlibExt.reverseESymm","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#reverse-esymm","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $S$ be a finite multiset of elements of $R$ with $n$ elements counted with multiplicity, and let $i$ be a natural number. The reversed elementary symmetric function $\\hat\\sigma_i(S)$ is $\\sigma_{n-i}(S)$, the elementary symmetric function of degree $n - i$ in the elements of $S$. Here $n - i$ is truncated subtraction, so $\\hat\\sigma_i(S) = \\sigma_0(S) = 1$ whenever $i \\ge n$.","title":"Reversed elementary symmetric functions"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/reverse-esymm-eq","kind":"node","lean":"MetaMathlibExt.reverseESymm_eq","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#reverse-esymm-eq","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For every commutative semiring $R$, natural number $i$ and finite multiset $S$ of elements of $R$ with $n$ elements counted with multiplicity, the reversed elementary symmetric function satisfies $\\hat\\sigma_i(S) = \\sigma_{n-i}(S)$, the elementary symmetric function of $S$ of degree $n - i$, where $n - i$ is truncated subtraction.","title":"Unfolding the reversed elementary symmetric function"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-double-factorial","kind":"node","lean":"MetaMathlibExt.oddDoubleFactorial","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-double-factorial","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For a natural number $e$, $(2^e-1)!!$ is the product of the odd natural numbers in the closed interval $[1, 2^e-1]$. The parameter is the exponent $e$, not the odd endpoint $2^e-1$.","title":"Odd double factorial below a power of two"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-double-factorial-eq","kind":"node","lean":"MetaMathlibExt.oddDoubleFactorial_eq","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-double-factorial-eq","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For every natural number $e$, $(2^e-1)!!$ equals the product of the odd natural numbers in the closed interval $[1, 2^e-1]$.","title":"Odd double factorial below a power of two as an odd-range product"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/one-mod-eight-below-two-pow","kind":"node","lean":"MetaMathlibExt.oneModEightBelowTwoPow","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Numbers below "],["math","\\(2^e\\)"],["text"," congruent to "],["math","\\(1\\)"],["text"," modulo "],["math","\\(8\\)"]],"source":"Numbers below $2^e$ congruent to $1$ modulo $8$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#one-mod-eight-below-two-pow","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For a natural number $e$, this is the finite set of natural numbers $n$ with $n < 2^e$ and $n \\equiv 1 \\pmod 8$.","title":"Numbers below $2^e$ congruent to $1$ modulo $8$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/mem-one-mod-eight","kind":"node","lean":"MetaMathlibExt.mem_oneModEight","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#mem-one-mod-eight","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For all natural numbers $e$ and $n$, $n$ lies in the set of natural numbers below $2^e$ that are congruent to $1$ modulo $8$ if and only if $n < 2^e$ and $n \\equiv 1 \\pmod 8$.","title":"Membership in the residues one modulo eight below a power of two"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/four-copies-one-mod-eight-below-two-pow","kind":"node","lean":"MetaMathlibExt.fourCopiesOneModEightBelowTwoPow","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Four copies of the residues "],["math","\\(1 \\bmod 8\\)"],["text"," below "],["math","\\(2^e\\)"]],"source":"Four copies of the residues $1 \\bmod 8$ below $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#four-copies-one-mod-eight-below-two-pow","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For a natural number $e$, this is the multiset of integers containing exactly four copies of each natural number $n < 2^e$ with $n \\equiv 1 \\pmod 8$, that is, of each of $1, 9, \\dots, 2^e - 7$ when $e \\ge 3$.","title":"Four copies of the residues $1 \\bmod 8$ below $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/four-copies-one-mod-eight-eq","kind":"node","lean":"MetaMathlibExt.fourCopiesOneModEight_eq","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#four-copies-one-mod-eight-eq","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"fully_proved","summary":"For every natural number $e$, the multiset of four copies of the residues $1 \\bmod 8$ below $2^e$ equals $4 \\cdot M$, the sum of four copies of $M$, where $M$ is the integer multiset of the set of natural numbers $n < 2^e$ with $n \\equiv 1 \\pmod 8$.","title":"Four copies as a multiset scalar multiple"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/exists-two-adic-unstable-add-eq-stable","kind":"node","lean":"MetaMathlibExt.exists_twoAdic_unstable_add_eq_stable","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#exists-two-adic-unstable-add-eq-stable","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Write $\\mathrm{od}(n)$ for the odd part of a natural number $n$, and for natural numbers $d$ and $e$ let $\\mathrm{uns}(e, d) = \\lfloor \\mathrm{od}((2^e)!) / 2^{e+1} \\rfloor \\bmod 2^d$ and $\\mathrm{stab}(e + 1, d) = \\lfloor \\mathrm{od}((2^{e+d})!) / 2^{e+1} \\rfloor \\bmod 2^d$. There is a 2-adic integer $K$ such that for all natural numbers $d < e$, $2^d$ divides $\\mathrm{uns}(e, d) + K - \\mathrm{stab}(e + 1, d)$ in\u2026","title":"A 2-adic constant relating unstable and stable bits"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-part-factorial-eq-prod-h","kind":"node","lean":"MetaMathlibExt.oddPart_factorial_eq_prod_h","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-part-factorial-eq-prod-h","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For every natural number $e \\ge 1$,","title":"Odd part of a dyadic factorial as a product of odd blocks"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-product-block-congr-initial-segment","kind":"node","lean":"MetaMathlibExt.oddProductBlock_congr_initial_segment","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-product-block-congr-initial-segment","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For natural numbers $l$ and $u$ write $\\operatorname{odpr}(l, u)$ for the product of the odd integers $j$ with $l \\le j \\le u$, and let $h(m) = \\operatorname{odpr}(2^{m-1}+1, 2^m-1)$. Let $m \\ge 4$ and $B$ be natural numbers with $m - 1 \\le B \\le 3m - 7$, and put $d = \\lfloor (B + 4 - m)/2 \\rfloor$, that is, $d = 2 + \\lfloor (B - m)/2 \\rfloor$ (so $d = 1$ when $B = m - 1$). Then","title":"Initial-segment congruence for the odd product block"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-part-bij-on-upper-half-odd-set","kind":"node","lean":"MetaMathlibExt.oddPart_bijOn_upperHalf_oddSet","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-part-bij-on-upper-half-odd-set","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For every natural number $e \\ge 1$, the map sending each natural number to its odd part restricts to a bijection from $I_e = \\{i \\in \\mathbb{N} : 2^{e-1} < i \\le 2^e\\}$ onto the set $S_e$ of odd natural numbers $i$ with $1 \\le i < 2^e$.","title":"Odd parts biject the upper dyadic half onto the odd numbers below a power of two"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-set-product-congr","kind":"node","lean":"MetaMathlibExt.oddSet_product_congr","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Product of the odd numbers below "],["math","\\(2^e\\)"]],"source":"Product of the odd numbers below $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-set-product-congr","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For every natural number $e \\ge 3$, the product of the odd integers $i$ with $1 \\le i < 2^e$ is congruent to $2^e + 1$ modulo $2^{e+1}$.","title":"Product of the odd numbers below $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-part-factorial-congr-previous","kind":"node","lean":"MetaMathlibExt.oddPart_factorial_congr_previous","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-part-factorial-congr-previous","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For every natural number $e \\ge 3$, $\\mathrm{od}((2^{e-1})!) \\equiv \\mathrm{od}((2^e)!) \\pmod{2^e}$, where $\\mathrm{od}(n)$ denotes the odd part of $n$.","title":"Odd parts of consecutive dyadic factorials agree modulo a power of two"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/exists-two-adic-odd-part-factorial-limit","kind":"node","lean":"MetaMathlibExt.exists_twoAdic_oddPart_factorial_limit","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","A 2-adic limit of the odd parts of "],["math","\\((2^e)!\\)"]],"source":"A 2-adic limit of the odd parts of $(2^e)!$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#exists-two-adic-odd-part-factorial-limit","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"There is a 2-adic integer $z \\in \\mathbb{Z}_2$ such that $z \\equiv \\operatorname{od}((2^{e-1})!) \\pmod{2^e}$ in $\\mathbb{Z}_2$ for every natural number $e \\ge 1$, where $\\operatorname{od}(n)$ is the odd part of $n$.","title":"A 2-adic limit of the odd parts of $(2^e)!$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-set-product-congr-shift","kind":"node","lean":"MetaMathlibExt.oddSet_product_congr_shift","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shifting the odd numbers below "],["math","\\(2^e\\)"],["text"," by "],["math","\\(2^e\\)"]],"source":"Shifting the odd numbers below $2^e$ by $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-set-product-congr-shift","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For every natural number $e \\ge 2$, let $S_e$ be the set of odd integers $i$ with $1 \\le i < 2^e$. Then","title":"Shifting the odd numbers below $2^e$ by $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-double-factorial-normalized-congr","kind":"node","lean":"MetaMathlibExt.oddDoubleFactorial_normalized_congr","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-double-factorial-normalized-congr","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For every natural number $e \\ge 3$, writing $(2^k-1)!!$ for the product of the odd numbers from $1$ to $2^k-1$, the integer $2^e$ divides $(2^e-1)!! - 1$, the integer $2^{e+1}$ divides $(2^{e+1}-1)!! - 1$, and","title":"Congruence of normalized odd double factorials"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/exists-two-adic-odd-double-factorial-lim","kind":"node","lean":"MetaMathlibExt.exists_twoAdic_oddDoubleFactorial_limit","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#exists-two-adic-odd-double-factorial-lim","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Write $(2^e - 1)!!$ for the product of the odd natural numbers from $1$ to $2^e - 1$. There is a 2-adic integer $w$ such that for every natural number $e \\ge 3$, $2^{e-1}$ divides $w - \\lfloor ((2^e - 1)!! - 1) / 2^e \\rfloor$ in $\\mathbb{Z}_2$.","title":"A 2-adic limit of normalized odd double factorials"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-part-factorial-difference-congr-prod","kind":"node","lean":"MetaMathlibExt.oddPart_factorial_difference_congr_product","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-part-factorial-difference-congr-prod","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Write $\\mathrm{od}(n)$ for the odd part of $n$, $(2^e-1)!!$ for the product of the odd numbers from $1$ to $2^e-1$, and $\\lfloor x \\rfloor$ for the floor of $x$; congruences are taken in the $2$-adic integers $\\mathbb{Z}_2$. Let $z, w \\in \\mathbb{Z}_2$ satisfy $z \\equiv \\mathrm{od}((2^{e-1})!) \\pmod{2^e}$ for every $e \\ge 1$ and $w \\equiv \\lfloor ((2^e-1)!! - 1)/2^e \\rfloor \\pmod{2^{e-1}}$ for every $e \\ge 3$. Then\u2026","title":"Normalized difference of odd parts of dyadic factorials"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/unstable-translation-eq-neg-product","kind":"node","lean":"MetaMathlibExt.unstable_translation_eq_neg_product","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","The unstable translation constant equals "],["math","\\(-zw\\)"]],"source":"The unstable translation constant equals $-zw$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#unstable-translation-eq-neg-product","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Write $\\operatorname{od}(n)$ for the odd part of a natural number $n$, $(2^e-1)!!$ for the product of the odd numbers from $1$ to $2^e-1$, $\\operatorname{uns}(e,d) = \\lfloor \\operatorname{od}((2^e)!)/2^{e+1} \\rfloor \\bmod 2^d$ and $\\operatorname{stab}(e+1,d) = \\lfloor \\operatorname{od}((2^{e+d})!)/2^{e+1} \\rfloor \\bmod 2^d$. Let $K$, $z$ and $w$ be $2$-adic integers such that, with congruences in $\\mathbb{Z}_2$,\u2026","title":"The unstable translation constant equals $-zw$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-set-product-linear-shift-congr","kind":"node","lean":"MetaMathlibExt.oddSet_product_linear_shift_congr","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shifting the odd numbers below "],["math","\\(2^e\\)"],["text"," by a multiple of "],["math","\\(2^e\\)"]],"source":"Shifting the odd numbers below $2^e$ by a multiple of $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-set-product-linear-shift-congr","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For every integer $A$ and every natural number $e \\ge 2$, let $S_e$ be the set of odd integers $i$ with $1 \\le i < 2^e$. Then","title":"Shifting the odd numbers below $2^e$ by a multiple of $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/odd-set-product-power-shift-congr","kind":"node","lean":"MetaMathlibExt.oddSet_product_power_shift_congr","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of shifted products over the odd numbers below "],["math","\\(2^e\\)"]],"source":"Powers of shifted products over the odd numbers below $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#odd-set-product-power-shift-congr","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"For all integers $A$ and $B$ and all natural numbers $j$ and $e$ with $e \\ge 2$, let $S_e$ be the set of odd integers $i$ with $1 \\le i < 2^e$. Then","title":"Powers of shifted products over the odd numbers below $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/reverse-esymm-one-odd-set-congr","kind":"node","lean":"MetaMathlibExt.reverseESymm_one_oddSet_congr","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\hat\\sigma_1\\)"],["text"," of the odd numbers below "],["math","\\(2^e\\)"]],"source":"$\\hat\\sigma_1$ of the odd numbers below $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#reverse-esymm-one-odd-set-congr","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Let $e \\ge 2$ and let $S_e$ be the set of odd natural numbers $i$ with $1 \\le i < 2^e$, regarded as a multiset of integers with $n$ elements, and write $\\sigma_k$ for the elementary symmetric function of degree $k$. Then the reversed elementary symmetric function $\\hat\\sigma_1(S_e) = \\sigma_{n-1}(S_e)$ satisfies $\\hat\\sigma_1(S_e) \\equiv 2^{2e-2} \\pmod{2^{2e-1}}$ in $\\mathbb{Z}$.","title":"$\\hat\\sigma_1$ of the odd numbers below $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/reverse-esymm-two-odd-set-congr","kind":"node","lean":"MetaMathlibExt.reverseESymm_two_oddSet_congr","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\hat\\sigma_2\\)"],["text"," of the odd numbers below "],["math","\\(2^e\\)"]],"source":"$\\hat\\sigma_2$ of the odd numbers below $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#reverse-esymm-two-odd-set-congr","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Let $e \\ge 2$ and let $S_e$ be the set of odd natural numbers $i$ with $1 \\le i < 2^e$, regarded as a multiset of integers with $n$ elements, and write $\\sigma_k$ for the elementary symmetric function of degree $k$. Then the reversed elementary symmetric function $\\hat\\sigma_2(S_e) = \\sigma_{n-2}(S_e)$ satisfies $\\hat\\sigma_2(S_e) \\equiv 2^{e-2} \\pmod{2^{e-1}}$ in $\\mathbb{Z}$.","title":"$\\hat\\sigma_2$ of the odd numbers below $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/square-mod-two-power-fiber-card","kind":"node","lean":"MetaMathlibExt.square_mod_twoPower_fiber_card","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Each residue "],["math","\\(1 \\bmod 8\\)"],["text"," has four odd square roots modulo "],["math","\\(2^e\\)"]],"source":"Each residue $1 \\bmod 8$ has four odd square roots modulo $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#square-mod-two-power-fiber-card","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Let $e \\ge 3$ and let $S_e$ be the set of odd natural numbers $i$ with $1 \\le i < 2^e$. For every natural number $v < 2^e$ with $v \\equiv 1 \\pmod 8$, exactly four elements $i \\in S_e$ satisfy $i^2 \\bmod 2^e = v$.","title":"Each residue $1 \\bmod 8$ has four odd square roots modulo $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/reverse-esymm-four-copies-one-mod-eight","kind":"node","lean":"MetaMathlibExt.reverseESymm_fourCopies_oneModEight_congr","parent":5801,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\hat\\sigma_1\\)"],["text"," of four copies of the numbers "],["math","\\(1 \\bmod 8\\)"],["text"," below "],["math","\\(2^e\\)"]],"source":"$\\hat\\sigma_1$ of four copies of the numbers $1 \\bmod 8$ below $2^e$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#reverse-esymm-four-copies-one-mod-eight","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Let $e \\ge 3$ and let $M_e$ be the multiset of integers containing each natural number $n < 2^e$ with $n \\equiv 1 \\pmod 8$ (that is, $1, 9, \\dots, 2^e - 7$) exactly four times, let $N$ be the number of elements of $M_e$ counted with multiplicity, and write $\\sigma_k$ for the elementary symmetric function of degree $k$. Then the reversed elementary symmetric function $\\hat\\sigma_1(M_e) = \\sigma_{N-1}(M_e)$ satisfies\u2026","title":"$\\hat\\sigma_1$ of four copies of the numbers $1 \\bmod 8$ below $2^e$"},{"id":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/sum-square-erased-odd-set-product-congr","kind":"node","lean":"MetaMathlibExt.sum_square_erased_oddSet_product_congr","parent":5801,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two/index.html#sum-square-erased-odd-set-product-congr","scope":"NumberTheory/p-adic-numbers/2-adic-odd-part-of-factorials-of-powers-of-two","status":"can_prove","summary":"Let $e \\ge 1$ and let $S_e$ be the set of odd positive integers below $2^e$. Then, as integers,","title":"Squared products of the odd residues omitting one factor"},{"data_url":"scopes/NumberTheory/p-adic-numbers/absolute-values-on-fields.json","id":"NumberTheory/p-adic-numbers/absolute-values-on-fields","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","title":"Absolute values on fields"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/is-nonarchimedean-of-char-p","kind":"node","lean":"AbsoluteValue.isNonarchimedean_of_charP","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#is-nonarchimedean-of-char-p","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $K$ be a field of characteristic $p > 0$. Then every real-valued absolute value $v$ on $K$ is nonarchimedean: $v(x + y) \\le \\max(v(x), v(y))$ for all $x, y \\in K$.","title":"Absolute values in positive characteristic are nonarchimedean"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/eq-one-of-ne-zero-of-finite","kind":"node","lean":"AbsoluteValue.eq_one_of_ne_zero_of_finite","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#eq-one-of-ne-zero-of-finite","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $K$ be a finite field and $v$ a real-valued absolute value on $K$. Then $v(x) = 1$ for every nonzero $x \\in K$.","title":"Absolute values on finite fields are trivial"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/induced-topology","kind":"node","lean":"AbsoluteValue.inducedTopology","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#induced-topology","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $K$ be a field and $v$ a real-valued absolute value on $K$. The topology induced by $v$ on $K$ is the topology pulled back along the identity map from $K$ to $K$ equipped with the norm $v$; equivalently, it is the metric topology of the distance $d(x, y) = v(x - y)$ on $K$.","title":"Topology induced by an absolute value"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/induced-topology-eq-iff-is-equiv","kind":"node","lean":"AbsoluteValue.inducedTopology_eq_iff_isEquiv","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#induced-topology-eq-iff-is-equiv","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $K$ be a field and $v$, $w$ real-valued absolute values on $K$. The topologies induced on $K$ by $v$ and by $w$ are equal if and only if $v$ and $w$ are equivalent.","title":"Absolute values inducing the same topology are equivalent"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/is-nonarchimedean-iff-nat-cast-le-one","kind":"node","lean":"AbsoluteValue.isNonarchimedean_iff_natCast_le_one","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#is-nonarchimedean-iff-nat-cast-le-one","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $K$ be a field and $v$ a real-valued absolute value on $K$. Then $v$ is nonarchimedean, that is, $v(x + y) \\le \\max(v(x), v(y))$ for all $x, y \\in K$, if and only if $v(n) \\le 1$ for every natural number $n$, viewed as the element $n \\cdot 1$ of $K$.","title":"Nonarchimedean criterion for absolute values"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/is-nonarchimedean-of-nat-cast-le-one","kind":"node","lean":"AbsoluteValue.isNonarchimedean_of_natCast_le_one","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#is-nonarchimedean-of-nat-cast-le-one","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $K$ be a field and $v$ a real-valued absolute value on $K$. If $v(n) \\le 1$ for every natural number $n$, viewed as the element $n \\cdot 1$ of $K$, then $v$ is nonarchimedean: $v(x + y) \\le \\max(v(x), v(y))$ for all $x, y \\in K$.","title":"Absolute values bounded by one on the natural numbers are nonarchimedean"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/is-nonarchimedean-iff-nat-cast-le-one-of","kind":"node","lean":"AbsoluteValue.isNonarchimedean_iff_natCast_le_one_of_pos","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#is-nonarchimedean-iff-nat-cast-le-one-of","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $K$ be a field and $v$ a real-valued absolute value on $K$. Then $v$ is nonarchimedean, that is, $v(x + y) \\le \\max(v(x), v(y))$ for all $x, y \\in K$, if and only if $v(n) \\le 1$ for every positive integer $n$, viewed as the element $n \\cdot 1$ of $K$.","title":"Nonarchimedean criterion using positive integers"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/rpow","kind":"node","lean":"AbsoluteValue.rpow","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#rpow","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $R$ be a semiring, $v$ a real-valued absolute value on $R$ and $c$ a real number with $0 < c \\le 1$. The $c$-th power of $v$ is the real-valued absolute value $x \\mapsto v(x)^c$ on $R$.","title":"Powers of an absolute value"},{"id":"NumberTheory/p-adic-numbers/absolute-values-on-fields/rpow-apply","kind":"node","lean":"AbsoluteValue.rpow_apply","parent":5847,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/absolute-values-on-fields/index.html#rpow-apply","scope":"NumberTheory/p-adic-numbers/absolute-values-on-fields","status":"fully_proved","summary":"Let $R$ be a semiring, $v$ a real-valued absolute value on $R$ and $c$ a real number with $0 < c \\le 1$. The $c$-th power of $v$ takes the value $v(x)^c$ at every $x \\in R$.","title":"Values of a power of an absolute value"},{"data_url":"scopes/NumberTheory/p-adic-numbers/hilbert-symbol.json","id":"NumberTheory/p-adic-numbers/hilbert-symbol","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","title":"Hilbert symbol"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/is-solvable","kind":"node","lean":"HilbertSymbol.IsSolvable","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solvability of "],["math","\\(ax^2 + by^2 = 1\\)"]],"source":"Solvability of $ax^2 + by^2 = 1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#is-solvable","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For a field $F$ and nonzero elements $a, b \\in F$, the equation $ax^2 + by^2 = 1$ is solvable over $F$ if there exist $x, y \\in F$ with $ax^2 + by^2 = 1$.","title":"Solvability of $ax^2 + by^2 = 1$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/hilbert-symbol","kind":"node","lean":"hilbertSymbol","parent":5857,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#hilbert-symbol","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For a field $F$ and nonzero elements $a, b \\in F$, the Hilbert symbol $(a, b)_F$ is the integer $1$ if there exist $x, y \\in F$ with $ax^2 + by^2 = 1$, and $-1$ otherwise.","title":"Hilbert symbol of a field"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/eq-one-or-neg-one","kind":"node","lean":"hilbertSymbol.eq_one_or_neg_one","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Hilbert symbol of a field is "],["math","\\(\\pm 1\\)"]],"source":"The Hilbert symbol of a field is $\\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#eq-one-or-neg-one","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For every field $F$ and all nonzero $a, b \\in F$, the Hilbert symbol $(a, b)_F$ of $a$ and $b$ over $F$ equals $1$ or $-1$.","title":"The Hilbert symbol of a field is $\\pm 1$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/padic-hilbert-symbol","kind":"node","lean":"padicHilbertSymbol","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-adic Hilbert symbol"]],"source":"The $p$-adic Hilbert symbol"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#padic-hilbert-symbol","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For a prime $p$ and nonzero $a, b \\in \\mathbb{Q}_p$, the $p$-adic Hilbert symbol $(a, b)_p$ is the integer $1$ if there exist $x, y \\in \\mathbb{Q}_p$ with $ax^2 + by^2 = 1$, and $-1$ otherwise.","title":"The $p$-adic Hilbert symbol"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/eq-one-iff","kind":"node","lean":"padicHilbertSymbol.eq_one_iff","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","When the "],["math","\\(p\\)"],["text","-adic Hilbert symbol is "],["math","\\(1\\)"]],"source":"When the $p$-adic Hilbert symbol is $1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#eq-one-iff","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For a prime $p$ and nonzero $a, b \\in \\mathbb{Q}_p$, the $p$-adic Hilbert symbol $(a, b)_p$ equals $1$ if and only if there exist $x, y \\in \\mathbb{Q}_p$ with $ax^2 + by^2 = 1$.","title":"When the $p$-adic Hilbert symbol is $1$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/eq-neg-one-iff","kind":"node","lean":"padicHilbertSymbol.eq_neg_one_iff","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","When the "],["math","\\(p\\)"],["text","-adic Hilbert symbol is "],["math","\\(-1\\)"]],"source":"When the $p$-adic Hilbert symbol is $-1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#eq-neg-one-iff","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For a prime $p$ and nonzero $a, b \\in \\mathbb{Q}_p$, the $p$-adic Hilbert symbol $(a, b)_p$ equals $-1$ if and only if there are no $x, y \\in \\mathbb{Q}_p$ with $ax^2 + by^2 = 1$.","title":"When the $p$-adic Hilbert symbol is $-1$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/eq-one-or-neg-one-2","kind":"node","lean":"padicHilbertSymbol.eq_one_or_neg_one","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-adic Hilbert symbol is "],["math","\\(\\pm 1\\)"]],"source":"The $p$-adic Hilbert symbol is $\\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#eq-one-or-neg-one-2","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For a prime $p$ and all nonzero $a, b \\in \\mathbb{Q}_p$, the $p$-adic Hilbert symbol $(a, b)_p$ equals $1$ or $-1$.","title":"The $p$-adic Hilbert symbol is $\\pm 1$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/diagonal-unit-isotropic","kind":"node","lean":"HilbertSymbol.diagonal_unit_isotropic","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit diagonal forms in at least three variables over "],["math","\\(\\mathbb{Q}_p\\)"],["text"," are isotropic for odd "],["math","\\(p\\)"]],"source":"Unit diagonal forms in at least three variables over $\\mathbb{Q}_p$ are isotropic for odd $p$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#diagonal-unit-isotropic","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"Unit diagonal forms in at least three variables over $\\mathbb{Q}_p$ are isotropic for odd $p$. Let $p$ be an odd prime, let $n > 2$, and let $a_1, \\dots, a_n$ be units of $\\mathbb{Z}_p$. Then there is a nonzero vector $(x_1, \\dots, x_n) \\in \\mathbb{Q}_p^n$ with $a_1 x_1^2 + \\dots + a_n x_n^2 = 0$.","title":"Unit diagonal forms in at least three variables over $\\mathbb{Q}_p$ are isotropic for odd $p$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/real-hilbert-symbol","kind":"node","lean":"realHilbertSymbol","parent":5857,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#real-hilbert-symbol","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For nonzero real numbers $a$ and $b$, the real Hilbert symbol of $a$ and $b$ is the integer $1$ if there are real numbers $x, y$ with $a x^2 + b y^2 = 1$, and the integer $-1$ otherwise.","title":"Real Hilbert symbol"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/eq-one-iff-2","kind":"node","lean":"realHilbertSymbol.eq_one_iff","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","Real Hilbert symbol equal to "],["math","\\(1\\)"]],"source":"Real Hilbert symbol equal to $1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#eq-one-iff-2","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For nonzero real numbers $a$ and $b$, the real Hilbert symbol of $a$ and $b$ equals $1$ if and only if there are real numbers $x, y$ with $a x^2 + b y^2 = 1$.","title":"Real Hilbert symbol equal to $1$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/eq-neg-one-iff-2","kind":"node","lean":"realHilbertSymbol.eq_neg_one_iff","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","Real Hilbert symbol equal to "],["math","\\(-1\\)"]],"source":"Real Hilbert symbol equal to $-1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#eq-neg-one-iff-2","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For nonzero real numbers $a$ and $b$, the real Hilbert symbol of $a$ and $b$ equals $-1$ if and only if there are no real numbers $x, y$ with $a x^2 + b y^2 = 1$.","title":"Real Hilbert symbol equal to $-1$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/eq-neg-one-iff-both-neg","kind":"node","lean":"realHilbertSymbol.eq_neg_one_iff_both_neg","parent":5857,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#eq-neg-one-iff-both-neg","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For nonzero real numbers $a$ and $b$, the real Hilbert symbol of $a$ and $b$ equals $-1$ if and only if $a < 0$ and $b < 0$.","title":"Sign rule for the real Hilbert symbol"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/eq-one-or-neg-one-3","kind":"node","lean":"realHilbertSymbol.eq_one_or_neg_one","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","The real Hilbert symbol is "],["math","\\(\\pm 1\\)"]],"source":"The real Hilbert symbol is $\\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#eq-one-or-neg-one-3","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For nonzero real numbers $a$ and $b$, the real Hilbert symbol of $a$ and $b$ equals $1$ or $-1$.","title":"The real Hilbert symbol is $\\pm 1$"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/unit-zp-to-qp","kind":"node","lean":"HilbertSymbol.unitZpToQp","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inclusion of "],["math","\\(p\\)"],["text","-adic integer units"]],"source":"Inclusion of $p$-adic integer units"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#unit-zp-to-qp","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For a prime $p$, this is the map from the units of $\\mathbb{Z}_p$ to the units of $\\mathbb{Q}_p$ induced by the inclusion $\\mathbb{Z}_p \\subseteq \\mathbb{Q}_p$.","title":"Inclusion of $p$-adic integer units"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/unit-zp-to-qp-coe","kind":"node","lean":"HilbertSymbol.unitZpToQp_coe","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of the inclusion of "],["math","\\(p\\)"],["text","-adic integer units"]],"source":"Value of the inclusion of $p$-adic integer units"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#unit-zp-to-qp-coe","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"For a prime $p$ and a unit $u$ of $\\mathbb{Z}_p$, the image of $u$ under the inclusion of the units of $\\mathbb{Z}_p$ into the units of $\\mathbb{Q}_p$, viewed as an element of $\\mathbb{Q}_p$, equals $u$ viewed as an element of $\\mathbb{Q}_p$.","title":"Value of the inclusion of $p$-adic integer units"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/chevalley-warning-hensel-lift","kind":"node","lean":"HilbertSymbol.chevalley_warning_hensel_lift","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","For odd "],["math","\\(p\\)"],["text",", a binary unit form over "],["math","\\(\\mathbb{Z}_p\\)"],["text"," represents a unit square"]],"source":"For odd $p$, a binary unit form over $\\mathbb{Z}_p$ represents a unit square"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#chevalley-warning-hensel-lift","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"Let $p$ be an odd prime and let $u, v$ be units of $\\mathbb{Z}_p$. Then there are $x_0, y_0 \\in \\mathbb{Z}_p$ and a unit $z_0$ of $\\mathbb{Z}_p$ with $z_0^2 = u x_0^2 + v y_0^2$.","title":"For odd $p$, a binary unit form over $\\mathbb{Z}_p$ represents a unit square"},{"id":"NumberTheory/p-adic-numbers/hilbert-symbol/hilbert-symbol-units-eq-one-of-odd","kind":"node","lean":"HilbertSymbol.hilbert_symbol_units_eq_one_of_odd","parent":5857,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Hilbert symbol of two "],["math","\\(p\\)"],["text","-adic units is trivial for odd "],["math","\\(p\\)"]],"source":"The Hilbert symbol of two $p$-adic units is trivial for odd $p$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/hilbert-symbol/index.html#hilbert-symbol-units-eq-one-of-odd","scope":"NumberTheory/p-adic-numbers/hilbert-symbol","status":"fully_proved","summary":"Let $p$ be an odd prime and let $u, v$ be units of $\\mathbb{Z}_p$. Then the $p$-adic Hilbert symbol of $u$ and $v$, viewed as units of $\\mathbb{Q}_p$, equals $1$.","title":"The Hilbert symbol of two $p$-adic units is trivial for odd $p$"},{"data_url":"scopes/NumberTheory/p-adic-numbers/krasner-s-lemma.json","id":"NumberTheory/p-adic-numbers/krasner-s-lemma","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/krasner-s-lemma/index.html","scope":"NumberTheory/p-adic-numbers/krasner-s-lemma","status":"fully_proved","title":"Krasner's lemma"},{"id":"NumberTheory/p-adic-numbers/krasner-s-lemma/is-closest-conjugate","kind":"node","lean":"AlgebraicClosure.IsClosestConjugate","parent":5875,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/krasner-s-lemma/index.html#is-closest-conjugate","scope":"NumberTheory/p-adic-numbers/krasner-s-lemma","status":"fully_proved","summary":"Let $K$ be a normed field with algebraic closure $\\overline{K}$, and let $\\lVert \\cdot \\rVert$ be the spectral norm on $\\overline{K}$ over $K$. For $\\alpha, \\beta \\in \\overline{K}$, $\\alpha$ is the closest conjugate to $\\beta$ if $\\lVert \\beta - \\alpha \\rVert < \\lVert \\beta - \\sigma(\\alpha) \\rVert$ for every $K$-algebra automorphism $\\sigma$ of $\\overline{K}$ with $\\sigma(\\alpha) \\ne \\alpha$.","title":"Closest conjugate"},{"id":"NumberTheory/p-adic-numbers/krasner-s-lemma/lt-spectral-norm-sub-of-is-conj-root","kind":"node","lean":"AlgebraicClosure.IsClosestConjugate.lt_spectralNorm_sub_of_isConjRoot","parent":5875,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/krasner-s-lemma/index.html#lt-spectral-norm-sub-of-is-conj-root","scope":"NumberTheory/p-adic-numbers/krasner-s-lemma","status":"fully_proved","summary":"Let $K$ be a complete nontrivially normed field whose norm is ultrametric, with algebraic closure $\\overline{K}$, let $\\lVert \\cdot \\rVert$ be the spectral norm on $\\overline{K}$ over $K$, and let $\\alpha, \\beta, \\alpha' \\in \\overline{K}$. If $\\alpha$ is the closest conjugate to $\\beta$ and $\\alpha' \\ne \\alpha$ is a conjugate of $\\alpha$ over $K$, then $\\lVert \\alpha - \\beta \\rVert < \\lVert \\alpha - \\alpha' \\rVert$.","title":"Closest conjugates satisfy Krasner's hypothesis"},{"id":"NumberTheory/p-adic-numbers/krasner-s-lemma/adjoin-le","kind":"node","lean":"AlgebraicClosure.IsClosestConjugate.adjoin_le","parent":5875,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/krasner-s-lemma/index.html#adjoin-le","scope":"NumberTheory/p-adic-numbers/krasner-s-lemma","status":"fully_proved","summary":"Let $K$ be a complete nontrivially normed field whose norm is ultrametric, with algebraic closure $\\overline{K}$, and let $\\alpha, \\beta \\in \\overline{K}$. If $\\alpha$ is the closest conjugate to $\\beta$ and $\\alpha$ is separable over $K$, then $K(\\alpha) \\subseteq K(\\beta)$.","title":"Krasner's lemma for closest conjugates"},{"id":"NumberTheory/p-adic-numbers/krasner-s-lemma/exists-is-closest-conjugate-root-of-l1-n","kind":"node","lean":"Polynomial.exists_isClosestConjugate_root_of_l1Norm_sub_lt","parent":5875,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/krasner-s-lemma/index.html#exists-is-closest-conjugate-root-of-l1-n","scope":"NumberTheory/p-adic-numbers/krasner-s-lemma","status":"fully_proved","summary":"Let $K$ be a complete nontrivially normed field whose norm is ultrametric, with algebraic closure $\\overline{K}$, and let $f \\in K[X]$ be monic, irreducible and separable. Then there is a real $\\delta > 0$ such that for every monic $g \\in K[X]$ with $\\lVert f - g \\rVert_1 < \\delta$, where $\\lVert \\cdot \\rVert_1$ is the $\\ell^1$ norm (the sum of the norms of the coefficients), and every root $\\beta \\in \\overline{K}$\u2026","title":"Closest conjugate roots of nearby monic polynomials"},{"id":"NumberTheory/p-adic-numbers/krasner-s-lemma/exists-pos-irreducible-and-separable-of","kind":"node","lean":"Polynomial.exists_pos_irreducible_and_separable_of_l1Norm_sub_lt","parent":5875,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/krasner-s-lemma/index.html#exists-pos-irreducible-and-separable-of","scope":"NumberTheory/p-adic-numbers/krasner-s-lemma","status":"fully_proved","summary":"Let $K$ be a complete nontrivially normed field whose norm is ultrametric, and let $f \\in K[X]$ be monic, irreducible and separable. Then there is a real $\\delta > 0$ such that every monic $g \\in K[X]$ with $\\lVert f - g \\rVert_1 < \\delta$, where $\\lVert \\cdot \\rVert_1$ is the $\\ell^1$ norm (the sum of the norms of the coefficients), is irreducible and separable.","title":"Small monic perturbations stay irreducible and separable"},{"id":"NumberTheory/p-adic-numbers/krasner-s-lemma/exists-monic-and-nat-degree-eq-and-l1-no","kind":"node","lean":"Polynomial.exists_monic_and_natDegree_eq_and_l1Norm_sub_lt_and_map_irreducible_and_separable","parent":5875,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/krasner-s-lemma/index.html#exists-monic-and-nat-degree-eq-and-l1-no","scope":"NumberTheory/p-adic-numbers/krasner-s-lemma","status":"fully_proved","summary":"Let $K$ be a field and $L$ a complete nontrivially normed field whose norm is ultrametric, with a $K$-algebra structure such that the image of $K$ in $L$ is dense. Let $f \\in L[X]$ be monic, irreducible and separable, and let $\\varepsilon > 0$ be real. Then there is a monic $g \\in K[X]$ with $\\deg g = \\deg f$ such that, writing $\\bar g \\in L[X]$ for the image of $g$, $\\lVert f - \\bar g \\rVert_1 < \\varepsilon$,\u2026","title":"Approximation by irreducible separable polynomials over a dense field"},{"data_url":"scopes/NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function.json","id":"NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function/index.html","scope":"NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function","status":"fully_proved","title":"Morita's p-adic gamma function"},{"id":"NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function/morita-gamma-nat","kind":"node","lean":"MetaMathlibExt.moritaGammaNat","parent":5882,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function/index.html#morita-gamma-nat","scope":"NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function","status":"fully_proved","summary":"For a prime $p$ and a natural number $n$, this is the $p$-adic integer $(-1)^n \\prod_{0 < j < n,\\ p \\nmid j} j$: $(-1)^n$ times the product of the positive integers below $n$ that are not divisible by $p$. At $n = 0$ the product is empty and the value is $1$. These are the finite values that Morita's $p$-adic gamma function interpolates.","title":"Morita's p-adic gamma values at natural numbers"},{"id":"NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function/is-morita-padic-gamma","kind":"node","lean":"MetaMathlibExt.IsMoritaPadicGamma","parent":5882,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function/index.html#is-morita-padic-gamma","scope":"NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function","status":"fully_proved","summary":"For a prime $p$, a continuous function $\\Gamma : \\mathbb{Z}_p \\to \\mathbb{Z}_p$ is a Morita $p$-adic gamma function when $\\Gamma(0) = 1$, $\\Gamma(n)$ equals Morita's finite value $(-1)^n \\prod_{0 < j < n,\\ p \\nmid j} j$ for every positive integer $n$, and $\\Gamma(x)$ is a unit of $\\mathbb{Z}_p$ for every $x \\in \\mathbb{Z}_p$.","title":"Morita's p-adic gamma function"},{"id":"NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function/exists-unique-morita-padic-gamma","kind":"node","lean":"MetaMathlibExt.existsUnique_moritaPadicGamma","parent":5882,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function/index.html#exists-unique-morita-padic-gamma","scope":"NumberTheory/p-adic-numbers/morita-s-p-adic-gamma-function","status":"fully_proved","summary":"For every odd prime $p$ there is exactly one continuous function $\\Gamma : \\mathbb{Z}_p \\to \\mathbb{Z}_p$ that is a Morita $p$-adic gamma function, that is, $\\Gamma(0) = 1$, $\\Gamma(n) = (-1)^n \\prod_{0 < j < n,\\ p \\nmid j} j$ for every positive integer $n$, and $\\Gamma(x)$ is a unit of $\\mathbb{Z}_p$ for every $x \\in \\mathbb{Z}_p$.","title":"Existence and uniqueness of Morita's p-adic gamma function"},{"data_url":"scopes/NumberTheory/p-adic-numbers/p-adic-expansions.json","id":"NumberTheory/p-adic-numbers/p-adic-expansions","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","title":"p-adic expansions"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/padic-digit","kind":"node","lean":"PadicInt.padicDigit","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#padic-digit","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$, and for each $n \\ge 0$ let $a_n$ be the unique natural number with $a_n < p^n$ and $a \\equiv a_n \\pmod{p^n \\mathbb{Z}_p}$. The $n$-th $p$-adic digit of $a$ is the natural number $(a_{n+1} - a_n)/p^n$.","title":"Digits of a p-adic integer"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/padic-digit-lt","kind":"node","lean":"PadicInt.padicDigit_lt","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#padic-digit-lt","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$. For every $n \\ge 0$, the $n$-th $p$-adic digit of $a$ is strictly less than $p$.","title":"p-adic digits are less than p"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/padic-expansion","kind":"node","lean":"PadicInt.padicExpansion","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#padic-expansion","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$. The $p$-adic expansion of $a$ is the sequence whose $n$-th term is the $n$-th $p$-adic digit of $a$, viewed as an element of $\\{0, 1, \\dots, p - 1\\}$.","title":"p-adic expansion"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/padic-expansion-val","kind":"node","lean":"PadicInt.padicExpansion_val","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#padic-expansion-val","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$. For every $n \\ge 0$, the $n$-th term of the $p$-adic expansion of $a$, read as a natural number, is the $n$-th $p$-adic digit of $a$.","title":"Terms of the p-adic expansion are the digits"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/padic-digit-mul-pow","kind":"node","lean":"PadicInt.padicDigit_mul_pow","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#padic-digit-mul-pow","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$, and for each $n \\ge 0$ let $a_n$ be the unique natural number with $a_n < p^n$ and $a \\equiv a_n \\pmod{p^n \\mathbb{Z}_p}$. For every $n \\ge 0$, the $n$-th $p$-adic digit $d_n$ of $a$ satisfies $d_n p^n = a_{n+1} - a_n$.","title":"p-adic digits measure the gaps between approximations"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/appr-succ-eq","kind":"node","lean":"PadicInt.appr_succ_eq","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#appr-succ-eq","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$, and for each $n \\ge 0$ let $a_n$ be the unique natural number with $a_n < p^n$ and $a \\equiv a_n \\pmod{p^n \\mathbb{Z}_p}$. For every $n \\ge 0$, the $n$-th $p$-adic digit $d_n$ of $a$ satisfies $a_{n+1} = a_n + d_n p^n$.","title":"One-step reconstruction from p-adic digits"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/appr-succ-eq-val","kind":"node","lean":"PadicInt.appr_succ_eq_val","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#appr-succ-eq-val","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$, and for each $n \\ge 0$ let $a_n$ be the unique natural number with $a_n < p^n$ and $a \\equiv a_n \\pmod{p^n \\mathbb{Z}_p}$. For every $n \\ge 0$, $a_{n+1} = a_n + e_n p^n$, where $e_n$ is the $n$-th term of the $p$-adic expansion of $a$, read as a natural number.","title":"One-step reconstruction from the p-adic expansion"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/appr-eq-sum-padic-digit","kind":"node","lean":"PadicInt.appr_eq_sum_padicDigit","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#appr-eq-sum-padic-digit","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$, and for each $n \\ge 0$ let $a_n$ be the unique natural number with $a_n < p^n$ and $a \\equiv a_n \\pmod{p^n \\mathbb{Z}_p}$. For every $n \\ge 0$, $a_n = \\sum_{k=0}^{n-1} d_k p^k$, where $d_k$ is the $k$-th $p$-adic digit of $a$.","title":"Approximations as sums of p-adic digits"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/appr-eq-sum-expansion-val","kind":"node","lean":"PadicInt.appr_eq_sum_expansion_val","parent":5886,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#appr-eq-sum-expansion-val","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime and $a \\in \\mathbb{Z}_p$, and for each $n \\ge 0$ let $a_n$ be the unique natural number with $a_n < p^n$ and $a \\equiv a_n \\pmod{p^n \\mathbb{Z}_p}$. For every $n \\ge 0$, $a_n = \\sum_{k=0}^{n-1} e_k p^k$, where $e_k$ is the $k$-th term of the $p$-adic expansion of $a$, read as a natural number.","title":"Approximations as truncations of the p-adic expansion"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/eq-of-padic-digit-eq","kind":"node","lean":"PadicInt.eq_of_padicDigit_eq","parent":5886,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-adic digits determine a "],["math","\\(p\\)"],["text","-adic integer"]],"source":"$p$-adic digits determine a $p$-adic integer"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#eq-of-padic-digit-eq","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime. If two $p$-adic integers $a, a' \\in \\mathbb{Z}_p$ have the same $n$-th $p$-adic digit for every $n \\ge 0$, then $a = a'$.","title":"$p$-adic digits determine a $p$-adic integer"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/exists-padic-digit-eq","kind":"node","lean":"PadicInt.exists_padicDigit_eq","parent":5886,"reading_title":{"format":"inline-parts-v1","parts":[["text","Existence of a "],["math","\\(p\\)"],["text","-adic integer with prescribed digits"]],"source":"Existence of a $p$-adic integer with prescribed digits"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#exists-padic-digit-eq","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime. For every sequence $(b_n)_{n \\ge 0}$ of natural numbers with $b_n < p$ for all $n$, there is a $p$-adic integer $a \\in \\mathbb{Z}_p$ whose $n$-th $p$-adic digit is $b_n$ for every $n \\ge 0$.","title":"Existence of a $p$-adic integer with prescribed digits"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/exists-unique-padic-digit-eq","kind":"node","lean":"PadicInt.existsUnique_padicDigit_eq","parent":5886,"reading_title":{"format":"inline-parts-v1","parts":[["text","Existence and uniqueness of "],["math","\\(p\\)"],["text","-adic integers with prescribed digits"]],"source":"Existence and uniqueness of $p$-adic integers with prescribed digits"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#exists-unique-padic-digit-eq","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime. For every sequence $(b_n)_{n \\ge 0}$ of natural numbers with $b_n < p$ for all $n$, there is exactly one $p$-adic integer $a \\in \\mathbb{Z}_p$ whose $n$-th $p$-adic digit is $b_n$ for every $n \\ge 0$.","title":"Existence and uniqueness of $p$-adic integers with prescribed digits"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/padic-expansion-injective","kind":"node","lean":"PadicInt.padicExpansion_injective","parent":5886,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-adic expansion map is injective"]],"source":"The $p$-adic expansion map is injective"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#padic-expansion-injective","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime. If two $p$-adic integers $a, a' \\in \\mathbb{Z}_p$ have the same $p$-adic expansion, then $a = a'$.","title":"The $p$-adic expansion map is injective"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/padic-expansion-surjective","kind":"node","lean":"PadicInt.padicExpansion_surjective","parent":5886,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-adic expansion map is surjective"]],"source":"The $p$-adic expansion map is surjective"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#padic-expansion-surjective","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime. Every sequence $(b_n)_{n \\ge 0}$ with values in $\\{0, 1, \\dots, p - 1\\}$ is the $p$-adic expansion of some $p$-adic integer $a \\in \\mathbb{Z}_p$.","title":"The $p$-adic expansion map is surjective"},{"id":"NumberTheory/p-adic-numbers/p-adic-expansions/padic-expansion-bijective","kind":"node","lean":"PadicInt.padicExpansion_bijective","parent":5886,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-adic expansion map is bijective"]],"source":"The $p$-adic expansion map is bijective"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-expansions/index.html#padic-expansion-bijective","scope":"NumberTheory/p-adic-numbers/p-adic-expansions","status":"fully_proved","summary":"Let $p$ be a prime. The map sending a $p$-adic integer $a \\in \\mathbb{Z}_p$ to its $p$-adic expansion is a bijection from $\\mathbb{Z}_p$ onto the set of all sequences $(b_n)_{n \\ge 0}$ with values in $\\{0, 1, \\dots, p - 1\\}$.","title":"The $p$-adic expansion map is bijective"},{"data_url":"scopes/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals.json","id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","title":"p-adic Pontryagin duals"},{"id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/padic-quotient-int","kind":"node","lean":"MetaMathlibExt.PadicQuotientInt","parent":5902,"reading_title":{"format":"inline-parts-v1","parts":[["text","The quotient "],["math","\\(\\mathbb{Q}_p/\\mathbb{Z}_p\\)"]],"source":"The quotient $\\mathbb{Q}_p/\\mathbb{Z}_p$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html#padic-quotient-int","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","summary":"For a prime $p$, $\\mathbb{Q}_p/\\mathbb{Z}_p$ is the quotient of $\\mathbb{Q}_p$, viewed as a $\\mathbb{Z}_p$-module, by the image of $\\mathbb{Z}_p$ under the canonical inclusion $\\mathbb{Z}_p \\to \\mathbb{Q}_p$.","title":"The quotient $\\mathbb{Q}_p/\\mathbb{Z}_p$"},{"id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/padic-pontryagin-dual","kind":"node","lean":"MetaMathlibExt.PadicPontryaginDual","parent":5902,"reading_title":{"format":"inline-parts-v1","parts":[["text","Algebraic "],["math","\\(p\\)"],["text","-adic Pontryagin dual"]],"source":"Algebraic $p$-adic Pontryagin dual"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html#padic-pontryagin-dual","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","summary":"For a prime $p$ and a $\\mathbb{Z}_p$-module $A$, the algebraic $p$-adic Pontryagin dual of $A$ is the $\\mathbb{Z}_p$-module $\\operatorname{Hom}_{\\mathbb{Z}_p}(A, \\mathbb{Q}_p/\\mathbb{Z}_p)$ of all $\\mathbb{Z}_p$-linear maps from $A$ to $\\mathbb{Q}_p/\\mathbb{Z}_p$; no topology or continuity condition is imposed.","title":"Algebraic $p$-adic Pontryagin dual"},{"id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/padic-pontryagin-dual-ext","kind":"node","lean":"MetaMathlibExt.PadicPontryaginDual_ext","parent":5902,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extensionality for the "],["math","\\(p\\)"],["text","-adic Pontryagin dual"]],"source":"Extensionality for the $p$-adic Pontryagin dual"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html#padic-pontryagin-dual-ext","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","summary":"Let $p$ be a prime and $A$ a $\\mathbb{Z}_p$-module. If two elements $\\varphi, \\psi$ of the algebraic $p$-adic Pontryagin dual of $A$ satisfy $\\varphi(a) = \\psi(a)$ for every $a \\in A$, then $\\varphi = \\psi$.","title":"Extensionality for the $p$-adic Pontryagin dual"},{"id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/padic-quotient-map","kind":"node","lean":"MetaMathlibExt.padicQuotientMap","parent":5902,"reading_title":{"format":"inline-parts-v1","parts":[["text","The quotient map "],["math","\\(\\mathbb{Q}_p \\to \\mathbb{Q}_p/\\mathbb{Z}_p\\)"]],"source":"The quotient map $\\mathbb{Q}_p \\to \\mathbb{Q}_p/\\mathbb{Z}_p$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html#padic-quotient-map","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","summary":"For a prime $p$, this is the canonical projection $\\mathbb{Q}_p \\to \\mathbb{Q}_p/\\mathbb{Z}_p$, $x \\mapsto x + \\mathbb{Z}_p$, as a $\\mathbb{Z}_p$-linear map.","title":"The quotient map $\\mathbb{Q}_p \\to \\mathbb{Q}_p/\\mathbb{Z}_p$"},{"id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/padic-quotient-map-algebra-map-eq-zero","kind":"node","lean":"MetaMathlibExt.padicQuotientMap_algebraMap_eq_zero","parent":5902,"reading_title":{"format":"inline-parts-v1","parts":[["text","The quotient map vanishes on "],["math","\\(\\mathbb{Z}_p\\)"]],"source":"The quotient map vanishes on $\\mathbb{Z}_p$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html#padic-quotient-map-algebra-map-eq-zero","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","summary":"For a prime $p$ and every $z \\in \\mathbb{Z}_p$, the quotient map $\\mathbb{Q}_p \\to \\mathbb{Q}_p/\\mathbb{Z}_p$ sends the image of $z$ in $\\mathbb{Q}_p$ to $0$.","title":"The quotient map vanishes on $\\mathbb{Z}_p$"},{"id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/padic-quotient-map-mk-q-eq","kind":"node","lean":"MetaMathlibExt.padicQuotientMap_mkQ_eq","parent":5902,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html#padic-quotient-map-mk-q-eq","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","summary":"For a prime $p$ and every $x \\in \\mathbb{Q}_p$, the quotient map $\\mathbb{Q}_p \\to \\mathbb{Q}_p/\\mathbb{Z}_p$ sends $x$ to its class $x + \\mathbb{Z}_p$ in the quotient module.","title":"The quotient map is the quotient projection"},{"id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/contragredient-smul","kind":"node","lean":"MetaMathlibExt.contragredientSMul","parent":5902,"reading_title":{"format":"inline-parts-v1","parts":[["text","Contragredient action on the "],["math","\\(p\\)"],["text","-adic Pontryagin dual"]],"source":"Contragredient action on the $p$-adic Pontryagin dual"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html#contragredient-smul","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","summary":"Let $p$ be a prime, $G$ a group and $A$ a $\\mathbb{Z}_p$-module on which $G$ acts by additive maps commuting with the $\\mathbb{Z}_p$-action. For $g \\in G$ and $\\varphi$ in the algebraic $p$-adic Pontryagin dual of $A$, $g \\cdot \\varphi$ is the element $a \\mapsto \\varphi(g^{-1} a)$ of the dual.","title":"Contragredient action on the $p$-adic Pontryagin dual"},{"id":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/smul-apply","kind":"node","lean":"MetaMathlibExt.smul_apply","parent":5902,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-pontryagin-duals/index.html#smul-apply","scope":"NumberTheory/p-adic-numbers/p-adic-pontryagin-duals","status":"fully_proved","summary":"Let $p$ be a prime, $G$ a group and $A$ a $\\mathbb{Z}_p$-module on which $G$ acts by additive maps commuting with the $\\mathbb{Z}_p$-action. For $g \\in G$, $\\varphi$ in the algebraic $p$-adic Pontryagin dual of $A$ and $a \\in A$, the contragredient action satisfies $(g \\cdot \\varphi)(a) = \\varphi(g^{-1} a)$.","title":"Evaluating the contragredient action"},{"data_url":"scopes/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials.json","id":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/index.html","scope":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","status":"fully_proved","title":"p-adic valuations of polynomials"},{"id":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/padic-polynomial-dilatation","kind":"node","lean":"PadicPolynomialDilatation","parent":5911,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/index.html#padic-polynomial-dilatation","scope":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","status":"fully_proved","summary":"Let $p$ be a prime, $f \\in \\mathbb{Z}_p[X]$ and $x_0 \\in \\mathbb{Z}_p$. Dilatation data for $f$ at $x_0$ consist of a natural number $e$ and a polynomial $g \\in \\mathbb{Z}_p[X]$ such that $g(X) = p^{-e} f(x_0 + pX)$ as polynomials over $\\mathbb{Q}_p$, and the reduction of $g$ modulo $p$ is nonzero.","title":"Dilatation of a p-adic polynomial at a point"},{"id":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/padic-polynomial-dilatation-one-zero","kind":"node","lean":"padicPolynomialDilatation_one_zero","parent":5911,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/index.html#padic-polynomial-dilatation-one-zero","scope":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","status":"fully_proved","summary":"For every prime $p$, the exponent $e = 0$ and the polynomial $g = 1$ form dilatation data for the constant polynomial $1 \\in \\mathbb{Z}_p[X]$ at the point $x_0 = 0$.","title":"Dilatation of the constant polynomial one at zero"},{"id":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/padic-polynomial-dilatation-exists","kind":"node","lean":"padicPolynomialDilatation_exists","parent":5911,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/index.html#padic-polynomial-dilatation-exists","scope":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","status":"fully_proved","summary":"Let $p$ be a prime and let $f \\in \\mathbb{Z}_p[X]$ have nonzero reduction modulo $p$. Then for every $x_0 \\in \\mathbb{Z}_p$ there are a natural number $e$ and a polynomial $g \\in \\mathbb{Z}_p[X]$ with nonzero reduction modulo $p$ such that $g(X) = p^{-e} f(x_0 + pX)$ as polynomials over $\\mathbb{Q}_p$; that is, dilatation data for $f$ at $x_0$ exist.","title":"Existence of dilatations of p-adic polynomials"},{"id":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/padic-valuation","kind":"node","lean":"Polynomial.padicValuation","parent":5911,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-adic valuation of an integer polynomial"]],"source":"The $p$-adic valuation of an integer polynomial"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/index.html#padic-valuation","scope":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","status":"fully_proved","summary":"Let $p$ be a prime and let $f \\in \\mathbb{Z}[X]$ be a nonzero polynomial, with $a_n$ the coefficient of $X^n$ in $f$. Writing $v_p(a)$ for the $p$-adic valuation of an integer $a$, the $p$-adic valuation of $f$ is the natural number $v_p(f) = \\min \\{ v_p(a_n) : a_n \\neq 0 \\}$.","title":"The $p$-adic valuation of an integer polynomial"},{"id":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/padic-valuation-le-coeff","kind":"node","lean":"Polynomial.padicValuation_le_coeff","parent":5911,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/index.html#padic-valuation-le-coeff","scope":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","status":"fully_proved","summary":"Let $p$ be a prime, write $v_p(a)$ for the $p$-adic valuation of an integer $a$, let $f \\in \\mathbb{Z}[X]$ be a nonzero polynomial with $a_n$ the coefficient of $X^n$ in $f$, and let $v_p(f) = \\min \\{ v_p(a_n) : a_n \\neq 0 \\}$ be its $p$-adic valuation. If $a_n \\neq 0$, then $v_p(f) \\le v_p(a_n)$.","title":"The valuation of a polynomial is at most that of each nonzero coefficient"},{"id":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/le-padic-valuation","kind":"node","lean":"Polynomial.le_padicValuation","parent":5911,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/index.html#le-padic-valuation","scope":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","status":"fully_proved","summary":"Let $p$ be a prime, write $v_p(a)$ for the $p$-adic valuation of an integer $a$, let $f \\in \\mathbb{Z}[X]$ be a nonzero polynomial with $a_n$ the coefficient of $X^n$ in $f$, and let $v_p(f) = \\min \\{ v_p(a_n) : a_n \\neq 0 \\}$ be its $p$-adic valuation. If $k$ is a natural number with $k \\le v_p(a_n)$ for every $n$ with $a_n \\neq 0$, then $k \\le v_p(f)$.","title":"Lower bounds for the valuation of a polynomial"},{"id":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/padic-valuation-c","kind":"node","lean":"Polynomial.padicValuation_C","parent":5911,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials/index.html#padic-valuation-c","scope":"NumberTheory/p-adic-numbers/p-adic-valuations-of-polynomials","status":"fully_proved","summary":"Let $p$ be a prime and let $a$ be a nonzero integer. The $p$-adic valuation of the constant polynomial $a \\in \\mathbb{Z}[X]$, the minimum of the $p$-adic valuations of its nonzero coefficients, equals the $p$-adic valuation $v_p(a)$ of the integer $a$.","title":"The valuation of a nonzero constant polynomial"},{"data_url":"scopes/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi.json","id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","title":"Points of valuation one of two-variable polynomials"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/card-lift-padic-val-gt-one-le-one","kind":"node","lean":"MetaMathlibExt.card_lift_padicVal_gt_one_le_one","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#card-lift-padic-val-gt-one-le-one","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $p$ be a prime, $g \\in \\mathbb{Z}[x]$ and $r$ an integer with $0 \\le r < p$ such that $g(r)$ and $g'(r)$ are not both divisible by $p$. Then at most one $k \\in \\{0, \\dots, p - 1\\}$ satisfies $p^2 \\mid g(r + kp)$.","title":"At most one lift has valuation above one"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/card-padic-val-gt-one-le-nat-degree","kind":"node","lean":"MetaMathlibExt.card_padicVal_gt_one_le_natDegree","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Few residues modulo "],["math","\\(p^2\\)"],["text"," with valuation above one"]],"source":"Few residues modulo $p^2$ with valuation above one"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#card-padic-val-gt-one-le-nat-degree","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $g \\in \\mathbb{Z}[x]$ be a non-zero polynomial of degree $d$ with non-zero discriminant. Then there is a natural number $P$ such that for every prime $p \\ge P$, at most $d$ integers $x$ with $0 \\le x < p^2$ satisfy $p^2 \\mid g(x)$.","title":"Few residues modulo $p^2$ with valuation above one"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/card-lift-padic-val-eq-one","kind":"node","lean":"MetaMathlibExt.card_lift_padicVal_eq_one","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#card-lift-padic-val-eq-one","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $p$ be a prime, let $f \\in \\mathbb{Z}[x, y]$ have partial derivatives $f_x$ and $f_y$, and let $r, s$ be integers in $\\{0, \\dots, p - 1\\}$ with $p \\mid f(r, s)$ such that $f_x(r, s)$ and $f_y(r, s)$ are not both divisible by $p$. Then exactly $p(p - 1)$ of the pairs $(k, l)$ with $0 \\le k, l < p$ satisfy $p \\mid f(r + kp, s + lp)$ and $p^2 \\nmid f(r + kp, s + lp)$, that is, $\\nu_p(f(r + kp, s + lp)) = 1$.","title":"Lifts of a nonsingular point with valuation one"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/card-common-lift-padic-val-of-gradients","kind":"node","lean":"MetaMathlibExt.card_common_lift_padicVal_of_gradients_independent","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#card-common-lift-padic-val-of-gradients","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $p$ be a prime, let $f \\in \\mathbb{Z}[x, y]$ have partial derivatives $f_x$ and $f_y$, and let $(r_1, s_1)$ and $(r_2, s_2)$ be pairs of integers in $\\{0, \\dots, p - 1\\}$ with $p \\mid f(r_i, s_i)$ for $i = 1, 2$ and","title":"Common lifts of two points with independent gradients"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/card-padic-val-eq-one-mod-sq-eq","kind":"node","lean":"MetaMathlibExt.card_padicVal_eq_one_mod_sq_eq","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of points of valuation one modulo "],["math","\\(p^2\\)"]],"source":"Number of points of valuation one modulo $p^2$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#card-padic-val-eq-one-mod-sq-eq","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $p$ be a prime and let $f \\in \\mathbb{Z}[x, y]$ have partial derivatives $f_x$ and $f_y$. Suppose that for all integers $r, s \\in \\{0, \\dots, p - 1\\}$ with $p \\mid f(r, s)$, the integers $f_x(r, s)$ and $f_y(r, s)$ are not both divisible by $p$. Let $m$ be the number of pairs $(r, s)$ with $0 \\le r, s < p$ and $p \\mid f(r, s)$. Then the number of pairs $(x, y)$ with $0 \\le x, y < p^2$, $p \\mid f(x, y)$ and\u2026","title":"Number of points of valuation one modulo $p^2$"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/card-lift-padic-val-eq-one-of-singular","kind":"node","lean":"MetaMathlibExt.card_lift_padicVal_eq_one_of_singular","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#card-lift-padic-val-eq-one-of-singular","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $p$ be a prime, let $f \\in \\mathbb{Z}[x, y]$ have partial derivatives $f_x$ and $f_y$, and let $r, s$ be integers in $\\{0, \\dots, p - 1\\}$ such that $p$ divides $f(r, s)$, $f_x(r, s)$ and $f_y(r, s)$. Then the number of pairs $(k, l)$ with $0 \\le k, l < p$, $p \\mid f(r + kp, s + lp)$ and $p^2 \\nmid f(r + kp, s + lp)$ is $0$ if $p^2 \\mid f(r, s)$, and $p^2$ otherwise.","title":"Lifts of a singular point with valuation one"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/card-common-lift-padic-val-of-gradients-2","kind":"node","lean":"MetaMathlibExt.card_common_lift_padicVal_of_gradients_dependent","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#card-common-lift-padic-val-of-gradients-2","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $p$ be a prime, let $f \\in \\mathbb{Z}[x, y]$ have partial derivatives $f_x$ and $f_y$, and let $(r_1, s_1)$ and $(r_2, s_2)$ be pairs of integers in $\\{0, \\dots, p - 1\\}$ with $p \\mid f(r_i, s_i)$ for $i = 1, 2$. Write $a_i = f_x(r_i, s_i)$, $b_i = f_y(r_i, s_i)$ and $c_i = f(r_i, s_i)/p$. Suppose that for each $i$ the integers $a_i$ and $b_i$ are not both divisible by $p$, and that $p \\mid a_1 b_2 - b_1 a_2$.\u2026","title":"Common lifts of two points with dependent gradients"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-curve-points","kind":"node","lean":"MetaMathlibExt.valuationCurvePoints","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Points of "],["math","\\(C_f(\\mathbb{F}_p)\\)"]],"source":"Points of $C_f(\\mathbb{F}_p)$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-curve-points","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For $f \\in \\mathbb{Z}[x, y]$ and a natural number $p$, $C_f(\\mathbb{F}_p)$ is the set of pairs $(r, s)$ of natural numbers with $r, s < p$ such that $p$ divides $f(r, s)$.","title":"Points of $C_f(\\mathbb{F}_p)$"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-curve-points-card-le","kind":"node","lean":"MetaMathlibExt.valuationCurvePoints_card_le","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(C_f(\\mathbb{F}_p)\\)"],["text"," has at most "],["math","\\(p^2\\)"],["text"," points"]],"source":"$C_f(\\mathbb{F}_p)$ has at most $p^2$ points"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-curve-points-card-le","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For every $f \\in \\mathbb{Z}[x, y]$ and every natural number $p$, the set $C_f(\\mathbb{F}_p)$ of pairs $(r, s)$ of natural numbers with $r, s < p$ such that $p$ divides $f(r, s)$ has at most $p^2$ elements.","title":"$C_f(\\mathbb{F}_p)$ has at most $p^2$ points"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-row","kind":"node","lean":"MetaMathlibExt.valuationRow","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-row","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For $f \\in \\mathbb{Z}[x, y]$ with partial derivatives $f_x$ and $f_y$, a natural number $p$ and a pair $(r, s)$ of natural numbers, the valuation row at $(r, s)$ is the vector $(f_x(r, s), f_y(r, s), q)$ reduced modulo $p$, an element of $(\\mathbb{Z}/p\\mathbb{Z})^3$, where $q$ is $f(r, s)/p$ rounded down ($q = 0$ when $p = 0$). These are the rows of the matrix in the paper's eq. (2); when $p$ divides $f(r, s)$, the\u2026","title":"Valuation row of a point"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-distinct-tuples","kind":"node","lean":"MetaMathlibExt.valuationDistinctTuples","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Distinct "],["math","\\(k\\)"],["text","-tuples of points of "],["math","\\(C_f(\\mathbb{F}_p)\\)"]],"source":"Distinct $k$-tuples of points of $C_f(\\mathbb{F}_p)$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-distinct-tuples","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For $f \\in \\mathbb{Z}[x, y]$ and natural numbers $p$ and $k$, let $C_f(\\mathbb{F}_p)$ be the set of pairs $(r, s)$ of natural numbers with $r, s < p$ such that $p$ divides $f(r, s)$. The distinct $k$-tuples of points of $C_f(\\mathbb{F}_p)$ are the $k$-tuples $(P_1, \\dots, P_k)$ with every $P_i$ in $C_f(\\mathbb{F}_p)$ and $P_i \\ne P_j$ for $i \\ne j$.","title":"Distinct $k$-tuples of points of $C_f(\\mathbb{F}_p)$"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-rank-two-tuple-count","kind":"node","lean":"MetaMathlibExt.valuationRankTwoTupleCount","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank-two tuple count "],["math","\\(m_k(f, p)\\)"]],"source":"Rank-two tuple count $m_k(f, p)$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-rank-two-tuple-count","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For $f \\in \\mathbb{Z}[x, y]$ and natural numbers $p$ and $k$, let $C_f(\\mathbb{F}_p)$ be the set of pairs $(r, s)$ of natural numbers with $r, s < p$ such that $p$ divides $f(r, s)$. Then $m_k(f, p)$ is the number of $k$-tuples $P = (P_1, \\dots, P_k)$ of pairwise distinct points of $C_f(\\mathbb{F}_p)$ such that the $k \\times 3$ matrix $B_k(P)$ over $\\mathbb{Z}/p\\mathbb{Z}$ whose $i$-th row is the valuation row at\u2026","title":"Rank-two tuple count $m_k(f, p)$"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-rank-two-tuple-count-one","kind":"node","lean":"MetaMathlibExt.valuationRankTwoTupleCount_one","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","The rank-two tuple count vanishes for "],["math","\\(k = 1\\)"]],"source":"The rank-two tuple count vanishes for $k = 1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-rank-two-tuple-count-one","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For every $f \\in \\mathbb{Z}[x, y]$ and every prime $p$, the rank-two tuple count $m_1(f, p)$ is $0$: a matrix with a single row has rank at most $1$.","title":"The rank-two tuple count vanishes for $k = 1$"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-rank-one-tuple-count","kind":"node","lean":"MetaMathlibExt.valuationRankOneTupleCount","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank-one tuple count "],["math","\\(o_k(f, p)\\)"]],"source":"Rank-one tuple count $o_k(f, p)$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-rank-one-tuple-count","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For $f \\in \\mathbb{Z}[x, y]$ and natural numbers $p$ and $k$, let $C_f(\\mathbb{F}_p)$ be the set of pairs $(r, s)$ of natural numbers with $r, s < p$ such that $p$ divides $f(r, s)$. Then $o_k(f, p)$ is the number of $k$-tuples $P = (P_1, \\dots, P_k)$ of pairwise distinct points of $C_f(\\mathbb{F}_p)$ such that the $k \\times 3$ matrix $B_k(P)$ over $\\mathbb{Z}/p\\mathbb{Z}$ whose $i$-th row is the valuation row at\u2026","title":"Rank-one tuple count $o_k(f, p)$"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-common-lift-count","kind":"node","lean":"MetaMathlibExt.valuationCommonLiftCount","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-common-lift-count","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"Let $f \\in \\mathbb{Z}[x, y]$, let $p$ and $k$ be natural numbers and let $P = ((r_1, s_1), \\dots, (r_k, s_k))$ be a $k$-tuple of pairs of natural numbers. The common lift count of $P$ is the number of pairs $(a, b)$ of natural numbers with $a, b < p$ such that $p^2$ divides $f(r_i + ap, s_i + bp)$ for every $i$.","title":"Common lift count"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-common-lift-count-le","kind":"node","lean":"MetaMathlibExt.valuationCommonLiftCount_le","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","The common lift count is at most "],["math","\\(p^2\\)"]],"source":"The common lift count is at most $p^2$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-common-lift-count-le","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For every $f \\in \\mathbb{Z}[x, y]$, all natural numbers $p$ and $k$ and every $k$-tuple $P$ of pairs of natural numbers, the common lift count of $P$ is at most $p^2$.","title":"The common lift count is at most $p^2$"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-square-count","kind":"node","lean":"MetaMathlibExt.valuationSquareCount","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-square-count","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For a polynomial $f \\in \\mathbb{Z}[x, y]$ and natural numbers $p$, $k$ and $l$, $X(k, l)$ is the number of points $(x, y)$ of the square $\\{kp, \\dots, kp + p - 1\\} \\times \\{lp, \\dots, lp + p - 1\\}$ with $p^2 \\mid f(x, y)$. For a prime $p$ this is the condition $\\nu_p(f(x, y)) > 1$, where points with $f(x, y) = 0$ count.","title":"Count of points of high valuation in a square"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-square-count-le","kind":"node","lean":"MetaMathlibExt.valuationSquareCount_le","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-square-count-le","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For every polynomial $f \\in \\mathbb{Z}[x, y]$ and all natural numbers $p$, $k$ and $l$, the number $X(k, l)$ of points $(x, y)$ of the square $\\{kp, \\dots, kp + p - 1\\} \\times \\{lp, \\dots, lp + p - 1\\}$ with $p^2 \\mid f(x, y)$ is at most $p^2$.","title":"Trivial bound for the square count"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/valuation-conjecture-one","kind":"node","lean":"MetaMathlibExt.ValuationConjectureOne","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#valuation-conjecture-one","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"fully_proved","summary":"For $f \\in \\mathbb{Z}[x, y]$, the proposition that for every $k \\ge 2$, $m_k(f, p)/p^2 \\to 1$ as $p \\to \\infty$ through the primes, where $m_k(f, p)$ is the rank-two tuple count.","title":"Rajkumar\u2013Shubham Conjecture 1"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/sum-expectation-distinct-tuples-eq","kind":"node","lean":"MetaMathlibExt.sum_expectation_distinct_tuples_eq","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#sum-expectation-distinct-tuples-eq","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $p$ be a prime, let $f \\in \\mathbb{Z}[x, y]$ have partial derivatives $f_x$ and $f_y$, and let $C_f(\\mathbb{F}_p)$ be the set of pairs of integers $0 \\le r, s < p$ with $p \\mid f(r, s)$. Suppose that no $(r, s) \\in C_f(\\mathbb{F}_p)$ has $p \\mid f_x(r, s)$ and $p \\mid f_y(r, s)$, and let $k \\ge 1$. For a $k$-tuple $P$ of points of $C_f(\\mathbb{F}_p)$, let $N(P)$ be the number of pairs $(a, b)$ with\u2026","title":"Sum of expectations over distinct tuples of points"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/tendsto-sum-expectation-distinct-tuples","kind":"node","lean":"MetaMathlibExt.tendsto_sum_expectation_distinct_tuples","parent":5919,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#tendsto-sum-expectation-distinct-tuples","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x, y]$ have partial derivatives $f_x$ and $f_y$, and for a prime $p$ let $C_f(\\mathbb{F}_p)$ be the set of pairs of integers $0 \\le r, s < p$ with $p \\mid f(r, s)$. Suppose that there is a natural number $P_0$ such that for every prime $p \\ge P_0$ and all integers $x, y$, the integers $f(x, y)$, $f_x(x, y)$ and $f_y(x, y)$ are not all divisible by $p$; that $|C_f(\\mathbb{F}_p)|/p \\to 1$ as\u2026","title":"Sums of expectations over distinct tuples tend to one"},{"id":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/tendsto-valuation-square-count-poisson","kind":"node","lean":"MetaMathlibExt.tendsto_valuationSquareCount_poisson","parent":5919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Poisson limit for points of valuation "],["math","\\(> 1\\)"]],"source":"Poisson limit for points of valuation $> 1$"},"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi/index.html#tendsto-valuation-square-count-poisson","scope":"NumberTheory/p-adic-numbers/points-of-valuation-one-of-two-variable-polynomi","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x, y]$ have partial derivatives $f_x$ and $f_y$, and for a prime $p$ let $C_f(\\mathbb{F}_p)$ be the set of pairs of integers $0 \\le r, s < p$ with $p \\mid f(r, s)$. Suppose that there is a natural number $P_0$ such that for every prime $p \\ge P_0$ and all integers $x, y$, the integers $f(x, y)$, $f_x(x, y)$ and $f_y(x, y)$ are not all divisible by $p$; that $|C_f(\\mathbb{F}_p)|/p \\to 1$ as\u2026","title":"Poisson limit for points of valuation $> 1$"},{"data_url":"scopes/NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions.json","id":"NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions","kind":"scope","navigation":"module","parent":5797,"result":false,"route":"../roadmap/NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions/index.html","scope":"NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions","status":"fully_proved","title":"Spectral norm on algebraic extensions"},{"id":"NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions/spectral-norm-le-one-iff-is-integral","kind":"node","lean":"MetaMathlibExt.spectralNorm_le_one_iff_isIntegral","parent":5942,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions/index.html#spectral-norm-le-one-iff-is-integral","scope":"NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions","status":"fully_proved","summary":"Let $A$ be an integral domain, $K$ a complete nontrivially normed field whose norm is ultrametric, and $L$ a field extension of $K$ of finite degree, where $K$ and $L$ are $A$-algebras and the algebra maps $A \\to K \\to L$ compose to $A \\to L$. Suppose that for every $y \\in K$, $\\|y\\| \\le 1$ if and only if $y$ is integral over $A$. Then for every $x \\in L$, the spectral norm of $x$ over $K$ is at most $1$ if and\u2026","title":"Spectral-norm unit ball is the integral closure"},{"id":"NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions/spectral-norm-le-one-iff-exists-integral","kind":"node","lean":"MetaMathlibExt.spectralNorm_le_one_iff_exists_integralClosure","parent":5942,"result":true,"route":"../roadmap/NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions/index.html#spectral-norm-le-one-iff-exists-integral","scope":"NumberTheory/p-adic-numbers/spectral-norm-on-algebraic-extensions","status":"fully_proved","summary":"Let $A$ be an integral domain, $K$ a complete nontrivially normed field whose norm is ultrametric, and $L$ a field extension of $K$ of finite degree, where $K$ and $L$ are $A$-algebras and the algebra maps $A \\to K \\to L$ compose to $A \\to L$. Let $B$ be an integral domain that is an $A$-algebra with an algebra map $B \\to L$ such that $A \\to B \\to L$ composes to $A \\to L$, and suppose $B$ is an integral closure of\u2026","title":"Spectral-norm unit ball as the image of an integral closure"},{"data_url":"scopes/NumberTheory/arithmetic-geometry.json","id":"NumberTheory/arithmetic-geometry","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/arithmetic-geometry/index.html","scope":"NumberTheory/arithmetic-geometry","status":"can_prove","summary":"Rational points of varieties over number fields: torsion and rank of elliptic curves, naive heights, Hasse principle, local-global failures.","title":"Arithmetic geometry"},{"data_url":"scopes/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces.json","id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","kind":"scope","navigation":"module","parent":5945,"result":false,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"can_prove","title":"Hasse principle for random Fano hypersurfaces"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/untitled","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.\ud835\udd38","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#untitled","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"fully_proved","summary":"$\\mathbb{A}_\\mathbb{Q}$, also written $\\mathbb{A}$, is the adele ring of $\\mathbb{Q}$.","title":"Adele ring of the rationals"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/height","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.height","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#height","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"fully_proved","summary":"Let $f$ be a multivariate polynomial with rational coefficients. Multiply the coefficients of $f$ by the least common multiple of their denominators and divide the resulting integers by their greatest common divisor; this gives the primitive integer coefficient vector $(a_m)_m$ of $f$, indexed by the monomials $m$ in the support of $f$. The height of $f$ is the Euclidean norm $\\sqrt{\\sum_m a_m^2}$ of this vector.\u2026","title":"Height of a rational polynomial"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/has-solutions-in","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.HasSolutionsIn","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#has-solutions-in","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"fully_proved","summary":"Let $R$ and $S$ be commutative semirings with $S$ an $R$-algebra, and let $f$ be a multivariate polynomial over $R$ in variables indexed by a type $\\sigma$. Then $f$ has a solution in $S$ if there is a vector $x \\in S^\\sigma$ whose components generate the unit ideal of $S$ and such that $f(x) = 0$, where $f$ is evaluated at $x$ through the algebra map $R \\to S$.","title":"Unimodular solutions of a polynomial"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/untitled-2","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.\ud835\udd4d","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#untitled-2","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"fully_proved","summary":"$\\mathbb{V}_{d,n}$ is the projectivization of the $\\mathbb{Q}$-vector space of homogeneous polynomials of degree $d$ in $n + 1$ variables with rational coefficients; its points are the degree-$d$ hypersurfaces in $\\mathbb{P}^n$ defined over $\\mathbb{Q}$. Since this space has dimension $N_{d,n} = \\binom{n+d}{d}$, the number of degree-$d$ monomials in $n + 1$ variables, $\\mathbb{V}_{d,n}$ is a copy of\u2026","title":"Parameter space of hypersurfaces"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/height-le","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.heightLE","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#height-le","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"fully_proved","summary":"For a real number $A$, $\\mathbb{V}_{d,n}(A)$ is the set of degree-$d$ hypersurfaces $V \\in \\mathbb{V}_{d,n}$ in $\\mathbb{P}^n$ whose height is at most $A$, where the height of $V$ is the height of the chosen nonzero homogeneous polynomial of degree $d$ that represents $V$.","title":"Hypersurfaces of bounded height"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/loc","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.\ud835\udd4dLoc","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#loc","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"fully_proved","summary":"$\\mathbb{V}^{\\mathrm{loc}}_{d,n}$ is the set of degree-$d$ hypersurfaces $V \\in \\mathbb{V}_{d,n}$ in $\\mathbb{P}^n$ that are everywhere locally soluble, that is, with $V(\\mathbb{A}_\\mathbb{Q}) \\ne \\emptyset$: the chosen homogeneous polynomial representing $V$ has a solution in the adele ring $\\mathbb{A}_\\mathbb{Q}$, namely a zero given by $n + 1$ adeles that generate the unit ideal of $\\mathbb{A}_\\mathbb{Q}$.","title":"Everywhere locally soluble hypersurfaces"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/loc-2","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.\u03c1Loc","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#loc-2","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"fully_proved","summary":"For natural numbers $d$ and $n$ and a real number $A$, let $\\mathbb{V}_{d,n}(A)$ be the set of degree-$d$ hypersurfaces in $\\mathbb{P}^n$ of height at most $A$, and let $\\mathbb{V}^{\\mathrm{loc}}_{d,n}$ be the set of everywhere locally soluble ones. The local density $\\rho^{\\mathrm{loc}}_{d,n}$ is the limit, as $A \\to \\infty$, of the number of elements of $\\mathbb{V}^{\\mathrm{loc}}_{d,n} \\cap \\mathbb{V}_{d,n}(A)$\u2026","title":"Local density of everywhere locally soluble hypersurfaces"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/untitled-3","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.\u03c1","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#untitled-3","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"fully_proved","summary":"For a real number $A$, $\\rho_{d,n}(A)$ is the proportion of hypersurfaces in $\\mathbb{V}_{d,n}(A)$, the degree-$d$ hypersurfaces in $\\mathbb{P}^n$ of height at most $A$, that have a rational point: it is the number of $V \\in \\mathbb{V}_{d,n}(A)$ with $V(\\mathbb{Q}) \\ne \\emptyset$ divided by the number of elements of $\\mathbb{V}_{d,n}(A)$, taken to be $0$ when $\\mathbb{V}_{d,n}(A)$ is empty. Here\u2026","title":"Proportion of soluble hypersurfaces of bounded height"},{"id":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/theorem-1-1","kind":"node","lean":"MathlibExt.NumberTheory.HassePrincipleRandomFanoWanted.HassePrincipleRandomFano.theorem_1_1","parent":5946,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces/index.html#theorem-1-1","scope":"NumberTheory/arithmetic-geometry/hasse-principle-for-random-fano-hypersurfaces","status":"can_prove","summary":"Let $d$ and $n$ be natural numbers with $d \\ge 2$, $n \\ge d$ and $(d, n) \\ne (3, 3)$. Then the proportion $\\rho_{d,n}(A)$ of degree-$d$ hypersurfaces in $\\mathbb{P}^n$ of height at most $A$ that have a rational point tends to the local density $\\rho^{\\mathrm{loc}}_{d,n}$ as the real number $A$ tends to $\\infty$.","title":"Density of soluble Fano hypersurfaces"},{"data_url":"scopes/NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves.json","id":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves","kind":"scope","navigation":"module","parent":5945,"result":false,"route":"../roadmap/NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/index.html","scope":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves","status":"can_prove","title":"Heights of rational points on elliptic curves"},{"id":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/naive-height-lower-bound-of-unit-coeff","kind":"node","lean":"MetaMathlibExt.naiveHeight_lower_bound_of_unit_coeff","parent":5956,"reading_title":{"format":"inline-parts-v1","parts":[["text","Height lower bounds on "],["math","\\(y^2 = x^3 + ax + b\\)"],["text"," when "],["math","\\(a = \\pm 1\\)"],["text"," or "],["math","\\(b = \\pm 1\\)"]],"source":"Height lower bounds on $y^2 = x^3 + ax + b$ when $a = \\pm 1$ or $b = \\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/index.html#naive-height-lower-bound-of-unit-coeff","scope":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves","status":"can_prove","summary":"Let $a, b \\in \\mathbb{Z}$ with $4a^3 + 27b^2 \\ne 0$, and for $u, v \\in \\mathbb{Q}$ let $H(u, v) = \\max(|U|, |V|, |W|)$, where $U, V, W$ are coprime integers with $(U : V : W) = (u : v : 1)$. Consider the rational solutions $(u, v)$ of $v^2 = u^3 + au + b$. (a) If $a = \\pm 1$ and $b \\notin \\{0, 1, -1\\}$, every solution has $H(u, v) \\ge (|b|/5)^{1/3}$. (b) If $a \\notin \\{0, 1, -1\\}$ and $b = -1$, every solution has\u2026","title":"Height lower bounds on $y^2 = x^3 + ax + b$ when $a = \\pm 1$ or $b = \\pm 1$"},{"id":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/naive-height-lower-bound-of-special-coef","kind":"node","lean":"MetaMathlibExt.naiveHeight_lower_bound_of_special_coeff","parent":5956,"reading_title":{"format":"inline-parts-v1","parts":[["text","Height lower bounds on "],["math","\\(y^2 = x^3 + ax + b\\)"],["text"," when "],["math","\\(b = \\pm a\\)"],["text",", "],["math","\\(b = 0\\)"],["text"," or "],["math","\\(a = 0\\)"]],"source":"Height lower bounds on $y^2 = x^3 + ax + b$ when $b = \\pm a$, $b = 0$ or $a = 0$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/index.html#naive-height-lower-bound-of-special-coef","scope":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves","status":"can_prove","summary":"Let $a, b \\in \\mathbb{Z}$ with $4a^3 + 27b^2 \\ne 0$, and for $u, v \\in \\mathbb{Q}$ let $H(u, v) = \\max(|U|, |V|, |W|)$, where $U, V, W$ are coprime integers with $(U : V : W) = (u : v : 1)$. Consider the rational solutions $(u, v)$ of $v^2 = u^3 + au + b$. (a) If $b = a$, every solution has $H(u, v) \\ge (|a|/4)^{1/3}$. (b) If $b = -a$, every solution with $H(u, v) < (|a|/4)^{1/3}$ is $(1, 1)$ or $(1, -1)$. (c) If\u2026","title":"Height lower bounds on $y^2 = x^3 + ax + b$ when $b = \\pm a$, $b = 0$ or $a = 0$"},{"id":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/naive-height-lt-sqrt-mul-naive-height-x","kind":"node","lean":"MetaMathlibExt.naiveHeight_lt_sqrt_mul_naiveHeight_x_rpow","parent":5956,"reading_title":{"format":"inline-parts-v1","parts":[["text","Height of a point on "],["math","\\(y^2 = x^3 + ax - b\\)"],["text"," versus the height of its "],["math","\\(x\\)"],["text","-coordinate"]],"source":"Height of a point on $y^2 = x^3 + ax - b$ versus the height of its $x$-coordinate"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/index.html#naive-height-lt-sqrt-mul-naive-height-x","scope":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves","status":"can_prove","summary":"Let $a, b \\in \\mathbb{Z}$ with $a \\ne 0$, $b > 0$ and $4a^3 + 27b^2 \\ne 0$. For $u \\in \\mathbb{Q}$ let $H(u) = \\max(|u_0|, u_1)$, where $u = u_0/u_1$ in lowest terms with $u_1 > 0$, and for $u, v \\in \\mathbb{Q}$ let $H(u, v) = \\max(|U|, |V|, |W|)$, where $U, V, W$ are coprime integers with $(U : V : W) = (u : v : 1)$. Then every rational solution $(u, v)$ of $v^2 = u^3 + au - b$ satisfies\u2026","title":"Height of a point on $y^2 = x^3 + ax - b$ versus the height of its $x$-coordinate"},{"id":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/naive-height-gt-of-small-linear-coeff","kind":"node","lean":"MetaMathlibExt.naiveHeight_gt_of_small_linear_coeff","parent":5956,"reading_title":{"format":"inline-parts-v1","parts":[["text","Height lower bound on "],["math","\\(y^2 = x^3 + ax + b\\)"],["text"," when "],["math","\\(|a|\\)"],["text"," is small"]],"source":"Height lower bound on $y^2 = x^3 + ax + b$ when $|a|$ is small"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves/index.html#naive-height-gt-of-small-linear-coeff","scope":"NumberTheory/arithmetic-geometry/heights-of-rational-points-on-elliptic-curves","status":"can_prove","summary":"Let $a, b \\in \\mathbb{Z}$ with $ab \\ne 0$ and $4a^3 + 27b^2 \\ne 0$, let $\\lVert p \\rVert = \\sqrt{2 + a^2 + b^2}$, and suppose $1 + |a| < \\lVert p \\rVert^{1/2}/1.782$. For $u, v \\in \\mathbb{Q}$ let $H(u, v) = \\max(|U|, |V|, |W|)$, where $U, V, W$ are coprime integers with $(U : V : W) = (u : v : 1)$. Then every rational solution $(u, v)$ of $v^2 = u^3 + au + b$ with $uv \\ne 0$ satisfies\u2026","title":"Height lower bound on $y^2 = x^3 + ax + b$ when $|a|$ is small"},{"data_url":"scopes/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa.json","id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","kind":"scope","navigation":"module","parent":5945,"result":false,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"can_prove","title":"Local-global conjecture for Apollonian circle packings"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/general-linear-group-action","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.GeneralLinearGroupAction","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#general-linear-group-action","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"Call $v \\in \\mathbb{Z}^4$ primitive when the greatest common divisor of its four entries is $1$. For every $X \\in \\mathrm{GL}_4(\\mathbb{Z})$ and every primitive $v$, the vector $Xv$ is primitive, so $\\mathrm{GL}_4(\\mathbb{Z})$ acts on the primitive vectors of $\\mathbb{Z}^4$ by $v \\mapsto Xv$.","title":"General linear group action on primitive vectors"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/descartes-matrix","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.DescartesMatrix","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#descartes-matrix","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"The Descartes matrix is the $4 \\times 4$ integer matrix $M$ with $M_{ii} = 1$ for every $i$ and $M_{ij} = -1$ for $i \\ne j$. It is the matrix of the Descartes form: for $v = (a, b, c, d) \\in \\mathbb{Z}^4$ one has $v^{\\mathsf T} M v = 2(a^2 + b^2 + c^2 + d^2) - (a + b + c + d)^2$.","title":"Descartes matrix"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/descartes-orthogonal-group","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.DescartesOrthogonalGroup","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#descartes-orthogonal-group","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"Let $M$ be the Descartes matrix. The Descartes orthogonal group is the subgroup of $\\mathrm{GL}_4(\\mathbb{Z})$ consisting of the matrices $X$ with $X^{\\mathsf T} M X = M$, that is, the invertible integer matrices that preserve the Descartes form $2(a^2 + b^2 + c^2 + d^2) - (a + b + c + d)^2$.","title":"Descartes orthogonal group"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/descartes-orthogonal-group-action","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.DescartesOrthogonalGroupAction","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#descartes-orthogonal-group-action","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"Let $Q$ be the set of vectors $(a, b, c, d) \\in \\mathbb{Z}^4$ with $(a + b + c + d)^2 = 2(a^2 + b^2 + c^2 + d^2)$ and $\\gcd(a, b, c, d) = 1$. For every matrix $X$ in the Descartes orthogonal group and every $v \\in Q$, the vector $Xv$ lies in $Q$, so the Descartes orthogonal group acts on $Q$ by $v \\mapsto Xv$.","title":"Descartes orthogonal group action"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/integral-primitive-descartes-quadruple","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.integralPrimitiveDescartesQuadruple","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#integral-primitive-descartes-quadruple","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"An integral primitive Descartes quadruple is a vector $(a, b, c, d) \\in \\mathbb{Z}^4$ such that $a + b + c + d > 0$, the Descartes equation $(a + b + c + d)^2 = 2(a^2 + b^2 + c^2 + d^2)$ holds, and $\\gcd(a, b, c, d) = 1$.","title":"Integral primitive Descartes quadruple"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/s","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.S\u2081","parent":5961,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Apollonian generator "],["math","\\(S_1\\)"]],"source":"The Apollonian generator $S_1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#s","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"$S_1$ is the element of $\\mathrm{GL}_4(\\mathbb{Z})$ given by the $4 \\times 4$ integer matrix whose first row is $(-1, 2, 2, 2)$ and whose other rows are those of the identity matrix; it is its own inverse. Acting on column vectors, it sends $(a, b, c, d)$ to $(-a + 2b + 2c + 2d, b, c, d)$.","title":"The Apollonian generator $S_1$"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/s-2","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.S\u2082","parent":5961,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Apollonian generator "],["math","\\(S_2\\)"]],"source":"The Apollonian generator $S_2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#s-2","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"$S_2$ is the element of $\\mathrm{GL}_4(\\mathbb{Z})$ given by the $4 \\times 4$ integer matrix whose second row is $(2, -1, 2, 2)$ and whose other rows are those of the identity matrix; it is its own inverse. Acting on column vectors, it sends $(a, b, c, d)$ to $(a, 2a - b + 2c + 2d, c, d)$.","title":"The Apollonian generator $S_2$"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/s-3","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.S\u2083","parent":5961,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Apollonian generator "],["math","\\(S_3\\)"]],"source":"The Apollonian generator $S_3$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#s-3","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"$S_3$ is the element of $\\mathrm{GL}_4(\\mathbb{Z})$ given by the $4 \\times 4$ integer matrix whose third row is $(2, 2, -1, 2)$ and whose other rows are those of the identity matrix; it is its own inverse. Acting on column vectors, it sends $(a, b, c, d)$ to $(a, b, 2a + 2b - c + 2d, d)$.","title":"The Apollonian generator $S_3$"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/s-4","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.S\u2084","parent":5961,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Apollonian generator "],["math","\\(S_4\\)"]],"source":"The Apollonian generator $S_4$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#s-4","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"$S_4$ is the element of $\\mathrm{GL}_4(\\mathbb{Z})$ given by the $4 \\times 4$ integer matrix whose fourth row is $(2, 2, 2, -1)$ and whose other rows are those of the identity matrix; it is its own inverse. Acting on column vectors, it sends $(a, b, c, d)$ to $(a, b, c, 2a + 2b + 2c - d)$.","title":"The Apollonian generator $S_4$"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/apollonian-group","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.ApollonianGroup","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#apollonian-group","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"The Apollonian group is the subgroup of $\\mathrm{GL}_4(\\mathbb{Z})$ generated by the four matrices $S_1, S_2, S_3, S_4$, where $S_i$ agrees with the $4 \\times 4$ identity matrix except in row $i$, which has entry $-1$ in column $i$ and entry $2$ in each of the other three columns.","title":"Apollonian group"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/transformation-group","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.TransformationGroup","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#transformation-group","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"The transformation group is the subgroup of $\\mathrm{GL}_4(\\mathbb{Z})$ generated by the Apollonian group together with all $4 \\times 4$ permutation matrices.","title":"The transformation group"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/transformation-group-smul-integral-primi","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.transformationGroup_smul_integralPrimitiveDescartesQuadruple","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#transformation-group-smul-integral-primi","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"For every $g$ in the transformation group, $g$ maps the set of integral primitive Descartes quadruples into itself: if $v \\in \\mathbb{Z}^4$ is an integral primitive Descartes quadruple, then so is $g v$.","title":"The transformation group preserves integral primitive Descartes quadruples"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/transformation-group-action","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.TransformationGroupAction","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#transformation-group-action","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"Each element $g$ of the transformation group sends an integral primitive Descartes quadruple $v$ to $g v$, by matrix\u2013vector multiplication: this is the restriction of the action of the transformation group on $\\mathbb{Z}^4$ to the set of integral primitive Descartes quadruples, which is closed under the action.","title":"Action of the transformation group on integral primitive Descartes quadruples"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/primitive-apollonian-circle-packing","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.PrimitiveApollonianCirclePacking","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#primitive-apollonian-circle-packing","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"A primitive Apollonian circle packing is an orbit of the transformation group acting on the set of integral primitive Descartes quadruples.","title":"Primitive Apollonian circle packing"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/curvature-set","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.curvatureSet","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#curvature-set","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"Let $A$ be a primitive Apollonian circle packing, viewed as an orbit of integral primitive Descartes quadruples. The curvature set of $A$ is the set of all integers that occur as an entry of some quadruple in $A$.","title":"Curvature set"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/theorem-1-6","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.theorem_1_6","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#theorem-1-6","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"can_prove","summary":"Let $\\mathcal{A}$ be the primitive Apollonian circle packing generated by the integral primitive Descartes quadruple $(-3, 5, 8, 8)$. For every integer $n$, $n^2$ does not belong to the curvature set of $\\mathcal{A}$.","title":"A packing with no square curvatures"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/missing-curvature-set","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.missingCurvatureSet","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#missing-curvature-set","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"Let $A$ be a primitive Apollonian circle packing. The missing curvature set of $A$ is the set of positive integers $n$ that do not lie in the curvature set of $A$ but are congruent modulo $24$ to some element of the curvature set of $A$.","title":"Missing curvature set"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/missing-curvatures","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.missingCurvatures","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#missing-curvatures","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"fully_proved","summary":"For a primitive Apollonian circle packing $A$ and a real number $N$, the number of missing curvatures of $A$ up to $N$ is the number of elements $n$ of the missing curvature set of $A$ with $n \\le N$, regarded as a real number.","title":"Number of missing curvatures"},{"id":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/theorem-1-3","kind":"node","lean":"MathlibExt.NumberTheory.LocalGlobalApollonianCirclePackingsWanted.LocalGlobalApollonianCirclePackings.theorem_1_3","parent":5961,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa/index.html#theorem-1-3","scope":"NumberTheory/arithmetic-geometry/local-global-conjecture-for-apollonian-circle-pa","status":"can_prove","summary":"For a primitive Apollonian circle packing $\\mathcal{A}$ and a real number $N$, let $M_{\\mathcal{A}}(N)$ be the number of missing curvatures of $\\mathcal{A}$ up to $N$. There are infinitely many primitive Apollonian circle packings $\\mathcal{A}$ such that $\\sqrt{N} = O(M_{\\mathcal{A}}(N))$ as $N$ tends to infinity. In other words, for these packings the number of missing curvatures up to $N$ is $\\Omega(\\sqrt{N})$.","title":"Infinitely many packings with many missing curvatures"},{"data_url":"scopes/NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem.json","id":"NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem","kind":"scope","navigation":"module","parent":5945,"result":false,"route":"../roadmap/NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem/index.html","scope":"NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem","status":"can_prove","title":"Mazur's torsion theorem"},{"id":"NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem/mazur-torsion-ncard-le","kind":"node","lean":"MetaMathlibExt.mazur_torsion_ncard_le","parent":5981,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem/index.html#mazur-torsion-ncard-le","scope":"NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem","status":"can_prove","summary":"Let $E$ be an elliptic curve over $\\mathbb{Q}$. The torsion subgroup of the group $E(\\mathbb{Q})$ of rational points of $E$ is finite and has at most $16$ elements.","title":"Mazur's torsion theorem"},{"id":"NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem/is-in-mazur-class","kind":"node","lean":"MathlibExt.NumberTheory.MazurTorsionWanted.NumberTheory.MazurTorsion.IsInMazurClass","parent":5981,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem/index.html#is-in-mazur-class","scope":"NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem","status":"fully_proved","summary":"An abelian group $T$ is in the Mazur class when it is isomorphic to $\\mathbb{Z}/n\\mathbb{Z}$ for some $n \\in \\{1, 2, \\ldots, 10, 12\\}$, or to $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2m\\mathbb{Z}$ for some $m \\in \\{1, 2, 3, 4\\}$.","title":"The Mazur class of abelian groups"},{"id":"NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem/mazur-torsion","kind":"node","lean":"MathlibExt.NumberTheory.MazurTorsionWanted.NumberTheory.MazurTorsion.mazur_torsion","parent":5981,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem/index.html#mazur-torsion","scope":"NumberTheory/arithmetic-geometry/mazur-s-torsion-theorem","status":"can_prove","summary":"Let $E$ be an elliptic curve over $\\mathbb{Q}$, given by a Weierstrass equation. The torsion subgroup of the group $E(\\mathbb{Q})$ of rational points of $E$ is in the Mazur class: it is isomorphic to $\\mathbb{Z}/n\\mathbb{Z}$ for some $n \\in \\{1, 2, \\ldots, 10, 12\\}$, or to $\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2m\\mathbb{Z}$ for some $m \\in \\{1, 2, 3, 4\\}$.","title":"Mazur's torsion theorem"},{"data_url":"scopes/NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx.json","id":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx","kind":"scope","navigation":"module","parent":5945,"result":false,"route":"../roadmap/NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/index.html","scope":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx","status":"can_prove","title":"Ranks of the elliptic curves y\u00b2 = x\u00b3 \u2212 2pqx"},{"id":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/exists-mv-polynomial-eval-eq-zero-of-nor","kind":"node","lean":"MetaMathlibExt.exists_mvPolynomial_eval_eq_zero_of_norm_lt_norm_pderiv_sq","parent":5985,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/index.html#exists-mv-polynomial-eval-eq-zero-of-nor","scope":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx","status":"can_prove","summary":"Let $p$ be a prime, let $f \\in \\mathbb{Z}_p[x_1, \\dots, x_n]$ and $a \\in \\mathbb{Z}_p^n$, and suppose that for some index $i$ the $p$-adic absolute values satisfy $|f(a)|_p < |(\\partial f / \\partial x_i)(a)|_p^2$, that is, $\\operatorname{ord}_p f(a) > 2 \\operatorname{ord}_p (\\partial f / \\partial x_i)(a)$. Then there is a point $x \\in \\mathbb{Z}_p^n$ with $f(x) = 0$.","title":"Multivariable Hensel lemma over the p-adic integers"},{"id":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/finrank-point-two-mul-mul-eq-one","kind":"node","lean":"MetaMathlibExt.finrank_point_two_mul_mul_eq_one","parent":5985,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/index.html#finrank-point-two-mul-mul-eq-one","scope":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx","status":"can_prove","summary":"Let $p \\ne q$ be primes and let $E$ be the elliptic curve $y^2 = x^3 - 2pqx$ over $\\mathbb{Q}$. Suppose that $(p \\bmod 8, q \\bmod 8)$ is one of $(3, 3)$, $(3, 5)$, $(3, 7)$, $(5, 5)$, $(5, 7)$, or that $p \\equiv 1 \\pmod 8$, $q \\equiv 3, 5$ or $7 \\pmod 8$ and the Legendre symbol $\\left(\\frac{p}{q}\\right)$ equals $-1$. If the rank of the group $E(\\mathbb{Q})$ of rational points is odd, then it equals $1$.","title":"Odd-rank curves y\u00b2 = x\u00b3 \u2212 2pqx: rank one cases"},{"id":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/finrank-point-two-mul-mul-eq-one-or-thre","kind":"node","lean":"MetaMathlibExt.finrank_point_two_mul_mul_eq_one_or_three","parent":5985,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/index.html#finrank-point-two-mul-mul-eq-one-or-thre","scope":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx","status":"can_prove","summary":"Let $p \\ne q$ be primes and let $E$ be the elliptic curve $y^2 = x^3 - 2pqx$ over $\\mathbb{Q}$, and write $\\left(\\frac{p}{q}\\right)$ for the Legendre symbol. Suppose that $p \\equiv q \\equiv 1 \\pmod 8$ and $\\left(\\frac{p}{q}\\right) = -1$, or that $p \\equiv q \\equiv 7 \\pmod 8$, or that $p \\equiv 1 \\pmod 8$, $q \\equiv 3, 5$ or $7 \\pmod 8$ and $\\left(\\frac{p}{q}\\right) = 1$. If the rank of the group $E(\\mathbb{Q})$ of\u2026","title":"Odd-rank curves y\u00b2 = x\u00b3 \u2212 2pqx: rank one or three cases"},{"id":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/finrank-point-two-mul-mul-eq-one-or-thre-2","kind":"node","lean":"MetaMathlibExt.finrank_point_two_mul_mul_eq_one_or_three_or_five","parent":5985,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/index.html#finrank-point-two-mul-mul-eq-one-or-thre-2","scope":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx","status":"can_prove","summary":"Let $p \\ne q$ be primes with $p \\equiv q \\equiv 1 \\pmod 8$ and Legendre symbol $\\left(\\frac{p}{q}\\right) = 1$, and let $E$ be the elliptic curve $y^2 = x^3 - 2pqx$ over $\\mathbb{Q}$. If the rank of the group $E(\\mathbb{Q})$ of rational points is odd, then it equals $1$, $3$ or $5$.","title":"Odd-rank curves y\u00b2 = x\u00b3 \u2212 2pqx: rank one, three or five case"},{"id":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/finrank-point-two-mul-mul-eq-three","kind":"node","lean":"MetaMathlibExt.finrank_point_two_mul_mul_eq_three","parent":5985,"result":true,"route":"../roadmap/NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx/index.html#finrank-point-two-mul-mul-eq-three","scope":"NumberTheory/arithmetic-geometry/ranks-of-the-elliptic-curves-y-x-2pqx","status":"can_prove","summary":"Let $r$ be a positive integer such that $p = 272r^2 + 120r + 9$ and $q = 2p + 9 = 544r^2 + 240r + 27$ are both prime, and let $E$ be the elliptic curve $y^2 = x^3 - 2pqx$ over $\\mathbb{Q}$. If the rank of the group $E(\\mathbb{Q})$ of rational points is odd, then it equals $3$.","title":"Odd-rank curves y\u00b2 = x\u00b3 \u2212 2pqx: a rank three family"},{"data_url":"scopes/NumberTheory/linear-recurrences.json","id":"NumberTheory/linear-recurrences","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/index.html","scope":"NumberTheory/linear-recurrences","status":"can_state","summary":"Fibonacci, Lucas, Pell, Tribonacci and other linear recurrences: identities, divisibility, periods, primitive divisors and Diophantine properties.","title":"Linear recurrence sequences"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-lucas-numbers.json","id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers","kind":"scope","parent":5991,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers","status":"can_prove","summary":"Identities, sums, reciprocal series, determinants and generating functions specifically for Fibonacci and Lucas numbers.","title":"Fibonacci and Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/alfred-consecutive-fibonacci-powers-dete","kind":"node","lean":"MetaMathlibExt.alfred_consecutive_fibonacci_powers_determinant","parent":5992,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/index.html#alfred-consecutive-fibonacci-powers-dete","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers","status":"fully_proved","summary":"Let $F_n$ denote the Fibonacci numbers extended to all integers $n$ by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$, so that $F_{-n} = (-1)^{n+1} F_n$. For every natural number $r$ and all integers $s$ and $k$, the $(r+1) \\times (r+1)$ integer matrix whose entry in row $i$ and column $j$, for $0 \\le i, j \\le r$, is $F_{s+ik}^{\\,r-j} \\, F_{s+(i+1)k}^{\\,j}$ has determinant\u2026","title":"Alfred's determinant of consecutive Fibonacci powers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/brousseau-fibonacci-reciprocal-sum","kind":"node","lean":"MetaMathlibExt.brousseau_fibonacci_reciprocal_sum","parent":5992,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/index.html#brousseau-fibonacci-reciprocal-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers","status":"fully_proved","summary":"Let $F_n$ denote the Fibonacci numbers, with $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$. For every positive integer $q$, the real series $\\sum_{k \\ge 1} (-1)^{k-1} \\frac{F_{2k+q}}{F_k^2 \\, F_{k+q}^2}$ is unconditionally summable, with sum $\\frac{1}{F_q} \\sum_{k=1}^{q} \\frac{(-1)^{k-1}}{F_k^2}$.","title":"Brousseau's alternating series of reciprocal Fibonacci squares"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums.json","id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","kind":"scope","navigation":"module","parent":5992,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","title":"Bernoulli\u2013Fibonacci and Euler\u2013Lucas double sums"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sqrt5","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sqrt5","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sqrt5","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"fully_proved","summary":"This is the real number $\\sqrt{5}$, the nonnegative real square root of $5$.","title":"The square root of five"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/fib-alpha","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.fibAlpha","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","The golden ratio "],["math","\\(\\alpha\\)"]],"source":"The golden ratio $\\alpha$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#fib-alpha","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"fully_proved","summary":"The golden ratio is the real number $\\alpha = (1 + \\sqrt{5})/2$.","title":"The golden ratio $\\alpha$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/fib-beta","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.fibBeta","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","The conjugate golden ratio "],["math","\\(\\beta\\)"]],"source":"The conjugate golden ratio $\\beta$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#fib-beta","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"fully_proved","summary":"The real number $\\beta = (1 - \\sqrt{5})/2$, the conjugate of the golden ratio $(1 + \\sqrt{5})/2$.","title":"The conjugate golden ratio $\\beta$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/euler-sec-number","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.eulerSecNumber","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#euler-sec-number","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"fully_proved","summary":"For a natural number $n$, the Euler number $E_n \\in \\mathbb{Q}$ is $0$ when $n$ is odd, and when $n = 2k$ is even it is the even-index Euler number $E_{2k}$, where $E_0 = 1$ and $\\sum_{i=0}^{k} \\binom{2k}{2i} E_{2i} = 0$ for $k \\ge 1$. The first values are $E_0 = 1$, $E_2 = -1$, $E_4 = 5$.","title":"Euler numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/gen-lucas-coeff","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.genLucasCoeff","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#gen-lucas-coeff","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"fully_proved","summary":"For a natural number $n$, the Genocchi\u2013Lucas coefficient is the real number","title":"Genocchi\u2013Lucas coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/add-fib-beta-pow-eq-sum","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.add_fibBeta_pow_eq_sum","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli\u2013Fibonacci expansion of "],["math","\\((x + \\beta)^m\\)"]],"source":"Bernoulli\u2013Fibonacci expansion of $(x + \\beta)^m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#add-fib-beta-pow-eq-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, let $B_k$ be the Bernoulli numbers, normalized so that $B_1 = -1/2$, and let $F_k$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). Then for every natural number $m$ and every real number $x$,","title":"Bernoulli\u2013Fibonacci expansion of $(x + \\beta)^m$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-bernoulli-fib-neg-fib-beta","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_bernoulli_fib_neg_fibBeta","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-bernoulli-fib-neg-fib-beta","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, let $F_n$ be the Fibonacci numbers, and let $B_k$ be the Bernoulli numbers, with $B_1 = -\\tfrac{1}{2}$. For every natural number $m \\ge 1$,","title":"A vanishing double sum of Bernoulli and Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-neg-two-pow-bernoulli-fib","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_neg_two_pow_bernoulli_fib","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli\u2013Fibonacci double sum with weights "],["math","\\((-2)^n\\)"]],"source":"Bernoulli\u2013Fibonacci double sum with weights $(-2)^n$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-neg-two-pow-bernoulli-fib","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $B_k$ be the Bernoulli numbers, with $B_1 = -1/2$, and let $F_k$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). For every natural number $m$,","title":"Bernoulli\u2013Fibonacci double sum with weights $(-2)^n$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/desc-factorial-mul-add-fib-beta-pow","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.descFactorial_mul_add_fibBeta_pow","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivatives of the Bernoulli\u2013Fibonacci expansion of "],["math","\\((x + \\beta)^m\\)"]],"source":"Derivatives of the Bernoulli\u2013Fibonacci expansion of $(x + \\beta)^m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#desc-factorial-mul-add-fib-beta-pow","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, let $B_k$ be the Bernoulli numbers with $B_1 = -1/2$, let $F_k$ be the Fibonacci numbers, and write $a^{\\underline{s}} = a(a-1)\\cdots(a-s+1)$ for the falling factorial. Then for all natural numbers $s \\le m$ and every real number $x$,","title":"Derivatives of the Bernoulli\u2013Fibonacci expansion of $(x + \\beta)^m$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/add-half-pow-eq-sum","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.add_half_pow_eq_sum","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler\u2013Lucas expansion of "],["math","\\((x + 1/2)^m\\)"]],"source":"Euler\u2013Lucas expansion of $(x + 1/2)^m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#add-half-pow-eq-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $E_k$ be the Euler (secant) numbers, which vanish at odd $k$ and begin $E_0 = 1$, $E_2 = -1$, $E_4 = 5$, and let $L_k$ be the Lucas numbers. Then for every natural number $m$ and every real number $x$,","title":"Euler\u2013Lucas expansion of $(x + 1/2)^m$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-neg-one-pow-euler-sec-lucas","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_neg_one_pow_eulerSec_lucas","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-neg-one-pow-euler-sec-lucas","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $E_k$ be the Euler (secant) numbers, with exponential generating function $2/(e^t + e^{-t})$, and let $L_k$ be the Lucas numbers ($L_0 = 2$, $L_1 = 1$, $L_{k+2} = L_{k+1} + L_k$). For every natural number $m \\ge 1$,","title":"The alternating Euler\u2013Lucas double sum vanishes"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-euler-sec-lucas","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_eulerSec_lucas","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-euler-sec-lucas","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $L_n$ be the Lucas numbers and let $E_k$ be the Euler (secant) numbers, given by $2/(e^t + e^{-t}) = \\sum_{k \\ge 0} E_k t^k / k!$. For every natural number $m$,","title":"A double sum of Euler and Lucas numbers equal to two"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-euler-sec-lucas-eq-two-fib-alpha-pow","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_eulerSec_lucas_eq_two_fibAlpha_pow","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-euler-sec-lucas-eq-two-fib-alpha-pow","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\alpha = (1 + \\sqrt{5})/2$, let $L_n$ be the Lucas numbers, and let $E_k$ be the Euler (secant) numbers, given by $2/(e^t + e^{-t}) = \\sum_{k \\ge 0} E_k t^k / k!$. For every natural number $m$,","title":"Powers of the golden ratio from Euler and Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-euler-sec-lucas-eq-two-fib-beta-pow","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_eulerSec_lucas_eq_two_fibBeta_pow","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-euler-sec-lucas-eq-two-fib-beta-pow","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, let $L_n$ be the Lucas numbers, and let $E_k$ be the Euler (secant) numbers, given by $2/(e^t + e^{-t}) = \\sum_{k \\ge 0} E_k t^k / k!$. For every natural number $m$,","title":"Powers of the conjugate golden ratio from Euler and Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-choose-euler-sec-lucas","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_choose_eulerSec_lucas","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-choose-euler-sec-lucas","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $L_n$ be the Lucas numbers and let $E_k$ be the Euler (secant) numbers, given by $2/(e^t + e^{-t}) = \\sum_{k \\ge 0} E_k t^k / k!$. For every natural number $n$,","title":"A binomial sum of Euler and Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/add-fib-beta-pow-eq-sum-gen-lucas-coeff","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.add_fibBeta_pow_eq_sum_genLucasCoeff","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","Genocchi\u2013Lucas expansion of "],["math","\\((x + \\beta)^m\\)"]],"source":"Genocchi\u2013Lucas expansion of $(x + \\beta)^m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#add-fib-beta-pow-eq-sum-gen-lucas-coeff","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, and for every natural number $n$ let","title":"Genocchi\u2013Lucas expansion of $(x + \\beta)^m$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-gen-lucas-coeff-neg-fib-beta","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_genLucasCoeff_neg_fibBeta","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-gen-lucas-coeff-neg-fib-beta","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, let $G_k$ be the Genocchi numbers and $L_n$ the Lucas numbers, and for each natural number $n$ let","title":"A vanishing sum of the corrected Genocchi\u2013Lucas coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-gen-lucas-coeff-weighted","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_genLucasCoeff_weighted","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-gen-lucas-coeff-weighted","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $G_k$ be the Genocchi numbers and $L_n$ the Lucas numbers, and for each natural number $n$ let","title":"A weighted sum of the corrected Genocchi\u2013Lucas coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-genocchi-lucas-add","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_genocchi_lucas_add","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-genocchi-lucas-add","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, let $G_k$ be the Genocchi numbers and let $L_k$ be the Lucas numbers ($L_0 = 2$, $L_1 = 1$, $L_{k+2} = L_{k+1} + L_k$). For every natural number $n$,","title":"Genocchi\u2013Lucas sum for powers of the conjugate golden ratio"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/add-mul-fib-beta-pow-eq-sum","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.add_mul_fibBeta_pow_eq_sum","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multinomial Bernoulli\u2013Fibonacci expansion of "],["math","\\((x + q\\beta)^m\\)"]],"source":"Multinomial Bernoulli\u2013Fibonacci expansion of $(x + q\\beta)^m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#add-mul-fib-beta-pow-eq-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, let $B_k$ be the Bernoulli numbers with $B_1 = -1/2$, and let $F_k$ be the Fibonacci numbers. For every natural number $q \\ge 1$, every natural number $m$ and every real number $x$, the power $(x + q\\beta)^m$ equals the sum, over $0 \\le n \\le m$ and over all pairs of tuples of natural numbers $(j_1, \\dots, j_q)$ and $(\\ell_1, \\dots, \\ell_q)$ with\u2026","title":"Multinomial Bernoulli\u2013Fibonacci expansion of $(x + q\\beta)^m$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/sum-bernoulli-fib-two-eq-one","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.sum_bernoulli_fib_two_eq_one","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#sum-bernoulli-fib-two-eq-one","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers and let $B_k$ be the Bernoulli numbers, with $B_1 = -\\tfrac{1}{2}$. For every natural number $m$,","title":"A multinomial sum of Bernoulli and Fibonacci numbers equal to one"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/two-pow-mul-add-half-pow","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.two_pow_mul_add_half_pow","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler\u2013Lucas multinomial expansion of "],["math","\\((x + q/2)^m\\)"]],"source":"Euler\u2013Lucas multinomial expansion of $(x + q/2)^m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#two-pow-mul-add-half-pow","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $E_k$ be the Euler (secant) numbers, with exponential generating function $2/(e^t + e^{-t})$, and let $L_k$ be the Lucas numbers ($L_0 = 2$, $L_1 = 1$, $L_{k+2} = L_{k+1} + L_k$). For all natural numbers $q \\ge 1$ and $m$ and every real number $x$,","title":"Euler\u2013Lucas multinomial expansion of $(x + q/2)^m$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/two-sqrt5-pow-mul-add","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.two_sqrt5_pow_mul_add","parent":5995,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multinomial Genocchi\u2013Lucas expansion of "],["math","\\((x + q\\beta)^m\\)"]],"source":"Multinomial Genocchi\u2013Lucas expansion of $(x + q\\beta)^m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#two-sqrt5-pow-mul-add","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"Let $\\beta = (1 - \\sqrt{5})/2$, let $G_k$ be the Genocchi numbers and $L_k$ the Lucas numbers, and for every natural number $n$ let","title":"Multinomial Genocchi\u2013Lucas expansion of $(x + q\\beta)^m$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/iterate-forward-diff-pow-eq-sum","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.iterate_forwardDiff_pow_eq_sum","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#iterate-forward-diff-pow-eq-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"For a real number $r$, let $\\Delta_r$ be the forward difference operator with step $r$, sending a function $f : \\mathbb{R} \\to \\mathbb{R}$ to $y \\mapsto f(y + r) - f(y)$, and let $\\Delta_r^k$ be its $k$-fold iterate (the identity when $k = 0$). For all real numbers $r$ and $x$ and all natural numbers $k$ and $n$, applying $\\Delta_r^k$ to the function $y \\mapsto y^n$ and evaluating at $x$ gives\u2026","title":"Iterated forward differences of a power as an alternating binomial sum"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/genocchi-number-div-eq-sum","kind":"node","lean":"MathlibExt.NumberTheory.FiniteSumsFibLucasWanted.genocchiNumber_div_eq_sum","parent":5995,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums/index.html#genocchi-number-div-eq-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/bernoulli-fibonacci-and-euler-lucas-double-sums","status":"can_prove","summary":"For every natural number $n \\ge 1$, the Genocchi number $G_n$ satisfies, in $\\mathbb{Q}$,","title":"Genocchi numbers via Stirling numbers of the second kind"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers.json","id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","kind":"scope","navigation":"module","parent":5992,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","status":"fully_proved","title":"Binomial sums of Fibonacci and Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/choose-succ-int","kind":"node","lean":"MetaMathlibExt.choose_succ_int","parent":6021,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/index.html#choose-succ-int","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","status":"fully_proved","summary":"For all natural numbers $n$ and $i$, $\\binom{n+1}{i+1} = \\binom{n}{i} + \\binom{n}{i+1}$, as an identity of integers.","title":"Pascal's rule for binomial coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/fib-conv-step","kind":"node","lean":"MetaMathlibExt.fib_conv_step","parent":6021,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/index.html#fib-conv-step","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","status":"fully_proved","summary":"Let $F_m$ denote the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For natural numbers $n$, $k$ and $i$ with $i \\le n$, $F_{2n+2-i+k} - F_{2n+1-i+k} = F_{2n-i+k}$, as an identity of integers.","title":"Fibonacci step in the alternating binomial sum"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/binomial-fibonacci-convolution","kind":"node","lean":"MetaMathlibExt.binomial_fibonacci_convolution","parent":6021,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/index.html#binomial-fibonacci-convolution","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","status":"fully_proved","summary":"Let $F_m$ denote the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For all natural numbers $n$ and $k$,","title":"Alternating binomial sum of Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/sqrt5-ne","kind":"node","lean":"MetaMathlibExt.sqrt5_ne","parent":6021,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/index.html#sqrt5-ne","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","status":"fully_proved","summary":"The real square root $\\sqrt{5}$ is not equal to $0$.","title":"The square root of 5 is nonzero"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/cb-lucas-binet","kind":"node","lean":"MetaMathlibExt.cb_lucas_binet","parent":6021,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/index.html#cb-lucas-binet","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","status":"fully_proved","summary":"For every natural number $n$, $\\varphi^n + \\psi^n = L_n$ as real numbers, where $\\varphi = (1 + \\sqrt{5})/2$ is the golden ratio, $\\psi = (1 - \\sqrt{5})/2$ is its conjugate, and $L_n$ is the $n$-th Lucas number.","title":"Binet's formula for the Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/cb-key-r","kind":"node","lean":"MetaMathlibExt.cb_keyR","parent":6021,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/index.html#cb-key-r","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","status":"fully_proved","summary":"For every natural number $n$,","title":"Church\u2013Bicknell convolution over the reals"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/church-bicknell-fibonacci-convolution","kind":"node","lean":"MetaMathlibExt.church_bicknell_fibonacci_convolution","parent":6021,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers/index.html#church-bicknell-fibonacci-convolution","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/binomial-sums-of-fibonacci-and-lucas-numbers","status":"fully_proved","summary":"For every natural number $n$,","title":"Church\u2013Bicknell Fibonacci binomial convolution"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions.json","id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","kind":"scope","navigation":"module","parent":5992,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","title":"Fibonacci identities via Fibonacci functions"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/lucas-fib-weighted-binomial-sum-even-ste","kind":"node","lean":"MetaMathlibExt.lucas_fib_weighted_binomial_sum_even_step","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#lucas-fib-weighted-binomial-sum-even-ste","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ be the Fibonacci numbers extended to all integers $m$ ($F_0 = 0$, $F_1 = 1$, $F_{m+2} = F_{m+1} + F_m$) and $L_m = F_{m-1} + F_{m+1}$ the Lucas numbers. For every natural number $n$ and all integers $k$, $r$ with $k$ even,","title":"Weighted binomial sums of Lucas and Fibonacci numbers with even step"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/lucas-div-fib-sq-add-one-telescoping","kind":"node","lean":"MetaMathlibExt.lucas_div_fib_sq_add_one_telescoping","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#lucas-div-fib-sq-add-one-telescoping","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ be the Fibonacci numbers extended to all integers $m$ ($F_0 = 0$, $F_1 = 1$, $F_{m+2} = F_{m+1} + F_m$) and $L_m = F_{m-1} + F_{m+1}$ the Lucas numbers. Then for every integer $k$,","title":"Two telescoping relations between Lucas and Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/sum-lucas-div-fib-sq-add-one","kind":"node","lean":"MetaMathlibExt.sum_lucas_div_fib_sq_add_one","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#sum-lucas-div-fib-sq-add-one","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ denote the Fibonacci numbers and $L_m$ the Lucas numbers ($L_0 = 2$, $L_1 = 1$, $L_{m+2} = L_{m+1} + L_m$). For every natural number $n$,","title":"Telescoping sums of Lucas numbers over Fibonacci squares plus one"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/fib-cube-mul-sum-choose-lucas-pow","kind":"node","lean":"MetaMathlibExt.fib_cube_mul_sum_choose_lucas_pow","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#fib-cube-mul-sum-choose-lucas-pow","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ and $L_m$ denote the Fibonacci and Lucas numbers. For all natural numbers $k$ and $n$,","title":"Binomial sum with even powers of a Lucas number"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/ten-mul-fib-cube-dvd","kind":"node","lean":"MetaMathlibExt.ten_mul_fib_cube_dvd","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#ten-mul-fib-cube-dvd","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ denote the Fibonacci numbers and $L_m$ the Lucas numbers. For all natural numbers $k$ and $n$, the integer $10 F_k^3$ divides $(2n+1) F_{2k} L_{(2n+1)k} - F_{(2n+1)k} L_k^2$.","title":"Ten times a Fibonacci cube divides a Fibonacci\u2013Lucas combination"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-candido-derivative-identity","kind":"node","lean":"MetaMathlibExt.gibonacci_candido_derivative_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-candido-derivative-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G : \\mathbb{Z} \\to \\mathbb{R}$ be a gibonacci sequence, that is, $G_{j+2} = G_{j+1} + G_j$ for every integer $j$. Then for every integer $k$,","title":"Cubic gibonacci identity from Candido's identity"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-sum-sq-mul-sum-lucas-like","kind":"node","lean":"MetaMathlibExt.gibonacci_sum_sq_mul_sum_lucas_like","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-sum-sq-mul-sum-lucas-like","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ for every integer $j$, and let $F_m$ be the Fibonacci numbers. Then for every natural number $n$ and every integer $k$ (the sums below are empty when $n = 0$),","title":"Companion of Melham's sum-of-squares identity for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/generalized-fibonacci-gelin-cesaro-deriv","kind":"node","lean":"MetaMathlibExt.generalized_fibonacci_gelin_cesaro_derivative","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#generalized-fibonacci-gelin-cesaro-deriv","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $p$ be a real number, let $W : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_{j+2} = p W_{j+1} + W_j$ for every integer $j$, and put $\\gamma_W = (p W_0 W_1 + W_0^2 - W_1^2)(p^2 - 1)$. Then for every integer $k$,","title":"Derivative form of the Gelin\u2013Ces\u00e0ro identity for generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/alternating-sum-fib-sq-fib-div-fib-pow-f","kind":"node","lean":"MetaMathlibExt.alternating_sum_fib_sq_fib_div_fib_pow_four_sub","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#alternating-sum-fib-sq-fib-div-fib-pow-f","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ and $L_m$ denote the Fibonacci and Lucas numbers. For every natural number $n$,","title":"Alternating Fibonacci and Lucas sums from the Gelin\u2013Ces\u00e0ro identity"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/generalized-fibonacci-gelin-cesaro-alter","kind":"node","lean":"MetaMathlibExt.generalized_fibonacci_gelin_cesaro_alternating_sum","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#generalized-fibonacci-gelin-cesaro-alter","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $p$ be a real number, let $W : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_{j+2} = p W_{j+1} + W_j$ for every integer $j$, and put $\\gamma_W = (p W_0 W_1 + W_0^2 - W_1^2)(p^2 - 1)$. Let $n \\ge 0$ and $k$ be integers such that $W_{j+k-2} W_{j+k-1} W_{j+k+1} W_{j+k+2} \\ne 0$ for $1 \\le j \\le n$ and $W_{n+k-1}$, $W_{k-1}$, $W_{n+k+2}$, $W_{k+2}$ are nonzero. Then","title":"Alternating Gelin\u2013Ces\u00e0ro sum for generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/sum-two-pow-lucas-v-div-lucas-u-sq-doubl","kind":"node","lean":"MetaMathlibExt.sum_two_pow_lucasV_div_lucasU_sq_doubling","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#sum-two-pow-lucas-v-div-lucas-u-sq-doubl","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $P \\ne 0$ be an integer and let $U_m = U_m(P, -1)$ and $V_m = V_m(P, -1)$ be the Lucas sequences with parameters $(P, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = P$, and both satisfy $X_{m+2} = P X_{m+1} + X_m$. For every integer $k \\ge 1$ and every integer $n \\ge 1$,","title":"Reciprocal series of Lucas sequences with doubling subscripts"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/five-fib-mul-gibonacci-sub-lucas-mul","kind":"node","lean":"MetaMathlibExt.five_fib_mul_gibonacci_sub_lucas_mul","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#five-fib-mul-gibonacci-sub-lucas-mul","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ denote the Fibonacci numbers, extended to all integers $m$ by $F_{-m} = (-1)^{m+1} F_m$, and let $L_m = F_{m-1} + F_{m+1}$ be the Lucas numbers, for every integer $m$. If $G : \\mathbb{Z} \\to \\mathbb{R}$ satisfies $G_{j+2} = G_{j+1} + G_j$ for every integer $j$, then for all integers $k$ and $r$,","title":"A Fibonacci\u2013Lucas identity for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/five-gibonacci-mul-sub-two-gibonacci","kind":"node","lean":"MetaMathlibExt.five_gibonacci_mul_sub_two_gibonacci","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#five-gibonacci-mul-sub-two-gibonacci","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$. Then for all integers $k$, $r$ and $s$,","title":"An identity for products of two gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/fib-mul-gibonacci-add-eq-gibonacci","kind":"node","lean":"MetaMathlibExt.fib_mul_gibonacci_add_eq_gibonacci","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#fib-mul-gibonacci-add-eq-gibonacci","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ denote the Fibonacci numbers, extended to all integers $m$ by $F_{-m} = (-1)^{m+1} F_m$. If $G : \\mathbb{Z} \\to \\mathbb{R}$ satisfies $G_{j+2} = G_{j+1} + G_j$ for every integer $j$, then $F_{k+1} G_{s+1} + F_k G_s = G_{k+s+1}$ for all integers $k$ and $s$.","title":"Addition formula for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-mul-gibonacci-add-eq-seed-comb","kind":"node","lean":"MetaMathlibExt.gibonacci_mul_gibonacci_add_eq_seed_combination","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-mul-gibonacci-add-eq-seed-comb","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$ and $j \\mapsto H_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$. Then for all integers $k$ and $s$,","title":"A sum of two gibonacci products in terms of the seeds"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-d-ocagne-type-identity","kind":"node","lean":"MetaMathlibExt.gibonacci_dOcagne_type_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-d-ocagne-type-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G$ and $H$ be gibonacci sequences, that is, functions $X : \\mathbb{Z} \\to \\mathbb{R}$ with $X_{j+2} = X_{j+1} + X_j$ for every integer $j$. Then for all integers $r$, $s$, $k$,","title":"d'Ocagne-type identity for two gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-product-shift-identity","kind":"node","lean":"MetaMathlibExt.gibonacci_product_shift_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-product-shift-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$ and $j \\mapsto H_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$. Then for all integers $s$ and $k$,","title":"Shifted product identity for two gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-triple-product-recurrence","kind":"node","lean":"MetaMathlibExt.gibonacci_triple_product_recurrence","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-triple-product-recurrence","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H, I : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$, $j \\mapsto H_j$ and $j \\mapsto I_j$, satisfy $G_{j+2} = G_{j+1} + G_j$, $H_{j+2} = H_{j+1} + H_j$ and $I_{j+2} = I_{j+1} + I_j$ for every integer $j$. Then for all integers $k$, $r$ and $s$,","title":"Product recurrence for three gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-two-mul-add-eq-fib-sq-combinat","kind":"node","lean":"MetaMathlibExt.gibonacci_two_mul_add_eq_fib_sq_combination","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-two-mul-add-eq-fib-sq-combinat","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ be the Fibonacci numbers extended to all integers $m$ ($F_0 = 0$, $F_1 = 1$, $F_{m+2} = F_{m+1} + F_m$), and let $G \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $G_{j+2} = G_{j+1} + G_j$ for every integer $j$. Then for all integers $k$, $s$,","title":"Gibonacci index doubling via Fibonacci squares"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-triple-product-seed-identity","kind":"node","lean":"MetaMathlibExt.gibonacci_triple_product_seed_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-triple-product-seed-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H, I \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $G_{j+2} = G_{j+1} + G_j$, $H_{j+2} = H_{j+1} + H_j$ and $I_{j+2} = I_{j+1} + I_j$ for every integer $j$. Then for all integers $k$, $r$, $s$,","title":"Seed expansion of a triple gibonacci product"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-add-gibonacci-eq-lucas-combina","kind":"node","lean":"MetaMathlibExt.gibonacci_add_gibonacci_eq_lucas_combination","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-add-gibonacci-eq-lucas-combina","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ be the Fibonacci numbers, extended to all integers $m$ by $F_{m+2} = F_{m+1} + F_m$, and let $L_m = F_{m-1} + F_{m+1}$ be the Lucas numbers. Let $G : \\mathbb{Z} \\to \\mathbb{R}$ be a gibonacci sequence, that is, $G_{j+2} = G_{j+1} + G_j$ for every integer $j$. Then for all integers $k$ and $r$, $G_{k+r+1} + G_{k+r-1} = L_k G_{r+1} + L_{k-1} G_r$.","title":"Lucas expansion of shifted gibonacci sums"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-seed-lucas-like-identity","kind":"node","lean":"MetaMathlibExt.gibonacci_seed_lucas_like_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-seed-lucas-like-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$ and $j \\mapsto H_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$. Then for all integers $k$ and $r$,","title":"Seed form of a Lucas-type product identity for two gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/two-mul-sum-gibonacci-mul-gibonacci","kind":"node","lean":"MetaMathlibExt.two_mul_sum_gibonacci_mul_gibonacci","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#two-mul-sum-gibonacci-mul-gibonacci","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$. For every natural number $n$ and all integers $k$ and $s$,","title":"Sum of products of two gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/alternating-binomial-sum-fib-cube-mul-gi","kind":"node","lean":"MetaMathlibExt.alternating_binomial_sum_fib_cube_mul_gibonacci","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#alternating-binomial-sum-fib-cube-mul-gi","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ denote the Fibonacci numbers, extended to all integers $m$ by $F_{-m} = (-1)^{m+1} F_m$. If $G : \\mathbb{Z} \\to \\mathbb{R}$ satisfies $G_{j+2} = G_{j+1} + G_j$ for every integer $j$, then for every natural number $n$ and all integers $k$ and $r$,","title":"Hoggatt\u2013Bicknell-type binomial sum with Fibonacci cubes"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/alternating-binomial-sum-four-gibonacci","kind":"node","lean":"MetaMathlibExt.alternating_binomial_sum_four_gibonacci_product","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#alternating-binomial-sum-four-gibonacci","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $E, G, H, I : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $E_{j+2} = E_{j+1} + E_j$ for every integer $j$, and likewise $G$, $H$ and $I$. Then for every natural number $n$ and all integers $k$, $r$, $s$ and $t$,","title":"Hoggatt\u2013Bicknell-type binomial sum of four gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/six-mul-sum-gibonacci-sq-mul-sum-mixed","kind":"node","lean":"MetaMathlibExt.six_mul_sum_gibonacci_sq_mul_sum_mixed","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#six-mul-sum-gibonacci-sq-mul-sum-mixed","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{m+2} = F_{m+1} + F_m$), and let $G, H \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$. Then for every natural number $n$ and all integers $k$, $r$,","title":"Mixed Melham-type sum for two gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/four-gibonacci-melham-type-identity","kind":"node","lean":"MetaMathlibExt.four_gibonacci_melham_type_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#four-gibonacci-melham-type-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H, I, J : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $G_{j+2} = G_{j+1} + G_j$ for every integer $j$, and likewise $H$, $I$ and $J$, and let $F_m$ denote the Fibonacci numbers. For every natural number $n$ and all integers $k$, $r$, $s$ and $t$, writing $S(X, x; Y, y) = \\sum_{j=0}^{2n-1} X_{x+j} Y_{y+j}$ for sequences $X, Y$ and integers $x, y$,","title":"Four-sequence Melham-type identity for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-mul-add-neg-one-pow-eq-fib-mul","kind":"node","lean":"MetaMathlibExt.gibonacci_mul_add_neg_one_pow_eq_fib_mul","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-mul-add-neg-one-pow-eq-fib-mul","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$ and $j \\mapsto H_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$, and let $F_m$ be the Fibonacci numbers, extended to negative indices by $F_{-m} = (-1)^{m+1} F_m$. Then for all integers $r$, $s$ and $k$,","title":"Two-sequence gibonacci identity with a Fibonacci factor"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-product-four-index-identity","kind":"node","lean":"MetaMathlibExt.gibonacci_product_four_index_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-product-four-index-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$ and $j \\mapsto H_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$, and let $F_m$ be the Fibonacci numbers, extended to negative indices by $F_{-m} = (-1)^{m+1} F_m$. Then for all integers $k$, $r$, $s$ and $t$,","title":"Four-index product identity for two gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-candido-two-sequence","kind":"node","lean":"MetaMathlibExt.gibonacci_candido_two_sequence","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-candido-two-sequence","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G$ and $H$ be gibonacci sequences, that is, functions $X : \\mathbb{Z} \\to \\mathbb{R}$ with $X_{j+2} = X_{j+1} + X_j$ for every integer $j$. Then for all integers $r$ and $k$,","title":"Two-sequence Candido-type identity for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-candido-four-sequence","kind":"node","lean":"MetaMathlibExt.gibonacci_candido_four_sequence","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-candido-four-sequence","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G$, $H$, $M$, $N$ be gibonacci sequences, that is, functions $X : \\mathbb{Z} \\to \\mathbb{R}$ with $X_{j+2} = X_{j+1} + X_j$ for every integer $j$. For two such sequences $X$, $Y$ and integers $x$, $y$ write $T(X, x; Y, y) = X_x Y_y + X_{x+1} Y_{y+1} + X_{x+2} Y_{y+2}$. Then for all integers $k$, $r$, $s$, $t$,","title":"Four-sequence Candido-type identity for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-add-neg-one-pow-mul-eq-lucas-m","kind":"node","lean":"MetaMathlibExt.gibonacci_add_neg_one_pow_mul_eq_lucas_mul","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-add-neg-one-pow-mul-eq-lucas-m","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ be the Fibonacci numbers, extended to all integers $m$ by $F_{m+2} = F_{m+1} + F_m$, and let $L_m = F_{m-1} + F_{m+1}$ be the Lucas numbers. Let $G : \\mathbb{Z} \\to \\mathbb{R}$ be a gibonacci sequence, that is, $G_{j+2} = G_{j+1} + G_j$ for every integer $j$. Then for all integers $r$ and $s$, $G_{s+r} + (-1)^r G_{s-r} = L_r G_s$.","title":"Lucas multiplier identity for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-lucas-like-mul-add-eq-fib-mul","kind":"node","lean":"MetaMathlibExt.gibonacci_lucas_like_mul_add_eq_fib_mul","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-lucas-like-mul-add-eq-fib-mul","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$ and $j \\mapsto H_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$, and let $F_m$ be the Fibonacci numbers, extended to negative indices by $F_{-m} = (-1)^{m+1} F_m$. Then for all integers $r$, $k$ and $s$,","title":"Two-sequence gibonacci identity with a Fibonacci factor and Lucas-type sums"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-gelin-cesaro-two-sequence","kind":"node","lean":"MetaMathlibExt.gibonacci_gelin_cesaro_two_sequence","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-gelin-cesaro-two-sequence","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G, H : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$ and $j \\mapsto H_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$, and put $e_G = G_0^2 - G_1^2 + G_0 G_1$. Then for all integers $k$ and $r$,","title":"Two-sequence Gelin\u2013Ces\u00e0ro derivative identity for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/fib-gibonacci-catalan-type-identity","kind":"node","lean":"MetaMathlibExt.fib_gibonacci_catalan_type_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#fib-gibonacci-catalan-type-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ denote the Fibonacci numbers, extended to all integers $m$ by $F_{-m} = (-1)^{m+1} F_m$. If $G : \\mathbb{Z} \\to \\mathbb{R}$ satisfies $G_{j+2} = G_{j+1} + G_j$ for every integer $j$, then for all integers $k$, $r$ and $s$,","title":"Catalan-type identity for Fibonacci and gibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/lucas-mul-gibonacci-seed-sub-eq","kind":"node","lean":"MetaMathlibExt.lucas_mul_gibonacci_seed_sub_eq","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#lucas-mul-gibonacci-seed-sub-eq","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ be the Fibonacci numbers extended to all integers $m$ ($F_0 = 0$, $F_1 = 1$, $F_{m+2} = F_{m+1} + F_m$) and $L_m = F_{m-1} + F_{m+1}$ the Lucas numbers, and let $G, H \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $G_{j+2} = G_{j+1} + G_j$ and $H_{j+2} = H_{j+1} + H_j$ for every integer $j$. Then for all integers $r$, $s$, $t$,","title":"Lucas-weighted identity for two gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-catalan-type-two-sequence","kind":"node","lean":"MetaMathlibExt.gibonacci_catalan_type_two_sequence","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-catalan-type-two-sequence","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $F_m$ be the Fibonacci numbers, extended to all integers $m$ by $F_{m+2} = F_{m+1} + F_m$, and let $G$ and $H$ be gibonacci sequences, that is, functions $X : \\mathbb{Z} \\to \\mathbb{R}$ with $X_{j+2} = X_{j+1} + X_j$ for every integer $j$. Then for all integers $k$, $r$, $s$, $t$,","title":"Two-sequence Catalan-type identity for gibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/gibonacci-div-fib-sq-add-one-identity","kind":"node","lean":"MetaMathlibExt.gibonacci_div_fib_sq_add_one_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#gibonacci-div-fib-sq-add-one-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $G : \\mathbb{Z} \\to \\mathbb{R}$, written $j \\mapsto G_j$, satisfy $G_{j+2} = G_{j+1} + G_j$ for every integer $j$, and let $F_m$ be the Fibonacci numbers, extended to negative indices by $F_{-m} = (-1)^{m+1} F_m$. Then for all integers $r$ and $k$,","title":"Gibonacci identity with Fibonacci-square denominators"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/horadam-sub-root-mul-div-eq-zpow","kind":"node","lean":"MetaMathlibExt.horadam_sub_root_mul_div_eq_zpow","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#horadam-sub-root-mul-div-eq-zpow","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $P$, $Q$, $\\sigma$, $\\tau$ be real numbers with $Q \\neq 0$, $\\sigma + \\tau = P$ and $\\sigma \\tau = Q$, and let $W \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_{j+2} = P W_{j+1} - Q W_j$ for every integer $j$. Then for every integer $k$: if $W_1 \\neq \\sigma W_0$, then","title":"Ratio form of the Binet formula for Horadam sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/horadam-product-exchange-identity","kind":"node","lean":"MetaMathlibExt.horadam_product_exchange_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#horadam-product-exchange-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $P$, $Q$ be real numbers with $Q \\neq 0$, and let $W, Z \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_{j+2} = P W_{j+1} - Q W_j$ and $Z_{j+2} = P Z_{j+1} - Q Z_j$ for every integer $j$. Then for all integers $r$, $k$,","title":"Product exchange identity for two Horadam sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/horadam-companion-lucas-mul-identity","kind":"node","lean":"MetaMathlibExt.horadam_companion_lucas_mul_identity","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#horadam-companion-lucas-mul-identity","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $P$, $Q$ be real numbers with $Q \\neq 0$. Let $W \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_{j+2} = P W_{j+1} - Q W_j$ for every integer $j$, and let $V \\colon \\mathbb{Z} \\to \\mathbb{R}$ (the companion Lucas sequence) satisfy the same recurrence with $V_0 = 2$ and $V_1 = P$. Then for all integers $k$, $r$,","title":"Companion Lucas sequence identity for Horadam sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/generalized-fibonacci-binet-add-eq-add","kind":"node","lean":"MetaMathlibExt.generalizedFibonacci_binet_add_eq_add","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#generalized-fibonacci-binet-add-eq-add","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $p$ be a real number and let $W : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_{j+2} = p W_{j+1} + W_j$ for every integer $j$. Put $\\sigma = (p + \\sqrt{p^2 + 4})/2$ and $\\tau = (p - \\sqrt{p^2 + 4})/2$. Then for every integer $j$,","title":"Companion Binet formula for generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/has-deriv-at-neg-rpow-branch","kind":"node","lean":"MetaMathlibExt.hasDerivAt_neg_rpow_branch","parent":6029,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative of a branch of "],["math","\\(\\lambda^x\\)"],["text"," for negative "],["math","\\(\\lambda\\)"]],"source":"Derivative of a branch of $\\lambda^x$ for negative $\\lambda$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#has-deriv-at-neg-rpow-branch","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $\\lambda < 0$ be real and $m$ an integer, and consider the branch $\\lambda^y = (-\\lambda)^y e^{i\\pi(2m+1)y}$, a function of real $y$ with complex values, where $(-\\lambda)^y$ is the real power of the positive number $-\\lambda$. For every real $x$ this function has derivative $\\lambda^x \\bigl(i\\pi(2m+1) + \\ln(-\\lambda)\\bigr)$ at $x$. The branch corresponds to the journal proof's evaluation $\\ln(-1) = i\\pi(2m+1)$.","title":"Derivative of a branch of $\\lambda^x$ for negative $\\lambda$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/generalized-fibonacci-function-deriv-re","kind":"node","lean":"MetaMathlibExt.generalizedFibonacci_function_deriv_re","parent":6029,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions/index.html#generalized-fibonacci-function-deriv-re","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-identities-via-fibonacci-functions","status":"can_prove","summary":"Let $p$ be a real number and let $W : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_{j+2} = p W_{j+1} + W_j$ for every integer $j$. Put $\\delta = \\sqrt{p^2 + 4}$, $\\sigma = (p + \\delta)/2$, $\\tau = (p - \\delta)/2$, $A = W_1 - W_0 \\tau$ and $B = W_1 - W_0 \\sigma$. For an integer $m$, define $w : \\mathbb{R} \\to \\mathbb{C}$ by","title":"Real part of the derivative of the generalized Fibonacci function at integers"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi.json","id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi","kind":"scope","navigation":"module","parent":5992,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi","status":"can_prove","title":"Fibonacci representations with polynomial coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/fib-linear-coeff-is-int-iff","kind":"node","lean":"MetaMathlibExt.fib_linear_coeff_isInt_iff","parent":6074,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/index.html#fib-linear-coeff-is-int-iff","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi","status":"can_prove","summary":"Let $F_n$ be the $n$-th Fibonacci number and put $F_{-1} = 1$, so that $F_{n-1} = F_{n+1} - F_n$ for every $n \\ge 0$. For rational numbers $a, b, c, d$, the number $w_n = (an + b)F_n + (cn + d)F_{n-1}$ is an integer for every natural number $n$ if and only if $d$ is an integer and there are integers $z_1, z_2, z_3$ with $a = (-z_1 - 3z_2 + 2z_3)/5$, $b = (6z_1 + 3z_2 - 2z_3)/5$ and $c = (-2z_1 + 4z_2 - z_3)/5$.","title":"Integer sequences with linear Fibonacci coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/fib-quadratic-coeff-is-int-iff","kind":"node","lean":"MetaMathlibExt.fib_quadratic_coeff_isInt_iff","parent":6074,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/index.html#fib-quadratic-coeff-is-int-iff","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi","status":"can_prove","summary":"Let $F_n$ be the $n$-th Fibonacci number and put $F_{-1} = 1$, so that $F_{n-1} = F_{n+1} - F_n$ for every $n \\ge 0$. For rational numbers $a, b, c, d, e, f$, the number $w_n = (an^2 + bn + c)F_n + (dn^2 + en + f)F_{n-1}$ is an integer for every natural number $n$ if and only if $f$ is an integer and there are integers $z_1, \\dots, z_5$ with $a = (-z_1 + 3z_2 + z_3 - 3z_4 + z_5)/10$,\u2026","title":"Integer sequences with quadratic Fibonacci coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/fib-quadratic-linear-coeff-is-int-iff","kind":"node","lean":"MetaMathlibExt.fib_quadratic_linear_coeff_isInt_iff","parent":6074,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/index.html#fib-quadratic-linear-coeff-is-int-iff","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi","status":"can_prove","summary":"Let $F_n$ be the $n$-th Fibonacci number and put $F_{-1} = 1$, so that $F_{n-1} = F_{n+1} - F_n$ for every $n \\ge 0$. For rational numbers $a, b, c, d, e$, the number $w_n = (an^2 + bn + c)F_n + (dn + e)F_{n-1}$ is an integer for every natural number $n$ if and only if $e$ is an integer and there are integers $z_1, z_2, z_3, z_4$ with $a = (2z_1 - z_2 - 2z_3 + z_4)/10$, $b = (-56z_1 - 7z_2 + 66z_3 - 23z_4)/50$,\u2026","title":"Integer sequences with quadratic and linear Fibonacci coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/fib-linear-coeff-add-alternating-is-int","kind":"node","lean":"MetaMathlibExt.fib_linear_coeff_add_alternating_isInt_iff","parent":6074,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi/index.html#fib-linear-coeff-add-alternating-is-int","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-representations-with-polynomial-coeffi","status":"can_prove","summary":"Let $F_n$ be the $n$-th Fibonacci number and put $F_{-1} = 1$, so that $F_{n-1} = F_{n+1} - F_n$ for every $n \\ge 0$. For rational numbers $a, b, c, d, e, f$, the number $w_n = (an + b)F_n + (cn + d)F_{n-1} + e + f(-1)^n$ is an integer for every natural number $n$ if and only if there are integers $v_0, \\dots, v_5$ with $a = (3v_0 + 2v_1 - 7v_2 - v_3 + 4v_4 - v_5)/5$,\u2026","title":"Integer sequences with linear Fibonacci coefficients and an alternating term"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for.json","id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for","kind":"scope","navigation":"module","parent":5992,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for","status":"can_prove","title":"Fibonacci series for \u03c0\u00b2"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/pi-sq-eq-fibonacci-factorial-series","kind":"node","lean":"MetaMathlibExt.pi_sq_eq_fibonacci_factorial_series","parent":6079,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci series for "],["math","\\(\\pi^2\\)"],["text"," with polynomial weights"]],"source":"Fibonacci series for $\\pi^2$ with polynomial weights"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/index.html#pi-sq-eq-fibonacci-factorial-series","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for","status":"can_prove","summary":"Let $p_1, p_2, \\dots$ be polynomials with integer coefficients such that $p_1 = 1$ and, for every $n \\ge 1$, $p_{n+1}(x) = (2x + 2n + 1)(2x + 2n + 2)\\, p_n(x) - (x + 1)^2 p_n(x + 1)$. Let $(a_n)_{n \\ge 0}$ be real numbers with $a_0 = -8\\sqrt{5}/250$, $a_1 = (25 - 4\\sqrt{5})/250$ and $a_{n+1} = a_n + a_{n-1}$ for $n \\ge 1$, and let $(b_n)_{n \\ge 0}$ be real numbers with $b_0 = b_1 = 0$ and\u2026","title":"Fibonacci series for $\\pi^2$ with polynomial weights"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/pi-sq-eq-fibonacci-central-series","kind":"node","lean":"MetaMathlibExt.pi_sq_eq_fibonacci_central_series","parent":6079,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two Fibonacci series for "],["math","\\(\\pi^2\\)"]],"source":"Two Fibonacci series for $\\pi^2$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/index.html#pi-sq-eq-fibonacci-central-series","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for","status":"can_prove","summary":"Writing $F_m$ for the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$),","title":"Two Fibonacci series for $\\pi^2$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/fib-pi-squared-coeff-eq-fib","kind":"node","lean":"MetaMathlibExt.fibPiSquared_coeff_eq_fib","parent":6079,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci closed form of the sequence "],["math","\\(a_n\\)"]],"source":"Fibonacci closed form of the sequence $a_n$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/index.html#fib-pi-squared-coeff-eq-fib","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be real numbers with $a_0 = -8\\sqrt{5}/250$, $a_1 = (25 - 4\\sqrt{5})/250$ and $a_{n+1} = a_n + a_{n-1}$ for $n \\ge 1$, and write $F_m$ for the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). Then for every $n \\ge 0$,","title":"Fibonacci closed form of the sequence $a_n$"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/fib-pi-squared-poly-eq-sum","kind":"node","lean":"MetaMathlibExt.fibPiSquared_poly_eq_sum","parent":6079,"reading_title":{"format":"inline-parts-v1","parts":[["text","Explicit formula for the polynomials "],["math","\\(p_n\\)"]],"source":"Explicit formula for the polynomials $p_n$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for/index.html#fib-pi-squared-poly-eq-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/fibonacci-series-for","status":"can_prove","summary":"Let $p_1, p_2, \\dots$ be polynomials with integer coefficients such that $p_1 = 1$ and, for every $n \\ge 1$, $p_{n+1}(x) = (2x + 2n + 1)(2x + 2n + 2)\\, p_n(x) - (x + 1)^2 p_n(x + 1)$. Then for every $n \\ge 1$, as polynomials in $x$,","title":"Explicit formula for the polynomials $p_n$"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions.json","id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","kind":"scope","navigation":"module","parent":5992,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"can_prove","title":"Sury's identity and its extensions"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/fib-at-int","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SuryIdentityExtensionsWanted.fibAtInt","parent":6084,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html#fib-at-int","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"fully_proved","summary":"For an integer $n$, the Fibonacci number $F_n$ as a rational number, where $F_0 = 0$, $F_1 = 1$, $F_{m+2} = F_{m+1} + F_m$ for natural numbers $m$, and $F_{-m} = (-1)^{m+1} F_m$ for natural numbers $m$.","title":"Fibonacci numbers extended to negative indices"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/lucas-at-int","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SuryIdentityExtensionsWanted.lucasAtInt","parent":6084,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html#lucas-at-int","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"fully_proved","summary":"For an integer $n$, the Lucas number $L_n$ as a rational number, where $L_0 = 2$, $L_1 = 1$, $L_{m+2} = L_{m+1} + L_m$ for natural numbers $m$, and $L_{-m} = (-1)^m L_m$ for natural numbers $m$.","title":"Lucas numbers at integer indices"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/second-order-recurrence-initial-one-sury","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SuryIdentityExtensionsWanted.second_order_recurrence_initial_one_sury_sum","parent":6084,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html#second-order-recurrence-initial-one-sury","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"can_prove","summary":"Let $c_1, c_2$ be rational numbers with $c_2 \\neq 0$, and let $(A_n)_{n \\ge 0}$ be a sequence of rational numbers with $A_0 = 1$ and $A_{n+2} = c_1 A_{n+1} + c_2 A_n$ for all $n \\ge 0$. Put $t = c_1 - A_1$ and assume $t \\neq 0$. Then for every natural number $n$,","title":"Sury's identity for a second-order recurrence starting at 1"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/second-order-recurrence-offset-sury-sum","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SuryIdentityExtensionsWanted.second_order_recurrence_offset_sury_sum","parent":6084,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html#second-order-recurrence-offset-sury-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"can_prove","summary":"Let $c_1, c_2$ be rational numbers with $c_2 \\neq 0$, and let $(X_n)_{n \\in \\mathbb{Z}}$ be a sequence of rational numbers with $X_{n+2} = c_1 X_{n+1} + c_2 X_n$ for all integers $n$. Let $k$ be an integer with $X_k \\neq 0$ and $X_{k-1} \\neq 0$. Then for every natural number $n$,","title":"Offset form of Sury's identity for a second-order recurrence"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/fib-step-sury-sum","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SuryIdentityExtensionsWanted.fib_step_sury_sum","parent":6084,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html#fib-step-sury-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"can_prove","summary":"Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers at integer indices. For all natural numbers $j \\ge 1$ and $n$,","title":"Step extension of Sury's identity for Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/lucas-step-sury-sum","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SuryIdentityExtensionsWanted.lucas_step_sury_sum","parent":6084,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html#lucas-step-sury-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"can_prove","summary":"Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers at integer indices. For all natural numbers $j \\ge 1$ and $n$, in the rational numbers,","title":"Step extension of Sury's identity for Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/fib-step-offset-sury-sum","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SuryIdentityExtensionsWanted.fib_step_offset_sury_sum","parent":6084,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html#fib-step-offset-sury-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers at integer indices. For all natural numbers $j$ and $n$ and every integer $k$ with $0 < |k| < j$,","title":"Offset step extension of Sury's identity, Fibonacci form"},{"id":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/lucas-step-offset-sury-fib-sum","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SuryIdentityExtensionsWanted.lucas_step_offset_sury_fib_sum","parent":6084,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions/index.html#lucas-step-offset-sury-fib-sum","scope":"NumberTheory/linear-recurrences/fibonacci-lucas-numbers/sury-s-identity-and-its-extensions","status":"can_prove","summary":"Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers at integer indices. For all natural numbers $j$ and $n$ and every integer $k$ with $|k| < j$,","title":"Offset step extension of Sury's identity, Lucas form"},{"data_url":"scopes/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors.json","id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors","kind":"scope","parent":5991,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/index.html","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors","status":"can_prove","summary":"General Lucas sequences U_n, V_n, Lucas pairs, divisibility, Bang\u2013Zsigmondy and primitive-divisor theorems, Fibonacci prime divisibility.","title":"Lucas sequences and primitive divisors"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/fib-prime-dvd-fib-of-legendre-sub","kind":"node","lean":"MetaMathlibExt.fib_prime_dvd_fib_of_legendre_sub","parent":6093,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/index.html#fib-prime-dvd-fib-of-legendre-sub","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors","status":"fully_proved","summary":"Let $p$ be a prime with $p > 2$ and $p \\ne 5$, and let $\\left(\\frac{5}{p}\\right)$ be the Legendre symbol of $5$ modulo $p$. Then $p$ divides the Fibonacci number $F_{p-1}$ if $\\left(\\frac{5}{p}\\right) = 1$, and $p$ divides the Fibonacci number $F_{p+1}$ otherwise. Since $\\left(\\frac{5}{p}\\right) = \\pm 1$, this says that $p$ divides $F_{p - \\left(\\frac{5}{p}\\right)}$.","title":"Fibonacci prime divisibility"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lengyel-fibonacci-five-adic","kind":"node","lean":"MetaMathlibExt.lengyel_fibonacci_five_adic","parent":6093,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(5\\)"],["text","-adic valuation of Fibonacci numbers"]],"source":"The $5$-adic valuation of Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/index.html#lengyel-fibonacci-five-adic","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors","status":"fully_proved","summary":"Writing $F_n$ for the $n$-th Fibonacci number and $v_5$ for the $5$-adic valuation, $v_5(F_n) = v_5(n)$ for every integer $n \\ge 1$.","title":"The $5$-adic valuation of Fibonacci numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences.json","id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","kind":"scope","navigation":"module","parent":6093,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"can_prove","title":"Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-sequence-params","kind":"node","lean":"MetaMathlibExt.LucasSequenceParams","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-sequence-params","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"Lucas sequence parameters are a pair of integers $(P, Q)$ with $Q \\ne 0$.","title":"Parameters of a Lucas sequence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/is-coprime","kind":"node","lean":"MetaMathlibExt.LucasSequenceParams.IsCoprime","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#is-coprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"Lucas sequence parameters $(P, Q)$ are coprime when there exist integers $x$ and $y$ with $xP + yQ = 1$.","title":"Coprime Lucas sequence parameters"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-u","kind":"node","lean":"MetaMathlibExt.lucasU","parent":6096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas sequence "],["math","\\(U(P, Q)\\)"]],"source":"Lucas sequence $U(P, Q)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-u","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For Lucas sequence parameters $(P, Q)$, the Lucas sequence $U(P, Q)$ is the integer sequence $(U_n)_{n \\ge 0}$ with $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = P U_{n+1} - Q U_n$ for every natural number $n$.","title":"Lucas sequence $U(P, Q)$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-v","kind":"node","lean":"MetaMathlibExt.lucasV","parent":6096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Companion Lucas sequence "],["math","\\(V(P, Q)\\)"]],"source":"Companion Lucas sequence $V(P, Q)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-v","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For Lucas sequence parameters $(P, Q)$, the companion Lucas sequence $V(P, Q)$ is the integer sequence $(V_n)_{n \\ge 0}$ with $V_0 = 2$, $V_1 = P$ and $V_{n+2} = P V_{n+1} - Q V_n$ for every natural number $n$.","title":"Companion Lucas sequence $V(P, Q)$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/fibonacci-lucas-params","kind":"node","lean":"MetaMathlibExt.fibonacciLucasParams","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#fibonacci-lucas-params","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"The Fibonacci parameters are the Lucas sequence parameters $P = 1$ and $Q = -1$; they satisfy the required condition $Q \\ne 0$.","title":"Fibonacci parameters for Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-number","kind":"node","lean":"MetaMathlibExt.lucasNumber","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-number","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"The Lucas numbers are the natural numbers $L_n$ defined by $L_0 = 2$, $L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n$ for every natural number $n$. They begin $2, 1, 3, 4, 7, \\dots$.","title":"Lucas numbers"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-u-fibonacci-lucas-params","kind":"node","lean":"MetaMathlibExt.lucasU_fibonacciLucasParams","parent":6096,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lucas sequence "],["math","\\(U(1, -1)\\)"],["text"," is the Fibonacci sequence"]],"source":"The Lucas sequence $U(1, -1)$ is the Fibonacci sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-u-fibonacci-lucas-params","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For every natural number $n$, the $n$-th term of the Lucas sequence $U(1, -1)$ equals the Fibonacci number $F_n$, where $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$.","title":"The Lucas sequence $U(1, -1)$ is the Fibonacci sequence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-u-add-two","kind":"node","lean":"MetaMathlibExt.lucasU_add_two","parent":6096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the Lucas sequence "],["math","\\(U(P, Q)\\)"]],"source":"Recurrence for the Lucas sequence $U(P, Q)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-u-add-two","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For Lucas sequence parameters $(P, Q)$ and every natural number $n$, the Lucas sequence $U = U(P, Q)$ satisfies $U_{n+2} = P U_{n+1} - Q U_n$.","title":"Recurrence for the Lucas sequence $U(P, Q)$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-u-cassini","kind":"node","lean":"MetaMathlibExt.lucasU_cassini","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-u-cassini","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For Lucas sequence parameters $(P, Q)$ and every natural number $n$, the Lucas sequence $U = U(P, Q)$ satisfies $U_{n+1}^2 - U_n U_{n+2} = Q^n$.","title":"Cassini's identity for Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-v-fibonacci-lucas-params","kind":"node","lean":"MetaMathlibExt.lucasV_fibonacciLucasParams","parent":6096,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lucas sequence "],["math","\\(V(1, -1)\\)"],["text"," gives the Lucas numbers"]],"source":"The Lucas sequence $V(1, -1)$ gives the Lucas numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-v-fibonacci-lucas-params","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For every natural number $n$, the $n$-th term of the companion Lucas sequence $V(1, -1)$ equals the $n$-th Lucas number $L_n$.","title":"The Lucas sequence $V(1, -1)$ gives the Lucas numbers"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/is-lucas-special-prime","kind":"node","lean":"MetaMathlibExt.IsLucasSpecialPrime","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#is-lucas-special-prime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For integers $P$ and $Q$, a natural number $p$ is a special prime for the Lucas parameters $(P, Q)$ if $p$ is prime and $p$ divides $\\gcd(P, Q)$.","title":"Special primes of a Lucas pair"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/is-lucas-special-prime-iff","kind":"node","lean":"MetaMathlibExt.isLucasSpecialPrime_iff","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#is-lucas-special-prime-iff","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For integers $P$ and $Q$ and a natural number $p$, $p$ is a special prime for the Lucas parameters $(P, Q)$ if and only if $p$ is prime and $p$ divides both $P$ and $Q$ in $\\mathbb{Z}$.","title":"Special primes are the common prime divisors of the parameters"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-binet-formulas-of-field","kind":"node","lean":"MetaMathlibExt.lucas_binet_formulas_of_field","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-binet-formulas-of-field","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"Let $P$ and $Q$ be integers with $Q \\ne 0$, let $U_n = U_n(P, Q)$ and $V_n = V_n(P, Q)$ be the associated Lucas sequences, and let $\\alpha$ and $\\beta$ be elements of a field $K$ with $\\alpha + \\beta = P$ and $\\alpha \\beta = Q$ in $K$. If $\\alpha \\ne \\beta$, then for every natural number $n$, $U_n = (\\alpha^n - \\beta^n) / (\\alpha - \\beta)$ and $V_n = \\alpha^n + \\beta^n$ in $K$. If $\\beta = \\alpha$, then for every\u2026","title":"Binet formulas for Lucas sequences over any field"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/lucas-binet-formulas","kind":"node","lean":"MetaMathlibExt.lucas_binet_formulas","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#lucas-binet-formulas","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"Let $P$ and $Q$ be integers with $Q \\ne 0$, let $U_n = U_n(P, Q)$ and $V_n = V_n(P, Q)$ be the associated Lucas sequences, and let $\\alpha$ and $\\beta$ be elements of a field $K$ of characteristic zero with $\\alpha + \\beta = P$ and $\\alpha \\beta = Q$ in $K$. If $\\alpha \\ne \\beta$, then for every natural number $n$, $U_n = (\\alpha^n - \\beta^n) / (\\alpha - \\beta)$ and $V_n = \\alpha^n + \\beta^n$ in $K$. If\u2026","title":"Binet formulas for Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/is-lucas-sequence-first-kind","kind":"node","lean":"MetaMathlibExt.IsLucasSequenceFirstKind","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#is-lucas-sequence-first-kind","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"For integers $P$ and $Q$, a sequence of integers $(u_n)_{n \\ge 0}$ is the Lucas sequence of the first kind with parameters $P$ and $Q$ when $u_0 = 0$, $u_1 = 1$, and $u_{n+2} = P u_{n+1} - Q u_n$ for every natural number $n$.","title":"Lucas sequence of the first kind"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/initial-zero","kind":"node","lean":"MetaMathlibExt.IsLucasSequenceFirstKind.initialZero","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#initial-zero","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"If $(u_n)_{n \\ge 0}$ is the Lucas sequence of the first kind with integer parameters $P$ and $Q$, then $u_0 = 0$.","title":"Lucas sequence of the first kind: value at zero"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/initial-one","kind":"node","lean":"MetaMathlibExt.IsLucasSequenceFirstKind.initialOne","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#initial-one","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"If $(u_n)_{n \\ge 0}$ is the Lucas sequence of the first kind with integer parameters $P$ and $Q$, then $u_1 = 1$.","title":"Lucas sequence of the first kind: value at one"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/recurrence-step","kind":"node","lean":"MetaMathlibExt.IsLucasSequenceFirstKind.recurrenceStep","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#recurrence-step","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"If $(u_n)_{n \\ge 0}$ is the Lucas sequence of the first kind with integer parameters $P$ and $Q$, then $u_{n+2} = P u_{n+1} - Q u_n$ for every natural number $n$.","title":"Lucas sequence of the first kind: the recurrence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/is-nondegenerate-lucas-pair","kind":"node","lean":"MetaMathlibExt.IsNondegenerateLucasPair","parent":6096,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#is-nondegenerate-lucas-pair","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"fully_proved","summary":"A pair of integers $(P, Q)$ is nondegenerate when $Q \\ne 0$ and, for all complex numbers $\\alpha$ and $\\beta$ with $\\alpha + \\beta = P$ and $\\alpha \\beta = Q$, the quotient $\\alpha / \\beta$ is not a root of unity. Such $\\alpha$ and $\\beta$ are the two roots of $x^2 - P x + Q$, in either order.","title":"Nondegenerate Lucas pair"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/two-mul-lucas-u-add","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.KruchininPerminovaHadamardWanted.two_mul_lucasU_add","parent":6096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Addition formula "],["math","\\(2U_{m+n} = U_m V_n + V_m U_n\\)"],["text"," for Lucas sequences"]],"source":"Addition formula $2U_{m+n} = U_m V_n + V_m U_n$ for Lucas sequences"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences/index.html#two-mul-lucas-u-add","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/lucas-sequences","status":"can_prove","summary":"For all integers $P$ and $Q$ with $Q \\neq 0$ and all natural numbers $m$ and $n$, the Lucas sequences $U_n = U_n(P, Q)$ and $V_n = V_n(P, Q)$ satisfy $2U_{m+n} = U_m V_n + V_m U_n$.","title":"Addition formula $2U_{m+n} = U_m V_n + V_m U_n$ for Lucas sequences"},{"data_url":"scopes/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences.json","id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences","kind":"scope","navigation":"module","parent":6093,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/index.html","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences","status":"can_prove","title":"Composite terms of shifted Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/lucas-u-three-neg-four-composite","kind":"node","lean":"MetaMathlibExt.lucasU_three_neg_four_composite","parent":6117,"reading_title":{"format":"inline-parts-v1","parts":[["text","Composite terms of the Lucas sequence "],["math","\\(U_n(3, -4)\\)"]],"source":"Composite terms of the Lucas sequence $U_n(3, -4)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/index.html#lucas-u-three-neg-four-composite","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences","status":"can_prove","summary":"Let $U_n = U_n(3, -4)$ be the Lucas sequence with $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = 3U_{n+1} + 4U_n$, so that $U_n = 0, 1, 3, 13, 51, 205, 819, \\ldots$. For every natural number $n \\ge 4$, $U_n$ is composite: $U_n > 1$ and $U_n$ is not prime.","title":"Composite terms of the Lucas sequence $U_n(3, -4)$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/lucas-u-q-one-shifts-eventually-composit","kind":"node","lean":"MetaMathlibExt.lucasU_q_one_shifts_eventually_composite","parent":6117,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit shifts of "],["math","\\(U_n(P, 1)\\)"],["text"," are eventually composite"]],"source":"Unit shifts of $U_n(P, 1)$ are eventually composite"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/index.html#lucas-u-q-one-shifts-eventually-composit","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences","status":"can_prove","summary":"Let $P \\ge 3$ be an integer and let $U_n = U_n(P, 1)$ be the Lucas sequence with $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = PU_{n+1} - U_n$. Then there is a natural number $N$ such that for every $n \\ge N$ the three integers $U_n - 1$, $U_n$ and $U_n + 1$ are all composite, that is, each is greater than $1$ and not prime.","title":"Unit shifts of $U_n(P, 1)$ are eventually composite"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/exists-shift-lucas-u-q-neg-one-composite","kind":"node","lean":"MetaMathlibExt.exists_shift_lucasU_q_neg_one_composite","parent":6117,"reading_title":{"format":"inline-parts-v1","parts":[["text","A shift of "],["math","\\(U_n(P, -1)\\)"],["text"," away from primes"]],"source":"A shift of $U_n(P, -1)$ away from primes"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/index.html#exists-shift-lucas-u-q-neg-one-composite","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences","status":"can_prove","summary":"Let $P \\ge 1$ be an integer and let $U_n = U_n(P, -1)$ be the Lucas sequence with $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = PU_{n+1} + U_n$. Then there is an integer $K > 0$ such that for every natural number $n$ the three integers $U_n + K - 1$, $U_n + K$ and $U_n + K + 1$ are all composite, that is, each is greater than $1$ and not prime.","title":"A shift of $U_n(P, -1)$ away from primes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/lucas-u-square-q-finite-primes","kind":"node","lean":"MetaMathlibExt.lucasU_square_q_finite_primes","parent":6117,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas sequences with square "],["math","\\(Q\\)"],["text"," have finitely many prime terms"]],"source":"Lucas sequences with square $Q$ have finitely many prime terms"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/index.html#lucas-u-square-q-finite-primes","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $P \\ge 1$ and $P^2 - 4Q > 0$, where $Q \\ne 0$ is a perfect square, and let $U_n = U_n(P, Q)$ be the Lucas sequence with $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = PU_{n+1} - QU_n$. Then $|U_n|$ is prime for only finitely many natural numbers $n$.","title":"Lucas sequences with square $Q$ have finitely many prime terms"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/lucas-u-power-params-finite-primes","kind":"node","lean":"MetaMathlibExt.lucasU_power_params_finite_primes","parent":6117,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/index.html#lucas-u-power-params-finite-primes","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences","status":"can_prove","summary":"Let $x \\ne y$ be real numbers such that $x + y$ and $xy$ are integers, and let $r > 1$ be a natural number. Let $P$ and $Q$ be integers with $P = x^r + y^r$ and $Q = x^r y^r$, and suppose $Q \\ne 0$, $P \\ge 1$ and $P^2 - 4Q > 0$. Let $U_n = U_n(P, Q)$ be the Lucas sequence with $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = PU_{n+1} - QU_n$. Then $|U_n|$ is prime for only finitely many natural numbers $n$.","title":"Lucas sequences with power parameters have finitely many prime terms"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/fib-add-14475-not-prime","kind":"node","lean":"MetaMathlibExt.fib_add_14475_not_prime","parent":6117,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci numbers shifted by "],["math","\\(14475\\)"],["text"," are never prime"]],"source":"Fibonacci numbers shifted by $14475$ are never prime"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences/index.html#fib-add-14475-not-prime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/composite-terms-of-shifted-lucas-sequences","status":"can_prove","summary":"Let $F_n$ be the $n$-th Fibonacci number, with $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$. For every natural number $n$, the number $F_n + 14475$ is not prime.","title":"Fibonacci numbers shifted by $14475$ are never prime"},{"data_url":"scopes/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors.json","id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","kind":"scope","navigation":"module","parent":6093,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"can_prove","title":"Primitive divisors"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/is-primitive-divisor","kind":"node","lean":"IntegerSequence.IsPrimitiveDivisor","parent":6124,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html#is-primitive-divisor","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"fully_proved","summary":"Let $(a_s)_{s \\ge 0}$ be a sequence of integers and let $m$ and $u$ be natural numbers. Then $u$ is a primitive divisor of $a_m$ when $m \\ge 1$, $u \\ge 2$, $u$ divides $a_m$, and $u$ divides no $a_s$ with $1 \\le s < m$.","title":"Primitive divisor of an integer sequence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/is-primitive-prime-divisor","kind":"node","lean":"IntegerSequence.IsPrimitivePrimeDivisor","parent":6124,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html#is-primitive-prime-divisor","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"fully_proved","summary":"Let $(a_s)_{s \\ge 0}$ be a sequence of integers and let $m$ and $u$ be natural numbers. Then $u$ is a primitive prime divisor of $a_m$ when $u$ is prime and $u$ is a primitive divisor of $a_m$.","title":"Primitive prime divisor of an integer sequence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/le-of-dvd","kind":"node","lean":"IntegerSequence.IsPrimitiveDivisor.le_of_dvd","parent":6124,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html#le-of-dvd","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"fully_proved","summary":"Let $(a_s)_{s \\ge 0}$ be a sequence of integers and let the natural number $u$ be a primitive divisor of $a_m$. If $n \\ge 1$ and $u$ divides $a_n$, then $m \\le n$.","title":"A primitive divisor appears at the least positive index"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/unique","kind":"node","lean":"IntegerSequence.IsPrimitiveDivisor.unique","parent":6124,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html#unique","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"fully_proved","summary":"Let $(a_s)_{s \\ge 0}$ be a sequence of integers. If a natural number $u$ is a primitive divisor of both $a_m$ and $a_n$, then $m = n$.","title":"A primitive divisor occurs at only one index"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/unique-2","kind":"node","lean":"IntegerSequence.IsPrimitivePrimeDivisor.unique","parent":6124,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html#unique-2","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"fully_proved","summary":"Let $(a_s)_{s \\ge 0}$ be a sequence of integers. If a natural number $u$ is a primitive prime divisor of both $a_m$ and $a_n$, then $m = n$.","title":"A primitive prime divisor occurs at only one index"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/is-primitive-divisor-2","kind":"node","lean":"LucasSequence.IsPrimitiveDivisor","parent":6124,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html#is-primitive-divisor-2","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"fully_proved","summary":"Let $u_0, u_1, u_2, \\dots$ be a sequence of integers and let $D$ be an integer (for a Lucas pair $(\\alpha, \\beta)$, the discriminant $D = (\\alpha - \\beta)^2$). A natural number $p$ is a primitive divisor of $u_n$ when $n \\ge 1$, $p$ is prime, $p$ divides $u_n$, and $p$ does not divide $D \\, u_1 u_2 \\cdots u_{n-1}$ (for $n = 1$ the product is empty, and the condition is that $p$ does not divide $D$).","title":"Primitive divisor relative to a discriminant"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/exists-is-primitive-divisor-fib","kind":"node","lean":"MetaMathlibExt.exists_isPrimitiveDivisor_fib","parent":6124,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html#exists-is-primitive-divisor-fib","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"fully_proved","summary":"Write $F_j$ for the $j$-th Fibonacci number. For every natural number $n \\ge 7$ with $n \\ne 12$, $F_n$ has a primitive divisor with respect to the discriminant $5$: there is a prime $p$ that divides $F_n$ but does not divide $5 F_1 F_2 \\cdots F_{n-1}$.","title":"Primitive divisors of Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/somer-eventually-has-primitive-prime-div","kind":"node","lean":"MetaMathlibExt.somer_eventually_has_primitive_prime_divisor","parent":6124,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors/index.html#somer-eventually-has-primitive-prime-div","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/primitive-divisors","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $Q \\ne 0$, and let $(u_n)_{n \\ge 0}$ be the integer sequence with $u_0 = 0$, $u_1 = 1$ and $u_{n+2} = Pu_{n+1} - Qu_n$ (the Lucas sequence of the first kind). Suppose the sequence is nondegenerate: whenever complex numbers $\\alpha$ and $\\beta$ satisfy $\\alpha + \\beta = P$, $\\alpha\\beta = Q$ and $\\beta \\ne 0$, no power $(\\alpha / \\beta)^k$ with $k \\ge 1$ equals $1$. Then for all\u2026","title":"Somer's primitive divisor theorem"},{"data_url":"scopes/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences.json","id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","kind":"scope","navigation":"module","parent":6093,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","title":"Divisibility properties of Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-two-mul-eq-lucas-u-mul-lucas-v","kind":"node","lean":"MetaMathlibExt.lucasU_two_mul_eq_lucasU_mul_lucasV","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Doubling formula "],["math","\\(U_{2n} = U_n V_n\\)"],["text"," for Lucas sequences"]],"source":"Doubling formula $U_{2n} = U_n V_n$ for Lucas sequences"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-two-mul-eq-lucas-u-mul-lucas-v","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $Q \\ne 0$, and let $U$ and $V$ be the Lucas sequences $U(P, Q)$ and $V(P, Q)$. Then $U_{2n} = U_n V_n$ for every natural number $n$.","title":"Doubling formula $U_{2n} = U_n V_n$ for Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-dvd-and-lucas-v-dvd-of-dvd","kind":"node","lean":"MetaMathlibExt.lucasU_dvd_and_lucasV_dvd_of_dvd","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-dvd-and-lucas-v-dvd-of-dvd","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U = U(P, Q)$ and $V = V(P, Q)$ be the Lucas sequences with integer parameters $P$ and $Q \\ne 0$, and let $m$ and $n$ be natural numbers with $m \\mid n$. Then $U_m \\mid U_n$, and if moreover $n/m$ is odd, then $V_m \\mid V_n$.","title":"Divisibility of Lucas sequences at divisible indices"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/p-dvd-lucas-u-two-mul-and-lucas-v-of-odd","kind":"node","lean":"MetaMathlibExt.p_dvd_lucasU_two_mul_and_lucasV_of_odd","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(P\\)"],["text"," divides "],["math","\\(U_{2n}\\)"],["text",", and "],["math","\\(V_n\\)"],["text"," for odd "],["math","\\(n\\)"]],"source":"$P$ divides $U_{2n}$, and $V_n$ for odd $n$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#p-dvd-lucas-u-two-mul-and-lucas-v-of-odd","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $Q \\ne 0$, and let $U$ and $V$ be the Lucas sequences $U(P, Q)$ and $V(P, Q)$. Then $P \\mid U_{2n}$ for every natural number $n$, and $P \\mid V_n$ for every odd natural number $n$.","title":"$P$ divides $U_{2n}$, and $V_n$ for odd $n$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-lucas-v-neg-p","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_neg_p","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-lucas-v-neg-p","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"For integers $P$ and $Q \\ne 0$ and every natural number $n$, the Lucas sequences $U$ and $V$ satisfy $U_n(-P, Q) = (-1)^{n+1} U_n(P, Q)$ and $V_n(-P, Q) = (-1)^n V_n(P, Q)$.","title":"Lucas sequences under negation of the first parameter"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-recurrence-add-mod-eq-of-dvd-lucas","kind":"node","lean":"MetaMathlibExt.lucas_recurrence_add_modEq_of_dvd_lucasU","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-recurrence-add-mod-eq-of-dvd-lucas","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P$ and $Q \\ne 0$, and let $(W_n)_{n \\ge 0}$ be a sequence of integers with $W_{n+2} = P W_{n+1} - Q W_n$ for every $n \\ge 0$. If $t \\ge 0$ and $m$ are integers with $m \\mid U_t$ and $\\gcd(m, Q) = 1$, then $W_{n+t} \\equiv U_{t+1} W_n \\pmod{m}$ for every $n \\ge 0$.","title":"Shift congruence for the Lucas recurrence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/prime-pow-succ-dvd-lucas-u-two-mul-iff","kind":"node","lean":"MetaMathlibExt.prime_pow_succ_dvd_lucasU_two_mul_iff","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","When "],["math","\\(p^{i+1}\\)"],["text"," divides "],["math","\\(U_{2n}\\)"],["text",", for a prime power "],["math","\\(p^i\\)"],["text"," exactly dividing "],["math","\\(P\\)"]],"source":"When $p^{i+1}$ divides $U_{2n}$, for a prime power $p^i$ exactly dividing $P$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#prime-pow-succ-dvd-lucas-u-two-mul-iff","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be coprime integers with $Q \\ne 0$, and let $U_n$ be the Lucas sequence $U(P, Q)$. Let $p$ be a prime and $i \\ge 1$ such that $p^i \\mid P$ and $p^{i+1} \\nmid P$. Then for every positive integer $n$, $p^{i+1} \\mid U_{2n}$ if and only if $p \\mid n$.","title":"When $p^{i+1}$ divides $U_{2n}$, for a prime power $p^i$ exactly dividing $P$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/gcd-lucas-u-two-mul-div-p-eq-one","kind":"node","lean":"MetaMathlibExt.gcd_lucasU_two_mul_div_p_eq_one","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coprimality of "],["math","\\(U_{2n}/P\\)"],["text"," and "],["math","\\(P\\)"]],"source":"Coprimality of $U_{2n}/P$ and $P$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#gcd-lucas-u-two-mul-div-p-eq-one","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $Q \\ne 0$, $P \\ne 0$ and $\\gcd(P, Q) = 1$, and let $(U_n)$ be the Lucas sequence $U(P, Q)$. If $n$ is a positive integer with $\\gcd(n, P) = 1$, then $\\gcd(U_{2n}/P, P) = 1$.","title":"Coprimality of $U_{2n}/P$ and $P$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-mod-two","kind":"node","lean":"MetaMathlibExt.lucasU_mod_two","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-mod-two","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P$ and $Q \\ne 0$, let $D = P^2 - 4Q$, and suppose $2 \\nmid \\gcd(P, Q)$. Then $U$ is periodic modulo $2$ from index $1$ on: there is $k > 0$ with $U_{n+k} \\equiv U_n \\pmod 2$ for every $n \\ge 1$. If $P$ is odd and $Q$ is even, then $U_n$ is odd for every $n \\ge 1$. If $P$ is even and $Q$ is odd, then for every $n \\ge 0$, $2 \\mid U_n$ if and only if\u2026","title":"Lucas sequences modulo 2"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-mod-three","kind":"node","lean":"MetaMathlibExt.lucasU_mod_three","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-mod-three","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P$ and $Q \\ne 0$, let $D = P^2 - 4Q$, and suppose $3 \\nmid \\gcd(P, Q)$. Then $U$ is periodic modulo $3$ from index $1$ on: there is $k > 0$ with $U_{n+k} \\equiv U_n \\pmod 3$ for every $n \\ge 1$. If $3 \\mid Q$, then $3 \\nmid U_n$ for every $n \\ge 1$. If $3 \\mid P$, then for every $n \\ge 0$, $3 \\mid U_n$ if and only if $2 \\mid n$. If $3 \\nmid P$ and\u2026","title":"Lucas sequences modulo 3"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-abs-strict-mono-of-discr-pos","kind":"node","lean":"MetaMathlibExt.lucas_abs_strictMono_of_discr_pos","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-abs-strict-mono-of-discr-pos","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers with $P \\neq 0$, $Q \\neq 0$ and $D = P^2 - 4Q > 0$, and let $U = U(P, Q)$ and $V = V(P, Q)$ be the Lucas sequences. Then $|U_n| < |U_{n+1}|$ for every $n \\ge 2$ and $|V_n| < |V_{n+1}|$ for every $n \\ge 1$; if $P > 0$, then $U_n > 0$ for every $n \\ge 1$ and $V_n > 0$ for every $n \\ge 0$; and $|U_3| \\ge 3$ unless $|P| = 1$ and $Q = -1$.","title":"Growth and positivity of Lucas sequences with positive discriminant"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/not-lucas-u-dvd-lucas-u-of-lt","kind":"node","lean":"MetaMathlibExt.not_lucasU_dvd_lucasU_of_lt","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(U_n\\)"],["text"," divides none of "],["math","\\(U_1, \\dots, U_{n-1}\\)"]],"source":"$U_n$ divides none of $U_1, \\dots, U_{n-1}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#not-lucas-u-dvd-lucas-u-of-lt","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $P \\ne 0$, $Q \\ne 0$ and $D = P^2 - 4Q > 0$, and let $U_n$ be the Lucas sequence $U(P, Q)$. Then for every $n \\ge 3$ and every $k$ with $0 < k < n$, $U_n$ does not divide $U_k$.","title":"$U_n$ divides none of $U_1, \\dots, U_{n-1}$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/gcd-p-discr-dvd-four","kind":"node","lean":"MetaMathlibExt.gcd_p_discr_dvd_four","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Common divisors of "],["math","\\(P\\)"],["text"," and the discriminant"]],"source":"Common divisors of $P$ and the discriminant"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#gcd-p-discr-dvd-four","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be coprime integers and let $D = P^2 - 4Q$. Then $\\gcd(P, D)$ divides $4$, and $\\gcd(P, D') = 1$ for every odd integer $D'$ dividing $D$.","title":"Common divisors of $P$ and the discriminant"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/odd-lucas-u-coprime-discr","kind":"node","lean":"MetaMathlibExt.odd_lucasU_coprime_discr","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(U_m\\)"],["text"," and "],["math","\\(U_{2m}/P\\)"],["text"," and their coprimality to "],["math","\\(D\\)"]],"source":"Parity of $U_m$ and $U_{2m}/P$ and their coprimality to $D$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#odd-lucas-u-coprime-discr","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be coprime integers with $P \\ne 0$ and $Q \\ne 0$, let $U_n$ be the Lucas sequence $U(P, Q)$, and let $D = P^2 - 4Q$. Let $m$ be an odd natural number with $\\gcd(m, PQD) = 1$, and with $3 \\nmid m$ if $P$ and $Q$ are both odd. Then $U_m$ and $U_{2m}/P$ are odd, and each of them is coprime to $D$.","title":"Parity of $U_m$ and $U_{2m}/P$ and their coprimality to $D$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-composite-of-discr-pos","kind":"node","lean":"MetaMathlibExt.lucasU_composite_of_discr_pos","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-composite-of-discr-pos","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P \\ne 0$ and $Q \\ne 0$ such that $\\gcd(P, Q) = 1$ and $D = P^2 - 4Q > 0$, and let $m \\ge 5$ be an odd integer. If $m$ is not prime, then $U_m$ is composite; and $U_{2m}/P$ is always composite. Here an integer $N$ is composite when $|N| > 1$ and $|N|$ is not prime.","title":"Composite terms of Lucas sequences with positive discriminant"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-pseudoprime-candidates-jacobi-sy","kind":"node","lean":"MetaMathlibExt.lucasU_pseudoprime_candidates_jacobiSym","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-pseudoprime-candidates-jacobi-sy","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P \\ne 0$ and $Q \\ne 0$ such that $\\gcd(P, Q) = 1$ and $D = P^2 - 4Q > 0$. Let $m > 1$ be an odd integer with $\\gcd(m, PQD) = 1$, and assume $3 \\nmid m$ when $P$ and $Q$ are both odd. Put $N_1 = U_m$ and $N_2 = U_{2m}/P$. Then $N_1$ and $N_2$ are positive and odd, and $\\gcd(N_1 N_2, PQD) = 1$. If $m$ is not prime, then $N_1$ and $N_2$ are composite,\u2026","title":"Pseudoprime candidates from a Lucas sequence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-degenerate-cases","kind":"node","lean":"MetaMathlibExt.lucas_degenerate_cases","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-degenerate-cases","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers, let $\\alpha, \\beta \\in \\mathbb{C}$ satisfy $\\alpha + \\beta = P$ and $\\alpha\\beta = Q$ (so they are the roots of $x^2 - Px + Q$), and let $D = P^2 - 4Q$. Let $(u_n)$ be the Lucas sequence of the first kind with parameters $P, Q$, and let $(v_n)$ be the integer sequence with $v_0 = 2$, $v_1 = P$ and $v_{n+2} = P v_{n+1} - Q v_n$ for all $n$. Suppose that $Q = 0$ or that $\\alpha/\\beta$ is a\u2026","title":"Classification of degenerate Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-rank-of-appearance","kind":"node","lean":"MetaMathlibExt.lucasRankOfAppearance","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-rank-of-appearance","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P$ and $Q \\ne 0$. The rank of appearance $\\rho(m)$ of a natural number $m$ in $U$ is the least positive integer $r$ with $m \\mid U_r$; when no positive $r$ has $m \\mid U_r$, $\\rho(m) = 0$.","title":"Rank of appearance in a Lucas sequence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-rank-of-appearance-le-of-dvd","kind":"node","lean":"MetaMathlibExt.lucasRankOfAppearance_le_of_dvd","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-rank-of-appearance-le-of-dvd","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $P, Q$ be integers with $Q \\neq 0$, let $U = U(P, Q)$ be the Lucas sequence, and write $\\rho(m)$ for the rank of appearance of a natural number $m$ in $U$. If $r > 0$ and $m \\mid U_r$, then $\\rho(m) \\le r$.","title":"Rank of appearance is at most any positive index of divisibility"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-rank-of-appearance-spec","kind":"node","lean":"MetaMathlibExt.lucasRankOfAppearance_spec","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-rank-of-appearance-spec","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $U = U(P, Q)$ be a Lucas sequence and $m$ a natural number. If $m \\mid U_r$ for some positive integer $r$, then the rank of appearance $\\rho(m)$ of $m$ in $U$ is positive and $m \\mid U_{\\rho(m)}$.","title":"The rank of appearance in a Lucas sequence is attained"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-rank-of-appearance-one","kind":"node","lean":"MetaMathlibExt.lucasRankOfAppearance_one","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank of appearance of "],["math","\\(1\\)"]],"source":"Rank of appearance of $1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-rank-of-appearance-one","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"For all integers $P, Q$ with $Q \\neq 0$, the rank of appearance of $1$ in the Lucas sequence $U(P, Q)$ is $1$.","title":"Rank of appearance of $1$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-pseudoprime-index","kind":"node","lean":"MetaMathlibExt.lucasPseudoprimeIndex","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-pseudoprime-index","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $P, Q$ be integers with $Q \\neq 0$ and $D = P^2 - 4Q$. For a natural number $N$, the index is $N - (D/N)$, where $(D/N)$ is the Jacobi symbol, replaced by $0$ when this difference is negative. For $N > 1$ the difference is positive.","title":"Index of the Lucas pseudoprime congruences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-pseudoprime","kind":"node","lean":"MetaMathlibExt.IsLucasPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-pseudoprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $P, Q$ be integers with $Q \\neq 0$, let $U_n$ be the Lucas sequence of $(P, Q)$ and let $D = P^2 - 4Q$. A natural number $N$ is a Lucas pseudoprime with parameters $P, Q$ if $N$ is odd and composite, $\\gcd(N, QD) = 1$ and $N$ divides $U_{N - (D/N)}$, where $(D/N)$ is the Jacobi symbol.","title":"Lucas pseudoprimes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-pseudoprime-second-kind","kind":"node","lean":"MetaMathlibExt.IsLucasPseudoprimeSecondKind","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-pseudoprime-second-kind","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $P, Q$ be integers with $Q \\neq 0$, let $U_n$ be the Lucas sequence of $(P, Q)$ and let $D = P^2 - 4Q$. A natural number $N$ is a Lucas pseudoprime of the second kind with parameters $P, Q$ if $N$ is odd and composite, $\\gcd(N, QD) = 1$ and $U_N \\equiv (D/N) \\pmod{N}$, where $(D/N)$ is the Jacobi symbol.","title":"Lucas pseudoprimes of the second kind"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-dickson-pseudoprime","kind":"node","lean":"MetaMathlibExt.IsDicksonPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-dickson-pseudoprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $P$ and $Q \\ne 0$ be integers with discriminant $D = P^2 - 4Q$, and let $(V_n)$ be the companion Lucas sequence $V(P, Q)$. A natural number $N$ is a Dickson pseudoprime with parameters $P$ and $Q$ if $N$ is odd, $N > 1$, $N$ is not prime, $\\gcd(N, QD) = 1$, and $V_N \\equiv P \\pmod N$.","title":"Dickson pseudoprimes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-dickson-pseudoprime-second-kind","kind":"node","lean":"MetaMathlibExt.IsDicksonPseudoprimeSecondKind","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-dickson-pseudoprime-second-kind","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $P$ and $Q \\ne 0$ be integers with discriminant $D = P^2 - 4Q$, and let $(V_n)$ be the companion Lucas sequence $V(P, Q)$. A natural number $N$ is a Dickson pseudoprime of the second kind with parameters $P$ and $Q$ if $N$ is odd, $N > 1$, $N$ is not prime, $\\gcd(N, QD) = 1$, and $V_{N - (D/N)} \\equiv 2Q^{(1 - (D/N))/2} \\pmod N$, where $(D/N)$ is the Jacobi symbol.","title":"Dickson pseudoprimes of the second kind"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-frobenius-pseudoprime","kind":"node","lean":"MetaMathlibExt.IsLucasFrobeniusPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-frobenius-pseudoprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $U$ and $V$ be the Lucas sequences with integer parameters $P$ and $Q \\neq 0$, and let $D = P^2 - 4Q$. A natural number $N$ is a Frobenius pseudoprime with parameters $P, Q$ when $\\gcd(N, PQD) = 1$ and $N$ is a Lucas pseudoprime, a Lucas pseudoprime of the second kind, a Dickson pseudoprime and a Dickson pseudoprime of the second kind with parameters $P, Q$. Equivalently, $\\gcd(N, PQD) = 1$, $N$ is odd and\u2026","title":"Frobenius pseudoprime for Lucas parameters"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-euler-pseudoprime","kind":"node","lean":"MetaMathlibExt.IsEulerPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-euler-pseudoprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $a$ be an integer. A natural number $N$ is an Euler pseudoprime to the base $a$ when $N$ is odd and composite, $\\gcd(a, N) = 1$, and $a^{(N-1)/2} \\equiv (a/N) \\pmod N$, where $(a/N)$ is the Jacobi symbol.","title":"Euler pseudoprime"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-euler-lucas-pseudoprime","kind":"node","lean":"MetaMathlibExt.IsEulerLucasPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-euler-lucas-pseudoprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $U$ and $V$ be the Lucas sequences with integer parameters $P$ and $Q \\neq 0$, and let $D = P^2 - 4Q$. A natural number $N$ is an Euler\u2013Lucas pseudoprime with parameters $P, Q$ when $N$ is odd and composite, $\\gcd(N, QD) = 1$, and, writing $\\varepsilon = (D/N)$ for the Jacobi symbol, $N \\mid U_{(N-\\varepsilon)/2}$ if the Jacobi symbol $(Q/N)$ equals $1$, and $N \\mid V_{(N-\\varepsilon)/2}$ if $(Q/N) = -1$.","title":"Euler\u2013Lucas pseudoprime"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-strong-lucas-pseudoprime","kind":"node","lean":"MetaMathlibExt.IsStrongLucasPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-strong-lucas-pseudoprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $P, Q$ be integers with $Q \\neq 0$, let $U_n$ and $V_n$ be the Lucas sequences of $(P, Q)$ and let $D = P^2 - 4Q$. A natural number $N$ is a strong Lucas pseudoprime with parameters $P, Q$ if $N$ is odd and composite, $\\gcd(N, QD) = 1$, and, writing $N - (D/N) = 2^s d$ with $d$ odd, where $(D/N)$ is the Jacobi symbol, either $N \\mid U_d$ or $N \\mid V_{2^r d}$ for some $r$ with $0 \\le r < s$.","title":"Strong Lucas pseudoprimes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-super-pseudoprime","kind":"node","lean":"MetaMathlibExt.IsSuperPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-super-pseudoprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"Let $P, Q$ be integers with $Q \\neq 0$, let $D = P^2 - 4Q$, and let $S$ be a property of natural numbers. A natural number $N$ is a super pseudoprime of type $S$ with parameters $(P, Q)$ when $N$ is odd, $N > 1$, $N$ is not prime, $\\gcd(N, QD) = 1$, and every divisor $d > 1$ of $N$ is prime or has property $S$.","title":"Super pseudoprime"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-pseudoprime-one-five-nine","kind":"node","lean":"MetaMathlibExt.isLucasPseudoprime_one_five_nine","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(9\\)"],["text"," is a Lucas pseudoprime for "],["math","\\((P, Q) = (1, 5)\\)"]],"source":"$9$ is a Lucas pseudoprime for $(P, Q) = (1, 5)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-pseudoprime-one-five-nine","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"The number $9$ is a Lucas pseudoprime with parameters $P = 1$ and $Q = 5$.","title":"$9$ is a Lucas pseudoprime for $(P, Q) = (1, 5)$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-strong-lucas-pseudoprime-one-five-nin","kind":"node","lean":"MetaMathlibExt.isStrongLucasPseudoprime_one_five_nine","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(9\\)"],["text"," is a strong Lucas pseudoprime for "],["math","\\((P, Q) = (1, 5)\\)"]],"source":"$9$ is a strong Lucas pseudoprime for $(P, Q) = (1, 5)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-strong-lucas-pseudoprime-one-five-nin","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"The number $9$ is a strong Lucas pseudoprime with parameters $P = 1$ and $Q = 5$.","title":"$9$ is a strong Lucas pseudoprime for $(P, Q) = (1, 5)$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-euler-lucas-pseudoprime-one-five-nine","kind":"node","lean":"MetaMathlibExt.isEulerLucasPseudoprime_one_five_nine","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-euler-lucas-pseudoprime-one-five-nine","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"For the Lucas sequences $U$ and $V$ with parameters $P = 1$ and $Q = 5$, the number $9$ is an Euler\u2013Lucas pseudoprime.","title":"9 is an Euler\u2013Lucas pseudoprime for P = 1, Q = 5"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-frobenius-pseudoprime-four-neg","kind":"node","lean":"MetaMathlibExt.isLucasFrobeniusPseudoprime_four_neg_one_nine","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-frobenius-pseudoprime-four-neg","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"For the Lucas sequences with parameters $P = 4$ and $Q = -1$, so that $D = P^2 - 4Q = 20$, the number $9$ is a Frobenius pseudoprime.","title":"9 is a Frobenius pseudoprime for P = 4, Q = -1"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-euler-pseudoprime-two-561","kind":"node","lean":"MetaMathlibExt.isEulerPseudoprime_two_561","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-euler-pseudoprime-two-561","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"The composite number $561 = 3 \\cdot 11 \\cdot 17$ is an Euler pseudoprime to the base $2$.","title":"561 is an Euler pseudoprime to the base 2"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-super-pseudoprime-one-five-nine","kind":"node","lean":"MetaMathlibExt.isSuperPseudoprime_one_five_nine","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(9\\)"],["text"," is a super Lucas pseudoprime for "],["math","\\((P, Q) = (1, 5)\\)"]],"source":"$9$ is a super Lucas pseudoprime for $(P, Q) = (1, 5)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-super-pseudoprime-one-five-nine","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"fully_proved","summary":"The number $9$ is a super Lucas pseudoprime with parameters $P = 1$, $Q = 5$: its divisors greater than $1$ are $3$, a prime, and $9$, a Lucas pseudoprime with these parameters.","title":"$9$ is a super Lucas pseudoprime for $(P, Q) = (1, 5)$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-rank-of-appearance-properties","kind":"node","lean":"MetaMathlibExt.lucasRankOfAppearance_properties","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-rank-of-appearance-properties","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P$ and $Q \\ne 0$, let $D = P^2 - 4Q$, let $\\rho(m)$ be the rank of appearance of $m$ in $U$, and write $(a/n)$ for the Jacobi symbol. Let $r$ and $s$ range over positive integers and $p$ over primes. Then: (i) if $\\gcd(r, Q) = 1$, then $r \\mid U_s$ if and only if $\\rho(r) \\mid s$; (ii) if $p$ is odd and $p \\nmid Q$, then $p \\mid U_{p - (D/p)}$; (iii)\u2026","title":"Properties of the rank of appearance in a Lucas sequence"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-euler-lucas-pseudoprime-of-is-strong","kind":"node","lean":"MetaMathlibExt.isEulerLucasPseudoprime_of_isStrongLucasPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-euler-lucas-pseudoprime-of-is-strong","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers. Every strong Lucas pseudoprime with parameters $P$ and $Q$ is an Euler\u2013Lucas pseudoprime with parameters $P$ and $Q$.","title":"Strong Lucas pseudoprimes are Euler\u2013Lucas pseudoprimes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-strong-lucas-pseudoprime-iff-padic-va","kind":"node","lean":"MetaMathlibExt.isStrongLucasPseudoprime_iff_padicValNat_two_eq","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Strong Lucas pseudoprimes via the "],["math","\\(2\\)"],["text","-adic valuation of ranks"]],"source":"Strong Lucas pseudoprimes via the $2$-adic valuation of ranks"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-strong-lucas-pseudoprime-iff-padic-va","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers such that the Lucas pair $(P, Q)$ is nondegenerate, let $D = P^2 - 4Q$, and let $\\rho(m)$ be the rank of appearance of $m$ in the Lucas sequence $U(P, Q)$. Let $N$ be a Lucas pseudoprime with parameters $P, Q$ such that $\\gcd(N, PQD) = 1$. Then $\\rho(p^k) = \\rho(p)$ for every prime $p$ dividing $N$, where $p^k$ is the exact power of $p$ dividing $N$; and $N$ is a strong Lucas pseudoprime with\u2026","title":"Strong Lucas pseudoprimes via the $2$-adic valuation of ranks"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-strong-lucas-pseudoprime-of-odd-lucas","kind":"node","lean":"MetaMathlibExt.isStrongLucasPseudoprime_of_odd_lucasRankOfAppearance","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-strong-lucas-pseudoprime-of-odd-lucas","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers such that the Lucas pair $(P, Q)$ is nondegenerate, and let $\\rho(N)$ be the rank of appearance of $N$ in the Lucas sequence $U(P, Q)$. If $N$ is a Lucas pseudoprime with parameters $P, Q$ and $\\rho(N)$ is odd, then $N$ is a strong Lucas pseudoprime with parameters $P, Q$.","title":"Lucas pseudoprimes of odd rank are strong"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-rank-of-appearance-eq-lcm-and-is-s","kind":"node","lean":"MetaMathlibExt.lucasRankOfAppearance_eq_lcm_and_isSuperPseudoprime_iff","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-rank-of-appearance-eq-lcm-and-is-s","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers with $Q \\neq 0$ such that the pair $(P, Q)$ is nondegenerate, let $D = P^2 - 4Q$, and write $\\rho(m)$ for the rank of appearance of $m$ in the Lucas sequence $U(P, Q)$. Let $N > 1$ be an odd number that is not prime, with $\\gcd(N, QD) = 1$, and for each prime $p \\mid N$ let $k_p$ be the exponent of $p$ in $N$ and let $m_p \\ge k_p$ satisfy $\\rho(p^{m_p}) = \\rho(p)$ and\u2026","title":"Rank of appearance and super Lucas pseudoprimes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-super-pseudoprime-of-lucas-rank-of-ap","kind":"node","lean":"MetaMathlibExt.isSuperPseudoprime_of_lucasRankOfAppearance_eq","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-super-pseudoprime-of-lucas-rank-of-ap","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers with $Q \\neq 0$ such that the pair $(P, Q)$ is nondegenerate, let $D = P^2 - 4Q$, and write $\\rho(m)$ for the rank of appearance of $m$ in the Lucas sequence $U(P, Q)$. Let $N > 1$ be an odd number that is not prime, with $\\gcd(N, PQD) = 1$, such that for every prime $p \\mid N$, with $p^{k}$ the exact power of $p$ dividing $N$, $\\rho(p^{k}) = \\rho(p)$, and $\\rho(p) = \\rho(q)$ for all primes\u2026","title":"Super strong Lucas pseudoprimes from equal ranks of appearance"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-frobenius-pseudoprime-of-is-eul","kind":"node","lean":"MetaMathlibExt.isLucasFrobeniusPseudoprime_of_isEulerLucasPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-frobenius-pseudoprime-of-is-eul","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q \\neq 0$ be integers forming a nondegenerate Lucas pair, and let $D = P^2 - 4Q$. If $N$ is an Euler\u2013Lucas pseudoprime with parameters $P, Q$ and an Euler pseudoprime to the base $Q$, and $\\gcd(N, PQD) = 1$, then $N$ is a Frobenius pseudoprime with parameters $P, Q$.","title":"From Euler\u2013Lucas to Frobenius pseudoprimes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-frobenius-pseudoprime-of-is-str","kind":"node","lean":"MetaMathlibExt.isLucasFrobeniusPseudoprime_of_isStrongLucasPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-frobenius-pseudoprime-of-is-str","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers forming a nondegenerate Lucas pair with $Q = 1$ or $Q = -1$, and let $D = P^2 - 4Q$. If $N$ is a strong Lucas pseudoprime with parameters $P, Q$ and $\\gcd(N, PD) = 1$, then $N$ is a Frobenius pseudoprime with parameters $P, Q$.","title":"From strong Lucas to Frobenius pseudoprimes when Q = \u00b11"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/exists-is-primitive-prime-divisor-lucas","kind":"node","lean":"MetaMathlibExt.exists_isPrimitivePrimeDivisor_lucasU_of_discr_pos","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#exists-is-primitive-prime-divisor-lucas","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be coprime integers forming a nondegenerate Lucas pair with discriminant $D = P^2 - 4Q > 0$, and let $(U_n)$ be the Lucas sequence $U(P, Q)$. Then for every $n \\ge 1$ with $n \\notin \\{1, 2, 6, 12\\}$, the term $U_n$ has a primitive prime divisor: a prime $p$ that divides $U_n$ but divides no $U_s$ with $1 \\le s < n$.","title":"Carmichael's primitive divisor theorem"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/exists-is-primitive-prime-divisor-lucas-2","kind":"node","lean":"MetaMathlibExt.exists_isPrimitivePrimeDivisor_lucasU","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#exists-is-primitive-prime-divisor-lucas-2","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be coprime integers forming a nondegenerate Lucas pair, and let $(U_n)$ be the Lucas sequence $U(P, Q)$. Then for every $n \\ge 1$ with $n \\notin \\{1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 18, 30\\}$, the term $U_n$ has a primitive prime divisor: a prime $p$ that divides $U_n$ but divides no $U_s$ with $1 \\le s < n$.","title":"Bilu\u2013Hanrot\u2013Voutier theorem for Lucas sequences"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/infinite-strong-frobenius-pseudoprimes-t","kind":"node","lean":"MetaMathlibExt.infinite_strong_frobenius_pseudoprimes_two_primes","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#infinite-strong-frobenius-pseudoprimes-t","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers forming a nondegenerate Lucas pair with $Q = \\pm 1$. Then infinitely many products $N = p_1 p_2$ of two distinct odd primes $p_1$ and $p_2$ are both strong Lucas pseudoprimes and Frobenius pseudoprimes with parameters $P$ and $Q$.","title":"Rotkiewicz: strong Lucas and Frobenius pseudoprimes with two prime factors"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/infinite-frobenius-pseudoprimes-mod-eq","kind":"node","lean":"MetaMathlibExt.infinite_frobenius_pseudoprimes_modEq","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#infinite-frobenius-pseudoprimes-mod-eq","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers forming a nondegenerate Lucas pair with $Q = \\pm 1$, and let $a$ and $b$ be coprime positive integers. Then infinitely many natural numbers $N \\equiv b \\pmod a$ are Frobenius pseudoprimes with parameters $P$ and $Q$.","title":"Rotkiewicz: Frobenius pseudoprimes in arithmetic progressions"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/pseudoprime-iff-neg-p","kind":"node","lean":"MetaMathlibExt.pseudoprime_iff_neg_p","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas-type pseudoprime properties are unchanged by "],["math","\\(P \\mapsto -P\\)"]],"source":"Lucas-type pseudoprime properties are unchanged by $P \\mapsto -P$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#pseudoprime-iff-neg-p","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers such that the Lucas pair $(P, Q)$ is nondegenerate, let $D = P^2 - 4Q$, and let $N > 1$ be an odd natural number with $\\gcd(N, QD) = 1$. Then $N$ is a Lucas pseudoprime for $(P, Q)$ if and only if it is one for $(-P, Q)$, and the same holds for Lucas pseudoprimes of the second kind, Dickson pseudoprimes, Dickson pseudoprimes of the second kind, and Frobenius pseudoprimes.","title":"Lucas-type pseudoprime properties are unchanged by $P \\mapsto -P$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-super-pseudoprime-nat-abs-lucas-u-pri","kind":"node","lean":"MetaMathlibExt.isSuperPseudoprime_natAbs_lucasU_prime","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Composite terms "],["math","\\(|U_p|\\)"],["text"," are super strong Lucas pseudoprimes"]],"source":"Composite terms $|U_p|$ are super strong Lucas pseudoprimes"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-super-pseudoprime-nat-abs-lucas-u-pri","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers with $\\gcd(P, Q) = 1$ such that the Lucas pair $(P, Q)$ is nondegenerate, let $U_n$ be the Lucas sequence of $(P, Q)$ and let $D = P^2 - 4Q$. Let $p \\ge 5$ be a prime with $\\gcd(p, PQD) = 1$ such that $|U_p|$ is composite. Then $|U_p|$ is a super strong Lucas pseudoprime with parameters $P, Q$, and if $Q = \\pm 1$ it is also a super Frobenius pseudoprime with parameters $P, Q$.","title":"Composite terms $|U_p|$ are super strong Lucas pseudoprimes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/nat-abs-lucas-u-prime-composite-criteria","kind":"node","lean":"MetaMathlibExt.natAbs_lucasU_prime_composite_criteria","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositeness criteria for "],["math","\\(|U_p|\\)"]],"source":"Compositeness criteria for $|U_p|$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#nat-abs-lucas-u-prime-composite-criteria","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be coprime integers such that the Lucas pair $(P, Q)$ is nondegenerate, let $U_n$ be the Lucas sequence $U(P, Q)$, let $D = P^2 - 4Q$, and write $\\left(\\frac{a}{b}\\right)$ for the Jacobi symbol. Call a prime $p$ admissible if $p \\ge 5$ and $\\gcd(p, PQD) = 1$. Then the following hold.","title":"Compositeness criteria for $|U_p|$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-super-pseudoprime-nat-abs-lucas-u-pri-2","kind":"node","lean":"MetaMathlibExt.isSuperPseudoprime_natAbs_lucasU_prime_of_lucasV","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-super-pseudoprime-nat-abs-lucas-u-pri-2","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P_1, Q_1$ be integers with $\\gcd(P_1, Q_1) = 1$ such that the Lucas pair $(P_1, Q_1)$ is nondegenerate, with Lucas sequences $U_n(P_1, Q_1)$, $V_n(P_1, Q_1)$ and discriminant $D_1 = P_1^2 - 4Q_1$. Let $k \\ge 2$ with $\\gcd(D_1, k) = 1$, let $\\varepsilon = \\pm 1$, and let $P = \\varepsilon V_k(P_1, Q_1)$, $Q = Q_1^k$ and $D = P^2 - 4Q$. Let $p$ be a prime with $p \\nmid k D_1$ and $\\gcd(p, PQD) = 1$, such that\u2026","title":"Super strong Lucas pseudoprimes from composed Lucas pairs"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-super-pseudoprime-nat-abs-lucas-u-two","kind":"node","lean":"MetaMathlibExt.isSuperPseudoprime_natAbs_lucasU_two_mul_prime_div","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Super Lucas pseudoprimes of the form "],["math","\\(|U_{2p}/P|\\)"]],"source":"Super Lucas pseudoprimes of the form $|U_{2p}/P|$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-super-pseudoprime-nat-abs-lucas-u-two","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers with $Q \\neq 0$ such that the pair $(P, Q)$ is nondegenerate and $\\gcd(P, Q) = 1$, let $D = P^2 - 4Q$, and let $U = U(P, Q)$ be the Lucas sequence. Let $p \\ge 5$ be a prime with $\\gcd(p, PQD) = 1$ such that $(p, |P|, Q)$ is none of $(5, 1, 2)$, $(5, 1, 3)$, $(5, 2, 3)$, $(5, 12, 55)$ and $(13, 1, 2)$. Then $|U_{2p}/P|$, with $U_{2p}/P$ the integer quotient, is a super Lucas pseudoprime with\u2026","title":"Super Lucas pseudoprimes of the form $|U_{2p}/P|$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-strong-lucas-pseudoprime-nat-abs-luca","kind":"node","lean":"MetaMathlibExt.isStrongLucasPseudoprime_natAbs_lucasU","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Strong Lucas pseudoprimes among the terms "],["math","\\(|U_m|\\)"]],"source":"Strong Lucas pseudoprimes among the terms $|U_m|$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-strong-lucas-pseudoprime-nat-abs-luca","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P, Q$ be integers with $\\gcd(P, Q) = 1$ such that the Lucas pair $(P, Q)$ is nondegenerate and $D = P^2 - 4Q > 0$, and let $U_n$ be the Lucas sequence of $(P, Q)$. Suppose that $P$ is odd, or $P \\equiv 0 \\pmod 4$ and $Q \\equiv -1 \\pmod 4$, or $P \\equiv 2 \\pmod 4$ and $Q \\equiv -1 \\pmod 8$. Let $m \\ge 5$ be an odd prime or a Lucas pseudoprime of the second kind with parameters $P, Q$, such that\u2026","title":"Strong Lucas pseudoprimes among the terms $|U_m|$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-strong-frobenius-of-q-eq-one-or","kind":"node","lean":"MetaMathlibExt.lucasU_strong_frobenius_of_q_eq_one_or_neg_one","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(|U_m|\\)"],["text"," is a strong Lucas and Frobenius pseudoprime when "],["math","\\(Q = \\pm 1\\)"]],"source":"$|U_m|$ is a strong Lucas and Frobenius pseudoprime when $Q = \\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-strong-frobenius-of-q-eq-one-or","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $Q = 1$ or $Q = -1$ such that the Lucas pair $(P, Q)$ is nondegenerate, let $U_n$ be the Lucas sequence $U(P, Q)$, and let $D = P^2 - 4Q$. Let $m \\ge 5$ be either an odd prime or a Lucas pseudoprime of the second kind for $(P, Q)$, with $\\gcd(m, PD) = 1$ and with $3 \\nmid m$ if $P$ is odd. Then $D > 0$, $\\gcd(U_m, PD) = 1$, and $3 \\nmid U_m$ if $P$ is odd. Suppose moreover that\u2026","title":"$|U_m|$ is a strong Lucas and Frobenius pseudoprime when $Q = \\pm 1$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-pseudoprime-nat-abs-lucas-u-two","kind":"node","lean":"MetaMathlibExt.isLucasPseudoprime_natAbs_lucasU_two_mul_div","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-pseudoprime-nat-abs-lucas-u-two","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U$ be the Lucas sequence with parameters $P$ and $Q$ forming a nondegenerate Lucas pair, with $\\gcd(P, Q) = 1$ and $D = P^2 - 4Q > 0$. Let $m \\ge 5$ be either an odd prime or a Frobenius pseudoprime with parameters $P, Q$, such that $\\gcd(m, PQD) = 1$, and $3 \\nmid m$ if $P$ and $Q$ are both odd. Suppose that $P$ is odd, or $P \\equiv 0 \\pmod 4$ and $Q \\equiv -1 \\pmod 4$, or $P \\equiv 2 \\pmod 4$ and\u2026","title":"Lucas pseudoprimes from even-index Lucas terms divided by P"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-two-mul-div-is-lucas-pseudoprime","kind":"node","lean":"MetaMathlibExt.lucasU_two_mul_div_isLucasPseudoprime_of_q_eq_one_or_neg_one","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(|U_{2m}/P|\\)"],["text"," is a Lucas pseudoprime when "],["math","\\(Q = \\pm 1\\)"]],"source":"$|U_{2m}/P|$ is a Lucas pseudoprime when $Q = \\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-two-mul-div-is-lucas-pseudoprime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $Q = 1$ or $Q = -1$ such that the Lucas pair $(P, Q)$ is nondegenerate, let $U_n$ be the Lucas sequence $U(P, Q)$, and let $D = P^2 - 4Q$. Let $m \\ge 5$ be either an odd prime or a Lucas pseudoprime for $(P, Q)$, with $\\gcd(m, PD) = 1$ and with $3 \\nmid m$ if $P$ is odd, and put $N_2 = U_{2m}/P$. Then $D > 0$, $\\gcd(N_2, PD) = 1$, and $3 \\nmid N_2$ if $P$ is odd; and if $P$ is odd,\u2026","title":"$|U_{2m}/P|$ is a Lucas pseudoprime when $Q = \\pm 1$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/is-lucas-frobenius-pseudoprime-iff-of-no","kind":"node","lean":"MetaMathlibExt.isLucasFrobeniusPseudoprime_iff_of_not_isStrongLucasPseudoprime","parent":6133,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#is-lucas-frobenius-pseudoprime-iff-of-no","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $U$ be the Lucas sequence with parameters $P$ and $Q$, where $Q = 1$ or $Q = -1$, and let $D = P^2 - 4Q$. Let $N$ be a Lucas pseudoprime with parameters $P, Q$ such that $\\gcd(N, D) = 1$ and $N$ is not a strong Lucas pseudoprime, let $\\rho(N)$ be the rank of appearance of $N$ in $U$, and let $k$ be a natural number with $2^k \\mid \\rho(N)$ and $2^{k+1} \\nmid \\rho(N)$. Write $(D/N)$ for the Jacobi symbol. If\u2026","title":"Frobenius criterion for Lucas pseudoprimes that are not strong"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/lucas-u-two-mul-div-is-lucas-frobenius-p","kind":"node","lean":"MetaMathlibExt.lucasU_two_mul_div_isLucasFrobeniusPseudoprime_iff","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","When "],["math","\\(|U_{2m}/P|\\)"],["text"," is a Frobenius pseudoprime, for "],["math","\\(Q = \\pm 1\\)"]],"source":"When $|U_{2m}/P|$ is a Frobenius pseudoprime, for $Q = \\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#lucas-u-two-mul-div-is-lucas-frobenius-p","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers with $Q = 1$ or $Q = -1$ such that the Lucas pair $(P, Q)$ is nondegenerate, let $U_n$ be the Lucas sequence $U(P, Q)$, and let $D = P^2 - 4Q$. Let $m \\ge 5$ be either an odd prime or a Lucas pseudoprime for $(P, Q)$, with $\\gcd(m, PD) = 1$ and with $3 \\nmid m$ if $P$ is odd, and write $\\left(\\frac{D}{m}\\right)$ for the Jacobi symbol. Suppose that $P$ is odd, or $P$ is even and $Q = -1$,\u2026","title":"When $|U_{2m}/P|$ is a Frobenius pseudoprime, for $Q = \\pm 1$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/infinite-lucas-u-two-mul-div-frobenius-a","kind":"node","lean":"MetaMathlibExt.infinite_lucasU_two_mul_div_frobenius_and_not","parent":6133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Frobenius and non-Frobenius pseudoprimes of the form "],["math","\\(U_{2m}/P\\)"]],"source":"Frobenius and non-Frobenius pseudoprimes of the form $U_{2m}/P$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences/index.html#infinite-lucas-u-two-mul-div-frobenius-a","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/divisibility-properties-of-lucas-sequences","status":"can_prove","summary":"Let $P$ and $Q$ be integers forming a nondegenerate Lucas pair with $Q = \\pm 1$, with $P$ odd if $Q = 1$, and let $(U_n)$ be the Lucas sequence $U(P, Q)$. Call $m$ admissible if it is an odd prime or a Lucas pseudoprime with parameters $P$ and $Q$. Then infinitely many natural numbers $N = |U_{2m}/P|$ with $m$ admissible are both Lucas pseudoprimes and Frobenius pseudoprimes with parameters $P$ and $Q$, and\u2026","title":"Frobenius and non-Frobenius pseudoprimes of the form $U_{2m}/P$"},{"data_url":"scopes/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes.json","id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","kind":"scope","navigation":"module","parent":6093,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","title":"Large Zsigmondy primes"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/is-large-zsigmondy-prime","kind":"node","lean":"MetaMathlibExt.IsLargeZsigmondyPrime","parent":6194,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#is-large-zsigmondy-prime","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"fully_proved","summary":"Let $a$, $b$, $n$ and $p$ be natural numbers. Then $p$ is a large Zsigmondy prime for $(a, b, n)$ when $p$ is a primitive prime divisor of the term $a^n - b^n$ of the integer sequence $(a^m - b^m)_{m \\ge 0}$, that is, $n \\ge 1$, $p$ is prime, $p$ divides $a^n - b^n$ and $p$ divides no $a^m - b^m$ with $1 \\le m < n$, and moreover $p^2$ divides $a^n - b^n$ or $p > n + 1$.","title":"Large Zsigmondy prime"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/is-large-zsigmondy-prime-two-one-three-s","kind":"node","lean":"MetaMathlibExt.isLargeZsigmondyPrime_two_one_three_seven","parent":6194,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(7\\)"],["text"," is a large Zsigmondy prime for "],["math","\\((2, 1, 3)\\)"]],"source":"$7$ is a large Zsigmondy prime for $(2, 1, 3)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#is-large-zsigmondy-prime-two-one-three-s","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"fully_proved","summary":"The prime $7$ is a large Zsigmondy prime for $(a, b, n) = (2, 1, 3)$: it divides $2^3 - 1 = 7$, divides neither $2^1 - 1$ nor $2^2 - 1$, and $7 > 3 + 1$.","title":"$7$ is a large Zsigmondy prime for $(2, 1, 3)$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/sqrt-div-two-le-totient","kind":"node","lean":"MetaMathlibExt.sqrt_div_two_le_totient","parent":6194,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#sqrt-div-two-le-totient","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"For every positive integer $n$, $\\varphi(n) \\ge \\sqrt{n}/2$, where $\\varphi$ is Euler's totient function.","title":"Square-root lower bound for Euler's totient"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/homogenized-cyclotomic-prime-pow-mul","kind":"node","lean":"MetaMathlibExt.homogenizedCyclotomic_prime_pow_mul","parent":6194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Homogenized cyclotomic values at "],["math","\\(p^\\beta k\\)"]],"source":"Homogenized cyclotomic values at $p^\\beta k$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#homogenized-cyclotomic-prime-pow-mul","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $p$ be a prime, let $\\beta \\ge 1$ and $k \\ge 1$ be integers with $p \\nmid k$, and let $a, b$ be integers; write $\\Phi_m(a, b)$ for the homogenized cyclotomic value. Then $\\Phi_{p^\\beta k}(a, b) = \\Phi_{pk}(a^{p^{\\beta - 1}}, b^{p^{\\beta - 1}})$ and $\\Phi_{p^\\beta k}(a, b) \\cdot \\Phi_k(a^{p^{\\beta - 1}}, b^{p^{\\beta - 1}}) = \\Phi_k(a^{p^\\beta}, b^{p^\\beta})$.","title":"Homogenized cyclotomic values at $p^\\beta k$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/prod-divisors-homogenized-cyclotomic","kind":"node","lean":"MetaMathlibExt.prod_divisors_homogenizedCyclotomic","parent":6194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Factorization of "],["math","\\(a^n - b^n\\)"],["text"," into homogenized cyclotomic values"]],"source":"Factorization of $a^n - b^n$ into homogenized cyclotomic values"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#prod-divisors-homogenized-cyclotomic","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"For every positive integer $n$ and all integers $a$ and $b$,","title":"Factorization of $a^n - b^n$ into homogenized cyclotomic values"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/homogenized-cyclotomic-mem-icc","kind":"node","lean":"MetaMathlibExt.homogenizedCyclotomic_mem_Icc","parent":6194,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#homogenized-cyclotomic-mem-icc","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a > b$ be positive integers and let $n \\ge 1$. Then the homogenized cyclotomic value $\\Phi_n(a, b)$ satisfies $(a - b)^{\\varphi(n)} \\le \\Phi_n(a, b) \\le (a + b)^{\\varphi(n)}$, where $\\varphi$ is Euler's totient function.","title":"Bounds for homogenized cyclotomic values"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/homogenized-cyclotomic-prime-mul","kind":"node","lean":"MetaMathlibExt.homogenizedCyclotomic_prime_mul","parent":6194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Homogenized cyclotomic values at "],["math","\\(pn\\)"]],"source":"Homogenized cyclotomic values at $pn$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#homogenized-cyclotomic-prime-mul","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $p$ be a prime, $n$ a positive integer and $a, b$ integers, and write $\\Phi_m(a, b)$ for the homogenized cyclotomic value. If $p \\mid n$, then $\\Phi_{pn}(a, b) = \\Phi_n(a^p, b^p)$. If $p \\nmid n$, then $\\Phi_{pn}(a, b) \\cdot \\Phi_n(a, b) = \\Phi_n(a^p, b^p)$.","title":"Homogenized cyclotomic values at $pn$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/padic-val-int-homogenized-cyclotomic-odd","kind":"node","lean":"MetaMathlibExt.padicValInt_homogenizedCyclotomic_odd","parent":6194,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-adic valuation of homogenized cyclotomic values at an odd prime"]],"source":"$p$-adic valuation of homogenized cyclotomic values at an odd prime"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#padic-val-int-homogenized-cyclotomic-odd","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a > b$ be positive integers, let $p$ be an odd prime dividing neither $a$ nor $b$, and let $k$ be the least positive integer such that $p$ divides $a^k - b^k$. Write $v_p$ for the $p$-adic valuation and $\\Phi_n(a, b)$ for the homogenized cyclotomic value. Then for every positive integer $n$: $v_p(\\Phi_n(a, b)) = v_p(a^k - b^k)$ if $n = k$; $v_p(\\Phi_n(a, b)) = 1$ if $n = p^\\beta k$ for some $\\beta \\ge 1$; and\u2026","title":"$p$-adic valuation of homogenized cyclotomic values at an odd prime"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/padic-val-int-two-homogenized-cyclotomic","kind":"node","lean":"MetaMathlibExt.padicValInt_two_homogenizedCyclotomic","parent":6194,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-adic valuation of homogenized cyclotomic values"]],"source":"$2$-adic valuation of homogenized cyclotomic values"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#padic-val-int-two-homogenized-cyclotomic","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a > b$ be odd positive integers. Write $v_2$ for the $2$-adic valuation and $\\Phi_n(a, b)$ for the homogenized cyclotomic value. Then for every positive integer $n$: $v_2(\\Phi_n(a, b)) = v_2(a - b)$ if $n = 1$; $v_2(\\Phi_n(a, b)) = v_2(a + b)$ if $n = 2$; $v_2(\\Phi_n(a, b)) = 1$ if $n = 2^\\beta$ for some $\\beta \\ge 2$; and $v_2(\\Phi_n(a, b)) = 0$ if $n$ is not a power of $2$.","title":"$2$-adic valuation of homogenized cyclotomic values"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/prime-dvd-homogenized-cyclotomic-cases","kind":"node","lean":"MetaMathlibExt.prime_dvd_homogenizedCyclotomic_cases","parent":6194,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#prime-dvd-homogenized-cyclotomic-cases","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a > b$ be coprime positive integers, let $n \\ge 2$, let $\\Phi_n(a, b)$ be the homogenized cyclotomic value, let $p$ be a prime dividing $\\Phi_n(a, b)$, and let $k$ be the least positive integer such that $p$ divides $a^k - b^k$. Then one of the following holds: (i) $p = 2$ and $n = 2^\\beta$ for some $\\beta \\ge 1$; (ii) $p \\ge 3$, $n = k$, and $p$ is a primitive prime divisor of the term $a^n - b^n$ of the\u2026","title":"Prime divisors of homogenized cyclotomic values"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/dvd-homogenized-cyclotomic-of-is-primiti","kind":"node","lean":"MetaMathlibExt.dvd_homogenizedCyclotomic_of_isPrimitivePrimeDivisor","parent":6194,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#dvd-homogenized-cyclotomic-of-is-primiti","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a$ and $b$ be distinct coprime positive integers and let $n \\ge 2$. If $p$ is a primitive prime divisor of the term $a^n - b^n$ of the integer sequence $(a^m - b^m)_{m \\ge 0}$ (a Zsigmondy prime for $(a, b, n)$), then $p$ divides the homogenized cyclotomic value $\\Phi_n(a, b)$.","title":"Zsigmondy primes divide the homogenized cyclotomic value"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/eq-sub-one-of-not-is-large-zsigmondy-pri","kind":"node","lean":"MetaMathlibExt.eq_sub_one_of_not_isLargeZsigmondyPrime","parent":6194,"reading_title":{"format":"inline-parts-v1","parts":[["text","A Zsigmondy prime "],["math","\\(q\\)"],["text"," that is not large satisfies "],["math","\\(n = q - 1\\)"]],"source":"A Zsigmondy prime $q$ that is not large satisfies $n = q - 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#eq-sub-one-of-not-is-large-zsigmondy-pri","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a > b$ be coprime positive integers and let $n \\ge 2$. If $q$ is a primitive prime divisor of the term $a^n - b^n$ of the integer sequence $(a^m - b^m)_{m \\ge 0}$ and $q$ is not a large Zsigmondy prime for $(a, b, n)$, then $n = q - 1$.","title":"A Zsigmondy prime $q$ that is not large satisfies $n = q - 1$"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/exists-is-large-zsigmondy-prime-of-lt","kind":"node","lean":"MetaMathlibExt.exists_isLargeZsigmondyPrime_of_lt","parent":6194,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#exists-is-large-zsigmondy-prime-of-lt","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a$ and $b$ be distinct coprime positive integers and let $n \\ge 3$. If $(n + 1) q < \\Phi_n(a, b)$ for every prime factor $q$ of $n$, where $\\Phi_n(a, b)$ is the homogenized cyclotomic value, then there is a large Zsigmondy prime for $(a, b, n)$.","title":"A size criterion for a large Zsigmondy prime"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/exists-is-large-zsigmondy-prime-of-not-m","kind":"node","lean":"MetaMathlibExt.exists_isLargeZsigmondyPrime_of_not_mem","parent":6194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Large Zsigmondy primes for exponents "],["math","\\(n > 1\\)"],["text"," outside six values"]],"source":"Large Zsigmondy primes for exponents $n > 1$ outside six values"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#exists-is-large-zsigmondy-prime-of-not-m","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a > b$ be coprime positive integers and let $n > 1$ with $n \\notin \\{2, 4, 6, 10, 12, 18\\}$. Then there is a large Zsigmondy prime for $(a, b, n)$.","title":"Large Zsigmondy primes for exponents $n > 1$ outside six values"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/exists-is-large-zsigmondy-prime-iff","kind":"node","lean":"MetaMathlibExt.exists_isLargeZsigmondyPrime_iff","parent":6194,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes/index.html#exists-is-large-zsigmondy-prime-iff","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/large-zsigmondy-primes","status":"can_prove","summary":"Let $a > b$ be coprime positive integers and let $n > 1$. There is a large Zsigmondy prime for $(a, b, n)$ if and only if none of the following holds: $n = 2$ and $a + b = 2^s$ or $a + b = 3 \\cdot 2^s$ for some natural number $s$; $n = 4$ and $(a, b) \\in \\{(2, 1), (3, 1)\\}$; $n = 6$ and $(a, b) \\in \\{(2, 1), (3, 1), (3, 2), (5, 1), (5, 4)\\}$; $n = 10$ and $(a, b) \\in \\{(2, 1), (3, 2)\\}$; $n = 12$ and\u2026","title":"Avc\u0131's classification of large Zsigmondy primes"},{"data_url":"scopes/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem.json","id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem","kind":"scope","navigation":"module","parent":6093,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem/index.html","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem","status":"fully_proved","title":"Zsigmondy's theorem"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem/is-bang-zsigmondy-exceptional","kind":"node","lean":"MetaMathlibExt.IsBangZsigmondyExceptional","parent":6210,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem/index.html#is-bang-zsigmondy-exceptional","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem","status":"fully_proved","summary":"A triple $(a, b, n)$ of natural numbers is a Bang\u2013Zsigmondy exceptional case when either $(a, b, n) = (2, 1, 6)$, or $n = 2$ and $a + b = 2^k$ for some natural number $k$.","title":"Exceptional cases of Zsigmondy's theorem"},{"id":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem/bang-zsigmondy-exists-primitive-prime-di","kind":"node","lean":"MetaMathlibExt.bangZsigmondy_exists_primitivePrimeDivisor","parent":6210,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem/index.html#bang-zsigmondy-exists-primitive-prime-di","scope":"NumberTheory/linear-recurrences/lucas-sequences-primitive-divisors/zsigmondy-s-theorem","status":"fully_proved","summary":"Let $a$, $b$ and $n$ be natural numbers with $0 < b < a$, $a$ and $b$ coprime, and $n \\ge 2$, and suppose that $(a, b, n)$ is not a Bang\u2013Zsigmondy exceptional case: neither $(a, b, n) = (2, 1, 6)$, nor $n = 2$ with $a + b$ a power of two. Then $a^n - b^n$ has a primitive prime divisor, that is, a prime $p$ that divides $a^n - b^n$ and divides no $a^m - b^m$ with $1 \\le m < n$.","title":"Zsigmondy's theorem"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences.json","id":"NumberTheory/linear-recurrences/second-order-sequences","kind":"scope","parent":5991,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences","status":"can_prove","summary":"Pell, Pell\u2013Lucas, Jacobsthal, gibonacci, Horadam, Leonardo, Oresme, bi-periodic and (a,b)-Fibonacci sequences.","title":"Pell, Jacobsthal and other second-order sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/associated-pell-cassini-identity","kind":"node","lean":"MetaMathlibExt.associated_pell_cassini_identity","parent":6213,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/index.html#associated-pell-cassini-identity","scope":"NumberTheory/linear-recurrences/second-order-sequences","status":"fully_proved","summary":"Let $(Q_n)_{n \\ge 0}$ be a sequence of integers with $Q_0 = Q_1 = 1$ and $Q_{n+2} = 2Q_{n+1} + Q_n$ for every $n \\ge 0$. Then for every natural number $k \\ge 1$,","title":"Cassini's identity for associated Pell numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","title":"(a, b)-Fibonacci and (a, b)-Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/ab-generalized-fibonacci-sequence","kind":"node","lean":"MetaMathlibExt.AbGeneralizedFibonacciSequence","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#ab-generalized-fibonacci-sequence","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"A Kalman\u2013Mena $(a,b)$-generalized Fibonacci sequence consists of real parameters $a$ and $b$ together with a real sequence $(F_n)_{n \\ge 0}$ such that $F_0 = 0$, $F_1 = 1$, $F_{n+2} = a F_{n+1} + b F_n$ for every $n \\ge 0$, and its ordinary generating series in the formal variable $x$ satisfies the identity of formal power series $(1 - a x - b x^2) \\sum_{n \\ge 0} F_n x^n = x$, that is,\u2026","title":"Kalman\u2013Mena generalized Fibonacci sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/ext","kind":"node","lean":"MetaMathlibExt.AbGeneralizedFibonacciSequence.ext","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#ext","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"Two Kalman\u2013Mena $(a,b)$-generalized Fibonacci sequences are equal when they have the same parameter $a$, the same parameter $b$ and the same sequence $(F_n)_{n \\ge 0}$.","title":"Extensionality for Kalman\u2013Mena generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/fib-ab","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.fibAB","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#fib-ab","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"For complex numbers $a$ and $b$, the $(a,b)$-Fibonacci numbers $F^{a,b}_n$, $n \\in \\mathbb{N}$, are the complex numbers given by $F^{a,b}_0 = 0$, $F^{a,b}_1 = 1$ and $F^{a,b}_{n+2} = a F^{a,b}_{n+1} + b F^{a,b}_n$ for every $n$.","title":"The (a, b)-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/fib-ab-zero","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.fibAB_zero","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#fib-ab-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"For all complex numbers $a$ and $b$, the $(a,b)$-Fibonacci number $F^{a,b}_0$ equals $0$.","title":"Zeroth (a, b)-Fibonacci number"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/fib-ab-one","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.fibAB_one","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#fib-ab-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"For all complex numbers $a$ and $b$, the $(a,b)$-Fibonacci number $F^{a,b}_1$ equals $1$.","title":"First (a, b)-Fibonacci number"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/fib-ab-add-two","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.fibAB_add_two","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#fib-ab-add-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"For all complex numbers $a$ and $b$ and every natural number $n$, the $(a,b)$-Fibonacci numbers satisfy $F^{a,b}_{n+2} = a F^{a,b}_{n+1} + b F^{a,b}_n$.","title":"Recurrence for the (a, b)-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/lucas-ab","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.lucasAB","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#lucas-ab","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"For complex numbers $a$ and $b$, the $(a, b)$-Lucas numbers are the sequence $(L_n)_{n \\ge 0}$ of complex numbers defined by $L_0 = 2$, $L_1 = a$ and $L_{n+2} = a L_{n+1} + b L_n$ for all $n \\ge 0$.","title":"(a, b)-Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/lucas-ab-zero","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.lucasAB_zero","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#lucas-ab-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"For all complex numbers $a$ and $b$, the $(a,b)$-Lucas number $L^{a,b}_0$ equals $2$.","title":"Zeroth (a, b)-Lucas number"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/lucas-ab-one","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.lucasAB_one","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#lucas-ab-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"For all complex numbers $a$ and $b$, the $(a,b)$-Lucas number $L^{a,b}_1$ equals $a$.","title":"First (a, b)-Lucas number"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/lucas-ab-add-two","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.lucasAB_add_two","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#lucas-ab-add-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"fully_proved","summary":"For all complex numbers $a$ and $b$ and every natural number $n$, the $(a,b)$-Lucas numbers satisfy $L^{a,b}_{n+2} = a L^{a,b}_{n+1} + b L^{a,b}_n$.","title":"Recurrence for the (a, b)-Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/disc-mul-fib-ab","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.disc_mul_fibAB","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#disc-mul-fib-ab","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $a$ and $b$ be complex numbers with $a^2 + 4b \\ne 0$, and write $F_n$ and $L_n$ for the $(a,b)$-Fibonacci and $(a,b)$-Lucas numbers. Then for every natural number $m$,","title":"Discriminant times an (a, b)-Fibonacci number"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-neg-b-pow-fib-ab-even-prod","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_neg_b_pow_fibAB_even_prod","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-neg-b-pow-fib-ab-even-prod","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $a, b$ be complex numbers and let $(F_n)$ be the $(a, b)$-Fibonacci numbers. Then for every integer $n \\ge 0$,","title":"Sum of even-indexed (a, b)-Fibonacci numbers weighted by powers of -b"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-neg-b-pow-lucas-ab","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_neg_b_pow_lucasAB","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-neg-b-pow-lucas-ab","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $a, b$ be complex numbers with $b \\ne 0$, and let $(F_n)$ and $(L_n)$ be the $(a, b)$-Fibonacci and $(a, b)$-Lucas numbers. Then for every integer $n \\ge 0$,","title":"Sums of (a, b)-Lucas numbers weighted by powers of -b"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-neg-b-pow-lucas-ab-shift","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_neg_b_pow_lucasAB_shift","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-neg-b-pow-lucas-ab-shift","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $a, b$ be complex numbers with $b \\ne 0$, and let $(F_n)$ and $(L_n)$ be the $(a, b)$-Fibonacci and $(a, b)$-Lucas numbers. Then for all integers $m, n \\ge 0$,","title":"Shifted sums of (a, b)-Lucas numbers weighted by powers of -b"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-neg-b-pow-fib-ab-disc","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_neg_b_pow_fibAB_disc","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-neg-b-pow-fib-ab-disc","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $a, b$ be complex numbers with $\\Delta = a^2 + 4b \\ne 0$, and let $(F_n)$ and $(L_n)$ be the $(a, b)$-Fibonacci and $(a, b)$-Lucas numbers. Then for every integer $n \\ge 0$,","title":"Sums of (a, b)-Fibonacci numbers weighted by powers of -b, via Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-neg-one-pow-fib","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_neg_one_pow_fib","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-neg-one-pow-fib","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $(F_n)$ and $(L_n)$ be the Fibonacci and Lucas numbers, that is, the $(1, 1)$-Fibonacci and $(1, 1)$-Lucas numbers, viewed as complex numbers. Then for every integer $n \\ge 0$,","title":"Alternating sums of Fibonacci and Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-neg-one-pow-pell","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_neg_one_pow_pell","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-neg-one-pow-pell","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $(P_n)$ and $(Q_n)$ be the Pell and Pell\u2013Lucas numbers, that is, the $(2, 1)$-Fibonacci and $(2, 1)$-Lucas numbers, viewed as complex numbers. Then for every integer $n \\ge 0$,","title":"Alternating sums of Pell and Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-neg-one-pow-jacobsthal","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_neg_one_pow_jacobsthal","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-neg-one-pow-jacobsthal","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $(J_n)$ and $(j_n)$ be the Jacobsthal and Jacobsthal\u2013Lucas numbers, that is, the $(1, 2)$-Fibonacci and $(1, 2)$-Lucas numbers, viewed as complex numbers. Then for every integer $n \\ge 0$,","title":"Weighted alternating sums of Jacobsthal and Jacobsthal\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-neg-one-pow-k-lucas-ab","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_neg_one_pow_k_lucasAB","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-neg-one-pow-k-lucas-ab","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $k$ be a complex number, and let $(F_{k,n})$ and $(L_{k,n})$ be the $k$-Fibonacci and $k$-Lucas numbers, that is, the $(k, 1)$-Fibonacci and $(k, 1)$-Lucas numbers. Then for all integers $m \\ge 1$ and $n \\ge 0$,","title":"Alternating sums of even-indexed k-Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/sum-c-pow-fib-ab","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.sum_c_pow_fibAB","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#sum-c-pow-fib-ab","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $a, b, c$ be complex numbers with $c \\ne 0$ and $c^2 - ac - b \\ne 0$, and let $(F_n)$ be the $(a, b)$-Fibonacci numbers. Then for every integer $n \\ge 1$,","title":"Sums of (a, b)-Fibonacci numbers weighted by powers of c"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/convolution-lucas-ab-fib-ab","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.convolution_lucasAB_fibAB","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#convolution-lucas-ab-fib-ab","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Let $a, b, d$ be complex numbers with $b \\ne d$, and write $L^{a,b}_n$ for the $(a,b)$-Lucas numbers and $F^{a,b}_n$ for the $(a,b)$-Fibonacci numbers. Then for every natural number $n$,","title":"A convolution of (a, b)-Lucas and (a, d)-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/convolution-specializations","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.ABFibonacciIdentitiesWanted.convolution_specializations","parent":6215,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers/index.html#convolution-specializations","scope":"NumberTheory/linear-recurrences/second-order-sequences/a-b-fibonacci-and-a-b-lucas-numbers","status":"can_prove","summary":"Write $F_n$ and $L_n$ for the Fibonacci and Lucas numbers (the $(1,1)$-Fibonacci and $(1,1)$-Lucas numbers), $J_n$ and $j_n$ for the Jacobsthal and Jacobsthal\u2013Lucas numbers (the $(1,2)$-Fibonacci and $(1,2)$-Lucas numbers), and $P_n$ and $Q_n$ for the Pell and Pell\u2013Lucas numbers (the $(2,1)$-Fibonacci and $(2,1)$-Lucas numbers), all as complex numbers. Then for every natural number $n$: (i)\u2026","title":"Fibonacci, Jacobsthal and Pell convolution identities"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers","status":"fully_proved","title":"Gibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/gibonacci-number","kind":"node","lean":"MetaMathlibExt.gibonacciNumber","parent":6238,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/index.html#gibonacci-number","scope":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers","status":"fully_proved","summary":"For integers $G_0$ and $G_1$, the Gibonacci sequence with initial values $G_0$ and $G_1$ is the integer sequence $(G_n)_{n \\ge 0}$ that starts with $G_0, G_1$ and satisfies $G_{n+2} = G_{n+1} + G_n$ for every $n \\ge 0$.","title":"Gibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/gibonacci-zero-one-eq-fib","kind":"node","lean":"MetaMathlibExt.gibonacci_zero_one_eq_fib","parent":6238,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/index.html#gibonacci-zero-one-eq-fib","scope":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers","status":"fully_proved","summary":"For every natural number $n$, the $n$-th Gibonacci number with initial values $G_0 = 0$ and $G_1 = 1$ equals the $n$-th Fibonacci number $F_n$ (where $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$).","title":"Gibonacci numbers starting from 0 and 1 are the Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/gibonacci-cassini-of-pos","kind":"node","lean":"MetaMathlibExt.gibonacci_cassini_of_pos","parent":6238,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/index.html#gibonacci-cassini-of-pos","scope":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers","status":"fully_proved","summary":"Let $G_0$ and $G_1$ be integers and let $(G_n)$ be the Gibonacci sequence with these initial values, so that $G_{n+2} = G_{n+1} + G_n$ for all $n \\ge 0$. Then for every natural number $n \\ge 1$, $G_{n+1} G_{n-1} - G_n^2 = (-1)^n (G_1^2 - G_0 G_1 - G_0^2)$.","title":"Cassini identity for Gibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/melham-fibonacci-gibonacci-sum-of-square","kind":"node","lean":"MetaMathlibExt.melham_fibonacci_gibonacci_sum_of_squares_general","parent":6238,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/index.html#melham-fibonacci-gibonacci-sum-of-square","scope":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers","status":"fully_proved","summary":"Let $G : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $G(j+2) = G(j+1) + G(j)$ for every integer $j$, and let $F_m$ denote the $m$-th Fibonacci number. Then for every integer $k$ and every natural number $n$,","title":"Melham's gibonacci sum-of-squares identity, general form"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/melham-fibonacci-gibonacci-sum-of-square-2","kind":"node","lean":"MetaMathlibExt.melham_fibonacci_gibonacci_sum_of_squares","parent":6238,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/index.html#melham-fibonacci-gibonacci-sum-of-square-2","scope":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers","status":"fully_proved","summary":"Let $G : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $G(j+2) = G(j+1) + G(j)$ for every integer $j$, with $G(0) \\ne 0$ or $G(1) \\ne 0$, and let $F_m$ denote the $m$-th Fibonacci number. Then for every integer $k$ and every natural number $n \\ge 1$,","title":"Melham's gibonacci sum-of-squares identity"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/gibonacci-sq-recurrence","kind":"node","lean":"MetaMathlibExt.gibonacci_sq_recurrence","parent":6238,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers/index.html#gibonacci-sq-recurrence","scope":"NumberTheory/linear-recurrences/second-order-sequences/gibonacci-numbers","status":"fully_proved","summary":"Let $g : \\mathbb{N} \\to \\mathbb{Z}$ satisfy $g(n+2) = g(n+1) + g(n)$ for every $n \\in \\mathbb{N}$, and let $F_3 = 2$ be the third Fibonacci number. Then for every $n \\in \\mathbb{N}$,","title":"Gibonacci squares recurrence"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences.json","id":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences","status":"fully_proved","title":"Horadam and generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/horadam-generalized-catalan-identity","kind":"node","lean":"MetaMathlibExt.horadam_generalized_catalan_identity","parent":6245,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/index.html#horadam-generalized-catalan-identity","scope":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $c_1, c_2 \\in R$, and let $W, Y, U : \\mathbb{N} \\to R$ be sequences that each satisfy $x_{n+2} = c_1 x_{n+1} + c_2 x_n$ for every $n \\in \\mathbb{N}$, with $U_0 = 0$ and $U_1 = 1$. Then for all $s, i, j \\in \\mathbb{N}$,","title":"Generalized Catalan identity for Horadam sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/generalized-fibonacci-binet","kind":"node","lean":"MetaMathlibExt.generalizedFibonacci_binet","parent":6245,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/index.html#generalized-fibonacci-binet","scope":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences","status":"fully_proved","summary":"Let $p$ be a real number and let $(W_k)_{k \\ge 0}$ be a real sequence with $W_{k+2} = p W_{k+1} + W_k$ for all $k \\ge 0$. Put $s = \\sqrt{p^2 + 4}$, and let $\\alpha = (p + s)/2$ and $\\beta = (p - s)/2$, the roots of $z^2 - pz - 1$. Then for every $n \\ge 0$, $W_n = \\frac{W_1 - W_0 \\beta}{s} \\alpha^n + \\frac{W_0 \\alpha - W_1}{s} \\beta^n$.","title":"Binet formula for generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/generalized-fibonacci-mixed-identity","kind":"node","lean":"MetaMathlibExt.generalizedFibonacci_mixed_identity","parent":6245,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/index.html#generalized-fibonacci-mixed-identity","scope":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences","status":"fully_proved","summary":"Let $p$ be a real number and let $W, F : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_{k+2} = p W_{k+1} + W_k$ and $F_{k+2} = p F_{k+1} + F_k$ for every integer $k$, with $F_0 = 0$ and $F_1 = 1$. Then for all integers $m$ and $n$, $F_{m-2} W_{n-2} - F_{m-3} W_{n-1} = (-1)^{m-1} W_{n-m+1}$.","title":"Mixed identity for generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/generalized-fibonacci-cassini","kind":"node","lean":"MetaMathlibExt.generalizedFibonacci_cassini","parent":6245,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/index.html#generalized-fibonacci-cassini","scope":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $p \\in R$, and let $(W_k)_{k \\ge 0}$ be a sequence in $R$ with $W_{k+2} = p W_{k+1} + W_k$ for all $k \\ge 0$. Then for every $n \\ge 0$, $W_n W_{n+2} - W_{n+1}^2 = (-1)^n (W_0^2 + p W_0 W_1 - W_1^2)$.","title":"Cassini identity for generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/generalized-fibonacci-tendsto-ratio","kind":"node","lean":"MetaMathlibExt.generalizedFibonacci_tendsto_ratio","parent":6245,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/index.html#generalized-fibonacci-tendsto-ratio","scope":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences","status":"fully_proved","summary":"Let $p > 0$ be a real number and let $(W_k)_{k \\ge 0}$ be a real sequence with $W_0 \\ge 0$, $W_1 > 0$ and $W_{k+2} = p W_{k+1} + W_k$ for all $k \\ge 0$. Then $W_{k+1}/W_k$ tends to $(p + \\sqrt{p^2 + 4})/2$ as $k \\to \\infty$.","title":"Ratio limit for generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/generalized-fibonacci-binet-cassini-and","kind":"node","lean":"MetaMathlibExt.generalizedFibonacci_binet_cassini_and_ratio_limit","parent":6245,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences/index.html#generalized-fibonacci-binet-cassini-and","scope":"NumberTheory/linear-recurrences/second-order-sequences/horadam-and-generalized-fibonacci-sequences","status":"fully_proved","summary":"Let $p \\ge 1$, $a \\ge 0$ and $b > 0$ be integers, and let $m \\ge 0$ and $n \\ge 0$ be integers. Let $W, F : \\mathbb{Z} \\to \\mathbb{R}$ satisfy $W_0 = a$, $W_1 = b$, $F_0 = 0$, $F_1 = 1$, $W_{k+2} = p W_{k+1} + W_k$ and $F_{k+2} = p F_{k+1} + F_k$ for every integer $k$. Put $s = \\sqrt{p^2 + 4}$, $\\alpha = (p + s)/2$ and $\\beta = (p - s)/2$. Then $W_n = \\frac{b - a \\beta}{s} \\alpha^n + \\frac{a \\alpha - b}{s} \\beta^n$,\u2026","title":"Binet, Cassini, mixed identity and ratio limit for generalized Fibonacci sequences"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function.json","id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","title":"Integer values of recurrence generating functions"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/gibonacci-generating-value","kind":"node","lean":"MetaMathlibExt.gibonacciGeneratingValue","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#gibonacci-generating-value","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For integers $a$ and $b$ and a rational number $x$, the value of the Gibonacci generating function at $x$ is the rational number","title":"Gibonacci generating function"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/gibonacci-generating-value-zero","kind":"node","lean":"MetaMathlibExt.gibonacciGeneratingValue_zero","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#gibonacci-generating-value-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $a$ and $b$, the Gibonacci generating function $G^{(a,b)}(x) = \\frac{(b - a)x + a}{1 - x - x^2}$ takes the value $G^{(a,b)}(0) = a$ at $x = 0$.","title":"Value of the Gibonacci generating function at zero"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/gibonacci-generating-value-one","kind":"node","lean":"MetaMathlibExt.gibonacciGeneratingValue_one","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#gibonacci-generating-value-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $a$ and $b$, the Gibonacci generating function $G^{(a,b)}(x) = \\frac{(b - a)x + a}{1 - x - x^2}$ takes the value $G^{(a,b)}(1) = -b$ at $x = 1$.","title":"Value of the Gibonacci generating function at one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good","kind":"node","lean":"MetaMathlibExt.IsGibonacciGood","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"Let $a$ and $b$ be integers and let $G^{(a,b)}(x) = \\frac{(b - a)x + a}{1 - x - x^2}$ be the generating function of the Gibonacci sequence with initial terms $a$ and $b$. A rational number $x$ is good for $(a, b)$ when $1 - x - x^2 \\ne 0$ and $G^{(a,b)}(x)$ is an integer.","title":"Good rationals for a Gibonacci generating function"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good-zero","kind":"node","lean":"MetaMathlibExt.isGibonacciGood_zero","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $a$ and $b$, the rational number $0$ is good for $(a, b)$.","title":"Zero is good for every pair"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good-one","kind":"node","lean":"MetaMathlibExt.isGibonacciGood_one","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $a$ and $b$, the rational number $1$ is good for $(a, b)$.","title":"One is good for every pair"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/gibonacci-power-series-mul","kind":"node","lean":"MetaMathlibExt.gibonacci_powerSeries_mul","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#gibonacci-power-series-mul","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a$ and $b$ be integers and let $(G_n)_{n \\ge 0}$ be the Gibonacci sequence with $G_0 = a$ and $G_1 = b$. In the ring $\\mathbb{Q}[[X]]$ of formal power series with rational coefficients,","title":"Generating series of a Gibonacci sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good-transform","kind":"node","lean":"MetaMathlibExt.isGibonacciGood_transform","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good-transform","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a$, $b$, $p$, $q$ be integers with $q \\ne 0$ and $\\gcd(p, q) = 1$, and suppose that $p/q$ is good for $(a, b)$. If $p + q \\ne 0$, then $q/(p + q)$ is good for $(a, b)$; and if $p \\ne 0$, then $(q - p)/p$ is good for $(a, b)$.","title":"Goodness passes to two transformed fractions"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good-iff","kind":"node","lean":"MetaMathlibExt.isGibonacciGood_iff","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good-iff","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a, b, p, q$ be integers with $(a, b) \\ne (0, 0)$, $q \\ne 0$ and $\\gcd(p, q) = 1$, and put $m = |q^2 - pq - p^2|$. Then $p / q$ is good for $(a, b)$ if and only if there are integers $u$ and $v$ with $(q - p) u \\equiv 1$, $p v \\equiv 1$, $a \\equiv -b p u$ and $a \\equiv -b q v \\pmod{m}$.","title":"Congruence criterion for good fractions"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good-fib-div","kind":"node","lean":"MetaMathlibExt.isGibonacciGood_fib_div","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good-fib-div","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"For all integers $a$ and $b$ and every natural number $n$, the ratio $F_n / F_{n+1}$ of consecutive Fibonacci numbers is good for $(a, b)$.","title":"Consecutive Fibonacci ratios are good"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good-standard-values","kind":"node","lean":"MetaMathlibExt.isGibonacciGood_standard_values","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good-standard-values","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a$ and $b$ be integers and let $(G_n)_{n \\ge 0}$ be the Gibonacci sequence with $G_0 = a$ and $G_1 = b$. For every natural number $n$, the ratio $F_n / F_{n+1}$ of consecutive Fibonacci numbers is good for $(a, b)$, and if $G_n \\ne 0$, then $-G_{n+1} / G_n$ is good for $(a, b)$.","title":"Two standard families of good values"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good-shifted-ratio","kind":"node","lean":"MetaMathlibExt.isGibonacciGood_shifted_ratio","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good-shifted-ratio","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a$ and $b$ be integers and let $(H_n)_{n \\ge 0}$ be the Gibonacci sequence with $H_0 = -b$ and $H_1 = a$. For every natural number $n$ with $H_{n+1} \\ne 0$, the ratio $H_n / H_{n+1}$ is good for $(a, b)$.","title":"Consecutive ratios of a shifted Gibonacci sequence are good"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-gibonacci-good-ratios-infinite-iff","kind":"node","lean":"MetaMathlibExt.isGibonacciGood_ratios_infinite_iff","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-gibonacci-good-ratios-infinite-iff","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $p$ and $q$ be coprime integers, let $(H_n)_{n \\ge 0}$ be the Gibonacci sequence with $H_0 = p$ and $H_1 = q$, and put $m = |q^2 - pq - p^2|$. Call a pair of integers $(a, b)$ admissible when $H_n / H_{n+1}$ is good for $(a, b)$ for every natural number $n$ with $H_{n+1} \\ne 0$. Then there are infinitely many admissible pairs, and a pair $(a, b)$ is admissible if and only if there are integers $u$ and $v$ with\u2026","title":"Good consecutive ratios of a Gibonacci sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the recurrence "],["math","\\(A_{n+1} = pA_n + A_{n-1}\\)"]],"source":"Generating function of the recurrence $A_{n+1} = pA_n + A_{n-1}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For integers $a$, $b$, $p$ and a rational number $x$, $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$, with the convention that a fraction with denominator $0$ is $0$. As a rational function, $A$ is the generating function $\\sum_{n \\ge 0} A_n x^n$ of the sequence with $A_0 = a$, $A_1 = b$ and $A_{n+1} = pA_n + A_{n-1}$; for $p \\ne 0$ the denominator $1 - px - x^2$ has no rational root.","title":"Generating function of the recurrence $A_{n+1} = pA_n + A_{n-1}$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-zero","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_zero","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $a, b, p$, the function $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$ on the rationals satisfies $A(0) = a$.","title":"Value of the generating function at zero"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-fwd","kind":"node","lean":"MetaMathlibExt.secondOrderFwd","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-fwd","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For integers $p, r, s$, this is the sequence $(A^{(r,s)}_n)_{n \\ge 0}$ of integers with $A^{(r,s)}_0 = r$, $A^{(r,s)}_1 = s$ and $A^{(r,s)}_{n+2} = pA^{(r,s)}_{n+1} + A^{(r,s)}_n$ for every natural number $n$.","title":"Forward part of a second-order sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-bwd","kind":"node","lean":"MetaMathlibExt.secondOrderBwd","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-bwd","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For integers $p$, $r$ and $s$, the backward part is the sequence $(B_k)_{k \\ge 0}$ with $B_0 = s$, $B_1 = r$ and $B_{k+2} = B_k - pB_{k+1}$. Thus $B_k$ is the term of index $1 - k$ of the two-sided sequence with $A_0 = r$, $A_1 = s$ and $A_{n+1} = pA_n + A_{n-1}$, computed backwards by $A_n = A_{n+2} - pA_{n+1}$.","title":"Backward part of the two-sided sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-bi-seq","kind":"node","lean":"MetaMathlibExt.secondOrderBiSeq","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-bi-seq","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For integers $p$, $r$ and $s$, the two-sided sequence $(A^{(r,s)}_n)_{n \\in \\mathbb{Z}}$ for $p$ is the integer sequence indexed by $\\mathbb{Z}$ with $A^{(r,s)}_0 = r$, $A^{(r,s)}_1 = s$ and $A^{(r,s)}_{n+1} = pA^{(r,s)}_n + A^{(r,s)}_{n-1}$ for every integer $n$.","title":"Two-sided second-order sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-bi-seq-small","kind":"node","lean":"MetaMathlibExt.secondOrderBiSeq_small","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-bi-seq-small","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $p$, $r$, $s$, the two-sided sequence $(A^{(r,s)}_n)_{n \\in \\mathbb{Z}}$ for $p$ with initial values $r$ and $s$ satisfies $A^{(r,s)}_0 = r$, $A^{(r,s)}_1 = s$, $A^{(r,s)}_2 = ps + r$ and $A^{(r,s)}_{-1} = s - pr$.","title":"First values of the two-sided sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-bi-seq-zero","kind":"node","lean":"MetaMathlibExt.secondOrderBiSeq_zero","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-bi-seq-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $p$, $r$, $s$, the two-sided sequence $(A^{(r,s)}_n)_{n \\in \\mathbb{Z}}$ for $p$ with initial values $r$ and $s$ satisfies $A^{(r,s)}_0 = r$.","title":"First initial value of the two-sided sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-bi-seq-one","kind":"node","lean":"MetaMathlibExt.secondOrderBiSeq_one","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-bi-seq-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $p$, $r$, $s$, the two-sided sequence $(A^{(r,s)}_n)_{n \\in \\mathbb{Z}}$ for $p$ with initial values $r$ and $s$ satisfies $A^{(r,s)}_1 = s$.","title":"Second initial value of the two-sided sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-bi-seq-add-two","kind":"node","lean":"MetaMathlibExt.secondOrderBiSeq_add_two","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-bi-seq-add-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"Let $p$, $r$, $s$ be integers and $(A^{(r,s)}_n)_{n \\in \\mathbb{Z}}$ the two-sided sequence for $p$ with initial values $A^{(r,s)}_0 = r$ and $A^{(r,s)}_1 = s$. Then $A^{(r,s)}_{n+2} = pA^{(r,s)}_{n+1} + A^{(r,s)}_n$ for every integer $n$.","title":"The two-sided sequence satisfies its recurrence at every index"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-mvalue","kind":"node","lean":"MetaMathlibExt.secondOrderMValue","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","The quantity "],["math","\\(M(r, s)\\)"],["text"," for the recurrence "],["math","\\(A_{n+1} = pA_n + A_{n-1}\\)"]],"source":"The quantity $M(r, s)$ for the recurrence $A_{n+1} = pA_n + A_{n-1}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-mvalue","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For integers $a$, $b$, $p$, $r$, $s$, $M(r, s) = \\frac{br^2 + bs^2 - apr^2 + 2ars}{s^2 - prs - r^2}$, a rational number, with the convention that a fraction with denominator $0$ is $0$.","title":"The quantity $M(r, s)$ for the recurrence $A_{n+1} = pA_n + A_{n-1}$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-mvalue-zero-one","kind":"node","lean":"MetaMathlibExt.secondOrderMValue_zero_one","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of "],["math","\\(M\\)"],["text"," at "],["math","\\((0, 1)\\)"]],"source":"Value of $M$ at $(0, 1)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-mvalue-zero-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For all integers $a$, $b$, $p$, the quantity $M(r, s) = \\frac{br^2 + bs^2 - apr^2 + 2ars}{s^2 - prs - r^2}$ satisfies $M(0, 1) = b$.","title":"Value of $M$ at $(0, 1)$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-in-second-order-family","kind":"node","lean":"MetaMathlibExt.IsInSecondOrderFamily","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-in-second-order-family","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"Let $p$ be an integer and $x_0$ a rational number, written $x_0 = r/s$ in lowest terms with $s > 0$, and let $(A^{(r,s)}_n)_{n \\in \\mathbb{Z}}$ be the two-sided sequence with $A^{(r,s)}_0 = r$, $A^{(r,s)}_1 = s$ and $A^{(r,s)}_{n+1} = pA^{(r,s)}_n + A^{(r,s)}_{n-1}$. A rational number $x$ belongs to the family generated by $x_0$ when there is an integer $n$ with $A^{(r,s)}_{n+1} \\ne 0$ and\u2026","title":"Family generated by a rational number"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/is-in-second-order-family-self","kind":"node","lean":"MetaMathlibExt.isInSecondOrderFamily_self","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#is-in-second-order-family-self","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"fully_proved","summary":"For every integer $p$ and every rational number $x_0$, $x_0$ belongs to the family generated by $x_0$ for $p$.","title":"Every rational number lies in its own family"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-family-integer","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_family_integer","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-family-integer","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a, b, p$ be integers with $p \\ne 0$, let $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$ for rational $x$, and for integers $r, s$ let $A^{(r,s)}$ be the two-sided integer sequence with $A^{(r,s)}_0 = r$, $A^{(r,s)}_1 = s$ and $A^{(r,s)}_{n+2} = pA^{(r,s)}_{n+1} + A^{(r,s)}_n$ for all integers $n$. If $r$ and $s$ are coprime integers with $s \\ne 0$ and $A(r/s)$ is an integer, then\u2026","title":"Integer values propagate along a family"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-trivial-families","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_trivial_families_integer","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-trivial-families","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a, b, p$ be integers with $p \\ne 0$, let $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$ for rational $x$, and for integers $r, s$ let $A^{(r,s)}$ be the two-sided integer sequence with $A^{(r,s)}_0 = r$, $A^{(r,s)}_1 = s$ and $A^{(r,s)}_{n+2} = pA^{(r,s)}_{n+1} + A^{(r,s)}_n$ for all integers $n$. Then $A(A^{(0,1)}_n / A^{(0,1)}_{n+1})$ is an integer for every integer $n \\ne -1$, and\u2026","title":"Two families of integer values"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-integer-iff","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_integer_iff","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-integer-iff","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a, b, p$ be integers with $p \\ne 0$ and let $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$ for rational $x$. Let $r, s$ be coprime integers with $s \\ne 0$ and put $D = s^2 - prs - r^2$. Then $A(r/s)$ is an integer if and only if $b \\equiv -art \\pmod{D}$ for every integer $t$ with $st \\equiv 1 \\pmod{D}$.","title":"Integrality criterion at a rational point"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-neighbors-integer","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_neighbors_integer","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-neighbors-integer","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a, b, p$ be integers with $p \\ne 0$ and let $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$ for rational $x$. If $r, s$ are coprime integers with $r \\ne 0$, $s \\ne 0$ and $r + ps \\ne 0$, and $A(r/s)$ is an integer, then $A(s/(r + ps))$ and $A((s - pr)/r)$ are integers.","title":"Neighbours of an integer value"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-mvalue-relations","kind":"node","lean":"MetaMathlibExt.secondOrderMValue_relations","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relations between values of "],["math","\\(M\\)"],["text"," at neighbouring pairs"]],"source":"Relations between values of $M$ at neighbouring pairs"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-mvalue-relations","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a$, $b$, $p$ be integers with $p \\ne 0$, and for integers $r$, $s$ let $M(r, s) = \\frac{br^2 + bs^2 - apr^2 + 2ars}{s^2 - prs - r^2}$. Then for all integers $r$, $s$ that are not both zero, $M(s - pr, r) + (p^2 + 2)M(r, s) + M(s, r + ps) = 0$ and $M(r, s)^2 + (p^2 + 2)M(r, s)M(s, r + ps) + M(s, r + ps)^2 = abp^3 + (a^2 - b^2)p^2$.","title":"Relations between values of $M$ at neighbouring pairs"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/exists-small-second-order-mvalue-of-neg","kind":"node","lean":"MetaMathlibExt.exists_small_secondOrderMValue_of_neg","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","A small value of "],["math","\\(M\\)"],["text"," in the family of "],["math","\\(r/s\\)"],["text"," when "],["math","\\(abp + a^2 - b^2 < 0\\)"]],"source":"A small value of $M$ in the family of $r/s$ when $abp + a^2 - b^2 < 0$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#exists-small-second-order-mvalue-of-neg","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a, b, p$ be integers with $p \\ne 0$ and $abp + a^2 - b^2 < 0$. Let $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$ for rational $x$, and for integers $u, v$ let $M(u, v) = \\frac{bu^2 + bv^2 - apu^2 + 2auv}{v^2 - puv - u^2}$, where in both a fraction with denominator $0$ is read as $0$. Let $r, s$ be coprime integers with $s \\ne 0$ such that $A(r/s)$ is an integer, and let $(A^{(r,s)}_n)_{n \\in \\mathbb{Z}}$ be the\u2026","title":"A small value of $M$ in the family of $r/s$ when $abp + a^2 - b^2 < 0$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/exists-small-second-order-mvalue-of-nonn","kind":"node","lean":"MetaMathlibExt.exists_small_secondOrderMValue_of_nonneg","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","A small value of "],["math","\\(M\\)"],["text"," in the family of "],["math","\\(r/s\\)"],["text"," when "],["math","\\(abp + a^2 - b^2 \\ge 0\\)"]],"source":"A small value of $M$ in the family of $r/s$ when $abp + a^2 - b^2 \\ge 0$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#exists-small-second-order-mvalue-of-nonn","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a, b, p$ be integers with $p \\ne 0$ and $abp + a^2 - b^2 \\ge 0$. Let $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$ for rational $x$, and for integers $u, v$ let $M(u, v) = \\frac{bu^2 + bv^2 - apu^2 + 2auv}{v^2 - puv - u^2}$, where in both a fraction with denominator $0$ is read as $0$. Let $r, s$ be coprime integers with $s \\ne 0$ such that $A(r/s)$ is an integer, and let $(A^{(r,s)}_n)_{n \\in \\mathbb{Z}}$ be\u2026","title":"A small value of $M$ in the family of $r/s$ when $abp + a^2 - b^2 \\ge 0$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-algorithm","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_algorithm","parent":6252,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-algorithm","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $a, b, p$ be integers with $p \\ne 0$ and $(a, b) \\ne (0, 0)$, let $A(x) = \\frac{a(1 - px) + bx}{1 - px - x^2}$ for rational $x$, and for integers $r, s$ let $A^{(r,s)}$ be the two-sided integer sequence with $A^{(r,s)}_0 = r$, $A^{(r,s)}_1 = s$ and $A^{(r,s)}_{n+2} = pA^{(r,s)}_{n+1} + A^{(r,s)}_n$ for all integers $n$. Put $E = abp^3 + (a^2 - b^2)p^2$, and let $B = E/(p^2 + 4)$ if $E \\ge 0$ and\u2026","title":"Algorithm for integer values of a second-order generating function"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-zero-one-integer","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_zero_one_integer_set","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rationals where "],["math","\\(x/(1 - px - x^2)\\)"],["text"," is an integer"]],"source":"Rationals where $x/(1 - px - x^2)$ is an integer"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-zero-one-integer","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $p \\ne 0$ be an integer, let $G(x) = \\frac{x}{1 - px - x^2}$ for rational $x$, and let $A^{(0,1)}$ be the two-sided integer sequence with $A^{(0,1)}_0 = 0$, $A^{(0,1)}_1 = 1$ and $A^{(0,1)}_{n+2} = pA^{(0,1)}_{n+1} + A^{(0,1)}_n$ for all integers $n$. Then the set of rational numbers $x$ for which $G(x)$ is an integer is $\\{A^{(0,1)}_n / A^{(0,1)}_{n+1} : n \\in \\mathbb{Z},\\ n \\ne -1\\}$.","title":"Rationals where $x/(1 - px - x^2)$ is an integer"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-lucas-u","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_lucasU","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer values of "],["math","\\(x/(1 - px - x^2)\\)"],["text"," come from Lucas ratios"]],"source":"Integer values of $x/(1 - px - x^2)$ come from Lucas ratios"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-lucas-u","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $p \\ne 0$ be an integer, let $(U_n)_{n \\ge 0}$ be the Lucas sequence $U(p, -1)$, so $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = pU_{n+1} + U_n$, and let $G(x) = \\frac{x}{1 - px - x^2}$ for rational $x$. Then $G(U_n/U_{n+1})$ is an integer for every natural number $n$; and if $G(x) = k$ for a rational number $x$ and an integer $k$, then $G(U_n/U_{n+1}) = k$ for some natural number $n$.","title":"Integer values of $x/(1 - px - x^2)$ come from Lucas ratios"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/second-order-gen-value-one-one-families","kind":"node","lean":"MetaMathlibExt.secondOrderGenValue_one_one_families","parent":6252,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer values of "],["math","\\((1 - px + x)/(1 - px - x^2)\\)"]],"source":"Integer values of $(1 - px + x)/(1 - px - x^2)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function/index.html#second-order-gen-value-one-one-families","scope":"NumberTheory/linear-recurrences/second-order-sequences/integer-values-of-recurrence-generating-function","status":"can_prove","summary":"Let $p \\ne 0$ be an integer, let $A(x) = \\frac{1 - px + x}{1 - px - x^2}$ for rational $x$, and for integers $r, s$ let $A^{(r,s)}$ be the two-sided integer sequence with $A^{(r,s)}_0 = r$, $A^{(r,s)}_1 = s$ and $A^{(r,s)}_{n+2} = pA^{(r,s)}_{n+1} + A^{(r,s)}_n$ for all integers $n$. If $x$ is a rational number for which $A(x)$ is an integer, then there is an integer $n$ with $A^{(0,1)}_{n+1} \\ne 0$ and\u2026","title":"Integer values of $(1 - px + x)/(1 - px - x^2)$"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences.json","id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","title":"Binomial sums with second-order recurrences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-sq-alternatin","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_sq_alternating","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-sq-alternatin","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$,","title":"Alternating squared-binomial sum with Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-cube-alternat","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_cube_alternating","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-cube-alternat","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$,","title":"Dixon-type alternating sum with Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-sq","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_sq","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-sq","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U$ and $V$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so that $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$, and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$","title":"Squared-binomial convolution of the Lucas sequences U and V"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-catalan-even","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_catalan_even","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-catalan-even","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer, let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$, and let $C_j$ be the $j$-th Catalan number. Then for every natural number $n$, in the rationals,","title":"Even-index Catalan convolution with Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-catalan-shifted","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_catalan_shifted","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-catalan-shifted","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer, let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$, and let $C_j$ be the $j$-th Catalan number. Then for all natural numbers $n$ and $m$ with $m \\ge 1$, in the rationals,","title":"Shifted Catalan convolution with Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-two-n","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_two_n","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-two-n","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U$ and $V$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so that $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$, and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$","title":"Convolution of U and V with shifted binomial weights"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-four-n-even-a","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_four_n_even_alternating","parent":6290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sum of "],["math","\\(\\binom{2n}{k} \\binom{4n}{2k}\\)"],["text"," with Lucas sequences"]],"source":"Alternating sum of $\\binom{2n}{k} \\binom{4n}{2k}$ with Lucas sequences"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-four-n-even-a","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$, in the rationals,","title":"Alternating sum of $\\binom{2n}{k} \\binom{4n}{2k}$ with Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-triple-altern","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_triple_alternating","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-triple-altern","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U$ and $V$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so that $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$, and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$","title":"Alternating triple-binomial convolution of U and V"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-inv-alternati","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_inv_alternating","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-inv-alternati","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$, in the rationals,","title":"Alternating sum with reciprocal binomial coefficients and Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-n-plus-k","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_n_plus_k","parent":6290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\binom{n}{k} \\binom{2n}{n+k} \\binom{n+k}{k}\\)"],["text"," with Lucas sequences"]],"source":"Sum of $\\binom{n}{k} \\binom{2n}{n+k} \\binom{n+k}{k}$ with Lucas sequences"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-n-plus-k","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$,","title":"Sum of $\\binom{n}{k} \\binom{2n}{n+k} \\binom{n+k}{k}$ with Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-catalan","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_catalan","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-catalan","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer, let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$, and let $C_j$ be the $j$-th Catalan number. Then for every natural number $n$,","title":"Binomial-Catalan sum with Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-catalan-alter","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_catalan_alternating","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-catalan-alter","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer, let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$, and let $C_j$ be the $j$-th Catalan number. Then for every natural number $n$,","title":"Alternating binomial-Catalan sum with Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-sq-catalan-al","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_sq_catalan_alternating","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-sq-catalan-al","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U$ and $V$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so that $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$, and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Writing $C_j$ for the $j$-th Catalan number, for every natural number $n$","title":"Alternating two-binomial Catalan convolution of U and V"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/lucas-u-lucas-v-sum-choose-sign-choose-t","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_sum_choose_sign_choose_two","parent":6290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial sums with signs "],["math","\\((-1)^{\\binom{k}{2}}\\)"],["text"," and Lucas sequences"]],"source":"Binomial sums with signs $(-1)^{\\binom{k}{2}}$ and Lucas sequences"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#lucas-u-lucas-v-sum-choose-sign-choose-t","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer and let $U_n$ and $V_n$ be the Lucas sequences $U(p, -1)$ and $V(p, -1)$, so $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = p$ and both satisfy $X_{j+2} = p X_{j+1} + X_j$. Then for every natural number $n$,","title":"Binomial sums with signs $(-1)^{\\binom{k}{2}}$ and Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/second-order-bi-seq-sum-central-choose-m","kind":"node","lean":"MetaMathlibExt.secondOrderBiSeq_sum_central_choose_mul","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#second-order-bi-seq-sum-central-choose-m","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer, let $U$ and $V$ be the two-sided second-order sequences for $p$ with $U_0 = 0$, $U_1 = 1$, $V_0 = 2$ and $V_1 = p$, both satisfying $X_{j+1} = p X_j + X_{j-1}$ for every integer $j$, and put $\\Delta = p^2 + 4$. Then for every natural number $n > 0$ and all integers $m$ and $r$,","title":"Two-sided binomial sums of products of Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/second-order-bi-seq-sum-central-choose-m-2","kind":"node","lean":"MetaMathlibExt.secondOrderBiSeq_sum_central_choose_mul_weighted","parent":6290,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences/index.html#second-order-bi-seq-sum-central-choose-m-2","scope":"NumberTheory/linear-recurrences/second-order-sequences/binomial-sums-with-second-order-recurrences","status":"can_prove","summary":"Let $p$ be an integer, let $U$ and $V$ be the two-sided second-order sequences for $p$ with $U_0 = 0$, $U_1 = 1$, $V_0 = 2$ and $V_1 = p$, both satisfying $X_{j+1} = p X_j + X_{j-1}$ for every integer $j$, and put $\\Delta = p^2 + 4$. Then for every natural number $n > 0$ and all integers $m$ and $r$,","title":"Weighted two-sided binomial sums of products of Lucas sequences"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","title":"Integral representations of k-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/k-jacobsthal-params","kind":"node","lean":"MetaMathlibExt.kJacobsthalParams","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#k-jacobsthal-params","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"fully_proved","summary":"For a natural number $k$, the Lucas-sequence parameters $(P, Q) = (k, -2)$. The Lucas sequences $J_{k,n} = U_n(k, -2)$ and $j_{k,n} = V_n(k, -2)$ with these parameters are the $k$-Jacobsthal numbers ($J_{k,0} = 0$, $J_{k,1} = 1$) and the $k$-Jacobsthal\u2013Lucas numbers ($j_{k,0} = 2$, $j_{k,1} = k$); both satisfy $x_{n+2} = k x_{n+1} + 2 x_n$.","title":"Lucas parameters of the k-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/lucas-u-k-jacobsthal-params-two","kind":"node","lean":"MetaMathlibExt.lucasU_kJacobsthalParams_two","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#lucas-u-k-jacobsthal-params-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"fully_proved","summary":"For every natural number $k$, the $k$-Jacobsthal number $J_{k,2} = U_2(k, -2)$ equals $k$.","title":"The second k-Jacobsthal number"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/lucas-v-k-jacobsthal-params-two","kind":"node","lean":"MetaMathlibExt.lucasV_kJacobsthalParams_two","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-Jacobsthal-Lucas number "],["math","\\(j_{k,2}\\)"]],"source":"The $k$-Jacobsthal-Lucas number $j_{k,2}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#lucas-v-k-jacobsthal-params-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"fully_proved","summary":"For every natural number $k$, the $k$-Jacobsthal-Lucas numbers $j_{k,n} = V_n(k, -2)$, the Lucas sequence $V$ with parameters $(P, Q) = (k, -2)$, satisfy $j_{k,2} = k^2 + 4$.","title":"The $k$-Jacobsthal-Lucas number $j_{k,2}$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/companion-kjacobsthal","kind":"node","lean":"MetaMathlibExt.companionKJacobsthal","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Companion "],["math","\\(k\\)"],["text","-Jacobsthal numbers"]],"source":"Companion $k$-Jacobsthal numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#companion-kjacobsthal","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"fully_proved","summary":"For a natural number $k$ and integers $a, b$, the companion $k$-Jacobsthal numbers $\\mathcal{J}_{k,n}(a, b)$, $n \\in \\mathbb{N}$, are the integers defined by $\\mathcal{J}_{k,0}(a, b) = a$, $\\mathcal{J}_{k,1}(a, b) = b$ and $\\mathcal{J}_{k,n+2}(a, b) = k\\,\\mathcal{J}_{k,n+1}(a, b) + 2\\,\\mathcal{J}_{k,n}(a, b)$.","title":"Companion $k$-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/companion-kjacobsthal-add-two","kind":"node","lean":"MetaMathlibExt.companionKJacobsthal_add_two","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence of the companion "],["math","\\(k\\)"],["text","-Jacobsthal numbers"]],"source":"Recurrence of the companion $k$-Jacobsthal numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#companion-kjacobsthal-add-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"fully_proved","summary":"For every natural number $k$, all integers $a, b$ and every natural number $n$, the companion $k$-Jacobsthal numbers satisfy $\\mathcal{J}_{k,n+2}(a, b) = k\\,\\mathcal{J}_{k,n+1}(a, b) + 2\\,\\mathcal{J}_{k,n}(a, b)$.","title":"Recurrence of the companion $k$-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/companion-kjacobsthal-two","kind":"node","lean":"MetaMathlibExt.companionKJacobsthal_two","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","The second companion "],["math","\\(k\\)"],["text","-Jacobsthal number"]],"source":"The second companion $k$-Jacobsthal number"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#companion-kjacobsthal-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"fully_proved","summary":"For every natural number $k$ and all integers $a, b$, the companion $k$-Jacobsthal numbers satisfy $\\mathcal{J}_{k,2}(a, b) = bk + 2a$.","title":"The second companion $k$-Jacobsthal number"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/companion-kjacobsthal-zero-one","kind":"node","lean":"MetaMathlibExt.companionKJacobsthal_zero_one","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-Jacobsthal numbers as companion numbers"]],"source":"The $k$-Jacobsthal numbers as companion numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#companion-kjacobsthal-zero-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"fully_proved","summary":"For a natural number $k$, let $\\mathcal{J}_{k,n}(a, b)$ denote the companion $k$-Jacobsthal numbers and $J_{k,n} = U_n(k, -2)$ the $k$-Jacobsthal numbers, the Lucas sequence with parameters $(k, -2)$. Then for every natural number $n$, $\\mathcal{J}_{k,n}(0, 1) = J_{k,n}$: the companion numbers with initial values $0$ and $1$ are the $k$-Jacobsthal numbers.","title":"The $k$-Jacobsthal numbers as companion numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/k-jacobsthal-binet-identities","kind":"node","lean":"MetaMathlibExt.kJacobsthal_binet_identities","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#k-jacobsthal-binet-identities","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $k$ and $n$ be natural numbers, let $J_{k,n} = U_n(k, -2)$ and $j_{k,n} = V_n(k, -2)$ be the $k$-Jacobsthal and $k$-Jacobsthal\u2013Lucas numbers, and let $\\Delta_k = \\sqrt{k^2 + 8}$ and $\\sigma_k = (k + \\Delta_k)/2$. Then, in $\\mathbb{R}$, $j_{k,n} + \\Delta_k J_{k,n} = 2\\sigma_k^n$ and $j_{k,n} - \\Delta_k J_{k,n} = 2(-2)^n / \\sigma_k^n$, and, in $\\mathbb{Z}$, $j_{k,n}^2 - (k^2 + 8) J_{k,n}^2 = (-2)^{n+2}$.","title":"Binet-type identities for k-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/k-jacobsthal-add-formulas","kind":"node","lean":"MetaMathlibExt.kJacobsthal_add_formulas","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#k-jacobsthal-add-formulas","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $k$, $m$ and $n$ be natural numbers and let $J_{k,n} = U_n(k, -2)$ and $j_{k,n} = V_n(k, -2)$ be the $k$-Jacobsthal and $k$-Jacobsthal\u2013Lucas numbers. Then $2J_{k,m+n} = J_{k,m} j_{k,n} + J_{k,n} j_{k,m}$ and $2j_{k,m+n} = j_{k,m} j_{k,n} + (k^2 + 8) J_{k,m} J_{k,n}$.","title":"Addition formulas for k-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/lucas-u-k-jacobsthal-mul-eq-integral","kind":"node","lean":"MetaMathlibExt.lucasU_kJacobsthal_mul_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#lucas-u-k-jacobsthal-mul-eq-integral","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $k$, $\\ell$ and $n$ be natural numbers, let $J_{k,n} = U_n(k, -2)$ and $j_{k,n} = V_n(k, -2)$ be the $k$-Jacobsthal and $k$-Jacobsthal\u2013Lucas numbers, and let $\\Delta_k = \\sqrt{k^2 + 8}$. Then","title":"Integral representation of k-Jacobsthal numbers at multiples"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/lucas-u-k-jacobsthal-even-odd-eq-integra","kind":"node","lean":"MetaMathlibExt.lucasU_kJacobsthal_even_odd_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#lucas-u-k-jacobsthal-even-odd-eq-integra","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $k$ and $n$ be natural numbers, let $J_{k,n} = U_n(k, -2)$ be the $k$-Jacobsthal numbers and let $\\Delta_k = \\sqrt{k^2 + 8}$. Then","title":"Integral representation of even- and odd-indexed k-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/lucas-v-k-jacobsthal-mul-eq-integral","kind":"node","lean":"MetaMathlibExt.lucasV_kJacobsthal_mul_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#lucas-v-k-jacobsthal-mul-eq-integral","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $k$, $\\ell$ and $n$ be natural numbers, let $J_{k,n} = U_n(k, -2)$ and $j_{k,n} = V_n(k, -2)$ be the $k$-Jacobsthal and $k$-Jacobsthal\u2013Lucas numbers, and let $\\Delta_k = \\sqrt{k^2 + 8}$. Then","title":"Integral representation of k-Jacobsthal\u2013Lucas numbers at multiples"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/lucas-v-k-jacobsthal-even-odd-eq-integra","kind":"node","lean":"MetaMathlibExt.lucasV_kJacobsthal_even_odd_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#lucas-v-k-jacobsthal-even-odd-eq-integra","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $k$ and $n$ be natural numbers, let $j_{k,n} = V_n(k, -2)$ be the $k$-Jacobsthal\u2013Lucas numbers and let $\\Delta_k = \\sqrt{k^2 + 8}$. Then","title":"Integral representation of even- and odd-indexed k-Jacobsthal\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/jacobsthal-mul-eq-integral","kind":"node","lean":"MetaMathlibExt.jacobsthal_mul_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#jacobsthal-mul-eq-integral","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $\\ell$ and $n$ be natural numbers and let $J_n = U_n(1, -2)$ and $j_n = V_n(1, -2)$ be the Jacobsthal and Jacobsthal\u2013Lucas numbers (the case $k = 1$). Then","title":"Integral representation of Jacobsthal numbers at multiples"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/jacobsthal-even-odd-eq-integral","kind":"node","lean":"MetaMathlibExt.jacobsthal_even_odd_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#jacobsthal-even-odd-eq-integral","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $J_n = U_n(1, -2)$ be the Jacobsthal numbers ($J_0 = 0$, $J_1 = 1$, $J_{n+2} = J_{n+1} + 2J_n$). For every natural number $n$, in $\\mathbb{R}$,","title":"Integral formulas for even- and odd-indexed Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/jacobsthal-lucas-mul-eq-integral","kind":"node","lean":"MetaMathlibExt.jacobsthalLucas_mul_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#jacobsthal-lucas-mul-eq-integral","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $J_m = U_m(1, -2)$ be the Jacobsthal numbers and $j_m = V_m(1, -2)$ the Jacobsthal-Lucas numbers ($j_0 = 2$, $j_1 = 1$, $j_{m+2} = j_{m+1} + 2j_m$). For all natural numbers $\\ell$ and $n$, in $\\mathbb{R}$,","title":"Integral representation of the Jacobsthal-Lucas numbers at multiples of an index"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/jacobsthal-mul-add-eq-integral","kind":"node","lean":"MetaMathlibExt.jacobsthal_mul_add_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#jacobsthal-mul-add-eq-integral","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $\\ell$, $n$ and $r$ be natural numbers and let $J_n = U_n(1, -2)$ and $j_n = V_n(1, -2)$ be the Jacobsthal and Jacobsthal\u2013Lucas numbers (the case $k = 1$). Then","title":"Integral representation of Jacobsthal numbers with an offset"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/jacobsthal-lucas-mul-add-eq-integral","kind":"node","lean":"MetaMathlibExt.jacobsthalLucas_mul_add_eq_integral","parent":6307,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#jacobsthal-lucas-mul-add-eq-integral","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $J_m = U_m(1, -2)$ be the Jacobsthal numbers and $j_m = V_m(1, -2)$ the Jacobsthal-Lucas numbers ($j_0 = 2$, $j_1 = 1$, $j_{m+2} = j_{m+1} + 2j_m$). For all natural numbers $\\ell$, $n$ and $r$, in $\\mathbb{R}$,","title":"Integral representation of the Jacobsthal-Lucas numbers along arithmetic progressions"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/tendsto-lucas-u-k-jacobsthal-ratio","kind":"node","lean":"MetaMathlibExt.tendsto_lucasU_kJacobsthal_ratio","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ratio of consecutive "],["math","\\(k\\)"],["text","-Jacobsthal numbers"]],"source":"Ratio of consecutive $k$-Jacobsthal numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#tendsto-lucas-u-k-jacobsthal-ratio","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"For every positive integer $k$, the $k$-Jacobsthal numbers $J_{k,n} = U_n(k, -2)$, the Lucas sequence $U$ with parameters $(P, Q) = (k, -2)$, satisfy $J_{k,n+1}/J_{k,n} \\to (k + \\sqrt{k^2 + 8})/2$ as $n \\to \\infty$, the quotients taken in $\\mathbb{R}$.","title":"Ratio of consecutive $k$-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/two-mul-companion-kjacobsthal","kind":"node","lean":"MetaMathlibExt.two_mul_companionKJacobsthal","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Companion "],["math","\\(k\\)"],["text","-Jacobsthal numbers via Lucas sequences"]],"source":"Companion $k$-Jacobsthal numbers via Lucas sequences"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#two-mul-companion-kjacobsthal","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"For a natural number $k$ and integers $a, b$, let $\\mathcal{J}_{k,n}(a, b)$ be the companion $k$-Jacobsthal numbers, and let $J_{k,n} = U_n(k, -2)$ and $j_{k,n} = V_n(k, -2)$ be the $k$-Jacobsthal and $k$-Jacobsthal\u2013Lucas numbers, the Lucas sequences $U$ and $V$ with parameters $(P, Q) = (k, -2)$. Then for every natural number $n$, $2\\,\\mathcal{J}_{k,n}(a, b) = a\\,j_{k,n} + (2b - ak)\\,J_{k,n}$.","title":"Companion $k$-Jacobsthal numbers via Lucas sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/companion-kjacobsthal-generating-functio","kind":"node","lean":"MetaMathlibExt.companionKJacobsthal_generatingFunction","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the companion "],["math","\\(k\\)"],["text","-Jacobsthal numbers"]],"source":"Generating function of the companion $k$-Jacobsthal numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#companion-kjacobsthal-generating-functio","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"For every natural number $k$ and all integers $a, b$, the companion $k$-Jacobsthal numbers $\\mathcal{J}_{k,n} = \\mathcal{J}_{k,n}(a, b)$ satisfy, in the ring $\\mathbb{Z}[[x]]$ of formal power series,","title":"Generating function of the companion $k$-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/companion-kjacobsthal-catalan","kind":"node","lean":"MetaMathlibExt.companionKJacobsthal_catalan","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan's identity for the companion "],["math","\\(k\\)"],["text","-Jacobsthal numbers"]],"source":"Catalan's identity for the companion $k$-Jacobsthal numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#companion-kjacobsthal-catalan","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $k$ be a natural number, $a, b$ integers, $\\mathcal{J}_{k,m} = \\mathcal{J}_{k,m}(a, b)$ the companion $k$-Jacobsthal numbers and $J_{k,m} = U_m(k, -2)$ the $k$-Jacobsthal numbers. For all natural numbers $r \\le n$,","title":"Catalan's identity for the companion $k$-Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/companion-kjacobsthal-mul-eq-integral","kind":"node","lean":"MetaMathlibExt.companionKJacobsthal_mul_eq_integral","parent":6307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral representation of the companion "],["math","\\(k\\)"],["text","-Jacobsthal numbers"]],"source":"Integral representation of the companion $k$-Jacobsthal numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers/index.html#companion-kjacobsthal-mul-eq-integral","scope":"NumberTheory/linear-recurrences/second-order-sequences/integral-representations-of-k-jacobsthal-numbers","status":"can_prove","summary":"Let $k$, $\\ell$, $n$ be natural numbers and $a, b$ integers. Write $\\mathcal{J}_{k,m} = \\mathcal{J}_{k,m}(a, b)$ for the companion $k$-Jacobsthal numbers, $J_{k,m} = U_m(k, -2)$ and $j_{k,m} = V_m(k, -2)$ for the $k$-Jacobsthal and $k$-Jacobsthal-Lucas numbers, and $\\Delta_k = \\sqrt{k^2 + 8}$. Then, in $\\mathbb{R}$,","title":"Integral representation of the companion $k$-Jacobsthal numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers","status":"fully_proved","title":"Jacobsthal and Jacobsthal\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers/jacobsthal-lucas-number","kind":"node","lean":"MetaMathlibExt.jacobsthalLucasNumber","parent":6331,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers/index.html#jacobsthal-lucas-number","scope":"NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers","status":"fully_proved","summary":"The Jacobsthal\u2013Lucas numbers $j_0, j_1, j_2, \\dots$ are the natural numbers defined by $j_0 = 2$, $j_1 = 1$ and $j_{n+2} = j_{n+1} + 2 j_n$ for all $n \\ge 0$. The sequence begins $2, 1, 5, 7, 17, \\dots$ (OEIS A014551).","title":"Jacobsthal\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers/jacobsthal-closed-form","kind":"node","lean":"MetaMathlibExt.jacobsthal_closedForm","parent":6331,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers/index.html#jacobsthal-closed-form","scope":"NumberTheory/linear-recurrences/second-order-sequences/jacobsthal-and-jacobsthal-lucas-numbers","status":"fully_proved","summary":"Let $J : \\mathbb{N} \\to \\mathbb{N}$ satisfy $J(0) = J(1) = 1$ and $J(n+2) = J(n+1) + 2J(n)$ for every $n \\in \\mathbb{N}$. Then for every $m \\in \\mathbb{N}$,","title":"Jacobsthal closed form"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","title":"Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo","kind":"node","lean":"MetaMathlibExt.genLeonardo","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"fully_proved","summary":"For an integer $k$, the generalized Leonardo numbers $L_{k,0}, L_{k,1}, L_{k,2}, \\ldots$ are the integers defined by $L_{k,0} = L_{k,1} = 1$ and $L_{k,n+2} = L_{k,n+1} + L_{k,n} + k$ for all $n \\ge 0$.","title":"Generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-two","kind":"node","lean":"MetaMathlibExt.genLeonardo_two","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"fully_proved","summary":"For every integer $k$, the generalized Leonardo number $L_{k,2}$ equals $2 + k$.","title":"Generalized Leonardo number of index two"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-eq-leonardo-pk-one","kind":"node","lean":"MetaMathlibExt.genLeonardo_eq_leonardoPK_one","parent":6334,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Leonardo numbers as Leonardo "],["math","\\(p\\)"],["text","-numbers with "],["math","\\(p = 1\\)"]],"source":"Generalized Leonardo numbers as Leonardo $p$-numbers with $p = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-eq-leonardo-pk-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"For all natural numbers $k$ and $n$, the generalized Leonardo number $L_{k,n}$ equals the generalized Leonardo $p$-number $\\mathcal{L}_{1,k}(n)$; both sequences start with $1, 1$ and satisfy the recurrence $a_{n+2} = a_{n+1} + a_n + k$.","title":"Generalized Leonardo numbers as Leonardo $p$-numbers with $p = 1$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-companion","kind":"node","lean":"MetaMathlibExt.genLeonardoCompanion","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-companion","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"fully_proved","summary":"For an integer $k$, let $L_{k,n}$ denote the generalized Leonardo numbers. The dual generalized Leonardo numbers $M_{k,n}$, for natural numbers $n$, are defined by $M_{k,0} = 1 - k$ and $M_{k,n+1} = L_{k,n} + L_{k,n+2}$, so that $M_{k,n} = L_{k,n-1} + L_{k,n+1}$ for $n \\ge 1$.","title":"Dual generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-companion-one","kind":"node","lean":"MetaMathlibExt.genLeonardoCompanion_one","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-companion-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"fully_proved","summary":"For every integer $k$, the dual generalized Leonardo number $M_{k,1}$ equals $3 + k$.","title":"The dual generalized Leonardo number of index one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-mul-companion","kind":"node","lean":"MetaMathlibExt.genLeonardo_mul_companion","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-mul-companion","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k$ be an integer, and let $L_{k,n}$ and $M_{k,n}$ denote the generalized Leonardo numbers and the dual generalized Leonardo numbers. For every natural number $n$,","title":"Product of a generalized Leonardo number and its dual"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-cross","kind":"node","lean":"MetaMathlibExt.genLeonardo_cross","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-cross","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k$ be an integer, and let $L_{k,n}$ and $M_{k,n}$ denote the generalized Leonardo numbers and the dual generalized Leonardo numbers. For every natural number $n \\ge 1$,","title":"Cross-product identity for generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-sq-sub-sq","kind":"node","lean":"MetaMathlibExt.genLeonardo_sq_sub_sq","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-sq-sub-sq","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"For every integer $k$ and every natural number $n \\ge 1$, the generalized Leonardo numbers $L_{k,n}$ and the dual generalized Leonardo numbers $M_{k,n}$ satisfy","title":"Square-difference identity for generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-shift","kind":"node","lean":"MetaMathlibExt.genLeonardo_shift","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-shift","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k$ be an integer and let $L_{k,n}$ denote the generalized Leonardo numbers. For all natural numbers $m$, $n$, $t$ with $1 \\le t \\le n$ and $n + 1 \\le m + t$,","title":"Shift identity for generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-sq-sum-diff","kind":"node","lean":"MetaMathlibExt.genLeonardo_sq_sum_diff","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-sq-sum-diff","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"For every integer $k$ and all natural numbers $m$, $n$ with $1 \\le n < m$, the generalized Leonardo numbers $L_{k,n}$ satisfy","title":"Sum-and-difference square identities for generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-binomial-sum","kind":"node","lean":"MetaMathlibExt.genLeonardo_binomial_sum","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-binomial-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k$ be an integer, and let $L_{k,n}$ and $M_{k,n}$ denote the generalized Leonardo numbers and the dual generalized Leonardo numbers. For all natural numbers $n$ and $r$,","title":"Binomial convolution identities for generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-binomial-sum-two-mul","kind":"node","lean":"MetaMathlibExt.genLeonardo_binomial_sum_two_mul","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-binomial-sum-two-mul","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k$ be an integer, and let $L_{k,n}$ and $M_{k,n}$ denote the generalized Leonardo numbers and the dual generalized Leonardo numbers. For all natural numbers $m$ and $r$,","title":"Even-order binomial identities for generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-binomial-alternating","kind":"node","lean":"MetaMathlibExt.genLeonardo_binomial_alternating","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-binomial-alternating","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k$ be an integer, and let $L_{k,n}$ and $M_{k,n}$ denote the generalized Leonardo numbers and the dual generalized Leonardo numbers. For all natural numbers $n \\ge 1$ and $r$,","title":"Alternating binomial identities for generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-binomial-steps","kind":"node","lean":"MetaMathlibExt.genLeonardo_binomial_steps","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-binomial-steps","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k$ be an integer, and let $L_{k,n}$ and $M_{k,n}$ denote the generalized Leonardo numbers and the dual generalized Leonardo numbers. For all natural numbers $n$ and $r$,","title":"Binomial identities with steps three and four for generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/log-concave-of-linear-recurrence-of-init","kind":"node","lean":"MetaMathlibExt.logConcave_of_linearRecurrence_of_initial_logConcave","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#log-concave-of-linear-recurrence-of-init","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k \\ge 2$ be an integer, let $c_1, \\dots, c_k$ be real numbers with $c_j \\ge 0$ for $2 \\le j \\le k$, and let $(z_n)_{n \\ge 0}$ be a sequence of positive real numbers with","title":"Log-concavity propagates along a linear recurrence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/log-convex-of-linear-recurrence-of-initi","kind":"node","lean":"MetaMathlibExt.logConvex_of_linearRecurrence_of_initial_logConvex","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#log-convex-of-linear-recurrence-of-initi","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k \\ge 2$ be an integer, let $c_1, \\dots, c_k$ be real numbers with $c_j \\ge 0$ for $2 \\le j \\le k$, and let $(z_n)_{n \\ge 0}$ be a sequence of positive real numbers with","title":"Log-convexity propagates along a linear recurrence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/inv-log-balanced-of-recurrence-of-initia","kind":"node","lean":"MetaMathlibExt.inv_logBalanced_of_recurrence_of_initial","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#inv-log-balanced-of-recurrence-of-initia","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $c_1$ and $c_2$ be real numbers with $c_2 \\ge 0$, and let $(z_n)_{n \\ge 0}$ be a sequence of positive real numbers with $z_{n+1} = c_1 z_n - c_2 z_{n-2}$ for every $n \\ge 2$. Write $w_n = 1/z_n$ and $x_n = z_{n+1}/z_n$, and let $n_0 \\ge 0$ be an integer. Suppose that for $n = n_0 + 1$ and $n = n_0 + 2$","title":"Log-balancedness of reciprocals along a three-term recurrence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/inv-gen-leonardo-log-balanced","kind":"node","lean":"MetaMathlibExt.inv_genLeonardo_logBalanced","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#inv-gen-leonardo-log-balanced","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k \\ge 1$ be an integer and let $L_{k,n}$ be the generalized Leonardo numbers. Then the sequence $(1/L_{k,n})_{n \\ge 3}$ is log-balanced (Do\u0161li\u0107): for every integer $n \\ge 4$,","title":"Log-balancedness of the reciprocals of generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-rpow-inv-log-convex","kind":"node","lean":"MetaMathlibExt.genLeonardo_rpow_inv_logConvex","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-rpow-inv-log-convex","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k \\ge 1$ be an integer and let $L_{k,n}$ be the generalized Leonardo numbers. Then the sequence $(L_{k,n}^{1/n})_{n \\ge 9}$ is log-convex: $(L_{k,n}^{1/n})^2 \\le L_{k,n-1}^{1/(n-1)} L_{k,n+1}^{1/(n+1)}$ for every integer $n \\ge 10$.","title":"Log-convexity of the n-th roots of generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-partial-sum-log-concave","kind":"node","lean":"MetaMathlibExt.genLeonardo_partialSum_logConcave","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-partial-sum-log-concave","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k \\ge 1$ be an integer, let $L_{k,n}$ be the generalized Leonardo numbers, and let $R_n = \\sum_{i=0}^{n} L_{k,i}$. Then $R_{n-1} R_{n+1} \\le R_n^2$ for every integer $n \\ge 2$.","title":"Log-concavity of partial sums of generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-rpow-inv-log-balanced","kind":"node","lean":"MetaMathlibExt.genLeonardo_rpow_inv_logBalanced","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-rpow-inv-log-balanced","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k \\ge 1$ be an integer, let $L_{k,n}$ be the generalized Leonardo numbers, and let $n_k$ be an integer with $n_k \\ge 9$ and $n_k \\ge (1 + \\sqrt{64k - 7})/4$. Then for every integer $n \\ge n_k + 1$,","title":"Log-balancedness of the n-th roots of generalized Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/gen-leonardo-pow-self-log-convex","kind":"node","lean":"MetaMathlibExt.genLeonardo_pow_self_logConvex","parent":6334,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers/index.html#gen-leonardo-pow-self-log-convex","scope":"NumberTheory/linear-recurrences/second-order-sequences/leonardo-numbers","status":"can_prove","summary":"Let $k \\ge 1$ be an integer and let $L_{k,n}$ be the generalized Leonardo numbers. Then the sequence $(L_{k,n}^n)_{n \\ge 3}$ is log-convex: $L_{k,n}^{2n} \\le L_{k,n-1}^{n-1} L_{k,n+1}^{n+1}$ for every integer $n \\ge 4$.","title":"Log-convexity of the self-powers of generalized Leonardo numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","title":"Hyper-Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo","kind":"node","lean":"MetaMathlibExt.hyperLeonardo","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"Let $Le_n$ be the Leonardo numbers, defined by $Le_0 = Le_1 = 1$ and $Le_{n+2} = Le_{n+1} + Le_n + 1$. The hyper-Leonardo numbers are the integers $Le^{(r)}_n$, for $r, n \\ge 0$, defined by $Le^{(0)}_n = Le_n$ and $Le^{(r+1)}_n = \\sum_{k=0}^{n} Le^{(r)}_k$.","title":"Hyper-Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-zero","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_zero","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"For every natural number $n$, the hyper-Leonardo number $Le^{(0)}_n$ equals the Leonardo number $L_{1,n}$, the generalized Leonardo number with $k = 1$, given by $L_{1,0} = L_{1,1} = 1$ and $L_{1,n+2} = L_{1,n+1} + L_{1,n} + 1$.","title":"Hyper-Leonardo numbers of order zero"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-succ","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_succ","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-succ","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"For all natural numbers $r$ and $n$, the hyper-Leonardo numbers $Le^{(r)}_n$ satisfy $Le^{(r+1)}_n = \\sum_{k=0}^{n} Le^{(r)}_k$.","title":"Hyper-Leonardo numbers as partial sums"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-one-small","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_one_small","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-one-small","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"The hyper-Leonardo numbers $Le^{(1)}_n$ of order $1$ satisfy $Le^{(1)}_0 = 1$, $Le^{(1)}_1 = 2$, $Le^{(1)}_2 = 5$ and $Le^{(1)}_3 = 10$.","title":"First hyper-Leonardo numbers of order one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-fibonacci","kind":"node","lean":"MetaMathlibExt.hyperFibonacci","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-fibonacci","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"The hyper-Fibonacci numbers $F^{(r)}_n$, for natural numbers $r$ and $n$, are the natural numbers defined by $F^{(0)}_n = F_n$, the $n$-th Fibonacci number (with $F_0 = 0$ and $F_1 = 1$), and $F^{(r+1)}_n = \\sum_{k=0}^{n} F^{(r)}_k$.","title":"Hyper-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-fibonacci-zero","kind":"node","lean":"MetaMathlibExt.hyperFibonacci_zero","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-fibonacci-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"For every natural number $n$, the hyper-Fibonacci number $F^{(0)}_n$ equals the $n$-th Fibonacci number $F_n$ (with $F_0 = 0$ and $F_1 = 1$).","title":"Hyper-Fibonacci numbers of order zero"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-fibonacci-succ","kind":"node","lean":"MetaMathlibExt.hyperFibonacci_succ","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-fibonacci-succ","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"For all natural numbers $r$ and $n$, the hyper-Fibonacci numbers $F^{(r)}_n$ satisfy $F^{(r+1)}_n = \\sum_{k=0}^{n} F^{(r)}_k$.","title":"Hyper-Fibonacci numbers as partial sums"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-fibonacci-one-small","kind":"node","lean":"MetaMathlibExt.hyperFibonacci_one_small","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-fibonacci-one-small","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"The hyper-Fibonacci numbers $F^{(1)}_n$ of order $1$ satisfy $F^{(1)}_0 = 0$, $F^{(1)}_1 = 1$, $F^{(1)}_2 = 2$ and $F^{(1)}_3 = 4$.","title":"Small values of the hyper-Fibonacci numbers of order one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-recurrence-small-order","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_recurrence_small_order","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-recurrence-small-order","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"The hyper-Leonardo numbers $Le^{(r)}_n$ satisfy $Le^{(0)}_0 = Le^{(0)}_1 = 1$, $Le^{(1)}_0 = 1$, $Le^{(1)}_1 = 2$, $Le^{(2)}_0 = 1$ and $Le^{(2)}_1 = 3$, and for every natural number $n \\ge 2$,","title":"Recurrences for the hyper-Leonardo numbers of orders zero, one and two"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-recurrence","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_recurrence","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-recurrence","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For every natural number $r$, the hyper-Leonardo numbers $Le^{(r)}_n$ of order $r$ satisfy $Le^{(r)}_0 = 1$, $Le^{(r)}_1 = r + 1$ and, for every natural number $n \\ge 2$, $Le^{(r)}_n = r + Le^{(r)}_{n-1} + Le^{(r)}_{n-2} + \\binom{n+r-2}{r} + \\sum_{j=1}^{r-1} j \\binom{n+r-2-j}{r-j}$.","title":"Recurrence for the hyper-Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-succ-succ","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_succ_succ","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-succ-succ","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For all natural numbers $r \\ge 1$ and $n \\ge 1$, the hyper-Leonardo numbers $Le^{(r)}_n$ satisfy $Le^{(r)}_n = Le^{(r)}_{n-1} + Le^{(r-1)}_n$.","title":"Recurrence across consecutive orders of hyper-Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-one-succ","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_one_succ","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-one-succ","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For every natural number $n$, the hyper-Leonardo numbers $Le^{(1)}_n$ of order $1$ satisfy $Le^{(1)}_{n+1} = Le^{(1)}_n + L_{1,n+1}$, where $L_{1,n}$ is the generalized Leonardo number with $k = 1$, that is, the Leonardo number $Le_n$.","title":"Consecutive hyper-Leonardo numbers of order one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/choose-add-power-series-mul-one-sub-pow","kind":"node","lean":"MetaMathlibExt.choose_add_powerSeries_mul_one_sub_pow","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#choose-add-power-series-mul-one-sub-pow","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"fully_proved","summary":"For every natural number $r$, the formal power series $\\sum_{n \\ge 0} \\binom{n+r}{r} t^n$ with integer coefficients satisfies $\\left(\\sum_{n \\ge 0} \\binom{n+r}{r} t^n\\right)(1-t)^{r+1} = 1$ in $\\mathbb{Z}[[t]]$, that is, $\\sum_{n \\ge 0} \\binom{n+r}{r} t^n = 1/(1-t)^{r+1}$.","title":"Generating function of shifted binomial coefficients"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/log-concave-convolution","kind":"node","lean":"MetaMathlibExt.logConcave_convolution","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#log-concave-convolution","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"Let $(x_n)_{n \\ge 0}$ and $(y_n)_{n \\ge 0}$ be sequences of positive real numbers that are log-concave, that is, $x_{j-1} x_{j+1} \\le x_j^2$ and $y_{j-1} y_{j+1} \\le y_j^2$ for every $j \\ge 1$. Let $z_n = \\sum_{k=0}^{n} x_k y_{n-k}$ be their convolution. Then $z_{j-1} z_{j+1} \\le z_j^2$ for every $j \\ge 1$.","title":"The convolution of log-concave sequences is log-concave"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/leonardo-generating-function","kind":"node","lean":"MetaMathlibExt.leonardo_generatingFunction","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#leonardo-generating-function","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"Let $Le_n$ be the Leonardo numbers ($Le_0 = Le_1 = 1$ and $Le_{n+2} = Le_{n+1} + Le_n + 1$), and let $L(t) = \\sum_{n \\ge 0} Le_n t^n$ in the ring $\\mathbb{Z}[[t]]$ of formal power series. Then $L(t)(1 - 2t + t^3) = 1 - t + t^2$ and $L(t)(1 - t - t^2)(1 - t) = 2(1 - t) - (1 - t - t^2)$.","title":"Generating function of the Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-one-two-generating-functi","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_one_two_generatingFunction","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-one-two-generating-functi","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"In the ring $\\mathbb{Z}[[t]]$ of formal power series, the hyper-Leonardo numbers $Le^{(r)}_n$ satisfy $\\left(\\sum_{n \\ge 0} Le^{(1)}_n t^n\\right)(1 - 2t + t^3)(1 - t) = 1 - t + t^2$ and $\\left(\\sum_{n \\ge 0} Le^{(2)}_n t^n\\right)(1 - 2t + t^3)(1 - t)^2 = 1 - t + t^2$.","title":"Generating functions of the hyper-Leonardo numbers of orders one and two"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-generating-function","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_generatingFunction","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-generating-function","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For every natural number $r$, the formal power series $\\sum_{n \\ge 0} \\mathrm{Le}^{(r)}_n t^n$ of the hyper-Leonardo numbers of order $r$ satisfies $\\left(\\sum_{n \\ge 0} \\mathrm{Le}^{(r)}_n t^n\\right)(1-2t+t^3)(1-t)^r = 1-t+t^2$ in $\\mathbb{Z}[[t]]$, that is, $\\sum_{n \\ge 0} \\mathrm{Le}^{(r)}_n t^n = \\frac{1-t+t^2}{(1-2t+t^3)(1-t)^r}$.","title":"Generating function of the hyper-Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-eq-hyper-fibonacci","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_eq_hyperFibonacci","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-eq-hyper-fibonacci","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For all natural numbers $r$ and $n$, the hyper-Leonardo number $\\mathrm{Le}^{(r)}_n$ and the hyper-Fibonacci number $F^{(r)}_{n+1}$ satisfy $\\mathrm{Le}^{(r)}_n = 2F^{(r)}_{n+1} - \\binom{n+r}{r}$ as integers.","title":"Hyper-Leonardo numbers in terms of hyper-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/leonardo-log-convex-log-concave-pointwis","kind":"node","lean":"MetaMathlibExt.leonardo_logConvex_logConcave_pointwise","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#leonardo-log-convex-log-concave-pointwis","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"Let $Le_n$ be the Leonardo numbers ($Le_0 = Le_1 = 1$ and $Le_{n+2} = Le_{n+1} + Le_n + 1$). Then $Le_n^2 \\le Le_{n-1} Le_{n+1}$ for $n = 1$ and $n = 3$, $Le_5^2 = Le_4 Le_6$, and $Le_{n-1} Le_{n+1} \\le Le_n^2$ for $n = 2$, $n = 4$ and every $n \\ge 6$.","title":"Log-convexity and log-concavity of the Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/leonardo-log-balanced-three","kind":"node","lean":"MetaMathlibExt.leonardo_logBalanced_three","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#leonardo-log-balanced-three","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"Let $Le_n$ be the Leonardo numbers ($Le_0 = Le_1 = 1$ and $Le_{n+2} = Le_{n+1} + Le_n + 1$). Then $Le_3^2 \\le Le_2 Le_4$ and $\\frac{Le_2}{2!} \\cdot \\frac{Le_4}{4!} \\le \\left(\\frac{Le_3}{3!}\\right)^2$, the second inequality in the real numbers. That is, $(Le_n)$ satisfies the log-convexity inequality at $n = 3$ and $(Le_n / n!)$ satisfies the log-concavity inequality at $n = 3$.","title":"Leonardo numbers are log-balanced at index three"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-log-convex-log-concave-on","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_logConvex_logConcave_one","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-log-convex-log-concave-on","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"The hyper-Leonardo numbers $\\mathrm{Le}^{(r)}_n$ satisfy $\\left(\\mathrm{Le}^{(1)}_1\\right)^2 \\le \\mathrm{Le}^{(1)}_0 \\mathrm{Le}^{(1)}_2$, and for every natural number $r \\ge 2$ they satisfy $\\mathrm{Le}^{(r)}_0 \\mathrm{Le}^{(r)}_2 \\le \\left(\\mathrm{Le}^{(r)}_1\\right)^2$.","title":"Log-convexity and log-concavity of the hyper-Leonardo numbers at index one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-one-log-concave","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_one_logConcave","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-one-log-concave","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For every natural number $n \\ge 2$, the hyper-Leonardo numbers $Le^{(1)}_n$ of order $1$ satisfy $Le^{(1)}_{n-1} Le^{(1)}_{n+1} \\le (Le^{(1)}_n)^2$.","title":"Eventual log-concavity of the hyper-Leonardo numbers of order one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-mul-sub-sq","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_mul_sub_sq","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-mul-sub-sq","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For all natural numbers $r \\ge 1$ and $n \\ge 1$, the hyper-Leonardo numbers $Le^{(r)}_n$ satisfy $Le^{(r)}_{n-1} Le^{(r)}_{n+1} - (Le^{(r)}_n)^2 = Le^{(r)}_n Le^{(r-1)}_{n+1} - Le^{(r-1)}_n Le^{(r)}_{n+1}$.","title":"Cassini-type identity for hyper-Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-log-concave","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_logConcave","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-log-concave","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For all natural numbers $r \\ge 1$ and $n \\ge 2$, the hyper-Leonardo numbers $Le^{(r)}_n$ of order $r$ satisfy $Le^{(r)}_{n-1} Le^{(r)}_{n+1} \\le (Le^{(r)}_n)^2$.","title":"Eventual log-concavity of the hyper-Leonardo numbers of positive order"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/hyper-leonardo-convex","kind":"node","lean":"MetaMathlibExt.hyperLeonardo_convex","parent":6357,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers/index.html#hyper-leonardo-convex","scope":"NumberTheory/linear-recurrences/second-order-sequences/hyper-leonardo-numbers","status":"can_prove","summary":"For all natural numbers $r$ and $n \\ge 1$, the hyper-Leonardo numbers $\\mathrm{Le}^{(r)}_n$ satisfy $2 \\mathrm{Le}^{(r)}_n \\le \\mathrm{Le}^{(r)}_{n-1} + \\mathrm{Le}^{(r)}_{n+1}$.","title":"Convexity of the hyper-Leonardo numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences.json","id":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","status":"fully_proved","title":"Normal second-order sequences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/normal-seq","kind":"node","lean":"MetaMathlibExt.normalSeq","parent":6383,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/index.html#normal-seq","scope":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","status":"fully_proved","summary":"For integers $a_1$ and $a_2$, the normal second-order number sequence $(u_n)_{n \\ge 0}$ with coefficients $a_1, a_2$ is the integer sequence defined by $u_0 = 0$, $u_1 = 1$ and $u_{n+2} = a_1 u_{n+1} + a_2 u_n$ for every $n \\ge 0$.","title":"Normal second-order number sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/normal-seq-one","kind":"node","lean":"MetaMathlibExt.normalSeq_one","parent":6383,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/index.html#normal-seq-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","status":"fully_proved","summary":"For all integers $a_1$ and $a_2$, the normal second-order number sequence $(u_n)_{n \\ge 0}$ with coefficients $a_1, a_2$ has $u_1 = 1$.","title":"First term of a normal number sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/normal-seq-two","kind":"node","lean":"MetaMathlibExt.normalSeq_two","parent":6383,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/index.html#normal-seq-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","status":"fully_proved","summary":"For all integers $a_1$ and $a_2$, the normal second-order number sequence $(u_n)_{n \\ge 0}$ with coefficients $a_1, a_2$ has $u_2 = a_1$.","title":"Second term of a normal number sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/normal-seq-recurrence","kind":"node","lean":"MetaMathlibExt.normalSeq_recurrence","parent":6383,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/index.html#normal-seq-recurrence","scope":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","status":"fully_proved","summary":"For all integers $a_1$ and $a_2$ and every natural number $n$, the normal second-order number sequence $(u_n)_{n \\ge 0}$ with coefficients $a_1, a_2$ satisfies $u_{n+2} = a_1 u_{n+1} + a_2 u_n$.","title":"Recurrence for a normal number sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/normal-seq-fib","kind":"node","lean":"MetaMathlibExt.normalSeqFib","parent":6383,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/index.html#normal-seq-fib","scope":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","status":"fully_proved","summary":"For an integer $a$, this sequence is the normal second-order number sequence $(u_n)_{n \\ge 0}$ with coefficients $a_1 = a$ and $a_2 = 1$; that is, $u_0 = 0$, $u_1 = 1$ and $u_{n+2} = a u_{n+1} + u_n$ for every $n \\ge 0$.","title":"Normal number sequence with second coefficient one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/normal-seq-fib-add","kind":"node","lean":"MetaMathlibExt.normalSeqFib_add","parent":6383,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/index.html#normal-seq-fib-add","scope":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","status":"fully_proved","summary":"Let $a$ be an integer and let $(u_n)_{n \\ge 0}$ be the normal second-order number sequence with coefficients $a_1 = a$ and $a_2 = 1$. Then $u_{m+n+1} = u_{m+1} u_{n+1} + u_m u_n$ for all natural numbers $m$ and $n$.","title":"Addition formula for the normal sequence with second coefficient one"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/normal-seq-odd-identity","kind":"node","lean":"MetaMathlibExt.normalSeq_odd_identity","parent":6383,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences/index.html#normal-seq-odd-identity","scope":"NumberTheory/linear-recurrences/second-order-sequences/normal-second-order-sequences","status":"fully_proved","summary":"Let $a$ be an integer and let $(u_n)_{n \\ge 0}$ be the normal second-order number sequence with coefficients $a_1 = a$ and $a_2 = 1$. Then $u_{2n-1} = u_n^2 + u_{n-1}^2$ for every natural number $n \\ge 2$.","title":"Odd-index sum of two squares for the normal sequence with second coefficient one"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers","status":"fully_proved","title":"Pell and Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/pell-lucas","kind":"node","lean":"MetaMathlibExt.pellLucas","parent":6391,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/index.html#pell-lucas","scope":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers","status":"fully_proved","summary":"The Pell\u2013Lucas numbers are the natural numbers $Q_n$ defined by $Q_0 = Q_1 = 2$ and $Q_{n+2} = 2 Q_{n+1} + Q_n$ for every natural number $n$.","title":"Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/pell-cassini-identity","kind":"node","lean":"MetaMathlibExt.pell_cassini_identity","parent":6391,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/index.html#pell-cassini-identity","scope":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers","status":"fully_proved","summary":"Let $(P_n)_{n \\ge 0}$ be the Pell numbers, the natural numbers with $P_0 = 0$, $P_1 = 1$ and $P_{n+2} = 2P_{n+1} + P_n$ for every $n \\ge 0$. Then for every natural number $k \\ge 1$,","title":"Cassini's identity for Pell numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/pell-lucas-sequence","kind":"node","lean":"MetaMathlibExt.pellLucasSequence","parent":6391,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/index.html#pell-lucas-sequence","scope":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers","status":"fully_proved","summary":"The Pell\u2013Lucas (associated Pell) numbers $Q_n$, for natural numbers $n$, are defined by $Q_0 = 1$, $Q_1 = 1$ and $Q_{n+2} = 2Q_{n+1} + Q_n$ for every $n$. They begin $1, 1, 3, 7, 17, \\dots$ (OEIS A001333).","title":"Pell\u2013Lucas sequence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/pell-lucas-sequence-succ-succ","kind":"node","lean":"MetaMathlibExt.pellLucasSequence_succ_succ","parent":6391,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/index.html#pell-lucas-sequence-succ-succ","scope":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers","status":"fully_proved","summary":"Write $Q_n$ for the $n$-th Pell\u2013Lucas (associated Pell) number. For every natural number $n$, $Q_{n+2} = 2Q_{n+1} + Q_n$.","title":"Recurrence for the Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/pell-number","kind":"node","lean":"MetaMathlibExt.pellNumber","parent":6391,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/index.html#pell-number","scope":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers","status":"fully_proved","summary":"The Pell numbers $P_n$, for natural numbers $n$, are defined by $P_0 = 0$, $P_1 = 1$ and $P_{n+2} = 2P_{n+1} + P_n$ for every $n$. They begin $0, 1, 2, 5, 12, \\dots$","title":"Pell numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/pell-binomial-transform","kind":"node","lean":"MetaMathlibExt.pell_binomial_transform","parent":6391,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers/index.html#pell-binomial-transform","scope":"NumberTheory/linear-recurrences/second-order-sequences/pell-and-pell-lucas-numbers","status":"fully_proved","summary":"Let $P \\colon \\mathbb{N} \\to \\mathbb{Z}$ satisfy $P_0 = 0$, $P_1 = 1$ and $P_{n+2} = 2 P_{n+1} + P_n$ for every $n$, so that $P_n$ is the $n$-th Pell number. Then for every natural number $n$,","title":"Binomial transform of the Pell numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"can_prove","title":"Brousseau sums of k-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/brousseau-a","kind":"node","lean":"MetaMathlibExt.brousseauA","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The numbers "],["math","\\(A_k^{(p)}\\)"],["text"," for "],["math","\\(k\\)"],["text","-Fibonacci Brousseau sums"]],"source":"The numbers $A_k^{(p)}$ for $k$-Fibonacci Brousseau sums"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#brousseau-a","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"fully_proved","summary":"For a rational number $k$, the rational numbers $A_k^{(p)}$, $p \\ge 0$, are defined by $A_k^{(0)} = 1$ and, for $p \\ge 1$,","title":"The numbers $A_k^{(p)}$ for $k$-Fibonacci Brousseau sums"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/brousseau-a-zero","kind":"node","lean":"MetaMathlibExt.brousseauA_zero","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The initial value "],["math","\\(A_k^{(0)} = 1\\)"]],"source":"The initial value $A_k^{(0)} = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#brousseau-a-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"fully_proved","summary":"For every rational number $k$, the number $A_k^{(0)}$ for $k$-Fibonacci Brousseau sums equals $1$.","title":"The initial value $A_k^{(0)} = 1$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/brousseau-a-one","kind":"node","lean":"MetaMathlibExt.brousseauA_one","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(A_k^{(1)} = 2/k\\)"]],"source":"The value $A_k^{(1)} = 2/k$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#brousseau-a-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"fully_proved","summary":"For every rational number $k$, the number $A_k^{(1)}$ for $k$-Fibonacci Brousseau sums equals $2/k$, where $2/0$ is read as $0$.","title":"The value $A_k^{(1)} = 2/k$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/brousseau-a-two","kind":"node","lean":"MetaMathlibExt.brousseauA_two","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(A_k^{(2)} = 8/k^2\\)"]],"source":"The value $A_k^{(2)} = 8/k^2$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#brousseau-a-two","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"fully_proved","summary":"For every rational number $k$, the number $A_k^{(2)}$ for $k$-Fibonacci Brousseau sums equals $8/k^2$, where $8/0$ is read as $0$.","title":"The value $A_k^{(2)} = 8/k^2$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/brousseau-coeff","kind":"node","lean":"MetaMathlibExt.brousseauCoeff","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient polynomials "],["math","\\(C_k^{(p)}\\)"],["text"," for "],["math","\\(k\\)"],["text","-Fibonacci Brousseau sums"]],"source":"The coefficient polynomials $C_k^{(p)}$ for $k$-Fibonacci Brousseau sums"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#brousseau-coeff","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"fully_proved","summary":"For a rational number $k$, let $A_k^{(r)}$ be the numbers for $k$-Fibonacci Brousseau sums. For a natural number $p$ and a rational number $n$,","title":"The coefficient polynomials $C_k^{(p)}$ for $k$-Fibonacci Brousseau sums"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/brousseau-coeff-zero","kind":"node","lean":"MetaMathlibExt.brousseauCoeff_zero","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomial "],["math","\\(C_k^{(0)}\\)"],["text"," is constant "],["math","\\(1\\)"]],"source":"The polynomial $C_k^{(0)}$ is constant $1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#brousseau-coeff-zero","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"fully_proved","summary":"For all rational numbers $k$ and $n$, the coefficient polynomial $C_k^{(0)}$ for $k$-Fibonacci Brousseau sums satisfies $C_k^{(0)}(n) = 1$.","title":"The polynomial $C_k^{(0)}$ is constant $1$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/brousseau-coeff-one","kind":"node","lean":"MetaMathlibExt.brousseauCoeff_one","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomial "],["math","\\(C_k^{(1)}(n) = n - 2/k\\)"]],"source":"The polynomial $C_k^{(1)}(n) = n - 2/k$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#brousseau-coeff-one","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"fully_proved","summary":"For all rational numbers $k$ and $n$, the coefficient polynomial $C_k^{(1)}$ for $k$-Fibonacci Brousseau sums satisfies $C_k^{(1)}(n) = n - 2/k$, where $2/0$ is read as $0$.","title":"The polynomial $C_k^{(1)}(n) = n - 2/k$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/k-fibonacci-eq-pow-add-convolution","kind":"node","lean":"MetaMathlibExt.kFibonacci_eq_pow_add_convolution","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-Fibonacci numbers as a power-weighted convolution"]],"source":"$k$-Fibonacci numbers as a power-weighted convolution"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#k-fibonacci-eq-pow-add-convolution","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"can_prove","summary":"For an integer $k$, let $F_{k,n}$ be the $k$-Fibonacci numbers, the Lucas sequence $U(k, -1)$: $F_{k,0} = 0$, $F_{k,1} = 1$ and $F_{k,n+2} = k F_{k,n+1} + F_{k,n}$. For every integer $k$, every $n \\ge 0$ and every $p \\ge 1$,","title":"$k$-Fibonacci numbers as a power-weighted convolution"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/k-fibonacci-power-convolution","kind":"node","lean":"MetaMathlibExt.kFibonacci_power_convolution","parent":6398,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#k-fibonacci-power-convolution","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"can_prove","summary":"Let $k$ be a nonzero integer and let $F_{k,n}$ be the $k$-Fibonacci numbers, the Lucas sequence $U(k, -1)$: $F_{k,0} = 0$, $F_{k,1} = 1$ and $F_{k,n+2} = k F_{k,n+1} + F_{k,n}$. Let $A_k^{(0)} = 1$ and $A_k^{(p)} = \\frac{2}{k} \\sum_{j=1}^{\\lceil p/2 \\rceil} \\binom{p}{2j-1} A_k^{(p-2j+1)}$ for $p \\ge 1$. Then for all natural numbers $n$ and $p$, with $0^0 = 1$,","title":"Power-weighted convolution of k-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/k-fibonacci-brousseau-sum","kind":"node","lean":"MetaMathlibExt.kFibonacci_brousseau_sum","parent":6398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Brousseau sums of "],["math","\\(k\\)"],["text","-Fibonacci numbers"]],"source":"Brousseau sums of $k$-Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#k-fibonacci-brousseau-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"can_prove","summary":"For an integer $k$, let $F_{k,n}$ be the $k$-Fibonacci numbers, the Lucas sequence $U(k, -1)$ ($F_{k,0} = 0$, $F_{k,1} = 1$, $F_{k,n+2} = k F_{k,n+1} + F_{k,n}$), and let $C_k^{(p)}$ be the coefficient polynomials. For every nonzero integer $k$ and all $n, p \\ge 0$,","title":"Brousseau sums of $k$-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/fibonacci-brousseau-sum-mul-odd","kind":"node","lean":"MetaMathlibExt.fibonacci_brousseau_sum_mul_odd","parent":6398,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#fibonacci-brousseau-sum-mul-odd","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers, $L_n$ the Lucas numbers and $C_k^{(p)}$ the coefficient polynomials. For every odd natural number $m$ and all $n, p \\ge 0$,","title":"Brousseau sums of Fibonacci numbers along an odd step"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/pell-brousseau-sum-mul-odd","kind":"node","lean":"MetaMathlibExt.pell_brousseau_sum_mul_odd","parent":6398,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#pell-brousseau-sum-mul-odd","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"can_prove","summary":"Let $P_n$ be the Pell numbers and $Q_n$ the Pell\u2013Lucas numbers, the Lucas sequence $V(2, -1)$: $Q_0 = Q_1 = 2$ and $Q_{n+2} = 2 Q_{n+1} + Q_n$. Let $m$ be an odd natural number and write $k = Q_m$. Let $A_k^{(0)} = 1$ and $A_k^{(p)} = \\frac{2}{k} \\sum_{j=1}^{\\lceil p/2 \\rceil} \\binom{p}{2j-1} A_k^{(p-2j+1)}$ for $p \\ge 1$, and let $C_k^{(p)}(x) = \\sum_{r=0}^{p} (-1)^r A_k^{(r)} \\binom{p}{r} x^{p-r}$, with\u2026","title":"Power-weighted Pell sums along odd multiples"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/k-fibonacci-shifted-brousseau-sum","kind":"node","lean":"MetaMathlibExt.kFibonacci_shifted_brousseau_sum","parent":6398,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers/index.html#k-fibonacci-shifted-brousseau-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/brousseau-sums-of-k-fibonacci-numbers","status":"can_prove","summary":"Let $k$ be a nonzero integer and let $F_{k,n}$ be the $k$-Fibonacci numbers, the Lucas sequence $U(k, -1)$: $F_{k,0} = 0$, $F_{k,1} = 1$ and $F_{k,n+2} = k F_{k,n+1} + F_{k,n}$. Let $A_k^{(0)} = 1$ and $A_k^{(p)} = \\frac{2}{k} \\sum_{j=1}^{\\lceil p/2 \\rceil} \\binom{p}{2j-1} A_k^{(p-2j+1)}$ for $p \\ge 1$, and let $C_k^{(p)}(x) = \\sum_{r=0}^{p} (-1)^r A_k^{(r)} \\binom{p}{r} x^{p-r}$, with $0^0 = 1$. Then for all\u2026","title":"Shifted power-weighted sums of k-Fibonacci numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers.json","id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","title":"Oresme numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/oresme-convolution-self","kind":"node","lean":"MetaMathlibExt.oresme_convolution_self","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#oresme-convolution-self","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $O_m = m/2^m$ denote the Oresme numbers. For every natural number $n$, $\\sum_{k=0}^{n} O_k O_{n-k} = \\frac{(n+1) n (n-1)}{6 \\cdot 2^n}$.","title":"Self-convolution of the Oresme numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/fib-oresme-convolution","kind":"node","lean":"MetaMathlibExt.fib_oresme_convolution","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#fib-oresme-convolution","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $F_m$ denote the Fibonacci numbers, with $F_0 = 0$, $F_1 = 1$ and $F_{m+2} = F_{m+1} + F_m$, and let $O_m = m/2^m$ denote the Oresme numbers. For every natural number $n$, $\\sum_{k=0}^{n} F_k O_{n-k} = \\frac{2}{5} F_{n+1} - \\frac{2(n+1)}{5 \\cdot 2^n}$.","title":"Fibonacci\u2013Oresme convolution"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/two-pow-pred-mul-fib-mod-eq-five","kind":"node","lean":"MetaMathlibExt.two_pow_pred_mul_fib_modEq_five","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#two-pow-pred-mul-fib-mod-eq-five","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers, with $F_0 = 0$ and $F_1 = 1$. For every natural number $n$, $2^{n-1} F_n \\equiv n \\pmod 5$, where at $n = 0$ the exponent $n - 1$ is taken to be $0$ (both sides are then $0$).","title":"Oresme\u2013Fibonacci congruence modulo 5"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/lucas-number-oresme-convolution","kind":"node","lean":"MetaMathlibExt.lucasNumber_oresme_convolution","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#lucas-number-oresme-convolution","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $L_m$ denote the Lucas numbers and let $O_m = m/2^m$ denote the Oresme numbers. For every natural number $n$, $\\sum_{k=0}^{n} L_k O_{n-k} = \\frac{2}{5} L_{n+1} - \\frac{2}{5 \\cdot 2^n}$.","title":"Lucas\u2013Oresme convolution"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/two-pow-mul-lucas-number-succ-mod-eq-fiv","kind":"node","lean":"MetaMathlibExt.two_pow_mul_lucasNumber_succ_modEq_five","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#two-pow-mul-lucas-number-succ-mod-eq-fiv","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $L_m$ denote the Lucas numbers. For every natural number $n$, $2^n L_{n+1} \\equiv 1 \\pmod{5}$.","title":"Powers of two times Lucas numbers modulo 5"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/pell-number-oresme-convolution","kind":"node","lean":"MetaMathlibExt.pellNumber_oresme_convolution","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#pell-number-oresme-convolution","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $P_m$ denote the Pell numbers and let $O_m = m/2^m$ denote the Oresme numbers. For every natural number $n$, $\\sum_{k=0}^{n} P_k O_{n-k} = \\frac{2}{49}(5 P_{n+1} - 4 P_n) - \\frac{2(7n + 5)}{49 \\cdot 2^n}$.","title":"Pell\u2013Oresme convolution"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/two-pow-mul-pell-number-combination-mod","kind":"node","lean":"MetaMathlibExt.two_pow_mul_pellNumber_combination_modEq","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#two-pow-mul-pell-number-combination-mod","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"For every natural number $n$, the Pell numbers $P_n$ satisfy $2^n (5 P_{n+1} - 4 P_n) \\equiv 7n + 5 \\pmod{49}$ as integers.","title":"Oresme\u2013Pell congruence modulo 49"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/pell-lucas-sequence-oresme-convolution","kind":"node","lean":"MetaMathlibExt.pellLucasSequence_oresme_convolution","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#pell-lucas-sequence-oresme-convolution","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $Q_m$ denote the Pell\u2013Lucas numbers, with $Q_0 = Q_1 = 1$ and $Q_{m+2} = 2Q_{m+1} + Q_m$, and let $O_m = m/2^m$ denote the Oresme numbers. For every natural number $n$, $\\sum_{k=0}^{n} Q_k O_{n-k} = \\frac{2}{49}(5 Q_{n+1} - 4 Q_n) + \\frac{7n - 2}{49 \\cdot 2^n}$.","title":"Pell\u2013Lucas\u2013Oresme convolution"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/two-pow-succ-mul-pell-lucas-sequence-com","kind":"node","lean":"MetaMathlibExt.two_pow_succ_mul_pellLucasSequence_combination_modEq","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#two-pow-succ-mul-pell-lucas-sequence-com","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $Q_n$ denote the Pell\u2013Lucas numbers, with $Q_0 = Q_1 = 1$. For every natural number $n$, $2^{n+1} (5 Q_{n+1} - 4 Q_n) \\equiv 2 - 7n \\pmod{49}$ as integers.","title":"Oresme\u2013Pell\u2013Lucas congruence modulo 49"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/jacobsthal-oresme-convolution","kind":"node","lean":"MetaMathlibExt.jacobsthal_oresme_convolution","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#jacobsthal-oresme-convolution","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $J_m$ denote the Jacobsthal numbers, the terms of the Lucas sequence $U(1, -2)$, and let $O_m = m/2^m$ denote the Oresme numbers. For every natural number $n$, $\\sum_{k=0}^{n} J_k O_{n-k} = \\frac{2}{9} J_{n+1} - \\frac{2(n+1)}{9 \\cdot 2^n}$.","title":"Jacobsthal\u2013Oresme convolution"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/two-pow-mul-jacobsthal-succ-mod-eq-nine","kind":"node","lean":"MetaMathlibExt.two_pow_mul_jacobsthal_succ_modEq_nine","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#two-pow-mul-jacobsthal-succ-mod-eq-nine","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $J_m$ denote the Jacobsthal numbers, the terms of the Lucas sequence $U(1, -2)$. For every natural number $n$, $2^n J_{n+1} \\equiv n + 1 \\pmod{9}$.","title":"Powers of two times Jacobsthal numbers modulo 9"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/jacobsthal-lucas-number-oresme-convoluti","kind":"node","lean":"MetaMathlibExt.jacobsthalLucasNumber_oresme_convolution","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#jacobsthal-lucas-number-oresme-convoluti","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $j_m$ denote the Jacobsthal\u2013Lucas numbers and let $O_m = m/2^m$ denote the Oresme numbers. For every natural number $n$, $\\sum_{k=0}^{n} j_k O_{n-k} = \\frac{2}{9} j_{n+1} - \\frac{2}{9 \\cdot 2^n}$.","title":"Jacobsthal\u2013Lucas\u2013Oresme convolution"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/two-pow-mul-jacobsthal-lucas-number-succ","kind":"node","lean":"MetaMathlibExt.two_pow_mul_jacobsthalLucasNumber_succ_modEq_nine","parent":6412,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers/index.html#two-pow-mul-jacobsthal-lucas-number-succ","scope":"NumberTheory/linear-recurrences/second-order-sequences/oresme-numbers","status":"can_prove","summary":"Let $j_m$ denote the Jacobsthal\u2013Lucas numbers. For every natural number $n$, $2^n j_{n+1} \\equiv 1 \\pmod{9}$.","title":"Powers of two times Jacobsthal\u2013Lucas numbers modulo 9"},{"data_url":"scopes/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences.json","id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","kind":"scope","navigation":"module","parent":6213,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","title":"Toeplitz\u2013Hessenberg determinants of recurrences"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/toeplitz-hessenberg-det-generating-funct","kind":"node","lean":"MetaMathlibExt.toeplitzHessenberg_det_generatingFunction","parent":6426,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#toeplitz-hessenberg-det-generating-funct","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $K$ be a field and $a_0, a_1, a_2, \\dots$ a sequence in $K$ with $a_0 \\ne 0$. For $n \\ge 1$ let $A_n$ be the $n \\times n$ Toeplitz\u2013Hessenberg matrix whose $(i, j)$ entry is $a_{i-j+1}$ if $j \\le i + 1$ and $0$ otherwise. In the ring of formal power series over $K$, let $f(x) = \\sum_{n \\ge 1} \\det(A_n) x^n$, $g(x) = \\sum_{i \\ge 1} a_i x^i$ and $h(x) = a_0^{-1} g(-a_0 x)$. Then $f(x)(1 + h(x)) = -h(x)$, that is,\u2026","title":"Trudi's formula as a generating function"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/toeplitz-hessenberg-det-linear-mul-pow-g","kind":"node","lean":"MetaMathlibExt.toeplitzHessenberg_det_linear_mul_pow_generatingFunction","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of Toeplitz\u2013Hessenberg determinants with entries "],["math","\\((bk + c)d^k\\)"]],"source":"Generating function of Toeplitz\u2013Hessenberg determinants with entries $(bk + c)d^k$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#toeplitz-hessenberg-det-linear-mul-pow-g","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring and $a, b, c, d \\in R$. Put $t_0 = a$ and $t_k = (bk + c) d^k$ for $k \\ge 1$, and for $n \\ge 1$ let $u_n$ be the determinant of the $n \\times n$ matrix whose $(i, j)$ entry is $t_{i-j+1}$ if $j \\le i + 1$ and $0$ otherwise. Then the formal power series $u(x) = \\sum_{n \\ge 1} u_n x^n$ over $R$ satisfies","title":"Generating function of Toeplitz\u2013Hessenberg determinants with entries $(bk + c)d^k$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/fib-pell-composition-sums","kind":"node","lean":"MetaMathlibExt.fib_pell_composition_sums","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","Composition-sum formulas for "],["math","\\(F_{2n}\\)"],["text",", "],["math","\\(F_{3n-1}\\)"],["text"," and "],["math","\\(P_{n-1}\\)"]],"source":"Composition-sum formulas for $F_{2n}$, $F_{3n-1}$ and $P_{n-1}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#fib-pell-composition-sums","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers and $P_k$ the Pell numbers. For every $n \\ge 1$, with each sum running over all compositions $(c_1, \\dots, c_r)$ of $n$ (tuples of positive integers with $c_1 + \\dots + c_r = n$), $F_{2n} = \\sum \\prod_{i=1}^{r} c_i$, $F_{3n-1} = 2^{n-1} \\sum \\prod_{i=1}^{r} \\frac{2 c_i - 1}{2^{c_i - 1}}$ and $P_{n-1} = 2^{n-1} \\sum \\prod_{i=1}^{r} \\frac{c_i - 1}{2^{c_i - 1}}$.","title":"Composition-sum formulas for $F_{2n}$, $F_{3n-1}$ and $P_{n-1}$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/composition-sum-succ-mul-two-zpow","kind":"node","lean":"MetaMathlibExt.composition_sum_succ_mul_two_zpow","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","A composition sum with weights "],["math","\\((c_i + 1) 2^{c_i - 2}\\)"],["text"," in terms of Fibonacci and Lucas numbers"]],"source":"A composition sum with weights $(c_i + 1) 2^{c_i - 2}$ in terms of Fibonacci and Lucas numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#composition-sum-succ-mul-two-zpow","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"A composition sum with weights $(c_i + 1) 2^{c_i - 2}$ in terms of Fibonacci and Lucas numbers. Let $F_k$ be the Fibonacci numbers and $L_k$ the Lucas numbers. For every $n \\ge 1$, the sum of $\\prod_{i=1}^{r} (c_i + 1) 2^{c_i - 2}$ over all compositions $(c_1, \\dots, c_r)$ of $n$ (tuples of positive integers with $c_1 + \\dots + c_r = n$) equals $5^{(n-1)/2} F_{n+1}$ if $n$ is odd and $5^{(n-2)/2} L_{n+1}$ if $n$ is\u2026","title":"A composition sum with weights $(c_i + 1) 2^{c_i - 2}$ in terms of Fibonacci and Lucas numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/fib-jacobsthal-pell-composition-sums","kind":"node","lean":"MetaMathlibExt.fib_jacobsthal_pell_composition_sums","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","Composition-sum formulas for "],["math","\\(F_{n+3}\\)"],["text",", "],["math","\\(F_{3n+1}\\)"],["text"," and Jacobsthal and Pell numbers"]],"source":"Composition-sum formulas for $F_{n+3}$, $F_{3n+1}$ and Jacobsthal and Pell numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#fib-jacobsthal-pell-composition-sums","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers, $P_k$ the Pell numbers and $J_k$ the Jacobsthal numbers, that is, the Lucas sequence $U(1, -2)$, with $J_0 = 0$, $J_1 = 1$ and $J_{k+2} = J_{k+1} + 2 J_k$. For every $n \\ge 1$, with each sum running over all compositions $(c_1, \\dots, c_r)$ of $n$ (tuples of positive integers with $c_1 + \\dots + c_r = n$), $\\sum \\prod_{i=1}^{r} (-1)^{c_i - 1} (c_i + 2) = F_{n+3}$,\u2026","title":"Composition-sum formulas for $F_{n+3}$, $F_{3n+1}$ and Jacobsthal and Pell numbers"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/fib-mul-add-eq-toeplitz-hessenberg-det-i","kind":"node","lean":"MetaMathlibExt.fib_mul_add_eq_toeplitzHessenberg_det_iff","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci subsequences "],["math","\\(F_{\\ell n + m}\\)"],["text"," that are Toeplitz\u2013Hessenberg determinants"]],"source":"Fibonacci subsequences $F_{\\ell n + m}$ that are Toeplitz\u2013Hessenberg determinants"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#fib-mul-add-eq-toeplitz-hessenberg-det-i","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"For complex numbers $a, b, c, d$ and $n \\ge 1$, let $u_n(a, b, c, d)$ be the determinant of the $n \\times n$ matrix whose entry in row $i$ and column $j$ is $a$ if $j = i + 1$, is $(bp + c) d^p$ with $p = i - j + 1$ if $j \\le i$, and is $0$ if $j > i + 1$. Let $F_k$ be the Fibonacci numbers and let $\\ell \\ge 1$ and $m \\ge 0$ be integers. There are complex numbers $a, b, c, d$ with $F_{\\ell n + m} = u_n(a, b, c, d)$\u2026","title":"Fibonacci subsequences $F_{\\ell n + m}$ that are Toeplitz\u2013Hessenberg determinants"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/pell-pell-lucas-composition-sums-sqrt-tw","kind":"node","lean":"MetaMathlibExt.pell_pellLucas_composition_sums_sqrt_two","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pell and Pell\u2013Lucas numbers as composition sums with "],["math","\\(\\sqrt 2\\)"],["text"," weights"]],"source":"Pell and Pell\u2013Lucas numbers as composition sums with $\\sqrt 2$ weights"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#pell-pell-lucas-composition-sums-sqrt-tw","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $P_n$ be the Pell numbers and $Q_n$ the Pell\u2013Lucas numbers ($Q_0 = Q_1 = 2$, $Q_{k+2} = 2Q_{k+1} + Q_k$). For every $n \\ge 1$, with $c$ running over the compositions of $n$ (ordered tuples of positive integers with sum $n$) and the products running over the parts $j$ of $c$, the real number","title":"Pell and Pell\u2013Lucas numbers as composition sums with $\\sqrt 2$ weights"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/oresme-toeplitz-hessenberg-det-formulas","kind":"node","lean":"MetaMathlibExt.oresme_toeplitzHessenberg_det_formulas","parent":6426,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#oresme-toeplitz-hessenberg-det-formulas","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"For numbers $a_0, a_1, \\dots, a_n$, let $\\det(a_0; a_1, \\dots, a_n)$ be the determinant of the $n \\times n$ matrix whose $(i, j)$ entry is $a_{i-j+1}$ if $j \\le i + 1$ and $0$ otherwise, so that $a_0$ fills the superdiagonal. Let $O_k = k/2^k$ be the Oresme numbers, $F_n$ the Fibonacci numbers, $L_n$ the Lucas numbers, $P_n$ the Pell numbers, $Q_n$ the Pell\u2013Lucas numbers ($Q_0 = Q_1 = 2$) and $J_n$ the Jacobsthal\u2026","title":"Toeplitz\u2013Hessenberg determinants with Oresme entries"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/horadam-eq-composition-sum","kind":"node","lean":"MetaMathlibExt.horadam_eq_composition_sum","parent":6426,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#horadam-eq-composition-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $p, q, r, s$ be complex numbers with $(q - r)^2 = 4s(p - 1)$ and $4s(p - 1) \\ne 0$, and let $(v_k)_{k \\ge 0}$ be the complex sequence with $v_0 = p$, $v_1 = q$ and $v_{k+2} = r v_{k+1} + s v_k$ for every $k \\ge 0$. Put $e = 2ps/(r - q)$. Then for every $n \\ge 1$,","title":"Horadam sequences as composition sums"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/horadam-composition-sum-recurrence","kind":"node","lean":"MetaMathlibExt.horadam_composition_sum_recurrence","parent":6426,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#horadam-composition-sum-recurrence","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $p, q, r, s$ be complex numbers with $(q - r)^2 = 4s(p - 1)$ and $4s(p - 1) \\ne 0$, put $e = 2ps/(r - q)$, and for every natural number $m$ let","title":"Horadam composition sums satisfy the Horadam recurrence"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/gibonacci-add-eq-composition-sum","kind":"node","lean":"MetaMathlibExt.gibonacci_add_eq_composition_sum","parent":6426,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#gibonacci-add-eq-composition-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $y$ and $z$ be non-negative integers, not both zero, and let $(G_k)$ be the Gibonacci sequence with $G_0 = y$ and $G_1 = z$. Let $m \\ge 3$ and $n \\ge 1$, and let $D$ be a complex number with $D^2 = (-1)^m (y^2 + yz - z^2)$. Then","title":"Translates of a Gibonacci sequence as composition sums"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/gibonacci-mul-add-eq-toeplitz-hessenberg","kind":"node","lean":"MetaMathlibExt.gibonacci_mul_add_eq_toeplitzHessenberg_det_iff","parent":6426,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#gibonacci-mul-add-eq-toeplitz-hessenberg","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $y$ and $z$ be non-negative integers, not both zero, and let $(G_k)$ be the Gibonacci sequence with $G_0 = y$ and $G_1 = z$. For complex numbers $a, b, c, d$ and $n \\ge 1$, let $u_n(a, b, c, d)$ be the determinant of the $n \\times n$ matrix whose entry in row $i$ and column $j$ is $a$ if $j = i + 1$, is $(bp + c) d^p$ with $p = i - j + 1$ if $j \\le i$, and is $0$ if $j > i + 1$. Let $m \\ge 3$ and let $k$ be a\u2026","title":"Multiples of a shifted Gibonacci sequence that are Toeplitz\u2013Hessenberg determinants"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/gibonacci-mul-add-eq-composition-sum","kind":"node","lean":"MetaMathlibExt.gibonacci_mul_add_eq_composition_sum","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subsequences "],["math","\\(G_{\\ell n + m}\\)"],["text"," of a Gibonacci sequence as composition sums"]],"source":"Subsequences $G_{\\ell n + m}$ of a Gibonacci sequence as composition sums"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#gibonacci-mul-add-eq-composition-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $y$ and $z$ be non-negative integers, not both zero, let $(G_k)$ be the Gibonacci sequence with $G_0 = y$ and $G_1 = z$, and let $L_k$ be the Lucas numbers. Let $\\ell \\ge 1$, $m \\ge 0$ and $n \\ge 1$, and let $E$ be a complex number with $E^2 = (-1)^\\ell G_\\ell (G_\\ell - L_\\ell y) + y^2$ if $m$ is even and $E^2 = (-1)^\\ell G_{\\ell+1} (G_{\\ell+1} - L_\\ell z) + z^2$ if $m$ is odd. Write\u2026","title":"Subsequences $G_{\\ell n + m}$ of a Gibonacci sequence as composition sums"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/fib-lucas-add-eq-composition-sum","kind":"node","lean":"MetaMathlibExt.fib_lucas_add_eq_composition_sum","parent":6426,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#fib-lucas-add-eq-composition-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers and $L_k$ the Lucas numbers, let $m \\ge 0$ and $n \\ge 1$, and write $\\sum_c$ for the sum over all compositions $c = (c_1, \\dots, c_r)$ of $n$ (tuples of positive integers with $c_1 + \\dots + c_r = n$), where $r$ is the number of parts of $c$. For every complex number $\\lambda$ with $\\lambda^2 = (-1)^{m+1}$, and with $\\lambda = 1$ if $m = 1$,","title":"Fibonacci and Lucas translates as composition sums"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/fib-add-three-companion-composition-sum","kind":"node","lean":"MetaMathlibExt.fib_add_three_companion_composition_sum","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","Companion composition-sum formula for "],["math","\\(F_{n+3}\\)"]],"source":"Companion composition-sum formula for $F_{n+3}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#fib-add-three-companion-composition-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers. For every $n \\ge 1$,","title":"Companion composition-sum formula for $F_{n+3}$"},{"id":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/gibonacci-two-mul-add-eq-composition-sum","kind":"node","lean":"MetaMathlibExt.gibonacci_two_mul_add_eq_composition_sum","parent":6426,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gibonacci subsequences "],["math","\\(G_{2n+m}\\)"],["text"," as composition sums"]],"source":"Gibonacci subsequences $G_{2n+m}$ as composition sums"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences/index.html#gibonacci-two-mul-add-eq-composition-sum","scope":"NumberTheory/linear-recurrences/second-order-sequences/toeplitz-hessenberg-determinants-of-recurrences","status":"can_prove","summary":"Let $y$ and $z$ be non-negative integers, not both zero, and let $(G_k)$ be the Gibonacci sequence with $G_0 = y$ and $G_1 = z$. Let $m \\ge 0$ and $n \\ge 1$, and let $F$ be a complex number with $F^2 = (-1)^{m+1} (y^2 + yz - z^2)$ such that $m \\ge 4$ or $F \\ne G_m$. Then","title":"Gibonacci subsequences $G_{2n+m}$ as composition sums"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences.json","id":"NumberTheory/linear-recurrences/higher-order-sequences","kind":"scope","parent":5991,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences","status":"can_prove","summary":"Tribonacci, tetranacci, pentanacci, k-generalized Fibonacci, Padovan, Perrin and Narayana sequences, dominant roots and Binet forms.","title":"Tribonacci and higher-order sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-pell-eq-two-colored-composit","kind":"node","lean":"MetaMathlibExt.generalized_pell_eq_two_colored_compositions","parent":6443,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/index.html#generalized-pell-eq-two-colored-composit","scope":"NumberTheory/linear-recurrences/higher-order-sequences","status":"fully_proved","summary":"Let $k \\ge 2$ and $n \\ge 1$ be natural numbers, and let $(P_m)_{m \\in \\mathbb{Z}}$ be natural numbers with $P_m = 0$ for $2 - k \\le m \\le 0$, $P_1 = 1$, and $P_m = 2P_{m-1} + P_{m-2} + \\dots + P_{m-k}$ for all $m \\ge 2$. Then the number of compositions of $n$ into parts from $\\{1, \\dots, k\\}$, where each part equal to $1$ carries one of two colors, equals $P_{n+1}$.","title":"Generalized Pell numbers count compositions with two-colored ones"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/narayana-cows-sequence","kind":"node","lean":"MetaMathlibExt.narayanaCowsSequence","parent":6443,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/index.html#narayana-cows-sequence","scope":"NumberTheory/linear-recurrences/higher-order-sequences","status":"fully_proved","summary":"The sequence $(N_n)_{n \\ge 0}$ of natural numbers defined by $N_0 = 0$, $N_1 = N_2 = 1$ and $N_n = N_{n-1} + N_{n-3}$ for $n \\ge 3$.","title":"Narayana's cows sequence"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences","status":"fully_proved","title":"Cubic analogues of Lucas sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/cubic-recurrence-root-condition","kind":"node","lean":"MetaMathlibExt.cubicRecurrenceRootCondition","parent":6446,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/index.html#cubic-recurrence-root-condition","scope":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences","status":"fully_proved","summary":"For integers $S_1, S_2, R$ and a complex number $\\gamma$, put $\\rho = R(\\gamma + \\gamma^{-1})$ (with the convention $0^{-1} = 0$) and $S_3 = RS_1^2 - 2RS_2 - 4R^3$. The root condition holds when $\\rho$ is a root of the cubic $x^3 - S_1x^2 + S_2x - S_3$, that is, when $\\rho^3 - S_1\\rho^2 + S_2\\rho - S_3 = 0$.","title":"Root condition for the Roettger\u2013Williams cubic"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/is-cubic-recurrence-pair","kind":"node","lean":"MetaMathlibExt.IsCubicRecurrencePair","parent":6446,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/index.html#is-cubic-recurrence-pair","scope":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences","status":"fully_proved","summary":"Integer sequences $U$ and $W$ form a cubic recurrence pair for integers $S_1, S_2, R$ when there are pairwise distinct complex numbers $\\gamma_1, \\gamma_2, \\gamma_3$, each different from $1$, with $\\gamma_1\\gamma_2\\gamma_3 = 1$, such that each $\\gamma_i$ satisfies the cubic recurrence root condition for $(S_1, S_2, R)$, $U_0 = 0$,\u2026","title":"Cubic recurrence pair"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/cubic-recurrence-discriminant","kind":"node","lean":"MetaMathlibExt.cubicRecurrenceDiscriminant","parent":6446,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/index.html#cubic-recurrence-discriminant","scope":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences","status":"fully_proved","summary":"For an integer $R$ and integer sequences $U$ and $W$, the discriminant sequence is $D_n = \\gcd(W_n - 6R^n, U_n)$ for $n \\ge 0$: the greatest common divisor of the natural numbers $|W_n - 6R^n|$ and $|U_n|$.","title":"Cubic-recurrence discriminant sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/cubic-recurrence-discriminant-dvd-mul","kind":"node","lean":"MetaMathlibExt.cubicRecurrenceDiscriminant_dvd_mul","parent":6446,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences/index.html#cubic-recurrence-discriminant-dvd-mul","scope":"NumberTheory/linear-recurrences/higher-order-sequences/cubic-analogues-of-lucas-sequences","status":"fully_proved","summary":"Let $S_1, S_2, R$ be integers, let $U$ and $W$ be integer sequences forming a cubic recurrence pair for $(S_1, S_2, R)$, and let $D_n = \\gcd(W_n - 6R^n, U_n)$ be their discriminant sequence. Then for all integers $n \\ge 1$ and $m \\ge 1$, $D_n$ divides $D_{nm}$.","title":"Divisibility of the cubic-recurrence discriminants"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers","status":"fully_proved","title":"Fibonacci p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/fibonacci-p","kind":"node","lean":"MetaMathlibExt.fibonacciP","parent":6451,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/index.html#fibonacci-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers","status":"fully_proved","summary":"For a natural number $p$, the Fibonacci $p$-numbers are the natural numbers $F_p(r)$, $r \\in \\mathbb{N}$, given by $F_p(0) = 0$, $F_p(r) = 1$ for $1 \\le r \\le p$, and $F_p(r) = F_p(r-1) + F_p(r-p-1)$ for $r > p$.","title":"Fibonacci p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/fibonacci-p-zero","kind":"node","lean":"MetaMathlibExt.fibonacciP_zero","parent":6451,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/index.html#fibonacci-p-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers","status":"fully_proved","summary":"For every natural number $p$, the Fibonacci $p$-number $F_p(0)$ equals $0$.","title":"Zeroth Fibonacci p-number"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/fibonacci-p-eq-one","kind":"node","lean":"MetaMathlibExt.fibonacciP_eq_one","parent":6451,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/index.html#fibonacci-p-eq-one","scope":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$ and $r$ with $1 \\le r \\le p$, the Fibonacci $p$-number $F_p(r)$ equals $1$.","title":"Initial Fibonacci p-numbers equal one"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/fibonacci-p-rec","kind":"node","lean":"MetaMathlibExt.fibonacciP_rec","parent":6451,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/index.html#fibonacci-p-rec","scope":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$ and $r$ with $p < r$, the Fibonacci $p$-numbers satisfy $F_p(r) = F_p(r-1) + F_p(r-p-1)$.","title":"Recurrence for Fibonacci p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/fibonacci-p-one","kind":"node","lean":"MetaMathlibExt.fibonacciP_one","parent":6451,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/index.html#fibonacci-p-one","scope":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers","status":"fully_proved","summary":"For every natural number $n$, the Fibonacci $1$-number $F_1(n)$ equals the $n$-th Fibonacci number $f_n$, where $f_0 = 0$, $f_1 = 1$ and $f_{k+2} = f_{k+1} + f_k$.","title":"Fibonacci 1-numbers are the Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/honsberger-fibonacci-p","kind":"node","lean":"MetaMathlibExt.honsberger_fibonacciP","parent":6451,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers/index.html#honsberger-fibonacci-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/fibonacci-p-numbers","status":"fully_proved","summary":"Let $p$, $m$ and $n$ be positive integers with $p \\le m + 1$ and $p \\le n + 2$, and write $F_p(r)$ for the Fibonacci $p$-numbers. Suppose that $F_p(0) = 0$, that $F_p(r) = 1$ for $1 \\le r \\le p$, and that $F_p(r) = F_p(r-1) + F_p(r-p-1)$ for $r > p$ (these hold by definition). Then","title":"Honsberger identity for Fibonacci p-numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","title":"Generalized Padovan\u2013Perrin recurrences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-char-poly-separable","kind":"node","lean":"MetaMathlibExt.padovanPerrin_charPoly_separable","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-char-poly-separable","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $k_1, k_2, p, s$ be natural numbers with $k_1 > 0$, $p \\ge 2$, $p < s$ and $k_1^2 < k_2$. Then the characteristic polynomial $P(z) = z^{s+p} - k_1 z^p - k_2$ of the $([k]; p; s)$-generalized Padovan\u2013Perrin recurrence is separable over $\\mathbb{C}$, that is, it has only simple roots.","title":"Simple roots of the Padovan\u2013Perrin characteristic polynomial"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-recurrence","kind":"node","lean":"MetaMathlibExt.padovanPerrinRecurrence","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-recurrence","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $k_1, k_2 \\in R$ and $p, s \\in \\mathbb{N}$. The $([k]; p; s)$-generalized Padovan\u2013Perrin recurrence is the linear recurrence of order $s + p$ with coefficients $c_i = [i = 0] \\, k_2 + [i = p] \\, k_1$ for $0 \\le i < s + p$, where $[\\cdot]$ is $1$ if the condition holds and $0$ otherwise: a sequence $v \\colon \\mathbb{N} \\to R$ solves it when\u2026","title":"Generalized Padovan\u2013Perrin recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-recurrence-is-solution-if","kind":"node","lean":"MetaMathlibExt.padovanPerrinRecurrence_isSolution_iff","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-recurrence-is-solution-if","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $k_1, k_2 \\in R$, and $p, s \\in \\mathbb{N}$ with $s > 0$. A sequence $v \\colon \\mathbb{N} \\to R$ is a solution of the $([k]; p; s)$-Padovan\u2013Perrin recurrence if and only if $v_{n+s+p} = k_1 v_{n+p} + k_2 v_n$ for every $n \\ge 0$.","title":"Characterization of solutions of the generalized Padovan\u2013Perrin recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-fundamental","kind":"node","lean":"MetaMathlibExt.padovanPerrinFundamental","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-fundamental","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $k_1, k_2 \\in R$ and $p, s \\in \\mathbb{N}$. For a natural number $j$, the $j$-th fundamental sequence $v^{(j)} \\colon \\mathbb{N} \\to R$ is the solution of the $([k]; p; s)$-Padovan\u2013Perrin recurrence with initial values $v^{(j)}_n = 1$ if $n + 1 = j$ and $v^{(j)}_n = 0$ otherwise, for $0 \\le n < s + p$.","title":"Fibonacci fundamental system of the generalized Padovan\u2013Perrin recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-fundamental-of-lt","kind":"node","lean":"MetaMathlibExt.padovanPerrinFundamental_of_lt","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-fundamental-of-lt","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $k_1, k_2 \\in R$ and $p, s, j \\in \\mathbb{N}$. For every $n < s + p$, the $j$-th sequence $v^{(j)}$ of the Fibonacci fundamental system satisfies $v^{(j)}_n = 1$ if $n + 1 = j$ and $v^{(j)}_n = 0$ otherwise.","title":"Initial values of the generalized Padovan\u2013Perrin fundamental system"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-fundamental-is-solution","kind":"node","lean":"MetaMathlibExt.padovanPerrinFundamental_isSolution","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-fundamental-is-solution","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $k_1, k_2 \\in R$ and $p, s \\in \\mathbb{N}$. For every natural number $j$, the $j$-th sequence $v^{(j)}$ of the Fibonacci fundamental system is a solution of the $([k]; p; s)$-Padovan\u2013Perrin recurrence.","title":"The generalized Padovan\u2013Perrin fundamental sequences solve the recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-rho","kind":"node","lean":"MetaMathlibExt.padovanPerrinRho","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-rho","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $k_1, k_2 \\in R$ and let $p, s, n$ be natural numbers. The coefficient $\\rho(n, s + p)$ attached to $k_1, k_2, p, s$ is $0$ when $n < s + p$; when $n \\ge s + p$ it is the sum of $\\binom{a + b}{a} k_1^a k_2^b$ over all pairs of natural numbers $a, b \\le n$ with $s a + (s + p) b = n - (s + p)$. For $s > 0$ the bounds $a, b \\le n$ hold automatically.","title":"Padovan\u2013Perrin coefficients"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-rho-of-lt","kind":"node","lean":"MetaMathlibExt.padovanPerrinRho_of_lt","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-rho-of-lt","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $k_1, k_2 \\in R$ and let $p, s, n$ be natural numbers with $n < s + p$. Then the Padovan\u2013Perrin coefficient $\\rho(n, s + p)$ attached to $k_1, k_2, p, s$ equals $0$.","title":"The Padovan\u2013Perrin coefficients vanish below the order"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-rho-self","kind":"node","lean":"MetaMathlibExt.padovanPerrinRho_self","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-rho-self","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $k_1, k_2 \\in R$ and let $p, s$ be natural numbers with $s > 0$. Then the Padovan\u2013Perrin coefficient $\\rho(s + p, s + p)$ attached to $k_1, k_2, p, s$ equals $1$.","title":"Normalization of the Padovan\u2013Perrin coefficients"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-rho-add-succ","kind":"node","lean":"MetaMathlibExt.padovanPerrinRho_add_succ","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-rho-add-succ","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $k_1, k_2 \\in R$, and $p, s \\in \\mathbb{N}$ with $s > 0$, and write $\\rho(m, s+p)$ for the Padovan\u2013Perrin coefficient. For every $n \\ge 0$,","title":"Recurrence for the generalized Padovan\u2013Perrin coefficients \u03c1"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-rho-one-one-one-two-twelv","kind":"node","lean":"MetaMathlibExt.padovanPerrinRho_one_one_one_two_twelve","parent":6458,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Padovan\u2013Perrin coefficient "],["math","\\(\\rho(12, 3)\\)"],["text"," equals "],["math","\\(5\\)"]],"source":"The Padovan\u2013Perrin coefficient $\\rho(12, 3)$ equals $5$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-rho-one-one-one-two-twelv","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"fully_proved","summary":"Over the integers, with $k_1 = k_2 = 1$, $p = 1$ and $s = 2$ (the Padovan\u2013Perrin recurrence $v_n = v_{n-2} + v_{n-3}$), the coefficient $\\rho(12, 3)$ equals $5$.","title":"The Padovan\u2013Perrin coefficient $\\rho(12, 3)$ equals $5$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-eq-sum-fundamental","kind":"node","lean":"MetaMathlibExt.padovanPerrin_eq_sum_fundamental","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-eq-sum-fundamental","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $k_1, k_2 \\in R$ and $p, s \\in \\mathbb{N}$, and for $1 \\le j \\le s + p$ let $v^{(j)}$ be the solution of the $([k]; p; s)$-Padovan\u2013Perrin recurrence with $v^{(j)}(m) = 1$ if $m + 1 = j$ and $v^{(j)}(m) = 0$ otherwise, for $0 \\le m < s + p$. Then every solution $v \\colon \\mathbb{N} \\to R$ of the recurrence satisfies, for every $n \\ge 0$,","title":"Expansion in the Fibonacci fundamental system"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-shift-eq-of-window","kind":"node","lean":"MetaMathlibExt.padovanPerrin_shift_eq_of_window","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-shift-eq-of-window","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $k_1, k_2 \\in R$, let $p, s$ be natural numbers and let $v, w \\colon \\mathbb{N} \\to R$ be solutions of the Padovan\u2013Perrin recurrence with parameters $k_1, k_2, p, s$. Let $n_0, m_0, N$ be natural numbers such that $v(j + n_0) = w(j + m_0)$ for every $j$ with $N \\le j < N + s + p$. Then $v(n + n_0) = w(n + m_0)$ for every $n \\ge N$.","title":"Solutions agreeing on a window agree afterwards"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-sol-space-eq-span-fundame","kind":"node","lean":"MetaMathlibExt.padovanPerrin_solSpace_eq_span_fundamental","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-sol-space-eq-span-fundame","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $k_1, k_2 \\in R$ and let $p, s$ be natural numbers. For $1 \\le j \\le s + p$ let $v^{(j)}$ be the solution of the Padovan\u2013Perrin recurrence with parameters $k_1, k_2, p, s$ whose initial values are $v^{(j)}(n) = 1$ if $n + 1 = j$ and $v^{(j)}(n) = 0$ otherwise, for $n < s + p$. Then the submodule of $R^{\\mathbb{N}}$ formed by all solutions of this recurrence is the submodule\u2026","title":"The Fibonacci fundamental system spans the solutions"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-fundamental-eq-shift","kind":"node","lean":"MetaMathlibExt.padovanPerrinFundamental_eq_shift","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-fundamental-eq-shift","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, $k_1, k_2 \\in R$, and let $p, s, j$ be natural numbers with $p \\ge 1$ and $p + 1 \\le j \\le s + p - 1$. Write $v^{(\\ell)}$ for the $\\ell$-th sequence of the Fibonacci fundamental system of the $([k]; p; s)$-Padovan\u2013Perrin recurrence. Then $v^{(j)}_n = v^{(s+p)}_{n+s+p-j}$ for every $n \\ge 0$, and $v^{(j)}_n = k_1 v^{(s+p)}_{n-j+p} + k_2 v^{(s+p)}_{n-j}$ for every $n \\ge j$.","title":"Shift property of the generalized Padovan\u2013Perrin fundamental system"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-companion-pow-apply","kind":"node","lean":"MetaMathlibExt.padovanPerrin_companion_pow_apply","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-companion-pow-apply","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $k_1, k_2 \\in R$, let $p, s$ be positive integers, and write $r = s + p$. Let $A$ be the $r \\times r$ matrix over $R$, with rows and columns indexed by $0, \\dots, r - 1$, whose row $0$ has $k_1$ in column $s - 1$, $k_2$ in column $r - 1$ and $0$ elsewhere, and whose entry in row $a \\ge 1$ and column $b$ is $1$ if $a = b + 1$ and $0$ otherwise. For $1 \\le \\ell \\le r$ let\u2026","title":"Powers of the Padovan\u2013Perrin companion matrix"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-fundamental-mul-sum","kind":"node","lean":"MetaMathlibExt.padovanPerrinFundamental_mul_sum","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-fundamental-mul-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, $k_1, k_2 \\in R$, and $p, s \\ge 1$. Write $r = s + p$ and $v^{(\\ell)}$ for the $\\ell$-th sequence of the Fibonacci fundamental system of the $([k]; p; s)$-Padovan\u2013Perrin recurrence. Then for all $m, n \\ge 0$ and all $1 \\le i, j \\le r$,","title":"Product identity for the generalized Padovan\u2013Perrin fundamental system"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-eq-sum-rho","kind":"node","lean":"MetaMathlibExt.padovanPerrin_eq_sum_rho","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-eq-sum-rho","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $k_1, k_2 \\in R$, let $p, s$ be positive integers, and let $v \\colon \\mathbb{N} \\to R$ satisfy $v(n + s + p) = k_1 v(n + p) + k_2 v(n)$ for every $n$. For $0 \\le m \\le s + p - 1$ put $A_m = k_2 v(m) + k_1 v(m + p)$ if $m < s$ and $A_m = k_2 v(m)$ if $m \\ge s$. Let $\\rho(N, s + p)$ be the Padovan\u2013Perrin coefficient: $0$ for $N < s + p$, and for $N \\ge s + p$ the sum of\u2026","title":"Combinatorial expression of the Padovan\u2013Perrin sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-fundamental-last-eq-rho","kind":"node","lean":"MetaMathlibExt.padovanPerrinFundamental_last_eq_rho","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-fundamental-last-eq-rho","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, $k_1, k_2 \\in R$, and $p, s \\ge 1$. Write $v^{(s+p)}$ for the $(s+p)$-th sequence of the Fibonacci fundamental system of the $([k]; p; s)$-Padovan\u2013Perrin recurrence and $\\rho(m, s+p)$ for the Padovan\u2013Perrin coefficient. Then $v^{(s+p)}_n = \\rho(n + 1, s + p)$ for every $n \\ge s + p$.","title":"The last generalized Padovan\u2013Perrin fundamental sequence is a shifted \u03c1"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-fundamental-eq-rho","kind":"node","lean":"MetaMathlibExt.padovanPerrinFundamental_eq_rho","parent":6458,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fundamental Padovan\u2013Perrin solutions as shifted coefficients "],["math","\\(\\rho\\)"]],"source":"Fundamental Padovan\u2013Perrin solutions as shifted coefficients $\\rho$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-fundamental-eq-rho","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $k_1, k_2 \\in R$, let $p \\ge 1$ and $s$ be natural numbers, and let $j$ satisfy $p + 1 \\le j \\le s + p - 1$. For $1 \\le \\ell \\le s + p$ let $v^{(\\ell)}$ be the solution of the $([k]; p; s)$-Padovan\u2013Perrin recurrence with $v^{(\\ell)}(m) = 1$ if $m + 1 = \\ell$ and $v^{(\\ell)}(m) = 0$ otherwise, for $0 \\le m < s + p$. Let $\\rho(N, s + p)$ be the Padovan\u2013Perrin coefficient: $0$ for\u2026","title":"Fundamental Padovan\u2013Perrin solutions as shifted coefficients $\\rho$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-char-poly-separable-of-p","kind":"node","lean":"MetaMathlibExt.padovanPerrin_charPoly_separable_of_p_eq_one","parent":6458,"reading_title":{"format":"inline-parts-v1","parts":[["text","Simple roots of the Padovan\u2013Perrin polynomial for "],["math","\\(p = 1\\)"]],"source":"Simple roots of the Padovan\u2013Perrin polynomial for $p = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-char-poly-separable-of-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $k_1, k_2, s$ be positive integers with $\\left(-\\frac{k_2 (s + 1)}{k_1 s}\\right)^s \\neq \\frac{k_1}{s + 1}$. Then the polynomial $P(z) = z^{s+1} - k_1 z - k_2$ is separable over $\\mathbb{C}$, that is, it has only simple roots.","title":"Simple roots of the Padovan\u2013Perrin polynomial for $p = 1$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-rho-eq-sum-roots","kind":"node","lean":"MetaMathlibExt.padovanPerrinRho_eq_sum_roots","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-rho-eq-sum-roots","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $k_1, k_2$ be complex numbers with $k_2 \\neq 0$, let $p, s \\ge 1$, and let $P(z) = z^{s+p} - k_1 z^p - k_2$. If $P$ is separable, that is, has $s + p$ distinct roots $\\lambda_1, \\dots, \\lambda_{s+p}$ in $\\mathbb{C}$, then for every $n \\ge s + p$ the Padovan\u2013Perrin coefficient satisfies","title":"Root formula for the generalized Padovan\u2013Perrin coefficients \u03c1"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/padovan-perrin-eq-sum-roots","kind":"node","lean":"MetaMathlibExt.padovanPerrin_eq_sum_roots","parent":6458,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#padovan-perrin-eq-sum-roots","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $k_1, k_2, p, s$ be natural numbers with $k_1 > 0$, $p \\ge 2$, $p < s$ and $k_1^2 < k_2$, let $P(z) = z^{s+p} - k_1 z^p - k_2$, and let $v \\colon \\mathbb{N} \\to \\mathbb{C}$ satisfy $v(n + s + p) = k_1 v(n + p) + k_2 v(n)$ for every $n$. For $0 \\le m \\le s + p - 1$ put $A_m = k_2 v(m) + k_1 v(m + p)$ if $m < s$ and $A_m = k_2 v(m)$ if $m \\ge s$. Then for every $n \\ge s + p$,","title":"Analytic expression of the Padovan\u2013Perrin sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/k-padovan-perrin-eq-sum-rho","kind":"node","lean":"MetaMathlibExt.kPadovanPerrin_eq_sum_rho","parent":6458,"reading_title":{"format":"inline-parts-v1","parts":[["text","Combinatorial formula for the "],["math","\\(k\\)"],["text","-Padovan\u2013Perrin numbers"]],"source":"Combinatorial formula for the $k$-Padovan\u2013Perrin numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#k-padovan-perrin-eq-sum-rho","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $k \\in R$, and let $v \\colon \\mathbb{N} \\to R$ satisfy $v(n + 3) = v(n + 1) + k\\,v(n)$ for every $n$, that is, $v$ solves the Padovan\u2013Perrin recurrence with $k_1 = 1$, $k_2 = k$, $p = 1$, $s = 2$. Let $\\rho(m) = \\rho(m, 3)$ be the Padovan\u2013Perrin coefficient for these parameters: $\\rho(m) = 0$ for $m < 3$, and for $m \\ge 3$ it is the sum of $\\binom{a+b}{a} k^b$ over all\u2026","title":"Combinatorial formula for the $k$-Padovan\u2013Perrin numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/k-padovan-perrin-fundamental-eq-rho","kind":"node","lean":"MetaMathlibExt.kPadovanPerrinFundamental_eq_rho","parent":6458,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fundamental "],["math","\\(k\\)"],["text","-Padovan\u2013Perrin solutions as coefficients "],["math","\\(\\rho\\)"]],"source":"Fundamental $k$-Padovan\u2013Perrin solutions as coefficients $\\rho$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#k-padovan-perrin-fundamental-eq-rho","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring and $k \\in R$, and take the Padovan\u2013Perrin recurrence with $k_1 = 1$, $k_2 = k$, $p = 1$, $s = 2$, that is, $v(n + 3) = v(n + 1) + k\\,v(n)$. Let $v^{(2)}$ and $v^{(3)}$ be its solutions with initial values $(v^{(2)}(0), v^{(2)}(1), v^{(2)}(2)) = (0, 1, 0)$ and $(v^{(3)}(0), v^{(3)}(1), v^{(3)}(2)) = (0, 0, 1)$. Let $\\rho(m) = \\rho(m, 3)$ be the Padovan\u2013Perrin coefficient for these\u2026","title":"Fundamental $k$-Padovan\u2013Perrin solutions as coefficients $\\rho$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/k-padovan-perrin-eq-sum-roots","kind":"node","lean":"MetaMathlibExt.kPadovanPerrin_eq_sum_roots","parent":6458,"reading_title":{"format":"inline-parts-v1","parts":[["text","Root formula for the "],["math","\\(k\\)"],["text","-Padovan\u2013Perrin numbers"]],"source":"Root formula for the $k$-Padovan\u2013Perrin numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences/index.html#k-padovan-perrin-eq-sum-roots","scope":"NumberTheory/linear-recurrences/higher-order-sequences/generalized-padovan-perrin-recurrences","status":"can_prove","summary":"Let $k \\ge 2$ be an integer, let $P(z) = z^3 - z - k$, and let $v \\colon \\mathbb{N} \\to \\mathbb{C}$ satisfy $v(n + 3) = v(n + 1) + k\\,v(n)$ for every $n$, that is, $v$ solves the Padovan\u2013Perrin recurrence with $k_1 = 1$, $k_2 = k$, $p = 1$, $s = 2$. Put $A_0 = k\\,v(0) + v(1)$, $A_1 = k\\,v(1) + v(2)$ and $A_2 = k\\,v(2)$. Then for every $n \\ge 3$,","title":"Root formula for the $k$-Padovan\u2013Perrin numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","title":"k-generalized Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/octanacci-window","kind":"node","lean":"MetaMathlibExt.octanacciWindow","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#octanacci-window","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"The octanacci window is the sequence $(w_n)_{n \\ge 0}$ of $8$-tuples of natural numbers with $w_0 = (1, 1, 2, 4, 8, 16, 32, 64)$ and, if $w_n = (a_1, \\dots, a_8)$, then $w_{n+1} = (a_2, \\dots, a_8, a_1 + a_2 + \\dots + a_8)$. The octanacci numbers $f_n$ are the first entries of these tuples, so $w_n = (f_n, f_{n+1}, \\dots, f_{n+7})$.","title":"Octanacci window"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/octanacci-number","kind":"node","lean":"MetaMathlibExt.octanacciNumber","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#octanacci-number","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"The octanacci sequence $(f_n)_{n \\ge 0}$ of natural numbers is defined by $f_0 = f_1 = 1$, $f_n = 2^{n-1}$ for $2 \\le n \\le 7$, and $f_n = f_{n-1} + f_{n-2} + \\dots + f_{n-8}$ for $n \\ge 8$. It begins $1, 1, 2, 4, 8, 16, 32, 64, 128, 255, \\dots$, and its generating function is $1/(1 - t - t^2 - \\dots - t^8)$.","title":"Octanacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/n-step-fibonacci","kind":"node","lean":"MetaMathlibExt.nStepFibonacci","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#n-step-fibonacci","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"For a natural number $N$, the $N$-step Fibonacci numbers $F^{(N)}_n$ ($n \\ge 0$), in the convention of Noe and Post, are defined by $F^{(N)}_0 = 0$, $F^{(N)}_1 = 1$ and, for $n \\ge 2$, $F^{(N)}_n = \\sum_{i=1}^{N} F^{(N)}_{n-i}$, where a term whose index $n - i$ is negative counts as $0$. In particular $F^{(0)}_n = 0$ for $n \\ge 2$.","title":"N-step Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/n-step-fibonacci-add-two","kind":"node","lean":"MetaMathlibExt.nStepFibonacci_add_two","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#n-step-fibonacci-add-two","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"For all natural numbers $N$ and $n$, the $N$-step Fibonacci numbers $F^{(N)}$ satisfy $F^{(N)}_{n+2} = \\sum_{i=0}^{N-1} F^{(N)}_{\\max(n+1-i,\\,0)}$.","title":"Recurrence for the N-step Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/n-step-fibonacci-zero","kind":"node","lean":"MetaMathlibExt.nStepFibonacci_zero","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#n-step-fibonacci-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"For every natural number $N$, the $N$-step Fibonacci number $F^{(N)}_0$ equals $0$.","title":"Zeroth N-step Fibonacci number"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/n-step-fibonacci-one","kind":"node","lean":"MetaMathlibExt.nStepFibonacci_one","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#n-step-fibonacci-one","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"For every natural number $N$, the $N$-step Fibonacci number $F^{(N)}_1$ equals $1$.","title":"First N-step Fibonacci number"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/n-step-fibonacci-three-five","kind":"node","lean":"MetaMathlibExt.nStepFibonacci_three_five","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#n-step-fibonacci-three-five","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"The $3$-step Fibonacci number $F^{(3)}_5$ equals $7$.","title":"The tribonacci number with index 5"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/dominant-root-bounds-k-generalized-fibon","kind":"node","lean":"MetaMathlibExt.dominant_root_bounds_k_generalized_fibonacci","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#dominant-root-bounds-k-generalized-fibon","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $k$ be a natural number and let $\\alpha > 1$ be a real number with $\\alpha^k = \\sum_{i=0}^{k-1} \\alpha^i$. Then $2 - 1/k < \\alpha < 2$, and if $k \\ge 4$, moreover $2 - 1/(3k) < \\alpha < 2$.","title":"Bounds on the dominant root of the k-generalized Fibonacci polynomial"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/dresden-du-simplified-binet-of-initial-w","kind":"node","lean":"MetaMathlibExt.dresden_du_simplified_binet_of_initial_window","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#dresden-du-simplified-binet-of-initial-w","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $k \\ge 2$ and let $F : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $F(n) = 0$ for all integers $n$ with $2 - k \\le n < 1$, $F(1) = 1$, and $F(n) = \\sum_{i=0}^{k-1} F(n - 1 - i)$ for all integers $n > 1$. Let $\\alpha$ be a positive real number with $\\alpha^k = \\sum_{i=0}^{k-1} \\alpha^i$, and put $c = (\\alpha - 1)/(2 + (k + 1)(\\alpha - 2))$. Then for every integer $n \\ge 2 - k$,\u2026","title":"Simplified Binet formula for k-generalized Fibonacci numbers from an initial window"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/dresden-du-simplified-binet","kind":"node","lean":"MetaMathlibExt.dresden_du_simplified_binet","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#dresden-du-simplified-binet","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $k \\ge 2$ and let $F : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $F(n) = 0$ for all integers $n < 1$, $F(1) = 1$, and $F(n) = \\sum_{i=0}^{k-1} F(n - 1 - i)$ for all integers $n > 1$. Let $\\alpha$ be a positive real number with $\\alpha^k = \\sum_{i=0}^{k-1} \\alpha^i$, and suppose that every positive real $\\beta$ with $\\beta^k = \\sum_{i=0}^{k-1} \\beta^i$ equals $\\alpha$. Put $c = (\\alpha - 1)/(2 + (k + 1)(\\alpha - 2))$.\u2026","title":"Dresden\u2013Du simplified Binet formula"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-pow-convolution-mstep-fibonacci","kind":"node","lean":"MetaMathlibExt.two_pow_convolution_mstep_fibonacci","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power-weighted convolution of Fibonacci "],["math","\\(m\\)"],["text","-step numbers"]],"source":"Power-weighted convolution of Fibonacci $m$-step numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-pow-convolution-mstep-fibonacci","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $m \\ge 1$ and let $F \\colon \\mathbb{N} \\to \\mathbb{N}$ satisfy $F(0) = 0$, $F(1) = 1$ and $F(k) = \\sum_{j=1}^{m} F(k - j)$ for every $k \\ge 2$, where a term $F(k - j)$ with $j > k$ is read as $F(0) = 0$. Then for every natural number $n$,","title":"Power-weighted convolution of Fibonacci $m$-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/binet-general","kind":"node","lean":"MetaMathlibExt.binet_general","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#binet-general","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $r \\ge 2$, let $A_1, \\dots, A_r$ and $c_1, \\dots, c_r$ be nonnegative real numbers with $A_r > 0$ and $c_j > 0$ for some $j$, and suppose that the indices $j$ with $A_j > 0$ have greatest common divisor $1$. Let $a_0, a_1, a_2, \\dots$ be the real sequence with $a_n = c_{n+1}$ for $0 \\le n < r$ and $a_n = \\sum_{j=1}^{r} A_j a_{n-j}$ for $n \\ge r$, and let $p(x) = x^r - \\sum_{j=1}^{r} A_j x^{r-j}$ be its\u2026","title":"Binet asymptotics for weighted generalized Fibonacci sequences, positive-root form"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/binet","kind":"node","lean":"MetaMathlibExt.binet","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#binet","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $r \\ge 2$, let $A_1, \\dots, A_r$ and $c_1, \\dots, c_r$ be nonnegative real numbers with $A_r > 0$ and $c_j > 0$ for some $j$, and suppose that the indices $j$ with $A_j > 0$ have greatest common divisor $1$. Let $a_0, a_1, a_2, \\dots$ be the real sequence with $a_n = c_{n+1}$ for $0 \\le n < r$ and $a_n = \\sum_{j=1}^{r} A_j a_{n-j}$ for $n \\ge r$, and let $p(x) = x^r - \\sum_{j=1}^{r} A_j x^{r-j}$ be its\u2026","title":"Binet asymptotics for weighted generalized Fibonacci sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-step-seq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.fibStepSeq","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial segments of the Fibonacci "],["math","\\(m\\)"],["text","-step sequence"]],"source":"Initial segments of the Fibonacci $m$-step sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-step-seq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"For natural numbers $m$ and $n$, this is the list of integers $[F_0, F_1, \\dots, F_n]$, built recursively: the list for $n = 0$ is $[0]$, the list for $n = 1$ is $[0, 1]$, and for $n \\ge 2$ the list for $n$ is the list for $n - 1$ followed by $F_n = \\sum_{j=1}^{m} F_{n-j}$, where $F_i = 0$ for $i < 0$.","title":"Initial segments of the Fibonacci $m$-step sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-step","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.fibStep","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci "],["math","\\(m\\)"],["text","-step numbers"]],"source":"Fibonacci $m$-step numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-step","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"For natural numbers $m$ and $n$, the integer $F^{(m)}_n$ is defined by $F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$ and $F^{(m)}_n = \\sum_{j=1}^{m} F^{(m)}_{n-j}$ for $n \\ge 2$, where $F^{(m)}_i = 0$ for $i < 0$.","title":"Fibonacci $m$-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/jacobsthal-seq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.jacobsthalSeq","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#jacobsthal-seq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"The Jacobsthal numbers are the integers $J_n$ defined by $J_0 = 0$, $J_1 = 1$ and $J_{n+2} = J_{n+1} + 2 J_n$ for all $n \\ge 0$.","title":"Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-step-gf","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.fibStepGF","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the Fibonacci "],["math","\\(m\\)"],["text","-step numbers"]],"source":"Generating function of the Fibonacci $m$-step numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-step-gf","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"For a natural number $m$, $F^{(m)}(X)$ is the formal power series $\\sum_{n \\ge 0} F^{(m)}_n X^n$ with rational coefficients, where $F^{(m)}_n$ is the $n$-th Fibonacci $m$-step number.","title":"Generating function of the Fibonacci $m$-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"Given integer sequences $a_0, \\dots, a_{r-1}$ and a natural number $n$, the sum over the compositions $K(r, n)$ of $n$ into $r$ nonnegative parts is the sum, over all $(k_0, \\dots, k_{r-1}) \\in \\mathbb{N}^r$ with $k_0 + \\dots + k_{r-1} = n$, of $\\prod_{i=0}^{r-1} a_i(k_i)$.","title":"Sums over compositions"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-step-mul-succ","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fibStep_mul_succ","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncated convolution of the "],["math","\\(m\\)"],["text","-step and "],["math","\\((m+1)\\)"],["text","-step Fibonacci numbers"]],"source":"Truncated convolution of the $m$-step and $(m+1)$-step Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-step-mul-succ","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(m)}_n$ for the Fibonacci $m$-step numbers ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For all natural numbers $m \\ge 1$ and $n \\ge m$,","title":"Truncated convolution of the $m$-step and $(m+1)$-step Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-mul-trib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fib_mul_trib","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-mul-trib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F_n$ and $T_n$ for the Fibonacci and tribonacci numbers, that is, the Fibonacci $m$-step numbers $F^{(m)}_n$ for $m = 2, 3$ ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For every natural number $n$,","title":"Convolution of the Fibonacci and tribonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-trib-mul-tetra","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_trib_mul_tetra","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-trib-mul-tetra","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $T_j$ and $Q_j$ be the Fibonacci $3$-step and $4$-step numbers, where the Fibonacci $k$-step numbers start $0, 1$ and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For every natural number $n$,","title":"Convolution of the Fibonacci 3-step and 4-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-tetra-mul-penta","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_tetra_mul_penta","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-tetra-mul-penta","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $Q_j$ and $P_j$ be the Fibonacci $4$-step and $5$-step numbers, where the Fibonacci $k$-step numbers start $0, 1$ and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For every natural number $n$,","title":"Convolution of the Fibonacci 4-step and 5-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-sum-fib-step-mul","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_sum_fibStep_mul","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterated convolution of Fibonacci "],["math","\\(m\\)"],["text","-step and "],["math","\\((m+p)\\)"],["text","-step numbers"]],"source":"Iterated convolution of Fibonacci $m$-step and $(m+p)$-step numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-sum-fib-step-mul","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $F^{(k)}_j$ be the $j$-th Fibonacci $k$-step number: the sequence starts $F^{(k)}_0 = 0$, $F^{(k)}_1 = 1$, and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For all natural numbers $m$, $p$ and $n$ with $m \\ge 1$ and $n \\ge m$,","title":"Iterated convolution of Fibonacci $m$-step and $(m+p)$-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-step-mul-add","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fibStep_mul_add","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution of the "],["math","\\(m\\)"],["text","-step Fibonacci numbers with consecutive sums of the "],["math","\\((m+2)\\)"],["text","-step ones"]],"source":"Convolution of the $m$-step Fibonacci numbers with consecutive sums of the $(m+2)$-step ones"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-step-mul-add","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Convolution of the $m$-step Fibonacci numbers with consecutive sums of the $(m+2)$-step ones. Write $F^{(m)}_n$ for the Fibonacci $m$-step numbers ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For all natural numbers $m \\ge 1$ and $n \\ge m$,","title":"Convolution of the $m$-step Fibonacci numbers with consecutive sums of the $(m+2)$-step ones"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-step-sub-one-eq-sum-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.fibStep_sub_one_eq_sum_sum","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-step-sub-one-eq-sum-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"For natural numbers $p$ and $n$ with $1 \\le p \\le n$, the Fibonacci $(p+1)$-step numbers $F^{(p+1)}_j$ satisfy","title":"Double partial sums of Fibonacci step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/mul-sum-fib-step","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.mul_sum_fibStep","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sums of the Fibonacci "],["math","\\(m\\)"],["text","-step numbers"]],"source":"Partial sums of the Fibonacci $m$-step numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#mul-sum-fib-step","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"For natural numbers $m \\ge 2$ and $n$, the Fibonacci $m$-step numbers $F^{(m)}_k$ satisfy","title":"Partial sums of the Fibonacci $m$-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-mul-tetra","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fib_mul_tetra","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-mul-tetra","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F_n$ and $Q_n$ for the Fibonacci and tetranacci numbers, that is, the Fibonacci $m$-step numbers $F^{(m)}_n$ for $m = 2, 4$ ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For every natural number $n$,","title":"Convolution of the Fibonacci and tetranacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-sum-fib-mul-penta","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_sum_fib_mul_penta","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-sum-fib-mul-penta","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F_n$ for the Fibonacci numbers and $P_n$ for the pentanacci numbers (the Fibonacci $m$-step numbers $F^{(m)}_n$ with $m = 2$ and $m = 5$). For every integer $n \\ge 0$, $2 \\sum_{j=0}^{n} F_j P_{n-j} = P_{n+2} + P_{n-1} - F_{n+2}$, where $P_{n-1}$ is read as $P_0 = 0$ when $n = 0$.","title":"Convolution of the Fibonacci and pentanacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-sum-penta-mul-trib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_sum_penta_mul_trib","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-sum-penta-mul-trib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $P_n$ for the pentanacci numbers and $T_n$ for the tribonacci numbers (the Fibonacci $m$-step numbers $F^{(m)}_n$ with $m = 5$ and $m = 3$). For every integer $n \\ge 0$, $2 \\sum_{j=0}^{n} P_j T_{n-j} = P_{n+3} + P_{n+1} + P_{n-1} - T_{n+3} - T_{n+1}$, where $P_{n-1}$ is read as $P_0 = 0$ when $n = 0$.","title":"Convolution of the pentanacci and tribonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-step-sub-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fibStep_sub_eq","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exchange identity for convolutions of "],["math","\\((m-2)\\)"],["text","-, "],["math","\\((m-1)\\)"],["text","- and "],["math","\\(m\\)"],["text","-step Fibonacci numbers"]],"source":"Exchange identity for convolutions of $(m-2)$-, $(m-1)$- and $m$-step Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-step-sub-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(m)}_n$ for the Fibonacci $m$-step numbers ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For all natural numbers $m \\ge 3$ and $n \\ge 1$,","title":"Exchange identity for convolutions of $(m-2)$-, $(m-1)$- and $m$-step Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/trib-sub-fib-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.trib_sub_fib_eq_sum","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#trib-sub-fib-eq-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $F_j$ and $T_j$ be the Fibonacci $2$-step and $3$-step numbers, where the Fibonacci $k$-step numbers start $0, 1$ and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For every natural number $n \\ge 1$,","title":"Difference of the Fibonacci 3-step and 2-step numbers as a convolution"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-mul-sub-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fib_mul_sub_eq","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-mul-sub-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F_n$, $T_n$ and $Q_n$ for the Fibonacci, tribonacci and tetranacci numbers, that is, the Fibonacci $m$-step numbers $F^{(m)}_n$ for $m = 2, 3, 4$ ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For every natural number $n \\ge 1$,","title":"Convolution of the Fibonacci numbers with tribonacci minus tetranacci"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-trib-mul-sub-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_trib_mul_sub_eq","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-trib-mul-sub-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $T_j$, $Q_j$ and $P_j$ be the Fibonacci $3$-, $4$- and $5$-step numbers, where the Fibonacci $k$-step numbers start $0, 1$ and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For every natural number $n \\ge 1$,","title":"Convolution of the Fibonacci 3-step numbers with differences of 4- and 5-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-step-pred-mul","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fibStep_pred_mul","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution of the "],["math","\\((m-1)\\)"],["text","-step and "],["math","\\(m\\)"],["text","-step Fibonacci numbers against powers of two"]],"source":"Convolution of the $(m-1)$-step and $m$-step Fibonacci numbers against powers of two"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-step-pred-mul","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(m)}_n$ for the Fibonacci $m$-step numbers ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For all natural numbers $m \\ge 2$ and $n \\ge 2$,","title":"Convolution of the $(m-1)$-step and $m$-step Fibonacci numbers against powers of two"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-trib-mul-tetra-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_trib_mul_tetra_eq","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-trib-mul-tetra-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $T_j$ and $Q_j$ be the Fibonacci $3$-step and $4$-step numbers, where the Fibonacci $k$-step numbers start $0, 1$ and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For every natural number $n$,","title":"Convolution of the Fibonacci 3-step and 4-step numbers as a sum weighted by powers of 2"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-neg-one-pow-sub-even","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_neg_one_pow_sub_even","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sum of differences of "],["math","\\(2m\\)"],["text","-step and "],["math","\\((2m-2)\\)"],["text","-step Fibonacci numbers"]],"source":"Alternating sum of differences of $2m$-step and $(2m-2)$-step Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-neg-one-pow-sub-even","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(m)}_n$ for the Fibonacci $m$-step numbers ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For all natural numbers $m \\ge 2$ and $n \\ge 2m - 1$,","title":"Alternating sum of differences of $2m$-step and $(2m-2)$-step Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-neg-one-pow-sub-odd","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_neg_one_pow_sub_odd","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-neg-one-pow-sub-odd","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $F^{(k)}_j$ be the $j$-th Fibonacci $k$-step number: the sequence starts $F^{(k)}_0 = 0$, $F^{(k)}_1 = 1$, and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For all natural numbers $m$ and $n$ with $m \\ge 1$ and $n \\ge 2m$,","title":"Alternating sums of differences of odd-order Fibonacci step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-sum-neg-one-pow-trib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_sum_neg_one_pow_trib","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-sum-neg-one-pow-trib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $T_n$ for the tribonacci numbers (the Fibonacci $3$-step numbers $F^{(3)}_n$). For every integer $n \\ge 0$, $2 \\sum_{j=0}^{n} (-1)^j T_j = (-1)^n (T_{n+1} - T_{n-1}) - 1$, where $T_{n-1}$ is read as $T_0 = 0$ when $n = 0$.","title":"Alternating sum of the tribonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-sum-penta-mul-trib-2","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_sum_penta_mul_trib'","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-sum-penta-mul-trib-2","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $P_n$ for the pentanacci numbers and $T_n$ for the tribonacci numbers (the Fibonacci $m$-step numbers $F^{(m)}_n$ with $m = 5$ and $m = 3$). For every integer $n \\ge 0$, $2 \\sum_{j=0}^{n} P_j T_{n-j} = -P_{n+7} + 2P_{n+6} - P_{n+5} + 2P_{n+4} + P_{n+3} - T_{n+4} + T_{n+2}$.","title":"Convolution of the pentanacci and tribonacci numbers, second closed form"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-step-mul-sub-two","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fibStep_mul_sub_two","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution of the "],["math","\\(m\\)"],["text","-step and "],["math","\\((m-2)\\)"],["text","-step Fibonacci numbers via Jacobsthal numbers"]],"source":"Convolution of the $m$-step and $(m-2)$-step Fibonacci numbers via Jacobsthal numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-step-mul-sub-two","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(m)}_n$ for the Fibonacci $m$-step numbers ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$), and $J_n$ for the Jacobsthal numbers ($J_0 = 0$, $J_1 = 1$, $J_{n+2} = J_{n+1} + 2 J_n$). For all natural numbers $m \\ge 3$ and $n \\ge 2$,","title":"Convolution of the $m$-step and $(m-2)$-step Fibonacci numbers via Jacobsthal numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-sum-jacobsthal-mul-trib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_sum_jacobsthal_mul_trib","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-sum-jacobsthal-mul-trib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $J_n$ for the Jacobsthal numbers ($J_0 = 0$, $J_1 = 1$, $J_{n+2} = J_{n+1} + 2J_n$) and $T_n$ for the tribonacci numbers (the Fibonacci $3$-step numbers $F^{(3)}_n$). For every integer $n \\ge 0$, $2 \\sum_{j=0}^{n} J_j T_{n-j} = 2J_{n+1} + J_{n+2} - T_{n+3} - T_{n+1}$.","title":"Convolution of the Jacobsthal and tribonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum-pell-fib-step","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum_pell_fibStep","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum-pell-fib-step","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $P_k$ for the Pell numbers ($P_0 = 0$, $P_1 = 1$, $P_{k+2} = 2P_{k+1} + P_k$), $F^{(s)}_k$ for the Fibonacci $s$-step numbers, and $\\sum_{K(3,N)}$ for a sum over all $(k_1, k_2, k_3) \\in \\mathbb{N}^3$ with $k_1 + k_2 + k_3 = N$. For all natural numbers $m \\ge 2$ and $n \\ge 1$,","title":"Triple convolution of Pell numbers with two consecutive Fibonacci step sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-pell-mul-fib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_pell_mul_fib","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-pell-mul-fib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $\\mathcal{P}_j$ be the Pell numbers ($\\mathcal{P}_0 = 0$, $\\mathcal{P}_1 = 1$, $\\mathcal{P}_{j+2} = 2\\mathcal{P}_{j+1} + \\mathcal{P}_j$) and $F_j$ the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{j+2} = F_{j+1} + F_j$), that is, the Fibonacci $2$-step numbers. For every natural number $n$,","title":"Convolution of the Pell numbers with the Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum-pell-fib-trib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum_pell_fib_trib","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum-pell-fib-trib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $P_k$ for the Pell numbers ($P_0 = 0$, $P_1 = 1$, $P_{k+2} = 2P_{k+1} + P_k$), $F_k$ and $T_k$ for the Fibonacci $2$- and $3$-step numbers, and $\\sum_{K(3,n)}$ for a sum over all $(k_1, k_2, k_3) \\in \\mathbb{N}^3$ with $k_1 + k_2 + k_3 = n$. For every natural number $n$,","title":"Pell\u2013Fibonacci\u2013tribonacci triple sum via a Pell\u2013tribonacci convolution"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-comp-prod-sum-pell-fib-trib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_compProdSum_pell_fib_trib","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-comp-prod-sum-pell-fib-trib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $\\mathcal{P}_n$ for the Pell numbers, $F_n$ for the Fibonacci numbers and $T_n$ for the tribonacci numbers (the Fibonacci $m$-step numbers $F^{(m)}_n$ with $m = 2$ and $m = 3$). For every integer $n \\ge 0$, $2 \\sum \\mathcal{P}_{k_1} F_{k_2} T_{k_3} = \\mathcal{P}_{n+1} - T_{n+3} - T_{n+2} + 2F_{n+2}$, where the sum runs over all triples $(k_1, k_2, k_3)$ of nonnegative integers with $k_1 + k_2 + k_3 = n$.","title":"Closed form for the Pell\u2013Fibonacci\u2013tribonacci triple convolution"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-step-gf-triple-identity","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.fibStepGF_triple_identity","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","A product identity for Fibonacci step generating functions of orders "],["math","\\(m\\)"],["text",", "],["math","\\(m+p\\)"],["text",", "],["math","\\(m+p+q\\)"]],"source":"A product identity for Fibonacci step generating functions of orders $m$, $m+p$, $m+p+q$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-step-gf-triple-identity","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"A product identity for Fibonacci step generating functions of orders $m$, $m+p$, $m+p+q$. Let $m, p, q \\ge 1$ be natural numbers. For each $k$, let $F^{(k)} = \\sum_{n \\ge 0} F^{(k)}_n X^n$ in $\\mathbb{Q}[[X]]$ be the ordinary generating function of the Fibonacci $k$-step numbers $F^{(k)}_n$. Then","title":"A product identity for Fibonacci step generating functions of orders $m$, $m+p$, $m+p+q$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-step-gf-triple-consecutive","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.fibStepGF_triple_consecutive","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-step-gf-triple-consecutive","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $m \\ge 1$ be a natural number. For each $k$, let $F^{(k)} = \\sum_{n \\ge 0} F^{(k)}_n X^n$ in $\\mathbb{Q}[[X]]$ be the ordinary generating function of the Fibonacci $k$-step numbers $F^{(k)}_n$. Then","title":"A product identity for three consecutive Fibonacci step generating functions"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-step-gf-mixed-triple","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.fibStepGF_mixed_triple","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-step-gf-mixed-triple","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"For a natural number $s$ let $F^{(s)}(X) = \\sum_{n \\ge 0} F^{(s)}_n X^n$ be the ordinary generating function, with rational coefficients, of the Fibonacci $s$-step numbers $F^{(s)}_n$. For all natural numbers $m \\ge 1$ and $p \\ge 1$, the formal power series over $\\mathbb{Q}$ satisfy","title":"Generating-function form of the triple convolution with two consecutive step sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum-fib-step-triple","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum_fibStep_triple","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum-fib-step-triple","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(s)}_k$ for the Fibonacci $s$-step numbers, and write $\\sum_{K(3,N)}$ for a sum over all $(k_1, k_2, k_3) \\in \\mathbb{N}^3$ with $k_1 + k_2 + k_3 = N$. For all natural numbers $m \\ge 1$, $p \\ge 1$ and $n \\ge m$,","title":"Triple convolution of two consecutive Fibonacci step sequences with a third"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum-fib-trib-tetra","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum_fib_trib_tetra","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum-fib-trib-tetra","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F_k$, $T_k$ and $Q_k$ for the Fibonacci $2$-, $3$- and $4$-step numbers, and write $\\sum_{K(3,n)}$ for a sum over all $(k_1, k_2, k_3) \\in \\mathbb{N}^3$ with $k_1 + k_2 + k_3 = n$. For every natural number $n$,","title":"Closed form of the Fibonacci\u2013tribonacci\u2013tetranacci triple sum"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-comp-prod-sum-trib-tetra-penta","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_compProdSum_trib_tetra_penta","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-comp-prod-sum-trib-tetra-penta","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $T_n$, $Q_n$ and $P_n$ for the Fibonacci $m$-step numbers $F^{(m)}_n$ with $m = 3, 4, 5$ respectively. For every integer $n \\ge 7$, $2 \\sum T_{k_1} Q_{k_2} P_{k_3} = P_n + P_{n-2} + P_{n-4} + T_n - T_{n-2} - 2Q_n$, where the sum runs over all triples $(k_1, k_2, k_3)$ of nonnegative integers with $k_1 + k_2 + k_3 = n - 7$.","title":"Triple convolution of the tribonacci, tetranacci and pentanacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-comp-prod-sum-fib-tetra-penta","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_compProdSum_fib_tetra_penta","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-comp-prod-sum-fib-tetra-penta","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $F_j$, $Q_j$ and $P_j$ be the Fibonacci $2$-, $4$- and $5$-step numbers, where the Fibonacci $k$-step numbers start $0, 1$ and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For every natural number $n \\ge 5$,","title":"Triple convolution of the Fibonacci 2-, 4- and 5-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-comp-prod-sum-fib-trib-penta","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_compProdSum_fib_trib_penta","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-comp-prod-sum-fib-trib-penta","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $F_j$, $T_j$ and $P_j$ be the Fibonacci $2$-, $3$- and $5$-step numbers, where the Fibonacci $k$-step numbers start $0, 1$ and each later term is the sum of the $k$ terms before it (terms of negative index being $0$). For every natural number $n \\ge 5$,","title":"Triple convolution of the Fibonacci 2-, 3- and 5-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-step-gf-quadruple-consecutive","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.fibStepGF_quadruple_consecutive","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-step-gf-quadruple-consecutive","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $m \\ge 1$ be a natural number. For each $k$, let $F^{(k)} = \\sum_{n \\ge 0} F^{(k)}_n X^n$ in $\\mathbb{Q}[[X]]$ be the ordinary generating function of the Fibonacci $k$-step numbers $F^{(k)}_n$. Then","title":"A product identity for four consecutive Fibonacci step generating functions"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/two-mul-comp-prod-sum-four","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.two_mul_compProdSum_four","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#two-mul-comp-prod-sum-four","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F_n$, $T_n$, $Q_n$ and $P_n$ for the Fibonacci $m$-step numbers $F^{(m)}_n$ with $m = 2, 3, 4, 5$ respectively. For every integer $n \\ge 5$, $2 \\sum F_{k_1} T_{k_2} Q_{k_3} P_{k_4} = P_{n+4} - P_{n+3} + P_{n+2} + T_{n+3} + T_{n+1} - F_n - 2Q_{n+3} - 2Q_{n+1}$, where the sum runs over all $4$-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers with $k_1 + k_2 + k_3 + k_4 = n - 5$.","title":"Four-fold convolution of the Fibonacci, tribonacci, tetranacci and pentanacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum-four-consecutive","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum_four_consecutive","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum-four-consecutive","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(s)}_k$ for the Fibonacci $s$-step numbers, and write $\\sum_{K(4,N)}$ for a sum over all $(k_1, k_2, k_3, k_4) \\in \\mathbb{N}^4$ with $k_1 + k_2 + k_3 + k_4 = N$. For all natural numbers $m \\ge 1$ and $n \\ge 2m + 2$,","title":"Convolution of four consecutive Fibonacci step sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum-even","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum_even","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum-even","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(s)}_k$ for the Fibonacci $s$-step numbers, and write $\\sum_{K(r,N)}$ for a sum over all $(k_1, \\dots, k_r) \\in \\mathbb{N}^r$ with $k_1 + \\dots + k_r = N$. For all natural numbers $\\ell \\ge 1$, $m \\ge 1$ and $n \\ge \\ell(m + \\ell - 1)$,","title":"Convolution of an even number of consecutive Fibonacci step sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum-odd","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum_odd","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum-odd","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(s)}_k$ for the Fibonacci $s$-step numbers, and write $\\sum_{K(r,N)}$ for a sum over all $(k_1, \\dots, k_r) \\in \\mathbb{N}^r$ with $k_1 + \\dots + k_r = N$. For all natural numbers $\\ell \\ge 1$, $m \\ge 1$ and $n \\ge \\ell(m + \\ell - 1)$,","title":"Convolution of an odd number of consecutive Fibonacci step sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/comp-prod-sum-fib-trib-tetra-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.compProdSum_fib_trib_tetra_eq","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#comp-prod-sum-fib-trib-tetra-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F_k$, $T_k$ and $Q_k$ for the Fibonacci $2$-, $3$- and $4$-step numbers, and write $\\sum_{K(3,N)}$ for a sum over all $(k_1, k_2, k_3) \\in \\mathbb{N}^3$ with $k_1 + k_2 + k_3 = N$. For every natural number $n \\ge 2$,","title":"Two convolution forms of the Fibonacci\u2013tribonacci\u2013tetranacci triple sum"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/sum-fib-step-window","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.sum_fibStep_window","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(2m + 2\\)"],["text"," consecutive Fibonacci "],["math","\\(m\\)"],["text","-step numbers"]],"source":"Sum of $2m + 2$ consecutive Fibonacci $m$-step numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#sum-fib-step-window","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $F^{(m)}_n$ for the Fibonacci $m$-step numbers ($F^{(m)}_0 = 0$, $F^{(m)}_1 = 1$, and each later term is the sum of the $m$ terms before it, terms at negative indices being $0$). For every natural number $m \\ge 2$ and every natural number $n$,","title":"Sum of $2m + 2$ consecutive Fibonacci $m$-step numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/five-mul-sum-fib-step6-mul-fib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.FibStepConvolutionWanted.five_mul_sum_fibStep6_mul_fib","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution of the Fibonacci "],["math","\\(6\\)"],["text","-step numbers with the Fibonacci numbers"]],"source":"Convolution of the Fibonacci $6$-step numbers with the Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#five-mul-sum-fib-step6-mul-fib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Write $s_n = F^{(6)}_n$ for the Fibonacci $6$-step numbers and $F_n = F^{(2)}_n$ for the Fibonacci numbers. For every natural number $n \\ge 3$,","title":"Convolution of the Fibonacci $6$-step numbers with the Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-binomial-truncation","kind":"node","lean":"MetaMathlibExt.kFibonacciBinomialTruncation","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncated binomial sum "],["math","\\(G_n^{(k)}\\)"]],"source":"Truncated binomial sum $G_n^{(k)}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-binomial-truncation","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"For an integer $x$ and a natural number $j$ let $\\binom{x}{j} = x(x-1)\\cdots(x-j+1)/j!$. For natural numbers $k$ and $n$, the rational number $G_n^{(k)}$ is","title":"Truncated binomial sum $G_n^{(k)}$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-binomial-truncation-two-two","kind":"node","lean":"MetaMathlibExt.kFibonacciBinomialTruncation_two_two","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value "],["math","\\(G_2^{(2)} = 1\\)"]],"source":"Value $G_2^{(2)} = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-binomial-truncation-two-two","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"The truncated binomial sum satisfies $G_2^{(2)} = 1$.","title":"Value $G_2^{(2)} = 1$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-binomial-truncation-two-four","kind":"node","lean":"MetaMathlibExt.kFibonacciBinomialTruncation_two_four","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value "],["math","\\(G_4^{(2)} = 3\\)"]],"source":"Value $G_4^{(2)} = 3$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-binomial-truncation-two-four","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"fully_proved","summary":"The truncated binomial sum satisfies $G_4^{(2)} = 3$.","title":"Value $G_4^{(2)} = 3$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/lagrange-inversion-formula-analytic","kind":"node","lean":"MetaMathlibExt.lagrange_inversion_formula_analytic","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#lagrange-inversion-formula-analytic","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $f, g : \\mathbb{C} \\to \\mathbb{C}$ be analytic at $z_0$. There are $\\varepsilon > 0$ and $\\delta > 0$ such that for every $\\alpha \\in \\mathbb{C}$ with $|\\alpha| < \\delta$ the equation $w = z_0 + \\alpha f(w)$ has exactly one solution $w$ with $|w - z_0| < \\varepsilon$, and for this solution the series $\\sum_{n \\ge 1} \\frac{\\alpha^n}{n!} \\frac{d^{n-1}}{dz^{n-1}} \\bigl(f(z)^n g'(z)\\bigr)\\big|_{z=z_0}$ converges\u2026","title":"Lagrange inversion formula"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-root-pow-eq-binomial-series","kind":"node","lean":"MetaMathlibExt.kFibonacci_root_pow_eq_binomial_series","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial series for the powers of the dominant "],["math","\\(k\\)"],["text","-Fibonacci root"]],"source":"Binomial series for the powers of the dominant $k$-Fibonacci root"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-root-pow-eq-binomial-series","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 2$ and let $\\rho_k > 0$ be a real number with $\\rho_k^k = 1 + \\rho_k + \\dots + \\rho_k^{k-1}$. For integers $a$ and $j \\ge 0$ write $\\binom{a}{j} = a(a-1)\\cdots(a-j+1)/j!$. Then for every $n \\ge 1$ the series $S = \\sum_{\\ell \\ge 0} \\frac{1}{\\ell+1} \\binom{k(\\ell+1)+\\ell-n}{\\ell} 2^{-(k+1)\\ell}$ converges absolutely and $\\rho_k^n = 2^n - n\\, 2^{n-k-1} S$.","title":"Binomial series for the powers of the dominant $k$-Fibonacci root"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/golden-ratio-pow-eq-binomial-series","kind":"node","lean":"MetaMathlibExt.goldenRatio_pow_eq_binomial_series","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#golden-ratio-pow-eq-binomial-series","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $\\varphi = (1 + \\sqrt{5})/2$ be the golden ratio, and for an integer $x$ and a natural number $j$ let $\\binom{x}{j} = x(x-1)\\cdots(x-j+1)/j!$. For every natural number $n \\ge 1$ the series $S_n = \\sum_{\\ell \\ge 0} \\frac{1}{\\ell + 1} \\binom{3\\ell + 2 - n}{\\ell} 2^{-3\\ell}$ converges absolutely and $\\varphi^n = 2^n - n \\, 2^{n-3} S_n$. Moreover, the series\u2026","title":"Binomial series for the powers of the golden ratio"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/has-sum-choose-div-two-pow-k-fibonacci-r","kind":"node","lean":"MetaMathlibExt.hasSum_choose_div_two_pow_kFibonacci_root","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial series summed at the "],["math","\\(k\\)"],["text","-Fibonacci root"]],"source":"Binomial series summed at the $k$-Fibonacci root"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#has-sum-choose-div-two-pow-k-fibonacci-r","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and $\\rho > 0$ a real number with $\\rho^k = 1 + \\rho + \\cdots + \\rho^{k-1}$, and for an integer $x$ and a natural number $j$ let $\\binom{x}{j} = x(x-1)\\cdots(x-j+1)/j!$. Then for every integer $a$ the series below converges absolutely and","title":"Binomial series summed at the $k$-Fibonacci root"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/has-sum-binomial-series-k-fibonacci-asym","kind":"node","lean":"MetaMathlibExt.hasSum_binomial_series_kFibonacci_asymptotic","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Difference binomial series at the "],["math","\\(k\\)"],["text","-Fibonacci root"]],"source":"Difference binomial series at the $k$-Fibonacci root"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#has-sum-binomial-series-k-fibonacci-asym","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and $\\rho > 0$ a real number with $\\rho^k = 1 + \\rho + \\cdots + \\rho^{k-1}$, and for an integer $x$ and a natural number $j$ let $\\binom{x}{j} = x(x-1)\\cdots(x-j+1)/j!$. For every natural number $n \\ne 1$ the series below converges absolutely and","title":"Difference binomial series at the $k$-Fibonacci root"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-eq-k-fibonacci-binomial-trun","kind":"node","lean":"MetaMathlibExt.kFibonacci_eq_kFibonacciBinomialTruncation","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-Fibonacci numbers as truncated binomial sums"]],"source":"$k$-Fibonacci numbers as truncated binomial sums"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-eq-k-fibonacci-binomial-trun","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 2$ and let $(F_n)_{n \\ge 0}$ be the $k$-Fibonacci sequence, given by $F_0 = \\dots = F_{k-2} = 0$, $F_{k-1} = 1$ and $F_{m+k} = F_m + F_{m+1} + \\dots + F_{m+k-1}$ for all $m \\ge 0$. For integers $a$ and $j \\ge 0$ write $\\binom{a}{j} = a(a-1)\\cdots(a-j+1)/j!$, and put $\\binom{a}{-1} = 0$. For $n \\ge 2$ let $G_n^{(k)}$ be the truncated binomial sum\u2026","title":"$k$-Fibonacci numbers as truncated binomial sums"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-binomial-truncation-initial","kind":"node","lean":"MetaMathlibExt.kFibonacciBinomialTruncation_initial_values","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-binomial-truncation-initial","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and let $G_n^{(k)}$ be the truncated binomial sum. Then $G_n^{(k)} = 2^{n-2}$ for every natural number $n$ with $2 \\le n \\le k + 1$, and $G_{k+2}^{(k)} = 2^k - 1$.","title":"Initial values of the truncated binomial sums"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-binomial-truncation-recurren","kind":"node","lean":"MetaMathlibExt.kFibonacciBinomialTruncation_recurrence","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-binomial-truncation-recurren","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $G_n^{(k)}$ be the truncated binomial sum. For all natural numbers $k \\ge 2$ and $n \\ge 2$, $G_{n+k+1}^{(k)} = 2 G_{n+k}^{(k)} - G_n^{(k)}$.","title":"Recurrence for the truncated binomial sums"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-eq-binomial-sum","kind":"node","lean":"MetaMathlibExt.kFibonacci_eq_binomial_sum","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial-sum formula for the "],["math","\\(k\\)"],["text","-Fibonacci numbers"]],"source":"Binomial-sum formula for the $k$-Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-eq-binomial-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 2$ and let $(F_n)_{n \\ge 0}$ be the $k$-Fibonacci sequence, given by $F_0 = \\dots = F_{k-2} = 0$, $F_{k-1} = 1$ and $F_{m+k} = F_m + F_{m+1} + \\dots + F_{m+k-1}$ for all $m \\ge 0$. For integers $a$ and $j \\ge 0$ write $\\binom{a}{j} = a(a-1)\\cdots(a-j+1)/j!$, and put $\\binom{a}{-1} = 0$. Then for every $n \\ge k$,","title":"Binomial-sum formula for the $k$-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/fib-eq-binomial-sum","kind":"node","lean":"MetaMathlibExt.fib_eq_binomial_sum","parent":6485,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#fib-eq-binomial-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$), and for an integer $x$ and a natural number $j$ let $\\binom{x}{j} = x(x-1)\\cdots(x-j+1)/j!$. For every natural number $n \\ge 2$,","title":"Fibonacci numbers as a truncated binomial sum"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/k-fibonacci-eq-alternating-binomial-sum","kind":"node","lean":"MetaMathlibExt.kFibonacci_eq_alternating_binomial_sum","parent":6485,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating binomial formula for the "],["math","\\(k\\)"],["text","-Fibonacci numbers"]],"source":"Alternating binomial formula for the $k$-Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers/index.html#k-fibonacci-eq-alternating-binomial-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and let $F_n^{(k)}$ be the $k$-Fibonacci numbers: $F_i^{(k)} = 0$ for $i < k - 1$, $F_{k-1}^{(k)} = 1$ and $F_{m+k}^{(k)} = F_m^{(k)} + F_{m+1}^{(k)} + \\cdots + F_{m+k-1}^{(k)}$ for every $m \\ge 0$. For an integer $x$ and a natural number $j$ let $\\binom{x}{j} = x(x-1)\\cdots(x-j+1)/j!$. Then for every natural number $n \\ge k$,","title":"Alternating binomial formula for the $k$-Fibonacci numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","title":"k-generalized Lucas numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/k-generalized-lucas","kind":"node","lean":"Nat.kGeneralizedLucas","parent":6561,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-generalized Lucas numbers"]],"source":"$k$-generalized Lucas numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html#k-generalized-lucas","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","summary":"For a natural number $k \\ge 2$, the $k$-generalized Lucas numbers $L^{(k)}_n$ are the natural numbers indexed by the integers $n \\ge 2 - k$ given by $L^{(k)}_n = 0$ for $2 - k \\le n \\le -1$, $L^{(k)}_0 = 2$, $L^{(k)}_1 = 1$, and $L^{(k)}_n = L^{(k)}_{n-1} + L^{(k)}_{n-2} + \\dots + L^{(k)}_{n-k}$ for every integer $n \\ge 2$. No value is assigned at indices below $2 - k$.","title":"$k$-generalized Lucas numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/k-generalized-lucas-zero","kind":"node","lean":"Nat.kGeneralizedLucas_zero","parent":6561,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-generalized Lucas numbers at index zero"]],"source":"$k$-generalized Lucas numbers at index zero"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html#k-generalized-lucas-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","summary":"For every natural number $k \\ge 2$, the $k$-generalized Lucas number $L^{(k)}_0$ equals $2$.","title":"$k$-generalized Lucas numbers at index zero"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/k-generalized-lucas-one","kind":"node","lean":"Nat.kGeneralizedLucas_one","parent":6561,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-generalized Lucas numbers at index one"]],"source":"$k$-generalized Lucas numbers at index one"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html#k-generalized-lucas-one","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","summary":"For every natural number $k \\ge 2$, the $k$-generalized Lucas number $L^{(k)}_1$ equals $1$.","title":"$k$-generalized Lucas numbers at index one"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/k-generalized-lucas-neg-eq-zero","kind":"node","lean":"Nat.kGeneralizedLucas_neg_eq_zero","parent":6561,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-generalized Lucas numbers vanish at negative indices"]],"source":"$k$-generalized Lucas numbers vanish at negative indices"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html#k-generalized-lucas-neg-eq-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","summary":"For every natural number $k \\ge 3$ and every integer $n$ with $2 - k \\le n \\le -1$, the $k$-generalized Lucas number $L^{(k)}_n$ equals $0$.","title":"$k$-generalized Lucas numbers vanish at negative indices"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/k-generalized-lucas-lower-zero","kind":"node","lean":"Nat.kGeneralizedLucas_lower_zero","parent":6561,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-generalized Lucas numbers at the lowest index"]],"source":"$k$-generalized Lucas numbers at the lowest index"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html#k-generalized-lucas-lower-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","summary":"For every natural number $k \\ge 3$, the $k$-generalized Lucas number $L^{(k)}_{2-k}$ equals $0$.","title":"$k$-generalized Lucas numbers at the lowest index"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/k-generalized-lucas-recurrence","kind":"node","lean":"Nat.kGeneralizedLucas_recurrence","parent":6561,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the "],["math","\\(k\\)"],["text","-generalized Lucas numbers"]],"source":"Recurrence for the $k$-generalized Lucas numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html#k-generalized-lucas-recurrence","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","summary":"For every natural number $k \\ge 2$ and every integer $n \\ge 2$, the $k$-generalized Lucas numbers satisfy $L^{(k)}_n = \\sum_{i=0}^{k-1} L^{(k)}_{n-k+i}$, that is, $L^{(k)}_n$ is the sum $L^{(k)}_{n-1} + \\dots + L^{(k)}_{n-k}$ of the $k$ preceding values.","title":"Recurrence for the $k$-generalized Lucas numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/n-step-lucas-list","kind":"node","lean":"MetaMathlibExt.nStepLucasList","parent":6561,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html#n-step-lucas-list","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $m$, this is the list $[L^{(n)}_1, \\dots, L^{(n)}_m]$ of length $m$ (empty for $m = 0$), where $L^{(n)}_k = 2^k - 1$ for $1 \\le k \\le n$ and $L^{(n)}_k$ is the sum of the $n$ preceding terms for $k \\ge n + 1$. For $n = 0$ every term is $0$.","title":"Finite prefixes of the n-step Lucas numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/n-step-lucas","kind":"node","lean":"MetaMathlibExt.nStepLucas","parent":6561,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers/index.html#n-step-lucas","scope":"NumberTheory/linear-recurrences/higher-order-sequences/k-generalized-lucas-numbers","status":"fully_proved","summary":"For natural numbers $n \\ge 1$ and $m$, the $n$-step Lucas number $L^{(n)}_m$ is defined by $L^{(n)}_0 = 0$, $L^{(n)}_m = 2^m - 1$ for $1 \\le m \\le n$, and $L^{(n)}_m = \\sum_{i=1}^{n} L^{(n)}_{m-i}$ for $m \\ge n + 1$. For $n = 0$ we set $L^{(0)}_m = 0$ for every $m$.","title":"n-step Lucas numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","title":"Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/leonardo-pkseq","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.leonardoPKSeq","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial values of the generalized Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Initial values of the generalized Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#leonardo-pkseq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$, $k$ and $n$, this is the list of integers $(\\mathcal{L}_{p,k}(0), \\mathcal{L}_{p,k}(1), \\dots, \\mathcal{L}_{p,k}(n))$ of length $n + 1$, where $\\mathcal{L}_{p,k}(i) = 1$ for $0 \\le i \\le p$ and $\\mathcal{L}_{p,k}(i) = \\mathcal{L}_{p,k}(i-1) + \\mathcal{L}_{p,k}(i-p-1) + k$ for $i > p$.","title":"Initial values of the generalized Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/leonardo-pk","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.leonardoPK","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Generalized Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#leonardo-pk","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $k$, the generalized Leonardo $p$-numbers are the integers $\\mathcal{L}_{p,k}(n)$, $n \\in \\mathbb{N}$, given by $\\mathcal{L}_{p,k}(n) = 1$ for $0 \\le n \\le p$ and $\\mathcal{L}_{p,k}(n) = \\mathcal{L}_{p,k}(n-1) + \\mathcal{L}_{p,k}(n-p-1) + k$ for $n > p$.","title":"Generalized Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/h-seq-forward","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.hSeqForward","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Forward evaluation of the "],["math","\\(H(p)\\)"],["text"," recurrence"]],"source":"Forward evaluation of the $H(p)$ recurrence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#h-seq-forward","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p$, a function $s : \\mathbb{Z} \\to \\mathbb{Z}$ and $n \\in \\mathbb{N}$, this is the list of the $2p + n + 1$ values at $-p, -p+1, \\dots, p+n$ (the entry at position $i$ is the value at $i - p$) of the sequence $H$ that equals $s$ on $[-p, p]$ and continues by the $H(p)$ recurrence $H(t) = H(t-1) + H(t-p) - H(t-2p-1)$ for $t > p$. Each step appends one value to the previous list, so evaluation\u2026","title":"Forward evaluation of the $H(p)$ recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/h-seq-backward","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.hSeqBackward","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Backward evaluation of the "],["math","\\(H(p)\\)"],["text"," recurrence"]],"source":"Backward evaluation of the $H(p)$ recurrence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#h-seq-backward","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p$, a function $s : \\mathbb{Z} \\to \\mathbb{Z}$ and $n \\in \\mathbb{N}$, this is the list of the values at $-p-n, \\dots, p$ of the sequence $H$ that equals $s$ on $[-p, p]$ and continues to the left by $H(t) = H(t+2p) + H(t+p+1) - H(t+2p+1)$ for $t < -p$, which is the $H(p)$ recurrence $H(u) = H(u-1) + H(u-p) - H(u-2p-1)$ at $u = t+2p+1$ solved for $H(t)$. Each step prepends one value to the\u2026","title":"Backward evaluation of the $H(p)$ recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/h-seq-z","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.hSeqZ","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-sided extension of initial values by the "],["math","\\(H(p)\\)"],["text"," recurrence"]],"source":"Two-sided extension of initial values by the $H(p)$ recurrence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#h-seq-z","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p \\ge 1$ and a function $s : \\mathbb{Z} \\to \\mathbb{Z}$, this is the unique function $H : \\mathbb{Z} \\to \\mathbb{Z}$ that agrees with $s$ on $[-p, p]$ and satisfies $H(t) = H(t-1) + H(t-p) - H(t-2p-1)$ for every integer $t$: its values at $t > p$ are computed forward and its values at $t < -p$ backward from this recurrence. For $p = 0$ its value is $s(0)$ at every $t \\le 0$ and $0$ at every\u2026","title":"Two-sided extension of initial values by the $H(p)$ recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/seq-az","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.seqAZ","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sequence "],["math","\\(A_p\\)"]],"source":"The sequence $A_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#seq-az","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p \\ge 1$, $A_p : \\mathbb{Z} \\to \\mathbb{Z}$ is the unique sequence with $A_p(n) = 0$ for $-p \\le n \\le 0$ and $A_p(n) = 1$ for $1 \\le n \\le p$ that satisfies $A_p(n) = A_p(n-1) + A_p(n-p) - A_p(n-2p-1)$ for every integer $n$. For $p = 0$ it is the zero sequence.","title":"The sequence $A_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/seq-bz","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.seqBZ","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-sided sequence "],["math","\\(B_p\\)"]],"source":"The two-sided sequence $B_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#seq-bz","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p \\ge 1$, the sequence $B_p$ of integers indexed by $\\mathbb{Z}$ is the unique solution of the recurrence $B_p(n) = B_p(n-1) + B_p(n-p) - B_p(n-2p-1)$ for every integer $n$ (the defining recurrence of the paper's class $H(p)$) with initial values $B_p(n) = 1$ for $1 \\le n \\le p$, $B_p(-p+1) = 1$, and $B_p(n) = 0$ for the other integers $n$ with $-p \\le n \\le 0$.","title":"The two-sided sequence $B_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/companion-q","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.companionQ","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","The companion matrix "],["math","\\(Q_p\\)"]],"source":"The companion matrix $Q_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#companion-q","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p$, $Q_p$ is the $(2p+1) \\times (2p+1)$ integer matrix with rows and columns indexed by $0, 1, \\dots, 2p$ whose entry $(2p, 0)$ is $-1$, whose other entries $(0, 0)$, $(p-1, 0)$ and $(i, i+1)$ for $0 \\le i < 2p$ are $1$, and whose remaining entries are $0$.","title":"The companion matrix $Q_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/q-pow-entry","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.qPowEntry","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entry formula for the powers of "],["math","\\(Q_p\\)"]],"source":"Entry formula for the powers of $Q_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#q-pow-entry","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $n$ and indices $a, b \\in \\{0, 1, \\dots, 2p\\}$, the value is the integer $A_p(n + 1 - b)$ if $a = 0$, the integer $B_p(n - p + 1 + a - b)$ if $1 \\le a \\le p - 1$, and the integer $-A_p(n - 2p + a - b)$ otherwise, where $A_p$ and $B_p$ are the two-sided auxiliary sequences of the generalized Leonardo $p$-numbers.","title":"Entry formula for the powers of $Q_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/r-matrix","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.rMatrix","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Hankel matrix "],["math","\\(R_{p,m}\\)"]],"source":"The Hankel matrix $R_{p,m}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#r-matrix","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p$, a sequence $H : \\mathbb{Z} \\to \\mathbb{Z}$ and a natural number $m$, $R_{p,m}$ is the $(2p+1) \\times (2p+1)$ integer matrix, with rows and columns indexed by $0, 1, \\dots, 2p$, whose $(a, b)$ entry is $H(m - a - b)$.","title":"The Hankel matrix $R_{p,m}$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/leonardo-pk-eq","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.leonardoPK_eq","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Leonardo "],["math","\\(p\\)"],["text","-numbers in terms of Fibonacci "],["math","\\(p\\)"],["text","-numbers"]],"source":"Generalized Leonardo $p$-numbers in terms of Fibonacci $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#leonardo-pk-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $p$, $k$ and $n$ with $p \\ge 1$ and $k \\ge 1$, the generalized Leonardo $p$-number $\\mathcal{L}_{p,k}(n)$ and the Fibonacci $p$-number $F_p(n+1)$ satisfy $\\mathcal{L}_{p,k}(n) = (k+1)F_p(n+1) - k$.","title":"Generalized Leonardo $p$-numbers in terms of Fibonacci $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/sum-leonardo-pk","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.sum_leonardoPK","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sums of the generalized Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Partial sums of the generalized Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#sum-leonardo-pk","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 1$ and $k \\ge 1$ be natural numbers, and let $\\mathcal{L}_{p,k}$ denote the generalized Leonardo $p$-numbers ($\\mathcal{L}_{p,k}(j) = 1$ for $0 \\le j \\le p$ and $\\mathcal{L}_{p,k}(j) = \\mathcal{L}_{p,k}(j-1) + \\mathcal{L}_{p,k}(j-p-1) + k$ for $j > p$). Then for every natural number $n$, $\\sum_{i=0}^{n} \\mathcal{L}_{p,k}(i) = \\mathcal{L}_{p,k}(n+p+1) - k(n+1) - 1$.","title":"Partial sums of the generalized Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/sum-leonardo-pk-stride","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.sum_leonardoPK_stride","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stride sums of the generalized Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Stride sums of the generalized Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#sum-leonardo-pk-stride","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 1$ and $k \\ge 1$ be natural numbers, and let $\\mathcal{L}_{p,k}$ denote the generalized Leonardo $p$-numbers ($\\mathcal{L}_{p,k}(j) = 1$ for $0 \\le j \\le p$ and $\\mathcal{L}_{p,k}(j) = \\mathcal{L}_{p,k}(j-1) + \\mathcal{L}_{p,k}(j-p-1) + k$ for $j > p$). Then for all natural numbers $m \\le p$ and $n$, the sum $S = \\sum_{i=0}^{n} \\mathcal{L}_{p,k}((p+1)i + m)$ satisfies\u2026","title":"Stride sums of the generalized Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/fibonacci-p-eq-seq-az-sub","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.fibonacciP_eq_seqAZ_sub","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci "],["math","\\(p\\)"],["text","-numbers as differences of "],["math","\\(A_p\\)"]],"source":"Fibonacci $p$-numbers as differences of $A_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#fibonacci-p-eq-seq-az-sub","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $p \\ge 1$ and $n \\ge 1$, the Fibonacci $p$-number $F_p(n)$ equals $A_p(n) - A_p(n-p)$. Here $A_p : \\mathbb{Z} \\to \\mathbb{Z}$ is the sequence with $A_p(t) = 0$ for $-p \\le t \\le 0$, $A_p(t) = 1$ for $1 \\le t \\le p$, and $A_p(t) = A_p(t-1) + A_p(t-p) - A_p(t-2p-1)$ for every integer $t$.","title":"Fibonacci $p$-numbers as differences of $A_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/seq-az-eq-sum-fibonacci-p","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.seqAZ_eq_sum_fibonacciP","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sequence "],["math","\\(A_p\\)"],["text"," as a sum of Fibonacci "],["math","\\(p\\)"],["text","-numbers"]],"source":"The sequence $A_p$ as a sum of Fibonacci $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#seq-az-eq-sum-fibonacci-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p$, $j$ and $m$ with $j < p$, the auxiliary sequence $A_p$ and the Fibonacci $p$-numbers $F_p$ satisfy $A_p(mp + j) = \\sum_{i=0}^{m} F_p(ip + j)$.","title":"The sequence $A_p$ as a sum of Fibonacci $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/seq-az-recurrence","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.seqAZ_recurrence","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the sequence "],["math","\\(A_p\\)"]],"source":"Recurrence for the sequence $A_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#seq-az-recurrence","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 2$ and $n \\ge 1$, the auxiliary sequence $A_p$ satisfies $A_p(n) = A_p(n-1) + A_p(n-p-1) + 1$ if $n \\equiv 1 \\pmod p$, and $A_p(n) = A_p(n-1) + A_p(n-p-1)$ otherwise.","title":"Recurrence for the sequence $A_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/seq-bz-recurrence","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.seqBZ_recurrence","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the sequence "],["math","\\(B_p\\)"]],"source":"Recurrence for the sequence $B_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#seq-bz-recurrence","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 2$ and $n \\ge 1$, the auxiliary sequence $B_p$ satisfies $B_p(n) = B_p(n-1) + B_p(n-p-1) + \\varepsilon$, where $\\varepsilon = 1$ if $n \\equiv 1 \\pmod p$, $\\varepsilon = -1$ if $n \\equiv 2 \\pmod p$, and $\\varepsilon = 0$ otherwise. For $p = 2$ this means $\\varepsilon = 1$ for odd $n$ and $\\varepsilon = -1$ for even $n$.","title":"Recurrence for the sequence $B_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/seq-az-eq-seq-bz","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.seqAZ_eq_seqBZ","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relation between the sequences "],["math","\\(A_p\\)"],["text"," and "],["math","\\(B_p\\)"]],"source":"Relation between the sequences $A_p$ and $B_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#seq-az-eq-seq-bz","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 2$ and $n$, the auxiliary sequences $A_p$ and $B_p$ satisfy $A_p(n) = B_p(n+1) - 1$ if $p$ divides $n$, and $A_p(n) = B_p(n+1)$ otherwise.","title":"Relation between the sequences $A_p$ and $B_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/companion-q-pow-apply","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.companionQ_pow_apply","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entries of the powers of the companion matrix "],["math","\\(Q_p\\)"]],"source":"Entries of the powers of the companion matrix $Q_p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#companion-q-pow-apply","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 2$ and $n$ be natural numbers, and index the rows and columns of the $(2p+1) \\times (2p+1)$ companion matrix $Q_p$ by $0, 1, \\dots, 2p$. Then for all indices $a$ and $b$, the entry $(a, b)$ of $Q_p^n$ equals $A_p(n + 1 - b)$ if $a = 0$, equals $B_p(n - p + 1 + a - b)$ if $1 \\le a \\le p - 1$, and equals $-A_p(n - (2p - a) - b)$ if $p \\le a \\le 2p$. Here $A_p, B_p : \\mathbb{Z} \\to \\mathbb{Z}$ are the\u2026","title":"Entries of the powers of the companion matrix $Q_p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/r-matrix-mul-companion-q-pow","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.rMatrix_mul_companionQ_pow","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of "],["math","\\(Q_p\\)"],["text"," shift the Hankel matrices "],["math","\\(R_{p,m}\\)"]],"source":"Powers of $Q_p$ shift the Hankel matrices $R_{p,m}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#r-matrix-mul-companion-q-pow","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 2$ be a natural number and let $H : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $H(t) = H(t-1) + H(t-p) - H(t-2p-1)$ for every integer $t$. For a natural number $m$, let $R_{p,m}$ be the $(2p+1) \\times (2p+1)$ integer matrix whose $(a, b)$ entry is $H(m - a - b)$ for $0 \\le a, b \\le 2p$, and let $Q_p$ be the $(2p+1) \\times (2p+1)$ companion matrix with entries $1$ at $(0, 0)$, at $(p-1, 0)$ and on the\u2026","title":"Powers of $Q_p$ shift the Hankel matrices $R_{p,m}$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/h-add-eq","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.h_add_eq","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution identity for sequences in "],["math","\\(H(p)\\)"]],"source":"Convolution identity for sequences in $H(p)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#h-add-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 2$ be a natural number and let $\\mathcal{H} : \\mathbb{Z} \\to \\mathbb{Z}$ be a sequence in $H(p)$, that is, $\\mathcal{H}(t) = \\mathcal{H}(t-1) + \\mathcal{H}(t-p) - \\mathcal{H}(t-2p-1)$ for every integer $t$. Then for all natural numbers $m \\ge 1$ and $n \\ge 1$,","title":"Convolution identity for sequences in $H(p)$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/succ-mul-sum-fibonacci-p","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.succ_mul_sum_fibonacciP","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of "],["math","\\(p+1\\)"],["text"," consecutive Fibonacci "],["math","\\(p\\)"],["text","-numbers"]],"source":"Sums of $p+1$ consecutive Fibonacci $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#succ-mul-sum-fibonacci-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 1$ and $n \\ge 1$ be natural numbers and let $k$ be a natural number. Let $F_p$ denote the Fibonacci $p$-numbers ($F_p(0) = 0$, $F_p(m) = 1$ for $1 \\le m \\le p$, and $F_p(m) = F_p(m-1) + F_p(m-p-1)$ for $m > p$), and let $\\mathcal{L}_{p,k}$ denote the generalized Leonardo $p$-numbers ($\\mathcal{L}_{p,k}(m) = 1$ for $0 \\le m \\le p$ and\u2026","title":"Sums of $p+1$ consecutive Fibonacci $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/leonardo-pkz","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.leonardoPKZ","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Leonardo "],["math","\\(p\\)"],["text","-numbers at integer arguments"]],"source":"Generalized Leonardo $p$-numbers at integer arguments"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#leonardo-pkz","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $k$ and an integer $z$, the value is the generalized Leonardo $p$-number $\\mathcal{L}_{p,k}(z)$ if $z \\ge 0$, and $-k$ if $z < 0$.","title":"Generalized Leonardo $p$-numbers at integer arguments"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/leonardo-pk-honsberger","kind":"node","lean":"MathlibExt.NumberTheory.LeonardoPWanted.leonardoPK_honsberger","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Honsberger-type identity for generalized Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Honsberger-type identity for generalized Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#leonardo-pk-honsberger","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p$, $k$, $m$ and $n$ be natural numbers with $p, k, m, n \\ge 1$. Write $\\mathcal{L}$ for the generalized Leonardo $p$-numbers $\\mathcal{L}_{p,k}$, extended to negative integers by $\\mathcal{L}(t) = -k$ for $t < 0$. Then","title":"Honsberger-type identity for generalized Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-type-seq","kind":"node","lean":"MetaMathlibExt.lucasTypeSeq","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-type-seq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"Let $\\alpha$ be a type with a zero, a one and an addition, let $p$ be a natural number and let $a_0, c \\in \\alpha$. Define $a_i = 1$ for $1 \\le i \\le p$ and $a_n = a_{n-1} + a_{n-p-1} + c$ for $n > p$. For every natural number $n$, the Lucas-type list with parameters $p$, $a_0$, $c$ ending at index $n$ is $[a_0, a_1, \\dots, a_n]$.","title":"Lucas-type recurrence lists"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-type-seq-length","kind":"node","lean":"MetaMathlibExt.lucasTypeSeq_length","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-type-seq-length","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every type $\\alpha$ with a zero, a one and an addition, every natural number $p$, all $a_0, c \\in \\alpha$ and every natural number $n$, the Lucas-type list with parameters $p$, $a_0$, $c$ ending at index $n$ has exactly $n + 1$ entries.","title":"Length of a Lucas-type list"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-type-seq-get-d","kind":"node","lean":"MetaMathlibExt.lucasTypeSeq_getD","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-type-seq-get-d","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"Let $\\alpha$ be a type with a zero, a one and an addition, let $p$ be a natural number and let $a_0, c \\in \\alpha$. For natural numbers $i \\le n$, the entry at index $i$ of the Lucas-type list with parameters $p$, $a_0$, $c$ ending at index $n$ equals the entry at index $i$ of the Lucas-type list ending at index $i$.","title":"Entries of a Lucas-type list do not depend on its length"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-type-seq-get-d-succ","kind":"node","lean":"MetaMathlibExt.lucasTypeSeq_getD_succ","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-type-seq-get-d-succ","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"Let $\\alpha$ be a type with a zero, a one and an addition, let $p$ be a natural number and let $a_0, c \\in \\alpha$. Write $S_m$ for the Lucas-type list with parameters $p$, $a_0$, $c$ ending at index $m$, and $S_m[i]$ for its entry at index $i$. Then for every natural number $n$, $S_{n+1}[n+1] = 1$ if $n + 1 \\le p$, and otherwise $S_{n+1}[n+1] = S_n[n] + S_{n-p}[n-p] + c$.","title":"Last entry of a Lucas-type list"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-p","kind":"node","lean":"MetaMathlibExt.lucasP","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas "],["math","\\(p\\)"],["text","-numbers"]],"source":"Lucas $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p$, the Lucas $p$-numbers are the natural numbers $L_{p,n}$ given by $L_{p,0} = p + 1$, $L_{p,i} = 1$ for $1 \\le i \\le p$, and $L_{p,n} = L_{p,n-1} + L_{p,n-p-1}$ for $n > p$.","title":"Lucas $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-p-zero","kind":"node","lean":"MetaMathlibExt.lucasP_zero","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial Lucas "],["math","\\(p\\)"],["text","-number"]],"source":"Initial Lucas $p$-number"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-p-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every natural number $p$, the Lucas $p$-number $L_{p,0}$ equals $p + 1$.","title":"Initial Lucas $p$-number"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-p-of-le","kind":"node","lean":"MetaMathlibExt.lucasP_of_le","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas "],["math","\\(p\\)"],["text","-numbers at indices "],["math","\\(1\\)"],["text"," to "],["math","\\(p\\)"]],"source":"Lucas $p$-numbers at indices $1$ to $p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-p-of-le","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every natural number $p$ and every $n$ with $1 \\le n \\le p$, the Lucas $p$-number $L_{p,n}$ equals $1$.","title":"Lucas $p$-numbers at indices $1$ to $p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-p-of-lt","kind":"node","lean":"MetaMathlibExt.lucasP_of_lt","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the Lucas "],["math","\\(p\\)"],["text","-numbers"]],"source":"Recurrence for the Lucas $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-p-of-lt","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every natural number $p$, write $L_{p,n}$ for the Lucas $p$-numbers. Then $L_{p,n} = L_{p,n-1} + L_{p,n-p-1}$ for every $n > p$.","title":"Recurrence for the Lucas $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-leonardo-p","kind":"node","lean":"MetaMathlibExt.lucasLeonardoP","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas-Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Lucas-Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p$, the Lucas-Leonardo $p$-numbers are the integers $R_{p,n}$ given by $R_{p,0} = p^2 + p + 1$, $R_{p,i} = 1$ for $1 \\le i \\le p$, and $R_{p,n} = R_{p,n-1} + R_{p,n-p-1} + p$ for $n > p$.","title":"Lucas-Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-leonardo-p-zero","kind":"node","lean":"MetaMathlibExt.lucasLeonardoP_zero","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial Lucas-Leonardo "],["math","\\(p\\)"],["text","-number"]],"source":"Initial Lucas-Leonardo $p$-number"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-leonardo-p-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every natural number $p$, the Lucas-Leonardo $p$-number $R_{p,0}$ equals $p^2 + p + 1$.","title":"Initial Lucas-Leonardo $p$-number"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-leonardo-p-of-le","kind":"node","lean":"MetaMathlibExt.lucasLeonardoP_of_le","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas-Leonardo "],["math","\\(p\\)"],["text","-numbers at indices "],["math","\\(1\\)"],["text"," to "],["math","\\(p\\)"]],"source":"Lucas-Leonardo $p$-numbers at indices $1$ to $p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-leonardo-p-of-le","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every natural number $p$ and every $n$ with $1 \\le n \\le p$, the Lucas-Leonardo $p$-number $R_{p,n}$ equals $1$.","title":"Lucas-Leonardo $p$-numbers at indices $1$ to $p$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-leonardo-p-of-lt","kind":"node","lean":"MetaMathlibExt.lucasLeonardoP_of_lt","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the Lucas-Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Recurrence for the Lucas-Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-leonardo-p-of-lt","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every natural number $p$, write $R_{p,n}$ for the Lucas-Leonardo $p$-numbers. Then $R_{p,n} = R_{p,n-1} + R_{p,n-p-1} + p$ for every $n > p$.","title":"Recurrence for the Lucas-Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-p-lucas-leonardo-p-one-values","kind":"node","lean":"MetaMathlibExt.lucasP_lucasLeonardoP_one_values","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","First values of the Lucas and Lucas-Leonardo "],["math","\\(1\\)"],["text","-numbers"]],"source":"First values of the Lucas and Lucas-Leonardo $1$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-p-lucas-leonardo-p-one-values","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"The Lucas $1$-numbers $L_{1,0}, \\dots, L_{1,5}$ are $2, 1, 3, 4, 7, 11$, and the Lucas-Leonardo $1$-numbers $R_{1,0}, \\dots, R_{1,5}$ are $3, 1, 5, 7, 13, 21$.","title":"First values of the Lucas and Lucas-Leonardo $1$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/incomplete-leonardo-p","kind":"node","lean":"MetaMathlibExt.incompleteLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#incomplete-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$, $n$ and $k$, the incomplete Leonardo $p$-number is the integer","title":"Incomplete Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/incomplete-leonardo-p-zero","kind":"node","lean":"MetaMathlibExt.incompleteLeonardoP_zero","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#incomplete-leonardo-p-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$ and $n$, the incomplete Leonardo $p$-number $\\mathcal{L}_{p,n}(0)$ equals $1$.","title":"Incomplete Leonardo p-numbers at zero"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/incomplete-lucas-leonardo-p","kind":"node","lean":"MetaMathlibExt.incompleteLucasLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#incomplete-lucas-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$, $n$ and $k$, the incomplete Lucas-Leonardo $p$-number is the rational number","title":"Incomplete Lucas-Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/incomplete-lucas-leonardo-p-zero","kind":"node","lean":"MetaMathlibExt.incompleteLucasLeonardoP_zero","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#incomplete-lucas-leonardo-p-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every natural number $p$ and every natural number $n \\ge 1$, the incomplete Lucas-Leonardo $p$-number $R_{p,n}(0)$ equals $1$.","title":"Incomplete Lucas-Leonardo p-numbers at zero"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-leonardo-p","kind":"node","lean":"MetaMathlibExt.hyperLeonardoP","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hyper Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Hyper Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$, $n$ and $k$, the hyper Leonardo $p$-number is the integer","title":"Hyper Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-leonardo-p-zero-left","kind":"node","lean":"MetaMathlibExt.hyperLeonardoP_zero_left","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hyper Leonardo "],["math","\\(p\\)"],["text","-numbers at "],["math","\\(n = 0\\)"]],"source":"Hyper Leonardo $p$-numbers at $n = 0$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-leonardo-p-zero-left","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$ and $k$, the hyper Leonardo $p$-number $\\mathcal{L}^{(k)}_{p,0}$ equals $1$.","title":"Hyper Leonardo $p$-numbers at $n = 0$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-lucas-leonardo-p","kind":"node","lean":"MetaMathlibExt.hyperLucasLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-lucas-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$, $n$ and $k$, the hyper Lucas-Leonardo $p$-number $R^{(k)}_{p,n}$ is the rational number defined by $R^{(0)}_{p,n} = R_{p,n}$, the Lucas-Leonardo $p$-number; $R^{(k)}_{p,0} = p^2 + p + 1$ for $k \\ge 1$; and, for $n \\ge 1$ and $k \\ge 1$,","title":"Hyper Lucas-Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-lucas-leonardo-p-zero-left","kind":"node","lean":"MetaMathlibExt.hyperLucasLeonardoP_zero_left","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hyper Lucas\u2013Leonardo "],["math","\\(p\\)"],["text","-numbers at "],["math","\\(n = 0\\)"]],"source":"Hyper Lucas\u2013Leonardo $p$-numbers at $n = 0$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-lucas-leonardo-p-zero-left","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$ and $k$, the hyper Lucas\u2013Leonardo $p$-number $R^{(k)}_{p,0}$ equals $p^2 + p + 1$.","title":"Hyper Lucas\u2013Leonardo $p$-numbers at $n = 0$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-lucas-leonardo-p-zero-right","kind":"node","lean":"MetaMathlibExt.hyperLucasLeonardoP_zero_right","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-lucas-leonardo-p-zero-right","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$ and $n$, the hyper Lucas-Leonardo $p$-number of order $0$ equals the Lucas-Leonardo $p$-number: $R^{(0)}_{p,n} = R_{p,n}$.","title":"Hyper Lucas-Leonardo p-numbers of order zero"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/leonardo-ptilings","kind":"node","lean":"MetaMathlibExt.leonardoPTilings","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Leonardo "],["math","\\(p\\)"],["text","-tilings"]],"source":"Leonardo $p$-tilings"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#leonardo-ptilings","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $n$, the tiling set $K_{p,n}$ consists of the linear tilings of an $n$-board by squares (length $1$), $p$-minos (length $p + 1$) and at most one $p$-tile, which comes in one of $p$ colors, has length at least $p + 1$, and can only be the initial piece. A tiling is recorded as its initial $p$-tile, given by its color and length, if there is one, followed by the sequence of the remaining\u2026","title":"Leonardo $p$-tilings"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/replicate-mem-leonardo-ptilings","kind":"node","lean":"MetaMathlibExt.replicate_mem_leonardoPTilings","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","The all-squares tiling is a Leonardo "],["math","\\(p\\)"],["text","-tiling"]],"source":"The all-squares tiling is a Leonardo $p$-tiling"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#replicate-mem-leonardo-ptilings","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$ and $n$, the linear tiling of an $n$-board by $n$ squares, with no $p$-tile, lies in the tiling set $K_{p,n}$.","title":"The all-squares tiling is a Leonardo $p$-tiling"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/bracelet-first-tile","kind":"node","lean":"MetaMathlibExt.BraceletFirstTile","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#bracelet-first-tile","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p$, a first tile records the tile of a bracelet that covers cell $1$ of the circular board, together with its phase (which cell of the tile covers cell $1$). It is one of: a curved square, in phase $1$; a curved $p$-mino of length $p + 1$ in phase $r + 1$ for some $0 \\le r \\le p$; a curved $p$-tile of length $p + 1$ in color $c$ for some $0 \\le c < p$ and phase $r + 1$ for some\u2026","title":"First tile of a bracelet"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/length","kind":"node","lean":"MetaMathlibExt.BraceletFirstTile.length","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#length","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For a natural number $p$, the length of the first tile of a bracelet is $1$ for a curved square, $p + 1$ for a curved $p$-mino in any phase, $p + 1$ for a curved $p$-tile of length $p + 1$ in any color and phase, and $l$ for a curved $p$-tile of length $l$ (in any color) starting at cell $1$.","title":"Length of the first tile of a bracelet"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-leonardo-bracelets","kind":"node","lean":"MetaMathlibExt.lucasLeonardoBracelets","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-leonardo-bracelets","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $n$, the bracelet set $B_{p,n}$ consists of the tilings of a circular $n$-board by curved squares (length $1$), curved $p$-minos (length $p + 1$) and at most one curved $p$-tile, which comes in one of $p$ colors and has either length $p + 1$, in any of its $p + 1$ phases, or length $l \\ge p + 2$, starting at cell $1$; a $p$-tile, if present, is the first tile. A bracelet is recorded as\u2026","title":"Lucas\u2013Leonardo bracelets"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/square-replicate-mem-lucas-leonardo-brac","kind":"node","lean":"MetaMathlibExt.square_replicate_mem_lucasLeonardoBracelets","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#square-replicate-mem-lucas-leonardo-brac","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For every natural number $p$ and every natural number $n \\ge 1$, the bracelet whose first tile is a curved square covering cell $1$ and whose remaining $n - 1$ pieces are all curved squares lies in the bracelet set $B_{p,n}$.","title":"The all-squares bracelet is a Lucas\u2013Leonardo bracelet"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/card-leonardo-ptilings","kind":"node","lean":"MetaMathlibExt.card_leonardoPTilings","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting Leonardo "],["math","\\(p\\)"],["text","-tilings"]],"source":"Counting Leonardo $p$-tilings"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#card-leonardo-ptilings","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $p \\ge 1$ and $n$, let $K_{p,n}$ be the set of linear tilings of an $n$-board by squares (length $1$), $p$-minos (length $p + 1$) and at most one $p$-tile, which comes in one of $p$ colors, has length at least $p + 1$ and can only be the initial piece. Then the number of tilings in $K_{p,n}$ equals the Leonardo $p$-number $\\mathcal{L}_{p,n}$, the generalized Leonardo $p$-number\u2026","title":"Counting Leonardo $p$-tilings"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/leonardo-pk-self-homogeneous","kind":"node","lean":"MetaMathlibExt.leonardoPK_self_homogeneous","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Homogeneous recurrence for the Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Homogeneous recurrence for the Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#leonardo-pk-self-homogeneous","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 1$ and write $\\mathcal{L}_{p,n}$ for the Leonardo $p$-number $\\mathcal{L}_{p,p}(n)$, the generalized Leonardo $p$-number with $k = p$. Then $\\mathcal{L}_{p,n} = \\mathcal{L}_{p,n-1} + \\mathcal{L}_{p,n-p} - \\mathcal{L}_{p,n-2p-1}$ for every $n \\ge 2p + 1$, $\\mathcal{L}_{p,i} = 1$ for every $i \\le p$, and $\\mathcal{L}_{p,p+j} = (j + 1) + jp$ for every $j$ with $1 \\le j \\le p$.","title":"Homogeneous recurrence for the Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/incomplete-leonardo-p-succ","kind":"node","lean":"MetaMathlibExt.incompleteLeonardoP_succ","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#incomplete-leonardo-p-succ","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n > p$ and $k \\le \\lfloor n/(p+1) \\rfloor$, the incomplete Leonardo $p$-numbers $\\mathcal{L}_{p,m}(j)$ satisfy","title":"Recurrence for incomplete Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/sum-choose-mul-incomplete-leonardo-p","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_incompleteLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#sum-choose-mul-incomplete-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n$, $r$ and $k$ with $n \\ge p + r$ and $k \\le \\lfloor (n - p - r)/(p + 1) \\rfloor$, the incomplete Leonardo $p$-numbers $\\mathcal{L}_{p,m}(j)$ satisfy","title":"Binomial sum identity for incomplete Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/sum-incomplete-leonardo-p","kind":"node","lean":"MetaMathlibExt.sum_incompleteLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#sum-incomplete-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n$, $k$ and $r$ with $n \\ge (p + 1)(k + 1)$, the incomplete Leonardo $p$-numbers $\\mathcal{L}_{p,m}(j)$ satisfy","title":"Partial sums of incomplete Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-leonardo-p-add","kind":"node","lean":"MetaMathlibExt.hyperLeonardoP_add","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial addition formula for hyper Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Binomial addition formula for hyper Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-leonardo-p-add","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n \\ge 1$, $m \\ge 1$ and $k \\ge m$, the hyper Leonardo $p$-numbers satisfy","title":"Binomial addition formula for hyper Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-leonardo-p-rec","kind":"node","lean":"MetaMathlibExt.hyperLeonardoP_rec","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the hyper Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Recurrence for the hyper Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-leonardo-p-rec","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 1$ be a natural number. For all natural numbers $n \\ge 1$ and $k \\ge 1$, the hyper Leonardo $p$-numbers satisfy $\\mathcal{L}^{(k)}_{p,n} = \\mathcal{L}^{(k)}_{p,n-1} + \\mathcal{L}^{(k-1)}_{p,n}$, and for every natural number $n$, $\\mathcal{L}^{(0)}_{p,n} = \\mathcal{L}_{p,n}$, where $\\mathcal{L}_{p,n}$ is the Leonardo $p$-number: $\\mathcal{L}_{p,n} = 1$ for $n \\le p$ and\u2026","title":"Recurrence for the hyper Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/leonardo-pk-eq-incomplete-add-hyper","kind":"node","lean":"MetaMathlibExt.leonardoPK_eq_incomplete_add_hyper","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#leonardo-pk-eq-incomplete-add-hyper","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n > p$ and $k \\ge 1$, let $N = n + (p + 1)k$. Then","title":"Leonardo p-numbers split into incomplete and hyper parts"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/card-lucas-leonardo-bracelets","kind":"node","lean":"MetaMathlibExt.card_lucasLeonardoBracelets","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#card-lucas-leonardo-bracelets","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $p \\ge 1$ and $n \\ge 1$, let $B_{p,n}$ be the set of tilings of a circular $n$-board by curved squares (length $1$), curved $p$-minos (length $p + 1$) and at most one curved $p$-tile in one of $p$ colors, which if present is the tile covering cell $1$ and either has length $p + 1$, in any of its $p + 1$ phases, or has length at least $p + 2$ and starts at cell $1$. Then the number of\u2026","title":"Counting Lucas\u2013Leonardo bracelets"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-leonardo-p-eq-lucas-p","kind":"node","lean":"MetaMathlibExt.lucasLeonardoP_eq_lucasP","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas-Leonardo "],["math","\\(p\\)"],["text","-numbers from Lucas "],["math","\\(p\\)"],["text","-numbers"]],"source":"Lucas-Leonardo $p$-numbers from Lucas $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-leonardo-p-eq-lucas-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For every $p \\ge 1$ and every natural number $n$, the Lucas-Leonardo $p$-number $R_{p,n}$ and the Lucas $p$-number $L_{p,n}$ satisfy $R_{p,n} = (p + 1) L_{p,n} - p$.","title":"Lucas-Leonardo $p$-numbers from Lucas $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/lucas-leonardo-p-eq-leonardo-pk","kind":"node","lean":"MetaMathlibExt.lucasLeonardoP_eq_leonardoPK","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas-Leonardo "],["math","\\(p\\)"],["text","-numbers from Leonardo "],["math","\\(p\\)"],["text","-numbers"]],"source":"Lucas-Leonardo $p$-numbers from Leonardo $p$-numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#lucas-leonardo-p-eq-leonardo-pk","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 1$, let $R_{p,n}$ denote the Lucas-Leonardo $p$-numbers, and let $\\mathcal{L}_{p,n}$ denote the Leonardo $p$-numbers $\\mathcal{L}_{p,p}(n)$ (the generalized Leonardo $p$-numbers with $k = p$), extended to $n = -1$ by $\\mathcal{L}_{p,-1} = -p$. Then $R_{p,n} = (p + 1) \\mathcal{L}_{p,n} - p \\mathcal{L}_{p,n-1}$ for every natural number $n$.","title":"Lucas-Leonardo $p$-numbers from Leonardo $p$-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/sum-lucas-leonardo-p","kind":"node","lean":"MetaMathlibExt.sum_lucasLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#sum-lucas-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For every natural number $p \\ge 1$ and every natural number $n$, the Lucas-Leonardo $p$-numbers $R_{p,m}$ satisfy $\\sum_{i=0}^{n} R_{p,i} = R_{p,n+p+1} - p(n + 1) - 1$.","title":"Sum of the Lucas-Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/sum-lucas-leonardo-p-stride","kind":"node","lean":"MetaMathlibExt.sum_lucasLeonardoP_stride","parent":6570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of the Lucas-Leonardo p-numbers at multiples of "],["math","\\(p + 1\\)"]],"source":"Sum of the Lucas-Leonardo p-numbers at multiples of $p + 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#sum-lucas-leonardo-p-stride","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For every natural number $p \\ge 1$ and every natural number $n$, the Lucas-Leonardo $p$-numbers $R_{p,m}$ satisfy $\\sum_{i=0}^{n} R_{p,(p+1)i} = R_{p,(p+1)n+1} - p(n - p - 1)$, an identity of integers.","title":"Sum of the Lucas-Leonardo p-numbers at multiples of $p + 1$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/incomplete-lucas-leonardo-p-succ","kind":"node","lean":"MetaMathlibExt.incompleteLucasLeonardoP_succ","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#incomplete-lucas-leonardo-p-succ","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n > p + 1$ and $k \\le \\lfloor n/(p+1) \\rfloor$, the incomplete Lucas-Leonardo $p$-numbers $R_{p,m}(j)$ satisfy","title":"Recurrence for incomplete Lucas-Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/sum-choose-mul-incomplete-lucas-leonardo","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_incompleteLucasLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#sum-choose-mul-incomplete-lucas-leonardo","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n$, $r$ and $k$ with $n \\ge p + r$ and $k \\le \\lfloor (n - p - r)/(p + 1) \\rfloor$, the incomplete Lucas-Leonardo $p$-numbers $R_{p,m}(j)$ satisfy, as rational numbers,","title":"Binomial sum identity for incomplete Lucas-Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-lucas-leonardo-p-eq","kind":"node","lean":"MetaMathlibExt.hyperLucasLeonardoP_eq","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-lucas-leonardo-p-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n > p$ and $k \\ge 1$, write $R_{p,m}$ for the Lucas\u2013Leonardo $p$-numbers and $R_{p,m}(j)$ for the incomplete Lucas\u2013Leonardo $p$-numbers. Then the hyper Lucas\u2013Leonardo $p$-number $R^{(k)}_{p,n}$ satisfies","title":"Hyper Lucas\u2013Leonardo numbers from incomplete ones"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/hyper-lucas-leonardo-p-eq-hyper-leonardo","kind":"node","lean":"MetaMathlibExt.hyperLucasLeonardoP_eq_hyperLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#hyper-lucas-leonardo-p-eq-hyper-leonardo","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For natural numbers $p \\ge 1$, $n > p$ and $k \\ge 1$, the hyper Lucas\u2013Leonardo $p$-number $R^{(k)}_{p,n}$ and the hyper Leonardo $p$-numbers $\\mathcal{L}^{(k)}_{p,n}$ satisfy","title":"Hyper Lucas\u2013Leonardo numbers from hyper Leonardo numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/super-leonardo-p","kind":"node","lean":"MetaMathlibExt.superLeonardoP","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#super-leonardo-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $k$ and a real sequence $x = (x_n)_{n \\in \\mathbb{N}}$, the supergeneralized Leonardo $p$-numbers are the real numbers $\\mathcal{L}_{p,k,x}(n)$, $n \\in \\mathbb{N}$, given by $\\mathcal{L}_{p,k,x}(n) = x_n$ for $n \\le p$ and $\\mathcal{L}_{p,k,x}(n) = \\mathcal{L}_{p,k,x}(n-1) + \\mathcal{L}_{p,k,x}(n-p-1) + k$ for $n > p$. Only the initial values $x_0, \\dots, x_p$ are used.","title":"Supergeneralized Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/super-leonardo-p-of-le","kind":"node","lean":"MetaMathlibExt.superLeonardoP_of_le","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#super-leonardo-p-of-le","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$, $k$ and $n$ with $n \\le p$ and every real sequence $x$, the supergeneralized Leonardo $p$-number $\\mathcal{L}_{p,k,x}(n)$ equals $x_n$.","title":"Initial values of the supergeneralized Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/super-leonardo-p-of-lt","kind":"node","lean":"MetaMathlibExt.superLeonardoP_of_lt","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#super-leonardo-p-of-lt","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"fully_proved","summary":"For all natural numbers $p$, $k$ and $n$ with $n > p$ and every real sequence $x$, the supergeneralized Leonardo $p$-numbers $\\mathcal{L}_{p,k,x}$ satisfy $\\mathcal{L}_{p,k,x}(n) = \\mathcal{L}_{p,k,x}(n-1) + \\mathcal{L}_{p,k,x}(n-p-1) + k$.","title":"Recurrence for the supergeneralized Leonardo p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/super-leonardo-p-eq-add-mul","kind":"node","lean":"MetaMathlibExt.superLeonardoP_eq_add_mul","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#super-leonardo-p-eq-add-mul","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 1$ and $k$ be natural numbers, $x$ a real sequence, $y$ the real sequence $y_i = i + 1$, and write $\\mathcal{L}_{p,k,x}(n)$ for the supergeneralized Leonardo $p$-numbers. For every natural number $n$, $\\mathcal{L}_{p,k,x}(n) = \\mathcal{L}_{p,0,x}(n)$ if $n \\le p$, and $\\mathcal{L}_{p,k,x}(n) = \\mathcal{L}_{p,0,x}(n) + \\mathcal{L}_{p,1,y}(n-(p+1)) \\cdot k$ if $n > p$.","title":"Reduction of supergeneralized Leonardo p-numbers to the homogeneous case"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/super-leonardo-p-zero-eq","kind":"node","lean":"MetaMathlibExt.superLeonardoP_zero_eq","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#super-leonardo-p-zero-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $p \\ge 1$ and $n$ and every real sequence $x$, the supergeneralized Leonardo $p$-numbers $\\mathcal{L}_{p,0,x}$ with $k = 0$ satisfy $\\mathcal{L}_{p,0,x}(n) = x_n$ if $n \\le p$, and if $n > p$, with $F_p$ denoting the Fibonacci $p$-numbers,","title":"Homogeneous supergeneralized Leonardo p-numbers via Fibonacci p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/super-leonardo-p-one-succ-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.superLeonardoP_one_succ_eq_sum_choose","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#super-leonardo-p-one-succ-eq-sum-choose","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"Let $p \\ge 1$ be a natural number and $y$ the real sequence $y_i = i + 1$, so that the initial values are $y_0, \\dots, y_p = 1, \\dots, p + 1$. For every natural number $n$, the supergeneralized Leonardo $p$-numbers $\\mathcal{L}_{p,1,y}$ with $k = 1$ and initial values $y$ satisfy","title":"Binomial sum for supergeneralized Leonardo p-numbers with linear initial values"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/one-add-sum-choose-add-sum-choose","kind":"node","lean":"MetaMathlibExt.one_add_sum_choose_add_sum_choose","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#one-add-sum-choose-add-sum-choose","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $J$, $n$ and $p$,","title":"Sch\u00fctz\u2013Kelly binomial sum identity"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/super-leonardo-p-eq-fibonacci-p-sum","kind":"node","lean":"MetaMathlibExt.superLeonardoP_eq_fibonacciP_sum","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#super-leonardo-p-eq-fibonacci-p-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $p \\ge 1$, $k$ and $n$ with $n > p$ and every real sequence $x$, the supergeneralized Leonardo $p$-numbers $\\mathcal{L}_{p,k,x}$ and the Fibonacci $p$-numbers $F_p$ satisfy","title":"Supergeneralized Leonardo p-numbers via Fibonacci p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/fibonacci-p-succ-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.fibonacciP_succ_eq_sum_choose","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#fibonacci-p-succ-eq-sum-choose","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $p \\ge 1$ and $n$, the Fibonacci $p$-numbers $F_p$ satisfy","title":"Binomial sum formula for Fibonacci p-numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/super-leonardo-p-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.superLeonardoP_eq_sum_choose","parent":6570,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers/index.html#super-leonardo-p-eq-sum-choose","scope":"NumberTheory/linear-recurrences/higher-order-sequences/leonardo-p-numbers","status":"can_prove","summary":"For all natural numbers $p \\ge 1$, $k$ and $n$ with $n > p$ and every real sequence $x$, the supergeneralized Leonardo $p$-numbers $\\mathcal{L}_{p,k,x}$ satisfy","title":"Explicit formula for the supergeneralized Leonardo p-numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences","status":"fully_proved","title":"Padovan and Perrin sequences"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences/shifted-padovan-sequence","kind":"node","lean":"MetaMathlibExt.shiftedPadovanSequence","parent":6650,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences/index.html#shifted-padovan-sequence","scope":"NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences","status":"fully_proved","summary":"The shifted Padovan sequence is the sequence of natural numbers $(a_n)_{n \\ge 0}$ with $a_0 = 1$, $a_1 = 0$, $a_2 = 1$ and $a_{n+3} = a_{n+1} + a_n$ for every $n \\ge 0$.","title":"Shifted Padovan sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences/shifted-padovan-sequence-add-three","kind":"node","lean":"MetaMathlibExt.shiftedPadovanSequence_add_three","parent":6650,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences/index.html#shifted-padovan-sequence-add-three","scope":"NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences","status":"fully_proved","summary":"For every natural number $n$, the shifted Padovan sequence $(a_n)_{n \\ge 0}$ satisfies $a_{n+3} = a_{n+1} + a_n$.","title":"Recurrence of the shifted Padovan sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences/perrin-sequence","kind":"node","lean":"MetaMathlibExt.perrinSequence","parent":6650,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences/index.html#perrin-sequence","scope":"NumberTheory/linear-recurrences/higher-order-sequences/padovan-and-perrin-sequences","status":"fully_proved","summary":"The Perrin sequence $E_n$, for natural numbers $n$, is defined by $E_0 = 3$, $E_1 = 0$, $E_2 = 2$ and $E_n = E_{n-2} + E_{n-3}$ for every $n \\ge 3$.","title":"Perrin sequence"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","title":"Self-convolutions of m-step Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/is-higher-fib-recurrence","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.IsHigherFibRecurrence","parent":6654,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(m\\)"],["text","-step Fibonacci recurrence"]],"source":"The $m$-step Fibonacci recurrence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#is-higher-fib-recurrence","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"A sequence $F : \\mathbb{Z} \\to \\mathbb{Q}$ satisfies the $m$-step Fibonacci recurrence if $F_n = F_{n-1} + F_{n-2} + \\dots + F_{n-m}$ for every integer $n$, that is, $F_n = \\sum_{i=0}^{m-1} F_{n-i-1}$. For $m = 0$ this says $F_n = 0$ for all $n$.","title":"The $m$-step Fibonacci recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/is-standard-higher-fib","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.IsStandardHigherFib","parent":6654,"reading_title":{"format":"inline-parts-v1","parts":[["text","The standard order-"],["math","\\(m\\)"],["text"," Fibonacci sequence"]],"source":"The standard order-$m$ Fibonacci sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#is-standard-higher-fib","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"A sequence $F : \\mathbb{Z} \\to \\mathbb{Q}$ is the standard order-$m$ Fibonacci sequence if it satisfies the $m$-step Fibonacci recurrence, $F_i = 0$ for every natural number $i < m - 1$, and $F_{m-1} = 1$, where $m - 1$ is computed in the integers.","title":"The standard order-$m$ Fibonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/weighted-higher-fib-sum","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.weightedHigherFibSum","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#weighted-higher-fib-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"For a sequence $F : \\mathbb{Z} \\to \\mathbb{Q}$, a natural number $m$, integers $N$ and $l$, and a rational number $q$, $S(m,N,l,q) = \\sum_{k=0}^{N} q^k F_{k+l}$; for $N < 0$ this is the empty sum $0$.","title":"Weighted sum of a sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/weighted-higher-fib-moment-sum","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.weightedHigherFibMomentSum","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#weighted-higher-fib-moment-sum","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"For a sequence $F : \\mathbb{Z} \\to \\mathbb{Q}$, a natural number $m$, integers $N$ and $l$, and a rational number $q$, $S_2(m,N,l,q) = \\sum_{k=0}^{N} k q^k F_{k+l}$; for $N < 0$ this is the empty sum $0$. The value does not depend on $m$.","title":"First-moment weighted sum"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/higher-fib-walk","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.higherFibWalk","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#higher-fib-walk","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"For a sequence $F : \\mathbb{Z} \\to \\mathbb{Q}$ and natural numbers $m$ and $n$, the shifted self-convolution is $w_n^{(m)} = \\sum_{j=0}^{n} F_{j+m-1} F_{n-j+m-1}$, with the indices computed in the integers.","title":"Shifted self-convolution"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/fib-q","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.fibQ","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#fib-q","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"For an integer $n$, $F_n$ is the $n$-th Fibonacci number as a rational number: $F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$ for $n \\ge 0$, and $F_{-n} = (-1)^{n+1} F_n$ for $n > 0$.","title":"Fibonacci numbers at integer indices"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/fib-walk-count","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.fibWalkCount","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#fib-walk-count","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"For $n \\ge 0$, $v_n$ is the rational number $v_n = \\sum_{j=0}^{n} F_{j+1} F_{n-j+1}$, where $F_i$ is the $i$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$).","title":"Fibonacci self-convolution"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/even-fib-walk-count","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.evenFibWalkCount","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#even-fib-walk-count","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"For $n \\ge 0$, $E_n$ is the rational number $\\sum F_{j+1} F_{n-j+1}$, the sum taken over the even integers $j$ with $0 \\le j \\le n$, where $F_i$ is the $i$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$).","title":"Even part of the Fibonacci self-convolution"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/odd-fib-walk-count","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.oddFibWalkCount","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#odd-fib-walk-count","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"fully_proved","summary":"For a natural number $n$, this is the sum of $F_{j+1} F_{n-j+1}$ over the odd $j$ with $0 \\le j \\le n$, where $F_k$ are the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{k+2} = F_{k+1} + F_k$). It is the part of $v_n = \\sum_{j=0}^{n} F_{j+1} F_{n-j+1}$ that comes from odd $j$.","title":"Odd part of the Fibonacci walk count"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/higher-fib-walk-recurrence","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.higherFibWalk_recurrence","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#higher-fib-walk-recurrence","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","summary":"Let $m \\ge 1$ and let $(F_n)_{n \\in \\mathbb{Z}}$ be a sequence of rational numbers with $F_n = F_{n-1} + \\dots + F_{n-m}$ for every integer $n$, $F_i = 0$ for $0 \\le i < m - 1$ and $F_{m-1} = 1$, and for $n \\ge 0$ let $w_n = \\sum_{j=0}^{n} F_{j+m-1} F_{n-j+m-1}$. Then for every integer $n \\ge 0$, $(m+1) w_{n+1} - 2m w_n = (n+m+2) F_{n+m} - (n+m) F_{n+m-1}$.","title":"Recurrence for the shifted self-convolution of higher-order Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/weighted-higher-fib-sum-eq","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.weightedHigherFibSum_eq","parent":6654,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form for weighted sums of an "],["math","\\(m\\)"],["text","-step Fibonacci sequence"]],"source":"Closed form for weighted sums of an $m$-step Fibonacci sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#weighted-higher-fib-sum-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","summary":"Let $m \\ge 1$ and let $F : \\mathbb{Z} \\to \\mathbb{Q}$ satisfy the $m$-step Fibonacci recurrence $F_k = F_{k-1} + \\dots + F_{k-m}$ for every integer $k$. For every natural number $n$, every integer $l$ and every rational number $q$ with $q^{m+1} - 2q + 1 \\ne 0$, the weighted sum $S(m,n,l,q) = \\sum_{k=0}^{n} q^k F_{k+l}$ equals","title":"Closed form for weighted sums of an $m$-step Fibonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/higher-fib-walk-eq-weighted-sums","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.higherFibWalk_eq_weighted_sums","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#higher-fib-walk-eq-weighted-sums","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","summary":"Let $m \\ge 1$ and let $(F_n)_{n \\in \\mathbb{Z}}$ be a sequence of rational numbers with $F_n = F_{n-1} + \\dots + F_{n-m}$ for every integer $n$, $F_i = 0$ for $0 \\le i < m - 1$ and $F_{m-1} = 1$. For $n \\ge 0$ let $w_n = \\sum_{j=0}^{n} F_{j+m-1} F_{n-j+m-1}$; for integers $N$, $l$ and rational $q$ let $S(N, l, q) = \\sum_{k=0}^{N} q^k F_{k+l}$ and $S_2(N, l, q) = \\sum_{k=0}^{N} k q^k F_{k+l}$, both equal to $0$ when\u2026","title":"Explicit formula for the shifted self-convolution of higher-order Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/higher-fib-self-convolution-eq-weighted","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.higherFib_selfConvolution_eq_weighted_sums","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#higher-fib-self-convolution-eq-weighted","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","summary":"Let $m \\ge 1$ and let $(F_n)_{n \\in \\mathbb{Z}}$ be a sequence of rational numbers with $F_n = F_{n-1} + \\dots + F_{n-m}$ for every integer $n$, $F_i = 0$ for $0 \\le i < m - 1$ and $F_{m-1} = 1$. For $p \\ge 0$ let $w_p = \\sum_{j=0}^{p} F_{j+m-1} F_{p-j+m-1}$; for integers $N$, $l$ and rational $q$ let $S(N, l, q) = \\sum_{k=0}^{N} q^k F_{k+l}$ and $S_2(N, l, q) = \\sum_{k=0}^{N} k q^k F_{k+l}$, both equal to $0$ when\u2026","title":"Self-convolution of higher-order Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/even-fib-walk-count-odd-length-eq-odd-fi","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.evenFibWalkCount_odd_length_eq_oddFibWalkCount","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#even-fib-walk-count-odd-length-eq-odd-fi","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$) and, for $n \\ge 0$, let $E_n$ and $O_n$ be the sums of $F_{j+1} F_{n-j+1}$ over the even and over the odd integers $j$ with $0 \\le j \\le n$. For every integer $k \\ge 1$, $E_{2k-1} = O_{2k-1}$.","title":"Even and odd parts of the Fibonacci self-convolution agree at odd length"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/even-fib-walk-count-even-length-eq-odd-a","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.evenFibWalkCount_even_length_eq_odd_add_fib","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#even-fib-walk-count-even-length-eq-odd-a","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$) and, for $n \\ge 0$, let $E_n$ and $O_n$ be the sums of $F_{j+1} F_{n-j+1}$ over the even and over the odd integers $j$ with $0 \\le j \\le n$. For every integer $k \\ge 1$, $E_{2k} = O_{2k} + F_{2k+2}$.","title":"Even and odd parts of the Fibonacci self-convolution at even length"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/fib-walk-counts-odd-length-eq-half","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.fibWalkCounts_odd_length_eq_half","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#fib-walk-counts-odd-length-eq-half","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$) and, for $n \\ge 0$, let $v_n = \\sum_{j=0}^{n} F_{j+1} F_{n-j+1}$, and let $E_n$ and $O_n$ be the sums of the same terms $F_{j+1} F_{n-j+1}$ over the even and over the odd integers $j$ with $0 \\le j \\le n$. For every integer $k \\ge 1$, $E_{2k-1} = v_{2k-1}/2$ and $O_{2k-1} = v_{2k-1}/2$.","title":"Halving the Fibonacci self-convolution at odd length"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/fib-walk-counts-even-length-eq","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.HigherFibonacciWalksWanted.fibWalkCounts_even_length_eq","parent":6654,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers/index.html#fib-walk-counts-even-length-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/self-convolutions-of-m-step-fibonacci-numbers","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$) and, for $n \\ge 0$, let $v_n = \\sum_{j=0}^{n} F_{j+1} F_{n-j+1}$, and let $E_n$ and $O_n$ be the sums of the same terms $F_{j+1} F_{n-j+1}$ over the even and over the odd integers $j$ with $0 \\le j \\le n$. For every integer $k \\ge 1$, $E_{2k} = (v_{2k} + F_{2k+2})/2$ and $O_{2k} = (v_{2k} - F_{2k+2})/2$.","title":"Halving the Fibonacci self-convolution at even length"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num","status":"can_prove","title":"Sums of squares of Tribonacci and Tetranacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/four-mul-sum-tribonacci-mul-sub-two","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.TribonacciTetranacciSumsOfSquaresWanted.four_mul_sum_tribonacci_mul_sub_two","parent":6672,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/index.html#four-mul-sum-tribonacci-mul-sub-two","scope":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num","status":"can_prove","summary":"Let $T \\colon \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $T(m+3) = T(m+2) + T(m+1) + T(m)$ for every integer $m$. Then for every natural number $n$,","title":"Sum of products of Tribonacci terms two apart"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/four-mul-sum-tribonacci-sq","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.TribonacciTetranacciSumsOfSquaresWanted.four_mul_sum_tribonacci_sq","parent":6672,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/index.html#four-mul-sum-tribonacci-sq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num","status":"can_prove","summary":"Let $T \\colon \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $T(m+3) = T(m+2) + T(m+1) + T(m)$ for every integer $m$. Then for every natural number $n$,","title":"Sum of squares of a Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/three-mul-sum-tetranacci-mul-sub-two-add","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.TribonacciTetranacciSumsOfSquaresWanted.three_mul_sum_tetranacci_mul_sub_two_add_sub_three","parent":6672,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/index.html#three-mul-sum-tetranacci-mul-sub-two-add","scope":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num","status":"can_prove","summary":"Let $Q \\colon \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $Q(m+4) = Q(m+3) + Q(m+2) + Q(m+1) + Q(m)$ for every integer $m$. Then for every natural number $n$,","title":"Sum of products of Tetranacci terms two and three apart"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/three-mul-sum-tetranacci-sq","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.TribonacciTetranacciSumsOfSquaresWanted.three_mul_sum_tetranacci_sq","parent":6672,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num/index.html#three-mul-sum-tetranacci-sq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/sums-of-squares-of-tribonacci-and-tetranacci-num","status":"can_prove","summary":"Let $Q \\colon \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $Q(m+4) = Q(m+3) + Q(m+2) + Q(m+1) + Q(m)$ for every integer $m$. Then for every natural number $n$,","title":"Sum of squares of a Tetranacci sequence"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers","status":"fully_proved","title":"Tetranacci and pentanacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers/pentanacci-number","kind":"node","lean":"MetaMathlibExt.pentanacciNumber","parent":6677,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers/index.html#pentanacci-number","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers","status":"fully_proved","summary":"The pentanacci numbers $a_n$, for natural numbers $n$, are defined by $a_0 = 1$, $a_1 = 1$, $a_2 = 2$, $a_3 = 4$, $a_4 = 8$ and $a_{n+5} = a_{n+4} + a_{n+3} + a_{n+2} + a_{n+1} + a_n$ for every $n$, so each term from index $5$ on is the sum of the five terms before it.","title":"Pentanacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers/tetranacci","kind":"node","lean":"MetaMathlibExt.tetranacci","parent":6677,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers/index.html#tetranacci","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tetranacci-and-pentanacci-numbers","status":"fully_proved","summary":"The Tetranacci sequence `0, 0, 0, 1, 1, 2, 4, 8, \u2026` (OEIS A000078), in which each term after the first four is the sum of its four predecessors.","title":"Tetranacci sequence"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","title":"Tribonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci","kind":"node","lean":"MetaMathlibExt.tribonacci","parent":6680,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"The Tribonacci sequence $(T_n)_{n \\ge 0}$ is defined by $T_0 = 0$, $T_1 = 0$, $T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$; it begins $0, 0, 1, 1, 2, 4, \\dots$ (OEIS A000073).","title":"Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-with-initial-values111","kind":"node","lean":"MetaMathlibExt.tribonacciWithInitialValues111","parent":6680,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-with-initial-values111","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"The sequence $(T_n)_{n \\ge 0}$ of natural numbers defined by $T_0 = T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for every $n \\ge 0$. It begins $1, 1, 1, 3, 5, 9, 17, \\dots$ (OEIS A000213).","title":"Tribonacci sequence with initial values 1, 1, 1"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/generalized-tribonacci","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci","parent":6680,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#generalized-tribonacci","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"For integers $s_0, s_1, s_2$, the generalized Tribonacci sequence $(T_n)_{n \\ge 0}$ with initial values $s_0, s_1, s_2$ is the sequence of integers defined by $T_0 = s_0$, $T_1 = s_1$, $T_2 = s_2$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for every $n \\ge 0$.","title":"Generalized Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/generalized-tribonacci-zero","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci_zero","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial value "],["math","\\(T_0 = s_0\\)"],["text"," of the generalized Tribonacci sequence"]],"source":"Initial value $T_0 = s_0$ of the generalized Tribonacci sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#generalized-tribonacci-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"For all integers $s_0, s_1, s_2$, the generalized Tribonacci sequence $(T_n)_{n \\ge 0}$ with initial values $s_0, s_1, s_2$ satisfies $T_0 = s_0$.","title":"Initial value $T_0 = s_0$ of the generalized Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/generalized-tribonacci-one","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci_one","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial value "],["math","\\(T_1 = s_1\\)"],["text"," of the generalized Tribonacci sequence"]],"source":"Initial value $T_1 = s_1$ of the generalized Tribonacci sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#generalized-tribonacci-one","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"For all integers $s_0, s_1, s_2$, the generalized Tribonacci sequence $(T_n)_{n \\ge 0}$ with initial values $s_0, s_1, s_2$ satisfies $T_1 = s_1$.","title":"Initial value $T_1 = s_1$ of the generalized Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/generalized-tribonacci-two","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci_two","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial value "],["math","\\(T_2 = s_2\\)"],["text"," of the generalized Tribonacci sequence"]],"source":"Initial value $T_2 = s_2$ of the generalized Tribonacci sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#generalized-tribonacci-two","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"For all integers $s_0, s_1, s_2$, the generalized Tribonacci sequence $(T_n)_{n \\ge 0}$ with initial values $s_0, s_1, s_2$ satisfies $T_2 = s_2$.","title":"Initial value $T_2 = s_2$ of the generalized Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/generalized-tribonacci-add-three","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci_add_three","parent":6680,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#generalized-tribonacci-add-three","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"For all integers $s_0, s_1, s_2$ and every $n \\ge 0$, the generalized Tribonacci sequence $(T_n)_{n \\ge 0}$ with initial values $s_0, s_1, s_2$ satisfies $T_{n+3} = T_{n+2} + T_{n+1} + T_n$.","title":"Recurrence of the generalized Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/generalized-tribonacci-zero-zero-one-eq","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci_zero_zero_one_eq_tribonacci","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generalized Tribonacci sequence with initial values "],["math","\\(0, 0, 1\\)"],["text"," is the Tribonacci sequence"]],"source":"The generalized Tribonacci sequence with initial values $0, 0, 1$ is the Tribonacci sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#generalized-tribonacci-zero-zero-one-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"The generalized Tribonacci sequence with initial values $0, 0, 1$ is the Tribonacci sequence. For every $n \\ge 0$, the $n$-th term of the generalized Tribonacci sequence with initial values $0, 0, 1$ equals the $n$-th Tribonacci number $t_n$ (where $t_0 = t_1 = 0$ and $t_2 = 1$), viewed as an integer.","title":"The generalized Tribonacci sequence with initial values $0, 0, 1$ is the Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/generalized-tribonacci-one-one-one-eq-tr","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci_one_one_one_eq_tribonacciWithInitialValues111","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer and natural Tribonacci sequences with initial values "],["math","\\(1, 1, 1\\)"],["text"," agree"]],"source":"Integer and natural Tribonacci sequences with initial values $1, 1, 1$ agree"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#generalized-tribonacci-one-one-one-eq-tr","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"For every $n \\ge 0$, the $n$-th term of the generalized Tribonacci sequence with initial values $1, 1, 1$ equals the $n$-th term of the Tribonacci sequence of natural numbers with initial values $1, 1, 1$, viewed as an integer.","title":"Integer and natural Tribonacci sequences with initial values $1, 1, 1$ agree"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/generalized-tribonacci-zero-one-one-eq-t","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci_zero_one_one_eq_tribonacci_succ","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generalized Tribonacci sequence with initial values "],["math","\\(0, 1, 1\\)"],["text"," is the shifted Tribonacci sequence"]],"source":"The generalized Tribonacci sequence with initial values $0, 1, 1$ is the shifted Tribonacci sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#generalized-tribonacci-zero-one-one-eq-t","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"The generalized Tribonacci sequence with initial values $0, 1, 1$ is the shifted Tribonacci sequence. For every $n \\ge 0$, the $n$-th term of the generalized Tribonacci sequence with initial values $0, 1, 1$ equals the Tribonacci number $t_{n+1}$, viewed as an integer, where $t_0 = t_1 = 0$ and $t_2 = 1$.","title":"The generalized Tribonacci sequence with initial values $0, 1, 1$ is the shifted Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-backward","kind":"node","lean":"MetaMathlibExt.tribonacciBackward","parent":6680,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-backward","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"The integer sequence $(B_m)_{m \\ge 0}$ is defined by $B_0 = 0$, $B_1 = 0$, $B_2 = 1$ and $B_{m+3} = B_m - B_{m+1} - B_{m+2}$ for every $m \\ge 0$.","title":"Backward half of the two-sided Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int","kind":"node","lean":"MetaMathlibExt.tribonacciInt","parent":6680,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"The integer sequence $(T_n)_{n \\in \\mathbb{Z}}$ is defined on the nonnegative indices by $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, and on the negative indices by $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$, the recurrence solved for its lowest term.","title":"Two-sided Tribonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-zero","kind":"node","lean":"MetaMathlibExt.tribonacciInt_zero","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-sided Tribonacci sequence at "],["math","\\(0\\)"]],"source":"The two-sided Tribonacci sequence at $0$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Then $T_0 = 0$.","title":"The two-sided Tribonacci sequence at $0$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-one","kind":"node","lean":"MetaMathlibExt.tribonacciInt_one","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-sided Tribonacci sequence at "],["math","\\(1\\)"]],"source":"The two-sided Tribonacci sequence at $1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-one","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Then $T_1 = 1$.","title":"The two-sided Tribonacci sequence at $1$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-two","kind":"node","lean":"MetaMathlibExt.tribonacciInt_two","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-sided Tribonacci sequence at "],["math","\\(2\\)"]],"source":"The two-sided Tribonacci sequence at $2$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-two","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Then $T_2 = 1$.","title":"The two-sided Tribonacci sequence at $2$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-neg-one","kind":"node","lean":"MetaMathlibExt.tribonacciInt_neg_one","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-sided Tribonacci sequence at "],["math","\\(-1\\)"]],"source":"The two-sided Tribonacci sequence at $-1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-neg-one","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Then $T_{-1} = 0$.","title":"The two-sided Tribonacci sequence at $-1$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-neg-two","kind":"node","lean":"MetaMathlibExt.tribonacciInt_neg_two","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-sided Tribonacci sequence at "],["math","\\(-2\\)"]],"source":"The two-sided Tribonacci sequence at $-2$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-neg-two","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Then $T_{-2} = 1$.","title":"The two-sided Tribonacci sequence at $-2$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-neg-three","kind":"node","lean":"MetaMathlibExt.tribonacciInt_neg_three","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-sided Tribonacci sequence at "],["math","\\(-3\\)"]],"source":"The two-sided Tribonacci sequence at $-3$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-neg-three","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Then $T_{-3} = -1$.","title":"The two-sided Tribonacci sequence at $-3$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-neg-seven","kind":"node","lean":"MetaMathlibExt.tribonacciInt_neg_seven","parent":6680,"reading_title":{"format":"inline-parts-v1","parts":[["text","The two-sided Tribonacci sequence at "],["math","\\(-7\\)"]],"source":"The two-sided Tribonacci sequence at $-7$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-neg-seven","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Then $T_{-7} = 1$.","title":"The two-sided Tribonacci sequence at $-7$"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-add-three","kind":"node","lean":"MetaMathlibExt.tribonacciInt_add_three","parent":6680,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-add-three","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Then $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for every integer $n$, including the three indices $n = -3, -2, -1$ at which the recurrence crosses zero.","title":"Tribonacci recurrence for the two-sided sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/tribonacci-int-nat-eq","kind":"node","lean":"MetaMathlibExt.tribonacciInt_nat_eq","parent":6680,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers/index.html#tribonacci-int-nat-eq","scope":"NumberTheory/linear-recurrences/higher-order-sequences/tribonacci-numbers","status":"fully_proved","summary":"On nonnegative indices the two-sided Tribonacci sequence is the shifted Tribonacci sequence. Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence: $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$, while $T_{-1} = 0$, $T_{-2} = 1$ and $T_{-m-3} = T_{-m} - T_{-m-1} - T_{-m-2}$ for $m \\ge 0$. Let $(t_n)_{n \\ge 0}$ be the Tribonacci sequence with $t_0 = t_1 = 0$, $t_2 = 1$\u2026","title":"On nonnegative indices the two-sided Tribonacci sequence is the shifted Tribonacci sequence"},{"data_url":"scopes/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers.json","id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","kind":"scope","navigation":"module","parent":6443,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"can_prove","title":"Zeta functions of generalized Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/is-lgeneralized-fibonacci","kind":"node","lean":"MetaMathlibExt.IsLGeneralizedFibonacci","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#is-lgeneralized-fibonacci","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $\\ell$ be a natural number. A function $F : \\mathbb{Z} \\to \\mathbb{Z}$ is an $\\ell$-generalized Fibonacci sequence when $F(n) = 0$ for every integer $n$ with $2 - \\ell \\le n \\le 0$, $F(1) = 1$, and $F(n) = \\sum_{i=0}^{\\ell-1} F(n-1-i)$ for every integer $n > 1$.","title":"\u2113-generalized Fibonacci sequence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-characteristic","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciCharacteristic","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-characteristic","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"For a natural number $\\ell$, the characteristic polynomial of the $\\ell$-generalized Fibonacci recurrence is the polynomial $\\varphi_\\ell(X) = X^\\ell - \\sum_{i=0}^{\\ell-1} X^i$ in $\\mathbb{C}[X]$; for $\\ell \\ge 2$ this is $X^\\ell - X^{\\ell-1} - \\cdots - X - 1$.","title":"Characteristic polynomial of the \u2113-generalized Fibonacci recurrence"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-coefficient","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciCoefficient","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-coefficient","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"For a natural number $\\ell$ and a complex number $\\beta$, the Dresden\u2013Du coefficient is the complex number $c(\\beta) = \\frac{\\beta - 1}{2 + (\\ell + 1)(\\beta - 2)}$, taken to be $0$ when the denominator vanishes.","title":"Dresden\u2013Du coefficient"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-other-roots","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciOtherRoots","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-other-roots","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"For a natural number $\\ell$ and a real number $\\alpha$, this is the finite set of distinct complex roots of the characteristic polynomial $\\varphi_\\ell(X) = X^\\ell - \\sum_{i=0}^{\\ell-1} X^i$ other than $\\alpha$. In applications $\\alpha$ is the dominant real root of $\\varphi_\\ell$.","title":"Other roots of the \u2113-generalized Fibonacci characteristic polynomial"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-pole","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZetaPole","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-pole","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $\\ell$ be a natural number, let $\\alpha$ be a real number, and let $R$ be the set of complex roots of $X^\\ell - \\sum_{i\\lt \\ell} X^i$ other than $\\alpha$. For a function $\\kappa$ from $\\mathbb{C}$ to the natural numbers and an integer $n$, the possible pole indexed by $(\\kappa, n)$ is","title":"Possible pole of the \u2113-generalized Fibonacci zeta function at a given index"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-characteristic-t","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciCharacteristic_two","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-characteristic-t","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"For $\\ell = 2$ the characteristic polynomial $\\varphi_2$ of the $\\ell$-generalized Fibonacci recurrence is the Fibonacci polynomial $X^2 - X - 1$ in $\\mathbb{C}[X]$.","title":"The \u2113-generalized Fibonacci characteristic polynomial for \u2113 = 2"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-pole-zero","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZetaPole_zero","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-pole-zero","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"For every natural number $\\ell$ and every real number $\\alpha$, the possible pole $s_{\\kappa,n}$ of the $\\ell$-generalized Fibonacci zeta function indexed by the identically zero function $\\kappa$ and $n = 0$ equals $0$.","title":"The possible pole at the zero index is zero"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-pole-set","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZetaPoleSet","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-pole-set","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"For a natural number $\\ell$ and a real number $\\alpha$, let $R$ be the set of distinct complex roots of the characteristic polynomial $X^\\ell - \\sum_{i=0}^{\\ell-1} X^i$ other than $\\alpha$. The set $S$ of possible poles is the set of all complex numbers $-K + (2n\\pi i + L)/\\log\\alpha$, where $n$ is an integer, $(\\kappa_\\beta)_{\\beta \\in R}$ is a family of natural numbers, $K = \\sum_{\\beta \\in R} \\kappa_\\beta$ and\u2026","title":"Set of possible poles of the \u2113-generalized Fibonacci zeta function"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-residue","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZetaResidue","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-residue","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"Let $\\ell$ be a natural number, let $\\alpha$ be a real number, let $R$ be the set of complex roots of $X^\\ell - \\sum_{i\\lt \\ell} X^i$ other than $\\alpha$, let $c(\\beta) = (\\beta - 1)/(2 + (\\ell + 1)(\\beta - 2))$ be the Dresden\u2013Du coefficient, and put $d = c(\\alpha)/\\alpha$. For a function $\\kappa$ from $\\mathbb{C}$ to the natural numbers and an integer $n$, let $k = \\sum_{\\beta \\in R} \\kappa(\\beta)$ and let $s_0$\u2026","title":"Residue expression of the \u2113-generalized Fibonacci zeta function"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-corollary-p","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZetaCorollaryPole","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-corollary-p","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"fully_proved","summary":"For natural numbers $\\ell$ and $k$, a real number $\\alpha$ and an integer $n$, let $q$ be the quotient of $k$ by $\\ell - 1$ in natural-number division, with $q = 0$ when $\\ell \\le 1$ (so $q = k/(\\ell - 1)$ when $\\ell \\ge 2$ and $\\ell - 1$ divides $k$). The corollary pole is the complex number $-(k + q) + i\\pi(2n + q)/\\log \\alpha$ if $\\ell$ is even and $-(k + q) + 2\\pi i n/\\log \\alpha$ if $\\ell$ is odd, where $\\log$\u2026","title":"Corollary pole of the \u2113-generalized Fibonacci zeta function"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-summable","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZeta_summable","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-summable","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $\\ell \\ge 2$ and let $F$ be an $\\ell$-generalized Fibonacci sequence. For every complex number $s$ with $\\operatorname{Re} s > 0$, the series $\\sum_{n \\ge 1} |F_n^{-s}|$ converges, where $F_n^{-s}$ is the principal complex power of $F_n$ viewed as a complex number.","title":"Absolute convergence of the \u2113-generalized Fibonacci zeta series"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-meromorphic","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZeta_meromorphicContinuation","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-meromorphic","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $\\ell \\ge 2$, let $F$ be an $\\ell$-generalized Fibonacci sequence, let $\\alpha > 1$ be a real number with $\\alpha^\\ell = \\sum_{i=0}^{\\ell-1} \\alpha^i$, and let $S$ be the set of possible poles determined by $\\ell$ and $\\alpha$. Then there is a function $f : \\mathbb{C} \\to \\mathbb{C}$ such that $f(s) = \\sum_{n \\ge 1} F(n)^{-s}$ for every complex $s$ with $\\operatorname{Re} s > 0$, $f$ is complex differentiable\u2026","title":"Meromorphic continuation of the \u2113-generalized Fibonacci zeta function"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-residue-2","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZeta_residue","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-residue-2","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $\\ell \\ge 2$, let $F$ be an $\\ell$-generalized Fibonacci sequence, and let $\\alpha > 1$ be a real number with $\\alpha^\\ell = \\sum_{i\\lt \\ell} \\alpha^i$. Let $R$ be the set of complex roots of $X^\\ell - \\sum_{i\\lt \\ell} X^i$ other than $\\alpha$, let $s_{\\kappa,n}$ be the possible pole and $\\rho_{\\kappa,n}$ the residue expression indexed by a function $\\kappa$ from $\\mathbb{C}$ to the natural numbers and an\u2026","title":"Residues of the \u2113-generalized Fibonacci zeta function"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-corollary-p-2","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZeta_corollaryPole_mem","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-corollary-p-2","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $\\ell \\ge 2$, let $\\alpha > 1$ be a real number with $\\alpha^\\ell = \\sum_{i=0}^{\\ell-1} \\alpha^i$, and let $k$ be a natural number divisible by $\\ell - 1$; write $q = k/(\\ell - 1)$. Then for every integer $n$, the point $-(k + q) + i\\pi(2n + q)/\\log \\alpha$ if $\\ell$ is even, and $-(k + q) + 2\\pi i n/\\log \\alpha$ if $\\ell$ is odd, belongs to the set $S$ of possible poles of the $\\ell$-generalized Fibonacci zeta\u2026","title":"Explicit possible poles of the \u2113-generalized Fibonacci zeta function"},{"id":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/l-generalized-fibonacci-zeta-neg-int-rat","kind":"node","lean":"MetaMathlibExt.lGeneralizedFibonacciZeta_negInt_rational","parent":6702,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers/index.html#l-generalized-fibonacci-zeta-neg-int-rat","scope":"NumberTheory/linear-recurrences/higher-order-sequences/zeta-functions-of-generalized-fibonacci-numbers","status":"can_prove","summary":"Let $\\ell \\ge 2$, let $F$ be an $\\ell$-generalized Fibonacci sequence, let $\\alpha > 1$ be a real number with $\\alpha^\\ell = \\sum_{i=0}^{\\ell-1} \\alpha^i$, and let $S$ be the set of possible poles determined by $\\ell$ and $\\alpha$. Let $m$ be a positive integer and let $f : \\mathbb{C} \\to \\mathbb{C}$ be a function that satisfies $f(s) = \\sum_{n \\ge 1} F(n)^{-s}$ whenever $\\operatorname{Re} s > 0$ and is complex\u2026","title":"Rational values of the \u2113-generalized Fibonacci zeta function at negative integers"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-polynomials.json","id":"NumberTheory/linear-recurrences/fibonacci-polynomials","kind":"scope","parent":5991,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials","status":"can_prove","summary":"Fibonacci, Lucas, Jacobsthal and gibonacci polynomials, their sums, squares and generating function values.","title":"Fibonacci and Lucas polynomials"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials.json","id":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials","kind":"scope","navigation":"module","parent":6718,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials","status":"fully_proved","title":"Generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/generalized-fibonacci-polynomial","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciPolynomial","parent":6719,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/index.html#generalized-fibonacci-polynomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials","status":"fully_proved","summary":"Given polynomials $p_0, p_1, d, g \\in \\mathbb{Z}[X]$, the generalized Fibonacci polynomial sequence $(G_n)_{n \\ge 0}$ in $\\mathbb{Z}[X]$ is defined by $G_0 = p_0$, $G_1 = p_1$ and $G_{n+2} = d \\, G_{n+1} + g \\, G_n$ for every natural number $n$.","title":"Generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/generalized-fibonacci-polynomial-zero","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciPolynomial_zero","parent":6719,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/index.html#generalized-fibonacci-polynomial-zero","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials","status":"fully_proved","summary":"For all polynomials $p_0, p_1, d, g \\in \\mathbb{Z}[X]$, the generalized Fibonacci polynomial sequence $(G_n)$ with initial terms $p_0, p_1$ and recurrence coefficients $d, g$ has $G_0 = p_0$.","title":"Index-zero term of a generalized Fibonacci polynomial sequence"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/generalized-fibonacci-polynomial-one","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciPolynomial_one","parent":6719,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/index.html#generalized-fibonacci-polynomial-one","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials","status":"fully_proved","summary":"For all polynomials $p_0, p_1, d, g \\in \\mathbb{Z}[X]$, the generalized Fibonacci polynomial sequence $(G_n)$ with initial terms $p_0, p_1$ and recurrence coefficients $d, g$ has $G_1 = p_1$.","title":"Index-one term of a generalized Fibonacci polynomial sequence"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/generalized-fibonacci-polynomial-add-two","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciPolynomial_add_two","parent":6719,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials/index.html#generalized-fibonacci-polynomial-add-two","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/generalized-fibonacci-polynomials","status":"fully_proved","summary":"For all polynomials $p_0, p_1, d, g$ in $\\mathbb{Z}[X]$ and every natural number $n$, the generalized Fibonacci polynomial sequence $(G_m)$ with initial terms $p_0, p_1$ and recurrence coefficients $d, g$ satisfies $G_{n+2} = d \\, G_{n+1} + g \\, G_n$.","title":"Recurrence for generalized Fibonacci polynomials"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol.json","id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","kind":"scope","navigation":"module","parent":6718,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","title":"Catalan identities for generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/gfp-fib-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.gfp_fib_eq_sum_choose","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#gfp-fib-eq-sum-choose","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $d$ and $g$ be polynomials with integer coefficients, and let $F_n$ be the generalized Fibonacci polynomials defined by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = d F_{n+1} + g F_n$. Then for every $n \\ge 1$,","title":"Explicit formula for Fibonacci-type generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/gfp-lucas-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.gfp_lucas_eq_sum_choose","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#gfp-lucas-eq-sum-choose","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $\\alpha$ and $p_0$ be integers with $\\alpha p_0 = 2$, let $p_1$ and $g$ be polynomials with integer coefficients, and put $d = \\alpha p_1$. Let $L_n$ be the generalized Fibonacci polynomials defined by $L_0 = p_0$, $L_1 = p_1$ and $L_{n+2} = d L_{n+1} + g L_n$. Then for every $n \\ge 1$, writing $c_i = \\lfloor n \\binom{n-i}{i} / (n-i) \\rfloor$,","title":"Explicit formula for Lucas-type generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/choose-sub-sub-one-eq-two-zpow-mul-sum","kind":"node","lean":"MetaMathlibExt.choose_sub_sub_one_eq_two_zpow_mul_sum","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#choose-sub-sub-one-eq-two-zpow-mul-sum","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"For natural numbers $m < n$,","title":"Webb\u2013Parberry binomial identity"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/choose-two-mul-add-one-eq-alternating-su","kind":"node","lean":"MetaMathlibExt.choose_two_mul_add_one_eq_alternating_sum","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#choose-two-mul-add-one-eq-alternating-su","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"For natural numbers $n$ and $k$ with $k < \\lfloor (n-1)/2 \\rfloor$,","title":"Binomial coefficients of odd lower index as an alternating sum"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/choose-two-mul-eq-alternating-sum","kind":"node","lean":"MetaMathlibExt.choose_two_mul_eq_alternating_sum","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#choose-two-mul-eq-alternating-sum","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"For natural numbers $n$ and $k$ with $k < \\lfloor n/2 \\rfloor$,","title":"Binomial coefficients of even lower index as an alternating sum"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/gfp-fib-catalan","kind":"node","lean":"MetaMathlibExt.gfp_fib_catalan","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#gfp-fib-catalan","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $d$ and $g$ be polynomials with integer coefficients, and let $F_n$ be the generalized Fibonacci polynomials defined by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = d F_{n+1} + g F_n$. Then for every natural number $n$ (for $n = 0$ both sides are $0$),","title":"Catalan identity for Fibonacci-type generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/gfp-lucas-catalan","kind":"node","lean":"MetaMathlibExt.gfp_lucas_catalan","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#gfp-lucas-catalan","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $\\alpha$ and $p_0$ be integers with $\\alpha p_0 = 2$, let $p_1$ and $g$ be polynomials with integer coefficients, and put $d = \\alpha p_1$. Let $L_n$ be the generalized Fibonacci polynomials defined by $L_0 = p_0$, $L_1 = p_1$ and $L_{n+2} = d L_{n+1} + g L_n$. Then for every natural number $n$,","title":"Catalan identity for Lucas-type generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/alternating-sum-choose-mul-choose-sub","kind":"node","lean":"MetaMathlibExt.alternating_sum_choose_mul_choose_sub","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#alternating-sum-choose-mul-choose-sub","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $\\ell$, $n$, $j$ and $s$ be natural numbers with $1 \\le \\ell \\le n$, $j \\ge 1$ and $\\ell j + s \\le n$. Then","title":"Alternating sum of shifted binomial coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/floor-pow-binomial-sum-recurrence","kind":"node","lean":"MetaMathlibExt.floorPowBinomialSum_recurrence","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#floor-pow-binomial-sum-recurrence","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $R$ be a commutative ring, $t \\in R$ and $\\ell \\ge 1$ an integer, and for $m \\ge 0$ let $T_\\ell(m; t) = \\sum_{j=0}^{m} \\binom{m}{j} t^{\\lfloor j/\\ell \\rfloor}$. Then for every integer $n \\ge 0$,","title":"Recurrence for the floor-exponent binomial sums"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/alternating-choose-mul-sum-choose","kind":"node","lean":"MetaMathlibExt.alternating_choose_mul_sum_choose","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#alternating-choose-mul-sum-choose","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"For all natural numbers $\\ell$ and $n$,","title":"Alternating sum of binomial row sums"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/floor-pow-binomial-sum-generating-functi","kind":"node","lean":"MetaMathlibExt.floorPowBinomialSum_generatingFunction","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#floor-pow-binomial-sum-generating-functi","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $R$ be a commutative ring, $t \\in R$ and $\\ell \\ge 1$ an integer, and for $n \\ge 0$ let $T_\\ell(n; t) = \\sum_{i=0}^{n} \\binom{n}{i} t^{\\lfloor i/\\ell \\rfloor}$. Then in the ring $R[[x]]$ of formal power series,","title":"Generating function of the floor-exponent binomial sums"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/section-binomial-sum-generating-function","kind":"node","lean":"MetaMathlibExt.sectionBinomialSum_generatingFunction","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#section-binomial-sum-generating-function","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $R$ be a commutative ring, $t \\in R$, and let $\\ell$ and $s$ be natural numbers with $s < \\ell$. Put $T_{\\ell,s}(n; t) = \\sum_{i=0}^{n} \\binom{n}{\\ell i + s} t^i$. Then, as formal power series in $x$ over $R$,","title":"Generating function of the binomial section sums"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/gfp-fib-eval-eq-integral","kind":"node","lean":"MetaMathlibExt.gfp_fib_eval_eq_integral","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#gfp-fib-eval-eq-integral","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $d$ and $g$ be polynomials with integer coefficients, and let $F_n$ be the generalized Fibonacci polynomials defined by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = d F_{n+1} + g F_n$. If $x$ is a real number with $d(x)^2 + 4 g(x) \\ge 0$, then for every $n \\ge 2$,","title":"Integral representation of Fibonacci-type generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/fib-eq-integral-one-add-mul-sqrt-five","kind":"node","lean":"MetaMathlibExt.fib_eq_integral_one_add_mul_sqrt_five","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#fib-eq-integral-one-add-mul-sqrt-five","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"For every integer $n \\ge 1$, the $n$-th Fibonacci number $F_n$ (with $F_0 = 0$ and $F_1 = 1$) satisfies","title":"Stewart's integral formula for the Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/gfp-fib-binomial-sum-eq-integral","kind":"node","lean":"MetaMathlibExt.gfp_fib_binomial_sum_eq_integral","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#gfp-fib-binomial-sum-eq-integral","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $d$ and $g$ be polynomials with integer coefficients, let $F_m$ be the generalized Fibonacci polynomials with $F_0 = 0$, $F_1 = 1$ and $F_{m+2} = d F_{m+1} + g F_m$, and let $x$ be a real number with $d(x)^2 + 4 g(x) \\ge 0$ and $g(x) \\ne 0$. Let $\\Phi \\colon \\mathbb{Z} \\to \\mathbb{R}$ satisfy $\\Phi(m) = F_m(x)$ for $m \\ge 0$ and $\\Phi(m+2) = d(x) \\Phi(m+1) + g(x) \\Phi(m)$ for every integer $m$. Then for every\u2026","title":"Binomial sum of generalized Fibonacci polynomials as an integral"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/fib-binomial-sum-eq-integral","kind":"node","lean":"MetaMathlibExt.fib_binomial_sum_eq_integral","parent":6724,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol/index.html#fib-binomial-sum-eq-integral","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/catalan-identities-for-generalized-fibonacci-pol","status":"can_prove","summary":"Let $F \\colon \\mathbb{Z} \\to \\mathbb{R}$ be a function whose value $F_m$ at every $m \\ge 0$ is the $m$-th Fibonacci number (with $F_0 = 0$ and $F_1 = 1$) and which satisfies $F_{m+2} = F_{m+1} + F_m$ for every integer $m$. Then for every real $k \\ge 2$ and every integer $n \\ge 1$,","title":"Binomial sum of Fibonacci numbers as an integral"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin.json","id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","kind":"scope","navigation":"module","parent":6718,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","title":"Integer values of Fibonacci polynomial generating functions"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/fibonacci-poly-gen-fun-mem-int-poly-iff","kind":"node","lean":"MetaMathlibExt.fibonacciPolyGenFun_mem_intPoly_iff","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#fibonacci-poly-gen-fun-mem-int-poly-iff","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Let $a$ and $b$ be positive integers with $b \\mid a$, let $F_i(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$ and $F_{i+2} = ax\\,F_{i+1} + b\\,F_i$ for all $i \\ge 0$, and let $f(x, t) = t / (1 - axt - bt^2)$. For a rational function $q(x) \\in \\mathbb{Q}(x)$, the rational function $f(x, q(x))$ is a polynomial with integer coefficients if and only if $q(x) = F_i(x) / F_{i+1}(x)$ for some\u2026","title":"Integer-polynomial values of the Fibonacci polynomial generating function"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/lucas-poly-gen-fun-mem-int-poly-iff","kind":"node","lean":"MetaMathlibExt.lucasPolyGenFun_mem_intPoly_iff","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#lucas-poly-gen-fun-mem-int-poly-iff","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Let $a$ and $b$ be positive integers with $b \\mid a$, let $F_i(x), L_i(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, $L_1 = ax$ and $G_{i+2} = ax\\,G_{i+1} + b\\,G_i$ for $G = F, L$ and all $i \\ge 0$, and let $l(x, t) = (2 - axt) / (1 - axt - bt^2)$. For a rational function $q(x) \\in \\mathbb{Q}(x)$, the rational function $l(x, q(x))$ is a polynomial with integer coefficients if and\u2026","title":"Integer-polynomial values of the Lucas polynomial generating function"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/fibonacci-gen-fun-eval-is-int-iff","kind":"node","lean":"MetaMathlibExt.fibonacciGenFun_eval_isInt_iff","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#fibonacci-gen-fun-eval-is-int-iff","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Let $a$ and $b$ be positive integers with $b \\mid a$, and let $F_i(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$ and $F_{i+2} = ax\\,F_{i+1} + b\\,F_i$ for all $i \\ge 0$. For a rational number $q$, the number $q / (1 - aq - bq^2)$ is an integer if and only if $q = F_i(1) / F_{i+1}(1)$ for some $i \\in \\mathbb{N}$ or $q = -F_{i+1}(1) / (b F_i(1))$ for some $i \\ge 1$.","title":"Integer values of the generalized Fibonacci generating function at rationals"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/lucas-gen-fun-eval-is-int-iff","kind":"node","lean":"MetaMathlibExt.lucasGenFun_eval_isInt_iff","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#lucas-gen-fun-eval-is-int-iff","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Let $a$ and $b$ be positive integers with $b \\mid a$, and let $F_i(x), L_i(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, $L_1 = ax$ and $G_{i+2} = ax\\,G_{i+1} + b\\,G_i$ for $G = F, L$ and all $i \\ge 0$. For a rational number $q$, the number $(2 - aq) / (1 - aq - bq^2)$ is an integer if and only if $q$ is one of $F_i(1) / F_{i+1}(1)$, $L_i(1) / L_{i+1}(1)$ and\u2026","title":"Integer values of the generalized Lucas generating function at rationals"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/fibonacci-gen-fun-eval-is-int-iff-of-abs","kind":"node","lean":"MetaMathlibExt.fibonacciGenFun_eval_isInt_iff_of_abs_lt_radius","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#fibonacci-gen-fun-eval-is-int-iff-of-abs","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Integer values of the generalized Fibonacci generating function in its interval of convergence. Let $a$ and $b$ be positive integers with $b \\mid a$, and let $F_i(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$ and $F_{i+2} = ax\\,F_{i+1} + b\\,F_i$ for all $i \\ge 0$. For a rational number $q$ with $|q| < 2 / (a + \\sqrt{a^2 + 4b})$, the radius of convergence of the power series of\u2026","title":"Integer values of the generalized Fibonacci generating function in its interval of convergence"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/lucas-gen-fun-eval-is-int-iff-of-abs-lt","kind":"node","lean":"MetaMathlibExt.lucasGenFun_eval_isInt_iff_of_abs_lt_radius","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#lucas-gen-fun-eval-is-int-iff-of-abs-lt","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Integer values of the generalized Lucas generating function in its interval of convergence. Let $a$ and $b$ be positive integers with $a \\ne 1$ and $b \\mid a$, and let $F_i(x), L_i(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, $L_1 = ax$ and $G_{i+2} = ax\\,G_{i+1} + b\\,G_i$ for $G = F, L$ and all $i \\ge 0$. For a rational number $q$ with $|q| < 2 / (a + \\sqrt{a^2 + 4b})$, the radius\u2026","title":"Integer values of the generalized Lucas generating function in its interval of convergence"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/lucas-number-gen-fun-eval-is-int-iff-of","kind":"node","lean":"MetaMathlibExt.lucasNumberGenFun_eval_isInt_iff_of_abs_lt_radius","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#lucas-number-gen-fun-eval-is-int-iff-of","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers and $L_n$ the Lucas numbers, given by $L_0 = 2$, $L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n$. For a rational number $q$ with $|q| < 2 / (1 + \\sqrt 5)$, the number $(2 - q) / (1 - q - q^2)$ is an integer if and only if $q = -1/2$, or $q = F_{2i} / F_{2i+1}$ or $q = L_{2i+1} / L_{2i+2}$ for some $i \\in \\mathbb{N}$.","title":"Integer values of the Lucas number generating function in its interval of convergence"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/fib-lucas-poly-of-pell-eq","kind":"node","lean":"MetaMathlibExt.fibLucasPoly_of_pell_eq","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#fib-lucas-poly-of-pell-eq","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Let $a$ and $b$ be positive integers, and let $F_n(x), L_n(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, $L_1 = ax$ and $G_{n+2} = ax\\,G_{n+1} + b\\,G_n$ for $G = F, L$ and all $n \\ge 0$. Let $r_0 \\in \\{0, 1\\}$ and let $P(x), Q(x) \\in \\mathbb{Q}[x]$ have nonnegative leading coefficients (the zero polynomial has leading coefficient $0$) and satisfy\u2026","title":"Polynomial Pell-type solutions as rescaled Fibonacci\u2013Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/fib-lucas-odd-of-pell-eq-of-squarefree","kind":"node","lean":"MetaMathlibExt.fibLucas_odd_of_pell_eq_of_squarefree","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#fib-lucas-odd-of-pell-eq-of-squarefree","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Let $a$ and $b$ be positive integers with $b \\mid a$ and $b$ squarefree, and let $F_i(x), L_i(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, $L_1 = ax$ and $G_{i+2} = ax\\,G_{i+1} + b\\,G_i$ for $G = F, L$ and all $i \\ge 0$. If natural numbers $P$ and $Q$ satisfy $P^2 - (a^2 + 4b) Q^2 = -4b$, then there is $n \\in \\mathbb{N}$ with $b^n P = L_{2n+1}(1)$ and $b^n Q = F_{2n+1}(1)$.","title":"Pell-type solutions as rescaled odd Fibonacci\u2013Lucas values, squarefree case"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/fib-lucas-odd-of-pell-eq","kind":"node","lean":"MetaMathlibExt.fibLucas_odd_of_pell_eq","parent":6741,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin/index.html#fib-lucas-odd-of-pell-eq","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/integer-values-of-fibonacci-polynomial-generatin","status":"can_prove","summary":"Let $a$ and $b$ be positive integers with $b \\mid a$, and let $F_i(x), L_i(x) \\in \\mathbb{Z}[x]$ be the polynomials with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, $L_1 = ax$ and $G_{i+2} = ax\\,G_{i+1} + b\\,G_i$ for $G = F, L$ and all $i \\ge 0$. If natural numbers $P$ and $Q$ satisfy $P^2 - (a^2 + 4b) Q^2 = -4b$, then there is $n \\in \\mathbb{N}$ with $b^n P = L_{2n+1}(1)$ and $b^n Q = F_{2n+1}(1)$.","title":"Pell-type solutions as rescaled odd Fibonacci\u2013Lucas values"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials.json","id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","kind":"scope","navigation":"module","parent":6718,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","title":"r-Fibonacci and r-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-fibonacci-polynomial","kind":"node","lean":"MetaMathlibExt.rFibonacciPolynomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(r\\)"],["text","-Fibonacci polynomials"]],"source":"The $r$-Fibonacci polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-fibonacci-polynomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $r$ be a natural number and let $x, y \\in R$. The $r$-Fibonacci polynomials $U^{(r)}_n(x, y) \\in R$, $n \\ge 0$, are defined by $U^{(r)}_0 = 0$, $U^{(r)}_k = x^{k-1}$ for $1 \\le k \\le r$, and $U^{(r)}_{n+1} = x \\, U^{(r)}_n + y \\, U^{(r)}_{n-r}$ for $n \\ge r$.","title":"The $r$-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-fibonacci-polynomial-zero","kind":"node","lean":"MetaMathlibExt.rFibonacciPolynomial_zero","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zeroth "],["math","\\(r\\)"],["text","-Fibonacci polynomial vanishes"]],"source":"The zeroth $r$-Fibonacci polynomial vanishes"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-fibonacci-polynomial-zero","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For every natural number $r$ and all $x, y$ in a commutative semiring $R$, the $r$-Fibonacci polynomial $U^{(r)}_0(x, y)$ is $0$.","title":"The zeroth $r$-Fibonacci polynomial vanishes"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-fibonacci-polynomial-succ-of-lt","kind":"node","lean":"MetaMathlibExt.rFibonacciPolynomial_succ_of_lt","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial values of the "],["math","\\(r\\)"],["text","-Fibonacci polynomials"]],"source":"Initial values of the $r$-Fibonacci polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-fibonacci-polynomial-succ-of-lt","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $x, y \\in R$, and let $n < r$ be natural numbers. Then the $r$-Fibonacci polynomial $U^{(r)}_{n+1}(x, y)$ equals $x^n$.","title":"Initial values of the $r$-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-fibonacci-polynomial-succ-of-le","kind":"node","lean":"MetaMathlibExt.rFibonacciPolynomial_succ_of_le","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the "],["math","\\(r\\)"],["text","-Fibonacci polynomials"]],"source":"Recurrence for the $r$-Fibonacci polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-fibonacci-polynomial-succ-of-le","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $x, y \\in R$, and let $r \\le n$ be natural numbers. Then the $r$-Fibonacci polynomials $U_m = U^{(r)}_m(x, y)$ satisfy $U_{n+1} = x \\, U_n + y \\, U_{n-r}$.","title":"Recurrence for the $r$-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-fibonacci-polynomial-one-one-one","kind":"node","lean":"MetaMathlibExt.rFibonacciPolynomial_one_one_one","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(1\\)"],["text","-Fibonacci polynomials at "],["math","\\(x = y = 1\\)"],["text"," are the Fibonacci numbers"]],"source":"The $1$-Fibonacci polynomials at $x = y = 1$ are the Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-fibonacci-polynomial-one-one-one","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For every natural number $n$, the $r$-Fibonacci polynomial with $r = 1$, evaluated at $x = y = 1$ in the natural numbers, is the $n$-th Fibonacci number: $U^{(1)}_n(1, 1) = F_n$, where $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$.","title":"The $1$-Fibonacci polynomials at $x = y = 1$ are the Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/sum-pow-mul-r-fibonacci-polynomial","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_rFibonacciPolynomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Geometrically weighted sum of "],["math","\\(r\\)"],["text","-Fibonacci polynomials"]],"source":"Geometrically weighted sum of $r$-Fibonacci polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#sum-pow-mul-r-fibonacci-polynomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$ be a natural number, let $x, y, m \\in K$ with $1 - m x - m^{r+1} y \\ne 0$, and write $U_j$ for the $r$-Fibonacci polynomial $U^{(r)}_j(x, y)$. Then for every natural number $n$,","title":"Geometrically weighted sum of $r$-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/sum-r-fibonacci-polynomial","kind":"node","lean":"MetaMathlibExt.sum_rFibonacciPolynomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of the first "],["math","\\(n\\)"],["text"," "],["math","\\(r\\)"],["text","-Fibonacci polynomials"]],"source":"Sum of the first $n$ $r$-Fibonacci polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#sum-r-fibonacci-polynomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$ be a natural number, let $x, y \\in K$ with $1 - x - y \\ne 0$, and write $U_j$ for the $r$-Fibonacci polynomial $U^{(r)}_j(x, y)$. Then for every natural number $n$,","title":"Sum of the first $n$ $r$-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/sum-neg-one-pow-mul-r-fibonacci-polynomi","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_mul_rFibonacciPolynomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sum of "],["math","\\(r\\)"],["text","-Fibonacci polynomials"]],"source":"Alternating sum of $r$-Fibonacci polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#sum-neg-one-pow-mul-r-fibonacci-polynomi","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$ be a natural number, let $x, y \\in K$ with $1 + x + (-1)^r y \\ne 0$, and write $U_j$ for the $r$-Fibonacci polynomial $U^{(r)}_j(x, y)$. Then for every natural number $n$,","title":"Alternating sum of $r$-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-lucas-polynomial","kind":"node","lean":"MetaMathlibExt.rLucasPolynomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(r\\)"],["text","-Lucas polynomials of type "],["math","\\(s\\)"]],"source":"The $r$-Lucas polynomials of type $s$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-lucas-polynomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $r$ and $s$ be natural numbers and let $x, y \\in R$. The $r$-Lucas polynomials of type $s$ are the sequence $V_n = V^{(r,s)}_n(x, y)$, $n \\ge 0$, of elements of $R$ given by $V_0 = s + 1$, $V_k = x^k$ for $1 \\le k \\le r$, and $V_{n+1} = x V_n + y V_{n-r}$ for $n \\ge r$.","title":"The $r$-Lucas polynomials of type $s$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-lucas-polynomial-zero","kind":"node","lean":"MetaMathlibExt.rLucasPolynomial_zero","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial value of the "],["math","\\(r\\)"],["text","-Lucas polynomials"]],"source":"Initial value of the $r$-Lucas polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-lucas-polynomial-zero","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For every commutative semiring $R$, all natural numbers $r$ and $s$ and all $x, y \\in R$, the $r$-Lucas polynomial of type $s$ satisfies $V^{(r,s)}_0(x, y) = s + 1$.","title":"Initial value of the $r$-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-lucas-polynomial-succ-of-lt","kind":"node","lean":"MetaMathlibExt.rLucasPolynomial_succ_of_lt","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial values of the "],["math","\\(r\\)"],["text","-Lucas polynomials"]],"source":"Initial values of the $r$-Lucas polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-lucas-polynomial-succ-of-lt","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $s$ be a natural number, let $x, y \\in R$, and let $n < r$ be natural numbers. Then the $r$-Lucas polynomial of type $s$ satisfies $V^{(r,s)}_{n+1}(x, y) = x^{n+1}$.","title":"Initial values of the $r$-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-lucas-polynomial-succ-of-le","kind":"node","lean":"MetaMathlibExt.rLucasPolynomial_succ_of_le","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the "],["math","\\(r\\)"],["text","-Lucas polynomials"]],"source":"Recurrence for the $r$-Lucas polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-lucas-polynomial-succ-of-le","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $s$ be a natural number, let $x, y \\in R$, and let $r \\le n$ be natural numbers. Then the $r$-Lucas polynomials of type $s$, $V_m = V^{(r,s)}_m(x, y)$, satisfy $V_{n+1} = x \\, V_n + y \\, V_{n-r}$.","title":"Recurrence for the $r$-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-fibonacci-polynomial-eq-sum-roots","kind":"node","lean":"MetaMathlibExt.rFibonacciPolynomial_eq_sum_roots","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binet formula for the "],["math","\\(r\\)"],["text","-Fibonacci polynomials (distinct zeros)"]],"source":"Binet formula for the $r$-Fibonacci polynomials (distinct zeros)"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-fibonacci-polynomial-eq-sum-roots","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$ be a natural number and let $x, y \\in K$. Suppose that $t^{r+1} - x t^r - y = \\prod_{k=0}^{r} (t - \\alpha_k)$ in $K[t]$, where $\\alpha_0, \\dots, \\alpha_r \\in K$ are pairwise distinct. Then for every $n \\ge 0$ the $r$-Fibonacci polynomials $U^{(r)}_n(x, y)$ satisfy","title":"Binet formula for the $r$-Fibonacci polynomials (distinct zeros)"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-lucas-polynomial-eq-sum-roots","kind":"node","lean":"MetaMathlibExt.rLucasPolynomial_eq_sum_roots","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binet formula for the "],["math","\\(r\\)"],["text","-Lucas polynomials (distinct zeros)"]],"source":"Binet formula for the $r$-Lucas polynomials (distinct zeros)"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-lucas-polynomial-eq-sum-roots","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$ and $s$ be natural numbers, and let $x, y \\in K$. Suppose that $t^{r+1} - x t^r - y = \\prod_{k=0}^{r} (t - \\alpha_k)$ in $K[t]$, where $\\alpha_0, \\dots, \\alpha_r \\in K$ are pairwise distinct. Then for every $n \\ge 0$ the $r$-Lucas polynomials of type $s$ satisfy","title":"Binet formula for the $r$-Lucas polynomials (distinct zeros)"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-fibonacci-polynomial-eq-of-double-root","kind":"node","lean":"MetaMathlibExt.rFibonacciPolynomial_eq_of_double_root","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-fibonacci-polynomial-eq-of-double-root","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, let $r \\ge 1$ be a natural number, and let $x, y, \\alpha \\in K$. Let $\\beta_1, \\dots, \\beta_{r-1} \\in K$ be pairwise distinct and all different from $\\alpha$, put $\\tilde P(t) = \\prod_{k=1}^{r-1} (t - \\beta_k)$, and suppose that $t^{r+1} - x t^r - y = (t - \\alpha)^2 \\tilde P(t)$ as polynomials over $K$. Then for every natural number $n$ the $r$-Fibonacci polynomial\u2026","title":"Binet formula for the r-Fibonacci polynomials with one double zero"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/r-lucas-polynomial-eq-of-double-root","kind":"node","lean":"MetaMathlibExt.rLucasPolynomial_eq_of_double_root","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binet formula for the "],["math","\\(r\\)"],["text","-Lucas polynomials (one double zero)"]],"source":"Binet formula for the $r$-Lucas polynomials (one double zero)"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#r-lucas-polynomial-eq-of-double-root","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, let $r \\ge 1$ and $s$ be natural numbers, and let $x, y, \\alpha \\in K$ with $\\alpha \\ne 0$. Suppose that $t^{r+1} - x t^r - y = (t - \\alpha)^2 \\tilde P(t)$ in $K[t]$, where $\\tilde P(t) = \\prod_{k=1}^{r-1} (t - \\beta_k)$ for pairwise distinct $\\beta_1, \\dots, \\beta_{r-1} \\in K$, all different from $\\alpha$. Then for every $n \\ge 0$ the $r$-Lucas polynomials of type $s$\u2026","title":"Binet formula for the $r$-Lucas polynomials (one double zero)"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-binomial","kind":"node","lean":"MetaMathlibExt.pqBinomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((p,q)\\)"],["text","-binomial coefficient"]],"source":"$(p,q)$-binomial coefficient"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-binomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring and $p, q \\in R$. For natural numbers $n$ and $k$, the $(p,q)$-binomial coefficient $\\binom{n}{k}_{p,q} \\in R$ is defined recursively by $\\binom{n}{0}_{p,q} = 1$, $\\binom{0}{k+1}_{p,q} = 0$ and $\\binom{n+1}{k+1}_{p,q} = p^{n-k} \\binom{n}{k}_{p,q} + q^{k+1} \\binom{n}{k+1}_{p,q}$, where $n - k$ is truncated subtraction ($0$ when $k > n$).","title":"$(p,q)$-binomial coefficient"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-binomial-zero-right","kind":"node","lean":"MetaMathlibExt.pqBinomial_zero_right","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((p,q)\\)"],["text","-binomial coefficients with lower index zero"]],"source":"$(p,q)$-binomial coefficients with lower index zero"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-binomial-zero-right","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For all elements $p, q$ of a commutative semiring and every natural number $n$, the $(p,q)$-binomial coefficient $\\binom{n}{0}_{p,q}$ equals $1$.","title":"$(p,q)$-binomial coefficients with lower index zero"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-binomial-zero-succ","kind":"node","lean":"MetaMathlibExt.pqBinomial_zero_succ","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((p,q)\\)"],["text","-binomial coefficients with upper index zero and positive lower index"]],"source":"$(p,q)$-binomial coefficients with upper index zero and positive lower index"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-binomial-zero-succ","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For all elements $p, q$ of a commutative semiring and every natural number $k$, the $(p,q)$-binomial coefficient $\\binom{0}{k+1}_{p,q}$ equals $0$.","title":"$(p,q)$-binomial coefficients with upper index zero and positive lower index"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-binomial-succ-succ","kind":"node","lean":"MetaMathlibExt.pqBinomial_succ_succ","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pascal recursion for "],["math","\\((p,q)\\)"],["text","-binomial coefficients"]],"source":"Pascal recursion for $(p,q)$-binomial coefficients"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-binomial-succ-succ","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For all elements $p, q$ of a commutative semiring and all natural numbers $n$ and $k$, the $(p,q)$-binomial coefficients satisfy $\\binom{n+1}{k+1}_{p,q} = p^{n-k} \\binom{n}{k}_{p,q} + q^{k+1} \\binom{n}{k+1}_{p,q}$, where $n - k$ is truncated subtraction ($0$ when $k > n$).","title":"Pascal recursion for $(p,q)$-binomial coefficients"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-binomial-one-left","kind":"node","lean":"MetaMathlibExt.pqBinomial_one_left","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\((p, q)\\)"],["text","-binomial coefficients at "],["math","\\(p = 1\\)"]],"source":"The $(p, q)$-binomial coefficients at $p = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-binomial-one-left","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring and $q \\in R$. For all natural numbers $n$ and $k$, the $(p, q)$-binomial coefficient $[n, k]_{p,q}$ at $p = 1$ equals the Gaussian binomial coefficient $\\binom{n}{k}_q$.","title":"The $(p, q)$-binomial coefficients at $p = 1$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rfibonacci","kind":"node","lean":"MetaMathlibExt.pqRFibonacci","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((p,q)\\)"],["text","-analogue of the "],["math","\\(r\\)"],["text","-Fibonacci polynomial"]],"source":"$(p,q)$-analogue of the $r$-Fibonacci polynomial"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rfibonacci","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $r$ and $m$ be natural numbers and let $x, y, p, q \\in R$. The $(p,q)$-analogue $U^{(r)}_n(x, y, p, q, m) \\in R$ of the $r$-Fibonacci polynomial is defined by $U^{(r)}_0(x, y, p, q, m) = 0$ and, for every natural number $n$,","title":"$(p,q)$-analogue of the $r$-Fibonacci polynomial"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rfibonacci-zero","kind":"node","lean":"MetaMathlibExt.pqRFibonacci_zero","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zeroth value of the "],["math","\\((p,q)\\)"],["text","-analogue of the "],["math","\\(r\\)"],["text","-Fibonacci polynomial"]],"source":"Zeroth value of the $(p,q)$-analogue of the $r$-Fibonacci polynomial"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rfibonacci-zero","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For all natural numbers $r$ and $m$ and all elements $x, y, p, q$ of a commutative semiring, the $(p,q)$-analogue of the $r$-Fibonacci polynomial satisfies $U^{(r)}_0(x, y, p, q, m) = 0$.","title":"Zeroth value of the $(p,q)$-analogue of the $r$-Fibonacci polynomial"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rfibonacci-one","kind":"node","lean":"MetaMathlibExt.pqRFibonacci_one","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","First value of the "],["math","\\((p,q)\\)"],["text","-analogue of the "],["math","\\(r\\)"],["text","-Fibonacci polynomial"]],"source":"First value of the $(p,q)$-analogue of the $r$-Fibonacci polynomial"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rfibonacci-one","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For all natural numbers $r$ and $m$ and all elements $x, y, p, q$ of a commutative semiring, the $(p,q)$-analogue of the $r$-Fibonacci polynomial satisfies $U^{(r)}_1(x, y, p, q, m) = 1$.","title":"First value of the $(p,q)$-analogue of the $r$-Fibonacci polynomial"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rfibonacci-succ-eq-left","kind":"node","lean":"MetaMathlibExt.pqRFibonacci_succ_eq_left","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","First recurrence for the "],["math","\\((p,q)\\)"],["text","-analogue of the "],["math","\\(r\\)"],["text","-Fibonacci polynomial"]],"source":"First recurrence for the $(p,q)$-analogue of the $r$-Fibonacci polynomial"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rfibonacci-succ-eq-left","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $R$ be a commutative semiring, let $r \\ge 1$, $n \\ge r$ and $m$ be natural numbers and let $x, y, p, q \\in R$. Write $U_j(x, y)$ for $U^{(r)}_j(x, y, p, q, m)$, the $(p,q)$-analogue of the $r$-Fibonacci polynomial with $r$, $p$, $q$ and $m$ fixed. Then","title":"First recurrence for the $(p,q)$-analogue of the $r$-Fibonacci polynomial"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rfibonacci-succ-eq-right","kind":"node","lean":"MetaMathlibExt.pqRFibonacci_succ_eq_right","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second recurrence for the "],["math","\\((p,q)\\)"],["text","-analogue of the "],["math","\\(r\\)"],["text","-Fibonacci polynomial"]],"source":"Second recurrence for the $(p,q)$-analogue of the $r$-Fibonacci polynomial"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rfibonacci-succ-eq-right","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $R$ be a commutative semiring, let $r \\ge 1$, $n \\ge r$ and $m$ be natural numbers and let $x, y, p, q \\in R$. Write $U_j(x, y)$ for $U^{(r)}_j(x, y, p, q, m)$, the $(p,q)$-analogue of the $r$-Fibonacci polynomial with $r$, $p$, $q$ and $m$ fixed. Then","title":"Second recurrence for the $(p,q)$-analogue of the $r$-Fibonacci polynomial"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas","kind":"node","lean":"MetaMathlibExt.pqRLucas","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $r$, $s$ and $m$ be natural numbers, let $x, y, p, q \\in R$, and write $[a, b]_{p,q}$ for the $(p, q)$-binomial coefficients. The first-kind $(p, q)$-analogue $V^{(r,s)}_n(x, y, p, q, m)$ of the $r$-Lucas polynomial of type $s$, written $V_n$, is defined by $V_0 = s + 1$ and, for $n \\ge 1$,","title":"First-kind (p, q)-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-zero","kind":"node","lean":"MetaMathlibExt.pqRLucas_zero","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-zero","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $r$, $s$ and $m$ be natural numbers, and let $x, y, p, q \\in R$. The first-kind $(p, q)$-analogue $V^{(r,s)}_n(x, y, p, q, m)$ of the $r$-Lucas polynomial of type $s$ satisfies $V^{(r,s)}_0(x, y, p, q, m) = s + 1$.","title":"Initial value of the first-kind (p, q)-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-second","kind":"node","lean":"MetaMathlibExt.pqRLucasSecond","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-second","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $K$ be a field, let $r$, $s$ and $m$ be natural numbers, let $x, y, p, q \\in K$, and write $[a, b]_{p,q}$ for the $(p, q)$-binomial coefficients. The second-kind $(p, q)$-analogue $\\mathbb{V}^{(r,s)}_n(x, y, p, q, m)$ of the $r$-Lucas polynomial of type $s$, written $\\mathbb{V}_n$, is defined by $\\mathbb{V}_0 = s + 1$ and, for $n \\ge 1$,","title":"Second-kind (p, q)-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-second-zero","kind":"node","lean":"MetaMathlibExt.pqRLucasSecond_zero","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-second-zero","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $K$ be a field, let $r$, $s$ and $m$ be natural numbers, and let $x, y, p, q \\in K$. The second-kind $(p, q)$-analogue $\\mathbb{V}^{(r,s)}_n(x, y, p, q, m)$ of the $r$-Lucas polynomial of type $s$ satisfies $\\mathbb{V}^{(r,s)}_0(x, y, p, q, m) = s + 1$.","title":"Initial value of the second-kind (p, q)-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-eq-pq-rfibonacci-add","kind":"node","lean":"MetaMathlibExt.pqRLucas_eq_pqRFibonacci_add","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-eq-pq-rfibonacci-add","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$, $s$ and $m$ be natural numbers, and let $x, y, p, q \\in K$ with $p \\ne 0$ and $q \\ne 0$. Write $U_n(x, y)$ for the $(p, q)$-analogue of the $r$-Fibonacci polynomial with parameters $p, q, m$, and $V_n(x, y)$ for the first-kind $(p, q)$-analogue of the $r$-Lucas polynomial of type $s$ with the same parameters. Then for every natural number $n \\ge r$,","title":"Splitting formula for first-kind (p, q)-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-second-eq-pq-rfibonacci-add","kind":"node","lean":"MetaMathlibExt.pqRLucasSecond_eq_pqRFibonacci_add","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-second-eq-pq-rfibonacci-add","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$, $s$ and $m$ be natural numbers, and let $x, y, p, q \\in K$ with $p \\ne 0$ and $q \\ne 0$. Write $U_n(x, y)$ for the $(p, q)$-analogue of the $r$-Fibonacci polynomial with parameters $p, q, m$, and $\\mathbb{V}_n(x, y)$ for the second-kind $(p, q)$-analogue of the $r$-Lucas polynomial of type $s$ with the same parameters. Then for every natural number $n \\ge r$,","title":"Splitting formula for second-kind (p, q)-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-eq-succ-mul-sub","kind":"node","lean":"MetaMathlibExt.pqRLucas_eq_succ_mul_sub","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-eq-succ-mul-sub","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$, $s$ and $m$ be natural numbers, and let $x, y, p, q \\in K$ with $p \\ne 0$ and $q \\ne 0$. Write $U_n(x, y)$ for the $(p, q)$-analogue of the $r$-Fibonacci polynomial with parameters $p, q, m$, and $V_n(x, y)$ for the first-kind $(p, q)$-analogue of the $r$-Lucas polynomial of type $s$ with the same parameters. Then for every natural number $n$,","title":"First-kind (p, q)-Lucas polynomials as a difference of (p, q)-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-second-eq-succ-mul-sub","kind":"node","lean":"MetaMathlibExt.pqRLucasSecond_eq_succ_mul_sub","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-second-eq-succ-mul-sub","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$, $s$ and $m$ be natural numbers, and let $x, y, p, q \\in K$ with $p \\ne 0$. Write $U_n(x, y)$ for the $(p, q)$-analogue of the $r$-Fibonacci polynomial with parameters $p, q, m$, and $\\mathbb{V}_n(x, y)$ for the second-kind $(p, q)$-analogue of the $r$-Lucas polynomial of type $s$ with the same parameters. Then for every natural number $n$,","title":"Second-kind (p, q)-Lucas polynomials as a difference of (p, q)-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-eq-mul-add","kind":"node","lean":"MetaMathlibExt.pqRLucas_eq_mul_add","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","First-kind "],["math","\\((p,q)\\)"],["text","-analogue of the "],["math","\\(r\\)"],["text","-Lucas polynomial via two "],["math","\\(r\\)"],["text","-Fibonacci terms"]],"source":"First-kind $(p,q)$-analogue of the $r$-Lucas polynomial via two $r$-Fibonacci terms"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-eq-mul-add","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $R$ be a commutative semiring, let $r \\ge 1$, $n \\ge r$, $s$ and $m$ be natural numbers and let $x, y, p, q \\in R$. Write $U_j(x, y)$ for the $(p,q)$-analogue $U^{(r)}_j(x, y, p, q, m)$ of the $r$-Fibonacci polynomial and $V_n(x, y)$ for the first-kind $(p,q)$-analogue $V^{(r,s)}_n(x, y, p, q, m)$ of the $r$-Lucas polynomial of type $s$, with $r$, $s$, $p$, $q$ and $m$ fixed. Then","title":"First-kind $(p,q)$-analogue of the $r$-Lucas polynomial via two $r$-Fibonacci terms"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/pq-rlucas-second-eq-mul-add","kind":"node","lean":"MetaMathlibExt.pqRLucasSecond_eq_mul_add","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#pq-rlucas-second-eq-mul-add","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$, $s$ and $m$ be natural numbers, and let $x, y, p, q \\in K$ with $p \\ne 0$ and $q \\ne 0$. Write $U_n(x, y)$ for the $(p, q)$-analogue of the $r$-Fibonacci polynomial with parameters $p, q, m$, and $\\mathbb{V}_n(x, y)$ for the second-kind $(p, q)$-analogue of the $r$-Lucas polynomial of type $s$ with the same parameters. Then for every natural number $n \\ge r$,","title":"Recurrence-type formula for second-kind (p, q)-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/sum-pow-mul-r-lucas-polynomial","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_rLucasPolynomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Geometrically weighted sum of "],["math","\\(r\\)"],["text","-Lucas polynomials"]],"source":"Geometrically weighted sum of $r$-Lucas polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#sum-pow-mul-r-lucas-polynomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$ and $s$ be natural numbers, let $x, y, m \\in K$ with $1 - m x - m^{r+1} y \\ne 0$, and write $V_j$ for the $r$-Lucas polynomial of type $s$, $V_j = V^{(r,s)}_j(x, y)$. Then for every natural number $n \\ge r - 1$,","title":"Geometrically weighted sum of $r$-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/sum-r-lucas-polynomial","kind":"node","lean":"MetaMathlibExt.sum_rLucasPolynomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of the first "],["math","\\(n\\)"],["text"," "],["math","\\(r\\)"],["text","-Lucas polynomials"]],"source":"Sum of the first $n$ $r$-Lucas polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#sum-r-lucas-polynomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$ and $s$ be natural numbers, let $x, y \\in K$ with $1 - x - y \\ne 0$, and write $V_j$ for the $r$-Lucas polynomial of type $s$, $V_j = V^{(r,s)}_j(x, y)$. Then for every natural number $n$ with $n \\ge 1$ and $n \\ge r - 1$,","title":"Sum of the first $n$ $r$-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/sum-neg-one-pow-mul-r-lucas-polynomial","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_mul_rLucasPolynomial","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sum of "],["math","\\(r\\)"],["text","-Lucas polynomials"]],"source":"Alternating sum of $r$-Lucas polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#sum-neg-one-pow-mul-r-lucas-polynomial","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, let $r \\ge 1$ and $s$ be natural numbers, let $x, y \\in K$ with $1 + x + (-1)^r y \\ne 0$, and write $V_j$ for the $r$-Lucas polynomial of type $s$, $V_j = V^{(r,s)}_j(x, y)$. Then for every natural number $n \\ge r - 1$,","title":"Alternating sum of $r$-Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci","kind":"node","lean":"MetaMathlibExt.negRFibonacci","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $A$ be a commutative ring, $r$ a natural number, $x \\in A$ and $y$ a unit of $A$. The values $U_n = U^{(-r)}_n(x, y) \\in A$ of the $(-r)$-Fibonacci polynomials are defined for $n \\ge 0$ by $U_0 = 0$, $U_1 = 1$, $U_n = 0$ for $2 \\le n \\le r$, and, when $r \\ge 1$, $U_{n+1} = y^{-1} U_{n-r+1} - y^{-1} x U_{n-r}$ for $n \\ge r$. When $r = 0$, $U_n = 0$ for every $n \\ge 2$.","title":"The (-r)-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci-zero","kind":"node","lean":"MetaMathlibExt.negRFibonacci_zero","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci-zero","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For every commutative ring $A$, natural number $r$, $x \\in A$ and unit $y$ of $A$, the $(-r)$-Fibonacci polynomial satisfies $U^{(-r)}_0(x, y) = 0$.","title":"The zeroth (-r)-Fibonacci polynomial"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci-one","kind":"node","lean":"MetaMathlibExt.negRFibonacci_one","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci-one","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"For every commutative ring $A$, natural number $r$, $x \\in A$ and unit $y$ of $A$, the $(-r)$-Fibonacci polynomial satisfies $U^{(-r)}_1(x, y) = 1$.","title":"The first (-r)-Fibonacci polynomial"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci-succ-of-lt","kind":"node","lean":"MetaMathlibExt.negRFibonacci_succ_of_lt","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci-succ-of-lt","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $A$ be a commutative ring, $x \\in A$ and $y$ a unit of $A$. For natural numbers $r$ and $n$ with $1 \\le n < r$, the $(-r)$-Fibonacci polynomial satisfies $U^{(-r)}_{n+1}(x, y) = 0$.","title":"The (-r)-Fibonacci polynomials vanish at indices 2 to r"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci-succ-of-le","kind":"node","lean":"MetaMathlibExt.negRFibonacci_succ_of_le","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci-succ-of-le","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $A$ be a commutative ring, $x \\in A$ and $y$ a unit of $A$, let $r$ and $n$ be natural numbers with $1 \\le r \\le n$, and write $U_m = U^{(-r)}_m(x, y)$. Then $U_{n+1} = y^{-1} U_{n-r+1} - y^{-1} x U_{n-r}$.","title":"Recurrence for the (-r)-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas","kind":"node","lean":"MetaMathlibExt.negRLucas","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Companion sequence of the "],["math","\\((-r)\\)"],["text","-Fibonacci polynomials"]],"source":"Companion sequence of the $(-r)$-Fibonacci polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $A$ be a commutative ring, $r$ a natural number, $x \\in A$ and $y$ a unit of $A$. The companion sequence $V^{(-r)}_n(x, y) \\in A$, $n \\ge 0$, is defined by $V^{(-r)}_0 = r + 1$ and, if $r \\ge 1$, by $V^{(-r)}_n = 0$ for $1 \\le n < r$, $V^{(-r)}_r = r y^{-1}$ and $V^{(-r)}_{n+1} = y^{-1} V^{(-r)}_{n-r+1} - y^{-1} x V^{(-r)}_{n-r}$ for $n \\ge r$. If $r = 0$, then $V^{(-r)}_n = 0$ for all $n \\ge 1$.","title":"Companion sequence of the $(-r)$-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-zero","kind":"node","lean":"MetaMathlibExt.negRLucas_zero","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zeroth term of the companion sequence "],["math","\\(V^{(-r)}\\)"]],"source":"Zeroth term of the companion sequence $V^{(-r)}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-zero","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $A$ be a commutative ring, $r$ a natural number, $x \\in A$ and $y$ a unit of $A$. The companion sequence of the $(-r)$-Fibonacci polynomials satisfies $V^{(-r)}_0(x, y) = r + 1$.","title":"Zeroth term of the companion sequence $V^{(-r)}$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-succ-of-lt","kind":"node","lean":"MetaMathlibExt.negRLucas_succ_of_lt","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing initial terms of the companion sequence "],["math","\\(V^{(-r)}\\)"]],"source":"Vanishing initial terms of the companion sequence $V^{(-r)}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-succ-of-lt","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $A$ be a commutative ring, $x \\in A$ and $y$ a unit of $A$. For natural numbers $r$ and $n$ with $n + 1 < r$, the companion sequence of the $(-r)$-Fibonacci polynomials satisfies $V^{(-r)}_{n+1}(x, y) = 0$; that is, $V^{(-r)}_m(x, y) = 0$ for $1 \\le m < r$.","title":"Vanishing initial terms of the companion sequence $V^{(-r)}$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-self","kind":"node","lean":"MetaMathlibExt.negRLucas_self","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(r\\)"],["text","-th term of the companion sequence "],["math","\\(V^{(-r)}\\)"]],"source":"The $r$-th term of the companion sequence $V^{(-r)}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-self","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $A$ be a commutative ring, $r \\ge 1$ a natural number, $x \\in A$ and $y$ a unit of $A$. The companion sequence of the $(-r)$-Fibonacci polynomials satisfies $V^{(-r)}_r(x, y) = r y^{-1}$.","title":"The $r$-th term of the companion sequence $V^{(-r)}$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-succ-of-le","kind":"node","lean":"MetaMathlibExt.negRLucas_succ_of_le","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the companion sequence "],["math","\\(V^{(-r)}\\)"]],"source":"Recurrence for the companion sequence $V^{(-r)}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-succ-of-le","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"fully_proved","summary":"Let $A$ be a commutative ring, $x \\in A$ and $y$ a unit of $A$. For natural numbers $r$ and $n$ with $1 \\le r \\le n$, the companion sequence of the $(-r)$-Fibonacci polynomials satisfies $V^{(-r)}_{n+1}(x, y) = y^{-1} V^{(-r)}_{n-r+1}(x, y) - y^{-1} x V^{(-r)}_{n-r}(x, y)$.","title":"Recurrence for the companion sequence $V^{(-r)}$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci-succ-eq-sum","kind":"node","lean":"MetaMathlibExt.negRFibonacci_succ_eq_sum","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci-succ-eq-sum","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $A$ be a commutative ring, $r \\ge 2$, $x \\in A$ and $y$ a unit of $A$, and write $U_n = U^{(-r)}_n(x, y)$. For every $n \\ge 0$,","title":"Explicit sum for the (-r)-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci-succ-eq-sum-2","kind":"node","lean":"MetaMathlibExt.negRFibonacci_succ_eq_sum'","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci-succ-eq-sum-2","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $A$ be a commutative ring, $r \\ge 2$, $x \\in A$ and $y$ a unit of $A$, and write $U_n = U^{(-r)}_n(x, y)$. For every $n \\ge 0$, $U_{n+1}$ is the sum of $\\binom{(n-k)/r}{k} y^{-(n-k)/r} (-x)^k$ over the integers $k$ with $0 \\le k \\le \\lfloor n/r \\rfloor$ and $r \\mid n - k$.","title":"Explicit sum for the (-r)-Fibonacci polynomials, second form"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-eq","kind":"node","lean":"MetaMathlibExt.negRLucas_eq","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The companion sequence in terms of the "],["math","\\((-r)\\)"],["text","-Fibonacci polynomials"]],"source":"The companion sequence in terms of the $(-r)$-Fibonacci polynomials"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-eq","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $A$ be a commutative ring, $r \\ge 2$ a natural number, $x \\in A$ and $y$ a unit of $A$, and let $U^{(-r)}_n(x, y)$ be the $(-r)$-Fibonacci polynomials and $V^{(-r)}_n(x, y)$ their companion sequence. For every natural number $n \\ge r$, $V^{(-r)}_n(x, y) = r \\, U^{(-r)}_{n+1}(x, y) - x y^{-1} U^{(-r)}_{n-r}(x, y)$.","title":"The companion sequence in terms of the $(-r)$-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-eq-sum","kind":"node","lean":"MetaMathlibExt.negRLucas_eq_sum","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Explicit sum for the companion sequence "],["math","\\(V^{(-r)}\\)"]],"source":"Explicit sum for the companion sequence $V^{(-r)}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-eq-sum","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, $r \\ge 2$ and $n \\ge 1$ natural numbers, $x \\in K$ and $y$ a unit of $K$, and let $V^{(-r)}_n(x, y)$ be the companion sequence of the $(-r)$-Fibonacci polynomials. Then","title":"Explicit sum for the companion sequence $V^{(-r)}$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-eq-sum-2","kind":"node","lean":"MetaMathlibExt.negRLucas_eq_sum'","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-eq-sum-2","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, $r \\ge 2$, $x \\in K$ and $y \\in K$ nonzero, and write $V_n = V^{(-r)}_n(x, y)$ for the companion $(-r)$-Lucas sequence. For every $n \\ge 1$, $V_n$ is the sum of","title":"Explicit sum for the (-r)-Lucas polynomials, second form"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci-generating-function","kind":"node","lean":"MetaMathlibExt.negRFibonacci_generatingFunction","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci-generating-function","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $A$ be a commutative ring, $r \\ge 2$, $x \\in A$ and $y$ a unit of $A$, and write $U_n = U^{(-r)}_n(x, y)$. In the ring $A[[z]]$ of formal power series,","title":"Generating function of the (-r)-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-generating-function","kind":"node","lean":"MetaMathlibExt.negRLucas_generatingFunction","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the companion sequence "],["math","\\(V^{(-r)}\\)"]],"source":"Generating function of the companion sequence $V^{(-r)}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-generating-function","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $A$ be a commutative ring, $r \\ge 2$ a natural number, $x \\in A$ and $y$ a unit of $A$, and let $V^{(-r)}_n(x, y)$ be the companion sequence of the $(-r)$-Fibonacci polynomials. In the ring $A[[z]]$ of formal power series,","title":"Generating function of the companion sequence $V^{(-r)}$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/separable-neg-rfibonacci-char-poly","kind":"node","lean":"MetaMathlibExt.separable_negRFibonacci_charPoly","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Separability of "],["math","\\(t^{r+1} - y^{-1} t + x y^{-1}\\)"]],"source":"Separability of $t^{r+1} - y^{-1} t + x y^{-1}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#separable-neg-rfibonacci-char-poly","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, let $r \\ge 1$, let $x \\in K$ and let $y \\in K$ be nonzero with $(r+1)^{r+1} x^r y \\ne r^r$. Then the polynomial $t^{r+1} - y^{-1} t + x y^{-1} \\in K[t]$ is separable, that is, coprime to its derivative; equivalently, it has no multiple zeros in an algebraic closure of $K$.","title":"Separability of $t^{r+1} - y^{-1} t + x y^{-1}$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rfibonacci-eq-sum-roots","kind":"node","lean":"MetaMathlibExt.negRFibonacci_eq_sum_roots","parent":6752,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rfibonacci-eq-sum-roots","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, $r \\ge 2$, $x \\in K$ and $y \\in K$ nonzero, and write $U_n = U^{(-r)}_n(x, y)$. Suppose that $\\beta_0, \\dots, \\beta_r \\in K$ are pairwise distinct and that $t^{r+1} - y^{-1} t + x y^{-1} = \\prod_{k=0}^{r} (t - \\beta_k)$ as polynomials over $K$. Then for every $n \\ge 0$,","title":"Binet formula for the (-r)-Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/neg-rlucas-eq-sum-roots","kind":"node","lean":"MetaMathlibExt.negRLucas_eq_sum_roots","parent":6752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binet formula for the companion sequence "],["math","\\(V^{(-r)}\\)"]],"source":"Binet formula for the companion sequence $V^{(-r)}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials/index.html#neg-rlucas-eq-sum-roots","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/r-fibonacci-and-r-lucas-polynomials","status":"can_prove","summary":"Let $K$ be a field, $r \\ge 2$ a natural number, $x \\in K$ and $y$ a unit of $K$, and let $V^{(-r)}_n(x, y)$ be the companion sequence of the $(-r)$-Fibonacci polynomials. Suppose $\\beta_0, \\dots, \\beta_r \\in K$ are pairwise distinct and $t^{r+1} - y^{-1} t + x y^{-1} = \\prod_{k=0}^{r} (t - \\beta_k)$ in the polynomial ring $K[t]$. Then $V^{(-r)}_n(x, y) = \\sum_{k=0}^{r} \\beta_k^n$ for every natural number $n$.","title":"Binet formula for the companion sequence $V^{(-r)}$"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi.json","id":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi","kind":"scope","navigation":"module","parent":6718,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi","status":"can_prove","title":"Reciprocal sums of squares of gibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/has-sum-gibonacci-poly-pow-ratio-telesco","kind":"node","lean":"MetaMathlibExt.hasSum_gibonacciPoly_pow_ratio_telescoping","parent":6812,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/index.html#has-sum-gibonacci-poly-pow-ratio-telesco","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi","status":"can_prove","summary":"Let $x > 0$ be real, let $\\alpha = (x + \\sqrt{x^2 + 4})/2$, and let $g_n$ be either the Fibonacci polynomials $f_n(x)$, given by $f_0 = 0$ and $f_1 = 1$, or the Lucas polynomials $l_n(x)$, given by $l_0 = 2$ and $l_1 = x$, where both satisfy $z_{n+2} = x z_{n+1} + z_n$. Let $k$, $r$, $t$, $\\lambda$ be positive integers and $p$ a natural number with $t \\le 2p$. Then the series","title":"A telescoping series of powers of gibonacci polynomial ratios"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/has-sum-fib-poly-sq-family","kind":"node","lean":"MetaMathlibExt.hasSum_fibPoly_sq_family","parent":6812,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/index.html#has-sum-fib-poly-sq-family","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi","status":"can_prove","summary":"Let $x > 0$ be real, let $\\alpha = (x + \\sqrt{x^2 + 4})/2$, and let $f_n = f_n(x)$ be the Fibonacci polynomials, given by $f_0 = 0$, $f_1 = 1$ and $f_{n+2} = x f_{n+1} + f_n$, evaluated at $x$. Let $k$, $r$, $t$ be positive integers and $p$ a natural number with $t \\le 2p$. Then the series","title":"Series over a family of Fibonacci polynomial squares"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/has-sum-lucas-poly-sq-family","kind":"node","lean":"MetaMathlibExt.hasSum_lucasPoly_sq_family","parent":6812,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/index.html#has-sum-lucas-poly-sq-family","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi","status":"can_prove","summary":"Let $x > 0$ be real, let $\\Delta = \\sqrt{x^2 + 4}$ and $\\alpha = (x + \\Delta)/2$, and let $f_n$ and $l_n$ be the Fibonacci and Lucas polynomials evaluated at $x$, given by $f_0 = 0$, $f_1 = 1$, $l_0 = 2$, $l_1 = x$ and the recurrence $z_{n+2} = x z_{n+1} + z_n$ for both. Let $k$, $r$, $t$ be positive integers and $p$ a natural number with $t \\le 2p$. Then the series","title":"Series over a family of Lucas polynomial squares"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/tsum-gibonacci-poly-sq-sub-family","kind":"node","lean":"MetaMathlibExt.tsum_gibonacciPoly_sq_sub_family","parent":6812,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/index.html#tsum-gibonacci-poly-sq-sub-family","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi","status":"can_prove","summary":"Reciprocal sums of shifted squares of Fibonacci and Lucas polynomials (Koshy\u2013Gao, Theorem 2.1). Let $f_n(x)$ and $l_n(x)$ be the Fibonacci and Lucas polynomials, given by $z_{n+2} = x z_{n+1} + z_n$ with initial values $f_0 = 0$, $f_1 = 1$ and $l_0 = 2$, $l_1 = x$. Let $x > 0$ be real, let $k$, $p$, $r$, $t$ be positive integers with $t \\le 2p$, and let $\\alpha = (x + \\sqrt{x^2 + 4})/2$. Then","title":"Reciprocal sums of shifted squares of Fibonacci and Lucas polynomials (Koshy\u2013Gao, Theorem 2.1)"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/tsum-jacobsthal-poly-sq-sub-family","kind":"node","lean":"MetaMathlibExt.tsum_jacobsthalPoly_sq_sub_family","parent":6812,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi/index.html#tsum-jacobsthal-poly-sq-sub-family","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/reciprocal-sums-of-squares-of-gibonacci-polynomi","status":"can_prove","summary":"Reciprocal sums of shifted squares of Jacobsthal and Jacobsthal\u2013Lucas polynomials (Koshy\u2013Gao, Theorem 3.1). Let $J_n(x)$ and $j_n(x)$ be the Jacobsthal and Jacobsthal\u2013Lucas polynomials, given by $z_{n+2} = z_{n+1} + x z_n$ with initial values $J_0 = 0$, $J_1 = 1$ and $j_0 = 2$, $j_1 = 1$. Let $x > 0$ be real, let $k$, $p$, $r$, $t$ be positive integers with $t \\le 2p$, let $w = (1 + \\sqrt{4x + 1})/2$, and write\u2026","title":"Reciprocal sums of shifted squares of Jacobsthal and Jacobsthal\u2013Lucas polynomials (Koshy\u2013Gao, Theorem 3.1)"},{"data_url":"scopes/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials.json","id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","kind":"scope","navigation":"module","parent":6718,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","title":"Sums of powers of gibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/gibonacci-poly-mul-sub-sq","kind":"node","lean":"MetaMathlibExt.gibonacciPoly_mul_sub_sq","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#gibonacci-poly-mul-sub-sq","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_i$ and $\\ell_i$ be the Fibonacci and Lucas polynomials in $\\mathbb{Z}[x]$, defined by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{i+2} = x g_{i+1} + g_i$ for $g = f, \\ell$. For all natural numbers $k \\le n$,","title":"Catalan-type identities for Fibonacci and Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-inv-pow-gibonacci-poly-sub-odd","kind":"node","lean":"MetaMathlibExt.tsum_inv_pow_gibonacciPoly_sub_odd","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-inv-pow-gibonacci-poly-sub-odd","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $x > 0$ be real, and let $g_n$ denote either the Fibonacci polynomials, given by $g_0 = 0$ and $g_1 = 1$, or the Lucas polynomials, given by $g_0 = 2$ and $g_1 = x$, both satisfying $g_{n+2} = x g_{n+1} + g_n$ and evaluated at $x$. Let $k$ be an odd natural number and $\\lambda$ a positive integer. Then","title":"Telescoping inverse-power series of gibonacci polynomials at odd indices, odd shift"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-inv-pow-gibonacci-poly-sub-even","kind":"node","lean":"MetaMathlibExt.tsum_inv_pow_gibonacciPoly_sub_even","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-inv-pow-gibonacci-poly-sub-even","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $x > 0$ be real, and let $g_n$ denote either the Fibonacci polynomials, given by $g_0 = 0$ and $g_1 = 1$, or the Lucas polynomials, given by $g_0 = 2$ and $g_1 = x$, both satisfying $g_{n+2} = x g_{n+1} + g_n$ and evaluated at $x$. Let $k$ be an even natural number and $\\lambda$ a positive integer. Then","title":"Telescoping inverse-power series of gibonacci polynomials at even indices, even shift"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-inv-pow-gibonacci-poly-succ-sub-odd","kind":"node","lean":"MetaMathlibExt.tsum_inv_pow_gibonacciPoly_succ_sub_odd","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-inv-pow-gibonacci-poly-succ-sub-odd","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $x > 0$ be real, and let $g_n$ denote either the Fibonacci polynomials, given by $g_0 = 0$ and $g_1 = 1$, or the Lucas polynomials, given by $g_0 = 2$ and $g_1 = x$, both satisfying $g_{n+2} = x g_{n+1} + g_n$ and evaluated at $x$. Let $k$ be an odd natural number and $\\lambda$ a positive integer. Then","title":"Telescoping inverse-power series of gibonacci polynomials at even indices, odd shift"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-inv-pow-gibonacci-poly-succ-sub-eve","kind":"node","lean":"MetaMathlibExt.tsum_inv_pow_gibonacciPoly_succ_sub_even","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-inv-pow-gibonacci-poly-succ-sub-eve","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $x > 0$ be real, and let $g_n$ denote either the Fibonacci polynomials, given by $g_0 = 0$ and $g_1 = 1$, or the Lucas polynomials, given by $g_0 = 2$ and $g_1 = x$, both satisfying $g_{n+2} = x g_{n+1} + g_n$ and evaluated at $x$. Let $k$ be an even natural number and $\\lambda$ a positive integer. Then","title":"Telescoping inverse-power series of gibonacci polynomials at odd indices, even shift"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/fib-poly-pow-even-sub","kind":"node","lean":"MetaMathlibExt.fibPoly_pow_even_sub","parent":6818,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even powers of Fibonacci polynomials at "],["math","\\(n \\pm k\\)"]],"source":"Even powers of Fibonacci polynomials at $n \\pm k$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#fib-poly-pow-even-sub","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_i \\in \\mathbb{Z}[x]$ be the Fibonacci polynomials, defined by $f_0 = 0$, $f_1 = 1$ and $f_{i+2} = x f_{i+1} + f_i$. For all natural numbers $k \\le n$ and every natural number $m$,","title":"Even powers of Fibonacci polynomials at $n \\pm k$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/lucas-poly-pow-even-sub","kind":"node","lean":"MetaMathlibExt.lucasPoly_pow_even_sub","parent":6818,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even powers of Lucas polynomials at "],["math","\\(n \\pm k\\)"]],"source":"Even powers of Lucas polynomials at $n \\pm k$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#lucas-poly-pow-even-sub","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_i$ and $\\ell_i$ be the Fibonacci and Lucas polynomials in $\\mathbb{Z}[x]$, defined by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{i+2} = x g_{i+1} + g_i$ for $g = f, \\ell$. For all natural numbers $k \\le n$ and every natural number $m$,","title":"Even powers of Lucas polynomials at $n \\pm k$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/fib-poly-pow-odd-sub","kind":"node","lean":"MetaMathlibExt.fibPoly_pow_odd_sub","parent":6818,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd powers of Fibonacci polynomials at "],["math","\\(n \\pm k\\)"]],"source":"Odd powers of Fibonacci polynomials at $n \\pm k$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#fib-poly-pow-odd-sub","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_i$ and $\\ell_i$ be the Fibonacci and Lucas polynomials in $\\mathbb{Z}[x]$, defined by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{i+2} = x g_{i+1} + g_i$ for $g = f, \\ell$. Let $k \\le n$ and $m$ be natural numbers. In $\\mathbb{Z}[x]$, if $k$ is odd then","title":"Odd powers of Fibonacci polynomials at $n \\pm k$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/lucas-poly-pow-odd-sub","kind":"node","lean":"MetaMathlibExt.lucasPoly_pow_odd_sub","parent":6818,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd powers of Lucas polynomials at "],["math","\\(n \\pm k\\)"]],"source":"Odd powers of Lucas polynomials at $n \\pm k$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#lucas-poly-pow-odd-sub","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_i$ and $\\ell_i$ be the Fibonacci and Lucas polynomials in $\\mathbb{Z}[x]$, defined by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{i+2} = x g_{i+1} + g_i$ for $g = f, \\ell$. Let $k \\le n$ and $m$ be natural numbers. In $\\mathbb{Z}[x]$, if $k$ is odd then","title":"Odd powers of Lucas polynomials at $n \\pm k$"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-poly-cube-even-index","kind":"node","lean":"MetaMathlibExt.tsum_fibPoly_cube_even_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-poly-cube-even-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_i$ and $\\ell_i$ be the Fibonacci and Lucas polynomials, defined by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{i+2} = x g_{i+1} + g_i$ for $g = f, \\ell$. For every real $x > 0$,","title":"Cubic-denominator series for even-index Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-lucas-poly-cube-even-index","kind":"node","lean":"MetaMathlibExt.tsum_lucasPoly_cube_even_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-lucas-poly-cube-even-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $x > 0$ be real, and let $f_n$ and $\\ell_n$ denote the Fibonacci and Lucas polynomials evaluated at $x$, given by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and the recurrence $z_{n+2} = x z_{n+1} + z_n$ for both. Then","title":"Cubic series over even-indexed Lucas polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-poly-cube-odd-index","kind":"node","lean":"MetaMathlibExt.tsum_fibPoly_cube_odd_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-poly-cube-odd-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_i$ and $\\ell_i$ be the Fibonacci and Lucas polynomials, defined by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{i+2} = x g_{i+1} + g_i$ for $g = f, \\ell$. For every real $x > 0$,","title":"Cubic-denominator series for odd-index Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-lucas-poly-cube-odd-index","kind":"node","lean":"MetaMathlibExt.tsum_lucasPoly_cube_odd_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-lucas-poly-cube-odd-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $\\ell_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, \\ell\\}$, evaluated at a real number $x > 0$. Then","title":"Cubic sums with odd-index Lucas polynomial denominators"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-cube-even-index","kind":"node","lean":"MetaMathlibExt.tsum_fib_cube_even_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-cube-even-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$) and $L_n$ the Lucas numbers. Then, as series of real numbers,","title":"Cubic-denominator series for even-index Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-lucas-cube-even-index","kind":"node","lean":"MetaMathlibExt.tsum_lucas_cube_even_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-lucas-cube-even-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers, given by $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, $L_1 = 1$ and the recurrence $z_{n+2} = z_{n+1} + z_n$ for both. Then","title":"Cubic series over even-indexed Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-cube-odd-index","kind":"node","lean":"MetaMathlibExt.tsum_fib_cube_odd_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-cube-odd-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$) and $L_n$ the Lucas numbers. Then, as series of real numbers,","title":"Cubic-denominator series for odd-index Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-lucas-cube-odd-index","kind":"node","lean":"MetaMathlibExt.tsum_lucas_cube_odd_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-lucas-cube-odd-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers, given by $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, $L_1 = 1$ and the recurrence $z_{n+2} = z_{n+1} + z_n$ for both. Then","title":"Cubic series over odd-indexed Lucas numbers"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-poly-even-pow-even-index","kind":"node","lean":"MetaMathlibExt.tsum_fibPoly_even_pow_even_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-poly-even-pow-even-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_i$ be the Fibonacci polynomials, defined by $f_0 = 0$, $f_1 = 1$ and $f_{i+2} = x f_{i+1} + f_i$. Let $x > 0$ be real and let $k$ and $m$ be positive integers. If $k$ is odd, put $L = (k + 1)/2$ and $s_r = 2r - 1$; if $k$ is even, put $L = k/2 + 1$ and $s_r = 2r$. Then","title":"Series of even powers of Fibonacci polynomials at even indices"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-lucas-poly-even-pow-even-index","kind":"node","lean":"MetaMathlibExt.tsum_lucasPoly_even_pow_even_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-lucas-poly-even-pow-even-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $\\ell_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, \\ell\\}$, evaluated at a real number $x > 0$, and let $k$ and $m$ be positive integers. Put $L = (k + 1)/2$ and $s_r = 2r - 1$ if $k$ is odd, and $L = k/2 + 1$ and $s_r = 2r$ if $k$ is even. Then","title":"Even-power sums with even-index Lucas polynomial denominators"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-poly-even-pow-odd-index","kind":"node","lean":"MetaMathlibExt.tsum_fibPoly_even_pow_odd_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-poly-even-pow-odd-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $x > 0$ be real, let $f_n$ denote the Fibonacci polynomials evaluated at $x$, given by $f_0 = 0$, $f_1 = 1$ and $f_{n+2} = x f_{n+1} + f_n$, and let $k$ and $m$ be positive integers. Put $M = (k + 1)/2$ and $t_r = 2r$ if $k$ is odd, and $M = k/2$ and $t_r = 2r - 1$ if $k$ is even. Then","title":"Even-power series over odd-indexed Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-lucas-poly-even-pow-odd-index","kind":"node","lean":"MetaMathlibExt.tsum_lucasPoly_even_pow_odd_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-lucas-poly-even-pow-odd-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $\\ell_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, \\ell\\}$, evaluated at a real number $x > 0$, and let $k$ and $m$ be positive integers. Put $M = (k + 1)/2$ and $t_r = 2r$ if $k$ is odd, and $M = k/2$ and $t_r = 2r - 1$ if $k$ is even. Then","title":"Even-power sums with odd-index Lucas polynomial denominators"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-poly-odd-pow-even-index","kind":"node","lean":"MetaMathlibExt.tsum_fibPoly_odd_pow_even_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-poly-odd-pow-even-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $x > 0$ be real, let $f_n$ and $\\ell_n$ denote the Fibonacci and Lucas polynomials evaluated at $x$, given by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and the recurrence $z_{n+2} = x z_{n+1} + z_n$ for both, and let $k$ be a positive integer and $m$ a natural number. Put $L = (k + 1)/2$ if $k$ is odd and $L = k/2 + 1$ if $k$ is even. If $k$ is odd, then","title":"Odd-power series over even-indexed Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-lucas-poly-odd-pow-even-index","kind":"node","lean":"MetaMathlibExt.tsum_lucasPoly_odd_pow_even_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-lucas-poly-odd-pow-even-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $\\ell_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, \\ell\\}$, evaluated at a real number $x > 0$, let $k$ be a positive integer and $m$ a natural number, and put $L = (k + 1)/2$ if $k$ is odd and $L = k/2 + 1$ if $k$ is even. If $k$ is odd, then","title":"Odd-power sums with even-index Lucas polynomial denominators"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-poly-odd-pow-odd-index","kind":"node","lean":"MetaMathlibExt.tsum_fibPoly_odd_pow_odd_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-poly-odd-pow-odd-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $x > 0$ be real, let $f_n$ and $\\ell_n$ denote the Fibonacci and Lucas polynomials evaluated at $x$, given by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and the recurrence $z_{n+2} = x z_{n+1} + z_n$ for both, and let $k$ be a positive integer and $m$ a natural number. Put $M = (k + 1)/2$ if $k$ is odd and $M = k/2$ if $k$ is even. If $k$ is odd, then","title":"Odd-power series over odd-indexed Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-lucas-poly-odd-pow-odd-index","kind":"node","lean":"MetaMathlibExt.tsum_lucasPoly_odd_pow_odd_index","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-lucas-poly-odd-pow-odd-index","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $\\ell_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $\\ell_0 = 2$, $\\ell_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, \\ell\\}$, evaluated at a real number $x > 0$, let $k$ be a positive integer and $m$ a natural number, and put $M = (k + 1)/2$ if $k$ is odd and $M = k/2$ if $k$ is even. If $k$ is odd, then","title":"Odd-power sums with odd-index Lucas polynomial denominators"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-gibonacci-poly-pow-ratio-telescope","kind":"node","lean":"MetaMathlibExt.tsum_gibonacciPoly_pow_ratio_telescope","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-gibonacci-poly-pow-ratio-telescope","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $l_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $l_0 = 2$, $l_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, l\\}$, evaluated at a real number $x > 0$, and let $\\alpha = (x + \\sqrt{x^2 + 4})/2$. Then for $g \\in \\{f, l\\}$, every positive integer $k$ and all natural numbers $r$ and $\\lambda$,","title":"Telescoping sum of powers of gibonacci polynomial ratios"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-gibonacci-poly-pow-inv-ratio-telesc","kind":"node","lean":"MetaMathlibExt.tsum_gibonacciPoly_pow_inv_ratio_telescope","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-gibonacci-poly-pow-inv-ratio-telesc","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $l_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $l_0 = 2$, $l_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, l\\}$, evaluated at a real number $x > 0$, and let $\\beta = (x - \\sqrt{x^2 + 4})/2$. Then for $g \\in \\{f, l\\}$, every positive integer $k$ and all natural numbers $r$ and $\\lambda$,","title":"Telescoping sum of powers of inverse gibonacci polynomial ratios"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-mul-div-gibonacci-poly-sq-add","kind":"node","lean":"MetaMathlibExt.tsum_fib_mul_div_gibonacciPoly_sq_add","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-mul-div-gibonacci-poly-sq-add","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $l_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $l_0 = 2$, $l_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, l\\}$, evaluated at a real number $x > 0$, and let $\\alpha = (x + \\sqrt{x^2 + 4})/2$. Let $g$ be $f$ with $(\\mu, \\nu) = (1, -1)$, or $l$ with $(\\mu, \\nu) = (x^2 + 4, 1)$. Then for every positive integer $k$ and every natural number $r$,","title":"Gibonacci polynomial square sums at even multiples of the index"},{"id":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/tsum-fib-mul-div-gibonacci-poly-shift-sq","kind":"node","lean":"MetaMathlibExt.tsum_fib_mul_div_gibonacciPoly_shift_sq_add","parent":6818,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials/index.html#tsum-fib-mul-div-gibonacci-poly-shift-sq","scope":"NumberTheory/linear-recurrences/fibonacci-polynomials/sums-of-powers-of-gibonacci-polynomials","status":"can_prove","summary":"Let $f_n$ and $l_n$ denote the Fibonacci and Lucas polynomials, given by $f_0 = 0$, $f_1 = 1$, $l_0 = 2$, $l_1 = x$ and $g_{n+2} = x g_{n+1} + g_n$ for $g \\in \\{f, l\\}$, evaluated at a real number $x > 0$, and let $\\beta = (x - \\sqrt{x^2 + 4})/2$. Let $g$ be $f$ with $(\\mu, \\nu) = (1, -1)$, or $l$ with $(\\mu, \\nu) = (x^2 + 4, 1)$. Then for every positive integer $k$ and every natural number $r$,","title":"Gibonacci polynomial square sums at shifted even multiples of the index"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory.json","id":"NumberTheory/linear-recurrences/recurrence-theory","kind":"scope","parent":5991,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory","status":"can_prove","summary":"C-finite, P-recursive and MC-finite sequences, Binet formulas, Skolem\u2013Mahler\u2013Lech, growth, generating functions and decomposable forms.","title":"General theory of recurrences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/is-precursive-sequence","kind":"node","lean":"MetaMathlibExt.IsPRecursiveSequence","parent":6848,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/index.html#is-precursive-sequence","scope":"NumberTheory/linear-recurrences/recurrence-theory","status":"fully_proved","summary":"Let $r$ be a natural number, let $P_0, \\dots, P_r$ be polynomials with integer coefficients, and let $(u_n)_{n \\ge 0}$ be a sequence of integers. The sequence $u$ is P-recursive of order $r$ with coefficient polynomials $P_0, \\dots, P_r$ if for every $n \\ge r$,","title":"P-recursive sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/binet-closed-form-second-order","kind":"node","lean":"MetaMathlibExt.binet_closed_form_second_order","parent":6848,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/index.html#binet-closed-form-second-order","scope":"NumberTheory/linear-recurrences/recurrence-theory","status":"fully_proved","summary":"Let $K$ be a field, let $P, Q, \\alpha, \\beta \\in K$ with $\\alpha \\ne \\beta$, and suppose that $\\alpha$ and $\\beta$ are both roots of $z^2 - Pz + Q$. If a sequence $u : \\mathbb{N} \\to K$ satisfies $u_{n+2} = P u_{n+1} - Q u_n$ for all $n$, then for every natural number $n$,","title":"Binet formula for second-order linear recurrences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/skolem-mahler-lech","kind":"node","lean":"MetaMathlibExt.skolemMahlerLech","parent":6848,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/index.html#skolem-mahler-lech","scope":"NumberTheory/linear-recurrences/recurrence-theory","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero, let $k$ be a natural number, let $c_0, \\dots, c_{k-1} \\in K$, and let $u_0, u_1, u_2, \\dots$ be a sequence in $K$ satisfying the linear recurrence $u_{n+k} = \\sum_{i=0}^{k-1} c_i u_{n+i}$ for all $n \\ge 0$ (for $k = 0$ the sum is empty and every $u_n$ is $0$). Then there are a finite set $F \\subseteq \\mathbb{N}$ and finitely many pairs $(a_1, d_1), \\dots, (a_m, d_m)$ of\u2026","title":"Skolem\u2013Mahler\u2013Lech theorem"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array.json","id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","title":"Annihilators and recursions of the circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/annihilates-seq","kind":"node","lean":"MetaMathlibExt.AnnihilatesSeq","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#annihilates-seq","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Let $K$ be a semiring, $p = \\sum_i a_i X^i \\in K[X]$ and $G \\colon \\mathbb{N} \\to K$ a sequence, and let $E$ be the translation operator $E(G_s) = G_{s+1}$. The operator $p(E)$ annihilates $G$ when $\\sum_i a_i G_{s+i} = 0$ for every $s \\ge 0$.","title":"Annihilator of a sequence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/annihilates-seq-zero","kind":"node","lean":"MetaMathlibExt.annihilatesSeq_zero","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#annihilates-seq-zero","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"For every semiring $K$ and every polynomial $p \\in K[X]$, the operator $p(E)$ annihilates the zero sequence, where $E$ is the translation operator $E(G_s) = G_{s+1}$.","title":"Every operator annihilates the zero sequence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/annihilates-seq-char-poly-of-is-solution","kind":"node","lean":"MetaMathlibExt.annihilatesSeq_charPoly_of_isSolution","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#annihilates-seq-char-poly-of-is-solution","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"Let $K$ be a field, $d \\ge 0$ and $a_0, \\dots, a_{d-1} \\in K$. If a sequence $G \\colon \\mathbb{N} \\to K$ satisfies $G_{n+d} = \\sum_{i=0}^{d-1} a_i G_{n+i}$ for every $n \\ge 0$, then the characteristic polynomial $X^d - \\sum_{i=0}^{d-1} a_i X^i$ of this recurrence annihilates $G$.","title":"The characteristic polynomial annihilates the solutions of its recurrence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/annihilates-seq-x-sub-c-pow","kind":"node","lean":"MetaMathlibExt.annihilatesSeq_X_sub_C_pow","parent":6852,"reading_title":{"format":"inline-parts-v1","parts":[["text","The operator "],["math","\\(E - c\\)"],["text"," annihilates the powers of "],["math","\\(c\\)"]],"source":"The operator $E - c$ annihilates the powers of $c$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#annihilates-seq-x-sub-c-pow","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"Let $K$ be a field, $c \\in K$, and let $E$ be the translation operator $E(G_s) = G_{s+1}$ on sequences $G \\colon \\mathbb{N} \\to K$. Then the operator $E - c$, that is, $p(E)$ for the polynomial $p = X - c \\in K[X]$, annihilates the sequence $s \\mapsto c^s$.","title":"The operator $E - c$ annihilates the powers of $c$"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/annihilates-seq-sum-prod-and-mul","kind":"node","lean":"MetaMathlibExt.annihilatesSeq_sum_prod_and_mul","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#annihilates-seq-sum-prod-and-mul","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"Let $E$ be the translation operator $E(G_s) = G_{s+1}$ on sequences. (i) Let $K$ be a field, $I$ a finite index set, and for each $i \\in I$ let $A_i \\in K[X]$, $G_i \\colon \\mathbb{N} \\to K$ and $c_i \\in K$. If $A_i(E)$ annihilates $G_i$ for every $i \\in I$, then $\\prod_{i \\in I} A_i(E)$ annihilates the sequence $s \\mapsto \\sum_{i \\in I} c_i G_i(s)$. (ii) Let $p_1, p_2 \\in \\mathbb{C}[X]$ be nonzero polynomials and\u2026","title":"The Addition-Multiplication Lemma for annihilators"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-edge","kind":"node","lean":"MetaMathlibExt.CircuitEdge","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-edge","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"An upright triangle of the grid has three edges: the left edge $L$, the right edge $R$ and the base $B$.","title":"Edges of an upright triangle"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-delta","kind":"node","lean":"MetaMathlibExt.circuitDelta","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-delta","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"For elements $x$, $y$, $z$ of a field $K$, the delta-wye resistance function is $\\Delta(x, y, z) = \\frac{xy}{x + y + z}$.","title":"Delta-wye resistance function"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-y","kind":"node","lean":"MetaMathlibExt.circuitY","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-y","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"For elements $a$, $b$, $c$ of a field $K$, the $Y$-$\\Delta$ resistance function is $Y(a, b, c) = (ab + bc + ca)/a$.","title":"The Y-\u0394 resistance function"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-top-arm","kind":"node","lean":"MetaMathlibExt.circuitTopArm","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-top-arm","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"For an upright triangle whose left, right and base edges carry the values $L$, $R$, $B$ in a field $K$, the top arm is $\\Delta(L, R, B) = \\frac{LR}{L + R + B}$, where $\\Delta$ is the delta-wye resistance function: the arm of the equivalent star from the centre of the triangle to its top vertex.","title":"Top delta-wye arm"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-right-arm","kind":"node","lean":"MetaMathlibExt.circuitRightArm","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-right-arm","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"For an upright triangle whose left, right and base edges carry the values $L$, $R$, $B$ in a field $K$, the bottom-right arm is $\\Delta(R, B, L) = \\frac{RB}{R + B + L}$, where $\\Delta$ is the delta-wye resistance function: the arm of the equivalent star from the centre of the triangle to its bottom-right vertex.","title":"Bottom-right delta-wye arm"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-left-arm","kind":"node","lean":"MetaMathlibExt.circuitLeftArm","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-left-arm","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"For an upright triangle whose left, right and base edges carry the values $L$, $R$, $B$ in a field $K$, the bottom-left arm is $\\Delta(B, L, R) = \\frac{BL}{B + L + R}$, where $\\Delta$ is the delta-wye resistance function: the arm of the equivalent star from the centre of the triangle to its bottom-left vertex.","title":"Bottom-left delta-wye arm"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-perimeter","kind":"node","lean":"MetaMathlibExt.circuitPerimeter","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-perimeter","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Let $K$ be a field, let $\\Delta(x, y, z) = xy/(x + y + z)$ be the delta-wye resistance function, and for an upright triangle $t$ write $L_t$, $R_t$, $B_t \\in K$ for the values on its left, right and base edges. For triangles $t$ and $u$, the left-perimeter function is $\\mathcal{P}(t, u) = \\Delta(B_t, L_t, R_t) + \\Delta(L_u, R_u, B_u)$, the series resistance of the bottom-left arm of $t$ and the top arm of $u$.","title":"Left-perimeter function"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-left-edge","kind":"node","lean":"MetaMathlibExt.circuitLeftEdge","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-left-edge","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Let $K$ be a field, let $\\Delta(x, y, z) = xy/(x + y + z)$ and $Y(a, b, c) = (ab + bc + ca)/a$ be the delta-wye and wye-delta resistance functions, and for an upright triangle $t$ write $L_t$, $R_t$, $B_t \\in K$ for the values on its left, right and base edges. For triangles $t$, $u$, $v$, the left-edge function is $\\mathcal{L}(t, u, v) = Y(\\Delta(R_t, B_t, L_t), \\Delta(B_u, L_u, R_u), \\Delta(L_v, R_v, B_v))$, the\u2026","title":"Left-edge function"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-right-edge","kind":"node","lean":"MetaMathlibExt.circuitRightEdge","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-right-edge","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Let $K$ be a field, let $\\Delta(x, y, z) = xy/(x + y + z)$ and $Y(a, b, c) = (ab + bc + ca)/a$ be the delta-wye and wye-delta resistance functions, and for an upright triangle $t$ write $L_t$, $R_t$, $B_t \\in K$ for the values on its left, right and base edges. For triangles $t$, $u$, $v$, the right-edge function is $\\mathcal{R}(t, u, v) = Y(\\Delta(B_u, L_u, R_u), \\Delta(R_t, B_t, L_t), \\Delta(L_v, R_v, B_v))$, the\u2026","title":"Right-edge function"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-base-edge","kind":"node","lean":"MetaMathlibExt.circuitBaseEdge","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-base-edge","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Let $K$ be a field. For a triangle with left, right and base resistances $L$, $R$, $B$ in $K$, its top arm is $\\Delta(L, R, B)$, its bottom-right arm is $\\Delta(R, B, L)$ and its bottom-left arm is $\\Delta(B, L, R)$, where $\\Delta(x, y, z) = xy/(x + y + z)$. For triangles $t$, $u$, $v$, the base-edge function is $\\mathcal{B}(t, u, v) = Y(a, b, c) = (ab + bc + ca)/a$, where $a$ is the top arm of $t$, $b$ is the\u2026","title":"Base-edge function"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-reduce","kind":"node","lean":"MetaMathlibExt.circuitReduce","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-reduce","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Let $K$ be a field and let $T$ be a labelling that assigns a value $T_{r,d,e} \\in K$ to the edge $e$ (left $L$, right $R$ or base $B$) of the upright triangle in row $r$ (from the top) and diagonal $d$ (from the left); write $T_{r,d}$ for that triangle and $\\Delta(x, y, z) = xy/(x + y + z)$ for the delta-wye resistance function. The reduced labelling $T'$ is given by $T'_{r,1,L} = \\mathcal{P}(T_{r,1}, T_{r+1,1})$\u2026","title":"Row reduction of a labelled triangular grid"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-grid","kind":"node","lean":"MetaMathlibExt.circuitGrid","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-grid","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Let $K$ be a field, and for each $m$ let either a boundary value $\\beta_m \\in K$ be given or none. The labellings $T^m$, which assign a value $T^m_{r,d,e} \\in K$ to the edge $e$ (left $L$, right $R$ or base $B$) of the upright triangle in row $r$ and diagonal $d$, are defined by recursion on $m$: $T^0$ is identically $1$, and $T^{m+1}$ is the row reduction of $T^m$, except that if $\\beta_{m+1}$ is given then every\u2026","title":"Reduced circuit grids"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-of","kind":"node","lean":"MetaMathlibExt.circuitArrayOf","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-of","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Let $K$ be a field and let $\\beta$ assign to each natural number $m$ either nothing or a value $\\beta(m) \\in K$. Let $T^0$ be the all-one triangular grid and, for $m \\ge 1$, let $T^m$ be obtained from $T^{m-1}$ by one row reduction followed, when $\\beta(m)$ is given, by setting every boundary edge (left edges on diagonal $1$, right edges on diagonal $r$ in row $r$) to $\\beta(m)$. For natural numbers $i$ and $j$,\u2026","title":"Circuit array of a boundary assignment"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array","kind":"node","lean":"MetaMathlibExt.circuitArray","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"The circuit array $C_{i,j} \\in \\mathbb{Q}$ is the circuit array of the reductions of the all-one triangular grid with no boundary edge replaced: $C_{i,j}$ is the left edge (for even $i$) or the right edge (for odd $i$) of the triangle in row $2j - 1$ and diagonal $j - \\lfloor (i+1)/2 \\rfloor$ of the all-one grid reduced $j$ times, with subtraction truncated at $0$.","title":"Circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-x","kind":"node","lean":"MetaMathlibExt.circuitArrayX","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-x","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"The one-variable circuit array $C^X_{i,j}$, with entries in the field $\\mathbb{Q}(X)$ of rational functions in one variable over $\\mathbb{Q}$, is the circuit array of the reductions of the all-one triangular grid in which, after the first reduction, every boundary edge is set to $(X - 3)/X$, and no boundary edge is replaced after any later reduction. Thus $C^X_{i,j}$ is the left edge (for even $i$) or the right\u2026","title":"One-variable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-var","kind":"node","lean":"MetaMathlibExt.circuitVar","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-var","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"For a natural number $m$, $X_m$ is the indeterminate $X_m$ regarded as an element of the field of fractions $\\mathbb{Q}(X_0, X_1, X_2, \\dots)$ of the polynomial ring $\\mathbb{Q}[X_0, X_1, X_2, \\dots]$ in countably many variables.","title":"Variables of the multivariable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-m","kind":"node","lean":"MetaMathlibExt.circuitArrayM","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-m","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"Reduce the all-one triangular grid row by row as in Section 2, and after the $m$-th reduction, for every $m \\ge 1$, set every boundary edge (the left edges on the first diagonal and the right edges on the last diagonal) to the variable $X_m$ of the field of rational functions over $\\mathbb{Q}$ in $X_0, X_1, \\dots$. The multivariable circuit array $C^M_{i,j}$ (Table 3) is the circuit array read off these grids: for\u2026","title":"Multivariable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-small","kind":"node","lean":"MetaMathlibExt.circuitArray_small","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-small","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"The circuit array satisfies $C_{0,1} = 2/3$, $C_{0,2} = 26/27$, $C_{1,2} = 13/12$ and $C_{2,2} = 1/2$.","title":"First entries of the circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-x-zero-one","kind":"node","lean":"MetaMathlibExt.circuitArrayX_zero_one","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-x-zero-one","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"The one-variable circuit array $C^X$, with entries in the field $\\mathbb{Q}(X)$ of rational functions over $\\mathbb{Q}$, satisfies $C^X_{0,1} = (X - 3)/X$.","title":"First entry of the one-variable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-m-diagonal","kind":"node","lean":"MetaMathlibExt.circuitArrayM_diagonal","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-m-diagonal","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"fully_proved","summary":"The multivariable circuit array $C^M$, with values in the field of rational functions over $\\mathbb{Q}$ in the variables $X_0, X_1, \\dots$, satisfies $C^M_{0,1} = X_1$ and $C^M_{2,2} = X_2$.","title":"First entries of the left diagonal of the multivariable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-x-inductive","kind":"node","lean":"MetaMathlibExt.circuitArrayX_inductive","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-x-inductive","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"Extend the one-variable circuit array $C^X$, with entries in the field $\\mathbb{Q}(X)$ of rational functions over $\\mathbb{Q}$, to integer row indices by $C_{i,j} = C^X_{i,j}$ for $i \\ge 0$ and $C_{i,j} = 1$ for $i < 0$. For an integer $i$ and a natural number $j$, let $\\tau_{i,j}$ be the triangle with left edge $C_{i,j}$ and right and base edges $C_{i-1,j}$. Let $\\mathcal{P}$, $\\mathcal{L}$ and $\\mathcal{R}$ be\u2026","title":"Column recursion for the one-variable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-x-row-zero","kind":"node","lean":"MetaMathlibExt.circuitArrayX_row_zero","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-x-row-zero","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"For every natural number $s \\ge 1$, the one-variable circuit array $C^X$, with entries in the field $\\mathbb{Q}(X)$ of rational functions over $\\mathbb{Q}$, satisfies $C^X_{0,s} = (3 \\cdot 9^{s-2} X - 1)/(3 \\cdot 9^{s-2} X)$, with an integer exponent (so $9^{s-2} = 1/9$ when $s = 1$), and the circuit array $C$ satisfies $C_{0,s} = (9^s/3 - 1)/(9^s/3)$ in $\\mathbb{Q}$.","title":"Row zero of the one-variable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-x-row-one","kind":"node","lean":"MetaMathlibExt.circuitArrayX_row_one","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-x-row-one","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"For every natural number $s \\ge 2$, the one-variable circuit array $C^X$, with entries in the field $\\mathbb{Q}(X)$ of rational functions over $\\mathbb{Q}$, satisfies $C^X_{1,s} = (3 \\cdot 9^{s-2} X - 1)/(3 \\cdot 9^{s-2} X - 3)$.","title":"Row one of the one-variable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-x-row-two","kind":"node","lean":"MetaMathlibExt.circuitArrayX_row_two","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-x-row-two","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"For every natural number $s \\ge 2$, the one-variable circuit array $C^X$, with entries in the field $\\mathbb{Q}(X)$ of rational functions over $\\mathbb{Q}$, satisfies $C^X_{2,s} = (c_{2,0}(s) - c_{2,1}(s) X + c_{2,2}(s) X^2)/(9^{s-2} X - 1)^2$, where $c_{2,0}(s) = 27 \\cdot 9^{s-3} - 1$, $c_{2,1}(s) = 24(2s - 3) \\cdot 9^{s-3}$ and $c_{2,2}(s) = 81 \\cdot 81^{s-3} - 3 \\cdot 9^{s-3}$, with integer exponents (so\u2026","title":"Row two of the one-variable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-m-row-zero","kind":"node","lean":"MetaMathlibExt.circuitArrayM_row_zero","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-m-row-zero","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"Let $C^M$ be the multivariable circuit array, with values in the field of rational functions over $\\mathbb{Q}$ in the variables $X_0, X_1, \\dots$. For every integer $s \\ge 1$, $C^M_{0,s} = (9^{s-1} - 1 + X_1)/9^{s-1}$.","title":"Row 0 of the multivariable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-m-row-one","kind":"node","lean":"MetaMathlibExt.circuitArrayM_row_one","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-m-row-one","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"Let $C^M$ be the multivariable circuit array, with values in the field of rational functions over $\\mathbb{Q}$ in the variables $X_0, X_1, \\dots$. For every integer $s \\ge 2$, $C^M_{1,s} = (9^{s-1} - 1 + X_1)/\\bigl(3(X_1 + 3 \\cdot 9^{s-2} - 1)\\bigr)$.","title":"Row 1 of the multivariable circuit array"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/circuit-array-m-row-two","kind":"node","lean":"MetaMathlibExt.circuitArrayM_row_two","parent":6852,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array/index.html#circuit-array-m-row-two","scope":"NumberTheory/linear-recurrences/recurrence-theory/annihilators-and-recursions-of-the-circuit-array","status":"can_prove","summary":"Let $C^M$ be the multivariable circuit array, with values in the field of rational functions over $\\mathbb{Q}$ in the variables $X_0, X_1, \\dots$. For every integer $s \\ge 2$,","title":"Row 2 of the multivariable circuit array"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences.json","id":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences","status":"fully_proved","title":"C-finite sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/is-integer-cfinite-sequence","kind":"node","lean":"MetaMathlibExt.IsIntegerCFiniteSequence","parent":6885,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/index.html#is-integer-cfinite-sequence","scope":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences","status":"fully_proved","summary":"An integer sequence $s \\colon \\mathbb{N} \\to \\mathbb{Z}$ is C-finite if there are $p, q \\in \\mathbb{N}$ and integers $c_0, \\dots, c_{p-1}$ such that $s(n + p) = \\sum_{i=0}^{p-1} c_i \\, s(n + i)$ for every $n \\ge q$.","title":"C-finite integer sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/is-integer-cfinite-sequence-of-is-soluti","kind":"node","lean":"MetaMathlibExt.isIntegerCFiniteSequence_of_isSolution","parent":6885,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/index.html#is-integer-cfinite-sequence-of-is-soluti","scope":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences","status":"fully_proved","summary":"Let $p \\in \\mathbb{N}$ and let $c_0, \\dots, c_{p-1}$ be integers. If an integer sequence $s$ satisfies $s(n + p) = \\sum_{i=0}^{p-1} c_i \\, s(n + i)$ for every $n \\in \\mathbb{N}$, then $s$ is C-finite.","title":"Solutions of linear recurrences are C-finite"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/is-solution-tail-of-c-finite-witness","kind":"node","lean":"MetaMathlibExt.isSolution_tail_of_cFiniteWitness","parent":6885,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/index.html#is-solution-tail-of-c-finite-witness","scope":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences","status":"fully_proved","summary":"Let $s$ be an integer sequence, let $p, q \\in \\mathbb{N}$ and let $c_0, \\dots, c_{p-1}$ be integers. If $s(n + p) = \\sum_{i=0}^{p-1} c_i \\, s(n + i)$ for every $n \\ge q$, then the shifted sequence $t(n) = s(n + q)$ satisfies $t(n + p) = \\sum_{i=0}^{p-1} c_i \\, t(n + i)$ for every $n \\in \\mathbb{N}$.","title":"Discarding the exceptional prefix of a C-finite recurrence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/exists-is-solution-tail","kind":"node","lean":"MetaMathlibExt.IsIntegerCFiniteSequence.exists_isSolution_tail","parent":6885,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences/index.html#exists-is-solution-tail","scope":"NumberTheory/linear-recurrences/recurrence-theory/c-finite-sequences","status":"fully_proved","summary":"If an integer sequence $s$ is C-finite, then there are a shift $q \\in \\mathbb{N}$, an order $p \\in \\mathbb{N}$ and integers $c_0, \\dots, c_{p-1}$ such that the shifted sequence $t(n) = s(n + q)$ satisfies $t(n + p) = \\sum_{i=0}^{p-1} c_i \\, t(n + i)$ for every $n \\in \\mathbb{N}$.","title":"A tail of a C-finite sequence satisfies a linear recurrence"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences.json","id":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences","status":"can_prove","title":"Decomposable forms on linear recurrences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/exists-homogeneous-form-eval-eq-pow-of-l","kind":"node","lean":"MetaMathlibExt.exists_homogeneous_form_eval_eq_pow_of_linearRecurrence","parent":6890,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/index.html#exists-homogeneous-form-eval-eq-pow-of-l","scope":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences","status":"can_prove","summary":"Let $\\gamma_0, \\dots, \\gamma_{k-1} \\in \\mathbb{C}$ and let $\\chi(x) = x^k - \\sum_{i\\lt k} \\gamma_i x^i$ be the characteristic polynomial of the recurrence $G_{n+k} = \\sum_{i\\lt k} \\gamma_i G_{n+i}$. Let $G^{(1)}, \\dots, G^{(k)} \\colon \\mathbb{Z} \\to \\mathbb{C}$ be $\\mathbb{C}$-linearly independent sequences, each satisfying this recurrence for every $n \\in \\mathbb{Z}$. Then there is a homogeneous polynomial $F$ of\u2026","title":"Homogeneous forms taking the values of a power on recurrence sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/exists-decomposable-form-eval-eq-pow-of","kind":"node","lean":"MetaMathlibExt.exists_decomposable_form_eval_eq_pow_of_linearRecurrence","parent":6890,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/index.html#exists-decomposable-form-eval-eq-pow-of","scope":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences","status":"can_prove","summary":"Let $\\gamma_0, \\dots, \\gamma_{k-1} \\in \\mathbb{C}$, let $\\chi(x) = x^k - \\sum_{i\\lt k} \\gamma_i x^i$, and let $G^{(1)}, \\dots, G^{(k)} \\colon \\mathbb{Z} \\to \\mathbb{C}$ be $\\mathbb{C}$-linearly independent sequences, each satisfying $G_{n+k} = \\sum_{i\\lt k} \\gamma_i G_{n+i}$ for every $n \\in \\mathbb{Z}$. Then there are linear forms $L_1, \\dots, L_k$ in $k$ variables with complex coefficients whose product\u2026","title":"Decomposable forms taking the values of a power on recurrence sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/exists-decomposable-form-eval-shifts-eq","kind":"node","lean":"MetaMathlibExt.exists_decomposable_form_eval_shifts_eq_pow","parent":6890,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/index.html#exists-decomposable-form-eval-shifts-eq","scope":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences","status":"can_prove","summary":"Let $\\gamma_0, \\dots, \\gamma_{k-1} \\in \\mathbb{C}$, let $\\chi(x) = x^k - \\sum_{i\\lt k} \\gamma_i x^i$, and let $G \\colon \\mathbb{Z} \\to \\mathbb{C}$ satisfy $G_{n+k} = \\sum_{i\\lt k} \\gamma_i G_{n+i}$ for every $n \\in \\mathbb{Z}$. Suppose that the $k$ vectors $(G_j, G_{j+1}, \\dots, G_{j+k-1}) \\in \\mathbb{C}^k$ for $0 \\le j < k$ are linearly independent. Then there are linear forms $L_1, \\dots, L_k$ in $k$ variables\u2026","title":"Decomposable forms on consecutive terms of one recurrence sequence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/not-algebraic-independent-linear-recurre","kind":"node","lean":"MetaMathlibExt.not_algebraicIndependent_linearRecurrence_solutions_pow","parent":6890,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/index.html#not-algebraic-independent-linear-recurre","scope":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences","status":"can_prove","summary":"Let $k$ be a natural number, let $\\gamma_0, \\dots, \\gamma_{k-1} \\in \\mathbb{C}$ and let $\\chi(x) = x^k - \\sum_{i\\lt k} \\gamma_i x^i$. Let $G^{(1)}, \\dots, G^{(k)} \\colon \\mathbb{Z} \\to \\mathbb{C}$ be sequences such that $G^{(j)}_{n+k} = \\sum_{i\\lt k} \\gamma_i G^{(j)}_{n+i}$ for every $j$ and every $n \\in \\mathbb{Z}$. Then the family of the $k + 1$ sequences $G^{(1)}, \\dots, G^{(k)}$ and $n \\mapsto (-\\chi(0))^n$\u2026","title":"Algebraic dependence of recurrence sequences and a power sequence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/exists-int-decomposable-form-eval-eq-det","kind":"node","lean":"MetaMathlibExt.exists_int_decomposable_form_eval_eq_det_pow_mul_pow","parent":6890,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/index.html#exists-int-decomposable-form-eval-eq-det","scope":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences","status":"can_prove","summary":"Let $\\gamma_0, \\dots, \\gamma_{k-1} \\in \\mathbb{Z}$, let $\\chi(x) = x^k - \\sum_{i\\lt k} \\gamma_i x^i$, suppose $\\chi(0) \\neq 0$, and let $G^{(1)}, \\dots, G^{(k)} \\colon \\mathbb{N} \\to \\mathbb{Z}$ each satisfy $G_{n+k} = \\sum_{i\\lt k} \\gamma_i G_{n+i}$ for every $n \\ge 0$. Suppose that the determinant $\\Delta$ of the $k \\times k$ matrix of initial values $(G^{(j)}_i)$, with rows $1 \\le j \\le k$ and columns\u2026","title":"Integer decomposable forms on integer recurrence sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/binary-recurrence-quadratic-form-eq-neg","kind":"node","lean":"MetaMathlibExt.binary_recurrence_quadratic_form_eq_neg_pow_mul_sq","parent":6890,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences/index.html#binary-recurrence-quadratic-form-eq-neg","scope":"NumberTheory/linear-recurrences/recurrence-theory/decomposable-forms-on-linear-recurrences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $A, B \\in R$, and let $(G_n)_{n \\ge 0}$ and $(H_n)_{n \\ge 0}$ be sequences in $R$ with $G_{n+2} = A G_{n+1} + B G_n$ and $H_{n+2} = A H_{n+1} + B H_n$ for all $n \\ge 0$. Put $C_G = G_1^2 - A G_0 G_1 - B G_0^2$, $C_H = H_1^2 - A H_0 H_1 - B H_0^2$, $E_{10} = G_1 H_1 - A G_1 H_0 - B G_0 H_0$, $E_{01} = G_1 H_1 - A G_0 H_1 - B G_0 H_0$ and $\\Delta = G_0 H_1 - G_1 H_0$. Then for every\u2026","title":"Quadratic form identity for binary recurrences"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences.json","id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","title":"Divisibility runs between recurrence sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/is-div-run-up-to","kind":"node","lean":"MetaMathlibExt.IsDivRunUpTo","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#is-div-run-up-to","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"fully_proved","summary":"For integers $x, y, a, b, c$ and a natural number $L$, let $(P_n)_{n \\ge 0}$ be the Gibonacci sequence with $P_0 = x$ and $P_1 = y$, and let $(Q_n)_{n \\ge 0}$ be the generalized Tribonacci sequence with $Q_0 = a$, $Q_1 = b$ and $Q_2 = c$. The configuration $(x, y; a, b, c)$ has a divisibility run of length at least $L$ when $P_i \\ne 0$ and $P_i \\mid Q_i$ for every $i < L$.","title":"Divisibility runs"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/is-div-run-up-to-zero","kind":"node","lean":"MetaMathlibExt.isDivRunUpTo_zero","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#is-div-run-up-to-zero","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"fully_proved","summary":"For integers $x, y, a, b, c$, let $(P_n)_{n \\ge 0}$ be the Gibonacci sequence with $P_0 = x$ and $P_1 = y$, let $(Q_n)_{n \\ge 0}$ be the generalized Tribonacci sequence with $Q_0 = a$, $Q_1 = b$ and $Q_2 = c$, and say $(x, y; a, b, c)$ has a divisibility run of length at least $L$ when $P_i \\ne 0$ and $P_i \\mid Q_i$ for every natural number $i < L$. Then $(x, y; a, b, c)$ always has a divisibility run of length at\u2026","title":"Divisibility runs of length zero"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/int-gcd-gibonacci-number-sub-two","kind":"node","lean":"MetaMathlibExt.int_gcd_gibonacciNumber_sub_two","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#int-gcd-gibonacci-number-sub-two","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"Let $x, y$ be integers and let $(P_n)_{n \\ge 0}$ be the Gibonacci sequence with $P_0 = x$ and $P_1 = y$. For every $n \\ge 2$, the nonnegative greatest common divisor of $P_n$ and $P_{n-2}$ equals that of $x$ and $y$: $\\gcd(P_n, P_{n-2}) = \\gcd(x, y)$.","title":"Gibonacci terms two apart have the gcd of the initial values"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/gibonacci-number-backward","kind":"node","lean":"MetaMathlibExt.gibonacciNumber_backward","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#gibonacci-number-backward","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"Let $x, y$ be integers and let $(P_n)_{n \\ge 0}$ be the Gibonacci sequence with $P_0 = x$ and $P_1 = y$. For every $n \\ge 3$, $P_{n-1} = P_n - P_{n-2}$ and $P_{n-3} = P_n - 2 P_{n-2}$.","title":"Backward recurrence for Gibonacci numbers"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/gibonacci-number-eq-fib","kind":"node","lean":"MetaMathlibExt.gibonacciNumber_eq_fib","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#gibonacci-number-eq-fib","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"Let $x, y$ be integers, let $(P_n)_{n \\ge 0}$ be the Gibonacci sequence with $P_0 = x$ and $P_1 = y$, and let $(F_n)_{n \\ge 0}$ be the Fibonacci sequence with $F_0 = 0$ and $F_1 = 1$. For every $n \\ge 1$, $P_n = F_{n-1}\\,x + F_n\\,y$.","title":"Gibonacci numbers in terms of Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/exists-generalized-tribonacci-basis","kind":"node","lean":"MetaMathlibExt.exists_generalizedTribonacci_basis","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#exists-generalized-tribonacci-basis","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"There are integer sequences $(u_n)_{n \\ge 0}$, $(v_n)_{n \\ge 0}$ and $(w_n)_{n \\ge 0}$, each satisfying the Tribonacci recurrence $s_{n+3} = s_{n+2} + s_{n+1} + s_n$ for all $n \\ge 0$, with initial values $(u_0, u_1, u_2) = (1, 0, 0)$, $(v_0, v_1, v_2) = (0, 1, 0)$ and $(w_0, w_1, w_2) = (0, 0, 1)$, such that for all integers $a, b, c$ and every $n \\ge 0$ the $n$-th term of the generalized Tribonacci sequence with\u2026","title":"Basis sequences for generalized Tribonacci sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/is-div-run-up-to-small","kind":"node","lean":"MetaMathlibExt.isDivRunUpTo_small","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#is-div-run-up-to-small","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"For integers $x, y, a, b, c$, let $(P_n)$ be the Gibonacci sequence with $P_0 = x$, $P_1 = y$ and $(Q_n)$ the generalized Tribonacci sequence with $Q_0 = a$, $Q_1 = b$, $Q_2 = c$, and say $(x, y; a, b, c)$ has a divisibility run of length at least $L$ when $P_i \\ne 0$ and $P_i \\mid Q_i$ for every $i < L$. Then there is a run of length at least $1$ if and only if $x \\ne 0$ and $x \\mid a$; a run of length at least\u2026","title":"Divisibility runs of length one, two and three"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/generalized-tribonacci-eq-zero-of-dvd","kind":"node","lean":"MetaMathlibExt.generalizedTribonacci_eq_zero_of_dvd","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#generalized-tribonacci-eq-zero-of-dvd","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"Let $x, y, a, b, c$ be integers, let $(P_n)_{n \\ge 0}$ be the Gibonacci sequence with $P_0 = x$ and $P_1 = y$, and let $(Q_n)_{n \\ge 0}$ be the generalized Tribonacci sequence with $Q_0 = a$, $Q_1 = b$ and $Q_2 = c$. If $P_n \\ne 0$ and $P_n \\mid Q_n$ for every $n \\ge 0$, then $Q_n = 0$ for every $n \\ge 0$.","title":"Divisibility by a Gibonacci sequence forces a Tribonacci sequence to vanish"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/gibonacci-generalized-tribonacci-growth","kind":"node","lean":"MetaMathlibExt.gibonacci_generalizedTribonacci_growth_bounds","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#gibonacci-generalized-tribonacci-growth","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. For integers $x, y$ let $(P_n)_{n \\ge 0}$ be the Gibonacci sequence with $P_0 = x$ and $P_1 = y$, and for integers $a, b, c$ let $(Q_n)_{n \\ge 0}$ be the generalized Tribonacci sequence with $Q_0 = a$, $Q_1 = b$ and $Q_2 = c$. Then (i) for all integers $x, y$ and every $n \\ge 0$, $|P_n| \\le (|x| + |y|)\\,\\varphi^n$; (ii) there are real constants $c_0 > 0$ and $c_1 > 0$ such that\u2026","title":"Growth bounds for Gibonacci and generalized Tribonacci sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-count","kind":"node","lean":"MetaMathlibExt.divRunCount","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-count","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"fully_proved","summary":"For natural numbers $B$ and $r$, $A_B(r)$ is the number of integer tuples $(x, y, a, b, c)$ with all five entries in $[-B, B]$, $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$ that have a divisibility run of length at least $r$: for every $i < r$, the Gibonacci number $P_i$ with initial values $P_0 = x$, $P_1 = y$ is nonzero and divides the generalized Tribonacci number $Q_i$ with initial values $Q_0 = a$,\u2026","title":"Counting configurations with long divisibility runs"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-count-one-one","kind":"node","lean":"MetaMathlibExt.divRunCount_one_one","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-count-one-one","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"fully_proved","summary":"Exactly $156$ integer tuples $(x, y, a, b, c)$ with all entries in $\\{-1, 0, 1\\}$, $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$ have a divisibility run of length at least $1$; that is, $A_1(1) = 156$.","title":"Run count at bound 1 and length 1"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-length-le-log","kind":"node","lean":"MetaMathlibExt.divRun_length_le_log","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-length-le-log","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"There is a real constant $C > 0$ such that for every natural number $B \\ge 2$, all integers $x, y, a, b, c \\in [-B, B]$ with $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$, and every natural number $L$ such that $(x, y, a, b, c)$ has a divisibility run of length at least $L$, we have $L \\le C \\log B$. Here a divisibility run of length at least $L$ means that for every $i < L$ the Gibonacci number $P_i$ with\u2026","title":"Divisibility runs have logarithmic length"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-count-one","kind":"node","lean":"MetaMathlibExt.divRunCount_one","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-count-one","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"For every natural number $B$, let $D(B) = \\sum_{m=1}^{B} \\lfloor B/m \\rfloor$ and let $A_B(1)$ be the number of integer tuples $(x, y, a, b, c) \\in [-B, B]^5$ with $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$ that have a divisibility run of length at least $1$. Then","title":"Exact formula for the length-one run count"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-count-one-asymptotic","kind":"node","lean":"MetaMathlibExt.divRunCount_one_asymptotic","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-count-one-asymptotic","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"There is a real constant $K$ such that for every natural number $B \\ge 2$ the number $A_B(1)$ of integer tuples $(x, y, a, b, c) \\in [-B, B]^5$ with $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$ that have a divisibility run of length at least $1$ satisfies $|A_B(1) - 32 B^4 \\log B| \\le K B^4$; that is, $A_B(1) = 32 B^4 \\log B + O(B^4)$.","title":"Asymptotics of the length-one run count"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-count-two","kind":"node","lean":"MetaMathlibExt.divRunCount_two","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-count-two","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"For every natural number $B$, let $S(B) = \\sum_{m=1}^{B} \\left(2\\lfloor B/m \\rfloor + 1\\right)$ and let $A_B(2)$ be the number of integer tuples $(x, y, a, b, c) \\in [-B, B]^5$ with $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$ that have a divisibility run of length at least $2$. Then $S(B) = 2 \\sum_{m=1}^{B} \\lfloor B/m \\rfloor + B$ and $A_B(2) = 4(2B+1) S(B)^2 - 4B^2$.","title":"Exact formula for the length-two run count"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-count-two-asymptotic","kind":"node","lean":"MetaMathlibExt.divRunCount_two_asymptotic","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-count-two-asymptotic","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"There is a real constant $K$ such that for every natural number $B \\ge 2$ the number $A_B(2)$ of integer tuples $(x, y, a, b, c) \\in [-B, B]^5$ with $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$ that have a divisibility run of length at least $2$ satisfies $|A_B(2) - 32 B^3 (\\log B)^2| \\le K B^3 \\log B$; that is, $A_B(2) = 32 B^3 (\\log B)^2 + O(B^3 \\log B)$.","title":"Asymptotics of the length-two run count"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-count-three","kind":"node","lean":"MetaMathlibExt.divRunCount_three","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-count-three","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"For a natural number $B$, let $A_B(3)$ be the number of integer tuples $(x, y, a, b, c) \\in [-B, B]^5$ with $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$ that have a divisibility run of length at least $3$, and for integers $m \\ge 1$ let $d_B(m) = 2\\lfloor B/m \\rfloor + 1$. Then","title":"Exact formula for the length-three run count"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/div-run-count-four","kind":"node","lean":"MetaMathlibExt.divRunCount_four","parent":6897,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences/index.html#div-run-count-four","scope":"NumberTheory/linear-recurrences/recurrence-theory/divisibility-runs-between-recurrence-sequences","status":"can_prove","summary":"For a natural number $B$, let $A_B(4)$ be the number of integer tuples $(x, y, a, b, c) \\in [-B, B]^5$ with $(x, y) \\ne (0, 0)$ and $(a, b, c) \\ne (0, 0, 0)$ that have a divisibility run of length at least $4$. For integers $x, y$ let $T_B(x, y)$ be the number of integer triples $(a, b, c) \\in [-B, B]^3$ with $x \\mid a$, $y \\mid b$, $x + y \\mid c$ and $x + 2y \\mid a + b + c$, and let $T'_B(x, y)$ be the number of\u2026","title":"Divisor-sum formula for the length-four run count"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence.json","id":"NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence","status":"can_prove","title":"Dominant roots and reciprocal sums of recurrences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence/norm-le-one-of-aeval-eq-zero-of-monotone","kind":"node","lean":"MetaMathlibExt.norm_le_one_of_aeval_eq_zero_of_monotone_coeff","parent":6916,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence/index.html#norm-le-one-of-aeval-eq-zero-of-monotone","scope":"NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence","status":"can_prove","summary":"Let $a_0 + a_1 x + \\cdots + a_n x^n$ be a nonzero real polynomial of degree $n$ with $0 \\le a_0 \\le a_1 \\le \\cdots \\le a_n$. Then every complex root $z$ of this polynomial satisfies $|z| \\le 1$.","title":"Enestr\u00f6m\u2013Kakeya theorem"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence/exists-dominant-root-of-pos-coeff-char-p","kind":"node","lean":"MetaMathlibExt.exists_dominant_root_of_pos_coeff_charPoly","parent":6916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dominant root of "],["math","\\(x^m - a_1 x^{m-1} - \\cdots - a_m\\)"],["text"," with positive coefficients"]],"source":"Dominant root of $x^m - a_1 x^{m-1} - \\cdots - a_m$ with positive coefficients"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence/index.html#exists-dominant-root-of-pos-coeff-char-p","scope":"NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence","status":"can_prove","summary":"Let $m \\ge 2$ and let $a_1, \\dots, a_m$ be positive integers. Then the polynomial $\\varphi(x) = x^m - a_1 x^{m-1} - \\cdots - a_m$ has exactly one positive real root $\\lambda$, this root satisfies $\\lambda > 1$, and the complex roots of $\\varphi$, counted with multiplicity, are $\\lambda$ (a simple root) together with $m - 1$ roots of modulus less than $\\lambda$.","title":"Dominant root of $x^m - a_1 x^{m-1} - \\cdots - a_m$ with positive coefficients"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence/tprod-one-sub-inv-add-eq-one-sub-tsum-ad","kind":"node","lean":"MetaMathlibExt.tprod_one_sub_inv_add_eq_one_sub_tsum_add_isBigO","parent":6916,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence/index.html#tprod-one-sub-inv-add-eq-one-sub-tsum-ad","scope":"NumberTheory/linear-recurrences/recurrence-theory/dominant-roots-and-reciprocal-sums-of-recurrence","status":"can_prove","summary":"Let $\\mu$ and $\\lambda$ be real numbers with $1 < \\mu < \\lambda$, let $c \\ne 0$ be a real number, and let $(e_k)_{k \\ge 0}$ be a real sequence with $e_k = O\\bigl((\\mu/\\lambda^3)^{k/2}\\bigr)$ as $k \\to \\infty$. Then, as $n \\to \\infty$, $\\prod_{k \\ge n} \\bigl(1 - 1/(c\\lambda^k) + e_k\\bigr) = 1 - \\sum_{k \\ge n} 1/(c\\lambda^k) + O\\bigl((\\mu/\\lambda^3)^{n/2}\\bigr)$.","title":"Asymptotic expansion of a perturbed tail product"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences.json","id":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences","status":"can_prove","title":"General conditional sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/decimation-coeff","kind":"node","lean":"MetaMathlibExt.decimationCoeff","parent":6920,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/index.html#decimation-coeff","scope":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences","status":"fully_proved","summary":"Let $(A_j)_{j \\ge 0}$ and $(s_n)_{n \\ge 0}$ be sequences in a commutative ring and let $m$, $r$ be natural numbers. For natural numbers $l$ and $i$, the decimation coefficient is","title":"Decimation coefficients"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/decimation-coeff-zero","kind":"node","lean":"MetaMathlibExt.decimationCoeff_zero","parent":6920,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/index.html#decimation-coeff-zero","scope":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences","status":"fully_proved","summary":"For sequences $(A_j)_{j \\ge 0}$ and $(s_n)_{n \\ge 0}$ in a commutative ring and natural numbers $m$, $r$, $i$, the decimation coefficient $v_0^{(i)}$ equals $s_i$.","title":"Zeroth decimation coefficient"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/decimation-coeff-one","kind":"node","lean":"MetaMathlibExt.decimationCoeff_one","parent":6920,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/index.html#decimation-coeff-one","scope":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences","status":"fully_proved","summary":"For sequences $(A_j)_{j \\ge 0}$ and $(s_n)_{n \\ge 0}$ in a commutative ring and natural numbers $m$, $r$, $i$, the decimation coefficient $v_1^{(i)}$ equals $s_{r+i} - A_{m-1} s_i$, where $A_{m-1}$ is read as $A_0$ when $m = 0$.","title":"First decimation coefficient"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/decimation-generating-function","kind":"node","lean":"MetaMathlibExt.decimation_generating_function","parent":6920,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/index.html#decimation-generating-function","scope":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences","status":"can_prove","summary":"Let $R$ be a commutative ring, let $m$ and $r$ be natural numbers, and let $(A_j)_{j \\ge 0}$ and $(s_n)_{n \\ge 0}$ be sequences in $R$ such that $s_{n+mr} = \\sum_{j\\lt m} A_j s_{n+jr}$ for all $n \\ge 0$. Then, in the ring $R[[x]]$ of formal power series,","title":"Generating function of a decimated linear recurrence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/decimation-binet","kind":"node","lean":"MetaMathlibExt.decimation_binet","parent":6920,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences/index.html#decimation-binet","scope":"NumberTheory/linear-recurrences/recurrence-theory/general-conditional-sequences","status":"can_prove","summary":"Let $m$ and $r$ be natural numbers and let $(A_j)_{j \\ge 0}$ and $(s_n)_{n \\ge 0}$ be sequences of complex numbers such that $s_{n+mr} = \\sum_{j\\lt m} A_j s_{n+jr}$ for all $n \\ge 0$. Let $f(x) = x^m - \\sum_{j\\lt m} A_j x^j$ and let $\\alpha_1, \\ldots, \\alpha_m$ be pairwise distinct roots of $f$ in $\\mathbb{C}$. Then for every $n \\ge 0$","title":"Binet formula for decimated linear recurrences"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences.json","id":"NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences","status":"fully_proved","title":"Growth of linear recurrence sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences/exists-norm-lt-pow-of-is-solution","kind":"node","lean":"MetaMathlibExt.exists_norm_lt_pow_of_isSolution","parent":6926,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences/index.html#exists-norm-lt-pow-of-is-solution","scope":"NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences","status":"fully_proved","summary":"Let $k$ be a natural number, let $c_0, \\dots, c_{k-1}$ be complex numbers, and let $t : \\mathbb{N} \\to \\mathbb{C}$ satisfy $t(n+k) = \\sum_{i=0}^{k-1} c_i \\, t(n+i)$ for all $n \\in \\mathbb{N}$. Then there is an integer $g \\ge 1$ such that $|t(n)| < g^{n+1}$ for all $n \\in \\mathbb{N}$.","title":"Exponential growth bound for solutions of linear recurrences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences/c-recursive-exponential-growth-bound","kind":"node","lean":"MetaMathlibExt.c_recursive_exponential_growth_bound","parent":6926,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences/index.html#c-recursive-exponential-growth-bound","scope":"NumberTheory/linear-recurrences/recurrence-theory/growth-of-linear-recurrence-sequences","status":"fully_proved","summary":"Let $t : \\mathbb{N} \\to \\mathbb{C}$ be C-recursive: there are an order $k \\ge 1$ and complex numbers $c_0, \\dots, c_{k-1}$ with $t(n+k) = \\sum_{i=0}^{k-1} c_i \\, t(n+i)$ for all $n \\in \\mathbb{N}$. Then there is an integer $g \\ge 1$ such that $|t(n)| < g^{n+1}$ for all $n \\in \\mathbb{N}$.","title":"Exponential growth bound for C-recursive sequences"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials.json","id":"NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials","status":"fully_proved","title":"Isobaric polynomials"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials/isobaric-weight","kind":"node","lean":"MetaMathlibExt.isobaricWeight","parent":6929,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials/index.html#isobaric-weight","scope":"NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials","status":"fully_proved","summary":"For a natural number $k$ and an exponent vector $\\alpha = (\\alpha_1, \\dots, \\alpha_k) \\in \\mathbb{N}^k$, the isobaric weight of $\\alpha$ is $\\sum_{j=1}^k j \\alpha_j$.","title":"Isobaric weight of an exponent vector"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials/is-isobaric","kind":"node","lean":"MetaMathlibExt.IsIsobaric","parent":6929,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials/index.html#is-isobaric","scope":"NumberTheory/linear-recurrences/recurrence-theory/isobaric-polynomials","status":"fully_proved","summary":"Let $k$ and $n$ be natural numbers. A polynomial $P \\in \\mathbb{Z}[t_1, \\dots, t_k]$ is isobaric of degree $n$ if every monomial $t_1^{\\alpha_1} \\cdots t_k^{\\alpha_k}$ occurring in $P$ with nonzero coefficient satisfies $\\sum_{j=1}^k j \\alpha_j = n$; that is, $P$ is weighted homogeneous of degree $n$ when the variable $t_j$ has weight $j$. In particular, the zero polynomial is isobaric of every degree.","title":"Isobaric polynomial"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences.json","id":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences","status":"fully_proved","title":"MC-finite sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/is-mcfinite","kind":"node","lean":"MetaMathlibExt.IsMCFinite","parent":6932,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/index.html#is-mcfinite","scope":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences","status":"fully_proved","summary":"An integer sequence $s$ is MC-finite if for every natural number $m > 0$ it eventually satisfies, modulo $m$, a linear recurrence with constant integer coefficients: there are natural numbers $p > 0$ and $q > 0$ and integers $c_0, \\dots, c_{p-1}$ such that $s(n + p) \\equiv \\sum_{i=0}^{p-1} c_i\\, s(n + i) \\pmod{m}$ for all $n \\ge q$.","title":"MC-finite sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/is-mcfinite-of-mv-polynomial-recursion","kind":"node","lean":"MetaMathlibExt.isMCFinite_of_mvPolynomial_recursion","parent":6932,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/index.html#is-mcfinite-of-mv-polynomial-recursion","scope":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences","status":"fully_proved","summary":"Let $u_0, \\dots, u_k$ be integer sequences and let $P_0, \\dots, P_k$ be polynomials with integer coefficients in $k + 2$ variables such that $u_i(n+1) = P_i(u_0(n), \\dots, u_k(n), n)$ for every $i \\le k$ and every $n \\ge 0$. Then the sequence $u_0$ is MC-finite.","title":"Polynomial recursions give MC-finite sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/is-mcfinite-2","kind":"node","lean":"MetaMathlibExt.IsPRecursiveSequence.isMCFinite","parent":6932,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/index.html#is-mcfinite-2","scope":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences","status":"fully_proved","summary":"Let $s$ be an integer sequence that is P-recursive of order $k$ with coefficient polynomials $Q_0, \\dots, Q_k$ with integer coefficients, that is, $Q_0(n)\\, s_n = \\sum_{i=1}^{k} Q_i(n)\\, s_{n-i}$ for every $n \\ge k$. If $Q_0 = 1$, then $s$ is MC-finite.","title":"P-recursive sequences with leading coefficient 1 are MC-finite"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/polynomially-recursive-mc-finite","kind":"node","lean":"MetaMathlibExt.polynomially_recursive_MC_finite","parent":6932,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences/index.html#polynomially-recursive-mc-finite","scope":"NumberTheory/linear-recurrences/recurrence-theory/mc-finite-sequences","status":"fully_proved","summary":"Call an integer sequence $s$ MC-finite if for every natural number $m > 0$ there are natural numbers $p > 0$ and $q > 0$ and integers $c_0, \\dots, c_{p-1}$ such that $m$ divides $s(n + p) - \\sum_{i=0}^{p-1} c_i\\, s(n + i)$ for all $n \\ge q$. Every integer sequence $s$ has the following two properties.","title":"Polynomially recursive integer sequences are MC-finite"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences.json","id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","title":"Mod-mod representations of C-recursive sequences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/generating-function-term-extraction-of-o","kind":"node","lean":"MetaMathlibExt.generating_function_term_extraction_of_one_third_growth","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#generating-function-term-extraction-of-o","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $t_0, t_1, t_2, \\dots$ be natural numbers and let $c \\ge 8$ be a natural number. Suppose that the real series $\\sum_{k \\ge 0} t_k c^{-k}$ is summable and that $t_n < c^{n/3}$, a real power, for every $n \\ge 1$. Then for every $n \\ge 1$,","title":"Term extraction from the generating function under cube-root growth"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/generating-function-term-representation","kind":"node","lean":"MetaMathlibExt.generating_function_term_representation_mono_base","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#generating-function-term-representation","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $t_0, t_1, t_2, \\dots$ be natural numbers and let $c$, $m$ and $r$ be natural numbers with $c \\ge 2$, $m \\ge 2$ and $c \\le r$. Suppose that for every $n \\ge m$ the real series $\\sum_{k \\ge 0} t_k \\, (c^{-n})^k$ is summable and","title":"Term extraction from the generating function persists in larger bases"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/neg-ediv-eq-ediv-add-one-of-not-dvd","kind":"node","lean":"MetaMathlibExt.neg_ediv_eq_ediv_add_one_of_not_dvd","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#neg-ediv-eq-ediv-add-one-of-not-dvd","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $A$ and $B$ be integers with $A \\ge 1$ and $B \\ge 2$ such that $B$ does not divide $A$. Then $-\\lfloor -A/B \\rfloor = \\lfloor A/B \\rfloor + 1$.","title":"Floor quotient of a negated non-multiple"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/mul-mod-eq-mul-mod-of-mul-lt","kind":"node","lean":"MetaMathlibExt.mul_mod_eq_mul_mod_of_mul_lt","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#mul-mod-eq-mul-mod-of-mul-lt","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $a$, $y$ and $C$ be natural numbers with $ay < C$. Then $(ay) \\bmod C = a \\cdot (y \\bmod C)$.","title":"Pulling a factor out of a remainder"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/add-one-mod-eq-mod-add-one-of-ne","kind":"node","lean":"MetaMathlibExt.add_one_mod_eq_mod_add_one_of_ne","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#add-one-mod-eq-mod-add-one-of-ne","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $x$ and $C$ be natural numbers with $C \\ge 2$. If $x \\bmod C \\ne (C - 1) \\bmod C$, that is, if $x$ is not congruent to $C - 1$ modulo $C$, then $(x + 1) \\bmod C = (x \\bmod C) + 1$.","title":"Adding one to a remainder without wrap-around"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/emod-emod-eq-abs-mul-ediv-emod-of-neg","kind":"node","lean":"MetaMathlibExt.emod_emod_eq_abs_mul_ediv_emod_of_neg","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#emod-emod-eq-abs-mul-ediv-emod-of-neg","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $a$, $A$, $B$ and $C$ be integers with $A > 0$, $B > 0$ and $C \\ge 2$ such that $C$ divides $A$, $B$ does not divide $A$, $a < 0$ and $B \\equiv a \\pmod C$. Write $\\lfloor A/B \\rfloor$ for the floor of $A/B$ and $X \\bmod M$ for the remainder of $X$ modulo $M$ in $\\{0, 1, \\dots, M - 1\\}$. If $|a| \\cdot (\\lfloor A/B \\rfloor \\bmod C) < C$, then $(A \\bmod B) \\bmod C = |a| \\cdot (\\lfloor A/B \\rfloor \\bmod C)$.","title":"Remainder of a remainder, negative residue case"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/neg-emod-emod-eq-mul-one-add-ediv-emod-o","kind":"node","lean":"MetaMathlibExt.neg_emod_emod_eq_mul_one_add_ediv_emod_of_pos","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#neg-emod-emod-eq-mul-one-add-ediv-emod-o","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $a$, $A$, $B$ and $C$ be integers with $A > 0$, $B > 0$ and $C \\ge 2$ such that $C$ divides $A$, $B$ does not divide $A$, $a > 0$ and $B \\equiv a \\pmod C$. Write $\\lfloor A/B \\rfloor$ for the floor of $A/B$ and $X \\bmod M$ for the remainder of $X$ modulo $M$ in $\\{0, 1, \\dots, M - 1\\}$. If $a + a \\cdot (\\lfloor A/B \\rfloor \\bmod C) < C$, then\u2026","title":"Remainder of a remainder, positive residue case"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/exists-mod-mod-representation-of-neg-fre","kind":"node","lean":"MetaMathlibExt.exists_mod_mod_representation_of_neg_free_term","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#exists-mod-mod-representation-of-neg-fre","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $t_0, t_1, t_2, \\dots$ be natural numbers, let $c$ be a natural number, and let $A$ and $B$ be polynomials with integer coefficients such that $A \\ne 0$, $B(0) = 1$ and $k = \\deg A < d = \\deg B$. Suppose that $\\bigl(\\sum_{n \\ge 0} t_n z^n\\bigr) B(z) = A(z)$ as formal power series. Let $\\tilde A(x) = x^k A(1/x)$ and $\\tilde B(x) = x^d B(1/x)$ be the reversed polynomials, let $\\alpha$ be the constant term of\u2026","title":"Mod-mod representation of C-recursive sequences, negative free term"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/exists-mod-mod-representation-of-pos-fre","kind":"node","lean":"MetaMathlibExt.exists_mod_mod_representation_of_pos_free_term","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#exists-mod-mod-representation-of-pos-fre","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $t_0, t_1, t_2, \\dots$ be natural numbers, let $c$ be a natural number, and let $A$ and $B$ be polynomials with integer coefficients such that $A \\ne 0$, $B(0) = 1$ and $k = \\deg A < d = \\deg B$. Suppose that $\\bigl(\\sum_{n \\ge 0} t_n z^n\\bigr) B(z) = A(z)$ as formal power series. Let $\\tilde A(x) = x^k A(1/x)$ and $\\tilde B(x) = x^d B(1/x)$ be the reversed polynomials, let $\\alpha$ be the constant term of\u2026","title":"Mod-mod representation of C-recursive sequences, positive free term"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/exists-mod-mod-representation-sign","kind":"node","lean":"MetaMathlibExt.exists_mod_mod_representation_sign","parent":6937,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences/index.html#exists-mod-mod-representation-sign","scope":"NumberTheory/linear-recurrences/recurrence-theory/mod-mod-representations-of-c-recursive-sequences","status":"can_prove","summary":"Let $t_0, t_1, t_2, \\dots$ be natural numbers, let $c$ be a natural number, and let $A$ and $B$ be polynomials with integer coefficients such that $A \\ne 0$, $B(0) = 1$ and $k = \\deg A < d = \\deg B$. Suppose that $\\bigl(\\sum_{n \\ge 0} t_n z^n\\bigr) B(z) = A(z)$ as formal power series. Let $\\tilde A(x) = x^k A(1/x)$ and $\\tilde B(x) = x^d B(1/x)$ be the reversed polynomials, let $\\alpha$ be the constant term of\u2026","title":"Single-formula mod-mod representation of C-recursive sequences"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re.json","id":"NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re","status":"can_prove","title":"Nonlinear floor recurrences satisfying linear recurrences"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re/a","kind":"node","lean":"MathlibExt.NumberTheory.OeisA022030Wanted.a","parent":6948,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re/index.html#a","scope":"NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re","status":"fully_proved","summary":"The sequence of natural numbers with $a(0) = 4$, $a(1) = 16$ and, for every natural number $n$, $a(n + 2) = \\lfloor (a(n + 1)^2 + a(n) - 1) / a(n) \\rfloor - 1$ if $n$ is even and $a(n + 2) = \\lfloor a(n + 1)^2 / a(n) \\rfloor + 1$ if $n$ is odd.","title":"The sequence OEIS A022030"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.OeisA022030Wanted.conjecture","parent":6948,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re/index.html#conjecture","scope":"NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re","status":"fully_proved","summary":"The statement that for every natural number $n \\ge 4$, the sequence $a$ of OEIS A022030 satisfies $a(n) = 4a(n - 1) - a(n - 3) + a(n - 4)$, with the subtraction taken in the natural numbers (truncated at $0$).","title":"Barker's linear recurrence for A022030"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.OeisA022030Wanted.conjecture_holds","parent":6948,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re/index.html#conjecture-holds","scope":"NumberTheory/linear-recurrences/recurrence-theory/nonlinear-floor-recurrences-satisfying-linear-re","status":"can_prove","summary":"For every natural number $n \\ge 4$, the sequence $a$ of OEIS A022030 satisfies $a(n) = 4a(n - 1) - a(n - 3) + a(n - 4)$, with the subtraction taken in the natural numbers (truncated at $0$).","title":"Barker's linear recurrence for A022030 holds"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples.json","id":"NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples","kind":"scope","navigation":"module","parent":6848,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples/index.html","scope":"NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples","status":"fully_proved","title":"Second-order recurrences from seed triples"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples/formal-alpha","kind":"node","lean":"MetaMathlibExt.formalAlpha","parent":6952,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples/index.html#formal-alpha","scope":"NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples","status":"fully_proved","summary":"For elements $a$, $b$, $c$ of a field $F$, the coefficient of the formal-sequence recurrence determined by the seed triplet $(a, b, c)$ is $\\alpha = \\frac{ab + bc + ca}{b^2} - 1$.","title":"Coefficient of the formal-sequence recurrence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples/formal-beta","kind":"node","lean":"MetaMathlibExt.formalBeta","parent":6952,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples/index.html#formal-beta","scope":"NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples","status":"fully_proved","summary":"For elements $a$, $b$, $c$ of a field $F$, the constant term of the formal-sequence recurrence determined by the seed triplet $(a, b, c)$ is $\\beta = \\frac{b^2 - ac}{b}$.","title":"Constant term of the formal-sequence recurrence"},{"id":"NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples/is-formal-sequence","kind":"node","lean":"MetaMathlibExt.IsFormalSequence","parent":6952,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples/index.html#is-formal-sequence","scope":"NumberTheory/linear-recurrences/recurrence-theory/second-order-recurrences-from-seed-triples","status":"fully_proved","summary":"Let $F$ be a field and $a, b, c \\in F$. A sequence $u : \\mathbb{N} \\to F$ is a formal sequence with seed triplet $(a, b, c)$ when $b \\neq 0$, $u_0 = a$, $u_1 = b$, $u_2 = c$, and for every $n \\ge 0$","title":"Formal sequence from a seed triplet"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrences-modulo-m.json","id":"NumberTheory/linear-recurrences/recurrences-modulo-m","kind":"scope","parent":5991,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/index.html","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m","status":"can_prove","summary":"Pisano periods, periods and zero indices of recurrences modulo primes and prime powers, residue distribution.","title":"Recurrences modulo m"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/subsequence-second-order-recurrence","kind":"node","lean":"MetaMathlibExt.subsequence_secondOrderRecurrence","parent":6956,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/index.html#subsequence-second-order-recurrence","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m","status":"can_prove","summary":"Let $(P, Q)$ be Lucas sequence parameters (integers with $Q \\ne 0$), let $U_n$ and $V_n$ be the terms of the Lucas sequence $U(P, Q)$ and of the companion Lucas sequence $V(P, Q)$, and let $(w_n)_{n \\ge 0}$ be an integer sequence with $w_{n+2} = P w_{n+1} - Q w_n$ for every natural number $n$. Then for all natural numbers $t > 0$ and $r$, the subsequence $w'_n = w_{tn + r}$ satisfies\u2026","title":"Subsequences of a second-order linear recurrence"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes.json","id":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes","kind":"scope","navigation":"module","parent":6956,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/index.html","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes","status":"fully_proved","title":"Bi-periodic Fibonacci sequences modulo primes"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/generalized-bi-periodic-fibonacci-period","kind":"node","lean":"MetaMathlibExt.generalizedBiPeriodicFibonacci_period_dvd_general","parent":6958,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/index.html#generalized-bi-periodic-fibonacci-period","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes","status":"fully_proved","summary":"Let $p$ be an odd prime and let $a, b, c, d \\in \\mathbb{Z}/p\\mathbb{Z}$ with $a \\ne 0$, $c \\ne 0$, $d \\ne 0$, and $a \\ne b$ or $c \\ne d$. Let $F : \\mathbb{N} \\to \\mathbb{Z}/p\\mathbb{Z}$ satisfy $F_0 = 0$, $F_1 = 1$ and, for every $n \\ge 2$, $F_n = a F_{n-1} + c F_{n-2}$ if $n$ is even and $F_n = b F_{n-1} + d F_{n-2}$ if $n$ is odd. Put $A = ab + c + d$ and $\\Delta = A^2 - 4cd$, and let\u2026","title":"Period of a generalized bi-periodic Fibonacci sequence modulo a prime, given the roots"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/generalized-bi-periodic-fibonacci-period-2","kind":"node","lean":"MetaMathlibExt.generalizedBiPeriodicFibonacci_period_dvd","parent":6958,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/index.html#generalized-bi-periodic-fibonacci-period-2","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes","status":"fully_proved","summary":"Let $p$ be an odd prime and let $a, b, c, d \\in \\mathbb{Z}/p\\mathbb{Z}$ with $a \\ne 0$, $c \\ne 0$, $d \\ne 0$, and $a \\ne b$ or $c \\ne d$. Let $F : \\mathbb{N} \\to \\mathbb{Z}/p\\mathbb{Z}$ satisfy $F_0 = 0$, $F_1 = 1$ and, for every $n \\ge 2$, $F_n = a F_{n-1} + c F_{n-2}$ if $n$ is even and $F_n = b F_{n-1} + d F_{n-2}$ if $n$ is odd. Put $A = ab + c + d$ and $\\Delta = A^2 - 4cd$, and let\u2026","title":"Period of a generalized bi-periodic Fibonacci sequence modulo a prime"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/generalized-biperiodic-fibonacci-zero-in","kind":"node","lean":"MetaMathlibExt.generalized_biperiodic_fibonacci_zero_indices_general","parent":6958,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero indices of a generalized bi-periodic Fibonacci sequence with "],["math","\\(c = d\\)"]],"source":"Zero indices of a generalized bi-periodic Fibonacci sequence with $c = d$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/index.html#generalized-biperiodic-fibonacci-zero-in","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes","status":"fully_proved","summary":"Let $a, b, c, d$ be natural numbers with $c = d$, and let $m \\ge 2$ be a natural number coprime to $a$ and to $c$. Let $(F_n)_{n \\ge 0}$ be a sequence of natural numbers with $F_0 = 0$, $F_1 = 1$, $F_{2n+2} = a F_{2n+1} + c F_{2n}$ and $F_{2n+3} = b F_{2n+2} + d F_{2n+1}$ for all $n \\ge 0$. Then there is an integer $l > 0$ such that, for every $n \\ge 0$, $m$ divides $F_n$ if and only if $l$ divides $n$, and\u2026","title":"Zero indices of a generalized bi-periodic Fibonacci sequence with $c = d$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/generalized-biperiodic-fibonacci-zero-in-2","kind":"node","lean":"MetaMathlibExt.generalized_biperiodic_fibonacci_zero_indices","parent":6958,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes/index.html#generalized-biperiodic-fibonacci-zero-in-2","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/bi-periodic-fibonacci-sequences-modulo-primes","status":"fully_proved","summary":"Let $a, b, c, d$ be positive natural numbers with $c = d$, and let $m \\ge 2$ be a natural number coprime to $a$, to $c$ and to $d$. Let $(F_n)_{n \\ge 0}$ be a sequence of natural numbers with $F_0 = 0$, $F_1 = 1$, $F_{2n+2} = a F_{2n+1} + c F_{2n}$ and $F_{2n+3} = b F_{2n+2} + d F_{2n+1}$ for all $n \\ge 0$. Then there is a natural number $l > 0$ such that, for every $n \\ge 0$, $m$ divides $F_n$ if and only if $l$\u2026","title":"Belbachir\u2013Salhi theorem on zero indices of generalized bi-periodic Fibonacci sequences"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n.json","id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","kind":"scope","navigation":"module","parent":6956,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","title":"Periods and averages of Lucas sequences modulo n"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod","kind":"node","lean":"MetaMathlibExt.lucasUPeriod","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period of a Lucas sequence modulo "],["math","\\(m\\)"]],"source":"Period of a Lucas sequence modulo $m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For a Lucas sequence $U = U(P, Q)$ and a natural number $m$, the least period $\\pi_U(m)$ of $U$ modulo $m$ is the minimal period of the pair $(0, 1)$ under the map $(x, y) \\mapsto (y, P y - Q x)$ on pairs of residues modulo $m$, that is, the least $r > 0$ with $U_r \\equiv 0 \\pmod m$ and $U_{r+1} \\equiv 1 \\pmod m$, and $0$ if there is no such $r$.","title":"Period of a Lucas sequence modulo $m$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-urestricted-period","kind":"node","lean":"MetaMathlibExt.lucasURestrictedPeriod","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Restricted period of a Lucas sequence modulo "],["math","\\(m\\)"]],"source":"Restricted period of a Lucas sequence modulo $m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-urestricted-period","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For a Lucas sequence $U = U(P, Q)$ and a natural number $m$, the least restricted period $\\rho_U(m)$ of $U$ modulo $m$ is the least $s > 0$ such that $m \\mid U_s$ and $U_{s+1}$ is coprime to $m$, and $0$ if there is no such $s$.","title":"Restricted period of a Lucas sequence modulo $m$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-set","kind":"node","lean":"MetaMathlibExt.lucasPeriodSet","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For the Lucas sequence $U$ with integer parameters $P$ and $Q \\ne 0$ ($U_0 = 0$, $U_1 = 1$, $U_{j+2} = P U_{j+1} - Q U_j$), $S(U)$ is the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$.","title":"Moduli divisible by their Lucas period"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-average-set","kind":"node","lean":"MetaMathlibExt.lucasAverageSet","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-average-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"Let $U$ be the Lucas sequence with parameters $P$ and $Q \\ne 0$, given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$, and let $S(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$. The average set $S'(U)$ is the set of $n \\in S(U)$ with $n \\mid U_1 + \\dots + U_n$, that is, those $n \\in S(U)$ for which the average $(U_1 + \\dots + U_n)/n$\u2026","title":"Average set of a Lucas sequence"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-all-averages-integral","kind":"node","lean":"MetaMathlibExt.lucasAllAveragesIntegral","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-all-averages-integral","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For integers $P$ and $Q \\ne 0$ and a natural number $n$, the property that $n$ has integral averages means that every integer sequence $(w_k)_{k \\ge 0}$ with $w_{k+2} = P w_{k+1} - Q w_k$ for all $k$ satisfies $n \\mid w_1 + \\dots + w_n$; for $n > 0$ this says that the average $A_W(n) = (w_1 + \\dots + w_n)/n$ is an integer.","title":"Integral averages of all recurrences"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-one","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_one","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For every Lucas sequence $U = U(P, Q)$, the least period $\\pi_U(1)$ of $U$ modulo $1$ equals $1$.","title":"Period of a Lucas sequence modulo 1"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-fibonacci-lucas-params-two","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_fibonacciLucasParams_two","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-fibonacci-lucas-params-two","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"The Fibonacci sequence, that is, the Lucas sequence with parameters $(P, Q) = (1, -1)$, has period $3$ modulo $2$.","title":"Period of the Fibonacci sequence modulo 2"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-urestricted-period-one","kind":"node","lean":"MetaMathlibExt.lucasURestrictedPeriod_one","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-urestricted-period-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For every Lucas sequence $U = U(P, Q)$, the least restricted period $\\rho_U(1)$ of $U$ modulo $1$ equals $1$.","title":"Restricted period of a Lucas sequence modulo 1"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-urestricted-period-fibonacci-lucas","kind":"node","lean":"MetaMathlibExt.lucasURestrictedPeriod_fibonacciLucasParams_two","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-urestricted-period-fibonacci-lucas","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"The least restricted period modulo $2$ of the Fibonacci sequence $F = U(1, -1)$ equals $3$ (note $F_3 = 2$).","title":"Restricted period of the Fibonacci sequence modulo 2"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/one-mem-lucas-period-set","kind":"node","lean":"MetaMathlibExt.one_mem_lucasPeriodSet","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lucas period set contains "],["math","\\(1\\)"]],"source":"The Lucas period set contains $1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#one-mem-lucas-period-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For all integers $P, Q$ with $Q \\ne 0$, the number $1$ lies in the Lucas period set $S(U)$ of $U = U(P, Q)$: the set of positive integers $n$ coprime to $Q$ whose period $\\pi_U(n)$ modulo $n$ divides $n$.","title":"The Lucas period set contains $1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/one-mem-lucas-average-set","kind":"node","lean":"MetaMathlibExt.one_mem_lucasAverageSet","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lucas average set contains "],["math","\\(1\\)"]],"source":"The Lucas average set contains $1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#one-mem-lucas-average-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For all integers $P, Q$ with $Q \\ne 0$, the number $1$ lies in the Lucas average set $S'(U)$ of $U = U(P, Q)$: the set of $n$ in the Lucas period set $S(U)$ with $n \\mid U_1 + \\dots + U_n$.","title":"The Lucas average set contains $1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-index-divisible-set","kind":"node","lean":"MetaMathlibExt.lucasIndexDivisibleSet","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-index-divisible-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P$ and $Q \\ne 0$, given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$ (the paper's $U(a, b)$ with $a = P$ and $b = -Q$). The set $T(U)$ consists of the positive integers $n$ with $\\gcd(n, Q) = 1$ and $n \\mid U_n$.","title":"Index-divisible set of a Lucas sequence"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/one-mem-lucas-index-divisible-set","kind":"node","lean":"MetaMathlibExt.one_mem_lucasIndexDivisibleSet","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The index-divisible set of a Lucas sequence contains "],["math","\\(1\\)"]],"source":"The index-divisible set of a Lucas sequence contains $1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#one-mem-lucas-index-divisible-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For all integers $P, Q$ with $Q \\ne 0$, the number $1$ lies in the set $T(U)$ of positive integers $n$ coprime to $Q$ with $n \\mid U_n$, where $U = U(P, Q)$ is their Lucas sequence.","title":"The index-divisible set of a Lucas sequence contains $1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-all-averages-integral-one","kind":"node","lean":"MetaMathlibExt.lucasAllAveragesIntegral_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The average at "],["math","\\(n = 1\\)"],["text"," is integral"]],"source":"The average at $n = 1$ is integral"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-all-averages-integral-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For integers $P$ and $Q \\ne 0$, every integer sequence $(w_k)_{k \\ge 0}$ with $w_{k+2} = P w_{k+1} - Q w_k$ for all $k$ satisfies $1 \\mid w_1$; that is, the average $A_W(1) = w_1$ is an integer.","title":"The average at $n = 1$ is integral"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-u-two-mul-and-lucas-u-dvd","kind":"node","lean":"MetaMathlibExt.lucasU_two_mul_and_lucasU_dvd","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-u-two-mul-and-lucas-u-dvd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ and $V$ be the Lucas sequences with integer parameters $P$ and $Q \\neq 0$. Then $U_{2n} = U_n V_n$ and $U_{2n+1} = -Q U_n^2 + U_{n+1}^2$ for every natural number $n$, and $U_m \\mid U_n$ for all natural numbers $m$ and $n$ with $m \\mid n$.","title":"Doubling formulas and divisibility for Lucas sequences"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-is-period-of-second-order","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_isPeriod_of_secondOrder","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periods of second-order recurrences modulo "],["math","\\(m\\)"]],"source":"Periods of second-order recurrences modulo $m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-is-period-of-second-order","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with integer parameters $P$ and $Q \\neq 0$, let $m$ be a positive integer with $\\gcd(m, Q) = 1$, and write $\\pi(m)$ and $\\rho(m)$ for the period and the restricted period of $U$ modulo $m$. Then $\\pi(m) > 0$ and $\\rho(m) > 0$, and every integer sequence $w$ with $w_{k+2} = P w_{k+1} - Q w_k$ for all $k$ satisfies $w_{n + \\pi(m)} \\equiv w_n \\pmod m$ and\u2026","title":"Periods of second-order recurrences modulo $m$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-lcm","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_lcm","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-lcm","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with integer parameters $P$ and $Q \\neq 0$, and write $\\pi(m)$ and $\\rho(m)$ for its period and restricted period modulo $m$. Let $m$ and $n$ be positive integers with $\\gcd(mn, Q) = 1$. Then $\\pi(\\operatorname{lcm}(m, n)) = \\operatorname{lcm}(\\pi(m), \\pi(n))$ and $\\rho(\\operatorname{lcm}(m, n)) = \\operatorname{lcm}(\\rho(m), \\rho(n))$, and for every $i \\ge 1$ the $i$-fold iterates\u2026","title":"Periods of Lucas sequences modulo a least common multiple"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/degenerate-second-order-closed-forms","kind":"node","lean":"MetaMathlibExt.degenerate_secondOrder_closedForms","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#degenerate-second-order-closed-forms","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"For every integer sequence $(w_j)_{j \\ge 0}$ and every $n \\in \\mathbb{N}$: if $w_{j+2} = w_j$ for all $j$, then $w_{2n} = w_0$ and $w_{2n+1} = w_1$; if $w_{j+2} = -w_j$ for all $j$, then $w_{4n} = w_0$, $w_{4n+1} = w_1$, $w_{4n+2} = -w_0$ and $w_{4n+3} = -w_1$; if $w_{j+2} = w_{j+1} - w_j$ for all $j$, then $w_{6n} = w_0$, $w_{6n+1} = w_1$, $w_{6n+2} = w_1 - w_0$, $w_{6n+3} = -w_0$, $w_{6n+4} = -w_1$ and\u2026","title":"Closed forms of eight degenerate second-order recurrences"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/degenerate-second-order-sums","kind":"node","lean":"MetaMathlibExt.degenerate_secondOrder_sums","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#degenerate-second-order-sums","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"For an integer sequence $(w_j)_{j \\ge 0}$ write $B_w(n) = w_1 + \\dots + w_n$. Let $u$ be the sequence with $u_0 = 0$, $u_1 = 1$ and $u_{j+2} = u_j$, and $v$ the sequence with $v_0 = 0$, $v_1 = 1$ and $v_{j+2} = -2 v_{j+1} - v_j$. For every $n \\ge 1$: $B_u(n) = \\lfloor (n + 1) / 2 \\rfloor$ and $B_v(n) = (-1)^{n+1} \\lfloor (n + 1) / 2 \\rfloor$; $n$ divides $B_w(n)$ for every integer sequence $w$ with $w_{j+2} = w_j$\u2026","title":"Sums of eight degenerate second-order recurrences"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/second-order-sum-formula","kind":"node","lean":"MetaMathlibExt.secondOrder_sum_formula","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#second-order-sum-formula","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P, Q$ be integers with $Q \\ne 0$ and $P^2 - 4Q \\ne 0$, let $U = U(P, Q)$ be their Lucas sequence, write $a = P$ and $b = -Q$, and let $w$ be an integer sequence with $w_{k+2} = a w_{k+1} + b w_k$ for all $k \\ge 0$. For $n \\ge 0$ let $B(n) = w_1 + \\dots + w_n$, and for $n > 0$ let $\\pi_U(n)$ be the period of $U$ modulo $n$: the least $r > 0$ with $U_r \\equiv 0$ and $U_{r+1} \\equiv 1 \\pmod n$, or $0$ if there is\u2026","title":"Sum formula for second-order recurrences"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-u-sum-formula","kind":"node","lean":"MetaMathlibExt.lucasU_sum_formula","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of the first "],["math","\\(n\\)"],["text"," terms of a Lucas sequence"]],"source":"Sum of the first $n$ terms of a Lucas sequence"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-u-sum-formula","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence $U_0 = 0$, $U_1 = 1$, $U_{j+2} = P U_{j+1} - Q U_j$ with integer parameters $P$ and $Q \\ne 0$, and let $B(n) = U_1 + \\cdots + U_n$. For every $n \\ge 0$: if $P - Q - 1 \\ne 0$, then $(P - Q - 1) B(n) = U_{n+1} - 1 - Q U_n$; if $P - Q - 1 = 0$, then $(Q - 1) B(n) = U_{n+1} - 1 - n$.","title":"Sum of the first $n$ terms of a Lucas sequence"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/second-order-sum-dvd-of-period-dvd","kind":"node","lean":"MetaMathlibExt.secondOrder_sum_dvd_of_period_dvd","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#second-order-sum-dvd-of-period-dvd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P, Q$ be integers with $Q \\ne 0$, let $U = U(P, Q)$ be their Lucas sequence, write $a = P$ and $b = -Q$, and suppose $a + b - 1 \\ne 0$. Let $n > 0$ be coprime to $b(a + b - 1)$, and let $\\pi_U(n)$ be the period of $U$ modulo $n$: the least $r > 0$ with $U_r \\equiv 0$ and $U_{r+1} \\equiv 1 \\pmod n$, or $0$ if there is none. Then $\\pi_U(n) > 0$, and for every $r \\ge 0$ divisible by $\\pi_U(n)$ and every integer\u2026","title":"Sums of a second-order recurrence over whole periods"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/prime-mul-mem-lucas-period-set","kind":"node","lean":"MetaMathlibExt.prime_mul_mem_lucasPeriodSet","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplying a member of "],["math","\\(S(U)\\)"],["text"," by one of its prime factors"]],"source":"Multiplying a member of $S(U)$ by one of its prime factors"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#prime-mul-mem-lucas-period-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P, Q$ be integers with $Q \\ne 0$, let $U = U(P, Q)$ be their Lucas sequence, write $a = P$ and $b = -Q$, and let $S(U)$ be the set of positive integers $n$ coprime to $Q$ whose period $\\pi_U(n)$ modulo $n$ divides $n$. If $n \\in S(U)$ and $\\ell$ is a prime dividing $n$, then $\\ell n \\in S(U)$. If moreover $a + b - 1 \\ne 0$ and $\\gcd(a + b - 1, n) = 1$, or $a + b - 1 = 0$ and $\\gcd(-b - 1, n) = 1$, then\u2026","title":"Multiplying a member of $S(U)$ by one of its prime factors"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/mul-mem-lucas-period-set-of-prime-factor","kind":"node","lean":"MetaMathlibExt.mul_mem_lucasPeriodSet_of_primeFactors","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplying a member of "],["math","\\(S(U)\\)"],["text"," by a number with no new prime factors"]],"source":"Multiplying a member of $S(U)$ by a number with no new prime factors"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#mul-mem-lucas-period-set-of-prime-factor","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P, Q$ be integers with $Q \\ne 0$, let $U = U(P, Q)$ be their Lucas sequence, write $a = P$ and $b = -Q$, and let $S(U)$ be the set of positive integers $n$ coprime to $Q$ whose period $\\pi_U(n)$ modulo $n$ divides $n$. Let $m > 0$ and $n \\in S(U)$ be such that every prime divisor of $m$ divides $n$. Then $m n \\in S(U)$. If moreover $a + b - 1 \\ne 0$ and $\\gcd(a + b - 1, n) = 1$, or $a + b - 1 = 0$ and\u2026","title":"Multiplying a member of $S(U)$ by a number with no new prime factors"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lcm-mem-lucas-period-set","kind":"node","lean":"MetaMathlibExt.lcm_mem_lucasPeriodSet","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lcm-mem-lucas-period-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with parameters $P$ and $Q \\ne 0$, given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $S(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$, and let $S'(U)$ be the set of $n \\in S(U)$ with $n \\mid U_1 + \\dots + U_n$. Say that a natural number $x$ has integral averages if $x \\mid w_1 + \\dots + w_x$ for\u2026","title":"Closure of the period set under lcm and products"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lcm-mem-lucas-average-set","kind":"node","lean":"MetaMathlibExt.lcm_mem_lucasAverageSet","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lcm-mem-lucas-average-set","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers and $U$ the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $S'(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$ and $n \\mid U_1 + \\dots + U_n$, and say that a natural number $x$ has integral averages if $x \\mid w_1 + \\dots + w_x$ for every integer sequence $(w_k)_{k \\ge 0}$ with\u2026","title":"The average set is closed under lcm"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-u-sum-dvd-iff-of-p-eq-one","kind":"node","lean":"MetaMathlibExt.lucasU_sum_dvd_iff_of_p_eq_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility of Lucas sums when "],["math","\\(P = 1\\)"]],"source":"Divisibility of Lucas sums when $P = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-u-sum-dvd-iff-of-p-eq-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence $U_0 = 0$, $U_1 = 1$, $U_{j+2} = P U_{j+1} - Q U_j$ with $P = 1$ and $Q \\ne 0$. For every positive integer $n$ coprime to $Q$, $n \\mid U_1 + \\cdots + U_n$ if and only if $U_{n+2} \\equiv 1 \\pmod n$.","title":"Divisibility of Lucas sums when $P = 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-all-averages-integral-lcm","kind":"node","lean":"MetaMathlibExt.lucasAllAveragesIntegral_lcm","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-all-averages-integral-lcm","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with parameters $P$ and $Q \\ne 0$, given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$, and let $S'(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$ and $n \\mid U_1 + \\dots + U_n$. Call an integer sequence $(w_k)_{k \\ge 0}$ a recurrence if $w_{k+2} = P w_{k+1} - Q w_k$ for all $k$, and say that a natural\u2026","title":"Integral averages extend to least common multiples"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-all-averages-integral-of-dvd-q","kind":"node","lean":"MetaMathlibExt.lucasAllAveragesIntegral_of_dvd_q","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral averages when "],["math","\\(n\\)"],["text"," divides "],["math","\\(Q\\)"]],"source":"Integral averages when $n$ divides $Q$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-all-averages-integral-of-dvd-q","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with parameters $P$ and $Q \\ne 0$, given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$, and let $S'(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$ and $n \\mid U_1 + \\dots + U_n$. Say that a natural number $x$ has integral averages if $x \\mid w_1 + \\dots + w_x$ for every integer sequence $(w_k)_{k \\ge 0}$\u2026","title":"Integral averages when $n$ divides $Q$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/not-dvd-lucas-u-sum-of-gcd","kind":"node","lean":"MetaMathlibExt.not_dvd_lucasU_sum_of_gcd","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Non-integer period average of a Lucas sequence when "],["math","\\(\\gcd(n, a, b) > 1\\)"]],"source":"Non-integer period average of a Lucas sequence when $\\gcd(n, a, b) > 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#not-dvd-lucas-u-sum-of-gcd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Theorem 2.13: for nondegenerate `U(a, b)` with `d = gcd(a, b) > 1` and `gcd(n, d) > 1`, `A_U(n)` is not an integer.","title":"Non-integer period average of a Lucas sequence when $\\gcd(n, a, b) > 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-u-sum-odd-of-even-p","kind":"node","lean":"MetaMathlibExt.lucasU_sum_odd_of_even_p","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas sums and powers of two for even "],["math","\\(P\\)"],["text"," and odd "],["math","\\(Q\\)"]],"source":"Lucas sums and powers of two for even $P$ and odd $Q$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-u-sum-odd-of-even-p","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence $U_0 = 0$, $U_1 = 1$, $U_{j+2} = P U_{j+1} - Q U_j$ with $(P, Q)$ nondegenerate, $P$ even and $Q$ odd, and let $S$ be the set of positive integers $m$ coprime to $Q$ such that the period of $U$ modulo $m$ divides $m$. If $n \\equiv 2 \\pmod 4$, then $U_1 + \\cdots + U_n$ is odd and not divisible by $n$. For every $k \\ge 0$, $2^k$ is the least element of $S$ divisible by $2^k$.","title":"Lucas sums and powers of two for even $P$ and odd $Q$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-average-set-two-pow-of-p-two-mod-f","kind":"node","lean":"MetaMathlibExt.lucasAverageSet_two_pow_of_p_two_mod_four","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of two when "],["math","\\(P \\equiv 2\\)"],["text"," and "],["math","\\(-Q \\equiv 1 \\pmod 4\\)"]],"source":"Powers of two when $P \\equiv 2$ and $-Q \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-average-set-two-pow-of-p-two-mod-f","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers forming a nondegenerate Lucas pair with $P \\equiv 2 \\pmod 4$ and $-Q \\equiv 1 \\pmod 4$, and let $U$ be the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $S(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$, let $S'(U)$ be the set of $n \\in S(U)$ with $n \\mid U_1 + \\dots + U_n$,\u2026","title":"Powers of two when $P \\equiv 2$ and $-Q \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-u-sum-not-dvd-of-p-zero-mod-four","kind":"node","lean":"MetaMathlibExt.lucasU_sum_not_dvd_of_p_zero_mod_four","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas sums not divisible by "],["math","\\(2^k\\)"],["text"," when "],["math","\\(P \\equiv 0\\)"],["text"," and "],["math","\\(-Q \\equiv 1 \\pmod 4\\)"]],"source":"Lucas sums not divisible by $2^k$ when $P \\equiv 0$ and $-Q \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-u-sum-not-dvd-of-p-zero-mod-four","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers forming a nondegenerate Lucas pair with $P \\equiv 0 \\pmod 4$ and $-Q \\equiv 1 \\pmod 4$, and let $U$ be the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $S'(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$ and $n \\mid U_1 + \\dots + U_n$. Then for every $k \\ge 1$ and every\u2026","title":"Lucas sums not divisible by $2^k$ when $P \\equiv 0$ and $-Q \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-average-set-two-pow-of-p-zero-mod","kind":"node","lean":"MetaMathlibExt.lucasAverageSet_two_pow_of_p_zero_mod_four","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of two when "],["math","\\(P \\equiv 0\\)"],["text"," and "],["math","\\(-Q \\equiv -1 \\pmod 4\\)"]],"source":"Powers of two when $P \\equiv 0$ and $-Q \\equiv -1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-average-set-two-pow-of-p-zero-mod","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers forming a nondegenerate Lucas pair with $P \\equiv 0 \\pmod 4$ and $-Q \\equiv -1 \\pmod 4$, and let $U$ be the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $S(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$, let $S'(U)$ be the set of $n \\in S(U)$ with $n \\mid U_1 + \\dots + U_n$,\u2026","title":"Powers of two when $P \\equiv 0$ and $-Q \\equiv -1 \\pmod 4$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-u-sum-not-dvd-of-p-two-mod-four","kind":"node","lean":"MetaMathlibExt.lucasU_sum_not_dvd_of_p_two_mod_four","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas sums not divisible by "],["math","\\(2^k\\)"],["text"," when "],["math","\\(P \\equiv 2\\)"],["text"," and "],["math","\\(-Q \\equiv -1 \\pmod 4\\)"]],"source":"Lucas sums not divisible by $2^k$ when $P \\equiv 2$ and $-Q \\equiv -1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-u-sum-not-dvd-of-p-two-mod-four","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers forming a nondegenerate Lucas pair with $P \\equiv 2 \\pmod 4$ and $-Q \\equiv -1 \\pmod 4$, and let $U$ be the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $S'(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$ and $n \\mid U_1 + \\dots + U_n$. Then for every $k \\ge 1$ and every\u2026","title":"Lucas sums not divisible by $2^k$ when $P \\equiv 2$ and $-Q \\equiv -1 \\pmod 4$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-eq-lcm-order-of","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_eq_lcm_orderOf","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-eq-lcm-order-of","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with integer parameters $P$ and $Q$, where the pair $(P, Q)$ is nondegenerate, and write $\\pi(m)$ and $\\rho(m)$ for the period and the restricted period of $U$ modulo $m$. Let $m \\ge 2$ with $\\gcd(m, Q) = 1$, let $h$ be the multiplicative order of $Q$ modulo $m$, and put $H = \\operatorname{lcm}(h, \\rho(m))$. Then $\\pi(m) = H$ or $\\pi(m) = 2H$. If moreover $m = \\ell^i$ for an odd prime\u2026","title":"Wyler's theorem on the period of a Lucas sequence"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-of-q-eq-neg-one","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_of_q_eq_neg_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period and restricted period of Lucas sequences with "],["math","\\(Q = -1\\)"]],"source":"Period and restricted period of Lucas sequences with $Q = -1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-of-q-eq-neg-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with integer parameters $P$ and $Q = -1$, where the pair $(P, Q)$ is nondegenerate, and write $\\pi(m)$ and $\\rho(m)$ for the period and the restricted period of $U$ modulo $m$. Let $m \\ge 2$. Then $\\pi(m)$ equals $\\rho(m)$, $2\\rho(m)$ or $4\\rho(m)$. If $m = \\ell^i$ for an odd prime $\\ell$ and some $i \\ge 1$, then $\\pi(m) = \\rho(m)$ when $\\rho(m) \\equiv 2 \\pmod 4$, $\\pi(m) = 2\\rho(m)$\u2026","title":"Period and restricted period of Lucas sequences with $Q = -1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-of-q-eq-one","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_of_q_eq_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period and restricted period of Lucas sequences with "],["math","\\(Q = 1\\)"]],"source":"Period and restricted period of Lucas sequences with $Q = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-of-q-eq-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with integer parameters $P$ and $Q = 1$, where the pair $(P, Q)$ is nondegenerate, and write $\\pi(m)$ and $\\rho(m)$ for the period and the restricted period of $U$ modulo $m$. Let $m \\ge 2$. Then $\\pi(m) = \\rho(m)$ or $\\pi(m) = 2\\rho(m)$, and if $m = \\ell^i$ for an odd prime $\\ell$ and some $i \\ge 1$ and $\\rho(m)$ is even, then $\\pi(m) = 2\\rho(m)$.","title":"Period and restricted period of Lucas sequences with $Q = 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-prime-dvd-discr","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_prime_dvd_discr","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes dividing the discriminant when "],["math","\\(Q = \\pm 1\\)"]],"source":"Primes dividing the discriminant when $Q = \\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-prime-dvd-discr","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U = U(P, Q)$ be a Lucas sequence and let $\\ell$ be a prime dividing its discriminant $D = P^2 - 4Q$. Write $\\pi_U(m)$ and $\\rho_U(m)$ for the least period and the least restricted period of $U$ modulo $m$. If $Q = -1$, then $\\ell = 2$ or $\\ell \\equiv 1 \\pmod 4$. If $Q = 1$, then $P \\equiv 2 \\pmod \\ell$ or $P \\equiv -2 \\pmod \\ell$; moreover, if $\\ell > 2$ and $P \\equiv 2 \\pmod \\ell$, then\u2026","title":"Primes dividing the discriminant when $Q = \\pm 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-eq-restricted-of-p-eq-q-ad","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_eq_restricted_of_p_eq_q_add_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas sequences with "],["math","\\(P = Q + 1\\)"]],"source":"Lucas sequences with $P = Q + 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-eq-restricted-of-p-eq-q-ad","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with integer parameters $P$ and $Q$, where $P = Q + 1$, $Q \\neq \\pm 1$ and the pair $(P, Q)$ is nondegenerate, and write $\\pi(n)$ and $\\rho(n)$ for the period and the restricted period of $U$ modulo $n$. Then $P^2 - 4Q = (Q - 1)^2$; for every natural number $n$, $(Q - 1) U_n = Q^n - 1$ and $U_{n+1} = Q U_n + 1$; $U_i \\equiv 1 \\pmod Q$ for every $i \\ge 1$; and $\\pi(n) = \\rho(n)$ for\u2026","title":"Lucas sequences with $P = Q + 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-of-p-eq-neg-q-sub-one","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_of_p_eq_neg_q_sub_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas sequences with "],["math","\\(P = -Q - 1\\)"]],"source":"Lucas sequences with $P = -Q - 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-of-p-eq-neg-q-sub-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with integer parameters $P$ and $Q$, where $P = -Q - 1$, $Q \\neq \\pm 1$ and the pair $(P, Q)$ is nondegenerate, and write $\\pi(n)$ and $\\rho(n)$ for the period and the restricted period of $U$ modulo $n$. Then $P^2 - 4Q = (Q - 1)^2$, and for every natural number $n$, $(1 - Q) U_n = (-Q)^n - (-1)^n$ and $U_{n+1} = -Q U_n + (-1)^n$. Moreover, for every positive integer $n$ with\u2026","title":"Lucas sequences with $P = -Q - 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-u-sum-two-mul-prime-pow-dvd","kind":"node","lean":"MetaMathlibExt.lucasU_sum_two_mul_prime_pow_dvd","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-u-sum-two-mul-prime-pow-dvd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence $U_0 = 0$, $U_1 = 1$, $U_{j+2} = P U_{j+1} - Q U_j$ with $(P, Q)$ nondegenerate. Let $\\ell$ be an odd prime and $e \\ge 1$ with $\\ell^e \\mid P$, $\\ell^{e+1} \\nmid P$ and $Q \\equiv -1 \\pmod{\\ell^{e+1}}$. Then $\\ell^e \\mid P - Q - 1$ and $\\ell^{e+1} \\nmid P - Q - 1$, and $\\ell^i \\mid U_1 + U_2 + \\cdots + U_{2m\\ell^i}$ for all $m \\ge 1$ and $i \\ge 1$.","title":"Prime-power divisibility of Lucas sums"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/mem-lucas-period-set-of-two-three-smooth","kind":"node","lean":"MetaMathlibExt.mem_lucasPeriodSet_of_two_three_smooth","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Members of "],["math","\\(S(U)\\)"],["text"," of the form "],["math","\\(2^i 3^j\\)"]],"source":"Members of $S(U)$ of the form $2^i 3^j$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#mem-lucas-period-set-of-two-three-smooth","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P, Q$ be integers with $Q \\ne 0$ such that the Lucas pair $(P, Q)$ is nondegenerate, let $U = U(P, Q)$ be its Lucas sequence, write $a = P$ and $b = -Q$, and suppose $a + b - 1 \\ne 0$. Call a positive integer $n$ good if $n$ lies in the Lucas period set $S(U)$ (the positive integers $n$ coprime to $Q$ whose period $\\pi_U(n)$ modulo $n$ divides $n$) and $n$ divides $w_1 + \\dots + w_n$ for every integer sequence\u2026","title":"Members of $S(U)$ of the form $2^i 3^j$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/is-least-lucas-average-set-of-odd","kind":"node","lean":"MetaMathlibExt.isLeast_lucasAverageSet_of_odd","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd moduli in the average set when "],["math","\\(P \\equiv 2\\)"],["text"," and "],["math","\\(Q \\equiv 1\\)"]],"source":"Odd moduli in the average set when $P \\equiv 2$ and $Q \\equiv 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#is-least-lucas-average-set-of-odd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $(P, Q)$ be a nondegenerate Lucas pair with $P - Q - 1 \\ne 0$, Lucas sequence $U$, period set $S(U)$ and average set $S'(U)$. Let $n \\ge 1$ be odd, and suppose that $P \\equiv 2$ and $Q \\equiv 1 \\pmod{\\ell}$ for every prime $\\ell$ dividing $n$. Then every prime dividing $n$ divides $P^2 - 4Q$, $n$ is the least element of $S'(U)$ divisible by $n$ and the least element of $S(U)$ divisible by $n$, and $n$ divides\u2026","title":"Odd moduli in the average set when $P \\equiv 2$ and $Q \\equiv 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-set-is-least-of-even-p","kind":"node","lean":"MetaMathlibExt.lucasPeriodSet_isLeast_of_even_p","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Least multiples of "],["math","\\(2^k n_1\\)"],["text"," in the period set for even "],["math","\\(P\\)"]],"source":"Least multiples of $2^k n_1$ in the period set for even $P$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-set-is-least-of-even-p","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence $U_0 = 0$, $U_1 = 1$, $U_{j+2} = P U_{j+1} - Q U_j$ with nondegenerate parameters $P, Q \\in \\mathbb{Z}$. Let $S$ be the set of positive integers $m$ coprime to $Q$ such that the period of $U$ modulo $m$ divides $m$, and let $S'$ be the set of $m \\in S$ with $m \\mid U_1 + \\cdots + U_m$. Call $m$ *integral for all recurrences* if every integer sequence $w$ with\u2026","title":"Least multiples of $2^k n_1$ in the period set for even $P$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/is-least-lucas-average-set-of-six-dvd","kind":"node","lean":"MetaMathlibExt.isLeast_lucasAverageSet_of_six_dvd","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiples of "],["math","\\(6\\)"],["text"," in the average set when "],["math","\\(P \\equiv 1\\)"],["text"," and "],["math","\\(Q \\equiv 1\\)"]],"source":"Multiples of $6$ in the average set when $P \\equiv 1$ and $Q \\equiv 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#is-least-lucas-average-set-of-six-dvd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $(P, Q)$ be a nondegenerate Lucas pair with Lucas sequence $U$, period set $S(U)$ and average set $S'(U)$. Let $n \\ge 1$ be divisible by $6$, and suppose that $P \\equiv 1$ and $Q \\equiv 1 \\pmod{\\ell}$ for every prime $\\ell$ dividing $n$. Then $n$ is the least element of $S'(U)$ divisible by $n$ and the least element of $S(U)$ divisible by $n$, and $n$ divides $w_1 + \\dots + w_n$ for every integer sequence $w$\u2026","title":"Multiples of $6$ in the average set when $P \\equiv 1$ and $Q \\equiv 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/is-least-lucas-average-set-of-three-dvd","kind":"node","lean":"MetaMathlibExt.isLeast_lucasAverageSet_of_three_dvd","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiples of "],["math","\\(3\\)"],["text"," in the average set when "],["math","\\(P \\equiv -1\\)"],["text"," and "],["math","\\(Q \\equiv 1\\)"]],"source":"Multiples of $3$ in the average set when $P \\equiv -1$ and $Q \\equiv 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#is-least-lucas-average-set-of-three-dvd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $(P, Q)$ be a nondegenerate Lucas pair with Lucas sequence $U$, period set $S(U)$ and average set $S'(U)$. Let $n \\ge 1$ be divisible by $3$, and suppose that $P \\equiv -1$ and $Q \\equiv 1 \\pmod{\\ell}$ for every prime $\\ell$ dividing $n$. Then $n$ is the least element of $S'(U)$ divisible by $n$ and the least element of $S(U)$ divisible by $n$, and $n$ divides $w_1 + \\dots + w_n$ for every integer sequence $w$\u2026","title":"Multiples of $3$ in the average set when $P \\equiv -1$ and $Q \\equiv 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/is-least-lucas-average-set-of-four-dvd","kind":"node","lean":"MetaMathlibExt.isLeast_lucasAverageSet_of_four_dvd","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiples of "],["math","\\(4\\)"],["text"," in the average set when "],["math","\\(P \\equiv 0\\)"],["text"," and "],["math","\\(Q \\equiv 1\\)"]],"source":"Multiples of $4$ in the average set when $P \\equiv 0$ and $Q \\equiv 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#is-least-lucas-average-set-of-four-dvd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $(P, Q)$ be a nondegenerate Lucas pair with Lucas sequence $U$, period set $S(U)$ and average set $S'(U)$. Let $k \\ge 2$, let $n_1$ be odd and put $n = 2^k n_1$. Suppose that $P \\equiv 0$ and $Q \\equiv 1 \\pmod 4$, and that $P \\equiv 0$ and $Q \\equiv 1 \\pmod{\\ell}$ for every prime $\\ell$ dividing $n_1$. Then $n$ is the least element of $S'(U)$ divisible by $n$ and the least element of $S(U)$ divisible by $n$,\u2026","title":"Multiples of $4$ in the average set when $P \\equiv 0$ and $Q \\equiv 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-set-eq-lucas-index-divisibl","kind":"node","lean":"MetaMathlibExt.lucasPeriodSet_eq_lucasIndexDivisibleSet","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period set equals index-divisible set for nondegenerate "],["math","\\(U(P, Q)\\)"],["text"," with "],["math","\\(P = Q + 1\\)"]],"source":"Period set equals index-divisible set for nondegenerate $U(P, Q)$ with $P = Q + 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-set-eq-lucas-index-divisibl","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U = U(P, Q)$ be a nondegenerate Lucas sequence with integer parameters $P$ and $Q \\ne 0$ and $P = Q + 1$ (in the paper's notation $U(a, b)$ with $a = P$ and $b = -Q$, this is $a = -b + 1$). Then $S(U) = T(U)$, where $S(U)$ is the set of positive integers $n$ with $\\gcd(n, Q) = 1$ and $\\pi_U(n) \\mid n$, $\\pi_U(n)$ being the period of $U$ modulo $n$, and $T(U)$ is the set of positive integers $n$ with\u2026","title":"Period set equals index-divisible set for nondegenerate $U(P, Q)$ with $P = Q + 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-orbit-closes","kind":"node","lean":"MetaMathlibExt.LucasPeriodOrbitCloses","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-orbit-closes","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"Let $U = U(P, Q)$ be the Lucas sequence with integer parameters $P$ and $Q \\ne 0$ (the paper's $U(a, b)$ with $a = P$ and $b = -Q$). For $m \\ge 0$ let $\\pi_U(m)$ be the least $r > 0$ with $U_r \\equiv 0$ and $U_{r+1} \\equiv 1 \\pmod m$ (congruence modulo $0$ being equality), or $0$ if there is none, and let $\\pi_U^i$ be its $i$-th iterate, with $\\pi_U^0(n) = n$. For natural numbers $n$ and $k$, the period orbit of\u2026","title":"Closing of the period orbit"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-orbit-index","kind":"node","lean":"MetaMathlibExt.lucasPeriodOrbitIndex","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The orbit index "],["math","\\(\\omega(n)\\)"]],"source":"The orbit index $\\omega(n)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-orbit-index","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For a Lucas sequence $U = U(P, Q)$ and a natural number $n$, $\\omega(n)$ is the least natural number $k$ such that the period orbit of $n$ closes at step $k$, and $\\omega(n) = 0$ when there is no such $k$. Here $\\pi_U(m)$ is the period of $U$ modulo $m$, $\\pi_U^i$ is its $i$-th iterate with $\\pi_U^0(n) = n$, and the orbit of $n$ closes at step $k$ when $\\pi_U^i(n)$ is coprime to $Q$ for all $i \\le k$ and\u2026","title":"The orbit index $\\omega(n)$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-orbit-lcm","kind":"node","lean":"MetaMathlibExt.lucasPeriodOrbitLcm","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The orbit lcm "],["math","\\(L(n)\\)"]],"source":"The orbit lcm $L(n)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-orbit-lcm","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For a Lucas sequence $U = U(P, Q)$ and a natural number $n$, let $\\pi_U(m)$ be the period of $U$ modulo $m$, with iterates $\\pi_U^i$ and $\\pi_U^0(n) = n$, and let $\\omega(n)$ be the orbit index of $n$ (the least step at which the period orbit of $n$ closes, or $0$ if there is none). Then $L(n)$ is defined by","title":"The orbit lcm $L(n)$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-orbit-closes-one-zero","kind":"node","lean":"MetaMathlibExt.lucasPeriodOrbitCloses_one_zero","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The period orbit of "],["math","\\(1\\)"],["text"," closes at step "],["math","\\(0\\)"]],"source":"The period orbit of $1$ closes at step $0$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-orbit-closes-one-zero","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For every Lucas sequence $U = U(P, Q)$, let $\\pi_U(m)$ be the period of $U$ modulo $m$ and $\\pi_U^i$ its $i$-th iterate, with $\\pi_U^0(n) = n$. Then the period orbit of $1$ closes at step $0$: $1$ is coprime to $Q$ and $\\pi_U^1(1) = \\pi_U^0(1)$, that is, $\\pi_U(1) = 1$.","title":"The period orbit of $1$ closes at step $0$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-orbit-index-one","kind":"node","lean":"MetaMathlibExt.lucasPeriodOrbitIndex_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The orbit index of "],["math","\\(1\\)"],["text"," is zero"]],"source":"The orbit index of $1$ is zero"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-orbit-index-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For every Lucas sequence $U = U(P, Q)$, the orbit index of $1$ is $\\omega(1) = 0$.","title":"The orbit index of $1$ is zero"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-orbit-lcm-one","kind":"node","lean":"MetaMathlibExt.lucasPeriodOrbitLcm_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The orbit lcm of "],["math","\\(1\\)"],["text"," is "],["math","\\(1\\)"]],"source":"The orbit lcm of $1$ is $1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-orbit-lcm-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"For every Lucas sequence $U = U(P, Q)$ with integer parameters $P$ and $Q \\ne 0$, $L(1) = 1$, where $L(n) = \\operatorname{lcm}(\\pi_U^0(n), \\dots, \\pi_U^{\\omega(n)}(n))$ is the lcm of the iterates $\\pi_U^i(n)$ of the period map $\\pi_U$ of $U$, with $\\pi_U^0(n) = n$, up to the orbit index $\\omega(n)$ of $n$.","title":"The orbit lcm of $1$ is $1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/dvd-lucas-period-orbit-lcm","kind":"node","lean":"MetaMathlibExt.dvd_lucasPeriodOrbitLcm","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(n\\)"],["text"," divides "],["math","\\(L(n)\\)"]],"source":"$n$ divides $L(n)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#dvd-lucas-period-orbit-lcm","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"fully_proved","summary":"Let $P, Q$ be integers with $Q \\ne 0$ and $U$ the Lucas sequence with $U_0 = 0$, $U_1 = 1$ and $U_{j+2} = P U_{j+1} - Q U_j$. For $m \\in \\mathbb{N}$ let $\\pi_U(m)$ be the least $r \\ge 1$ with $U_r \\equiv 0$ and $U_{r+1} \\equiv 1 \\pmod m$, or $0$ if there is none, and let $\\pi_U^i$ be its $i$-th iterate, so that $\\pi_U^0(m) = m$. Let $\\omega(n)$ be the least $k$ such that $\\pi_U^i(n)$ is coprime to $Q$ for all\u2026","title":"$n$ divides $L(n)$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-prime-properties","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_prime_properties","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-prime-properties","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U = U(P, Q)$ be a nondegenerate Lucas sequence with discriminant $D = P^2 - 4Q$, and let $p$ be a prime. Write $\\pi_U(m)$ and $\\rho_U(m)$ for the least period and the least restricted period of $U$ modulo $m$, and $\\left(\\frac{x}{p}\\right)$ for the Jacobi symbol. (i) If $p \\nmid 2QD$, then $\\rho_U(p) > 1$, $\\rho_U(p) \\mid p - \\left(\\frac{D}{p}\\right)$, $\\rho_U(p) \\mid \\pi_U(p)$ and\u2026","title":"Periods of a Lucas sequence modulo a prime"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-u-uniform-of-prime-dvd-discr","kind":"node","lean":"MetaMathlibExt.lucasU_uniform_of_prime_dvd_discr","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-u-uniform-of-prime-dvd-discr","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence with integer parameters $P$ and $Q$, where the pair $(P, Q)$ is nondegenerate, and write $\\pi(m)$ and $\\rho(m)$ for the period and the restricted period of $U$ modulo $m$; for a positive integer $N$ and a residue $r$ modulo $N$, let $c_N(r)$ be the number of $k < \\pi(N)$ with $U_k \\equiv r \\pmod N$. Let $p$ be a prime with $p \\nmid Q$ and $p \\mid P^2 - 4Q$, and put\u2026","title":"Uniform distribution of Lucas sequences modulo primes dividing the discriminant"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/is-least-lucas-period-set-eq-lucas-perio","kind":"node","lean":"MetaMathlibExt.isLeast_lucasPeriodSet_eq_lucasPeriodOrbitLcm","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Least multiples in "],["math","\\(S(U)\\)"],["text"," and the orbit lcm "],["math","\\(L(n)\\)"]],"source":"Least multiples in $S(U)$ and the orbit lcm $L(n)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#is-least-lucas-period-set-eq-lucas-perio","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers and $U$ the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $\\pi_U(m)$ be the period of $U$ modulo $m$ and $\\pi_U^i$ its $i$-th iterate, with $\\pi_U^0(m) = m$. Let $S(U)$ be the set of positive integers $m$ coprime to $Q$ with $\\pi_U(m) \\mid m$, let $S'(U)$ be the set of $m \\in S(U)$ with $m \\mid U_1 + \\dots + U_m$, and say that a natural number\u2026","title":"Least multiples in $S(U)$ and the orbit lcm $L(n)$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-orbit-lcm-dvd-of-dvd","kind":"node","lean":"MetaMathlibExt.lucasPeriodOrbitLcm_dvd_of_dvd","parent":6963,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-orbit-lcm-dvd-of-dvd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers and $U$ the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $\\pi_U(m)$ be the period of $U$ modulo $m$ and $\\pi_U^i$ its $i$-th iterate, with $\\pi_U^0(m) = m$. Say that $\\omega(m)$ exists if the period orbit of $m$ closes at some step $k$, meaning that $\\pi_U^i(m)$ is coprime to $Q$ for all $i \\le k$ and $\\pi_U^{k+1}(m) = \\pi_U^j(m)$ for some\u2026","title":"Divisibility and multiplicativity of the orbit lcm"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-orbit-lcm-two-three","kind":"node","lean":"MetaMathlibExt.lucasPeriodOrbitLcm_two_three","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The orbit lcms "],["math","\\(L(2)\\)"],["text"," and "],["math","\\(L(3)\\)"]],"source":"The orbit lcms $L(2)$ and $L(3)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-orbit-lcm-two-three","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers, write $a = P$ and $b = -Q$, and let $U$ be the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $\\pi_U(m)$ be the period of $U$ modulo $m$ and $\\pi_U^i$ its $i$-th iterate, with $\\pi_U^0(m) = m$. Say that $L(m)$ exists if the period orbit of $m$ closes at some step $k$, meaning that $\\pi_U^i(m)$ is coprime to $Q$ for all $i \\le k$ and\u2026","title":"The orbit lcms $L(2)$ and $L(3)$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/exists-lucas-period-set-multiple","kind":"node","lean":"MetaMathlibExt.exists_lucasPeriodSet_multiple","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Least multiples in "],["math","\\(S(U)\\)"],["text"," for nondegenerate "],["math","\\(U(P, Q)\\)"]],"source":"Least multiples in $S(U)$ for nondegenerate $U(P, Q)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#exists-lucas-period-set-multiple","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers forming a nondegenerate Lucas pair, and let $U$ be the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Let $S(U)$ be the set of positive integers $n$ coprime to $Q$ such that the period $\\pi_U(n)$ of $U$ modulo $n$ divides $n$, and say that a natural number $x$ has integral averages if $x \\mid w_1 + \\dots + w_x$ for every integer sequence\u2026","title":"Least multiples in $S(U)$ for nondegenerate $U(P, Q)$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/mem-lucas-average-set-of-q-eq-neg-one","kind":"node","lean":"MetaMathlibExt.mem_lucasAverageSet_of_q_eq_neg_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral averages for "],["math","\\(U(P, -1)\\)"]],"source":"Integral averages for $U(P, -1)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#mem-lucas-average-set-of-q-eq-neg-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U = U(P, Q)$ be a Lucas sequence with $Q = -1$, so that its discriminant is $D = P^2 + 4$, and let $\\pi_U(n)$ be the least period of $U$ modulo $n$. Call a natural number $n$ good if $n$ lies in the set $S'(U)$ of positive integers $n$ coprime to $Q$ with $\\pi_U(n) \\mid n$ and $n \\mid U_1 + \\dots + U_n$, and every integer sequence $w$ with $w_{k+2} = P w_{k+1} + w_k$ for all $k$ satisfies\u2026","title":"Integral averages for $U(P, -1)$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/mem-lucas-average-set-of-q-eq-one","kind":"node","lean":"MetaMathlibExt.mem_lucasAverageSet_of_q_eq_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral averages for "],["math","\\(U(P, 1)\\)"]],"source":"Integral averages for $U(P, 1)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#mem-lucas-average-set-of-q-eq-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U = U(P, Q)$ be a nondegenerate Lucas sequence with $Q = 1$, so that its discriminant is $D = P^2 - 4$, and let $\\pi_U(n)$ be the least period of $U$ modulo $n$. Call a natural number $n$ good if $n$ lies in the set $S'(U)$ of positive integers $n$ coprime to $Q$ with $\\pi_U(n) \\mid n$ and $n \\mid U_1 + \\dots + U_n$, and every integer sequence $w$ with $w_{k+2} = P w_{k+1} - w_k$ for all $k$ satisfies\u2026","title":"Integral averages for $U(P, 1)$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-set-of-q-eq-neg-one","kind":"node","lean":"MetaMathlibExt.lucasPeriodSet_of_q_eq_neg_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Least period and average multiples for "],["math","\\(Q = -1\\)"]],"source":"Least period and average multiples for $Q = -1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-set-of-q-eq-neg-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence $U_0 = 0$, $U_1 = 1$, $U_{j+2} = P U_{j+1} - Q U_j$ with $Q = -1$ and $(P, Q)$ nondegenerate. Write $\\pi(m)$ for the period of $U$ modulo $m$ and $\\pi^i$ for the $i$-th iterate of $\\pi$ ($\\pi^0(m) = m$). Let $S$ be the set of positive integers $m$ coprime to $Q$ with $\\pi(m) \\mid m$, and $S'$ the set of $m \\in S$ with $m \\mid U_1 + \\cdots + U_m$. Say the orbit of $n$ closes at $k$ if\u2026","title":"Least period and average multiples for $Q = -1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-period-set-of-q-eq-one","kind":"node","lean":"MetaMathlibExt.lucasPeriodSet_of_q_eq_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Least period and average multiples for "],["math","\\(Q = 1\\)"]],"source":"Least period and average multiples for $Q = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-period-set-of-q-eq-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U$ be the Lucas sequence $U_0 = 0$, $U_1 = 1$, $U_{j+2} = P U_{j+1} - Q U_j$ with $Q = 1$ and $(P, Q)$ nondegenerate. Let $S$ be the set of positive integers $m$ coprime to $Q$ such that the period of $U$ modulo $m$ divides $m$, and $S'$ the set of $m \\in S$ with $m \\mid U_1 + \\cdots + U_m$. For $n \\ge 1$ let $C(n)$ and $C'(n)$ be the least elements of $S$ and of $S'$ divisible by $n$, when they exist. Call\u2026","title":"Least period and average multiples for $Q = 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-average-set-of-p-eq-q-add-one","kind":"node","lean":"MetaMathlibExt.lucasAverageSet_of_p_eq_q_add_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period and average sets when "],["math","\\(P = Q + 1\\)"]],"source":"Period and average sets when $P = Q + 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-average-set-of-p-eq-q-add-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers forming a nondegenerate Lucas pair with $P = Q + 1$, so that the roots of $x^2 - Px + Q$ are $Q$ and $1$, and let $U$ be the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Write $\\pi_U(n)$ for the period of $U$ modulo $n$, $\\rho_U(n)$ for the least $s > 0$ with $n \\mid U_s$ and $U_{s+1}$ coprime to $n$, $S(U)$ for the set of positive integers $n$\u2026","title":"Period and average sets when $P = Q + 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-average-set-of-p-eq-neg-q-sub-one","kind":"node","lean":"MetaMathlibExt.lucasAverageSet_of_p_eq_neg_q_sub_one","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period and average sets when "],["math","\\(P = -Q - 1\\)"]],"source":"Period and average sets when $P = -Q - 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-average-set-of-p-eq-neg-q-sub-one","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P$ and $Q \\ne 0$ be integers forming a nondegenerate Lucas pair with $P = -Q - 1$, so that the roots of $x^2 - Px + Q$ are $-Q$ and $-1$, and let $U$ be the Lucas sequence given by $U_0 = 0$, $U_1 = 1$ and $U_{k+2} = P U_{k+1} - Q U_k$. Write $\\pi_U(n)$ for the period of $U$ modulo $n$, $S(U)$ for the set of positive integers $n$ coprime to $Q$ with $\\pi_U(n) \\mid n$, $S'(U)$ for the set of $n \\in S(U)$ with\u2026","title":"Period and average sets when $P = -Q - 1$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/mem-lucas-index-divisible-set-iff-chain","kind":"node","lean":"MetaMathlibExt.mem_lucasIndexDivisibleSet_iff_chain","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Smyth's description of the indices "],["math","\\(n\\)"],["text"," with "],["math","\\(n \\mid U_n\\)"]],"source":"Smyth's description of the indices $n$ with $n \\mid U_n$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#mem-lucas-index-divisible-set-iff-chain","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $P, Q$ be coprime integers with $Q \\ne 0$ such that the Lucas pair $(P, Q)$ is nondegenerate, let $U = U(P, Q)$ be its Lucas sequence, write $a = P$, $b = -Q$ (so that $U_{k+2} = a U_{k+1} + b U_k$) and $D = P^2 - 4Q$, and let $T(U)$ be the set of positive integers $n$ coprime to $Q$ with $n \\mid U_n$. Call $q_0$ admissible if $q_0 = 1$, or $q_0 = 6$ with $a \\equiv 3$ and $b \\equiv \\pm 1 \\pmod 6$, or $q_0 = 12$\u2026","title":"Smyth's description of the indices $n$ with $n \\mid U_n$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/lucas-uperiod-quotient-dvd-and-seed-mem","kind":"node","lean":"MetaMathlibExt.lucasUPeriod_quotient_dvd_and_seed_mem_lucasPeriodSet","parent":6963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period quotients and seeds of "],["math","\\(S(U)\\)"]],"source":"Period quotients and seeds of $S(U)$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n/index.html#lucas-uperiod-quotient-dvd-and-seed-mem","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/periods-and-averages-of-lucas-sequences-modulo-n","status":"can_prove","summary":"Let $U = U(P, Q)$ be a nondegenerate Lucas sequence with $Q = -1$, or $Q = 1$, or $Q$ odd and $P = -Q - 1$. Write $\\pi_U(m)$ and $\\rho_U(m)$ for the least period and the least restricted period of $U$ modulo $m$, and $S(U)$ for the set of positive integers $n$ coprime to $Q$ with $\\pi_U(n) \\mid n$. For every prime $p \\nmid Q$, $\\rho_U(p)$ divides $\\pi_U(p)$, and the quotient $\\pi_U(p) / \\rho_U(p)$ divides $4$ if\u2026","title":"Period quotients and seeds of $S(U)$"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods.json","id":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","kind":"scope","navigation":"module","parent":6956,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/index.html","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","status":"can_prove","title":"Pisano periods"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/fulton-morris-pisano-fixed-point","kind":"node","lean":"MetaMathlibExt.fulton_morris_pisano_fixed_point","parent":7034,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/index.html#fulton-morris-pisano-fixed-point","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","status":"fully_proved","summary":"Write $F_t$ for the $t$-th Fibonacci number. Let $m > 1$ be a natural number and let $\\pi$ be a positive integer with $F_\\pi \\equiv 0 \\pmod{m}$ and $F_{\\pi+1} \\equiv 1 \\pmod{m}$ such that $\\pi \\le t$ for every positive integer $t$ with $F_t \\equiv 0 \\pmod{m}$ and $F_{t+1} \\equiv 1 \\pmod{m}$; that is, $\\pi$ is the Pisano period of $m$. Then $\\pi = m$ if and only if $m = 24 \\cdot 5^{\\lambda - 1}$ for some integer\u2026","title":"Fulton\u2013Morris theorem on Pisano periods equal to the modulus"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/pisano-period","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.FibonacciSumDivisibilityWanted.pisanoPeriod","parent":7034,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/index.html#pisano-period","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","status":"fully_proved","summary":"For a natural number $n$, the Pisano period $\\pi(n)$ is the minimal period of the pair $(0, 1)$ under the Fibonacci step $(a, b) \\mapsto (b, a + b)$ on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$: the least $k \\ge 1$ such that $k$ applications of the step map $(0, 1)$ back to $(0, 1)$, or $0$ if there is no such $k$ (as for $n = 0$, where $\\mathbb{Z}/0\\mathbb{Z} = \\mathbb{Z}$).","title":"Pisano period"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/pisano-period-fib-dvd-two-mul","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.FibonacciSumDivisibilityWanted.pisanoPeriod_fib_dvd_two_mul","parent":7034,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Pisano period of "],["math","\\(F_n\\)"],["text"," divides "],["math","\\(2n\\)"],["text"," for even "],["math","\\(n\\)"]],"source":"The Pisano period of $F_n$ divides $2n$ for even $n$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/index.html#pisano-period-fib-dvd-two-mul","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","status":"can_prove","summary":"Let $F_n$ denote the $n$-th Fibonacci number and $\\pi(m)$ the Pisano period of $m$. For every even natural number $n > 1$, $\\pi(F_n)$ divides $2n$.","title":"The Pisano period of $F_n$ divides $2n$ for even $n$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/tendsto-i-inf-pisano-period-div-at-top","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.FibonacciSumDivisibilityWanted.tendsto_iInf_pisanoPeriod_div_atTop","parent":7034,"reading_title":{"format":"inline-parts-v1","parts":[["text","The minimum of "],["math","\\(\\pi(n)/n\\)"],["text"," over "],["math","\\(1 \\le n < N\\)"],["text"," tends to zero"]],"source":"The minimum of $\\pi(n)/n$ over $1 \\le n < N$ tends to zero"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/index.html#tendsto-i-inf-pisano-period-div-at-top","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","status":"can_prove","summary":"Let $\\pi(n)$ denote the Pisano period of $n$. As $N \\to \\infty$, the minimum of the real numbers $\\pi(n)/n$ over the integers $1 \\le n \\le N - 1$ (defined for $N \\ge 2$) tends to $0$.","title":"The minimum of $\\pi(n)/n$ over $1 \\le n < N$ tends to zero"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/sum-fib-mod-prime-eq-and-not-dvd","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.FibonacciSumDivisibilityWanted.sum_fib_mod_prime_eq_and_not_dvd","parent":7034,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of the first "],["math","\\(p\\)"],["text"," Fibonacci numbers modulo a prime "],["math","\\(p\\)"]],"source":"Sum of the first $p$ Fibonacci numbers modulo a prime $p$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/index.html#sum-fib-mod-prime-eq-and-not-dvd","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","status":"can_prove","summary":"Let $p > 5$ be a prime, let $F_i$ denote the $i$-th Fibonacci number, with $F_1 = F_2 = 1$, and let $S = F_1 + F_2 + \\dots + F_p$. If $p \\equiv 1$ or $4 \\pmod 5$, then $S$ leaves remainder $1$ on division by $p$; if $p \\equiv 2$ or $3 \\pmod 5$, then $S$ leaves remainder $p - 2$ on division by $p$; and in every case $p$ does not divide $S$.","title":"Sum of the first $p$ Fibonacci numbers modulo a prime $p$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/fib-two-mul-prime-odd-and-fib-fib-add-tw","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.FibonacciSumDivisibilityWanted.fib_two_mul_prime_odd_and_fib_fib_add_two_mod","parent":7034,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd Fibonacci numbers "],["math","\\(F_n\\)"],["text"," with "],["math","\\(F_{F_n + 2} \\equiv 1 \\pmod{F_n}\\)"]],"source":"Odd Fibonacci numbers $F_n$ with $F_{F_n + 2} \\equiv 1 \\pmod{F_n}$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/index.html#fib-two-mul-prime-odd-and-fib-fib-add-tw","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","status":"can_prove","summary":"Let $F_m$ denote the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. If $p$ is an odd prime with $p \\equiv 2 \\pmod 3$ and $p \\equiv 2$ or $3 \\pmod 5$, and $n = 2p$ or $n = 4p$, then $F_n$ is odd and $F_{F_n + 2}$ leaves remainder $1$ on division by $F_n$.","title":"Odd Fibonacci numbers $F_n$ with $F_{F_n + 2} \\equiv 1 \\pmod{F_n}$"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/infinite-set-of-odd-dvd-sum-fib","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.FibonacciSumDivisibilityWanted.infinite_setOf_odd_dvd_sum_fib","parent":7034,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many odd "],["math","\\(k\\)"],["text"," divide the sum of the first "],["math","\\(k\\)"],["text"," Fibonacci numbers"]],"source":"Infinitely many odd $k$ divide the sum of the first $k$ Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods/index.html#infinite-set-of-odd-dvd-sum-fib","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/pisano-periods","status":"can_prove","summary":"Let $F_i$ denote the $i$-th Fibonacci number, with $F_1 = F_2 = 1$. There are infinitely many odd positive integers $k$ such that $k$ divides $F_1 + F_2 + \\dots + F_k$.","title":"Infinitely many odd $k$ divide the sum of the first $k$ Fibonacci numbers"},{"data_url":"scopes/NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers.json","id":"NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers","kind":"scope","navigation":"module","parent":6956,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers/index.html","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers","status":"can_prove","title":"Residues of Lucas numbers"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers/statement","kind":"node","lean":"MathlibExt.NumberTheory.LucasNumbersModuloMWanted.Statement","parent":7042,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers/index.html#statement","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers","status":"fully_proved","summary":"The proposition that for every integer $m > 1$, with $L_n$ the Lucas numbers, every residue modulo $m$ is attained (for every $r$ with $0 \\le r < m$ there is $n \\ge 0$ with $L_n \\equiv r \\pmod m$) if and only if $m \\in \\{2, 4, 6, 7, 14\\}$ or $m = 3^k$ for some integer $k \\ge 1$.","title":"Classification of moduli for which the Lucas numbers attain every residue"},{"id":"NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers/statement-holds","kind":"node","lean":"MathlibExt.NumberTheory.LucasNumbersModuloMWanted.Statement_holds","parent":7042,"reading_title":{"format":"inline-parts-v1","parts":[["text","Avila\u2013Chen theorem on Lucas numbers modulo "],["math","\\(m\\)"]],"source":"Avila\u2013Chen theorem on Lucas numbers modulo $m$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers/index.html#statement-holds","scope":"NumberTheory/linear-recurrences/recurrences-modulo-m/residues-of-lucas-numbers","status":"can_prove","summary":"Let $L_n$ denote the Lucas numbers. For every integer $m > 1$, every residue modulo $m$ is attained by the Lucas numbers (for every $r$ with $0 \\le r < m$ there is $n \\ge 0$ with $L_n \\equiv r \\pmod m$) if and only if $m \\in \\{2, 4, 6, 7, 14\\}$ or $m = 3^k$ for some integer $k \\ge 1$.","title":"Avila\u2013Chen theorem on Lucas numbers modulo $m$"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences","kind":"scope","parent":5991,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences","status":"can_state","summary":"Squares, perfect powers, repdigits, factorials and powers of two among terms of recurrence sequences.","title":"Diophantine problems in recurrences"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/tribonacci-factorial-product-index-mem","kind":"node","lean":"MetaMathlibExt.tribonacci_factorialProduct_index_mem","parent":7045,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/index.html#tribonacci-factorial-product-index-mem","scope":"NumberTheory/linear-recurrences/diophantine-recurrences","status":"can_prove","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the integer sequence with $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{j+3} = T_{j+2} + T_{j+1} + T_j$ for every integer $j$. If $n$ is an integer such that $|T_n| = m_1! \\, m_2! \\cdots m_k!$ for some $k \\ge 1$ and some positive integers $m_1, \\dots, m_k$, then $n \\in \\{-9, -8, -7, -5, -3, -2, 1, 2, 3, 4, 7\\}$.","title":"Tribonacci numbers that are products of factorials"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc","status":"can_prove","title":"Brocard\u2013Ramanujan equation in recurrence sequences"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/tribonacci-int-neg-ten","kind":"node","lean":"MetaMathlibExt.tribonacciInt_neg_ten","parent":7047,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Tribonacci number of index "],["math","\\(-10\\)"]],"source":"The Tribonacci number of index $-10$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/index.html#tribonacci-int-neg-ten","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc","status":"fully_proved","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence, determined by $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for all integers $n$. Then $T_{-10} = 5$.","title":"The Tribonacci number of index $-10$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/brocard-ramanujan-not-tribonacci","kind":"node","lean":"MetaMathlibExt.brocardRamanujan_not_tribonacci","parent":7047,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/index.html#brocard-ramanujan-not-tribonacci","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc","status":"can_prove","summary":"Let $(T_n)_{n \\ge 0}$ be the Tribonacci sequence, given by $T_0 = T_1 = 0$, $T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for $n \\ge 0$. For all natural numbers $m$ and $n$, $m! + 1 \\ne T_n^2$.","title":"The Brocard\u2013Ramanujan equation has no Tribonacci solutions"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/brocard-ramanujan-not-tetranacci","kind":"node","lean":"MetaMathlibExt.brocardRamanujan_not_tetranacci","parent":7047,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/index.html#brocard-ramanujan-not-tetranacci","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc","status":"can_prove","summary":"Let $(Q_n)_{n \\ge 0}$ be the Tetranacci sequence, given by $Q_0 = Q_1 = Q_2 = 0$, $Q_3 = 1$ and $Q_{n+4} = Q_{n+3} + Q_{n+2} + Q_{n+1} + Q_n$ for $n \\ge 0$. For all natural numbers $m$ and $n$, $m! + 1 \\ne Q_n^2$.","title":"The Brocard\u2013Ramanujan equation has no Tetranacci solutions"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/tribonacci-padic-val-int-forty-seven","kind":"node","lean":"MetaMathlibExt.tribonacci_padicValInt_fortySeven","parent":7047,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(47\\)"],["text","-adic valuation of Tribonacci numbers near "],["math","\\(\\pm 1\\)"]],"source":"The $47$-adic valuation of Tribonacci numbers near $\\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc/index.html#tribonacci-padic-val-int-forty-seven","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/brocard-ramanujan-equation-in-recurrence-sequenc","status":"can_prove","summary":"Let $(T_n)_{n \\in \\mathbb{Z}}$ be the two-sided Tribonacci sequence, determined by $T_0 = 0$, $T_1 = T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$ for all integers $n$. Let $\\nu_{47}$ be the $47$-adic valuation on the integers, with the convention $\\nu_{47}(0) = 0$, and let $J = \\{1, 2, -2, -3, -7\\}$. First, $T_j = 1$ or $T_j = -1$ for every $j \\in J$. Second, for every $j \\in J$ and every nonzero integer $r$,\u2026","title":"The $47$-adic valuation of Tribonacci numbers near $\\pm 1$"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom","status":"can_prove","title":"Discriminants of k-generalized Fibonacci polynomials"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom/abs-discr-k-generalized-fibonacci-polyno","kind":"node","lean":"MetaMathlibExt.abs_discr_kGeneralizedFibonacciPolynomial_mem_iff","parent":7052,"reading_title":{"format":"inline-parts-v1","parts":[["text","When the absolute discriminant of a "],["math","\\(k\\)"],["text","-generalized Fibonacci polynomial is a "],["math","\\(k\\)"],["text","-generalized Fibonacci number"]],"source":"When the absolute discriminant of a $k$-generalized Fibonacci polynomial is a $k$-generalized Fibonacci number"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom/index.html#abs-discr-k-generalized-fibonacci-polyno","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom","status":"can_prove","summary":"When the absolute discriminant of a $k$-generalized Fibonacci polynomial is a $k$-generalized Fibonacci number. Let $k \\ge 2$ be a natural number and let $F : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $F(n) = 0$ for every integer $n$ with $2 - k \\le n \\le 0$, $F(1) = 1$, and $F(n) = F(n-1) + F(n-2) + \\cdots + F(n-k)$ for every integer $n \\ge 2$, so that $F(n)$ for $n \\ge 0$ is the $k$-generalized Fibonacci sequence. Then\u2026","title":"When the absolute discriminant of a $k$-generalized Fibonacci polynomial is a $k$-generalized Fibonacci number"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom/k-generalized-fibonacci-discr-eq-index-b","kind":"node","lean":"MetaMathlibExt.kGeneralizedFibonacci_discr_eq_index_bounds","parent":7052,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom/index.html#k-generalized-fibonacci-discr-eq-index-b","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom","status":"can_prove","summary":"Let $k > 100$ be a natural number and let $F : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $F(n) = 0$ for every integer $n$ with $2 - k \\le n \\le 0$, $F(1) = 1$, and $F(n) = F(n-1) + F(n-2) + \\cdots + F(n-k)$ for every integer $n \\ge 2$ (the $k$-generalized Fibonacci sequence). Write $L = (k \\log k - 2 \\log(k-1)) / \\log 2$, where $\\log$ is the natural logarithm. If a natural number $n$ satisfies\u2026","title":"Bounds on the index in Luca's discriminant equation"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom/k-generalized-fibonacci-discr-eq-bounds","kind":"node","lean":"MetaMathlibExt.kGeneralizedFibonacci_discr_eq_bounds","parent":7052,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom/index.html#k-generalized-fibonacci-discr-eq-bounds","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/discriminants-of-k-generalized-fibonacci-polynom","status":"can_prove","summary":"Let $k > 100$ be a natural number and let $F : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $F(n) = 0$ for every integer $n$ with $2 - k \\le n \\le 0$, $F(1) = 1$, and $F(n) = F(n-1) + F(n-2) + \\cdots + F(n-k)$ for every integer $n \\ge 2$ (the $k$-generalized Fibonacci sequence). If a natural number $n$ satisfies $F(n)\\,(k-1)^2 = 2^{k+1} k^k - (k+1)^{k+1}$, then $k < 2 \\cdot 10^{16}$ and $n < 1.5 \\cdot 10^{18}$.","title":"Absolute bounds on solutions of Luca's discriminant equation"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"can_prove","title":"Factorials that are Fibonacci integers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/factorial-mem-fibonacci-integers-iff","kind":"node","lean":"MetaMathlibExt.factorial_mem_fibonacciIntegers_iff","parent":7056,"reading_title":{"format":"inline-parts-v1","parts":[["text","The largest factorial that is a Fibonacci integer is "],["math","\\(36!\\)"]],"source":"The largest factorial that is a Fibonacci integer is $36!$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#factorial-mem-fibonacci-integers-iff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"can_prove","summary":"For every natural number $n$, the factorial $n!$ lies in the subgroup of the multiplicative group $\\mathbb{Q}^\\times$ generated by the Fibonacci numbers $F_m$ with $m \\ge 1$ if and only if $n \\le 36$.","title":"The largest factorial that is a Fibonacci integer is $36!$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/primitive-prime-dvd-fib-mod-eq-one-or-ne","kind":"node","lean":"MetaMathlibExt.primitive_prime_dvd_fib_modEq_one_or_neg_one","parent":7056,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#primitive-prime-dvd-fib-mod-eq-one-or-ne","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"can_prove","summary":"Let $F_n$ denote the $n$-th Fibonacci number. For every natural number $m > 12$, every primitive prime divisor $p$ of $F_m$ (a prime $p$ that divides $F_m$ but divides no $F_n$ with $0 < n < m$) satisfies $p \\equiv \\pm 1 \\pmod{m}$, that is, $p \\equiv 1 \\pmod{m}$ or $p + 1 \\equiv 0 \\pmod{m}$.","title":"Carmichael's congruence for primitive prime divisors of Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/fib-entry-point","kind":"node","lean":"MetaMathlibExt.fibEntryPoint","parent":7056,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#fib-entry-point","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"fully_proved","summary":"The entry point (rank of apparition) $z(k)$ of a natural number $k$ is the least $m > 0$ such that $k$ divides the Fibonacci number $F_m$, and $z(k) = 0$ if there is no such $m$.","title":"Entry point in the Fibonacci sequence"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/fib-entry-point-two","kind":"node","lean":"MetaMathlibExt.fibEntryPoint_two","parent":7056,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci entry point of "],["math","\\(2\\)"]],"source":"Fibonacci entry point of $2$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#fib-entry-point-two","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"fully_proved","summary":"The least $m > 0$ such that $2$ divides the Fibonacci number $F_m$ is $3$, that is, the Fibonacci entry point of $2$ is $z(2) = 3$.","title":"Fibonacci entry point of $2$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/fib-entry-point-five","kind":"node","lean":"MetaMathlibExt.fibEntryPoint_five","parent":7056,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci entry point of "],["math","\\(5\\)"]],"source":"Fibonacci entry point of $5$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#fib-entry-point-five","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"fully_proved","summary":"The least $m > 0$ such that $5$ divides the Fibonacci number $F_m$ is $5$, that is, the Fibonacci entry point of $5$ is $z(5) = 5$.","title":"Fibonacci entry point of $5$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/fib-cyclotomic-value","kind":"node","lean":"MetaMathlibExt.fibCyclotomicValue","parent":7056,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#fib-cyclotomic-value","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"fully_proved","summary":"For a natural number $m$, $\\Phi_m$ is the rational number $\\prod_{d \\mid m} F_{m/d}^{\\mu(d)}$, where $d$ runs over the positive divisors of $m$, $F_k$ is the $k$-th Fibonacci number and $\\mu$ is the M\u00f6bius function. For $m > 1$ this is the value $\\Phi_m(\\alpha, \\beta)$ of the homogenized $m$-th cyclotomic polynomial at the roots $\\alpha, \\beta$ of $x^2 - x - 1$.","title":"Fibonacci cyclotomic values"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/fib-cyclotomic-value-two","kind":"node","lean":"MetaMathlibExt.fibCyclotomicValue_two","parent":7056,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci cyclotomic value at "],["math","\\(2\\)"]],"source":"Fibonacci cyclotomic value at $2$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#fib-cyclotomic-value-two","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"fully_proved","summary":"The Fibonacci cyclotomic value $\\Phi_2 = \\prod_{d \\mid 2} F_{2/d}^{\\mu(d)} = F_2 / F_1$, where $F_k$ is the $k$-th Fibonacci number and $\\mu$ is the M\u00f6bius function, equals $1$.","title":"Fibonacci cyclotomic value at $2$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/fib-primitive-part","kind":"node","lean":"MetaMathlibExt.fibPrimitivePart","parent":7056,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#fib-primitive-part","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"fully_proved","summary":"For a natural number $m$, the primitive part of the Fibonacci number $F_m$ is $\\Psi_m = \\prod_p p^{a_p}$, where the product runs over the primes $p$ dividing $F_m$ whose Fibonacci entry point is $z(p) = m$, and $a_p$ is the exponent of $p$ in $F_m$.","title":"Primitive part of a Fibonacci number"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/fib-primitive-part-six","kind":"node","lean":"MetaMathlibExt.fibPrimitivePart_six","parent":7056,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primitive part of "],["math","\\(F_6\\)"]],"source":"Primitive part of $F_6$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#fib-primitive-part-six","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"fully_proved","summary":"The primitive part of the Fibonacci number $F_6$ is $\\Psi_6 = 1$: $F_6 = 8 = 2^3$ and the Fibonacci entry point of $2$ is $z(2) = 3 \\ne 6$, so $F_6$ has no prime factor $p$ with $z(p) = 6$.","title":"Primitive part of $F_6$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/exists-injection-prime-factors-fib-primi","kind":"node","lean":"MetaMathlibExt.exists_injection_primeFactors_fibPrimitivePart","parent":7056,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#exists-injection-prime-factors-fib-primi","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"can_prove","summary":"Let $I$ and $J$ be finite multisets of natural numbers, every element of which is at least $3$ and different from $6$ and $12$, such that no element of $I$ equals an element of $J$ and $\\prod_{a \\in I} \\Phi_a / \\prod_{b \\in J} \\Phi_b$ is a natural number, where $\\Phi_m$ is the $m$-th Fibonacci cyclotomic value. Let $\\Psi_m$ be the primitive part of the Fibonacci number $F_m$ and $z(p)$ the Fibonacci entry point of\u2026","title":"Prime factors of the primitive parts of the denominator indices"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/card-padic-val-rat-fib-delta-eq-one-lt","kind":"node","lean":"MetaMathlibExt.card_padicValRat_fibDelta_eq_one_lt","parent":7056,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on denominator indices of "],["math","\\(p\\)"],["text","-adic valuation one"]],"source":"Bound on denominator indices of $p$-adic valuation one"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers/index.html#card-padic-val-rat-fib-delta-eq-one-lt","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/factorials-that-are-fibonacci-integers","status":"can_prove","summary":"Let $n \\ge 37$ and let $I$ and $J$ be finite multisets of natural numbers, every element of which is at least $3$ and different from $6$ and $12$, such that no element of $I$ equals an element of $J$ and $n! = \\prod_{a \\in I} \\Phi_a / \\prod_{b \\in J} \\Phi_b$, where $\\Phi_m$ is the $m$-th Fibonacci cyclotomic value. Let $p$ be a prime whose Fibonacci entry point $z(p)$ exceeds $12$, and let $N_p$ be the number of\u2026","title":"Bound on denominator indices of $p$-adic valuation one"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_state","title":"k-Pell\u2013Lucas numbers close to powers of two"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas","kind":"node","lean":"MetaMathlibExt.kPellLucas","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-generalized Pell\u2013Lucas numbers"]],"source":"$k$-generalized Pell\u2013Lucas numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the $k$-generalized Pell\u2013Lucas number $Q^{(k)}_n$ is the natural number defined by $Q^{(k)}_0 = Q^{(k)}_1 = 2$ and","title":"$k$-generalized Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-zero","kind":"node","lean":"MetaMathlibExt.kPellLucas_zero","parent":7068,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-zero","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"fully_proved","summary":"For every natural number $k$, the $k$-generalized Pell\u2013Lucas number $Q^{(k)}_0$ equals $2$.","title":"Initial value of the generalized Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-one","kind":"node","lean":"MetaMathlibExt.kPellLucas_one","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-Pell\u2013Lucas number of index one"]],"source":"The $k$-Pell\u2013Lucas number of index one"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-one","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"fully_proved","summary":"For every natural number $k$, the $k$-Pell\u2013Lucas number $Q^{(k)}_1$ equals $2$.","title":"The $k$-Pell\u2013Lucas number of index one"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-add-two","kind":"node","lean":"MetaMathlibExt.kPellLucas_add_two","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the "],["math","\\(k\\)"],["text","-Pell\u2013Lucas numbers"]],"source":"Recurrence for the $k$-Pell\u2013Lucas numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-add-two","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"fully_proved","summary":"Let $Q^{(k)}_n$ denote the $k$-generalized Pell\u2013Lucas numbers. For all natural numbers $k$ and $n$ with $k \\le n + 2$, $Q^{(k)}_{n+2} = 2 Q^{(k)}_{n+1} + \\sum_{j=0}^{k-2} Q^{(k)}_{n-j}$, where the sum is empty when $k < 2$. So once all $k - 1$ earlier terms have nonnegative index, the full recurrence holds.","title":"Recurrence for the $k$-Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-two","kind":"node","lean":"MetaMathlibExt.kPellLucas_two","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(2\\)"],["text","-generalized Pell\u2013Lucas numbers are the Pell\u2013Lucas numbers"]],"source":"The $2$-generalized Pell\u2013Lucas numbers are the Pell\u2013Lucas numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-two","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"fully_proved","summary":"For every natural number $n$, the $2$-generalized Pell\u2013Lucas number $Q^{(2)}_n$ equals the $n$th Pell\u2013Lucas number $Q_n$; this is the sequence $2, 2, 6, 14, 34, \\ldots$.","title":"The $2$-generalized Pell\u2013Lucas numbers are the Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-three-small","kind":"node","lean":"MetaMathlibExt.kPellLucas_three_small","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two values of the "],["math","\\(3\\)"],["text","-generalized Pell\u2013Lucas numbers"]],"source":"Two values of the $3$-generalized Pell\u2013Lucas numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-three-small","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"fully_proved","summary":"The $3$-generalized Pell\u2013Lucas numbers $Q^{(3)}_n$ satisfy $Q^{(3)}_3 = 16$ and $Q^{(3)}_6 = 260$.","title":"Two values of the $3$-generalized Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-eq-two-pow-iff","kind":"node","lean":"MetaMathlibExt.kPellLucas_eq_two_pow_iff","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-Pell\u2013Lucas numbers that are powers of two (Corollary 1)"]],"source":"$k$-Pell\u2013Lucas numbers that are powers of two (Corollary 1)"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-eq-two-pow-iff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"Let $k \\ge 2$, $n \\ge 1$ and $m \\ge 1$ be natural numbers, and let $Q^{(k)}_n$ be the $n$th $k$-Pell\u2013Lucas number. Then $Q^{(k)}_n = 2^m$ if and only if either $n = 1$ and $m = 1$, or $n = 3$, $m = 4$ and $k \\ge 3$.","title":"$k$-Pell\u2013Lucas numbers that are powers of two (Corollary 1)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-dominant-root-bounds","kind":"node","lean":"MetaMathlibExt.kPellLucas_dominantRoot_bounds","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds for the dominant root of the "],["math","\\(k\\)"],["text","-Pell\u2013Lucas polynomial (Lemma 1)"]],"source":"Bounds for the dominant root of the $k$-Pell\u2013Lucas polynomial (Lemma 1)"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-dominant-root-bounds","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and let $\\alpha > 0$ be a real number with $\\alpha^k = 2\\alpha^{k-1} + \\sum_{i=0}^{k-2} \\alpha^i$, that is, a positive real root of $\\Phi_k(x) = x^k - 2x^{k-1} - \\sum_{i=0}^{k-2} x^i$. Write $g_k(z) = (z - 1) / ((k + 1) z^2 - 3kz + k - 1)$ and let $\\varphi$ be the golden ratio. Then (a) $0.276 < g_k(\\alpha) < 0.5$, and $|g_k(\\beta)| < 1$ for every complex root $\\beta \\ne \\alpha$ of\u2026","title":"Bounds for the dominant root of the $k$-Pell\u2013Lucas polynomial (Lemma 1)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-dominant-root-golden-approx","kind":"node","lean":"MetaMathlibExt.kPellLucas_dominantRoot_golden_approx","parent":7068,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-dominant-root-golden-approx","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio, let $k$ and $n$ be natural numbers with $k \\ge 50$ and $1 < n < \\varphi^{k/2}$, and let $\\alpha > 0$ be a real number with $\\alpha^k = 2\\alpha^{k-1} + \\sum_{i=0}^{k-2} \\alpha^i$. Write $g_k(z) = (z - 1) / ((k + 1) z^2 - 3kz + k - 1)$. Then (i) $|(2\\alpha - 2)\\alpha^n - 2\\varphi^{2n+1}| < 4\\varphi^{2n} / \\varphi^{k/2}$ and (ii) $|g_k(\\alpha) - g_k(\\varphi^2)| < 4k / \\varphi^k$.","title":"Golden-ratio approximations at the dominant root (Lemma 2)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-dominant-root-main-term","kind":"node","lean":"MetaMathlibExt.kPellLucas_dominantRoot_main_term","parent":7068,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-dominant-root-main-term","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio, let $k$ and $n$ be natural numbers with $k \\ge 50$ and $n < \\varphi^{k/2}$, and let $\\alpha > 0$ be a real number with $\\alpha^k = 2\\alpha^{k-1} + \\sum_{i=0}^{k-2} \\alpha^i$. Write $g_k(z) = (z - 1) / ((k + 1) z^2 - 3kz + k - 1)$. Then there is a real number $\\xi$ with $|\\xi| < 1.25 / \\varphi^{k/2}$ such that\u2026","title":"Main term of the dominant root (Lemma 3)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/lt-abs-sub-div-of-max-partial-quotient","kind":"node","lean":"MetaMathlibExt.lt_abs_sub_div_of_max_partialQuotient","parent":7068,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#lt-abs-sub-div-of-max-partial-quotient","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"Let $\\tau$ be an irrational real number with simple continued fraction expansion $[a_0; a_1, a_2, \\ldots]$, so that $a_0 = \\lfloor \\tau \\rfloor$, and let $q_0, q_1, \\ldots$ be the denominators of its convergents. Let $M > 0$ be a real number, let $N$ be the least positive integer with $q_N > M$, and put $a(M) = \\max \\{a_k : 0 \\le k \\le N\\}$. Then","title":"Continued-fraction lower bound for rational approximations"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/abs-log-one-add-lt-and-abs-lt-exp-sub-on","kind":"node","lean":"MetaMathlibExt.abs_log_one_add_lt_and_abs_lt_exp_sub_one","parent":7068,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#abs-log-one-add-lt-and-abs-lt-exp-sub-on","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"Let $a$ and $x$ be real numbers with $0 < a < 1$ and $0 < |x| < a$. Then $|\\log(1 + x)| < \\frac{-\\log(1 - a)}{a} |x|$ and $|x| < \\frac{a}{1 - e^{-a}} |e^x - 1|$.","title":"Linear bounds for the logarithm and the exponential near zero (Lemma 6)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/lt-two-rpow-mul-mul-log-rpow","kind":"node","parent":7068,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#lt-two-rpow-mul-mul-log-rpow","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_state","summary":"If `\u2113 \u2265 1`, `T > (4\u2113\u00b2)^\u2113` and `T > x / (log x)^\u2113`, then `x < 2^\u2113 T (log T)^\u2113`; stated for `x > 1`, where `(log x)^\u2113` is a real power of a positive number.","title":"Lemma 7"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-close-two-pow-exponent-boun","kind":"node","lean":"MetaMathlibExt.kPellLucas_close_two_pow_exponent_bounds","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exponent bounds for "],["math","\\(k\\)"],["text","-Pell\u2013Lucas numbers close to a power of two (Lemma 8)"]],"source":"Exponent bounds for $k$-Pell\u2013Lucas numbers close to a power of two (Lemma 8)"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-close-two-pow-exponent-boun","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"Let $k$, $n$ and $m$ be natural numbers with $k \\ge 2$, $n \\ge 1$, $m \\ge 2$ and $n \\ge k + 1$, and let $Q^{(k)}_n$ be the $n$th $k$-Pell\u2013Lucas number. If $|Q^{(k)}_n - 2^m| < 2^{m/2}$, then $0.69n - 1.69 < m < 1.39n + 3.78$.","title":"Exponent bounds for $k$-Pell\u2013Lucas numbers close to a power of two (Lemma 8)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-close-two-pow-lt-bounds","kind":"node","lean":"MetaMathlibExt.kPellLucas_close_two_pow_lt_bounds","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["text","Absolute bounds for "],["math","\\(k\\)"],["text","-Pell\u2013Lucas numbers close to a power of two (Lemma 9)"]],"source":"Absolute bounds for $k$-Pell\u2013Lucas numbers close to a power of two (Lemma 9)"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-close-two-pow-lt-bounds","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"Let $k$, $n$ and $m$ be natural numbers with $k \\ge 2$, $m \\ge 1$ and $n \\ge k + 1$, and let $Q^{(k)}_n$ be the $n$th $k$-Pell\u2013Lucas number. If $|Q^{(k)}_n - 2^m| < 2^{m/2}$, then $n < 1.7 \\cdot 10^{15} \\, k^4 (\\log k)^3$ and $m < 2.37 \\cdot 10^{15} \\, k^4 (\\log k)^3$.","title":"Absolute bounds for $k$-Pell\u2013Lucas numbers close to a power of two (Lemma 9)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/k-pell-lucas-not-close-two-pow-of-large","kind":"node","lean":"MetaMathlibExt.kPellLucas_not_close_two_pow_of_large_order","parent":7068,"reading_title":{"format":"inline-parts-v1","parts":[["text","No "],["math","\\(k\\)"],["text","-Pell\u2013Lucas number with "],["math","\\(k > 350\\)"],["text"," and index "],["math","\\(n \\ge k + 1\\)"],["text"," is close to a power of two (Lemma 10)"]],"source":"No $k$-Pell\u2013Lucas number with $k > 350$ and index $n \\ge k + 1$ is close to a power of two (Lemma 10)"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two/index.html#k-pell-lucas-not-close-two-pow-of-large","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/k-pell-lucas-numbers-close-to-powers-of-two","status":"can_prove","summary":"No $k$-Pell\u2013Lucas number with $k > 350$ and index $n \\ge k + 1$ is close to a power of two (Lemma 10). For all natural numbers $k$, $n$ and $m$ with $k > 350$ and $n \\ge k + 1$, the $n$th $k$-Pell\u2013Lucas number $Q^{(k)}_n$ satisfies $|Q^{(k)}_n - 2^m| \\ge 2^{m/2}$.","title":"No $k$-Pell\u2013Lucas number with $k > 350$ and index $n \\ge k + 1$ is close to a power of two (Lemma 10)"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits","status":"can_state","title":"Narayana numbers as concatenations of repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/is-concat-two-repdigits-base","kind":"node","lean":"MetaMathlibExt.IsConcatTwoRepdigitsBase","parent":7085,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/index.html#is-concat-two-repdigits-base","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits","status":"fully_proved","summary":"A natural number $n$ is a concatenation of two repdigits in base $b$ if there are digits $a_1, a_2$ with $1 \\le a_1 \\le b - 1$, $0 \\le a_2 \\le b - 1$ and $a_1 \\ne a_2$, and block lengths $l, m \\ge 1$, such that $(b - 1) n = a_1 (b^l - 1) b^m + a_2 (b^m - 1)$. The digit conditions force $b \\ge 2$, and the equation says that the base-$b$ expansion of $n$ is $l$ copies of $a_1$ followed by $m$ copies of $a_2$.","title":"Concatenations of two repdigits in a general base"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/is-concat-two-repdigits-base-examples","kind":"node","lean":"MetaMathlibExt.isConcatTwoRepdigitsBase_examples","parent":7085,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/index.html#is-concat-two-repdigits-base-examples","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits","status":"fully_proved","summary":"The number $4$ is a concatenation of two repdigits in base $2$, since $4 = (100)_2$, and the number $406$ is a concatenation of two repdigits in base $5$, since $406 = (3111)_5$.","title":"Examples of two-repdigit concatenations in bases 2 and 5"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/narayana-cows-sequence-is-concat-two-rep","kind":"node","lean":"MetaMathlibExt.narayanaCowsSequence_isConcatTwoRepdigitsBase_iff","parent":7085,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/index.html#narayana-cows-sequence-is-concat-two-rep","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits","status":"can_prove","summary":"Let $(N_n)$ be Narayana's cows sequence and let $2 \\le b \\le 9$. Then $N_n$ is a concatenation of two repdigits in base $b$ if and only if $(n, b)$ is one of the following $43$ pairs: $n = 4$ and $b = 2$; $n = 5$ and $b = 3$; $n = 6$ and $b \\in \\{2, 4\\}$; $n = 7$ and $b \\in \\{2, 3, 4, 6\\}$; $n = 8$ and $b \\in \\{3, 4, 5, 6, 7, 9\\}$; $n = 9$ and $b \\in \\{4, 5, 6, 7, 8, 9\\}$; $n = 10$ and $b \\in \\{5, 6, 7, 8, 9\\}$;\u2026","title":"Narayana numbers that concatenate two repdigits in bases 2 to 9"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/lt-two-rpow-mul-mul-log-rpow","kind":"node","parent":7085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Guzm\u00e1n S\u00e1nchez\u2013Luca lemma: "],["math","\\(L < 2^r H (\\log H)^r\\)"]],"source":"Guzm\u00e1n S\u00e1nchez\u2013Luca lemma: $L < 2^r H (\\log H)^r$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/index.html#lt-two-rpow-mul-mul-log-rpow","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits","status":"can_state","summary":"Lemma 3 (Guzm\u00e1n S\u00e1nchez and Luca): for real `r \u2265 1` and `H` with `H > (4r\u00b2)^r` and `H > L / (log L)^r`, we have `L < 2^r H (log H)^r`. The hypothesis `H > 0` is implied by `H > (4r\u00b2)^r` and is dropped.","title":"Guzm\u00e1n S\u00e1nchez\u2013Luca lemma: $L < 2^r H (\\log H)^r$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/narayana-cows-sequence-concat-log-bounds","kind":"node","lean":"MetaMathlibExt.narayanaCowsSequence_concat_log_bounds","parent":7085,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits/index.html#narayana-cows-sequence-concat-log-bounds","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/narayana-numbers-as-concatenations-of-repdigits","status":"can_prove","summary":"Let $(N_n)$ be Narayana's cows sequence and let $\\alpha$ be a real number with $\\alpha^3 = \\alpha^2 + 1$ (so $\\alpha \\approx 1.46557$ is the unique real root of $x^3 - x^2 - 1$). Let $b \\ge 2$, $n \\ge 1$ and $l, m \\ge 1$ be natural numbers, and let $a_1, a_2$ be digits with $1 \\le a_1 \\le b - 1$, $0 \\le a_2 \\le b - 1$ and $a_1 \\ne a_2$. If $(b - 1) N_n = a_1 (b^l - 1) b^m + a_2 (b^m - 1)$, that is, the base-$b$\u2026","title":"Logarithmic bounds for Narayana numbers that concatenate two repdigits"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","title":"Perrin numbers that are Perrin squares"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-neg","kind":"node","lean":"MetaMathlibExt.perrinNeg","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-neg","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"fully_proved","summary":"The Perrin sequence $R_k$, given by $R_0 = 3$, $R_1 = 0$, $R_2 = 2$ and $R_{k+3} = R_{k+1} + R_k$, extends uniquely to all integers $k$ by running the recurrence backward. This defines the integers $R_{-n}$ for natural numbers $n$: $R_0 = 3$, $R_{-1} = -1$, $R_{-2} = 1$ and $R_{-(n+3)} = R_{-n} - R_{-(n+2)}$ for every $n \\ge 0$.","title":"Perrin sequence at negative indices"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-neg-bridge","kind":"node","lean":"MetaMathlibExt.perrinNeg_bridge","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-neg-bridge","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"fully_proved","summary":"Write $R_n$ for the Perrin sequence and $R_{-n}$ for its extension to negative indices, for natural numbers $n$. The two definitions give the same value at index $0$, and the Perrin recurrence holds across index $0$: $R_2 = R_0 + R_{-1}$ and $R_1 = R_{-1} + R_{-2}$.","title":"The Perrin sequence and its backward extension fit together at index zero"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-neg-rec","kind":"node","lean":"MetaMathlibExt.perrinNeg_rec","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-neg-rec","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"fully_proved","summary":"Let $R_{-k}$ denote the Perrin sequence extended to negative indices. For every natural number $n$, $R_{-n} = R_{-(n+2)} + R_{-(n+3)}$.","title":"Perrin recurrence at negative indices"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-neg-small-values","kind":"node","lean":"MetaMathlibExt.perrinNeg_small_values","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-neg-small-values","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"fully_proved","summary":"The Perrin sequence $R_k$ extended to negative indices satisfies $R_{-5} = 4$, $R_{-18} = 25$, $R_{-1} = R_{-7} = R_{-11} = R_{-29} = -1$ and $R_{-2} = 1$.","title":"Some values of the Perrin sequence at negative indices"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-sequence-eq-sq-iff","kind":"node","lean":"MetaMathlibExt.perrinSequence_eq_sq_iff","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-sequence-eq-sq-iff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $R_n$ denote the $n$-th term of the Perrin sequence. For positive integers $n$ and $m$, $R_n = R_m^2$ holds if and only if $n = m = 1$ (where $R_1 = 0 = R_1^2$).","title":"Perrin numbers that are squares of Perrin numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-neg-eq-pm-perrin-sequence-sq","kind":"node","lean":"MetaMathlibExt.perrinNeg_eq_pm_perrinSequence_sq","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-neg-eq-pm-perrin-sequence-sq","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $R_k$ denote the Perrin sequence, extended to negative indices. The pairs of positive integers $(n, m)$ with $R_{-n} = R_m^2$ or $R_{-n} = -R_m^2$ are exactly $(5, 2)$, $(5, 4)$, $(18, 5)$ and $(18, 6)$; that is, $R_{-5} = 4 = R_2^2 = R_4^2$ and $R_{-18} = 25 = R_5^2 = R_6^2$.","title":"Negative-index Perrin numbers equal to plus or minus a Perrin square"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-sequence-ne-perrin-neg-sq","kind":"node","lean":"MetaMathlibExt.perrinSequence_ne_perrinNeg_sq","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-sequence-ne-perrin-neg-sq","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $R_n$ denote the $n$-th term of the Perrin sequence and $R_{-m}$ the term at index $-m$ of its backward extension to negative indices. For all positive integers $n$ and $m$, $R_n \\ne R_{-m}^2$.","title":"No positive-index Perrin number is the square of a negative-index one"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-neg-eq-pm-perrin-neg-sq","kind":"node","lean":"MetaMathlibExt.perrinNeg_eq_pm_perrinNeg_sq","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-neg-eq-pm-perrin-neg-sq","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $R_{-k}$ denote the Perrin sequence extended to negative indices. The pairs of positive integers $(n, m)$ with $R_{-n} = R_{-m}^2$ or $R_{-n} = -R_{-m}^2$ are exactly the pairs with $n, m \\in \\{1, 2, 7, 11, 29\\}$, together with $(5, 3)$, $(5, 6)$, $(5, 20)$, $(18, 8)$ and $(18, 16)$.","title":"Negative-index Perrin numbers equal to plus or minus a negative-index Perrin square"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-zero-unique","kind":"node","lean":"MetaMathlibExt.perrin_zero_unique","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-zero-unique","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $R_n$ denote the $n$-th term of the Perrin sequence and $R_{-n}$ the term at index $-n$ of its backward extension to negative indices. For every natural number $n$, $R_n = 0$ if and only if $n = 1$, and $R_{-n} \\ne 0$. Thus $R_1 = 0$ is the only zero of the sequence $(R_k)_{k \\in \\mathbb{Z}}$.","title":"The only zero of the two-sided Perrin sequence"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/perrin-neg-abs-bounds","kind":"node","lean":"MetaMathlibExt.perrinNeg_abs_bounds","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#perrin-neg-abs-bounds","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $\\alpha$ be a real number with $\\alpha^3 = \\alpha + 1$ (the real root of $X^3 - X - 1$), and let $R_{-k}$ denote the Perrin sequence extended to negative indices. For every integer $k \\ge 6$,","title":"Size of the Perrin numbers at negative indices"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/matveev-two-logs-complex-lower-bound","kind":"node","lean":"MetaMathlibExt.matveev_two_logs_complex_lower_bound","parent":7091,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#matveev-two-logs-complex-lower-bound","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $K$ be a number field of degree $D = [K : \\mathbb{Q}]$ with a field embedding $f : K \\to \\mathbb{C}$, let $\\eta_1, \\eta_2 \\in K$ be nonzero, and let $b_1, b_2$ be integers. Let $B$ be a real number with $|b_1| \\le B$ and $|b_2| \\le B$, and let $A_1, A_2$ be real numbers with $A_j \\ge \\max(D\\,h(\\eta_j), |\\log f(\\eta_j)|, 0.16)$ for $j = 1, 2$, where $h$ is the absolute logarithmic Weil height and $\\log$ is the\u2026","title":"Matveev's lower bound for two logarithms (complex case)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/one-div-partial-quotient-bound-lt-abs-su","kind":"node","lean":"MetaMathlibExt.one_div_partialQuotient_bound_lt_abs_sub","parent":7091,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for "],["math","\\(|x - p/q|\\)"],["text"," from bounded partial quotients"]],"source":"Lower bound for $|x - p/q|$ from bounded partial quotients"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#one-div-partial-quotient-bound-lt-abs-su","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $x$ be an irrational real number with continued fraction expansion $[a_0; a_1, a_2, \\dots]$, where $a_0 = \\lfloor x \\rfloor$, and with convergents $p_k / q_k$. Let $M \\ge 1$ and $N \\ge 0$ be integers with $q_N \\le M < q_{N+1}$, and let $a$ be a real number with $a \\ge a_j$ for every $0 \\le j \\le N + 1$. Then","title":"Lower bound for $|x - p/q|$ from bounded partial quotients"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/abs-convergent-num-sub-lt-of-le-den","kind":"node","lean":"MetaMathlibExt.abs_convergent_num_sub_lt_of_le_den","parent":7091,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergents minimize "],["math","\\(|p - qx|\\)"],["text"," up to their denominator"]],"source":"Convergents minimize $|p - qx|$ up to their denominator"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares/index.html#abs-convergent-num-sub-lt-of-le-den","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/perrin-numbers-that-are-perrin-squares","status":"can_prove","summary":"Let $x$ be an irrational real number with continued fraction convergents $p_k / q_k$, where $p_0 / q_0 = \\lfloor x \\rfloor / 1$. If $n > 1$ is a natural number and $p, q$ are integers with $0 < q \\le q_n$ and $p / q \\ne p_n / q_n$, then $|p_n - q_n x| < |p - q x|$.","title":"Convergents minimize $|p - qx|$ up to their denominator"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"can_prove","title":"Recurrence terms that are not sums of two prime powers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/is-sum-of-two-prime-powers","kind":"node","lean":"MetaMathlibExt.IsSumOfTwoPrimePowers","parent":7105,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html#is-sum-of-two-prime-powers","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"fully_proved","summary":"An integer $n$ is a sum of two prime powers if $n = p^a + q^b$ for some primes $p$ and $q$ and some natural numbers $a$ and $b$. Since the exponents may be $0$, the number $1 = p^0$ counts as a prime power here.","title":"Sums of two prime powers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/is-sum-of-two-prime-powers-two","kind":"node","lean":"MetaMathlibExt.isSumOfTwoPrimePowers_two","parent":7105,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text"," is a sum of two prime powers"]],"source":"$2$ is a sum of two prime powers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html#is-sum-of-two-prime-powers-two","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"fully_proved","summary":"The integer $2$ is a sum of two prime powers, since $2 = 2^0 + 2^0$.","title":"$2$ is a sum of two prime powers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/not-is-sum-of-two-prime-powers-of-lt-two","kind":"node","lean":"MetaMathlibExt.not_isSumOfTwoPrimePowers_of_lt_two","parent":7105,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integers below "],["math","\\(2\\)"],["text"," are not sums of two prime powers"]],"source":"Integers below $2$ are not sums of two prime powers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html#not-is-sum-of-two-prime-powers-of-lt-two","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"fully_proved","summary":"If $n$ is an integer with $n < 2$, then $n$ is not a sum of two prime powers, since every prime power $p^a$ is at least $1$.","title":"Integers below $2$ are not sums of two prime powers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/tribonacci-infinite-not-is-sum-of-two-pr","kind":"node","lean":"MetaMathlibExt.tribonacci_infinite_not_isSumOfTwoPrimePowers","parent":7105,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html#tribonacci-infinite-not-is-sum-of-two-pr","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"can_prove","summary":"Let $(T_n)_{n \\ge 0}$ be the Tribonacci sequence, given by $T_0 = 0$, $T_1 = 0$, $T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$. There are infinitely many natural numbers $k$ such that $T_k$ is not a sum of two prime powers.","title":"Infinitely many Tribonacci numbers are not sums of two prime powers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/shifted-padovan-infinite-not-is-sum-of-t","kind":"node","lean":"MetaMathlibExt.shiftedPadovan_infinite_not_isSumOfTwoPrimePowers","parent":7105,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html#shifted-padovan-infinite-not-is-sum-of-t","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be the shifted Padovan sequence, given by $a_0 = 1$, $a_1 = 0$, $a_2 = 1$ and $a_{n+3} = a_{n+1} + a_n$, so that $a_{n+2} = P_n$ is the $n$-th Padovan number, where $P_0 = P_1 = P_2 = 1$ and $P_{n+3} = P_{n+1} + P_n$. There are infinitely many natural numbers $k$ such that $a_k$ is not a sum of two prime powers.","title":"Infinitely many Padovan numbers are not sums of two prime powers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/perrin-infinite-not-is-sum-of-two-prime","kind":"node","lean":"MetaMathlibExt.perrin_infinite_not_isSumOfTwoPrimePowers","parent":7105,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html#perrin-infinite-not-is-sum-of-two-prime","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"can_prove","summary":"Let $(E_n)_{n \\ge 0}$ be the Perrin sequence, given by $E_0 = 3$, $E_1 = 0$, $E_2 = 2$ and $E_{n+3} = E_{n+1} + E_n$. There are infinitely many natural numbers $k$ such that $E_k$ is not a sum of two prime powers.","title":"Infinitely many Perrin numbers are not sums of two prime powers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/leonardo-infinite-not-is-sum-of-two-prim","kind":"node","lean":"MetaMathlibExt.leonardo_infinite_not_isSumOfTwoPrimePowers","parent":7105,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html#leonardo-infinite-not-is-sum-of-two-prim","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"can_prove","summary":"Let $(L_n)_{n \\ge 0}$ be the Leonardo numbers, the generalized Leonardo numbers $L_{1,n}$ with parameter $1$, given by $L_0 = L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n + 1$. There are infinitely many natural numbers $k$ such that $L_k$ is not a sum of two prime powers.","title":"Infinitely many Leonardo numbers are not sums of two prime powers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/infinite-third-order-recurrence-with-inf","kind":"node","lean":"MetaMathlibExt.infinite_thirdOrderRecurrence_with_infinite_not_isSum","parent":7105,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime/index.html#infinite-third-order-recurrence-with-inf","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/recurrence-terms-that-are-not-sums-of-two-prime","status":"can_prove","summary":"There are infinitely many sequences $(s_n)_{n \\ge 0}$ of positive integers with all of the following properties: each value is taken by $s_n$ for only finitely many $n$; there are integers $c_0, c_1, c_2$ with $c_0 \\ne 0$ and $s_{n+3} = c_0 s_n + c_1 s_{n+1} + c_2 s_{n+2}$ for all $n \\ge 0$; there are no integers $d_0, d_1$ with $s_{n+2} = d_0 s_n + d_1 s_{n+1}$ for all $n \\ge 0$; and $s_k$ is not a sum of two\u2026","title":"Third-order recurrences with infinitely many terms not sums of two prime powers"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"can_prove","title":"Repdigit concatenations in balancing numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/balancing-num","kind":"node","lean":"MetaMathlibExt.balancingNum","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#balancing-num","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"The balancing numbers are the integers $B_n$ for $n \\in \\mathbb{N}$ defined by $B_0 = 0$, $B_1 = 1$ and $B_{n+2} = 6 B_{n+1} - B_n$ for all $n$.","title":"Balancing numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/balancing-num-two","kind":"node","lean":"MetaMathlibExt.balancingNum_two","parent":7114,"reading_title":{"format":"inline-parts-v1","parts":[["text","The balancing number "],["math","\\(B_2\\)"],["text"," is "],["math","\\(6\\)"]],"source":"The balancing number $B_2$ is $6$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#balancing-num-two","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"The balancing number $B_2$ equals $6$.","title":"The balancing number $B_2$ is $6$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/balancing-num-three","kind":"node","lean":"MetaMathlibExt.balancingNum_three","parent":7114,"reading_title":{"format":"inline-parts-v1","parts":[["text","The balancing number "],["math","\\(B_3\\)"],["text"," is "],["math","\\(35\\)"]],"source":"The balancing number $B_3$ is $35$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#balancing-num-three","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"The balancing number $B_3$ equals $35$.","title":"The balancing number $B_3$ is $35$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/lucas-balancing-num","kind":"node","lean":"MetaMathlibExt.lucasBalancingNum","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#lucas-balancing-num","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"The sequence of integers $(C_n)_{n \\ge 0}$ defined by $C_0 = 3$, $C_1 = 1$ and $C_{n+2} = 6 C_{n+1} - C_n$ for all $n \\ge 0$.","title":"Lucas-balancing numbers (shifted indexing)"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/lucas-balancing-num-two","kind":"node","lean":"MetaMathlibExt.lucasBalancingNum_two","parent":7114,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lucas-balancing number "],["math","\\(C_2\\)"],["text"," is "],["math","\\(3\\)"]],"source":"The Lucas-balancing number $C_2$ is $3$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#lucas-balancing-num-two","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"For the Lucas-balancing numbers defined by $C_0 = 3$, $C_1 = 1$ and $C_{n+2} = 6C_{n+1} - C_n$, we have $C_2 = 3$.","title":"The Lucas-balancing number $C_2$ is $3$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/lucas-balancing-num-three","kind":"node","lean":"MetaMathlibExt.lucasBalancingNum_three","parent":7114,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lucas-balancing number "],["math","\\(C_3\\)"],["text"," is "],["math","\\(17\\)"]],"source":"The Lucas-balancing number $C_3$ is $17$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#lucas-balancing-num-three","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"For the Lucas-balancing numbers defined by $C_0 = 3$, $C_1 = 1$ and $C_{n+2} = 6C_{n+1} - C_n$, we have $C_3 = 17$.","title":"The Lucas-balancing number $C_3$ is $17$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-repdigit","kind":"node","lean":"MetaMathlibExt.IsRepdigit","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-repdigit","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"A natural number $n$ is a repdigit if there are natural numbers $d$ and $m$ with $1 \\le d \\le 9$ and $m \\ge 1$ such that $9n = d(10^m - 1)$. Equivalently, $n$ is the positive integer whose decimal expansion consists of $m$ copies of the digit $d$.","title":"Decimal repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-repdigit-eleven","kind":"node","lean":"MetaMathlibExt.isRepdigit_eleven","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-repdigit-eleven","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"The number $11$ is a repdigit.","title":"Eleven is a repdigit"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-repdigit-of-le","kind":"node","lean":"MetaMathlibExt.isRepdigit_of_le","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-repdigit-of-le","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"Every natural number $d$ with $1 \\le d \\le 9$ is a repdigit.","title":"Nonzero single digits are repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-concat-two-repdigits","kind":"node","lean":"MetaMathlibExt.IsConcatTwoRepdigits","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-concat-two-repdigits","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"A natural number $n$ is a concatenation of two repdigits if there are digits $d_1, d_2$ with $1 \\le d_1 \\le 9$, $0 \\le d_2 \\le 9$ and $d_1 \\ne d_2$, and block lengths $m_1, m_2 \\ge 1$, such that the decimal expansion of $n$ is $m_1$ copies of $d_1$ followed by $m_2$ copies of $d_2$; that is, $9n = d_1 (10^{m_1} - 1) 10^{m_2} + d_2 (10^{m_2} - 1)$. Equivalently, the decimal expansion of $n$ has exactly two runs of\u2026","title":"Concatenations of two decimal repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-concat-three-repdigits","kind":"node","lean":"MetaMathlibExt.IsConcatThreeRepdigits","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-concat-three-repdigits","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"A natural number $n$ is a concatenation of three repdigits if there are digits $d_1, d_2, d_3$ with $1 \\le d_1 \\le 9$, $0 \\le d_2 \\le 9$, $0 \\le d_3 \\le 9$, $d_1 \\ne d_2$ and $d_2 \\ne d_3$, and block lengths $m_1, m_2, m_3 \\ge 1$, such that the decimal expansion of $n$ is $m_1$ copies of $d_1$, then $m_2$ copies of $d_2$, then $m_3$ copies of $d_3$; that is,","title":"Concatenations of three decimal repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-concat-two-repdigits-thirty-five","kind":"node","lean":"MetaMathlibExt.isConcatTwoRepdigits_thirtyFive","parent":7114,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(35\\)"],["text"," is a concatenation of two repdigits"]],"source":"$35$ is a concatenation of two repdigits"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-concat-two-repdigits-thirty-five","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"The number $35$, whose decimal expansion consists of the one-digit blocks $3$ and $5$, is a concatenation of two repdigits.","title":"$35$ is a concatenation of two repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-concat-three-repdigits-two-zero-four","kind":"node","lean":"MetaMathlibExt.isConcatThreeRepdigits_twoZeroFour","parent":7114,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(204\\)"],["text"," is a concatenation of three repdigits"]],"source":"$204$ is a concatenation of three repdigits"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-concat-three-repdigits-two-zero-four","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"The number $204$, whose decimal expansion consists of the one-digit blocks $2$, $0$ and $4$, is a concatenation of three repdigits.","title":"$204$ is a concatenation of three repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-repdigit-diff","kind":"node","lean":"MetaMathlibExt.IsRepdigitDiff","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-repdigit-diff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"A natural number $n$ is a difference of two repdigits if there are repdigits $a$ and $b$ with $a = n + b$, that is, $n = a - b$.","title":"Differences of two repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/is-repdigit-diff-six","kind":"node","lean":"MetaMathlibExt.isRepdigitDiff_six","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#is-repdigit-diff-six","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"fully_proved","summary":"The number $6$ is a difference of two repdigits: $6 = 11 - 5$, where $11$ and $5$ are repdigits.","title":"Six is a difference of two repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/balancing-num-concat-two-repdigits","kind":"node","lean":"MetaMathlibExt.balancingNum_concatTwoRepdigits","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#balancing-num-concat-two-repdigits","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"can_prove","summary":"For every natural number $k$, if the balancing number $B_k$ is a concatenation of two repdigits, then $B_k = 35$.","title":"Balancing numbers that are concatenations of two repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/balancing-num-concat-three-repdigits","kind":"node","lean":"MetaMathlibExt.balancingNum_concatThreeRepdigits","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#balancing-num-concat-three-repdigits","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"can_prove","summary":"For every natural number $k$, if the balancing number $B_k$ is a concatenation of three repdigits, then $B_k = 204$ or $B_k = 1189$.","title":"Balancing numbers that are concatenations of three repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/lucas-balancing-num-concat-two-repdigits","kind":"node","lean":"MetaMathlibExt.lucasBalancingNum_concatTwoRepdigits","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#lucas-balancing-num-concat-two-repdigits","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"can_prove","summary":"For every natural number $k$, if the decimal expansion of the Lucas-balancing number $C_k$ consists of exactly two runs of equal digits, that is, $C_k$ is a block of one repeated digit followed by a block of a different repeated digit, then $C_k = 17$ or $C_k = 577$.","title":"Lucas-balancing numbers that are concatenations of two repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/lucas-balancing-num-concat-three-repdigi","kind":"node","lean":"MetaMathlibExt.lucasBalancingNum_concatThreeRepdigits","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#lucas-balancing-num-concat-three-repdigi","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"can_prove","summary":"For every natural number $k$, if the decimal expansion of the Lucas-balancing number $C_k$ consists of exactly three runs of equal digits, that is, $C_k$ is three blocks of repeated digits in which adjacent blocks have different digits, then $C_k = 3363$.","title":"Lucas-balancing numbers that are concatenations of three repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/balancing-num-repdigit-diff","kind":"node","lean":"MetaMathlibExt.balancingNum_repdigitDiff","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#balancing-num-repdigit-diff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number. If the balancing number $B_k$ is a difference of two repdigits, then $B_k = 6$ or $B_k = 35$.","title":"Balancing numbers that are differences of two repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/lucas-balancing-num-repdigit-diff","kind":"node","lean":"MetaMathlibExt.lucasBalancingNum_repdigitDiff","parent":7114,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers/index.html#lucas-balancing-num-repdigit-diff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-concatenations-in-balancing-numbers","status":"can_prove","summary":"For every natural number $k \\ge 2$, if the Lucas-balancing number $C_k$ is a difference of two repdigits, then $C_k = 3$ or $C_k = 17$.","title":"Lucas-balancing numbers that are differences of two repdigits"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers","status":"can_prove","title":"Repdigit products of k-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers/k-fibonacci-mul-eq-repdigit-iff","kind":"node","lean":"MetaMathlibExt.kFibonacci_mul_eq_repdigit_iff","parent":7136,"reading_title":{"format":"inline-parts-v1","parts":[["text","Products of two "],["math","\\(k\\)"],["text","-Fibonacci numbers that are repdigits"]],"source":"Products of two $k$-Fibonacci numbers that are repdigits"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers/index.html#k-fibonacci-mul-eq-repdigit-iff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 3$ and let $(F_n)_{n \\ge 2-k}$ be the $k$-generalized Fibonacci sequence, given by $F_{2-k} = \\dots = F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2} + \\dots + F_{n-k}$ for $n \\ge 2$. Let $a, \\ell, m, n$ be natural numbers with $3 \\le m \\le n$, $\\ell \\ge 2$ and $1 \\le a \\le 9$. Then $F_n F_m = a (10^\\ell - 1)/9$ (the $\\ell$-digit repdigit with digit $a$) if and only if\u2026","title":"Products of two $k$-Fibonacci numbers that are repdigits"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers/k-fibonacci-repdigit-index-lt","kind":"node","lean":"MetaMathlibExt.kFibonacci_repdigit_index_lt","parent":7136,"reading_title":{"format":"inline-parts-v1","parts":[["text","Index bound for repdigit products of "],["math","\\(k\\)"],["text","-Fibonacci numbers"]],"source":"Index bound for repdigit products of $k$-Fibonacci numbers"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers/index.html#k-fibonacci-repdigit-index-lt","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers","status":"can_prove","summary":"Let $k \\ge 3$ and let $(F_n)_{n \\ge 2-k}$ be the $k$-generalized Fibonacci sequence, given by $F_{2-k} = \\dots = F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2} + \\dots + F_{n-k}$ for $n \\ge 2$. Let $a, \\ell, m, n$ be natural numbers with $3 \\le m \\le n$, $\\ell \\ge 2$, $1 \\le a \\le 9$ and $n \\ge k + 2$. If $F_n F_m = a (10^\\ell - 1)/9$, then $n < 1.64 \\cdot 10^{29} k^8 (\\log k)^5$, where $\\log$ is the natural\u2026","title":"Index bound for repdigit products of $k$-Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers/one-div-partial-quotient-bound-mul-lt-ab","kind":"node","lean":"MetaMathlibExt.one_div_partialQuotient_bound_mul_lt_abs_mul_sub","parent":7136,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers/index.html#one-div-partial-quotient-bound-mul-lt-ab","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/repdigit-products-of-k-fibonacci-numbers","status":"can_prove","summary":"Let $\\gamma$ be an irrational real number with continued fraction $[a_0; a_1, a_2, \\dots]$, so that $a_0 = \\lfloor \\gamma \\rfloor$, and let $q_0 = 1, q_1, q_2, \\dots$ be the denominators of its convergents. Let $M \\ge 1$ and $N \\ge 0$ be integers with $q_N \\le M < q_{N+1}$, and let $A$ be a real number with $a_i \\le A$ for $0 \\le i \\le N + 1$. Then $|x \\gamma - y| > 1/((A + 2) x)$ for all integers $x, y$ with\u2026","title":"Partial-quotient lower bound for rational approximation"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers","status":"can_prove","title":"Squares among Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/pell-lucas-sequence-is-square-iff","kind":"node","lean":"MathlibExt.NumberTheory.PellLucasDiophantineWanted.pellLucasSequence_isSquare_iff","parent":7140,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/index.html#pell-lucas-sequence-is-square-iff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers","status":"can_prove","summary":"For every natural number $n$, the Pell\u2013Lucas number $Q_n$ is a perfect square if and only if $n = 0$ or $n = 1$; thus the only square terms of the sequence are $Q_0 = Q_1 = 1$.","title":"Square Pell\u2013Lucas numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/gcd-pell-number-pell-lucas-sequence","kind":"node","lean":"MathlibExt.NumberTheory.PellLucasDiophantineWanted.gcd_pellNumber_pellLucasSequence","parent":7140,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/index.html#gcd-pell-number-pell-lucas-sequence","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers","status":"can_prove","summary":"For every natural number $n$, the Pell number $P_n$ and the Pell\u2013Lucas number $Q_n$ satisfy $\\gcd(P_n, Q_n) = 1$.","title":"Pell and Pell\u2013Lucas numbers are coprime"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/gcd-pell-lucas-sequence-succ","kind":"node","lean":"MathlibExt.NumberTheory.PellLucasDiophantineWanted.gcd_pellLucasSequence_succ","parent":7140,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/index.html#gcd-pell-lucas-sequence-succ","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers","status":"can_prove","summary":"For every natural number $n$, the Pell\u2013Lucas numbers $Q_n$ and $Q_{n+1}$ satisfy $\\gcd(Q_n, Q_{n+1}) = 1$.","title":"Consecutive Pell\u2013Lucas numbers are coprime"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/pow-add-pow-ne-sq","kind":"node","lean":"MathlibExt.NumberTheory.PellLucasDiophantineWanted.pow_add_pow_ne_sq","parent":7140,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of two coprime positive "],["math","\\(n\\)"],["text","-th powers are not squares for "],["math","\\(n \\ge 4\\)"]],"source":"Sums of two coprime positive $n$-th powers are not squares for $n \\ge 4$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/index.html#pow-add-pow-ne-sq","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers","status":"can_prove","summary":"Let $n \\ge 4$ be a natural number and let $x$ and $y$ be coprime positive integers. Then $x^n + y^n$ is not a perfect square: there is no natural number $z$ with $x^n + y^n = z^2$.","title":"Sums of two coprime positive $n$-th powers are not squares for $n \\ge 4$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/pell-lucas-sequence-pow-add-pow-eq-sq","kind":"node","lean":"MathlibExt.NumberTheory.PellLucasDiophantineWanted.pellLucasSequence_pow_add_pow_eq_sq","parent":7140,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers/index.html#pell-lucas-sequence-pow-add-pow-eq-sq","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-among-pell-lucas-numbers","status":"can_prove","summary":"Let $Q_n$ denote the Pell\u2013Lucas numbers. For natural numbers $n$, $x$ and $y$, the equation $Q_n^x + Q_{n+1}^x = y^2$ holds if and only if $n = 1$, $x = 1$ and $y = 2$; that is, its only solution in natural numbers is $Q_1 + Q_2 = 1 + 3 = 2^2$.","title":"Sums of powers of consecutive Pell\u2013Lucas numbers that are squares"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences","status":"fully_proved","title":"Squares in Fibonacci and Lucas sequences"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/cohn-twice-square-lucas","kind":"node","lean":"MetaMathlibExt.cohn_twice_square_lucas","parent":7146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cohn's theorem on Lucas numbers of the form "],["math","\\(2x^2\\)"]],"source":"Cohn's theorem on Lucas numbers of the form $2x^2$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/index.html#cohn-twice-square-lucas","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences","status":"fully_proved","summary":"Let $L_n$ denote the Lucas numbers, with $L_0 = 2$, $L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n$. If $n$ and $x$ are natural numbers with $n \\ge 1$ and $L_n = 2x^2$, then $n = 6$.","title":"Cohn's theorem on Lucas numbers of the form $2x^2$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/cohn-square-lucas","kind":"node","lean":"MetaMathlibExt.cohn_square_lucas","parent":7146,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/index.html#cohn-square-lucas","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences","status":"fully_proved","summary":"Let $L_0 = 2$, $L_1 = 1$ and $L_{n+2} = L_{n+1} + L_n$ be the Lucas numbers. If $n \\ge 1$ and $L_n = x^2$ for some natural number $x$, then $n = 1$ or $n = 3$.","title":"Cohn's theorem on square Lucas numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/cohn-square-fibonacci","kind":"node","lean":"MetaMathlibExt.cohn_square_fibonacci","parent":7146,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/index.html#cohn-square-fibonacci","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences","status":"fully_proved","summary":"Let $F_n$ denote the Fibonacci numbers, with $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$. If $n$ and $x$ are natural numbers with $n \\ge 1$ and $F_n = x^2$, then $n \\in \\{1, 2, 12\\}$.","title":"Cohn's theorem on square Fibonacci numbers"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/cohn-twice-square-fibonacci","kind":"node","lean":"MetaMathlibExt.cohn_twice_square_fibonacci","parent":7146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cohn's theorem on Fibonacci numbers of the form "],["math","\\(2x^2\\)"]],"source":"Cohn's theorem on Fibonacci numbers of the form $2x^2$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences/index.html#cohn-twice-square-fibonacci","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/squares-in-fibonacci-and-lucas-sequences","status":"fully_proved","summary":"Let $F_n$ denote the Fibonacci numbers, with $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$. If $n$ and $x$ are natural numbers with $n \\ge 1$ and $F_n = 2x^2$, then $n \\in \\{3, 6\\}$.","title":"Cohn's theorem on Fibonacci numbers of the form $2x^2$"},{"data_url":"scopes/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas.json","id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","kind":"scope","navigation":"module","parent":7045,"result":false,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","title":"Weighted power sums and perfect powers in Lucas sequences"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/lucas-u-root-pow","kind":"node","lean":"MetaMathlibExt.lucasU_root_pow","parent":7151,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#lucas-u-root-pow","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ and $y \\neq 0$ be integers and let $(U_n)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = x U_{n+1} - y U_n$. If $\\alpha$ is a complex number with $\\alpha^2 - x\\alpha + y = 0$, then $\\alpha^n = \\alpha U_n - y U_{n-1}$ for every $n \\geq 1$.","title":"Powers of a root of the Lucas characteristic polynomial"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/lucas-v-eq-lucas-u-succ-sub","kind":"node","lean":"MetaMathlibExt.lucasV_eq_lucasU_succ_sub","parent":7151,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#lucas-v-eq-lucas-u-succ-sub","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ and $y \\neq 0$ be integers and let $(U_n)$ and $(V_n)$ be the Lucas sequences with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = x$ and $W_{n+2} = x W_{n+1} - y W_n$ for $W = U, V$. Then $V_n = U_{n+1} - y U_{n-1}$ for every $n \\geq 1$.","title":"The companion Lucas sequence in terms of the fundamental one"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/lucas-u-add-eq-mul-lucas-v-sub","kind":"node","lean":"MetaMathlibExt.lucasU_add_eq_mul_lucasV_sub","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Addition formula for the Lucas sequence "],["math","\\(U\\)"]],"source":"Addition formula for the Lucas sequence $U$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#lucas-u-add-eq-mul-lucas-v-sub","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ and $y \\neq 0$ be integers and let $(U_n)$ and $(V_n)$ be the Lucas sequences with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = x$ and $W_{n+2} = x W_{n+1} - y W_n$ for $W = U, V$. Then for all natural numbers $n \\leq m$, $U_{m+n} = U_m V_n - y^n U_{m-n}$.","title":"Addition formula for the Lucas sequence $U$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/lucas-v-sq-eq","kind":"node","lean":"MetaMathlibExt.lucasV_sq_eq","parent":7151,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#lucas-v-sq-eq","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ and $y \\neq 0$ be integers and let $(U_n)$ and $(V_n)$ be the Lucas sequences with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = x$ and $W_{n+2} = x W_{n+1} - y W_n$ for $W = U, V$. Then for every natural number $n$, $V_n^2 = (x^2 - 4y) U_n^2 + 4 y^n$.","title":"Square of the companion Lucas sequence"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/lucas-u-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.lucasU_eq_sum_choose","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial-sum formula for the Lucas sequence "],["math","\\(U\\)"]],"source":"Binomial-sum formula for the Lucas sequence $U$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#lucas-u-eq-sum-choose","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ and $y \\neq 0$ be integers and let $(U_n)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = x U_{n+1} - y U_n$. Then for every $r \\geq 1$,","title":"Binomial-sum formula for the Lucas sequence $U$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/lucas-u-lucas-v-alpha-bounds","kind":"node","lean":"MetaMathlibExt.lucasU_lucasV_alpha_bounds","parent":7151,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#lucas-u-lucas-v-alpha-bounds","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ and $y \\neq 0$ be integers with $x \\geq 1$, $|y + 1| \\leq x$ and $D = x^2 - 4y > 0$, let $(U_n)$ and $(V_n)$ be the Lucas sequences with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$, $V_0 = 2$, $V_1 = x$ and $W_{n+2} = x W_{n+1} - y W_n$ for $W = U, V$, and put $\\alpha = (x + \\sqrt{D})/2$. Then for every $n \\geq 2$ we have $x \\alpha^{n-2} \\leq U_n$; if $y \\leq 0$, then $U_n \\leq \\alpha^{n-1}$ and\u2026","title":"Bounds on Lucas sequences by powers of the dominant root"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/lucas-u-ratio-lower-bound","kind":"node","lean":"MetaMathlibExt.lucasU_ratio_lower_bound","parent":7151,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#lucas-u-ratio-lower-bound","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ and $y \\neq 0$ be integers with $x \\geq 1$ and $|y + 1| \\leq x$, and let $(U_n)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = x U_{n+1} - y U_n$. If $y < 0$, then $U_n / U_{n-1} \\geq x - xy/(x^2 - y)$ for every $n > 2$. If $y > 0$, then $U_n / U_{n-1} \\geq (x + \\sqrt{x^2 - 4y})/2$ for every $n \\geq 2$.","title":"Lower bounds for ratios of consecutive Lucas terms"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/weighted-power-sum-log-bounds-of-neg","kind":"node","lean":"MetaMathlibExt.weighted_power_sum_log_bounds_of_neg","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logarithmic bounds for weighted Lucas power sums when "],["math","\\(y < 0\\)"]],"source":"Logarithmic bounds for weighted Lucas power sums when $y < 0$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#weighted-power-sum-log-bounds-of-neg","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x \\ge 1$ and $y < 0$ be integers with $|y + 1| \\le x$, let $(U_m)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{m+2} = x U_{m+1} - y U_m$, and let $\\alpha = (x + \\sqrt{x^2 - 4y})/2$. Let $k$ be a natural number and $n$, $p$, $q$ positive integers with $\\sum_{j=1}^k j U_j^p = U_n^q$. If $k \\ge 2$, then $\\log_\\alpha k + p \\log_\\alpha x + (k - 2)p < (n - 1)q$. If\u2026","title":"Logarithmic bounds for weighted Lucas power sums when $y < 0$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/weighted-power-sum-log-bounds-of-pos","kind":"node","lean":"MetaMathlibExt.weighted_power_sum_log_bounds_of_pos","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logarithmic bounds for weighted Lucas power sums when "],["math","\\(y > 0\\)"]],"source":"Logarithmic bounds for weighted Lucas power sums when $y > 0$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#weighted-power-sum-log-bounds-of-pos","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ and $y > 0$ be integers with $|y + 1| \\le x$ and $D = x^2 - 4y > 0$, let $(U_m)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{m+2} = x U_{m+1} - y U_m$, and let $\\alpha = (x + \\sqrt{D})/2$. Let $k$ be a natural number and $n$, $p$, $q$ positive integers with $\\sum_{j=1}^k j U_j^p = U_n^q$. If $k \\ge 2$, then\u2026","title":"Logarithmic bounds for weighted Lucas power sums when $y > 0$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/padic-val-int-lucas-u-of-dvd-discr","kind":"node","lean":"MetaMathlibExt.padicValInt_lucasU_of_dvd_discr","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Valuation of "],["math","\\(U_n\\)"],["text"," at a prime dividing the discriminant"]],"source":"Valuation of $U_n$ at a prime dividing the discriminant"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#padic-val-int-lucas-u-of-dvd-discr","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x \\ge 1$ and $y \\ne 0$ be coprime integers with $|y + 1| \\le x$, let $(U_m)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{m+2} = x U_{m+1} - y U_m$, and let $\\ell$ be a prime dividing $D = x^2 - 4y$. Write $\\nu_\\ell$ for the $\\ell$-adic valuation of an integer, with $\\nu_\\ell(0) = 0$. Then for every $n \\ge 1$, $\\nu_\\ell(U_n) = \\nu_\\ell(n) + \\nu_\\ell(U_\\ell) - 1$ if\u2026","title":"Valuation of $U_n$ at a prime dividing the discriminant"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/lucas-v-quadratic-eq-pm-discr-zpow","kind":"node","lean":"MetaMathlibExt.lucasV_quadratic_eq_pm_discr_zpow","parent":7151,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#lucas-v-quadratic-eq-pm-discr-zpow","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ be a positive integer, let $(V_n)$ be the companion Lucas sequence with parameters $(x, -1)$, given by $V_0 = 2$, $V_1 = x$ and $V_{n+2} = x V_{n+1} + V_n$, and let $D = x^2 + 4$. Let $p, q, k$ be natural numbers with $p$ odd and $q, k$ positive. If","title":"A quadratic equation in the companion Lucas sequence"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/one-add-two-mul-pow-eq-lucas-u-pow-iff","kind":"node","lean":"MetaMathlibExt.one_add_two_mul_pow_eq_lucasU_pow_iff","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(1 + 2x^p = U_n^q\\)"],["text"," for Lucas sequences with "],["math","\\(y = \\pm 1\\)"]],"source":"Solutions of $1 + 2x^p = U_n^q$ for Lucas sequences with $y = \\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#one-add-two-mul-pow-eq-lucas-u-pow-iff","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ be a positive integer, let $y \\in \\{1, -1\\}$, and let $(U_m)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{m+2} = x U_{m+1} - y U_m$. For positive integers $n$, $p$, $q$, the equation $1 + 2x^p = U_n^q$ holds if and only if $(n, p, q, x, y)$ is one of $(4, p, 1, 1, -1)$, $(1 + 2^{p+1}, p, 1, 2, 1)$, $(3, 2, 2, 2, 1)$, $(3, 1, 1, 2, -1)$ and $(5, 3, 1, 3, 1)$, where\u2026","title":"Solutions of $1 + 2x^p = U_n^q$ for Lucas sequences with $y = \\pm 1$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/weighted-power-sum-ne-lucas-u-pow","kind":"node","lean":"MetaMathlibExt.weighted_power_sum_ne_lucasU_pow","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","No solutions of "],["math","\\(\\sum_{j=1}^k j U_j^p = U_n^q\\)"],["text"," when "],["math","\\(y = -1\\)"],["text"," and "],["math","\\(x \\ge 2\\)"],["text"," or "],["math","\\(y = 1\\)"],["text"," and "],["math","\\(x \\ge 3\\)"],["text",", with "],["math","\\(k \\ge 3\\)"],["text"," and exponents at most "],["math","\\(11\\)"]],"source":"No solutions of $\\sum_{j=1}^k j U_j^p = U_n^q$ when $y = -1$ and $x \\ge 2$ or $y = 1$ and $x \\ge 3$, with $k \\ge 3$ and exponents at most $11$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#weighted-power-sum-ne-lucas-u-pow","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"No solutions of $\\sum_{j=1}^k j U_j^p = U_n^q$ when $y = -1$ and $x \\ge 2$ or $y = 1$ and $x \\ge 3$, with $k \\ge 3$ and exponents at most $11$. Let $y \\in \\{1, -1\\}$, let $x$ be an integer with $x \\ge 2$ if $y = -1$ and $x \\ge 3$ if $y = 1$, and let $(U_m)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{m+2} = x U_{m+1} - y U_m$. If $n$, $p$, $q$ are positive integers with\u2026","title":"No solutions of $\\sum_{j=1}^k j U_j^p = U_n^q$ when $y = -1$ and $x \\ge 2$ or $y = 1$ and $x \\ge 3$, with $k \\ge 3$ and exponents at most $11$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/exists-remainders-weighted-power-sum","kind":"node","lean":"MetaMathlibExt.exists_remainders_weighted_power_sum","parent":7151,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#exists-remainders-weighted-power-sum","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ be a positive integer, let $y \\in \\{1, -1\\}$, and let $(U_n)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = x U_{n+1} - y U_n$. Suppose $D = x^2 - 4y > 0$ and put $\\alpha = (x + \\sqrt{D})/2$. Let $n, p, q$ be positive integers and $k \\geq 3$ an integer with","title":"Remainder form of the weighted power-sum equation for Lucas sequences"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/le-of-weighted-power-sum-eq","kind":"node","lean":"MetaMathlibExt.le_of_weighted_power_sum_eq","parent":7151,"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#le-of-weighted-power-sum-eq","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ be a positive integer, let $y \\in \\{1, -1\\}$, and let $(U_n)$ be the Lucas sequence with parameters $(x, y)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{n+2} = x U_{n+1} - y U_n$. If $n, p, q$ are positive integers and $k \\geq 3$ is an integer with","title":"Bound on the parameter in the weighted power-sum equation for Lucas sequences"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/remainder-bound-of-neg","kind":"node","lean":"MetaMathlibExt.remainder_bound_of_neg","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Remainder bound for weighted Lucas power sums when "],["math","\\(y = -1\\)"]],"source":"Remainder bound for weighted Lucas power sums when $y = -1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#remainder-bound-of-neg","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ be a positive integer, let $(U_m)$ be the Lucas sequence with parameters $(x, -1)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{m+2} = x U_{m+1} + U_m$, and let $D = x^2 + 4$ and $\\alpha = (x + \\sqrt{D})/2$. If positive integers $n$, $p$, $q$ and an integer $k \\ge 3$ satisfy $\\sum_{j=1}^k j U_j^p = U_n^q$, then the remainder $R_2 = U_n^q - \\alpha^{nq}/D^{q/2}$ satisfies\u2026","title":"Remainder bound for weighted Lucas power sums when $y = -1$"},{"id":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/remainder-bound-of-pos","kind":"node","lean":"MetaMathlibExt.remainder_bound_of_pos","parent":7151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Remainder bound for weighted Lucas power sums when "],["math","\\(y = 1\\)"]],"source":"Remainder bound for weighted Lucas power sums when $y = 1$"},"result":true,"route":"../roadmap/NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas/index.html#remainder-bound-of-pos","scope":"NumberTheory/linear-recurrences/diophantine-recurrences/weighted-power-sums-and-perfect-powers-in-lucas","status":"can_prove","summary":"Let $x$ be a positive integer with $D = x^2 - 4 > 0$, let $(U_m)$ be the Lucas sequence with parameters $(x, 1)$, given by $U_0 = 0$, $U_1 = 1$ and $U_{m+2} = x U_{m+1} - U_m$, let $V_2 = x^2 - 2$ be the term of index $2$ of the companion Lucas sequence, and let $\\alpha = (x + \\sqrt{D})/2$. If positive integers $n$, $p$, $q$ and an integer $k \\ge 3$ satisfy $\\sum_{j=1}^k j U_j^p = U_n^q$, then the remainder\u2026","title":"Remainder bound for weighted Lucas power sums when $y = 1$"},{"data_url":"scopes/NumberTheory/primes-primality.json","id":"NumberTheory/primes-primality","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/primes-primality/index.html","scope":"NumberTheory/primes-primality","status":"can_prove","summary":"Individual primes and prime tests: special prime families, primality criteria, pseudoprimes, Carmichael numbers, Fermat and Mersenne numbers.","title":"Primes and primality"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes","kind":"scope","parent":7169,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes","status":"can_prove","summary":"Primality criteria (Proth, Pocklington, Lucas\u2013Lehmer), Fermat, Lucas, Frobenius and strong pseudoprimes, Carmichael and Korselt numbers.","title":"Primality tests and pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/infinitely-many-primes-fibonacci-pseudop","kind":"node","lean":"MetaMathlibExt.infinitely_many_primes_fibonacci_pseudoprime","parent":7170,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/index.html#infinitely-many-primes-fibonacci-pseudop","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes","status":"fully_proved","summary":"Let $F_m$ denote the $m$-th Fibonacci number. There are infinitely many primes $p$ such that $n = F_{2p}$ satisfies $n > 1$, $n$ is not prime, and $n$ divides $F_{n - \\left(\\frac{5}{n}\\right)}$, where $\\left(\\frac{5}{n}\\right)$ is the Jacobi symbol. In other words, $F_{2p}$ is a Fibonacci pseudoprime: a composite number $n$ that divides $F_{n - \\left(\\frac{5}{n}\\right)}$.","title":"Infinitely many Fibonacci pseudoprimes at twice-prime indices"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/pocklington-prime-power-criterion","kind":"node","lean":"MetaMathlibExt.pocklington_prime_power_criterion","parent":7170,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pocklington's criterion for "],["math","\\(N = Kp^e + 1\\)"]],"source":"Pocklington's criterion for $N = Kp^e + 1$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/index.html#pocklington-prime-power-criterion","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes","status":"fully_proved","summary":"Let $N$, $K$ and $e$ be natural numbers and let $p$ be a prime, with $0 < K < p^e$ and $N = Kp^e + 1$. If some natural number $a$ satisfies $a^{N-1} \\equiv 1 \\pmod{N}$ and $\\gcd\\big(a^{(N-1)/p} - 1, N\\big) = 1$, where $(N-1)/p$ is the quotient of $N - 1$ by $p$ rounded down and $a^{(N-1)/p} - 1$ is truncated at $0$, then $N$ is prime.","title":"Pocklington's criterion for $N = Kp^e + 1$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/proth-theorem","kind":"node","lean":"MetaMathlibExt.proth_theorem","parent":7170,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/index.html#proth-theorem","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes","status":"fully_proved","summary":"Let $h$ and $k$ be natural numbers with $h$ odd and $h < 2^k$, and let $N = h \\cdot 2^k + 1$. Then $N$ is prime if and only if there is a residue class $a$ modulo $N$ with $a^{(N-1)/2} = -1$ in $\\mathbb{Z}/N\\mathbb{Z}$.","title":"Proth's theorem"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","title":"Absolute Lucas pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/jacobi-sym-prime-add-mul-eq","kind":"node","lean":"MetaMathlibExt.jacobiSym_prime_add_mul_eq","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Jacobi symbols of a discriminant "],["math","\\(D\\)"],["text"," agree at odd primes congruent modulo "],["math","\\(D\\)"]],"source":"Jacobi symbols of a discriminant $D$ agree at odd primes congruent modulo $D$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#jacobi-sym-prime-add-mul-eq","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $p$ and $p'$ be odd primes and let $D$ be an integer with $D \\equiv 0$ or $1 \\pmod 4$ that is coprime to $p$. If $p' = p + D x$ for some integer $x$, then the Jacobi symbols agree: $\\left(\\frac{D}{p'}\\right) = \\left(\\frac{D}{p}\\right)$.","title":"Jacobi symbols of a discriminant $D$ agree at odd primes congruent modulo $D$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/mul-dvd-prod-sub-sign-iff","kind":"node","lean":"MetaMathlibExt.mul_dvd_prod_sub_sign_iff","parent":7174,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#mul-dvd-prod-sub-sign-iff","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $\\varepsilon_1, \\varepsilon_2, \\varepsilon_3 \\in \\{1, -1\\}$, let $c_1, c_2, c_3$ be pairwise coprime positive integers, let $g \\ne 0$ be an integer, and put $n = (c_1 g + \\varepsilon_1)(c_2 g + \\varepsilon_2)(c_3 g + \\varepsilon_3)$ and $\\varepsilon_n = \\varepsilon_1 \\varepsilon_2 \\varepsilon_3$. Then $c_i g \\mid n - \\varepsilon_n$ for $i = 1, 2, 3$ if and only if","title":"Ballot's Lemma 2: three divisibilities as one congruence"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-dcarmichael","kind":"node","lean":"MetaMathlibExt.IsDCarmichael","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(D\\)"],["text","-Carmichael number"]],"source":"$D$-Carmichael number"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-dcarmichael","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"Let $D$ be an integer. A natural number $n$ is a $D$-Carmichael number when $D \\equiv 0$ or $1 \\pmod 4$, $n > 1$ is odd and composite, $D$ and $n$ are coprime, and for all integers $P$ and $Q \\ne 0$ with $P$ and $Q$ coprime, $P^2 - 4Q = D$ and $Q$ coprime to $n$, the number $n$ divides the term $U_{n - \\varepsilon_n}(P, Q)$ of the Lucas sequence $U(P, Q)$, where $\\varepsilon_n = \\left(\\frac{D}{n}\\right)$ is the\u2026","title":"$D$-Carmichael number"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/not-prime","kind":"node","lean":"MetaMathlibExt.IsDCarmichael.not_prime","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(D\\)"],["text","-Carmichael numbers are not prime"]],"source":"$D$-Carmichael numbers are not prime"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#not-prime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"For every integer $D$ and every natural number $n$, if $n$ is a $D$-Carmichael number, then $n$ is not prime.","title":"$D$-Carmichael numbers are not prime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/not-is-dcarmichael-five-nine","kind":"node","lean":"MetaMathlibExt.not_isDCarmichael_five_nine","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(9\\)"],["text"," is not a "],["math","\\(5\\)"],["text","-Carmichael number"]],"source":"$9$ is not a $5$-Carmichael number"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#not-is-dcarmichael-five-nine","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"The number $9$ is not a $D$-Carmichael number for $D = 5$. Indeed, the Jacobi symbol $(5 \\mid 9)$ equals $1$, and the Fibonacci sequence $U(1, -1)$ has $F_8 = 21$, which $9$ does not divide.","title":"$9$ is not a $5$-Carmichael number"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/lucas-digit-sum","kind":"node","lean":"MetaMathlibExt.lucasDigitSum","parent":7174,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#lucas-digit-sum","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"Let $D$ be an integer and let $p \\ge 2$ and $n$ be natural numbers, with base-$p$ expansion $n = \\sum_i c_i p^i$, $0 \\le c_i < p$. With $\\varepsilon_p = \\left(\\frac{D}{p}\\right)$ the Jacobi symbol, the extended digit sum of $n$ is $s_p(n) = \\sum_i c_i \\varepsilon_p^i$.","title":"Extended digit sum"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/lucas-digit-sum-five-seven-6601","kind":"node","lean":"MetaMathlibExt.lucasDigitSum_five_seven_6601","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","The extended digit sum of "],["math","\\(6601\\)"],["text"," in base "],["math","\\(7\\)"],["text"," for "],["math","\\(D = 5\\)"]],"source":"The extended digit sum of $6601$ in base $7$ for $D = 5$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#lucas-digit-sum-five-seven-6601","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"The base-$7$ expansion is $6601 = 5 \\cdot 7 + 7^2 + 5 \\cdot 7^3 + 2 \\cdot 7^4$ and the Jacobi symbol is $\\left(\\frac{5}{7}\\right) = -1$, so the extended digit sum with $D = 5$ is $s_7(6601) = -5 + 1 - 5 + 2 = -7$.","title":"The extended digit sum of $6601$ in base $7$ for $D = 5$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-primary-dcarmichael","kind":"node","lean":"MetaMathlibExt.IsPrimaryDCarmichael","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primary "],["math","\\(D\\)"],["text","-Carmichael number"]],"source":"Primary $D$-Carmichael number"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-primary-dcarmichael","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"Let $D$ be an integer. A natural number $n$ is a primary $D$-Carmichael number when $n$ is a $D$-Carmichael number and $|s_p(n) - \\varepsilon_n| \\le p - \\varepsilon_p$ for every prime $p \\mid n$, where $\\varepsilon_m = \\left(\\frac{D}{m}\\right)$ is the Jacobi symbol and $s_p(n) = \\sum_i c_i \\varepsilon_p^i$ is the extended digit sum of the base-$p$ expansion $n = \\sum_i c_i p^i$.","title":"Primary $D$-Carmichael number"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-secondary-dcarmichael","kind":"node","lean":"MetaMathlibExt.IsSecondaryDCarmichael","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Secondary "],["math","\\(D\\)"],["text","-Carmichael number"]],"source":"Secondary $D$-Carmichael number"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-secondary-dcarmichael","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"Let $D$ be an integer. A natural number $n$ is a secondary $D$-Carmichael number when $n$ is a $D$-Carmichael number, $|s_p(n) - \\varepsilon_n| \\le 2 (p - \\varepsilon_p)$ for every prime $p \\mid n$, and $|s_p(n) - \\varepsilon_n| > p - \\varepsilon_p$ for some prime $p \\mid n$, where $\\varepsilon_m = \\left(\\frac{D}{m}\\right)$ is the Jacobi symbol and $s_p(n) = \\sum_i c_i \\varepsilon_p^i$ is the extended digit sum of\u2026","title":"Secondary $D$-Carmichael number"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-dcarmichael-2","kind":"node","lean":"MetaMathlibExt.IsPrimaryDCarmichael.isDCarmichael","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primary "],["math","\\(D\\)"],["text","-Carmichael numbers are "],["math","\\(D\\)"],["text","-Carmichael"]],"source":"Primary $D$-Carmichael numbers are $D$-Carmichael"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-dcarmichael-2","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"For every integer $D$ and natural number $n$, if $n$ is a primary $D$-Carmichael number, then $n$ is a $D$-Carmichael number.","title":"Primary $D$-Carmichael numbers are $D$-Carmichael"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-signature-carmichael","kind":"node","lean":"MetaMathlibExt.IsSignatureCarmichael","parent":7174,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-signature-carmichael","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"Let $\\sigma = (\\varepsilon_1, \\dots, \\varepsilon_r)$ be a list of integers. A natural number $n$ is $\\sigma$-Carmichael when every $\\varepsilon_i$ is $1$ or $-1$, $r \\ge 2$, $n$ is odd and squarefree with exactly $r$ prime factors $p_1 < \\dots < p_r$, and $n$ is $D$-Carmichael for every integer $D \\equiv 0$ or $1 \\pmod 4$ whose Jacobi symbols satisfy $\\left(\\frac{D}{p_i}\\right) = \\varepsilon_i$ for all $i$.","title":"Carmichael numbers of a signature"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/not-is-signature-carmichael-four","kind":"node","lean":"MetaMathlibExt.not_isSignatureCarmichael_four","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(4\\)"],["text"," is not a signature Carmichael number"]],"source":"$4$ is not a signature Carmichael number"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#not-is-signature-carmichael-four","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"For every finite sequence $\\sigma$ of integers, $4$ is not a $\\sigma$-Carmichael number, because $4$ is not squarefree.","title":"$4$ is not a signature Carmichael number"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-universal-linear-dform","kind":"node","lean":"MetaMathlibExt.IsUniversalLinearDForm","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Universal "],["math","\\(D\\)"],["text","-forms"]],"source":"Universal $D$-forms"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-universal-linear-dform","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"Let $D$ be an integer, $\\sigma$ a list of integers, and $f(x) = (a_1 x + b_1) \\cdots (a_k x + b_k)$ a product of linear factors with integer coefficients. Then $f$ is a universal $D$-form with respect to $\\sigma$ when: every factor has $a_i \\ne 0$ and $\\gcd(a_i, b_i) = 1$; for every prime $\\ell$ there is an integer $x$ with $\\ell \\nmid f(x)$; and for every integer $x$ at which every factor $a_i x + b_i$ is a\u2026","title":"Universal $D$-forms"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/not-is-universal-linear-dform-mul-succ","kind":"node","lean":"MetaMathlibExt.not_isUniversalLinearDForm_mul_succ","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(x(x+1)\\)"],["text"," is never a universal form"]],"source":"$x(x+1)$ is never a universal form"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#not-is-universal-linear-dform-mul-succ","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"fully_proved","summary":"For every integer $D$ and every finite sequence $\\sigma$ of integers, the product of the linear factors $x$ and $x + 1$ is not a universal $D$-form with respect to $\\sigma$: since $x(x+1)$ is even for every integer $x$, the prime $2$ divides every value of the product.","title":"$x(x+1)$ is never a universal form"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-dcarmichael-iff-korselt","kind":"node","lean":"MetaMathlibExt.isDCarmichael_iff_korselt","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Williams' Korselt criterion for "],["math","\\(D\\)"],["text","-Carmichael numbers"]],"source":"Williams' Korselt criterion for $D$-Carmichael numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-dcarmichael-iff-korselt","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $D$ be an integer with $D \\equiv 0$ or $1 \\pmod 4$, and let $n > 1$ be an odd composite natural number coprime to $D$. Then $n$ is a $D$-Carmichael number if and only if $n$ is squarefree and $p - \\left(\\frac{D}{p}\\right)$ divides $n - \\left(\\frac{D}{n}\\right)$ for every prime $p \\mid n$, where $\\left(\\frac{D}{m}\\right)$ is the Jacobi symbol.","title":"Williams' Korselt criterion for $D$-Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/infinite-primary-carmichael-of-prime-ktu","kind":"node","lean":"MetaMathlibExt.infinite_primary_carmichael_of_primeKTuples","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime "],["math","\\(k\\)"],["text","-tuples give infinitely many primary Carmichael numbers (Wagstaff)"]],"source":"Prime $k$-tuples give infinitely many primary Carmichael numbers (Wagstaff)"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#infinite-primary-carmichael-of-prime-ktu","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Suppose the prime $k$-tuples conjecture holds in the following form: for every finite list of integer pairs $(a_1, b_1), \\dots, (a_k, b_k)$ with all $a_i \\ge 1$ such that for every prime $\\ell$ there is an integer $x$ with $\\ell \\nmid \\prod_i (a_i x + b_i)$, there are infinitely many natural numbers $x$ for which every $a_i x + b_i$ is a positive prime. Then there are infinitely many composite natural numbers\u2026","title":"Prime $k$-tuples give infinitely many primary Carmichael numbers (Wagstaff)"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-signature-carmichael-iff","kind":"node","lean":"MetaMathlibExt.isSignatureCarmichael_iff","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Korselt criterion for "],["math","\\(\\sigma\\)"],["text","-Carmichael numbers"]],"source":"Korselt criterion for $\\sigma$-Carmichael numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-signature-carmichael-iff","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $\\sigma = (\\varepsilon_1, \\dots, \\varepsilon_r)$ be a list of signs $\\varepsilon_i \\in \\{1, -1\\}$ of length $r \\ge 2$, put $\\varepsilon_\\sigma = \\varepsilon_1 \\cdots \\varepsilon_r$, and let $n$ be an odd natural number. Then $n$ is $\\sigma$-Carmichael if and only if $n$ is squarefree, has exactly $r$ prime factors $p_1 < \\dots < p_r$, and satisfies $p_i - \\varepsilon_i \\mid n - \\varepsilon_\\sigma$ for every $i$.","title":"Korselt criterion for $\\sigma$-Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/two-factor-is-dcarmichael","kind":"node","lean":"MetaMathlibExt.twoFactor_isDCarmichael","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-prime-factor "],["math","\\(D\\)"],["text","-Carmichael numbers"]],"source":"Two-prime-factor $D$-Carmichael numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#two-factor-is-dcarmichael","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $D$ be an integer and assume the prime $k$-tuples conjecture (the paper's Conjecture 1) in the following form: for every finite list of linear forms $a_i x + b_i$ with integers $a_i \\ge 1$ and $b_i$ such that for every prime $\\ell$ there is an integer $x$ with $\\ell \\nmid \\prod_i (a_i x + b_i)$, there are infinitely many natural numbers $x$ at which every $a_i x + b_i$ is a positive prime. Then, first, for all\u2026","title":"Two-prime-factor $D$-Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-universal-linear-dform-chernick","kind":"node","lean":"MetaMathlibExt.isUniversalLinearDForm_chernick","parent":7174,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-universal-linear-dform-chernick","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $D$ be an integer and let $3 < p_1 < p_2 < p_3$ be primes such that $p_1 p_2 p_3$ is $D$-Carmichael, and write $\\varepsilon_i = \\left(\\frac{D}{p_i}\\right)$ for the Jacobi symbols. Let $g$ be the greatest common divisor of $p_1 - \\varepsilon_1$, $p_2 - \\varepsilon_2$ and $p_3 - \\varepsilon_3$, and let $c_1, c_2, c_3$ be pairwise coprime integers with $g c_i = p_i - \\varepsilon_i$ for each $i$. Then","title":"Chernick-type universal forms for three prime factors"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/infinite-primary-five-carmichael-of-prim","kind":"node","lean":"MetaMathlibExt.infinite_primary_fiveCarmichael_of_primeKTuples","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime "],["math","\\(k\\)"],["text","-tuples give primary "],["math","\\(5\\)"],["text","-Carmichael numbers of every three-term signature"]],"source":"Prime $k$-tuples give primary $5$-Carmichael numbers of every three-term signature"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#infinite-primary-five-carmichael-of-prim","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Suppose the prime $k$-tuples conjecture holds in the following form: for every finite list of integer pairs $(a_1, b_1), \\dots, (a_k, b_k)$ with all $a_i \\ge 1$ such that for every prime $\\ell$ there is an integer $x$ with $\\ell \\nmid \\prod_i (a_i x + b_i)$, there are infinitely many natural numbers $x$ for which every $a_i x + b_i$ is a positive prime. Then for all signs\u2026","title":"Prime $k$-tuples give primary $5$-Carmichael numbers of every three-term signature"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/three-qr-is-dcarmichael-iff","kind":"node","lean":"MetaMathlibExt.threeQR_isDCarmichael_iff","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(D\\)"],["text","-Carmichael numbers of the form "],["math","\\(3qr\\)"]],"source":"The $D$-Carmichael numbers of the form $3qr$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#three-qr-is-dcarmichael-iff","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $q < r$ be primes with $3 < q$. Then $3qr$ is a $D$-Carmichael number for some integer $D$ if and only if $3qr$ is one of $105$, $195$, $255$, $399$, $561$ and $2703$. Moreover, for all integers $D$ and $D'$ such that $3qr$ is both $D$-Carmichael and $D'$-Carmichael, the Jacobi symbols agree at $3$, $q$ and $r$: $(D \\mid 3) = (D' \\mid 3)$, $(D \\mid q) = (D' \\mid q)$ and $(D \\mid r) = (D' \\mid r)$.","title":"The $D$-Carmichael numbers of the form $3qr$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/primary-five-carmichael-pairs-of-prime-k","kind":"node","lean":"MetaMathlibExt.primaryFiveCarmichael_pairs_of_primeKTuples","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primary and secondary "],["math","\\(5\\)"],["text","-Carmichael numbers from prime quadruples"]],"source":"Primary and secondary $5$-Carmichael numbers from prime quadruples"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#primary-five-carmichael-pairs-of-prime-k","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Assume the prime $k$-tuples conjecture (the paper's Conjecture 1) in the following form: for every finite list of linear forms $a_i x + b_i$ with integers $a_i \\ge 1$ and $b_i$ such that for every prime $\\ell$ there is an integer $x$ with $\\ell \\nmid \\prod_i (a_i x + b_i)$, there are infinitely many natural numbers $x$ at which every $a_i x + b_i$ is a positive prime. For a natural number $x$ put $p = 30x + 17$,\u2026","title":"Primary and secondary $5$-Carmichael numbers from prime quadruples"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/primary-thirteen-carmichael-pairs-of-pri","kind":"node","lean":"MetaMathlibExt.primaryThirteenCarmichael_pairs_of_primeKTuples","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primary and secondary "],["math","\\(13\\)"],["text","-Carmichael numbers from prime quadruples"]],"source":"Primary and secondary $13$-Carmichael numbers from prime quadruples"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#primary-thirteen-carmichael-pairs-of-pri","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Assume the prime $k$-tuples conjecture (the paper's Conjecture 1) in the following form: for every finite list of linear forms $a_i x + b_i$ with integers $a_i \\ge 1$ and $b_i$ such that for every prime $\\ell$ there is an integer $x$ with $\\ell \\nmid \\prod_i (a_i x + b_i)$, there are infinitely many natural numbers $x$ at which every $a_i x + b_i$ is a positive prime. For a natural number $x$ put $p = 78x + 137$,\u2026","title":"Primary and secondary $13$-Carmichael numbers from prime quadruples"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/universal-five-thirteen-forms","kind":"node","lean":"MetaMathlibExt.universal_fiveThirteen_forms","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Universal "],["math","\\(5\\)"],["text","- and "],["math","\\(13\\)"],["text","-forms from prime quadruples"]],"source":"Universal $5$- and $13$-forms from prime quadruples"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#universal-five-thirteen-forms","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Assume the prime $k$-tuples conjecture (the paper's Conjecture 1) in the following form: for every finite list of linear forms $a_i x + b_i$ with integers $a_i \\ge 1$ and $b_i$ such that for every prime $\\ell$ there is an integer $x$ with $\\ell \\nmid \\prod_i (a_i x + b_i)$, there are infinitely many natural numbers $x$ at which every $a_i x + b_i$ is a positive prime. Then, first, for $D = 5$ and for $D = 13$: the\u2026","title":"Universal $5$- and $13$-forms from prime quadruples"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/infinite-primary-five-neg-eleven-of-schi","kind":"node","lean":"MetaMathlibExt.infinite_primary_five_negEleven_of_schinzelH","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Schinzel's Hypothesis H gives numbers that are primary "],["math","\\(5\\)"],["text","- and "],["math","\\((-11)\\)"],["text","-Carmichael"]],"source":"Schinzel's Hypothesis H gives numbers that are primary $5$- and $(-11)$-Carmichael"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#infinite-primary-five-neg-eleven-of-schi","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Suppose Schinzel's Hypothesis H holds in the following form: for every finite set $F$ of integer polynomials, each satisfying the Bunyakovsky condition, such that $F$ satisfies the Schinzel condition, there are infinitely many natural numbers $n$ for which $|f(n)|$ is prime for every $f \\in F$. Then there are infinitely many natural numbers $n$ that are both primary $5$-Carmichael and primary $(-11)$-Carmichael and\u2026","title":"Schinzel's Hypothesis H gives numbers that are primary $5$- and $(-11)$-Carmichael"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-dcarmichael-opposite-signs","kind":"node","lean":"MetaMathlibExt.isDCarmichael_opposite_signs","parent":7174,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-dcarmichael-opposite-signs","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $D$ and $D'$ be integers and let $n$ be both a $D$-Carmichael and a $D'$-Carmichael number whose Jacobi symbols satisfy $\\left(\\frac{D}{n}\\right) \\ne \\left(\\frac{D'}{n}\\right)$. Then $\\left(\\frac{D'}{p}\\right) = -\\left(\\frac{D}{p}\\right)$ for every prime $p \\mid n$, the number of distinct prime factors of $n$ is odd, and $3 \\nmid n$.","title":"Bi-Carmichael numbers for signatures of opposite signs"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/jacobi-sym-eq-of-is-dcarmichael-three","kind":"node","lean":"MetaMathlibExt.jacobiSym_eq_of_isDCarmichael_three","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-prime-factor "],["math","\\(D\\)"],["text","-Carmichael numbers have a single Jacobi sign"]],"source":"Three-prime-factor $D$-Carmichael numbers have a single Jacobi sign"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#jacobi-sym-eq-of-is-dcarmichael-three","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $D$ and $D'$ be integers and let $n$ be a natural number with exactly three distinct prime factors. If $n$ is both $D$-Carmichael and $D'$-Carmichael, then the Jacobi symbols agree: $\\left(\\frac{D}{n}\\right) = \\left(\\frac{D'}{n}\\right)$.","title":"Three-prime-factor $D$-Carmichael numbers have a single Jacobi sign"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/mul-dvd-prod-sub-sign-of-mod-eq","kind":"node","lean":"MetaMathlibExt.mul_dvd_prod_sub_sign_of_modEq","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ballot's Lemma 3: the three divisibilities persist modulo "],["math","\\(c_1 c_2 c_3\\)"]],"source":"Ballot's Lemma 3: the three divisibilities persist modulo $c_1 c_2 c_3$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#mul-dvd-prod-sub-sign-of-mod-eq","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $\\varepsilon_1, \\varepsilon_2, \\varepsilon_3 \\in \\{1, -1\\}$, let $c_1, c_2, c_3$ be pairwise coprime positive integers, let $\\varepsilon = \\varepsilon_1 \\varepsilon_2 \\varepsilon_3$, and let $g \\ne 0$ be an integer such that $c_i g$ divides $(c_1 g + \\varepsilon_1)(c_2 g + \\varepsilon_2)(c_3 g + \\varepsilon_3) - \\varepsilon$ for $i = 1, 2, 3$. Then every nonzero integer $h \\equiv g \\pmod{c_1 c_2 c_3}$ satisfies\u2026","title":"Ballot's Lemma 3: the three divisibilities persist modulo $c_1 c_2 c_3$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/three-qr-is-dcarmichael-divisibility","kind":"node","lean":"MetaMathlibExt.threeQR_isDCarmichael_divisibility","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility conditions for "],["math","\\(D\\)"],["text","-Carmichael numbers "],["math","\\(3qr\\)"]],"source":"Divisibility conditions for $D$-Carmichael numbers $3qr$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#three-qr-is-dcarmichael-divisibility","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $D$ be an integer, let $q < r$ be primes with $3 < q$, and write $\\alpha = (D \\mid 3)$, $\\beta = (D \\mid q)$ and $\\gamma = (D \\mid r)$ for the Jacobi symbols. If $3qr$ is a $D$-Carmichael number, then $q - \\beta$ divides $3r - \\alpha\\gamma$ and $r - \\gamma$ divides $3q - \\alpha\\beta$; moreover $r = 3q + 2\\gamma$ if $\\gamma = -\\alpha\\beta$, and $2r = 3q + \\gamma$ if $\\gamma = \\alpha\\beta$.","title":"Divisibility conditions for $D$-Carmichael numbers $3qr$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/is-dcarmichael-mul-iff","kind":"node","lean":"MetaMathlibExt.isDCarmichael_mul_iff","parent":7174,"reading_title":{"format":"inline-parts-v1","parts":[["text","Williams' characterization of two-prime "],["math","\\(D\\)"],["text","-Carmichael numbers"]],"source":"Williams' characterization of two-prime $D$-Carmichael numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes/index.html#is-dcarmichael-mul-iff","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/absolute-lucas-pseudoprimes","status":"can_prove","summary":"Let $D$ be an integer with $D \\equiv 0$ or $1 \\pmod 4$ and let $p_1 < p_2$ be primes. Then $p_1 p_2$ is a $D$-Carmichael number if and only if $p_2 = p_1 + 2$, $\\left(\\frac{D}{p_1}\\right) = -1$ and $\\left(\\frac{D}{p_2}\\right) = 1$, where $\\left(\\frac{D}{p}\\right)$ is the Jacobi symbol.","title":"Williams' characterization of two-prime $D$-Carmichael numbers"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index","status":"can_prove","title":"Carmichael index"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/carmichael-index","kind":"node","lean":"MetaMathlibExt.carmichaelIndex","parent":7205,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/index.html#carmichael-index","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index","status":"fully_proved","summary":"For a natural number $n$, the Carmichael index $N(n)$ is the number of integers $a$ with $0 \\le a \\le n - 1$ and $a^n \\equiv a \\pmod{n}$.","title":"Carmichael index"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/carmichael-index-one","kind":"node","lean":"MetaMathlibExt.carmichaelIndex_one","parent":7205,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/index.html#carmichael-index-one","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index","status":"fully_proved","summary":"The Carmichael index of $1$ is $N(1) = 1$.","title":"The Carmichael index of 1"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/carmichael-index-four-and-sixty-six","kind":"node","lean":"MetaMathlibExt.carmichaelIndex_four_and_sixtySix","parent":7205,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/index.html#carmichael-index-four-and-sixty-six","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index","status":"fully_proved","summary":"The Carmichael index satisfies $N(4) = 2$ and $N(66) = 24$.","title":"The Carmichael indices of 4 and 66"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/carmichael-index-eq-prod","kind":"node","lean":"MetaMathlibExt.carmichaelIndex_eq_prod","parent":7205,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/index.html#carmichael-index-eq-prod","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index","status":"can_prove","summary":"Let $n \\ge 2$ and, for each prime $p$ dividing $n$, let $p^{e_p}$ be the largest power of $p$ dividing $n$. Then the Carmichael index of $n$ is $N(n) = \\prod_p \\bigl(N(n; p^{e_p}) + 1\\bigr)$, the product running over the primes $p$ dividing $n$, where $N(n; p^{e_p})$ is the number of units of $\\mathbb{Z}/p^{e_p}\\mathbb{Z}$ whose multiplicative order divides $\\gcd(p - 1, n/p^{e_p} - 1)$.","title":"Product formula for the Carmichael index"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/carmichael-index-ne-sub-one","kind":"node","lean":"MetaMathlibExt.carmichaelIndex_ne_sub_one","parent":7205,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Carmichael index of "],["math","\\(n\\)"],["text"," is never "],["math","\\(n - 1\\)"]],"source":"The Carmichael index of $n$ is never $n - 1$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index/index.html#carmichael-index-ne-sub-one","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-index","status":"can_prove","summary":"For every positive integer $n$, the Carmichael index $N(n)$, the number of integers $a$ with $0 \\le a \\le n - 1$ and $a^n \\equiv a \\pmod{n}$, is not equal to $n - 1$.","title":"The Carmichael index of $n$ is never $n - 1$"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"can_prove","title":"Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/is-carmichael-number","kind":"node","lean":"Nat.IsCarmichaelNumber","parent":7211,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html#is-carmichael-number","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"fully_proved","summary":"A natural number $n$ is a Carmichael number if $n > 1$, $n$ is not prime, and $a^n \\equiv a \\pmod{n}$ for every natural number $a$.","title":"Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/is-carmichael-number-iff-zmod","kind":"node","lean":"Nat.isCarmichaelNumber_iff_zmod","parent":7211,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html#is-carmichael-number-iff-zmod","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"fully_proved","summary":"A natural number $n$ is a Carmichael number if and only if $n > 1$, $n$ is not prime, and $x^n = x$ for every element $x$ of $\\mathbb{Z}/n\\mathbb{Z}$.","title":"Carmichael numbers via the residue ring"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/is-carmichael-number-iff-korselt","kind":"node","lean":"Nat.isCarmichaelNumber_iff_korselt","parent":7211,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html#is-carmichael-number-iff-korselt","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"fully_proved","summary":"A natural number $n > 1$ is a Carmichael number if and only if $n$ is squarefree, $p - 1$ divides $n - 1$ for every prime divisor $p$ of $n$, and $n$ is not prime.","title":"Korselt's criterion"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/infinite-carmichael-prime-card-factors-o","kind":"node","lean":"MetaMathlibExt.infinite_carmichael_prime_cardFactors_of_heathBrown","parent":7211,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html#infinite-carmichael-prime-card-factors-o","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"can_prove","summary":"Suppose there is a real constant $C$ such that for every natural number $d \\ge 2$ and every natural number $c$ coprime to $d$ there is a prime $p \\equiv c \\pmod{d}$ with $p \\le C d (\\log d)^2$ (Heath-Brown's conjecture on the least prime in an arithmetic progression). Then there are infinitely many Carmichael numbers $n$ whose number of prime factors counted with multiplicity, $\\Omega(n)$, is prime.","title":"Infinitely many Carmichael numbers with a prime number of prime factors (conditional)"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/choose-le-choose-mul-card-prod-eq-one","kind":"node","lean":"MetaMathlibExt.choose_le_choose_mul_card_prod_eq_one","parent":7211,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html#choose-le-choose-mul-card-prod-eq-one","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"can_prove","summary":"Let $G$ be an abelian group and let $n$ be the least natural number $m$ such that every sequence of $m$ elements of $G$ has a nonempty subsequence with product $1$ (the Davenport constant of $G$). Let $r$ and $t$ be natural numbers with $n < t < r$. Then for every sequence $s_1, \\dots, s_r$ of elements of $G$, the number $M$ of sets $S \\subseteq \\{1, \\dots, r\\}$ with $t - n \\le |S| \\le t$ and\u2026","title":"Alford\u2013Granville\u2013Pomerance count of zero-sum subsequences"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/exists-prime-mod-eq-le-rpow-five-point-t","kind":"node","lean":"MetaMathlibExt.exists_prime_modEq_le_rpow_five_point_two","parent":7211,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html#exists-prime-mod-eq-le-rpow-five-point-t","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"can_prove","summary":"There is a real constant $C$ such that for every natural number $d \\ge 1$ and every natural number $c$ coprime to $d$, there is a prime $p \\equiv c \\pmod{d}$ with $p \\le C d^{5.2}$.","title":"Linnik's theorem with exponent 5.2 (Xylouris)"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/exists-card-smooth-shifted-primes-ge","kind":"node","lean":"MetaMathlibExt.exists_card_smooth_shifted_primes_ge","parent":7211,"reading_title":{"format":"inline-parts-v1","parts":[["text","Many primes "],["math","\\(q\\)"],["text"," with "],["math","\\(q - 1\\)"],["text"," smooth"]],"source":"Many primes $q$ with $q - 1$ smooth"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html#exists-card-smooth-shifted-primes-ge","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"can_prove","summary":"Let $1 < \\theta < 2$. There are real constants $\\gamma > 0$ and $Y$ such that for every real $y > Y$, the number of primes $q$ with $y^\\theta / \\log y \\le q \\le y^\\theta$ such that every prime divisor of $q - 1$ is at most $y$ is at least $\\gamma y^\\theta / \\log(y^\\theta)$.","title":"Many primes $q$ with $q - 1$ smooth"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/exists-carmichael-prime-card-factors-bet","kind":"node","lean":"MetaMathlibExt.exists_carmichael_prime_cardFactors_between_of_heathBrown","parent":7211,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers/index.html#exists-carmichael-prime-card-factors-bet","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/carmichael-numbers","status":"can_prove","summary":"Assume Heath-Brown's conjecture in the following form: there is a real constant $C$ such that for every natural number $d \\ge 2$ and every natural number $c$ coprime to $d$ there is a prime $p \\equiv c \\pmod{d}$ with $p \\le C d (\\log d)^2$. Let $1 < \\theta < 2$, and let $\\gamma > 0$ be such that for all sufficiently large real $y$, the number of primes $q$ with $y^\\theta / \\log y \\le q \\le y^\\theta$ such that every\u2026","title":"Carmichael numbers with a prime number of prime factors in a range"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","title":"Even pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/is-even-pseudoprime","kind":"node","lean":"MathlibExt.NumberTheory.IsEvenPseudoprime","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#is-even-pseudoprime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"fully_proved","summary":"A natural number $N$ is an even pseudoprime if $N$ is even, $N > 2$ and $2^N \\equiv 2 \\pmod{N}$.","title":"Even pseudoprime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/even-pseudoprimes","kind":"node","lean":"MathlibExt.NumberTheory.evenPseudoprimes","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#even-pseudoprimes","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"fully_proved","summary":"The set of even pseudoprimes is the set of natural numbers $N$ such that $N$ is even, $N > 2$ and $2^N \\equiv 2 \\pmod{N}$.","title":"The set of even pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/even","kind":"node","lean":"MathlibExt.NumberTheory.IsEvenPseudoprime.even","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#even","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"fully_proved","summary":"Every even pseudoprime $N$ is an even natural number.","title":"Even pseudoprimes are even"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/two-lt","kind":"node","lean":"MathlibExt.NumberTheory.IsEvenPseudoprime.two_lt","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#two-lt","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"fully_proved","summary":"Every even pseudoprime $N$ satisfies $N > 2$.","title":"Even pseudoprimes exceed two"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/modeq","kind":"node","lean":"MathlibExt.NumberTheory.IsEvenPseudoprime.modeq","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#modeq","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"fully_proved","summary":"If $N$ is an even pseudoprime, then $2^N \\equiv 2 \\pmod{N}$.","title":"Fermat congruence for even pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/not-prime","kind":"node","lean":"MathlibExt.NumberTheory.IsEvenPseudoprime.not_prime","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#not-prime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"fully_proved","summary":"No even pseudoprime is a prime number: the only even prime is $2$, and every even pseudoprime is greater than $2$.","title":"Even pseudoprimes are not prime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/even-pseudoprime-mod-eq-and-order-of-odd","kind":"node","lean":"MetaMathlibExt.evenPseudoprime_modEq_and_orderOf_odd","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#even-pseudoprime-mod-eq-and-order-of-odd","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Let $N$ be an even pseudoprime and $p$ an odd prime dividing $N$, and write $\\ell_2(p)$ for the multiplicative order of $2$ modulo $p$. Then $N \\equiv p \\pmod{p\\,\\ell_2(p)}$, $\\ell_2(p)$ is odd, $p \\equiv 1$ or $7 \\pmod 8$, and $14p \\le N$.","title":"Odd prime divisors of an even pseudoprime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/even-pseudoprime-mod-sixteen","kind":"node","lean":"MetaMathlibExt.evenPseudoprime_mod_sixteen","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#even-pseudoprime-mod-sixteen","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Every even pseudoprime $N$ satisfies $N \\equiv 2 \\pmod{16}$ or $N \\equiv 14 \\pmod{16}$.","title":"Even pseudoprimes modulo 16"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/even-pseudoprime-wieferich-of-sq-dvd","kind":"node","lean":"MetaMathlibExt.evenPseudoprime_wieferich_of_sq_dvd","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#even-pseudoprime-wieferich-of-sq-dvd","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Let $N$ be an even pseudoprime and $p$ a prime with $p^2 \\mid N$. Then $p$ is a Wieferich prime, that is $2^{p-1} \\equiv 1 \\pmod{p^2}$, and $p \\ge 3511$.","title":"Square prime factors of even pseudoprimes are Wieferich primes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/sum-inv-mod-eq-order-of-le","kind":"node","lean":"MetaMathlibExt.sum_inv_modEq_orderOf_le","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#sum-inv-mod-eq-order-of-le","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Write $\\ell_2(p)$ for the multiplicative order of $2$ modulo a prime $p$. Let $p$ be an odd prime and $k$ a natural number, and let $F_p(k)$ be the sum of $1/N$ over the integers $N$ with $e^k < N \\le e^{k+1}$, $N \\equiv p \\pmod{p\\,\\ell_2(p)}$ and $N > p$. Then $F_p(k) \\le 2 / (p\\,\\ell_2(p))$.","title":"Reciprocal sum over a residue class in an exponential window"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/sum-inv-mod-eq-order-of-mod-sixteen-le","kind":"node","lean":"MetaMathlibExt.sum_inv_modEq_orderOf_mod_sixteen_le","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#sum-inv-mod-eq-order-of-mod-sixteen-le","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Write $\\ell_2(p)$ for the multiplicative order of $2$ modulo a prime $p$. Let $p$ be an odd prime with $\\ell_2(p)$ odd, let $k$ be a natural number, and let $S$ be the sum of $1/N$ over the integers $N$ with $e^k < N \\le e^{k+1}$, $N \\equiv p \\pmod{p\\,\\ell_2(p)}$ and $N \\equiv \\pm 2 \\pmod{16}$. Then $S \\le 1 / (4p\\,\\ell_2(p))$ whenever $16p\\,\\ell_2(p) \\le e^k$, and in all cases\u2026","title":"Reciprocal sum over a residue class restricted modulo 16"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/smooth-reciprocal-tail-le-rpow-mul-smoot","kind":"node","lean":"MetaMathlibExt.smoothReciprocalTail_le_rpow_mul_smoothZeta","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#smooth-reciprocal-tail-le-rpow-mul-smoot","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Let $x, y, s$ be real numbers with $2 \\le y < x$ and $0 < s < 1$. Call a positive integer $y$-smooth if all its prime factors are at most $y$ (so $1$ is $y$-smooth). Let $S(x, y)$ be the sum of $1/n$ over the $y$-smooth integers $n > x$, and let $\\zeta(\\sigma, y)$ be the sum of $n^{-\\sigma}$ over all $y$-smooth integers $n$. Then $S(x, y) \\le x^{-s} \\zeta(1 - s, y)$, where both series converge.","title":"Rankin's bound for reciprocal sums of smooth numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/smooth-reciprocal-tail-lt-nguyen-pomeran","kind":"node","lean":"MetaMathlibExt.smoothReciprocalTail_lt_nguyenPomerance","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#smooth-reciprocal-tail-lt-nguyen-pomeran","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Let $x, y$ be real numbers with $2 \\le y < x$. Call a positive integer $y$-smooth if all its prime factors are at most $y$, let $S(x, y)$ be the sum of $1/n$ over the $y$-smooth integers $n > x$, and let $\\zeta(\\sigma, y)$ be the sum of $n^{-\\sigma}$ over all $y$-smooth integers $n$. Put $u = \\log x / \\log y$ and $s = \\log(u \\log u) / \\log y$. If $u \\ge 3$ and $s \\le 1/3$, then","title":"Nguyen\u2013Pomerance bound for reciprocal sums of smooth numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/order-of-two-odd-small-count-lt","kind":"node","lean":"MetaMathlibExt.orderOfTwo_odd_small_count_lt","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#order-of-two-odd-small-count-lt","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Write $\\ell_2(p)$ for the multiplicative order of $2$ modulo a prime $p$. For every real $x > 0$, the number of primes $p$ with $2 < p \\le x$ such that $\\ell_2(p)$ is odd and $1 < \\ell_2(p) \\le p^{2/5}$ is less than $0.00754\\, x^{4/5}$.","title":"Primes with odd order of 2 of small size"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/order-of-two-odd-medium-count-lt","kind":"node","lean":"MetaMathlibExt.orderOfTwo_odd_medium_count_lt","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#order-of-two-odd-medium-count-lt","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"Write $\\ell_2(p)$ for the multiplicative order of $2$ modulo a prime $p$. For every real $x \\ge 337$, the number of primes $p$ with $2 < p \\le x$ such that $\\ell_2(p)$ is odd and $p^{2/5} < \\ell_2(p) \\le p^{12/25}$ is less than $0.00754\\, x^{24/25}$.","title":"Primes with odd order of 2 of medium size"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/even-pseudoprimes-infinite","kind":"node","lean":"MathlibExt.NumberTheory.EvenPseudoprimesWanted.evenPseudoprimes_infinite","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#even-pseudoprimes-infinite","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"There are infinitely many even pseudoprimes.","title":"Beeger's theorem"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/even-pseudoprime-reciprocal-sum-bounds","kind":"node","lean":"MathlibExt.NumberTheory.EvenPseudoprimesWanted.evenPseudoprime_reciprocal_sum_bounds","parent":7220,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes/index.html#even-pseudoprime-reciprocal-sum-bounds","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/even-pseudoprimes","status":"can_prove","summary":"The sum of the reciprocals of all even pseudoprimes lies strictly between `0.000011` and `0.0059`.","title":"Cheng\u2013Kinlaw\u2013Nguyen reciprocal-sum bounds"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g","status":"can_prove","title":"Fermat congruence criteria for twin and Sophie Germain primes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/twin-prime-iff-forall-fermat-congruence","kind":"node","lean":"MetaMathlibExt.twinPrime_iff_forall_fermat_congruence","parent":7238,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/index.html#twin-prime-iff-forall-fermat-congruence","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g","status":"can_prove","summary":"Let $n \\ge 3$ be odd, and suppose neither $n$ nor $n + 2$ is a Carmichael number. Then $n$ and $n + 2$ are both prime if and only if $2a^{n+1} \\equiv (n + 2)\\,a^2 - n \\pmod{n(n + 2)}$ for every integer $a$ coprime to $n(n + 2)$; and $n$ and $n + 2$ are both prime if and only if $n\\,a^{n+1} \\equiv (n + 2)\\,a^{n-1} - 2 \\pmod{n(n + 2)}$ for every integer $a$ coprime to $n(n + 2)$.","title":"Fermat-type characterizations of twin primes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/not-is-carmichael-number-two-pow-mul-pri","kind":"node","lean":"MetaMathlibExt.not_isCarmichaelNumber_two_pow_mul_prime_add_one","parent":7238,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/index.html#not-is-carmichael-number-two-pow-mul-pri","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g","status":"can_prove","summary":"For every natural number $n$ and every odd prime $p$, the number $2^n p + 1$ is not a Carmichael number.","title":"Alahmadi\u2013Luca theorem on Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/is-sophie-germain-prime-iff-forall-ferma","kind":"node","lean":"MetaMathlibExt.isSophieGermainPrime_iff_forall_fermat_congruence","parent":7238,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/index.html#is-sophie-germain-prime-iff-forall-ferma","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g","status":"can_prove","summary":"Let $p$ be a prime. Then $p$ is a Sophie Germain prime if and only if $a^{2p} \\equiv (2p + 1)\\,a^{p+1} - 2p \\pmod{p(2p + 1)}$ for every integer $a$ coprime to $p(2p + 1)$; and $p$ is a Sophie Germain prime if and only if $(2p + 1)\\,a^{p-1} \\equiv p\\,a^{2p} + p + 1 \\pmod{p(2p + 1)}$ for every integer $a$ coprime to $p(2p + 1)$.","title":"Fermat-type characterizations of Sophie Germain primes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/sophie-germain-chain-fermat-congruence","kind":"node","lean":"MetaMathlibExt.sophieGermainChain_fermat_congruence","parent":7238,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/index.html#sophie-germain-chain-fermat-congruence","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g","status":"can_prove","summary":"Let $p$ and $j$ be natural numbers and put $q_{i+1} = 2^i (p + 1) - 1$ for $i \\ge 0$, so that $q_1 = p$ and $q_{i+2} = 2q_{i+1} + 1$. Suppose $q_1, \\dots, q_{j+2}$ are all prime, and write $q = q_{j+1}$ and $q' = q_{j+2} = 2q + 1$. Then for every integer $a$ coprime to $qq'$, $a^{q'-1} \\equiv q'\\,a^{q+1} - 2q \\pmod{qq'}$ and $q'\\,a^{q-1} \\equiv q\\,a^{q'-1} + q + 1 \\pmod{qq'}$.","title":"Fermat-type congruences along a Sophie Germain prime chain"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/sophie-germain-chain-length-le-order-of","kind":"node","lean":"MetaMathlibExt.sophieGermainChain_length_le_orderOf_two","parent":7238,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/index.html#sophie-germain-chain-length-le-order-of","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g","status":"can_prove","summary":"Let $p$ be an odd prime and let $\\mathrm{ord}_p(2)$ be the multiplicative order of $2$ modulo $p$. Then there is some $j \\le \\mathrm{ord}_p(2)$ such that $2^j (p + 1) - 1$ is not prime. Thus the Sophie Germain prime chain $q_1 = p$, $q_{i+1} = 2q_i + 1 = 2^i (p + 1) - 1$ begins with at most $\\mathrm{ord}_p(2)$ consecutive primes.","title":"Length of a Sophie Germain prime chain"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/forall-fermat-congruence-iff-of-coprime","kind":"node","lean":"MetaMathlibExt.forall_fermat_congruence_iff_of_coprime","parent":7238,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g/index.html#forall-fermat-congruence-iff-of-coprime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-congruence-criteria-for-twin-and-sophie-g","status":"can_prove","summary":"Let $n$ and $m$ be coprime natural numbers with $0 < n < m$. Since $\\gcd(m - n, nm) = 1$, the residue of $m - n$ is invertible in $\\mathbb{Z}/nm\\mathbb{Z}$; write $u$ for its inverse. Then the equalities $a^{m-1} = (a^{m-n} - 1)\\,u\\,n + a^{m-n}$ and $(a^{m-n} - 1)\\,u\\,n + a^{m-n} = (a^{m-n} - 1)\\,u\\,m + 1$ in $\\mathbb{Z}/nm\\mathbb{Z}$ hold for every integer $a$ coprime to $nm$ if and only if both $n$ and $m$ have\u2026","title":"Fermat's congruence for a product of coprime moduli"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes","status":"can_prove","title":"Fermat pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes/cipolla-pseudoprime-construction","kind":"node","lean":"MetaMathlibExt.cipolla_pseudoprime_construction","parent":7245,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes/index.html#cipolla-pseudoprime-construction","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes","status":"fully_proved","summary":"Let $a > 1$ be a natural number and let $p$ be a prime that does not divide $a(a^2 - 1)$. Put $n_1 = (a^p - 1)/(a - 1)$, $n_2 = (a^p + 1)/(a + 1)$ and $n = n_1 n_2$. Then $n$ is a Fermat pseudoprime to base $a$: $n > 1$, $n$ is not prime, and $n$ divides $a^{n-1} - 1$.","title":"Cipolla's pseudoprime construction"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.NonSquarefreeBaseTwoPseudoprimeWanted.conjecture","parent":7245,"reading_title":{"format":"inline-parts-v1","parts":[["text","Guy's question on non-squarefree base-two pseudoprimes beyond "],["math","\\(10^{13}\\)"]],"source":"Guy's question on non-squarefree base-two pseudoprimes beyond $10^{13}$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes/index.html#conjecture","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes","status":"fully_proved","summary":"The statement that there is a natural number $n > 10^{13}$ that is a Fermat pseudoprime to base $2$ (a composite number $n > 1$ with $n \\mid 2^{n-1} - 1$) and is not squarefree.","title":"Guy's question on non-squarefree base-two pseudoprimes beyond $10^{13}$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.NonSquarefreeBaseTwoPseudoprimeWanted.conjecture_holds","parent":7245,"reading_title":{"format":"inline-parts-v1","parts":[["text","A non-squarefree base-two pseudoprime beyond "],["math","\\(10^{13}\\)"],["text"," exists"]],"source":"A non-squarefree base-two pseudoprime beyond $10^{13}$ exists"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes/index.html#conjecture-holds","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/fermat-pseudoprimes","status":"can_prove","summary":"There is a natural number $n > 10^{13}$ that is a Fermat pseudoprime to base $2$ (a composite number $n > 1$ with $n \\mid 2^{n-1} - 1$) and is not squarefree.","title":"A non-squarefree base-two pseudoprime beyond $10^{13}$ exists"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","title":"Frobenius pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/is-greatest-common-monic-divisor","kind":"node","lean":"Polynomial.IsGreatestCommonMonicDivisor","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#is-greatest-common-monic-divisor","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $h, g_1, g_2 \\in R[x]$. Then $h$ is a greatest common monic divisor of $g_1$ and $g_2$ when $h$, $g_1$ and $g_2$ are monic and the ideal generated by $g_1$ and $g_2$ equals the principal ideal generated by $h$.","title":"Greatest common monic divisor"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/unfold","kind":"node","lean":"Polynomial.IsGreatestCommonMonicDivisor.unfold","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#unfold","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"For polynomials $h$, $g_1$ and $g_2$ over a commutative ring $R$, $h$ is a greatest common monic divisor of $g_1$ and $g_2$ if and only if $h$, $g_1$ and $g_2$ are monic and the ideal of $R[X]$ generated by $g_1$ and $g_2$ equals the principal ideal generated by $h$.","title":"Unfolding the greatest common monic divisor"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/is-frobenius-probable-prime","kind":"node","lean":"Nat.IsFrobeniusProbablePrime","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#is-frobenius-probable-prime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"Let $f \\in \\mathbb{Z}[x]$ be a polynomial of degree $d$ with discriminant $\\Delta$, and let $n$ be a natural number. Then $n$ is a Frobenius probable prime with respect to $f$ when $f$ is monic, $n$ is odd, $n > 1$, $n$ is coprime to $|f(0)\\Delta|$, and there exist polynomials $F_i, f_i \\in (\\mathbb{Z}/n\\mathbb{Z})[x]$ and natural numbers $q_i$, indexed by $i \\in \\mathbb{N}$, such that: and $f_{i-1}$, and\u2026","title":"Frobenius probable prime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/is-frobenius-pseudoprime","kind":"node","lean":"Nat.IsFrobeniusPseudoprime","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#is-frobenius-pseudoprime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"A natural number $n$ is a Frobenius pseudoprime with respect to a polynomial $f \\in \\mathbb{Z}[x]$ when $n$ is not prime and $n$ is a Frobenius probable prime with respect to $f$.","title":"Frobenius pseudoprime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/unfold-2","kind":"node","lean":"Nat.IsFrobeniusProbablePrime.unfold","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#unfold-2","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"Let $f \\in \\mathbb{Z}[x]$ be a polynomial of degree $d$ with discriminant $\\Delta$, and let $n$ be a natural number. Then $n$ is a Frobenius probable prime with respect to $f$ if and only if $f$ is monic, $n$ is odd, $n > 1$, $n$ is coprime to $|f(0)\\Delta|$, and there exist polynomials $F_i, f_i \\in (\\mathbb{Z}/n\\mathbb{Z})[x]$ and natural numbers $q_i$, indexed by $i \\in \\mathbb{N}$, such that: and $f_{i-1}$, and\u2026","title":"Characterization of Frobenius probable primes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/monic","kind":"node","lean":"Nat.IsFrobeniusProbablePrime.monic","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#monic","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"If $n$ is a Frobenius probable prime with respect to a polynomial $f \\in \\mathbb{Z}[x]$, then $f$ is monic.","title":"Frobenius probable primes use monic polynomials"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/odd","kind":"node","lean":"Nat.IsFrobeniusProbablePrime.odd","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#odd","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"If $n$ is a Frobenius probable prime with respect to a polynomial $f \\in \\mathbb{Z}[x]$, then $n$ is odd.","title":"Frobenius probable primes are odd"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/one-lt","kind":"node","lean":"Nat.IsFrobeniusProbablePrime.one_lt","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#one-lt","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"If $n$ is a Frobenius probable prime with respect to a polynomial $f \\in \\mathbb{Z}[x]$, then $n > 1$.","title":"Frobenius probable primes exceed one"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/unfold-3","kind":"node","lean":"Nat.IsFrobeniusPseudoprime.unfold","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#unfold-3","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"For every natural number $n$ and every polynomial $f \\in \\mathbb{Z}[x]$, $n$ is a Frobenius pseudoprime with respect to $f$ if and only if $n$ is not prime and $n$ is a Frobenius probable prime with respect to $f$.","title":"Characterization of Frobenius pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/composite","kind":"node","lean":"Nat.IsFrobeniusPseudoprime.composite","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#composite","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"If $n$ is a Frobenius pseudoprime with respect to a polynomial $f \\in \\mathbb{Z}[x]$, then $n$ is not prime.","title":"Frobenius pseudoprimes are not prime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/probable-prime","kind":"node","lean":"Nat.IsFrobeniusPseudoprime.probablePrime","parent":7249,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes/index.html#probable-prime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/frobenius-pseudoprimes","status":"fully_proved","summary":"If $n$ is a Frobenius pseudoprime with respect to a polynomial $f \\in \\mathbb{Z}[x]$, then $n$ is a Frobenius probable prime with respect to $f$.","title":"Frobenius pseudoprimes are Frobenius probable primes"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers","status":"can_prove","title":"Generalized Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/is-aacarmichael","kind":"node","lean":"MetaMathlibExt.IsAACarmichael","parent":7261,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((a, a)\\)"],["text","-Carmichael numbers"]],"source":"$(a, a)$-Carmichael numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/index.html#is-aacarmichael","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers","status":"fully_proved","summary":"Let $a$ be an integer. A natural number $n$ is an $(a, a)$-Carmichael number if $n$ is squarefree and $p - a$ divides $n - a$ in $\\mathbb{Z}$ for every prime $p$ dividing $n$.","title":"$(a, a)$-Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/is-aacarmichael-of-prime","kind":"node","lean":"MetaMathlibExt.isAACarmichael_of_prime","parent":7261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes are "],["math","\\((a, a)\\)"],["text","-Carmichael numbers"]],"source":"Primes are $(a, a)$-Carmichael numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/index.html#is-aacarmichael-of-prime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers","status":"fully_proved","summary":"For every integer $a$, every prime $p$ is an $(a, a)$-Carmichael number.","title":"Primes are $(a, a)$-Carmichael numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/aa-carmichael-count-ge-of-least-prime-in","kind":"node","lean":"MetaMathlibExt.aaCarmichael_count_ge_of_least_prime_in_progression_bound","parent":7261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional lower bound for "],["math","\\((a, a)\\)"],["text","-Carmichael numbers with prescribed gcd"]],"source":"Conditional lower bound for $(a, a)$-Carmichael numbers with prescribed gcd"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/index.html#aa-carmichael-count-ge-of-least-prime-in","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers","status":"can_prove","summary":"Assume Conjecture 2: there are a real number $A \\ge 2$ and a natural number $l_0$ such that for every natural number $l \\ge l_0$ and every integer $b$ with $\\gcd(b, l) = 1$ there is a prime $p$ with $p \\equiv b \\pmod{l}$ and $p < l (\\log l)^A$. Let $a$ be an integer and $\\nu$ a positive integer with $\\gcd(a, \\nu) = 1$ and $a + \\nu$ odd. For real $X$ let $C_\\nu(X, a)$ be the number of $(a, a)$-Carmichael numbers\u2026","title":"Conditional lower bound for $(a, a)$-Carmichael numbers with prescribed gcd"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/rpow-log-le-prime-of-eq-mul-add-one-of-d","kind":"node","lean":"MetaMathlibExt.rpow_log_le_prime_of_eq_mul_add_one_of_dvd_prod_primes","parent":7261,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/index.html#rpow-log-le-prime-of-eq-mul-add-one-of-d","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers","status":"can_prove","summary":"For a natural number $z$, let $J$ be the product of the primes $r$ with $z/2 \\le r \\le z$, and write $\\omega(g)$ for the number of distinct prime factors of $g$. For every real $A$ and all sufficiently large natural numbers $z$ the following holds: if $j$, $g$ and $q$ are natural numbers with $1 \\le j \\le (\\log z)^{2A}$, $g \\mid J$, $\\omega(g) = \\lfloor \\log z \\rfloor$, $q$ prime and $q = gj + 1$, then","title":"Size of the auxiliary primes in the construction"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/carmichael-prod-primes-eq-mul-add-one-le","kind":"node","lean":"MetaMathlibExt.carmichael_prod_primes_eq_mul_add_one_le_exp","parent":7261,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/index.html#carmichael-prod-primes-eq-mul-add-one-le","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers","status":"can_prove","summary":"For a natural number $z$, let $J$ be the product of the primes $r$ with $z/2 \\le r \\le z$, and write $\\omega(g)$ for the number of distinct prime factors of $g$. For every real $A$ and all sufficiently large natural numbers $z$ the following holds: for every natural number $j$ with $1 \\le j \\le (\\log z)^{2A}$ and every finite set $S$ of primes such that each $q \\in S$ equals $gj + 1$ for some natural number $g$\u2026","title":"The Carmichael function of a product of auxiliary primes is small"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/int-gcd-sub-eq-of-eq-mul-mul-add","kind":"node","lean":"MetaMathlibExt.int_gcd_sub_eq_of_eq_mul_mul_add","parent":7261,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers/index.html#int-gcd-sub-eq-of-eq-mul-mul-add","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/generalized-carmichael-numbers","status":"can_prove","summary":"Let $a$ be an integer and let $\\nu$, $L_1$, $L_2$, $k_1$, $k_2$, $d_1$, $d_2$ be natural numbers with $\\gcd(L_1, L_2) = 1$, $\\gcd(k_1, \\nu L_1 L_2) = \\gcd(k_2, \\nu L_1 L_2) = 1$, $\\gcd(k_1, k_2) = 1$, $d_1 \\mid L_1$ and $d_2 \\mid L_2$. If $p_1 = d_1 k_1 \\nu + a$ and $p_2 = d_2 k_2 \\nu + a$, then $\\gcd(p_1 - a, p_2 - a) = \\nu$.","title":"Exact gcd of two shifted values in the construction"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","title":"Lucas\u2013Lehmer test"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/acceptable-seed-of-jacobi","kind":"node","lean":"MetaMathlibExt.acceptableSeed_of_jacobi","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#acceptable-seed-of-jacobi","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"Let $n \\ge 3$ be an odd natural number, let $M_n = 2^n - 1$, and let $s_1$ be an integer. Define $s_{k+1} = s_k^2 - 2$ for $k \\ge 1$. If the Jacobi symbols satisfy $\\left(\\frac{s_1 + 2}{M_n}\\right) = -1$ and $\\left(\\frac{s_1 - 2}{M_n}\\right) = 1$, then $M_n$ is prime if and only if $M_n$ divides $s_{n-1}$.","title":"Lucas\u2013Lehmer test with a general seed"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-prime-iff-lucas-lehmer-six-div","kind":"node","lean":"MetaMathlibExt.mersenne_prime_iff_lucasLehmer_six_div_five","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas\u2013Lehmer test with the rational seed "],["math","\\(6/5\\)"]],"source":"Lucas\u2013Lehmer test with the rational seed $6/5$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-prime-iff-lucas-lehmer-six-div","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $n \\ge 3$, let $M_n = 2^n - 1$, and define $s_1 = 6 \\cdot 5^{-1}$ and $s_{k+1} = s_k^2 - 2$ in $\\mathbb{Z}/M_n\\mathbb{Z}$. If $n \\equiv 3 \\pmod 4$, then $M_n$ is prime if and only if $s_{n-1} = 0$. If $n \\equiv 1 \\pmod 4$, then $M_n$ is prime if and only if $s_{n-2} = 0$.","title":"Lucas\u2013Lehmer test with the rational seed $6/5$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/lehmer-v","kind":"node","lean":"MetaMathlibExt.lehmerV","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#lehmer-v","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"For integers $R$ and $Q$, Lehmer's companion sequence $\\bar V_k(R, Q)$, $k = 0, 1, 2, \\ldots$, is the integer sequence with $\\bar V_0 = 2$, $\\bar V_1 = 1$, $\\bar V_2 = R - 2Q$, $\\bar V_3 = R - 3Q$ and $\\bar V_{k+4} = (R - 2Q)\\,\\bar V_{k+2} - Q^2\\,\\bar V_k$ for every $k \\ge 0$.","title":"Lehmer's companion sequence"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/lehmer-v-add-four","kind":"node","lean":"MetaMathlibExt.lehmerV_add_four","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#lehmer-v-add-four","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"For all integers $R$ and $Q$ and every natural number $k$, Lehmer's companion sequence satisfies $\\bar V_{k+4}(R, Q) = (R - 2Q)\\,\\bar V_{k+2}(R, Q) - Q^2\\,\\bar V_k(R, Q)$.","title":"Fourth-order recurrence of Lehmer's companion sequence"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/lehmer-v-two-neg-one","kind":"node","lean":"MetaMathlibExt.lehmerV_two_neg_one","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lehmer's companion sequence for "],["math","\\(R = 2\\)"],["text"," and "],["math","\\(Q = -1\\)"]],"source":"Lehmer's companion sequence for $R = 2$ and $Q = -1$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#lehmer-v-two-neg-one","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"For $R = 2$ and $Q = -1$, Lehmer's companion sequence has $\\bar V_2(2, -1) = 4$, $\\bar V_4(2, -1) = 14$ and $\\bar V_8(2, -1) = 194$.","title":"Lehmer's companion sequence for $R = 2$ and $Q = -1$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-prime-of-dvd-lehmer-v","kind":"node","lean":"MetaMathlibExt.mersenne_prime_of_dvd_lehmerV","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-prime-of-dvd-lehmer-v","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $n \\ge 3$ be odd, let $M_n = 2^n - 1$, and let $R$ and $Q$ be integers with $\\gcd(M_n, Q) = 1$. If $M_n$ divides Lehmer's companion term $\\bar V_{2^{n-1}}(R, Q)$, then $M_n$ is prime.","title":"Lehmer-sequence primality criterion for Mersenne numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-dvd-lehmer-v-of-jacobi","kind":"node","lean":"MetaMathlibExt.mersenne_dvd_lehmerV_of_jacobi","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-dvd-lehmer-v-of-jacobi","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $n \\ge 3$ be an odd natural number such that $M_n = 2^n - 1$ is prime, let $R$ and $Q$ be integers, let $\\bar V_k(R, Q)$ denote Lehmer's companion sequence, and put $D = R - 4Q$. If the Jacobi symbols $\\left(\\frac{DR}{M_n}\\right)$ and $\\left(\\frac{QR}{M_n}\\right)$ are both $-1$, then $M_n$ divides $\\bar V_{(M_n+1)/2}(R, Q) = \\bar V_{2^{n-1}}(R, Q)$.","title":"Divisibility of Lehmer's companion sequence by a Mersenne prime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-dvd-lehmer-v-of-diophantine","kind":"node","lean":"MetaMathlibExt.mersenne_dvd_lehmerV_of_diophantine","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-dvd-lehmer-v-of-diophantine","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $n \\ge 3$ be an odd natural number such that $M_n = 2^n - 1$ is prime, and let $r$, $d$, $q$, $X$, $Y$, $Z$ be integers with $\\gcd(M_n, Y) = 1$ and $4qY^2 = rX^2 - dZ^2$. Put $R = rX^2$ and $Q = qY^2$, and let $\\bar V_k(R, Q)$ denote Lehmer's companion sequence. If the Jacobi symbols $\\left(\\frac{rd}{M_n}\\right)$ and $\\left(\\frac{qr}{M_n}\\right)$ are both $-1$, then $M_n$ divides\u2026","title":"Divisibility of Lehmer's companion sequence by a Mersenne prime, Diophantine form"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/pell-euler-solutions","kind":"node","lean":"MetaMathlibExt.pell_euler_solutions","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler's family of solutions of "],["math","\\(aT^2 - bU^2 = c\\)"]],"source":"Euler's family of solutions of $aT^2 - bU^2 = c$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#pell-euler-solutions","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $a$, $b$, $c$ be integers with $ab > 1$, $ab$ not a perfect square and $c \\ne 0$, and let $T_1$, $U_1$ be integers with $aT_1^2 - bU_1^2 = c$. Let $\\varepsilon = x_1 + y_1\\sqrt{ab}$ be the fundamental solution of $x^2 - ab\\,y^2 = 1$ and, for every natural number $j$, write $\\varepsilon^j = x_j + y_j\\sqrt{ab}$. Then for every $j \\ge 0$ the integers $T = T_1 x_j + b U_1 y_j$ and $U = a T_1 y_j + U_1 x_j$ satisfy\u2026","title":"Euler's family of solutions of $aT^2 - bU^2 = c$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/is-odd-squarefree-part","kind":"node","lean":"MetaMathlibExt.IsOddSquarefreePart","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#is-odd-squarefree-part","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"For an integer $N$ and a natural number $k$, $k$ is an odd squarefree part of $N$, written $N^* = k$, when $k$ is odd and squarefree and $N = 2^a m^2 k$ for some natural number $a$ and some integer $m$.","title":"Odd squarefree part"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/is-odd-squarefree-part-360","kind":"node","lean":"MetaMathlibExt.isOddSquarefreePart_360","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odd squarefree part of "],["math","\\(360\\)"]],"source":"The odd squarefree part of $360$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#is-odd-squarefree-part-360","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"The number $5$ is the odd squarefree part of $360$: $5$ is odd and squarefree, and $360 = 2^a m^2 \\cdot 5$ for some natural number $a$ and some integer $m$.","title":"The odd squarefree part of $360$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-omega","kind":"node","lean":"MetaMathlibExt.mersenneOmega","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","The order "],["math","\\(\\Omega(m)\\)"],["text"," of "],["math","\\(2\\)"],["text",", doubled when odd"]],"source":"The order $\\Omega(m)$ of $2$, doubled when odd"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-omega","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"For a natural number $m$, let $\\omega(m)$ be the multiplicative order of $2$ modulo $m$, taken to be $0$ when no positive power of $2$ is congruent to $1$ modulo $m$ (that is, when $m$ is even). Then $\\Omega(m) = 2\\omega(m)$ if $\\omega(m)$ is odd, and $\\Omega(m) = \\omega(m)$ otherwise.","title":"The order $\\Omega(m)$ of $2$, doubled when odd"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-omega-1365","kind":"node","lean":"MetaMathlibExt.mersenneOmega_1365","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\Omega\\)"],["text"," for the seed "],["math","\\(37\\)"]],"source":"$\\Omega$ for the seed $37$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-omega-1365","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"We have $\\Omega(1365) = 12$, where $\\Omega(m)$ is the multiplicative order of $2$ modulo $m$, doubled when that order is odd. Here $1365 = 35 \\cdot 39$ is the product of the odd squarefree parts $35$ of $37 - 2$ and $39$ of $37 + 2$, the parts $s_1^-$ and $s_1^+$ for the seed $s_1 = 37$.","title":"$\\Omega$ for the seed $37$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-eta-condition","kind":"node","lean":"MetaMathlibExt.MersenneEtaCondition","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Condition (etar) on Jacobi symbols of "],["math","\\(M_n\\)"]],"source":"Condition (etar) on Jacobi symbols of $M_n$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-eta-condition","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"Let $s^-$, $s^+$ and $n$ be natural numbers and let $M_n = 2^n - 1$. For a natural number $s$ write $\\left(\\frac{M_n}{s}\\right)$ for the Jacobi symbol, the product of the Legendre symbols $\\left(\\frac{M_n}{p}\\right)$ over the prime factors $p$ of $s$ counted with multiplicity (so it is $1$ for $s = 0$ and $s = 1$). Condition (etar) holds for $s^-$, $s^+$ and $n$ when $s^- = 1$ or\u2026","title":"Condition (etar) on Jacobi symbols of $M_n$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-eta-condition-37","kind":"node","lean":"MetaMathlibExt.mersenneEtaCondition_37","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Condition (etar) for the seed "],["math","\\(37\\)"]],"source":"Condition (etar) for the seed $37$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-eta-condition-37","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"fully_proved","summary":"For the seed $s_1 = 37$, the odd squarefree parts of $s_1 - 2 = 35$ and $s_1 + 2 = 39$ are $s_1^- = 35$ and $s_1^+ = 39$. Condition (etar) for $s_1^- = 35$ and $s_1^+ = 39$ holds for $n = 7$ and fails for $n = 1$.","title":"Condition (etar) for the seed $37$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/infinite-acceptable-seeds-same-parts","kind":"node","lean":"MetaMathlibExt.infinite_acceptableSeeds_same_parts","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#infinite-acceptable-seeds-same-parts","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $n \\ge 3$ be a natural number and $M_n = 2^n - 1$. For an integer $S$ put $S_1 = S$ and $S_{k+1} = S_k^2 - 2$, and call $S$ an acceptable seed for $M_n$ if $M_n$ is prime exactly when $M_n$ divides $S_{n-1}$. Let $s_1 \\ge 3$ be an integer that is an acceptable seed for $M_n$, and let $s_1^-$ and $s_1^+$ be the odd squarefree parts of $s_1 - 2$ and $s_1 + 2$. Then there are infinitely many integers $S \\ge 3$\u2026","title":"Infinitely many acceptable seeds with the same odd squarefree parts"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/order-of-two-mul-coprime","kind":"node","lean":"MetaMathlibExt.orderOf_two_mul_coprime","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Order of "],["math","\\(2\\)"],["text"," modulo a product of coprime odd numbers"]],"source":"Order of $2$ modulo a product of coprime odd numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#order-of-two-mul-coprime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $m_1$ and $m_2$ be coprime odd natural numbers, and write $\\omega(m)$ for the multiplicative order of $2$ modulo $m$. Then $\\omega(m_1 m_2) = \\operatorname{lcm}(\\omega(m_1), \\omega(m_2))$.","title":"Order of $2$ modulo a product of coprime odd numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/one-lt-gcd-mersenne-omega-of-legendre-sy","kind":"node","lean":"MetaMathlibExt.one_lt_gcd_mersenneOmega_of_legendreSym_eq_zero","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","A zero Legendre symbol forces "],["math","\\(\\gcd(n, \\Omega) > 1\\)"]],"source":"A zero Legendre symbol forces $\\gcd(n, \\Omega) > 1$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#one-lt-gcd-mersenne-omega-of-legendre-sy","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $s_1 \\ge 3$ be an integer, let $s_1^-$ and $s_1^+$ be the odd squarefree parts of $s_1 - 2$ and $s_1 + 2$, and let $\\Omega = \\Omega(s_1^- s_1^+)$. Let $p$ be a prime dividing $s_1^- s_1^+$, let $n$ be a natural number and let $M_n = 2^n - 1$. If the Legendre symbol $\\left(\\frac{M_n}{p}\\right)$ is $0$, then $\\gcd(n, \\Omega) > 1$.","title":"A zero Legendre symbol forces $\\gcd(n, \\Omega) > 1$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/exists-not-mersenne-eta-condition-of-min","kind":"node","lean":"MetaMathlibExt.exists_not_mersenneEtaCondition_of_minus","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Condition (etar) fails for some reduced residue when "],["math","\\(s_1^- \\equiv 3 \\pmod 4\\)"]],"source":"Condition (etar) fails for some reduced residue when $s_1^- \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#exists-not-mersenne-eta-condition-of-min","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $s_1 \\ge 3$ be an integer, let $s_1^-$ and $s_1^+$ be the odd squarefree parts of $s_1 - 2$ and $s_1 + 2$, and let $\\Omega = \\Omega(s_1^- s_1^+)$ be the order of $2$ modulo $s_1^- s_1^+$, doubled when it is odd. If $s_1^- \\equiv 3 \\pmod 4$, then there is a natural number $n$ with $1 \\le n < \\Omega$ and $\\gcd(n, \\Omega) = 1$ for which condition (etar) for $s_1^-$, $s_1^+$ and $n$ fails.","title":"Condition (etar) fails for some reduced residue when $s_1^- \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/exists-not-mersenne-eta-condition-of-plu","kind":"node","lean":"MetaMathlibExt.exists_not_mersenneEtaCondition_of_plus","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Condition (etar) fails for some reduced residue when "],["math","\\(s_1^+ \\equiv 1 \\pmod 4\\)"]],"source":"Condition (etar) fails for some reduced residue when $s_1^+ \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#exists-not-mersenne-eta-condition-of-plu","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $s_1 \\ge 3$ be an integer, let $s_1^-$ and $s_1^+$ be the odd squarefree parts of $s_1 - 2$ and $s_1 + 2$, and let $\\Omega = \\Omega(s_1^- s_1^+)$ be the order of $2$ modulo $s_1^- s_1^+$, doubled when it is odd. If $s_1^+ \\equiv 1 \\pmod 4$, then there is a natural number $n$ with $1 \\le n < \\Omega$ and $\\gcd(n, \\Omega) = 1$ for which condition (etar) for $s_1^-$, $s_1^+$ and $n$ fails.","title":"Condition (etar) fails for some reduced residue when $s_1^+ \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/minus-mod-four-of-mersenne-eta-condition","kind":"node","lean":"MetaMathlibExt.minus_mod_four_of_mersenneEtaCondition_all","parent":7268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residues mod "],["math","\\(4\\)"],["text"," of "],["math","\\(s_1^-\\)"],["text"," and "],["math","\\(s_1^+\\)"],["text"," when (etar) always holds"]],"source":"Residues mod $4$ of $s_1^-$ and $s_1^+$ when (etar) always holds"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#minus-mod-four-of-mersenne-eta-condition","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $s_1 \\ge 3$ be an integer, let $s_1^-$ and $s_1^+$ be the odd squarefree parts of $s_1 - 2$ and $s_1 + 2$, and let $\\Omega = \\Omega(s_1^- s_1^+)$. If the number of integers $n$ with $1 \\le n < \\Omega$ and $\\gcd(n, \\Omega) = 1$ for which condition (etar) holds for $s_1^-$, $s_1^+$ and $n$ equals $\\varphi(\\Omega)$, then $s_1^- \\equiv 1 \\pmod 4$ and $s_1^+ \\equiv 3 \\pmod 4$.","title":"Residues mod $4$ of $s_1^-$ and $s_1^+$ when (etar) always holds"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/mersenne-eta-condition-not-all-of-odd","kind":"node","lean":"MetaMathlibExt.mersenneEtaCondition_not_all_of_odd","parent":7268,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test/index.html#mersenne-eta-condition-not-all-of-odd","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/lucas-lehmer-test","status":"can_prove","summary":"Let $s_1 \\ge 3$ be an odd integer, let $s_1^-$ and $s_1^+$ be the odd squarefree parts of $s_1 - 2$ and $s_1 + 2$, and let $\\Omega = \\Omega(s_1^- s_1^+)$. Then the number of integers $n$ with $1 \\le n < \\Omega$ and $\\gcd(n, \\Omega) = 1$ for which condition (etar) holds for $s_1^-$, $s_1^+$ and $n$ is not equal to $\\varphi(\\Omega)$.","title":"Condition (etar) fails for some reduced residue when the seed is odd"},{"data_url":"scopes/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes.json","id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","kind":"scope","navigation":"module","parent":7170,"result":false,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","title":"Strong pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/is-strong-pseudoprime","kind":"node","lean":"MetaMathlibExt.IsStrongPseudoprime","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#is-strong-pseudoprime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"fully_proved","summary":"Let $n$ be a natural number and $b$ a residue class modulo $n$. Then $n$ is a strong pseudoprime to the base $b$ if $n$ is an odd composite number (that is, $n > 1$, $n$ is odd and $n$ is not prime) and there are natural numbers $k$ and $d$ with $d$ odd and $n - 1 = 2^k d$ such that either $b^d = 1$ or $b^{2^i d} = -1$ for some $i < k$, where the powers are computed in $\\mathbb{Z}/n\\mathbb{Z}$.","title":"Strong pseudoprime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/is-strong-pseudoprime-2047-two","kind":"node","lean":"MetaMathlibExt.isStrongPseudoprime_2047_two","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#is-strong-pseudoprime-2047-two","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"fully_proved","summary":"The number $2047 = 23 \\cdot 89$ is a strong pseudoprime to the base $2$.","title":"2047 is a strong pseudoprime to base 2"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/is-strong-pseudoprime-one","kind":"node","lean":"MetaMathlibExt.isStrongPseudoprime_one","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#is-strong-pseudoprime-one","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"fully_proved","summary":"Every odd composite number $n$ is a strong pseudoprime to the base $1$.","title":"Every odd composite is a strong pseudoprime to base 1"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/is-strong-pseudoprime-iff","kind":"node","lean":"MetaMathlibExt.isStrongPseudoprime_iff","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#is-strong-pseudoprime-iff","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"fully_proved","summary":"Let $n$ be a natural number and $b$ a residue class modulo $n$. Then $n$ is a strong pseudoprime to the base $b$ if and only if $n > 1$, $n$ is not prime, $n$ is odd, and there are natural numbers $k$ and $d$ with $n - 1 = 2^k d$ and $d$ odd such that $b^d = 1$ or $b^{2^i d} = -1$ in $\\mathbb{Z}/n\\mathbb{Z}$ for some $i < k$.","title":"Unfolded definition of strong pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/one-lt","kind":"node","lean":"MetaMathlibExt.IsStrongPseudoprime.one_lt","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#one-lt","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"fully_proved","summary":"If $n$ is a strong pseudoprime to some base $b$ modulo $n$, then $n > 1$.","title":"Strong pseudoprimes exceed one"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/not-prime","kind":"node","lean":"MetaMathlibExt.IsStrongPseudoprime.not_prime","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#not-prime","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"fully_proved","summary":"If $n$ is a strong pseudoprime to some base $b$ modulo $n$, then $n$ is not prime.","title":"Strong pseudoprimes are not prime"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/odd","kind":"node","lean":"MetaMathlibExt.IsStrongPseudoprime.odd","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#odd","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"fully_proved","summary":"If $n$ is a strong pseudoprime to some base $b$ modulo $n$, then $n$ is odd.","title":"Strong pseudoprimes are odd"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/card-fermat-psp-bases-eq-prod-gcd","kind":"node","lean":"MetaMathlibExt.card_fermatPsp_bases_eq_prod_gcd","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#card-fermat-psp-bases-eq-prod-gcd","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Let $n > 1$ be a composite natural number. Then the number of integers $b$ with $1 \\le b < n$ such that $n$ is a Fermat pseudoprime to the base $b$, that is, $b^{n-1} \\equiv 1 \\pmod{n}$, equals $\\prod_p \\gcd(n - 1, p - 1)$, where $p$ runs over the distinct prime divisors of $n$.","title":"Number of Fermat pseudoprime bases (Monier; Baillie\u2013Wagstaff)"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/is-square-card-fermat-psp-bases-mul-of-p","kind":"node","lean":"MetaMathlibExt.isSquare_card_fermatPsp_bases_mul_of_prime","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#is-square-card-fermat-psp-bases-mul-of-p","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Let $p$ and $q$ be distinct primes. Then the number of integers $b$ with $1 \\le b < pq$ such that $pq$ is a Fermat pseudoprime to the base $b$, that is, $b^{pq-1} \\equiv 1 \\pmod{pq}$, is a perfect square.","title":"Products of two distinct primes have a square number of Fermat pseudoprime bases"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/card-strong-pseudoprime-bases","kind":"node","lean":"MetaMathlibExt.card_strongPseudoprime_bases","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#card-strong-pseudoprime-bases","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Let $n$ be an odd composite natural number, let $r$ be the number of distinct prime divisors of $n$, and let $v$ be the least of the $2$-adic valuations of $p - 1$ over the prime divisors $p$ of $n$. Write $d$ for the odd part of $n - 1$ and $d_p$ for the odd part of $p - 1$. Then the number of integers $b$ with $1 \\le b < n$ such that $n$ is a strong pseudoprime to the base $b$ equals, as a rational number,","title":"Monier's formula for the number of strong pseudoprime bases"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/strong-pseudoprime-bases-subgroup-tfae","kind":"node","lean":"MetaMathlibExt.strongPseudoprime_bases_subgroup_tfae","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#strong-pseudoprime-bases-subgroup-tfae","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Let $n > 1$ be an odd number that is not prime, let $r$ be the number of distinct prime factors of $n$, and let $v$ be the least exponent of $2$ in $p - 1$ over the prime factors $p$ of $n$. Then the following are equivalent: (a) there is a subgroup of the unit group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ whose elements are exactly the units $u$ such that $n$ is a strong pseudoprime to the base $u$; (b) the number of\u2026","title":"When the strong pseudoprime bases form a subgroup"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/has-density-zero-strong-pseudoprime-base","kind":"node","lean":"MetaMathlibExt.hasDensity_zero_strongPseudoprime_bases_not_subgroup","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#has-density-zero-strong-pseudoprime-base","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"The set of odd composite natural numbers $n$ for which the units $b$ of $\\mathbb{Z}/n\\mathbb{Z}$ such that $n$ is a strong pseudoprime to the base $b$ do not form a subgroup of the unit group has natural density $0$.","title":"Strong pseudoprime bases almost always form a subgroup"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/has-density-zero-strong-pseudoprime-base-2","kind":"node","lean":"MetaMathlibExt.hasDensity_zero_strongPseudoprime_bases_conditions_fail","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#has-density-zero-strong-pseudoprime-base-2","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"For an odd composite natural number $n$, let $r$ be the number of distinct prime divisors of $n$, let $v$ be the least of the $2$-adic valuations of $p - 1$ over the prime divisors $p$ of $n$, and consider the conditions: (a) the units $b$ of $\\mathbb{Z}/n\\mathbb{Z}$ such that $n$ is a strong pseudoprime to the base $b$ form a subgroup of the unit group; (b) the number of integers $b$ with $1 \\le b < n$ to which\u2026","title":"The subgroup criteria for strong pseudoprime bases hold almost always"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/is-strong-pseudoprime-of-dvd-primitive-p","kind":"node","lean":"MetaMathlibExt.isStrongPseudoprime_of_dvd_primitivePart","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#is-strong-pseudoprime-of-dvd-primitive-p","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Composite divisors of primitive parts without their intrinsic factor are strong pseudoprimes. Let $b > 1$ and $m > 2$ be natural numbers, and let $\\Phi_j$ denote the $j$-th cyclotomic polynomial. If a composite natural number $c > 1$ divides $\\Phi_m(b) / \\gcd(\\Phi_m(b), m)$ or $\\Phi_{2m}(b) / \\gcd(\\Phi_{2m}(b), m)$, then $c$ is a strong pseudoprime to the base $b$.","title":"Composite divisors of primitive parts without their intrinsic factor are strong pseudoprimes"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/is-strong-pseudoprime-zpow-of-dvd-primit","kind":"node","lean":"MetaMathlibExt.isStrongPseudoprime_zpow_of_dvd_primitivePart","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#is-strong-pseudoprime-zpow-of-dvd-primit","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Composite divisors of primitive parts without the intrinsic factor are strong pseudoprimes to every power of the base. Let $b > 1$ and $m > 2$ be natural numbers, write $\\Phi_j(b)$ for the value at $b$ of the $j$-th cyclotomic polynomial over $\\mathbb{Z}$, and let $c > 1$ be a natural number that is not prime. If $c$ divides $\\Phi_m(b) / \\gcd(\\Phi_m(b), m)$ or $\\Phi_{2m}(b) / \\gcd(\\Phi_{2m}(b), m)$, then for every\u2026","title":"Composite divisors of primitive parts without the intrinsic factor are strong pseudoprimes to every power of the base"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/is-strong-pseudoprime-zpow-and-neg","kind":"node","lean":"MetaMathlibExt.isStrongPseudoprime_zpow_and_neg","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#is-strong-pseudoprime-zpow-and-neg","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Let $n$ be a natural number and $b$ a unit of $\\mathbb{Z}/n\\mathbb{Z}$ such that $n$ is a strong pseudoprime to the base $b$. Then for every integer $t$, $n$ is a strong pseudoprime to the base $b^t$, and $n$ is a strong pseudoprime to the base $-b$.","title":"Strong pseudoprime bases are closed under powers and negation"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/fermat-number-is-strong-pseudoprime-two","kind":"node","lean":"MetaMathlibExt.fermatNumber_isStrongPseudoprime_two_zpow","parent":7291,"reading_title":{"format":"inline-parts-v1","parts":[["text","Composite Fermat numbers are strong pseudoprimes to every power of "],["math","\\(2\\)"]],"source":"Composite Fermat numbers are strong pseudoprimes to every power of $2$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#fermat-number-is-strong-pseudoprime-two","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Let $k$ be a natural number such that the Fermat number $F_k = 2^{2^k} + 1$ is composite. Then for every integer $t$, $F_k$ is a strong pseudoprime to the base $2^t$, where the power is taken in the unit group of $\\mathbb{Z}/F_k\\mathbb{Z}$ (in which $2$ is a unit because $F_k$ is odd).","title":"Composite Fermat numbers are strong pseudoprimes to every power of $2$"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/fermat-number-is-strong-pseudoprime-two-2","kind":"node","lean":"MetaMathlibExt.fermatNumber_isStrongPseudoprime_two_pow_add_sub","parent":7291,"reading_title":{"format":"inline-parts-v1","parts":[["text","Strong pseudoprime bases "],["math","\\(2^{2^{k-1}+t} \\pm 2^t\\)"],["text"," of composite Fermat numbers"]],"source":"Strong pseudoprime bases $2^{2^{k-1}+t} \\pm 2^t$ of composite Fermat numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#fermat-number-is-strong-pseudoprime-two-2","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Let $k$ be a natural number such that the Fermat number $F_k = 2^{2^k} + 1$ is composite. Then for every natural number $t$, $F_k$ is a strong pseudoprime to both bases $2^{2^{k-1}+t} + 2^t$ and $2^{2^{k-1}+t} - 2^t$, computed modulo $F_k$.","title":"Strong pseudoprime bases $2^{2^{k-1}+t} \\pm 2^t$ of composite Fermat numbers"},{"id":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/fermat-number-strong-pseudoprime-bases-s","kind":"node","lean":"MetaMathlibExt.fermatNumber_strongPseudoprime_bases_subgroup","parent":7291,"result":true,"route":"../roadmap/NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes/index.html#fermat-number-strong-pseudoprime-bases-s","scope":"NumberTheory/primes-primality/primality-tests-pseudoprimes/strong-pseudoprimes","status":"can_prove","summary":"Let $k$ be a natural number such that the Fermat number $F_k = 2^{2^k} + 1$ is composite, and let $S_k$ be the set of residues modulo $F_k$ of the form $2^t$, $2^{2^{k-1}+t} + 2^t$ or $2^{2^{k-1}+t} - 2^t$ with $t$ a natural number. Then the units of $\\mathbb{Z}/F_k\\mathbb{Z}$ that lie in $S_k$ form a subgroup of the unit group of order $2^{k+2}$, and $F_k$ is a strong pseudoprime to every base in $S_k$.","title":"A subgroup of strong pseudoprime bases of a composite Fermat number"},{"data_url":"scopes/NumberTheory/primes-primality/special-primes.json","id":"NumberTheory/primes-primality/special-primes","kind":"scope","parent":7169,"result":false,"route":"../roadmap/NumberTheory/primes-primality/special-primes/index.html","scope":"NumberTheory/primes-primality/special-primes","status":"can_prove","summary":"Fermat, Mersenne, Sophie Germain, Wieferich, Cullen and other named prime families, Fermat numbers, Sierpi\u0144ski\u2013Riesel numbers, prime-producing formulas.","title":"Special classes of primes"},{"id":"NumberTheory/primes-primality/special-primes/is-fortunate-number","kind":"node","lean":"MetaMathlibExt.IsFortunateNumber","parent":7311,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/index.html#is-fortunate-number","scope":"NumberTheory/primes-primality/special-primes","status":"fully_proved","summary":"Write $q\\#$ for the primorial of a natural number $q$, the product of all primes $p \\le q$. For natural numbers $q$ and $m$, $m$ is the Fortunate number of $q$ when $q$ is prime, $m > 1$, $q\\# + m$ is prime, and $q\\# + k$ is not prime for any $k$ with $1 < k < m$.","title":"Fortunate numbers"},{"id":"NumberTheory/primes-primality/special-primes/gandhi-nth-prime-formula","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.GandhiNthPrimeFormulaWanted.gandhi_nthPrimeFormula","parent":7311,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gandhi's formula for the "],["math","\\(n\\)"],["text","th prime"]],"source":"Gandhi's formula for the $n$th prime"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/index.html#gandhi-nth-prime-formula","scope":"NumberTheory/primes-primality/special-primes","status":"fully_proved","summary":"For $i \\ge 1$ let $p_i$ be the $i$th prime, so that $p_1 = 2$, and for $n \\ge 1$ let $P_{n-1} = p_1 p_2 \\cdots p_{n-1}$, the empty product $1$ when $n = 1$. Let $\\mu$ be the M\u00f6bius function. Then for every $n \\ge 1$,","title":"Gandhi's formula for the $n$th prime"},{"id":"NumberTheory/primes-primality/special-primes/generalized-fermat-number","kind":"node","lean":"MetaMathlibExt.generalizedFermatNumber","parent":7311,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/index.html#generalized-fermat-number","scope":"NumberTheory/primes-primality/special-primes","status":"fully_proved","summary":"For an even natural number $b$ and a positive integer $n$, the generalized Fermat number is $F_n(b) = b^{2^n} + 1$ (also written $F_{b,n}$).","title":"Generalized Fermat number"},{"data_url":"scopes/NumberTheory/primes-primality/special-primes/cullen-numbers.json","id":"NumberTheory/primes-primality/special-primes/cullen-numbers","kind":"scope","navigation":"module","parent":7311,"result":false,"route":"../roadmap/NumberTheory/primes-primality/special-primes/cullen-numbers/index.html","scope":"NumberTheory/primes-primality/special-primes/cullen-numbers","status":"fully_proved","title":"Cullen numbers"},{"id":"NumberTheory/primes-primality/special-primes/cullen-numbers/s-cullen-number","kind":"node","lean":"MetaMathlibExt.sCullenNumber","parent":7315,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(s\\)"],["text","-Cullen value"]],"source":"The $s$-Cullen value"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/cullen-numbers/index.html#s-cullen-number","scope":"NumberTheory/primes-primality/special-primes/cullen-numbers","status":"fully_proved","summary":"For natural numbers $s$ and $n$, the $s$-Cullen value is $C_{n,s} = n s^n + 1$.","title":"The $s$-Cullen value"},{"id":"NumberTheory/primes-primality/special-primes/cullen-numbers/is-scullen-number","kind":"node","lean":"MetaMathlibExt.IsSCullenNumber","parent":7315,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(s\\)"],["text","-Cullen numbers"]],"source":"$s$-Cullen numbers"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/cullen-numbers/index.html#is-scullen-number","scope":"NumberTheory/primes-primality/special-primes/cullen-numbers","status":"fully_proved","summary":"For natural numbers $s$ and $n$, write $C_{n,s} = n s^n + 1$ for the $s$-Cullen value. A natural number $m$ is an $s$-Cullen number when $s \\ge 2$ and $m = C_{n,s}$ for some natural number $n \\ge 1$.","title":"$s$-Cullen numbers"},{"data_url":"scopes/NumberTheory/primes-primality/special-primes/fermat-numbers.json","id":"NumberTheory/primes-primality/special-primes/fermat-numbers","kind":"scope","navigation":"module","parent":7311,"result":false,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","title":"Fermat numbers"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/is-fermat-prime","kind":"node","lean":"MetaMathlibExt.IsFermatPrime","parent":7318,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#is-fermat-prime","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"fully_proved","summary":"A natural number $p$ is a Fermat prime if $p$ is prime and $p = 2^{2^k} + 1$ for some natural number $k$, that is, $p$ is a prime Fermat number.","title":"Fermat prime"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/fermat-number-mod-prime-exact-preperiod","kind":"node","lean":"Nat.fermatNumber_mod_prime_exact_preperiod_period","parent":7318,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#fermat-number-mod-prime-exact-preperiod","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"fully_proved","summary":"Let $p$ be an odd prime and write the multiplicative order of $2$ modulo $p$ as $2^v w$ with $v, w$ natural numbers and $w$ odd. Let $k$ be the multiplicative order of $2$ modulo $w$ (so $k = 1$ when $w = 1$), and let $F_m = 2^{2^m} + 1$ be the $m$-th Fermat number. Then the sequence $n \\mapsto F_{v+n} \\bmod p$ is periodic with period $k$; for every $h < v$, the sequence $n \\mapsto F_{h+n} \\bmod p$ has no positive\u2026","title":"Preperiod and period of Fermat numbers modulo an odd prime"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/fermat-number-mod-eq-exists-repeat","kind":"node","lean":"MetaMathlibExt.fermatNumber_modEq_exists_repeat","parent":7318,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two of "],["math","\\(F_0, \\dots, F_p\\)"],["text"," are congruent modulo "],["math","\\(p\\)"]],"source":"Two of $F_0, \\dots, F_p$ are congruent modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#fermat-number-mod-eq-exists-repeat","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Write $F_n = 2^{2^n} + 1$. For every positive integer $p$ there are natural numbers $i < j \\le p$ with $F_i \\equiv F_j \\pmod p$.","title":"Two of $F_0, \\dots, F_p$ are congruent modulo $p$"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/fermat-number-mod-eq-add","kind":"node","lean":"MetaMathlibExt.fermatNumber_modEq_add","parent":7318,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#fermat-number-mod-eq-add","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Write $F_n = 2^{2^n} + 1$. Let $m$, $i$ and $j$ be natural numbers with $F_i \\equiv F_j \\pmod m$. Then $F_{i+t} \\equiv F_{j+t} \\pmod m$ for every natural number $t$.","title":"Congruences between Fermat numbers propagate"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/fermat-number-not-mod-eq-add-of-le","kind":"node","lean":"MetaMathlibExt.fermatNumber_not_modEq_add_of_le","parent":7318,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#fermat-number-not-mod-eq-add-of-le","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Write $F_n = 2^{2^n} + 1$ for the $n$-th Fermat number. Let $m$ be a natural number and let $i \\le j$. If $F_j \\not\\equiv F_{j+t} \\pmod{m}$ for every $t \\ge 1$, then $F_i \\not\\equiv F_{i+t} \\pmod{m}$ for every $t \\ge 1$.","title":"Non-congruence of Fermat numbers passes to smaller indices"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/fermat-number-mod-eq-iff-preperiod-perio","kind":"node","lean":"MetaMathlibExt.fermatNumber_modEq_iff_preperiod_period","parent":7318,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#fermat-number-mod-eq-iff-preperiod-perio","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Write $F_n = 2^{2^n} + 1$. Let $p$ be a prime, and let $h$ and $k$ be the preperiod and the period of $n \\mapsto F_n \\bmod p$ in the following sense: $k > 0$, the sequence $n \\mapsto F_{h+n} \\bmod p$ is periodic with period $k$, every positive period of it is a multiple of $k$, and for no $h' < h$ is the sequence $n \\mapsto F_{h'+n} \\bmod p$ periodic with a positive period. Then for all natural numbers $i < j$,\u2026","title":"Coincidences among Fermat residues"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/gcd-card-sub-one-exponent-eq-div-order-o","kind":"node","lean":"MetaMathlibExt.gcd_card_sub_one_exponent_eq_div_orderOf","parent":7318,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#gcd-card-sub-one-exponent-eq-div-order-o","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $g$ be a primitive root modulo $p$ (an element of $\\mathbb{Z}/p\\mathbb{Z}$ of multiplicative order $p - 1$), and let $a \\in \\mathbb{Z}/p\\mathbb{Z}$. If $g^i = a$ for a natural number $i$, then $\\gcd(p - 1, i) = (p - 1)/\\operatorname{ord}_p(a)$, where $\\operatorname{ord}_p(a)$ is the multiplicative order of $a$ modulo $p$.","title":"The index of a residue and its multiplicative order"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/exists-coprime-mod-eq-of-coprime","kind":"node","lean":"MetaMathlibExt.exists_coprime_modEq_of_coprime","parent":7318,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#exists-coprime-mod-eq-of-coprime","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Let $\\alpha$, $\\beta$ and $\\gamma$ be natural numbers with $\\gamma > 0$ and $\\gcd(\\alpha, \\beta) = 1$. Then there is a natural number $c > 0$ with $\\gcd(c, \\gamma) = 1$ and $c \\equiv \\beta \\pmod{\\alpha}$.","title":"Coprime representatives of a residue class"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/exists-primitive-root-pow-card-sub-one-d","kind":"node","lean":"MetaMathlibExt.exists_primitiveRoot_pow_card_sub_one_div_orderOf","parent":7318,"reading_title":{"format":"inline-parts-v1","parts":[["text","A primitive root in which a residue has index "],["math","\\((p-1)/\\operatorname{ord}_p(a)\\)"]],"source":"A primitive root in which a residue has index $(p-1)/\\operatorname{ord}_p(a)$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#exists-primitive-root-pow-card-sub-one-d","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Let $p$ be a prime and let $a$ be a nonzero residue modulo $p$, of multiplicative order $\\operatorname{ord}_p(a)$. Then there is a primitive root $g$ modulo $p$ (an element of order $p - 1$ in $\\mathbb{Z}/p\\mathbb{Z}$) with $g^{(p-1)/\\operatorname{ord}_p(a)} = a$.","title":"A primitive root in which a residue has index $(p-1)/\\operatorname{ord}_p(a)$"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/exponent-dvd-card-sub-one-iff","kind":"node","lean":"MetaMathlibExt.exponent_dvd_card_sub_one_iff","parent":7318,"reading_title":{"format":"inline-parts-v1","parts":[["text","Indices dividing "],["math","\\(p - 1\\)"]],"source":"Indices dividing $p - 1$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#exponent-dvd-card-sub-one-iff","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $g$ be a primitive root modulo $p$ (an element of order $p - 1$ in $\\mathbb{Z}/p\\mathbb{Z}$), let $a \\in \\mathbb{Z}/p\\mathbb{Z}$ and let $i$ be a natural number with $g^i = a$. Then $i$ divides $p - 1$ if and only if $i = (p - 1)/\\operatorname{ord}_p(a)$, where $\\operatorname{ord}_p(a)$ is the multiplicative order of $a$.","title":"Indices dividing $p - 1$"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/exists-unique-two-adic-index-of-two","kind":"node","lean":"MetaMathlibExt.existsUnique_twoAdic_index_of_two","parent":7318,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniqueness of the "],["math","\\(2\\)"],["text","-adic decomposition of the index of "],["math","\\(2\\)"]],"source":"Uniqueness of the $2$-adic decomposition of the index of $2$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#exists-unique-two-adic-index-of-two","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Let $p$ be an odd prime and write $p - 1 = 2^r u$ with $u$ odd. Then there is exactly one pair $(s, t)$ of natural numbers with $s \\le r$ and $t \\mid u$ such that $g^{2^s t} = 2$ for some primitive root $g$ modulo $p$ (an element of order $p - 1$ in $\\mathbb{Z}/p\\mathbb{Z}$).","title":"Uniqueness of the $2$-adic decomposition of the index of $2$"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/fermat-number-mod-prime-preperiod-period","kind":"node","lean":"MetaMathlibExt.fermatNumber_mod_prime_preperiod_period_of_index","parent":7318,"reading_title":{"format":"inline-parts-v1","parts":[["text","Preperiod and period of Fermat residues from the index of "],["math","\\(2\\)"]],"source":"Preperiod and period of Fermat residues from the index of $2$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#fermat-number-mod-prime-preperiod-period","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Write $F_n = 2^{2^n} + 1$. Let $p$ be an odd prime, write $p - 1 = 2^r u$ with $u$ odd, and let $s \\le r$ and $t \\mid u$ be natural numbers such that $g^{2^s t} = 2$ for some primitive root $g$ modulo $p$ (an element of order $p - 1$ in $\\mathbb{Z}/p\\mathbb{Z}$). Let $k$ be the multiplicative order of $2$ modulo $u/t$, with $k = 1$ when $t = u$. Then the sequence $n \\mapsto F_{r-s+n} \\bmod p$ is periodic with\u2026","title":"Preperiod and period of Fermat residues from the index of $2$"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/prime-dvd-fermat-number-iff-order-of-two","kind":"node","lean":"MetaMathlibExt.prime_dvd_fermatNumber_iff_orderOf_two","parent":7318,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime divisors of Fermat numbers and the order of "],["math","\\(2\\)"]],"source":"Prime divisors of Fermat numbers and the order of $2$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#prime-dvd-fermat-number-iff-order-of-two","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Write $F_n = 2^{2^n} + 1$ for the $n$-th Fermat number. An odd prime $p$ divides $F_n$ if and only if the multiplicative order of $2$ modulo $p$ equals $2^{n+1}$.","title":"Prime divisors of Fermat numbers and the order of $2$"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/exists-prime-dvd-fermat-number-iff-perio","kind":"node","lean":"MetaMathlibExt.exists_prime_dvd_fermatNumber_iff_period_eq_one","parent":7318,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#exists-prime-dvd-fermat-number-iff-perio","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Write $F_n = 2^{2^n} + 1$ for the $n$-th Fermat number. Let $p$ be an odd prime, let $h$ be a natural number and let $k$ be a positive integer such that the sequence $n \\mapsto F_{h+n} \\bmod p$ is periodic with period $k$ and every positive period of it is a multiple of $k$. Then $p$ divides $F_n$ for some natural number $n$ if and only if $k = 1$.","title":"Prime divisors of Fermat numbers and the period of Fermat residues"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/order-of-two-two-adic-of-mod-eight","kind":"node","lean":"MetaMathlibExt.orderOf_two_twoAdic_of_mod_eight","parent":7318,"reading_title":{"format":"inline-parts-v1","parts":[["text","The power of "],["math","\\(2\\)"],["text"," dividing the order of "],["math","\\(2\\)"],["text"," modulo a prime"]],"source":"The power of $2$ dividing the order of $2$ modulo a prime"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#order-of-two-two-adic-of-mod-eight","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Let $p$ be a prime and let $\\operatorname{ord}_p(2)$ be the multiplicative order of $2$ modulo $p$. If $p \\equiv 7 \\pmod 8$, then $\\operatorname{ord}_p(2)$ is odd. If $p \\equiv 3 \\pmod 8$, then $\\operatorname{ord}_p(2)$ is divisible by $2$ but not by $4$. If $p \\equiv 5 \\pmod 8$, then $\\operatorname{ord}_p(2)$ is divisible by $4$ but not by $8$.","title":"The power of $2$ dividing the order of $2$ modulo a prime"},{"id":"NumberTheory/primes-primality/special-primes/fermat-numbers/exists-prime-one-mod-eight-fermat-preper","kind":"node","lean":"MetaMathlibExt.exists_prime_one_mod_eight_fermat_preperiod","parent":7318,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every preperiod of Fermat residues occurs for a prime "],["math","\\(p \\equiv 1 \\pmod 8\\)"]],"source":"Every preperiod of Fermat residues occurs for a prime $p \\equiv 1 \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/fermat-numbers/index.html#exists-prime-one-mod-eight-fermat-preper","scope":"NumberTheory/primes-primality/special-primes/fermat-numbers","status":"can_prove","summary":"Write $F_n = 2^{2^n} + 1$. For every natural number $v$ there is a prime $p \\equiv 1 \\pmod 8$ such that the sequence $n \\mapsto F_n \\bmod p$ has preperiod exactly $v$: the sequence $n \\mapsto F_{v+n} \\bmod p$ is periodic with some positive period, and for no $h < v$ is the sequence $n \\mapsto F_{h+n} \\bmod p$ periodic with a positive period.","title":"Every preperiod of Fermat residues occurs for a prime $p \\equiv 1 \\pmod 8$"},{"data_url":"scopes/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers.json","id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","kind":"scope","navigation":"module","parent":7311,"result":false,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","title":"Sierpi\u0144ski and Riesel numbers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/exists-arith-prog-sierpinski-add-mersenn","kind":"node","lean":"MathlibExt.NumberTheory.SierpinskiMersenneWanted.exists_arithProg_sierpinski_add_mersenne","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#exists-arith-prog-sierpinski-add-mersenn","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"For a natural number $j$ let $M_j = 2^j - 1$ be the $j$-th Mersenne number (so $M_0 = 0$). For every natural number $t$ there are natural numbers $a$ and $d > 0$ such that, for every natural number $r$, every natural number $j \\le t$ and every natural number $n$, the number $(a + rd + M_j) \\cdot 2^n + 1$ is composite.","title":"Composite numbers from Mersenne shifts of an arithmetic progression"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-sierpinski-number","kind":"node","lean":"MetaMathlibExt.IsSierpinskiNumber","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-sierpinski-number","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"A natural number $k$ is a Sierpi\u0144ski number if $k$ is odd and, for every integer $n \\ge 1$, the number $k \\cdot 2^n + 1$ is not prime.","title":"Sierpi\u0144ski number"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-riesel-number","kind":"node","lean":"MetaMathlibExt.IsRieselNumber","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-riesel-number","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"A natural number $k$ is a Riesel number if $k$ is odd and, for every integer $n \\ge 1$, the number $k \\cdot 2^n - 1$ is not prime.","title":"Riesel number"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/not-is-sierpinski-number-one","kind":"node","lean":"MetaMathlibExt.not_isSierpinskiNumber_one","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(1\\)"],["text"," is not a Sierpi\u0144ski number"]],"source":"$1$ is not a Sierpi\u0144ski number"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#not-is-sierpinski-number-one","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"The number $1$ is not a Sierpi\u0144ski number, since $1 \\cdot 2 + 1 = 3$ is prime.","title":"$1$ is not a Sierpi\u0144ski number"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/not-is-riesel-number-one","kind":"node","lean":"MetaMathlibExt.not_isRieselNumber_one","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#not-is-riesel-number-one","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"The number $1$ is not a Riesel number, since $1 \\cdot 2^2 - 1 = 3$ is prime.","title":"One is not a Riesel number"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/repstring","kind":"node","lean":"MetaMathlibExt.repstring","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#repstring","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"For natural numbers $b$, $k$, $z$ and $t$, let $\\ell$ be the number of base-$b$ digits of $k$. The $b$-repstring is","title":"Repstrings"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/repstring-ten-one-two-three","kind":"node","lean":"MetaMathlibExt.repstring_ten_one_two_three","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(10\\)"],["text","-repstring"]],"source":"A $10$-repstring"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#repstring-ten-one-two-three","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"The $10$-repstring with $k = 1$, $z = 2$ and $t = 3$ is $1_{10}^{(2;3)} = 1001001$.","title":"A $10$-repstring"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/repstring-two-one-zero-three","kind":"node","lean":"MetaMathlibExt.repstring_two_one_zero_three","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","The three-digit "],["math","\\(2\\)"],["text","-repunit"]],"source":"The three-digit $2$-repunit"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#repstring-two-one-zero-three","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"The $2$-repstring with $k = 1$, $z = 0$ and $t = 3$, that is, the $2$-repunit $1_2^{(3)}$ with three digits, equals $7$.","title":"The three-digit $2$-repunit"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-sierpinski-covering-data","kind":"node","lean":"MetaMathlibExt.IsSierpinskiCoveringData","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-sierpinski-covering-data","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"Let $k$ be a natural number and $S$ a finite set of triples $(r, m, p)$ of natural numbers. Then $S$ is Sierpi\u0144ski covering data for $k$ if every integer is congruent to $r$ modulo $m$ for some $(r, m, p) \\in S$, distinct triples in $S$ have distinct third entries, and every $(r, m, p) \\in S$ has $m \\ge 1$, $p$ prime, $p \\mid 2^m - 1$ and $p \\mid k \\cdot 2^r + 1$.","title":"Sierpi\u0144ski covering data"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-riesel-covering-data","kind":"node","lean":"MetaMathlibExt.IsRieselCoveringData","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-riesel-covering-data","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"Let $k$ be a natural number and let $S$ be a finite set of triples $(r, m, p)$ of natural numbers. Then $S$ is Riesel covering data for $k$ (data producing $k$ as a Riesel number by the covering system method) when every integer is congruent to $r$ modulo $m$ for some $(r, m, p) \\in S$, distinct triples in $S$ have distinct third entries $p$, and every $(r, m, p) \\in S$ has $m \\ge 1$, $p$ prime, $p \\mid 2^m - 1$\u2026","title":"Riesel covering data"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/not-is-sierpinski-covering-data-empty","kind":"node","lean":"MetaMathlibExt.not_isSierpinskiCoveringData_empty","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#not-is-sierpinski-covering-data-empty","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"For every natural number $k$, the empty set is not Sierpi\u0144ski covering data for $k$: the empty family of residue classes does not cover $\\mathbb{Z}$.","title":"Empty Sierpi\u0144ski covering data"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/not-is-riesel-covering-data-empty","kind":"node","lean":"MetaMathlibExt.not_isRieselCoveringData_empty","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#not-is-riesel-covering-data-empty","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"fully_proved","summary":"For every natural number $k$, the empty set is not Riesel covering data for $k$, because its residue classes do not cover $\\mathbb{Z}$.","title":"The empty set is not Riesel covering data"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/not-is-sierpinski-covering-data-two-pow","kind":"node","lean":"MetaMathlibExt.not_isSierpinskiCoveringData_two_pow_sub_one","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Theorem 3.2 (no Sierpi\u0144ski "],["math","\\(2\\)"],["text","-repdigits from coverings)"]],"source":"Theorem 3.2 (no Sierpi\u0144ski $2$-repdigits from coverings)"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#not-is-sierpinski-covering-data-two-pow","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"For every natural number $t \\ge 1$, the covering system method cannot produce the $2$-repdigit $2^t - 1$ as a Sierpi\u0144ski number: no finite set of triples $(r_j, m_j, p_j)$ is Sierpi\u0144ski covering data for $2^t - 1$.","title":"Theorem 3.2 (no Sierpi\u0144ski $2$-repdigits from coverings)"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-sierpinski-repintegers","kind":"node","lean":"MetaMathlibExt.infinite_sierpinski_repintegers","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-sierpinski-repintegers","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $k$ be a Sierpi\u0144ski number that has Sierpi\u0144ski covering data, so that $k$ is produced as a Sierpi\u0144ski number by the covering system method. Then for every natural number $b \\ge 2$ there are infinitely many $t \\ge 1$ such that the $b$-repinteger $k_b^{(t)} = k\\,(b^{\\ell t} - 1)/(b^{\\ell} - 1)$, where $\\ell$ is the number of base-$b$ digits of $k$ (so $k_b^{(t)}$ is $k$ written $t$ times in base $b$), is a\u2026","title":"Covering-produced Sierpi\u0144ski numbers give infinitely many Sierpi\u0144ski repintegers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-riesel-repintegers","kind":"node","lean":"MetaMathlibExt.infinite_riesel_repintegers","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-riesel-repintegers","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $k$ be a Riesel number that has Riesel covering data, so that $k$ is produced as a Riesel number by the covering system method. Then for every natural number $b \\ge 2$ there are infinitely many $t \\ge 1$ such that the $b$-repinteger $k_b^{(t)} = k\\,(b^{\\ell t} - 1)/(b^{\\ell} - 1)$, where $\\ell$ is the number of base-$b$ digits of $k$ (so $k_b^{(t)}$ is $k$ written $t$ times in base $b$), is a Riesel number.","title":"Covering-produced Riesel numbers give infinitely many Riesel repintegers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-sierpinski-number-repstring-18107","kind":"node","lean":"MetaMathlibExt.isSierpinskiNumber_repstring_18107","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-sierpinski-number-repstring-18107","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"For every natural number $t \\equiv 25 \\pmod{56}$, the binary repinteger $18107_2^{(t)} = 18107\\,(2^{15t} - 1)/(2^{15} - 1)$, whose binary expansion is that of $18107$ (fifteen binary digits) written $t$ times, is a Sierpi\u0144ski number.","title":"Binary repintegers of 18107 that are Sierpi\u0144ski numbers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-riesel-number-repstring-18107","kind":"node","lean":"MetaMathlibExt.isRieselNumber_repstring_18107","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-riesel-number-repstring-18107","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"For every natural number $t \\equiv 31 \\pmod{56}$, the binary repinteger $18107_2^{(t)} = 18107\\,(2^{15t} - 1)/(2^{15} - 1)$, whose binary expansion is that of $18107$ (fifteen binary digits) written $t$ times, is a Riesel number.","title":"Binary repintegers of 18107 that are Riesel numbers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-sierpinski-repdigits","kind":"node","lean":"MetaMathlibExt.infinite_sierpinski_repdigits","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many Sierpi\u0144ski repdigits in every base "],["math","\\(b > 78557\\)"]],"source":"Infinitely many Sierpi\u0144ski repdigits in every base $b > 78557$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-sierpinski-repdigits","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $b > 78557$ be a natural number. Then there are infinitely many Sierpi\u0144ski numbers of the form $k(b^t - 1)/(b - 1)$ with $1 \\le k < b$ and $t \\ge 1$, that is, $b$-repdigits: the digit $k$ written $t$ times in base $b$.","title":"Infinitely many Sierpi\u0144ski repdigits in every base $b > 78557$"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-riesel-repdigits","kind":"node","lean":"MetaMathlibExt.infinite_riesel_repdigits","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many Riesel repdigits in every base "],["math","\\(b > 509203\\)"]],"source":"Infinitely many Riesel repdigits in every base $b > 509203$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-riesel-repdigits","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $b > 509203$ be a natural number. Then there are infinitely many Riesel numbers of the form $k(b^t - 1)/(b - 1)$ with $1 \\le k < b$ and $t \\ge 1$, that is, $b$-repdigits: the digit $k$ written $t$ times in base $b$.","title":"Infinitely many Riesel repdigits in every base $b > 509203$"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-sierpinski-number-repstring-41","kind":"node","lean":"MetaMathlibExt.isSierpinskiNumber_repstring_41","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-sierpinski-number-repstring-41","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $b$ and $t$ be natural numbers with $b \\equiv 87 \\pmod{11184810}$ and $t \\equiv 59 \\pmod{120}$. Then the base-$b$ repdigit $41_b^{(t)} = 41\\,(b^t - 1)/(b - 1)$ is a Sierpi\u0144ski number. Since $b \\ge 87 > 41$, its $t$ base-$b$ digits all equal $41$.","title":"Repdigits with digit 41 that are Sierpi\u0144ski numbers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-riesel-number-repstring-101","kind":"node","lean":"MetaMathlibExt.isRieselNumber_repstring_101","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Riesel repdigits with digit "],["math","\\(101\\)"],["text"," in bases "],["math","\\(b \\equiv 180 \\pmod{11184810}\\)"]],"source":"Riesel repdigits with digit $101$ in bases $b \\equiv 180 \\pmod{11184810}$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-riesel-number-repstring-101","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $b$ and $t$ be natural numbers with $b \\equiv 180 \\pmod{11184810}$ and $t \\equiv 171 \\pmod{240}$. Then the $b$-repinteger $101_b^{(t)}$, which is the repdigit $101\\,(b^t - 1)/(b - 1)$ because $101 < b$, is a Riesel number.","title":"Riesel repdigits with digit $101$ in bases $b \\equiv 180 \\pmod{11184810}$"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-sierpinski-repunits-of-primitiv","kind":"node","lean":"MetaMathlibExt.infinite_sierpinski_repunits_of_primitive_divisors","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many Sierpi\u0144ski repunits from primitive divisors of "],["math","\\(2^{2^j} - 1\\)"]],"source":"Infinitely many Sierpi\u0144ski repunits from primitive divisors of $2^{2^j} - 1$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-sierpinski-repunits-of-primitiv","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $\\tau$ be a natural number. For $1 \\le j \\le \\tau$ let $p_j$ be a primitive prime divisor of $2^{2^j} - 1$ in the sequence $(2^m - 1)_{m \\ge 0}$, and let $q \\ne p_\\tau$ be a primitive prime divisor of $2^{2^\\tau} - 1$ in the same sequence. Let $b > 2$ be a natural number with $q \\nmid b$. For $1 \\le j \\le \\tau$ put $\\ell_j = 1$ if $p_j \\mid b$, $\\ell_j = p_j$ if $p_j \\nmid b$ and $b \\equiv 1 \\pmod{p_j}$, and\u2026","title":"Infinitely many Sierpi\u0144ski repunits from primitive divisors of $2^{2^j} - 1$"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-sierpinski-repunits-mod-641","kind":"node","lean":"MetaMathlibExt.infinite_sierpinski_repunits_mod_641","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many Sierpi\u0144ski repunits in bases "],["math","\\(b \\equiv 1, 147, 265, 378 \\pmod{641}\\)"],["text"," with "],["math","\\(b \\not\\equiv 1 \\pmod 5\\)"]],"source":"Infinitely many Sierpi\u0144ski repunits in bases $b \\equiv 1, 147, 265, 378 \\pmod{641}$ with $b \\not\\equiv 1 \\pmod 5$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-sierpinski-repunits-mod-641","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Infinitely many Sierpi\u0144ski repunits in bases $b \\equiv 1, 147, 265, 378 \\pmod{641}$ with $b \\not\\equiv 1 \\pmod 5$. Let $b > 2$ be a natural number with $b \\equiv c \\pmod{641}$ for some $c \\in \\{1, 147, 265, 378\\}$ and $b \\not\\equiv 1 \\pmod 5$. Then there are infinitely many Sierpi\u0144ski numbers among the $b$-repunits $(b^t - 1)/(b - 1)$ with $t \\ge 1$.","title":"Infinitely many Sierpi\u0144ski repunits in bases $b \\equiv 1, 147, 265, 378 \\pmod{641}$ with $b \\not\\equiv 1 \\pmod 5$"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-riesel-repunits-of-primitive-di","kind":"node","lean":"MetaMathlibExt.infinite_riesel_repunits_of_primitive_divisors","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many Riesel repunits from primitive divisors of "],["math","\\(2^{2^j} - 1\\)"]],"source":"Infinitely many Riesel repunits from primitive divisors of $2^{2^j} - 1$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-riesel-repunits-of-primitive-di","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $\\tau$ be a natural number. For $1 \\le j \\le \\tau$ let $p_j$ be a primitive prime divisor of $2^{2^j} - 1$ in the sequence $(2^m - 1)_{m \\ge 0}$, let $q \\ne p_\\tau$ be a primitive prime divisor of $2^{2^\\tau} - 1$ in the same sequence, and put $P = p_1 \\cdots p_\\tau$. Let $b > 2$ be a natural number with $b \\equiv 1 \\pmod P$, and suppose that (i) $q \\mid b$, or (ii) $b \\equiv 1 \\pmod q$, or (iii) $q \\nmid b$,\u2026","title":"Infinitely many Riesel repunits from primitive divisors of $2^{2^j} - 1$"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-riesel-repunits-mod-two-pow-64","kind":"node","lean":"MetaMathlibExt.infinite_riesel_repunits_mod_two_pow_64_sub_one","parent":7335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many Riesel repunits in bases "],["math","\\(b \\equiv 16518444216571 \\pmod{2^{64} - 1}\\)"]],"source":"Infinitely many Riesel repunits in bases $b \\equiv 16518444216571 \\pmod{2^{64} - 1}$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-riesel-repunits-mod-two-pow-64","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $b$ be a natural number with $b \\equiv 16518444216571 \\pmod{18446744073709551615}$, where $18446744073709551615 = 2^{64} - 1$. Then there are infinitely many Riesel numbers among the $b$-repunits $(b^t - 1)/(b - 1)$ with $t \\ge 1$.","title":"Infinitely many Riesel repunits in bases $b \\equiv 16518444216571 \\pmod{2^{64} - 1}$"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-sierpinski-number-repstring-659","kind":"node","lean":"MetaMathlibExt.isSierpinskiNumber_repstring_659","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-sierpinski-number-repstring-659","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"For every natural number $t$ with $t \\equiv 131 \\pmod{2730}$ or $t \\equiv 1361 \\pmod{2730}$, the binary repstring $659_2^{(2;t)} = 659\\,(2^{12t} - 1)/(2^{12} - 1)$ is a Sierpi\u0144ski number. Its binary expansion is that of $659$ (ten binary digits) written $t$ times with two zeros between consecutive copies.","title":"Binary repstrings of 659 that are Sierpi\u0144ski numbers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-riesel-number-repstring-659","kind":"node","lean":"MetaMathlibExt.isRieselNumber_repstring_659","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-riesel-number-repstring-659","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"For every natural number $t$ with $t \\equiv 1369 \\pmod{2730}$ or $t \\equiv 2599 \\pmod{2730}$, the binary repstring $659_2^{(2;t)} = 659\\,(2^{12t} - 1)/(2^{12} - 1)$ is a Riesel number. Its binary expansion is that of $659$ (ten binary digits) written $t$ times with two zeros between consecutive copies.","title":"Binary repstrings of 659 that are Riesel numbers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/sierpinski-covering-data-repstring-iff-r","kind":"node","lean":"MetaMathlibExt.sierpinskiCoveringData_repstring_iff_riesel","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#sierpinski-covering-data-repstring-iff-r","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $k \\ge 1$, $\\mathfrak{t} \\ge 1$, $w \\ge 1$ and $z \\ge 0$ be natural numbers. There is a single finite set of triples that is Sierpi\u0144ski covering data for the $2$-repstring $k_2^{(z;t)}$ for every $t \\ge 1$ with $t \\equiv \\mathfrak{t} \\pmod{w}$ if and only if there is a single finite set of triples that is Riesel covering data for $k_2^{(z;t')}$ for every $t' \\ge 1$ with $t' \\equiv -\\mathfrak{t} \\pmod{w}$.","title":"Theorem 4.6 (Sierpi\u0144ski and Riesel repstring coverings correspond)"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/is-riesel-number-narayana-cows-sequence","kind":"node","lean":"MetaMathlibExt.isRieselNumber_narayanaCowsSequence","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#is-riesel-number-narayana-cows-sequence","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"Let $(N_t)$ be Narayana's cows sequence, with $N_0 = 0$ and $N_1 = N_2 = 1$. If $t$ is congruent modulo $718391520$ to one of the $27$ residues $3629412$, $13386852$, $73151172$, $96325092$, $101203812$, $119499012$, $175604292$, $198778212$, $212194692$, $240247332$, $268299972$, $328064292$, $337821732$, $360995652$, $384169572$, $402464772$, $425638692$, $467107812$, $476865252$, $495160452$, $564682212$,\u2026","title":"Narayana numbers that are Riesel numbers"},{"id":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/infinite-set-of-is-sierpinski-number-nar","kind":"node","lean":"MetaMathlibExt.infinite_setOf_isSierpinskiNumber_narayanaCowsSequence","parent":7335,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers/index.html#infinite-set-of-is-sierpinski-number-nar","scope":"NumberTheory/primes-primality/special-primes/sierpi-ski-and-riesel-numbers","status":"can_prove","summary":"There are infinitely many natural numbers $k$ that are Sierpi\u0144ski numbers and terms of Narayana's cows sequence $(N_t)$, that is, $k = N_t$ for some $t$; here a Sierpi\u0144ski number is an odd $k$ such that $k \\cdot 2^n + 1$ is not prime for every $n \\ge 1$.","title":"Infinitely many Narayana numbers are Sierpi\u0144ski numbers"},{"data_url":"scopes/NumberTheory/primes-primality/special-primes/socialist-primes.json","id":"NumberTheory/primes-primality/special-primes/socialist-primes","kind":"scope","navigation":"module","parent":7311,"result":false,"route":"../roadmap/NumberTheory/primes-primality/special-primes/socialist-primes/index.html","scope":"NumberTheory/primes-primality/special-primes/socialist-primes","status":"fully_proved","title":"Socialist primes"},{"id":"NumberTheory/primes-primality/special-primes/socialist-primes/is-socialist-prime","kind":"node","lean":"Nat.IsSocialistPrime","parent":7366,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/socialist-primes/index.html#is-socialist-prime","scope":"NumberTheory/primes-primality/special-primes/socialist-primes","status":"fully_proved","summary":"A natural number $p$ is a socialist prime if $p$ is prime, $p > 5$, and the factorials $2!, 3!, \\ldots, (p-1)!$ have pairwise distinct remainders modulo $p$.","title":"Socialist primes"},{"id":"NumberTheory/primes-primality/special-primes/socialist-primes/factorial-mod-ne","kind":"node","lean":"Nat.IsSocialistPrime.factorial_mod_ne","parent":7366,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/socialist-primes/index.html#factorial-mod-ne","scope":"NumberTheory/primes-primality/special-primes/socialist-primes","status":"fully_proved","summary":"Let $p$ be a socialist prime. If $a$ and $b$ are distinct natural numbers with $2 \\le a \\le p - 1$ and $2 \\le b \\le p - 1$, then $a!$ and $b!$ have different remainders modulo $p$.","title":"Distinct factorial residues modulo a socialist prime"},{"data_url":"scopes/NumberTheory/primes-primality/special-primes/sophie-germain-primes.json","id":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","kind":"scope","navigation":"module","parent":7311,"result":false,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sophie-germain-primes/index.html","scope":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","status":"can_prove","title":"Sophie Germain primes"},{"id":"NumberTheory/primes-primality/special-primes/sophie-germain-primes/is-sophie-germain-prime","kind":"node","lean":"Nat.IsSophieGermainPrime","parent":7369,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sophie-germain-primes/index.html#is-sophie-germain-prime","scope":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","status":"fully_proved","summary":"A natural number $p$ is a Sophie Germain prime if $p$ is prime and $2p + 1$ is prime. The case $p = 2$ is not excluded.","title":"Sophie Germain primes"},{"id":"NumberTheory/primes-primality/special-primes/sophie-germain-primes/is-sophie-germain-prime-iff","kind":"node","lean":"Nat.isSophieGermainPrime_iff","parent":7369,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sophie-germain-primes/index.html#is-sophie-germain-prime-iff","scope":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","status":"fully_proved","summary":"A natural number $p$ is a Sophie Germain prime if and only if $p$ is prime and $2p + 1$ is prime.","title":"Characterization of Sophie Germain primes"},{"id":"NumberTheory/primes-primality/special-primes/sophie-germain-primes/prime","kind":"node","lean":"Nat.IsSophieGermainPrime.prime","parent":7369,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sophie-germain-primes/index.html#prime","scope":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","status":"fully_proved","summary":"If $p$ is a Sophie Germain prime, then $p$ is prime.","title":"Sophie Germain primes are prime"},{"id":"NumberTheory/primes-primality/special-primes/sophie-germain-primes/prime-two-mul-add-one","kind":"node","lean":"Nat.IsSophieGermainPrime.prime_two_mul_add_one","parent":7369,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sophie-germain-primes/index.html#prime-two-mul-add-one","scope":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","status":"fully_proved","summary":"If $p$ is a Sophie Germain prime, then $2p + 1$ is prime.","title":"Safe prime of a Sophie Germain prime"},{"id":"NumberTheory/primes-primality/special-primes/sophie-germain-primes/infinite-set-of-prime-not-is-sophie-germ","kind":"node","lean":"MetaMathlibExt.infinite_setOf_prime_not_isSophieGermainPrime","parent":7369,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sophie-germain-primes/index.html#infinite-set-of-prime-not-is-sophie-germ","scope":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","status":"can_prove","summary":"The set of primes $q$ that are not Sophie Germain primes, that is, the primes $q$ for which $2q + 1$ is not prime, is infinite.","title":"Infinitely many primes are not Sophie Germain primes"},{"id":"NumberTheory/primes-primality/special-primes/sophie-germain-primes/set-of-prime-not-is-sophie-germain-prime","kind":"node","lean":"MetaMathlibExt.setOf_prime_not_isSophieGermainPrime_eq_inter","parent":7369,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sophie-germain-primes/index.html#set-of-prime-not-is-sophie-germain-prime","scope":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","status":"can_prove","summary":"For an odd prime $p$, let $A(p) = \\{pr + (p - 1)/2 : r = 1, 2, 3, \\dots\\}$, and let $A$ be the union of the sets $A(p)$ over all odd primes $p$. A prime is not a Sophie Germain prime if and only if it lies in $A$; equivalently, the Sophie Germain primes are exactly the primes outside $A$.","title":"Primes that are not Sophie Germain primes"},{"id":"NumberTheory/primes-primality/special-primes/sophie-germain-primes/is-sophie-germain-prime-ne-two-iff-facto","kind":"node","lean":"MetaMathlibExt.isSophieGermainPrime_ne_two_iff_factorial_modEq","parent":7369,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/sophie-germain-primes/index.html#is-sophie-germain-prime-ne-two-iff-facto","scope":"NumberTheory/primes-primality/special-primes/sophie-germain-primes","status":"can_prove","summary":"Let $a \\ge 2$ be a natural number. Then $a$ is a Sophie Germain prime other than $2$ if and only if $(a - 1)! \\equiv 2a - 1 \\pmod{a(2a + 1)}$ or $(a - 1)! \\equiv -(6a + 1) \\pmod{a(2a + 1)}$, the congruences being taken in the integers.","title":"Wilson-type characterization of Sophie Germain primes"},{"data_url":"scopes/NumberTheory/primes-primality/special-primes/wieferich-primes.json","id":"NumberTheory/primes-primality/special-primes/wieferich-primes","kind":"scope","navigation":"module","parent":7311,"result":false,"route":"../roadmap/NumberTheory/primes-primality/special-primes/wieferich-primes/index.html","scope":"NumberTheory/primes-primality/special-primes/wieferich-primes","status":"fully_proved","title":"Wieferich primes"},{"id":"NumberTheory/primes-primality/special-primes/wieferich-primes/is-wieferich-prime","kind":"node","lean":"Nat.IsWieferichPrime","parent":7377,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/wieferich-primes/index.html#is-wieferich-prime","scope":"NumberTheory/primes-primality/special-primes/wieferich-primes","status":"fully_proved","summary":"A natural number $p$ is a Wieferich prime (to base $2$) if $p$ is prime and $p^2$ divides $2^{p-1} - 1$.","title":"Wieferich primes"},{"id":"NumberTheory/primes-primality/special-primes/wieferich-primes/is-wieferich-prime-iff-mod-eq","kind":"node","lean":"Nat.isWieferichPrime_iff_modEq","parent":7377,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/wieferich-primes/index.html#is-wieferich-prime-iff-mod-eq","scope":"NumberTheory/primes-primality/special-primes/wieferich-primes","status":"fully_proved","summary":"For every natural number $p$, $p$ is a Wieferich prime if and only if $p$ is prime and $2^{p-1} \\equiv 1 \\pmod{p^2}$.","title":"Wieferich primes via congruences"},{"id":"NumberTheory/primes-primality/special-primes/wieferich-primes/is-wieferich-prime-iff-mod","kind":"node","lean":"Nat.isWieferichPrime_iff_mod","parent":7377,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/wieferich-primes/index.html#is-wieferich-prime-iff-mod","scope":"NumberTheory/primes-primality/special-primes/wieferich-primes","status":"fully_proved","summary":"For every natural number $p$, $p$ is a Wieferich prime if and only if $p$ is prime and the remainder of $2^{p-1}$ on division by $p^2$ is $1$.","title":"Wieferich primes via remainders"},{"data_url":"scopes/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers.json","id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","kind":"scope","navigation":"module","parent":7311,"result":false,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","title":"Square prime factors of Fermat numbers"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/two-pow-half-mod-eq-one-fermat-number","kind":"node","lean":"MetaMathlibExt.two_pow_half_modEq_one_fermatNumber","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#two-pow-half-mod-eq-one-fermat-number","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"Let $F_n = 2^{2^n} + 1$ be the $n$-th Fermat number. For every $n \\ge 2$ and every odd prime $p$ dividing $F_n$, we have $2^{(p-1)/2} \\equiv 1 \\pmod{F_n}$. This result is due to Warren and Bray (1967).","title":"Warren\u2013Bray congruence for prime divisors of Fermat numbers"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/sq-dvd-fermat-number-iff","kind":"node","lean":"MetaMathlibExt.sq_dvd_fermatNumber_iff","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#sq-dvd-fermat-number-iff","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"Let $F_n = 2^{2^n} + 1$ be the $n$-th Fermat number, and let $p$ be a prime dividing $F_n$. Then $p^2$ divides $F_n$ if and only if $2^{p-1} \\equiv 1 \\pmod{p^2}$.","title":"Squared prime factors of Fermat numbers"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/sq-dvd-fermat-number-iff-wieferich-order","kind":"node","lean":"MetaMathlibExt.sq_dvd_fermatNumber_iff_wieferich_orderOf","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#sq-dvd-fermat-number-iff-wieferich-order","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"Let $F_n = 2^{2^n} + 1$ be the $n$-th Fermat number. For every natural number $n$ and every odd prime $p$, $p^2$ divides $F_n$ if and only if $p$ is a Wieferich prime (to base $2$) and the multiplicative order of $2$ modulo $p$ is $2^{n+1}$.","title":"Squared prime factors of Fermat numbers via Wieferich primes and orders"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/gcd-pow-pow-add-one","kind":"node","lean":"MetaMathlibExt.gcd_pow_pow_add_one","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#gcd-pow-pow-add-one","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"For natural numbers $a$, $b$ and $k$ write $F_k(a, b) = a^{b^k} + 1$, and let $a, b \\ge 2$ and $m$, $n$ be natural numbers. If $n < m$ and $a$ and $b$ are both even, then $\\gcd(F_m(a, b), F_n(a, b)) = 1$. If $n < m$, $a$ is odd and $b$ is even, then $\\gcd(F_m(a, b), F_n(a, b)) = 2$. If $n \\le m$ and $b$ is odd, then $\\gcd(F_m(a, b), F_n(a, b)) = F_n(a, b)$.","title":"Greatest common divisors of generalized Fermat numbers"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/monotone-card-factors-pow-pow-add-one","kind":"node","lean":"MetaMathlibExt.monotone_cardFactors_pow_pow_add_one","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#monotone-card-factors-pow-pow-add-one","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"For natural numbers $a$, $b$ and $n$ write $F_n(a, b) = a^{b^n} + 1$, and let $\\Omega(N)$ be the number of prime factors of $N$ counted with multiplicity. If $a, b \\ge 2$ and $b$ is odd, then $\\Omega(F_n(a, b)) \\le \\Omega(F_m(a, b))$ for all natural numbers $n \\le m$.","title":"Monotonicity of the number of prime factors of generalized Fermat numbers"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/not-pow-pow-mod-eq-one-of-lt","kind":"node","lean":"MetaMathlibExt.not_pow_pow_modEq_one_of_lt","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#not-pow-pow-mod-eq-one-of-lt","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"Let $a, b, m \\ge 2$ and $n \\ge 1$ be natural numbers with $a^{b^n} \\not\\equiv 1 \\pmod m$. Then $a^{b^t} \\not\\equiv 1 \\pmod m$ for every natural number $t < n$.","title":"Smaller iterated powers are not congruent to one"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/sq-dvd-pow-pow-add-one-wieferich","kind":"node","lean":"MetaMathlibExt.sq_dvd_pow_pow_add_one_wieferich","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#sq-dvd-pow-pow-add-one-wieferich","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"For natural numbers $a$, $b$ and $n$ write $F_n(a, b) = a^{b^n} + 1$. Let $a, b \\ge 2$, let $n$ be a natural number, let $p$ be an odd prime coprime to $ab$, and write $\\operatorname{ord}_p(a)$ for the multiplicative order of $a$ modulo $p$. If $p^2$ divides $F_n(a, b)$, then $a^{p-1} \\equiv 1 \\pmod{p^2}$ (that is, $p$ is a Wieferich prime to base $a$) and $\\operatorname{ord}_p(a)$ divides $2b^n$. Conversely, if\u2026","title":"Squared prime factors of generalized Fermat numbers"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/sq-dvd-pow-two-pow-add-one-iff-wieferich","kind":"node","lean":"MetaMathlibExt.sq_dvd_pow_two_pow_add_one_iff_wieferich","parent":7381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squared prime factors of "],["math","\\(a^{2^n} + 1\\)"]],"source":"Squared prime factors of $a^{2^n} + 1$"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#sq-dvd-pow-two-pow-add-one-iff-wieferich","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"Let $p$ be an odd prime, let $a$ be a natural number not divisible by $p$, and let $n$ be a natural number. Then $p^2$ divides $a^{2^n} + 1$ if and only if $a^{p-1} \\equiv 1 \\pmod{p^2}$ (that is, $p$ is a Wieferich prime to base $a$) and the multiplicative order of $a$ modulo $p$ is $2^{n+1}$.","title":"Squared prime factors of $a^{2^n} + 1$"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/pow-sub-one-mod-eq-one-sq-iff-order-of-e","kind":"node","lean":"MetaMathlibExt.pow_sub_one_modEq_one_sq_iff_orderOf_eq","parent":7381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wieferich primes to base "],["math","\\(a\\)"],["text"," via multiplicative orders"]],"source":"Wieferich primes to base $a$ via multiplicative orders"},"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#pow-sub-one-mod-eq-one-sq-iff-order-of-e","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"Let $p$ be a prime and $a$ a natural number not divisible by $p$. Then $a^{p-1} \\equiv 1 \\pmod{p^2}$, that is, $p$ is a Wieferich prime to base $a$, if and only if the multiplicative order of $a$ modulo $p^2$ equals the multiplicative order of $a$ modulo $p$.","title":"Wieferich primes to base $a$ via multiplicative orders"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/dvd-pow-pow-add-one-shift","kind":"node","lean":"MetaMathlibExt.dvd_pow_pow_add_one_shift","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#dvd-pow-pow-add-one-shift","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"For natural numbers $a$, $b$ and $m$ write $F_m(a, b) = a^{b^m} + 1$. Let $a, b, D \\ge 2$ and $n$ be natural numbers such that $D$ divides $F_n(a, b)$. If $b$ is odd, then $D$ divides $F_{n+k}(a, b)$ for every $k \\ge 1$. If $b$ is even and $D \\ge 3$, then $D$ does not divide $F_{n+k}(a, b)$ for any $k \\ge 1$.","title":"Divisibility of later generalized Fermat numbers"},{"id":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/is-least-periodic-start-pow-pow-add-one","kind":"node","lean":"MetaMathlibExt.isLeast_periodic_start_pow_pow_add_one_mod_sq","parent":7381,"result":true,"route":"../roadmap/NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers/index.html#is-least-periodic-start-pow-pow-add-one","scope":"NumberTheory/primes-primality/special-primes/square-prime-factors-of-fermat-numbers","status":"can_prove","summary":"For natural numbers $a$, $b$ and $k$ write $F_k(a, b) = a^{b^k} + 1$. Let $a, b \\ge 2$, let $p$ be a prime, and let $n_0$ be the least natural number with $p^2 \\mid F_{n_0}(a, b)$. Say that the sequence $(F_k(a, b))_k$ is periodic modulo $p^2$ from index $m$ if there is a $T > 0$ with $F_{m+k+T}(a, b) \\equiv F_{m+k}(a, b) \\pmod{p^2}$ for every $k \\ge 0$. If $b$ is odd, then $n_0$ is the least index from which the\u2026","title":"Start of the period of generalized Fermat numbers modulo the square of a prime"},{"data_url":"scopes/NumberTheory/primes-primality/composite-numbers.json","id":"NumberTheory/primes-primality/composite-numbers","kind":"scope","navigation":"module","parent":7169,"result":false,"route":"../roadmap/NumberTheory/primes-primality/composite-numbers/index.html","scope":"NumberTheory/primes-primality/composite-numbers","status":"fully_proved","title":"Composite numbers"},{"id":"NumberTheory/primes-primality/composite-numbers/is-composite","kind":"node","lean":"Nat.IsComposite","parent":7393,"result":true,"route":"../roadmap/NumberTheory/primes-primality/composite-numbers/index.html#is-composite","scope":"NumberTheory/primes-primality/composite-numbers","status":"fully_proved","summary":"A natural number $n$ is composite if $n > 1$ and $n$ is not prime.","title":"Composite numbers"},{"id":"NumberTheory/primes-primality/composite-numbers/one-lt","kind":"node","lean":"Nat.IsComposite.one_lt","parent":7393,"result":true,"route":"../roadmap/NumberTheory/primes-primality/composite-numbers/index.html#one-lt","scope":"NumberTheory/primes-primality/composite-numbers","status":"fully_proved","summary":"Every composite natural number $n$ satisfies $1 < n$.","title":"Composite numbers exceed one"},{"id":"NumberTheory/primes-primality/composite-numbers/not-prime","kind":"node","lean":"Nat.IsComposite.not_prime","parent":7393,"result":true,"route":"../roadmap/NumberTheory/primes-primality/composite-numbers/index.html#not-prime","scope":"NumberTheory/primes-primality/composite-numbers","status":"fully_proved","summary":"A composite natural number is not prime.","title":"Composite numbers are not prime"},{"data_url":"scopes/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes.json","id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","kind":"scope","navigation":"module","parent":7169,"result":false,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"can_prove","title":"Topological proofs of the infinitude of primes"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/coprime-set","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.coprimeSet","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#coprime-set","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"fully_proved","summary":"For a positive integer $n$, $\\Omega_n$ is the set of positive integers $m$ with $\\gcd(n, m) = 1$.","title":"Positive integers coprime to a given integer"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/macias-basis","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.maciasBasis","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#macias-basis","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"fully_proved","summary":"For a positive integer $n$ let $\\Omega_n$ be the set of positive integers coprime to $n$. Mac\u00edas's coprime basis is the family $\\beta = \\{\\Omega_n : n \\ge 1\\}$ of subsets of the positive integers.","title":"Mac\u00edas's coprime basis"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/macias-topology","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.maciasTopology","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#macias-topology","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"fully_proved","summary":"For a positive integer $n$ let $\\Omega_n$ be the set of positive integers coprime to $n$. Mac\u00edas's coprime topology $\\tau$ is the topology on the positive integers generated by the sets $\\Omega_n$, $n \\ge 1$, that is, the coarsest topology in which every $\\Omega_n$ is open.","title":"Mac\u00edas's coprime topology"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/macias-topology-2","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.maciasTopology\u2081","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#macias-topology-2","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"fully_proved","summary":"The topology $\\tau_1$ on the set of positive integers different from $1$ is the subspace topology induced by Mac\u00edas's coprime topology $\\tau$ on the positive integers.","title":"Mac\u00edas's coprime topology without 1"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/prime-set","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.primeSet","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#prime-set","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"fully_proved","summary":"The set of positive integers that are prime numbers.","title":"Primes among the positive integers"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/prime-set-2","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.primeSet\u2081","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#prime-set-2","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"fully_proved","summary":"Within the set of positive integers different from $1$, this is the subset consisting of the prime numbers.","title":"Primes among the positive integers different from 1"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/macias-basis-is-topological-basis","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.maciasBasis_isTopologicalBasis","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#macias-basis-is-topological-basis","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"can_prove","summary":"For a positive integer $n$ let $\\Omega_n$ be the set of positive integers coprime to $n$. The family $\\beta = \\{\\Omega_n : n \\ge 1\\}$ is a basis of the topology on the positive integers that it generates (Mac\u00edas's coprime topology).","title":"Mac\u00edas's coprime sets form a basis"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/closure-singleton-prime-eq-multiples","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.closure_singleton_prime_eq_multiples","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#closure-singleton-prime-eq-multiples","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"can_prove","summary":"If $p$ is a prime number, then the closure of $\\{p\\}$ in the positive integers with Mac\u00edas's coprime topology is the set of positive integers divisible by $p$.","title":"Closure of a prime in Mac\u00edas's coprime topology"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/infinite-primes-iff-dense-prime-set","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.infinite_primes_iff_dense_primeSet","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#infinite-primes-iff-dense-prime-set","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"can_prove","summary":"There are infinitely many prime numbers if and only if the set of prime numbers is dense in the positive integers with Mac\u00edas's coprime topology.","title":"Infinitude of the primes as density"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/dense-prime-set-of-dense-prime-set","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.dense_primeSet_of_dense_primeSet\u2081","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#dense-prime-set-of-dense-prime-set","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"can_prove","summary":"Let $X$ be the positive integers with Mac\u00edas's coprime topology and $X_1$ the subspace of positive integers different from $1$. If the set of prime numbers is dense in $X_1$, then it is dense in $X$.","title":"Density of the primes passes from the subspace without 1"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/dense-prime-set","kind":"node","lean":"MathlibExt.Topology.MaciasCoprimeTopologyWanted.dense_primeSet\u2081","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#dense-prime-set","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"can_prove","summary":"The set of prime numbers is dense in the set of positive integers different from $1$, with the subspace topology induced by Mac\u00edas's coprime topology on the positive integers.","title":"The primes are dense in the coprime topology without 1"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/infinite-iff-dense-macias-topology-of-su","kind":"node","lean":"MetaMathlibExt.infinite_iff_dense_maciasTopology_of_subset_primes","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#infinite-iff-dense-macias-topology-of-su","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"can_prove","summary":"Equip the positive integers with the comaximal topology, the topology generated by the sets $\\{m : \\gcd(n, m) = 1\\}$ for positive integers $n$. A nonempty set $A$ of primes is infinite if and only if it is dense in this topology.","title":"A nonempty set of primes is dense in the comaximal space iff it is infinite"},{"id":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/dense-macias-topology-arithmetic-progres","kind":"node","lean":"MetaMathlibExt.dense_maciasTopology_arithmeticProgression","parent":7397,"result":true,"route":"../roadmap/NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes/index.html#dense-macias-topology-arithmetic-progres","scope":"NumberTheory/primes-primality/topological-proofs-of-the-infinitude-of-primes","status":"can_prove","summary":"Equip the positive integers with the comaximal topology, the topology generated by the sets $\\{m : \\gcd(n, m) = 1\\}$ for positive integers $n$. If $a$ and $b$ are coprime positive integers, then the arithmetic progression $\\{ak + b : k \\in \\mathbb{N}\\} = \\{b, a + b, 2a + b, \\dots\\}$, with $k$ ranging over the natural numbers including $0$, is dense in this topology.","title":"Coprime arithmetic progressions are dense in the comaximal space"},{"data_url":"scopes/NumberTheory/distribution-of-primes.json","id":"NumberTheory/distribution-of-primes","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/index.html","scope":"NumberTheory/distribution-of-primes","status":"can_prove","summary":"Counting primes: prime number theorem, explicit Chebyshev-type bounds, prime gaps and short intervals, primes in progressions, sieves, prime patterns.","title":"Distribution of primes"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem","kind":"scope","parent":7411,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem","status":"can_prove","summary":"PNT and its proofs, Chebyshev theta and psi, explicit bounds for \u03c0(x) and the nth prime (Dusart, Axler, Rosser), Mertens constants.","title":"Prime number theorem and explicit bounds"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds","status":"fully_proved","title":"Chebyshev function bounds"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds/chebyshev-theta-explicit-bounds","kind":"node","lean":"MetaMathlibExt.chebyshev_theta_explicit_bounds","parent":7413,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds/index.html#chebyshev-theta-explicit-bounds","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$, the sum running over primes, and let","title":"Chebyshev's explicit bounds for the theta function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds/chebyshev-theta-gt-one-third","kind":"node","lean":"MetaMathlibExt.chebyshev_theta_gt_one_third","parent":7413,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds/index.html#chebyshev-theta-gt-one-third","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/chebyshev-function-bounds","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$, the sum running over primes, be the first Chebyshev function. Then $\\theta(x) > x/3$ for every real $x \\ge 3$.","title":"Chebyshev's lower bound for the theta function"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem","status":"can_prove","title":"Error term in the prime number theorem"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem/prime-number-theorem-chebyshev-theta-err","kind":"node","lean":"MetaMathlibExt.PrimeNumberTheorem.prime_number_theorem_chebyshev_theta_error","parent":7416,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime number theorem with error term for Chebyshev's "],["math","\\(\\theta\\)"]],"source":"Prime number theorem with error term for Chebyshev's $\\theta$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem/index.html#prime-number-theorem-chebyshev-theta-err","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem","status":"can_prove","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$ be the sum of $\\log p$ over the primes $p \\le x$. For every real $\\varepsilon > 0$, $\\theta(x) - x = O(x \\exp(-(\\log x)^{3/5 - \\varepsilon}))$ as $x \\to \\infty$.","title":"Prime number theorem with error term for Chebyshev's $\\theta$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem/dusart2016-tail-theorem","kind":"node","lean":"MetaMathlibExt.dusart2016_tail_theorem","parent":7416,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's bound for "],["math","\\(\\vartheta\\)"],["text"," and "],["math","\\(\\psi\\)"],["text"," from a zero-free region"]],"source":"Dusart's bound for $\\vartheta$ and $\\psi$ from a zero-free region"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem/index.html#dusart2016-tail-theorem","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/error-term-in-the-prime-number-theorem","status":"can_prove","summary":"Let $R = 5.69693$, let $\\log$ be the natural logarithm, and let $\\vartheta(x) = \\sum_{p \\le x} \\log p$ (over primes $p$) and $\\psi(x) = \\sum_{n \\le x} \\Lambda(n)$, with $\\Lambda$ the von Mangoldt function, be Chebyshev's functions. Suppose the Riemann zeta function has no zero $s$ with $|\\operatorname{Im} s| \\ge 2\\pi$ and $\\operatorname{Re} s \\ge 1 - \\frac{1}{R \\log |\\operatorname{Im} s|}$. Then for every real\u2026","title":"Dusart's bound for $\\vartheta$ and $\\psi$ from a zero-free region"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","title":"Explicit bounds for the nth prime"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/axler-nth-prime-lower-bound","kind":"node","lean":"MetaMathlibExt.axler_nth_prime_lower_bound","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Axler's lower bound for the "],["math","\\(n\\)"],["text","th prime"]],"source":"Axler's lower bound for the $n$th prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#axler-nth-prime-lower-bound","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every integer $n \\ge 2$, the $n$th prime $p_n$ (so that $p_1 = 2$) satisfies $p_n > n \\left(\\log n + \\log\\log n - 1 + \\frac{\\log\\log n - 2}{\\log n} - \\frac{(\\log\\log n)^2 - 6 \\log\\log n + 11.321}{2 \\log^2 n}\\right)$.","title":"Axler's lower bound for the $n$th prime"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/axler-nth-prime-upper-bound","kind":"node","lean":"MetaMathlibExt.axler_nth_prime_upper_bound","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Axler's upper bound for the "],["math","\\(n\\)"],["text","th prime"]],"source":"Axler's upper bound for the $n$th prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#axler-nth-prime-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every integer $n \\ge 46254381$, the $n$th prime $p_n$ (so that $p_1 = 2$) satisfies $p_n < n \\left(\\log n + \\log\\log n - 1 + \\frac{\\log\\log n - 2}{\\log n} - \\frac{(\\log\\log n)^2 - 6 \\log\\log n + 10.667}{2 \\log^2 n}\\right)$.","title":"Axler's upper bound for the $n$th prime"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/mandl-prime-sum-inequality","kind":"node","lean":"MetaMathlibExt.mandl_prime_sum_inequality","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mandl's inequality for the sum of the first "],["math","\\(n\\)"],["text"," primes"]],"source":"Mandl's inequality for the sum of the first $n$ primes"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#mandl-prime-sum-inequality","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_k$ denote the $k$-th prime, so that $p_1 = 2$. For every natural number $n \\ge 9$, twice the sum of the first $n$ primes is at most $n$ times the $n$-th prime: $2(p_1 + \\cdots + p_n) \\le n \\, p_n$.","title":"Mandl's inequality for the sum of the first $n$ primes"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/prime-counting-gt-div-log-of-seventeen-l","kind":"node","lean":"MetaMathlibExt.primeCounting_gt_div_log_of_seventeen_le","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Rosser\u2013Schoenfeld lower bound "],["math","\\(x / \\log x\\)"],["text"," for the prime-counting function"]],"source":"The Rosser\u2013Schoenfeld lower bound $x / \\log x$ for the prime-counting function"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#prime-counting-gt-div-log-of-seventeen-l","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every real number $x \\ge 17$, the number $\\pi(x)$ of primes at most $x$ satisfies $x / \\log x < \\pi(x)$.","title":"The Rosser\u2013Schoenfeld lower bound $x / \\log x$ for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/nth-prime-lt-mul-log-nth-prime","kind":"node","lean":"MetaMathlibExt.nthPrime_lt_mul_log_nthPrime","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","For "],["math","\\(n \\ge 7\\)"],["text",", the "],["math","\\(n\\)"],["text","th prime is less than "],["math","\\(n\\)"],["text"," times its logarithm"]],"source":"For $n \\ge 7$, the $n$th prime is less than $n$ times its logarithm"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#nth-prime-lt-mul-log-nth-prime","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every integer $n \\ge 7$, the $n$th prime $p_n$ (counting from $p_1 = 2$) satisfies $p_n < n \\log p_n$.","title":"For $n \\ge 7$, the $n$th prime is less than $n$ times its logarithm"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/prime-counting-lt-div-log-sub-eleven-ten","kind":"node","lean":"MetaMathlibExt.primeCounting_lt_div_log_sub_eleven_tenths","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's upper bound "],["math","\\(x / (\\log x - 1.1)\\)"],["text"," for the prime-counting function"]],"source":"Dusart's upper bound $x / (\\log x - 1.1)$ for the prime-counting function"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#prime-counting-lt-div-log-sub-eleven-ten","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every real number $x \\ge 60184$, the number $\\pi(x)$ of primes at most $x$ satisfies $\\pi(x) < x / (\\log x - 1.1)$.","title":"Dusart's upper bound $x / (\\log x - 1.1)$ for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/nth-prime-lt-mul-log-nth-prime-sub-one","kind":"node","lean":"MetaMathlibExt.nthPrime_lt_mul_log_nthPrime_sub_one","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","An upper bound for the "],["math","\\(n\\)"],["text","-th prime in terms of its logarithm"]],"source":"An upper bound for the $n$-th prime in terms of its logarithm"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#nth-prime-lt-mul-log-nth-prime-sub-one","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_n$ denote the $n$-th prime, so that $p_1 = 2$. For every integer $n \\ge 711$, $p_n < n(\\log p_n - 1)$.","title":"An upper bound for the $n$-th prime in terms of its logarithm"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/prime-counting-gt-div-log-sub-one-sub-in","kind":"node","lean":"MetaMathlibExt.primeCounting_gt_div_log_sub_one_sub_inv_log","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound "],["math","\\(x / (\\log x - 1 - 1 / \\log x)\\)"],["text"," for the prime-counting function"]],"source":"Lower bound $x / (\\log x - 1 - 1 / \\log x)$ for the prime-counting function"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#prime-counting-gt-div-log-sub-one-sub-in","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every real number $x \\ge 468049$, the number $\\pi(x)$ of primes at most $x$ satisfies $\\pi(x) > x / (\\log x - 1 - 1 / \\log x)$.","title":"Lower bound $x / (\\log x - 1 - 1 / \\log x)$ for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/nth-prime-lt-mul-log-nth-prime-sub-one-s","kind":"node","lean":"MetaMathlibExt.nthPrime_lt_mul_log_nthPrime_sub_one_sub_inv_log","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","A second-order upper bound for the "],["math","\\(n\\)"],["text","-th prime in terms of its logarithm"]],"source":"A second-order upper bound for the $n$-th prime in terms of its logarithm"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#nth-prime-lt-mul-log-nth-prime-sub-one-s","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_n$ denote the $n$-th prime, so that $p_1 = 2$. For every integer $n \\ge 39071$,","title":"A second-order upper bound for the $n$-th prime in terms of its logarithm"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/mul-log-nth-prime-le-log","kind":"node","lean":"MetaMathlibExt.mul_log_nthPrime_le_log","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for "],["math","\\(\\log n\\)"],["text"," in terms of the logarithm of the "],["math","\\(n\\)"],["text","-th prime"]],"source":"Lower bound for $\\log n$ in terms of the logarithm of the $n$-th prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#mul-log-nth-prime-le-log","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_n$ denote the $n$-th prime, so that $p_1 = 2$. For every integer $n \\ge 234057667276344608$, $\\log n \\ge 0.914 \\log p_n$.","title":"Lower bound for $\\log n$ in terms of the logarithm of the $n$-th prime"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/nth-prime-sq-log-bound","kind":"node","lean":"MetaMathlibExt.nthPrime_sq_log_bound","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","An explicit inequality between "],["math","\\(n\\)"],["text"," and the "],["math","\\(n\\)"],["text","th prime"]],"source":"An explicit inequality between $n$ and the $n$th prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#nth-prime-sq-log-bound","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every integer $n \\ge 1$, the $n$th prime $p_n$ (counting from $p_1 = 2$) satisfies","title":"An explicit inequality between $n$ and the $n$th prime"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/loglog-ratio-sum-le-one","kind":"node","lean":"MetaMathlibExt.loglog_ratio_sum_le_one","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","An auxiliary inequality in "],["math","\\(\\log n\\)"],["text"," and "],["math","\\(\\log\\log n\\)"]],"source":"An auxiliary inequality in $\\log n$ and $\\log\\log n$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#loglog-ratio-sum-le-one","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every integer $n \\ge 2$, with $L = \\log n$,","title":"An auxiliary inequality in $\\log n$ and $\\log\\log n$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/nth-prime-gt-mul-log-sub-corollary-one","kind":"node","lean":"MetaMathlibExt.nthPrime_gt_mul_log_sub_corollary_one","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bounds for the "],["math","\\(n\\)"],["text","-th prime in terms of its logarithm"]],"source":"Lower bounds for the $n$-th prime in terms of its logarithm"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#nth-prime-gt-mul-log-sub-corollary-one","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_n$ denote the $n$-th prime, so that $p_1 = 2$, and put $L = \\log p_n$. Then $p_n > n\\left(L - 1 - \\frac{1.109}{L}\\right)$ for every integer $n \\ge 2621506158680$, $p_n > n\\left(L - 1 - \\frac{1}{L} - \\frac{3.48}{L^2}\\right)$ for every integer $n \\ge 4605081012945$, $p_n > n\\left(L - 1 - \\frac{1}{L} - \\frac{3.024334}{L^2} - \\frac{14.893}{L^3}\\right)$ for every integer $n \\ge 4514634967572$, and\u2026","title":"Lower bounds for the $n$-th prime in terms of its logarithm"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/nth-prime-lt-mul-log-sub-corollary-two","kind":"node","lean":"MetaMathlibExt.nthPrime_lt_mul_log_sub_corollary_two","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zhang\u2013Xiong upper bounds for the "],["math","\\(n\\)"],["text","th prime"]],"source":"Zhang\u2013Xiong upper bounds for the $n$th prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#nth-prime-lt-mul-log-sub-corollary-two","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_n$ be the $n$th prime (counting from $p_1 = 2$) and $L = \\log p_n$. Set $c_2 = 2.975666$, $c_3 = 13.024334$, $c_4 = 70.951332$, $c_5 = 460.634397856444$ and $c_6 = 3444.031844143556$, and set $N_2 = 3278841$, $N_3 = 355193892$, $N_4 = 6226419706$, $N_5 = 38900733192$ and $N_6 = 64497259289$. Then for each $k \\in \\{2, 3, 4, 5, 6\\}$ and every integer $n \\ge N_k$,","title":"Zhang\u2013Xiong upper bounds for the $n$th prime"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/log-nth-prime-ge-log-add-loglog","kind":"node","lean":"MetaMathlibExt.log_nthPrime_ge_log_add_loglog","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for the logarithm of the "],["math","\\(n\\)"],["text","-th prime"]],"source":"Lower bound for the logarithm of the $n$-th prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#log-nth-prime-ge-log-add-loglog","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_n$ denote the $n$-th prime, so that $p_1 = 2$. For every integer $n \\ge 3$,","title":"Lower bound for the logarithm of the $n$-th prime"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/mul-nth-prime-sub-sum-primes-bounds","kind":"node","lean":"MetaMathlibExt.mul_nthPrime_sub_sum_primes_bounds","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds for "],["math","\\(n\\)"],["text"," times the "],["math","\\(n\\)"],["text","-th prime minus the sum of the first "],["math","\\(n\\)"],["text"," primes"]],"source":"Bounds for $n$ times the $n$-th prime minus the sum of the first $n$ primes"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#mul-nth-prime-sub-sum-primes-bounds","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_k$ denote the $k$-th prime, so that $p_1 = 2$, and for $n \\ge 1$ put $L = \\log p_n$ and $S(n) = n p_n - \\sum_{k=1}^{n} p_k$. For every integer $n \\ge 440200309$,","title":"Bounds for $n$ times the $n$-th prime minus the sum of the first $n$ primes"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/mandl-sum-gt-omega-bound","kind":"node","lean":"MetaMathlibExt.mandl_sum_gt_omega_bound","parent":7419,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#mandl-sum-gt-omega-bound","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_k$ denote the $k$-th prime, so that $p_1 = 2$. For every integer $n \\ge 4605081012945$, put $\\ell = \\log n$, $L = \\log p_n$, $M(n) = \\frac{n p_n}{2} - \\sum_{k=1}^{n} p_k$ and","title":"Explicit lower bound in Mandl's inequality"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/mandl-sum-lt-omega-bound","kind":"node","lean":"MetaMathlibExt.mandl_sum_lt_Omega_bound","parent":7419,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#mandl-sum-lt-omega-bound","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_k$ denote the $k$-th prime, so that $p_1 = 2$. For every integer $n \\ge 64497259289$, put $\\ell = \\log n$, $L = \\log p_n$, $M(n) = \\frac{n p_n}{2} - \\sum_{k=1}^{n} p_k$,","title":"Explicit upper bound in Mandl's inequality"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/mandl-sum-between-b-alpha-b-beta","kind":"node","lean":"MetaMathlibExt.mandl_sum_between_B_alpha_B_beta","parent":7419,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#mandl-sum-between-b-alpha-b-beta","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"Let $p_k$ denote the $k$-th prime, so that $p_1 = 2$, and let $M(n) = \\frac{n p_n}{2} - \\sum_{k=1}^{n} p_k$. For real $\\eta$ let","title":"Zhang\u2013Xiong refinement of Mandl's inequality"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/rosser-nth-prime-lower-bound-zero-based","kind":"node","lean":"MetaMathlibExt.rosser_nthPrime_lower_bound_zeroBased","parent":7419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rosser's lower bound for the "],["math","\\(n\\)"],["text","th prime"]],"source":"Rosser's lower bound for the $n$th prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime/index.html#rosser-nth-prime-lower-bound-zero-based","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-nth-prime","status":"can_prove","summary":"For every positive integer $n$, the $n$th prime $p_n$ (counting from $p_1 = 2$) satisfies $p_n > n \\log n$.","title":"Rosser's lower bound for the $n$th prime"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems","status":"can_prove","title":"Mertens' theorems"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/mertens-constant","kind":"node","lean":"MetaMathlibExt.mertensConstant","parent":7440,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/index.html#mertens-constant","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems","status":"fully_proved","summary":"With $p$ ranging over the primes and $\\log$ the natural logarithm, this is the real number","title":"The constant in Mertens' first theorem"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/dusart-log-prime-sum","kind":"node","lean":"MetaMathlibExt.dusart_log_prime_sum","parent":7440,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/index.html#dusart-log-prime-sum","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems","status":"can_prove","summary":"Let $p$ range over the primes, let $\\log$ be the natural logarithm, and let $M = \\lim_{n \\to \\infty} \\bigl(\\sum_{p \\le n} \\frac{\\log p}{p} - \\log n\\bigr)$ be the constant in Mertens' first theorem. For every real number $x \\ge 912560$ there is a real number $E$ with $|E| \\le \\frac{0.3}{(\\log x)^2}$ such that","title":"Dusart's explicit Mertens first theorem"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/meissel-mertens-constant","kind":"node","lean":"MetaMathlibExt.meisselMertensConstant","parent":7440,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/index.html#meissel-mertens-constant","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems","status":"fully_proved","summary":"With $p$ ranging over the primes and $\\log$ the natural logarithm, the Meissel\u2013Mertens constant is the real number","title":"The Meissel\u2013Mertens constant"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/dusart-reciprocal-prime-sum","kind":"node","lean":"MetaMathlibExt.dusart_reciprocal_prime_sum","parent":7440,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems/index.html#dusart-reciprocal-prime-sum","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/mertens-theorems","status":"can_prove","summary":"Let $p$ range over the primes, let $\\log$ be the natural logarithm, and let $B = \\lim_{n \\to \\infty} \\bigl(\\sum_{p \\le n} \\frac{1}{p} - \\log \\log n\\bigr)$ be the Meissel\u2013Mertens constant. For every real number $x \\ge 2278383$ there is a real number $E$ with $|E| \\le \\frac{0.2}{(\\log x)^3}$ such that","title":"Dusart's explicit Mertens second theorem"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","title":"Explicit bounds for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-counting-lt-div-log-sub-six-terms","kind":"node","lean":"MetaMathlibExt.primeCounting_lt_div_log_sub_six_terms","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-counting-lt-div-log-sub-six-terms","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$ and $\\log$ the natural logarithm. For every real $x \\ge 48$,","title":"Axler's six-term upper bound for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/div-log-sub-six-terms-lt-prime-counting","kind":"node","lean":"MetaMathlibExt.div_log_sub_six_terms_lt_primeCounting","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#div-log-sub-six-terms-lt-prime-counting","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"For every real $x \\ge 1751189194177$ (the prime $p_{64497259289}$),","title":"Axler's six-term lower bound for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/div-log-sub-three-terms-lt-prime-countin","kind":"node","lean":"MetaMathlibExt.div_log_sub_three_terms_lt_primeCounting","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#div-log-sub-three-terms-lt-prime-countin","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$ and $\\log$ the natural logarithm. For every real $x \\ge 65405887$,","title":"Axler's unconditional three-term lower bound for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-counting-lt-div-log-sub-four-terms","kind":"node","lean":"MetaMathlibExt.primeCounting_lt_div_log_sub_four_terms","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-counting-lt-div-log-sub-four-terms","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$ and $\\log$ the natural logarithm. For every real $x \\ge 29.53$,","title":"Axler's upper bound for the prime-counting function above 29.53"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-counting-lt-eight-term-series","kind":"node","lean":"MetaMathlibExt.primeCounting_lt_eight_term_series","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-counting-lt-eight-term-series","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"For every real $x > 1$, the number $\\pi(x)$ of primes $p \\le x$ satisfies $\\pi(x) < \\frac{x}{\\log x} + \\frac{x}{\\log^2 x} + \\frac{2x}{\\log^3 x} + \\frac{6.024334\\,x}{\\log^4 x} + \\frac{24.024334\\,x}{\\log^5 x} + \\frac{120.12167\\,x}{\\log^6 x} + \\frac{720.73002\\,x}{\\log^7 x} + \\frac{6098\\,x}{\\log^8 x}$.","title":"Eight-term upper bound for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/four-term-series-lt-prime-counting","kind":"node","lean":"MetaMathlibExt.four_term_series_lt_primeCounting","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#four-term-series-lt-prime-counting","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$ and $\\log$ the natural logarithm. For every real $x \\ge 10384261$,","title":"Axler's four-term series lower bound for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/exists-prime-mem-ioc-mul-one-add-div-log","kind":"node","lean":"MetaMathlibExt.exists_prime_mem_Ioc_mul_one_add_div_log_pow","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#exists-prime-mem-ioc-mul-one-add-div-log","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\log$ denote the natural logarithm, and put $a = 1/(1.42969 \\cdot 10^{12} - 1)$ and $b = 1/(1.59753 \\cdot 10^{12} - 1)$. For every real $x \\ge 952527672606693$ there is a prime $p$ with $x < p \\le x\\,(1 + 43a/\\log x)$. Moreover, for each of the pairs $(n, X_n)$ equal to $(2, 684943746324434)$, $(3, 543684371469081)$, $(4, 336149866771577)$ and $(5, 246782656239427)$, and every real $x \\ge X_n$, there is a\u2026","title":"Axler's primes in short intervals"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-counting-lt-div-log-sub-table-two","kind":"node","lean":"MetaMathlibExt.primeCounting_lt_div_log_sub_table_two","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-counting-lt-div-log-sub-table-two","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$ and $\\log$ the natural logarithm. For each of the three rows $(a_0, a_1, a_2, x_0)$ equal to $(1.0343, 0, 0, 106640139304611)$, $(1, 1.109, 0, 81250795096339)$ and $(1, 1, 3.48, 145413088724077)$, every real $x \\ge x_0$ satisfies","title":"Axler's Table 2 upper bounds for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-counting-lt-div-log-sub-table-thre","kind":"node","lean":"MetaMathlibExt.primeCounting_lt_div_log_sub_table_three","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-counting-lt-div-log-sub-table-thre","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$ and $\\log$ the natural logarithm. For each of the three rows $(a_3, a_4, a_5, x_0)$ equal to $(14.893, 0, 0, 142464507937911)$, $(12.975666, 79.962, 0, 22)$ and $(12.975666, 71.048668, 533.594, 32)$, every real $x \\ge x_0$ satisfies","title":"Axler's Table 3 upper bounds for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-counting-lt-three-term-series-tabl","kind":"node","lean":"MetaMathlibExt.primeCounting_lt_three_term_series_table_four","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-counting-lt-three-term-series-tabl","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$. For each of the twelve pairs $(\\varepsilon, x_0)$ listed below and every real $x \\ge x_0$, $\\pi(x) < \\frac{x}{\\log x} + \\frac{x}{\\log^2 x} + \\frac{(2 + \\varepsilon)\\,x}{\\log^3 x}$. The pairs are $(0.21, 160930932942272)$, $(0.215, 83016503500865)$, $(0.22, 43999690220699)$, $(0.225, 23824649646672)$, $(0.23, 13279102022111)$, $(0.24, 4511700549332)$,\u2026","title":"Three-term upper bounds for the prime-counting function (Table 4)"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/div-log-sub-table-five-lt-prime-counting","kind":"node","lean":"MetaMathlibExt.div_log_sub_table_five_lt_primeCounting","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#div-log-sub-table-five-lt-prime-counting","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$ and $\\log$ the natural logarithm. For each of the four rows $(a_3, a_4, a_5, x_0)$ equal to $(13.024334, 70.951332, 460.634397856444, 1035745443241)$, $(13.024334, 70.951332, 0, 153887581621)$, $(13.024334, 0, 0, 7713187213)$ and $(0, 0, 0, 54941209)$, every real $x \\ge x_0$ satisfies","title":"Axler's Table 5 lower bounds for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/eight-term-series-table-six-lt-prime-cou","kind":"node","lean":"MetaMathlibExt.eight_term_series_table_six_lt_primeCounting","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#eight-term-series-table-six-lt-prime-cou","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$ and $\\log$ the natural logarithm. For each of the five rows $(b_5, b_6, b_7, b_8, x_0)$ equal to $(23.975666, 119.87833, 719.26998, 5034.88986, 1681111802141)$, $(23.975666, 119.87833, 719.26998, 0, 721733241667)$, $(23.975666, 119.87833, 0, 0, 110838719141)$, $(23.975666, 0, 0, 0, 1331691853)$ and $(0, 0, 0, 0, 10383799)$, every real $x \\ge x_0$ satisfies","title":"Axler's Table 6 lower bounds for the prime-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/abs-theta-sub-lt-div-log-cube","kind":"node","lean":"MetaMathlibExt.abs_theta_sub_lt_div_log_cube","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#abs-theta-sub-lt-div-log-cube","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"For every real $x \\ge 1757126630797$ (the prime $p_{64707865143}$),","title":"Axler's cubic-logarithm bound for Chebyshev's theta function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/abs-theta-sub-lt-axler-four-bounds","kind":"node","lean":"MetaMathlibExt.abs_theta_sub_lt_axler_four_bounds","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#abs-theta-sub-lt-axler-four-bounds","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\vartheta(x) = \\sum_{p \\le x} \\log p$ be Chebyshev's function, the sum over the primes $p \\le x$. Then for real $x$ we have $|\\vartheta(x) - x| < 0.043\\, x / \\log^3 x$ for $x \\ge e^{40}$, $|\\vartheta(x) - x| < 0.15\\, x / \\log^3 x$ for $e^{35} \\le x < e^{5000}$, $|\\vartheta(x) - x| < 99.07\\, x / \\log^4 x$ for $x \\ge e^{25}$, and $|\\vartheta(x) - x| < 100\\, x / \\log^4 x$ for $x \\ge 70111$.","title":"Axler's four bounds for Chebyshev's theta function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/abs-sum-inv-prime-sub-log-log-sub-le","kind":"node","lean":"MetaMathlibExt.abs_sum_inv_prime_sub_log_log_sub_le","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#abs-sum-inv-prime-sub-log-log-sub-le","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"For every real $x \\ge 1757126630797$,","title":"Axler's explicit Mertens second theorem"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/abs-sum-log-div-prime-sub-log-sub-le","kind":"node","lean":"MetaMathlibExt.abs_sum_log_div_prime_sub_log_sub_le","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#abs-sum-log-div-prime-sub-log-sub-le","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"For every real $x \\ge 1757126630797$,","title":"Axler's explicit Mertens first theorem"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prod-one-sub-inv-prime-bounds","kind":"node","lean":"MetaMathlibExt.prod_one_sub_inv_prime_bounds","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prod-one-sub-inv-prime-bounds","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\gamma$ be the Euler\u2013Mascheroni constant and $f(x) = \\frac{0.024334}{3 \\log^3 x} \\left(1 + \\frac{15}{4 \\log x}\\right)$. For every real $x \\ge 1757126630797$, the product over the primes $p \\le x$ satisfies $\\frac{e^{-\\gamma}}{\\log x} \\exp(-f(x)) < \\prod_{p \\le x} \\left(1 - \\frac{1}{p}\\right) < \\frac{e^{-\\gamma}}{\\log x} \\exp\\left(f(x) + \\frac{1.02}{(x - 1) \\log x}\\right)$.","title":"Explicit Mertens third theorem"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prod-one-add-inv-prime-bounds","kind":"node","lean":"MetaMathlibExt.prod_one_add_inv_prime_bounds","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Explicit bounds for "],["math","\\(\\prod_{p \\le x} (1 + 1/p)\\)"]],"source":"Explicit bounds for $\\prod_{p \\le x} (1 + 1/p)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prod-one-add-inv-prime-bounds","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\gamma$ be the Euler\u2013Mascheroni constant and $f(x) = \\frac{0.024334}{3 \\log^3 x} \\left(1 + \\frac{15}{4 \\log x}\\right)$. For every real $x \\ge 1757126630797$, the product over the primes $p \\le x$ satisfies\u2026","title":"Explicit bounds for $\\prod_{p \\le x} (1 + 1/p)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/series-lt-prime-counting-of-lt-prime-cou","kind":"node","lean":"MetaMathlibExt.series_lt_primeCounting_of_lt_primeCounting","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Converting a lower bound for "],["math","\\(\\pi(x)\\)"],["text"," into a truncated series"]],"source":"Converting a lower bound for $\\pi(x)$ into a truncated series"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#series-lt-prime-counting-of-lt-prime-cou","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$. Let $n \\ge 1$ be an integer, let $a_0, \\dots, a_n$ be reals with $a_0 > 0$ and $a_1, \\dots, a_n < 0$, and let $x_0 > 0$. Suppose that for every real $x \\ge x_0$ the denominator $D(x) = a_0 \\log x + \\sum_{i=1}^{n} a_i / \\log^{i-1} x$ is positive and $\\pi(x) > x / D(x)$. Let $b_0 = 1/a_0$ and $b_k = -\\frac{1}{a_0} \\sum_{i=1}^{k} a_i b_{k-i}$ for $1 \\le k \\le n$. Then\u2026","title":"Converting a lower bound for $\\pi(x)$ into a truncated series"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-counting-mem-icc-aux-j","kind":"node","lean":"MetaMathlibExt.primeCounting_mem_Icc_auxJ","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime counting between the auxiliary functions "],["math","\\(J\\)"]],"source":"Prime counting between the auxiliary functions $J$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-counting-mem-icc-aux-j","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and $\\vartheta(x) = \\sum_{p \\le x} \\log p$ Chebyshev's function. Let $k \\ge 1$ be an integer and let $\\eta > 0$ and $x_1 > 1$ be reals with $|\\vartheta(y) - y| < \\eta y / \\log^k y$ for every real $y \\ge x_1$. For reals $e$ and $x \\ge x_1$ set\u2026","title":"Prime counting between the auxiliary functions $J$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-counting-le-logarithmic-integral","kind":"node","lean":"MetaMathlibExt.primeCounting_le_logarithmicIntegral","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-counting-le-logarithmic-integral","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ denote the number of primes $p \\le x$, and let $\\operatorname{li}(x)$ be the logarithmic integral, the principal value $\\mathrm{PV}\\int_0^x \\frac{dt}{\\log t}$. For every real $x$ with $2 \\le x \\le 10^{19}$, $\\pi(x) \\le \\operatorname{li}(x)$.","title":"B\u00fcthe's bound of the prime-counting function by the logarithmic integral"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/abs-prime-counting-sub-logarithmic-integ","kind":"node","lean":"MetaMathlibExt.abs_primeCounting_sub_logarithmicIntegral_lt","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#abs-prime-counting-sub-logarithmic-integ","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"For every real $x$ with $2657 \\le x \\le 1.101 \\cdot 10^{26}$,","title":"Johnston's range for Schoenfeld's inequality"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/abs-logarithmic-integral-add-tsum-inv-pr","kind":"node","lean":"MetaMathlibExt.abs_logarithmicIntegral_add_tsum_inv_prime_pow_lt","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#abs-logarithmic-integral-add-tsum-inv-pr","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $k \\ge 1$ be an integer and let $\\eta > 0$ and $x_0 > 1$ be reals such that $|\\vartheta(y) - y| < \\eta y / \\log^k y$ for every real $y \\ge x_0$, where $\\vartheta(y) = \\sum_{p \\le y} \\log p$ is Chebyshev's function. Then for every integer $n \\ge 2$ and every real $x \\ge x_0$,","title":"Tails of prime power sums from a Chebyshev bound"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/abs-mertens-remainder-sub-tsum-logarithm","kind":"node","lean":"MetaMathlibExt.abs_mertensRemainder_sub_tsum_logarithmicIntegral_lt","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#abs-mertens-remainder-sub-tsum-logarithm","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $k \\ge 1$ be an integer and let $\\eta > 0$ and $x_0 > 1$ be reals such that $|\\vartheta(y) - y| < \\eta y / \\log^k y$ for every real $y \\ge x_0$, where $\\vartheta(y) = \\sum_{p \\le y} \\log p$ is Chebyshev's function. For real $x$ let $S(x) = \\sum_{p > x} \\big(\\log(1 - 1/p) + 1/p\\big)$, the sum over the primes $p > x$. Then for every real $x \\ge x_0$,","title":"The Mertens remainder from a Chebyshev bound"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/abs-logarithmic-integral-add-tsum-inv-pr-2","kind":"node","lean":"MetaMathlibExt.abs_logarithmicIntegral_add_tsum_inv_prime_pow_lt_of_RH","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#abs-logarithmic-integral-add-tsum-inv-pr-2","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Assume the Riemann hypothesis. Then for every integer $n \\ge 2$ and every real $x \\ge 599$,","title":"Tails of prime power sums under the Riemann hypothesis"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/axler-theta-error-factor-tail-antitone","kind":"node","lean":"MetaMathlibExt.axlerThetaErrorFactor_tail_antitone","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#axler-theta-error-factor-tail-antitone","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $R = 5.5666305$ and for real $y \\ge e^{673}$ let","title":"The last branch of Axler's error factor is nonincreasing"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/dusart-2010-prime-interval-bound","kind":"node","lean":"dusart_2010_prime_interval_bound","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#dusart-2010-prime-interval-bound","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"For every real number $x \\ge 396738$ there is a prime $p$ with $x < p \\le x \\bigl(1 + \\frac{1}{25 (\\log x)^2}\\bigr)$, where $\\log$ is the natural logarithm.","title":"Dusart's primes in short intervals"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/dusart-pi-upper","kind":"node","lean":"dusart_pi_upper","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's upper bound for "],["math","\\(\\pi(x)\\)"],["text"," for all "],["math","\\(x > 1\\)"]],"source":"Dusart's upper bound for $\\pi(x)$ for all $x > 1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#dusart-pi-upper","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and $\\log$ the natural logarithm. For every real number $x > 1$, $\\pi(x) \\le \\frac{x}{\\log x} \\bigl(1 + \\frac{1.2762}{\\log x}\\bigr)$.","title":"Dusart's upper bound for $\\pi(x)$ for all $x > 1$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/dusart-pi-lower","kind":"node","lean":"dusart_pi_lower","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's lower bound for "],["math","\\(\\pi(x)\\)"],["text"," for "],["math","\\(x \\ge 599\\)"]],"source":"Dusart's lower bound for $\\pi(x)$ for $x \\ge 599$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#dusart-pi-lower","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and $\\log$ the natural logarithm. For every real number $x \\ge 599$, $\\frac{x}{\\log x} \\bigl(1 + \\frac{1}{\\log x}\\bigr) \\le \\pi(x)$.","title":"Dusart's lower bound for $\\pi(x)$ for $x \\ge 599$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/dusart-pi-asymptotic-expansion","kind":"node","lean":"MetaMathlibExt.dusart_pi_asymptotic_expansion","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's asymptotic expansion of "],["math","\\(\\pi(x)\\)"],["text"," with explicit error"]],"source":"Dusart's asymptotic expansion of $\\pi(x)$ with explicit error"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#dusart-pi-asymptotic-expansion","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and $\\log$ the natural logarithm. For every real number $x \\ge 4 \\cdot 10^9$ there is a real number $E$ with $|E| \\le \\frac{7.32}{(\\log x)^3}$ such that","title":"Dusart's asymptotic expansion of $\\pi(x)$ with explicit error"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/dusart-pi-lower-second-order","kind":"node","lean":"MetaMathlibExt.dusart_pi_lower_second_order","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's second-order lower bound for "],["math","\\(\\pi(x)\\)"]],"source":"Dusart's second-order lower bound for $\\pi(x)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#dusart-pi-lower-second-order","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and let $\\log$ be the natural logarithm. For every real number $x \\ge 88789$,","title":"Dusart's second-order lower bound for $\\pi(x)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/dusart-pi-lower-shifted-log","kind":"node","lean":"MetaMathlibExt.dusart_pi_lower_shifted_log","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's shifted-logarithm lower bound for "],["math","\\(\\pi(x)\\)"]],"source":"Dusart's shifted-logarithm lower bound for $\\pi(x)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#dusart-pi-lower-shifted-log","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and let $\\log$ be the natural logarithm. For every real number $x \\ge 5393$, $\\pi(x) \\ge \\frac{x}{\\log x - 1}$.","title":"Dusart's shifted-logarithm lower bound for $\\pi(x)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/dusart-pi-upper-second-order","kind":"node","lean":"MetaMathlibExt.dusart_pi_upper_second_order","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's second-order upper bound for "],["math","\\(\\pi(x)\\)"]],"source":"Dusart's second-order upper bound for $\\pi(x)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#dusart-pi-upper-second-order","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and let $\\log$ be the natural logarithm. For every real number $x > 1$,","title":"Dusart's second-order upper bound for $\\pi(x)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/dusart-pi-upper-shifted-log","kind":"node","lean":"MetaMathlibExt.dusart_pi_upper_shifted_log","parent":7445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's shifted-logarithm upper bound for "],["math","\\(\\pi(x)\\)"]],"source":"Dusart's shifted-logarithm upper bound for $\\pi(x)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#dusart-pi-upper-shifted-log","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and let $\\log$ be the natural logarithm. For every real number $x > e^{1.112}$, $\\pi(x) \\le \\frac{x}{\\log x - 1.112}$.","title":"Dusart's shifted-logarithm upper bound for $\\pi(x)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/prime-prime-counting-asymptotic","kind":"node","lean":"MetaMathlibExt.prime_prime_counting_asymptotic","parent":7445,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function/index.html#prime-prime-counting-asymptotic","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-bounds-for-the-prime-counting-function","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes at most $x$, and let $\\operatorname{Li}(x) = \\int_2^x dt / \\log t$. Then $\\pi(\\pi(x)) \\sim x / (\\log x)^2$ as the real number $x$ tends to infinity, and there is a constant $A > 0$ such that, as $x \\to \\infty$,","title":"Counting primes with prime subscripts"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","title":"Explicit Mertens-type prime sum estimates"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/dickman-ge-rpow-neg-mul","kind":"node","lean":"MetaMathlibExt.dickman_ge_rpow_neg_mul","parent":7481,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html#dickman-ge-rpow-neg-mul","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","summary":"Let $\\rho$ be a Dickman\u2013de Bruijn function. Then $\\rho(x) \\ge x^{-1.42 x}$ for every real $x \\ge 1$.","title":"Explicit lower bound for the Dickman function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/exists-norm-log-deriv-riemann-zeta-le-in","kind":"node","lean":"MetaMathlibExt.exists_norm_logDeriv_riemannZeta_le_inv_norm_add","parent":7481,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html#exists-norm-log-deriv-riemann-zeta-le-in","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","summary":"Let $\\zeta$ be the Riemann zeta function. There are real numbers $C$ and $\\alpha_0 > 0$ such that for all real $\\alpha$ and $\\tau$ with $0 < \\alpha < \\alpha_0$ and $|\\tau| \\le 1/2$, the point $s = 1 + \\alpha + i\\tau$ satisfies $|\\zeta'(s)/\\zeta(s)| \\le 1/|s - 1| + C$.","title":"Logarithmic derivative of the zeta function near its pole"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/abs-sum-inv-primes-sub-meissel-mertens-c","kind":"node","lean":"MetaMathlibExt.abs_sum_inv_primes_sub_meisselMertensConstant_le","parent":7481,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html#abs-sum-inv-primes-sub-meissel-mertens-c","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","summary":"Let $M$ be the Meissel\u2013Mertens constant. Then $0.2614 \\le M < 0.2615$, and for every real $x > 1$, $\\left|\\sum_{p \\le x} 1/p - \\log \\log x - M\\right| \\le 1/(\\log x)^2$, where the sum runs over the primes $p \\le x$.","title":"Mertens' second theorem with the Rosser\u2013Schoenfeld error term"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/log-log-add-le-sum-inv-primes-le","kind":"node","lean":"MetaMathlibExt.log_log_add_le_sum_inv_primes_le","parent":7481,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html#log-log-add-le-sum-inv-primes-le","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","summary":"For every real $x \\ge 2$, $\\log \\log x + 0.2614 \\le \\sum_{p \\le x} 1/p \\le \\log \\log x + 0.8666$, where the sum runs over the primes $p \\le x$.","title":"Two-sided explicit Mertens bound"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/sum-abs-cos-sub-div-primes-le","kind":"node","lean":"MetaMathlibExt.sum_abs_cos_sub_div_primes_le","parent":7481,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html#sum-abs-cos-sub-div-primes-le","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","summary":"Let $K$ be a real number with $\\frac{1}{2\\pi} \\int_0^{2\\pi} |\\cos t - K| \\, dt = 1 - K$ (this equation has exactly one real solution, $K \\approx 0.3286$). Then for every real $\\alpha$ with $0 < \\alpha \\le 1$ and every real $\\tau$, $\\sum_{p \\le e^{1/\\alpha}} |\\cos(\\tau \\log p) - K|/p \\le (1 - K) \\log(1/\\alpha) + (2 + 2K) \\log \\log(|\\tau| + 3) + 7.28$, where the sum runs over the primes $p \\le e^{1/\\alpha}$.","title":"Explicit bound for a cosine-weighted sum over primes"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/re-sum-div-primes-cpow-le","kind":"node","lean":"MetaMathlibExt.re_sum_div_primes_cpow_le","parent":7481,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html#re-sum-div-primes-cpow-le","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","summary":"Throughout, sums over $p$ run over primes. Let $K$ be a real number with $\\frac{1}{2\\pi} \\int_0^{2\\pi} |\\cos t - K| \\, dt = 1 - K$. Let $g$ be a real-valued function on the natural numbers with $|g(p)| \\le 1$ for every prime $p$, let $\\alpha$ be a real number with $0 < \\alpha \\le 1$, put $x = e^{1/\\alpha}$, and let $\\lambda$ be a real number with $\\sum_{p \\le x} (1 - g(p))/p = \\lambda \\sum_{p \\le x} 1/p$. Let $M$\u2026","title":"Upper bound for twisted prime sums with real coefficients"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/norm-lseries-le-exp-nu-mul-exp-re-sum-pr","kind":"node","lean":"MetaMathlibExt.norm_LSeries_le_exp_nu\u2082_mul_exp_re_sum_primes","parent":7481,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html#norm-lseries-le-exp-nu-mul-exp-re-sum-pr","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","summary":"Let $g$ be a complex-valued completely multiplicative arithmetic function with $g(1) = 1$ and $|g(n)| \\le 1$ for every $n$, and let $F(s) = \\sum_{n \\ge 1} g(n) n^{-s}$ be its Dirichlet series. Let $\\nu_2 = \\sum_p (-\\log(1 - 1/p) - 1/p)$, summed over all primes $p$. Then $\\nu_2 \\le 0.316$, and for every complex $s$ with $\\operatorname{Re} s > 1$, $|F(s)| \\le e^{\\nu_2} \\exp(\\operatorname{Re} \\sum_p g(p) p^{-s})$,\u2026","title":"Bound for a Dirichlet series by its prime sum"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/tsum-rpow-neg-primes-gt-exp-inv-le","kind":"node","lean":"MetaMathlibExt.tsum_rpow_neg_primes_gt_exp_inv_le","parent":7481,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates/index.html#tsum-rpow-neg-primes-gt-exp-inv-le","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/explicit-mertens-type-prime-sum-estimates","status":"can_prove","summary":"For every real $\\alpha$ with $0 < \\alpha \\le 1$, $\\sum_{p > e^{1/\\alpha}} p^{-(1 + \\alpha)} \\le 0.9235$, where the sum runs over the primes $p > e^{1/\\alpha}$.","title":"Uniform tail bound for the prime zeta series"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","title":"Prime number theorem"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-series","kind":"node","lean":"Nat.Primes.logDirichletSeries","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-series","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"For a complex number $s$, the prime logarithmic Dirichlet series is $\\Phi(s) = \\sum_{p} \\frac{\\log p}{p^s}$, summed over all primes $p$ (the sum is taken to be $0$ where the family is not summable).","title":"Prime logarithmic Dirichlet series"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-correction-term","kind":"node","lean":"Nat.Primes.logDirichletCorrectionTerm","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-correction-term","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"For a prime $p$, a natural number $k$ and a complex number $s$, the correction summand is $\\frac{\\log p}{(p^{k+2})^s}$, the term of the prime-power correction series that belongs to the prime power $p^{k+2}$.","title":"Summand of the prime-power correction series"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-correction","kind":"node","lean":"Nat.Primes.logDirichletCorrection","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-correction","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"For a complex number $s$, the prime-power correction series is $C(s) = \\sum_{(p, k)} \\frac{\\log p}{(p^{k+2})^s}$, summed over all pairs of a prime $p$ and a natural number $k \\ge 0$ (the sum is taken to be $0$ where the family is not summable).","title":"Prime-power correction series"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-continuation","kind":"node","lean":"Nat.Primes.logDirichletContinuation","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-continuation","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"For a complex number $s$, the prime logarithmic continuation is $G(s) = -\\frac{\\zeta_1'(s)}{\\zeta_1(s)} - C(s)$, where $\\zeta_1$ is the entire function with $\\zeta_1(s) = (s - 1)\\zeta(s)$ for $s \\ne 1$ and $\\zeta_1(1) = 1$, the quotient is taken to be $0$ where $\\zeta_1(s) = 0$, and $C$ is the prime-power correction series.","title":"The canonical prime logarithmic continuation"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-correction-analytic-on","kind":"node","lean":"Nat.Primes.logDirichletCorrection_analyticOn","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-correction-analytic-on","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"The prime-power correction series $C(s) = \\sum_{(p, k)} \\frac{\\log p}{(p^{k+2})^s}$, summed over all primes $p$ and natural numbers $k \\ge 0$, is analytic at every point of the open half-plane $\\{s \\in \\mathbb{C} : \\operatorname{Re} s > 1/2\\}$.","title":"Analyticity of the prime-power correction series"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-von-mangoldt-split","kind":"node","lean":"Nat.Primes.logDirichlet_vonMangoldt_split","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-von-mangoldt-split","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the L-series $\\sum_{n \\ge 1} \\Lambda(n) n^{-s}$ of the von Mangoldt function $\\Lambda$ equals $\\Phi(s) + C(s)$, where $\\Phi(s) = \\sum_p \\log p / p^s$ is the prime logarithmic Dirichlet series, summed over all primes $p$, and $C(s) = \\sum_{(p, k)} \\log p / (p^{k+2})^s$ is the higher-prime-power correction series, summed over all pairs of a prime $p$ and a\u2026","title":"Splitting the von Mangoldt L-series over prime powers"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-series-eq","kind":"node","lean":"Nat.Primes.logDirichletSeries_eq","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-series-eq","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, $\\Phi(s) = -\\frac{\\zeta'(s)}{\\zeta(s)} - C(s)$, where $\\Phi$ is the prime logarithmic Dirichlet series, $\\zeta$ is the Riemann zeta function and $C$ is the prime-power correction series.","title":"Prime logarithmic series via the logarithmic derivative of zeta"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/riemann-zeta-ne-zero-of-one-le-re","kind":"node","lean":"Nat.Primes.riemannZeta\u2081_ne_zero_of_one_le_re","parent":7490,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonvanishing of the Riemann zeta function times "],["math","\\(s - 1\\)"],["text"," on "],["math","\\(\\operatorname{Re} s \\ge 1\\)"]],"source":"Nonvanishing of the Riemann zeta function times $s - 1$ on $\\operatorname{Re} s \\ge 1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#riemann-zeta-ne-zero-of-one-le-re","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\zeta$ be the Riemann zeta function and let $\\zeta_1 \\colon \\mathbb{C} \\to \\mathbb{C}$ be given by $\\zeta_1(s) = (s - 1)\\zeta(s)$ for $s \\ne 1$ and $\\zeta_1(1) = 1$. Then $\\zeta_1(s) \\ne 0$ for every complex number $s$ with $\\operatorname{Re} s \\ge 1$.","title":"Nonvanishing of the Riemann zeta function times $s - 1$ on $\\operatorname{Re} s \\ge 1$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-continuation-meromorphic","kind":"node","lean":"Nat.Primes.logDirichletContinuation_meromorphic","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-continuation-meromorphic","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"The prime logarithmic continuation $G$ is meromorphic on the open half-plane $\\{s \\in \\mathbb{C} : \\operatorname{Re} s > 1/2\\}$.","title":"Meromorphy of the prime logarithmic continuation"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-continuation-analytic-on","kind":"node","lean":"Nat.Primes.logDirichletContinuation_analyticOn","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-continuation-analytic-on","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"The prime logarithmic continuation $G$ is analytic at every point of the closed half-plane $\\{s \\in \\mathbb{C} : \\operatorname{Re} s \\ge 1\\}$.","title":"Analyticity of the prime logarithmic continuation"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-continuation-agree","kind":"node","lean":"Nat.Primes.logDirichletContinuation_agree","parent":7490,"reading_title":{"format":"inline-parts-v1","parts":[["text","The prime logarithmic continuation on the half-plane "],["math","\\(\\operatorname{Re} s > 1\\)"]],"source":"The prime logarithmic continuation on the half-plane $\\operatorname{Re} s > 1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-continuation-agree","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\Phi(s) = \\sum_p \\frac{\\log p}{p^s}$ be the prime logarithmic Dirichlet series, summed over all primes $p$, and let $G$ be the prime logarithmic continuation. Then $G(s) = \\Phi(s) - \\frac{1}{s - 1}$ for every complex number $s$ with $\\operatorname{Re} s > 1$.","title":"The prime logarithmic continuation on the half-plane $\\operatorname{Re} s > 1$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/log-dirichlet-meromorphic-continuation","kind":"node","lean":"Nat.Primes.logDirichlet_meromorphicContinuation","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#log-dirichlet-meromorphic-continuation","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\Phi(s) = \\sum_p \\frac{\\log p}{p^s}$ be the prime logarithmic Dirichlet series. There is a function $g : \\mathbb{C} \\to \\mathbb{C}$ that is meromorphic on the open half-plane $\\operatorname{Re} s > 1/2$, analytic at every point of the closed half-plane $\\operatorname{Re} s \\ge 1$, and satisfies $g(s) = \\Phi(s) - \\frac{1}{s-1}$ whenever $\\operatorname{Re} s > 1$.","title":"Meromorphic continuation of the prime logarithmic Dirichlet series"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/prime-counting-sub-theta-div-log-is-litt","kind":"node","lean":"Chebyshev.primeCounting_sub_theta_div_log_isLittleO","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#prime-counting-sub-theta-div-log-is-litt","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and let $\\theta(x) = \\sum_{p \\le x} \\log p$, the sum running over primes. As the real number $x$ tends to $+\\infty$,","title":"Prime counting versus theta over log"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/prime-counting-is-equivalent-of-theta-is","kind":"node","lean":"Chebyshev.primeCounting_isEquivalent_of_theta_isEquivalent","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#prime-counting-is-equivalent-of-theta-is","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and let $\\theta(x) = \\sum_{p \\le x} \\log p$, the sum running over primes. If $\\theta(x) \\sim x$ as the real number $x$ tends to $+\\infty$, then $\\pi(x) \\sim x / \\log x$ as $x \\to +\\infty$.","title":"Theta asymptotic implies the prime number theorem"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/theta-is-equivalent-of-prime-counting-is","kind":"node","lean":"Chebyshev.theta_isEquivalent_of_primeCounting_isEquivalent","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#theta-is-equivalent-of-prime-counting-is","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and let $\\theta(x) = \\sum_{p \\le x} \\log p$, the sum running over primes. If $\\pi(x) \\sim x / \\log x$ as the real number $x$ tends to $+\\infty$, then $\\theta(x) \\sim x$ as $x \\to +\\infty$.","title":"Prime number theorem implies the theta asymptotic"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/prime-counting-is-equivalent-iff-theta-i","kind":"node","lean":"Chebyshev.primeCounting_isEquivalent_iff_theta_isEquivalent","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#prime-counting-is-equivalent-iff-theta-i","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$ and let $\\theta(x) = \\sum_{p \\le x} \\log p$ be Chebyshev's theta function, the sum running over primes. Then, as the real number $x$ tends to $+\\infty$, $\\pi(x) \\sim x / \\log x$ holds if and only if $\\theta(x) \\sim x$.","title":"Prime number theorem is equivalent to the theta asymptotic"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/prime-log-series","kind":"node","lean":"Chebyshev.primeLogSeries","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#prime-log-series","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$, the prime logarithm series at $s$ is the complex number $\\sum_p \\log p \\cdot p^{-s}$, where the sum runs over all primes $p$ and $p^{-s}$ is the complex power. The sum is unconditional and is taken to be $0$ by convention when the family $(\\log p \\cdot p^{-s})_p$ is not summable.","title":"The prime logarithm series"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/laplace-theta-exp-eq-prime-log-series-di","kind":"node","lean":"Chebyshev.laplace_theta_exp_eq_primeLogSeries_div","parent":7490,"reading_title":{"format":"inline-parts-v1","parts":[["text","Laplace transform of "],["math","\\(\\theta(e^t)\\)"]],"source":"Laplace transform of $\\theta(e^t)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#laplace-theta-exp-eq-prime-log-series-di","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$ be Chebyshev's function, the sum running over the primes $p \\le x$. For every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > 1$, the Laplace transform $\\int_{(0, \\infty)} e^{-st}\\,\\theta(e^t)\\,dt$ of $t \\mapsto \\theta(e^t)$ at $s$ equals $\\frac{1}{s} \\sum_p \\log p \\cdot p^{-s}$, the prime logarithm series at $s$ divided by $s$, where the sum runs over all primes $p$.","title":"Laplace transform of $\\theta(e^t)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/has-laplace-theta-exp","kind":"node","lean":"Chebyshev.hasLaplace_theta_exp","parent":7490,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergence of the Laplace transform of "],["math","\\(\\theta(e^t)\\)"]],"source":"Convergence of the Laplace transform of $\\theta(e^t)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#has-laplace-theta-exp","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$ be Chebyshev's function, the sum running over the primes $p \\le x$. For every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > 1$, the function $t \\mapsto \\theta(e^t)$ has Laplace transform $\\frac{1}{s} \\sum_p \\log p \\cdot p^{-s}$ at $s$, where the sum runs over all primes $p$ (this is the prime logarithm series at $s$ divided by $s$): the function\u2026","title":"Convergence of the Laplace transform of $\\theta(e^t)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/differentiable-on-laplace-theta-exp","kind":"node","lean":"Chebyshev.differentiableOn_laplace_theta_exp","parent":7490,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holomorphy of the Laplace transform of "],["math","\\(\\theta(e^t)\\)"]],"source":"Holomorphy of the Laplace transform of $\\theta(e^t)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#differentiable-on-laplace-theta-exp","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$ be Chebyshev's function, the sum running over the primes $p \\le x$. Then the Laplace transform $F(s) = \\int_{(0, \\infty)} e^{-st}\\,\\theta(e^t)\\,dt$ of $t \\mapsto \\theta(e^t)$ is complex differentiable at every point of the open half-plane $\\{s \\in \\mathbb{C} : \\operatorname{Re} s > 1\\}$, so it is holomorphic there.","title":"Holomorphy of the Laplace transform of $\\theta(e^t)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/shifted-log-dirichlet-continuation","kind":"node","lean":"Nat.Primes.shiftedLogDirichletContinuation","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#shifted-log-dirichlet-continuation","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\Phi(s) = \\sum_p \\log p / p^s$, the sum over primes, and let $g$ be the canonical continuation of $\\Phi(s) - 1/(s - 1)$ to $\\operatorname{Re} s > 1/2$. For $s \\in \\mathbb{C}$, the shifted continuation is $g_1(s) = g(s + 1)$.","title":"The shifted log-Dirichlet continuation"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/shifted-log-dirichlet-continuation-merom","kind":"node","lean":"Nat.Primes.shiftedLogDirichletContinuation_meromorphic","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#shifted-log-dirichlet-continuation-merom","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"The shifted continuation $g_1$ is meromorphic at every point $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > -1/2$.","title":"Meromorphy of the shifted continuation"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/shifted-log-dirichlet-continuation-analy","kind":"node","lean":"Nat.Primes.shiftedLogDirichletContinuation_analyticOn","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#shifted-log-dirichlet-continuation-analy","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"The shifted continuation $g_1$ is complex analytic at every point $s \\in \\mathbb{C}$ with $\\operatorname{Re} s \\ge 0$.","title":"Holomorphy of the shifted continuation on the closed half-plane"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/shifted-log-dirichlet-continuation-agree","kind":"node","lean":"Nat.Primes.shiftedLogDirichletContinuation_agree","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#shifted-log-dirichlet-continuation-agree","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\Phi(s) = \\sum_p \\log p / p^s$ be the prime logarithmic Dirichlet series and $g_1$ the shifted continuation. For every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > 0$, $g_1(s) = \\Phi(s + 1) - 1/s$.","title":"The shifted continuation agrees with the shifted series"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/normalized-theta-error","kind":"node","lean":"Chebyshev.normalizedThetaError","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#normalized-theta-error","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"For real $t$, let $H(t) = \\theta(e^t) e^{-t} - 1$, where $\\theta(x) = \\sum_{p \\le x} \\log p$ is the Chebyshev function, the sum running over primes $p \\le x$.","title":"The normalized Chebyshev error"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/normalized-theta-error-continuation","kind":"node","lean":"Chebyshev.normalizedThetaErrorContinuation","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#normalized-theta-error-continuation","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\Phi(s) = \\sum_p \\log p / p^s$, the sum over primes, let $g$ be the canonical continuation of $\\Phi(s) - 1/(s - 1)$, and let $g_1(s) = g(s + 1)$ be the shifted continuation. For $s \\in \\mathbb{C}$, the normalized-error continuation is $F(s) = (g_1(s) - 1)/(s + 1)$, with $F(-1) = 0$ since division by zero gives $0$.","title":"The normalized-error continuation"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/normalized-theta-error-continuation-mero","kind":"node","lean":"Chebyshev.normalizedThetaErrorContinuation_meromorphic","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#normalized-theta-error-continuation-mero","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"The normalized-error continuation $F$ is meromorphic at every point $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > -1/2$.","title":"Meromorphy of the normalized-error continuation"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/normalized-theta-error-continuation-anal","kind":"node","lean":"Chebyshev.normalizedThetaErrorContinuation_analyticOn","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#normalized-theta-error-continuation-anal","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"The normalized-error continuation $F$ is complex analytic at every point $s \\in \\mathbb{C}$ with $\\operatorname{Re} s \\ge 0$.","title":"Holomorphy of the normalized-error continuation on the closed half-plane"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/has-laplace-normalized-theta-error","kind":"node","lean":"Chebyshev.hasLaplace_normalizedThetaError","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#has-laplace-normalized-theta-error","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$ be the Chebyshev function (the sum over primes $p \\le x$), let $H(t) = \\theta(e^t) e^{-t} - 1$ for real $t$ be the normalized Chebyshev error, viewed as a complex-valued function, and let $F$ be the normalized-error continuation. For every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > 0$, the function $t \\mapsto e^{-st} H(t)$ is integrable on $(0, \\infty)$ and\u2026","title":"Laplace transform of the normalized Chebyshev error"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/measurable-normalized-theta-error","kind":"node","lean":"Chebyshev.measurable_normalizedThetaError","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#measurable-normalized-theta-error","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$ be Chebyshev's function, the sum running over primes $p$. The normalized theta error $H(t) = \\theta(e^t)e^{-t} - 1$ is a Borel measurable function of $t \\in \\mathbb{R}$.","title":"Measurability of the normalized theta error"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/abs-normalized-theta-error-le","kind":"node","lean":"Chebyshev.abs_normalizedThetaError_le","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#abs-normalized-theta-error-le","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$ be Chebyshev's function, the sum running over primes $p$, and let $H(t) = \\theta(e^t)e^{-t} - 1$ be the normalized theta error. Then $|H(t)| \\le \\log 4 + 1$ for every real $t$.","title":"Explicit bound for the normalized theta error"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/normalized-theta-error-bounded","kind":"node","lean":"Chebyshev.normalizedThetaError_bounded","parent":7490,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundedness of the normalized theta error on "],["math","\\((0, \\infty)\\)"]],"source":"Boundedness of the normalized theta error on $(0, \\infty)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#normalized-theta-error-bounded","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\theta(x) = \\sum_{p \\le x} \\log p$ be Chebyshev's function, the sum running over primes $p$, and let $H(t) = \\theta(e^t)e^{-t} - 1$ be the normalized theta error. There is a real number $M$ such that $|H(t)| \\le M$ for every real $t \\ge 0$.","title":"Boundedness of the normalized theta error on $(0, \\infty)$"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/prime-number-theorem","kind":"node","lean":"MathlibExt.NumberTheory.PrimeNumberTheoremWanted.prime_number_theorem","parent":7490,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem/index.html#prime-number-theorem","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-number-theorem","status":"fully_proved","summary":"Let $\\pi(n)$ be the number of primes $p \\le n$. Then $\\pi(n) \\sim n / \\log n$ as $n \\to \\infty$ through the natural numbers, that is, $\\pi(n) - n / \\log n = o(n / \\log n)$, where $\\log$ is the natural logarithm.","title":"Prime number theorem"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","status":"can_prove","title":"Prime-counting formulas from divisor functions"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/abs-exp-approx-sub-dvd-indicator-le","kind":"node","lean":"MetaMathlibExt.abs_exp_approx_sub_dvd_indicator_le","parent":7524,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/index.html#abs-exp-approx-sub-dvd-indicator-le","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","status":"can_prove","summary":"Let $A$ be a real-valued function of two natural numbers such that $A(k, n) \\sin^2(n\\pi/k) \\ge 1$ for all natural numbers $k, n$ with $1 \\le k \\le n$ and $k \\nmid n$. Then for all natural numbers $k, n$ with $1 \\le k \\le n$,","title":"Exponential approximation of the divisibility indicator"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/divisor-count-approx-sub-is-big-o","kind":"node","lean":"MetaMathlibExt.divisorCountApprox_sub_isBigO","parent":7524,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/index.html#divisor-count-approx-sub-is-big-o","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","status":"can_prove","summary":"Let $A$ be a real-valued function of two natural numbers such that $A(k, n) \\sin^2(n\\pi/k) \\ge 1$ for all natural numbers $k, n$ with $1 \\le k \\le n$ and $k \\nmid n$. For a natural number $n$ let $\\tau(n)$ be the number of positive divisors of $n$ and","title":"Exponential approximation of the divisor-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/prime-counting-approx-sub-lt","kind":"node","lean":"MetaMathlibExt.primeCountingApprox_sub_lt","parent":7524,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/index.html#prime-counting-approx-sub-lt","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","status":"can_prove","summary":"For every natural number $n \\ge 2$,","title":"Prime counting from the exact divisor count"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/one-le-pi-sq-mul-sq-div-two-mul-sin-sq","kind":"node","lean":"MetaMathlibExt.one_le_pi_sq_mul_sq_div_two_mul_sin_sq","parent":7524,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/index.html#one-le-pi-sq-mul-sq-div-two-mul-sin-sq","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","status":"can_prove","summary":"For all positive integers $k$ and $n$ with $k \\nmid n$,","title":"Sine lower bound at non-divisors"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/divisor-count-approx-one-sub-is-big-o","kind":"node","lean":"MetaMathlibExt.divisorCountApprox_one_sub_isBigO","parent":7524,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/index.html#divisor-count-approx-one-sub-is-big-o","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","status":"can_prove","summary":"For a natural number $k$ let $\\tau(k)$ be the number of positive divisors of $k$ and","title":"Trigonometric approximation of the divisor-counting function"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/prime-counting-approx-abs-sub-lt","kind":"node","lean":"MetaMathlibExt.primeCountingApprox_abs_sub_lt","parent":7524,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/index.html#prime-counting-approx-abs-sub-lt","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","status":"can_prove","summary":"For a natural number $k$ let","title":"Prime-counting function from the smooth divisor count"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/divisor-sum-approx-sub-lt","kind":"node","lean":"MetaMathlibExt.divisorSumApprox_sub_lt","parent":7524,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions/index.html#divisor-sum-approx-sub-lt","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-counting-formulas-from-divisor-functions","status":"can_prove","summary":"Let $A$ be a real-valued function of two natural numbers such that $A(k, n) \\sin^2(n\\pi/k) \\ge 1$ for all natural numbers $k, n$ with $1 \\le k \\le n$ and $k \\nmid n$. For a natural number $n \\ge 1$ let $\\sigma(n)$ be the sum of the positive divisors of $n$ and","title":"Exponential approximation of the sum-of-divisors function"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes.json","id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes","kind":"scope","navigation":"module","parent":7412,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes/index.html","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes","status":"fully_proved","title":"Prime-indexed primes"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes/prime-primes-ratio-tendsto-one","kind":"node","lean":"MetaMathlibExt.primePrimes_ratio_tendsto_one'","parent":7532,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes/index.html#prime-primes-ratio-tendsto-one","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes","status":"fully_proved","summary":"Let $p_0 = 2$, $p_1 = 3$, $p_2 = 5$, $\\dots$ be the primes in increasing order, indexed from zero, and let $q_n = p_{p_n}$. Then $q_n / q_{n+1} \\to 1$ as $n \\to \\infty$.","title":"Ratio of consecutive zero-based prime-indexed primes tends to one"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes/prime-primes-one-based-ratio-tendsto-one","kind":"node","lean":"MetaMathlibExt.primePrimes_oneBased_ratio_tendsto_one","parent":7532,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes/index.html#prime-primes-one-based-ratio-tendsto-one","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes","status":"fully_proved","summary":"Let $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $\\dots$ be the primes in increasing order and let $q_n = p_{p_n}$. Then $q_n / q_{n+1} \\to 1$ as $n \\to \\infty$.","title":"Ratio of consecutive prime-indexed primes tends to one"},{"id":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes/prime-primes-ratio-tendsto-one-2","kind":"node","lean":"MetaMathlibExt.primePrimes_ratio_tendsto_one","parent":7532,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes/index.html#prime-primes-ratio-tendsto-one-2","scope":"NumberTheory/distribution-of-primes/prime-number-theorem/prime-indexed-primes","status":"fully_proved","summary":"Assume that there are infinitely many primes (which always holds). Let $p_0 = 2$, $p_1 = 3$, $p_2 = 5$, $\\dots$ be the primes in increasing order, indexed from zero, and let $q_n = p_{p_n}$. Then $q_n / q_{n+1} \\to 1$ as $n \\to \\infty$.","title":"Broughan\u2013Barnett ratio limit, zero-based form"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals","kind":"scope","parent":7411,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals","status":"can_prove","summary":"Bertrand-type and Breusch theorems, Ramanujan primes, prime gaps, bounded gaps, Sylvester\u2013Schur on prime factors of consecutive products.","title":"Prime gaps and primes in short intervals"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/dusart-prime-gap","kind":"node","lean":"MetaMathlibExt.dusart_prime_gap","parent":7536,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/index.html#dusart-prime-gap","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals","status":"can_prove","summary":"Let $\\log$ be the natural logarithm. For every real number $x \\ge 468991632$ there is a prime $p$ with $x < p \\le x + \\frac{x}{5000 (\\log x)^2}$.","title":"Dusart's short-interval prime gap bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-two-primes-between","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.ramanujan_two_primes_between","parent":7536,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/index.html#ramanujan-two-primes-between","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals","status":"fully_proved","summary":"For every natural number $n \\ge 6$ there are primes $p$ and $q$ with $n < p < q < 2n$; that is, the open interval $(n, 2n)$ contains two distinct primes.","title":"Ramanujan's two-primes refinement of Bertrand's postulate"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"can_prove","title":"Bonse's inequality"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/bonse-primes-infinite","kind":"node","lean":"MetaMathlibExt.bonse_primes_infinite","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#bonse-primes-infinite","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"fully_proved","summary":"The set of prime natural numbers is infinite.","title":"Infinitely many primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/bonse-nth-strict-mono","kind":"node","lean":"MetaMathlibExt.bonse_nth_strictMono","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#bonse-nth-strict-mono","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"fully_proved","summary":"For $k \\ge 1$ let $p_k$ be the $k$-th smallest prime number. If $i < j$, then $p_i < p_j$.","title":"The enumeration of the primes is strictly increasing"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/bonse-nth-prime","kind":"node","lean":"MetaMathlibExt.bonse_nth_prime","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#bonse-nth-prime","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"fully_proved","summary":"For every $k \\ge 1$, the $k$-th smallest prime number exists and is prime.","title":"The enumeration of the primes takes prime values"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/bonse-nth-five","kind":"node","lean":"MetaMathlibExt.bonse_nth_five","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#bonse-nth-five","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"fully_proved","summary":"The sixth prime number, counting $2$ as the first, is $13$.","title":"The sixth prime is thirteen"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/bonse-nth-succ-lt","kind":"node","lean":"MetaMathlibExt.bonse_nth_succ_lt","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#bonse-nth-succ-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"fully_proved","summary":"Let $p_1 < p_2 < p_3 < \\cdots$ be the prime numbers in increasing order. For every $k \\ge 1$, $p_{k+1} < 2 p_k$.","title":"Bertrand's postulate for consecutive primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/bonse-inequality","kind":"node","lean":"MetaMathlibExt.bonse_inequality","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#bonse-inequality","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"fully_proved","summary":"Let $p_1 = 2, p_2 = 3, p_3 = 5, \\dots$ be the prime numbers in increasing order. For every natural number $k > 4$, the square of the $(k+1)$-st prime is smaller than the product of the first $k$ primes: $p_{k+1}^2 < p_1 p_2 \\cdots p_k$.","title":"Bonse's inequality"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/prime-nth","kind":"node","lean":"MathlibExt.NumberTheory.BonseWanted.primeNth","parent":7539,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(m\\)"],["text","-th prime, counted from one"]],"source":"The $m$-th prime, counted from one"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#prime-nth","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"fully_proved","summary":"For $m \\ge 1$, $p_m$ is the $m$-th prime, so that $p_1 = 2$, $p_2 = 3$, $p_3 = 5$; at $m = 0$ the value is also $2$.","title":"The $m$-th prime, counted from one"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/is-bonse-seq","kind":"node","lean":"MathlibExt.NumberTheory.BonseWanted.IsBonseSeq","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#is-bonse-seq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"fully_proved","summary":"For $m \\ge 1$ let $p_m$ be the $m$-th prime, so that $p_1 = 2$, and set $p_0 = 2$. A sequence $a_0, a_1, a_2, \\ldots$ of natural numbers is a Bonse sequence if $p_{a_n}^2 < p_{a_0} p_{a_1} \\cdots p_{a_{n-1}}$ for all sufficiently large $n$.","title":"Bonse sequence of prime indices"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/is-bonse-seq-of-liminf-gt","kind":"node","lean":"MathlibExt.NumberTheory.BonseWanted.isBonseSeq_of_liminf_gt","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#is-bonse-seq-of-liminf-gt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"can_prove","summary":"Let $a_0 < a_1 < a_2 < \\cdots$ be a strictly increasing sequence of positive integers and write $A_n = a_0 a_1 \\cdots a_{n-1}$ for the product of its first $n$ terms. If there is a real number $c > 2$ such that $\\log A_n / \\log a_n > c$ for all sufficiently large $n$, then $a$ is a Bonse sequence.","title":"A sufficient condition for a Bonse sequence"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/not-is-bonse-seq-of-liminf-lt","kind":"node","lean":"MathlibExt.NumberTheory.BonseWanted.not_isBonseSeq_of_liminf_lt","parent":7539,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality/index.html#not-is-bonse-seq-of-liminf-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bonse-s-inequality","status":"can_prove","summary":"Let $a_0 < a_1 < a_2 < \\cdots$ be a strictly increasing sequence of positive integers and write $A_n = a_0 a_1 \\cdots a_{n-1}$ for the product of its first $n$ terms. Suppose that $(n + 1) \\log \\log a_n / \\log a_{n+1} \\to 0$ as $n \\to \\infty$, and that there is a real number $c < 2$ such that $\\log A_n / \\log a_n < c$ for infinitely many $n$. Then $a$ is not a Bonse sequence.","title":"A sufficient condition for a non-Bonse sequence"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8","status":"fully_proved","title":"Breusch's theorem on primes between x and 9x/8"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","The modulus "],["math","\\(Q_9 = 210\\)"],["text"," to The grid function reaches "],["math","\\(2\\)"],["text"," at "],["math","\\(657\\)"]],"source":"The modulus $Q_9 = 210$ to The grid function reaches $2$ at $657$"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","title":"The modulus $Q_9 = 210$ to The grid function reaches $2$ at $657$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/q9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.Q9","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","The modulus "],["math","\\(Q_9 = 210\\)"]],"source":"The modulus $Q_9 = 210$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#q9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For the $k = 9$ case of Breusch's argument, the truncation modulus is the natural number $Q_9 = 2 \\cdot 3 \\cdot 5 \\cdot 7 = 210$.","title":"The modulus $Q_9 = 210$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/w9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","The M\u00f6bius divisor table for "],["math","\\(210\\)"]],"source":"The M\u00f6bius divisor table for $210$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#w9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"$W_9$ is the list of pairs of a natural number and an integer $(1, 1), (2, -1), (3, -1), (5, -1), (6, 1), (7, -1), (10, 1), (14, 1), (15, 1), (21, 1),$ $(30, -1), (35, 1), (42, -1), (70, -1), (105, -1), (210, 1)$, in this order. It pairs each divisor $d$ of $210$ with its M\u00f6bius value $\\mu(d)$.","title":"The M\u00f6bius divisor table for $210$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/w9-length","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9_length","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor table for "],["math","\\(210\\)"],["text"," has 16 entries"]],"source":"The divisor table for $210$ has 16 entries"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#w9-length","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"The M\u00f6bius divisor table $W_9$ for $210$ is a list of length $16$.","title":"The divisor table for $210$ has 16 entries"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/w9-complete","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9_complete","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor table for "],["math","\\(210\\)"],["text"," lists every divisor"]],"source":"The divisor table for $210$ lists every divisor"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#w9-complete","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"The set of first components $d$ of the pairs $(d, \\varepsilon)$ in the M\u00f6bius divisor table $W_9$ is exactly the set of divisors of $Q_9 = 210$.","title":"The divisor table for $210$ lists every divisor"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/w9-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9_pos","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisors in the table for "],["math","\\(210\\)"],["text"," are positive"]],"source":"Divisors in the table for $210$ are positive"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#w9-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every pair $(d, \\varepsilon)$ in the M\u00f6bius divisor table $W_9$ for $210$, $d \\ge 1$.","title":"Divisors in the table for $210$ are positive"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/w9-dvd","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9_dvd","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entries of the divisor table for "],["math","\\(210\\)"],["text"," divide "],["math","\\(210\\)"]],"source":"Entries of the divisor table for $210$ divide $210$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#w9-dvd","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every pair $(d, \\varepsilon)$ in the M\u00f6bius divisor table $W_9$, $d$ divides $Q_9 = 210$.","title":"Entries of the divisor table for $210$ divide $210$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/g9term","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9term","parent":7551,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#g9term","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For a natural number $d$, an integer $m$ and a natural number $j$, the summand of the pair $(d, m)$ at $j$ is the integer","title":"Summand of the Breusch grid function"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/g9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Breusch grid function for "],["math","\\(k = 9\\)"]],"source":"Breusch grid function for $k = 9$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#g9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For a natural number $j$, $G_9(j)$ is the integer","title":"Breusch grid function for $k = 9$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/m9-block0","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M9_block0","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid bound "],["math","\\(G_9 \\le 3\\)"],["text"," for "],["math","\\(0 \\le j < 2700\\)"]],"source":"Grid bound $G_9 \\le 3$ for $0 \\le j < 2700$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#m9-block0","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $k < 2700$, the grid function satisfies $G_9(0 + k) \\le 3$; that is, $G_9(j) \\le 3$ for all natural numbers $j < 2700$.","title":"Grid bound $G_9 \\le 3$ for $0 \\le j < 2700$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/m9-block1","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M9_block1","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid bound "],["math","\\(G_9 \\le 3\\)"],["text"," for "],["math","\\(2700 \\le j < 5400\\)"]],"source":"Grid bound $G_9 \\le 3$ for $2700 \\le j < 5400$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#m9-block1","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $k < 2700$, the grid function satisfies $G_9(2700 + k) \\le 3$; that is, $G_9(j) \\le 3$ for all natural numbers $j$ with $2700 \\le j < 5400$.","title":"Grid bound $G_9 \\le 3$ for $2700 \\le j < 5400$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/m9-block2","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M9_block2","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid bound "],["math","\\(G_9 \\le 3\\)"],["text"," for "],["math","\\(5400 \\le j < 8100\\)"]],"source":"Grid bound $G_9 \\le 3$ for $5400 \\le j < 8100$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#m9-block2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $k < 2700$, the grid function satisfies $G_9(5400 + k) \\le 3$; that is, $G_9(j) \\le 3$ for all natural numbers $j$ with $5400 \\le j < 8100$.","title":"Grid bound $G_9 \\le 3$ for $5400 \\le j < 8100$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/m9-block3","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M9_block3","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid bound "],["math","\\(G_9 \\le 3\\)"],["text"," for "],["math","\\(8100 \\le j < 10800\\)"]],"source":"Grid bound $G_9 \\le 3$ for $8100 \\le j < 10800$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#m9-block3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $k < 2700$, the grid function satisfies $G_9(8100 + k) \\le 3$; that is, $G_9(j) \\le 3$ for all natural numbers $j$ with $8100 \\le j < 10800$.","title":"Grid bound $G_9 \\le 3$ for $8100 \\le j < 10800$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/m9-block4","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M9_block4","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid bound "],["math","\\(G_9 \\le 3\\)"],["text"," for "],["math","\\(10800 \\le j < 13500\\)"]],"source":"Grid bound $G_9 \\le 3$ for $10800 \\le j < 13500$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#m9-block4","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $k < 2700$, the grid function satisfies $G_9(10800 + k) \\le 3$; that is, $G_9(j) \\le 3$ for all natural numbers $j$ with $10800 \\le j < 13500$.","title":"Grid bound $G_9 \\le 3$ for $10800 \\le j < 13500$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/m9-block5","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M9_block5","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid bound "],["math","\\(G_9 \\le 3\\)"],["text"," for "],["math","\\(13500 \\le j < 16200\\)"]],"source":"Grid bound $G_9 \\le 3$ for $13500 \\le j < 16200$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#m9-block5","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $k < 2700$, the grid function satisfies $G_9(13500 + k) \\le 3$; that is, $G_9(j) \\le 3$ for all natural numbers $j$ with $13500 \\le j < 16200$.","title":"Grid bound $G_9 \\le 3$ for $13500 \\le j < 16200$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/m9-block6","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M9_block6","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid bound "],["math","\\(G_9 \\le 3\\)"],["text"," for "],["math","\\(16200 \\le j < 18900\\)"]],"source":"Grid bound $G_9 \\le 3$ for $16200 \\le j < 18900$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#m9-block6","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $k < 2700$, the grid function satisfies $G_9(16200 + k) \\le 3$; that is, $G_9(j) \\le 3$ for all natural numbers $j$ with $16200 \\le j < 18900$.","title":"Grid bound $G_9 \\le 3$ for $16200 \\le j < 18900$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/m9-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M9_le","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid bound "],["math","\\(G_9 \\le 3\\)"]],"source":"Grid bound $G_9 \\le 3$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#m9-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $j < 18900$, the grid function satisfies $G_9(j) \\le 3$.","title":"Grid bound $G_9 \\le 3$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/jmin1-attained","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin1_attained","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grid function reaches "],["math","\\(1\\)"],["text"," at "],["math","\\(279\\)"]],"source":"The grid function reaches $1$ at $279$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#jmin1-attained","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"The Breusch grid function $G_9$ for $k = 9$ satisfies $G_9(279) \\ge 1$.","title":"The grid function reaches $1$ at $279$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/jmin1-core","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin1_core","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grid function stays below "],["math","\\(1\\)"],["text"," on "],["math","\\((10, 279)\\)"]],"source":"The grid function stays below $1$ on $(10, 279)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#jmin1-core","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"For every natural number $j$ with $10 \\le j < 279$, the Breusch grid function $G_9$ for $k = 9$ satisfies $G_9(j) < 1$.","title":"The grid function stays below $1$ on $(10, 279)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/jmin1","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin1","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","First index from "],["math","\\(10\\)"],["text"," where the grid function reaches "],["math","\\(1\\)"]],"source":"First index from $10$ where the grid function reaches $1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#jmin1","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"The Breusch grid function $G_9$ for $k = 9$ satisfies $G_9(279) \\ge 1$, and $G_9(j) < 1$ for every natural number $j$ with $10 \\le j < 279$. So $279$ is the least $j \\ge 10$ with $G_9(j) \\ge 1$.","title":"First index from $10$ where the grid function reaches $1$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/jmin2-attained","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin2_attained","parent":7551,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grid function reaches "],["math","\\(2\\)"],["text"," at "],["math","\\(657\\)"]],"source":"The grid function reaches $2$ at $657$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01/index.html#jmin2-attained","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-01","status":"fully_proved","summary":"The Breusch grid function $G_9$ for $k = 9$ satisfies $G_9(657) \\ge 2$.","title":"The grid function reaches $2$ at $657$"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grid function stays below "],["math","\\(2\\)"],["text"," on "],["math","\\((10, 657)\\)"],["text"," to Quotients of Q\u2089 by its divisors are positive"]],"source":"The grid function stays below $2$ on $(10, 657)$ to Quotients of Q\u2089 by its divisors are positive"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","title":"The grid function stays below $2$ on $(10, 657)$ to Quotients of Q\u2089 by its divisors are positive"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/jmin2-core","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin2_core","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grid function stays below "],["math","\\(2\\)"],["text"," on "],["math","\\((10, 657)\\)"]],"source":"The grid function stays below $2$ on $(10, 657)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#jmin2-core","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For every natural number $j$ with $10 \\le j < 657$, the Breusch grid function $G_9$ for $k = 9$ satisfies $G_9(j) < 2$.","title":"The grid function stays below $2$ on $(10, 657)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/jmin2","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin2","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","First index from "],["math","\\(10\\)"],["text"," where the grid function reaches "],["math","\\(2\\)"]],"source":"First index from $10$ where the grid function reaches $2$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#jmin2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"The Breusch grid function $G_9$ for $k = 9$ satisfies $G_9(657) \\ge 2$, and $G_9(j) < 2$ for every natural number $j$ with $10 \\le j < 657$. So $657$ is the least $j \\ge 10$ with $G_9(j) \\ge 2$.","title":"First index from $10$ where the grid function reaches $2$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/jmin3-attained","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin3_attained","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grid function reaches "],["math","\\(3\\)"],["text"," at "],["math","\\(963\\)"]],"source":"The grid function reaches $3$ at $963$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#jmin3-attained","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"The Breusch grid function $G_9$ for $k = 9$ satisfies $G_9(963) \\ge 3$.","title":"The grid function reaches $3$ at $963$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/jmin3-core","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin3_core","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grid function stays below "],["math","\\(3\\)"],["text"," on "],["math","\\((10, 963)\\)"]],"source":"The grid function stays below $3$ on $(10, 963)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#jmin3-core","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For every natural number $j$ with $10 \\le j < 963$, the Breusch grid function $G_9$ for $k = 9$ satisfies $G_9(j) < 3$.","title":"The grid function stays below $3$ on $(10, 963)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/jmin3","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.jmin3","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#jmin3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"The grid function $G_9$ satisfies $G_9(963) \\ge 3$, and $G_9(j) < 3$ for every natural number $j$ with $10 \\le j < 963$.","title":"First index from 10 on with grid value at least 3"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/w9-mu-sign","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9_mu_sign","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signs in the divisor table for "],["math","\\(210\\)"]],"source":"Signs in the divisor table for $210$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#w9-mu-sign","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For every pair $(d, \\varepsilon)$ in the M\u00f6bius divisor table $W_9$ for $210$, $\\varepsilon = 1$ or $\\varepsilon = -1$.","title":"Signs in the divisor table for $210$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/w9-nodup","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9_nodup","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor table for "],["math","\\(210\\)"],["text"," has no repeated entries"]],"source":"The divisor table for $210$ has no repeated entries"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#w9-nodup","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"No pair $(d, \\varepsilon)$ occurs more than once in the list $W_9$, the M\u00f6bius divisor table for $210$.","title":"The divisor table for $210$ has no repeated entries"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/map-sum-to-finset","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.map_sum_toFinset","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#map-sum-to-finset","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, let $l$ be a list of elements of $\\mathbb{N} \\times \\mathbb{Z}$ with no repeated entries, and let $f : \\mathbb{N} \\times \\mathbb{Z} \\to M$. Then the sum of the values of $f$ along $l$ equals the sum of $f$ over the finite set of entries of $l$.","title":"Sum over a duplicate-free list"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/exp-le-inv-one-sub","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.exp_le_inv_one_sub","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#exp-le-inv-one-sub","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For every real number $u$ with $0 \\le u < 1$, $e^u \\le \\frac{1}{1 - u}$.","title":"Exponential bound by the geometric series"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/one-sub-inv-le-log","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.one_sub_inv_le_log","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#one-sub-inv-le-log","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For every real number $t \\ge 1$, $1 - 1/t \\le \\log t$.","title":"Logarithm is at least 1 \u2212 1/t"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/stirling-lower","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.stirling_lower","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#stirling-lower","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For every natural number $n \\ge 1$, $n^n e^{1-n} \\le n!$.","title":"Weak Stirling lower bound for factorials"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/stirling-upper","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.stirling_upper","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#stirling-upper","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For every natural number $n \\ge 1$, $n! \\le n^{n+1} e^{1-n}$.","title":"Weak Stirling upper bound for factorials"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/n10m","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N10m","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","Scaled factorial argument "],["math","\\(N_{10}\\)"]],"source":"Scaled factorial argument $N_{10}$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#n10m","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"Let $Q_9 = 2 \\cdot 3 \\cdot 5 \\cdot 7 = 210$. For natural numbers $m$ and $n$, $N_{10}(m, n)$ is the natural number $10 n \\lfloor Q_9 / m \\rfloor$, where $\\lfloor Q_9 / 0 \\rfloor$ is taken to be $0$.","title":"Scaled factorial argument $N_{10}$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/n9m","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N9m","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#n9m","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For natural numbers $m$ and $n$, $N_9(m, n) = 9n \\lfloor Q_9 / m \\rfloor$, where $Q_9 = 210$ and $\\lfloor Q_9 / m \\rfloor$ is the quotient of natural-number division, taken to be $0$ when $m = 0$.","title":"Scaled factorial argument N\u2089"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/n1m","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N1m","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","Scaled factorial argument "],["math","\\(N_1\\)"]],"source":"Scaled factorial argument $N_1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#n1m","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"Let $Q_9 = 2 \\cdot 3 \\cdot 5 \\cdot 7 = 210$. For natural numbers $m$ and $n$, $N_1(m, n)$ is the natural number $n \\lfloor Q_9 / m \\rfloor$, where $\\lfloor Q_9 / 0 \\rfloor$ is taken to be $0$.","title":"Scaled factorial argument $N_1$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/ratio9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.ratio9","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#ratio9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For natural numbers $m$ and $n$, $\\mathrm{ratio}_9(m, n) = N_{10}(m, n)! / (N_9(m, n)! \\cdot N_1(m, n)!)$, with natural-number division, where $N_k(m, n) = kn \\lfloor Q_9 / m \\rfloor$ for $k = 1, 9, 10$ and $Q_9 = 210$ (the quotient $\\lfloor Q_9 / m \\rfloor$ is taken to be $0$ when $m = 0$).","title":"Binomial factor ratio\u2089"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/log-f9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.logF9","parent":7572,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logarithm of Breusch's product "],["math","\\(F_9(n)\\)"]],"source":"Logarithm of Breusch's product $F_9(n)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#log-f9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"Let $w_9$ be the table of divisors $d$ of $Q_9 = 210$ paired with their M\u00f6bius values $\\varepsilon$. For a natural number $n$, writing $k_d = 210/d$,","title":"Logarithm of Breusch's product $F_9(n)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/q9-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.Q9_pos","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#q9-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"The truncation modulus $Q_9$ of the $k = 9$ case satisfies $0 < Q_9$.","title":"Q\u2089 is positive"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/q9-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.Q9_eq","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#q9-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"The truncation modulus $Q_9 = 2 \\cdot 3 \\cdot 5 \\cdot 7$ of the $k = 9$ case equals $210$.","title":"Value of Q\u2089"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/q9div-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.Q9div_pos","parent":7572,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02/index.html#q9div-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-02","status":"fully_proved","summary":"For every natural number $m \\ge 1$ dividing $Q_9 = 210$, the quotient $Q_9 / m$ is at least $1$.","title":"Quotients of Q\u2089 by its divisors are positive"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positivity of the scaled factorial argument "],["math","\\(N_{10}\\)"],["text"," to Exact formula for the scaled factorial argument "],["math","\\(N_{10}\\)"]],"source":"Positivity of the scaled factorial argument $N_{10}$ to Exact formula for the scaled factorial argument $N_{10}$"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","title":"Positivity of the scaled factorial argument $N_{10}$ to Exact formula for the scaled factorial argument $N_{10}$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/n10-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N10_pos","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positivity of the scaled factorial argument "],["math","\\(N_{10}\\)"]],"source":"Positivity of the scaled factorial argument $N_{10}$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#n10-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $Q_9 = 210$ and, for natural numbers $m$ and $n$, let $N_{10}(m, n) = 10 n \\lfloor Q_9 / m \\rfloor$. If $n \\ge 1$, $m \\ge 1$ and $m$ divides $Q_9$, then $N_{10}(m, n) \\ge 1$.","title":"Positivity of the scaled factorial argument $N_{10}$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/n9-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N9_pos","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#n9-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"For natural numbers $m$ and $n$ with $n \\ge 1$, $m \\ge 1$ and $m$ dividing $Q_9 = 210$, the scaled factorial argument $N_9(m, n) = 9n \\lfloor Q_9 / m \\rfloor$ satisfies $N_9(m, n) \\ge 1$.","title":"Positivity of the scaled argument N\u2089"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/n1-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N1_pos","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positivity of the scaled factorial argument "],["math","\\(N_1\\)"]],"source":"Positivity of the scaled factorial argument $N_1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#n1-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $Q_9 = 210$ and, for natural numbers $m$ and $n$, let $N_1(m, n) = n \\lfloor Q_9 / m \\rfloor$. If $n \\ge 1$, $m \\ge 1$ and $m$ divides $Q_9$, then $N_1(m, n) \\ge 1$.","title":"Positivity of the scaled factorial argument $N_1$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/n10-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N10_le","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for the scaled factorial argument "],["math","\\(N_{10}\\)"]],"source":"Upper bound for the scaled factorial argument $N_{10}$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#n10-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $Q_9 = 210$ and, for natural numbers $m$ and $n$, let $N_{10}(m, n) = 10 n \\lfloor Q_9 / m \\rfloor$, where $\\lfloor Q_9 / 0 \\rfloor$ is taken to be $0$. Then $N_{10}(m, n) \\le 10 Q_9 n$ for all natural numbers $m$ and $n$.","title":"Upper bound for the scaled factorial argument $N_{10}$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/n9-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N9_le","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for the scaled factorial argument "],["math","\\(N_9\\)"]],"source":"Upper bound for the scaled factorial argument $N_9$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#n9-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $Q_9 = 210$ and, for natural numbers $m$ and $n$, let $N_9(m, n) = 9 n \\lfloor Q_9 / m \\rfloor$, where $\\lfloor Q_9 / 0 \\rfloor$ is taken to be $0$. Then $N_9(m, n) \\le 10 Q_9 n$ for all natural numbers $m$ and $n$.","title":"Upper bound for the scaled factorial argument $N_9$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/n1-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N1_le","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for the scaled factorial argument "],["math","\\(N_1\\)"]],"source":"Upper bound for the scaled factorial argument $N_1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#n1-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $Q_9 = 210$ and, for natural numbers $m$ and $n$, let $N_1(m, n) = n \\lfloor Q_9 / m \\rfloor$, where $\\lfloor Q_9 / 0 \\rfloor$ is taken to be $0$. Then $N_1(m, n) \\le 10 Q_9 n$ for all natural numbers $m$ and $n$.","title":"Upper bound for the scaled factorial argument $N_1$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/nadd9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.Nadd9","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#nadd9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"For natural numbers $m$ and $n$ with $m \\ge 1$ and $m$ dividing $Q_9 = 210$, write $N_k(m, n) = kn \\lfloor Q_9 / m \\rfloor$ for $k = 1, 9, 10$. Then $N_{10}(m, n) = N_9(m, n) + N_1(m, n)$.","title":"N\u2081\u2080 splits as N\u2089 + N\u2081"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/ratio9-dvd","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.ratio9_dvd","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#ratio9-dvd","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"For natural numbers $m$ and $n$ with $m \\ge 1$ and $m$ dividing $Q_9 = 210$, write $N_k(m, n) = kn \\lfloor Q_9 / m \\rfloor$ for $k = 1, 9, 10$. Then $N_9(m, n)! \\cdot N_1(m, n)!$ divides $N_{10}(m, n)!$.","title":"Factorial divisibility behind ratio\u2089"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/ratio9-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.ratio9_pos","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#ratio9-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"For natural numbers $m$ and $n$ with $n \\ge 1$, $m \\ge 1$ and $m$ dividing $Q_9 = 210$, the binomial factor $\\mathrm{ratio}_9(m, n) = N_{10}(m, n)! / (N_9(m, n)! \\cdot N_1(m, n)!)$ is positive, where $N_k(m, n) = kn \\lfloor Q_9 / m \\rfloor$ for $k = 1, 9, 10$.","title":"Positivity of ratio\u2089"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/log-ratio9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log_ratio9","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#log-ratio9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $m$ and $n$ be natural numbers with $n \\ge 1$, $m \\ge 1$ and $m \\mid 210$, and put $k = 210/m$. Then","title":"Logarithm of the binomial factor"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/w9-mu-sum-q","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9_mu_sum_Q","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#w9-mu-sum-q","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $w_9$ be the table of pairs $(d, \\mu(d))$, where $d$ runs over the divisors of $210$ and $\\mu$ is the M\u00f6bius function. Then the sum of $\\mu(d)/d$ over the pairs of $w_9$, computed in $\\mathbb{Q}$, is $8/35$; that is, $\\sum_{d \\mid 210} \\mu(d)/d = 8/35$.","title":"M\u00f6bius reciprocal sum over the divisors of 210, in the rationals"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/w9-mu-sum-r","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.w9_mu_sum_R","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#w9-mu-sum-r","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $w_9$ be the table of pairs $(d, \\mu(d))$, where $d$ runs over the divisors of $210$ and $\\mu$ is the M\u00f6bius function. Then the sum of $\\mu(d)/d$ over the pairs of $w_9$, computed in $\\mathbb{R}$, is $8/35$; that is, $\\sum_{d \\mid 210} \\mu(d)/d = 8/35$.","title":"M\u00f6bius reciprocal sum over the divisors of 210, in the reals"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/log-fact-lower","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log_fact_lower","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for "],["math","\\(\\log N!\\)"]],"source":"Lower bound for $\\log N!$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#log-fact-lower","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"For every natural number $N \\ge 1$, $N \\log N - N + 1 \\le \\log N!$.","title":"Lower bound for $\\log N!$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/log-fact-upper","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log_fact_upper","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for "],["math","\\(\\log N!\\)"]],"source":"Upper bound for $\\log N!$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#log-fact-upper","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"For every natural number $N \\ge 1$, $\\log N! \\le (N + 1) \\log N - N + 1$.","title":"Upper bound for $\\log N!$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/fmain","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.fmain","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#fmain","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"For a real number $x$, $f_{\\mathrm{main}}(x) = x \\log x - x$.","title":"Main Stirling term"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/log-f9-term-lower","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.logF9_term_lower","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#log-f9-term-lower","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $(d, \\varepsilon)$ be a pair in the M\u00f6bius divisor table $W_9$ for $Q_9 = 210$ and let $n \\ge 1$ be a natural number. Write $q = \\lfloor Q_9 / d \\rfloor$ for the natural-number quotient, and let $A = N_{10}(d, n) = 10nq$, $B = N_9(d, n) = 9nq$ and $C = N_1(d, n) = nq$ be the scaled factorial arguments and $f_{\\mathrm{main}}(x) = x \\log x - x$ the main Stirling term. Then\u2026","title":"Termwise Stirling lower bound in Breusch's argument"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/cndvd9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.CNdvd9","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#cndvd9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $w_9$ be the table of divisors $d$ of $Q_9 = 210$ paired with their M\u00f6bius values. If $C$ is a natural number with $Q_9 \\mid C$, then for every pair $(d, \\varepsilon)$ in $w_9$ and every natural number $n$, $d \\mid C n$.","title":"Divisors of 210 divide multiples of 210"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/k9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.K9","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","The constant "],["math","\\(K_9\\)"]],"source":"The constant $K_9$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#k9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $w_9$ be the table of divisors $d$ of $Q_9 = 210$ paired with their M\u00f6bius values $\\varepsilon$. Then","title":"The constant $K_9$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/z9-value","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.Z9_value","parent":7593,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#z9-value","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $w_9$ be the table of pairs $(d, \\mu(d))$ for the divisors $d$ of $210$, let $f(x) = x \\log x - x$, and let $K_9 = \\sum_{(d, \\mu(d)) \\in w_9} \\mu(d) \\log d / d$. For natural numbers $C > 0$ and $n \\ge 1$ with $210 \\mid C$, writing $N = Cn$ (so that $N/d$ is an integer for each $d$ in $w_9$), we have $\\sum_{(d, \\mu(d)) \\in w_9} \\mu(d) f(N/d) = \\frac{8}{35} N (\\log N - 1) - K_9 N$.","title":"Closed form of the M\u00f6bius-weighted main-term sum"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/n10-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N10_eq","parent":7593,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exact formula for the scaled factorial argument "],["math","\\(N_{10}\\)"]],"source":"Exact formula for the scaled factorial argument $N_{10}$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03/index.html#n10-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-03","status":"fully_proved","summary":"Let $Q_9 = 210$ and, for natural numbers $m$ and $n$, let $N_{10}(m, n) = 10 n \\lfloor Q_9 / m \\rfloor$. If $m \\ge 1$ and $m$ divides $Q_9$, then $N_{10}(m, n) = \\lfloor 10 Q_9 n / m \\rfloor$.","title":"Exact formula for the scaled factorial argument $N_{10}$"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exact formula for the scaled factorial argument "],["math","\\(N_9\\)"],["text"," to Power bound for the multinomial coefficient "],["math","\\(M(t)\\)"],["text",", small "],["math","\\(t\\)"]],"source":"Exact formula for the scaled factorial argument $N_9$ to Power bound for the multinomial coefficient $M(t)$, small $t$"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","title":"Exact formula for the scaled factorial argument $N_9$ to Power bound for the multinomial coefficient $M(t)$, small $t$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/n9-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N9_eq","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exact formula for the scaled factorial argument "],["math","\\(N_9\\)"]],"source":"Exact formula for the scaled factorial argument $N_9$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#n9-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $Q_9 = 210$ and, for natural numbers $m$ and $n$, let $N_9(m, n) = 9 n \\lfloor Q_9 / m \\rfloor$. If $m \\ge 1$ and $m$ divides $Q_9$, then $N_9(m, n) = \\lfloor 9 Q_9 n / m \\rfloor$.","title":"Exact formula for the scaled factorial argument $N_9$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/n1-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.N1_eq","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exact formula for the scaled factorial argument "],["math","\\(N_1\\)"]],"source":"Exact formula for the scaled factorial argument $N_1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#n1-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $Q_9 = 210$ and, for natural numbers $m$ and $n$, let $N_1(m, n) = n \\lfloor Q_9 / m \\rfloor$. If $m \\ge 1$ and $m$ divides $Q_9$, then $N_1(m, n) = \\lfloor Q_9 n / m \\rfloor$.","title":"Exact formula for the scaled factorial argument $N_1$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/z9-combined","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.Z9_combined","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#z9-combined","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $w_9$ be the table of pairs $(d, \\mu(d))$ for the divisors $d$ of $210$, and let $f(x) = x \\log x - x$. For every natural number $n \\ge 1$, writing $N = 210n$ (so that $N/d$ is an integer for each $d$ in $w_9$), we have $\\sum_{(d, \\mu(d)) \\in w_9} \\mu(d) \\bigl(f(10N/d) - f(9N/d) - f(N/d)\\bigr) = 48 n (10 \\log 10 - 9 \\log 9)$.","title":"Combined main term of the M\u00f6bius-weighted Stirling sums"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/log-f9-lower","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.logF9_lower","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for "],["math","\\(\\log F_9(n)\\)"]],"source":"Lower bound for $\\log F_9(n)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#log-f9-lower","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $w_9$ be the table of divisors $d$ of $Q_9 = 210$ paired with their M\u00f6bius values $\\varepsilon$, and for a natural number $n$ let","title":"Lower bound for $\\log F_9(n)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/theta34-of-interval","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.theta34_of_interval","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#theta34-of-interval","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $a \\le m \\le b$ be natural numbers, and write $n\\#$ for the primorial of $n$ (the product of all primes $p \\le n$). If $10^a \\cdot b\\# \\le 34^a$, then $10^m \\cdot m\\# \\le 34^m$.","title":"Extending the primorial bound leftward across an interval"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/m34","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","The multinomial coefficient "],["math","\\(M(t)\\)"],["text"," for the 3.4 primorial bound"]],"source":"The multinomial coefficient $M(t)$ for the 3.4 primorial bound"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#m34","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"For a natural number $t$, $M(t) = \\frac{(6t)!}{(3t)!\\,(2t)!\\,t!}$, defined as the quotient of $(6t)!$ by $(3t)!\\,(2t)!\\,t!$ in natural-number division.","title":"The multinomial coefficient $M(t)$ for the 3.4 primorial bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/m34-dvd","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34_dvd","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#m34-dvd","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"For every natural number $t$, the product $(3t)!\\,(2t)!\\,t!$ divides $(6t)!$.","title":"Factorial divisibility behind the multinomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/m34-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34_pos","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positivity of the multinomial coefficient "],["math","\\(M(t)\\)"]],"source":"Positivity of the multinomial coefficient $M(t)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#m34-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $M(t)$ be the natural-number quotient of $(6t)!$ by $(3t)!\\,(2t)!\\,t!$. For every natural number $t$, $M(t) > 0$.","title":"Positivity of the multinomial coefficient $M(t)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/m34-eq-mul","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34_eq_mul","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cleared-denominator identity for the multinomial coefficient "],["math","\\(M(t)\\)"]],"source":"Cleared-denominator identity for the multinomial coefficient $M(t)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#m34-eq-mul","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $M(t)$ be the natural-number quotient of $(6t)!$ by $(3t)!\\,(2t)!\\,t!$. For every natural number $t$, $M(t) \\cdot \\big((3t)!\\,(2t)!\\,t!\\big) = (6t)!$.","title":"Cleared-denominator identity for the multinomial coefficient $M(t)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/m34-eq-choose-mul","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34_eq_choose_mul","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","The multinomial coefficient "],["math","\\(M(t)\\)"],["text"," as a product of binomial coefficients"]],"source":"The multinomial coefficient $M(t)$ as a product of binomial coefficients"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#m34-eq-choose-mul","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $M(t) = \\frac{(6t)!}{(3t)!\\,(2t)!\\,t!}$. For every natural number $t$, $M(t) = \\binom{6t}{3t} \\binom{3t}{t}$.","title":"The multinomial coefficient $M(t)$ as a product of binomial coefficients"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/choose-middle-half","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.choose_middle_half","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#choose-middle-half","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"For every natural number $m \\ge 1$, $\\binom{2m}{m} \\le 2^{2m-1}$.","title":"Half-power bound for the central binomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/multinomial3-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.multinomial3_le","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multinomial bound for "],["math","\\((3t)!\\)"]],"source":"Multinomial bound for $(3t)!$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#multinomial3-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"For every natural number $t$, $(3t)! \\le 3^{3t} (t!)^3$.","title":"Multinomial bound for $(3t)!$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/choose3-mul-choose2","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.choose3_mul_choose2","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#choose3-mul-choose2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"For every natural number $t$, $\\binom{3t}{t} \\binom{2t}{t} (t!)^3 = (3t)!$.","title":"Binomial factorization of the factorial of a multiple of three"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/four-pow-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.four_pow_le","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#four-pow-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"For every natural number $t$, $4^t \\le (2t+1)\\binom{2t}{t}$.","title":"Four-power lower bound for the central binomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/m34-key","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34_key","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Key estimate for the multinomial coefficient "],["math","\\(M(t)\\)"]],"source":"Key estimate for the multinomial coefficient $M(t)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#m34-key","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $M(t) = \\frac{(6t)!}{(3t)!\\,(2t)!\\,t!}$. For every natural number $t \\ge 1$, $2\\,M(t) \\le 432^t \\, (2t + 1)$.","title":"Key estimate for the multinomial coefficient $M(t)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/p34-base","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.P34_base","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#p34-base","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"The inequality $10^{1145} \\cdot 432^{230} \\cdot 461 \\le 2 \\cdot 34^{1145}$ holds; this is the inequality $10^{5t-5} \\cdot 432^{t} \\cdot (2t+1) \\le 2 \\cdot 34^{5t-5}$ at $t = 230$.","title":"Base case of the threshold inequality"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/p34-step","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.P34_step","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#p34-step","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $t \\ge 230$ be a natural number. If $10^{5t-5} \\cdot 432^{t} \\cdot (2t+1) \\le 2 \\cdot 34^{5t-5}$, then $10^{5t} \\cdot 432^{t+1} \\cdot (2t+3) \\le 2 \\cdot 34^{5t}$, which is the same inequality with $t+1$ in place of $t$.","title":"Induction step of the threshold inequality"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/p34","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.P34","parent":7614,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#p34","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"For every natural number $t \\ge 230$, $10^{5t-5} \\cdot 432^{t} \\cdot (2t+1) \\le 2 \\cdot 34^{5t-5}$.","title":"Threshold inequality for the multinomial bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/m34-bound-large","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34_bound_large","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power bound for the multinomial coefficient "],["math","\\(M(t)\\)"],["text",", large "],["math","\\(t\\)"]],"source":"Power bound for the multinomial coefficient $M(t)$, large $t$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#m34-bound-large","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $M(t) = \\frac{(6t)!}{(3t)!\\,(2t)!\\,t!}$. For every natural number $t \\ge 230$, $10^{5t-5} \\cdot M(t) \\le 34^{5t-5}$, that is, $M(t) \\le 3.4^{5t-5}$.","title":"Power bound for the multinomial coefficient $M(t)$, large $t$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/m34-bound-small","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34_bound_small","parent":7614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power bound for the multinomial coefficient "],["math","\\(M(t)\\)"],["text",", small "],["math","\\(t\\)"]],"source":"Power bound for the multinomial coefficient $M(t)$, small $t$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04/index.html#m34-bound-small","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-04","status":"fully_proved","summary":"Let $M(t) = \\frac{(6t)!}{(3t)!\\,(2t)!\\,t!}$. For every natural number $k < 207$, writing $t = 23 + k$ (so $23 \\le t < 230$), we have $10^{5t-5} \\cdot M(t) \\le 34^{5t-5}$.","title":"Power bound for the multinomial coefficient $M(t)$, small $t$"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power bound for the multinomial coefficient "],["math","\\(M(t)\\)"],["text"," to Upper bound for log 3"]],"source":"Power bound for the multinomial coefficient $M(t)$ to Upper bound for log 3"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","title":"Power bound for the multinomial coefficient $M(t)$ to Upper bound for log 3"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/m34-bound","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.M34_bound","parent":7635,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power bound for the multinomial coefficient "],["math","\\(M(t)\\)"]],"source":"Power bound for the multinomial coefficient $M(t)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#m34-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $M(t) = \\frac{(6t)!}{(3t)!\\,(2t)!\\,t!}$. For every natural number $t \\ge 23$, $10^{5t-5} \\cdot M(t) \\le 34^{5t-5}$, that is, $M(t) \\le 3.4^{5t-5}$.","title":"Power bound for the multinomial coefficient $M(t)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/div-add-div-add-div-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.div_add_div_add_div_le","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#div-add-div-add-div-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"For natural numbers $a, b, c$ and $q > 0$, $\\lfloor a/q \\rfloor + \\lfloor b/q \\rfloor + \\lfloor c/q \\rfloor \\le \\lfloor (a+b+c)/q \\rfloor$.","title":"Superadditivity of floor division for three terms"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/vp-term-case1","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.vp_term_case1","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#vp-term-case1","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $t \\ge 1$ and $p$ be natural numbers with $3t < p \\le 6t$. Then $\\lfloor 6t/p \\rfloor - \\lfloor 3t/p \\rfloor - \\lfloor 2t/p \\rfloor \\ge 1$.","title":"Floor-difference bound on the range 3t < p \u2264 6t"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/vp-term-case2","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.vp_term_case2","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#vp-term-case2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $t \\ge 1$ and $p$ be natural numbers with $2t < p \\le 3t$. Then $\\lfloor 6t/p \\rfloor - \\lfloor 3t/p \\rfloor - \\lfloor 2t/p \\rfloor \\ge 1$.","title":"Floor-difference bound on the range 2t < p \u2264 3t"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/vp-term-case3","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.vp_term_case3","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#vp-term-case3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $t \\ge 1$ and $p$ be natural numbers with $3t < 2p$ and $p \\le 2t$. Then $\\lfloor 6t/p \\rfloor - \\lfloor 3t/p \\rfloor - \\lfloor 2t/p \\rfloor \\ge 1$.","title":"Floor-difference bound on the range 3t/2 < p \u2264 2t"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/vp-term-case4","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.vp_term_case4","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#vp-term-case4","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $t \\ge 1$ and $p$ be natural numbers with $6t < 5p$ and $2p \\le 3t$. Then $\\lfloor 6t/p \\rfloor - \\lfloor 3t/p \\rfloor - \\lfloor 2t/p \\rfloor \\ge 1$.","title":"Floor-difference bound on the range 6t/5 < p \u2264 3t/2"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/vp-term-case5","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.vp_term_case5","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#vp-term-case5","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $t \\ge 1$ and $p$ be natural numbers with $t < p$ and $5p \\le 6t$. Then $\\lfloor 6t/p \\rfloor - \\lfloor 3t/p \\rfloor - \\lfloor 2t/p \\rfloor \\ge 1$.","title":"Floor-difference bound on the range t < p \u2264 6t/5"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/vp-term1","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.vp_term1","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#vp-term1","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $t \\ge 1$ and $p$ be natural numbers with $t < p \\le 6t$. Then $\\lfloor 6t/p \\rfloor - \\lfloor 3t/p \\rfloor - \\lfloor 2t/p \\rfloor \\ge 1$.","title":"Floor-difference bound on the range t < p \u2264 6t"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/vp-m34","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.vp_M34","parent":7635,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes in "],["math","\\((t, 6t)\\)"],["text"," occur in "],["math","\\(M(t)\\)"]],"source":"Primes in $(t, 6t)$ occur in $M(t)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#vp-m34","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"For every natural number $t \\ge 1$ and every prime $p$ with $t < p \\le 6t$, the exponent of $p$ in the prime factorization of the multinomial coefficient $M(t) = (6t)!/((3t)!\\,(2t)!\\,t!)$ is at least $1$.","title":"Primes in $(t, 6t)$ occur in $M(t)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/theta-le-m34","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.theta_le_M34","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#theta-le-m34","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"For a natural number $n$ let $n\\#$ be the primorial, the product of the primes $p \\le n$, and let $M(t) = \\frac{(6t)!}{(3t)!\\,(2t)!\\,t!}$. For every natural number $m \\ge 1$, with $t = \\lfloor (m+5)/6 \\rfloor$, we have $m\\# \\le t\\# \\cdot M(t)$.","title":"Splitting the primorial with a multinomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/theta34-base","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.theta34_base","parent":7635,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primorial bound "],["math","\\(3.4^m\\)"],["text"," below "],["math","\\(133\\)"]],"source":"Primorial bound $3.4^m$ below $133$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#theta34-base","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"For a natural number $n$ let $n\\#$ be the primorial, the product of the primes $p \\le n$. For every natural number $m < 133$, $10^m \\cdot m\\# \\le 34^m$.","title":"Primorial bound $3.4^m$ below $133$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/theta34-nat","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.theta34_nat","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#theta34-nat","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"For every natural number $m$, the primorial $m\\#$ (the product of all primes $p \\le m$) satisfies $10^m \\cdot m\\# \\le 34^m$.","title":"Primorial bound by powers of 3.4, integer form"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/theta34-real","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.theta34_real","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#theta34-real","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"For every real number $x \\ge 0$, the primorial of $\\lfloor x \\rfloor$ (the product of all primes $p \\le \\lfloor x \\rfloor$) is at most $3.4^x$, where $3.4^x$ is the real power.","title":"Primorial bound by powers of 3.4, real form"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/log-lt-of-exp-gt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log_lt_of_exp_gt","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#log-lt-of-exp-gt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $a$ and $r$ be real numbers with $a > 0$. If $a < e^r$, then $\\log a < r$, where $\\log$ is the natural logarithm.","title":"Bounding a logarithm above by the exponential"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/log-gt-of-exp-lt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log_gt_of_exp_lt","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#log-gt-of-exp-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"For real numbers $a > 0$ and $r$, if $e^r < a$ then $r < \\log a$, where $\\log$ is the natural logarithm.","title":"Logarithm lower bound from an exponential bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/lt-exp-of-pow-lt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.lt_exp_of_pow_lt","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#lt-exp-of-pow-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $a$ and $c$ be nonnegative real numbers and let $n$ and $m$ be positive integers. If $c < e$ and $a^n < c^m$, then $a < e^{m/n}$, where $m/n$ is real division.","title":"Exponential lower bound from a power comparison"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/exp-lt-of-pow-lt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.exp_lt_of_pow_lt","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#exp-lt-of-pow-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"Let $e$ be Euler's number, let $a$ and $c$ be real numbers and let $m, n$ be nonzero natural numbers, with $a \\ge 0$, $e^{m/n} \\ge 0$ and $c > e$. If $c^m < a^n$, then $e^{m/n} < a$.","title":"Power comparison above e bounds an exponential"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/log2-gt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log2_gt","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#log2-gt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"The natural logarithm of $2$ satisfies $0.6931 < \\log 2$.","title":"Lower bound for log 2"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/log5-gt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log5_gt","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#log5-gt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"The natural logarithm of $5$ satisfies $1.6094 < \\log 5$.","title":"Lower bound for log 5"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/log3-lt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log3_lt","parent":7635,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05/index.html#log3-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-05","status":"fully_proved","summary":"The natural logarithm of $3$ satisfies $\\log 3 < 1.0987$.","title":"Upper bound for log 3"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for log 17 to One-period shift of "],["math","\\(G_9\\)"]],"source":"Upper bound for log 17 to One-period shift of $G_9$"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","title":"Upper bound for log 17 to One-period shift of $G_9$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/log17-lt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log17_lt","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#log17-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"The natural logarithm of $17$ satisfies $\\log 17 < 2.8333$.","title":"Upper bound for log 17"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/log10-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log10_eq","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#log10-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"The natural logarithm satisfies $\\log 10 = \\log 2 + \\log 5$.","title":"Logarithm of 10"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/log9-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log9_eq","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#log9-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"The natural logarithm satisfies $\\log 9 = 2 \\log 3$.","title":"Logarithm of 9"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/log34div10-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log34div10_eq","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#log34div10-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"The natural logarithm satisfies $\\log(34/10) = \\log 17 - \\log 5$.","title":"Logarithm of 3.4"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/c9-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.c9_eq","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#c9-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"In the real numbers, $\\frac{18900}{279} + \\frac{18900}{657} + \\frac{18900}{963} = \\frac{28121100}{242141}$.","title":"Evaluation of a three-term rational sum"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/t03c-eqmain-margin","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.T03c_eqmain_margin","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#t03c-eqmain-margin","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"With $\\log$ the natural logarithm, $48\\,(10 \\log 10 - 9 \\log 9) - \\left(\\frac{18900}{279} + \\frac{18900}{657} + \\frac{18900}{963}\\right) \\log \\frac{34}{10} > 13$.","title":"Numerical margin of the main term"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/pi-third-small","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.pi_third_small","parent":7656,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-counting bound "],["math","\\(\\pi(m) \\le m/3\\)"],["text"," for "],["math","\\(36 \\le m < 156\\)"]],"source":"Prime-counting bound $\\pi(m) \\le m/3$ for $36 \\le m < 156$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#pi-third-small","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"Let $\\pi(m)$ denote the number of primes $p \\le m$. For every natural number $m$ with $36 \\le m < 156$, $3\\pi(m) \\le m$.","title":"Prime-counting bound $\\pi(m) \\le m/3$ for $36 \\le m < 156$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/pi-third-large","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.pi_third_large","parent":7656,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-counting bound "],["math","\\(\\pi(m) \\le m/3\\)"],["text"," from "],["math","\\(156\\)"],["text"," on"]],"source":"Prime-counting bound $\\pi(m) \\le m/3$ from $156$ on"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#pi-third-large","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"Let $\\pi(m)$ denote the number of primes $p \\le m$. For every natural number $m \\ge 156$, $3\\pi(m) \\le m$.","title":"Prime-counting bound $\\pi(m) \\le m/3$ from $156$ on"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/pi-third-nat","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.pi_third_nat","parent":7656,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-counting bound "],["math","\\(\\pi(m) \\le m/3\\)"],["text"," for natural "],["math","\\(m \\ge 36\\)"]],"source":"Prime-counting bound $\\pi(m) \\le m/3$ for natural $m \\ge 36$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#pi-third-nat","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"Let $\\pi(m)$ denote the number of primes $p \\le m$. For every natural number $m \\ge 36$, $3\\pi(m) \\le m$.","title":"Prime-counting bound $\\pi(m) \\le m/3$ for natural $m \\ge 36$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/pi-third","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.pi_third","parent":7656,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-counting bound "],["math","\\(\\pi(x) \\le x/3\\)"],["text"," for real "],["math","\\(x \\ge 36\\)"]],"source":"Prime-counting bound $\\pi(x) \\le x/3$ for real $x \\ge 36$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#pi-third","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"Let $\\pi(n)$ denote the number of primes $p \\le n$. For every real number $x \\ge 36$, $\\pi(\\lfloor x \\rfloor) \\le x/3$.","title":"Prime-counting bound $\\pi(x) \\le x/3$ for real $x \\ge 36$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/log10-lt","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log10_lt","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#log10-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"The natural logarithm of $10$ satisfies $\\log 10 < 2.4$.","title":"Upper bound for log 10"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/log-div-sqrt-decay","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log_div_sqrt_decay","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#log-div-sqrt-decay","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"Let $a$ and $t$ be real numbers with $a > 0$, $t > 1$ and $\\log a > 2$, where $\\log$ is the natural logarithm. Then $\\frac{\\log(at)}{\\sqrt{at}} < \\frac{\\log a}{\\sqrt{a}}$.","title":"Strict decay of log over square root beyond e squared"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/log-le-sqrt06","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.log_le_sqrt06","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#log-le-sqrt06","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"For every real number $x \\ge 10^7$, $\\log x \\le 0.006\\sqrt{x}$, where $\\log$ is the natural logarithm.","title":"Logarithm bounded by a multiple of the square root"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/gap-pos","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.gap_pos","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#gap-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"For every natural number $n \\ge 4762$, $13n - 16 - 48\\log(2100n) - \\sqrt{2100n}\\,\\log(2100n) > 0$, where $\\log$ is the natural logarithm.","title":"Positivity of the gap in Breusch's prime interval argument"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/g9r","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9r","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#g9r","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"For real $x$, the continuous Breusch function is the integer","title":"Continuous Breusch function"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/nat-floor-nat-div","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.nat_floor_nat_div","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#nat-floor-nat-div","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"For all natural numbers $a$ and $b$, the floor of the real number $a / b$ equals the quotient of $a$ by $b$ in integer division. When $b = 0$ both sides are $0$, since division by zero gives $0$ in both the reals and the natural numbers.","title":"Floor of a quotient of natural numbers"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/nat-mod-le-of-dvd","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.nat_mod_le_of_dvd","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#nat-mod-le-of-dvd","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"Let $a$, $b$, $d$ be natural numbers with $b > 0$, $d > 0$, $d \\mid b$ and $d \\mid a$. Then the remainder of $a$ modulo $b$ is at most $b - d$.","title":"Remainder bound for common multiples"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/floor-grid-match","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.floor_grid_match","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#floor-grid-match","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"Let $C$ and $L$ be positive natural numbers, let $m$ be a divisor of $210$ with $C \\mid mL$, and let $x \\ge 0$ be a real number. Then $\\lfloor Cx/m \\rfloor = \\lfloor C \\lfloor Lx \\rfloor / (mL) \\rfloor$.","title":"Real floors at divisors of 210 agree with grid floors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/div-shift189","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.div_shift189","parent":7656,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#div-shift189","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"Let $C$ and $m$ be natural numbers with $m > 0$ and $m \\mid C$. Then for every natural number $j$, $\\lfloor C(j + 18900)/(18900m) \\rfloor = \\lfloor Cj/(18900m) \\rfloor + C/m$.","title":"Shifting a floor quotient by one period"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/g9-shift","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9_shift","parent":7656,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-period shift of "],["math","\\(G_9\\)"]],"source":"One-period shift of $G_9$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06/index.html#g9-shift","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-06","status":"fully_proved","summary":"For every natural number $j$, the grid $G$-function for $k = 9$ satisfies $G_9(j + 18900) = G_9(j)$.","title":"One-period shift of $G_9$"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periodicity of "],["math","\\(G_9\\)"],["text"," in multiples of "],["math","\\(18900\\)"],["text"," to The continuous Breusch function vanishes at small ratios"]],"source":"Periodicity of $G_9$ in multiples of $18900$ to The continuous Breusch function vanishes at small ratios"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","title":"Periodicity of $G_9$ in multiples of $18900$ to The continuous Breusch function vanishes at small ratios"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9-period-iter","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9_period_iter","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periodicity of "],["math","\\(G_9\\)"],["text"," in multiples of "],["math","\\(18900\\)"]],"source":"Periodicity of $G_9$ in multiples of $18900$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9-period-iter","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For all natural numbers $o$ and $r$, the grid $G$-function for $k = 9$ satisfies $G_9(18900\\,o + r) = G_9(r)$.","title":"Periodicity of $G_9$ in multiples of $18900$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9-eq-mod","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9_eq_mod","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(G_9\\)"],["text"," depends only on the residue modulo "],["math","\\(18900\\)"]],"source":"$G_9$ depends only on the residue modulo $18900$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9-eq-mod","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For every natural number $j$, the grid $G$-function for $k = 9$ satisfies $G_9(j) = G_9(j \\bmod 18900)$.","title":"$G_9$ depends only on the residue modulo $18900$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9r-eq-grid","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9r_eq_grid","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9r-eq-grid","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For every real $x \\ge 0$, the continuous Breusch function $G$ and the Breusch grid function $G_9$ satisfy $G(x) = G_9(\\lfloor 18900x \\rfloor)$.","title":"The continuous Breusch function agrees with the grid function"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9-zero9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9_zero9","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9-zero9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For every natural number $j < 9$, the Breusch grid function $G_9$ satisfies $G_9(j) = 0$.","title":"The Breusch grid function vanishes below 9"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9-at9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9_at9","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of "],["math","\\(G_9\\)"],["text"," at "],["math","\\(9\\)"]],"source":"Value of $G_9$ at $9$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9-at9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"The grid $G$-function for $k = 9$ satisfies $G_9(9) = 1$.","title":"Value of $G_9$ at $9$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9-small-le1","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9_small_le1","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound "],["math","\\(G_9 \\le 1\\)"],["text"," below "],["math","\\(10\\)"]],"source":"Bound $G_9 \\le 1$ below $10$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9-small-le1","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For every natural number $j < 10$, the grid $G$-function for $k = 9$ satisfies $G_9(j) \\le 1$.","title":"Bound $G_9 \\le 1$ below $10$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9r-le3","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9r_le3","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9r-le3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For every real $x \\ge 0$, the continuous Breusch function $G$ satisfies $G(x) \\le 3$.","title":"The continuous Breusch function is at most 3"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9r-vanish","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9r_vanish","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9r-vanish","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For every real $x$ with $0 \\le x < 1/2100$, the continuous Breusch function $G$ satisfies $G(x) = 0$.","title":"The continuous Breusch function vanishes near zero"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9r-level1","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9r_level1","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Level-"],["math","\\(1\\)"],["text"," threshold for the continuous Breusch function"]],"source":"Level-$1$ threshold for the continuous Breusch function"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9r-level1","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $Q_9 = 210$. For every real $x \\ge 1/(9 Q_9) = 1/1890$, if the continuous Breusch function $G$ satisfies $G(x) \\ge 1$, then $x \\ge 279/18900$.","title":"Level-$1$ threshold for the continuous Breusch function"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9r-level2","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9r_level2","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Level-"],["math","\\(2\\)"],["text"," threshold for the continuous Breusch function"]],"source":"Level-$2$ threshold for the continuous Breusch function"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9r-level2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $Q_9 = 210$. For every real $x \\ge 1/(9 Q_9) = 1/1890$, if the continuous Breusch function $G$ satisfies $G(x) \\ge 2$, then $x \\ge 657/18900$.","title":"Level-$2$ threshold for the continuous Breusch function"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9r-level3","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9r_level3","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Level-"],["math","\\(3\\)"],["text"," threshold for the continuous Breusch function"]],"source":"Level-$3$ threshold for the continuous Breusch function"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9r-level3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $Q_9 = 210$. For every real $x \\ge 1/(9 Q_9) = 1/1890$, if the continuous Breusch function $G$ satisfies $G(x) \\ge 3$, then $x \\ge 963/18900$.","title":"Level-$3$ threshold for the continuous Breusch function"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/int-floor-cast","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.intFloor_cast","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#int-floor-cast","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For natural numbers $t$ and $d$, the floor of the real number $t/d$, as an integer, equals the natural-number quotient of $t$ by $d$ (rounded down), as an integer. Both sides are $0$ when $d = 0$, since division by zero is $0$ for both real numbers and natural numbers.","title":"Integer cast of the floor of a quotient of natural numbers"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/floor-n10","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.floor_N10","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Legendre terms of "],["math","\\(N_{10}\\)"],["text"," as real floors"]],"source":"Legendre terms of $N_{10}$ as real floors"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#floor-n10","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $m \\ge 1$ divide $Q_9 = 210$, and let $p \\ge 2$, $n$ and $j$ be natural numbers. Write $N_{10}(m, n) = 10n \\cdot (Q_9/m)$. Then $\\lfloor N_{10}(m, n)/p^j \\rfloor$ equals the floor of the real number $10 Q_9 (n/p^j)/m$.","title":"Legendre terms of $N_{10}$ as real floors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/floor-n9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.floor_N9","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Legendre terms of "],["math","\\(N_9\\)"],["text"," as real floors"]],"source":"Legendre terms of $N_9$ as real floors"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#floor-n9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $m \\ge 1$ divide $Q_9 = 210$, and let $p \\ge 2$, $n$ and $j$ be natural numbers. Write $N_9(m, n) = 9n \\cdot (Q_9/m)$. Then $\\lfloor N_9(m, n)/p^j \\rfloor$ equals the floor of the real number $9 Q_9 (n/p^j)/m$.","title":"Legendre terms of $N_9$ as real floors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/floor-n1","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.floor_N1","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Legendre terms of "],["math","\\(N_1\\)"],["text"," as real floors"]],"source":"Legendre terms of $N_1$ as real floors"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#floor-n1","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $m \\ge 1$ divide $Q_9 = 210$, and let $p \\ge 2$, $n$ and $j$ be natural numbers. Write $N_1(m, n) = n \\cdot (Q_9/m)$. Then $\\lfloor N_1(m, n)/p^j \\rfloor$ equals the floor of the real number $Q_9 (n/p^j)/m$.","title":"Legendre terms of $N_1$ as real floors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/legendre-sum9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.legendre_sum9","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#legendre-sum9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $p$ be a prime and let $N$ and $b$ be natural numbers with $\\lfloor \\log_p N \\rfloor < b$, where $\\lfloor \\log_p N \\rfloor$ is taken to be $0$ when $N = 0$. Then the exponent of $p$ in $N!$, as an integer, equals $\\sum_{j=1}^{b-1} \\lfloor N / p^j \\rfloor$.","title":"Legendre's formula with integer summands"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/vp-ratio9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.vp_ratio9","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#vp-ratio9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $Q = 210$, let $m \\ge 1$ be a divisor of $Q$, and let $n$, $p$ be natural numbers. Write $N = n \\cdot (Q/m)$ and $R_m(n) = (10N)! / ((9N)! \\, N!)$. Then, as integers, $v_p(R_m(n)) = v_p((10N)!) - v_p((9N)!) - v_p(N!)$, where $v_p(k)$ is the exponent of $p$ in the prime factorization of $k$ (zero when $p$ is not prime).","title":"Valuation of a factorial ratio as a difference of valuations"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/nu9term","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.nu9term","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#nu9term","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For a pair $(m, \\varepsilon)$ of a natural number $m$ and an integer $\\varepsilon$, and natural numbers $p$ and $n$, let $a = \\lfloor 210/m \\rfloor$ (so $a = 0$ when $m = 0$). The term is the integer $\\varepsilon\\bigl(v_p((10na)!) - v_p((9na)!) - v_p((na)!)\\bigr)$, where $v_p(k)$ is the exponent of $p$ in the prime factorization of $k$ (zero when $p$ is not prime).","title":"One divisor's contribution to the valuation"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/nu9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.nu9","parent":7677,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-adic valuation of the weighted product"]],"source":"The $p$-adic valuation of the weighted product"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#nu9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"For natural numbers $p$ and $n$, $\\nu_p(n)$ is the integer $\\sum_{d \\mid 210} \\mu(d)\\bigl(v_p((2100n/d)!) - v_p((1890n/d)!) - v_p((210n/d)!)\\bigr)$, where $\\mu$ is the M\u00f6bius function and $v_p(m)$ is the exponent of $p$ in the prime factorization of $m$ (zero when $p$ is not prime).","title":"The $p$-adic valuation of the weighted product"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/g9r-vanish-of","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.G9r_vanish_of","parent":7677,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07/index.html#g9r-vanish-of","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-07","status":"fully_proved","summary":"Let $p$ be a prime and let $n \\ge 1$ and $j$ be natural numbers. If $p^j > 2100n$, then the continuous Breusch function $G$ satisfies $G(n/p^j) = 0$.","title":"The continuous Breusch function vanishes at small ratios"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","kind":"scope","navigation":"transparent","parent":7550,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","title":"Valuation identity for the weighted product to The primorial as a product of primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/nu9-eq-sum-g","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.nu9_eq_sum_G","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#nu9-eq-sum-g","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Write $v_p(m)$ for the exponent of $p$ in $m$ and $\\mu$ for the M\u00f6bius function, and set $\\nu_p(n) = \\sum_{d \\mid 210} \\mu(d)\\bigl(v_p((2100n/d)!) - v_p((1890n/d)!) - v_p((210n/d)!)\\bigr)$ and, for real $x \\ge 0$, $G(x) = \\sum_{d \\mid 210} \\mu(d)\\bigl(\\lfloor 2100x/d \\rfloor - \\lfloor 1890x/d \\rfloor - \\lfloor 210x/d \\rfloor\\bigr)$. If $p$ is a prime and $n$, $b$ are natural numbers with\u2026","title":"Valuation identity for the weighted product"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/nu9-eq-single","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.nu9_eq_single","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#nu9-eq-single","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Write $v_p(m)$ for the exponent of $p$ in $m$ and $\\mu$ for the M\u00f6bius function, and set $\\nu_p(n) = \\sum_{d \\mid 210} \\mu(d)\\bigl(v_p((2100n/d)!) - v_p((1890n/d)!) - v_p((210n/d)!)\\bigr)$ and, for real $x \\ge 0$, $G(x) = \\sum_{d \\mid 210} \\mu(d)\\bigl(\\lfloor 2100x/d \\rfloor - \\lfloor 1890x/d \\rfloor - \\lfloor 210x/d \\rfloor\\bigr)$. If $p$ is a prime and $n \\ge 1$ is a natural number with $2100n < p^2$, then\u2026","title":"Single-term valuation identity for large primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-10-q9","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_10Q9","parent":7698,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of "],["math","\\(10Qn\\)"]],"source":"Value of $10Qn$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-10-q9","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $Q = 210$ be the truncation modulus. For every natural number $n$, $10 \\cdot Q \\cdot n = 2100n$.","title":"Value of $10Qn$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-log-fact","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_log_fact","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-log-fact","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"For every nonzero natural number $K$, $\\log K = \\sum_{p} v_p(K) \\log p$, where $p$ runs over the prime divisors of $K$ and $v_p(K)$ is the exponent of $p$ in the prime factorization of $K$.","title":"Logarithm of a natural number from its factorization"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-supp-sub","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_supp_sub","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-supp-sub","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $(m, \\mu)$ be an entry of the divisor table $w_9$ (so $m$ divides $210$ and $\\mu$ is the M\u00f6bius value of $m$), let $n$ be a natural number and put $d = 210/m$. Then every prime factor of $(10nd)!/((9nd)!\\,(nd)!)$ is at most $2100n$.","title":"Bound on the prime factors of the ratio factors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-nu9term-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_nu9term_eq","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-nu9term-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $Q = 210$, let $m \\ge 1$ be a divisor of $Q$, let $w$ be an integer, and let $p$ and $n$ be natural numbers. Put $N = n \\cdot (Q/m)$ and $R = (10N)!/((9N)!\\,N!)$. For a natural number $a$, write $v_p(a)$ for the exponent of $p$ in the prime factorization of $a$ (zero when $p$ is not prime). Then $w \\, (v_p((10N)!) - v_p((9N)!) - v_p(N!)) = w \\, v_p(R)$.","title":"Valuation term of one binomial factor"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-nu9term-zero","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_nu9term_zero","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-nu9term-zero","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $m \\ge 1$ be a divisor of $Q_9 = 210$, let $\\mu$ be an integer, let $p$ and $n$ be natural numbers, and put $d = 210/m$. For a natural number $k$ write $v_p(k)$ for the exponent of $p$ in the prime factorization of $k$ (zero when $p$ is not prime). If $v_p\\bigl((10nd)!/((9nd)!\\,(nd)!)\\bigr) = 0$, then the valuation term $\\mu\\,\\bigl(v_p((10nd)!) - v_p((9nd)!) - v_p((nd)!)\\bigr)$ of the pair $(m, \\mu)$ at $p$ and\u2026","title":"Valuation terms vanish off the prime support"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-q-expand","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_q_expand","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-q-expand","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $(m, \\mu)$ be an entry of the divisor table $w_9$ (so $m$ divides $210$ and $\\mu$ is the M\u00f6bius value of $m$), let $n \\ge 1$ be a natural number, put $d = 210/m$ and $R = (10nd)!/((9nd)!\\,(nd)!)$, and for a prime $p$ write $\\nu_p = \\mu\\,\\bigl(v_p((10nd)!) - v_p((9nd)!) - v_p((nd)!)\\bigr)$, where $v_p(k)$ is the exponent of $p$ in the prime factorization of $k$. Then $\\mu \\log R$ equals the sum of $\\nu_p \\log p$\u2026","title":"A weighted log-factor as a sum over primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-bridge","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_bridge","parent":7698,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime factorization of "],["math","\\(\\log F_9(n)\\)"]],"source":"Prime factorization of $\\log F_9(n)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-bridge","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $n \\ge 1$ be a natural number. For each divisor $m$ of $210$ put $N_m = 210n/m$, let $\\mu$ be the M\u00f6bius function, and let $F_9(n) = \\prod_{m \\mid 210} \\binom{10N_m}{N_m}^{\\mu(m)}$. For a prime $p$ let $v_p$ denote the $p$-adic valuation and put $\\nu_9(p, n) = \\sum_{m \\mid 210} \\mu(m) (v_p((10N_m)!) - v_p((9N_m)!) - v_p(N_m!))$, the $p$-adic valuation of $F_9(n)$. Then\u2026","title":"Prime factorization of $\\log F_9(n)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-logp-nonneg","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_logp_nonneg","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-logp-nonneg","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"For every prime $p$, $\\log p \\ge 0$.","title":"Logarithms of primes are nonnegative"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-g-to-nat","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_G_toNat","parent":7698,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(G\\)"],["text"," is at most its positive part"]],"source":"$G$ is at most its positive part"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-g-to-nat","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"For real $y$ let $G(y) = \\sum_{m \\mid 210} \\mu(m) (\\lfloor 2100y/m \\rfloor_+ - \\lfloor 1890y/m \\rfloor_+ - \\lfloor 210y/m \\rfloor_+)$, where $\\mu$ is the M\u00f6bius function and $\\lfloor t \\rfloor_+ = \\max(\\lfloor t \\rfloor, 0)$. For every real number $x$, $G(x) \\le \\max(G(x), 0)$.","title":"$G$ is at most its positive part"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-to-nat-le3","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_toNat_le3","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-to-nat-le3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"For every real $x \\ge 0$, the continuous Breusch function $G$ satisfies $\\max(G(x), 0) \\le 3$.","title":"The truncated continuous Breusch function is at most 3"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-of-to-nat","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_of_toNat","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-of-to-nat","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $z$ be an integer and let $i \\ge 1$ be a natural number. If $i \\le \\max(z, 0)$, then $i \\le z$.","title":"Positive lower bounds on a truncation lift to the integer"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-layercake","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_layercake","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-layercake","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $B$ be a finite set of natural numbers, let $M : \\mathbb{N} \\to \\mathbb{N}$ and $c : \\mathbb{N} \\to \\mathbb{R}$, and suppose $M(p) \\le 3$ for every $p \\in B$. Then $\\sum_{p \\in B} M(p) c(p) = \\sum_{i=1}^{3} \\sum_{p \\in B,\\, M(p) \\ge i} c(p)$.","title":"Layer-cake decomposition with three levels"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-ple-of-g","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_ple_of_G","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-ple-of-g","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $n \\ge 1$ be a natural number, $p$ a prime and $j \\ge 1$ a real number. If $j / 18900 \\le n / p$, then $p \\le 18900\\,n / j$.","title":"A level bound turns into a bound on the prime"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-xlo","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_xlo","parent":7698,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes up to "],["math","\\(1890n\\)"],["text"," have "],["math","\\(n/p \\ge 1/1890\\)"]],"source":"Primes up to $1890n$ have $n/p \\ge 1/1890$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-xlo","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"Let $Q = 210$ be the truncation modulus, let $n \\ge 1$ be a natural number and let $p$ be a prime with $p \\le 1890 n$. Then $1/(9Q) \\le n/p$, where $9Q = 1890$.","title":"Primes up to $1890n$ have $n/p \\ge 1/1890$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-ple1","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_ple1","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-ple1","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"For real $x \\ge 0$ let $G(x) = \\sum_{m \\mid 210} \\mu(m)\\bigl(\\lfloor 2100x/m \\rfloor - \\lfloor 1890x/m \\rfloor - \\lfloor 210x/m \\rfloor\\bigr)$, where $\\mu$ is the M\u00f6bius function. Let $n \\ge 1$ be a natural number and $p \\le 1890n$ a prime. If $G(n/p) \\ge 1$, then $p \\le 18900n/279$.","title":"Upper bound for primes where a M\u00f6bius-weighted floor sum is at least 1"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-ple2","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_ple2","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-ple2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"For real $x \\ge 0$ let $G(x) = \\sum_{m \\mid 210} \\mu(m)\\bigl(\\lfloor 2100x/m \\rfloor - \\lfloor 1890x/m \\rfloor - \\lfloor 210x/m \\rfloor\\bigr)$, where $\\mu$ is the M\u00f6bius function. Let $n \\ge 1$ be a natural number and $p \\le 1890n$ a prime. If $G(n/p) \\ge 2$, then $p \\le 18900n/657$.","title":"Upper bound for primes where a M\u00f6bius-weighted floor sum is at least 2"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-ple3","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_ple3","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-ple3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"For real $x \\ge 0$ let $G(x) = \\sum_{m \\mid 210} \\mu(m)\\bigl(\\lfloor 2100x/m \\rfloor - \\lfloor 1890x/m \\rfloor - \\lfloor 210x/m \\rfloor\\bigr)$, where $\\mu$ is the M\u00f6bius function. Let $n \\ge 1$ be a natural number and $p \\le 1890n$ a prime. If $G(n/p) \\ge 3$, then $p \\le 18900n/963$.","title":"Upper bound for primes where a M\u00f6bius-weighted floor sum is at least 3"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/u9-primorial-eq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_primorial_eq","parent":7698,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08/index.html#u9-primorial-eq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-08","status":"fully_proved","summary":"For every natural number $k$, the primorial $k\\#$ equals the product of all primes $p$ with $p < k + 1$.","title":"The primorial as a product of primes"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Chebyshev-type bound for the sum of prime logarithms to Primes in "],["math","\\((X, 9X/8)\\)"],["text"," for large "],["math","\\(X\\)"],["text",", assuming the upper bound"]],"source":"Chebyshev-type bound for the sum of prime logarithms to Primes in $(X, 9X/8)$ for large $X$, assuming the upper bound"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","title":"Chebyshev-type bound for the sum of prime logarithms to Primes in $(X, 9X/8)$ for large $X$, assuming the upper bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/u9-theta-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_theta_le","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#u9-theta-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"For every real number $Y \\ge 0$, the sum of $\\log p$ over all primes $p \\le \\lfloor Y \\rfloor$ is at most $Y \\log 3.4$.","title":"Chebyshev-type bound for the sum of prime logarithms"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/u9-bi-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_Bi_le","parent":7719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-logarithm bound for one level of "],["math","\\(G\\)"]],"source":"Prime-logarithm bound for one level of $G$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#u9-bi-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"For real $x \\ge 0$ let $G(x) = \\sum_{m \\mid 210} \\mu(m) (\\lfloor 2100x/m \\rfloor - \\lfloor 1890x/m \\rfloor - \\lfloor 210x/m \\rfloor)$, where $\\mu$ is the M\u00f6bius function. Let $n \\ge 1$ be a natural number, let $B$ be a finite set of primes $p$ with $p \\le 1890n$, let $i \\ge 1$ be a natural number, and let $j \\ge 1$ be a real number with $18900n/j \\ge 0$. Suppose that every prime $p \\le 1890n$ with $G(n/p) \\ge i$\u2026","title":"Prime-logarithm bound for one level of $G$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/u9-b-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_B_le","parent":7719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Contribution of primes with "],["math","\\(p^2 > 2100n\\)"]],"source":"Contribution of primes with $p^2 > 2100n$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#u9-b-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Write $v_p(m)$ for the exponent of $p$ in $m$ and $\\mu$ for the M\u00f6bius function, and set $\\nu_p(n) = \\sum_{d \\mid 210} \\mu(d)\\bigl(v_p((2100n/d)!) - v_p((1890n/d)!) - v_p((210n/d)!)\\bigr)$. For every natural number $n \\ge 1$, the sum of $\\nu_p(n) \\log p$ over the primes $p \\le 1890n$ with $p^2 > 2100n$ satisfies","title":"Contribution of primes with $p^2 > 2100n$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/u9-cterm-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_Cterm_le","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#u9-cterm-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Let $n \\ge 1$ be a natural number and let $p$ be a prime with $p^2 \\le 2100n$. For each divisor $m$ of $210$ put $N_m = 210n/m$, let $\\mu$ be the M\u00f6bius function and $v_p$ the $p$-adic valuation, and put $\\nu_9(p, n) = \\sum_{m \\mid 210} \\mu(m) (v_p((10N_m)!) - v_p((9N_m)!) - v_p(N_m!))$. Then $\\nu_9(p, n) \\log p \\le 3 \\log(2100n)$.","title":"Bound for one small-square prime term"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/u9-ccard-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_Ccard_le","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#u9-ccard-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Let $n \\ge 1$ be a natural number and let $C$ be a finite set of primes $p$ with $p^2 \\le 2100n$. Then $C$ has at most $\\sqrt{2100n}/3$ elements.","title":"Counting small-square primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/u9-c-le","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.U9_C_le","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#u9-c-le","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Let $n \\ge 1$ be a natural number. For each divisor $m$ of $210$ put $N_m = 210n/m$, let $\\mu$ be the M\u00f6bius function, and for a prime $p$ with $p$-adic valuation $v_p$ put $\\nu_9(p, n) = \\sum_{m \\mid 210} \\mu(m) (v_p((10N_m)!) - v_p((9N_m)!) - v_p(N_m!))$. Then the sum of $\\nu_9(p, n) \\log p$ over the primes $p \\le 1890n$ with $p^2 \\le 2100n$ is at most $\\sqrt{2100n} \\log(2100n)$.","title":"Contribution of the small-square primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/log-f9-upper","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.logF9_upper","parent":7719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for "],["math","\\(\\log F_9(n)\\)"],["text"," when "],["math","\\((1890n, 2100n)\\)"],["text"," has no prime"]],"source":"Upper bound for $\\log F_9(n)$ when $(1890n, 2100n)$ has no prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#log-f9-upper","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Let $n \\ge 1$ be a natural number such that no prime $p$ satisfies $1890n < p \\le 2100n$, and let $\\log F_9(n) = \\sum_{d \\mid 210} \\mu(d) \\log \\frac{(2100n/d)!}{(1890n/d)!\\,(210n/d)!}$, where $\\mu$ is the M\u00f6bius function. Then","title":"Upper bound for $\\log F_9(n)$ when $(1890n, 2100n)$ has no prime"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/upper-ineq","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.UpperIneq","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#upper-ineq","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"For a natural number $n$ write $L(n) = \\sum_{d \\mid 210} \\mu(d) \\log \\binom{2100n/d}{210n/d}$, where $\\mu$ is the M\u00f6bius function; this is the logarithm of Breusch's product $F_9(n)$. The endgame upper-bound inequality at $n$ is the statement","title":"Breusch's endgame upper-bound inequality"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/chain","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.chain","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#chain","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"The chain is the list of the following 111 natural numbers, in order:","title":"Breusch's prime chain"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/chain-prime","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.chain_prime","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#chain-prime","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Every entry of the chain is a prime number.","title":"Entries of Breusch's prime chain are prime"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/chain-sorted","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.chain_sorted","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#chain-sorted","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Each entry of the chain is less than every later entry.","title":"Breusch's prime chain is strictly increasing"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/chain-ratio","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.chain_ratio","parent":7719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Consecutive entries of Breusch's prime chain are within a factor "],["math","\\(9/8\\)"]],"source":"Consecutive entries of Breusch's prime chain are within a factor $9/8$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#chain-ratio","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"If $b$ immediately follows $a$ in the chain, then $8b < 9a$.","title":"Consecutive entries of Breusch's prime chain are within a factor $9/8$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/chain-get-last","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.chain_getLast","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#chain-get-last","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"The last entry of the chain is $9820997$.","title":"Last entry of Breusch's prime chain"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/bracket","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.bracket","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#bracket","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Let $X$ be a real number and let $c_0 < c_1 < \\dots < c_k$ be natural numbers with $c_0 \\le X < c_k$. Then there is an index $i < k$ with $c_i \\le X < c_{i+1}$.","title":"Bracketing a real number in an increasing list"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/zip-cons-mem","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.zip_cons_mem","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#zip-cons-mem","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Let $R$ be a list of natural numbers and let $a, b, d$ be natural numbers. If $(a, b)$ is a pair of consecutive entries of $R$ (an element of the list obtained by zipping $R$ with its tail), then $(a, b)$ is also a pair of consecutive entries of the list obtained by prepending $d$ to $R$.","title":"Consecutive pairs survive prepending"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/zip-mem-of-consec","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.zip_mem_of_consec","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#zip-mem-of-consec","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Let $l$, $l_1$, $l_2$ be lists of natural numbers and let $a, b$ be natural numbers. If $l$ is the concatenation of $l_1$, the two-entry list $a, b$, and $l_2$, then $(a, b)$ is a pair of consecutive entries of $l$ (an element of the list obtained by zipping $l$ with its tail).","title":"Adjacent entries form a consecutive pair"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/chain-cover","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.chain_cover","parent":7719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes in "],["math","\\((X, 9X/8)\\)"],["text"," for "],["math","\\(53 \\le X < 9820997\\)"]],"source":"Primes in $(X, 9X/8)$ for $53 \\le X < 9820997$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#chain-cover","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"For every real number $X$ with $53 \\le X < 9820997$ there is a prime $p$ with $X < p < \\frac{9}{8}X$.","title":"Primes in $(X, 9X/8)$ for $53 \\le X < 9820997$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/overlap","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.overlap","parent":7719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every large real lies in a window "],["math","\\((5600n/3, 1890n)\\)"]],"source":"Every large real lies in a window $(5600n/3, 1890n)$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#overlap","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"For every real number $X \\ge 5600 \\cdot 4762 / 3$ there is a natural number $n \\ge 4762$ with $5600n/3 \\le X \\le 1890n$.","title":"Every large real lies in a window $(5600n/3, 1890n)$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/main-contra","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.main_contra","parent":7719,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#main-contra","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"Let $n \\ge 4762$ be a natural number. If there is no prime $p$ with $1890n < p \\le 2100n$ and $n$ satisfies the endgame upper-bound inequality, then we have a contradiction.","title":"Endgame contradiction in Breusch's theorem"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/analytic-at-x","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.analytic_at_X","parent":7719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes in "],["math","\\((X, 9X/8)\\)"],["text"," for large "],["math","\\(X\\)"],["text",", assuming the upper bound"]],"source":"Primes in $(X, 9X/8)$ for large $X$, assuming the upper bound"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09/index.html#analytic-at-x","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-09","status":"fully_proved","summary":"For a natural number $n$ write $L(n) = \\sum_{m \\mid 210} \\mu(m) \\log \\binom{2100n/m}{210n/m}$, where $\\mu$ is the M\u00f6bius function. Let $X \\ge 5600 \\cdot 4762 / 3$ be a real number, and suppose that every natural number $n \\ge 1$ such that no prime $p$ satisfies $1890n < p \\le 2100n$ also satisfies\u2026","title":"Primes in $(X, 9X/8)$ for large $X$, assuming the upper bound"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10","kind":"scope","navigation":"transparent","parent":7550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Breusch's theorem from the upper bound to Breusch's theorem on primes between "],["math","\\(x\\)"],["text"," and "],["math","\\(9x/8\\)"]],"source":"Breusch's theorem from the upper bound to Breusch's theorem on primes between $x$ and $9x/8$"},"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10","status":"fully_proved","title":"Breusch's theorem from the upper bound to Breusch's theorem on primes between $x$ and $9x/8$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10/breusch-prime-interval-of-upper","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.Internal.breusch_prime_interval_of_upper","parent":7740,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10/index.html#breusch-prime-interval-of-upper","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10","status":"fully_proved","summary":"For a natural number $n$ write $L(n) = \\sum_{m \\mid 210} \\mu(m) \\log \\binom{2100n/m}{210n/m}$, where $\\mu$ is the M\u00f6bius function. Let $x \\ge 48$ be a real number, and suppose that every natural number $n \\ge 1$ such that no prime $p$ satisfies $1890n < p \\le 2100n$ also satisfies $L(n) \\le \\left(\\frac{18900}{279} + \\frac{18900}{657} + \\frac{18900}{963}\\right) n \\log 3.4 + \\sqrt{2100n}\\,\\log(2100n)$. Then there is\u2026","title":"Breusch's theorem from the upper bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10/breusch-prime-interval","kind":"node","lean":"MathlibExt.NumberTheory.BreuschWanted.breusch_prime_interval","parent":7740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Breusch's theorem on primes between "],["math","\\(x\\)"],["text"," and "],["math","\\(9x/8\\)"]],"source":"Breusch's theorem on primes between $x$ and $9x/8$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10/index.html#breusch-prime-interval","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/breusch-s-theorem-on-primes-between-x-and-9x-8/part-10","status":"fully_proved","summary":"For every real number $x \\ge 48$ there is a prime $p$ with $x < p < \\frac{9}{8}x$.","title":"Breusch's theorem on primes between $x$ and $9x/8$"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture","status":"fully_proved","title":"Oppermann's conjecture"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture/oppermann-lower-prime-count","kind":"node","lean":"MetaMathlibExt.oppermannLowerPrimeCount","parent":7743,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture/index.html#oppermann-lower-prime-count","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture","status":"fully_proved","summary":"For a natural number $n$, this is the number of primes $p$ with $n^2 - n \\le p \\le n^2$.","title":"Primes in the lower Oppermann interval"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture/oppermann-upper-prime-count","kind":"node","lean":"MetaMathlibExt.oppermannUpperPrimeCount","parent":7743,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture/index.html#oppermann-upper-prime-count","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture","status":"fully_proved","summary":"For a natural number $n$, this is the number of primes $p$ with $n^2 \\le p \\le n^2 + n$.","title":"Primes in the upper Oppermann interval"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture/oppermann-asymptotic","kind":"node","lean":"MetaMathlibExt.oppermannAsymptotic","parent":7743,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture/index.html#oppermann-asymptotic","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/oppermann-s-conjecture","status":"fully_proved","summary":"For a natural number $n$, this is the real number $n / (2 \\log n)$, where $\\log$ is the natural logarithm. It is used as the asymptotic main term for the number of primes in each of the intervals $[n^2 - n, n^2]$ and $[n^2, n^2 + n]$.","title":"Main term for the Oppermann prime counts"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"can_prove","title":"Prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/are-consecutive-primes","kind":"node","lean":"MetaMathlibExt.AreConsecutivePrimes","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#are-consecutive-primes","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Natural numbers $p$ and $q$ are consecutive primes when $p$ and $q$ are prime, $p < q$, and every prime $r$ satisfies $r \\le p$ or $q \\le r$; that is, no prime lies strictly between $p$ and $q$.","title":"Consecutive primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/is-consecutive-prime-block","kind":"node","lean":"MetaMathlibExt.IsConsecutivePrimeBlock","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#is-consecutive-prime-block","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"A list $l$ of natural numbers is a block of consecutive primes when $l$ is nonempty, every entry of $l$ is prime, $l$ is strictly increasing, and every two adjacent entries of $l$ are consecutive primes.","title":"Block of consecutive primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/is-restricted-consecutive-prime-block","kind":"node","lean":"MetaMathlibExt.IsRestrictedConsecutivePrimeBlock","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#is-restricted-consecutive-prime-block","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $P$ be a predicate on natural numbers. A list $l$ of natural numbers is a block of consecutive primes restricted by $P$ when $l$ is a block of consecutive primes and every entry of $l$ satisfies $P$.","title":"Restricted block of consecutive primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/prime-gap","kind":"node","lean":"primeGap","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#prime-gap","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 = 2$, $p_1 = 3$, $p_2 = 5, \\dots$ be the primes in increasing order, indexed from $0$. For a natural number $n$, the prime gap at $n$ is $p_{n+1} - p_n$.","title":"Prime gap"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/prime-gap-liminf","kind":"node","lean":"MathlibExt.NumberTheory.primeGapLiminf","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#prime-gap-liminf","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order and let $g_n = p_{n+1} - p_n$ be the $n$-th prime gap. The liminf of prime gaps is $\\liminf_{n \\to \\infty} g_n$, taken in $\\mathbb{N} \\cup \\{\\infty\\}$; the value $\\infty$ is allowed.","title":"Liminf of prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/consecutive-nth-prime","kind":"node","lean":"MathlibExt.NumberTheory.consecutive_nth_prime","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#consecutive-nth-prime","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. For every $n \\in \\mathbb{N}$, $p_n$ and $p_{n+1}$ are consecutive primes: both are prime, $p_n < p_{n+1}$, and no prime lies strictly between them.","title":"Successive primes in the enumeration are consecutive"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/prime-gap-pos","kind":"node","lean":"MathlibExt.NumberTheory.primeGap_pos","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#prime-gap-pos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order and let $g_n = p_{n+1} - p_n$ be the $n$-th prime gap. Then $g_n > 0$ for every $n \\in \\mathbb{N}$, including $n = 0$.","title":"Prime gaps are positive"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/two-le-prime-gap","kind":"node","lean":"MathlibExt.NumberTheory.two_le_primeGap","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#two-le-prime-gap","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order and let $g_n = p_{n+1} - p_n$ be the $n$-th prime gap. For every $n \\ge 1$, $g_n \\ge 2$.","title":"Prime gaps after the first are at least two"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/two-le-prime-gap-liminf","kind":"node","lean":"MathlibExt.NumberTheory.two_le_primeGapLiminf","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#two-le-prime-gap-liminf","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order and let $g_n = p_{n+1} - p_n$ be the $n$-th prime gap. Then $\\liminf_{n \\to \\infty} g_n \\ge 2$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"The liminf of prime gaps is at least two"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/prime-gap-liminf-le-iff-infinite","kind":"node","lean":"MathlibExt.NumberTheory.primeGapLiminf_le_iff_infinite","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#prime-gap-liminf-le-iff-infinite","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order, let $g_n = p_{n+1} - p_n$ be the $n$-th prime gap, and let $L = \\liminf_{n \\to \\infty} g_n$, taken in $\\mathbb{N} \\cup \\{\\infty\\}$. For every natural number $B$, $L \\le B$ if and only if $g_n \\le B$ for infinitely many $n$.","title":"When the liminf of prime gaps is at most a given bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/prime-gap-liminf-ne-top-iff-bounded-pair","kind":"node","lean":"MathlibExt.NumberTheory.primeGapLiminf_ne_top_iff_bounded_pairs","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#prime-gap-liminf-ne-top-iff-bounded-pair","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order, let $g_n = p_{n+1} - p_n$ be the $n$-th prime gap, and let $L = \\liminf_{n \\to \\infty} g_n$, taken in $\\mathbb{N} \\cup \\{\\infty\\}$. Then $L \\ne \\infty$ if and only if there is a natural number $B > 0$ such that infinitely many primes $p$ admit a prime $q$ with $p < q$ and $q - p \\le B$.","title":"Finite liminf of prime gaps and bounded gaps between primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/prime-gap-eq-two-iff","kind":"node","lean":"MathlibExt.NumberTheory.primeGap_eq_two_iff","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#prime-gap-eq-two-iff","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order and let $g_n = p_{n+1} - p_n$ be the $n$-th prime gap. For every $n \\in \\mathbb{N}$, $g_n = 2$ if and only if $p_n + 2$ is prime.","title":"A prime gap equals two exactly at a twin prime"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/prime-gap-liminf-eq-two-iff","kind":"node","lean":"MathlibExt.NumberTheory.primeGapLiminf_eq_two_iff","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#prime-gap-liminf-eq-two-iff","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order and let $g_n = p_{n+1} - p_n$ be the $n$-th prime gap. Then $\\liminf_{n \\to \\infty} g_n = 2$, taken in $\\mathbb{N} \\cup \\{\\infty\\}$, if and only if there are infinitely many primes $p$ such that $p + 2$ is also prime.","title":"The liminf of prime gaps is two exactly when there are infinitely many twin primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/sup-prime-gap","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1138Wanted.supPrimeGap","parent":7747,"reading_title":{"format":"inline-parts-v1","parts":[["text","Largest prime gap below "],["math","\\(x\\)"]],"source":"Largest prime gap below $x$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#sup-prime-gap","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 = 2 < p_1 = 3 < p_2 = 5 < \\dots$ be the primes in increasing order. For a real number $x$, $d(x)$ is the largest prime gap $p_{n+1} - p_n$ over all $n$ with $p_n < x$ (the next prime $p_{n+1}$ may exceed $x$), and $d(x) = 0$ when no prime is less than $x$.","title":"Largest prime gap below $x$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/snd-gt-half-fst","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1138Wanted.sndGtHalfFst","parent":7747,"reading_title":{"format":"inline-parts-v1","parts":[["text","The regime "],["math","\\(x \\to \\infty\\)"],["text"," with "],["math","\\(x/2 < y < x\\)"]],"source":"The regime $x \\to \\infty$ with $x/2 < y < x$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#snd-gt-half-fst","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"The filter on pairs $(x, y)$ of real numbers along which $x \\to \\infty$ while $x/2 < y < x$: a set of pairs belongs to it exactly when it contains every pair $(x, y)$ with $x/2 < y < x$ and $x$ sufficiently large.","title":"The regime $x \\to \\infty$ with $x/2 < y < x$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/prime-count-ioc-mul-const","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1138Wanted.primeCountIocMulConst","parent":7747,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes in a window of "],["math","\\(C\\)"],["text"," maximal prime gaps"]],"source":"Primes in a window of $C$ maximal prime gaps"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#prime-count-ioc-mul-const","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $p_0 = 2 < p_1 = 3 < p_2 = 5 < \\dots$ be the primes in increasing order, let $\\pi$ be the prime-counting function, and for real $x$ let $d(x)$ be the largest prime gap $p_{n+1} - p_n$ over all $n$ with $p_n < x$, with $d(x) = 0$ when no prime is less than $x$. For a real number $C$, this is the function on pairs $(x, y)$ of real numbers given by $\\pi(\\lfloor y + C\\,d(x) \\rfloor) - \\pi(\\lfloor y \\rfloor)$, where\u2026","title":"Primes in a window of $C$ maximal prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/vardi-asymptotic","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1138Wanted.vardiAsymptotic","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#vardi-asymptotic","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"Let $\\pi$ be the prime-counting function and, for real $x$, let $d(x)$ be the largest prime gap $p_{n+1} - p_n$ over the primes $p_n < x$. The proposition states that for every real $C > 1$,","title":"Vardi's prime-gap asymptotic"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/disproved-vardi","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1138Wanted.disprovedVardi","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#disproved-vardi","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"fully_proved","summary":"The proposition that Vardi's prime-gap asymptotic does not hold.","title":"Negation of Vardi's prime-gap asymptotic"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/vardi-asymptotic-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1138Wanted.vardiAsymptotic_refuted","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#vardi-asymptotic-refuted","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"can_prove","summary":"Let $\\pi$ be the prime-counting function and, for real $x$, let $d(x)$ be the largest prime gap $p_{n+1} - p_n$ over the primes $p_n < x$. It is not true that for every real $C > 1$, $\\pi(\\lfloor y + C\\,d(x) \\rfloor) - \\pi(\\lfloor y \\rfloor)$ is asymptotically equivalent to $C\\,d(x) / \\log y$ as $x \\to \\infty$ over pairs $(x, y)$ with $x/2 < y < x$.","title":"Vardi's prime-gap asymptotic is false"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem6Wanted.conjecture_holds","parent":7747,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps/index.html#conjecture-holds","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/prime-gaps","status":"can_prove","summary":"Let $p_0 = 2$, $p_1 = 3$, $p_2 = 5, \\dots$ be the primes in increasing order and let $d_n = p_{n+1} - p_n$ be the prime gap at $n$. There are infinitely many natural numbers $n$ with $d_n < d_{n+1} < d_{n+2}$.","title":"Erd\u0151s Problem 6 (increasing prime gaps)"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","title":"Bounded gaps between primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/subpower-bound","kind":"node","lean":"WantedExt.NumberTheory.Polymath8b.SubpowerBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#subpower-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"For functions $f, g : \\mathbb{R} \\to \\mathbb{R}$, $f$ is bounded by $g$ up to subpower losses if for every real $\\delta > 0$ we have $f(x) = O(x^{\\delta} g(x))$ as $x \\to \\infty$.","title":"Subpower bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/dyadic-support","kind":"node","lean":"WantedExt.NumberTheory.Polymath8b.DyadicSupport","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#dyadic-support","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"A finitely supported sequence $\\alpha : \\mathbb{N} \\to \\mathbb{R}$ is supported in the dyadic interval at scale $N \\in \\mathbb{R}$ if every $n$ with $\\alpha(n) \\ne 0$ satisfies $n \\ge 1$ and $N \\le n \\le 2N$.","title":"Support in a dyadic interval"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/divisor-log-bound","kind":"node","lean":"WantedExt.NumberTheory.Polymath8b.DivisorLogBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#divisor-log-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"A family $(\\alpha_x)_{x \\in \\mathbb{R}}$ of finitely supported sequences $\\alpha_x : \\mathbb{N} \\to \\mathbb{R}$ satisfies the divisor\u2013logarithm bound if there are a real number $C > 0$ and natural numbers $k$ and $l$ such that for all sufficiently large real $x$ and all integers $n \\ge 1$,","title":"Divisor\u2013logarithm bound for a family of sequences"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/siegel-walfisz","kind":"node","lean":"WantedExt.NumberTheory.Polymath8b.SiegelWalfisz","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#siegel-walfisz","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"Let $(\\beta_x)_{x \\in \\mathbb{R}}$ be a family of finitely supported sequences $\\beta_x : \\mathbb{N} \\to \\mathbb{R}$ and let $M : \\mathbb{R} \\to \\mathbb{R}$. The family satisfies the Siegel\u2013Walfisz hypothesis at scale $M$ if for every real $A \\ge 1$ there are a real number $C > 0$ and a natural number $k$ such that, for all sufficiently large real $x$ and all natural numbers $q \\ge 1$, $r \\ge 1$ and $a$ with\u2026","title":"Siegel\u2013Walfisz hypothesis for a family of sequences"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/elliott-halberstam-at","kind":"node","lean":"WantedExt.NumberTheory.Polymath8b.ElliottHalberstamAt","parent":7768,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elliott\u2013Halberstam estimate at level "],["math","\\(\\theta\\)"]],"source":"Elliott\u2013Halberstam estimate at level $\\theta$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#elliott-halberstam-at","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"For a real number $\\theta$, the Elliott\u2013Halberstam estimate at level $\\theta$ is the following statement. Let $Q : \\mathbb{R} \\to \\mathbb{N}$ be such that $Q(x) = O(x^{\\delta} x^{\\theta})$ as $x \\to \\infty$ for every real $\\delta > 0$, and let $A \\ge 1$ be real. Then there is a real number $C > 0$ such that for all sufficiently large real $x$, the averaged discrepancy of $\\Lambda_x$ up to $Q(x)$ is at most\u2026","title":"Elliott\u2013Halberstam estimate at level $\\theta$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/generalized-elliott-halberstam-at","kind":"node","lean":"WantedExt.NumberTheory.Polymath8b.GeneralizedElliottHalberstamAt","parent":7768,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Elliott\u2013Halberstam estimate at level "],["math","\\(\\theta\\)"]],"source":"Generalized Elliott\u2013Halberstam estimate at level $\\theta$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#generalized-elliott-halberstam-at","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"For a real number $\\theta$, the generalized Elliott\u2013Halberstam estimate at level $\\theta$ is the following statement. Let $\\varepsilon > 0$ be real, let $N, M : \\mathbb{R} \\to \\mathbb{R}$, and let $(\\alpha_x)_{x \\in \\mathbb{R}}$ and $(\\beta_x)_{x \\in \\mathbb{R}}$ be families of finitely supported sequences $\\mathbb{N} \\to \\mathbb{R}$. Assume that for all sufficiently large real $x$ we have $N(x) > 0$ and\u2026","title":"Generalized Elliott\u2013Halberstam estimate at level $\\theta$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/elliott-halberstam","kind":"node","lean":"WantedExt.NumberTheory.Polymath8b.ElliottHalberstam","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#elliott-halberstam","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"The Elliott\u2013Halberstam estimate at level $\\theta$ holds for every real number $\\theta$ with $0 < \\theta < 1$.","title":"Elliott\u2013Halberstam conjecture (dyadic form)"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/generalized-elliott-halberstam","kind":"node","lean":"WantedExt.NumberTheory.Polymath8b.GeneralizedElliottHalberstam","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#generalized-elliott-halberstam","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"The generalized Elliott\u2013Halberstam estimate at level $\\theta$ holds for every real number $\\theta$ with $0 < \\theta < 1$.","title":"Generalized Elliott\u2013Halberstam conjecture (dyadic form)"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/zhang2014-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.zhang2014UpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#zhang2014-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. Then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 70\\,000\\,000$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Zhang's 2014 bound on prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/polymath8a2014-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.polymath8a2014UpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#polymath8a2014-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. Then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 4680$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Polymath8a bound 4680 on small prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/maynard2015-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.maynard2015UpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#maynard2015-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. Then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 600$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Maynard's bound 600 on small prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/polymath8b2014-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.polymath8b2014UpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#polymath8b2014-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. Then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 246$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Polymath8b bound 246 on small prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/zhang2026-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.zhang2026UpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#zhang2026-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. Then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 240$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Zhang's 2026 bound of 240 on prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/openai2026-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.openai2026UpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#openai2026-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. Then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 186$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"OpenAI's bound 186 on small prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/stadlmann2026-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.stadlmann2026UpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#stadlmann2026-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. Then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 240$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Stadlmann's bound 240 on small prime gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/maynard-elliott-halberstam-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.maynardElliottHalberstamUpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#maynard-elliott-halberstam-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. If the Elliott\u2013Halberstam conjecture holds at every level $\\theta$ with $0 < \\theta < 1$, in Polymath8b's dyadic form, then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 12$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Maynard's bound under Elliott\u2013Halberstam"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/gpy-elliott-halberstam-upper-bound","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.gpyElliottHalberstamUpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#gpy-elliott-halberstam-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. If the Elliott\u2013Halberstam conjecture holds at every level $\\theta$ with $0 < \\theta < 1$, in Polymath8b's dyadic form, then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 16$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Goldston\u2013Pintz\u2013Y\u0131ld\u0131r\u0131m bound under Elliott\u2013Halberstam"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/polymath8b-generalized-elliott-halbersta","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.polymath8bGeneralizedElliottHalberstamUpperBound","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#polymath8b-generalized-elliott-halbersta","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $p_0 < p_1 < p_2 < \\cdots$ be the primes in increasing order. If the generalized Elliott\u2013Halberstam conjecture holds at every level $\\theta$ with $0 < \\theta < 1$, in Polymath8b's dyadic form, then $\\liminf_{n \\to \\infty} (p_{n+1} - p_n) \\le 6$, where the liminf is taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Polymath8b bound under the generalized Elliott\u2013Halberstam conjecture"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/openai2026-dhl","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.openai2026DHL","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#openai2026-dhl","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"can_prove","summary":"Let $h_1, \\dots, h_{40}$ be distinct integers forming an admissible tuple: for every prime $p$, some residue class modulo $p$ contains none of the $h_i$. Then there are infinitely many natural numbers $n$ such that at least two of the integers $n + h_1, \\dots, n + h_{40}$ are prime.","title":"Two primes in infinitely many translates of an admissible 40-tuple"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/shifts186","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.shifts186","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#shifts186","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"The list of the forty integers $0, 2, 6, 12, 20, 26, 30, 32, 36, 42, 48, 50, 56, 60, 68, 72, 78, 86, 90, 92, 98, 102, 110, 116, 120, 126, 132, 138, 140, 146, 152, 156, 158, 162, 168, 170, 176, 180, 182, 186$, in this order.","title":"The forty shifts behind the bound 186"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/tuple186","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.tuple186","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#tuple186","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"The $40$-tuple of integers $h = (h_0, \\dots, h_{39})$ whose entries, in order, are $0, 2, 6, 12, 20, 26, 30, 32, 36, 42, 48, 50, 56, 60, 68, 72, 78, 86, 90, 92, 98, 102, 110, 116, 120, 126, 132, 138, 140, 146, 152, 156, 158, 162, 168, 170, 176, 180, 182, 186$.","title":"The explicit 40-tuple for the gap bound 186"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/tuple186-injective","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.tuple186_injective","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#tuple186-injective","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"Let $h = (h_0, \\dots, h_{39})$ be the explicit $40$-tuple of integers $(0, 2, 6, \\dots, 182, 186)$. Then the entries of $h$ are pairwise distinct: $h_i = h_j$ implies $i = j$.","title":"The explicit 40-tuple has distinct entries"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/tuple186-bounds","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.tuple186_bounds","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#tuple186-bounds","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"Let $h = (h_0, \\dots, h_{39})$ be the explicit $40$-tuple of integers $(0, 2, 6, \\dots, 182, 186)$. Then $0 \\le h_i \\le 186$ for every $i \\in \\{0, \\dots, 39\\}$.","title":"The entries of the explicit 40-tuple lie between 0 and 186"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/tuple186-diameter","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.tuple186_diameter","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#tuple186-diameter","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"Let $h = (h_0, \\dots, h_{39})$ be the explicit $40$-tuple of integers $(0, 2, 6, \\dots, 182, 186)$. The largest value in the set $\\{h_0, \\dots, h_{39}\\}$ minus the smallest value in it equals $186$.","title":"The explicit 40-tuple has diameter 186"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/tuple186-admissible","kind":"node","lean":"WantedExt.NumberTheory.PrimeGapBounds.tuple186_admissible","parent":7768,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes/index.html#tuple186-admissible","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/bounded-gaps-between-primes","status":"fully_proved","summary":"Let $h = (h_0, \\dots, h_{39})$ be the explicit $40$-tuple of integers $(0, 2, 6, \\dots, 182, 186)$. Then $h$ is Hardy\u2013Littlewood admissible: for every prime $p$, the number of residue classes modulo $p$ that contain an entry of $h$ is not $p$.","title":"The explicit 40-tuple is admissible"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials","status":"fully_proved","title":"Primes just above factorials and primorials"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/prime-gap-factorial-general","kind":"node","lean":"MetaMathlibExt.prime_gap_factorial_general","parent":7794,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime gaps near "],["math","\\(n!\\)"],["text",", general form"]],"source":"Prime gaps near $n!$, general form"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/index.html#prime-gap-factorial-general","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials","status":"fully_proved","summary":"Let $n$, $p$ and $r$ be natural numbers such that every prime $q > n$ satisfies $q \\ge r$. If $p$ is a prime with $n! + 1 < p < n! + r^2$, then $p - n!$ is prime.","title":"Prime gaps near $n!$, general form"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/prime-gap-factorial","kind":"node","lean":"MetaMathlibExt.prime_gap_factorial","parent":7794,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime gaps near "],["math","\\(n!\\)"]],"source":"Prime gaps near $n!$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/index.html#prime-gap-factorial","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials","status":"fully_proved","summary":"Let $n$ be a natural number and let $r$ be the smallest prime greater than $n$. If $p$ is a prime with $n! + 1 < p < n! + r^2$, then $p - n!$ is prime.","title":"Prime gaps near $n!$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/prime-sub-primorial-prime-of-between-con","kind":"node","lean":"MetaMathlibExt.prime_sub_primorial_prime_of_between_consecutive_primes_general","parent":7794,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/index.html#prime-sub-primorial-prime-of-between-con","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials","status":"fully_proved","summary":"Let $q$ and $r$ be natural numbers such that no prime $s$ satisfies $q < s < r$, and let $q\\#$ be the primorial of $q$, the product of all primes at most $q$. If $p$ is a prime with $q\\# + 1 < p < q\\# + r^2$, then $p - q\\#$ is prime.","title":"Primes just above a primorial, general form"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/prime-sub-primorial-prime-of-between-con-2","kind":"node","lean":"MetaMathlibExt.prime_sub_primorial_prime_of_between_consecutive_primes","parent":7794,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/index.html#prime-sub-primorial-prime-of-between-con-2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials","status":"fully_proved","summary":"Let $q < r$ be consecutive primes, so that no prime lies strictly between them, and let $q\\#$ be the primorial of $q$, the product of all primes at most $q$. If $p$ is a prime with $q\\# + 1 < p < q\\# + r^2$, then $p - q\\#$ is prime.","title":"Primes just above a primorial (\u010cejchan\u2013K\u0159\u00ed\u017eek\u2013Somer)"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/prime-sub-primorial-prime","kind":"node","lean":"MetaMathlibExt.AreConsecutivePrimes.prime_sub_primorial_prime","parent":7794,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials/index.html#prime-sub-primorial-prime","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-just-above-factorials-and-primorials","status":"fully_proved","summary":"Let $q < r$ be consecutive primes and let $q\\#$ be the primorial of $q$, the product of all primes at most $q$. If $p$ is a prime with $q\\# + 1 < p < q\\# + r^2$, then $p - q\\#$ is prime.","title":"Primes just above a primorial, for a consecutive-prime pair"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1","status":"can_prove","title":"Primes of the form 6k \u2212 1"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1/exists-prime-mod-eq-five-between","kind":"node","lean":"MathlibExt.NumberTheory.DistinctEisensteinPrimesWanted.exists_prime_modEq_five_between","parent":7800,"reading_title":{"format":"inline-parts-v1","parts":[["text","Breusch's theorem for primes of the form "],["math","\\(6k - 1\\)"]],"source":"Breusch's theorem for primes of the form $6k - 1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1/index.html#exists-prime-mod-eq-five-between","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1","status":"can_prove","summary":"For every natural number $x \\ge 7$ there is a prime $p$ with $x < p < 2x$ and $p \\equiv 5 \\pmod 6$, that is, a prime of the form $6k - 1$ strictly between $x$ and $2x$.","title":"Breusch's theorem for primes of the form $6k - 1$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1/eq-sum-distinct-primes-mod-eq-five","kind":"node","lean":"MathlibExt.NumberTheory.DistinctEisensteinPrimesWanted.eq_sum_distinct_primes_modEq_five","parent":7800,"reading_title":{"format":"inline-parts-v1","parts":[["text","Makowski's theorem on sums of distinct primes of the form "],["math","\\(6k - 1\\)"]],"source":"Makowski's theorem on sums of distinct primes of the form $6k - 1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1/index.html#eq-sum-distinct-primes-mod-eq-five","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/primes-of-the-form-6k-1","status":"can_prove","summary":"Every natural number $n > 161$ is the sum of the elements of some finite set of primes $p$ with $p \\equiv 5 \\pmod 6$; that is, $n$ is a sum of distinct primes of the form $6k - 1$.","title":"Makowski's theorem on sums of distinct primes of the form $6k - 1$"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes","status":"can_prove","title":"Ramanujan primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/is-labos-prime","kind":"node","lean":"MetaMathlibExt.IsLabosPrime","parent":7803,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/index.html#is-labos-prime","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes","status":"fully_proved","summary":"Write $\\pi(x)$ for the number of primes $p \\le x$. For natural numbers $n$ and $L$, $L$ is the $n$-th Labos prime when $n \\ge 1$, $L \\ge 1$, $\\pi(L) - \\pi(\\lfloor L/2 \\rfloor) = n$, and no integer $m$ with $1 \\le m < L$ satisfies $\\pi(m) - \\pi(\\lfloor m/2 \\rfloor) = n$; that is, $L$ is the least positive integer with $\\pi(L) - \\pi(\\lfloor L/2 \\rfloor) = n$.","title":"Labos prime"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/is-ramanujan-prime","kind":"node","lean":"MetaMathlibExt.IsRamanujanPrime","parent":7803,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/index.html#is-ramanujan-prime","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes","status":"fully_proved","summary":"Let $\\pi$ be the prime-counting function. For natural numbers $R$ and $n$, $R$ satisfies the Ramanujan counting condition for $n$ when $R$ is prime and $\\pi(R) - \\pi(R/2) = n$.","title":"Primes satisfying the Ramanujan counting condition"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/is-nth-ramanujan-prime","kind":"node","lean":"MetaMathlibExt.IsNthRamanujanPrime","parent":7803,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text","-th Ramanujan prime"]],"source":"The $n$-th Ramanujan prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/index.html#is-nth-ramanujan-prime","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes","status":"fully_proved","summary":"Let $\\pi$ be the prime-counting function. For natural numbers $R$ and $n$, $R$ is the $n$-th Ramanujan prime when $n \\ge 1$, $R$ is a prime with $\\pi(R) - \\pi(R/2) = n$, and every prime $p$ with $\\pi(p) - \\pi(p/2) = n$ satisfies $p \\le R$. That is, $R$ is the largest prime satisfying the Ramanujan counting condition for $n$.","title":"The $n$-th Ramanujan prime"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/prime-counting-within","kind":"node","lean":"MetaMathlibExt.primeCountingWithin","parent":7803,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/index.html#prime-counting-within","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes","status":"fully_proved","summary":"For a set $P$ of natural numbers and a real number $x$, $\\pi_P(x)$ is the number of members of $P$ that are at most $x$.","title":"Counting members of a set up to a real bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/is-vpramanujan-number","kind":"node","lean":"MetaMathlibExt.IsVPRamanujanNumber","parent":7803,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/index.html#is-vpramanujan-number","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes","status":"fully_proved","summary":"Let $P$ be a set of natural numbers, $v$ a real number and $m$, $R$ natural numbers, and write $\\pi_P(x)$ for the number of members of $P$ that are at most $x$. Then $R$ is the $(v, P)$-Ramanujan number of index $m$ when $P$ is an infinite set of primes, $v > 1$, and $R$ is the least natural number such that $\\pi_P(x) - \\pi_P(x/v) \\ge m$ for every real $x \\ge R$, that is, every interval $(x/v, x]$ with $x \\ge R$\u2026","title":"Generalized Ramanujan numbers"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/laishram-ramanujan-prime-bound","kind":"node","lean":"MetaMathlibExt.laishram_ramanujan_prime_bound","parent":7803,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes/index.html#laishram-ramanujan-prime-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/ramanujan-primes","status":"can_prove","summary":"Let $p_k$ denote the $k$-th prime, so that $p_1 = 2$. For all natural numbers $n$ and $R$, if $R$ is the $n$-th Ramanujan prime (which entails $n \\ge 1$), then $R < p_{3n}$.","title":"Laishram's upper bound for Ramanujan primes"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","kind":"scope","navigation":"module","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"can_prove","title":"Smooth gaps between primes"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/is-difference-smooth","kind":"node","lean":"MetaMathlibExt.IsDifferenceSmooth","parent":7810,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html#is-difference-smooth","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"fully_proved","summary":"Let $y$ be a natural number. A finite set $H$ of integers is difference $y$-smooth if for all distinct $h, h' \\in H$ every prime factor of $h - h'$ is at most $y$.","title":"Difference-smooth sets of integers"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/is-difference-smooth-singleton","kind":"node","lean":"MetaMathlibExt.isDifferenceSmooth_singleton","parent":7810,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html#is-difference-smooth-singleton","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"fully_proved","summary":"For every natural number $y$ and every integer $a$, the set $\\{a\\}$ is difference $y$-smooth.","title":"One-element sets are difference smooth"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/infinite-prime-pairs-smooth-47-gap","kind":"node","lean":"MetaMathlibExt.infinite_prime_pairs_smooth_47_gap","parent":7810,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many prime pairs with "],["math","\\(47\\)"],["text","-smooth difference"]],"source":"Infinitely many prime pairs with $47$-smooth difference"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html#infinite-prime-pairs-smooth-47-gap","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"can_prove","summary":"There are infinitely many pairs of primes $p > q$ such that every prime factor of $p - q$ is at most $47$.","title":"Infinitely many prime pairs with $47$-smooth difference"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/maynard-tao-admissible-tuples","kind":"node","lean":"MetaMathlibExt.maynard_tao_admissible_tuples","parent":7810,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html#maynard-tao-admissible-tuples","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"can_prove","summary":"For every natural number $m \\ge 2$ there is a natural number $k$ such that for every admissible finite set $H$ of at least $k$ integers there are infinitely many integers $n$ for which $n + h$ is a (positive) prime for at least $m$ elements $h \\in H$.","title":"Maynard\u2013Tao theorem"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/infinite-prime-tuples-smooth-gaps","kind":"node","lean":"MetaMathlibExt.infinite_prime_tuples_smooth_gaps","parent":7810,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime "],["math","\\(m\\)"],["text","-tuples with smooth gaps"]],"source":"Prime $m$-tuples with smooth gaps"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html#infinite-prime-tuples-smooth-gaps","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"can_prove","summary":"Let $m \\ge 2$ and $k$ be natural numbers such that every admissible finite set $H$ of at least $k$ integers has infinitely many integers $n$ for which $n + h$ is a (positive) prime for at least $m$ elements $h \\in H$. Then there are infinitely many $m$-tuples of primes $p_1 < p_2 < \\dots < p_m$ such that for all $j < i$ every prime factor of $p_i - p_j$ is at most $k$.","title":"Prime $m$-tuples with smooth gaps"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/infinite-prime-pairs-smooth-5-gap-of-ell","kind":"node","lean":"MetaMathlibExt.infinite_prime_pairs_smooth_5_gap_of_elliottHalberstam","parent":7810,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime pairs with "],["math","\\(5\\)"],["text","-smooth difference under Elliott\u2013Halberstam"]],"source":"Prime pairs with $5$-smooth difference under Elliott\u2013Halberstam"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html#infinite-prime-pairs-smooth-5-gap-of-ell","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"can_prove","summary":"If the Elliott\u2013Halberstam conjecture holds, then there are infinitely many pairs of primes $p > q$ such that every prime factor of $p - q$ is at most $5$.","title":"Prime pairs with $5$-smooth difference under Elliott\u2013Halberstam"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/exists-admissible-difference-smooth","kind":"node","lean":"MetaMathlibExt.exists_admissible_differenceSmooth","parent":7810,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html#exists-admissible-difference-smooth","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"can_prove","summary":"For every natural number $k$ there is an admissible set $H$ of exactly $k$ integers that is difference $k$-smooth: for all distinct $h, h' \\in H$, every prime factor of $h - h'$ is at most $k$.","title":"Admissible difference-smooth sets of every size"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/not-is-difference-smooth-of-is-admissibl","kind":"node","lean":"MetaMathlibExt.not_isDifferenceSmooth_of_isAdmissibleFinset","parent":7810,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes/index.html#not-is-difference-smooth-of-is-admissibl","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/smooth-gaps-between-primes","status":"can_prove","summary":"Let $H$ be an admissible finite set of $k \\ge 2$ integers, let $z \\le k$ be a prime and let $\\ell < z$ be a natural number. Then $H$ is not difference $\\ell$-smooth. Equivalently, $H$ is not difference $\\ell$-smooth for any $\\ell$ less than the largest prime at most $k$.","title":"Admissible sets are not difference smooth at small bounds"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem","kind":"scope","parent":7536,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem","status":"fully_proved","title":"Sylvester\u2013Schur theorem"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/second-residual-mid-prime-counting-lt-st","kind":"node","lean":"Nat.SylvesterSchur.Internal.second_residual_mid_primeCounting_lt_start_pow","parent":7819,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-counting estimate for Sylvester\u2013Schur when "],["math","\\(38 \\le n < 945\\)"]],"source":"Prime-counting estimate for Sylvester\u2013Schur when $38 \\le n < 945$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/index.html#second-residual-mid-prime-counting-lt-st","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem","status":"fully_proved","summary":"For all natural numbers $n$ and $k$ with $38 \\le n < 945$ and $k \\ge 1291$, we have $n! \\cdot (k + n - 1)^{\\pi(n)} < k^n$, where $\\pi(n)$ denotes the number of primes at most $n$.","title":"Prime-counting estimate for Sylvester\u2013Schur when $38 \\le n < 945$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erdos-choose-prime-factor-gt","kind":"node","lean":"Nat.SylvesterSchur.Internal.erdos_choose_prime_factor_gt","parent":7819,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/index.html#erdos-choose-prime-factor-gt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem","status":"fully_proved","summary":"For all natural numbers $n$ and $k$ with $1 \\le n < k$, the binomial coefficient $\\binom{k + n - 1}{n}$ has a prime factor $p > n$.","title":"Erd\u0151s key lemma"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/exists-prime-gt-and-dvd-asc-factorial","kind":"node","lean":"Nat.exists_prime_gt_and_dvd_ascFactorial","parent":7819,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/index.html#exists-prime-gt-and-dvd-asc-factorial","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $0 < k < n$. Then some prime $p > k$ divides the product $n(n+1)\\cdots(n+k-1)$ of the $k$ consecutive integers starting at $n$.","title":"Sylvester\u2013Schur theorem"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms","status":"fully_proved","title":"Sylvester-Schur: binomial and consecutive forms"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms/prime-dvd-choose-of-dvd-consecutive","kind":"node","lean":"Nat.SylvesterSchur.prime_dvd_choose_of_dvd_consecutive","parent":7823,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms/index.html#prime-dvd-choose-of-dvd-consecutive","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms","status":"fully_proved","summary":"Let $n$, $k$ and $j$ be natural numbers with $j < n$, and let $p$ be a prime with $p > n$. If $p$ divides $k + j$, one of the $n$ consecutive integers $k, \\dots, k+n-1$, then $p$ divides $\\binom{k+n-1}{n}$.","title":"Large prime divisors of a block term divide the binomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms/exists-dvd-consecutive-of-prime-dvd-choo","kind":"node","lean":"Nat.SylvesterSchur.exists_dvd_consecutive_of_prime_dvd_choose","parent":7823,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms/index.html#exists-dvd-consecutive-of-prime-dvd-choo","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers and let $p$ be a prime. If $p$ divides $\\binom{k+n-1}{n}$, then $p$ divides $k + i$ for some $i$ with $0 \\le i < n$.","title":"Prime divisors of the block binomial coefficient divide a block term"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms/consecutive-of-exists-prime-dvd-choose","kind":"node","lean":"Nat.SylvesterSchur.consecutive_of_exists_prime_dvd_choose","parent":7823,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms/index.html#consecutive-of-exists-prime-dvd-choose","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/sylvester-schur-binomial-and-consecutive-forms","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers. If some prime $p > n$ divides $\\binom{k+n-1}{n}$, then some prime $p > n$ divides one of the integers $k + i$ with $0 \\le i < n$.","title":"From a large prime divisor of the binomial coefficient to one in the block"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur","status":"fully_proved","title":"Power-of-four bounds for Sylvester-Schur"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur/central-erdos-layers-two-mul-add-two-lt","kind":"node","lean":"Nat.SylvesterSchur.central_erdos_layers_two_mul_add_two_lt_four_pow","parent":7827,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur/index.html#central-erdos-layers-two-mul-add-two-lt","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur","status":"fully_proved","summary":"Let $n \\ge 512$ be a natural number and $N = 2n + 2$. For a natural number $x$ write $x\\#$ for the primorial of $x$, the product of all primes $p \\le x$, and let $L = \\lfloor \\log_2 N \\rfloor$. Then $n \\cdot \\lfloor N/3 \\rfloor\\# \\cdot \\prod_{m=2}^{L} \\lfloor N^{1/m} \\rfloor\\# < 4^n$.","title":"Central Erd\u0151s layer estimate"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur/erdos-layers-le-four-pow","kind":"node","lean":"Nat.SylvesterSchur.erdos_layers_le_four_pow","parent":7827,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur/index.html#erdos-layers-le-four-pow","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers. For a natural number $x$ write $x\\#$ for the primorial of $x$, the product of all primes $p \\le x$. Let $L = \\lfloor \\log_2 N \\rfloor$, with $L = 0$ when $N = 0$, and let $P = 1$ if $L = 0$ and $P = r\\# \\cdot \\prod_{m=2}^{L} \\lfloor N^{1/m} \\rfloor\\#$ if $L \\ge 1$. Then $P \\le 4^{r + \\lfloor \\sqrt{N} \\rfloor L}$.","title":"Power-of-four bound for the Erd\u0151s layer product"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur/erdos-layers-le-four-pow-three-layers","kind":"node","lean":"Nat.SylvesterSchur.erdos_layers_le_four_pow_three_layers","parent":7827,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur/index.html#erdos-layers-le-four-pow-three-layers","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/power-of-four-bounds-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers. For a natural number $x$ write $x\\#$ for the primorial of $x$, the product of all primes $p \\le x$. Let $L = \\lfloor \\log_2 N \\rfloor$, with $L = 0$ when $N = 0$, and let $P = 1$ if $L = 0$ and $P = r\\# \\cdot \\prod_{m=2}^{L} \\lfloor N^{1/m} \\rfloor\\#$ if $L \\ge 1$. Then $P \\le 4^{r + \\lfloor \\sqrt{N} \\rfloor + \\lfloor N^{1/3} \\rfloor L}$.","title":"Three-layer power-of-four bound for the Erd\u0151s layer product"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large","kind":"scope","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large","status":"fully_proved","title":"Large residual estimates for the Sylvester-Schur theorem"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/second-residual-large-central-offset-bou","kind":"node","lean":"Nat.SylvesterSchur.Internal.second_residual_large_central_offset_bound","parent":7831,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/index.html#second-residual-large-central-offset-bou","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $945 \\le n < k$ and $k + n - 1 < 3n$, and put $N = k + n - 1$ and $i = k - n - 1$. For a natural number $m$ let $m\\#$ denote the product of all primes $p \\le m$, and let $P_0 = \\lfloor N/3 \\rfloor\\#$ and $P_j = \\lfloor N^{1/(j+1)} \\rfloor\\#$ for $j \\ge 1$. Then","title":"Central-offset residual bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/second-residual-tiny-prime-counting-lt-s","kind":"node","lean":"Nat.SylvesterSchur.Internal.second_residual_tiny_primeCounting_lt_start_pow","parent":7831,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/index.html#second-residual-tiny-prime-counting-lt-s","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $7 \\le n < 38$ and $k \\ge 100$, and let $\\pi(n)$ be the number of primes $p \\le n$. Then $n! \\cdot (k + n - 1)^{\\pi(n)} < k^{n}$.","title":"Tiny-range prime-counting bound"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur","kind":"scope","navigation":"module","parent":7831,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur","status":"fully_proved","title":"Power bounds for large residual Sylvester-Schur estimates"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur/self-le-four-pow-div-one-twenty-eight-ad","kind":"node","lean":"Nat.SylvesterSchur.Internal.self_le_four_pow_div_one_twenty_eight_add_one","parent":7834,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur/index.html#self-le-four-pow-div-one-twenty-eight-ad","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur","status":"fully_proved","summary":"For every natural number $n \\ge 512$, $n \\le 4^{\\lfloor n/128 \\rfloor + 1}$.","title":"A linear bound by a power of four"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur/two-pow-nineteen-mul-div-twelve-le-three","kind":"node","lean":"Nat.SylvesterSchur.Internal.two_pow_nineteen_mul_div_twelve_le_three_pow","parent":7834,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur/index.html#two-pow-nineteen-mul-div-twelve-le-three","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur","status":"fully_proved","summary":"For every natural number $n$, $2^{19 \\lfloor n/12 \\rfloor} \\le 3^{n}$.","title":"Powers of two against powers of three"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur/four-pow-mul-two-pow-mul-four-pow-lt-thr","kind":"node","lean":"Nat.SylvesterSchur.Internal.four_pow_mul_two_pow_mul_four_pow_lt_three_pow_mul_four_pow","parent":7834,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur/index.html#four-pow-mul-two-pow-mul-four-pow-lt-thr","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/power-bounds-for-large-residual-sylvester-schur","status":"fully_proved","summary":"Let $D$, $E$, $i$, $n$ be natural numbers with $2D + i + 2E < 19 \\lfloor i/12 \\rfloor + 2n$. Then $4^{D} \\cdot 2^{i} \\cdot 4^{E} < 3^{i} \\cdot 4^{n}$.","title":"A power comparison for the central-offset estimate"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar","kind":"scope","navigation":"module","parent":7831,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar","status":"fully_proved","title":"Scaled ternary bound for the Sylvester-Schur large residual estimate"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar/sqrt-le-div-fifteen-of-le-four-mul","kind":"node","lean":"Nat.SylvesterSchur.Internal.sqrt_le_div_fifteen_of_le_four_mul","parent":7838,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar/index.html#sqrt-le-div-fifteen-of-le-four-mul","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar","status":"fully_proved","summary":"Let $n$ and $N$ be natural numbers with $900 \\le n$ and $N \\le 4n$. Then $\\lfloor \\sqrt{N} \\rfloor \\le \\lfloor n/15 \\rfloor$.","title":"Square root bound on a linear range"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar/nth-root-three-mul-log-le-one-fifth-of-l","kind":"node","lean":"Nat.SylvesterSchur.Internal.nthRoot_three_mul_log_le_one_fifth_of_le_four_mul","parent":7838,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar/index.html#nth-root-three-mul-log-le-one-fifth-of-l","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar","status":"fully_proved","summary":"Let $n$ and $N$ be natural numbers with $945 \\le n$ and $N \\le 4n$. Then $\\lfloor N^{1/3} \\rfloor \\cdot \\lfloor \\log_2 N \\rfloor \\le \\lfloor n/5 \\rfloor$, where $\\lfloor \\log_2 N \\rfloor$ is read as $0$ when $N = 0$.","title":"Cube root times logarithm on a linear range"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar/first-residual-large-scaled-ternary-boun","kind":"node","lean":"Nat.SylvesterSchur.Internal.first_residual_large_scaled_ternary_bound","parent":7838,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar/index.html#first-residual-large-scaled-ternary-boun","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/residual-large/scaled-ternary-bound-for-the-sylvester-schur-lar","status":"fully_proved","summary":"Let $n$ and $N$ be natural numbers with $945 \\le n$ and $3n \\le N \\le n^2 + n$. Then","title":"Scaled ternary bound for the first residual estimate"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur","status":"fully_proved","title":"Prime-counting estimates for Sylvester-Schur"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/exists-large-prime-dvd-choose-of-not-for","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_not_forall_prime_dvd_le","parent":7842,"reading_title":{"format":"inline-parts-v1","parts":[["text","Negating the no-large-prime condition for "],["math","\\(\\binom{N}{r}\\)"]],"source":"Negating the no-large-prime condition for $\\binom{N}{r}$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-not-for","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers. If it is not the case that every prime dividing $\\binom{N}{r}$ is at most $r$, then some prime $p$ with $r < p$ divides $\\binom{N}{r}$.","title":"Negating the no-large-prime condition for $\\binom{N}{r}$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/exists-large-prime-dvd-choose-of-upper-l","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_upper_lt_lower","parent":7842,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prime divisor of "],["math","\\(\\binom{N}{r}\\)"],["text"," above "],["math","\\(r\\)"],["text"," from an upper bound below the lower bound"]],"source":"A prime divisor of $\\binom{N}{r}$ above $r$ from an upper bound below the lower bound"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-upper-l","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur","status":"fully_proved","summary":"Let $N$, $r$ and $U$ be natural numbers, and let $m = \\max(N + 1 - r, 0)$. Suppose that $\\binom{N}{r} \\le U$ whenever every prime dividing $\\binom{N}{r}$ is at most $r$, and that $U < m^r / r!$ as rational numbers. Then some prime $p$ with $r < p$ divides $\\binom{N}{r}$.","title":"A prime divisor of $\\binom{N}{r}$ above $r$ from an upper bound below the lower bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/prime-counting-le-sub-one","kind":"node","lean":"Nat.SylvesterSchur.primeCounting_le_sub_one","parent":7842,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most "],["math","\\(r - 1\\)"],["text"," primes up to "],["math","\\(r\\)"]],"source":"At most $r - 1$ primes up to $r$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/index.html#prime-counting-le-sub-one","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur","status":"fully_proved","summary":"For every natural number $r \\ge 2$, the number $\\pi(r)$ of primes $p \\le r$ satisfies $\\pi(r) \\le r - 1$, since all such primes lie in $\\{2, \\dots, r\\}$.","title":"At most $r - 1$ primes up to $r$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/prime-counting-le-half-add-one","kind":"node","lean":"Nat.SylvesterSchur.primeCounting_le_half_add_one","parent":7842,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most "],["math","\\(\\lfloor r/2 \\rfloor + 1\\)"],["text"," primes up to "],["math","\\(r\\)"]],"source":"At most $\\lfloor r/2 \\rfloor + 1$ primes up to $r$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/index.html#prime-counting-le-half-add-one","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur","status":"fully_proved","summary":"For every natural number $r$, the number $\\pi(r)$ of primes $p \\le r$ satisfies $\\pi(r) \\le \\lfloor r/2 \\rfloor + 1$.","title":"At most $\\lfloor r/2 \\rfloor + 1$ primes up to $r$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/prime-counting-le-three-mul-div-eight","kind":"node","lean":"Nat.SylvesterSchur.primeCounting_le_three_mul_div_eight","parent":7842,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s's "],["math","\\(3r/8\\)"],["text"," prime-counting bound"]],"source":"Erd\u0151s's $3r/8$ prime-counting bound"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur/index.html#prime-counting-le-three-mul-div-eight","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/prime-counting-estimates-for-sylvester-schur","status":"fully_proved","summary":"For every natural number $r \\ge 38$, the number $\\pi(r)$ of primes $p \\le r$ satisfies $\\pi(r) \\le \\lfloor 3r/8 \\rfloor$.","title":"Erd\u0151s's $3r/8$ prime-counting bound"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","title":"Binomial factorization bounds for Sylvester-Schur"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/choose-eq-prod-small-prime-powers-of-no","kind":"node","lean":"Nat.SylvesterSchur.choose_eq_prod_small_prime_powers_of_no_large","parent":7848,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html#choose-eq-prod-small-prime-powers-of-no","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","summary":"Let $r \\le N$ be natural numbers such that every prime divisor of $\\binom{N}{r}$ is at most $r$. Then $\\prod_{p=0}^{r} p^{v_p(\\binom{N}{r})} = \\binom{N}{r}$, where $v_p(m)$ is the exponent of $p$ in the prime factorization of $m$ (zero when $p$ is not prime).","title":"Truncating the prime factorization of a binomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/choose-le-top-pow-prime-counting-of-no-l","kind":"node","lean":"Nat.SylvesterSchur.choose_le_top_pow_primeCounting_of_no_large","parent":7848,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html#choose-le-top-pow-prime-counting-of-no-l","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","summary":"Let $0 < N$ and $r \\le N$ be natural numbers such that every prime divisor of $\\binom{N}{r}$ is at most $r$. Then $\\binom{N}{r} \\le N^{\\pi(r)}$, where $\\pi(r)$ is the number of primes $p \\le r$.","title":"Prime-counting bound for binomial coefficients without large prime factors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/choose-le-top-pow-sub-one-of-no-large","kind":"node","lean":"Nat.SylvesterSchur.choose_le_top_pow_sub_one_of_no_large","parent":7848,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html#choose-le-top-pow-sub-one-of-no-large","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","summary":"Let $0 < N$ and $2 \\le r \\le N$ be natural numbers such that every prime divisor of $\\binom{N}{r}$ is at most $r$. Then $\\binom{N}{r} \\le N^{r-1}$.","title":"Elementary power bound for binomial coefficients without large prime factors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/choose-le-top-pow-half-add-one-of-no-lar","kind":"node","lean":"Nat.SylvesterSchur.choose_le_top_pow_half_add_one_of_no_large","parent":7848,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html#choose-le-top-pow-half-add-one-of-no-lar","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","summary":"Let $0 < N$ and $r \\le N$ be natural numbers such that every prime divisor of $\\binom{N}{r}$ is at most $r$. Then $\\binom{N}{r} \\le N^{\\lfloor r/2 \\rfloor + 1}$.","title":"Half-exponent bound for binomial coefficients without large prime factors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/choose-le-top-pow-three-mul-div-eight-of","kind":"node","lean":"Nat.SylvesterSchur.choose_le_top_pow_three_mul_div_eight_of_no_large","parent":7848,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html#choose-le-top-pow-three-mul-div-eight-of","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","summary":"Let $0 < N$ and $38 \\le r \\le N$ be natural numbers such that every prime divisor of $\\binom{N}{r}$ is at most $r$. Then $\\binom{N}{r} \\le N^{\\lfloor 3r/8 \\rfloor}$.","title":"Three-eighths exponent bound for binomial coefficients without large prime factors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/choose-le-prod-primorial-nth-root-of-no","kind":"node","lean":"Nat.SylvesterSchur.choose_le_prod_primorial_nthRoot_of_no_large","parent":7848,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html#choose-le-prod-primorial-nth-root-of-no","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","summary":"Let $0 < N$ and $r \\le N$ be natural numbers such that every prime divisor of $\\binom{N}{r}$ is at most $r$. Write $P(m)$ for the product of all primes $p \\le m$ and let $L = \\lfloor \\log_2 N \\rfloor$. Then $\\binom{N}{r} \\le \\prod_{k=1}^{L} P(\\lfloor N^{1/k} \\rfloor)$.","title":"Erd\u0151s root-layer bound for binomial coefficients without large prime factors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/choose-le-prod-erdos-layers-of-no-large","kind":"node","lean":"Nat.SylvesterSchur.choose_le_prod_erdos_layers_of_no_large","parent":7848,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html#choose-le-prod-erdos-layers-of-no-large","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","summary":"Let $0 < N$ and $r \\le N$ be natural numbers such that every prime divisor of $\\binom{N}{r}$ is at most $r$. Write $P(m)$ for the product of all primes $p \\le m$ and let $L = \\lfloor \\log_2 N \\rfloor$. Then $\\binom{N}{r} \\le \\prod_{k=1}^{L} a_k$, where $a_1 = P(r)$ and $a_k = P(\\lfloor N^{1/k} \\rfloor)$ for $k \\ge 2$.","title":"Erd\u0151s layer bound for binomial coefficients without large prime factors"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/choose-le-prod-erdos-div-three-layers-of","kind":"node","lean":"Nat.SylvesterSchur.choose_le_prod_erdos_div_three_layers_of_no_large","parent":7848,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu/index.html#choose-le-prod-erdos-div-three-layers-of","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/binomial-factorization-bounds-for-sylvester-schu","status":"fully_proved","summary":"Let $N \\ge 6$ and $r$ be natural numbers with $2r \\le N$ (hence $r \\le N$) such that every prime divisor of $\\binom{N}{r}$ is at most $r$. Write $P(m)$ for the product of all primes $p \\le m$ and let $L = \\lfloor \\log_2 N \\rfloor$. Then $\\binom{N}{r} \\le \\prod_{k=1}^{L} a_k$, where $a_1 = P(\\lfloor N/3 \\rfloor)$ and $a_k = P(\\lfloor N^{1/k} \\rfloor)$ for $k \\ge 2$.","title":"Erd\u0151s layer bound with a one-third first layer"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur","status":"fully_proved","title":"Erd\u0151s layer products for Sylvester-Schur"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur/pow-mul-choose-le-pow-mul-choose","kind":"node","lean":"Nat.SylvesterSchur.pow_mul_choose_le_pow_mul_choose","parent":7857,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur/index.html#pow-mul-choose-le-pow-mul-choose","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur","status":"fully_proved","summary":"For natural numbers $r$, $M$ and $N$ with $r \\le M \\le N$, we have $N^r \\binom{M}{r} \\le M^r \\binom{N}{r}$.","title":"Scaled monotonicity of binomial coefficients"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur/exists-large-prime-dvd-choose-of-factori","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_factorial_mul_prod_erdos_div_three_layers_lt_lower_pow","parent":7857,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-factori","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$, $r \\le N$, $2r \\le N$ and $6 \\le N$. For a natural number $x$ write $x\\#$ for the primorial of $x$, the product of all primes $p \\le x$, and let $L = \\lfloor \\log_2 N \\rfloor$. If $r! \\cdot \\lfloor N/3 \\rfloor\\# \\cdot \\prod_{m=2}^{L} \\lfloor N^{1/m} \\rfloor\\# < (N + 1 - r)^r$, then the binomial coefficient $\\binom{N}{r}$ has a prime factor $p > r$.","title":"Erd\u0151s layer criterion, factorial form"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur/exists-large-prime-dvd-choose-of-mul-pro","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_mul_prod_erdos_div_three_layers_lt_four_pow","parent":7857,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-mul-pro","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/erd-s-layer-products-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $4 \\le r$, $2r \\le N$ and $6 \\le N$. For a natural number $x$ write $x\\#$ for the primorial of $x$, the product of all primes $p \\le x$, and let $L = \\lfloor \\log_2 N \\rfloor$. If $r \\cdot \\lfloor N/3 \\rfloor\\# \\cdot \\prod_{m=2}^{L} \\lfloor N^{1/m} \\rfloor\\# < 4^r$, then the binomial coefficient $\\binom{N}{r}$ has a prime factor $p > r$.","title":"Erd\u0151s layer criterion, power-of-four form"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","title":"Large-prime criteria for Sylvester-Schur"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-factori","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_factorial_mul_prod_erdos_layers_lt_lower_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-factori","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$ and $r \\le N$, write $m\\#$ for the product of the primes at most $m$, and let $L = \\lfloor \\log_2 N \\rfloor$. Put $P_1 = r\\#$ and $P_k = \\lfloor N^{1/k} \\rfloor\\#$ for $k \\ge 2$. If","title":"Large-prime criterion from the Erd\u0151s layer bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-factori-2","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_factorial_mul_prod_primorial_nthRoot_lt_lower_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-factori-2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$ and $r \\le N$, write $m\\#$ for the product of the primes at most $m$, and let $L = \\lfloor \\log_2 N \\rfloor$. If","title":"Large-prime criterion from the prime-power layer bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-mul-top","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_mul_top_pow_sqrt_mul_primorial_lt_four_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-mul-top","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $4 \\le r$ and $2r \\le N$, and write $r\\#$ for the product of the primes at most $r$. If $r \\cdot N^{\\lfloor \\sqrt{N} \\rfloor + 1} \\cdot r\\# < 4^r$, then $\\binom{N}{r}$ has a prime divisor $p$ with $p > r$.","title":"Central-range large-prime criterion from the square-root split bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-factori-3","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_factorial_mul_top_pow_sqrt_mul_primorial_lt_lower_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-factori-3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$ and $r \\le N$, and write $r\\#$ for the product of the primes at most $r$. If $r! \\cdot N^{\\lfloor \\sqrt{N} \\rfloor + 1} \\cdot r\\# < (N + 1 - r)^r$, then $\\binom{N}{r}$ has a prime divisor $p$ with $p > r$.","title":"Large-prime criterion from the square-root split bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-factori-4","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_factorial_mul_top_pow_primeCounting_lt_lower_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-factori-4","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$ and $r \\le N$, and let $\\pi(r)$ be the number of primes at most $r$. If $r! \\cdot N^{\\pi(r)} < (N + 1 - r)^r$, then $\\binom{N}{r}$ has a prime divisor $p$ with $p > r$.","title":"Large-prime criterion from the prime-counting bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-factori-5","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_factorial_mul_top_pow_half_add_one_lt_lower_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-factori-5","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$ and $r \\le N$. If $r! \\cdot N^{\\lfloor r/2 \\rfloor + 1} < (N + 1 - r)^r$, then $\\binom{N}{r}$ has a prime divisor $p$ with $p > r$.","title":"Large-prime criterion from the half-plus-one exponent bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-factori-6","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_factorial_mul_top_pow_three_mul_div_eight_lt_lower_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-factori-6","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$, $38 \\le r$ and $r \\le N$. If $r! \\cdot N^{\\lfloor 3r/8 \\rfloor} < (N + 1 - r)^r$, then $\\binom{N}{r}$ has a prime divisor $p$ with $p > r$.","title":"Large-prime criterion from the 3r/8 exponent bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-self-po","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_self_pow_mul_top_pow_three_mul_div_eight_lt_lower_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-self-po","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$, $38 \\le r$ and $r \\le N$. If $r^r \\cdot N^{\\lfloor 3r/8 \\rfloor} < (N + 1 - r)^r$, then $\\binom{N}{r}$ has a prime divisor $p$ with $p > r$.","title":"Power-only large-prime criterion from the 3r/8 exponent bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/exists-large-prime-dvd-choose-of-factori-7","kind":"node","lean":"Nat.SylvesterSchur.exists_large_prime_dvd_choose_of_factorial_mul_top_pow_sub_one_lt_lower_pow","parent":7861,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur/index.html#exists-large-prime-dvd-choose-of-factori-7","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/large-prime-criteria-for-sylvester-schur","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $0 < N$, $2 \\le r$ and $r \\le N$. If $r! \\cdot N^{r-1} < (N + 1 - r)^r$, then $\\binom{N}{r}$ has a prime divisor $p$ with $p > r$.","title":"Large-prime criterion from the r \u2212 1 exponent bound"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","title":"Conditional Sylvester-Schur consequences"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-two","kind":"node","lean":"Nat.SylvesterSchur.consecutive_two","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-two","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"For every natural number $k > 2$, one of the two integers $k$, $k + 1$ is divisible by a prime $p > 2$.","title":"Sylvester\u2013Schur theorem for two consecutive integers"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-three","kind":"node","lean":"Nat.SylvesterSchur.consecutive_three","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-three","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"For every natural number $k > 3$, one of the three integers $k$, $k + 1$, $k + 2$ is divisible by a prime $p > 3$.","title":"Sylvester\u2013Schur theorem for three consecutive integers"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-four","kind":"node","lean":"Nat.SylvesterSchur.consecutive_four","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-four","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"For every natural number $k > 4$, some prime $p > 4$ divides one of $k$, $k+1$, $k+2$, $k+3$.","title":"Four consecutive integers above four have a prime factor above four"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-five","kind":"node","lean":"Nat.SylvesterSchur.consecutive_five","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-five","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"For every natural number $k > 5$, some prime $p > 5$ divides one of $k$, $k+1$, $k+2$, $k+3$, $k+4$.","title":"Five consecutive integers above five have a prime factor above five"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-six","kind":"node","lean":"Nat.SylvesterSchur.consecutive_six","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-six","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"For every natural number $k > 6$, some prime $p > 6$ divides one of $k$, $k+1$, $k+2$, $k+3$, $k+4$, $k+5$.","title":"Six consecutive integers above six have a prime factor above six"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-of-factorial-mul-top-pow-pri","kind":"node","lean":"Nat.SylvesterSchur.consecutive_of_factorial_mul_top_pow_primeCounting_lt_start_pow","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-of-factorial-mul-top-pow-pri","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers and let $\\pi(n)$ be the number of primes at most $n$. If $n! \\cdot (k+n-1)^{\\pi(n)} < k^n$, then some prime $p > n$ divides one of the integers $k, k+1, \\dots, k+n-1$.","title":"Prime-counting criterion for a large prime in a consecutive block"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-of-factorial-mul-top-pow-hal","kind":"node","lean":"Nat.SylvesterSchur.consecutive_of_factorial_mul_top_pow_half_add_one_lt_start_pow","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-of-factorial-mul-top-pow-hal","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers. If $n! \\cdot (k+n-1)^{\\lfloor n/2 \\rfloor + 1} < k^n$, then some prime $p > n$ divides one of the integers $k, k+1, \\dots, k+n-1$.","title":"Half-exponent criterion for a large prime in a consecutive block"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-of-quadratic-start","kind":"node","lean":"Nat.SylvesterSchur.consecutive_of_quadratic_start","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-of-quadratic-start","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"Let $n \\ge 38$ and $k > n^2$ be natural numbers. Then some prime $p > n$ divides one of the integers $k, k+1, \\dots, k+n-1$.","title":"Quadratic starts give a large prime in a consecutive block"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-of-factorial-mul-top-pow-sqr","kind":"node","lean":"Nat.SylvesterSchur.consecutive_of_factorial_mul_top_pow_sqrt_mul_primorial_lt_start_pow","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-of-factorial-mul-top-pow-sqr","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers, write $N = k + n - 1$, and let $P(n)$ denote the product of all primes at most $n$. If $n! \\cdot N^{\\lfloor \\sqrt{N} \\rfloor + 1} \\cdot P(n) < k^n$, then some prime $p > n$ divides one of the integers $k, k+1, \\dots, k+n-1$.","title":"Square-root and primorial criterion for a large prime in a consecutive block"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-of-factorial-mul-prod-erdos","kind":"node","lean":"Nat.SylvesterSchur.consecutive_of_factorial_mul_prod_erdos_layers_lt_start_pow","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-of-factorial-mul-prod-erdos","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers, write $N = k + n - 1$ and $L = \\lfloor \\log_2 N \\rfloor$, and for a natural number $m$ let $P(m)$ denote the product of all primes at most $m$. If $n! \\cdot P(n) \\cdot \\prod_{i=2}^{L} P(\\lfloor N^{1/i} \\rfloor) < k^n$, then some prime $p > n$ divides one of the integers $k, k+1, \\dots, k+n-1$.","title":"Erd\u0151s layer criterion for a large prime in a consecutive block"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-of-factorial-mul-prod-erdos-2","kind":"node","lean":"Nat.SylvesterSchur.consecutive_of_factorial_mul_prod_erdos_div_three_layers_lt_start_pow","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-of-factorial-mul-prod-erdos-2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"Let $n < k$ be natural numbers, write $N = k + n - 1$ and $L = \\lfloor \\log_2 N \\rfloor$, assume $N \\ge 6$, and for a natural number $m$ let $P(m)$ denote the product of all primes at most $m$. If $n! \\cdot P(\\lfloor N/3 \\rfloor) \\cdot \\prod_{i=2}^{L} P(\\lfloor N^{1/i} \\rfloor) < k^n$, then some prime $p > n$ divides one of the integers $k, k+1, \\dots, k+n-1$.","title":"Erd\u0151s layer criterion with a one-third first layer for a large prime in a consecutive block"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/consecutive-of-large-start","kind":"node","lean":"Nat.SylvesterSchur.consecutive_of_large_start","parent":7871,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences/index.html#consecutive-of-large-start","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/conditional-sylvester-schur-consequences","status":"fully_proved","summary":"Let $n \\ge 2$ and $k > n$ be natural numbers with $n! \\cdot 2^{n-1} < k$. Then some prime $p > n$ divides one of the integers $k, k+1, \\dots, k+n-1$.","title":"Large starts give a large prime in a consecutive block"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","title":"Block-level reductions for the Sylvester-Schur theorem"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/bertrand-base","kind":"node","lean":"Nat.SylvesterSchur.Internal.bertrand_base","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#bertrand-base","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"For every natural number $n \\ge 1$ there are $i < n$ and a prime $p > n$ such that $p$ divides $n + 1 + i$. Thus one of the $n$ consecutive integers $n + 1, \\dots, 2n$ has a prime divisor greater than $n$.","title":"Bertrand's postulate in block form"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/prime-dvd-seq-implies-choose","kind":"node","lean":"Nat.SylvesterSchur.Internal.prime_dvd_seq_implies_choose","parent":7884,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime divisors above "],["math","\\(n\\)"],["text"," of a block term divide the binomial coefficient"]],"source":"Prime divisors above $n$ of a block term divide the binomial coefficient"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#prime-dvd-seq-implies-choose","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n$, $k$ and $j$ be natural numbers with $j < n$, and let $r$ be a prime with $r > n$. If $r$ divides $k + j$, then $r$ divides $\\binom{k+n-1}{n}$.","title":"Prime divisors above $n$ of a block term divide the binomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/prime-dvd-choose-of-mem-interval","kind":"node","lean":"Nat.SylvesterSchur.Internal.prime_dvd_choose_of_mem_interval","parent":7884,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes above "],["math","\\(n\\)"],["text"," in the block divide the block binomial coefficient"]],"source":"Primes above $n$ in the block divide the block binomial coefficient"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#prime-dvd-choose-of-mem-interval","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers and let $p$ be a prime with $p > n$ and $k \\le p \\le k + n - 1$. Then $p$ divides $\\binom{k+n-1}{n}$.","title":"Primes above $n$ in the block divide the block binomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-large-prime-d","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_large_prime_dvd_block","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-large-prime-d","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers. If some prime $p > n$ divides $k + i$ for some $i < n$, then some prime $q > n$ divides $\\binom{k + n - 1}{n}$.","title":"Large prime divisors pass from a block to its binomial coefficient"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-dvd-block-of-succ-succ-prime-not","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_dvd_block_of_succ_succ_prime_not_dvd_pred","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-dvd-block-of-succ-succ-prime-not","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n \\ge 1$ and $k$ be natural numbers and let $s = n + 2$. If $s$ divides neither $k - 1$ nor $k - 2$ (natural-number differences, truncated at $0$), then $s$ divides $k + i$ for some $i < n$.","title":"A block term divisible by the block length plus two"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-large-start","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_large_start","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-large-start","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $2 \\le n < k$ be natural numbers with $n! \\cdot 2^{n-1} < k$. Then some prime $p > n$ divides $\\binom{k + n - 1}{n}$.","title":"Sylvester\u2013Schur criterion for blocks with a large start"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-half-bound","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_half_bound","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-half-bound","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers and put $N = k + n - 1$. If $n! \\cdot N^{\\lfloor n/2 \\rfloor + 1} < k^n$, then some prime $p > n$ divides $\\binom{N}{n}$.","title":"Sylvester\u2013Schur criterion from the half-exponent bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/start-le-quadratic-of-no-large-prime-dvd","kind":"node","lean":"Nat.SylvesterSchur.Internal.start_le_quadratic_of_no_large_prime_dvd_block","parent":7884,"reading_title":{"format":"inline-parts-v1","parts":[["text","Blocks without large prime divisors start at most at "],["math","\\(n^2\\)"]],"source":"Blocks without large prime divisors start at most at $n^2$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#start-le-quadratic-of-no-large-prime-dvd","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n \\ge 38$ and $k$ be natural numbers. If none of the $n$ consecutive integers $k, k+1, \\dots, k+n-1$ has a prime divisor greater than $n$, then $k \\le n^2$.","title":"Blocks without large prime divisors start at most at $n^2$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-prime-countin","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_primeCounting_bound","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-prime-countin","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers, put $N = k + n - 1$, and let $\\pi(n)$ be the number of primes at most $n$. If $n! \\cdot N^{\\pi(n)} < k^n$, then some prime $p > n$ divides $\\binom{N}{n}$.","title":"Sylvester\u2013Schur criterion from the prime-counting bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-sqrt-primoria","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_sqrt_primorial_bound","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-sqrt-primoria","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Large prime divisor of the block binomial coefficient from the square-root primorial bound. Let $n \\ge 1$ and $k > n$ be natural numbers, write $N = k + n - 1$, let $\\lfloor\\sqrt{N}\\rfloor$ be the integer square root of $N$ and let $P$ be the product of all primes at most $n$. If $n! \\cdot N^{\\lfloor\\sqrt{N}\\rfloor + 1} \\cdot P < k^n$, then some prime $p > n$ divides $\\binom{N}{n}$.","title":"Large prime divisor of the block binomial coefficient from the square-root primorial bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-erdos-layers","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_erdos_layers_bound","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-erdos-layers","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers, put $N = k + n - 1$ and $L = \\lfloor \\log_2 N \\rfloor$, and write $m\\#$ for the product of the primes at most $m$. If","title":"Sylvester\u2013Schur criterion from the Erd\u0151s layer bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-erdos-div-thr","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_erdos_div_three_layers_bound","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-erdos-div-thr","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n < k$ be natural numbers, put $N = k + n - 1$ and $L = \\lfloor \\log_2 N \\rfloor$, and assume $N \\ge 6$. Write $m\\#$ for the product of the primes at most $m$. If","title":"Sylvester\u2013Schur criterion from the Erd\u0151s layer bound with a one-third first layer"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-erdos-layers-2","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_erdos_layers_four_pow_bound","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-erdos-layers-2","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers and put $N = k + n - 1$. If","title":"Sylvester\u2013Schur criterion from the coarse power-of-four Erd\u0151s bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-erdos-layers-3","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_erdos_layers_three_layers_bound","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-erdos-layers-3","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $1 \\le n < k$ be natural numbers and put $N = k + n - 1$. If","title":"Sylvester\u2013Schur criterion from the three-layer power-of-four Erd\u0151s bound"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-smooth-count","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_smooth_count_bound","parent":7884,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-smooth-count","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/block-level-reductions-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers, let $\\pi(n)$ be the number of primes at most $n$, and let $\\lfloor\\sqrt{k+n-1}\\rfloor$ be the integer square root. If $2^{\\pi(n)} \\lfloor\\sqrt{k+n-1}\\rfloor < n$, then some prime $p > n$ divides $\\binom{k+n-1}{n}$.","title":"Large prime divisor of the block binomial coefficient from the smooth-number count"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t.json","id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t","kind":"scope","navigation":"module","parent":7819,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/index.html","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t","status":"fully_proved","title":"Finite residual checks for the Sylvester-Schur theorem"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-scaled-ternar","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_scaled_ternary_erdos_layers_bound","parent":7900,"reading_title":{"format":"inline-parts-v1","parts":[["text","A numerical criterion for a prime divisor of "],["math","\\(\\binom{N}{r}\\)"],["text"," above "],["math","\\(r\\)"],["text"," when "],["math","\\(N \\ge 3r\\)"]],"source":"A numerical criterion for a prime divisor of $\\binom{N}{r}$ above $r$ when $N \\ge 3r$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-scaled-ternar","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $N$ and $r$ be natural numbers with $r \\ge 4$ and $3r \\le N$, and let $e = r + \\lfloor \\sqrt{N} \\rfloor + \\lfloor N^{1/3} \\rfloor \\cdot \\lfloor \\log_2 N \\rfloor$. If $r \\cdot 2^r \\cdot (3r)^r \\cdot 4^e < 3^r \\cdot 4^r \\cdot N^r$, then some prime $p$ with $r < p$ divides $\\binom{N}{r}$.","title":"A numerical criterion for a prime divisor of $\\binom{N}{r}$ above $r$ when $N \\ge 3r$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/exists-prime-dvd-choose-of-central-offse","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_of_central_offset_erdos_div_three_layers_bound","parent":7900,"reading_title":{"format":"inline-parts-v1","parts":[["text","A numerical criterion for a prime divisor of "],["math","\\(\\binom{N}{r}\\)"],["text"," above "],["math","\\(r\\)"],["text"," when "],["math","\\(N = 2r + i\\)"]],"source":"A numerical criterion for a prime divisor of $\\binom{N}{r}$ above $r$ when $N = 2r + i$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-of-central-offse","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $r$, $i$ and $N$ be natural numbers with $r \\ge 4$, $i \\le r$, $N = 2r + i$ and $N \\ge 6$. For a natural number $n$ write $n\\#$ for the product of the primes $p \\le n$. Let $L = \\lfloor \\log_2 N \\rfloor$, let $a_0 = \\lfloor N/3 \\rfloor\\#$ and $a_j = \\lfloor N^{1/(j+1)} \\rfloor\\#$ for $j \\ge 1$, and put $P = \\prod_{j=0}^{L-1} a_j$. If $r \\cdot 2^i \\cdot P < 3^i \\cdot 4^r$, then some prime $p$ with $r < p$\u2026","title":"A numerical criterion for a prime divisor of $\\binom{N}{r}$ above $r$ when $N = 2r + i$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/exists-prime-dvd-choose-two-mul-add-two","kind":"node","lean":"Nat.SylvesterSchur.Internal.exists_prime_dvd_choose_two_mul_add_two_small","parent":7900,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prime divisor of "],["math","\\(\\binom{2n+2}{n}\\)"],["text"," above "],["math","\\(n\\)"],["text"," for small "],["math","\\(n\\)"]],"source":"A prime divisor of $\\binom{2n+2}{n}$ above $n$ for small $n$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/index.html#exists-prime-dvd-choose-two-mul-add-two","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t","status":"fully_proved","summary":"For every natural number $n$ with $7 \\le n < 512$, some prime $p$ with $n < p$ divides $\\binom{2n+2}{n}$.","title":"A prime divisor of $\\binom{2n+2}{n}$ above $n$ for small $n$"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/second-residual-tiny-has-large-prime-fac","kind":"node","lean":"Nat.SylvesterSchur.Internal.second_residual_tiny_has_large_prime_factor","parent":7900,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/index.html#second-residual-tiny-has-large-prime-fac","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $7 \\le n < 38$ and $n + 3 \\le k < 100$. Then there are a natural number $i < n$ and a prime $p > n$ such that $p$ divides $k + i$.","title":"A block term with a large prime factor, small range"},{"id":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/second-residual-mid-has-large-prime-fact","kind":"node","lean":"Nat.SylvesterSchur.Internal.second_residual_mid_has_large_prime_factor","parent":7900,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t/index.html#second-residual-mid-has-large-prime-fact","scope":"NumberTheory/distribution-of-primes/prime-gaps-short-intervals/sylvester-schur-theorem/finite-residual-checks-for-the-sylvester-schur-t","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $38 \\le n < 945$ and $n + 3 \\le k < 1291$. Then there are a natural number $i < n$ and a prime $p > n$ such that $p$ divides $k + i$.","title":"A block term with a large prime factor, middle range"},{"data_url":"scopes/NumberTheory/distribution-of-primes/primes-in-progressions.json","id":"NumberTheory/distribution-of-primes/primes-in-progressions","kind":"scope","parent":7411,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/index.html","scope":"NumberTheory/distribution-of-primes/primes-in-progressions","status":"can_prove","summary":"Dirichlet density, PNT for progressions, Siegel\u2013Walfisz, Bombieri\u2013Vinogradov, Brun\u2013Titchmarsh, prime races.","title":"Primes in arithmetic progressions"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/dirichlet-density-primes-arithmetic-prog","kind":"node","lean":"MetaMathlibExt.NumberTheory.PrimeCounting.dirichlet_density_primes_arithmetic_progression","parent":7906,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/index.html#dirichlet-density-primes-arithmetic-prog","scope":"NumberTheory/distribution-of-primes/primes-in-progressions","status":"fully_proved","summary":"Let $a$ and $m$ be natural numbers with $m > 0$ and $\\gcd(a, m) = 1$, and let $\\varphi$ be Euler's totient function. As the real number $s$ tends to $1$ from above,","title":"Dirichlet density of primes in an arithmetic progression"},{"data_url":"scopes/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes.json","id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","kind":"scope","navigation":"module","parent":7906,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","title":"Discrepancy of sequences in residue classes"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/discrepancy","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.discrepancy","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#discrepancy","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero, let $\\alpha : \\mathbb{N} \\to K$ be finitely supported, let $q$ be a positive integer and let $a$ be a natural number. The discrepancy of $\\alpha$ in the residue class of $a$ modulo $q$ is","title":"Discrepancy in a residue class"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/discrepancy-single-zero","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.discrepancy_single_zero","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#discrepancy-single-zero","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero, let $v \\in K$, let $q$ be a positive integer and let $a \\in \\mathbb{N}$. The sequence with value $v$ at index $0$ and $0$ at every other index has discrepancy $0$ at modulus $q$ and residue $a$.","title":"A sequence supported at index zero has zero discrepancy"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/discrepancy-single","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.discrepancy_single","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#discrepancy-single","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero, let $n$ be a positive integer, let $v \\in K$, let $q$ be a positive integer and let $a \\in \\mathbb{N}$. Let $e$ be the sequence with value $v$ at index $n$ and $0$ at every other index. Then the discrepancy of $e$ at modulus $q$ and residue $a$ equals $A - B/\\varphi(q)$, where $A = v$ if $n \\equiv a \\pmod{q}$ and $A = 0$ otherwise, $B = v$ if $\\gcd(n, q) = 1$ and $B = 0$\u2026","title":"Discrepancy of a single positive term"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/primitive-discrepancy","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.primitiveDiscrepancy","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#primitive-discrepancy","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"For a finitely supported sequence $\\alpha : \\mathbb{N} \\to \\mathbb{R}$ and a positive integer $q$, the primitive discrepancy of $\\alpha$ modulo $q$ is the maximum, over the residue classes $a$ modulo $q$ that are coprime to $q$, of $|\\Delta(\\alpha; a \\bmod q)|$, where $\\Delta(\\alpha; a \\bmod q)$ is the discrepancy of $\\alpha$ in the residue class of $a$ modulo $q$.","title":"Primitive discrepancy"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/primitive-discrepancy-attained","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.primitiveDiscrepancy_attained","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#primitive-discrepancy-attained","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"For every finitely supported sequence $\\alpha : \\mathbb{N} \\to \\mathbb{R}$ and every positive integer $q$, there is a residue class $a$ modulo $q$ coprime to $q$ such that the primitive discrepancy of $\\alpha$ modulo $q$ equals $|\\Delta(\\alpha; a \\bmod q)|$, the absolute value of the discrepancy of $\\alpha$ in the residue class of $a$ modulo $q$.","title":"The primitive discrepancy is attained"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/abs-discrepancy-le-primitive-discrepancy","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.abs_discrepancy_le_primitiveDiscrepancy","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#abs-discrepancy-le-primitive-discrepancy","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $\\alpha$ be a finitely supported real sequence indexed by $\\mathbb{N}$, let $q$ be a positive integer, and let $a$ be a unit of $\\mathbb{Z}/q\\mathbb{Z}$, represented by its least nonnegative residue. Then the absolute value of the discrepancy of $\\alpha$ at modulus $q$ and residue $a$ is at most the primitive discrepancy of $\\alpha$ at modulus $q$.","title":"Primitive residue-class discrepancies are bounded by the primitive discrepancy"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/averaged-discrepancy","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.averagedDiscrepancy","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#averaged-discrepancy","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"For a finitely supported real sequence $\\alpha$ indexed by $\\mathbb{N}$ and a natural number $Q$, the averaged discrepancy of $\\alpha$ up to $Q$ is the sum $\\sum_{q=1}^{Q} \\Delta^*(\\alpha; q)$, where $\\Delta^*(\\alpha; q)$ is the primitive discrepancy of $\\alpha$ at modulus $q$.","title":"Averaged discrepancy"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/averaged-discrepancy-zero","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.averagedDiscrepancy_zero","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#averaged-discrepancy-zero","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"For every finitely supported real sequence $\\alpha$ indexed by $\\mathbb{N}$, the averaged discrepancy of $\\alpha$ with cutoff $Q = 0$ equals $0$, since there are no positive moduli $q \\le 0$.","title":"The averaged discrepancy with cutoff zero vanishes"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/convolution","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.convolution","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#convolution","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $R$ be a semiring and let $\\alpha, \\beta \\colon \\mathbb{N} \\to R$ be finitely supported. Their convolution $\\alpha \\star \\beta$ is the finitely supported sequence whose value at $n$ is $\\sum_{a, b \\ge 1,\\ ab = n} \\alpha(a)\\beta(b)$; the values of $\\alpha$ and $\\beta$ at index $0$ are discarded.","title":"Multiplicative convolution of finitely supported sequences"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/convolution-apply","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.convolution_apply","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#convolution-apply","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $R$ be a semiring and let $\\alpha, \\beta \\colon \\mathbb{N} \\to R$ be finitely supported, with convolution $\\alpha \\star \\beta$. For every $n \\in \\mathbb{N}$,","title":"Coefficients of the multiplicative convolution"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/convolution-apply-zero","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.convolution_apply_zero","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#convolution-apply-zero","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $R$ be a semiring and let $\\alpha, \\beta \\colon \\mathbb{N} \\to R$ be finitely supported. Then their convolution $\\alpha \\star \\beta$ has value $0$ at index $0$.","title":"The multiplicative convolution vanishes at index zero"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/convolution-single-zero-left","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.convolution_single_zero_left","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#convolution-single-zero-left","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $R$ be a semiring, let $v \\in R$, and let $\\beta \\colon \\mathbb{N} \\to R$ be finitely supported. If $e$ is the sequence with value $v$ at index $0$ and $0$ at every other index, then the convolution $e \\star \\beta$ is the zero sequence.","title":"Convolving a sequence supported at zero gives zero"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/convolution-single-zero-right","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.convolution_single_zero_right","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#convolution-single-zero-right","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"Let $R$ be a semiring, let $\\alpha \\colon \\mathbb{N} \\to R$ be finitely supported, and let $v \\in R$. If $e$ is the sequence with value $v$ at index $0$ and $0$ at every other index, then the convolution $\\alpha \\star e$ is the zero sequence.","title":"Convolving with a sequence supported at zero gives zero"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/dyadic-von-mangoldt","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.dyadicVonMangoldt","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#dyadic-von-mangoldt","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"For a real number $x$, this is the finitely supported function on $\\mathbb{N}$ whose value at $n$ is $\\Lambda(n)$ when $n \\ge 1$ and $x \\le n \\le 2x$, and $0$ otherwise, where $\\Lambda$ is the von Mangoldt function.","title":"Von Mangoldt function on a dyadic interval"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/dyadic-von-mangoldt-apply","kind":"node","lean":"MathlibExt.NumberTheory.PrimeDistribution.dyadicVonMangoldt_apply","parent":7908,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes/index.html#dyadic-von-mangoldt-apply","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/discrepancy-of-sequences-in-residue-classes","status":"fully_proved","summary":"For every real $x > 0$ and every natural number $n$, the von Mangoldt function on the dyadic interval for $x$ takes at $n$ the value $\\Lambda(n)$ if $x \\le n \\le 2x$, and $0$ otherwise, where $\\Lambda$ is the von Mangoldt function.","title":"Values of the dyadic von Mangoldt function"},{"data_url":"scopes/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions.json","id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","kind":"scope","navigation":"module","parent":7906,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","title":"Prime number theorem for arithmetic progressions"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/prime-counting-mod","kind":"node","lean":"Nat.primeCountingMod","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#prime-counting-mod","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"For natural numbers $X$, $q$ and $a$, $\\pi(X; q, a)$ is the number of primes $p \\le X$ with $p \\equiv a \\pmod q$. The modulus $q = 1$ is allowed, and for $q = 0$ the congruence means $p = a$.","title":"Prime counting in a residue class"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/prime-counting-mod-one","kind":"node","lean":"Nat.primeCountingMod_one","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#prime-counting-mod-one","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"For all natural numbers $X$ and $a$, the number $\\pi(X; 1, a)$ of primes $p \\le X$ with $p \\equiv a \\pmod 1$ equals $\\pi(X)$, the number of primes $p \\le X$.","title":"Prime counting modulo one"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/prime-counting-mod-eq-card-filter-range","kind":"node","lean":"Nat.primeCountingMod_eq_card_filter_range","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#prime-counting-mod-eq-card-filter-range","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"For all natural numbers $n$, $q$ and $a$, the number $\\pi(n; q, a)$ of primes $p \\le n$ with $p \\equiv a \\pmod q$ equals the number of integers $p$ with $0 \\le p \\le n$ such that $p$ is prime and $p \\equiv a \\pmod q$.","title":"Residue-class prime count as a filtered range"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/psi-residue-class","kind":"node","lean":"Chebyshev.psiResidueClass","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#psi-residue-class","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"For a natural number $q$, a residue class $a \\in \\mathbb{Z}/q\\mathbb{Z}$ and a real number $x$, $\\psi(x; q, a)$ is the sum of the von Mangoldt function $\\Lambda(n)$ over the positive integers $n \\le x$ with $n \\equiv a \\pmod q$.","title":"Chebyshev's function in a residue class"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/psi-residue-class-of-lt-one","kind":"node","lean":"Chebyshev.psiResidueClass_of_lt_one","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#psi-residue-class-of-lt-one","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"For every natural number $q$, every residue class $a \\in \\mathbb{Z}/q\\mathbb{Z}$ and every real number $x < 1$, we have $\\psi(x; q, a) = 0$, where $\\psi(x; q, a)$ is the sum of the von Mangoldt function $\\Lambda(n)$ over the positive integers $n \\le x$ with $n \\equiv a \\pmod q$.","title":"The residue-class Chebyshev function vanishes below one"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/tendsto-psi-residue-class-div-self","kind":"node","lean":"Chebyshev.tendsto_psiResidueClass_div_self","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#tendsto-psi-residue-class-div-self","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"Let $q \\ge 1$ be an integer, let $b$ be a unit of $\\mathbb{Z}/q\\mathbb{Z}$, let $\\Lambda$ be the von Mangoldt function and let $\\varphi$ be Euler's totient function. For real $x$ let $\\psi(x; q, b)$ be the sum of $\\Lambda(n)$ over the integers $1 \\le n \\le x$ with $n \\equiv b \\pmod{q}$. Then $\\psi(x; q, b)/x \\to 1/\\varphi(q)$ as the real number $x$ tends to infinity.","title":"Prime number theorem for Chebyshev's function in a reduced residue class"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/exists-abs-sum-residue-class-div-sub-log","kind":"node","lean":"Chebyshev.exists_abs_sum_residueClass_div_sub_log_le","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#exists-abs-sum-residue-class-div-sub-log","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"Let $q \\ge 1$ be an integer, let $b$ be a unit of $\\mathbb{Z}/q\\mathbb{Z}$, let $\\Lambda$ be the von Mangoldt function and let $\\varphi$ be Euler's totient function. There is a real constant $C$ such that for every real $x \\ge 1$,","title":"Mertens' theorem in arithmetic progressions"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/weak-pnt-arithmetic-progression","kind":"node","lean":"Chebyshev.weakPNT_arithmeticProgression","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#weak-pnt-arithmetic-progression","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"Let $q$ and $a$ be natural numbers with $a < q$ and $\\gcd(a, q) = 1$, let $\\Lambda$ be the von Mangoldt function and let $\\varphi$ be Euler's totient function. Then, as $N \\to \\infty$ through the natural numbers,","title":"Weak prime number theorem in arithmetic progressions"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/prime-number-theorem-arithmetic-progress","kind":"node","lean":"MathlibExt.NumberTheory.PrimeNumberTheoremAPWanted.prime_number_theorem_arithmetic_progression","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#prime-number-theorem-arithmetic-progress","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"Let $q \\ge 2$ and $a$ be natural numbers with $\\gcd(a, q) = 1$, and for each natural number $n$ let $N(n)$ be the number of primes $p \\le n$ with $p \\equiv a \\pmod q$. Then $N(n) \\log n / n \\to 1 / \\varphi(q)$ as $n \\to \\infty$, where $\\varphi$ is Euler's totient function and $\\log$ is the natural logarithm.","title":"Prime number theorem in arithmetic progressions"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/tendsto-prime-counting-mod-mul-log-div","kind":"node","lean":"MathlibExt.NumberTheory.PrimeNumberTheoremAPWanted.tendsto_primeCountingMod_mul_log_div","parent":7924,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions/index.html#tendsto-prime-counting-mod-mul-log-div","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/prime-number-theorem-for-arithmetic-progressions","status":"fully_proved","summary":"Let $q \\ge 2$ and $a$ be natural numbers with $\\gcd(a, q) = 1$, and let $\\pi(n; q, a)$ be the number of primes $p \\le n$ with $p \\equiv a \\pmod q$. Then $\\pi(n; q, a) \\log n / n \\to 1 / \\varphi(q)$ as $n \\to \\infty$, where $\\varphi$ is Euler's totient function and $\\log$ is the natural logarithm.","title":"Prime number theorem for the residue-class prime count"},{"data_url":"scopes/NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem.json","id":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem","kind":"scope","navigation":"module","parent":7906,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/index.html","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem","status":"can_prove","title":"Bombieri\u2013Vinogradov theorem"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/bombieri-vinogradov-residue-error","kind":"node","lean":"MetaMathlibExt.NumberTheory.PrimeCounting.bombieriVinogradovResidueError","parent":7935,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/index.html#bombieri-vinogradov-residue-error","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem","status":"fully_proved","summary":"For natural numbers $y$, $q$ and $a$, the residue-class error is $E(y; q, a) = \\left|\\psi(y; q, a) - y/\\varphi(q)\\right|$, where $\\psi(y; q, a)$ is Chebyshev's function in the residue class of $a$ modulo $q$ and $\\varphi$ is Euler's totient function.","title":"Residue-class error of Chebyshev's function"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/bombieri-vinogradov-max-residue-error","kind":"node","lean":"MetaMathlibExt.NumberTheory.PrimeCounting.bombieriVinogradovMaxResidueError","parent":7935,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/index.html#bombieri-vinogradov-max-residue-error","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem","status":"fully_proved","summary":"For natural numbers $y$ and $q$, $E^{\\ast}(y; q)$ is the maximum of the residue-class error $E(y; q, a)$ over the natural numbers $a < q$ coprime to $q$, and $0$ if there is no such $a$.","title":"Maximal residue-class error"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/bombieri-vinogradov-max-up-to-error","kind":"node","lean":"MetaMathlibExt.NumberTheory.PrimeCounting.bombieriVinogradovMaxUpToError","parent":7935,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/index.html#bombieri-vinogradov-max-up-to-error","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem","status":"fully_proved","summary":"For natural numbers $x$ and $q$, $E^{\\ast\\ast}(x; q)$ is the maximum of the maximal residue-class error $E^{\\ast}(y; q)$ over the natural numbers $y \\le x$.","title":"Maximal residue-class error up to a level"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/bombieri-vinogradov-sum","kind":"node","lean":"MetaMathlibExt.NumberTheory.PrimeCounting.bombieriVinogradovSum","parent":7935,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/index.html#bombieri-vinogradov-sum","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem","status":"fully_proved","summary":"For a natural number $x$ and a real number $A$, the Bombieri\u2013Vinogradov sum $S(x, A)$ is the sum of the maximal residue-class error $E^{\\ast\\ast}(x; q)$ up to $x$ over the integers $q$ with $1 \\le q \\le x$ and $q \\le \\sqrt{x} / (\\log x)^{A + 5}$.","title":"Bombieri\u2013Vinogradov sum"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/bombieri-vinogradov","kind":"node","lean":"MetaMathlibExt.NumberTheory.PrimeCounting.bombieriVinogradov","parent":7935,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem/index.html#bombieri-vinogradov","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/bombieri-vinogradov-theorem","status":"can_prove","summary":"Let $\\psi(y; q, a)$ denote Chebyshev's function in the residue class of $a$ modulo $q$, and let $\\varphi$ be Euler's totient function. For a natural number $x$ and a real number $A$, let $S(x, A)$ be the sum, over the integers $q$ with $1 \\le q \\le x$ and $q \\le \\sqrt{x} / (\\log x)^{A + 5}$, of the maximum of $\\left|\\psi(y; q, a) - y/\\varphi(q)\\right|$ over the natural numbers $y \\le x$ and $a < q$ coprime to $q$.\u2026","title":"Bombieri\u2013Vinogradov theorem"},{"data_url":"scopes/NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem.json","id":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem","kind":"scope","navigation":"module","parent":7906,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/index.html","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem","status":"fully_proved","title":"Brun\u2013Titchmarsh theorem"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/brun-titchmarsh-uniform","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.BrunTitchmarshUniformWanted.brun_titchmarsh_uniform","parent":7941,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/index.html#brun-titchmarsh-uniform","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem","status":"fully_proved","summary":"Write $\\pi(X; k, \\ell)$ for the number of primes $p \\le X$ with $p \\equiv \\ell \\pmod{k}$, and $\\varphi$ for Euler's totient function. For every real $\\beta$ with $0 < \\beta < 1$ there is a real constant $\\xi_1 > 0$ such that for all natural numbers $X$, $k$ and $\\ell$ with $X \\ge 2$, $k \\ge 1$ and $k \\le X^{\\beta}$,","title":"Uniform Brun\u2013Titchmarsh bound"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/moebius-sq-div-totient-sum-gt-log","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.moebiusSq_div_totient_sum_gt_log","parent":7941,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/index.html#moebius-sq-div-totient-sum-gt-log","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem","status":"fully_proved","summary":"For every natural number $z \\ge 6$, $\\log(z + 1) + 107/100 < \\sum_{r=1}^{z} \\mu(r)^2 / \\varphi(r)$, where $\\mu$ is the M\u00f6bius function and $\\varphi$ is Euler's totient function.","title":"Lower bound for the squarefree totient sum"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/squarefree-totient-weighted-lower","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.squarefreeTotientWeightedLower","parent":7941,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/index.html#squarefree-totient-weighted-lower","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem","status":"fully_proved","summary":"For every natural number $z \\ge 100$, $\\log z + 36/100 < \\sum_{q=1}^{z} (1 + q/z)^{-1} \\, \\mu(q)^2 / \\varphi(q)$, where $\\mu$ is the M\u00f6bius function and $\\varphi$ is Euler's totient function.","title":"Weighted lower bound for the squarefree totient sum"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/brun-titchmarsh","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.BrunTitchmarshWanted.brun_titchmarsh","parent":7941,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem/index.html#brun-titchmarsh","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/brun-titchmarsh-theorem","status":"fully_proved","summary":"Let $X$, $q$ and $a$ be natural numbers such that $a$ and $q$ are coprime and $1 \\le q < X$. Write $\\pi(X; q, a)$ for the number of primes $p \\le X$ with $p \\equiv a \\pmod{q}$, and $\\varphi$ for Euler's totient function. Then","title":"Brun\u2013Titchmarsh inequality"},{"data_url":"scopes/NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias.json","id":"NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias","kind":"scope","navigation":"module","parent":7906,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias/index.html","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias","status":"can_prove","title":"Chebyshev's bias"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias/chebyshev-lead","kind":"node","lean":"MathlibExt.NumberTheory.ChebyshevSignChangeWanted.NumberTheory.ChebyshevSignChangeProblem.chebyshevLead","parent":7946,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set where primes "],["math","\\(\\equiv 3 \\pmod 4\\)"],["text"," lead"]],"source":"The set where primes $\\equiv 3 \\pmod 4$ lead"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias/index.html#chebyshev-lead","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias","status":"fully_proved","summary":"This is the set of natural numbers $n$ for which the number of primes $p \\le n$ with $p \\equiv 3 \\pmod 4$ is strictly greater than the number of primes $p \\le n$ with $p \\equiv 1 \\pmod 4$.","title":"The set where primes $\\equiv 3 \\pmod 4$ lead"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias/chebyshev-sign-change","kind":"node","lean":"MathlibExt.NumberTheory.ChebyshevSignChangeWanted.NumberTheory.ChebyshevSignChangeProblem.chebyshev_sign_change","parent":7946,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias/index.html#chebyshev-sign-change","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/chebyshev-s-bias","status":"can_prove","summary":"There are infinitely many natural numbers $n$ for which the primes $p \\le n$ with $p \\equiv 3 \\pmod 4$ strictly outnumber the primes $p \\le n$ with $p \\equiv 1 \\pmod 4$, and infinitely many natural numbers $n$ for which the primes $p \\le n$ with $p \\equiv 1 \\pmod 4$ strictly outnumber the primes $p \\le n$ with $p \\equiv 3 \\pmod 4$.","title":"Hardy\u2013Littlewood sign changes in the prime race modulo 4"},{"data_url":"scopes/NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem.json","id":"NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem","kind":"scope","navigation":"module","parent":7906,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem/index.html","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem","status":"can_prove","title":"Siegel\u2013Walfisz theorem"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem/siegel-walfisz","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.SiegelWalfiszWanted.siegel_walfisz","parent":7949,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem/index.html#siegel-walfisz","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem","status":"can_prove","summary":"For every real $N > 0$ there are real constants $c > 0$, $C > 0$ and $X_0 > 1$ such that for every real $x \\ge X_0$, every positive integer $q$ with $q \\le (\\log x)^N$ and every unit $a$ of $\\mathbb{Z}/q\\mathbb{Z}$,","title":"Siegel\u2013Walfisz theorem for Chebyshev's function in residue classes"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem/uniform-prime-counting-arithmetic-progre","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.UniformPrimeCountingWanted.uniform_prime_counting_arithmetic_progression","parent":7949,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem/index.html#uniform-prime-counting-arithmetic-progre","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/siegel-walfisz-theorem","status":"can_prove","summary":"Let $\\varphi$ be Euler's totient function and let $\\mathrm{Li}(x) = \\int_2^x du / \\log u$. For every real $B > 0$ there are real constants $c > 0$, $C > 0$ and $X_0 \\ge 2$ such that for every real $x \\ge X_0$ and all natural numbers $q > 0$ and $b$ with $\\gcd(b, q) = 1$ and $q \\le (\\log x)^B$, the number $\\pi(x; q, b)$ of primes $p \\le x$ with $p \\equiv b \\pmod q$ satisfies","title":"Siegel\u2013Walfisz theorem for primes in arithmetic progressions"},{"data_url":"scopes/NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes.json","id":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes","kind":"scope","navigation":"module","parent":7906,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/index.html","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes","status":"can_prove","title":"Wallis-type prime products in residue classes"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/wallis-prime-factor-eq","kind":"node","lean":"MetaMathlibExt.wallis_prime_factor_eq","parent":7952,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/index.html#wallis-prime-factor-eq","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes","status":"can_prove","summary":"For every odd prime $p$,","title":"Factor identity for Wallis-type prime products"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/thinned-wallis-prime-product-asymptotic","kind":"node","lean":"MetaMathlibExt.thinned_wallis_prime_product_asymptotic","parent":7952,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/index.html#thinned-wallis-prime-product-asymptotic","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes","status":"can_prove","summary":"Let $m \\ge 2$, let $q = 2^m$, and let $S$ be a set of units of $\\mathbb{Z}/q\\mathbb{Z}$. Let $\\varphi$ be Euler's totient function and $\\chi_4$ the nontrivial Dirichlet character modulo $4$, put $\\mu(S) = \\sum_{a \\in S} \\chi_4(a)$, where $\\chi_4(a)$ means $\\chi_4$ of the reduction of $a$ modulo $4$, and for real $x$ let","title":"Asymptotic for thinned Wallis-type prime products"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/thinned-wallis-prime-product-tendsto-pos","kind":"node","lean":"MetaMathlibExt.thinned_wallis_prime_product_tendsto_pos_iff","parent":7952,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/index.html#thinned-wallis-prime-product-tendsto-pos","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes","status":"can_prove","summary":"Let $m \\ge 2$, let $q = 2^m$, and let $S$ be a set of units of $\\mathbb{Z}/q\\mathbb{Z}$. Let $\\chi_4$ be the nontrivial Dirichlet character modulo $4$, put $\\mu(S) = \\sum_{a \\in S} \\chi_4(a)$, where $\\chi_4(a)$ means $\\chi_4$ of the reduction of $a$ modulo $4$, and for real $x$ let","title":"Convergence of thinned Wallis-type prime products"},{"id":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/thinned-wallis-prime-product-constant","kind":"node","lean":"MetaMathlibExt.thinned_wallis_prime_product_constant","parent":7952,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes/index.html#thinned-wallis-prime-product-constant","scope":"NumberTheory/distribution-of-primes/primes-in-progressions/wallis-type-prime-products-in-residue-classes","status":"can_prove","summary":"Let $m \\ge 2$, let $q = 2^m$, and let $S$ be a set of units of $\\mathbb{Z}/q\\mathbb{Z}$. Let $\\varphi$ be Euler's totient function and $\\chi_4$ the nontrivial Dirichlet character modulo $4$, put $\\mu(S) = \\sum_{a \\in S} \\chi_4(a)$, where $\\chi_4(a)$ means $\\chi_4$ of the reduction of $a$ modulo $4$, and for real $x$ let","title":"The constant in the thinned Wallis-type prime product asymptotic"},{"data_url":"scopes/NumberTheory/distribution-of-primes/sieve-methods.json","id":"NumberTheory/distribution-of-primes/sieve-methods","kind":"scope","parent":7411,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/index.html","scope":"NumberTheory/distribution-of-primes/sieve-methods","status":"can_prove","summary":"Selberg, Brun and large sieves, almost primes, parity problem, counting integers with restricted prime factors via sieves.","title":"Sieve methods"},{"data_url":"scopes/NumberTheory/distribution-of-primes/sieve-methods/almost-primes.json","id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes","kind":"scope","navigation":"module","parent":7957,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes/index.html","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes","status":"fully_proved","title":"Almost primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes/almost-prime-counting","kind":"node","lean":"MetaMathlibExt.almostPrimeCounting","parent":7958,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes/index.html#almost-prime-counting","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes","status":"fully_proved","summary":"For natural numbers $k$ and $x$, $\\pi_k(x)$ is the number of integers $n$ with $1 \\le n \\le x$ that have exactly $k$ prime factors, counted with multiplicity. This is the counting function in Landau's theorem on almost primes.","title":"Counting almost primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes/landau-almost-prime-counting-is-equivale","kind":"node","lean":"MetaMathlibExt.landau_almostPrimeCounting_isEquivalent","parent":7958,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes/index.html#landau-almost-prime-counting-is-equivale","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes","status":"fully_proved","summary":"Let $k \\ge 1$ be a natural number and, for a natural number $x$, let $\\pi_k(x)$ be the number of integers $n$ with $1 \\le n \\le x$ that have exactly $k$ prime factors, counted with multiplicity. Then, as the natural number $x$ tends to infinity,","title":"Landau's theorem on almost primes"},{"data_url":"scopes/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index.json","id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","kind":"scope","navigation":"module","parent":7957,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","title":"Almost primes of almost prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/almost-prime-index-set","kind":"node","lean":"MetaMathlibExt.almostPrimeIndexSet","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#almost-prime-index-set","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"fully_proved","summary":"For natural numbers $k_1, \\dots, k_m$ with $m \\ge 1$, the set $N_{k_1 \\cdots k_m}$ of natural numbers is defined by recursion on $m$: $N_{k_1}$ is the set of $k_1$-almost primes (the $n \\ne 0$ with exactly $k_1$ prime factors counted with multiplicity), and for $m \\ge 2$, $N_{k_1 \\cdots k_m}$ is the set of the $i$th $k_1$-almost primes, in increasing order and counted from $1$, for $i \\ge 1$ in\u2026","title":"Almost primes of almost prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/almost-prime-index-set-nil","kind":"node","lean":"MetaMathlibExt.almostPrimeIndexSet_nil","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#almost-prime-index-set-nil","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"fully_proved","summary":"For every natural number $k$, the almost prime index set $N_k$ with the single index $k$ is the set of $k$-almost primes, the natural numbers $n \\ne 0$ with exactly $k$ prime factors counted with multiplicity.","title":"Almost prime index sets with one index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/mem-almost-prime-index-set-cons","kind":"node","lean":"MetaMathlibExt.mem_almostPrimeIndexSet_cons","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#mem-almost-prime-index-set-cons","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"fully_proved","summary":"For natural numbers $k$, $j$, $n$ and $\\ell_1, \\dots, \\ell_r$ (with $r \\ge 0$), $n$ lies in the almost prime index set $N_{k j \\ell_1 \\cdots \\ell_r}$ if and only if there is an $i \\ge 1$ in $N_{j \\ell_1 \\cdots \\ell_r}$ such that $n$ is the $i$th $k$-almost prime in increasing order, counted from $1$ (taken to be $0$ if there are fewer than $i$ of them).","title":"Membership in an almost prime index set"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/nth-prime-dusart-bounds","kind":"node","lean":"MetaMathlibExt.nth_prime_dusart_bounds","parent":7961,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dusart's bounds for the "],["math","\\(n\\)"],["text","th prime"]],"source":"Dusart's bounds for the $n$th prime"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#nth-prime-dusart-bounds","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"For $n \\ge 1$ let $p_n$ denote the $n$th prime, so that $p_1 = 2$, and let $\\log$ be the natural logarithm. For every natural number $n \\ge 2$,","title":"Dusart's bounds for the $n$th prime"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/summable-inv-almost-prime-index-set","kind":"node","lean":"MetaMathlibExt.summable_inv_almostPrimeIndexSet","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#summable-inv-almost-prime-index-set","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A $k$-almost prime is a positive integer with exactly $k$ prime factors counted with multiplicity. For positive integers $k_1, \\dots, k_m$ let $N_{k_1}$ be the set of $k_1$-almost primes and, for $m \\ge 2$, let $N_{k_1 k_2 \\cdots k_m}$ be the set of $k_1$-almost primes whose index in the increasing enumeration of the $k_1$-almost primes (starting at $1$) lies in $N_{k_2 \\cdots k_m}$. Then for every $m \\ge 2$ and\u2026","title":"Convergence of reciprocal sums over almost primes of almost prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/tsum-inv-primes-of-semiprime-index-bound","kind":"node","lean":"MetaMathlibExt.tsum_inv_primes_of_semiprime_index_bounds","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#tsum-inv-primes-of-semiprime-index-bound","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"Let $N_{1,2}$ be the set of primes $p_i$ (indexed from $p_1 = 2$) whose index $i$ is a semiprime, that is, a product of two not necessarily distinct primes. Then $0.9910 < \\sum_{n \\in N_{1,2}} 1/n < 0.9915$.","title":"Reciprocal sum of the primes of semiprime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/semiprime-count-asymptotic-expansion","kind":"node","lean":"MetaMathlibExt.semiprime_count_asymptotic_expansion","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#semiprime-count-asymptotic-expansion","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A semiprime is a product of exactly two primes counted with multiplicity; for real $x$ let $N_2(x)$ be the number of semiprimes at most $x$. Let $\\beta$ be the Meissel\u2013Mertens constant, $\\gamma$ the Euler\u2013Mascheroni constant and $\\log$ the natural logarithm. There are real constants $c_0, c_1, c_2, \\dots$ with $c_0 = \\beta$ and","title":"Cri\u015fan\u2013Erban expansion of the semiprime counting function"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/semiprime-reciprocal-sum-explicit-estima","kind":"node","lean":"MetaMathlibExt.semiprime_reciprocal_sum_explicit_estimate","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#semiprime-reciprocal-sum-explicit-estima","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A semiprime is a product of exactly two primes counted with multiplicity; for real $x$ let $S(x)$ be the sum of $1/n$ over the semiprimes $n \\le x$. Let $\\beta$ be the Meissel\u2013Mertens constant, $\\gamma$ the Euler\u2013Mascheroni constant and $\\log$ the natural logarithm, and put","title":"Explicit estimate for the reciprocal sum of the semiprimes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/almost-prime-count-le-erdos-sarkozy","kind":"node","lean":"MetaMathlibExt.almostPrime_count_le_erdos_sarkozy","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#almost-prime-count-le-erdos-sarkozy","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"There is a real constant $C$ such that for every integer $k \\ge 1$ and every real $t \\ge 1$, the number $\\pi_{k+1}(t)$ of integers $n$ with $1 \\le n \\le t$ that have exactly $k + 1$ prime factors, counted with multiplicity, satisfies $\\pi_{k+1}(t) \\le C \\cdot \\frac{k^4}{2^k} \\, t \\log t$.","title":"Erd\u0151s\u2013S\u00e1rk\u00f6zy bound for almost primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/log-prime-counting-asymptotics","kind":"node","lean":"MetaMathlibExt.log_primeCounting_asymptotics","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#log-prime-counting-asymptotics","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"Let $\\pi(x)$ be the number of primes $p \\le x$. As $x \\to \\infty$,","title":"Logarithmic expansions of the prime counting function"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/inv-log-semiprime-count-asymptotic","kind":"node","lean":"MetaMathlibExt.inv_log_semiprime_count_asymptotic","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#inv-log-semiprime-count-asymptotic","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"Let $\\pi_2(x)$ be the number of integers $n$ with $1 \\le n \\le x$ that have exactly two prime factors counted with multiplicity, and let $\\beta$ be the Meissel\u2013Mertens constant. Then, as $x \\to \\infty$,","title":"Expansion of the reciprocal logarithm of the semiprime count"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/primes-of-semiprime-index-count-asymptot","kind":"node","lean":"MetaMathlibExt.primes_of_semiprime_index_count_asymptotic","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#primes-of-semiprime-index-count-asymptot","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A semiprime is a product of exactly two primes counted with multiplicity. Let $p_i$ denote the $i$th prime ($p_1 = 2$), and for real $x$ let $N_{1,2}(x)$ be the number of primes $p_i \\le x$ whose index $i$ is a semiprime. Let $\\beta$ be the Meissel\u2013Mertens constant, $\\gamma$ the Euler\u2013Mascheroni constant, $\\log$ the natural logarithm, and","title":"Counting primes of semiprime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/semiprimes-of-prime-index-count-asymptot","kind":"node","lean":"MetaMathlibExt.semiprimes_of_prime_index_count_asymptotic","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#semiprimes-of-prime-index-count-asymptot","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A semiprime is a product of exactly two primes counted with multiplicity. Let $s_i$ denote the $i$th semiprime ($s_1 = 4$), and for real $x$ let $N_{2,1}(x)$ be the number of semiprimes $s_i \\le x$ whose index $i$ is prime. Let $\\beta$ be the Meissel\u2013Mertens constant and $\\log$ the natural logarithm. Then, as $x \\to \\infty$,","title":"Counting semiprimes of prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/primes-of-semiprime-index-count-sub-semi","kind":"node","lean":"MetaMathlibExt.primes_of_semiprime_index_count_sub_semiprimes_of_prime_index","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#primes-of-semiprime-index-count-sub-semi","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A semiprime is a product of exactly two primes counted with multiplicity. Let $p_i$ denote the $i$th prime ($p_1 = 2$) and $s_i$ the $i$th semiprime ($s_1 = 4$). For real $x$ let $N_{1,2}(x)$ be the number of primes $p_i \\le x$ whose index $i$ is a semiprime, and $N_{2,1}(x)$ the number of semiprimes $s_i \\le x$ whose index $i$ is prime. Then $N_{2,1}(x) < N_{1,2}(x)$ for all sufficiently large $x$, and, with\u2026","title":"Primes of semiprime index eventually outnumber semiprimes of prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/almost-prime-index-set-pair-count-asympt","kind":"node","lean":"MetaMathlibExt.almostPrimeIndexSet_pair_count_asymptotic","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#almost-prime-index-set-pair-count-asympt","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"Let $j, k \\ge 1$ be integers and, for real $x$, let $N_{jk}(x)$ be the number of $j$-almost primes $n \\le x$ whose index in the increasing enumeration of the $j$-almost primes, counted from $1$, is a $k$-almost prime. Writing $M(x) = \\frac{x \\, (\\log \\log x)^{j+k-2}}{(\\log x)^2 \\, (j-1)! \\, (k-1)!}$, we have $N_{jk}(x) - M(x) = O(M(x) / \\log \\log x)$ as $x \\to \\infty$.","title":"Counting almost primes of almost prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/almost-prime-index-set-count-asymptotic","kind":"node","lean":"MetaMathlibExt.almostPrimeIndexSet_count_asymptotic","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#almost-prime-index-set-count-asymptotic","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"Let $m \\ge 1$, let $k_1, \\dots, k_m$ be positive integers, let $N_{k_1 \\cdots k_m}$ be the almost prime index set (for $m = 1$ the $k_1$-almost primes, and for $m \\ge 2$ the $k_1$-almost primes whose index in increasing order, counted from $1$, lies in $N_{k_2 \\cdots k_m}$), and for real $x$ let $N_{k_1 \\cdots k_m}(x)$ be the number of its elements $n \\le x$. Writing\u2026","title":"Counting almost primes of iterated almost prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/nth-almost-prime-index-set-pair-asymptot","kind":"node","lean":"MetaMathlibExt.nth_almostPrimeIndexSet_pair_asymptotic","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#nth-almost-prime-index-set-pair-asymptot","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"Let $j, k \\ge 1$ be integers and, for $n \\ge 1$, let $p_{j,k,n}$ be the $n$th element, in increasing order, of the set $N_{jk}$ of $j$-almost primes whose index in the increasing enumeration of the $j$-almost primes, counted from $1$, is a $k$-almost prime. Writing $P(n) = \\frac{n (\\log n)^2 \\, (j-1)! \\, (k-1)!}{(\\log \\log n)^{j+k-2}}$, we have $p_{j,k,n} - P(n) = O(P(n) / \\log \\log n)$ as $n \\to \\infty$ through\u2026","title":"Size of the nth almost prime of almost prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/nth-almost-prime-index-set-asymptotic","kind":"node","lean":"MetaMathlibExt.nth_almostPrimeIndexSet_asymptotic","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#nth-almost-prime-index-set-asymptotic","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"Let $m \\ge 1$, let $k_1, \\dots, k_m$ be positive integers, and for $n \\ge 1$ let $p_{k_1, \\dots, k_m, n}$ be the $n$th element, in increasing order, of the almost prime index set $N_{k_1 \\cdots k_m}$. Writing $P(n) = n (\\log n)^m \\prod_{i=1}^m \\frac{(k_i - 1)!}{(\\log \\log n)^{k_i - 1}}$, we have $p_{k_1, \\dots, k_m, n} - P(n) = O(P(n) / \\log \\log n)$ as $n \\to \\infty$ through the integers.","title":"Size of the nth element of an iterated almost prime index set"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/tsum-inv-primes-of-almost-prime-index-su","kind":"node","lean":"MetaMathlibExt.tsum_inv_primes_of_almostPrime_index_sub_one_isBigO","parent":7961,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reciprocal sum of the primes of "],["math","\\(k\\)"],["text","-almost prime index"]],"source":"Reciprocal sum of the primes of $k$-almost prime index"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#tsum-inv-primes-of-almost-prime-index-su","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"For $k \\ge 0$ let $N_{1k}$ be the set of primes $p_i$ (indexed from $p_1 = 2$) whose index $i$ is a $k$-almost prime, that is, has exactly $k$ prime factors counted with multiplicity. Then $\\sum_{n \\in N_{1k}} 1/n = 1 + O(e^{-k/2})$ as $k \\to \\infty$.","title":"Reciprocal sum of the primes of $k$-almost prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/tsum-inv-almost-prime-index-set-pair-low","kind":"node","lean":"MetaMathlibExt.tsum_inv_almostPrimeIndexSet_pair_lower_bound","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#tsum-inv-almost-prime-index-set-pair-low","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A $k$-almost prime is a positive integer with exactly $k$ prime factors counted with multiplicity. For positive integers $j$ and $k$ let $N_{jk}$ be the set of $j$-almost primes whose index in the increasing enumeration of the $j$-almost primes (starting at $1$) is a $k$-almost prime. Fix an integer $j \\ge 1$. There is a real constant $c > 0$ such that","title":"Lower bound for reciprocal sums over almost primes of almost prime index"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/tsum-inv-almost-prime-index-set-triple-i","kind":"node","lean":"MetaMathlibExt.tsum_inv_almostPrimeIndexSet_triple_isBigO","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#tsum-inv-almost-prime-index-set-triple-i","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A $k$-almost prime is a positive integer with exactly $k$ prime factors counted with multiplicity. For positive integers $a$ and $b$ let $N_{ab}$ be the set of $a$-almost primes whose index in the increasing enumeration of the $a$-almost primes (starting at $1$) is a $b$-almost prime, and for positive integers $k_1, k_2, k$ let $N_{k_1 k_2 k}$ be the set of $k_1$-almost primes whose index in the increasing\u2026","title":"Exponential decay of reciprocal sums over triple almost prime index sets"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/tendsto-tsum-inv-almost-prime-index-set","kind":"node","lean":"MetaMathlibExt.tendsto_tsum_inv_almostPrimeIndexSet_append_zero","parent":7961,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index/index.html#tendsto-tsum-inv-almost-prime-index-set","scope":"NumberTheory/distribution-of-primes/sieve-methods/almost-primes-of-almost-prime-index","status":"can_prove","summary":"A $k$-almost prime is a positive integer with exactly $k$ prime factors counted with multiplicity. For positive integers $k_1, \\dots, k_m$ let $N_{k_1}$ be the set of $k_1$-almost primes and, for $m \\ge 2$, let $N_{k_1 k_2 \\cdots k_m}$ be the set of $k_1$-almost primes whose index in the increasing enumeration of the $k_1$-almost primes (starting at $1$) lies in $N_{k_2 \\cdots k_m}$. Fix $m \\ge 2$ and positive\u2026","title":"Reciprocal sums over iterated almost prime index sets tend to zero"},{"data_url":"scopes/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim.json","id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","kind":"scope","navigation":"module","parent":7957,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","title":"Counting integers in residue classes modulo primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/ones-diag-det","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.onesDiagDet","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#ones-diag-det","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"fully_proved","summary":"For natural numbers $p_1, \\dots, p_k$, the determinant $A_k(p_1, \\dots, p_k)$ is the determinant of the $k \\times k$ integer matrix with $p_1, \\dots, p_k$ on the diagonal and $1$ in every other entry.","title":"Ones-diagonal determinant"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/bordered-ones-diag","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.borderedOnesDiag","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#bordered-ones-diag","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"fully_proved","summary":"For natural numbers $p_1, \\dots, p_k$, the $(k+1) \\times (k+1)$ integer matrix whose first row consists of ones and whose row $i + 1$, for $1 \\le i \\le k$, has $p_i$ in column $i$ and $1$ in every other column. Equivalently, it is the $k \\times k$ matrix with diagonal entries $p_1, \\dots, p_k$ and all other entries $1$, with a column of ones appended on the right and then a row of ones placed on top.","title":"Bordered ones-diagonal matrix"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/bordered-ones-diag-det","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.borderedOnesDiagDet","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#bordered-ones-diag-det","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"fully_proved","summary":"For natural numbers $p_1, \\dots, p_k$, the integer $F_k(p_1, \\dots, p_k)$ is $(-1)^k$ times the determinant of the bordered ones-diagonal matrix of $p_1, \\dots, p_k$: the $(k+1) \\times (k+1)$ integer matrix whose first row consists of ones and whose row $i + 1$, for $1 \\le i \\le k$, has $p_i$ in column $i$ and $1$ in every other column.","title":"Bordered ones-diagonal determinant"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/first-primes","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.firstPrimes","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#first-primes","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"fully_proved","summary":"For a natural number $k$, the $k$-tuple whose entry with index $i$, for $0 \\le i < k$, is the $(i+1)$-st prime; that is, the first $k$ primes $2, 3, 5, \\dots$ in increasing order.","title":"The first primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/residue-hit-count","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.residueHitCount","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#residue-hit-count","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"fully_proved","summary":"For natural numbers $p_1, \\dots, p_k$, $r_1, \\dots, r_k$ and $n$, the residue hit count $\\gamma(n)$ is the number of indices $i$ with $n \\equiv r_i \\pmod{p_i}$, that is, the number of the residue classes $r_i \\bmod p_i$ that contain $n$.","title":"Residue hit count"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/available-count","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.availableCount","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#available-count","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"fully_proved","summary":"For natural numbers $p_1, \\dots, p_k$ and $r_1, \\dots, r_k$, let $\\gamma(n)$ be the number of indices $i$ with $n \\equiv r_i \\pmod{p_i}$, and call $n$ available if $\\gamma(n) \\le 1$. This is the number of available integers $n$ with $1 \\le n \\le p_1 p_2 \\cdots p_k$. For the first $k$ primes it is the count written $\\mathrm{av}(k)$.","title":"Number of available integers"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/free-count","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.freeCount","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#free-count","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"fully_proved","summary":"For natural numbers $p_1, \\dots, p_k$ and $r_1, \\dots, r_k$, let $\\gamma(n)$ be the number of indices $i$ with $n \\equiv r_i \\pmod{p_i}$. The free count is the number of integers $n$ with $1 \\le n \\le p_1 \\cdots p_k$ and $\\gamma(n) = 0$.","title":"Free count"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/occupied-count","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.occupiedCount","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#occupied-count","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"fully_proved","summary":"For natural numbers $p_1, \\dots, p_k$ and $r_1, \\dots, r_k$, let $\\gamma(n)$ be the number of indices $i$ with $n \\equiv r_i \\pmod{p_i}$. The occupied count is the number of integers $n$ with $1 \\le n \\le p_1 \\cdots p_k$ and $\\gamma(n) > 1$.","title":"Occupied count"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/bordered-ones-diag-det-first-primes-recu","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.borderedOnesDiagDet_firstPrimes_recurrence","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#bordered-ones-diag-det-first-primes-recu","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $k \\ge 2$ and let $p_1, \\dots, p_k$ be the first $k$ primes. For $m \\le k$ let $f_m = F_m(p_1, \\dots, p_m)$, where $F_m(p_1, \\dots, p_m)$ is $(-1)^m$ times the determinant of the $(m+1) \\times (m+1)$ matrix whose first row consists of ones and whose row $i + 1$, for $1 \\le i \\le m$, has $p_i$ in column $i$ and $1$ in every other column. Then $f_k = (p_k - 1) f_{k-1}$.","title":"Recurrence for the bordered determinants of the first primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/ones-diag-det-first-primes-recurrence","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.onesDiagDet_firstPrimes_recurrence","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#ones-diag-det-first-primes-recurrence","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $p_1 < p_2 < \\cdots$ be the primes in increasing order. For $m \\ge 0$ let $a_m$ be the determinant of the $m \\times m$ integer matrix with $p_1, \\dots, p_m$ on the diagonal and $1$ in every other entry, and let $f_m = (-1)^m \\det B_m$, where $B_m$ is the $(m+1) \\times (m+1)$ integer matrix whose first row consists of ones and whose row $i + 1$, for $1 \\le i \\le m$, has $p_i$ in column $i$ and $1$ in every other\u2026","title":"Lemma 2, determinant recurrence"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/occupied-count-first-primes-succ-add-fre","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.occupiedCount_firstPrimes_succ_add_free","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#occupied-count-first-primes-succ-add-fre","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $k \\ge 1$, let $p_1 < p_2 < \\cdots$ be the primes in increasing order, and let $r_1, \\dots, r_{k+1}$ be natural numbers. For $m \\in \\{k, k + 1\\}$ write $P_m = p_1 \\cdots p_m$, let $\\gamma_m(n)$ be the number of $i \\le m$ with $n \\equiv r_i \\pmod{p_i}$, let $\\mathrm{occ}(m)$ be the number of integers $n$ with $1 \\le n \\le P_m$ and $\\gamma_m(n) > 1$, and let $\\mathrm{free}(m)$ be the number of those with\u2026","title":"Lemma 3, occupied recurrence"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/free-count-first-primes-succ","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.freeCount_firstPrimes_succ","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#free-count-first-primes-succ","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $k \\ge 1$, let $p_1 < p_2 < \\cdots$ be the primes in increasing order, and let $r_1, \\dots, r_{k+1}$ be natural numbers. For $m \\in \\{k, k + 1\\}$ write $\\mathrm{free}(m)$ for the number of integers $n$ with $1 \\le n \\le p_1 \\cdots p_m$ such that $n \\not\\equiv r_i \\pmod{p_i}$ for every $i \\le m$. Then $\\mathrm{free}(k+1) = (p_{k+1} - 1) \\cdot \\mathrm{free}(k)$.","title":"Lemma 3, free recurrence"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/available-count-first-primes-succ","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.availableCount_firstPrimes_succ","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#available-count-first-primes-succ","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $k \\ge 1$, let $p_1, \\dots, p_{k+1}$ be the first $k + 1$ primes, and let $r_1, \\dots, r_{k+1}$ be any natural numbers. For $m = k$ and $m = k + 1$, call an integer $n$ free if $n \\not\\equiv r_i \\pmod{p_i}$ for every $i \\le m$, and available if $n \\equiv r_i \\pmod{p_i}$ for at most one $i \\le m$; let $\\mathrm{av}(m)$ and $\\mathrm{free}(m)$ be the numbers of available and of free integers $n$ with\u2026","title":"Recurrence for the number of available integers"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/available-count-first-primes","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.availableCount_firstPrimes","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#available-count-first-primes","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $k$ be a natural number, let $p_1, \\dots, p_k$ be the first $k$ primes, and let $r_1, \\dots, r_k$ be any natural numbers. Call an integer $n$ available if $n \\equiv r_i \\pmod{p_i}$ holds for at most one index $i$. Then the number of available integers $n$ with $1 \\le n \\le p_1 p_2 \\cdots p_k$ equals $a_k = A_k(p_1, \\dots, p_k)$, the determinant of the $k \\times k$ matrix with diagonal entries $p_1, \\dots, p_k$\u2026","title":"Available integers for the first primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/free-count-first-primes","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.freeCount_firstPrimes","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#free-count-first-primes","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $k$ be a natural number, let $p_1 < \\cdots < p_k$ be the first $k$ primes and let $r_1, \\dots, r_k$ be natural numbers. Then the number of integers $n$ with $1 \\le n \\le p_1 \\cdots p_k$ such that $n \\not\\equiv r_i \\pmod{p_i}$ for every $i$ equals $f_k = (-1)^k \\det B$, where $B$ is the $(k+1) \\times (k+1)$ integer matrix whose first row consists of ones and whose row $i + 1$, for $1 \\le i \\le k$, has $p_i$ in\u2026","title":"Theorem 1, free count"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/available-count-two-primes","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.availableCount_two_primes","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#available-count-two-primes","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $p \\ne q$ be primes and let $r_1, r_2$ be any natural numbers. Call an integer $n$ available if it satisfies at most one of the congruences $n \\equiv r_1 \\pmod p$ and $n \\equiv r_2 \\pmod q$. Then the number of available integers $n$ with $1 \\le n \\le pq$ equals $A_2(p, q)$, the determinant of the $2 \\times 2$ matrix with diagonal entries $p, q$ and both other entries $1$.","title":"Available integers for two distinct primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/free-count-two-primes","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.freeCount_two_primes","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#free-count-two-primes","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $p$ and $q$ be distinct primes and let $r_1, r_2$ be natural numbers. Then the number of integers $n$ with $1 \\le n \\le pq$ such that $n \\not\\equiv r_1 \\pmod p$ and $n \\not\\equiv r_2 \\pmod q$ equals $F_2(p, q)$, where","title":"Lemma 5, free count"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/available-count-three-primes","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.availableCount_three_primes","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#available-count-three-primes","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $p$, $q$, $s$ be pairwise distinct primes and let $r_1, r_2, r_3$ be any natural numbers. Call an integer $n$ available if it satisfies at most one of the congruences $n \\equiv r_1 \\pmod p$, $n \\equiv r_2 \\pmod q$ and $n \\equiv r_3 \\pmod s$. Then the number of available integers $n$ with $1 \\le n \\le pqs$ equals $A_3(p, q, s)$, the determinant of the $3 \\times 3$ matrix with diagonal entries $p, q, s$ and all\u2026","title":"Available integers for three distinct primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/available-count-distinct-primes","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.availableCount_distinct_primes","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#available-count-distinct-primes","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $k \\ge 2$, let $p_1, \\dots, p_k$ be pairwise distinct primes, and let $r_1, \\dots, r_k$ be any natural numbers. Call an integer $n$ available if $n \\equiv r_i \\pmod{p_i}$ holds for at most one index $i$. Then the number of available integers $n$ with $1 \\le n \\le p_1 p_2 \\cdots p_k$ equals $A_k(p_1, \\dots, p_k)$, the determinant of the $k \\times k$ matrix with diagonal entries $p_1, \\dots, p_k$ and all other\u2026","title":"Available integers for distinct primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/free-count-distinct-primes","kind":"node","lean":"MathlibExt.NumberTheory.CountingAppearancesProgressionsWanted.freeCount_distinct_primes","parent":7984,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim/index.html#free-count-distinct-primes","scope":"NumberTheory/distribution-of-primes/sieve-methods/counting-integers-in-residue-classes-modulo-prim","status":"can_prove","summary":"Let $k \\ge 2$, let $p_1, \\dots, p_k$ be distinct primes and let $r_1, \\dots, r_k$ be natural numbers. Then the number of integers $n$ with $1 \\le n \\le p_1 p_2 \\cdots p_k$ such that $n \\not\\equiv r_i \\pmod{p_i}$ for every $i$ equals $F_k(p_1, \\dots, p_k) = (-1)^k \\det B$, where $B$ is the $(k+1) \\times (k+1)$ integer matrix whose first row consists of ones and whose row $i + 1$, for $1 \\le i \\le k$, has $p_i$ in\u2026","title":"Theorem 2, free count"},{"data_url":"scopes/NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes.json","id":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes","kind":"scope","navigation":"module","parent":7957,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/index.html","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes","status":"can_prove","title":"Large prime factors of shifted primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/exists-card-max-prime-fac-prime-add-gt-r","kind":"node","lean":"MetaMathlibExt.exists_card_maxPrimeFac_prime_add_gt_rpow","parent":8005,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/index.html#exists-card-max-prime-fac-prime-add-gt-r","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes","status":"can_prove","summary":"For every integer $a$ and every real $\\theta$ with $0 < \\theta \\le 0.677$ there is a real $\\delta$ with $0 < \\delta < 1$ such that, for all sufficiently large real $x$, the number of primes $p \\le x$ for which the largest prime factor $P(|p + a|)$ of $|p + a|$ exceeds $x^\\theta$ is greater than $\\delta x / \\log x$. Here $P(0) = 0$ and $P(1) = 1$.","title":"Baker\u2013Harman: shifted primes with a large prime factor"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/exists-rpow-div-log-le-mul-card-prime-dv","kind":"node","lean":"MetaMathlibExt.exists_rpow_div_log_le_mul_card_prime_dvd_prime_sub_one","parent":8005,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/index.html#exists-rpow-div-log-le-mul-card-prime-dv","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes","status":"can_prove","summary":"There is a real number $C > 0$ such that for all sufficiently large real $x$ we have $x^{0.677} / \\log x \\le C \\cdot N(x)$, where $N(x)$ is the number of primes $q \\le x$ with $q > x^{0.677}$ such that $q \\mid p - 1$ for some prime $p \\le x$.","title":"Large prime divisors of shifted primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/carmichael-constant-run-length-is-big-o","kind":"node","lean":"MetaMathlibExt.carmichael_constant_run_length_isBigO","parent":8005,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/index.html#carmichael-constant-run-length-is-big-o","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes","status":"can_prove","summary":"For real $x$, let $F_\\lambda(x)$ be the largest natural number $k$ for which there is a natural number $n \\ge 0$ with $n + k \\le x$ and $\\lambda(n + 1) = \\lambda(n + 2) = \\cdots = \\lambda(n + k)$ (and $0$ if there is no such $k$), where $\\lambda$ is the Carmichael function. Then $F_\\lambda(x) = O\\left((\\log x)^{1000/677}\\right)$ as $x \\to \\infty$; that is, $F_\\lambda(x) = O\\left((\\log x)^{1/c}\\right)$ with\u2026","title":"Runs of constant Carmichael values"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/totient-nonincreasing-run-length-is-equi","kind":"node","lean":"MetaMathlibExt.totient_nonincreasing_run_length_isEquivalent","parent":8005,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes/index.html#totient-nonincreasing-run-length-is-equi","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-prime-factors-of-shifted-primes","status":"can_prove","summary":"For real $x$, let $G_\\varphi(x)$ be the largest natural number $k$ for which there is a natural number $n \\ge 0$ with $n + k \\le x$ and $\\varphi(n + 1) \\ge \\varphi(n + 2) \\ge \\cdots \\ge \\varphi(n + k)$ (and $0$ if there is no such $k$), where $\\varphi$ is Euler's totient function. Then $G_\\varphi(x) \\sim \\log_3 x / \\log_6 x$ as $x \\to \\infty$, where $\\log_j$ denotes the $j$-fold iterate of the natural logarithm.","title":"Pollack\u2013Pomerance\u2013Trevi\u00f1o: nonincreasing runs of Euler's function"},{"data_url":"scopes/NumberTheory/distribution-of-primes/sieve-methods/large-sieve.json","id":"NumberTheory/distribution-of-primes/sieve-methods/large-sieve","kind":"scope","navigation":"module","parent":7957,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-sieve/index.html","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-sieve","status":"fully_proved","title":"Large sieve"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/large-sieve/weighted-hilbert-cosecant","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.weightedHilbertCosecant","parent":8010,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-sieve/index.html#weighted-hilbert-cosecant","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-sieve","status":"fully_proved","summary":"Let $\\iota$ be a finite index set and for $r \\in \\iota$ let $x_r$ and $\\delta_r > 0$ be real numbers and $z_r$ a complex number. Write $\\|t\\|$ for the distance from a real number $t$ to the nearest integer, and suppose that $\\delta_r \\le \\|x_r - x_s\\|$ for all $r \\ne s$. Then","title":"The Montgomery\u2013Vaughan weighted Hilbert inequality"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/large-sieve/weighted-large-sieve","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.weightedLargeSieve","parent":8010,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-sieve/index.html#weighted-large-sieve","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-sieve","status":"fully_proved","summary":"Let $N$ be a natural number, let $a_n$ be complex numbers, let $\\iota$ be a finite index set and for $r \\in \\iota$ let $x_r$ and $\\delta_r > 0$ be real numbers. Write $\\|t\\|$ for the distance from a real number $t$ to the nearest integer, and suppose that $\\delta_r \\le \\|x_r - x_s\\|$ for all $r \\ne s$. Then","title":"The Montgomery\u2013Vaughan weighted large sieve"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/large-sieve/arithmetic-large-sieve-progression","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.arithmeticLargeSieve_progression","parent":8010,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/large-sieve/index.html#arithmetic-large-sieve-progression","scope":"NumberTheory/distribution-of-primes/sieve-methods/large-sieve","status":"fully_proved","summary":"Let $N$, $z$, $q$ and $a$ be natural numbers with $z > 0$ and $q > 0$, and let $S$ be a finite set of natural numbers less than $N$ such that for every $n \\in S$ the number $qn + a$ is prime and greater than $z$. Write $\\mu$ for the M\u00f6bius function and $\\varphi$ for Euler's totient function. Then","title":"Arithmetic large sieve for primes in a progression"},{"data_url":"scopes/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve.json","id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","kind":"scope","navigation":"module","parent":7957,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","title":"Selberg sieve"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-set","kind":"node","lean":"MathlibExt.SelbergSieve.levelSet","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-set","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For a bounding sieve whose sifting primes have product $P$ and a natural number $R$, the level-$R$ set is the set of divisors $l$ of $P$ with $l \\le R$.","title":"Selberg level set"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-sum","kind":"node","lean":"MathlibExt.SelbergSieve.levelSum","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-sum","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For a bounding sieve whose sifting primes have product $P$, with multiplicative density function $\\nu$, and a natural number $R$, the level-$R$ Selberg sum is $G = \\sum_{l \\mid P,\\ l \\le R} g(l)$, where $g(l) = \\nu(l) \\prod_{p \\mid l} (1 - \\nu(p))^{-1}$, the product running over the primes $p$ dividing $l$.","title":"Selberg level sum"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-y","kind":"node","lean":"MathlibExt.SelbergSieve.levelY","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-y","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"Let $s$ be a bounding sieve with density function $\\nu$ and Selberg terms $g(l) = \\nu(l) \\prod_{p \\mid l} (1 - \\nu(p))^{-1}$, the product running over the primes $p$ dividing $l$, and let $R$ be a natural number. Write $D_R$ for the level set of $s$ at level $R$ (the divisors of the sifting product of $s$ that are at most $R$) and $G(R) = \\sum_{l \\in D_R} g(l)$ for the level sum. The normalized Selberg coefficient\u2026","title":"Normalized Selberg coefficients"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-weights","kind":"node","lean":"MathlibExt.SelbergSieve.levelWeights","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-weights","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"Let $s$ be a bounding sieve with sifting product $P$ (the product of its sifting primes) and density function $\\nu$, let $R$ be a natural number, and let $y_m$ denote the normalized Selberg coefficients of $s$ at level $R$. For a natural number $d$, the level-$R$ Selberg weight of $s$ at $d$ is","title":"Selberg sieve weights"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-sum-ge-one","kind":"node","lean":"MathlibExt.SelbergSieve.levelSum_ge_one","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-sum-ge-one","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For every bounding sieve and every natural number $R \\ge 1$, the level-$R$ Selberg sum is at least $1$.","title":"The Selberg level sum is at least one"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-weights-inner","kind":"node","lean":"MathlibExt.SelbergSieve.levelWeights_inner","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-weights-inner","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For a bounding sieve whose sifting primes have product $P$, with density function $\\nu$, a natural number $R$ and a divisor $l$ of $P$, the level-$R$ Selberg weights $\\lambda_d$ satisfy $\\sum_{d \\mid P,\\ l \\mid d} \\nu(d)\\,\\lambda_d = y_l$, where $y_l$ is the normalized level-$R$ Selberg $y$-coefficient at $l$.","title":"Inversion formula for the Selberg level weights"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-weights-one","kind":"node","lean":"MathlibExt.SelbergSieve.levelWeights_one","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-weights-one","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For every bounding sieve and every natural number $R \\ge 1$, the level-$R$ Selberg weight at $1$ equals $1$.","title":"The Selberg level weight at one"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-weights-main-sum","kind":"node","lean":"MathlibExt.SelbergSieve.levelWeights_mainSum","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-weights-main-sum","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For a bounding sieve whose sifting primes have product $P$, with density function $\\nu$, and a natural number $R \\ge 1$, let $\\lambda_d$ be the level-$R$ Selberg weights and $\\Lambda_d = \\sum_{d_1, d_2 \\mid d,\\ \\operatorname{lcm}(d_1, d_2) = d} \\lambda_{d_1} \\lambda_{d_2}$ their $\\Lambda^2$ coefficients. Then $\\sum_{d \\mid P} \\Lambda_d\\,\\nu(d) = 1/G$, where $G$ is the level-$R$ Selberg sum.","title":"Main term of the Selberg level weights"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-weights-nu-mul-abs-le","kind":"node","lean":"MathlibExt.SelbergSieve.levelWeights_nu_mul_abs_le","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-weights-nu-mul-abs-le","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For a bounding sieve whose sifting primes have product $P$, with density function $\\nu$, a natural number $R \\ge 1$ and a divisor $d$ of $P$, the level-$R$ Selberg weight $\\lambda_d$ at $d$ satisfies $|\\nu(d)\\,\\lambda_d| \\le 1$.","title":"Weighted bound for the Selberg level weights"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-weights-abs-le","kind":"node","lean":"MathlibExt.SelbergSieve.levelWeights_abs_le","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-weights-abs-le","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For a bounding sieve whose sifting primes have product $P$, a natural number $R \\ge 1$ and a divisor $d$ of $P$, the level-$R$ Selberg weight $\\lambda_d$ at $d$ satisfies $|\\lambda_d| \\le 1$.","title":"Selberg level weights are bounded by one"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-weights-support","kind":"node","lean":"MathlibExt.SelbergSieve.levelWeights_support","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-weights-support","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"For every bounding sieve and all natural numbers $R$ and $d$, if the level-$R$ Selberg weight $\\lambda_d$ at $d$ is nonzero, then $d \\le R$.","title":"Support of the Selberg level weights"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/level-weights-err-sum-le","kind":"node","lean":"MathlibExt.SelbergSieve.levelWeights_errSum_le","parent":8014,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve/index.html#level-weights-err-sum-le","scope":"NumberTheory/distribution-of-primes/sieve-methods/selberg-sieve","status":"fully_proved","summary":"Take a bounding sieve whose sifting primes have product $P$, with density function $\\nu$ and remainders $r_d = A_d - \\nu(d) X$, where $A_d$ is the total weight of the multiples of $d$ in the set to be sifted and $X$ approximates its total weight. Let $R \\ge 1$ be a natural number and suppose that $d\\,\\nu(d) \\ge 1$ and $|r_d| \\le d$ for every divisor $d$ of $P$. Let $\\lambda_d$ be the level-$R$ Selberg weights and\u2026","title":"Error bound for the Selberg level weights"},{"data_url":"scopes/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem.json","id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","kind":"scope","navigation":"module","parent":7957,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","title":"Sieving shifted primes and the parity problem"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-sum-inv-totient-fifth-to-half-te","kind":"node","lean":"MetaMathlibExt.calzado_sum_inv_totient_fifth_to_half_tendsto","parent":8027,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reciprocal totients of the primes between "],["math","\\(x^{1/5}\\)"],["text"," and "],["math","\\(x^{1/2}\\)"]],"source":"Reciprocal totients of the primes between $x^{1/5}$ and $x^{1/2}$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-sum-inv-totient-fifth-to-half-te","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"As the real number $x$ tends to infinity, the sum of $1/\\varphi(p)$ over the primes $p$ with $x^{1/5} < p \\le x^{1/2}$ tends to $\\log(5/2)$, where $\\varphi$ is Euler's totient function.","title":"Reciprocal totients of the primes between $x^{1/5}$ and $x^{1/2}$"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-sum-inv-totient-near-sqrt-is-big","kind":"node","lean":"MetaMathlibExt.calzado_sum_inv_totient_near_sqrt_isBigO","parent":8027,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reciprocal totients of the primes just below "],["math","\\(x^{1/2}\\)"]],"source":"Reciprocal totients of the primes just below $x^{1/2}$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-sum-inv-totient-near-sqrt-is-big","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"For every real $B > 0$ there is a constant $C$ such that for all sufficiently large real $x$, the sum of $1/\\varphi(p)$ over the primes $p$ with $x^{1/2} (\\log x)^{-B} < p \\le x^{1/2}$ is at most $C \\log\\log x / \\log x$, where $\\varphi$ is Euler's totient function.","title":"Reciprocal totients of the primes just below $x^{1/2}$"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-sieve-euler-product-asymp","kind":"node","lean":"MetaMathlibExt.calzado_sieve_euler_product_asymp","parent":8027,"reading_title":{"format":"inline-parts-v1","parts":[["text","Size of the Euler product of a sieve of dimension "],["math","\\(k\\)"]],"source":"Size of the Euler product of a sieve of dimension $k$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-sieve-euler-product-asymp","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"Let $P$ be a set of primes, $k > 0$ a real number, and $\\omega$ a real-valued function on the natural numbers with $0 \\le \\omega(p) < p$ for $p \\in P$. Suppose there is a constant $C_0$ with $|\\sum_{p \\in P,\\ t \\le p < z} \\omega(p) \\log p / p - k \\log(z/t)| \\le C_0$ for all real $t$ and $z$ with $2 \\le t \\le z$. Then there are constants $c > 0$ and $C > 0$ such that for every real $z \\ge 2$,\u2026","title":"Size of the Euler product of a sieve of dimension $k$"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-shifted-primes-sifted-multiple-l","kind":"node","lean":"MetaMathlibExt.calzado_shifted_primes_sifted_multiple_le","parent":8027,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-shifted-primes-sifted-multiple-l","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"Let $a$ be a nonzero even integer. For real $x$ and a prime $\\ell$ let $S_\\ell(x)$ be the number of primes $n \\le x$ such that $\\ell$ divides $n + a$ and no prime $q < x^{1/5}$ divides $n + a$. There is a constant $C$ such that for all sufficiently large $x$ and every prime $\\ell$ with $x^{1/5} < \\ell \\le x^{1/2}$, $S_\\ell(x) \\le C x / (\\varphi(\\ell) (\\log x)^2)$, where $\\varphi$ is Euler's totient function.","title":"Upper bound for sifted shifted primes divisible by a large prime"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-shifted-primes-sifted-asymp","kind":"node","lean":"MetaMathlibExt.calzado_shifted_primes_sifted_asymp","parent":8027,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-shifted-primes-sifted-asymp","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"Let $a$ be a nonzero even integer, and for real $x$ let $S(x)$ be the number of primes $n \\le x$ such that no prime $q < x^{1/5}$ divides $n + a$. There are constants $0 < C_1 < C_2$ such that $C_1 x / (\\log x)^2 < S(x) < C_2 x / (\\log x)^2$ for all sufficiently large $x$.","title":"Order of magnitude of the sifted shifted primes"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-shifted-primes-sifted-multiple-l-2","kind":"node","lean":"MetaMathlibExt.calzado_shifted_primes_sifted_multiple_le_sifted","parent":8027,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-shifted-primes-sifted-multiple-l-2","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"Let $a$ be a nonzero even integer. For real $x$ let $S(x)$ be the number of primes $n \\le x$ such that no prime $q < x^{1/5}$ divides $n + a$, and for a prime $\\ell$ let $S_\\ell(x)$ be the number of those $n$ for which moreover $\\ell$ divides $n + a$. There is a constant $C$ such that for all sufficiently large $x$ and every prime $\\ell$ with $x^{1/5} < \\ell \\le x^{1/2}$, $S_\\ell(x) \\le C S(x) / \\varphi(\\ell)$,\u2026","title":"Relative bound for sifted shifted primes divisible by a large prime"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-selberg-upper-bound-sieve","kind":"node","lean":"MetaMathlibExt.calzado_selberg_upper_bound_sieve","parent":8027,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-selberg-upper-bound-sieve","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"For all real numbers $\\kappa > 0$ and $C_0$ there is a constant $C$ with the following property. Let $P$ be a set of primes, $\\omega$ a non-negative multiplicative arithmetic function with $\\omega(p) < p$ for $p \\in P$, $A$ a finite multiset of positive integers, and $X > 0$ and $z \\ge 2$ real numbers such that $|\\sum_{p \\in P,\\ t \\le p < z} \\omega(p) \\log p / p - \\kappa \\log(z/t)| \\le C_0$ for every real $t$ with\u2026","title":"Selberg's upper bound sieve"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-linear-sieve","kind":"node","lean":"MetaMathlibExt.calzado_linear_sieve","parent":8027,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-linear-sieve","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"Let $\\gamma$ be the Euler\u2013Mascheroni constant, $C_0$ a real number, and $F, f \\colon \\mathbb{R} \\to \\mathbb{R}$ functions continuous on $(0, \\infty)$ with $F(s) = 2e^{\\gamma}/s$ for $0 < s \\le 3$ and $f(s) = 0$ for $0 < s \\le 2$, such that $u \\mapsto uF(u)$ has derivative $f(s - 1)$ at every $s > 3$ and $u \\mapsto uf(u)$ has derivative $F(s - 1)$ at every $s > 2$. Then $f(s) = 2e^{\\gamma}\\log(s - 1)/s$ for\u2026","title":"Rosser\u2013Iwaniec linear sieve"},{"id":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/calzado-shifted-primes-prime-count-lower","kind":"node","lean":"MetaMathlibExt.calzado_shifted_primes_prime_count_lower","parent":8027,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for primes "],["math","\\(n\\)"],["text"," with "],["math","\\(n + a\\)"],["text"," prime, up to a sieve error"]],"source":"Lower bound for primes $n$ with $n + a$ prime, up to a sieve error"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem/index.html#calzado-shifted-primes-prime-count-lower","scope":"NumberTheory/distribution-of-primes/sieve-methods/sieving-shifted-primes-and-the-parity-problem","status":"can_prove","summary":"Let $a$ be a nonzero even integer, and call a prime $n$ admissible at level $x$ if no prime $q < x^{1/5}$ divides $n + a$. For a prime $p$ and real $x$ let $\\bar\\Delta_p(x) = \\#\\{n \\le x \\text{ admissible} : n \\equiv -a \\pmod p\\} - \\#\\{n \\le x \\text{ admissible} : \\gcd(n, p) = 1\\} / \\varphi(p)$, where $\\varphi$ is Euler's totient function. There is $C > 0$ such that for every real $B > 0$ and all sufficiently large\u2026","title":"Lower bound for primes $n$ with $n + a$ prime, up to a sieve error"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-patterns.json","id":"NumberTheory/distribution-of-primes/prime-patterns","kind":"scope","parent":7411,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/index.html","scope":"NumberTheory/distribution-of-primes/prime-patterns","status":"can_prove","summary":"Twin primes and Brun's constant, Hardy\u2013Littlewood tuples, Goldbach/Lemoine-type statements, primes represented by polynomials (Friedlander\u2013Iwaniec).","title":"Prime patterns and conjectures"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/clement-twin-prime-criterion","kind":"node","lean":"MetaMathlibExt.clement_twin_prime_criterion","parent":8037,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/index.html#clement-twin-prime-criterion","scope":"NumberTheory/distribution-of-primes/prime-patterns","status":"fully_proved","summary":"For every natural number $p \\ge 2$, the numbers $p$ and $p + 2$ are both prime if and only if","title":"Clement's criterion for twin primes"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/friedlander-iwaniec","kind":"node","lean":"MathlibExt.NumberTheory.FriedlanderIwaniecWanted.NumberTheory.FriedlanderIwaniec.friedlander_iwaniec","parent":8037,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/index.html#friedlander-iwaniec","scope":"NumberTheory/distribution-of-primes/prime-patterns","status":"can_prove","summary":"There are infinitely many primes $p$ of the form $p = a^2 + b^4$ with $a$ and $b$ natural numbers.","title":"Friedlander\u2013Iwaniec theorem"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/lemoine-of-le-ten-pow-thirteen","kind":"node","lean":"MetaMathlibExt.lemoine_of_le_ten_pow_thirteen","parent":8037,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lemoine's conjecture up to "],["math","\\(10^{13}\\)"]],"source":"Lemoine's conjecture up to $10^{13}$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/index.html#lemoine-of-le-ten-pow-thirteen","scope":"NumberTheory/distribution-of-primes/prime-patterns","status":"can_prove","summary":"Every odd natural number $n$ with $5 < n \\le 10^{13}$ can be written as $n = p + 2q$ with $p$ and $q$ prime.","title":"Lemoine's conjecture up to $10^{13}$"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate.json","id":"NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate","kind":"scope","navigation":"module","parent":8037,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate/index.html","scope":"NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate","status":"can_prove","title":"Admissible sets with no prime translate"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate/erdos429-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem429Wanted.erdos429_refuted","parent":8041,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate/index.html#erdos429-refuted","scope":"NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate","status":"can_prove","summary":"For every function $f : \\mathbb{N} \\to \\mathbb{N}$ with $f(N) \\to \\infty$ as $N \\to \\infty$, there is an infinite set $A$ of natural numbers such that $|A \\cap \\{1, \\ldots, N\\}| \\le f(N)$ for every natural number $N$, for every prime $p$ some residue class modulo $p$ contains no element of $A$, and for every integer $n$ some $a \\in A$ makes $n + a$ not a positive prime.","title":"Sparse admissible sets without prime translates (Erd\u0151s Problem 429)"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate/exists-sparse-subset-pow-no-prime-transl","kind":"node","lean":"MetaMathlibExt.exists_sparse_subset_pow_no_prime_translate","parent":8041,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate/index.html#exists-sparse-subset-pow-no-prime-transl","scope":"NumberTheory/distribution-of-primes/prime-patterns/admissible-sets-with-no-prime-translate","status":"can_prove","summary":"Let $a$ be a natural number that is a primitive root modulo infinitely many primes, and let $S = \\{a^k : k \\ge 1\\}$. For every non-decreasing unbounded function $f : \\mathbb{N} \\to \\mathbb{N}$ there is an infinite set $A \\subseteq S$ such that $|A \\cap \\{1, \\dots, N\\}| \\le f(N)$ for every natural number $N$, and such that no translate $A + n$ with $n \\in \\mathbb{Z}$ consists entirely of (positive) primes.","title":"Sparse subsets of powers with no prime translate"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem.json","id":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","kind":"scope","navigation":"module","parent":8037,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/index.html","scope":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","status":"can_prove","title":"Brun's theorem"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/brun-twin-prime-reciprocal-series-summab","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.BrunTwinPrimeReciprocalsWanted.brun_twin_prime_reciprocal_series_summable","parent":8044,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/index.html#brun-twin-prime-reciprocal-series-summab","scope":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","status":"fully_proved","summary":"The series $\\sum_{p} \\left(\\frac{1}{p} + \\frac{1}{p + 2}\\right)$, taken over all primes $p$ such that $p + 2$ is also prime, converges in $\\mathbb{R}$.","title":"Brun's theorem on twin primes"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/brun-twin-prime-reciprocal-term","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.brunTwinPrimeReciprocalTerm","parent":8044,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/index.html#brun-twin-prime-reciprocal-term","scope":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","status":"fully_proved","summary":"For a natural number $p$, the twin-prime reciprocal term $t(p)$ is the real number $\\frac{1}{p} + \\frac{1}{p + 2}$ when $p$ and $p + 2$ are both prime, and $0$ otherwise.","title":"Twin-prime reciprocal term"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/summable-brun-twin-prime-reciprocal-term","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.summable_brunTwinPrimeReciprocalTerm","parent":8044,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/index.html#summable-brun-twin-prime-reciprocal-term","scope":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","status":"fully_proved","summary":"For a natural number $p$, let $t(p)$ be $\\frac{1}{p} + \\frac{1}{p + 2}$ when $p$ and $p + 2$ are both prime and $0$ otherwise. Then the series $\\sum_{p \\ge 0} t(p)$ converges unconditionally in $\\mathbb{R}$.","title":"Summability of the twin-prime reciprocal terms"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/brun-constant","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.brunConstant","parent":8044,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/index.html#brun-constant","scope":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","status":"fully_proved","summary":"Brun's constant is the real number $B = \\sum_{p} \\left(\\frac{1}{p} + \\frac{1}{p + 2}\\right)$, the sum over all primes $p$ such that $p + 2$ is also prime. A prime lying in two twin-prime pairs, such as $5$ in $(3, 5)$ and $(5, 7)$, is counted once for each pair.","title":"Brun's constant"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/has-sum-brun-constant","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.hasSum_brunConstant","parent":8044,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/index.html#has-sum-brun-constant","scope":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","status":"fully_proved","summary":"For a natural number $p$, let $t(p)$ be $\\frac{1}{p} + \\frac{1}{p + 2}$ when $p$ and $p + 2$ are both prime and $0$ otherwise. Then the series $\\sum_{p \\ge 0} t(p)$ over the natural numbers converges unconditionally in $\\mathbb{R}$, and its sum is Brun's constant.","title":"Brun's constant is the sum of the twin-prime reciprocal series"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/brun-constant-lower-bound","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.brunConstant_lower_bound","parent":8044,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/index.html#brun-constant-lower-bound","scope":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","status":"can_prove","summary":"Brun's constant $B$, the sum of $\\frac{1}{p} + \\frac{1}{p + 2}$ over all primes $p$ such that $p + 2$ is also prime, satisfies $B > 1.840503$.","title":"Lower bound for Brun's constant"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/brun-constant-upper-bound","kind":"node","lean":"MathlibExt.NumberTheory.PrimeCounting.brunConstant_upper_bound","parent":8044,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem/index.html#brun-constant-upper-bound","scope":"NumberTheory/distribution-of-primes/prime-patterns/brun-s-theorem","status":"can_prove","summary":"Brun's constant $B$, the sum of $\\frac{1}{p} + \\frac{1}{p + 2}$ over all primes $p$ such that $p + 2$ is also prime, satisfies $B < 2.288513$.","title":"Upper bound for Brun's constant"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture.json","id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","kind":"scope","navigation":"module","parent":8037,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"can_prove","title":"Hardy\u2013Littlewood prime tuples conjecture"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/hardy-littlewood-residue-count","kind":"node","lean":"MetaMathlibExt.hardyLittlewoodResidueCount","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#hardy-littlewood-residue-count","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"For an $m$-tuple $b = (b_1, \\dots, b_m)$ of integers and a natural number $p$, $v_p(b)$ is the number of distinct values among the remainders $b_1 \\bmod p, \\dots, b_m \\bmod p$, that is, the number of residue classes modulo $p$ that contain an entry of $b$.","title":"Number of residue classes met by an integer tuple"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/is-hardy-littlewood-admissible","kind":"node","lean":"MetaMathlibExt.IsHardyLittlewoodAdmissible","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#is-hardy-littlewood-admissible","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"An $m$-tuple $b$ of integers is admissible when $v_p(b) \\ne p$ for every prime $p$, where $v_p(b)$ is the number of residue classes modulo $p$ that contain an entry of $b$.","title":"Hardy\u2013Littlewood admissible tuples"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/hardy-littlewood-prime-tuple-count","kind":"node","lean":"MetaMathlibExt.hardyLittlewoodPrimeTupleCount","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#hardy-littlewood-prime-tuple-count","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"For an $m$-tuple $b = (b_1, \\dots, b_m)$ of integers and a natural number $x$, $P(x; b)$ is the number of integers $n$ with $1 \\le n \\le x$ such that each of $n + b_1, \\dots, n + b_m$ is a (positive) prime number.","title":"The prime-tuple counting function"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/hardy-littlewood-euler-factor","kind":"node","lean":"MetaMathlibExt.hardyLittlewoodEulerFactor","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#hardy-littlewood-euler-factor","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"For an $m$-tuple $b$ of integers and a natural number $p$, the Euler factor of $b$ at $p$ is the real number","title":"Euler factors of the Hardy\u2013Littlewood singular series"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/hardy-littlewood-singular-series","kind":"node","lean":"MetaMathlibExt.hardyLittlewoodSingularSeries","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#hardy-littlewood-singular-series","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"For an $m$-tuple $b$ of integers, the singular series $\\mathfrak{S}(b)$ is the product over all primes $p$ of the Euler factors","title":"The Hardy\u2013Littlewood singular series"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/is-admissible-finset","kind":"node","lean":"MetaMathlibExt.IsAdmissibleFinset","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#is-admissible-finset","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"A finite set $H$ of integers is admissible if for every prime $p$ there is a residue class $r$ modulo $p$ that contains no element of $H$.","title":"Admissible finite sets of integers"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/hardy-littlewood-residue-count-ne-iff","kind":"node","lean":"MetaMathlibExt.hardyLittlewoodResidueCount_ne_iff","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#hardy-littlewood-residue-count-ne-iff","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"Let $b = (b_1, \\dots, b_m)$ be an $m$-tuple of integers, let $p$ be a prime, and let $v_p(b)$ be the number of residue classes modulo $p$ that contain an entry of $b$. Then $v_p(b) \\ne p$ if and only if there is a residue class $r$ modulo $p$ with $b_i \\not\\equiv r \\pmod p$ for every $i$.","title":"Missing a residue class modulo a prime"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/is-hardy-littlewood-admissible-iff-is-ad","kind":"node","lean":"MetaMathlibExt.isHardyLittlewoodAdmissible_iff_isAdmissibleFinset","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#is-hardy-littlewood-admissible-iff-is-ad","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"A tuple of integers is Hardy\u2013Littlewood admissible exactly when its set of entries is admissible.","title":"Hardy\u2013Littlewood admissibility via the set of entries"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/is-admissible-finset-empty","kind":"node","lean":"MetaMathlibExt.isAdmissibleFinset_empty","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#is-admissible-finset-empty","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"The empty set of integers is admissible: for every prime $p$, some residue class modulo $p$ contains none of its elements.","title":"The empty set is admissible"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/is-admissible-finset-singleton-zero","kind":"node","lean":"MetaMathlibExt.isAdmissibleFinset_singleton_zero","parent":8052,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\{0\\}\\)"],["text"," is admissible"]],"source":"The set $\\{0\\}$ is admissible"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#is-admissible-finset-singleton-zero","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"fully_proved","summary":"The one-element set $\\{0\\}$ of integers is admissible: for every prime $p$ it misses the residue class of $1$ modulo $p$.","title":"The set $\\{0\\}$ is admissible"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/hardy-littlewood-prime-tuple-conjecture","kind":"node","lean":"MetaMathlibExt.hardyLittlewoodPrimeTupleConjecture","parent":8052,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture/index.html#hardy-littlewood-prime-tuple-conjecture","scope":"NumberTheory/distribution-of-primes/prime-patterns/hardy-littlewood-prime-tuples-conjecture","status":"can_prove","summary":"Let $m \\ge 1$ and let $b = (b_1, \\dots, b_m)$ be an admissible $m$-tuple of pairwise distinct integers. Then the singular series $\\mathfrak{S}(b)$ is nonzero, and the prime-tuple count $P(x; b)$, the number of integers $n$ with $1 \\le n \\le x$ such that $n + b_1, \\dots, n + b_m$ are all prime, satisfies","title":"Hardy\u2013Littlewood prime-tuple conjecture"},{"data_url":"scopes/NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials.json","id":"NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials","kind":"scope","navigation":"module","parent":8037,"result":false,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials/index.html","scope":"NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials","status":"can_prove","title":"Prime values of polynomials"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials/bunyakovsky-condition","kind":"node","lean":"Polynomial.BunyakovskyCondition","parent":8064,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials/index.html#bunyakovsky-condition","scope":"NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials","status":"fully_proved","summary":"An integer polynomial $f \\in \\mathbb{Z}[x]$ satisfies the Bunyakovsky condition when it has degree at least $1$, is irreducible in $\\mathbb{Z}[x]$, and has positive leading coefficient.","title":"Bunyakovsky condition"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials/schinzel-condition","kind":"node","lean":"Polynomial.SchinzelCondition","parent":8064,"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials/index.html#schinzel-condition","scope":"NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials","status":"fully_proved","summary":"A finite set $F$ of integer polynomials satisfies the Schinzel condition when it has no fixed prime divisor: for every prime $p$ there is a natural number $n$ such that $p$ divides none of the values $f(n)$ for $f \\in F$.","title":"Schinzel condition"},{"id":"NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials/conjecture-a-prime-values-four-square","kind":"node","lean":"MetaMathlibExt.conjectureA_prime_values_four_square","parent":8064,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many primes of the form "],["math","\\(4k^2 + 2k + 1\\)"]],"source":"Infinitely many primes of the form $4k^2 + 2k + 1$"},"result":true,"route":"../roadmap/NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials/index.html#conjecture-a-prime-values-four-square","scope":"NumberTheory/distribution-of-primes/prime-patterns/prime-values-of-polynomials","status":"can_prove","summary":"There are infinitely many natural numbers $k$ such that $4k^2 + 2k + 1$ is prime.","title":"Infinitely many primes of the form $4k^2 + 2k + 1$"},{"data_url":"scopes/NumberTheory/arithmetic-functions.json","id":"NumberTheory/arithmetic-functions","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/index.html","scope":"NumberTheory/arithmetic-functions","status":"can_prove","summary":"Divisor sums, totients, M\u00f6bius and multiplicative functions, their mean values and extremal orders, and integers classified by them.","title":"Arithmetic functions"},{"id":"NumberTheory/arithmetic-functions/lcm-product-identity","kind":"node","lean":"MetaMathlibExt.lcm_product_identity","parent":8068,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/index.html#lcm-product-identity","scope":"NumberTheory/arithmetic-functions","status":"fully_proved","summary":"For every natural number $n$, the product of $p^{\\lfloor n/(p-1) \\rfloor}$ over the primes $p \\le n + 1$ equals the least common multiple of all products $i_1 i_2 \\cdots i_n$ of $n$ positive integers with $i_1 + \\cdots + i_n \\le 2n$.","title":"The LCM product identity"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers.json","id":"NumberTheory/arithmetic-functions/perfect-numbers","kind":"scope","parent":8068,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers","status":"can_prove","summary":"Perfect, multiperfect, odd perfect, abundant, practical, Zumkeller, Giuga, amicable numbers, spoof factorizations, Lehmer's totient problem.","title":"Perfect numbers and their relatives"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/euler-characterization-of-even-perfect-n","kind":"node","lean":"MetaMathlibExt.euler_characterization_of_even_perfect_numbers","parent":8070,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/index.html#euler-characterization-of-even-perfect-n","scope":"NumberTheory/arithmetic-functions/perfect-numbers","status":"fully_proved","summary":"Every even perfect number $N$ is of the form $N = 2^{p-1}(2^p - 1)$ for some natural number $p$ such that $p$ and $2^p - 1$ are both prime.","title":"Euler's theorem on even perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/is-extremely-abundant-number","kind":"node","lean":"MetaMathlibExt.IsExtremelyAbundantNumber","parent":8070,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/index.html#is-extremely-abundant-number","scope":"NumberTheory/arithmetic-functions/perfect-numbers","status":"fully_proved","summary":"Let $\\sigma(n)$ denote the sum of the positive divisors of $n$. A natural number $n$ is extremely abundant if either $n = 10080$, or $n > 10080$ and $\\sigma(n) / (n \\log \\log n) > \\sigma(m) / (m \\log \\log m)$ for every natural number $m$ with $10080 \\le m < n$, where $\\log$ is the natural logarithm.","title":"Extremely abundant number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/is-giuga-number","kind":"node","lean":"MetaMathlibExt.IsGiugaNumber","parent":8070,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/index.html#is-giuga-number","scope":"NumberTheory/arithmetic-functions/perfect-numbers","status":"fully_proved","summary":"A natural number $n$ is a Giuga number if it is composite (that is, $n \\ge 2$ and $n$ is not prime) and $p$ divides $n/p - 1$ for every prime divisor $p$ of $n$.","title":"Giuga number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/is-practical-number","kind":"node","lean":"MetaMathlibExt.IsPracticalNumber","parent":8070,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/index.html#is-practical-number","scope":"NumberTheory/arithmetic-functions/perfect-numbers","status":"fully_proved","summary":"A natural number $n$ is practical if $n > 0$ and every integer $m$ with $1 \\le m \\le n$ is the sum of a set of distinct positive divisors of $n$.","title":"Practical number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/thabit-ibn-qurra","kind":"node","lean":"MetaMathlibExt.thabit_ibn_qurra","parent":8070,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/index.html#thabit-ibn-qurra","scope":"NumberTheory/arithmetic-functions/perfect-numbers","status":"fully_proved","summary":"Let $n \\ge 2$ be a natural number and put $p = 3 \\cdot 2^n - 1$, $q = 3 \\cdot 2^{n-1} - 1$ and $r = 9 \\cdot 2^{2n-1} - 1$. If $p$, $q$ and $r$ are all prime, then $2^n p q < 2^n r$, the sum of the proper divisors of $2^n p q$ equals $2^n r$, and the sum of the proper divisors of $2^n r$ equals $2^n p q$; that is, $2^n p q$ and $2^n r$ are distinct amicable numbers.","title":"Th\u0101bit ibn Qurra's theorem"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers.json","id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","kind":"scope","navigation":"module","parent":8070,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","title":"Biunitary triperfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/sigma-biunitary-eq-three-mul-iff-of-twen","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_eq_three_mul_iff_of_twentySeven_dvd","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","The only biunitary triperfect number divisible by "],["math","\\(27\\)"],["text"," is "],["math","\\(2160\\)"]],"source":"The only biunitary triperfect number divisible by $27$ is $2160$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#sigma-biunitary-eq-three-mul-iff-of-twen","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"For every positive integer $N$ divisible by $27$, $\\sigma^{**}(N) = 3N$ if and only if $N = 2160$, where $\\sigma^{**}$ is the biunitary divisor sum.","title":"The only biunitary triperfect number divisible by $27$ is $2160$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/odd-sigma-biunitary-iff","kind":"node","lean":"MetaMathlibExt.odd_sigmaBiunitary_iff","parent":8076,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#odd-sigma-biunitary-iff","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"For every positive integer $n$, the biunitary divisor sum $\\sigma^{**}(n)$ is odd if and only if $n = 2^k$ for some natural number $k$ (so $n = 1$ is included).","title":"Parity of the biunitary divisor sum"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/two-pow-card-prime-factors-dvd-sigma-biu","kind":"node","lean":"MetaMathlibExt.two_pow_card_primeFactors_dvd_sigmaBiunitary_of_odd","parent":8076,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#two-pow-card-prime-factors-dvd-sigma-biu","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"For every odd natural number $n$, $2^{\\omega(n)}$ divides $\\sigma^{**}(n)$, where $\\omega(n)$ is the number of distinct prime factors of $n$ and $\\sigma^{**}$ is the biunitary divisor sum.","title":"Powers of two dividing the biunitary divisor sum of an odd number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/two-pow-card-prime-factors-sub-one-dvd-s","kind":"node","lean":"MetaMathlibExt.two_pow_card_primeFactors_sub_one_dvd_sigmaBiunitary_of_even","parent":8076,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#two-pow-card-prime-factors-sub-one-dvd-s","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"For every positive even natural number $n$, writing $\\omega(n)$ for the number of distinct prime factors of $n$, the power $2^{\\omega(n) - 1}$ divides the biunitary divisor sum $\\sigma^{**}(n)$.","title":"Divisibility of the biunitary divisor sum of an even number by a power of two"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/sigma-biunitary-prime-pow-div-ge","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_prime_pow_div_ge","parent":8076,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#sigma-biunitary-prime-pow-div-ge","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $p$ be a prime and $e \\ge 1$, and write $\\sigma^{**}$ for the biunitary divisor sum. Then, in $\\mathbb{Q}$, $\\sigma^{**}(p^e)/p^e \\ge 1 + 1/p^2$; if moreover $e \\ne 2$, then $\\sigma^{**}(p^e)/p^e \\ge 1 + 1/p$; and if $e \\ge 3$, then $\\sigma^{**}(p^e)/p^e \\ge (1 + 1/p)(1 + 1/p^3)$.","title":"Lower bounds for the biunitary divisor sum of a prime power"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/sigma-biunitary-prime-pow-div-ge-of-two","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_prime_pow_div_ge_of_two_mul_sub_one_le","parent":8076,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#sigma-biunitary-prime-pow-div-ge-of-two","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $p$ be a prime and let $m \\ge 1$ and $e \\ge 2m - 1$ be natural numbers, and write $\\sigma^{**}$ for the biunitary divisor sum. Then, in $\\mathbb{Q}$, $\\sigma^{**}(p^e)/p^e \\ge \\sigma^{**}(p^{2m})/p^{2m}$, and if moreover $e \\ne 2m$, then $\\sigma^{**}(p^e)/p^e \\ge \\sum_{i=0}^{m} 1/p^i$.","title":"Comparison bounds for the biunitary divisor sum of a prime power"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/primitive-prime-divisor-pow-sub-one-mod","kind":"node","lean":"MetaMathlibExt.primitivePrimeDivisor_pow_sub_one_mod_eq_one","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primitive prime divisors of "],["math","\\(a^n - 1\\)"],["text"," are "],["math","\\(1\\)"],["text"," modulo "],["math","\\(n\\)"]],"source":"Primitive prime divisors of $a^n - 1$ are $1$ modulo $n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#primitive-prime-divisor-pow-sub-one-mod","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $a \\ge 2$ and $n \\ge 2$ be natural numbers, and let $p$ be a primitive prime divisor of the integer sequence $m \\mapsto a^m - 1$ at index $n$, that is, a prime that divides $a^n - 1$ and divides no $a^m - 1$ with $1 \\le m < n$. Then $p \\equiv 1 \\pmod n$.","title":"Primitive prime divisors of $a^n - 1$ are $1$ modulo $n$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/exists-primitive-prime-divisor-pow-add-o","kind":"node","lean":"MetaMathlibExt.exists_primitivePrimeDivisor_pow_add_one","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primitive prime factors of "],["math","\\(a^n + 1\\)"]],"source":"Primitive prime factors of $a^n + 1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#exists-primitive-prime-divisor-pow-add-o","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $a \\ge 2$ and $n \\ge 1$ be integers with $(a, n) \\ne (2, 3)$, and if $n = 1$ suppose that $a + 1$ is not a power of $2$. Then $a^n + 1$ has a primitive prime factor $p$, that is, a prime $p$ that divides $a^n + 1$ but divides no $a^m + 1$ with $1 \\le m < n$, such that $p \\equiv 1 \\pmod{2n}$.","title":"Primitive prime factors of $a^n + 1$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/sigma-biunitary-three-pow-cases","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_three_pow_cases","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime factors of the biunitary divisor sum of a power of "],["math","\\(3\\)"]],"source":"Prime factors of the biunitary divisor sum of a power of $3$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#sigma-biunitary-three-pow-cases","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $f$, $t$ and $\\eta$ be natural numbers with $\\eta \\in \\{0, 1\\}$ and $f = 2t - 1 - \\eta$, suppose $f \\ge 5$, and write $\\sigma^{**}$ for the biunitary divisor sum. Then at least one of the following holds: (A) $\\eta = 0$ and $\\sigma^{**}(3^f)$ has a prime factor $p$ with $5 < p \\le (3^t - 1)/2$; (B) $5 \\mid \\sigma^{**}(3^f)$, and $f \\equiv 2 \\pmod 4$ or $f \\in \\{7, 8\\}$; (C) $\\sigma^{**}(3^f)$ has a prime factor\u2026","title":"Prime factors of the biunitary divisor sum of a power of $3$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/sigma-biunitary-two-pow-cases","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_two_pow_cases","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime factors of the biunitary divisor sum of a power of "],["math","\\(2\\)"]],"source":"Prime factors of the biunitary divisor sum of a power of $2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#sigma-biunitary-two-pow-cases","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $e$, $s$ and $\\delta$ be natural numbers with $\\delta \\in \\{0, 1\\}$ and $e = 2s - 1 - \\delta$, suppose $e \\ge 6$ and $e \\notin \\{8, 12\\}$, and write $\\sigma^{**}$ for the biunitary divisor sum. Then at least one of the following holds: (a) $\\delta = 0$ and $\\sigma^{**}(2^e)$ has two distinct prime factors $q_1, q_2$ with $5 < q_i \\le 2^s - 1$ for $i = 1, 2$; (b) $5 \\mid \\sigma^{**}(2^e)$; (c) $\\sigma^{**}(2^e)$\u2026","title":"Prime factors of the biunitary divisor sum of a power of $2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/exists-prime-ge-five-dvd-sigma-biunitary","kind":"node","lean":"MetaMathlibExt.exists_prime_ge_five_dvd_sigmaBiunitary_prime_pow","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime factors at least "],["math","\\(5\\)"],["text"," of biunitary divisor sums of prime powers"]],"source":"Prime factors at least $5$ of biunitary divisor sums of prime powers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#exists-prime-ge-five-dvd-sigma-biunitary","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"For every prime $p$ and every integer $g \\ge 2$ with $(p, g) \\ne (2, 4)$, the biunitary divisor sum $\\sigma^{**}(p^g)$ has a prime factor $q \\ge 5$.","title":"Prime factors at least $5$ of biunitary divisor sums of prime powers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/exists-prime-ge-seven-dvd-sigma-biunitar","kind":"node","lean":"MetaMathlibExt.exists_prime_ge_seven_dvd_sigmaBiunitary_prime_pow","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime factors at least "],["math","\\(7\\)"],["text"," of biunitary divisor sums of odd prime powers"]],"source":"Prime factors at least $7$ of biunitary divisor sums of odd prime powers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#exists-prime-ge-seven-dvd-sigma-biunitar","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"For every odd prime $p$ and every integer $g \\ge 4$, the biunitary divisor sum $\\sigma^{**}(p^g)$ has a prime factor $q \\ge 7$.","title":"Prime factors at least $7$ of biunitary divisor sums of odd prime powers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/exists-two-primes-dvd-sigma-biunitary-th","kind":"node","lean":"MetaMathlibExt.exists_two_primes_dvd_sigmaBiunitary_thirteen_pow","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two prime factors of the biunitary divisor sum of "],["math","\\(13^g\\)"]],"source":"Two prime factors of the biunitary divisor sum of $13^g$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#exists-two-primes-dvd-sigma-biunitary-th","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"For every integer $g \\ge 2$, the biunitary divisor sum $\\sigma^{**}(13^g)$ has two distinct prime factors $q_1$ and $q_2$, neither of which lies in $\\{2, 3, 41, 547\\}$.","title":"Two prime factors of the biunitary divisor sum of $13^g$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/exists-prime-dvd-sigma-biunitary-forty-o","kind":"node","lean":"MetaMathlibExt.exists_prime_dvd_sigmaBiunitary_fortyOne_pow","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime factors of the biunitary divisor sum of "],["math","\\(41^g\\)"]],"source":"Prime factors of the biunitary divisor sum of $41^g$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#exists-prime-dvd-sigma-biunitary-forty-o","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"For every integer $g \\ge 1$, the biunitary divisor sum $\\sigma^{**}(41^g)$ has a prime factor $q$ with $q \\notin \\{2, 3, 5, 13\\}$.","title":"Prime factors of the biunitary divisor sum of $41^g$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/four-le-factorization-two-of-biunitary-t","kind":"node","lean":"MetaMathlibExt.four_le_factorization_two_of_biunitaryTriperfect","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Biunitary triperfect numbers divisible by "],["math","\\(27\\)"],["text"," are divisible by "],["math","\\(16\\)"]],"source":"Biunitary triperfect numbers divisible by $27$ are divisible by $16$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#four-le-factorization-two-of-biunitary-t","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $N$ be a biunitary triperfect number, that is, a positive integer with $\\sigma^{**}(N) = 3N$, where $\\sigma^{**}$ is the biunitary divisor sum. If $3^3$ divides $N$, then $2^4$ divides $N$.","title":"Biunitary triperfect numbers divisible by $27$ are divisible by $16$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/factorization-two-ne-five-of-biunitary-t","kind":"node","lean":"MetaMathlibExt.factorization_two_ne_five_of_biunitaryTriperfect","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","No biunitary triperfect number divisible by "],["math","\\(27\\)"],["text"," is exactly divisible by "],["math","\\(2^5\\)"]],"source":"No biunitary triperfect number divisible by $27$ is exactly divisible by $2^5$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#factorization-two-ne-five-of-biunitary-t","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $N$ be a biunitary triperfect number, that is, a positive integer with $\\sigma^{**}(N) = 3N$, where $\\sigma^{**}$ is the biunitary divisor sum. If $3^3$ divides $N$, then the exact power of $2$ dividing $N$ is not $2^5$.","title":"No biunitary triperfect number divisible by $27$ is exactly divisible by $2^5$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/factorization-two-ne-six-of-biunitary-tr","kind":"node","lean":"MetaMathlibExt.factorization_two_ne_six_of_biunitaryTriperfect","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","No biunitary triperfect number divisible by "],["math","\\(27\\)"],["text"," is exactly divisible by "],["math","\\(2^6\\)"]],"source":"No biunitary triperfect number divisible by $27$ is exactly divisible by $2^6$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#factorization-two-ne-six-of-biunitary-tr","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $N$ be a biunitary triperfect number, that is, a positive integer with $\\sigma^{**}(N) = 3N$, where $\\sigma^{**}$ is the biunitary divisor sum. If $3^3$ divides $N$, then the exact power of $2$ dividing $N$ is not $2^6$.","title":"No biunitary triperfect number divisible by $27$ is exactly divisible by $2^6$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/not-seven-le-and-factorization-three-eq","kind":"node","lean":"MetaMathlibExt.not_seven_le_and_factorization_three_eq_three","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","No biunitary triperfect number has "],["math","\\(3\\)"],["text","-exponent "],["math","\\(3\\)"],["text"," and "],["math","\\(2\\)"],["text","-exponent at least "],["math","\\(7\\)"]],"source":"No biunitary triperfect number has $3$-exponent $3$ and $2$-exponent at least $7$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#not-seven-le-and-factorization-three-eq","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $N$ be a positive integer with $\\sigma^{**}(N) = 3N$, where $\\sigma^{**}$ is the biunitary divisor sum, and let $e$ and $f$ be the exponents of $2$ and $3$ in the prime factorization of $N$. If $f \\ge 3$, then $e \\ge 7$ and $f = 3$ do not both hold.","title":"No biunitary triperfect number has $3$-exponent $3$ and $2$-exponent at least $7$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/not-seven-le-and-factorization-three-eq-2","kind":"node","lean":"MetaMathlibExt.not_seven_le_and_factorization_three_eq_four","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","No biunitary triperfect number has "],["math","\\(3\\)"],["text","-exponent "],["math","\\(4\\)"],["text"," and "],["math","\\(2\\)"],["text","-exponent at least "],["math","\\(7\\)"]],"source":"No biunitary triperfect number has $3$-exponent $4$ and $2$-exponent at least $7$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#not-seven-le-and-factorization-three-eq-2","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $N$ be a positive integer with $\\sigma^{**}(N) = 3N$, where $\\sigma^{**}$ is the biunitary divisor sum, and let $e$ and $f$ be the exponents of $2$ and $3$ in the prime factorization of $N$. If $f \\ge 3$, then $e \\ge 7$ and $f = 4$ do not both hold.","title":"No biunitary triperfect number has $3$-exponent $4$ and $2$-exponent at least $7$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/eq-2160-of-factorization-two-eq-four","kind":"node","lean":"MetaMathlibExt.eq_2160_of_factorization_two_eq_four","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Biunitary triperfect numbers exactly divisible by "],["math","\\(2^4\\)"],["text"," and by "],["math","\\(3^3\\)"],["text"," or "],["math","\\(3^4\\)"]],"source":"Biunitary triperfect numbers exactly divisible by $2^4$ and by $3^3$ or $3^4$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#eq-2160-of-factorization-two-eq-four","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Let $N$ be a biunitary triperfect number, that is, a positive integer with $\\sigma^{**}(N) = 3N$, where $\\sigma^{**}$ is the biunitary divisor sum. If the exact power of $2$ dividing $N$ is $2^4$ and the exact power of $3$ dividing $N$ is $3^3$ or $3^4$, then $N = 2160$.","title":"Biunitary triperfect numbers exactly divisible by $2^4$ and by $3^3$ or $3^4$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/not-five-dvd-of-factorization-two-eq-fou","kind":"node","lean":"MetaMathlibExt.not_five_dvd_of_factorization_two_eq_four","parent":8076,"reading_title":{"format":"inline-parts-v1","parts":[["text","Biunitary triperfect numbers exactly divisible by "],["math","\\(2^4\\)"],["text"," and divisible by "],["math","\\(3^5\\)"],["text"," are prime to "],["math","\\(5\\)"]],"source":"Biunitary triperfect numbers exactly divisible by $2^4$ and divisible by $3^5$ are prime to $5$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers/index.html#not-five-dvd-of-factorization-two-eq-fou","scope":"NumberTheory/arithmetic-functions/perfect-numbers/biunitary-triperfect-numbers","status":"can_prove","summary":"Biunitary triperfect numbers exactly divisible by $2^4$ and divisible by $3^5$ are prime to $5$. Let $N$ be a biunitary triperfect number, that is, a positive integer with $\\sigma^{**}(N) = 3N$, where $\\sigma^{**}$ is the biunitary divisor sum. If the exact power of $2$ dividing $N$ is $2^4$ and $3^5$ divides $N$, then $5$ does not divide $N$.","title":"Biunitary triperfect numbers exactly divisible by $2^4$ and divisible by $3^5$ are prime to $5$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers.json","id":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers","kind":"scope","navigation":"module","parent":8070,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers","status":"fully_proved","title":"Multiperfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/is-kperfect","kind":"node","lean":"Nat.IsKPerfect","parent":8098,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-perfect numbers"]],"source":"$k$-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/index.html#is-kperfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the number $n$ is $k$-perfect when $n > 0$ and $\\sigma(n) = kn$, where $\\sigma(n)$ is the sum of the positive divisors of $n$.","title":"$k$-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/pos-k","kind":"node","lean":"Nat.IsKPerfect.pos_k","parent":8098,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parameter of a "],["math","\\(k\\)"],["text","-perfect number is positive"]],"source":"The parameter of a $k$-perfect number is positive"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/index.html#pos-k","scope":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers","status":"fully_proved","summary":"For natural numbers $k$ and $n$, if $n$ is $k$-perfect, then $k > 0$.","title":"The parameter of a $k$-perfect number is positive"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/is-kperfect-two-iff","kind":"node","lean":"Nat.isKPerfect_two_iff","parent":8098,"reading_title":{"format":"inline-parts-v1","parts":[["text","Perfect numbers are the "],["math","\\(2\\)"],["text","-perfect numbers"]],"source":"Perfect numbers are the $2$-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/index.html#is-kperfect-two-iff","scope":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers","status":"fully_proved","summary":"A natural number $n$ is $2$-perfect if and only if it is perfect, that is, $n > 0$ and the sum of the proper divisors of $n$ equals $n$.","title":"Perfect numbers are the $2$-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/pillai-equations-two-pow-three-pow","kind":"node","lean":"MetaMathlibExt.pillai_equations_two_pow_three_pow","parent":8098,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/index.html#pillai-equations-two-pow-three-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers","status":"fully_proved","summary":"For natural numbers $a$ and $b$, the integer equation $2^a - 3^b = -1$ holds if and only if $(a, b) = (1, 1)$ or $(a, b) = (3, 2)$. For natural numbers $a$, $b$ and $c$, the integer equation $2^a - 3^b = 2^c - 1$ holds if and only if $(a, b, c) = (2, 1, 1)$, or $(a, b, c) = (4, 2, 3)$, or $b = 0$ and $c = a$.","title":"Two Pillai equations in powers of two and three"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/steuerwald-three-perfect-exponents","kind":"node","lean":"MetaMathlibExt.steuerwald_three_perfect_exponents","parent":8098,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers/index.html#steuerwald-three-perfect-exponents","scope":"NumberTheory/arithmetic-functions/perfect-numbers/multiperfect-numbers","status":"fully_proved","summary":"Let $N$, $a$, $b$ and $m$ be natural numbers with $N = 2^a 3^b m$, $a \\ge 1$, $b \\ge 1$, and $m$ coprime to $6$. If $\\sigma(N) = 3N$, where $\\sigma(N)$ is the sum of the divisors of $N$, then either $a = 1$ and $b \\ne 1$, or $a \\ne 1$ and $b = 1$.","title":"Steuerwald's theorem on 3-perfect numbers"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers.json","id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","kind":"scope","navigation":"module","parent":8070,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","title":"Near-perfect and deficient-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/kldeficient-perfect-with","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.KLDeficientPerfectWith","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((k,\\ell)\\)"],["text","-deficient-perfect with a given divisor"]],"source":"$(k,\\ell)$-deficient-perfect with a given divisor"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#kldeficient-perfect-with","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"fully_proved","summary":"For natural numbers $k$, $\\ell$, $n$ and $d$, the number $n$ is $[k,\\ell]$-deficient-perfect with divisor $d$ when $d$ is a proper divisor of $n$ ($d \\mid n$ and $d < n$) and $\\sigma_k(n) + d^k = n^k + \\ell n$, where $\\sigma_k(n) = \\sum_{e \\mid n} e^k$ is the sum of the $k$-th powers of the divisors of $n$.","title":"$(k,\\ell)$-deficient-perfect with a given divisor"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-kldeficient-perfect","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.IsKLDeficientPerfect","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((k,\\ell)\\)"],["text","-deficient-perfect numbers"]],"source":"$(k,\\ell)$-deficient-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-kldeficient-perfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"fully_proved","summary":"For natural numbers $k$, $\\ell$ and $n$, write $\\sigma_k(n) = \\sum_{e \\mid n} e^k$. The number $n$ is $[k,\\ell]$-deficient-perfect when there is a divisor $d$ of $n$ with $d < n$ such that $\\sigma_k(n) + d^k = n^k + \\ell n$.","title":"$(k,\\ell)$-deficient-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/klnear-perfect-with","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.KLNearPerfectWith","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((k,\\ell)\\)"],["text","-near-perfect with a given divisor"]],"source":"$(k,\\ell)$-near-perfect with a given divisor"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#klnear-perfect-with","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"fully_proved","summary":"For natural numbers $k$, $\\ell$, $n$ and $d$, the number $n$ is $[k,\\ell]$-near-perfect with divisor $d$ when $d$ is a proper divisor of $n$ ($d \\mid n$ and $d < n$) and $\\sigma_k(n) = n^k + d^k + \\ell n$, where $\\sigma_k(n) = \\sum_{e \\mid n} e^k$ is the sum of the $k$-th powers of the divisors of $n$.","title":"$(k,\\ell)$-near-perfect with a given divisor"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-klnear-perfect","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.IsKLNearPerfect","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((k,\\ell)\\)"],["text","-near-perfect numbers"]],"source":"$(k,\\ell)$-near-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-klnear-perfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"fully_proved","summary":"For natural numbers $k$, $\\ell$ and $n$, write $\\sigma_k(n) = \\sum_{e \\mid n} e^k$. The number $n$ is $[k,\\ell]$-near-perfect when there is a divisor $d$ of $n$ with $d < n$ such that $\\sigma_k(n) = n^k + d^k + \\ell n$.","title":"$(k,\\ell)$-near-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-deficient-fk-perfect","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.IsDeficientFkPerfect","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Deficient "],["math","\\(F_k\\)"],["text","-perfect numbers"]],"source":"Deficient $F_k$-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-deficient-fk-perfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"fully_proved","summary":"For natural numbers $k$ and $n$, write $\\sigma_k(n) = \\sum_{e \\mid n} e^k$. The number $n$ is deficient $F_k$-perfect when it is $[k,3]$-deficient-perfect, that is, when there is a divisor $d$ of $n$ with $d < n$ such that $\\sigma_k(n) + d^k = n^k + 3n$.","title":"Deficient $F_k$-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-near-fk-perfect","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.IsNearFkPerfect","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Near "],["math","\\(F_k\\)"],["text","-perfect numbers"]],"source":"Near $F_k$-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-near-fk-perfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the number $n$ is near $F_k$-perfect when it is $[k,3]$-near-perfect, that is, when some proper divisor $d$ of $n$ ($d \\mid n$ and $d < n$) satisfies $\\sigma_k(n) = n^k + d^k + 3n$, where $\\sigma_k(n) = \\sum_{e \\mid n} e^k$ is the sum of the $k$-th powers of the divisors of $n$.","title":"Near $F_k$-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-deficient-fperfect","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.IsDeficientFPerfect","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Deficient "],["math","\\(F\\)"],["text","-perfect numbers"]],"source":"Deficient $F$-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-deficient-fperfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"fully_proved","summary":"For a natural number $n$, write $\\sigma_2(n) = \\sum_{e \\mid n} e^2$. The number $n$ is deficient $F$-perfect when it is $[2,3]$-deficient-perfect, that is, when there is a divisor $d$ of $n$ with $d < n$ such that $\\sigma_2(n) + d^2 = n^2 + 3n$.","title":"Deficient $F$-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-near-fperfect","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.IsNearFPerfect","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Near "],["math","\\(F\\)"],["text","-perfect numbers"]],"source":"Near $F$-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-near-fperfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"fully_proved","summary":"A natural number $n$ is near $F$-perfect when it is $[2,3]$-near-perfect, that is, when some proper divisor $d$ of $n$ ($d \\mid n$ and $d < n$) satisfies $\\sigma_2(n) = n^2 + d^2 + 3n$, where $\\sigma_2(n) = \\sum_{e \\mid n} e^2$ is the sum of the squares of the divisors of $n$.","title":"Near $F$-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-kldeficient-perfect-of-prime-odd","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isKLDeficientPerfect_of_prime_odd","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd primes are not "],["math","\\((k,\\ell)\\)"],["text","-deficient-perfect"]],"source":"Odd primes are not $(k,\\ell)$-deficient-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-kldeficient-perfect-of-prime-odd","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be an odd prime. Then for all natural numbers $k$ and $\\ell$, $p$ is not $[k,\\ell]$-deficient-perfect.","title":"Odd primes are not $(k,\\ell)$-deficient-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-kldeficient-perfect-two","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.isKLDeficientPerfect_two","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text"," is "],["math","\\((k,1)\\)"],["text","-deficient-perfect"]],"source":"$2$ is $(k,1)$-deficient-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-kldeficient-perfect-two","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"For every natural number $k$, the number $2$ is $[k,1]$-deficient-perfect.","title":"$2$ is $(k,1)$-deficient-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-deficient-fperfect-of-prime","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isDeficientFPerfect_of_prime","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No prime is deficient "],["math","\\(F\\)"],["text","-perfect"]],"source":"No prime is deficient $F$-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-deficient-fperfect-of-prime","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"If $p$ is a prime, then $p$ is not deficient $F$-perfect.","title":"No prime is deficient $F$-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-klnear-perfect-of-prime","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isKLNearPerfect_of_prime","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes are not "],["math","\\((k,\\ell)\\)"],["text","-near-perfect"]],"source":"Primes are not $(k,\\ell)$-near-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-klnear-perfect-of-prime","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k$ and $\\ell$ be natural numbers with $\\ell \\ge 1$ and let $p$ be a prime. Then $p$ is not $[k,\\ell]$-near-perfect.","title":"Primes are not $(k,\\ell)$-near-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/prime-pow-not-f-perfect","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.prime_pow_not_fPerfect","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No prime power is deficient "],["math","\\(F\\)"],["text","-perfect or near "],["math","\\(F\\)"],["text","-perfect"]],"source":"No prime power is deficient $F$-perfect or near $F$-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#prime-pow-not-f-perfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"For every prime $p$ and every natural number $a$, the number $n = p^a$ is neither deficient $F$-perfect nor near $F$-perfect: there is no divisor $d$ of $n$ with $d < n$ and $\\sigma_2(n) + d^2 = n^2 + 3n$, and no divisor $d$ of $n$ with $d < n$ and $\\sigma_2(n) = n^2 + d^2 + 3n$, where $\\sigma_2(n)$ is the sum of the squares of the divisors of $n$.","title":"No prime power is deficient $F$-perfect or near $F$-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/kl-near-perfect-with-prime-pow-two-iff","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.klNearPerfectWith_prime_pow_two_iff","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime powers that are "],["math","\\((2,\\ell)\\)"],["text","-near-perfect"]],"source":"Prime powers that are $(2,\\ell)$-near-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#kl-near-perfect-with-prime-pow-two-iff","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $a$, $\\ell$ and $d$ be natural numbers with $\\ell \\ge 1$, and write $\\sigma_2(m)$ for the sum of the squares of the divisors of $m$. Then $d$ is a proper divisor of $p^a$ ($d \\mid p^a$ and $d < p^a$) with $\\sigma_2(p^a) = (p^a)^2 + d^2 + \\ell p^a$ if and only if $a = 2$, $\\ell = 1$ and $d = 1$.","title":"Prime powers that are $(2,\\ell)$-near-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-kldeficient-perfect-prime-pow-two-imp","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.isKLDeficientPerfect_prime_pow_two_imp","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((2,\\ell)\\)"],["text","-deficient-perfect prime powers (corrected)"]],"source":"$(2,\\ell)$-deficient-perfect prime powers (corrected)"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-kldeficient-perfect-prime-pow-two-imp","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be a prime and let $a$ and $\\ell$ be natural numbers. If $p^a$ is $[2,\\ell]$-deficient-perfect, then $\\ell = 1$, and either $a = 2$, or $p = 2$ and $a = 1$.","title":"$(2,\\ell)$-deficient-perfect prime powers (corrected)"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/kl-near-perfect-with-prime-pow-iff","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.klNearPerfectWith_prime_pow_iff","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime powers that are "],["math","\\((k,\\ell)\\)"],["text","-near-perfect"]],"source":"Prime powers that are $(k,\\ell)$-near-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#kl-near-perfect-with-prime-pow-iff","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $a$, $k$, $\\ell$ and $d$ be natural numbers with $k \\ge 1$ and $\\ell \\ge 1$, and write $\\sigma_k(m) = \\sum_{e \\mid m} e^k$. Then $d$ is a proper divisor of $p^a$ ($d \\mid p^a$ and $d < p^a$) with $\\sigma_k(p^a) = (p^a)^k + d^k + \\ell p^a$ if and only if $2 \\le a \\le k$, $d = 1$ and $\\ell = p^{k-a} \\sum_{i=0}^{a-2} p^{ik}$.","title":"Prime powers that are $(k,\\ell)$-near-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-kldeficient-perfect-prime-pow-imp","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.isKLDeficientPerfect_prime_pow_imp","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((k,\\ell)\\)"],["text","-deficient-perfect prime powers for "],["math","\\(k \\ge 2\\)"]],"source":"$(k,\\ell)$-deficient-perfect prime powers for $k \\ge 2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-kldeficient-perfect-prime-pow-imp","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be a prime and let $a$, $k$ and $\\ell$ be natural numbers with $k \\ge 2$. If $p^a$ is $[k,\\ell]$-deficient-perfect, then $p = 2$ and $a = 1$.","title":"$(k,\\ell)$-deficient-perfect prime powers for $k \\ge 2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/kl-deficient-perfect-with-one-prime-pow","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.klDeficientPerfectWith_one_prime_pow_iff","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime powers that are "],["math","\\((1,\\ell)\\)"],["text","-deficient-perfect with a given divisor"]],"source":"Prime powers that are $(1,\\ell)$-deficient-perfect with a given divisor"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#kl-deficient-perfect-with-one-prime-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $a$, $\\ell$ and $d$ be natural numbers, and write $\\sigma_1(m)$ for the sum of the divisors of $m$. Then $d$ is a proper divisor of $p^a$ ($d \\mid p^a$ and $d < p^a$) with $\\sigma_1(p^a) + d = p^a + \\ell p^a$ if and only if $p = 2$, $\\ell = 1$, $d = 1$ and $a \\ge 1$.","title":"Prime powers that are $(1,\\ell)$-deficient-perfect with a given divisor"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-klnear-perfect-two-mul-prime","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isKLNearPerfect_two_mul_prime","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Twice an odd prime is not "],["math","\\((k,\\ell)\\)"],["text","-near-perfect"]],"source":"Twice an odd prime is not $(k,\\ell)$-near-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-klnear-perfect-two-mul-prime","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k$ and $\\ell$ be natural numbers with $\\ell \\ge 1$ and let $p$ be an odd prime. Then $2p$ is not $[k,\\ell]$-near-perfect.","title":"Twice an odd prime is not $(k,\\ell)$-near-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-deficient-fperfect-two-pow-mul-prime","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.isDeficientFPerfect_two_pow_mul_prime_imp","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Deficient "],["math","\\(F\\)"],["text","-perfect numbers of the form "],["math","\\(2^a p\\)"]],"source":"Deficient $F$-perfect numbers of the form $2^a p$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-deficient-fperfect-two-pow-mul-prime","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be a prime and $a$ a natural number. If $2^a p$ is deficient $F$-perfect, then $p = 3$ and $a = 1$, so the number is $6$.","title":"Deficient $F$-perfect numbers of the form $2^a p$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-kl-deficient-perfect-with-two-pow-mu","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_klDeficientPerfectWith_two_pow_mul","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2^a p\\)"],["text"," is not "],["math","\\((k,\\ell)\\)"],["text","-deficient-perfect with divisor "],["math","\\(1\\)"],["text"," or "],["math","\\(p\\)"]],"source":"$2^a p$ is not $(k,\\ell)$-deficient-perfect with divisor $1$ or $p$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-kl-deficient-perfect-with-two-pow-mu","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k$, $\\ell$, $a$ be natural numbers with $\\ell \\ge 1$ and let $p$ be an odd prime. Call $n$ $[k,\\ell]$-deficient-perfect with divisor $d$ when $d$ divides $n$, $d < n$ and $\\sigma_k(n) + d^k = n^k + \\ell n$, where $\\sigma_k(n)$ is the sum of the $k$-th powers of the divisors of $n$. Then $2^a p$ is not $[k,\\ell]$-deficient-perfect with divisor $1$, and not with divisor $p$.","title":"$2^a p$ is not $(k,\\ell)$-deficient-perfect with divisor $1$ or $p$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-near-fperfect-two-pow-mul-prime","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isNearFPerfect_two_pow_mul_prime","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No near "],["math","\\(F\\)"],["text","-perfect number of the form "],["math","\\(2^a p\\)"]],"source":"No near $F$-perfect number of the form $2^a p$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-near-fperfect-two-pow-mul-prime","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"For every prime $p$ and every natural number $a$, the number $n = 2^a p$ is not near $F$-perfect: there is no divisor $d$ of $n$ with $d < n$ and $\\sigma_2(n) = n^2 + d^2 + 3n$, where $\\sigma_2(n)$ is the sum of the squares of the divisors of $n$.","title":"No near $F$-perfect number of the form $2^a p$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-klnear-perfect-two-two-pow-mul","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isKLNearPerfect_two_two_pow_mul","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No "],["math","\\((2,\\ell)\\)"],["text","-near-perfect number of the form "],["math","\\(2^a p\\)"]],"source":"No $(2,\\ell)$-near-perfect number of the form $2^a p$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-klnear-perfect-two-two-pow-mul","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $\\ell$ and $a$ be natural numbers with $\\ell \\ge 1$, and let $p$ be an odd prime. Then $2^a p$ is not $[2,\\ell]$-near-perfect.","title":"No $(2,\\ell)$-near-perfect number of the form $2^a p$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-kl-near-perfect-with-two-pow-mul","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_klNearPerfectWith_two_pow_mul","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","For "],["math","\\(a \\ge 2\\)"],["text",", "],["math","\\(2^a p\\)"],["text"," is not "],["math","\\((k,\\ell)\\)"],["text","-near-perfect with divisor "],["math","\\(1\\)"],["text"," or "],["math","\\(p\\)"]],"source":"For $a \\ge 2$, $2^a p$ is not $(k,\\ell)$-near-perfect with divisor $1$ or $p$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-kl-near-perfect-with-two-pow-mul","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k$, $\\ell$, $a$ be natural numbers with $\\ell \\ge 1$ and $a \\ge 2$, and let $p$ be an odd prime. Call $n$ $[k,\\ell]$-near-perfect with divisor $d$ when $d$ divides $n$, $d < n$ and $\\sigma_k(n) = n^k + d^k + \\ell n$, where $\\sigma_k(n)$ is the sum of the $k$-th powers of the divisors of $n$. Then $2^a p$ is not $[k,\\ell]$-near-perfect with divisor $1$, and not with divisor $p$.","title":"For $a \\ge 2$, $2^a p$ is not $(k,\\ell)$-near-perfect with divisor $1$ or $p$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-deficient-fperfect-two-mul-prime-pow","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.isDeficientFPerfect_two_mul_prime_pow","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","The only deficient "],["math","\\(F\\)"],["text","-perfect number of the form "],["math","\\(2p^a\\)"],["text"," is "],["math","\\(6\\)"]],"source":"The only deficient $F$-perfect number of the form $2p^a$ is $6$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-deficient-fperfect-two-mul-prime-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"The number $6$ is deficient $F$-perfect, and for every prime $p$ and every natural number $a$, if $2p^a$ is deficient $F$-perfect then $2p^a = 6$.","title":"The only deficient $F$-perfect number of the form $2p^a$ is $6$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-near-fperfect-two-mul-prime-pow","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isNearFPerfect_two_mul_prime_pow","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No near "],["math","\\(F\\)"],["text","-perfect number of the form "],["math","\\(2p^a\\)"]],"source":"No near $F$-perfect number of the form $2p^a$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-near-fperfect-two-mul-prime-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"For every prime $p$ and every natural number $a$, the number $n = 2p^a$ is not near $F$-perfect: there is no divisor $d$ of $n$ with $d < n$ and $\\sigma_2(n) = n^2 + d^2 + 3n$, where $\\sigma_2(n)$ is the sum of the squares of the divisors of $n$.","title":"No near $F$-perfect number of the form $2p^a$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/is-klnear-perfect-one-two-mul-prime-pow","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.isKLNearPerfect_one_two_mul_prime_pow","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","The only "],["math","\\((1,\\ell)\\)"],["text","-near-perfect number of the form "],["math","\\(2p^a\\)"],["text"," is "],["math","\\(18\\)"]],"source":"The only $(1,\\ell)$-near-perfect number of the form $2p^a$ is $18$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#is-klnear-perfect-one-two-mul-prime-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"For natural numbers $\\ell$ and $n$, say that $n$ is $[1,\\ell]$-near-perfect when some divisor $d$ of $n$ with $d < n$ satisfies $\\sigma(n) = n + d + \\ell n$, where $\\sigma(n)$ is the sum of the divisors of $n$. Then $18$ is $[1,1]$-near-perfect, and for every odd prime $p$ and all natural numbers $a$ and $\\ell \\ge 1$, if $2p^a$ is $[1,\\ell]$-near-perfect then $2p^a = 18$.","title":"The only $(1,\\ell)$-near-perfect number of the form $2p^a$ is $18$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/kl-near-perfect-with-two-mul-prime-pow-i","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.klNearPerfectWith_two_mul_prime_pow_imp","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","A divisibility condition for near-perfect numbers "],["math","\\(2p^a\\)"]],"source":"A divisibility condition for near-perfect numbers $2p^a$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#kl-near-perfect-with-two-mul-prime-pow-i","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be an odd prime, let $a$, $k$, $\\ell$ and $d$ be natural numbers with $k > 1$ and $\\ell \\ge 1$, and write $\\sigma_k(m) = \\sum_{e \\mid m} e^k$. If $d$ is a proper divisor of $2p^a$ ($d \\mid 2p^a$ and $d < 2p^a$) with $\\sigma_k(2p^a) = (2p^a)^k + d^k + \\ell \\cdot 2p^a$, then $p^2 \\mid 1 + 2^k$, and $a$ is even if and only if $d$ is odd.","title":"A divisibility condition for near-perfect numbers $2p^a$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-near-fperfect-two-pow-mul-prime-p","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isNearFPerfect_two_pow_mul_prime_pow","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No near "],["math","\\(F\\)"],["text","-perfect number of the form "],["math","\\(2^a p^b\\)"],["text"," with "],["math","\\(b\\)"],["text"," even"]],"source":"No near $F$-perfect number of the form $2^a p^b$ with $b$ even"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-near-fperfect-two-pow-mul-prime-p","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"For every prime $p$ and all natural numbers $a$ and $b$ with $b$ even, the number $n = 2^a p^b$ is not near $F$-perfect: there is no divisor $d$ of $n$ with $d < n$ and $\\sigma_2(n) = n^2 + d^2 + 3n$, where $\\sigma_2(n)$ is the sum of the squares of the divisors of $n$.","title":"No near $F$-perfect number of the form $2^a p^b$ with $b$ even"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-deficient-fk-perfect-prime-pow-mu","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isDeficientFkPerfect_prime_pow_mul","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No deficient "],["math","\\(F_k\\)"],["text","-perfect number "],["math","\\(p_1^a p_2\\)"],["text"," with distinct odd primes"]],"source":"No deficient $F_k$-perfect number $p_1^a p_2$ with distinct odd primes"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-deficient-fk-perfect-prime-pow-mu","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k > 1$ be a natural number, let $p_1 \\ne p_2$ be odd primes and let $a$ be a natural number. Then $p_1^a p_2$ is not deficient $F_k$-perfect.","title":"No deficient $F_k$-perfect number $p_1^a p_2$ with distinct odd primes"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-near-fk-perfect-prime-pow-mul","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isNearFkPerfect_prime_pow_mul","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No near "],["math","\\(F_k\\)"],["text","-perfect number of the form "],["math","\\(p_1^a p_2\\)"]],"source":"No near $F_k$-perfect number of the form $p_1^a p_2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-near-fk-perfect-prime-pow-mul","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k > 1$, let $p_1$ and $p_2$ be distinct odd primes and let $a$ be a natural number. Then $p_1^a p_2$ is not near $F_k$-perfect.","title":"No near $F_k$-perfect number of the form $p_1^a p_2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-deficient-fk-perfect-two-mul-pow","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isDeficientFkPerfect_two_mul_pow","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No deficient "],["math","\\(F_k\\)"],["text","-perfect number of the form "],["math","\\(2p^a\\)"],["text"," for "],["math","\\(k > 2\\)"]],"source":"No deficient $F_k$-perfect number of the form $2p^a$ for $k > 2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-deficient-fk-perfect-two-mul-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k > 2$, let $p$ be an odd prime and let $a$ be a natural number. Then $2p^a$ is not deficient $F_k$-perfect.","title":"No deficient $F_k$-perfect number of the form $2p^a$ for $k > 2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-deficient-fk-perfect-pow-mul-pow","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isDeficientFkPerfect_pow_mul_pow","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No deficient "],["math","\\(F_k\\)"],["text","-perfect number "],["math","\\(p_1^a p_2^b\\)"],["text"," with "],["math","\\(a + b\\)"],["text"," odd or "],["math","\\(a, b\\)"],["text"," both odd"]],"source":"No deficient $F_k$-perfect number $p_1^a p_2^b$ with $a + b$ odd or $a, b$ both odd"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-deficient-fk-perfect-pow-mul-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k > 1$ be a natural number, let $p_1 \\ne p_2$ be odd primes and let $a, b \\ge 1$ be natural numbers such that $a + b$ is odd or both $a$ and $b$ are odd. Then $n = p_1^a p_2^b$ is not deficient $F_k$-perfect: there is no divisor $d$ of $n$ with $d < n$ and $\\sigma_k(n) + d^k = n^k + 3n$, where $\\sigma_k(n) = \\sum_{e \\mid n} e^k$.","title":"No deficient $F_k$-perfect number $p_1^a p_2^b$ with $a + b$ odd or $a, b$ both odd"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-near-fk-perfect-pow-mul-pow","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isNearFkPerfect_pow_mul_pow","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No near "],["math","\\(F_k\\)"],["text","-perfect number "],["math","\\(p_1^a p_2^b\\)"],["text"," with "],["math","\\(a + b\\)"],["text"," odd or "],["math","\\(a, b\\)"],["text"," both odd"]],"source":"No near $F_k$-perfect number $p_1^a p_2^b$ with $a + b$ odd or $a, b$ both odd"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-near-fk-perfect-pow-mul-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k > 1$, let $p_1 \\ne p_2$ be odd primes and let $a, b \\ge 1$ be natural numbers such that $a + b$ is odd or both $a$ and $b$ are odd. Then $p_1^a p_2^b$ is not near $F_k$-perfect.","title":"No near $F_k$-perfect number $p_1^a p_2^b$ with $a + b$ odd or $a, b$ both odd"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-deficient-fk-perfect-prod-primes","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isDeficientFkPerfect_prod_primes","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No product of distinct odd primes is deficient "],["math","\\(F_k\\)"],["text","-perfect"]],"source":"No product of distinct odd primes is deficient $F_k$-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-deficient-fk-perfect-prod-primes","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k > 1$ and let $s$ be a finite set of odd primes. Then $\\prod_{p \\in s} p$ is not deficient $F_k$-perfect.","title":"No product of distinct odd primes is deficient $F_k$-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-is-near-fk-perfect-prod-primes","kind":"node","lean":"MathlibExt.NumberTheory.FkPerfectWanted.not_isNearFkPerfect_prod_primes","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No product of distinct odd primes is near "],["math","\\(F_k\\)"],["text","-perfect"]],"source":"No product of distinct odd primes is near $F_k$-perfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-is-near-fk-perfect-prod-primes","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k > 1$ and let $s$ be a finite set of odd primes. Then $\\prod_{p \\in s} p$ is not near $F_k$-perfect.","title":"No product of distinct odd primes is near $F_k$-perfect"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/two-near-perfect-two-pow-mul-prime-cases","kind":"node","lean":"MetaMathlibExt.two_near_perfect_two_pow_mul_prime_cases","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-near perfect numbers of the form "],["math","\\(2^k p\\)"]],"source":"$2$-near perfect numbers of the form $2^k p$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#two-near-perfect-two-pow-mul-prime-cases","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $k \\ge 1$, let $p$ be a prime and let $n = 2^k p$. Suppose $d_1 \\ne d_2$ are divisors of $n$ with $\\sigma(n) = 2n + d_1 + d_2$, where $\\sigma$ is the sum-of-divisors function (that is, $n$ is $2$-near perfect with omitted divisors $d_1, d_2$). Then, after possibly swapping $d_1$ and $d_2$, one of the following holds, where all equations are between integers: (1) $p = 2^k - 1$, $d_1 = 1$ and $d_2 = p$; (2) for\u2026","title":"$2$-near perfect numbers of the form $2^k p$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/two-near-perfect-two-pow-mul-prime-sq","kind":"node","lean":"MetaMathlibExt.two_near_perfect_two_pow_mul_prime_sq","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-near perfect numbers of the form "],["math","\\(2^k p^2\\)"]],"source":"$2$-near perfect numbers of the form $2^k p^2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#two-near-perfect-two-pow-mul-prime-sq","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $k$ be a natural number and let $n = 2^k p^2$. If $n$ has divisors $d_1 \\ne d_2$ with $\\sigma(n) = 2n + d_1 + d_2$, where $\\sigma$ is the sum-of-divisors function (that is, $n$ is $2$-near perfect with omitted divisors $d_1, d_2$), then $n$ is $18$, $36$ or $200$.","title":"$2$-near perfect numbers of the form $2^k p^2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/eq-one-of-is-square-discriminant","kind":"node","lean":"MetaMathlibExt.eq_one_of_isSquare_discriminant","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square values of the discriminant "],["math","\\(2^{2k+2} + 2^{k+2} - 2^{a+2} - 7\\)"]],"source":"Square values of the discriminant $2^{2k+2} + 2^{k+2} - 2^{a+2} - 7$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#eq-one-of-is-square-discriminant","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $a$ and $k$ be natural numbers with $1 \\le a \\le k$. If $2^{2k+2} + 2^{k+2} - 2^{a+2} - 7$ is the square of an integer, then $k = a = 1$.","title":"Square values of the discriminant $2^{2k+2} + 2^{k+2} - 2^{a+2} - 7$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/dvd-of-two-near-perfect-eq-six","kind":"node","lean":"MetaMathlibExt.dvd_of_two_near_perfect_eq_six","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility constraints on odd solutions of "],["math","\\((2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1\\)"]],"source":"Divisibility constraints on odd solutions of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#dvd-of-two-near-perfect-eq-six","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Divisibility constraints on odd solutions of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1$. Let $b$ and $k$ be natural numbers and let $p$ be an odd natural number such that $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1$. Then, as integers, $p$ divides $2^k - 1$ and $p + 1$ divides $2^b - 2$.","title":"Divisibility constraints on odd solutions of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-two-near-perfect-eq-seven","kind":"node","lean":"MetaMathlibExt.not_two_near_perfect_eq_seven","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No odd solutions of "],["math","\\((2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1\\)"],["text"," for "],["math","\\(2 \\le b \\le k - 1\\)"]],"source":"No odd solutions of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1$ for $2 \\le b \\le k - 1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-two-near-perfect-eq-seven","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"No odd solutions of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1$ for $2 \\le b \\le k - 1$. Let $b$ and $k$ be natural numbers with $2 \\le b$ and $b + 1 \\le k$, and let $p$ be an odd natural number. Then $(2^{k+1} - 1)(p^2 + p + 1) \\ne 2^{k+1} p^2 + 2^b p + 1$.","title":"No odd solutions of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p + 1$ for $2 \\le b \\le k - 1$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/omitted-divisors-sum-two-pow-mul-prime-s","kind":"node","lean":"MetaMathlibExt.omitted_divisors_sum_two_pow_mul_prime_sq","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum and parity of the omitted divisors of a "],["math","\\(2\\)"],["text","-near perfect "],["math","\\(2^k p^2\\)"]],"source":"Sum and parity of the omitted divisors of a $2$-near perfect $2^k p^2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#omitted-divisors-sum-two-pow-mul-prime-s","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be an odd prime, let $k$ be a natural number, and let $d_1 \\ne d_2$ be divisors of $n = 2^k p^2$ with $\\sigma(n) = 2n + d_1 + d_2$, where $\\sigma$ denotes the sum of divisors. Then $d_1 + d_2 = -p^2 + (2^{k+1} - 1) p + (2^{k+1} - 1)$, and $d_1$ and $d_2$ have opposite parity.","title":"Sum and parity of the omitted divisors of a $2$-near perfect $2^k p^2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/dvd-of-two-near-perfect-eq-twelve","kind":"node","lean":"MetaMathlibExt.dvd_of_two_near_perfect_eq_twelve","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd solutions "],["math","\\(p\\)"],["text"," of "],["math","\\((2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1\\)"],["text"," divide "],["math","\\(2^k - 1\\)"]],"source":"Odd solutions $p$ of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1$ divide $2^k - 1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#dvd-of-two-near-perfect-eq-twelve","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Odd solutions $p$ of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1$ divide $2^k - 1$. Let $k$ and $b$ be positive natural numbers and let $p$ be an odd natural number such that $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1$. Then $p$ divides $2^k - 1$.","title":"Odd solutions $p$ of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1$ divide $2^k - 1$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/succ-dvd-of-two-near-perfect-eq-twelve","kind":"node","lean":"MetaMathlibExt.succ_dvd_of_two_near_perfect_eq_twelve","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd solutions "],["math","\\(p\\)"],["text"," of "],["math","\\((2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1\\)"],["text"," satisfy "],["math","\\(p + 1 \\mid 2^b + 2\\)"]],"source":"Odd solutions $p$ of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1$ satisfy $p + 1 \\mid 2^b + 2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#succ-dvd-of-two-near-perfect-eq-twelve","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Odd solutions $p$ of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1$ satisfy $p + 1 \\mid 2^b + 2$. Let $k$ and $b$ be positive natural numbers and let $p$ be an odd natural number such that $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1$. Then, as integers, $p + 1$ divides $2^b + 2$.","title":"Odd solutions $p$ of $(2^{k+1} - 1)(p^2 + p + 1) = 2^{k+1} p^2 + 2^b p^2 + 1$ satisfy $p + 1 \\mid 2^b + 2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/strong-two-near-perfect-two-pow-mul-prim","kind":"node","lean":"MetaMathlibExt.strong_two_near_perfect_two_pow_mul_prime_form","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Strongly "],["math","\\(2\\)"],["text","-near perfect numbers of the form "],["math","\\(2^k p\\)"]],"source":"Strongly $2$-near perfect numbers of the form $2^k p$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#strong-two-near-perfect-two-pow-mul-prim","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be an odd prime, let $k$ be a natural number and let $n = 2^k p$. Suppose that $n$ is strongly $2$-near perfect, that is, $\\sigma(n) = 2n + d + n/d$ for some divisor $d$ of $n$, where $\\sigma$ is the sum-of-divisors function. Then there is a natural number $a \\le k$ with $p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1$, that is, $p = (2^{k+1} - 2^a - 1)/(1 + 2^{k-a})$.","title":"Strongly $2$-near perfect numbers of the form $2^k p$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/lt-two-mul-of-strong-two-near-perfect-fo","kind":"node","lean":"MetaMathlibExt.lt_two_mul_of_strong_two_near_perfect_form","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime solutions of "],["math","\\(p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1\\)"],["text"," satisfy "],["math","\\(k < 2a\\)"]],"source":"Prime solutions of $p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1$ satisfy $k < 2a$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#lt-two-mul-of-strong-two-near-perfect-fo","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $a \\le k$ be natural numbers and let $p$ be a prime with $p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1$, that is, $p = (2^{k+1} - 2^a - 1)/(1 + 2^{k-a})$. Then $k < 2a$.","title":"Prime solutions of $p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1$ satisfy $k < 2a$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/eq-two-of-two-pow-add-one-dvd","kind":"node","lean":"MetaMathlibExt.eq_two_of_two_pow_add_one_dvd","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","If "],["math","\\(b \\ge 1\\)"],["text"," and "],["math","\\(2^b + 1\\)"],["text"," divides "],["math","\\(3 \\cdot 2^a + 1\\)"],["text",", then "],["math","\\(b = 2\\)"]],"source":"If $b \\ge 1$ and $2^b + 1$ divides $3 \\cdot 2^a + 1$, then $b = 2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#eq-two-of-two-pow-add-one-dvd","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $a$ and $b$ be natural numbers with $b \\ge 1$. If $2^b + 1$ divides $3 \\cdot 2^a + 1$, then $b = 2$.","title":"If $b \\ge 1$ and $2^b + 1$ divides $3 \\cdot 2^a + 1$, then $b = 2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/two-near-perfect-two-pow-mul-prime-sq-ca","kind":"node","lean":"MetaMathlibExt.two_near_perfect_two_pow_mul_prime_sq_case_one","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-near perfect "],["math","\\(2^k p^2\\)"],["text"," whose omitted divisors are powers of "],["math","\\(2\\)"]],"source":"$2$-near perfect $2^k p^2$ whose omitted divisors are powers of $2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#two-near-perfect-two-pow-mul-prime-sq-ca","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be an odd prime, let $k$ be a natural number and let $i \\ne j$ be natural numbers with $i \\le k$ and $j \\le k$. If $n = 2^k p^2$ satisfies $\\sigma(n) = 2n + 2^i + 2^j$, where $\\sigma$ is the sum-of-divisors function (that is, $n$ is $2$-near perfect with omitted divisors $2^i$ and $2^j$), then $n = 18$ and either $i = 0, j = 1$ or $i = 1, j = 0$, so the omitted divisors are $1$ and $2$.","title":"$2$-near perfect $2^k p^2$ whose omitted divisors are powers of $2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/not-two-near-perfect-two-pow-mul-prime-s","kind":"node","lean":"MetaMathlibExt.not_two_near_perfect_two_pow_mul_prime_sq_case_two","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","No "],["math","\\(2\\)"],["text","-near perfect "],["math","\\(2^k p^2\\)"],["text"," with omitted divisors "],["math","\\(2^i\\)"],["text"," and "],["math","\\(p\\)"],["text"," or "],["math","\\(p^2\\)"]],"source":"No $2$-near perfect $2^k p^2$ with omitted divisors $2^i$ and $p$ or $p^2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#not-two-near-perfect-two-pow-mul-prime-s","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be an odd prime, let $k$ and $i$ be natural numbers with $i \\le k$, let $m$ be $1$ or $2$, and let $n = 2^k p^2$. Then $\\sigma(n) \\ne 2n + 2^i + p^m$, where $\\sigma$ is the sum-of-divisors function; that is, $n$ is not $2$-near perfect with omitted divisors $2^i$ and $p^m$.","title":"No $2$-near perfect $2^k p^2$ with omitted divisors $2^i$ and $p$ or $p^2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/two-near-perfect-two-pow-mul-prime-sq-ca-2","kind":"node","lean":"MetaMathlibExt.two_near_perfect_two_pow_mul_prime_sq_case_three","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-near perfect "],["math","\\(2^k p^2\\)"],["text"," with omitted divisors "],["math","\\(1\\)"],["text"," and "],["math","\\(2^b p^2\\)"]],"source":"$2$-near perfect $2^k p^2$ with omitted divisors $1$ and $2^b p^2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#two-near-perfect-two-pow-mul-prime-sq-ca-2","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be an odd prime and let $b, k$ be integers with $1 \\le b \\le k$. If $n = 2^k p^2$ satisfies $\\sigma(n) = 2n + 1 + 2^b p^2$, where $\\sigma$ is the sum-of-divisors function (that is, $n$ is $2$-near perfect with omitted divisors $1$ and $2^b p^2$), then $n = 36$ and $2^b p^2 = 18$.","title":"$2$-near perfect $2^k p^2$ with omitted divisors $1$ and $2^b p^2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/two-near-perfect-two-pow-mul-prime-sq-ca-3","kind":"node","lean":"MetaMathlibExt.two_near_perfect_two_pow_mul_prime_sq_case_five","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-near perfect "],["math","\\(2^k p^2\\)"],["text"," with omitted divisors "],["math","\\(p^m\\)"],["text"," and "],["math","\\(2^b p^j\\)"]],"source":"$2$-near perfect $2^k p^2$ with omitted divisors $p^m$ and $2^b p^j$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#two-near-perfect-two-pow-mul-prime-sq-ca-3","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $p$ be an odd prime, let each of $m$ and $j$ be $1$ or $2$, and let $b, k$ be integers with $1 \\le b \\le k$. If $n = 2^k p^2$ satisfies $\\sigma(n) = 2n + p^m + 2^b p^j$, where $\\sigma$ is the sum-of-divisors function (that is, $n$ is $2$-near perfect with omitted divisors $p^m$ and $2^b p^j$), then $n = 200$.","title":"$2$-near perfect $2^k p^2$ with omitted divisors $p^m$ and $2^b p^j$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/strong-two-near-perfect-two-pow-mul-prim-2","kind":"node","lean":"MetaMathlibExt.strong_two_near_perfect_two_pow_mul_prime_classification","parent":8104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classification of prime solutions of "],["math","\\(p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1\\)"]],"source":"Classification of prime solutions of $p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers/index.html#strong-two-near-perfect-two-pow-mul-prim-2","scope":"NumberTheory/arithmetic-functions/perfect-numbers/near-perfect-and-deficient-perfect-numbers","status":"can_prove","summary":"Let $a \\le k$ be natural numbers and let $p$ be a prime with $p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1$, that is, $p = (2^{k+1} - 2^a - 1)/(1 + 2^{k-a})$. Then $k = a + 2$, $2^k p / 2^a = 4p$, $5p = 2^{a+3} - 2^a - 1$, and $a \\equiv 3 \\pmod 4$.","title":"Classification of prime solutions of $p(1 + 2^{k-a}) = 2^{k+1} - 2^a - 1$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers.json","id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","kind":"scope","navigation":"module","parent":8070,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","title":"Primitive non-deficient numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/primitive-nondeficient-prime-power-bound","kind":"node","lean":"MetaMathlibExt.primitive_nondeficient_prime_power_bound","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#primitive-nondeficient-prime-power-bound","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"fully_proved","summary":"Let $\\sigma(m)$ denote the sum of the divisors of $m$. Let $n$ be a positive integer that is non-deficient, $\\sigma(n) \\ge 2n$, and primitive, meaning that every divisor $m$ of $n$ with $m < n$ is deficient, $\\sigma(m) < 2m$. Let $k$ be the number of distinct prime factors of $n$ and $R$ their product. Then $n$ has a prime factor $p$ with $p^{v_p(n) + 1} < 2kR$, where $v_p(n)$ is the exponent of $p$ in $n$.","title":"Prime-power bound for primitive non-deficient numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/is-nondeficient","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.IsNondeficient","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#is-nondeficient","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"fully_proved","summary":"A natural number $n$ is non-deficient when $n \\ne 0$ and $\\sigma(n) \\ge 2n$, where $\\sigma(n)$ is the sum of the positive divisors of $n$.","title":"Non-deficient number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/is-primitive-nondeficient","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.IsPrimitiveNondeficient","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#is-primitive-nondeficient","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"fully_proved","summary":"A natural number $n$ is primitive non-deficient when it is non-deficient and every proper divisor $m$ of $n$ (a divisor with $m < n$) is deficient, that is, $\\sigma(m) < 2m$, where $\\sigma(m)$ is the sum of the positive divisors of $m$.","title":"Primitive non-deficient number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/abundancy-index","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.abundancyIndex","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#abundancy-index","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"fully_proved","summary":"For a natural number $n$, the abundancy index is the rational number $h(n) = \\sigma(n)/n$, where $\\sigma(n)$ is the sum of the positive divisors of $n$.","title":"Abundancy index"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/abundancy-sup","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.abundancySup","parent":8157,"reading_title":{"format":"inline-parts-v1","parts":[["text","Product of "],["math","\\(p/(p-1)\\)"],["text"," over the prime factors"]],"source":"Product of $p/(p-1)$ over the prime factors"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#abundancy-sup","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"fully_proved","summary":"For a natural number $n$, let $H(n) = \\prod_p p/(p-1)$ in $\\mathbb{Q}$, the product over the distinct primes $p$ dividing $n$. For $n = 0$ the product is empty ($0$ is taken to have no prime factors), so $H(0) = 1$.","title":"Product of $p/(p-1)$ over the prime factors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/is-primitive-nondeficient-two-pow-mul-pr","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.isPrimitiveNondeficient_two_pow_mul_prime_iff","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#is-primitive-nondeficient-two-pow-mul-pr","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"Let $p$ be an odd prime and $a$ a natural number. Then $2^a p$ is primitive non-deficient if and only if $2^a < p \\le 2^{a+1} - 1$.","title":"Even primitive non-deficient numbers with two prime factors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/deficient-of-odd-card-prime-factors-eq-t","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.deficient_of_odd_card_primeFactors_eq_two","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#deficient-of-odd-card-prime-factors-eq-t","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"If $n$ is an odd natural number with exactly two distinct prime divisors, then $\\sigma(n) < 2n$, where $\\sigma(n)$ is the sum of the positive divisors of $n$.","title":"Odd numbers with two prime factors are deficient"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/exists-pow-factorization-add-one-lt-two","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.exists_pow_factorization_add_one_lt_two_mul_radical","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#exists-pow-factorization-add-one-lt-two","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"Let $n$ be an odd primitive non-deficient number with exactly $3$ or exactly $4$ distinct prime factors, and let $R$ be the product of the distinct prime factors of $n$. Then some prime factor $p$ of $n$ satisfies $p^{a+1} < 2R$, where $a$ is the exponent of $p$ in $n$.","title":"Small component for odd primitive non-deficient numbers with three or four prime factors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/grun-bounds-of-is-nondeficient","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.grun_bounds_of_isNondeficient","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#grun-bounds-of-is-nondeficient","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"Let $n$ be a non-deficient number with $k$ distinct prime factors, the least of which is $p_1$. Then $\\tfrac{3}{2} p_1 - 2 < k$, and if $p_1 \\ge 5$ then $p_1 + 2 \\le k$.","title":"Gr\u00fcn's bound for non-deficient numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/two-pow-factorization-add-one-lt-radical","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.two_pow_factorization_add_one_lt_radical","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#two-pow-factorization-add-one-lt-radical","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"Let $n$ be an even primitive non-deficient number, that is, a positive integer with $\\sigma(n) \\ge 2n$ all of whose proper divisors $m$ satisfy $\\sigma(m) < 2m$, where $\\sigma$ is the sum of divisors. If $2^a$ is the largest power of $2$ dividing $n$ and $R$ is the product of the distinct primes dividing $n$, then $2^{a+1} < R$.","title":"The power of 2 in an even primitive non-deficient number is below its radical"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/one-div-one-sub-mul-lt-add","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.one_div_one_sub_mul_lt_add","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#one-div-one-sub-mul-lt-add","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"For real numbers $k$ and $m$ with $k \\ge 2$ and $0 < m < 1/(k(k+1))$, we have $1/(1 - km) < 1 + (k+1)m$.","title":"Zelinsky's reciprocal bound"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/abundancy-index-div-le","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.abundancyIndex_div_le","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#abundancy-index-div-le","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"Let $n$ be a positive integer, $p$ a prime and $a \\ge 1$ an integer such that $p^a$ exactly divides $n$ (that is, $p^a \\mid n$ and $p^{a+1} \\nmid n$). Then $h(n)/h(n/p) \\le 1 + 1/p^a$, where $h(m) = \\sigma(m)/m$ is the abundancy index and $\\sigma(m)$ is the sum of the positive divisors of $m$.","title":"Abundancy ratio at an exactly dividing prime power"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/exists-pow-factorization-add-one-lt-max","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.exists_pow_factorization_add_one_lt_max","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#exists-pow-factorization-add-one-lt-max","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"Let $n$ be a primitive non-deficient number with $k$ distinct prime factors, the least of which is $p_1$, and suppose that $(p_1, k) \\ne (3, 3)$. Let $\\alpha$ be a rational number with $0 < \\alpha < 1$ such that $H(n) \\ge 2 + \\alpha$, where $H(n) = \\prod_p p/(p-1)$ is the product over the distinct primes $p$ dividing $n$. Then some prime factor $p$ of $n$ satisfies\u2026","title":"Generalized Puchta lemma"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/max-prime-factor-lt-sqrt-radical-or-exis","kind":"node","lean":"MathlibExt.NumberTheory.PrimitiveNondeficientWanted.max_primeFactor_lt_sqrt_radical_or_exists_pow_lt","parent":8157,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers/index.html#max-prime-factor-lt-sqrt-radical-or-exis","scope":"NumberTheory/arithmetic-functions/perfect-numbers/primitive-non-deficient-numbers","status":"can_prove","summary":"Let $n$ be an odd primitive non-deficient number, that is, a positive integer with $\\sigma(n) \\ge 2n$ all of whose proper divisors $m$ satisfy $\\sigma(m) < 2m$, where $\\sigma$ is the sum of divisors. Let $k$ be the number of distinct prime factors of $n$, let $p_1$ and $p_k$ be its least and greatest prime factors, and let $R$ be the product of the distinct primes dividing $n$. Then either $p_k < \\sqrt{R}$, or some\u2026","title":"Dichotomy for odd primitive non-deficient numbers"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers.json","id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","kind":"scope","navigation":"module","parent":8070,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","title":"Odd perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/eulerian-form","kind":"node","lean":"MetaMathlibExt.EulerianForm","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#eulerian-form","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"fully_proved","summary":"For natural numbers $N$, $q$, $k$ and $n$, the quadruple $(N, q, k, n)$ is in Eulerian form when $N$ is an odd perfect number, $q$ is prime, $N = q^k n^2$, $q \\equiv 1 \\pmod 4$, $k \\equiv 1 \\pmod 4$, and $q$ and $n$ are coprime.","title":"Eulerian form of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/is-euler-prime","kind":"node","lean":"MetaMathlibExt.IsEulerPrime","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#is-euler-prime","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"fully_proved","summary":"A natural number $q$ is an Euler prime of a natural number $N$ when there are natural numbers $k$ and $n$ such that $(N, q, k, n)$ is in Eulerian form: $N$ is an odd perfect number, $q$ is prime, $N = q^k n^2$, $q \\equiv 1 \\pmod 4$, $k \\equiv 1 \\pmod 4$, and $q$ and $n$ are coprime.","title":"Euler prime of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/euler-odd-perfect-form","kind":"node","lean":"MetaMathlibExt.euler_odd_perfect_form","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#euler-odd-perfect-form","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"fully_proved","summary":"Every odd perfect number $n$ can be written as $n = q^s m^2$ with natural numbers $q$, $s$ and $m$, where $q$ is prime, $q \\equiv 1 \\pmod 4$, $s \\equiv 1 \\pmod 4$, and $q$ and $m$ are coprime.","title":"Euler's theorem on odd perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/prod-one-sub-inv-le-of-prefix-prod-le","kind":"node","lean":"MetaMathlibExt.prod_one_sub_inv_le_of_prefix_prod_le","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Comparison of products of "],["math","\\(1 - 1/x\\)"],["text"," under prefix-product domination"]],"source":"Comparison of products of $1 - 1/x$ under prefix-product domination"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#prod-one-sub-inv-le-of-prefix-prod-le","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $k \\ge 0$ and let $z_1 \\le \\dots \\le z_k$ and $y_1 \\le \\dots \\le y_k$ be real numbers, all greater than $1$, such that $z_1 \\cdots z_l \\le y_1 \\cdots y_l$ for every $l$ with $0 \\le l \\le k$ (for $l = 0$ both sides are the empty product). Then $\\prod_{i=1}^k (1 - 1/z_i) \\le \\prod_{i=1}^k (1 - 1/y_i)$, with equality if and only if $z_i = y_i$ for all $i$.","title":"Comparison of products of $1 - 1/x$ under prefix-product domination"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/hbc-admissible-half-first-le","kind":"node","lean":"MetaMathlibExt.hbcAdmissible_half_first_le","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","First-element bounds for HBC-admissible tuples at "],["math","\\(1/2\\)"]],"source":"First-element bounds for HBC-admissible tuples at $1/2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#hbc-admissible-half-first-le","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $k \\ge 1$ and let $y_1 < y_2 < \\dots < y_k$ be positive integers with $\\prod_{i=1}^{k} (1 - 1/y_i) \\le 1/2 < \\prod_{i=1}^{k-1} (1 - 1/y_i)$. Then $y_1 \\le k + 1$; if every $y_i$ is odd, then $y_1 < 2(k + 3)/3$; and if every $y_i$ is prime, then $y_1 < 3k/7 + 3$.","title":"First-element bounds for HBC-admissible tuples at $1/2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-min-fac-lt","kind":"node","lean":"MetaMathlibExt.oddPerfect_minFac_lt","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-min-fac-lt","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"If $N$ is an odd perfect number with $m$ distinct prime divisors, then the smallest prime divisor $p_1$ of $N$ satisfies $p_1 < 3m/7 + 3$.","title":"A linear bound on the smallest prime divisor of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-five-le-card-prime-factors","kind":"node","lean":"MetaMathlibExt.oddPerfect_five_le_card_primeFactors","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-five-le-card-prime-factors","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Every odd perfect number has at least five distinct prime divisors.","title":"Sylvester's bound on the number of prime divisors of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/finite-set-of-odd-perfect-card-prime-fac","kind":"node","lean":"MetaMathlibExt.finite_setOf_oddPerfect_card_primeFactors_eq","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#finite-set-of-odd-perfect-card-prime-fac","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"For every natural number $m$, there are only finitely many odd perfect numbers with exactly $m$ distinct prime divisors.","title":"Dickson's finiteness theorem for odd perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-exists-even-factorization-ne","kind":"node","lean":"MetaMathlibExt.oddPerfect_exists_even_factorization_ne_two","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-exists-even-factorization-ne","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"If $N$ is an odd perfect number, then $N$ has a prime divisor whose exponent in the prime factorization of $N$ is even and different from $2$.","title":"Steuerwald's theorem on odd perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-exists-six-le-factorization","kind":"node","lean":"MetaMathlibExt.oddPerfect_exists_six_le_factorization_of_lt","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","McDaniel's exponent bound for odd perfect numbers below "],["math","\\(10^{9118}\\)"]],"source":"McDaniel's exponent bound for odd perfect numbers below $10^{9118}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-exists-six-le-factorization","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"If $N$ is an odd perfect number with $N < 10^{9118}$, then $N$ has a prime divisor whose exponent in the prime factorization of $N$ is at least $6$.","title":"McDaniel's exponent bound for odd perfect numbers below $10^{9118}$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-eleven-le-card-prime-factors","kind":"node","lean":"MetaMathlibExt.oddPerfect_eleven_le_card_primeFactors_of_not_three_dvd","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-eleven-le-card-prime-factors","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Every odd perfect number that is not divisible by $3$ has at least $11$ distinct prime divisors. This is due to Kishore (1981) and Hagis (1983).","title":"Odd perfect numbers not divisible by three have at least eleven prime divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-ten-le-card-prime-factors","kind":"node","lean":"MetaMathlibExt.oddPerfect_ten_le_card_primeFactors","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-ten-le-card-prime-factors","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Every odd perfect number $N$ has at least ten distinct prime divisors, and at least twelve if $3$ does not divide $N$.","title":"Nielsen's bound on the number of prime divisors of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-min-fac-lt-card-sub-one-div","kind":"node","lean":"MetaMathlibExt.oddPerfect_minFac_lt_card_sub_one_div_two","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-min-fac-lt-card-sub-one-div","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"If $N$ is an odd perfect number with $m$ distinct prime divisors, then the smallest prime divisor $p$ of $N$ satisfies $p < (m - 1)/2$.","title":"Zelinsky's bound on the smallest prime divisor of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/ten-pow-fifteen-hundred-lt-of-odd-perfec","kind":"node","lean":"MetaMathlibExt.ten_pow_fifteen_hundred_lt_of_oddPerfect","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#ten-pow-fifteen-hundred-lt-of-odd-perfec","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Every odd perfect number exceeds $10^{1500}$.","title":"Ochem\u2013Rao lower bound for odd perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-exists-three-large-prime-div","kind":"node","lean":"MetaMathlibExt.oddPerfect_exists_three_large_prime_divisors","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-exists-three-large-prime-div","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"If $N$ is an odd perfect number, then $N$ has three distinct prime divisors $p$, $q$, $r$ with $p \\ge 10^8$, $q \\ge 10^4$ and $r \\ge 100$. Equivalently, the largest, second largest and third largest prime divisors of $N$ are at least $10^8$, $10^4$ and $100$ respectively.","title":"Goto\u2013Ohno\u2013Iannucci bounds on the largest prime divisors of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/hbc-calibrating-prod-identities","kind":"node","lean":"MetaMathlibExt.hbcCalibrating_prod_identities","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#hbc-calibrating-prod-identities","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $a$ and $d$ be positive integers and let $k \\ge 1$. Put $\\alpha = (a + d)/d$, and let $n_i = 1 + \\alpha^{2^{i-1}}$ for $1 \\le i < k$ and $n_k = \\alpha^{2^{k-1}}$ (the calibrating numbers for $a/(a + d)$). Then $\\prod_{i=1}^{k} (1 - 1/n_i) = a/(a + d)$; for every $j$ with $1 \\le j < k$, $\\prod_{i=1}^{j} n_i = (d/a)(\\alpha^{2^j} - 1)$; and $\\prod_{i=1}^{k} n_i = (d/a)(\\alpha^{2^k} - \\alpha^{2^{k-1}})$.","title":"Product identities for calibrating numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/hbc-admissible-exists-prefix-le-and-prod","kind":"node","lean":"MetaMathlibExt.hbcAdmissible_exists_prefix_le_and_prod_le","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#hbc-admissible-exists-prefix-le-and-prod","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $a$ and $d$ be coprime positive integers, let $k \\ge 1$, and let $y_1 < y_2 < \\dots < y_k$ be positive integers with $\\prod_{i=1}^{k} (1 - 1/y_i) \\le a/(a + d) < \\prod_{i=1}^{k-1} (1 - 1/y_i)$ (the source calls such a tuple HBC-admissible for $a/(a + d)$). Put $\\alpha = (a + d)/d$, let $n_i = 1 + \\alpha^{2^{i-1}}$ for $1 \\le i < k$ and $n_k = \\alpha^{2^{k-1}}$, and let $F_k(x) = x^{2^k} - x^{2^{k-1}}$. Then\u2026","title":"Stone's product bound for HBC-admissible tuples"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/hbc-admissible-prod-le-sqrt","kind":"node","lean":"MetaMathlibExt.hbcAdmissible_prod_le_sqrt","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stone's "],["math","\\(\\sqrt{d}\\)"],["text"," product bound for HBC-admissible tuples"]],"source":"Stone's $\\sqrt{d}$ product bound for HBC-admissible tuples"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#hbc-admissible-prod-le-sqrt","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $a$ and $d$ be coprime positive integers, let $k \\ge 1$, and let $y_1 < y_2 < \\dots < y_k$ be positive integers with $\\prod_{i=1}^{k} (1 - 1/y_i) \\le a/(a + d) < \\prod_{i=1}^{k-1} (1 - 1/y_i)$. Put $\\beta = (a + d)/\\sqrt{d}$. Then $a \\prod_{i=1}^{k} y_i \\le \\beta^{2^k} - \\beta^{2^{k-1}}$.","title":"Stone's $\\sqrt{d}$ product bound for HBC-admissible tuples"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/hbc-admissible-prod-le-of-ne-floor","kind":"node","lean":"MetaMathlibExt.hbcAdmissible_prod_le_of_ne_floor","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stone's "],["math","\\(d^{3/4}\\)"],["text"," product bound for HBC-admissible tuples"]],"source":"Stone's $d^{3/4}$ product bound for HBC-admissible tuples"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#hbc-admissible-prod-le-of-ne-floor","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $a$ and $d$ be coprime positive integers, let $k \\ge 2$, and let $y_1 < y_2 < \\dots < y_k$ be positive integers with $\\prod_{i=1}^{k} (1 - 1/y_i) \\le a/(a + d) < \\prod_{i=1}^{k-1} (1 - 1/y_i)$ and $y_1 \\ne \\lfloor 1 + (a + d)/d \\rfloor$. Put $\\beta = (a + d)/d^{3/4}$. Then $a \\prod_{i=1}^{k} y_i \\le \\beta^{2^k} - \\beta^{2^{k-1}}$.","title":"Stone's $d^{3/4}$ product bound for HBC-admissible tuples"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/hbc-admissible-seven-tenths-prod-le","kind":"node","lean":"MetaMathlibExt.hbcAdmissible_seven_tenths_prod_le","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Product bound for HBC-admissible tuples at "],["math","\\(7/10\\)"]],"source":"Product bound for HBC-admissible tuples at $7/10$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#hbc-admissible-seven-tenths-prod-le","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $k \\ge 3$ and let $y_1 < y_2 < \\dots < y_k$ be positive integers with $\\prod_{i=1}^{k} (1 - 1/y_i) \\le 7/10 < \\prod_{i=1}^{k-1} (1 - 1/y_i)$. Then $\\prod_{i=1}^{k} y_i \\le \\frac{2}{7}\\bigl(\\sqrt{15}^{\\,2^k} - \\sqrt{15}^{\\,2^{k-1}}\\bigr)$.","title":"Product bound for HBC-admissible tuples at $7/10$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/hbc-admissible-half-prime-prod-le-of-fiv","kind":"node","lean":"MetaMathlibExt.hbcAdmissible_half_prime_prod_le_of_five_le","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Product bounds for prime tuples at "],["math","\\(1/2\\)"],["text"," by size of the first prime"]],"source":"Product bounds for prime tuples at $1/2$ by size of the first prime"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#hbc-admissible-half-prime-prod-le-of-fiv","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $p_1 < p_2 < \\dots < p_k$ be primes with $\\prod_{i=1}^{k} (1 - 1/p_i) \\le 1/2 < \\prod_{i=1}^{k-1} (1 - 1/p_i)$ and $p_1 \\ge 5$, and let $F_k(x) = x^{2^k} - x^{2^{k-1}}$ and $P = \\prod_{i=1}^{k} p_i$. Then $P \\le F_k(\\sqrt{8/3^{7/8}})$. Moreover, if $6 \\le p_1 < 18$, then $P \\le F_k(\\sqrt{12/5^{15/16}})$; if $18 \\le p_1 < 258$, then $P \\le F_k(\\sqrt{512/255^{31/32}})$; and if $p_1 \\ge 258$, then\u2026","title":"Product bounds for prime tuples at $1/2$ by size of the first prime"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/hbc-admissible-half-prime-prod-le-of-eq","kind":"node","lean":"MetaMathlibExt.hbcAdmissible_half_prime_prod_le_of_eq_five","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Product bound for prime tuples at "],["math","\\(1/2\\)"],["text"," starting at "],["math","\\(5\\)"]],"source":"Product bound for prime tuples at $1/2$ starting at $5$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#hbc-admissible-half-prime-prod-le-of-eq","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $p_1 < p_2 < \\dots < p_k$ be primes with $\\prod_{i=1}^{k} (1 - 1/p_i) \\le 1/2 < \\prod_{i=1}^{k-1} (1 - 1/p_i)$ and $p_1 = 5$. Put $x = \\sqrt{8/3^{31/32}}$. Then $\\prod_{i=1}^{k} p_i \\le x^{2^k} - x^{2^{k-1}}$.","title":"Product bound for prime tuples at $1/2$ starting at $5$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-prime-reciprocal-sum-lt-log","kind":"node","lean":"MetaMathlibExt.oddPerfect_primeReciprocalSum_lt_log_two_sub","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-prime-reciprocal-sum-lt-log","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $n$ be an odd perfect number whose least prime factor $p_1$ satisfies $p_1 \\ge 11$, and let $T(n) = \\sum_{p \\mid n} 1/p$, the sum over the primes $p$ dividing $n$. Then $T(n) < \\log 2 - \\frac{11}{50 p_1^2}$.","title":"Zelinsky's upper bound on the prime reciprocal sum of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-abundancy-sup-sub-two-lt","kind":"node","lean":"MetaMathlibExt.oddPerfect_abundancySup_sub_two_lt","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-abundancy-sup-sub-two-lt","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $n$ be an odd perfect number whose least prime factor $p_1$ satisfies $p_1 \\ge 11$, and let $H(n) = \\prod_{p \\mid n} p/(p-1)$, the product over the primes $p$ dividing $n$. Then its surplus $H(n) - 2$ satisfies $H(n) - 2 < \\frac{25}{8 p_1 \\log p_1}$.","title":"Surplus of an odd perfect number with least prime factor at least 11"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/exp-add-two-fifths-sq-le-one-add-add-sq","kind":"node","lean":"MetaMathlibExt.exp_add_two_fifths_sq_le_one_add_add_sq","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exponential lower bound for "],["math","\\(1 + x + x^2\\)"],["text"," on "],["math","\\((0, 1/11)\\)"]],"source":"Exponential lower bound for $1 + x + x^2$ on $(0, 1/11)$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#exp-add-two-fifths-sq-le-one-add-add-sq","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"For every real $x$ with $0 \\le x \\le 1/11$, $\\exp(x + 2x^2/5) \\le 1 + x + x^2$, with equality if and only if $x = 0$.","title":"Exponential lower bound for $1 + x + x^2$ on $(0, 1/11)$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/div-log-add-one-le-prime-counting","kind":"node","lean":"MetaMathlibExt.div_log_add_one_le_primeCounting","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for the prime-counting function at primes "],["math","\\(p \\ge 19\\)"]],"source":"Lower bound for the prime-counting function at primes $p \\ge 19$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#div-log-add-one-le-prime-counting","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"For every prime $p \\ge 19$, the number $\\pi(p)$ of primes at most $p$ satisfies $\\pi(p) \\ge p/\\log p + 1$, where $\\log$ is the natural logarithm.","title":"Lower bound for the prime-counting function at primes $p \\ge 19$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/tsum-one-div-sq-primes-ge-lt","kind":"node","lean":"MetaMathlibExt.tsum_one_div_sq_primes_ge_lt","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tail bound for the sum of "],["math","\\(1/p^2\\)"],["text"," over primes"]],"source":"Tail bound for the sum of $1/p^2$ over primes"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#tsum-one-div-sq-primes-ge-lt","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $q \\ge 3$ be a prime and let $M(q) = \\sum_{p \\ge q} 1/p^2$ be the sum over all primes $p \\ge q$. Then $M(q) < \\frac{1}{q \\log q}\\left(1 + \\frac{2 \\log \\log q}{\\log q}\\right)$, where $\\log$ is the natural logarithm.","title":"Tail bound for the sum of $1/p^2$ over primes"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/tsum-log-sq-div-sq-sub-one-primes-ge-le","kind":"node","lean":"MetaMathlibExt.tsum_log_sq_div_sq_sub_one_primes_ge_le","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on the tail of the prime product of "],["math","\\(p^2/(p^2-1)\\)"]],"source":"Bound on the tail of the prime product of $p^2/(p^2-1)$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#tsum-log-sq-div-sq-sub-one-primes-ge-le","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $q$ be an odd prime. Then $\\sum_{p \\ge q} \\log \\frac{p^2}{p^2-1} \\le \\frac{1}{q \\log q}\\left(1 + \\frac{2 \\log \\log q}{\\log q}\\right) + \\frac{4}{q^3}$, where the sum runs over the primes $p \\ge q$. Equivalently, $\\log E$ is at most this bound, where $E = \\prod_{p \\ge q} \\frac{p^2}{p^2-1}$ is the product over the primes $p \\ge q$.","title":"Bound on the tail of the prime product of $p^2/(p^2-1)$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/log-one-add-one-div-sub-one-le","kind":"node","lean":"MetaMathlibExt.log_one_add_one_div_sub_one_le","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for "],["math","\\(\\log(1 + 1/(x-1))\\)"]],"source":"Upper bound for $\\log(1 + 1/(x-1))$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#log-one-add-one-div-sub-one-le","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"For every integer $x \\ge 2$, $\\log(1 + 1/(x-1)) \\le 1/x + 25/(32x^2)$, where $\\log$ is the natural logarithm.","title":"Upper bound for $\\log(1 + 1/(x-1))$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/abundancy-sup-div-abundancy-index-le-of","kind":"node","lean":"MetaMathlibExt.abundancySup_div_abundancyIndex_le_of_single_odd_exponent","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Suryanarayana's bound for "],["math","\\(H(n)/h(n)\\)"],["text"," with one odd exponent"]],"source":"Suryanarayana's bound for $H(n)/h(n)$ with one odd exponent"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#abundancy-sup-div-abundancy-index-le-of","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"For a positive integer $n$ write $h(n) = \\sigma(n)/n$ for its abundancy index, where $\\sigma(n)$ is the sum of the divisors of $n$, and $H(n) = \\prod_{p \\mid n} p/(p-1)$ for the product over its prime divisors. Let $n$ be odd and let $j$ be a prime divisor of $n$ whose exponent in $n$ is odd, while the exponent of every other prime divisor of $n$ is even. Then","title":"Suryanarayana's bound for $H(n)/h(n)$ with one odd exponent"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-factorization-five-eq-one-of","kind":"node","lean":"MetaMathlibExt.oddPerfect_factorization_five_eq_one_of_dvd","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-factorization-five-eq-one-of","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"If $n$ is an odd perfect number divisible by $165$, then $5$ divides $n$ exactly once: $5 \\mid n$ but $25 \\nmid n$.","title":"Odd perfect numbers divisible by 165 contain 5 exactly once"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/le-two-mul-prod-sub-one-sub-one-of-two-l","kind":"node","lean":"MetaMathlibExt.le_two_mul_prod_sub_one_sub_one_of_two_lt","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on a prime that pushes "],["math","\\(\\prod p/(p-1)\\)"],["text"," past "],["math","\\(2\\)"]],"source":"Bound on a prime that pushes $\\prod p/(p-1)$ past $2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#le-two-mul-prod-sub-one-sub-one-of-two-l","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $s$ be a finite set of odd primes with $\\prod_{p \\in s} p/(p-1) \\le 2$, and let $q$ be an odd prime with $\\left(\\prod_{p \\in s} p/(p-1)\\right) \\cdot q/(q-1) > 2$. Then $q \\le 2 \\prod_{p \\in s} (p - 1) - 1$.","title":"Bound on a prime that pushes $\\prod p/(p-1)$ past $2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-prime-reciprocal-sum-le-of-g","kind":"node","lean":"MetaMathlibExt.oddPerfect_primeReciprocalSum_le_of_gcd_fifteen","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-prime-reciprocal-sum-le-of-g","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $n$ be an odd perfect number and $T(n) = \\sum_{p \\mid n} 1/p$, the sum over the primes $p$ dividing $n$. If $\\gcd(15, n) = 1$ then $T(n) \\le 0.693148$; if $\\gcd(15, n) = 3$ then $T(n) \\le 0.657304$; if $\\gcd(15, n) = 5$ then $T(n) \\le 0.678036$; and if $\\gcd(15, n) = 15$ then $T(n) \\le 0.673770$.","title":"Cohen\u2013Hagis\u2013Suryanarayana bounds on the prime reciprocal sum of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-abundancy-sup-le-of-gcd-fift","kind":"node","lean":"MetaMathlibExt.oddPerfect_abundancySup_le_of_gcd_fifteen","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-abundancy-sup-le-of-gcd-fift","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $n$ be an odd perfect number and $H(n) = \\prod_{p \\mid n} p/(p-1)$, the product over the primes $p$ dividing $n$. If $\\gcd(15, n) = 1$ then $H(n) \\le 2.014754$; if $\\gcd(15, n) = 3$ then $H(n) \\le 2.096234$; if $\\gcd(15, n) = 5$ then $H(n) \\le 2.031002$; and if $\\gcd(15, n) = 15$ then $H(n) \\le 2.165439$.","title":"Suryanarayana's bounds on the prime product of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/primitive-nondeficient-prime-reciprocal","kind":"node","lean":"MetaMathlibExt.primitiveNondeficient_primeReciprocalSum_ge","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#primitive-nondeficient-prime-reciprocal","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $n$ be a primitive non-deficient number with least prime factor $p_1$, let $T(n) = \\sum_{p \\mid n} 1/p$ and $H(n) = \\prod_{p \\mid n} p/(p-1)$, with the sum and product over the primes $p$ dividing $n$, and let $S = H(n) - 2$ be the surplus of $n$. Then $T(n) \\ge \\log 2 - \\frac{25}{64 p_1} + \\frac{S}{2} - \\frac{S^2}{4}$.","title":"Lower bound on the prime reciprocal sum of a primitive non-deficient number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/primitive-nondeficient-prime-reciprocal-2","kind":"node","lean":"MetaMathlibExt.primitiveNondeficient_primeReciprocalSum_lt_of_single_odd_exponent","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#primitive-nondeficient-prime-reciprocal-2","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Upper bound on the prime reciprocal sum of a primitive non-deficient number with one odd exponent. Let $n$ be a primitive non-deficient number with least prime factor $p_1 \\ge 11$, and let $t$ be a prime factor of $n$ such that the exponent of $t$ in $n$ is odd and the exponent of every other prime factor of $n$ is even. Suppose $2 p_1 \\le t + 2$, and let $T(n) = \\sum_{p \\mid n} 1/p$, the sum over the primes $p$\u2026","title":"Upper bound on the prime reciprocal sum of a primitive non-deficient number with one odd exponent"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-perfect-abundancy-sup-le","kind":"node","lean":"MetaMathlibExt.oddPerfect_abundancySup_le","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-perfect-abundancy-sup-le","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $n$ be an odd perfect number with least prime factor $p_1$, and let $H(n) = \\prod_{p \\mid n} p/(p-1)$, the product over the primes $p$ dividing $n$. Then $H(n) \\le 2 + \\frac{9}{4 p_1^2}$.","title":"Bound on the prime product of an odd perfect number via its least prime factor"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/odd-primitive-nondeficient-prime-recipro","kind":"node","lean":"MetaMathlibExt.odd_primitiveNondeficient_primeReciprocalSum_ge","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#odd-primitive-nondeficient-prime-recipro","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"Let $n$ be an odd primitive non-deficient number with least prime factor $p_1$, let $T(n) = \\sum_{p \\mid n} 1/p$ and $H(n) = \\prod_{p \\mid n} p/(p-1)$, with the sum and product over the primes $p$ dividing $n$, and let $S = H(n) - 2$ be the surplus of $n$. Then $T(n) \\ge \\log 2 - \\frac{25}{32 p_1 \\log p_1} \\left(1 + \\frac{2 \\log \\log p_1}{\\log p_1}\\right) + \\frac{S}{2} - \\frac{S^2}{4}$.","title":"Lower bound on the prime reciprocal sum of an odd primitive non-deficient number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/exists-abundancy-sup-le-two-add-div-min","kind":"node","lean":"MetaMathlibExt.exists_abundancySup_le_two_add_div_minFac","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on "],["math","\\(H(n)\\)"],["text"," for primitive non-deficient numbers by the least prime factor"]],"source":"Bound on $H(n)$ for primitive non-deficient numbers by the least prime factor"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#exists-abundancy-sup-le-two-add-div-min","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"For a positive integer $n$ write $H(n) = \\prod_{p \\mid n} p/(p-1)$, the product over the prime divisors of $n$. There is a real constant $C$ such that every primitive non-deficient number $n$ with least prime factor $p_1$ satisfies $H(n) \\le 2 + C/p_1$.","title":"Bound on $H(n)$ for primitive non-deficient numbers by the least prime factor"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/not-is-euler-prime-of-mod-165-eq","kind":"node","lean":"MetaMathlibExt.not_isEulerPrime_of_mod_165_eq","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes congruent to "],["math","\\(-1\\)"],["text"," modulo "],["math","\\(165\\)"],["text"," are not Euler primes"]],"source":"Primes congruent to $-1$ modulo $165$ are not Euler primes"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#not-is-euler-prime-of-mod-165-eq","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"For all natural numbers $q$ and $N$ with $q \\equiv -1 \\pmod{165}$, $q$ is not an Euler prime of $N$. In particular, no prime $q \\equiv -1 \\pmod{165}$ is the Euler prime of an odd perfect number.","title":"Primes congruent to $-1$ modulo $165$ are not Euler primes"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/not-abundancy-sup-odd-part-succ-lt-two-a","kind":"node","lean":"MetaMathlibExt.not_abundancySup_oddPart_succ_lt_two_and_two_lt","parent":8172,"reading_title":{"format":"inline-parts-v1","parts":[["text","No odd prime "],["math","\\(q\\)"],["text"," lifts "],["math","\\(H\\)"],["text"," of the odd part of "],["math","\\(q + 1\\)"],["text"," across "],["math","\\(2\\)"]],"source":"No odd prime $q$ lifts $H$ of the odd part of $q + 1$ across $2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#not-abundancy-sup-odd-part-succ-lt-two-a","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"For a positive integer $m$ write $H(m) = \\prod_{p \\mid m} p/(p-1)$, the product over the prime divisors of $m$, and $\\operatorname{odd}(m)$ for the largest odd divisor of $m$. There is no odd prime $q$ with $H(\\operatorname{odd}(q+1)) < 2$ and $H(q \\cdot \\operatorname{odd}(q+1)) > 2$.","title":"No odd prime $q$ lifts $H$ of the odd part of $q + 1$ across $2$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/eulerian-form-exponent-eq-one-or-pow-fiv","kind":"node","lean":"MetaMathlibExt.eulerianForm_exponent_eq_one_or_pow_five_lt","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#eulerian-form-exponent-eq-one-or-pow-fiv","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"If $(n, q, e, m)$ is in Eulerian form, that is, $n$ is an odd perfect number, $q$ is a prime coprime to $m$, $n = q^e m^2$ and $q \\equiv e \\equiv 1 \\pmod 4$, then $e = 1$ or $q^5 < 2m^2$, that is, $q < 2^{1/5} m^{2/5}$.","title":"Fifth-power bound on the Euler prime of an odd perfect number"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/eulerian-form-exponent-eq-one-or-mul-pow","kind":"node","lean":"MetaMathlibExt.eulerianForm_exponent_eq_one_or_mul_pow_five_lt","parent":8172,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers/index.html#eulerian-form-exponent-eq-one-or-mul-pow","scope":"NumberTheory/arithmetic-functions/perfect-numbers/odd-perfect-numbers","status":"can_prove","summary":"If $(n, q, e, m)$ is in Eulerian form, that is, $n$ is an odd perfect number, $q$ is a prime coprime to $m$, $n = q^e m^2$ and $q \\equiv e \\equiv 1 \\pmod 4$, then $e = 1$ or $3^2 \\cdot 5^2 \\cdot 11^2 \\cdot 13^2 \\cdot q^5 < 2m^2$, that is, $q < \\left(2/(3^2 \\cdot 5^2 \\cdot 11^2 \\cdot 13^2)\\right)^{1/5} m^{2/5}$.","title":"Refined fifth-power bound on the Euler prime of an odd perfect number"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers.json","id":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers","kind":"scope","navigation":"module","parent":8070,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers","status":"can_prove","title":"Pseudoperfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/in-a","kind":"node","lean":"MathlibExt.NumberTheory.DivisorSumReciprocalConvergenceWanted.InA","parent":8216,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/index.html#in-a","scope":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers","status":"fully_proved","summary":"A natural number $n$ lies in $A$, the set of primitive pseudoperfect numbers, when $n > 0$, $n$ is pseudoperfect (it is positive and equal to the sum of some set of distinct proper divisors of itself), and no proper divisor of $n$ (a divisor $m \\mid n$ with $m < n$) is pseudoperfect.","title":"Primitive pseudoperfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.DivisorSumReciprocalConvergenceWanted.conjecture","parent":8216,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/index.html#conjecture","scope":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers","status":"fully_proved","summary":"The proposition that the reciprocals of the primitive pseudoperfect numbers have finite sum: the family $(1/n)$, indexed by the natural numbers $n$ that are primitive pseudoperfect, is summable in $\\mathbb{R}$. Here $n$ is pseudoperfect if $n > 0$ and $n$ is the sum of some set of distinct proper divisors of $n$, and primitive pseudoperfect if moreover no proper divisor of $n$ is pseudoperfect.","title":"Erd\u0151s Problem 469 (statement)"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.DivisorSumReciprocalConvergenceWanted.conjecture_holds","parent":8216,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/index.html#conjecture-holds","scope":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers","status":"can_prove","summary":"Call a natural number $n$ pseudoperfect if $n > 0$ and $n$ is the sum of some set of distinct proper divisors of $n$, and primitive pseudoperfect if moreover no proper divisor of $n$ is pseudoperfect. Then the family $(1/n)$, indexed by the primitive pseudoperfect numbers $n$, is summable in $\\mathbb{R}$, that is, the sum of $1/n$ over all primitive pseudoperfect numbers $n$ is finite.","title":"Reciprocals of primitive pseudoperfect numbers have finite sum"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/conjecture-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem825Wanted.conjecture","parent":8216,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/index.html#conjecture-2","scope":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers","status":"fully_proved","summary":"Write $\\sigma(n)$ for the sum of the positive divisors of a positive integer $n$, and call $n$ pseudoperfect if $n$ is the sum of the elements of some set of proper divisors of $n$. The statement is the conjunction of two claims: there is a real $C > 0$ such that every positive integer $n$ with $\\sigma(n) > Cn$ is pseudoperfect, and every real $C > 0$ with this property satisfies $C > 2$.","title":"Erd\u0151s Problem 825"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/conjecture-holds-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem825Wanted.conjecture_holds","parent":8216,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers/index.html#conjecture-holds-2","scope":"NumberTheory/arithmetic-functions/perfect-numbers/pseudoperfect-numbers","status":"can_prove","summary":"Write $\\sigma(n)$ for the sum of the positive divisors of a positive integer $n$, and call $n$ pseudoperfect if $n$ is the sum of the elements of some set of proper divisors of $n$. Then there is a real $C > 0$ such that every positive integer $n$ with $\\sigma(n) > Cn$ is pseudoperfect, and every real $C > 0$ with this property satisfies $C > 2$.","title":"Erd\u0151s Problem 825 is true"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations.json","id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","kind":"scope","navigation":"module","parent":8070,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","title":"Spoof Lehmer factorizations"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-spoof-factorization","kind":"node","lean":"MetaMathlibExt.IsSpoofFactorization","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-spoof-factorization","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"A finite multiset $F = \\{(x_1, a_1), \\dots, (x_r, a_r)\\}$ of pairs of natural numbers, read as the formal product $x_1^{a_1} \\cdots x_r^{a_r}$, is a spoof factorization if every base $x_i$ and every exponent $a_i$ is positive.","title":"Spoof factorizations"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-eval","kind":"node","lean":"MetaMathlibExt.spoofEval","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-eval","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Let $F = x_1^{a_1} \\cdots x_r^{a_r}$ be a finite multiset of pairs $(x_i, a_i)$ of natural numbers, each a base $x_i$ with an exponent $a_i$. The spoof evaluation of $F$ is the integer $\\tilde e(F) = x_1^{a_1} \\cdots x_r^{a_r} = \\prod_{i=1}^{r} x_i^{a_i}$.","title":"Spoof evaluation"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-totient","kind":"node","lean":"MetaMathlibExt.spoofTotient","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-totient","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Let $F$ be a finite multiset of pairs $(x_i, a_i)$ of natural numbers, $i = 1, \\dots, r$, viewed as the formal product $x_1^{a_1} \\cdots x_r^{a_r}$. The spoof totient of $F$ is the integer $\\tilde\\varphi(F) = \\prod_{i=1}^{r} (x_i^{a_i} - x_i^{a_i - 1})$, where the exponent $a_i - 1$ is truncated at $0$, so a pair with $a_i = 0$ contributes the factor $0$.","title":"Spoof totient"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-eval-spoof-totient-three-three","kind":"node","lean":"MetaMathlibExt.spoofEval_spoofTotient_three_three","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spoof evaluation and spoof totient of "],["math","\\(3 \\cdot 3\\)"]],"source":"Spoof evaluation and spoof totient of $3 \\cdot 3$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-eval-spoof-totient-three-three","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Let $3 \\cdot 3$ denote the spoof factorization $3^1 \\cdot 3^1$, with two factors of base $3$ and exponent $1$. Its spoof evaluation is $\\tilde e(3 \\cdot 3) = 9$ and its spoof totient is $\\tilde\\varphi(3 \\cdot 3) = 4$.","title":"Spoof evaluation and spoof totient of $3 \\cdot 3$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-spoof-lehmer","kind":"node","lean":"MetaMathlibExt.IsSpoofLehmer","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-spoof-lehmer","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Let $k$ be an integer and $F$ a finite multiset of pairs $(x, a)$ of natural numbers. Write $\\tilde e(F) = \\prod x^a$ for the spoof evaluation and $\\tilde\\varphi(F) = \\prod (x^a - x^{a-1})$ for the spoof totient of $F$, the products running over the pairs of $F$ with multiplicity. Then $F$ is a (spoof) $k$-Lehmer factorization if $F$ is a spoof factorization (every base and exponent is positive),\u2026","title":"Spoof Lehmer factorizations"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-spoof-lehmer-three-three","kind":"node","lean":"MetaMathlibExt.isSpoofLehmer_three_three","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(3 \\cdot 3\\)"],["text"," is a "],["math","\\(2\\)"],["text","-Lehmer factorization"]],"source":"$3 \\cdot 3$ is a $2$-Lehmer factorization"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-spoof-lehmer-three-three","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"The spoof factorization $3^1 \\cdot 3^1$ is a $2$-Lehmer factorization; indeed $2 \\cdot (3 - 1)(3 - 1) = 3 \\cdot 3 - 1$.","title":"$3 \\cdot 3$ is a $2$-Lehmer factorization"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-spoof-lehmer-one-one","kind":"node","lean":"MetaMathlibExt.isSpoofLehmer_one_one","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","The factorizations "],["math","\\(1^a \\cdot 1^b\\)"],["text"," are Lehmer for every "],["math","\\(k\\)"]],"source":"The factorizations $1^a \\cdot 1^b$ are Lehmer for every $k$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-spoof-lehmer-one-one","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"For all positive integers $a$ and $b$ and every integer $k$, the spoof factorization $1^a \\cdot 1^b$ is a $k$-Lehmer factorization.","title":"The factorizations $1^a \\cdot 1^b$ are Lehmer for every $k$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-trivial-spoof","kind":"node","lean":"MetaMathlibExt.IsTrivialSpoof","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-trivial-spoof","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"A finite multiset $F$ of pairs $(x, a)$ of natural numbers, read as a spoof factorization with bases $x$ and exponents $a$, is trivial if every base $x$ occurring in $F$ equals $1$.","title":"Trivial spoof factorizations"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-odd-spoof","kind":"node","lean":"MetaMathlibExt.IsOddSpoof","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-odd-spoof","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"A finite multiset $F$ of pairs $(x, a)$ of natural numbers, read as a spoof factorization with bases $x$ and exponents $a$, is odd if every base $x$ occurring in $F$ is odd.","title":"Odd spoof factorizations"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-of-bases","kind":"node","lean":"MetaMathlibExt.spoofOfBases","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-of-bases","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"For a finite multiset $x$ of natural numbers $x_1, \\dots, x_r$, this is the formal product $x_1 \\cdots x_r = x_1^1 \\cdots x_r^1$, that is, the multiset of pairs $(x_1, 1), \\dots, (x_r, 1)$, each consisting of a base $x_i$ and the exponent $1$.","title":"Factorization with all exponents one"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-of-bases-three-three","kind":"node","lean":"MetaMathlibExt.spoofOfBases_three_three","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-of-bases-three-three","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"The factorization with all exponents one on the multiset of bases consisting of two copies of $3$ is the multiset consisting of two copies of the pair $(3, 1)$, that is, $3 \\cdot 3 = 3^1 \\cdot 3^1$.","title":"The exponent-one factorization on the bases 3 and 3"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-lehmer-equation-solution","kind":"node","lean":"MetaMathlibExt.IsLehmerEquationSolution","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-lehmer-equation-solution","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"A finite multiset $x$ of natural numbers solves Equation (1) if every element $x_j$ of $x$ satisfies $x_j \\ge 2$ and there is a positive integer $k$ with $k \\prod_j (x_j - 1) = \\prod_j x_j - 1$, where the products run over the elements of $x$ with multiplicity and are computed in $\\mathbb{Z}$.","title":"Solutions of the Lehmer equation"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-lehmer-equation-solution-three-three","kind":"node","lean":"MetaMathlibExt.isLehmerEquationSolution_three_three","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","The multiset "],["math","\\(\\{3, 3\\}\\)"],["text"," solves the Lehmer equation"]],"source":"The multiset $\\{3, 3\\}$ solves the Lehmer equation"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-lehmer-equation-solution-three-three","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Every element of the multiset $\\{3, 3\\}$ is at least $2$, and there is a positive integer $k$ with $k \\cdot (3 - 1)(3 - 1) = 3 \\cdot 3 - 1$ (namely $k = 2$). So $\\{3, 3\\}$ solves Equation (1), the equation $k \\prod_j (x_j - 1) = \\prod_j x_j - 1$ in the elements $x_j$ of a multiset, counted with multiplicity.","title":"The multiset $\\{3, 3\\}$ solves the Lehmer equation"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-kappa","kind":"node","lean":"MetaMathlibExt.lehmerKappa","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lehmer function "],["math","\\(\\kappa_r\\)"]],"source":"The Lehmer function $\\kappa_r$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-kappa","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"For a list $t = (t_1, \\dots, t_r)$ of real numbers,","title":"The Lehmer function $\\kappa_r$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-kappa-two-two","kind":"node","lean":"MetaMathlibExt.lehmerKappa_two_two","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lehmer function at "],["math","\\((2, 2)\\)"]],"source":"The Lehmer function at $(2, 2)$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-kappa-two-two","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"The Lehmer function $\\kappa_2(t_1, t_2) = (t_1 t_2 - 1) / ((t_1 - 1)(t_2 - 1))$ takes the value $\\kappa_2(2, 2) = 3$.","title":"The Lehmer function at $(2, 2)$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-completions","kind":"node","lean":"MetaMathlibExt.lehmerCompletions","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-completions","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Let $t = (t_1, \\dots, t_s)$ be a list of real numbers and let $r$ be a natural number. The admissible completions of $t$ to length $r$ are the lists $u = (u_1, \\dots, u_m)$ of real numbers with $s + m = r$ such that $t_1, \\dots, t_s, u_1, \\dots, u_m$ is nondecreasing and $u_j \\ge 2$ for every $j$.","title":"Admissible completions for the Lehmer bounds"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-lower","kind":"node","lean":"MetaMathlibExt.lehmerLower","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-lower","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Let $t = (t_1, \\dots, t_s)$ be a list of real numbers and let $r$ be a natural number. An admissible completion of $t$ to length $r$ is a list $u = (u_1, \\dots, u_m)$ of real numbers with $s + m = r$ such that $t_1, \\dots, t_s, u_1, \\dots, u_m$ is nondecreasing and $u_j \\ge 2$ for every $j$. The lower Lehmer bound $L(t; r)$ is the infimum of $\\kappa_r(t_1, \\dots, t_s, u_1, \\dots, u_m)$ over all admissible\u2026","title":"The lower Lehmer bound"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-upper","kind":"node","lean":"MetaMathlibExt.lehmerUpper","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper Lehmer bound "],["math","\\(U(t; r)\\)"]],"source":"Upper Lehmer bound $U(t; r)$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-upper","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"For real numbers $y_1, \\dots, y_r$ let $\\kappa_r(y_1, \\dots, y_r) = (y_1 \\cdots y_r - 1) / ((y_1 - 1) \\cdots (y_r - 1))$, where division by $0$ gives $0$. For a list $t = (t_1, \\dots, t_s)$ of real numbers and a natural number $r$, an admissible completion of $t$ is a list $u = (u_{s+1}, \\dots, u_r)$ of $r - s$ real numbers (there is none if $s > r$) such that $t_1, \\dots, t_s, u_{s+1}, \\dots, u_r$ is nondecreasing\u2026","title":"Upper Lehmer bound $U(t; r)$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-completions-length","kind":"node","lean":"MetaMathlibExt.lehmerCompletions_length","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-completions-length","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Let $t = (t_1, \\dots, t_r)$ be a nondecreasing list of $r$ real numbers. Then the empty list is the only admissible completion of $t$ to length $r$, that is, the only list $u$ of real numbers such that $r + |u| = r$, the concatenation of $t$ and $u$ is nondecreasing, and every entry of $u$ is at least $2$.","title":"A full-length nondecreasing list has only the empty completion"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-lower-upper-length","kind":"node","lean":"MetaMathlibExt.lehmerLower_upper_length","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-lower-upper-length","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"Let $L(t; r)$ and $U(t; r)$ be the lower and upper Lehmer bounds and $\\kappa_r$ the Lehmer function. For every nondecreasing list $t = (t_1, \\dots, t_r)$ of real numbers, $L(t; r) = U(t; r) = \\kappa_r(t_1, \\dots, t_r)$.","title":"The Lehmer bounds of a full-length list"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/is-trivial-spoof-of-is-spoof-lehmer-of-o","kind":"node","lean":"MetaMathlibExt.isTrivialSpoof_of_isSpoofLehmer_of_one_mem","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spoof Lehmer factorizations with a base "],["math","\\(1\\)"],["text"," are trivial"]],"source":"Spoof Lehmer factorizations with a base $1$ are trivial"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#is-trivial-spoof-of-is-spoof-lehmer-of-o","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"If a finite multiset $F$ of pairs $(x, a)$ of natural numbers is a $k$-Lehmer factorization for some integer $k$, and some pair in $F$ has base $x = 1$, then every base of $F$ equals $1$, that is, $F$ is trivial.","title":"Spoof Lehmer factorizations with a base $1$ are trivial"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-lehmer-exponents-eq-one","kind":"node","lean":"MetaMathlibExt.spoofLehmer_exponents_eq_one","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nontrivial spoof Lehmer factorizations have all exponents "],["math","\\(1\\)"]],"source":"Nontrivial spoof Lehmer factorizations have all exponents $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-lehmer-exponents-eq-one","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"If $F$ is a nontrivial spoof $k$-Lehmer factorization for some integer $k$, then every exponent of $F$ is $1$.","title":"Nontrivial spoof Lehmer factorizations have all exponents $1$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-lehmer-base-not-mod-eq-one","kind":"node","lean":"MetaMathlibExt.spoofLehmer_base_not_modEq_one","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-lehmer-base-not-mod-eq-one","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $x_1, \\dots, x_r$ be natural numbers and let $F = x_1 \\cdots x_r$ be the spoof factorization with these bases, all with exponent $1$. Suppose $F$ is a nontrivial spoof $k$-Lehmer factorization for some integer $k$. If $n$ is an integer with $n \\ne \\pm 1$ that divides the spoof evaluation $\\tilde e(F)$, then $x_i \\not\\equiv 1 \\pmod n$ for every $i$.","title":"Bases of a spoof Lehmer factorization modulo divisors of its evaluation"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-lehmer-base-not-mod-eq-one-base","kind":"node","lean":"MetaMathlibExt.spoofLehmer_base_not_modEq_one_base","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bases of a nontrivial spoof Lehmer factorization are not "],["math","\\(1\\)"],["text"," modulo one another"]],"source":"Bases of a nontrivial spoof Lehmer factorization are not $1$ modulo one another"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-lehmer-base-not-mod-eq-one-base","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $x_1, \\dots, x_r$ be natural numbers and let $F = x_1 \\cdots x_r$ be the spoof factorization with these bases, all with exponent $1$. If $F$ is a nontrivial spoof $k$-Lehmer factorization for some integer $k$, then $x_i \\not\\equiv 1 \\pmod{x_j}$ for all $i$ and $j$, including $i = j$.","title":"Bases of a nontrivial spoof Lehmer factorization are not $1$ modulo one another"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-lehmer-base-not-mod-eq-one-prime","kind":"node","lean":"MetaMathlibExt.spoofLehmer_base_not_modEq_one_prime","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-lehmer-base-not-mod-eq-one-prime","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $x_1, \\dots, x_r$ be natural numbers and let $F = x_1 \\cdots x_r$ be the spoof factorization with these bases, all with exponent $1$. Suppose $F$ is a nontrivial spoof $k$-Lehmer factorization for some integer $k$. If $p$ is a prime dividing the spoof evaluation $\\tilde e(F)$, then $x_i \\not\\equiv 1 \\pmod p$ for every $i$. Moreover, if $F$ is even (not every base is odd), then every base $x_i$ is even.","title":"Primes dividing the evaluation of a spoof Lehmer factorization"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-kappa-antitone-bounds","kind":"node","lean":"MetaMathlibExt.lehmerKappa_antitone_bounds","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-kappa-antitone-bounds","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"For real numbers $t_1, \\dots, t_r$ let $\\kappa_r(t_1, \\dots, t_r) = (t_1 \\cdots t_r - 1) / ((t_1 - 1) \\cdots (t_r - 1))$. Let $r \\ge 2$ and let $t_1, \\dots, t_r \\ge 2$ be real numbers. Then $1 < \\kappa_r(t_1, \\dots, t_r) \\le 2^r - 1$. Moreover, for every index $j$ and every real $y > t_j$, replacing $t_j$ by $y$ strictly decreases $\\kappa_r$, that is,\u2026","title":"Bounds and monotonicity of the Lehmer function"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-lower-upper-formula","kind":"node","lean":"MetaMathlibExt.lehmerLower_upper_formula","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-lower-upper-formula","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $L(t; r)$ and $U(t; r)$ be the lower and upper Lehmer bounds of a list $t$ of real numbers, and write $\\emptyset$ for the empty list. Then $L(\\emptyset; r) = 1$ for every $r \\ge 1$, and $U(\\emptyset; r) = 2^r - 1$ for every natural number $r$. Let $t = (t_1, \\dots, t_s)$ with $s \\ge 1$ and $2 \\le t_1 \\le \\dots \\le t_s$, and let $r \\ge s$ be a natural number. Then","title":"Formulas for the Lehmer bounds"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-lower-upper-antitone","kind":"node","lean":"MetaMathlibExt.lehmerLower_upper_antitone","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-lower-upper-antitone","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $L(t; r)$ and $U(t; r)$ be the lower and upper Lehmer bounds: the infimum and the supremum of $\\kappa_r(t_1, \\dots, t_s, u_1, \\dots, u_{r-s})$ over the admissible completions $u$ of $t = (t_1, \\dots, t_s)$, where $\\kappa_r$ is the Lehmer function. Let $r$ be a natural number and let $t = (t_1, \\dots, t_s)$ and $t' = (t'_1, \\dots, t'_s)$ be nondecreasing lists of real numbers of the same length $s \\le r$ with\u2026","title":"Monotonicity of the Lehmer bounds"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-lehmer-lower-le-le-upper","kind":"node","lean":"MetaMathlibExt.spoofLehmer_lower_le_le_upper","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-lehmer-lower-le-le-upper","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $x_1 \\le \\dots \\le x_r$ be natural numbers and let $F = x_1 \\cdots x_r$ be the spoof factorization with these bases, all with exponent $1$. Suppose $F$ is a nontrivial spoof $k$-Lehmer factorization for an integer $k$. For a list $t = (t_1, \\dots, t_s)$ of real numbers, let $L(t; r)$ and $U(t; r)$ be the lower and upper Lehmer bounds: the infimum and the supremum in $\\mathbb{R}$ of\u2026","title":"Lehmer bounds at the initial segments of a spoof Lehmer factorization"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-equation-solutions-finite","kind":"node","lean":"MetaMathlibExt.lehmerEquation_solutions_finite","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finitely many solutions of the Lehmer equation in "],["math","\\(r\\)"],["text"," variables"]],"source":"Finitely many solutions of the Lehmer equation in $r$ variables"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-equation-solutions-finite","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"For every natural number $r \\ge 2$, there are only finitely many pairs $(k, x)$ of an integer $k > 0$ and a tuple $x = (x_1, \\dots, x_r)$ of natural numbers with $x_i \\ge 2$ for every $i$ such that $k (x_1 - 1) \\cdots (x_r - 1) = x_1 \\cdots x_r - 1$.","title":"Finitely many solutions of the Lehmer equation in $r$ variables"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/lehmer-equation-solutions-count","kind":"node","lean":"MetaMathlibExt.lehmerEquation_solutions_count","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#lehmer-equation-solutions-count","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Call a finite multiset $\\{x_1, \\dots, x_r\\}$ of natural numbers a solution of the Lehmer equation if $x_i \\ge 2$ for every $i$ and $k (x_1 - 1) \\cdots (x_r - 1) = x_1 \\cdots x_r - 1$ for some integer $k > 0$. There are exactly $76$ solutions with $2 \\le r \\le 6$ (counted over all such $r$ together). Exactly $31$ of them have every $x_i$ odd, and exactly $45$ have every $x_i$ even.","title":"Solutions of the Lehmer equation with at most six variables"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-lehmer-finite","kind":"node","lean":"MetaMathlibExt.spoofLehmer_finite","parent":8222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finitely many nontrivial spoof Lehmer factorizations with "],["math","\\(r\\)"],["text"," factors"]],"source":"Finitely many nontrivial spoof Lehmer factorizations with $r$ factors"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-lehmer-finite","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"For every natural number $r$, there are only finitely many finite multisets $F$ of pairs (base, exponent) of natural numbers with exactly $r$ elements such that $F$ is a nontrivial spoof $k$-Lehmer factorization for some integer $k$.","title":"Finitely many nontrivial spoof Lehmer factorizations with $r$ factors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-lehmer-at-most-six-tables","kind":"node","lean":"MetaMathlibExt.spoofLehmer_atMostSix_tables","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-lehmer-at-most-six-tables","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $F$ be a finite multiset of pairs (base, exponent) of natural numbers with at most $6$ elements. Then $F$ is a nontrivial odd spoof $k$-Lehmer factorization for some integer $k$ if and only if every exponent of $F$ is $1$ and the multiset of bases of $F$ is one of the following $31$, written as tuples: $(3, 3)$, $(3, 5, 15)$, $(3, 5, 17, 255)$, $(3, 3, 3, 3)$, $(3, 5, 17, 257, 65535)$, $(3, 5, 17, 285, 2507)$,\u2026","title":"Spoof Lehmer factorizations with at most six factors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-totient-spoof-of-bases","kind":"node","lean":"MetaMathlibExt.spoofTotient_spoofOfBases","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-totient-spoof-of-bases","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"fully_proved","summary":"For natural numbers $x_1, \\dots, x_r$, the spoof totient of the spoof factorization $x_1 \\cdots x_r$ with every exponent $1$ is $\\prod_{i=1}^{r} (x_i - 1)$, computed in the integers. This is Hasanalizade's spoof unitary totient $\\tilde{\\varphi}^*(x_1 \\cdots x_r)$.","title":"Spoof totient of a unitary factorization"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/nielsen-prod-one-sub-inv-bound","kind":"node","lean":"MetaMathlibExt.nielsen_prod_one_sub_inv_bound","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#nielsen-prod-one-sub-inv-bound","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $r$, $a$ and $b$ be natural numbers and let $x_1 \\le \\dots \\le x_r$ be integers with every $x_j > 1$. If $\\prod_{j=1}^{r} (1 - 1/x_j) \\le a/b < \\prod_{j=1}^{r-1} (1 - 1/x_j)$, then $a \\prod_{j=1}^{r} x_j \\le (a + 1)^{2^r} - (a + 1)^{2^{r-1}}$.","title":"Nielsen's bound"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/spoof-eval-le-of-is-spoof-lehmer-spoof-o","kind":"node","lean":"MetaMathlibExt.spoofEval_le_of_isSpoofLehmer_spoofOfBases","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#spoof-eval-le-of-is-spoof-lehmer-spoof-o","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $k$ be an integer and let $x_1, \\dots, x_r$ be natural numbers. If the spoof factorization $x_1 \\cdots x_r$ with every exponent $1$ is a spoof $k$-Lehmer factorization (a unitary $k$-Lehmer factorization), then its spoof evaluation $\\tilde{e}(x_1 \\cdots x_r) = \\prod_{i=1}^{r} x_i$ is at most $2^{2^r} - 2^{2^{r-1}}$.","title":"Bound on unitary Lehmer factorizations"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/eq-two-or-eq-three-of-is-spoof-lehmer-re","kind":"node","lean":"MetaMathlibExt.eq_two_or_eq_three_of_isSpoofLehmer_replicate","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#eq-two-or-eq-three-of-is-spoof-lehmer-re","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $k$ be an integer and let $x \\ge 2$ and $s$ be natural numbers. If the spoof factorization $x \\cdots x$ with $s$ factors, each with base $x$ and exponent $1$, is a spoof $k$-Lehmer factorization (a unitary $k$-Lehmer factorization), then either $x = 2$ and $s \\ge 2$, or $x = 3$ and $s \\in \\{2, 4\\}$.","title":"Repeated unitary Lehmer factorizations"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/unitary-spoof-lehmer-lemma","kind":"node","lean":"MetaMathlibExt.unitary_spoof_lehmer_lemma","parent":8222,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations/index.html#unitary-spoof-lehmer-lemma","scope":"NumberTheory/arithmetic-functions/perfect-numbers/spoof-lehmer-factorizations","status":"can_prove","summary":"Let $F$ be a finite multiset of integers, write $E = \\prod_{f \\in F} f$ and $P = \\prod_{f \\in F} (f - 1)$, and let $k$, $\\alpha$, $x$ and $y$ be integers with $\\alpha = \\pm 1$. If $k P = E + \\alpha$ and $k P (x - 1)(y - 1) = E x y - 1$, then $(\\alpha x - E - \\alpha)(\\alpha y - E - \\alpha) - (E^2 + \\alpha E - \\alpha) = 0$.","title":"Lehmer's lemma for unitary spoof factorizations"},{"data_url":"scopes/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers.json","id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","kind":"scope","navigation":"module","parent":8070,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","title":"Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-szumkeller","kind":"node","lean":"MetaMathlibExt.IsSZumkeller","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-szumkeller","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"A natural number $n$ is s-Zumkeller if its proper divisors can be split into two disjoint nonempty sets whose elements have equal sums of squares. Each of the two sums is then $(\\sigma_2(n) - n^2)/2$, where $\\sigma_2(n)$ is the sum of the squares of the positive divisors of $n$.","title":"s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-unitary-szumkeller","kind":"node","lean":"MetaMathlibExt.IsUnitarySZumkeller","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-unitary-szumkeller","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"A natural number $n$ is a unitary s-Zumkeller number when its proper unitary divisors (the unitary divisors of $n$ other than $n$) split into two nonempty disjoint sets $A$ and $B$ with equal sums of squares, $\\sum_{d \\in A} d^2 = \\sum_{d \\in B} d^2$.","title":"Unitary s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-szumkeller-sixty","kind":"node","lean":"MetaMathlibExt.isSZumkeller_sixty","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(60\\)"],["text"," is s-Zumkeller"]],"source":"$60$ is s-Zumkeller"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-szumkeller-sixty","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"The number $60$ is s-Zumkeller: the squares of its proper divisors $1, 2, 5, 30$ sum to $1^2 + 2^2 + 5^2 + 30^2 = 930$, which is half of the sum of the squares of all proper divisors of $60$.","title":"$60$ is s-Zumkeller"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/not-is-szumkeller-six","kind":"node","lean":"MetaMathlibExt.not_isSZumkeller_six","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#not-is-szumkeller-six","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"The number $6$ is not an s-Zumkeller number: the squares of its proper divisors sum to $1^2 + 2^2 + 3^2 = 14$, and they admit no split into two parts with equal sums.","title":"The number 6 is not s-Zumkeller"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-unitary-szumkeller-sixty","kind":"node","lean":"MetaMathlibExt.isUnitarySZumkeller_sixty","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-unitary-szumkeller-sixty","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"The number $60$ is a unitary s-Zumkeller number: its proper unitary divisors are $1, 3, 4, 5, 12, 15, 20$, and $20^2 + 3^2 + 1^2 = 410$ is half the sum of their squares.","title":"The number 60 is unitary s-Zumkeller"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/not-is-unitary-szumkeller-six","kind":"node","lean":"MetaMathlibExt.not_isUnitarySZumkeller_six","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#not-is-unitary-szumkeller-six","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"The number $6$ is not a unitary s-Zumkeller number.","title":"The number 6 is not unitary s-Zumkeller"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/sigma-one-prod-formulas","kind":"node","lean":"MetaMathlibExt.sigma_one_prod_formulas","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#sigma-one-prod-formulas","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n \\ne 0$ be a natural number, let $\\sigma(n)$ be the sum of the divisors of $n$, and for each prime $p$ dividing $n$ let $k_p$ be the exponent of $p$ in $n$. Then, with products over the primes $p$ dividing $n$,","title":"Product formulas for the sum of divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/sigma-two-prod-formulas","kind":"node","lean":"MetaMathlibExt.sigma_two_prod_formulas","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#sigma-two-prod-formulas","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n \\ne 0$ be a natural number, let $\\sigma_2(n)$ be the sum of the squares of the divisors of $n$, and for each prime $p$ dividing $n$ let $k_p$ be the exponent of $p$ in $n$. Then, with products over the primes $p$ dividing $n$,","title":"Product formulas for the sum of squared divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/card-unitary-divisors","kind":"node","lean":"MetaMathlibExt.card_unitaryDivisors","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#card-unitary-divisors","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"A positive integer $n$ with $m$ distinct prime factors has exactly $2^m$ unitary divisors.","title":"Number of unitary divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/sum-sq-unitary-divisors","kind":"node","lean":"MetaMathlibExt.sum_sq_unitaryDivisors","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#sum-sq-unitary-divisors","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"For every positive integer $n$, the sum of the squares of the unitary divisors of $n$ equals $\\prod_{p} \\left(1 + p^{2k_p}\\right)$, where $p$ ranges over the distinct prime factors of $n$ and $k_p$ is the exponent of $p$ in $n$.","title":"Sum of squares of unitary divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-szumkeller-even-props","kind":"node","lean":"MetaMathlibExt.isSZumkeller_even_props","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-szumkeller-even-props","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n$ be an even s-Zumkeller number and let $\\sigma_2(n)$ be the sum of the squares of the positive divisors of $n$. Then $\\sigma_2(n)$ is even, some odd prime $p$ divides $n$ with an odd exponent in the prime factorization of $n$, and $3n^2 \\le 2\\sigma_2(n)$.","title":"Properties of even s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-szumkeller-iff-of-even","kind":"node","lean":"MetaMathlibExt.isSZumkeller_iff_of_even","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-szumkeller-iff-of-even","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n$ be an even positive integer and let $\\sigma_2(n)$ be the sum of the squares of the positive divisors of $n$. Then $n$ is s-Zumkeller if and only if there is a (possibly empty) set $S$ of proper divisors of $n$, not containing $n/2$, such that $4 \\sum_{d \\in S} d^2 = 2\\sigma_2(n) - 3n^2$.","title":"Criterion for even s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/twelve-le-card-divisors-of-is-szumkeller","kind":"node","lean":"MetaMathlibExt.twelve_le_card_divisors_of_isSZumkeller","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#twelve-le-card-divisors-of-is-szumkeller","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"If $n$ is an even s-Zumkeller number, then $n$ has at least $12$ positive divisors.","title":"Even s-Zumkeller numbers have at least twelve divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/three-dvd-of-is-szumkeller","kind":"node","lean":"MetaMathlibExt.three_dvd_of_isSZumkeller","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#three-dvd-of-is-szumkeller","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"If $n$ is an even s-Zumkeller number, then $3$ divides $n$.","title":"Even s-Zumkeller numbers are divisible by 3"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/six-dvd-of-is-szumkeller","kind":"node","lean":"MetaMathlibExt.six_dvd_of_isSZumkeller","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#six-dvd-of-is-szumkeller","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Every even s-Zumkeller number is divisible by $6$.","title":"Even s-Zumkeller numbers are divisible by 6"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/two-div-min-fac-sq-add-one-lt-prod","kind":"node","lean":"MetaMathlibExt.two_div_minFac_sq_add_one_lt_prod","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#two-div-min-fac-sq-add-one-lt-prod","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n$ be an s-Zumkeller number and let $p_1$ be its least prime factor. Then","title":"A prime-factor product bound for s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/three-le-card-prime-factors-of-is-szumke","kind":"node","lean":"MetaMathlibExt.three_le_card_primeFactors_of_isSZumkeller_even","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#three-le-card-prime-factors-of-is-szumke","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"If $n$ is an even s-Zumkeller number, then $n$ has at least $3$ distinct prime factors.","title":"Even s-Zumkeller numbers have at least three prime factors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/eight-le-card-prime-factors-of-is-szumke","kind":"node","lean":"MetaMathlibExt.eight_le_card_primeFactors_of_isSZumkeller_odd","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd s-Zumkeller numbers divisible by "],["math","\\(3\\)"],["text"," have at least eight prime factors"]],"source":"Odd s-Zumkeller numbers divisible by $3$ have at least eight prime factors"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#eight-le-card-prime-factors-of-is-szumke","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Every odd s-Zumkeller number divisible by $3$ has at least $8$ distinct prime factors.","title":"Odd s-Zumkeller numbers divisible by $3$ have at least eight prime factors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-square-of-is-szumkeller-odd","kind":"node","lean":"MetaMathlibExt.isSquare_of_isSZumkeller_odd","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-square-of-is-szumkeller-odd","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Every odd s-Zumkeller number is a perfect square.","title":"Odd s-Zumkeller numbers are perfect squares"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/le-of-is-szumkeller-odd-three-dvd","kind":"node","lean":"MetaMathlibExt.le_of_isSZumkeller_odd_three_dvd","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#le-of-is-szumkeller-odd-three-dvd","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Every odd s-Zumkeller number $n$ divisible by $3$ satisfies $n \\ge 12442607161209225 = (3 \\cdot 5 \\cdot 7 \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 19 \\cdot 23)^2$.","title":"Lower bound for odd s-Zumkeller numbers divisible by 3"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/harmonic-mean-sq-le-of-is-szumkeller","kind":"node","lean":"MetaMathlibExt.harmonicMeanSq_le_of_isSZumkeller","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#harmonic-mean-sq-le-of-is-szumkeller","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n$ be an even s-Zumkeller number and let $\\tau(n)$ be the number of positive divisors of $n$. Then $\\tau(n) / \\sum_{d \\mid n} d^{-2} \\le 2\\tau(n)/3$, where the sum runs over the positive divisors $d$ of $n$.","title":"Harmonic mean of squared divisors of even s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/harmonic-mean-sq-lt-of-is-szumkeller-mer","kind":"node","lean":"MetaMathlibExt.harmonicMeanSq_lt_of_isSZumkeller_mersenne","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Harmonic mean of squared divisors of s-Zumkeller numbers "],["math","\\(2^\\alpha(2^{\\alpha+1}-1)\\)"]],"source":"Harmonic mean of squared divisors of s-Zumkeller numbers $2^\\alpha(2^{\\alpha+1}-1)$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#harmonic-mean-sq-lt-of-is-szumkeller-mer","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $\\alpha \\ge 1$ be an integer, let $n = 2^\\alpha (2^{\\alpha+1} - 1)$ and let $\\tau(n)$ be the number of positive divisors of $n$. If $n$ is s-Zumkeller, then $\\tau(n) / \\sum_{d \\mid n} d^{-2} < 2^{2\\alpha+3}/3$, where the sum runs over the positive divisors $d$ of $n$.","title":"Harmonic mean of squared divisors of s-Zumkeller numbers $2^\\alpha(2^{\\alpha+1}-1)$"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-unitary-szumkeller-bounds","kind":"node","lean":"MetaMathlibExt.isUnitarySZumkeller_bounds","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-unitary-szumkeller-bounds","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n$ be a unitary s-Zumkeller number, let $q$ be the least unitary divisor of $n$ greater than $1$, and for each prime $p$ dividing $n$ let $k_p$ be the exponent of $p$ in $n$. Then, with both sums over all unitary divisors $d$ of $n$ (including $n$ itself), $n^2 (2/q^2 + 1) \\le \\sum_d d^2$ and $2/q^2 + 1 \\le \\sum_d 1/d^2$. Moreover $2/q^2 + 1 \\le \\prod_{p \\mid n} (1 + p^{2k_p})/p^{2k_p}$, the product running\u2026","title":"Lower bounds for unitary s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/card-unitary-divisors-of-is-unitary-szum","kind":"node","lean":"MetaMathlibExt.card_unitaryDivisors_of_isUnitarySZumkeller","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#card-unitary-divisors-of-is-unitary-szum","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n$ be a unitary s-Zumkeller number and let $q$ be the least unitary divisor of $n$ greater than $1$. If $q = 2$, then $n$ has more than $2^6$ unitary divisors; if $q = 3$ or $q = 4$, then $n$ has more than $2^2$ unitary divisors.","title":"Unitary divisor counts of unitary s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/not-is-unitary-szumkeller-of-odd","kind":"node","lean":"MetaMathlibExt.not_isUnitarySZumkeller_of_odd","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#not-is-unitary-szumkeller-of-odd","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"There is no odd unitary s-Zumkeller number.","title":"No odd unitary s-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/divisor-prod","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.divisorProd","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#divisor-prod","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"For a natural number $n$, $T(n)$ is the product of all positive divisors of $n$. For $n = 0$ the set of divisors is taken to be empty, so $T(0) = 1$.","title":"Product of the divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-super-mzumkeller","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.IsSuperMZumkeller","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-super-mzumkeller","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"For a natural number $n$, let $T(n)$ be the product of the positive divisors of $n$ (for $n = 0$ the divisor set is taken to be empty, so $T(0) = 1$). Then $n$ is *super-m-Zumkeller* if there is a set $A$ of positive divisors of $T(n)$ such that the product of the elements of $A$ equals the product of the positive divisors of $T(n)$ that are not in $A$; that is, the divisors of $T(n)$ split into two complementary\u2026","title":"Super-m-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-kmperfect","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.IsKMPerfect","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-m-perfect numbers"]],"source":"$k$-m-perfect numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-kmperfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $n$ is $k$-m-perfect if $T(n) = n^k$, where $T(n)$ is the product of the divisors of $n$.","title":"$k$-m-perfect numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/divisor-prod-sq","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.divisorProd_sq","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#divisor-prod-sq","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"For every positive integer $n$, let $T(n)$ be the product of the positive divisors of $n$ and $\\tau(n)$ their number. Then $T(n)^2 = n^{\\tau(n)}$.","title":"Square of the product of the divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/three-le-card-divisors-of-is-super-mzumk","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.three_le_card_divisors_of_isSuperMZumkeller","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Super-m-Zumkeller numbers "],["math","\\(n \\ge 2\\)"],["text"," have at least three divisors"]],"source":"Super-m-Zumkeller numbers $n \\ge 2$ have at least three divisors"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#three-le-card-divisors-of-is-super-mzumk","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"If a natural number $n \\ge 2$ is super-m-Zumkeller, then the number $\\tau(n)$ of positive divisors of $n$ satisfies $\\tau(n) \\ge 3$.","title":"Super-m-Zumkeller numbers $n \\ge 2$ have at least three divisors"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-super-mzumkeller-iff","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.isSuperMZumkeller_iff","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-super-mzumkeller-iff","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $n \\ge 2$ be an integer, let $\\tau(n)$ be the number of positive divisors of $n$ and $T(n)$ their product. Then $n$ is super-m-Zumkeller if and only if $8$ divides $\\alpha \\cdot \\tau(n) \\cdot \\tau(T(n))$ for every prime $p$ dividing $n$, where $\\alpha$ is the exponent of $p$ in $n$.","title":"Criterion for super-m-Zumkeller numbers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-super-mzumkeller-prime-pow-iff","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.isSuperMZumkeller_prime_pow_iff","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-super-mzumkeller-prime-pow-iff","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $p$ be a prime and $\\alpha \\ge 1$ an integer. Then $p^\\alpha$ is super-m-Zumkeller if and only if $\\alpha \\equiv 0, 2, 5, 7, 8, 10, 13$ or $15 \\pmod{16}$.","title":"Super-m-Zumkeller prime powers"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-super-mzumkeller-prod-primes","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.isSuperMZumkeller_prod_primes","parent":8261,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-super-mzumkeller-prod-primes","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"For every finite set $S$ of at least three primes, the product $\\prod_{p \\in S} p$ is super-m-Zumkeller.","title":"Products of at least three distinct primes are super-m-Zumkeller"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-super-mzumkeller-two-pow-mul-two-prim","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.isSuperMZumkeller_two_pow_mul_two_primes","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd powers of "],["math","\\(2\\)"],["text"," times two distinct odd primes are super-m-Zumkeller"]],"source":"Odd powers of $2$ times two distinct odd primes are super-m-Zumkeller"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-super-mzumkeller-two-pow-mul-two-prim","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $k$ be an odd natural number and let $p$ and $q$ be distinct odd primes. Then $2^k p q$ is super-m-Zumkeller.","title":"Odd powers of $2$ times two distinct odd primes are super-m-Zumkeller"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-super-mzumkeller-two-pow-mul-prod-pri","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.isSuperMZumkeller_two_pow_mul_prod_primes","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of "],["math","\\(2\\)"],["text"," times at least three distinct odd primes are super-m-Zumkeller"]],"source":"Powers of $2$ times at least three distinct odd primes are super-m-Zumkeller"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-super-mzumkeller-two-pow-mul-prod-pri","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $k$ be a natural number and $S$ a finite set of at least three odd primes. Then $2^k \\prod_{p \\in S} p$ is super-m-Zumkeller.","title":"Powers of $2$ times at least three distinct odd primes are super-m-Zumkeller"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-super-mzumkeller-of-is-kmperfect","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.isSuperMZumkeller_of_isKMPerfect","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-m-perfect numbers with "],["math","\\(k\\)"],["text"," a positive multiple of "],["math","\\(4\\)"],["text"," are super-m-Zumkeller"]],"source":"$k$-m-perfect numbers with $k$ a positive multiple of $4$ are super-m-Zumkeller"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-super-mzumkeller-of-is-kmperfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number with $k \\equiv 0 \\pmod 4$. Then every natural number that is $k$-m-perfect is super-m-Zumkeller.","title":"$k$-m-perfect numbers with $k$ a positive multiple of $4$ are super-m-Zumkeller"},{"id":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/is-super-mzumkeller-four-not-km-perfect","kind":"node","lean":"MathlibExt.NumberTheory.SuperMZumkellerWanted.isSuperMZumkeller_four_not_kmPerfect","parent":8261,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(4\\)"],["text"," is super-m-Zumkeller but not "],["math","\\(k\\)"],["text","-m-perfect for "],["math","\\(k \\ge 2\\)"]],"source":"$4$ is super-m-Zumkeller but not $k$-m-perfect for $k \\ge 2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers/index.html#is-super-mzumkeller-four-not-km-perfect","scope":"NumberTheory/arithmetic-functions/perfect-numbers/zumkeller-numbers","status":"can_prove","summary":"The number $4$ is super-m-Zumkeller, and for no natural number $k \\ge 2$ is $4$ a $k$-m-perfect number. This is the paper's counterexample to the converse of Theorem 6.","title":"$4$ is super-m-Zumkeller but not $k$-m-perfect for $k \\ge 2$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions","kind":"scope","parent":8068,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions","status":"can_prove","summary":"\u03c3, \u03c4, \u03c6, Jordan and Dedekind \u03c8, unitary and biunitary divisors, gcd-sums, Ramanujan sums, totient multiplicities and values.","title":"Divisor and totient functions"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01","kind":"scope","navigation":"transparent","parent":8299,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01","status":"can_prove","title":"Part 1 of 2"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ford-totient-multiplicity","kind":"node","lean":"MetaMathlibExt.ford_totient_multiplicity","parent":8300,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/index.html#ford-totient-multiplicity","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01","status":"can_prove","summary":"For every natural number $k \\ge 2$ there is a natural number $m$ such that exactly $k$ natural numbers $x$ satisfy $\\varphi(x) = m$, where $\\varphi$ is Euler's totient function.","title":"Ford's theorem on totient multiplicities"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function","status":"fully_proved","title":"Alder's totient function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function/alder-totient-two","kind":"node","lean":"ArithmeticFunction.alderTotientTwo","parent":8302,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function/index.html#alder-totient-two","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function","status":"fully_proved","summary":"Alder's extension of Euler's totient with second argument $2$ is the arithmetic function $g : \\mathbb{N} \\to \\mathbb{N}$ sending a natural number $k$ to the number of natural numbers $r < k$ with $\\gcd(k, r) = 1$ and $\\gcd(k, 2r+1) = 1$. In particular $g(0) = 0$, since there is no natural number $r < 0$.","title":"Alder's totient with second argument 2"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function/alder-totient-two-succ-iic-apply","kind":"node","lean":"ArithmeticFunction.alderTotientTwo_succ_Iic_apply","parent":8302,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function/index.html#alder-totient-two-succ-iic-apply","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function","status":"fully_proved","summary":"For every natural number $k$, Alder's totient $g$ with second argument $2$ satisfies that $g(k+1)$ is the number of natural numbers $r \\le k$ with $\\gcd(r, k+1) = 1$ and $\\gcd(2r+1, k+1) = 1$.","title":"Alder's totient at a successor as a count over an inclusive range"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function/alder-totient-two-le-totient","kind":"node","lean":"ArithmeticFunction.alderTotientTwo_le_totient","parent":8302,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alder's totient with second argument "],["math","\\(2\\)"],["text"," is at most Euler's totient"]],"source":"Alder's totient with second argument $2$ is at most Euler's totient"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function/index.html#alder-totient-two-le-totient","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/alder-s-totient-function","status":"fully_proved","summary":"For every natural number $k$, Alder's totient $g(k)$ with second argument $2$ is at most Euler's totient $\\varphi(k)$.","title":"Alder's totient with second argument $2$ is at most Euler's totient"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum","status":"can_prove","title":"Average and divisibility of the divisor sum"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum/tendsto-density-not-dvd-sigma","kind":"node","lean":"MathlibExt.NumberTheory.AmicableWanted.tendsto_density_not_dvd_sigma","parent":8306,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum/index.html#tendsto-density-not-dvd-sigma","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum","status":"can_prove","summary":"For a natural number $l$ let $\\sigma(l)$ denote the sum of the positive divisors of $l$, with $\\sigma(0) = 0$. Then for every positive integer $N$, the set of natural numbers $l$ such that $N$ does not divide $\\sigma(l)$ has natural density $0$.","title":"A fixed positive integer divides the divisor sum of almost every natural number"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum/sum-sigma-div-lt","kind":"node","lean":"MathlibExt.NumberTheory.AmicableWanted.sum_sigma_div_lt","parent":8306,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum/index.html#sum-sigma-div-lt","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/average-and-divisibility-of-the-divisor-sum","status":"can_prove","summary":"For a positive integer $n$ let $\\sigma(n)$ denote the sum of the positive divisors of $n$, and let $\\zeta(2) = \\sum_{n=1}^{\\infty} 1/n^2$. Then for every real number $x \\ge 1$,","title":"Linear bound for the partial sums of the abundancy index"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function","status":"fully_proved","title":"Dedekind psi function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/dedekind-psi","kind":"node","lean":"ArithmeticFunction.dedekindPsi","parent":8309,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/index.html#dedekind-psi","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function","status":"fully_proved","summary":"The Dedekind psi function $\\psi$ is the arithmetic function with values in $\\mathbb{N}$ given by $\\psi(0) = 0$ and, for $n > 0$,","title":"Dedekind psi function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/dedekind-psi-eq-prod-factorization","kind":"node","lean":"ArithmeticFunction.dedekindPsi_eq_prod_factorization","parent":8309,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/index.html#dedekind-psi-eq-prod-factorization","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function","status":"fully_proved","summary":"For every natural number $n \\ne 0$, the Dedekind psi function $\\psi$ satisfies","title":"Dedekind psi function as a product over prime powers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/dedekind-psi-prime-pow","kind":"node","lean":"ArithmeticFunction.dedekindPsi_prime_pow","parent":8309,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/index.html#dedekind-psi-prime-pow","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function","status":"fully_proved","summary":"For every prime $p$ and every natural number $k > 0$, the Dedekind psi function $\\psi$ satisfies $\\psi(p^k) = p^{k-1} (p + 1)$.","title":"Dedekind psi function at prime powers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/is-multiplicative-dedekind-psi","kind":"node","lean":"ArithmeticFunction.isMultiplicative_dedekindPsi","parent":8309,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/index.html#is-multiplicative-dedekind-psi","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function","status":"fully_proved","summary":"The Dedekind psi function $\\psi$ is multiplicative: $\\psi(1) = 1$ and $\\psi(mn) = \\psi(m) \\psi(n)$ for all coprime natural numbers $m$ and $n$.","title":"Multiplicativity of the Dedekind psi function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/dedekind-psi-mul-prod-prime-factors","kind":"node","lean":"ArithmeticFunction.dedekindPsi_mul_prod_primeFactors","parent":8309,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/index.html#dedekind-psi-mul-prod-prime-factors","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function","status":"fully_proved","summary":"For every natural number $n \\ne 0$, the Dedekind psi function $\\psi$ satisfies","title":"Cleared-denominator product formula for the Dedekind psi function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/dedekind-psi-eq-rational-prod","kind":"node","lean":"ArithmeticFunction.dedekindPsi_eq_rational_prod","parent":8309,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function/index.html#dedekind-psi-eq-rational-prod","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/dedekind-psi-function","status":"fully_proved","summary":"For every natural number $n \\ne 0$, the Dedekind psi function $\\psi$ satisfies, as an identity of rational numbers,","title":"Product formula for the Dedekind psi function"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums","status":"fully_proved","title":"Divisor-power sums"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/sigma-r","kind":"node","lean":"ArithmeticFunction.sigmaR","parent":8316,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/index.html#sigma-r","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums","status":"fully_proved","summary":"For a complex number $s$, $\\sigma_s$ is the complex-valued arithmetic function with $\\sigma_s(n) = \\sum_{d \\mid n} d^s$ for every natural number $n \\ge 1$, where $d^s$ is the complex power, and $\\sigma_s(0) = 0$.","title":"Divisor-power sum with complex exponent"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/sigma-r-apply","kind":"node","lean":"ArithmeticFunction.sigmaR_apply","parent":8316,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/index.html#sigma-r-apply","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums","status":"fully_proved","summary":"For every natural number $n$ and complex number $s$, the divisor-power sum $\\sigma_s$ with complex exponent $s$ satisfies $\\sigma_s(n) = \\sum_{d \\mid n} d^s$, the sum running over the positive divisors $d$ of $n$; for $n = 0$ the sum is empty, so $\\sigma_s(0) = 0$.","title":"Divisor-sum formula for the complex-exponent divisor sum"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/sigma-r-apply-one","kind":"node","lean":"ArithmeticFunction.sigmaR_apply_one","parent":8316,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/index.html#sigma-r-apply-one","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums","status":"fully_proved","summary":"For every complex number $s$, the divisor-power sum $\\sigma_s(n) = \\sum_{d \\mid n} d^s$ with complex exponent $s$ satisfies $\\sigma_s(1) = 1$.","title":"The complex-exponent divisor sum at one"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/sigma-r-zero-apply","kind":"node","lean":"ArithmeticFunction.sigmaR_zero_apply","parent":8316,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/index.html#sigma-r-zero-apply","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums","status":"fully_proved","summary":"Let $\\sigma_s(n) = \\sum_{d \\mid n} d^s$ be the divisor-power sum with complex exponent $s$. For every natural number $n$, $\\sigma_0(n)$ equals the number of positive divisors of $n$, with the convention that $0$ has no divisors, so that $\\sigma_0(0) = 0$.","title":"The complex-exponent divisor sum at exponent zero counts divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/sigma-r-nat-cast-apply","kind":"node","lean":"ArithmeticFunction.sigmaR_natCast_apply","parent":8316,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums/index.html#sigma-r-nat-cast-apply","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisor-power-sums","status":"fully_proved","summary":"For all natural numbers $k$ and $n$, the value at $n$ of the complex-exponent divisor sum $\\sum_{d \\mid n} d^s$ with $s = k$ equals the natural number $\\sigma_k(n) = \\sum_{d \\mid n} d^k$, viewed as a complex number.","title":"Natural exponents give the classical divisor function"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","title":"Divisors in short intervals"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/short-interval-divisor-count","kind":"node","lean":"MetaMathlibExt.shortIntervalDivisorCount","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#short-interval-divisor-count","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"fully_proved","summary":"For a natural number $n$ and real numbers $X$ and $Y$, Letendre's $D_n(X, Y)$ is the number of positive divisors $d$ of $n$ with $X \\le d \\le X + Y$. Here $0$ is taken to have no divisors, so $D_0(X, Y) = 0$.","title":"Number of divisors in a short interval"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/short-interval-divisor-count-le-card-div","kind":"node","lean":"MetaMathlibExt.shortIntervalDivisorCount_le_card_divisors","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#short-interval-divisor-count-le-card-div","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"fully_proved","summary":"For every natural number $n$ and all real numbers $X$ and $Y$, the number $D_n(X, Y)$ of positive divisors of $n$ in $[X, X + Y]$ is at most the number $\\tau(n)$ of positive divisors of $n$. Here $0$ is taken to have no divisors, so $D_0(X, Y) = \\tau(0) = 0$.","title":"Trivial bound for the short-interval divisor count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/letendre-xi","kind":"node","lean":"MetaMathlibExt.letendreXi","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#letendre-xi","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"fully_proved","summary":"For real numbers $\\theta$ and $\\eta$, Letendre's saving exponent $\\xi(\\theta, \\eta)$ is defined as follows. If $\\eta \\le \\theta^2$, then $\\xi(\\theta, \\eta) = 1$. Otherwise, if $\\theta \\le 1/2$, then $\\xi(\\theta, \\eta) = \\theta^2/\\eta$ when $2\\eta \\le \\theta$ and $\\xi(\\theta, \\eta) = 4(\\theta - \\eta)$ when $2\\eta > \\theta$; and if $\\theta > 1/2$, then\u2026","title":"Letendre's saving exponent"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/letendre-xi-of-le-sq","kind":"node","lean":"MetaMathlibExt.letendreXi_of_le_sq","parent":8322,"reading_title":{"format":"inline-parts-v1","parts":[["text","Letendre's exponent equals "],["math","\\(1\\)"],["text"," when "],["math","\\(\\eta \\le \\theta^2\\)"]],"source":"Letendre's exponent equals $1$ when $\\eta \\le \\theta^2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#letendre-xi-of-le-sq","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"fully_proved","summary":"For all real numbers $\\theta$ and $\\eta$ with $\\eta \\le \\theta^2$, Letendre's saving exponent satisfies $\\xi(\\theta, \\eta) = 1$.","title":"Letendre's exponent equals $1$ when $\\eta \\le \\theta^2$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/letendre-xi-of-small-theta-large-eta","kind":"node","lean":"MetaMathlibExt.letendreXi_of_small_theta_large_eta","parent":8322,"reading_title":{"format":"inline-parts-v1","parts":[["text","Letendre's exponent for "],["math","\\(\\theta^2 < \\eta\\)"],["text",", "],["math","\\(\\theta \\le 1/2\\)"],["text"," and "],["math","\\(\\theta < 2\\eta\\)"]],"source":"Letendre's exponent for $\\theta^2 < \\eta$, $\\theta \\le 1/2$ and $\\theta < 2\\eta$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#letendre-xi-of-small-theta-large-eta","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"fully_proved","summary":"For all real numbers $\\theta$ and $\\eta$ with $\\theta^2 < \\eta$, $\\theta \\le 1/2$ and $\\theta < 2\\eta$, Letendre's saving exponent satisfies $\\xi(\\theta, \\eta) = 4(\\theta - \\eta)$.","title":"Letendre's exponent for $\\theta^2 < \\eta$, $\\theta \\le 1/2$ and $\\theta < 2\\eta$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/prod-zpow-le-lcm-zpow-mul-prod-gcd","kind":"node","lean":"MetaMathlibExt.prod_zpow_le_lcm_zpow_mul_prod_gcd","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#prod-zpow-le-lcm-zpow-mul-prod-gcd","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"Let $k$ be a natural number, let $d_1, \\dots, d_k$ be positive integers and let $t$ be an integer. Then, as an inequality of rational numbers,","title":"Cohen's lcm\u2013gcd inequality"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/exists-perm-prefix-sum-le","kind":"node","lean":"MetaMathlibExt.exists_perm_prefix_sum_le","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#exists-perm-prefix-sum-le","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"Let $k$ be a natural number and let $x_1, \\dots, x_k$ and $y_1, \\dots, y_k$ be real numbers with $\\sum_i x_i \\le \\sum_i y_i$. Then there is a permutation $\\sigma$ of $\\{1, \\dots, k\\}$ such that $\\sum_{i \\le s} x_{\\sigma(i)} \\le \\sum_{i \\le s} y_{\\sigma(i)}$ for every $s$ with $0 \\le s \\le k$.","title":"Reordering so that prefix sums compare"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/card-le-of-avoids-residues-prime-pow","kind":"node","lean":"MetaMathlibExt.card_le_of_avoids_residues_primePow","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#card-le-of-avoids-residues-prime-pow","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"Let $N \\ge 1$ be an integer and let $M$ be a finite set of integers contained in $[a, a + N)$ for some integer $a$. Let $S$ be a finite set of primes and, for each $p \\in S$, let $\\beta_p \\ge 1$ be an integer and let $\\Omega_p$ be a set of fewer than $p^{\\beta_p}$ integers in $[0, p^{\\beta_p})$ such that no $m \\in M$ has its remainder modulo $p^{\\beta_p}$ in $\\Omega_p$. Then for every real $Q \\ge 1$,","title":"Arithmetic large sieve with prime-power moduli"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/card-unsolvable-sq-add-prime-pow","kind":"node","lean":"MetaMathlibExt.card_unsolvable_sq_add_primePow","parent":8322,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(x\\)"],["text"," with "],["math","\\(x^2 + p^v u\\)"],["text"," not a square modulo "],["math","\\(p^{v+1}\\)"]],"source":"Counting $x$ with $x^2 + p^v u$ not a square modulo $p^{v+1}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#card-unsolvable-sq-add-prime-pow","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"Let $p \\ge 3$ be a prime and let $u$ be an integer not divisible by $p$. For $v \\ge 0$ let $S_v$ be the number of integers $0 \\le x < p^{v+1}$ for which there is no integer $y$ with $x^2 + p^v u \\equiv y^2 \\pmod{p^{v+1}}$. Then $2 S_0 = p - \\left(\\frac{-u}{p}\\right)$, where $\\left(\\frac{-u}{p}\\right)$ is the Legendre symbol, $S_1 = p$, and $S_v = p S_{v-2}$ for every $v \\ge 2$.","title":"Counting $x$ with $x^2 + p^v u$ not a square modulo $p^{v+1}$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/exists-short-interval-divisor-count-le-o","kind":"node","lean":"MetaMathlibExt.exists_shortIntervalDivisorCount_le_of_window_theta_sq","parent":8322,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisors in windows of length "],["math","\\(n^{\\theta^2 - \\varepsilon}\\)"]],"source":"Divisors in windows of length $n^{\\theta^2 - \\varepsilon}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#exists-short-interval-divisor-count-le-o","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"For a natural number $n$ and real numbers $X$, $Y$, let $D_n(X, Y)$ be the number of divisors $d$ of $n$ with $X \\le d \\le X + Y$. There is an absolute constant $C$ such that for every integer $n \\ge 1$ and all real numbers $\\theta$, $\\varepsilon$ with $0 < \\theta < 1$ and $0 < \\varepsilon < \\theta^2$,\u2026","title":"Divisors in windows of length $n^{\\theta^2 - \\varepsilon}$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/is-greatest-letendre-xi","kind":"node","lean":"MetaMathlibExt.isGreatest_letendreXi","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#is-greatest-letendre-xi","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"Let $\\theta$ and $\\eta$ be real numbers with $\\theta^2 \\le \\eta < \\theta$. Then Letendre's saving exponent $\\xi(\\theta, \\eta)$ is the largest real number $A$ with $0 < A \\le 1$ such that","title":"Letendre's exponent as the largest admissible exponent"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/exists-alpha-with-margin","kind":"node","lean":"MetaMathlibExt.exists_alpha_with_margin","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#exists-alpha-with-margin","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"Let $\\xi(\\theta, \\eta)$ denote Letendre's saving exponent, and let $\\theta$, $\\eta$ and $\\delta$ be real numbers with $0 \\le \\delta \\le 1$, $\\eta < \\theta < 1$ and $0 < \\theta^2 - (\\theta(1 - \\theta))^2 \\delta \\le \\eta$. Then there is a real number $A > 0$ with $\\xi(\\theta, \\eta) - \\delta \\le A \\le \\xi(\\theta, \\eta)$ such that $((\\theta - t)/\\alpha)^2 - (\\eta - t)/\\alpha \\ge (\\theta(1 - \\theta))^2 \\delta$ for all\u2026","title":"Exponents with a margin in Letendre's inequality"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/exists-short-interval-divisor-count-le-l","kind":"node","lean":"MetaMathlibExt.exists_shortIntervalDivisorCount_le_letendre","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#exists-short-interval-divisor-count-le-l","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"For a natural number $n$ and real numbers $X$, $Y$, let $D_n(X, Y)$ be the number of divisors $d$ of $n$ with $X \\le d \\le X + Y$, let $\\tau(n)$ be the number of divisors of $n$, let $V(n)$ be the largest exponent in the prime factorization of $n$, and let $\\xi(\\theta, \\eta)$ be Letendre's saving exponent. There is an absolute constant $C$ such that for all real numbers $\\theta$, $\\eta$ with $0 < \\eta < \\theta < 1$\u2026","title":"Letendre's bound for divisors in short intervals"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/exists-large-short-interval-divisor-coun","kind":"node","lean":"MetaMathlibExt.exists_large_shortIntervalDivisorCount_pos","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#exists-large-short-interval-divisor-coun","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"Let $\\theta$ and $\\varepsilon$ be real numbers with $0 < \\theta < 1$ and $0 < \\varepsilon < \\theta$. Then for every natural number $N$ there is a natural number $n \\ge N$ with at least one divisor $d$ satisfying $n^\\theta \\le d \\le n^\\theta + n^{\\theta - \\varepsilon}$.","title":"Arbitrarily large integers with a divisor in a short window"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/exists-lower-bound-conjecture-constant","kind":"node","lean":"MetaMathlibExt.exists_lower_bound_conjecture_constant","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#exists-lower-bound-conjecture-constant","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"There is an absolute constant $c > 0$ with the following property. Let $\\theta$, $\\varepsilon$ and $k$ be real numbers with $0 < \\theta < 1$ and $0 < \\varepsilon < \\theta$, and suppose that for every integer $n \\ge 1$ the number of divisors $d$ of $n$ with $n^\\theta \\le d \\le n^\\theta + n^{\\theta - \\varepsilon}$ is at most $k$. Then\u2026","title":"Lower bound for constants bounding divisors in short windows"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/exists-short-interval-divisor-count-le-s","kind":"node","lean":"MetaMathlibExt.exists_shortIntervalDivisorCount_le_sqrt_letendre","parent":8322,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#exists-short-interval-divisor-count-le-s","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"For a natural number $n$ and real numbers $X$, $Y$, let $D_n(X, Y)$ be the number of divisors $d$ of $n$ with $X \\le d \\le X + Y$, let $\\tau(n)$ be the number of divisors of $n$, and let $\\xi(\\theta, \\eta)$ be Letendre's saving exponent. There is an absolute constant $C$ such that for all real numbers $\\theta$, $\\eta$ with $0 < \\eta < \\theta < 1$ and every integer $n \\ge 2$,\u2026","title":"Letendre's square-root bound for divisors in short intervals"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/exists-short-interval-divisor-count-sqrt","kind":"node","lean":"MetaMathlibExt.exists_shortIntervalDivisorCount_sqrt_window_le","parent":8322,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisors near "],["math","\\(\\sqrt{an/b}\\)"]],"source":"Divisors near $\\sqrt{an/b}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals/index.html#exists-short-interval-divisor-count-sqrt","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/divisors-in-short-intervals","status":"can_prove","summary":"Let $a$ and $b$ be coprime positive integers. For a natural number $n$ and real numbers $X$, $Y$, let $D_n(X, Y)$ be the number of divisors $d$ of $n$ with $X \\le d \\le X + Y$, and let $\\omega(m)$ be the number of distinct prime factors of $m$. There is a constant $C$, depending on $a$ and $b$, such that for all integers $n \\ge 1$ and $i \\ge 1$, writing $x_0 = \\sqrt{an/b}$ and $w = \\omega(4abn)$,","title":"Divisors near $\\sqrt{an/b}$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","status":"can_prove","title":"Euler's function and divisor sums along linear recurrences"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/totient-div-gg-and-sigma-div-ll-log-log","kind":"node","lean":"MetaMathlibExt.totient_div_gg_and_sigma_div_ll_log_log","parent":8340,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/index.html#totient-div-gg-and-sigma-div-ll-log-log","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","status":"can_prove","summary":"Let $\\varphi$ be Euler's totient function and $\\sigma(n)$ the sum of the positive divisors of $n$. There is a constant $C > 0$ such that $\\varphi(n)/n \\ge C/\\log\\log n$ for every integer $n \\ge 3$, and there is a constant $C'$ such that $\\sigma(n)/n \\le C' \\log\\log n$ for every integer $n \\ge 3$.","title":"Lower bound for Euler's function and upper bound for the sum of divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/card-min-fac-gt-le-div-log","kind":"node","lean":"MetaMathlibExt.card_minFac_gt_le_div_log","parent":8340,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/index.html#card-min-fac-gt-le-div-log","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","status":"can_prove","summary":"Write $p(n)$ for the least prime factor of an integer $n \\ge 2$, and set $p(1) = 1$. There is a constant $C$ such that for all real $x$ and $y$ with $2 \\le y \\le x$, the number of integers $1 \\le n \\le x$ with $p(n) > y$ is at most $C x / \\log y$.","title":"Sieve bound for integers without small prime factors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/card-card-factors-ge-ten-log-log-le","kind":"node","lean":"MetaMathlibExt.card_cardFactors_ge_ten_log_log_le","parent":8340,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/index.html#card-card-factors-ge-ten-log-log-le","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","status":"can_prove","summary":"Let $\\Omega(n)$ be the number of prime factors of $n$ counted with multiplicity. There is a constant $C$ such that for every real $x \\ge 10$, the number of integers $1 \\le n \\le x$ with $\\Omega(n) \\ge 10 \\log\\log x$ is at most $C x / (\\log x)^2$.","title":"Integers with many prime factors are rare"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/card-divisors-sigma-gt-exp-sqrt-log-le","kind":"node","lean":"MetaMathlibExt.card_divisors_sigma_gt_exp_sqrt_log_le","parent":8340,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/index.html#card-divisors-sigma-gt-exp-sqrt-log-le","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","status":"can_prove","summary":"Let $\\sigma(n)$ be the sum of the positive divisors of $n$ and $\\tau(m)$ the number of positive divisors of $m$. There is a constant $C$ such that for every real $x \\ge 10$, the number of integers $1 \\le n \\le x$ with $\\tau(\\sigma(n)) > \\exp(\\sqrt{\\log x})$ is at most $C x / (\\log x)^2$.","title":"Integers whose sum of divisors has many divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/zero-set-card-le-exp-exp-exp","kind":"node","lean":"MetaMathlibExt.zero_set_card_le_exp_exp_exp","parent":8340,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/index.html#zero-set-card-le-exp-exp-exp","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","status":"can_prove","summary":"Let $(U_n)_{n \\ge 1}$ be a sequence of complex numbers satisfying $U_{m+k} = c_0 U_m + c_1 U_{m+1} + \\dots + c_{k-1} U_{m+k-1}$ for all $m \\ge 1$, with complex coefficients $c_i$, where the order $k$ of this recurrence is minimal among all linear recurrences with complex coefficients satisfied by $(U_n)_{n \\ge 1}$, and $k \\ge 2$. Suppose the recurrence is nondegenerate: no root of its characteristic polynomial\u2026","title":"Schmidt's bound on the zeros of a linear recurrence"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/card-abs-le-dominant-root-pow-le-sqrt","kind":"node","lean":"MetaMathlibExt.card_abs_le_dominant_root_pow_le_sqrt","parent":8340,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/index.html#card-abs-le-dominant-root-pow-le-sqrt","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","status":"can_prove","summary":"Let $(U_n)_{n \\ge 1}$ be a sequence of integers satisfying $U_{m+k} = c_0 U_m + c_1 U_{m+1} + \\dots + c_{k-1} U_{m+k-1}$ for all $m \\ge 1$, with integer coefficients $c_i$, where the order $k$ of this recurrence is minimal among all linear recurrences with integer coefficients satisfied by $(U_n)_{n \\ge 1}$. Suppose the recurrence is nondegenerate: no complex root of its characteristic polynomial\u2026","title":"Few small values of a nondegenerate integer linear recurrence"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/card-totient-sigma-inequality-fails-le-d","kind":"node","lean":"MetaMathlibExt.card_totient_sigma_inequality_fails_le_div_log","parent":8340,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r/index.html#card-totient-sigma-inequality-fails-le-d","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/euler-s-function-and-divisor-sums-along-linear-r","status":"can_prove","summary":"Let $(U_n)_{n \\ge 1}$ be a sequence of integers satisfying $U_{m+k} = c_0 U_m + c_1 U_{m+1} + \\dots + c_{k-1} U_{m+k-1}$ for all $m \\ge 1$, with integer coefficients $c_i$, where the order $k$ of this recurrence is minimal among all linear recurrences with integer coefficients satisfied by $(U_n)_{n \\ge 1}$. Suppose the recurrence is nondegenerate: no complex root of its characteristic polynomial\u2026","title":"Euler's function and the sum of divisors along a linear recurrence"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors","status":"can_prove","title":"Gaps between divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors/statement","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem884Wanted.Statement","parent":8348,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors/index.html#statement","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors","status":"fully_proved","summary":"The statement that there is a real constant $C > 0$ such that for every positive integer $n$, if $d_1 < d_2 < \\dots < d_t$ are all the divisors of $n$ in increasing order, then $\\sum_{1 \\le i < j \\le t} \\frac{1}{d_j - d_i} \\le C \\left(1 + \\sum_{1 \\le i < t} \\frac{1}{d_{i+1} - d_i}\\right)$.","title":"Divisor-gap sums (Erd\u0151s Problem 884)"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors/statement-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem884Wanted.Statement_refuted","parent":8348,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors/index.html#statement-refuted","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/gaps-between-divisors","status":"can_prove","summary":"For every real constant $C > 0$ there is a positive integer $n$ whose divisors $d_1 < d_2 < \\dots < d_t$, listed in increasing order, satisfy $\\sum_{1 \\le i < j \\le t} \\frac{1}{d_j - d_i} > C \\left(1 + \\sum_{1 \\le i < t} \\frac{1}{d_{i+1} - d_i}\\right)$.","title":"Erd\u0151s Problem 884 is false (Larsen)"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","title":"Jordan's totient function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient","kind":"node","lean":"ArithmeticFunction.jordanTotient","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For a natural number $k$, Jordan's totient function $J_k$ is the arithmetic function with natural-number values given by $J_k(0) = 0$ and, for $n \\ne 0$,","title":"Jordan's totient function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-zero","kind":"node","lean":"ArithmeticFunction.jordanTotient_zero","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-zero","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For every natural number $k$, the Jordan totient function satisfies $J_k(0) = 0$; this is the convention extending $J_k$ to $0$.","title":"Jordan's totient at zero"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-apply-one","kind":"node","lean":"ArithmeticFunction.jordanTotient_apply_one","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-apply-one","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For every natural number $k$, the Jordan totient function satisfies $J_k(1) = 1$.","title":"Jordan's totient at one"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-eq-prod-factorization","kind":"node","lean":"ArithmeticFunction.jordanTotient_eq_prod_factorization","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-eq-prod-factorization","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For all natural numbers $k$ and $n \\neq 0$, the Jordan totient function satisfies","title":"Prime-factorization formula for Jordan's totient"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-prime-pow","kind":"node","lean":"ArithmeticFunction.jordanTotient_prime_pow","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-prime-pow","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For every natural number $k$, every prime $p$ and every integer $e \\ge 1$, the Jordan totient function satisfies $J_k(p^e) = p^{k(e-1)}(p^k - 1)$.","title":"Jordan's totient at prime powers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-prime","kind":"node","lean":"ArithmeticFunction.jordanTotient_prime","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-prime","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For every natural number $k$ and every prime $p$, the Jordan totient function satisfies $J_k(p) = p^k - 1$.","title":"Jordan's totient at primes"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/is-multiplicative-jordan-totient","kind":"node","lean":"ArithmeticFunction.isMultiplicative_jordanTotient","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#is-multiplicative-jordan-totient","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For every natural number $k$, the Jordan totient function $J_k$ is multiplicative: $J_k(1) = 1$, and $J_k(mn) = J_k(m) J_k(n)$ for all coprime natural numbers $m$ and $n$.","title":"Jordan's totient function is multiplicative"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-one-apply","kind":"node","lean":"ArithmeticFunction.jordanTotient_one_apply","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-one-apply","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For every natural number $n$, $J_1(n) = \\varphi(n)$, where $J_1$ is the Jordan totient function $J_k$ with $k = 1$ and $\\varphi$ is Euler's totient function; at $n = 0$ both sides are $0$.","title":"Jordan's totient of order one is Euler's totient"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-eq-sum-moebius","kind":"node","lean":"ArithmeticFunction.jordanTotient_eq_sum_moebius","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-eq-sum-moebius","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For all natural numbers $k$ and $n \\neq 0$, the Jordan totient function satisfies, as an identity of integers,","title":"M\u00f6bius sum for Jordan's totient"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-mul-prod-prime-factors-po","kind":"node","lean":"ArithmeticFunction.jordanTotient_mul_prod_primeFactors_pow","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-mul-prod-prime-factors-po","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For all natural numbers $k$ and $n \\neq 0$, the Jordan totient function satisfies","title":"Cleared-denominator Euler product for Jordan's totient"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/jordan-totient-eq-rational-prod","kind":"node","lean":"ArithmeticFunction.jordanTotient_eq_rational_prod","parent":8351,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function/index.html#jordan-totient-eq-rational-prod","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/jordan-s-totient-function","status":"fully_proved","summary":"For all natural numbers $k$ and $n \\neq 0$, the Jordan totient function satisfies, as an identity of rational numbers,","title":"Euler product for Jordan's totient"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity","status":"fully_proved","title":"Liouville's divisor-function identity"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity/liouville-divisor-function-identity-gene","kind":"node","lean":"MetaMathlibExt.liouville_divisor_function_identity_general","parent":8363,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity/index.html#liouville-divisor-function-identity-gene","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity","status":"fully_proved","summary":"For every natural number $n$, writing $\\tau(d)$ for the number of positive divisors of $d$,","title":"Liouville's divisor-function identity for all natural numbers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity/liouville-divisor-function-identity","kind":"node","lean":"MetaMathlibExt.liouville_divisor_function_identity","parent":8363,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity/index.html#liouville-divisor-function-identity","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/liouville-s-divisor-function-identity","status":"fully_proved","summary":"For every positive integer $n$, writing $\\tau(d)$ for the number of positive divisors of $d$,","title":"Liouville's divisor-function identity"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers","status":"fully_proved","title":"Nicol numbers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers/is-nicol-number","kind":"node","lean":"MetaMathlibExt.IsNicolNumber","parent":8366,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers/index.html#is-nicol-number","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers","status":"fully_proved","summary":"A Nicol number is a positive integer $n$ that divides $\\varphi(n) + \\sigma(n)$, where $\\varphi$ is Euler's totient function and $\\sigma(n)$ is the sum of the positive divisors of $n$.","title":"Nicol numbers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers/is-tnicol-number","kind":"node","lean":"MetaMathlibExt.IsTNicolNumber","parent":8366,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(t\\)"],["text","-Nicol numbers"]],"source":"$t$-Nicol numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers/index.html#is-tnicol-number","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers","status":"fully_proved","summary":"For a natural number $t$, a $t$-Nicol number is a positive integer $n$ such that $\\varphi(n) + \\sigma(n) = t n$, where $\\varphi$ is Euler's totient function and $\\sigma(n)$ is the sum of the positive divisors of $n$.","title":"$t$-Nicol numbers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers/is-nicol-number-2","kind":"node","lean":"MetaMathlibExt.IsTNicolNumber.isNicolNumber","parent":8366,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every "],["math","\\(t\\)"],["text","-Nicol number is a Nicol number"]],"source":"Every $t$-Nicol number is a Nicol number"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers/index.html#is-nicol-number-2","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/nicol-numbers","status":"fully_proved","summary":"For all natural numbers $t$ and $n$, if $n$ is a $t$-Nicol number, then $n$ is a Nicol number.","title":"Every $t$-Nicol number is a Nicol number"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","title":"Parity of divisor counts"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/exists-sum-eq-two-mul-of-involution","kind":"node","lean":"Finset.exists_sum_eq_two_mul_of_involution","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#exists-sum-eq-two-mul-of-involution","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"Let $s$ be a finite subset of a type $\\alpha$, let $f : \\alpha \\to \\alpha$ satisfy $f(a) \\in s$, $f(f(a)) = a$ and $f(a) \\neq a$ for every $a \\in s$, and let $g : \\alpha \\to \\mathbb{Z}$ satisfy $g(f(a)) = g(a)$ for every $a \\in s$. Then $\\sum_{a \\in s} g(a) = 2k$ for some integer $k$.","title":"Sums invariant under a fixed-point-free involution are even"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/even-card-of-involution","kind":"node","lean":"Finset.even_card_of_involution","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#even-card-of-involution","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"Let $s$ be a finite subset of a type $\\alpha$ and let $f : \\alpha \\to \\alpha$ satisfy $f(a) \\in s$, $f(f(a)) = a$ and $f(a) \\neq a$ for every $a \\in s$. Then $s$ has an even number of elements.","title":"A fixed-point-free involution forces even cardinality"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/sum-div-two-mod-two-eq-card-filter","kind":"node","lean":"Finset.sum_div_two_mod_two_eq_card_filter","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#sum-div-two-mod-two-eq-card-filter","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"Let $s$ be a finite set of natural numbers and let $f : \\mathbb{N} \\to \\mathbb{N}$ be a function such that $f(k)$ is even for every $k \\in s$. Then the natural number $\\frac{1}{2} \\sum_{k \\in s} f(k)$ is congruent modulo $2$ to the number of elements $k \\in s$ with $f(k) \\equiv 2 \\pmod 4$.","title":"Half a sum of even numbers modulo 2"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/div-mem-divisors","kind":"node","lean":"Nat.div_mem_divisors","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#div-mem-divisors","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"If $n$ is a nonzero natural number and $d$ is a divisor of $n$, then $n/d$ is also a divisor of $n$.","title":"Complementary divisor"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/div-div-self-of-mem-divisors","kind":"node","lean":"Nat.div_div_self_of_mem_divisors","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#div-div-self-of-mem-divisors","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"If $n$ is a nonzero natural number and $d$ is a divisor of $n$, then $n/(n/d) = d$.","title":"The complementary divisor of the complementary divisor"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/div-ne-of-mem-divisors-of-not-is-square","kind":"node","lean":"Nat.div_ne_of_mem_divisors_of_not_isSquare","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#div-ne-of-mem-divisors-of-not-is-square","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"If $n$ is a natural number that is not a perfect square and $d$ is a divisor of $n$, then $n/d \\neq d$.","title":"Complementary divisors of a nonsquare are distinct"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/card-divisors-two-mul-of-odd","kind":"node","lean":"Nat.card_divisors_two_mul_of_odd","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#card-divisors-two-mul-of-odd","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"If $u$ is an odd natural number, then $2u$ has exactly twice as many positive divisors as $u$.","title":"Divisor count of twice an odd number"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/even-card-divisors-of-not-is-square","kind":"node","lean":"Nat.even_card_divisors_of_not_isSquare","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#even-card-divisors-of-not-is-square","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"If a natural number $n$ is not a perfect square, then $n$ has an even number of positive divisors.","title":"A nonsquare has an even number of divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/odd-card-divisors-iff-is-square","kind":"node","lean":"Nat.odd_card_divisors_iff_isSquare","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#odd-card-divisors-iff-is-square","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"A nonzero natural number $u$ has an odd number of positive divisors if and only if $u$ is a perfect square.","title":"Squares are exactly the numbers with an odd number of divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/nat-eq-one-or-neg-one-of-odd","kind":"node","lean":"ZMod.\u03c7\u2084_nat_eq_one_or_neg_one_of_odd","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#nat-eq-one-or-neg-one-of-odd","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"Let $\\chi_4$ be the nontrivial character modulo $4$ with values in $\\mathbb{Z}$: $\\chi_4(n) = 1$ for $n \\equiv 1 \\pmod 4$, $\\chi_4(n) = -1$ for $n \\equiv 3 \\pmod 4$, and $\\chi_4(n) = 0$ for even $n$. For every odd natural number $d$, $\\chi_4(d) = 1$ or $\\chi_4(d) = -1$.","title":"The quadratic character mod 4 is \u00b11 on odd numbers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/nat-eq-one-or-neg-one-of-odd-2","kind":"node","lean":"ZMod.\u03c7\u2088'_nat_eq_one_or_neg_one_of_odd","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#nat-eq-one-or-neg-one-of-odd-2","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"Let $\\chi_8'$ be the primitive quadratic character modulo $8$ with values in $\\mathbb{Z}$ (the character of $\\mathbb{Q}(\\sqrt{-2})$): $\\chi_8'(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_8'(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$, and $\\chi_8'(n) = 0$ for even $n$. For every odd natural number $d$, $\\chi_8'(d) = 1$ or $\\chi_8'(d) = -1$.","title":"The second quadratic character mod 8 is \u00b11 on odd numbers"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/sum-divisors-emod-four-eq-card","kind":"node","lean":"MulChar.sum_divisors_emod_four_eq_card","parent":8370,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts/index.html#sum-divisors-emod-four-eq-card","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/parity-of-divisor-counts","status":"fully_proved","summary":"Let $N$ be a natural number and let $\\chi$ be a multiplicative character on $\\mathbb{Z}/N\\mathbb{Z}$ with values in $\\mathbb{Z}$, applied to natural numbers through their residues modulo $N$. Let $n$ be a natural number that is not a perfect square such that $\\chi(n) = 1$ and $\\chi(d) \\in \\{1, -1\\}$ for every positive divisor $d$ of $n$, and let $\\tau(n)$ be the number of positive divisors of $n$. Then\u2026","title":"Character divisor sums modulo 4"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","title":"Pillai's gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/dirichlet-series-gcd-sum","kind":"node","lean":"MetaMathlibExt.dirichlet_series_gcd_sum","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#dirichlet-series-gcd-sum","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"fully_proved","summary":"For $n \\ge 1$ let $g(n) = \\sum_{i=1}^{n} \\gcd(n, i)$. For every complex number $s$ with $\\operatorname{Re} s > 2$, the series $\\sum_{n \\ge 1} g(n)/n^s$ sums to $\\zeta(s-1)^2/\\zeta(s)$, where $\\zeta$ is the Riemann zeta function.","title":"Dirichlet series of the gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/gcd-sum-eq-totient-sum","kind":"node","lean":"MetaMathlibExt.gcd_sum_eq_totient_sum","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#gcd-sum-eq-totient-sum","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"fully_proved","summary":"For every positive integer $n$, $\\sum_{j=1}^{n} \\gcd(j, n) = n \\sum_{d \\mid n} \\varphi(d)/d$, where $\\varphi$ is Euler's totient function and $d$ runs over the positive divisors of $n$.","title":"The gcd-sum as a divisor sum of totients"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/gcd-sum-mean","kind":"node","lean":"MetaMathlibExt.gcdSumMean","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#gcd-sum-mean","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"fully_proved","summary":"For a natural number $n$, let $P(n) = \\sum_{j=1}^{n} \\gcd(j, n)$ be Pillai's gcd-sum function. The mean gcd-sum function is the real-valued function given by $A(n) = P(n)/n$ for $n \\ge 1$ and $A(0) = 0$.","title":"The mean gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/gcd-sum-mean-one","kind":"node","lean":"MetaMathlibExt.gcdSumMean_one","parent":8383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The mean gcd-sum function at "],["math","\\(1\\)"]],"source":"The mean gcd-sum function at $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#gcd-sum-mean-one","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"fully_proved","summary":"The mean gcd-sum function $A$ satisfies $A(1) = 1$.","title":"The mean gcd-sum function at $1$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/gcd-sum-mean-two","kind":"node","lean":"MetaMathlibExt.gcdSumMean_two","parent":8383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The mean gcd-sum function at "],["math","\\(2\\)"]],"source":"The mean gcd-sum function at $2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#gcd-sum-mean-two","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"fully_proved","summary":"The mean gcd-sum function $A$ satisfies $A(2) = 3/2$, since $A(2) = P(2)/2$, where $P(n) = \\sum_{j=1}^{n} \\gcd(j, n)$ is Pillai's gcd-sum function and $P(2) = \\gcd(1, 2) + \\gcd(2, 2) = 3$.","title":"The mean gcd-sum function at $2$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/gcd-sum-mean-eq-sum-totient-div","kind":"node","lean":"MetaMathlibExt.gcdSumMean_eq_sum_totient_div","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#gcd-sum-mean-eq-sum-totient-div","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the mean gcd-sum function $A$ satisfies $A(n) = \\sum_{d \\mid n} \\varphi(d)/d$, where $\\varphi$ is Euler's totient function and the sum runs over the positive divisors $d$ of $n$.","title":"Divisor-sum formula for the mean gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/sum-gcd-sum-mean-sq-asymptotic","kind":"node","lean":"MetaMathlibExt.sum_gcdSumMean_sq_asymptotic","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#sum-gcd-sum-mean-sq-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"For $n \\ge 1$ let $A(n) = \\frac{1}{n} \\sum_{j=1}^{n} \\gcd(j, n)$ be the mean gcd-sum function. There are a real polynomial $P$ and a real constant $C$ such that, as $x \\to \\infty$,","title":"Asymptotic formula for the second moment of the mean gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/sum-gcd-sum-mean-cube-asymptotic","kind":"node","lean":"MetaMathlibExt.sum_gcdSumMean_cube_asymptotic","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#sum-gcd-sum-mean-cube-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"For $n \\ge 1$ let $A(n) = \\frac{1}{n} \\sum_{j=1}^{n} \\gcd(j, n)$ be the mean gcd-sum function. There are a real polynomial $P$ and a real constant $C$ such that, as $x \\to \\infty$,","title":"Asymptotic formula for the third moment of the mean gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/sum-gcd-sum-mean-pow-four-asymptotic","kind":"node","lean":"MetaMathlibExt.sum_gcdSumMean_pow_four_asymptotic","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#sum-gcd-sum-mean-pow-four-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"For $n \\ge 1$ let $A(n) = \\frac{1}{n} \\sum_{j=1}^{n} \\gcd(j, n)$ be the mean gcd-sum function. There is a real polynomial $P$ such that for every $\\varepsilon > 0$, as $x \\to \\infty$,","title":"Asymptotic formula for the fourth moment of the mean gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/sum-gcd-sum-mean-pow-asymptotic","kind":"node","lean":"MetaMathlibExt.sum_gcdSumMean_pow_asymptotic","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#sum-gcd-sum-mean-pow-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"For $n \\ge 1$ let $A(n) = \\frac{1}{n} \\sum_{j=1}^{n} \\gcd(j, n)$ be the mean gcd-sum function, and let $k \\ge 5$ be an integer. There is a real polynomial $P$ such that for every $\\varepsilon > 0$, as $x \\to \\infty$,","title":"Asymptotic formula for the higher moments of the mean gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/lseries-gcd-sum-mean-pow-eq-riemann-zeta","kind":"node","lean":"MetaMathlibExt.LSeries_gcdSumMean_pow_eq_riemannZeta_pow_mul","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#lseries-gcd-sum-mean-pow-eq-riemann-zeta","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"Let $k$ be a natural number and let $A$ be the mean gcd-sum function. There are functions $g, h \\colon \\mathbb{N} \\to \\mathbb{C}$ such that the Dirichlet series $G_k(s) = \\sum_{n \\ge 1} g(n) n^{-s}$ converges absolutely for every complex $s$ with $\\operatorname{Re} s > 1/2$, the Dirichlet series $\\tilde G_{k,2}(s) = \\sum_{n \\ge 1} h(n) n^{-s}$ converges absolutely for every complex $s$ with\u2026","title":"Factorization of the Dirichlet series of powers of the mean gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/alternating-choose-two-pow-mul-pow-eq-ze","kind":"node","lean":"MetaMathlibExt.alternating_choose_two_pow_mul_pow_eq_zero","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#alternating-choose-two-pow-mul-pow-eq-ze","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"Let $k$ and $m$ be natural numbers with $k \\ge 2$ and $m \\ge 2^k - 1$. Then, computed in the integers,","title":"An alternating binomial sum vanishes"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/lseries-gcd-sum-mean-sq-eq-riemann-zeta","kind":"node","lean":"MetaMathlibExt.LSeries_gcdSumMean_sq_eq_riemannZeta_pow_four_div","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#lseries-gcd-sum-mean-sq-eq-riemann-zeta","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"Let $A$ be the mean gcd-sum function. There is a function $h \\colon \\mathbb{N} \\to \\mathbb{C}$ such that the Dirichlet series $\\tilde G_{2,2}(s) = \\sum_{n \\ge 1} h(n) n^{-s}$ converges absolutely for every complex $s$ with $\\operatorname{Re} s > 0$, and for every complex $s$ with $\\operatorname{Re} s > 1$,","title":"Dirichlet series of the square of the mean gcd-sum function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/riemann-zeta-eighth-moment-five-eighths","kind":"node","lean":"MetaMathlibExt.riemannZeta_eighth_moment_five_eighths_le","parent":8383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ivi\u0107's eighth-moment bound on the line "],["math","\\(\\operatorname{Re} s = 5/8\\)"]],"source":"Ivi\u0107's eighth-moment bound on the line $\\operatorname{Re} s = 5/8$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#riemann-zeta-eighth-moment-five-eighths","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"There are real constants $C$ and $T_0$ such that for every real $T \\ge T_0$,","title":"Ivi\u0107's eighth-moment bound on the line $\\operatorname{Re} s = 5/8$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/riemann-zeta-twelfth-moment-asymptotic","kind":"node","lean":"MetaMathlibExt.riemannZeta_twelfth_moment_asymptotic","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#riemann-zeta-twelfth-moment-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"Lemma 3 (quoted twelfth-moment asymptotic). For fixed `5/7 < \u03c3 < 1` and every `\u03b5 > 0`, `\u222b_1^T |\u03b6(\u03c3 + i t)|^{12} dt = T \u2211_{n \u2265 1} d_6(n)^2 n^{-2\u03c3} + O(T^{(9 - 7\u03c3)/4 + \u03b5})`, where `d_6 = \u03b6^{*6}` is the six-fold divisor function. The printed side condition `|t| \u2265 1` is implied by the integration range and omitted.","title":"Lemma 3"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/riemann-zeta-fourth-moment-div-le","kind":"node","lean":"MetaMathlibExt.riemannZeta_fourth_moment_div_le","parent":8383,"reading_title":{"format":"inline-parts-v1","parts":[["text","A weighted fourth-moment bound for "],["math","\\(\\zeta\\)"],["text"," on the critical line"]],"source":"A weighted fourth-moment bound for $\\zeta$ on the critical line"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#riemann-zeta-fourth-moment-div-le","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"There are real constants $C$ and $T_0$ such that for every real $T \\ge T_0$,","title":"A weighted fourth-moment bound for $\\zeta$ on the critical line"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/exists-riemann-zeta-norm-le-exp-loglog-s","kind":"node","lean":"MetaMathlibExt.exists_riemannZeta_norm_le_exp_loglog_sq","parent":8383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramachandra\u2013Sankaranarayanan bound for "],["math","\\(\\zeta\\)"],["text"," at a well-chosen height"]],"source":"Ramachandra\u2013Sankaranarayanan bound for $\\zeta$ at a well-chosen height"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#exists-riemann-zeta-norm-le-exp-loglog-s","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"Let $\\zeta$ be the Riemann zeta function. There are real constants $C$ and $U_0$ such that for every real $U \\ge U_0$ there is a real number $T$ with $U < T < 2U$ such that for every real $\\sigma \\ge 1/2$,","title":"Ramachandra\u2013Sankaranarayanan bound for $\\zeta$ at a well-chosen height"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/gcd-sum-mean-pow-error-sq-integral-le","kind":"node","lean":"MetaMathlibExt.gcdSumMean_pow_error_sq_integral_le","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#gcd-sum-mean-pow-error-sq-integral-le","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"Let $A(n) = \\frac{1}{n} \\sum_{m=1}^{n} \\gcd(m, n)$ be the mean gcd-sum function. Assume the Riemann Hypothesis and let $k \\ge 2$ be an integer. Then there is a real polynomial $P$ with the following property. Write $E(u) = \\sum_{1 \\le n \\le u} A(n)^k - u P(\\log u)$ for real $u$. For every real $\\eta$ with $\\frac{1}{2^k}\\left(4^k - 3^k - \\binom{2^k}{2}\\right) + \\frac{1}{2^{k+1}} + 1 \\le \\eta$ there are real\u2026","title":"Mean square of the error term for gcd-sum moments under RH"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/gcd-sum-mean-pow-error-integral-le","kind":"node","lean":"MetaMathlibExt.gcdSumMean_pow_error_integral_le","parent":8383,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function/index.html#gcd-sum-mean-pow-error-integral-le","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/pillai-s-gcd-sum-function","status":"can_prove","summary":"Let $A(n) = \\frac{1}{n} \\sum_{m=1}^{n} \\gcd(m, n)$ be the mean gcd-sum function. Assume the Riemann Hypothesis and let $k \\ge 2$ be an integer. Then there is a real polynomial $P$ with the following property. Write $E(u) = \\sum_{1 \\le n \\le u} A(n)^k - u P(\\log u)$ for real $u$. For every real $\\eta$ with $\\frac{1}{2^k}\\left(4^k - 3^k - \\binom{2^k}{2}\\right) + \\frac{1}{2^{k+1}} + 1 \\le \\eta$ there are real\u2026","title":"Integral of the error term for gcd-sum moments under RH"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums","status":"fully_proved","title":"Ramanujan sums"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/ramanujan-sum","kind":"node","lean":"MetaMathlibExt.ramanujanSum","parent":8403,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/index.html#ramanujan-sum","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums","status":"fully_proved","summary":"For a positive integer $n$ and an integer $t$, the Ramanujan sum of modulus $n$ at $t$ is the complex number","title":"Ramanujan sum"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/ramanujan-sum-zero","kind":"node","lean":"MetaMathlibExt.ramanujanSum_zero","parent":8403,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/index.html#ramanujan-sum-zero","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums","status":"fully_proved","summary":"For every positive integer $n$, the Ramanujan sum $c_n(0)$ equals Euler's totient $\\varphi(n)$.","title":"Ramanujan sum at zero"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/ramanujan-sum-modulus-one","kind":"node","lean":"MetaMathlibExt.ramanujanSum_modulus_one","parent":8403,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/index.html#ramanujan-sum-modulus-one","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums","status":"fully_proved","summary":"For every integer $t$, the Ramanujan sum $c_1(t)$ equals $1$.","title":"Ramanujan sums of modulus one"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/ramanujan-sum-add-mul-modulus","kind":"node","lean":"MetaMathlibExt.ramanujanSum_add_mul_modulus","parent":8403,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums/index.html#ramanujan-sum-add-mul-modulus","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/ramanujan-sums","status":"fully_proved","summary":"For every positive integer $n$ and all integers $t$ and $k$, the Ramanujan sum $c_n$ of modulus $n$ satisfies $c_n(t + kn) = c_n(t)$.","title":"Periodicity of Ramanujan sums"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","title":"Residues of rounded quotients n/k"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/floor-quotient-odd-count","kind":"node","lean":"MetaMathlibExt.floorQuotientOddCount","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#floor-quotient-odd-count","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"fully_proved","summary":"For a natural number $n$, the floor sequence parity count $\\mathcal{F}_n$ is the number of integers $k$ with $1 \\le k \\le n$ for which $\\lfloor n/k \\rfloor$ is odd (OEIS A059851).","title":"Floor sequence parity count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/floor-quotient-odd-count-ten","kind":"node","lean":"MetaMathlibExt.floorQuotientOddCount_ten","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","The floor sequence parity count at "],["math","\\(10\\)"]],"source":"The floor sequence parity count at $10$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#floor-quotient-odd-count-ten","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"fully_proved","summary":"The number $\\mathcal{F}_{10}$ of integers $k$ with $1 \\le k \\le 10$ for which $\\lfloor 10/k \\rfloor$ is odd equals $7$: the quotients $\\lfloor 10/k \\rfloor$ for $k = 1, \\dots, 10$ are $10, 5, 3, 2, 2, 1, 1, 1, 1, 1$, and seven of them are odd.","title":"The floor sequence parity count at $10$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/ceil-quotient-odd-count","kind":"node","lean":"MetaMathlibExt.ceilQuotientOddCount","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#ceil-quotient-odd-count","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"fully_proved","summary":"For a natural number $n$, the ceiling parity count $\\mathcal{C}_n$ is the number of integers $k$ with $1 \\le k \\le n$ for which $\\lceil n/k \\rceil$ is odd (OEIS A330926).","title":"Ceiling parity count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/ceil-quotient-odd-count-nine","kind":"node","lean":"MetaMathlibExt.ceilQuotientOddCount_nine","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ceiling parity count at "],["math","\\(9\\)"]],"source":"The ceiling parity count at $9$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#ceil-quotient-odd-count-nine","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"fully_proved","summary":"The ceiling parity count $\\mathcal{C}_9$, the number of integers $k$ with $1 \\le k \\le 9$ for which $\\lceil 9/k \\rceil$ is odd, equals $5$.","title":"The ceiling parity count at $9$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/round-quotient-odd-count","kind":"node","lean":"MetaMathlibExt.roundQuotientOddCount","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#round-quotient-odd-count","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"fully_proved","summary":"For a natural number $n$, the rounding sequence parity count $\\mathcal R_n$ is the number of integers $k$ with $1 \\le k \\le n$ for which the nearest integer to $n/k$ is odd, where half-integers are rounded up: the nearest integer to a rational number $x$ is taken to be $\\lfloor x + 1/2 \\rfloor$.","title":"Rounding sequence parity count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/round-quotient-odd-count-ten","kind":"node","lean":"MetaMathlibExt.roundQuotientOddCount_ten","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","The rounding sequence parity count at "],["math","\\(10\\)"]],"source":"The rounding sequence parity count at $10$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#round-quotient-odd-count-ten","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"fully_proved","summary":"The number $\\mathcal{R}_{10}$ of integers $k$ with $1 \\le k \\le 10$ for which the nearest integer to $10/k$ (half-integers rounded up) is odd equals $7$.","title":"The rounding sequence parity count at $10$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/shifted-floor-residue-count","kind":"node","lean":"MetaMathlibExt.shiftedFloorResidueCount","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#shifted-floor-residue-count","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"fully_proved","summary":"For a natural number $n$, real numbers $\\alpha$ and $\\nu$, an integer $r$ and a natural number $m$, the count $\\mathcal N_{n,\\alpha,\\nu,r,m}$ is the number of integers $k$ with $1 \\le k \\le n$ such that $\\lfloor (n - \\nu)/k + \\alpha \\rfloor \\equiv r \\pmod m$. In other words, it counts the terms of the $\\alpha$-shifted, $\\nu$-offset floor sequence\u2026","title":"Residue counts of the shifted floor sequence"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/shifted-floor-residue-count-modulus-one","kind":"node","lean":"MetaMathlibExt.shiftedFloorResidueCount_modulus_one","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#shifted-floor-residue-count-modulus-one","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"fully_proved","summary":"For a natural number $n$, real numbers $\\alpha$ and $\\nu$, an integer $r$ and a natural number $m$, let $\\mathcal N_{n,\\alpha,\\nu,r,m}$ be the number of integers $k$ with $1 \\le k \\le n$ such that $\\lfloor (n - \\nu)/k + \\alpha \\rfloor \\equiv r \\pmod m$. Then $\\mathcal N_{n,\\alpha,\\nu,r,1} = n$ for every natural number $n$, all real numbers $\\alpha$ and $\\nu$ and every integer $r$.","title":"Every term is counted modulo one"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/floor-div-sequence-card-ge","kind":"node","lean":"MetaMathlibExt.floor_div_sequence_card_ge","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#floor-div-sequence-card-ge","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every natural number $n$ and every integer $k \\ge 1$, exactly $\\lfloor n/k \\rfloor$ of the integers $d$ with $1 \\le d \\le n$ satisfy $\\lfloor n/d \\rfloor \\ge k$; that is, exactly $\\lfloor n/k \\rfloor$ terms of the floor sequence $\\lfloor n/1 \\rfloor, \\dots, \\lfloor n/n \\rfloor$ are at least $k$.","title":"Counting large terms of the floor sequence"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/floor-quotient-odd-count-eq-alternating","kind":"node","lean":"MetaMathlibExt.floorQuotientOddCount_eq_alternating_sum","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#floor-quotient-odd-count-eq-alternating","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $\\mathcal{F}_n$ be the number of integers $k$ with $1 \\le k \\le n$ for which $\\lfloor n/k \\rfloor$ is odd. For every natural number $n$, $\\mathcal{F}_n = \\sum_{d=1}^{n} \\lfloor n/d \\rfloor (-1)^{d+1}$.","title":"The floor parity count as an alternating sum"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/floor-quotient-odd-count-eq-divisor-summ","kind":"node","lean":"MetaMathlibExt.floorQuotientOddCount_eq_divisor_summatory_sub","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#floor-quotient-odd-count-eq-divisor-summ","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $\\tau(m)$ be the number of positive divisors of $m$, let $D(x) = \\sum_{1 \\le m \\le x} \\tau(m)$, and let $\\mathcal{F}_n$ be the number of integers $k$ with $1 \\le k \\le n$ for which $\\lfloor n/k \\rfloor$ is odd. For every natural number $n$, $\\mathcal{F}_n = D(n) - 2D(n/2)$, where $D(n/2) = D(\\lfloor n/2 \\rfloor)$.","title":"The floor parity count through the divisor summatory function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/floor-quotient-odd-count-eq-hyperbolic-l","kind":"node","lean":"MetaMathlibExt.floorQuotientOddCount_eq_hyperbolic_lattice_difference","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#floor-quotient-odd-count-eq-hyperbolic-l","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every natural number $n$, the floor sequence parity count $\\mathcal{F}_n$ equals the number of integer pairs $(x, y)$ with $x, y \\ge 1$ and $n/2 < xy \\le n$ minus the number of integer pairs $(x, y)$ with $x, y \\ge 1$ and $xy \\le n/2$.","title":"Odd floor quotients as a hyperbolic lattice-point difference"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/divisor-summatory-asymptotic","kind":"node","lean":"MetaMathlibExt.divisor_summatory_asymptotic","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#divisor-summatory-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $\\tau(m)$ be the number of positive divisors of $m$, let $D(x) = \\sum_{1 \\le m \\le x} \\tau(m)$ for real $x$, and let $\\gamma$ be the Euler\u2013Mascheroni constant. There is a real constant $C$ such that $|D(x) - (x \\log x + (2\\gamma - 1)\\, x)| \\le C \\sqrt{x}$ for every real $x \\ge 1$.","title":"Dirichlet's asymptotic for the divisor summatory function"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/floor-quotient-odd-count-asymptotic","kind":"node","lean":"MetaMathlibExt.floorQuotientOddCount_asymptotic","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#floor-quotient-odd-count-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $\\mathcal{F}_n$ be the number of integers $k$ with $1 \\le k \\le n$ for which $\\lfloor n/k \\rfloor$ is odd. There is a real constant $C$ such that $|\\mathcal{F}_n - n \\log 2| \\le C \\sqrt{n}$ for every integer $n \\ge 1$, and $\\mathcal{F}_n / n \\to \\log 2$ as $n \\to \\infty$.","title":"Asymptotics of the floor parity count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/ceil-quotient-odd-count-succ-eq","kind":"node","lean":"MetaMathlibExt.ceilQuotientOddCount_succ_eq","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#ceil-quotient-odd-count-succ-eq","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For a natural number $m$, let $\\mathcal{C}_m$ and $\\mathcal{F}_m$ be the numbers of integers $k$ with $1 \\le k \\le m$ for which $\\lceil m/k \\rceil$, respectively $\\lfloor m/k \\rfloor$, is odd. Then for every natural number $n$, $\\mathcal{C}_{n+1} = (n + 1) - \\mathcal{F}_n$.","title":"Ceiling parity count from the floor parity count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/ceil-quotient-odd-count-asymptotic","kind":"node","lean":"MetaMathlibExt.ceilQuotientOddCount_asymptotic","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#ceil-quotient-odd-count-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $\\mathcal{C}_n$ be the number of integers $k$ with $1 \\le k \\le n$ for which $\\lceil n/k \\rceil$ is odd. There is a real constant $C$ such that $|\\mathcal{C}_n - (1 - \\log 2)\\, n| \\le C \\sqrt{n}$ for every integer $n \\ge 1$, and $\\mathcal{C}_n / n \\to 1 - \\log 2$ as $n \\to \\infty$.","title":"Asymptotics of the ceiling parity count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-divisors-le-two-mul-sqrt","kind":"node","lean":"MetaMathlibExt.card_divisors_le_two_mul_sqrt","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-divisors-le-two-mul-sqrt","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every natural number $n$, the number of positive divisors of $n$ is at most $2\\sqrt{n}$. For $n = 0$ the set of divisors is taken to be empty, so the bound holds trivially.","title":"Square-root bound for the number of divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/round-div-sequence-card-ge","kind":"node","lean":"MetaMathlibExt.round_div_sequence_card_ge","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#round-div-sequence-card-ge","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $n$ be a natural number. Every $d$ with $1 \\le d \\le n$ satisfies $\\lfloor n/d + 1/2 \\rfloor \\ge 1$, and for every integer $k \\ge 2$ the number of $d$ with $1 \\le d \\le n$ and $\\lfloor n/d + 1/2 \\rfloor \\ge k$ is $\\lfloor 2n/(2k-1) \\rfloor$.","title":"Counting large terms of the rounded quotient sequence"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/round-quotient-odd-count-eq-alternating","kind":"node","lean":"MetaMathlibExt.roundQuotientOddCount_eq_alternating_sum","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#round-quotient-odd-count-eq-alternating","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every natural number $n$, the rounding sequence parity count $\\mathcal{R}_n$ satisfies $\\mathcal{R}_n = -n + \\sum_{d=1}^{n} (-1)^{d+1} \\lfloor 2n/(2d-1) \\rfloor$.","title":"Alternating-sum formula for the rounding sequence parity count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/round-quotient-odd-count-eq-sum-divisor","kind":"node","lean":"MetaMathlibExt.roundQuotientOddCount_eq_sum_divisor_difference","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#round-quotient-odd-count-eq-sum-divisor","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For a positive integer $k$ and $r \\in \\{1, 3\\}$, let $d_{r,4}(k)$ be the number of positive divisors of $k$ congruent to $r$ modulo $4$. Then for every natural number $n$, the rounding sequence parity count $\\mathcal{R}_n$ satisfies $\\mathcal{R}_n = -n + \\sum_{k=1}^{2n} (d_{1,4}(k) - d_{3,4}(k))$.","title":"Divisor-sum formula for the rounding sequence parity count"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-sq-add-sq-eq","kind":"node","lean":"MetaMathlibExt.card_sq_add_sq_eq","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-sq-add-sq-eq","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every integer $n \\ge 1$, the number $r_2(n)$ of pairs $(a, b) \\in \\mathbb{Z}^2$ with $a^2 + b^2 = n$ equals $4\\,(d_{1,4}(n) - d_{3,4}(n))$, where $d_{r,4}(n)$ is the number of positive divisors of $n$ congruent to $r$ modulo $4$.","title":"Jacobi's formula for the number of representations as a sum of two squares"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-lattice-disc-two-mul","kind":"node","lean":"MetaMathlibExt.card_lattice_disc_two_mul","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lattice points in the disc of radius "],["math","\\(\\sqrt{2n}\\)"]],"source":"Lattice points in the disc of radius $\\sqrt{2n}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-lattice-disc-two-mul","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every natural number $n$, the number of pairs $(x, y) \\in \\mathbb{Z}^2$ with $x^2 + y^2 \\le 2n$ equals $4\\,\\mathcal{R}_n + 4n + 1$, where $\\mathcal{R}_n$ is the number of $k \\in \\{1, \\dots, n\\}$ for which the nearest integer to $n/k$, with half-integers rounded up, is odd. At $n = 0$ both sides are $1$.","title":"Lattice points in the disc of radius $\\sqrt{2n}$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/round-quotient-odd-count-sub-one-le-succ","kind":"node","lean":"MetaMathlibExt.roundQuotientOddCount_sub_one_le_succ","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#round-quotient-odd-count-sub-one-le-succ","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every natural number $n$, the rounding sequence parity count $\\mathcal{R}$ satisfies $\\mathcal{R}_{n+1} \\ge \\mathcal{R}_n - 1$.","title":"The rounding sequence parity count drops by at most one"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/gauss-circle-asymptotic","kind":"node","lean":"MetaMathlibExt.gauss_circle_asymptotic","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#gauss-circle-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"There is a constant $C$ such that for every real $x \\ge 1$, the number $N(x)$ of integer pairs $(a, b)$ with $a^2 + b^2 \\le x$ satisfies $|N(x) - \\pi x| \\le C \\sqrt{x}$.","title":"Gauss's bound for lattice points in a disc"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/round-quotient-odd-count-asymptotic","kind":"node","lean":"MetaMathlibExt.roundQuotientOddCount_asymptotic","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#round-quotient-odd-count-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"There is a constant $C$ such that the rounding sequence parity count $\\mathcal{R}_n$ satisfies $|\\mathcal{R}_n - (\\pi/2 - 1)n| \\le C\\sqrt{n}$ for every natural number $n \\ge 1$; moreover $\\mathcal{R}_n / n \\to \\pi/2 - 1$ as $n \\to \\infty$.","title":"Asymptotic density of odd rounded quotients"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/shifted-floor-sequence-card-ge","kind":"node","lean":"MetaMathlibExt.shifted_floor_sequence_card_ge","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#shifted-floor-sequence-card-ge","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $n$ be a natural number and let $\\alpha$ and $\\nu$ be real numbers with $0 \\le \\alpha < 1$, $0 \\le \\nu < 1$ and $\\nu \\le n\\alpha$. Then $\\lfloor (n - \\nu)/d + \\alpha \\rfloor \\ge 1$ for all $n$ integers $d$ with $1 \\le d \\le n$, and for every natural number $k \\ge 2$ the number of integers $d$ with $1 \\le d \\le n$ and $\\lfloor (n - \\nu)/d + \\alpha \\rfloor \\ge k$ is $\\lfloor (n - \\nu)/(k - \\alpha) \\rfloor$.","title":"Tail counts of the shifted floor sequence"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/shifted-floor-residue-count-eq-sum-floor","kind":"node","lean":"MetaMathlibExt.shiftedFloorResidueCount_eq_sum_floor_sub","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#shifted-floor-residue-count-eq-sum-floor","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $n$ and $m$ be natural numbers with $m \\ge 2$, and let $\\alpha$ and $\\nu$ be real numbers with $0 \\le \\alpha < 1$, $0 \\le \\nu < 1$ and $\\nu \\le n\\alpha$. For an integer $r$ let $\\mathcal N_{n,\\alpha,\\nu,r,m}$ be the number of integers $k$ with $1 \\le k \\le n$ such that $\\lfloor (n - \\nu)/k + \\alpha \\rfloor \\equiv r \\pmod m$. Then","title":"Floor-sum formulas for the shifted floor residue counts"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/alternating-sum-floor-div-eq-add-big-o","kind":"node","lean":"MetaMathlibExt.alternating_sum_floor_div_eq_add_bigO","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#alternating-sum-floor-div-eq-add-big-o","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $0 \\le \\nu < 1$ and let $b_0, b_1, b_2, \\dots$ be a strictly increasing sequence of real numbers with $b_k \\ge 1$ and $b_k \\ge k$ for every $k \\ge 0$. Then there is a real constant $C$ such that for every integer $n \\ge 1$, with both sums taken over the integers $k$ satisfying $0 \\le k \\le n - \\nu$,","title":"Alternating sums of floor quotients along an increasing sequence"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/shifted-floor-residue-count-asymptotic-s","kind":"node","lean":"MetaMathlibExt.shiftedFloorResidueCount_asymptotic_series","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Series asymptotics for the shifted floor residue counts with "],["math","\\(2 \\le r \\le m\\)"]],"source":"Series asymptotics for the shifted floor residue counts with $2 \\le r \\le m$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#shifted-floor-residue-count-asymptotic-s","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $r$ and $m$ be natural numbers with $2 \\le r \\le m$, and let $\\alpha$ and $\\nu$ be real numbers with $0 \\le \\alpha < 1$ and $0 \\le \\nu < 1$. For a natural number $n$ let $\\mathcal N_{n,\\alpha,\\nu,r,m}$ be the number of integers $k$ with $1 \\le k \\le n$ such that $\\lfloor (n - \\nu)/k + \\alpha \\rfloor \\equiv r \\pmod m$, and put","title":"Series asymptotics for the shifted floor residue counts with $2 \\le r \\le m$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/shifted-floor-residue-count-one-asymptot","kind":"node","lean":"MetaMathlibExt.shiftedFloorResidueCount_one_asymptotic_series","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Series asymptotics for the shifted floor residue count with "],["math","\\(r = 1\\)"]],"source":"Series asymptotics for the shifted floor residue count with $r = 1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#shifted-floor-residue-count-one-asymptot","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $m \\ge 2$ be a natural number and let $\\alpha$ and $\\nu$ be real numbers with $0 \\le \\alpha < 1$ and $0 \\le \\nu < 1$. For a natural number $n$ let $\\mathcal N_{n,\\alpha,\\nu,1,m}$ be the number of integers $k$ with $1 \\le k \\le n$ such that $\\lfloor (n - \\nu)/k + \\alpha \\rfloor \\equiv 1 \\pmod m$, and put","title":"Series asymptotics for the shifted floor residue count with $r = 1$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/tsum-alternating-inv-eq-integral","kind":"node","lean":"MetaMathlibExt.tsum_alternating_inv_eq_integral","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#tsum-alternating-inv-eq-integral","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $r$ and $m$ be natural numbers with $2 \\le r \\le m$ and let $\\alpha$ be a real number with $0 \\le \\alpha < 1$. Then","title":"Integral formula for an alternating series of reciprocals"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/shifted-floor-residue-count-asymptotic-i","kind":"node","lean":"MetaMathlibExt.shiftedFloorResidueCount_asymptotic_integral","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#shifted-floor-residue-count-asymptotic-i","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $m \\ge 1$ be a natural number and let $\\alpha$ and $\\nu$ be real numbers with $0 \\le \\alpha < 1$ and $0 \\le \\nu < 1$. For natural numbers $n$ and $r$ let $\\mathcal N_{n,\\alpha,\\nu,r,m}$ be the number of integers $k$ with $1 \\le k \\le n$ such that $\\lfloor (n - \\nu)/k + \\alpha \\rfloor \\equiv r \\pmod m$, and put","title":"Integral asymptotics for the shifted floor residue counts"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/shifted-floor-residue-count-div-tendsto","kind":"node","lean":"MetaMathlibExt.shiftedFloorResidueCount_div_tendsto","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#shifted-floor-residue-count-div-tendsto","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $r$ and $m$ be natural numbers with $1 \\le r \\le m$, and let $\\alpha$ and $\\nu$ be real numbers with $0 \\le \\alpha < 1$ and $0 \\le \\nu < 1$. For a natural number $n$ let $\\mathcal N_{n,\\alpha,\\nu,r,m}$ be the number of integers $k$ with $1 \\le k \\le n$ such that $\\lfloor (n - \\nu)/k + \\alpha \\rfloor \\equiv r \\pmod m$, and for a natural number $s$ put","title":"Limiting densities of the shifted floor residue counts"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-lattice-disc-eq-shifted-floor-resid","kind":"node","lean":"MetaMathlibExt.card_lattice_disc_eq_shiftedFloorResidueCount","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lattice points in the disc of radius "],["math","\\(\\sqrt{n}\\)"],["text"," via shifted floor sequences"]],"source":"Lattice points in the disc of radius $\\sqrt{n}$ via shifted floor sequences"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-lattice-disc-eq-shifted-floor-resid","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every natural number $n$, the number of pairs $(x, y) \\in \\mathbb{Z}^2$ with $x^2 + y^2 \\le n$ equals $4\\,\\mathcal{N}_{\\lceil n/2 \\rceil, 1/2, \\{n/2\\}, 1, 2} + 4\\lfloor n/2 \\rfloor + 1$, where $\\{n/2\\}$ is the fractional part of $n/2$ and $\\mathcal{N}_{N,\\alpha,\\nu,r,m}$ is the number of $k \\in \\{1, \\dots, N\\}$ with $\\lfloor (N - \\nu)/k + \\alpha \\rfloor \\equiv r \\pmod m$. At $n = 0$ both sides are $1$.","title":"Lattice points in the disc of radius $\\sqrt{n}$ via shifted floor sequences"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-eisenstein-form-eq","kind":"node","lean":"MetaMathlibExt.card_eisenstein_form_eq","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of representations by "],["math","\\(x^2 + xy + y^2\\)"]],"source":"Number of representations by $x^2 + xy + y^2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-eisenstein-form-eq","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every integer $n \\ge 1$, the number of pairs $(x, y) \\in \\mathbb{Z}^2$ with $x^2 + xy + y^2 = n$ equals $6\\,(d_{1,3}(n) - d_{2,3}(n))$, where $d_{r,3}(n)$ is the number of positive divisors of $n$ congruent to $r$ modulo $3$.","title":"Number of representations by $x^2 + xy + y^2$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-eisenstein-ellipse-eq","kind":"node","lean":"MetaMathlibExt.card_eisenstein_ellipse_eq","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lattice points in the ellipse "],["math","\\(x^2 + xy + y^2 \\le n\\)"]],"source":"Lattice points in the ellipse $x^2 + xy + y^2 \\le n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-eisenstein-ellipse-eq","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every natural number $n$, the number of pairs $(x, y) \\in \\mathbb{Z}^2$ with $x^2 + xy + y^2 \\le n$ equals $6\\,\\mathcal{N}_{n,0,0,1,3} + 1$, where $\\mathcal{N}_{n,0,0,1,3}$ is the number of $k \\in \\{1, \\dots, n\\}$ with $\\lfloor n/k \\rfloor \\equiv 1 \\pmod 3$. At $n = 0$ both sides are $1$.","title":"Lattice points in the ellipse $x^2 + xy + y^2 \\le n$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-eisenstein-ellipse-asymptotic","kind":"node","lean":"MetaMathlibExt.card_eisenstein_ellipse_asymptotic","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lattice points in the ellipse "],["math","\\(x^2 + xy + y^2 \\le n\\)"],["text",", asymptotically"]],"source":"Lattice points in the ellipse $x^2 + xy + y^2 \\le n$, asymptotically"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-eisenstein-ellipse-asymptotic","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"There is a real constant $C$ such that for every integer $n \\ge 1$, the number $E(n)$ of pairs $(x, y) \\in \\mathbb{Z}^2$ with $x^2 + xy + y^2 \\le n$ satisfies $\\bigl| E(n) - \\tfrac{2}{3}\\sqrt{3}\\,\\pi n \\bigr| \\le C\\sqrt{n}$.","title":"Lattice points in the ellipse $x^2 + xy + y^2 \\le n$, asymptotically"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-hexagonal-lattice-disc-eq","kind":"node","lean":"MetaMathlibExt.card_hexagonal_lattice_disc_eq","parent":8408,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-hexagonal-lattice-disc-eq","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"Let $\\omega = e^{2\\pi i/3}$, so that $\\mathbb{Z} + \\mathbb{Z}\\omega$ is the hexagonal (triangular) lattice whose neighboring points are one unit apart. For every natural number $n$, the number of pairs $(x, y) \\in \\mathbb{Z}^2$ with $|x + y\\omega| \\le \\sqrt{n}$ equals $1 + 6\\,\\mathcal{N}_{n,0,0,1,3}$, where $\\mathcal{N}_{n,\\alpha,\\nu,r,m}$ is the number of $k \\in \\{1, \\dots, n\\}$ with\u2026","title":"Lattice points in a disc of the hexagonal lattice"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-sq-add-two-mul-sq-eq","kind":"node","lean":"MetaMathlibExt.card_sq_add_two_mul_sq_eq","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dickson's divisor formula for "],["math","\\(x^2 + 2y^2\\)"]],"source":"Dickson's divisor formula for $x^2 + 2y^2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-sq-add-two-mul-sq-eq","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For every integer $n \\ge 1$, the number of pairs $(x, y) \\in \\mathbb{Z}^2$ with $x^2 + 2y^2 = n$ equals $2\\,(d_1(n) + d_3(n) - d_5(n) - d_7(n))$, where $d_r(n)$ denotes the number of positive divisors $d$ of $n$ with $d \\equiv r \\pmod 8$.","title":"Dickson's divisor formula for $x^2 + 2y^2$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/card-sq-add-two-mul-sq-ellipse-eq","kind":"node","lean":"MetaMathlibExt.card_sq_add_two_mul_sq_ellipse_eq","parent":8408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lattice points in the ellipse "],["math","\\(x^2 + 2y^2 \\le n\\)"]],"source":"Lattice points in the ellipse $x^2 + 2y^2 \\le n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k/index.html#card-sq-add-two-mul-sq-ellipse-eq","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/residues-of-rounded-quotients-n-k","status":"can_prove","summary":"For a natural number $n$, let $F(n)$ be the number of pairs $(x, y) \\in \\mathbb{Z}^2$ with $x^2 + 2y^2 \\le n$, and let $\\mathcal{N}_{N,\\alpha,\\nu,r,m}$ be the number of $k \\in \\{1, \\dots, N\\}$ with $\\lfloor (N - \\nu)/k + \\alpha \\rfloor \\equiv r \\pmod m$. Then $F(0) = 1$, $F(1) = 3$, $F(2) = 5$ and $F(5) = 11$. For every natural number $n \\notin \\{1, 2, 5\\}$, writing $N = \\lceil n/4 \\rceil$ and $\\theta = \\{-n/4\\}$\u2026","title":"Lattice points in the ellipse $x^2 + 2y^2 \\le n$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors","status":"can_prove","title":"Totient of the sum of divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/card-free-of-prime-set-le-exp-neg-sum-in","kind":"node","lean":"MetaMathlibExt.card_free_of_prime_set_le_exp_neg_sum_inv","parent":8449,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/index.html#card-free-of-prime-set-le-exp-neg-sum-in","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors","status":"can_prove","summary":"There is an absolute constant $C > 0$ such that for every set $P$ of primes and every real $x > 1$, the number of integers $n$ with $1 \\le n \\le x$ that are divisible by no prime in $P$ is at most $C x \\exp(-A_P(x))$, where $A_P(x) = \\sum_{p \\in P,\\ p \\le x} 1/p$.","title":"Pollack\u2013Pomerance sieve bound"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/card-not-dvd-sigma-le","kind":"node","lean":"MetaMathlibExt.card_not_dvd_sigma_le","parent":8449,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integers whose sum of divisors is not divisible by "],["math","\\(p\\)"]],"source":"Integers whose sum of divisors is not divisible by $p$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/index.html#card-not-dvd-sigma-le","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors","status":"can_prove","summary":"There is an absolute constant $C > 0$ such that for every prime $p$ and every real $x \\ge e^p$, the number of integers $n$ with $1 \\le n \\le x$ for which $p$ does not divide the sum of divisors $\\sigma(n)$ is at most $C x (\\log \\log x / \\log x)^{1/(p-1)}$.","title":"Integers whose sum of divisors is not divisible by $p$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/card-totient-sigma-ge-mul-le","kind":"node","lean":"MetaMathlibExt.card_totient_sigma_ge_mul_le","parent":8449,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for the number of "],["math","\\(n\\)"],["text"," with "],["math","\\(\\varphi(\\sigma(n)) \\ge c n\\)"]],"source":"Upper bound for the number of $n$ with $\\varphi(\\sigma(n)) \\ge c n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/index.html#card-totient-sigma-ge-mul-le","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors","status":"can_prove","summary":"Write $\\log_k$ for the $k$-fold iterated natural logarithm, so $\\log_3 x = \\log \\log \\log x$, and let $\\varphi$ be Euler's totient function and $\\sigma$ the sum of divisors. For every real $c > 0$ there is a constant $C$ such that for all sufficiently large $x$, the number of integers $n$ with $1 \\le n \\le x$ and $\\varphi(\\sigma(n)) \\ge c n$ is at most","title":"Upper bound for the number of $n$ with $\\varphi(\\sigma(n)) \\ge c n$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/card-totient-sigma-ge-div-is-big-o","kind":"node","lean":"MetaMathlibExt.card_totient_sigma_ge_div_isBigO","parent":8449,"reading_title":{"format":"inline-parts-v1","parts":[["text","How often "],["math","\\(\\varphi(\\sigma(n)) \\ge n / f(n)\\)"]],"source":"How often $\\varphi(\\sigma(n)) \\ge n / f(n)$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/index.html#card-totient-sigma-ge-div-is-big-o","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors","status":"can_prove","summary":"Write $\\log_k$ for the $k$-fold iterated natural logarithm, so $\\log_3 x = \\log \\log \\log x$, and let $\\varphi$ be Euler's totient function and $\\sigma$ the sum of divisors. Let $f$ be a real function that is positive and non-decreasing on $(0, \\infty)$ with $f(x) = o(\\log_4 x)$ as $x \\to \\infty$, and let $P_f(x)$ be the number of integers $n$ with $1 \\le n \\le x$ and $\\varphi(\\sigma(n)) \\ge n / f(n)$. Then, as\u2026","title":"How often $\\varphi(\\sigma(n)) \\ge n / f(n)$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/card-totient-sigma-gt-div-log-five-is-bi","kind":"node","lean":"MetaMathlibExt.card_totient_sigma_gt_div_log_five_isBigO","parent":8449,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exceptions to "],["math","\\(\\varphi(\\sigma(n)) \\le n / \\log_5 n\\)"]],"source":"Exceptions to $\\varphi(\\sigma(n)) \\le n / \\log_5 n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors/index.html#card-totient-sigma-gt-div-log-five-is-bi","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/totient-of-the-sum-of-divisors","status":"can_prove","summary":"Write $\\log_k$ for the $k$-fold iterated natural logarithm, and let $\\varphi$ be Euler's totient function and $\\sigma$ the sum of divisors. The number of integers $n$ with $1 \\le n \\le x$ and $\\varphi(\\sigma(n)) > n / \\log_5 n$ is $O(x \\log_5 x / \\log_4 x)$ as $x \\to \\infty$.","title":"Exceptions to $\\varphi(\\sigma(n)) \\le n / \\log_5 n$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors","status":"fully_proved","title":"Unitary divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors/unitary-divisors","kind":"node","lean":"Nat.unitaryDivisors","parent":8455,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors/index.html#unitary-divisors","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors","status":"fully_proved","summary":"For a natural number $n$, the unitary divisors of $n$ are the divisors $d$ of $n$ that are coprime to the complementary divisor $n/d$, that is, with $\\gcd(d, n/d) = 1$. For $n = 0$ there are no unitary divisors.","title":"Unitary divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors/sigma-star","kind":"node","lean":"Nat.sigmaStar","parent":8455,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors/index.html#sigma-star","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors","status":"fully_proved","summary":"For a natural number $n$, $\\sigma^*(n)$ denotes the sum of the unitary divisors of $n$.","title":"Sum of unitary divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors/is-primitive-by-divisibility","kind":"node","lean":"Set.IsPrimitiveByDivisibility","parent":8455,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors/index.html#is-primitive-by-divisibility","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/unitary-divisors","status":"fully_proved","summary":"Let $S$ be a set of natural numbers. A natural number $n$ is primitive in $S$ under divisibility when $n \\in S$ and no proper divisor of $n$, that is, no divisor $d$ of $n$ with $d < n$, belongs to $S$.","title":"Elements primitive under divisibility"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","title":"Biunitary divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/gcd-unitary","kind":"node","lean":"MetaMathlibExt.gcdUnitary","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#gcd-unitary","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For natural numbers $a$ and $b$, $\\gcd_1(a, b)$ is the largest natural number that is a unitary divisor of both $a$ and $b$, where a unitary divisor of $n$ is a divisor $d$ of $n \\neq 0$ with $\\gcd(d, n / d) = 1$. If $a = 0$ or $b = 0$, there is no common unitary divisor and $\\gcd_1(a, b) = 0$.","title":"Greatest common unitary divisor"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/gcd-unitary-comm","kind":"node","lean":"MetaMathlibExt.gcdUnitary_comm","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#gcd-unitary-comm","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For all natural numbers $a$ and $b$, the greatest common unitary divisor satisfies $\\gcd_1(a, b) = \\gcd_1(b, a)$.","title":"Symmetry of the greatest common unitary divisor"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/gcd-unitary-zero-left","kind":"node","lean":"MetaMathlibExt.gcdUnitary_zero_left","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#gcd-unitary-zero-left","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For every natural number $b$, the greatest common unitary divisor satisfies $\\gcd_1(0, b) = 0$, since $0$ has no unitary divisors.","title":"Greatest common unitary divisor with zero on the left"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/gcd-unitary-zero-right","kind":"node","lean":"MetaMathlibExt.gcdUnitary_zero_right","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#gcd-unitary-zero-right","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For every natural number $a$, the greatest common unitary divisor satisfies $\\gcd_1(a, 0) = 0$, since $0$ has no unitary divisors.","title":"Greatest common unitary divisor with zero on the right"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/le-gcd-unitary","kind":"node","lean":"MetaMathlibExt.le_gcdUnitary","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#le-gcd-unitary","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For natural numbers $a$, $b$ and $d$, if $d$ is a unitary divisor of both $a$ and $b$, then $d \\le \\gcd_1(a, b)$, where $\\gcd_1(a, b)$ is the greatest common unitary divisor of $a$ and $b$.","title":"Common unitary divisors are at most the greatest one"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/gcd-unitary-mem-unitary-divisors","kind":"node","lean":"MetaMathlibExt.gcdUnitary_mem_unitaryDivisors","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#gcd-unitary-mem-unitary-divisors","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For positive natural numbers $a$ and $b$, the greatest common unitary divisor $\\gcd_1(a, b)$ is a unitary divisor of $a$ and a unitary divisor of $b$.","title":"The greatest common unitary divisor is a common unitary divisor"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/gcd-unitary-dvd-left","kind":"node","lean":"MetaMathlibExt.gcdUnitary_dvd_left","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#gcd-unitary-dvd-left","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For positive natural numbers $a$ and $b$, the greatest common unitary divisor $\\gcd_1(a, b)$ divides $a$.","title":"The greatest common unitary divisor divides its first argument"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/gcd-unitary-dvd-right","kind":"node","lean":"MetaMathlibExt.gcdUnitary_dvd_right","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#gcd-unitary-dvd-right","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For positive natural numbers $a$ and $b$, the greatest common unitary divisor $\\gcd_1(a, b)$ divides $b$.","title":"The greatest common unitary divisor divides its second argument"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/gcd-unitary-pos","kind":"node","lean":"MetaMathlibExt.gcdUnitary_pos","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#gcd-unitary-pos","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For positive natural numbers $a$ and $b$, the greatest common unitary divisor $\\gcd_1(a, b)$ is positive.","title":"Positivity of the greatest common unitary divisor"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/biunitary-divisors","kind":"node","lean":"MetaMathlibExt.biunitaryDivisors","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#biunitary-divisors","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For a natural number $n$, the biunitary divisors of $n$ are the divisors $d$ of $n$ with $\\gcd_1(d, n/d) = 1$, where $\\gcd_1(a, b)$ denotes the greatest common unitary divisor of $a$ and $b$. For $n = 0$ the set is empty.","title":"Biunitary divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/sigma-biunitary","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#sigma-biunitary","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For a natural number $n$, $\\sigma^{**}(n)$ is the sum of the biunitary divisors of $n$, that is, of the divisors $d$ of $n$ with $\\gcd_1(d, n / d) = 1$, where $\\gcd_1$ denotes the greatest common unitary divisor. Since $0$ has no biunitary divisors, $\\sigma^{**}(0) = 0$.","title":"Biunitary divisor sum"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/mem-biunitary-divisors","kind":"node","lean":"MetaMathlibExt.mem_biunitaryDivisors","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#mem-biunitary-divisors","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For natural numbers $n$ and $d$, $d$ is a biunitary divisor of $n$ if and only if $d$ divides $n$, $n \\neq 0$ and $\\gcd_1(d, n / d) = 1$, where $\\gcd_1$ denotes the greatest common unitary divisor.","title":"Characterization of biunitary divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/biunitary-divisors-zero","kind":"node","lean":"MetaMathlibExt.biunitaryDivisors_zero","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#biunitary-divisors-zero","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"The set of biunitary divisors of $0$ is empty.","title":"Zero has no biunitary divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/sigma-biunitary-zero","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_zero","parent":8459,"reading_title":{"format":"inline-parts-v1","parts":[["text","The biunitary divisor sum of "],["math","\\(0\\)"]],"source":"The biunitary divisor sum of $0$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#sigma-biunitary-zero","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"The sum $\\sigma^{**}(0)$ of the biunitary divisors of $0$ equals $0$.","title":"The biunitary divisor sum of $0$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/dvd-of-mem-biunitary-divisors","kind":"node","lean":"MetaMathlibExt.dvd_of_mem_biunitaryDivisors","parent":8459,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#dvd-of-mem-biunitary-divisors","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"For natural numbers $n$ and $d$, if $d$ is a biunitary divisor of $n$, then $d$ divides $n$.","title":"Biunitary divisors divide"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/sigma-biunitary-one","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_one","parent":8459,"reading_title":{"format":"inline-parts-v1","parts":[["text","The biunitary divisor sum of "],["math","\\(1\\)"]],"source":"The biunitary divisor sum of $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#sigma-biunitary-one","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"The sum $\\sigma^{**}(1)$ of the biunitary divisors of $1$ equals $1$.","title":"The biunitary divisor sum of $1$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/sigma-biunitary-four","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_four","parent":8459,"reading_title":{"format":"inline-parts-v1","parts":[["text","The biunitary divisor sum of "],["math","\\(4\\)"]],"source":"The biunitary divisor sum of $4$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#sigma-biunitary-four","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"The sum $\\sigma^{**}(4)$ of the biunitary divisors of $4$ equals $5$.","title":"The biunitary divisor sum of $4$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/sigma-biunitary-sixteen","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_sixteen","parent":8459,"reading_title":{"format":"inline-parts-v1","parts":[["text","The biunitary divisor sum of "],["math","\\(16\\)"]],"source":"The biunitary divisor sum of $16$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#sigma-biunitary-sixteen","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"The sum $\\sigma^{**}(16)$ of the biunitary divisors of $16$ equals $1 + 2 + 8 + 16 = 27$.","title":"The biunitary divisor sum of $16$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/sigma-biunitary-twenty-seven","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_twentySeven","parent":8459,"reading_title":{"format":"inline-parts-v1","parts":[["text","The biunitary divisor sum of "],["math","\\(27\\)"]],"source":"The biunitary divisor sum of $27$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#sigma-biunitary-twenty-seven","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"The sum $\\sigma^{**}(27)$ of the biunitary divisors of $27$ equals $1 + 3 + 9 + 27 = 40$.","title":"The biunitary divisor sum of $27$"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/sigma-biunitary-one-hundred-twenty","kind":"node","lean":"MetaMathlibExt.sigmaBiunitary_oneHundredTwenty","parent":8459,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(120\\)"],["text"," is biunitary triperfect"]],"source":"$120$ is biunitary triperfect"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors/index.html#sigma-biunitary-one-hundred-twenty","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/biunitary-divisors","status":"fully_proved","summary":"The sum $\\sigma^{**}(120)$ of the biunitary divisors of $120$ equals $3 \\cdot 120 = 360$.","title":"$120$ is biunitary triperfect"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n","status":"can_prove","title":"Regular integers modulo n"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n/ncard-regular-eq-sum-totient-unitary-div","kind":"node","lean":"MetaMathlibExt.ncard_regular_eq_sum_totient_unitaryDivisors","parent":8480,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n/index.html#ncard-regular-eq-sum-totient-unitary-div","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n","status":"fully_proved","summary":"For every positive integer $n$, the number of residue classes $m \\in \\mathbb{Z}/n\\mathbb{Z}$ for which the equation $m^2 x = m$ has a solution $x \\in \\mathbb{Z}/n\\mathbb{Z}$ equals the sum of $\\varphi(d)$ over the unitary divisors $d$ of $n$ (the positive divisors $d$ of $n$ with $\\gcd(d, n/d) = 1$), where $\\varphi$ is Euler's totient function.","title":"Morgado's formula as a sum over unitary divisors"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n/morgado-card-regular-modulo","kind":"node","lean":"MetaMathlibExt.morgado_card_regular_modulo","parent":8480,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n/index.html#morgado-card-regular-modulo","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n","status":"fully_proved","summary":"Let $n$ be a positive integer. Call a residue class $m \\in \\mathbb{Z}/n\\mathbb{Z}$ regular if the equation $m^2 x = m$ has a solution $x \\in \\mathbb{Z}/n\\mathbb{Z}$, and let $\\varrho(n)$ be the number of regular residue classes. Let $U(n)$ be the set of positive divisors $d$ of $n$ with $\\gcd(d, n/d) = 1$. Then","title":"Morgado's formula for the number of regular residues"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n/is-regular-mod-iff-gcd-sq-eq-gcd","kind":"node","lean":"isRegularMod_iff_gcd_sq_eq_gcd","parent":8480,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n/index.html#is-regular-mod-iff-gcd-sq-eq-gcd","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/regular-integers-modulo-n","status":"can_prove","summary":"Let $n$ be a positive integer and $m$ an integer, and write $\\gcd$ for the nonnegative greatest common divisor. The congruence $m^2 x \\equiv m \\pmod{n}$ has a solution $x \\in \\mathbb{Z}$ if and only if $\\gcd(m^2, n) = \\gcd(m, n)$, and $\\gcd(m^2, n) = \\gcd(m, n)$ holds if and only if $\\gcd(m, n)$ and $n / \\gcd(m, n)$ are coprime, that is, $\\gcd(m, n)$ is a unitary divisor of $n$.","title":"Morgado's characterization of regular integers"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n","kind":"scope","navigation":"module","parent":8300,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n","status":"can_prove","title":"Values of n + \u03c6(n)"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem822Wanted.conjecture","parent":8484,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n/index.html#conjecture","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n","status":"fully_proved","summary":"The statement that the set $\\{n + \\varphi(n) : n \\in \\mathbb{N}\\}$, where $\\varphi$ is Euler's totient function, has positive lower density in $\\mathbb{N}$.","title":"Erd\u0151s Problem 822"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem822Wanted.conjecture_holds","parent":8484,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n/index.html#conjecture-holds","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-01/values-of-n-n","status":"can_prove","summary":"The set $\\{n + \\varphi(n) : n \\in \\mathbb{N}\\}$, where $\\varphi$ is Euler's totient function, has positive lower density in $\\mathbb{N}$.","title":"Erd\u0151s Problem 822 is true"},{"data_url":"scopes/NumberTheory/arithmetic-functions/divisor-totient-functions/part-02.json","id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-02","kind":"scope","navigation":"transparent","parent":8299,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-02/index.html","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-02","status":"fully_proved","title":"Part 2 of 2"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-02/gronwall-sigma-limsup","kind":"node","lean":"MathlibExt.NumberTheory.ArithmeticFunction.GronwallSigmaLimsupWanted.gronwall_sigma_limsup","parent":8487,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-02/index.html#gronwall-sigma-limsup","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-02","status":"fully_proved","summary":"Let $\\sigma(n)$ be the sum of the positive divisors of $n$ and let $\\gamma$ be the Euler\u2013Mascheroni constant. Then, as $n$ runs over the natural numbers, $\\limsup_{n \\to \\infty} \\sigma(n) / (n \\log \\log n) = e^{\\gamma}$.","title":"Gronwall's theorem"},{"id":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-02/mobius-formula-relatively-prime-subset-c","kind":"node","lean":"MetaMathlibExt.mobius_formula_relatively_prime_subset_counts_interval","parent":8487,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/divisor-totient-functions/part-02/index.html#mobius-formula-relatively-prime-subset-c","scope":"NumberTheory/arithmetic-functions/divisor-totient-functions/part-02","status":"fully_proved","summary":"Let $\\ell < m$ and $k \\ge 1$ be natural numbers, and let $\\mu$ be the M\u00f6bius function. The number of nonempty subsets of $\\{\\ell + 1, \\dots, m\\}$ whose elements have greatest common divisor $1$ is","title":"Relatively prime subset counts"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions","kind":"scope","parent":8068,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions","status":"can_prove","summary":"Dirichlet and unitary convolution, completely multiplicative and additive functions, Wintner and Euler-product mean values, additive uniqueness sets, Mertens bounds.","title":"Multiplicative functions and mean values"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/summatory-moebius-div-bound","kind":"node","lean":"MetaMathlibExt.summatory_moebius_div_bound","parent":8490,"reading_title":{"format":"inline-parts-v1","parts":[["text","Explicit bounds for the summatory function of "],["math","\\(\\mu(n)/n\\)"]],"source":"Explicit bounds for the summatory function of $\\mu(n)/n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/index.html#summatory-moebius-div-bound","scope":"NumberTheory/arithmetic-functions/multiplicative-functions","status":"can_prove","summary":"Let $\\mu$ be the M\u00f6bius function. For every real $x \\ge 33$,","title":"Explicit bounds for the summatory function of $\\mu(n)/n$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/prime-power-factorization-sum-eq-iff","kind":"node","lean":"MetaMathlibExt.prime_power_factorization_sum_eq_iff","parent":8490,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/index.html#prime-power-factorization-sum-eq-iff","scope":"NumberTheory/arithmetic-functions/multiplicative-functions","status":"fully_proved","summary":"For positive integers $y$ and $s$, let $T_s(y) = \\sum_q e_q q^s$, where $q$ runs over the primes dividing $y$ and $e_q$ is the exponent of $q$ in $y$, so that each prime factor is counted with multiplicity. Let $p$ be a prime, let $m$ be a positive integer and let $y = p^m$. Then for every positive integer $s$, $T_s(y) = y$ if and only if $m = p^k$ and $s = p^k - k$ for some natural number $k$.","title":"Prime powers fixed by a uniform power sum of their prime factors"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","title":"Additive uniqueness of multiplicative functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-on-squares","kind":"node","lean":"MetaMathlibExt.multiplicative_additive_on_squares","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-on-squares","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(1) = 1$ and $f(ab) = f(a) f(b)$ for coprime $a$ and $b$) such that $f(m^2 + n^2) = f(m^2) + f(n^2)$ for all positive integers $m$ and $n$. Then $f(4)$ is $0$ or $4$; $f(q^{2k}) = f(q^2)^k$ for all positive integers $q$ and $k$; and if $f(4) = 4$, then $f(m^2) = m^2$ for every positive integer $m$.","title":"Multiplicative functions additive on pairs of squares"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/consecutive-square-sums-k-additive-uniqu","kind":"node","lean":"MetaMathlibExt.consecutiveSquareSums_kAdditiveUniqueness","parent":8493,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of consecutive squares form a "],["math","\\(k\\)"],["text","-additive uniqueness set for every "],["math","\\(k \\ge 2\\)"]],"source":"Sums of consecutive squares form a $k$-additive uniqueness set for every $k \\ge 2$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#consecutive-square-sums-k-additive-uniqu","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $k \\ge 2$, and call a natural number a sum of consecutive squares if it equals $(n+1)^2 + (n+2)^2 + \\dots + m^2$ for some natural numbers $n < m$. Let $f$ be a complex-valued multiplicative arithmetic function (so $f(1) = 1$ and $f(ab) = f(a) f(b)$ for coprime $a$ and $b$). If $f(x_1 + \\dots + x_k) = f(x_1) + \\dots + f(x_k)$ for all sums of consecutive squares $x_1, \\dots, x_k$, then $f(n) = n$ for every\u2026","title":"Sums of consecutive squares form a $k$-additive uniqueness set for every $k \\ge 2$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/is-sum-two-positive-triangular","kind":"node","lean":"MathlibExt.NumberTheory.ArithmeticFunction.MultiplicativeTriangularAdditiveWanted.IsSumTwoPositiveTriangular","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#is-sum-two-positive-triangular","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"fully_proved","summary":"A natural number $n$ is a sum of two positive triangular numbers if there are natural numbers $a \\ge 1$ and $b \\ge 1$ with $2n = a(a+1) + b(b+1)$, that is, $n = \\frac{a(a+1)}{2} + \\frac{b(b+1)}{2}$.","title":"Sums of two positive triangular numbers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-on-sum-two-trian","kind":"node","lean":"MathlibExt.NumberTheory.ArithmeticFunction.MultiplicativeTriangularAdditiveWanted.multiplicative_additive_on_sum_two_triangular_classification","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-on-sum-two-trian","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ (so $f$ is defined on the natural numbers with $f(0) = 0$) such that $f(a + b) = f(a) + f(b)$ whenever $a$ and $b$ are both sums of two positive triangular numbers. Then either $f(n) = n$ for every natural number $n$, or $f(n) = 0$ for every natural number $n \\notin \\{1, 3, 5\\}$ and $f(3) f(5) = 0$.","title":"Park's theorem on multiplicative functions additive on sums of two triangular numbers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-on-sum-two-trian-2","kind":"node","lean":"MathlibExt.NumberTheory.ArithmeticFunction.MultiplicativeTriangularAdditiveWanted.multiplicative_additive_on_sum_two_triangular_eq_id_le_21","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-on-sum-two-trian-2","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ whenever $a$ and $b$ are both sums of two positive triangular numbers. If $f(2) \\neq 0$, then $f(n) = n$ for every natural number $n \\le 21$.","title":"Park's Lemma 2: the identity up to 21 when the value at 2 is nonzero"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-on-sum-two-trian-3","kind":"node","lean":"MathlibExt.NumberTheory.ArithmeticFunction.MultiplicativeTriangularAdditiveWanted.multiplicative_additive_on_sum_two_triangular_zero_of_three_ne_zero","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-on-sum-two-trian-3","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ whenever $a$ and $b$ are both sums of two positive triangular numbers. If $f(2) = 0$ and $f(3) \\neq 0$, then $f(n) = 0$ for every natural number $n$ with $4 \\le n \\le 21$.","title":"Park's Lemma 3: vanishing from 4 to 21 when the value at 2 is zero and at 3 is not"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-on-sum-two-trian-4","kind":"node","lean":"MathlibExt.NumberTheory.ArithmeticFunction.MultiplicativeTriangularAdditiveWanted.multiplicative_additive_on_sum_two_triangular_zero_of_five_ne_zero","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-on-sum-two-trian-4","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ whenever $a$ and $b$ are both sums of two positive triangular numbers. If $f(2) = 0$ and $f(5) \\neq 0$, then $f(n) = 0$ for every natural number $n$ with $2 \\le n \\le 21$ and $n \\neq 5$.","title":"Park's Lemma 4: vanishing from 2 to 21 except at 5"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-on-sum-two-trian-5","kind":"node","lean":"MathlibExt.NumberTheory.ArithmeticFunction.MultiplicativeTriangularAdditiveWanted.multiplicative_additive_on_sum_two_triangular_zero_of_all_zero","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-on-sum-two-trian-5","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ whenever $a$ and $b$ are both sums of two positive triangular numbers. If $f(2) = f(3) = f(5) = 0$, then $f(n) = 0$ for every natural number $n$ with $2 \\le n \\le 21$.","title":"Park's Lemma 5: vanishing from 2 to 21 when the values at 2, 3 and 5 are zero"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-nonagonal-additive-eq-id","kind":"node","lean":"MetaMathlibExt.multiplicative_nonagonal_additive_eq_id","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-nonagonal-additive-eq-id","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ for all positive integers $a$ and $b$ that are generalized nonagonal numbers, that is, $2a = m(7m - 5)$ and $2b = k(7k - 5)$ for some integers $m$ and $k$. Then $f(n) = n$ for every natural number $n$.","title":"Additive uniqueness of generalized nonagonal numbers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-nonagonal-additive-eq-sel","kind":"node","lean":"MetaMathlibExt.multiplicative_nonagonal_additive_eq_self_of_le_eleven","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-nonagonal-additive-eq-sel","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ for all positive integers $a$ and $b$ that are generalized nonagonal numbers, that is, $2a = m(7m - 5)$ and $2b = k(7k - 5)$ for some integers $m$ and $k$. Then $f(n) = n$ for every integer $n$ with $1 \\le n \\le 11$.","title":"Nonagonal additive uniqueness: the values up to 11"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-nonagonal-additive-three","kind":"node","lean":"MetaMathlibExt.multiplicative_nonagonal_additive_three_pow","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-nonagonal-additive-three","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ for all positive integers $a$ and $b$ that are generalized nonagonal numbers, that is, $2a = m(7m - 5)$ and $2b = k(7k - 5)$ for some integers $m$ and $k$. Let $r$ be a natural number and put $\\alpha(x) = \\frac{14}{17} \\cdot \\frac{2}{3} x - \\frac{12}{17}$. If $f(n) = n$ for every natural number\u2026","title":"Nonagonal additive uniqueness: the step at powers of 3"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-nonagonal-additive-five-p","kind":"node","lean":"MetaMathlibExt.multiplicative_nonagonal_additive_five_pow","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-nonagonal-additive-five-p","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ for all positive integers $a$ and $b$ that are generalized nonagonal numbers, that is, $2a = m(7m - 5)$ and $2b = k(7k - 5)$ for some integers $m$ and $k$. Let $r \\ge 3$ be a natural number and put $\\beta(x) = x/4$. If $f(n) = n$ for every natural number $n \\le \\beta(5^r)$, then $f(5^r) = 5^r$.","title":"Nonagonal additive uniqueness: the step at powers of 5"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-nonagonal-additive-seven","kind":"node","lean":"MetaMathlibExt.multiplicative_nonagonal_additive_seven_pow","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-nonagonal-additive-seven","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ for all positive integers $a$ and $b$ that are generalized nonagonal numbers, that is, $2a = m(7m - 5)$ and $2b = k(7k - 5)$ for some integers $m$ and $k$. Let $r$ be a natural number. If $f(n) = n$ for every natural number $n < 7^r$, then $f(7^r) = 7^r$.","title":"Nonagonal additive uniqueness: the step at powers of 7"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-nonagonal-additive-two-po","kind":"node","lean":"MetaMathlibExt.multiplicative_nonagonal_additive_two_pow","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-nonagonal-additive-two-po","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a multiplicative arithmetic function with values in $\\mathbb{C}$ such that $f(a + b) = f(a) + f(b)$ for all positive integers $a$ and $b$ that are generalized nonagonal numbers, that is, $2a = m(7m - 5)$ and $2b = k(7k - 5)$ for some integers $m$ and $k$. Let $r$ be a natural number. If $f(n) = n$ for every natural number $n < 2^r$, then $f(2^r) = 2^r$.","title":"Nonagonal additive uniqueness: the step at powers of 2"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-parallelogram-squares-eq","kind":"node","lean":"MetaMathlibExt.multiplicative_parallelogram_squares_eq_sq","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-parallelogram-squares-eq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$, and suppose $f(4) \\ne 0$. Then $f(n) = n^2$ for every natural number $n$.","title":"Parallelogram uniqueness of squares for multiplicative functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-parallelogram-squares-of","kind":"node","lean":"MetaMathlibExt.multiplicative_parallelogram_squares_of_f_four_eq_zero","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-parallelogram-squares-of","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$, and suppose $f(4) = 0$. Then one of the following holds: (1) for every natural number $n$, $f(n) = 0$ if $n \\equiv 0 \\pmod 4$, $f(n) = 4$ if $n \\equiv 2 \\pmod 4$, and $f(n) = 1$ otherwise; (2) $f(2) = 4$, $f(2^r) = 0$ for every $r \\ge 2$, and\u2026","title":"Multiplicative parallelogram solutions vanishing at 4"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-two-pow","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_two_pow","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-two-pow","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. If $f(4) = 0$, then $f(2^r) = 0$ for every integer $r \\ge 2$. If $f(4) \\ne 0$, then $f(2^r) = 2^{2r}$ for every integer $r \\ge 1$.","title":"Values of a parallelogram-equation multiplicative function at powers of two"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-sq-add-one","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_sq_add_one","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-sq-add-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. Let $p$ be a prime such that $f(q^s) = q^{2s}$ for every prime $q < p$ and every integer $s \\ge 1$. If $p$ is odd and $f(p) = p^2$, then $f(p^2 + 1) = (p^2 + 1)^2$.","title":"Parallelogram induction: the value at p\u00b2 + 1"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-add-two","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_add_two","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-add-two","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. Let $p$ be a prime such that $f(q^s) = q^{2s}$ for every prime $q < p$ and every integer $s \\ge 1$. If $p \\equiv 3 \\pmod 4$, then $f(p + 2) = (p + 2)^2$ and $f((p + 2)^2) = (p + 2)^4$.","title":"Parallelogram induction: the values at p + 2 and (p + 2)\u00b2"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-k-sq-add-on","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_k_sq_add_one","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-k-sq-add-on","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. Let $p$ be a prime such that $f(q^s) = q^{2s}$ for every prime $q < p$ and every integer $s \\ge 1$. If $p = 4k - 1$ for a natural number $k$, then $f(k^2 + 1) = (k^2 + 1)^2$ and $f(4k^2 + 1) = (4k^2 + 1)^2$.","title":"Parallelogram induction: the values at k\u00b2 + 1 and 4k\u00b2 + 1"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-five-k-add","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_five_k_add_three","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-five-k-add","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. Let $p$ be a prime such that $f(q^s) = q^{2s}$ for every prime $q < p$ and every integer $s \\ge 1$. If $p = 4k - 1$ for a natural number $k$ with $4 \\nmid k$, and $p \\ge 19$, then $f(5k + 3) = (5k + 3)^2$.","title":"Parallelogram induction: the value at 5k + 3"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-six-k-add-o","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_six_k_add_one","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-six-k-add-o","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. Let $p$ be a prime such that $f(q^s) = q^{2s}$ for every prime $q < p$ and every integer $s \\ge 1$. If $p = 4k - 1$ for a natural number $k$ with $k \\not\\equiv 3 \\pmod 4$, and $p \\ge 19$, then $f(6k + 1) = (6k + 1)^2$.","title":"Parallelogram induction: the value at 6k + 1"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-prime-pow","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_prime_pow","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-prime-pow","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. Let $p$ be a prime such that $f(q^s) = q^{2s}$ for every prime $q < p$ and every integer $s \\ge 1$. If $p$ is odd and $f(p) = p^2$, then $f(p^r) = p^{2r}$ for every integer $r \\ge 2$.","title":"Parallelogram induction: odd prime powers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-prime-one-m","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_prime_one_mod_four","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-prime-one-m","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. Let $p$ be a prime such that $f(q^s) = q^{2s}$ for every prime $q < p$ and every integer $s \\ge 1$. If $p \\equiv 1 \\pmod 4$, then $f(p) = p^2$.","title":"Parallelogram induction: primes congruent to 1 mod 4"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/parallelogram-multiplicative-prime-three","kind":"node","lean":"MetaMathlibExt.parallelogram_multiplicative_prime_three_mod_four","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#parallelogram-multiplicative-prime-three","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $f$ be a complex-valued multiplicative arithmetic function (so $f(0) = 0$) such that $f(m^2 + n^2) + f(m^2 - n^2) = 2f(m^2) + 2f(n^2)$ for all integers $m > n \\ge 1$. Let $p$ be a prime such that $f(q^s) = q^{2s}$ for every prime $q < p$ and every integer $s \\ge 1$. If $p = 4k - 1$ for some natural number $k$ and $p \\ge 19$, then $f(p) = p^2$.","title":"Parallelogram induction: primes congruent to 3 mod 4 from 19 on"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/sq-add-triangular-sub-two-eq","kind":"node","lean":"MetaMathlibExt.sq_add_triangular_sub_two_eq","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#sq-add-triangular-sub-two-eq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $t_v = v(v + 1)/2$ denote the $v$-th triangular number. For every natural number $n \\ge 2$, $n^2 + t_{n-2} = (n - 1)^2 + t_n$.","title":"A square\u2013triangular number identity"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-sq-triangular-additive-sq","kind":"node","lean":"MetaMathlibExt.multiplicative_sq_triangular_additive_sq","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-sq-triangular-additive-sq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $t_v = v(v + 1)/2$ denote the $v$-th triangular number, and let $f$ be a complex-valued multiplicative arithmetic function with $f(u^2 + t_v) = f(u^2) + f(t_v)$ for all positive integers $u$ and $v$. Let $N \\ge 3$ be an integer such that $f(n) = n$ for every integer $n$ with $1 \\le n \\le N - 1$. Then $f(m^2) = m^2$ for every integer $m$ with $1 \\le m \\le N - 2$.","title":"Square\u2013triangular additivity fixes small squares"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-sq-triangular-additive-tw","kind":"node","lean":"MetaMathlibExt.multiplicative_sq_triangular_additive_two_pow_odd","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-sq-triangular-additive-tw","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $t_v = v(v + 1)/2$ denote the $v$-th triangular number, and let $f$ be a complex-valued multiplicative arithmetic function with $f(u^2 + t_v) = f(u^2) + f(t_v)$ for all positive integers $u$ and $v$. Let $N \\ge 4$ be an integer such that $f(n) = n$ for every integer $n$ with $1 \\le n \\le N - 1$. Then $f(2^{2k-1}) = 2^{2k-1}$ for every positive integer $k$ with $2k \\le N - 2$.","title":"Square\u2013triangular additivity fixes odd powers of two"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-sq-triangular-additive-eq","kind":"node","lean":"MetaMathlibExt.multiplicative_sq_triangular_additive_eq_id","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-sq-triangular-additive-eq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $t_v = v(v + 1)/2$ denote the $v$-th triangular number. If $f$ is a complex-valued multiplicative arithmetic function with $f(u^2 + t_v) = f(u^2) + f(t_v)$ for all positive integers $u$ and $v$, then $f(n) = n$ for every natural number $n$.","title":"Squares and triangular numbers are an additive uniqueness pair"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/sum-nonzero-squares-iff-not-mem-exceptio","kind":"node","lean":"MetaMathlibExt.sum_nonzero_squares_iff_not_mem_exceptional","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#sum-nonzero-squares-iff-not-mem-exceptio","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $m \\ge 4$ and let $n$ be a positive integer. Then $n$ is a sum of $m$ squares of positive integers if and only if $n$ does not lie in the exceptional set $E_m$, where $E_4 = \\{1, 3, 5, 9, 11, 17, 29, 41\\} \\cup \\{2 \\cdot 4^r, 6 \\cdot 4^r, 14 \\cdot 4^r : r \\ge 0\\}$, $E_5 = \\{1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33\\}$, and for $m \\ge 6$,\u2026","title":"Dubouis's theorem on sums of nonzero squares"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/sum-six-and-eight-nonzero-squares","kind":"node","lean":"MetaMathlibExt.sum_six_and_eight_nonzero_squares","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#sum-six-and-eight-nonzero-squares","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Every integer $n \\ge 20$ is a sum of $6$ squares of positive integers, and every integer $n \\ge 22$ is a sum of $8$ squares of positive integers.","title":"Sums of six and of eight nonzero squares"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-three-additive-two-square","kind":"node","lean":"MetaMathlibExt.multiplicative_three_additive_two_squares_base","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-three-additive-two-square","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $E_2 = \\{a^2 + b^2 : a, b \\ge 1\\}$ be the set of sums of two squares of positive integers. If $f$ is a complex-valued multiplicative function with $f(2) \\ne 0$ such that $f(A_1 + A_2 + A_3) = f(A_1) + f(A_2) + f(A_3)$ for all $A_1, A_2, A_3 \\in E_2$, then $f(2) = 2$, $f(3) = 3$ and $f(5) = 5$.","title":"Small values of a multiplicative function 3-additive on sums of two squares"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-four-additive-two-squares","kind":"node","lean":"MetaMathlibExt.multiplicative_four_additive_two_squares_base","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-four-additive-two-squares","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $E_2 = \\{a^2 + b^2 : a, b \\ge 1\\}$ be the set of sums of two squares of positive integers. If $f$ is a complex-valued multiplicative function with $f(2) \\ne 0$ such that $f(A_1 + A_2 + A_3 + A_4) = f(A_1) + f(A_2) + f(A_3) + f(A_4)$ for all $A_1, \\dots, A_4 \\in E_2$, then $f(n) = n$ for every $n \\in \\{2, 3, 4, 5, 6, 7, 8, 10\\}$.","title":"Small values of a multiplicative function 4-additive on sums of two squares"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-paired-squares-e","kind":"node","lean":"MetaMathlibExt.multiplicative_additive_paired_squares_eq_id_of_three_or_four","parent":8493,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicative functions additive on six or eight paired squares and nonzero at "],["math","\\(2\\)"],["text"," are the identity"]],"source":"Multiplicative functions additive on six or eight paired squares and nonzero at $2$ are the identity"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-paired-squares-e","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Multiplicative functions additive on six or eight paired squares and nonzero at $2$ are the identity. Let $k \\in \\{3, 4\\}$ and let $f$ be a complex-valued multiplicative function with $f(2) \\ne 0$ such that","title":"Multiplicative functions additive on six or eight paired squares and nonzero at $2$ are the identity"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-paired-squares-d","kind":"node","lean":"MetaMathlibExt.multiplicative_additive_paired_squares_dichotomy","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-paired-squares-d","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $k \\ge 5$ and let $f$ be a complex-valued multiplicative function such that","title":"Dichotomy for multiplicative functions additive on paired squares"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-paired-squares-e-2","kind":"node","lean":"MetaMathlibExt.multiplicative_additive_paired_squares_eq_id_of_ne_zero","parent":8493,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-paired-squares-e-2","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Let $k \\ge 5$ and let $f$ be a complex-valued multiplicative function such that","title":"A nonzero value in the vanishing range of the dichotomy forces the identity"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-six-squares-eq-i","kind":"node","lean":"MetaMathlibExt.multiplicative_additive_six_squares_eq_id","parent":8493,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicative functions additive on six paired squares and nonzero at "],["math","\\(2\\)"],["text"," are the identity"]],"source":"Multiplicative functions additive on six paired squares and nonzero at $2$ are the identity"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-six-squares-eq-i","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Multiplicative functions additive on six paired squares and nonzero at $2$ are the identity. Let $f$ be a complex-valued multiplicative function with $f(2) \\ne 0$ such that","title":"Multiplicative functions additive on six paired squares and nonzero at $2$ are the identity"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/multiplicative-additive-eight-squares-eq","kind":"node","lean":"MetaMathlibExt.multiplicative_additive_eight_squares_eq_id","parent":8493,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicative functions additive on eight paired squares and nonzero at "],["math","\\(2\\)"],["text"," are the identity"]],"source":"Multiplicative functions additive on eight paired squares and nonzero at $2$ are the identity"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions/index.html#multiplicative-additive-eight-squares-eq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/additive-uniqueness-of-multiplicative-functions","status":"can_prove","summary":"Multiplicative functions additive on eight paired squares and nonzero at $2$ are the identity. Let $f$ be a complex-valued multiplicative function with $f(2) \\ne 0$ such that","title":"Multiplicative functions additive on eight paired squares and nonzero at $2$ are the identity"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","title":"Completely additive height functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/is-completely-additive-nat","kind":"node","lean":"MetaMathlibExt.IsCompletelyAdditiveNat","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#is-completely-additive-nat","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"A function $H : \\mathbb{N} \\to \\mathbb{N}$ is completely additive on the positive integers if $H(ab) = H(a) + H(b)$ for all integers $a, b \\ge 1$.","title":"Completely additive functions on the positive integers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/map-one","kind":"node","lean":"MetaMathlibExt.IsCompletelyAdditiveNat.map_one","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#map-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"If $H : \\mathbb{N} \\to \\mathbb{N}$ is completely additive on the positive integers, that is, $H(ab) = H(a) + H(b)$ for all $a, b \\ge 1$, then $H(1) = 0$.","title":"A completely additive function vanishes at one"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/is-height-function","kind":"node","lean":"MetaMathlibExt.IsHeightFunction","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#is-height-function","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"A function $H : \\mathbb{N} \\to \\mathbb{N}$ is a height function if it is completely additive on the positive integers, that is $H(ab) = H(a) + H(b)$ for all integers $a, b \\ge 1$; if $H(p) \\ge 1$ for every prime $p$; and if for every $n \\ge 1$ only finitely many primes $p$ have $H(p) = n$.","title":"Height functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/nth-prime-one","kind":"node","lean":"MetaMathlibExt.nthPrimeOne","parent":8532,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(i\\)"],["text","-th prime"]],"source":"The $i$-th prime"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#nth-prime-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For a natural number $i \\ge 1$, $p_i$ denotes the $i$-th prime in increasing order, so that $p_1 = 2$. For $i = 0$ the definition also gives $p_0 = 2$.","title":"The $i$-th prime"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/nth-prime-one-one-two-three","kind":"node","lean":"MetaMathlibExt.nthPrimeOne_one_two_three","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#nth-prime-one-one-two-three","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"With primes indexed from one, the first three primes are $p_1 = 2$, $p_2 = 3$ and $p_3 = 5$.","title":"The first three primes"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/additive-of-prime-values","kind":"node","lean":"MetaMathlibExt.additiveOfPrimeValues","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#additive-of-prime-values","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"Given a function $v : \\mathbb{N} \\to \\mathbb{N}$, this is the function $\\mathbb{N} \\to \\mathbb{N}$ that sends each $m \\ge 1$ with prime factorization $m = \\prod_p p^{a_p}$ to $\\sum_p a_p\\, v(p)$, and sends $0$ to $0$. Only the values of $v$ at primes matter.","title":"Completely additive function with prescribed prime values"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/additive-of-prime-values-one","kind":"node","lean":"MetaMathlibExt.additiveOfPrimeValues_one","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#additive-of-prime-values-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For every function $v : \\mathbb{N} \\to \\mathbb{N}$, the completely additive function with prime values $v$ (which sends $m = \\prod_p p^{a_p}$ to $\\sum_p a_p\\, v(p)$) takes the value $0$ at $1$.","title":"The completely additive extension vanishes at one"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/additive-of-prime-values-prime","kind":"node","lean":"MetaMathlibExt.additiveOfPrimeValues_prime","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#additive-of-prime-values-prime","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For every function $v : \\mathbb{N} \\to \\mathbb{N}$ and every prime $p$, the completely additive function with prime values $v$ (which sends $m = \\prod_q q^{a_q}$ to $\\sum_q a_q\\, v(q)$) takes the value $v(p)$ at $p$.","title":"The completely additive extension at a prime"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/is-completely-additive-nat-additive-of-p","kind":"node","lean":"MetaMathlibExt.isCompletelyAdditiveNat_additiveOfPrimeValues","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#is-completely-additive-nat-additive-of-p","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For every function $v : \\mathbb{N} \\to \\mathbb{N}$, let $H_v$ be the function with $H_v(m) = \\sum_p a_p v(p)$ for $m = \\prod_p p^{a_p}$. Then $H_v$ is completely additive on the positive integers: $H_v(ab) = H_v(a) + H_v(b)$ for all integers $a, b \\ge 1$.","title":"The extension from prime values is completely additive"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/height-level-eq-products-union-primes","kind":"node","lean":"MetaMathlibExt.heightLevel_eq_products_union_primes","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#height-level-eq-products-union-primes","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H$ be a height function and let $n \\ge 1$, and for $k \\ge 0$ write $\\mathcal{H}_k = \\{m \\ge 1 : H(m) = k\\}$. Then $\\mathcal{H}_n$ is the union of two sets: the set of products $p m'$, where $p$ is a prime with $H(p) = \\ell$ for some integer $\\ell$ with $1 \\le \\ell \\le \\lfloor n/2 \\rfloor$ and $m' \\in \\mathcal{H}_{n - \\ell}$; and the set of primes $p$ with $H(p) = n$.","title":"Level construction for a height function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/height-level-card-eq-multipartitions","kind":"node","lean":"MetaMathlibExt.heightLevel_card_eq_multipartitions","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#height-level-card-eq-multipartitions","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H$ be a height function and, for $\\ell \\ge 1$, let $\\pi_\\ell$ be the number of primes $p$ with $H(p) = \\ell$. For every $n \\ge 0$, the number of positive integers $m$ with $H(m) = n$ equals the number of families $(\\alpha_{\\ell,j})$ of nonnegative integers indexed by $1 \\le \\ell \\le n$ and $0 \\le j < \\pi_\\ell$ with $\\sum_{\\ell,j} \\ell\\, \\alpha_{\\ell,j} = n$, that is, the number of partitions of $n$ in which a\u2026","title":"Height levels are counted by coloured partitions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/exists-height-function-with-profile","kind":"node","lean":"MetaMathlibExt.exists_heightFunction_with_profile","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#exists-height-function-with-profile","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $(\\pi_k)_{k \\ge 1}$ be nonnegative integers with $\\pi_k \\ne 0$ for infinitely many $k$ (equivalently, $\\sum_k \\pi_k = \\infty$). Then there is a height function $H$ such that for every $k \\ge 1$ exactly $\\pi_k$ primes $p$ satisfy $H(p) = k$.","title":"Realizing a prescribed profile"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/is-height-function-of-finite-fibers","kind":"node","lean":"MetaMathlibExt.isHeightFunction_of_finite_fibers","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#is-height-function-of-finite-fibers","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(i) \\ge 1$ for every $i \\ge 1$, and suppose that for every $t \\ge 1$ only finitely many $i \\ge 1$ have $f(i) = t$. Let $H$ be the completely additive function on the positive integers with $H(p) = f(i)$ when $p$ is the $i$-th prime (so $H(2) = f(1)$). Then $H$ is a height function, and for every $n \\ge 1$ the number of primes $p$ with $H(p) = n$ equals the number of\u2026","title":"Height functions with prescribed values at the primes"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/is-greatest-prime-index-height-level","kind":"node","lean":"MetaMathlibExt.isGreatest_primeIndexHeight_level","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#is-greatest-prime-index-height-level","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H$ be the completely additive function on the positive integers with $H(p) = i$ when $p$ is the $i$-th prime, so that $H(2) = 1$, $H(3) = 2$ and $H(5) = 3$. For every natural number $n$, $2^n$ is the largest integer $m \\ge 1$ with $H(m) = n$.","title":"The largest integer at each height for the prime-index height"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/matula-max","kind":"node","lean":"MetaMathlibExt.matulaMax","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#matula-max","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"Write $p_i$ for the $i$-th prime, indexed from one ($p_1 = 2$). For a natural number $n$, $P(n)$ is the result of applying $i \\mapsto p_i$ to $8$ exactly $n - 3$ times, so $P(4) = p_8$ and $P(n) = p_{P(n-1)}$ for $n \\ge 5$. For $n \\le 3$ there are no applications and $P(n) = 8$.","title":"Gutman\u2013Ivi\u0107 maximal Matula numbers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/matula-max-four","kind":"node","lean":"MetaMathlibExt.matulaMax_four","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#matula-max-four","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"The Gutman\u2013Ivi\u0107 maximal Matula number $P(4)$ equals $p_8$, the eighth prime.","title":"The first maximal Matula number"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/matula-max-succ","kind":"node","lean":"MetaMathlibExt.matulaMax_succ","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#matula-max-succ","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"Write $p_i$ for the $i$-th prime, indexed from one ($p_1 = 2$). For every $n \\ge 4$ the Gutman\u2013Ivi\u0107 maximal Matula numbers satisfy $P(n + 1) = p_{P(n)}$.","title":"Recursion for the maximal Matula numbers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/matula-is-greatest-level","kind":"node","lean":"MetaMathlibExt.matula_isGreatest_level","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#matula-is-greatest-level","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Write $p_i$ for the $i$-th prime, indexed from one ($p_1 = 2$), and let $H$ be a function from the natural numbers to themselves that is completely additive on the positive integers and satisfies $H(p_i) = H(i) + 1$ for every $i \\ge 1$ (Matula's height function). Then for every $n \\ge 4$ the largest positive integer $m$ with $H(m) = n$ is the Gutman\u2013Ivi\u0107 maximal Matula number $P(n)$.","title":"Largest number at a given Matula height"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/matula-is-least-level","kind":"node","lean":"MetaMathlibExt.matula_isLeast_level","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#matula-is-least-level","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Write $p_i$ for the $i$-th prime, indexed from one ($p_1 = 2$), and let $H$ be a function from the natural numbers to themselves that is completely additive on the positive integers and satisfies $H(p_i) = H(i) + 1$ for every $i \\ge 1$ (Matula's height function). Then for every $n > 1$ the smallest positive integer $m$ with $H(m) = n$ is $3 \\cdot 5^{(n-2)/3}$ if $n \\equiv 2 \\pmod 3$, $5^{n/3}$ if\u2026","title":"Smallest number at a given Matula height"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/nth-prime-one-rosser-schoenfeld","kind":"node","lean":"MetaMathlibExt.nthPrimeOne_rosser_schoenfeld","parent":8532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rosser\u2013Schoenfeld bounds for the "],["math","\\(k\\)"],["text","-th prime"]],"source":"Rosser\u2013Schoenfeld bounds for the $k$-th prime"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#nth-prime-one-rosser-schoenfeld","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Write $p_k$ for the $k$-th prime, indexed from one ($p_1 = 2$). There is a natural number $k_0$ such that every $k \\ge k_0$ satisfies $k (\\log k + \\log \\log k - 1) \\le p_k \\le k (\\log k + \\log \\log k)$.","title":"Rosser\u2013Schoenfeld bounds for the $k$-th prime"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/two-mul-lt-nth-prime-one","kind":"node","lean":"MetaMathlibExt.two_mul_lt_nthPrimeOne","parent":8532,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-th prime exceeds "],["math","\\(2k\\)"],["text"," for "],["math","\\(k \\ge 6\\)"]],"source":"The $k$-th prime exceeds $2k$ for $k \\ge 6$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#two-mul-lt-nth-prime-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"For every natural number $k \\ge 6$, the $k$-th prime $p_k$, indexed from one so that $p_1 = 2$, satisfies $p_k > 2k$.","title":"The $k$-th prime exceeds $2k$ for $k \\ge 6$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/sum-height-eq-sum-prime-powers","kind":"node","lean":"MetaMathlibExt.sum_height_eq_sum_prime_powers","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#sum-height-eq-sum-prime-powers","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H : \\mathbb{N} \\to \\mathbb{N}$ be completely additive on the positive integers, and let $x \\ge 2$ be a natural number. Then","title":"Prime-power decomposition of summatory heights"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/sum-height-asymptotic","kind":"node","lean":"MetaMathlibExt.sum_height_asymptotic","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#sum-height-asymptotic","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H : \\mathbb{N} \\to \\mathbb{N}$ be completely additive on the positive integers, and for real $x$ let $F(x) = \\sum_{1 \\le m \\le x} H(m)$. Then there is a real constant $C$ such that for every real $x \\ge 2$,","title":"Prime reduction for summatory heights"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/is-height-function-of-iterative","kind":"node","lean":"MetaMathlibExt.isHeightFunction_of_iterative","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#is-height-function-of-iterative","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Write $p_i$ for the $i$-th prime, so $p_1 = 2$. Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(1) = 1$; suppose that for all integers $i, r \\ge 1$, if $p_r$ divides $f(i)$ then $r < i$ (this forces $f(i) \\ge 1$ for $i \\ge 1$); and suppose that for every $t \\ge 1$ only finitely many $i \\ge 1$ have $f(i) = t$. Let $j \\ge 1$. Then there is a function $H : \\mathbb{N} \\to \\mathbb{N}$ that is completely additive on the\u2026","title":"Iteratively defined height functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/weighted-partition-coeff","kind":"node","lean":"MetaMathlibExt.weightedPartitionCoeff","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#weighted-partition-coeff","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"Given real weights $(b_k)_{k \\ge 0}$, the coefficients $a_n$ of the weighted partition product $\\prod_{k \\ge 1} (1 - q^k)^{-b_k}$ are defined by $a_0 = 1$ and, for $n \\ge 1$, the logarithmic-derivative recurrence $n a_n = \\sum_{m=1}^{n} \\big(\\sum_{k \\mid m} k b_k\\big) a_{n-m}$ (valid for real exponents $b_k$). The value $b_0$ is unused.","title":"Weighted partition coefficients"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/weighted-partition-coeff-zero","kind":"node","lean":"MetaMathlibExt.weightedPartitionCoeff_zero","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#weighted-partition-coeff-zero","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For every sequence of real weights $(b_k)$, the weighted partition coefficient $a_0$ attached to $(b_k)$ equals $1$.","title":"Zeroth weighted partition coefficient"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/weighted-partition-coeff-one","kind":"node","lean":"MetaMathlibExt.weightedPartitionCoeff_one","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#weighted-partition-coeff-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For every sequence of real weights $(b_k)$, the weighted partition coefficient $a_1$ attached to $(b_k)$ equals $b_1$.","title":"First weighted partition coefficient"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/meinardus-hypotheses","kind":"node","lean":"MetaMathlibExt.MeinardusHypotheses","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#meinardus-hypotheses","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"Let $b_1, b_2, \\dots$ be real weights (the value $b_0$ is unused), let $\\alpha$ and $A$ be reals and let $G : \\mathbb{C} \\to \\mathbb{C}$. The weights satisfy Meinardus' hypotheses (M1)\u2013(M3) with exponent $\\alpha$, residue $A$ and continuation $G$ if there is $C_0$ with $0 < C_0 < 1$ such that: (M1) for $\\operatorname{Re} s > \\alpha$ the Dirichlet series $B(s) = \\sum_{k \\ge 1} b_k k^{-s}$ converges absolutely and\u2026","title":"Meinardus' hypotheses"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/meinardus-asymptotic","kind":"node","lean":"MetaMathlibExt.meinardus_asymptotic","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#meinardus-asymptotic","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $b_0, b_1, b_2, \\dots$ be nonnegative reals with $b_k \\ne 0$ for some $k \\ge 1$, and let $a_n$ be the coefficients of the weighted partition product $\\prod_{k \\ge 1} (1 - q^k)^{-b_k}$. Suppose that the weights satisfy Meinardus' hypotheses (M1)\u2013(M3) with exponent $\\alpha > 0$, residue $A > 0$ and continuation $G$, and put $\\kappa = (\\operatorname{Re} G(0) - 1 - \\alpha/2)/(\\alpha + 1)$ and\u2026","title":"Meinardus' theorem"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/lseries-profile-simple-pole","kind":"node","lean":"MetaMathlibExt.lseries_profile_simple_pole","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#lseries-profile-simple-pole","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $(\\pi_k)_{k \\ge 0}$ be nonnegative real numbers and let $C$, $\\alpha$, $\\eta$ be positive reals. Suppose there is a constant $K$ with $\\bigl|\\sum_{k=1}^{\\lfloor x \\rfloor} \\pi_k - (C/\\alpha) x^{\\alpha}\\bigr| \\le K x^{\\alpha - \\eta}$ for every real $x \\ge 1$. Then the Dirichlet series $B(s) = \\sum_{k \\ge 1} \\pi_k k^{-s}$ converges absolutely for $\\operatorname{Re} s > \\alpha$, and there is a function $R$,\u2026","title":"A simple pole of a Dirichlet series from the growth of its partial sums"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/height-level-card-meinardus","kind":"node","lean":"MetaMathlibExt.heightLevel_card_meinardus","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#height-level-card-meinardus","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H$ be a height function, for $k \\ge 0$ let $\\pi_k$ be the number of primes $p$ with $H(p) = k$, and for $n \\ge 0$ let $N_n$ be the number of integers $m \\ge 1$ with $H(m) = n$. Let $\\alpha > 0$ and $C > 0$ be real numbers and suppose that the weights $(\\pi_k)$ satisfy Meinardus' hypotheses with exponent $\\alpha$, residue $C$ and some continuation $G$. Put\u2026","title":"Meinardus asymptotics for the levels of a height function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/height-profile-identification","kind":"node","lean":"MetaMathlibExt.heightProfile_identification","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#height-profile-identification","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H$ be a height function, for $k \\ge 0$ let $\\pi_k$ be the number of primes $p$ with $H(p) = k$, and for $n \\ge 0$ let $N_n$ be the number of integers $m \\ge 1$ with $H(m) = n$. Let $r > 0$ and $A > 0$ be real numbers and suppose that the weights $(\\pi_k)$ satisfy Meinardus' hypotheses with exponent $r$, residue $A$ and some continuation $G$. Suppose further that $N_n \\sim C_1 n^{\\kappa} \\exp(C_2 n^{\\beta})$ as\u2026","title":"Identifying the profile from the growth of the levels"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/profile-partial-sum","kind":"node","lean":"MetaMathlibExt.profilePartialSum","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#profile-partial-sum","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For a sequence $\\pi : \\mathbb{N} \\to \\mathbb{N}$ and a natural number $k$, the cumulative profile is $\\Pi_k = \\sum_{j=1}^{k} \\pi_j$. The value $\\pi_0$ is not used, and $\\Pi_0 = 0$.","title":"Cumulative profile"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/profile-partial-sum-succ","kind":"node","lean":"MetaMathlibExt.profilePartialSum_succ","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#profile-partial-sum-succ","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For every sequence $\\pi : \\mathbb{N} \\to \\mathbb{N}$ with cumulative profile $\\Pi_k = \\sum_{j=1}^{k} \\pi_j$ and every natural number $k$, $\\Pi_{k+1} = \\Pi_k + \\pi_{k+1}$.","title":"Recursion for the cumulative profile"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/seq-height-index","kind":"node","lean":"MetaMathlibExt.seqHeightIndex","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#seq-height-index","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"Let $\\pi : \\mathbb{N} \\to \\mathbb{N}$ be a sequence with cumulative profile $\\Pi_k = \\sum_{j=1}^{k} \\pi_j$. For a natural number $i$, the block index $k(i)$ is the least natural number $k$ with $i \\le \\Pi_k$, and $k(i) = 0$ if there is no such $k$. When $i \\ge 1$ and $\\Pi$ is unbounded, $k(i)$ is the unique $k$ with $\\Pi_{k-1} < i \\le \\Pi_k$.","title":"Block index of a profile"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/seq-height-index-const-one","kind":"node","lean":"MetaMathlibExt.seqHeightIndex_const_one","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#seq-height-index-const-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"For the constant profile $\\pi_k = 1$, whose cumulative profile is $\\Pi_k = \\sum_{j=1}^{k} \\pi_j = k$, the block index $k(i)$, the least $k$ with $i \\le \\Pi_k$, equals $i$ for every natural number $i$.","title":"Block index of the constant profile"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/canonical-seq-height","kind":"node","lean":"MetaMathlibExt.canonicalSeqHeight","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#canonical-seq-height","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"fully_proved","summary":"Let $(\\pi_k)_{k \\ge 1}$ be a sequence of nonnegative integers with partial sums $\\Pi_k = \\sum_{1 \\le j \\le k} \\pi_j$. For $i \\ge 1$ let $k(i)$ be the least $k \\ge 0$ with $i \\le \\Pi_k$, or $0$ if there is none, and let $p_1 = 2, p_2 = 3, \\dots$ be the primes in increasing order. The canonical sequential realization $H_{\\mathrm{seq}}$ of $\\pi$ is the function $\\mathbb{N} \\to \\mathbb{N}$ that sends $m \\ge 1$ to\u2026","title":"Canonical sequential realization of a profile"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/seq-height-index-is-equivalent","kind":"node","lean":"MetaMathlibExt.seqHeightIndex_isEquivalent","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#seq-height-index-is-equivalent","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $\\pi : \\mathbb{N} \\to \\mathbb{N}$ be a sequence with cumulative profile $\\Pi_k = \\sum_{j=1}^{k} \\pi_j$, let $k(i)$ be its block index (the least $k$ with $i \\le \\Pi_k$, and $0$ if there is none), let $H_{\\mathrm{seq}}$ be its canonical sequential realization, and write $p_i$ for the $i$-th prime, indexed from one. Let $C > 0$ and $\\alpha > 0$ be real numbers with $\\Pi_k \\sim C k^\\alpha$ as $k \\to \\infty$. Then\u2026","title":"Inverting a polynomial profile"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/sum-rpow-div-log-is-equivalent","kind":"node","lean":"MetaMathlibExt.sum_rpow_div_log_isEquivalent","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#sum-rpow-div-log-is-equivalent","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"For every real $\\beta > 0$,","title":"A logarithmic-integral sum"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/canonical-seq-prime-harmonic-sum","kind":"node","lean":"MetaMathlibExt.canonicalSeq_primeHarmonicSum","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#canonical-seq-prime-harmonic-sum","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $(\\pi_k)_{k \\ge 1}$ be a sequence of nonnegative integers with partial sums $\\Pi_k = \\sum_{1 \\le j \\le k} \\pi_j$, let $H_{\\mathrm{seq}}$ be its canonical sequential realization, and suppose that $C > 0$ and $\\alpha > 0$ are real numbers with $\\Pi_k \\sim C k^\\alpha$ as $k \\to \\infty$. Write $S(x) = \\sum_{p \\le x} H_{\\mathrm{seq}}(p)/p$, the sum over primes $p \\le x$, and let $m(x)$ be the number of primes\u2026","title":"Prime-harmonic growth of the canonical sequential realization"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/canonical-seq-average-order","kind":"node","lean":"MetaMathlibExt.canonicalSeq_averageOrder","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#canonical-seq-average-order","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $(\\pi_k)_{k \\ge 1}$ be a sequence of nonnegative integers with partial sums $\\Pi_k = \\sum_{1 \\le j \\le k} \\pi_j$, let $H_{\\mathrm{seq}}$ be its canonical sequential realization, and suppose that $C > 0$ and $\\alpha > 0$ are real numbers with $\\Pi_k \\sim C k^\\alpha$ as $k \\to \\infty$. Then, as $x \\to \\infty$ over the reals, each of the three functions $x \\sum_{p \\le x} H_{\\mathrm{seq}}(p)/p$,\u2026","title":"Average order of the canonical sequential realization"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/canonical-seq-partition-profile-growth","kind":"node","lean":"MetaMathlibExt.canonicalSeq_partitionProfile_growth","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#canonical-seq-partition-profile-growth","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H_{\\mathrm{seq}}$ be the canonical sequential realization of the constant profile $\\pi_k = 1$ for all $k \\ge 1$ (so $H_{\\mathrm{seq}}(p_i) = i$ for the $i$-th prime $p_i$). Then, as $x \\to \\infty$ over the reals, $\\sum_{p \\le x} H_{\\mathrm{seq}}(p)/p = \\Theta\\bigl(x / (\\log x)^2\\bigr)$, the sum running over primes $p \\le x$, and $\\sum_{1 \\le n \\le x} H_{\\mathrm{seq}}(n) = \\Theta\\bigl(x^2 / (\\log x)^2\\bigr)$,\u2026","title":"Growth for the ordinary partition profile"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/canonical-seq-plane-partition-profile-gr","kind":"node","lean":"MetaMathlibExt.canonicalSeq_planePartitionProfile_growth","parent":8532,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions/index.html#canonical-seq-plane-partition-profile-gr","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-additive-height-functions","status":"can_prove","summary":"Let $H_{\\mathrm{seq}}$ be the canonical sequential realization of the profile $\\pi_k = k$ for $k \\ge 1$, let $S(x) = \\sum_{p \\le x} H_{\\mathrm{seq}}(p)/p$ (the sum over primes $p \\le x$) and $F(x) = \\sum_{1 \\le n \\le x} H_{\\mathrm{seq}}(n)$, and let $m(x)$ be the number of primes $p \\le x$. Then, as $x \\to \\infty$ over the reals, $S(x) \\sim 2\\sqrt{2}\\, \\sqrt{m(x)} / \\log m(x)$,\u2026","title":"Growth for the plane partition profile"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions","status":"fully_proved","title":"Completely multiplicative functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions/is-completely-multiplicative","kind":"node","lean":"ArithmeticFunction.IsCompletelyMultiplicative","parent":8576,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions/index.html#is-completely-multiplicative","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions","status":"fully_proved","summary":"Let $R$ be a type with a multiplication and a zero. An arithmetic function $f$ with values in $R$ (so $f(0) = 0$) is completely multiplicative if $f(mn) = f(m) f(n)$ for all $m, n \\in \\mathbb{N} = \\{0, 1, 2, \\dots\\}$.","title":"Completely multiplicative arithmetic function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions/map-mul","kind":"node","lean":"ArithmeticFunction.IsCompletelyMultiplicative.map_mul","parent":8576,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions/index.html#map-mul","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions","status":"fully_proved","summary":"Let $R$ be a type with a multiplication and a zero, and let $f$ be a completely multiplicative arithmetic function with values in $R$. Then $f(mn) = f(m) f(n)$ for all $m, n \\in \\mathbb{N}$.","title":"Completely multiplicative functions preserve products"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions/is-multiplicative","kind":"node","lean":"ArithmeticFunction.IsCompletelyMultiplicative.isMultiplicative","parent":8576,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions/index.html#is-multiplicative","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/completely-multiplicative-functions","status":"fully_proved","summary":"Let $R$ be a monoid with zero and let $f$ be a completely multiplicative arithmetic function with values in $R$ such that $f(1) = 1$. Then $f$ is multiplicative: $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all coprime natural numbers $m$ and $n$.","title":"Normalized completely multiplicative functions are multiplicative"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions","status":"can_prove","title":"Hyperbolic floor sums of arithmetic functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions/summable-abs-mul-log-pow-div-sub","kind":"node","lean":"MetaMathlibExt.summable_abs_mul_log_pow_div_sub","parent":8580,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions/index.html#summable-abs-mul-log-pow-div-sub","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions","status":"can_prove","summary":"Let $f$ be a real function on the natural numbers with $f(n) = O(n^\\alpha)$ as $n \\to \\infty$ for some real $\\alpha$ with $0 \\le \\alpha < 1$. Then for every natural number $i$ the series","title":"Absolute convergence of a log-weighted difference series"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions/hyperbolic-floor-sum-asymptotic","kind":"node","lean":"MetaMathlibExt.hyperbolic_floor_sum_asymptotic","parent":8580,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions/index.html#hyperbolic-floor-sum-asymptotic","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions","status":"can_prove","summary":"Let $r \\ge 2$ and let $\\tau_r(n)$ be the number of ordered factorizations $n = n_1 \\cdots n_r$ of $n$ into $r$ positive factors. Let $P(t) = \\sum_j a_j t^j$ be a real polynomial with $\\sum_{1 \\le n \\le x} \\tau_r(n) - x P(\\log x) = O(x^{1 - 1/r} (\\log x)^{r-2})$ as $x \\to \\infty$. Let $f$ be a nonnegative real function on the natural numbers such that, for some real $\\alpha$ with $0 \\le \\alpha < 1$,\u2026","title":"Asymptotics of hyperbolic floor sums"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions/hyperbolic-floor-sum-asymptotic-additive","kind":"node","lean":"MetaMathlibExt.hyperbolic_floor_sum_asymptotic_additive","parent":8580,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions/index.html#hyperbolic-floor-sum-asymptotic-additive","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/hyperbolic-floor-sums-of-arithmetic-functions","status":"can_prove","summary":"Let $r \\ge 2$ and let $\\tau_r(n)$ be the number of ordered factorizations $n = n_1 \\cdots n_r$ of $n$ into $r$ positive factors. Let $P(t) = \\sum_j a_j t^j$ be a real polynomial with $\\sum_{1 \\le n \\le x} \\tau_r(n) - x P(\\log x) = O(x^{1 - 1/r} (\\log x)^{r-2})$ as $x \\to \\infty$. Let $g$ be a real function on the natural numbers with $g(1) \\ne 0$ and $g(a) = O(2^{a/2})$ as $a \\to \\infty$. For $n \\ge 1$ let\u2026","title":"Hyperbolic floor sums of additive functions of prime exponents"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals","status":"can_prove","title":"Multiplicative functions in short intervals"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/int-extension","kind":"node","lean":"ArithmeticFunction.intExtension","parent":8584,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/index.html#int-extension","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals","status":"fully_proved","summary":"Let $R$ be a type with a zero element $0$ and let $f$ be an arithmetic function with values in $R$. The extension of $f$ to the integers is the function on $\\mathbb{Z}$ that sends an integer $z$ to $f(z)$ if $z > 0$ and to $0$ if $z \\le 0$.","title":"Extension of an arithmetic function to the integers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/is-one-bounded","kind":"node","lean":"MathlibExt.NumberTheory.BoundedMultiplicativeFunctionsWanted.BoundedMultiplicativeFunctions.IsOneBounded","parent":8584,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/index.html#is-one-bounded","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals","status":"fully_proved","summary":"A function $f \\colon \\mathbb{N} \\to \\mathbb{C}$ is 1-bounded if $|f(n)| \\le 1$ for every natural number $n$.","title":"1-bounded function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/theorem-1-3","kind":"node","lean":"MathlibExt.NumberTheory.BoundedMultiplicativeFunctionsWanted.BoundedMultiplicativeFunctions.theorem_1_3","parent":8584,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/index.html#theorem-1-3","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals","status":"can_prove","summary":"Non-pretentious multiplicative functions do not correlate with polynomial phases on short intervals on average. For a function $g$ on the integers, a natural number $k$, a real $x$ and a real $H > 0$ write","title":"Non-pretentious multiplicative functions do not correlate with polynomial phases on short intervals on average"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/corollary-1-1","kind":"node","lean":"MathlibExt.NumberTheory.BoundedMultiplicativeFunctionsWanted.BoundedMultiplicativeFunctions.corollary_1_1","parent":8584,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals/index.html#corollary-1-1","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/multiplicative-functions-in-short-intervals","status":"can_prove","summary":"Let $\\lambda$ be the Liouville function, extended to the integers by $\\lambda(n) = 0$ for $n \\le 0$, let $k$ be a natural number and let $\\theta$ be a real number with $0 < \\theta < 1$. For real $x$ and $H > 0$ write","title":"Liouville's function does not correlate with polynomial phases on short intervals on average"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","title":"M\u00f6bius\u2013Fermat function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/fermat-unit-function","kind":"node","lean":"MetaMathlibExt.fermatUnitFunction","parent":8589,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#fermat-unit-function","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"fully_proved","summary":"The Fermat unit function $u_M$ is the integer-valued arithmetic function with $u_M(l) = 1$ when $l = 2^n - 1$ for some integer $n \\ge 1$, and $u_M(l) = 0$ otherwise (in particular $u_M(0) = 0$). Equivalently, $u_M(l) = 1$ exactly when $l \\ge 1$ and $l + 1$ is a power of $2$.","title":"Fermat unit function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/fermat-unit-function-two-pow-sub-one","kind":"node","lean":"MetaMathlibExt.fermatUnitFunction_two_pow_sub_one","parent":8589,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Fermat unit function at "],["math","\\(2^n - 1\\)"]],"source":"The Fermat unit function at $2^n - 1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#fermat-unit-function-two-pow-sub-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"fully_proved","summary":"For every integer $n \\ge 1$, the Fermat unit function $u_M$ satisfies $u_M(2^n - 1) = 1$.","title":"The Fermat unit function at $2^n - 1$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/fermat-unit-function-small-values","kind":"node","lean":"MetaMathlibExt.fermatUnitFunction_small_values","parent":8589,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#fermat-unit-function-small-values","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"fully_proved","summary":"The Fermat unit function $u_M$ satisfies $u_M(1) = u_M(3) = u_M(7) = u_M(63) = 1$ and $u_M(2) = u_M(9) = 0$.","title":"Small values of the Fermat unit function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/mobius-fermat-one-and-vanishing","kind":"node","lean":"MetaMathlibExt.mobiusFermat_one_and_vanishing","parent":8589,"reading_title":{"format":"inline-parts-v1","parts":[["text","The M\u00f6bius\u2013Fermat and von Mangoldt\u2013Fermat functions at "],["math","\\(1\\)"],["text"," and off the Fermat semigroup"]],"source":"The M\u00f6bius\u2013Fermat and von Mangoldt\u2013Fermat functions at $1$ and off the Fermat semigroup"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#mobius-fermat-one-and-vanishing","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","summary":"The M\u00f6bius\u2013Fermat and von Mangoldt\u2013Fermat functions at $1$ and off the Fermat semigroup. Let $u_M$ be the Fermat unit function and let $\\mu_M$ be an integer-valued arithmetic function with $\\mu_M * u_M = \\varepsilon$, where $*$ is Dirichlet convolution and $\\varepsilon$ is its identity, with $\\varepsilon(1) = 1$ and $\\varepsilon(l) = 0$ for $l \\ne 1$; thus $\\mu_M$ is the Dirichlet inverse of $u_M$, the\u2026","title":"The M\u00f6bius\u2013Fermat and von Mangoldt\u2013Fermat functions at $1$ and off the Fermat semigroup"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/mobius-fermat-eq-multinomial","kind":"node","lean":"MetaMathlibExt.mobiusFermat_eq_multinomial","parent":8589,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#mobius-fermat-eq-multinomial","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","summary":"Let $u_M$ be the Fermat unit function and let $\\mu_M$ be an integer-valued arithmetic function with $\\mu_M * u_M = \\varepsilon$, where $*$ is Dirichlet convolution and $\\varepsilon$ is its identity, with $\\varepsilon(1) = 1$ and $\\varepsilon(l) = 0$ for $l \\ne 1$; thus $\\mu_M$ is the Dirichlet inverse of $u_M$, the M\u00f6bius\u2013Fermat function. Let $n_1, \\dots, n_k$ be finitely many (possibly no) distinct integers with\u2026","title":"Multinomial formula for the M\u00f6bius\u2013Fermat function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/von-mangoldt-fermat-eq-multinomial","kind":"node","lean":"MetaMathlibExt.vonMangoldtFermat_eq_multinomial","parent":8589,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#von-mangoldt-fermat-eq-multinomial","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","summary":"Let $u_M$ be the Fermat unit function, and let $\\mu$ be an integer-valued arithmetic function with $\\mu * u_M = \\varepsilon$, where $*$ is Dirichlet convolution and $\\varepsilon$ is its identity (so $\\mu$ is the M\u00f6bius\u2013Fermat function $\\mu_M$). Let $\\Lambda_M = \\mu * u'_M$, computed with real values, where $u'_M(l) = u_M(l) \\log l$. Let $f : \\mathbb{N} \\to \\mathbb{N}$ be finitely supported, with $n \\ge 2$ and\u2026","title":"Multinomial formula for the von Mangoldt\u2013Fermat function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/mobius-fermat-mul-factorial-eq-squarefre","kind":"node","lean":"MetaMathlibExt.mobiusFermat_mul_factorial_eq_squarefree_part","parent":8589,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#mobius-fermat-mul-factorial-eq-squarefre","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","summary":"Let $u_M$ be the Fermat unit function and let $\\mu_M$ be an integer-valued arithmetic function with $\\mu_M * u_M = \\varepsilon$, where $*$ is Dirichlet convolution and $\\varepsilon$ is its identity, with $\\varepsilon(1) = 1$ and $\\varepsilon(l) = 0$ for $l \\ne 1$; thus $\\mu_M$ is the Dirichlet inverse of $u_M$, the M\u00f6bius\u2013Fermat function. Let $m_1, m_2 \\ge 0$ be integers, let $n_3, \\dots, n_k$ be finitely many\u2026","title":"Reduction of the M\u00f6bius\u2013Fermat function to distinct factors"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/mobius-fermat-three-pow-mul-seven-pow","kind":"node","lean":"MetaMathlibExt.mobiusFermat_three_pow_mul_seven_pow","parent":8589,"reading_title":{"format":"inline-parts-v1","parts":[["text","The M\u00f6bius\u2013Fermat function at "],["math","\\(3^m 7^n\\)"]],"source":"The M\u00f6bius\u2013Fermat function at $3^m 7^n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#mobius-fermat-three-pow-mul-seven-pow","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","summary":"Let $u_M$ be the Fermat unit function and let $\\mu_M$ be an integer-valued arithmetic function with $\\mu_M * u_M = \\varepsilon$, where $*$ is Dirichlet convolution and $\\varepsilon$ is its identity, with $\\varepsilon(1) = 1$ and $\\varepsilon(l) = 0$ for $l \\ne 1$; thus $\\mu_M$ is the Dirichlet inverse of $u_M$, the M\u00f6bius\u2013Fermat function. Then for all integers $m, n \\ge 0$,","title":"The M\u00f6bius\u2013Fermat function at $3^m 7^n$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/mobius-fermat-three-pow-mul-seven-pow-mu","kind":"node","lean":"MetaMathlibExt.mobiusFermat_three_pow_mul_seven_pow_mul_pow","parent":8589,"reading_title":{"format":"inline-parts-v1","parts":[["text","The M\u00f6bius\u2013Fermat function at "],["math","\\(3^m 7^n q^k\\)"]],"source":"The M\u00f6bius\u2013Fermat function at $3^m 7^n q^k$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#mobius-fermat-three-pow-mul-seven-pow-mu","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","summary":"Let $u_M$ be the Fermat unit function and let $\\mu_M$ be an integer-valued arithmetic function with $\\mu_M * u_M = \\varepsilon$, where $*$ is Dirichlet convolution and $\\varepsilon$ is its identity, with $\\varepsilon(1) = 1$ and $\\varepsilon(l) = 0$ for $l \\ne 1$; thus $\\mu_M$ is the Dirichlet inverse of $u_M$, the M\u00f6bius\u2013Fermat function. Let $t \\ge 4$ be an integer with $t \\ne 6$ and put $q = 2^t - 1$. Then for\u2026","title":"The M\u00f6bius\u2013Fermat function at $3^m 7^n q^k$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/mobius-fermat-closed-form","kind":"node","lean":"MetaMathlibExt.mobiusFermat_closed_form","parent":8589,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#mobius-fermat-closed-form","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","summary":"Let $u_M$ be the Fermat unit function and let $\\mu_M$ be an integer-valued arithmetic function with $\\mu_M * u_M = \\varepsilon$, where $*$ is Dirichlet convolution and $\\varepsilon$ is its identity, with $\\varepsilon(1) = 1$ and $\\varepsilon(l) = 0$ for $l \\ne 1$; thus $\\mu_M$ is the Dirichlet inverse of $u_M$, the M\u00f6bius\u2013Fermat function. Let $m_1, m_2 \\ge 0$ be integers, let $n_3, \\dots, n_k$ be finitely many\u2026","title":"Closed form of the M\u00f6bius\u2013Fermat function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/von-mangoldt-fermat-closed-form","kind":"node","lean":"MetaMathlibExt.vonMangoldtFermat_closed_form","parent":8589,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function/index.html#von-mangoldt-fermat-closed-form","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/m-bius-fermat-function","status":"can_prove","summary":"Let $u_M$ be the Fermat unit function and let $\\mu_M$ be an integer-valued arithmetic function with $\\mu_M * u_M = \\varepsilon$, where $*$ is Dirichlet convolution and $\\varepsilon$ is its identity, with $\\varepsilon(1) = 1$ and $\\varepsilon(l) = 0$ for $l \\ne 1$; thus $\\mu_M$ is the Dirichlet inverse of $u_M$, the M\u00f6bius\u2013Fermat function. Let $\\Lambda_M = \\mu_M * u_M'$, computed with real values, where\u2026","title":"Closed form of the von Mangoldt\u2013Fermat function"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"can_prove","title":"Random multiplicative functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/rademacher-measure","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.rademacherMeasure","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#rademacher-measure","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"The Rademacher measure is the uniform probability measure on the two integer units $\\{-1, 1\\}$.","title":"Rademacher measure on the integer units"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/rademacher-measure-singleton","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.rademacherMeasure_singleton","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#rademacher-measure-singleton","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"For each integer unit $x \\in \\{-1, 1\\}$, the Rademacher measure of the singleton $\\{x\\}$ is $1/2$.","title":"Each Rademacher sign has probability one half"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/is-rademacher-family","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.IsRademacherFamily","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#is-rademacher-family","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$. A family $(X_p)$, indexed by the primes $p$, of maps $X_p \\colon \\Omega \\to \\{-1, 1\\}$ into the units of $\\mathbb{Z}$ is a Rademacher family for $\\mu$ when every $X_p$ is measurable, every $X_p$ has law the Rademacher measure (the uniform measure on $\\{-1, 1\\}$) under $\\mu$, and the family is mutually independent under $\\mu$.","title":"Rademacher family"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/exists-rademacher-family","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.exists_rademacherFamily","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#exists-rademacher-family","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"There exist a measurable space $\\Omega$, a probability measure $\\mu$ on $\\Omega$ and a family $(X_p)$, indexed by the primes $p$, of maps $X_p \\colon \\Omega \\to \\{-1, 1\\}$ into the units of $\\mathbb{Z}$ that is a Rademacher family for $\\mu$.","title":"Existence of a Rademacher family"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/prime-sign-nat","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.primeSignNat","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#prime-sign-nat","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $X = (X_p)$ be a family, indexed by the primes $p$, of maps $X_p \\colon \\Omega \\to \\{-1, 1\\}$ into the units of $\\mathbb{Z}$. For a natural number $n$ and $\\omega \\in \\Omega$, the prime sign of $n$ at $\\omega$ is the integer $X_n(\\omega)$ if $n$ is prime and $1$ otherwise.","title":"Prime sign of a natural number"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/measurable-prime-sign-nat","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.IsRademacherFamily.measurable_primeSignNat","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#measurable-prime-sign-nat","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $(X_p)$ be a Rademacher family for a measure $\\mu$ on a measurable space $\\Omega$. For every natural number $n$, the map $\\omega \\mapsto \\varepsilon_n(\\omega)$ from $\\Omega$ to $\\mathbb{Z}$ is measurable, where the prime sign $\\varepsilon_n(\\omega)$ is $X_n(\\omega)$ if $n$ is prime and $1$ otherwise.","title":"Measurability of the prime signs"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/rademacher-function","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.rademacherFunction","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#rademacher-function","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $(X_p)_p$ be a family of maps $X_p : \\Omega \\to \\{-1, 1\\}$ indexed by the primes, and let $\\omega \\in \\Omega$. The Rademacher random multiplicative function at $\\omega$ is the arithmetic function $f : \\mathbb{N} \\to \\mathbb{Z}$ with $f(0) = 0$ and","title":"Rademacher random multiplicative function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/rademacher-function-apply","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.rademacherFunction_apply","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#rademacher-function-apply","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $X = (X_p)$ be a family, indexed by the primes $p$, of maps $X_p \\colon \\Omega \\to \\{-1, 1\\}$, let $\\omega \\in \\Omega$, and let $f$ be the Rademacher random multiplicative function of $X$ at $\\omega$. For every natural number $n \\neq 0$, $f(n) = \\mu(n)^2 \\prod_{p \\mid n} \\varepsilon_p(\\omega)$, where $\\mu$ is the M\u00f6bius function, the product runs over the distinct primes $p$ dividing $n$, and\u2026","title":"Formula for the Rademacher random multiplicative function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/rademacher-function-eq-prod-of-squarefre","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.rademacherFunction_eq_prod_of_squarefree","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#rademacher-function-eq-prod-of-squarefre","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $X = (X_p)$ be a family, indexed by the primes $p$, of maps $X_p \\colon \\Omega \\to \\{-1, 1\\}$, let $\\omega \\in \\Omega$, and let $f$ be the Rademacher random multiplicative function of $X$ at $\\omega$. For every squarefree natural number $n$, $f(n) = \\prod_{p \\mid n} \\varepsilon_p(\\omega)$, the product of the prime signs $\\varepsilon_p(\\omega)$ at $\\omega$ over the distinct primes $p$ dividing $n$.","title":"The Rademacher random multiplicative function on squarefree numbers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/rademacher-function-eq-zero-of-not-squar","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.rademacherFunction_eq_zero_of_not_squarefree","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#rademacher-function-eq-zero-of-not-squar","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $X = (X_p)$ be a family, indexed by the primes $p$, of maps $X_p \\colon \\Omega \\to \\{-1, 1\\}$, let $\\omega \\in \\Omega$, and let $f$ be the Rademacher random multiplicative function of $X$ at $\\omega$. Then $f(n) = 0$ for every natural number $n$ that is not squarefree, including $n = 0$.","title":"The Rademacher random multiplicative function vanishes off squarefree numbers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/is-multiplicative-rademacher-function","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.isMultiplicative_rademacherFunction","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#is-multiplicative-rademacher-function","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $X = (X_p)$ be a family, indexed by the primes $p$, of maps $X_p \\colon \\Omega \\to \\{-1, 1\\}$, and let $\\omega \\in \\Omega$. The Rademacher random multiplicative function $f$ of $X$ at $\\omega$ satisfies $f(1) = 1$ and $f(mn) = f(m) f(n)$ for all coprime natural numbers $m$ and $n$.","title":"The Rademacher random multiplicative function is multiplicative"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/rademacher-function-prime","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.rademacherFunction_prime","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#rademacher-function-prime","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $(X_p)_p$ be a family of maps $X_p : \\Omega \\to \\{-1, 1\\}$ indexed by the primes, let $\\omega \\in \\Omega$, and let $f$ be the associated Rademacher random multiplicative function at $\\omega$. For every prime $p$, the value $f(p)$ is the sign assigned to $p$, that is, $f(p) = X_p(\\omega)$.","title":"Value of the Rademacher function at a prime"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/rademacher-function-prime-sq","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.rademacherFunction_prime_sq","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#rademacher-function-prime-sq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $X = (X_p)$ be a family, indexed by the primes $p$, of maps $X_p \\colon \\Omega \\to \\{-1, 1\\}$, let $\\omega \\in \\Omega$, and let $f$ be the Rademacher random multiplicative function of $X$ at $\\omega$. Then $f(p^2) = 0$ for every prime $p$.","title":"The Rademacher random multiplicative function vanishes at prime squares"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/measurable-rademacher-function","kind":"node","lean":"MetaMathlibExt.RandomMultiplicativeFunction.IsRademacherFamily.measurable_rademacherFunction","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#measurable-rademacher-function","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Let $(X_p)$ be a Rademacher family for a measure $\\mu$ on a measurable space $\\Omega$, and for $\\omega \\in \\Omega$ let $f_\\omega$ be the Rademacher random multiplicative function of $(X_p)$ at $\\omega$. For every natural number $n$, the map $\\omega \\mapsto f_\\omega(n)$ from $\\Omega$ to $\\mathbb{Z}$ is measurable.","title":"Measurability of the Rademacher random multiplicative function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/partial-sum","kind":"node","lean":"MathlibExt.NumberTheory.RademacherLILConstantWanted.partialSum","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#partial-sum","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"Given signs $X_p(\\omega) \\in \\{-1, 1\\}$ indexed by the primes $p$ and a sample point $\\omega$, and given $N \\in \\mathbb{N}$, the partial sum $S_N(\\omega)$ is the real number $\\sum_{m=1}^{N} f_\\omega(m)$, where $f_\\omega$ is the Rademacher multiplicative function of the signs $X_p(\\omega)$: $f_\\omega(m) = \\prod_{p \\mid m} X_p(\\omega)$ for squarefree $m$ and $f_\\omega(m) = 0$ otherwise.","title":"Partial sums of the Rademacher multiplicative function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.RademacherLILConstantWanted.conjecture","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#conjecture","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"fully_proved","summary":"The statement that there is a real constant $c > 0$ such that, for every probability space $(\\Omega, \\mu)$ and every family $(X_p)_p$ of independent, measurable random signs indexed by the primes, each uniform on $\\{-1, 1\\}$, for $\\mu$-almost every $\\omega$ the limit superior as $N \\to \\infty$ of $S_N(\\omega) / \\sqrt{N \\log \\log N}$ equals $c$. Here $S_N(\\omega) = \\sum_{m=1}^{N} f_\\omega(m)$ is the partial sum of\u2026","title":"Rademacher LIL constant conjecture"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.RademacherLILConstantWanted.conjecture_refuted","parent":8601,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions/index.html#conjecture-refuted","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/random-multiplicative-functions","status":"can_prove","summary":"The Rademacher LIL constant conjecture is false: there is no real constant $c > 0$ such that, for every probability space $(\\Omega, \\mu)$ and every family $(X_p)_p$ of independent, measurable random signs indexed by the primes, each uniform on $\\{-1, 1\\}$, for $\\mu$-almost every $\\omega$ the limit superior as $N \\to \\infty$ of $S_N(\\omega) / \\sqrt{N \\log \\log N}$ equals $c$. Here\u2026","title":"No iterated-logarithm constant for Rademacher multiplicative functions"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm","status":"fully_proved","title":"Rearick logarithm"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm/rearick-log","kind":"node","lean":"MetaMathlibExt.rearickLog","parent":8619,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm/index.html#rearick-log","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm","status":"fully_proved","summary":"Let $f$ be a real-valued arithmetic function with $f(1) > 0$, and let $f^{-1}$ be its Dirichlet inverse. The Rearick logarithm of $f$ is the arithmetic function $L(f)$ with $L(f)(0) = 0$, $L(f)(1) = \\log f(1)$, and $L(f)(n) = \\sum_{d \\mid n} f(d) \\, f^{-1}(n/d) \\log d$ for $n > 1$.","title":"Rearick logarithm"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm/rearick-log-eq-pmul-log-mul-dirichlet-in","kind":"node","lean":"MetaMathlibExt.rearickLog_eq_pmul_log_mul_dirichletInverse","parent":8619,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm/index.html#rearick-log-eq-pmul-log-mul-dirichlet-in","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm","status":"fully_proved","summary":"Let $f$ be a real-valued arithmetic function with $f(1) = 1$, and let $f^{-1}$ be its Dirichlet inverse. Then the Rearick logarithm of $f$ is the Dirichlet convolution of the pointwise product $n \\mapsto f(n) \\log n$ with $f^{-1}$.","title":"The Rearick logarithm as a Dirichlet convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm/rearick-log-eq-zero-iff-is-multiplicativ","kind":"node","lean":"MetaMathlibExt.rearick_log_eq_zero_iff_isMultiplicative","parent":8619,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm/index.html#rearick-log-eq-zero-iff-is-multiplicativ","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/rearick-logarithm","status":"fully_proved","summary":"Let $f$ be a real-valued arithmetic function with $f(1) > 0$, and let $L(f)$ be its Rearick logarithm. Then $f$ is multiplicative (that is, $f(1) = 1$ and $f(mn) = f(m) f(n)$ whenever $m$ and $n$ are coprime) if and only if $L(f)(m) = 0$ for every natural number $m$ that is not a prime power, including $m = 0$ and $m = 1$.","title":"Rearick's characterization of multiplicative functions"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","status":"can_prove","title":"Antimultiplicative arithmetic functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/is-antimultiplicative","kind":"node","lean":"MetaMathlibExt.IsAntimultiplicative","parent":8623,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/index.html#is-antimultiplicative","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","status":"fully_proved","summary":"Let $R$ be a type with a zero and a one. An arithmetical function $f$ with values in $R$ is antimultiplicative if $f(1) = 1$ and $f(p^k) = 0$ for every prime $p$ and every integer $k \\ge 1$.","title":"Antimultiplicative arithmetical functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/is-antimultiplicative-one","kind":"node","lean":"MetaMathlibExt.isAntimultiplicative_one","parent":8623,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/index.html#is-antimultiplicative-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","status":"fully_proved","summary":"For every semiring $R$, the identity $\\delta$ of Dirichlet convolution on arithmetical functions with values in $R$, given by $\\delta(1) = 1$ and $\\delta(n) = 0$ for $n \\ne 1$, is antimultiplicative.","title":"The Dirichlet identity is antimultiplicative"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/apply-prime","kind":"node","lean":"MetaMathlibExt.IsAntimultiplicative.apply_prime","parent":8623,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/index.html#apply-prime","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","status":"fully_proved","summary":"Let $R$ be a type with a zero and a one, and let $f$ be an antimultiplicative arithmetical function with values in $R$. Then $f(p) = 0$ for every prime $p$.","title":"Antimultiplicative functions vanish at primes"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/dirichlet-units-multiplicative-antimulti","kind":"node","lean":"MetaMathlibExt.dirichletUnits_multiplicative_antimultiplicative_subgroups","parent":8623,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/index.html#dirichlet-units-multiplicative-antimulti","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","status":"can_prove","summary":"Let $U$ be the group of units of the ring of real arithmetical functions under Dirichlet convolution. The units $f$ with $f(1) = 1$, the multiplicative units and the antimultiplicative units form subgroups $U_1$, $U_M$ and $U_A$ of $U$, respectively. Moreover $U_M \\le U_1$, $U_A \\le U_1$ and $U_M \\cap U_A = \\{\\delta\\}$, where $\\delta$ is the Dirichlet identity.","title":"Multiplicative and antimultiplicative units form subgroups"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/exists-unique-multiplicative-mul-antimul","kind":"node","lean":"MetaMathlibExt.existsUnique_multiplicative_mul_antimultiplicative","parent":8623,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/index.html#exists-unique-multiplicative-mul-antimul","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","status":"can_prove","summary":"Every real arithmetical function $f$ with $f(1) = 1$ can be written in exactly one way as a Dirichlet convolution $f = g * h$ of a multiplicative real arithmetical function $g$ and an antimultiplicative real arithmetical function $h$.","title":"Unique splitting into multiplicative and antimultiplicative parts"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/dirichlet-real-power-vector-space","kind":"node","lean":"MetaMathlibExt.dirichletRealPower_vectorSpace","parent":8623,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/index.html#dirichlet-real-power-vector-space","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","status":"can_prove","summary":"Let $L$ denote the Rearick logarithm and $*$ the Dirichlet convolution. There is a map $(\\alpha, f) \\mapsto f^{(\\alpha)}$, sending a real number $\\alpha$ and a real arithmetical function $f$ to a real arithmetical function, such that: for every real $\\alpha$ and every real arithmetical function $f$ with $f(1) = 1$, we have $f^{(\\alpha)}(1) = 1$ and $L(f^{(\\alpha)}) = \\alpha L(f)$; and for all real arithmetical\u2026","title":"Rearick's real power and its vector space laws"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/dirichlet-real-power-multiplicative-anti","kind":"node","lean":"MetaMathlibExt.dirichletRealPower_multiplicative_antimultiplicative_subspaces","parent":8623,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions/index.html#dirichlet-real-power-multiplicative-anti","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/antimultiplicative-arithmetic-functions","status":"can_prove","summary":"Let $L$ denote the Rearick logarithm. Let $f$ and $F$ be real arithmetical functions with $f(1) = F(1) = 1$, and let $\\alpha$ be a real number with $L(F) = \\alpha L(f)$. If $f$ is multiplicative, then $F$ is multiplicative, and if $f$ is antimultiplicative, then $F$ is antimultiplicative. Moreover, every real arithmetical function $f$ with $f(1) = 1$ can be written in exactly one way as a Dirichlet convolution\u2026","title":"Real powers preserve multiplicativity and antimultiplicativity"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","title":"Square-splitting convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-u","kind":"node","lean":"MetaMathlibExt.squareSplitU","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-u","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"For a natural number $n \\ge 1$ with prime factorization $n = \\prod_p p^{e_p}$, its squarefree part is $u(n) = \\prod_p p^{e_p \\bmod 2}$, the product of the primes dividing $n$ to an odd power. By convention $u(0) = 1$.","title":"Squarefree part"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-v","kind":"node","lean":"MetaMathlibExt.squareSplitV","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-v","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"For $n \\ge 1$ with prime factorization $n = \\prod_p p^{e_p}$, the square-root part of $n$ is $v(n) = \\prod_p p^{\\lfloor e_p/2 \\rfloor}$; by convention $v(0) = 1$. For $n \\ge 1$, $v(n)^2$ is the largest square dividing $n$, and $n = u(n)\\, v(n)^2$ where $u(n) = n / v(n)^2$ is squarefree.","title":"The square-root part of a natural number"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-u-one","kind":"node","lean":"MetaMathlibExt.squareSplitU_one","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-u-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"We have $u(1) = 1$, where $u(n)$ is the product of the primes dividing $n$ to an odd power.","title":"Squarefree part of one"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-v-one","kind":"node","lean":"MetaMathlibExt.squareSplitV_one","parent":8631,"reading_title":{"format":"inline-parts-v1","parts":[["text","The square-root part of "],["math","\\(1\\)"],["text"," is "],["math","\\(1\\)"]],"source":"The square-root part of $1$ is $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-v-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"We have $v(1) = 1$, where for $n \\ge 1$ the square-root part $v(n)$ is the product of $p^{\\lfloor e/2 \\rfloor}$ over the prime powers $p^e$ exactly dividing $n$.","title":"The square-root part of $1$ is $1$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-u-mul-square-split-v-sq","kind":"node","lean":"MetaMathlibExt.squareSplitU_mul_squareSplitV_sq","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-u-mul-square-split-v-sq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"For every natural number $n \\ne 0$ with prime factorization $n = \\prod_p p^{e_p}$, we have $n = u(n)\\,v(n)^2$, where $u(n) = \\prod_p p^{e_p \\bmod 2}$ and $v(n) = \\prod_p p^{\\lfloor e_p/2 \\rfloor}$.","title":"Square-splitting decomposition"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-divisors","kind":"node","lean":"MetaMathlibExt.squareSplitDivisors","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-divisors","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"For a natural number $n \\ge 1$ with prime factorization $n = \\prod_p p^{e_p}$, let $u(n) = \\prod_p p^{e_p \\bmod 2}$ and $v(n) = \\prod_p p^{\\lfloor e_p/2 \\rfloor}$, and let $u(0) = v(0) = 1$. The set of square-splitting divisors of $n$ is $A(n) = \\{u_1 v_1^2 : u_1 \\mid u(n),\\ v_1 \\mid v(n)\\}$, where $u_1$ and $v_1$ range over the positive divisors of $u(n)$ and $v(n)$.","title":"Square-splitting divisors"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-divisors-one","kind":"node","lean":"MetaMathlibExt.squareSplitDivisors_one","parent":8631,"reading_title":{"format":"inline-parts-v1","parts":[["text","The square-splitting divisor set of "],["math","\\(1\\)"]],"source":"The square-splitting divisor set of $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-divisors-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"Let $A(n) = \\{u_1 v_1^2 : u_1 \\mid u(n),\\ v_1 \\mid v(n)\\}$ be the square-splitting divisor set of $n \\ge 1$, where $u(n)$ is the squarefree part of $n$ and $v(n)$ is the positive integer with $n = u(n) v(n)^2$. Then $A(1) = \\{1\\}$.","title":"The square-splitting divisor set of $1$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-conv","kind":"node","lean":"MetaMathlibExt.squareSplitConv","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-conv","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"For complex arithmetic functions $f$ and $g$, the square-splitting $A$-convolution of Narkiewicz type is the arithmetic function $f \\star g$ given by $(f \\star g)(n) = \\sum_{d \\in A(n)} f(d)\\, g(n/d)$ for $n \\ge 1$ and $(f \\star g)(0) = 0$. Here $A(n) = \\{u_1 v_1^2 : u_1 \\mid u(n),\\ v_1 \\mid v(n)\\}$ is the square-splitting divisor set of $n$, where $u(n)$ is the squarefree part of $n$ and $n = u(n) v(n)^2$.","title":"Square-splitting convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-conv-apply","kind":"node","lean":"MetaMathlibExt.squareSplitConv_apply","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-conv-apply","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"fully_proved","summary":"For complex arithmetic functions $f$ and $g$ and every natural number $n \\ne 0$, the square-splitting convolution of $f$ and $g$ takes the value $(f \\star g)(n) = \\sum_{d \\in A(n)} f(d)\\, g(n/d)$ at $n$, where $A(n)$ is the square-splitting divisor set of $n$.","title":"Value of the square-splitting convolution at a positive integer"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-conv-is-multiplicative","kind":"node","lean":"MetaMathlibExt.squareSplitConv_isMultiplicative","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-conv-is-multiplicative","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"If $f$ and $g$ are multiplicative complex arithmetic functions, then their square-splitting convolution $f \\star g$ is multiplicative. Here an arithmetic function $h$ is multiplicative when $h(1) = 1$ and $h(mn) = h(m)\\, h(n)$ for all coprime $m$ and $n$.","title":"The square-splitting convolution preserves multiplicativity"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-divisors-properties","kind":"node","lean":"MetaMathlibExt.squareSplitDivisors_properties","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-divisors-properties","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $A(n) = \\{u_1 v_1^2 : u_1 \\mid u(n),\\ v_1 \\mid v(n)\\}$ be the square-splitting divisor set of $n$, where $u(n)$ is the squarefree part of $n$ and $n = u(n) v(n)^2$ for $n \\ge 1$. Then (i) for every $n \\ge 1$, both $1$ and $n$ belong to $A(n)$, and every element of $A(n)$ divides $n$; (ii) for all coprime natural numbers $m$ and $n$, $A(mn) = \\{ab : a \\in A(m),\\ b \\in A(n)\\}$; and (iii) for every $n \\ge 1$,\u2026","title":"Basic properties of the square-splitting divisor sets"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-conv-comm-ring-axioms","kind":"node","lean":"MetaMathlibExt.squareSplitConv_commRing_axioms","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-conv-comm-ring-axioms","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"For all complex arithmetic functions $f$, $g$ and $h$, the square-splitting convolution $\\star$ satisfies $f \\star g = g \\star f$, $(f \\star g) \\star h = f \\star (g \\star h)$, $f \\star (g + h) = f \\star g + f \\star h$ and $\\varepsilon \\star f = f$, where $\\varepsilon$ is the arithmetic function with $\\varepsilon(1) = 1$ and $\\varepsilon(n) = 0$ for $n \\ne 1$.","title":"Ring axioms for the square-splitting convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-conv-bivariate-iso","kind":"node","lean":"MetaMathlibExt.squareSplitConv_bivariate_iso","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-conv-bivariate-iso","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $\\star$ be the square-splitting convolution. For all complex arithmetic functions $f$ and $g$, every squarefree natural number $u$ and every natural number $v \\ge 1$,","title":"Square-splitting convolution as a bivariate Dirichlet convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-moebius-formula","kind":"node","lean":"MetaMathlibExt.squareSplitMoebius_formula","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-moebius-formula","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $\\star$ be the square-splitting convolution, $\\mathbf{1}$ the constant function $1$ on the positive integers, $\\varepsilon$ the identity element ($\\varepsilon(1) = 1$ and $\\varepsilon(n) = 0$ for $n \\ne 1$), and let $\\mu_\\star$ be any complex-valued arithmetic function with $\\mathbf{1} \\star \\mu_\\star = \\varepsilon$. Then $\\mu_\\star(n) = \\mu(u(n))\\,\\mu(v(n))$ for every $n \\ge 1$, where $\\mu$ is the M\u00f6bius\u2026","title":"M\u00f6bius function of the square-splitting convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-moebius-inversion","kind":"node","lean":"MetaMathlibExt.squareSplit_moebius_inversion","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-moebius-inversion","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $\\star$ be the square-splitting convolution, $\\mathbf{1}$ the constant function $1$ on the positive integers, $\\varepsilon$ the identity element ($\\varepsilon(1) = 1$ and $\\varepsilon(n) = 0$ for $n \\ne 1$), and let $\\mu_\\star$ be any complex-valued arithmetic function with $\\mathbf{1} \\star \\mu_\\star = \\varepsilon$. Then for all complex-valued arithmetic functions $f$ and $F$, $F = \\mathbf{1} \\star f$ if and\u2026","title":"M\u00f6bius inversion for the square-splitting convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-conv-inverse-of-is-multipli","kind":"node","lean":"MetaMathlibExt.squareSplitConv_inverse_of_isMultiplicative","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-conv-inverse-of-is-multipli","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $f$ be a multiplicative complex arithmetic function (an arithmetic function $h$ is multiplicative when $h(1) = 1$ and $h(mn) = h(m)\\, h(n)$ for coprime $m$ and $n$), let $\\star$ be the square-splitting convolution, and let $\\varepsilon$ be the arithmetic function with $\\varepsilon(1) = 1$ and $\\varepsilon(n) = 0$ for $n \\ne 1$. Then there is a complex arithmetic function $g$ with $f \\star g = \\varepsilon$ such\u2026","title":"Inverse of a multiplicative function under the square-splitting convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-totient-formula","kind":"node","lean":"MetaMathlibExt.squareSplitTotient_formula","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-totient-formula","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $A(n)$ be the set of square-splitting divisors of $n$, and let $\\varphi_\\star$ be any integer-valued arithmetic function with $\\sum_{d \\in A(n)} \\varphi_\\star(d) = n$ for every $n \\ge 1$. Then $\\varphi_\\star$ is multiplicative and $\\varphi_\\star(1) = 1$. For every prime $p$, $\\varphi_\\star(p) = p - 1$, and for every $b \\ge 1$, $\\varphi_\\star(p^{2b}) = (p^2 - 1)\\,p^{2b-2}$ and\u2026","title":"The square-splitting totient function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-totient-sum-asymptotic","kind":"node","lean":"MetaMathlibExt.squareSplitTotient_sum_asymptotic","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-totient-sum-asymptotic","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $A(n)$ be the set of square-splitting divisors of $n$, and let $\\varphi_\\star$ be any integer-valued arithmetic function with $\\sum_{d \\in A(n)} \\varphi_\\star(d) = n$ for every $n \\ge 1$. Then the product $M = \\prod_p \\left(1 - \\frac{1}{p^2} + \\frac{(p-1)^2}{p^5}\\right)$ converges absolutely, that is, $\\sum_p \\left|\\frac{(p-1)^2}{p^5} - \\frac{1}{p^2}\\right| < \\infty$, and for every $0 < \\varepsilon < 1/2$,","title":"Average order of the square-splitting totient function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-divisor-count-lseries","kind":"node","lean":"MetaMathlibExt.squareSplitDivisorCount_LSeries","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-divisor-count-lseries","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $\\tau^\\star(n) = |A(n)|$ be the number of elements of the square-splitting divisor set $A(n)$, and let $H(s) = \\prod_p (1 - 3p^{-2s} + 2p^{-3s})$, where $p$ runs over the primes. For every complex number $s$ with $\\operatorname{Re} s > 1$, the series $\\sum_{k \\ge 0} \\tau^\\star(p^k) p^{-ks}$ converges to $(1 + 2p^{-s})/(1 - p^{-2s})^2$ for each prime $p$, and the Dirichlet series\u2026","title":"Dirichlet series of the square-splitting divisor function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/sum-asymptotic-of-lseries-eq-zeta-sq-div","kind":"node","lean":"MetaMathlibExt.sum_asymptotic_of_LSeries_eq_zeta_sq_div_zeta_two_mul","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#sum-asymptotic-of-lseries-eq-zeta-sq-div","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $f, g : \\mathbb{N} \\to \\mathbb{C}$ and let $\\sigma_0 < 1/2$ be a real number. Suppose that the Dirichlet series $G(s) = \\sum_{n \\ge 1} g(n) n^{-s}$ converges absolutely for $\\operatorname{Re} s > \\sigma_0$, that $G(1) \\ne 0$, and that for every complex $s$ with $\\operatorname{Re} s > 1$ the Dirichlet series $F(s) = \\sum_{n \\ge 1} f(n) n^{-s}$ converges absolutely and $F(s) = \\zeta(s)^2 G(s) / \\zeta(2s)$, where\u2026","title":"Partial-sum asymptotics for a Riemann zeta quotient times a Dirichlet series"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-divisor-count-sum-asymptoti","kind":"node","lean":"MetaMathlibExt.squareSplitDivisorCount_sum_asymptotic","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-divisor-count-sum-asymptoti","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $\\tau^\\star(n) = |A(n)|$ be the number of elements of the square-splitting divisor set $A(n)$, let $\\gamma$ be the Euler\u2013Mascheroni constant, and let $A_\\tau = \\prod_p (1 - 1/(p+1)^2)$ and $B_\\tau = 6 \\sum_p \\log p / ((p-1)(p+1)(p+2))$, where $p$ runs over the primes. Then, as the real number $x$ tends to infinity, $\\sum_{1 \\le n \\le x} \\tau^\\star(n) - A_\\tau x \\log x - A_\\tau (2\\gamma - 1 - B_\\tau - c) x$ is\u2026","title":"Average order of the square-splitting divisor function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-moebius-support-density","kind":"node","lean":"MetaMathlibExt.squareSplitMoebius_support_density","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-moebius-support-density","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $\\star$ be the square-splitting convolution, $\\mathbf{1}$ the constant function $1$ on the positive integers, $\\varepsilon$ the identity element ($\\varepsilon(1) = 1$ and $\\varepsilon(n) = 0$ for $n \\ne 1$), and let $\\mu_\\star$ be any complex-valued arithmetic function with $\\mathbf{1} \\star \\mu_\\star = \\varepsilon$. Then for every $n \\ge 1$, $\\mu_\\star(n) \\ne 0$ if and only if every prime exponent of $n$ is at\u2026","title":"Support and density of the square-splitting M\u00f6bius function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/square-split-sigma-multiplicative-euler","kind":"node","lean":"MetaMathlibExt.squareSplitSigma_multiplicative_eulerProduct","parent":8631,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution/index.html#square-split-sigma-multiplicative-euler","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/square-splitting-convolution","status":"can_prove","summary":"Let $\\alpha \\in \\mathbb{C}$, let $\\sigma_\\beta(m) = \\sum_{d \\mid m} d^\\beta$ for $\\beta \\in \\mathbb{C}$ (complex powers), and let $f(n) = \\sigma_\\alpha(u(n))\\,\\sigma_{2\\alpha}(v(n))$, where $u(n)$ is the product of the primes dividing $n$ to an odd power and $v(n)$ is the positive integer with $n = u(n)\\,v(n)^2$. Then $f(1) = 1$ and $f(mn) = f(m)\\,f(n)$ for all coprime $m$ and $n$. For every prime $p$ and every\u2026","title":"A square-splitting divisor function and its Euler product"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","title":"Tuples coprime along the edges of a graph"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-indicator","kind":"node","lean":"MetaMathlibExt.coprimalityIndicator","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-indicator","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"fully_proved","summary":"Let $E$ be a set of ordered pairs $(i, j)$ of indices in $\\{0, \\dots, k-1\\}$, the edge set of a graph $G$ on these $k$ vertices. For $n = (n_0, \\dots, n_{k-1}) \\in \\mathbb{N}^k$, the coprimality indicator $\\delta_G(n)$ is the natural number $1$ if $\\gcd(n_i, n_j) = 1$ for every $(i, j) \\in E$, and $0$ otherwise.","title":"Coprimality indicator of a graph"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-indicator-empty","kind":"node","lean":"MetaMathlibExt.coprimalityIndicator_empty","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-indicator-empty","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"fully_proved","summary":"Let $k \\in \\mathbb{N}$ and let $G$ be the graph on $k$ vertices with no edges, and let $\\delta_G$ be its coprimality indicator. Then $\\delta_G(n) = 1$ for every $n = (n_0, \\dots, n_{k-1}) \\in \\mathbb{N}^k$.","title":"Coprimality indicator of the empty graph"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-indicator-small-values","kind":"node","lean":"MetaMathlibExt.coprimalityIndicator_small_values","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-indicator-small-values","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"fully_proved","summary":"Let $G$ be the graph on the two vertices $0, 1$ with the single edge $(0, 1)$, with coprimality indicator $\\delta_G$. Then $\\delta_G(2, 3) = 1$ and $\\delta_G(2, 4) = 0$.","title":"Coprimality indicator of a single edge: examples"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprime-pair-count","kind":"node","lean":"MetaMathlibExt.coprimePairCount","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprime-pair-count","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"fully_proved","summary":"For $n = (n_0, \\dots, n_{k-1}) \\in \\mathbb{N}^k$, the number of coprime pairs of $n$ is the number of index pairs $(i, j)$ with $0 \\le i < j < k$ and $\\gcd(n_i, n_j) = 1$.","title":"Number of coprime pairs"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprime-pair-count-small-values","kind":"node","lean":"MetaMathlibExt.coprimePairCount_small_values","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprime-pair-count-small-values","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"fully_proved","summary":"The triple $(2, 3, 4)$ has exactly two coprime pairs of components, and the triple $(6, 10, 15)$ has none, that is, its components are pairwise non-coprime.","title":"Number of coprime pairs: examples"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-density","kind":"node","lean":"MetaMathlibExt.coprimalityDensity","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-density","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"fully_proved","summary":"Let $k$ be a natural number and let $E$ be a set of ordered pairs $(a, b)$ of vertices in $\\{1, \\dots, k\\}$, the edge set of a graph $G$. For $F \\subseteq E$ let $v(F)$ be the number of distinct vertices that occur in some pair of $F$. The coprimality density of $G$ is the real number","title":"Coprimality density of a graph"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-density-empty","kind":"node","lean":"MetaMathlibExt.coprimalityDensity_empty","parent":8655,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coprimality density of the empty graph is "],["math","\\(1\\)"]],"source":"The coprimality density of the empty graph is $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-density-empty","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"fully_proved","summary":"For every natural number $k$, the coprimality density of the graph on $\\{1, \\dots, k\\}$ with no edges equals $1$.","title":"The coprimality density of the empty graph is $1$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-graph-count-asymp","kind":"node","lean":"MetaMathlibExt.coprimality_graph_count_asymp","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-graph-count-asymp","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$ and let $E$ be a set of pairs $(i, j)$ with $0 \\le i < j < k$, the edge set of a graph $G$ on the vertices $0, \\dots, k-1$. Let $d_G$ be the maximum degree of $G$, that is, the largest number of pairs in $E$ that contain a single vertex; let $\\delta_G$ be the coprimality indicator and $A_G$ the coprimality density of $G$. Then there is a real constant $C$ such that for every integer $N \\ge 2$,","title":"Asymptotic count of tuples coprime along the edges of a graph"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-density-eq-independent-set-p","kind":"node","lean":"MetaMathlibExt.coprimalityDensity_eq_independent_set_product","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-density-eq-independent-set-p","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k$ be a natural number, let $E$ be a set of pairs $(a, b)$ with $1 \\le a < b \\le k$, let $G$ be the graph on $\\{1, \\dots, k\\}$ with edge set $E$, and let $A_G$ be its coprimality density. For $0 \\le m \\le k$, let $i_m(G)$ be the number of independent sets of $G$ of size $m$, that is, of sets $S \\subseteq \\{1, \\dots, k\\}$ with $|S| = m$ such that no edge of $E$ has both endpoints in $S$. Then","title":"Hu's independent-set formula for the coprimality density"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-indicator-mul","kind":"node","lean":"MetaMathlibExt.coprimalityIndicator_mul","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-indicator-mul","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $G$ be a graph on the vertices $0, \\dots, k-1$ given by an arbitrary set $E$ of ordered pairs of vertices, with coprimality indicator $\\delta_G$. If $m, n \\in \\mathbb{N}^k$ satisfy $\\gcd(m_0 m_1 \\cdots m_{k-1}, n_0 n_1 \\cdots n_{k-1}) = 1$, then $\\delta_G(m_0 n_0, \\dots, m_{k-1} n_{k-1}) = \\delta_G(m)\\, \\delta_G(n)$.","title":"Multiplicativity of the coprimality indicator"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/exactly-coprime-pairs-inclusion-exclusio","kind":"node","lean":"MetaMathlibExt.exactly_coprime_pairs_inclusion_exclusion","parent":8655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inclusion\u2013exclusion for tuples with exactly "],["math","\\(r\\)"],["text"," coprime pairs"]],"source":"Inclusion\u2013exclusion for tuples with exactly $r$ coprime pairs"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#exactly-coprime-pairs-inclusion-exclusio","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$, $r \\ge 0$ and $n \\in \\mathbb{N}^k$, and let $\\beta_r(n)$ be $1$ if $n$ has exactly $r$ coprime pairs and $0$ otherwise. For a set $E$ of pairs $(a, b)$ with $0 \\le a < b < k$, let $\\delta_E$ be the coprimality indicator of the graph with edge set $E$. Then, as integers,","title":"Inclusion\u2013exclusion for tuples with exactly $r$ coprime pairs"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/at-least-coprime-pairs-inclusion-exclusi","kind":"node","lean":"MetaMathlibExt.at_least_coprime_pairs_inclusion_exclusion","parent":8655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inclusion\u2013exclusion for at least "],["math","\\(r\\)"],["text"," coprime pairs"]],"source":"Inclusion\u2013exclusion for at least $r$ coprime pairs"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#at-least-coprime-pairs-inclusion-exclusi","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$ and $r \\ge 1$, and let $n = (n_1, \\dots, n_k)$ be a tuple of natural numbers (zero allowed). Let $\\beta'_r(n)$ be $1$ if $\\gcd(n_a, n_b) = 1$ for at least $r$ pairs $a < b$, and $0$ otherwise. For a set $E$ of pairs $(a, b)$ with $1 \\le a < b \\le k$, let $\\delta_E(n)$ be $1$ if $\\gcd(n_a, n_b) = 1$ for every $(a, b) \\in E$, and $0$ otherwise. Then","title":"Inclusion\u2013exclusion for at least $r$ coprime pairs"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-dirichlet-series-eq","kind":"node","lean":"MetaMathlibExt.coprimality_dirichlet_series_eq","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-dirichlet-series-eq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$, let $E$ be a set of pairs $(a, b)$ with $1 \\le a < b \\le k$, let $G$ be the graph on $\\{1, \\dots, k\\}$ with edge set $E$, and let $J$ be a vertex cover of $G$, that is, every edge of $E$ has an endpoint in $J$. Let $\\delta_G(n_1, \\dots, n_k)$ be $1$ if $\\gcd(n_a, n_b) = 1$ for every $(a, b) \\in E$, and $0$ otherwise. For $L \\subseteq J$ let $N(L)$ be the set of vertices adjacent to some vertex of\u2026","title":"Dirichlet series of the coprimality indicator"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-euler-factor-summable","kind":"node","lean":"MetaMathlibExt.coprimality_euler_factor_summable","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-euler-factor-summable","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$, let $E$ be a set of pairs $(a, b)$ with $1 \\le a < b \\le k$, let $G$ be the graph on $\\{1, \\dots, k\\}$ with edge set $E$, and let $J$ be a vertex cover of $G$, that is, every edge of $E$ has an endpoint in $J$. For $L \\subseteq J$ let $N(L)$ be the set of vertices adjacent to some vertex of $L$. For complex numbers $s = (s_1, \\dots, s_k)$ and a prime $p$ put","title":"Absolute convergence of the vertex-cover Euler product"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-density-eq-tsum-moebius-conv","kind":"node","lean":"MetaMathlibExt.coprimalityDensity_eq_tsum_moebius_conv","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-density-eq-tsum-moebius-conv","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$, let $E$ be a set of pairs $(a, b)$ with $1 \\le a < b \\le k$, let $G$ be the graph on $\\{1, \\dots, k\\}$ with edge set $E$, and let $A_G$ be its coprimality density. Let $\\delta_G(n_1, \\dots, n_k)$ be $1$ if $\\gcd(n_a, n_b) = 1$ for every $(a, b) \\in E$, and $0$ otherwise, and for a tuple $n = (n_1, \\dots, n_k)$ of positive integers let","title":"M\u00f6bius-convolution series for the coprimality density"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-density-eq-prime-power-produ","kind":"node","lean":"MetaMathlibExt.coprimalityDensity_eq_prime_power_product","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-density-eq-prime-power-produ","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$, let $E$ be a set of pairs $(a, b)$ with $1 \\le a < b \\le k$, let $G$ be the graph on $\\{1, \\dots, k\\}$ with edge set $E$, and let $A_G$ be its coprimality density. For natural numbers $n_1, \\dots, n_k$ let $\\delta_G(n_1, \\dots, n_k)$ be $1$ if $\\gcd(n_a, n_b) = 1$ for every $(a, b) \\in E$, and $0$ otherwise. Then","title":"Prime-power Euler product for the coprimality density"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/coprimality-density-eq-vertex-cover-prod","kind":"node","lean":"MetaMathlibExt.coprimalityDensity_eq_vertex_cover_product","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#coprimality-density-eq-vertex-cover-prod","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$, let $E$ be a set of pairs $(a, b)$ with $1 \\le a < b \\le k$, let $G$ be the graph on $\\{1, \\dots, k\\}$ with edge set $E$, and let $A_G$ be its coprimality density. Let $J \\subseteq \\{1, \\dots, k\\}$ be a vertex cover of $G$, that is, every edge of $E$ has an endpoint in $J$. For $L \\subseteq J$ let $N(L)$ be the set of vertices adjacent to some vertex of $L$. Then","title":"Vertex-cover Euler product for the coprimality density"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/exactly-coprime-pairs-count-asymp","kind":"node","lean":"MetaMathlibExt.exactly_coprime_pairs_count_asymp","parent":8655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic count of tuples with exactly "],["math","\\(r\\)"],["text"," coprime pairs"]],"source":"Asymptotic count of tuples with exactly $r$ coprime pairs"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#exactly-coprime-pairs-count-asymp","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$ and $r \\ge 0$. For a set $E$ of pairs $(a, b)$ with $0 \\le a < b < k$, let $A_E$ be the coprimality density of the graph with edge set $E$, and put","title":"Asymptotic count of tuples with exactly $r$ coprime pairs"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/at-least-coprime-pairs-count-asymp","kind":"node","lean":"MetaMathlibExt.at_least_coprime_pairs_count_asymp","parent":8655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic count of "],["math","\\(k\\)"],["text","-tuples with at least "],["math","\\(r\\)"],["text"," coprime pairs"]],"source":"Asymptotic count of $k$-tuples with at least $r$ coprime pairs"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#at-least-coprime-pairs-count-asymp","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$ and $r \\ge 1$. For a set $E$ of pairs $(a, b)$ with $1 \\le a < b \\le k$, let $A_E$ be the coprimality density of the graph on $\\{1, \\dots, k\\}$ with edge set $E$, and put","title":"Asymptotic count of $k$-tuples with at least $r$ coprime pairs"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/pairwise-non-coprime-density-eq-tsum-moe","kind":"node","lean":"MetaMathlibExt.pairwise_non_coprime_density_eq_tsum_moebius_conv","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#pairwise-non-coprime-density-eq-tsum-moe","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $k \\ge 2$. For a $k$-tuple $n = (n_1, \\dots, n_k)$ of positive integers let $\\beta(n) = 1$ if the number of coprime pairs of $n$ is $0$, that is, if $\\gcd(n_i, n_j) \\ne 1$ for all $1 \\le i < j \\le k$, and $\\beta(n) = 0$ otherwise. Let $(\\mu * \\beta)(n) = \\sum_{d_1 \\mid n_1, \\dots, d_k \\mid n_k} \\mu(d_1) \\cdots \\mu(d_k)\\, \\beta(n_1/d_1, \\dots, n_k/d_k)$, where $\\mu$ is the M\u00f6bius function. Then","title":"The M\u00f6bius series for pairwise non-coprime tuples equals the inclusion\u2013exclusion sum"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/euler-product-mean-value-of-coprime-prod","kind":"node","lean":"MetaMathlibExt.euler_product_mean_value_of_coprime_prod","parent":8655,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph/index.html#euler-product-mean-value-of-coprime-prod","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/tuples-coprime-along-the-edges-of-a-graph","status":"can_prove","summary":"Let $f : \\mathbb{N}^k \\to \\mathbb{C}$ satisfy $f(a_0 b_0, \\dots, a_{k-1} b_{k-1}) = f(a) f(b)$ for all $a, b \\in \\mathbb{N}^k$ with $\\gcd(a_0 \\cdots a_{k-1}, b_0 \\cdots b_{k-1}) = 1$. Let $\\mu$ be the M\u00f6bius function, and for $n \\in \\mathbb{N}^k$ with every $n_i \\ge 1$ put $(\\mu * f)(n) = \\sum_{d} \\mu(d_0) \\cdots \\mu(d_{k-1}) f(n_0/d_0, \\dots, n_{k-1}/d_{k-1})$, the sum over all $d \\in \\mathbb{N}^k$ with each $d_i$\u2026","title":"Euler product for the mean value of a multiplicative function of several variables"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","title":"Unitary convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/unitary-divisors-prime-pow","kind":"node","lean":"Nat.unitaryDivisors_prime_pow","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#unitary-divisors-prime-pow","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"For every prime $p$ and every natural number $m$, the unitary divisors of $p^m$, that is, the divisors $d$ of $p^m$ with $\\gcd(d, p^m/d) = 1$, are exactly $1$ and $p^m$.","title":"Unitary divisors of a prime power"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/mul-mem-unitary-divisors-mul","kind":"node","lean":"Nat.mul_mem_unitaryDivisors_mul","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#mul-mem-unitary-divisors-mul","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $m$ and $n$ be coprime natural numbers. If $a$ is a unitary divisor of $m$ and $b$ is a unitary divisor of $n$, then $ab$ is a unitary divisor of $mn$.","title":"Products of unitary divisors of coprime numbers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/gcd-mem-unitary-divisors-of-mem-mul","kind":"node","lean":"Nat.gcd_mem_unitaryDivisors_of_mem_mul","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#gcd-mem-unitary-divisors-of-mem-mul","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $m$ and $n$ be coprime natural numbers. If $d$ is a unitary divisor of $mn$, then $\\gcd(d, m)$ is a unitary divisor of $m$.","title":"Gcd with a coprime factor preserves unitary divisors"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/pow-omega","kind":"node","lean":"MetaMathlibExt.powOmega","parent":8677,"reading_title":{"format":"inline-parts-v1","parts":[["text","The arithmetic function "],["math","\\(c^{\\Omega}\\)"]],"source":"The arithmetic function $c^{\\Omega}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#pow-omega","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a monoid with zero and let $c \\in R$. This is the arithmetic function $c^{\\Omega}$ with values in $R$ given by $c^{\\Omega}(0) = 0$ and $c^{\\Omega}(n) = c^{\\Omega(n)}$ for $n \\ge 1$, where $\\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. It is completely multiplicative.","title":"The arithmetic function $c^{\\Omega}$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/pow-omega-apply","kind":"node","lean":"MetaMathlibExt.powOmega_apply","parent":8677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of "],["math","\\(c^{\\Omega}\\)"],["text"," at a positive integer"]],"source":"Value of $c^{\\Omega}$ at a positive integer"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#pow-omega-apply","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a monoid with zero, let $c \\in R$, and let $c^{\\Omega}$ be the arithmetic function with values in $R$ that is $0$ at $0$ and $c^{\\Omega(n)}$ at each $n \\ge 1$, where $\\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. For every natural number $n \\ne 0$, its value at $n$ is $c^{\\Omega(n)}$.","title":"Value of $c^{\\Omega}$ at a positive integer"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/pow-omega-one","kind":"node","lean":"MetaMathlibExt.powOmega_one","parent":8677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of "],["math","\\(c^{\\Omega}\\)"],["text"," at "],["math","\\(1\\)"]],"source":"Value of $c^{\\Omega}$ at $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#pow-omega-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a monoid with zero, let $c \\in R$, and let $c^{\\Omega}$ be the arithmetic function with values in $R$ that is $0$ at $0$ and $c^{\\Omega(n)}$ at each $n \\ge 1$, where $\\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. Its value at $1$ is $1$.","title":"Value of $c^{\\Omega}$ at $1$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/pow-omega-prime-pow","kind":"node","lean":"MetaMathlibExt.powOmega_prime_pow","parent":8677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of "],["math","\\(c^{\\Omega}\\)"],["text"," at prime powers"]],"source":"Value of $c^{\\Omega}$ at prime powers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#pow-omega-prime-pow","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a monoid with zero, let $c \\in R$, and let $c^{\\Omega}$ be the arithmetic function with values in $R$ that is $0$ at $0$ and $c^{\\Omega(n)}$ at each $n \\ge 1$, where $\\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. For every prime $p$ and every natural number $m$, its value at $p^m$ is $c^m$.","title":"Value of $c^{\\Omega}$ at prime powers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/is-completely-multiplicative-pow-omega","kind":"node","lean":"MetaMathlibExt.isCompletelyMultiplicative_powOmega","parent":8677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Complete multiplicativity of "],["math","\\(c^{\\Omega}\\)"]],"source":"Complete multiplicativity of $c^{\\Omega}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#is-completely-multiplicative-pow-omega","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a monoid with zero and $c \\in R$. Let $c^{\\Omega}$ be the arithmetic function $n \\mapsto c^{\\Omega(n)}$ for $n \\ge 1$, where $\\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity. Then $c^{\\Omega}$ is completely multiplicative: for all natural numbers $m$ and $n$, its value at $mn$ is the product of its values at $m$ and at $n$.","title":"Complete multiplicativity of $c^{\\Omega}$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/is-multiplicative-pow-omega","kind":"node","lean":"MetaMathlibExt.isMultiplicative_powOmega","parent":8677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicativity of "],["math","\\(c^{\\Omega}\\)"]],"source":"Multiplicativity of $c^{\\Omega}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#is-multiplicative-pow-omega","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a monoid with zero and $c \\in R$. The arithmetic function $n \\mapsto c^{\\Omega(n)}$ for $n \\ge 1$, where $\\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity, is multiplicative.","title":"Multiplicativity of $c^{\\Omega}$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/unitary-conv","kind":"node","lean":"MetaMathlibExt.unitaryConv","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#unitary-conv","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a semiring. The unitary convolution of two arithmetic functions $f$ and $g$ with values in $R$ is the arithmetic function $f \\sqcup g$ given by $(f \\sqcup g)(n) = \\sum_{d \\parallel n} f(d)\\, g(n/d)$, where the sum runs over the unitary divisors $d$ of $n$: for $n \\ge 1$ these are the divisors $d$ of $n$ with $\\gcd(d, n/d) = 1$, and $0$ has no unitary divisors. In particular $(f \\sqcup g)(0) = 0$.","title":"Unitary convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/unitary-conv-apply","kind":"node","lean":"MetaMathlibExt.unitaryConv_apply","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#unitary-conv-apply","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a semiring and let $f$ and $g$ be arithmetic functions with values in $R$. For every natural number $n$, their unitary convolution $f \\sqcup g$ satisfies $(f \\sqcup g)(n) = \\sum_{d \\parallel n} f(d)\\, g(n/d)$, where the sum runs over the unitary divisors $d$ of $n$: for $n \\ge 1$ these are the divisors $d$ of $n$ with $\\gcd(d, n/d) = 1$, and for $n = 0$ there are none, so the sum is $0$.","title":"Unitary convolution as a sum over unitary divisors"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/unitary-conv-one","kind":"node","lean":"MetaMathlibExt.unitaryConv_one","parent":8677,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unitary convolution at "],["math","\\(1\\)"]],"source":"Unitary convolution at $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#unitary-conv-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a semiring and let $f$ and $g$ be arithmetic functions with values in $R$. Their unitary convolution $f \\sqcup g$, defined by $(f \\sqcup g)(n) = \\sum_{d \\parallel n} f(d)\\, g(n/d)$ over the unitary divisors $d$ of $n$, satisfies $(f \\sqcup g)(1) = f(1)\\, g(1)$.","title":"Unitary convolution at $1$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/unitary-conv-prime-pow","kind":"node","lean":"MetaMathlibExt.unitaryConv_prime_pow","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#unitary-conv-prime-pow","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a semiring and let $f$ and $g$ be arithmetic functions with values in $R$. For every prime $p$ and every natural number $m \\ge 1$, their unitary convolution $f \\sqcup g$ satisfies $(f \\sqcup g)(p^m) = f(1)\\, g(p^m) + f(p^m)\\, g(1)$.","title":"Unitary convolution at prime powers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/is-multiplicative-unitary-conv","kind":"node","lean":"MetaMathlibExt.isMultiplicative_unitaryConv","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#is-multiplicative-unitary-conv","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a commutative semiring. If $f$ and $g$ are multiplicative arithmetic functions with values in $R$, then their unitary convolution $f \\sqcup g$, given by $(f \\sqcup g)(n) = \\sum_{d} f(d)\\, g(n/d)$ with $d$ running over the unitary divisors of $n$, is multiplicative.","title":"The unitary convolution of multiplicative functions is multiplicative"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/multiplicative-of-prime-powers","kind":"node","lean":"MetaMathlibExt.multiplicativeOfPrimePowers","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#multiplicative-of-prime-powers","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero and let $g : \\mathbb{N} \\to R$. This is the arithmetic function $f$ with values in $R$ given by $f(0) = 0$ and $f(n) = \\prod_{p \\mid n} g(v_p(n))$ for $n \\ge 1$, where the product runs over the primes $p$ dividing $n$ and $v_p(n)$ is the exponent of $p$ in $n$. It is multiplicative, it takes the value $g(m)$ at every prime power $p^m$ with $m \\ge 1$, and its value at $1$ is\u2026","title":"Multiplicative function with prescribed values at prime powers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/multiplicative-of-prime-powers-apply","kind":"node","lean":"MetaMathlibExt.multiplicativeOfPrimePowers_apply","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#multiplicative-of-prime-powers-apply","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero, let $g : \\mathbb{N} \\to R$, and let $n \\ne 0$ be a natural number. The multiplicative function with value $g(m)$ at every prime power $p^m$ with $m \\ge 1$ takes the value $\\prod_{q \\mid n} g(v_q(n))$ at $n$, where $q$ runs over the primes dividing $n$ and $v_q(n)$ is the exponent of $q$ in the factorization of $n$.","title":"Value of the function with prescribed prime-power values"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/multiplicative-of-prime-powers-one","kind":"node","lean":"MetaMathlibExt.multiplicativeOfPrimePowers_one","parent":8677,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function with prescribed prime-power values is "],["math","\\(1\\)"],["text"," at "],["math","\\(1\\)"]],"source":"The function with prescribed prime-power values is $1$ at $1$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#multiplicative-of-prime-powers-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero and $g : \\mathbb{N} \\to R$. The multiplicative function with prime-power values $g$, given by $n \\mapsto \\prod_{q \\mid n} g(v_q(n))$ for $n \\ge 1$ (with $q$ running over the primes dividing $n$ and $v_q(n)$ the exponent of $q$ in $n$), takes the value $1$ at $n = 1$, whatever $g(0)$ is.","title":"The function with prescribed prime-power values is $1$ at $1$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/multiplicative-of-prime-powers-prime-pow","kind":"node","lean":"MetaMathlibExt.multiplicativeOfPrimePowers_prime_pow","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#multiplicative-of-prime-powers-prime-pow","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero and $g : \\mathbb{N} \\to R$. For every prime $p$ and every $m \\ge 1$, the multiplicative function with prime-power values $g$, given by $n \\mapsto \\prod_{q \\mid n} g(v_q(n))$ for $n \\ge 1$ (with $q$ running over the primes dividing $n$ and $v_q(n)$ the exponent of $q$ in $n$), takes the value $g(m)$ at $p^m$.","title":"The function with prescribed prime-power values at a prime power"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/is-multiplicative-multiplicative-of-prim","kind":"node","lean":"MetaMathlibExt.isMultiplicative_multiplicativeOfPrimePowers","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#is-multiplicative-multiplicative-of-prim","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero and $g : \\mathbb{N} \\to R$. The arithmetic function $n \\mapsto \\prod_{q \\mid n} g(v_q(n))$ for $n \\ge 1$, where $q$ runs over the primes dividing $n$ and $v_q(n)$ is the exponent of $q$ in $n$, is multiplicative.","title":"Multiplicativity of the function with prescribed prime-power values"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/eq-multiplicative-of-prime-powers","kind":"node","lean":"MetaMathlibExt.eq_multiplicativeOfPrimePowers","parent":8677,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution/index.html#eq-multiplicative-of-prime-powers","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/unitary-convolution","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero, let $g : \\mathbb{N} \\to R$, and let $f$ be a multiplicative arithmetic function with values in $R$. If $f(p^m) = g(m)$ for every prime $p$ and every $m \\ge 1$, then $f$ is the multiplicative function with prime-power values $g$, that is, the arithmetic function $n \\mapsto \\prod_{q \\mid n} g(v_q(n))$ for $n \\ge 1$, where $q$ runs over the primes dividing $n$ and $v_q(n)$ is\u2026","title":"A multiplicative function with prescribed prime-power values"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity","status":"fully_proved","title":"Vaughan's identity"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity/vaughan-identity-general","kind":"node","lean":"MetaMathlibExt.vaughan_identity_general","parent":8698,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity/index.html#vaughan-identity-general","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\mu$ and $\\Lambda$ be arithmetic functions with values in $R$ (so $\\mu(0) = \\Lambda(0) = 0$), and let $L : \\mathbb{N} \\to R$ be any function. Suppose that for every natural number $m$, $\\sum_{d \\mid m} \\mu(d)$ is $1$ if $m = 1$ and $0$ otherwise, and $\\Lambda(m) = \\sum_{d \\mid m} \\mu(d) \\, L(m/d)$, where both sums run over the $d$ with $0 \\le d \\le m$ and $d \\mid m$. Then for all\u2026","title":"Vaughan's identity in a commutative ring"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity/vaughan-identity","kind":"node","lean":"MetaMathlibExt.vaughan_identity","parent":8698,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity/index.html#vaughan-identity","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/vaughan-s-identity","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\mu$ and $\\Lambda$ be arithmetic functions with values in $R$ (so $\\mu(0) = \\Lambda(0) = 0$), and let $L : \\mathbb{N} \\to R$ be any function. Suppose that for every natural number $m$, $\\sum_{d \\mid m} \\mu(d)$ is $1$ if $m = 1$ and $0$ otherwise, and $\\Lambda(m) = \\sum_{d \\mid m} \\mu(d) \\, L(m/d)$, where both sums run over the $d$ with $0 \\le d \\le m$ and $d \\mid m$. Then for all\u2026","title":"Vaughan's identity"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","title":"Weighted sums of Dirichlet convolutions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/b-free-indicator","kind":"node","lean":"MetaMathlibExt.bFreeIndicator","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Indicator of "],["math","\\(B_\\kappa\\)"],["text","-free integers"]],"source":"Indicator of $B_\\kappa$-free integers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#b-free-indicator","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"fully_proved","summary":"Let $\\kappa$ be a function from the natural numbers to the natural numbers (only its values $\\kappa(p)$ at primes $p$ matter) and let $B_\\kappa$ be the set of prime powers $p^{\\kappa(p)}$. A natural number $n$ is $B_\\kappa$-free if $n \\ge 1$ and no $p^{\\kappa(p)}$ with $p$ prime divides $n$. The indicator $q_{B_\\kappa}$ is the integer-valued arithmetic function with $q_{B_\\kappa}(n) = 1$ if $n$ is $B_\\kappa$-free\u2026","title":"Indicator of $B_\\kappa$-free integers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/b-free-indicator-apply","kind":"node","lean":"MetaMathlibExt.bFreeIndicator_apply","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of the "],["math","\\(B_\\kappa\\)"],["text","-free indicator"]],"source":"Values of the $B_\\kappa$-free indicator"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#b-free-indicator-apply","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"fully_proved","summary":"Let $\\kappa$ be a function from the natural numbers to the natural numbers. For every natural number $n$, the indicator $q_{B_\\kappa}$ of $B_\\kappa$-free integers satisfies $q_{B_\\kappa}(n) = 1$ if $n \\ne 0$ and no $p^{\\kappa(p)}$ with $p$ prime divides $n$, and $q_{B_\\kappa}(n) = 0$ otherwise.","title":"Values of the $B_\\kappa$-free indicator"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/b-free-indicator-one","kind":"node","lean":"MetaMathlibExt.bFreeIndicator_one","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","The integer "],["math","\\(1\\)"],["text"," is "],["math","\\(B_\\kappa\\)"],["text","-free"]],"source":"The integer $1$ is $B_\\kappa$-free"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#b-free-indicator-one","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"fully_proved","summary":"Let $\\kappa$ be a function from the natural numbers to the natural numbers with $\\kappa(p) \\ge 1$ for every prime $p$. Then the indicator $q_{B_\\kappa}$ of $B_\\kappa$-free integers satisfies $q_{B_\\kappa}(1) = 1$.","title":"The integer $1$ is $B_\\kappa$-free"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/b-free-indicator-prime-pow","kind":"node","lean":"MetaMathlibExt.bFreeIndicator_prime_pow","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","The prime power "],["math","\\(p^{\\kappa(p)}\\)"],["text"," is not "],["math","\\(B_\\kappa\\)"],["text","-free"]],"source":"The prime power $p^{\\kappa(p)}$ is not $B_\\kappa$-free"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#b-free-indicator-prime-pow","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"fully_proved","summary":"Let $\\kappa$ be a function from the natural numbers to the natural numbers and let $p$ be a prime. Then the indicator $q_{B_\\kappa}$ of $B_\\kappa$-free integers satisfies $q_{B_\\kappa}(p^{\\kappa(p)}) = 0$.","title":"The prime power $p^{\\kappa(p)}$ is not $B_\\kappa$-free"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/sum-pow-mul-pow-add-eq","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_pow_add_eq","parent":8701,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#sum-pow-mul-pow-add-eq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $R$ be a commutative ring, let $a, b \\in R$ and let $\\tau, \\xi$ be natural numbers. Then","title":"Product formula for homogeneous sums of powers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/dirichlet-convolution-mul-eq-sum-gcd","kind":"node","lean":"MetaMathlibExt.dirichletConvolution_mul_eq_sum_gcd","parent":8701,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#dirichlet-convolution-mul-eq-sum-gcd","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $u$ and $v$ be complex-valued completely multiplicative arithmetic functions, meaning $u(mn) = u(m) u(n)$ and $v(mn) = v(m) v(n)$ for all natural numbers $m$ and $n$. Let $u * v$ be their Dirichlet convolution and $\\mu$ the M\u00f6bius function. Then for all natural numbers $r$ and $s$,","title":"Gcd expansion of a convolution of completely multiplicative functions at a product"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/sum-dirichlet-convolution-mul-eq","kind":"node","lean":"MetaMathlibExt.sum_dirichletConvolution_mul_eq","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of a Dirichlet convolution over multiples of "],["math","\\(r\\)"]],"source":"Sums of a Dirichlet convolution over multiples of $r$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#sum-dirichlet-convolution-mul-eq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $u$ and $v$ be completely multiplicative arithmetic functions with complex values, let $u * v$ be their Dirichlet convolution, let $\\mu$ be the M\u00f6bius function, and for real $y$ put $D(y) = \\sum_{1 \\le j \\le y} (u * v)(j)$. Then for every natural number $r$ and every real $x$,","title":"Sums of a Dirichlet convolution over multiples of $r$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/sum-mul-dirichlet-convolution-eq-sum-moe","kind":"node","lean":"MetaMathlibExt.sum_mul_dirichletConvolution_eq_sum_moebius","parent":8701,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#sum-mul-dirichlet-convolution-eq-sum-moe","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $w$, $u$ and $v$ be arithmetic functions with complex values, let $*$ denote Dirichlet convolution, let $\\mu$ be the M\u00f6bius function and put $g = w * \\mu$. Then for every real $x$,","title":"M\u00f6bius expansion of weighted sums of a Dirichlet convolution"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/weighted-sum-dirichlet-convolution-eq","kind":"node","lean":"MetaMathlibExt.weighted_sum_dirichletConvolution_eq","parent":8701,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#weighted-sum-dirichlet-convolution-eq","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $u$ and $v$ be completely multiplicative arithmetic functions with complex values, let $w$ be any arithmetic function with complex values, let $*$ denote Dirichlet convolution, let $\\mu$ be the M\u00f6bius function, put $g = w * \\mu$, and for real $y$ put $D(y) = \\sum_{1 \\le j \\le y} (u * v)(j)$. Then for every real $x$,","title":"Weighted summation of Dirichlet convolutions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/b-free-indicator-mul-moebius-apply","kind":"node","lean":"MetaMathlibExt.bFreeIndicator_mul_moebius_apply","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(B_\\kappa\\)"],["text","-free indicator convolved with the M\u00f6bius function"]],"source":"The $B_\\kappa$-free indicator convolved with the M\u00f6bius function"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#b-free-indicator-mul-moebius-apply","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $\\kappa$ be a function from the natural numbers to the natural numbers with $\\kappa(p) \\ge 1$ for every prime $p$, let $q_{B_\\kappa}$ be the indicator of $B_\\kappa$-free integers, let $\\mu$ be the M\u00f6bius function and let $*$ denote Dirichlet convolution. Let $n \\ge 1$ be a natural number with prime factorization $n = \\prod_p p^{e_p}$. If $\\kappa(p)$ divides $e_p$ for every prime $p$, then\u2026","title":"The $B_\\kappa$-free indicator convolved with the M\u00f6bius function"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/b-free-coprime-indicator-mul-moebius","kind":"node","lean":"MetaMathlibExt.bFreeCoprimeIndicator_mul_moebius","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(B_\\kappa\\)"],["text","-free indicator twisted by the principal character, convolved with "],["math","\\(\\mu\\)"]],"source":"The $B_\\kappa$-free indicator twisted by the principal character, convolved with $\\mu$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#b-free-coprime-indicator-mul-moebius","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $\\kappa$ be a function from the natural numbers to the natural numbers with $\\kappa(p) \\ge 1$ for every prime $p$, let $\\xi$ be a natural number, let $q_{B_\\kappa}$ be the indicator of $B_\\kappa$-free integers and let $\\mu$ be the M\u00f6bius function. Put $w(d) = q_{B_\\kappa}(d)$ if $\\gcd(d, \\xi) = 1$ and $w(d) = 0$ otherwise, so that $w = q_{B_\\kappa} \\chi_\\xi$ with $\\chi_\\xi$ the principal character modulo $\\xi$,\u2026","title":"The $B_\\kappa$-free indicator twisted by the principal character, convolved with $\\mu$"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/k-free-divisor-sum-asymptotic","kind":"node","lean":"MetaMathlibExt.kFree_divisor_sum_asymptotic","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic for the divisor function summed over "],["math","\\(k\\)"],["text","-free integers"]],"source":"Asymptotic for the divisor function summed over $k$-free integers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#k-free-divisor-sum-asymptotic","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number, let $q_k$ be the indicator of $k$-free integers (the indicator of $B_\\kappa$-free integers for the constant function $\\kappa = k$), let $\\tau(n)$ be the number of positive divisors of $n$, let $\\gamma$ be the Euler\u2013Mascheroni constant, let $\\Delta(x) = \\sum_{1 \\le n \\le x} \\tau(n) - x \\log x - (2\\gamma - 1) x$ be the Dirichlet divisor error term, and put","title":"Asymptotic for the divisor function summed over $k$-free integers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/b-free-divisor-sum-asymptotic","kind":"node","lean":"MetaMathlibExt.bFree_divisor_sum_asymptotic","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic for the divisor function summed over "],["math","\\(B_\\kappa\\)"],["text","-free integers"]],"source":"Asymptotic for the divisor function summed over $B_\\kappa$-free integers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#b-free-divisor-sum-asymptotic","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $\\kappa$ be a function from the natural numbers to the natural numbers and let $K \\ge 2$ be the minimum of $\\kappa(p)$ over the primes $p$: $\\kappa(p) \\ge K$ for every prime $p$, with equality for some prime. Let $q_{B_\\kappa}$ be the indicator of $B_\\kappa$-free integers, $\\tau(n)$ the number of positive divisors of $n$, $\\gamma$ the Euler\u2013Mascheroni constant and\u2026","title":"Asymptotic for the divisor function summed over $B_\\kappa$-free integers"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/b-free-divisor-sum-progression-asymptoti","kind":"node","lean":"MetaMathlibExt.bFree_divisor_sum_progression_asymptotic","parent":8701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic for the divisor function summed over "],["math","\\(B_\\kappa\\)"],["text","-free integers in a progression"]],"source":"Asymptotic for the divisor function summed over $B_\\kappa$-free integers in a progression"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions/index.html#b-free-divisor-sum-progression-asymptoti","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/weighted-sums-of-dirichlet-convolutions","status":"can_prove","summary":"Let $\\kappa$ be a function from the natural numbers to the natural numbers and let $K$ be the minimum of $\\kappa(p)$ over the primes $p$: $\\kappa(p) \\ge K$ for every prime $p$, with equality for some prime. Assume $K \\ge 2$ and $K \\ne 3$. Let $a$ and $\\xi$ be natural numbers with $a < \\xi$ and $\\gcd(a, \\xi) = 1$. Let $q_{B_\\kappa}$ be the indicator of $B_\\kappa$-free integers, $\\tau(n)$ the number of positive\u2026","title":"Asymptotic for the divisor function summed over $B_\\kappa$-free integers in a progression"},{"data_url":"scopes/NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem.json","id":"NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem","kind":"scope","navigation":"module","parent":8490,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem/index.html","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem","status":"fully_proved","title":"Wintner's mean-value theorem"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem/wintner-mean-value","kind":"node","lean":"MetaMathlibExt.wintner_mean_value","parent":8716,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem/index.html#wintner-mean-value","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem","status":"fully_proved","summary":"Let $f$ be a complex-valued arithmetic function, let $\\mu$ be the M\u00f6bius function, and let $\\mu * f$ be their Dirichlet convolution, $(\\mu * f)(n) = \\sum_{d \\mid n} \\mu(d) f(n/d)$. If the series $\\sum_{n \\ge 1} |(\\mu * f)(n)|/n$ converges, then","title":"Wintner's mean-value theorem"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem/euler-product-mean-value","kind":"node","lean":"MetaMathlibExt.euler_product_mean_value","parent":8716,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem/index.html#euler-product-mean-value","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem","status":"fully_proved","summary":"Let $k$ be a natural number, write $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$, and let $f \\colon \\mathbb{N}^k \\to \\mathbb{C}$ satisfy $f(a_1 b_1, \\dots, a_k b_k) = f(a) f(b)$ for all $a, b \\in \\mathbb{N}^k$ with $\\gcd(a_i, b_i) = 1$ for every $i$. For a prime $p$ and $\\nu = (\\nu_1, \\dots, \\nu_k) \\in \\mathbb{N}^k$ write $p^\\nu = (p^{\\nu_1}, \\dots, p^{\\nu_k})$, $|\\nu| = \\nu_1 + \\dots + \\nu_k$ and","title":"Euler product for mean values of coordinatewise multiplicative functions"},{"id":"NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem/wintner-mean-value-multivariable","kind":"node","lean":"MetaMathlibExt.wintner_mean_value_multivariable","parent":8716,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem/index.html#wintner-mean-value-multivariable","scope":"NumberTheory/arithmetic-functions/multiplicative-functions/wintner-s-mean-value-theorem","status":"fully_proved","summary":"Let $k \\ge 1$ and let $f$ be a complex-valued function on $k$-tuples of natural numbers. For a $k$-tuple $n = (n_1, \\dots, n_k)$ of positive integers put","title":"Wintner's mean-value theorem in several variables"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes.json","id":"NumberTheory/arithmetic-functions/special-integer-classes","kind":"scope","parent":8068,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes","status":"can_prove","summary":"Squarefree, k-free, k-full and powerful, smooth and rough numbers, highly composite numbers, integers with prescribed prime factors.","title":"Integers with restricted factorization"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/average-card-factors-over-h-free","kind":"node","lean":"MetaMathlibExt.average_cardFactors_over_h_free","parent":8720,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/index.html#average-card-factors-over-h-free","scope":"NumberTheory/arithmetic-functions/special-integer-classes","status":"fully_proved","summary":"Let $h \\ge 2$ be an integer. Call a positive integer $n$ $h$-free if every prime occurs in the factorization of $n$ with exponent less than $h$, and write $\\Omega(n)$ for the number of prime factors of $n$ counted with multiplicity and $\\zeta$ for the Riemann zeta function. Then, as the real number $x$ tends to infinity,","title":"Average number of prime factors of h-free integers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/is-consecutive-polynomial-family","kind":"node","lean":"MetaMathlibExt.IsConsecutivePolynomialFamily","parent":8720,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/index.html#is-consecutive-polynomial-family","scope":"NumberTheory/arithmetic-functions/special-integer-classes","status":"fully_proved","summary":"Let $k$ be a natural number. A family $L_0, \\dots, L_{k-1}$ of polynomials in $\\mathbb{Z}[X]$ is consecutive when $L_{i+1} - L_i = 1$ for every $i$ with $i + 1 < k$.","title":"Consecutive polynomial family"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/same-prime-support","kind":"node","lean":"Nat.SamePrimeSupport","parent":8720,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/index.html#same-prime-support","scope":"NumberTheory/arithmetic-functions/special-integer-classes","status":"fully_proved","summary":"Natural numbers $a$ and $b$ have the same prime support when every prime $p$ divides $a$ if and only if it divides $b$.","title":"Natural numbers with the same prime divisors"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs","status":"can_prove","title":"Benelux pairs"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs/benelux-pair-first-kind-iff","kind":"node","lean":"MetaMathlibExt.benelux_pair_first_kind_iff","parent":8724,"reading_title":{"format":"inline-parts-v1","parts":[["text","Benelux pairs of the first kind below "],["math","\\(1.4 \\cdot 10^{12}\\)"]],"source":"Benelux pairs of the first kind below $1.4 \\cdot 10^{12}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs/index.html#benelux-pair-first-kind-iff","scope":"NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs","status":"can_prove","summary":"Let $m < n$ be non-negative integers with $n < 1.4 \\cdot 10^{12}$. Then $m$ and $n$ have the same prime factors and $m + 1$ and $n + 1$ have the same prime factors (that is, $(m, n)$ is a Benelux pair of the first kind) if and only if $(m, n) = (75, 1215)$ or $(m, n) = (2^k - 2, 2^{2k} - 2^{k+1})$ for some integer $k$ with $2 \\le k \\le 20$.","title":"Benelux pairs of the first kind below $1.4 \\cdot 10^{12}$"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs/benelux-pair-second-kind-iff","kind":"node","lean":"MetaMathlibExt.benelux_pair_second_kind_iff","parent":8724,"reading_title":{"format":"inline-parts-v1","parts":[["text","Benelux pairs of the second kind below "],["math","\\(1.4 \\cdot 10^{12}\\)"]],"source":"Benelux pairs of the second kind below $1.4 \\cdot 10^{12}$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs/index.html#benelux-pair-second-kind-iff","scope":"NumberTheory/arithmetic-functions/special-integer-classes/benelux-pairs","status":"can_prove","summary":"Let $m < n$ be non-negative integers with $n < 1.4 \\cdot 10^{12}$. Then $m$ and $n + 1$ have the same prime factors and $m + 1$ and $n$ have the same prime factors (that is, $(m, n)$ is a Benelux pair of the second kind) if and only if $(m, n) = (35, 4374)$ or $(m, n) = (2^k + 1, 2^{2k} + 2^{k+1})$ for some integer $k$ with $0 \\le k \\le 20$.","title":"Benelux pairs of the second kind below $1.4 \\cdot 10^{12}$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers","status":"can_prove","title":"Consecutive smooth numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.SmoothConsecutiveIntegersWanted.conjecture","parent":8727,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers/index.html#conjecture","scope":"NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers","status":"fully_proved","summary":"This is the statement that for every real $\\varepsilon > 0$ and every natural number $k \\ge 2$ there is a natural number $N$ such that for every natural number $n \\ge N$ there is a natural number $a$ with $\\lfloor n/2 \\rfloor \\le a + 1$ and $a + k \\le n$ for which every prime factor $p$ of each of the $k$ consecutive integers $a + 1, \\dots, a + k$ satisfies $p \\le n^\\varepsilon$.","title":"Erd\u0151s Problem 369, location variant"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.SmoothConsecutiveIntegersWanted.conjecture_holds","parent":8727,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers/index.html#conjecture-holds","scope":"NumberTheory/arithmetic-functions/special-integer-classes/consecutive-smooth-numbers","status":"can_prove","summary":"For every real $\\varepsilon > 0$ and every natural number $k \\ge 2$ there is a natural number $N$ such that for every natural number $n \\ge N$ there is a natural number $a$ with $\\lfloor n/2 \\rfloor \\le a + 1$ and $a + k \\le n$ for which every prime factor $p$ of each of the $k$ consecutive integers $a + 1, \\dots, a + k$ satisfies $p \\le n^\\varepsilon$.","title":"Solution of Erd\u0151s Problem 369, location variant"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime","status":"can_prove","title":"Density of integers by their k-th smallest prime factor"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/is-kth-smallest-prime-factor","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem690Wanted.IsKthSmallestPrimeFactor","parent":8730,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-th smallest prime factor"]],"source":"The $k$-th smallest prime factor"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/index.html#is-kth-smallest-prime-factor","scope":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime","status":"fully_proved","summary":"For natural numbers $k$, $n$ and $p$, $p$ is the $k$-th smallest prime factor of $n$ when $n \\ne 0$, $p$ is a prime dividing $n$, and exactly $k - 1$ primes dividing $n$ are smaller than $p$.","title":"The $k$-th smallest prime factor"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/kth-prime-factor-density","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem690Wanted.kthPrimeFactorDensity","parent":8730,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-th prime-factor density"]],"source":"The $k$-th prime-factor density"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/index.html#kth-prime-factor-density","scope":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime","status":"fully_proved","summary":"For natural numbers $k$ and $p$, let $S_{k,p}$ be the set of natural numbers whose $k$-th smallest prime factor is $p$. The $k$-th prime-factor density $d_k(p)$ is the natural density of $S_{k,p}$, that is, the limit of $\\lvert S_{k,p} \\cap \\{0, 1, \\dots, N - 1\\} \\rvert / N$ as $N \\to \\infty$, when this limit exists; otherwise $d_k(p)$ is an unspecified real number.","title":"The $k$-th prime-factor density"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/is-unimodal-on-primes","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem690Wanted.IsUnimodalOnPrimes","parent":8730,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/index.html#is-unimodal-on-primes","scope":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime","status":"fully_proved","summary":"A function $f \\colon \\mathbb{N} \\to \\mathbb{R}$ is unimodal on the primes if there is a prime $p_0$ such that $f(p) \\le f(q)$ for all primes $p$ and $q$ with $p \\le q \\le p_0$, and $f(q) \\le f(p)$ for all primes $p$ and $q$ with $p_0 \\le p \\le q$.","title":"Unimodality on the primes"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem690Wanted.conjecture","parent":8730,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/index.html#conjecture","scope":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime","status":"fully_proved","summary":"The proposition that for every natural number $k \\ge 1$ the $k$-th prime-factor density $d_k$ is unimodal on the primes: there is a prime $p_0$ such that $d_k(p) \\le d_k(q)$ for all primes $p$ and $q$ with $p \\le q \\le p_0$, and $d_k(q) \\le d_k(p)$ for all primes $p$ and $q$ with $p_0 \\le p \\le q$. Here $d_k(p)$ is the natural density of the set of natural numbers whose $k$-th smallest prime factor is $p$ (an\u2026","title":"Erd\u0151s Problem 690"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem690Wanted.conjecture_refuted","parent":8730,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime/index.html#conjecture-refuted","scope":"NumberTheory/arithmetic-functions/special-integer-classes/density-of-integers-by-their-k-th-smallest-prime","status":"can_prove","summary":"There is a natural number $k \\ge 1$ such that the $k$-th prime-factor density $d_k$ is not unimodal on the primes, that is, there is no prime $p_0$ such that $d_k(p) \\le d_k(q)$ for all primes $p$ and $q$ with $p \\le q \\le p_0$ and $d_k(q) \\le d_k(p)$ for all primes $p$ and $q$ with $p_0 \\le p \\le q$. Here $d_k(p)$ is the natural density of the set of natural numbers whose $k$-th smallest prime factor is $p$ (an\u2026","title":"Erd\u0151s Problem 690 is false"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","title":"Distribution of cyclic numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/is-cyclic-number","kind":"node","lean":"MetaMathlibExt.IsCyclicNumber","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#is-cyclic-number","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"fully_proved","summary":"A positive integer $n$ is cyclic when $\\gcd(n, \\varphi(n)) = 1$, where $\\varphi$ is Euler's totient function.","title":"Cyclic numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/oppermann-cyclic-count-left","kind":"node","lean":"MetaMathlibExt.oppermannCyclicCountLeft","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#oppermann-cyclic-count-left","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"fully_proved","summary":"For a natural number $n$, this is the number of cyclic numbers $k$ with $n^2 - n \\le k \\le n^2$.","title":"Cyclic numbers in the left Oppermann interval"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/oppermann-cyclic-count-right","kind":"node","lean":"MetaMathlibExt.oppermannCyclicCountRight","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#oppermann-cyclic-count-right","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"fully_proved","summary":"For a natural number $n$, this is the number of cyclic numbers $k$ with $n^2 \\le k \\le n^2 + n$.","title":"Cyclic numbers in the right Oppermann interval"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/oppermann-cyclic-left-approximation","kind":"node","lean":"MetaMathlibExt.oppermannCyclicLeftApproximation","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#oppermann-cyclic-left-approximation","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"fully_proved","summary":"Let $\\gamma$ be the Euler\u2013Mascheroni constant and, for real $x$, let $L(x) = \\log \\log \\log (x^2)$. The left-hand approximation is the function of the real variable $x$ given by","title":"Left-hand approximation for cyclic numbers near squares"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/oppermann-cyclic-approximation","kind":"node","lean":"MetaMathlibExt.oppermannCyclicApproximation","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#oppermann-cyclic-approximation","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"fully_proved","summary":"Let $\\gamma$ be the Euler\u2013Mascheroni constant and, for real $x$, let $L(x) = \\log \\log \\log x$. The simplified approximation is the function of the real variable $x$ given by","title":"Simplified approximation for cyclic numbers near squares"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/oppermann-cyclic-right-approximation","kind":"node","lean":"MetaMathlibExt.oppermannCyclicRightApproximation","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#oppermann-cyclic-right-approximation","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"fully_proved","summary":"Let $\\gamma$ be the Euler\u2013Mascheroni constant and, for real $x$, let $L(x) = \\log \\log \\log (x(x+1))$. The right-hand approximation is the function of the real variable $x$ given by","title":"Right-hand approximation for cyclic numbers near squares"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/cyclic-count-is-equivalent-composite-cyc","kind":"node","lean":"MetaMathlibExt.cyclicCount_isEquivalent_compositeCyclicCount","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#cyclic-count-is-equivalent-composite-cyc","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"For real $x$, let $A(x)$ be the number of cyclic numbers $m$ with $1 \\le m \\le x$, and let $cc(x)$ be the number of composite cyclic numbers $m \\le x$, that is, cyclic numbers $m > 1$ that are not prime. Then, as $x \\to \\infty$, $A(x) \\sim cc(x)$ and $cc(x) \\sim x e^{-\\gamma} / \\log \\log \\log x$, where $\\gamma$ is the Euler\u2013Mascheroni constant.","title":"Asymptotic count of cyclic and composite cyclic numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/cyclic-counting-functions-regularly-vary","kind":"node","lean":"MetaMathlibExt.cyclicCountingFunctions_regularlyVarying_index_one","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#cyclic-counting-functions-regularly-vary","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"For real $x$, let $A(x)$ be the number of cyclic numbers $m$ with $1 \\le m \\le x$, let $\\pi(x)$ be the number of primes $p \\le x$, and let $cc(x)$ be the number of composite cyclic numbers $m \\le x$, that is, cyclic numbers $m > 1$ that are not prime. Then for every real $\\lambda > 0$, each of $A(\\lambda x) / A(x)$, $\\pi(\\lambda x) / \\pi(x)$ and $cc(\\lambda x) / cc(x)$ tends to $\\lambda$ as $x \\to \\infty$.","title":"The cyclic counting functions vary regularly with index one"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/cyclic-numbers-variance-div-mean-rpow-te","kind":"node","lean":"MetaMathlibExt.cyclicNumbers_variance_div_mean_rpow_tendsto","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#cyclic-numbers-variance-div-mean-rpow-te","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"Let $c_1 < c_2 < \\cdots$ be the cyclic numbers in increasing order, so $c_1 = 1$. For $n \\ge 1$ let $m(c,n) = \\frac{1}{n} \\sum_{i=1}^{n} c_i$ be the mean of $c_1, \\dots, c_n$, and for $n \\ge 2$ let $v(c,n) = \\frac{1}{n-1} \\sum_{i=1}^{n} (c_i - m(c,n))^2$ be their variance. Then for every real $b$, as $n \\to \\infty$, $v(c,n) / m(c,n)^b$ tends to $\\infty$ if $b < 2$, to $1/3$ if $b = 2$, and to $0$ if $b > 2$.","title":"Taylor's law for cyclic numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/cyclic-numbers-mean-variance-is-equivale","kind":"node","lean":"MetaMathlibExt.cyclicNumbers_mean_variance_isEquivalent","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#cyclic-numbers-mean-variance-is-equivale","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"Let $c_1 < c_2 < \\cdots$ be the cyclic numbers in increasing order, so $c_1 = 1$. For $n \\ge 1$ let $m(c,n) = \\frac{1}{n} \\sum_{i=1}^{n} c_i$ be the mean of $c_1, \\dots, c_n$, and for $n \\ge 2$ let $v(c,n) = \\frac{1}{n-1} \\sum_{i=1}^{n} (c_i - m(c,n))^2$ be their variance. Then, as $n \\to \\infty$, $m(c,n) \\sim c_n / 2$ and $v(c,n) \\sim c_n^2 / 12$.","title":"Mean and variance of the cyclic numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/cyclic-numbers-moment-is-equivalent-mome","kind":"node","lean":"MetaMathlibExt.cyclicNumbers_moment_isEquivalent_moment_rpow","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#cyclic-numbers-moment-is-equivalent-mome","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"Let $c_1 < c_2 < \\cdots$ be the cyclic numbers in increasing order, so $c_1 = 1$, and for $n \\ge 1$ and a natural number $k$ let $\\mu'(c,n,k) = \\frac{1}{n} \\sum_{i=1}^{n} c_i^k$ be the $k$-th moment of $c_1, \\dots, c_n$. Then for all natural numbers $j \\ge 1$ and $k$, as $n \\to \\infty$,","title":"Relation between the moments of the cyclic numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/cyclic-numbers-moment-is-equivalent-mean","kind":"node","lean":"MetaMathlibExt.cyclicNumbers_moment_isEquivalent_mean_pow","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#cyclic-numbers-moment-is-equivalent-mean","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"Let $c_1 < c_2 < \\cdots$ be the cyclic numbers in increasing order, so $c_1 = 1$. For $n \\ge 1$ and a natural number $k$ let $\\mu'(c,n,k) = \\frac{1}{n} \\sum_{i=1}^{n} c_i^k$ be the $k$-th moment of $c_1, \\dots, c_n$, and let $m(c,n) = \\mu'(c,n,1)$ be their mean. Then for every natural number $k$, as $n \\to \\infty$,","title":"Generalized Taylor's law for cyclic numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/cyclic-number-gap-is-little-o-of-riemann","kind":"node","lean":"MetaMathlibExt.cyclicNumber_gap_isLittleO_of_riemannHypothesis","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#cyclic-number-gap-is-little-o-of-riemann","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"Assume the Riemann hypothesis. Let $c_1 < c_2 < \\cdots$ be the cyclic numbers in increasing order and $p_1 < p_2 < \\cdots$ the primes in increasing order. Then the gaps $d_n = c_{n+1} - c_n$ between consecutive cyclic numbers satisfy $d_n = o(\\sqrt{p_n} \\log p_n)$ as $n \\to \\infty$.","title":"Gaps between cyclic numbers under the Riemann hypothesis"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/oppermann-analog-for-cyclic-numbers","kind":"node","lean":"MetaMathlibExt.oppermannAnalogForCyclicNumbers","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#oppermann-analog-for-cyclic-numbers","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"For every natural number $n > 1$ there are cyclic numbers $c$ and $c'$ with $n^2 - n < c < n^2 < c' < n^2 + n$. Moreover, let $C_L(n)$ and $C_R(n)$ be the numbers of cyclic numbers in $[n^2 - n, n^2]$ and in $[n^2, n^2 + n]$, and let $A_L$, $A_R$ and $A$ be the left-hand, right-hand and simplified approximations for cyclic numbers near squares. Then, as $n \\to \\infty$ through the natural numbers,\u2026","title":"Cohen's Oppermann conjecture for cyclic numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/sum-pow-cyclic-numbers-is-equivalent","kind":"node","lean":"MetaMathlibExt.sum_pow_cyclicNumbers_isEquivalent","parent":8736,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers/index.html#sum-pow-cyclic-numbers-is-equivalent","scope":"NumberTheory/arithmetic-functions/special-integer-classes/distribution-of-cyclic-numbers","status":"can_prove","summary":"Let $k \\ge 1$ be an integer. Let $c_n$ be the $n$-th cyclic number ($c_1 = 1$), let $C(x)$ be the number of cyclic numbers $m \\le x$, and let $\\gamma$ be the Euler\u2013Mascheroni constant. Then, as $n \\to \\infty$ through the natural numbers,","title":"Power sums of cyclic numbers"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers","status":"fully_proved","title":"Highly composite numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers/is-highly-composite","kind":"node","lean":"Nat.IsHighlyComposite","parent":8752,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers/index.html#is-highly-composite","scope":"NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers","status":"fully_proved","summary":"A positive natural number is highly composite if it has strictly more divisors than every smaller positive natural number.","title":"Highly composite numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers/factorization-antitone","kind":"node","lean":"Nat.IsHighlyComposite.factorization_antitone","parent":8752,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers/index.html#factorization-antitone","scope":"NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers","status":"fully_proved","summary":"If $H$ is a highly composite number and $p \\le q$ are primes, then the exponent of $q$ in the prime factorization of $H$ is at most the exponent of $p$.","title":"Exponents of highly composite numbers are non-increasing"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers/dvd-of-prime-le","kind":"node","lean":"Nat.IsHighlyComposite.dvd_of_prime_le","parent":8752,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers/index.html#dvd-of-prime-le","scope":"NumberTheory/arithmetic-functions/special-integer-classes/highly-composite-numbers","status":"fully_proved","summary":"If $H$ is a highly composite number and $p \\le q$ are primes with $q \\mid H$, then $p \\mid H$. In other words, the set of primes dividing $H$ is an initial segment of the primes.","title":"Prime divisors of highly composite numbers form an initial segment"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4","status":"can_prove","title":"Integers with all prime factors 3 mod 4"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4/is-equivalent-sum-of-lseries-eq-div-cpow","kind":"node","lean":"MetaMathlibExt.isEquivalent_sum_of_LSeries_eq_div_cpow","parent":8756,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4/index.html#is-equivalent-sum-of-lseries-eq-div-cpow","scope":"NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4","status":"can_prove","summary":"Let $a_1, a_2, \\ldots$ be nonnegative real numbers whose Dirichlet series $F(s) = \\sum_{n \\ge 1} a_n n^{-s}$ converges absolutely for $\\mathrm{Re}\\, s > 1$, and let $\\alpha < 1$ be real. Suppose that $H$ is a function holomorphic on an open set containing the half-plane $\\mathrm{Re}\\, s \\ge 1$ and nonzero on that half-plane, and that $F(s) = H(s) / (s - 1)^{1 - \\alpha}$ for $\\mathrm{Re}\\, s > 1$, with the principal\u2026","title":"Raikov's Tauberian theorem"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4/is-equivalent-card-prime-factors-three-m","kind":"node","lean":"MetaMathlibExt.isEquivalent_card_primeFactors_three_mod_four","parent":8756,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting integers whose prime factors are all "],["math","\\(\\equiv 3 \\pmod 4\\)"]],"source":"Counting integers whose prime factors are all $\\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4/index.html#is-equivalent-card-prime-factors-three-m","scope":"NumberTheory/arithmetic-functions/special-integer-classes/integers-with-all-prime-factors-3-mod-4","status":"can_prove","summary":"Let $C$ be a real number such that $\\sqrt{\\ln x} \\prod_{q \\le x} (1 - 1/q) \\to C$ as the natural number $x$ tends to infinity, where $q$ runs over the primes with $q \\equiv 1 \\pmod 4$. Let $g(x)$ be the number of integers $n$ with $1 \\le n \\le x$ all of whose prime factors are $\\equiv 3 \\pmod 4$ (so $n = 1$ is counted). Then","title":"Counting integers whose prime factors are all $\\equiv 3 \\pmod 4$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","title":"Power-free integers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/is-power-free","kind":"node","lean":"Nat.IsPowerFree","parent":8759,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html#is-power-free","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","summary":"Let $k$ be a natural number. A natural number $n$ is $k$-free when $k \\ge 2$, $n > 0$, and $p^k$ does not divide $n$ for any prime $p$.","title":"Power-free natural numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/is-power-free-iff","kind":"node","lean":"Nat.isPowerFree_iff","parent":8759,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html#is-power-free-iff","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $n$ is $k$-free if and only if $k \\ge 2$, $n > 0$, and $p^k$ does not divide $n$ for any prime $p$.","title":"Characterization of power-free numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/two-le","kind":"node","lean":"Nat.IsPowerFree.two_le","parent":8759,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html#two-le","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","summary":"If a natural number $n$ is $k$-free, then $k \\ge 2$.","title":"The exponent of a power-free number is at least two"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/pos","kind":"node","lean":"Nat.IsPowerFree.pos","parent":8759,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html#pos","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","summary":"If a natural number $n$ is $k$-free, then $n > 0$.","title":"Power-free numbers are positive"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/not-dvd","kind":"node","lean":"Nat.IsPowerFree.not_dvd","parent":8759,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html#not-dvd","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","summary":"If a natural number $n$ is $k$-free and $p$ is a prime, then $p^k$ does not divide $n$.","title":"A k-free number has no prime k-th power divisor"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/of-dvd","kind":"node","lean":"Nat.IsPowerFree.of_dvd","parent":8759,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html#of-dvd","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","summary":"If a natural number $n$ is $k$-free and $m$ divides $n$, then $m$ is $k$-free.","title":"Divisors of k-free numbers are k-free"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/mono","kind":"node","lean":"Nat.IsPowerFree.mono","parent":8759,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html#mono","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","summary":"If a natural number $n$ is $k$-free and $k \\le l$, then $n$ is $l$-free.","title":"Power-freeness is monotone in the exponent"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/is-power-free-two","kind":"node","lean":"Nat.isPowerFree_two","parent":8759,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers/index.html#is-power-free-two","scope":"NumberTheory/arithmetic-functions/special-integer-classes/power-free-integers","status":"fully_proved","summary":"For every natural number $n$, $n$ is $2$-free if and only if $n$ is squarefree, that is, $d^2$ divides $n$ only for $d = 1$.","title":"A natural number is 2-free if and only if it is squarefree"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","title":"Powerful and k-full numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/is-kfull","kind":"node","lean":"Nat.IsKFull","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-full numbers"]],"source":"$k$-full numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#is-kfull","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"Let $k$ be a natural number. A natural number $n$ is $k$-full if $n \\ne 0$ and $p^k$ divides $n$ for every prime $p$ dividing $n$.","title":"$k$-full numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/powerful-part","kind":"node","lean":"Nat.powerfulPart","parent":8768,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#powerful-part","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"For a natural number $n \\ge 1$ and a prime $p$, let $v_p(n)$ be the exponent of $p$ in $n$. The powerful (or $2$-full) part of $n$ is the product of the prime powers $p^{v_p(n)}$ over the primes $p$ dividing $n$ with $v_p(n) \\ge 2$ (and $1$ if there are none). The powerful part of $0$ is defined to be $1$.","title":"Powerful part of a natural number"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/powerful-part-factorization","kind":"node","lean":"Nat.powerfulPart_factorization","parent":8768,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#powerful-part-factorization","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $p$, write $v_p(n)$ for the exponent of $p$ in the prime factorization of $n$ (it is $0$ when $p$ is not a prime factor of $n$). Then the exponent of $p$ in the powerful part of $n$ equals $v_p(n)$ if $v_p(n) \\ge 2$, and $0$ otherwise.","title":"Prime exponents of the powerful part"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/powerful-part-dvd","kind":"node","lean":"Nat.powerfulPart_dvd","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","The powerful part of "],["math","\\(n\\)"],["text"," divides "],["math","\\(n\\)"]],"source":"The powerful part of $n$ divides $n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#powerful-part-dvd","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"For every natural number $n$, the powerful part of $n$ divides $n$.","title":"The powerful part of $n$ divides $n$"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/is-kfull-iff-factorization","kind":"node","lean":"Nat.isKFull_iff_factorization","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","Factorization criterion for "],["math","\\(k\\)"],["text","-full numbers"]],"source":"Factorization criterion for $k$-full numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#is-kfull-iff-factorization","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"Let $k$ and $n$ be natural numbers with $n \\ne 0$, and write $v_p(n)$ for the exponent of a prime $p$ in the prime factorization of $n$. Then $n$ is $k$-full if and only if $k \\le v_p(n)$ for every prime $p$ dividing $n$.","title":"Factorization criterion for $k$-full numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/not-is-kfull-zero","kind":"node","lean":"Nat.not_isKFull_zero","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero is not "],["math","\\(k\\)"],["text","-full"]],"source":"Zero is not $k$-full"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#not-is-kfull-zero","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"For every natural number $k$, the number $0$ is not $k$-full.","title":"Zero is not $k$-full"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/is-kfull-one","kind":"node","lean":"Nat.isKFull_one","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","One is "],["math","\\(k\\)"],["text","-full"]],"source":"One is $k$-full"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#is-kfull-one","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"For every natural number $k$, the number $1$ is $k$-full: it is nonzero and has no prime divisors.","title":"One is $k$-full"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/is-kfull-k-eq-zero","kind":"node","lean":"Nat.isKFull_k_eq_zero","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(0\\)"],["text","-full numbers"]],"source":"The $0$-full numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#is-kfull-k-eq-zero","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"A natural number $n$ is $0$-full if and only if $n \\ne 0$, since $p^0 = 1$ divides $n$ for every prime $p$.","title":"The $0$-full numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/is-kfull-k-eq-one","kind":"node","lean":"Nat.isKFull_k_eq_one","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(1\\)"],["text","-full numbers"]],"source":"The $1$-full numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#is-kfull-k-eq-one","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"A natural number $n$ is $1$-full if and only if $n \\ne 0$, since $p^1 = p$ divides $n$ for every prime $p$ dividing $n$.","title":"The $1$-full numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/is-kfull-mono","kind":"node","lean":"Nat.isKFull_mono","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity of "],["math","\\(k\\)"],["text","-fullness in "],["math","\\(k\\)"]],"source":"Monotonicity of $k$-fullness in $k$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#is-kfull-mono","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"Let $j \\le k$ be natural numbers. Every $k$-full natural number is $j$-full.","title":"Monotonicity of $k$-fullness in $k$"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/is-kfull-of-two-le","kind":"node","lean":"Nat.isKFull_of_two_le","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","For "],["math","\\(k \\ge 2\\)"],["text",", "],["math","\\(k\\)"],["text","-full numbers are "],["math","\\(2\\)"],["text","-full"]],"source":"For $k \\ge 2$, $k$-full numbers are $2$-full"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#is-kfull-of-two-le","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"Let $k \\ge 2$ be a natural number. Every $k$-full natural number is $2$-full.","title":"For $k \\ge 2$, $k$-full numbers are $2$-full"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/is-kfull-two","kind":"node","lean":"Nat.isKFull_two","parent":8768,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(2\\)"],["text","-full numbers"]],"source":"The $2$-full numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers/index.html#is-kfull-two","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-and-k-full-numbers","status":"fully_proved","summary":"A natural number $n$ is $2$-full if and only if $n \\ne 0$ and $p^2$ divides $n$ for every prime $p$ dividing $n$. These are the squarefull (also called powerful) numbers.","title":"The $2$-full numbers"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","title":"Mean value of 2^\u03c9(n) over squareful numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/squareful-two-pow-omega-lseries-eq-zeta","kind":"node","lean":"MetaMathlibExt.squareful_two_pow_omega_LSeries_eq_zeta_quotient_fifteen","parent":8781,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fine zeta factorization of the squareful "],["math","\\(2^{\\omega(n)}\\)"],["text"," Dirichlet series"]],"source":"Fine zeta factorization of the squareful $2^{\\omega(n)}$ Dirichlet series"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html#squareful-two-pow-omega-lseries-eq-zeta","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","summary":"Let $\\omega(n)$ be the number of distinct prime factors of $n$, call $n$ squareful when $n \\ne 0$ and $p^2 \\mid n$ for every prime $p \\mid n$ (that is, $n$ is $2$-full), and let $F(s) = \\sum_{n \\text{ squareful}} 2^{\\omega(n)} n^{-s}$. There is a sequence $h : \\mathbb{N} \\to \\mathbb{C}$ whose Dirichlet series $H(s) = \\sum_{n \\ge 1} h(n) n^{-s}$ converges absolutely for $\\operatorname{Re} s > 1/15$, such that for\u2026","title":"Fine zeta factorization of the squareful $2^{\\omega(n)}$ Dirichlet series"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/squareful-two-pow-omega-lseries-eq-zeta-2","kind":"node","lean":"MetaMathlibExt.squareful_two_pow_omega_LSeries_eq_zeta_quotient_five","parent":8781,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coarse zeta factorization of the squareful "],["math","\\(2^{\\omega(n)}\\)"],["text"," Dirichlet series"]],"source":"Coarse zeta factorization of the squareful $2^{\\omega(n)}$ Dirichlet series"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html#squareful-two-pow-omega-lseries-eq-zeta-2","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","summary":"Let $\\omega(n)$ be the number of distinct prime factors of $n$, call $n$ squareful when $n \\ne 0$ and $p^2 \\mid n$ for every prime $p \\mid n$ (that is, $n$ is $2$-full), and let $F(s) = \\sum_{n \\text{ squareful}} 2^{\\omega(n)} n^{-s}$. There is a sequence $h : \\mathbb{N} \\to \\mathbb{C}$ whose Dirichlet series $H(s) = \\sum_{n \\ge 1} h(n) n^{-s}$ converges absolutely for $\\operatorname{Re} s > 1/5$, such that for\u2026","title":"Coarse zeta factorization of the squareful $2^{\\omega(n)}$ Dirichlet series"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/zeta-fourth-moment-le-uniform","kind":"node","lean":"MetaMathlibExt.zeta_fourth_moment_le_uniform","parent":8781,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html#zeta-fourth-moment-le-uniform","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","summary":"There are real constants $C$ and $T_0$ such that for every real $\\sigma$ with $1/2 \\le \\sigma \\le 1$ and every real $T \\ge T_0$,","title":"Uniform fourth-moment bound for the Riemann zeta function"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/inv-zeta-le-vinogradov-korobov","kind":"node","lean":"MetaMathlibExt.inv_zeta_le_vinogradov_korobov","parent":8781,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vinogradov\u2013Korobov bound for "],["math","\\(1/\\zeta\\)"]],"source":"Vinogradov\u2013Korobov bound for $1/\\zeta$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html#inv-zeta-le-vinogradov-korobov","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","summary":"Write $s = \\sigma + it$ and $g(T) = (\\log T)^{2/3} (\\log \\log T)^{1/3}$. There are real constants $C > 0$, $K$ and $T_0 \\ge 3$ such that for every real $T$ and every complex $s$ with $T_0 < t \\le T$ and $\\sigma \\ge 1 - C / g(T)$, we have $\\zeta(s) \\ne 0$ and $|1/\\zeta(s)| \\le K g(T)$.","title":"Vinogradov\u2013Korobov bound for $1/\\zeta$"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/zeta-ne-zero-vinogradov-korobov","kind":"node","lean":"MetaMathlibExt.zeta_ne_zero_vinogradov_korobov","parent":8781,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html#zeta-ne-zero-vinogradov-korobov","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","summary":"There are real constants $C > 0$ and $t_0 \\ge 3$ such that $\\zeta(\\sigma + it) \\ne 0$ for all real $\\sigma$ and $t$ with $|t| \\ge t_0$ and $\\sigma \\ge 1 - C / \\bigl((\\log |t|)^{2/3} (\\log \\log |t|)^{1/3}\\bigr)$.","title":"Vinogradov\u2013Korobov zero-free region"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/zeta-le-rpow-of-riemann-hypothesis","kind":"node","lean":"MetaMathlibExt.zeta_le_rpow_of_riemannHypothesis","parent":8781,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html#zeta-le-rpow-of-riemann-hypothesis","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","summary":"Assume the Riemann hypothesis, and let $\\varepsilon > 0$ be real. Then there are real constants $C$ and $t_0$ such that $|\\zeta(\\sigma + it)| \\le C |t|^{\\varepsilon}$ for all real $\\sigma$ and $t$ with $1/2 \\le \\sigma \\le 1$ and $|t| \\ge t_0$.","title":"Lindel\u00f6f bound from the Riemann hypothesis"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/inv-zeta-le-rpow-of-riemann-hypothesis","kind":"node","lean":"MetaMathlibExt.inv_zeta_le_rpow_of_riemannHypothesis","parent":8781,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound for "],["math","\\(1/\\zeta\\)"],["text"," from the Riemann hypothesis"]],"source":"Bound for $1/\\zeta$ from the Riemann hypothesis"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html#inv-zeta-le-rpow-of-riemann-hypothesis","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","summary":"Assume the Riemann hypothesis, and let $1/2 < \\sigma \\le 1$ and $\\varepsilon > 0$ be real. Then there are real constants $C$ and $t_0$, depending on $\\sigma$ and $\\varepsilon$, such that $|1/\\zeta(\\sigma + it)| \\le C |t|^{\\varepsilon}$ for every real $t$ with $|t| \\ge t_0$.","title":"Bound for $1/\\zeta$ from the Riemann hypothesis"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/squareful-two-pow-omega-sum-asymptotic","kind":"node","lean":"MetaMathlibExt.squareful_two_pow_omega_sum_asymptotic","parent":8781,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic for the sum of "],["math","\\(2^{\\omega(n)}\\)"],["text"," over squareful "],["math","\\(n\\)"]],"source":"Asymptotic for the sum of $2^{\\omega(n)}$ over squareful $n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers/index.html#squareful-two-pow-omega-sum-asymptotic","scope":"NumberTheory/arithmetic-functions/special-integer-classes/mean-value-of-2-n-over-squareful-numbers","status":"can_prove","summary":"Let $\\omega(n)$ be the number of distinct prime factors of $n$, and let $f(n) = 2^{\\omega(n)}$ if $n$ is squareful ($n \\ne 0$ and $p^2 \\mid n$ for every prime $p \\mid n$; so $f(1) = 1$) and $f(n) = 0$ otherwise. There are real constants $C_1$, $C_2$, $C_3$, $C_4$, $c > 0$, $K$ and $x_0$ such that for every real $x \\ge x_0$,","title":"Asymptotic for the sum of $2^{\\omega(n)}$ over squareful $n$"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers","status":"can_prove","title":"Odd and even k-full and l-free numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers/tendsto-card-odd-k-full-div-card-even-k","kind":"node","lean":"MetaMathlibExt.tendsto_card_odd_kFull_div_card_even_kFull","parent":8790,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd-to-even ratio of "],["math","\\(k\\)"],["text","-full numbers"]],"source":"Odd-to-even ratio of $k$-full numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers/index.html#tendsto-card-odd-k-full-div-card-even-k","scope":"NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers","status":"can_prove","summary":"Let $k \\ge 2$. For a natural number $N$ let $K_o(N)$ and $K_e(N)$ be the numbers of odd and of even $k$-full numbers $n \\le N$. Then $K_o(N)/K_e(N) \\to 2 - 2^{1 - 1/k}$ as $N \\to \\infty$.","title":"Odd-to-even ratio of $k$-full numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers/tendsto-card-odd-k-full-l-free-div-card","kind":"node","lean":"MetaMathlibExt.tendsto_card_odd_kFull_lFree_div_card_even_kFull_lFree","parent":8790,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd-to-even ratio of "],["math","\\(k\\)"],["text","-full, "],["math","\\(l\\)"],["text","-free numbers"]],"source":"Odd-to-even ratio of $k$-full, $l$-free numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers/index.html#tendsto-card-odd-k-full-l-free-div-card","scope":"NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers","status":"can_prove","summary":"Let $2 \\le k < l$. Call a natural number $n$ a $[k, l]$-integer if it is $k$-full and $l$-free, that is, $n \\ge 1$ and every prime $p$ dividing $n$ satisfies $p^k \\mid n$ and $p^l \\nmid n$. For a natural number $N$ let $O_{k,l}(N)$ and $E_{k,l}(N)$ be the numbers of odd and of even $[k, l]$-integers $n \\le N$. Then $O_{k,l}(N)/E_{k,l}(N) \\to (2 - 2^{1 - 1/k})/(1 - 2^{1 - l/k})$ as $N \\to \\infty$.","title":"Odd-to-even ratio of $k$-full, $l$-free numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers/tendsto-card-odd-l-free-div-card-even-l","kind":"node","lean":"MetaMathlibExt.tendsto_card_odd_lFree_div_card_even_lFree","parent":8790,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd-to-even ratio of "],["math","\\(l\\)"],["text","-free numbers"]],"source":"Odd-to-even ratio of $l$-free numbers"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers/index.html#tendsto-card-odd-l-free-div-card-even-l","scope":"NumberTheory/arithmetic-functions/special-integer-classes/odd-and-even-k-full-and-l-free-numbers","status":"can_prove","summary":"Let $l \\ge 2$. For a natural number $N$ let $O(N)$ and $E(N)$ be the numbers of odd and of even $l$-free positive integers $n \\le N$. Then $O(N)/E(N) \\to 2^l/(2^l - 2)$ as $N \\to \\infty$.","title":"Odd-to-even ratio of $l$-free numbers"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions","status":"can_prove","title":"Powerful numbers in arithmetic progressions"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/arbitrarily-long-arithmetic-progressions","kind":"node","lean":"MetaMathlibExt.arbitrarily_long_arithmetic_progressions_of_powerful_numbers_general","parent":8794,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/index.html#arbitrarily-long-arithmetic-progressions","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions","status":"fully_proved","summary":"For every natural number $k$ there are positive integers $a$ and $d$ such that $a + id$ is $2$-full (powerful) for every natural number $i < k$.","title":"Arithmetic progressions of 2-full numbers of every length"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/arbitrarily-long-arithmetic-progressions-2","kind":"node","lean":"MetaMathlibExt.arbitrarily_long_arithmetic_progressions_of_powerful_numbers","parent":8794,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/index.html#arbitrarily-long-arithmetic-progressions-2","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions","status":"fully_proved","summary":"For every integer $k \\ge 3$ there are positive integers $a$ and $d$ such that each of the $k$ numbers $a + id$ with $0 \\le i < k$ is powerful: every prime that divides $a + id$ has its square dividing $a + id$.","title":"Arbitrarily long arithmetic progressions of powerful numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/is-coprime-powerful-ap4","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem937Wanted.IsCoprimePowerfulAP4","parent":8794,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/index.html#is-coprime-powerful-ap4","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions","status":"fully_proved","summary":"For natural numbers $a$ and $d$, the numbers $a$, $a + d$, $a + 2d$, $a + 3d$ form a four-term arithmetic progression of pairwise coprime powerful numbers when $d > 0$, each of the four numbers is powerful (that is, $2$-full: nonzero and divisible by $p^2$ for every prime $p$ dividing it), and any two of the four numbers are coprime.","title":"Four-term progressions of pairwise coprime powerful numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem937Wanted.conjecture","parent":8794,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/index.html#conjecture","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions","status":"fully_proved","summary":"There are infinitely many pairs of natural numbers $(a, d)$ with $d > 0$ such that the four numbers $a$, $a + d$, $a + 2d$, $a + 3d$ are powerful (each is nonzero and divisible by $p^2$ for every prime $p$ dividing it) and pairwise coprime; that is, infinitely many four-term arithmetic progressions of pairwise coprime powerful numbers.","title":"Statement of Erd\u0151s Problem 937"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem937Wanted.conjecture_holds","parent":8794,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions/index.html#conjecture-holds","scope":"NumberTheory/arithmetic-functions/special-integer-classes/powerful-numbers-in-arithmetic-progressions","status":"can_prove","summary":"There are infinitely many pairs of natural numbers $(a, d)$ with $d > 0$ such that the four numbers $a$, $a + d$, $a + 2d$, $a + 3d$ are powerful (each is nonzero and divisible by $p^2$ for every prime $p$ dividing it) and pairwise coprime. That is, there are infinitely many four-term arithmetic progressions of pairwise coprime powerful numbers.","title":"Erd\u0151s Problem 937 holds (Bajpai\u2013Bennett\u2013Chan)"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","title":"Rough numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/is-rough","kind":"node","lean":"Nat.IsRough","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#is-rough","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers. Then $n$ is $k$-rough if $n \\neq 0$ and every prime $p$ dividing $n$ satisfies $k < p$; that is, $n$ is positive and has no prime factor at most $k$.","title":"Rough numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/is-rough-iff","kind":"node","lean":"Nat.isRough_iff","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#is-rough-iff","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $n$ is $k$-rough if and only if $n \\neq 0$ and every prime $p$ dividing $n$ satisfies $k < p$.","title":"Characterization of rough numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/ne-zero","kind":"node","lean":"Nat.IsRough.ne_zero","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#ne-zero","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"For every natural number $k$, every $k$-rough natural number $n$ satisfies $n \\ne 0$.","title":"Rough numbers are nonzero"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/pos","kind":"node","lean":"Nat.IsRough.pos","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#pos","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"For every natural number $k$, every $k$-rough natural number $n$ satisfies $n > 0$.","title":"Rough numbers are positive"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/lt-of-prime-dvd","kind":"node","lean":"Nat.IsRough.lt_of_prime_dvd","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#lt-of-prime-dvd","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"Let $n$, $k$ and $p$ be natural numbers. If $n$ is $k$-rough and $p$ is a prime dividing $n$, then $k < p$.","title":"Prime divisors of a rough number exceed the bound"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/is-rough-iff-prime-factors-list","kind":"node","lean":"Nat.isRough_iff_primeFactorsList","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#is-rough-iff-prime-factors-list","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $n$ is $k$-rough if and only if $n \\neq 0$ and every prime in the list of prime factors of $n$ (with multiplicity) is greater than $k$.","title":"Roughness through the list of prime factors"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/is-rough-one","kind":"node","lean":"Nat.isRough_one","parent":8800,"reading_title":{"format":"inline-parts-v1","parts":[["text","The number "],["math","\\(1\\)"],["text"," is rough"]],"source":"The number $1$ is rough"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#is-rough-one","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"For every natural number $k$, the number $1$ is $k$-rough.","title":"The number $1$ is rough"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/is-rough-iff-2","kind":"node","lean":"Nat.Prime.isRough_iff","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#is-rough-iff-2","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $k$ with $p$ prime, $p$ is $k$-rough if and only if $k < p$.","title":"A prime is rough exactly above the bound"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/of-dvd","kind":"node","lean":"Nat.IsRough.of_dvd","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#of-dvd","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"Let $k$ be a natural number. If a natural number $m$ is $k$-rough and $d$ is a nonzero divisor of $m$, then $d$ is $k$-rough.","title":"Divisors of rough numbers are rough"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/is-rough-mul","kind":"node","lean":"Nat.isRough_mul","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#is-rough-mul","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"For natural numbers $m$, $n$ and $k$, the product $mn$ is $k$-rough if and only if both $m$ and $n$ are $k$-rough.","title":"A product is rough exactly when both factors are"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/mul","kind":"node","lean":"Nat.IsRough.mul","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#mul","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"For natural numbers $m$, $n$ and $k$, if $m$ and $n$ are both $k$-rough, then $mn$ is $k$-rough.","title":"Products of rough numbers are rough"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/of-le","kind":"node","lean":"Nat.IsRough.of_le","parent":8800,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#of-le","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"Let $k$ and $l$ be natural numbers with $l \\le k$. Every $k$-rough natural number is $l$-rough.","title":"Roughness is antitone in the bound"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/eq-one-of-mem-smooth-numbers","kind":"node","lean":"Nat.IsRough.eq_one_of_mem_smoothNumbers","parent":8800,"reading_title":{"format":"inline-parts-v1","parts":[["text","Only "],["math","\\(1\\)"],["text"," is rough and smooth at the same bound"]],"source":"Only $1$ is rough and smooth at the same bound"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#eq-one-of-mem-smooth-numbers","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers. If $n$ is $k$-rough and $n$ is positive with every prime factor less than $k + 1$, then $n = 1$.","title":"Only $1$ is rough and smooth at the same bound"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/mem-rough-numbers-up-to","kind":"node","lean":"Nat.IsRough.mem_roughNumbersUpTo","parent":8800,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rough numbers other than "],["math","\\(1\\)"],["text"," lie in Mathlib's rough numbers up to a bound"]],"source":"Rough numbers other than $1$ lie in Mathlib's rough numbers up to a bound"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers/index.html#mem-rough-numbers-up-to","scope":"NumberTheory/arithmetic-functions/special-integer-classes/rough-numbers","status":"fully_proved","summary":"Let $n$, $N$ and $k$ be natural numbers. If $n$ is $k$-rough, $n \\le N$ and $n \\ne 1$, then $n$ belongs to the set of natural numbers $m$ with $1 \\le m \\le N$ that are not $(k+1)$-smooth, where $m$ is $(k+1)$-smooth when $m > 0$ and every prime factor of $m$ is less than $k + 1$; for $m \\ge 1$, not being $(k+1)$-smooth means having a prime factor $p \\ge k + 1$.","title":"Rough numbers other than $1$ lie in Mathlib's rough numbers up to a bound"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"can_prove","title":"Squarefree numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-local-coeff","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30LocalCoeff","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-local-coeff","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"Given $a : \\mathbb{N} \\to \\mathbb{C}$ and natural numbers $p$ and $e$, the local factor of $a$ at $p$ with exponent $e$ is $1$ if $e = 0$, $a(p)$ if $e = 1$, and $0$ if $e \\ge 2$.","title":"Local factor"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For a function $a : \\mathbb{N} \\to \\mathbb{C}$ and $n \\in \\mathbb{N}$, the Entry 30 coefficient $c_a(n) \\in \\mathbb{C}$ is $0$ when $n = 0$. When $n \\ne 0$ it is the product, over the prime factors $p$ of $n$, of the local factor of $a$ at $p$ with exponent the multiplicity of $p$ in $n$. The local factor at $p$ with exponent $e$ is $1$ for $e = 0$, $a(p)$ for $e = 1$, and $0$ for $e \\ge 2$.","title":"Coefficients for Entry 30"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-non-squarefree-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30NonSquarefreeTerm","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-non-squarefree-term","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$ and a natural number $j$, put $n = j + 1$; the $j$-th term is $0$ if $n$ is squarefree and $n^{-s}$ otherwise. As $j$ ranges over the natural numbers, these are the terms of $\\sum n^{-s}$ over non-squarefree $n \\ge 1$.","title":"Terms of the Dirichlet series over non-squarefree integers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-odd-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30OddTerm","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-odd-term","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$ and a natural number $j$, put $n = j + 1$; the $j$-th term is $n^{-s}$ if $n$ is squarefree and has an odd number of distinct prime factors, and $0$ otherwise. As $j$ ranges over the natural numbers, these are the terms of $\\sum n^{-s}$ over squarefree $n \\ge 1$ with an odd number of prime factors.","title":"Terms over squarefree integers with an odd number of prime factors"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-signed-coeff","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30SignedCoeff","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-signed-coeff","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$ and a natural number $n$, let $a(p) = -p^{-s}$ for natural numbers $p$. The signed coefficient of $n$ is $0$ if $n = 0$ and, for $n \\ne 0$, the product over the prime factors $p$ of $n$ of the local factor of $a$ at $p$ with exponent the multiplicity of $p$ in $n$, where the local factor of $a$ at $p$ with exponent $e$ is $1$ for $e = 0$, $a(p)$ for $e = 1$, and $0$ for $e \\ge 2$.","title":"Signed multiplicative coefficient"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-squarefree-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30SquarefreeTerm","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-squarefree-term","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For a complex number $s$ and a natural number $j$, let $n = j + 1$; the $j$-th squarefree term is $n^{-s}$ if $n$ is squarefree and $0$ otherwise, so the series of these terms over $j \\ge 0$ is $\\sum n^{-s}$ over the squarefree $n \\ge 1$.","title":"Terms of the Dirichlet series over squarefree integers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-local-coeff-zero","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30LocalCoeff_zero","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-local-coeff-zero","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every $a : \\mathbb{N} \\to \\mathbb{C}$ and every natural number $p$, the local factor of $a$ at $p$ with exponent $0$ is $1$.","title":"Local factor at exponent zero"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-local-coeff-one","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30LocalCoeff_one","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-local-coeff-one","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every $a : \\mathbb{N} \\to \\mathbb{C}$ and every natural number $p$, the local factor of $a$ at $p$ with exponent $1$ is $a(p)$.","title":"Local factor at exponent one"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-local-coeff-of-two-le","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30LocalCoeff_of_two_le","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-local-coeff-of-two-le","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$, every $p \\in \\mathbb{N}$ and every $e \\ge 2$, the Entry 30 local factor of $a$ at $p$ with exponent $e$ is $0$.","title":"Entry 30 local factors vanish at exponents at least two"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff-of-squarefree","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff_of_squarefree","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff-of-squarefree","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$ and every squarefree $n \\in \\mathbb{N}$ with $n \\ne 0$, the Entry 30 coefficient $c_a(n)$ is the product of $a(p)$ over the prime factors $p$ of $n$.","title":"Entry 30 coefficients at squarefree integers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff-of-not-squarefree","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff_of_not_squarefree","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff-of-not-squarefree","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$ and every natural number $n \\ne 0$ that is not squarefree, the Entry 30 coefficient $c_a(n)$ is $0$.","title":"Entry 30 coefficients vanish at nonzero non-squarefree natural numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff-of-ne-zero","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff_of_ne_zero","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff-of-ne-zero","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$ and every $n \\in \\mathbb{N}$ with $n \\ne 0$, the Entry 30 coefficient $c_a(n)$ equals the product of $a(p)$ over the prime factors $p$ of $n$ when $n$ is squarefree, and $c_a(n) = 0$ otherwise.","title":"Closed form of the Entry 30 coefficients"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/summable-chapter5-entry30-coeff-add-two","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.summable_chapter5Entry30Coeff_add_two","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#summable-chapter5-entry30-coeff-add-two","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$ and let $c > 1$ be real with $|a(p)| \\le p^{-c}$ for every prime $p$. Then the Entry 30 coefficients $f_a(n)$ of $a$ (equal to $0$ at $n = 0$; for $n \\ge 1$, the product of $a(p)$ over the primes $p \\mid n$ if $n$ is squarefree, and $0$ otherwise) form a summable family over $n \\ge 2$.","title":"Summability of the Entry 30 coefficients"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff-zero","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff_zero","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff-zero","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$, the Entry 30 coefficient $c_a(0)$ is $0$.","title":"The Entry 30 coefficient at zero"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff-one","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff_one","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff-one","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$, the Entry 30 coefficient $c_a(1)$ is $1$.","title":"The Entry 30 coefficient at one"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff-prime","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff_prime","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff-prime","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$ and every prime $p$, the Entry 30 coefficient $c_a(p)$ equals $a(p)$.","title":"Entry 30 coefficients at primes"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff-prime-pow-of-two","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff_prime_pow_of_two_le","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff-prime-pow-of-two","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$, every prime $p$ and every $e \\ge 2$, the Entry 30 coefficient satisfies $c_a(p^e) = 0$.","title":"Entry 30 coefficients vanish at higher prime powers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-coeff-mul-of-coprime","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30Coeff_mul_of_coprime","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-coeff-mul-of-coprime","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every function $a : \\mathbb{N} \\to \\mathbb{C}$ and all coprime $m, n \\in \\mathbb{N}$, the Entry 30 coefficients $c_a$ satisfy $c_a(mn) = c_a(m)\\,c_a(n)$.","title":"Entry 30 coefficients are multiplicative"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/has-prod-one-add-chapter5-entry30-coeff","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.hasProd_one_add_chapter5Entry30Coeff","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#has-prod-one-add-chapter5-entry30-coeff","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$ and let $c > 1$ be real with $|a(p)| \\le p^{-c}$ for every prime $p$. Write $f_a(n)$ for the Entry 30 coefficient of $a$ at $n$ (it is $0$ at $n = 0$; for $n \\ge 1$ it is the product of $a(p)$ over the primes $p \\mid n$ if $n$ is squarefree, and $0$ otherwise). Then the product of $1 + a(p)$ over all primes $p$ converges unconditionally to $1 + \\sum_{n \\ge 2} f_a(n)$.","title":"Euler product for the Entry 30 coefficients"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-squarefree-term-def","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30SquarefreeTerm_def","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-squarefree-term-def","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$ and a natural number $j$, the $j$-th term of the Dirichlet series over squarefree integers equals $(j + 1)^{-s}$ if $j + 1$ is squarefree and $0$ otherwise.","title":"Squarefree term, explicit form"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/summable-chapter5-entry30-squarefree-ter","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.summable_chapter5Entry30SquarefreeTerm","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#summable-chapter5-entry30-squarefree-ter","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the series of $n^{-s}$ over the squarefree $n \\ge 1$ is summable.","title":"Summability of the squarefree terms"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/tsum-chapter5-entry30-squarefree-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.tsum_chapter5Entry30SquarefreeTerm","parent":8815,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(n^{-s}\\)"],["text"," over squarefree "],["math","\\(n\\)"]],"source":"Sum of $n^{-s}$ over squarefree $n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#tsum-chapter5-entry30-squarefree-term","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the sum of $n^{-s}$ over the squarefree $n \\ge 1$ equals $\\zeta(s)/\\zeta(2s)$.","title":"Sum of $n^{-s}$ over squarefree $n$"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-odd-term-def","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30OddTerm_def","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-odd-term-def","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$ and a natural number $j$, the $j$-th term of the Dirichlet series over squarefree integers with an odd number of prime factors equals $(j + 1)^{-s}$ if $j + 1$ is squarefree and has an odd number of distinct prime factors, and $0$ otherwise.","title":"Odd-count squarefree term, explicit form"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/summable-chapter5-entry30-odd-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.summable_chapter5Entry30OddTerm","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#summable-chapter5-entry30-odd-term","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the series of $n^{-s}$ over the squarefree $n \\ge 1$ with an odd number of distinct prime factors is summable.","title":"Summability of the odd squarefree terms"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/tsum-chapter5-entry30-odd-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.tsum_chapter5Entry30OddTerm","parent":8815,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(n^{-s}\\)"],["text"," over squarefree "],["math","\\(n\\)"],["text"," with an odd number of prime factors"]],"source":"Sum of $n^{-s}$ over squarefree $n$ with an odd number of prime factors"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#tsum-chapter5-entry30-odd-term","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the sum of $n^{-s}$ over the squarefree $n \\ge 1$ with an odd number of distinct prime factors equals $(\\zeta(s)^2 - \\zeta(2s))/(2\\zeta(s)\\zeta(2s))$.","title":"Sum of $n^{-s}$ over squarefree $n$ with an odd number of prime factors"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-non-squarefree-term-def","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30NonSquarefreeTerm_def","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-non-squarefree-term-def","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$ and a natural number $j$, the $j$-th term of the Dirichlet series over non-squarefree integers equals $0$ if $j + 1$ is squarefree and $(j + 1)^{-s}$ otherwise.","title":"Non-squarefree term, explicit form"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/summable-chapter5-entry30-non-squarefree","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.summable_chapter5Entry30NonSquarefreeTerm","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#summable-chapter5-entry30-non-squarefree","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the series of $n^{-s}$ over the non-squarefree $n \\ge 1$ is summable.","title":"Summability of the non-squarefree terms"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/tsum-chapter5-entry30-non-squarefree-ter","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.tsum_chapter5Entry30NonSquarefreeTerm","parent":8815,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(n^{-s}\\)"],["text"," over non-squarefree "],["math","\\(n\\)"]],"source":"Sum of $n^{-s}$ over non-squarefree $n$"},"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#tsum-chapter5-entry30-non-squarefree-ter","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the sum of $n^{-s}$ over the non-squarefree $n \\ge 1$ equals $\\zeta(s)(\\zeta(2s) - 1)/\\zeta(2s)$.","title":"Sum of $n^{-s}$ over non-squarefree $n$"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/chapter5-entry30-signed-coeff-of-ne-zero","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.chapter5Entry30SignedCoeff_of_ne_zero","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#chapter5-entry30-signed-coeff-of-ne-zero","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$ and a natural number $n \\ne 0$, the signed coefficient of $n$ equals $-n^{-s}$ if $n$ is squarefree with an odd number of distinct prime factors, $n^{-s}$ if $n$ is squarefree with an even number of distinct prime factors, and $0$ if $n$ is not squarefree.","title":"Explicit form of the signed coefficient"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/ramanujan-part1-ch5-entry30-secondordere","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry30Secondordereulerian.ramanujan_part1_ch5_entry30_secondordereulerian","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#ramanujan-part1-ch5-entry30-secondordere","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"(i) Let $a : \\mathbb{N} \\to \\mathbb{C}$ and let $c > 1$ be real with $|a(p)| \\le p^{-c}$ for every prime $p$, and write $f_a(n)$ for the Entry 30 coefficient of $a$ at $n$. Then the family $(f_a(n))_{n \\ge 2}$ is summable; for every $n \\ge 1$, $f_a(n)$ is the product of $a(p)$ over the primes $p \\mid n$ if $n$ is squarefree and $0$ otherwise; and the product of $1 + a(p)$ over all primes $p$ converges\u2026","title":"Ramanujan's squarefree Dirichlet series (Entry 30)"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/even-or-odd-non-squarefree","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem844Wanted.evenOrOddNonSquarefree","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#even-or-odd-non-squarefree","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For a natural number $N$, the set of integers $n$ with $1 \\le n \\le N$ that are even or not squarefree, that is, the even numbers in $\\{1, \\dots, N\\}$ together with the odd non-squarefree numbers in $\\{1, \\dots, N\\}$.","title":"Even or odd non-squarefree numbers up to a bound"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem844Wanted.conjecture","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#conjecture","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"fully_proved","summary":"For a natural number $N$, let $E_N$ be the set of integers $n$ with $1 \\le n \\le N$ that are even or not squarefree. The statement that for every natural number $N$, the largest size of a set $A \\subseteq \\{1, \\dots, N\\}$ such that $ab$ is not squarefree for all $a, b \\in A$ (including $a = b$) is the size of $E_N$.","title":"Erd\u0151s Problem 844"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem844Wanted.conjecture_holds","parent":8815,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers/index.html#conjecture-holds","scope":"NumberTheory/arithmetic-functions/special-integer-classes/squarefree-numbers","status":"can_prove","summary":"For every natural number $N$, the largest size of a set $A \\subseteq \\{1, \\dots, N\\}$ such that $ab$ is not squarefree for all $a, b \\in A$ (including $a = b$) is the number of integers $n$ with $1 \\le n \\le N$ that are even or not squarefree.","title":"Erd\u0151s Problem 844 is true"},{"data_url":"scopes/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers.json","id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","kind":"scope","navigation":"module","parent":8720,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"can_prove","title":"Weakly prime-additive numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/is-prime-divisor-power-sum","kind":"node","lean":"MetaMathlibExt.IsPrimeDivisorPowerSum","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#is-prime-divisor-power-sum","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $t$, $n$ is a sum of positive powers of $t$ distinct prime divisors of $n$ when there are a set $s$ of $t$ distinct primes, each dividing $n$, and positive integers $e_p$ for $p \\in s$ such that $n = \\sum_{p \\in s} p^{e_p}$.","title":"Sums of powers of distinct prime divisors"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/is-weakly-prime-additive","kind":"node","lean":"MetaMathlibExt.IsWeaklyPrimeAdditive","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#is-weakly-prime-additive","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"fully_proved","summary":"A natural number $n$ is weakly prime-additive if it has at least two distinct prime divisors and is a sum $n = \\sum_{p \\in s} p^{e_p}$ of positive powers $p^{e_p}$, $e_p \\ge 1$, over some set $s$ of distinct prime divisors of $n$.","title":"Weakly prime-additive numbers"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/weakly-prime-additive-length","kind":"node","lean":"MetaMathlibExt.weaklyPrimeAdditiveLength","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#weakly-prime-additive-length","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"fully_proved","summary":"For a natural number $n$, the length $\\kappa_n$ is the least $t$ such that $n$ is a sum of positive powers of $t$ distinct prime divisors of $n$, and $\\kappa_n = 0$ when $n$ has no such representation.","title":"Weakly prime-additive length"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/is-prime-divisor-power-sum-thirty","kind":"node","lean":"MetaMathlibExt.isPrimeDivisorPowerSum_thirty","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#is-prime-divisor-power-sum-thirty","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"fully_proved","summary":"The number $30$ is a sum of positive powers of three distinct prime divisors of $30$, namely $30 = 2 + 3 + 5^2$.","title":"Thirty is a sum of powers of three of its prime divisors"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/is-weakly-prime-additive-thirty","kind":"node","lean":"MetaMathlibExt.isWeaklyPrimeAdditive_thirty","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#is-weakly-prime-additive-thirty","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"fully_proved","summary":"The number $30$ is weakly prime-additive.","title":"Thirty is weakly prime-additive"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/weakly-prime-additive-length-le","kind":"node","lean":"MetaMathlibExt.weaklyPrimeAdditiveLength_le","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#weakly-prime-additive-length-le","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $t$, if $n$ is a sum of positive powers of $t$ distinct prime divisors of $n$, then the length $\\kappa_n$ of $n$ is at most $t$.","title":"The length is at most the size of any representation"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/infinite-weakly-prime-additive-length-fo","kind":"node","lean":"MetaMathlibExt.infinite_weaklyPrimeAdditive_length_four_dvd","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#infinite-weakly-prime-additive-length-fo","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"can_prove","summary":"Assume $(\\ast)$: for all natural numbers $a$, $f$, $g$ with $1 \\le a \\le f$, $\\gcd(a, f) = 1$ and $4 \\mid f$, and $g$ an odd prime dividing $f$ with Jacobi symbol $\\left(\\frac{g}{a}\\right) = -1$, there is a prime $\\ell \\equiv a \\pmod f$ such that $g$ is a primitive root modulo $\\ell$. Then for every positive integer $m$ there are infinitely many weakly prime-additive numbers $n$ with $m \\mid n$ and length\u2026","title":"Weakly prime-additive multiples of length four"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/infinite-primes-dvd-four-prime-power-sum","kind":"node","lean":"MetaMathlibExt.infinite_primes_dvd_four_prime_power_sum","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#infinite-primes-dvd-four-prime-power-sum","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"can_prove","summary":"Assume $(\\ast)$: for all natural numbers $a$, $f$, $g$ with $1 \\le a \\le f$, $\\gcd(a, f) = 1$ and $4 \\mid f$, and $g$ an odd prime dividing $f$ with Jacobi symbol $\\left(\\frac{g}{a}\\right) = -1$, there is a prime $\\ell \\equiv a \\pmod f$ such that $g$ is a primitive root modulo $\\ell$. Let $p$, $q$, $r$ be distinct odd primes, at least one of which is congruent to $3$ or $5$ modulo $8$. Then there are infinitely\u2026","title":"Primes completing a four-term prime-power divisibility"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/infinite-shortest-weakly-prime-additive","kind":"node","lean":"MetaMathlibExt.infinite_shortest_weaklyPrimeAdditive_dvd_iff","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#infinite-shortest-weakly-prime-additive","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"can_prove","summary":"For every positive integer $m$, there are infinitely many weakly prime-additive numbers $n$ with $m \\mid n$ and length $\\kappa_n = 3$ if and only if $8 \\nmid m$.","title":"Fang\u2013Chen theorem on shortest weakly prime-additive multiples"},{"id":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/infinite-weakly-prime-additive-length-le","kind":"node","lean":"MetaMathlibExt.infinite_weaklyPrimeAdditive_length_le_four_dvd","parent":8849,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers/index.html#infinite-weakly-prime-additive-length-le","scope":"NumberTheory/arithmetic-functions/special-integer-classes/weakly-prime-additive-numbers","status":"can_prove","summary":"Assume $(\\ast)$: for all natural numbers $a$, $f$, $g$ with $1 \\le a \\le f$, $\\gcd(a, f) = 1$ and $4 \\mid f$, and $g$ an odd prime dividing $f$ with Jacobi symbol $\\left(\\frac{g}{a}\\right) = -1$, there is a prime $\\ell \\equiv a \\pmod f$ such that $g$ is a primitive root modulo $\\ell$. Then for every positive integer $m$ there are infinitely many weakly prime-additive numbers $n$ with $m \\mid n$ and length\u2026","title":"Weakly prime-additive multiples of length at most four"},{"data_url":"scopes/NumberTheory/arithmetic-functions/arithmetic-derivative.json","id":"NumberTheory/arithmetic-functions/arithmetic-derivative","kind":"scope","navigation":"module","parent":8068,"result":false,"route":"../roadmap/NumberTheory/arithmetic-functions/arithmetic-derivative/index.html","scope":"NumberTheory/arithmetic-functions/arithmetic-derivative","status":"fully_proved","title":"Arithmetic derivative"},{"id":"NumberTheory/arithmetic-functions/arithmetic-derivative/arithmetic-derivative","kind":"node","lean":"MetaMathlibExt.arithmeticDerivative","parent":8860,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/arithmetic-derivative/index.html#arithmetic-derivative","scope":"NumberTheory/arithmetic-functions/arithmetic-derivative","status":"fully_proved","summary":"For a natural number $n$, the arithmetic derivative of $n$ is","title":"Arithmetic derivative"},{"id":"NumberTheory/arithmetic-functions/arithmetic-derivative/arithmetic-derivative-zero","kind":"node","lean":"MetaMathlibExt.arithmeticDerivative_zero","parent":8860,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/arithmetic-derivative/index.html#arithmetic-derivative-zero","scope":"NumberTheory/arithmetic-functions/arithmetic-derivative","status":"fully_proved","summary":"The arithmetic derivative of $0$ is $0$: $D(0) = 0$.","title":"Arithmetic derivative of zero"},{"id":"NumberTheory/arithmetic-functions/arithmetic-derivative/arithmetic-derivative-one","kind":"node","lean":"MetaMathlibExt.arithmeticDerivative_one","parent":8860,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/arithmetic-derivative/index.html#arithmetic-derivative-one","scope":"NumberTheory/arithmetic-functions/arithmetic-derivative","status":"fully_proved","summary":"The arithmetic derivative of $1$ is $0$: $D(1) = 0$.","title":"Arithmetic derivative of one"},{"id":"NumberTheory/arithmetic-functions/arithmetic-derivative/arithmetic-derivative-prime","kind":"node","lean":"MetaMathlibExt.arithmeticDerivative_prime","parent":8860,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/arithmetic-derivative/index.html#arithmetic-derivative-prime","scope":"NumberTheory/arithmetic-functions/arithmetic-derivative","status":"fully_proved","summary":"For every prime $p$, the arithmetic derivative of $p$ is $1$: $D(p) = 1$.","title":"Arithmetic derivative of a prime"},{"id":"NumberTheory/arithmetic-functions/arithmetic-derivative/arithmetic-derivative-eq-sum-of-subset","kind":"node","lean":"MetaMathlibExt.arithmeticDerivative_eq_sum_of_subset","parent":8860,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/arithmetic-derivative/index.html#arithmetic-derivative-eq-sum-of-subset","scope":"NumberTheory/arithmetic-functions/arithmetic-derivative","status":"fully_proved","summary":"Let $n$ be a natural number and let $s$ be a finite set of natural numbers containing every prime factor of $n$. Then the arithmetic derivative of $n$ is $D(n) = \\sum_{p \\in s} \\nu_p(n) \\cdot (n / p)$, where $\\nu_p(n)$ is the exponent of $p$ in the prime factorization of $n$ (zero when $p$ is not a prime factor of $n$) and $n / p$ is division in the natural numbers, rounded down.","title":"Arithmetic derivative over a superset of the prime factors"},{"id":"NumberTheory/arithmetic-functions/arithmetic-derivative/arithmetic-derivative-mul","kind":"node","lean":"MetaMathlibExt.arithmeticDerivative_mul","parent":8860,"result":true,"route":"../roadmap/NumberTheory/arithmetic-functions/arithmetic-derivative/index.html#arithmetic-derivative-mul","scope":"NumberTheory/arithmetic-functions/arithmetic-derivative","status":"fully_proved","summary":"For all natural numbers $m$ and $n$, the arithmetic derivative $D$ satisfies $D(mn) = D(m)\\,n + m\\,D(n)$.","title":"Leibniz rule for the arithmetic derivative"},{"data_url":"scopes/NumberTheory/zeta-l-functions.json","id":"NumberTheory/zeta-l-functions","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/index.html","scope":"NumberTheory/zeta-l-functions","status":"can_prove","summary":"Riemann zeta and xi, Dirichlet characters and L-functions, Dedekind and Artin L-functions, Riemann hypothesis criteria, multiple zeta values.","title":"Zeta and L-functions"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta.json","id":"NumberTheory/zeta-l-functions/riemann-zeta","kind":"scope","parent":8867,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta","status":"can_prove","summary":"Continuation and zeros of \u03b6, zero-free regions, moments and subconvexity, RH criteria (Robin, Lagarias), de Bruijn\u2013Newman constant, zeta values.","title":"Riemann zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/franel-landau","kind":"node","lean":"MetaMathlibExt.franel_landau","parent":8868,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/index.html#franel-landau","scope":"NumberTheory/zeta-l-functions/riemann-zeta","status":"can_prove","summary":"For a natural number $N$, let $F_N(0) < F_N(1) < \\dots < F_N(L_N - 1)$ be the terms of the Farey sequence of order $N$, and for $k < L_N$ let $\\delta_{N,k} = F_N(k) - k/(L_N - 1)$. The Riemann hypothesis holds if and only if for every real $\\varepsilon > 0$, $\\sum_{k=0}^{L_N - 1} \\delta_{N,k}^2 = O(N^{-1+\\varepsilon})$ as $N \\to \\infty$.","title":"Franel\u2013Landau theorem"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/zeta-three-infinite-product-formulas","kind":"node","lean":"MetaMathlibExt.zeta_three_infinite_product_formulas","parent":8868,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kachi\u2013Tzermias infinite products for "],["math","\\(\\zeta(3)\\)"]],"source":"Kachi\u2013Tzermias infinite products for $\\zeta(3)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/index.html#zeta-three-infinite-product-formulas","scope":"NumberTheory/zeta-l-functions/riemann-zeta","status":"fully_proved","summary":"Let $\\zeta$ be the Riemann zeta function and, for $n \\ge 1$, put $a_n = \\left(1 - \\frac{1}{n+1}\\right)^{(-1)^n n(n+1)/2}$. Then, as $m \\to \\infty$,","title":"Kachi\u2013Tzermias infinite products for $\\zeta(3)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-hypothesis-iff-lagarias-elementa","kind":"node","lean":"MathlibExt.NumberTheory.LagariasWanted.NumberTheory.riemann_hypothesis_iff_lagarias_elementary_criterion","parent":8868,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/index.html#riemann-hypothesis-iff-lagarias-elementa","scope":"NumberTheory/zeta-l-functions/riemann-zeta","status":"can_prove","summary":"The Riemann hypothesis holds (every zero $s$ of the Riemann zeta function with $s \\ne 1$ that is not one of the trivial zeros $-2, -4, -6, \\dots$ has real part $1/2$) if and only if, for every positive integer $n$,","title":"Lagarias' elementary criterion for the Riemann hypothesis"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/meromorphic-on-riemann-zeta","kind":"node","lean":"meromorphicOn_riemannZeta","parent":8868,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/index.html#meromorphic-on-riemann-zeta","scope":"NumberTheory/zeta-l-functions/riemann-zeta","status":"fully_proved","summary":"The Riemann zeta function $\\zeta$ is meromorphic on all of $\\mathbb{C}$.","title":"Meromorphy of the Riemann zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/ramanujan-part1-ch7-entry2-faulhaber","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry2Faulhaber.ramanujan_part1_ch7_entry2_faulhaber","parent":8868,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's formula for the alternating zeta function at "],["math","\\(-r\\)"]],"source":"Ramanujan's formula for the alternating zeta function at $-r$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/index.html#ramanujan-part1-ch7-entry2-faulhaber","scope":"NumberTheory/zeta-l-functions/riemann-zeta","status":"fully_proved","summary":"Let $\\eta(s) = (1 - 2^{1-s})\\,\\zeta(s)$ for $s \\ne 1$ and $\\eta(1) = \\log 2$, where $\\zeta$ is the Riemann zeta function, and let $B^*(z) = 2\\,\\Gamma(z + 1)\\,\\zeta(z) / (2\\pi)^z$. Then for every $s \\in \\mathbb{C}$ with $\\mathrm{Re}\\, s > 0$ the partial sums $\\sum_{j=0}^{N-1} (-1)^j / (j+1)^s$ tend to $\\eta(s)$ as $N \\to \\infty$; the functions $r \\mapsto \\eta(-r)$ and\u2026","title":"Ramanujan's formula for the alternating zeta function at $-r$"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","status":"can_prove","title":"De Bruijn\u2013Newman constant"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/phi-term","kind":"node","lean":"MathlibExt.NumberTheory.DeBruijnNewman.phiTerm","parent":8874,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/index.html#phi-term","scope":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","status":"fully_proved","summary":"For real $u$ and a natural number $n$, the $n$-th summand of Newman's kernel, counted from $n = 0$, is","title":"Summands of Newman's kernel"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/phi","kind":"node","lean":"MathlibExt.NumberTheory.DeBruijnNewman.phi","parent":8874,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/index.html#phi","scope":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","status":"fully_proved","summary":"For real $u$, Newman's kernel is the sum of the series","title":"Newman's kernel"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/h","kind":"node","lean":"MathlibExt.NumberTheory.DeBruijnNewman.H","parent":8874,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/index.html#h","scope":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","status":"fully_proved","summary":"For real $\\lambda$ and complex $z$,","title":"The de Bruijn\u2013Newman deformation"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/zeros-all-real","kind":"node","lean":"MathlibExt.NumberTheory.DeBruijnNewman.ZerosAllReal","parent":8874,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/index.html#zeros-all-real","scope":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","status":"fully_proved","summary":"For real $\\lambda$, this is the statement that every zero of the de Bruijn\u2013Newman deformation $H_\\lambda$ is real: every complex $z$ with $H_\\lambda(z) = 0$ has imaginary part $0$.","title":"Real-rootedness of the de Bruijn\u2013Newman deformation"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/de-bruijn-newman-constant","kind":"node","lean":"MathlibExt.NumberTheory.DeBruijnNewman.deBruijnNewmanConstant","parent":8874,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/index.html#de-bruijn-newman-constant","scope":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","status":"fully_proved","summary":"The de Bruijn\u2013Newman constant $\\Lambda$ is the infimum of the set of real numbers $\\lambda$ such that every zero of the de Bruijn\u2013Newman deformation $H_\\lambda$ is real. The infimum is taken in $\\mathbb{R}$, with the convention that it is $0$ when this set is empty or not bounded below.","title":"The de Bruijn\u2013Newman constant"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/rodgers-tao-negative-time-wanted","kind":"node","lean":"MathlibExt.NumberTheory.DeBruijnNewmanWanted.rodgersTaoNegativeTime_wanted","parent":8874,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/index.html#rodgers-tao-negative-time-wanted","scope":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","status":"can_prove","summary":"For every real $\\lambda < 0$, not every zero of the de Bruijn\u2013Newman deformation $H_\\lambda$ is real: there is a complex $z$ with $H_\\lambda(z) = 0$ and nonzero imaginary part.","title":"Nonreal zeros of the de Bruijn\u2013Newman deformation at negative times"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/platt-trudgian-upper-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.DeBruijnNewmanWanted.plattTrudgianUpperBound_wanted","parent":8874,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant/index.html#platt-trudgian-upper-bound-wanted","scope":"NumberTheory/zeta-l-functions/riemann-zeta/de-bruijn-newman-constant","status":"can_prove","summary":"Every zero of the de Bruijn\u2013Newman deformation $H_{1/5}$ is real: every complex $z$ with $H_{1/5}(z) = 0$ has imaginary part $0$.","title":"All zeros of the de Bruijn\u2013Newman deformation are real at time 1/5"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be","status":"can_prove","title":"Euler's evaluations of odd zeta and Dirichlet beta series"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/has-sum-one-div-odd-pow","kind":"node","lean":"MetaMathlibExt.hasSum_one_div_odd_pow","parent":8882,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/index.html#has-sum-one-div-odd-pow","scope":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be","status":"can_prove","summary":"Let $\\zeta$ be the Riemann zeta function. For every natural number $k \\ge 2$, the series $\\sum_{n \\ge 0} 1/(2n+1)^k$ converges in $\\mathbb{C}$ and","title":"Zeta series over odd denominators"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/dirichlet-beta-two-mul-add-one","kind":"node","lean":"MetaMathlibExt.dirichletBeta_two_mul_add_one","parent":8882,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/index.html#dirichlet-beta-two-mul-add-one","scope":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be","status":"can_prove","summary":"Let $\\beta$ be the Dirichlet $L$-function of the nontrivial character modulo $4$, which for $\\operatorname{Re} s > 1$ is the series $\\beta(s) = \\sum_{n \\ge 1} (-1)^{n-1}/(2n-1)^s$, and let $E_0 = 1, E_2 = -1, E_4 = 5, E_6 = -61, \\dots$ be the even-index Euler numbers. Then for every natural number $m$,","title":"Dirichlet beta at odd positive integers"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/tendsto-integral-sinc-pow","kind":"node","lean":"MetaMathlibExt.tendsto_integral_sinc_pow","parent":8882,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrals of powers of "],["math","\\(\\sin x / x\\)"]],"source":"Integrals of powers of $\\sin x / x$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/index.html#tendsto-integral-sinc-pow","scope":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be","status":"can_prove","summary":"For every natural number $k \\ge 1$, the improper integral $\\lim_{R \\to \\infty} \\int_0^R (\\sin x / x)^k \\, dx$ exists and","title":"Integrals of powers of $\\sin x / x$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/tendsto-sum-sin-pow-div-pow","kind":"node","lean":"MetaMathlibExt.tendsto_sum_sin_pow_div_pow","parent":8882,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of powers of "],["math","\\(\\sin(nt)/n\\)"]],"source":"Sums of powers of $\\sin(nt)/n$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be/index.html#tendsto-sum-sin-pow-div-pow","scope":"NumberTheory/zeta-l-functions/riemann-zeta/euler-s-evaluations-of-odd-zeta-and-dirichlet-be","status":"can_prove","summary":"Let $k \\ge 1$ be a natural number, suppose that the limit $L = \\lim_{R \\to \\infty} \\int_0^R (\\sin x / x)^k \\, dx$ exists, and let $t$ be a real number with $0 < t < 2\\pi$ if $k = 1$ and $0 \\le t \\le 2\\pi/k$ if $k \\ge 2$. Then the partial sums of $\\sum_{n \\ge 1} \\sin^k(nt)/n^k$ converge and","title":"Sums of powers of $\\sin(nt)/n$"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function","status":"can_prove","title":"Moments and subconvexity of the zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/riemann-zeta-twelfth-moment-le","kind":"node","lean":"MetaMathlibExt.riemannZeta_twelfth_moment_le","parent":8887,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/index.html#riemann-zeta-twelfth-moment-le","scope":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function","status":"can_prove","summary":"For every $\\varepsilon > 0$ there is a constant $C$ such that for every real $T \\ge 1$,","title":"Twelfth moment of the Riemann zeta function on the critical line"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/riemann-zeta-twelfth-moment-le-log","kind":"node","lean":"MetaMathlibExt.riemannZeta_twelfth_moment_le_log","parent":8887,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/index.html#riemann-zeta-twelfth-moment-le-log","scope":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function","status":"can_prove","summary":"There are constants $C$ and $T_0$ such that for every real $T \\ge T_0$,","title":"Heath-Brown's twelfth moment bound with the logarithmic factor"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/riemann-zeta-twelfth-moment-five-seventh","kind":"node","lean":"MetaMathlibExt.riemannZeta_twelfth_moment_five_sevenths_le","parent":8887,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/index.html#riemann-zeta-twelfth-moment-five-seventh","scope":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function","status":"can_prove","summary":"For every $\\varepsilon > 0$ there is a constant $C$ such that for every real $T \\ge 1$,","title":"Twelfth moment of the Riemann zeta function on the line of real part 5/7"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/riemann-zeta-norm-le-subconvex","kind":"node","lean":"MetaMathlibExt.riemannZeta_norm_le_subconvex","parent":8887,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/index.html#riemann-zeta-norm-le-subconvex","scope":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function","status":"can_prove","summary":"For every $\\varepsilon > 0$ there is a constant $C$ such that for all real $\\sigma$ and $t$ with $1/2 \\le \\sigma \\le 1 + \\varepsilon$ and $|t| \\ge 1$,","title":"Subconvexity bound for the Riemann zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/dirichlet-lfunction-mod-four-norm-le-sub","kind":"node","lean":"MetaMathlibExt.dirichletLFunction_mod_four_norm_le_subconvex","parent":8887,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function/index.html#dirichlet-lfunction-mod-four-norm-le-sub","scope":"NumberTheory/zeta-l-functions/riemann-zeta/moments-and-subconvexity-of-the-zeta-function","status":"can_prove","summary":"Let $\\varepsilon > 0$ and let $\\chi$ be a non-principal Dirichlet character modulo $4$ (there is exactly one). Then there is a constant $C$ such that for all real $\\sigma$ and $t$ with $1/2 \\le \\sigma \\le 1 + \\varepsilon$ and $|t| \\ge 1$,","title":"Subconvexity bound for the Dirichlet L-function modulo 4"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values","status":"can_prove","title":"Paperfolding series for odd zeta values"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/hurwitz-zeta-three-quarters-two-mul-add","kind":"node","lean":"MetaMathlibExt.hurwitzZeta_three_quarters_two_mul_add_one","parent":8893,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hurwitz zeta at "],["math","\\(3/4\\)"],["text"," and odd arguments"]],"source":"Hurwitz zeta at $3/4$ and odd arguments"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/index.html#hurwitz-zeta-three-quarters-two-mul-add","scope":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values","status":"can_prove","summary":"For every integer $k \\ge 1$,","title":"Hurwitz zeta at $3/4$ and odd arguments"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/riemann-zeta-two-mul-add-one-sub-eq-tsum","kind":"node","lean":"MetaMathlibExt.riemannZeta_two_mul_add_one_sub_eq_tsum_paperfolding_thueMorse","parent":8893,"reading_title":{"format":"inline-parts-v1","parts":[["text","T\u00f3th's paperfolding and Thue\u2013Morse series for "],["math","\\(\\zeta(2k+1)\\)"]],"source":"T\u00f3th's paperfolding and Thue\u2013Morse series for $\\zeta(2k+1)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/index.html#riemann-zeta-two-mul-add-one-sub-eq-tsum","scope":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values","status":"can_prove","summary":"Let $(f_n)_{n \\ge 0}$ be the regular paperfolding sequence and $(t_n)_{n \\ge 0}$ the Thue\u2013Morse sequence, both indexed from $0$, and let $E_{2k}$ be the even-index Euler number. For every integer $k \\ge 1$,","title":"T\u00f3th's paperfolding and Thue\u2013Morse series for $\\zeta(2k+1)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/tsum-paperfolding-two-mul-add-one","kind":"node","lean":"MetaMathlibExt.tsum_paperfolding_two_mul_add_one","parent":8893,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/index.html#tsum-paperfolding-two-mul-add-one","scope":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values","status":"can_prove","summary":"Let $(f_n)_{n \\ge 0}$ be the regular paperfolding sequence, indexed from $0$, and let $E_{2k}$ be the even-index Euler number. For every integer $k \\ge 1$,","title":"A paperfolding Dirichlet series at odd integers"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/tsum-neg-one-pow-paperfolding","kind":"node","lean":"MetaMathlibExt.tsum_neg_one_pow_paperfolding","parent":8893,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\pm 1\\)"],["text"," paperfolding Dirichlet series at odd integers"]],"source":"The $\\pm 1$ paperfolding Dirichlet series at odd integers"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values/index.html#tsum-neg-one-pow-paperfolding","scope":"NumberTheory/zeta-l-functions/riemann-zeta/paperfolding-series-for-odd-zeta-values","status":"can_prove","summary":"Let $(f_n)_{n \\ge 0}$ be the regular paperfolding sequence, indexed from $0$, and let $E_{2k}$ be the even-index Euler number. For every integer $k \\ge 1$,","title":"The $\\pm 1$ paperfolding Dirichlet series at odd integers"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","title":"Riemann xi function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/riemann-xi","kind":"node","lean":"RiemannXi.riemannXi","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#riemann-xi","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"Let $\\Lambda$ be the completed Riemann zeta function, the meromorphic continuation of $\\pi^{-s/2}\\Gamma(s/2)\\zeta(s)$, which has simple poles at $s = 0$ and $s = 1$, and let $\\Lambda_0$ be the entire function with $\\Lambda_0(s) = \\Lambda(s) + \\frac{1}{s} + \\frac{1}{1 - s}$ for $s \\ne 0, 1$. The Riemann xi function is $\\xi(s) = \\frac{s(s - 1)}{2}\\Lambda_0(s) + \\frac{1}{2}$ for $s \\in \\mathbb{C}$.","title":"Riemann xi function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/differentiable-riemann-xi","kind":"node","lean":"RiemannXi.differentiable_riemannXi","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#differentiable-riemann-xi","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"The Riemann xi function $\\xi$ is complex differentiable at every point of $\\mathbb{C}$.","title":"The Riemann xi function is entire"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/riemann-xi-one-sub","kind":"node","lean":"RiemannXi.riemannXi_one_sub","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#riemann-xi-one-sub","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"Let $\\xi$ be the Riemann xi function. For every complex number $s$, $\\xi(1 - s) = \\xi(s)$.","title":"Functional equation of the Riemann xi function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/riemann-xi-eq-mul-completed-riemann-zeta","kind":"node","lean":"RiemannXi.riemannXi_eq_mul_completedRiemannZeta","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#riemann-xi-eq-mul-completed-riemann-zeta","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"Let $\\xi$ be the Riemann xi function and let $\\Lambda$ be the completed Riemann zeta function, the meromorphic continuation of $\\pi^{-s/2}\\Gamma(s/2)\\zeta(s)$, which has simple poles at $s = 0$ and $s = 1$. For every complex number $s$ with $s \\ne 0$ and $s \\ne 1$, $\\xi(s) = \\frac{s(s - 1)}{2}\\Lambda(s)$.","title":"The xi function in terms of the completed zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/riemann-xi-ne-zero-of-one-le-re","kind":"node","lean":"RiemannXi.riemannXi_ne_zero_of_one_le_re","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#riemann-xi-ne-zero-of-one-le-re","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"Let $\\xi$ be the Riemann xi function. For every complex number $s$ with $\\operatorname{Re}(s) \\ge 1$, $\\xi(s) \\ne 0$.","title":"The xi function has no zeros with real part at least one"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/re-pos-of-riemann-xi-eq-zero","kind":"node","lean":"RiemannXi.re_pos_of_riemannXi_eq_zero","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#re-pos-of-riemann-xi-eq-zero","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"Let $\\xi$ be the Riemann xi function. If $s$ is a complex number with $\\xi(s) = 0$, then $\\operatorname{Re}(s) > 0$.","title":"Zeros of the xi function have positive real part"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/re-lt-one-of-riemann-xi-eq-zero","kind":"node","lean":"RiemannXi.re_lt_one_of_riemannXi_eq_zero","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#re-lt-one-of-riemann-xi-eq-zero","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"Let $\\xi$ be the Riemann xi function. If $s$ is a complex number with $\\xi(s) = 0$, then $\\operatorname{Re}(s) < 1$.","title":"Zeros of the xi function have real part less than one"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/riemann-xi-mem-strip","kind":"node","lean":"RiemannXi.riemannXi_mem_strip","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#riemann-xi-mem-strip","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"Let $\\xi$ be the Riemann xi function. If $s$ is a complex number with $\\xi(s) = 0$, then $0 < \\operatorname{Re}(s) < 1$.","title":"Zeros of the xi function lie in the critical strip"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/riemann-xi-entire-functional-equation-an","kind":"node","lean":"RiemannXi.riemannXi_entire_functional_equation_and_critical_strip","parent":8898,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function/index.html#riemann-xi-entire-functional-equation-an","scope":"NumberTheory/zeta-l-functions/riemann-zeta/riemann-xi-function","status":"fully_proved","summary":"The Riemann xi function $\\xi$ is entire, satisfies $\\xi(1 - s) = \\xi(s)$ for every complex number $s$, and every complex number $s$ with $\\xi(s) = 0$ satisfies $0 < \\operatorname{Re}(s) < 1$.","title":"Basic properties of the Riemann xi function"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","title":"Robin's inequality"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/robin-criterion-for-riemann-hypothesis","kind":"node","lean":"MetaMathlibExt.robin_criterion_for_riemann_hypothesis","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#robin-criterion-for-riemann-hypothesis","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"The Riemann hypothesis holds if and only if $\\sigma(n) < e^{\\gamma}\\, n \\log\\log n$ for every natural number $n > 5040$, where $\\sigma(n)$ is the sum of the positive divisors of $n$, $\\gamma$ is the Euler\u2013Mascheroni constant, and $\\log$ is the natural logarithm.","title":"Robin's criterion for the Riemann hypothesis"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/robin-exceptional-set","kind":"node","lean":"MetaMathlibExt.robinExceptionalSet","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#robin-exceptional-set","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"fully_proved","summary":"For a real number $x$, the Robin exceptional set up to $x$ is the set of natural numbers $n$ with $7! = 5040 < n \\le x$ that fail Robin's inequality, that is, with $\\sigma(n) \\ge e^\\gamma n \\log\\log n$, where $\\sigma(n)$ is the sum of the divisors of $n$ and $\\gamma$ is the Euler\u2013Mascheroni constant. Its cardinality is the counting function $N_R(x)$.","title":"Robin exceptional set"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/robin-exceptional-set-eq-empty-of-le","kind":"node","lean":"MetaMathlibExt.robinExceptionalSet_eq_empty_of_le","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Robin exceptional set is empty up to "],["math","\\(7!\\)"]],"source":"The Robin exceptional set is empty up to $7!$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#robin-exceptional-set-eq-empty-of-le","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"fully_proved","summary":"For every real $x \\le 7! = 5040$, the Robin exceptional set up to $x$ is empty; that is, its cardinality $N_R(x)$ is $0$.","title":"The Robin exceptional set is empty up to $7!$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/card-robin-exceptional-set-is-big-o-div","kind":"node","lean":"MetaMathlibExt.card_robinExceptionalSet_isBigO_div_log_log","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Failures of Robin's inequality number "],["math","\\(O(x/\\log\\log x)\\)"]],"source":"Failures of Robin's inequality number $O(x/\\log\\log x)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#card-robin-exceptional-set-is-big-o-div","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $N_R(x)$ be the number of elements of the Robin exceptional set up to $x$. Then $N_R(x) = O(x / \\log\\log x)$ as the real number $x$ tends to infinity.","title":"Failures of Robin's inequality number $O(x/\\log\\log x)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/card-robin-exceptional-set-is-big-o-div-2","kind":"node","lean":"MetaMathlibExt.card_robinExceptionalSet_isBigO_div_sqrt_log","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Failures of Robin's inequality number "],["math","\\(O(x/\\sqrt{\\log x})\\)"]],"source":"Failures of Robin's inequality number $O(x/\\sqrt{\\log x})$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#card-robin-exceptional-set-is-big-o-div-2","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $N_R(x)$ be the number of elements of the Robin exceptional set up to $x$. Then $N_R(x) = O(x / \\sqrt{\\log x})$ as the real number $x$ tends to infinity.","title":"Failures of Robin's inequality number $O(x/\\sqrt{\\log x})$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/card-robin-exceptional-set-is-big-o-rpow","kind":"node","lean":"MetaMathlibExt.card_robinExceptionalSet_isBigO_rpow","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Failures of Robin's inequality number "],["math","\\(O(x^\\varepsilon)\\)"]],"source":"Failures of Robin's inequality number $O(x^\\varepsilon)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#card-robin-exceptional-set-is-big-o-rpow","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $N_R(x)$ be the number of elements of the Robin exceptional set up to $x$. For every real $\\varepsilon > 0$, $N_R(x) = O(x^\\varepsilon)$ as the real number $x$ tends to infinity.","title":"Failures of Robin's inequality number $O(x^\\varepsilon)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/harmonic-extension","kind":"node","lean":"MetaMathlibExt.harmonicExtension","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#harmonic-extension","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"fully_proved","summary":"For a real number $x$, let","title":"The real extension of the harmonic numbers"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/harmonic-extension-zero","kind":"node","lean":"MetaMathlibExt.harmonicExtension_zero","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#harmonic-extension-zero","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"fully_proved","summary":"The real extension $H(x) = \\int_0^1 \\frac{t^x - 1}{t - 1}\\,dt$ of the harmonic numbers satisfies $H(0) = 0$.","title":"The real extension of the harmonic numbers vanishes at zero"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/harmonic-extension-nat-cast","kind":"node","lean":"MetaMathlibExt.harmonicExtension_natCast","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#harmonic-extension-nat-cast","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $H(x) = \\int_0^1 \\frac{t^x - 1}{t - 1}\\,dt$ for real $x$ be the real extension of the harmonic numbers. For every natural number $n$, $H(n) = H_n$, where $H_n = \\sum_{i=1}^n 1/i$ is the $n$-th harmonic number.","title":"The real extension agrees with the harmonic numbers"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/nth-prime-bounds","kind":"node","lean":"MetaMathlibExt.nth_prime_bounds","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rosser\u2013Schoenfeld and Dusart bounds for the "],["math","\\(k\\)"],["text","-th prime"]],"source":"Rosser\u2013Schoenfeld and Dusart bounds for the $k$-th prime"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#nth-prime-bounds","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $p_k$ be the $k$-th prime ($p_1 = 2$) and $\\log$ the natural logarithm. For every natural number $k \\ge 6$,","title":"Rosser\u2013Schoenfeld and Dusart bounds for the $k$-th prime"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/log-log-primorial-div-prime-gt","kind":"node","lean":"MetaMathlibExt.log_log_primorial_div_prime_gt","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#log-log-primorial-div-prime-gt","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $p_\\ell$ be the $\\ell$-th prime ($p_1 = 2$) and $\\log$ the natural logarithm. For natural numbers $k \\ge 99$ and $j$ with $1 \\le j \\le k + 1$, let $m = p_1 p_2 \\cdots p_{k+1} / p_j$. Then","title":"Lower bound for the double logarithm of a primorial divided by one of its primes"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/robin-inequality-of-not-dvd","kind":"node","lean":"MetaMathlibExt.robin_inequality_of_not_dvd","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Robin's inequality for "],["math","\\(n > 5040\\)"],["text"," when one of "],["math","\\(2, 3, 5\\)"],["text"," does not divide "],["math","\\(n\\)"]],"source":"Robin's inequality for $n > 5040$ when one of $2, 3, 5$ does not divide $n$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#robin-inequality-of-not-dvd","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $\\sigma(n)$ denote the sum of the divisors of $n$ and $\\gamma$ the Euler\u2013Mascheroni constant. Let $p \\in \\{2, 3, 5\\}$ and let $n > 5040$ be a natural number not divisible by $p$. Then Robin's inequality $\\sigma(n) < e^{\\gamma} n \\log \\log n$ holds.","title":"Robin's inequality for $n > 5040$ when one of $2, 3, 5$ does not divide $n$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/robin-inequality-two-pow-mul-odd","kind":"node","lean":"MetaMathlibExt.robin_inequality_two_pow_mul_odd","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Robin's inequality for "],["math","\\(2^k n\\)"],["text"," with "],["math","\\(n\\)"],["text"," odd and large"]],"source":"Robin's inequality for $2^k n$ with $n$ odd and large"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#robin-inequality-two-pow-mul-odd","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $\\sigma(N)$ denote the sum of the divisors of $N$ and $\\gamma$ the Euler\u2013Mascheroni constant. Let $k \\ge 1$ and let $n$ be an odd natural number with $n > b(k) = e^{e^{\\sqrt{2.51(2^{k+1} - 1)/e^{\\gamma}}}} / 2^k$. Then Robin's inequality $\\sigma(2^k n) < e^{\\gamma} 2^k n \\log \\log (2^k n)$ holds.","title":"Robin's inequality for $2^k n$ with $n$ odd and large"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/robin-inequality-two-mul-odd-and-primori","kind":"node","lean":"MetaMathlibExt.robin_inequality_two_mul_odd_and_primorial","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Robin's inequality for "],["math","\\(2n\\)"],["text"," with "],["math","\\(n \\ge 620\\)"],["text"," odd and for primorials above "],["math","\\(30\\)"]],"source":"Robin's inequality for $2n$ with $n \\ge 620$ odd and for primorials above $30$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#robin-inequality-two-mul-odd-and-primori","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $\\sigma(N)$ denote the sum of the divisors of $N$ and $\\gamma$ the Euler\u2013Mascheroni constant. For every odd natural number $n \\ge 620$, Robin's inequality $\\sigma(2n) < e^{\\gamma} \\cdot 2n \\log \\log (2n)$ holds. For every natural number $m$ whose primorial $m\\# = \\prod_{p \\le m,\\ p \\text{ prime}} p$ exceeds $30$, Robin's inequality $\\sigma(m\\#) < e^{\\gamma} \\, m\\# \\log \\log (m\\#)$ holds.","title":"Robin's inequality for $2n$ with $n \\ge 620$ odd and for primorials above $30$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/robin-inequality-two-pow-mul-odd-and-twe","kind":"node","lean":"MetaMathlibExt.robin_inequality_two_pow_mul_odd_and_twentyOne_free","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Robin's inequality for "],["math","\\(2^k n > 5040\\)"],["text"," with "],["math","\\(n\\)"],["text"," odd under a bound on "],["math","\\(k\\)"],["text",", and for "],["math","\\(21\\)"],["text","-free "],["math","\\(n > 5040\\)"]],"source":"Robin's inequality for $2^k n > 5040$ with $n$ odd under a bound on $k$, and for $21$-free $n > 5040$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#robin-inequality-two-pow-mul-odd-and-twe","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Robin's inequality for $2^k n > 5040$ with $n$ odd under a bound on $k$, and for $21$-free $n > 5040$. Let $\\sigma(N)$ denote the sum of the divisors of $N$ and $\\gamma$ the Euler\u2013Mascheroni constant, and let $C = 10^{10^{13.11485}}$. First, let $k \\ge 1$ be such that $2^k \\tilde b(k) = e^{e^{(0.0168(2^{k+1} - 1)/e^{\\gamma})^{1/3}}} < C$. Then for every odd natural number $n$ with $2^k n > 5040$, Robin's inequality\u2026","title":"Robin's inequality for $2^k n > 5040$ with $n$ odd under a bound on $k$, and for $21$-free $n > 5040$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/robin-inequality-density-one","kind":"node","lean":"MetaMathlibExt.robin_inequality_density_one","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#robin-inequality-density-one","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $\\sigma(n)$ be the sum of the divisors of $n$, $\\gamma$ the Euler\u2013Mascheroni constant and $\\log$ the natural logarithm. The proportion of the integers $n$ with $1 \\le n \\le N$ that satisfy Robin's inequality $\\sigma(n) < e^{\\gamma} n \\log \\log n$ tends to $1$ as $N \\to \\infty$.","title":"Robin's inequality holds on a set of natural density one"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/harmonic-extension-lt-log-add","kind":"node","lean":"MetaMathlibExt.harmonicExtension_lt_log_add","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound "],["math","\\(H(x) < \\log x + \\gamma + 1/(2x)\\)"]],"source":"Upper bound $H(x) < \\log x + \\gamma + 1/(2x)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#harmonic-extension-lt-log-add","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"With $H$ the real extension of the harmonic numbers, $\\log$ the natural logarithm and $\\gamma$ the Euler\u2013Mascheroni constant, every real $x \\ge 1$ satisfies $H(x) < \\log x + \\gamma + \\frac{1}{2x}$.","title":"Upper bound $H(x) < \\log x + \\gamma + 1/(2x)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/harmonic-extension-lt-two-log","kind":"node","lean":"MetaMathlibExt.harmonicExtension_lt_two_log","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logarithmic upper bound for "],["math","\\(H(x)\\)"],["text"," with a "],["math","\\(\\pi^2\\)"],["text"," correction"]],"source":"Logarithmic upper bound for $H(x)$ with a $\\pi^2$ correction"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#harmonic-extension-lt-two-log","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"With $H$ the real extension of the harmonic numbers and $\\log$ the natural logarithm, every real $x \\ge 4$ satisfies","title":"Logarithmic upper bound for $H(x)$ with a $\\pi^2$ correction"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/harmonic-succ-le-and-log-harmonic-extens","kind":"node","lean":"MetaMathlibExt.harmonic_succ_le_and_log_harmonicExtension_le","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#harmonic-succ-le-and-log-harmonic-extens","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $H_m = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{m}$ be the $m$-th harmonic number, $H(x)$ the real extension $\\int_0^1 \\frac{t^x - 1}{t - 1}\\,dt$ of the harmonic numbers, and $\\log$ the natural logarithm. Then $H_{n+1} \\le n / \\log n$ for every natural number $n > 1$, and $\\log H(x) \\le x / (2 \\log x)$ for every real number $x \\ge 4$.","title":"Upper bounds for harmonic numbers and for the logarithm of their real extension"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/harmonic-extension-mul-log-lt","kind":"node","lean":"MetaMathlibExt.harmonicExtension_mul_log_lt","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for "],["math","\\(H(x) \\log H(x)\\)"]],"source":"Upper bound for $H(x) \\log H(x)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#harmonic-extension-mul-log-lt","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"With $H$ the real extension of the harmonic numbers and $\\log$ the natural logarithm, every real $x \\ge 4$ satisfies","title":"Upper bound for $H(x) \\log H(x)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/harmonic-extension-mul-log-div-sq-lt-der","kind":"node","lean":"MetaMathlibExt.harmonicExtension_mul_log_div_sq_lt_deriv","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for the derivative "],["math","\\(H'(x)\\)"]],"source":"Lower bound for the derivative $H'(x)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#harmonic-extension-mul-log-div-sq-lt-der","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"With $H$ the real extension of the harmonic numbers and $\\log$ the natural logarithm, every real $x \\ge 4$ satisfies","title":"Lower bound for the derivative $H'(x)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/exp-harmonic-extension-mul-log-div-monot","kind":"node","lean":"MetaMathlibExt.exp_harmonicExtension_mul_log_div_monotoneOn","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity of "],["math","\\(e^{H(x)} \\log H(x) / x\\)"]],"source":"Monotonicity of $e^{H(x)} \\log H(x) / x$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#exp-harmonic-extension-mul-log-div-monot","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"With $H$ the real extension of the harmonic numbers and $\\log$ the natural logarithm, the function $g(x) = e^{H(x)} \\log H(x) / x$ is non-decreasing on $[4, \\infty)$: $g(x) \\le g(y)$ whenever $4 \\le x \\le y$.","title":"Monotonicity of $e^{H(x)} \\log H(x) / x$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/exp-harmonic-mul-log-div-monotone-on","kind":"node","lean":"MetaMathlibExt.exp_harmonic_mul_log_div_monotoneOn","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity of "],["math","\\(e^{H_n} \\log H_n / n\\)"]],"source":"Monotonicity of $e^{H_n} \\log H_n / n$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#exp-harmonic-mul-log-div-monotone-on","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"With $H_n = \\sum_{i=1}^n 1/i$ the $n$-th harmonic number and $\\log$ the natural logarithm, the sequence $e^{H_n} \\log H_n / n$ is non-decreasing for $n \\ge 1$: its value at $m$ is at most its value at $n$ whenever $1 \\le m \\le n$.","title":"Monotonicity of $e^{H_n} \\log H_n / n$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/least-kaneko-lagarias-counterexample-sup","kind":"node","lean":"MetaMathlibExt.least_kaneko_lagarias_counterexample_superabundant","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#least-kaneko-lagarias-counterexample-sup","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $\\sigma(n)$ be the sum of the divisors of $n$, $H_n = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n}$ the $n$-th harmonic number and $\\log$ the natural logarithm. Let $n_0 > 60$ be a natural number for which the Kaneko\u2013Lagarias inequality $\\sigma(n_0) < e^{H_{n_0}} \\log H_{n_0}$ fails, while $\\sigma(n) < e^{H_n} \\log H_n$ holds for every natural number $n$ with $60 < n < n_0$. Then $n_0$ is superabundant:\u2026","title":"A least counterexample to the Kaneko\u2013Lagarias inequality is superabundant"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/kaneko-lagarias-of-robin","kind":"node","lean":"MetaMathlibExt.kaneko_lagarias_of_robin","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#kaneko-lagarias-of-robin","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $\\sigma(n)$ be the sum of the divisors of $n$, $\\gamma$ the Euler\u2013Mascheroni constant, $\\log$ the natural logarithm and $H_n = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n}$ the $n$-th harmonic number. For every natural number $n$, if Robin's inequality $\\sigma(n) < e^{\\gamma} n \\log \\log n$ holds, then so does the Kaneko\u2013Lagarias inequality $\\sigma(n) < e^{H_n} \\log H_n$.","title":"Robin's inequality implies the Kaneko\u2013Lagarias inequality"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/prod-succ-div-prime-le","kind":"node","lean":"MetaMathlibExt.prod_succ_div_prime_le","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#prod-succ-div-prime-le","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $p_\\ell$ be the $\\ell$-th prime ($p_1 = 2$), $\\log$ the natural logarithm and $B \\approx 0.2615$ the Meissel\u2013Mertens constant. For all natural numbers $k$ and $j$ with $1 \\le j \\le k + 1$,","title":"Mertens-type bound for a product over the first primes with one prime omitted"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/prod-succ-div-prime-lt-bound","kind":"node","lean":"MetaMathlibExt.prod_succ_div_prime_lt_bound","parent":8908,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#prod-succ-div-prime-lt-bound","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $p_\\ell$ be the $\\ell$-th prime ($p_1 = 2$), $\\log$ the natural logarithm and $B$ the Meissel\u2013Mertens constant. For natural numbers $k \\ge 6$ and $j$ with $1 \\le j \\le k + 1$, put $U = (k+1)(\\log(k+1) + \\log \\log(k+1))$ and $L = (k+1)(\\log(k+1) + \\log \\log(k+1) - 1)$. Then","title":"Explicit bound for a product over the first primes with one prime omitted"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/exists-bound-mul-prod-sq-lt-exp-euler-ma","kind":"node","lean":"MetaMathlibExt.exists_bound_mul_prod_sq_lt_exp_eulerMascheroni_mul","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual bound for "],["math","\\(C(k)B(k)\\)"],["text"," by an explicit logarithm"]],"source":"Eventual bound for $C(k)B(k)$ by an explicit logarithm"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#exists-bound-mul-prod-sq-lt-exp-euler-ma","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $p_\\ell$ denote the $\\ell$-th prime ($p_1 = 2$), $\\log$ the natural logarithm, $\\gamma$ the Euler\u2013Mascheroni constant and $c_1$ the Meissel\u2013Mertens constant. For $j \\in \\{1, 2, 3\\}$ and $k \\in \\mathbb{N}$ put $L(k) = (k+1)(\\log(k+1) + \\log\\log(k+1))$, $L^-(k) = (k+1)(\\log(k+1) + \\log\\log(k+1) - 1)$ and","title":"Eventual bound for $C(k)B(k)$ by an explicit logarithm"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/exists-bound-mul-prod-sq-lt-exp-euler-ma-2","kind":"node","lean":"MetaMathlibExt.exists_bound_mul_prod_sq_lt_exp_eulerMascheroni_mul_log_log","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual bound for "],["math","\\(C(k)B(k)\\)"],["text"," by "],["math","\\(e^\\gamma \\log\\log\\)"],["text"," of a primorial quotient"]],"source":"Eventual bound for $C(k)B(k)$ by $e^\\gamma \\log\\log$ of a primorial quotient"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#exists-bound-mul-prod-sq-lt-exp-euler-ma-2","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $p_\\ell$ denote the $\\ell$-th prime ($p_1 = 2$), $\\log$ the natural logarithm, $\\gamma$ the Euler\u2013Mascheroni constant, $c_1$ the Meissel\u2013Mertens constant and $p\\#$ the product of all primes at most $p$. For $j \\in \\{1, 2, 3\\}$ and $k \\in \\mathbb{N}$ put $L(k) = (k+1)(\\log(k+1) + \\log\\log(k+1))$, $L^-(k) = (k+1)(\\log(k+1) + \\log\\log(k+1) - 1)$ and","title":"Eventual bound for $C(k)B(k)$ by $e^\\gamma \\log\\log$ of a primorial quotient"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/exp-five-div-log-bound-lt","kind":"node","lean":"MetaMathlibExt.exp_five_div_log_bound_lt","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","The factor "],["math","\\(\\hat C(k)\\)"],["text"," is below "],["math","\\(1.525\\)"]],"source":"The factor $\\hat C(k)$ is below $1.525$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#exp-five-div-log-bound-lt","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"Let $\\log$ be the natural logarithm. For every natural number $k \\ge 13042$, with $L^-(k) = (k+1)(\\log(k+1) + \\log\\log(k+1) - 1)$,","title":"The factor $\\hat C(k)$ is below $1.525$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/sigma-div-lt-totient-ratio-le-prod-prime","kind":"node","lean":"MetaMathlibExt.sigma_div_lt_totient_ratio_le_prod_primes","parent":8908,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds for "],["math","\\(\\sigma(n)/n\\)"],["text"," over primes avoiding "],["math","\\(p_j\\)"]],"source":"Bounds for $\\sigma(n)/n$ over primes avoiding $p_j$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality/index.html#sigma-div-lt-totient-ratio-le-prod-prime","scope":"NumberTheory/zeta-l-functions/riemann-zeta/robin-s-inequality","status":"can_prove","summary":"For $\\ell \\ge 1$ let $p_\\ell$ be the $\\ell$-th prime, so that $p_1 = 2$. Let $1 \\le j \\le k$, let $q_1 < \\cdots < q_k$ be primes different from $p_j$, let $\\alpha_1, \\dots, \\alpha_k \\ge 1$, and put $n = \\prod_{i=1}^k q_i^{\\alpha_i}$. Let $\\sigma(n)$ be the sum of the divisors of $n$ and let $\\varphi$ be Euler's totient function. Then","title":"Bounds for $\\sigma(n)/n$ over primes avoiding $p_j$"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","status":"fully_proved","title":"Stieltjes constants"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/ramanujan-part1-ch7-entry13-bernoullipol","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry13Bernoullipolyadd.ramanujan_part1_ch7_entry13_bernoullipolyadd","parent":8940,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/index.html#ramanujan-part1-ch7-entry13-bernoullipol","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","status":"fully_proved","summary":"There is exactly one sequence of real numbers $(c_k)_{k \\ge 0}$ such that, for every $k \\ge 0$,","title":"Stieltjes constants exist, and the zeroth is Euler's constant"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/ramanujan-part1-ch7-entry13-bernoullirec","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry13Bernoullirecurrence.ramanujan_part1_ch7_entry13_bernoullirecurrence","parent":8940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Laurent expansion of "],["math","\\(\\zeta\\)"],["text"," at "],["math","\\(s = 1\\)"]],"source":"Laurent expansion of $\\zeta$ at $s = 1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/index.html#ramanujan-part1-ch7-entry13-bernoullirec","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","status":"fully_proved","summary":"Let $(c_k)_{k \\ge 0}$ be real numbers such that, for every $k \\ge 0$, $\\sum_{j=1}^{m} (\\log j)^k / j - (\\log m)^{k+1} / (k+1)$ tends to $c_k$ as $m \\to \\infty$. Then for every complex $s \\ne 1$ the series","title":"Laurent expansion of $\\zeta$ at $s = 1$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/ramanujan-part1-ch7-entry14-bernoulliasy","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry14Bernoulliasymptotic.ramanujan_part1_ch7_entry14_bernoulliasymptotic","parent":8940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Expansion of "],["math","\\(\\sum 1/(n^{s+1} \\log n)\\)"],["text"," in the Stieltjes constants"]],"source":"Expansion of $\\sum 1/(n^{s+1} \\log n)$ in the Stieltjes constants"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/index.html#ramanujan-part1-ch7-entry14-bernoulliasy","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","status":"fully_proved","summary":"Let $(c_k)_{k \\ge 0}$ be real numbers such that, for every $k \\ge 0$, $\\sum_{j=1}^{m} (\\log j)^k / j - (\\log m)^{k+1} / (k+1)$ tends to $c_k$ as $m \\to \\infty$, and put $A_k = (-1)^k c_k / k!$. Then the series $\\sum_{n \\ge 2} \\frac{1}{n^2 \\log n}$ and $\\sum_{k \\ge 1} \\frac{A_k}{k+1}$ converge absolutely, and for every real $s > 0$ the series $\\sum_{n \\ge 2} \\frac{1}{n^{s+1} \\log n}$ and\u2026","title":"Expansion of $\\sum 1/(n^{s+1} \\log n)$ in the Stieltjes constants"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/chapter7-stieltjes-a-eq-entry14-bernoull","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry14Bernoullipolymult.chapter7StieltjesA_eq_entry14Bernoulliasymptotic","parent":8940,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/index.html#chapter7-stieltjes-a-eq-entry14-bernoull","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","status":"fully_proved","summary":"For every real sequence $(c_k)_{k \\ge 0}$, the coefficients $A_k = (-1)^k c_k / k!$ defined for the asymptotic expansion of $\\sum 1/(n(n^x - 1))$ are, by definition, the coefficients $A_k$ used in the expansion of $\\sum 1/(n^{s+1} \\log n)$ in the Stieltjes constants.","title":"Two definitions of the normalized Stieltjes coefficients agree"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/summable-chapter7-entry14-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry14Bernoullipolymult.summable_chapter7Entry14Term","parent":8940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergence of "],["math","\\(\\sum 1/(n(n^x - 1))\\)"]],"source":"Convergence of $\\sum 1/(n(n^x - 1))$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/index.html#summable-chapter7-entry14-term","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","status":"fully_proved","summary":"For every real $x > 0$ the series $\\sum_{n \\ge 2} \\frac{1}{n(n^x - 1)}$ converges absolutely.","title":"Convergence of $\\sum 1/(n(n^x - 1))$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/chapter7-stieltjes-approx-eq-entry13-ber","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry14Bernoullipolymult.chapter7StieltjesApprox_eq_entry13Bernoullipolyadd","parent":8940,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/index.html#chapter7-stieltjes-approx-eq-entry13-ber","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","status":"fully_proved","summary":"For all $k, m \\ge 0$, the approximant $\\sum_{j=1}^{m} (\\log j)^k / j - (\\log m)^{k+1} / (k+1)$ defined for the asymptotic expansion of $\\sum 1/(n(n^x - 1))$ is, by definition, the approximant used for the existence of the Stieltjes constants.","title":"Two definitions of the Stieltjes approximants agree"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/ramanujan-part1-ch7-entry14-bernoullipol","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry14Bernoullipolymult.ramanujan_part1_ch7_entry14_bernoullipolymult","parent":8940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic expansion of "],["math","\\(\\sum 1/(n(n^x - 1))\\)"],["text"," as "],["math","\\(x \\to 0^+\\)"]],"source":"Asymptotic expansion of $\\sum 1/(n(n^x - 1))$ as $x \\to 0^+$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants/index.html#ramanujan-part1-ch7-entry14-bernoullipol","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stieltjes-constants","status":"fully_proved","summary":"Let $a_0$ be a real number such that $\\sum_{n=2}^{m} \\frac{1}{n \\log n} - \\log \\log m$ tends to $a_0$ as $m \\to \\infty$, and let $(c_k)_{k \\ge 0}$ be real numbers such that, for every $k \\ge 0$, $\\sum_{j=1}^{m} (\\log j)^k / j - (\\log m)^{k+1} / (k+1)$ tends to $c_k$ as $m \\to \\infty$. Then for every real $x > 0$ the series $F(x) = \\sum_{n \\ge 2} \\frac{1}{n(n^x - 1)}$ converges absolutely, and for every $N \\ge 0$,","title":"Asymptotic expansion of $\\sum 1/(n(n^x - 1))$ as $x \\to 0^+$"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","title":"Stride series for zeta and eta values"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/stride-alternating-sum","kind":"node","lean":"MetaMathlibExt.strideAlternatingSum","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#stride-alternating-sum","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"fully_proved","summary":"For natural numbers $q$, $t$ and $n$, let $K = \\lfloor (n + q - 1)/q \\rfloor$ if $q \\ge 1$ and $K = 0$ if $q = 0$. The alternating stride sum is the real number","title":"Alternating stride sum"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/stride-signed-sum","kind":"node","lean":"MetaMathlibExt.strideSignedSum","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#stride-signed-sum","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"fully_proved","summary":"For natural numbers $q$, $t$ and $n$, let $K = \\lfloor (n + q - 1)/q \\rfloor$ if $q \\ge 1$ and $K = 0$ if $q = 0$. The signed stride sum is the real number","title":"Signed stride sum"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/stride-alternating-sum-of-le","kind":"node","lean":"MetaMathlibExt.strideAlternatingSum_of_le","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#stride-alternating-sum-of-le","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"fully_proved","summary":"For natural numbers $q$, $t$ and $n$ with $1 \\le n \\le q$, the alternating stride sum satisfies $S_{q,t}(n) = 1/n^t$.","title":"Alternating stride sum with a single term"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/stride-signed-sum-of-le","kind":"node","lean":"MetaMathlibExt.strideSignedSum_of_le","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#stride-signed-sum-of-le","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"fully_proved","summary":"For natural numbers $q$, $t$ and $n$ with $1 \\le n \\le q$, the signed stride sum satisfies $T_{q,t}(n) = 1/n^t$.","title":"Signed stride sum with a single term"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/stride-alternating-sum-add","kind":"node","lean":"MetaMathlibExt.strideAlternatingSum_add","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#stride-alternating-sum-add","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"fully_proved","summary":"Let $q \\ge 1$, $n \\ge 1$ and $t \\ge 0$ be integers and let $S_{q,t}$ denote the alternating stride sum. Then $S_{q,t}(n + q) = 1/(n + q)^t - S_{q,t}(n)$.","title":"Recurrence for the alternating stride sum"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/stride-signed-sum-add","kind":"node","lean":"MetaMathlibExt.strideSignedSum_add","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#stride-signed-sum-add","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"fully_proved","summary":"Let $q \\ge 1$, $n \\ge 1$ and $t \\ge 0$ be integers and let $T_{q,t}$ denote the signed stride sum. Then $T_{q,t}(n + q) = 1/(n + q)^t + (-1)^{q+1} T_{q,t}(n)$.","title":"Recurrence for the signed stride sum"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tendsto-stride-alternating-sum-riemann-z","kind":"node","lean":"MetaMathlibExt.tendsto_strideAlternatingSum_riemannZeta","parent":8948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating stride sum series for "],["math","\\(\\zeta(t + 1)\\)"]],"source":"Alternating stride sum series for $\\zeta(t + 1)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tendsto-stride-alternating-sum-riemann-z","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"Let $q \\ge 1$ and $t \\ge 1$ be integers and let $S_{q,t}$ denote the alternating stride sum. Then the partial sums","title":"Alternating stride sum series for $\\zeta(t + 1)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tendsto-stride-signed-sum-digamma","kind":"node","lean":"MetaMathlibExt.tendsto_strideSignedSum_digamma","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tendsto-stride-signed-sum-digamma","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"Let $q \\ge 1$ and $t \\ge 1$ be integers, let $a$ be a real number with $-1 \\le a < 1$, let $T_{q,t}$ denote the signed stride sum and $\\psi$ the digamma function. As $N \\to \\infty$, the partial sums","title":"Digamma series for the signed stride sum"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tendsto-stride-signed-sum-dirichlet-eta","kind":"node","lean":"MetaMathlibExt.tendsto_strideSignedSum_dirichletEta","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tendsto-stride-signed-sum-dirichlet-eta","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"Let $q, t \\ge 1$ be integers and let $T_{q,t}(n)$ denote the signed stride sum. Then the partial sums","title":"Stride series for the Dirichlet eta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tendsto-stride-sums-digamma-odd","kind":"node","lean":"MetaMathlibExt.tendsto_strideSums_digamma_odd","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tendsto-stride-sums-digamma-odd","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"Let $q, t \\ge 1$ be integers, let $a$ be a real number with $-1 \\le a < 1$, let $S_{q,t}(n)$ denote the alternating stride sum and $T_{q,t}(n)$ the signed stride sum, and let $\\psi$ be the digamma function. Then the partial sums","title":"Digamma identity for stride sums over odd indices"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tendsto-stride-sums-dirichlet-lambda","kind":"node","lean":"MetaMathlibExt.tendsto_strideSums_dirichletLambda","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tendsto-stride-sums-dirichlet-lambda","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"Let $q, t \\ge 1$ be integers, and let $S_{q,t}(n)$ denote the alternating stride sum and $T_{q,t}(n)$ the signed stride sum. Then the partial sums","title":"Stride series for the Dirichlet lambda function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tendsto-stride-alternating-sum-riemann-z-2","kind":"node","lean":"MetaMathlibExt.tendsto_strideAlternatingSum_riemannZeta_add","parent":8948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating stride sum series for "],["math","\\(\\zeta(p + t + 1)\\)"]],"source":"Alternating stride sum series for $\\zeta(p + t + 1)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tendsto-stride-alternating-sum-riemann-z-2","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"Let $q \\ge 1$ and $t \\ge 1$ be integers, let $p \\ge 0$ be an even integer and let $S_{q,t}$ denote the alternating stride sum. Then the partial sums","title":"Alternating stride sum series for $\\zeta(p + t + 1)$"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tendsto-stride-signed-sum-dirichlet-eta-2","kind":"node","lean":"MetaMathlibExt.tendsto_strideSignedSum_dirichletEta_add","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tendsto-stride-signed-sum-dirichlet-eta-2","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"Let $q, t \\ge 1$ be integers, let $p \\ge 0$ be an even integer, and let $T_{q,t}(n)$ denote the signed stride sum. Then the partial sums","title":"Higher-power stride series for the Dirichlet eta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tendsto-stride-alternating-sum-digamma-h","kind":"node","lean":"MetaMathlibExt.tendsto_strideAlternatingSum_digamma_half","parent":8948,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tendsto-stride-alternating-sum-digamma-h","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"Let $q \\ge 1$ and $t \\ge 1$ be integers, let $S_{q,t}$ denote the alternating stride sum and $\\psi$ the digamma function. As $N \\to \\infty$, the partial sums","title":"Digamma series for the alternating stride sum"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/tsum-inv-pow-mul-two-mul-sub-one","kind":"node","lean":"MetaMathlibExt.tsum_inv_pow_mul_two_mul_sub_one","parent":8948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form of "],["math","\\(\\sum_{n \\ge 1} 1/(n^t(2n-1))\\)"]],"source":"Closed form of $\\sum_{n \\ge 1} 1/(n^t(2n-1))$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values/index.html#tsum-inv-pow-mul-two-mul-sub-one","scope":"NumberTheory/zeta-l-functions/riemann-zeta/stride-series-for-zeta-and-eta-values","status":"can_prove","summary":"For every integer $t \\ge 1$,","title":"Closed form of $\\sum_{n \\ge 1} 1/(n^t(2n-1))$"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function","status":"can_prove","title":"Zero-free regions for the Riemann zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function/kadiri-zero-free-region","kind":"node","lean":"MetaMathlibExt.kadiri_zero_free_region","parent":8964,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function/index.html#kadiri-zero-free-region","scope":"NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function","status":"can_prove","summary":"Let $\\zeta$ be the Riemann zeta function and $\\log$ the natural logarithm. For every complex number $s = \\sigma + it$ with $|t| \\ge 2$ and $\\sigma \\ge 1 - \\frac{1}{5.70176 \\log |t|}$, we have $\\zeta(s) \\ne 0$.","title":"Kadiri's zero-free region for the Riemann zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function/kadiri-zero-free-region-refined","kind":"node","lean":"MetaMathlibExt.kadiri_zero_free_region_refined","parent":8964,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function/index.html#kadiri-zero-free-region-refined","scope":"NumberTheory/zeta-l-functions/riemann-zeta/zero-free-regions-for-the-riemann-zeta-function","status":"can_prove","summary":"Let $\\zeta$ be the Riemann zeta function and $\\log$ the natural logarithm. For every complex number $s = \\sigma + it$ with $|t| \\ge 2$ and $\\sigma \\ge 1 - \\frac{1}{5.69693 \\log |t|}$, we have $\\zeta(s) \\ne 0$.","title":"Kadiri's refined zero-free region for the Riemann zeta function"},{"data_url":"scopes/NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function.json","id":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function","kind":"scope","navigation":"module","parent":8868,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/index.html","scope":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function","status":"can_prove","title":"Zeros of the Riemann zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/zeta-zero-count","kind":"node","lean":"MetaMathlibExt.zetaZeroCount","parent":8967,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/index.html#zeta-zero-count","scope":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function","status":"fully_proved","summary":"For a real number $T$, $N(T)$ is the number of zeros $\\rho = \\beta + i\\gamma$ of the Riemann zeta function with $0 < \\beta < 1$ and $0 < \\gamma \\le T$, each counted with its multiplicity as a zero.","title":"Zero-counting function of the Riemann zeta function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/bellotti-wong-zero-count-bound","kind":"node","lean":"MetaMathlibExt.bellotti_wong_zero_count_bound","parent":8967,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/index.html#bellotti-wong-zero-count-bound","scope":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function","status":"can_prove","summary":"For every real number $T \\ge e$, the number $N(T)$ of zeros $\\rho = \\beta + i\\gamma$ of the Riemann zeta function with $0 < \\beta < 1$ and $0 < \\gamma \\le T$, counted with multiplicity, satisfies","title":"First Bellotti\u2013Wong bound on the zero-counting function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/bellotti-wong-zero-count-bound-small","kind":"node","lean":"MetaMathlibExt.bellotti_wong_zero_count_bound_small","parent":8967,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/index.html#bellotti-wong-zero-count-bound-small","scope":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function","status":"can_prove","summary":"For every real number $T \\ge e$, the number $N(T)$ of zeros $\\rho = \\beta + i\\gamma$ of the Riemann zeta function with $0 < \\beta < 1$ and $0 < \\gamma \\le T$, counted with multiplicity, satisfies","title":"Second Bellotti\u2013Wong bound on the zero-counting function"},{"id":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/platt-trudgian-rh-to-height","kind":"node","lean":"MetaMathlibExt.platt_trudgian_rh_to_height","parent":8967,"reading_title":{"format":"inline-parts-v1","parts":[["text","Platt\u2013Trudgian verification of the Riemann hypothesis to height "],["math","\\(3 \\cdot 10^{12}\\)"]],"source":"Platt\u2013Trudgian verification of the Riemann hypothesis to height $3 \\cdot 10^{12}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function/index.html#platt-trudgian-rh-to-height","scope":"NumberTheory/zeta-l-functions/riemann-zeta/zeros-of-the-riemann-zeta-function","status":"can_prove","summary":"Every zero $\\rho$ of the Riemann zeta function with $0 < \\operatorname{Im} \\rho \\le 3000175332800$ satisfies $\\operatorname{Re} \\rho = 1/2$.","title":"Platt\u2013Trudgian verification of the Riemann hypothesis to height $3 \\cdot 10^{12}$"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions","kind":"scope","parent":8867,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions","status":"can_prove","summary":"Dirichlet characters, conductors, primitive and quadratic characters, Dirichlet series continuation, L-function bounds, Artin and Selberg-class functions.","title":"Dirichlet characters and L-functions"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/ramanujan-part1-ch7-entry18-bernoullidif","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry18Bernoullidiff.ramanujan_part1_ch7_entry18_bernoullidiff","parent":8972,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/index.html#ramanujan-part1-ch7-entry18-bernoullidif","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions","status":"fully_proved","summary":"Let $\\chi$ be the non-principal Dirichlet character modulo $4$, so $\\chi(1) = 1$, $\\chi(3) = -1$ and $\\chi(0) = \\chi(2) = 0$; let $\\beta(s) = L(s, \\chi)$ be its Dirichlet $L$-function (the Dirichlet beta function, continued to all of $\\mathbb{C}$); and put $E^*(s) = 2\\,\\Gamma(s)\\,\\beta(s) / (\\pi/2)^s$. Then the functions $\\cos(\\pi r/2)\\, E^*(1 - r)$, $(\\pi/2)^r E^*(r) / \\Gamma(r)$ and\u2026","title":"Functional equation of the Dirichlet beta function"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","kind":"scope","navigation":"module","parent":8972,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","title":"a-points of Selberg class functions"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/selberg-class-function","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#selberg-class-function","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"fully_proved","summary":"A function in the Selberg class consists of Dirichlet coefficients $a \\colon \\mathbb{N} \\to \\mathbb{C}$, a function $F \\colon \\mathbb{C} \\to \\mathbb{C}$, a number $r \\in \\mathbb{N}$ of Gamma factors, a conductor $Q > 0$, real numbers $\\lambda_1, \\dots, \\lambda_r > 0$, complex numbers $\\mu_1, \\dots, \\mu_r$ with $\\operatorname{Re} \\mu_j \\ge 0$ and a root number $\\omega \\in \\mathbb{C}$ with $|\\omega| = 1$, such that:","title":"Functions in the Selberg class"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/degree","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.degree","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#degree","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"fully_proved","summary":"For a function $F$ in the Selberg class whose functional equation has Gamma-factor scalings $\\lambda_1, \\dots, \\lambda_r$, the degree of $F$ is $d_F = 2 \\sum_{j=1}^{r} \\lambda_j$.","title":"Degree of a function in the Selberg class"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/degree-nonneg","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.degree_nonneg","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#degree-nonneg","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"fully_proved","summary":"Let $F$ be a function in the Selberg class with Gamma-factor scalings $\\lambda_1, \\dots, \\lambda_r$. Then the degree $d_F = 2 \\sum_{j=1}^{r} \\lambda_j$ satisfies $d_F \\ge 0$.","title":"Nonnegativity of the degree"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/differentiable-at-to-fun","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.differentiableAt_toFun","parent":8974,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holomorphy of Selberg class functions away from "],["math","\\(s = 1\\)"]],"source":"Holomorphy of Selberg class functions away from $s = 1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#differentiable-at-to-fun","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"fully_proved","summary":"For a function $F$ in the Selberg class, $F$ is complex differentiable at every $s \\in \\mathbb{C}$ with $s \\ne 1$.","title":"Holomorphy of Selberg class functions away from $s = 1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/a-point-count","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.aPointCount","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#a-point-count","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"fully_proved","summary":"Let $F$ be a function in the Selberg class, $k \\in \\mathbb{N}$, $a \\in \\mathbb{C}$ and $E_2, E_1, T \\in \\mathbb{R}$. Then $N_k(a; T, F)$ is the number of points $\\rho$ with $F^{(k)}(\\rho) = a$, where $F^{(k)}$ is the $k$-th derivative of $F$, in the rectangle $E_2 < \\operatorname{Re} \\rho < E_1$, $0 < \\operatorname{Im} \\rho < T$, each counted with its multiplicity as a zero of $F^{(k)} - a$.","title":"Counting function of a-points of derivatives"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/a-point-count-of-nonpos","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.aPointCount_of_nonpos","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#a-point-count-of-nonpos","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"fully_proved","summary":"Let $F$ be a function in the Selberg class, $k \\in \\mathbb{N}$, $a \\in \\mathbb{C}$ and $E_2, E_1 \\in \\mathbb{R}$. If $T \\le 0$, then $N_k(a; T, F) = 0$, where $N_k(a; T, F)$ is the number of a-points of the $k$-th derivative $F^{(k)}$, counted with multiplicity, in $E_2 < \\operatorname{Re} s < E_1$, $0 < \\operatorname{Im} s < T$.","title":"The a-point count vanishes at nonpositive height"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/exists-no-a-point-right","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.exists_no_aPoint_right","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#exists-no-a-point-right","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with Dirichlet coefficients $a(n)$, let $k \\ge 1$, and let $a \\in \\mathbb{C}$ be such that $a \\ne 0$ or $a(n) \\ne 0$ for some $n \\ge 2$. Then there is a real number $E_1 \\ge 1$ such that the $k$-th derivative $F^{(k)}$ satisfies $F^{(k)}(s) \\ne a$ for every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s \\ge E_1$.","title":"Lemma 1 (no a-points of derivatives far to the right)"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/iterated-deriv-reflected-asymptotic","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.iteratedDeriv_reflected_asymptotic","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#iterated-deriv-reflected-asymptotic","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with Gamma-factor data $\\lambda_j$, $\\mu_j$ for $1 \\le j \\le r$, conductor $Q$, root number $\\omega$ and degree $d_F = 2 \\sum_j \\lambda_j$, let $k \\ge 1$ and let $c > 1$. Write $\\log$ for the principal branch of the logarithm and put","title":"Asymptotics of the derivatives of the reflected function (Lemma 2)"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/exists-no-a-point-left","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.exists_no_aPoint_left","parent":8974,"reading_title":{"format":"inline-parts-v1","parts":[["text","No "],["math","\\(a\\)"],["text","-points of derivatives far to the left (Lemma 3)"]],"source":"No $a$-points of derivatives far to the left (Lemma 3)"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#exists-no-a-point-left","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with Dirichlet coefficients $a(n)$, let $k \\ge 1$, and let $a \\in \\mathbb{C}$ be such that $a \\ne 0$ or $a(n) \\ne 0$ for some $n \\ge 2$. Then there are real numbers $E_2 \\le 0$ and $\\tau$ such that the $k$-th derivative $F^{(k)}$ satisfies $F^{(k)}(s) \\ne a$ for every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s \\le E_2$ and $|\\operatorname{Im} s| \\ge \\tau$.","title":"No $a$-points of derivatives far to the left (Lemma 3)"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/a-point-count-unit-window","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.aPointCount_unit_window","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#a-point-count-unit-window","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with Dirichlet coefficients $a(n)$, let $k \\ge 1$, and let $a \\in \\mathbb{C}$ be such that $a \\ne 0$ or $a(n) \\ne 0$ for some $n \\ge 2$. Let $E_2, E_1$ be real numbers and let $N_k(a; T, F)$ be the number of a-points of the $k$-th derivative $F^{(k)}$, counted with multiplicity, in $E_2 < \\operatorname{Re} s < E_1$, $0 < \\operatorname{Im} s < T$. Then\u2026","title":"Lemma 4 (a-points of derivatives in windows of unit height)"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/log-deriv-a-point-partial-fraction","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.logDeriv_aPoint_partialFraction","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#log-deriv-a-point-partial-fraction","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class, let $k \\ge 1$, write $F^{(k)}$ for the $k$-th derivative of $F$, and let $a \\in \\mathbb{C}$ and let $\\sigma_1$, $\\sigma_2$ be real numbers. Then there are real numbers $t_0$ and $C$ such that for every $s \\in \\mathbb{C}$ with $\\sigma_1 < \\operatorname{Re} s < \\sigma_2$, $\\operatorname{Im} s \\ge t_0$ and $F^{(k)}(s) \\ne a$,","title":"Local partial-fraction formula at the a-points of the k-th derivative (Lemma 5)"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/has-sum-log-deriv-a-point-of-ne-zero","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.hasSum_logDeriv_aPoint_of_ne_zero","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#has-sum-log-deriv-a-point-of-ne-zero","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with Dirichlet coefficients $a(n)$, let $k \\ge 1$ and let $a \\in \\mathbb{C}$ with $a \\ne 0$. Then there is a real number $E$ such that for every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s \\ge E$,","title":"Lemma 6, case a \u2260 0 (expansion of the logarithmic derivative)"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/has-sum-log-deriv-zero","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.hasSum_logDeriv_zero","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#has-sum-log-deriv-zero","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with Dirichlet coefficients $a(n)$, write $F^{(k)}$ for the $k$-th derivative of $F$, let $k \\ge 1$, and let $\\ell \\ge 2$ be such that $a(\\ell) \\ne 0$ and $a(n) = 0$ for $2 \\le n < \\ell$. Then there is a real number $E$ such that for every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s \\ge E$,","title":"Expansion of the logarithmic derivative of the k-th derivative (Lemma 6, zeros case)"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/a-point-count-asymptotic","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.aPointCount_asymptotic","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#a-point-count-asymptotic","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with conductor $Q$, Gamma-factor scalings $\\lambda_1, \\dots, \\lambda_r$ and degree $d_F$, let $k \\ge 1$ and $a \\in \\mathbb{C}$ with $a \\ne 0$, and put $\\lambda = \\prod_{j=1}^{r} \\lambda_j^{2\\lambda_j}$. Let $E_1, E_2, \\tau$ be real numbers such that the $k$-th derivative $F^{(k)}$ satisfies $F^{(k)}(s) \\ne a$ whenever $\\operatorname{Re} s \\ge E_1$, and $F^{(k)}(s) \\ne a$\u2026","title":"Asymptotic count of a-points of derivatives in the Selberg class"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/sum-a-point-pow-of-ne-zero","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.sum_aPoint_pow_of_ne_zero","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#sum-a-point-pow-of-ne-zero","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with Dirichlet coefficients $a_F(n)$, write $F^{(k)}$ for the $k$-th derivative of $F$, let $k \\ge 1$, let $a \\ne 0$ be a complex number and let $x > 1$ be real. Let $E_1$, $E_2$ and $\\tau$ be real numbers such that $F^{(k)}(s) \\ne a$ whenever $\\operatorname{Re} s \\ge E_1$, and whenever $\\operatorname{Re} s \\le E_2$ and $|\\operatorname{Im} s| \\ge \\tau$. Put","title":"Landau-type formula for the a-points of the k-th derivative (Theorem 2, a \u2260 0)"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/sum-zero-pow","kind":"node","lean":"MetaMathlibExt.SelbergClassFunction.sum_zero_pow","parent":8974,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions/index.html#sum-zero-pow","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/a-points-of-selberg-class-functions","status":"can_prove","summary":"Let $F$ be a function in the Selberg class with Dirichlet coefficients $a(n)$, write $F^{(k)}$ for the $k$-th derivative of $F$, let $k \\ge 1$, let $\\ell \\ge 2$ be such that $a(\\ell) \\ne 0$ and $a(n) = 0$ for $2 \\le n < \\ell$, and let $x > 1$ be real. Let $E_1$, $E_2$ and $\\tau$ be real numbers such that $F^{(k)}(s) \\ne 0$ whenever $\\operatorname{Re} s \\ge E_1$, and whenever $\\operatorname{Re} s \\le E_2$ and\u2026","title":"Landau-type formula for the zeros of the k-th derivative (Theorem 2, zeros case)"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","kind":"scope","navigation":"module","parent":8972,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","title":"Analytic continuation of Dirichlet series"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/abel-partial-sum","kind":"node","lean":"abelPartialSum","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#abel-partial-sum","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"For $f : \\mathbb{N} \\to \\mathbb{C}$ and $n \\in \\mathbb{N}$, the one-based partial sum is $A(n) = \\sum_{k=1}^{n} f(k)$; in particular $A(0) = 0$ and the value $f(0)$ is never used.","title":"Partial sums of a coefficient sequence"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/abel-summatory","kind":"node","lean":"abelSummatory","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#abel-summatory","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"For $f : \\mathbb{N} \\to \\mathbb{C}$ with partial sums $A(n) = \\sum_{k=1}^{n} f(k)$, the summatory step function of $f$ is the function $S : \\mathbb{R} \\to \\mathbb{C}$ with $S(t) = A(\\lfloor t \\rfloor)$ for $t \\ge 0$ and $S(t) = 0$ for $t < 0$. Equivalently, $S(t) = \\sum_{1 \\le k \\le t} f(k)$ for every real $t$.","title":"Summatory step function"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/abel-continuation","kind":"node","lean":"abelContinuation","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#abel-continuation","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"For $f : \\mathbb{N} \\to \\mathbb{C}$ with partial sums $A(n) = \\sum_{k=1}^{n} f(k)$, the Abel continuation of $f$ is the function $F_f : \\mathbb{C} \\to \\mathbb{C}$ given by","title":"Abel continuation of a Dirichlet series"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/abel-summatory-eq-zero-of-lt-one","kind":"node","lean":"abelSummatory_eq_zero_of_lt_one","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#abel-summatory-eq-zero-of-lt-one","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"For $f : \\mathbb{N} \\to \\mathbb{C}$, the summatory step function $S(t) = \\sum_{1 \\le k \\le t} f(k)$ satisfies $S(t) = 0$ for every real $t < 1$.","title":"The summatory step function vanishes below one"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/locally-integrable-on-abel-summatory","kind":"node","lean":"locallyIntegrableOn_abelSummatory","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#locally-integrable-on-abel-summatory","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"For every $f : \\mathbb{N} \\to \\mathbb{C}$, the summatory step function $t \\mapsto \\sum_{k=1}^{\\lfloor t \\rfloor} f(k)$ on $\\mathbb{R}$ is locally integrable on the open half-line $(0, \\infty)$: every $t > 0$ has a neighbourhood within $(0, \\infty)$ on which this function is Lebesgue integrable.","title":"Local integrability of the summatory function"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/is-big-o-abel-summatory-at-top","kind":"node","lean":"isBigO_abelSummatory_atTop","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#is-big-o-abel-summatory-at-top","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{C}$ have partial sums $A(n) = \\sum_{k=1}^{n} f(k)$ and summatory step function $S(t) = \\sum_{1 \\le k \\le t} f(k)$, and let $\\sigma$ be any real number with $A(n) = O(n^\\sigma)$ as $n \\to \\infty$. Then $S(t) = O(t^\\sigma)$ as $t \\to \\infty$ through the reals.","title":"Growth of the summatory step function"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/is-big-o-abel-summatory-nhds-gt-zero","kind":"node","lean":"isBigO_abelSummatory_nhdsGT_zero","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#is-big-o-abel-summatory-nhds-gt-zero","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{C}$ have summatory step function $S(t) = \\sum_{1 \\le k \\le t} f(k)$. For every real $b$, $S(t) = O(t^{-b})$ as $t \\to 0^+$.","title":"The summatory step function is negligible near zero"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/abel-continuation-eq-mul-mellin","kind":"node","lean":"abelContinuation_eq_mul_mellin","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#abel-continuation-eq-mul-mellin","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{C}$ have partial sums $A(n) = \\sum_{k=1}^{n} f(k)$ and summatory step function $S(t) = \\sum_{1 \\le k \\le t} f(k)$ for real $t$, and let $\\sigma$ be a real number with $A(n) = O(n^\\sigma)$ as $n \\to \\infty$. Then for every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > \\sigma$, the Abel continuation satisfies $F_f(s) = s\\,\\mathcal{M}S(-s)$, where\u2026","title":"The Abel continuation as a Mellin transform"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/differentiable-at-abel-continuation","kind":"node","lean":"differentiableAt_abelContinuation","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#differentiable-at-abel-continuation","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{C}$ have partial sums $A(n) = \\sum_{k=1}^{n} f(k)$, and let $\\sigma$ be a real number with $A(n) = O(n^\\sigma)$ as $n \\to \\infty$. Then the Abel continuation $F_f$ of $f$ is complex differentiable at every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > \\sigma$.","title":"Complex differentiability of the Abel continuation"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/analytic-on-abel-continuation","kind":"node","lean":"analyticOn_abelContinuation","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#analytic-on-abel-continuation","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{C}$ have partial sums $A(n) = \\sum_{k=1}^{n} f(k)$, and let $\\sigma$ be a real number with $A(n) = O(n^\\sigma)$ as $n \\to \\infty$. Then the Abel continuation $F_f$ of $f$ is analytic on the half-plane $\\{s \\in \\mathbb{C} : \\operatorname{Re} s > \\sigma\\}$.","title":"Analyticity of the Abel continuation"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/abel-continuation-eq-lseries","kind":"node","lean":"abelContinuation_eq_LSeries","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#abel-continuation-eq-lseries","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{C}$ have partial sums $A(n) = \\sum_{k=1}^{n} f(k)$, let $\\sigma \\ge 0$ be a real number with $A(n) = O(n^\\sigma)$ as $n \\to \\infty$, and let $s$ be a complex number with $\\operatorname{Re} s > \\sigma$ at which the Dirichlet series $\\sum_{n \\ge 1} f(n) n^{-s}$ converges absolutely. Then the Abel continuation $F_f$ of $f$ satisfies $F_f(s) = \\sum_{n=1}^{\\infty} f(n) n^{-s}$.","title":"Abel continuation agrees with the Dirichlet series"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/tendsto-abel-continuation-icc","kind":"node","lean":"tendsto_abelContinuation_Icc","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#tendsto-abel-continuation-icc","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{C}$ with partial sums $A(n) = \\sum_{k=1}^{n} f(k)$, let $\\sigma$ be a real number and let $s$ be a complex number with $\\sigma < \\operatorname{Re} s$. If $A(n) = O(n^\\sigma)$ as $n \\to \\infty$, then the partial sums $\\sum_{n=1}^{N} f(n) n^{-s}$ converge, as $N \\to \\infty$, to the Abel continuation $F_f(s)$ of $f$ at $s$.","title":"Abel continuation as the limit of Dirichlet partial sums"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/linear-asymptotic-dirichlet-continuation","kind":"node","lean":"linearAsymptoticDirichletContinuation","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#linear-asymptotic-dirichlet-continuation","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"For $a : \\mathbb{N} \\to \\mathbb{C}$ and $\\rho \\in \\mathbb{C}$, the linear asymptotic Dirichlet continuation of $a$ with constant $\\rho$ is the function $L_{a,\\rho} : \\mathbb{C} \\to \\mathbb{C}$ given by $L_{a,\\rho}(s) = \\rho\\,\\zeta(s) + F_{a-\\rho}(s)$, where $\\zeta$ is the Riemann zeta function and $F_{a-\\rho}$ is the Abel continuation of the sequence $n \\mapsto a(n) - \\rho$.","title":"Linear asymptotic Dirichlet continuation"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/abel-partial-sum-sub-const","kind":"node","lean":"abelPartialSum_sub_const","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#abel-partial-sum-sub-const","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"For $a : \\mathbb{N} \\to \\mathbb{C}$, $\\rho \\in \\mathbb{C}$ and $n \\in \\mathbb{N}$, the one-based partial sum $\\sum_{k=1}^{n} (a(k) - \\rho)$ equals $\\sum_{k=1}^{n} a(k) - n\\rho$.","title":"Partial sums after subtracting a constant"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/is-big-o-abel-partial-sum-sub-const","kind":"node","lean":"isBigO_abelPartialSum_sub_const","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#is-big-o-abel-partial-sum-sub-const","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$, $\\rho \\in \\mathbb{C}$ and $\\sigma \\in \\mathbb{R}$. If the one-based partial sums $A(n) = \\sum_{k=1}^{n} a(k)$ satisfy $A(n) - n\\rho = O(n^{\\sigma})$ as $n \\to \\infty$, then $\\sum_{k=1}^{n} (a(k) - \\rho) = O(n^{\\sigma})$ as $n \\to \\infty$.","title":"Growth bound for the partial sums of a shifted sequence"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/analytic-at-linear-asymptotic-dirichlet","kind":"node","lean":"analyticAt_linearAsymptoticDirichletContinuation_of_ne_one","parent":8991,"reading_title":{"format":"inline-parts-v1","parts":[["text","Analyticity of the linear asymptotic continuation away from "],["math","\\(1\\)"]],"source":"Analyticity of the linear asymptotic continuation away from $1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#analytic-at-linear-asymptotic-dirichlet","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$, $\\rho \\in \\mathbb{C}$ and $\\sigma \\in \\mathbb{R}$, and suppose that the one-based partial sums $A(n) = \\sum_{k=1}^{n} a(k)$ satisfy $A(n) - n\\rho = O(n^{\\sigma})$ as $n \\to \\infty$. Then the linear asymptotic Dirichlet continuation $L_{a,\\rho}$ of $a$ with constant $\\rho$ is complex analytic at every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > \\sigma$ and $s \\neq 1$.","title":"Analyticity of the linear asymptotic continuation away from $1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/linear-asymptotic-dirichlet-continuation-2","kind":"node","lean":"linearAsymptoticDirichletContinuation_eq_inv_sub_mul","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#linear-asymptotic-dirichlet-continuation-2","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$ and $\\rho \\in \\mathbb{C}$, let $L_{a,\\rho}$ be the linear asymptotic Dirichlet continuation of $a$ with constant $\\rho$, and let $F_{a-\\rho}$ be the Abel continuation of the sequence $n \\mapsto a(n) - \\rho$. Let $\\zeta_1 : \\mathbb{C} \\to \\mathbb{C}$ be the function with $\\zeta_1(s) = (s - 1)\\,\\zeta(s)$ for $s \\neq 1$ and $\\zeta_1(1) = 1$, where $\\zeta$ is the Riemann zeta\u2026","title":"Cleared-denominator form of the linear asymptotic continuation"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/meromorphic-at-linear-asymptotic-dirichl","kind":"node","lean":"meromorphicAt_linearAsymptoticDirichletContinuation_one","parent":8991,"reading_title":{"format":"inline-parts-v1","parts":[["text","Meromorphy of the linear asymptotic continuation at "],["math","\\(1\\)"]],"source":"Meromorphy of the linear asymptotic continuation at $1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#meromorphic-at-linear-asymptotic-dirichl","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$, $\\rho \\in \\mathbb{C}$ and $\\sigma \\in \\mathbb{R}$ with $\\sigma < 1$, and suppose that the one-based partial sums $A(n) = \\sum_{k=1}^{n} a(k)$ satisfy $A(n) - n\\rho = O(n^{\\sigma})$ as $n \\to \\infty$. Then the linear asymptotic Dirichlet continuation $L_{a,\\rho}$ of $a$ with constant $\\rho$ is meromorphic at $s = 1$.","title":"Meromorphy of the linear asymptotic continuation at $1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/meromorphic-order-at-linear-asymptotic-d","kind":"node","lean":"meromorphicOrderAt_linearAsymptoticDirichletContinuation_one","parent":8991,"reading_title":{"format":"inline-parts-v1","parts":[["text","The linear asymptotic continuation has a simple pole at "],["math","\\(1\\)"]],"source":"The linear asymptotic continuation has a simple pole at $1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#meromorphic-order-at-linear-asymptotic-d","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$, $\\rho \\in \\mathbb{C}$ with $\\rho \\neq 0$ and $\\sigma \\in \\mathbb{R}$ with $\\sigma < 1$, and suppose that the one-based partial sums $A(n) = \\sum_{k=1}^{n} a(k)$ satisfy $A(n) - n\\rho = O(n^{\\sigma})$ as $n \\to \\infty$. Then the meromorphic order at $s = 1$ of the linear asymptotic Dirichlet continuation $L_{a,\\rho}$ of $a$ with constant $\\rho$ is $-1$.","title":"The linear asymptotic continuation has a simple pole at $1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/tendsto-sub-one-mul-linear-asymptotic-di","kind":"node","lean":"tendsto_sub_one_mul_linearAsymptoticDirichletContinuation","parent":8991,"reading_title":{"format":"inline-parts-v1","parts":[["text","Limit of the linear asymptotic continuation times "],["math","\\(s - 1\\)"],["text"," at "],["math","\\(1\\)"]],"source":"Limit of the linear asymptotic continuation times $s - 1$ at $1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#tendsto-sub-one-mul-linear-asymptotic-di","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$, $\\rho \\in \\mathbb{C}$ and $\\sigma \\in \\mathbb{R}$ with $\\sigma < 1$, and suppose that the one-based partial sums $A(n) = \\sum_{k=1}^{n} a(k)$ satisfy $A(n) - n\\rho = O(n^{\\sigma})$ as $n \\to \\infty$. Let $L_{a,\\rho}$ be the linear asymptotic Dirichlet continuation of $a$ with constant $\\rho$. Then $(s - 1)\\,L_{a,\\rho}(s) \\to \\rho$ as $s \\to 1$ with $s \\neq 1$.","title":"Limit of the linear asymptotic continuation times $s - 1$ at $1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/meromorphic-on-linear-asymptotic-dirichl","kind":"node","lean":"meromorphicOn_linearAsymptoticDirichletContinuation","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#meromorphic-on-linear-asymptotic-dirichl","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$, $\\rho \\in \\mathbb{C}$ and $\\sigma \\in \\mathbb{R}$ with $\\sigma < 1$, and suppose that the one-based partial sums $A(n) = \\sum_{k=1}^{n} a(k)$ satisfy $A(n) - n\\rho = O(n^{\\sigma})$ as $n \\to \\infty$. Then the linear asymptotic Dirichlet continuation $L_{a,\\rho}$ of $a$ with constant $\\rho$ is meromorphic at every point of the half-plane\u2026","title":"Meromorphy of the linear asymptotic continuation on a half-plane"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/tendsto-linear-asymptotic-dirichlet-cont","kind":"node","lean":"tendsto_linearAsymptoticDirichletContinuation_Icc","parent":8991,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series/index.html#tendsto-linear-asymptotic-dirichlet-cont","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/analytic-continuation-of-dirichlet-series","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{C}$, $\\rho \\in \\mathbb{C}$ and $\\sigma \\in \\mathbb{R}$ with $\\sigma < 1$, and suppose that the one-based partial sums $A(n) = \\sum_{k=1}^{n} a(k)$ satisfy $A(n) - n\\rho = O(n^{\\sigma})$ as $n \\to \\infty$. Then for every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > 1$, the partial sums $\\sum_{n=1}^{N} a(n)\\, n^{-s}$ converge, as $N \\to \\infty$, to $L_{a,\\rho}(s)$, where $L_{a,\\rho}$\u2026","title":"Convergence of the Dirichlet series to the linear asymptotic continuation"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","kind":"scope","navigation":"module","parent":8972,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","title":"Artin L-functions"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/artin-local-datum","kind":"node","lean":"Representation.ArtinLocalDatum","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#artin-local-datum","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"For a finite group $G$, an Artin local datum consists of a subgroup $D \\le G$ (the decomposition group), a normal subgroup $I$ of $D$ (the inertia group), and an element $\\sigma$ of the quotient group $D/I$ (the Frobenius coset).","title":"Artin local datum"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/artin-local-denominator","kind":"node","lean":"Representation.artinLocalDenominator","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#artin-local-denominator","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, let $p$ be a prime, let $(D, I, \\sigma)$ be an Artin local datum for $G$, and let $s \\in \\mathbb{C}$. Let $V^{I}$ be the subspace of vectors fixed by $\\rho(I)$, on which $D/I$ acts, and let $\\rho(\\sigma)$ be the action of $\\sigma$ on $V^{I}$. The Artin local denominator is","title":"Artin local denominator"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/artin-local-factor-regular","kind":"node","lean":"Representation.ArtinLocalFactorRegular","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#artin-local-factor-regular","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, let $p$ be a prime, let $d$ be an Artin local datum for $G$, and let $s \\in \\mathbb{C}$. The Artin local factor of $\\rho$ at $p$, $d$ is regular at $s$ when the Artin local denominator of $\\rho$ at $p$, $d$ and $s$ is nonzero, that is, when the local factor has no pole at $s$.","title":"Regularity of the Artin local factor"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/artin-local-factor-raw","kind":"node","lean":"Representation.artinLocalFactorRaw","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#artin-local-factor-raw","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, let $p$ be a prime, let $(D, I, \\sigma)$ be an Artin local datum for $G$, and let $s \\in \\mathbb{C}$. Let $V^{I}$ be the subspace of vectors fixed by $\\rho(I)$ and $\\rho(\\sigma)$ the induced action of $\\sigma \\in D/I$ on it. The Artin local factor is the inverse of the Artin local denominator,","title":"Artin local factor"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/artin-euler-product-of-local-data","kind":"node","lean":"Representation.artinEulerProductOfLocalData","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#artin-euler-product-of-local-data","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, and suppose that for every prime $p$ an Artin local datum $d_p$ for $G$ is given. For $s \\in \\mathbb{C}$, this is the product over all primes $p$ of the Artin local factors of $\\rho$ at $p$, $d_p$ and $s$, taken as an unconditional infinite product (the limit of the products over finite sets of primes); when this\u2026","title":"Euler product of Artin local factors"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/has-artin-euler-product","kind":"node","lean":"Representation.HasArtinEulerProduct","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#has-artin-euler-product","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, let $d_p$ be an Artin local datum for $G$ for every prime $p$, and let $s, z \\in \\mathbb{C}$. The Artin Euler product of $\\rho$ and $(d_p)$ has value $z$ at $s$ when the products of the Artin local factors of $\\rho$ at $p$, $d_p$ and $s$ over finite sets of primes $p$ converge to $z$ as the finite set grows to all\u2026","title":"Convergent Artin Euler product"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/has-nonzero-artin-euler-product","kind":"node","lean":"Representation.HasNonzeroArtinEulerProduct","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#has-nonzero-artin-euler-product","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, let $d_p$ be an Artin local datum for $G$ for every prime $p$, and let $s, z \\in \\mathbb{C}$. The Artin Euler product of $\\rho$ and $(d_p)$ has nonzero value $z$ at $s$ when it has value $z$ at $s$ and $z \\ne 0$.","title":"Nonzero convergent Artin Euler product"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/artin-local-factor-raw-of-action-eq-id-g","kind":"node","lean":"Representation.artinLocalFactorRaw_of_action_eq_id_general","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#artin-local-factor-raw-of-action-eq-id-g","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, let $p$ be a prime, let $(D, I, \\sigma)$ be an Artin local datum for $G$, and let $s \\in \\mathbb{C}$. Let $V^{I}$ be the subspace of vectors fixed by $\\rho(I)$ and $n = \\dim V^{I}$. If $\\sigma$ acts as the identity on $V^{I}$, then the Artin local factor of $\\rho$ at $p$, $(D, I, \\sigma)$ and $s$ equals\u2026","title":"Artin local factor when Frobenius acts trivially"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/artin-local-factor-raw-of-action-eq-id","kind":"node","lean":"Representation.artinLocalFactorRaw_of_action_eq_id","parent":9014,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions/index.html#artin-local-factor-raw-of-action-eq-id","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/artin-l-functions","status":"fully_proved","summary":"Let $\\rho$ be a representation of a finite group $G$ on a finite-dimensional complex vector space $V$, let $p$ be a prime, let $(D, I, \\sigma)$ be an Artin local datum for $G$, and let $s \\in \\mathbb{C}$. Let $V^{I}$ be the subspace of vectors fixed by $\\rho(I)$. If $\\sigma$ acts as the identity on $V^{I}$ and $V^{I}$ is one-dimensional, then the Artin local factor of $\\rho$ at $p$, $(D, I, \\sigma)$ and $s$ equals\u2026","title":"Artin local factor of a one-dimensional trivial Frobenius action"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","kind":"scope","navigation":"module","parent":8972,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","title":"Conductors and primitive Dirichlet characters"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/is-dirichlet-character-modulus","kind":"node","lean":"MonoidHom.IsDirichletCharacterModulus","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#is-dirichlet-character-modulus","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a monoid with zero, let $\\chi \\colon \\mathbb{Z} \\to R$ be a multiplicative map with $\\chi(1) = 1$, and let $q$ be a natural number. Then $\\chi$ is a Dirichlet character of modulus $q$ if $\\chi(n + q) = \\chi(n)$ for every integer $n$, and, for every integer $n$, $\\chi(n) \\neq 0$ if and only if $q$ and $n$ are coprime.","title":"Dirichlet character of a given modulus"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/is-dirichlet-character-modulus-iff","kind":"node","lean":"MonoidHom.isDirichletCharacterModulus_iff","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#is-dirichlet-character-modulus-iff","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a nontrivial monoid with zero in which nonzero elements can be cancelled on both sides, and let $\\chi \\colon \\mathbb{Z} \\to R$ be a multiplicative map with $\\chi(1) = 1$. Suppose $m \\ge 1$ is the least positive period of $\\chi$: $\\chi(n + m) = \\chi(n)$ for every integer $n$, and every positive integer $d$ with $\\chi(n + d) = \\chi(n)$ for all $n$ satisfies $m \\le d$. Then for every integer $m' \\ge 1$,\u2026","title":"Moduli of a character with a given least period"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/is-dirichlet-character-modulus-self","kind":"node","lean":"MonoidHom.isDirichletCharacterModulus_self","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#is-dirichlet-character-modulus-self","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a nontrivial monoid with zero in which nonzero elements can be cancelled on both sides, and let $\\chi \\colon \\mathbb{Z} \\to R$ be a multiplicative map with $\\chi(1) = 1$. If $m \\ge 1$ is the least positive period of $\\chi$ (that is, $\\chi(n + m) = \\chi(n)$ for every integer $n$, and every positive integer $d$ with $\\chi(n + d) = \\chi(n)$ for all $n$ satisfies $m \\le d$), then $\\chi$ is a Dirichlet\u2026","title":"A character is a Dirichlet character modulo its least period"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/primitive-inducer","kind":"node","lean":"DirichletCharacter.PrimitiveInducer","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#primitive-inducer","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero and $m$ a natural number. A primitive inducer of modulus $m$ with values in $R$ consists of a natural number $d$, its level, with $d \\mid m$, together with a Dirichlet character $\\psi$ of level $d$ with values in $R$ that is primitive, that is, whose conductor is $d$. These are the elements of the set $X(m)$ of Corollary 18.16; changing the level of $\\psi$ from $d$ to $m$\u2026","title":"Primitive characters at divisor levels"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/ext","kind":"node","lean":"DirichletCharacter.PrimitiveInducer.ext","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#ext","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero and $m$ a natural number, and let $p$ and $q$ be primitive inducers of modulus $m$ with values in $R$. If $p$ and $q$ have the same level, and the character of $p$, transported along this equality of levels, equals the character of $q$, then $p = q$.","title":"Equality of primitive characters at divisor levels"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/equiv-primitive-inducer","kind":"node","lean":"DirichletCharacter.equivPrimitiveInducer","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#equiv-primitive-inducer","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero and $m \\ge 1$ a natural number. This is the bijection from Dirichlet characters of level $m$ with values in $R$ to primitive inducers of modulus $m$ (primitive Dirichlet characters with values in $R$ whose level divides $m$) that sends $\\chi$ to the primitive character associated to $\\chi$, whose level is the conductor of $\\chi$. Its inverse sends a primitive character\u2026","title":"Dirichlet characters correspond to primitive characters at divisor levels"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/equiv-primitive-inducer-level","kind":"node","lean":"DirichletCharacter.equivPrimitiveInducer_level","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#equiv-primitive-inducer-level","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero, let $m \\ge 1$, and let $\\chi$ be a Dirichlet character of level $m$ with values in $R$. The level of the primitive inducer that the bijection between Dirichlet characters of level $m$ and primitive inducers of modulus $m$ assigns to $\\chi$ is the conductor of $\\chi$.","title":"Level of the primitive inducer attached to a character"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/equiv-primitive-inducer-char","kind":"node","lean":"DirichletCharacter.equivPrimitiveInducer_char","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#equiv-primitive-inducer-char","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero, let $m \\ge 1$, and let $\\chi$ be a Dirichlet character of level $m$ with values in $R$. The character of the primitive inducer that the bijection between Dirichlet characters of level $m$ and primitive inducers of modulus $m$ assigns to $\\chi$, a Dirichlet character whose level is the conductor of $\\chi$, is the primitive character associated to $\\chi$.","title":"Character of the primitive inducer attached to a character"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/equiv-unit-hom","kind":"node","lean":"DirichletCharacter.PrimitiveInducer.equivUnitHom","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#equiv-unit-hom","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero and $m \\ge 1$ a natural number. This is the bijection from primitive inducers of modulus $m$ with values in $R$ to monoid homomorphisms $(\\mathbb{Z}/m\\mathbb{Z})^\\times \\to R^\\times$ that sends a primitive character $\\psi$ of level $d \\mid m$ to the homomorphism $u \\mapsto \\chi(u)$, where $\\chi$ is the Dirichlet character of level $m$ obtained from $\\psi$ by changing the\u2026","title":"Primitive characters at divisor levels correspond to unit-group characters"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/equiv-unit-hom-apply","kind":"node","lean":"DirichletCharacter.PrimitiveInducer.equivUnitHom_apply","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#equiv-unit-hom-apply","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero, let $m \\ge 1$, and let $p$ be a primitive inducer of modulus $m$ with values in $R$, with level $d$ and character $\\psi$. For every unit $u$ of $\\mathbb{Z}/m\\mathbb{Z}$, the value at $u$ of the homomorphism $(\\mathbb{Z}/m\\mathbb{Z})^\\times \\to R^\\times$ corresponding to $p$, viewed as an element of $R$, equals the value at $u$ of the Dirichlet character of level $m$\u2026","title":"Values of the unit-group character of a primitive inducer"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/conductor-eq-of-is-primitive-change-leve","kind":"node","lean":"DirichletCharacter.conductor_eq_of_isPrimitive_changeLevel","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#conductor-eq-of-is-primitive-change-leve","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero, let $n \\ge 1$ and $d$ be natural numbers with $d \\mid n$, and let $\\chi$ be a Dirichlet character of level $n$ with values in $R$. If $\\chi'$ is a primitive Dirichlet character of level $d$ whose change of level to $n$ is $\\chi$, then $d$ equals the conductor of $\\chi$.","title":"The level of a primitive inducer is the conductor"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/eq-primitive-character-of-change-level-e","kind":"node","lean":"DirichletCharacter.eq_primitiveCharacter_of_changeLevel_eq","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#eq-primitive-character-of-change-level-e","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero, let $n \\ge 1$ and $d$ be natural numbers with $d \\mid n$, and let $\\chi$ be a Dirichlet character of level $n$ with values in $R$ whose conductor is $d$. If $\\chi'$ is a Dirichlet character of level $d$ whose change of level to $n$ is $\\chi$, then $\\chi'$ is the primitive character associated to $\\chi$.","title":"An inducer at the conductor level is the primitive character"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/primitive-character-is-primitive-and-uni","kind":"node","lean":"DirichletCharacter.primitiveCharacter_isPrimitive_and_unique","parent":9024,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters/index.html#primitive-character-is-primitive-and-uni","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/conductors-and-primitive-dirichlet-characters","status":"fully_proved","summary":"Let $R$ be a commutative monoid with zero, let $n \\ge 1$, and let $\\chi$ be a Dirichlet character of level $n$ with values in $R$. The primitive character associated to $\\chi$, whose level is the conductor of $\\chi$, is primitive, and its change of level to $n$ is $\\chi$. Moreover, if $\\chi'$ is a primitive Dirichlet character of some level $d$ with $d \\mid n$ whose change of level to $n$ is $\\chi$, then $d$ is the\u2026","title":"Existence and uniqueness of the primitive character inducing a Dirichlet character"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density","kind":"scope","navigation":"module","parent":8972,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density","status":"fully_proved","title":"Dedekind zeta function and polar density"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/summable-height-one-spectrum-abs-norm-rp","kind":"node","lean":"NumberField.summable_heightOneSpectrum_absNorm_rpow_neg_of_one_lt","parent":9038,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/index.html#summable-height-one-spectrum-abs-norm-rp","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density","status":"fully_proved","summary":"For `\u03c3 > 1`, the Dedekind zeta terms `(N v) ^ (-\u03c3)` are summable over the height-one primes of the ring of integers.","title":"Summability of Dedekind zeta terms for \u03c3 > 1"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/partial-dedekind-zeta","kind":"node","lean":"NumberField.partialDedekindZeta","parent":9038,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/index.html#partial-dedekind-zeta","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density","status":"fully_proved","summary":"Let $K$ be a number field and $S$ a set of nonzero prime ideals of its ring of integers $\\mathcal{O}_K$. For such a prime $\\mathfrak{p}$ let $N(\\mathfrak{p}) = |\\mathcal{O}_K / \\mathfrak{p}|$ be its absolute norm. The partial Dedekind zeta function of $S$ is the function $\\zeta_{K,S} : \\mathbb{C} \\to \\mathbb{C}$ given by the Euler product","title":"Partial Dedekind zeta function"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/has-polar-density","kind":"node","lean":"NumberField.Set.HasPolarDensity","parent":9038,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/index.html#has-polar-density","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density","status":"fully_proved","summary":"Let $K$ be a number field, let $S$ be a set of height-one primes of the ring of integers of $K$, and let $\\zeta_S$ be the partial Dedekind zeta function of $S$. A rational number $\\rho$ is a polar density of $S$ when there exist a positive integer $n$, an integer $m$ and a function $F : \\mathbb{C} \\to \\mathbb{C}$ such that $F$ is meromorphic at $1$, $F(s) = \\zeta_S(s)^n$ for all $s$ with $\\operatorname{Re} s > 1$\u2026","title":"Polar density"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/unique","kind":"node","lean":"NumberField.Set.HasPolarDensity.unique","parent":9038,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/index.html#unique","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density","status":"fully_proved","summary":"Let $K$ be a number field and let $S$ be a set of height-one primes of the ring of integers of $K$. If $\\rho_1$ and $\\rho_2$ are both polar densities of $S$, then $\\rho_1 = \\rho_2$.","title":"Uniqueness of the polar density"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/has-polar-density-empty","kind":"node","lean":"NumberField.Set.hasPolarDensity_empty","parent":9038,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density/index.html#has-polar-density-empty","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/dedekind-zeta-function-and-polar-density","status":"fully_proved","summary":"For every number field $K$, the empty set of height-one primes of the ring of integers of $K$ has polar density $0$.","title":"The empty set has polar density zero"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","kind":"scope","navigation":"module","parent":8972,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"can_prove","title":"Moments of Dedekind zeta coefficients over sums of six squares"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/dirichlet-lfunction-mod-four-fourth-mome","kind":"node","lean":"MetaMathlibExt.dirichletLFunction_mod_four_fourth_moment_le","parent":9044,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourth moment of the "],["math","\\(L\\)"],["text","-function modulo "],["math","\\(4\\)"]],"source":"Fourth moment of the $L$-function modulo $4$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#dirichlet-lfunction-mod-four-fourth-mome","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"can_prove","summary":"Let $\\chi$ be a nontrivial Dirichlet character modulo $4$ and let $\\varepsilon > 0$. Then there is a real constant $C$ such that for every real $T \\ge 1$, $\\int_1^T |L(1/2 + it, \\chi)|^4 \\, dt \\le C\\, T^{1 + \\varepsilon}$.","title":"Fourth moment of the $L$-function modulo $4$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/dedekind-zeta-coeff","kind":"node","lean":"MetaMathlibExt.dedekindZetaCoeff","parent":9044,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#dedekind-zeta-coeff","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"fully_proved","summary":"For a number field $K$ with ring of integers $\\mathcal{O}_K$ and a natural number $n$, $a_K(n)$ is the number of ideals of $\\mathcal{O}_K$ of absolute norm $n$. These are the coefficients of $\\zeta_K(s) = \\sum_{n \\ge 1} a_K(n)\\, n^{-s}$.","title":"Coefficients of the Dedekind zeta function"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/dedekind-zeta-coeff-one","kind":"node","lean":"MetaMathlibExt.dedekindZetaCoeff_one","parent":9044,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Dedekind zeta coefficient at "],["math","\\(1\\)"]],"source":"The Dedekind zeta coefficient at $1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#dedekind-zeta-coeff-one","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"fully_proved","summary":"For every number field $K$, $a_K(1) = 1$, where $a_K(n)$ is the number of ideals of the ring of integers of $K$ of absolute norm $n$: the unit ideal is the only ideal of norm $1$.","title":"The Dedekind zeta coefficient at $1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/six-squares-dedekind-moment","kind":"node","lean":"MetaMathlibExt.sixSquaresDedekindMoment","parent":9044,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#six-squares-dedekind-moment","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"fully_proved","summary":"For a number field $K$, a natural number $\\ell$ and a real number $x$, the moment is the real number $S_\\ell(x) = \\sum_{1 \\le n \\le x} r_6(n)\\, a_K(n)^\\ell$, where $n$ runs over the integers in $[1, x]$, $a_K(n)$ is the number of ideals of the ring of integers of $K$ of absolute norm $n$, and $r_6(n)$ is the coefficient of $q^n$ in $\\theta(q)^6$, with $\\theta(q) = \\sum_{m \\in \\mathbb{Z}} q^{m^2}$; that is, $r_6(n)$\u2026","title":"Six-squares moments of Dedekind zeta coefficients"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/six-squares-dedekind-moment-of-lt-one","kind":"node","lean":"MetaMathlibExt.sixSquaresDedekindMoment_of_lt_one","parent":9044,"reading_title":{"format":"inline-parts-v1","parts":[["text","The six-squares moment vanishes below "],["math","\\(1\\)"]],"source":"The six-squares moment vanishes below $1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#six-squares-dedekind-moment-of-lt-one","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"fully_proved","summary":"For every number field $K$, every natural number $\\ell$ and every real $x < 1$, the six-squares moment $S_\\ell(x) = \\sum_{1 \\le n \\le x} r_6(n)\\, a_K(n)^\\ell$ equals $0$, where $r_6(n)$ is the number of representations of $n$ as a sum of six squares of integers and $a_K(n)$ is the number of ideals of the ring of integers of $K$ of absolute norm $n$.","title":"The six-squares moment vanishes below $1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/dedekind-six-squares-a0","kind":"node","lean":"MetaMathlibExt.dedekindSixSquaresA0","parent":9044,"reading_title":{"format":"inline-parts-v1","parts":[["text","The exponent "],["math","\\(a_{0,\\ell}\\)"]],"source":"The exponent $a_{0,\\ell}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#dedekind-six-squares-a0","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"fully_proved","summary":"For a natural number $\\ell$, define","title":"The exponent $a_{0,\\ell}$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/dedekind-six-squares-a3","kind":"node","lean":"MetaMathlibExt.dedekindSixSquaresA3","parent":9044,"reading_title":{"format":"inline-parts-v1","parts":[["text","The exponent "],["math","\\(a_{3,\\ell}\\)"]],"source":"The exponent $a_{3,\\ell}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#dedekind-six-squares-a3","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"fully_proved","summary":"For a natural number $\\ell$, $a_{3,\\ell}$ is the sum over the integers $1 \\le i \\le \\lfloor (\\ell - 1)/2 \\rfloor$ of $\\binom{\\ell}{2i+1} q_i$, where $q_i$ is the integer part of $4\\,(2i+1)!/((i-1)!\\,(i+3)!)$.","title":"The exponent $a_{3,\\ell}$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/dedekind-six-squares-a0-a3-small","kind":"node","lean":"MetaMathlibExt.dedekindSixSquaresA0_A3_small","parent":9044,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of "],["math","\\(a_{0,\\ell}\\)"],["text"," and "],["math","\\(a_{3,\\ell}\\)"],["text"," for "],["math","\\(\\ell = 3, 4, 5\\)"]],"source":"Values of $a_{0,\\ell}$ and $a_{3,\\ell}$ for $\\ell = 3, 4, 5$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#dedekind-six-squares-a0-a3-small","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"fully_proved","summary":"For a natural number $\\ell$ let $a_{0,\\ell} = 1 + \\sum_{i=1}^{\\lfloor \\ell/2 \\rfloor} \\binom{\\ell}{2i} C_i$, with $C_i$ the $i$-th Catalan number, and let $a_{3,\\ell}$ be the sum over the integers $1 \\le i \\le \\lfloor (\\ell - 1)/2 \\rfloor$ of $\\binom{\\ell}{2i+1}$ times the integer part of $4\\,(2i+1)!/((i-1)!\\,(i+3)!)$. Then $(a_{0,3}, a_{3,3}) = (4, 1)$, $(a_{0,4}, a_{3,4}) = (9, 4)$ and\u2026","title":"Values of $a_{0,\\ell}$ and $a_{3,\\ell}$ for $\\ell = 3, 4, 5$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/six-squares-dedekind-moment-two-three","kind":"node","lean":"MetaMathlibExt.sixSquaresDedekindMoment_two_three","parent":9044,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#six-squares-dedekind-moment-two-three","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"can_prove","summary":"Let $K$ be a number field of degree $3$ over $\\mathbb{Q}$ with negative discriminant. For $n \\ge 1$ let $a_K(n)$ be the number of ideals of the ring of integers of $K$ of absolute norm $n$ and $r_6(n)$ the number of representations of $n$ as a sum of six squares of integers, and for real $x$ and natural $\\ell$ let $S_\\ell(x) = \\sum_{1 \\le n \\le x} r_6(n)\\, a_K(n)^\\ell$. Then there are real constants $c_1, c_2$ such\u2026","title":"Second and third six-squares moments over a cubic field of negative discriminant"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/six-squares-dedekind-moment-ge-four","kind":"node","lean":"MetaMathlibExt.sixSquaresDedekindMoment_ge_four","parent":9044,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums/index.html#six-squares-dedekind-moment-ge-four","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/moments-of-dedekind-zeta-coefficients-over-sums","status":"can_prove","summary":"Let $K$ be a number field of degree $3$ over $\\mathbb{Q}$ with negative discriminant and let $\\ell \\ge 4$. For $n \\ge 1$ let $a_K(n)$ be the number of ideals of the ring of integers of $K$ of absolute norm $n$ and $r_6(n)$ the number of representations of $n$ as a sum of six squares of integers, and for real $x$ let $S_\\ell(x) = \\sum_{1 \\le n \\le x} r_6(n)\\, a_K(n)^\\ell$. Let\u2026","title":"Higher six-squares moments over a non-normal cubic field"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","kind":"scope","navigation":"module","parent":8972,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","title":"Perron's formula"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/perron-kernel","kind":"node","lean":"PerronKernel.perronKernel","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#perron-kernel","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For real numbers $y$, $c$ and $T$, the truncated Perron kernel is the complex number","title":"Truncated Perron kernel"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/continuous-perron-kernel-integrand","kind":"node","lean":"PerronKernel.continuous_perronKernel_integrand","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#continuous-perron-kernel-integrand","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y > 0$ and $c \\ne 0$ be real numbers. Then the function from $\\mathbb{R}$ to $\\mathbb{C}$ sending $t$ to $y^{c+it}/(c+it)$, where $y^{c+it}$ is the complex power, is continuous.","title":"Continuity of the Perron kernel integrand"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/exp-sub-one-eq","kind":"node","lean":"PerronKernel.exp_sub_one_eq","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral formula for "],["math","\\(e^w - 1\\)"]],"source":"Integral formula for $e^w - 1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#exp-sub-one-eq","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For every complex number $w$,","title":"Integral formula for $e^w - 1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/e1","kind":"node","lean":"PerronKernel.E1","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","The averaging operator "],["math","\\(E_1\\)"]],"source":"The averaging operator $E_1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#e1","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For a complex number $u$, define $E_1(u) = \\int_0^1 e^{tu}\\,dt$, the integral being over real $t$ from $0$ to $1$.","title":"The averaging operator $E_1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/e1-zero","kind":"node","lean":"PerronKernel.E1_zero","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#e1-zero","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $E_1(u) = \\int_0^1 e^{tu}\\,dt$ for $u \\in \\mathbb{C}$. Then $E_1(0) = 1$.","title":"Value of the averaging operator at zero"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/e1-eq-div","kind":"node","lean":"PerronKernel.E1_eq_div","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#e1-eq-div","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $E_1(u) = \\int_0^1 e^{tu}\\,dt$ for $u \\in \\mathbb{C}$. Then $E_1(u) = (e^u - 1)/u$ for every nonzero complex number $u$.","title":"The averaging operator as a difference quotient"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/e1-sub-one-le","kind":"node","lean":"PerronKernel.E1_sub_one_le","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#e1-sub-one-le","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $E_1(u) = \\int_0^1 e^{tu}\\,dt$ for $u \\in \\mathbb{C}$. For every complex number $v$ with $|v| \\le 1$, we have $|E_1(v) - 1| \\le |v|$.","title":"Linear bound for the averaging operator near zero"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/e1-continuous-at-zero","kind":"node","lean":"PerronKernel.E1_continuousAt_zero","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#e1-continuous-at-zero","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"The function $E_1 : \\mathbb{C} \\to \\mathbb{C}$ given by $E_1(u) = \\int_0^1 e^{tu}\\,dt$ is continuous at $0$.","title":"Continuity of the averaging operator at zero"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/e1-continuous","kind":"node","lean":"PerronKernel.E1_continuous","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#e1-continuous","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"The function $E_1 : \\mathbb{C} \\to \\mathbb{C}$ given by $E_1(u) = \\int_0^1 e^{tu}\\,dt$ is continuous.","title":"Continuity of the averaging operator"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/e1-has-deriv-at","kind":"node","lean":"PerronKernel.E1_hasDerivAt","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#e1-has-deriv-at","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $E_1(u) = \\int_0^1 e^{tu}\\,dt$ for $u \\in \\mathbb{C}$. For every $x_0 \\in \\mathbb{C}$, the function $E_1$ has complex derivative $\\int_0^1 t\\,e^{t x_0}\\,dt$ at $x_0$.","title":"Derivative of the averaging operator"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/e1-differentiable","kind":"node","lean":"PerronKernel.E1_differentiable","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#e1-differentiable","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"The function $E_1 : \\mathbb{C} \\to \\mathbb{C}$ given by $E_1(u) = \\int_0^1 e^{tu}\\,dt$ is complex differentiable at every point of $\\mathbb{C}$.","title":"The averaging operator is entire"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/perron-remainder","kind":"node","lean":"PerronKernel.perronRemainder","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#perron-remainder","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For a real number $y$ and a complex number $s$, the Perron remainder is $R_y(s) = \\log y \\cdot E_1(s \\log y)$, where $E_1(u) = \\int_0^1 e^{tu}\\,dt$ is the averaging operator and $\\log$ is the real logarithm (extended by $\\log 0 = 0$ and $\\log y = \\log |y|$ for $y < 0$).","title":"Perron remainder"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/perron-remainder-differentiable","kind":"node","lean":"PerronKernel.perronRemainder_differentiable","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#perron-remainder-differentiable","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For a real number $y$, the Perron remainder is the function $R_y(s) = \\log y \\int_0^1 e^{ts\\log y}\\,dt$ of $s \\in \\mathbb{C}$, where $\\log y = \\log|y|$ for $y \\ne 0$ and $\\log 0 = 0$. For every real $y$, the function $R_y$ is complex differentiable at every point of $\\mathbb{C}$.","title":"The Perron remainder is entire"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/perron-remainder-eq-div","kind":"node","lean":"PerronKernel.perronRemainder_eq_div","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#perron-remainder-eq-div","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For every real $y > 0$ and every complex $s \\neq 0$, the Perron remainder $R_y$ satisfies $R_y(s) = (y^s - 1)/s$, where $y^s = e^{s \\log y}$.","title":"The Perron remainder as a difference quotient"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/perron-remainder-slope","kind":"node","lean":"PerronKernel.perronRemainder_slope","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#perron-remainder-slope","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For every real $y$ and every complex $h \\neq 0$, the Perron remainder $R_y$ satisfies","title":"Slope of the Perron remainder at the origin"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/norm-cpow-of-real","kind":"node","lean":"PerronKernel.norm_cpow_ofReal","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#norm-cpow-of-real","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For every real $y > 0$ and every complex number $s$, $|y^s| = \\exp(\\operatorname{Re}(s) \\log y)$, where $y^s$ is the principal complex power.","title":"Modulus of a complex power of a positive real"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/cpow-div-eq-remainder-add","kind":"node","lean":"PerronKernel.cpow_div_eq_remainder_add","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#cpow-div-eq-remainder-add","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y > 0$ be real and let $s \\ne 0$ be complex. Then $y^s/s = R_y(s) + 1/s$, where $R_y(s) = \\log y \\cdot E_1(s \\log y)$ is the Perron remainder and $E_1(u) = \\int_0^1 e^{tu}\\,dt$.","title":"Splitting off the pole of the Perron kernel quotient"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/integral-exp-mul","kind":"node","lean":"PerronKernel.integral_exp_mul","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(\\exp(Lx)\\)"]],"source":"Integral of $\\exp(Lx)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#integral-exp-mul","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"For real numbers $L \\ne 0$, $a$ and $b$,","title":"Integral of $\\exp(Lx)$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/horiz-side-bound-gt-one","kind":"node","lean":"PerronKernel.horiz_side_bound_gt_one","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Horizontal side bound for the Perron integrand when "],["math","\\(y > 1\\)"]],"source":"Horizontal side bound for the Perron integrand when $y > 1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#horiz-side-bound-gt-one","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y > 1$, let $U > 0$, $c > 0$ and $T > 0$ be real numbers, and let $\\tau = T$ or $\\tau = -T$. Then","title":"Horizontal side bound for the Perron integrand when $y > 1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/vert-side-bound","kind":"node","lean":"PerronKernel.vert_side_bound","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound for the integral of "],["math","\\(y^s/s\\)"],["text"," along a vertical segment"]],"source":"Bound for the integral of $y^s/s$ along a vertical segment"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#vert-side-bound","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y, d, T$ be real numbers with $y > 0$, $d \\neq 0$ and $T > 0$, and write $y^s = e^{s \\log y}$. Then","title":"Bound for the integral of $y^s/s$ along a vertical segment"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/vert-side-bound-left","kind":"node","lean":"PerronKernel.vert_side_bound_left","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound for the integral of "],["math","\\(y^s/s\\)"],["text"," along the left vertical side"]],"source":"Bound for the integral of $y^s/s$ along the left vertical side"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#vert-side-bound-left","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y, U, T$ be real numbers with $y > 0$, $U > 0$ and $T > 0$, and write $y^s = e^{s \\log y}$. Then","title":"Bound for the integral of $y^s/s$ along the left vertical side"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/vert-side-bound-right","kind":"node","lean":"PerronKernel.vert_side_bound_right","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound for the integral of "],["math","\\(y^s/s\\)"],["text"," along the right vertical side"]],"source":"Bound for the integral of $y^s/s$ along the right vertical side"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#vert-side-bound-right","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y, U, T$ be real numbers with $y > 0$, $U > 0$ and $T > 0$, and write $y^s = e^{s \\log y}$. Then","title":"Bound for the integral of $y^s/s$ along the right vertical side"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/rect-identity-gt-one","kind":"node","lean":"PerronKernel.rect_identity_gt_one","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangle integral of "],["math","\\(y^s/s\\)"],["text"," around the origin"]],"source":"Rectangle integral of $y^s/s$ around the origin"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#rect-identity-gt-one","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y, U, c, T$ be real numbers with $y > 0$, $U > 0$, $c > 0$ and $T > 0$, and write $y^s = e^{s \\log y}$. Then","title":"Rectangle integral of $y^s/s$ around the origin"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/horiz-side-bound-exact","kind":"node","lean":"PerronKernel.horiz_side_bound_exact","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Horizontal side bound for the Perron integrand on "],["math","\\((c, U)\\)"]],"source":"Horizontal side bound for the Perron integrand on $(c, U)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#horiz-side-bound-exact","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y > 0$ with $\\log y \\ne 0$, let $c \\le U$ and $T > 0$ be real numbers, and let $\\tau = T$ or $\\tau = -T$. Then","title":"Horizontal side bound for the Perron integrand on $(c, U)$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/rect-identity-lt-one","kind":"node","lean":"PerronKernel.rect_identity_lt_one","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangle integral of "],["math","\\(y^s/s\\)"],["text"," with the pole outside"]],"source":"Rectangle integral of $y^s/s$ with the pole outside"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#rect-identity-lt-one","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y, U, c, T$ be real numbers with $y > 0$, $0 < c < U$ and $T > 0$, and write $y^s = e^{s \\log y}$. Then","title":"Rectangle integral of $y^s/s$ with the pole outside"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/perron-bound-gt-one","kind":"node","lean":"PerronKernel.perron_bound_gt_one","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncated Perron bound for "],["math","\\(y > 1\\)"]],"source":"Truncated Perron bound for $y > 1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#perron-bound-gt-one","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y > 1$, $c > 0$ and $T > 0$ be real numbers, and let $K(y, c, T) = \\frac{1}{2\\pi}\\int_{-T}^{T} \\frac{y^{c+it}}{c+it}\\,dt$ be the truncated Perron kernel. Then","title":"Truncated Perron bound for $y > 1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/perron-bound-lt-one","kind":"node","lean":"PerronKernel.perron_bound_lt_one","parent":9055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncated Perron bound for "],["math","\\(0 < y < 1\\)"]],"source":"Truncated Perron bound for $0 < y < 1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#perron-bound-lt-one","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $0 < y < 1$, $c > 0$ and $T > 0$ be real numbers, and let $K(y, c, T) = \\frac{1}{2\\pi}\\int_{-T}^{T} \\frac{y^{c+it}}{c+it}\\,dt$ be the truncated Perron kernel. Then","title":"Truncated Perron bound for $0 < y < 1$"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/perron-bound","kind":"node","lean":"PerronKernel.perron_bound","parent":9055,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula/index.html#perron-bound","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/perron-s-formula","status":"fully_proved","summary":"Let $y > 0$ with $y \\ne 1$, and let $c > 0$ and $T > 0$ be real numbers. Let $K(y, c, T) = \\frac{1}{2\\pi}\\int_{-T}^{T} \\frac{y^{c+it}}{c+it}\\,dt$ be the truncated Perron kernel, and let $\\delta(y) = 1$ if $y > 1$ and $\\delta(y) = 0$ if $y < 1$. Then","title":"Truncated Perron formula"},{"data_url":"scopes/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions.json","id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","kind":"scope","navigation":"module","parent":8972,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"can_prove","title":"Subconvexity bounds for Dirichlet L-functions"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/is-central-value-exponent-on","kind":"node","lean":"MathlibExt.NumberTheory.DirichletSubconvexity.IsCentralValueExponentOn","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#is-central-value-exponent-on","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"fully_proved","summary":"Let $S$ be a set of natural numbers and $\\theta$ a real number. Then $\\theta$ is a central-value exponent on $S$ if $\\theta \\ge 0$ and for every $\\varepsilon > 0$ there are a real number $C > 0$ and a natural number $Q$ such that for every positive $q \\in S$ with $q \\ge Q$ and every primitive Dirichlet character $\\chi$ modulo $q$, the analytically continued Dirichlet $L$-function satisfies\u2026","title":"Central-value exponents on a set of levels"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/is-dirichlet-subconvexity-exponent","kind":"node","lean":"MathlibExt.NumberTheory.DirichletSubconvexity.IsDirichletSubconvexityExponent","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#is-dirichlet-subconvexity-exponent","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"fully_proved","summary":"A real number $\\theta$ is an admissible conductor-aspect exponent for primitive Dirichlet characters if it is a central-value exponent on the set of all natural numbers: $\\theta \\ge 0$, and for every $\\varepsilon > 0$ there are a real number $C > 0$ and a natural number $Q$ such that for every positive integer $q \\ge Q$ and every primitive Dirichlet character $\\chi$ modulo $q$, the analytically continued Dirichlet\u2026","title":"Admissible exponents for Dirichlet central values"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/dirichlet-subconvexity-exponent","kind":"node","lean":"MathlibExt.NumberTheory.DirichletSubconvexity.dirichletSubconvexityExponent","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#dirichlet-subconvexity-exponent","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"fully_proved","summary":"The infimum, taken in the extended real numbers, of the set of all real $\\theta$ that are admissible conductor-aspect exponents for primitive Dirichlet characters. If no real number is admissible, this infimum is $+\\infty$.","title":"The conductor-aspect Dirichlet subconvexity exponent"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/dirichlet-subconvexity-exponent-nonneg","kind":"node","lean":"MathlibExt.NumberTheory.DirichletSubconvexity.dirichletSubconvexityExponent_nonneg","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#dirichlet-subconvexity-exponent-nonneg","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"fully_proved","summary":"The conductor-aspect Dirichlet subconvexity exponent, an extended real number, is at least $0$. This holds by construction: every admissible exponent is nonnegative, and if no real exponent is admissible, the exponent is the infimum of the empty set, which is $+\\infty$.","title":"The Dirichlet subconvexity exponent is nonnegative"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/dirichlet-subconvexity-exponent-le-of-ad","kind":"node","lean":"MathlibExt.NumberTheory.DirichletSubconvexity.dirichletSubconvexityExponent_le_of_admissible","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#dirichlet-subconvexity-exponent-le-of-ad","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"fully_proved","summary":"If a real number $\\theta$ is an admissible conductor-aspect exponent for primitive Dirichlet characters, then the conductor-aspect Dirichlet subconvexity exponent is at most $\\theta$, as extended real numbers.","title":"Admissible exponents bound the Dirichlet subconvexity exponent"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/burgess-three-sixteenths-admissible","kind":"node","lean":"MathlibExt.NumberTheory.DirichletSubconvexity.burgessThreeSixteenths_admissible","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#burgess-three-sixteenths-admissible","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"can_prove","summary":"The number $3/16$ is an admissible conductor-aspect exponent for primitive Dirichlet characters: for every $\\varepsilon > 0$ there are a real number $C > 0$ and a natural number $Q$ such that for every positive integer $q \\ge Q$ and every primitive Dirichlet character $\\chi$ modulo $q$, the analytically continued Dirichlet $L$-function satisfies $|L(1/2, \\chi)| \\le C q^{3/16 + \\varepsilon}$.","title":"Burgess's bound for Dirichlet central values"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/dirichlet-subconvexity-exponent-le-three","kind":"node","lean":"MathlibExt.NumberTheory.DirichletSubconvexity.dirichletSubconvexityExponent_le_threeSixteenths","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#dirichlet-subconvexity-exponent-le-three","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"can_prove","summary":"The conductor-aspect Dirichlet subconvexity exponent, an extended real number, is at most $3/16$.","title":"The Dirichlet subconvexity exponent is at most 3/16"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/petrow-young-cube-free-one-sixth-admissi","kind":"node","lean":"MathlibExt.NumberTheory.DirichletSubconvexity.petrowYoungCubeFreeOneSixth_admissible","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#petrow-young-cube-free-one-sixth-admissi","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"can_prove","summary":"The number $1/6$ is a central-value exponent on the set of cube-free levels, the natural numbers $q > 0$ such that $p^3$ does not divide $q$ for any prime $p$: for every $\\varepsilon > 0$ there are a real number $C > 0$ and a natural number $Q$ such that for every cube-free $q \\ge Q$ and every primitive Dirichlet character $\\chi$ modulo $q$, the analytically continued Dirichlet $L$-function satisfies\u2026","title":"Petrow\u2013Young Weyl bound for central values"},{"id":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/corollary-1-3","kind":"node","lean":"MathlibExt.NumberTheory.DirichletWeylBoundWanted.DirichletWeylBound.corollary_1_3","parent":9084,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions/index.html#corollary-1-3","scope":"NumberTheory/zeta-l-functions/dirichlet-l-functions/subconvexity-bounds-for-dirichlet-l-functions","status":"can_prove","summary":"For every real $\\varepsilon > 0$ there is a real number $C$ such that for every cube-free natural number $q$ (that is, $q > 0$ and $p^3$ does not divide $q$ for any prime $p$), every primitive Dirichlet character $\\chi$ modulo $q$ and every real $t$, the analytically continued Dirichlet $L$-function satisfies","title":"Petrow\u2013Young Weyl bound on the critical line"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values","kind":"scope","parent":8867,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values","status":"can_prove","summary":"Multiple zeta and zeta-star values, Hoffman bases, word algebras, cyclic sum formulas, finite and q-analogue multiple zeta values.","title":"Multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/summable-strict-summand","kind":"node","lean":"MetaMathlibExt.MultipleZeta.summable_strictSummand","parent":9094,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/index.html#summable-strict-summand","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values","status":"fully_proved","summary":"Let $\\mathbf{k} = (k_1, \\dots, k_r)$ be an admissible index, that is, a nonempty index of positive integers with $k_1 \\ge 2$. Then the strict multiple-zeta series of $\\mathbf{k}$, the family of terms $(m_1^{k_1} \\cdots m_r^{k_r})^{-1}$ indexed by the strictly decreasing tuples $m_1 > \\cdots > m_r$ of positive integers, is summable.","title":"Convergence of multiple zeta series"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","title":"Multiple-zeta indices"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index-2","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#index-2","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"An index is a finite sequence $(k_1, \\ldots, k_r)$ of positive integers with $r \\ge 0$, so the empty sequence is allowed. It is represented as a natural number $n$ together with a composition of $n$, that is, a sequence of positive integers with sum $n$.","title":"Multiple-zeta index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/entries","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.entries","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#entries","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"The entries of an index $(k_1, \\ldots, k_r)$ form the list $[k_1, \\ldots, k_r]$ of its terms, in order.","title":"Entries of an index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/weight","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.weight","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#weight","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"The weight of an index $(k_1, \\ldots, k_r)$ is the sum $k_1 + \\cdots + k_r$ of its entries.","title":"Weight of an index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/depth","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.depth","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#depth","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"The depth of an index $(k_1, \\ldots, k_r)$ is its number of entries $r$.","title":"Depth of an index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/height","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.height","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#height","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"The height of an index $(k_1, \\ldots, k_r)$ is the number of positions $i$ with $1 \\le i \\le r$ and $k_i \\ge 2$.","title":"Height of an index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/is-admissible","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.IsAdmissible","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#is-admissible","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"An index $(k_1, \\ldots, k_r)$ is admissible when it is nonempty ($r \\ge 1$) and its first entry satisfies $k_1 \\ge 2$.","title":"Admissible index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/is-admissible-2","kind":"node","lean":"MetaMathlibExt.MultipleZeta.IsAdmissible","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#is-admissible-2","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"A list $[k_1, \\ldots, k_r]$ of natural numbers is admissible when it is nonempty ($r \\ge 1$), $k_1 \\ge 2$, and $k_i > 0$ for every $i$ with $2 \\le i \\le r$.","title":"Admissible list"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/is-admissible-iff","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.isAdmissible_iff","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#is-admissible-iff","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"An index is admissible if and only if the list of its entries is an admissible list.","title":"Admissibility of an index via its entries"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/hoffman-block","kind":"node","lean":"MetaMathlibExt.MultipleZeta.hoffmanBlock","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#hoffman-block","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"For a positive integer $k$, the Hoffman block $z_k = x^{k-1} y$ is the word in the free monoid on two letters $x$ and $y$ consisting of $k - 1$ copies of $x$ followed by one $y$.","title":"Hoffman block"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/to-list-hoffman-block","kind":"node","lean":"MetaMathlibExt.MultipleZeta.toList_hoffmanBlock","parent":9096,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices/index.html#to-list-hoffman-block","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-indices","status":"fully_proved","summary":"For every positive integer $k$, the list of letters of the Hoffman block $z_k$ consists of $k - 1$ copies of $x$ (false) followed by one $y$ (true).","title":"Hoffman block as a list"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","title":"Multiple-zeta and multiple-zeta-star series"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/empty","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.empty","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#empty","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"The empty multiple-zeta index, which has no entries and weight zero.","title":"Empty multiple-zeta index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/entry","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.entry","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#entry","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"For a multiple-zeta index $k = (k_0, \\dots, k_{d-1})$ of depth $d$ and $0 \\le i < d$, the entry of $k$ at $i$ is the positive natural number $k_i$.","title":"Entry of a multiple-zeta index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/strict-decreasing-tuple","kind":"node","lean":"MetaMathlibExt.MultipleZeta.StrictDecreasingTuple","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#strict-decreasing-tuple","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"For a natural number $r$, a strictly decreasing $r$-tuple is a tuple $(m_0, \\dots, m_{r-1})$ of positive integers with $m_0 > m_1 > \\dots > m_{r-1}$.","title":"Strictly decreasing tuples"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/empty-2","kind":"node","lean":"MetaMathlibExt.MultipleZeta.StrictDecreasingTuple.empty","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#empty-2","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"The unique strictly decreasing tuple of positive natural numbers of length zero, namely the empty tuple.","title":"Empty strictly decreasing tuple"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/strict-summand","kind":"node","lean":"MetaMathlibExt.MultipleZeta.strictSummand","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#strict-summand","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"Let $\\mathbf{k} = (k_0, \\dots, k_{r-1})$ be an index of depth $r$ and let $(m_0, \\dots, m_{r-1})$ be a strictly decreasing $r$-tuple of positive integers. The summand attached to $\\mathbf{k}$ and this tuple is the real number $\\left(\\prod_{i=0}^{r-1} m_i^{k_i}\\right)^{-1}$.","title":"Multiple zeta summand"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/is-admissible-or-empty","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.IsAdmissibleOrEmpty","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#is-admissible-or-empty","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"A multiple-zeta index is admissible or empty when it is admissible (nonempty with first entry at least two) or has depth zero, that is, has no entries.","title":"Admissible or empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/strict-value","kind":"node","lean":"MetaMathlibExt.MultipleZeta.strictValue","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#strict-value","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"Let $\\mathbf{k} = (k_0, \\dots, k_{r-1})$ be an index of depth $r$ that is admissible ($r \\ge 1$ and $k_0 \\ge 2$) or empty. Its multiple zeta value is the real number","title":"Multiple zeta value"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/weakly-decreasing-tuple","kind":"node","lean":"MetaMathlibExt.MultipleZeta.WeaklyDecreasingTuple","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#weakly-decreasing-tuple","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"For a natural number $r$, a weakly decreasing $r$-tuple is a tuple $(m_0, \\dots, m_{r-1})$ of positive integers with $m_0 \\ge m_1 \\ge \\dots \\ge m_{r-1}$.","title":"Weakly decreasing tuples"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/empty-3","kind":"node","lean":"MetaMathlibExt.MultipleZeta.WeaklyDecreasingTuple.empty","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#empty-3","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"The unique weakly decreasing tuple of positive natural numbers of length zero, namely the empty tuple.","title":"Empty weakly decreasing tuple"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/to-weakly-decreasing","kind":"node","lean":"MetaMathlibExt.MultipleZeta.StrictDecreasingTuple.toWeaklyDecreasing","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#to-weakly-decreasing","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"For a natural number $r$, every strictly decreasing $r$-tuple $(m_0, \\dots, m_{r-1})$ of positive integers is weakly decreasing; this map sends it to the same tuple, regarded as a weakly decreasing $r$-tuple.","title":"Strictly decreasing tuples as weakly decreasing tuples"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/star-summand","kind":"node","lean":"MetaMathlibExt.MultipleZeta.starSummand","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#star-summand","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"For a multiple-zeta index $k = (k_0, \\dots, k_{d-1})$ of depth $d$ and a weakly decreasing tuple $m = (m_0, \\dots, m_{d-1})$ of positive natural numbers, the star summand of $k$ at $m$ is the real number $\\left(\\prod_{i\\lt d} m_i^{k_i}\\right)^{-1}$.","title":"Multiple-zeta-star summand"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/star-summand-to-weakly-decreasing","kind":"node","lean":"MetaMathlibExt.MultipleZeta.starSummand_toWeaklyDecreasing","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#star-summand-to-weakly-decreasing","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"For a multiple-zeta index $k$ of depth $d$ and a strictly decreasing tuple $m$ of $d$ positive natural numbers, the star summand of $k$ at $m$, with $m$ regarded as a weakly decreasing tuple, equals the strict summand of $k$ at $m$.","title":"Star and strict summands agree on strictly decreasing tuples"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/star-value","kind":"node","lean":"MetaMathlibExt.MultipleZeta.starValue","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#star-value","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"Let $\\mathbf{k} = (k_0, \\dots, k_{r-1})$ be an index of depth $r$ that is admissible ($r \\ge 1$ and $k_0 \\ge 2$) or empty. Its multiple zeta-star value is the real number","title":"Multiple zeta-star value"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/star-value-empty","kind":"node","lean":"MetaMathlibExt.MultipleZeta.starValue_empty","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#star-value-empty","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"The multiple-zeta-star value of the empty index is $1$.","title":"Multiple-zeta-star value of the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/replicate","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.replicate","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#replicate","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"For a natural number $k$ and a positive natural number $e$, the index $\\{e\\}^k = (e, \\dots, e)$ consisting of $k$ entries all equal to $e$, so of weight $ke$.","title":"Repeated index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/replicate-is-admissible-or-empty","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.replicate_isAdmissibleOrEmpty","parent":9107,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series/index.html#replicate-is-admissible-or-empty","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/multiple-zeta-and-multiple-zeta-star-series","status":"fully_proved","summary":"For every natural number $k$ and every positive natural number $e \\ge 2$, the index $\\{e\\}^k$ consisting of $k$ entries all equal to $e$ is admissible or empty.","title":"Repeated indices with entry at least two are admissible or empty"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","title":"Reversing multiple-zeta summation conventions"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/reverse","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.reverse","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#reverse","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"The reverse of an index $(k_0, \\dots, k_{d-1})$ is the index $(k_{d-1}, \\dots, k_0)$ with the same entries in the opposite order.","title":"Reversed index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/strict-increasing-tuple","kind":"node","lean":"MetaMathlibExt.MultipleZeta.StrictIncreasingTuple","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#strict-increasing-tuple","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"For a natural number $d$, a strictly increasing tuple of length $d$ is a tuple $(m_0, \\dots, m_{d-1})$ of positive natural numbers with $m_0 < m_1 < \\cdots < m_{d-1}$. These are the summation variables of the increasing multiple-zeta convention.","title":"Strictly increasing tuples"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/is-increasing-admissible","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.IsIncreasingAdmissible","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#is-increasing-admissible","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"An index $(k_0, \\dots, k_{d-1})$ is admissible in the increasing convention when its list of entries read in reverse order, $(k_{d-1}, \\dots, k_0)$, is admissible: it is nonempty, its first element $k_{d-1}$ is at least two, and its other elements are positive. Since the entries of an index are positive, this says that the index is nonempty and its last entry is at least two.","title":"Admissibility in the increasing convention"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/is-increasing-admissible-iff-reverse-is","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.isIncreasingAdmissible_iff_reverse_isAdmissible","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#is-increasing-admissible-iff-reverse-is","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"An index is admissible in the increasing convention (nonempty with last entry at least two) if and only if its reverse is admissible (nonempty with first entry at least two).","title":"Increasing admissibility is admissibility of the reverse"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/reverse-entry","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.reverse_entry","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#reverse-entry","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"Let $k = (k_0, \\dots, k_{d-1})$ be an index of depth $d$; its reverse also has depth $d$. For every $i < d$, the $i$-th entry of the reverse of $k$ is $k_{d-1-i}$.","title":"Entries of the reversed index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/increasing-summand","kind":"node","lean":"MetaMathlibExt.MultipleZeta.increasingSummand","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#increasing-summand","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"Let $k = (k_0, \\dots, k_{d-1})$ be an index of depth $d$. To a strictly increasing tuple $m = (m_0, \\dots, m_{d-1})$ of positive natural numbers it attaches the real number","title":"Summand of the increasing multiple-zeta series"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/increasing-value","kind":"node","lean":"MetaMathlibExt.MultipleZeta.increasingValue","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#increasing-value","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"Let $k = (k_0, \\dots, k_{d-1})$ be an index of depth $d$ that is admissible in the increasing convention. Its value in the increasing convention is the real number","title":"Multiple zeta value in the increasing convention"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/increasing-equiv-decreasing","kind":"node","lean":"MetaMathlibExt.MultipleZeta.increasingEquivDecreasing","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#increasing-equiv-decreasing","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"Let $k$ be an index of depth $d$; its reverse also has depth $d$. Sending a strictly increasing tuple $(m_0, \\dots, m_{d-1})$ of positive natural numbers to the reversed tuple $(m_{d-1}, \\dots, m_0)$ is a bijection from the strictly increasing tuples of length $d$ onto the strictly decreasing tuples of length $d$, whose inverse again reverses the order of the coordinates.","title":"Reversal of strictly increasing tuples"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/increasing-summand-eq-strict-summand-rev","kind":"node","lean":"MetaMathlibExt.MultipleZeta.increasingSummand_eq_strictSummand_reverse","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#increasing-summand-eq-strict-summand-rev","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"Let $k = (k_0, \\dots, k_{d-1})$ be an index of depth $d$ and let $m = (m_0, \\dots, m_{d-1})$ be a strictly increasing tuple of positive natural numbers. Then the summand of the increasing multiple-zeta series of $k$ at $m$ equals the summand of the strict multiple-zeta series of the reversed index $(k_{d-1}, \\dots, k_0)$ at the strictly decreasing tuple $(m_{d-1}, \\dots, m_0)$.","title":"Reversal preserves each summand"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/increasing-value-eq-strict-value-reverse","kind":"node","lean":"MetaMathlibExt.MultipleZeta.increasingValue_eq_strictValue_reverse","parent":9124,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions/index.html#increasing-value-eq-strict-value-reverse","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/reversing-multiple-zeta-summation-conventions","status":"fully_proved","summary":"Let $k = (k_0, \\dots, k_{d-1})$ be an index of depth $d$ that is admissible in the increasing convention, so that its reverse $(k_{d-1}, \\dots, k_0)$ is admissible. Then the value of $k$ in the increasing convention equals the strict multiple zeta value of the reverse of $k$:","title":"Increasing values agree with strict values of the reverse"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro","status":"can_prove","title":"Fa\u00e0 di Bruno's formula and nested sums of reciprocal squares"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/taylor-coeff-comp-eq-sum-partitions","kind":"node","lean":"MetaMathlibExt.taylorCoeff_comp_eq_sum_partitions","parent":9135,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/index.html#taylor-coeff-comp-eq-sum-partitions","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro","status":"can_prove","summary":"Let $f, g : \\mathbb{R} \\to \\mathbb{R}$, let $a \\in \\mathbb{R}$ and let $n$ be a natural number. Suppose that $g$ is $n$ times continuously differentiable on a neighbourhood of $a$ and that $f$ is $n$ times continuously differentiable on a neighbourhood of $g(a)$. For a partition $\\pi$ of $n$, let $m_i(\\pi)$ be the number of parts of $\\pi$ equal to $i$, and let $\\ell(\\pi) = m_1(\\pi) + \\cdots + m_n(\\pi)$ be its\u2026","title":"Fa\u00e0 di Bruno's formula for Taylor coefficients"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/tsum-weakly-decreasing-prod-inv-odd-sq-e","kind":"node","lean":"MetaMathlibExt.tsum_weaklyDecreasing_prod_inv_odd_sq_eq","parent":9135,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/index.html#tsum-weakly-decreasing-prod-inv-odd-sq-e","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro","status":"can_prove","summary":"For every natural number $k$,","title":"Nested sums of reciprocal odd squares"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/star-value-replicate-two-eq","kind":"node","lean":"MetaMathlibExt.starValue_replicate_two_eq","parent":9135,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/index.html#star-value-replicate-two-eq","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro","status":"can_prove","summary":"For every natural number $k$, the multiple zeta-star value","title":"Nested sums of reciprocal squares"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/tsum-strict-decreasing-prod-inv-odd-sq-e","kind":"node","lean":"MetaMathlibExt.tsum_strictDecreasing_prod_inv_odd_sq_eq","parent":9135,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro/index.html#tsum-strict-decreasing-prod-inv-odd-sq-e","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/fa-di-bruno-s-formula-and-nested-sums-of-recipro","status":"can_prove","summary":"For every natural number $k$,","title":"Strictly nested sums of reciprocal odd squares"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","title":"Finite multiple zeta values of level N"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/colored-finite-mzv","kind":"node","lean":"MetaMathlibExt.coloredFiniteMZV","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#colored-finite-mzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"fully_proved","summary":"Let $p$ and $N$ be natural numbers, let $k = (k_1, \\dots, k_r)$ be a finite sequence of natural numbers and let $\\alpha = (\\alpha_1, \\dots, \\alpha_r) \\in (\\mathbb{Z}/N\\mathbb{Z})^r$. The colored finite multiple harmonic sum $\\zeta^{\\alpha}_{p,N}(k) \\in \\mathbb{Z}/p\\mathbb{Z}$ is the sum of $m_1^{-k_1} \\cdots m_r^{-k_r}$ over all integers $0 < m_1 < \\dots < m_r < p$ with $m_i \\equiv \\alpha_i \\pmod N$ for every $i$.","title":"Colored finite multiple harmonic sums"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/colored-finite-mzv-zero","kind":"node","lean":"MetaMathlibExt.coloredFiniteMZV_zero","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#colored-finite-mzv-zero","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"fully_proved","summary":"For all natural numbers $p$ and $N$, the colored finite multiple harmonic sum $\\zeta^{\\varnothing}_{p,N}(\\varnothing)$ of the empty index, with the empty tuple of colors, equals $1$ in $\\mathbb{Z}/p\\mathbb{Z}$.","title":"Colored finite multiple harmonic sum of the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/sum-ioo-eq-pow-mul-colored-finite-mzv","kind":"node","lean":"MetaMathlibExt.sum_Ioo_eq_pow_mul_coloredFiniteMZV","parent":9140,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums over a subinterval of "],["math","\\((0, p)\\)"],["text"," as colored finite multiple zeta values"]],"source":"Sums over a subinterval of $(0, p)$ as colored finite multiple zeta values"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#sum-ioo-eq-pow-mul-colored-finite-mzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"Let $N \\ge 1$, $0 \\le j < N$, and let $k = (k_1, \\dots, k_r)$ be a tuple of natural numbers of weight $w = k_1 + \\dots + k_r$. For a prime $p$ and $\\alpha \\in (\\mathbb{Z}/N\\mathbb{Z})^r$ let $\\zeta^{\\alpha}_{p,N}(k) \\in \\mathbb{Z}/p\\mathbb{Z}$ be the colored finite multiple harmonic sum, the sum of $1/(m_1^{k_1} \\cdots m_r^{k_r})$ over the integers $0 < m_1 < \\dots < m_r < p$ with $m_i \\equiv \\alpha_i \\pmod{N}$ for\u2026","title":"Sums over a subinterval of $(0, p)$ as colored finite multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/two-mul-colored-finite-mzv-twelve","kind":"node","lean":"MetaMathlibExt.two_mul_coloredFiniteMZV_twelve","parent":9140,"reading_title":{"format":"inline-parts-v1","parts":[["text","Level-"],["math","\\(12\\)"],["text"," colored finite zeta values and Bernoulli numbers"]],"source":"Level-$12$ colored finite zeta values and Bernoulli numbers"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#two-mul-colored-finite-mzv-twelve","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"Let $k \\ge 3$ be odd. For a prime $p > k$ write $Z_p(k) = B_{p-k}/k \\in \\mathbb{Z}/p\\mathbb{Z}$, where $B_n$ is the $n$-th Bernoulli number, and for $\\alpha \\in \\mathbb{Z}/12\\mathbb{Z}$ let $\\zeta^{\\alpha}_{p,12}(k) \\in \\mathbb{Z}/p\\mathbb{Z}$ be the sum of $m^{-k}$ over $0 < m < p$ with $m \\equiv \\alpha \\pmod{12}$. Then for all but finitely many primes $p$, in $\\mathbb{Z}/p\\mathbb{Z}$,\u2026","title":"Level-$12$ colored finite zeta values and Bernoulli numbers"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/fermat-last-theorem-for-of-regular","kind":"node","lean":"MetaMathlibExt.fermatLastTheoremFor_of_regular","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#fermat-last-theorem-for-of-regular","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"Let $p$ be an odd prime and let $K$ be a $p$-th cyclotomic field over $\\mathbb{Q}$ (a number field generated over $\\mathbb{Q}$ by a primitive $p$-th root of unity). If $p$ does not divide the class number of $K$, then Fermat's Last Theorem holds for the exponent $p$: there are no positive integers $a, b, c$ with $a^p + b^p = c^p$.","title":"Kummer's theorem for regular primes"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/fermat-first-case-of-not-is-wieferich-pr","kind":"node","lean":"MetaMathlibExt.fermat_first_case_of_not_isWieferichPrime","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#fermat-first-case-of-not-is-wieferich-pr","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"Let $p$ be a prime that is not a Wieferich prime, that is, $p^2 \\nmid 2^{p-1} - 1$. Then there are no integers $x, y, z$ with $p \\nmid xyz$ and $x^p + y^p = z^p$.","title":"Wieferich's criterion for the first case of Fermat's Last Theorem"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/fermat-quotient-two-mul-eq-neg-sum","kind":"node","lean":"MetaMathlibExt.fermatQuotient_two_mul_eq_neg_sum","parent":9140,"reading_title":{"format":"inline-parts-v1","parts":[["text","Skula\u2013Dobson\u2013Ichimura congruence for the Fermat quotient of "],["math","\\(2\\)"]],"source":"Skula\u2013Dobson\u2013Ichimura congruence for the Fermat quotient of $2$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#fermat-quotient-two-mul-eq-neg-sum","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"Let $p$ be an odd prime and $N$ a positive integer with $p \\nmid N$. Write $q_p(2) = (2^{p-1} - 1)/p$ for the Fermat quotient of $2$, and for integers $i \\ge 0$ and $M \\ge 1$ let $s_p(i, M) \\in \\mathbb{Z}/p\\mathbb{Z}$ be the sum of $m^{-1}$ over the integers $m$ with $0 < m < p$ and $ip/M < m < (i+1)p/M$. Then $(N + 1)\\, q_p(2) \\equiv -\\sum_{0 \\le j < N/2} s_p(2j, 2N) \\pmod{p}$.","title":"Skula\u2013Dobson\u2013Ichimura congruence for the Fermat quotient of $2$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/exists-odd-not-dvd-bernoulli-num-le","kind":"node","lean":"MetaMathlibExt.exists_odd_not_dvd_bernoulli_num_le","parent":9140,"reading_title":{"format":"inline-parts-v1","parts":[["text","A small odd "],["math","\\(k\\)"],["text"," with "],["math","\\(p \\nmid B_{p-k}\\)"]],"source":"A small odd $k$ with $p \\nmid B_{p-k}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#exists-odd-not-dvd-bernoulli-num-le","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"For every prime $p \\ge 11$ there is an odd integer $k \\ge 3$ such that $p$ does not divide the numerator of the Bernoulli number $B_{p-k}$ and $k \\le (p-3)/2$ if $p \\equiv 1 \\pmod 4$, while $k \\le (p-5)/2$ if $p \\equiv 3 \\pmod 4$.","title":"A small odd $k$ with $p \\nmid B_{p-k}$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/finite-mzvcolored","kind":"node","lean":"MetaMathlibExt.finiteMZVColored","parent":9140,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite multiple zeta value of level "],["math","\\(N\\)"],["text"," with a color map"]],"source":"Finite multiple zeta value of level $N$ with a color map"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#finite-mzvcolored","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"fully_proved","summary":"Let $p$ and $N$ be natural numbers, $k = (k_1, \\dots, k_r)$ a tuple of natural numbers, and $c$ a color map, which sends each $a \\in \\mathbb{Z}/N\\mathbb{Z}$ either to $\\emptyset$ or to a tuple in $(\\mathbb{Z}/N\\mathbb{Z})^r$. The $p$-component $\\zeta^c_{\\mathcal{A},N}(k)_p \\in \\mathbb{Z}/p\\mathbb{Z}$ is the colored finite multiple harmonic sum $\\zeta^{\\alpha}_{p,N}(k)$ when $c(p \\bmod N) = \\alpha$ is a tuple, and\u2026","title":"Finite multiple zeta value of level $N$ with a color map"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/finite-mzvcolored-none","kind":"node","lean":"MetaMathlibExt.finiteMZVColored_none","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#finite-mzvcolored-none","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"fully_proved","summary":"For all natural numbers $p$ and $N$ and every tuple $k = (k_1, \\dots, k_r)$ of natural numbers, if $c$ is the color map sending every $a \\in \\mathbb{Z}/N\\mathbb{Z}$ to $\\emptyset$, then the $p$-component $\\zeta^c_{\\mathcal{A},N}(k)_p$ of the finite multiple zeta value of level $N$ with color map $c$ is $0$.","title":"The empty color map gives zero"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/finite-mzvcolored-some","kind":"node","lean":"MetaMathlibExt.finiteMZVColored_some","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#finite-mzvcolored-some","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"fully_proved","summary":"For all natural numbers $p$ and $N$, every tuple $k = (k_1, \\dots, k_r)$ of natural numbers and every $\\alpha \\in (\\mathbb{Z}/N\\mathbb{Z})^r$, if $c$ is the color map sending every $a \\in \\mathbb{Z}/N\\mathbb{Z}$ to $\\alpha$, then the $p$-component $\\zeta^c_{\\mathcal{A},N}(k)_p$ of the finite multiple zeta value of level $N$ with color map $c$ equals the colored finite multiple harmonic sum $\\zeta^{\\alpha}_{p,N}(k)$.","title":"A constant color map gives the colored harmonic sum"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/finite-mzvcolored-eq-neg-one-pow-mul-rev","kind":"node","lean":"MetaMathlibExt.finiteMZVColored_eq_neg_one_pow_mul_reverse","parent":9140,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reversal for finite multiple zeta values of level "],["math","\\(N\\)"]],"source":"Reversal for finite multiple zeta values of level $N$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#finite-mzvcolored-eq-neg-one-pow-mul-rev","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"Let $N \\ge 1$, let $k = (k_1, \\dots, k_r)$ be a tuple of natural numbers of weight $w = k_1 + \\dots + k_r$, and let $c$ be a color map, which sends each $a \\in \\mathbb{Z}/N\\mathbb{Z}$ either to $\\emptyset$ or to a tuple in $(\\mathbb{Z}/N\\mathbb{Z})^r$. Write $k^{\\leftarrow} = (k_r, \\dots, k_1)$, and let $c^{\\leftarrow}$ be the color map with $c^{\\leftarrow}(a) = (a - \\alpha_r, \\dots, a - \\alpha_1)$ when\u2026","title":"Reversal for finite multiple zeta values of level $N$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/colored-finite-mzv-one-eq-pow-mul-sum-pr","kind":"node","lean":"MetaMathlibExt.coloredFiniteMZV_one_eq_pow_mul_sum_prod","parent":9140,"reading_title":{"format":"inline-parts-v1","parts":[["text","Decomposition of finite multiple zeta values into level-"],["math","\\(N\\)"],["text"," pieces"]],"source":"Decomposition of finite multiple zeta values into level-$N$ pieces"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#colored-finite-mzv-one-eq-pow-mul-sum-pr","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"Let $N \\ge 1$ and let $k = (k_1, \\dots, k_r)$ be a finite sequence of natural numbers (entries equal to $0$ are allowed) of weight $w = k_1 + \\dots + k_r$. For a prime $p$ put $\\zeta_p(k) = \\sum_{0 < m_1 < \\dots < m_r < p} m_1^{-k_1} \\cdots m_r^{-k_r}$ in $\\mathbb{Z}/p\\mathbb{Z}$, and for $0 \\le j < N$ and a finite sequence $l$ let $\\zeta^{[j]}_{p,N}(l)$ be the colored finite multiple harmonic sum of level $N$ of\u2026","title":"Decomposition of finite multiple zeta values into level-$N$ pieces"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/exists-finite-mzvcolored-eq-sum-of-dvd","kind":"node","lean":"MetaMathlibExt.exists_finiteMZVColored_eq_sum_of_dvd","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#exists-finite-mzvcolored-eq-sum-of-dvd","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"Let $N$ and $M$ be positive integers with $N \\mid M$, let $k = (k_1, \\dots, k_r)$ be an index of positive integers of weight $w = k_1 + \\dots + k_r$, and let $c$ be a color map from $\\mathbb{Z}/N\\mathbb{Z}$ to $(\\mathbb{Z}/N\\mathbb{Z})^r \\cup \\{\\varnothing\\}$. Write $\\zeta^{c}_{p,N}(k)$ for the $p$-component of the finite multiple zeta value of level $N$ with color map $c$. Then there are rationals\u2026","title":"Raising the level of finite multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/exists-bound-fermat-quotient-iff-finite","kind":"node","lean":"MetaMathlibExt.exists_bound_fermatQuotient_iff_finite_iInter","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#exists-bound-fermat-quotient-iff-finite","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"The following are equivalent: (a) there is a natural number $M$ such that for every odd prime $p$ there is an integer $N$ with $1 \\le N \\le M$, $p \\nmid N$ and $p^2 \\nmid N^{p-1} - 1$; (b) there is a natural number $M \\ge 3$ such that only finitely many primes $p$ satisfy $p^2 \\mid N^{p-1} - 1$ for every integer $N$ with $2 \\le N \\le M$.","title":"Criterion for a uniform bound on the least non-Wieferich base"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/exists-bound-bernoulli-iff-finite-i-inte","kind":"node","lean":"MetaMathlibExt.exists_bound_bernoulli_iff_finite_iInter","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#exists-bound-bernoulli-iff-finite-i-inte","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"The following are equivalent: (a) there is a natural number $M$ such that for every prime $p \\ge 5$ there is an odd integer $k$ with $3 \\le k \\le M$, $k < p$ and $p \\nmid \\operatorname{num}(B_{p-k})$; (b) there is a natural number $M \\ge 5$ such that only finitely many primes $p$ satisfy $k < p$ and $p \\mid \\operatorname{num}(B_{p-k})$ for every odd integer $k$ with $3 \\le k \\le M$.","title":"Criterion for a uniform bound on the least non-vanishing Bernoulli index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/exists-finite-mzv-ne-zero-of-conjectures","kind":"node","lean":"MetaMathlibExt.exists_finiteMZV_ne_zero_of_conjectures","parent":9140,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n/index.html#exists-finite-mzv-ne-zero-of-conjectures","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/finite-multiple-zeta-values-of-level-n","status":"can_prove","summary":"(i) Suppose that infinitely many odd primes $p$ are regular, in the sense that $p$ does not divide the class number of any $p$-th cyclotomic field over $\\mathbb{Q}$. Then for every natural number $w \\notin \\{1, 2, 4\\}$ there is an index $k = (k_1, \\dots, k_r)$ of positive integers with $k_1 + \\dots + k_r = w$ such that $\\zeta_p(k) = \\sum_{0 < m_1 < \\dots < m_r < p} m_1^{-k_1} \\cdots m_r^{-k_r}$ is non-zero in\u2026","title":"Regular and non-Wieferich primes give non-vanishing finite multiple zeta values"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"can_prove","title":"Height one multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/height-one-multiple-zeta","kind":"node","lean":"MetaMathlibExt.heightOneMultipleZeta","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#height-one-multiple-zeta","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"fully_proved","summary":"For a complex number $s$ and an integer $j \\ge 0$, the height one multiple zeta function is","title":"Height one multiple zeta function"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/height-one-multiple-zeta-zero","kind":"node","lean":"MetaMathlibExt.heightOneMultipleZeta_zero","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#height-one-multiple-zeta-zero","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"fully_proved","summary":"For every complex number $s$, the height one multiple zeta function $\\zeta(s, \\{1\\}^j)$ at $j = 0$ is the Dirichlet series of the Riemann zeta function: $\\zeta(s, \\{1\\}^0) = \\sum_{n \\ge 0} (n + 1)^{-s}$.","title":"Height one multiple zeta function of depth one"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/multiple-hurwitz-zeta","kind":"node","lean":"MetaMathlibExt.multipleHurwitzZeta","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#multiple-hurwitz-zeta","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"fully_proved","summary":"For complex numbers $r$ and $s$, the multiple Hurwitz zeta function of order $r$, with Hurwitz parameter $a = 1$, is the Dirichlet series","title":"Multiple Hurwitz zeta function"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/multiple-hurwitz-zeta-zero","kind":"node","lean":"MetaMathlibExt.multipleHurwitzZeta_zero","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#multiple-hurwitz-zeta-zero","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"fully_proved","summary":"For every complex number $s$, the multiple Hurwitz zeta function of order $0$ is identically one: $\\zeta_0(s) = 1$.","title":"Multiple Hurwitz zeta function of order zero"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/has-sum-height-one-multiple-zeta-mul-pow","kind":"node","lean":"MetaMathlibExt.hasSum_heightOneMultipleZeta_mul_pow","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#has-sum-height-one-multiple-zeta-mul-pow","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"can_prove","summary":"Let $r$ and $s$ be complex numbers with $\\operatorname{Re} s > 1$, $|r - 1| < |s - 1|$ and $\\operatorname{Re} r < \\operatorname{Re} s$. Write $\\zeta(s, \\{1\\}^j)$ for the height one multiple zeta function and $\\zeta_r(s)$ for the multiple Hurwitz zeta function of order $r$. Then the series $\\sum_{j \\ge 0} \\zeta(s, \\{1\\}^j) \\, (r - 1)^j$ is summable with sum $\\zeta_r(s)$.","title":"Expansion of the multiple Hurwitz zeta function in height one multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/has-sum-height-one-multiple-zeta-mul-nat","kind":"node","lean":"MetaMathlibExt.hasSum_heightOneMultipleZeta_mul_nat_pow","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#has-sum-height-one-multiple-zeta-mul-nat","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"can_prove","summary":"Let $n \\ge 0$ be an integer. Write $\\zeta(s, \\{1\\}^j)$ for the height one multiple zeta function, $\\zeta$ for the Riemann zeta function, $c(n, m)$ for the unsigned Stirling numbers of the first kind and $B_j^{(k)}(x)$ for the poly-Bernoulli polynomials. For every complex $s$ with $\\operatorname{Re} s > n + 1$, the series $\\sum_{j \\ge 0} \\zeta(s, \\{1\\}^j) \\, n^j$ (with $0^0 = 1$) is summable with sum\u2026","title":"Height one multiple zeta series weighted by powers of a natural number"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/tsum-height-one-multiple-zeta-mul-neg-na","kind":"node","lean":"MetaMathlibExt.tsum_heightOneMultipleZeta_mul_neg_nat_pow","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#tsum-height-one-multiple-zeta-mul-neg-na","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"can_prove","summary":"Let $r \\ge 1$ be an integer and let $\\zeta(s, \\{1\\}^j)$ denote the height one multiple zeta function. For every complex number $s$ with $\\operatorname{Re} s > r + 1$,","title":"Height one multiple zeta series weighted by powers of a negative integer"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/has-sum-height-one-multiple-zeta-goldbac","kind":"node","lean":"MetaMathlibExt.hasSum_heightOneMultipleZeta_goldbach","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#has-sum-height-one-multiple-zeta-goldbac","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"can_prove","summary":"Let $k \\ge 1$ be an integer. Write $\\zeta(s, \\{1\\}^j)$ for the height one multiple zeta function, $\\zeta$ for the Riemann zeta function, $c(k, m)$ for the unsigned Stirling numbers of the first kind, $H_k = \\sum_{i=1}^{k} 1/i$ for the $k$-th harmonic number and $H_{k,k} = \\sum_{i=1}^{k} (-1)^{i-1} \\binom{k}{i} i^{-k}$ for the Roman harmonic number. Then the series\u2026","title":"Generalized Goldbach series for height one multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/has-sum-succ-mul-height-one-multiple-zet","kind":"node","lean":"MetaMathlibExt.hasSum_succ_mul_heightOneMultipleZeta_mul_pow","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#has-sum-succ-mul-height-one-multiple-zet","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"can_prove","summary":"Let $r$ and $s$ be complex numbers with $\\operatorname{Re} s > 0$, $|r| < |s|$ and $\\operatorname{Re} r < \\operatorname{Re} s$. Write $\\zeta(s, \\{1\\}^j)$ for the height one multiple zeta function, $H_m^{[r]}(k)$ for the generalized hyperharmonic numbers and $H_m^{[r]} = H_m^{[r]}(1)$; equivalently, $H_m^{[r]}$ is the derivative of $\\binom{m + r - 1}{m}$ with respect to $r$. Then the series\u2026","title":"Expansion of the hyperharmonic zeta function in height one multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/tendsto-alternating-height-one-multiple","kind":"node","lean":"MetaMathlibExt.tendsto_alternating_heightOneMultipleZeta_integral","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#tendsto-alternating-height-one-multiple","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"can_prove","summary":"Let $k \\ge -1$ and $p \\ge 0$ be integers, write $\\zeta(s, \\{1\\}^p)$ for the height one multiple zeta function and $\\zeta_r(s)$ for the multiple Hurwitz zeta function of order $r$. Then the partial sums of the series $\\nu_{k,p} = \\sum_{n \\ge 2} \\frac{(-1)^n \\, \\zeta(n, \\{1\\}^p)}{n + k}$ converge to $\\int_0^1 (1 - r)^{k+1} \\, \\zeta_r(p + 2) \\, dr$.","title":"Shifted alternating series of height one multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/height-one-multiple-zeta-laurent-linear","kind":"node","lean":"MetaMathlibExt.heightOneMultipleZeta_laurent_linear_coeff","parent":9158,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values/index.html#height-one-multiple-zeta-laurent-linear","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/height-one-multiple-zeta-values","status":"can_prove","summary":"Let $j \\ge 1$ be an integer, write $\\zeta(s, \\{1\\}^{j-1})$ for the height one multiple zeta function and $\\zeta_r(s)$ for the multiple Hurwitz zeta function of order $r$. There are complex numbers $c_0, \\dots, c_{j-1}$ and a function $F$ analytic at $s = 1$ such that $\\zeta(s, \\{1\\}^{j-1}) = \\sum_{i=0}^{j-1} c_i (s - 1)^{-(i+1)} + F(s)$ for all $s$ near $1$ with $\\operatorname{Re} s > 1$, and","title":"Linear Laurent coefficient of the height one multiple zeta function"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","title":"Cyclic sum formula"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/to-index","kind":"node","lean":"MetaMathlibExt.CyclicSumFormula.toIndex","parent":9170,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html#to-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","summary":"A list $(l_1, \\dots, l_r)$ of positive integers determines the multiple-zeta index whose entries are $l_1, \\dots, l_r$ in this order; its weight is $l_1 + \\cdots + l_r$.","title":"The index of a list of positive integers"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/left-index","kind":"node","lean":"MetaMathlibExt.CyclicSumFormula.leftIndex","parent":9170,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html#left-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","summary":"Let $k = (k_1, \\dots, k_l)$ be a list of positive integers, with subscripts read modulo $l$. For $1 \\le j \\le l$ and a natural number $i$, the left-hand index of $k$ at $(j, i)$ is the index $(k_j - i + 1, k_{j+1}, \\dots, k_{j+l-1}, i)$ when $1 \\le i \\le k_j - 1$, and the index $(2)$ otherwise. The run $k_{j+1}, \\dots, k_{j+l-1}$ is empty when $l = 1$.","title":"Left-hand indices of the cyclic sum formula"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/right-index","kind":"node","lean":"MetaMathlibExt.CyclicSumFormula.rightIndex","parent":9170,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html#right-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","summary":"Let $k = (k_1, \\dots, k_l)$ be a list of positive integers, with subscripts read modulo $l$. For $1 \\le j \\le l$, the right-hand index of $k$ at $j$ is the index $(k_j + 1, k_{j+1}, \\dots, k_{j+l-1})$, where the run $k_{j+1}, \\dots, k_{j+l-1}$ is empty when $l = 1$.","title":"Right-hand indices of the cyclic sum formula"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/left-index-is-admissible-or-empty","kind":"node","lean":"MetaMathlibExt.CyclicSumFormula.leftIndex_isAdmissibleOrEmpty","parent":9170,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html#left-index-is-admissible-or-empty","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","summary":"For every list $k = (k_1, \\dots, k_l)$ of positive integers, every $1 \\le j \\le l$ and every natural number $i$, the left-hand index of the cyclic sum formula of $k$ at $(j, i)$ is admissible or empty.","title":"Left-hand indices of the cyclic sum formula are admissible or empty"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/right-index-is-admissible-or-empty","kind":"node","lean":"MetaMathlibExt.CyclicSumFormula.rightIndex_isAdmissibleOrEmpty","parent":9170,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html#right-index-is-admissible-or-empty","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","summary":"For every list $k = (k_1, \\dots, k_l)$ of positive integers and every $1 \\le j \\le l$, the right-hand index of the cyclic sum formula of $k$ at $j$ is admissible or empty.","title":"Right-hand indices of the cyclic sum formula are admissible or empty"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/hoffman-ohno-cyclic-sum-formula-strict","kind":"node","lean":"MetaMathlibExt.hoffman_ohno_cyclic_sum_formula_strict","parent":9170,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html#hoffman-ohno-cyclic-sum-formula-strict","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","summary":"Let $k = (k_1, \\dots, k_l)$ be a nonempty list of positive integers, not all equal to $1$, with subscripts read modulo $l$, and write $\\zeta$ for the strict multiple zeta value. For $1 \\le j \\le l$ and $1 \\le i \\le k_j - 1$ the left-hand index of the cyclic sum formula of $k$ at $(j, i)$ is $(k_j - i + 1, k_{j+1}, \\dots, k_{j+l-1}, i)$, and its right-hand index at $j$ is $(k_j + 1, k_{j+1}, \\dots, k_{j+l-1})$; the\u2026","title":"Cyclic sum formula for strict multiple zeta values of indices"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/hoffman-ohno-cyclic-sum-formula","kind":"node","lean":"MetaMathlibExt.hoffman_ohno_cyclic_sum_formula","parent":9170,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html#hoffman-ohno-cyclic-sum-formula","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","summary":"For a finite list $(n_1, \\dots, n_r)$ of natural numbers let $\\zeta(n_1, \\dots, n_r)$ be the sum of $(m_1^{n_1} \\cdots m_r^{n_r})^{-1}$ over all integers $m_1 > m_2 > \\cdots > m_r > 0$, taken to be $0$ when this family is not summable. Let $k = (k_1, \\dots, k_l)$ be a nonempty list of positive integers, not all equal to $1$, with subscripts read modulo $l$. Then","title":"Cyclic sum formula (Hoffman\u2013Ohno)"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/cyclic-sum-formula-multiple-zeta-star-va","kind":"node","lean":"MetaMathlibExt.cyclic_sum_formula_multiple_zeta_star_values","parent":9170,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula/index.html#cyclic-sum-formula-multiple-zeta-star-va","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/cyclic-sum-formula","status":"fully_proved","summary":"Let $l \\ge 1$ and let $k_0, \\dots, k_{l-1}$ be positive integers, not all equal to $1$, with subscripts read modulo $l$, and put $K = k_0 + \\cdots + k_{l-1}$. Write $\\zeta^\\star$ for the multiple zeta-star value and $\\zeta$ for the strict multiple zeta value. Suppose that for each $0 \\le j \\le l - 1$ and each natural number $i$ an index $\\mathbf{c}_{j,i}$ is given that is admissible or empty, and that for\u2026","title":"Cyclic sum formula for multiple zeta-star values"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","title":"Hoffman algebra and its bases"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/x-word","kind":"node","lean":"MetaMathlibExt.MultipleZeta.xWord","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","The letter "],["math","\\(x\\)"],["text"," as a word"]],"source":"The letter $x$ as a word"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#x-word","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"In the free monoid on the two letters $x$ and $y$, with $x$ represented by the Boolean value false and $y$ by true, this is the word of length one consisting of the letter $x$.","title":"The letter $x$ as a word"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/y-word","kind":"node","lean":"MetaMathlibExt.MultipleZeta.yWord","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","The letter "],["math","\\(y\\)"],["text"," as a word"]],"source":"The letter $y$ as a word"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#y-word","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"In the free monoid on the two letters $x$ and $y$, this is the word of length one consisting of the single letter $y$.","title":"The letter $y$ as a word"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/pnat-entries","kind":"node","lean":"MetaMathlibExt.MultipleZeta.Index.pnatEntries","parent":9179,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#pnat-entries","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"For an index $(k_1, \\dots, k_r)$, this is the list $(k_1, \\dots, k_r)$ of its entries in order, each regarded as a positive integer.","title":"Entries of an index as positive integers"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/z-word-pnat","kind":"node","lean":"MetaMathlibExt.MultipleZeta.zWordPNat","parent":9179,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#z-word-pnat","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"For a finite list $(k_1, \\dots, k_r)$ of positive integers, this is the word $z_{k_1} z_{k_2} \\cdots z_{k_r}$ in the free monoid on the two letters $x$ and $y$, where $z_k = x^{k-1} y$ is the Hoffman block of $k$. The empty list gives the empty word.","title":"Hoffman word of a list of positive integers"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/z-word","kind":"node","lean":"MetaMathlibExt.MultipleZeta.zWord","parent":9179,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#z-word","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"For an index $\\mathbf{k} = (k_1, \\dots, k_r)$ of positive integers, the Hoffman word of $\\mathbf{k}$ is $z_{k_1} z_{k_2} \\cdots z_{k_r}$ in the free monoid on the two letters $x$ and $y$, where $z_k = x^{k-1} y$. The empty index gives the empty word.","title":"Hoffman word of an index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/z-monomial","kind":"node","lean":"MetaMathlibExt.MultipleZeta.zMonomial","parent":9179,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#z-monomial","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"For an index $\\mathbf{k} = (k_1, \\dots, k_r)$ of positive integers, this is the monomial $z_{k_1} z_{k_2} \\cdots z_{k_r}$ with coefficient $1$ in the free associative algebra $\\mathbb{Q}\\langle x, y \\rangle$ over $\\mathbb{Q}$ on two noncommuting letters $x$ and $y$, where $z_k = x^{k-1} y$. In other words, it is the Hoffman word of $\\mathbf{k}$ viewed as an element of $\\mathbb{Q}\\langle x, y \\rangle$. The empty\u2026","title":"Hoffman monomial of an index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/is-hone-word","kind":"node","lean":"MetaMathlibExt.MultipleZeta.IsHOneWord","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","Words of "],["math","\\(\\mathfrak{H}^1\\)"]],"source":"Words of $\\mathfrak{H}^1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#is-hone-word","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"A word $w$ in the two letters $x$ and $y$ is an $\\mathfrak{H}^1$-word when $w$ is empty or $w = uy$ for some word $u$, that is, when $w$ is empty or ends in $y$.","title":"Words of $\\mathfrak{H}^1$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/is-hzero-word","kind":"node","lean":"MetaMathlibExt.MultipleZeta.IsHZeroWord","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","Words of "],["math","\\(\\mathfrak{H}^0\\)"]],"source":"Words of $\\mathfrak{H}^0$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#is-hzero-word","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"A word $w$ in the two letters $x$ and $y$ is an $\\mathfrak{H}^0$-word when $w$ is empty or $w = xuy$ for some word $u$, that is, when $w$ is empty or begins with $x$ and ends in $y$.","title":"Words of $\\mathfrak{H}^0$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/word-submonoid-hone","kind":"node","lean":"MetaMathlibExt.MultipleZeta.wordSubmonoidHOne","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","The monoid of "],["math","\\(\\mathfrak{H}^1\\)"],["text","-words"]],"source":"The monoid of $\\mathfrak{H}^1$-words"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#word-submonoid-hone","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"This is the submonoid of the free monoid on the two letters $x$ and $y$ consisting of the words that are empty or end in $y$.","title":"The monoid of $\\mathfrak{H}^1$-words"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/word-submonoid-hzero","kind":"node","lean":"MetaMathlibExt.MultipleZeta.wordSubmonoidHZero","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","The monoid of "],["math","\\(\\mathfrak{H}^0\\)"],["text","-words"]],"source":"The monoid of $\\mathfrak{H}^0$-words"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#word-submonoid-hzero","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"This is the submonoid of the free monoid on the two letters $x$ and $y$ consisting of the words that are empty or begin with $x$ and end in $y$.","title":"The monoid of $\\mathfrak{H}^0$-words"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/h-one","kind":"node","lean":"MetaMathlibExt.MultipleZeta.hOne","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subalgebra "],["math","\\(\\mathfrak{H}^1\\)"]],"source":"The subalgebra $\\mathfrak{H}^1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#h-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"Let $\\mathbb{Q}\\langle x, y\\rangle$ be the free associative $\\mathbb{Q}$-algebra on two letters $x$ and $y$, whose monomials are the words in $x$ and $y$. The subalgebra $\\mathfrak{H}^1$ of $\\mathbb{Q}\\langle x, y\\rangle$ is the $\\mathbb{Q}$-linear span of the words that are empty or end in $y$; equivalently, it consists of the polynomials in which every word with nonzero coefficient is empty or ends in $y$.","title":"The subalgebra $\\mathfrak{H}^1$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/h-zero","kind":"node","lean":"MetaMathlibExt.MultipleZeta.hZero","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subalgebra "],["math","\\(\\mathfrak{H}^0\\)"]],"source":"The subalgebra $\\mathfrak{H}^0$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#h-zero","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"Let $\\mathbb{Q}\\langle x, y\\rangle$ be the free associative $\\mathbb{Q}$-algebra on two letters $x$ and $y$, whose monomials are the words in $x$ and $y$. The subalgebra $\\mathfrak{H}^0$ of $\\mathbb{Q}\\langle x, y\\rangle$ is the $\\mathbb{Q}$-linear span of the words that are empty or begin with $x$ and end in $y$; equivalently, it consists of the polynomials in which every word with nonzero coefficient is empty or\u2026","title":"The subalgebra $\\mathfrak{H}^0$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/is-hzero-index","kind":"node","lean":"MetaMathlibExt.MultipleZeta.IsHZeroIndex","parent":9179,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#is-hzero-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"An index is empty-or-admissible when it is admissible, that is, nonempty with first entry at least $2$, or when it has no entries.","title":"Empty or admissible indices"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/hzero-index","kind":"node","lean":"MetaMathlibExt.MultipleZeta.HZeroIndex","parent":9179,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#hzero-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"The type whose elements are the indices that are admissible, that is, nonempty with first entry at least $2$, or that have no entries.","title":"The type of empty or admissible indices"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/h-one-basis","kind":"node","lean":"MathlibExt.NumberTheory.MultipleZeta.WordAlgebraWanted.hOne_basis","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hoffman basis of "],["math","\\(\\mathfrak{H}^1\\)"]],"source":"Hoffman basis of $\\mathfrak{H}^1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#h-one-basis","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"Let $\\mathfrak{H} = \\mathbb{Q}\\langle x, y \\rangle$ be the free associative $\\mathbb{Q}$-algebra on two generators $x$ and $y$, let $\\mathfrak{H}^1$ be the $\\mathbb{Q}$-linear span of the empty word and the words ending in $y$ (a subalgebra), and for each positive integer $k$ let $z_k = x^{k-1} y$. The family of monomials $z_{k_1} z_{k_2} \\cdots z_{k_r}$, indexed by all indices $(k_1, \\dots, k_r)$ of positive\u2026","title":"Hoffman basis of $\\mathfrak{H}^1$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/h-zero-basis","kind":"node","lean":"MathlibExt.NumberTheory.MultipleZeta.WordAlgebraWanted.hZero_basis","parent":9179,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hoffman monomial basis of "],["math","\\(\\mathfrak{H}^0\\)"]],"source":"Hoffman monomial basis of $\\mathfrak{H}^0$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases/index.html#h-zero-basis","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/hoffman-algebra-and-its-bases","status":"fully_proved","summary":"Let $\\mathbb{Q}\\langle x, y\\rangle$ be the free associative $\\mathbb{Q}$-algebra on two letters $x$ and $y$, and let $\\mathfrak{H}^0$ be its subalgebra consisting of the $\\mathbb{Q}$-linear combinations of the empty word and of words that begin with $x$ and end with $y$. For $k \\ge 1$ put $z_k = x^{k-1} y$, and for an index $\\mathbf{k} = (k_1, \\dots, k_r)$ of positive integers (possibly empty) put\u2026","title":"Hoffman monomial basis of $\\mathfrak{H}^0$"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values","kind":"scope","navigation":"module","parent":9094,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values","status":"can_prove","title":"q-analogues of multiple zeta values"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","kind":"scope","navigation":"transparent","parent":9196,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"can_prove","title":"Partial sums of Schlesinger\u2013Zudilin qMZVs to Partial sums of general qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sz-qmzvpartial","kind":"node","lean":"MetaMathlibExt.szQMZVPartial","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sz-qmzvpartial","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For an index $(k_1, \\dots, k_r)$ of natural numbers and a natural number $M$, the partial sum is the formal power series in $q$ over $\\mathbb{Q}$","title":"Partial sums of Schlesinger\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sz-qmzv","kind":"node","lean":"MetaMathlibExt.szQMZV","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sz-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For an index $k = (k_1, \\dots, k_r)$ of natural numbers, the Schlesinger\u2013Zudilin qMZV $\\zeta_q^{SZ}(k)$ is the formal power series in $q$ over $\\mathbb{Q}$ whose coefficient of $q^N$, for each $N \\ge 0$, is the coefficient of $q^N$ in the partial sum","title":"Schlesinger\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sz-qmzv-nil","kind":"node","lean":"MetaMathlibExt.szQMZV_nil","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sz-qmzv-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"The Schlesinger\u2013Zudilin qMZV of the empty index is $1$: $\\zeta_q^{SZ}(\\emptyset) = 1$.","title":"The empty Schlesinger\u2013Zudilin qMZV"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sz-qmzvpartial-cons","kind":"node","lean":"MetaMathlibExt.szQMZVPartial_cons","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sz-qmzvpartial-cons","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For natural numbers $k_1$ and $M$ and an index $\\kappa = (k_2, \\dots, k_r)$ (possibly empty), write $Z_M(k_1, \\kappa)$ and $Z_m(\\kappa)$ for the partial sums of Schlesinger\u2013Zudilin qMZVs with bounds $M$ and $m$. Then","title":"Recursion for partial sums of Schlesinger\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/coeff-zero-sz-qmzv-cons","kind":"node","lean":"MetaMathlibExt.coeff_zero_szQMZV_cons","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#coeff-zero-sz-qmzv-cons","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For every nonempty index $(k_1, \\dots, k_r)$ of natural numbers, the constant term of the Schlesinger\u2013Zudilin qMZV $\\zeta_q^{\\mathrm{SZ}}(k_1, \\dots, k_r)$ is $0$.","title":"Nonempty Schlesinger\u2013Zudilin qMZVs have no constant term"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/coeff-one-sz-qmzv-singleton-one","kind":"node","lean":"MetaMathlibExt.coeff_one_szQMZV_singleton_one","parent":9197,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient of "],["math","\\(q\\)"],["text"," in "],["math","\\(\\zeta_q^{\\mathrm{SZ}}(1)\\)"]],"source":"The coefficient of $q$ in $\\zeta_q^{\\mathrm{SZ}}(1)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#coeff-one-sz-qmzv-singleton-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"The coefficient of $q$ in the Schlesinger\u2013Zudilin qMZV $\\zeta_q^{\\mathrm{SZ}}(1) = \\sum_{m > 0} q^m / (1 - q^m)$ is $1$.","title":"The coefficient of $q$ in $\\zeta_q^{\\mathrm{SZ}}(1)$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sz-block-index","kind":"node","lean":"MetaMathlibExt.szBlockIndex","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sz-block-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For a list of pairs of natural numbers $(k_1, d_1), \\dots, (k_r, d_r)$, the block index is the index $(k_1, \\{0\\}^{d_1 - 1}, \\dots, k_r, \\{0\\}^{d_r - 1})$ obtained by concatenating, for $j = 1, \\dots, r$, the entry $k_j$ followed by $d_j - 1$ zeros (no zeros when $d_j = 0$).","title":"Block indices for Schlesinger\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sz-block-index-nil","kind":"node","lean":"MetaMathlibExt.szBlockIndex_nil","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sz-block-index-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"The block index of the empty list of pairs is the empty index.","title":"Block index of the empty list"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sz-block-index-cons","kind":"node","lean":"MetaMathlibExt.szBlockIndex_cons","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sz-block-index-cons","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For natural numbers $k$, $d$ and a list $L$ of pairs of natural numbers, the block index of the list with first pair $(k, d)$ followed by $L$ is $k$, followed by $d - 1$ zeros (none when $d = 0$), followed by the block index of $L$.","title":"Recursion for block indices"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/ooz-qmzvpartial","kind":"node","lean":"MetaMathlibExt.oozQMZVPartial","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#ooz-qmzvpartial","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For an index $(k_1, \\dots, k_r)$ of natural numbers and a natural number $M$, the partial OOZ sum is the formal power series in $q$ over $\\mathbb{Q}$","title":"Partial Ohno\u2013Okuda\u2013Zudilin sums"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/ooz-qmzv","kind":"node","lean":"MetaMathlibExt.oozQMZV","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#ooz-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For an index $(k_1, \\dots, k_r)$ of natural numbers with $r \\ge 1$, the (extended) Ohno\u2013Okuda\u2013Zudilin qMZV is the formal power series in $q$ over $\\mathbb{Q}$","title":"Ohno\u2013Okuda\u2013Zudilin q-multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/ooz-qmzv-nil","kind":"node","lean":"MetaMathlibExt.oozQMZV_nil","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#ooz-qmzv-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"The Ohno\u2013Okuda\u2013Zudilin qMZV of the empty index is $1$: $\\zeta_q^{\\mathrm{OOZ}}(\\varnothing) = 1$.","title":"The OOZ-qMZV of the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/coeff-zero-ooz-qmzv-cons","kind":"node","lean":"MetaMathlibExt.coeff_zero_oozQMZV_cons","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#coeff-zero-ooz-qmzv-cons","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"For every nonempty index $(k_1, \\dots, k_r)$ of natural numbers, the constant term of the Ohno\u2013Okuda\u2013Zudilin qMZV $\\zeta_q^{\\mathrm{OOZ}}(k_1, \\dots, k_r)$ is $0$.","title":"Nonempty Ohno\u2013Okuda\u2013Zudilin qMZVs have no constant term"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/coeff-sz-qmzv-sz-block-index-eq-sum-choo","kind":"node","lean":"MetaMathlibExt.coeff_szQMZV_szBlockIndex_eq_sum_choose","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#coeff-sz-qmzv-sz-block-index-eq-sum-choo","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"can_prove","summary":"Let $r \\ge 0$, and let $k_1, \\dots, k_r$ and $d_1, \\dots, d_r$ be positive integers. For every natural number $N$, the coefficient of $q^N$ in the Schlesinger\u2013Zudilin qMZV $\\zeta_q^{\\mathrm{SZ}}(k_1, \\{0\\}^{d_1 - 1}, \\dots, k_r, \\{0\\}^{d_r - 1})$, whose index lists each $k_j$ followed by $d_j - 1$ zeros, equals","title":"Coefficients of Schlesinger\u2013Zudilin qMZVs as sums of binomial products"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sz-qmzv-sz-block-index-dual","kind":"node","lean":"MetaMathlibExt.szQMZV_szBlockIndex_dual","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sz-qmzv-sz-block-index-dual","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"can_prove","summary":"Let $\\zeta_q^{SZ}$ denote the Schlesinger\u2013Zudilin qMZV, let $(k_1, d_1), \\dots, (k_r, d_r)$ be pairs of natural numbers with $k_j \\ge 1$ and $d_j \\ge 1$ for all $j$, and let $\\{0\\}^n$ denote $n$ zeros. Then","title":"Schlesinger\u2013Zudilin duality"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/sum-coeff-sz-qmzv-replicate-one-eq-card","kind":"node","lean":"MetaMathlibExt.sum_coeff_szQMZV_replicate_one_eq_card_partition","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#sum-coeff-sz-qmzv-replicate-one-eq-card","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"can_prove","summary":"Let $\\zeta_q^{SZ}$ denote the Schlesinger\u2013Zudilin qMZV, $\\{1\\}^r$ the index of $r$ ones and $p(N)$ the number of partitions of $N$. For all natural numbers $N \\ge 1$ and $R \\ge N$, the coefficient of $q^N$ in $\\sum_{r=1}^{R} \\zeta_q^{SZ}(\\{1\\}^r)$ equals $p(N)$.","title":"Partitions from Schlesinger\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/exists-conj-classes-perm-equiv-partition","kind":"node","lean":"MetaMathlibExt.exists_conjClasses_perm_equiv_partition","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#exists-conj-classes-perm-equiv-partition","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"can_prove","summary":"For every natural number $n$, there is a bijection from the set of conjugacy classes of the symmetric group $S_n$ to the set of partitions of $n$ that sends the class of each permutation $\\sigma$ to the cycle type of $\\sigma$: the partition of $n$ formed by the lengths of the cycles of $\\sigma$, fixed points counting as cycles of length $1$.","title":"Cycle type classifies conjugacy classes of permutations"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/card-conj-classes-perm-eq-card-partition","kind":"node","lean":"MetaMathlibExt.card_conjClasses_perm_eq_card_partition","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#card-conj-classes-perm-eq-card-partition","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"can_prove","summary":"For every natural number $n$, the number of conjugacy classes of the symmetric group $S_n$ equals the number $p(n)$ of partitions of $n$.","title":"Number of conjugacy classes of the symmetric group"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/card-conj-classes-gl-eq-coeff-sum-ooz-qm","kind":"node","lean":"MetaMathlibExt.card_conjClasses_GL_eq_coeff_sum_oozQMZV","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#card-conj-classes-gl-eq-coeff-sum-ooz-qm","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"can_prove","summary":"Let $K$ be a finite field with $c$ elements and, for $n \\ge 0$, let $a_{n,K}$ be the number of conjugacy classes of $\\mathrm{GL}_n(K)$, so that $a_{0,K} = 1$ because $\\mathrm{GL}_0(K)$ is the trivial group. Write $\\{1\\}^r$ for the index consisting of $r$ ones and $\\zeta_q^{\\mathrm{OOZ}}$ for the Ohno\u2013Okuda\u2013Zudilin qMZV. Then for all natural numbers $N \\le R$, $a_{N,K}$ equals the coefficient of $q^N$ in\u2026","title":"Class numbers of finite general linear groups via OOZ-qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/q-mzvpartial","kind":"node","lean":"MetaMathlibExt.qMZVPartial","parent":9197,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01/index.html#q-mzvpartial","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-01","status":"fully_proved","summary":"Let $(k_1, Q_1), \\dots, (k_r, Q_r)$ be pairs of natural numbers $k_j$ and polynomials $Q_j \\in \\mathbb{Q}[X]$, and let $M$ be a natural number. The partial sum is the formal power series in $q$ over $\\mathbb{Q}$","title":"Partial sums of general qMZVs"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","kind":"scope","navigation":"transparent","parent":9196,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"can_prove","title":"General q-multiple zeta values to Sums over partitions are invariant under transposition"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/q-mzv","kind":"node","lean":"MetaMathlibExt.qMZV","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#q-mzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For natural numbers $k_1, \\dots, k_r$ and polynomials $Q_1, \\dots, Q_r \\in \\mathbb{Q}[X]$, the general qMZV $\\zeta_q(k_1, \\dots, k_r; Q_1, \\dots, Q_r)$ is the formal power series in $q$ over $\\mathbb{Q}$ whose coefficient of $q^N$, for every $N$, is the coefficient of $q^N$ in the finite sum $\\sum_{N \\ge m_1 > m_2 > \\dots > m_r > 0} \\prod_{j=1}^{r} Q_j(q^{m_j}) / (1 - q^{m_j})^{k_j}$ over integers\u2026","title":"General q-multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/q-mzv-nil","kind":"node","lean":"MetaMathlibExt.qMZV_nil","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#q-mzv-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"The general q-multiple zeta value of the empty data is $\\zeta_q(\\varnothing; \\varnothing) = 1$.","title":"General qMZV of the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/coeff-zero-q-mzv-cons","kind":"node","lean":"MetaMathlibExt.coeff_zero_qMZV_cons","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#coeff-zero-q-mzv-cons","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For every nonempty list $(k_1, Q_1), \\dots, (k_r, Q_r)$ of natural numbers $k_j$ and polynomials $Q_j \\in \\mathbb{Q}[X]$, the constant coefficient of the general q-multiple zeta value $\\zeta_q(k_1, \\dots, k_r; Q_1, \\dots, Q_r)$ is $0$. No condition on $Q_1(0)$ is assumed.","title":"Nonempty general qMZVs have no constant term"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/q-mzvspace","kind":"node","lean":"MetaMathlibExt.qMZVSpace","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#q-mzvspace","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For a natural number $d$, $\\mathcal{Z}_{q,d}$ is the $\\mathbb{Q}$-subspace of the formal power series $\\mathbb{Q}[[q]]$ spanned by the general qMZVs $\\zeta_q(k_1, \\dots, k_r; Q_1, \\dots, Q_r)$ over all $r \\ge 0$, natural numbers $k_1, \\dots, k_r$ and polynomials $Q_1, \\dots, Q_r \\in \\mathbb{Q}[X]$ with $\\deg Q_j + d \\le k_j$ for every $j$ (the zero polynomial counting as degree $0$) and $Q_1(0) = 0$ when $r \\ge 1$.","title":"Spaces of general qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/one-mem-q-mzvspace","kind":"node","lean":"MetaMathlibExt.one_mem_qMZVSpace","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#one-mem-q-mzvspace","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For every natural number $d$, the constant power series $1$ lies in the $\\mathbb{Q}$-span $\\mathcal{Z}_{q,d}$ of the general q-multiple zeta values $\\zeta_q(k_1, \\dots, k_r; Q_1, \\dots, Q_r)$ with $Q_1(0) = 0$ and $\\deg Q_j + d \\le k_j$ for all $j$, where the zero polynomial is given degree $0$; it is the qMZV of the empty index.","title":"The constant series one lies in every space of qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/bz-qmzv","kind":"node","lean":"MetaMathlibExt.bzQMZV","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#bz-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For an index $k = (k_1, \\dots, k_r)$ of natural numbers, the Bradley\u2013Zhao q-multiple zeta value is the general qMZV $\\zeta_q^{BZ}(k) = \\zeta_q(k_1, \\dots, k_r; X^{k_1 - 1}, \\dots, X^{k_r - 1})$, a formal power series in $q$ over $\\mathbb{Q}$, where $k_j - 1$ is truncated at $0$. For an admissible index ($k_1 \\ge 2$ and all $k_i \\ge 1$) it equals","title":"Bradley\u2013Zhao q-multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/bz-qmzv-nil","kind":"node","lean":"MetaMathlibExt.bzQMZV_nil","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#bz-qmzv-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"The Bradley\u2013Zhao q-multiple zeta value of the empty index is $\\zeta_q^{BZ}(\\varnothing) = 1$.","title":"Bradley\u2013Zhao qMZV of the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/coeff-zero-bz-qmzv-cons","kind":"node","lean":"MetaMathlibExt.coeff_zero_bzQMZV_cons","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#coeff-zero-bz-qmzv-cons","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For every nonempty index $(k_1, \\dots, k_r)$ of natural numbers, the constant coefficient of the Bradley\u2013Zhao q-multiple zeta value $\\zeta_q^{BZ}(k_1, \\dots, k_r)$ is $0$.","title":"Nonempty Bradley\u2013Zhao qMZVs have no constant term"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/bz-block-index","kind":"node","lean":"MetaMathlibExt.bzBlockIndex","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#bz-block-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"The block index of a list $[(a_1, b_1), \\dots, (a_r, b_r)]$ of pairs of natural numbers is the index $(a_1 + 1, \\{1\\}^{b_1 - 1}, \\dots, a_r + 1, \\{1\\}^{b_r - 1})$, where $\\{1\\}^n$ stands for $n$ consecutive entries equal to $1$ and $b_j - 1$ is truncated at $0$.","title":"Block index of a list of pairs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/bz-block-index-cons","kind":"node","lean":"MetaMathlibExt.bzBlockIndex_cons","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#bz-block-index-cons","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For natural numbers $a, b$ and a list $L$ of pairs of natural numbers, the block index of the list obtained by prepending $(a, b)$ to $L$ consists of the entry $a + 1$, followed by $b - 1$ entries equal to $1$ (none when $b = 0$), followed by the block index of $L$.","title":"Block index of a prepended pair"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/q-mzvspace-zero-eq-span-sz-qmzv","kind":"node","lean":"MetaMathlibExt.qMZVSpace_zero_eq_span_szQMZV","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#q-mzvspace-zero-eq-span-sz-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"can_prove","summary":"Let $\\mathcal{Z}_q$ be the space $\\mathcal{Z}_{q,d}$ of general qMZVs for $d = 0$. Then $\\mathcal{Z}_q$ is the $\\mathbb{Q}$-span of the Schlesinger\u2013Zudilin qMZVs $\\zeta_q^{\\mathrm{SZ}}(k_1, \\dots, k_r)$ over all indices of natural numbers with $k_1 \\ge 1$ when $r \\ge 1$ (including the empty index, with $\\zeta_q^{\\mathrm{SZ}}(\\varnothing) = 1$). Moreover, $\\mathcal{Z}_q$ is closed under the operator\u2026","title":"Spanning set of Schlesinger\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/q-mzvspace-one-eq-span-bz-qmzv","kind":"node","lean":"MetaMathlibExt.qMZVSpace_one_eq_span_bzQMZV","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#q-mzvspace-one-eq-span-bz-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"can_prove","summary":"Let $\\mathcal{Z}_{q,1}$ be the space $\\mathcal{Z}_{q,d}$ of general qMZVs for $d = 1$. Then $\\mathcal{Z}_{q,1}$ is the $\\mathbb{Q}$-span of the Bradley\u2013Zhao qMZVs $\\zeta_q^{\\mathrm{BZ}}(k_1, \\dots, k_r)$ over all admissible indices, that is, all indices of natural numbers with $k_i \\ge 1$ for every $i$ and $k_1 \\ge 2$ when $r \\ge 1$ (including the empty index, with $\\zeta_q^{\\mathrm{BZ}}(\\varnothing) = 1$).","title":"Spanning set of Bradley\u2013Zhao qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/q-mzvspace-eq-span-ooz-qmzv","kind":"node","lean":"MetaMathlibExt.qMZVSpace_eq_span_oozQMZV","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#q-mzvspace-eq-span-ooz-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"can_prove","summary":"Let $\\mathcal{Z}_{q,1}$ and $\\mathcal{Z}_q$ be the spaces $\\mathcal{Z}_{q,d}$ of general qMZVs for $d = 1$ and $d = 0$. Then (i) $\\mathcal{Z}_{q,1}$ is the $\\mathbb{Q}$-span of the Ohno\u2013Okuda\u2013Zudilin qMZVs $\\zeta_q^{\\mathrm{OOZ}}(k_1, \\dots, k_r)$ over all indices of natural numbers with $k_i \\ge 1$ for every $i$ and $k_1 \\ge 2$ when $r \\ge 1$; and (ii) $\\mathcal{Z}_q$ is the $\\mathbb{Q}$-span of the\u2026","title":"Spanning sets of Ohno\u2013Okuda\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/bz-qmzv-bz-block-index-dual","kind":"node","lean":"MetaMathlibExt.bzQMZV_bzBlockIndex_dual","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#bz-qmzv-bz-block-index-dual","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"can_prove","summary":"For every $r \\ge 0$ and all natural numbers $a_1, b_1, \\dots, a_r, b_r \\ge 1$, the Bradley\u2013Zhao q-multiple zeta values satisfy","title":"Bradley\u2013Zhao duality for q-multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/sum-bz-qmzv-add-eq-sum-bz-qmzv-dual-add","kind":"node","lean":"MetaMathlibExt.sum_bzQMZV_add_eq_sum_bzQMZV_dual_add","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#sum-bz-qmzv-add-eq-sum-bz-qmzv-dual-add","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"can_prove","summary":"Let $(a_1, b_1), \\dots, (a_r, b_r)$ be pairs of positive integers, write $\\{1\\}^{n}$ for $n$ repetitions of $1$, let $\\mathbf{k} = (a_1 + 1, \\{1\\}^{b_1 - 1}, \\dots, a_r + 1, \\{1\\}^{b_r - 1})$ be the admissible index they determine, of depth $\\ell$, and let $\\mathbf{k}^\\vee = (b_r + 1, \\{1\\}^{a_r - 1}, \\dots, b_1 + 1, \\{1\\}^{a_1 - 1})$ be its dual, of depth $\\ell^\\vee$. Then for every natural number $c$,\u2026","title":"q-Ohno relations for Bradley\u2013Zhao qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/stanley-partitions","kind":"node","lean":"MetaMathlibExt.stanleyPartitions","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#stanley-partitions","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For natural numbers $r$ and $N$, $\\mathcal{P}_r(N)$ is the finite set of partitions of $N$ of length $r$ in Stanley coordinates: the pairs of tuples $((m_1, \\dots, m_r), (n_1, \\dots, n_r))$ of positive integers with $m_1 > m_2 > \\dots > m_r$ and $m_1 n_1 + \\dots + m_r n_r = N$.","title":"Partitions in Stanley coordinates"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/mem-stanley-partitions","kind":"node","lean":"MetaMathlibExt.mem_stanleyPartitions","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#mem-stanley-partitions","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For natural numbers $r$ and $N$, a pair $((m_1, \\dots, m_r), (n_1, \\dots, n_r))$ of $r$-tuples of natural numbers lies in the set $\\mathcal{P}_r(N)$ of partitions of $N$ of length $r$ in Stanley coordinates if and only if $m_1 > m_2 > \\dots > m_r$, every $m_j$ and every $n_j$ is positive, and $m_1 n_1 + \\dots + m_r n_r = N$.","title":"Membership criterion for partitions in Stanley coordinates"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/stanley-transpose","kind":"node","lean":"MetaMathlibExt.stanleyTranspose","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#stanley-transpose","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For tuples $\\mathbf{m} = (m_1, \\dots, m_r)$ and $\\mathbf{n} = (n_1, \\dots, n_r)$ of natural numbers, the transpose $\\rho(\\mathbf{m}, \\mathbf{n})$ is the pair of tuples $((n_1 + \\dots + n_r, \\dots, n_1 + n_2, n_1), (m_r, m_{r-1} - m_r, \\dots, m_1 - m_2))$: for $1 \\le j \\le r$, the $j$-th entry of the first tuple is $n_1 + \\dots + n_{r+1-j}$ and the $j$-th entry of the second tuple is $m_{r+1-j} - m_{r+2-j}$, where\u2026","title":"Transposed partition in Stanley coordinates"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/stanley-transpose-fin-one","kind":"node","lean":"MetaMathlibExt.stanleyTranspose_fin_one","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#stanley-transpose-fin-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"fully_proved","summary":"For all natural numbers $m$ and $n$, the transpose $\\rho$ in Stanley coordinates sends the pair of one-entry tuples $((m), (n))$ to $((n), (m))$.","title":"Transpose of a one-part partition"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/sum-stanley-partitions-comp-stanley-tran","kind":"node","lean":"MetaMathlibExt.sum_stanleyPartitions_comp_stanleyTranspose","parent":9218,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02/index.html#sum-stanley-partitions-comp-stanley-tran","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-02","status":"can_prove","summary":"For all natural numbers $r$ and $N$ and every function $a$ from pairs of $r$-tuples of natural numbers to $\\mathbb{Q}$, $\\sum_{p \\in \\mathcal{P}_r(N)} a(p) = \\sum_{p \\in \\mathcal{P}_r(N)} a(\\rho(p))$, where $\\mathcal{P}_r(N)$ is the set of partitions of $N$ of length $r$ in Stanley coordinates and $\\rho$ is the transpose map.","title":"Sums over partitions are invariant under transposition"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","kind":"scope","navigation":"transparent","parent":9196,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"can_prove","title":"Coefficients of qMZVs as polynomial functions on partitions to Takeyama\u2013Bradley\u2013Zhao qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/exists-mv-polynomial-coeff-q-mzv-eq-sum","kind":"node","lean":"MetaMathlibExt.exists_mvPolynomial_coeff_qMZV_eq_sum_stanleyPartitions","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#exists-mv-polynomial-coeff-q-mzv-eq-sum","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"can_prove","summary":"Let $r \\ge 1$ and let $(k_1, Q_1), \\dots, (k_r, Q_r)$ be pairs of a natural number $k_j$ and a polynomial $Q_j \\in \\mathbb{Q}[X]$ with $\\deg Q_j \\le k_j$ for every $j$ and $Q_1(0) = 0$. For $s, N \\ge 0$ let $\\mathcal{P}_s(N)$ be the set of pairs $(\\mathbf{m}, \\mathbf{n})$ of tuples $\\mathbf{m} = (m_1, \\dots, m_s)$ and $\\mathbf{n} = (n_1, \\dots, n_s)$ of positive integers with $m_1 > \\cdots > m_s$ and\u2026","title":"Coefficients of qMZVs as polynomial functions on partitions"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/mem-q-mzvspace-zero-iff-exists-mv-polyno","kind":"node","lean":"MetaMathlibExt.mem_qMZVSpace_zero_iff_exists_mvPolynomial","parent":9239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial functions on partitions characterize "],["math","\\(\\mathcal{Z}_q\\)"]],"source":"Polynomial functions on partitions characterize $\\mathcal{Z}_q$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#mem-q-mzvspace-zero-iff-exists-mv-polyno","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"can_prove","summary":"Let $\\mathcal{Z}_q = \\mathcal{Z}_{q,0}$ be the space of general qMZVs, and for natural numbers $r$ and $N$ let $\\mathcal{P}_r(N)$ be the set of partitions of $N$ of length $r$ in Stanley coordinates. A power series $S = \\sum_{N \\ge 0} a_N q^N \\in \\mathbb{Q}[[q]]$ lies in $\\mathcal{Z}_q$ if and only if there are $f_0 \\in \\mathbb{Q}$, polynomials $f_r \\in \\mathbb{Q}[X_1, \\dots, X_r, Y_1, \\dots, Y_r]$ for $r \\ge 1$\u2026","title":"Polynomial functions on partitions characterize $\\mathcal{Z}_q$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/quasi-shuffle","kind":"node","lean":"MetaMathlibExt.quasiShuffle","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#quasi-shuffle","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"Fix an alphabet and a product of letters $a \\diamond b$, a formal sum of letters given as a finite list (possibly empty, possibly with repetitions). The quasi-shuffle product $u \\ast v$ of two words $u$ and $v$ is the formal sum of words, recorded as a list, defined recursively by $\\mathbf{1} \\ast w = w \\ast \\mathbf{1} = w$ for the empty word $\\mathbf{1}$ and by\u2026","title":"Quasi-shuffle product"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/quasi-shuffle-nil-left","kind":"node","lean":"MetaMathlibExt.quasiShuffle_nil_left","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#quasi-shuffle-nil-left","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"For every alphabet, every product of letters and every word $w$, the quasi-shuffle product of the empty word $\\mathbf{1}$ with $w$ is $\\mathbf{1} \\ast w = w$, the formal sum consisting of the single word $w$.","title":"The empty word is a left unit for the quasi-shuffle product"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/quasi-shuffle-singleton-singleton","kind":"node","lean":"MetaMathlibExt.quasiShuffle_singleton_singleton","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#quasi-shuffle-singleton-singleton","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"For natural numbers $a$ and $b$, viewed as one-letter words, the quasi-shuffle product with letter product $a \\diamond b = a + b$ (the stuffle product) is $a \\ast b = ab + ba + (a + b)$: the two-letter words $ab$ and $ba$ and the one-letter word $(a + b)$, listed in this order.","title":"The stuffle product of two letters"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-shuffle","kind":"node","lean":"MetaMathlibExt.szShuffle","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-shuffle","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"The SZ-qshuffle product $\\text{\u0448}$ of two words in the letters $p$ and $y$ is the formal sum of words, counted with multiplicity, defined recursively by $\\mathbf{1} \\mathbin{\\text{\u0448}} w = w \\mathbin{\\text{\u0448}} \\mathbf{1} = w$ for the empty word $\\mathbf{1}$, and, for words $u, v$, $y u \\mathbin{\\text{\u0448}} v = y (u \\mathbin{\\text{\u0448}} v)$, $p u \\mathbin{\\text{\u0448}} y v = y (p u \\mathbin{\\text{\u0448}} v)$ and\u2026","title":"SZ-qshuffle product"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-shuffle-y-y","kind":"node","lean":"MetaMathlibExt.szShuffle_y_y","parent":9239,"reading_title":{"format":"inline-parts-v1","parts":[["text","SZ-qshuffle of "],["math","\\(y\\)"],["text"," with itself"]],"source":"SZ-qshuffle of $y$ with itself"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-shuffle-y-y","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"The SZ-qshuffle product of the one-letter word $y$ with itself is the single word $yy$: $y \\mathbin{\\text{\u0448}} y = yy$.","title":"SZ-qshuffle of $y$ with itself"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-word","kind":"node","lean":"MetaMathlibExt.szWord","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-word","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"The word of an index $(k_1, \\dots, k_r)$ of natural numbers is the word $p^{k_1} y \\cdots p^{k_r} y$ in the letters $p$ and $y$.","title":"Word of an index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-word-nil","kind":"node","lean":"MetaMathlibExt.szWord_nil","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-word-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"The word of the empty index is the empty word.","title":"Word of the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-index-of-word-aux","kind":"node","lean":"MetaMathlibExt.szIndexOfWordAux","parent":9239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reading an index off a word in "],["math","\\(p\\)"],["text"," and "],["math","\\(y\\)"]],"source":"Reading an index off a word in $p$ and $y$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-index-of-word-aux","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"For $c \\in \\mathbb{N}$ and a word $w$ in the letters $p$ and $y$, the index $R_c(w)$, a finite sequence of natural numbers, is defined recursively by $R_c(\\mathbf{1}) = ()$ for the empty word $\\mathbf{1}$, $R_c(p w) = R_{c+1}(w)$, and $R_c(y w) = (c, R_0(w))$, the index with first entry $c$ followed by the entries of $R_0(w)$. Thus $R_c(p^{k_1} y\\, p^{k_2} y \\cdots p^{k_r} y\\, p^j) = (c + k_1, k_2, \\dots, k_r)$ for\u2026","title":"Reading an index off a word in $p$ and $y$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-index-of-word-aux-replicate-append","kind":"node","lean":"MetaMathlibExt.szIndexOfWordAux_replicate_append","parent":9239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reading off a leading block of "],["math","\\(p\\)"],["text","s"]],"source":"Reading off a leading block of $p$s"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-index-of-word-aux-replicate-append","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"Write $R_c(w)$ for the index read off a word $w$ in the letters $p$ and $y$ when $c$ letters $p$ have already been read. For all $c, n \\in \\mathbb{N}$ and every word $w$, $R_c(p^n y w)$ is the index whose first entry is $c + n$ and whose remaining entries are those of $R_0(w)$.","title":"Reading off a leading block of $p$s"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-qmzvword","kind":"node","lean":"MetaMathlibExt.szQMZVWord","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-qmzvword","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"Write $\\zeta_q^{SZ}$ for the Schlesinger\u2013Zudilin qMZV of an index of natural numbers. For a word $w$ in the letters $p$ and $y$ with $r$ letters $y$, let $k_j$ be the number of letters $p$ between the $(j-1)$-st and the $j$-th letter $y$ of $w$ (for $j = 1$, before the first $y$); letters $p$ after the last $y$ are ignored. The value of $\\zeta_q^{SZ}$ on $w$ is $\\zeta_q^{SZ}(k_1, \\dots, k_r)$. In particular the\u2026","title":"Schlesinger\u2013Zudilin qMZVs of words"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-qmzvword-sz-word-singleton","kind":"node","lean":"MetaMathlibExt.szQMZVWord_szWord_singleton","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-qmzvword-sz-word-singleton","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"Write $\\zeta_q^{SZ}$ for the Schlesinger\u2013Zudilin qMZV, extended to words in the letters $p$ and $y$ by sending a word to $\\zeta_q^{SZ}$ of the index read off it: the number of letters $p$ before each letter $y$ since the previous $y$, with letters $p$ after the last $y$ dropped. For every natural number $k$, the value of $\\zeta_q^{SZ}$ on the word $p^k y$ of the one-entry index $(k)$ is $\\zeta_q^{SZ}(k)$.","title":"SZ-qMZV of the word of a one-entry index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/q-mzv-mul-eq-sum-quasi-shuffle","kind":"node","lean":"MetaMathlibExt.qMZV_mul_eq_sum_quasiShuffle","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#q-mzv-mul-eq-sum-quasi-shuffle","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"can_prove","summary":"Consider words in letters $(k, Q)$ with $k \\in \\mathbb{N}$ and $Q \\in \\mathbb{Q}[X]$, and let $u$ and $v$ be words in which every letter satisfies $\\deg Q \\le k$ and the first letter, if any, satisfies $Q(0) = 0$. Let $u \\ast v$ be the quasi-shuffle product of $u$ and $v$ with letter product $(k_1, Q_1) \\diamond (k_2, Q_2) = (k_1 + k_2, Q_1 Q_2)$, a formal sum of words. Then\u2026","title":"Quasi-shuffle relations for general qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-qmzv-mul-eq-sum-stuffle","kind":"node","lean":"MetaMathlibExt.szQMZV_mul_eq_sum_stuffle","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-qmzv-mul-eq-sum-stuffle","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"can_prove","summary":"Write $\\zeta_q^{SZ}$ for the Schlesinger\u2013Zudilin qMZV. Let $v$ and $w$ be finite lists of natural numbers that are SZ-admissible, that is, each is empty or has first entry at least $1$. Then","title":"Stuffle product formula for Schlesinger\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/sz-qmzv-mul-eq-sum-sz-shuffle","kind":"node","lean":"MetaMathlibExt.szQMZV_mul_eq_sum_szShuffle","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#sz-qmzv-mul-eq-sum-sz-shuffle","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"can_prove","summary":"Write $\\zeta_q^{SZ}$ for the Schlesinger\u2013Zudilin qMZV. Let $u$ and $v$ be finite lists of natural numbers that are SZ-admissible, that is, each is empty or has first entry at least $1$, and let $w_u$ and $w_v$ be their words in the letters $p$ and $y$ (the index $(k_1, \\dots, k_r)$ has word $p^{k_1} y \\cdots p^{k_r} y$). Then","title":"Shuffle product formula for Schlesinger\u2013Zudilin qMZVs (Singer)"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/bz-qmzv-mul-eq-sum-quasi-shuffle","kind":"node","lean":"MetaMathlibExt.bzQMZV_mul_eq_sum_quasiShuffle","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#bz-qmzv-mul-eq-sum-quasi-shuffle","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"can_prove","summary":"Identify an index $(k_1, \\dots, k_r)$ with the word $z_{k_1} \\cdots z_{k_r}$, and let $u$ and $v$ be admissible indices: finite, possibly empty, sequences of positive integers whose first entry, if any, is at least $2$. Let $u \\ast_{BZ} v$ be the quasi-shuffle product of $u$ and $v$ with letter product $z_a \\diamond z_b = z_{a+b} + z_{a+b-1}$, a formal sum of indices. Then\u2026","title":"Quasi-shuffle relations for Bradley\u2013Zhao qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/q-mzv-summable-differentiable-on-tendsto","kind":"node","lean":"MetaMathlibExt.qMZV_summable_differentiableOn_tendsto","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#q-mzv-summable-differentiable-on-tendsto","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"can_prove","summary":"Let $(k_1, Q_1), \\dots, (k_r, Q_r)$ be pairs of an integer $k_j \\ge 1$ and a polynomial $Q_j \\in \\mathbb{Q}[X]$ with $\\deg Q_j \\le k_j$ for every $j$ and $Q_1(0) = 0$, and write the general qMZV as $\\zeta_q(k_1, \\dots, k_r; Q_1, \\dots, Q_r) = \\sum_{n \\ge 0} c_n q^n$. Then (i) for every complex $q$ with $|q| < 1$ the series $\\sum_{n \\ge 0} c_n q^n$ converges absolutely; (ii) the function\u2026","title":"Analytic properties of general qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/tbz-letter","kind":"node","lean":"MetaMathlibExt.tbzLetter","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#tbz-letter","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"The letters of the TBZ model are the symbol $\\bar 1$ and the natural numbers $k \\ge 1$. The letter $\\bar 1$, with summand $f(\\bar 1, m) = q^m / (1 - q^m)$, is sent to the pair $(1, X)$, and the letter $k$, with summand $f(k, m) = q^{(k-1)m} / (1 - q^m)^k$, is sent to the pair $(k, X^{k-1})$. Here a pair $(k, Q)$ of a natural number and a rational polynomial stands for the summand $Q(q^m) / (1 - q^m)^k$ of a general\u2026","title":"Letters of the Takeyama\u2013Bradley\u2013Zhao model as qMZV data"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/tbz-qmzv","kind":"node","lean":"MetaMathlibExt.tbzQMZV","parent":9239,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03/index.html#tbz-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-03","status":"fully_proved","summary":"For an index $(k_1, \\dots, k_r)$ whose entries are the symbol $\\bar 1$ or natural numbers, the TBZ-qMZV is the general qMZV of the data obtained by replacing each entry by its pair in the Takeyama\u2013Bradley\u2013Zhao letter encoding. When every natural-number entry is at least $1$ and $k_1 \\ne 1$, this is","title":"Takeyama\u2013Bradley\u2013Zhao qMZVs"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","kind":"scope","navigation":"transparent","parent":9196,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"can_prove","title":"TBZ-qMZV of the empty index to Coefficients of the SZ generating series"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/tbz-qmzv-nil","kind":"node","lean":"MetaMathlibExt.tbzQMZV_nil","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#tbz-qmzv-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"The Takeyama\u2013Bradley\u2013Zhao qMZV of the empty index is $1$: $\\zeta_q^{\\mathrm{TBZ}}(\\varnothing) = 1$.","title":"TBZ-qMZV of the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/tbz-qmzv-map-some","kind":"node","lean":"MetaMathlibExt.tbzQMZV_map_some","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#tbz-qmzv-map-some","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For every index $(k_1, \\dots, k_r)$ of natural numbers, read as a TBZ index without the symbol $\\bar 1$, the TBZ-qMZV equals the Bradley\u2013Zhao qMZV: $\\zeta_q^{\\mathrm{TBZ}}(k_1, \\dots, k_r) = \\zeta_q^{\\mathrm{BZ}}(k_1, \\dots, k_r)$.","title":"The TBZ model extends the BZ model"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/q-mzvspace-zero-eq-span-tbz-qmzv","kind":"node","lean":"MetaMathlibExt.qMZVSpace_zero_eq_span_tbzQMZV","parent":9260,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Takeyama\u2013Bradley\u2013Zhao model spans "],["math","\\(\\mathcal{Z}_q\\)"]],"source":"The Takeyama\u2013Bradley\u2013Zhao model spans $\\mathcal{Z}_q$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#q-mzvspace-zero-eq-span-tbz-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"can_prove","summary":"Let $\\overline{\\mathbb{N}}$ consist of the positive integers and one further symbol $\\bar 1$. The space $\\mathcal{Z}_q = \\mathcal{Z}_{q,0}$ equals the $\\mathbb{Q}$-linear span of the TBZ-qMZVs $\\zeta_q^{\\mathrm{TBZ}}(k_1, \\dots, k_r)$ over all $r \\ge 0$ and all $k_1, \\dots, k_r \\in \\overline{\\mathbb{N}}$ with $k_1 \\ne 1$ when $r \\ge 1$.","title":"The Takeyama\u2013Bradley\u2013Zhao model spans $\\mathcal{Z}_q$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/q-mzvspace-circ","kind":"node","lean":"MetaMathlibExt.qMZVSpaceCirc","parent":9260,"reading_title":{"format":"inline-parts-v1","parts":[["text","The spaces "],["math","\\(\\mathcal{Z}^\\circ_{q,d}\\)"]],"source":"The spaces $\\mathcal{Z}^\\circ_{q,d}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#q-mzvspace-circ","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For a natural number $d$, the space $\\mathcal{Z}^\\circ_{q,d}$ is the $\\mathbb{Q}$-linear span, in the formal power series ring $\\mathbb{Q}[[q]]$, of the general qMZVs $\\zeta_q(k_1, \\dots, k_r; Q_1, \\dots, Q_r)$ over all $r \\ge 0$, natural numbers $k_1, \\dots, k_r$ and polynomials $Q_1, \\dots, Q_r \\in \\mathbb{Q}[X]$ such that for every $j$ we have $k_j \\ge 1$, $\\deg Q_j + d \\le k_j$ and $Q_j(0) = 0$. Here the zero\u2026","title":"The spaces $\\mathcal{Z}^\\circ_{q,d}$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/one-mem-q-mzvspace-circ","kind":"node","lean":"MetaMathlibExt.one_mem_qMZVSpaceCirc","parent":9260,"reading_title":{"format":"inline-parts-v1","parts":[["text","The constant "],["math","\\(1\\)"],["text"," lies in "],["math","\\(\\mathcal{Z}^\\circ_{q,d}\\)"]],"source":"The constant $1$ lies in $\\mathcal{Z}^\\circ_{q,d}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#one-mem-q-mzvspace-circ","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For every natural number $d$, the constant power series $1$ (the qMZV of the empty index) lies in the space $\\mathcal{Z}^\\circ_{q,d}$.","title":"The constant $1$ lies in $\\mathcal{Z}^\\circ_{q,d}$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/q-mzvspace-circ-le-q-mzvspace","kind":"node","lean":"MetaMathlibExt.qMZVSpaceCirc_le_qMZVSpace","parent":9260,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathcal{Z}^\\circ_{q,d}\\)"],["text"," is contained in "],["math","\\(\\mathcal{Z}_{q,d}\\)"]],"source":"$\\mathcal{Z}^\\circ_{q,d}$ is contained in $\\mathcal{Z}_{q,d}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#q-mzvspace-circ-le-q-mzvspace","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For every natural number $d$, the space $\\mathcal{Z}^\\circ_{q,d}$ is a subspace of $\\mathcal{Z}_{q,d}$.","title":"$\\mathcal{Z}^\\circ_{q,d}$ is contained in $\\mathcal{Z}_{q,d}$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/okounkov-poly","kind":"node","lean":"MetaMathlibExt.okounkovPoly","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#okounkov-poly","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, the Okounkov polynomial $p_k \\in \\mathbb{Q}[X]$ is $p_k(X) = X^{k/2}$ if $k$ is even and $p_k(X) = X^{(k-1)/2}(1 + X)$ if $k$ is odd.","title":"Okounkov polynomials"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/okounkov-poly-two-three","kind":"node","lean":"MetaMathlibExt.okounkovPoly_two_three","parent":9260,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Okounkov polynomials "],["math","\\(p_2\\)"],["text"," and "],["math","\\(p_3\\)"]],"source":"The Okounkov polynomials $p_2$ and $p_3$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#okounkov-poly-two-three","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"The Okounkov polynomials of index $2$ and $3$ are $p_2(X) = X$ and $p_3(X) = X(1 + X)$.","title":"The Okounkov polynomials $p_2$ and $p_3$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/okounkov-qmzv","kind":"node","lean":"MetaMathlibExt.okounkovQMZV","parent":9260,"reading_title":{"format":"inline-parts-v1","parts":[["text","Okounkov "],["math","\\(q\\)"],["text","-multiple zeta values"]],"source":"Okounkov $q$-multiple zeta values"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#okounkov-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For a natural number $k$ let $p_k$ be Okounkov's polynomial: $p_k(X) = X^{k/2}$ if $k$ is even and $p_k(X) = X^{(k-1)/2}(1 + X)$ if $k$ is odd. For an index $(k_1, \\dots, k_r)$ of natural numbers, the Okounkov qMZV is the general qMZV $\\zeta_q^{\\mathrm{Oko}}(k_1, \\dots, k_r) = \\zeta_q(k_1, \\dots, k_r; p_{k_1}, \\dots, p_{k_r})$, a formal power series in $q$ over $\\mathbb{Q}$.","title":"Okounkov $q$-multiple zeta values"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/okounkov-qmzv-nil","kind":"node","lean":"MetaMathlibExt.okounkovQMZV_nil","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#okounkov-qmzv-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"The Okounkov $q$-multiple zeta value of the empty index is $1$: $\\zeta_q^{\\mathrm{Oko}}(\\varnothing) = 1$.","title":"Okounkov qMZV of the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/q-mzvspace-circ-one-eq-span-okounkov-qmz","kind":"node","lean":"MetaMathlibExt.qMZVSpaceCirc_one_eq_span_okounkovQMZV","parent":9260,"reading_title":{"format":"inline-parts-v1","parts":[["text","Okounkov qMZVs span "],["math","\\(\\mathcal{Z}^\\circ_{q,1}\\)"]],"source":"Okounkov qMZVs span $\\mathcal{Z}^\\circ_{q,1}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#q-mzvspace-circ-one-eq-span-okounkov-qmz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"can_prove","summary":"The space $\\mathcal{Z}^\\circ_{q,1}$ equals the $\\mathbb{Q}$-linear span of the Okounkov qMZVs $\\zeta_q^{\\mathrm{Oko}}(k_1, \\dots, k_r)$ over all indices of length $r \\ge 0$ with every $k_i \\ge 2$.","title":"Okounkov qMZVs span $\\mathcal{Z}^\\circ_{q,1}$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/eulerian-qseries","kind":"node","lean":"MetaMathlibExt.eulerianQSeries","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#eulerian-qseries","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For natural numbers $k$ and $m \\ge 1$, the Eulerian $q$-series is","title":"Eulerian q-series"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/coeff-mul-eulerian-qseries","kind":"node","lean":"MetaMathlibExt.coeff_mul_eulerianQSeries","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#coeff-mul-eulerian-qseries","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For natural numbers $k$, $m$ and $n$ with $m > 0$ and $n > 0$, the coefficient of $q^{mn}$ in the Eulerian $q$-series $\\sum_{\\ell \\ge 1} \\frac{\\ell^{k-1}}{(k-1)!} q^{m\\ell}$ is $\\frac{n^{k-1}}{(k-1)!}$, where $k - 1$ is read as $0$ when $k = 0$.","title":"Coefficients of the Eulerian q-series"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/bi-bracket-partial","kind":"node","lean":"MetaMathlibExt.biBracketPartial","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#bi-bracket-partial","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For pairs $(k_1, d_1), \\dots, (k_r, d_r)$ of natural numbers and a bound $M \\in \\mathbb{N}$, the partial bi-bracket is the $q$-series","title":"Partial sums of bi-brackets"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/bi-bracket","kind":"node","lean":"MetaMathlibExt.biBracket","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#bi-bracket","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For integers $k_1, \\dots, k_r \\ge 1$ and $d_1, \\dots, d_r \\ge 0$, the bi-bracket is the $q$-series","title":"Bi-brackets"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/bi-bracket-nil","kind":"node","lean":"MetaMathlibExt.biBracket_nil","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#bi-bracket-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"The bi-bracket of the empty index is $1$: $g(\\varnothing; \\varnothing) = 1$.","title":"Empty bi-bracket"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/bi-bracket-gen","kind":"node","lean":"MetaMathlibExt.biBracketGen","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#bi-bracket-gen","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For $r \\in \\mathbb{N}$, the generating series of the bi-brackets of depth $r$ is the formal power series in the variables $X_1, \\dots, X_r$, $Y_1, \\dots, Y_r$ with coefficients in $\\mathbb{Q}[[q]]$","title":"Generating series of bi-brackets"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/coeff-bi-bracket-gen","kind":"node","lean":"MetaMathlibExt.coeff_biBracketGen","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#coeff-bi-bracket-gen","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For $r \\in \\mathbb{N}$ and exponents $a_1, \\dots, a_r, b_1, \\dots, b_r \\ge 0$, the coefficient of $\\prod_{j=1}^{r} X_j^{a_j} Y_j^{b_j}$ in the generating series $g(X_1, \\dots, X_r; Y_1, \\dots, Y_r)$ of bi-brackets of depth $r$ is the bi-bracket $g(a_1 + 1, \\dots, a_r + 1; b_1, \\dots, b_r)$.","title":"Coefficients of the bi-bracket generating series"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/sz-gen","kind":"node","lean":"MetaMathlibExt.szGen","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#sz-gen","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For a natural number $r$, the generating series of SZ-qMZVs of depth $r$ is the formal power series in the $2r$ variables $X_1, \\dots, X_r, Y_1, \\dots, Y_r$ with coefficients in $\\mathbb{Q}[[q]]$","title":"Generating series of Schlesinger\u2013Zudilin qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/coeff-sz-gen","kind":"node","lean":"MetaMathlibExt.coeff_szGen","parent":9260,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04/index.html#coeff-sz-gen","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-04","status":"fully_proved","summary":"For $r \\in \\mathbb{N}$ and exponents $a_1, \\dots, a_r, b_1, \\dots, b_r \\ge 0$, the coefficient of $\\prod_{j=1}^{r} X_j^{a_j} Y_j^{b_j}$ in the generating series $s(X_1, \\dots, X_r; Y_1, \\dots, Y_r)$ of Schlesinger\u2013Zudilin $q$-multiple zeta values of depth $r$ is $\\zeta_q^{\\mathrm{SZ}}(a_1 + 1, \\{0\\}^{b_1}, \\dots, a_r + 1, \\{0\\}^{b_r})$, where $\\{0\\}^b$ stands for $b$ consecutive zeros.","title":"Coefficients of the SZ generating series"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","kind":"scope","navigation":"transparent","parent":9196,"reading_title":{"format":"inline-parts-v1","parts":[["text","The logarithm series "],["math","\\(\\ln(1 + T)\\)"],["text"," to Weighted chain sums"]],"source":"The logarithm series $\\ln(1 + T)$ to Weighted chain sums"},"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"can_prove","title":"The logarithm series $\\ln(1 + T)$ to Weighted chain sums"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/log-one-add-series","kind":"node","lean":"MetaMathlibExt.logOneAddSeries","parent":9281,"reading_title":{"format":"inline-parts-v1","parts":[["text","The logarithm series "],["math","\\(\\ln(1 + T)\\)"]],"source":"The logarithm series $\\ln(1 + T)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#log-one-add-series","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"The formal power series $\\ln(1 + T)$ over $\\mathbb{Q}$ is $\\sum_{n \\ge 1} (-1)^{n+1} T^n / n$; its constant term is $0$.","title":"The logarithm series $\\ln(1 + T)$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/coeff-log-one-add-series-zero-one","kind":"node","lean":"MetaMathlibExt.coeff_logOneAddSeries_zero_one","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#coeff-log-one-add-series-zero-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"The power series $\\ln(1 + T) = \\sum_{n \\ge 1} (-1)^{n+1} T^n / n$ over $\\mathbb{Q}$ has constant coefficient $0$ and coefficient $1$ at $T$.","title":"Low-order coefficients of the logarithm series"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/coeff-bi-bracket-gen-eq-sum-exp","kind":"node","lean":"MetaMathlibExt.coeff_biBracketGen_eq_sum_exp","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#coeff-bi-bracket-gen-eq-sum-exp","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"can_prove","summary":"Let $r$ and $N$ be natural numbers and let $g(X_1, \\dots, X_r; Y_1, \\dots, Y_r)$ be the generating series of bi-brackets of depth $r$, a power series in the $2r$ variables over $\\mathbb{Q}[[q]]$. The power series over $\\mathbb{Q}$ obtained by taking the coefficient of $q^N$ in every coefficient of $g$ equals the finite sum","title":"Bi-bracket generating series as a sum of exponentials"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/bi-bracket-gen-partition-relation","kind":"node","lean":"MetaMathlibExt.biBracketGen_partition_relation","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#bi-bracket-gen-partition-relation","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"can_prove","summary":"For every natural number $r$, the generating series $g(X_1, \\dots, X_r; Y_1, \\dots, Y_r)$ of bi-brackets of depth $r$ satisfies","title":"Partition relation for bi-brackets"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/bi-bracket-gen-eq-sz-gen-subst","kind":"node","lean":"MetaMathlibExt.biBracketGen_eq_szGen_subst","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#bi-bracket-gen-eq-sz-gen-subst","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"can_prove","summary":"Let $r$ be a natural number, let $g(X_1, \\dots, X_r; Y_1, \\dots, Y_r)$ be the generating series of bi-brackets of depth $r$ and $s(X_1, \\dots, X_r; Y_1, \\dots, Y_r)$ the generating series of Schlesinger\u2013Zudilin qMZVs of depth $r$, both power series in the $2r$ variables over $\\mathbb{Q}[[q]]$. For a power series $Z$ without constant term, write $e^Z$ and $\\ln(1 + Z)$ for the substitution of $Z$ into the exponential\u2026","title":"Translation between bi-brackets and the Schlesinger\u2013Zudilin model"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/coeff-ooz-qmzv-sz-block-index-eq-sum-cho","kind":"node","lean":"MetaMathlibExt.coeff_oozQMZV_szBlockIndex_eq_sum_choose","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#coeff-ooz-qmzv-sz-block-index-eq-sum-cho","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"can_prove","summary":"Let $r$ and $N$ be natural numbers and let $k_1, \\dots, k_r \\ge 1$ and $d_1, \\dots, d_r \\ge 0$ be integers. Writing $\\{0\\}^d$ for $d$ consecutive zeros, the coefficient of $q^N$ in the Ohno\u2013Okuda\u2013Zudilin qMZV $\\zeta_q^{\\mathrm{OOZ}}(k_1, \\{0\\}^{d_1}, \\dots, k_r, \\{0\\}^{d_r})$ equals the coefficient of $q^N$ in","title":"Generating series of OOZ-qMZVs, coefficientwise"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/q-mzvspace-circ-one-eq-span-bracket-okou","kind":"node","lean":"MetaMathlibExt.qMZVSpaceCirc_one_eq_span_bracket_okounkov_bz","parent":9281,"reading_title":{"format":"inline-parts-v1","parts":[["text","Brackets, Okounkov qMZVs and Bradley\u2013Zhao qMZVs span "],["math","\\(\\mathcal{Z}^\\circ_{q,1}\\)"]],"source":"Brackets, Okounkov qMZVs and Bradley\u2013Zhao qMZVs span $\\mathcal{Z}^\\circ_{q,1}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#q-mzvspace-circ-one-eq-span-bracket-okou","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"can_prove","summary":"For an index $k = (k_1, \\dots, k_r)$ of natural numbers let $g(k) = g(k_1, \\dots, k_r; 0, \\dots, 0)$ be the bracket, that is, the bi-bracket with all $d_j = 0$, and let $\\zeta_q^{\\mathrm{Oko}}(k)$ and $\\zeta_q^{BZ}(k)$ be the Okounkov and Bradley\u2013Zhao qMZVs. Then the space $\\mathcal{Z}^\\circ_{q,1}$ equals the $\\mathbb{Q}$-linear span of the brackets $g(k)$, equals the $\\mathbb{Q}$-linear span of the Okounkov qMZVs\u2026","title":"Brackets, Okounkov qMZVs and Bradley\u2013Zhao qMZVs span $\\mathcal{Z}^\\circ_{q,1}$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/weighted-quasi-shuffle","kind":"node","lean":"MetaMathlibExt.weightedQuasiShuffle","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#weighted-quasi-shuffle","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"Let $A$ be an alphabet and let $d$ assign to any two letters $a, b \\in A$ a finite formal $\\mathbb{Q}$-linear combination $d(a, b) = \\sum_c d(a, b)_c \\, c$ of letters. The weighted quasi-shuffle product $u \\ast v$ of two words over $A$ is the finite formal $\\mathbb{Q}$-linear combination of words defined by recursion on total length: $\\mathbf{1} \\ast w = w \\ast \\mathbf{1} = w$ for the empty word $\\mathbf{1}$, and\u2026","title":"Weighted quasi-shuffle product"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/weighted-quasi-shuffle-nil-left","kind":"node","lean":"MetaMathlibExt.weightedQuasiShuffle_nil_left","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#weighted-quasi-shuffle-nil-left","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"For every alphabet $A$, every weighted letter product $d$ on $A$ (assigning to two letters a finite formal $\\mathbb{Q}$-linear combination of letters) and every word $v$ over $A$, the weighted quasi-shuffle product of the empty word $\\mathbf{1}$ with $v$ is $v$: $\\mathbf{1} \\ast v = v$.","title":"The empty word is a left unit for the weighted quasi-shuffle"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/bi-bracket-lambda","kind":"node","lean":"MetaMathlibExt.biBracketLambda","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#bi-bracket-lambda","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"For natural numbers $a$, $b$ and $j$,","title":"Bachmann's coefficients"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/bi-bracket-lambda-one-one-one","kind":"node","lean":"MetaMathlibExt.biBracketLambda_one_one_one","parent":9281,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bachmann's coefficient "],["math","\\(\\lambda_{1,1}^1\\)"]],"source":"Bachmann's coefficient $\\lambda_{1,1}^1$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#bi-bracket-lambda-one-one-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"Bachmann's coefficient $\\lambda_{1,1}^1$ equals $-1/2$, the Bernoulli number $B_1$.","title":"Bachmann's coefficient $\\lambda_{1,1}^1$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/bi-bracket-diamond","kind":"node","lean":"MetaMathlibExt.biBracketDiamond","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#bi-bracket-diamond","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"Write $z_{k,d}$ for the letter indexed by a pair $(k, d)$ of natural numbers, and let $\\lambda_{a,b}^j$ be Bachmann's coefficients. The product of the letters $z_{k_1,d_1}$ and $z_{k_2,d_2}$ is the formal $\\mathbb{Q}$-linear combination of letters","title":"Bachmann's product of bi-bracket letters"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/bi-bracket-mul-eq-sum-weighted-quasi-shu","kind":"node","lean":"MetaMathlibExt.biBracket_mul_eq_sum_weightedQuasiShuffle","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#bi-bracket-mul-eq-sum-weighted-quasi-shu","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"can_prove","summary":"Let $u$ and $v$ be words in the letters $z_{k,d}$, indexed by pairs $(k, d)$ of natural numbers, with $k \\ge 1$ in every letter. Write $\\sum_w c_w w$ for the quasi-shuffle product of $u$ and $v$ induced by Bachmann's letter product $\\boxtimes$, a formal $\\mathbb{Q}$-linear combination of words, and $g(w)$ for the bi-bracket of a word $w$. Then $g(u) g(v) = \\sum_w c_w g(w)$.","title":"Bi-brackets form a quasi-shuffle algebra"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/ooz-t","kind":"node","lean":"MetaMathlibExt.oozT","parent":9281,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(T\\)"],["text"," of the OOZ quasi-shuffle product"]],"source":"The map $T$ of the OOZ quasi-shuffle product"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#ooz-t","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"Words are finite sequences of letters $z_k$ with $k$ a natural number. The map $T$ sends each word to a finite formal $\\mathbb{Q}$-linear combination of words: it sends the empty word $\\mathbf{1}$ to $\\mathbf{1}$, and a word $z_n v$ with first letter $z_n$ to $z_n v - z_{n-1} v$, where $n - 1$ is truncated at $0$ (so $T(z_0 v) = 0$).","title":"The map $T$ of the OOZ quasi-shuffle product"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/ooz-t-nil","kind":"node","lean":"MetaMathlibExt.oozT_nil","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#ooz-t-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"Let $T$ be the map of the OOZ quasi-shuffle product, which sends words in letters $z_k$ ($k$ a natural number) to finite formal $\\mathbb{Q}$-linear combinations of words. Then $T$ sends the empty word $\\mathbf{1}$ to itself: $T(\\mathbf{1}) = \\mathbf{1}$.","title":"The OOZ map fixes the empty word"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/ooz-t-singleton-two","kind":"node","lean":"MetaMathlibExt.oozT_singleton_two","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#ooz-t-singleton-two","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"Let $T$ be the map of the OOZ quasi-shuffle product, which sends words in letters $z_k$ ($k$ a natural number) to finite formal $\\mathbb{Q}$-linear combinations of words. Then $T$ sends the one-letter word $z_2$ to $z_2 - z_1$: $T(z_2) = z_2 - z_1$.","title":"The OOZ map on a one-letter word"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/stuffle-comb","kind":"node","lean":"MetaMathlibExt.stuffleComb","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#stuffle-comb","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"On words in letters $z_k$ ($k$ a natural number), the stuffle product $\\ast$ is the quasi-shuffle product with letter product $z_m \\diamond z_n = z_{m+n}$: $\\mathbf{1} \\ast w = w \\ast \\mathbf{1} = w$ for the empty word $\\mathbf{1}$, and $z_m u \\ast z_n v = z_m (u \\ast z_n v) + z_n (z_m u \\ast v) + z_{m+n} (u \\ast v)$, counted with multiplicity. It is extended $\\mathbb{Q}$-bilinearly to finite formal\u2026","title":"Stuffle product of formal linear combinations of words"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/stuffle-comb-single-nil","kind":"node","lean":"MetaMathlibExt.stuffleComb_single_nil","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#stuffle-comb-single-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"For every word $b$ in letters $z_k$ ($k$ a natural number), the stuffle product of the empty word $\\mathbf{1}$ with $b$ is $b$: $\\mathbf{1} \\ast b = b$, as formal $\\mathbb{Q}$-linear combinations of words.","title":"The empty word is a left unit for the stuffle product"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/ooz-qmzv-mul-eq-of-ooz-t-eq-stuffle-comb","kind":"node","lean":"MetaMathlibExt.oozQMZV_mul_eq_of_oozT_eq_stuffleComb","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#ooz-qmzv-mul-eq-of-ooz-t-eq-stuffle-comb","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"can_prove","summary":"Identify words in letters $z_k$ with lists $(k_1, \\dots, k_r)$ of natural numbers, and call such a word admissible if every $k_i \\ge 1$ and, when $r \\ge 1$, $k_1 \\ge 2$. Let $T$ send the empty word $\\mathbf{1}$ to $\\mathbf{1}$ and a word $z_n v$ with first letter $z_n$ to the formal $\\mathbb{Q}$-linear combination $z_n v - z_{n-1} v$ (with $n - 1$ truncated at $0$). Extend $T$ and the Ohno\u2013Okuda\u2013Zudilin qMZV\u2026","title":"OOZ qMZVs respect the OOZ quasi-shuffle product"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/bz-chain-sum","kind":"node","lean":"MetaMathlibExt.bzChainSum","parent":9281,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05/index.html#bz-chain-sum","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-05","status":"fully_proved","summary":"Let $o_1, \\dots, o_\\ell$ be a word of markers, each either a natural number (a weighted position) or a blank (an unweighted position); let $W$ be the set of weighted positions and $k_t$ the number at position $t \\in W$. For natural numbers $M$ and $R$, the weighted chain sum of the word at bound $M$ and target $R$ is the natural number","title":"Weighted chain sums"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","kind":"scope","navigation":"transparent","parent":9196,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"can_prove","title":"Marker word of a list of blocks to Duality for OOZ-qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/bz-chain-word","kind":"node","lean":"MetaMathlibExt.bzChainWord","parent":9302,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#bz-chain-word","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"A marker is either a weight $k \\in \\mathbb{N}$ (a weighted position) or a blank (an unweighted position). The marker word of a list of blocks $(s_1, k_1), \\dots, (s_r, k_r)$ of natural numbers is the concatenation, in order, of the words consisting of the weight $k_t$ followed by $s_t - 1$ blanks (no blanks when $s_t \\le 1$).","title":"Marker word of a list of blocks"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/bz-chain-sum-bz-chain-word-one-one","kind":"node","lean":"MetaMathlibExt.bzChainSum_bzChainWord_one_one","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","Chain sum of the single block "],["math","\\((1, 1)\\)"]],"source":"Chain sum of the single block $(1, 1)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#bz-chain-sum-bz-chain-word-one-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"The marker word of the single block $(1, 1)$ is one position weighted by $1$, and its chain sum with bound $M = 3$ and target $R = 2$ equals $3$: it is the sum of $\\binom{j}{1} = j$ over $0 < m < 3$ and $j \\ge 0$ with $m j = 2$, which is $2 + 1 = 3$.","title":"Chain sum of the single block $(1, 1)$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/bz-chain-sum-dual","kind":"node","lean":"MetaMathlibExt.bzChainSum_dual","parent":9302,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#bz-chain-sum-dual","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"can_prove","summary":"For a list of blocks $(s_1, w_1), \\dots, (s_r, w_r)$ of positive integers and an integer $N \\ge 1$, let $S_N((s_1, w_1), \\dots, (s_r, w_r))$ be the sum of $\\binom{j_1}{w_1} \\cdots \\binom{j_r}{w_r}$ over all strictly decreasing chains $N + 1 > a_1 > \\dots > a_n > 0$ of length $n = s_1 + \\dots + s_r$ and all multiplicities $c_1, \\dots, c_n \\ge 0$ with $a_1 c_1 + \\dots + a_n c_n = N$, where the chain is cut into\u2026","title":"Combinatorial form of Bradley\u2013Zhao duality"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/sz-dual-index","kind":"node","lean":"MetaMathlibExt.szDualIndex","parent":9302,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#sz-dual-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"Identify an index $k = (k_1, \\dots, k_r)$ of natural numbers with the word $p^{k_1} y \\cdots p^{k_r} y$ in the letters $p$ and $y$, and let $\\tilde\\tau$ be the anti-automorphism of words with $\\tilde\\tau(p) = y$ and $\\tilde\\tau(y) = p$ (reverse the word and swap the letters). Write $\\tilde\\tau(p^{k_1} y \\cdots p^{k_r} y)$ as $p^{a_1} y \\cdots p^{a_s} y$ followed by a possibly empty run of letters $p$ (nonempty only\u2026","title":"The SZ dual index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/sz-dual-index-two","kind":"node","lean":"MetaMathlibExt.szDualIndex_two","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The SZ dual of "],["math","\\((2)\\)"]],"source":"The SZ dual of $(2)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#sz-dual-index-two","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"The SZ dual index of the one-entry index $(2)$ is $(1, 0)$, that is, $(2)^\\dagger = (1, 0)$.","title":"The SZ dual of $(2)$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/bz-dual-index","kind":"node","lean":"MetaMathlibExt.bzDualIndex","parent":9302,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#bz-dual-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"For an index $k = (k_1, \\dots, k_r)$ of natural numbers, form the word $x^{k_1 - 1} y \\cdots x^{k_r - 1} y$ in two letters $x$ and $y$ (each $k_i - 1$ truncated at $0$); when all $k_i \\ge 1$ this is $z_{k_1} \\cdots z_{k_r}$ with $z_k = x^{k-1} y$. Apply the anti-automorphism $\\tau$ of words with $\\tau(x) = y$ and $\\tau(y) = x$, which reverses a word and swaps its letters, and write the result as\u2026","title":"Bradley\u2013Zhao dual index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/bz-dual-index-three","kind":"node","lean":"MetaMathlibExt.bzDualIndex_three","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bradley\u2013Zhao dual of "],["math","\\((3)\\)"]],"source":"Bradley\u2013Zhao dual of $(3)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#bz-dual-index-three","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"The Bradley\u2013Zhao dual index of $(3)$ is $(2, 1)$. In terms of words in the letters $x$ and $y$, with $z_k = x^{k-1} y$ and $\\tau$ the anti-automorphism of words that reverses a word and swaps $x$ and $y$, this says $\\tau(z_3) = \\tau(x x y) = x y y = z_2 z_1$.","title":"Bradley\u2013Zhao dual of $(3)$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-to-bztail","kind":"node","lean":"MetaMathlibExt.oozToBZTail","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The tail part of the map "],["math","\\(U\\)"]],"source":"The tail part of the map $U$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-to-bztail","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"A word is a finite sequence $z_{k_1} \\cdots z_{k_s}$ of letters $z_k$, $k \\in \\mathbb{N}$. This map sends the empty word to itself and, for $s \\ge 1$, sends $z_{k_1} \\cdots z_{k_s}$ to the formal $\\mathbb{Q}$-linear combination of words","title":"The tail part of the map $U$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-to-bz","kind":"node","lean":"MetaMathlibExt.oozToBZ","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(U\\)"],["text"," from OOZ to BZ words"]],"source":"The map $U$ from OOZ to BZ words"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-to-bz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"A word is a finite sequence $z_{k_1} \\cdots z_{k_r}$ of letters $z_k$, $k \\in \\mathbb{N}$. The map $U$ sends each word to a formal $\\mathbb{Q}$-linear combination of words: $U(\\mathbf{1}) = \\mathbf{1}$ for the empty word $\\mathbf{1}$, and for $r \\ge 1$","title":"The map $U$ from OOZ to BZ words"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-to-bz-nil","kind":"node","lean":"MetaMathlibExt.oozToBZ_nil","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(U\\)"],["text"," fixes the empty index"]],"source":"$U$ fixes the empty index"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-to-bz-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"Write $\\mathbf{1}$ for the empty index. The map $U$ from OOZ to BZ indices sends $\\mathbf{1}$ to the $\\mathbb{Q}$-linear combination of indices consisting of $\\mathbf{1}$ with coefficient $1$; that is, $U(\\mathbf{1}) = \\mathbf{1}$.","title":"$U$ fixes the empty index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-to-bz-two","kind":"node","lean":"MetaMathlibExt.oozToBZ_two","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(U\\)"],["text"," fixes "],["math","\\(z_2\\)"]],"source":"The map $U$ fixes $z_2$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-to-bz-two","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"The map $U$ from OOZ to BZ words sends the one-letter word $z_2$ to itself with coefficient $1$: $U(z_2) = z_2$.","title":"The map $U$ fixes $z_2$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-to-sztail","kind":"node","lean":"MetaMathlibExt.oozToSZTail","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The tail part of the map "],["math","\\(V\\)"]],"source":"The tail part of the map $V$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-to-sztail","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"Identify an index $(k_1, \\dots, k_s)$ of natural numbers with the word $p^{k_1} y \\cdots p^{k_s} y$ in the letters $p$ and $y$. This map sends the empty word to itself and, for $s \\ge 1$, sends $p^{k_1} y \\cdots p^{k_s} y$ to the formal $\\mathbb{Q}$-linear combination of words","title":"The tail part of the map $V$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-to-sz","kind":"node","lean":"MetaMathlibExt.oozToSZ","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(V\\)"],["text"," from OOZ to SZ words"]],"source":"The map $V$ from OOZ to SZ words"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-to-sz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"Identify an index $(k_1, \\dots, k_r)$ of natural numbers with the word $p^{k_1} y \\cdots p^{k_r} y$ in the letters $p$ and $y$. The map $V$ sends each such word to a formal $\\mathbb{Q}$-linear combination of words: $V(\\mathbf{1}) = \\mathbf{1}$ for the empty word $\\mathbf{1}$, and for $r \\ge 1$","title":"The map $V$ from OOZ to SZ words"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-to-sz-one","kind":"node","lean":"MetaMathlibExt.oozToSZ_one","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(V\\)"],["text"," fixes "],["math","\\(py\\)"]],"source":"The map $V$ fixes $py$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-to-sz-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"fully_proved","summary":"The map $V$ from OOZ to SZ words sends the word $py$ (the index $(1)$) to itself with coefficient $1$: $V(py) = py$.","title":"The map $V$ fixes $py$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-qmzv-eq-sum-bz-qmzv-ooz-to-bz","kind":"node","lean":"MetaMathlibExt.oozQMZV_eq_sum_bzQMZV_oozToBZ","parent":9302,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-qmzv-eq-sum-bz-qmzv-ooz-to-bz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"can_prove","summary":"Write $\\zeta_q^{\\mathrm{OOZ}}$ and $\\zeta_q^{\\mathrm{BZ}}$ for the Ohno\u2013Okuda\u2013Zudilin and Bradley\u2013Zhao qMZVs. Let $k = (k_1, \\dots, k_r)$ be an index of natural numbers with all $k_i \\ge 1$ and, when $r \\ge 1$, $k_1 \\ge 2$ (the empty index is allowed). Write the image of $k$ under the map $U$ from OOZ to BZ indices as a $\\mathbb{Q}$-linear combination $U(k) = \\sum_w c_w \\, w$ of indices. Then\u2026","title":"OOZ-qMZVs of admissible indices via BZ-qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/bij-on-linear-combination-ooz-to-bz","kind":"node","lean":"MetaMathlibExt.bijOn_linearCombination_oozToBZ","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(U\\)"],["text"," is a bijection on combinations of admissible indices"]],"source":"The map $U$ is a bijection on combinations of admissible indices"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#bij-on-linear-combination-ooz-to-bz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"can_prove","summary":"Call an index $(k_1, \\dots, k_r)$ of natural numbers admissible if all $k_i \\ge 1$ and, when $r \\ge 1$, $k_1 \\ge 2$; the empty index is admissible. Extend the map $U$ from OOZ to BZ indices $\\mathbb{Q}$-linearly to finite $\\mathbb{Q}$-linear combinations of indices. This extension maps the set of $\\mathbb{Q}$-linear combinations of admissible indices bijectively onto itself.","title":"The map $U$ is a bijection on combinations of admissible indices"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/ooz-qmzv-eq-sum-sz-qmzv-ooz-to-sz","kind":"node","lean":"MetaMathlibExt.oozQMZV_eq_sum_szQMZV_oozToSZ","parent":9302,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#ooz-qmzv-eq-sum-sz-qmzv-ooz-to-sz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"can_prove","summary":"Write $\\zeta_q^{\\mathrm{OOZ}}$ and $\\zeta_q^{\\mathrm{SZ}}$ for the Ohno\u2013Okuda\u2013Zudilin and Schlesinger\u2013Zudilin qMZVs. Let $k = (k_1, \\dots, k_r)$ be an index of natural numbers that is empty or has $k_1 \\ge 1$ (the entries $k_2, \\dots, k_r$ may be $0$). Write the image of $k$ under the map $V$ from OOZ to SZ indices as a $\\mathbb{Q}$-linear combination $V(k) = \\sum_w c_w \\, w$ of indices. Then\u2026","title":"OOZ-qMZVs via SZ-qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/bij-on-linear-combination-ooz-to-sz","kind":"node","lean":"MetaMathlibExt.bijOn_linearCombination_oozToSZ","parent":9302,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(V\\)"],["text"," is a bijection on combinations of SZ-admissible indices"]],"source":"The map $V$ is a bijection on combinations of SZ-admissible indices"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#bij-on-linear-combination-ooz-to-sz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"can_prove","summary":"Call an index $(k_1, \\dots, k_r)$ of natural numbers SZ-admissible if it is empty or $k_1 \\ge 1$ (the entries $k_2, \\dots, k_r$ may be $0$). Extend the map $V$ from OOZ to SZ indices $\\mathbb{Q}$-linearly to finite $\\mathbb{Q}$-linear combinations of indices. This extension maps the set of $\\mathbb{Q}$-linear combinations of SZ-admissible indices bijectively onto itself.","title":"The map $V$ is a bijection on combinations of SZ-admissible indices"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/sum-ooz-qmzv-eq-of-ooz-to-bz-eq-bz-dual","kind":"node","lean":"MetaMathlibExt.sum_oozQMZV_eq_of_oozToBZ_eq_bzDual","parent":9302,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#sum-ooz-qmzv-eq-of-ooz-to-bz-eq-bz-dual","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"can_prove","summary":"Call an index $(k_1, \\dots, k_r)$ of natural numbers admissible if $k_j \\ge 1$ for all $j$ and, when $r \\ge 1$, $k_1 \\ge 2$. Extend $\\zeta_q^{\\mathrm{OOZ}}$ and the map $U$ linearly to formal $\\mathbb{Q}$-linear combinations of indices, and let $\\tau$ act on such a combination by replacing each index with its BZ dual index. Let $w$ be an admissible index and let $x$ be a $\\mathbb{Q}$-linear combination of\u2026","title":"Duality for restricted OOZ-qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/sum-ooz-qmzv-eq-of-ooz-to-sz-eq-sz-dual","kind":"node","lean":"MetaMathlibExt.sum_oozQMZV_eq_of_oozToSZ_eq_szDual","parent":9302,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06/index.html#sum-ooz-qmzv-eq-of-ooz-to-sz-eq-sz-dual","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-06","status":"can_prove","summary":"Call an index $(k_1, \\dots, k_r)$ of natural numbers SZ-admissible if it is empty or $k_1 \\ge 1$. Extend $\\zeta_q^{\\mathrm{OOZ}}$ and the map $V$ linearly to formal $\\mathbb{Q}$-linear combinations of indices, and let $\\tilde\\tau$ act on such a combination by replacing each index with its SZ dual index. Let $w$ be an SZ-admissible index and let $x$ be a $\\mathbb{Q}$-linear combination of SZ-admissible indices with\u2026","title":"Duality for OOZ-qMZVs"},{"data_url":"scopes/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07.json","id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","kind":"scope","navigation":"transparent","parent":9196,"result":false,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"can_prove","title":"Partial sums of SZ-star qMZVs to Duality for TBZ-qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/sz-star-qmzvpartial","kind":"node","lean":"MetaMathlibExt.szStarQMZVPartial","parent":9323,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#sz-star-qmzvpartial","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"fully_proved","summary":"For an index $k = (k_1, \\dots, k_r)$ of natural numbers and a bound $M \\in \\mathbb{N}$, the partial sum is the formal power series in $q$ over $\\mathbb{Q}$","title":"Partial sums of SZ-star qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/sz-star-qmzv","kind":"node","lean":"MetaMathlibExt.szStarQMZV","parent":9323,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#sz-star-qmzv","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"fully_proved","summary":"For an index $k = (k_1, \\dots, k_r)$ of natural numbers, the SZ-star qMZV $\\zeta_q^{SZ,\\star}(k)$ is the formal power series in $q$ over $\\mathbb{Q}$","title":"SZ-star qMZVs"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/sz-star-qmzv-nil","kind":"node","lean":"MetaMathlibExt.szStarQMZV_nil","parent":9323,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#sz-star-qmzv-nil","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"fully_proved","summary":"The SZ-star qMZV of the empty index is $1$: $\\zeta_q^{SZ,\\star}(\\varnothing) = 1$.","title":"The empty SZ-star qMZV"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/coeff-one-sz-star-qmzv-singleton-one","kind":"node","lean":"MetaMathlibExt.coeff_one_szStarQMZV_singleton_one","parent":9323,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient of "],["math","\\(q\\)"],["text"," in "],["math","\\(\\zeta_q^{SZ,\\star}(1)\\)"]],"source":"The coefficient of $q$ in $\\zeta_q^{SZ,\\star}(1)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#coeff-one-sz-star-qmzv-singleton-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"fully_proved","summary":"The coefficient of $q$ in the SZ-star qMZV $\\zeta_q^{SZ,\\star}(1) = \\sum_{m > 0} q^m / (1 - q^m)$ is $1$.","title":"The coefficient of $q$ in $\\zeta_q^{SZ,\\star}(1)$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/ooz-qmzv-eq-sz-star-qmzv-sz-dual-index","kind":"node","lean":"MetaMathlibExt.oozQMZV_eq_szStarQMZV_szDualIndex","parent":9323,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#ooz-qmzv-eq-sz-star-qmzv-sz-dual-index","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"can_prove","summary":"For every index $k = (k_1, \\dots, k_r)$ of natural numbers that is empty or has $k_1 \\ge 1$, the Ohno\u2013Okuda\u2013Zudilin qMZV of $k$ equals the SZ-star qMZV of the SZ dual index $k^\\dagger$: $\\zeta_q^{OOZ}(k) = \\zeta_q^{SZ,\\star}(k^\\dagger)$.","title":"OOZ-qMZVs are SZ-star qMZVs of the dual index"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/tbz-to-sz","kind":"node","lean":"MetaMathlibExt.tbzToSZ","parent":9323,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#tbz-to-sz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"fully_proved","summary":"Words in the letters $\\bar{1}$ and $k \\in \\mathbb{N}$ are sent to formal $\\mathbb{Q}$-linear combinations of finite sequences of natural numbers (SZ indices) letter by letter: the empty word goes to the empty index with coefficient $1$, and a word $x w$ with first letter $x$ goes to the image of $w$ with a new first entry prepended to each index, where $\\bar{1}$ contributes the entry $1$ and a letter $k$\u2026","title":"The map from TBZ words to SZ indices"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/tbz-to-sz-bar","kind":"node","lean":"MetaMathlibExt.tbzToSZ_bar","parent":9323,"reading_title":{"format":"inline-parts-v1","parts":[["text","The TBZ-to-SZ map on the letter "],["math","\\(\\bar{1}\\)"]],"source":"The TBZ-to-SZ map on the letter $\\bar{1}$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#tbz-to-sz-bar","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"fully_proved","summary":"The map $U_{\\mathrm{TBZ}}$ sends the one-letter word $\\bar{1}$ to the index $(1)$ with coefficient $1$.","title":"The TBZ-to-SZ map on the letter $\\bar{1}$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/sz-to-tbz","kind":"node","lean":"MetaMathlibExt.szToTBZ","parent":9323,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(V_{TBZ}\\)"],["text"," from SZ indices to TBZ words"]],"source":"The map $V_{TBZ}$ from SZ indices to TBZ words"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#sz-to-tbz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"fully_proved","summary":"For an index $k = (k_1, \\dots, k_r)$ of natural numbers, $V_{TBZ}(k)$ is the formal $\\mathbb{Q}$-linear combination of TBZ words (words in the letters $\\bar 1$ and $j \\ge 1$; write $z_a$ for the one-letter word $a$) obtained by replacing each letter $k_i = 0$ by $z_1 - z_{\\bar 1}$ and each letter $k_i \\ge 1$ by $\\sum_{j=1}^{k_i} (-1)^{k_i - j} z_{j'}$, where $j' = \\bar 1$ for $j = 1$ and $j' = j$ otherwise, and\u2026","title":"The map $V_{TBZ}$ from SZ indices to TBZ words"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/sz-to-tbz-one","kind":"node","lean":"MetaMathlibExt.szToTBZ_one","parent":9323,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(V_{TBZ}\\)"],["text"," on the index "],["math","\\((1)\\)"]],"source":"The map $V_{TBZ}$ on the index $(1)$"},"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#sz-to-tbz-one","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"fully_proved","summary":"The map $V_{TBZ}$ sends the SZ index $(1)$ to the one-letter TBZ word $\\bar 1$ with coefficient $1$.","title":"The map $V_{TBZ}$ on the index $(1)$"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/tbz-qmzv-eq-sum-sz-qmzv-tbz-to-sz","kind":"node","lean":"MetaMathlibExt.tbzQMZV_eq_sum_szQMZV_tbzToSZ","parent":9323,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#tbz-qmzv-eq-sum-sz-qmzv-tbz-to-sz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"can_prove","summary":"Let $w = (w_1, \\dots, w_r)$ be a TBZ word: each letter is either $\\bar 1$ or a natural number $k \\ge 1$, and $w_1$ is not the number $1$. Let $U_{TBZ}(w) = \\sum_v c_v\\, v$ be the formal $\\mathbb{Q}$-linear combination of SZ indices obtained by replacing each letter $\\bar 1$ by $1$ and each letter $k$ by $k$ or by $k - 1$, summed over all choices. Then the Takeyama\u2013Bradley\u2013Zhao qMZV of $w$ is the corresponding\u2026","title":"TBZ-qMZVs in the SZ model"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/sz-qmzv-eq-sum-tbz-qmzv-sz-to-tbz","kind":"node","lean":"MetaMathlibExt.szQMZV_eq_sum_tbzQMZV_szToTBZ","parent":9323,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#sz-qmzv-eq-sum-tbz-qmzv-sz-to-tbz","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"can_prove","summary":"Let $k = (k_1, \\dots, k_r)$ be an index of natural numbers that is empty or has $k_1 \\ge 1$. Let $V_{TBZ}(k) = \\sum_w c_w\\, w$ be the formal $\\mathbb{Q}$-linear combination of TBZ words (words in the letters $\\bar 1$ and $j \\ge 1$; write $z_a$ for the one-letter word $a$) obtained by replacing each letter $k_i = 0$ by $z_1 - z_{\\bar 1}$ and each letter $k_i \\ge 1$ by $\\sum_{j=1}^{k_i} (-1)^{k_i - j} z_{j'}$, where\u2026","title":"SZ-qMZVs in the TBZ model"},{"id":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/tbz-qmzv-eq-sum-tbz-qmzv-sz-to-tbz-sz-du","kind":"node","lean":"MetaMathlibExt.tbzQMZV_eq_sum_tbzQMZV_szToTBZ_szDual","parent":9323,"result":true,"route":"../roadmap/NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07/index.html#tbz-qmzv-eq-sum-tbz-qmzv-sz-to-tbz-sz-du","scope":"NumberTheory/zeta-l-functions/multiple-zeta-values/q-analogues-of-multiple-zeta-values/part-07","status":"can_prove","summary":"Let $w$ be a TBZ word: each letter is either $\\bar 1$ or a natural number $k \\ge 1$, and the first letter is not the number $1$. Let $U_{TBZ}(w) = \\sum_v c_v\\, v$ be the formal $\\mathbb{Q}$-linear combination of SZ indices obtained by replacing each letter $\\bar 1$ by $1$ and each letter $k$ by $k$ or by $k - 1$, summed over all choices. For an SZ index $u = (u_1, \\dots, u_s)$ of natural numbers, let\u2026","title":"Duality for TBZ-qMZVs"},{"data_url":"scopes/NumberTheory/exponential-sums.json","id":"NumberTheory/exponential-sums","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/exponential-sums/index.html","scope":"NumberTheory/exponential-sums","status":"can_prove","summary":"Gauss sums, Weyl and Vinogradov sums, incomplete exponential sums, Burgess bounds, Dirichlet polynomials, trigonometric and cotangent sums.","title":"Exponential and character sums"},{"id":"NumberTheory/exponential-sums/landsberg-schaar","kind":"node","lean":"MetaMathlibExt.landsberg_schaar","parent":9336,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/index.html#landsberg-schaar","scope":"NumberTheory/exponential-sums","status":"fully_proved","summary":"For all positive integers $p$ and $q$,","title":"Landsberg\u2013Schaar relation"},{"id":"NumberTheory/exponential-sums/congruent-pair-quadratic-sum-eq-sub-dede","kind":"node","lean":"MetaMathlibExt.congruentPairQuadraticSum_eq_sub_dedekindSum","parent":9336,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/index.html#congruent-pair-quadratic-sum-eq-sub-dede","scope":"NumberTheory/exponential-sums","status":"can_prove","summary":"Let $p$ be a prime and $l$ a positive integer with $\\gcd(p, l) = 1$. Let $F(p, l)$ be the sum of $\\left(\\frac{a}{p} - \\frac{1}{2}\\right)\\left(\\frac{b}{p} - \\frac{1}{2}\\right)$ over all pairs of integers $a, b$ with $1 \\le a, b \\le p - 1$ and $b \\equiv a l \\pmod{p}$, and let","title":"Mittou's evaluation of a quadratic sum over congruent pairs"},{"data_url":"scopes/NumberTheory/exponential-sums/cotangent-power-sums.json","id":"NumberTheory/exponential-sums/cotangent-power-sums","kind":"scope","navigation":"module","parent":9336,"result":false,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"can_prove","title":"Cotangent power sums"},{"id":"NumberTheory/exponential-sums/cotangent-power-sums/cotangent-power-coeff","kind":"node","lean":"MetaMathlibExt.cotangentPowerCoeff","parent":9339,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html#cotangent-power-coeff","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"fully_proved","summary":"For natural numbers $r$ and $j$, the rational number $c_{r,j}$ is","title":"Girstmair's cotangent power coefficients"},{"id":"NumberTheory/exponential-sums/cotangent-power-sums/cotangent-power-coeff-one-one","kind":"node","lean":"MetaMathlibExt.cotangentPowerCoeff_one_one","parent":9339,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html#cotangent-power-coeff-one-one","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"fully_proved","summary":"Girstmair's cotangent power coefficient $c_{1,1}$ equals $1$, matching the identity $i \\cot = i \\cot$.","title":"Girstmair's coefficient at index (1, 1)"},{"id":"NumberTheory/exponential-sums/cotangent-power-sums/cotangent-power-coeff-two-two","kind":"node","lean":"MetaMathlibExt.cotangentPowerCoeff_two_two","parent":9339,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html#cotangent-power-coeff-two-two","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"fully_proved","summary":"Girstmair's cotangent power coefficient $c_{2,2}$ equals $-1$, matching the identity $(i \\cot)^2 = 1 - i^2 \\cot'$ with $\\cot' = -1 - \\cot^2$.","title":"Girstmair's coefficient at index (2, 2)"},{"id":"NumberTheory/exponential-sums/cotangent-power-sums/i-mul-cot-pow-eq-sum-iterated-deriv-cot","kind":"node","lean":"MetaMathlibExt.I_mul_cot_pow_eq_sum_iteratedDeriv_cot","parent":9339,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html#i-mul-cot-pow-eq-sum-iterated-deriv-cot","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"can_prove","summary":"Let $c_{r,j}$ be Girstmair's cotangent power coefficients and $\\cot^{(m)}$ the $m$-th derivative of the real cotangent function. For every natural number $r \\ge 1$ and every real number $x$ with $\\sin x \\ne 0$,","title":"Cotangent powers through cotangent derivatives"},{"id":"NumberTheory/exponential-sums/cotangent-power-sums/cotangent-power-coeff-eq-zero-of-mod-two","kind":"node","lean":"MetaMathlibExt.cotangentPowerCoeff_eq_zero_of_mod_two_ne","parent":9339,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html#cotangent-power-coeff-eq-zero-of-mod-two","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"can_prove","summary":"For natural numbers $j$ and $r$ with $1 \\le j \\le r$ and $r \\not\\equiv j \\pmod 2$, Girstmair's cotangent power coefficient $c_{r,j}$ is $0$.","title":"Parity of Girstmair's cotangent power coefficients"},{"id":"NumberTheory/exponential-sums/cotangent-power-sums/sum-conj-dirichlet-character-mul-i-mul-c","kind":"node","lean":"MetaMathlibExt.sum_conj_dirichletCharacter_mul_I_mul_cot_pow_of_odd","parent":9339,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html#sum-conj-dirichlet-character-mul-i-mul-c","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"can_prove","summary":"Let $n \\ge 2$, let $r$ be odd, and let $\\chi$ be a complex Dirichlet character modulo $n$ with $\\chi(-1) = -1$. Let $f$ be the conductor of $\\chi$, $\\chi_f$ the primitive character modulo $f$ attached to $\\chi$, and $c_{r,j}$ Girstmair's cotangent power coefficients. Write $\\tau(\\overline{\\chi_f}) = \\sum_{k=1}^{f} \\overline{\\chi_f(k)} e^{2 \\pi i k/f}$ for the Gauss sum and\u2026","title":"Girstmair's cotangent power sums, odd case"},{"id":"NumberTheory/exponential-sums/cotangent-power-sums/sum-conj-dirichlet-character-mul-i-mul-c-2","kind":"node","lean":"MetaMathlibExt.sum_conj_dirichletCharacter_mul_I_mul_cot_pow_of_even","parent":9339,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html#sum-conj-dirichlet-character-mul-i-mul-c-2","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"can_prove","summary":"Let $n \\ge 2$, let $r \\ge 2$ be even, and let $\\chi$ be a complex Dirichlet character modulo $n$ with $\\chi(-1) = 1$. Let $f$ be the conductor of $\\chi$, $\\chi_f$ the primitive character modulo $f$ attached to $\\chi$, and $c_{r,j}$ Girstmair's cotangent power coefficients. Write $\\tau(\\overline{\\chi_f}) = \\sum_{k=1}^{f} \\overline{\\chi_f(k)} e^{2 \\pi i k/f}$ for the Gauss sum,\u2026","title":"Girstmair's cotangent power sums, even case"},{"id":"NumberTheory/exponential-sums/cotangent-power-sums/cotangent-power-coeff-eq-bernoulli-tuple","kind":"node","lean":"MetaMathlibExt.cotangentPowerCoeff_eq_bernoulli_tuple_sum","parent":9339,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/cotangent-power-sums/index.html#cotangent-power-coeff-eq-bernoulli-tuple","scope":"NumberTheory/exponential-sums/cotangent-power-sums","status":"can_prove","summary":"Let $c_{r,j}$ be Girstmair's cotangent power coefficients and $B_m$ the $m$-th Bernoulli number. For natural numbers $j$ and $r$ with $1 \\le j \\le r$ and $j \\equiv r \\pmod 2$,","title":"Cotangent power coefficients through even Bernoulli numbers"},{"data_url":"scopes/NumberTheory/exponential-sums/incomplete-exponential-sums.json","id":"NumberTheory/exponential-sums/incomplete-exponential-sums","kind":"scope","navigation":"module","parent":9336,"result":false,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","title":"Incomplete exponential sums"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/one-div-sin-pi-lt","kind":"node","lean":"MetaMathlibExt.one_div_sin_pi_lt","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#one-div-sin-pi-lt","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"For every real number $x$ with $0 < x < 1$,","title":"Partial-fraction bound for the cosecant"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/sum-one-div-abs-sin-lt","kind":"node","lean":"MetaMathlibExt.sum_one_div_abs_sin_lt","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#sum-one-div-abs-sin-lt","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, let $a$ be an integer not divisible by $p$, let $B$ be an integer with $1 \\le B < p/2$, and let $\\gamma$ be the Euler\u2013Mascheroni constant. Then","title":"Bound for a sum of reciprocal sines"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/sum-abs-sin-mul-div-lt","kind":"node","lean":"MetaMathlibExt.sum_abs_sin_mul_div_lt","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#sum-abs-sin-mul-div-lt","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $K$ be a positive integer, let $\\xi$ be a real number, and let $\\gamma$ be the Euler\u2013Mascheroni constant. Then","title":"Bound for a weighted sum of absolute sines"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/exists-trig-poly-minorant-majorant","kind":"node","lean":"MetaMathlibExt.exists_trigPoly_minorant_majorant","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#exists-trig-poly-minorant-majorant","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $a \\le b$ be real numbers with $b - a < 1$ and let $K \\ge 1$ be an integer. Let $\\chi(x) = 1$ if $a \\le x - n \\le b$ for some integer $n$, and $\\chi(x) = 0$ otherwise. Then there are complex numbers $\\alpha_-(k)$ and $\\alpha_+(k)$, for $k \\in \\mathbb{Z}$, with $\\alpha_\\pm(0) = b - a \\pm \\frac{1}{K+1}$, such that for every real $x$ the trigonometric polynomials\u2026","title":"Trigonometric polynomials bounding the indicator of an arc"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/norm-incomplete-quadratic-sum-lt","kind":"node","lean":"MetaMathlibExt.norm_incomplete_quadratic_sum_lt","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#norm-incomplete-quadratic-sum-lt","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, write $e_p(t) = e^{2\\pi i t/p}$, let $a, b, c$ be integers with $p \\nmid a$, and let $B$ be an integer with $\\sqrt{p} < B < p/2$. Then $\\left|\\sum_{x=1}^{B} e_p(a x^2 + b x + c)\\right| < \\sqrt{\\frac{2}{\\pi} \\, p \\log \\frac{16.6 B^2}{p}}$.","title":"Incomplete quadratic exponential sums of medium length"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/norm-incomplete-monic-cubic-sum-lt","kind":"node","lean":"MetaMathlibExt.norm_incomplete_monic_cubic_sum_lt","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#norm-incomplete-monic-cubic-sum-lt","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, write $e_p(t) = e^{2\\pi i t/p}$, let $b, c$ be integers, let $B$ be an integer with $4 p^{1/3} < B < \\sqrt{p/12}$, and put $S = \\sum_{x=1}^{B} e_p(x^3 + b x^2 + c x)$. Then $|S| < (Bp)^{1/4} \\left(\\log \\frac{B^3}{p}\\right)^{1/4} (\\log (Bp))^{1/4}$ and $|S| < (Bp)^{1/4} (\\log p)^{1/2}$.","title":"Bounds for short incomplete exponential sums of monic cubics"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/exists-incomplete-sum-eq-complete-sum-ad","kind":"node","lean":"MetaMathlibExt.exists_incompleteSum_eq_completeSum_add","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#exists-incomplete-sum-eq-complete-sum-ad","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, write $e_p(t) = e^{2\\pi i t/p}$, let $f$ be a polynomial with integer coefficients, and let $B$ be an integer with $1 \\le B \\le p$. Put $T_p(B) = \\frac{1}{p} \\sum_{y=1}^{p-1} \\left|\\frac{\\sin(\\pi B y/p)}{\\sin(\\pi y/p)}\\right|$ and $M = \\max_{1 \\le y \\le p-1} \\left|\\sum_{x=1}^{p} e_p(f(x) + yx)\\right|$. Then there is a complex number $\\theta$ with $|\\theta| \\le 1$ such that\u2026","title":"Completing an incomplete exponential sum"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/sqrt-le-sup-norm-complete-sum","kind":"node","lean":"MetaMathlibExt.sqrt_le_sup_norm_completeSum","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#sqrt-le-sup-norm-complete-sum","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, let $f$ be a polynomial with integer coefficients, and write $e_p(t) = e^{2\\pi i t / p}$ for integers $t$. Then","title":"Lower bound for the maximal twisted complete exponential sum"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/norm-incomplete-sum-le-min-mul-sup","kind":"node","lean":"MetaMathlibExt.norm_incompleteSum_le_min_mul_sup","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#norm-incomplete-sum-le-min-mul-sup","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, write $e_p(t) = e^{2\\pi i t/p}$, let $f$ be a polynomial with integer coefficients whose reduction modulo $p$ has degree at least $2$, and let $B$ be an integer with $1 \\le B \\le p$. Put $\\Phi(f, p) = \\max_{0 \\le y \\le p-1} \\left|\\sum_{x=1}^{p} e_p(f(x) + yx)\\right|$. Then\u2026","title":"Explicit completion bound for incomplete exponential sums"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/norm-incomplete-sum-lt-sup-mul-log","kind":"node","lean":"MetaMathlibExt.norm_incompleteSum_lt_sup_mul_log","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#norm-incomplete-sum-lt-sup-mul-log","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p \\ge 5$ be a prime, write $e_p(t) = e^{2\\pi i t/p}$, let $f$ be a polynomial with integer coefficients whose reduction modulo $p$ has degree at least $2$, and let $B$ be an integer with $1 \\le B \\le p$. Put $\\Phi(f, p) = \\max_{0 \\le y \\le p-1} \\left|\\sum_{x=1}^{p} e_p(f(x) + yx)\\right|$. Then $\\left|\\sum_{x=1}^{B} e_p(f(x))\\right| < \\Phi(f, p) \\log p$.","title":"Logarithmic completion bound for incomplete exponential sums"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/sum-min-one-div-abs-sin-lt","kind":"node","lean":"MetaMathlibExt.sum_min_one_div_abs_sin_lt","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#sum-min-one-div-abs-sin-lt","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, let $a$ be an integer not divisible by $p$, and let $B$ be an integer with $\\sqrt{p/\\pi} < B < p/2$. Then","title":"Bound for a truncated sum of reciprocal sines"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/sum-abs-sin-div-abs-sin-lt","kind":"node","lean":"MetaMathlibExt.sum_abs_sin_div_abs_sin_lt","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#sum-abs-sin-div-abs-sin-lt","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $q$, $K$ and $B$ be natural numbers with $1 \\le K < q$ and $B \\ge 1$, and let $\\gamma$ be the Euler\u2013Mascheroni constant. Then","title":"Bound for a sum of sine ratios"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/norm-sq-incomplete-quadratic-sum-le","kind":"node","lean":"MetaMathlibExt.norm_sq_incomplete_quadratic_sum_le","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#norm-sq-incomplete-quadratic-sum-le","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, write $e_p(t) = e^{2\\pi i t/p}$, let $a, b, c$ be integers with $p \\nmid a$, let $B$ be an integer with $\\sqrt{p} < B < p/2$, and put $S = \\sum_{x=1}^{B} e_p(a x^2 + b x + c)$ and $R = \\frac{2pB(B+1)}{\\pi(p^2 - B^2)}$. Then $|S|^2$ is at most the minimum of $\\frac{2p}{\\pi} \\log \\frac{B^2}{p} + 1.366 p + 3B + R$ and $\\frac{2p}{\\pi} \\log B + 0.368 p + B + \\frac{p}{\\pi B} + R$.","title":"Square of an incomplete quadratic exponential sum"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/norm-incomplete-quadratic-sum-lt-sqrt-ad","kind":"node","lean":"MetaMathlibExt.norm_incomplete_quadratic_sum_lt_sqrt_add_sqrt","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#norm-incomplete-quadratic-sum-lt-sqrt-ad","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p$ be a prime, write $e_p(t) = e^{2\\pi i t/p}$, let $a, b, c$ be integers with $p \\nmid a$, and let $B$ be an integer with $1 \\le B \\le p$. Then $\\left|\\sum_{x=1}^{B} e_p(a x^2 + b x + c)\\right| < \\sqrt{p} + \\sqrt{\\frac{2}{\\pi} p \\log p + \\frac{p}{2}}$.","title":"Uniform bound for incomplete quadratic exponential sums"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/norm-incomplete-cubic-sum-le","kind":"node","lean":"MetaMathlibExt.norm_incomplete_cubic_sum_le","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#norm-incomplete-cubic-sum-le","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $p \\ge 3$ be a prime, write $e_p(t) = e^{2\\pi i t/p}$ and $\\delta_p = \\frac{1.0661}{\\log \\log p}$, let $a, b, c$ be integers with $p \\nmid a$, let $B$ be a positive integer, and put $S = \\sum_{x=1}^{B} e_p(a x^3 + b x^2 + c x)$. If $B < \\sqrt{p/2}$, then $|S| \\le 1.1 \\, B^{1/4} p^{1/4 + \\delta_p/4} (\\log p)^{1/4}$. If $\\sqrt{p/2} < B \\le p$, then\u2026","title":"Explicit bounds for incomplete cubic exponential sums"},{"id":"NumberTheory/exponential-sums/incomplete-exponential-sums/exists-sum-card-divisors-div-sub-le","kind":"node","lean":"MetaMathlibExt.exists_sum_card_divisors_div_sub_le","parent":9348,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/incomplete-exponential-sums/index.html#exists-sum-card-divisors-div-sub-le","scope":"NumberTheory/exponential-sums/incomplete-exponential-sums","status":"can_prove","summary":"Let $\\tau(n)$ be the number of positive divisors of $n$ and let $\\gamma$ be the Euler\u2013Mascheroni constant. There is a real constant $C$ such that $\\sum_{k=1}^{N} \\frac{\\log k}{k} - \\frac{(\\log N)^2}{2}$ tends to $\\frac{\\gamma^2 - C}{2}$ as $N \\to \\infty$ and such that for every real $x \\ge 2$,\u2026","title":"Explicit asymptotic for the logarithmic mean of the divisor function"},{"data_url":"scopes/NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials.json","id":"NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials","kind":"scope","navigation":"module","parent":9336,"result":false,"route":"../roadmap/NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials/index.html","scope":"NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials","status":"can_prove","title":"Large values of Dirichlet polynomials"},{"id":"NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials/bound","kind":"node","lean":"MathlibExt.NumberTheory.LargeValueEstimatesWanted.LargeValueEstimates.bound","parent":9365,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials/index.html#bound","scope":"NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials","status":"fully_proved","summary":"For a natural number $N$ and real numbers $V$ and $T$, this is the real number $N^2 V^{-2} + N^{18/5} V^{-4} + T N^{12/5} V^{-4}$, the factor multiplying $T^{o(T)}$ on the right-hand side of the Guth\u2013Maynard large values estimate for Dirichlet polynomials. Here $V^{-2}$ and $V^{-4}$ are read as $0$ when $V = 0$.","title":"The Guth\u2013Maynard large values bound"},{"id":"NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials/theorem-1-1","kind":"node","lean":"MathlibExt.NumberTheory.LargeValueEstimatesWanted.LargeValueEstimates.theorem_1_1","parent":9365,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials/index.html#theorem-1-1","scope":"NumberTheory/exponential-sums/large-values-of-dirichlet-polynomials","status":"can_prove","summary":"There is a function $o : \\mathbb{R} \\to \\mathbb{R}$ with $o(T) \\to 0$ as $T \\to \\infty$ such that the following holds. Let $(b_n)_{n \\ge 0}$ be complex numbers with $|b_n| \\le 1$ for all $n$, let $N \\ge 1$ and $R$ be natural numbers, let $V > 0$ and $T > 1$ be real numbers, and let $t_0, \\dots, t_{R-1}$ be points of $[0, T]$ with $|t_r - t_s| \\ge 1$ whenever $r \\ne s$. Suppose that, with $i$ the imaginary unit,","title":"Guth\u2013Maynard large values estimate"},{"data_url":"scopes/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses.json","id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","kind":"scope","navigation":"module","parent":9336,"result":false,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","title":"Mean values of products of modular inverses"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/modular-product-sum","kind":"node","lean":"MetaMathlibExt.modularProductSum","parent":9368,"reading_title":{"format":"inline-parts-v1","parts":[["text","The modular product sum "],["math","\\(S_k(n, m)\\)"]],"source":"The modular product sum $S_k(n, m)$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#modular-product-sum","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"fully_proved","summary":"For natural numbers $k$, $n$ and $m$, the modular product sum $S_k(n, m)$ is the natural number","title":"The modular product sum $S_k(n, m)$"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/modular-product-sum-two-three-one","kind":"node","lean":"MetaMathlibExt.modularProductSum_two_three_one","parent":9368,"reading_title":{"format":"inline-parts-v1","parts":[["text","The modular product sum "],["math","\\(S_2(3, 1)\\)"]],"source":"The modular product sum $S_2(3, 1)$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#modular-product-sum-two-three-one","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"fully_proved","summary":"The modular product sum satisfies $S_2(3, 1) = 5$: the pairs $(a_1, a_2) \\in \\{1, 2, 3\\}^2$ of numbers coprime to $3$ with $a_1 a_2 \\equiv 1 \\pmod 3$ are $(1, 1)$ and $(2, 2)$, and $1 \\cdot 1 + 2 \\cdot 2 = 5$.","title":"The modular product sum $S_2(3, 1)$"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/modular-product-sum-one-five-three","kind":"node","lean":"MetaMathlibExt.modularProductSum_one_five_three","parent":9368,"reading_title":{"format":"inline-parts-v1","parts":[["text","The modular product sum "],["math","\\(S_1(5, 3)\\)"]],"source":"The modular product sum $S_1(5, 3)$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#modular-product-sum-one-five-three","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"fully_proved","summary":"The modular product sum satisfies $S_1(5, 3) = 3$: the only $a \\in \\{1, \\dots, 5\\}$ coprime to $5$ with $a \\equiv 3 \\pmod 5$ is $a = 3$.","title":"The modular product sum $S_1(5, 3)$"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/xu-zhang-constant","kind":"node","lean":"MetaMathlibExt.xuZhangConstant","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#xu-zhang-constant","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"fully_proved","summary":"For a natural number $q$, the Xu\u2013Zhang constant $C_q$ is the product over the prime factors $p$ of $q$, with $p^\\alpha$ the exact power of $p$ dividing $q$,","title":"Xu\u2013Zhang constant"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/xu-zhang-constant-prime","kind":"node","lean":"MetaMathlibExt.xuZhangConstant_prime","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#xu-zhang-constant-prime","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"fully_proved","summary":"For every prime $p$, the Xu\u2013Zhang constant of $p$ is","title":"Xu\u2013Zhang constant at a prime"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/modular-product-sum-explicit-estimate","kind":"node","lean":"MetaMathlibExt.modularProductSum_explicit_estimate","parent":9368,"reading_title":{"format":"inline-parts-v1","parts":[["text","Explicit estimate for the modular product sum "],["math","\\(S_k(n, m)\\)"]],"source":"Explicit estimate for the modular product sum $S_k(n, m)$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#modular-product-sum-explicit-estimate","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $\\varphi$ be Euler's totient function and $S_k(n, m)$ the modular product sum. For all natural numbers $k \\ge 3$, $n \\ge 2$ and $m$ with $\\gcd(m, n) = 1$,","title":"Explicit estimate for the modular product sum $S_k(n, m)$"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/modular-product-sum-odd-estimate","kind":"node","lean":"MetaMathlibExt.modularProductSum_odd_estimate","parent":9368,"reading_title":{"format":"inline-parts-v1","parts":[["text","Estimate for the modular product sum "],["math","\\(S_k(n, m)\\)"],["text"," at odd moduli"]],"source":"Estimate for the modular product sum $S_k(n, m)$ at odd moduli"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#modular-product-sum-odd-estimate","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $\\varphi$ be Euler's totient function and $S_k(n, m)$ the modular product sum. For every natural number $k \\ge 4$ there is a real constant $C_k$ such that for every odd natural number $n \\ge 3$ and every natural number $m$ with $\\gcd(m, n) = 1$,","title":"Estimate for the modular product sum $S_k(n, m)$ at odd moduli"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/modular-product-sum-two-explicit-estimat","kind":"node","lean":"MetaMathlibExt.modularProductSum_two_explicit_estimate","parent":9368,"reading_title":{"format":"inline-parts-v1","parts":[["text","Explicit estimate for the modular product sum "],["math","\\(S_2(n, m)\\)"]],"source":"Explicit estimate for the modular product sum $S_2(n, m)$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#modular-product-sum-two-explicit-estimat","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $\\varphi$ be Euler's totient function, $\\tau(n)$ the number of divisors of $n$, $\\sigma_{1/2}(n) = \\sum_{d \\mid n} \\sqrt{d}$ and $S_2(n, m)$ the modular product sum. For all natural numbers $n \\ge 2$ and $m$ with $\\gcd(m, n) = 1$,","title":"Explicit estimate for the modular product sum $S_2(n, m)$"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/modular-product-sum-two-mean-square","kind":"node","lean":"MetaMathlibExt.modularProductSum_two_mean_square","parent":9368,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean square of the error in the modular product sum "],["math","\\(S_2(n, m)\\)"]],"source":"Mean square of the error in the modular product sum $S_2(n, m)$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#modular-product-sum-two-mean-square","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $\\varphi$ be Euler's totient function, $S_2(n, m)$ the modular product sum and $C_n$ the Xu\u2013Zhang constant of $n$. For every real $\\varepsilon > 0$ there is a real constant $C$ such that for every odd natural number $n \\ge 3$,","title":"Mean square of the error in the modular product sum $S_2(n, m)$"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/modular-product-sum-two-error-large","kind":"node","lean":"MetaMathlibExt.modularProductSum_two_error_large","parent":9368,"reading_title":{"format":"inline-parts-v1","parts":[["text","Large values of the error in the modular product sum "],["math","\\(S_2(n, m)\\)"]],"source":"Large values of the error in the modular product sum $S_2(n, m)$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#modular-product-sum-two-error-large","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $\\varphi$ be Euler's totient function and $S_2(n, m)$ the modular product sum. There are a real number $c > 0$ and a natural number $N$ such that every odd natural number $n \\ge N$ admits a natural number $m$ with $1 \\le m \\le n$ and $\\gcd(m, n) = 1$ for which","title":"Large values of the error in the modular product sum $S_2(n, m)$"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/odd-character-twisted-weighted-sum-fourt","kind":"node","lean":"MetaMathlibExt.odd_character_twisted_weighted_sum_fourth_moment","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#odd-character-twisted-weighted-sum-fourt","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $\\varphi$ be Euler's totient function and $C_q$ the Xu\u2013Zhang constant of $q$. For every natural number $\\ell$ and every real $\\varepsilon > 0$ there is a real constant $C$ such that for every odd natural number $q \\ge 3$,","title":"Xu\u2013Zhang twisted fourth moment of weighted character sums"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/odd-character-weighted-sum-fourth-moment","kind":"node","lean":"MetaMathlibExt.odd_character_weighted_sum_fourth_moment","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#odd-character-weighted-sum-fourth-moment","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"There is a real constant $C$ such that for every odd integer $n \\ge 3$,","title":"Fourth moment of weighted odd character sums"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/weighted-kloosterman-sum-bound","kind":"node","lean":"MetaMathlibExt.weighted_kloosterman_sum_bound","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#weighted-kloosterman-sum-bound","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $n \\ge 1$ and $k \\ge 1$ be integers and let $m$ be a natural number coprime to $n$. Write $e_n(x) = e^{2\\pi i x / n}$, and for $a$ coprime to $n$ let $\\bar a \\in \\{0, \\dots, n - 1\\}$ be the inverse of $a$ modulo $n$. Then","title":"Bound for a weighted Kloosterman-type sum"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/sum-mul-additive-character","kind":"node","lean":"MetaMathlibExt.sum_mul_additive_character","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#sum-mul-additive-character","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $q \\ge 2$, let $k$ be a natural number not divisible by $q$, and write $e_q(x) = e^{2\\pi i x / q}$. Then","title":"Weighted sum of an additive character"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/coprime-sum-mul-additive-character","kind":"node","lean":"MetaMathlibExt.coprime_sum_mul_additive_character","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#coprime-sum-mul-additive-character","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"For a positive integer $r$ and an integer $x$ write $e_r(x) = \\exp(2\\pi i x / r)$, and let $\\mu$ be the M\u00f6bius function. For all natural numbers $n \\ge 2$ and $k \\ge 1$,","title":"Weighted additive character sum over reduced residues"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/sum-icc-eq-sum-divisors-coprime","kind":"node","lean":"MetaMathlibExt.sum_Icc_eq_sum_divisors_coprime","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#sum-icc-eq-sum-divisors-coprime","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"For every function $f : \\mathbb{N} \\to \\mathbb{C}$ and every natural number $n$,","title":"Splitting a sum by greatest common divisor"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/sum-inv-norm-one-sub-additive-character","kind":"node","lean":"MetaMathlibExt.sum_inv_norm_one_sub_additive_character","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#sum-inv-norm-one-sub-additive-character","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $q \\ge 2$ and $\\ell \\ge 1$ be integers, let $s \\in \\{1, -1\\}$, and write $e_q(x) = e^{2\\pi i x / q}$. Then","title":"Sum of reciprocal distances from roots of unity to 1"},{"id":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/sum-sqrt-gcd-div-norm-one-sub-additive-c","kind":"node","lean":"MetaMathlibExt.sum_sqrt_gcd_div_norm_one_sub_additive_character","parent":9368,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses/index.html#sum-sqrt-gcd-div-norm-one-sub-additive-c","scope":"NumberTheory/exponential-sums/mean-values-of-products-of-modular-inverses","status":"can_prove","summary":"Let $n \\ge 2$, let $d$ be a divisor of $n$, and write $e_m(x) = e^{2\\pi i x / m}$. Then","title":"Gcd-weighted sum of reciprocal distances from roots of unity to 1"},{"data_url":"scopes/NumberTheory/exponential-sums/quadratic-gauss-sums.json","id":"NumberTheory/exponential-sums/quadratic-gauss-sums","kind":"scope","navigation":"module","parent":9336,"result":false,"route":"../roadmap/NumberTheory/exponential-sums/quadratic-gauss-sums/index.html","scope":"NumberTheory/exponential-sums/quadratic-gauss-sums","status":"fully_proved","title":"Quadratic Gauss sums"},{"id":"NumberTheory/exponential-sums/quadratic-gauss-sums/quadratic-gauss-sum5","kind":"node","lean":"MetaMathlibExt.quadraticGaussSum5","parent":9387,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quadratic Gauss sum modulo "],["math","\\(5\\)"]],"source":"Quadratic Gauss sum modulo $5$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/quadratic-gauss-sums/index.html#quadratic-gauss-sum5","scope":"NumberTheory/exponential-sums/quadratic-gauss-sums","status":"fully_proved","summary":"For an integer $k$, the quadratic Gauss sum $g(k; 5)$ is the complex number $g(k; 5) = \\sum_{n=0}^{4} e^{2\\pi i k n^2 / 5}$.","title":"Quadratic Gauss sum modulo $5$"},{"id":"NumberTheory/exponential-sums/quadratic-gauss-sums/quadratic-gauss-sum5-eval","kind":"node","lean":"MetaMathlibExt.quadraticGaussSum5_eval","parent":9387,"reading_title":{"format":"inline-parts-v1","parts":[["text","Evaluation of the quadratic Gauss sum modulo "],["math","\\(5\\)"]],"source":"Evaluation of the quadratic Gauss sum modulo $5$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/quadratic-gauss-sums/index.html#quadratic-gauss-sum5-eval","scope":"NumberTheory/exponential-sums/quadratic-gauss-sums","status":"fully_proved","summary":"For every integer $k$ not divisible by $5$, the quadratic Gauss sum $g(k; 5) = \\sum_{n=0}^{4} e^{2\\pi i k n^2 / 5}$ equals $\\left(\\frac{k}{5}\\right)\\sqrt{5}$, where $\\left(\\frac{k}{5}\\right)$ is the Legendre symbol.","title":"Evaluation of the quadratic Gauss sum modulo $5$"},{"data_url":"scopes/NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem.json","id":"NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem","kind":"scope","navigation":"module","parent":9336,"result":false,"route":"../roadmap/NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem/index.html","scope":"NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem","status":"can_prove","title":"Vinogradov's mean value theorem"},{"id":"NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem/j","kind":"node","lean":"MathlibExt.NumberTheory.VinogradovMeanValueWanted.NumberTheory.VinogradovMeanValue.J","parent":9390,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem/index.html#j","scope":"NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem","status":"fully_proved","summary":"For natural numbers $s$, $k$ and $X$, $J_{s,k}(X)$ is the number of pairs of tuples $(x_1, \\dots, x_s)$ and $(y_1, \\dots, y_s)$ of integers in $\\{1, \\dots, X\\}$ such that $x_1^j + \\dots + x_s^j = y_1^j + \\dots + y_s^j$ for every natural number $j \\le k$, regarded as a real number.","title":"The Vinogradov mean value count"},{"id":"NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem/vinogradov-mean-value","kind":"node","lean":"MathlibExt.NumberTheory.VinogradovMeanValueWanted.NumberTheory.VinogradovMeanValue.vinogradov_mean_value","parent":9390,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem/index.html#vinogradov-mean-value","scope":"NumberTheory/exponential-sums/vinogradov-s-mean-value-theorem","status":"can_prove","summary":"For all natural numbers $s$ and $k$ and every real $\\varepsilon > 0$ there is a constant $C$ such that $J_{s,k}(X) \\le C (X^{s + \\varepsilon} + X^{2s - k(k+1)/2 + \\varepsilon})$ for all sufficiently large natural numbers $X$, where $J_{s,k}(X)$ is the number of pairs of tuples $(x_1, \\dots, x_s)$ and $(y_1, \\dots, y_s)$ of integers in $\\{1, \\dots, X\\}$ with $x_1^j + \\dots + x_s^j = y_1^j + \\dots + y_s^j$ for every\u2026","title":"Vinogradov's mean value theorem"},{"data_url":"scopes/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums.json","id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","kind":"scope","navigation":"module","parent":9336,"result":false,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","title":"Watson's cosecant and cosine sums"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-cos-sum","kind":"node","lean":"MetaMathlibExt.watsonCosSum","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-cos-sum","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"fully_proved","summary":"For natural numbers $n$ and $\\nu$, $C_n(\\nu)$ is the real number","title":"Generalized Watson trigonometric sum"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-alternating-cosecant-sum","kind":"node","lean":"MetaMathlibExt.watsonAlternatingCosecantSum","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-alternating-cosecant-sum","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"fully_proved","summary":"For a natural number $n$, the alternating cosecant sum is the real number","title":"Watson's alternating cosecant sum"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-cos-sum-two-one","kind":"node","lean":"MetaMathlibExt.watsonCosSum_two_one","parent":9393,"reading_title":{"format":"inline-parts-v1","parts":[["text","Watson's cosine sum at "],["math","\\(n = 2\\)"],["text",", "],["math","\\(\\nu = 1\\)"]],"source":"Watson's cosine sum at $n = 2$, $\\nu = 1$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-cos-sum-two-one","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"fully_proved","summary":"The cosine sum $C_n(\\nu) = \\sum_{l=1}^{n-1} \\cos(2 \\pi \\nu l / n) / \\sin(\\pi l / n)$ satisfies $C_2(1) = -1$.","title":"Watson's cosine sum at $n = 2$, $\\nu = 1$"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-alternating-cosecant-sum-two","kind":"node","lean":"MetaMathlibExt.watsonAlternatingCosecantSum_two","parent":9393,"reading_title":{"format":"inline-parts-v1","parts":[["text","Watson's alternating cosecant sum at "],["math","\\(n = 2\\)"]],"source":"Watson's alternating cosecant sum at $n = 2$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-alternating-cosecant-sum-two","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"fully_proved","summary":"The alternating cosecant sum $C_2 = \\sum_{l=1}^{1} (-1)^{l+1} / \\sin(\\pi l / 2)$ equals $1$.","title":"Watson's alternating cosecant sum at $n = 2$"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-odd-cotangent-deriv","kind":"node","lean":"MetaMathlibExt.watsonOddCotangentDeriv","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-odd-cotangent-deriv","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"fully_proved","summary":"For natural numbers $r$, $n$ and $\\nu$, $\\mathcal{H}_{2r-1}(\\nu/n)$ is the real number $\\cot^{(2r-1)}(\\pi\\nu/n)$, the $(2r-1)$st derivative of the cotangent function evaluated at $\\pi\\nu/n$. Here $2r - 1$ is computed in the natural numbers, so for $r = 0$ this is $\\cot(\\pi\\nu/n)$ itself.","title":"Odd cotangent derivatives"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-cosh-integral-digamma","kind":"node","lean":"MetaMathlibExt.watson_cosh_integral_digamma","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-cosh-integral-digamma","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","summary":"Let $b > 0$ be a real number and let $\\alpha, \\beta$ be complex numbers with $\\operatorname{Re}(\\alpha + b - \\beta) > 0$ and $\\operatorname{Re}(\\alpha + b + \\beta) > 0$. Then the function $x \\mapsto e^{-\\alpha x} \\cosh(\\beta x) / \\cosh(bx)$ is integrable on $(0, \\infty)$, and, writing $\\psi$ for the digamma function,","title":"Digamma evaluation of a hyperbolic Laplace integral"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-expansion-term","kind":"node","lean":"MetaMathlibExt.watsonExpansionTerm","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-expansion-term","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"fully_proved","summary":"For natural numbers $r$, $n$ and $\\nu$, let $B_{2r}$ be the $2r$th Bernoulli number and let $\\mathcal{H}_{2r-1}(\\nu/n) = \\cot^{(2r-1)}(\\pi\\nu/n)$ be the $(2r-1)$st derivative of the cotangent evaluated at $\\pi\\nu/n$. The $r$th Watson expansion term is the real number","title":"Watson expansion terms"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-cos-sum-almost-asymptotic","kind":"node","lean":"MetaMathlibExt.watson_cos_sum_almost_asymptotic","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-cos-sum-almost-asymptotic","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","summary":"Let $n \\ge 2$, $N \\ge 2$ and $\\nu$ be natural numbers with $1 \\le \\nu \\le n - 1$, and let","title":"Almost-asymptotic expansion of Watson's cosine sum"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-odd-cotangent-deriv-neg","kind":"node","lean":"MetaMathlibExt.watson_odd_cotangent_deriv_neg","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-odd-cotangent-deriv-neg","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","summary":"Let $r$, $n$ and $\\nu$ be natural numbers with $r \\ge 1$ and $1 \\le \\nu \\le n - 1$. Then $\\mathcal{H}_{2r-1}(\\nu/n) = \\cot^{(2r-1)}(\\pi\\nu/n)$, the $(2r-1)$st derivative of the cotangent evaluated at $\\pi\\nu/n$, is strictly negative.","title":"The odd cotangent derivatives in Watson's expansion are negative"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-alternating-main","kind":"node","lean":"MetaMathlibExt.watsonAlternatingMain","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-alternating-main","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"fully_proved","summary":"For natural numbers $N$ and $n$, the main part is the real number","title":"Main part of the asymptotic expansion of Watson's alternating cosecant sum"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-alternating-complete-asymptotic","kind":"node","lean":"MetaMathlibExt.watson_alternating_complete_asymptotic","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-alternating-complete-asymptotic","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","summary":"For natural numbers $N$ and $n$ let $C_n = \\sum_{l=1}^{n-1} (-1)^{l+1} / \\sin(\\pi l / n)$ and","title":"Complete asymptotic expansion of Watson's alternating cosecant sum"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-alternating-odd-zero","kind":"node","lean":"MetaMathlibExt.watson_alternating_odd_zero","parent":9393,"reading_title":{"format":"inline-parts-v1","parts":[["text","Watson's alternating cosecant sum vanishes for odd "],["math","\\(n\\)"]],"source":"Watson's alternating cosecant sum vanishes for odd $n$"},"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-alternating-odd-zero","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","summary":"For every odd natural number $n$,","title":"Watson's alternating cosecant sum vanishes for odd $n$"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-alternating-sharp-bounds","kind":"node","lean":"MetaMathlibExt.watson_alternating_sharp_bounds","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-alternating-sharp-bounds","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","summary":"For every even natural number $n \\ge 2$, the alternating cosecant sum $C_n = \\sum_{l=1}^{n-1} (-1)^{l+1} / \\sin(\\pi l / n)$ satisfies the strict inequalities","title":"Two-sided bounds for Watson's alternating cosecant sum"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-alternating-simple-bounds","kind":"node","lean":"MetaMathlibExt.watson_alternating_simple_bounds","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-alternating-simple-bounds","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","summary":"For every even natural number $n \\ge 4$, the alternating cosecant sum $C_n = \\sum_{l=1}^{n-1} (-1)^{l+1} / \\sin(\\pi l / n)$ satisfies the strict inequalities","title":"Decimal bounds for Watson's alternating cosecant sum"},{"id":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/watson-cos-sum-sixth-limits","kind":"node","lean":"MetaMathlibExt.watson_cos_sum_sixth_limits","parent":9393,"result":true,"route":"../roadmap/NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums/index.html#watson-cos-sum-sixth-limits","scope":"NumberTheory/exponential-sums/watson-s-cosecant-and-cosine-sums","status":"can_prove","summary":"For natural numbers $n$ and $\\nu$ let $C_n(\\nu) = \\sum_{l=1}^{n-1} \\cos(2 \\pi \\nu l / n) / \\sin(\\pi l / n)$. As $m \\to \\infty$ through the natural numbers, $C_{6m}(m) \\to 0$ and $C_{6m}(5m) \\to 0$.","title":"Vanishing limits of Watson's cosine sum at one sixth and five sixths"},{"data_url":"scopes/NumberTheory/diophantine-equations.json","id":"NumberTheory/diophantine-equations","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/index.html","scope":"NumberTheory/diophantine-equations","status":"can_state","summary":"Integer and rational solutions of polynomial and exponential equations, Pell equations, Thue equations, Diophantine tuples, Egyptian fractions.","title":"Diophantine equations"},{"data_url":"scopes/NumberTheory/diophantine-equations/exponential-diophantine.json","id":"NumberTheory/diophantine-equations/exponential-diophantine","kind":"scope","parent":9409,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/index.html","scope":"NumberTheory/diophantine-equations/exponential-diophantine","status":"can_prove","summary":"Equations with unknown exponents: Pillai, Catalan, St\u00f8rmer, Nagell\u2013Ljunggren, Ramanujan\u2013Nagell and Terai-type equations, powers plus one.","title":"Exponential Diophantine equations"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/stormer-theorem","kind":"node","lean":"MetaMathlibExt.stormer_theorem","parent":9410,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/index.html#stormer-theorem","scope":"NumberTheory/diophantine-equations/exponential-diophantine","status":"fully_proved","summary":"For every natural number $y \\ge 2$, there is a finite set $S$ of pairs of natural numbers such that, for every positive integer $n$ such that all prime factors of $n$ and all prime factors of $n + 1$ are at most $y$, the pair $(n, n + 1)$ lies in $S$.","title":"St\u00f8rmer's theorem"},{"data_url":"scopes/NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation.json","id":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation","kind":"scope","navigation":"module","parent":9410,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/index.html","scope":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation","status":"can_prove","title":"Goormaghtigh equation"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/goormaghtigh-eq-of-gcd-sub-one-gt-one-of","kind":"node","lean":"MetaMathlibExt.goormaghtigh_eq_of_gcd_sub_one_gt_one_of_le_fifty","parent":9412,"reading_title":{"format":"inline-parts-v1","parts":[["text","Goormaghtigh solutions with "],["math","\\(\\gcd(m - 1, n - 1) > 1\\)"],["text"," and "],["math","\\(m \\le 50\\)"]],"source":"Goormaghtigh solutions with $\\gcd(m - 1, n - 1) > 1$ and $m \\le 50$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/index.html#goormaghtigh-eq-of-gcd-sub-one-gt-one-of","scope":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation","status":"can_prove","summary":"Let $x, y, m, n$ be natural numbers with $2 \\le x < y$, $n > 2$, $m \\le 50$ and $\\gcd(m - 1, n - 1) > 1$, and suppose","title":"Goormaghtigh solutions with $\\gcd(m - 1, n - 1) > 1$ and $m \\le 50$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/goormaghtigh-prime-lt-ten-pow-seven-hund","kind":"node","lean":"MetaMathlibExt.goormaghtigh_prime_lt_ten_pow_seven_hundred","parent":9412,"reading_title":{"format":"inline-parts-v1","parts":[["text","No Goormaghtigh primes below "],["math","\\(10^{700}\\)"],["text"," besides "],["math","\\(31\\)"],["text"," and "],["math","\\(8191\\)"]],"source":"No Goormaghtigh primes below $10^{700}$ besides $31$ and $8191$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/index.html#goormaghtigh-prime-lt-ten-pow-seven-hund","scope":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation","status":"can_prove","summary":"Let $p$ be a prime with $p < 10^{700}$, and let $x, y, m, n$ be natural numbers with $2 \\le x < y$ and $n \\ge 3$ such that","title":"No Goormaghtigh primes below $10^{700}$ besides $31$ and $8191$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/goormaghtigh-numbers-length-ge-five-fini","kind":"node","lean":"MetaMathlibExt.goormaghtigh_numbers_length_ge_five_finite_of_abc","parent":9412,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/index.html#goormaghtigh-numbers-length-ge-five-fini","scope":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation","status":"can_prove","summary":"Assume the abc conjecture in the following form: for every real $\\varepsilon > 0$, there are only finitely many triples $(a, b, c)$ of positive integers with $a + b = c$, $\\gcd(a, b) = 1$ and $\\operatorname{rad}(abc)^{1 + \\varepsilon} \\le c$, where $\\operatorname{rad}(k)$ is the product of the distinct primes dividing $k$. Then there are only finitely many natural numbers $N$ for which there exist natural numbers\u2026","title":"Under abc, finitely many Goormaghtigh numbers have both lengths at least five"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/goormaghtigh-primes-finite-of-abc","kind":"node","lean":"MetaMathlibExt.goormaghtigh_primes_finite_of_abc","parent":9412,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation/index.html#goormaghtigh-primes-finite-of-abc","scope":"NumberTheory/diophantine-equations/exponential-diophantine/goormaghtigh-equation","status":"can_prove","summary":"Assume the abc conjecture in the following form: for every real $\\varepsilon > 0$, there are only finitely many triples $(a, b, c)$ of positive integers with $a + b = c$, $\\gcd(a, b) = 1$ and $\\operatorname{rad}(abc)^{1 + \\varepsilon} \\le c$, where $\\operatorname{rad}(k)$ is the product of the distinct primes dividing $k$. Then there are only finitely many primes $p$ for which there exist natural numbers\u2026","title":"Under abc, there are only finitely many Goormaghtigh primes"},{"data_url":"scopes/NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation.json","id":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation","kind":"scope","navigation":"module","parent":9410,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/index.html","scope":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation","status":"can_prove","title":"Nagell\u2013Ljunggren equation"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/nagell-ljunggren-cube-no-solution-of-mod","kind":"node","lean":"MetaMathlibExt.nagell_ljunggren_cube_no_solution_of_mod_six_eq_five","parent":9417,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Nagell\u2013Ljunggren cube equation for "],["math","\\(n \\equiv 5 \\pmod 6\\)"]],"source":"The Nagell\u2013Ljunggren cube equation for $n \\equiv 5 \\pmod 6$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/index.html#nagell-ljunggren-cube-no-solution-of-mod","scope":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation","status":"can_prove","summary":"Let $n$ be a natural number with $n \\equiv 5 \\pmod 6$. Then there are no integers $x$ and $y$ with $x \\equiv 2, 3, 5$ or $6 \\pmod 9$ such that $x^n - 1 = (x - 1) y^3$. Since such an $x$ is not $1$, this says that $(x^n - 1)/(x - 1) = y^3$ has no such integer solution.","title":"The Nagell\u2013Ljunggren cube equation for $n \\equiv 5 \\pmod 6$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/nagell-ljunggren-cube-no-solution","kind":"node","lean":"MetaMathlibExt.nagell_ljunggren_cube_no_solution","parent":9417,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Nagell\u2013Ljunggren cube equation for "],["math","\\(x \\equiv 2, 3, 5, 6 \\pmod 9\\)"]],"source":"The Nagell\u2013Ljunggren cube equation for $x \\equiv 2, 3, 5, 6 \\pmod 9$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/index.html#nagell-ljunggren-cube-no-solution","scope":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation","status":"can_prove","summary":"Let $n \\ge 3$ be a natural number. Then there are no integers $x$ and $y$ with $x \\equiv 2, 3, 5$ or $6 \\pmod 9$ such that $x^n - 1 = (x - 1) y^3$. Since such an $x$ is not $1$, this says that $(x^n - 1)/(x - 1) = y^3$ has no such integer solution.","title":"The Nagell\u2013Ljunggren cube equation for $x \\equiv 2, 3, 5, 6 \\pmod 9$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/nagell-ljunggren-solutions-of-conditions","kind":"node","lean":"MetaMathlibExt.nagell_ljunggren_solutions_of_conditions","parent":9417,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/index.html#nagell-ljunggren-solutions-of-conditions","scope":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation","status":"can_prove","summary":"Let $x$ and $y$ be integers and $n$ and $q$ natural numbers with $|x| > 1$, $y > 1$, $n > 2$ and $q \\ge 2$, and suppose that $x^n - 1 = (x - 1) y^q$, that is, $(x^n - 1)/(x - 1) = y^q$ (as $x \\ne 1$). If $q = 2$, or $3 \\mid n$, or $4 \\mid n$, or $q = 3$ and $n \\not\\equiv 5 \\pmod 6$, then $(x, y, n, q)$ is one of $(3, 11, 5, 2)$, $(7, 20, 4, 2)$, $(18, 7, 3, 3)$ and $(-19, 7, 3, 3)$.","title":"Ljunggren's theorem on the Nagell\u2013Ljunggren equation"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/nagell-ljunggren-exponent-structure","kind":"node","lean":"MetaMathlibExt.nagell_ljunggren_exponent_structure","parent":9417,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/index.html#nagell-ljunggren-exponent-structure","scope":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation","status":"can_prove","summary":"Let $x, y, n, q$ be natural numbers with $x \\ge 2$, $y \\ge 2$, $n \\ge 3$ and $q \\ge 2$ such that $(x^n - 1)/(x - 1) = y^q$, and suppose that $(x, y, n, q)$ is none of the known solutions $(3, 11, 5, 2)$, $(7, 20, 4, 2)$ and $(18, 7, 3, 3)$. Then the least prime divisor of $n$ is at least $29$, $n$ has at most four prime factors counted with multiplicity, $n$ is prime if $q = 3$, and $n = q$ if $q \\mid n$.","title":"Structure of a further solution of the Nagell\u2013Ljunggren equation"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/nagell-ljunggren-cube-of-five-dvd","kind":"node","lean":"MetaMathlibExt.nagell_ljunggren_cube_of_five_dvd","parent":9417,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Nagell\u2013Ljunggren cube equation when "],["math","\\(5 \\mid n\\)"]],"source":"The Nagell\u2013Ljunggren cube equation when $5 \\mid n$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation/index.html#nagell-ljunggren-cube-of-five-dvd","scope":"NumberTheory/diophantine-equations/exponential-diophantine/nagell-ljunggren-equation","status":"can_prove","summary":"There are no natural numbers $x \\ge 2$, $y \\ge 2$ and $n \\ge 3$ with $5 \\mid n$ such that $(x^n - 1)/(x - 1) = y^3$.","title":"The Nagell\u2013Ljunggren cube equation when $5 \\mid n$"},{"data_url":"scopes/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers.json","id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","kind":"scope","navigation":"module","parent":9410,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","title":"Squares as sums or differences of powers"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/eq-zero-of-three-pow-succ-add-one-eq-sq","kind":"node","lean":"Nat.eq_zero_of_three_pow_succ_add_one_eq_sq","parent":9423,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#eq-zero-of-three-pow-succ-add-one-eq-sq","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"fully_proved","summary":"For natural numbers $y$ and $z$, if $3^{y+1} + 1 = z^2$, then $y = 0$.","title":"Powers of three plus one that are perfect squares"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/pow-add-pow-eq-sq","kind":"node","lean":"MathlibExt.NumberTheory.ExponentialDiophantineWanted.pow_add_pow_eq_sq","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(p^x + (p + a)^y = z^2\\)"],["text"," for odd primes "],["math","\\(p\\)"],["text"," and "],["math","\\(p + a\\)"]],"source":"Solutions of $p^x + (p + a)^y = z^2$ for odd primes $p$ and $p + a$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#pow-add-pow-eq-sq","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\ne 2$ and $p \\ne 3$, and let $a$ be a natural number such that $p + a$ is prime, $a \\equiv 0 \\pmod 4$ and $p$ divides $a + 1$. If natural numbers $x$, $y$ and $z$ satisfy $p^x + (p + a)^y = z^2$, then $x = 0$, $y = 1$ and $z^2 = p + a + 1$.","title":"Solutions of $p^x + (p + a)^y = z^2$ for odd primes $p$ and $p + a$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/pow19-add-pow151-ne-sq","kind":"node","lean":"MathlibExt.NumberTheory.ExponentialDiophantineWanted.pow19_add_pow151_ne_sq","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(19^x + 151^y\\)"],["text"," is never a perfect square"]],"source":"$19^x + 151^y$ is never a perfect square"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#pow19-add-pow151-ne-sq","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"For all natural numbers $x$, $y$ and $z$, $19^x + 151^y \\ne z^2$.","title":"$19^x + 151^y$ is never a perfect square"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/two-pow-add-one-eq-sq","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.two_pow_add_one_eq_sq","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(2^x + 1 = z^2\\)"]],"source":"Solutions of $2^x + 1 = z^2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#two-pow-add-one-eq-sq","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"For natural numbers $x$ and $z$, $2^x + 1 = z^2$ holds if and only if $x = 3$ and $z = 3$.","title":"Solutions of $2^x + 1 = z^2$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/pow-add-one-eq-sq","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.pow_add_one_eq_sq","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(n^x + 1 = z^2\\)"],["text"," for "],["math","\\(n \\ne 2\\)"]],"source":"Solutions of $n^x + 1 = z^2$ for $n \\ne 2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#pow-add-one-eq-sq","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $n \\ge 1$ be a natural number with $n \\ne 2$. For natural numbers $x$ and $z$, $n^x + 1 = z^2$ holds if and only if $x = 1$ and $z^2 = n + 1$.","title":"Solutions of $n^x + 1 = z^2$ for $n \\ne 2$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/one-add-pow-eq-sq","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.one_add_pow_eq_sq","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","The case "],["math","\\(x = 0\\)"],["text"," of "],["math","\\(n^x + (5p)^y = z^2\\)"]],"source":"The case $x = 0$ of $n^x + (5p)^y = z^2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#one-add-pow-eq-sq","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\ne 2$. For natural numbers $y$ and $z$, $1 + (5p)^y = z^2$ holds if and only if $y = 1$ and $z^2 = 5p + 1$.","title":"The case $x = 0$ of $n^x + (5p)^y = z^2$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/even-of-pos-exp","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.even_of_pos_exp","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Evenness of "],["math","\\(x\\)"],["text"," in "],["math","\\(n^x + (5p)^y = z^2\\)"],["text"," when "],["math","\\(n \\equiv 2, 3 \\pmod 5\\)"]],"source":"Evenness of $x$ in $n^x + (5p)^y = z^2$ when $n \\equiv 2, 3 \\pmod 5$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#even-of-pos-exp","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime and $n$ a natural number with $n \\equiv 2 \\pmod 5$ or $n \\equiv 3 \\pmod 5$. If natural numbers $x$, $y$, $z$ with $y > 0$ satisfy $n^x + (5p)^y = z^2$, then $x$ is even.","title":"Evenness of $x$ in $n^x + (5p)^y = z^2$ when $n \\equiv 2, 3 \\pmod 5$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/opposite-parity","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.opposite_parity","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Opposite parity of the exponents in "],["math","\\(n^x + (5p)^y = z^2\\)"],["text"," when "],["math","\\(n \\equiv p \\equiv 3 \\pmod 4\\)"]],"source":"Opposite parity of the exponents in $n^x + (5p)^y = z^2$ when $n \\equiv p \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#opposite-parity","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime and $n$ a natural number with $n \\equiv 3 \\pmod 4$ and $p \\equiv 3 \\pmod 4$. If natural numbers $x$, $y$, $z$ satisfy $n^x + (5p)^y = z^2$, then $x$ and $y$ have opposite parity.","title":"Opposite parity of the exponents in $n^x + (5p)^y = z^2$ when $n \\equiv p \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/no-solution-mod-four-one","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.no_solution_mod_four_one","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","No solutions of "],["math","\\(n^x + (5p)^y = z^2\\)"],["text"," when "],["math","\\(n \\equiv p \\equiv 1 \\pmod 4\\)"]],"source":"No solutions of $n^x + (5p)^y = z^2$ when $n \\equiv p \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#no-solution-mod-four-one","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime and $n$ a natural number with $n \\equiv 1 \\pmod 4$ and $p \\equiv 1 \\pmod 4$. Then there are no natural numbers $x$, $y$, $z$ with $n^x + (5p)^y = z^2$.","title":"No solutions of $n^x + (5p)^y = z^2$ when $n \\equiv p \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/two-pow-solutions","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.two_pow_solutions","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(2^x + (5p)^y = z^2\\)"],["text"," for "],["math","\\(p \\equiv 3 \\pmod 4\\)"]],"source":"Solutions of $2^x + (5p)^y = z^2$ for $p \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#two-pow-solutions","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 4$. For natural numbers $x$, $y$, $z$, $2^x + (5p)^y = z^2$ holds if and only if $(x, y, z) = (3, 0, 3)$, or $p = 3$ and $(x, y, z) = (6, 2, 17)$, or $x = 0$, $y = 1$ and $z^2 = 5p + 1$.","title":"Solutions of $2^x + (5p)^y = z^2$ for $p \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/solutions-mod-sixty","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.solutions_mod_sixty","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(n^x + (5p)^y = z^2\\)"],["text"," for "],["math","\\(n \\equiv 3, 27 \\pmod{60}\\)"],["text"," and "],["math","\\(p \\equiv 7 \\pmod{12}\\)"]],"source":"Solutions of $n^x + (5p)^y = z^2$ for $n \\equiv 3, 27 \\pmod{60}$ and $p \\equiv 7 \\pmod{12}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#solutions-mod-sixty","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 7 \\pmod{12}$, and let $n$ be a natural number with $n \\equiv 3 \\pmod{60}$ or $n \\equiv 27 \\pmod{60}$ such that $\\gcd(n, p) = 1$ and $\\gcd(n, 5p - 1) = 1$. For natural numbers $x$, $y$, $z$, $n^x + (5p)^y = z^2$ holds if and only if either $x = 0$, $y = 1$ and $z^2 = 5p + 1$, or $x = 1$, $y = 0$ and $z^2 = n + 1$.","title":"Solutions of $n^x + (5p)^y = z^2$ for $n \\equiv 3, 27 \\pmod{60}$ and $p \\equiv 7 \\pmod{12}$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/two-pow-add-fifteen-pow","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.two_pow_add_fifteen_pow","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(2^x + 15^y = z^2\\)"]],"source":"Solutions of $2^x + 15^y = z^2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#two-pow-add-fifteen-pow","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"The natural numbers $x$, $y$, $z$ with $2^x + 15^y = z^2$ are exactly $(x, y, z) = (3, 0, 3)$, $(0, 1, 4)$ and $(6, 2, 17)$.","title":"Solutions of $2^x + 15^y = z^2$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/three-pow-add-thirtyfive-pow","kind":"node","lean":"MathlibExt.NumberTheory.PowAddPowEqSqWanted.three_pow_add_thirtyfive_pow","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(3^x + 35^y = z^2\\)"]],"source":"Solutions of $3^x + 35^y = z^2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#three-pow-add-thirtyfive-pow","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"The natural numbers $x$, $y$, $z$ with $3^x + 35^y = z^2$ are exactly $(x, y, z) = (1, 0, 2)$ and $(x, y, z) = (0, 1, 6)$.","title":"Solutions of $3^x + 35^y = z^2$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/prime-pow-add-one-eq-sq-iff","kind":"node","lean":"MetaMathlibExt.prime_pow_add_one_eq_sq_iff","parent":9423,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#prime-pow-add-one-eq-sq-iff","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime. Then non-negative integers $x$ and $y$ satisfy $p^x + 1 = y^2$ if and only if $(p, x, y) = (2, 3, 3)$ or $(p, x, y) = (3, 1, 2)$.","title":"Prime powers one less than a square"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/one-add-two-pow-mul-pow-eq-sq-iff","kind":"node","lean":"MetaMathlibExt.one_add_two_pow_mul_pow_eq_sq_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squares of the form "],["math","\\(1 + (2^k(2p - 1))^y\\)"],["text"," for prime pairs"]],"source":"Squares of the form $1 + (2^k(2p - 1))^y$ for prime pairs"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#one-add-two-pow-mul-pow-eq-sq-iff","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be an odd prime such that $2p - 1$ is prime, and let $k \\ge 1$ be an integer. Then non-negative integers $y$ and $z$ satisfy $1 + (2^k(2p - 1))^y = z^2$ if and only if $k = 4$, $p = 3$, $y = 1$ and $z = 9$.","title":"Squares of the form $1 + (2^k(2p - 1))^y$ for prime pairs"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/terai-pythagorean-of-prime-of-mod-eight","kind":"node","lean":"MetaMathlibExt.terai_pythagorean_of_prime_of_mod_eight","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terai's conjecture when "],["math","\\(c\\)"],["text"," is prime and "],["math","\\(b \\equiv \\pm 5 \\pmod 8\\)"]],"source":"Terai's conjecture when $c$ is prime and $b \\equiv \\pm 5 \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#terai-pythagorean-of-prime-of-mod-eight","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $a, b, c$ be natural numbers with $a^2 + b^2 = c^2$, $\\gcd(a, b, c) = 1$, $b \\equiv \\pm 5 \\pmod 8$ (that is, $b$ is $3$ or $5$ modulo $8$) and $c$ prime. Then for all positive integers $x, y, z$, the equation $x^2 + b^y = c^z$ holds if and only if $(x, y, z) = (a, 2, 2)$.","title":"Terai's conjecture when $c$ is prime and $b \\equiv \\pm 5 \\pmod 8$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/mordell-curves-two-eighteen-one-six-two","kind":"node","lean":"MetaMathlibExt.mordell_curves_two_eighteen_one_six_two_integer_points","parent":9423,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#mordell-curves-two-eighteen-one-six-two","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"The integer solutions of $y^2 = x^3 - 2$ are exactly $(x, y) = (3, 5)$ and $(3, -5)$, the integer solutions of $y^2 = x^3 - 18$ are exactly $(x, y) = (3, 3)$ and $(3, -3)$, and $y^2 = x^3 - 162$ has no integer solutions.","title":"Integer points on three Mordell curves"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/three-pow-eq-sq-add-two-iff","kind":"node","lean":"MetaMathlibExt.three_pow_eq_sq_add_two_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(3^x = y^2 + 2\\)"]],"source":"Solutions of $3^x = y^2 + 2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#three-pow-eq-sq-add-two-iff","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"For natural numbers $x, y$, we have $3^x = y^2 + 2$ if and only if $(x, y)$ is $(1, 1)$ or $(3, 5)$.","title":"Solutions of $3^x = y^2 + 2$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/two-mul-sq-add-one-eq-three-pow-iff","kind":"node","lean":"MetaMathlibExt.two_mul_sq_add_one_eq_three_pow_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(2x^2 + 1 = 3^y\\)"]],"source":"Solutions of $2x^2 + 1 = 3^y$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#two-mul-sq-add-one-eq-three-pow-iff","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"For natural numbers $x, y$, we have $2x^2 + 1 = 3^y$ if and only if $(x, y)$ is $(0, 0)$, $(1, 1)$, $(2, 2)$ or $(11, 5)$.","title":"Solutions of $2x^2 + 1 = 3^y$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/prime-pair-pow-add-pow-eq-sq-iff","kind":"node","lean":"MetaMathlibExt.prime_pair_pow_add_pow_eq_sq_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squares of the form "],["math","\\(p^x + (2p - 1)^y\\)"],["text"," for prime pairs"]],"source":"Squares of the form $p^x + (2p - 1)^y$ for prime pairs"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#prime-pair-pow-add-pow-eq-sq-iff","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be an odd prime such that $2p - 1$ is prime. Then non-negative integers $x$, $y$ and $z$ satisfy $p^x + (2p - 1)^y = z^2$ if and only if $p = 3$ and $(x, y, z) = (1, 0, 2)$.","title":"Squares of the form $p^x + (2p - 1)^y$ for prime pairs"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/prime-pair-pow-add-two-pow-odd-mul-pow-n","kind":"node","lean":"MetaMathlibExt.prime_pair_pow_add_two_pow_odd_mul_pow_ne_sq","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","No squares of the form "],["math","\\(p^x + (2^{2k+1}(2p - 1))^y\\)"],["text"," for prime pairs with "],["math","\\(p \\equiv 5, 7 \\pmod 8\\)"]],"source":"No squares of the form $p^x + (2^{2k+1}(2p - 1))^y$ for prime pairs with $p \\equiv 5, 7 \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#prime-pair-pow-add-two-pow-odd-mul-pow-n","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"No squares of the form $p^x + (2^{2k+1}(2p - 1))^y$ for prime pairs with $p \\equiv 5, 7 \\pmod 8$. Let $p$ be a prime with $p \\equiv 5 \\pmod 8$ or $p \\equiv 7 \\pmod 8$ such that $2p - 1$ is prime. Then for all non-negative integers $k$, $x$, $y$ and $z$, $p^x + (2^{2k+1}(2p - 1))^y \\ne z^2$.","title":"No squares of the form $p^x + (2^{2k+1}(2p - 1))^y$ for prime pairs with $p \\equiv 5, 7 \\pmod 8$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/prime-pair-pow-add-two-pow-even-mul-pow","kind":"node","lean":"MetaMathlibExt.prime_pair_pow_add_two_pow_even_mul_pow_eq_sq_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squares of the form "],["math","\\(p^x + (2^{2k}(2p - 1))^y\\)"],["text"," for prime pairs"]],"source":"Squares of the form $p^x + (2^{2k}(2p - 1))^y$ for prime pairs"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#prime-pair-pow-add-two-pow-even-mul-pow","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 4$ such that $2p - 1$ is prime, and let $k \\ge 1$ be an integer. Then non-negative integers $x$, $y$ and $z$ satisfy $p^x + (2^{2k}(2p - 1))^y = z^2$ if and only if either $p = 3$ and $(x, y, z) = (1, 0, 2)$, or $p = 3$, $k = 2$ and $(x, y, z) = (0, 1, 9)$.","title":"Squares of the form $p^x + (2^{2k}(2p - 1))^y$ for prime pairs"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/prime-pair-pow-sub-pow-eq-sq-iff","kind":"node","lean":"MetaMathlibExt.prime_pair_pow_sub_pow_eq_sq_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squares of the form "],["math","\\((2p - 1)^x - p^y\\)"],["text"," for prime pairs"]],"source":"Squares of the form $(2p - 1)^x - p^y$ for prime pairs"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#prime-pair-pow-sub-pow-eq-sq-iff","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 4$ such that $2p - 1$ is prime. Then non-negative integers $x$, $y$ and $z$ satisfy $(2p - 1)^x - p^y = z^2$ if and only if $(x, y, z) = (0, 0, 0)$, or $p = 3$ and $(x, y, z) = (2, 2, 4)$, or $p = 2t^2 + 1$ for some non-negative integer $t$ and $(x, y, z) = (1, 0, 2t)$.","title":"Squares of the form $(2p - 1)^x - p^y$ for prime pairs"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/prime-pair-two-pow-even-mul-pow-sub-pow","kind":"node","lean":"MetaMathlibExt.prime_pair_two_pow_even_mul_pow_sub_pow_eq_sq_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squares of the form "],["math","\\((2^{2k}(2p - 1))^x - p^y\\)"],["text"," for prime pairs"]],"source":"Squares of the form $(2^{2k}(2p - 1))^x - p^y$ for prime pairs"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#prime-pair-two-pow-even-mul-pow-sub-pow","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 4$ such that $2p - 1$ is prime, and let $k \\ge 1$ be an integer. Then non-negative integers $x$, $y$ and $z$ satisfy $(2^{2k}(2p - 1))^x - p^y = z^2$ if and only if $(x, y, z) = (0, 0, 0)$.","title":"Squares of the form $(2^{2k}(2p - 1))^x - p^y$ for prime pairs"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/prime-pair-two-pow-odd-mul-pow-sub-pow-e","kind":"node","lean":"MetaMathlibExt.prime_pair_two_pow_odd_mul_pow_sub_pow_eq_sq_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squares of the form "],["math","\\((2^{2k+1}(2p - 1))^x - p^y\\)"],["text"," for prime pairs"]],"source":"Squares of the form $(2^{2k+1}(2p - 1))^x - p^y$ for prime pairs"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#prime-pair-two-pow-odd-mul-pow-sub-pow-e","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 5 \\pmod 8$ such that $2p - 1$ is prime, and let $k$ be a non-negative integer. Then non-negative integers $x$, $y$ and $z$ satisfy $(2^{2k+1}(2p - 1))^x - p^y = z^2$ if and only if $(x, y, z) = (0, 0, 0)$, or $k = 0$, $x = 1$, $y = 0$ and $4p = z^2 + 3$.","title":"Squares of the form $(2^{2k+1}(2p - 1))^x - p^y$ for prime pairs"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/prime-pair-pow-sub-two-pow-odd-mul-pow-e","kind":"node","lean":"MetaMathlibExt.prime_pair_pow_sub_two_pow_odd_mul_pow_eq_sq_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squares of the form "],["math","\\(p^x - (2^{2k+1}(2p - 1))^y\\)"],["text"," for prime pairs"]],"source":"Squares of the form $p^x - (2^{2k+1}(2p - 1))^y$ for prime pairs"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#prime-pair-pow-sub-two-pow-odd-mul-pow-e","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 8$ such that $2p - 1$ is prime, and let $k$ be a non-negative integer. Then non-negative integers $x$, $y$ and $z$ satisfy $p^x - (2^{2k+1}(2p - 1))^y = z^2$ if and only if $(x, y, z) = (0, 0, 0)$.","title":"Squares of the form $p^x - (2^{2k+1}(2p - 1))^y$ for prime pairs"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/two-pow-three-pow-six-pow-eq-sq-solution","kind":"node","lean":"MetaMathlibExt.two_pow_three_pow_six_pow_eq_sq_solutions","parent":9423,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#two-pow-three-pow-six-pow-eq-sq-solution","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"The six equations $(-1)^\\alpha 2^x + (-1)^\\beta (2^k \\cdot 3)^y = z^2$ with $\\alpha, \\beta, k \\in \\{0, 1\\}$ and $\\alpha\\beta = 0$ have exactly the following solutions in natural numbers $x, y, z$, where the equations are read in the integers. The equation $2^x + 3^y = z^2$ holds if and only if $(x, y, z)$ is $(0, 1, 2)$, $(3, 0, 3)$ or $(4, 2, 5)$. The equation $2^x - 3^y = z^2$ holds if and only if $(x, y, z)$ is\u2026","title":"The exponential equations for the prime pair (2, 3)"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/two-add-seven-pow-eq-sq-iff","kind":"node","lean":"MetaMathlibExt.two_add_seven_pow_eq_sq_iff","parent":9423,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(2 + 7^y = z^2\\)"]],"source":"Solutions of $2 + 7^y = z^2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers/index.html#two-add-seven-pow-eq-sq-iff","scope":"NumberTheory/diophantine-equations/exponential-diophantine/squares-as-sums-or-differences-of-powers","status":"can_prove","summary":"For natural numbers $y, z$, we have $2 + 7^y = z^2$ if and only if $(y, z) = (1, 3)$.","title":"Solutions of $2 + 7^y = z^2$"},{"data_url":"scopes/NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture.json","id":"NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture","kind":"scope","navigation":"module","parent":9410,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture/index.html","scope":"NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture","status":"can_prove","title":"Terai's conjecture"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture/terai-ten-fifteen-five-eq-one-one-two","kind":"node","lean":"MetaMathlibExt.terai_ten_fifteen_five_eq_one_one_two","parent":9452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Terai equation "],["math","\\((10m^2 + 1)^x + (15m^2 - 1)^y = (5m)^z\\)"]],"source":"The Terai equation $(10m^2 + 1)^x + (15m^2 - 1)^y = (5m)^z$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture/index.html#terai-ten-fifteen-five-eq-one-one-two","scope":"NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture","status":"can_prove","summary":"For every integer $m > 1$, the only solution of $(10m^2 + 1)^x + (15m^2 - 1)^y = (5m)^z$ in positive integers $x, y, z$ is $(x, y, z) = (1, 1, 2)$.","title":"The Terai equation $(10m^2 + 1)^x + (15m^2 - 1)^y = (5m)^z$"},{"id":"NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture/gauss-class-number-neg-four-mul-lt","kind":"node","lean":"MetaMathlibExt.gaussClassNumber_neg_four_mul_lt","parent":9452,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture/index.html#gauss-class-number-neg-four-mul-lt","scope":"NumberTheory/diophantine-equations/exponential-diophantine/terai-s-conjecture","status":"can_prove","summary":"For every positive integer $D$, the Gauss class number $h(-4D)$, the number of primitive, reduced, positive definite binary quadratic forms of discriminant $-4D$, satisfies $h(-4D) < \\frac{4}{\\pi} \\sqrt{D} \\log(2e\\sqrt{D})$, where $\\log$ is the natural logarithm.","title":"Hua's class number bound"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations","kind":"scope","parent":9409,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations","status":"can_prove","summary":"Pell and negative Pell equations, norm equations, Pythagorean triples, Markov triples and quadratic equations in two or three variables.","title":"Pell and quadratic Diophantine equations"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/dvd-card-connected-component-markoff-gra","kind":"node","lean":"MathlibExt.Combinatorics.MarkoffGraphWanted.Combinatorics.dvd_card_connectedComponent_markoffGraph","parent":9455,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/index.html#dvd-card-connected-component-markoff-gra","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations","status":"can_prove","summary":"Let $p > 3$ be a prime. The Markoff graph over $\\mathbb{Z}/p\\mathbb{Z}$ has as vertices the nonzero triples $(x_0, x_1, x_2)$ of elements of $\\mathbb{Z}/p\\mathbb{Z}$ with $x_0^2 + x_1^2 + x_2^2 = 3 x_0 x_1 x_2$, and two distinct triples are adjacent when one is obtained from the other by a Vieta involution, which replaces one coordinate $x_i$ by $3 x_j x_k - x_i$ and keeps the other two coordinates $x_j$, $x_k$,\u2026","title":"Chen's theorem for Markoff graphs"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/fibonacci-norm-eq-one-or-neg-one-iff","kind":"node","lean":"FibonacciNormEquation.fibonacci_norm_eq_one_or_neg_one_iff","parent":9455,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer solutions of "],["math","\\(x^2 - xy - y^2 = \\pm 1\\)"]],"source":"Integer solutions of $x^2 - xy - y^2 = \\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/index.html#fibonacci-norm-eq-one-or-neg-one-iff","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations","status":"fully_proved","summary":"For integers $x$ and $y$, we have $x^2 - xy - y^2 = 1$ or $x^2 - xy - y^2 = -1$ if and only if there are an integer $n$ and a sign $s \\in \\{1, -1\\}$ with $x = s F_n$ and $y = s F_{n-1}$. Here $(F_n)_{n \\in \\mathbb{Z}}$ is the Fibonacci sequence extended to all integers by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for every integer $n$.","title":"Integer solutions of $x^2 - xy - y^2 = \\pm 1$"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem","kind":"scope","navigation":"module","parent":9455,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem","status":"can_prove","title":"B\u00fcchi's problem"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/is-buchi-sequence","kind":"node","lean":"MathlibExt.NumberTheory.BuchiSquareSequenceBoundWanted.IsBuchiSequence","parent":9458,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/index.html#is-buchi-sequence","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem","status":"fully_proved","summary":"Let $M$ be a natural number. A sequence of integers $u_0, u_1, u_2, \\dots$ is a B\u00fcchi sequence of length $M$ when the second differences of its squares equal $2$ along its first $M$ terms, that is, $u_{j+2}^2 - 2u_{j+1}^2 + u_j^2 = 2$ for every $j$ with $j + 2 < M$.","title":"B\u00fcchi sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/is-trivial","kind":"node","lean":"MathlibExt.NumberTheory.BuchiSquareSequenceBoundWanted.IsTrivial","parent":9458,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/index.html#is-trivial","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem","status":"fully_proved","summary":"Let $M$ be a natural number. A sequence of integers $u_0, u_1, u_2, \\dots$ is trivial at length $M$ when there is an integer $x$ with $u_j^2 = (x + j)^2$ for every $j < M$; equivalently, after changing the signs of some of its terms, its first $M$ terms are the consecutive integers $x, x + 1, \\dots, x + M - 1$.","title":"Trivial sequences in B\u00fcchi's problem"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.BuchiSquareSequenceBoundWanted.conjecture","parent":9458,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/index.html#conjecture","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem","status":"fully_proved","summary":"There is a natural number $M \\ge 5$ such that every integer B\u00fcchi sequence of length $M$ is trivial: for every sequence of integers $u_0, u_1, u_2, \\dots$ with $u_{j+2}^2 - 2u_{j+1}^2 + u_j^2 = 2$ for all $j$ with $j + 2 < M$, there is an integer $x$ with $u_j^2 = (x + j)^2$ for all $j < M$.","title":"B\u00fcchi's square-sequence conjecture"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.BuchiSquareSequenceBoundWanted.conjecture_holds","parent":9458,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem/index.html#conjecture-holds","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/b-chi-s-problem","status":"can_prove","summary":"There is a natural number $M \\ge 5$ such that every integer B\u00fcchi sequence of length $M$ is trivial: for every sequence of integers $u_0, u_1, u_2, \\dots$ with $u_{j+2}^2 - 2u_{j+1}^2 + u_j^2 = 2$ for all $j$ with $j + 2 < M$, there is an integer $x$ with $u_j^2 = (x + j)^2$ for all $j < M$.","title":"Resolution of B\u00fcchi's square-sequence conjecture"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","kind":"scope","navigation":"module","parent":9455,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","title":"Generalized Markov triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-triple","kind":"node","lean":"MetaMathlibExt.IsGenMarkovTriple","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-triple","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For a natural number $k$, a $k$-generalized Markov triple is a triple $(a, b, c)$ of positive integers satisfying the $k$-generalized Markov equation $\\mathrm{GME}(k)$:","title":"Generalized Markov triple"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-triple-one-one-one","kind":"node","lean":"MetaMathlibExt.isGenMarkovTriple_one_one_one","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((1, 1, 1)\\)"],["text"," is a generalized Markov triple"]],"source":"$(1, 1, 1)$ is a generalized Markov triple"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-triple-one-one-one","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For every natural number $k$, the triple $(1, 1, 1)$ is a $k$-generalized Markov triple.","title":"$(1, 1, 1)$ is a generalized Markov triple"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-triple-zero-one-one-two","kind":"node","lean":"MetaMathlibExt.isGenMarkovTriple_zero_one_one_two","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Markov triple "],["math","\\((1, 1, 2)\\)"]],"source":"The Markov triple $(1, 1, 2)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-triple-zero-one-one-two","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"The triple $(1, 1, 2)$ is a $0$-generalized Markov triple, that is, a classical Markov triple: $1^2 + 1^2 + 2^2 = 3 \\cdot 1 \\cdot 1 \\cdot 2$.","title":"The Markov triple $(1, 1, 2)$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-number","kind":"node","lean":"MetaMathlibExt.IsGenMarkovNumber","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-number","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ be a natural number. A natural number $b$ is a $k$-generalized Markov number when there are natural numbers $a$ and $c$ such that $(a, b, c)$ is a $k$-generalized Markov triple.","title":"Generalized Markov number"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/has-unique-max-gen-markov-triple","kind":"node","lean":"MetaMathlibExt.HasUniqueMaxGenMarkovTriple","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#has-unique-max-gen-markov-triple","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ and $b$ be natural numbers. The number $b$ is the maximum of a unique $k$-generalized Markov triple up to order if there is a $k$-generalized Markov triple $(a, b, c)$ with $a \\le b$ and $c \\le b$, and every $k$-generalized Markov triple $(a', b, c')$ with $a' \\le b$ and $c' \\le b$ satisfies $(a', c') = (a, c)$ or $(a', c') = (c, a)$.","title":"Unique maximal generalized Markov triple"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-number-2","kind":"node","lean":"MetaMathlibExt.HasUniqueMaxGenMarkovTriple.isGenMarkovNumber","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-number-2","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ and $b$ be natural numbers. If $b$ is the maximum of a unique $k$-generalized Markov triple up to order, then $b$ is a $k$-generalized Markov number.","title":"A unique maximum is a generalized Markov number"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-congruence-count","kind":"node","lean":"MetaMathlibExt.genMarkovCongruenceCount","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of solutions of "],["math","\\(x^2 + kx + 1 \\equiv 0 \\pmod b\\)"]],"source":"Number of solutions of $x^2 + kx + 1 \\equiv 0 \\pmod b$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-congruence-count","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For natural numbers $k$ and $b$, this is the number of integers $x$ with $0 \\le x < b$ such that $b$ divides $x^2 + kx + 1$. For $b = 0$ it is $0$.","title":"Number of solutions of $x^2 + kx + 1 \\equiv 0 \\pmod b$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-congruence-count-zero-five","kind":"node","lean":"MetaMathlibExt.genMarkovCongruenceCount_zero_five","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence "],["math","\\(x^2 + 1 \\equiv 0 \\pmod 5\\)"],["text"," has two solutions"]],"source":"The congruence $x^2 + 1 \\equiv 0 \\pmod 5$ has two solutions"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-congruence-count-zero-five","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Exactly two integers $x$ with $0 \\le x < 5$ satisfy $x^2 + 1 \\equiv 0 \\pmod 5$.","title":"The congruence $x^2 + 1 \\equiv 0 \\pmod 5$ has two solutions"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/has-unique-max-gen-markov-triple-of-cong","kind":"node","lean":"MetaMathlibExt.hasUniqueMaxGenMarkovTriple_of_congruenceCount_le_two","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniqueness criterion via the congruence "],["math","\\(x^2 + kx + 1 \\equiv 0 \\pmod{b}\\)"]],"source":"Uniqueness criterion via the congruence $x^2 + kx + 1 \\equiv 0 \\pmod{b}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#has-unique-max-gen-markov-triple-of-cong","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number and let $b$ be a $k$-generalized Markov number with $b \\ne 1$ and $b \\ne k + 2$. If the congruence $x^2 + kx + 1 \\equiv 0 \\pmod{b}$ has at most two solutions $x$ with $0 \\le x < b$, then $b$ is the maximum of a unique $k$-generalized Markov triple up to order.","title":"Uniqueness criterion via the congruence $x^2 + kx + 1 \\equiv 0 \\pmod{b}$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/has-unique-max-gen-markov-triple-prime","kind":"node","lean":"MetaMathlibExt.hasUniqueMaxGenMarkovTriple_prime","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Markov numbers "],["math","\\(p\\)"],["text"," and "],["math","\\(2p\\)"],["text"," are unique maxima"]],"source":"Generalized Markov numbers $p$ and $2p$ are unique maxima"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#has-unique-max-gen-markov-triple-prime","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number and $p$ a prime. If $b$ is a $k$-generalized Markov number with $b = p$ or $b = 2p$, then $b$ is the maximum of a unique $k$-generalized Markov triple up to order.","title":"Generalized Markov numbers $p$ and $2p$ are unique maxima"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/has-unique-max-gen-markov-triple-prime-p","kind":"node","lean":"MetaMathlibExt.hasUniqueMaxGenMarkovTriple_prime_pow","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniqueness for prime powers "],["math","\\(p^m\\)"],["text"," and "],["math","\\(2p^m\\)"],["text"," under conditions on "],["math","\\(k\\)"]],"source":"Uniqueness for prime powers $p^m$ and $2p^m$ under conditions on $k$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#has-unique-max-gen-markov-triple-prime-p","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ and $m$ be natural numbers with $m \\ge 2$ and let $p$ be a prime. Suppose that $k = 2$; or that $k \\ge 4$ is even and $p^2$ divides neither $k/2 + 1$ nor $k/2 - 1$; or that $k$ is odd and $p^2$ divides neither $k + 2$ nor $k - 2$ in $\\mathbb{Z}$. If $b$ is a $k$-generalized Markov number with $b = p^m$ or $b = 2p^m$, then $b$ is the maximum of a unique $k$-generalized Markov triple up to order.","title":"Uniqueness for prime powers $p^m$ and $2p^m$ under conditions on $k$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/has-unique-max-gen-markov-triple-odd-pri","kind":"node","lean":"MetaMathlibExt.hasUniqueMaxGenMarkovTriple_odd_prime_pow","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniqueness for odd prime powers "],["math","\\(p^m\\)"],["text"," and "],["math","\\(2p^m\\)"],["text"," with "],["math","\\(p \\nmid k^2 - 4\\)"]],"source":"Uniqueness for odd prime powers $p^m$ and $2p^m$ with $p \\nmid k^2 - 4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#has-unique-max-gen-markov-triple-odd-pri","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ and $m$ be natural numbers with $m \\ge 2$, and let $p$ be an odd prime that does not divide $k^2 - 4$ in $\\mathbb{Z}$. If $b$ is a $k$-generalized Markov number with $b = p^m$ or $b = 2p^m$, then $b$ is the maximum of a unique $k$-generalized Markov triple up to order.","title":"Uniqueness for odd prime powers $p^m$ and $2p^m$ with $p \\nmid k^2 - 4$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/card-max-gen-markov-triples-le","kind":"node","lean":"MetaMathlibExt.card_maxGenMarkovTriples_le","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#card-max-gen-markov-triples-le","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number such that either $k \\ge 4$ is even and $k/2 + 1$ and $k/2 - 1$ are squarefree, or $k$ is odd and the integers $k + 2$ and $k - 2$ are squarefree. Let $b$ be a $k$-generalized Markov number that is not divisible by $4$, and suppose that the number $n$ of odd primes dividing $b$ is at least $1$; equivalently, $b = p_1^{a_1} \\cdots p_n^{a_n}$ or $b = 2 p_1^{a_1} \\cdots p_n^{a_n}$ for\u2026","title":"Bound on the number of generalized Markov triples with a given maximum"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/pairwise-coprime","kind":"node","lean":"MetaMathlibExt.IsGenMarkovTriple.pairwise_coprime","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#pairwise-coprime","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"If $(a, b, c)$ is a $k$-generalized Markov triple, then $\\gcd(a, b) = \\gcd(b, c) = \\gcd(a, c) = 1$.","title":"Generalized Markov triples are pairwise coprime"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-iff-gen-second-markov","kind":"node","lean":"MetaMathlibExt.genMarkov_iff_genSecondMarkov","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-iff-gen-second-markov","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k \\in \\mathbb{N}$ and let $a, b, c$ be rational numbers. Put $x = (3 + 3k)a - k$, $y = (3 + 3k)b - k$ and $z = (3 + 3k)c - k$. Then $a^2 + b^2 + c^2 + k(bc + ca + ab) = (3 + 3k)abc$ if and only if $x^2 + y^2 + z^2 + (k^2 + 2k)(x + y + z) + 2k^3 + 3k^2 = xyz$. The first equation is the $k$-generalized Markov equation $\\mathrm{GME}(k)$ and the second the $k$-generalized second Markov equation $\\mathrm{GSME}(k)$.","title":"The generalized Markov equation and the generalized second Markov equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-second-markov-vieta","kind":"node","lean":"MetaMathlibExt.genSecondMarkov_vieta","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-second-markov-vieta","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number and let $a, b, c$ be integers with","title":"Vieta jumping for the second generalized Markov equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/special-linear-group-two-trace-identitie","kind":"node","lean":"MetaMathlibExt.specialLinearGroup_two_trace_identities","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trace identities in "],["math","\\(\\mathrm{SL}(2, \\mathbb{Z})\\)"]],"source":"Trace identities in $\\mathrm{SL}(2, \\mathbb{Z})$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#special-linear-group-two-trace-identitie","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"For all $A, B \\in \\mathrm{SL}(2, \\mathbb{Z})$ we have $\\operatorname{tr} A = \\operatorname{tr} A^{-1}$, $\\operatorname{tr}(AB) = \\operatorname{tr} A \\cdot \\operatorname{tr} B - \\operatorname{tr}(AB^{-1})$, and $A^2 = (\\operatorname{tr} A) A - I$, where $I$ is the identity matrix.","title":"Trace identities in $\\mathrm{SL}(2, \\mathbb{Z})$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-congruence-count-prime-le-two","kind":"node","lean":"MetaMathlibExt.genMarkovCongruenceCount_prime_le_two","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most two solutions of "],["math","\\(x^2 + kx + 1 \\equiv 0\\)"],["text"," modulo a prime"]],"source":"At most two solutions of $x^2 + kx + 1 \\equiv 0$ modulo a prime"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-congruence-count-prime-le-two","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"For every prime $p$ and every $k \\in \\mathbb{N}$, at most two integers $x$ with $0 \\le x < p$ satisfy $x^2 + kx + 1 \\equiv 0 \\pmod p$.","title":"At most two solutions of $x^2 + kx + 1 \\equiv 0$ modulo a prime"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-congruence-count-prime-pow-le","kind":"node","lean":"MetaMathlibExt.genMarkovCongruenceCount_prime_pow_le_two","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most two solutions of "],["math","\\(x^2 + kx + 1 \\equiv 0\\)"],["text"," modulo an odd prime power"]],"source":"At most two solutions of $x^2 + kx + 1 \\equiv 0$ modulo an odd prime power"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-congruence-count-prime-pow-le","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $p$ be an odd prime, let $m \\ge 2$ and let $k \\in \\mathbb{N}$ with $k^2 - 4 \\not\\equiv 0 \\pmod p$. Then at most two integers $x$ with $0 \\le x < p^m$ satisfy $x^2 + kx + 1 \\equiv 0 \\pmod{p^m}$.","title":"At most two solutions of $x^2 + kx + 1 \\equiv 0$ modulo an odd prime power"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/even-of-even-gen-markov-number","kind":"node","lean":"MetaMathlibExt.even_of_even_genMarkovNumber","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","An even generalized Markov number forces "],["math","\\(k\\)"],["text"," to be even"]],"source":"An even generalized Markov number forces $k$ to be even"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#even-of-even-gen-markov-number","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number. If some $k$-generalized Markov number $b$ is even, then $k$ is even.","title":"An even generalized Markov number forces $k$ to be even"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/mod-four-eq-two-of-four-dvd-gen-markov-n","kind":"node","lean":"MetaMathlibExt.mod_four_eq_two_of_four_dvd_genMarkovNumber","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Markov numbers divisible by "],["math","\\(4\\)"]],"source":"Generalized Markov numbers divisible by $4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#mod-four-eq-two-of-four-dvd-gen-markov-n","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number. If some $k$-generalized Markov number is divisible by $4$, then $k \\equiv 2 \\pmod{4}$.","title":"Generalized Markov numbers divisible by $4$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/not-dvd-sq-sub-four-of-large-prime","kind":"node","lean":"MetaMathlibExt.not_dvd_sq_sub_four_of_large_prime","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Large odd primes do not divide "],["math","\\(k^2 - 4\\)"]],"source":"Large odd primes do not divide $k^2 - 4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#not-dvd-sq-sub-four-of-large-prime","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $p$ be an odd prime and $k$ a natural number. If either $k \\ge 4$ is even and $p > k/2 + 1$, or $k$ is odd and $p > k + 2$, then $p$ does not divide $k^2 - 4$.","title":"Large odd primes do not divide $k^2 - 4$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/not-dvd-sq-sub-four-of-gen-markov-number","kind":"node","lean":"MetaMathlibExt.not_dvd_sq_sub_four_of_genMarkovNumber_prime_pow","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#not-dvd-sq-sub-four-of-gen-markov-number","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $p$ be an odd prime and $m \\ge 2$, and suppose that some $k$-generalized Markov number $b$ equals $p^m$ or $2p^m$. Suppose moreover that either $k \\ge 4$ is even and $p^2$ divides neither $k/2 + 1$ nor $k/2 - 1$, or $k$ is odd and $p^2$ divides neither $k + 2$ nor $k - 2$. Then $p$ does not divide $k^2 - 4$.","title":"Discriminant condition for prime-power generalized Markov numbers"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-triple-two-iff-sq","kind":"node","lean":"MetaMathlibExt.genMarkovTriple_two_iff_sq","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-triple-two-iff-sq","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"If $(a, b, c)$ is a Markov triple, that is, a $0$-generalized Markov triple, then $(a^2, b^2, c^2)$ is a $2$-generalized Markov triple. Conversely, for every $2$-generalized Markov triple $(A, B, C)$ there is a Markov triple $(a, b, c)$ with $A = a^2$, $B = b^2$ and $C = c^2$.","title":"2-generalized Markov triples are squares of Markov triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/markov-has-unique-max-triple-of-congruen","kind":"node","lean":"MetaMathlibExt.markov_hasUniqueMaxTriple_of_congruenceCount_le_two","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#markov-has-unique-max-triple-of-congruen","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $b \\ne 1, 2$ be a Markov number (a $0$-generalized Markov number). If the congruence $x^2 + 1 \\equiv 0 \\pmod{b}$ has at most two solutions $x$ with $0 \\le x < b$, then $b$ is the maximum of a unique Markov triple up to order.","title":"Uniqueness criterion for classical Markov numbers"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/markov-has-unique-max-triple-odd-prime-p","kind":"node","lean":"MetaMathlibExt.markov_hasUniqueMaxTriple_odd_prime_pow","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniqueness for Markov numbers "],["math","\\(p^m\\)"],["text"," and "],["math","\\(2p^m\\)"],["text"," with "],["math","\\(p\\)"],["text"," an odd prime"]],"source":"Uniqueness for Markov numbers $p^m$ and $2p^m$ with $p$ an odd prime"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#markov-has-unique-max-triple-odd-prime-p","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $p$ be an odd prime and $m \\ge 1$. If $b = p^m$ or $b = 2p^m$ is a Markov number (a $0$-generalized Markov number), then $b$ is the maximum of a unique Markov triple up to order.","title":"Uniqueness for Markov numbers $p^m$ and $2p^m$ with $p$ an odd prime"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/markov-eq-of-characteristic-number-eq","kind":"node","lean":"MetaMathlibExt.markov_eq_of_characteristic_number_eq","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#markov-eq-of-characteristic-number-eq","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $m \\ge 5$ and let $(m, m_2, m_3)$ and $(m, n_2, n_3)$ be Markov triples (that is, $0$-generalized Markov triples) with $m_2, m_3, n_2, n_3 \\le m$. Suppose some integer $x$ with $0 < x < m$ satisfies $m_2 x \\equiv \\pm m_3 \\pmod{m}$ and $n_2 x \\equiv \\pm n_3 \\pmod{m}$, where the two signs are chosen independently. Then $(m_2, m_3) = (n_2, n_3)$ or $(m_2, m_3) = (n_3, n_2)$.","title":"Markov triples with a common characteristic number agree up to order"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-vieta","kind":"node","lean":"MetaMathlibExt.genMarkovVieta","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-vieta","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For a natural number $k$, the left Vieta jumping sends a triple $(a, b, c)$ of natural numbers to $(a, \\lfloor (a^2 + kab + b^2)/c \\rfloor, b)$ and the right Vieta jumping sends it to $(b, \\lfloor (b^2 + kbc + c^2)/a \\rfloor, c)$, where division by $0$ gives $0$. The left jumping is indexed by the boolean value false and the right one by true.","title":"Vieta jumping of generalized Markov triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/wide-gen-markov-tree","kind":"node","lean":"MetaMathlibExt.wideGenMarkovTree","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#wide-gen-markov-tree","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Fix a natural number $k$. The wide $k$-generalized Markov tree $\\mathrm{WMT}(k)$ is the binary tree of triples of natural numbers with root $(1, 1, 1)$ in which a vertex $(a, b, c)$ has left child $(a, \\lfloor (a^2 + kab + b^2)/c \\rfloor, b)$ and right child $(b, \\lfloor (b^2 + kbc + c^2)/a \\rfloor, c)$, the left and right Vieta jumpings. For a path $w$, a finite sequence of left and right steps, the vertex of\u2026","title":"The wide generalized Markov tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-tree","kind":"node","lean":"MetaMathlibExt.genMarkovTree","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-tree","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For a natural number $k$ and a finite sequence $w$ of left and right steps, the vertex of the $k$-generalized Markov tree $MT(k)$ at $w$ is the vertex of the wide $k$-generalized Markov tree $WMT(k)$ reached from its root by one left step followed by the steps of $w$ in order. Thus $MT(k)$ is the full subtree of $WMT(k)$ rooted at the left child of the root.","title":"Generalized Markov tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/wide-gen-markov-tree-append-singleton","kind":"node","lean":"MetaMathlibExt.wideGenMarkovTree_append_singleton","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#wide-gen-markov-tree-append-singleton","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For every natural number $k$, every path $w$ and every direction $d$ (left or right), the vertex of the wide $k$-generalized Markov tree $\\mathrm{WMT}(k)$ at the path $w$ followed by one step in direction $d$ is the image of the vertex at $w$ under the Vieta jumping in direction $d$.","title":"Children in the wide generalized Markov tree are Vieta jumpings"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-tree-nil","kind":"node","lean":"MetaMathlibExt.genMarkovTree_nil","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-tree-nil","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For every natural number $k$, the root of the $k$-generalized Markov tree $MT(k)$, its vertex at the empty path, is $(1, k + 2, 1)$.","title":"Root of the generalized Markov tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-tree-zero-false","kind":"node","lean":"MetaMathlibExt.genMarkovTree_zero_false","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-tree-zero-false","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For $k = 0$, the vertex of the Markov tree $MT(0)$ reached from the root by one left step is $(1, 5, 2)$.","title":"Left child of the root of the classical Markov tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/wide-gen-markov-tree-is-gen-markov-tripl","kind":"node","lean":"MetaMathlibExt.wideGenMarkovTree_isGenMarkovTriple_and_twice","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#wide-gen-markov-tree-is-gen-markov-tripl","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number. Every vertex of the wide $k$-generalized Markov tree $\\mathrm{WMT}(k)$ is a $k$-generalized Markov triple. Moreover, every $k$-generalized Markov triple $(a, b, c)$ with $a < b$ and $c < b$ is the vertex of $\\mathrm{WMT}(k)$ at exactly one path that starts with a left step, and at exactly one path that starts with a right step.","title":"Generalized Markov triples in the wide Markov tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-tree-is-gen-markov-triple-and","kind":"node","lean":"MetaMathlibExt.genMarkovTree_isGenMarkovTriple_and_once","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-tree-is-gen-markov-triple-and","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"For every natural number $k$, every vertex of the $k$-generalized Markov tree $MT(k)$ is a $k$-generalized Markov triple, and for every $k$-generalized Markov triple $(a, b, c)$ with $a < b$ and $c < b$ there is exactly one path $w$ such that the vertex of $MT(k)$ at $w$ is $(a, b, c)$.","title":"Vertices of the generalized Markov tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/vieta-eq","kind":"node","lean":"MetaMathlibExt.IsGenMarkovTriple.vieta_eq","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#vieta-eq","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"If $(a, b, c)$ is a $k$-generalized Markov triple, then, as integers, $a^2 + kab + b^2 = c\\,((3 + 3k)ab - c - k(a + b))$ and $b^2 + kbc + c^2 = a\\,((3 + 3k)bc - a - k(b + c))$.","title":"Vieta jumping identities for generalized Markov triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-induced-gen-second-markov-solution","kind":"node","lean":"MetaMathlibExt.IsInducedGenSecondMarkovSolution","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-induced-gen-second-markov-solution","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For a natural number $k$, a triple $(a, b, c)$ of integers is an induced positive solution of the $k$-generalized second Markov equation $\\mathrm{GSME}(k)$ when $a, b, c > 0$,","title":"Induced solution of the generalized second Markov equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-induced-gen-second-markov-solution-ro","kind":"node","lean":"MetaMathlibExt.isInducedGenSecondMarkovSolution_root","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-induced-gen-second-markov-solution-ro","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For every natural number $k$, the triple $(3 + 2k, 3 + 2k, 3 + 2k)$, the image of $(1, 1, 1)$ under $x \\mapsto (3 + 3k)x - k$, is an induced positive solution of the $k$-generalized second Markov equation $\\mathrm{GSME}(k)$.","title":"The root induced solution of the generalized second Markov equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/vieta","kind":"node","lean":"MetaMathlibExt.IsInducedGenSecondMarkovSolution.vieta","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#vieta","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"If $(a, b, c)$ is an induced solution of the $k$-generalized second Markov equation $\\mathrm{GSME}(k)$, then so are $(a, ab - c - k^2 - 2k, b)$ and $(b, bc - a - k^2 - 2k, c)$.","title":"Vieta jumping preserves induced solutions of the generalized second Markov equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-second-markov-tree-induced-and-once","kind":"node","lean":"MetaMathlibExt.genSecondMarkovTree_induced_and_once","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-second-markov-tree-induced-and-once","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"For a natural number $k$, let $SMT(k)$ be the tree obtained from the $k$-generalized Markov tree $MT(k)$ by replacing each vertex $(a, b, c)$ with the integer triple $((3 + 3k)a - k, (3 + 3k)b - k, (3 + 3k)c - k)$. Every vertex of $SMT(k)$ is an induced solution of the $k$-generalized second Markov equation, and for every induced solution $(a, b, c)$ with $a < b$ and $c < b$ there is exactly one path $w$ such that\u2026","title":"Vertices of the second generalized Markov tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-s","kind":"node","lean":"MetaMathlibExt.genCohnS","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(S\\)"],["text"," of the generalized Cohn triple relation"]],"source":"The matrix $S$ of the generalized Cohn triple relation"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-s","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For a natural number $k$, $S$ is the $2 \\times 2$ integer matrix","title":"The matrix $S$ of the generalized Cohn triple relation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-s-zero","kind":"node","lean":"MetaMathlibExt.genCohnS_zero","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(S\\)"],["text"," vanishes for "],["math","\\(k = 0\\)"]],"source":"The matrix $S$ vanishes for $k = 0$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-s-zero","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For $k = 0$, the matrix $S$ of the generalized Cohn triple relation is the zero $2 \\times 2$ matrix, so the relation $Q = PR - S$ reduces to the classical Cohn relation $Q = PR$.","title":"The matrix $S$ vanishes for $k = 0$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-cohn-matrix","kind":"node","lean":"MetaMathlibExt.IsGenCohnMatrix","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-cohn-matrix","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ be a natural number. An integer $2 \\times 2$ matrix $P = (p_{ij})$ is a $k$-generalized Cohn matrix if $\\det P = 1$, its entry $p_{12}$ is a $k$-generalized Markov number, and its trace equals $(3 + 3k) p_{12} - k$.","title":"Generalized Cohn matrices"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-cohn-triple","kind":"node","lean":"MetaMathlibExt.IsGenCohnTriple","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-cohn-triple","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ be a natural number and let $S$ be the integer matrix with rows $(k, 0)$ and $(3k^2 + 3k, k)$. A triple $(P, Q, R)$ of integer $2 \\times 2$ matrices is a $k$-generalized Cohn triple associated with natural numbers $(a, b, c)$ when $P$, $Q$ and $R$ are $k$-generalized Cohn matrices, $Q = PR - S$, the $(1, 2)$-entries of $P$, $Q$ and $R$ are $a$, $b$ and $c$ respectively, and $(a, b, c)$ is a $k$-generalized\u2026","title":"Generalized Cohn triple"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-triple-2","kind":"node","lean":"MetaMathlibExt.IsGenCohnTriple.isGenMarkovTriple","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-triple-2","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ be a natural number. If $(P, Q, R)$ is a $k$-generalized Cohn triple associated with $(a, b, c)$, then $(a, b, c)$ is a $k$-generalized Markov triple.","title":"Generalized Cohn triples give generalized Markov triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-p1","kind":"node","lean":"MetaMathlibExt.genCohnP1","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(P_{1;\\ell}\\)"]],"source":"The matrix $P_{1;\\ell}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-p1","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For a natural number $k$ and an integer $\\ell$, $P_{1;\\ell}$ is the integer $2 \\times 2$ matrix with first row $(\\ell, 1)$ and second row $(-\\ell^2 + 2k\\ell + 3\\ell - 1, -\\ell + 2k + 3)$.","title":"The matrix $P_{1;\\ell}$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-q1","kind":"node","lean":"MetaMathlibExt.genCohnQ1","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(Q_{1;\\ell}\\)"]],"source":"The matrix $Q_{1;\\ell}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-q1","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For a natural number $k$ and an integer $\\ell$, $Q_{1;\\ell}$ is the $2 \\times 2$ integer matrix","title":"The matrix $Q_{1;\\ell}$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-r1","kind":"node","lean":"MetaMathlibExt.genCohnR1","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(R_{1;\\ell}\\)"]],"source":"The matrix $R_{1;\\ell}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-r1","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For a natural number $k$ and an integer $\\ell$, $R_{1;\\ell}$ is the $2 \\times 2$ integer matrix","title":"The matrix $R_{1;\\ell}$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-pqr1-det","kind":"node","lean":"MetaMathlibExt.genCohnPQR1_det","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrices "],["math","\\(P_{1;\\ell}\\)"],["text",", "],["math","\\(Q_{1;\\ell}\\)"],["text"," and "],["math","\\(R_{1;\\ell}\\)"],["text"," have determinant "],["math","\\(1\\)"]],"source":"The matrices $P_{1;\\ell}$, $Q_{1;\\ell}$ and $R_{1;\\ell}$ have determinant $1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-pqr1-det","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For every natural number $k$ and every integer $\\ell$, the $2 \\times 2$ integer matrices $P_{1;\\ell}$, $Q_{1;\\ell}$ and $R_{1;\\ell}$ (which depend on $k$) all have determinant $1$.","title":"The matrices $P_{1;\\ell}$, $Q_{1;\\ell}$ and $R_{1;\\ell}$ have determinant $1$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-step","kind":"node","lean":"MetaMathlibExt.genCohnStep","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-step","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Fix a natural number $k$ and let $S$ be the matrix of the $k$-generalized Cohn triple relation. On triples $(P, Q, R)$ of $2 \\times 2$ integer matrices, the left step (encoded by the Boolean false) sends $(P, Q, R)$ to its left child $(P, PQ - S, Q)$, and the right step (encoded by true) sends it to its right child $(Q, QR - S, R)$.","title":"One step of the wide generalized Cohn tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-path","kind":"node","lean":"MetaMathlibExt.genCohnPath","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-path","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Fix a natural number $k$ and let $S$ be the integer $2 \\times 2$ matrix with rows $(k, 0)$ and $(3k^2 + 3k, k)$. For a triple $T$ of integer $2 \\times 2$ matrices and a path $w$, a finite sequence of left and right moves, this is the triple obtained from $T$ by applying the moves of $w$ in order, where a left move sends $(P, Q, R)$ to $(P, PQ - S, Q)$ and a right move sends $(P, Q, R)$ to $(Q, QR - S, R)$.","title":"Paths of generalized Cohn steps"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/wide-gen-cohn-tree","kind":"node","lean":"MetaMathlibExt.wideGenCohnTree","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#wide-gen-cohn-tree","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Fix a natural number $k$ and an integer $\\ell$, and let $S = \\begin{pmatrix} k & 0 \\\\ 3k^2 + 3k & k \\end{pmatrix}$. The wide $k$-generalized Cohn tree $\\mathrm{WGCT}(k, \\ell)$ is the binary tree of triples of $2 \\times 2$ integer matrices whose root is $(P_{1;\\ell}, Q_{1;\\ell}, R_{1;\\ell})$, where","title":"The wide generalized Cohn tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-tree","kind":"node","lean":"MetaMathlibExt.genCohnTree","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-generalized Cohn tree"]],"source":"The $k$-generalized Cohn tree"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-tree","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$ and $\\ell \\in \\mathbb{Z}$, the $k$-generalized Cohn tree $\\mathrm{GCT}(k, \\ell)$ is the full subtree of the wide $k$-generalized Cohn tree $\\mathrm{WGCT}(k, \\ell)$ rooted at the left child of its root. For a finite sequence $w$ of left and right moves, this is the vertex of $\\mathrm{GCT}(k, \\ell)$ reached from its root along $w$, that is, the vertex of $\\mathrm{WGCT}(k, \\ell)$ reached from\u2026","title":"The $k$-generalized Cohn tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/wide-gen-cohn-tree-append-singleton","kind":"node","lean":"MetaMathlibExt.wideGenCohnTree_append_singleton","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#wide-gen-cohn-tree-append-singleton","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ be a natural number, $\\ell$ an integer, and $S$ the integer matrix with rows $(k, 0)$ and $(3k^2 + 3k, k)$. For every path $w$ in the wide $k$-generalized Cohn tree $\\mathrm{WGCT}(k, \\ell)$, if $(P, Q, R)$ is the vertex at $w$, then the vertex at $w$ followed by a left step is $(P, PQ - S, Q)$ and the vertex at $w$ followed by a right step is $(Q, QR - S, R)$; that is, each child is obtained from its parent\u2026","title":"Children in the wide generalized Cohn tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/cohn-triple-reachable","kind":"node","lean":"MetaMathlibExt.cohnTriple_reachable","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#cohn-triple-reachable","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Call a $0$-generalized Cohn triple a Cohn triple; for $k = 0$ the matrix $S$ of the generalized Cohn relation is zero, so the relation reads $Q = PR$. Call $(P, Q, R) \\mapsto (P, PQ, Q)$ the left step and $(P, Q, R) \\mapsto (Q, QR, R)$ the right step. First, if $(P, Q, R)$ is a Cohn triple associated with $(a, b, c)$ and $a \\le b$, $c \\le b$, then $(P, Q, R)$ is obtained by applying finitely many left and right\u2026","title":"Cohn steps reach every classical Cohn triple whose middle entry is largest"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-triple-reachable","kind":"node","lean":"MetaMathlibExt.genCohnTriple_reachable","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-triple-reachable","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k \\in \\mathbb{N}$ and let $S$ be the integer matrix with rows $(k, 0)$ and $(3k^2 + 3k, k)$. Call $(P, Q, R) \\mapsto (P, PQ - S, Q)$ the left step and $(P, Q, R) \\mapsto (Q, QR - S, R)$ the right step. First, if $(P, Q, R)$ is a $k$-generalized Cohn triple associated with $(a, b, c)$ and $a \\le b$, $c \\le b$, then $(P, Q, R)$ is obtained by applying finitely many left and right steps to some $k$-generalized\u2026","title":"Cohn steps reach every generalized Cohn triple whose middle entry is largest"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-cohn-triple-one-iff","kind":"node","lean":"MetaMathlibExt.isGenCohnTriple_one_iff","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Cohn triples associated with "],["math","\\((1, 1, 1)\\)"]],"source":"Generalized Cohn triples associated with $(1, 1, 1)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-cohn-triple-one-iff","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number. For an integer $\\ell$, let $P_{1;\\ell}$ be the integer matrix with rows $(\\ell, 1)$ and $(-\\ell^2 + 2k\\ell + 3\\ell - 1, -\\ell + 2k + 3)$, let $Q_{1;\\ell}$ be the one with rows $(k + \\ell + 1, 1)$ and $(k^2 - \\ell^2 + 3k + \\ell + 1, k - \\ell + 2)$, and let $R_{1;\\ell}$ be the one with rows $(2k + \\ell + 2, 1)$ and $(-\\ell^2 - 2k\\ell + 2k - \\ell + 1, -\\ell + 1)$. Then for every integer\u2026","title":"Generalized Cohn triples associated with $(1, 1, 1)$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/step","kind":"node","lean":"MetaMathlibExt.IsGenCohnTriple.step","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#step","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number, let $S$ be the integer matrix with rows $(k, 0)$ and $(3k^2 + 3k, k)$, and let $(P, Q, R)$ be a $k$-generalized Cohn triple associated with $(a, b, c)$. Then $(P, PQ - S, Q)$ is a $k$-generalized Cohn triple associated with $(a, \\lfloor (a^2 + kab + b^2)/c \\rfloor, b)$, and $(Q, QR - S, R)$ is a $k$-generalized Cohn triple associated with $(b, \\lfloor (b^2 + kbc + c^2)/a \\rfloor, c)$.","title":"Children of a generalized Cohn triple"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/mul-e-mul","kind":"node","lean":"MetaMathlibExt.IsGenCohnMatrix.mul_E_mul","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#mul-e-mul","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number, let $M$ be a $k$-generalized Cohn matrix, and let $E$ be the $2 \\times 2$ integer matrix whose only nonzero entry is $3 + 3k$, in row $2$ and column $1$. Then $MEM = (\\operatorname{tr} M + k)M + E$ and $M^{-1} E M^{-1} = -(\\operatorname{tr} M^{-1} + k)M^{-1} + E$.","title":"A product identity for generalized Cohn matrices"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/wide-gen-cohn-tree-is-gen-cohn-triple","kind":"node","lean":"MetaMathlibExt.wideGenCohnTree_isGenCohnTriple","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#wide-gen-cohn-tree-is-gen-cohn-triple","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number. For every integer $\\ell$ and every path $w$, the vertex of the wide $k$-generalized Cohn tree $\\mathrm{WGCT}(k, \\ell)$ at $w$ is a $k$-generalized Cohn triple associated with the vertex of the wide $k$-generalized Markov tree $\\mathrm{WMT}(k)$ at $w$. Moreover, every $k$-generalized Markov triple $(a, b, c)$ with $a \\le b$ and $c \\le b$ has an associated $k$-generalized Cohn triple.","title":"Vertices of the wide generalized Cohn tree are generalized Cohn triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/mem-wide-gen-cohn-tree","kind":"node","lean":"MetaMathlibExt.IsGenCohnTriple.mem_wideGenCohnTree","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#mem-wide-gen-cohn-tree","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number and let $(P, Q, R)$ be a $k$-generalized Cohn triple associated with $(a, b, c)$, where $a \\le b$ and $c \\le b$. Then there are an integer $\\ell$ and a path $w$ such that $(P, Q, R)$ is the vertex of the wide $k$-generalized Cohn tree $\\mathrm{WGCT}(k, \\ell)$ reached along $w$.","title":"Generalized Cohn triples with largest middle entry lie in wide generalized Cohn trees"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/parent","kind":"node","lean":"MetaMathlibExt.IsGenCohnTriple.parent","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#parent","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number and let $S$ be the integer matrix with rows $(k, 0)$ and $(3k^2 + 3k, k)$. Let $(P, Q, R)$ be a $k$-generalized Cohn triple associated with $(a, b, c)$, and let $b' = \\lfloor (a^2 + kac + c^2)/b \\rfloor$. Then $(P, R, P^{-1}(R + S))$ is a $k$-generalized Cohn triple associated with $(a, c, b')$, and $((P + S)R^{-1}, P, R)$ is a $k$-generalized Cohn triple associated with $(b', a, c)$.","title":"Parent triples of a generalized Cohn triple"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/wide-gen-cohn-tree-snd-injective","kind":"node","lean":"MetaMathlibExt.wideGenCohnTree_snd_injective","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#wide-gen-cohn-tree-snd-injective","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"For every natural number $k$ and every integer $\\ell$, if the vertices $(P, Q, R)$ and $(P', Q', R')$ of the wide $k$-generalized Cohn tree $\\mathrm{WGCT}(k, \\ell)$ at paths $w$ and $w'$ (finite sequences of left and right steps) satisfy $Q = Q'$, then $w = w'$.","title":"Distinct paths give distinct middle matrices in the wide generalized Cohn tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-tree-is-gen-cohn-triple","kind":"node","lean":"MetaMathlibExt.genCohnTree_isGenCohnTriple","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-tree-is-gen-cohn-triple","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"For every natural number $k$, every integer $\\ell$ and every path $w$ (a finite sequence of left and right steps), the vertex $(P, Q, R)$ of the $k$-generalized Cohn tree $\\mathrm{GCT}(k, \\ell)$ at $w$ is a $k$-generalized Cohn triple associated with the vertex $(a, b, c)$ of the $k$-generalized Markov tree $\\mathrm{MT}(k)$ at the same path $w$.","title":"Vertices of the generalized Cohn tree are Cohn triples over the Markov tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-tree-snd-injective","kind":"node","lean":"MetaMathlibExt.genCohnTree_snd_injective","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-tree-snd-injective","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"For every natural number $k$ and every integer $\\ell$, vertices of the $k$-generalized Cohn tree $\\mathrm{GCT}(k, \\ell)$ at different paths have different middle matrices.","title":"Middle matrices of the generalized Cohn tree are distinct"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/exists-unique-mem-gen-cohn-tree","kind":"node","lean":"MetaMathlibExt.IsGenCohnTriple.existsUnique_mem_genCohnTree","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#exists-unique-mem-gen-cohn-tree","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $(P, Q, R)$ be a $k$-generalized Cohn triple associated with $(a, b, c)$, where $b > \\max\\{a, c\\}$. Then there is exactly one pair $(\\ell, w)$ of an integer $\\ell$ and a path $w$ (a finite sequence of left and right moves) such that $(P, Q, R)$ is the vertex of the $k$-generalized Cohn tree $\\mathrm{GCT}(k, \\ell)$ reached along $w$.","title":"Cohn triples sit at a unique vertex of a Cohn tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/farey-tree","kind":"node","lean":"MetaMathlibExt.fareyTree","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#farey-tree","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Write a fraction $n/d$, with $n$ and $d$ natural numbers, for the pair $(n, d)$, so that $1/0$ stands for $\\infty$. The Farey tree $\\mathrm{FT}$ is the infinite binary tree of triples of fractions with root $(0/1, 1/1, 1/0)$ in which a vertex $(a/b, c/d, e/f)$ has left child $(a/b, (a+c)/(b+d), c/d)$ and right child $(c/d, (c+e)/(d+f), e/f)$. For a path $w$, a finite sequence of left and right moves, this gives the\u2026","title":"The Farey tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/farey-tree-false","kind":"node","lean":"MetaMathlibExt.fareyTree_false","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#farey-tree-false","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"The left child of the root $(0/1, 1/1, 1/0)$ of the Farey tree is $(0/1, 1/2, 1/1)$.","title":"Left child of the root of the Farey tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-farey-triple","kind":"node","lean":"MetaMathlibExt.IsFareyTriple","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-farey-triple","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Write a pair $(a, b)$ of natural numbers as the fraction $a/b$, so that $1/0$ is allowed. Three such fractions $(a/b, c/d, e/f)$ form a Farey triple if $\\gcd(a, b) = \\gcd(c, d) = \\gcd(e, f) = 1$ and","title":"Farey triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-farey-triple-root","kind":"node","lean":"MetaMathlibExt.isFareyTriple_root","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The triple "],["math","\\((0/1, 1/1, 1/0)\\)"],["text"," is a Farey triple"]],"source":"The triple $(0/1, 1/1, 1/0)$ is a Farey triple"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-farey-triple-root","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"The fractions $0/1$, $1/1$ and $1/0$, given as the pairs of natural numbers $(0, 1)$, $(1, 1)$ and $(1, 0)$, form a Farey triple.","title":"The triple $(0/1, 1/1, 1/0)$ is a Farey triple"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/farey-tree-properties","kind":"node","lean":"MetaMathlibExt.fareyTree_properties","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#farey-tree-properties","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Write a fraction $a/b$, with $a$ and $b$ natural numbers, for the pair $(a, b)$, so that $1/0$ stands for $\\infty$, and compare fractions by cross multiplication: $a/b < c/d$ means $ad < cb$. (1) If $(a/b, c/d, e/f)$ is a Farey triple, then so are $(a/b, (a+c)/(b+d), c/d)$ and $(c/d, (c+e)/(d+f), e/f)$; and the vertex of the Farey tree reached along every path is a Farey triple. (2) For all coprime positive natural\u2026","title":"Basic properties of the Farey tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-cohn-label","kind":"node","lean":"MetaMathlibExt.IsGenCohnLabel","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-cohn-label","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ be a natural number and $\\ell$ an integer, and write a pair $(a, b)$ of natural numbers as the fraction $a/b$. An integer $2 \\times 2$ matrix $C$ is labeled by the fraction $q$, written $C = C_q(k, \\ell)$, if there is a path $w$ from the root such that, for one of the three positions (first, second or third), $C$ is the entry in that position of the vertex of the generalized Cohn tree\u2026","title":"Fraction labels of generalized Cohn matrices"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-cohn-label-one","kind":"node","lean":"MetaMathlibExt.isGenCohnLabel_one","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The middle matrix of the root of "],["math","\\(\\mathrm{GCT}(k, \\ell)\\)"],["text"," has label "],["math","\\(1/1\\)"]],"source":"The middle matrix of the root of $\\mathrm{GCT}(k, \\ell)$ has label $1/1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-cohn-label-one","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For every natural number $k$ and every integer $\\ell$, the second entry of the root vertex of the generalized Cohn tree $\\mathrm{GCT}(k, \\ell)$ is labeled by the fraction $1/1$, that is, it is $C_{1/1}(k, \\ell)$.","title":"The middle matrix of the root of $\\mathrm{GCT}(k, \\ell)$ has label $1/1$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-label-injective-index-strict-mo","kind":"node","lean":"MetaMathlibExt.genCohnLabel_injective_index_strictMono","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-label-injective-index-strict-mo","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Fix a natural number $k$ and an integer $\\ell$. Write fractions $a/b$, with $a$ and $b$ natural numbers, as pairs $(a, b)$, compared by cross multiplication ($a/b < c/d$ means $ad < cb$), and write $C = C_q(k, \\ell)$ when the matrix $C$ carries the fraction label $q$ in the $k$-generalized Cohn tree $\\mathrm{GCT}(k, \\ell)$. For an integer $2 \\times 2$ matrix $M = (m_{ij})$ let $I_M = m_{11}/m_{12} \\in \\mathbb{Q}$.\u2026","title":"The Cohn labeling is injective and increasing"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-label","kind":"node","lean":"MetaMathlibExt.IsGenMarkovLabel","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-label","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"Let $k$ be a natural number. Write a fraction $n/d$ as the pair $(n, d)$ of natural numbers, with $1/0$ standing for $\\infty$. Let $FT$ be the Farey tree and $MT(k)$ the $k$-generalized Markov tree, with vertices of both indexed by finite paths of left and right steps from the root. A natural number $m$ carries the label $q$, written $m = m_q$, when there are a path $w$ and a position $i \\in \\{1, 2, 3\\}$ such that\u2026","title":"Fraction labeling of generalized Markov numbers"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-label-one","kind":"node","lean":"MetaMathlibExt.isGenMarkovLabel_one","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The fraction "],["math","\\(1/1\\)"],["text"," labels "],["math","\\(k + 2\\)"]],"source":"The fraction $1/1$ labels $k + 2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-label-one","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For every natural number $k$, the number $k + 2$ carries the label $1/1$ in the fraction labeling of $k$-generalized Markov numbers, that is, $m_{1/1} = k + 2$.","title":"The fraction $1/1$ labels $k + 2$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/has-unique-max-gen-markov-triple-iff-exi","kind":"node","lean":"MetaMathlibExt.hasUniqueMaxGenMarkovTriple_iff_existsUnique_label","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#has-unique-max-gen-markov-triple-iff-exi","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number and $b$ a $k$-generalized Markov number. Then $b$ is the maximum of a unique $k$-generalized Markov triple up to order if and only if there is exactly one pair $(n, d)$ of natural numbers with $n \\le d$ such that $b$ is a label of the fraction $n/d$ in the fraction labeling of $k$-generalized Markov numbers.","title":"Unique maximal triple via the fraction labeling"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-uniqueness-iff-label-injectiv","kind":"node","lean":"MetaMathlibExt.genMarkovUniqueness_iff_label_injective","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-uniqueness-iff-label-injectiv","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"For every natural number $k$, the following are equivalent: every $k$-generalized Markov number is the maximum of a unique $k$-generalized Markov triple up to order (Conjecture 1.8 for $k$); and the fraction labeling of $k$-generalized Markov numbers is injective on $[0, 1]$, that is, whenever $(n, d)$ and $(n', d')$ are pairs of natural numbers with $n \\le d$ and $n' \\le d'$ and a number $m$ is a label of both\u2026","title":"Uniqueness of maximal triples and injectivity of the fraction labeling"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-char-number","kind":"node","lean":"MetaMathlibExt.IsGenMarkovCharNumber","parent":9463,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-char-number","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"For natural numbers $m_r$, $m_t$, $m_s$, a natural number $x$ is a characteristic number of $(m_r, m_t, m_s)$ when $0 < x < m_t/2$ and $m_r x \\equiv \\pm m_s \\pmod{m_t}$, that is, $m_t$ divides $m_r x - m_s$ or $m_r x + m_s$ in $\\mathbb{Z}$.","title":"Characteristic number"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/is-gen-markov-char-number-one-five-two","kind":"node","lean":"MetaMathlibExt.isGenMarkovCharNumber_one_five_two","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characteristic number of "],["math","\\((1, 5, 2)\\)"]],"source":"Characteristic number of $(1, 5, 2)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#is-gen-markov-char-number-one-five-two","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"fully_proved","summary":"The number $2$ is a characteristic number of $(m_r, m_t, m_s) = (1, 5, 2)$: indeed $0 < 2 < 5/2$ and $1 \\cdot 2 \\equiv 2 \\pmod 5$.","title":"Characteristic number of $(1, 5, 2)$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-tree-neg-snd-entry-pos","kind":"node","lean":"MetaMathlibExt.genCohnTree_neg_snd_entry_pos","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positive corner entry of the generalized Cohn tree for "],["math","\\(\\ell = -k\\)"]],"source":"Positive corner entry of the generalized Cohn tree for $\\ell = -k$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-tree-neg-snd-entry-pos","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number and $w$ a path, and suppose that the middle fraction $n/d$ of the vertex of the Farey tree at $w$ satisfies $0 < n < d$. Then the $(1,1)$-entry of the middle matrix of the vertex of the $k$-generalized Cohn tree $\\mathrm{GCT}(k, -k)$ at $w$ is positive.","title":"Positive corner entry of the generalized Cohn tree for $\\ell = -k$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-cohn-tree-neg-snd-eq","kind":"node","lean":"MetaMathlibExt.genCohnTree_neg_snd_eq","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shape of the generalized Cohn matrices for "],["math","\\(\\ell = -k\\)"]],"source":"Shape of the generalized Cohn matrices for $\\ell = -k$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-cohn-tree-neg-snd-eq","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k$ be a natural number and $w$ a path, and suppose that the middle fraction $n/d$ of the vertex of the Farey tree at $w$ satisfies $0 < n < d$. Let $(a, b, c)$ be the vertex of the $k$-generalized Markov tree $\\mathrm{MT}(k)$ at $w$, and let $C$ be the middle matrix of the vertex of the $k$-generalized Cohn tree $\\mathrm{GCT}(k, -k)$ at $w$. Then there is a natural number $u$ that is a characteristic number of\u2026","title":"Shape of the generalized Cohn matrices for $\\ell = -k$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-char-number-dvd-sub","kind":"node","lean":"MetaMathlibExt.genMarkovCharNumber_dvd_sub","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The characteristic number solves "],["math","\\(m_r x \\equiv m_s \\pmod{m_t}\\)"]],"source":"The characteristic number solves $m_r x \\equiv m_s \\pmod{m_t}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-char-number-dvd-sub","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k \\in \\mathbb{N}$, let $w$ be a finite sequence of left and right moves, let $(r, t, s)$ be the vertex of the Farey tree at $w$, and let $(m_r, m_t, m_s)$ be the vertex of the $k$-generalized Markov tree $\\mathrm{MT}(k)$ at $w$. Write $t = n/d$. If $0 < n < d$ and $u$ is a characteristic number of $(m_r, m_t, m_s)$, then $m_t$ divides $m_r u - m_s$ in $\\mathbb{Z}$, that is, $m_r u \\equiv m_s \\pmod{m_t}$.","title":"The characteristic number solves $m_r x \\equiv m_s \\pmod{m_t}$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-char-number-sq-add","kind":"node","lean":"MetaMathlibExt.genMarkovCharNumber_sq_add","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The characteristic number solves "],["math","\\(x^2 + kx + 1 \\equiv 0 \\pmod{m_t}\\)"]],"source":"The characteristic number solves $x^2 + kx + 1 \\equiv 0 \\pmod{m_t}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-char-number-sq-add","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k \\in \\mathbb{N}$, let $w$ be a finite sequence of left and right moves, let $(r, t, s)$ be the vertex of the Farey tree at $w$, and let $(m_r, m_t, m_s)$ be the vertex of the $k$-generalized Markov tree $\\mathrm{MT}(k)$ at $w$. Write $t = n/d$. If $0 < n < d$ and $u$ is a characteristic number of $(m_r, m_t, m_s)$, then $m_t$ divides $u^2 + ku + 1$, that is, $u^2 + ku + 1 \\equiv 0 \\pmod{m_t}$.","title":"The characteristic number solves $x^2 + kx + 1 \\equiv 0 \\pmod{m_t}$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/gen-markov-char-number-add-lt","kind":"node","lean":"MetaMathlibExt.genMarkovCharNumber_add_lt","parent":9463,"reading_title":{"format":"inline-parts-v1","parts":[["text","The characteristic number is less than "],["math","\\(m_t/2 - k\\)"]],"source":"The characteristic number is less than $m_t/2 - k$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples/index.html#gen-markov-char-number-add-lt","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/generalized-markov-triples","status":"can_prove","summary":"Let $k \\in \\mathbb{N}$, let $w$ be a finite sequence of left and right moves, let $(r, t, s)$ be the vertex of the Farey tree at $w$, and let $(m_r, m_t, m_s)$ be the vertex of the $k$-generalized Markov tree $\\mathrm{MT}(k)$ at $w$. Write $t = n/d$. If $0 < n < d$ and $u$ is a characteristic number of $(m_r, m_t, m_s)$, then $2(u + k) < m_t$, that is, $u + k < m_t/2$.","title":"The characteristic number is less than $m_t/2 - k$"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","kind":"scope","navigation":"module","parent":9455,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","title":"Negative Pell equation and angle bisectors"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-fund-y","kind":"node","lean":"MetaMathlibExt.negPellFundY","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-fund-y","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"For a natural number $d$, $Y$ is the least positive integer $y$ such that $x^2 - d y^2 = -1$ for some integer $x$, and $Y = 0$ when there is no such $y$.","title":"Second coordinate of the fundamental negative Pell solution"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-fund-x","kind":"node","lean":"MetaMathlibExt.negPellFundX","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-fund-x","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"For a natural number $d$, let $Y$ be the least positive integer $y$ such that $x^2 - d y^2 = -1$ for some integer $x$, or $0$ if there is no such $y$. Then $X = \\lfloor \\sqrt{d Y^2 - 1} \\rfloor$, where $d Y^2 - 1$ is read as $0$ when $d Y^2 = 0$. So $X$ is the nonnegative integer with $X^2 - d Y^2 = -1$ when a solution exists, and $X = 0$ otherwise.","title":"First coordinate of the fundamental negative Pell solution"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-fund-spec","kind":"node","lean":"MetaMathlibExt.negPellFund_spec","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-fund-spec","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"Let $d$ be a natural number such that $x^2 - d y^2 = -1$ has a solution in a natural number $x$ and a positive integer $y$. Then the fundamental solution $(X, Y)$, the one with the least positive $y$, satisfies $Y > 0$ and $X^2 + 1 = d Y^2$.","title":"The fundamental negative Pell solution solves the equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-fund-y-le","kind":"node","lean":"MetaMathlibExt.negPellFundY_le","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-fund-y-le","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"Let $d$ be a natural number and let $Y$ be the second coordinate of its fundamental negative Pell solution. If $y$ is a positive integer such that $x^2 - d y^2 = -1$ for some natural number $x$, then $Y \\le y$.","title":"Minimality of the fundamental negative Pell solution"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-fund-two","kind":"node","lean":"MetaMathlibExt.negPellFund_two","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-fund-two","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"The least positive integer $y$ such that $x^2 - 2 y^2 = -1$ for some integer $x$ is $Y = 1$, and the corresponding first coordinate is $X = 1$; that is, the fundamental solution for $d = 2$ is $(1, 1)$.","title":"The fundamental solution of the negative Pell equation for d = 2"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-f","kind":"node","lean":"MetaMathlibExt.negPellF","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-f","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"For a natural number $d$, let $Y$ be the least positive integer $y$ such that $x^2 - d y^2 = -1$ for some integer $x$, let $X$ be the nonnegative integer with $X^2 - d Y^2 = -1$, and set $X = Y = 0$ when there is no such $y$. In the ring $\\mathbb{Z}[\\sqrt{d}]$ of formal expressions $a + b \\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ and $(\\sqrt{d})^2 = d$, put $\\varepsilon = X + Y \\sqrt{d}$. For $n \\ge 0$, $f_n$ is the\u2026","title":"First coordinates of the powers of the fundamental negative Pell solution"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-g","kind":"node","lean":"MetaMathlibExt.negPellG","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-g","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"For a natural number $d$, let $Y$ be the least positive integer $y$ such that $x^2 - d y^2 = -1$ for some integer $x$, let $X$ be the nonnegative integer with $X^2 - d Y^2 = -1$, and set $X = Y = 0$ when there is no such $y$. In the ring $\\mathbb{Z}[\\sqrt{d}]$ of formal expressions $a + b \\sqrt{d}$ with $a, b \\in \\mathbb{Z}$ and $(\\sqrt{d})^2 = d$, put $\\varepsilon = X + Y \\sqrt{d}$. For $n \\ge 0$, $g_n$ is the\u2026","title":"Second coordinates of the powers of the fundamental negative Pell solution"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-f-zero-neg-pell-g-zero","kind":"node","lean":"MetaMathlibExt.negPellF_zero_negPellG_zero","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-f-zero-neg-pell-g-zero","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"For every natural number $d$, the sequences $f_n$ and $g_n$ of the negative Pell equation $x^2 - d y^2 = -1$ satisfy $f_0 = 1$ and $g_0 = 0$.","title":"Initial values of the negative Pell sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-succ","kind":"node","lean":"MetaMathlibExt.negPell_succ","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-succ","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"Let $d$ be a natural number, let $(X, Y)$ be the solution of $x^2 - d y^2 = -1$ in natural numbers with least positive $Y$ (with $X = Y = 0$ if there is none), and define integers $f_n, g_n$ by $(X + Y\\sqrt{d})^n = f_n + g_n\\sqrt{d}$, computed formally with $(\\sqrt{d})^2 = d$. Then for every $n \\ge 0$,","title":"Recurrence for the negative Pell sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-f-two-one","kind":"node","lean":"MetaMathlibExt.negPellF_two_one","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-f-two-one","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"Let $(X, Y)$ be the solution of $x^2 - 2y^2 = -1$ in natural numbers with the least positive $Y$, and for each $n$ let $f_n$ and $g_n$ be the integers with $(X + Y\\sqrt 2)^n = f_n + g_n \\sqrt 2$. Then $f_1 = g_1 = 1$.","title":"The negative Pell sequences for d = 2 at n = 1"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-slope-equation","kind":"node","lean":"MetaMathlibExt.bisector_slope_equation","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-slope-equation","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $a, b, c$ be real numbers with $|a| \\ne |b|$ and $c \\ne 0$, and suppose that the lines $y = cx$ and $y = -c^{-1} x$ both bisect the angles between the lines $y = ax$ and $y = bx$, that is,","title":"Angle bisectors satisfy the bisector equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-rat-solution","kind":"node","lean":"MetaMathlibExt.bisector_equation_rat_solution","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-rat-solution","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $a, b, c$ be rational numbers with $|a| \\ne |b|$ and $(a - c)^2 (b^2 + 1) = (b - c)^2 (a^2 + 1)$. Then there are a square-free natural number $d > 0$ and rational numbers $a_2, b_2$ with $a^2 - d a_2^2 = -1$ and $b^2 - d b_2^2 = -1$ such that either $a_2 + b_2 \\ne 0$ and $c = (a b_2 + a_2 b) / (a_2 + b_2)$, or $b_2 - a_2 \\ne 0$ and $c = (a b_2 - a_2 b) / (b_2 - a_2)$.","title":"Rational solutions of the bisector equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-symm","kind":"node","lean":"MetaMathlibExt.bisector_equation_symm","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-symm","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $a, b, c$ be real numbers with $(a - c)^2 (b^2 + 1) = (b - c)^2 (a^2 + 1)$. Then the triples $(-a, -b, -c)$ and $(b, a, c)$ satisfy the same equation, and so does $(a, b, -c^{-1})$ when $c \\ne 0$.","title":"Symmetries of the bisector equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-int-zero","kind":"node","lean":"MetaMathlibExt.bisector_equation_int_zero","parent":9549,"reading_title":{"format":"inline-parts-v1","parts":[["text","No nontrivial integer solutions of the bisector equation with "],["math","\\(a = 0\\)"],["text"," or "],["math","\\(b = 0\\)"]],"source":"No nontrivial integer solutions of the bisector equation with $a = 0$ or $b = 0$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-int-zero","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $a, b, c$ be integers with $a = 0$ or $b = 0$ and $(a - c)^2 (b^2 + 1) = (b - c)^2 (a^2 + 1)$. Then $|a| = |b|$.","title":"No nontrivial integer solutions of the bisector equation with $a = 0$ or $b = 0$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-int-solution","kind":"node","lean":"MetaMathlibExt.bisector_equation_int_solution","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-int-solution","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"For a natural number $d > 1$ such that $x^2 - d y^2 = -1$ has an integer solution, let $(X, Y)$ be its solution in positive integers with $Y$ least, and define integers $f^{(d)}_k$ and $g^{(d)}_k$ by $f^{(d)}_k + g^{(d)}_k \\sqrt{d} = (X + Y \\sqrt{d})^k$ for natural numbers $k$. Let $a, b, c$ be integers with $|a| \\ne |b|$ and","title":"Integer solutions of the bisector equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-form-nine","kind":"node","lean":"MetaMathlibExt.bisector_equation_form_nine","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-form-nine","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a natural number such that $x^2 - d y^2 = -1$ has an integer solution, let $(X, Y)$ be its solution in positive integers with $Y$ least, and define integers $f_k$ and $g_k$ by $f_k + g_k \\sqrt{d} = (X + Y \\sqrt{d})^k$ for natural numbers $k$. Let $m, n > 0$ be natural numbers and put $a = f_{(2m-1)(2n-1)}$, $b = f_{(2m-1)(2n+1)}$ and $c = g_{(2m-1) 2n} / g_{2m-1}$. Then $g_{2m-1}$ divides\u2026","title":"Bisector-equation solutions from the negative Pell sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-form-ten","kind":"node","lean":"MetaMathlibExt.bisector_equation_form_ten","parent":9549,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bisector-equation solutions from the powers of "],["math","\\(1 + \\sqrt{2}\\)"]],"source":"Bisector-equation solutions from the powers of $1 + \\sqrt{2}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-form-ten","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"The least solution of $x^2 - 2 y^2 = -1$ in positive integers is $(1, 1)$; define integers $f_k$ and $g_k$ by $f_k + g_k \\sqrt{2} = (1 + \\sqrt{2})^k$ for natural numbers $k$. For every natural number $n > 0$, the integers $a = f_{2n-1}$, $b = -f_{2n+1}$ and $c = f_{2n}$ satisfy","title":"Bisector-equation solutions from the powers of $1 + \\sqrt{2}$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/general-pell-finitely-many-classes","kind":"node","lean":"MetaMathlibExt.generalPell_finitely_many_classes","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#general-pell-finitely-many-classes","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 0$ be an integer that is not a perfect square, let $\\varepsilon$ be the fundamental solution of $x^2 - dy^2 = 1$, viewed as a unit of $\\mathbb{Z}[\\sqrt d]$, and let $z \\ne 0$ be an integer such that $x^2 - dy^2 = z$ has an integral solution. Then there is a finite set $S$ of integral solutions $(a, b)$ of $x^2 - dy^2 = z$ such that every integral solution $(x, y)$ satisfies\u2026","title":"Finitely many classes of solutions of a general Pell equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-rat-solvable-iff","kind":"node","lean":"MetaMathlibExt.negPell_rat_solvable_iff","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-rat-solvable-iff","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"For a positive squarefree integer $d$, the equation $x^2 - d y^2 = -1$ has a rational solution if and only if no prime divisor of $d$ is congruent to $3$ modulo $4$.","title":"Rational solvability of the negative Pell equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-sequences-basic","kind":"node","lean":"MetaMathlibExt.negPell_sequences_basic","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-sequences-basic","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a natural number such that $x^2 - d y^2 = -1$ has an integral solution, let $(X, Y)$ be its solution in natural numbers with least positive $Y$, and define integers $f_n, g_n$ by $(X + Y\\sqrt{d})^n = f_n + g_n\\sqrt{d}$. Then $f_n^2 - d g_n^2 = (-1)^n$ for every $n \\ge 0$, the sequence $(g_n)_{n \\ge 0}$ is strictly increasing, the sequence $(f_n)_{n \\ge 1}$ is strictly increasing, and, with the real\u2026","title":"Basic properties of the negative Pell sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-addition-formulas","kind":"node","lean":"MetaMathlibExt.negPell_addition_formulas","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-addition-formulas","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d$ be a natural number such that $x^2 - dy^2 = -1$ has an integral solution, let $(X, Y)$ be the solution of $x^2 - dy^2 = -1$ in natural numbers with the least positive $Y$, and let $f_n$ and $g_n$ be the integers defined by $f_0 = 1$, $g_0 = 0$, $f_{n+1} = X f_n + dY g_n$ and $g_{n+1} = Y f_n + X g_n$, so that $(X + Y\\sqrt d)^n = f_n + g_n \\sqrt d$. Then for all natural numbers $n \\le m$,","title":"Addition and doubling formulas for the negative Pell sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-sum-difference-formulas","kind":"node","lean":"MetaMathlibExt.negPell_sum_difference_formulas","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-sum-difference-formulas","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d$ be a natural number such that $x^2 - d y^2 = -1$ has an integral solution, let $(X, Y)$ be its solution in natural numbers with least positive $Y$, and define integers $f_n, g_n$ by $(X + Y\\sqrt{d})^n = f_n + g_n\\sqrt{d}$, computed formally with $(\\sqrt{d})^2 = d$. Let $m > n$ be natural numbers. If $n$ is even, then","title":"Sum and difference formulas for the negative Pell sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-g-le-neg-pell-f","kind":"node","lean":"MetaMathlibExt.negPellG_le_negPellF","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-g-le-neg-pell-f","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a natural number such that $x^2 - d y^2 = -1$ has an integer solution, let $(X, Y)$ be its fundamental solution (the one with $X, Y > 0$ and $Y$ least), and write $(X + Y \\sqrt{d})^n = f_n + g_n \\sqrt{d}$. Then for every $n \\ge 0$, $g_n \\le f_n$, with equality if and only if $d = 2$ and $n = 1$.","title":"Comparison of the negative Pell sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-gcd-eq-one","kind":"node","lean":"MetaMathlibExt.negPell_gcd_eq_one","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-gcd-eq-one","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a natural number such that $x^2 - d y^2 = -1$ has an integral solution, let $(X, Y)$ be its solution in natural numbers with least positive $Y$, and define integers $f_n, g_n$ by $(X + Y\\sqrt{d})^n = f_n + g_n\\sqrt{d}$. Then for every $n > 0$, $\\gcd(d, f_n) = 1$ and $\\gcd(f_n, g_n) = 1$.","title":"Coprimality of the negative Pell sequences"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-f-gcd-of-even-quotient","kind":"node","lean":"MetaMathlibExt.negPellF_gcd_of_even_quotient","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-f-gcd-of-even-quotient","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a natural number such that $x^2 - dy^2 = -1$ has an integral solution, let $(X, Y)$ be the solution of $x^2 - dy^2 = -1$ in natural numbers with the least positive $Y$, and for each $n$ let $f_n$ and $g_n$ be the integers with $(X + Y\\sqrt d)^n = f_n + g_n \\sqrt d$. If $0 < n \\le m$ are natural numbers such that $n$ divides $m$ and $m / n$ is even, then $\\gcd(f_m, f_n) = 1$.","title":"Coprimality of negative Pell terms whose indices have an even ratio"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-f-dvd-iff","kind":"node","lean":"MetaMathlibExt.negPellF_dvd_iff","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-f-dvd-iff","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a natural number such that $x^2 - dy^2 = -1$ has an integral solution, let $(X, Y)$ be the solution of $x^2 - dy^2 = -1$ in natural numbers with the least positive $Y$, and for each $n$ let $f_n$ and $g_n$ be the integers with $(X + Y\\sqrt d)^n = f_n + g_n \\sqrt d$. For natural numbers $0 < n \\le m$, $f_n$ divides $f_m$ if and only if either $d = 2$ and $n = 1$, or $n$ divides $m$ and $m / n$ is odd.","title":"Divisibility between terms of the negative Pell sequence"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/neg-pell-g-dvd-iff","kind":"node","lean":"MetaMathlibExt.negPellG_dvd_iff","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#neg-pell-g-dvd-iff","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a natural number such that $x^2 - d y^2 = -1$ has an integer solution, let $(X, Y)$ be its fundamental solution (the one with $X, Y > 0$ and $Y$ least), and write $(X + Y \\sqrt{d})^n = f_n + g_n \\sqrt{d}$. For natural numbers $0 < n \\le m$, $g_n$ divides $g_m$ if and only if $n$ divides $m$.","title":"Divisibility of the second coordinates of negative Pell powers"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/abs-sq-sub-mul-sq-ne-of-two-mod-four","kind":"node","lean":"MetaMathlibExt.abs_sq_sub_mul_sq_ne_of_two_mod_four","parent":9549,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(|x^2 - d y^2|\\)"],["text"," is never "],["math","\\(2 \\bmod 4\\)"],["text"," when "],["math","\\(d \\equiv 1 \\pmod 4\\)"]],"source":"$|x^2 - d y^2|$ is never $2 \\bmod 4$ when $d \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#abs-sq-sub-mul-sq-ne-of-two-mod-four","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d$ and $z$ be integers with $d \\equiv 1 \\pmod 4$ and $z \\equiv 2 \\pmod 4$. Then $|x^2 - d y^2| \\ne z$ for all integers $x$ and $y$.","title":"$|x^2 - d y^2|$ is never $2 \\bmod 4$ when $d \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/even-of-abs-sq-sub-mul-sq-four-mod-eight","kind":"node","lean":"MetaMathlibExt.even_of_abs_sq_sub_mul_sq_four_mod_eight","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#even-of-abs-sq-sub-mul-sq-four-mod-eight","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d$ and $z$ be integers with $d \\equiv 1 \\pmod 8$ and $z \\equiv 4 \\pmod 8$. If integers $x$ and $y$ satisfy $|x^2 - dy^2| = z$, then $x$ and $y$ are both even.","title":"Solutions of |x\u00b2 \u2212 dy\u00b2| = z are even when d \u2261 1 and z \u2261 4 (mod 8)"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/even-of-abs-sq-sub-mul-sq-zero-mod-eight","kind":"node","lean":"MetaMathlibExt.even_of_abs_sq_sub_mul_sq_zero_mod_eight","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#even-of-abs-sq-sub-mul-sq-zero-mod-eight","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d$ and $z$ be integers with $d \\equiv 5 \\pmod 8$ and $z \\equiv 0 \\pmod 8$. If integers $x$ and $y$ satisfy $|x^2 - dy^2| = z$, then $x$ and $y$ are both even.","title":"Solutions of |x\u00b2 \u2212 dy\u00b2| = z are even when d \u2261 5 and z \u2261 0 (mod 8)"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-primes","kind":"node","lean":"MetaMathlibExt.strictPrimitivePrimes","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-primes","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"For an integer $d$, $S(d)$ is the set of primes $p$ for which there is an integer $l > 0$ such that $|x^2 - d y^2| = p^l$ has a strictly primitive integral solution, that is, integers $x, y$ with $x$ and $d y$ coprime.","title":"Primes with strictly primitive representations"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-exponent","kind":"node","lean":"MetaMathlibExt.strictPrimitiveExponent","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-exponent","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"Let $d$ be an integer and $p$ a natural number, and let $S(d)$ be the set of primes $q$ for which $|x^2 - d y^2| = q^l$ has a strictly primitive integral solution (integers $x, y$ with $x$ and $d y$ coprime) for some integer $l > 0$. If $p \\in S(d)$, then $l_p$ is the least integer $l > 0$ such that $|x^2 - d y^2| = p^l$ has a strictly primitive integral solution; if $p \\notin S(d)$, then $l_p = 0$.","title":"Strictly primitive exponent"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-exponent-of-not-mem","kind":"node","lean":"MetaMathlibExt.strictPrimitiveExponent_of_not_mem","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-exponent-of-not-mem","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"Let $d$ be an integer and let $S(d)$ be the set of primes $p$ for which $|x^2 - d y^2| = p^l$ has a strictly primitive integral solution ($x$ and $d y$ coprime) for some integer $l > 0$. Let $l_p$ be the least such $l$ when $p \\in S(d)$, and $l_p = 0$ otherwise. Then $l_p = 0$ for every natural number $p \\notin S(d)$, in particular for every $p$ that is not prime.","title":"Vanishing of the strictly primitive exponent"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-exponent-spec","kind":"node","lean":"MetaMathlibExt.strictPrimitiveExponent_spec","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-exponent-spec","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"Let $d$ be an integer and let $S(d)$ be the set of primes $p$ for which $|x^2 - d y^2| = p^l$ has a strictly primitive integral solution ($x$ and $d y$ coprime) for some integer $l > 0$; for $p \\in S(d)$ let $l_p$ be the least such $l$. Then for every $p \\in S(d)$, $l_p > 0$ and $|x^2 - d y^2| = p^{l_p}$ has a strictly primitive integral solution.","title":"The strictly primitive exponent is attained"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-exponent-le","kind":"node","lean":"MetaMathlibExt.strictPrimitiveExponent_le","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-exponent-le","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"Let $d$ be an integer and let $S(d)$ be the set of primes $p$ for which $|x^2 - d y^2| = p^l$ has a strictly primitive integral solution ($x$ and $d y$ coprime) for some integer $l > 0$; for $p \\in S(d)$ let $l_p$ be the least such $l$. If $p \\in S(d)$ and $l > 0$ is an integer for which $|x^2 - d y^2| = p^l$ has a strictly primitive integral solution, then $l_p \\le l$.","title":"Minimality of the strictly primitive exponent"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/seven-mem-strict-primitive-primes-two","kind":"node","lean":"MetaMathlibExt.seven_mem_strictPrimitivePrimes_two","parent":9549,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(7\\)"],["text"," is a strictly primitive prime for "],["math","\\(d = 2\\)"]],"source":"$7$ is a strictly primitive prime for $d = 2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#seven-mem-strict-primitive-primes-two","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"The prime $7$ belongs to $S(2)$, the set of primes $p$ for which $|x^2 - 2 y^2| = p^l$ has a strictly primitive integral solution ($x$ and $2 y$ coprime) for some integer $l > 0$. A witness is $3^2 - 2 \\cdot 1^2 = 7$.","title":"$7$ is a strictly primitive prime for $d = 2$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-exponent-two-seven","kind":"node","lean":"MetaMathlibExt.strictPrimitiveExponent_two_seven","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-exponent-two-seven","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"fully_proved","summary":"For $d = 2$ and $p = 7$, the strictly primitive exponent $l_7$ equals $1$: the least integer $l > 0$ such that $|x^2 - 2 y^2| = 7^l$ has an integral solution with $x$ and $2 y$ coprime exists and equals $1$.","title":"The strictly primitive exponent of 7 for d = 2"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-rat-solution-pythagore","kind":"node","lean":"MetaMathlibExt.bisector_equation_rat_solution_pythagorean","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-rat-solution-pythagore","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $a, b, c$ be rational numbers with $|a| \\ne |b|$ and $(a - c)^2 (b^2 + 1) = (b - c)^2 (a^2 + 1)$, and suppose that $a^2 + 1$ and $b^2 + 1$ are squares of rational numbers. Then there are integers $l, m, n$ with $|l| \\ne |m|$, $l m \\ne n^2$ and $l m n \\ne 0$ such that","title":"Pythagorean parametrization of rational solutions of the bisector equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-form-two","kind":"node","lean":"MetaMathlibExt.bisector_equation_form_two","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-form-two","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $l, m, n$ be integers with $|l| \\ne |m|$, $l m \\ne n^2$ and $l m n \\ne 0$, and put","title":"Parametric rational solutions of the bisector equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-rat-solution-neg-pell","kind":"node","lean":"MetaMathlibExt.bisector_equation_rat_solution_negPell","parent":9549,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rational solutions of the bisector equation for square-free "],["math","\\(d > 1\\)"]],"source":"Rational solutions of the bisector equation for square-free $d > 1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-rat-solution-neg-pell","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a square-free integer and let $a, b, c$ be rational numbers with $|a| \\ne |b|$ and","title":"Rational solutions of the bisector equation for square-free $d > 1$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/bisector-equation-form-three","kind":"node","lean":"MetaMathlibExt.bisector_equation_form_three","parent":9549,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bisector-equation solutions from two solutions of "],["math","\\(x^2 - d y^2 = -1\\)"]],"source":"Bisector-equation solutions from two solutions of $x^2 - d y^2 = -1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#bisector-equation-form-three","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d$ be an integer and let $a, a_2, b, b_2$ be rational numbers with $a^2 - d a_2^2 = -1$ and $b^2 - d b_2^2 = -1$. If $a_2 + b_2 \\ne 0$, then $c = (a b_2 + a_2 b) / (a_2 + b_2)$ satisfies","title":"Bisector-equation solutions from two solutions of $x^2 - d y^2 = -1$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strictly-primitive-prod","kind":"node","lean":"MetaMathlibExt.strictlyPrimitive_prod","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strictly-primitive-prod","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a square-free integer. Write elements of $\\mathbb{Z}[\\sqrt{d}]$ as $u + v\\sqrt{d}$ with integers $u, v$, and let $N(u + v\\sqrt{d}) = u^2 - d v^2$ be the norm. Let $p_1, \\dots, p_r$ be distinct primes and $n_1, \\dots, n_r$ natural numbers, and suppose that each $a_i = u_i + v_i\\sqrt{d}$ satisfies $|N(a_i)| = p_i^{n_i}$ with $u_i$ and $d v_i$ coprime. Then $a = a_1 \\cdots a_r = u + v\\sqrt{d}$ satisfies\u2026","title":"Products of strictly primitive solutions"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strictly-primitive-pow","kind":"node","lean":"MetaMathlibExt.strictlyPrimitive_pow","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strictly-primitive-pow","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a square-free integer, $p$ an odd prime, and $m, n > 0$ natural numbers. Write elements of $\\mathbb{Z}[\\sqrt{d}]$ as $u + v\\sqrt{d}$ with integers $u, v$, and let $N(u + v\\sqrt{d}) = u^2 - d v^2$ be the norm. If $a = u + v\\sqrt{d}$ satisfies $|N(a)| = p^m$ with $u$ and $d v$ coprime, then $a^n = u' + v'\\sqrt{d}$ satisfies $|N(a^n)| = p^{mn}$ with $u'$ and $d v'$ coprime; that is, $a^n$ gives a\u2026","title":"Powers of strictly primitive solutions"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strictly-primitive-neg","kind":"node","lean":"MetaMathlibExt.strictlyPrimitive_neg","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strictly-primitive-neg","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a square-free integer such that $x^2 - d y^2 = -1$ has an integral solution, and let $z \\ne 0$ be an integer. If $x^2 - d y^2 = z$ has an integral solution with $x$ and $d y$ coprime, then so does $x^2 - d y^2 = -z$.","title":"Negating the value of a strictly primitive solution"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/exists-strictly-primitive-dvd-iff-split","kind":"node","lean":"MetaMathlibExt.exists_strictlyPrimitive_dvd_iff_split","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#exists-strictly-primitive-dvd-iff-split","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a square-free integer, let $K$ be a number field of degree $2$ over $\\mathbb{Q}$ containing an element $s$ with $s^2 = d$ (so $K = \\mathbb{Q}(\\sqrt d)$), and let $p$ be a prime such that $p \\ne 2$, or $d \\equiv 1 \\pmod 8$, or $d \\equiv 2 \\pmod 4$. Then there are integers $x, y$ with $x$ and $dy$ coprime and $p \\mid x^2 - dy^2$ if and only if $p$ splits in $K$, that is, exactly two prime ideals of the\u2026","title":"Strictly primitive solutions modulo p and splitting of p"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-primes-eq-split","kind":"node","lean":"MetaMathlibExt.strictPrimitivePrimes_eq_split","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-primes-eq-split","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a square-free integer, let $K$ be any number field of degree $2$ over $\\mathbb{Q}$ containing an element $s$ with $s^2 = d$, and let $\\mathcal{O}_K$ be its ring of integers. Suppose that $d \\equiv 1 \\pmod 8$, or that $d \\equiv 5 \\pmod 8$ and every unit of $\\mathcal{O}_K$ lies in $\\mathbb{Z}[s]$, or that $d \\equiv 2 \\pmod 4$. Then $S(d)$, the set of primes $p$ for which $|x^2 - d y^2| = p^l$ has an\u2026","title":"Strictly primitive primes are the split primes"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-primes-eq-split-insert","kind":"node","lean":"MetaMathlibExt.strictPrimitivePrimes_eq_split_insert_two","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-primes-eq-split-insert","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a square-free integer, let $K$ be any number field of degree $2$ over $\\mathbb{Q}$ containing an element $s$ with $s^2 = d$, and let $\\mathcal{O}_K$ be its ring of integers. Suppose that $d \\equiv 5 \\pmod 8$ and that some unit of $\\mathcal{O}_K$ does not lie in $\\mathbb{Z}[s]$ (equivalently, the fundamental unit of $K$ does not lie in $\\mathbb{Z}[s]$). Then $S(d)$, the set of primes $p$ for which\u2026","title":"Strictly primitive primes are 2 and the split primes"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/strict-primitive-exponent-le-class-numbe","kind":"node","lean":"MetaMathlibExt.strictPrimitiveExponent_le_classNumber","parent":9549,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors/index.html#strict-primitive-exponent-le-class-numbe","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/negative-pell-equation-and-angle-bisectors","status":"can_prove","summary":"Let $d > 1$ be a squarefree integer with $d \\equiv 2$ or $d \\equiv 3 \\pmod 4$, and let $K$ be a number field of degree $2$ containing a square root of $d$ (so $K \\cong \\mathbb{Q}(\\sqrt{d})$). Let $S(d)$ be the set of primes $p$ for which $|x^2 - d y^2| = p^l$ has a strictly primitive integral solution ($x$ and $d y$ coprime) for some integer $l > 0$, and for $p \\in S(d)$ let $l_p$ be the least such $l$. Then\u2026","title":"The strictly primitive exponent is at most the class number"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","kind":"scope","navigation":"module","parent":9455,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","title":"Pell equation solutions in a rotated square"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-u","kind":"node","lean":"MetaMathlibExt.pellU","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-u","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"fully_proved","summary":"For integers $D$, $\\alpha$, $\\beta$, the function $u : \\mathbb{R} \\to \\mathbb{R}$ is $u(x) = ((\\alpha + \\beta\\sqrt D)^x + (\\alpha - \\beta\\sqrt D)^x)/2$, where $\\sqrt D$ is the real square root (zero when $D < 0$) and $(\\cdot)^x$ is the real power.","title":"The Pell interpolation function u"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-v","kind":"node","lean":"MetaMathlibExt.pellV","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-v","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"fully_proved","summary":"For integers $D$, $\\alpha$, $\\beta$, the function $v : \\mathbb{R} \\to \\mathbb{R}$ is $v(x) = ((\\alpha + \\beta\\sqrt D)^x - (\\alpha - \\beta\\sqrt D)^x)/(2\\sqrt D)$, where $\\sqrt D$ is the real square root (zero when $D < 0$), $(\\cdot)^x$ is the real power and division by zero gives $0$; in particular $v$ is identically $0$ when $D \\le 0$.","title":"The Pell interpolation function v"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-u-zero","kind":"node","lean":"MetaMathlibExt.pellU_zero","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-u-zero","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"fully_proved","summary":"For all integers $D$, $\\alpha$, $\\beta$, the function $u(x) = ((\\alpha + \\beta\\sqrt D)^x + (\\alpha - \\beta\\sqrt D)^x)/2$ of a real variable $x$ satisfies $u(0) = 1$.","title":"Value of the Pell interpolation function u at 0"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-v-zero","kind":"node","lean":"MetaMathlibExt.pellV_zero","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-v-zero","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"fully_proved","summary":"For all integers $D$, $\\alpha$, $\\beta$, the function $v(x) = ((\\alpha + \\beta\\sqrt D)^x - (\\alpha - \\beta\\sqrt D)^x)/(2\\sqrt D)$ of a real variable $x$, where $\\sqrt D$ is the real square root (zero when $D < 0$) and division by zero gives $0$, satisfies $v(0) = 0$.","title":"Value of the Pell interpolation function v at 0"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-solutions-eq-signed-powers","kind":"node","lean":"MetaMathlibExt.pell_solutions_eq_signed_powers","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-solutions-eq-signed-powers","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer and let $(\\alpha, \\beta)$ be a fundamental solution of $x^2 - Dy^2 = 1$, that is, an integer solution with $\\alpha > 1$ and $\\beta > 0$ such that $\\alpha \\le x$ for every integer solution $(x, y)$ with $x > 1$ (one exists only when $D$ is positive and not a square). For $n \\ge 0$ let $u_n, v_n$ be the integers with $u_n + v_n\\sqrt D = (\\alpha + \\beta\\sqrt D)^n$. Then integers $x, y$ satisfy\u2026","title":"Pell solutions are signed powers of the fundamental solution"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-u-pell-v-nat-cast","kind":"node","lean":"MetaMathlibExt.pellU_pellV_natCast","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-u-pell-v-nat-cast","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer and let $(\\alpha, \\beta)$ be a fundamental solution of $x^2 - Dy^2 = 1$, that is, an integer solution with $\\alpha > 1$ and $\\beta > 0$ such that $\\alpha \\le x$ for every integer solution $(x, y)$ with $x > 1$ (one exists only when $D$ is positive and not a square). For $n \\ge 0$ let $u_n, v_n$ be the integers with $u_n + v_n\\sqrt D = (\\alpha + \\beta\\sqrt D)^n$, and for real $t$ let\u2026","title":"The Pell interpolation functions agree with the solution sequence at natural numbers"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-inverse-closed-form","kind":"node","lean":"MetaMathlibExt.pell_inverse_closed_form","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-inverse-closed-form","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer, let $(\\alpha, \\beta)$ be the fundamental solution of the Pell equation $X^2 - D Y^2 = 1$, put $\\varepsilon = \\alpha + \\beta \\sqrt{D}$ and $\\bar\\varepsilon = \\alpha - \\beta \\sqrt{D}$, and let $u(t) = (\\varepsilon^t + \\bar\\varepsilon^t) / 2$ and $v(t) = (\\varepsilon^t - \\bar\\varepsilon^t) / (2 \\sqrt{D})$ for real $t$. For every real $x \\ge 1$, put $z = (D x - \\sqrt{D x^2 - D + 1}) / (D - 1)$\u2026","title":"Closed form for inverting the sum of the Pell interpolating functions"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-inverse-floor","kind":"node","lean":"MetaMathlibExt.pell_inverse_floor","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-inverse-floor","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer, let $(\\alpha, \\beta)$ be the fundamental solution of the Pell equation $X^2 - D Y^2 = 1$, put $\\varepsilon = \\alpha + \\beta \\sqrt{D}$ and $\\bar\\varepsilon = \\alpha - \\beta \\sqrt{D}$, and let $u(s) = (\\varepsilon^s + \\bar\\varepsilon^s) / 2$ and $v(s) = (\\varepsilon^s - \\bar\\varepsilon^s) / (2 \\sqrt{D})$ for real $s$. If $x \\ge 1$ and $t \\ge 0$ are real numbers with $u(t) + v(t) = x$, then\u2026","title":"Floor of the inverse of the sum of the Pell interpolating functions"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-solutions-in-rotated-square","kind":"node","lean":"MetaMathlibExt.pell_solutions_in_rotated_square","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-solutions-in-rotated-square","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer and let $(\\alpha, \\beta)$ be a fundamental solution of $x^2 - Dy^2 = 1$, that is, an integer solution with $\\alpha > 1$ and $\\beta > 0$ such that $\\alpha \\le x$ for every integer solution $(x, y)$ with $x > 1$ (one exists only when $D$ is positive and not a square). For $n \\ge 0$ let $u_n, v_n$ be the integers with $u_n + v_n\\sqrt D = (\\alpha + \\beta\\sqrt D)^n$, and for real $t$ let\u2026","title":"Pell solutions in a closed rotated square"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-solutions-in-rotated-square-explici","kind":"node","lean":"MetaMathlibExt.pell_solutions_in_rotated_square_explicit","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-solutions-in-rotated-square-explici","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer and let $(\\alpha, \\beta)$ be a fundamental solution of $x^2 - Dy^2 = 1$, that is, an integer solution with $\\alpha > 1$ and $\\beta > 0$ such that $\\alpha \\le x$ for every integer solution $(x, y)$ with $x > 1$ (one exists only when $D$ is positive and not a square). For $n \\ge 0$ let $u_n, v_n$ be the integers with $u_n + v_n\\sqrt D = (\\alpha + \\beta\\sqrt D)^n$. For real $\\lambda$ let\u2026","title":"Explicit count of Pell solutions in a closed rotated square"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shifted-real-solutions-eventually-i","kind":"node","lean":"MetaMathlibExt.pell_shifted_real_solutions_eventually_in_quadrants","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shifted-real-solutions-eventually-i","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer and let $(\\alpha, \\beta)$ be a fundamental solution of $X^2 - DY^2 = 1$, that is, an integer solution with $\\alpha > 1$ and $\\beta > 0$ such that $\\alpha \\le X$ for every integer solution $(X, Y)$ with $X > 1$ (one exists only when $D$ is positive and not a square). For real $x$ let $u(x) = ((\\alpha + \\beta\\sqrt D)^x + (\\alpha - \\beta\\sqrt D)^x)/2$ and\u2026","title":"Shifted real Pell solutions eventually lie in the closed quadrants"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shifted-intercepts-max","kind":"node","lean":"MetaMathlibExt.pell_shifted_intercepts_max","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shifted-intercepts-max","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D > 0$, $a \\ne 0$ and $b$ be integers. The intercepts of the shifted Pell curve $(x - a)^2 - D (y - b)^2 = 1$ with the axes are $(a \\pm \\sqrt{1 + D b^2}, 0)$ and $(0, b \\pm \\sqrt{(a^2 - 1) / D})$, with $L^1$ norms $|a \\pm \\sqrt{1 + D b^2}|$ and $|b \\pm \\sqrt{(a^2 - 1) / D}|$. The largest of these four norms is $|a| + \\sqrt{1 + D b^2}$.","title":"Largest intercept norm of a shifted Pell curve"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shifted-norm","kind":"node","lean":"MetaMathlibExt.pellShiftedNorm","parent":9598,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(L^1\\)"],["text"," norm along a shifted Pell curve"]],"source":"The $L^1$ norm along a shifted Pell curve"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shifted-norm","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"fully_proved","summary":"For integers $D$, $\\alpha$, $\\beta$, $a$, $b$, $i$ and $j$, put $\\varepsilon = \\alpha + \\beta \\sqrt{D}$ and $\\bar\\varepsilon = \\alpha - \\beta \\sqrt{D}$, and let $u(x) = (\\varepsilon^x + \\bar\\varepsilon^x) / 2$ and $v(x) = (\\varepsilon^x - \\bar\\varepsilon^x) / (2 \\sqrt{D})$ for real $x$. The function $f_{i,j}$ is defined on the reals by $f_{i,j}(x) = |i u(x) + a| + |j v(x) + b|$.","title":"The $L^1$ norm along a shifted Pell curve"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shifted-norm-zero","kind":"node","lean":"MetaMathlibExt.pellShiftedNorm_zero","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shifted-norm-zero","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"fully_proved","summary":"Let $D$, $\\alpha$, $\\beta$, $a$, $b$, $i$ and $j$ be integers. Put $\\varepsilon = \\alpha + \\beta \\sqrt{D}$ and $\\bar\\varepsilon = \\alpha - \\beta \\sqrt{D}$, let $u(x) = (\\varepsilon^x + \\bar\\varepsilon^x) / 2$ and $v(x) = (\\varepsilon^x - \\bar\\varepsilon^x) / (2 \\sqrt{D})$ for real $x$, and let $f_{i,j}(x) = |i u(x) + a| + |j v(x) + b|$. Then $f_{i,j}(0) = |i + a| + |b|$.","title":"Value at zero of the shifted Pell norm function"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shift-bound","kind":"node","lean":"MetaMathlibExt.pellShiftBound","parent":9598,"reading_title":{"format":"inline-parts-v1","parts":[["text","The bound "],["math","\\(K\\)"],["text"," for the shifted Pell equation"]],"source":"The bound $K$ for the shifted Pell equation"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shift-bound","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"fully_proved","summary":"For integers $D$, $a$ and $b$, the constant attached to the shifted Pell equation $(x - a)^2 - D (y - b)^2 = 1$ is the real number $K = |a| + \\max(|b| + 1, \\sqrt{1 + D b^2})$, where $\\sqrt{s}$ is taken to be $0$ for $s < 0$.","title":"The bound $K$ for the shifted Pell equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shift-bound-zero-zero","kind":"node","lean":"MetaMathlibExt.pellShiftBound_zero_zero","parent":9598,"reading_title":{"format":"inline-parts-v1","parts":[["text","The bound "],["math","\\(K\\)"],["text"," of the unshifted Pell equation"]],"source":"The bound $K$ of the unshifted Pell equation"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shift-bound-zero-zero","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"fully_proved","summary":"For every integer $D$, the constant $K = |a| + \\max(|b| + 1, \\sqrt{1 + D b^2})$ with $a = b = 0$ equals $1$.","title":"The bound $K$ of the unshifted Pell equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shifted-norm-zero-le-pell-shift-bou","kind":"node","lean":"MetaMathlibExt.pellShiftedNorm_zero_le_pellShiftBound","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shifted-norm-zero-le-pell-shift-bou","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$, $\\alpha$, $\\beta$, $a$ and $b$ be integers and let $i, j \\in \\{1, -1\\}$. Put $\\varepsilon = \\alpha + \\beta \\sqrt{D}$ and $\\bar\\varepsilon = \\alpha - \\beta \\sqrt{D}$, let $u(x) = (\\varepsilon^x + \\bar\\varepsilon^x) / 2$ and $v(x) = (\\varepsilon^x - \\bar\\varepsilon^x) / (2 \\sqrt{D})$ for real $x$, let $f_{i,j}(x) = |i u(x) + a| + |j v(x) + b|$ and let $K = |a| + \\max(|b| + 1, \\sqrt{1 + D b^2})$. Then\u2026","title":"The shifted Pell norm function at zero is at most the bound"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shifted-norm-le-pell-shift-bound","kind":"node","lean":"MetaMathlibExt.pellShiftedNorm_le_pellShiftBound","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shifted-norm-le-pell-shift-bound","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer, let $(\\alpha, \\beta)$ be the fundamental solution of the Pell equation $X^2 - D Y^2 = 1$, put $\\varepsilon = \\alpha + \\beta \\sqrt{D}$ and $\\bar\\varepsilon = \\alpha - \\beta \\sqrt{D}$, and let $u(x) = (\\varepsilon^x + \\bar\\varepsilon^x) / 2$ and $v(x) = (\\varepsilon^x - \\bar\\varepsilon^x) / (2 \\sqrt{D})$ for real $x$. Let $a$ and $b$ be integers, let $i, j \\in \\{1, -1\\}$, let\u2026","title":"The shifted Pell norm function is at most the bound up to its last crossing"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/pell-shifted-norm-bij-on-ici","kind":"node","lean":"MetaMathlibExt.pellShiftedNorm_bijOn_Ici","parent":9598,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square/index.html#pell-shifted-norm-bij-on-ici","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-equation-solutions-in-a-rotated-square","status":"can_prove","summary":"Let $D$ be an integer, let $(\\alpha, \\beta)$ be the fundamental solution of the Pell equation $X^2 - D Y^2 = 1$, put $\\varepsilon = \\alpha + \\beta \\sqrt{D}$ and $\\bar\\varepsilon = \\alpha - \\beta \\sqrt{D}$, and let $u(x) = (\\varepsilon^x + \\bar\\varepsilon^x) / 2$ and $v(x) = (\\varepsilon^x - \\bar\\varepsilon^x) / (2 \\sqrt{D})$ for real $x$. Let $a$ and $b$ be integers, let $i, j \\in \\{1, -1\\}$, let\u2026","title":"The shifted Pell norm function is increasing past its last crossing of the bound"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation","kind":"scope","navigation":"module","parent":9455,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation","status":"can_prove","title":"Pell's equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation/pell-type-solution-mul-unit","kind":"node","lean":"MetaMathlibExt.pellType_solution_mul_unit","parent":9618,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation/index.html#pell-type-solution-mul-unit","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation","status":"fully_proved","summary":"Let $a, b, c, T, U, x, y$ be integers with $a T^2 - b U^2 = c$ and $x^2 - ab y^2 = 1$. Then the pair $(T x + b U y, a T y + U x)$ is again a solution: $a (T x + b U y)^2 - b (a T y + U x)^2 = c$.","title":"Composing a Pell-type solution with a norm-one unit"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation/pell-solution-is-convergent","kind":"node","lean":"MathlibExt.NumberTheory.PellConvergentWanted.NumberTheory.pell_solution_is_convergent","parent":9618,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation/index.html#pell-solution-is-convergent","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pell-s-equation","status":"can_prove","summary":"Let $d$ be a positive squarefree integer and let $x$ and $y$ be positive integers with $x^2 - d y^2 = 1$. Then $x / y$ is a convergent of the regular continued fraction expansion of $\\sqrt{d}$: there is a natural number $n$ such that the $n$-th convergent of $\\sqrt{d}$ (counting from the $0$-th) equals $x / y$.","title":"Solutions of Pell's equation are continued-fraction convergents"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","kind":"scope","navigation":"module","parent":9455,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","title":"Pythagorean triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/primitive-pythagorean-sides-sum-two-squa","kind":"node","lean":"MathlibExt.NumberTheory.PrimitivePythagoreanFormsWanted.primitive_pythagorean_sides_sum_two_squares_infinite","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#primitive-pythagorean-sides-sum-two-squa","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"There are infinitely many triples $(X, Y, Z)$ of natural numbers with $X^2 + Y^2 = Z^2$ and $\\gcd(X, Y) = 1$ such that each of $X$, $Y$ and $Z$ is a sum of two squares of positive integers.","title":"Infinitely many primitive Pythagorean triples with sides sums of two squares"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/primitive-pythagorean-sides-square-middl","kind":"node","lean":"MathlibExt.NumberTheory.PrimitivePythagoreanFormsWanted.primitive_pythagorean_sides_square_middle_infinite","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#primitive-pythagorean-sides-square-middl","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"There are infinitely many triples $(X, Y, Z)$ of natural numbers with $X^2 + Y^2 = Z^2$ and $\\gcd(X, Y) = 1$ such that $X$ and $Z$ are each a sum of two squares of positive integers and $Y$ is the square of a positive integer.","title":"Infinitely many primitive Pythagorean triples with a square leg"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/primitive-pythagorean-sides-sum-two-cube","kind":"node","lean":"MathlibExt.NumberTheory.PrimitivePythagoreanFormsWanted.primitive_pythagorean_sides_sum_two_cubes_infinite","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#primitive-pythagorean-sides-sum-two-cube","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"There are infinitely many triples $(X, Y, Z)$ of natural numbers with $X^2 + Y^2 = Z^2$ and $\\gcd(X, Y) = 1$ such that each of $X$, $Y$ and $Z$ is a sum of two cubes of positive integers.","title":"Infinitely many primitive Pythagorean triples with sides sums of two cubes"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/rational-right-triangles-reciprocal-sums","kind":"node","lean":"MathlibExt.NumberTheory.PrimitivePythagoreanFormsWanted.rational_right_triangles_reciprocal_sums_infinite","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#rational-right-triangles-reciprocal-sums","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"There are infinitely many triples $(X, Y, Z)$ of positive rational numbers with $X^2 + Y^2 = Z^2$ such that each of $X$, $Y$ and $Z$ has the form $1/u + 1/v$ for some positive integers $u$ and $v$.","title":"Infinitely many rational right triangles with sides sums of unit fractions"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/rational-right-triangles-reciprocal-diff","kind":"node","lean":"MathlibExt.NumberTheory.PrimitivePythagoreanFormsWanted.rational_right_triangles_reciprocal_differences_infinite","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#rational-right-triangles-reciprocal-diff","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"There are infinitely many triples $(X, Y, Z)$ of positive rational numbers with $X^2 + Y^2 = Z^2$ such that each of $X$, $Y$ and $Z$ has the form $1/u - 1/v$ for some positive integers $u$ and $v$.","title":"Infinitely many rational right triangles with sides differences of unit fractions"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/amato-cathetus-params","kind":"node","lean":"MetaMathlibExt.amatoCathetusParams","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#amato-cathetus-params","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"For a natural number $x$, let $n$ be the exponent of $2$ in $x$ (taken to be $0$ for $x = 0$). The set $C(x)$ consists of the natural numbers $d$ with $d \\le x$ and $d \\mid x^2$ such that, if $x$ is even, $d = 2^s l$ for some natural numbers $s$ and $l$ with $1 \\le s \\le n - 1$ and $l \\mid x^2$.","title":"Amato's cathetus parameter set"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/one-mem-amato-cathetus-params-three","kind":"node","lean":"MetaMathlibExt.one_mem_amatoCathetusParams_three","parent":9621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Amato's parameter set for the cathetus "],["math","\\(3\\)"],["text"," contains "],["math","\\(1\\)"]],"source":"Amato's parameter set for the cathetus $3$ contains $1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#one-mem-amato-cathetus-params-three","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"We have $1 \\in C(3)$, where $C(x)$ is Amato's cathetus parameter set.","title":"Amato's parameter set for the cathetus $3$ contains $1$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/two-mem-amato-cathetus-params-four","kind":"node","lean":"MetaMathlibExt.two_mem_amatoCathetusParams_four","parent":9621,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text"," is an Amato parameter for the cathetus "],["math","\\(4\\)"]],"source":"$2$ is an Amato parameter for the cathetus $4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#two-mem-amato-cathetus-params-four","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"The number $2$ belongs to Amato's parameter set $C(4)$ for the cathetus $4$.","title":"$2$ is an Amato parameter for the cathetus $4$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/pythagorean-cathetus-product","kind":"node","lean":"MetaMathlibExt.pythagorean_cathetus_product","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#pythagorean-cathetus-product","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"Let $a$ and $b$ be positive natural numbers, and let $d_1 \\in C(a)$ and $d_2 \\in C(b)$, where $C(x)$ is Amato's cathetus parameter set. Define the rational numbers $a_1 = a^2/(2 d_1) - d_1/2$, $a_2 = a^2/(2 d_1) + d_1/2$, $b_1 = b^2/(2 d_2) - d_2/2$, $b_2 = b^2/(2 d_2) + d_2/2$, $y = (ab)^2/(2 d_1 d_2) - d_1 d_2/2$ and $z = (ab)^2/(2 d_1 d_2) + d_1 d_2/2$, so that $(a, a_1, a_2)$, $(b, b_1, b_2)$ and $(ab, y, z)$\u2026","title":"Products of Pythagorean triples through a cathetus"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/pythagorean-triple-mul","kind":"node","lean":"MetaMathlibExt.pythagoreanTripleMul","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#pythagorean-triple-mul","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"For a commutative ring $R$, the product of two triples $(x, y, z)$ and $(x', y', z')$ in $R^3$ is","title":"Amato's product of triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/pythagorean-triple-mul-three-four-five","kind":"node","lean":"MetaMathlibExt.pythagoreanTripleMul_three_four_five","parent":9621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Amato's product of "],["math","\\((3, 4, 5)\\)"],["text"," with itself"]],"source":"Amato's product of $(3, 4, 5)$ with itself"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#pythagorean-triple-mul-three-four-five","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"For integer triples, $(3, 4, 5) \\star (3, 4, 5) = (9, 40, 41)$, where $\\star$ is Amato's product of triples.","title":"Amato's product of $(3, 4, 5)$ with itself"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/pythagorean-triple-mul-one","kind":"node","lean":"MetaMathlibExt.pythagoreanTripleMul_one","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#pythagorean-triple-mul-one","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"For every commutative ring $R$ and every triple $u \\in R^3$, we have $u \\star (1, 0, 1) = u$, where $\\star$ is Amato's product of triples.","title":"Right identity for Amato's product"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/rat-pythagorean-triples","kind":"node","lean":"MetaMathlibExt.ratPythagoreanTriples","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#rat-pythagorean-triples","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"The set $E$ consists of the triples $(x, y, z)$ of rational numbers with $x \\ne 0$ and $x^2 + y^2 = z^2$.","title":"Rational Pythagorean triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/three-four-five-mem-rat-pythagorean-trip","kind":"node","lean":"MetaMathlibExt.three_four_five_mem_ratPythagoreanTriples","parent":9621,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((3, 4, 5)\\)"],["text"," is a rational Pythagorean triple"]],"source":"$(3, 4, 5)$ is a rational Pythagorean triple"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#three-four-five-mem-rat-pythagorean-trip","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"The triple $(3, 4, 5)$ belongs to the set $E$ of rational triples $(x, y, z)$ with $x \\neq 0$ and $x^2 + y^2 = z^2$.","title":"$(3, 4, 5)$ is a rational Pythagorean triple"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/pythagorean-triple-mul-mem-rat-pythagore","kind":"node","lean":"MetaMathlibExt.pythagoreanTripleMul_mem_ratPythagoreanTriples","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#pythagorean-triple-mul-mem-rat-pythagore","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"Let $E$ be the set of rational triples $(x, y, z)$ with $x \\ne 0$ and $x^2 + y^2 = z^2$, and let $\\star$ be Amato's product of triples. If $u \\in E$ and $v \\in E$, then $u \\star v \\in E$; that is, $\\star$ is a binary operation on $E$.","title":"Amato's product preserves rational Pythagorean triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/rat-pythagorean-triples-comm-group","kind":"node","lean":"MetaMathlibExt.ratPythagoreanTriples_commGroup","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#rat-pythagorean-triples-comm-group","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"Let $E$ be the set of rational triples $(x, y, z)$ with $x \\ne 0$ and $x^2 + y^2 = z^2$, and let $\\star$ be Amato's product of triples. Then $(1, 0, 1) \\in E$; $(u \\star v) \\star w = u \\star (v \\star w)$ for all $u, v, w \\in E$; $u \\star v = v \\star u$ for all $u, v \\in E$; $u \\star (1, 0, 1) = u$ for every $u \\in E$; every $u \\in E$ has some $v \\in E$ with $u \\star v = (1, 0, 1)$; and $E$ is infinite.","title":"Group laws for Amato's product on rational Pythagorean triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/int-pythagorean-triples-comm-monoid","kind":"node","lean":"MetaMathlibExt.intPythagoreanTriples_commMonoid","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#int-pythagorean-triples-comm-monoid","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"Let $M$ be the set of integer triples $(a, b, c)$ with $a^2 + b^2 = c^2$, let $N$ be the set of triples in $M$ whose three entries are nonnegative, and let $\\star$ be Amato's product of triples. Then the product of two triples in $M$ lies in $M$, $\\star$ is associative and commutative on $M$, and $u \\star (1, 0, 1) = u$ for every $u \\in M$. Moreover $(1, 0, 1) \\in N$, the product of two triples in $N$ lies in $N$,\u2026","title":"Integral Pythagorean triples form a commutative monoid"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/unit-circle-rat-mul","kind":"node","lean":"MetaMathlibExt.unitCircleRatMul","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#unit-circle-rat-mul","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"For rational pairs $(x, y)$ and $(x', y')$, the product is $(x, y) \\odot (x', y') = (x x' / (1 + y y'), (y + y') / (1 + y y'))$, where division by $0$ gives $0$; so the product is $(0, 0)$ when $1 + y y' = 0$.","title":"Amato's product on rational points of the unit circle"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/unit-circle-rat-mul-three-four-five","kind":"node","lean":"MetaMathlibExt.unitCircleRatMul_three_four_five","parent":9621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squaring "],["math","\\((3/5, 4/5)\\)"],["text"," under Amato's product"]],"source":"Squaring $(3/5, 4/5)$ under Amato's product"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#unit-circle-rat-mul-three-four-five","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"fully_proved","summary":"For Amato's product $(x, y) \\odot (x', y') = (x x' / (1 + y y'), (y + y') / (1 + y y'))$ on $\\mathbb{Q}^2$, we have $(3/5, 4/5) \\odot (3/5, 4/5) = (9/41, 40/41)$.","title":"Squaring $(3/5, 4/5)$ under Amato's product"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/unit-circle-rat-comm-group","kind":"node","lean":"MetaMathlibExt.unitCircleRat_commGroup","parent":9621,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples/index.html#unit-circle-rat-comm-group","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/pythagorean-triples","status":"can_prove","summary":"Let $P = \\{(x, y) \\in \\mathbb{Q}^2 : x \\neq 0,\\ x^2 + y^2 = 1\\}$ and let $\\odot$ be Amato's product $(x, y) \\odot (x', y') = (x x' / (1 + y y'), (y + y') / (1 + y y'))$. Then $(1, 0) \\in P$; for all $p, q \\in P$ we have $p \\odot q \\in P$; $\\odot$ is associative and commutative on $P$; for every $(x, y) \\in P$ we have $(x, y) \\odot (1, 0) = (x, y)$ and $(x, y) \\odot (x, -y) = (1, 0)$; and $P$ is infinite.","title":"Amato's group law on rational points of the unit circle"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","kind":"scope","navigation":"module","parent":9455,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","status":"can_prove","title":"Shadow Markov triples"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/shadow-markoff-mutate","kind":"node","lean":"MetaMathlibExt.shadowMarkoffMutate","parent":9642,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/index.html#shadow-markoff-mutate","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","status":"fully_proved","summary":"A state is a pair $(x, y)$ of a classical part $x \\in \\mathbb{Q}^3$ and a shadow part $y \\in \\mathbb{Q}^3$, with coordinates indexed by the positions $0, 1, 2$. The shadow Markoff mutation at position $i$ replaces $x_i$ by $x_i' = \\bigl(\\sum_{j \\ne i} x_j^2\\bigr) / x_i$ and $y_i$ by $\\bigl(-x_i' y_i + 2 \\sum_{j \\ne i} x_j y_j\\bigr) / x_i$, and leaves every other coordinate of $x$ and $y$ unchanged.","title":"Shadow Markoff mutation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/shadow-markoff-root","kind":"node","lean":"MetaMathlibExt.shadowMarkoffRoot","parent":9642,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/index.html#shadow-markoff-root","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","status":"fully_proved","summary":"For $v = (\\alpha, \\beta, \\gamma) \\in \\mathbb{Z}^3$, the initial triple with shadow part $v$ is $(1 + \\alpha\\varepsilon, 1 + \\beta\\varepsilon, 1 + \\gamma\\varepsilon)$, where $\\varepsilon^2 = 0$. It is stored as the state $(x, y)$ with classical part $x = (1, 1, 1) \\in \\mathbb{Q}^3$ and shadow part $y = (\\alpha, \\beta, \\gamma) \\in \\mathbb{Q}^3$.","title":"Initial triple of a shadow Markoff tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/shadow-markoff-tree-node","kind":"node","lean":"MetaMathlibExt.ShadowMarkoffTreeNode","parent":9642,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/index.html#shadow-markoff-tree-node","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","status":"fully_proved","summary":"Fix $v \\in \\mathbb{Z}^3$ and let $r$ be the initial triple with shadow part $v$, viewed as a state (a pair of a classical part and a shadow part, both in $\\mathbb{Q}^3$, with coordinates indexed by the positions $0, 1, 2$). The lower nodes of the shadow Markoff tree of $v$ are pairs $(s, l)$ of a state $s$ and a position $l$, the position mutated last. They are defined inductively: the state obtained from $r$ by\u2026","title":"Lower nodes of a shadow Markoff tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/shadow-markoff-tree","kind":"node","lean":"MetaMathlibExt.shadowMarkoffTree","parent":9642,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/index.html#shadow-markoff-tree","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","status":"fully_proved","summary":"For $v \\in \\mathbb{Z}^3$, the shadow Markoff tree of $v$ is the set of states $(x, y) \\in \\mathbb{Q}^3 \\times \\mathbb{Q}^3$ (classical part $x$, shadow part $y$) consisting of the initial triple $r$ with shadow part $v$, the shadow Markoff mutation of $r$ at position $0$, and the state $s$ of every lower node $(s, l)$ of the shadow Markoff tree of $v$.","title":"Shadow Markoff tree"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/shadow-markoff-mutate-vertex-001","kind":"node","lean":"MetaMathlibExt.shadowMarkoffMutate_vertex_001","parent":9642,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first two mutations in the tree of the vertex "],["math","\\((0:0:1)\\)"]],"source":"The first two mutations in the tree of the vertex $(0:0:1)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/index.html#shadow-markoff-mutate-vertex-001","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","status":"fully_proved","summary":"Number the positions $0, 1, 2$ and write a state as $(x \\mid y)$, with classical part $x \\in \\mathbb{Q}^3$ and shadow part $y \\in \\mathbb{Q}^3$. The initial triple $(1, 1, 1 + \\varepsilon)$, where $\\varepsilon^2 = 0$, has shadow part $(0, 0, 1)$ (the paper's vertex $(0:0:1)$) and is the state $(1, 1, 1 \\mid 0, 0, 1)$. Its shadow Markoff mutation at position $0$ is $(2, 1, 1 \\mid 2, 0, 1)$, and the mutation of that\u2026","title":"The first two mutations in the tree of the vertex $(0:0:1)$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/exists-polygon-shadow-markoff-tree-nonne","kind":"node","lean":"MetaMathlibExt.exists_polygon_shadowMarkoffTree_nonneg_iff","parent":9642,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/index.html#exists-polygon-shadow-markoff-tree-nonne","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","status":"can_prove","summary":"There is a finite set $V \\subseteq \\mathbb{Q}^3$ such that for every $v = (\\alpha, \\beta, \\gamma) \\in \\mathbb{Z}^3$ the following are equivalent: every state $(x, y)$ of the shadow Markoff tree of $v$, the tree of the initial triple $(1 + \\alpha\\varepsilon, 1 + \\beta\\varepsilon, 1 + \\gamma\\varepsilon)$ with $\\varepsilon^2 = 0$, has all coordinates of its classical part $x$ and of its shadow part $y$ nonnegative;\u2026","title":"Nonnegativity on a shadow Markoff tree is governed by finitely many points"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/shadow-markoff-tree-one-shadow-eq","kind":"node","lean":"MetaMathlibExt.shadowMarkoffTree_one_shadow_eq","parent":9642,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shadow parts equal classical parts when "],["math","\\(v = (1, 1, 1)\\)"]],"source":"Shadow parts equal classical parts when $v = (1, 1, 1)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples/index.html#shadow-markoff-tree-one-shadow-eq","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/shadow-markov-triples","status":"can_prove","summary":"In the shadow Markoff tree of $v = (1, 1, 1)$, the tree of the initial triple $(1 + \\varepsilon, 1 + \\varepsilon, 1 + \\varepsilon)$ with $\\varepsilon^2 = 0$, every solution $(x, y)$ has shadow part $y$ equal to its classical part $x$.","title":"Shadow parts equal classical parts when $v = (1, 1, 1)$"},{"data_url":"scopes/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares.json","id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","kind":"scope","navigation":"module","parent":9455,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"can_prove","title":"Sums of consecutive squares"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/so-cseq","kind":"node","lean":"MetaMathlibExt.SoCSEq","parent":9650,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#so-cseq","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"fully_proved","summary":"Integers $a$, $b$, $c$ satisfy the sum of consecutive squares (SoCS) equation when $2\\,b(b+1)(2b+1) = a(a+1)(2a+1) + c(c+1)(2c+1)$.","title":"The sum of consecutive squares equation"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/is-so-cssolution","kind":"node","lean":"MetaMathlibExt.IsSoCSSolution","parent":9650,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#is-so-cssolution","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"fully_proved","summary":"A triple of integers $(a, b, c)$ is a SoCS solution when $0 < a + 1 < b < c$ and $(a, b, c)$ satisfies the sum of consecutive squares equation $2\\,b(b+1)(2b+1) = a(a+1)(2a+1) + c(c+1)(2c+1)$, that is, $(a+1)^2 + \\dots + b^2 = (b+1)^2 + \\dots + c^2$.","title":"Solutions of the sum of consecutive squares problem"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/is-so-cssolution-two-four-five","kind":"node","lean":"MetaMathlibExt.isSoCSSolution_two_four_five","parent":9650,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((2, 4, 5)\\)"],["text"," solves the sum of consecutive squares problem"]],"source":"$(2, 4, 5)$ solves the sum of consecutive squares problem"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#is-so-cssolution-two-four-five","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"fully_proved","summary":"The triple $(2, 4, 5)$ is a solution of the sum of consecutive squares problem: $0 < 3 < 4 < 5$ and $3^2 + 4^2 = 5^2$.","title":"$(2, 4, 5)$ solves the sum of consecutive squares problem"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/so-cs-ratio-bounds","kind":"node","lean":"MetaMathlibExt.soCS_ratio_bounds","parent":9650,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#so-cs-ratio-bounds","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"can_prove","summary":"For every solution $(a, b, c)$ of the sum of consecutive squares problem, $1 < (b - a)/(c - b) < 1 + 2^{1/3} + 2^{2/3}$.","title":"Bounds on the ratio of the two spans"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/so-cs-pell-param-pos-nonsquare","kind":"node","lean":"MetaMathlibExt.soCS_pell_param_pos_nonsquare","parent":9650,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#so-cs-pell-param-pos-nonsquare","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"can_prove","summary":"For every solution $(a, b, c)$ of the sum of consecutive squares problem, the integer $A = 3(b - c)^2\\bigl(12(b - a)^2(c - b)^2 - (a - 2b + c)^4\\bigr)$ is positive and is not a perfect square.","title":"The Pell-type parameter is a positive nonsquare"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/so-cs-infinite-family-plane","kind":"node","lean":"MetaMathlibExt.soCS_infinite_family_plane","parent":9650,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#so-cs-infinite-family-plane","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"can_prove","summary":"Let $(a, b, c)$ be a solution of the sum of consecutive squares problem. Then there are infinitely many distinct integer triples $(x, y, z)$ satisfying the sum of consecutive squares equation $2\\,y(y+1)(2y+1) = x(x+1)(2x+1) + z(z+1)(2z+1)$, with no order or sign condition, such that $(b - c)x + (c - a)y + (a - b)z = 0$ and $z - x \\equiv c - a \\pmod 2$.","title":"Infinite families of solutions in a plane"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/so-cs-odd-span","kind":"node","lean":"MetaMathlibExt.soCS_odd_span","parent":9650,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions with odd span "],["math","\\(c - a\\)"]],"source":"Solutions with odd span $c - a$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#so-cs-odd-span","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"can_prove","summary":"For every odd integer $N \\ge 3$ there is a solution $(a, b, c)$ of the sum of consecutive squares problem with $c - a = N$, together with infinitely many distinct solutions $(x, y, z)$ of that problem with $z - x$ odd, all lying in the plane $(b - c)x + (c - a)y + (a - b)z = 0$, which passes through $(a, b, c)$ and contains the line $x = y = z$.","title":"Solutions with odd span $c - a$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/so-cs-even-span-infinite","kind":"node","lean":"MetaMathlibExt.soCS_even_span_infinite","parent":9650,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many solutions with "],["math","\\(c - a\\)"],["text"," even"]],"source":"Infinitely many solutions with $c - a$ even"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#so-cs-even-span-infinite","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"can_prove","summary":"There are infinitely many solutions $(a, b, c)$ of the sum of consecutive squares problem with $c - a$ even.","title":"Infinitely many solutions with $c - a$ even"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/so-cs-card-le","kind":"node","lean":"MetaMathlibExt.soCS_card_le","parent":9650,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most "],["math","\\(2(N - 1)\\)"],["text"," solutions with "],["math","\\(c - a = N\\)"]],"source":"At most $2(N - 1)$ solutions with $c - a = N$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#so-cs-card-le","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"can_prove","summary":"For every integer $N > 1$, there are at most $2(N - 1)$ solutions $(a, b, c)$ of the sum of consecutive squares problem with $c - a = N$.","title":"At most $2(N - 1)$ solutions with $c - a = N$"},{"id":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/so-cs-not-finite-planes","kind":"node","lean":"MetaMathlibExt.soCS_not_finite_planes","parent":9650,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares/index.html#so-cs-not-finite-planes","scope":"NumberTheory/diophantine-equations/pell-quadratic-equations/sums-of-consecutive-squares","status":"can_prove","summary":"Let $S$ be a finite set of affine planes $ux + vy + wz = t$ in $\\mathbb{R}^3$, with $u, v, w, t$ real and $(u, v, w) \\ne (0, 0, 0)$. Then some solution $(a, b, c)$ of the sum of consecutive squares problem lies on none of the planes in $S$, that is, $ua + vb + wc \\ne t$ for every plane in $S$.","title":"No finitely many planes contain all solutions"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations.json","id":"NumberTheory/diophantine-equations/higher-degree-equations","kind":"scope","parent":9409,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations","status":"can_state","summary":"Thue equations, generalized Fermat equations, sums of cubes, x^y = y^x, abc conjecture, Brocard\u2013Ramanujan and symmetric-function equations.","title":"Higher-degree Diophantine equations"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-taxicab-seed-exists","kind":"node","lean":"MetaMathlibExt.generalized_taxicab_seed_exists","parent":9661,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/index.html#generalized-taxicab-seed-exists","scope":"NumberTheory/diophantine-equations/higher-degree-equations","status":"fully_proved","summary":"Fix natural numbers $m \\ge 1$ and $t \\ge 1$, and let $T(n, m, t)$ be, for each natural number $n$, either a natural number or undefined, such that for all natural numbers $n$ and $v$, $T(n, m, t)$ is defined and equals $v$ exactly when $v$ is the least natural number with at least $t$ representations as a sum of exactly $n$ positive $m$-th powers, representations being counted as unordered multisets of summands.\u2026","title":"Existence of seeds for generalized taxicab numbers"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/rational-parametrization-pow-swap","kind":"node","lean":"MetaMathlibExt.rational_parametrization_pow_swap","parent":9661,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rational solutions of "],["math","\\(x^y = y^x\\)"]],"source":"Rational solutions of $x^y = y^x$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/index.html#rational-parametrization-pow-swap","scope":"NumberTheory/diophantine-equations/higher-degree-equations","status":"fully_proved","summary":"If $x$ and $y$ are positive rational numbers with $y < x$ and $x^y = y^x$ (as real powers), then there is a positive integer $v$ such that $x = (1 + 1/v)^{v+1}$ and $y = (1 + 1/v)^{v}$.","title":"Rational solutions of $x^y = y^x$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/sophie-germain","kind":"node","lean":"MetaMathlibExt.sophieGermain","parent":9661,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/index.html#sophie-germain","scope":"NumberTheory/diophantine-equations/higher-degree-equations","status":"fully_proved","summary":"Let $p$ be an odd prime such that $2p + 1$ is also prime. If $x$, $y$, $z$ are pairwise coprime positive integers with $x^p + y^p = z^p$, then $p$ divides $xyz$.","title":"Sophie Germain's theorem"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/abc-conjecture","kind":"node","lean":"MetaMathlibExt.abc_conjecture","parent":9661,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/index.html#abc-conjecture","scope":"NumberTheory/diophantine-equations/higher-degree-equations","status":"can_prove","summary":"For every real $\\varepsilon > 0$ there is a real $C_\\varepsilon > 0$ such that for all integers $a$, $b$, $c$ with $a + b = c$ and $\\gcd(a, b) = 1$,","title":"The abc conjecture"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation.json","id":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","kind":"scope","navigation":"module","parent":9661,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","status":"can_prove","title":"Brocard\u2013Ramanujan equation"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/brocard-ramanujan-power-free-finite","kind":"node","lean":"MetaMathlibExt.brocardRamanujan_powerFree_finite","parent":9666,"reading_title":{"format":"inline-parts-v1","parts":[["text","Brocard\u2013Ramanujan solutions with "],["math","\\(x - 1\\)"],["text"," or "],["math","\\(x + 1\\)"],["text"," power-free are finite"]],"source":"Brocard\u2013Ramanujan solutions with $x - 1$ or $x + 1$ power-free are finite"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/index.html#brocard-ramanujan-power-free-finite","scope":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","status":"can_prove","summary":"For every natural number $k \\ge 2$, there are only finitely many pairs of natural numbers $(n, x)$ with $n! + 1 = x^2$ such that $x - 1$ or $x + 1$ is $k$-free.","title":"Brocard\u2013Ramanujan solutions with $x - 1$ or $x + 1$ power-free are finite"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/brocard-ramanujan-seven-free-subset","kind":"node","lean":"MetaMathlibExt.brocardRamanujan_sevenFree_subset","parent":9666,"reading_title":{"format":"inline-parts-v1","parts":[["text","Brocard\u2013Ramanujan solutions with "],["math","\\(x - 1\\)"],["text"," or "],["math","\\(x + 1\\)"],["text"," seven-free"]],"source":"Brocard\u2013Ramanujan solutions with $x - 1$ or $x + 1$ seven-free"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/index.html#brocard-ramanujan-seven-free-subset","scope":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","status":"can_prove","summary":"If natural numbers $n$ and $x$ satisfy $n! + 1 = x^2$ and $x - 1$ or $x + 1$ is $7$-free, then $(n, x)$ is one of $(4, 5)$, $(5, 11)$ and $(7, 71)$.","title":"Brocard\u2013Ramanujan solutions with $x - 1$ or $x + 1$ seven-free"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/brocard-ramanujan-few-prime-factors-fini","kind":"node","lean":"MetaMathlibExt.brocardRamanujan_fewPrimeFactors_finite","parent":9666,"reading_title":{"format":"inline-parts-v1","parts":[["text","Brocard\u2013Ramanujan solutions with "],["math","\\(x - 1\\)"],["text"," or "],["math","\\(x + 1\\)"],["text"," having few prime factors are finite"]],"source":"Brocard\u2013Ramanujan solutions with $x - 1$ or $x + 1$ having few prime factors are finite"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/index.html#brocard-ramanujan-few-prime-factors-fini","scope":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","status":"can_prove","summary":"For every natural number $l \\ge 2$, there are only finitely many pairs of natural numbers $(n, x)$ with $n! + 1 = x^2$ such that $x - 1$ or $x + 1$ has fewer than $l$ distinct prime divisors.","title":"Brocard\u2013Ramanujan solutions with $x - 1$ or $x + 1$ having few prime factors are finite"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/brocard-ramanujan-prime-pow-subset","kind":"node","lean":"MetaMathlibExt.brocardRamanujan_primePow_subset","parent":9666,"reading_title":{"format":"inline-parts-v1","parts":[["text","Brocard\u2013Ramanujan solutions with "],["math","\\(x - 1\\)"],["text"," or "],["math","\\(x + 1\\)"],["text"," a prime power"]],"source":"Brocard\u2013Ramanujan solutions with $x - 1$ or $x + 1$ a prime power"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/index.html#brocard-ramanujan-prime-pow-subset","scope":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","status":"can_prove","summary":"If natural numbers $n$ and $x$ satisfy $n! + 1 = x^2$ and $x - 1$ or $x + 1$ is a prime power, then $(n, x) = (4, 5)$.","title":"Brocard\u2013Ramanujan solutions with $x - 1$ or $x + 1$ a prime power"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/le-exp-of-dvd-factorial","kind":"node","lean":"MetaMathlibExt.le_exp_of_dvd_factorial","parent":9666,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/index.html#le-exp-of-dvd-factorial","scope":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","status":"can_prove","summary":"Let $n \\ge 1$ and let $x$ be a positive integer dividing $n!$. Write $K(x)$ for the largest exponent in the prime factorization of $x$, with $K(1) = 0$. Then $x \\le \\exp(3 n K(x) / 2)$.","title":"Divisors of a factorial bounded by their largest prime exponent"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/le-rpow-mul-pow-of-dvd-factorial","kind":"node","lean":"MetaMathlibExt.le_rpow_mul_pow_of_dvd_factorial","parent":9666,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/index.html#le-rpow-mul-pow-of-dvd-factorial","scope":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","status":"can_prove","summary":"Let $n \\ge 1$ and let $x$ be a positive integer dividing $n!$, and write $\\omega(x)$ for the number of distinct prime divisors of $x$. Then for every positive integer $a$, $x \\le n^{n/a} (a \\, \\omega(x) + 1)^{n \\, \\omega(x)}$, where $n^{n/a}$ is a real power.","title":"Divisors of a factorial bounded by their number of prime factors"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/brocard-ramanujan-polynomial-power-free","kind":"node","lean":"MetaMathlibExt.brocardRamanujan_polynomial_powerFree_mul_fewPrimeFactors_finite","parent":9666,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation/index.html#brocard-ramanujan-polynomial-power-free","scope":"NumberTheory/diophantine-equations/higher-degree-equations/brocard-ramanujan-equation","status":"can_prove","summary":"Let $k \\ge 2$, $l \\ge 2$ and $m \\ge 1$ be natural numbers and let $P \\in \\mathbb{Z}[X_1, \\ldots, X_m]$. Write $F_k P_l$ for the set of products $y z$ of a $k$-free number $y$ and a positive integer $z$ with fewer than $l$ distinct prime divisors. Then there are only finitely many pairs $(n, x)$ of a natural number $n$ and an integer $x$ such that $n! + 1 = x^2$ and $x = x_1 \\cdots x_m P(x_1, \\ldots, x_m) + 1$ or\u2026","title":"Finitely many Brocard\u2013Ramanujan solutions of polynomial shape"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers.json","id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","kind":"scope","navigation":"module","parent":9661,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","title":"Consecutive powerful numbers"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/cube-split-of-mul-eq-prime-sq-mul-cube-o","kind":"node","lean":"MetaMathlibExt.cube_split_of_mul_eq_prime_sq_mul_cube_of_gcd_eq_one","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Splitting "],["math","\\(RS = p^2 C^3\\)"],["text"," when "],["math","\\(R\\)"],["text"," and "],["math","\\(S\\)"],["text"," are coprime"]],"source":"Splitting $RS = p^2 C^3$ when $R$ and $S$ are coprime"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#cube-split-of-mul-eq-prime-sq-mul-cube-o","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"Let $p$ be a prime and let $R$, $S$, $C$ be integers with $RS = p^2 C^3$ and $\\gcd(R, S) = 1$. Then there are integers $C_1$, $C_2$ such that either $R = C_1^3$ and $S = p^2 C_2^3$, or $R = p^2 C_1^3$ and $S = C_2^3$.","title":"Splitting $RS = p^2 C^3$ when $R$ and $S$ are coprime"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/cube-split-of-mul-eq-prime-sq-mul-cube-o-2","kind":"node","lean":"MetaMathlibExt.cube_split_of_mul_eq_prime_sq_mul_cube_of_gcd_prime","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Splitting "],["math","\\(RS = p^2 C^3\\)"],["text"," when "],["math","\\(\\gcd(R, S)\\)"],["text"," is prime"]],"source":"Splitting $RS = p^2 C^3$ when $\\gcd(R, S)$ is prime"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#cube-split-of-mul-eq-prime-sq-mul-cube-o-2","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"Let $p$ be a prime and let $R$, $S$, $C$ be integers with $RS = p^2 C^3$ such that $g = \\gcd(R, S)$ is prime. Then there are integers $C_1$, $C_2$ such that one of the following holds: $R = g C_1^3$ and $S = g C_2^3$; or $R = g C_1^3$ and $S = g^2 p^2 C_2^3$; or $S = g C_1^3$ and $R = g^2 p^2 C_2^3$; or $R = g p^2 C_1^3$ and $S = g^2 C_2^3$; or $S = g p^2 C_1^3$ and $R = g^2 C_2^3$.","title":"Splitting $RS = p^2 C^3$ when $\\gcd(R, S)$ is prime"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/sq-add-add-one-eq-three-mul-cube-iff","kind":"node","lean":"MetaMathlibExt.sq_add_add_one_eq_three_mul_cube_iff","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer solutions of "],["math","\\(u^2 \\pm u + 1 = 3v^3\\)"]],"source":"Integer solutions of $u^2 \\pm u + 1 = 3v^3$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#sq-add-add-one-eq-three-mul-cube-iff","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For all integers $u$ and $v$, $u^2 + u + 1 = 3v^3$ if and only if $(u, v) = (-2, 1)$ or $(u, v) = (1, 1)$, and $u^2 - u + 1 = 3v^3$ if and only if $(u, v) = (2, 1)$ or $(u, v) = (-1, 1)$.","title":"Integer solutions of $u^2 \\pm u + 1 = 3v^3$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/sq-add-add-one-eq-cube-iff","kind":"node","lean":"MetaMathlibExt.sq_add_add_one_eq_cube_iff","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","When "],["math","\\(u^2 \\pm u + 1\\)"],["text"," is a cube"]],"source":"When $u^2 \\pm u + 1$ is a cube"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#sq-add-add-one-eq-cube-iff","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For every integer $u$, $u^2 + u + 1$ is the cube of an integer if and only if $u \\in \\{-19, -1, 0, 18\\}$, and $u^2 - u + 1$ is the cube of an integer if and only if $u \\in \\{-18, 0, 1, 19\\}$.","title":"When $u^2 \\pm u + 1$ is a cube"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/gcd-sub-one-sq-add-add-one","kind":"node","lean":"MetaMathlibExt.gcd_sub_one_sq_add_add_one","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","The gcd of "],["math","\\(x \\mp 1\\)"],["text"," and "],["math","\\(x^2 \\pm x + 1\\)"]],"source":"The gcd of $x \\mp 1$ and $x^2 \\pm x + 1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#gcd-sub-one-sq-add-add-one","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For every integer $x$, $\\gcd(x - 1, x^2 + x + 1) = \\gcd(x - 1, 3)$ and $\\gcd(x + 1, x^2 - x + 1) = \\gcd(x + 1, 3)$.","title":"The gcd of $x \\mp 1$ and $x^2 \\pm x + 1$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/cube-sub-cube-eq-two-or-one-iff","kind":"node","lean":"MetaMathlibExt.cube_sub_cube_eq_two_or_one_iff","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer solutions of "],["math","\\(u^3 - v^3 = 2\\)"],["text"," and "],["math","\\(u^3 - v^3 = 1\\)"]],"source":"Integer solutions of $u^3 - v^3 = 2$ and $u^3 - v^3 = 1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#cube-sub-cube-eq-two-or-one-iff","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For all integers $u$ and $v$, $u^3 - v^3 = 2$ if and only if $(u, v) = (1, -1)$, and $u^3 - v^3 = 1$ if and only if $(u, v) = (1, 0)$ or $(u, v) = (0, -1)$.","title":"Integer solutions of $u^3 - v^3 = 2$ and $u^3 - v^3 = 1$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/cube-sub-two-mul-cube-eq-one-iff","kind":"node","lean":"MetaMathlibExt.cube_sub_two_mul_cube_eq_one_iff","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer solutions of "],["math","\\(u^3 - 2v^3 = 1\\)"]],"source":"Integer solutions of $u^3 - 2v^3 = 1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#cube-sub-two-mul-cube-eq-one-iff","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For all integers $u$ and $v$, $u^3 - 2v^3 = 1$ if and only if $(u, v) = (1, 0)$ or $(u, v) = (-1, -1)$.","title":"Integer solutions of $u^3 - 2v^3 = 1$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/cube-sub-mul-cube-eq-one-iff-of-mem","kind":"node","lean":"MetaMathlibExt.cube_sub_mul_cube_eq_one_iff_of_mem","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer solutions of "],["math","\\(u^3 - d v^3 = 1\\)"],["text"," for "],["math","\\(d \\in \\{4, 18, 36\\}\\)"]],"source":"Integer solutions of $u^3 - d v^3 = 1$ for $d \\in \\{4, 18, 36\\}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#cube-sub-mul-cube-eq-one-iff-of-mem","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"Let $d$ be one of the integers $4$, $18$, $36$. For all integers $u$ and $v$, $u^3 - d v^3 = 1$ if and only if $(u, v) = (1, 0)$.","title":"Integer solutions of $u^3 - d v^3 = 1$ for $d \\in \\{4, 18, 36\\}$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/not-cube-sub-one-eq-prime-sq-mul-cube-an","kind":"node","lean":"MetaMathlibExt.not_cube_sub_one_eq_prime_sq_mul_cube_and_cube_add_one","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(x^3 - 1\\)"],["text"," and "],["math","\\(x^3 + 1\\)"],["text"," are not both a prime square times a cube"]],"source":"$x^3 - 1$ and $x^3 + 1$ are not both a prime square times a cube"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#not-cube-sub-one-eq-prime-sq-mul-cube-an","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"There are no primes $p$, $q$ and integers $x$, $a$, $b$ with $x^3 - 1 = p^2 a^3$ and $x^3 + 1 = q^2 b^3$.","title":"$x^3 - 1$ and $x^3 + 1$ are not both a prime square times a cube"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/pow-six-sub-one-ne-prime-sq-mul-prime-sq","kind":"node","lean":"MetaMathlibExt.pow_six_sub_one_ne_prime_sq_mul_prime_sq_mul_cube","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(x^6 - 1\\)"],["text"," is not "],["math","\\(p^2 q^2\\)"],["text"," times a nonzero cube"]],"source":"$x^6 - 1$ is not $p^2 q^2$ times a nonzero cube"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#pow-six-sub-one-ne-prime-sq-mul-prime-sq","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For all primes $p$, $q$ and all integers $x$ and $a$ with $a \\ne 0$, $x^6 - 1 \\ne p^2 q^2 a^3$.","title":"$x^6 - 1$ is not $p^2 q^2$ times a nonzero cube"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/int-sq-sub-int-sq-ne-two","kind":"node","lean":"MathlibExt.NumberTheory.PowerfulNumbersWanted.int_sq_sub_int_sq_ne_two","parent":9674,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#int-sq-sub-int-sq-ne-two","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For all integers $a$ and $b$, $a^2 - b^2 \\neq 2$.","title":"Two is not a difference of two squares"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/three-exact-dvd-sq-add-self-add-one","kind":"node","lean":"MathlibExt.NumberTheory.PowerfulNumbersWanted.three_exact_dvd_sq_add_self_add_one","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exact divisibility of "],["math","\\(x^2 + x + 1\\)"],["text"," by "],["math","\\(3\\)"]],"source":"Exact divisibility of $x^2 + x + 1$ by $3$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#three-exact-dvd-sq-add-self-add-one","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"If $x$ is an integer with $x \\equiv 1 \\pmod 3$, then $3$ divides $x^2 + x + 1$ but $9$ does not.","title":"Exact divisibility of $x^2 + x + 1$ by $3$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/three-exact-dvd-sq-sub-self-add-one","kind":"node","lean":"MathlibExt.NumberTheory.PowerfulNumbersWanted.three_exact_dvd_sq_sub_self_add_one","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exact divisibility of "],["math","\\(x^2 - x + 1\\)"],["text"," by "],["math","\\(3\\)"]],"source":"Exact divisibility of $x^2 - x + 1$ by $3$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#three-exact-dvd-sq-sub-self-add-one","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"If $x$ is an integer with $x \\equiv 2 \\pmod 3$, then $3$ divides $x^2 - x + 1$ but $9$ does not.","title":"Exact divisibility of $x^2 - x + 1$ by $3$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/exists-int-point-elliptic-of-exists-int","kind":"node","lean":"MathlibExt.NumberTheory.PowerfulNumbersWanted.exists_int_point_elliptic_of_exists_int_point_quartic","parent":9674,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#exists-int-point-elliptic-of-exists-int","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"Let $a$, $c$ and $e$ be integers with $a \\neq 0$. If there are integers $x \\neq 0$ and $y$ with $y^2 = a x^4 + c x^2 + e$, then there are integers $X \\neq 0$ and $Y$ with $Y^2 = X^3 + c X^2 + a e X$.","title":"Integer points on a quartic give integer points on a cubic"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/pell-three-eq-one-positive-solutions-iff","kind":"node","lean":"MathlibExt.NumberTheory.PowerfulNumbersWanted.pell_three_eq_one_positive_solutions_iff_and_recurrence","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positive solutions of "],["math","\\(x^2 - 3y^2 = 1\\)"]],"source":"Positive solutions of $x^2 - 3y^2 = 1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#pell-three-eq-one-positive-solutions-iff","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For integers $x$ and $y$, we have $x > 0$, $y > 0$ and $x^2 - 3y^2 = 1$ if and only if $x + y\\sqrt{3} = (2 + \\sqrt{3})^k$ in $\\mathbb{Z}[\\sqrt{3}]$ for some natural number $k \\ge 1$. Writing $(2 + \\sqrt{3})^k = u_k + v_k \\sqrt{3}$ for $k \\ge 0$, we have $(u_1, v_1) = (2, 1)$, $(u_2, v_2) = (7, 4)$, and $u_{k+2} = 4u_{k+1} - u_k$ and $v_{k+2} = 4v_{k+1} - v_k$ for every $k \\ge 0$.","title":"Positive solutions of $x^2 - 3y^2 = 1$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/pell-three-eq-neg-two-positive-solutions","kind":"node","lean":"MathlibExt.NumberTheory.PowerfulNumbersWanted.pell_three_eq_neg_two_positive_solutions_iff_and_recurrence","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positive solutions of "],["math","\\(x^2 - 3y^2 = -2\\)"]],"source":"Positive solutions of $x^2 - 3y^2 = -2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#pell-three-eq-neg-two-positive-solutions","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"For integers $x$ and $y$, we have $x > 0$, $y > 0$ and $x^2 - 3y^2 = -2$ if and only if $x + y\\sqrt{3} = (1 + \\sqrt{3})(2 + \\sqrt{3})^k$ in $\\mathbb{Z}[\\sqrt{3}]$ for some integer $k \\ge 0$. Writing $(1 + \\sqrt{3})(2 + \\sqrt{3})^k = u_k + v_k \\sqrt{3}$ for $k \\ge 0$, we have $(u_0, v_0) = (1, 1)$, $(u_1, v_1) = (5, 3)$, and $u_{k+2} = 4u_{k+1} - u_k$ and $v_{k+2} = 4v_{k+1} - v_k$ for every $k \\ge 0$.","title":"Positive solutions of $x^2 - 3y^2 = -2$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/not-exists-prime-cube-square-neighbors-o","kind":"node","lean":"MathlibExt.NumberTheory.PowerfulNumbersWanted.not_exists_prime_cube_square_neighbors_of_cube","parent":9674,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#not-exists-prime-cube-square-neighbors-o","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"There are no primes $p$, $q$ and positive integers $x$, $y$, $z$ with $x^3 - 1 = p^3 y^2$ and $x^3 + 1 = q^3 z^2$.","title":"No cube whose neighbours are prime cubes times squares"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/not-exists-sixty-four-mul-pow-six-sub-on","kind":"node","lean":"MathlibExt.NumberTheory.PowerfulNumbersWanted.not_exists_sixty_four_mul_pow_six_sub_one_eq","parent":9674,"reading_title":{"format":"inline-parts-v1","parts":[["text","No integer solutions of "],["math","\\(64x^6 - 1 = p^3 q^3 y^2\\)"],["text"," with "],["math","\\(p\\)"],["text",", "],["math","\\(q\\)"],["text"," prime"]],"source":"No integer solutions of $64x^6 - 1 = p^3 q^3 y^2$ with $p$, $q$ prime"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers/index.html#not-exists-sixty-four-mul-pow-six-sub-on","scope":"NumberTheory/diophantine-equations/higher-degree-equations/consecutive-powerful-numbers","status":"can_prove","summary":"There are no integers $x$, $y$ and primes $p$, $q$ with $64 x^6 - 1 = p^3 q^3 y^2$.","title":"No integer solutions of $64x^6 - 1 = p^3 q^3 y^2$ with $p$, $q$ prime"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation.json","id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","kind":"scope","navigation":"module","parent":9661,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","status":"can_prove","title":"Generalized Fermat equation"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/generalized-fermat-solvable","kind":"node","lean":"MetaMathlibExt.GeneralizedFermatSolvable","parent":9693,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/index.html#generalized-fermat-solvable","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","status":"fully_proved","summary":"For natural numbers $m$, $n$, $r$, the equation $x^m + y^n = z^r$ is called solvable when there are nonzero integers $x$, $y$, $z$ with $x^m + y^n = z^r$.","title":"Solvable generalized Fermat equation"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/generalized-fermat-solvable-three-six-tw","kind":"node","lean":"MetaMathlibExt.generalizedFermatSolvable_three_six_two","parent":9693,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/index.html#generalized-fermat-solvable-three-six-tw","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","status":"fully_proved","summary":"The equation $x^3 + y^6 = z^2$ has a solution in nonzero integers, namely $2^3 + 1^6 = 3^2$.","title":"A solution of x\u00b3 + y\u2076 = z\u00b2"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/swap","kind":"node","lean":"MetaMathlibExt.GeneralizedFermatSolvable.swap","parent":9693,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/index.html#swap","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","status":"fully_proved","summary":"For natural numbers $m$, $n$, $r$, if $x^m + y^n = z^r$ has a solution in nonzero integers, then so does $x^n + y^m = z^r$.","title":"Swapping the first two exponents of a generalized Fermat equation"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/not-generalized-fermat-solvable-of-two-g","kind":"node","lean":"MetaMathlibExt.not_generalizedFermatSolvable_of_two_gcd_gt_one","parent":9693,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/index.html#not-generalized-fermat-solvable-of-two-g","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","status":"can_prove","summary":"Let $m, n, r \\ge 2$ be integers with $\\gcd(m, n, r) = 1$ such that at least two of $\\gcd(m, n)$, $\\gcd(m, r)$ and $\\gcd(n, r)$ are greater than $1$. Suppose there are no natural numbers $s, t, w$ that are pairwise coprime, with $s$ odd and $3 \\nmid w$, such that $(m, n, r)$ is one of $(3s, 6t, 2w)$, $(6t, 3s, 2w)$, $(2w, 3s, 6t)$ and $(3s, 2w, 6t)$. Then the equation $x^m + y^n = z^r$ has no solution in nonzero\u2026","title":"Generalized Fermat equations with two non-coprime pairs of exponents"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/pow-add-pow-ne-pow-of-prime-five-le","kind":"node","lean":"MetaMathlibExt.pow_add_pow_ne_pow_of_prime_five_le","parent":9693,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/index.html#pow-add-pow-ne-pow-of-prime-five-le","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","status":"can_prove","summary":"Let $p \\le q$ be primes with $p \\ge 5$, and let $x, y, z$ be pairwise coprime nonzero integers. Then $x^p + y^p \\ne z^q$.","title":"The Bartolom\u00e9\u2013Mih\u0103ilescu theorem for prime exponents"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/generalized-fermat-three-six-two-mul","kind":"node","lean":"MetaMathlibExt.generalizedFermat_three_six_two_mul","parent":9693,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/index.html#generalized-fermat-three-six-two-mul","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","status":"can_prove","summary":"Let $s, t, w$ be positive integers. If $s, t, w$ are pairwise coprime, $s$ is odd and $3 \\nmid w$, then the equation $x^{3s} + y^{6t} = z^{2w}$ has infinitely many solutions $(x, y, z)$ in nonzero integers; otherwise it has no solution in nonzero integers.","title":"Generalized Fermat equations with exponents 3s, 6t and 2w"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/generalized-fermat-mul-exponent","kind":"node","lean":"MetaMathlibExt.generalizedFermat_mul_exponent","parent":9693,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation/index.html#generalized-fermat-mul-exponent","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fermat-equation","status":"can_prove","summary":"Let $u, v \\ge 2$ be integers. (A) The equation $x^u + y^{uv} = z^v$ has a solution in nonzero integers $x, y, z$ if and only if it has one with $\\gcd(x, y, z) = 1$. (B) If $u$ or $v$ is even, $(u, v) \\ne (2, 2)$ and $(u, v) \\ne (3, 2)$, the equation has no solution in nonzero integers. (C) If $3 \\mid u$ or $3 \\mid v$, and $(u, v) \\ne (3, 2)$, the equation has no solution in nonzero integers.","title":"Generalized Fermat equations with exponents u, uv and v"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation.json","id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","kind":"scope","navigation":"module","parent":9661,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","title":"Generalized fruit equation"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/two-unramified-degree-one-primes","kind":"node","lean":"MetaMathlibExt.twoUnramifiedDegreeOnePrimes","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unramified primes of degree one above "],["math","\\(2\\)"]],"source":"Unramified primes of degree one above $2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#two-unramified-degree-one-primes","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"fully_proved","summary":"For a number field $K$ with ring of integers $\\mathcal{O}_K$, $T_K$ is the set of prime ideals $P$ of $\\mathcal{O}_K$ that contain $2$ and have ramification index $e(P \\mid 2) = 1$ and inertia degree $f(P \\mid 2) = 1$ over $\\mathbb{Z}$.","title":"Unramified primes of degree one above $2$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/mem-two-unramified-degree-one-primes","kind":"node","lean":"MetaMathlibExt.mem_twoUnramifiedDegreeOnePrimes","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in "],["math","\\(T_K\\)"]],"source":"Membership in $T_K$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#mem-two-unramified-degree-one-primes","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"fully_proved","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$, and let $T_K$ be the set of prime ideals of $\\mathcal{O}_K$ that contain $2$ and have ramification index $1$ and inertia degree $1$ over $\\mathbb{Z}$. Then an ideal $P$ of $\\mathcal{O}_K$ lies in $T_K$ if and only if $P$ is a prime ideal, $2 \\in P$, and $P$ has ramification index $1$ and inertia degree $1$ over $\\mathbb{Z}$.","title":"Membership in $T_K$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/ring-of-integers-quotient-pow-ring-equiv","kind":"node","lean":"MetaMathlibExt.ringOfIntegers_quotient_pow_ringEquiv_zmod_two_pow","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quotients by powers of a prime in "],["math","\\(T_K\\)"]],"source":"Quotients by powers of a prime in $T_K$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#ring-of-integers-quotient-pow-ring-equiv","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$, and let $P \\in T_K$, that is, $P$ is a prime ideal of $\\mathcal{O}_K$ lying over $2$ with ramification index $1$ and inertia degree $1$ over $\\mathbb{Z}$. Then for every $n \\ge 1$ the ring $\\mathcal{O}_K / P^n$ is isomorphic to $\\mathbb{Z} / 2^n \\mathbb{Z}$.","title":"Quotients by powers of a prime in $T_K$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/generalized-fruit-no-solution-two-dvd","kind":"node","lean":"MetaMathlibExt.generalizedFruit_no_solution_two_dvd","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","No solutions of the generalized fruit equation with even "],["math","\\(x\\)"]],"source":"No solutions of the generalized fruit equation with even $x$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#generalized-fruit-no-solution-two-dvd","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ such that some prime ideal of $\\mathcal{O}_K$ lying over $2$ has ramification index $1$ and inertia degree $1$ over $\\mathbb{Z}$. Let $a, b \\in \\mathcal{O}_K$ be nonzero, let $r \\ge 2$ and $d \\ge 3$ be odd natural numbers, and let $c = 2^d b - 3^r$. Then the equation $a x^d - y^2 - z^2 + x y z - c = 0$ has no solution $(x, y, z) \\in \\mathcal{O}_K^3$\u2026","title":"No solutions of the generalized fruit equation with even $x$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/two-unramified-degree-one-primes-nonempt","kind":"node","lean":"MetaMathlibExt.twoUnramifiedDegreeOnePrimes_nonempty_iff_quadratic","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unramified degree-one primes above "],["math","\\(2\\)"],["text"," in quadratic fields"]],"source":"Unramified degree-one primes above $2$ in quadratic fields"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#two-unramified-degree-one-primes-nonempt","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"Let $t \\ne 1$ be a squarefree integer, and let $K$ be a number field of degree $2$ over $\\mathbb{Q}$ that contains an element $\\alpha$ with $\\alpha^2 = t$, so that $K = \\mathbb{Q}(\\sqrt{t})$. Then the set $T_K$ of prime ideals of the ring of integers $\\mathcal{O}_K$ above $2$ with ramification index $1$ and inertia degree $1$ is nonempty if and only if $t \\equiv 1 \\pmod{8}$.","title":"Unramified degree-one primes above $2$ in quadratic fields"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/squarefree-one-mod-eight-relative-densit","kind":"node","lean":"MetaMathlibExt.squarefree_one_mod_eight_relativeDensity","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relative density of square-free integers congruent to "],["math","\\(1\\)"],["text"," modulo "],["math","\\(8\\)"]],"source":"Relative density of square-free integers congruent to $1$ modulo $8$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#squarefree-one-mod-eight-relative-densit","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"As $X \\to \\infty$ through the natural numbers, the number of square-free integers $t$ with $1 \\le t \\le X$ and $t \\equiv 1 \\pmod 8$, divided by the number of square-free integers $t$ with $1 \\le t \\le X$, tends to $1/6$.","title":"Relative density of square-free integers congruent to $1$ modulo $8$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/squarefree-count-arithmetic-progression","kind":"node","lean":"MetaMathlibExt.squarefree_count_arithmeticProgression_isEquivalent","parent":9701,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#squarefree-count-arithmetic-progression","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"Let $r \\in \\mathbb{Z}$ and let $N \\ge 1$ be an integer such that $s = \\gcd(r, N)$ is square-free. Then, as the real number $X \\to \\infty$, the number of square-free integers $t$ with $1 \\le t \\le X$ and $t \\equiv r \\pmod N$ is asymptotically equivalent to","title":"Square-free integers in an arithmetic progression"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/weierstrass-torsion-not-prime-pow-integr","kind":"node","lean":"MetaMathlibExt.weierstrass_torsion_not_primePow_integral","parent":9701,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#weierstrass-torsion-not-prime-pow-integr","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$, and let $E$ be a Weierstrass curve with coefficients in $\\mathcal{O}_K$ whose base change to $K$ is an elliptic curve. If $P = (x, y) \\in E(K)$ is a point of finite order $m \\ge 2$ and $m$ is not a prime power, then $x$ and $y$ lie in $\\mathcal{O}_K$.","title":"Torsion points of non-prime-power order are integral"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/short-weierstrass-torsion-integral","kind":"node","lean":"MetaMathlibExt.short_weierstrass_torsion_integral","parent":9701,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#short-weierstrass-torsion-integral","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"Let $E : y^2 = x^3 + A x + B$ be an elliptic curve over $\\mathbb{Q}$ with $A, B \\in \\mathbb{Z}$. Every point $P = (x, y) \\in E(\\mathbb{Q})$ of finite order $m \\ge 2$ has $x, y \\in \\mathbb{Z}$.","title":"Nagell\u2013Lutz integrality"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/fruit-weierstrass-curve","kind":"node","lean":"MetaMathlibExt.fruitWeierstrassCurve","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Weierstrass curve "],["math","\\(E_\\alpha\\)"],["text"," of the generalized fruit equation"]],"source":"The Weierstrass curve $E_\\alpha$ of the generalized fruit equation"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#fruit-weierstrass-curve","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"fully_proved","summary":"For a commutative ring $R$ and $\\alpha \\in R$, $E_\\alpha$ is the Weierstrass curve over $R$ with coefficients $a_1 = -\\alpha$, $a_2 = a_3 = a_4 = 0$ and $a_6 = -(\\alpha^2 + 5)$, that is, the curve $y^2 - \\alpha x y = x^3 - (\\alpha^2 + 5)$.","title":"The Weierstrass curve $E_\\alpha$ of the generalized fruit equation"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/fruit-weierstrass-curve-2","kind":"node","lean":"MetaMathlibExt.fruitWeierstrassCurve_\u0394","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Discriminant of the curve "],["math","\\(E_\\alpha\\)"]],"source":"Discriminant of the curve $E_\\alpha$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#fruit-weierstrass-curve-2","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"fully_proved","summary":"For every commutative ring $R$ and every $\\alpha \\in R$, the Weierstrass curve $E_\\alpha : y^2 - \\alpha x y = x^3 - (\\alpha^2 + 5)$ over $R$ has discriminant $\\alpha^8 + 5\\alpha^6 - 432\\alpha^4 - 4320\\alpha^2 - 10800$.","title":"Discriminant of the curve $E_\\alpha$"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/fruit-weierstrass-curve-not-is-of-fin-or","kind":"node","lean":"MetaMathlibExt.fruitWeierstrassCurve_not_isOfFinOrder_of_two_dvd","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Torsion points of "],["math","\\(E_\\alpha\\)"],["text"," with even integral "],["math","\\(x\\)"],["text","-coordinate have prime-power order"]],"source":"Torsion points of $E_\\alpha$ with even integral $x$-coordinate have prime-power order"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#fruit-weierstrass-curve-not-is-of-fin-or","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ such that some prime ideal of $\\mathcal{O}_K$ lying over $2$ has ramification index $1$ and inertia degree $1$ over $\\mathbb{Z}$. Let $\\alpha \\in \\mathcal{O}_K$ satisfy $\\alpha^8 + 5\\alpha^6 - 432\\alpha^4 - 4320\\alpha^2 - 10800 \\neq 0$, and let $E_\\alpha$ be the curve $y^2 - \\alpha x y = x^3 - (\\alpha^2 + 5)$ over $K$. If\u2026","title":"Torsion points of $E_\\alpha$ with even integral $x$-coordinate have prime-power order"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/fruit-weierstrass-curve-rat-not-is-of-fi","kind":"node","lean":"MetaMathlibExt.fruitWeierstrassCurve_rat_not_isOfFinOrder_of_two_dvd_num","parent":9701,"reading_title":{"format":"inline-parts-v1","parts":[["text","Points of "],["math","\\(E_\\alpha(\\mathbb{Q})\\)"],["text"," with even numerator have infinite order"]],"source":"Points of $E_\\alpha(\\mathbb{Q})$ with even numerator have infinite order"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation/index.html#fruit-weierstrass-curve-rat-not-is-of-fi","scope":"NumberTheory/diophantine-equations/higher-degree-equations/generalized-fruit-equation","status":"can_prove","summary":"Let $\\alpha$ be an integer with $\\alpha^8 + 5\\alpha^6 - 432\\alpha^4 - 4320\\alpha^2 - 10800 \\neq 0$, and let $E_\\alpha$ be the curve $y^2 - \\alpha x y = x^3 - (\\alpha^2 + 5)$ over $\\mathbb{Q}$. If $P = (x_0, y_0) \\in E_\\alpha(\\mathbb{Q})$ is a point such that $x_0$, written in lowest terms, has even numerator, then $P$ has infinite order.","title":"Points of $E_\\alpha(\\mathbb{Q})$ with even numerator have infinite order"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions.json","id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","kind":"scope","navigation":"module","parent":9661,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","title":"Integer triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.HasSquareElementarySymmetricFunctions","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"fully_proved","summary":"Integers $a$, $b$ and $c$ have square elementary symmetric functions when $a + b + c$, $ab + bc + ca$ and $abc$ are each the square of an integer.","title":"Integer triples with square elementary symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-2","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.HasSquareElementarySymmetricFunctions\u211a","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-2","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"fully_proved","summary":"Rational numbers $a$, $b$, $c$ have square elementary symmetric functions when each of $a + b + c$, $ab + bc + ca$ and $abc$ is the square of a rational number.","title":"Rationals with square elementary symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/is-sum-two-squares","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.IsSumTwoSquares\u211a","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#is-sum-two-squares","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"fully_proved","summary":"A rational number $q$ is a sum of two squares when $q = x^2 + y^2$ for some rational numbers $x$ and $y$.","title":"Sums of two rational squares"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/is-sum-two-squares-2","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.IsSumTwoSquares\u2124","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#is-sum-two-squares-2","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"fully_proved","summary":"An integer $z$ is a sum of two squares when $z = x^2 + y^2$ for some integers $x$ and $y$.","title":"Sums of two integer squares"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/is-sum-two-nonzero-squares","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.IsSumTwoNonzeroSquares\u2124","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#is-sum-two-nonzero-squares","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"fully_proved","summary":"An integer $z$ is a sum of two nonzero squares when $z = x^2 + y^2$ for some nonzero integers $x$ and $y$.","title":"Sums of two nonzero integer squares"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/sum-two-squares-div-sum-two-squares","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.sum_two_squares_div_sum_two_squares\u211a","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#sum-two-squares-div-sum-two-squares","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"If $\\alpha$ and $\\beta$ are nonzero rational numbers that are each a sum of two squares of rational numbers, then $\\alpha / \\beta$ is also a sum of two squares of rational numbers.","title":"Quotients of sums of two rational squares"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-3","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_pairwise_products","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-3","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"If $a$, $b$, $c$ are integers such that each of $a + b + c$, $ab + bc + ca$ and $abc$ is the square of an integer, then the same holds for the triple $ab$, $bc$, $ca$.","title":"Pairwise products preserve square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/is-sum-two-squares-of-has-square-element","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.isSumTwoSquares\u211a_of_hasSquareElementarySymmetricFunctions\u211a_and_scaling","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#is-sum-two-squares-of-has-square-element","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"Let $a$, $b$, $c$ be rational numbers such that $a + b + c$, $ab + bc + ca$ and $abc$ are squares of rational numbers. Then each of $a$, $b$ and $c$ is a sum of two squares of rational numbers. Moreover, there is a nonzero rational number $k$ such that $A = ka$, $B = kb$ and $C = kc$ are integers, $A + B + C$, $AB + BC + CA$ and $ABC$ are squares of integers, and each of $A$, $B$ and $C$ is a sum of two squares of\u2026","title":"Rational solutions are sums of two squares and rescale to integer solutions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-4","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family33","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-4","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all integers $s$ and $t$, put $a = t^2 (s^2 - t^2)^2 (s^2 + t^2)$, $b = s^2 (s^2 - t^2)^2 (s^2 + t^2)$ and $c = 4 s^4 t^4$. Then each of $a + b + c$, $ab + bc + ca$ and $abc$ is the square of an integer.","title":"Choudhry's family (3.3) of integer triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-5","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family34","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-5","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all integers $s$ and $t$, put $a = 4 s^4 t^2 (s^2 + t^2)$, $b = 4 s^2 t^4 (s^2 + t^2)$ and $c = (s^4 - t^4)^2$. Then each of $a + b + c$, $ab + bc + ca$ and $abc$ is the square of an integer.","title":"Choudhry's family (3.4) of integer triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-6","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family35","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-6","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all integers $s$ and $t$, let $f = s^6 - s^4 t^2 - 5 s^2 t^4 + t^6$ and $g = 3 s^6 + s^4 t^2 + s^2 t^4 - t^6$, and put $a = 4 s^4 t^2 (s^2 + t^2) f^2$, $b = (s^4 - t^4)^2 f^2$ and $c = 4 s^2 t^4 (s^2 + t^2) g^2$. Then each of $a + b + c$, $ab + bc + ca$ and $abc$ is the square of an integer.","title":"Choudhry's family (3.5) of integer triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-7","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family36","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-7","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all integers $s$ and $t$, let $f = s^6 - s^4 t^2 - 5 s^2 t^4 + t^6$ and $g = 3 s^6 + s^4 t^2 + s^2 t^4 - t^6$, and put $a = t^2 (s^2 - t^2)^2 (s^2 + t^2) g^2$, $b = 4 s^4 t^4 g^2$ and $c = s^2 (s^2 - t^2)^2 (s^2 + t^2) f^2$. Then each of $a + b + c$, $ab + bc + ca$ and $abc$ is the square of an integer.","title":"Choudhry's family (3.6) of integer triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-8","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family330_and_entries_square","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-8","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all integers $m$ and $n$, let $Q = m^4 - 6 m^2 n^2 + n^4$ and put $a = (m^4 - n^4)^2 Q^2$, $b = 4 m^2 n^2 (m^2 + n^2)^2 Q^2$ and $c = 64 m^4 n^4 (m^2 - n^2)^4$. Then each of $a$, $b$, $c$, $a + b + c$, $ab + bc + ca$ and $abc$ is the square of an integer.","title":"Choudhry's family (3.30) of square triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-9","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family331_and_entries_square","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-9","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all integers $m$ and $n$, let $a = 64 m^4 n^4 (m^2 - n^2)^2$, $b = 16 m^2 n^2 (m^2 - n^2)^4$ and $c = (m^2 + n^2)^2 (m^4 - 6 m^2 n^2 + n^4)^2$. Then $a + b + c$, $ab + bc + ca$ and $abc$ are squares of integers, and $a$, $b$ and $c$ are themselves squares of integers.","title":"Choudhry's family (3.31) of square triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-10","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family332_and_entries_square","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-10","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all integers $m$ and $n$, let $Q = m^4 - 6 m^2 n^2 + n^4$, $P = m^{12} - 26 m^{10} n^2 + 79 m^8 n^4 - 44 m^6 n^6 + 79 m^4 n^8 - 26 m^2 n^{10} + n^{12}$ and $R = m^{12} - 10 m^{10} n^2 + 15 m^8 n^4 - 204 m^6 n^6 + 15 m^4 n^8 - 10 m^2 n^{10} + n^{12}$, and put $a = 64 m^4 n^4 (m^2 - n^2)^2 P^2$, $b = (m^2 + n^2)^2 Q^2 P^2$ and $c = 16 m^2 n^2 (m^2 - n^2)^4 R^2$. Then each of $a$, $b$, $c$, $a + b + c$,\u2026","title":"Choudhry's family (3.32) of square triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-11","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family333_and_entries_square","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-11","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all integers $m$ and $n$, let $Q = m^4 - 6 m^2 n^2 + n^4$, $P = m^{12} - 26 m^{10} n^2 + 79 m^8 n^4 - 44 m^6 n^6 + 79 m^4 n^8 - 26 m^2 n^{10} + n^{12}$ and $R = m^{12} - 10 m^{10} n^2 + 15 m^8 n^4 - 204 m^6 n^6 + 15 m^4 n^8 - 10 m^2 n^{10} + n^{12}$, and put $a = (m^4 - n^4)^2 Q^2 R^2$, $b = 64 m^4 n^4 (m^2 - n^2)^4 R^2$ and $c = 4 m^2 n^2 (m^2 + n^2)^2 Q^2 P^2$. Then each of $a$, $b$, $c$, $a + b + c$,\u2026","title":"Choudhry's family (3.33) of square triples with square symmetric functions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/has-square-elementary-symmetric-function-12","kind":"node","lean":"MathlibExt.NumberTheory.ThreeIntegersSymmetricSquaresWanted.hasSquareElementarySymmetricFunctions_family334_and_entries_sum_two_nonzero_squares","parent":9715,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions/index.html#has-square-elementary-symmetric-function-12","scope":"NumberTheory/diophantine-equations/higher-degree-equations/integer-triples-with-square-symmetric-functions","status":"can_prove","summary":"For all nonzero integers $r$ and $s$, let $E = r^8 + 2 r^6 s^2 - 12 r^4 s^4 - 6 r^2 s^6 - s^8$, $T = r^{12} + 6 r^{10} s^2 + 87 r^8 s^4 + 108 r^6 s^6 + 55 r^4 s^8 + 14 r^2 s^{10} + s^{12}$ and $S = 5 r^6 + 3 r^4 s^2 + 3 r^2 s^4 + s^6$, and put $a = r^2 (r^2 + s^2) E^2 T$, $b = s^2 (r^2 + s^2) S^2 E^2$ and $c = 4 r^4 s^4 S^2 T$. Then each of $a + b + c$, $ab + bc + ca$ and $abc$ is the square of an integer, and each\u2026","title":"Choudhry's family (3.34) of sums of two nonzero squares"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres.json","id":"NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres","kind":"scope","navigation":"module","parent":9661,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres","status":"can_state","title":"Perfect powers in products of arithmetic progressions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres/solution","kind":"node","lean":"MathlibExt.NumberTheory.ErdosSupersingularPrimesWanted.ErdosSupersingularPrimes.Solution","parent":9733,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres/index.html#solution","scope":"NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres","status":"fully_proved","summary":"For a natural number $k$, a solution consists of a natural number $l$ and integers $n$, $d$, $y$ such that $\\gcd(n, d) = 1$ and","title":"Solutions of the Erd\u0151s equation for products of arithmetic progressions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres/k","kind":"node","lean":"MathlibExt.NumberTheory.ErdosSupersingularPrimesWanted.ErdosSupersingularPrimes.k\u2080","parent":9733,"reading_title":{"format":"inline-parts-v1","parts":[["text","The threshold constant "],["math","\\(k_0\\)"],["text"," in the Bennett\u2013Siksek theorem"]],"source":"The threshold constant $k_0$ in the Bennett\u2013Siksek theorem"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres/index.html#k","scope":"NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres","status":"can_state","summary":"A natural number $k_0$, to be supplied: the absolute constant from which Theorem 2 holds, that is, for every natural number $k \\ge k_0$, every solution in integers $n$, $d$, $y$ and a prime natural number $l$ of $\\prod_{i=0}^{k-1} (n + i d) = y^l$ with $\\gcd(n, d) = 1$ has $y = 0$ or $d = 0$ or $l \\le \\exp(10^k)$.","title":"The threshold constant $k_0$ in the Bennett\u2013Siksek theorem"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres/theorem-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosSupersingularPrimesWanted.ErdosSupersingularPrimes.theorem_2","parent":9733,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres/index.html#theorem-2","scope":"NumberTheory/diophantine-equations/higher-degree-equations/perfect-powers-in-products-of-arithmetic-progres","status":"stated","summary":"Let $k_0$ be the threshold constant of this module. For every natural number $k \\ge k_0$, every natural number $l$ that is prime, and all integers $n$, $d$, $y$ with $\\gcd(n, d) = 1$ and","title":"Bennett\u2013Siksek theorem on perfect powers in arithmetic progressions"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations.json","id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","kind":"scope","navigation":"module","parent":9661,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"can_prove","title":"Split families of Thue equations"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/is-split-thue-recurrence-family","kind":"node","lean":"MetaMathlibExt.IsSplitThueRecurrenceFamily","parent":9737,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html#is-split-thue-recurrence-family","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$, let $G_i(n)$ be integers for $i, n \\in \\mathbb{N}$, and let $\\gamma_i \\in \\mathbb{R}$ and $\\delta_i \\in \\mathbb{C}$ for $i \\in \\mathbb{N}$. Then $(d, G, \\gamma, \\delta)$ is a split Thue recurrence family when $d \\ge 3$ and there are finite sets $R_i \\subseteq \\mathbb{C}$ and complex numbers $c_i(r)$ ($i \\in \\mathbb{N}$, $r \\in \\mathbb{C}$) such that: for every $1 \\le i \\le d$, $c_i(r) \\ne 0$\u2026","title":"Split Thue recurrence family"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/three-le","kind":"node","lean":"MetaMathlibExt.IsSplitThueRecurrenceFamily.three_le","parent":9737,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html#three-le","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"fully_proved","summary":"If $(d, G, \\gamma, \\delta)$ is a split Thue recurrence family, then $d \\ge 3$.","title":"A split Thue recurrence family has at least three sequences"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/is-split-thue-recurrence-family-example","kind":"node","lean":"MetaMathlibExt.isSplitThueRecurrenceFamily_example","parent":9737,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html#is-split-thue-recurrence-family-example","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"fully_proved","summary":"Let $G_1(n) = 3^n$, $G_2(n) = 5^n + 1$ and $G_i(n) = 5^n + 2^n$ for $i \\notin \\{1, 2\\}$; let $\\gamma_1 = 3$ and $\\gamma_i = 5$ for $i \\ne 1$; and let $\\delta_2 = 1$ and $\\delta_i = 2$ for $i \\ne 2$. Then $(3, G, \\gamma, \\delta)$ is a split Thue recurrence family: the dominant roots of $G_1, G_2, G_3$ are $3, 5, 5$ and the second dominant roots of $G_2, G_3$ are $1, 2$.","title":"Example of a split Thue recurrence family"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/split-thue-recurrence-solutions-bounded","kind":"node","lean":"MetaMathlibExt.splitThue_recurrence_solutions_bounded","parent":9737,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html#split-thue-recurrence-solutions-bounded","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"can_prove","summary":"Let $(d, G, \\gamma, \\delta)$ be a split Thue recurrence family. Then there is $\\kappa \\in \\mathbb{R}$ such that for every $n \\in \\mathbb{N}$ and all integers $x, y$ with $|y| \\ge 2$ and","title":"Hilgart's theorem on split families of Thue equations"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/split-thue-recurrence-solutions-eventual","kind":"node","lean":"MetaMathlibExt.splitThue_recurrence_solutions_eventually_trivial","parent":9737,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html#split-thue-recurrence-solutions-eventual","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"can_prove","summary":"Let $(d, G, \\gamma, \\delta)$ be a split Thue recurrence family. Then there is $n_0$ such that for every $n \\ge n_0$ and all integers $x, y$,","title":"Split Thue families eventually have only trivial solutions"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/split-thue-recurrence-roots-asymptotic","kind":"node","lean":"MetaMathlibExt.splitThue_recurrence_roots_asymptotic","parent":9737,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html#split-thue-recurrence-roots-asymptotic","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"can_prove","summary":"Let $(d, G, \\gamma, \\delta)$ be a split Thue recurrence family. Call reals $\\alpha_i(n)$ ($i, n \\in \\mathbb{N}$) an admissible labelling if there is $N$ such that for all $n \\ge N$ and $1 \\le i \\le d$, $\\prod_{k=1}^{d} (\\alpha_i(n) - G_k(n)) = 1$ and $|\\alpha_i(n) - G_i(n)| < 1$. Then, first, there is $N$ such that for all $n \\ge N$ every $z \\in \\mathbb{C}$ with $\\prod_{k=1}^{d} (z - G_k(n)) = 1$ is real; second,\u2026","title":"Asymptotics of the roots of a split Thue family"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/split-thue-recurrence-roots-growth","kind":"node","lean":"MetaMathlibExt.splitThue_recurrence_roots_growth","parent":9737,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html#split-thue-recurrence-roots-growth","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"can_prove","summary":"Let $(d, G, \\gamma, \\delta)$ be a split Thue recurrence family, and let $\\alpha_i(n)$ ($i, n \\in \\mathbb{N}$) be reals for which there is $N$ such that for all $n \\ge N$ and $1 \\le i \\le d$, $\\prod_{k=1}^{d} (\\alpha_i(n) - G_k(n)) = 1$ and $|\\alpha_i(n) - G_i(n)| < 1$. Then there are $c > 0$ and $N'$ such that for all $n \\ge N'$ and $1 \\le i \\le d$ we have $|\\alpha_i(n)| \\ge c \\gamma_i^n$, and for every\u2026","title":"Growth of the roots of a split Thue family"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/split-thue-recurrence-log-unit-det-asymp","kind":"node","lean":"MetaMathlibExt.splitThue_recurrence_logUnitDet_asymptotic","parent":9737,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations/index.html#split-thue-recurrence-log-unit-det-asymp","scope":"NumberTheory/diophantine-equations/higher-degree-equations/split-families-of-thue-equations","status":"can_prove","summary":"Let $(d, G, \\gamma, \\delta)$ be a split Thue recurrence family, let $\\alpha_i(n)$ ($i, n \\in \\mathbb{N}$) be reals for which there is $N$ such that for all $n \\ge N$ and $1 \\le i \\le d$, $\\prod_{l=1}^{d} (\\alpha_i(n) - G_l(n)) = 1$ and $|\\alpha_i(n) - G_i(n)| < 1$, and let $1 \\le k \\le d - 1$. Let $B_k(n)$ be the $k \\times k$ matrix with entries $\\log |\\alpha_i(n) - G_j(n)|$ for $1 \\le i, j \\le k$, where\u2026","title":"Determinants of logarithms in a split Thue family"},{"data_url":"scopes/NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials.json","id":"NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials","kind":"scope","navigation":"module","parent":9661,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials/index.html","scope":"NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials","status":"can_prove","title":"Sums of cubes of polynomials"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials/int-sum-four-polynomial-cubes-of-cube-fo","kind":"node","lean":"MathlibExt.NumberTheory.PolynomialCubesWanted.int_sum_four_polynomial_cubes_of_cube_form","parent":9746,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integers of the form "],["math","\\(p^3 + q^3\\)"],["text"," or "],["math","\\(2(p^6 - q^6)\\)"],["text"," as sums of four polynomial cubes"]],"source":"Integers of the form $p^3 + q^3$ or $2(p^6 - q^6)$ as sums of four polynomial cubes"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials/index.html#int-sum-four-polynomial-cubes-of-cube-fo","scope":"NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials","status":"can_prove","summary":"Let $p$ and $q$ be integers and let $N = p^3 + q^3$ or $N = 2(p^6 - q^6)$. Then there are polynomials $f, g, h, k \\in \\mathbb{Z}[x]$, each of degree at least $1$, such that $f^3 + g^3 + h^3 + k^3 = N$ as an identity in $\\mathbb{Z}[x]$.","title":"Integers of the form $p^3 + q^3$ or $2(p^6 - q^6)$ as sums of four polynomial cubes"},{"id":"NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials/int-eq-sum-five-polynomial-cubes","kind":"node","lean":"MathlibExt.NumberTheory.PolynomialCubesWanted.int_eq_sum_five_polynomial_cubes","parent":9746,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials/index.html#int-eq-sum-five-polynomial-cubes","scope":"NumberTheory/diophantine-equations/higher-degree-equations/sums-of-cubes-of-polynomials","status":"can_prove","summary":"For every integer $N$ there are polynomials $f, g, h, k, l \\in \\mathbb{Z}[x]$, each of degree at least $1$, such that $f^3 + g^3 + h^3 + k^3 + l^3 = N$ as an identity in $\\mathbb{Z}[x]$.","title":"Every integer is a sum of five cubes of polynomials"},{"data_url":"scopes/NumberTheory/diophantine-equations/diophantine-tuples-egyptian.json","id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian","kind":"scope","parent":9409,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/index.html","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian","status":"can_prove","summary":"Diophantine m-tuples, Erd\u0151s\u2013Straus problem, Egyptian fraction representations and greedy unit-fraction algorithms.","title":"Diophantine tuples and Egyptian fractions"},{"data_url":"scopes/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples.json","id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples","kind":"scope","navigation":"module","parent":9749,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples/index.html","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples","status":"can_prove","title":"Diophantine tuples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples/is-diophantine-tuple","kind":"node","lean":"MathlibExt.NumberTheory.DiophantineQuintupleConjectureQuestion1Wanted.IsDiophantineTuple","parent":9750,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples/index.html#is-diophantine-tuple","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples","status":"fully_proved","summary":"A finite set $s$ of natural numbers is a Diophantine tuple when every element of $s$ is positive and, for all $a, b \\in s$ with $a < b$, the number $ab + 1$ is a perfect square.","title":"Diophantine tuples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples/statement","kind":"node","lean":"MathlibExt.NumberTheory.DiophantineQuintupleConjectureQuestion1Wanted.Statement","parent":9750,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples/index.html#statement","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples","status":"fully_proved","summary":"The proposition that no finite set of natural numbers with exactly five elements is a Diophantine tuple, that is, that there is no Diophantine quintuple.","title":"The statement that no Diophantine quintuple exists"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples/statement-holds","kind":"node","lean":"MathlibExt.NumberTheory.DiophantineQuintupleConjectureQuestion1Wanted.Statement_holds","parent":9750,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples/index.html#statement-holds","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/diophantine-tuples","status":"can_prove","summary":"The statement that no Diophantine quintuple exists holds: no finite set of natural numbers with exactly five elements is a Diophantine tuple.","title":"There is no Diophantine quintuple"},{"data_url":"scopes/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations.json","id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","kind":"scope","navigation":"module","parent":9749,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","title":"Egyptian fraction underapproximations"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/is-nondecreasing-egyptian","kind":"node","lean":"EgyptianFraction.IsNondecreasingEgyptian","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#is-nondecreasing-egyptian","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"A tuple $(a_1, \\dots, a_n)$ of natural numbers lies in $\\mathcal{E}_n^{\\le}$ when $2 \\le a_1 \\le a_2 \\le \\dots \\le a_n$, that is, every $a_i$ is at least $2$ and the tuple is nondecreasing.","title":"Nondecreasing Egyptian denominator tuples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/is-strict-egyptian","kind":"node","lean":"EgyptianFraction.IsStrictEgyptian","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#is-strict-egyptian","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"A tuple $(a_1, \\dots, a_n)$ of natural numbers lies in $\\mathcal{E}_n^{<}$ when $2 \\le a_1 < a_2 < \\dots < a_n$, that is, every $a_i$ is at least $2$ and the tuple is strictly increasing.","title":"Strictly increasing Egyptian denominator tuples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/is-nondecreasing-underapproximation","kind":"node","lean":"EgyptianFraction.IsNondecreasingUnderapproximation","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#is-nondecreasing-underapproximation","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $\\lambda$ be a real number and $a = (a_1, \\dots, a_n)$ a tuple of natural numbers. Then $a$ is a nondecreasing underapproximation of $\\lambda$ when $n \\ge 1$, $\\lambda > 0$, $a$ lies in $\\mathcal{E}_n^{\\le}$ (that is, $2 \\le a_1 \\le \\dots \\le a_n$), and $\\sum_{i=1}^{n} 1/a_i < \\lambda$, the sum taken in the real numbers.","title":"Egyptian underapproximations with nondecreasing denominators"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/is-strict-underapproximation","kind":"node","lean":"EgyptianFraction.IsStrictUnderapproximation","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#is-strict-underapproximation","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $\\lambda$ be a real number and $a = (a_1, \\dots, a_n)$ a tuple of natural numbers. Then $a$ is a strict underapproximation of $\\lambda$ when $n \\ge 1$, $\\lambda > 0$, $a$ lies in $\\mathcal{E}_n^{<}$ (that is, $2 \\le a_1 < \\dots < a_n$, so the unit fractions are distinct), and $\\sum_{i=1}^{n} 1/a_i < \\lambda$, the sum taken in the real numbers.","title":"Egyptian underapproximations with strictly increasing denominators"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/nonempty","kind":"node","lean":"EgyptianFraction.IsNondecreasingUnderapproximation.nonempty","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#nonempty","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a nondecreasing underapproximation of $\\lambda$, then $n \\ge 1$.","title":"Nondecreasing underapproximations have at least one term"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/target-pos","kind":"node","lean":"EgyptianFraction.IsNondecreasingUnderapproximation.target_pos","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#target-pos","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a nondecreasing underapproximation of $\\lambda$, then $\\lambda > 0$.","title":"The target of a nondecreasing underapproximation is positive"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/is-nondecreasing-egyptian-2","kind":"node","lean":"EgyptianFraction.IsNondecreasingUnderapproximation.isNondecreasingEgyptian","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#is-nondecreasing-egyptian-2","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"If $a = (a_1, \\dots, a_n)$ is a nondecreasing underapproximation of a real number $\\lambda$, then $a$ lies in $\\mathcal{E}_n^{\\le}$, that is, $2 \\le a_1 \\le \\dots \\le a_n$.","title":"Nondecreasing underapproximations have Egyptian denominator tuples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/denominator-ge-two","kind":"node","lean":"EgyptianFraction.IsNondecreasingUnderapproximation.denominator_ge_two","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#denominator-ge-two","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a nondecreasing underapproximation of $\\lambda$, then $a_i \\ge 2$ for every $1 \\le i \\le n$.","title":"Denominators of a nondecreasing underapproximation are at least two"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/monotone","kind":"node","lean":"EgyptianFraction.IsNondecreasingUnderapproximation.monotone","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#monotone","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a nondecreasing underapproximation of $\\lambda$, then $a_i \\le a_j$ whenever $1 \\le i \\le j \\le n$.","title":"Denominators of a nondecreasing underapproximation are nondecreasing"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/sum-lt","kind":"node","lean":"EgyptianFraction.IsNondecreasingUnderapproximation.sum_lt","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#sum-lt","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a nondecreasing underapproximation of $\\lambda$, then $\\sum_{i=1}^{n} 1/a_i < \\lambda$.","title":"A nondecreasing underapproximation stays below its target"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/nonempty-2","kind":"node","lean":"EgyptianFraction.IsStrictUnderapproximation.nonempty","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#nonempty-2","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a strict underapproximation of $\\lambda$, then $n \\ge 1$.","title":"Strict underapproximations have at least one term"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/target-pos-2","kind":"node","lean":"EgyptianFraction.IsStrictUnderapproximation.target_pos","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#target-pos-2","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a strict underapproximation of $\\lambda$, then $\\lambda > 0$.","title":"The target of a strict underapproximation is positive"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/is-strict-egyptian-2","kind":"node","lean":"EgyptianFraction.IsStrictUnderapproximation.isStrictEgyptian","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#is-strict-egyptian-2","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"If $a = (a_1, \\dots, a_n)$ is a strict underapproximation of a real number $\\lambda$, then $a$ lies in $\\mathcal{E}_n^{<}$, that is, $2 \\le a_1 < \\dots < a_n$.","title":"Strict underapproximations have strictly increasing Egyptian denominators"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/denominator-ge-two-2","kind":"node","lean":"EgyptianFraction.IsStrictUnderapproximation.denominator_ge_two","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#denominator-ge-two-2","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a strict underapproximation of $\\lambda$, then $a_i \\ge 2$ for every $1 \\le i \\le n$.","title":"Denominators of a strict underapproximation are at least two"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/strict-mono","kind":"node","lean":"EgyptianFraction.IsStrictUnderapproximation.strictMono","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#strict-mono","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a strict underapproximation of $\\lambda$, then $a_i < a_j$ whenever $1 \\le i < j \\le n$.","title":"Denominators of a strict underapproximation increase strictly"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/sum-lt-2","kind":"node","lean":"EgyptianFraction.IsStrictUnderapproximation.sum_lt","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#sum-lt-2","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a strict underapproximation of $\\lambda$, then $\\sum_{i=1}^{n} 1/a_i < \\lambda$, the sum taken in the real numbers.","title":"A strict underapproximation stays below its target"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/to-is-nondecreasing-underapproximation","kind":"node","lean":"EgyptianFraction.IsStrictUnderapproximation.toIsNondecreasingUnderapproximation","parent":9754,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations/index.html#to-is-nondecreasing-underapproximation","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/egyptian-fraction-underapproximations","status":"fully_proved","summary":"Let $n$ be a natural number, $\\lambda$ a real number and $a_1, \\dots, a_n$ natural numbers. If $(a_1, \\dots, a_n)$ is a strict underapproximation of $\\lambda$, then it is also a nondecreasing underapproximation of $\\lambda$.","title":"Strict underapproximations are nondecreasing underapproximations"},{"data_url":"scopes/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture.json","id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture","kind":"scope","navigation":"module","parent":9749,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/index.html","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture","status":"can_prove","title":"Erd\u0151s\u2013Straus conjecture"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/erdos-straus-type-i-of-dvd-sq","kind":"node","lean":"MetaMathlibExt.erdosStraus_typeI_of_dvd_sq","parent":9772,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/index.html#erdos-straus-type-i-of-dvd-sq","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture","status":"can_prove","summary":"Let $p$ be a prime and let $x$ be a natural number with $\\lceil p/4 \\rceil \\le x \\le \\lceil p/2 \\rceil$. If $d$ is a positive divisor of $x^2$ with $d \\equiv -px \\pmod{4x - p}$, then there are natural numbers $y$ and $z$ with $(4x - p)\\,y = px + d$ and $(4x - p)\\,d\\,z = px(px + d)$ such that $x \\le y \\le z$ and $p \\nmid y$.","title":"Type I Erd\u0151s\u2013Straus triples from divisors of a square"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/erdos-straus-type-i-exists-dvd-sq","kind":"node","lean":"MetaMathlibExt.erdosStraus_typeI_exists_dvd_sq","parent":9772,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/index.html#erdos-straus-type-i-exists-dvd-sq","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture","status":"can_prove","summary":"Let $p$ be a prime and let $x \\le y \\le z$ be positive integers with $4/p = 1/x + 1/y + 1/z$ and $p \\nmid y$. Then $\\lceil p/4 \\rceil \\le x \\le \\lceil p/2 \\rceil$, and there is a positive divisor $d$ of $x^2$ with $d \\equiv -px \\pmod{4x - p}$, $(4x - p)\\,y = px + d$ and $(4x - p)\\,d\\,z = px(px + d)$.","title":"Divisor criterion necessary for type I Erd\u0151s\u2013Straus solutions"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/erdos-straus-type-ii-exists-dvd-sq","kind":"node","lean":"MetaMathlibExt.erdosStraus_typeII_exists_dvd_sq","parent":9772,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/index.html#erdos-straus-type-ii-exists-dvd-sq","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture","status":"can_prove","summary":"Let $p$ be a prime and let $x \\le y \\le z$ be positive integers with $4/p = 1/x + 1/y + 1/z$ and $p \\mid y$. Then $\\lceil p/4 \\rceil \\le x \\le \\lceil p/2 \\rceil$, and there is a positive divisor $d$ of $x^2$ with $d \\le x$, $d \\equiv -x \\pmod{4x - p}$, $(4x - p)\\,y = p(x + d)$ and $(4x - p)\\,d\\,z = px(x + d)$.","title":"Divisor criterion necessary for type II Erd\u0151s\u2013Straus solutions"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/erdos-straus-prime-restricted-representa","kind":"node","lean":"MathlibExt.NumberTheory.ErdosStrausPrimeStructureWanted.erdos_straus_prime_restricted_representation_iff","parent":9772,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture/index.html#erdos-straus-prime-restricted-representa","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/erd-s-straus-conjecture","status":"can_prove","summary":"For a prime $p$, there are positive integers $a \\le b \\le c$ with $p \\nmid a$, $p \\mid b$, $p \\mid c$ and $4/p = 1/a + 1/b + 1/c$ if and only if there are positive integers $q$, $r$, $s_1$, $s_2$ with $q \\equiv 3 \\pmod 4$, both $s_1$ and $s_2$ dividing $(q + 1)/4$, and $p + 4 s_1 s_2 = qr$.","title":"Chamberland's characterization of restricted Erd\u0151s\u2013Straus representations"},{"data_url":"scopes/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions.json","id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","kind":"scope","navigation":"module","parent":9749,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"can_prove","title":"Odd greedy algorithm for Egyptian fractions"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/is-odd-greedy-choice","kind":"node","lean":"MetaMathlibExt.IsOddGreedyChoice","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#is-odd-greedy-choice","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"fully_proved","summary":"For a rational number $r$ and a natural number $d$, $d$ is the odd greedy choice at $r$ when $r > 0$, $d$ is positive and odd, $1/d \\le r$, and $d \\le e$ for every positive odd natural number $e$ with $1/e \\le r$. Equivalently, $1/d$ is the largest unit fraction with odd denominator that does not exceed $r$.","title":"Odd greedy choice"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/pos-r","kind":"node","lean":"MetaMathlibExt.IsOddGreedyChoice.pos_r","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#pos-r","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"fully_proved","summary":"If $d$ is the odd greedy choice at a rational number $r$, then $r > 0$.","title":"The odd greedy choice needs a positive remainder"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/pos-d","kind":"node","lean":"MetaMathlibExt.IsOddGreedyChoice.pos_d","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#pos-d","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"fully_proved","summary":"If $d$ is the odd greedy choice at a rational number $r$, then $d > 0$.","title":"The odd greedy choice is positive"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/odd","kind":"node","lean":"MetaMathlibExt.IsOddGreedyChoice.odd","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#odd","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"fully_proved","summary":"If $d$ is the odd greedy choice at a rational number $r$, then $d$ is odd.","title":"The odd greedy choice is odd"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/inv-le","kind":"node","lean":"MetaMathlibExt.IsOddGreedyChoice.inv_le","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#inv-le","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"fully_proved","summary":"If $d$ is the odd greedy choice at a rational number $r$, then $1/d \\le r$.","title":"The odd greedy unit fraction does not exceed the remainder"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/min","kind":"node","lean":"MetaMathlibExt.IsOddGreedyChoice.min","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#min","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"fully_proved","summary":"If $d$ is the odd greedy choice at a rational number $r$, then $d \\le e$ for every positive odd natural number $e$ with $1/e \\le r$.","title":"The odd greedy choice is minimal"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/unique","kind":"node","lean":"MetaMathlibExt.IsOddGreedyChoice.unique","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#unique","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"fully_proved","summary":"If $d$ and $d'$ are both odd greedy choices at the same rational number $r$, then $d = d'$.","title":"Uniqueness of the odd greedy choice"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/is-odd-greedy-choice-iff","kind":"node","lean":"MetaMathlibExt.isOddGreedyChoice_iff","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#is-odd-greedy-choice-iff","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"fully_proved","summary":"A natural number $d$ is the odd greedy choice at a rational number $r$ if and only if $r > 0$ and $d$ is the least element of the set of positive odd natural numbers $e$ with $1/e \\le r$.","title":"Characterization of the odd greedy choice"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/even-num-of-even-sum-unit-fractions","kind":"node","lean":"MathlibExt.NumberTheory.OddGreedyExpansionWanted.even_num_of_even_sum_unit_fractions","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#even-num-of-even-sum-unit-fractions","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"can_prove","summary":"Let $m$ be an even natural number and let $d_1, \\dots, d_m$ be odd natural numbers. If integers $a$ and $b$ with $b \\ne 0$ satisfy $a/b = \\sum_{i=1}^{m} 1/d_i$, then $a$ is even.","title":"Even sums of odd unit fractions have even numerator"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/is-odd-greedy-choice-iff-2","kind":"node","lean":"MathlibExt.NumberTheory.OddGreedyExpansionWanted.isOddGreedyChoice_iff","parent":9777,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions/index.html#is-odd-greedy-choice-iff-2","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/odd-greedy-algorithm-for-egyptian-fractions","status":"can_prove","summary":"Let $n$ and $d$ be positive natural numbers and let $x_1$ be an odd natural number. Then $x_1$ is the odd greedy choice at $n/d$ (the first denominator of the odd greedy expansion of $n/d$) if and only if $d \\le n x_1$ and $n x_1 < d + 2n$, that is, $0 \\le n x_1 - d < 2n$ in the integers.","title":"Odd greedy choice at a fraction"},{"data_url":"scopes/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples.json","id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","kind":"scope","navigation":"module","parent":9749,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","title":"S-Diophantine tuples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/is-sdiophantine-tuple","kind":"node","lean":"MetaMathlibExt.IsSDiophantineTuple","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(S\\)"],["text","-Diophantine tuples"]],"source":"$S$-Diophantine tuples"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#is-sdiophantine-tuple","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"fully_proved","summary":"Let $S$ be a finite set of natural numbers. A finite set $s$ of natural numbers is an $S$-Diophantine tuple if all its elements are positive and, for all distinct $a, b \\in s$, every prime factor of $ab + 1$ lies in $S$. A tuple $(a_1, \\dots, a_m)$ of distinct positive integers is encoded by the set of its entries, which has $m$ elements.","title":"$S$-Diophantine tuples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/is-sdiophantine-tuple-two-one-three","kind":"node","lean":"MetaMathlibExt.isSDiophantineTuple_two_one_three","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((1, 3)\\)"],["text"," is a "],["math","\\(\\{2\\}\\)"],["text","-Diophantine pair"]],"source":"$(1, 3)$ is a $\\{2\\}$-Diophantine pair"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#is-sdiophantine-tuple-two-one-three","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"fully_proved","summary":"The set $\\{1, 3\\}$ is a $\\{2\\}$-Diophantine tuple, since $1 \\cdot 3 + 1 = 2^2$.","title":"$(1, 3)$ is a $\\{2\\}$-Diophantine pair"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/is-sdiophantine-tuple-two-three-five-one","kind":"node","lean":"MetaMathlibExt.isSDiophantineTuple_two_three_five_one_three_eight","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((1, 3, 8)\\)"],["text"," is a "],["math","\\(\\{2, 3, 5\\}\\)"],["text","-Diophantine triple"]],"source":"$(1, 3, 8)$ is a $\\{2, 3, 5\\}$-Diophantine triple"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#is-sdiophantine-tuple-two-three-five-one","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"fully_proved","summary":"The set $\\{1, 3, 8\\}$ is a $\\{2, 3, 5\\}$-Diophantine tuple, since $1 \\cdot 3 + 1 = 2^2$, $1 \\cdot 8 + 1 = 3^2$ and $3 \\cdot 8 + 1 = 5^2$.","title":"$(1, 3, 8)$ is a $\\{2, 3, 5\\}$-Diophantine triple"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/not-mul-add-one-dvd-mul-add-one","kind":"node","lean":"MetaMathlibExt.not_mul_add_one_dvd_mul_add_one","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(ac + 1\\)"],["text"," does not divide "],["math","\\(bc + 1\\)"]],"source":"$ac + 1$ does not divide $bc + 1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#not-mul-add-one-dvd-mul-add-one","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"If $a < b < c$ are positive integers, then $ac + 1$ does not divide $bc + 1$.","title":"$ac + 1$ does not divide $bc + 1$"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/padic-val-nat-two-smallest-eq","kind":"node","lean":"MetaMathlibExt.padicValNat_two_smallest_eq","parent":9788,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#padic-val-nat-two-smallest-eq","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $\\ell$ be a prime and $a, b, c, d$ natural numbers. Put $s_1 = ab + 1$, $s_2 = ac + 1$, $s_3 = ad + 1$, $s_4 = bc + 1$, $s_5 = bd + 1$, $s_6 = cd + 1$, and let $v_i$ be the $\\ell$-adic valuation of $s_i$. Then in each of the tuples $(v_2, v_3, v_4, v_5)$, $(v_1, v_2, v_5, v_6)$ and $(v_1, v_3, v_4, v_6)$ the minimum value occurs at two different positions.","title":"The two smallest valuations coincide"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/dvd-gcd-sub-div-gcd","kind":"node","lean":"MetaMathlibExt.dvd_gcd_sub_div_gcd","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(a\\)"],["text",", "],["math","\\(b\\)"],["text",", "],["math","\\(c\\)"],["text",", "],["math","\\(d\\)"],["text"," divide gcds of reduced differences"]],"source":"$a$, $b$, $c$, $d$ divide gcds of reduced differences"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#dvd-gcd-sub-div-gcd","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $a, b, c, d$ be natural numbers and put $s_1 = ab + 1$, $s_2 = ac + 1$, $s_3 = ad + 1$, $s_4 = bc + 1$, $s_5 = bd + 1$, $s_6 = cd + 1$. For indices $i, j$ let $t_{ij}$ be the integer $(s_j - s_i) / \\gcd(s_j, s_i)$. Then $a \\mid \\gcd(t_{12}, t_{13}, t_{23})$, $b \\mid \\gcd(t_{14}, t_{15}, t_{45})$, $c \\mid \\gcd(t_{24}, t_{26}, t_{46})$ and $d \\mid \\gcd(t_{35}, t_{36}, t_{56})$ in $\\mathbb{Z}$.","title":"$a$, $b$, $c$, $d$ divide gcds of reduced differences"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/not-exists-s-diophantine-system-four","kind":"node","lean":"MetaMathlibExt.not_exists_sDiophantine_system_four","parent":9788,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#not-exists-s-diophantine-system-four","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $p \\ne q$ be odd primes and let $a, b, c, d$ be integers with $0 < a < b < c < d$. Then there are no natural numbers $\\beta_1, \\alpha_4, \\beta_4, \\alpha_2, \\alpha_5, \\alpha_3, \\beta_3, \\beta_6$ with $ab + 1 = q^{\\beta_1}$, $bc + 1 = p^{\\alpha_4} q^{\\beta_4}$, $ac + 1 = p^{\\alpha_2}$, $bd + 1 = p^{\\alpha_5}$, $ad + 1 = p^{\\alpha_3} q^{\\beta_3}$ and $cd + 1 = q^{\\beta_6}$.","title":"The Szalay\u2013Ziegler system (4) has no solutions"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/s-diophantine-quadruple-exponent-cases","kind":"node","lean":"MetaMathlibExt.sDiophantine_quadruple_exponent_cases","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exponent cases for "],["math","\\(\\{p, q\\}\\)"],["text","-Diophantine quadruples with "],["math","\\(p \\equiv 3 \\pmod 4\\)"]],"source":"Exponent cases for $\\{p, q\\}$-Diophantine quadruples with $p \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#s-diophantine-quadruple-exponent-cases","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $p \\ne q$ be primes with $p \\equiv 3 \\pmod 4$, and let $a < b < c < d$ be natural numbers such that $\\{a, b, c, d\\}$ is a $\\{p, q\\}$-Diophantine tuple. Write $s_1, \\dots, s_6$ for $ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1$, and let $\\alpha_i$ and $\\beta_i$ be the exponents of $p$ and $q$ in the prime factorization of $s_i$. Then one of the following holds: (I)\u2026","title":"Exponent cases for $\\{p, q\\}$-Diophantine quadruples with $p \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/not-exists-s-diophantine-quadruple-of-mo","kind":"node","lean":"MetaMathlibExt.not_exists_sDiophantine_quadruple_of_mod_four_eq_three","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["text","No "],["math","\\(\\{p, q\\}\\)"],["text","-Diophantine quadruple when "],["math","\\(p\\)"],["text"," or "],["math","\\(q\\)"],["text"," is "],["math","\\(3\\)"],["text"," modulo "],["math","\\(4\\)"]],"source":"No $\\{p, q\\}$-Diophantine quadruple when $p$ or $q$ is $3$ modulo $4$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#not-exists-s-diophantine-quadruple-of-mo","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $p \\ne q$ be primes with $p \\equiv 3 \\pmod 4$ or $q \\equiv 3 \\pmod 4$. Then no four-element set of natural numbers is a $\\{p, q\\}$-Diophantine tuple; that is, there is no $\\{p, q\\}$-Diophantine quadruple.","title":"No $\\{p, q\\}$-Diophantine quadruple when $p$ or $q$ is $3$ modulo $4$"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/not-exists-s-diophantine-quintuple","kind":"node","lean":"MetaMathlibExt.not_exists_sDiophantine_quintuple","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["text","There is no "],["math","\\(\\{p, q\\}\\)"],["text","-Diophantine quintuple"]],"source":"There is no $\\{p, q\\}$-Diophantine quintuple"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#not-exists-s-diophantine-quintuple","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $p < q$ be primes. Then no five-element set of natural numbers is a $\\{p, q\\}$-Diophantine tuple; that is, there is no $\\{p, q\\}$-Diophantine quintuple.","title":"There is no $\\{p, q\\}$-Diophantine quintuple"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/s-diophantine-quadruple-case-iii-bounds","kind":"node","lean":"MetaMathlibExt.sDiophantine_quadruple_caseIII_bounds","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds in Case III for "],["math","\\(\\{p, q\\}\\)"],["text","-Diophantine quadruples"]],"source":"Bounds in Case III for $\\{p, q\\}$-Diophantine quadruples"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#s-diophantine-quadruple-case-iii-bounds","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $p$ and $q$ be primes with $p \\equiv 3 \\pmod 4$, $p \\ge 7$ and $q \\equiv 1 \\pmod 4$, and let $a < b < c < d$ be natural numbers such that $\\{a, b, c, d\\}$ is a $\\{p, q\\}$-Diophantine tuple. Put $s_1 = ab + 1$, $s_2 = ac + 1$, $s_3 = ad + 1$, $s_4 = bc + 1$, $s_5 = bd + 1$, $s_6 = cd + 1$, and let $\\alpha_i$ and $\\beta_i$ be the $p$-adic and $q$-adic valuations of $s_i$. Suppose that\u2026","title":"Bounds in Case III for $\\{p, q\\}$-Diophantine quadruples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/s-diophantine-quadruple-case-iv-bound","kind":"node","lean":"MetaMathlibExt.sDiophantine_quadruple_caseIV_bound","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on "],["math","\\(b\\)"],["text"," in Case IV for "],["math","\\(\\{p, q\\}\\)"],["text","-Diophantine quadruples"]],"source":"Bound on $b$ in Case IV for $\\{p, q\\}$-Diophantine quadruples"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#s-diophantine-quadruple-case-iv-bound","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $p$ and $q$ be primes with $p \\equiv 3 \\pmod 4$, $p \\ge 7$ and $q \\equiv 1 \\pmod 4$, and let $a < b < c < d$ be natural numbers such that $\\{a, b, c, d\\}$ is a $\\{p, q\\}$-Diophantine tuple. Write $s_1, \\dots, s_6$ for $ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1$, and let $\\alpha_i$ and $\\beta_i$ be the exponents of $p$ and $q$ in the prime factorization of $s_i$. Suppose that\u2026","title":"Bound on $b$ in Case IV for $\\{p, q\\}$-Diophantine quadruples"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/s-diophantine-quadruple-fourteen-cases","kind":"node","lean":"MetaMathlibExt.sDiophantine_quadruple_fourteen_cases","parent":9788,"reading_title":{"format":"inline-parts-v1","parts":[["text","The fourteen exponent cases for "],["math","\\(\\{p, q\\}\\)"],["text","-Diophantine quadruples"]],"source":"The fourteen exponent cases for $\\{p, q\\}$-Diophantine quadruples"},"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples/index.html#s-diophantine-quadruple-fourteen-cases","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/s-diophantine-tuples","status":"can_prove","summary":"Let $p \\ne q$ be primes and let $a < b < c < d$ be natural numbers such that $\\{a, b, c, d\\}$ is a $\\{p, q\\}$-Diophantine tuple. Write $s_1, \\dots, s_6$ for $ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1$, and let $\\alpha_i$ and $\\beta_i$ be the exponents of $p$ and $q$ in the prime factorization of $s_i$. Then one of the following fourteen cases holds: (I) $\\alpha_1 = \\alpha_2 = \\alpha_3 < \\alpha_4 < \\alpha_5$,\u2026","title":"The fourteen exponent cases for $\\{p, q\\}$-Diophantine quadruples"},{"data_url":"scopes/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums.json","id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","kind":"scope","navigation":"module","parent":9749,"result":false,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","title":"Sets with square pairwise sums"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.PairwiseSumsSquares","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"fully_proved","summary":"Let $\\alpha$ be a type with an addition and a multiplication, and let $(a_i)_{i \\in I}$ and $(b_j)_{j \\in J}$ be families in $\\alpha$ indexed by arbitrary types $I$ and $J$. They have all pairwise sums square when, for every $i \\in I$ and $j \\in J$, the sum $a_i + b_j$ is a square in $\\alpha$, that is, $a_i + b_j = c \\cdot c$ for some $c \\in \\alpha$.","title":"Families whose pairwise sums are squares"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/is-semi-magic-roots","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.IsSemiMagicRoots","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#is-semi-magic-roots","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"fully_proved","summary":"A $3 \\times 3$ integer matrix $E = (e_{ij})$ has semi-magic squares when the matrix $(e_{ij}^2)$ of its squared entries is semi-magic: there is an integer $S$ such that $\\sum_{j=1}^{3} e_{ij}^2 = S$ for every row $i$ and $\\sum_{i=1}^{3} e_{ij}^2 = S$ for every column $j$.","title":"Semi-magic matrix of squares"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/exists-int-pairwise-sums-squares-scale-t","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.exists_int_pairwiseSumsSquares_scale_translate","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#exists-int-pairwise-sums-squares-scale-t","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"Let $a_1, \\dots, a_m$ and $b_1, \\dots, b_n$ be rational numbers such that every sum $a_i + b_j$ is the square of a rational number, and let $t$ be an integer and $k$ a rational number such that $k^2 a_i - t$ is an integer for every $i$ and $k^2 b_j + t$ is an integer for every $j$. Then the integers $a_i' = k^2 a_i - t$ and $b_j' = k^2 b_j + t$ are such that every sum $a_i' + b_j'$ is the square of an integer.","title":"Scaling rational families with square pairwise sums to integers"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-triple-of-is-semi","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_triple_of_isSemiMagicRoots","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-triple-of-is-semi","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"Let $e_{ij}$, $1 \\le i, j \\le 3$, be integers such that the matrix of squares $(e_{ij}^2)$ is semi-magic, that is, all its row sums and all its column sums are equal. Put $a_1 = e_{13}^2$, $a_2 = e_{31}^2$, $a_3 = e_{22}^2$ and $b_1 = 0$, $b_2 = e_{21}^2 - e_{13}^2$, $b_3 = e_{11}^2 - e_{22}^2$. Then $a_i + b_j$ is a perfect square for all $i, j \\in \\{1, 2, 3\\}$.","title":"Triples with square pairwise sums from a semi-magic square of squares"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/exists-is-semi-magic-roots-of-pairwise-s","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.exists_isSemiMagicRoots_of_pairwiseSumsSquares_triple","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#exists-is-semi-magic-roots-of-pairwise-s","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"Let $a_1, a_2, a_3$ and $b_1, b_2, b_3$ be integers such that every sum $a_i + b_j$ is the square of an integer. Then there is a $3 \\times 3$ integer matrix $E = (e_{ij})$ whose squared entries $e_{ij}^2$ have all three row sums and all three column sums equal to one common integer, and such that $e_{11}^2 = a_3 + b_3$, $e_{12}^2 = a_2 + b_2$, $e_{13}^2 = a_1 + b_1$, $e_{21}^2 = a_1 + b_2$, $e_{22}^2 = a_3 + b_1$,\u2026","title":"Triples with square pairwise sums come from semi-magic squares"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-quintuple-of-is-se","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_quintuple_of_isSemiMagicRoots","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-quintuple-of-is-se","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"Let $e_{ij}$ and $f_{ij}$, $1 \\le i, j \\le 3$, be integers such that the matrices of squares $(e_{ij}^2)$ and $(f_{ij}^2)$ are both semi-magic (in each matrix all row sums and all column sums are equal) and have the same diagonal, $e_{ii}^2 = f_{ii}^2$ for $i = 1, 2, 3$. Put $a_1 = e_{13}^2$, $a_2 = e_{31}^2$, $a_3 = e_{22}^2$, $a_4 = f_{13}^2$, $a_5 = f_{31}^2$ and $b_1 = 0$, $b_2 = e_{21}^2 - e_{13}^2$,\u2026","title":"Five integers with square sums from two semi-magic squares of squares"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/exists-is-semi-magic-roots-f-of-pairwise","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.exists_isSemiMagicRoots_f_of_pairwiseSumsSquares_quintuple","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#exists-is-semi-magic-roots-f-of-pairwise","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"Let $E = (e_{ij})$ be a $3 \\times 3$ integer matrix whose squared entries $e_{ij}^2$ have all three row sums and all three column sums equal to one common integer. Put $(a_1, a_2, a_3) = (e_{13}^2, e_{31}^2, e_{22}^2)$ and $(b_1, b_2, b_3) = (0, e_{21}^2 - e_{13}^2, e_{11}^2 - e_{22}^2)$, and let $a_4, a_5$ be integers such that every sum $a_i + b_j$ with $1 \\le i \\le 5$ and $1 \\le j \\le 3$ is the square of an\u2026","title":"Quintuple extensions come from a second semi-magic square"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-quadruple-of-is-se","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_quadruple_of_isSemiMagicRoots","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-quadruple-of-is-se","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"Let $E = (e_{ij})$, $G = (g_{ij})$ and $H = (h_{ij})$ be $3 \\times 3$ integer matrices, each of whose squared entries have all three row sums and all three column sums equal to one common integer, such that $e_{12}^2 = g_{12}^2$, $e_{13}^2 = g_{13}^2$, $e_{21}^2 = g_{21}^2$, $e_{31}^2 = g_{31}^2$, $e_{11}^2 = h_{11}^2$, $e_{22}^2 = h_{22}^2$, $g_{11}^2 = h_{12}^2$ and $g_{22}^2 = h_{31}^2$. Put\u2026","title":"Extending the triples to quadruples with square pairwise sums"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/exists-is-semi-magic-roots-gh-of-pairwis","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.exists_isSemiMagicRoots_gh_of_pairwiseSumsSquares_quadruple","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#exists-is-semi-magic-roots-gh-of-pairwis","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"Let $E = (e_{ij})$ be a $3 \\times 3$ integer matrix whose squared entries $e_{ij}^2$ have all three row sums and all three column sums equal to one common integer. Put $(a_1, a_2, a_3) = (e_{13}^2, e_{31}^2, e_{22}^2)$ and $(b_1, b_2, b_3) = (0, e_{21}^2 - e_{13}^2, e_{11}^2 - e_{22}^2)$, and let $a_4, b_4$ be integers such that every sum $a_i + b_j$ with $1 \\le i, j \\le 4$ is the square of an integer. Then there\u2026","title":"Quadruple extensions come from two further semi-magic squares"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-triple-family31","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_triple_family31","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-triple-family31","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"For all integers $p, q, r, s$, put $a_1 = 0$, $a_2 = 16pqrs$, $a_3 = (p+q+r+s)(p+q-r-s)(p-q+r-s)(p-q-r+s)$ and $b_1 = 4(pr - qs)^2$, $b_2 = 4(ps - qr)^2$, $b_3 = 4(pq - rs)^2$. Then $a_i + b_j$ is a perfect square for all $i, j \\in \\{1, 2, 3\\}$.","title":"Four-parameter triples with square pairwise sums, family (3.1)"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-triple-family32","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_triple_family32","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-triple-family32","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"For all integers $p, q, r, s$, put $a_1 = -(p+q+r+s)(p+q-r-s)(p-q+r-s)(p-q-r+s)$, $a_2 = -(p-q+r+s)(p-q-r-s)(p+q+r-s)(p+q-r+s)$, $a_3 = 0$ and $b_1 = (p^2 - q^2 + r^2 - s^2)^2$, $b_2 = (p^2 - q^2 - r^2 + s^2)^2$, $b_3 = (p^2 + q^2 - r^2 - s^2)^2$. Then $a_i + b_j$ is a perfect square for all $i, j \\in \\{1, 2, 3\\}$.","title":"Four-parameter triples with square pairwise sums, family (3.2)"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-triple-family33","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_triple_family33","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-triple-family33","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"For all integers $p, q, r, s$, put $a_1 = 4(ps - qr)^2$, $a_2 = 4(ps + qr)^2$, $a_3 = (p^2 - q^2 - r^2 + s^2)^2$ and $b_1 = 4(p^2 - q^2)(r^2 - s^2)$, $b_2 = 0$, $b_3 = 4(p^2 - r^2)(q^2 - s^2)$. Then $a_i + b_j$ is a perfect square for all $i, j \\in \\{1, 2, 3\\}$.","title":"Four-parameter triples with square pairwise sums, family (3.3)"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-quintuple-family41","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_quintuple_family41","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-quintuple-family41","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"For rational numbers $f, g, m, u_1, u_2, v_1, v_2$, put $\\alpha = m u_1 u_2 (u_1 v_1 - u_2 v_2)$, $\\beta = (u_1 v_2 - u_2 v_1)(m^2 u_1 u_2 v_1 v_2 - 1)$, $\\gamma = m v_1 v_2 (u_1 v_1 - u_2 v_2)$, $\\alpha' = m u_1 u_2 (u_1 v_1 + u_2 v_2)$, $\\beta' = (u_1 v_2 + u_2 v_1)(m^2 u_1 u_2 v_1 v_2 + 1)$ and $\\gamma' = m v_1 v_2 (u_1 v_1 + u_2 v_2)$, and let $a_1 = 4(\\alpha f^2 - \\beta f g - \\gamma g^2)^2$,\u2026","title":"A seven-parameter family of five and three rationals with square pairwise sums"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-quadruple-family51","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_quadruple_family51","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-quadruple-family51","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"For integers $u_1, u_2, v_1, v_2$, put $w = v_1^4 - 6 v_1^2 v_2^2 + v_2^4$, $s = v_1^2 + v_2^2$ and $d = v_1^2 - v_2^2$, and let $a_1 = w^2 d^2 (u_1^2 + u_2^2)^2$, $a_2 = w^2 s^2 (u_1^2 - u_2^2)^2$, $a_3 = d^2 (s^2 u_1^2 - 16 u_1 u_2 v_1^2 v_2^2 - s^2 u_2^2)^2$, $a_4 = d^2 (s^2 u_1^2 + 16 u_1 u_2 v_1^2 v_2^2 - s^2 u_2^2)^2$, $b_1 = 0$,\u2026","title":"A four-parameter family of integer quadruples with square pairwise sums"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-quadruple-family58","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_quadruple_family58","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-quadruple-family58","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"For rational numbers $m, q, r$, let $a_1 = 16 m^2 (m^4 - 1)^2 \\bigl((5m^4 - 2m^2 + 1) q^2 - 2(m^4 - 6m^2 + 1) q r + (m^4 - 2m^2 + 5) r^2\\bigr)^2$, $a_2 = 16 m^2 (m^4 - 1)^2 \\bigl((5m^4 - 2m^2 + 1) q^2 - (m^4 - 2m^2 + 5) r^2\\bigr)^2$, $a_3 = 16 m^2 (m^4 - 1)^2 \\bigl((7m^4 - 2m^2 - 1) q^2 - (2m^4 - 12m^2 + 2) q r - (m^4 + 2m^2 - 7) r^2\\bigr)^2$,\u2026","title":"First three-parameter family of rational quadruples with square pairwise sums"},{"id":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/pairwise-sums-squares-quadruple-family59","kind":"node","lean":"MathlibExt.NumberTheory.TwoSetsSumsetSquaresWanted.pairwiseSumsSquares_quadruple_family59","parent":9802,"result":true,"route":"../roadmap/NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums/index.html#pairwise-sums-squares-quadruple-family59","scope":"NumberTheory/diophantine-equations/diophantine-tuples-egyptian/sets-with-square-pairwise-sums","status":"can_prove","summary":"For rational numbers $m, q, r$, let $a_1 = 16 m^2 (m^4 - 1)^2 \\bigl((m^6 - 2m^4 + 5m^2) q^2 + (2m^5 - 12m^3 + 2m) q r + (5m^4 - 2m^2 + 1) r^2\\bigr)^2$, $a_2 = 16 m^2 (m^4 - 1)^2 \\bigl((m^6 - 2m^4 + 5m^2) q^2 - (5m^4 - 2m^2 + 1) r^2\\bigr)^2$, $a_3 = (m^2 - 1)^2 \\bigl((m^{10} + 26m^6 - 40m^4 - 3m^2) q^2 + (2m^9 - 24m^7 + 76m^5 - 24m^3 + 2m) q r - (3m^8 + 40m^6 - 26m^4 - 1) r^2\\bigr)^2$,\u2026","title":"Second three-parameter family of rational quadruples with square pairwise sums"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares.json","id":"NumberTheory/quadratic-forms-sums-of-squares","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares","status":"can_state","summary":"Representations by quadratic forms: sums of two, three and four squares, binary forms and class numbers, universality theorems, Waring's problem.","title":"Quadratic forms and sums of squares"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares","kind":"scope","parent":9819,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares","status":"can_state","summary":"Two-, three- and four-square theorems, Jacobi-type counts, Gaussian and Hurwitz integer methods, x\u00b2+2y\u00b2 and similar representations.","title":"Sums of squares and representation numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/landau-ramanujan-asymptotic","kind":"node","lean":"MetaMathlibExt.landau_ramanujan_asymptotic","parent":9820,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/index.html#landau-ramanujan-asymptotic","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares","status":"fully_proved","summary":"For real $x$, let $B(x)$ be the number of natural numbers $m \\le x$, including $m = 0$, that can be written as $m = a^2 + b^2$ with natural numbers $a$ and $b$. Then there is a constant $c_0 > 0$ such that $B(x)$ is asymptotically equivalent to $c_0 x / \\sqrt{\\log x}$ as $x \\to \\infty$.","title":"Landau\u2013Ramanujan theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/nat-eq-sq-sub-sq-iff-not-eq-four-mul-add","kind":"node","lean":"MetaMathlibExt.nat_eq_sq_sub_sq_iff_not_eq_four_mul_add_two","parent":9820,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/index.html#nat-eq-sq-sub-sq-iff-not-eq-four-mul-add","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares","status":"fully_proved","summary":"A positive natural number $n$ can be written as $n = a^2 - b^2$ with natural numbers $a$ and $b$ if and only if there is no natural number $k$ with $n = 4k + 2$.","title":"Positive integers that are differences of two squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sum-two-coprime-squares-characterization","kind":"node","lean":"MetaMathlibExt.SumTwoCoprimeSquares.sum_two_coprime_squares_characterization","parent":9820,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/index.html#sum-two-coprime-squares-characterization","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares","status":"fully_proved","summary":"Let $d > 2$ be a natural number. Then $d = a^2 + b^2$ for some coprime positive natural numbers $a$ and $b$ if and only if there is a natural number $q$, all of whose prime factors are congruent to $1$ modulo $4$, such that $d = q$ or $d = 2q$.","title":"Sums of two coprime squares"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","kind":"scope","navigation":"module","parent":9820,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","title":"Geometric sums as sums of two squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/phi4p-x1","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.phi4pX1","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","The alternating polynomial "],["math","\\(X_1\\)"]],"source":"The alternating polynomial $X_1$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#phi4p-x1","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"fully_proved","summary":"For a natural number $p$, $X_1$ is the polynomial with integer coefficients","title":"The alternating polynomial $X_1$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/phi4p-y1","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.phi4pY1","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","The alternating polynomial "],["math","\\(Y_1\\)"]],"source":"The alternating polynomial $Y_1$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#phi4p-y1","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"fully_proved","summary":"For a natural number $p$, $Y_1$ is the polynomial with integer coefficients","title":"The alternating polynomial $Y_1$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/cyclotomic-four-mul-eq-sq-add-sq","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.cyclotomicFourMul_eq_sq_add_sq","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","The cyclotomic polynomial "],["math","\\(\\Phi_{4p}\\)"],["text"," as a sum of two explicit squares"]],"source":"The cyclotomic polynomial $\\Phi_{4p}$ as a sum of two explicit squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#cyclotomic-four-mul-eq-sq-add-sq","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Let $p$ be an odd prime, let $X_1 = \\sum_{i=0}^{(p-1)/2} (-1)^i x^{p-1-2i}$ and $Y_1 = \\sum_{i=0}^{(p-3)/2} (-1)^i x^{p-2-2i}$ in $\\mathbb{Z}[x]$, and let $\\Phi_{4p}$ be the $4p$-th cyclotomic polynomial over $\\mathbb{Z}$. Then $\\Phi_{4p} = X_1^2 + Y_1^2$.","title":"The cyclotomic polynomial $\\Phi_{4p}$ as a sum of two explicit squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/cyclotomic-four-mul-pow-eq-sq-add-sq","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.cyclotomicFourMulPow_eq_sq_add_sq","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","The cyclotomic polynomial "],["math","\\(\\Phi_{4p^k}\\)"],["text"," as a sum of two explicit squares"]],"source":"The cyclotomic polynomial $\\Phi_{4p^k}$ as a sum of two explicit squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#cyclotomic-four-mul-pow-eq-sq-add-sq","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Let $p$ be an odd prime, let $k \\ge 1$, let $X_1 = \\sum_{i=0}^{(p-1)/2} (-1)^i x^{p-1-2i}$ and $Y_1 = \\sum_{i=0}^{(p-3)/2} (-1)^i x^{p-2-2i}$ in $\\mathbb{Z}[x]$, and let $\\Phi_{4p^k}$ be the $4p^k$-th cyclotomic polynomial over $\\mathbb{Z}$. Then $\\Phi_{4p^k}(x) = X_1\\bigl(x^{p^{k-1}}\\bigr)^2 + Y_1\\bigl(x^{p^{k-1}}\\bigr)^2$.","title":"The cyclotomic polynomial $\\Phi_{4p^k}$ as a sum of two explicit squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/exists-sq-add-sq-eq-cyclotomic","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.exists_sq_add_sq_eq_cyclotomic","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every cyclotomic polynomial "],["math","\\(\\Phi_{4q}\\)"],["text"," is a sum of two squares"]],"source":"Every cyclotomic polynomial $\\Phi_{4q}$ is a sum of two squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#exists-sq-add-sq-eq-cyclotomic","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"For every integer $q \\ge 1$ there are polynomials $A, B \\in \\mathbb{Z}[x]$ such that the $4q$-th cyclotomic polynomial $\\Phi_{4q}$ over $\\mathbb{Z}$ satisfies $\\Phi_{4q} = A^2 + B^2$.","title":"Every cyclotomic polynomial $\\Phi_{4q}$ is a sum of two squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/geom-sum-pythagorean","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.geomSum_pythagorean","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","Geometric sums for primes "],["math","\\(p \\equiv 3 \\pmod 4\\)"],["text"," and odd "],["math","\\(n\\)"],["text"," are sums of two squares"]],"source":"Geometric sums for primes $p \\equiv 3 \\pmod 4$ and odd $n$ are sums of two squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#geom-sum-pythagorean","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 4$, let $n$ be a positive odd integer, and let $S = \\sum_{j=0}^{p-1} p^{nj} = 1 + p^n + p^{2n} + \\cdots + p^{(p-1)n}$. Then every prime divisor of $S$ is $2$ or congruent to $1$ modulo $4$, and there are natural numbers $a, b$ with $S = a^2 + b^2$.","title":"Geometric sums for primes $p \\equiv 3 \\pmod 4$ and odd $n$ are sums of two squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/geom-sum-pow-eq-sq-add-sq","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.geomSumPow_eq_sq_add_sq","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","Geometric sums with "],["math","\\(p^k\\)"],["text"," terms, "],["math","\\(p \\equiv 3 \\pmod 4\\)"],["text"," and "],["math","\\(n\\)"],["text"," odd, are sums of two squares"]],"source":"Geometric sums with $p^k$ terms, $p \\equiv 3 \\pmod 4$ and $n$ odd, are sums of two squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#geom-sum-pow-eq-sq-add-sq","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 4$, let $n$ be a positive odd integer, let $k$ be a positive integer, and put $q = p^k$. Then there are integers $a, b$ with $\\sum_{j=0}^{q-1} p^{nj} = 1 + p^n + p^{2n} + \\cdots + p^{(q-1)n} = a^2 + b^2$.","title":"Geometric sums with $p^k$ terms, $p \\equiv 3 \\pmod 4$ and $n$ odd, are sums of two squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/alt-geom-sum-pythagorean","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.altGeomSum_pythagorean","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating geometric sums for odd primes "],["math","\\(p\\)"],["text"," and even "],["math","\\(n\\)"],["text"," are sums of two coprime squares"]],"source":"Alternating geometric sums for odd primes $p$ and even $n$ are sums of two coprime squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#alt-geom-sum-pythagorean","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Let $p$ be an odd prime, let $n$ be a positive even integer, and let $S = \\sum_{j=0}^{p-1} (-1)^j p^{nj} = 1 - p^n + p^{2n} - \\cdots + p^{(p-1)n}$. Then every prime divisor of $S$ is $2$ or congruent to $1$ modulo $4$, and there are coprime natural numbers $a, b$ with $S = a^2 + b^2$.","title":"Alternating geometric sums for odd primes $p$ and even $n$ are sums of two coprime squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/alt-geom-sum-pythagorean-one-mod-four","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.altGeomSum_pythagorean_one_mod_four","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating geometric sums for primes "],["math","\\(p \\equiv 1 \\pmod 4\\)"],["text"," are sums of two squares"]],"source":"Alternating geometric sums for primes $p \\equiv 1 \\pmod 4$ are sums of two squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#alt-geom-sum-pythagorean-one-mod-four","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, let $n \\ge 0$ be an integer, and let $S = \\sum_{j=0}^{p-1} (-1)^j p^{nj} = 1 - p^n + p^{2n} - \\cdots + p^{(p-1)n}$. Then every prime divisor of $S$ is $2$ or congruent to $1$ modulo $4$, and there are natural numbers $a, b$ with $S = a^2 + b^2$.","title":"Alternating geometric sums for primes $p \\equiv 1 \\pmod 4$ are sums of two squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/alt-geom-sum-pow-eq-sq-add-sq","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.altGeomSumPow_eq_sq_add_sq","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating geometric sums with "],["math","\\(p^k\\)"],["text"," terms, "],["math","\\(p \\equiv 1 \\pmod 4\\)"],["text",", as sums of two squares"]],"source":"Alternating geometric sums with $p^k$ terms, $p \\equiv 1 \\pmod 4$, as sums of two squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#alt-geom-sum-pow-eq-sq-add-sq","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, let $n$ be a positive odd integer, let $k$ be a positive integer, and put $q = p^k$. Then there are integers $a, b$ with $\\sum_{j=0}^{q-1} (-1)^j p^{nj} = 1 - p^n + p^{2n} - \\cdots + p^{(q-1)n} = a^2 + b^2$.","title":"Alternating geometric sums with $p^k$ terms, $p \\equiv 1 \\pmod 4$, as sums of two squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/alt-geom-sum-eq-sq-sub-sq","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.altGeomSum_eq_sq_sub_sq","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating geometric sums for primes "],["math","\\(p \\equiv 3 \\pmod 4\\)"],["text"," and odd "],["math","\\(n\\)"],["text"," are differences of two squares"]],"source":"Alternating geometric sums for primes $p \\equiv 3 \\pmod 4$ and odd $n$ are differences of two squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#alt-geom-sum-eq-sq-sub-sq","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Alternating geometric sums for primes $p \\equiv 3 \\pmod 4$ and odd $n$ are differences of two squares. Let $p$ be a prime with $p \\equiv 3 \\pmod 4$ and let $n$ be a positive odd integer. Then there are integers $a, b$ with $\\sum_{j=0}^{p-1} (-1)^j p^{nj} = 1 - p^n + p^{2n} - \\cdots + p^{(p-1)n} = a^2 - b^2$.","title":"Alternating geometric sums for primes $p \\equiv 3 \\pmod 4$ and odd $n$ are differences of two squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/alt-bigeom-pythagorean","kind":"node","lean":"MathlibExt.NumberTheory.GeometricSumsTwoSquaresWanted.altBigeom_pythagorean","parent":9824,"reading_title":{"format":"inline-parts-v1","parts":[["text","An alternating sum in coprime integers "],["math","\\(x, y\\)"],["text"," is a sum of two coprime squares"]],"source":"An alternating sum in coprime integers $x, y$ is a sum of two coprime squares"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares/index.html#alt-bigeom-pythagorean","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/geometric-sums-as-sums-of-two-squares","status":"can_prove","summary":"Let $x$ and $y$ be coprime integers, let $n$ be a positive integer, and let $S = \\sum_{j=0}^{2n} (-1)^j x^{4n-2j} y^{2j}$, that is, $S = x^{4n} - x^{4n-2} y^2 + \\cdots - x^2 y^{4n-2} + y^{4n}$. Then every prime divisor of $S$ is $2$ or congruent to $1$ modulo $4$, and there are coprime natural numbers $a, b$ with $S = a^2 + b^2$.","title":"An alternating sum in coprime integers $x, y$ is a sum of two coprime squares"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","kind":"scope","navigation":"module","parent":9820,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","title":"Representation numbers of sums of squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/solutions","kind":"node","lean":"SumSquares.solutions","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#solutions","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the solution set is the set of all integer vectors $x = (x_1, \\dots, x_k) \\in \\mathbb{Z}^k$ with $x_1^2 + \\cdots + x_k^2 = n$.","title":"Integer solutions of a sum of squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/representation-number","kind":"node","lean":"SumSquares.representationNumber","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#representation-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the representation number $r_k(n)$ is the number of elements of the solution set, that is, the number of integer vectors $(x_1, \\dots, x_k) \\in \\mathbb{Z}^k$ with $x_1^2 + \\cdots + x_k^2 = n$. Representations that differ in the order or the signs of the entries are counted separately.","title":"Representation numbers of sums of squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/solutions-finite","kind":"node","lean":"SumSquares.solutions_finite","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#solutions-finite","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"For all natural numbers $k$ and $n$, there are only finitely many integer vectors $(x_1, \\dots, x_k) \\in \\mathbb{Z}^k$ with $x_1^2 + \\cdots + x_k^2 = n$.","title":"Finitely many representations as a sum of squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/representation-number-eq-to-finset-card","kind":"node","lean":"SumSquares.representationNumber_eq_toFinset_card","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#representation-number-eq-to-finset-card","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"For all natural numbers $k$ and $n$, the representation number $r_k(n)$ equals the number of elements of the finite solution set of integer vectors $(x_1, \\dots, x_k) \\in \\mathbb{Z}^k$ with $x_1^2 + \\cdots + x_k^2 = n$.","title":"The representation number counts a finite set"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/zero-mem-solutions","kind":"node","lean":"SumSquares.zero_mem_solutions","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#zero-mem-solutions","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"For every $k \\in \\mathbb{N}$, the zero vector of $\\mathbb{Z}^k$ has squared norm $0$, that is, it lies in the set of integer vectors $x \\in \\mathbb{Z}^k$ with $x_1^2 + \\cdots + x_k^2 = 0$.","title":"The zero vector represents zero as a sum of squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/solutions-zero-zero","kind":"node","lean":"SumSquares.solutions_zero_zero","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#solutions-zero-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"The set of vectors $x \\in \\mathbb{Z}^0$ whose sum of squares (an empty sum) equals $0$ is all of $\\mathbb{Z}^0$.","title":"Solution set of the empty sum at zero"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/solutions-zero-succ","kind":"node","lean":"SumSquares.solutions_zero_succ","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#solutions-zero-succ","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"For every natural number $n \\ge 1$, no vector $x \\in \\mathbb{Z}^0$ has sum of squares equal to $n$; that is, the solution set is empty.","title":"Zero squares never sum to a positive integer"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/solutions-one-zero","kind":"node","lean":"SumSquares.solutions_one_zero","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#solutions-one-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"The set of vectors $x \\in \\mathbb{Z}^1$ with $x_1^2 = 0$ consists of the zero vector alone.","title":"Solution set of one square equal to zero"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/representation-number-zero-zero","kind":"node","lean":"SumSquares.representationNumber_zero_zero","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#representation-number-zero-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"The representation number $r_0(0)$, the number of integer vectors $x \\in \\mathbb{Z}^0$ whose sum of squares (an empty sum) equals $0$, is $1$.","title":"Zero as a sum of zero squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/representation-number-zero-succ","kind":"node","lean":"SumSquares.representationNumber_zero_succ","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#representation-number-zero-succ","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the representation number $r_0(n)$, the number of integer vectors $x \\in \\mathbb{Z}^0$ whose sum of squares (an empty sum) equals $n$, is $0$.","title":"Positive integers have no representation as a sum of zero squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/representation-number-one-zero","kind":"node","lean":"SumSquares.representationNumber_one_zero","parent":9837,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares/index.html#representation-number-one-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representation-numbers-of-sums-of-squares","status":"fully_proved","summary":"The representation number $r_1(0)$, the number of integer vectors $x \\in \\mathbb{Z}^1$ with $x_1^2 = 0$, equals $1$.","title":"Zero as a sum of one square"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","kind":"scope","navigation":"module","parent":9820,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","title":"Jacobi's two-square theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/div-chi-sum","kind":"node","lean":"SumSquares.divChiSum","parent":9849,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#div-chi-sum","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"For a natural number $n$, this is the integer $\\sum_{d \\mid n} \\chi_4(d)$, the sum over the positive divisors $d$ of $n$ of the nonprincipal Dirichlet character $\\chi_4$ modulo 4, which takes the value $0$ at even $d$, $1$ at $d \\equiv 1 \\pmod 4$ and $-1$ at $d \\equiv 3 \\pmod 4$. For $n = 0$ there are no divisors and the sum is $0$.","title":"Divisor sum of the character modulo 4"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/div-chi-sum-mul-of-coprime","kind":"node","lean":"SumSquares.divChiSum_mul_of_coprime","parent":9849,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_4\\)"],["text"," is multiplicative"]],"source":"The divisor sum of $\\chi_4$ is multiplicative"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#div-chi-sum-mul-of-coprime","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"Let $\\chi_4$ be the nonprincipal Dirichlet character modulo 4 and, for a natural number $n$, write $S(n) = \\sum_{d \\mid n} \\chi_4(d)$ for the sum over the positive divisors $d$ of $n$ (so $S(0) = 0$). If $m$ and $n$ are coprime natural numbers, then $S(mn) = S(m)\\,S(n)$.","title":"The divisor sum of $\\chi_4$ is multiplicative"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/div-chi-sum-one","kind":"node","lean":"SumSquares.divChiSum_one","parent":9849,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_4\\)"],["text"," at one"]],"source":"The divisor sum of $\\chi_4$ at one"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#div-chi-sum-one","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"For the nonprincipal Dirichlet character $\\chi_4$ modulo 4, the sum $\\sum_{d \\mid 1} \\chi_4(d)$ over the divisors of $1$ equals $1$.","title":"The divisor sum of $\\chi_4$ at one"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/div-chi-sum-prime-pow-two","kind":"node","lean":"SumSquares.divChiSum_prime_pow_two","parent":9849,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_4\\)"],["text"," at powers of two"]],"source":"The divisor sum of $\\chi_4$ at powers of two"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#div-chi-sum-prime-pow-two","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"Let $\\chi_4$ be the nonprincipal Dirichlet character modulo 4. For every natural number $e$, $\\sum_{d \\mid 2^e} \\chi_4(d) = 1$: only the divisor $1$ contributes.","title":"The divisor sum of $\\chi_4$ at powers of two"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/div-chi-sum-prime-pow-split","kind":"node","lean":"SumSquares.divChiSum_prime_pow_split","parent":9849,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_4\\)"],["text"," at powers of a prime "],["math","\\(p \\equiv 1 \\pmod 4\\)"]],"source":"The divisor sum of $\\chi_4$ at powers of a prime $p \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#div-chi-sum-prime-pow-split","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"Let $\\chi_4$ be the nonprincipal Dirichlet character modulo 4. If $p$ is a prime with $p \\equiv 1 \\pmod 4$ and $e$ is a natural number, then $\\sum_{d \\mid p^e} \\chi_4(d) = e + 1$.","title":"The divisor sum of $\\chi_4$ at powers of a prime $p \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/div-chi-sum-prime-pow-inert","kind":"node","lean":"SumSquares.divChiSum_prime_pow_inert","parent":9849,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_4\\)"],["text"," at powers of a prime "],["math","\\(q \\equiv 3 \\pmod 4\\)"]],"source":"The divisor sum of $\\chi_4$ at powers of a prime $q \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#div-chi-sum-prime-pow-inert","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"Let $\\chi_4$ be the nonprincipal Dirichlet character modulo 4. If $q$ is a prime with $q \\equiv 3 \\pmod 4$ and $f$ is a natural number, then $\\sum_{d \\mid q^f} \\chi_4(d)$ equals $1$ when $f$ is even and $0$ when $f$ is odd.","title":"The divisor sum of $\\chi_4$ at powers of a prime $q \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/pair-equiv-gaussian-norm-fiber","kind":"node","lean":"SumSquares.pairEquivGaussianNormFiber","parent":9849,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#pair-equiv-gaussian-norm-fiber","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"For a natural number $j$, the map $(a, b) \\mapsto a + bi$ is a bijection from the set of integer pairs $(a, b)$ with $a^2 + b^2 = j$ to the set of Gaussian integers $z$ with norm $N(z) = j$; its inverse sends $z$ to the pair of its real and imaginary parts.","title":"Sums of two squares as Gaussian norms"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/representation-number-two-eq","kind":"node","lean":"SumSquares.representationNumber_two_eq","parent":9849,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#representation-number-two-eq","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"For every natural number $j$, the representation number $r_2(j)$ equals the number of pairs $(a, b)$ of integers with $a^2 + b^2 = j$.","title":"Two-square representation numbers count integer pairs"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/representation-number-two-jacobi-count","kind":"node","lean":"SumSquares.representationNumber_two_jacobi_count","parent":9849,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem/index.html#representation-number-two-jacobi-count","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/jacobi-s-two-square-theorem","status":"fully_proved","summary":"Let $\\chi_4$ be the nonprincipal Dirichlet character modulo $4$, so $\\chi_4(d)$ is $0$ for even $d$, $1$ for $d \\equiv 1 \\pmod 4$ and $-1$ for $d \\equiv 3 \\pmod 4$. For every natural number $j \\ge 1$, the representation number $r_2(j)$, the number of integer pairs $(x_1, x_2)$ with $x_1^2 + x_2^2 = j$ counted with order and signs, satisfies $r_2(j) = 4 \\sum_{d \\mid j} \\chi_4(d)$, where $d$ runs over the positive\u2026","title":"Jacobi's two-square theorem"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","kind":"scope","navigation":"module","parent":9820,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","title":"Representations by squares and triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/square-theta-series","kind":"node","lean":"MetaMathlibExt.squareThetaSeries","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#square-theta-series","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"The square theta series is the formal power series $\\theta(q) = \\sum_{m \\in \\mathbb{Z}} q^{m^2}$ in $\\mathbb{Z}[[q]]$: for each $n \\ge 0$, its coefficient of $q^n$ is the number of integers $m$ with $m^2 = n$.","title":"The square theta series"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/triangular-theta-series","kind":"node","lean":"MetaMathlibExt.triangularThetaSeries","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#triangular-theta-series","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"The triangular theta series is the formal power series $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$ with integer coefficients: for each natural number $n$, its coefficient of $q^n$ is the number of natural numbers $m$ with $m(m+1)/2 = n$.","title":"Triangular theta series"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/coeff-square-theta-series-small","kind":"node","lean":"MetaMathlibExt.coeff_squareThetaSeries_small","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#coeff-square-theta-series-small","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"In the square theta series $\\theta(q) = \\sum_{m \\in \\mathbb{Z}} q^{m^2}$, the coefficients of $q^0$, $q^1$, $q^2$ and $q^4$ are $1$, $2$, $0$ and $2$ respectively.","title":"First coefficients of the square theta series"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/coeff-triangular-theta-series-small","kind":"node","lean":"MetaMathlibExt.coeff_triangularThetaSeries_small","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#coeff-triangular-theta-series-small","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"In the triangular theta series $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$, the coefficients of $q^0$, $q^1$, $q^2$ and $q^3$ are $1$, $1$, $0$ and $1$ respectively.","title":"First coefficients of the triangular theta series"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/is-unit-square-theta-series","kind":"node","lean":"MetaMathlibExt.isUnit_squareThetaSeries","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#is-unit-square-theta-series","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"The square theta series $\\theta(q) = \\sum_{m \\in \\mathbb{Z}} q^{m^2}$ is a unit of the ring $\\mathbb{Z}[[q]]$ of formal power series with integer coefficients, since its constant coefficient is $1$.","title":"The square theta series is a unit"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/is-unit-triangular-theta-series","kind":"node","lean":"MetaMathlibExt.isUnit_triangularThetaSeries","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#is-unit-triangular-theta-series","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"The triangular theta series $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$ is a unit of the ring $\\mathbb{Z}[[q]]$ of formal power series with integer coefficients, since its constant coefficient is $1$.","title":"The triangular theta series is a unit"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-squares-rep-count","kind":"node","lean":"MetaMathlibExt.sumOfSquaresRepCount","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-squares-rep-count","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"For an integer $k$ and a natural number $n$, $r_k(n)$ is the coefficient of $q^n$ in $\\theta(q)^k$, where $\\theta(q) = \\sum_{m \\in \\mathbb{Z}} q^{m^2}$. Since $\\theta$ has constant coefficient $1$, it is a unit of $\\mathbb{Z}[[q]]$, so $\\theta^k$ is a power series for negative $k$ as well.","title":"Representation numbers for sums of squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-triangular-rep-count","kind":"node","lean":"MetaMathlibExt.sumOfTriangularRepCount","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-triangular-rep-count","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"For an integer $k$ and a natural number $n$, $t_k(n)$ is the coefficient of $q^n$ in $\\psi(q)^k$, where $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$. Since $\\psi$ has constant coefficient $1$, it is a unit of $\\mathbb{Z}[[q]]$, so $\\psi^k$ is a power series for negative $k$ as well.","title":"Representation numbers for sums of triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-squares-rep-count-one","kind":"node","lean":"MetaMathlibExt.sumOfSquaresRepCount_one","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-squares-rep-count-one","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"For an integer $k$ and $n \\ge 0$, let $r_k(n)$ be the coefficient of $q^n$ in $\\theta(q)^k$, where $\\theta$ is the square theta series. Then for every $n \\ge 0$, $r_1(n)$ equals the coefficient of $q^n$ in $\\theta(q)$.","title":"The one-square counts are the coefficients of the square theta series"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-triangular-rep-count-one","kind":"node","lean":"MetaMathlibExt.sumOfTriangularRepCount_one","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-triangular-rep-count-one","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"For an integer $k$, let $t_k(n)$ be the coefficient of $q^n$ in $\\psi(q)^k$, where $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$. Then for every natural number $n$, $t_1(n)$ equals the coefficient of $q^n$ in $\\psi(q)$.","title":"Triangular representation numbers with one summand"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-squares-rep-count-add-mul-prime","kind":"node","lean":"MetaMathlibExt.sumOfSquaresRepCount_add_mul_prime","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-squares-rep-count-add-mul-prime","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$ and $n \\ge 0$, let $r_k(n)$ be the coefficient of $q^n$ in $\\theta(q)^k$, where $\\theta$ is the square theta series; for $k \\ge 1$ it is the number of representations of $n$ as an ordered sum of $k$ squares of integers. Let $n \\ge 1$ and $k \\ge 1$ be integers and let $p$ be a prime with $p \\ge n + 1$. If $p$ divides $r_k(n)$, then $p$ divides $r_{k+jp}(n)$ for every integer $j$.","title":"Divisibility of square representation counts by a large prime is periodic"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-triangular-rep-count-add-mul-prim","kind":"node","lean":"MetaMathlibExt.sumOfTriangularRepCount_add_mul_prime","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-triangular-rep-count-add-mul-prim","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$, let $t_k(n)$ be the coefficient of $q^n$ in $\\psi(q)^k$, where $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$. Let $n \\ge 1$ and $k \\ge 1$ be natural numbers and let $p$ be a prime with $p \\ge n + 1$. If $p$ divides $t_k(n)$, then $p$ divides $t_{k+rp}(n)$ for every integer $r$.","title":"Prime divisibility of triangular representation numbers persists under prime shifts"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-sum-of-squares-rep-count-window","kind":"node","lean":"MetaMathlibExt.sum_sumOfSquaresRepCount_window","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-sum-of-squares-rep-count-window","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$, let $r_k(n)$ be the coefficient of $q^n$ in $\\theta(q)^k$, where $\\theta(q) = \\sum_{m \\in \\mathbb{Z}} q^{m^2}$. Let $n \\ge 1$, let $p$ be an odd prime with $p - 1 \\ge n + 1$, and let $s \\ge p$ be an integer. Then $p$ divides $\\sum_{i=0}^{p-1} r_{s-i}(n)$.","title":"Window congruence for sums of squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-sum-of-triangular-rep-count-window","kind":"node","lean":"MetaMathlibExt.sum_sumOfTriangularRepCount_window","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-sum-of-triangular-rep-count-window","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$, let $t_k(n)$ be the coefficient of $q^n$ in $\\psi(q)^k$, where $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$. Let $n \\ge 1$, let $p$ be an odd prime with $p - 1 \\ge n + 1$, and let $s \\ge p$ be an integer. Then $p$ divides $\\sum_{i=0}^{p-1} t_{s-i}(n)$.","title":"Window congruence for sums of triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/rep-count-ne-zero-of-window","kind":"node","lean":"MetaMathlibExt.repCount_ne_zero_of_window","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#rep-count-ne-zero-of-window","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$ and $n \\ge 0$, let $r_k(n)$ and $t_k(n)$ be the coefficients of $q^n$ in $\\theta(q)^k$ and $\\psi(q)^k$, where $\\theta$ and $\\psi$ are the square and triangular theta series (units of $\\mathbb{Z}[[q]]$). Let $n \\ge 1$, let $p$ be an odd prime with $p - 1 \\ge n + 1$, and let $s$ be an integer. If $p$ does not divide $r_{s-1}(n) + r_{s-2}(n) + \\cdots + r_{s-(p-1)}(n)$, then $r_s(n) \\ne 0$. Likewise,\u2026","title":"Representability criterion from a window of counts"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-triangular-rep-count-ne-zero-of-w","kind":"node","lean":"MetaMathlibExt.sumOfTriangularRepCount_ne_zero_of_window","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-triangular-rep-count-ne-zero-of-w","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$, let $t_k(n)$ be the coefficient of $q^n$ in $\\psi(q)^k$, where $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$. Let $n \\ge 1$ and let $p$ be an odd prime with $p - 1 \\ge n + 1$. For every integer $s$, if $p$ does not divide $\\sum_{i=1}^{p-1} t_{s-i}(n)$, then $t_s(n) \\ne 0$.","title":"Window criterion for representability by triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/prime-dvd-rep-count-of-half-lt","kind":"node","lean":"MetaMathlibExt.prime_dvd_repCount_of_half_lt","parent":9859,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prime between "],["math","\\(p/2\\)"],["text"," and "],["math","\\(p\\)"],["text"," divides the representation counts of "],["math","\\(p\\)"]],"source":"A prime between $p/2$ and $p$ divides the representation counts of $p$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#prime-dvd-rep-count-of-half-lt","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For integers $k, n \\ge 1$, let $r_k(n)$ be the number of representations of $n$ as an ordered sum of $k$ squares of integers and $t_k(n)$ the number of representations of $n$ as an ordered sum of $k$ triangular numbers $m(m+1)/2$ with $m \\ge 0$. Let $p$ be an odd prime and $\\ell$ a prime with $p/2 < \\ell < p$. Then $\\ell$ divides both $r_\\ell(p)$ and $t_\\ell(p)$.","title":"A prime between $p/2$ and $p$ divides the representation counts of $p$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/triangular-log-coeff","kind":"node","lean":"MetaMathlibExt.triangularLogCoeff","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#triangular-log-coeff","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"For a natural number $n$, let $b_n$ be the rational number $b_n = \\sum_{d \\mid n,\\ n/d \\text{ odd}} 1/d - \\sum_{d \\mid n,\\ n/d \\text{ even}} 1/d$, where $d$ runs over the positive divisors of $n$; for $n = 0$ the sums are taken to be empty, so $b_0 = 0$.","title":"Logarithmic coefficients of the triangular theta series"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/triangular-log-coeff-small","kind":"node","lean":"MetaMathlibExt.triangularLogCoeff_small","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#triangular-log-coeff-small","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"fully_proved","summary":"The logarithmic coefficients $b_n$ of the triangular theta series satisfy $b_1 = 1$, $b_2 = -1/2$ and $b_3 = 4/3$; this matches the expansion $\\log(1 + q + q^3 + \\cdots) = q - q^2/2 + 4q^3/3 + \\cdots$.","title":"First logarithmic coefficients of the triangular theta series"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-triangular-rep-count-eq-sum-compo","kind":"node","lean":"MetaMathlibExt.sumOfTriangularRepCount_eq_sum_compositions","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-triangular-rep-count-eq-sum-compo","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$, let $t_k(n)$ be the coefficient of $q^n$ in $\\psi(q)^k$, where $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$. For $m \\ge 1$ let $b_m = \\sum_{d \\mid m,\\ m/d \\text{ odd}} 1/d - \\sum_{d \\mid m,\\ m/d \\text{ even}} 1/d$, the coefficient of $q^m$ in $\\log \\psi(q)$. Then for every integer $k$ and natural number $n$,","title":"Composition formula for triangular representation numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-triangular-rep-count-succ","kind":"node","lean":"MetaMathlibExt.sumOfTriangularRepCount_succ","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-triangular-rep-count-succ","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$, let $t_k(n)$ be the coefficient of $q^n$ in $\\psi(q)^k$, where $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$. For $m \\ge 1$ let $b_m = \\sum_{d \\mid m,\\ m/d \\text{ odd}} 1/d - \\sum_{d \\mid m,\\ m/d \\text{ even}} 1/d$, the coefficient of $q^m$ in $\\log \\psi(q)$. Then for every integer $k$ and natural number $n$,","title":"Recurrence for triangular representation numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-squares-rep-count-integer-root","kind":"node","lean":"MetaMathlibExt.sumOfSquaresRepCount_integer_root","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-squares-rep-count-integer-root","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$ and $n \\ge 0$, let $r_k(n)$ be the coefficient of $q^n$ in $\\theta(q)^k$, where $\\theta$ is the square theta series. For every $n \\ge 0$ there is a polynomial $P(z) = \\sum_i c_i z^i$ with integer coefficients such that $P(z) = n! \\, r_z(n)$ for every integer $z$, and every nonzero integer $k$ with $r_k(n) = 0$ divides $c_1$ and satisfies $|k| < 2 |c_i|^{1/(n-i)}$ for some $i$ with $0 \\le i < n$.","title":"Integer zeros of the sums-of-squares counting polynomial"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/sum-of-triangular-rep-count-integer-root","kind":"node","lean":"MetaMathlibExt.sumOfTriangularRepCount_integer_root","parent":9859,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number/index.html#sum-of-triangular-rep-count-integer-root","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-squares-and-triangular-number","status":"can_prove","summary":"For an integer $k$, let $t_k(n)$ be the coefficient of $q^n$ in $\\psi(q)^k$, where $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$. For every natural number $n$ there is a polynomial $P(z) = \\sum_i a_i z^i$ with integer coefficients such that $P(z) = n!\\, t_z(n)$ for every integer $z$, and every nonzero integer $k$ with $t_k(n) = 0$ divides $a_1$ and satisfies $|k| < 2 |a_i|^{1/(n-i)}$ for some $i < n$.","title":"Integer roots of triangular representation numbers"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","kind":"scope","navigation":"module","parent":9820,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","title":"Representations by x\u00b2 + 2y\u00b2"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/chi-neg2-one","kind":"node","lean":"SumSquares.chiNeg2_one","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","The character "],["math","\\(\\chi_{-2}\\)"],["text"," takes the value "],["math","\\(1\\)"],["text"," at "],["math","\\(1\\)"]],"source":"The character $\\chi_{-2}$ takes the value $1$ at $1$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#chi-neg2-one","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $\\chi_{-2}$ be the quadratic character modulo $8$ with values in $\\mathbb{Z}$ (the character of $\\mathbb{Q}(\\sqrt{-2})$): $\\chi_{-2}(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_{-2}(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$, and $\\chi_{-2}(n) = 0$ for even $n$. Then $\\chi_{-2}(1) = 1$.","title":"The character $\\chi_{-2}$ takes the value $1$ at $1$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/div-chi-neg2-sum","kind":"node","lean":"SumSquares.divChiNeg2Sum","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisor sum of the character "],["math","\\(\\chi_{-2}\\)"]],"source":"Divisor sum of the character $\\chi_{-2}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#div-chi-neg2-sum","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $\\chi_{-2}$ be the quadratic character modulo $8$ with values in $\\mathbb{Z}$ (the character of $\\mathbb{Q}(\\sqrt{-2})$): $\\chi_{-2}(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_{-2}(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$, and $\\chi_{-2}(n) = 0$ for even $n$. For a natural number $t$, the divisor sum of $\\chi_{-2}$ at $t$ is the integer $\\sum_{d \\mid t} \\chi_{-2}(d)$, the sum over the positive divisors $d$ of $t$.\u2026","title":"Divisor sum of the character $\\chi_{-2}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/div-chi-neg2-sum-mul-of-coprime","kind":"node","lean":"SumSquares.divChiNeg2Sum_mul_of_coprime","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicativity of the divisor sum of "],["math","\\(\\chi_{-2}\\)"]],"source":"Multiplicativity of the divisor sum of $\\chi_{-2}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#div-chi-neg2-sum-mul-of-coprime","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $\\chi_{-2}$ be the quadratic character modulo $8$ with $\\chi_{-2}(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_{-2}(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$ and $\\chi_{-2}(n) = 0$ for even $n$. For a natural number $t$, let $S(t) = \\sum_{d \\mid t} \\chi_{-2}(d)$, the sum over the positive divisors $d$ of $t$, with $S(0) = 0$ (the empty sum). For coprime natural numbers $m$ and $n$, $S(mn) = S(m)\\,S(n)$.","title":"Multiplicativity of the divisor sum of $\\chi_{-2}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/div-chi-neg2-sum-one","kind":"node","lean":"SumSquares.divChiNeg2Sum_one","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_{-2}\\)"],["text"," at "],["math","\\(1\\)"]],"source":"The divisor sum of $\\chi_{-2}$ at $1$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#div-chi-neg2-sum-one","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $\\chi_{-2}$ be the quadratic character modulo $8$ with $\\chi_{-2}(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_{-2}(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$ and $\\chi_{-2}(n) = 0$ for even $n$. For a positive integer $t$, let $S(t) = \\sum_{d \\mid t} \\chi_{-2}(d)$, the sum over the positive divisors $d$ of $t$. Then $S(1) = 1$.","title":"The divisor sum of $\\chi_{-2}$ at $1$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/div-chi-neg2-sum-prime-pow-two","kind":"node","lean":"SumSquares.divChiNeg2Sum_prime_pow_two","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_{-2}\\)"],["text"," at powers of "],["math","\\(2\\)"]],"source":"The divisor sum of $\\chi_{-2}$ at powers of $2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#div-chi-neg2-sum-prime-pow-two","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $\\chi_{-2}$ be the quadratic character modulo $8$ with $\\chi_{-2}(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_{-2}(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$ and $\\chi_{-2}(n) = 0$ for even $n$. For a positive integer $t$, let $S(t) = \\sum_{d \\mid t} \\chi_{-2}(d)$, the sum over the positive divisors $d$ of $t$. For every natural number $e$, $S(2^e) = 1$: only the divisor $1$ contributes.","title":"The divisor sum of $\\chi_{-2}$ at powers of $2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/div-chi-neg2-sum-prime-pow-split","kind":"node","lean":"SumSquares.divChiNeg2Sum_prime_pow_split","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_{-2}\\)"],["text"," at powers of a prime "],["math","\\(p \\equiv 1, 3 \\pmod 8\\)"]],"source":"The divisor sum of $\\chi_{-2}$ at powers of a prime $p \\equiv 1, 3 \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#div-chi-neg2-sum-prime-pow-split","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $\\chi_{-2}$ be the quadratic character modulo $8$ with $\\chi_{-2}(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_{-2}(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$ and $\\chi_{-2}(n) = 0$ for even $n$. For a positive integer $t$, let $S(t) = \\sum_{d \\mid t} \\chi_{-2}(d)$, the sum over the positive divisors $d$ of $t$. If $p$ is a prime with $p \\equiv 1 \\pmod 8$ or $p \\equiv 3 \\pmod 8$, then $S(p^e) = e + 1$ for every natural\u2026","title":"The divisor sum of $\\chi_{-2}$ at powers of a prime $p \\equiv 1, 3 \\pmod 8$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/div-chi-neg2-sum-prime-pow-inert","kind":"node","lean":"SumSquares.divChiNeg2Sum_prime_pow_inert","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor sum of "],["math","\\(\\chi_{-2}\\)"],["text"," at powers of a prime "],["math","\\(q \\equiv 5, 7 \\pmod 8\\)"]],"source":"The divisor sum of $\\chi_{-2}$ at powers of a prime $q \\equiv 5, 7 \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#div-chi-neg2-sum-prime-pow-inert","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $\\chi_{-2}$ be the quadratic character modulo $8$ with $\\chi_{-2}(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_{-2}(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$ and $\\chi_{-2}(n) = 0$ for even $n$. For a positive integer $t$, let $S(t) = \\sum_{d \\mid t} \\chi_{-2}(d)$, the sum over the positive divisors $d$ of $t$. If $q$ is a prime with $q \\equiv 5 \\pmod 8$ or $q \\equiv 7 \\pmod 8$, then for every natural number $f$,\u2026","title":"The divisor sum of $\\chi_{-2}$ at powers of a prime $q \\equiv 5, 7 \\pmod 8$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/rep-x2-y2-count","kind":"node","lean":"SumSquares.repX2Y2Count","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of elements of "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," of a given norm"]],"source":"Number of elements of $\\mathbb{Z}(\\sqrt{-2})$ of a given norm"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#rep-x2-y2-count","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"For a natural number $t$, the representation count of $t$ is the number of elements $z = a + b\\sqrt{-2}$ of $\\mathbb{Z}[\\sqrt{-2}]$, with $a, b$ integers, whose norm $N(z) = a^2 + 2b^2$ has absolute value $t$. Since the norm is never negative, this is the number of elements of norm $t$.","title":"Number of elements of $\\mathbb{Z}(\\sqrt{-2})$ of a given norm"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/rep-x2-y2-count-one","kind":"node","lean":"SumSquares.repX2Y2Count_one","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of norm one in "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"Elements of norm one in $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#rep-x2-y2-count-one","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Exactly two elements $a + b\\sqrt{-2}$ of $\\mathbb{Z}[\\sqrt{-2}]$, with $a, b$ integers, have norm $a^2 + 2b^2 = 1$.","title":"Elements of norm one in $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/rep-x2-y2-count-two-pow","kind":"node","lean":"SumSquares.repX2Y2Count_two_pow","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," whose norm is a power of "],["math","\\(2\\)"]],"source":"Elements of $\\mathbb{Z}(\\sqrt{-2})$ whose norm is a power of $2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#rep-x2-y2-count-two-pow","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"For every natural number $e$, exactly two elements $a + b\\sqrt{-2}$ of $\\mathbb{Z}[\\sqrt{-2}]$, with $a, b$ integers, have norm $a^2 + 2b^2 = 2^e$.","title":"Elements of $\\mathbb{Z}(\\sqrt{-2})$ whose norm is a power of $2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/rep-x2-y2-count-inert-pow","kind":"node","lean":"SumSquares.repX2Y2Count_inert_pow","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," whose norm is a power of an inert prime"]],"source":"Elements of $\\mathbb{Z}(\\sqrt{-2})$ whose norm is a power of an inert prime"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#rep-x2-y2-count-inert-pow","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $q$ be a prime with $q \\equiv 5$ or $q \\equiv 7 \\pmod 8$, and let $f$ be a natural number. The number of elements $a + b\\sqrt{-2}$ of $\\mathbb{Z}[\\sqrt{-2}]$, with $a, b$ integers, of norm $a^2 + 2b^2 = q^f$ is $2$ if $f$ is even and $0$ if $f$ is odd.","title":"Elements of $\\mathbb{Z}(\\sqrt{-2})$ whose norm is a power of an inert prime"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/rep-x2-y2-count-split-pow","kind":"node","lean":"SumSquares.repX2Y2Count_split_pow","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," whose norm is a power of a split prime"]],"source":"Elements of $\\mathbb{Z}(\\sqrt{-2})$ whose norm is a power of a split prime"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#rep-x2-y2-count-split-pow","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Write $N(a + b\\sqrt{-2}) = a^2 + 2b^2$ for the norm of an element of $\\mathbb{Z}[\\sqrt{-2}]$, where $a, b$ are integers. Let $p$ be a prime with $p \\equiv 1$ or $p \\equiv 3 \\pmod 8$, and suppose $\\pi$ is a prime element of $\\mathbb{Z}[\\sqrt{-2}]$ with $N(\\pi) = p$. Then for every natural number $e$, the number of elements of $\\mathbb{Z}[\\sqrt{-2}]$ of norm $p^e$ is $2(e + 1)$.","title":"Elements of $\\mathbb{Z}(\\sqrt{-2})$ whose norm is a power of a split prime"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/x2y2-count","kind":"node","lean":"SumSquares.x2y2_count","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"],["text"," of norm "],["math","\\(t\\)"],["text"," counted by a divisor sum"]],"source":"Elements of $\\mathbb{Z}(\\sqrt{-2})$ of norm $t$ counted by a divisor sum"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#x2y2-count","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"For every positive integer $t$, the number of elements $a + b\\sqrt{-2}$ of $\\mathbb{Z}[\\sqrt{-2}]$, with $a, b$ integers, of norm $a^2 + 2b^2 = t$ equals $2 \\sum_{d \\mid t} \\chi(d)$, where $d$ runs over the positive divisors of $t$ and $\\chi$ is the character modulo $8$ given by $\\chi(d) = 1$ if $d \\equiv 1, 3 \\pmod 8$, $\\chi(d) = -1$ if $d \\equiv 5, 7 \\pmod 8$, and $\\chi(d) = 0$ if $d$ is even.","title":"Elements of $\\mathbb{Z}(\\sqrt{-2})$ of norm $t$ counted by a divisor sum"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/re-mul-re-add-two-mul-im-mul-im","kind":"node","lean":"SumSquares.re_mul_re_add_two_mul_im_mul_im","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinates of an element of norm "],["math","\\(N\\)"],["text"," in "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"Coordinates of an element of norm $N$ in $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#re-mul-re-add-two-mul-im-mul-im","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $N$ be a natural number and let $z = a + b\\sqrt{-2}$, with $a, b \\in \\mathbb{Z}$, be an element of $\\mathbb{Z}[\\sqrt{-2}]$ whose norm has absolute value $N$. Then $a^2 + 2b^2 = N$.","title":"Coordinates of an element of norm $N$ in $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/rep-x2-y2-count-eq-four-mul-pos","kind":"node","lean":"SumSquares.repX2Y2Count_eq_four_mul_pos","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting representations by "],["math","\\(x^2 + 2y^2\\)"],["text"," through positive solutions"]],"source":"Counting representations by $x^2 + 2y^2$ through positive solutions"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#rep-x2-y2-count-eq-four-mul-pos","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $N$ be a natural number such that every element $a + b\\sqrt{-2}$ of $\\mathbb{Z}[\\sqrt{-2}]$ of norm $a^2 + 2b^2 = N$ has $a \\neq 0$ and $b \\neq 0$. Then the number of elements of $\\mathbb{Z}[\\sqrt{-2}]$ of norm $N$ is four times the number of pairs $(x, y)$ of positive integers with $x^2 + 2y^2 = N$.","title":"Counting representations by $x^2 + 2y^2$ through positive solutions"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/div-chi-neg2-sum-two-mul-of-odd","kind":"node","lean":"SumSquares.divChiNeg2Sum_two_mul_of_odd","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Doubling an odd argument leaves the divisor sum of "],["math","\\(\\chi_{-2}\\)"],["text"," unchanged"]],"source":"Doubling an odd argument leaves the divisor sum of $\\chi_{-2}$ unchanged"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#div-chi-neg2-sum-two-mul-of-odd","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"Let $\\chi_{-2}$ be the quadratic character modulo $8$ with $\\chi_{-2}(n) = 1$ for $n \\equiv 1, 3 \\pmod 8$, $\\chi_{-2}(n) = -1$ for $n \\equiv 5, 7 \\pmod 8$ and $\\chi_{-2}(n) = 0$ for even $n$. For a positive integer $t$, let $S(t) = \\sum_{d \\mid t} \\chi_{-2}(d)$, the sum over the positive divisors $d$ of $t$. For every odd natural number $m$, $S(2m) = S(m)$.","title":"Doubling an odd argument leaves the divisor sum of $\\chi_{-2}$ unchanged"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/pair-equiv-sqrt-neg-two-norm-fiber","kind":"node","lean":"SumSquares.pairEquivSqrtNegTwoNormFiber","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Representations by "],["math","\\(x^2 + 2y^2\\)"],["text"," as elements of "],["math","\\(\\mathbb{Z}(\\sqrt{-2})\\)"]],"source":"Representations by $x^2 + 2y^2$ as elements of $\\mathbb{Z}(\\sqrt{-2})$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#pair-equiv-sqrt-neg-two-norm-fiber","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"For every natural number $j$, the map $(x, y) \\mapsto x + y\\sqrt{-2}$ is a bijection from the set of integer pairs $(x, y)$ with $x^2 + 2y^2 = j$ to the set of elements $z \\in \\mathbb{Z}[\\sqrt{-2}]$ whose norm $N(z)$ has absolute value $j$.","title":"Representations by $x^2 + 2y^2$ as elements of $\\mathbb{Z}(\\sqrt{-2})$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/x2-add-two-sq-count","kind":"node","lean":"SumSquares.x2_add_two_sq_count","parent":9883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of representations by "],["math","\\(x^2 + 2y^2\\)"]],"source":"Number of representations by $x^2 + 2y^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y/index.html#x2-add-two-sq-count","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/representations-by-x-2y","status":"fully_proved","summary":"For every integer $j \\ge 1$, the number of pairs of integers $(x, y)$ with $x^2 + 2y^2 = j$ equals $2 \\sum_{d \\mid j} \\chi(d)$, where $d$ runs over the positive divisors of $j$ and $\\chi$ is the Dirichlet character modulo $8$ attached to $\\mathbb{Q}(\\sqrt{-2})$: $\\chi(d) = 1$ if $d \\equiv 1, 3 \\pmod 8$, $\\chi(d) = -1$ if $d \\equiv 5, 7 \\pmod 8$, and $\\chi(d) = 0$ if $d$ is even.","title":"Number of representations by $x^2 + 2y^2$"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","kind":"scope","navigation":"module","parent":9820,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"can_prove","title":"Sums of four squares via Hurwitz continued fractions"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-round","kind":"node","lean":"MetaMathlibExt.hurwitzRound","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-round","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"For a complex number $z$, let $[z] = m + n i$, where $m = \\lfloor \\operatorname{Re} z + 1/2 \\rfloor$ and $n = \\lfloor \\operatorname{Im} z + 1/2 \\rfloor$; that is, the real and imaginary parts of $z$ are rounded to the nearest integer, with halves rounded up. Then $[z]$ is a Gaussian integer nearest to $z$.","title":"Hurwitz rounding to the nearest Gaussian integer"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-iter","kind":"node","lean":"MetaMathlibExt.hurwitzIter","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-iter","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"For a complex number $z$, the Hurwitz iterates are $z_0 = z$ and $z_{n+1} = (z_n - [z_n])^{-1}$ for $n \\ge 0$, where $[u]$ is the Gaussian integer nearest to $u$, obtained by rounding the real and imaginary parts of $u$ to the nearest integer with halves rounded up, and where the inverse of $0$ is taken to be $0$.","title":"Iterates of the Hurwitz algorithm"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-digit","kind":"node","lean":"MetaMathlibExt.hurwitzDigit","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-digit","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"Let $z$ be a complex number with Hurwitz iterates $z_0 = z, z_1, z_2, \\dots$. For $n \\ge 0$, the $n$-th partial quotient is $a_n = [z_n]$, the Gaussian integer nearest to $z_n$, obtained by rounding the real and imaginary parts of $z_n$ to the nearest integer with halves rounded up.","title":"Partial quotients of the Hurwitz expansion"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-pq","kind":"node","lean":"MetaMathlibExt.hurwitzPQ","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-pq","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"Let $z$ be a complex number with Hurwitz partial quotients $a_0, a_1, a_2, \\dots$. This is the sequence whose term for $n \\ge 0$ is the pair of pairs of complex numbers $((P_{n-1}, Q_{n-1}), (P_n, Q_n))$, where $P_{-1} = 1$, $Q_{-1} = 0$, $P_0 = a_0$, $Q_0 = 1$, and $P_{n+1} = a_{n+1} P_n + P_{n-1}$ and $Q_{n+1} = a_{n+1} Q_n + Q_{n-1}$ for $n \\ge 0$.","title":"Consecutive pairs of Hurwitz convergents"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-p","kind":"node","lean":"MetaMathlibExt.hurwitzP","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-p","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"Let $z$ be a complex number with Hurwitz partial quotients $a_0, a_1, a_2, \\dots$. Its convergent numerators $P_n$, $n \\ge 0$, are given by $P_{-1} = 1$, $P_0 = a_0$ and $P_{n+1} = a_{n+1} P_n + P_{n-1}$ for $n \\ge 0$.","title":"Convergent numerators of the Hurwitz expansion"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-q","kind":"node","lean":"MetaMathlibExt.hurwitzQ","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-q","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"Let $z$ be a complex number with Hurwitz partial quotients $a_0, a_1, a_2, \\dots$. Its convergent denominators $Q_n$, $n \\ge 0$, are given by $Q_{-1} = 0$, $Q_0 = 1$ and $Q_{n+1} = a_{n+1} Q_n + Q_{n-1}$ for $n \\ge 0$.","title":"Convergent denominators of the Hurwitz expansion"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-legal","kind":"node","lean":"MetaMathlibExt.HurwitzLegal","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-legal","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"Let $z$ be a complex number with Hurwitz iterates $z_j$ and partial quotients $a_j$. A natural number $n$ is a legal index for $z$ when $z_j \\ne a_j$ for every $j < n$, that is, when the Hurwitz expansion of $z$ has not terminated before step $n$.","title":"Legal indices of the Hurwitz expansion"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/abs-re-sub-hurwitz-round-le","kind":"node","lean":"MetaMathlibExt.abs_re_sub_hurwitzRound_le","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#abs-re-sub-hurwitz-round-le","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"For every complex number $z$, let $[z]$ be the Gaussian integer nearest to $z$, obtained by rounding the real and imaginary parts to the nearest integer with halves rounded up. Then $|\\operatorname{Re}(z - [z])| \\le 1/2$ and $|\\operatorname{Im}(z - [z])| \\le 1/2$.","title":"Rounding error of the nearest Gaussian integer"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-pq-recursion","kind":"node","lean":"MetaMathlibExt.hurwitzPQ_recursion","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-pq-recursion","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"fully_proved","summary":"Let $z$ be a complex number with Hurwitz partial quotients $a_n$, convergent numerators $P_n$ and convergent denominators $Q_n$. Then $P_0 = a_0$, $Q_0 = 1$, $P_1 = a_1 a_0 + 1$ and $Q_1 = a_1$, and for every $n \\ge 0$, $P_{n+2} = a_{n+2} P_{n+1} + P_n$ and $Q_{n+2} = a_{n+2} Q_{n+1} + Q_n$.","title":"Recursion for the Hurwitz convergents"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-q-mul-sub-hurwitz-p","kind":"node","lean":"MetaMathlibExt.hurwitzQ_mul_sub_hurwitzP","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-q-mul-sub-hurwitz-p","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"can_prove","summary":"Let $z$ be a complex number with Hurwitz iterates $z_0 = z, z_1, z_2, \\dots$, convergent numerators $P_n$ and convergent denominators $Q_n$. For every legal index $n$ for $z$,","title":"Error of the Hurwitz convergents"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/norm-hurwitz-q-lt-succ","kind":"node","lean":"MetaMathlibExt.norm_hurwitzQ_lt_succ","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#norm-hurwitz-q-lt-succ","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"can_prove","summary":"Let $z$ be a complex number and let $Q_0, Q_1, \\dots$ be the convergent denominators of its Hurwitz continued fraction expansion. For every $n \\ge 0$ such that $n + 1$ is a legal index for $z$ (the expansion has not terminated before step $n + 1$), $|Q_n| < |Q_{n+1}|$.","title":"Hurwitz's theorem: the convergent denominators grow in absolute value"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/norm-sub-hurwitz-convergent-le","kind":"node","lean":"MetaMathlibExt.norm_sub_hurwitz_convergent_le","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#norm-sub-hurwitz-convergent-le","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"can_prove","summary":"Let $z$ be a complex number and let $P_k$ and $Q_k$ be the convergent numerators and denominators of its Hurwitz continued fraction expansion. For every legal index $k$ for $z$ (the expansion has not terminated before step $k$), $|z - P_k / Q_k| \\le 1 / |Q_k|^2$.","title":"Lakein's first theorem on Hurwitz convergents"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/hurwitz-four-squares","kind":"node","lean":"MetaMathlibExt.hurwitz_four_squares","parent":9902,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract/index.html#hurwitz-four-squares","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-four-squares-via-hurwitz-continued-fract","status":"can_prove","summary":"Let $w$, $x$, $y$ be natural numbers such that $w$ is odd and not a perfect square, $2x < w$, $2y < w$ and $w$ divides $x^2 + y^2 + 1$. Expand $z = (x + yi)/w$ by the Hurwitz algorithm, with convergent numerators $P_n$ and convergent denominators $Q_n$. Then there is exactly one natural number $n$ such that $n + 1$ is a legal index for $z$, $|Q_n| \\le \\sqrt{w} < |Q_{n+1}|$, and, if $|Q_n| \\ne \\sqrt{w}$,\u2026","title":"Sums of four squares via Hurwitz continued fractions"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","kind":"scope","navigation":"module","parent":9820,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"can_state","title":"Sums of three squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/exists-three-squares-of-not-four-pow-mul","kind":"node","lean":"Nat.exists_three_squares_of_not_four_pow_mul_eight_mul_add_seven","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#exists-three-squares-of-not-four-pow-mul","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"fully_proved","summary":"Every natural number $n$ that is not of the form $4^a (8b + 7)$ with natural numbers $a, b$ is a sum of three squares: there are natural numbers $x, y, z$ with $x^2 + y^2 + z^2 = n$.","title":"Legendre's three-square theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/not-exists-four-pow-mul-eight-mul-add-se","kind":"node","lean":"Nat.not_exists_four_pow_mul_eight_mul_add_seven_of_exists_three_squares","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#not-exists-four-pow-mul-eight-mul-add-se","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"fully_proved","summary":"If a natural number $n$ is a sum of three squares of natural numbers, then $n$ is not of the form $4^a (8b + 7)$ with natural numbers $a, b$.","title":"Converse of Legendre's three-square theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/every-nat-eq-sum-three-sq-div-eight","kind":"node","lean":"MetaMathlibExt.every_nat_eq_sum_three_sq_div_eight","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#every-nat-eq-sum-three-sq-div-eight","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"fully_proved","summary":"For every natural number $m$ there are natural numbers $a$, $b$, $c$ with $m = \\lfloor a^2/8 \\rfloor + \\lfloor b^2/8 \\rfloor + \\lfloor c^2/8 \\rfloor$.","title":"Sums of three terms of the sequence \u230an\u00b2/8\u230b"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/three-square-representation-count","kind":"node","lean":"Nat.threeSquareRepresentationCount","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#three-square-representation-count","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"fully_proved","summary":"For a natural number $n$, $r_3(n)$ is the number of ordered triples $(x, y, z)$ of integers in the interval $[-n, n]$ with $x^2 + y^2 + z^2 = n$. Every integer solution of $x^2 + y^2 + z^2 = n$ lies in this interval (also for $n = 0$), so $r_3(n)$ is the number of all ordered integer solutions.","title":"Number of representations as a sum of three squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/three-square-representation-count-eq-zer","kind":"node","lean":"Nat.threeSquareRepresentationCount_eq_zero_of_mod_eight_eq_seven","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#three-square-representation-count-eq-zer","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"fully_proved","summary":"For a natural number $m$, let $r_3(m)$ denote the number of ordered triples of integers $(x, y, z)$ with $x^2 + y^2 + z^2 = m$. For every positive natural number $n$ with $n \\equiv 7 \\pmod 8$, $r_3(n) = 0$. This is the case $n \\equiv 7 \\pmod 8$ of Gauss's formula for $r_3$.","title":"Numbers congruent to 7 modulo 8 are not sums of three squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/three-square-representation-count-eq-thr","kind":"node","lean":"Nat.threeSquareRepresentationCount_eq_threeSquareRepresentationCount_div_four","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#three-square-representation-count-eq-thr","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"fully_proved","summary":"For a natural number $m$, let $r_3(m)$ denote the number of ordered triples of integers $(x, y, z)$ with $x^2 + y^2 + z^2 = m$. For every positive natural number $n$ divisible by $4$, $r_3(n) = r_3(n/4)$. This is the case $4 \\mid n$ of Gauss's formula for $r_3$.","title":"Removing a factor of 4 preserves the three-square count"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/r3-legendre-recursion","kind":"node","lean":"MetaMathlibExt.r3_legendre_recursion","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#r3-legendre-recursion","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"fully_proved","summary":"For a natural number $m$, let $r_3(m)$ denote the number of ordered triples of integers $(x, y, z)$ with $x^2 + y^2 + z^2 = m$. Let $p \\ge 3$ be a prime and let $n \\ge 1$ and $\\alpha \\ge 0$ be natural numbers. Then","title":"Legendre-symbol recursion for three-square counts"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/hurwitz-class-number","kind":"node","lean":"Nat.hurwitzClassNumber","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#hurwitz-class-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"can_state","summary":"The Hurwitz class number is the function $H$ from the natural numbers to the rationals given by $H(0) = -1/12$; $H(N) = 0$ if $N \\equiv 1$ or $2 \\pmod 4$; and, for $N > 0$ with $N \\equiv 0$ or $3 \\pmod 4$, $H(N)$ is the number of equivalence classes of positive-definite integral binary quadratic forms of discriminant $-N$, primitive or not, where the class of a form equivalent to $a (x^2 + y^2)$ for some positive\u2026","title":"Hurwitz class number"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/three-square-representation-count-eq-twe","kind":"node","lean":"Nat.threeSquareRepresentationCount_eq_twelve_mul_hurwitzClassNumber_four_n","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#three-square-representation-count-eq-twe","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"stated","summary":"For every positive natural number $n$ with $n \\equiv 1$, $2$, $5$ or $6 \\pmod 8$, the number $r_3(n)$ of ordered triples $(x, y, z)$ of integers with $x^2 + y^2 + z^2 = n$ equals $12 H(4n)$, where $H$ is the Hurwitz class number.","title":"Gauss's three-square formula, residues 1, 2, 5, 6 modulo 8"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/three-square-representation-count-eq-twe-2","kind":"node","lean":"Nat.threeSquareRepresentationCount_eq_twenty_four_mul_hurwitzClassNumber","parent":9916,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares/index.html#three-square-representation-count-eq-twe-2","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-three-squares","status":"stated","summary":"For every positive natural number $n$ with $n \\equiv 3 \\pmod 8$, the number $r_3(n)$ of ordered triples $(x, y, z)$ of integers with $x^2 + y^2 + z^2 = n$ equals $24 H(n)$, where $H$ is the Hurwitz class number.","title":"Gauss's three-square formula, residue 3 modulo 8"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree.json","id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree","kind":"scope","navigation":"module","parent":9820,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree","status":"can_prove","title":"Sums of two squares via the Stern\u2013Brocot tree"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree/continued-fraction-path-eq-convergents","kind":"node","lean":"MetaMathlibExt.continuedFraction_path_eq_convergents","parent":9927,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree/index.html#continued-fraction-path-eq-convergents","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree","status":"can_prove","summary":"Let $R = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}$ and $L = \\begin{pmatrix} 1 & 0 \\\\ 1 & 1 \\end{pmatrix}$, as matrices over $\\mathbb{N}$. Let $q_0, q_1, q_2, \\dots$ be natural numbers, and define natural numbers $A_k$, $B_k$ for $k \\ge -1$ by $A_{-1} = 1$, $A_0 = q_0$, $B_{-1} = 0$, $B_0 = 1$ and, for $k \\ge 1$, $A_k = q_k A_{k-1} + A_{k-2}$ and $B_k = q_k B_{k-1} + B_{k-2}$. For $n \\ge 0$ let\u2026","title":"Continued fraction convergents as matrix paths"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree/exists-palindrome-continued-fraction-div","kind":"node","lean":"MetaMathlibExt.exists_palindrome_continuedFraction_div","parent":9927,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree/index.html#exists-palindrome-continued-fraction-div","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, and let $x_0$ be an integer with $0 < x_0 < p/2$ and $x_0^2 \\equiv -1 \\pmod p$. Then there are positive integers $q_0, q_1, \\dots, q_m$ with $m + 1$ even and $q_i = q_{m-i}$ for every $i$ such that $p / x_0 = [q_0; q_1, \\dots, q_m] = q_0 + 1/(q_1 + 1/(\\cdots + 1/q_m))$.","title":"Perron's lemma"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree/sq-add-sq-of-stern-brocot-path","kind":"node","lean":"MetaMathlibExt.sq_add_sq_of_sternBrocot_path","parent":9927,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree/index.html#sq-add-sq-of-stern-brocot-path","scope":"NumberTheory/quadratic-forms-sums-of-squares/sums-of-squares/sums-of-two-squares-via-the-stern-brocot-tree","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, and let $x_0$ be an integer with $0 < x_0 < p/2$ and $x_0^2 \\equiv -1 \\pmod p$. Let $n \\ge 0$ and let $q_0, q_1, \\dots, q_n$ be positive integers with $p / x_0 = [q_0; q_1, \\dots, q_n, q_n, \\dots, q_1, q_0]$, where $[a_0; a_1, \\dots, a_m] = a_0 + 1/(a_1 + 1/(\\cdots + 1/a_m))$. With $R = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}$ and\u2026","title":"Sums of two squares from the Stern\u2013Brocot tree"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms.json","id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms","kind":"scope","parent":9819,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms","status":"can_prove","summary":"Reduction theory, discriminants and class numbers of binary forms, quaternary forms, 15 and 290 theorems.","title":"Binary and higher quadratic forms"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems.json","id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","kind":"scope","navigation":"module","parent":9931,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"can_prove","title":"15 and 290 theorems"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/eval-q","kind":"node","lean":"MathlibExt.NumberTheory.ConwaySchneebergerFifteenWanted.NumberTheory.ConwaySchneebergerFifteenProblem.evalQ","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#eval-q","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"fully_proved","summary":"For an $n \\times n$ matrix $Q$ with integer entries, the quadratic form of $Q$ is the map $\\mathbb{Z}^n \\to \\mathbb{Z}$ sending $x$ to $Q(x) = \\sum_{i,j} Q_{ij} x_i x_j$, where $i$ and $j$ range over $\\{0, \\dots, n-1\\}$. No symmetry of $Q$ is assumed.","title":"Quadratic form of an integer matrix"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/represents","kind":"node","lean":"MathlibExt.NumberTheory.ConwaySchneebergerFifteenWanted.NumberTheory.ConwaySchneebergerFifteenProblem.Represents","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#represents","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"fully_proved","summary":"An $n \\times n$ matrix $Q$ with integer entries represents an integer $m$ if $Q(x) = m$ for some $x \\in \\mathbb{Z}^n$, where $Q(x) = \\sum_{i,j} Q_{ij} x_i x_j$.","title":"Integers represented by an integer-matrix quadratic form"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/is-universal","kind":"node","lean":"MathlibExt.NumberTheory.ConwaySchneebergerFifteenWanted.NumberTheory.ConwaySchneebergerFifteenProblem.IsUniversal","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#is-universal","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"fully_proved","summary":"An $n \\times n$ matrix $Q$ with integer entries is universal if its quadratic form $Q(x) = \\sum_{i,j} Q_{ij} x_i x_j$ represents every positive integer, that is, for every positive integer $m$ there is $x \\in \\mathbb{Z}^n$ with $Q(x) = m$.","title":"Universal integer-matrix quadratic forms"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/conway-schneeberger-fifteen","kind":"node","lean":"MathlibExt.NumberTheory.ConwaySchneebergerFifteenWanted.NumberTheory.ConwaySchneebergerFifteenProblem.conway_schneeberger_fifteen","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#conway-schneeberger-fifteen","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"can_prove","summary":"Let $Q$ be a positive definite symmetric $n \\times n$ matrix with integer entries, with quadratic form $Q(x) = \\sum_{i,j} Q_{ij} x_i x_j$ on $\\mathbb{Z}^n$. Then $Q$ is universal, that is, it represents every positive integer, if and only if it represents every integer $k$ with $1 \\le k \\le 15$.","title":"Conway\u2013Schneeberger fifteen theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/eval-q-2","kind":"node","lean":"MathlibExt.NumberTheory.TwoNinetyTheoremWanted.NumberTheory.TwoNinety.evalQ","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#eval-q-2","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"fully_proved","summary":"For a ring $R$ and an $n \\times n$ matrix $M$ with entries in $R$, the value of the quadratic form of $M$ at a vector $x \\in R^n$ is $x \\cdot (M x) = \\sum_{i,j} x_i M_{ij} x_j$, where $i$ and $j$ range over $\\{0, \\dots, n-1\\}$.","title":"Quadratic form of a matrix"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/represents-2","kind":"node","lean":"MathlibExt.NumberTheory.TwoNinetyTheoremWanted.NumberTheory.TwoNinety.Represents","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#represents-2","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"fully_proved","summary":"For a ring $R$, an $n \\times n$ matrix $M$ with entries in $R$ represents a natural number $m$ if there is an integer vector $x \\in \\mathbb{Z}^n$ with $\\sum_{i,j} x_i M_{ij} x_j = m$, where the entries of $x$ and the number $m$ are viewed as elements of $R$.","title":"Natural numbers represented by a quadratic form"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/is-universal-2","kind":"node","lean":"MathlibExt.NumberTheory.TwoNinetyTheoremWanted.NumberTheory.TwoNinety.IsUniversal","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#is-universal-2","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"fully_proved","summary":"For a ring $R$, an $n \\times n$ matrix $M$ with entries in $R$ is universal if it represents every positive integer: for every positive integer $m$ there is $x \\in \\mathbb{Z}^n$ with $\\sum_{i,j} x_i M_{ij} x_j = m$ in $R$.","title":"Universal quadratic forms over a ring"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/integral","kind":"node","lean":"MathlibExt.NumberTheory.TwoNinetyTheoremWanted.NumberTheory.TwoNinety.Integral","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#integral","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"fully_proved","summary":"For a ring $R$, an $n \\times n$ matrix $M$ with entries in $R$ is integral if its quadratic form takes integer values at integer vectors: for every $x \\in \\mathbb{Z}^n$, the value $\\sum_{i,j} x_i M_{ij} x_j$ computed in $R$ lies in the image of $\\mathbb{Z}$ in $R$.","title":"Integer-valued quadratic forms"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/critical-numbers","kind":"node","lean":"MathlibExt.NumberTheory.TwoNinetyTheoremWanted.NumberTheory.TwoNinety.criticalNumbers","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#critical-numbers","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"fully_proved","summary":"The critical numbers of the 290 theorem form the set of the 29 natural numbers $1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290$.","title":"Critical numbers of the 290 theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/two-ninety-theorem","kind":"node","lean":"MathlibExt.NumberTheory.TwoNinetyTheoremWanted.NumberTheory.TwoNinety.two_ninety_theorem","parent":9932,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems/index.html#two-ninety-theorem","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/15-and-290-theorems","status":"can_prove","summary":"Let $n$ be a natural number and let $M$ be a real $n \\times n$ matrix that is positive definite (symmetric, with $x^{\\top} M x > 0$ for every nonzero $x \\in \\mathbb{R}^n$) and integral (for every $x \\in \\mathbb{Z}^n$, the value $x^{\\top} M x$ is an integer). If $M$ represents each of the 29 critical numbers of the 290 theorem, that is, for each critical number $m$ there is $x \\in \\mathbb{Z}^n$ with\u2026","title":"The 290 theorem"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo.json","id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo","kind":"scope","navigation":"module","parent":9931,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo","status":"can_prove","title":"Arithmetic progressions represented by binary forms"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo/exists-int-representations-of-rat-arithm","kind":"node","lean":"MathlibExt.NumberTheory.QuadraticForms.BinaryQuadraticArithmeticProgressionsWanted.exists_int_representations_of_rat_arithmetic_progression","parent":9943,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo/index.html#exists-int-representations-of-rat-arithm","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo","status":"can_prove","summary":"Let $a, b, c$ be integers, $n$ a natural number and $u_0, d$ rational numbers. If for every natural number $i < n$ there are rational numbers $x, y$ with $ax^2 + bxy + cy^2 = u_0 + id$, then there is a nonzero integer $k$ such that for every natural number $i < n$ there are integers $x, y$ with $ax^2 + bxy + cy^2 = k^2(u_0 + id)$.","title":"Rational representations of an arithmetic progression scale to integral ones"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo/exists-representation-nine-term-diagonal","kind":"node","lean":"MathlibExt.NumberTheory.QuadraticForms.BinaryQuadraticArithmeticProgressionsWanted.exists_representation_nine_term_diagonal_quadratic","parent":9943,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nine-term arithmetic progressions represented by "],["math","\\(px^2 + qy^2\\)"]],"source":"Nine-term arithmetic progressions represented by $px^2 + qy^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo/index.html#exists-representation-nine-term-diagonal","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo","status":"can_prove","summary":"Let $p, q, g, h$ be integers such that $-4pq$ is not a perfect square, and put","title":"Nine-term arithmetic progressions represented by $px^2 + qy^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo/exists-nine-term-arithmetic-progression","kind":"node","lean":"MathlibExt.NumberTheory.QuadraticForms.BinaryQuadraticArithmeticProgressionsWanted.exists_nine_term_arithmetic_progression_binary_quadratic","parent":9943,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo/index.html#exists-nine-term-arithmetic-progression","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/arithmetic-progressions-represented-by-binary-fo","status":"can_prove","summary":"Let $a, b, c$ be integers such that the discriminant $b^2 - 4ac$ is not a perfect square. Then there are integers $u_0$ and $d \\ne 0$ such that for every natural number $i < 9$ there are integers $x, y$ with $u_0 + id = ax^2 + bxy + cy^2$.","title":"Binary quadratic forms with non-square discriminant represent nine-term progressions"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms.json","id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","kind":"scope","navigation":"module","parent":9931,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","title":"Binary quadratic forms"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/of-coefficients","kind":"node","lean":"BinaryQuadraticForm.ofCoefficients","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#of-coefficients","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"For integers $a, b, c$, this is the quadratic form on the real vector space $\\mathbb{R}^2$ given by $(x, y) \\mapsto a x^2 + b x y + c y^2$.","title":"Real quadratic form of an integer coefficient triple"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/of-coefficients-apply","kind":"node","lean":"BinaryQuadraticForm.ofCoefficients_apply","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#of-coefficients-apply","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"For integers $a, b, c$ and real numbers $x, y$, the real quadratic form of the triple $(a, b, c)$ takes the value $a x^2 + b x y + c y^2$ at the point $(x, y)$.","title":"Evaluation of the real quadratic form of a coefficient triple"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/of-coefficients-pos-def","kind":"node","lean":"BinaryQuadraticForm.ofCoefficients_posDef","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#of-coefficients-pos-def","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"Let $D$ be a positive integer and let $a, b, c$ be integers with $a > 0$ and $b^2 - 4ac = -D$. Then the real quadratic form $(x, y) \\mapsto a x^2 + b x y + c y^2$ on $\\mathbb{R}^2$ is positive definite.","title":"Positive definiteness for negative discriminant"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/imaginary-root","kind":"node","lean":"BinaryQuadraticForm.imaginaryRoot","parent":9947,"reading_title":{"format":"inline-parts-v1","parts":[["text","The imaginary square root of "],["math","\\(-D\\)"]],"source":"The imaginary square root of $-D$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#imaginary-root","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"For a natural number $D$, the imaginary root of $D$ is the complex number $i\\sqrt{D}$, where $\\sqrt{D}$ is the nonnegative real square root; it is the square root of $-D$ with nonnegative imaginary part.","title":"The imaginary square root of $-D$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/imaginary-root-sq","kind":"node","lean":"BinaryQuadraticForm.imaginaryRoot_sq","parent":9947,"reading_title":{"format":"inline-parts-v1","parts":[["text","The imaginary root squares to "],["math","\\(-D\\)"]],"source":"The imaginary root squares to $-D$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#imaginary-root-sq","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"For every natural number $D$, the complex number $i\\sqrt{D}$ satisfies $(i\\sqrt{D})^2 = -D$.","title":"The imaginary root squares to $-D$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/imaginary-root-is-integral","kind":"node","lean":"BinaryQuadraticForm.imaginaryRoot_isIntegral","parent":9947,"reading_title":{"format":"inline-parts-v1","parts":[["text","The imaginary root is integral over "],["math","\\(\\mathbb{Q}\\)"]],"source":"The imaginary root is integral over $\\mathbb{Q}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#imaginary-root-is-integral","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"For every natural number $D$, the complex number $i\\sqrt{D}$ is integral over $\\mathbb{Q}$.","title":"The imaginary root is integral over $\\mathbb{Q}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/imaginary-root-not-mem-bot","kind":"node","lean":"BinaryQuadraticForm.imaginaryRoot_not_mem_bot","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#imaginary-root-not-mem-bot","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"For every positive integer $D$, the complex number $i\\sqrt{D}$ does not lie in the subfield $\\mathbb{Q}$ of $\\mathbb{C}$.","title":"The imaginary root is not rational"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/imaginary-quadratic-field","kind":"node","lean":"BinaryQuadraticForm.imaginaryQuadraticField","parent":9947,"reading_title":{"format":"inline-parts-v1","parts":[["text","The imaginary quadratic field "],["math","\\(\\mathbb{Q}(\\sqrt{-D})\\)"]],"source":"The imaginary quadratic field $\\mathbb{Q}(\\sqrt{-D})$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#imaginary-quadratic-field","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"For a natural number $D$, this is the subfield $\\mathbb{Q}(i\\sqrt{D})$ of $\\mathbb{C}$ generated over $\\mathbb{Q}$ by the imaginary root $i\\sqrt{D}$.","title":"The imaginary quadratic field $\\mathbb{Q}(\\sqrt{-D})$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/imaginary-quadratic-field-finrank","kind":"node","lean":"BinaryQuadraticForm.imaginaryQuadraticField_finrank","parent":9947,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathbb{Q}(\\sqrt{-D})\\)"],["text"," is a quadratic extension"]],"source":"$\\mathbb{Q}(\\sqrt{-D})$ is a quadratic extension"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#imaginary-quadratic-field-finrank","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"For every positive integer $D$, the subfield $\\mathbb{Q}(i\\sqrt{D})$ of $\\mathbb{C}$ has degree $2$ over $\\mathbb{Q}$.","title":"$\\mathbb{Q}(\\sqrt{-D})$ is a quadratic extension"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/form-generator-is-integral","kind":"node","lean":"BinaryQuadraticForm.formGenerator_isIntegral","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#form-generator-is-integral","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"Let $D$ be a natural number and let $a, b, c$ be integers with $b^2 - 4ac = -D$. Then the element $(b + i\\sqrt{D})/2$ of the field $\\mathbb{Q}(i\\sqrt{D}) \\subseteq \\mathbb{C}$ is integral over $\\mathbb{Z}$.","title":"The form generator is an algebraic integer"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/form-ideal","kind":"node","lean":"BinaryQuadraticForm.formIdeal","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#form-ideal","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"Let $D$ be a natural number, let $K = \\mathbb{Q}(i\\sqrt{D}) \\subseteq \\mathbb{C}$ and let $\\mathcal{O}_K$ be its ring of integers. For integers $a, b, c$, the ideal of the triple $(a, b, c)$ is the fractional ideal of $\\mathcal{O}_K$ in $K$ generated by $a$ and $(b + i\\sqrt{D})/2$.","title":"Ideal of a binary quadratic form"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/form-ideal-ne-zero","kind":"node","lean":"BinaryQuadraticForm.formIdeal_ne_zero","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#form-ideal-ne-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"Let $D$ be a natural number and let $a, b, c$ be integers with $a \\ne 0$. Then the fractional ideal of the ring of integers of $\\mathbb{Q}(i\\sqrt{D}) \\subseteq \\mathbb{C}$ generated by $a$ and $(b + i\\sqrt{D})/2$ is nonzero.","title":"The ideal of a form is nonzero"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/class-of-form","kind":"node","lean":"BinaryQuadraticForm.classOfForm","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#class-of-form","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"Let $D$ be a natural number, let $K = \\mathbb{Q}(i\\sqrt{D}) \\subseteq \\mathbb{C}$ and let $\\mathcal{O}_K$ be its ring of integers. For integers $a, b, c$ with $a \\ne 0$, the ideal class of the triple $(a, b, c)$ is the class, in the class group of $\\mathcal{O}_K$, of the nonzero fractional ideal of $\\mathcal{O}_K$ in $K$ generated by $a$ and $(b + i\\sqrt{D})/2$.","title":"Ideal class of a binary quadratic form"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/is-gauss-reduced","kind":"node","lean":"BinaryQuadraticForm.IsGaussReduced","parent":9947,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms/index.html#is-gauss-reduced","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/binary-quadratic-forms","status":"fully_proved","summary":"The integral binary quadratic form $a x^2 + b x y + c y^2$ is Gauss reduced when $a > 0$, its discriminant $b^2 - 4ac$ is negative, $|b| \\le a \\le c$, and $b \\ge 0$ whenever $|b| = a$ or $a = c$.","title":"Gauss-reduced binary quadratic forms"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums.json","id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","kind":"scope","navigation":"module","parent":9931,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"can_prove","title":"Kronecker's class number sums"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/reduced-forms-of-discriminant","kind":"node","lean":"MetaMathlibExt.reducedFormsOfDiscriminant","parent":9962,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#reduced-forms-of-discriminant","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"For an integer $d$, $R(d)$ is the finite set of integer triples $(a, b, c)$ such that the binary quadratic form $a x^2 + b x y + c y^2$ is Gauss reduced (that is, $a > 0$, $b^2 - 4ac < 0$, $|b| \\le a \\le c$, and $b \\ge 0$ whenever $|b| = a$ or $a = c$) and has discriminant $b^2 - 4ac = d$. In particular $R(d)$ is empty when $d \\ge 0$.","title":"Reduced forms of a given discriminant"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/gauss-class-number","kind":"node","lean":"MetaMathlibExt.gaussClassNumber","parent":9962,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#gauss-class-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"For an integer $d$, the Gauss class number $h(d)$ is the number of integer triples $(a, b, c)$ with $b^2 - 4ac = d$ for which the binary quadratic form $ax^2 + bxy + cy^2$ is positive definite ($a > 0$ and $d < 0$), reduced ($|b| \\le a \\le c$, with $b \\ge 0$ whenever $|b| = a$ or $a = c$) and primitive ($\\gcd(a, b, c) = 1$).","title":"Gauss class number of a discriminant"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/unweighted-hurwitz-class-number","kind":"node","lean":"MetaMathlibExt.unweightedHurwitzClassNumber","parent":9962,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#unweighted-hurwitz-class-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"For an integer $d$, the unweighted Hurwitz class number $H(d)$ is the number of elements of the set $R(d)$ of reduced forms of discriminant $d$, that is, the number of integer triples $(a, b, c)$, primitive or not, for which $a x^2 + b x y + c y^2$ is a Gauss-reduced binary quadratic form of discriminant $b^2 - 4ac = d$. Every reduced form is counted with weight $1$; in particular $H(d) = 0$ when $d \\ge 0$.","title":"Unweighted Hurwitz class number"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/reduced-form-weight","kind":"node","lean":"MetaMathlibExt.reducedFormWeight","parent":9962,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#reduced-form-weight","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"The weight of an integer triple $(a, b, c)$, standing for the form $ax^2 + bxy + cy^2$, is $1/2$ if $a = c$ and $b = 0$, $1/3$ if $a = b = c \\ne 0$, and $1$ otherwise.","title":"Weight of a reduced binary quadratic form"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/kronecker-fodd","kind":"node","lean":"MetaMathlibExt.kroneckerFOdd","parent":9962,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kronecker's sum "],["math","\\(F_o(n)\\)"]],"source":"Kronecker's sum $F_o(n)$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#kronecker-fodd","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"For a natural number $n$, $F_o(n)$ is the sum of the weights of the reduced positive-definite integral binary quadratic forms $ax^2 + bxy + cy^2$ of discriminant $b^2 - 4ac = -4n$ in which $a$ or $c$ is odd. Here a form is reduced when $|b| \\le a \\le c$, with $b \\ge 0$ whenever $|b| = a$ or $a = c$, and its weight is $1/2$ if $a = c$ and $b = 0$, $1/3$ if $a = b = c$, and $1$ otherwise.","title":"Kronecker's sum $F_o(n)$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/kronecker-feven","kind":"node","lean":"MetaMathlibExt.kroneckerFEven","parent":9962,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kronecker's sum "],["math","\\(F_e(n)\\)"]],"source":"Kronecker's sum $F_e(n)$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#kronecker-feven","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"For a natural number $n$, $F_e(n)$ is the sum of the weights of the reduced positive-definite integral binary quadratic forms $ax^2 + bxy + cy^2$ of discriminant $b^2 - 4ac = -4n$ in which $a$ and $c$ are both even. Here a form is reduced when $|b| \\le a \\le c$, with $b \\ge 0$ whenever $|b| = a$ or $a = c$, and its weight is $1/2$ if $a = c$ and $b = 0$, $1/3$ if $a = b = c$, and $1$ otherwise.","title":"Kronecker's sum $F_e(n)$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/kronecker-discriminant","kind":"node","lean":"MetaMathlibExt.kroneckerDiscriminant","parent":9962,"reading_title":{"format":"inline-parts-v1","parts":[["text","The discriminant "],["math","\\(d(\\alpha, g)\\)"],["text"," attached to "],["math","\\(n = 2^\\alpha g h^2\\)"]],"source":"The discriminant $d(\\alpha, g)$ attached to $n = 2^\\alpha g h^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#kronecker-discriminant","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"For natural numbers $\\alpha$ and $g$, let $d(\\alpha, g) = -8g$ if $\\alpha$ is odd; if $\\alpha$ is even, let $d(\\alpha, g) = -4g$ when $g \\equiv 1 \\pmod 4$ and $d(\\alpha, g) = -g$ otherwise.","title":"The discriminant $d(\\alpha, g)$ attached to $n = 2^\\alpha g h^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/kronecker-class-multiplier","kind":"node","lean":"MetaMathlibExt.kroneckerClassMultiplier","parent":9962,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#kronecker-class-multiplier","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"For an integer $d$ and a natural number $h \\ge 1$, $\\ell(d, h)$ is the product, over the primes $p$ dividing $h$, of","title":"Kronecker's class-number multiplier"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/gauss-class-number-small","kind":"node","lean":"MetaMathlibExt.gaussClassNumber_small","parent":9962,"reading_title":{"format":"inline-parts-v1","parts":[["text","Class numbers of the discriminants "],["math","\\(-3\\)"],["text",", "],["math","\\(-4\\)"],["text"," and "],["math","\\(-12\\)"]],"source":"Class numbers of the discriminants $-3$, $-4$ and $-12$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#gauss-class-number-small","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"The Gauss class numbers satisfy $h(-3) = h(-4) = h(-12) = 1$, and the unweighted Hurwitz class number satisfies $H(-12) = 2$.","title":"Class numbers of the discriminants $-3$, $-4$ and $-12$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/kronecker-f-small","kind":"node","lean":"MetaMathlibExt.kroneckerF_small","parent":9962,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kronecker's sums at "],["math","\\(n = 1\\)"],["text"," and "],["math","\\(n = 3\\)"]],"source":"Kronecker's sums at $n = 1$ and $n = 3$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#kronecker-f-small","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"Kronecker's sums $F_o$ and $F_e$ satisfy $F_o(1) = 1/2$, $F_e(1) = 0$, $F_o(3) = 1$ and $F_e(3) = 1/3$.","title":"Kronecker's sums at $n = 1$ and $n = 3$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/kronecker-discriminant-kronecker-class-m","kind":"node","lean":"MetaMathlibExt.kroneckerDiscriminant_kroneckerClassMultiplier_small","parent":9962,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small values of the discriminant "],["math","\\(d(\\alpha, g)\\)"],["text"," and the multiplier "],["math","\\(\\ell(d, h)\\)"]],"source":"Small values of the discriminant $d(\\alpha, g)$ and the multiplier $\\ell(d, h)$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#kronecker-discriminant-kronecker-class-m","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"fully_proved","summary":"The discriminant attached to $n = 2^\\alpha g h^2$ satisfies $d(0, 1) = -4$, $d(0, 3) = -3$ and $d(1, 1) = -8$, and Kronecker's class-number multiplier satisfies $\\ell(-4, 1) = 1$ and $\\ell(-4, 3) = 5$.","title":"Small values of the discriminant $d(\\alpha, g)$ and the multiplier $\\ell(d, h)$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/three-square-representation-count-eq-kro","kind":"node","lean":"MetaMathlibExt.threeSquareRepresentationCount_eq_kronecker_classNumber","parent":9962,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#three-square-representation-count-eq-kro","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"can_prove","summary":"Let $n$, $\\alpha$, $g$, $h$ be natural numbers with $n = 2^\\alpha g h^2$, where $g$ and $h$ are odd and $g$ is squarefree. Let $d$ be the Kronecker discriminant of $(\\alpha, g)$, namely $d = -4g$ if $\\alpha$ is even and $g \\equiv 1 \\pmod 4$, $d = -g$ if $\\alpha$ is even and $g \\equiv 3 \\pmod 4$, and $d = -8g$ if $\\alpha$ is odd. Put $L = \\ell(d, h) \\, h(d)$, where $\\ell(d, h)$ is the Kronecker class multiplier and\u2026","title":"Gauss's three-square formula after Aygin and Williams"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/kronecker-fodd-kronecker-feven-eq-class","kind":"node","lean":"MetaMathlibExt.kroneckerFOdd_kroneckerFEven_eq_classNumber","parent":9962,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#kronecker-fodd-kronecker-feven-eq-class","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"can_prove","summary":"Let $n = 2^\\alpha g h^2$ with $g$ and $h$ odd natural numbers and $g$ squarefree. Let $d = -4g$ if $\\alpha$ is even and $g \\equiv 1 \\pmod 4$, $d = -g$ if $\\alpha$ is even and $g \\equiv 3 \\pmod 4$, and $d = -8g$ if $\\alpha$ is odd. Let $L = \\ell(d, h) \\cdot h(d)$, where $\\ell(d, h)$ is Kronecker's class-number multiplier and $h(d)$ is the Gauss class number of $d$, and for even $\\alpha$ let $t = 2^{\\alpha/2}$. Then\u2026","title":"Aygin\u2013Williams evaluation of Kronecker's sums"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/unweighted-hurwitz-class-number-neg-four","kind":"node","lean":"MetaMathlibExt.unweightedHurwitzClassNumber_neg_four_mul_eq_classNumber","parent":9962,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums/index.html#unweighted-hurwitz-class-number-neg-four","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/kronecker-s-class-number-sums","status":"can_prove","summary":"Let $n$, $\\alpha$, $g$, $h$ be natural numbers with $n = 2^\\alpha g h^2$, where $g$ and $h$ are odd and $g$ is squarefree. Let $d$ be the Kronecker discriminant of $(\\alpha, g)$, namely $d = -4g$ if $\\alpha$ is even and $g \\equiv 1 \\pmod 4$, $d = -g$ if $\\alpha$ is even and $g \\equiv 3 \\pmod 4$, and $d = -8g$ if $\\alpha$ is odd. Put $L = \\ell(d, h) \\, h(d)$, where $\\ell(d, h)$ is the Kronecker class multiplier and\u2026","title":"Unweighted Hurwitz class numbers via Gauss class numbers"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f.json","id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","kind":"scope","navigation":"module","parent":9931,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","title":"Representation numbers of quaternary quadratic forms"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/quaternary-rep-one-one-ten-ten-and-two-t","kind":"node","lean":"MetaMathlibExt.quaternaryRep_one_one_ten_ten_and_two_two_five_five","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","Representation numbers of the quaternary forms "],["math","\\((1,1,10,10)\\)"],["text"," and "],["math","\\((2,2,5,5)\\)"]],"source":"Representation numbers of the quaternary forms $(1,1,10,10)$ and $(2,2,5,5)$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#quaternary-rep-one-one-ten-ten-and-two-t","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"For a complex number $q$ write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. For integers $a, b, c, d$ and $n$, let $N(a,b,c,d;n)$ be the number of $(x_1, x_2, x_3, x_4) \\in \\mathbb{Z}^4$ with $a x_1^2 + b x_2^2 + c x_3^2 + d x_4^2 = n$, and let $\\sigma(n/k)$ be the sum of the positive divisors of $n/k$ if $k \\mid n$ and $0$ otherwise. Let $b_1, b_2, b_3$ be sequences of complex numbers indexed by $n \\ge 0$ such that for\u2026","title":"Representation numbers of the quaternary forms $(1,1,10,10)$ and $(2,2,5,5)$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/quaternary-rep-one-one-fifteen-fifteen-t","kind":"node","lean":"MetaMathlibExt.quaternaryRep_one_one_fifteen_fifteen_three_three_five_five_one_three_five_fifteen","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","Representation numbers of the quaternary forms "],["math","\\((1,1,15,15)\\)"],["text",", "],["math","\\((3,3,5,5)\\)"],["text"," and "],["math","\\((1,3,5,15)\\)"]],"source":"Representation numbers of the quaternary forms $(1,1,15,15)$, $(3,3,5,5)$ and $(1,3,5,15)$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#quaternary-rep-one-one-fifteen-fifteen-t","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Representation numbers of the quaternary forms $(1,1,15,15)$, $(3,3,5,5)$ and $(1,3,5,15)$. For a complex number $q$ write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. For integers $a, b, c, d$ and $n$, let $N(a,b,c,d;n)$ be the number of $(x_1, x_2, x_3, x_4) \\in \\mathbb{Z}^4$ with $a x_1^2 + b x_2^2 + c x_3^2 + d x_4^2 = n$, and let $\\sigma(n/k)$ be the sum of the positive divisors of $n/k$ if $k \\mid n$ and $0$\u2026","title":"Representation numbers of the quaternary forms $(1,1,15,15)$, $(3,3,5,5)$ and $(1,3,5,15)$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/phi-sq-phi-sq-ten-add-five-phi-sq-two-ph","kind":"node","lean":"MetaMathlibExt.phi_sq_phi_sq_ten_add_five_phi_sq_two_phi_sq_five","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","A theta identity for "],["math","\\(\\varphi(q)^2\\varphi(q^{10})^2 + 5\\varphi(q^2)^2\\varphi(q^5)^2\\)"]],"source":"A theta identity for $\\varphi(q)^2\\varphi(q^{10})^2 + 5\\varphi(q^2)^2\\varphi(q^5)^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#phi-sq-phi-sq-ten-add-five-phi-sq-two-ph","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $\\varphi(x) = \\sum_{n \\in \\mathbb{Z}} x^{n^2}$ and $\\psi(x) = \\sum_{n \\ge 0} x^{n(n+1)/2}$ for Ramanujan's theta functions, and $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then","title":"A theta identity for $\\varphi(q)^2\\varphi(q^{10})^2 + 5\\varphi(q^2)^2\\varphi(q^5)^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/phi-sq-mul-phi-sq-ten-eisenstein","kind":"node","lean":"MetaMathlibExt.phi_sq_mul_phi_sq_ten_eisenstein","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eisenstein-series formula for "],["math","\\(\\varphi(q)^2\\varphi(q^{10})^2\\)"]],"source":"Eisenstein-series formula for $\\varphi(q)^2\\varphi(q^{10})^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#phi-sq-mul-phi-sq-ten-eisenstein","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $\\varphi(x) = \\sum_{n \\in \\mathbb{Z}} x^{n^2}$ for Ramanujan's theta function, $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and $P_k = P(q^k)$, where $P(x) = 1 - 24 \\sum_{m \\ge 1} m x^m / (1 - x^m)$ is the weight-two Eisenstein series. Then","title":"Eisenstein-series formula for $\\varphi(q)^2\\varphi(q^{10})^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/phi-sq-two-mul-phi-sq-five-eisenstein","kind":"node","lean":"MetaMathlibExt.phi_sq_two_mul_phi_sq_five_eisenstein","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eisenstein-series formula for "],["math","\\(\\varphi(q^2)^2\\varphi(q^5)^2\\)"]],"source":"Eisenstein-series formula for $\\varphi(q^2)^2\\varphi(q^5)^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#phi-sq-two-mul-phi-sq-five-eisenstein","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $\\varphi(x) = \\sum_{n \\in \\mathbb{Z}} x^{n^2}$ for Ramanujan's theta function, $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and $P_k = P(q^k)$, where $P(x) = 1 - 24 \\sum_{m \\ge 1} m x^m / (1 - x^m)$ is the weight-two Eisenstein series. Then","title":"Eisenstein-series formula for $\\varphi(q^2)^2\\varphi(q^5)^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/twelve-phi-sq-phi-sq-ten-add-twelve-phi","kind":"node","lean":"MetaMathlibExt.twelve_phi_sq_phi_sq_ten_add_twelve_phi_sq_two_phi_sq_five","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","A theta identity for "],["math","\\(12\\varphi(q)^2\\varphi(q^{10})^2 + 12\\varphi(q^2)^2\\varphi(q^5)^2\\)"]],"source":"A theta identity for $12\\varphi(q)^2\\varphi(q^{10})^2 + 12\\varphi(q^2)^2\\varphi(q^5)^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#twelve-phi-sq-phi-sq-ten-add-twelve-phi","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $\\varphi(x) = \\sum_{n \\in \\mathbb{Z}} x^{n^2}$ for Ramanujan's theta function and $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then","title":"A theta identity for $12\\varphi(q)^2\\varphi(q^{10})^2 + 12\\varphi(q^2)^2\\varphi(q^5)^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/phi-sq-mul-phi-sq-fifteen-eisenstein","kind":"node","lean":"MetaMathlibExt.phi_sq_mul_phi_sq_fifteen_eisenstein","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eisenstein-series formula for "],["math","\\(\\varphi(q)^2\\varphi(q^{15})^2\\)"]],"source":"Eisenstein-series formula for $\\varphi(q)^2\\varphi(q^{15})^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#phi-sq-mul-phi-sq-fifteen-eisenstein","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $\\varphi(x) = \\sum_{n \\in \\mathbb{Z}} x^{n^2}$ for Ramanujan's theta function, $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and $P_k = P(q^k)$, where $P(x) = 1 - 24 \\sum_{m \\ge 1} m x^m / (1 - x^m)$ is the weight-two Eisenstein series. Then","title":"Eisenstein-series formula for $\\varphi(q)^2\\varphi(q^{15})^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/q-mul-ramanujan-f-one-three-five-fifteen","kind":"node","lean":"MetaMathlibExt.q_mul_ramanujanF_one_three_five_fifteen","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","A product identity for Ramanujan's "],["math","\\(f(q)f(q^3)f(q^5)f(q^{15})\\)"]],"source":"A product identity for Ramanujan's $f(q)f(q^3)f(q^5)f(q^{15})$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#q-mul-ramanujan-f-one-three-five-fifteen","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and let $f(x) = \\prod_{m \\ge 1} (1 - (-x)^m) = (-x; -x)_\\infty$ be Ramanujan's function. Then","title":"A product identity for Ramanujan's $f(q)f(q^3)f(q^5)f(q^{15})$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/phi-sq-three-mul-phi-sq-five-eisenstein","kind":"node","lean":"MetaMathlibExt.phi_sq_three_mul_phi_sq_five_eisenstein","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eisenstein-series formula for "],["math","\\(\\varphi(q^3)^2\\varphi(q^5)^2\\)"]],"source":"Eisenstein-series formula for $\\varphi(q^3)^2\\varphi(q^5)^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#phi-sq-three-mul-phi-sq-five-eisenstein","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $\\varphi(x) = \\sum_{n \\in \\mathbb{Z}} x^{n^2}$ for Ramanujan's theta function, $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and $P_k = P(q^k)$, where $P(x) = 1 - 24 \\sum_{m \\ge 1} m x^m / (1 - x^m)$ is the weight-two Eisenstein series. Then","title":"Eisenstein-series formula for $\\varphi(q^3)^2\\varphi(q^5)^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/phi-mul-phi-three-mul-phi-five-mul-phi-f","kind":"node","lean":"MetaMathlibExt.phi_mul_phi_three_mul_phi_five_mul_phi_fifteen_eisenstein","parent":9977,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eisenstein-series formula for "],["math","\\(\\varphi(q)\\varphi(q^3)\\varphi(q^5)\\varphi(q^{15})\\)"]],"source":"Eisenstein-series formula for $\\varphi(q)\\varphi(q^3)\\varphi(q^5)\\varphi(q^{15})$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f/index.html#phi-mul-phi-three-mul-phi-five-mul-phi-f","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representation-numbers-of-quaternary-quadratic-f","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $\\varphi(x) = \\sum_{n \\in \\mathbb{Z}} x^{n^2}$ for Ramanujan's theta function, $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and $P_k = P(q^k)$, where $P(x) = 1 - 24 \\sum_{m \\ge 1} m x^m / (1 - x^m)$ is the weight-two Eisenstein series. Then","title":"Eisenstein-series formula for $\\varphi(q)\\varphi(q^3)\\varphi(q^5)\\varphi(q^{15})$"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z.json","id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z","kind":"scope","navigation":"module","parent":9931,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z","status":"can_prove","title":"Representations by x\u00b2 + y\u00b2 \u2212 z\u00b2"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/erdos1148-prop","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1148Wanted.Erdos1148Prop","parent":9988,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded representation by "],["math","\\(x^2 + y^2 - z^2\\)"]],"source":"Bounded representation by $x^2 + y^2 - z^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/index.html#erdos1148-prop","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z","status":"fully_proved","summary":"A natural number $n$ has a bounded representation by $x^2 + y^2 - z^2$ if there are natural numbers $x, y, z$ with $n = x^2 + y^2 - z^2$ and $x^2 \\le n$, $y^2 \\le n$, $z^2 \\le n$, where the subtraction is truncated subtraction of natural numbers.","title":"Bounded representation by $x^2 + y^2 - z^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/weaker-prop","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1148Wanted.weaker_prop","parent":9988,"reading_title":{"format":"inline-parts-v1","parts":[["text","Representation by "],["math","\\(x^2 + y^2 - z^2\\)"],["text"," with the bound "],["math","\\(n + 2\\sqrt{n}\\)"]],"source":"Representation by $x^2 + y^2 - z^2$ with the bound $n + 2\\sqrt{n}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/index.html#weaker-prop","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z","status":"fully_proved","summary":"A natural number $n$ has this representation if there are natural numbers $x, y, z$ with $n = x^2 + y^2 - z^2$, where the subtraction is truncated subtraction of natural numbers, and each of $x^2$, $y^2$, $z^2$ is at most $n + 2\\sqrt{n}$ as a real number.","title":"Representation by $x^2 + y^2 - z^2$ with the bound $n + 2\\sqrt{n}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/eventual-representation","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1148Wanted.eventualRepresentation","parent":9988,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual bounded representation by "],["math","\\(x^2 + y^2 - z^2\\)"]],"source":"Eventual bounded representation by $x^2 + y^2 - z^2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/index.html#eventual-representation","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z","status":"fully_proved","summary":"The statement that every sufficiently large natural number $n$ can be written as $n = x^2 + y^2 - z^2$ with natural numbers $x, y, z$ satisfying $\\max(x^2, y^2, z^2) \\le n$.","title":"Eventual bounded representation by $x^2 + y^2 - z^2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/weaker-bound","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1148Wanted.weakerBound","parent":9988,"reading_title":{"format":"inline-parts-v1","parts":[["text","Representability of every "],["math","\\(n\\)"],["text"," with the bound "],["math","\\(n + 2\\sqrt{n}\\)"]],"source":"Representability of every $n$ with the bound $n + 2\\sqrt{n}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/index.html#weaker-bound","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z","status":"fully_proved","summary":"The statement that every natural number $n$ can be written as $n = x^2 + y^2 - z^2$ with natural numbers $x, y, z$, the subtraction being truncated subtraction of natural numbers, such that each of $x^2$, $y^2$, $z^2$ is at most $n + 2\\sqrt{n}$.","title":"Representability of every $n$ with the bound $n + 2\\sqrt{n}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/eventual-representation-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1148Wanted.eventualRepresentation_holds","parent":9988,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every large integer is a bounded value of "],["math","\\(x^2 + y^2 - z^2\\)"],["text"," (Erd\u0151s Problem 1148)"]],"source":"Every large integer is a bounded value of $x^2 + y^2 - z^2$ (Erd\u0151s Problem 1148)"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z/index.html#eventual-representation-holds","scope":"NumberTheory/quadratic-forms-sums-of-squares/binary-quaternary-forms/representations-by-x-y-z","status":"can_prove","summary":"For every sufficiently large natural number $n$ there are natural numbers $x, y, z$ with $n = x^2 + y^2 - z^2$ and $\\max(x^2, y^2, z^2) \\le n$.","title":"Every large integer is a bounded value of $x^2 + y^2 - z^2$ (Erd\u0151s Problem 1148)"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate.json","id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate","kind":"scope","parent":9819,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate","status":"can_prove","summary":"Hilbert\u2013Waring, g(3)=9, Fermat polygonal theorem, polygonal, triangular and polytopic numbers and their sums.","title":"Waring's problem and figurate numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/hilbert-waring","kind":"node","lean":"MathlibExt.NumberTheory.HilbertWaringWanted.hilbert_waring","parent":9994,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/index.html#hilbert-waring","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate","status":"fully_proved","summary":"For every integer $k \\ge 1$ there is a natural number $g$ such that every natural number $n$ is a sum of $g$ $k$-th powers of natural numbers, that is, $n = x_1^k + \\dots + x_g^k$ for some $x_1, \\dots, x_g \\in \\mathbb{N}$.","title":"Hilbert\u2013Waring theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/four-triangular-numbers-eq-divisor-sum","kind":"node","lean":"MetaMathlibExt.four_triangular_numbers_eq_divisor_sum","parent":9994,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/index.html#four-triangular-numbers-eq-divisor-sum","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate","status":"fully_proved","summary":"For $k \\ge 0$ let $T_k = k(k+1)/2$ be the $k$-th triangular number. For every natural number $j$, the number of ordered quadruples $(k_1, k_2, k_3, k_4)$ of integers with $0 \\le k_i \\le j$ and $T_{k_1} + T_{k_2} + T_{k_3} + T_{k_4} = j$ equals $\\sigma(2j+1)$, the sum of the positive divisors of $2j+1$.","title":"Legendre's four-triangular-numbers theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/hurwitz-quaternion-waring-bound","kind":"node","lean":"MetaMathlibExt.hurwitz_quaternion_waring_bound","parent":9994,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/index.html#hurwitz-quaternion-waring-bound","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate","status":"fully_proved","summary":"Call a rational quaternion a Hurwitz integer if its four coordinates are either all integers or all of the form $a + \\tfrac12$ with $a$ an integer. For every positive integer $\\ell$ with $\\ell \\equiv 1$ or $\\ell \\equiv 2 \\pmod 3$ there is a natural number $g$ such that every Hurwitz integer is a sum of at most $g$ $\\ell$-th powers of Hurwitz integers.","title":"Waring's problem for Hurwitz quaternion integers"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers.json","id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","kind":"scope","navigation":"module","parent":9994,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","title":"Lucas-polytopic numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-bracket-aux","kind":"node","lean":"MetaMathlibExt.lucasBracketAux","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-bracket-aux","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For real numbers $s$ and $t$, this is the sequence of real numbers $a_0, a_1, a_2, \\dots$ defined by $a_0 = 0$, $a_1 = 1$ and $a_{n+2} = s a_{n+1} + t a_n$ for every natural number $n$.","title":"Lucas brackets for arbitrary real parameters"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-parameters","kind":"node","lean":"MetaMathlibExt.LucasParameters","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-parameters","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For real numbers $s$ and $t$, let $\\{n\\}_{s,t}$ be the sequence of real numbers with $\\{0\\}_{s,t} = 0$, $\\{1\\}_{s,t} = 1$ and $\\{n+2\\}_{s,t} = s\\{n+1\\}_{s,t} + t\\{n\\}_{s,t}$ for every natural number $n$. Nondegenerate Lucas parameters are a pair of real numbers $(s, t)$ with $s \\ne 0$, $t \\ne 0$ and $\\{n\\}_{s,t} \\ne 0$ for every natural number $n \\ne 0$.","title":"Nondegenerate Lucas parameters"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-bracket","kind":"node","lean":"MetaMathlibExt.lucasBracket","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-bracket","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For nondegenerate Lucas parameters $(s, t)$, the Lucas brackets are the real numbers $\\{n\\}_{s,t}$, for natural numbers $n$, defined by $\\{0\\}_{s,t} = 0$, $\\{1\\}_{s,t} = 1$ and $\\{n+2\\}_{s,t} = s\\{n+1\\}_{s,t} + t\\{n\\}_{s,t}$ for every natural number $n$.","title":"Lucas brackets"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-factorial","kind":"node","lean":"MetaMathlibExt.lucasFactorial","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-factorial","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For nondegenerate Lucas parameters $(s, t)$, write $\\{k\\}_{s,t}$ for the $k$-th Lucas bracket. For a natural number $n$, the $s,t$-factorial is the real number $\\{n\\}_{s,t}! = \\prod_{k=1}^{n} \\{k\\}_{s,t}$, the product of $\\{1\\}_{s,t}, \\dots, \\{n\\}_{s,t}$; for $n = 0$ it is the empty product $1$.","title":"Factorial of Lucas brackets"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucasnomial","kind":"node","lean":"MetaMathlibExt.lucasnomial","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucasnomial","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"Let $(s, t)$ be nondegenerate Lucas parameters and write $\\{m\\}_{s,t}!$ for the $s,t$-factorial of a natural number $m$. For natural numbers $n$ and $k$, the Lucasnomial coefficient is the real number","title":"Lucasnomial coefficients"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-polytopic-number","kind":"node","lean":"MetaMathlibExt.lucasPolytopicNumber","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-polytopic-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For nondegenerate Lucas parameters $(s, t)$ and natural numbers $n$ and $d$, the $n$-th simplicial $d$-Lucas-polytopic number is the Lucasnomial coefficient $\\binom{n+d-1}{d}_{s,t}$, where $n + d - 1$ is read as $0$ when $n = d = 0$.","title":"Simplicial Lucas-polytopic numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-triangular-number","kind":"node","lean":"MetaMathlibExt.lucasTriangularNumber","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-triangular-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For nondegenerate Lucas parameters $(s, t)$ and a natural number $n$, the $n$-th Lucas triangular number is the Lucasnomial coefficient $\\binom{n+1}{2}_{s,t}$.","title":"Lucas triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-tetrahedral-number","kind":"node","lean":"MetaMathlibExt.lucasTetrahedralNumber","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-tetrahedral-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For nondegenerate Lucas parameters $(s, t)$ and a natural number $n$, the $n$-th Lucas tetrahedral number is the Lucasnomial coefficient $\\binom{n+2}{3}_{s,t}$.","title":"Lucas tetrahedral numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-pentachoron-number","kind":"node","lean":"MetaMathlibExt.lucasPentachoronNumber","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-pentachoron-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For nondegenerate Lucas parameters $(s, t)$ and a natural number $n$, the $n$-th Lucas pentachoron number is the Lucasnomial coefficient $\\binom{n+3}{4}_{s,t}$.","title":"Lucas pentachoron numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-hexateron-number","kind":"node","lean":"MetaMathlibExt.lucasHexateronNumber","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-hexateron-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For nondegenerate Lucas parameters $(s, t)$ and a natural number $n$, the $n$-th Lucas hexateron number is the Lucasnomial coefficient $\\binom{n+4}{5}_{s,t}$.","title":"Lucas hexateron numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucasnomial-succ-eq-sum-lucas-polytopic","kind":"node","lean":"MetaMathlibExt.lucasnomial_succ_eq_sum_lucasPolytopic","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucasnomial-succ-eq-sum-lucas-polytopic","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters, with Lucasnomial coefficients $\\binom{n}{k}_{s,t}$, and let $\\varphi$ and $\\varphi'$ be real numbers with $\\varphi + \\varphi' = s$ and $\\varphi \\varphi' = -t$. For all natural numbers $n \\ge 1$ and $d$,","title":"Sum formula for Lucasnomial coefficients"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-triangular-identities","kind":"node","lean":"MetaMathlibExt.lucasTriangular_identities","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-triangular-identities","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters and let $\\varphi, \\varphi'$ be real numbers with $\\varphi + \\varphi' = s$ and $\\varphi\\varphi' = -t$; write $\\{n\\}_{s,t}$ for the Lucas brackets and $\\binom{n}{k}_{s,t}$ for the Lucasnomial coefficients. Then $\\binom{n+1}{2}_{s,t} = t\\binom{n}{2}_{s,t} + \\{n\\}_{s,t}^2$ and $\\binom{n+1}{2}_{s,t} = \\sum_{k=1}^{n} t^{n-k}\\{k\\}_{s,t}^2$ for every natural number $n$, and\u2026","title":"Identities for the Lucas triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-triangular-sq-sub-sq","kind":"node","lean":"MetaMathlibExt.lucasTriangular_sq_sub_sq","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-triangular-sq-sub-sq","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters, with Lucas brackets $\\{n\\}_{s,t}$ and Lucasnomial coefficients $\\binom{n}{k}_{s,t}$. For every natural number $n$,","title":"Weighted difference of consecutive squared Lucas triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/sum-cubes-eq-lucas-triangular-sq","kind":"node","lean":"MetaMathlibExt.sum_cubes_eq_lucasTriangular_sq","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#sum-cubes-eq-lucas-triangular-sq","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters, with Lucas brackets $\\{n\\}_{s,t}$ and Lucasnomial coefficients $\\binom{n}{k}_{s,t}$. For every natural number $n$,","title":"Weighted sum of cubes of Lucas brackets"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/sum-cubes-specializations","kind":"node","lean":"MetaMathlibExt.sum_cubes_specializations","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#sum-cubes-specializations","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers, $P_n$ the Pell numbers, and $J_n$ the Jacobsthal numbers, that is, the Lucas sequence $U(1, -2)$, with $J_0 = 0$, $J_1 = 1$ and $J_{n+2} = J_{n+1} + 2 J_n$. For every natural number $n$,","title":"Sum-of-cubes identities for Fibonacci, Pell, Jacobsthal and Mersenne numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-tetrahedral-recurrence","kind":"node","lean":"MetaMathlibExt.lucasTetrahedral_recurrence","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-tetrahedral-recurrence","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters, and write $\\{n\\}_{s,t}$ for the Lucas brackets and $\\binom{n}{k}_{s,t}$ for the Lucasnomial coefficients. For every natural number $n$,","title":"Recurrence for the Lucas tetrahedral numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-derivative","kind":"node","lean":"MetaMathlibExt.lucasDerivative","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-derivative","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For real numbers $\\varphi$ and $\\varphi'$, the Lucas derivative $D_{\\varphi,\\varphi'}$ sends a function $f : \\mathbb{R} \\to \\mathbb{R}$ to the function whose value at $x \\ne 0$ is $(f(\\varphi x) - f(\\varphi' x))/((\\varphi - \\varphi')x)$, read as $0$ when $\\varphi = \\varphi'$, and whose value at $0$ is the derivative $f'(0)$, read as $0$ when $f$ is not differentiable at $0$.","title":"Lucas derivative"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-derivative-of-ne-zero","kind":"node","lean":"MetaMathlibExt.lucasDerivative_of_ne_zero","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-derivative-of-ne-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For all real numbers $\\varphi$, $\\varphi'$ and $x \\ne 0$ and every function $f : \\mathbb{R} \\to \\mathbb{R}$, the Lucas derivative $D_{\\varphi,\\varphi'}$ satisfies $D_{\\varphi,\\varphi'} f(x) = (f(\\varphi x) - f(\\varphi' x))/((\\varphi - \\varphi')x)$, where the quotient is read as $0$ when $\\varphi = \\varphi'$.","title":"The Lucas derivative away from the origin"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-derivative-zero","kind":"node","lean":"MetaMathlibExt.lucasDerivative_zero","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-derivative-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For all real numbers $\\varphi$, $\\varphi'$ and every function $f : \\mathbb{R} \\to \\mathbb{R}$, the Lucas derivative $D_{\\varphi,\\varphi'}$ satisfies $D_{\\varphi,\\varphi'} f(0) = f'(0)$, where $f'(0)$ is read as $0$ when $f$ is not differentiable at $0$.","title":"The Lucas derivative at the origin"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-derivative-id","kind":"node","lean":"MetaMathlibExt.lucasDerivative_id","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-derivative-id","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"fully_proved","summary":"For distinct real numbers $\\varphi \\ne \\varphi'$, the Lucas derivative $D_{\\varphi,\\varphi'}$ of the identity function $y \\mapsto y$ equals $1$ at every real number $x$.","title":"The Lucas derivative of the identity"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/iterate-lucas-derivative-inv-one-sub","kind":"node","lean":"MetaMathlibExt.iterate_lucasDerivative_inv_one_sub","parent":9998,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterated Lucas derivatives of "],["math","\\(1/(1 - x)\\)"]],"source":"Iterated Lucas derivatives of $1/(1 - x)$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#iterate-lucas-derivative-inv-one-sub","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters, let $\\varphi \\ne \\varphi'$ be real numbers with $\\varphi + \\varphi' = s$ and $\\varphi\\varphi' = -t$, write $D_{\\varphi,\\varphi'}$ for the Lucas derivative and $\\{n\\}_{s,t}!$ for the $s,t$-factorial. For every natural number $n$ and every real number $x$ with $\\prod_{k=0}^{n} (1 - \\varphi^{n-k}\\varphi'^k x) \\ne 0$, the $n$-fold iterate of $D_{\\varphi,\\varphi'}$ applied\u2026","title":"Iterated Lucas derivatives of $1/(1 - x)$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-polytopic-generating-function","kind":"node","lean":"MetaMathlibExt.lucasPolytopic_generatingFunction","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-polytopic-generating-function","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters, let $\\varphi, \\varphi'$ be real numbers with $\\varphi + \\varphi' = s$ and $\\varphi\\varphi' = -t$, let $d$ be a natural number and write $\\binom{n}{k}_{s,t}$ for the Lucasnomial coefficients, so that the $n$-th simplicial $d$-Lucas-polytopic number is $\\binom{n+d-1}{d}_{s,t}$. Then, as formal power series in $x$ with real coefficients,","title":"Generating function of the simplicial Lucas-polytopic numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/lucas-triangular-sq-generating-function","kind":"node","lean":"MetaMathlibExt.lucasTriangular_sq_generatingFunction","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#lucas-triangular-sq-generating-function","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters, with Lucas brackets $\\{n\\}_{s,t}$ and Lucasnomial coefficients $\\binom{n}{k}_{s,t}$, and let $\\varphi$ and $\\varphi'$ be real numbers with $\\varphi + \\varphi' = s$ and $\\varphi \\varphi' = -t$. Then, as formal power series in $x$ with real coefficients,","title":"Generating function of the squared Lucas triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/has-sum-pow-div-lucas-bracket-two-mul","kind":"node","lean":"MetaMathlibExt.hasSum_pow_div_lucasBracket_two_mul","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#has-sum-pow-div-lucas-bracket-two-mul","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters with $s > 0$ and $s^2 + 4t > 0$, write $\\{n\\}_{s,t}$ for the Lucas brackets, let $\\varphi' = (s - \\sqrt{s^2 + 4t})/2$ and let $L(x) = \\sum_{n \\ge 1} x^n/(1 - x^n)$ be the Lambert series. Then the series $\\sum_{n \\ge 1} t^n/\\{2n\\}_{s,t}$ converges absolutely, and its sum is $\\sqrt{s^2 + 4t}\\,\\bigl(L(\\varphi'^2/t) - L(\\varphi'^4/t^2)\\bigr)$.","title":"Even-index Lucas-bracket series as Lambert series"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/has-sum-inv-lucas-bracket-mul-succ","kind":"node","lean":"MetaMathlibExt.hasSum_inv_lucasBracket_mul_succ","parent":9998,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers/index.html#has-sum-inv-lucas-bracket-mul-succ","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/lucas-polytopic-numbers","status":"can_prove","summary":"Let $(s, t)$ be nondegenerate Lucas parameters with $s > 0$, $s^2 + 4t > 0$ and $|t| > 1$, write $\\{n\\}_{s,t}$ for the Lucas brackets and let $\\varphi = (s + \\sqrt{s^2 + 4t})/2$. Then the series $\\sum_{n \\ge 1} 1/(\\{n\\}_{s,t}\\{n+1\\}_{s,t})$ converges absolutely, and its sum is $\\varphi - (1 + t)\\ln_{s,t}(1 + \\varphi/t)$, where $\\ln_{s,t}(1 + \\varphi/t) = -\\sum_{n \\ge 1} (-\\varphi/t)^n/\\{n\\}_{s,t}$ is the Lucas\u2026","title":"Sum of reciprocals of products of consecutive Lucas brackets"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations.json","id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","kind":"scope","navigation":"module","parent":9994,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","title":"Nicomachus's theorem and its generalizations"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachean-nu","kind":"node","lean":"MetaMathlibExt.nicomacheanNu","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachean-nu","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a finite sequence $\\sigma = (a_1, \\ldots, a_n)$ of elements of a commutative ring, the Nicomachean defect of $\\sigma$ is $\\nu(\\sigma) = \\left(\\sum_{i=1}^n a_i\\right)^2 - \\sum_{i=1}^n a_i^3$.","title":"Nicomachean defect"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachean-nu-examples","kind":"node","lean":"MetaMathlibExt.nicomacheanNu_examples","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachean-nu-examples","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For the Nicomachean defect $\\nu$ of a sequence of integers, $\\nu(1, 2, 3, 4) = 0$, $\\nu(1, 2) = 0$, $\\nu(3, 3, 3) = 0$, $\\nu(1, 2, 2, 3, 4, 5, 6, 8) = 0$ and $\\nu(1, 2, 2, 3, 4, 6, 6, 8) = -28$.","title":"Examples of the Nicomachean defect"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachean-alpha","kind":"node","lean":"MetaMathlibExt.nicomacheanAlpha","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachean-alpha","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For rational numbers $n$, $s$ and $a$, define","title":"The Nicomachean progression factor"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachean-alpha-one-two-three","kind":"node","lean":"MetaMathlibExt.nicomacheanAlpha_one_two_three","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","Specializations of the Nicomachean factor at "],["math","\\(s = 1, 2, 3\\)"]],"source":"Specializations of the Nicomachean factor at $s = 1, 2, 3$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachean-alpha-one-two-three","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"Let $\\alpha(n, s, a) = 2n^2 + na - a^2 + n(n-1)(s - s^2)/2 - (n-1)as$. For all rational numbers $n$ and $a$, $\\alpha(n, 1, a) = 2n^2 + a - a^2$, $\\alpha(n, 2, a) = n^2 + (1 - a)n - a^2 + 2a$ and $\\alpha(n, 3, a) = -(a + n - 3)(a + n)$.","title":"Specializations of the Nicomachean factor at $s = 1, 2, 3$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachean-bag-square-root-second-diffe","kind":"node","lean":"MetaMathlibExt.nicomachean_bagSquare_root_secondDifference_tendsto","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachean-bag-square-root-second-diffe","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Write $\\nu(a_1, \\dots, a_k) = (a_1 + \\dots + a_k)^2 - (a_1^3 + \\dots + a_k^3)$ for the Nicomachean defect of a finite sequence. For natural numbers $m$ let","title":"Second differences of roots of translated Nicomachean bag squares"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/exists-pos-rat-nicomachean-nu-eq-zero-la","kind":"node","lean":"MetaMathlibExt.exists_pos_rat_nicomacheanNu_eq_zero_large_den","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#exists-pos-rat-nicomachean-nu-eq-zero-la","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Write $\\nu(a_1, \\dots, a_n) = (a_1 + \\dots + a_n)^2 - (a_1^3 + \\dots + a_n^3)$ for the Nicomachean defect of a finite sequence. For every natural number $Q$ there is a finite sequence $\\sigma$ of positive rational numbers with $\\nu(\\sigma) = 0$ having an entry whose denominator in lowest terms is greater than $Q$.","title":"Nicomachean sequences of rationals with large denominators"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/exists-two-valued-nicomachean-large-gap","kind":"node","lean":"MetaMathlibExt.exists_two_valued_nicomachean_large_gap","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#exists-two-valued-nicomachean-large-gap","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Write $\\nu(a_1, \\dots, a_n) = (a_1 + \\dots + a_n)^2 - (a_1^3 + \\dots + a_n^3)$ for the Nicomachean defect of a finite sequence. For every real $\\varepsilon > 0$ there are integers $0 < a < b$ and a finite sequence $\\sigma$ whose entries all equal $a$ or $b$, in which both $a$ and $b$ occur, such that $\\nu(\\sigma) = 0$ and $(b - a)/L > 1 - \\varepsilon$, where $L$ is the length of $\\sigma$.","title":"Two-valued Nicomachean sequences with a large gap"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachean-run-repeat-generating-functi","kind":"node","lean":"MetaMathlibExt.nicomachean_run_repeat_generatingFunctions","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachean-run-repeat-generating-functi","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"For integers $j, m \\ge 1$ let $\\sigma(j, m)$ be the integer sequence $(1, 2, \\ldots, j + 1, m, \\ldots, m)$ in which $m$ is repeated $2j$ times, and let $\\nu$ denote the Nicomachean defect. Call $j \\ge 1$ admissible if $\\nu(\\sigma(j, m)) = 0$ for some integer $m \\ge 1$, let $j_k$ be the $k$-th smallest admissible integer (counting from $k = 0$), and let $m_k = j_k + \\lfloor \\sqrt{2j_k^2 + 3j_k + 2} \\rfloor$. Then\u2026","title":"Generating functions for run-and-repeat Nicomachean sequences"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/exists-nicomachean-large-top-gap","kind":"node","lean":"MetaMathlibExt.exists_nicomachean_large_top_gap","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#exists-nicomachean-large-top-gap","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Write $\\nu(a_1, \\dots, a_n) = (a_1 + \\dots + a_n)^2 - (a_1^3 + \\dots + a_n^3)$ for the Nicomachean defect of a finite sequence. For every natural number $G$ there is a finite sequence $\\sigma$ of positive integers with $\\nu(\\sigma) = 0$ and entries $a < b$ of $\\sigma$ such that $b$ is the largest entry of $\\sigma$ and occurs exactly once, every entry other than $b$ is at most $a$, and $b - a > G$.","title":"Nicomachean sequences with a large gap at the top"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/exists-golden-ratio-nicomachean-sequence","kind":"node","lean":"MetaMathlibExt.exists_goldenRatio_nicomachean_sequence","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#exists-golden-ratio-nicomachean-sequence","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Let $\\varphi = (1 + \\sqrt 5)/2$ be the golden ratio and write $\\nu(a_1, \\dots, a_k) = (a_1 + \\dots + a_k)^2 - (a_1^3 + \\dots + a_k^3)$ for the Nicomachean defect of a finite sequence of real numbers. For every natural number $N$ there is a finite sequence $\\sigma$ of positive real numbers, each of the form $p + q\\varphi$ with $p, q \\in \\mathbb{Z}$, such that $\\nu(\\sigma) = 0$, $\\sigma$ contains $\\varphi^{-n}$ for\u2026","title":"Golden-ratio sequences with vanishing Nicomachean defect"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachean-adjoined-arithmetic-progress","kind":"node","lean":"MetaMathlibExt.nicomachean_adjoined_arithmeticProgression","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachean-adjoined-arithmetic-progress","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Write $\\nu(a_1, \\dots, a_n) = (a_1 + \\dots + a_n)^2 - (a_1^3 + \\dots + a_n^3)$ for the Nicomachean defect of a finite sequence. Let $(a_1, \\dots, a_n)$ be a finite, possibly empty, sequence of real numbers and let $x_1, x_2, x_3, \\dots$ be nonzero real numbers with $x_{j+1} \\ne -x_j$ for every $j \\ge 1$. If $\\nu(a_1, \\dots, a_n, x_1, \\dots, x_l) = 0$ for every $l \\ge 1$, then $x_{j+1} = x_j + 1$ for every\u2026","title":"Adjoined Nicomachean sequences are eventually arithmetic progressions"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachean-alpha-pos-int-root-iff","kind":"node","lean":"MetaMathlibExt.nicomacheanAlpha_posIntRoot_iff","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positive integer roots of the Nicomachean factor for "],["math","\\(s = 1, 2\\)"]],"source":"Positive integer roots of the Nicomachean factor for $s = 1, 2$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachean-alpha-pos-int-root-iff","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Let $\\alpha(n, s, a) = 2n^2 + na - a^2 + n(n-1)(s - s^2)/2 - (n-1)as$ and let $F_0 = 0$, $F_1 = 1$, $F_{k+2} = F_{k+1} + F_k$ be the Fibonacci numbers. Let $n \\ge 1$ be a natural number. Then $\\alpha(n, 1, h) = 0$ for some positive integer $h$ if and only if $2n^2 = k(k+1)$ for some natural number $k$, that is, $n^2$ is a triangular number; $\\alpha(n, 2, h) = 0$ for some positive integer $h$ if and only if\u2026","title":"Positive integer roots of the Nicomachean factor for $s = 1, 2$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/exists-nicomachean-family-same-quadratic","kind":"node","lean":"MetaMathlibExt.exists_nicomachean_family_same_quadratic_factor","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#exists-nicomachean-family-same-quadratic","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Write $\\nu(a_1, \\dots, a_n) = (a_1 + \\dots + a_n)^2 - (a_1^3 + \\dots + a_n^3)$ for the Nicomachean defect of a finite sequence. For every natural number $N$ there are finite sequences $\\sigma_1, \\dots, \\sigma_N$ of positive integers, pairwise distinct as multisets, and an integer $c$ such that for every $i$ we have $\\nu(\\sigma_i) = 0$ and, for all integers $a$ and $b$, the sequence $(\\sigma_i, a, b)$ obtained by\u2026","title":"Nicomachean families sharing a quadratic factor"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-triangular","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusTriangular","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-triangular","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a natural number $m$, $T_m = m(m+1)/2$ is the $m$-th triangular number, viewed as a real number.","title":"Triangular numbers as reals"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-lodd","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusLOdd","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odd-case cube sum "],["math","\\(L_{m,o}\\)"]],"source":"The odd-case cube sum $L_{m,o}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-lodd","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a natural number $m$ and a real number $x$,","title":"The odd-case cube sum $L_{m,o}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-rodd","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusROdd","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odd-case square "],["math","\\(R_{m,o}(x)\\)"]],"source":"The odd-case square $R_{m,o}(x)$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-rodd","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a natural number $m$ and a real number $x$,","title":"The odd-case square $R_{m,o}(x)$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-podd","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusPOdd","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odd-case polynomial "],["math","\\(P_{m,o}\\)"]],"source":"The odd-case polynomial $P_{m,o}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-podd","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a natural number $m$ and a real number $x$,","title":"The odd-case polynomial $P_{m,o}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-leven","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusLEven","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The even-case cube sum "],["math","\\(L_{m,e}\\)"]],"source":"The even-case cube sum $L_{m,e}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-leven","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a natural number $m$ and a real number $x$,","title":"The even-case cube sum $L_{m,e}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-reven","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusREven","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The even-case square "],["math","\\(R_{m,e}\\)"]],"source":"The even-case square $R_{m,e}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-reven","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a natural number $m$ and a real number $x$,","title":"The even-case square $R_{m,e}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-pclosed","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusPClosed","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-pclosed","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For real numbers $m$ and $x$, let","title":"Faulhaber expansion of the odd-case polynomial"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-peven","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusPEven","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The even-case polynomial "],["math","\\(P_{m,e}\\)"]],"source":"The even-case polynomial $P_{m,e}$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-peven","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a natural number $m$ and a real number $x$, $P_{m,e}(x) = \\widetilde{P}(-m - 1, x)$, where $\\widetilde{P}(m, x)$ is the Faulhaber expansion of the odd-case polynomial $P_{m,o}(x)$ as a polynomial in $m$ (the paper defines $P_{m,e}(x)$ by substituting $-m - 1$ for $m$ in that expansion).","title":"The even-case polynomial $P_{m,e}$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-alpha","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusAlpha","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Kim\u2013Stolarsky convergents "],["math","\\(\\alpha_k\\)"]],"source":"The Kim\u2013Stolarsky convergents $\\alpha_k$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-alpha","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"The rational numbers $\\alpha_k$, for natural numbers $k$, are defined by $\\alpha_0 = 0$, $\\alpha_1 = 1$ and $\\alpha_{k+1} = \\frac{15 + 7\\alpha_k}{13 + 6\\alpha_k}$ for $k \\ge 1$.","title":"The Kim\u2013Stolarsky convergents $\\alpha_k$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-u","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusU","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Kim\u2013Stolarsky integers "],["math","\\(u_k\\)"]],"source":"The Kim\u2013Stolarsky integers $u_k$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-u","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"fully_proved","summary":"For a natural number $k$, $u_k$ is the coefficient of $x^{k-1}$ in the power series expansion over $\\mathbb{Q}$ of","title":"The Kim\u2013Stolarsky integers $u_k$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-lodd-sub-rodd-succ-eq","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusLOdd_sub_ROdd_succ_eq","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-lodd-sub-rodd-succ-eq","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Let $L_{m,o}$ be the odd-case cube sum, $R_{m,o}$ the odd-case square and $P_{m,o}$ the odd-case polynomial. For every natural number $m \\ge 2$ and every real number $x$,","title":"Generalized Nicomachus identity, odd case"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-leven-sub-reven-eq","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusLEven_sub_REven_eq","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-leven-sub-reven-eq","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Let $L_{m,e}$ be the even-case cube sum, $R_{m,e}$ the even-case square and $P_{m,e}$ the even-case polynomial. For every natural number $m \\ge 2$ and every real number $x$,","title":"Generalized Nicomachus identity, even case"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-u-eq-closed-form","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusU_eq_closedForm","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form for the Kim\u2013Stolarsky integers "],["math","\\(u_k\\)"]],"source":"Closed form for the Kim\u2013Stolarsky integers $u_k$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-u-eq-closed-form","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Let $u_k$ be the coefficient of $x^{k-1}$ in the power series expansion of $(1 + 502x + x^2)/((1 - x)(1 - 398x + x^2))$. For every integer $k \\ge 1$,","title":"Closed form for the Kim\u2013Stolarsky integers $u_k$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-u-succ-eq","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusU_succ_eq","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the Kim\u2013Stolarsky integers "],["math","\\(u_k\\)"]],"source":"Recurrence for the Kim\u2013Stolarsky integers $u_k$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-u-succ-eq","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Let $u_k$ be the coefficient of $x^{k-1}$ in the power series expansion of $(1 + 502x + x^2)/((1 - x)(1 - 398x + x^2))$. For every integer $k \\ge 1$,","title":"Recurrence for the Kim\u2013Stolarsky integers $u_k$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/nicomachus-alpha-eq","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.nicomachusAlpha_eq","parent":10024,"reading_title":{"format":"inline-parts-v1","parts":[["text","The convergents "],["math","\\(\\alpha_k\\)"],["text"," in terms of the integers "],["math","\\(u_k\\)"]],"source":"The convergents $\\alpha_k$ in terms of the integers $u_k$"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#nicomachus-alpha-eq","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Let $\\alpha_k$ be the Kim\u2013Stolarsky convergents, with $\\alpha_1 = 1$ and $\\alpha_{k+1} = (15 + 7\\alpha_k)/(13 + 6\\alpha_k)$ for $k \\ge 1$, and let $u_k$ be the coefficient of $x^{k-1}$ in the power series expansion of $(1 + 502x + x^2)/((1 - x)(1 - 398x + x^2))$. For every integer $k \\ge 1$,","title":"The convergents $\\alpha_k$ in terms of the integers $u_k$"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/sum-quadratic-mul-quadratic-eq","kind":"node","lean":"MathlibExt.NumberTheory.NicomachusGeneralizationWanted.sum_quadratic_mul_quadratic_eq","parent":10024,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations/index.html#sum-quadratic-mul-quadratic-eq","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/nicomachus-s-theorem-and-its-generalizations","status":"can_prove","summary":"Let $k \\ge 1$, let $\\alpha_k$ be the numbers of Equation (12) (so $\\alpha_1 = 1$ and $\\alpha_{k+1} = (15 + 7\\alpha_k)/(13 + 6\\alpha_k)$), let $u_k$ be the integers of Proposition 1, and let $T_n = n(n+1)/2$. With $(a, b, c) = (1, \\alpha_k, -(1 + \\alpha_k))$ and $(d, e, f) = (-a, -c, -b)$ as in (17), for every natural number $m$,","title":"Kim\u2013Stolarsky balanced quadratic sum"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers.json","id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","kind":"scope","navigation":"module","parent":9994,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","title":"Polygonal numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number","kind":"node","lean":"Nat.polygonalNumber","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For natural numbers $s$ and $n$, the $n$-th $s$-gonal number is $p_s(n) = (s - 2)\\binom{n}{2} + n$, where $s - 2$ is truncated subtraction of natural numbers (so it is $0$ when $s < 2$).","title":"Polygonal numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-zero","kind":"node","lean":"Nat.polygonalNumber_zero","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $s$, $p_s(0) = 0$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Polygonal numbers at zero"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-one","kind":"node","lean":"Nat.polygonalNumber_one","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-one","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $s$, $p_s(1) = 1$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Polygonal numbers at one"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-at-two","kind":"node","lean":"Nat.polygonalNumber_at_two","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-at-two","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $s \\ge 3$, $p_s(2) = s$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Polygonal numbers at two"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-at-three","kind":"node","lean":"Nat.polygonalNumber_at_three","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-at-three","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $s \\ge 3$, $p_s(3) = 3s - 3$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Polygonal numbers at three"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-at-four","kind":"node","lean":"Nat.polygonalNumber_at_four","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-at-four","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $s \\ge 3$, $p_s(4) = 6s - 8$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Polygonal numbers at four"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-three","kind":"node","lean":"Nat.polygonalNumber_three","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-three","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $n$, the $n$-th triangular number is $p_3(n) = \\frac{n(n + 1)}{2}$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Closed form of the triangular numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-four","kind":"node","lean":"Nat.polygonalNumber_four","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-four","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $n$, the $n$-th square number is $p_4(n) = n^2$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Square numbers as polygonal numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-five","kind":"node","lean":"Nat.polygonalNumber_five","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-five","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $n$, the $n$-th pentagonal number is $p_5(n) = \\frac{3n^2 - n}{2}$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Closed form of the pentagonal numbers"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/polygonal-number-six","kind":"node","lean":"Nat.polygonalNumber_six","parent":10054,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers/index.html#polygonal-number-six","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/polygonal-numbers","status":"fully_proved","summary":"For every natural number $n$, the $n$-th hexagonal number is $p_6(n) = 2n^2 - n$, where $p_s(n)$ is the $n$-th $s$-gonal number.","title":"Closed form of the hexagonal numbers"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem.json","id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem","kind":"scope","navigation":"module","parent":9994,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem","status":"fully_proved","title":"Fermat polygonal number theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem/exists-eq-add-add-triangular","kind":"node","lean":"Nat.exists_eq_add_add_triangular","parent":10065,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem/index.html#exists-eq-add-add-triangular","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem","status":"fully_proved","summary":"Every natural number $n$ is a sum of three triangular numbers: there are natural numbers $x, y, z$ with $n = T_x + T_y + T_z$, where $T_m = m(m + 1)/2$ is the $m$-th triangular ($3$-gonal) number, so that $T_0 = 0$.","title":"Gauss's Eureka theorem"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem/exists-sum-four-sq-eq-and-sum-eq-of-odd","kind":"node","lean":"Nat.exists_sum_four_sq_eq_and_sum_eq_of_odd","parent":10065,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem/index.html#exists-sum-four-sq-eq-and-sum-eq-of-odd","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem","status":"fully_proved","summary":"Let $a$ and $b$ be odd natural numbers with $b^2 < 4a$ and $3a < b^2 + 2b + 4$. Then there are natural numbers $s, t, u, v$ with $a = s^2 + t^2 + u^2 + v^2$ and $b = s + t + u + v$.","title":"Cauchy's lemma"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem/fermat-polygonal-number-theorem","kind":"node","lean":"MetaMathlibExt.fermat_polygonal_number_theorem","parent":10065,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem/index.html#fermat-polygonal-number-theorem","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/fermat-polygonal-number-theorem","status":"fully_proved","summary":"For every natural number $s \\ge 3$, every natural number $n$ is a sum of at most $s$ $s$-gonal numbers: there are a finite set $T$ with at most $s$ elements and natural numbers $k_i$ for $i \\in T$ such that $n = \\sum_{i \\in T} P_s(k_i)$, where $P_s(m) = (s - 2)\\binom{m}{2} + m$ is the $m$-th $s$-gonal number. The $k_i$ may repeat, and since $P_s(0) = 0$ the case $n = 0$ is the empty sum.","title":"Fermat polygonal number theorem"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes.json","id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes","kind":"scope","navigation":"module","parent":9994,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes","status":"can_prove","title":"Waring's problem for cubes"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes/is-sum-of-cubes","kind":"node","lean":"MathlibExt.NumberTheory.WieferichNineCubesWanted.NumberTheory.IsSumOfCubes","parent":10069,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(k\\)"],["text"," cubes"]],"source":"Sum of $k$ cubes"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes/index.html#is-sum-of-cubes","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes","status":"fully_proved","summary":"A natural number $n$ is a sum of $k$ cubes if there are natural numbers $x_1, \\dots, x_k$ with $n = x_1^3 + \\dots + x_k^3$.","title":"Sum of $k$ cubes"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes/wieferich-g-three","kind":"node","lean":"MathlibExt.NumberTheory.WieferichNineCubesWanted.NumberTheory.wieferich_g_three","parent":10069,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes/index.html#wieferich-g-three","scope":"NumberTheory/quadratic-forms-sums-of-squares/waring-figurate/waring-s-problem-for-cubes","status":"can_prove","summary":"Every natural number is a sum of $9$ cubes of natural numbers, and $9$ cannot be replaced by $8$: some natural number is not a sum of $8$ cubes of natural numbers.","title":"Wieferich's nine cubes theorem"},{"data_url":"scopes/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials.json","id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","kind":"scope","navigation":"module","parent":9819,"result":false,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","title":"Quadratic packing polynomials"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/sector-lattice-points","kind":"node","lean":"MetaMathlibExt.sectorLatticePoints","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#sector-lattice-points","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"For a real number $\\alpha$, let $S(\\alpha) = \\{(x, y) \\in \\mathbb{R}^2 : 0 \\le y \\le \\alpha x\\}$ be the sector of slope $\\alpha$. Its lattice points are $I(\\alpha) = S(\\alpha) \\cap \\mathbb{Z}^2$, viewed as a subset of $\\mathbb{R}^2$.","title":"Lattice points of a sector"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/zero-mem-sector-lattice-points","kind":"node","lean":"MetaMathlibExt.zero_mem_sectorLatticePoints","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#zero-mem-sector-lattice-points","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"For every real number $\\alpha$, the origin $(0, 0)$ lies in the set $I(\\alpha)$ of lattice points of the sector of slope $\\alpha$.","title":"The origin is a lattice point of every sector"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/is-quadratic-packing-polynomial","kind":"node","lean":"MetaMathlibExt.IsQuadraticPackingPolynomial","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#is-quadratic-packing-polynomial","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"A real function $P(x, y)$ of two real variables is a quadratic packing polynomial (QPP) on a set $S \\subseteq \\mathbb{R}^2$ if it is a polynomial of total degree two, that is, $P(x, y) = a x^2 + b x y + c y^2 + d x + e y + f$ for some real $a, b, c, d, e, f$ with $a, b, c$ not all zero, and $P$ maps $S$ bijectively onto the set $\\mathbb{N}_0 = \\{0, 1, 2, \\dots\\}$ of natural numbers.","title":"Quadratic packing polynomial"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/exists-nat-eq","kind":"node","lean":"MetaMathlibExt.IsQuadraticPackingPolynomial.exists_nat_eq","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#exists-nat-eq","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"If $P$ is a quadratic packing polynomial on a set $S \\subseteq \\mathbb{R}^2$, then for every point $(x, y) \\in S$ the value $P(x, y)$ is a natural number.","title":"Quadratic packing polynomials take natural-number values"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/sector-staircase","kind":"node","lean":"MetaMathlibExt.sectorStaircase","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#sector-staircase","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"For natural numbers $m, n, i$, let $l = \\gcd(m - 1, n)$. The $i$-th staircase of the sector of slope $n/m$ is $J_i = \\{(x, y) \\in I(n/m) : nx - (m - 1)y = li\\}$, where $I(n/m)$ is the set of lattice points of that sector.","title":"Staircases of a rational sector"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/zero-mem-sector-staircase-zero","kind":"node","lean":"MetaMathlibExt.zero_mem_sectorStaircase_zero","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#zero-mem-sector-staircase-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"For all natural numbers $m, n$, the origin $(0, 0)$ lies on the staircase $J_0$ of the sector of slope $n/m$.","title":"The origin lies on the zeroth staircase"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/staircase-first-height","kind":"node","lean":"MetaMathlibExt.staircaseFirstHeight","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#staircase-first-height","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"For natural numbers $m, n, i$, the first height $\\bar y_i$ of the staircase $J_i$ of the sector of slope $n/m$ is the least $y$-coordinate of a point of $J_i$, that is, the height of its first step; it is $0$ if $J_i$ is empty.","title":"First height of a staircase"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/staircase-first-height-zero","kind":"node","lean":"MetaMathlibExt.staircaseFirstHeight_zero","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#staircase-first-height-zero","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"For all natural numbers $m, n$, the first height of the staircase $J_0$ of the sector of slope $n/m$ is $\\bar y_0 = 0$; indeed the origin lies on $J_0$.","title":"The zeroth staircase starts at height zero"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/consecutive-step-difference","kind":"node","lean":"MetaMathlibExt.consecutiveStepDifference","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#consecutive-step-difference","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"For natural numbers $m, n$ and a real function $P(x, y)$ of two real variables, let $l = \\gcd(m \\mathbin{\\dot-} 1, n)$, where $m \\mathbin{\\dot-} 1$ is subtraction truncated at $0$ (so $l = n$ when $m = 0$). The step difference of $P$ for the sector of slope $n/m$ is $k = P((m - 1)/l, n/l) - P(0, 0)$, with $m - 1$ here the real difference and division by $l = 0$ read as $0$: the change\u2026","title":"Step difference"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/consecutive-step-difference-example-twel","kind":"node","lean":"MetaMathlibExt.consecutiveStepDifference_example_twelve_sevenths","parent":10072,"reading_title":{"format":"inline-parts-v1","parts":[["text","Step difference of the slope-"],["math","\\(12/7\\)"],["text"," example"]],"source":"Step difference of the slope-$12/7$ example"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#consecutive-step-difference-example-twel","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"fully_proved","summary":"Let $P(x, y) = 6(x - y/2)(x - y/2 - 3/2) + x + y + 2$. For $m = 7$ and $n = 12$ (the sector of slope $12/7$), the step difference of $P$ is $k = 3$: with $l = \\gcd(6, 12) = 6$, one has $P(1, 2) - P(0, 0) = 3$.","title":"Step difference of the slope-$12/7$ example"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/quadratic-packing-polynomial-sector-form","kind":"node","lean":"MetaMathlibExt.quadraticPackingPolynomial_sector_form","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#quadratic-packing-polynomial-sector-form","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $\\alpha > 0$ be a real number and let $P$ be a quadratic packing polynomial on the lattice sector $I(\\alpha)$. Then there are integers $A, B, C, D, E, F$ with $A > 0$, $B^2 = AC$ and $\\alpha = A/(1 - B)$ such that for all real $x, y$","title":"Form of a quadratic packing polynomial on a sector (Gjaldb\u00e6k)"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/quadratic-packing-polynomial-rational-se","kind":"node","lean":"MetaMathlibExt.quadraticPackingPolynomial_rational_sector_form","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#quadratic-packing-polynomial-rational-se","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $m, n$ be coprime positive integers and let $P$ be a quadratic packing polynomial on the lattice points $I(n/m)$ of the sector of slope $n/m$. Then $n$ divides $(m - 1)^2$, and there are integers $D, E, F$ such that for all real $x, y$","title":"Stanton's form of a quadratic packing polynomial on a rational sector"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/is-quadratic-packing-polynomial-sheared","kind":"node","lean":"MetaMathlibExt.isQuadraticPackingPolynomial_sheared_example_twelve_sevenths","parent":10072,"reading_title":{"format":"inline-parts-v1","parts":[["text","The slope-"],["math","\\(12/7\\)"],["text"," example on the sheared sector"]],"source":"The slope-$12/7$ example on the sheared sector"},"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#is-quadratic-packing-polynomial-sheared","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $M(x, y) = (x - y/2, y)$ and let $\\hat I(12/7) = M(I(12/7))$ be the image under $M$ of the lattice points $I(12/7)$ of the sector of slope $12/7$. Then $\\hat P(x, y) = 6x(x - 3/2) + x + \\frac{3}{2} y + 2$ is a quadratic packing polynomial on $\\hat I(12/7)$.","title":"The slope-$12/7$ example on the sheared sector"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/quadratic-packing-polynomial-first-step","kind":"node","lean":"MetaMathlibExt.quadraticPackingPolynomial_first_step_value","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#quadratic-packing-polynomial-first-step","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $m, n$ be coprime positive integers, let $l = \\gcd(m - 1, n)$, and let $P$ be a quadratic packing polynomial on the lattice points $I(n/m)$ of the sector of slope $n/m$ whose step difference for $m, n$ is a positive integer $k$. Put $F = P(0, 0)$ and $\\hat P(x, y) = P(x + \\frac{m - 1}{n} y, y)$, and let $\\bar y_k$ be the height of the first step of the $k$-th staircase of the sector. Then\u2026","title":"First step value and sheared form of a quadratic packing polynomial"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/quadratic-packing-polynomial-staircase-c","kind":"node","lean":"MetaMathlibExt.quadraticPackingPolynomial_staircase_congruence","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#quadratic-packing-polynomial-staircase-c","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $m, n$ be positive coprime integers, let $P$ be a quadratic packing polynomial on the lattice sector $I(n/m)$, and suppose that the step difference of $P$ equals a positive integer $k$. Then for every natural number $i$, every point $p$ of the staircase $J_i$ and every point $q$ of the staircase $J_{i+k}$, the difference $P(q) - P(p)$ is an integer multiple of $k$.","title":"Staircase congruence for quadratic packing polynomials"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/quadratic-packing-polynomial-staircase-f","kind":"node","lean":"MetaMathlibExt.quadraticPackingPolynomial_staircaseFirstHeight","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#quadratic-packing-polynomial-staircase-f","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $m, n$ be positive coprime integers and $l = \\gcd(m - 1, n)$, let $P$ be a quadratic packing polynomial on the lattice sector $I(n/m)$, and suppose that the step difference of $P$ equals a positive integer $k$. Then the first height of the staircase $J_k$ is $\\bar y_k = n/l - 1$.","title":"Height of the first step of the k-th staircase"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/is-quadratic-packing-polynomial-sheared-2","kind":"node","lean":"MetaMathlibExt.isQuadraticPackingPolynomial_sheared_iff","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#is-quadratic-packing-polynomial-sheared-2","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $m, n$ be coprime positive integers, let $l = \\gcd(m - 1, n)$, and assume that $n/l$ divides $l$. Let $k$ be a positive integer with $k \\equiv (m - 1)/l \\pmod{n/l}$, let $F$ be a real number, and let $\\hat P(x, y) = \\frac{n}{2} x \\left(x - \\frac{k}{n/l}\\right) + x + \\frac{k}{n/l} y + F$. Let $M(x, y) = (x - \\frac{m - 1}{n} y, y)$ and let $\\hat I(n/m) = M(I(n/m))$ be the image under $M$ of the lattice points\u2026","title":"Sheared criterion for quadratic packing polynomials"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/quadratic-packing-polynomial-constant-te","kind":"node","lean":"MetaMathlibExt.quadraticPackingPolynomial_constant_term","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#quadratic-packing-polynomial-constant-te","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $m, n$ be coprime positive integers, let $l = \\gcd(m - 1, n)$, and let $P$ be a quadratic packing polynomial on the lattice points $I(n/m)$ of the sector of slope $n/m$. If the step difference of $P$ for $m, n$ is a positive integer $k$, then $P(0, 0) = \\frac{l^2}{n} (k - 1)(k + 1)/12$.","title":"Constant term of a quadratic packing polynomial on a rational sector"},{"id":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/quadratic-packing-polynomial-necessary-f","kind":"node","lean":"MetaMathlibExt.quadraticPackingPolynomial_necessary_form","parent":10072,"result":true,"route":"../roadmap/NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials/index.html#quadratic-packing-polynomial-necessary-f","scope":"NumberTheory/quadratic-forms-sums-of-squares/quadratic-packing-polynomials","status":"can_prove","summary":"Let $m, n$ be coprime positive integers, let $l = \\gcd(m - 1, n)$, and let $P$ be a quadratic packing polynomial on the lattice points $I(n/m)$ of the sector of slope $n/m$ whose step difference for $m, n$ is a positive integer $k$. Then $n/l$ divides $l$, $k \\equiv (m - 1)/l \\pmod{n/l}$, and for all real $x, y$ one has\u2026","title":"Step-difference form of a quadratic packing polynomial on a rational sector"},{"data_url":"scopes/NumberTheory/diophantine-approximation.json","id":"NumberTheory/diophantine-approximation","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/index.html","scope":"NumberTheory/diophantine-approximation","status":"can_prove","summary":"Approximating reals by rationals or algebraics, Roth and subspace theorems, linear forms in logarithms, transcendence, uniform distribution, geometry of numbers.","title":"Diophantine approximation and transcendence"},{"data_url":"scopes/NumberTheory/diophantine-approximation/transcendence-irrationality.json","id":"NumberTheory/diophantine-approximation/transcendence-irrationality","kind":"scope","parent":10092,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/index.html","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality","status":"can_prove","summary":"Hermite\u2013Lindemann, Lindemann\u2013Weierstrass, Gelfond\u2013Schneider, Mahler functions, irrationality and linear independence of constants.","title":"Transcendence and irrationality"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/cdt-linear-independent","kind":"node","lean":"MathlibExt.NumberTheory.CalegariDimitrovTangLinearIndependentWanted.NumberTheory.CalegariDimitrovTangLinearIndependent.cdt_linearIndependent","parent":10093,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/index.html#cdt-linear-independent","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality","status":"can_prove","summary":"Let $\\chi : \\mathbb{Z}/3\\mathbb{Z} \\to \\mathbb{C}$ be given by $\\chi(0) = 0$, $\\chi(1) = 1$ and $\\chi(2) = -1$ (the nontrivial Dirichlet character modulo $3$), let $\\zeta$ be the Riemann zeta function and let $L(s, \\chi)$ be the Dirichlet $L$-function of $\\chi$. Then the complex numbers $1$, $\\zeta(2)$ and $L(2, \\chi)$ are linearly independent over $\\mathbb{Q}$. Moreover, let $m$ and $n$ be integers, neither of\u2026","title":"Calegari\u2013Dimitrov\u2013Tang linear independence theorems"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/hermite-lindemann","kind":"node","lean":"MathlibExt.NumberTheory.LindemannWanted.hermite_lindemann","parent":10093,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/index.html#hermite-lindemann","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality","status":"fully_proved","summary":"If $\\alpha$ is a nonzero complex number that is algebraic over $\\mathbb{Q}$, then $e^{\\alpha}$ is transcendental over $\\mathbb{Q}$.","title":"Hermite\u2013Lindemann theorem"},{"data_url":"scopes/NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem.json","id":"NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem","kind":"scope","navigation":"module","parent":10093,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem/index.html","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem","status":"fully_proved","title":"Gelfond\u2013Schneider theorem"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem/gelfond-schneider-complex-exp","kind":"node","lean":"MathlibExt.NumberTheory.TranscendenceWanted.gelfondSchneider_complex_exp","parent":10096,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem/index.html#gelfond-schneider-complex-exp","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem","status":"fully_proved","summary":"Let $a, b \\in \\mathbb{C}$ be algebraic over $\\mathbb{Q}$ with $a \\neq 1$ and $b \\notin \\mathbb{Q}$, and let $z \\in \\mathbb{C}$ satisfy $e^z = a$. Then $e^{bz}$ is transcendental over $\\mathbb{Q}$.","title":"Gelfond\u2013Schneider theorem for every value of a complex power"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem/gelfond-schneider-real-rpow","kind":"node","lean":"MathlibExt.NumberTheory.TranscendenceWanted.gelfondSchneider_real_rpow","parent":10096,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem/index.html#gelfond-schneider-real-rpow","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/gelfond-schneider-theorem","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be real numbers that are algebraic over $\\mathbb{Q}$, with $\\alpha > 0$, $\\alpha \\neq 1$ and $\\beta$ irrational. Then the real power $\\alpha^\\beta$ is transcendental over $\\mathbb{Q}$.","title":"Gelfond\u2013Schneider theorem for real powers"},{"data_url":"scopes/NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem.json","id":"NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem","kind":"scope","navigation":"module","parent":10093,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem/index.html","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem","status":"fully_proved","title":"Lindemann\u2013Weierstrass theorem"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem/lindemann-weierstrass","kind":"node","lean":"MathlibExt.NumberTheory.LindemannWeierstrassWanted.lindemann_weierstrass'","parent":10099,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem/index.html#lindemann-weierstrass","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem","status":"fully_proved","summary":"Let $\\iota$ be a finite index set, let $\\alpha_i$ ($i \\in \\iota$) be pairwise distinct complex numbers that are algebraic over $\\mathbb{Q}$, and let $\\beta_i$ ($i \\in \\iota$) be complex numbers that are algebraic over $\\mathbb{Q}$ and not all zero. Then $\\sum_{i \\in \\iota} \\beta_i e^{\\alpha_i} \\neq 0$.","title":"Lindemann\u2013Weierstrass theorem (universe-polymorphic form)"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem/lindemann-weierstrass-2","kind":"node","lean":"MathlibExt.NumberTheory.LindemannWeierstrassWanted.lindemann_weierstrass","parent":10099,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem/index.html#lindemann-weierstrass-2","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/lindemann-weierstrass-theorem","status":"fully_proved","summary":"Let $\\iota$ be a finite index set, let $\\alpha_i$ ($i \\in \\iota$) be pairwise distinct complex numbers that are algebraic over $\\mathbb{Q}$, and let $\\beta_i$ ($i \\in \\iota$) be complex numbers that are algebraic over $\\mathbb{Q}$ and not all zero. Then $\\sum_{i \\in \\iota} \\beta_i e^{\\alpha_i} \\neq 0$.","title":"Lindemann\u2013Weierstrass theorem"},{"data_url":"scopes/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences.json","id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","kind":"scope","navigation":"module","parent":10093,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","title":"Reciprocal sums of doubly exponential sequences"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-remainder","kind":"node","lean":"MetaMathlibExt.pseudoGreedyRemainder","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-remainder","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"fully_proved","summary":"For a real number $r$, the remainder sequence $(x_n)_{n \\ge 0}$ of the pseudo-greedy expansion of $r$ is defined by $x_0 = r$ and $x_{n+1} = x_n - 1/a_n$, where $a_n = \\lfloor x_n^{-1} + 1 + 1/2 \\rfloor$ is the integer nearest to $x_n^{-1} + 1$, with halves rounded up.","title":"Pseudo-greedy remainder sequence"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-expansion","kind":"node","lean":"MetaMathlibExt.pseudoGreedyExpansion","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-expansion","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"fully_proved","summary":"Let $r$ be a real number with pseudo-greedy remainder sequence $(x_n)_{n \\ge 0}$. The pseudo-greedy expansion of $r$ is the integer sequence $a_n = \\lfloor x_n^{-1} + 1 + 1/2 \\rfloor$ for $n \\ge 0$; that is, $a_n$ is the integer nearest to $x_n^{-1} + 1$, with halves rounded up.","title":"Pseudo-greedy expansion"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-gap","kind":"node","lean":"MetaMathlibExt.pseudoGreedyGap","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-gap","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"fully_proved","summary":"Let $r$ be a real number, and let $(x_n)$ be the remainder sequence and $(a_n)$ the pseudo-greedy expansion of $r$. The gap sequence of $r$ is the real sequence $\\varepsilon_n = x_n^{-1} + 1 - a_n$ for $n \\ge 0$.","title":"Gap sequence of the pseudo-greedy expansion"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-remainder-succ","kind":"node","lean":"MetaMathlibExt.pseudoGreedyRemainder_succ","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-remainder-succ","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"fully_proved","summary":"For every real number $r$ and every $n \\ge 0$, the remainder sequence $(x_n)$ and the pseudo-greedy expansion $(a_n)$ of $r$ satisfy $x_{n+1} = x_n - 1/a_n$.","title":"Remainder recursion"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-expansion-one-zero","kind":"node","lean":"MetaMathlibExt.pseudoGreedyExpansion_one_zero","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-expansion-one-zero","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"fully_proved","summary":"The first term $a_0$ of the pseudo-greedy expansion of $1$ equals $2$, the first Sylvester number.","title":"The pseudo-greedy expansion of 1 starts with 2"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-expansion-one-one","kind":"node","lean":"MetaMathlibExt.pseudoGreedyExpansion_one_one","parent":10102,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second term of the pseudo-greedy expansion of "],["math","\\(1\\)"]],"source":"Second term of the pseudo-greedy expansion of $1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-expansion-one-one","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"fully_proved","summary":"The pseudo-greedy expansion $(a_n)_{n \\ge 0}$ of $1$ satisfies $a_1 = 3$.","title":"Second term of the pseudo-greedy expansion of $1$"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-gap-one-zero","kind":"node","lean":"MetaMathlibExt.pseudoGreedyGap_one_zero","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-gap-one-zero","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"fully_proved","summary":"The first term $\\varepsilon_0$ of the gap sequence of the pseudo-greedy expansion of $1$ is $0$: the first remainder is $x_0 = 1$ and the first term of the expansion is $a_0 = 2$, so $\\varepsilon_0 = x_0^{-1} + 1 - a_0 = 1 + 1 - 2 = 0$.","title":"The first gap of the expansion of 1 vanishes"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/eq-round-inv-remainder-add","kind":"node","lean":"MetaMathlibExt.eq_round_inv_remainder_add","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#eq-round-inv-remainder-add","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $\\beta \\ge 0$ and $r$ be real numbers, and let $(a_n)_{n \\ge 0}$ be a sequence of positive integers with $|a_n^2 / a_{n+1} - \\beta| \\le 1/3$ for all $n$ and $\\sum_n 1 / a_n = r$. Write $x_n = r - \\sum_{k\\lt n} 1 / a_k$. Then for every $n$ with $a_n \\ge 8 (\\beta + 1/3)^2$ we have $a_n = \\lfloor x_n^{-1} + \\beta + 1/2 \\rfloor$, that is, $a_n$ is the integer nearest to $x_n^{-1} + \\beta$, with halves rounded up.\u2026","title":"Recovering a doubly exponential sequence from its reciprocal sum"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/eq-sylvester-of-ratio-mem-of-has-sum-one","kind":"node","lean":"MetaMathlibExt.eq_sylvester_of_ratio_mem_of_hasSum_one","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#eq-sylvester-of-ratio-mem-of-has-sum-one","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be a sequence of positive integers with $2/3 \\le a_n^2 / a_{n+1} \\le 4/3$ for all $n$ and $\\sum_n 1 / a_n = 1$. Then $a_n = s_n(1)$ for all $n \\ge 0$, where $(s_n(1))_{n \\ge 0}$ is the generalized Sylvester sequence with parameter $1$, that is, the Sylvester sequence $2, 3, 7, 43, \\ldots$.","title":"Characterization of the Sylvester sequence"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/eq-fib-two-pow-of-ratio-le-of-has-sum-mi","kind":"node","lean":"MetaMathlibExt.eq_fib_two_pow_of_ratio_le_of_hasSum_millin","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#eq-fib-two-pow-of-ratio-le-of-has-sum-mi","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be a sequence of positive integers with $a_n^2 / a_{n+1} \\le 2/3$ for all $n$ and $\\sum_n 1 / a_n = (5 - \\sqrt 5) / 2$. Then $a_n = F_{2^{n+1}}$ for all $n \\ge 0$, where $F_k$ is the $k$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$); that is, $a$ is the sequence $F_2, F_4, F_8, \\ldots$.","title":"Characterization of the denominators of the Millin series"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/countable-not-type2-irrationality-sequen","kind":"node","lean":"MetaMathlibExt.countable_not_type2IrrationalitySequence_floor_pow_two_pow","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#countable-not-type2-irrationality-sequen","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Doubly exponential floors are Type 2 irrationality sequences for all but countably many bases. For all real numbers $\\alpha > 1$ outside a countable set, the sequence $(\\lfloor \\alpha^{2^{n+1}} \\rfloor)_{n \\ge 0}$ is a Type 2 irrationality sequence: for every sequence $(b_n)_{n \\ge 0}$ of positive integers with $b_n / \\lfloor \\alpha^{2^{n+1}} \\rfloor \\to 1$, the series $\\sum_n 1 / b_n$ does not converge to a\u2026","title":"Doubly exponential floors are Type 2 irrationality sequences for all but countably many bases"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/has-sum-one-div-pseudo-greedy-expansion","kind":"node","lean":"MetaMathlibExt.hasSum_one_div_pseudoGreedyExpansion","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#has-sum-one-div-pseudo-greedy-expansion","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"For every real number $r > 0$, the pseudo-greedy expansion $(a_n)_{n \\ge 0}$ of $r$ satisfies $\\sum_n 1 / a_n = r$.","title":"Convergence of the pseudo-greedy expansion"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-expansion-succ-eq","kind":"node","lean":"MetaMathlibExt.pseudoGreedyExpansion_succ_eq","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-expansion-succ-eq","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $r > 0$ be a real number, and let $(a_n)$ be its pseudo-greedy expansion and $(\\varepsilon_n)$ its gap sequence. Then for every $n \\ge 0$,","title":"Recursion for the pseudo-greedy expansion"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-gap-succ-eq-zero","kind":"node","lean":"MetaMathlibExt.pseudoGreedyGap_succ_eq_zero","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-gap-succ-eq-zero","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $r > 0$ be a real number and let $(\\varepsilon_n)$ be the gap sequence of its pseudo-greedy expansion. For every $n \\ge 0$, if $\\varepsilon_n = 0$, then $\\varepsilon_{n+1} = 0$.","title":"A vanishing gap stays zero"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-expansion-rational-structu","kind":"node","lean":"MetaMathlibExt.pseudoGreedyExpansion_rational_structure","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-expansion-rational-structu","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $p$ and $q$ be positive integers, let $r = p/q$, and let $(x_n)$, $(a_n)$ and $(\\varepsilon_n)$ be the remainder sequence, the pseudo-greedy expansion and the gap sequence of $r$. Put $d_n = q a_0 a_1 \\cdots a_{n-1}$ (so $d_0 = q$). Then there are positive integers $c_n$ and integers $e_n$ ($n \\ge 0$) such that for every $n$ we have $x_n = c_n / d_n$, $\\varepsilon_n = e_n / c_n$, $e_n \\equiv d_n \\pmod{c_n}$,\u2026","title":"Structure of the pseudo-greedy expansion of a rational"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/eventually-pseudo-greedy-gap-eq-zero","kind":"node","lean":"MetaMathlibExt.eventually_pseudoGreedyGap_eq_zero","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#eventually-pseudo-greedy-gap-eq-zero","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $r$ be a positive rational number and let $(\\varepsilon_n)_{n \\ge 0}$ be the gap sequence of the pseudo-greedy expansion of $r$. Suppose that $\\liminf_n \\varepsilon_n \\prod_{k\\lt n} (1 - \\varepsilon_k) \\ge 0$, that is, for every $\\delta > 0$ we have $\\varepsilon_n \\prod_{k\\lt n} (1 - \\varepsilon_k) \\ge -\\delta$ for all sufficiently large $n$; or that $\\varepsilon_n \\ge 0$ for all sufficiently large $n$. Then\u2026","title":"Eventual vanishing of the pseudo-greedy gaps of a rational number"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/exists-tail-eq-pseudo-greedy-expansion","kind":"node","lean":"MetaMathlibExt.exists_tail_eq_pseudoGreedyExpansion","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#exists-tail-eq-pseudo-greedy-expansion","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be a sequence of positive integers with $a_n^2 / a_{n+1} \\to 1$ such that $\\sum_n 1 / a_n$ converges. Then there is a natural number $N$ such that, with $r = \\sum_{k \\ge 0} 1 / a_{N+k}$, the tail $(a_{N+n})_{n \\ge 0}$ is the pseudo-greedy expansion of $r$, and the gap sequence of this expansion tends to $0$.","title":"Tails that are pseudo-greedy expansions"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/eventually-sylvester-recurrence-of-ratio","kind":"node","lean":"MetaMathlibExt.eventually_sylvester_recurrence_of_rational_sum","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#eventually-sylvester-recurrence-of-ratio","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be a sequence of positive integers with $a_n^2 / a_{n+1} \\to 1$ such that $\\sum_n 1 / a_n$ converges to a rational number. Suppose that $\\liminf_n (a_0 \\cdots a_{n-1} / a_n)(1 - a_n^2 / a_{n+1}) \\ge 0$, that is, for every $\\delta > 0$ this quantity is at least $-\\delta$ for all sufficiently large $n$; or that $a_{n+1} \\ge a_n^2 - a_n + 1$ for all sufficiently large $n$. Then\u2026","title":"Erd\u0151s\u2013Straus and Badea criterion for the Sylvester recurrence"},{"id":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/pseudo-greedy-conjecture-iff-erdos-graha","kind":"node","lean":"MetaMathlibExt.pseudoGreedy_conjecture_iff_erdos_graham_question","parent":10102,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences/index.html#pseudo-greedy-conjecture-iff-erdos-graha","scope":"NumberTheory/diophantine-approximation/transcendence-irrationality/reciprocal-sums-of-doubly-exponential-sequences","status":"can_prove","summary":"The following are equivalent. (1) For every positive rational number $r$ whose pseudo-greedy gap sequence $(\\varepsilon_n)_{n \\ge 0}$ tends to $0$, we have $\\varepsilon_n = 0$ for all sufficiently large $n$. (2) Every sequence $(a_n)_{n \\ge 0}$ of positive integers with $a_n^2 / a_{n+1} \\to 1$ such that $\\sum_n 1 / a_n$ converges to a rational number satisfies $a_{n+1} = a_n^2 - a_n + 1$ for all sufficiently large\u2026","title":"Koizumi's conjecture is equivalent to the Erd\u0151s\u2013Graham question"},{"data_url":"scopes/NumberTheory/diophantine-approximation/roth-linear-forms.json","id":"NumberTheory/diophantine-approximation/roth-linear-forms","kind":"scope","parent":10092,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/index.html","scope":"NumberTheory/diophantine-approximation/roth-linear-forms","status":"can_prove","summary":"Thue\u2013Siegel\u2013Roth, Ridout, Schmidt subspace, Baker, Baker\u2013W\u00fcstholz, Matveev bounds, Baker\u2013Davenport reduction, approximation by algebraic numbers.","title":"Roth's theorem and linear forms in logarithms"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/matveev-lower-bound-linear-forms-logarit","kind":"node","lean":"MetaMathlibExt.matveev_lower_bound_linear_forms_logarithms","parent":10122,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/index.html#matveev-lower-bound-linear-forms-logarit","scope":"NumberTheory/diophantine-approximation/roth-linear-forms","status":"can_prove","summary":"Let $L$ be a number field of degree $D = [L : \\mathbb{Q}]$ with a field embedding $f : L \\to \\mathbb{R}$, let $s \\ge 1$ be a natural number, and let $\\eta_1, \\dots, \\eta_s \\in L$ satisfy $\\eta_i \\ne 1$ and $f(\\eta_i) > 0$ for every $i$. Let $d_1, \\dots, d_s$ be nonzero integers and let $B$ be a real number with $B \\ge 1$ and $|d_i| \\le B$ for every $i$. Let $A_1, \\dots, A_s$ be real numbers with\u2026","title":"Matveev's lower bound for linear forms in logarithms (real case)"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/schmidt-subspace","kind":"node","lean":"MathlibExt.NumberTheory.SchmidtSubspaceWanted.NumberTheory.SchmidtSubspace.schmidt_subspace","parent":10122,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/index.html#schmidt-subspace","scope":"NumberTheory/diophantine-approximation/roth-linear-forms","status":"can_prove","summary":"Let $\\sigma$ be a finite set with at least two elements, and for $i \\in \\sigma$ let $L_i(x) = \\sum_{j \\in \\sigma} a_{ij} x_j$ be linear forms whose coefficients $a_{ij} \\in \\mathbb{C}$ are algebraic over $\\mathbb{Q}$ and whose coefficient vectors $(a_{ij})_{j \\in \\sigma}$ are linearly independent over $\\mathbb{C}$. Let $\\varepsilon > 0$ be real. Then there is a finite set $S$ of nonzero vectors in\u2026","title":"Schmidt's subspace theorem"},{"data_url":"scopes/NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de.json","id":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de","kind":"scope","navigation":"module","parent":10122,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/index.html","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de","status":"can_prove","title":"Approximation by algebraic numbers of bounded degree"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/height","kind":"node","lean":"MathlibExt.NumberTheory.OnApproximationOfRealsWanted.OnApproximationOfReals.height","parent":10125,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/index.html#height","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de","status":"fully_proved","summary":"Let $\\alpha$ be a real algebraic number, let $p$ be its monic minimal polynomial over $\\mathbb{Q}$, and let $D$ be the least common multiple of the denominators of the nonzero coefficients of $p$. The height $H(\\alpha)$ is the largest absolute value of a coefficient of the integer polynomial $D p$. For a real number that is not algebraic over $\\mathbb{Q}$, the height is $0$.","title":"Height of a real algebraic number"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/algebraic-deg-le","kind":"node","lean":"MathlibExt.NumberTheory.OnApproximationOfRealsWanted.OnApproximationOfReals.AlgebraicDegLE","parent":10125,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/index.html#algebraic-deg-le","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de","status":"fully_proved","summary":"For a natural number $n$, this is the type of real numbers $\\alpha$ that are algebraic over $\\mathbb{Q}$ and whose minimal polynomial over $\\mathbb{Q}$ has degree at most $n$.","title":"Real algebraic numbers of bounded degree"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/stars","kind":"node","lean":"MathlibExt.NumberTheory.OnApproximationOfRealsWanted.OnApproximationOfReals.\u03c9_stars","parent":10125,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/index.html#stars","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de","status":"fully_proved","summary":"Let $n$ be a natural number, $\\xi$ a real number, and $H(\\alpha)$ the height of a real algebraic number $\\alpha$. This is the set of real numbers $\\omega > 0$ for which there are infinitely many real algebraic numbers $\\alpha$ of degree at most $n$ with $0 < |\\xi - \\alpha| \\le H(\\alpha)^{-\\omega - 1}$.","title":"Approximation exponents admitting infinitely many solutions"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/classical-exponent","kind":"node","lean":"MathlibExt.NumberTheory.OnApproximationOfRealsWanted.OnApproximationOfReals.classicalExponent","parent":10125,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/index.html#classical-exponent","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de","status":"fully_proved","summary":"For a natural number $n$ and a real number $\\xi$, let $H(\\alpha)$ denote the height of a real algebraic number $\\alpha$. The classical exponent $\\omega^*_n(\\xi)$ is the supremum in $[0, \\infty]$ of the set of real numbers $\\omega > 0$ for which there are infinitely many real algebraic numbers $\\alpha$ of degree at most $n$ with $0 < |\\xi - \\alpha| \\le H(\\alpha)^{-\\omega - 1}$. It is $0$ when this set is empty.","title":"Classical exponent of approximation by algebraic numbers"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/theorem-1-1","kind":"node","lean":"MathlibExt.NumberTheory.OnApproximationOfRealsWanted.OnApproximationOfReals.theorem_1_1","parent":10125,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de/index.html#theorem-1-1","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/approximation-by-algebraic-numbers-of-bounded-de","status":"can_prove","summary":"Let $n \\ge 2$ be an integer and let $a = 1/(2 - \\log 2)$. For every real number $\\xi$ that is transcendental over $\\mathbb{Q}$, the classical exponent of approximation to $\\xi$ by real algebraic numbers of degree at most $n$ satisfies $\\omega^*_n(\\xi) \\ge a n$.","title":"Po\u00ebls' lower bound for the classical exponent"},{"data_url":"scopes/NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction.json","id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction","kind":"scope","navigation":"module","parent":10122,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/index.html","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction","status":"fully_proved","title":"Baker\u2013Davenport reduction"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/nearest-integer-reduction-core","kind":"node","lean":"MetaMathlibExt.nearestInteger_reduction_core","parent":10131,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/index.html#nearest-integer-reduction-core","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction","status":"fully_proved","summary":"For a real number $x$ write $\\lVert x \\rVert = |x - \\operatorname{round}(x)|$ for the distance from $x$ to a nearest integer. Let $q$, $M$, $u$ be natural numbers with $u \\le M$, let $v$ be an integer, and let $\\kappa$ and $\\mu$ be real numbers. If $\\varepsilon = \\lVert \\mu q \\rVert - M \\lVert \\kappa q \\rVert$, then $\\varepsilon \\le q \\, |u\\kappa - v + \\mu|$.","title":"Nearest-integer estimate in the Baker\u2013Davenport reduction"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/baker-davenport-reduction-absolute-gener","kind":"node","lean":"MetaMathlibExt.bakerDavenport_reduction_absolute_general","parent":10131,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/index.html#baker-davenport-reduction-absolute-gener","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction","status":"fully_proved","summary":"For a real number $x$ write $\\lVert x \\rVert = |x - \\operatorname{round}(x)|$ for the distance from $x$ to a nearest integer. Let $M$ and $q$ be natural numbers with $q > 0$, and let $\\kappa$, $\\mu$, $A$, $B$ be real numbers with $A > 0$ and $B > 1$. Put $\\varepsilon = \\lVert \\mu q \\rVert - M \\lVert \\kappa q \\rVert$ and suppose that $\\varepsilon > 0$. Then there are no natural numbers $r$, $s$, $w$ with $r \\le M$,\u2026","title":"Baker\u2013Davenport reduction for an arbitrary denominator, absolute-value form"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/baker-davenport-reduction-absolute","kind":"node","lean":"MetaMathlibExt.bakerDavenport_reduction_absolute","parent":10131,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/index.html#baker-davenport-reduction-absolute","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction","status":"fully_proved","summary":"For a real number $x$ write $\\lVert x \\rVert = |x - \\operatorname{round}(x)|$ for the distance from $x$ to a nearest integer. Let $M$ be a positive integer and $\\kappa$ an irrational real number, and let $q$ be the denominator of the $j$-th convergent of the continued fraction of $\\kappa$ for some natural number $j$, with $q > 6M$. Let $A$, $B$, $\\mu$ be real numbers with $A > 0$ and $B > 1$. Put\u2026","title":"Baker\u2013Davenport reduction lemma, absolute-value form"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/baker-davenport-reduction-one-sided-gene","kind":"node","lean":"MetaMathlibExt.bakerDavenport_reduction_oneSided_general","parent":10131,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/index.html#baker-davenport-reduction-one-sided-gene","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction","status":"fully_proved","summary":"For a real number $x$ write $\\lVert x \\rVert = |x - \\operatorname{round}(x)|$ for the distance from $x$ to a nearest integer. Let $N$ and $q$ be natural numbers with $q > 0$, and let $\\gamma$, $\\mu$, $A$, $B$ be real numbers with $A > 0$ and $B > 1$. Put $\\xi = \\lVert \\mu q \\rVert - N \\lVert \\gamma q \\rVert$ and suppose that $\\xi > 0$. Then there are no natural numbers $u$, $v$, $w$ with $u \\le N$,\u2026","title":"Baker\u2013Davenport reduction for an arbitrary denominator, one-sided form"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/baker-davenport-reduction-one-sided","kind":"node","lean":"MetaMathlibExt.bakerDavenport_reduction_oneSided","parent":10131,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction/index.html#baker-davenport-reduction-one-sided","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-davenport-reduction","status":"fully_proved","summary":"For a real number $x$ write $\\lVert x \\rVert = |x - \\operatorname{round}(x)|$ for the distance from $x$ to a nearest integer. Let $N$ be a positive integer and $\\gamma$ an irrational real number, and let $q$ be the denominator of the $j$-th convergent of the continued fraction of $\\gamma$ for some natural number $j$, with $q > 6N$. Let $A$, $B$, $\\mu$ be real numbers with $A > 0$ and $B > 1$. Put\u2026","title":"Baker\u2013Davenport reduction lemma, one-sided form"},{"data_url":"scopes/NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem.json","id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem","kind":"scope","navigation":"module","parent":10122,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem/index.html","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem","status":"can_prove","title":"Baker\u2013W\u00fcstholz theorem"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem/c","kind":"node","lean":"MathlibExt.NumberTheory.BakerWustholzWanted.BakerWustholz.C","parent":10137,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem/index.html#c","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem","status":"fully_proved","summary":"For natural numbers $n$ and $d$, the Baker\u2013W\u00fcstholz constant is the real number $C(n, d) = 18 \\cdot (n + 1)! \\cdot n^{n+1} \\cdot (32 d)^{n+2} \\cdot \\log(2nd)$.","title":"The Baker\u2013W\u00fcstholz constant"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem/modified-height","kind":"node","lean":"MathlibExt.NumberTheory.BakerWustholzWanted.BakerWustholz.modifiedHeight","parent":10137,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem/index.html#modified-height","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem","status":"fully_proved","summary":"Let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$ and let $\\varphi : K \\to \\mathbb{C}$ be a ring homomorphism. For $\\alpha \\in K$, the modified height of $\\alpha$ with respect to $\\varphi$ is","title":"Modified height"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem/baker-wustholz-linear-forms-logs","kind":"node","lean":"MathlibExt.NumberTheory.BakerWustholzWanted.bakerWustholz_linearForms_logs","parent":10137,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem/index.html#baker-wustholz-linear-forms-logs","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/baker-w-stholz-theorem","status":"can_prove","summary":"Let $n \\ge 1$, let $K$ be a number field of degree $d = [K : \\mathbb{Q}]$, let $\\varphi : K \\to \\mathbb{C}$ be a ring homomorphism, and let $\\alpha_1, \\dots, \\alpha_n$ be nonzero elements of $K$. Let $b_1, \\dots, b_n$ be integers and let $B \\ge 2$ be a natural number with $|b_i| \\le B$ for every $i$. With $\\log$ the principal branch of the complex logarithm, suppose that\u2026","title":"Baker\u2013W\u00fcstholz theorem on linear forms in logarithms"},{"data_url":"scopes/NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem.json","id":"NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem","kind":"scope","navigation":"module","parent":10122,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem/index.html","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem","status":"fully_proved","title":"Thue\u2013Siegel\u2013Roth theorem"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem/finite-of-divisible-approximations","kind":"node","lean":"MathlibExt.NumberTheory.RothAuxiliary.finite_of_divisible_approximations","parent":10141,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem/index.html#finite-of-divisible-approximations","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a real algebraic number, let $0 < \\varepsilon \\le 1$, and let $S$ be a finite index set carrying, for each $l \\in S$, a natural number $p_l$ and reals $\\mu_l, \\nu_l \\ge 0$ with $\\sigma = \\sum_{l \\in S} (\\mu_l + \\nu_l)$ less than $2 + \\varepsilon$. Then there are only finitely many rationals $q$, written in lowest terms with numerator $n$ and denominator $d \\ge 1$, such that\u2026","title":"Roth's theorem with divisibility conditions"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem/ridout-padic-roth","kind":"node","lean":"MetaMathlibExt.ridout_padic_roth","parent":10141,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ridout's "],["math","\\(p\\)"],["text","-adic extension of Roth's theorem"]],"source":"Ridout's $p$-adic extension of Roth's theorem"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem/index.html#ridout-padic-roth","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem","status":"fully_proved","summary":"Let $\\xi$ be an irrational real algebraic number, let $\\varepsilon > 0$, and let $S$ be a finite set of primes. For a rational number $r$, write $r = p/q$ in lowest terms with $p \\in \\mathbb{Z}$ and $q \\ge 1$, and let $H(r) = \\max(|p|, q)$ be its height. For a prime $\\ell$, let $|\\cdot|_\\ell$ be the $\\ell$-adic absolute value on $\\mathbb{Q}$, normalized by $|\\ell|_\\ell = \\ell^{-1}$. Then only finitely many\u2026","title":"Ridout's $p$-adic extension of Roth's theorem"},{"id":"NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem/thue-siegel-roth","kind":"node","lean":"MathlibExt.NumberTheory.ThueSiegelRothWanted.thue_siegel_roth","parent":10141,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem/index.html#thue-siegel-roth","scope":"NumberTheory/diophantine-approximation/roth-linear-forms/thue-siegel-roth-theorem","status":"fully_proved","summary":"Let $\\alpha$ be an irrational real number that is algebraic over $\\mathbb{Q}$, and let $\\varepsilon > 0$ be real. Then only finitely many rational numbers $q$ satisfy $|\\alpha - q| < 1 / b^{2 + \\varepsilon}$, where $b$ is the denominator of $q$ in lowest terms.","title":"Thue\u2013Siegel\u2013Roth theorem"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric","kind":"scope","parent":10092,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric","status":"can_prove","summary":"Weyl equidistribution, three-gap theorem, Duffin\u2013Schaeffer, normal numbers, Beatty sequences, discrepancy, lattice points and Minkowski-type theorems.","title":"Uniform distribution and metric approximation"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/weyl-criterion","kind":"node","lean":"MetaMathlibExt.weyl_criterion","parent":10145,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/index.html#weyl-criterion","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric","status":"fully_proved","summary":"Let $(x_n)_{n \\ge 0}$ be a sequence of real numbers and write $\\operatorname{frac}(t) = t - \\lfloor t \\rfloor$ for the fractional part of a real number $t$. Then the following are equivalent: (i) for all real numbers $a < b$ with $0 \\le a$ and $b \\le 1$, $\\frac{1}{N} \\#\\{n < N : a \\le \\operatorname{frac}(x_n) < b\\} \\to b - a$ as $N \\to \\infty$; (ii) for every nonzero integer $h$,\u2026","title":"Weyl's equidistribution criterion"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","title":"Beatty sequences"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-term","kind":"node","lean":"MetaMathlibExt.beattyTerm","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-term","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"fully_proved","summary":"For a positive integer $n$ and a real number $\\alpha$, the Beatty term $b_n(\\alpha)$ is the natural number $\\max(\\lfloor n/\\alpha \\rfloor, 0)$: the $n$-th term $\\lfloor n \\cdot (1/\\alpha) \\rfloor$ of the Beatty sequence of slope $1/\\alpha$, truncated at $0$. Division by zero is read as $0$, so $b_n(0) = 0$.","title":"Beatty term"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-term-eq-floor","kind":"node","lean":"MetaMathlibExt.beattyTerm_eq_floor","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-term-eq-floor","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"fully_proved","summary":"For every positive integer $n$ and every real number $\\alpha$, the Beatty term $b_n(\\alpha)$ equals the natural-number floor of $n/\\alpha$: the largest natural number $m$ with $m \\le n/\\alpha$ when $n/\\alpha \\ge 0$, and $0$ when $n/\\alpha < 0$. Division by zero is read as $0$, so $n/0 = 0$.","title":"Beatty term as a natural-number floor"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/is-almost-beatty-sequence","kind":"node","lean":"MetaMathlibExt.IsAlmostBeattySequence","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#is-almost-beatty-sequence","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"fully_proved","summary":"Let $\\alpha$ be a real number. A sequence $(s_n)_{n \\ge 1}$ of natural numbers indexed by the positive integers is an almost Beatty sequence for $\\alpha$ when $0 < \\alpha < 1$, the sequence is strictly increasing, $s_n > 0$ for every $n \\ge 1$, and there is a natural number $B$ with $|s_n - b_n(\\alpha)| \\le B$ for every $n \\ge 1$, where $b_n(\\alpha)$ is the Beatty term, equal to $\\lfloor n/\\alpha \\rfloor$ once\u2026","title":"Almost Beatty sequence"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/is-positive-nat-three-way-partition","kind":"node","lean":"MetaMathlibExt.IsPositiveNatThreeWayPartition","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#is-positive-nat-three-way-partition","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"fully_proved","summary":"Three sequences $s_1, s_2, s_3$ of natural numbers indexed by the positive integers partition the positive integers when their ranges are pairwise disjoint, that is, $s_i(m) \\ne s_j(n)$ for all $i \\ne j$ and all positive integers $m$ and $n$, and the union of their ranges is exactly the set of positive integers: a natural number $k$ is a value of some $s_i$ if and only if $k > 0$.","title":"Three-way partition of the positive integers"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/is-almost-beatty-partition","kind":"node","lean":"MetaMathlibExt.IsAlmostBeattyPartition","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#is-almost-beatty-partition","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"fully_proved","summary":"Given real numbers $\\alpha_1, \\alpha_2, \\alpha_3$, three sequences $s_1, s_2, s_3$ of natural numbers indexed by the positive integers form an almost Beatty partition when, for each $i$, the sequence $s_i$ is an almost Beatty sequence for $\\alpha_i$, and the three sequences form a three-way partition of the positive integers: their ranges are pairwise disjoint and their union is the set of positive integers.","title":"Almost Beatty partition"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-term-is-almost-beatty-sequence","kind":"node","lean":"MetaMathlibExt.beattyTerm_isAlmostBeattySequence","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-term-is-almost-beatty-sequence","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"fully_proved","summary":"For every real number $\\alpha$ with $0 < \\alpha < 1$, the sequence of Beatty terms $n \\mapsto b_n(\\alpha) = \\lfloor n/\\alpha \\rfloor$, indexed by the positive integers $n$, is an almost Beatty sequence for $\\alpha$.","title":"Beatty sequences are almost Beatty"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/uspensky-no-partition-into-three-or-more","kind":"node","lean":"MetaMathlibExt.uspensky_no_partition_into_three_or_more_beatty_sequences","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#uspensky-no-partition-into-three-or-more","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"fully_proved","summary":"For a real number $\\alpha$ with $0 < \\alpha < 1$, let $B_\\alpha = \\{\\lfloor n/\\alpha \\rfloor : n \\text{ an integer}, n \\ge 1\\}$ be its Beatty sequence. Let $k \\ge 3$ and let $\\alpha_1, \\dots, \\alpha_k$ be real numbers with $0 < \\alpha_i < 1$ for every $i$. Then $B_{\\alpha_1}, \\dots, B_{\\alpha_k}$ do not partition the positive integers: it is not the case both that $B_{\\alpha_i} \\cap B_{\\alpha_j} = \\emptyset$ for\u2026","title":"Uspensky's theorem on Beatty partitions"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-complementary-iff-irrational-inv","kind":"node","lean":"MetaMathlibExt.beatty_complementary_iff_irrational_inv_add_inv","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-complementary-iff-irrational-inv","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $\\alpha$ and $\\beta$ be positive real numbers. The set of the numbers $\\lfloor k\\alpha\\rfloor$ for positive integers $k$ and the set of the numbers $\\lfloor k\\beta\\rfloor$ for positive integers $k$ are disjoint and their union is the set of positive integers if and only if $\\alpha$ is irrational and $\\alpha^{-1} + \\beta^{-1} = 1$.","title":"Beatty's theorem and its converse"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-card-compl-between-eq-floor","kind":"node","lean":"MetaMathlibExt.beatty_card_compl_between_eq_floor","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(\\beta\\)"],["text","-Beatty terms between "],["math","\\(\\lfloor n\\alpha\\rfloor\\)"],["text"," and "],["math","\\(\\lfloor n\\beta\\rfloor\\)"]],"source":"Counting $\\beta$-Beatty terms between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-card-compl-between-eq-floor","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $\\alpha$ be an irrational number with $1 < \\alpha < 2$, let $\\beta$ be a real number with $\\alpha^{-1} + \\beta^{-1} = 1$, and let $B$ be the set of the numbers $\\lfloor k\\beta\\rfloor$ for positive integers $k$. Then for every positive integer $n$, the number of elements of $B$ in the open interval $(\\lfloor n\\alpha\\rfloor, \\lfloor n\\beta\\rfloor)$ equals $\\lfloor n(2 - \\alpha)\\rfloor$.","title":"Counting $\\beta$-Beatty terms between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-card-self-between-eq-floor","kind":"node","lean":"MetaMathlibExt.beatty_card_self_between_eq_floor","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(\\alpha\\)"],["text","-Beatty terms between "],["math","\\(\\lfloor n\\alpha\\rfloor\\)"],["text"," and "],["math","\\(\\lfloor n\\beta\\rfloor\\)"]],"source":"Counting $\\alpha$-Beatty terms between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-card-self-between-eq-floor","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $\\alpha$ be an irrational number with $1 < \\alpha < 2$, let $\\beta$ be a real number with $\\alpha^{-1} + \\beta^{-1} = 1$, and let $A$ be the set of the numbers $\\lfloor k\\alpha\\rfloor$ for positive integers $k$. Then for every positive integer $n$, the number of elements of $A$ in the open interval $(\\lfloor n\\alpha\\rfloor, \\lfloor n\\beta\\rfloor)$ equals $\\lfloor n(\\beta - 2)\\rfloor$.","title":"Counting $\\alpha$-Beatty terms between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-floor-inv-sub-eq-floor-two-sub","kind":"node","lean":"MetaMathlibExt.beatty_floor_inv_sub_eq_floor_two_sub","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polanco's floor identity for "],["math","\\(\\lfloor (2 - \\alpha) n \\rfloor\\)"]],"source":"Polanco's floor identity for $\\lfloor (2 - \\alpha) n \\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-floor-inv-sub-eq-floor-two-sub","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $\\alpha$ be an irrational real number with $1 < \\alpha < 2$, let $\\beta$ be a real number with $\\alpha^{-1} + \\beta^{-1} = 1$, and write $a_m = \\lfloor m\\alpha \\rfloor$ and $b_m = \\lfloor m\\beta \\rfloor$ for integers $m$ (so $b_0 = 0$). Then for every positive integer $n$,","title":"Polanco's floor identity for $\\lfloor (2 - \\alpha) n \\rfloor$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-floor-inv-sub-eq-floor-sub-two","kind":"node","lean":"MetaMathlibExt.beatty_floor_inv_sub_eq_floor_sub_two","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polanco's floor identity for "],["math","\\(\\lfloor (\\beta - 2) n \\rfloor\\)"]],"source":"Polanco's floor identity for $\\lfloor (\\beta - 2) n \\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-floor-inv-sub-eq-floor-sub-two","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $\\alpha$ be an irrational real number with $1 < \\alpha < 2$, let $\\beta$ be a real number with $\\alpha^{-1} + \\beta^{-1} = 1$, and write $a_n = \\lfloor n\\alpha \\rfloor$ and $b_n = \\lfloor n\\beta \\rfloor$. Then for every positive integer $n$,","title":"Polanco's floor identity for $\\lfloor (\\beta - 2) n \\rfloor$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-card-compl-between-is-beatty","kind":"node","lean":"MetaMathlibExt.beatty_card_compl_between_isBeatty","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\beta\\)"],["text","-Beatty counts between "],["math","\\(\\lfloor n\\alpha\\rfloor\\)"],["text"," and "],["math","\\(\\lfloor n\\beta\\rfloor\\)"],["text"," form a Beatty sequence"]],"source":"The $\\beta$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$ form a Beatty sequence"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-card-compl-between-is-beatty","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"The $\\beta$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$ form a Beatty sequence. Let $\\alpha$ be an irrational number with $1 < \\alpha < 2$, let $\\beta$ be a real number with $\\alpha^{-1} + \\beta^{-1} = 1$, and let $B$ be the set of the numbers $\\lfloor k\\beta\\rfloor$ for positive integers $k$. Then there is a real number $r$ with $0 < r < 1$ such that for every positive integer $n$,\u2026","title":"The $\\beta$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$ form a Beatty sequence"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-floor-fiber-card-eq-or-eq-succ","kind":"node","lean":"MetaMathlibExt.beatty_floor_fiber_card_eq_or_eq_succ","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicities of the values of "],["math","\\(\\lfloor n\\alpha\\rfloor\\)"],["text"," for "],["math","\\(1/(k + 1) < \\alpha < 1/k\\)"]],"source":"Multiplicities of the values of $\\lfloor n\\alpha\\rfloor$ for $1/(k + 1) < \\alpha < 1/k$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-floor-fiber-card-eq-or-eq-succ","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $k$ be a positive integer and let $\\alpha$ be a real number with $1/(k + 1) < \\alpha < 1/k$. Then every non-negative integer $t$ equals $\\lfloor n\\alpha\\rfloor$ for exactly $k$ or exactly $k + 1$ positive integers $n$, there are infinitely many $t$ for which it is exactly $k$, and there are infinitely many $t$ for which it is exactly $k + 1$.","title":"Multiplicities of the values of $\\lfloor n\\alpha\\rfloor$ for $1/(k + 1) < \\alpha < 1/k$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-floor-fiber-card-succ-iff","kind":"node","lean":"MetaMathlibExt.beatty_floor_fiber_card_succ_iff","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicity "],["math","\\(k + 1\\)"],["text"," for "],["math","\\(\\alpha\\)"],["text"," is multiplicity "],["math","\\(k\\)"],["text"," for "],["math","\\(\\alpha/(1 - \\alpha)\\)"]],"source":"Multiplicity $k + 1$ for $\\alpha$ is multiplicity $k$ for $\\alpha/(1 - \\alpha)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-floor-fiber-card-succ-iff","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $k$ be a positive integer, let $\\alpha$ be a real number with $1/(k + 1) < \\alpha < 1/k$, and let $\\beta = \\alpha/(1 - \\alpha)$. Then a non-negative integer $t$ equals $\\lfloor n\\alpha\\rfloor$ for exactly $k + 1$ positive integers $n$ if and only if it equals $\\lfloor n\\beta\\rfloor$ for exactly $k$ positive integers $n$.","title":"Multiplicity $k + 1$ for $\\alpha$ is multiplicity $k$ for $\\alpha/(1 - \\alpha)$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-floor-fiber-card-sets-is-beatty","kind":"node","lean":"MetaMathlibExt.beatty_floor_fiber_card_sets_isBeatty","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","The values of "],["math","\\(\\lfloor n\\alpha\\rfloor\\)"],["text"," of each multiplicity form Beatty sets"]],"source":"The values of $\\lfloor n\\alpha\\rfloor$ of each multiplicity form Beatty sets"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-floor-fiber-card-sets-is-beatty","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $k$ be a positive integer and let $\\alpha$ be an irrational number with $1/(k + 1) < \\alpha < 1/k$. Then there are real numbers $\\gamma > 0$ and $\\delta > 0$ such that the positive integers $t$ that equal $\\lfloor n\\alpha\\rfloor$ for exactly $k + 1$ positive integers $n$ are exactly the numbers $\\lfloor m\\gamma\\rfloor$ for positive integers $m$, and the positive integers $t$ that equal $\\lfloor n\\alpha\\rfloor$\u2026","title":"The values of $\\lfloor n\\alpha\\rfloor$ of each multiplicity form Beatty sets"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-floor-fiber-card-eq-add-indicator","kind":"node","lean":"MetaMathlibExt.beatty_floor_fiber_card_eq_add_indicator","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicities of "],["math","\\(\\lfloor n\\alpha\\rfloor\\)"],["text"," are "],["math","\\(k\\)"],["text"," plus the indicator of a Beatty set"]],"source":"Multiplicities of $\\lfloor n\\alpha\\rfloor$ are $k$ plus the indicator of a Beatty set"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-floor-fiber-card-eq-add-indicator","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $k$ be a positive integer and let $\\alpha$ be an irrational number with $1/(k + 1) < \\alpha < 1/k$. Then there is a real number $\\delta > 0$ such that, writing $D$ for the set of the numbers $\\lfloor m\\delta\\rfloor$ for positive integers $m$, every non-negative integer $t$ equals $\\lfloor n\\alpha\\rfloor$ for exactly $k + 1$ positive integers $n$ if $t \\in D$, and for exactly $k$ positive integers $n$ if\u2026","title":"Multiplicities of $\\lfloor n\\alpha\\rfloor$ are $k$ plus the indicator of a Beatty set"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-card-compl-between-succ-sub","kind":"node","lean":"MetaMathlibExt.beatty_card_compl_between_succ_sub","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Increments of the "],["math","\\(\\beta\\)"],["text","-Beatty counts between "],["math","\\(\\lfloor n\\alpha\\rfloor\\)"],["text"," and "],["math","\\(\\lfloor n\\beta\\rfloor\\)"]],"source":"Increments of the $\\beta$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-card-compl-between-succ-sub","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Increments of the $\\beta$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$. Let $\\alpha$ be an irrational number with $1 < \\alpha < 2$, let $\\beta$ be a real number with $\\alpha^{-1} + \\beta^{-1} = 1$, let $B$ be the set of the numbers $\\lfloor k\\beta\\rfloor$ for positive integers $k$, and for an integer $m$ let $r(m)$ be the number of elements of $B$ in the open interval\u2026","title":"Increments of the $\\beta$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-card-self-between-succ-sub","kind":"node","lean":"MetaMathlibExt.beatty_card_self_between_succ_sub","parent":10147,"reading_title":{"format":"inline-parts-v1","parts":[["text","Increments of the "],["math","\\(\\alpha\\)"],["text","-Beatty counts between "],["math","\\(\\lfloor n\\alpha\\rfloor\\)"],["text"," and "],["math","\\(\\lfloor n\\beta\\rfloor\\)"]],"source":"Increments of the $\\alpha$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-card-self-between-succ-sub","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Increments of the $\\alpha$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$. Let $\\beta$ be an irrational number with $\\beta > 2$, let $\\alpha$ be a real number with $\\alpha^{-1} + \\beta^{-1} = 1$, let $A$ be the set of the numbers $\\lfloor k\\alpha\\rfloor$ for positive integers $k$, and for an integer $m$ let $c(m)$ be the number of elements of $A$ in the open interval\u2026","title":"Increments of the $\\alpha$-Beatty counts between $\\lfloor n\\alpha\\rfloor$ and $\\lfloor n\\beta\\rfloor$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/beatty-skipping-card-eq-add-indicator","kind":"node","lean":"MetaMathlibExt.beatty_skipping_card_eq_add_indicator","parent":10147,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences/index.html#beatty-skipping-card-eq-add-indicator","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/beatty-sequences","status":"can_prove","summary":"Let $k \\ge 1$ be an integer, let $\\alpha$ be an irrational real number with $(2k - 1)/k < \\alpha < (2k + 1)/(k + 1)$, and let $\\beta$ be a real number with $\\alpha^{-1} + \\beta^{-1} = 1$. Let $A = \\{\\lfloor j\\alpha \\rfloor : j \\ge 1\\}$ and, for each positive integer $n$, let $c_n$ be the number of elements of $A$ strictly between $\\lfloor n\\alpha \\rfloor$ and $\\lfloor n\\beta \\rfloor$. Let $f(x) = 2 - 1/x$, let\u2026","title":"Multiplicities of Polanco's skipping sequence"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts","status":"can_prove","title":"Discrepancy of fractional parts"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/count-q","kind":"node","lean":"MathlibExt.NumberTheory.ErdosSequenceDiscrepancyWanted.CountQ","parent":10168,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/index.html#count-q","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts","status":"fully_proved","summary":"For a sequence of integers $(x_n)_{n \\ge 0}$, a real number $\\alpha$, a natural number $N$ and rationals $p, q$, the number of indices $n < N$ for which the fractional part $\\{\\alpha x_n\\}$ lies in the closed interval $[p, q]$.","title":"Counting fractional parts in a rational interval"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/discrep","kind":"node","lean":"MathlibExt.NumberTheory.ErdosSequenceDiscrepancyWanted.discrep","parent":10168,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/index.html#discrep","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts","status":"fully_proved","summary":"For a sequence of integers $(x_n)_{n \\ge 0}$, a real number $\\alpha$ and a natural number $N$, the discrepancy $D(N)$ is the supremum, over rationals $p, q$ with $0 \\le p \\le q \\le 1$, of $\\lvert A(p, q) - (q - p) N \\rvert$, where $A(p, q)$ is the number of indices $n < N$ for which the fractional part $\\{\\alpha x_n\\}$ lies in the closed interval $[p, q]$.","title":"Discrepancy of fractional parts over rational intervals"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosSequenceDiscrepancyWanted.conjecture","parent":10168,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/index.html#conjecture","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts","status":"fully_proved","summary":"For a strictly increasing sequence of integers $x_0 < x_1 < \\cdots$, a real number $\\alpha$ and a natural number $N$, let $D(N)$ be the supremum, over rationals $p, q$ with $0 \\le p \\le q \\le 1$, of $\\lvert A(p, q) - (q - p) N \\rvert$, where $A(p, q)$ is the number of indices $n < N$ for which the fractional part $\\{\\alpha x_n\\}$ lies in $[p, q]$. The statement is that for every such sequence, for almost every\u2026","title":"Erd\u0151s Problem 992 on the discrepancy of fractional parts"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosSequenceDiscrepancyWanted.conjecture_refuted","parent":10168,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts/index.html#conjecture-refuted","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/discrepancy-of-fractional-parts","status":"can_prove","summary":"For a sequence of integers $(x_n)_{n \\ge 0}$, a real number $\\alpha$ and a natural number $N$, let $D(N)$ be the supremum, over rationals $p, q$ with $0 \\le p \\le q \\le 1$, of $\\lvert A(p, q) - (q - p) N \\rvert$, where $A(p, q)$ is the number of indices $n < N$ for which the fractional part $\\{\\alpha x_n\\}$ lies in $[p, q]$. It is not true that for every strictly increasing sequence of integers\u2026","title":"Erd\u0151s Problem 992 has a negative answer"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"can_prove","title":"Duffin\u2013Schaeffer conjecture"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/inequality-1-7","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.inequality_1_7","parent":10173,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#inequality-1-7","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"fully_proved","summary":"For a function $\\psi$ from the positive integers to the nonnegative reals, a real number $\\alpha$, a natural number $a$ and a positive integer $q$, inequality (1.7) is the condition $|\\alpha - a/q| \\le \\psi(q)/q$.","title":"The approximation inequality (1.7)"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/untitled","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.\ud835\udc9c","parent":10173,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\mathcal{A}\\)"],["text"," of the Duffin\u2013Schaeffer conjecture"]],"source":"The set $\\mathcal{A}$ of the Duffin\u2013Schaeffer conjecture"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#untitled","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"fully_proved","summary":"For a function $\\psi$ from the positive integers to the nonnegative reals, $\\mathcal{A}(\\psi)$ is the set of real numbers $\\alpha \\in [0, 1]$ for which inequality (1.7), $|\\alpha - a/q| \\le \\psi(q)/q$, has infinitely many solutions $(a, q)$ with $a$ a natural number, $q$ a positive integer and $\\gcd(a, q) = 1$.","title":"The set $\\mathcal{A}$ of the Duffin\u2013Schaeffer conjecture"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/theorem-1","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.theorem_1","parent":10173,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#theorem-1","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"can_prove","summary":"Let $\\psi$ be a function from the positive integers to the nonnegative reals such that the series $\\sum_{q \\ge 1} \\psi(q)\\varphi(q)/q$ diverges, where $\\varphi$ is Euler's totient function. Let $\\mathcal{A}(\\psi)$ be the set of real numbers $\\alpha \\in [0, 1]$ for which $|\\alpha - a/q| \\le \\psi(q)/q$ has infinitely many solutions with $a$ a natural number, $q$ a positive integer and $\\gcd(a, q) = 1$. Then\u2026","title":"Koukoulopoulos\u2013Maynard Theorem 1 (Duffin\u2013Schaeffer conjecture)"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/untitled-2","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.\ud835\udca6","parent":10173,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\mathcal{K}\\)"],["text"," of Catlin's conjecture"]],"source":"The set $\\mathcal{K}$ of Catlin's conjecture"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#untitled-2","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"fully_proved","summary":"For a function $\\psi$ from the positive integers to the nonnegative reals, $\\mathcal{K}(\\psi)$ is the set of real numbers $\\alpha \\in [0, 1]$ for which inequality (1.7), $|\\alpha - a/q| \\le \\psi(q)/q$, has infinitely many solutions $(a, q)$ with $a$ a natural number and $q$ a positive integer such that $a \\le q$.","title":"The set $\\mathcal{K}$ of Catlin's conjecture"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/star","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.\u03c8_star","parent":10173,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function "],["math","\\(\\psi^*\\)"],["text"," of Catlin's conjecture"]],"source":"The function $\\psi^*$ of Catlin's conjecture"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#star","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"fully_proved","summary":"The function $\\psi^*$ of Catlin's conjecture. For a function $\\psi$ from the positive integers to the nonnegative reals, $\\psi^*$ sends each positive integer $q$ to $\\psi^*(q) = \\varphi(q) \\cdot \\sup \\{\\psi(n)/n : n \\text{ a positive integer with } q \\mid n\\}$, where $\\varphi$ is Euler's totient function and the supremum is taken in $[0, \\infty]$.","title":"The function $\\psi^*$ of Catlin's conjecture"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/theorem-2-a","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.theorem_2_a","parent":10173,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#theorem-2-a","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"can_prove","summary":"Let $\\psi$ be a function from the positive integers to the nonnegative reals, and for each positive integer $q$ let $\\psi^*(q) = \\varphi(q) \\cdot \\sup \\{\\psi(n)/n : n \\text{ a positive integer with } q \\mid n\\}$, where $\\varphi$ is Euler's totient function and the supremum is taken in $[0, \\infty]$. Let $\\mathcal{K}(\\psi)$ be the set of real numbers $\\alpha \\in [0, 1]$ for which $|\\alpha - a/q| \\le \\psi(q)/q$ has\u2026","title":"Koukoulopoulos\u2013Maynard Theorem 2(a) (Catlin's conjecture, convergent case)"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/theorem-2-b","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.theorem_2_b","parent":10173,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#theorem-2-b","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"can_prove","summary":"Let $\\psi$ be a function from the positive integers to the nonnegative reals, and for each positive integer $q$ let $\\psi^*(q) = \\varphi(q) \\cdot \\sup \\{\\psi(n)/n : n \\text{ a positive integer with } q \\mid n\\}$, where $\\varphi$ is Euler's totient function and the supremum is taken in $[0, \\infty]$. Let $\\mathcal{K}(\\psi)$ be the set of real numbers $\\alpha \\in [0, 1]$ for which $|\\alpha - a/q| \\le \\psi(q)/q$ has\u2026","title":"Koukoulopoulos\u2013Maynard Theorem 2(b) (Catlin's conjecture, divergent case)"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/s-inf","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.s_inf","parent":10173,"reading_title":{"format":"inline-parts-v1","parts":[["text","The convergence exponent "],["math","\\(s(\\psi)\\)"]],"source":"The convergence exponent $s(\\psi)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#s-inf","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"fully_proved","summary":"For a function $\\psi$ from the positive integers to the nonnegative reals, $s(\\psi)$ is the infimum, taken in the nonnegative reals, of the set of real numbers $\\beta \\ge 0$ for which the series $\\sum_{q \\ge 1} \\varphi(q) (\\psi(q)/q)^{\\beta}$ converges, where $\\varphi$ is Euler's totient function. If there is no such $\\beta$, then $s(\\psi) = 0$.","title":"The convergence exponent $s(\\psi)$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/corollary-3","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferConjectureWanted.DuffinSchaefferConjecture.corollary_3","parent":10173,"reading_title":{"format":"inline-parts-v1","parts":[["text","Koukoulopoulos\u2013Maynard Corollary 3 (Hausdorff dimension of "],["math","\\(\\mathcal{A}\\)"],["text",")"]],"source":"Koukoulopoulos\u2013Maynard Corollary 3 (Hausdorff dimension of $\\mathcal{A}$)"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#corollary-3","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"can_prove","summary":"Let $\\psi$ be a function from the positive integers to $[0, 1/2]$. Let $\\mathcal{A}(\\psi)$ be the set of real numbers $\\alpha \\in [0, 1]$ for which $|\\alpha - a/q| \\le \\psi(q)/q$ has infinitely many solutions with $a$ a natural number, $q$ a positive integer and $\\gcd(a, q) = 1$. Let $s(\\psi)$ be the infimum, taken in the nonnegative reals (where the empty set has infimum $0$), of the real numbers $\\beta \\ge 0$ for\u2026","title":"Koukoulopoulos\u2013Maynard Corollary 3 (Hausdorff dimension of $\\mathcal{A}$)"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/duffin-schaeffer","kind":"node","lean":"MathlibExt.NumberTheory.DuffinSchaefferWanted.NumberTheory.DuffinSchaeffer.duffin_schaeffer","parent":10173,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture/index.html#duffin-schaeffer","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/duffin-schaeffer-conjecture","status":"can_prove","summary":"Let $(\\delta_n)_{n \\in \\mathbb{N}}$ be a sequence of nonnegative real numbers. Let $W$ be the set of points $x$ of the circle $\\mathbb{R}/\\mathbb{Z}$ such that, for infinitely many positive integers $n$, the distance from $x$ to some point of additive order exactly $n$ is less than $\\delta_n$. Then $W$ has measure $1$ for the Haar measure of total mass $1$ on $\\mathbb{R}/\\mathbb{Z}$ if and only if the series\u2026","title":"Duffin\u2013Schaeffer conjecture on the unit circle"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","title":"Floor and ceiling function sums"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/alternating-sum-floor-sqrt-mul","kind":"node","lean":"MathlibExt.NumberTheory.AlternatingFloorSqrtSumWanted.alternating_sum_floor_sqrt_mul","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#alternating-sum-floor-sqrt-mul","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"For every odd positive integer $n$, $\\sum_{j=1}^{n} (-1)^{j+1} \\lfloor \\sqrt{jn} \\rfloor = (n+1)/2$.","title":"An alternating sum of floors of square roots"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/sum-ceil-add-div-eq-ceil-mul-add","kind":"node","lean":"MetaMathlibExt.sum_ceil_add_div_eq_ceil_mul_add","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#sum-ceil-add-div-eq-ceil-mul-add","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"For every real number $x$ and every natural number $n$, $\\sum_{k=1}^{n} \\lceil x + k/n \\rceil = \\lceil n x \\rceil + n$.","title":"Ceiling analogue of Hermite's identity"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"fully_proved","summary":"For a function $F : \\mathbb{R} \\to \\mathbb{Z}$ and real numbers $x_1, \\dots, x_n$, the inclusion-exclusion-like sum is the integer $g^{(F)}(x_1, \\dots, x_n) = \\sum_T (-1)^{|T| - 1} F\\left(\\sum_{i \\in T} x_i\\right)$, where $T$ ranges over the nonempty subsets of $\\{1, \\dots, n\\}$.","title":"Inclusion-exclusion-like sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum-one","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum_one","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum-one","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"fully_proved","summary":"For every function $F : \\mathbb{R} \\to \\mathbb{Z}$ and every real $x$, the inclusion-exclusion-like sum of $F$ in the single variable $x$ is $g^{(F)}(x) = F(x)$.","title":"The inclusion-exclusion-like sum in one variable"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum-two","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum_two","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum-two","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"fully_proved","summary":"For every function $F : \\mathbb{R} \\to \\mathbb{Z}$ and all real numbers $x$ and $y$, $g^{(F)}(x, y) = F(x) + F(y) - F(x + y)$.","title":"The inclusion-exclusion-like sum in two variables"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum","kind":"node","lean":"MetaMathlibExt.tverbergSum","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"fully_proved","summary":"Let $F : \\mathbb{R} \\to \\mathbb{Z}$, let $a = (a_1, \\dots, a_\\ell)$ be a tuple of integers, let $m$ be a natural number and let $k$ be an integer. Tverberg's sum is $f^{(F)}(a, m, k) = \\sum_{T \\subseteq \\{1, \\dots, \\ell\\}} (-1)^{\\ell - |T|} F((k + \\sum_{i \\in T} a_i) / m)$, where the division is in $\\mathbb{R}$ (with $x / 0 = 0$).","title":"Tverberg's sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum-zero","kind":"node","lean":"MetaMathlibExt.tverbergSum_zero","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum-zero","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"fully_proved","summary":"For every function $F : \\mathbb{R} \\to \\mathbb{Z}$, every natural number $m$ and every integer $k$, Tverberg's sum of the empty tuple of integers is $f^{(F)}(\\emptyset, m, k) = F(k / m)$, where $k / m$ is division in $\\mathbb{R}$ (with $x / 0 = 0$).","title":"Tverberg's sum of the empty tuple"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum-one","kind":"node","lean":"MetaMathlibExt.tverbergSum_one","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum-one","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"fully_proved","summary":"For every function $F : \\mathbb{R} \\to \\mathbb{Z}$, all integers $a$ and $k$ and every natural number $m$, Tverberg's sum of $F$ for the one-element family $(a)$, modulus $m$ and shift $k$ equals $F((k + a)/m) - F(k/m)$, with $x/0 = 0$.","title":"Tverberg's sum of a single integer"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum-relations-and-periodi","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum_relations_and_periodic","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum-relations-and-periodi","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $n \\ge 2$ and write $g_1$, $g_2$, $g_3$, $g_4$ for the inclusion-exclusion-like sums $g^{(F)}$ with $F$ the floor function $\\lfloor y \\rfloor$, the ceiling function $\\lceil y \\rceil$, the nearest-integer function $\\lfloor y + 1/2 \\rfloor$ (ties rounded up) and $\\lceil y - 1/2 \\rceil$ (ties rounded down), respectively. For every $x = (x_1, \\dots, x_n) \\in \\mathbb{R}^n$: (i)\u2026","title":"Relations and periodicity of the inclusion-exclusion-like sums"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum-eq-incl-excl-like-sum","kind":"node","lean":"MetaMathlibExt.tverbergSum_eq_inclExclLikeSum","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum-eq-incl-excl-like-sum","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $F$ be one of $\\lfloor \\cdot \\rfloor$, $\\lceil \\cdot \\rceil$, $r^+(y) = \\lfloor y + 1/2 \\rfloor$ and $r^-(y) = \\lceil y - 1/2 \\rceil$. For real numbers $x_1, \\dots, x_n$ let $g(x_1, \\dots, x_n) = \\sum_{\\emptyset \\ne U \\subseteq \\{1, \\dots, n\\}} (-1)^{|U| - 1} F(\\sum_{i \\in U} x_i)$ be the inclusion-exclusion-like sum of $F$. For every natural number $\\ell$, integers $a_1, \\dots, a_\\ell$, a natural number $m$\u2026","title":"Tverberg's sum in terms of the inclusion-exclusion-like sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum-periodic","kind":"node","lean":"MetaMathlibExt.tverbergSum_periodic","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum-periodic","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $\\ell \\ge 2$ and let $F$ be one of $\\lfloor \\cdot \\rfloor$, $\\lceil \\cdot \\rceil$, $r^+(y) = \\lfloor y + 1/2 \\rfloor$ and $r^-(y) = \\lceil y - 1/2 \\rceil$. For integers $a_1, \\dots, a_\\ell$, a natural number $m$ and an integer $k$, let $f(a, m, k) = \\sum_{U \\subseteq \\{1, \\dots, \\ell\\}} (-1)^{\\ell - |U|} F((k + \\sum_{i \\in U} a_i)/m)$ be Tverberg's sum of $F$, with $x/0 = 0$. For every index $i$ and every\u2026","title":"Periodicity of Tverberg's sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum-floor-bounds","kind":"node","lean":"MetaMathlibExt.tverbergSum_floor_bounds","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum-floor-bounds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $\\ell \\ge 2$ and let $F = \\lfloor \\cdot \\rfloor$. For integers $a_1, \\dots, a_\\ell$, a natural number $m$ and an integer $k$, let $f(a, m, k) = \\sum_{U \\subseteq \\{1, \\dots, \\ell\\}} (-1)^{\\ell - |U|} F((k + \\sum_{i \\in U} a_i)/m)$ be Tverberg's sum of $F$, with $x/0 = 0$. Then $-2^{\\ell-2} \\le f(a, m, k) \\le 2^{\\ell-2}$ for all such $a$, $m$ and $k$. Both bounds are attained: for every integer $r \\ge 1$, if\u2026","title":"Sharp bounds for Tverberg's floor sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum-floor-extrema","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum_floor_extrema","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum-floor-extrema","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $n \\ge 2$ and let $g_1$ be the inclusion-exclusion-like sum $g^{(F)}$ in $n$ real variables with $F$ the floor function. Then $-2^{n-2} \\le g_1(x) \\le 2^{n-2} - 1$ for every $x \\in \\mathbb{R}^n$; the minimum $-2^{n-2}$ is attained at the point with all coordinates $1/2$, and the maximum $2^{n-2} - 1$ at the point with all coordinates $1/2 - 1/n^2$.","title":"Extrema of the floor inclusion-exclusion-like sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-partial-sum-vanish-and-periodic","kind":"node","lean":"MetaMathlibExt.tverbergPartialSum_vanish_and_periodic","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-partial-sum-vanish-and-periodic","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $\\ell \\ge 2$, let $a = (a_1, \\dots, a_\\ell)$ be integers and let $m$ be a natural number. Let $F$ be one of $\\lfloor \\cdot \\rfloor$, $\\lceil \\cdot \\rceil$, $r^+(y) = \\lfloor y + 1/2 \\rfloor$ and $r^-(y) = \\lceil y - 1/2 \\rceil$, and for an integer $k$ let $f^F(a, k) = \\sum_{U \\subseteq \\{1, \\dots, \\ell\\}} (-1)^{\\ell - |U|} F((k + \\sum_{i \\in U} a_i)/m)$ be Tverberg's sum, with $x/0 = 0$. For natural $K$ put\u2026","title":"Vanishing and periodicity of partial Tverberg sums"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/floor-ceil-div-add-half","kind":"node","lean":"MetaMathlibExt.floor_ceil_div_add_half","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#floor-ceil-div-add-half","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"For every integer $q$ and every positive integer $m$, $\\lfloor q/m + 1/2 \\rfloor = \\lfloor (q + \\lfloor m/2 \\rfloor)/m \\rfloor$ and $\\lceil q/m + 1/2 \\rceil = \\lceil (q + \\lceil m/2 \\rceil)/m \\rceil$.","title":"Floor and ceiling of a quotient plus one half"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum-ceil-extrema","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum_ceil_extrema","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum-ceil-extrema","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $n \\ge 2$ and let $g_2$ be the inclusion-exclusion-like sum $g^{(F)}$ in $n$ real variables with $F$ the ceiling function. Then $-2^{n-2} + 1 \\le g_2(x) \\le 2^{n-2}$ for every $x \\in \\mathbb{R}^n$; the minimum $-2^{n-2} + 1$ is attained at the point with all coordinates $1/2 + 1/n^2$, and the maximum $2^{n-2}$ at the point with all coordinates $1/2$.","title":"Extrema of the ceiling inclusion-exclusion-like sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum-round-extrema","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum_round_extrema","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum-round-extrema","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $n \\ge 2$ and let $g_3$ be the inclusion-exclusion-like sum $g^{(F)}$ in $n$ real variables with $F(y) = \\lfloor y + 1/2 \\rfloor$, the nearest integer to $y$ with ties rounded up. Then $-2^{n-2} \\le g_3(x) \\le 2^{n-2}$ for every $x \\in \\mathbb{R}^n$; the minimum $-2^{n-2}$ is attained at the point with all coordinates $1/2 - 1/n^2$, and the maximum $2^{n-2}$ at the point with all coordinates $1/2$.","title":"Extrema of the round-half-up inclusion-exclusion-like sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum-round-down-extrema","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum_roundDown_extrema","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum-round-down-extrema","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $n \\ge 2$ and let $g_4$ be the inclusion-exclusion-like sum $g^{(F)}$ in $n$ real variables with $F(y) = \\lceil y - 1/2 \\rceil$, the nearest integer to $y$ with ties rounded down. Then $-2^{n-2} \\le g_4(x) \\le 2^{n-2}$ for every $x \\in \\mathbb{R}^n$; the minimum $-2^{n-2}$ is attained at the point with all coordinates $1/2$, and the maximum $2^{n-2}$ at the point with all coordinates $1/2 + 1/n^2$.","title":"Extrema of the round-half-down inclusion-exclusion-like sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/incl-excl-like-sum-bounds-summary","kind":"node","lean":"MetaMathlibExt.inclExclLikeSum_bounds_summary","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#incl-excl-like-sum-bounds-summary","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $n \\ge 2$ and write $g_1$, $g_2$, $g_3$, $g_4$ for the inclusion-exclusion-like sums $g^{(F)}$ in $n$ real variables with $F$ the floor function $\\lfloor y \\rfloor$, the ceiling function $\\lceil y \\rceil$, the nearest-integer function $\\lfloor y + 1/2 \\rfloor$ (ties rounded up) and $\\lceil y - 1/2 \\rceil$ (ties rounded down), respectively. Then for every $x \\in \\mathbb{R}^n$,\u2026","title":"Bounds for the four inclusion-exclusion-like sums"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum-ceil-bounds","kind":"node","lean":"MetaMathlibExt.tverbergSum_ceil_bounds","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum-ceil-bounds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $\\ell \\ge 2$ and let $F = \\lceil \\cdot \\rceil$. For integers $a_1, \\dots, a_\\ell$, a natural number $m$ and an integer $k$, let $f(a, m, k) = \\sum_{U \\subseteq \\{1, \\dots, \\ell\\}} (-1)^{\\ell - |U|} F((k + \\sum_{i \\in U} a_i)/m)$ be Tverberg's sum of $F$, with $x/0 = 0$. Then $-2^{\\ell-2} \\le f(a, m, k) \\le 2^{\\ell-2}$ for all such $a$, $m$ and $k$. Both bounds are attained: for every integer $r \\ge 1$, if\u2026","title":"Sharp bounds for Tverberg's ceiling sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum-round-bounds","kind":"node","lean":"MetaMathlibExt.tverbergSum_round_bounds","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum-round-bounds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $\\ell \\ge 2$ and let $F$ be rounding to the nearest integer with ties rounded up, $F(y) = \\lfloor y + 1/2 \\rfloor$. For integers $a_1, \\dots, a_\\ell$, a natural number $m$ and an integer $k$, let $f(a, m, k) = \\sum_{U \\subseteq \\{1, \\dots, \\ell\\}} (-1)^{\\ell - |U|} F((k + \\sum_{i \\in U} a_i)/m)$ be Tverberg's sum of $F$, with $x/0 = 0$. Then $-2^{\\ell-2} \\le f(a, m, k) \\le 2^{\\ell-2}$ for all such $a$, $m$ and\u2026","title":"Sharp bounds for Tverberg's sum with rounding half up"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-sum-round-down-bounds","kind":"node","lean":"MetaMathlibExt.tverbergSum_roundDown_bounds","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-sum-round-down-bounds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $\\ell \\ge 2$ and let $F$ be rounding to the nearest integer with ties rounded down, $F(y) = \\lceil y - 1/2 \\rceil$. For integers $a_1, \\dots, a_\\ell$, a natural number $m$ and an integer $k$, let $f(a, m, k) = \\sum_{U \\subseteq \\{1, \\dots, \\ell\\}} (-1)^{\\ell - |U|} F((k + \\sum_{i \\in U} a_i)/m)$ be Tverberg's sum of $F$, with $x/0 = 0$. Then $-2^{\\ell-2} \\le f(a, m, k) \\le 2^{\\ell-2}$ for all such $a$, $m$ and\u2026","title":"Sharp bounds for Tverberg's sum with rounding half down"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-tail-sum-ceil-two-bounds","kind":"node","lean":"MetaMathlibExt.tverbergTailSum_ceil_two_bounds","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-tail-sum-ceil-two-bounds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"For integers $a, b$ and natural numbers $m, K$, let $T(a, b, m, K) = \\sum_{k=1}^{K} f^{(\\lceil \\cdot \\rceil)}((a, b), m, k)$, where $f^{(\\lceil \\cdot \\rceil)}$ is Tverberg's sum with $F$ the ceiling function. Then $0 \\le T(a, b, m, K) \\le \\lfloor m/2 \\rfloor$ for all such $a, b, m, K$. Moreover, for every natural number $r$, $T(r, r, 2r, 2r) = 0$ and $T(r, r, 2r, r) = r$.","title":"Sharp bounds for two-term tail sums of Tverberg's ceiling sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-tail-sum-ceil-bounds","kind":"node","lean":"MetaMathlibExt.tverbergTailSum_ceil_bounds","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-tail-sum-ceil-bounds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $\\ell \\ge 2$. For a tuple $a = (a_1, \\dots, a_\\ell)$ of integers and natural numbers $m, K$, let $T(a, m, K) = \\sum_{k=1}^{K} f^{(\\lceil \\cdot \\rceil)}(a, m, k)$, where $f^{(\\lceil \\cdot \\rceil)}$ is Tverberg's sum with $F$ the ceiling function. Then $-2^{\\ell-2} \\lfloor m/2 \\rfloor \\le T(a, m, K) \\le 2^{\\ell-2} \\lfloor m/2 \\rfloor$ for all such $a, m, K$. Moreover, for every natural number $r$, if $a_i = r$\u2026","title":"Bounds for tail sums of Tverberg's ceiling sum"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-partial-sum-round-two-bounds","kind":"node","lean":"MetaMathlibExt.tverbergPartialSum_round_two_bounds","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-partial-sum-round-two-bounds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $r^+(y) = \\lfloor y + 1/2 \\rfloor$ be rounding to the nearest integer with ties rounded up. For integers $a, b, k$ and a natural number $m$, let $f(a, b; m, k) = r^+(k/m) - r^+((k + a)/m) - r^+((k + b)/m) + r^+((k + a + b)/m)$ be Tverberg's sum of $r^+$ at $(a, b)$, with $x/0 = 0$. For all integers $a, b$ and natural numbers $m, K$,\u2026","title":"Bounds for partial Tverberg sums of two integers under rounding half up"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/tverberg-tail-sum-round-down-two-bounds","kind":"node","lean":"MetaMathlibExt.tverbergTailSum_roundDown_two_bounds","parent":10184,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums/index.html#tverberg-tail-sum-round-down-two-bounds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/floor-and-ceiling-function-sums","status":"can_prove","summary":"Let $r^-(x) = \\lceil x - 1/2 \\rceil$. For integers $a, b$ and natural numbers $m, K$, let $T(a, b, m, K) = \\sum_{k=1}^{K} f^{(r^-)}((a, b), m, k)$, where $f^{(r^-)}$ is Tverberg's sum with $F = r^-$. Then $-\\lfloor m/2 \\rfloor \\le T(a, b, m, K) \\le \\lfloor m/2 \\rfloor$ for all such $a, b, m, K$, and the lower bound is attained: for every natural number $r$, $T(r, r, 2r, r) = -r$.","title":"Bounds for two-term tail sums of Tverberg's sum with half-down rounding"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes","status":"can_prove","title":"Fractional parts of multiples of primes"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/well-distributed","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem997Wanted.WellDistributed","parent":10211,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/index.html#well-distributed","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes","status":"fully_proved","summary":"A sequence $(x_m)_{m \\in \\mathbb{N}}$ of real numbers is well-distributed if for every $\\varepsilon > 0$ there is a $K$ such that for all $k \\ge K$, all $n \\in \\mathbb{N}$ and all real numbers $a, b$ with $0 \\le a \\le b \\le 1$, the number $N$ of indices $m$ with $n < m \\le n + k$ and $x_m \\in [a, b)$ satisfies $|N - (b - a)k| < \\varepsilon k$.","title":"Well-distributed sequences"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/prime-fract","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem997Wanted.primeFract","parent":10211,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/index.html#prime-fract","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes","status":"fully_proved","summary":"For a real number $\\alpha$, this is the sequence taking $n \\in \\mathbb{N}$ to the fractional part $\\{\\alpha p_n\\} = \\alpha p_n - \\lfloor \\alpha p_n \\rfloor$, where $p_0 = 2, p_1 = 3, p_2 = 5, \\dots$ are the primes in increasing order.","title":"Fractional parts of multiples of the primes"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/statement","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem997Wanted.Statement","parent":10211,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/index.html#statement","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes","status":"fully_proved","summary":"For every real number $\\alpha$, the sequence $n \\mapsto \\{\\alpha p_n\\}$ of fractional parts of $\\alpha$ times the primes $p_0 = 2, p_1 = 3, p_2 = 5, \\dots$ is not well-distributed.","title":"Statement of Erd\u0151s Problem 997"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/statement-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem997Wanted.Statement_holds","parent":10211,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes/index.html#statement-holds","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/fractional-parts-of-multiples-of-primes","status":"can_prove","summary":"For every real number $\\alpha$, the sequence $n \\mapsto \\{\\alpha p_n\\}$ of fractional parts of $\\alpha$ times the primes $p_0 = 2, p_1 = 3, p_2 = 5, \\dots$ is not well-distributed.","title":"Erd\u0151s Problem 997 holds"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs","status":"can_prove","title":"Lattice points in translated discs"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs/disc-lattice-count","kind":"node","lean":"MathlibExt.NumberTheory.DiophantineApproximation.ScottishBook152Wanted.DiscLatticeCount","parent":10216,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs/index.html#disc-lattice-count","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs","status":"fully_proved","summary":"For $w = (w_1, w_2) \\in \\mathbb{R}^2$ and a natural number $n$, the disc lattice count is the number of integer points $(p_1, p_2) \\in \\mathbb{Z}^2$ in the closed disc of radius $1$ centred at $n w$, that is, with $(p_1 - n w_1)^2 + (p_2 - n w_2)^2 \\le 1$.","title":"Lattice points in a translated unit disc"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs/steinhaus-disc-frequencies-exist","kind":"node","lean":"MathlibExt.NumberTheory.DiophantineApproximation.ScottishBook152Wanted.steinhausDiscFrequenciesExist","parent":10216,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs/index.html#steinhaus-disc-frequencies-exist","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/lattice-points-in-translated-discs","status":"can_prove","summary":"Let $w = (w_1, w_2) \\in \\mathbb{R}^2$ be such that $w_1$, $w_2$ and $w_1 / w_2$ are all irrational, and for each natural number $n$ let $c(n)$ be the disc lattice count of $w$ and $n$, the number of integer lattice points in the closed disc of radius $1$ centred at $n w$. Then for each $m$ equal to $2$, $3$ or $4$, the proportion of the integers $n$ with $0 \\le n < N$ and $c(n) = m$ converges to a real number as\u2026","title":"Steinhaus's disc frequencies exist"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture","status":"can_prove","title":"Littlewood conjecture"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture/dist-to-nearest-int","kind":"node","lean":"MathlibExt.Dynamics.EinsiedlerKatokLindenstraussWanted.Dynamics.EinsiedlerKatokLindenstrauss.distToNearestInt","parent":10219,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture/index.html#dist-to-nearest-int","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture","status":"fully_proved","summary":"For a real number $x$, this is $|x - r|$, where $r$ is the integer nearest to $x$ (halves rounded up); equivalently, it is the minimum of $|x - n|$ over all integers $n$.","title":"Distance to the nearest integer"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture/einsiedler-katok-lindenstrauss","kind":"node","lean":"MathlibExt.Dynamics.EinsiedlerKatokLindenstraussWanted.Dynamics.EinsiedlerKatokLindenstrauss.einsiedler_katok_lindenstrauss","parent":10219,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture/index.html#einsiedler-katok-lindenstrauss","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/littlewood-conjecture","status":"can_prove","summary":"For a real number $x$, write $\\lVert x \\rVert$ for the distance from $x$ to the nearest integer. The set of pairs $(\\alpha, \\beta)$ of real numbers with $\\liminf_{n \\to \\infty} n \\cdot \\lVert n\\alpha \\rVert \\cdot \\lVert n\\beta \\rVert > 0$, the lower limit taken over natural numbers $n$, has Hausdorff dimension $0$ in $\\mathbb{R}^2$.","title":"Einsiedler\u2013Katok\u2013Lindenstrauss theorem"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"can_prove","title":"Normal numbers"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/digit-seq","kind":"node","lean":"MetaMathlibExt.NormalNumber.digitSeq","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Base-"],["math","\\(b\\)"],["text"," digits of a real number"]],"source":"Base-$b$ digits of a real number"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#digit-seq","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For natural numbers $b$ and $n$ and a real number $x$, the $n$-th digit of $x$ in base $b$, counting from $n = 0$ just after the radix point, is $\\lfloor b^{n+1} \\{x\\} \\rfloor \\bmod b$, where $\\{x\\} = x - \\lfloor x \\rfloor$ is the fractional part of $x$.","title":"Base-$b$ digits of a real number"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/is-simply-normal-in-base","kind":"node","lean":"MetaMathlibExt.NormalNumber.IsSimplyNormalInBase","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#is-simply-normal-in-base","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For a natural number $b$ and a real number $x$, write $a_k$ for the $k$-th base-$b$ digit of $x$ after the radix point, counting from $k = 0$. Then $x$ is simply normal in base $b$ if for every natural number $d < b$ the proportion $\\#\\{k < n : a_k = d\\} / n$ tends to $1/b$ as $n \\to \\infty$.","title":"Simply normal numbers"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/is-normal-in-base","kind":"node","lean":"MetaMathlibExt.NormalNumber.IsNormalInBase","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#is-normal-in-base","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For a natural number $b$ and a real number $x$, write $a_i$ for the $i$-th base-$b$ digit of $x$ after the radix point, counting from $i = 0$. Then $x$ is normal in base $b$ if for every natural number $k$ and all natural numbers $w_0, \\dots, w_{k-1} < b$ the proportion $\\#\\{i < n : a_{i+j} = w_j \\text{ for all } j < k\\} / n$ tends to $1/b^k$ as $n \\to \\infty$.","title":"Normal numbers"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/is-simply-normal-in-base-2","kind":"node","lean":"MetaMathlibExt.NormalNumber.IsNormalInBase.isSimplyNormalInBase","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#is-simply-normal-in-base-2","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"If a real number $x$ is normal in base $b$, then $x$ is simply normal in base $b$.","title":"Normal numbers are simply normal"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/is-absolutely-normal","kind":"node","lean":"MetaMathlibExt.NormalNumber.IsAbsolutelyNormal","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#is-absolutely-normal","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"A real number is absolutely normal if it is normal in every base at least two.","title":"Absolutely normal numbers"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-h1","kind":"node","lean":"MetaMathlibExt.campbellH1","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Campbell's sequence "],["math","\\(h_1\\)"]],"source":"Campbell's sequence $h_1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-h1","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For a natural number $n$, $h_1(n) = 0$ if $n < 10$ and $h_1(n) = 10^{\\lfloor \\log_{10} n \\rfloor}$ if $n \\ge 10$; thus for $n \\ge 10$, $h_1(n)$ is the largest power of $10$ that is at most $n$.","title":"Campbell's sequence $h_1$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-h1-small","kind":"node","lean":"MetaMathlibExt.campbellH1_small","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small values of "],["math","\\(h_1\\)"]],"source":"Small values of $h_1$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-h1-small","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"Campbell's auxiliary sequence $h_1$ satisfies $h_1(9) = 0$, $h_1(10) = 10$, $h_1(99) = 10$ and $h_1(100) = 100$.","title":"Small values of $h_1$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-h2","kind":"node","lean":"MetaMathlibExt.campbellH2","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Campbell's sequence "],["math","\\(h_2\\)"]],"source":"Campbell's sequence $h_2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-h2","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For a natural number $n$, let $k = \\lfloor \\log_{10} n \\rfloor$ (with $k = 0$ for $n = 0$), let $h_1$ be Campbell's sequence and let $S_k = \\sum_{i=1}^{10^k} h_1(i)$. Then $h_2(n) = 10^k \\lfloor S_k / 10^k \\rfloor + \\sum_{i=10^k+1}^{n} h_1(i)$, where the last sum is empty when $n \\le 10^k$.","title":"Campbell's sequence $h_2$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-h2-small","kind":"node","lean":"MetaMathlibExt.campbellH2_small","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small values of Campbell's sequence "],["math","\\(h_2\\)"]],"source":"Small values of Campbell's sequence $h_2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-h2-small","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"The sequence $h_2$ satisfies $h_2(9) = 0$, $h_2(10) = 10$, $h_2(21) = 120$, $h_2(99) = 900$ and $h_2(100) = 1000$.","title":"Small values of Campbell's sequence $h_2$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-h3","kind":"node","lean":"MetaMathlibExt.campbellH3","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Campbell's auxiliary function "],["math","\\(h_3^{(m)}\\)"]],"source":"Campbell's auxiliary function $h_3^{(m)}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-h3","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For a natural number $m$, let $H(m) = h_2(10^m) - h_2(10^m - 1)$, where $h_2$ is Campbell's sequence (in the paper's notation $H(m) = y_1(m + 1) - y_2(m)$ with $y_1(n) = h_2(10^{n-1})$ and $y_2(n) = h_2(10^n - 1)$). Define $h_3^{(m)} : \\mathbb{R} \\to \\mathbb{R}$ by $h_3^{(m)}(t) = a(m)\\,t + b(m)$ for $t \\le 10^m - 1/10$ and $h_3^{(m)}(t) = c(m)\\,t + d(m)$ for $t > 10^m - 1/10$, where\u2026","title":"Campbell's auxiliary function $h_3^{(m)}$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-h3-of-le","kind":"node","lean":"MetaMathlibExt.campbellH3_of_le","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","First piece of Campbell's auxiliary function "],["math","\\(h_3^{(m)}\\)"]],"source":"First piece of Campbell's auxiliary function $h_3^{(m)}$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-h3-of-le","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For every natural number $m$ and every real number $t \\le 10^m - 1/10$, $h_3^{(m)}(t) = a(m)\\,t + b(m)$, where $H(m) = h_2(10^m) - h_2(10^m - 1)$ with $h_2$ Campbell's sequence, $a(m) = (20H(m) - 11 \\cdot 10^m)/9 + 9 \\cdot 10^{m-1}$ and $b(m) = (10^m - 1)(-20H(m) + 11 \\cdot 10^m)/9 + 10^m - 9 \\cdot 10^{2m-1}$, with the exponents $m - 1$ and $2m - 1$ read as $0$ when $m = 0$.","title":"First piece of Campbell's auxiliary function $h_3^{(m)}$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-f","kind":"node","lean":"MetaMathlibExt.campbellF","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-f","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"Let $h_2$ be Campbell's sequence and $h_3^{(m)}$ the auxiliary piecewise-linear functions of the construction, and for $n \\ge 1$ put $y_1(n) = h_2(10^{n-1})$ and $y_2(n) = h_2(10^n - 1)$. Campbell's interpolation is the function $f : \\mathbb{R} \\to \\mathbb{R}$ with $f(x) = \\frac{89}{110} e^{(100/89)(11x - 100)} + \\frac{1}{10}$ for $x \\le 100/11$ and $f(x) = 10x - 90$ for $100/11 < x \\le 10$. For $x > 10$ put\u2026","title":"Campbell's interpolation"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-f-of-mem","kind":"node","lean":"MetaMathlibExt.campbellF_of_mem","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Campbell's interpolation on "],["math","\\((100/11, 10)\\)"]],"source":"Campbell's interpolation on $(100/11, 10)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-f-of-mem","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For every real $x$ with $100/11 < x \\le 10$, Campbell's interpolation satisfies $f(x) = 10x - 90$.","title":"Campbell's interpolation on $(100/11, 10)$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-block","kind":"node","lean":"MetaMathlibExt.campbellBlock","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Decimal digit block of "],["math","\\(h_2(n)\\)"]],"source":"Decimal digit block of $h_2(n)$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-block","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"For a natural number $n$, the block of $n$ is the list of base-$10$ digits of $h_2(n)$, where $h_2$ is Campbell's sequence, written most significant digit first; when $h_2(n) = 0$ the block is the single digit $0$.","title":"Decimal digit block of $h_2(n)$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-concat-digit","kind":"node","lean":"MetaMathlibExt.campbellConcatDigit","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digits of the concatenation "],["math","\\(0.h_2(1)h_2(2)\\ldots\\)"]],"source":"Digits of the concatenation $0.h_2(1)h_2(2)\\ldots$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-concat-digit","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"Let $h_2$ be Campbell's sequence. For $j \\ge 0$, $d_j$ is the digit in position $j$ (counting from $0$) after the decimal point of $0.h_2(1)h_2(2)\\ldots$, the string obtained by writing the blocks of $h_2(1), h_2(2), \\ldots$ one after another, where the block of $h_2(n)$ is its list of base-$10$ digits, most significant first, with $0$ written as the single digit $0$.","title":"Digits of the concatenation $0.h_2(1)h_2(2)\\ldots$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-concat-digit-small","kind":"node","lean":"MetaMathlibExt.campbellConcatDigit_small","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first eleven digits of the concatenation "],["math","\\(0.h_2(1)h_2(2)\\ldots\\)"]],"source":"The first eleven digits of the concatenation $0.h_2(1)h_2(2)\\ldots$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-concat-digit-small","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"Let $h_2$ be Campbell's sequence and let $d_0, d_1, d_2, \\ldots$ be the digits after the decimal point of $0.h_2(1)h_2(2)\\ldots$, the concatenation of the base-$10$ digit blocks of $h_2(1), h_2(2), \\ldots$ (with $0$ written as the single digit $0$). Then $(d_0, d_1, \\ldots, d_{10}) = (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0)$: nine zero blocks from $h_2(1) = \\cdots = h_2(9) = 0$, then the block of $h_2(10) = 10$.","title":"The first eleven digits of the concatenation $0.h_2(1)h_2(2)\\ldots$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-constant","kind":"node","lean":"MetaMathlibExt.campbellConstant","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-constant","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"fully_proved","summary":"Let $h_2$ be Campbell's sequence. Campbell's constant is the real number $\\sum_{j \\ge 0} d_j \\, 10^{-(j+1)} = 0.d_0 d_1 d_2 \\ldots$, where $d_0, d_1, d_2, \\ldots$ are the digits after the decimal point of the concatenation $0.h_2(1)h_2(2)\\ldots$ of the base-$10$ digit blocks of $h_2(1), h_2(2), \\ldots$ (with $0$ written as the single digit $0$).","title":"Campbell's constant"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-h1-sum-and-y-relations","kind":"node","lean":"MetaMathlibExt.campbellH1_sum_and_y_relations","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of "],["math","\\(h_1\\)"],["text"," and the jumps of "],["math","\\(h_2\\)"],["text"," at powers of ten"]],"source":"Sums of $h_1$ and the jumps of $h_2$ at powers of ten"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-h1-sum-and-y-relations","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"can_prove","summary":"Let $h_1$ be Campbell's auxiliary sequence and $h_2$ Campbell's sequence. For every integer $\\ell \\ge 1$, $\\sum_{i=1}^{10^{\\ell+1}-1} h_1(i) = \\sum_{j=0}^{\\ell-1} 900 \\cdot 100^j$, the number written $9090\\cdots900$ ($\\ell$ blocks $90$, then $0$). Moreover, writing $y_1(n) = h_2(10^{n-1})$ and $y_2(n) = h_2(10^n - 1)$, for every integer $\\ell \\ge 2$ we have\u2026","title":"Sums of $h_1$ and the jumps of $h_2$ at powers of ten"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/floor-campbell-f-eq-campbell-h2","kind":"node","lean":"MetaMathlibExt.floor_campbellF_eq_campbellH2","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","The floor of Campbell's interpolation at positive integers is Campbell's sequence "],["math","\\(h_2\\)"]],"source":"The floor of Campbell's interpolation at positive integers is Campbell's sequence $h_2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#floor-campbell-f-eq-campbell-h2","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"can_prove","summary":"Let $f$ be Campbell's interpolation and $h_2$ Campbell's sequence. For every integer $n \\ge 1$, $\\lfloor f(n) \\rfloor = h_2(n)$, and for every integer $n \\ge 10$, $f(n) = h_2(n)$.","title":"The floor of Campbell's interpolation at positive integers is Campbell's sequence $h_2$"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-f-differentiable-on","kind":"node","lean":"MetaMathlibExt.campbellF_differentiableOn","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-f-differentiable-on","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"can_prove","summary":"Campbell's interpolation $f$ is differentiable at every real $x > 0$.","title":"Differentiability of Campbell's interpolation"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/campbell-f-monotone-on","kind":"node","lean":"MetaMathlibExt.campbellF_monotoneOn","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#campbell-f-monotone-on","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"can_prove","summary":"Campbell's interpolation $f$ is monotone (non-decreasing) on $[0, \\infty)$: for all real numbers $x$ and $y$ with $0 \\le x \\le y$, $f(x) \\le f(y)$.","title":"Monotonicity of Campbell's interpolation"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/tendsto-log-campbell-f-div-log","kind":"node","lean":"MetaMathlibExt.tendsto_log_campbellF_div_log","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#tendsto-log-campbell-f-div-log","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"can_prove","summary":"For Campbell's interpolation $f$, $\\log f(x) / \\log x \\to 2$ as $x \\to \\infty$ over the reals; that is, $\\eta(f) = 2$, where $\\eta(h) = \\lim_{x \\to \\infty} \\log h(x) / \\log x$.","title":"Growth exponent of Campbell's interpolation"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/tendsto-log-deriv-campbell-f-div-log","kind":"node","lean":"MetaMathlibExt.tendsto_log_deriv_campbellF_div_log","parent":10222,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#tendsto-log-deriv-campbell-f-div-log","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"can_prove","summary":"For Campbell's interpolation $f$, $\\log f'(x) / \\log x \\to 1$ as $x \\to \\infty$ over the reals; that is, $\\eta(f') = 1$, where $\\eta(h) = \\lim_{x \\to \\infty} \\log h(x) / \\log x$. Here $f'(x)$ is taken to be $0$ at any point where $f$ is not differentiable, and $\\log 0 = 0$.","title":"Growth exponent of the derivative of Campbell's interpolation"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/not-is-normal-in-base-campbell-constant","kind":"node","lean":"MetaMathlibExt.not_isNormalInBase_campbellConstant","parent":10222,"reading_title":{"format":"inline-parts-v1","parts":[["text","Campbell's concatenation of "],["math","\\(h_2\\)"],["text"," is not normal in base "],["math","\\(10\\)"]],"source":"Campbell's concatenation of $h_2$ is not normal in base $10$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers/index.html#not-is-normal-in-base-campbell-constant","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/normal-numbers","status":"can_prove","summary":"Let $h_2$ be Campbell's sequence and let $c_0, c_1, c_2, \\ldots$ be the base-$10$ digits of $h_2(1), h_2(2), \\ldots$, each number written most significant digit first and the numbers written one after another (with $h_2(n) = 0$ written as the single digit $0$). The real number $\\sum_{j \\ge 0} c_j 10^{-(j+1)} = 0.c_0c_1c_2\\ldots$ is not normal in base $10$.","title":"Campbell's concatenation of $h_2$ is not normal in base $10$"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem","status":"can_prove","title":"Reverse Minkowski theorem"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/untitled","kind":"node","lean":"MathlibExt.NumberTheory.ReverseMinkowskiWanted.ReverseMinkowski.\u03c1","parent":10247,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/index.html#untitled","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem","status":"fully_proved","summary":"For a real number $s$ and a $\\mathbb{Z}$-submodule $\\mathcal{L} \\subseteq \\mathbb{R}^n$, the Gaussian mass of $\\mathcal{L}$ at parameter $s$ is $\\rho_s(\\mathcal{L}) = \\sum_{y \\in \\mathcal{L}} e^{-\\pi \\|y\\|^2 / s^2}$, where $\\|y\\|$ is the Euclidean norm.","title":"Gaussian mass of a lattice"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/b","kind":"node","lean":"MathlibExt.NumberTheory.ReverseMinkowskiWanted.ReverseMinkowski.B","parent":10247,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/index.html#b","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem","status":"fully_proved","summary":"Let $\\mathcal{L} \\subseteq \\mathbb{R}^n$ be a discrete $\\mathbb{Z}$-submodule of rank $m$. Its basis matrix $B(\\mathcal{L})$ is the real $n \\times m$ matrix whose $j$-th column is the $j$-th vector of a fixed, arbitrarily chosen $\\mathbb{Z}$-basis of $\\mathcal{L}$.","title":"Basis matrix of a lattice"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/determinant","kind":"node","lean":"MathlibExt.NumberTheory.ReverseMinkowskiWanted.ReverseMinkowski.determinant","parent":10247,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/index.html#determinant","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem","status":"fully_proved","summary":"For a discrete $\\mathbb{Z}$-submodule $\\mathcal{L} \\subseteq \\mathbb{R}^n$ with basis matrix $B = B(\\mathcal{L})$ (the real matrix whose columns form a fixed $\\mathbb{Z}$-basis of $\\mathcal{L}$), the determinant of $\\mathcal{L}$ is $\\det(\\mathcal{L}) = \\sqrt{\\det(B^{T} B)}$.","title":"Determinant of a lattice"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/theorem-1-2","kind":"node","lean":"MathlibExt.NumberTheory.ReverseMinkowskiWanted.ReverseMinkowski.theorem_1_2","parent":10247,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem/index.html#theorem-1-2","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/reverse-minkowski-theorem","status":"can_prove","summary":"Let $\\mathcal{L} \\subseteq \\mathbb{R}^n$ be a full-rank lattice, that is, a discrete subgroup of $\\mathbb{R}^n$ that spans $\\mathbb{R}^n$, such that every subgroup $\\mathcal{L}' \\subseteq \\mathcal{L}$ (a lattice of possibly lower rank) has determinant $\\det(\\mathcal{L}') \\ge 1$. Let $t = 10(\\log n + 2)$, where $\\log$ is the natural logarithm. Then the Gaussian mass of $\\mathcal{L}$ at parameter $1/t$ satisfies\u2026","title":"Reverse Minkowski theorem"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"can_prove","title":"Smallest denominators in intervals"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/smallest-denominator","kind":"node","lean":"MetaMathlibExt.smallestDenominator","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#smallest-denominator","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"For real numbers $x$ and $\\delta$, the smallest denominator $q_{\\min}(x, \\delta)$ is the least positive integer $q$ for which there is an integer $p$ with $|x - p/q| < \\delta/2$, that is, with $p/q$ in the open interval $(x - \\delta/2, x + \\delta/2)$. If there is no such $q$, which happens only when $\\delta \\le 0$, then $q_{\\min}(x, \\delta) = 0$.","title":"Smallest denominator in an interval"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/smallest-denominator-le","kind":"node","lean":"MetaMathlibExt.smallestDenominator_le","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#smallest-denominator-le","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"Let $x$ and $\\delta$ be real numbers, let $q$ be a positive integer and let $p$ be an integer. If $|x - p/q| < \\delta/2$, then the smallest denominator satisfies $q_{\\min}(x, \\delta) \\le q$.","title":"Denominators of nearby rationals bound the smallest denominator"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/exists-denominator-mem-window","kind":"node","lean":"MetaMathlibExt.exists_denominator_mem_window","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#exists-denominator-mem-window","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"For every real number $x$ and every real $\\delta > 0$ there are a positive integer $q$ and an integer $p$ with $|x - p/q| < \\delta/2$.","title":"Every real number has arbitrarily close rationals"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/smallest-denominator-spec","kind":"node","lean":"MetaMathlibExt.smallestDenominator_spec","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#smallest-denominator-spec","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"For every real number $x$ and every real $\\delta > 0$, the smallest denominator $q = q_{\\min}(x, \\delta)$ is positive and there is an integer $p$ with $|x - p/q| < \\delta/2$.","title":"The smallest denominator is positive and attained"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/not-mem-window-of-lt-smallest-denominato","kind":"node","lean":"MetaMathlibExt.not_mem_window_of_lt_smallestDenominator","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#not-mem-window-of-lt-smallest-denominato","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"Let $x$ and $\\delta$ be real numbers and let $q$ be a positive integer with $q < q_{\\min}(x, \\delta)$, where $q_{\\min}(x, \\delta)$ is the smallest denominator. Then there is no integer $p$ with $|x - p/q| < \\delta/2$.","title":"Minimality of the smallest denominator"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/smallest-denominator-le-iff","kind":"node","lean":"MetaMathlibExt.smallestDenominator_le_iff","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#smallest-denominator-le-iff","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"Let $x$ be a real number, let $\\delta > 0$ be real, let $q$ be a natural number and let $q_{\\min}(x, \\delta)$ be the smallest denominator. Then $q_{\\min}(x, \\delta) \\le q$ if and only if there are a positive integer $q' \\le q$ and an integer $p$ with $|x - p/q'| < \\delta/2$.","title":"Upper bounds on the smallest denominator"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/smallest-denominator-eq-one","kind":"node","lean":"MetaMathlibExt.smallestDenominator_eq_one","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#smallest-denominator-eq-one","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"For every real number $x$ and every real $\\delta > 1$, the smallest denominator $q_{\\min}(x, \\delta)$ equals $1$.","title":"Wide intervals have smallest denominator one"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/log-smallest-denominator-density","kind":"node","lean":"MetaMathlibExt.logSmallestDenominatorDensity","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#log-smallest-denominator-density","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"The function $\\eta_{\\log}$ on the real line is defined by $\\eta_{\\log}(s) = \\frac{6}{\\pi^2} e^{2s}$ for $s \\le 0$, by $\\eta_{\\log}(s) = \\frac{6}{\\pi^2}\\left(-e^{2s} + 2 + 4s\\right)$ for $0 < s \\le \\log 2$, and by","title":"Density of the logarithmic smallest denominator"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/log-smallest-denominator-density-zero","kind":"node","lean":"MetaMathlibExt.logSmallestDenominatorDensity_zero","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#log-smallest-denominator-density-zero","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"The density $\\eta_{\\log}$ of the logarithmic smallest denominator satisfies $\\eta_{\\log}(0) = 6/\\pi^2$.","title":"Value of the log smallest-denominator density at zero"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/log-smallest-denominator-density-of-nonp","kind":"node","lean":"MetaMathlibExt.logSmallestDenominatorDensity_of_nonpos","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#log-smallest-denominator-density-of-nonp","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"fully_proved","summary":"For every real $s \\le 0$, the density $\\eta_{\\log}$ of the logarithmic smallest denominator satisfies $\\eta_{\\log}(s) = \\frac{6}{\\pi^2} e^{2s}$.","title":"The log smallest-denominator density for nonpositive arguments"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/tendsto-volume-log-smallest-denominator","kind":"node","lean":"MetaMathlibExt.tendsto_volume_log_smallestDenominator_gt","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#tendsto-volume-log-smallest-denominator","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"can_prove","summary":"For real $x$ and $\\delta > 0$ let $q_{\\min}(x, \\delta)$ be the smallest positive integer $q$ such that some rational $p/q$ lies in the open interval $(x - \\delta/2, x + \\delta/2)$, and let $\\eta_{\\log}$ be the log smallest denominator density. Then $\\eta_{\\log}$ is a probability density on $\\mathbb{R}$: $\\eta_{\\log}(s) \\ge 0$ for every real $s$ and $\\int_{-\\infty}^{\\infty} \\eta_{\\log}(s) \\, ds = 1$. Moreover, for\u2026","title":"Limit law of the smallest denominator"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/tendsto-integral-log-smallest-denominato","kind":"node","lean":"MetaMathlibExt.tendsto_integral_log_smallestDenominator_pow","parent":10252,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals/index.html#tendsto-integral-log-smallest-denominato","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/smallest-denominators-in-intervals","status":"can_prove","summary":"Let $0 \\le a \\le b \\le 1$ be real numbers and let $n \\ge 0$ be an integer. For real $x$ and $\\delta > 0$ let $q_{\\min}(x, \\delta)$ be the smallest positive integer $q$ such that some rational $p/q$ lies in the open interval $(x - \\delta/2, x + \\delta/2)$, and let $\\eta_{\\log}$ be the log smallest denominator density. Then, as $\\delta \\to 0^+$,","title":"Convergence of the log moments of smallest denominators"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"can_prove","title":"Square roots of cubic polynomials modulo one"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/norm-sum-exp-two-pi-i-le-of-iterated-der","kind":"node","lean":"MetaMathlibExt.norm_sum_exp_two_pi_I_le_of_iteratedDeriv_two","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#norm-sum-exp-two-pi-i-le-of-iterated-der","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"can_prove","summary":"Let $a < b$ be integers, let $\\rho > 0$, and let $f : \\mathbb{R} \\to \\mathbb{R}$ be twice continuously differentiable, with either $f''(t) \\ge \\rho$ for every $t \\in [a, b]$ or $f''(t) \\le -\\rho$ for every $t \\in [a, b]$. Then","title":"Second-derivative test for exponential sums"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/exists-sqrt-cubic-equidistributed-mod-on","kind":"node","lean":"MetaMathlibExt.exists_sqrt_cubic_equidistributed_mod_one","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#exists-sqrt-cubic-equidistributed-mod-on","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"can_prove","summary":"Square roots of cubics with positive leading coefficient are uniformly distributed modulo one. Let $q(t) = a t^3 + b t^2 + c t + d$ with real coefficients $a, b, c, d$ and $a > 0$, write $\\{x\\} = x - \\lfloor x \\rfloor$ for the fractional part of a real $x$, and let $e(x) = e^{2\\pi i x}$. Then there is an integer $T \\ge 1$ with three properties. First, $q(t) > 0$ for every real $t \\ge T$. Second, the sequence\u2026","title":"Square roots of cubics with positive leading coefficient are uniformly distributed modulo one"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/square-pyramidal-number","kind":"node","lean":"MetaMathlibExt.squarePyramidalNumber","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#square-pyramidal-number","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"fully_proved","summary":"For a natural number $n$, the $n$-th square pyramidal number is $P_n = 1^2 + 2^2 + \\dots + n^2 = \\sum_{i=1}^{n} i^2$ (so $P_0 = 0$).","title":"Square pyramidal number"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/square-pyramidal-square-distance","kind":"node","lean":"MetaMathlibExt.squarePyramidalSquareDistance","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#square-pyramidal-square-distance","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"fully_proved","summary":"For a natural number $n$, let $P_n$ be the $n$-th square pyramidal number and $s = \\lfloor \\sqrt{P_n} \\rfloor$. The distance from $P_n$ to the nearest square is","title":"Distance from a square pyramidal number to the nearest square"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/square-pyramidal-square-distance-example","kind":"node","lean":"MetaMathlibExt.squarePyramidalSquareDistance_examples","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#square-pyramidal-square-distance-example","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"fully_proved","summary":"Let $P_n$ be the $n$-th square pyramidal number and $a_n$ the distance from $P_n$ to the nearest square. Then $P_1, P_2, P_3, P_4$ are $1, 5, 14, 30$; $a_2 = 1$ (since $P_2 = 5$ and the nearest square is $4$); $P_{24} = 4900 = 70^2$; and $a_{24} = 0$ (the cannonball solution).","title":"Small square pyramidal numbers and their distances to squares"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/square-pyramidal-square-distance-average","kind":"node","lean":"MetaMathlibExt.squarePyramidalSquareDistance_average_asymptotic","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#square-pyramidal-square-distance-average","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"can_prove","summary":"Let $a_n$ be the distance from the $n$-th square pyramidal number to the nearest square, and for real $x \\ge 1$ let $A(x) = \\frac{1}{x} \\sum_{1 \\le n \\le x} a_n$. There is a real constant $C$ such that for every real $x \\ge 1$,","title":"Average distance from square pyramidal numbers to the nearest square"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/square-pyramidal-square-distance-moment","kind":"node","lean":"MetaMathlibExt.squarePyramidalSquareDistance_moment_asymptotic","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#square-pyramidal-square-distance-moment","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"can_prove","summary":"Let $a_n$ be the distance from the $n$-th square pyramidal number to the nearest square, and for a positive integer $k$ and real $x \\ge 1$ let $M_k(x) = \\sum_{1 \\le n \\le x} a_n^k$. For every positive integer $k$ there is a real constant $C_k$ such that for every real $x \\ge 1$,","title":"Moments of the distance from square pyramidal numbers to the nearest square"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/square-pyramidal-dist-nearest-square-roo","kind":"node","lean":"MetaMathlibExt.squarePyramidal_dist_nearest_square_root_equidistributed","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#square-pyramidal-dist-nearest-square-roo","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"can_prove","summary":"The distance from the square root of a square pyramidal number to the nearest integer is uniformly distributed. Let $P_n$ be the $n$-th square pyramidal number, write $\\|x\\| = |x - \\operatorname{round}(x)|$ for the distance from a real $x$ to the nearest integer, and let $(y_n)_{n \\ge 0}$ be natural numbers such that $|P_n - y_n^2| \\le |P_n - z^2|$ for all natural numbers $n$ and $z$ (so $y_n^2$ is a square nearest\u2026","title":"The distance from the square root of a square pyramidal number to the nearest integer is uniformly distributed"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/dist-round-pow-fourier-expansion","kind":"node","lean":"MetaMathlibExt.dist_round_pow_fourier_expansion","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#dist-round-pow-fourier-expansion","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"can_prove","summary":"Let $k$ be a positive integer, let $g(u) = |u - \\operatorname{round}(u)|^k$ be the $k$-th power of the distance from a real $u$ to the nearest integer, let $e(x) = e^{2\\pi i x}$, and for every integer $m$ let $c(m) = \\int_0^1 g(u)\\, e(-mu)\\, du$. Then $c(0) = 1/(2^k (k + 1))$, the mean of $g$ over a period; there is a real constant $C$ with $|c(m)| \\le C / m^2$ for every integer $m \\ne 0$; the series\u2026","title":"Fourier expansion of a power of the distance to the nearest integer"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/sum-dist-round-sqrt-square-pyramidal-pow","kind":"node","lean":"MetaMathlibExt.sum_dist_round_sqrt_squarePyramidal_pow","parent":10265,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one/index.html#sum-dist-round-sqrt-square-pyramidal-pow","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/square-roots-of-cubic-polynomials-modulo-one","status":"can_prove","summary":"Power sums of the distance from the square root of a square pyramidal number to the nearest integer. Let $P_n$ be the $n$-th square pyramidal number and write $\\|x\\| = |x - \\operatorname{round}(x)|$ for the distance from a real $x$ to the nearest integer. For every positive integer $k$ there is a real constant $C_k$ such that for every real $t \\ge 1$,","title":"Power sums of the distance from the square root of a square pyramidal number to the nearest integer"},{"data_url":"scopes/NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem.json","id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem","kind":"scope","navigation":"module","parent":10145,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem/index.html","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem","status":"fully_proved","title":"Three-gap theorem"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem/is-three-gap-point","kind":"node","lean":"MetaMathlibExt.IsThreeGapPoint","parent":10276,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem/index.html#is-three-gap-point","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem","status":"fully_proved","summary":"Let $n$ be a natural number and $\\alpha$ a real number, and write $\\operatorname{frac}(t) = t - \\lfloor t \\rfloor$ for the fractional part of a real number $t$. A real number $x$ is a three-gap point of $n$ and $\\alpha$ if $x = \\operatorname{frac}(k\\alpha)$ for some natural number $k < n$.","title":"Points of the three-gap configuration"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem/is-three-gap-length","kind":"node","lean":"MetaMathlibExt.IsThreeGapLength","parent":10276,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem/index.html#is-three-gap-length","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem","status":"fully_proved","summary":"Let $n$ be a natural number and $\\alpha$ a real number. In this definition a point means a three-gap point of $n$ and $\\alpha$, that is, a number $\\operatorname{frac}(k\\alpha) = k\\alpha - \\lfloor k\\alpha \\rfloor$ for a natural number $k < n$. A real number $d$ is a gap length of $n$ and $\\alpha$ if either $d = y - x$ for points $x < y$ with no point strictly between $x$ and $y$, or there are a least point\u2026","title":"Gap lengths of the three-gap configuration"},{"id":"NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem/three-gap","kind":"node","lean":"MetaMathlibExt.three_gap","parent":10276,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem/index.html#three-gap","scope":"NumberTheory/diophantine-approximation/uniform-distribution-metric/three-gap-theorem","status":"fully_proved","summary":"Let $n \\ge 1$ be a natural number and $\\alpha$ an irrational real number. Consider the points $\\operatorname{frac}(k\\alpha) = k\\alpha - \\lfloor k\\alpha \\rfloor$ for $0 \\le k < n$ on the circle $[0, 1)$. The set of gap lengths of $n$ and $\\alpha$, that is, the lengths of the gaps between cyclically consecutive points, including the gap that wraps around from the greatest point to the least, is finite and has at most\u2026","title":"Three-gap theorem"},{"data_url":"scopes/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra.json","id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","kind":"scope","navigation":"module","parent":10092,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","title":"n-Lagrange and n-Markoff spectra"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-lagrange-approximability","kind":"node","lean":"MetaMathlibExt.nLagrangeApproximability","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(n\\)"],["text","-Lagrange approximability"]],"source":"$n$-Lagrange approximability"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-lagrange-approximability","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"fully_proved","summary":"For a natural number $n$ and a real number $\\xi$, the $n$-Lagrange approximability of $\\xi$ is the element of the extended reals","title":"$n$-Lagrange approximability"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-lagrange-approximability-one","kind":"node","lean":"MetaMathlibExt.nLagrangeApproximability_one","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(1\\)"],["text","-Lagrange approximability is the classical one"]],"source":"The $1$-Lagrange approximability is the classical one"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-lagrange-approximability-one","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"fully_proved","summary":"For every real number $\\xi$, the $1$-Lagrange approximability of $\\xi$ is","title":"The $1$-Lagrange approximability is the classical one"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-markoff-approximability","kind":"node","lean":"MetaMathlibExt.nMarkoffApproximability","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(n\\)"],["text","-Markoff approximability"]],"source":"$n$-Markoff approximability"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-markoff-approximability","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"fully_proved","summary":"For a natural number $n$ and real numbers $\\xi$ and $\\xi'$, the $n$-Markoff approximability is the supremum in the extended reals","title":"$n$-Markoff approximability"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-markoff-approximability-comm","kind":"node","lean":"MetaMathlibExt.nMarkoffApproximability_comm","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Symmetry of the "],["math","\\(n\\)"],["text","-Markoff approximability"]],"source":"Symmetry of the $n$-Markoff approximability"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-markoff-approximability-comm","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"fully_proved","summary":"For every natural number $n$ and all real numbers $\\xi$ and $\\xi'$, the $n$-Markoff approximability satisfies $\\mu_n(\\xi, \\xi') = \\mu_n(\\xi', \\xi)$.","title":"Symmetry of the $n$-Markoff approximability"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-lagrange-spectrum","kind":"node","lean":"MetaMathlibExt.nLagrangeSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(n\\)"],["text","-Lagrange spectrum"]],"source":"$n$-Lagrange spectrum"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-lagrange-spectrum","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"fully_proved","summary":"For a natural number $n$, the $n$-Lagrange spectrum $L_n$ is the set of real numbers $\\alpha$ such that $\\lambda_n(\\xi) = \\alpha$ for some irrational real number $\\xi$, where $\\lambda_n$ is the $n$-Lagrange approximability; only finite values of $\\lambda_n$ count. For $n = 1$ this is the classical Lagrange spectrum $L$.","title":"$n$-Lagrange spectrum"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/mem-n-lagrange-spectrum","kind":"node","lean":"MetaMathlibExt.mem_nLagrangeSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in the "],["math","\\(n\\)"],["text","-Lagrange spectrum"]],"source":"Membership in the $n$-Lagrange spectrum"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#mem-n-lagrange-spectrum","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"fully_proved","summary":"For every natural number $n$ and real number $\\alpha$, $\\alpha$ lies in the $n$-Lagrange spectrum $L_n$ if and only if $\\alpha = \\lambda_n(\\xi)$ for some irrational real number $\\xi$, where $\\lambda_n$ is the $n$-Lagrange approximability.","title":"Membership in the $n$-Lagrange spectrum"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-markoff-spectrum","kind":"node","lean":"MetaMathlibExt.nMarkoffSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text","-Markoff spectrum"]],"source":"The $n$-Markoff spectrum"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-markoff-spectrum","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"fully_proved","summary":"For a natural number $n$ and irrational numbers $\\xi \\ne \\xi'$, the $n$-Markoff approximability $\\mu_n(\\xi, \\xi')$ is the extended real number $\\sup_{(s, t) \\in \\mathbb{Z}^2 \\setminus \\{0\\}} \\gcd(t, n) \\, |\\xi - \\xi'| / (|s - t\\xi| \\, |s - t\\xi'|)$. The $n$-Markoff spectrum $M_n$ is the set of real numbers $\\alpha$ such that $\\mu_n(\\xi, \\xi') = \\alpha$ for some irrational numbers $\\xi \\ne \\xi'$. For $n = 1$ this is\u2026","title":"The $n$-Markoff spectrum"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/mem-n-markoff-spectrum","kind":"node","lean":"MetaMathlibExt.mem_nMarkoffSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in the "],["math","\\(n\\)"],["text","-Markoff spectrum"]],"source":"Membership in the $n$-Markoff spectrum"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#mem-n-markoff-spectrum","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"fully_proved","summary":"For every natural number $n$ and real number $\\alpha$, $\\alpha$ lies in the $n$-Markoff spectrum $M_n$ if and only if $\\alpha = \\mu_n(\\xi, \\xi')$ for some distinct irrational real numbers $\\xi$ and $\\xi'$, where $\\mu_n$ is the $n$-Markoff approximability.","title":"Membership in the $n$-Markoff spectrum"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/convergent-quality-and-lagrange-limsup","kind":"node","lean":"MetaMathlibExt.convergent_quality_and_lagrange_limsup","parent":10280,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#convergent-quality-and-lagrange-limsup","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $\\xi$ be an irrational real number, with complete quotients $\\xi_0 = \\xi$ and $\\xi_{k+1} = 1/\\{\\xi_k\\}$, where $\\{x\\}$ is the fractional part of $x$, and digits $a_k = \\lfloor \\xi_k \\rfloor$, so that $\\xi = [a_0; a_1, a_2, \\dots]$. For $k \\ge 1$ let $c_k = a_k + [0; a_{k-1}, \\dots, a_1] + \\{\\xi_k\\}$, where the middle term is $0$ when $k = 1$. Then for every $k \\ge 1$ the convergent\u2026","title":"Convergents and the Lagrange approximability"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/abs-sub-le-of-cont-frac-prefix-eq","kind":"node","lean":"MetaMathlibExt.abs_sub_le_of_contFrac_prefix_eq","parent":10280,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#abs-sub-le-of-cont-frac-prefix-eq","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $\\xi$ and $\\xi'$ be real numbers with continued fraction expansions $[a_0; a_1, a_2, \\dots]$ and $[a_0'; a_1', a_2', \\dots]$, computed by $\\xi_0 = \\xi$, $a_i = \\lfloor \\xi_i \\rfloor$ and $\\xi_{i+1} = 1/(\\xi_i - a_i)$, stopping when $\\xi_i = a_i$ (so an expansion is finite exactly when the number is rational). Let $k \\ge 0$ be an integer, and suppose that $a_0 = a_0'$ and that for each $1 \\le i \\le k$ either\u2026","title":"Real numbers with a common continued fraction prefix are close"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-approximability-eq-sup-divisors","kind":"node","lean":"MetaMathlibExt.nApproximability_eq_sup_divisors","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text","-approximabilities as maxima over divisors"]],"source":"The $n$-approximabilities as maxima over divisors"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-approximability-eq-sup-divisors","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $n \\ge 1$, let $\\lambda_n$ and $\\mu_n$ be the $n$-Lagrange and $n$-Markoff approximabilities, and let $\\xi \\ne \\xi'$ be irrational real numbers. Then $\\lambda_n(\\xi) = \\max_{g \\mid n} \\lambda_1(g\\xi)$ and $\\mu_n(\\xi, \\xi') = \\max_{g \\mid n} \\mu_1(g\\xi, g\\xi')$, where $g$ runs over the positive divisors of $n$ and the maxima are taken in the extended reals.","title":"The $n$-approximabilities as maxima over divisors"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-spectrum-subset-of-dvd","kind":"node","lean":"MetaMathlibExt.nSpectrum_subset_of_dvd","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility inclusions of the "],["math","\\(n\\)"],["text","-Lagrange and "],["math","\\(n\\)"],["text","-Markoff spectra"]],"source":"Divisibility inclusions of the $n$-Lagrange and $n$-Markoff spectra"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-spectrum-subset-of-dvd","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $m$ and $n$ be natural numbers with $m \\ge 1$, $n \\ge 1$ and $m \\mid n$, and write $L_k$ and $M_k$ for the $k$-Lagrange and $k$-Markoff spectra. Then $L_n \\subseteq L_m$ and $M_n \\subseteq M_m$; in particular $L_n \\subseteq L_1$ and $M_n \\subseteq M_1$, where $L_1$ and $M_1$ are the classical Lagrange and Markoff spectra.","title":"Divisibility inclusions of the $n$-Lagrange and $n$-Markoff spectra"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-lagrange-spectrum-subset-n-markoff-spe","kind":"node","lean":"MetaMathlibExt.nLagrangeSpectrum_subset_nMarkoffSpectrum_attained","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text","-Lagrange spectrum lies in the "],["math","\\(n\\)"],["text","-Markoff spectrum, with attained suprema"]],"source":"The $n$-Lagrange spectrum lies in the $n$-Markoff spectrum, with attained suprema"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-lagrange-spectrum-subset-n-markoff-spe","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $n \\ge 1$, let $L_n$ and $M_n$ be the $n$-Lagrange and $n$-Markoff spectra, and let $\\mu_n$ be the $n$-Markoff approximability. Then $L_n \\subseteq M_n$. Moreover, for every $\\alpha \\in L_n$ there are distinct irrational real numbers $\\xi$ and $\\xi'$ with $\\mu_n(\\xi, \\xi') = \\alpha$ at which the supremum defining $\\mu_n(\\xi, \\xi')$ is attained: some pair of integers $(s, t) \\ne (0, 0)$ satisfies","title":"The $n$-Lagrange spectrum lies in the $n$-Markoff spectrum, with attained suprema"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-markoff-spectrum-attained","kind":"node","lean":"MetaMathlibExt.nMarkoffSpectrum_attained","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Attained suprema in the "],["math","\\(n\\)"],["text","-Markoff spectrum"]],"source":"Attained suprema in the $n$-Markoff spectrum"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-markoff-spectrum-attained","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $n \\ge 1$, let $M_n$ be the $n$-Markoff spectrum, and let $\\mu_n$ be the $n$-Markoff approximability. For every $\\alpha \\in M_n$ there are distinct irrational real numbers $\\xi$ and $\\xi'$ with $\\mu_n(\\xi, \\xi') = \\alpha$ at which the supremum defining $\\mu_n(\\xi, \\xi')$ is attained: some pair of integers $(s, t) \\ne (0, 0)$ satisfies","title":"Attained suprema in the $n$-Markoff spectrum"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/is-closed-n-markoff-spectrum","kind":"node","lean":"MetaMathlibExt.isClosed_nMarkoffSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text","-Markoff spectrum is closed"]],"source":"The $n$-Markoff spectrum is closed"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#is-closed-n-markoff-spectrum","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"For every integer $n \\ge 1$, the $n$-Markoff spectrum $M_n$ is a closed subset of $\\mathbb{R}$.","title":"The $n$-Markoff spectrum is closed"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/exists-is-least-n-lagrange-spectrum-n-ma","kind":"node","lean":"MetaMathlibExt.exists_isLeast_nLagrangeSpectrum_nMarkoffSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Common minimum of the "],["math","\\(n\\)"],["text","-Lagrange and "],["math","\\(n\\)"],["text","-Markoff spectra"]],"source":"Common minimum of the $n$-Lagrange and $n$-Markoff spectra"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#exists-is-least-n-lagrange-spectrum-n-ma","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"For every integer $n \\ge 1$ there is a real number $\\alpha$ that is the least element both of the $n$-Lagrange spectrum $L_n$ and of the $n$-Markoff spectrum $M_n$.","title":"Common minimum of the $n$-Lagrange and $n$-Markoff spectra"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/smul-spectrum-subset-n-spectrum","kind":"node","lean":"MetaMathlibExt.smul_spectrum_subset_nSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Scaled classical spectra lie in the "],["math","\\(n\\)"],["text","-spectra"]],"source":"Scaled classical spectra lie in the $n$-spectra"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#smul-spectrum-subset-n-spectrum","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $n \\ge 1$ be a natural number and write $L_k$ and $M_k$ for the $k$-Lagrange and $k$-Markoff spectra, so that $L_1$ and $M_1$ are the classical Lagrange and Markoff spectra. Then $n\\alpha \\in L_n$ for every $\\alpha \\in L_1$ and $n\\alpha \\in M_n$ for every $\\alpha \\in M_1$; that is, $n L_1 \\subseteq L_n$ and $n M_1 \\subseteq M_n$.","title":"Scaled classical spectra lie in the $n$-spectra"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/hall-ray-n-lagrange-spectrum","kind":"node","lean":"MetaMathlibExt.hall_ray_nLagrangeSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hall's ray in the "],["math","\\(n\\)"],["text","-Lagrange spectrum"]],"source":"Hall's ray in the $n$-Lagrange spectrum"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#hall-ray-n-lagrange-spectrum","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $L_1$ be the classical Lagrange spectrum and let $F$ (Fre\u012dman's constant) be the infimum of the real numbers $c$ with $[c, \\infty) \\subseteq L_1$. Then for every integer $n \\ge 1$, $[nF, \\infty) \\subseteq L_n \\subseteq M_n$, where $L_n$ is the $n$-Lagrange spectrum and $M_n$ is the $n$-Markoff spectrum.","title":"Hall's ray in the $n$-Lagrange spectrum"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/is-closed-n-lagrange-spectrum","kind":"node","lean":"MetaMathlibExt.isClosed_nLagrangeSpectrum","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text","-Lagrange spectrum is closed"]],"source":"The $n$-Lagrange spectrum is closed"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#is-closed-n-lagrange-spectrum","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"For every integer $n \\ge 1$, the $n$-Lagrange spectrum $L_n$ is a closed subset of $\\mathbb{R}$.","title":"The $n$-Lagrange spectrum is closed"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/n-lagrange-spectrum-eq-closure-quadratic","kind":"node","lean":"MetaMathlibExt.nLagrangeSpectrum_eq_closure_quadratic","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text","-Lagrange spectrum is the closure of its quadratic-irrational values"]],"source":"The $n$-Lagrange spectrum is the closure of its quadratic-irrational values"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#n-lagrange-spectrum-eq-closure-quadratic","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $n \\ge 1$, let $\\lambda_n$ be the $n$-Lagrange approximability, and let $P_n$ be the set of real numbers $\\alpha$ such that $\\alpha = \\lambda_n(\\xi)$ for some quadratic irrational $\\xi$, that is, an irrational real number with $a\\xi^2 + b\\xi + c = 0$ for some integers $a \\ne 0$, $b$, $c$. Then the $n$-Lagrange spectrum $L_n$ is the closure of $P_n$ in $\\mathbb{R}$.","title":"The $n$-Lagrange spectrum is the closure of its quadratic-irrational values"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/min-n-lagrange-spectrum-isolated-sqrt-ra","kind":"node","lean":"MetaMathlibExt.min_nLagrangeSpectrum_isolated_sqrt_rat_of_prime_lt_2000","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Isolated minimum of the "],["math","\\(p\\)"],["text","-Lagrange spectrum for primes below 2000"]],"source":"Isolated minimum of the $p$-Lagrange spectrum for primes below 2000"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#min-n-lagrange-spectrum-isolated-sqrt-ra","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"For every prime $p < 2000$, the $p$-Lagrange spectrum $L_p$ has a least element $\\alpha$ with the following properties: $\\alpha$ is an isolated point of $L_p$, that is, there is $\\varepsilon > 0$ such that every $\\beta \\in L_p$ with $|\\beta - \\alpha| < \\varepsilon$ equals $\\alpha$; $\\alpha$ is irrational; and $\\alpha = \\sqrt{r}$ for some rational number $r$.","title":"Isolated minimum of the $p$-Lagrange spectrum for primes below 2000"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/no-eventually-one-two-cont-frac-pair-iff","kind":"node","lean":"MetaMathlibExt.no_eventually_one_two_contFrac_pair_iff_eq_67","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","The prime "],["math","\\(67\\)"],["text"," and continued fraction digits "],["math","\\(1\\)"],["text"," and "],["math","\\(2\\)"]],"source":"The prime $67$ and continued fraction digits $1$ and $2$"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#no-eventually-one-two-cont-frac-pair-iff","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"For an irrational real number $\\xi$, define its complete quotients by $\\xi_0 = \\xi$ and $\\xi_{k+1} = 1/\\{\\xi_k\\}$, where $\\{\\cdot\\}$ denotes the fractional part, and its continued fraction digits by $a_k = \\lfloor \\xi_k \\rfloor$. Let $p < 2000$ be a prime. Then there is no irrational $\\xi$ such that the digits of both $\\xi$ and $p\\xi$ eventually all lie in $\\{1, 2\\}$ if and only if $p = 67$.","title":"The prime $67$ and continued fraction digits $1$ and $2$"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/markoff-point-mem-n-lagrange-spectrum-if","kind":"node","lean":"MetaMathlibExt.markoff_point_mem_nLagrangeSpectrum_iff_principal_norm","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Markoff points in the "],["math","\\(p\\)"],["text","-Lagrange spectrum"]],"source":"Markoff points in the $p$-Lagrange spectrum"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#markoff-point-mem-n-lagrange-spectrum-if","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $x \\le y \\le z$ be positive integers with $x^2 + y^2 + z^2 = 3xyz$ (a sorted Markoff triple), let $\\lambda_z = \\sqrt{9 - 4/z^2}$, let $D = 9z^2 - 4$ if $z$ is odd and $D = (9z^2 - 4)/4$ if $z$ is even, and let $p$ be a prime. Call $p$ a principal norm if $|u^2 + Duv + \\frac{D^2 - D}{4} v^2| = p$ for some integers $u, v$, that is, if some element $u + v \\frac{D + \\sqrt{D}}{2}$ of the quadratic order\u2026","title":"Markoff points in the $p$-Lagrange spectrum"},{"id":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/markoff-points-mem-n-lagrange-spectrum-i","kind":"node","lean":"MetaMathlibExt.markoff_points_mem_nLagrangeSpectrum_iff","parent":10280,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruence conditions for Markoff points in the "],["math","\\(p\\)"],["text","-Lagrange spectrum"]],"source":"Congruence conditions for Markoff points in the $p$-Lagrange spectrum"},"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra/index.html#markoff-points-mem-n-lagrange-spectrum-i","scope":"NumberTheory/diophantine-approximation/n-lagrange-and-n-markoff-spectra","status":"can_prove","summary":"Let $p$ be a prime and let $L_p$ be the $p$-Lagrange spectrum. Then: (a) $\\sqrt{5} \\in L_p$ if and only if $p \\equiv 0, \\pm 1 \\pmod 5$; (b) $2\\sqrt{2} \\in L_p$ if and only if $p \\equiv 2, \\pm 1 \\pmod 8$; (c) $\\sqrt{221}/5 \\in L_p$ if and only if $p$ is a square modulo $13$ and modulo $17$; (d) $\\sqrt{1517}/13 \\in L_p$ if and only if $p$ is a square modulo $37$ and modulo $41$; (e) $\\sqrt{7565}/29 \\in L_p$ if and\u2026","title":"Congruence conditions for Markoff points in the $p$-Lagrange spectrum"},{"data_url":"scopes/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces.json","id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","kind":"scope","navigation":"module","parent":10092,"result":false,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"can_prove","title":"Rational approximation of linear subspaces"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/dist-lines","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.distLines","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#dist-lines","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"For lines $x$ and $y$ through the origin in $\\mathbb{R}^d$, the distance $\\operatorname{dist}(x, y)$ is $\\sin \\theta$, where $\\theta \\in [0, \\pi]$ is the angle between nonzero vectors spanning $x$ and $y$. This is the usual distance on the projective space.","title":"Distance between lines"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/dist-line-subspace","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.distLineSubspace","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#dist-line-subspace","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"For a line $x$ through the origin in $\\mathbb{R}^d$ and a subspace $y$ of $\\mathbb{R}^d$, the distance $\\operatorname{dist}(x, y)$ is the infimum, over all lines $y'$ contained in $y$, of the distance $\\operatorname{dist}(x, y')$ between the lines $x$ and $y'$. When $y = 0$ there is no such line and the value is $0$.","title":"Distance from a line to a subspace"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/dist-subspaces","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.distSubspaces","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#dist-subspaces","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"For subspaces $x$ and $y$ of $\\mathbb{R}^d$, the distance $\\operatorname{dist}(x, y)$ is defined as follows. If $\\dim y \\le \\dim x$, it is the supremum, over all lines $y'$ contained in $y$, of the distance $\\operatorname{dist}(y', x)$ from the line $y'$ to the subspace $x$. If $\\dim y > \\dim x$, it is the supremum, over all lines $x'$ contained in $x$, of the distance $\\operatorname{dist}(x', y)$ from the line\u2026","title":"Distance between subspaces"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/int-points","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.intPoints","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#int-points","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"$\\mathbb{Z}^d$ is the $\\mathbb{Z}$-submodule of $\\mathbb{R}^d$ spanned by the standard basis vectors, that is, the lattice of points with integer coordinates.","title":"The integer lattice"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/int-points-subspace","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.intPointsSubspace","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#int-points-subspace","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"For a subspace $V$ of $\\mathbb{R}^d$, this is the set $V \\cap \\mathbb{Z}^d$ of integer points of $V$, as a $\\mathbb{Z}$-submodule of $V$.","title":"Integer points of a subspace"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/is-rational","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.IsRational","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#is-rational","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"A subspace $V$ of $\\mathbb{R}^d$ is rational if $V \\cap \\mathbb{Z}^d$ is a full-rank lattice in $V$, that is, a discrete subgroup of $V$ that spans $V$ over $\\mathbb{R}$.","title":"Rational subspace"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/chosen-basis","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.chosenBasis","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#chosen-basis","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"For a rational subspace $V$ of $\\mathbb{R}^d$, this is a fixed choice of a $\\mathbb{Z}$-basis of the lattice $V \\cap \\mathbb{Z}^d$, indexed by $\\{0, \\dots, \\dim V - 1\\}$ and viewed as an $\\mathbb{R}$-basis of $V$.","title":"Chosen basis of a rational subspace"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/height","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.height","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#height","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"For a rational subspace $V$ of $\\mathbb{R}^d$ of dimension $k$, let $v_1, \\dots, v_k$ be the chosen basis of the lattice $V \\cap \\mathbb{Z}^d$. The height of $V$ is $H(V) = \\lVert v_1 \\wedge \\dots \\wedge v_k \\rVert$, where the exterior power $\\bigwedge^k \\mathbb{R}^d$ carries the inner product whose value on decomposable elements $u_1 \\wedge \\dots \\wedge u_k$ and $w_1 \\wedge \\dots \\wedge w_k$ is the Gram\u2026","title":"Height of a rational subspace"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/diophantine-exponent","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.diophantineExponent","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#diophantine-exponent","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"fully_proved","summary":"Let $x$ be a subspace of $\\mathbb{R}^d$ and let $k$ be a natural number. For subspaces $v$ of $\\mathbb{R}^d$ write $\\operatorname{dist}(v, x)$ for the distance between the subspaces $v$ and $x$, and for a rational subspace $v$ write $H(v)$ for its height. The diophantine exponent $\\beta_k(x)$ of $x$ for approximation by $k$-dimensional rational subspaces is the infimum, taken in $[0, \\infty]$, of the set of real\u2026","title":"Diophantine exponent"},{"id":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/theorem-1","kind":"node","lean":"MathlibExt.NumberTheory.LinearSubspacesWanted.LinearSubspaces.theorem_1","parent":10305,"result":true,"route":"../roadmap/NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces/index.html#theorem-1","scope":"NumberTheory/diophantine-approximation/rational-approximation-of-linear-subspaces","status":"can_prove","summary":"Let $d$, $l$ and $k$ be natural numbers with $d \\ge 2$, $0 < l < d$ and $1 \\le k \\le l$, and for a subspace $x$ of $\\mathbb{R}^d$ write $\\beta_k(x)$ for the diophantine exponent of $x$ for approximation by $k$-dimensional rational subspaces. Then every $l$-dimensional subspace $x$ of $\\mathbb{R}^d$ satisfies $\\beta_k(x) \\ge \\frac{d}{k(d - l)}$. Moreover, some $l$-dimensional subspace $x$ of $\\mathbb{R}^d$ satisfies\u2026","title":"Dirichlet's principle in Grassmann varieties"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions.json","id":"NumberTheory/continued-fractions-expansions","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/index.html","scope":"NumberTheory/continued-fractions-expansions","status":"can_prove","summary":"Continued fractions, Farey sequences, Stern\u2013Brocot tree, Engel, Pierce, L\u00fcroth and similar series expansions of real numbers.","title":"Continued fractions and expansions of reals"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01.json","id":"NumberTheory/continued-fractions-expansions/part-01","kind":"scope","navigation":"transparent","parent":10316,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01","status":"can_prove","title":"Part 1 of 2"},{"id":"NumberTheory/continued-fractions-expansions/part-01/is-laplace-qfamily","kind":"node","lean":"MetaMathlibExt.IsLaplaceQFamily","parent":10317,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/index.html#is-laplace-qfamily","scope":"NumberTheory/continued-fractions-expansions/part-01","status":"fully_proved","summary":"Let $R$ be a semiring and let $(P_k)_{k \\ge 0}$ and $(Q_k)_{k \\ge 0}$ be sequences of polynomials in $R[X]$. The sequence $(Q_k)$ is a Laplace Q-polynomial family companion to $(P_k)$ when $Q_0 = 1$ and, for every natural number $k \\ge 1$,","title":"Laplace Q-polynomial families"},{"id":"NumberTheory/continued-fractions-expansions/part-01/lagrange-quadratic-convergent","kind":"node","lean":"MetaMathlibExt.lagrange_quadratic_convergent","parent":10317,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(P u^2 + Q u y + R y^2 = 1\\)"],["text"," give convergents of "],["math","\\(-Q/(2P)\\)"]],"source":"Solutions of $P u^2 + Q u y + R y^2 = 1$ give convergents of $-Q/(2P)$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/index.html#lagrange-quadratic-convergent","scope":"NumberTheory/continued-fractions-expansions/part-01","status":"fully_proved","summary":"Let $P, Q, R, u, y$ be integers with $P > 0$ and $y > 0$, and let $D = Q^2 - 4PR$. Assume that $D < 0$, that $D \\ne -3$, and that $P \\ne 2$ if $D = -4$. If $P u^2 + Q u y + R y^2 = 1$, then, writing $\\omega = -Q/(2P)$ with regular continued fraction expansion $\\omega = [a_0; a_1, a_2, \\dots]$, there is a natural number $n$ such that $u/y$ equals the $n$-th convergent of $\\omega$ and either $n = 0$ or the partial\u2026","title":"Solutions of $P u^2 + Q u y + R y^2 = 1$ give convergents of $-Q/(2P)$"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series.json","id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"can_prove","title":"Continued fractions of alternating series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/alternating-series-cont-fract-two-step-p","kind":"node","lean":"MetaMathlibExt.alternatingSeries_contFract_twoStepPattern","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#alternating-series-cont-fract-two-step-p","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"can_prove","summary":"Let $(u_i)_{i \\ge 0}$ be positive integers with $u_0 = 1$, and let $x$ be an irrational real number with $\\sum_{i \\ge 0} (-1)^i / u_i = x$. Write $x = [a_0; a_1, a_2, \\dots]$ for its continued fraction expansion and $p_n / q_n = [a_0; a_1, \\dots, a_n]$ for its convergents, with $q_0 = 1$ and $q_1 = a_1$. Suppose that for every $n \\ge 0$ we have $\\sum_{i=0}^{n} (-1)^i / u_i = p_{n+1} / q_{n+1}$,\u2026","title":"Alternating series with a two-step continued fraction pattern"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/alternating-series-cont-fract-three-step","kind":"node","lean":"MetaMathlibExt.alternatingSeries_contFract_threeStepPattern","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#alternating-series-cont-fract-three-step","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"can_prove","summary":"Let $(u_i)_{i \\ge 0}$ be positive integers with $u_0 = 1$, and let $x$ be an irrational real number with $\\sum_{i \\ge 0} (-1)^i / u_i = x$. Write $x = [a_0; a_1, a_2, \\dots]$ for its continued fraction expansion and $p_n / q_n = [a_0; a_1, \\dots, a_n]$ for its convergents, with $q_0 = 1$ and $q_1 = a_1$. Suppose that for every $n \\ge 0$ we have $\\sum_{i=0}^{n} (-1)^i / u_i = p_{n+1} / q_{n+1}$,\u2026","title":"Alternating series with a three-step continued fraction pattern"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/two-step-pattern-den","kind":"node","lean":"MetaMathlibExt.twoStepPatternDen","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#two-step-pattern-den","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"fully_proved","summary":"For a natural number $\\beta$ and a sequence $(\\omega_n)_{n \\ge 0}$ of natural numbers, define natural numbers $q_n$ by $q_0 = 1$, $q_1 = \\omega_0$, and for every $n \\ge 0$, $q_{2n+2} = q_{2n+1} + q_{2n}$ and $q_{2n+3} = \\omega_{n+1} q_{2n+1}^\\beta q_{2n+2} + q_{2n+1}$.","title":"Convergent denominators of the two-step pattern"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/two-step-pattern-den-one-one","kind":"node","lean":"MetaMathlibExt.twoStepPatternDen_one_one","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#two-step-pattern-den-one-one","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"fully_proved","summary":"For $\\beta = 1$ and $\\omega_n = 1$ for all $n$, the convergent denominators $q_0, \\dots, q_7$ of the two-step pattern are $1, 1, 2, 3, 5, 18, 23, 432$ (the paper's first example).","title":"Small values of the two-step pattern denominators"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/three-step-pattern-den","kind":"node","lean":"MetaMathlibExt.threeStepPatternDen","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#three-step-pattern-den","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"fully_proved","summary":"For a natural number $\\beta$ and a sequence $(\\omega_n)_{n \\ge 0}$ of natural numbers, define natural numbers $q_n$ by $q_0 = 1$, $q_1 = \\omega_0$, and for every $n \\ge 0$, $q_{3n+2} = q_{3n+1} + q_{3n}$, $q_{3n+3} = q_{3n+2} + q_{3n+1}$ and $q_{3n+4} = \\omega_{n+1} q_{3n+3} q_{3n+2}^\\beta + q_{3n+2}$.","title":"Convergent denominators of the three-step pattern"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/three-step-pattern-den-one-one","kind":"node","lean":"MetaMathlibExt.threeStepPatternDen_one_one","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#three-step-pattern-den-one-one","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"fully_proved","summary":"For $\\beta = 1$ and $\\omega_n = 1$ for all $n$, the convergent denominators $q_0, \\dots, q_7$ of the three-step pattern are $1, 1, 2, 3, 8, 11, 19, 220$.","title":"Small values of the three-step pattern denominators"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/two-step-pattern-alternating-series-cont","kind":"node","lean":"MetaMathlibExt.twoStepPattern_alternatingSeries_contFract","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#two-step-pattern-alternating-series-cont","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"can_prove","summary":"Let $\\beta$ be a natural number, let $(\\omega_n)_{n \\ge 0}$ be positive integers, and let $(q_n)_{n \\ge 0}$ be the convergent denominators of the two-step pattern for $\\beta$ and $(\\omega_n)$. Then the series $\\sum_{i \\ge 0} (-1)^i / (q_i q_{i+1})$ converges absolutely, and its sum $x$ has continued fraction expansion $[a_0; a_1, a_2, \\dots]$ with $a_0 = 0$, $a_1 = \\omega_0$, and $a_{2n+2} = 1$ and\u2026","title":"Two-step pattern as an alternating series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/two-step-pattern-irrationality-exponent","kind":"node","lean":"MetaMathlibExt.twoStepPattern_irrationalityExponent","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#two-step-pattern-irrationality-exponent","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"can_prove","summary":"Let $\\beta \\ge 1$ be a natural number, let $(\\omega_n)_{n \\ge 0}$ be positive integers, let $(q_n)_{n \\ge 0}$ be the convergent denominators of the two-step pattern for $\\beta$ and $(\\omega_n)$, and let $x = \\sum_{i \\ge 0} (-1)^i / (q_i q_{i+1})$. Let $L$ be the limit superior, in the extended real numbers, of $\\ln \\omega_n / \\ln q_{2n}$ as $n \\to \\infty$. Then $x$ is Liouville with exponent $p$ for every real\u2026","title":"Irrationality exponent of the two-step pattern"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/three-step-pattern-alternating-series-co","kind":"node","lean":"MetaMathlibExt.threeStepPattern_alternatingSeries_contFract","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#three-step-pattern-alternating-series-co","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"can_prove","summary":"Let $\\beta$ be a natural number, let $(\\omega_n)_{n \\ge 0}$ be positive integers, and let $(q_n)_{n \\ge 0}$ be the convergent denominators of the three-step pattern for $\\beta$ and $(\\omega_n)$. Then the series $\\sum_{i \\ge 0} (-1)^i / (q_i q_{i+1})$ converges absolutely, and its sum $x$ has continued fraction expansion $[a_0; a_1, a_2, \\dots]$ with $a_0 = 0$, $a_1 = \\omega_0$, and $a_{3n+2} = a_{3n+3} = 1$ and\u2026","title":"Three-step pattern as an alternating series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/three-step-pattern-irrationality-exponen","kind":"node","lean":"MetaMathlibExt.threeStepPattern_irrationalityExponent","parent":10320,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series/index.html#three-step-pattern-irrationality-exponen","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-alternating-series","status":"can_prove","summary":"Let $\\beta \\ge 1$ be a natural number, let $(\\omega_n)_{n \\ge 0}$ be positive integers, let $(q_n)_{n \\ge 0}$ be the convergent denominators of the three-step pattern for $\\beta$ and $(\\omega_n)$, and let $x = \\sum_{i \\ge 0} (-1)^i / (q_i q_{i+1})$. Let $L$ be the limit superior, in the extended real numbers, of $\\ln \\omega_n / \\ln q_{3n}$ as $n \\to \\infty$. Then $x$ is Liouville with exponent $p$ for every real\u2026","title":"Irrationality exponent of the three-step pattern"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series.json","id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","title":"Continued fractions of Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-inv-var","kind":"node","lean":"MetaMathlibExt.laurentInvVar","parent":10331,"reading_title":{"format":"inline-parts-v1","parts":[["text","The variable "],["math","\\(x\\)"],["text"," of "],["math","\\(K((x^{-1}))\\)"]],"source":"The variable $x$ of $K((x^{-1}))$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-inv-var","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field. Formal Laurent series in $x^{-1}$ over $K$ are realized as Laurent series $\\sum_n c_n X^n$ in $X = x^{-1}$, and the variable $x$ is the series $X^{-1}$, with coefficient $1$ at $X^{-1}$ and $0$ elsewhere.","title":"The variable $x$ of $K((x^{-1}))$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-inv-var-mul-single-one","kind":"node","lean":"MetaMathlibExt.laurentInvVar_mul_single_one","parent":10331,"reading_title":{"format":"inline-parts-v1","parts":[["text","The variable "],["math","\\(x\\)"],["text"," times "],["math","\\(x^{-1}\\)"],["text"," is one"]],"source":"The variable $x$ times $x^{-1}$ is one"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-inv-var-mul-single-one","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field. In $K((x^{-1}))$, the product of the variable $x$ and the series $x^{-1}$ (coefficient $1$ at $x^{-1}$ and $0$ elsewhere) is $1$.","title":"The variable $x$ times $x^{-1}$ is one"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/poly-to-laurent-inv","kind":"node","lean":"MetaMathlibExt.polyToLaurentInv","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#poly-to-laurent-inv","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field. The embedding $K[x] \\to K((x^{-1}))$ is the $K$-algebra homomorphism that sends $x$ to the element $x$ of $K((x^{-1}))$, that is, it evaluates a polynomial at $x$.","title":"Polynomials as Laurent series in the inverse variable"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/poly-to-laurent-inv-x","kind":"node","lean":"MetaMathlibExt.polyToLaurentInv_X","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#poly-to-laurent-inv-x","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field. The embedding $K[x] \\to K((x^{-1}))$ sends the polynomial $x$ to the element $x$ of $K((x^{-1}))$.","title":"The polynomial variable embeds as the Laurent variable"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-poly-part","kind":"node","lean":"MetaMathlibExt.laurentPolyPart","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-poly-part","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field and write a Laurent series $\\alpha \\in K((x^{-1}))$ as $\\alpha = \\sum_n c_n x^{-n}$. Its polynomial part is $[\\alpha] = \\sum_{n \\le 0} c_n x^{-n}$, the sum of the terms in nonnegative powers of $x$.","title":"Polynomial part of a Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-poly-part-coeff","kind":"node","lean":"MetaMathlibExt.laurentPolyPart_coeff","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-poly-part-coeff","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field and write $\\alpha \\in K((x^{-1}))$ as $\\alpha = \\sum_n c_n x^{-n}$. For every integer $n$, the coefficient of $x^{-n}$ in the polynomial part $[\\alpha]$ is $c_n$ if $n \\le 0$ and $0$ if $n > 0$.","title":"Coefficients of the polynomial part of a Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-poly-part-laurent-inv-var","kind":"node","lean":"MetaMathlibExt.laurentPolyPart_laurentInvVar","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-poly-part-laurent-inv-var","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field. In $K((x^{-1}))$ the polynomial part of $x$ is $x$, and the polynomial part of $x^{-1}$ is $0$.","title":"Polynomial parts of the variable and its inverse"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-complete-quotient","kind":"node","lean":"MetaMathlibExt.laurentCompleteQuotient","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-complete-quotient","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field and $\\alpha \\in K((x^{-1}))$. The complete quotients of the continued fraction expansion of $\\alpha$ are $\\alpha_0 = \\alpha$ and $\\alpha_{n+1} = (\\alpha_n - [\\alpha_n])^{-1}$ for $n \\ge 0$, where $[\\beta]$ is the polynomial part of $\\beta$ and $0^{-1} = 0$. So the complete quotients become $0$ after a finite expansion ends.","title":"Complete quotients of a Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-partial-quotient","kind":"node","lean":"MetaMathlibExt.laurentPartialQuotient","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-partial-quotient","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field, $\\alpha \\in K((x^{-1}))$ and $n \\ge 0$. The $n$-th partial quotient of $\\alpha$ is $b_n = [\\alpha_n]$, the polynomial part of the $n$-th complete quotient $\\alpha_n$ of $\\alpha$.","title":"Partial quotients of a Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-partial-quotient-zero","kind":"node","lean":"MetaMathlibExt.laurentPartialQuotient_zero","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-partial-quotient-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field and $\\alpha \\in K((x^{-1}))$. The partial quotient $b_0$ of $\\alpha$ is its polynomial part $[\\alpha]$, and the complete quotient $\\alpha_1$ is $(\\alpha - [\\alpha])^{-1}$, where $0^{-1} = 0$.","title":"First step of the continued fraction of a Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-has-bounded-partial-quotients","kind":"node","lean":"MetaMathlibExt.LaurentHasBoundedPartialQuotients","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-has-bounded-partial-quotients","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field. A Laurent series $\\alpha \\in K((x^{-1}))$ with partial quotients $b_0, b_1, \\dots$ has bounded partial quotients if there is an integer $B$ such that for every $n \\ge 0$ and every integer $k > B$ the coefficient of $x^k$ in $b_n$ is zero, that is, every nonzero $b_n$ is a polynomial in $x$ of degree at most $B$.","title":"Laurent series with bounded partial quotients"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-conv-num","kind":"node","lean":"MetaMathlibExt.laurentConvNum","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-conv-num","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field and let $\\alpha \\in K((x^{-1}))$ have partial quotients $b_0, b_1, \\dots$. The numerators of the convergents of $\\alpha$ are $P_{-1} = 1$, $P_0 = b_0$ and $P_{n+1} = b_{n+1} P_n + P_{n-1}$ for $n \\ge 0$. The sequence is indexed with a shift of one: its term at $n \\ge 0$ is $P_{n-1}$.","title":"Numerators of the convergents of a Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-conv-den","kind":"node","lean":"MetaMathlibExt.laurentConvDen","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-conv-den","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field, let $\\alpha \\in K((x^{-1}))$ and let $b_0, b_1, \\dots$ be its partial quotients. The denominators of the convergents of $\\alpha$ are given by $Q_{-1} = 0$, $Q_0 = 1$ and $Q_{n+1} = b_{n+1} Q_n + Q_{n-1}$ for $n \\ge 0$. They are indexed from $0$ here: the $k$-th term of the sequence is $Q_{k-1}$.","title":"Denominators of the convergents"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-conv-two","kind":"node","lean":"MetaMathlibExt.laurentConv_two","parent":10331,"reading_title":{"format":"inline-parts-v1","parts":[["text","The convergent "],["math","\\(P_1 / Q_1\\)"],["text"," of a Laurent series"]],"source":"The convergent $P_1 / Q_1$ of a Laurent series"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-conv-two","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field and let $\\alpha \\in K((x^{-1}))$ have partial quotients $b_0, b_1, \\dots$. Let $P_n$ and $Q_n$ be the numerators and denominators of its convergents, given by $P_{-1} = 1$, $Q_{-1} = 0$, $P_0 = b_0$, $Q_0 = 1$, $P_{n+1} = b_{n+1} P_n + P_{n-1}$ and $Q_{n+1} = b_{n+1} Q_n + Q_{n-1}$. Then $P_1 = b_1 b_0 + 1$ and $Q_1 = b_1$.","title":"The convergent $P_1 / Q_1$ of a Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-is-irrational","kind":"node","lean":"MetaMathlibExt.LaurentIsIrrational","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-is-irrational","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field. A Laurent series $\\alpha \\in K((x^{-1}))$ is irrational if $Q \\alpha \\ne P$ for all polynomials $P, Q \\in K[x]$ with $Q \\ne 0$, where polynomials are embedded in $K((x^{-1}))$ by sending $x$ to the variable $x$.","title":"Irrational Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/not-laurent-is-irrational-poly-to-lauren","kind":"node","lean":"MetaMathlibExt.not_laurentIsIrrational_polyToLaurentInv","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#not-laurent-is-irrational-poly-to-lauren","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"fully_proved","summary":"Let $K$ be a field. For every polynomial $P \\in K[x]$, the image of $P$ in $K((x^{-1}))$ is not irrational.","title":"Polynomials are not irrational"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/baum-sweet-cubic-bounded-partial-quotien","kind":"node","lean":"MetaMathlibExt.baumSweet_cubic_boundedPartialQuotients","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#baum-sweet-cubic-bounded-partial-quotien","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $p \\in \\mathbb{F}_2[x]$ have degree at least $1$. If $f \\in \\mathbb{F}_2((x^{-1}))$ satisfies $f^3 + p^{-1} f + 1 = 0$, then $f$ has bounded partial quotients.","title":"Baum\u2013Sweet: cubics with bounded partial quotients"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/baum-sweet-unbounded-partial-quotients","kind":"node","lean":"MetaMathlibExt.baumSweet_unboundedPartialQuotients","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#baum-sweet-unbounded-partial-quotients","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $n > 1$ and let $g \\in \\mathbb{F}_2[x]$ have degree at least $1$. If $f \\in \\mathbb{F}_2((x^{-1}))$ satisfies $f^{2^n+1} + g^{-1} f + 1 = 0$, then $f$ does not have bounded partial quotients.","title":"Baum\u2013Sweet: unbounded partial quotients for higher degrees"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/baum-sweet-exists-root-continued-fractio","kind":"node","lean":"MetaMathlibExt.baumSweet_exists_root_continuedFraction","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#baum-sweet-exists-root-continued-fractio","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $n \\ge 0$ and let $P, Q \\in \\mathbb{F}_2[x]$ with $P + Q^{2^n} \\ne 0$ and $P + Q^{2^n} \\ne 1$. Then some $f \\in \\mathbb{F}_2((x^{-1}))$ satisfies $f^{2^n+1} + Q f^{2^n} + P f + QP + 1 = 0$ and has partial quotients $b_0 = Q$ and $b_k = P^{2^{(k-1)n}} + Q^{2^{kn}}$ for all $k \\ge 1$, that is, $f = [Q; P + Q^{2^n}, P^{2^n} + Q^{2^{2n}}, P^{2^{2n}} + Q^{2^{3n}}, \\dots]$.","title":"Baum\u2013Sweet: explicit continued fractions of roots"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-convergent-properties","kind":"node","lean":"MetaMathlibExt.laurentConvergent_properties","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-convergent-properties","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $K$ be a field and let $\\alpha \\in K((x^{-1}))$ be irrational, with partial quotients $b_0, b_1, \\dots$ and convergents $P_n / Q_n$, where $P_{-1} = 1$, $Q_{-1} = 0$, $P_0 = b_0$, $Q_0 = 1$, $P_{n+1} = b_{n+1} P_n + P_{n-1}$ and $Q_{n+1} = b_{n+1} Q_n + Q_{n-1}$. For a Laurent series $y = \\sum_m c_m x^{-m}$ let $v(y)$ be its order, the least $m$ with $c_m \\ne 0$, and let $v(0) = 0$; thus $|y| = |x|^{-v(y)}$ for\u2026","title":"Basic properties of the convergents of a Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-partial-quotient-pow-card","kind":"node","lean":"MetaMathlibExt.laurentPartialQuotient_pow_card","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-partial-quotient-pow-card","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $K$ be a finite field with $q$ elements and let $\\alpha \\in K((x^{-1}))$, with partial quotients $b_0, b_1, \\dots$ and convergent numerators and denominators $P_n, Q_n$, where $P_{-1} = 1$ and $Q_{-1} = 0$. Then for every $n \\ge 0$ the $n$-th partial quotient of $\\alpha^q$ is $b_n^q$, and for every $n \\ge -1$ the convergent numerator and denominator of $\\alpha^q$ of index $n$ are $P_n^q$ and $Q_n^q$.","title":"Frobenius acts on continued fractions of Laurent series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-eq-convergent-of-order","kind":"node","lean":"MetaMathlibExt.laurent_eq_convergent_of_order","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-eq-convergent-of-order","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $K$ be a field and let $\\alpha \\in K((x^{-1}))$ be irrational, with partial quotients $b_0, b_1, \\dots$ and convergents $P_n / Q_n$, where $P_{-1} = 1$, $Q_{-1} = 0$, $P_0 = b_0$, $Q_0 = 1$, $P_{n+1} = b_{n+1} P_n + P_{n-1}$ and $Q_{n+1} = b_{n+1} Q_n + Q_{n-1}$. For a Laurent series $y = \\sum_m c_m x^{-m}$ let $v(y)$ be its order, the least $m$ with $c_m \\ne 0$, and let $v(0) = 0$. Let $P, Q \\in K[x]$ be\u2026","title":"Close rational approximations and convergents"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-sub-convergent-order","kind":"node","lean":"MetaMathlibExt.laurent_sub_convergent_order","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-sub-convergent-order","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $K$ be a field, let $\\alpha \\in K((x^{-1}))$ be irrational with convergents $P_n/Q_n$ ($n \\ge 0$), and for nonzero $y \\in K((x^{-1}))$ let $|y| = |x|^{\\deg y}$, where $\\deg y$ is the largest exponent of $x$ occurring in $y$ and $|x| > 1$ is fixed. (i) If $|\\alpha| \\ge 1$, then $|\\alpha - P_n/Q_n| < |\\alpha|$ for all $n \\ge 0$. (ii) If $|\\alpha| < 1$, then $|\\alpha - P_0/Q_0| = |\\alpha|$ and\u2026","title":"Distance from an irrational Laurent series to its convergents"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/lasjaunias-partial-quotient-bound","kind":"node","lean":"MetaMathlibExt.lasjaunias_partialQuotient_bound","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#lasjaunias-partial-quotient-bound","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $\\mathbb{F}_q$ be a finite field, let $A, B, C, D \\in \\mathbb{F}_q[x]$ with $\\Delta = AD - BC \\ne 0$ and $C \\ne 0$, and let $\\alpha \\in \\mathbb{F}_q((x^{-1}))$ be irrational with $\\alpha = (A\\alpha^q + B)/(C\\alpha^q + D)$. Write $|y| = |x|^{\\deg y}$ for the absolute value on $\\mathbb{F}_q((x^{-1}))$, where $\\deg y$ is the largest exponent of $x$ occurring in $y \\ne 0$, and write $P_n/Q_n$ for the convergents\u2026","title":"Lasjaunias's bound on the next partial quotient"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-mobius-sub-identities","kind":"node","lean":"MetaMathlibExt.laurentMobius_sub_identities","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-mobius-sub-identities","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $K$ be a finite field with $q$ elements, let $A, B, C, D \\in K[x]$ with $\\Delta = AD - BC \\ne 0$, and let $\\alpha \\in K((x^{-1}))$ be irrational with $\\alpha = (A \\alpha^q + B) / (C \\alpha^q + D)$. Let $P, Q \\in K[x]$ with $Q \\ne 0$, put $R_1 = DP - BQ$, $S_1 = -CP + AQ$, $R_2 = AP^q + BQ^q$ and $S_2 = CP^q + DQ^q$, and assume $S_1 \\ne 0$ and $S_2 \\ne 0$. Then\u2026","title":"Difference identities for a M\u00f6bius\u2013Frobenius fixed point"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/laurent-mobius-order-identities","kind":"node","lean":"MetaMathlibExt.laurentMobius_order_identities","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#laurent-mobius-order-identities","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $K$ be a finite field with $q$ elements, let $A, B, C, D \\in K[x]$ with $\\Delta = AD - BC \\ne 0$ and $C \\ne 0$, and let $\\alpha \\in K((x^{-1}))$ be irrational with $\\alpha = (A \\alpha^q + B) / (C \\alpha^q + D)$. Let $b_0, b_1, \\dots$ be the partial quotients of $\\alpha$ and $P_n / Q_n$ its convergents, where $P_{-1} = 1$, $Q_{-1} = 0$, $P_0 = b_0$, $Q_0 = 1$, $P_{n+1} = b_{n+1} P_n + P_{n-1}$ and\u2026","title":"Degree identities at convergents of a M\u00f6bius\u2013Frobenius fixed point"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/lasjaunias-unbounded-criterion","kind":"node","lean":"MetaMathlibExt.lasjaunias_unbounded_criterion","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#lasjaunias-unbounded-criterion","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $\\mathbb{F}_q$ be a finite field, let $A, B, C, D \\in \\mathbb{F}_q[x]$ with $\\Delta = AD - BC \\ne 0$ and $C \\ne 0$, and let $\\alpha \\in \\mathbb{F}_q((x^{-1}))$ be irrational with $\\alpha = (A\\alpha^q + B)/(C\\alpha^q + D)$. Write $|y| = |x|^{\\deg y}$ for the absolute value on $\\mathbb{F}_q((x^{-1}))$, where $\\deg y$ is the largest exponent of $x$ occurring in $y \\ne 0$, and write $P_n/Q_n$ for the convergents\u2026","title":"Lasjaunias's criterion for unbounded partial quotients"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/ayadi-cubic-bounded-partial-quotients","kind":"node","lean":"MetaMathlibExt.ayadi_cubic_boundedPartialQuotients","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#ayadi-cubic-bounded-partial-quotients","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $\\ell \\ge 2$ and let $A \\in \\mathbb{F}_2[x]$ be a polynomial of degree $2\\ell$ whose constant and linear coefficients are both $1$, that is, $A = x^{2\\ell} + \\sum_{i=2}^{2\\ell-1} e_i x^i + x + 1$ with $e_i \\in \\mathbb{F}_2$. If $\\alpha \\in \\mathbb{F}_2((x^{-1}))$ has positive degree in $x$ (the largest exponent of $x$ occurring in $\\alpha$ is positive) and satisfies $x^\\ell \\alpha^3 + A \\alpha^2 + 1 = 0$, then\u2026","title":"Bounded partial quotients for a cubic of Ayadi et al"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/baum-sweet-x-cubic-partial-quotient-degr","kind":"node","lean":"MetaMathlibExt.baumSweet_x_cubic_partialQuotient_degree_le_two","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#baum-sweet-x-cubic-partial-quotient-degr","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"If $\\alpha \\in \\mathbb{F}_2((x^{-1}))$ satisfies $x\\alpha^3 + \\alpha + x = 0$, then every partial quotient $b_n$ of $\\alpha$, $n \\ge 0$, has degree at most $2$ in $x$.","title":"Baum\u2013Sweet cubic: partial quotients of degree at most 2"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/lasjaunias-unbounded-partial-quotients-o","kind":"node","lean":"MetaMathlibExt.lasjaunias_unboundedPartialQuotients_of_small","parent":10331,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series/index.html#lasjaunias-unbounded-partial-quotients-o","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-laurent-series","status":"can_prove","summary":"Let $\\mathbb{F}_q$ be a finite field, let $\\ell \\ge 1$ and let $D \\in \\mathbb{F}_q[x]$ with $D(0) \\ne 0$. Write $|y| = |x|^{\\deg y}$ for the absolute value on $\\mathbb{F}_q((x^{-1}))$, where $\\deg y$ is the largest exponent of $x$ occurring in $y \\ne 0$. If $\\alpha \\in \\mathbb{F}_q((x^{-1}))$ is irrational, satisfies $x\\alpha^{q+1} + D\\alpha + x^\\ell = 0$ and $|\\alpha| < |x|^{-((q-1)\\ell + 1)}$, then $\\alpha$ does\u2026","title":"Small roots of a hyperquadratic equation have unbounded partial quotients"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals.json","id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","title":"Continued fractions of quadratic irrationals"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/convergent-den","kind":"node","lean":"MetaMathlibExt.convergentDen","parent":10362,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#convergent-den","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"fully_proved","summary":"For a sequence of natural numbers $a_0, a_1, a_2, \\dots$, its convergent denominators $q_0, q_1, q_2, \\dots$ are the natural numbers defined by $q_0 = 1$, $q_1 = a_1$ and $q_{i+2} = a_{i+2} q_{i+1} + q_i$ for $i \\ge 0$. They are the denominators of the convergents of $[a_0; a_1, a_2, \\dots]$, and $q_i$ is the continuant $K(a_1, \\dots, a_i)$; the term $a_0$ is never used.","title":"Convergent denominators as continuants"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/convergent-den-zero","kind":"node","lean":"MetaMathlibExt.convergentDen_zero","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","The convergent denominator "],["math","\\(q_0\\)"],["text"," equals "],["math","\\(1\\)"]],"source":"The convergent denominator $q_0$ equals $1$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#convergent-den-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"fully_proved","summary":"For every sequence of natural numbers $a_0, a_1, a_2, \\dots$ with convergent denominators $q_0, q_1, q_2, \\dots$, we have $q_0 = 1$.","title":"The convergent denominator $q_0$ equals $1$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/convergent-den-one","kind":"node","lean":"MetaMathlibExt.convergentDen_one","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","The convergent denominator "],["math","\\(q_1\\)"],["text"," equals "],["math","\\(a_1\\)"]],"source":"The convergent denominator $q_1$ equals $a_1$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#convergent-den-one","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"fully_proved","summary":"For every sequence of natural numbers $a_0, a_1, a_2, \\dots$ with convergent denominators $q_0, q_1, q_2, \\dots$, we have $q_1 = a_1$.","title":"The convergent denominator $q_1$ equals $a_1$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/convergent-den-add-two","kind":"node","lean":"MetaMathlibExt.convergentDen_add_two","parent":10362,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#convergent-den-add-two","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"fully_proved","summary":"For every sequence of natural numbers $a_0, a_1, a_2, \\dots$ with convergent denominators $q_0, q_1, q_2, \\dots$ and every natural number $n$, we have $q_{n+2} = a_{n+2} q_{n+1} + q_n$.","title":"Recurrence for the convergent denominators"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/convergent-den-two","kind":"node","lean":"MetaMathlibExt.convergentDen_two","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","The convergent denominator "],["math","\\(q_2\\)"],["text"," equals "],["math","\\(a_2 a_1 + 1\\)"]],"source":"The convergent denominator $q_2$ equals $a_2 a_1 + 1$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#convergent-den-two","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"fully_proved","summary":"For every sequence of natural numbers $a_0, a_1, a_2, \\dots$ with convergent denominators $q_0, q_1, q_2, \\dots$, we have $q_2 = a_2 a_1 + 1$.","title":"The convergent denominator $q_2$ equals $a_2 a_1 + 1$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/has-periodic-cf","kind":"node","lean":"MetaMathlibExt.HasPeriodicCF","parent":10362,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#has-periodic-cf","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"fully_proved","summary":"For a real number $x$, natural numbers $a_0$, $l$, $c$ and a sequence of natural numbers $a_1, a_2, \\dots$, we write $x = [a_0; \\overline{a_1, \\dots, a_{l-1}, c}]$ when the regular continued fraction of $x$ is the infinite expansion $[a_0; a_1, \\dots, a_{l-1}, c, a_1, \\dots, a_{l-1}, c, \\dots]$: its integer part is $a_0$, and for every $k \\ge 1$ its $k$-th partial quotient exists and equals $c$ when $l$ divides $k$\u2026","title":"Continued fractions with a prescribed period"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/floor-eq","kind":"node","lean":"MetaMathlibExt.HasPeriodicCF.floor_eq","parent":10362,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#floor-eq","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"fully_proved","summary":"Let $x$ be a real number and let $a_0$, $l$, $c$ and $a_1, a_2, \\dots$ be natural numbers. Suppose that the regular continued fraction of $x$ has integer part $a_0$ and that, for every $k \\ge 1$, its $k$-th partial quotient is $c$ when $l$ divides $k$ and $a_r$ otherwise, where $r$ is the remainder of $k$ modulo $l$ (so $r = k$ when $l = 0$). For $l \\ge 1$ this says that the continued fraction is\u2026","title":"The integer part of a periodic continued fraction is the floor"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/sqrt-has-periodic-cf-exists-iff-of-odd","kind":"node","lean":"MetaMathlibExt.sqrt_hasPeriodicCF_exists_iff_of_odd","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity criterion for "],["math","\\(\\sqrt{d}\\)"],["text"," with a given symmetric period, odd length"]],"source":"Parity criterion for $\\sqrt{d}$ with a given symmetric period, odd length"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#sqrt-has-periodic-cf-exists-iff-of-odd","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l$ be odd and let $a_1, \\dots, a_{l-1}$ be positive integers with $a_i = a_{l-i}$ for $1 \\le i \\le l - 1$. For $0 \\le i \\le l - 1$ let $q_i = K(a_1, \\dots, a_i)$ be the continuant, given by $q_0 = 1$, $q_1 = a_1$ and $q_{i+2} = a_{i+2} q_{i+1} + q_i$. Then there are a natural number $d$ that is not a perfect square and a natural number $a_0$ with $\\sqrt{d} = [a_0; \\overline{a_1, \\dots, a_{l-1}, 2a_0}]$ if and\u2026","title":"Parity criterion for $\\sqrt{d}$ with a given symmetric period, odd length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/sqrt-has-periodic-cf-exists-iff-of-even","kind":"node","lean":"MetaMathlibExt.sqrt_hasPeriodicCF_exists_iff_of_even","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity criterion for "],["math","\\(\\sqrt{d}\\)"],["text"," with a given symmetric period, even length"]],"source":"Parity criterion for $\\sqrt{d}$ with a given symmetric period, even length"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#sqrt-has-periodic-cf-exists-iff-of-even","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 2$ be even and let $a_1, \\dots, a_{l-1}$ be positive integers with $a_i = a_{l-i}$ for $1 \\le i \\le l - 1$. For $0 \\le i \\le l - 1$ let $q_i = K(a_1, \\dots, a_i)$ be the continuant, given by $q_0 = 1$, $q_1 = a_1$ and $q_{i+2} = a_{i+2} q_{i+1} + q_i$. Then there are a natural number $d$ that is not a perfect square and a natural number $a_0$ with $\\sqrt{d} = [a_0; \\overline{a_1, \\dots, a_{l-1}, 2a_0}]$\u2026","title":"Parity criterion for $\\sqrt{d}$ with a given symmetric period, even length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/convergent-den-symm-split","kind":"node","lean":"MetaMathlibExt.convergentDen_symm_split","parent":10362,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#convergent-den-symm-split","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l$ be a natural number and let $a_0, a_1, a_2, \\dots$ be natural numbers with convergent denominators $q_0, q_1, q_2, \\dots$, such that $a_i = a_{l-i}$ for $1 \\le i < l$. Then for every $i$ with $1 \\le i \\le l - 2$,","title":"Splitting the last convergent denominator of a symmetric sequence"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/shifted-convergent-den-symm-split","kind":"node","lean":"MetaMathlibExt.shifted_convergentDen_symm_split","parent":10362,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#shifted-convergent-den-symm-split","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l$ be a natural number and let $a_1, \\dots, a_{l-1}$ be natural numbers with $a_i = a_{l-i}$ for $1 \\le i \\le l - 1$. For $j \\ge 1$ let $r_j = K(a_2, \\dots, a_j)$ be the continuant, given by $r_1 = 1$, $r_2 = a_2$ and $r_{j+2} = a_{j+2} r_{j+1} + r_j$. Then for every $i$ with $2 \\le i \\le l - 3$,","title":"Splitting the shifted continuant of a symmetric sequence"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/convergent-den-parity-of-even","kind":"node","lean":"MetaMathlibExt.convergentDen_parity_of_even","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(q_{l-1}\\)"],["text"," for a symmetric sequence, even "],["math","\\(l\\)"]],"source":"Parity of $q_{l-1}$ for a symmetric sequence, even $l$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#convergent-den-parity-of-even","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 2$ be even and let $a_0, a_1, a_2, \\dots$ be natural numbers with convergent denominators $q_0, q_1, q_2, \\dots$, such that $a_i = a_{l-i}$ for $1 \\le i < l$. Then $q_{l-1} \\equiv a_{l/2} \\, q_{l/2-1} \\pmod 2$.","title":"Parity of $q_{l-1}$ for a symmetric sequence, even $l$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/shifted-convergent-den-parity-of-even","kind":"node","lean":"MetaMathlibExt.shifted_convergentDen_parity_of_even","parent":10362,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#shifted-convergent-den-parity-of-even","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 4$ be even and let $a_1, \\dots, a_{l-1}$ be natural numbers with $a_i = a_{l-i}$ for $1 \\le i \\le l - 1$. For $j \\ge 1$ let $r_j = K(a_2, \\dots, a_j)$ be the continuant, given by $r_1 = 1$, $r_2 = a_2$ and $r_{j+2} = a_{j+2} r_{j+1} + r_j$. Then $r_{l-2} \\equiv a_{l/2} \\, r_{l/2-1} \\pmod 2$.","title":"Parity of the shifted continuant of a symmetric sequence, even length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/convergent-den-parity-of-odd","kind":"node","lean":"MetaMathlibExt.convergentDen_parity_of_odd","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(q_{l-1}\\)"],["text"," for a symmetric sequence, odd "],["math","\\(l\\)"]],"source":"Parity of $q_{l-1}$ for a symmetric sequence, odd $l$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#convergent-den-parity-of-odd","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 3$ be odd and let $a_0, a_1, a_2, \\dots$ be natural numbers with convergent denominators $q_0, q_1, q_2, \\dots$, such that $a_i = a_{l-i}$ for $1 \\le i < l$. Then $q_{l-1} \\equiv q_{(l-1)/2} + q_{(l-3)/2} \\pmod 2$.","title":"Parity of $q_{l-1}$ for a symmetric sequence, odd $l$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/shifted-convergent-den-parity-of-odd","kind":"node","lean":"MetaMathlibExt.shifted_convergentDen_parity_of_odd","parent":10362,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#shifted-convergent-den-parity-of-odd","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 5$ be odd and let $a_1, \\dots, a_{l-1}$ be natural numbers with $a_i = a_{l-i}$ for $1 \\le i \\le l - 1$. For $j \\ge 1$ let $r_j = K(a_2, \\dots, a_j)$ be the continuant, given by $r_1 = 1$, $r_2 = a_2$ and $r_{j+2} = a_{j+2} r_{j+1} + r_j$. Then $r_{l-2} \\equiv r_{(l-1)/2} + r_{(l-3)/2} \\pmod 2$.","title":"Parity of the shifted continuant of a symmetric sequence, odd length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/sqrt-periodic-cf-obstruction-count-of-od","kind":"node","lean":"MetaMathlibExt.sqrt_periodicCF_obstruction_count_of_odd","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of unrealizable parity patterns for "],["math","\\(\\sqrt{d}\\)"],["text",", odd length"]],"source":"Number of unrealizable parity patterns for $\\sqrt{d}$, odd length"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#sqrt-periodic-cf-obstruction-count-of-od","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 3$ be odd. Call a parity pattern $(v_1, \\dots, v_{(l-1)/2}) \\in (\\mathbb{Z}/2)^{(l-1)/2}$ realizable if there are positive integers $a_1, \\dots, a_{l-1}$ with $a_i = a_{l-i}$ for $1 \\le i \\le l - 1$ and $a_j \\equiv v_j \\pmod 2$ for $1 \\le j \\le (l-1)/2$, a natural number $d$ that is not a perfect square and a natural number $a_0$ with $\\sqrt{d} = [a_0; \\overline{a_1, \\dots, a_{l-1}, 2a_0}]$. If $A$ is\u2026","title":"Number of unrealizable parity patterns for $\\sqrt{d}$, odd length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/sqrt-periodic-cf-obstruction-count-of-ev","kind":"node","lean":"MetaMathlibExt.sqrt_periodicCF_obstruction_count_of_even","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of unrealizable parity patterns for "],["math","\\(\\sqrt{d}\\)"],["text",", even length"]],"source":"Number of unrealizable parity patterns for $\\sqrt{d}$, even length"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#sqrt-periodic-cf-obstruction-count-of-ev","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 4$ be even. Call a parity pattern $(v_1, \\dots, v_{l/2}) \\in (\\mathbb{Z}/2)^{l/2}$ realizable if there are positive integers $a_1, \\dots, a_{l-1}$ with $a_i = a_{l-i}$ for $1 \\le i \\le l - 1$ and $a_j \\equiv v_j \\pmod 2$ for $1 \\le j \\le l/2$, a natural number $d$ that is not a perfect square and a natural number $a_0$ with $\\sqrt{d} = [a_0; \\overline{a_1, \\dots, a_{l-1}, 2a_0}]$. If $B$ is the number of\u2026","title":"Number of unrealizable parity patterns for $\\sqrt{d}$, even length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/half-one-add-sqrt-periodic-cf-obstructio","kind":"node","lean":"MetaMathlibExt.half_one_add_sqrt_periodicCF_obstruction_count_of_odd","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every parity pattern is realized by "],["math","\\((1 + \\sqrt{d})/2\\)"],["text"," for odd period length"]],"source":"Every parity pattern is realized by $(1 + \\sqrt{d})/2$ for odd period length"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#half-one-add-sqrt-periodic-cf-obstructio","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l$ be odd. For every parity pattern $(v_1, \\dots, v_{(l-1)/2}) \\in (\\mathbb{Z}/2)^{(l-1)/2}$ there are positive integers $a_1, \\dots, a_{l-1}$ with $a_i = a_{l-i}$ for $1 \\le i \\le l - 1$ and $a_j \\equiv v_j \\pmod 2$ for $1 \\le j \\le (l-1)/2$, a natural number $d$ that is not a perfect square and a natural number $a_0$ with $(1 + \\sqrt{d})/2 = [a_0; \\overline{a_1, \\dots, a_{l-1}, 2a_0 - 1}]$. That is, the\u2026","title":"Every parity pattern is realized by $(1 + \\sqrt{d})/2$ for odd period length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/half-one-add-sqrt-periodic-cf-obstructio-2","kind":"node","lean":"MetaMathlibExt.half_one_add_sqrt_periodicCF_obstruction_count_of_even","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unrealizable parity patterns for "],["math","\\((1 + \\sqrt d)/2\\)"],["text"," with even "],["math","\\(l\\)"]],"source":"Unrealizable parity patterns for $(1 + \\sqrt d)/2$ with even $l$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#half-one-add-sqrt-periodic-cf-obstructio-2","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 2$ be even. Call a parity pattern $(v_1, \\dots, v_{l/2}) \\in (\\mathbb{Z}/2)^{l/2}$ realizable if there are positive integers $a_1, \\dots, a_{l-1}$ with $a_i = a_{l-i}$ for $1 \\le i < l$ and $a_j \\equiv v_j \\pmod 2$ for $1 \\le j \\le l/2$, together with natural numbers $d$ and $a_0$ such that $d \\equiv 1 \\pmod 4$, $d$ is not a perfect square and the regular continued fraction of $(1 + \\sqrt d)/2$ is","title":"Unrealizable parity patterns for $(1 + \\sqrt d)/2$ with even $l$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/sqrt-has-periodic-cf-exists-iff-parity","kind":"node","lean":"MetaMathlibExt.sqrt_hasPeriodicCF_exists_iff_parity","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity criterion for "],["math","\\(\\sqrt{d}\\)"],["text"," with a given symmetric period"]],"source":"Parity criterion for $\\sqrt{d}$ with a given symmetric period"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#sqrt-has-periodic-cf-exists-iff-parity","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 3$ and let $a_1, \\dots, a_{l-1}$ be positive integers with $a_i = a_{l-i}$ for $1 \\le i < l$. Let $q_j$ be the denominator of the $j$-th convergent of $[a_0; a_1, a_2, \\dots]$ (so $q_0 = 1$, $q_1 = a_1$ and $q_{j+2} = a_{j+2} q_{j+1} + q_j$), and let $r_{l-2}$ be the denominator of the $(l-3)$-th convergent of $[a_1; a_2, a_3, \\dots]$. There are natural numbers $d$ and $a_0$ with $d$ not a perfect square\u2026","title":"Parity criterion for $\\sqrt{d}$ with a given symmetric period"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/half-one-add-sqrt-has-periodic-cf-exists","kind":"node","lean":"MetaMathlibExt.half_one_add_sqrt_hasPeriodicCF_exists_iff_parity","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity criterion for "],["math","\\((1 + \\sqrt{d})/2\\)"],["text"," with a given symmetric period"]],"source":"Parity criterion for $(1 + \\sqrt{d})/2$ with a given symmetric period"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#half-one-add-sqrt-has-periodic-cf-exists","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l \\ge 3$ and let $a_1, \\dots, a_{l-1}$ be positive integers with $a_i = a_{l-i}$ for $1 \\le i < l$. Let $q_j$ be the denominator of the $j$-th convergent of $[a_0; a_1, a_2, \\dots]$ (so $q_0 = 1$, $q_1 = a_1$ and $q_{j+2} = a_{j+2} q_{j+1} + q_j$), and let $r_{l-2}$ be the denominator of the $(l-3)$-th convergent of $[a_1; a_2, a_3, \\dots]$. There are natural numbers $d$ and $a_0$ with $d \\equiv 1 \\pmod 4$,\u2026","title":"Parity criterion for $(1 + \\sqrt{d})/2$ with a given symmetric period"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/half-one-add-sqrt-has-periodic-cf-exists-2","kind":"node","lean":"MetaMathlibExt.half_one_add_sqrt_hasPeriodicCF_exists_classification","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Existence of "],["math","\\((1 + \\sqrt{d})/2\\)"],["text"," with a given symmetric period"]],"source":"Existence of $(1 + \\sqrt{d})/2$ with a given symmetric period"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#half-one-add-sqrt-has-periodic-cf-exists-2","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $l$ be a natural number and $a_1, \\dots, a_{l-1}$ positive integers with $a_i = a_{l-i}$ for $1 \\le i < l$. If $l$ is odd, there are natural numbers $d$ and $a_0$ with $d \\equiv 1 \\pmod 4$, $d$ not a perfect square, and $(1 + \\sqrt{d})/2 = [a_0; \\overline{a_1, \\dots, a_{l-1}, 2a_0 - 1}]$. If $l$ is even and positive, such $d$ and $a_0$ exist if and only if $a_{l/2}$ is odd.","title":"Existence of $(1 + \\sqrt{d})/2$ with a given symmetric period"},{"id":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/fundamental-unit-eq-convergent-or-cube","kind":"node","lean":"MetaMathlibExt.fundamentalUnit_eq_convergent_or_cube","parent":10362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fundamental unit from the period of "],["math","\\(\\sqrt{d}\\)"]],"source":"Fundamental unit from the period of $\\sqrt{d}$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals/index.html#fundamental-unit-eq-convergent-or-cube","scope":"NumberTheory/continued-fractions-expansions/part-01/continued-fractions-of-quadratic-irrationals","status":"can_prove","summary":"Let $d \\ge 2$ be a squarefree integer, let $\\varepsilon$ be the least real number $u > 1$ of the form $u = (x + y\\sqrt{d})/2$ with $x, y \\in \\mathbb{Z}$ and $x^2 - d y^2 = \\pm 4$ (the fundamental unit of $\\mathbb{Q}(\\sqrt{d})$), let $\\sqrt{d} = [a_0; a_1, a_2, \\dots]$ be the regular continued fraction of $\\sqrt{d}$, with convergents $p_k / q_k$ (so $p_0 = a_0$ and $q_0 = 1$), and let $l$ be the least positive\u2026","title":"Fundamental unit from the period of $\\sqrt{d}$"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/engel-expansions.json","id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","title":"Engel expansions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/neg-rem","kind":"node","lean":"EngelExpansion.negRem","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#neg-rem","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"For natural numbers $n$ and $k$ with $k \\ge 1$, the negated remainder of $n$ modulo $k$ is the unique $u$ with $0 \\le u < k$ and $u \\equiv -n \\pmod{k}$: it equals $0$ when $k$ divides $n$ and $k - (n \\bmod k)$ otherwise. For $k = 0$ it is defined to be $0$.","title":"Negated remainder"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/neg-rem-eq-iff","kind":"node","lean":"EngelExpansion.negRem_eq_iff","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#neg-rem-eq-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"For natural numbers $n$, $k$ and $u$ with $k \\ne 0$, the negated remainder of $n$ modulo $k$ equals $u$ if and only if $u < k$ and $k$ divides $n + u$.","title":"Characterization of the negated remainder"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/traj","kind":"node","lean":"EngelExpansion.traj","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#traj","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"For natural numbers $n$ and $a$, the Engel remainder trajectory of $a$ with respect to $n$ is the sequence $t_0, t_1, t_2, \\dots$ of natural numbers with $t_0 = a$ and, for every $j \\ge 0$, $t_{j+1} = 0$ if $t_j = 0$, while if $t_j \\ne 0$ then $t_{j+1}$ is the negated remainder of $n$ modulo $t_j$: the unique natural number $u < t_j$ with $u \\equiv -n \\pmod{t_j}$. Once the trajectory reaches $0$ it stays there.","title":"Engel remainder trajectory"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/traj-zero","kind":"node","lean":"EngelExpansion.traj_zero","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#traj-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"For all natural numbers $n$ and $a$, the Engel remainder trajectory of $a$ with respect to $n$ has value $a$ at step $0$.","title":"Initial term of the Engel trajectory"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/traj-succ-of-eq-zero","kind":"node","lean":"EngelExpansion.traj_succ_of_eq_zero","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#traj-succ-of-eq-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$, $a$ and $j$ be natural numbers and let $t_0, t_1, \\dots$ be the Engel remainder trajectory of $a$ with respect to $n$. If $t_j = 0$, then $t_{j+1} = 0$.","title":"Zero is absorbing for the Engel trajectory"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/traj-succ-of-ne-zero","kind":"node","lean":"EngelExpansion.traj_succ_of_ne_zero","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#traj-succ-of-ne-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$, $a$ and $j$ be natural numbers and let $t_0, t_1, \\dots$ be the Engel remainder trajectory of $a$ with respect to $n$. If $t_j \\ne 0$, then $t_{j+1}$ is the negated remainder of $n$ modulo $t_j$: the unique natural number $u < t_j$ with $u \\equiv -n \\pmod{t_j}$.","title":"Step of the Engel trajectory at a nonzero term"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/traj-one","kind":"node","lean":"EngelExpansion.traj_one","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#traj-one","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$ and $t$ be natural numbers with $t \\ne 0$. Then the Engel remainder trajectory of $t$ with respect to $n$ is, at step $1$, the negated remainder of $n$ modulo $t$: the unique natural number $u < t$ with $u \\equiv -n \\pmod{t}$.","title":"First step of the Engel trajectory"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/exists-traj-eq-zero","kind":"node","lean":"EngelExpansion.exists_traj_eq_zero","parent":10386,"reading_title":{"format":"inline-parts-v1","parts":[["text","An Engel trajectory started at "],["math","\\(a\\)"],["text"," vanishes within "],["math","\\(a + 1\\)"],["text"," steps"]],"source":"An Engel trajectory started at $a$ vanishes within $a + 1$ steps"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#exists-traj-eq-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$ and $a$ be natural numbers and let $t_0 = a, t_1, t_2, \\dots$ be the Engel trajectory of $a$ with respect to $n$. Then $t_i = 0$ for some $i \\le a + 1$.","title":"An Engel trajectory started at $a$ vanishes within $a + 1$ steps"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/exists-traj-eq-zero-pos","kind":"node","lean":"EngelExpansion.exists_traj_eq_zero_pos","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#exists-traj-eq-zero-pos","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$ and $a$ be natural numbers and let $t_0 = a, t_1, t_2, \\dots$ be the Engel trajectory of $a$ with respect to $n$. Then $t_j = 0$ for some $j \\ge 1$.","title":"Engel trajectories vanish at a positive step"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/length","kind":"node","lean":"EngelExpansion.length","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#length","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"For natural numbers $n$ and $a$, let $t_0 = a, t_1, t_2, \\dots$ be the Engel trajectory of $a$ with respect to $n$. The Engel length $L(n, a)$ is the least $j \\ge 1$ with $t_j = 0$.","title":"Engel length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/length-pos","kind":"node","lean":"EngelExpansion.length_pos","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#length-pos","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"For all natural numbers $n$ and $a$, the Engel length $L(n, a)$ of $a$ with respect to $n$ satisfies $L(n, a) \\ge 1$.","title":"The Engel length is positive"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/traj-length-eq-zero","kind":"node","lean":"EngelExpansion.traj_length_eq_zero","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#traj-length-eq-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"For all natural numbers $n$ and $a$, let $L$ be the Engel length of $a$ with respect to $n$. Then the Engel remainder trajectory of $a$ with respect to $n$ is $0$ at step $L$.","title":"Engel trajectory vanishes at the Engel length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/length-le-of-traj-eq-zero","kind":"node","lean":"EngelExpansion.length_le_of_traj_eq_zero","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#length-le-of-traj-eq-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$, $a$ and $j$ be natural numbers with $j \\ge 1$, and let $t_0 = a, t_1, t_2, \\dots$ be the Engel trajectory of $a$ with respect to $n$. If $t_j = 0$, then the Engel length $L(n, a)$ satisfies $L(n, a) \\le j$.","title":"The Engel length is at most any positive vanishing step"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/traj-ne-zero-of-lt-length","kind":"node","lean":"EngelExpansion.traj_ne_zero_of_lt_length","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#traj-ne-zero-of-lt-length","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$, $a$ and $j$ be natural numbers and let $L$ be the Engel length of $a$ with respect to $n$. If $1 \\le j < L$, then the Engel remainder trajectory of $a$ with respect to $n$ is nonzero at step $j$.","title":"Engel trajectory is nonzero at positive steps before the Engel length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/max-length","kind":"node","lean":"EngelExpansion.maxLength","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#max-length","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"For a natural number $n$, the maximal Engel length is $M(n) = \\max_{1 \\le a \\le n} L(n, a)$, where $L(n, a)$ is the Engel length of $a$ with respect to $n$; for $n = 0$ the maximum is over the empty set and $M(0) = 0$.","title":"Maximal Engel length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/length-le-max-length","kind":"node","lean":"EngelExpansion.length_le_maxLength","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#length-le-max-length","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$ and $a$ be natural numbers with $1 \\le a \\le n$. Then the Engel length $L(n, a)$ of $a$ with respect to $n$ is at most the maximal Engel length $M(n) = \\max_{1 \\le b \\le n} L(n, b)$.","title":"Each Engel length is at most the maximal Engel length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions/max-length-le-of-forall","kind":"node","lean":"EngelExpansion.maxLength_le_of_forall","parent":10386,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/engel-expansions/index.html#max-length-le-of-forall","scope":"NumberTheory/continued-fractions-expansions/part-01/engel-expansions","status":"fully_proved","summary":"Let $n$ and $B$ be natural numbers. If the Engel length $L(n, a)$ of $a$ with respect to $n$ satisfies $L(n, a) \\le B$ for every $a$ with $1 \\le a \\le n$, then the maximal Engel length $M(n) = \\max_{1 \\le a \\le n} L(n, a)$ (with $M(0) = 0$) satisfies $M(n) \\le B$.","title":"A uniform bound on Engel lengths bounds the maximal Engel length"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f.json","id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","title":"Extended continuants and bi-infinite continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-fwd","kind":"node","lean":"MetaMathlibExt.extendedContinuantFwd","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-fwd","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"fully_proved","summary":"For an integer sequence $x : \\mathbb{Z} \\to \\mathbb{Z}$ and an integer $b$, the forward branch with start index $b$ is the sequence $(F_n)_{n \\ge 0}$ of integers defined by $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = x_{b+n} F_{n+1} + F_n$. Its value $F_n$ is the extended continuant $K_{n-2}(x)_b$.","title":"Forward branch of the extended continuant"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-bwd","kind":"node","lean":"MetaMathlibExt.extendedContinuantBwd","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-bwd","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"fully_proved","summary":"For an integer sequence $x : \\mathbb{Z} \\to \\mathbb{Z}$ and an integer $b$, the backward branch with start index $b$ is the sequence $(B_m)_{m \\ge 0}$ of integers defined by $B_0 = 1$, $B_1 = 0$ and $B_{m+2} = -x_{b-1-m} B_{m+1} + B_m$. Its value $B_m$ is the extended continuant $K_{-1-m}(x)_b$ of negative order $-1-m$.","title":"Backward branch of the extended continuant"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant","kind":"node","lean":"MetaMathlibExt.extendedContinuant","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"fully_proved","summary":"Let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be a bi-infinite integer sequence and $b$ an integer. The extended continuant $K_a(x)_b$ of order $a \\in \\mathbb{Z}$ with start index $b$ (the paper's $K_a(x)_b^{a+b}$) is the integer defined by $K_{-2}(x)_b = 0$, $K_{-1}(x)_b = 1$, the forward recurrence $K_a(x)_b = x_{b+a} K_{a-1}(x)_b + K_{a-2}(x)_b$ for $a \\ge 0$, and the backward recurrence\u2026","title":"Extended continuants"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-neg-two","kind":"node","lean":"MetaMathlibExt.extendedContinuant_neg_two","parent":10404,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extended continuants of order "],["math","\\(-2\\)"],["text"," vanish"]],"source":"Extended continuants of order $-2$ vanish"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-neg-two","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"fully_proved","summary":"For every bi-infinite integer sequence $x : \\mathbb{Z} \\to \\mathbb{Z}$ and every integer $b$, the extended continuant of order $-2$ with start index $b$ is zero: $K_{-2}(x)_b = 0$.","title":"Extended continuants of order $-2$ vanish"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-neg-one","kind":"node","lean":"MetaMathlibExt.extendedContinuant_neg_one","parent":10404,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extended continuant of order "],["math","\\(-1\\)"]],"source":"Extended continuant of order $-1$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-neg-one","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"fully_proved","summary":"For every integer sequence $x : \\mathbb{Z} \\to \\mathbb{Z}$ and every integer $b$, the extended continuant of $x$ of order $-1$ with start index $b$ equals $1$, that is, $K_{-1}(x)_b = 1$.","title":"Extended continuant of order $-1$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-zero","kind":"node","lean":"MetaMathlibExt.extendedContinuant_zero","parent":10404,"reading_title":{"format":"inline-parts-v1","parts":[["text","The extended continuant of order "],["math","\\(0\\)"],["text"," is a single term"]],"source":"The extended continuant of order $0$ is a single term"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"fully_proved","summary":"For every bi-infinite integer sequence $x : \\mathbb{Z} \\to \\mathbb{Z}$ and every integer $b$, the extended continuant of order $0$ with start index $b$ is $K_0(x)_b = x_b$.","title":"The extended continuant of order $0$ is a single term"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-recurrence","kind":"node","lean":"MetaMathlibExt.extendedContinuant_recurrence","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-recurrence","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"fully_proved","summary":"For every bi-infinite integer sequence $x : \\mathbb{Z} \\to \\mathbb{Z}$ and all integers $a$ and $b$, the extended continuants $K_a(x)_b$ of order $a$ with start index $b$ satisfy $K_a(x)_b = x_{b+a} K_{a-1}(x)_b + K_{a-2}(x)_b$.","title":"Recurrence for extended continuants"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-neg-three-to-six","kind":"node","lean":"MetaMathlibExt.extendedContinuant_neg_three_to_six","parent":10404,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extended continuants of orders "],["math","\\(-3\\)"],["text"," to "],["math","\\(-6\\)"]],"source":"Extended continuants of orders $-3$ to $-6$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-neg-three-to-six","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"fully_proved","summary":"Let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be an integer sequence and write $K_a(x)_b$ for its extended continuant of order $a$ with start index $b$. For every integer $h$, $K_{-3}(x)_h = 1$, $K_{-4}(x)_h = -x_{h-2}$, $K_{-5}(x)_h = 1 + x_{h-3} x_{h-2}$ and $K_{-6}(x)_h = -(x_{h-4} + x_{h-2} + x_{h-4} x_{h-3} x_{h-2})$.","title":"Extended continuants of orders $-3$ to $-6$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-westbound-matrix-pro","kind":"node","lean":"MetaMathlibExt.extendedContinuant_westbound_matrix_prod","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-westbound-matrix-pro","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be a bi-infinite integer sequence, write $K_a(x)_b$ for the extended continuant of order $a$ with start index $b$, and put $M_i = \\begin{pmatrix} 0 & 1 \\\\ 1 & -x_{-i} \\end{pmatrix}$. Then for every integer $n \\ge 1$,","title":"Westbound matrix product of extended continuants"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-neg-order","kind":"node","lean":"MetaMathlibExt.extendedContinuant_neg_order","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-neg-order","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be an integer sequence and write $K_a(x)_b$ for its extended continuant of order $a$ with start index $b$. For all integers $h$ and $n$,","title":"Reflection formula for the order of extended continuants"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-bennett-nat","kind":"node","lean":"MetaMathlibExt.extendedContinuant_bennett_nat","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-bennett-nat","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be an integer sequence and write $K_a(x)_b$ for its extended continuant of order $a$ with start index $b$. For all nonnegative integers $k$, $\\ell$, $m$, $n$,","title":"Bennett's continuant identity for nonnegative indices"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-bennett","kind":"node","lean":"MetaMathlibExt.extendedContinuant_bennett","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-bennett","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be an integer sequence and write $K_a(x)_b$ for its extended continuant of order $a$ with start index $b$. For all integers $k$, $\\ell$, $m$, $n$,","title":"Bennett's continuant identity for all integer indices"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-euler","kind":"node","lean":"MetaMathlibExt.extendedContinuant_euler","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-euler","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be an integer sequence and write $K_a(x)_b$ for its extended continuant of order $a$ with start index $b$. For all integers $h$, $k$, $\\ell$, $m$,","title":"Euler's continuant identity for extended continuants"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/extended-continuant-det","kind":"node","lean":"MetaMathlibExt.extendedContinuant_det","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#extended-continuant-det","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be an integer sequence and write $K_a(x)_b$ for its extended continuant of order $a$ with start index $b$. For every integer $n$,","title":"Determinant identity for extended continuants"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/not-silent-of-even","kind":"node","lean":"MetaMathlibExt.not_silent_of_even","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#not-silent-of-even","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $\\sigma \\ge 0$ be an even integer and write $K_a(x)_b$ for the extended continuant of order $a$ with start index $b$. For all bi-infinite integer sequences $x, z : \\mathbb{Z} \\to \\mathbb{Z}$ it is not the case that $K_{-i-\\sigma-1}(x)_0 = K_i(z)_0$ and $K_{-i-\\sigma-2}(x)_1 = K_{i-1}(z)_1$ for every natural number $i$.","title":"No silent sequences of even order"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/one-silent-terms","kind":"node","lean":"MetaMathlibExt.oneSilent_terms","parent":10404,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of a "],["math","\\(1\\)"],["text","-silent sequence"]],"source":"Terms of a $1$-silent sequence"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#one-silent-terms","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $x, y, z : \\mathbb{Z} \\to \\mathbb{Z}$ be bi-infinite integer sequences and write $K_a(x)_b$ for the extended continuant of order $a$ with start index $b$. Suppose that for every natural number $i$, $K_i(x)_0 = K_i(y)_0$, $K_{i-1}(x)_1 = K_{i-1}(y)_1$, $K_{-i-2}(x)_0 = K_i(z)_0$ and $K_{-i-3}(x)_1 = K_{i-1}(z)_1$. Then $x_n = y_n$ for every $n \\ge 0$, $x_{-n} = -z_n$ for every $n \\ge 1$, and $z_0 = 0$.","title":"Terms of a $1$-silent sequence"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/three-silent-terms","kind":"node","lean":"MetaMathlibExt.threeSilent_terms","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#three-silent-terms","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $x, y, z : \\mathbb{Z} \\to \\mathbb{Z}$ be bi-infinite integer sequences, and write $K_a(x)_b$ for the extended continuant of $x$ of order $a$ with start index $b$. Suppose that for every natural number $i$ we have $K_i(x)_0 = K_i(y)_0$, $K_{i-1}(x)_1 = K_{i-1}(y)_1$, $K_{-i-4}(x)_0 = K_i(z)_0$ and $K_{-i-5}(x)_1 = K_{i-1}(z)_1$, and that $x_{-1} = 0$. Then $x_n = y_n$ for every $n \\ge 0$, and $x_{-n} = -z_{n-2}$\u2026","title":"Terms of 3-silent sequences"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/odd-silent-terms","kind":"node","lean":"MetaMathlibExt.oddSilent_terms","parent":10404,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of a "],["math","\\(\\sigma\\)"],["text","-silent sequence for odd "],["math","\\(\\sigma \\ge 5\\)"]],"source":"Terms of a $\\sigma$-silent sequence for odd $\\sigma \\ge 5$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#odd-silent-terms","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $\\sigma \\ge 5$ be an odd integer, let $x, y, z : \\mathbb{Z} \\to \\mathbb{Z}$ be bi-infinite integer sequences, and write $K_a(x)_b$ for the extended continuant of order $a$ with start index $b$. Suppose that for every natural number $i$, $K_i(x)_0 = K_i(y)_0$, $K_{i-1}(x)_1 = K_{i-1}(y)_1$, $K_{-i-\\sigma-1}(x)_0 = K_i(z)_0$ and $K_{-i-\\sigma-2}(x)_1 = K_{i-1}(z)_1$. Then $x_n = y_n$ for every $n \\ge 0$,\u2026","title":"Terms of a $\\sigma$-silent sequence for odd $\\sigma \\ge 5$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/silent-terms-table-one-mod-six","kind":"node","lean":"MetaMathlibExt.silentTerms_table_one_mod_six","parent":10404,"reading_title":{"format":"inline-parts-v1","parts":[["text","Silent-term formulas on table (33), case "],["math","\\(\\sigma \\equiv 1 \\pmod 6\\)"]],"source":"Silent-term formulas on table (33), case $\\sigma \\equiv 1 \\pmod 6$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#silent-terms-table-one-mod-six","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $\\sigma \\ge 7$ be an integer with $\\sigma \\equiv 1 \\pmod 6$, let $c_1 \\ne c_2$ be integers, and let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be a bi-infinite integer sequence with $x_{-3} = x_0 + c_1$, $x_{-2} = 1$, $x_{-1} = -1$ and, for all integers $4 \\le m \\le \\sigma - 3$, $x_{-m} = -1$ if $m \\equiv 1 \\pmod 3$ and $x_{-m} = 1$ otherwise. Write $K_a(x)_b$ for the extended continuant of order $a$ with start index $b$,\u2026","title":"Silent-term formulas on table (33), case $\\sigma \\equiv 1 \\pmod 6$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/silent-terms-table-five-mod-six","kind":"node","lean":"MetaMathlibExt.silentTerms_table_five_mod_six","parent":10404,"reading_title":{"format":"inline-parts-v1","parts":[["text","Silent-term formulas on table (33), case "],["math","\\(\\sigma \\equiv 5 \\pmod 6\\)"]],"source":"Silent-term formulas on table (33), case $\\sigma \\equiv 5 \\pmod 6$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#silent-terms-table-five-mod-six","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $\\sigma \\ge 7$ be an integer with $\\sigma \\equiv 5 \\pmod 6$, let $c_1 \\ne c_2$ be integers, and let $x : \\mathbb{Z} \\to \\mathbb{Z}$ be a bi-infinite integer sequence with $x_{-3} = x_0 + c_1$, $x_{-2} = 1$, $x_{-1} = -1$ and, for all integers $4 \\le m \\le \\sigma - 3$, $x_{-m} = -1$ if $m \\equiv 2 \\pmod 3$ and $x_{-m} = 1$ otherwise. Write $K_a(x)_b$ for the extended continuant of order $a$ with start index $b$,\u2026","title":"Silent-term formulas on table (33), case $\\sigma \\equiv 5 \\pmod 6$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/conjugate-continued-fraction-not-purely","kind":"node","lean":"MetaMathlibExt.conjugate_continuedFraction_not_purelyPeriodic","parent":10404,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f/index.html#conjugate-continued-fraction-not-purely","scope":"NumberTheory/continued-fractions-expansions/part-01/extended-continuants-and-bi-infinite-continued-f","status":"can_prove","summary":"Let $P$, $Q$ and $D$ be integers with $D > 0$ not a perfect square and $Q > 0$, and let $\\bar\\alpha = (P - \\sqrt{D})/Q$. Then the simple continued fraction $[a_0; a_1, a_2, \\dots]$ of $\\bar\\alpha$, which is infinite because $\\bar\\alpha$ is irrational, is not purely periodic: there is no integer $s \\ge 1$ with $a_{n+s} = a_n$ for all $n \\ge 0$. Here $a_0 = \\lfloor \\bar\\alpha \\rfloor$ is included.","title":"The conjugate continued fraction is not purely periodic"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator.json","id":"NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator","status":"can_prove","title":"Farey fractions with a given numerator"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator/farey-numerator-count","kind":"node","lean":"MathlibExt.NumberTheory.FareyNumeratorWanted.fareyNumeratorCount","parent":10426,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator/index.html#farey-numerator-count","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator","status":"fully_proved","summary":"For natural numbers $n$ and $h$, $N_n(h)$ is the number of integers $b$ with $h \\le b \\le n$ and $\\gcd(h, b) = 1$; for $h \\ge 1$ this is the number of fractions with numerator $h$ in the Farey sequence of order $n$.","title":"Number of Farey fractions with a given numerator"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator/farey-between-count","kind":"node","lean":"MathlibExt.NumberTheory.FareyNumeratorWanted.farey_between_count","parent":10426,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator/index.html#farey-between-count","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator","status":"can_prove","summary":"Let $k$, $n$ and $h$ be natural numbers with $k > 1$, $h \\ge 1$ and $n \\ge kh - 1$, and let $\\varphi$ be Euler's totient function. The number of integers $b$ with $1 \\le b \\le n$, $\\gcd(h, b) = 1$ and $1/k < h/b \\le 1/(k-1)$ is $\\varphi(h)$. If moreover $h > 2$, then twice the number of integers $b$ with $1 \\le b \\le n$, $\\gcd(h, b) = 1$ and $1/k < h/b < 2/(2k-1)$ is $\\varphi(h)$.","title":"Farey fractions with equal numerators between consecutive unit fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator/farey-numerator-count-formula","kind":"node","lean":"MathlibExt.NumberTheory.FareyNumeratorWanted.fareyNumeratorCount_formula","parent":10426,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator/index.html#farey-numerator-count-formula","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-fractions-with-a-given-numerator","status":"can_prove","summary":"Let $h$ and $n$ be natural numbers with $0 < h < n$, and let $N_n(h)$ be the number of integers $b$ with $h \\le b \\le n$ and $\\gcd(h, b) = 1$ (the number of fractions with numerator $h$ in the Farey sequence of order $n$). Then","title":"Formula for the number of Farey fractions with a given numerator"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/farey-sequences.json","id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","title":"Farey sequences"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/farey-finset","kind":"node","lean":"Farey.fareyFinset","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#farey-finset","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"For a natural number $n$, the Farey set $F_n$ of order $n$ is the finite set of rationals $a/d$ with $a$ and $d$ natural numbers, $1 \\le d \\le n$ and $0 \\le a \\le d$.","title":"Farey set"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/farey-seq","kind":"node","lean":"Farey.fareySeq","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#farey-seq","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"For a natural number $n$, the Farey sequence of order $n$ is the list of the elements of the Farey set $F_n$ in increasing order.","title":"Farey sequence"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/mem-farey-finset-iff","kind":"node","lean":"Farey.mem_fareyFinset_iff","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#mem-farey-finset-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"For a natural number $n$ and a rational number $q$, $q$ belongs to the Farey set $F_n$ of order $n$ if and only if $0 \\le q \\le 1$ and the denominator of $q$ in lowest terms is at most $n$.","title":"Membership in the Farey set"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/farey-finset-mono","kind":"node","lean":"Farey.fareyFinset_mono","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#farey-finset-mono","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"If $m \\le n$ are natural numbers, then every element of the Farey set $F_m$ belongs to $F_n$, that is, $F_m \\subseteq F_n$.","title":"Farey sets grow with the order"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/is-farey-neighbor","kind":"node","lean":"Farey.IsFareyNeighbor","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#is-farey-neighbor","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"For a natural number $n$, rationals $x$ and $y$ are Farey neighbors of order $n$ when both lie in the Farey set $F_n$, $x < y$, and no element $z$ of $F_n$ satisfies $x < z < y$.","title":"Farey neighbors"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/mediant","kind":"node","lean":"Farey.mediant","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#mediant","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"For rational numbers $x = a/b$ and $y = c/d$ written in lowest terms with $b, d > 0$, the mediant of $x$ and $y$ is the rational number $(a + c)/(b + d)$.","title":"Mediant"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/neighbor-den-sum-gt","kind":"node","lean":"Farey.neighbor_den_sum_gt","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#neighbor-den-sum-gt","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"Let $n$ be a natural number and let $F_n$ be the Farey set of order $n$. If $x < y$ are neighbors in $F_n$, meaning that $x, y \\in F_n$ and no element of $F_n$ lies strictly between them, then the denominators $b$ of $x$ and $d$ of $y$ in lowest terms satisfy $b + d > n$.","title":"Farey neighbors have denominator sum exceeding the order"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/det-one-and-sum-gt-imp-neighbor","kind":"node","lean":"Farey.det_one_and_sum_gt_imp_neighbor","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#det-one-and-sum-gt-imp-neighbor","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"Let $n$ be a natural number and let $x = a/b$ and $y = c/d$ be elements of the Farey set $F_n$, written in lowest terms with $b, d > 0$. If $bc = ad + 1$ and $n < b + d$, then $x$ and $y$ are Farey neighbors of order $n$.","title":"Determinant one and large denominator sum give Farey neighbors"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/det-eq-one","kind":"node","lean":"Farey.IsFareyNeighbor.det_eq_one","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#det-eq-one","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"Let $n$ be a natural number. If $x$ and $y$ are Farey neighbors of order $n$, written in lowest terms as $x = a/b$ and $y = c/d$ with $b, d > 0$, then $bc = ad + 1$.","title":"Consecutive Farey fractions have determinant one"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/is-farey-neighbor-iff-det-and-sum","kind":"node","lean":"Farey.isFareyNeighbor_iff_det_and_sum","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#is-farey-neighbor-iff-det-and-sum","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"Let $n$ be a natural number and let $x = a/b$ and $y = c/d$ be elements of the Farey set $F_n$, written in lowest terms with $b, d > 0$. Then $x$ and $y$ are Farey neighbors of order $n$ if and only if $bc = ad + 1$ and $n < b + d$.","title":"Arithmetic characterization of Farey neighbors"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/mediant-not-mem","kind":"node","lean":"Farey.IsFareyNeighbor.mediant_not_mem","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#mediant-not-mem","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"Let $n$ be a natural number. If $x$ and $y$ are Farey neighbors of order $n$, then the mediant of $x$ and $y$ does not belong to the Farey set $F_n$.","title":"The mediant of Farey neighbors is not in the Farey set"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/mediant-mem-succ","kind":"node","lean":"Farey.IsFareyNeighbor.mediant_mem_succ","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#mediant-mem-succ","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"Let $n$ be a natural number and let $x$ and $y$ be Farey neighbors of order $n$ whose denominators in lowest terms sum to $n + 1$. Then the mediant of $x$ and $y$ belongs to the Farey set $F_{n+1}$ and does not belong to $F_n$.","title":"Mediants that enter the next Farey set"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences/farey-succ-generation","kind":"node","lean":"Farey.farey_succ_generation","parent":10430,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-sequences/index.html#farey-succ-generation","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-sequences","status":"fully_proved","summary":"Let $n \\ge 1$ be a natural number. A rational number $q$ belongs to the Farey set $F_{n+1}$ if and only if $q$ belongs to $F_n$, or there are Farey neighbors $x$ and $y$ of order $n$ whose denominators in lowest terms sum to $n + 1$ such that $q$ is the mediant of $x$ and $y$.","title":"Mediants generate the next Farey set"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/farey-staircase.json","id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","title":"Farey staircase"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-staircase","kind":"node","lean":"MetaMathlibExt.fareyStaircase","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-staircase","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"fully_proved","summary":"For a natural number $n$ and a real number $x$, the Farey staircase of order $n$ is $A_n(x) = \\frac{1}{n} \\sum_{k=1}^{n} \\lfloor kx \\rfloor / k$, the average of the first $n$ lower rational approximations $\\lfloor kx \\rfloor / k$ of $x$. For $n = 0$ the sum is empty and $A_0(x) = 0$.","title":"Farey staircase"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-staircase-zero","kind":"node","lean":"MetaMathlibExt.fareyStaircase_zero","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-staircase-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"fully_proved","summary":"For every real number $x$, the Farey staircase $A_0$ of order $0$ satisfies $A_0(x) = 0$.","title":"The Farey staircase of order zero vanishes"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-staircase-one","kind":"node","lean":"MetaMathlibExt.fareyStaircase_one","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-staircase-one","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"fully_proved","summary":"For every real number $x$, the Farey staircase $A_1$ of order $1$ satisfies $A_1(x) = \\lfloor x \\rfloor$.","title":"The Farey staircase of order one is the floor function"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-zoom","kind":"node","lean":"MetaMathlibExt.fareyZoom","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-zoom","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"fully_proved","summary":"For a natural number $n$ and a real number $x$, the zoomed Farey staircase of order $n$ is $B_0(x) = 0$ for $n = 0$ and $B_n(x) = n A_n(x/n)$ for $n \\ge 1$, where $A_n$ is the Farey staircase of order $n$. For $n \\ge 1$ this equals $\\sum_{k=1}^{n} \\lfloor (k/n) x \\rfloor / k$.","title":"Zoomed Farey staircase"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-zoom-eq-sum","kind":"node","lean":"MetaMathlibExt.fareyZoom_eq_sum","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-zoom-eq-sum","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"fully_proved","summary":"For every natural number $n \\ge 1$ and every real number $x$, the zoomed Farey staircase $B_n$ of order $n$ satisfies $B_n(x) = \\sum_{k=1}^{n} \\lfloor (k/n) x \\rfloor / k$.","title":"The zoomed Farey staircase as a sum"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-zoom-one","kind":"node","lean":"MetaMathlibExt.fareyZoom_one","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-zoom-one","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"fully_proved","summary":"For every real number $x$, the zoomed Farey staircase $B_1$ of order $1$ satisfies $B_1(x) = \\lfloor x \\rfloor$.","title":"The zoomed Farey staircase of order one is the floor function"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/tendsto-farey-zoom","kind":"node","lean":"MetaMathlibExt.tendsto_fareyZoom","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#tendsto-farey-zoom","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"Let $B_n$ be the zoomed Farey staircase of order $n$. For every real $x \\ge 0$, $B_n(x) \\to \\sum_{k=1}^{\\lfloor x \\rfloor} \\log(x/k)$ as $n \\to \\infty$.","title":"Limit of the Farey staircase zoomed at the origin"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/tendsto-farey-zoom-sub-at-rational","kind":"node","lean":"MetaMathlibExt.tendsto_fareyZoom_sub_at_rational","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#tendsto-farey-zoom-sub-at-rational","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"Let $B_n$ be the zoomed Farey staircase of order $n$, let $x \\ge 0$ be real, and let $p$ and $q$ be coprime positive integers. Then $B_n(x + pn/q) - B_n(pn/q) \\to \\frac{1}{q} \\sum_{k=1}^{\\lfloor qx \\rfloor} \\log(qx/k)$ as $n \\to \\infty$.","title":"Limit of the Farey staircase zoomed at a rational point"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/tendsto-sq-mul-farey-staircase-sub","kind":"node","lean":"MetaMathlibExt.tendsto_sq_mul_fareyStaircase_sub","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#tendsto-sq-mul-farey-staircase-sub","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"Let $A_n$ be the Farey staircase of order $n$ and $D_n = A_n - A_{n-1}$. For every real $x \\ge 0$, $n^2 D_n(x/n) \\to \\sum_{k=1}^{\\lfloor x \\rfloor} (1 - \\log(x/k))$ as $n \\to \\infty$.","title":"Limit of the rescaled Farey staircase increment"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/ici-one-sub-log-two-subset-closure-floor","kind":"node","lean":"MetaMathlibExt.Ici_one_sub_log_two_subset_closure_floor_gaps","parent":10444,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floor gaps of Farey staircases are dense above "],["math","\\(1 - \\log 2\\)"]],"source":"Floor gaps of Farey staircases are dense above $1 - \\log 2$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#ici-one-sub-log-two-subset-closure-floor","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"For $n \\ge 1$ and $x \\in \\mathbb{R}$ let $f_n(x) = \\lfloor nx \\rfloor - n A_n(x)$, where $A_n$ is the Farey staircase of order $n$; thus $f_n(x) = \\lfloor nx \\rfloor - \\sum_{k=1}^n \\lfloor kx \\rfloor / k$. Every real number $y \\ge 1 - \\log 2$ lies in the closure of the set $S = \\{f_n(x) : n \\ge 1,\\ x \\in \\mathbb{R}\\}$, that is, $S$ is dense in $[1 - \\log 2, \\infty)$.","title":"Floor gaps of Farey staircases are dense above $1 - \\log 2$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/floor-gaps-inter-iic-finite","kind":"node","lean":"MetaMathlibExt.floor_gaps_inter_Iic_finite","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#floor-gaps-inter-iic-finite","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"For a natural number $n \\ge 1$ and a real number $x$, let $f_n(x) = \\lfloor nx \\rfloor - n A_n(x)$, where $A_n$ is the Farey staircase of order $n$. Then for every real number $\\varepsilon > 0$, only finitely many distinct real numbers that are at most $1 - \\log 2 - \\varepsilon$ occur as a value $f_n(x)$ with $n \\ge 1$ and $x$ real; here $\\log$ is the natural logarithm.","title":"Finiteness of small floor gaps of the Farey staircase"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-staircase-sub-eq","kind":"node","lean":"MetaMathlibExt.fareyStaircase_sub_eq","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-staircase-sub-eq","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"Let $n \\ge 2$ be a natural number and $x$ a real number, write $A_m$ for the Farey staircase of order $m$, and let $D_n(x) = A_n(x) - A_{n-1}(x)$. Then $D_n(x) = \\frac{1}{n} \\left( \\lfloor nx \\rfloor / n - A_{n-1}(x) \\right)$ and $D_n(x) = \\frac{1}{n-1} \\left( \\lfloor nx \\rfloor / n - A_n(x) \\right)$.","title":"Recursion between consecutive Farey staircases"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-staircase-le-floor-div","kind":"node","lean":"MetaMathlibExt.fareyStaircase_le_floor_div","parent":10444,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds comparing the Farey staircase with "],["math","\\(\\lfloor nx \\rfloor / n\\)"]],"source":"Bounds comparing the Farey staircase with $\\lfloor nx \\rfloor / n$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-staircase-le-floor-div","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"For every natural number $n \\ge 1$ and every real number $x$, the Farey staircase $A_n$ of order $n$ satisfies $A_n(x) \\le \\lfloor nx \\rfloor / n \\le A_n(x) + (\\log n)/n$, where $\\log$ is the natural logarithm.","title":"Bounds comparing the Farey staircase with $\\lfloor nx \\rfloor / n$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/farey-staircase-sub-mem-icc","kind":"node","lean":"MetaMathlibExt.fareyStaircase_sub_mem_Icc","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#farey-staircase-sub-mem-icc","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"Let $n \\ge 2$ be a natural number and $x$ a real number, and write $A_m$ for the Farey staircase of order $m$. Then $0 \\le A_n(x) - A_{n-1}(x) \\le \\frac{\\log n}{n(n-1)}$, where $\\log$ is the natural logarithm.","title":"Bounds on the increment of the Farey staircase"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/tendsto-harmonic-block-sub-integral","kind":"node","lean":"MetaMathlibExt.tendsto_harmonic_block_sub_integral","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#tendsto-harmonic-block-sub-integral","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"Let $(a_n)$ and $(b_n)$ be sequences of natural numbers with $a_n \\le b_n$ for every $n$ and $a_n \\to \\infty$. Then $\\sum_{k=a_n}^{b_n} 1/k - \\int_{a_n}^{b_n} dt/t \\to 0$ as $n \\to \\infty$.","title":"Harmonic blocks versus the logarithmic integral"},{"id":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase/tendsto-dilation-quotient","kind":"node","lean":"MetaMathlibExt.tendsto_dilation_quotient","parent":10444,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/farey-staircase/index.html#tendsto-dilation-quotient","scope":"NumberTheory/continued-fractions-expansions/part-01/farey-staircase","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ have derivative $f'$ at the point $x \\in \\mathbb{R}$. Then the quotient $(\\lambda^{-1} f(\\lambda x) - f(x))/(\\lambda - 1)$ tends to $x f' - f(x)$ as $\\lambda \\to 1$ with $\\lambda \\ne 1$; that is, the dilation derivative of $f$ at $x$ exists and equals $x f' - f(x)$.","title":"Dilation derivative of a differentiable function"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio.json","id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"can_prove","title":"Folding lemma and paperfolding continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/finite-cont-frac","kind":"node","lean":"MetaMathlibExt.finiteContFrac","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#finite-cont-frac","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For a finite list $a_0, a_1, \\dots, a_t$ of integers (possibly zero or negative), the value $[a_0; a_1, \\dots, a_t]$ is the rational number $a_0 + 1/(a_1 + 1/(\\cdots + 1/a_t))$, computed from the right: the empty list has value $0$, and the list $a_0, a_1, \\dots, a_t$ has value $a_0 + v^{-1}$, where $v$ is the value of the list $a_1, \\dots, a_t$. In particular $[a] = a + 0^{-1} = a$; here and at any vanishing\u2026","title":"Value of a finite continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/finite-cont-frac-cons","kind":"node","lean":"MetaMathlibExt.finiteContFrac_cons","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#finite-cont-frac-cons","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For every integer $a$ and all integers $l_1, \\dots, l_n$ with $n \\ge 0$, $[a; l_1, \\dots, l_n] = a + [l_1; l_2, \\dots, l_n]^{-1}$, where the empty continued fraction (the case $n = 0$) has value $0$ and $0^{-1}$ is taken to be $0$ whenever $[l_1; l_2, \\dots, l_n] = 0$.","title":"Recursion for finite continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/finite-cont-frac-singleton","kind":"node","lean":"MetaMathlibExt.finiteContFrac_singleton","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#finite-cont-frac-singleton","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For every integer $a$, the finite continued fraction $[a]$ has value $a$.","title":"Continued fraction of length one"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/finite-cont-frac-one-two-two","kind":"node","lean":"MetaMathlibExt.finiteContFrac_one_two_two","parent":10461,"reading_title":{"format":"inline-parts-v1","parts":[["text","The continued fraction "],["math","\\((1; 2, 2)\\)"]],"source":"The continued fraction $(1; 2, 2)$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#finite-cont-frac-one-two-two","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"The finite continued fraction $[1; 2, 2]$ has value $7/5$.","title":"The continued fraction $(1; 2, 2)$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/finite-cont-frac-add-inv-mul-den-sq","kind":"node","lean":"MetaMathlibExt.finiteContFrac_add_inv_mul_den_sq","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#finite-cont-frac-add-inv-mul-den-sq","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"can_prove","summary":"Let $a_0$ be an integer, let $t$ be even and let $a_1, \\dots, a_t$ be integers with $a_i \\ge 1$ for every $i$, and let $q$ be the denominator in lowest terms of the value $[a_0; a_1, \\dots, a_t]$ of the finite continued fraction. Then for every integer $x \\ge 1$, $[a_0; a_1, \\dots, a_t] + 1/(x q^2) = [a_0; a_1, \\dots, a_t, x, -a_t, \\dots, -a_1]$.","title":"Folding lemma for finite continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/finite-cont-frac-zero-merge","kind":"node","lean":"MetaMathlibExt.finiteContFrac_zero_merge","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#finite-cont-frac-zero-merge","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"can_prove","summary":"For all integers $a$ and $b$ and all finite lists of integers $u_1, \\dots, u_m$ and $v_1, \\dots, v_n$ ($m, n \\ge 0$), the finite continued fractions satisfy $[u_1, \\dots, u_m, a, 0, b, v_1, \\dots, v_n] = [u_1, \\dots, u_m, a + b, v_1, \\dots, v_n]$.","title":"A zero partial quotient merges its neighbours"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/finite-cont-frac-neg-block","kind":"node","lean":"MetaMathlibExt.finiteContFrac_neg_block","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#finite-cont-frac-neg-block","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"can_prove","summary":"Let $x \\ge 0$ be an integer, let $t \\ge 2$ and let $a_1, \\dots, a_t$ be integers with $a_i \\ge 1$ for every $i$. Then for every finite list of integers $b_1, \\dots, b_m$ ($m \\ge 0$), the finite continued fractions satisfy $[b_1, \\dots, b_m, x, -a_t, -a_{t-1}, \\dots, -a_1] = [b_1, \\dots, b_m, x - 1, 1, a_t - 1, a_{t-1}, \\dots, a_1]$.","title":"Normalizing a negated block"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-sum-term","kind":"node","lean":"MetaMathlibExt.paperfoldingSumTerm","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-sum-term","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For a sequence $x = (x_k)_{k \\ge 0}$ of natural numbers, the real numbers $s_k$ are defined by $s_0 = 1$ and $s_{k+1} = s_k^2 / x_k$, where the quotient is read as $0$ when $x_k = 0$. They are the terms of the sums $S = \\sum_k s_k$.","title":"Terms of the paperfolding sums"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-sum-term-zero","kind":"node","lean":"MetaMathlibExt.paperfoldingSumTerm_zero","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-sum-term-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For every sequence $x$ of natural numbers, the term $s_0$ of the paperfolding sum terms defined by $x$ equals $1$.","title":"First term of the paperfolding sums"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-sum-term-succ","kind":"node","lean":"MetaMathlibExt.paperfoldingSumTerm_succ","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-sum-term-succ","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For every sequence $x$ of natural numbers and every $k \\in \\mathbb{N}$, the paperfolding sum terms defined by $x$ satisfy $s_{k+1} = s_k^2 / x_k$, where the quotient is read as $0$ when $x_k = 0$.","title":"Recursion for the paperfolding sum terms"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-sum-term-one","kind":"node","lean":"MetaMathlibExt.paperfoldingSumTerm_one","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-sum-term-one","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For every sequence $x$ of natural numbers with $x_0 = 2$, the term $s_1$ of the paperfolding sum terms defined by $x$ equals $1/2$.","title":"Second term of the paperfolding sums"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/cloitre-crease-interleave","kind":"node","lean":"MetaMathlibExt.cloitreCreaseInterleave","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#cloitre-crease-interleave","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"Given a sequence $c : \\mathbb{N} \\to \\mathbb{N}$ and a start index $j$, the crease interleaving of a finite word $w$ of natural numbers reads $w$ from left to right as letters $(2)$ and $(1, 1)$ and inserts $c_j, c_{j+1}, c_{j+2}, \\dots$ after the successive letters. Precisely, the empty word gives the empty word; a word $2, w'$ gives $2, c_j$ followed by the interleaving of $w'$ from index $j + 1$; a word\u2026","title":"Interleaving creases into a Cloitre word"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/cloitre-crease-interleave-cloitre-approx","kind":"node","lean":"MetaMathlibExt.cloitreCreaseInterleave_cloitreApprox_two","parent":10461,"reading_title":{"format":"inline-parts-v1","parts":[["text","Crease interleaving of the Cloitre approximant "],["math","\\(A_2\\)"]],"source":"Crease interleaving of the Cloitre approximant $A_2$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#cloitre-crease-interleave-cloitre-approx","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For every sequence $c : \\mathbb{N} \\to \\mathbb{N}$, the crease interleaving by $c$, from start index $1$, of the approximant $A_2 = (1, 1, 2, 1, 1, 1, 1)$ of Cloitre's sequence is $(1, 1, c_1, 2, c_2, 1, 1, c_3, 1, 1, c_4)$: each of the four letters $(1, 1)$, $(2)$, $(1, 1)$, $(1, 1)$ of $A_2$ is followed by its crease $c_1$, $c_2$, $c_3$, $c_4$ in turn.","title":"Crease interleaving of the Cloitre approximant $A_2$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-cont-frac-word","kind":"node","lean":"MetaMathlibExt.paperfoldingContFracWord","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-cont-frac-word","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For a sequence $x = (x_k)_{k \\ge 0}$ of natural numbers, the finite words $A_k$ of natural numbers are defined by $A_0 = (1, 1)$ and $A_{k+1} = A_k \\, (x_{k+1} - 1) \\, A_k^r$: the concatenation of $A_k$, the single letter $\\max(x_{k+1} - 1, 0)$ and the mirror image $A_k^r$ of $A_k$. The mirror image of a word reverses it and then replaces a leading $2$ by $1, 1$, or two leading letters $1, 1$ by $2$, leaving any\u2026","title":"Continued-fraction words of the paperfolding sums"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-cont-frac-word-one","kind":"node","lean":"MetaMathlibExt.paperfoldingContFracWord_one","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-cont-frac-word-one","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"For every sequence $x$ of natural numbers, the continued-fraction word $A_1$ defined by $x$ is $(1, 1, \\max(x_1 - 1, 0), 2)$.","title":"The second paperfolding continued-fraction word"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-cont-frac-word-const-six-tw","kind":"node","lean":"MetaMathlibExt.paperfoldingContFracWord_const_six_two","parent":10461,"reading_title":{"format":"inline-parts-v1","parts":[["text","The paperfolding continued-fraction word "],["math","\\(A_2\\)"],["text"," for constant "],["math","\\(x_k = 6\\)"]],"source":"The paperfolding continued-fraction word $A_2$ for constant $x_k = 6$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-cont-frac-word-const-six-tw","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"fully_proved","summary":"Let $x_k = 6$ for every $k$, and let $A_0 = (1, 1)$ and $A_{k+1} = A_k\\,(x_{k+1} - 1)\\,A_k^r$, where the mirror image $A^r$ of a word $A$ is obtained by reversing $A$ and then replacing a leading $2$ by $1, 1$, or two leading letters $1, 1$ by $2$, leaving any other word unchanged. Then $A_2 = (1, 1, 5, 2, 5, 1, 1, 5, 1, 1)$.","title":"The paperfolding continued-fraction word $A_2$ for constant $x_k = 6$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-sum-cont-frac-cloitre-ruler","kind":"node","lean":"MetaMathlibExt.paperfoldingSum_contFrac_cloitre_ruler","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-sum-cont-frac-cloitre-ruler","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"can_prove","summary":"Let $x = (x_k)_{k \\ge 0}$ be natural numbers with $x_0 = 2$ and $x_k \\ge 2$ for all $k$, let $s_k$ be the paperfolding sum terms defined by $x$, and let $S = \\sum_{k \\ge 0} s_k$, with continued fraction expansion $[a_0; a_1, a_2, \\ldots]$. For $j \\ge 1$ let $c_j = x_{\\nu_2(j) + 1} - 1$, where $\\nu_2(j)$ is the exponent of $2$ in $j$. Then $a_0 = 1$, and for every $k$ the partial quotients $a_1, a_2, \\ldots$ begin\u2026","title":"Continued fraction of a paperfolding sum in Cloitre and ruler order"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/paperfolding-sum-cont-frac-recursion","kind":"node","lean":"MetaMathlibExt.paperfoldingSum_contFrac_recursion","parent":10461,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continued fraction of a paperfolding sum via the words "],["math","\\(A_k\\)"]],"source":"Continued fraction of a paperfolding sum via the words $A_k$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#paperfolding-sum-cont-frac-recursion","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"can_prove","summary":"Let $x = (x_k)_{k \\ge 0}$ be natural numbers with $x_0 = 2$ and $x_k \\ge 2$ for all $k$, let $s_k$ be the paperfolding sum terms defined by $x$, and let $S = \\sum_{k \\ge 0} s_k$, with continued fraction expansion $[a_0; a_1, a_2, \\ldots]$. Let $A_k$ be the continued-fraction words defined by $x$, so that $A_0 = (1, 1)$ and $A_{k+1} = A_k \\, (x_{k+1} - 1) \\, A_k^r$, where the mirror image $A^r$ of a word $A$ is\u2026","title":"Continued fraction of a paperfolding sum via the words $A_k$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/tendsto-cloitre-approx-count-div-length","kind":"node","lean":"MetaMathlibExt.tendsto_cloitreApprox_count_div_length","parent":10461,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio/index.html#tendsto-cloitre-approx-count-div-length","scope":"NumberTheory/continued-fractions-expansions/part-01/folding-lemma-and-paperfolding-continued-fractio","status":"can_prove","summary":"Let $A_k$ be the $k$-th approximant of Cloitre's sequence. As $k \\to \\infty$, the proportion of entries of $A_k$ equal to $1$ tends to $2/3$, and the proportion of entries equal to $2$ tends to $1/3$.","title":"Digit frequencies in Cloitre's approximants"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string.json","id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","title":"Gauss\u2013Kuzmin masses of continued fraction strings"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/convergent-matrix","kind":"node","lean":"MetaMathlibExt.convergentMatrix","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#convergent-matrix","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"For a finite string $a = (a_1, \\ldots, a_n)$ of natural numbers, the convergent matrix $C(a)$ is the $2 \\times 2$ integer matrix given by the ordered product $C(a) = \\begin{pmatrix} 0 & 1 \\\\ 1 & a_1 \\end{pmatrix} \\cdots \\begin{pmatrix} 0 & 1 \\\\ 1 & a_n \\end{pmatrix}$ (the identity matrix when $n = 0$). Its entries are written $C(a) = \\begin{pmatrix} p' & p \\\\ q' & q \\end{pmatrix}$.","title":"Convergent matrix"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/convergent-matrix-nil","kind":"node","lean":"MetaMathlibExt.convergentMatrix_nil","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#convergent-matrix-nil","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"The convergent matrix of the empty string is the $2 \\times 2$ identity matrix.","title":"Convergent matrix of the empty string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/convergent-matrix-cons","kind":"node","lean":"MetaMathlibExt.convergentMatrix_cons","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#convergent-matrix-cons","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"For every natural number $t$ and every finite string $a$ of natural numbers, the convergent matrix of the string $(t, a)$, obtained by prepending $t$ to $a$, is $C(t, a) = \\begin{pmatrix} 0 & 1 \\\\ 1 & t \\end{pmatrix} C(a)$.","title":"Prepending a digit to a string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/cf-cylinder","kind":"node","lean":"MetaMathlibExt.cfCylinder","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#cf-cylinder","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"For a finite string $a = (a_1, \\ldots, a_n)$ of natural numbers, the fundamental interval $I(a)$ is the set of real numbers $x \\in [0, 1)$ whose continued fraction expansion $x = [0; a_1(x), a_2(x), \\ldots]$ has at least $n$ digits, with $a_i(x) = a_i$ for $1 \\le i \\le n$.","title":"Fundamental interval of a string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/cf-cylinder-nil","kind":"node","lean":"MetaMathlibExt.cfCylinder_nil","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#cf-cylinder-nil","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"The fundamental interval of the empty string is the whole interval: $I(()) = [0, 1)$.","title":"Fundamental interval of the empty string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/fund-interval","kind":"node","lean":"MetaMathlibExt.fundInterval","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#fund-interval","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"For real numbers $\\alpha$ and $\\beta$, the interval $\\langle \\alpha, \\beta \\rangle$ is $[\\alpha, \\beta)$ if $\\alpha < \\beta$, and $(\\beta, \\alpha]$ otherwise.","title":"Half-open interval between two reals"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/fund-interval-of-lt","kind":"node","lean":"MetaMathlibExt.fundInterval_of_lt","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#fund-interval-of-lt","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"For real numbers $\\alpha$ and $\\beta$, write $\\langle \\alpha, \\beta \\rangle$ for $[\\alpha, \\beta)$ if $\\alpha < \\beta$ and for $(\\beta, \\alpha]$ otherwise. If $\\alpha < \\beta$, then $\\langle \\alpha, \\beta \\rangle = [\\alpha, \\beta)$.","title":"Half-open interval between increasing endpoints"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/gauss-kuzmin-mass","kind":"node","lean":"MetaMathlibExt.gaussKuzminMass","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#gauss-kuzmin-mass","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"For a finite string $a$ of natural numbers with fundamental interval $I(a)$, the Gauss\u2013Kuzmin mass of $a$ is $P_{GK}(a) = \\frac{1}{\\log 2} \\int_{I(a)} \\frac{dx}{1 + x}$, the measure of $I(a)$ under the Gauss\u2013Kuzmin measure with density $\\frac{1}{(1 + x) \\log 2}$.","title":"Gauss\u2013Kuzmin mass of a string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/gauss-kuzmin-mass-nil","kind":"node","lean":"MetaMathlibExt.gaussKuzminMass_nil","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#gauss-kuzmin-mass-nil","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"The Gauss\u2013Kuzmin mass of the empty string is $P_{GK}(()) = \\frac{1}{\\log 2} \\int_{[0, 1)} \\frac{dx}{1 + x}$.","title":"Gauss\u2013Kuzmin mass of the empty string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/characteristic-number","kind":"node","lean":"MetaMathlibExt.characteristicNumber","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#characteristic-number","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"For a finite string $a$ of natural numbers with convergent matrix $C(a) = \\begin{pmatrix} p' & p \\\\ q' & q \\end{pmatrix}$, the characteristic number of $a$ is the integer $\\chi(a) = (p + q)(q' + q)$.","title":"Characteristic number of a string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/characteristic-number-nil","kind":"node","lean":"MetaMathlibExt.characteristicNumber_nil","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#characteristic-number-nil","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"The characteristic number of the empty string is $\\chi(()) = 1$.","title":"Characteristic number of the empty string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/is-stable-string","kind":"node","lean":"MetaMathlibExt.IsStableString","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#is-stable-string","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"For a finite string $a = (a_1, \\dots, a_n)$ of natural numbers, let $M(t) = \\begin{pmatrix} 0 & 1 \\\\ 1 & t \\end{pmatrix}$ and write the convergent matrix of $a$ as","title":"Stable strings"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/is-stable-string-nil","kind":"node","lean":"MetaMathlibExt.isStableString_nil","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#is-stable-string-nil","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"The empty string is a stable string.","title":"The empty string is stable"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/has-nontrivial-symmetry","kind":"node","lean":"MetaMathlibExt.HasNontrivialSymmetry","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#has-nontrivial-symmetry","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"A finite string $a$ of natural numbers has a nontrivial symmetry if there is a string $b$ that is a rearrangement of $a$ (the same entries with the same multiplicities, in some order) such that $b \\ne a$, $b$ is not the reverse of $a$, and $P_{GK}(b) = P_{GK}(a)$, where $P_{GK}$ denotes the Gauss\u2013Kuzmin measure of a string.","title":"Nontrivial symmetry of a string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/not-has-nontrivial-symmetry-nil","kind":"node","lean":"MetaMathlibExt.not_hasNontrivialSymmetry_nil","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#not-has-nontrivial-symmetry-nil","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"fully_proved","summary":"The empty string has no nontrivial symmetry.","title":"The empty string has no nontrivial symmetry"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/gauss-kuzmin-string-frequency","kind":"node","lean":"MetaMathlibExt.gaussKuzmin_string_frequency","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#gauss-kuzmin-string-frequency","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"For a real number $x$, let $a_1(x), a_2(x), \\dots$ be the partial denominators of the continued fraction expansion of $x$. For almost every real $x$, with respect to Lebesgue measure, the following holds: for every string $b = (b_1, \\dots, b_n)$ of positive integers, the number of indices $i \\in \\{0, \\dots, N - 1\\}$ such that $a_{i+j}(x) = b_j$ for all $j = 1, \\dots, n$, divided by $N$, tends to the Gauss\u2013Kuzmin\u2026","title":"Generalized Gauss\u2013Kuzmin theorem"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/gauss-kuzmin-mass-reverse","kind":"node","lean":"MetaMathlibExt.gaussKuzminMass_reverse","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#gauss-kuzmin-mass-reverse","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"For every string $a$ of positive integers, $P_{GK}(\\overleftarrow{a}) = P_{GK}(a)$, where $P_{GK}$ is the Gauss\u2013Kuzmin mass and $\\overleftarrow{a}$ is the reverse of $a$.","title":"Reversal symmetry of the Gauss\u2013Kuzmin mass"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/density-eq-gauss-kuzmin-of-reverse-symme","kind":"node","lean":"MetaMathlibExt.density_eq_gaussKuzmin_of_reverse_symmetric","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#density-eq-gauss-kuzmin-of-reverse-symme","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Let $f$ be a real function that is continuous and nonnegative on $[0, 1]$ with $\\int_0^1 f(x)\\,dx = 1$, and let $S$ be an infinite set of natural numbers. Write $I(a)$ for the fundamental interval of a string $a$ and $\\overleftarrow{a}$ for the reverse of $a$. If $\\int_{I(a)} f(x)\\,dx = \\int_{I(\\overleftarrow{a})} f(x)\\,dx$ for every string $a$ of positive integers whose length lies in $S$, then\u2026","title":"Reversal symmetry forces the Gauss\u2013Kuzmin density"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/exists-reverse-symmetric-density-ne-gaus","kind":"node","lean":"MetaMathlibExt.exists_reverse_symmetric_density_ne_gaussKuzmin","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#exists-reverse-symmetric-density-ne-gaus","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Write $I(a)$ for the fundamental interval of a string $a$ and $\\overleftarrow{a}$ for the reverse of $a$. For every integer $N \\ge 1$ there is a real function $f$, continuous and nonnegative on $[0, 1]$ with $\\int_0^1 f(x)\\,dx = 1$, such that $\\int_{I(a)} f(x)\\,dx = \\int_{I(\\overleftarrow{a})} f(x)\\,dx$ for every string $a$ of positive integers of length at most $N$, but $f(x) \\ne \\frac{1}{\\log 2 \\cdot (1 + x)}$\u2026","title":"Symmetry up to a fixed length does not force the Gauss\u2013Kuzmin density"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/convergent-matrix-reverse-append-cons","kind":"node","lean":"MetaMathlibExt.convergentMatrix_reverse_append_cons","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#convergent-matrix-reverse-append-cons","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Let $a = (a_1, \\ldots, a_n)$ be a finite string of positive integers with convergent matrix $C(a) = \\begin{pmatrix} p' & p \\\\ q' & q \\end{pmatrix}$, let $t \\ge 1$ be an integer, and write $\\overleftarrow{a} = (a_n, \\ldots, a_1)$ for the reversed string. Then (i) $C(\\overleftarrow{a}) = \\begin{pmatrix} p' & q' \\\\ p & q \\end{pmatrix}$; (ii)\u2026","title":"Convergent matrices of reversed and extended strings"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/cf-cylinder-reverse-append-cons-ae-eq","kind":"node","lean":"MetaMathlibExt.cfCylinder_reverse_append_cons_ae_eq","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#cf-cylinder-reverse-append-cons-ae-eq","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Let $a = (a_1, \\ldots, a_n)$ be a finite string of positive integers with convergent matrix $C(a) = \\begin{pmatrix} p' & p \\\\ q' & q \\end{pmatrix}$, let $t \\ge 1$ be an integer, write $\\overleftarrow{a} = (a_n, \\ldots, a_1)$ for the reversed string, and for real numbers $x$ and $y$ let $\\langle x, y \\rangle$ be $[x, y)$ if $x < y$ and $(y, x]$ otherwise. Write $I(b)$ for the fundamental interval of a string $b$,\u2026","title":"Fundamental intervals of reversed and extended strings"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/gauss-kuzmin-mass-eq-abs-logb","kind":"node","lean":"MetaMathlibExt.gaussKuzminMass_eq_abs_logb","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#gauss-kuzmin-mass-eq-abs-logb","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Let $a$ be a string of $n$ positive integers, let $p$, $q'$ and $q$ be the upper-right, lower-left and lower-right entries of its convergent matrix $C(a)$, and let $\\overleftarrow{a}$ be the reverse of $a$. Then the Gauss\u2013Kuzmin masses satisfy $P_{GK}(a) = P_{GK}(\\overleftarrow{a})$ and $P_{GK}(a) = \\left|\\log_2\\left(1 + \\frac{(-1)^n}{(p + q)(q' + q)}\\right)\\right|$.","title":"Explicit Gauss\u2013Kuzmin mass of a string"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/density-cf-value-eq-of-append-symmetric","kind":"node","lean":"MetaMathlibExt.density_cfValue_eq_of_append_symmetric","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#density-cf-value-eq-of-append-symmetric","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Let $f$ be a real function that is continuous and nonnegative on $[0, 1]$ with $\\int_0^1 f(x)\\,dx = 1$. Let $a = (a_1, \\dots, a_n)$ be a string of positive integers, let $p$ and $q$ be the upper-right and lower-right entries of its convergent matrix $C(a)$, so that $p/q = [0; a_1, \\dots, a_n]$, and let $\\overleftarrow{a}$ be the reverse of $a$. Write $I(b)$ for the fundamental interval of a string $b$, $(a, t)$ for\u2026","title":"Symmetric one-digit extensions fix the density at a rational point"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/reverse-symmetric-of-eq-gauss-kuzmin-at","kind":"node","lean":"MetaMathlibExt.reverse_symmetric_of_eq_gaussKuzmin_at_length","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#reverse-symmetric-of-eq-gauss-kuzmin-at","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ and let $N \\ge 1$. For a string $a$ of positive integers write $I(a)$ for its fundamental interval, $P_{GK}(a)$ for its Gauss\u2013Kuzmin measure and $a^{\\mathrm{rev}}$ for its reverse. If $\\mu(I(a)) = P_{GK}(a)$ for every string $a$ of exactly $N$ positive integers, then $\\mu(I(a)) = \\mu(I(a^{\\mathrm{rev}}))$ for every string $a$ of at most $N$ positive integers.","title":"Reversal symmetry from agreement with the Gauss\u2013Kuzmin measure"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/gauss-kuzmin-mass-eq-iff-characteristic","kind":"node","lean":"MetaMathlibExt.gaussKuzminMass_eq_iff_characteristicNumber_eq","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#gauss-kuzmin-mass-eq-iff-characteristic","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Write $P_{GK}$ for the Gauss\u2013Kuzmin mass and $\\chi$ for the characteristic number of a string. If $a$ and $b$ are strings of positive integers whose lengths have the same parity, then $P_{GK}(a) = P_{GK}(b)$ if and only if $\\chi(a) = \\chi(b)$.","title":"Gauss\u2013Kuzmin masses agree exactly when characteristic numbers do"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/characteristic-number-swap-ne","kind":"node","lean":"MetaMathlibExt.characteristicNumber_swap_ne","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#characteristic-number-swap-ne","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"For positive integers $a$, $b$, $c$ with $a \\ne b$, the strings $(a, b, c)$ and $(b, a, c)$ have different characteristic numbers: $\\chi(a, b, c) \\ne \\chi(b, a, c)$.","title":"Swapping the first two of three digits changes the characteristic number"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/is-stable-string-families","kind":"node","lean":"MetaMathlibExt.isStableString_families","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#is-stable-string-families","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"(i) For every positive integer $t$, the strings $(t, 2t)$ and $(t, 1, 2t + 1)$ are stable. (ii) If $a = (a_1, \\dots, a_m)$ is a stable string of positive integers, then for every positive integer $t$ the string $(t, a_m, \\dots, a_1, 2t)$, obtained from the reverse of $a$ by adding $t$ at the front and $2t$ at the end, is stable. (iii) For every $n \\ge 2$ there are infinitely many stable strings of $n$ positive\u2026","title":"Families of stable strings"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/stable-plus-nontrivial-symmetry","kind":"node","lean":"MetaMathlibExt.stable_plus_nontrivial_symmetry","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#stable-plus-nontrivial-symmetry","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"Let $n \\ge 1$ and let $a = (a_1, \\dots, a_n)$ be a stable string of positive integers. Put $a^+ = (2, 1, a_1, \\dots, a_{n-1}, a_n + 1)$ and $\\sigma(a^+) = (2, a_n + 1, a_{n-1}, \\dots, a_1, 1)$. Then $\\sigma(a^+)$ is a rearrangement of $a^+$, it differs both from $a^+$ and from the reverse of $a^+$, and $P_{GK}(\\sigma(a^+)) = P_{GK}(a^+)$, where $P_{GK}$ denotes the Gauss\u2013Kuzmin measure of a string.","title":"Nontrivial symmetries from stable strings"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/nontrivial-symmetry-length-three-and-inf","kind":"node","lean":"MetaMathlibExt.nontrivialSymmetry_length_three_and_infinite","parent":10481,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string/index.html#nontrivial-symmetry-length-three-and-inf","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-masses-of-continued-fraction-string","status":"can_prove","summary":"(i) No string of three positive integers has a nontrivial symmetry. (ii) For every $n \\ge 4$ there are infinitely many strings of $n$ positive integers that have a nontrivial symmetry.","title":"No nontrivial symmetries in length three, infinitely many from length four"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator.json","id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"can_prove","title":"Gauss\u2013Kuzmin\u2013Wirsing operator"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/gkw-term","kind":"node","lean":"MathlibExt.Analysis.WirsingConstant.gkwTerm","parent":10511,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html#gkw-term","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"fully_proved","summary":"For a function $f : \\mathbb{R} \\to \\mathbb{R}$, a real number $x$ and a natural number $k$, the $k$-th term of the Gauss\u2013Kuzmin\u2013Wirsing series of $f$ at $x$ is $\\frac{1}{(x+k+1)^2} f\\left(\\frac{1}{x+k+1}\\right)$, with the convention $1/0 = 0$.","title":"Terms of the Gauss\u2013Kuzmin\u2013Wirsing series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/is-eigenvalue","kind":"node","lean":"MathlibExt.Analysis.WirsingConstant.IsEigenvalue","parent":10511,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html#is-eigenvalue","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"fully_proved","summary":"A real number $\\lambda$ is an eigenvalue of the Gauss\u2013Kuzmin\u2013Wirsing operator in the continuous model when there is a function $f : \\mathbb{R} \\to \\mathbb{R}$, continuous on $[0, 1]$, bounded on $[0, 1]$ and not identically zero on $[0, 1]$, such that for every $x \\in [0, 1]$","title":"Eigenvalues of the Gauss\u2013Kuzmin\u2013Wirsing operator on continuous functions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/is-eigenvalue-one","kind":"node","lean":"MathlibExt.Analysis.WirsingConstant.isEigenvalue_one","parent":10511,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html#is-eigenvalue-one","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"fully_proved","summary":"There is a function $f : \\mathbb{R} \\to \\mathbb{R}$ that is continuous on $[0, 1]$, bounded on $[0, 1]$ and not identically zero on $[0, 1]$, such that $\\sum_{k \\ge 1} \\frac{1}{(x+k)^2} f\\left(\\frac{1}{x+k}\\right) = f(x)$ for every $x \\in [0, 1]$; that is, $1$ is an eigenvalue of the Gauss\u2013Kuzmin\u2013Wirsing operator in the continuous model.","title":"The Gauss\u2013Kuzmin\u2013Wirsing operator has eigenvalue one"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/gkw-term-hol","kind":"node","lean":"MathlibExt.Analysis.WirsingConstant.gkwTermHol","parent":10511,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html#gkw-term-hol","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$, a complex number $z$ and a natural number $k$, the $k$-th term of the Gauss\u2013Kuzmin\u2013Wirsing series of $f$ at $z$ is $\\frac{1}{(z+k+1)^2} f\\left(\\frac{1}{z+k+1}\\right)$, with the convention $1/0 = 0$.","title":"Terms of the complex Gauss\u2013Kuzmin\u2013Wirsing series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/is-holomorphic-eigenvalue","kind":"node","lean":"MathlibExt.Analysis.WirsingConstant.IsHolomorphicEigenvalue","parent":10511,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html#is-holomorphic-eigenvalue","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"fully_proved","summary":"Let $D = \\{z \\in \\mathbb{C} : |z - 1| < 3/2\\}$ be the Mayer disk. A complex number $\\lambda$ is a holomorphic eigenvalue of the Gauss\u2013Kuzmin\u2013Wirsing operator when there is a function $f : \\mathbb{C} \\to \\mathbb{C}$, holomorphic on $D$ and not identically zero on $D$, such that for every $z \\in D$","title":"Holomorphic eigenvalues of the Gauss\u2013Kuzmin\u2013Wirsing operator"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/wirsing-constant","kind":"node","lean":"MathlibExt.Analysis.WirsingConstant.wirsingConstant","parent":10511,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html#wirsing-constant","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"fully_proved","summary":"The Wirsing constant is the supremum in $\\mathbb{R}$ of $|\\lambda|$ over all holomorphic eigenvalues $\\lambda \\ne 1$ of the Gauss\u2013Kuzmin\u2013Wirsing operator on the Mayer disk $\\{z \\in \\mathbb{C} : |z - 1| < 3/2\\}$, with the convention that the supremum is $0$ when there is no such $\\lambda$ or when these absolute values are unbounded.","title":"Wirsing constant"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/mayer-roepstorff-lower-bound-wanted","kind":"node","lean":"MathlibExt.Analysis.WirsingConstantWanted.mayerRoepstorffLowerBound_wanted","parent":10511,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html#mayer-roepstorff-lower-bound-wanted","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"can_prove","summary":"The Wirsing constant is at least $0.30366299$.","title":"Mayer\u2013Roepstorff lower bound for the Wirsing constant"},{"id":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/mayer-roepstorff-upper-bound-wanted","kind":"node","lean":"MathlibExt.Analysis.WirsingConstantWanted.mayerRoepstorffUpperBound_wanted","parent":10511,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator/index.html#mayer-roepstorff-upper-bound-wanted","scope":"NumberTheory/continued-fractions-expansions/part-01/gauss-kuzmin-wirsing-operator","status":"can_prove","summary":"The Wirsing constant is at most $0.30366327$.","title":"Mayer\u2013Roepstorff upper bound for the Wirsing constant"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant.json","id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"can_prove","title":"Hata's Farey formula for Euler's constant"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/farey-interval-mediant-telescoping","kind":"node","lean":"MetaMathlibExt.farey_interval_mediant_telescoping","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#farey-interval-mediant-telescoping","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"can_prove","summary":"For an interval $[x/y, z/w]$ write $f([x/y, z/w]) = (xw + yz)/(xyzw)$. Let $a, b, c, d$ be natural numbers with $a > 0$, $d > 0$ and $bc = ad + 1$. Then, computing in the real numbers, the value of $f$ on $[a/b, c/d]$ minus its values on the two halves $[a/b, (a + c)/(b + d)]$ and $[(a + c)/(b + d), c/d]$ cut at the mediant equals","title":"Mediant telescoping identity for Farey intervals"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/farey-interval-subset-reciprocal-interva","kind":"node","lean":"MetaMathlibExt.farey_interval_subset_reciprocal_interval","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#farey-interval-subset-reciprocal-interva","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"can_prove","summary":"Let $a, b, c, d$ be natural numbers with $a > 0$, $c \\le d$ and $bc = ad + 1$, so that $[a/b, c/d] \\subseteq [0, 1]$ is a Farey interval with positive left endpoint. Then there is a natural number $n \\ge 1$ with $1/(n + 1) \\le a/b$ and $c/d \\le 1/n$, that is, $[a/b, c/d] \\subseteq [1/(n + 1), 1/n]$.","title":"Farey intervals lie in reciprocal intervals"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/has-sum-farey-subintervals-reciprocal-in","kind":"node","lean":"MetaMathlibExt.hasSum_farey_subintervals_reciprocal_interval","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#has-sum-farey-subintervals-reciprocal-in","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"can_prove","summary":"Let $n \\ge 1$ and $J_n = [1/(n + 1), 1/n]$. The family $1/(abcd(a + c)(b + d))$, indexed by the quadruples $(a, b, c, d)$ of natural numbers with $bc = ad + 1$, $b \\le (n + 1)a$ and $nc \\le d$ (that is, by the Farey intervals $[a/b, c/d] \\subseteq J_n$, including $J_n$ itself), is summable with sum","title":"Sum over the Farey intervals inside a reciprocal interval"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/euler-mascheroni-eq-hata-farey-sum","kind":"node","lean":"MetaMathlibExt.euler_mascheroni_eq_hata_farey_sum","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#euler-mascheroni-eq-hata-farey-sum","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"can_prove","summary":"Let $S$ be the set of quadruples $(a, b, c, d)$ of natural numbers with $bc = ad + 1$, $a > 0$ and $c \\le d$; these are the Farey intervals $[a/b, c/d] \\subseteq [0, 1]$ with positive left endpoint. The family $1/(abcd(a + c)(b + d))$, indexed by $(a, b, c, d) \\in S$, is summable, and Euler's constant $\\gamma$ satisfies","title":"Hata's formula for Euler's constant"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/farey-generation","kind":"node","lean":"MetaMathlibExt.fareyGeneration","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#farey-generation","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"fully_proved","summary":"The $n$-th Stern-Brocot generation $F_n$ is a finite set of intervals $[a/b, c/d]$, each stored as the quadruple $(a, b, c, d)$ of natural numbers. It is defined by $F_0 = \\{[0/1, 1/1]\\}$, and $F_{n+1}$ is obtained by replacing each $[a/b, c/d] \\in F_n$ with its two mediant halves $[a/b, (a + c)/(b + d)]$ and $[(a + c)/(b + d), c/d]$.","title":"Stern-Brocot generations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/farey-generation-one-two","kind":"node","lean":"MetaMathlibExt.fareyGeneration_one_two","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#farey-generation-one-two","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"fully_proved","summary":"Writing $[a/b, c/d]$ for the quadruple $(a, b, c, d)$, the first two Stern-Brocot generations are $F_1 = \\{[0/1, 1/2], [1/2, 1/1]\\}$ and $F_2 = \\{[0/1, 1/3], [1/3, 1/2], [1/2, 2/3], [2/3, 1/1]\\}$.","title":"The first two Stern-Brocot generations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/farey-generation-succ","kind":"node","lean":"MetaMathlibExt.fareyGeneration_succ","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#farey-generation-succ","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"fully_proved","summary":"For every natural number $n$, the generation $F_{n+1}$ is the union, over all intervals $[a/b, c/d] \\in F_n$, of the two mediant halves $[a/b, (a + c)/(b + d)]$ and $[(a + c)/(b + d), c/d]$, where each interval $[x/y, z/w]$ is stored as the quadruple $(x, y, z, w)$.","title":"Recurrence for the Stern-Brocot generations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/farey-generation-length-le","kind":"node","lean":"MetaMathlibExt.fareyGeneration_length_le","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#farey-generation-length-le","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"can_prove","summary":"For every natural number $n$, every interval $[a/b, c/d]$ of the $n$-th Stern-Brocot generation $F_n$, that is, every quadruple $(a, b, c, d) \\in F_n$, satisfies $c/d - a/b \\le 1/(n + 1)$.","title":"Length bound for the Stern-Brocot generations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/abs-sum-farey-partition-sub-integral-le","kind":"node","lean":"MetaMathlibExt.abs_sum_farey_partition_sub_integral_le","parent":10520,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant/index.html#abs-sum-farey-partition-sub-integral-le","scope":"NumberTheory/continued-fractions-expansions/part-01/hata-s-farey-formula-for-euler-s-constant","status":"can_prove","summary":"Let $a, b, c, d, m$ be natural numbers with $a > 0$, $bc = ad + 1$, $c \\le d$ and $m \\ge 2$. Let $A, B : \\mathbb{N} \\to \\mathbb{N}$ satisfy $A(1) = a$, $B(1) = b$, $A(m) = c$, $B(m) = d$ and $B(k) A(k+1) = A(k) B(k+1) + 1$ for $1 \\le k \\le m - 1$, and let $\\delta$ be a real number with $1/(B(k) B(k+1)) \\le \\delta$ for $1 \\le k \\le m - 1$. Then","title":"Riemann-sum bound for Farey partitions"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/l-roth-series.json","id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","title":"L\u00fcroth series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-digit","kind":"node","lean":"MetaMathlibExt.lurothDigit","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-digit","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a real number $x$, the first L\u00fcroth digit of $x$ is the natural number $d_1(x) = \\lfloor 1/x \\rfloor + 1$, where $\\lfloor \\cdot \\rfloor$ is the floor truncated at $0$ and $1/0 = 0$. For $x \\in (0, 1]$ this is the usual digit and is at least $2$; for $x \\le 0$ and for $x > 1$ it equals $1$.","title":"First L\u00fcroth digit"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-map","kind":"node","lean":"MetaMathlibExt.lurothMap","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-map","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a real number $x$ with first L\u00fcroth digit $d = d_1(x)$, the L\u00fcroth map is $L(x) = d(d - 1)x - (d - 1)$.","title":"L\u00fcroth map"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-digit-seq","kind":"node","lean":"MetaMathlibExt.lurothDigitSeq","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-digit-seq","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a real number $x$ and $n \\ge 1$, the $n$-th L\u00fcroth digit of $x$ is $d_n(x) = d_1(L^{n-1}(x))$, where $d_1$ is the first L\u00fcroth digit, $L$ is the L\u00fcroth map and $L^{n-1}$ is its $(n-1)$-fold iterate ($L^0$ is the identity).","title":"L\u00fcroth digits"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-digit-one","kind":"node","lean":"MetaMathlibExt.lurothDigit_one","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-digit-one","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"The first L\u00fcroth digit of $1$ is $d_1(1) = 2$.","title":"First L\u00fcroth digit of 1"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-digit-half","kind":"node","lean":"MetaMathlibExt.lurothDigit_half","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-digit-half","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"The first L\u00fcroth digit of $1/2$ is $d_1(1/2) = 3$.","title":"First L\u00fcroth digit of 1/2"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-map-one","kind":"node","lean":"MetaMathlibExt.lurothMap_one","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-map-one","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"The L\u00fcroth map $L$ satisfies $L(1) = 1$.","title":"The L\u00fcroth map fixes 1"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-map-half","kind":"node","lean":"MetaMathlibExt.lurothMap_half","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-map-half","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"The L\u00fcroth map $L$ sends $1/2$ to $1$: $L(1/2) = 1$.","title":"The L\u00fcroth map at 1/2"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-map-zero","kind":"node","lean":"MetaMathlibExt.lurothMap_zero","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-map-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"The L\u00fcroth map $L$ satisfies $L(0) = 0$.","title":"The L\u00fcroth map fixes 0"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-digit-seq-succ","kind":"node","lean":"MetaMathlibExt.lurothDigitSeq_succ","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-digit-seq-succ","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For every real number $x$ and every $n \\ge 1$, the $n$-th L\u00fcroth digit of $L(x)$, where $L$ is the L\u00fcroth map, is the $(n+1)$-st L\u00fcroth digit of $x$: $d_n(L(x)) = d_{n+1}(x)$.","title":"The L\u00fcroth map shifts the digits"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-luroth-cylinder","kind":"node","lean":"MetaMathlibExt.volume_lurothCylinder","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-luroth-cylinder","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Write $d_k(x)$ for the $k$-th L\u00fcroth digit of $x$. For every $n \\in \\mathbb{N}$ and all natural numbers $c_1, \\dots, c_n$, the set $\\{x \\in (0, 1] : d_j(x) = c_j \\text{ for } 1 \\le j \\le n\\}$ has Lebesgue measure $\\prod_{j=1}^{n} \\frac{1}{c_j (c_j - 1)}$, with the convention $1/0 = 0$; in particular both sides vanish when some $c_j < 2$.","title":"Measure of L\u00fcroth cylinders"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-luroth-digit-ge-zero-one","kind":"node","lean":"MetaMathlibExt.volume_lurothDigit_ge_zero_one","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-luroth-digit-ge-zero-one","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $(\\Psi(n))_{n \\ge 0}$ be a sequence of positive reals, write $d_k(x)$ for the $k$-th L\u00fcroth digit of $x$, and let $E$ be the set of $x \\in (0, 1]$ such that $d_{n+1}(x) \\ge \\Psi(n)$ for infinitely many $n$. If $\\sum_{n \\ge 0} 1 / \\Psi(n) < \\infty$ then $E$ has Lebesgue measure $0$, and if $\\sum_{n \\ge 0} 1 / \\Psi(n) = \\infty$ then $E$ has Lebesgue measure $1$.","title":"Borel\u2013Bernstein theorem for L\u00fcroth digits"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-luroth-weighted-product-ge-zero-o","kind":"node","lean":"MetaMathlibExt.volume_lurothWeightedProduct_ge_zero_one","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-luroth-weighted-product-ge-zero-o","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $m \\ge 1$, let $t_0, \\dots, t_{m-1}$ be positive reals, let $T = \\max_i t_i$ and let $\\ell$ be the number of indices $j$ with $t_j = T$. Let $(\\Psi(n))_{n \\ge 0}$ be a sequence of positive reals for which there is $c > 1$ with $\\Psi(n) \\ge c$ for all sufficiently large $n$ (equivalently, $\\liminf_n \\Psi(n) > 1$). Write $d_k(x)$ for the $k$-th L\u00fcroth digit of $x$ and let $E$ be the set of $x \\in (0, 1]$ such\u2026","title":"Zero-one law for weighted products of consecutive L\u00fcroth digits"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-luroth-consecutive-digits-zero-on","kind":"node","lean":"MetaMathlibExt.volume_lurothConsecutiveDigits_zero_one","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-luroth-consecutive-digits-zero-on","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $m \\in \\mathbb{N}$, let $F$ be a real-valued function on $m$-tuples of natural numbers and let $(\\Psi(n))_{n \\ge 0}$ be a real sequence. For $y \\in \\mathbb{R}$ put","title":"Zero-one law for functions of consecutive L\u00fcroth digits"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-luroth-consecutive-digits-zero-on-2","kind":"node","lean":"MetaMathlibExt.volume_lurothConsecutiveDigits_zero_one_integral","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-luroth-consecutive-digits-zero-on-2","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $m \\in \\mathbb{N}$ and let $F : \\mathbb{R}^m \\to \\mathbb{R}$ be a measurable function that is positive on $[2, \\infty)^m$ and non-decreasing there for the coordinatewise order. Let $M \\in \\mathbb{R}$ satisfy $F(x) \\le M F(\\lfloor x_1 \\rfloor, \\dots, \\lfloor x_m \\rfloor)$ for every $x \\in [2, \\infty)^m$, and for $y \\in \\mathbb{R}$ put","title":"Integral zero-one law for functions of consecutive L\u00fcroth digits"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-limsup-luroth-preimage-zero-one","kind":"node","lean":"MetaMathlibExt.volume_limsup_lurothPreimage_zero_one","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-limsup-luroth-preimage-zero-one","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $m \\in \\mathbb{N}$, let $(S_n)_{n \\ge 0}$ be a sequence of sets of $m$-tuples of natural numbers, write $d_k(x)$ for the $k$-th L\u00fcroth digit of $x$ and $\\lambda$ for Lebesgue measure. Put $A_n = \\{x \\in (0, 1] : (d_1(x), \\dots, d_m(x)) \\in S_n\\}$ (a union of cylinders of level $m$) and let $E$ be the set of $x \\in (0, 1]$ such that $(d_{n+1}(x), \\dots, d_{n+m}(x)) \\in S_n$ for infinitely many $n$. If\u2026","title":"Borel\u2013Cantelli dichotomy for L\u00fcroth cylinder events"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-luroth-digits-ge-le","kind":"node","lean":"MetaMathlibExt.volume_lurothDigits_ge_le","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-luroth-digits-ge-le","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $m \\in \\mathbb{N}$, let $F : \\mathbb{R}^m \\to \\mathbb{R}$ be non-decreasing on $[2, \\infty)^m$ for the coordinatewise order, let $y \\in \\mathbb{R}$, and write $d_k(x)$ for the $k$-th L\u00fcroth digit of $x$. Then the Lebesgue measure of $\\{x \\in (0, 1] : F(d_1(x), \\dots, d_m(x)) \\ge y\\}$ is at most","title":"Upper bound for the measure of a L\u00fcroth digit superlevel set"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/le-volume-luroth-digits-ge","kind":"node","lean":"MetaMathlibExt.le_volume_lurothDigits_ge","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#le-volume-luroth-digits-ge","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $m$ be a natural number, let $D = [2, \\infty)^m$, let $F : \\mathbb{R}^m \\to \\mathbb{R}$ satisfy $F(x) > 0$ for every $x \\in D$, and let $M$ be a real number with $F(x) \\le M\\,F(\\lfloor x_1 \\rfloor, \\ldots, \\lfloor x_m \\rfloor)$ for every $x \\in D$ (the paper's condition (6)). Put $G(y) = \\int_{\\{x \\in D \\,:\\, F(x) \\ge y\\}} \\prod_{i=1}^m x_i^{-2}\\,dx$ for real $y$, let $\\lambda$ be Lebesgue measure on\u2026","title":"Lower bound for the measure of large L\u00fcroth digit values"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/well-approximable-set","kind":"node","lean":"MetaMathlibExt.wellApproximableSet","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#well-approximable-set","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a function $\\psi : \\mathbb{N} \\to \\mathbb{R}$, the set $W(\\psi)$ of $\\psi$-well approximable numbers consists of the real numbers $x \\in [0, 1)$ such that for infinitely many natural numbers $q$ there is an integer $p$ with $|x - p/q| < \\psi(q)/q$.","title":"Well approximable numbers"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/well-approximable-set-subset-ico","kind":"node","lean":"MetaMathlibExt.wellApproximableSet_subset_Ico","parent":10530,"reading_title":{"format":"inline-parts-v1","parts":[["text","Well approximable numbers lie in "],["math","\\((0, 1)\\)"]],"source":"Well approximable numbers lie in $(0, 1)$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#well-approximable-set-subset-ico","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For every function $\\psi : \\mathbb{N} \\to \\mathbb{R}$, every point of the set $W(\\psi)$ of $\\psi$-well approximable numbers lies in the interval $[0, 1)$.","title":"Well approximable numbers lie in $(0, 1)$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/lower-order-at-infinity","kind":"node","lean":"MetaMathlibExt.lowerOrderAtInfinity","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#lower-order-at-infinity","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For $\\psi : \\mathbb{N} \\to \\mathbb{R}$, the lower order at infinity of $1/\\psi$ is","title":"Lower order at infinity"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/upper-order-at-infinity","kind":"node","lean":"MetaMathlibExt.upperOrderAtInfinity","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#upper-order-at-infinity","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a function $\\psi$ from the natural numbers to the reals, the upper order at infinity of $\\psi$ is the element","title":"Upper order at infinity"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/order-ratio-rpow-inv","kind":"node","lean":"MetaMathlibExt.orderRatio_rpow_inv","parent":10530,"reading_title":{"format":"inline-parts-v1","parts":[["text","The order ratio of "],["math","\\(q^{-\\tau}\\)"]],"source":"The order ratio of $q^{-\\tau}$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#order-ratio-rpow-inv","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For every real $\\tau$ and all sufficiently large natural numbers $q$, $\\max(0, -\\log((q^{\\tau})^{-1})/\\log q) = \\max(0, \\tau)$.","title":"The order ratio of $q^{-\\tau}$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/lower-order-at-infinity-rpow-inv","kind":"node","lean":"MetaMathlibExt.lowerOrderAtInfinity_rpow_inv","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#lower-order-at-infinity-rpow-inv","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a real number $\\tau$, let $\\psi : \\mathbb{N} \\to \\mathbb{R}$ be given by $\\psi(q) = q^{-\\tau}$ (the value at $q = 0$ does not matter). Then the lower order at infinity of $1/\\psi$ is $\\tau_\\psi = \\max(\\tau, 0)$, an element of $[0, \\infty]$; in particular $\\tau_\\psi = \\tau$ when $\\tau \\ge 0$.","title":"Lower order at infinity of a power function"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/upper-order-at-infinity-rpow-inv","kind":"node","lean":"MetaMathlibExt.upperOrderAtInfinity_rpow_inv","parent":10530,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper order at infinity of "],["math","\\(q^{-\\tau}\\)"]],"source":"Upper order at infinity of $q^{-\\tau}$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#upper-order-at-infinity-rpow-inv","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For every real $\\tau$, the upper order at infinity of $q \\mapsto (q^{\\tau})^{-1}$ is $\\max(0, \\tau)$; in particular it is $\\tau$ when $\\tau \\ge 0$.","title":"Upper order at infinity of $q^{-\\tau}$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-well-approximable-set-khintchine","kind":"node","lean":"MetaMathlibExt.volume_wellApproximableSet_khintchine","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-well-approximable-set-khintchine","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $\\psi : \\mathbb{N} \\to \\mathbb{R}$ be non-increasing on all of $\\mathbb{N}$ (including $\\psi(0) \\ge \\psi(1)$) with $0 \\le \\psi(q) \\le 1/2$ for every $q$. If the series $\\sum_q \\psi(q)$ converges, then the set $W(\\psi)$ of $\\psi$-well approximable numbers has Lebesgue measure $0$; if the series diverges, then $W(\\psi)$ has Lebesgue measure $1$.","title":"Khintchine's theorem"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/dim-h-well-approximable-set-rpow","kind":"node","lean":"MetaMathlibExt.dimH_wellApproximableSet_rpow","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#dim-h-well-approximable-set-rpow","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"For a real number $\\tau \\ge 1$, let $W(\\tau)$ be the set of $x \\in [0, 1)$ such that for infinitely many natural numbers $q$ there is an integer $p$ with $|x - p/q| < q^{-(1+\\tau)}$; this is the set of $\\psi$-well approximable numbers for $\\psi(q) = q^{-\\tau}$. Then the Hausdorff dimension of $W(\\tau)$ is $2/(1 + \\tau)$.","title":"Jarn\u00edk\u2013Besicovitch theorem"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/dim-h-well-approximable-set-dodson","kind":"node","lean":"MetaMathlibExt.dimH_wellApproximableSet_dodson","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#dim-h-well-approximable-set-dodson","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $\\psi$ be a non-increasing function from the natural numbers to $(0, 1/2]$, let $W(\\psi)$ be the set of $x \\in [0, 1)$ such that for infinitely many $q$ there is an integer $p$ with $|x - p/q| < \\psi(q)/q$, and let $\\tau_\\psi = \\liminf_{q \\to \\infty} (-\\log \\psi(q) / \\log q)$, taken in $[0, \\infty]$, be the lower order at infinity of $1/\\psi$. If $\\tau_\\psi \\ge 1$, then the Hausdorff dimension of $W(\\psi)$ is\u2026","title":"Dodson's theorem"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-denominator","kind":"node","lean":"MetaMathlibExt.lurothDenominator","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-denominator","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a sequence of natural numbers $d_0, d_1, d_2, \\dots$ (numbered from $0$), the $n$-th L\u00fcroth denominator is","title":"L\u00fcroth denominators"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-convergent","kind":"node","lean":"MetaMathlibExt.lurothConvergent","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-convergent","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a sequence of natural numbers $d_0, d_1, d_2, \\dots$ (numbered from $0$) with L\u00fcroth denominators $Q_0, Q_1, Q_2, \\dots$, the $n$-th L\u00fcroth convergent is the rational number","title":"L\u00fcroth convergents"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-digit-seq-one","kind":"node","lean":"MetaMathlibExt.lurothDigitSeq_one","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-digit-seq-one","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"Every L\u00fcroth digit of $1$ equals $2$, that is, $1 = [2, 2, 2, \\dots]$ in L\u00fcroth notation.","title":"The L\u00fcroth digits of 1"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-map-luroth-map-27-71","kind":"node","lean":"MetaMathlibExt.lurothMap_lurothMap_27_71","parent":10530,"reading_title":{"format":"inline-parts-v1","parts":[["text","The L\u00fcroth orbit of "],["math","\\(27/71\\)"],["text"," has period two"]],"source":"The L\u00fcroth orbit of $27/71$ has period two"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-map-luroth-map-27-71","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"The L\u00fcroth map $T$ sends $27/71$ to $20/71$ and sends $20/71$ to $27/71$; in particular $T(T(27/71)) = 27/71$.","title":"The L\u00fcroth orbit of $27/71$ has period two"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-denominator-succ","kind":"node","lean":"MetaMathlibExt.lurothDenominator_succ","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-denominator-succ","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For every sequence of natural numbers $d_0, d_1, d_2, \\dots$ with L\u00fcroth denominators $Q_0, Q_1, Q_2, \\dots$ and every $n \\ge 0$,","title":"Recurrence for L\u00fcroth denominators"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-denominator-two","kind":"node","lean":"MetaMathlibExt.lurothDenominator_two","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-denominator-two","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"If $d_j = 2$ for every $j \\ge 0$, then the L\u00fcroth denominators are $Q_n = 2^{n+1}$ for every $n \\ge 0$ (digits numbered from $0$; this is the paper's $Q_{n+1} = 2^{n+1}$).","title":"L\u00fcroth denominators of the constant digit sequence 2"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-convergent-succ","kind":"node","lean":"MetaMathlibExt.lurothConvergent_succ","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-convergent-succ","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For every sequence of natural numbers $d_0, d_1, d_2, \\dots$ with L\u00fcroth denominators $Q_n$ and L\u00fcroth convergents $x_n$, and every $n \\ge 0$, $x_{n+1} = x_n + 1/Q_{n+1}$, where $1/Q_{n+1}$ is read as $0$ if $Q_{n+1} = 0$.","title":"Recurrence for L\u00fcroth convergents"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-well-approximable-set","kind":"node","lean":"MetaMathlibExt.lurothWellApproximableSet","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-well-approximable-set","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For a function $\\psi$ from the natural numbers to the reals, $L(\\psi)$ is the set of $x \\in (0, 1]$ such that $|x - c_n(x)| < \\psi(Q_n(x))/Q_n(x)$ for infinitely many $n$, where $Q_n(x)$ and $c_n(x)$ are the $n$-th L\u00fcroth denominator and the $n$-th L\u00fcroth convergent of the L\u00fcroth digit sequence of $x$.","title":"L\u00fcroth well approximable numbers"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-well-approximable-set-subset-ioc","kind":"node","lean":"MetaMathlibExt.lurothWellApproximableSet_subset_Ioc","parent":10530,"reading_title":{"format":"inline-parts-v1","parts":[["text","L\u00fcroth well approximable numbers lie in "],["math","\\((0, 1)\\)"]],"source":"L\u00fcroth well approximable numbers lie in $(0, 1)$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-well-approximable-set-subset-ioc","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"For every function $\\psi$ from the natural numbers to the reals, the set $L(\\psi)$ of L\u00fcroth $\\psi$-well approximable numbers is contained in $(0, 1]$.","title":"L\u00fcroth well approximable numbers lie in $(0, 1)$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-triple-set","kind":"node","lean":"MetaMathlibExt.lurothTripleSet","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-triple-set","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"The set $S$ consists of the triples $(c_n(y), Q_n(y), d_n(y))$ for $y \\in (0, 1]$ and $n \\ge 0$, where $d_0(y), d_1(y), \\dots$ are the L\u00fcroth digits of $y$ (indexed from $0$), $Q_n(y) = d_n(y) \\prod_{j < n} d_j(y)(d_j(y) - 1)$ is the $n$-th L\u00fcroth denominator of this digit sequence, and $c_n(y) = \\sum_{k \\le n} 1/Q_k(y)$ is its $n$-th L\u00fcroth convergent. The first entry is the rational number $c_n(y)$, the second\u2026","title":"L\u00fcroth convergent triples"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/half-two-two-mem-luroth-triple-set","kind":"node","lean":"MetaMathlibExt.half_two_two_mem_lurothTripleSet","parent":10530,"reading_title":{"format":"inline-parts-v1","parts":[["text","A L\u00fcroth triple from the expansion of "],["math","\\(1\\)"]],"source":"A L\u00fcroth triple from the expansion of $1$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#half-two-two-mem-luroth-triple-set","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"fully_proved","summary":"Let $S$ be the set of triples $(P_n(y)/Q_n(y), Q_n(y), d_n(y))$ for $y \\in (0, 1]$ and $n \\ge 0$, where $P_n(y)/Q_n(y)$ is the $n$-th L\u00fcroth convergent of $y$, $Q_n(y)$ its L\u00fcroth denominator and $d_n(y)$ its $n$-th L\u00fcroth digit (indexed from $0$). Then $(1/2, 2, 2) \\in S$: it is the triple of the first convergent of $1 = [2, 2, 2, \\ldots]$.","title":"A L\u00fcroth triple from the expansion of $1$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-convergent-lt-le","kind":"node","lean":"MetaMathlibExt.lurothConvergent_lt_le","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-convergent-lt-le","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $x \\in (0, 1]$ have L\u00fcroth digits $d_0, d_1, d_2, \\dots$ (numbered from $0$), L\u00fcroth denominators $Q_0, Q_1, Q_2, \\dots$ and L\u00fcroth convergents $x_0, x_1, x_2, \\dots$. Then for every $n \\ge 0$,","title":"L\u00fcroth convergents approximate from below"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/hausdorff-measure-limsup-closed-ball-of","kind":"node","lean":"MetaMathlibExt.hausdorffMeasure_limsup_closedBall_of_volume","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#hausdorff-measure-limsup-closed-ball-of","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $(c_n)$ and $(r_n)$ be real sequences with $r_n \\ge 0$ for every $n$ and $r_n \\to 0$, and let $I_n = [c_n - r_n, c_n + r_n]$. Assume that $I_n \\subseteq [0, 1]$ for every $n$, let $0 < s < 1$, and let $I_n^s = [c_n - r_n^s, c_n + r_n^s]$. If $[0, 1] \\cap \\limsup_{n \\to \\infty} I_n^s$ has Lebesgue measure $1$, then the $s$-dimensional Hausdorff measure of $\\limsup_{n \\to \\infty} I_n$ is infinite and the\u2026","title":"Mass transference principle for intervals"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/exists-luroth-convergent-eq-of-mem-lurot","kind":"node","lean":"MetaMathlibExt.exists_lurothConvergent_eq_of_mem_lurothTripleSet","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#exists-luroth-convergent-eq-of-mem-lurot","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"For $y \\in (0, 1]$ and $n \\ge 0$ let $P_n(y)/Q_n(y)$ be the $n$-th L\u00fcroth convergent of $y$, $Q_n(y)$ its L\u00fcroth denominator and $d_n(y)$ the $n$-th L\u00fcroth digit of $y$, and let $S$ be the set of all triples $(P_n(y)/Q_n(y), Q_n(y), d_n(y))$ for $y \\in (0, 1]$ and $n \\ge 0$. If $x \\in (0, 1]$ and $(P/Q, Q, d) \\in S$ satisfy $0 < x - P/Q \\le 1/((d - 1) Q)$, then $P/Q$ is a L\u00fcroth convergent of $x$, that is,\u2026","title":"Recognising L\u00fcroth convergents"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-well-approximable-set-rpow-eq-lim","kind":"node","lean":"MetaMathlibExt.lurothWellApproximableSet_rpow_eq_limsup","parent":10530,"reading_title":{"format":"inline-parts-v1","parts":[["text","L\u00fcroth "],["math","\\(q^{-\\tau}\\)"],["text","-well approximable numbers as a limsup over convergent triples"]],"source":"L\u00fcroth $q^{-\\tau}$-well approximable numbers as a limsup over convergent triples"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-well-approximable-set-rpow-eq-lim","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $\\tau \\ge 0$ and let $L(\\tau)$ be the set of L\u00fcroth $\\psi$-well approximable numbers for $\\psi(q) = q^{-\\tau}$. For a triple $t = (c, Q, d)$ in the set $S$ of L\u00fcroth convergent triples, put $B_t = (c, c + Q^{-(1+\\tau)}) \\cap (c, c + 1/((d - 1)Q)]$. Then $L(\\tau)$ consists of the real numbers $x$ that lie in $B_t$ for infinitely many $t \\in S$.","title":"L\u00fcroth $q^{-\\tau}$-well approximable numbers as a limsup over convergent triples"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/luroth-well-approximable-set-eq-limsup","kind":"node","lean":"MetaMathlibExt.lurothWellApproximableSet_eq_limsup","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#luroth-well-approximable-set-eq-limsup","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $\\psi$ be a function from the natural numbers to $(0, 1]$ whose lower order at infinity $\\tau_\\psi$ is greater than $1$. Then the set $L(\\psi)$ of L\u00fcroth $\\psi$-well approximable numbers consists of the real numbers $x$ for which there are infinitely many triples $(c, Q, d)$ in the set $S$ of L\u00fcroth convergent triples with $c < x < c + \\psi(Q)/Q$.","title":"L\u00fcroth well approximable numbers as a limsup over convergent triples"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/volume-luroth-well-approximable-set","kind":"node","lean":"MetaMathlibExt.volume_lurothWellApproximableSet","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#volume-luroth-well-approximable-set","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $\\psi$ be a non-increasing function from the natural numbers (including $0$) to $(0, 1]$, and let $L(\\psi)$ be the set of L\u00fcroth $\\psi$-well approximable numbers. If the series $\\sum_{q \\ge 1} -\\psi(q) \\log \\psi(q) / q$ converges, then $L(\\psi)$ has Lebesgue measure $0$; if it diverges, then $L(\\psi)$ has Lebesgue measure $1$.","title":"Tan\u2013Zhou theorem (L\u00fcroth analogue of Khintchine's theorem)"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/dim-h-luroth-well-approximable-set-rpow","kind":"node","lean":"MetaMathlibExt.dimH_lurothWellApproximableSet_rpow_of_one_le","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#dim-h-luroth-well-approximable-set-rpow","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"For $x \\in (0, 1]$ let $P_n(x)/Q_n(x)$ be the $n$-th L\u00fcroth convergent of $x$, with L\u00fcroth denominator $Q_n(x)$. For a real number $\\tau \\ge 1$ let $L(\\tau)$ be the set of $x \\in (0, 1]$ such that $|x - P_n(x)/Q_n(x)| < Q_n(x)^{-\\tau}/Q_n(x)$ for infinitely many $n$, that is, the set of L\u00fcroth $\\psi$-well approximable numbers for $\\psi(q) = q^{-\\tau}$. Then the Hausdorff dimension of $L(\\tau)$ is $1/(1 + \\tau)$.","title":"Cao\u2013Wu\u2013Zhang theorem"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/dim-h-hausdorff-measure-luroth-well-appr","kind":"node","lean":"MetaMathlibExt.dimH_hausdorffMeasure_lurothWellApproximableSet_rpow","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#dim-h-hausdorff-measure-luroth-well-appr","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"For $x \\in (0, 1]$ let $P_n(x)/Q_n(x)$ be the $n$-th L\u00fcroth convergent of $x$, with L\u00fcroth denominator $Q_n(x)$. For a real number $\\tau \\ge 0$ let $L(\\tau)$ be the set of $x \\in (0, 1]$ such that $|x - P_n(x)/Q_n(x)| < Q_n(x)^{-\\tau}/Q_n(x)$ for infinitely many $n$, that is, the set of L\u00fcroth $\\psi$-well approximable numbers for $\\psi(q) = q^{-\\tau}$. Then the Hausdorff dimension of $L(\\tau)$ is $1/(1 + \\tau)$,\u2026","title":"Hausdorff dimension and measure of L\u00fcroth well approximable sets for powers"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/dim-h-luroth-well-approximable-set-bound","kind":"node","lean":"MetaMathlibExt.dimH_lurothWellApproximableSet_bounds","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#dim-h-luroth-well-approximable-set-bound","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $\\psi$ be a function from the natural numbers to $(0, 1]$. Let $L(\\psi)$ be the set of $x \\in (0, 1]$ such that $|x - P_n(x)/Q_n(x)| < \\psi(Q_n(x))/Q_n(x)$ for infinitely many $n$, where $P_n(x)/Q_n(x)$ is the $n$-th L\u00fcroth convergent of $x$ and $Q_n(x)$ its L\u00fcroth denominator. Let $\\tau_\\psi = \\liminf_{q \\to \\infty} (-\\log \\psi(q) / \\log q)$ and\u2026","title":"Dimension bounds for L\u00fcroth well approximable sets"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/dim-h-luroth-well-approximable-set-bound-2","kind":"node","lean":"MetaMathlibExt.dimH_lurothWellApproximableSet_bounds_denominators","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#dim-h-luroth-well-approximable-set-bound-2","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $\\psi$ be a function from the natural numbers to $(0, 1]$. Let $L(\\psi)$ be the set of $x \\in (0, 1]$ such that $|x - P_n(x)/Q_n(x)| < \\psi(Q_n(x))/Q_n(x)$ for infinitely many $n$, where $P_n(x)/Q_n(x)$ is the $n$-th L\u00fcroth convergent of $x$ and $Q_n(x)$ its L\u00fcroth denominator. For $k \\ge 0$ let $S_k$ be the set of numbers $d_k \\prod_{j < k} d_j (d_j - 1)$, where $d_0, \\ldots, d_k$ are integers at least $2$\u2026","title":"Dimension bounds for L\u00fcroth well approximable sets via denominators"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/exists-not-antitone-dim-h-luroth-well-ap","kind":"node","lean":"MetaMathlibExt.exists_not_antitone_dimH_lurothWellApproximableSet_eq_zero","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#exists-not-antitone-dim-h-luroth-well-ap","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"For every real number $\\tau \\ge 0$ there is a function $\\psi$ from the natural numbers to $(0, 1]$ such that: $\\psi$ is not non-increasing on $\\{q : q \\ge N\\}$ for any $N$; the lower order at infinity $\\liminf_{q \\to \\infty} (-\\log \\psi(q) / \\log q)$ of $1/\\psi$ equals $\\tau$; and the set $L(\\psi)$ of $x \\in (0, 1]$ such that $|x - P_n(x)/Q_n(x)| < \\psi(Q_n(x))/Q_n(x)$ for infinitely many $n$, where $P_n(x)/Q_n(x)$\u2026","title":"Without monotonicity the L\u00fcroth dimension can vanish"},{"id":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series/dim-h-luroth-well-approximable-set-of-te","kind":"node","lean":"MetaMathlibExt.dimH_lurothWellApproximableSet_of_tendsto_sum","parent":10530,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/l-roth-series/index.html#dim-h-luroth-well-approximable-set-of-te","scope":"NumberTheory/continued-fractions-expansions/part-01/l-roth-series","status":"can_prove","summary":"Let $\\psi$ be a function from the natural numbers to $(0, 1]$. Let $L(\\psi)$ be the set of $x \\in (0, 1]$ such that $|x - P_n(x)/Q_n(x)| < \\psi(Q_n(x))/Q_n(x)$ for infinitely many $n$, where $P_n(x)/Q_n(x)$ is the $n$-th L\u00fcroth convergent of $x$ and $Q_n(x)$ its L\u00fcroth denominator. Let $\\tau_\\psi = \\liminf_{q \\to \\infty} (-\\log \\psi(q) / \\log q)$, taken in $[0, \\infty]$, be the lower order at infinity of $1/\\psi$,\u2026","title":"Dimension of L\u00fcroth well approximable sets under a divergence condition"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions.json","id":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions","status":"fully_proved","title":"Periodic continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/continued-fraction-eventually-periodic","kind":"node","lean":"MetaMathlibExt.ContinuedFractionEventuallyPeriodic","parent":10581,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/index.html#continued-fraction-eventually-periodic","scope":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions","status":"fully_proved","summary":"Let $x$ be a real number with regular continued fraction $[a_0; a_1, a_2, \\dots]$. The continued fraction of $x$ is eventually periodic if it does not terminate and its sequence of partial denominators $(a_n)_{n \\ge 1}$, which omits the head coefficient $a_0$, is eventually periodic: there are natural numbers $p > 0$ and $n_0 \\ge 1$ such that $a_{n+p} = a_n$ for every $n \\ge n_0$.","title":"Eventually periodic continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/is-real-quadratic-irrational","kind":"node","lean":"MetaMathlibExt.IsRealQuadraticIrrational","parent":10581,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/index.html#is-real-quadratic-irrational","scope":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions","status":"fully_proved","summary":"A real number $x$ is a real quadratic irrational if it is integral (equivalently, algebraic) over $\\mathbb{Q}$ and its minimal polynomial over $\\mathbb{Q}$ has degree $2$.","title":"Real quadratic irrationals"},{"id":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/continued-fraction-eventually-periodic-o","kind":"node","lean":"MetaMathlibExt.continuedFractionEventuallyPeriodic_of_isRealQuadraticIrrational","parent":10581,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/index.html#continued-fraction-eventually-periodic-o","scope":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions","status":"fully_proved","summary":"If a real number $x$ is a real quadratic irrational, then the continued fraction of $x$ is eventually periodic.","title":"Lagrange's theorem on periodic continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/is-real-quadratic-irrational-of-continue","kind":"node","lean":"MetaMathlibExt.isRealQuadraticIrrational_of_continuedFractionEventuallyPeriodic","parent":10581,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/index.html#is-real-quadratic-irrational-of-continue","scope":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions","status":"fully_proved","summary":"If the continued fraction of a real number $x$ is eventually periodic, then $x$ is a real quadratic irrational.","title":"Euler's theorem on periodic continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/continued-fraction-eventually-periodic-i","kind":"node","lean":"MetaMathlibExt.continuedFractionEventuallyPeriodic_iff_isRealQuadraticIrrational","parent":10581,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions/index.html#continued-fraction-eventually-periodic-i","scope":"NumberTheory/continued-fractions-expansions/part-01/periodic-continued-fractions","status":"fully_proved","summary":"A real number $x$ has an eventually periodic continued fraction if and only if $x$ is a real quadratic irrational.","title":"Euler\u2013Lagrange theorem"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers.json","id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","title":"Mallows representation of irrational numbers"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-letter","kind":"node","lean":"MetaMathlibExt.MallowsLetter","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-letter","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The letters of Mallows representations are the three distinct letters $L$, $M$ and $R$, standing for \"left\", \"middle\" and \"right\".","title":"The Mallows alphabet"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-letter","kind":"node","lean":"MetaMathlibExt.SternBrocotLetter","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-letter","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The alphabet of Stern\u2013Brocot representations: the two letters $L$ (\"left\") and $R$ (\"right\").","title":"Stern\u2013Brocot letters"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-parity-letter","kind":"node","lean":"MetaMathlibExt.MallowsParityLetter","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-parity-letter","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The parity alphabet consists of the four letters $O_L$, $E_R$, $O_R$ and $E_L$. The intermediate set $I$ linking the Stern\u2013Brocot and Mallows representations is a set of infinite sequences over this alphabet.","title":"The parity alphabet"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/nat-pair-to-real","kind":"node","lean":"MetaMathlibExt.natPairToReal","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#nat-pair-to-real","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The pair $(n, m)$ of natural numbers encodes the fraction with numerator $n$ and denominator $m$, whose real value is $n / m$ (taken to be $0$ when $m = 0$).","title":"Real value of a fraction encoded as a pair"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/nat-pair-to-real-one-two","kind":"node","lean":"MetaMathlibExt.natPairToReal_one_two","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pair "],["math","\\((1, 2)\\)"],["text"," encodes "],["math","\\(1/2\\)"]],"source":"The pair $(1, 2)$ encodes $1/2$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#nat-pair-to-real-one-two","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The real value of the fraction encoded by the pair $(1, 2)$ is $1/2$.","title":"The pair $(1, 2)$ encodes $1/2$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/gap-mediant","kind":"node","lean":"MetaMathlibExt.gapMediant","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#gap-mediant","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For a gap bounded by two fractions $n_1/m_1$ and $n_2/m_2$, each encoded as the pair (numerator, denominator) of natural numbers, the mediant is the fraction $(n_1 + n_2)/(m_1 + m_2)$, encoded as the pair $(n_1 + n_2, m_1 + m_2)$.","title":"Mediant of a gap"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/gap-mediant-root","kind":"node","lean":"MetaMathlibExt.gapMediant_root","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The mediant of "],["math","\\(0/1\\)"],["text"," and "],["math","\\(1/1\\)"]],"source":"The mediant of $0/1$ and $1/1$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#gap-mediant-root","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"With fractions encoded as pairs (numerator, denominator) of natural numbers, the mediant of the gap bounded by $0/1$ and $1/1$ is $1/2$, that is, the pair $(1, 2)$.","title":"The mediant of $0/1$ and $1/1$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-insert","kind":"node","lean":"MetaMathlibExt.mallowsInsert","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-insert","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $n_1/m_1$ and $n_2/m_2$ be fractions, given as pairs of natural numbers (numerator, denominator). Mallows' insertion rule inserts between them the ordered pair of fractions $\\langle (n_1 + n_2)/(m_1 + m_2),\\ (n_1 + 2n_2)/(m_1 + 2m_2) \\rangle$ if $n_1$ is even, and $\\langle (2n_1 + n_2)/(2m_1 + m_2),\\ (n_1 + n_2)/(m_1 + m_2) \\rangle$ if $n_1$ is odd. The rule is applied to adjacent fractions $n_1/m_1 < n_2/m_2$.","title":"Mallows' insertion rule"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-insert-zero-one","kind":"node","lean":"MetaMathlibExt.mallowsInsert_zero_one","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mallows insertion between "],["math","\\(0/1\\)"],["text"," and "],["math","\\(1/1\\)"]],"source":"Mallows insertion between $0/1$ and $1/1$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-insert-zero-one","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Mallows' insertion rule inserts the pair $\\langle 1/2, 2/3 \\rangle$ between the fractions $0/1$ and $1/1$, so the first level of the Mallows tree is $MT_1 = \\langle 1/2, 2/3 \\rangle$.","title":"Mallows insertion between $0/1$ and $1/1$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-insert-half-two-thirds","kind":"node","lean":"MetaMathlibExt.mallowsInsert_half_twoThirds","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mallows insertion between "],["math","\\(1/2\\)"],["text"," and "],["math","\\(2/3\\)"]],"source":"Mallows insertion between $1/2$ and $2/3$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-insert-half-two-thirds","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Mallows' insertion rule inserts the pair $\\langle 4/7, 3/5 \\rangle$ between the fractions $1/2$ and $2/3$ (here the left fraction $1/2$ has odd numerator).","title":"Mallows insertion between $1/2$ and $2/3$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-refine","kind":"node","lean":"MetaMathlibExt.mallowsRefine","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-refine","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $a_1, a_2, \\ldots, a_k$ be a finite list of fractions, each fraction $n/m$ recorded as the pair $(n, m)$ of natural numbers. Its refinement is the list obtained by keeping every entry and inserting, between each two consecutive entries $a_t$ and $a_{t+1}$, the two fractions that Mallows' insertion rule places between them, in the order the rule gives them. A list with fewer than two entries is its own refinement.","title":"One refinement step of the Mallows tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-inserted","kind":"node","lean":"MetaMathlibExt.mallowsInserted","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-inserted","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Fractions $n/m$ are written as pairs $(n, m)$ of natural numbers. For a list of fractions $a_0, a_1, \\dots, a_k$, this is the list obtained by taking, for $j = 0, 1, \\dots, k - 1$ in turn, the two fractions $p_j, q_j$ (in this order) that Mallows' insertion rule inserts between $a_j$ and $a_{j+1}$, that is, the list $p_0, q_0, p_1, q_1, \\dots, p_{k-1}, q_{k-1}$. It is empty when the list has fewer than two entries.","title":"Fractions inserted by one Mallows step"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-level","kind":"node","lean":"MetaMathlibExt.mallowsLevel","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-level","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Fractions $n/m$ are written as pairs $(n, m)$ of natural numbers. Start from the list $\\langle 0/1, 1/1\\rangle$ and refine it repeatedly, each refinement step inserting between every two adjacent entries the pair of fractions given by Mallows' insertion rule. For $i \\ge 1$, $MT_i$ is the list of the fractions inserted at the $i$-th step: the fractions that Mallows' rule inserts between adjacent entries of the list\u2026","title":"Mallows levels"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-level-one","kind":"node","lean":"MetaMathlibExt.mallowsLevel_one","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-level-one","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The ordered list of fractions inserted at the second step of Mallows' construction is $MT_2 = \\langle 1/3, 2/5, 4/7, 3/5, 3/4, 4/5\\rangle$.","title":"The second Mallows level"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-child","kind":"node","lean":"MetaMathlibExt.mallowsChild","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-child","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let a vertex of the Mallows tree have gap $(a, b)$, where $a$ and $b$ are fractions, and let $\\langle p, q \\rangle$ be the pair of fractions that Mallows' insertion rule inserts between $a$ and $b$. The left, middle and right children of the vertex have gaps $(a, p)$, $(p, q)$ and $(q, b)$ respectively.","title":"Children of a Mallows-tree vertex"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-node","kind":"node","lean":"MetaMathlibExt.mallowsNode","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-node","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Fractions $n/m$ are written as pairs $(n, m)$ of natural numbers. The root of the Mallows tree has gap $(0/1, 1/1)$. If a vertex has gap $(a, b)$ and Mallows' insertion rule inserts $\\langle p, q\\rangle$ in $(a, b)$, then its children along the letters $L$, $M$ and $R$ have gaps $(a, p)$, $(p, q)$ and $(q, b)$. The node of a word $w$ over $L$, $M$, $R$ is the gap of the vertex reached from the root by following the\u2026","title":"Gaps of the Mallows tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-label","kind":"node","lean":"MetaMathlibExt.mallowsLabel","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-label","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For a word $w$ over the letters $L$, $M$, $R$, the label of the Mallows-tree vertex reached from the root by the path $w$ is the pair $\\langle p, q\\rangle$ of fractions that Mallows' insertion rule inserts in the gap $(a, b)$ of that vertex.","title":"Labels of the Mallows tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-label-small","kind":"node","lean":"MetaMathlibExt.mallowsLabel_small","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-label-small","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"In the Mallows tree, the root is labelled $\\langle 1/2, 2/3\\rangle$, its middle child is labelled $\\langle 4/7, 3/5\\rangle$ and its right child is labelled $\\langle 3/4, 4/5\\rangle$.","title":"Labels of the first Mallows-tree vertices"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-choose","kind":"node","lean":"MetaMathlibExt.mallowsChoose","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-choose","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let a vertex of the Mallows tree have gap $(a, b)$ and label $\\langle p, q \\rangle$, the pair of fractions that Mallows' insertion rule inserts between $a$ and $b$. For a real number $\\alpha$, the letter chosen at this vertex is $L$ if $\\alpha < p$; otherwise it is $M$ if $\\alpha < q$, and $R$ if $\\alpha \\ge q$.","title":"Letter chosen at a Mallows-tree vertex"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-interval","kind":"node","lean":"MetaMathlibExt.mallowsInterval","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-interval","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Fractions $n/m$ are written as pairs $(n, m)$ of natural numbers. For a gap $(a, b)$ let $\\langle p, q\\rangle$ be the pair that Mallows' insertion rule inserts in $(a, b)$. For a real number $\\alpha$, the gaps $I_0, I_1, \\dots$ are defined by $I_0 = (0/1, 1/1)$ and, if $I_i = (a, b)$, by $I_{i+1} = (a, p)$ when $\\alpha < p$, $I_{i+1} = (p, q)$ when $p \\le \\alpha < q$, and $I_{i+1} = (q, b)$ when $\\alpha \\ge q$. In\u2026","title":"Gaps along the Mallows path of a real number"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep","kind":"node","lean":"MetaMathlibExt.mallowsRep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For a real number $\\alpha$, the Mallows representation $M(\\alpha) = M_1(\\alpha) M_2(\\alpha) \\cdots$ is the infinite word over $\\{L, M, R\\}$ that records the path of $\\alpha$ down the Mallows tree. Fractions are written as numerator\u2013denominator pairs. Start with the gap between $0/1$ and $1/1$. At step $i \\ge 1$, if the current gap lies between $a = n_1/m_1$ and $b = n_2/m_2$, let $(p, q)$ be the pair that Mallows'\u2026","title":"Mallows representation"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-zero-of-lt","kind":"node","lean":"MetaMathlibExt.mallowsRep_zero_of_lt","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-zero-of-lt","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For every real number $\\alpha < 1/2$, the first letter $M_1(\\alpha)$ of the Mallows representation of $\\alpha$ is $L$.","title":"The first Mallows letter below one half"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-child","kind":"node","lean":"MetaMathlibExt.sternBrocotChild","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-child","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Fractions are pairs $(n, m)$ of natural numbers (numerator, denominator), and the mediant of $a = n_1/m_1$ and $b = n_2/m_2$ is $a \\oplus b = (n_1 + n_2)/(m_1 + m_2)$. The child of a gap $(a, b)$ in direction $L$ is the gap $(a, a \\oplus b)$, and its child in direction $R$ is the gap $(a \\oplus b, b)$. For the gap of a Stern\u2013Brocot vertex, whose label is $a \\oplus b$, these are the gaps of its left and right\u2026","title":"Children in the Stern\u2013Brocot tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-label","kind":"node","lean":"MetaMathlibExt.sternBrocotLabel","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-label","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Fractions are pairs (numerator, denominator) of natural numbers, and the mediant of $n_1/m_1$ and $n_2/m_2$ is $(n_1 + n_2)/(m_1 + m_2)$. For a word $w$ over $\\{L, R\\}$, start from the gap $(0/1, 1/1)$ and read $w$ from left to right, replacing the current gap $(a, b)$, with mediant $c$, by $(a, c)$ for each $L$ and by $(c, b)$ for each $R$. The label of $w$ is the mediant of the final gap: the label of the vertex\u2026","title":"Labels of the Stern\u2013Brocot tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-label-small","kind":"node","lean":"MetaMathlibExt.sternBrocotLabel_small","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-label-small","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"In the left subtree of the Stern\u2013Brocot tree, the root (reached by the empty path) is labelled $1/2$, and the vertex reached from the root by the path $LR$ is labelled $2/5$, as numerator\u2013denominator pairs.","title":"Small labels of the Stern\u2013Brocot tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-choose","kind":"node","lean":"MetaMathlibExt.sternBrocotChoose","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-choose","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $\\alpha$ be a real number and let $(a, b)$ be a gap with fractions $a = n_1/m_1$ and $b = n_2/m_2$ (pairs of natural numbers). The letter chosen at $(a, b)$ is $L$ if $\\alpha$ is less than the real number $(n_1 + n_2)/(m_1 + m_2)$, the value of the mediant of $a$ and $b$ (the label of the Stern\u2013Brocot vertex with this gap), and $R$ otherwise.","title":"The Stern\u2013Brocot choice at a vertex"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-interval","kind":"node","lean":"MetaMathlibExt.sternBrocotInterval","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-interval","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Fractions are pairs (numerator, denominator) of natural numbers, and the mediant $a \\oplus b$ of $a = n_1/m_1$ and $b = n_2/m_2$ is $(n_1 + n_2)/(m_1 + m_2)$. For a real number $\\alpha$, the gaps $I_0(\\alpha), I_1(\\alpha), \\dots$ are defined by $I_0(\\alpha) = (0/1, 1/1)$ and, if $I_j(\\alpha) = (a, b)$, by $I_{j+1}(\\alpha) = (a, a \\oplus b)$ when $\\alpha < a \\oplus b$ (as real numbers) and\u2026","title":"Gaps along the Stern\u2013Brocot path"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-frac","kind":"node","lean":"MetaMathlibExt.sternBrocotFrac","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-frac","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Fractions are pairs (numerator, denominator) of natural numbers, and the mediant $a \\oplus b$ of $a = n_1/m_1$ and $b = n_2/m_2$ is $(n_1 + n_2)/(m_1 + m_2)$. For a real number $\\alpha$, start with the gap $(0/1, 1/1)$; at each step pass from the gap $(a, b)$ to $(a, a \\oplus b)$ if $\\alpha < a \\oplus b$ (as real numbers) and to $(a \\oplus b, b)$ otherwise. For $j \\ge 0$, the fraction $f_{j+1}(\\alpha)$ is the\u2026","title":"Fractions on the Stern\u2013Brocot path"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-frac-zero","kind":"node","lean":"MetaMathlibExt.sternBrocotFrac_zero","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Stern\u2013Brocot path starts at "],["math","\\(1/2\\)"]],"source":"The Stern\u2013Brocot path starts at $1/2$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-frac-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For every real number $\\alpha$, the first fraction $f_1(\\alpha)$ on the Stern\u2013Brocot path of $\\alpha$ is $1/2$, as the pair with numerator $1$ and denominator $2$.","title":"The Stern\u2013Brocot path starts at $1/2$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-rep","kind":"node","lean":"MetaMathlibExt.sternBrocotRep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-rep","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For a real number $\\alpha$, the Stern\u2013Brocot representation $SB(\\alpha) = SB_1(\\alpha) SB_2(\\alpha) \\cdots$ is the infinite sequence over $\\{L, R\\}$ obtained by walking down the Stern\u2013Brocot tree from its vertex $1/2$, the vertex with gap $(0/1, 1/1)$. The vertex with gap $(a, b)$ has label the mediant $a \\oplus b$ and children with gaps $(a, a \\oplus b)$ (left) and $(a \\oplus b, b)$ (right). If $f_j(\\alpha)$ is\u2026","title":"Stern\u2013Brocot representation"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-rep-zero-of-lt","kind":"node","lean":"MetaMathlibExt.sternBrocotRep_zero_of_lt","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-rep-zero-of-lt","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For every real number $\\alpha < 1/2$, the first letter $SB_1(\\alpha)$ of the Stern\u2013Brocot representation of $\\alpha$ is $L$.","title":"First Stern\u2013Brocot letter below one half"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-phi","kind":"node","lean":"MetaMathlibExt.sternBrocotPhi","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(\\varphi_{SBT}\\)"]],"source":"The map $\\varphi_{SBT}$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-phi","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $a = n_1/m_1$ and $b = n_2/m_2$ be fractions, given by numerator and denominator, and consider the Stern\u2013Brocot vertex with gap $(a, b)$: its label is the mediant $f = a \\oplus b = (n_1 + n_2)/(m_1 + m_2)$, and its left and right children have labels $l = a \\oplus f$ and $r = f \\oplus b$. Then $\\varphi_{SBT}$ of this vertex is the pair $\\langle l, f \\rangle$ if the numerator of $l$ is even, and\u2026","title":"The map $\\varphi_{SBT}$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-psi","kind":"node","lean":"MetaMathlibExt.sternBrocotPsi","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(\\psi_{SBT}\\)"]],"source":"The map $\\psi_{SBT}$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-psi","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Encode a Stern\u2013Brocot vertex by its gap $(a, b)$ of fractions: its label is the mediant $a \\oplus b$ (numerators added, denominators added), and its left and right children are the vertices with gaps $(a, a \\oplus b)$ and $(a \\oplus b, b)$. Let $v_L$ and $v_R$ be the left and right children of a vertex $v$. If the label of $v_L$ has odd numerator, then $\\psi_{SBT}(v)$ is the list consisting of $v_L$ followed by the\u2026","title":"The map $\\psi_{SBT}$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-phi-psi-root","kind":"node","lean":"MetaMathlibExt.sternBrocotPhi_psi_root","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The maps "],["math","\\(\\varphi_{SBT}\\)"],["text"," and "],["math","\\(\\psi_{SBT}\\)"],["text"," at the root"]],"source":"The maps $\\varphi_{SBT}$ and $\\psi_{SBT}$ at the root"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-phi-psi-root","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"At the root $1/2$ of the Stern\u2013Brocot tree (the vertex with gap $(0/1, 1/1)$), $\\varphi_{SBT}(1/2) = \\langle 1/2, 2/3 \\rangle$ and $\\psi_{SBT}(1/2) = \\langle 1/3, 3/5, 3/4 \\rangle$, where the vertices $1/3$, $3/5$ and $3/4$ are the ones with gaps $(0/1, 1/2)$, $(1/2, 2/3)$ and $(2/3, 1/1)$.","title":"The maps $\\varphi_{SBT}$ and $\\psi_{SBT}$ at the root"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-qstep","kind":"node","lean":"MetaMathlibExt.mallowsQStep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-qstep","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The map $\\mathrm{step}$ sends a letter of the parity alphabet $\\{O_L, E_R, O_R, E_L\\}$ and a letter of $\\{L, R\\}$ to a letter of the parity alphabet, by $\\mathrm{step}(O_L, L) = O_L$, $\\mathrm{step}(O_L, R) = E_R$, $\\mathrm{step}(E_R, L) = O_R$, $\\mathrm{step}(E_R, R) = O_L$, $\\mathrm{step}(O_R, L) = E_L$, $\\mathrm{step}(O_R, R) = O_R$, $\\mathrm{step}(E_L, L) = O_R$ and $\\mathrm{step}(E_L, R) = O_L$.","title":"Transition rule of the parity sequence"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-q","kind":"node","lean":"MetaMathlibExt.mallowsQ","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-q","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $x = x_0 x_1 x_2 \\cdots$ be an infinite word over $\\{L, R\\}$. Its parity sequence $Q(x) = Q_0 Q_1 Q_2 \\cdots$ over the parity alphabet $\\{O_L, E_R, O_R, E_L\\}$ is defined by $Q_0 = O_L$ and $Q_{j+1} = \\mathrm{step}(Q_j, x_j)$ for every $j \\ge 0$, where $\\mathrm{step}$ is the transition rule of the parity sequence.","title":"The parity sequence of a Stern\u2013Brocot word"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-q-small","kind":"node","lean":"MetaMathlibExt.mallowsQ_small","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-q-small","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $x = x_0 x_1 x_2 \\cdots$ be an infinite word over $\\{L, R\\}$ with $x_0 = R$ and $x_1 = L$. Then the parity sequence $Q(x) = Q_0 Q_1 Q_2 \\cdots$ of $x$ satisfies $Q_0 = O_L$, $Q_1 = E_R$ and $Q_2 = O_R$.","title":"First values of the parity sequence"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-admissible","kind":"node","lean":"MetaMathlibExt.mallowsAdmissible","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The admissible parity sequences "],["math","\\(I\\)"]],"source":"The admissible parity sequences $I$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-admissible","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The set $I$ consists of the infinite sequences $x = (x_0, x_1, \\dots)$ over the alphabet $\\{O_L, E_R, O_R, E_L\\}$ such that $x_0 = O_L$; no two consecutive terms $(x_j, x_{j+1})$ form one of the pairs $(O_L, O_R)$, $(O_L, E_L)$, $(E_R, E_L)$, $(E_R, E_R)$, $(O_R, O_L)$, $(O_R, E_R)$, $(E_L, E_L)$, $(E_L, E_R)$; $x$ is not eventually constant equal to $O_L$, nor eventually constant equal to $O_R$; and no tail of $x$\u2026","title":"The admissible parity sequences $I$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/const-ol-not-mem-mallows-admissible","kind":"node","lean":"MetaMathlibExt.const_OL_not_mem_mallowsAdmissible","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The constant sequence "],["math","\\(O_L O_L O_L \\cdots\\)"],["text"," is not admissible"]],"source":"The constant sequence $O_L O_L O_L \\cdots$ is not admissible"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#const-ol-not-mem-mallows-admissible","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The constant sequence whose every term is $O_L$ does not belong to the set $I$ of admissible sequences over $\\{O_L, E_R, O_R, E_L\\}$.","title":"The constant sequence $O_L O_L O_L \\cdots$ is not admissible"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-odd-index","kind":"node","lean":"MetaMathlibExt.mallowsOddIndex","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-odd-index","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $x = x_0 x_1 x_2 \\cdots$ be an infinite sequence over the parity alphabet $\\{O_L, E_R, O_R, E_L\\}$, and call a position $j \\ge 0$ odd when $x_j \\in \\{O_L, O_R\\}$. For $i \\ge 0$, the $i$-th odd position of $x$ is the $(i+1)$-st smallest odd position of $x$, and it is $0$ by convention when $x$ has at most $i$ odd positions.","title":"Odd positions of a parity sequence"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-odd-index-zero","kind":"node","lean":"MetaMathlibExt.mallowsOddIndex_zero","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first odd position of a sequence starting with "],["math","\\(O_L\\)"]],"source":"The first odd position of a sequence starting with $O_L$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-odd-index-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $x = x_0 x_1 x_2 \\cdots$ be an infinite sequence over the parity alphabet $\\{O_L, E_R, O_R, E_L\\}$ with $x_0 = O_L$, and call a position $j \\ge 0$ odd when $x_j \\in \\{O_L, O_R\\}$. Then the $0$-th odd position of $x$, the smallest odd position, is $0$.","title":"The first odd position of a sequence starting with $O_L$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-tletter","kind":"node","lean":"MetaMathlibExt.mallowsTLetter","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The letter rule of "],["math","\\(T\\)"]],"source":"The letter rule of $T$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-tletter","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For a truth value $b$ and letters $a, a'$ of $\\{O_L, E_R, O_R, E_L\\}$, the letter $\\tau(b, a, a') \\in \\{L, M, R\\}$ is defined as follows. If $b$ is true, it is $L$ when $a = O_L$ and $R$ otherwise. If $b$ is false, it is $M$ when $a \\ne a'$, $L$ when $a = a' = O_R$, and $R$ when $a = a' \\ne O_R$.","title":"The letter rule of $T$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-t","kind":"node","lean":"MetaMathlibExt.mallowsT","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(T\\)"]],"source":"The map $T$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-t","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $x = \\langle x_1, x_2, \\dots \\rangle$ be a sequence over $\\{O_L, E_R, O_R, E_L\\}$ and let $o(1) < o(2) < \\cdots$ be the positions $j$ with $x_j \\in \\{O_L, O_R\\}$. The sequence $T(x) = \\langle T_1(x), T_2(x), \\dots \\rangle$ over $\\{L, M, R\\}$ is defined as follows. If $o(i + 1) = o(i) + 1$, then $T_i(x)$ is $L$ when $x_{o(i)} = O_L$ and $R$ otherwise. If $o(i + 1) \\ne o(i) + 1$, then $T_i(x)$ is $M$ when\u2026","title":"The map $T$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-tfin","kind":"node","lean":"MetaMathlibExt.mallowsTFin","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The finite version "],["math","\\(\\dot T\\)"],["text"," of "],["math","\\(T\\)"]],"source":"The finite version $\\dot T$ of $T$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-tfin","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"Let $y = \\langle y_1, \\dots, y_n \\rangle$ be a finite word over $\\{O_L, E_R, O_R, E_L\\}$ and let $p_1 < p_2 < \\cdots < p_k$ be the positions $j$ with $y_j \\in \\{O_L, O_R\\}$. Then $\\dot T(y) = \\langle t_1, \\dots, t_{k-1} \\rangle$ is the word over $\\{L, M, R\\}$ (empty if $k \\le 1$) given as follows. If $p_{i+1} = p_i + 1$, then $t_i$ is $L$ when $y_{p_i} = O_L$ and $R$ otherwise. If $p_{i+1} \\ne p_i + 1$, then $t_i$\u2026","title":"The finite version $\\dot T$ of $T$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-tfin-small","kind":"node","lean":"MetaMathlibExt.mallowsTFin_small","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two values of "],["math","\\(\\dot T\\)"]],"source":"Two values of $\\dot T$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-tfin-small","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"The finite version $\\dot T$ of $T$ satisfies $\\dot T(\\langle O_L, E_R, O_R \\rangle) = \\langle M \\rangle$ and $\\dot T(\\langle O_L, O_L \\rangle) = \\langle L \\rangle$.","title":"Two values of $\\dot T$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/is-positively-poisson-stable","kind":"node","lean":"MetaMathlibExt.IsPositivelyPoissonStable","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#is-positively-poisson-stable","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"A sequence $x = (x_0, x_1, x_2, \\dots)$ with values in a set $A$ is positively Poisson stable if there is an index $n_0$ such that for every length $\\ell$ there is a shift $p \\ge 1$ with $x_{n_0 + p + t} = x_{n_0 + t}$ for all $t < \\ell$; that is, every initial block of the tail $(x_{n_0}, x_{n_0 + 1}, \\dots)$ occurs again at a later position.","title":"Positively Poisson stable sequence"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/is-positively-poisson-stable-const","kind":"node","lean":"MetaMathlibExt.isPositivelyPoissonStable_const","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#is-positively-poisson-stable-const","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"fully_proved","summary":"For every set $A$ and every $c \\in A$, the constant sequence $(c, c, c, \\dots)$ is positively Poisson stable.","title":"Constant sequences are positively Poisson stable"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-tree-label-unique-order-pat","kind":"node","lean":"MetaMathlibExt.sternBrocotTree_label_unique_order_path","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-tree-label-unique-order-pat","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Consider the subtree of the Stern\u2013Brocot tree with root $1/2$, and identify each vertex with its path $w$ from the root, a finite word over $\\{L, R\\}$. Then: (i) every rational $q$ with $0 < q < 1$ is the label of exactly one vertex; (ii) for all words $u, v_1, v_2$ over $\\{L, R\\}$ with $v_1$ and $v_2$ of the same length, the label of $u L v_1$ is less than the label of $u R v_2$; (iii) if the label of the vertex\u2026","title":"Stern\u2013Brocot tree: labels, order and paths"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-rep-tendsto-and-cont-fract","kind":"node","lean":"MetaMathlibExt.sternBrocotRep_tendsto_and_contFract","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-rep-tendsto-and-cont-fract","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational with continued fraction expansion $\\alpha = [0; a_1, a_2, \\ldots]$, let $f_1(\\alpha), f_2(\\alpha), \\ldots$ be the fractions labelling the vertices visited by its Stern\u2013Brocot path, and let $SB(\\alpha) = SB_1(\\alpha) SB_2(\\alpha) \\cdots$ be its Stern\u2013Brocot representation. Then (i) $f_j(\\alpha) \\to \\alpha$ as $j \\to \\infty$; and (ii)\u2026","title":"Stern\u2013Brocot representation and continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/image-stern-brocot-rep","kind":"node","lean":"MetaMathlibExt.image_sternBrocotRep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#image-stern-brocot-rep","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"A sequence over $\\{L, R\\}$ is the Stern\u2013Brocot representation $\\mathrm{SB}(\\alpha)$ of some irrational $\\alpha \\in (0, 1)$ if and only if it is neither eventually constantly $L$ nor eventually constantly $R$.","title":"Characterization of Stern\u2013Brocot representations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-tree-label-unique-order","kind":"node","lean":"MetaMathlibExt.mallowsTree_label_unique_order","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Each rational in "],["math","\\((0, 1)\\)"],["text"," occurs once in the Mallows tree, in increasing order on each level"]],"source":"Each rational in $(0, 1)$ occurs once in the Mallows tree, in increasing order on each level"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-tree-label-unique-order","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Each rational in $(0, 1)$ occurs once in the Mallows tree, in increasing order on each level. For a finite word $w$ over $\\{L, M, R\\}$ let $\\langle p_w, q_w \\rangle$ be the label, a pair of fractions, of the vertex of the Mallows tree reached from the root by the path $w$; fractions are compared by their real values. Then: (i) for every rational number $r$ with $0 < r < 1$ there is exactly one pair $(w, s)$ of a\u2026","title":"Each rational in $(0, 1)$ occurs once in the Mallows tree, in increasing order on each level"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-level-eq-phi-psi-iterate","kind":"node","lean":"MetaMathlibExt.mallowsLevel_eq_phi_psi_iterate","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-level-eq-phi-psi-iterate","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Encode each vertex of the Stern\u2013Brocot tree (left subtree, root $1/2$) by its gap, so that the root is the gap $(0/1, 1/1)$, and extend $\\psi_{\\mathrm{SBT}}$ and $\\varphi_{\\mathrm{SBT}}$ to lists of vertices by applying them to each entry and concatenating the results in order. Then for every $i \\ge 1$ the ordered list $MT_i$ of fractions inserted at step $i$ of Mallows' construction is\u2026","title":"Mallows levels from the Stern\u2013Brocot maps"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/image-mallows-rep","kind":"node","lean":"MetaMathlibExt.image_mallowsRep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#image-mallows-rep","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"A sequence over $\\{L, M, R\\}$ is the Mallows representation $M(\\alpha)$ of some irrational $\\alpha \\in (0, 1)$ if and only if it is neither eventually constantly $L$ nor eventually constantly $R$.","title":"Characterization of Mallows representations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/bij-on-mallows-q","kind":"node","lean":"MetaMathlibExt.bijOn_mallowsQ","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hida's map "],["math","\\(Q\\)"],["text"," is a bijection onto "],["math","\\(I\\)"]],"source":"Hida's map $Q$ is a bijection onto $I$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#bij-on-mallows-q","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\mathrm{SB}(\\alpha)$ be the Stern\u2013Brocot representation of a real number $\\alpha$, let $Q$ be Hida's map from sequences over $\\{L, R\\}$ to sequences over $\\{O_L, E_R, O_R, E_L\\}$, and let $I$ be the set of admissible sequences over $\\{O_L, E_R, O_R, E_L\\}$. Then $Q$ maps the set of Stern\u2013Brocot representations $\\{\\mathrm{SB}(\\alpha) : \\alpha \\in (0, 1) \\text{ irrational}\\}$ bijectively onto $I$: for every\u2026","title":"Hida's map $Q$ is a bijection onto $I$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-frac-parity-side","kind":"node","lean":"MetaMathlibExt.sternBrocotFrac_parity_side","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-frac-parity-side","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational, let $f_1(\\alpha), f_2(\\alpha), \\dots$ be the fractions on its Stern\u2013Brocot path, and let $SB(\\alpha) = \\langle SB_1, SB_2, \\dots \\rangle$ be its Stern\u2013Brocot representation. Let $Q(SB(\\alpha)) = \\langle Q_1, Q_2, \\dots \\rangle$ be the sequence over $\\{O_L, E_R, O_R, E_L\\}$ with $Q_1 = O_L$ in which $Q_{j+1}$ is determined by $Q_j$ and $SB_j$ as follows: from $O_L$,\u2026","title":"Parity and side of the Stern\u2013Brocot fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-odd-index-gap","kind":"node","lean":"MetaMathlibExt.mallowsOddIndex_gap","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-odd-index-gap","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $x = x_0 x_1 x_2 \\cdots$ be a sequence in the admissible set $I$ over the letters $O_L$, $E_R$, $O_R$, $E_L$, and for $i \\ge 0$ let $p_i$ be the $i$-th position (counting from $0$) at which $x$ has one of the letters $O_L$, $O_R$. Then for every $i \\ge 0$, $p_{i+1} > p_i + 1$ holds if and only if $p_{i+1} = p_i + 2$.","title":"Gaps between odd letters of admissible sequences"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/bij-on-mallows-t","kind":"node","lean":"MetaMathlibExt.bijOn_mallowsT","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hida's map "],["math","\\(T\\)"],["text"," is a bijection from "],["math","\\(I\\)"],["text"," onto the Mallows representations"]],"source":"Hida's map $T$ is a bijection from $I$ onto the Mallows representations"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#bij-on-mallows-t","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $I$ be the set of admissible sequences over $\\{O_L, E_R, O_R, E_L\\}$, let $T$ be Hida's map from sequences over $\\{O_L, E_R, O_R, E_L\\}$ to sequences over $\\{L, M, R\\}$, and let $M(\\alpha)$ be the Mallows representation of a real number $\\alpha$. Then $T$ maps $I$ bijectively onto the set $\\{M(\\alpha) : \\alpha \\in (0, 1) \\text{ irrational}\\}$: for every $x \\in I$ the sequence $T(x)$ lies in this set, $T$ is\u2026","title":"Hida's map $T$ is a bijection from $I$ onto the Mallows representations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-phi-odd-index","kind":"node","lean":"MetaMathlibExt.sternBrocotPhi_oddIndex","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-phi-odd-index","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For irrational `\u03b1 \u2208 (0, 1)` and `i \u2265 1`, `\u03c6_SBT(f_{o(i)}(\u03b1)) = pf_i(\u03b1)`. The vertex `f_{o(i)}(\u03b1)` is encoded by its gap, `pf_i(\u03b1)` is the label `\u27e8p, q\u27e9` inserted in the gap `mallowsInterval \u03b1 (i - 1)`, written as the list `[p, q]`, and `o` is computed on `Q(SB(\u03b1))`.","title":"Stern\u2013Brocot labels of the Mallows representation vertices"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-t-mallows-q-stern-brocot-rep","kind":"node","lean":"MetaMathlibExt.mallowsT_mallowsQ_sternBrocotRep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-t-mallows-q-stern-brocot-rep","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For every irrational number $\\alpha$ with $0 < \\alpha < 1$, the Mallows representation $M(\\alpha)$ is obtained from the Stern\u2013Brocot representation $\\mathrm{SB}(\\alpha)$ by applying the map $Q$, which gives a sequence over $\\{O_L, E_R, O_R, E_L\\}$, and then the map $T$, which gives a sequence over $\\{L, M, R\\}$: $T(Q(\\mathrm{SB}(\\alpha))) = M(\\alpha)$.","title":"The Mallows representation from the Stern\u2013Brocot representation"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/block-match-stern-brocot-mallows-q","kind":"node","lean":"MetaMathlibExt.block_match_sternBrocot_mallowsQ","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#block-match-stern-brocot-mallows-q","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational, let $x = \\mathrm{SB}(\\alpha)$ be its Stern\u2013Brocot representation, let $q = Q(x)$ be the image of $x$ under Hida's map $Q$ to sequences over $\\{O_L, E_R, O_R, E_L\\}$, and let $T(q)$ be the image of $q$ under Hida's map $T$ to sequences over $\\{L, M, R\\}$; all sequences are indexed from $0$. Let $o(i)$ be the $i$-th position, counting from $0$, at which $q$ takes the value $O_L$\u2026","title":"Block-matching lemma"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-t-starts-with-iff","kind":"node","lean":"MetaMathlibExt.mallowsT_startsWith_iff","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Blocks of "],["math","\\(T(Q(\\mathrm{SB}(\\alpha)))\\)"],["text"," and of "],["math","\\(Q(\\mathrm{SB}(\\alpha))\\)"]],"source":"Blocks of $T(Q(\\mathrm{SB}(\\alpha)))$ and of $Q(\\mathrm{SB}(\\alpha))$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-t-starts-with-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $X$ be one of the letters $O_L, E_R, O_R, E_L$, let $u = u_1 \\cdots u_\\ell$ be a finite word over $\\{L, R\\}$, and let $\\dot Q^X(u) = \\langle X_0, X_1, \\dots, X_\\ell \\rangle$ be the word with $X_0 = X$ and $X_k = \\delta(X_{k-1}, u_k)$ for $1 \\le k \\le \\ell$, where $\\delta$ is the transition rule that defines the map $Q$ from sequences over $\\{L, R\\}$ to sequences over $\\{O_L, E_R, O_R, E_L\\}$. Assume that\u2026","title":"Blocks of $T(Q(\\mathrm{SB}(\\alpha)))$ and of $Q(\\mathrm{SB}(\\alpha))$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-t-eq-m-iff-run-end","kind":"node","lean":"MetaMathlibExt.mallowsT_eq_M_iff_runEnd","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","The letter "],["math","\\(M\\)"],["text"," in "],["math","\\(T(Q(\\mathrm{SB}(\\alpha)))\\)"],["text"," marks the ends of runs"]],"source":"The letter $M$ in $T(Q(\\mathrm{SB}(\\alpha)))$ marks the ends of runs"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-t-eq-m-iff-run-end","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational, let $x = Q(\\mathrm{SB}(\\alpha))$ be the sequence over $\\{O_L, E_R, O_R, E_L\\}$ obtained from the Stern\u2013Brocot representation $\\mathrm{SB}(\\alpha)$, and let $o(1) < o(2) < \\cdots$ be the positions $j$ with $x_j \\in \\{O_L, O_R\\}$. Let $e_1, e_2, \\ldots$ be positive integers such that $(x_{o(1)}, x_{o(2)}, \\ldots) = O_L^{e_1} O_R^{e_2} O_L^{e_3} \\cdots$, that is, for every\u2026","title":"The letter $M$ in $T(Q(\\mathrm{SB}(\\alpha)))$ marks the ends of runs"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/is-eventually-periodic-mallows-rep-iff-s","kind":"node","lean":"MetaMathlibExt.isEventuallyPeriodic_mallowsRep_iff_sternBrocotRep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#is-eventually-periodic-mallows-rep-iff-s","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For every irrational $\\alpha \\in (0, 1)$, the Mallows representation $M(\\alpha)$ is eventually periodic if and only if the Stern\u2013Brocot representation $\\mathrm{SB}(\\alpha)$ is eventually periodic.","title":"Mallows and Stern\u2013Brocot representations are eventually periodic together"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/is-eventually-periodic-mallows-rep-iff-q","kind":"node","lean":"MetaMathlibExt.isEventuallyPeriodic_mallowsRep_iff_quadratic","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#is-eventually-periodic-mallows-rep-iff-q","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For every irrational $\\alpha \\in (0, 1)$, the Mallows representation $M(\\alpha)$ is eventually periodic if and only if $\\alpha$ is a real quadratic irrational, that is, algebraic of degree $2$ over $\\mathbb{Q}$.","title":"Lagrange's theorem for Mallows representations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/is-positively-poisson-stable-mallows-rep","kind":"node","lean":"MetaMathlibExt.isPositivelyPoissonStable_mallowsRep_iff_sternBrocotRep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#is-positively-poisson-stable-mallows-rep","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For every irrational $\\alpha \\in (0, 1)$, the Mallows representation of $\\alpha$ is positively Poisson stable if and only if the Stern\u2013Brocot representation of $\\alpha$ is positively Poisson stable.","title":"Poisson stability of the Mallows and Stern\u2013Brocot representations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/is-positively-poisson-stable-mallows-rep-2","kind":"node","lean":"MetaMathlibExt.isPositivelyPoissonStable_mallowsRep_iff_contFract","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#is-positively-poisson-stable-mallows-rep-2","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational, with continued fraction expansion $\\alpha = [0; a_1, a_2, \\dots]$. Then the Mallows representation of $\\alpha$ is positively Poisson stable if and only if the sequence $(a_1, a_2, \\dots)$ of its partial quotients is positively Poisson stable.","title":"Poisson stability of the Mallows representation and the continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/transitive-mallows-rep-iff-stern-brocot","kind":"node","lean":"MetaMathlibExt.transitive_mallowsRep_iff_sternBrocotRep","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#transitive-mallows-rep-iff-stern-brocot","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational. Every finite word over the alphabet $\\{L, M, R\\}$ occurs as a factor of the Mallows representation of $\\alpha$ if and only if every finite word over the alphabet $\\{L, R\\}$ occurs as a factor of the Stern\u2013Brocot representation of $\\alpha$; that is, the Mallows representation of $\\alpha$ is transitive if and only if its Stern\u2013Brocot representation is. The empty word occurs\u2026","title":"Transitivity of the Mallows and Stern\u2013Brocot representations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/transitive-mallows-rep-iff-cont-fract","kind":"node","lean":"MetaMathlibExt.transitive_mallowsRep_iff_contFract","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#transitive-mallows-rep-iff-cont-fract","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational with continued fraction expansion $\\alpha = [0; a_1, a_2, \\ldots]$. The Mallows representation of $\\alpha$ is transitive, that is, every finite word over $\\{L, M, R\\}$ occurs in it as a factor, if and only if every finite word over the positive integers occurs as a factor of the sequence $a_1 a_2 a_3 \\cdots$.","title":"Transitive Mallows representations and continued fractions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-ne-m-iff","kind":"node","lean":"MetaMathlibExt.mallowsRep_ne_M_iff","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mallows representations without the letter "],["math","\\(M\\)"]],"source":"Mallows representations without the letter $M$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-ne-m-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational, with continued fraction expansion $\\alpha = [0; a_1, a_2, \\ldots]$. The letter $M$ does not occur in the Mallows representation $M(\\alpha)$ if and only if $a_{2k}$ is even for every $k \\ge 1$.","title":"Mallows representations without the letter $M$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-ne-r-iff","kind":"node","lean":"MetaMathlibExt.mallowsRep_ne_R_iff","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mallows representations without the letter "],["math","\\(R\\)"]],"source":"Mallows representations without the letter $R$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-ne-r-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational, with continued fraction expansion $\\alpha = [0; a_1, a_2, \\ldots]$. The letter $R$ does not occur in the Mallows representation $M(\\alpha)$ if and only if $a_2 = 1$, $a_3$ is even, and the sequence $a_4, a_5, \\ldots$ is a concatenation of blocks, each of the form $(2, b)$ or $(1, c, 1, b)$ with $b$ even. Equivalently,\u2026","title":"Mallows representations without the letter $R$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-ne-l-iff","kind":"node","lean":"MetaMathlibExt.mallowsRep_ne_L_iff","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mallows representations without the letter "],["math","\\(L\\)"]],"source":"Mallows representations without the letter $L$"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-ne-l-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational, with continued fraction expansion $\\alpha = [0; a_1, a_2, \\ldots]$. The letter $L$ does not occur in the Mallows representation $M(\\alpha)$ if and only if $a_1 = 1$, $a_2$ is odd, and the sequence $a_3, a_4, \\ldots$ is a concatenation of blocks, each of the form $(2, b)$ or $(1, c, 1, b)$ with $b$ even. Equivalently,\u2026","title":"Mallows representations without the letter $L$"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/stern-brocot-frac-parity-not-eventually","kind":"node","lean":"MetaMathlibExt.sternBrocotFrac_parity_not_eventually_const","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#stern-brocot-frac-parity-not-eventually","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational and let $f_1(\\alpha), f_2(\\alpha), \\dots$ be the fractions on its Stern\u2013Brocot path. Then there are no natural numbers $c$ and $j_0$ such that the numerator of $f_j(\\alpha)$ has remainder $c$ modulo $2$ for all $j \\ge j_0$: the parities of these numerators are not eventually constant.","title":"The numerator parities are not eventually constant"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/parity-eventually-alternating-iff","kind":"node","lean":"MetaMathlibExt.parity_eventually_alternating_iff","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#parity-eventually-alternating-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha$ be an irrational number with $0 < \\alpha < 1$, let $f_1(\\alpha), f_2(\\alpha), \\dots$ be the fractions on its Stern\u2013Brocot path, and let $M(\\alpha) = \\langle M_1(\\alpha), M_2(\\alpha), \\dots \\rangle$ be its Mallows representation. The following are equivalent: (a) there is $j_0$ such that for every $j \\ge j_0$ the numerators of $f_j(\\alpha)$ and $f_{j+1}(\\alpha)$ have different parities; (b) the letter\u2026","title":"Eventually alternating parities of Stern\u2013Brocot numerators"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/parity-eventually-period-three-iff","kind":"node","lean":"MetaMathlibExt.parity_eventually_period_three_iff","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#parity-eventually-period-three-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\alpha \\in (0, 1)$ be irrational, let $f_1(\\alpha), f_2(\\alpha), \\dots$ be the fractions on its Stern\u2013Brocot path, with numerators $p_1, p_2, \\dots$, and let $M(\\alpha) = \\langle M_1, M_2, \\dots \\rangle$ be its Mallows representation. Then there is $j_0 \\ge 1$ with $p_{j+3} \\equiv p_j \\pmod 2$ for all $j \\ge j_0$ if and only if one of the following holds: (a) there is $i_0 \\ge 1$ such that $M_{i_0 + 2t} = L$\u2026","title":"Eventual period three of the numerator parities"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-ae-properties","kind":"node","lean":"MetaMathlibExt.mallowsRep_ae_properties","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-ae-properties","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Each of the following holds for Lebesgue-almost every $\\alpha \\in (0, 1)$, provided $\\alpha$ is irrational: (i) the Mallows representation $M(\\alpha)$ is not eventually periodic; (ii) $M(\\alpha)$ is positively Poisson stable; (iii) $M(\\alpha)$ is transitive, that is, every finite word over $\\{L, M, R\\}$ occurs in $M(\\alpha)$.","title":"Typical properties of Mallows representations"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-ae-factor-complexity","kind":"node","lean":"MetaMathlibExt.mallowsRep_ae_factorComplexity","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-ae-factor-complexity","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For almost every irrational $\\alpha \\in (0, 1)$, with respect to Lebesgue measure, the factor complexity of the Mallows representation $M(\\alpha)$ satisfies $p_{M(\\alpha)}(n) = 3^n$ for every $n \\ge 0$.","title":"Factor complexity of almost every Mallows representation"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-ae-not-relatively-dense","kind":"node","lean":"MetaMathlibExt.mallowsRep_ae_not_relativelyDense","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-ae-not-relatively-dense","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For Lebesgue-almost every $\\alpha \\in (0, 1)$, if $\\alpha$ is irrational, then there is a finite word $w$ over $\\{L, M, R\\}$ that does not occur relatively densely in the Mallows representation $M(\\alpha) = M_1(\\alpha) M_2(\\alpha) \\cdots$: there is no $\\ell > 0$ such that, for every $i \\ge 0$, the window $M_{i+1}(\\alpha) \\cdots M_{i+\\ell}(\\alpha)$ contains an occurrence of $w$.","title":"Words without bounded gaps in almost every Mallows representation"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-ae-frequency-zero","kind":"node","lean":"MetaMathlibExt.mallowsRep_ae_frequency_zero","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-ae-frequency-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $w$ be a finite word over $\\{L, M, R\\}$ that contains the letter $M$ or contains both $L$ and $R$. Then for almost every irrational $\\alpha \\in (0, 1)$, with respect to Lebesgue measure, $w$ has frequency $0$ in the Mallows representation $M(\\alpha) = M_0(\\alpha) M_1(\\alpha) M_2(\\alpha) \\cdots$: if $N_n$ is the number of positions $i < n$ with $M_i(\\alpha) M_{i+1}(\\alpha) \\cdots M_{i+|w|-1}(\\alpha) = w$, then\u2026","title":"Words of frequency zero in almost every Mallows representation"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-ae-matching-fraction-one","kind":"node","lean":"MetaMathlibExt.mallowsRep_ae_matchingFraction_one","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-ae-matching-fraction-one","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For almost every irrational $\\alpha \\in (0, 1)$, with respect to Lebesgue measure, the Mallows representation $M(\\alpha) = M_0(\\alpha) M_1(\\alpha) M_2(\\alpha) \\cdots$ satisfies, for every $\\ell \\ge 0$,","title":"Matching fractions of almost every Mallows representation"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-cylinder-measure-extrema","kind":"node","lean":"MetaMathlibExt.mallowsRep_cylinder_measure_extrema","parent":10587,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-cylinder-measure-extrema","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"Let $\\ell \\ge 0$. For a word $w = w_1 \\cdots w_\\ell$ over $\\{L, M, R\\}$, let $C_w$ be the set of irrational $\\alpha \\in (0, 1)$ whose Mallows representation begins with $w$, that is, $M_t(\\alpha) = w_t$ for $1 \\le t \\le \\ell$. Over the $3^\\ell$ words $w$ of length $\\ell$, the Lebesgue measure of $C_w$ attains the maximum value $1/(\\ell + 1)$ and the minimum value","title":"Extremal measures of Mallows cylinder sets"},{"id":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/mallows-rep-eq-m-measure-lt","kind":"node","lean":"MetaMathlibExt.mallowsRep_eq_M_measure_lt","parent":10587,"reading_title":{"format":"inline-parts-v1","parts":[["text","Measure bound for the letter "],["math","\\(M\\)"],["text"," at a fixed position"]],"source":"Measure bound for the letter $M$ at a fixed position"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers/index.html#mallows-rep-eq-m-measure-lt","scope":"NumberTheory/continued-fractions-expansions/part-01/mallows-representation-of-irrational-numbers","status":"can_prove","summary":"For every $i \\ge 1$, the set of irrational $\\alpha \\in (0, 1)$ whose Mallows representation has $i$-th letter $M_i(\\alpha) = M$ has Lebesgue measure less than $3 - 2\\sqrt{2}$.","title":"Measure bound for the letter $M$ at a fixed position"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions.json","id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","title":"Pierce expansions"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/traj","kind":"node","lean":"PierceExpansion.traj","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#traj","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For natural numbers $n$ and $a$, the Pierce remainder trajectory of $a$ with respect to $n$ is the sequence $t_0, t_1, t_2, \\dots$ of natural numbers with $t_0 = a$ and, for every $j \\ge 0$, $t_{j+1} = 0$ if $t_j = 0$, while if $t_j \\ne 0$ then $t_{j+1} = n \\bmod t_j$ is the remainder of $n$ modulo $t_j$. Once the trajectory reaches $0$ it stays there.","title":"Pierce remainder trajectory"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/traj-zero","kind":"node","lean":"PierceExpansion.traj_zero","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#traj-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For all natural numbers $n$ and $a$, the Pierce remainder trajectory of $a$ with respect to $n$ has value $a$ at step $0$.","title":"Initial term of the Pierce trajectory"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/traj-succ-of-eq-zero","kind":"node","lean":"PierceExpansion.traj_succ_of_eq_zero","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#traj-succ-of-eq-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"Let $n$, $a$ and $j$ be natural numbers and let $t_0, t_1, \\dots$ be the Pierce remainder trajectory of $a$ with respect to $n$. If $t_j = 0$, then $t_{j+1} = 0$.","title":"Step of the Pierce trajectory at a zero term"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/traj-succ-of-ne-zero","kind":"node","lean":"PierceExpansion.traj_succ_of_ne_zero","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#traj-succ-of-ne-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"Let $n$, $a$ and $j$ be natural numbers and let $t_0, t_1, \\dots$ be the Pierce remainder trajectory of $a$ with respect to $n$. If $t_j \\ne 0$, then $t_{j+1} = n \\bmod t_j$, the remainder of $n$ modulo $t_j$.","title":"Step of the Pierce trajectory at a nonzero term"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/traj-one","kind":"node","lean":"PierceExpansion.traj_one","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#traj-one","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"Let $n$ and $t$ be natural numbers with $t \\ne 0$. Then the Pierce remainder trajectory of $t$ with respect to $n$ is, at step $1$, equal to $n \\bmod t$, the remainder of $n$ modulo $t$.","title":"First step of the Pierce trajectory"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/exists-traj-eq-zero","kind":"node","lean":"PierceExpansion.exists_traj_eq_zero","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#exists-traj-eq-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For natural numbers $n$ and $a$, let $(t_j)_{j \\ge 0}$ be the Pierce remainder trajectory: $t_0 = a$, and $t_{j+1} = n \\bmod t_j$ if $t_j \\ne 0$, $t_{j+1} = 0$ if $t_j = 0$. Then $t_i = 0$ for some $i \\le a + 1$.","title":"A bound on when a Pierce trajectory vanishes"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/exists-traj-eq-zero-pos","kind":"node","lean":"PierceExpansion.exists_traj_eq_zero_pos","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#exists-traj-eq-zero-pos","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For all natural numbers $n$ and $a$, the Pierce remainder trajectory $(t_j)_{j \\ge 0}$ of $n$ and $a$ (with $t_0 = a$) satisfies $t_j = 0$ for some $j \\ge 1$.","title":"Pierce trajectories vanish at a positive step"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/length","kind":"node","lean":"PierceExpansion.length","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#length","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For natural numbers $n$ and $a$, let $(t_j)_{j \\ge 0}$ be the Pierce remainder trajectory, with $t_0 = a$, $t_{j+1} = n \\bmod t_j$ if $t_j \\ne 0$ and $t_{j+1} = 0$ if $t_j = 0$. The Pierce length $\\ell(n, a)$ is the least $j \\ge 1$ with $t_j = 0$; such a $j$ always exists.","title":"Pierce length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/length-pos","kind":"node","lean":"PierceExpansion.length_pos","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#length-pos","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For all natural numbers $n$ and $a$, the Pierce length $\\ell(n, a)$ satisfies $\\ell(n, a) \\ge 1$.","title":"The Pierce length is positive"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/traj-length-eq-zero","kind":"node","lean":"PierceExpansion.traj_length_eq_zero","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#traj-length-eq-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For all natural numbers $n$ and $a$, the Pierce remainder trajectory $(t_j)_{j \\ge 0}$ of $n$ and $a$ satisfies $t_{\\ell(n, a)} = 0$, where $\\ell(n, a)$ is the Pierce length.","title":"The Pierce trajectory vanishes at the Pierce length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/length-le-of-traj-eq-zero","kind":"node","lean":"PierceExpansion.length_le_of_traj_eq_zero","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#length-le-of-traj-eq-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For natural numbers $n$, $a$ and $j$ with $j \\ge 1$, if the Pierce remainder trajectory $(t_i)_{i \\ge 0}$ of $n$ and $a$ satisfies $t_j = 0$, then the Pierce length $\\ell(n, a)$ satisfies $\\ell(n, a) \\le j$.","title":"The Pierce length is at most any positive vanishing step"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/traj-ne-zero-of-lt-length","kind":"node","lean":"PierceExpansion.traj_ne_zero_of_lt_length","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#traj-ne-zero-of-lt-length","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For natural numbers $n$, $a$ and $j$ with $1 \\le j < \\ell(n, a)$, where $\\ell(n, a)$ is the Pierce length, the Pierce remainder trajectory $(t_i)_{i \\ge 0}$ of $n$ and $a$ satisfies $t_j \\ne 0$.","title":"The Pierce trajectory is nonzero before the Pierce length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/max-length","kind":"node","lean":"PierceExpansion.maxLength","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#max-length","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For a natural number $n$, the maximal Pierce length $P(n)$ is the maximum of the Pierce lengths $\\ell(n, a)$ over $1 \\le a \\le n$, with $P(0) = 0$.","title":"Maximal Pierce length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/length-le-max-length","kind":"node","lean":"PierceExpansion.length_le_maxLength","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#length-le-max-length","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"For natural numbers $n$ and $a$ with $1 \\le a \\le n$, the Pierce length $\\ell(n, a)$ satisfies $\\ell(n, a) \\le P(n)$, where $P(n) = \\max_{1 \\le b \\le n} \\ell(n, b)$ is the maximal Pierce length.","title":"Each Pierce length is at most the maximal Pierce length"},{"id":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/max-length-le-of-forall","kind":"node","lean":"PierceExpansion.maxLength_le_of_forall","parent":10669,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/pierce-expansions/index.html#max-length-le-of-forall","scope":"NumberTheory/continued-fractions-expansions/part-01/pierce-expansions","status":"fully_proved","summary":"Let $n$ and $L$ be natural numbers. If the Pierce length $\\ell(n, a)$ satisfies $\\ell(n, a) \\le L$ for every $a$ with $1 \\le a \\le n$, then the maximal Pierce length $P(n) = \\max_{1 \\le a \\le n} \\ell(n, a)$ (which is $0$ when $n = 0$) satisfies $P(n) \\le L$.","title":"A uniform bound on Pierce lengths bounds the maximal Pierce length"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths.json","id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","title":"Polynomial continued fractions and Farey paths"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract","kind":"node","lean":"MetaMathlibExt.polyContFract","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field, let $a_0, a_1, a_2, \\ldots$ be polynomials in $F[X]$ and let $n \\in \\mathbb{N}$. The continued fraction $[a_0; a_1, \\ldots, a_n]$ is the generalized continued fraction over the field $F(X)$ of rational functions with integral part $a_0$ and, for $1 \\le i \\le n$, an $i$th term with partial numerator $1$ and partial denominator $a_i$; it has no further terms. For $i \\ge 0$ write $A_i$ and $B_i$\u2026","title":"Finite continued fraction with polynomial partial quotients"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-h","kind":"node","lean":"MetaMathlibExt.polyContFract_h","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-h","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field, let $n \\in \\mathbb N$, and let $a_0, a_1, \\dots, a_n \\in F[X]$. The integral part (head term) of the continued fraction $[a_0, a_1, \\dots, a_n]$ over $F(X)$ is $a_0$.","title":"Integral part of a polynomial continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-s-get","kind":"node","lean":"MetaMathlibExt.polyContFract_s_get?","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-s-get","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field, let $a_0, a_1, \\ldots$ be polynomials in $F[X]$ and let $n \\in \\mathbb{N}$. For every $i \\in \\mathbb{N}$, the term of index $i$ (counting from $0$) of the continued fraction $[a_0; a_1, \\ldots, a_n]$ over $F(X)$ is the pair with partial numerator $1$ and partial denominator $a_{i+1}$ if $i < n$, and does not exist if $i \\ge n$.","title":"Terms of a polynomial continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-nums-zero","kind":"node","lean":"MetaMathlibExt.polyContFract_nums_zero","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-nums-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field, let $a_0, a_1, \\ldots$ be polynomials in $F[X]$ and let $n \\in \\mathbb{N}$. The numerator $A_0$ of the zeroth convergent of the continued fraction $[a_0; a_1, \\ldots, a_n]$ over $F(X)$ is $a_0$.","title":"Zeroth convergent numerator of a polynomial continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-dens-zero","kind":"node","lean":"MetaMathlibExt.polyContFract_dens_zero","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-dens-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field, let $a_0, a_1, \\ldots$ be polynomials in $F[X]$ and let $n \\in \\mathbb{N}$. The denominator $B_0$ of the zeroth convergent of the continued fraction $[a_0; a_1, \\ldots, a_n]$ over $F(X)$ is $1$.","title":"Zeroth convergent denominator of a polynomial continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-nums-one","kind":"node","lean":"MetaMathlibExt.polyContFract_nums_one","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-nums-one","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field, let $a_0, a_1, \\ldots$ be polynomials in $F[X]$ and let $n \\ge 1$. The numerator $A_1$ of the first convergent of the continued fraction $[a_0; a_1, \\ldots, a_n]$ over $F(X)$ is $a_1 a_0 + 1$.","title":"First convergent numerator of a polynomial continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-dens-one","kind":"node","lean":"MetaMathlibExt.polyContFract_dens_one","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-dens-one","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field, let $a_0, a_1, \\ldots$ be polynomials in $F[X]$ and let $n \\ge 1$. The denominator $B_1$ of the first convergent of the continued fraction $[a_0; a_1, \\ldots, a_n]$ over $F(X)$ is $a_1$.","title":"First convergent denominator of a polynomial continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-convergents-farey-path","kind":"node","lean":"MetaMathlibExt.polyContFract_convergents_farey_path","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-convergents-farey-path","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $F$ be a field, let $n \\in \\mathbb N$, and let $a_0, a_1, \\dots, a_n \\in F[X]$ with $a_i$ nonconstant for $1 \\le i \\le n$. Write $A_i$ and $B_i$ for the numerator and the denominator of the $i$th convergent $A_i / B_i$ of the continued fraction $[a_0, a_1, \\dots, a_n]$ over $F(X)$. Then for every $i \\le n$, $A_i$ and $B_i$ are polynomials in $F[X]$, $B_i \\ne 0$, and $A_i$ and $B_i$ are coprime in $F[X]$; and\u2026","title":"Convergents of a polynomial continued fraction form a Farey path"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-extend-farey-path","kind":"node","lean":"MetaMathlibExt.polyContFract_extend_farey_path","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-extend-farey-path","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $F$ be a field, let $n \\ge 1$, and let $a_0, a_1, \\dots, a_n \\in F[X]$ with $a_i$ nonconstant for $1 \\le i \\le n$. Let $A_0, \\dots, A_n, B_0, \\dots, B_n \\in F[X]$ be such that, for every $i \\le n$, $A_i$ and $B_i$ are the numerator and the denominator of the $i$th convergent of the continued fraction $[a_0, a_1, \\dots, a_n]$ over $F(X)$. Let $u, v \\in F[X]$ be such that $u B_n - v A_n$ is a unit of $F[X]$ (a\u2026","title":"Extending a Farey path extends the continued fraction"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-of-farey-path","kind":"node","lean":"MetaMathlibExt.polyContFract_of_farey_path","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-of-farey-path","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $F$ be a field, let $n \\in \\mathbb N$, let $u_i, v_i \\in F[X]$ for $i \\ge 0$, and let $c_i \\in F$ for $i \\ge 1$. Suppose that $v_0 = 1$ and that for $1 \\le i \\le n$ we have $c_i \\ne 0$, $\\deg v_{i-1} < \\deg v_i$ and $u_i v_{i-1} - u_{i-1} v_i = c_i$. Let $a_0, a_1, \\ldots \\in F[X]$ satisfy $a_0 = u_0$, $a_1 = c_1^{-1} v_1$, $a_2 = -c_2^{-1} c_1 (u_2 v_0 - u_0 v_2)$ and, for $3 \\le i \\le n$,","title":"Continued fraction along a Farey path of polynomials"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-of-farey-path-det-neg-on","kind":"node","lean":"MetaMathlibExt.polyContFract_of_farey_path_det_neg_one_pow_pred","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-of-farey-path-det-neg-on","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $F$ be a field, let $n \\in \\mathbb{N}$, and let $u_i, v_i \\in F[X]$ for $i \\ge 0$. Suppose that $v_0 = 1$ and that for $1 \\le i \\le n$ we have $\\deg v_{i-1} < \\deg v_i$ and $u_i v_{i-1} - u_{i-1} v_i = (-1)^{i-1}$. Let $a_0, a_1, \\ldots \\in F[X]$ satisfy $a_0 = u_0$, $a_1 = v_1$ and $a_i = (-1)^i (u_i v_{i-2} - u_{i-2} v_i)$ for $2 \\le i \\le n$. Then $a_i$ is nonconstant for $1 \\le i \\le n$, and for every\u2026","title":"Farey paths of polynomials with alternating determinants starting at 1"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-of-farey-path-det-neg-on-2","kind":"node","lean":"MetaMathlibExt.polyContFract_of_farey_path_det_neg_one_pow","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-of-farey-path-det-neg-on-2","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $F$ be a field, let $n \\in \\mathbb{N}$, and let $u_i, v_i \\in F[X]$ for $i \\ge 0$. Suppose that $v_0 = 1$ and that for $1 \\le i \\le n$ we have $\\deg v_{i-1} < \\deg v_i$ and $u_i v_{i-1} - u_{i-1} v_i = (-1)^i$. Let $a_0, a_1, \\ldots \\in F[X]$ satisfy $a_0 = u_0$, $a_1 = -v_1$ and $a_i = (-1)^i (u_i v_{i-2} - u_{i-2} v_i)$ for $2 \\le i \\le n$. Then $a_i$ is nonconstant for $1 \\le i \\le n$, and for every\u2026","title":"Farey paths of polynomials with alternating determinants starting at \u22121"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-of-farey-path-det-one","kind":"node","lean":"MetaMathlibExt.polyContFract_of_farey_path_det_one","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-of-farey-path-det-one","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $F$ be a field, let $n \\in \\mathbb{N}$, and let $u_i, v_i \\in F[X]$ for $i \\ge 0$. Suppose that $v_0 = 1$ and that for $1 \\le i \\le n$ we have $\\deg v_{i-1} < \\deg v_i$ and $u_i v_{i-1} - u_{i-1} v_i = 1$. Let $a_0, a_1, \\ldots \\in F[X]$ satisfy $a_0 = u_0$, $a_1 = v_1$ and $a_i = (-1)^{i+1} (u_i v_{i-2} - u_{i-2} v_i)$ for $2 \\le i \\le n$. Then $a_i$ is nonconstant for $1 \\le i \\le n$, and for every $i \\le n$\u2026","title":"Farey paths of polynomials with all determinants 1"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-cont-fract-of-farey-path-det-neg-on-3","kind":"node","lean":"MetaMathlibExt.polyContFract_of_farey_path_det_neg_one","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-cont-fract-of-farey-path-det-neg-on-3","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $F$ be a field, let $n \\in \\mathbb{N}$, and let $u_i, v_i \\in F[X]$ for $i \\ge 0$. Suppose that $v_0 = 1$ and that for $1 \\le i \\le n$ we have $\\deg v_{i-1} < \\deg v_i$ and $u_i v_{i-1} - u_{i-1} v_i = -1$. Let $a_0, a_1, \\ldots \\in F[X]$ satisfy $a_0 = u_0$, $a_1 = -v_1$ and $a_i = (-1)^{i+1} (u_i v_{i-2} - u_{i-2} v_i)$ for $2 \\le i \\le n$. Then $a_i$ is nonconstant for $1 \\le i \\le n$, and for every\u2026","title":"Farey paths of polynomials with all determinants \u22121"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-farey-series","kind":"node","lean":"MetaMathlibExt.polyFareySeries","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-farey-series","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field and $n$ a natural number. The Farey series $\\mathcal{F}_n$ of order $n$ over $F$ is the set of pairs $(u, v)$ of polynomials in $F[x]$, read as the fraction $u / v$, such that $\\deg u < \\deg v \\le n$, $u$ and $v$ are coprime, and $v$ is monic. Here $\\deg 0 = -\\infty$, so $u = 0$ is allowed.","title":"Farey series over a field"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-frac","kind":"node","lean":"MetaMathlibExt.polyFrac","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-frac","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field. The value of a pair $(u, v)$ of polynomials in $F[x]$ is the rational function $u / v$ in the field $F(x)$ of rational functions over $F$ (it is $0$ when $v = 0$).","title":"The fraction of a pair of polynomials"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/zero-one-mem-poly-farey-series","kind":"node","lean":"MetaMathlibExt.zero_one_mem_polyFareySeries","parent":10685,"reading_title":{"format":"inline-parts-v1","parts":[["text","The fraction "],["math","\\(0/1\\)"],["text"," lies in every Farey series"]],"source":"The fraction $0/1$ lies in every Farey series"},"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#zero-one-mem-poly-farey-series","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"For every field $F$ and every natural number $n$, the pair $(0, 1)$, that is the fraction $0 / 1$, belongs to the Farey series $\\mathcal{F}_n$ of order $n$ over $F$.","title":"The fraction $0/1$ lies in every Farey series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/one-x-mem-poly-farey-series","kind":"node","lean":"MetaMathlibExt.one_X_mem_polyFareySeries","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#one-x-mem-poly-farey-series","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field. For every natural number $n \\ge 1$, the pair $(1, X)$, that is, the fraction $1/X$, belongs to the Farey series $\\mathcal F_n$ of order $n$ over $F$.","title":"The fraction 1/X in the polynomial Farey series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/is-poly-farey-deg-neighbor","kind":"node","lean":"MetaMathlibExt.IsPolyFareyDegNeighbor","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#is-poly-farey-deg-neighbor","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field, let $n \\in \\mathbb N$, and let $\\mathcal F_n$ be the Farey series of order $n$ over $F$. For pairs $(u, v)$ and $(h, k)$ of polynomials over $F$, read as the fractions $u/v$ and $h/k$, the fraction $h/k$ is a $(\\deg k)$-neighbour of $u/v$ in $\\mathcal F_n$ when both fractions lie in $\\mathcal F_n$, $h/k \\ne u/v$ as rational functions, and $|u/v - h/k| \\le |u/v - h'/k'|$ for every\u2026","title":"Degree neighbours in the polynomial Farey series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/not-is-poly-farey-deg-neighbor-self","kind":"node","lean":"MetaMathlibExt.not_isPolyFareyDegNeighbor_self","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#not-is-poly-farey-deg-neighbor-self","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field and let $n \\in \\mathbb N$. For every pair $(u, v)$ of polynomials over $F$, the fraction $u/v$ is not a $(\\deg v)$-neighbour of itself in the Farey series $\\mathcal F_n$ of order $n$ over $F$.","title":"No fraction is its own degree neighbour"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/are-poly-farey-neighbors","kind":"node","lean":"MetaMathlibExt.ArePolyFareyNeighbors","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#are-poly-farey-neighbors","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"fully_proved","summary":"Let $F$ be a field and let $n \\in \\mathbb N$. Two pairs $(u, v)$ and $(h, k)$ of polynomials over $F$, read as the fractions $u/v$ and $h/k$, are neighbours in the Farey series $\\mathcal F_n$ of order $n$ over $F$ when $u/v$ is a $(\\deg v)$-neighbour of $h/k$ in $\\mathcal F_n$ and $h/k$ is a $(\\deg k)$-neighbour of $u/v$ in $\\mathcal F_n$. In particular, both fractions then lie in $\\mathcal F_n$.","title":"Neighbouring fractions in the polynomial Farey series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/are-poly-farey-neighbors-iff","kind":"node","lean":"MetaMathlibExt.arePolyFareyNeighbors_iff","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#are-poly-farey-neighbors-iff","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $F$ be a field, let $n \\in \\mathbb N$, and let $(u, v)$ and $(h, k)$ be elements of the Farey series $\\mathcal F_n$ of order $n$ over $F$, read as the fractions $u/v$ and $h/k$. Then $u/v$ and $h/k$ are neighbours in $\\mathcal F_n$ if and only if the polynomial $uk - hv$ has degree $0$, that is, $uk - hv$ is a nonzero constant.","title":"Neighbour criterion for the polynomial Farey series"},{"id":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/poly-farey-neighbor-count","kind":"node","lean":"MetaMathlibExt.polyFarey_neighbor_count","parent":10685,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths/index.html#poly-farey-neighbor-count","scope":"NumberTheory/continued-fractions-expansions/part-01/polynomial-continued-fractions-and-farey-paths","status":"can_prove","summary":"Let $p$ be a prime, let $n \\in \\mathbb{N}$, and let $u / v$ be a fraction of the Farey series $\\mathcal{F}_n$ over $\\mathbb{F}_p$ with $q = \\deg v \\ge 1$. Then for every $t \\in \\mathbb{N}$ with $q + t \\le n$, exactly $(p - 1) p^t$ fractions $h / k \\in \\mathcal{F}_n$ with $\\deg k = q + t$ are neighbours of $u / v$ in $\\mathcal{F}_n$, and exactly one fraction $h / k \\in \\mathcal{F}_n$ with $\\deg k < q$ is a neighbour\u2026","title":"Number of neighbours of a polynomial Farey fraction"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences.json","id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","kind":"scope","navigation":"module","parent":10317,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","title":"TRIP-Stern sequences"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/trip-stern-tree","kind":"node","lean":"MetaMathlibExt.TripSternTree","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#trip-stern-tree","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"A TRIP-Stern tree with node values in a type $\\alpha$ consists of a root seed in $\\alpha$ together with two maps from $\\alpha$ to $\\alpha$, the left transition and the right transition. No condition is imposed on the seed or on the two transitions; in the source the seed is $(1,1,1)$ and the left and right transitions are induced by the maps $F_0$ and $F_1$ defined there.","title":"TRIP-Stern tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/node-at-path","kind":"node","lean":"MetaMathlibExt.TripSternTree.nodeAtPath","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#node-at-path","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"Let $T$ be a TRIP-Stern tree with seed $s$, left transition $L$ and right transition $R$. A binary path is a finite sequence of steps, each left or right. The node of $T$ selected by a path $(b_1, \\dots, b_k)$ is obtained by starting at $s$ and applying, for $i = 1, \\dots, k$ in order, $L$ if $b_i$ is a left step and $R$ if $b_i$ is a right step.","title":"Node selected by a binary path"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/node-at-empty-path","kind":"node","lean":"MetaMathlibExt.TripSternTree.nodeAtEmptyPath","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#node-at-empty-path","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"For every TRIP-Stern tree $T$, the node of $T$ selected by the empty binary path is the seed of $T$.","title":"The empty path selects the seed"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/node-at-path-append-left","kind":"node","lean":"MetaMathlibExt.TripSternTree.nodeAtPathAppendLeft","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#node-at-path-append-left","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"Let $T$ be a TRIP-Stern tree with left transition $L$. For every binary path $p$, the node of $T$ selected by $p$ followed by one left step equals $L$ applied to the node of $T$ selected by $p$.","title":"Extending a path by a left step"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/node-at-path-append-right","kind":"node","lean":"MetaMathlibExt.TripSternTree.nodeAtPathAppendRight","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#node-at-path-append-right","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"Let $T$ be a TRIP-Stern tree with right transition $R$. For every binary path $p$, the node of $T$ selected by $p$ followed by one right step equals $R$ applied to the node of $T$ selected by $p$.","title":"Extending a path by a right step"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/nodes-at-depth","kind":"node","lean":"MetaMathlibExt.TripSternTree.nodesAtDepth","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#nodes-at-depth","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"Let $T$ be a TRIP-Stern tree and $n$ a natural number. The set of nodes of $T$ at depth $n$ is the set of all values $y$ such that some binary path of length $n$ selects $y$ in $T$.","title":"Nodes at a given depth of a TRIP-Stern tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/node-membership-at-depth","kind":"node","lean":"MetaMathlibExt.TripSternTree.nodeMembershipAtDepth","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#node-membership-at-depth","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"For every TRIP-Stern tree $T$, natural number $n$ and node value $y$, $y$ lies in the set of nodes of $T$ at depth $n$ if and only if there is a binary path of length $n$ whose selected node in $T$ is $y$.","title":"Membership in a level of a TRIP-Stern tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/nodes-at-depth-zero","kind":"node","lean":"MetaMathlibExt.TripSternTree.nodesAtDepthZero","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#nodes-at-depth-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"For every TRIP-Stern tree $T$ and every node value $y$, $y$ lies in the set of nodes of $T$ at depth $0$ if and only if $y$ is the seed of $T$.","title":"Depth zero is the seed"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/leftmost-node","kind":"node","lean":"MetaMathlibExt.TripSternTree.leftmostNode","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#leftmost-node","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"Let $T$ be a TRIP-Stern tree with seed $s$ and left transition $L$. The left-most node of $T$ at depth $n$ is $L^n(s)$, the result of applying $L$ $n$ times to $s$; that is, it is $s$ at depth $0$, and at depth $n + 1$ it is $L$ applied to the left-most node at depth $n$.","title":"Left-most node of a TRIP-Stern tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/rightmost-node","kind":"node","lean":"MetaMathlibExt.TripSternTree.rightmostNode","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#rightmost-node","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"For a TRIP-Stern tree $T$ and a natural number $n$, the rightmost node of $T$ at depth $n$ is obtained from the seed of $T$ by applying the right transition of $T$ exactly $n$ times: at depth $0$ it is the seed, and at depth $n + 1$ it is the image of the rightmost node at depth $n$ under the right transition.","title":"Rightmost node of a TRIP-Stern tree"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/leftmost-node-at-zero","kind":"node","lean":"MetaMathlibExt.TripSternTree.leftmostNodeAtZero","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#leftmost-node-at-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"For every TRIP-Stern tree $T$, the left-most node of $T$ at depth $0$ is the seed of $T$.","title":"Left-most node at depth zero"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/leftmost-node-at-successor","kind":"node","lean":"MetaMathlibExt.TripSternTree.leftmostNodeAtSuccessor","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#leftmost-node-at-successor","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"Let $T$ be a TRIP-Stern tree with left transition $L$. For every natural number $n$, the left-most node of $T$ at depth $n + 1$ equals $L$ applied to the left-most node of $T$ at depth $n$.","title":"Recursion for the left-most node"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/rightmost-node-at-zero","kind":"node","lean":"MetaMathlibExt.TripSternTree.rightmostNodeAtZero","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#rightmost-node-at-zero","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"For every TRIP-Stern tree $T$, the rightmost node of $T$ at depth $0$ is the seed of $T$.","title":"Rightmost node at depth zero"},{"id":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/rightmost-node-at-successor","kind":"node","lean":"MetaMathlibExt.TripSternTree.rightmostNodeAtSuccessor","parent":10709,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences/index.html#rightmost-node-at-successor","scope":"NumberTheory/continued-fractions-expansions/part-01/trip-stern-sequences","status":"fully_proved","summary":"For every TRIP-Stern tree $T$ and every natural number $n$, the rightmost node of $T$ at depth $n + 1$ is the image of the rightmost node of $T$ at depth $n$ under the right transition of $T$.","title":"Successor step of the rightmost node"},{"data_url":"scopes/NumberTheory/continued-fractions-expansions/part-02.json","id":"NumberTheory/continued-fractions-expansions/part-02","kind":"scope","navigation":"transparent","parent":10316,"result":false,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-02/index.html","scope":"NumberTheory/continued-fractions-expansions/part-02","status":"fully_proved","title":"Part 2 of 2"},{"id":"NumberTheory/continued-fractions-expansions/part-02/of-den-succ-lt-succ-succ","kind":"node","lean":"GenContFract.of_den_succ_lt_succ_succ","parent":10724,"result":true,"route":"../roadmap/NumberTheory/continued-fractions-expansions/part-02/index.html#of-den-succ-lt-succ-succ","scope":"NumberTheory/continued-fractions-expansions/part-02","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function, let $v \\in K$, and write $v = [a_0; a_1, a_2, \\dots]$ for its regular continued fraction, with convergent denominators $q_0 = 1$, $q_1 = a_1$ and $q_{k+2} = a_{k+2} q_{k+1} + q_k$. For every natural number $n$, if the partial quotient $a_{n+2}$ exists (the continued fraction has not terminated before it), then $q_{n+1} < q_{n+2}$.","title":"Strict growth of continued-fraction denominators"},{"data_url":"scopes/NumberTheory/heights-mahler-measure.json","id":"NumberTheory/heights-mahler-measure","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/heights-mahler-measure/index.html","scope":"NumberTheory/heights-mahler-measure","status":"can_prove","summary":"Absolute heights of algebraic numbers, Mahler measure, Lehmer's problem, Dobrowolski bounds, Schinzel\u2013Zassenhaus, house and trace problems.","title":"Heights and Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/dimitrov","kind":"node","lean":"MathlibExt.NumberTheory.SchinzelZassenhausWanted.NumberTheory.dimitrov","parent":10726,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/index.html#dimitrov","scope":"NumberTheory/heights-mahler-measure","status":"can_prove","summary":"Let $K$ be a number field and let $\\alpha \\in K$ be a nonzero algebraic integer that is not a root of unity. Let $d'$ be the number of distinct values of the argument $\\arg \\sigma(\\alpha)$ as $\\sigma$ ranges over the ring homomorphisms $K \\to \\mathbb{C}$. Then the house of $\\alpha$, the maximum of $|\\sigma(\\alpha)|$ over these $\\sigma$, is at least $2^{1/(4d')}$.","title":"Dimitrov's lower bound for the house of an algebraic integer"},{"data_url":"scopes/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure.json","id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","kind":"scope","navigation":"module","parent":10726,"result":false,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","title":"Absolute height and Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/primitive-int-minpoly","kind":"node","lean":"NumberField.primitiveIntMinpoly","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#primitive-int-minpoly","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field that is an algebra over $\\mathbb{Q}$, let $x \\in K$, and let $m_x \\in \\mathbb{Q}[X]$ be the minimal polynomial of $x$ over $\\mathbb{Q}$, with $m_x = 0$ when $x$ is not algebraic over $\\mathbb{Q}$. Choose a nonzero integer $b$ such that $b\\,m_x$ has integer coefficients. The primitive integral minimal polynomial $P_x \\in \\mathbb{Z}[X]$ of $x$ is the primitive part of $b\\,m_x$: when $b\\,m_x \\neq 0$\u2026","title":"Primitive integral minimal polynomial"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/abs-log-height-eq-log-leading-coeff-add","kind":"node","lean":"MathlibExt.NumberTheory.Height.AbsoluteWanted.absLogHeight\u2081_eq_log_leadingCoeff_add_sum_log_roots_of_isIntegral","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#abs-log-height-eq-log-leading-coeff-add","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero and let $x \\in K$ be integral over $\\mathbb{Q}$. Let $d$ be the degree of the minimal polynomial of $x$ over $\\mathbb{Q}$, and let $P_x \\in \\mathbb{Z}[X]$ be the primitive integral minimal polynomial of $x$, with leading coefficient $a$. Then the absolute logarithmic height of $x$ is","title":"Logarithmic height from the roots of the minimal polynomial, for an integral element"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/abs-log-height-eq-log-leading-coeff-add-2","kind":"node","lean":"MathlibExt.NumberTheory.Height.AbsoluteWanted.absLogHeight\u2081_eq_log_leadingCoeff_add_sum_log_roots_of_isAlgebraic","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#abs-log-height-eq-log-leading-coeff-add-2","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero and let $x \\in K$ be algebraic over $\\mathbb{Q}$. Let $d$ be the degree of the minimal polynomial of $x$ over $\\mathbb{Q}$, and let $P_x \\in \\mathbb{Z}[X]$ be the primitive integral minimal polynomial of $x$, with leading coefficient $a$. Then the absolute logarithmic height of $x$ is","title":"Logarithmic height from the roots of the minimal polynomial, for an algebraic element"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/abs-log-height-eq-log-leading-coeff-add-3","kind":"node","lean":"MathlibExt.NumberTheory.Height.AbsoluteWanted.absLogHeight\u2081_eq_log_leadingCoeff_add_sum_log_roots","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#abs-log-height-eq-log-leading-coeff-add-3","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero that is algebraic over $\\mathbb{Q}$, and let $x \\in K$. Let $d$ be the degree of the minimal polynomial of $x$ over $\\mathbb{Q}$, and let $P_x \\in \\mathbb{Z}[X]$ be the primitive integral minimal polynomial of $x$, with leading coefficient $a$. Then the absolute logarithmic height of $x$ is","title":"Logarithmic height from the roots of the minimal polynomial"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/abs-mul-height-eq-map-mahler-measure-of","kind":"node","lean":"MathlibExt.NumberTheory.Height.AbsoluteWanted.absMulHeight\u2081_eq_mapMahlerMeasure_of_isIntegral","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#abs-mul-height-eq-map-mahler-measure-of","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero and let $x \\in K$ be integral over $\\mathbb{Q}$. Let $d$ be the degree of the minimal polynomial of $x$ over $\\mathbb{Q}$, and let $P_x \\in \\mathbb{Z}[X]$ be the primitive integral minimal polynomial of $x$. Then the absolute multiplicative height of $x$ is $H(x) = M(P_x)^{1/d}$, where $M(P_x)$ is the Mahler measure of $P_x$ regarded as a polynomial with complex coefficients.","title":"Multiplicative height as a Mahler measure, for an integral element"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/abs-mul-height-eq-map-mahler-measure","kind":"node","lean":"MathlibExt.NumberTheory.Height.AbsoluteWanted.absMulHeight\u2081_eq_mapMahlerMeasure","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#abs-mul-height-eq-map-mahler-measure","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero that is algebraic over $\\mathbb{Q}$, and let $x \\in K$. Let $d$ be the degree of the minimal polynomial of $x$ over $\\mathbb{Q}$, and let $P_x \\in \\mathbb{Z}[X]$ be the primitive integral minimal polynomial of $x$. Then the absolute multiplicative height of $x$ is $H(x) = M(P_x)^{1/d}$, where $M(P_x)$ is the Mahler measure of $P_x$ regarded as a polynomial with complex\u2026","title":"Absolute multiplicative height as a Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/positive-primitive-int-minpoly","kind":"node","lean":"NumberField.positivePrimitiveIntMinpoly","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#positive-primitive-int-minpoly","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field that is a $\\mathbb{Q}$-algebra, let $x \\in K$, and let $p$ be the canonical primitive integral minimal polynomial of $x$. The positive primitive integral minimal polynomial of $x$ is $-p$ if the leading coefficient of $p$ is negative, and $p$ otherwise.","title":"Positive primitive integral minimal polynomial"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/positive-primitive-int-minpoly-is-primit","kind":"node","lean":"NumberField.positivePrimitiveIntMinpoly_isPrimitive","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#positive-primitive-int-minpoly-is-primit","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"For every element $x$ of a field $K$ that is an algebra over $\\mathbb{Q}$, the positive primitive integral minimal polynomial of $x$ is primitive: the greatest common divisor of its integer coefficients is $1$.","title":"The positive primitive integral minimal polynomial is primitive"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/positive-primitive-int-minpoly-ne-zero","kind":"node","lean":"NumberField.positivePrimitiveIntMinpoly_ne_zero","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#positive-primitive-int-minpoly-ne-zero","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field that is a $\\mathbb{Q}$-algebra. For every $x \\in K$, the positive primitive integral minimal polynomial of $x$ is not the zero polynomial.","title":"The positive primitive integral minimal polynomial is nonzero"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/positive-primitive-int-minpoly-leading-c","kind":"node","lean":"NumberField.positivePrimitiveIntMinpoly_leadingCoeff_pos","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#positive-primitive-int-minpoly-leading-c","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"For every element $x$ of a field $K$ that is an algebra over $\\mathbb{Q}$, the leading coefficient of the positive primitive integral minimal polynomial of $x$ is a strictly positive integer.","title":"The positive primitive integral minimal polynomial has positive leading coefficient"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/positive-primitive-int-minpoly-nat-degre","kind":"node","lean":"NumberField.positivePrimitiveIntMinpoly_natDegree","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#positive-primitive-int-minpoly-nat-degre","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"For every element $x$ of a field $K$ that is an algebra over $\\mathbb{Q}$, the positive primitive integral minimal polynomial of $x$ has the same degree as the primitive integral minimal polynomial of $x$.","title":"Sign normalization preserves the degree"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/positive-primitive-int-minpoly-nat-degre-2","kind":"node","lean":"NumberField.positivePrimitiveIntMinpoly_natDegree_eq","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#positive-primitive-int-minpoly-nat-degre-2","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"The positive primitive integral minimal polynomial has the degree of the minimal polynomial. For every element $x$ of a field $K$ that is an algebra over $\\mathbb{Q}$, the positive primitive integral minimal polynomial of $x$ has the same degree as the minimal polynomial of $x$ over $\\mathbb{Q}$.","title":"The positive primitive integral minimal polynomial has the degree of the minimal polynomial"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/positive-primitive-int-minpoly-eval-at","kind":"node","lean":"NumberField.positivePrimitiveIntMinpoly_eval\u2082_at","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#positive-primitive-int-minpoly-eval-at","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero that is an algebra over $\\mathbb{Q}$, and let $x \\in K$ be algebraic over $\\mathbb{Q}$. Then the positive primitive integral minimal polynomial of $x$, with its integer coefficients mapped into $K$, vanishes at $x$.","title":"An algebraic element is a root of its positive primitive integral minimal polynomial"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/algebraic-mahler-measure","kind":"node","lean":"NumberField.algebraicMahlerMeasure","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#algebraic-mahler-measure","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"For an element $x$ of a field $K$ that is an algebra over $\\mathbb{Q}$, the algebraic Mahler measure $M(x)$ is the Mahler measure of the positive primitive integral minimal polynomial of $x$, regarded as a polynomial with complex coefficients. If $x$ is not algebraic over $\\mathbb{Q}$, this polynomial is the constant $1$, so $M(x) = 1$.","title":"Algebraic Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/algebraic-mahler-measure-eq","kind":"node","lean":"NumberField.algebraicMahlerMeasure_eq","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#algebraic-mahler-measure-eq","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"For every element $x$ of a field $K$ that is an algebra over $\\mathbb{Q}$, the algebraic Mahler measure of $x$, that is the Mahler measure of the positive primitive integral minimal polynomial of $x$, equals the Mahler measure of the primitive integral minimal polynomial of $x$, both polynomials being regarded as polynomials with complex coefficients.","title":"The sign normalization does not change the Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/one-le-algebraic-mahler-measure","kind":"node","lean":"NumberField.one_le_algebraicMahlerMeasure","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#one-le-algebraic-mahler-measure","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"For every element $x$ of a field $K$ that is an algebra over $\\mathbb{Q}$, the algebraic Mahler measure $M(x)$ of $x$ satisfies $1 \\le M(x)$.","title":"The algebraic Mahler measure is at least one"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/abs-log-height-eq-log-algebraic-mahler-m","kind":"node","lean":"NumberField.absLogHeight\u2081_eq_log_algebraicMahlerMeasure","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#abs-log-height-eq-log-algebraic-mahler-m","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero that is an algebra over $\\mathbb{Q}$, and let $x \\in K$ be algebraic over $\\mathbb{Q}$. Let $d$ be the degree of the minimal polynomial of $x$ over $\\mathbb{Q}$, and let $M(x)$ be the algebraic Mahler measure of $x$, the Mahler measure of the positive primitive integral minimal polynomial of $x$ regarded as a complex polynomial. Then the absolute logarithmic height of $x$\u2026","title":"Absolute logarithmic height from the algebraic Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/log-algebraic-mahler-measure-add-one-lt","kind":"node","lean":"NumberField.log_algebraicMahlerMeasure_add_one_lt","parent":10728,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure/index.html#log-algebraic-mahler-measure-add-one-lt","scope":"NumberTheory/heights-mahler-measure/absolute-height-and-mahler-measure","status":"fully_proved","summary":"Let $x$ be an element of a field $K$ that is an algebra over $\\mathbb{Q}$, and let $M(x)$ be its algebraic Mahler measure. If $M(x) > 1$, then $1 + \\log M(x) < M(x)$.","title":"One plus the logarithm of a Mahler measure above one is below it"},{"data_url":"scopes/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant.json","id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","kind":"scope","navigation":"module","parent":10726,"result":false,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"can_prove","title":"Asymptotic Dobrowolski constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/dobrowolski-scale","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.dobrowolskiScale","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#dobrowolski-scale","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"For a natural number $d$, the Dobrowolski scale is","title":"Dobrowolski scale"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/dobrowolski-scale-pos","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.dobrowolskiScale_pos","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#dobrowolski-scale-pos","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"For every natural number $d \\ge 3$, the Dobrowolski scale $\\left(\\frac{\\log \\log d}{\\log d}\\right)^3$ is strictly positive.","title":"Positivity of the Dobrowolski scale"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/is-asymptotic-dobrowolski-coefficient","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.IsAsymptoticDobrowolskiCoefficient","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#is-asymptotic-dobrowolski-coefficient","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"A real number $c$ is an asymptotic Dobrowolski coefficient if for every $\\varepsilon > 0$ there is a natural number $D \\ge 3$ such that every nonzero algebraic number $\\alpha \\in \\mathbb{C}$ that is not a root of unity and whose minimal polynomial over $\\mathbb{Q}$ has degree $d \\ge D$ satisfies","title":"Asymptotic Dobrowolski coefficients"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/asymptotic-dobrowolski-constant","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.asymptoticDobrowolskiConstant","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#asymptotic-dobrowolski-constant","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"The asymptotic Dobrowolski constant is the supremum, in the extended real numbers $[-\\infty, +\\infty]$, of the set of all real asymptotic Dobrowolski coefficients.","title":"Asymptotic Dobrowolski constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/coe-le-asymptotic-dobrowolski-constant","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.coe_le_asymptoticDobrowolskiConstant","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#coe-le-asymptotic-dobrowolski-constant","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"Every real asymptotic Dobrowolski coefficient $c$ satisfies $c \\le$ the asymptotic Dobrowolski constant, as extended real numbers.","title":"Coefficients are bounded by the asymptotic Dobrowolski constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/zero-is-asymptotic-dobrowolski-coefficie","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.zero_isAsymptoticDobrowolskiCoefficient","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#zero-is-asymptotic-dobrowolski-coefficie","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"The real number $0$ is an asymptotic Dobrowolski coefficient: for every real $\\varepsilon > 0$ there is a natural number $D \\ge 3$ such that every nonzero algebraic number $\\alpha \\in \\mathbb{C}$ that is not a root of unity and whose minimal polynomial over $\\mathbb{Q}$ has degree $d \\ge D$ satisfies $1 - \\varepsilon \\left(\\frac{\\log \\log d}{\\log d}\\right)^3 \\le M(\\alpha)$, where $M(\\alpha)$ is the Mahler measure\u2026","title":"Zero is an asymptotic Dobrowolski coefficient"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/asymptotic-dobrowolski-constant-le-iff","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.asymptoticDobrowolskiConstant_le_iff","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#asymptotic-dobrowolski-constant-le-iff","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"For every extended real number $b$, the asymptotic Dobrowolski constant is at most $b$ if and only if every real asymptotic Dobrowolski coefficient $c$ satisfies $c \\le b$.","title":"Upper bounds for the asymptotic Dobrowolski constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/is-asymptotic-dobrowolski-coefficient-if","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.isAsymptoticDobrowolskiCoefficient_iff","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#is-asymptotic-dobrowolski-coefficient-if","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"A real number $c$ is an asymptotic Dobrowolski coefficient if and only if $c$ is at most the asymptotic Dobrowolski constant, as extended real numbers.","title":"Coefficients are exactly the reals below the constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/is-asymptotic-dobrowolski-coefficient-if-2","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.isAsymptoticDobrowolskiCoefficient_iff_strict","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#is-asymptotic-dobrowolski-coefficient-if-2","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"A real number $c$ is an asymptotic Dobrowolski coefficient if and only if for every $\\varepsilon > 0$ there is a natural number $D \\ge 3$ such that every nonzero algebraic number $\\alpha \\in \\mathbb{C}$ that is not a root of unity and whose minimal polynomial over $\\mathbb{Q}$ has degree $d \\ge D$ satisfies","title":"Strict form of asymptotic Dobrowolski coefficients"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/asymptotic-dobrowolski-constant-nonneg","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.asymptoticDobrowolskiConstant_nonneg","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#asymptotic-dobrowolski-constant-nonneg","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"The asymptotic Dobrowolski constant is at least $0$.","title":"The asymptotic Dobrowolski constant is nonnegative"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/asymptotic-dobrowolski-constant-le-top","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.asymptoticDobrowolskiConstant_le_top","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#asymptotic-dobrowolski-constant-le-top","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"fully_proved","summary":"The asymptotic Dobrowolski constant is at most $+\\infty$.","title":"Trivial upper bound for the asymptotic Dobrowolski constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/dobrowolski-one-le-asymptotic-constant","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.dobrowolski_one_le_asymptoticConstant","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#dobrowolski-one-le-asymptotic-constant","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"can_prove","summary":"The asymptotic Dobrowolski constant is at least $1$, as an extended real number.","title":"Dobrowolski's lower bound for the asymptotic Dobrowolski constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/cantor-straus-two-le-asymptotic-constant","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.cantorStraus_two_le_asymptoticConstant","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#cantor-straus-two-le-asymptotic-constant","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"can_prove","summary":"The asymptotic Dobrowolski constant is at least $2$, as an extended real number.","title":"Cantor\u2013Straus lower bound for the asymptotic Dobrowolski constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/louboutin-nine-fourths-le-asymptotic-con","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.louboutin_nine_fourths_le_asymptoticConstant","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#louboutin-nine-fourths-le-asymptotic-con","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"can_prove","summary":"The asymptotic Dobrowolski constant is at least $9/4$, as an extended real number.","title":"Louboutin's lower bound for the asymptotic Dobrowolski constant"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/blanksby-montgomery-degree-bound","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.blanksbyMontgomery_degreeBound","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#blanksby-montgomery-degree-bound","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"can_prove","summary":"Let $\\alpha$ be a nonzero complex number that is algebraic over $\\mathbb{Q}$ and is not a root of unity, and let $d$ be the degree of its minimal polynomial over $\\mathbb{Q}$. If $d \\ge 2$, then the algebraic Mahler measure $M(\\alpha)$ satisfies $M(\\alpha) > 1 + \\frac{1}{52\\, d \\log(6d)}$, with $\\log$ the natural logarithm.","title":"Blanksby\u2013Montgomery lower bound for the Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/stewart-degree-bound","kind":"node","lean":"MathlibExt.NumberTheory.AsymptoticDobrowolski.stewart_degreeBound","parent":10747,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant/index.html#stewart-degree-bound","scope":"NumberTheory/heights-mahler-measure/asymptotic-dobrowolski-constant","status":"can_prove","summary":"Let $\\alpha$ be a nonzero complex number that is algebraic over $\\mathbb{Q}$ and is not a root of unity, and let $d$ be the degree of its minimal polynomial over $\\mathbb{Q}$. If $d \\ge 2$, then the algebraic Mahler measure $M(\\alpha)$ satisfies $M(\\alpha) > 1 + \\frac{1}{10000\\, d \\log d}$, with $\\log$ the natural logarithm.","title":"Stewart's lower bound for the Mahler measure"},{"data_url":"scopes/NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house.json","id":"NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house","kind":"scope","navigation":"module","parent":10726,"result":false,"route":"../roadmap/NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house/index.html","scope":"NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house","status":"can_prove","title":"Cyclotomic integers of small house"},{"id":"NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house/cyclotomic-integer-house-le-two","kind":"node","lean":"MathlibExt.NumberTheory.SmallHouseWanted.NumberTheory.cyclotomic_integer_house_le_two","parent":10764,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house/index.html#cyclotomic-integer-house-le-two","scope":"NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house","status":"can_prove","summary":"Let $n \\ge 1$, let $K$ be a number field that is a cyclotomic extension of $\\mathbb{Q}$ generated by the $n$-th roots of unity, and let $\\beta \\in K$ be an algebraic integer lying in the maximal real subfield of $K$. Write $\\operatorname{house}(\\beta)$ for the largest absolute value of the images of $\\beta$ under the complex embeddings of $K$. If $\\operatorname{house}(\\beta) \\le 2$, then\u2026","title":"Real cyclotomic integers with house at most 2"},{"id":"NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house/cyclotomic-integer-house-between-two-and","kind":"node","lean":"MathlibExt.NumberTheory.SmallHouseWanted.NumberTheory.cyclotomic_integer_house_between_two_and_76_33","parent":10764,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house/index.html#cyclotomic-integer-house-between-two-and","scope":"NumberTheory/heights-mahler-measure/cyclotomic-integers-of-small-house","status":"can_prove","summary":"Let $n \\ge 1$, let $K$ be a number field that is a cyclotomic extension of $\\mathbb{Q}$ generated by the $n$-th roots of unity, and let $\\beta \\in K$ be an algebraic integer lying in the maximal real subfield of $K$. Write $\\operatorname{house}(\\beta)$ for the largest absolute value of the images of $\\beta$ under the complex embeddings of $K$. If $2 < \\operatorname{house}(\\beta) < 76/33$, then\u2026","title":"Real cyclotomic integers with house between 2 and 76/33"},{"data_url":"scopes/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem.json","id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","kind":"scope","navigation":"module","parent":10726,"result":false,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"can_prove","title":"Lehmer's Mahler measure problem"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/int-mahler-measure","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.intMahlerMeasure","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#int-mahler-measure","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"The Mahler measure of $f \\in \\mathbb{Z}[X]$ is the Mahler measure of its image in $\\mathbb{C}[X]$. It equals $|a| \\prod_i \\max(1, |\\alpha_i|)$, where $a$ is the leading coefficient of $f$ and the $\\alpha_i$ are the complex roots of $f$ counted with multiplicity.","title":"Mahler measure of an integer polynomial"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/one-le-int-mahler-measure","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.one_le_intMahlerMeasure","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#one-le-int-mahler-measure","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"If $f \\in \\mathbb{Z}[X]$ is a nonzero polynomial, then its Mahler measure satisfies $M(f) \\ge 1$.","title":"Nonzero integer polynomials have Mahler measure at least one"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/int-mahler-measure-x-sub-c","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.intMahlerMeasure_X_sub_C","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#int-mahler-measure-x-sub-c","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"For every integer $a$, the Mahler measure of the integer polynomial $X - a$ is $\\max(1, |a|)$.","title":"Mahler measure of a monic linear integer polynomial"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/int-mahler-measure-x-sub-one","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.intMahlerMeasure_X_sub_one","parent":10767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mahler measure of "],["math","\\(X - 1\\)"]],"source":"Mahler measure of $X - 1$"},"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#int-mahler-measure-x-sub-one","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"The integer polynomial $X - 1$ has Mahler measure $1$.","title":"Mahler measure of $X - 1$"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/int-mahler-measure-x-sub-two","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.intMahlerMeasure_X_sub_two","parent":10767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mahler measure of "],["math","\\(X - 2\\)"]],"source":"Mahler measure of $X - 2$"},"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#int-mahler-measure-x-sub-two","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"The integer polynomial $X - 2$ has Mahler measure $2$.","title":"Mahler measure of $X - 2$"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/mahler-measures-above-one","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.mahlerMeasuresAboveOne","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#mahler-measures-above-one","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"This is the set of real numbers of the form $M(f)$, where $f \\in \\mathbb{Z}[X]$ is an integer polynomial whose Mahler measure $M(f)$ is strictly larger than $1$.","title":"Mahler measures above one"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/lehmer-constant","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.lehmerConstant","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#lehmer-constant","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"Writing $M(f)$ for the Mahler measure of an integer polynomial $f$, Lehmer's constant is $C_{40a} = \\inf \\{M(f) : f \\in \\mathbb{Z}[X],\\ M(f) > 1\\}$, the infimum of the Mahler measures of integer polynomials that are strictly larger than $1$.","title":"Lehmer's constant"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/lehmer-constant-le-int-mahler-measure","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.lehmerConstant_le_intMahlerMeasure","parent":10767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lehmer's constant bounds every Mahler measure above "],["math","\\(1\\)"]],"source":"Lehmer's constant bounds every Mahler measure above $1$"},"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#lehmer-constant-le-int-mahler-measure","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"For every integer polynomial $f$ whose Mahler measure is strictly larger than $1$, Lehmer's constant is at most the Mahler measure of $f$.","title":"Lehmer's constant bounds every Mahler measure above $1$"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/one-le-lehmer-constant","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.one_le_lehmerConstant","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#one-le-lehmer-constant","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"Lehmer's constant $L$, the infimum of the Mahler measures strictly larger than $1$ of polynomials in $\\mathbb{Z}[X]$, satisfies $1 \\le L$.","title":"Lehmer's constant is at least one"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/lehmer-constant-le-two","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.lehmerConstant_le_two","parent":10767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lehmer's constant is at most "],["math","\\(2\\)"]],"source":"Lehmer's constant is at most $2$"},"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#lehmer-constant-le-two","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"Lehmer's constant $C_{40a}$ satisfies $C_{40a} \\le 2$.","title":"Lehmer's constant is at most $2$"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/lehmer-question","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.LehmerQuestion","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#lehmer-question","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"The proposition that there are integer polynomials with Mahler measure arbitrarily close to $1$ from above: for every real $\\varepsilon > 0$ there is a polynomial $f \\in \\mathbb{Z}[X]$ whose Mahler measure $M(f)$ satisfies $1 < M(f) < 1 + \\varepsilon$.","title":"Lehmer's question"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/lehmer-constant-eq-one-iff","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.lehmerConstant_eq_one_iff","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#lehmer-constant-eq-one-iff","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"Lehmer's constant equals $1$ if and only if Lehmer's question has a positive answer, that is, for every $\\varepsilon > 0$ there is an integer polynomial whose Mahler measure lies strictly between $1$ and $1 + \\varepsilon$.","title":"Lehmer's question in terms of Lehmer's constant"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/lehmer-polynomial","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.lehmerPolynomial","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#lehmer-polynomial","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"Lehmer's polynomial is the integer polynomial $X^{10} + X^9 - X^7 - X^6 - X^5 - X^4 - X^3 + X + 1$.","title":"Lehmer's polynomial"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/smyth-polynomial","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.smythPolynomial","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#smyth-polynomial","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"Smyth's polynomial is $X^3 - X - 1 \\in \\mathbb{Z}[X]$.","title":"Smyth's polynomial"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/is-reciprocal","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.IsReciprocal","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#is-reciprocal","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"fully_proved","summary":"An integer polynomial $f$ is reciprocal when reversing its coefficients gives $f$ or $-f$, that is, when $X^d f(1/X) = f$ or $X^d f(1/X) = -f$, where $d$ is the degree of $f$, taken to be $0$ when $f = 0$.","title":"Reciprocal polynomial"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/int-mahler-measure-lehmer-polynomial-le","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.intMahlerMeasure_lehmerPolynomial_le","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#int-mahler-measure-lehmer-polynomial-le","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"can_prove","summary":"The Mahler measure of Lehmer's polynomial is at most $1176281/1000000 = 1.176281$.","title":"Mahler measure of Lehmer's polynomial"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/lehmer1933-upper-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.lehmer1933UpperBound_wanted","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#lehmer1933-upper-bound-wanted","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"can_prove","summary":"Lehmer's constant is at most $1176281/1000000 = 1.176281$.","title":"Lehmer's upper bound for Lehmer's constant"},{"id":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/smyth1971-nonreciprocal-wanted","kind":"node","lean":"MathlibExt.NumberTheory.LehmerConstant.smyth1971Nonreciprocal_wanted","parent":10767,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem/index.html#smyth1971-nonreciprocal-wanted","scope":"NumberTheory/heights-mahler-measure/lehmer-s-mahler-measure-problem","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[X]$ be an integer polynomial that is not reciprocal and whose constant term $f(0)$ is nonzero. Then the Mahler measure of $f$ is at least the Mahler measure of Smyth's polynomial $X^3 - X - 1$.","title":"Smyth's theorem for nonreciprocal polynomials"},{"data_url":"scopes/NumberTheory/heights-mahler-measure/multivariable-mahler-measure.json","id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","kind":"scope","navigation":"module","parent":10726,"result":false,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","title":"Multivariable Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure/real-sum","kind":"node","lean":"LogarithmicMahlerMeasure.realSum","parent":10786,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html#real-sum","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","summary":"For an angle vector $\\theta \\in \\mathbb{R}^n$ and an exponent vector $e \\in \\mathbb{Z}^n$, the real phase of $e$ at $\\theta$ is $\\sum_{j} e_j \\theta_j$.","title":"Phase of an exponent vector"},{"id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure/torus-character","kind":"node","lean":"LogarithmicMahlerMeasure.torusCharacter","parent":10786,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html#torus-character","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","summary":"For an angle vector $\\theta \\in \\mathbb{R}^n$, the torus character at $\\theta$ sends an exponent vector $e \\in \\mathbb{Z}^n$ to $\\exp\\bigl(i \\sum_{j} e_j \\theta_j\\bigr) \\in \\mathbb{C}$.","title":"Torus character"},{"id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure/torus-character-hom","kind":"node","lean":"LogarithmicMahlerMeasure.torusCharacterHom","parent":10786,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html#torus-character-hom","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","summary":"For an angle vector $\\theta \\in \\mathbb{R}^n$, the torus character $e \\mapsto \\exp\\bigl(i \\sum_{j} e_j \\theta_j\\bigr)$, viewed as a map from the additive group $\\mathbb{Z}^n$ written multiplicatively to the multiplicative monoid of $\\mathbb{C}$, is a monoid homomorphism: it sends $0$ to $1$ and sums to products.","title":"Torus character as a monoid homomorphism"},{"id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure/torus-eval","kind":"node","lean":"LogarithmicMahlerMeasure.torusEval","parent":10786,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html#torus-eval","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","summary":"For an angle vector $\\theta \\in \\mathbb{R}^n$, evaluation at the point $(\\exp(i\\theta_1), \\dots, \\exp(i\\theta_n))$ of the unit torus is the $\\mathbb{C}$-algebra homomorphism from complex Laurent polynomials in $n$ variables to $\\mathbb{C}$ that, writing $X^e$ for the monomial with exponent vector $e \\in \\mathbb{Z}^n$, sends $\\sum_e c_e X^e$ to $\\sum_e c_e \\exp\\bigl(i \\sum_{j} e_j \\theta_j\\bigr)$.","title":"Evaluation on the unit torus"},{"id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure/normalized-box-integral","kind":"node","lean":"LogarithmicMahlerMeasure.normalizedBoxIntegral","parent":10786,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html#normalized-box-integral","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","summary":"For a function $f \\colon \\mathbb{R}^m \\to \\mathbb{R}$, the normalized box integral of $f$ is $(2\\pi)^{-m} \\int_{[0, 2\\pi]^m} f(x) \\, dx$, the integral taken with respect to Lebesgue measure (and equal to $0$ when $f$ is not integrable on the box).","title":"Normalized box integral"},{"id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure/logarithmic-mahler-measure","kind":"node","lean":"LogarithmicMahlerMeasure.logarithmicMahlerMeasure","parent":10786,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html#logarithmic-mahler-measure","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","summary":"For a complex Laurent polynomial $F$ in $m$ variables, the logarithmic Mahler measure of $F$ is the real number $(2\\pi)^{-m} \\int_{[0, 2\\pi]^m} \\log |F(e^{i\\theta_1}, \\dots, e^{i\\theta_m})| \\, d\\theta$, the integral taken with respect to Lebesgue measure.","title":"Logarithmic Mahler measure"},{"id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure/fin-one-add-equiv","kind":"node","lean":"LogarithmicMahlerMeasure.finOneAddEquiv","parent":10786,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html#fin-one-add-equiv","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","summary":"The additive isomorphism from $\\mathbb{Z}^1$, the group of functions $\\{0\\} \\to \\mathbb{Z}$ under pointwise addition, to $\\mathbb{Z}$ that sends a function $e$ to its value $e(0)$.","title":"Integer vectors of length one"},{"id":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure/bridge-laurent","kind":"node","lean":"LogarithmicMahlerMeasure.bridgeLaurent","parent":10786,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/multivariable-mahler-measure/index.html#bridge-laurent","scope":"NumberTheory/heights-mahler-measure/multivariable-mahler-measure","status":"fully_proved","summary":"The $\\mathbb{C}$-algebra isomorphism from complex Laurent polynomials whose exponents are vectors $e \\in \\mathbb{Z}^1$ (functions $\\{0\\} \\to \\mathbb{Z}$) to complex Laurent polynomials $\\mathbb{C}[T, T^{-1}]$ in one variable, induced by the additive isomorphism $\\mathbb{Z}^1 \\cong \\mathbb{Z}$, $e \\mapsto e(0)$. Writing $X^{e}$ for the monomial with exponent vector $e$, it sends $c\\,X^{e}$ to $c\\,T^{e(0)}$.","title":"Identification of one-variable Laurent polynomials"},{"data_url":"scopes/NumberTheory/heights-mahler-measure/pisot-numbers.json","id":"NumberTheory/heights-mahler-measure/pisot-numbers","kind":"scope","navigation":"module","parent":10726,"result":false,"route":"../roadmap/NumberTheory/heights-mahler-measure/pisot-numbers/index.html","scope":"NumberTheory/heights-mahler-measure/pisot-numbers","status":"fully_proved","title":"Pisot numbers"},{"id":"NumberTheory/heights-mahler-measure/pisot-numbers/is-pisot-number","kind":"node","lean":"MetaMathlibExt.IsPisotNumber","parent":10795,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/pisot-numbers/index.html#is-pisot-number","scope":"NumberTheory/heights-mahler-measure/pisot-numbers","status":"fully_proved","summary":"A Pisot number is a real algebraic integer greater than one whose other complex conjugates all have norm less than one.","title":"Pisot number"},{"id":"NumberTheory/heights-mahler-measure/pisot-numbers/is-unique-pisot-root","kind":"node","lean":"MetaMathlibExt.IsUniquePisotRoot","parent":10795,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/pisot-numbers/index.html#is-unique-pisot-root","scope":"NumberTheory/heights-mahler-measure/pisot-numbers","status":"fully_proved","summary":"Let $P$ be a polynomial with real coefficients and $r$ a real number. Then $r$ is the unique Pisot root of $P$ when $r$ is a Pisot number with $P(r) = 0$ and every Pisot number $s$ with $P(s) = 0$ equals $r$.","title":"Unique Pisot root of a polynomial"},{"data_url":"scopes/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem.json","id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","kind":"scope","navigation":"module","parent":10726,"result":false,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","title":"Schur\u2013Siegel\u2013Smyth trace problem"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/conjugates","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.conjugates","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#conjugates","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"The conjugates of a complex number $\\alpha$ are the roots in $\\mathbb{C}$ of its minimal polynomial over $\\mathbb{Q}$, counted with multiplicity. If $\\alpha$ is transcendental, its minimal polynomial is taken to be $0$ and it has no conjugates.","title":"Conjugates"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/degree","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.degree","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#degree","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"The degree of a complex number $\\alpha$ is the degree of its minimal polynomial over $\\mathbb{Q}$. If $\\alpha$ is transcendental, its minimal polynomial is taken to be $0$, so its degree is $0$.","title":"Degree of a complex number"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/trace","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.trace","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#trace","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"For $\\alpha \\in \\mathbb{C}$, the trace of $\\alpha$ is the sum of its conjugates, that is, of the roots in $\\mathbb{C}$ of the minimal polynomial of $\\alpha$ over $\\mathbb{Q}$, counted with multiplicity. For transcendental $\\alpha$ there are no conjugates and the trace is $0$.","title":"Trace as the sum of conjugates"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/absolute-trace","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.absoluteTrace","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#absolute-trace","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"The absolute trace of a complex number $\\alpha$ is the real number obtained by dividing the real part of the trace of $\\alpha$ by the degree of $\\alpha$. If $\\alpha$ is transcendental, its trace and its degree are both $0$, and its absolute trace is taken to be $0$.","title":"Absolute trace"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/is-totally-positive","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.IsTotallyPositive","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#is-totally-positive","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"A complex number $\\alpha$ is a totally positive algebraic integer when it is integral over $\\mathbb{Z}$ and every conjugate of $\\alpha$ (every root in $\\mathbb{C}$ of its minimal polynomial over $\\mathbb{Q}$) is a strictly positive real number.","title":"Totally positive algebraic integers"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/trace-pos-re","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.trace_pos_re","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#trace-pos-re","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"If $\\alpha \\in \\mathbb{C}$ is a totally positive algebraic integer, then the real part of the trace of $\\alpha$ is strictly positive.","title":"The trace of a totally positive algebraic integer has positive real part"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/trace-im-eq-zero","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.trace_im_eq_zero","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#trace-im-eq-zero","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"If $\\alpha \\in \\mathbb{C}$ is a totally positive algebraic integer, then the imaginary part of the trace of $\\alpha$ is $0$.","title":"The trace of a totally positive algebraic integer is real"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/absolute-trace-pos","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.absoluteTrace_pos","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#absolute-trace-pos","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"If a complex number $\\alpha$ is a totally positive algebraic integer, then the absolute trace of $\\alpha$ is strictly positive.","title":"Positivity of the absolute trace"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/minpoly-rat-one-eq","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.minpoly_rat_one_eq","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#minpoly-rat-one-eq","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"The minimal polynomial of $1 \\in \\mathbb{C}$ over $\\mathbb{Q}$ is $X - 1$.","title":"Minimal polynomial of one"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/is-totally-positive-one","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.isTotallyPositive_one","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#is-totally-positive-one","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"The complex number $1$ is a totally positive algebraic integer.","title":"One is a totally positive algebraic integer"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/is-admissible","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.IsAdmissible","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#is-admissible","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"A real number $\\rho$ is admissible when $\\rho \\ge 0$ and only finitely many totally positive algebraic integers $\\alpha \\in \\mathbb{C}$ have absolute trace less than $\\rho$.","title":"Admissible trace bounds"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/schur-siegel-smyth-constant","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.schurSiegelSmythConstant","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#schur-siegel-smyth-constant","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"The Schur\u2013Siegel\u2013Smyth trace constant $C_{86}$ is the supremum, taken in the extended reals, of the admissible real bounds: the real numbers $\\rho \\ge 0$ such that only finitely many complex numbers that are totally positive algebraic integers have absolute trace less than $\\rho$.","title":"Schur\u2013Siegel\u2013Smyth trace constant"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/coe-le-schur-siegel-smyth-constant","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.coe_le_schurSiegelSmythConstant","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#coe-le-schur-siegel-smyth-constant","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"Let $C_{86}$ be the Schur\u2013Siegel\u2013Smyth trace constant, an extended real number. Every admissible real number $\\rho$ satisfies $\\rho \\le C_{86}$ in the extended reals.","title":"Admissible bounds lie below the Schur\u2013Siegel\u2013Smyth constant"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/schur-siegel-smyth-constant-le-iff","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.schurSiegelSmythConstant_le_iff","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#schur-siegel-smyth-constant-le-iff","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"For every extended real number $b$, the Schur\u2013Siegel\u2013Smyth trace constant is at most $b$ if and only if every admissible real number $c$ satisfies $c \\le b$.","title":"Upper bounds for the Schur\u2013Siegel\u2013Smyth constant"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/trace-2","kind":"node","lean":"MathlibExt.NumberTheory.AlgebraicIntegersWanted.AlgebraicIntegers.trace","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#trace-2","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"For a complex number $\\alpha$, the trace $\\operatorname{tr}(\\alpha) \\in \\mathbb{C}$ is the sum, counted with multiplicity, of the complex roots of the minimal polynomial of $\\alpha$ over $\\mathbb{Q}$. When $\\alpha$ is algebraic these roots are the conjugates $\\alpha_1, \\dots, \\alpha_n$ of $\\alpha$, so $\\operatorname{tr}(\\alpha) = \\alpha_1 + \\dots + \\alpha_n$; when $\\alpha$ is transcendental its minimal polynomial\u2026","title":"Sum of the conjugates of a complex number"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/degree-2","kind":"node","lean":"MathlibExt.NumberTheory.AlgebraicIntegersWanted.AlgebraicIntegers.degree","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#degree-2","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"For a complex number $\\alpha$, the degree $\\deg(\\alpha)$ is the natural number given by the degree of the minimal polynomial of $\\alpha$ over $\\mathbb{Q}$. When $\\alpha$ is transcendental its minimal polynomial is $0$ by convention, so $\\deg(\\alpha) = 0$.","title":"Degree of a complex number over the rationals"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/is-totally-positive-2","kind":"node","lean":"MathlibExt.NumberTheory.AlgebraicIntegersWanted.AlgebraicIntegers.IsTotallyPositive","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#is-totally-positive-2","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"A complex number $\\alpha$ is totally positive if it is algebraic over $\\mathbb{Q}$ and every complex root of its minimal polynomial over $\\mathbb{Q}$, that is, every conjugate of $\\alpha$, is a positive real number.","title":"Totally positive algebraic numbers"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/lambda-sss","kind":"node","lean":"MathlibExt.NumberTheory.AlgebraicIntegersWanted.AlgebraicIntegers.lambda_SSS","parent":10798,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Schur\u2013Siegel\u2013Smyth constant "],["math","\\(\\lambda_{\\mathrm{SSS}}\\)"]],"source":"The Schur\u2013Siegel\u2013Smyth constant $\\lambda_{\\mathrm{SSS}}$"},"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#lambda-sss","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"fully_proved","summary":"Let $\\lambda_{\\mathrm{SSS}}$ be the supremum in $\\mathbb{R}$ of the set of real numbers $\\lambda$ such that, for every $\\varepsilon > 0$, only finitely many complex numbers $\\alpha$ are totally positive algebraic integers with $\\operatorname{tr}(\\alpha) < (\\lambda - \\varepsilon) \\deg(\\alpha)$, where $\\operatorname{tr}(\\alpha)$ and $\\deg(\\alpha)$ are the trace and the degree of $\\alpha$ over $\\mathbb{Q}$ (the trace\u2026","title":"The Schur\u2013Siegel\u2013Smyth constant $\\lambda_{\\mathrm{SSS}}$"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/theorem-1-1","kind":"node","lean":"MathlibExt.NumberTheory.AlgebraicIntegersWanted.AlgebraicIntegers.theorem_1_1","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#theorem-1-1","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","summary":"The Schur\u2013Siegel\u2013Smyth constant satisfies $\\lambda_{\\mathrm{SSS}} < 1.89831$; this is Theorem 1.1 of the source.","title":"Smith's theorem on the Schur\u2013Siegel\u2013Smyth trace problem"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/absolute-trace-one-le-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.absoluteTrace_one_le_wanted","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#absolute-trace-one-le-wanted","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","summary":"For every complex number $\\alpha$ that is a totally positive algebraic integer, the absolute trace of $\\alpha$ is at least $1$.","title":"Absolute trace of a totally positive algebraic integer is at least one"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/schur1918-lower-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.schur1918LowerBound_wanted","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#schur1918-lower-bound-wanted","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","summary":"The Schur\u2013Siegel\u2013Smyth trace constant $C_{86}$, an extended real number, satisfies $\\sqrt{e} \\le C_{86}$.","title":"Schur's lower bound for the Schur\u2013Siegel\u2013Smyth constant"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/smyth1984-lower-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.smyth1984LowerBound_wanted","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#smyth1984-lower-bound-wanted","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","summary":"The Schur\u2013Siegel\u2013Smyth trace constant $C_{86}$, an extended real number, satisfies $1.7719 \\le C_{86}$.","title":"Smyth's lower bound for the Schur\u2013Siegel\u2013Smyth constant"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/orloski-sardari-smith2025-lower-bound-wa","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.orloskiSardariSmith2025LowerBound_wanted","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#orloski-sardari-smith2025-lower-bound-wa","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","summary":"The Schur\u2013Siegel\u2013Smyth trace constant $C_{86}$, an extended real number, satisfies $1.80203 \\le C_{86}$.","title":"Orloski\u2013Sardari\u2013Smith lower bound for the Schur\u2013Siegel\u2013Smyth constant"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/classical-upper-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.classicalUpperBound_wanted","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#classical-upper-bound-wanted","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","summary":"The Schur\u2013Siegel\u2013Smyth trace constant $C_{86}$, an extended real number, satisfies $C_{86} \\le 2$.","title":"Classical upper bound for the Schur\u2013Siegel\u2013Smyth constant"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/smith2024-upper-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.smith2024UpperBound_wanted","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#smith2024-upper-bound-wanted","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","summary":"The Schur\u2013Siegel\u2013Smyth trace constant $C_{86}$, an extended real number, satisfies $C_{86} < 1.89831$.","title":"Smith's upper bound for the Schur\u2013Siegel\u2013Smyth trace constant"},{"id":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/sardari-orloski2024-upper-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SchurSiegelSmyth.sardariOrloski2024UpperBound_wanted","parent":10798,"result":true,"route":"../roadmap/NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem/index.html#sardari-orloski2024-upper-bound-wanted","scope":"NumberTheory/heights-mahler-measure/schur-siegel-smyth-trace-problem","status":"can_prove","summary":"The Schur\u2013Siegel\u2013Smyth trace constant $C_{86}$, an extended real number, satisfies $C_{86} \\le 1.8216$.","title":"Sardari\u2013Orloski upper bound for the Schur\u2013Siegel\u2013Smyth constant"},{"data_url":"scopes/NumberTheory/modular-forms-q-series.json","id":"NumberTheory/modular-forms-q-series","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/index.html","scope":"NumberTheory/modular-forms-q-series","status":"can_prove","summary":"Modular forms and functions, congruence subgroups, Ramanujan tau, theta functions, mock theta functions, Lambert series, partition congruences.","title":"Modular forms, q-series and partitions"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms.json","id":"NumberTheory/modular-forms-q-series/modular-forms","kind":"scope","parent":10825,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms","status":"can_prove","summary":"Modular and cusp forms, congruence subgroups, cusps, Eisenstein series congruences, Ramanujan\u2013Petersson conjecture, spectral gaps.","title":"Modular forms"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f.json","id":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f","status":"can_prove","title":"Congruences between Eisenstein series and cusp forms"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/eisenstein-congr-sum-delta-pow-mul-eisen","kind":"node","lean":"MetaMathlibExt.eisenstein_congr_sum_delta_pow_mul_eisenstein","parent":10827,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lu's congruence for the Eisenstein series "],["math","\\(G_k\\)"]],"source":"Lu's congruence for the Eisenstein series $G_k$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/index.html#eisenstein-congr-sum-delta-pow-mul-eisen","scope":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f","status":"can_prove","summary":"Let $k \\ge 12$ be even with $k \\ne 14$. For $w, n \\in \\mathbb{N}$ let $\\sigma_{-1}(n)$ be $1$ if $n = 0$ and $0$ otherwise; for $w \\ge 1$ let $\\sigma_{w-1}(0) = -B_w/(2w)$, where $B_w$ is the $w$-th Bernoulli number, and $\\sigma_{w-1}(n) = \\sum_{d \\mid n} d^{w-1}$ for $n \\ge 1$. In $\\mathbb{Q}[[q]]$ let $G_w = \\sum_{n \\ge 0} \\sigma_{w-1}(n) q^n$, let $E_w = G_w/\\sigma_{w-1}(0)$ for even $w$, and let\u2026","title":"Lu's congruence for the Eisenstein series $G_k$"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/sigma-congr-sum-tau-conv-sigma","kind":"node","lean":"MetaMathlibExt.sigma_congr_sum_tau_conv_sigma","parent":10827,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lu's congruence for the divisor sums "],["math","\\(\\sigma_{k-1}(n)\\)"]],"source":"Lu's congruence for the divisor sums $\\sigma_{k-1}(n)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/index.html#sigma-congr-sum-tau-conv-sigma","scope":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f","status":"can_prove","summary":"Let $k \\ge 12$ be even with $k \\ne 14$. For $w, n \\in \\mathbb{N}$ let $\\sigma_{-1}(n)$ be $1$ if $n = 0$ and $0$ otherwise; for $w \\ge 1$ let $\\sigma_{w-1}(0) = -B_w/(2w)$, where $B_w$ is the $w$-th Bernoulli number, and $\\sigma_{w-1}(n) = \\sum_{d \\mid n} d^{w-1}$ for $n \\ge 1$. In $\\mathbb{Q}[[q]]$ let $G_w = \\sum_{n \\ge 0} \\sigma_{w-1}(n) q^n$ and $\\Delta = q \\prod_{j \\ge 1} (1 - q^j)^{24}$, and for\u2026","title":"Lu's congruence for the divisor sums $\\sigma_{k-1}(n)$"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/eisenstein-congr-delta-mul-eisenstein-di","kind":"node","lean":"MetaMathlibExt.eisenstein_congr_delta_mul_eisenstein_dim_one","parent":10827,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lu's one-term congruence for "],["math","\\(G_k\\)"]],"source":"Lu's one-term congruence for $G_k$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/index.html#eisenstein-congr-delta-mul-eisenstein-di","scope":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f","status":"can_prove","summary":"Let $k \\in \\{12, 16, 18, 20, 22, 26\\}$. For $w, n \\in \\mathbb{N}$ let $\\sigma_{-1}(n)$ be $1$ if $n = 0$ and $0$ otherwise; for $w \\ge 1$ let $\\sigma_{w-1}(0) = -B_w/(2w)$, where $B_w$ is the $w$-th Bernoulli number, and $\\sigma_{w-1}(n) = \\sum_{d \\mid n} d^{w-1}$ for $n \\ge 1$. In $\\mathbb{Q}[[q]]$ let $G_w = \\sum_{n \\ge 0} \\sigma_{w-1}(n) q^n$, let $E_w = G_w/\\sigma_{w-1}(0)$ for even $w$, and let\u2026","title":"Lu's one-term congruence for $G_k$"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/eisenstein-congr-delta-mul-eisenstein-di-2","kind":"node","lean":"MetaMathlibExt.eisenstein_congr_delta_mul_eisenstein_dim_two","parent":10827,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lu's two-term congruence for "],["math","\\(G_k\\)"]],"source":"Lu's two-term congruence for $G_k$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/index.html#eisenstein-congr-delta-mul-eisenstein-di-2","scope":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f","status":"can_prove","summary":"Let $k \\in \\{24, 28, 30, 32, 34, 38\\}$. For $w, n \\in \\mathbb{N}$ let $\\sigma_{-1}(n)$ be $1$ if $n = 0$ and $0$ otherwise; for $w \\ge 1$ let $\\sigma_{w-1}(0) = -B_w/(2w)$, where $B_w$ is the $w$-th Bernoulli number, and $\\sigma_{w-1}(n) = \\sum_{d \\mid n} d^{w-1}$ for $n \\ge 1$. In $\\mathbb{Q}[[q]]$ let $G_w = \\sum_{n \\ge 0} \\sigma_{w-1}(n) q^n$, let $E_w = G_w/\\sigma_{w-1}(0)$ for even $w$, and let\u2026","title":"Lu's two-term congruence for $G_k$"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/eisenstein-congr-delta-mul-eisenstein-di-3","kind":"node","lean":"MetaMathlibExt.eisenstein_congr_delta_mul_eisenstein_dim_three","parent":10827,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lu's three-term congruence for "],["math","\\(G_k\\)"]],"source":"Lu's three-term congruence for $G_k$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f/index.html#eisenstein-congr-delta-mul-eisenstein-di-3","scope":"NumberTheory/modular-forms-q-series/modular-forms/congruences-between-eisenstein-series-and-cusp-f","status":"can_prove","summary":"Let $k \\in \\{36, 40, 42, 44, 46, 50\\}$. For $w, n \\in \\mathbb{N}$ let $\\sigma_{-1}(n)$ be $1$ if $n = 0$ and $0$ otherwise; for $w \\ge 1$ let $\\sigma_{w-1}(0) = -B_w/(2w)$, where $B_w$ is the $w$-th Bernoulli number, and $\\sigma_{w-1}(n) = \\sum_{d \\mid n} d^{w-1}$ for $n \\ge 1$. In $\\mathbb{Q}[[q]]$ let $G_w = \\sum_{n \\ge 0} \\sigma_{w-1}(n) q^n$, let $E_w = G_w/\\sigma_{w-1}(0)$ for even $w$, and let\u2026","title":"Lu's three-term congruence for $G_k$"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/hecke-operators.json","id":"NumberTheory/modular-forms-q-series/modular-forms/hecke-operators","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/hecke-operators/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/hecke-operators","status":"fully_proved","title":"Hecke operators"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/hecke-operators/hecke-operator-to-fun","kind":"node","lean":"AbstractHeckeOperator.HeckeOperator_toFun","parent":10833,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/hecke-operators/index.html#hecke-operator-to-fun","scope":"NumberTheory/modular-forms-q-series/modular-forms/hecke-operators","status":"fully_proved","summary":"The Hecke operator `T(g)` as a function on fixed points.","title":"Hecke operator as a function on fixed points"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/hecke-operators/hecke-operator-add-monoid-hom","kind":"node","lean":"AbstractHeckeOperator.HeckeOperator_addMonoidHom","parent":10833,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/hecke-operators/index.html#hecke-operator-add-monoid-hom","scope":"NumberTheory/modular-forms-q-series/modular-forms/hecke-operators","status":"fully_proved","summary":"The Hecke operator `T(g)` as an additive monoid homomorphism.","title":"Hecke operator as an additive homomorphism"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/hecke-operators/hecke-operator","kind":"node","lean":"AbstractHeckeOperator.HeckeOperator","parent":10833,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/hecke-operators/index.html#hecke-operator","scope":"NumberTheory/modular-forms-q-series/modular-forms/hecke-operators","status":"fully_proved","summary":"The Hecke operator `T(g)` as a linear map on fixed points.","title":"Hecke operator"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/modular-functions.json","id":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-functions/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions","status":"fully_proved","title":"Modular functions"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions/is-meromorphic-at-infty","kind":"node","lean":"ModularFunction.IsMeromorphicAtInfty","parent":10837,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-functions/index.html#is-meromorphic-at-infty","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions","status":"fully_proved","summary":"A function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ on the upper half-plane is meromorphic at infinity when there are a positive integer $N$ and a function $g \\colon \\mathbb{C} \\to \\mathbb{C}$ meromorphic at $0$ such that $f(\\tau) = g(e^{2 \\pi i \\tau / N})$ for every $\\tau \\in \\mathbb{H}$ with sufficiently large imaginary part.","title":"Meromorphy at infinity"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions/is-meromorphic-at-cusps","kind":"node","lean":"ModularFunction.IsMeromorphicAtCusps","parent":10837,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-functions/index.html#is-meromorphic-at-cusps","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions","status":"fully_proved","summary":"A function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ on the upper half-plane is meromorphic at the cusps when, for every $\\gamma \\in \\mathrm{SL}(2, \\mathbb{Z})$, the function $\\tau \\mapsto f(\\gamma \\cdot \\tau)$ is meromorphic at infinity, where $\\gamma$ acts on $\\mathbb{H}$ by M\u00f6bius transformations.","title":"Meromorphy at the cusps"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions/is-meromorphic-on-h","kind":"node","lean":"ModularFunction.IsMeromorphicOnH","parent":10837,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-functions/index.html#is-meromorphic-on-h","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions","status":"fully_proved","summary":"A function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ on the upper half-plane is meromorphic when there is a function $g \\colon \\mathbb{C} \\to \\mathbb{C}$ that is meromorphic on the open set $\\{z \\in \\mathbb{C} : \\operatorname{Im} z > 0\\}$ and satisfies $g(\\tau) = f(\\tau)$ for every $\\tau \\in \\mathbb{H}$.","title":"Meromorphy on the upper half-plane"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions/is-invariant-under","kind":"node","lean":"ModularFunction.IsInvariantUnder","parent":10837,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-functions/index.html#is-invariant-under","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions","status":"fully_proved","summary":"Let $\\Gamma$ be a subgroup of $\\mathrm{SL}(2, \\mathbb{Z})$. A function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ on the upper half-plane is invariant under $\\Gamma$ when $f(\\gamma \\cdot \\tau) = f(\\tau)$ for every $\\gamma \\in \\Gamma$ and every $\\tau \\in \\mathbb{H}$, where $\\gamma$ acts on $\\mathbb{H}$ by M\u00f6bius transformations.","title":"Invariance under a subgroup of the modular group"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions/is-modular-function","kind":"node","lean":"IsModularFunction","parent":10837,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-functions/index.html#is-modular-function","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-functions","status":"fully_proved","summary":"Let $\\Gamma$ be a subgroup of $\\mathrm{SL}(2, \\mathbb{Z})$. A function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ on the upper half-plane is a modular function for $\\Gamma$ when it is meromorphic on $\\mathbb{H}$, invariant under $\\Gamma$, and meromorphic at the cusps.","title":"Modular functions"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups.json","id":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups","status":"fully_proved","title":"Modular group and congruence subgroups"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/mem-sup-zpowers-iff-of-central-sq-eq-one","kind":"node","lean":"Subgroup.mem_sup_zpowers_iff_of_central_sq_eq_one","parent":10843,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/index.html#mem-sup-zpowers-iff-of-central-sq-eq-one","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups","status":"fully_proved","summary":"Membership in the join of a subgroup with the cyclic subgroup generated by a central involution.","title":"Membership in a subgroup joined with a central involution"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/exists-mem-gamma-map-eq-of-not-dvd","kind":"node","lean":"CongruenceSubgroup.exists_mem_Gamma_map_eq_of_not_dvd","parent":10843,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/index.html#exists-mem-gamma-map-eq-of-not-dvd","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups","status":"fully_proved","summary":"If `p` does not divide `N`, then reduction modulo `p` maps `\u0393(N)` onto `SL(2, ZMod p)`.","title":"Reduction mod p maps \u0393(N) onto SL(2, \u2124/p\u2124) when p does not divide N"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/neg-one-mem-gamma1-iff","kind":"node","lean":"CongruenceSubgroup.neg_one_mem_Gamma1_iff","parent":10843,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/index.html#neg-one-mem-gamma1-iff","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups","status":"fully_proved","summary":"`-1` lies in `\u0393\u2081(N)` if and only if `N \u2264 2`.","title":"\u22121 lies in \u0393\u2081(N) exactly when N \u2264 2"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/exists-eq-conj-t-zpow-of-trace-sq-eq-fou","kind":"node","lean":"ModularGroup.exists_eq_conj_T_zpow_of_trace_sq_eq_four","parent":10843,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/index.html#exists-eq-conj-t-zpow-of-trace-sq-eq-fou","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups","status":"fully_proved","summary":"Every element of `SL(2, \u2124)` whose trace squares to `4` is, up to a sign, conjugate to a power of the translation `T`.","title":"Parabolic elements of SL(2, \u2124) are conjugate to \u00b1 powers of T"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/one-mem-strict-periods-gamma0","kind":"node","lean":"CongruenceSubgroup.one_mem_strictPeriods_Gamma0","parent":10843,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups/index.html#one-mem-strict-periods-gamma0","scope":"NumberTheory/modular-forms-q-series/modular-forms/modular-group-and-congruence-subgroups","status":"fully_proved","summary":"`1` is a strict period of `\u0393\u2080(N)`: the translation matrix `T` lies in `\u0393\u2080(N)`, and the strict periods then contain the integer multiples of `1`.","title":"1 is a strict period of \u0393\u2080(N)"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function.json","id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function","status":"can_prove","title":"Ramanujan tau function"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function/untitled","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanTauWanted.NumberTheory.RamanujanTauProblem.\u03c4","parent":10849,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function/index.html#untitled","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function","status":"fully_proved","summary":"For a natural number $n$, $\\tau(n)$ is the complex number given by the coefficient of $q^n$ in the $q$-expansion of the modular discriminant $\\Delta = \\eta^{24}$, where $\\eta$ is the Dedekind eta function and $q = e^{2\\pi i z}$.","title":"Ramanujan \u03c4-function"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function/one","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanTauWanted.NumberTheory.RamanujanTauProblem.\u03c4_one","parent":10849,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function/index.html#one","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function","status":"fully_proved","summary":"The Ramanujan \u03c4-function satisfies $\\tau(1) = 1$: the coefficient of $q$ in the $q$-expansion of the modular discriminant $\\Delta$ is $1$.","title":"Normalisation of the Ramanujan \u03c4-function"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function/ramanujan-petersson","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanTauWanted.NumberTheory.RamanujanTauProblem.ramanujan_petersson","parent":10849,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function/index.html#ramanujan-petersson","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-tau-function","status":"can_prove","summary":"For every prime $p$, the Ramanujan \u03c4-function satisfies $|\\tau(p)| \\le 2p^{11/2}$, where $|\\cdot|$ is the absolute value of a complex number.","title":"Ramanujan\u2013Petersson conjecture for \u03c4 (Deligne's theorem)"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre.json","id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"can_prove","title":"Rankin\u2013Selberg coefficients in arithmetic progressions"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/cusp-form-normalized-coeff","kind":"node","lean":"MetaMathlibExt.cuspFormNormalizedCoeff","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#cusp-form-normalized-coeff","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"fully_proved","summary":"Let $f$ be a cusp form of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with $q$-expansion $f(z) = \\sum_{n \\ge 0} a_f(n) e(nz)$, where $e(z) = e^{2\\pi i z}$. For each natural number $n$, the normalized Fourier coefficient of $f$ is the complex number $\\lambda_f(n) = a_f(n) / n^{(k-1)/2}$, where $n^{(k-1)/2}$ is the real power.","title":"Normalized Fourier coefficients of a cusp form"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/cusp-form-normalized-coeff-one","kind":"node","lean":"MetaMathlibExt.cuspFormNormalizedCoeff_one","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#cusp-form-normalized-coeff-one","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"fully_proved","summary":"Let $f$ be a cusp form of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with $q$-expansion $f(z) = \\sum_{n \\ge 0} a_f(n) e^{2\\pi i n z}$, and let $\\lambda_f$ be its normalized Fourier coefficients. Then $\\lambda_f(1) = a_f(1)$, the coefficient of $e^{2\\pi i z}$ in the $q$-expansion of $f$.","title":"Normalized cusp form coefficient at one"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/is-normalized-hecke-eigenform","kind":"node","lean":"MetaMathlibExt.IsNormalizedHeckeEigenform","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#is-normalized-hecke-eigenform","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"fully_proved","summary":"Let $f$ be a cusp form of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with normalized Fourier coefficients $\\lambda_f$. Then $f$ is a normalized Hecke eigenform when $\\lambda_f(1) = 1$ and, for all positive integers $m$ and $n$, $\\lambda_f(m) \\lambda_f(n) = \\sum_{d \\mid \\gcd(m, n)} \\lambda_f(mn/d^2)$, the sum running over the positive divisors $d$ of $\\gcd(m, n)$. The set of these forms is denoted\u2026","title":"Normalized Hecke eigenform"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/normalized-coeff-prime-sq","kind":"node","lean":"MetaMathlibExt.IsNormalizedHeckeEigenform.normalizedCoeff_prime_sq","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#normalized-coeff-prime-sq","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"fully_proved","summary":"Let $f$ be a normalized Hecke eigenform of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with normalized Fourier coefficients $\\lambda_f$. Then $\\lambda_f(p)^2 = \\lambda_f(p^2) + 1$ for every prime $p$.","title":"Hecke relation at the square of a prime"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/rankin-selberg-coeff","kind":"node","lean":"MetaMathlibExt.rankinSelbergCoeff","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#rankin-selberg-coeff","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"fully_proved","summary":"Let $f$ be a cusp form of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with normalized Fourier coefficients $\\lambda_f$. The Rankin\u2013Selberg coefficients of $f$ form the arithmetic function $\\lambda_{f \\times f}$ with $\\lambda_{f \\times f}(0) = 0$ and, for $n \\ge 1$, $\\lambda_{f \\times f}(n) = \\sum_{l^2 m = n} \\lambda_f(m)^2$, the sum running over the pairs of positive integers $(l, m)$ with\u2026","title":"Rankin\u2013Selberg coefficients"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/rankin-selberg-coeff-apply","kind":"node","lean":"MetaMathlibExt.rankinSelbergCoeff_apply","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#rankin-selberg-coeff-apply","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"fully_proved","summary":"Let $f$ be a cusp form of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with normalized Fourier coefficients $\\lambda_f$ and Rankin\u2013Selberg coefficients $\\lambda_{f \\times f}$. For every natural number $n$, $\\lambda_{f \\times f}(n) = \\sum_d \\lambda_f(n/d)^2$, where $d$ runs over the positive divisors of $n$ that are perfect squares (so the sum is empty when $n = 0$).","title":"Rankin\u2013Selberg coefficients as a sum over square divisors"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/rankin-selberg-coeff-one","kind":"node","lean":"MetaMathlibExt.rankinSelbergCoeff_one","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#rankin-selberg-coeff-one","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"fully_proved","summary":"Let $f$ be a cusp form of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with normalized Fourier coefficients $\\lambda_f$ and Rankin\u2013Selberg coefficients $\\lambda_{f \\times f}$. Then $\\lambda_{f \\times f}(1) = \\lambda_f(1)^2$.","title":"Rankin\u2013Selberg coefficient at one"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/rankin-selberg-coeff-prime","kind":"node","lean":"MetaMathlibExt.rankinSelbergCoeff_prime","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#rankin-selberg-coeff-prime","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"fully_proved","summary":"Let $f$ be a cusp form of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with normalized Fourier coefficients $\\lambda_f$ and Rankin\u2013Selberg coefficients $\\lambda_{f \\times f}$. Then $\\lambda_{f \\times f}(p) = \\lambda_f(p)^2$ for every prime $p$.","title":"Rankin\u2013Selberg coefficients at primes"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/sum-cusp-form-normalized-coeff-pow-sq-ar","kind":"node","lean":"MetaMathlibExt.sum_cuspFormNormalizedCoeff_pow_sq_arithmeticProgression","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#sum-cusp-form-normalized-coeff-pow-sq-ar","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"can_prove","summary":"Let $f$ be a normalized Hecke eigenform of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with normalized Fourier coefficients $\\lambda_f$, let $j \\ge 2$ be an integer, and put $\\delta = 92/597$ if $j = 2$ and $\\delta = 92/(69(j-1)(j+3) + 247)$ if $j \\ge 3$. Then there are real numbers $c_q > 0$, one for each prime $q$, with the following property: for every $\\varepsilon > 0$ there is a real constant\u2026","title":"Sums of squared normalized coefficients at perfect powers in arithmetic progressions"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/sum-rankin-selberg-coeff-sq-arithmetic-p","kind":"node","lean":"MetaMathlibExt.sum_rankinSelbergCoeff_sq_arithmeticProgression","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#sum-rankin-selberg-coeff-sq-arithmetic-p","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"can_prove","summary":"Let $f$ be a level-one cusp form of weight $k$ that is a normalized Hecke eigenform, let $\\lambda_{f\\times f}$ be its Rankin\u2013Selberg coefficients, let $\\varphi$ be Euler's totient function, and put $\\eta = 23/261$. There are real polynomials $P_q$, one for each prime $q$, each of degree $1$ with positive leading coefficient, such that for every $\\varepsilon > 0$ there is a constant $C$ with","title":"Hua's theorem on squared Rankin\u2013Selberg coefficients in arithmetic progressions"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/sum-rankin-selberg-coeff-pow-arithmetic","kind":"node","lean":"MetaMathlibExt.sum_rankinSelbergCoeff_pow_arithmeticProgression","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#sum-rankin-selberg-coeff-pow-arithmetic","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"can_prove","summary":"Let $f$ be a normalized Hecke eigenform of weight $k \\in \\mathbb{Z}$ for $\\mathrm{SL}_2(\\mathbb{Z})$, with Rankin\u2013Selberg coefficients $\\lambda_{f \\times f}$, let $w \\ge 2$ be an integer, write $\\lambda_{w, f \\times f}$ for the $w$-fold Dirichlet convolution power of $\\lambda_{f \\times f}$, and put $\\vartheta_w = 92/(247w + 10)$. Then there are real polynomials $P_q$, one for each prime $q$, each of degree $w - 1$\u2026","title":"Sums of Rankin\u2013Selberg convolution powers in arithmetic progressions"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/sum-rankin-selberg-coeff-sq-mul-primitiv","kind":"node","lean":"MetaMathlibExt.sum_rankinSelbergCoeff_sq_mul_primitive_character","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#sum-rankin-selberg-coeff-sq-mul-primitiv","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"can_prove","summary":"Let $f$ be a level-one cusp form of weight $k$ that is a normalized Hecke eigenform, let $\\lambda_{f\\times f}$ be its Rankin\u2013Selberg coefficients, and put $\\eta = 23/261$. For every $\\varepsilon > 0$ there is a constant $C$ such that for every prime $q$, every primitive Dirichlet character $\\chi$ modulo $q$ and every real $x \\ge 1$ with $q \\le x^{\\eta}$,","title":"Sums of squared Rankin\u2013Selberg coefficients twisted by a primitive character"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/sum-rankin-selberg-coeff-sq-mul-principa","kind":"node","lean":"MetaMathlibExt.sum_rankinSelbergCoeff_sq_mul_principal_character","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#sum-rankin-selberg-coeff-sq-mul-principa","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"can_prove","summary":"Let $f$ be a level-one cusp form of weight $k$ that is a normalized Hecke eigenform, let $\\lambda_{f\\times f}$ be its Rankin\u2013Selberg coefficients, and for a prime $q$ let $\\chi_0$ be the principal Dirichlet character modulo $q$. There are real polynomials $P_q$, one for each prime $q$, each of degree $1$ with positive leading coefficient, such that for every $\\varepsilon > 0$ there is a constant $C$ with","title":"Sums of squared Rankin\u2013Selberg coefficients twisted by the principal character"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/sum-rankin-selberg-coeff-pow-mul-primiti","kind":"node","lean":"MetaMathlibExt.sum_rankinSelbergCoeff_pow_mul_primitive_character","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#sum-rankin-selberg-coeff-pow-mul-primiti","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"can_prove","summary":"Let $f$ be a level-one cusp form of weight $k$ that is a normalized Hecke eigenform, let $\\lambda_{f\\times f}$ be its Rankin\u2013Selberg coefficients, let $w \\ge 2$, let $\\lambda_{w,f\\times f}$ be the $w$-th power of $\\lambda_{f\\times f}$ under Dirichlet convolution, and put $\\vartheta_w = 92/(247w + 10)$. For every $\\varepsilon > 0$ there is a constant $C$ such that for every prime $q$, every primitive Dirichlet\u2026","title":"Sums of Rankin\u2013Selberg convolution powers twisted by a primitive character"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/sum-rankin-selberg-coeff-pow-mul-princip","kind":"node","lean":"MetaMathlibExt.sum_rankinSelbergCoeff_pow_mul_principal_character","parent":10853,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre/index.html#sum-rankin-selberg-coeff-pow-mul-princip","scope":"NumberTheory/modular-forms-q-series/modular-forms/rankin-selberg-coefficients-in-arithmetic-progre","status":"can_prove","summary":"Let $f$ be a level-one cusp form of weight $k$ that is a normalized Hecke eigenform, let $\\lambda_{f\\times f}$ be its Rankin\u2013Selberg coefficients, let $w \\ge 2$, let $\\lambda_{w,f\\times f}$ be the $w$-th power of $\\lambda_{f\\times f}$ under Dirichlet convolution, and for a prime $q$ let $\\chi_0$ be the principal Dirichlet character modulo $q$. There are real polynomials $P_q$, one for each prime $q$, each of degree\u2026","title":"Sums of Rankin\u2013Selberg convolution powers twisted by the principal character"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture.json","id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"can_prove","title":"Selberg's eigenvalue conjecture"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/is-congruence-subgroup","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.IsCongruenceSubgroup","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#is-congruence-subgroup","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"A subgroup $\\Gamma$ of $\\mathrm{SL}(2, \\mathbb{Z})$ is a congruence subgroup if there is a positive integer $N$ such that every $\\gamma \\in \\mathrm{SL}(2, \\mathbb{Z})$ whose entries are congruent modulo $N$ to those of the identity matrix lies in $\\Gamma$; that is, $\\Gamma$ contains the principal congruence subgroup of level $N$.","title":"Congruence subgroup of the modular group"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/is-congruence-subgroup-top","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.isCongruenceSubgroup_top","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#is-congruence-subgroup-top","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"The full group $\\mathrm{SL}(2, \\mathbb{Z})$, as a subgroup of itself, is a congruence subgroup.","title":"The modular group is a congruence subgroup"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/ext","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.ext","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#ext","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For a function $f$ from the upper half plane to $\\mathbb{C}$, its zero extension is the function from $\\mathbb{C}$ to $\\mathbb{C}$ that takes the value $f(z)$ at every $z$ with $\\operatorname{Im} z > 0$ and the value $0$ at every other point.","title":"Zero extension from the upper half plane"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/partial-re","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.partialRe","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#partial-re","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For a function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ and a point $z \\in \\mathbb{H}$, $\\partial_x f(z)$ is the real Fr\u00e9chet derivative at $z$ of the zero extension of $f$ to $\\mathbb{C}$, applied to the vector $1$. When the zero extension is real-differentiable at $z$ this is the partial derivative of $f$ with respect to $x = \\operatorname{Re} z$; otherwise the value is $0$.","title":"Partial derivative in the real direction"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/partial-im","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.partialIm","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#partial-im","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For a function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ and a point $z \\in \\mathbb{H}$, $\\partial_y f(z)$ is the real Fr\u00e9chet derivative at $z$ of the zero extension of $f$ to $\\mathbb{C}$, applied to the vector $i$. When the zero extension is real-differentiable at $z$ this is the partial derivative of $f$ with respect to $y = \\operatorname{Im} z$; otherwise the value is $0$.","title":"Partial derivative in the imaginary direction"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/hyperbolic-laplacian","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.hyperbolicLaplacian","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#hyperbolic-laplacian","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For a function $f$ from the upper half plane to $\\mathbb{C}$ and a point $z = x + iy$ of the upper half plane, the hyperbolic Laplacian of $f$ at $z$ is","title":"Hyperbolic Laplacian"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/hyperbolic-laplacian-const","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.hyperbolicLaplacian_const","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#hyperbolic-laplacian-const","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For every complex number $c$ and every point $z$ of the upper half plane, the hyperbolic Laplacian of the constant function with value $c$ at $z$ is $0$.","title":"The hyperbolic Laplacian kills constants"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/hyperbolic-laplacian-im-sq","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.hyperbolicLaplacian_im_sq","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#hyperbolic-laplacian-im-sq","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"Let $f$ be the function on the upper half plane given by $f(u) = (\\operatorname{Im} u)^2$, viewed as a complex number. Then for every point $z$ of the upper half plane the hyperbolic Laplacian of $f$ at $z$ equals $-2 (\\operatorname{Im} z)^2$.","title":"Hyperbolic Laplacian of the squared imaginary part"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/is-smooth","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.IsSmooth","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#is-smooth","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"A function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ is smooth (of order two) when it is the restriction of a function $F \\colon \\mathbb{C} \\to \\mathbb{C}$ that is twice continuously differentiable, in the real sense, on the open set $\\{z \\in \\mathbb{C} : \\operatorname{Im} z > 0\\}$.","title":"Twice continuously differentiable functions on the upper half-plane"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/fundamental-domain","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.fundamentalDomain","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#fundamental-domain","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"The standard fundamental domain for $\\mathrm{SL}(2, \\mathbb{Z})$ is the set of points $z$ of the upper half plane with $|\\operatorname{Re} z| \\le 1/2$ and $|z| \\ge 1$.","title":"Standard fundamental domain of the modular group"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/is-coset-cover","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.IsCosetCover","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#is-coset-cover","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"A finite set $s \\subseteq \\mathrm{SL}(2, \\mathbb{Z})$ covers $\\mathrm{SL}(2, \\mathbb{Z})$ by cosets of a subgroup $\\Gamma$ when every $g \\in \\mathrm{SL}(2, \\mathbb{Z})$ can be written as $g = \\gamma a$ with $\\gamma \\in \\Gamma$ and $a \\in s$, that is, when the cosets $\\Gamma a$ with $a \\in s$ cover $\\mathrm{SL}(2, \\mathbb{Z})$. The elements of $s$ need not lie in distinct cosets.","title":"Finite coset covers of the modular group"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/is-l2-quotient","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.IsL2Quotient","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#is-l2-quotient","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For a subgroup $\\Gamma$ of $\\mathrm{SL}(2, \\mathbb{Z})$, a function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ is square-integrable on $\\Gamma \\backslash \\mathbb{H}$ when there is a finite set $s \\subseteq \\mathrm{SL}(2, \\mathbb{Z})$ whose cosets $\\Gamma a$, $a \\in s$, cover $\\mathrm{SL}(2, \\mathbb{Z})$, such that","title":"Square-integrable functions on a modular quotient"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/is-eigenvalue","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.IsEigenvalue","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#is-eigenvalue","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"Let $\\Gamma$ be a subgroup of $\\mathrm{SL}(2, \\mathbb{Z})$. A real number $\\lambda$ is an eigenvalue of the hyperbolic Laplacian $\\Delta$ for $\\Gamma$ when there is a function $f \\colon \\mathbb{H} \\to \\mathbb{C}$ on the upper half plane $\\mathbb{H}$ such that $f$ is not identically zero, $f$ is the restriction of a function on $\\mathbb{C}$ that is twice continuously differentiable in the real sense on the upper\u2026","title":"Eigenvalues of the hyperbolic Laplacian on a modular quotient"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/exceptional-gaps","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.exceptionalGaps","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#exceptional-gaps","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For a subgroup $\\Gamma$ of $\\mathrm{SL}(2, \\mathbb{Z})$, the set of gap contributions of $\\Gamma$ is the set of real numbers consisting of $0$ together with $\\sqrt{1/4 - \\lambda}$ for every real eigenvalue $\\lambda$ of the hyperbolic Laplacian for $\\Gamma$ with $0 < \\lambda < 1/4$ (an exceptional eigenvalue).","title":"Gap contributions of a subgroup of the modular group"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/exceptional-gap","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.exceptionalGap","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#exceptional-gap","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For a subgroup $\\Gamma$ of $\\mathrm{SL}(2, \\mathbb{Z})$, the gap parameter $\\theta(\\Gamma)$ is the supremum of the set of gap contributions of $\\Gamma$: the number $0$ together with $\\sqrt{1/4 - \\lambda}$ for every real eigenvalue $\\lambda$ of the hyperbolic Laplacian for $\\Gamma$ with $0 < \\lambda < 1/4$.","title":"Gap parameter of a subgroup of the modular group"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/selberg-constant","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.selbergConstant","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#selberg-constant","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"Selberg's constant $C_{61}$ is the supremum of the gap parameters $\\theta(\\Gamma)$ over all congruence subgroups $\\Gamma$ of $\\mathrm{SL}(2, \\mathbb{Z})$, where $\\theta(\\Gamma)$ is the supremum of $0$ and of the numbers $\\sqrt{1/4 - \\lambda}$ for the eigenvalues $\\lambda$ of $\\Gamma$ with $0 < \\lambda < 1/4$.","title":"Selberg's congruence spectral-gap constant"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/selberg-eigenvalue-conjecture","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.SelbergEigenvalueConjecture","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#selberg-eigenvalue-conjecture","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"No congruence subgroup $\\Gamma$ of $\\mathrm{SL}(2, \\mathbb{Z})$ has an eigenvalue $\\lambda$ with $0 < \\lambda < 1/4$: for every congruence subgroup $\\Gamma$ and every eigenvalue $\\lambda$ of $\\Gamma$, either $\\lambda \\le 0$ or $\\lambda \\ge 1/4$.","title":"Selberg's eigenvalue conjecture"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/sqrt-le-exceptional-gap","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.sqrt_le_exceptionalGap","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#sqrt-le-exceptional-gap","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"Let $\\Gamma$ be a subgroup of $\\mathrm{SL}(2, \\mathbb{Z})$ and let $\\lambda$ be a real eigenvalue of the hyperbolic Laplacian on $L^2(\\Gamma \\backslash \\mathbb{H})$ with $0 < \\lambda < 1/4$. Then $\\sqrt{1/4 - \\lambda}$ is at most the gap parameter $\\theta(\\Gamma)$ of $\\Gamma$.","title":"Exceptional eigenvalues bound the gap parameter"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/exceptional-gap-le-selberg-constant","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.exceptionalGap_le_selbergConstant","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#exceptional-gap-le-selberg-constant","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"For every congruence subgroup $\\Gamma$ of $\\mathrm{SL}(2, \\mathbb{Z})$, the gap parameter $\\theta(\\Gamma)$ is at most the Selberg constant $C_{61}$.","title":"Congruence gaps are bounded by Selberg's constant"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/selberg-constant-le-half","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.selbergConstant_le_half","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#selberg-constant-le-half","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"Selberg's constant $C_{61}$ satisfies $C_{61} \\le 1/2$.","title":"Selberg's constant is at most one half"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/sqrt-le-selberg-constant","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.sqrt_le_selbergConstant","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#sqrt-le-selberg-constant","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"Let $\\Gamma$ be a congruence subgroup of $\\mathrm{SL}(2, \\mathbb{Z})$ and let $\\lambda$ be a real eigenvalue of the hyperbolic Laplacian on $L^2(\\Gamma \\backslash \\mathbb{H})$ with $0 < \\lambda < 1/4$. Then $\\sqrt{1/4 - \\lambda}$ is at most Selberg's constant $C_{61}$.","title":"Exceptional eigenvalues bound Selberg's constant"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/selberg-constant-eq-zero-iff","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.selbergConstant_eq_zero_iff","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#selberg-constant-eq-zero-iff","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"fully_proved","summary":"Selberg's constant $C_{61}$ equals $0$ if and only if Selberg's eigenvalue conjecture holds, that is, if and only if no congruence subgroup of $\\mathrm{SL}(2, \\mathbb{Z})$ has an eigenvalue $\\lambda$ with $0 < \\lambda < 1/4$.","title":"Selberg's constant vanishes exactly under Selberg's eigenvalue conjecture"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/selberg-three-sixteenths-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.selberg_threeSixteenths_wanted","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#selberg-three-sixteenths-wanted","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"can_prove","summary":"Let $\\Gamma$ be a congruence subgroup of $\\mathrm{SL}(2, \\mathbb{Z})$ and let $\\lambda$ be a real eigenvalue of the hyperbolic Laplacian $\\Delta$ for $\\Gamma$, that is, $\\Delta f = \\lambda f$ for some nonzero, twice continuously differentiable, $\\Gamma$-invariant function $f$ on the upper half plane that is square-integrable on $\\Gamma \\backslash \\mathbb{H}$. Then $\\lambda$ does not lie in the open interval\u2026","title":"Selberg's 3/16 theorem"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/selberg-upper-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.selbergUpperBound_wanted","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#selberg-upper-bound-wanted","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"can_prove","summary":"Selberg's congruence spectral-gap constant $C_{61}$ is at most $1/4$.","title":"Selberg's bound on the spectral-gap constant"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/kim-sarnak975-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.kimSarnak975_wanted","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#kim-sarnak975-wanted","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"can_prove","summary":"Let $\\Gamma$ be a congruence subgroup of $\\mathrm{SL}(2, \\mathbb{Z})$ and let $\\lambda$ be a real eigenvalue of the hyperbolic Laplacian $\\Delta$ for $\\Gamma$, that is, $\\Delta f = \\lambda f$ for some nonzero, twice continuously differentiable, $\\Gamma$-invariant function $f$ on the upper half plane that is square-integrable on $\\Gamma \\backslash \\mathbb{H}$. Then $\\lambda$ does not lie in the open interval\u2026","title":"Kim\u2013Sarnak eigenvalue bound"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/kim-sarnak-upper-bound-wanted","kind":"node","lean":"MathlibExt.NumberTheory.SelbergSpectralGap.kimSarnakUpperBound_wanted","parent":10869,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture/index.html#kim-sarnak-upper-bound-wanted","scope":"NumberTheory/modular-forms-q-series/modular-forms/selberg-s-eigenvalue-conjecture","status":"can_prove","summary":"Selberg's congruence spectral-gap constant $C_{61}$ is at most $7/64$.","title":"Kim\u2013Sarnak bound on the spectral-gap constant"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture.json","id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","title":"Ramanujan\u2013Petersson conjecture"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/local-exponent-family","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.LocalExponentFamily","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#local-exponent-family","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"A local-exponent family indexed by a type $\\iota$ is a function $m : \\iota \\to \\mathbb{R}$, the local Langlands exponents, with $0 \\le m(i) \\le 1/2$ for every $i \\in \\iota$.","title":"Family of local exponents"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/exponent-set","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.exponentSet","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#exponent-set","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For a local-exponent family $F$ indexed by a type $\\iota$, with local exponents $m(i)$ for $i \\in \\iota$, the exponent set of $F$ is the set of real numbers $\\{0\\} \\cup \\{m(i) : i \\in \\iota\\}$.","title":"Exponent set of a local-exponent family"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/ramanujan-exponent-of","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.ramanujanExponentOf","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#ramanujan-exponent-of","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"The Ramanujan exponent of a local-exponent family $F$ with local exponents $m(i)$, $i \\in \\iota$, is the supremum of the set $\\{0\\} \\cup \\{m(i) : i \\in \\iota\\}$. It is the supremum of the local exponents when $\\iota$ is nonempty, and $0$ when $\\iota$ is empty.","title":"Ramanujan exponent of a local-exponent family"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/is-admissible","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.IsAdmissible","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#is-admissible","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"A local-exponent family $F$ is admissible at a real number $\\delta$ (Hypothesis $H_2(\\delta)$ for $F$) when every local exponent $m_i$ of $F$ ($i \\in \\iota$) is at most $\\delta$.","title":"Admissibility of an exponent for a local-exponent family"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/generalized-ramanujan-for","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.GeneralizedRamanujanFor","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#generalized-ramanujan-for","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"A local-exponent family $F$ satisfies the Generalized Ramanujan conclusion when every local exponent $m_i$ of $F$ ($i \\in \\iota$) is zero.","title":"Generalized Ramanujan conjecture for a family"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/exponent-le-ramanujan-exponent-of","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.exponent_le_ramanujanExponentOf","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#exponent-le-ramanujan-exponent-of","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For every local-exponent family $F$ and every index $i$, the local exponent $m_i$ of $F$ at $i$ is at most the Ramanujan exponent of $F$.","title":"Local exponents are bounded by the Ramanujan exponent"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/is-admissible-ramanujan-exponent-of","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.isAdmissible_ramanujanExponentOf","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#is-admissible-ramanujan-exponent-of","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For every local-exponent family $F$, $F$ is admissible at its Ramanujan exponent: every local exponent of $F$ is at most the Ramanujan exponent of $F$.","title":"The Ramanujan exponent is admissible"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/ramanujan-exponent-of-le-of-is-admissibl","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.ramanujanExponentOf_le_of_isAdmissible","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#ramanujan-exponent-of-le-of-is-admissibl","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"Let $F$ be a local-exponent family and let $\\delta \\ge 0$ be a real number. If $\\delta$ is admissible for $F$, that is, every local exponent of $F$ is at most $\\delta$, then the Ramanujan exponent of $F$ is at most $\\delta$.","title":"Admissible exponents bound the Ramanujan exponent"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/is-admissible-iff-ramanujan-exponent-of","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.isAdmissible_iff_ramanujanExponentOf_le","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#is-admissible-iff-ramanujan-exponent-of","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For every local-exponent family $F$ and every real $\\delta \\ge 0$, $F$ is admissible at $\\delta$ if and only if the Ramanujan exponent of $F$ is at most $\\delta$.","title":"Admissibility is a bound by the Ramanujan exponent"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/ramanujan-exponent-of-eq-zero-iff","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.ramanujanExponentOf_eq_zero_iff","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#ramanujan-exponent-of-eq-zero-iff","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For every local-exponent family $F$, the Ramanujan exponent of $F$ is $0$ if and only if $F$ satisfies the generalized Ramanujan conclusion, that is, every local exponent of $F$ is $0$.","title":"The Ramanujan exponent vanishes exactly in the tempered case"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/admissible-exponents","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.admissibleExponents","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#admissible-exponents","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For a local-exponent family $F$ with local exponents $m_i$ ($i \\in \\iota$), the set of admissible exponents of $F$ is the set of real numbers $\\delta \\ge 0$ such that $F$ is admissible at $\\delta$, that is, $m_i \\le \\delta$ for every index $i$.","title":"Admissible exponents of a local-exponent family"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/ramanujan-exponent-of-eq-s-inf-admissibl","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.ramanujanExponentOf_eq_sInf_admissibleExponents","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#ramanujan-exponent-of-eq-s-inf-admissibl","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For every local-exponent family $F$, the Ramanujan exponent of $F$ equals the infimum of the admissible exponents of $F$, that is, of the set of real numbers $\\delta \\ge 0$ such that every local exponent of $F$ is at most $\\delta$.","title":"The Ramanujan exponent is the infimum of the admissible exponents"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/is-satake-pair","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.IsSatakePair","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#is-satake-pair","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For real numbers $N$ and $t$, a pair $(\\alpha, \\beta)$ of complex numbers is in arithmetic Satake normalization with local exponent $t$ at residue cardinality $N$ when $|\\alpha\\beta| = 1$, $|\\alpha| = N^t$ and $|\\beta| = N^{-t}$.","title":"Satake pairs with a given local exponent"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/hecke-norm-le-two-mul-rpow-of-is-satake","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.hecke_norm_le_two_mul_rpow_of_isSatakePair","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#hecke-norm-le-two-mul-rpow-of-is-satake","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"Let $N \\ge 1$ and $t, \\delta$ be real numbers with $0 \\le t \\le \\delta$. If complex numbers $\\alpha, \\beta$ form a Satake pair with local exponent $t$ at residue cardinality $N$, then $|\\alpha + \\beta| \\le 2 N^\\delta$.","title":"Hecke bound from a Satake pair"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/hecke-norm-le-two-mul-residue-norm-rpow","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.hecke_norm_le_two_mul_residueNorm_rpow_of_isSatakePair","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#hecke-norm-le-two-mul-residue-norm-rpow","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"Let $K$ be a number field, $v$ a finite place of $K$ with residue norm $N(v)$, and $t, \\delta$ real numbers with $0 \\le t \\le \\delta$. If complex numbers $\\alpha, \\beta$ form a Satake pair with local exponent $t$ at residue cardinality $N(v)$, then $|\\alpha + \\beta| \\le 2 N(v)^\\delta$.","title":"Hecke bound from a Satake pair at a finite place"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/selberg-witness","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.SelbergWitness","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#selberg-witness","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"The type of pairs $(\\Gamma, \\lambda)$ where $\\Gamma$ is a congruence subgroup of $\\mathrm{SL}_2(\\mathbb{Z})$ and $\\lambda$ is a real eigenvalue of the hyperbolic Laplacian on $L^2(\\Gamma \\backslash \\mathbb{H})$ with $0 < \\lambda < 1/4$.","title":"Exceptional Selberg eigenvalue witnesses"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/selberg-gap","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.selbergGap","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#selberg-gap","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"For an exceptional-eigenvalue witness, a congruence subgroup $\\Gamma$ of $\\mathrm{SL}_2(\\mathbb{Z})$ with a Laplace eigenvalue $\\lambda \\in (0, 1/4)$, the Selberg gap is $\\sqrt{1/4 - \\lambda}$: the archimedean local exponent $\\theta$ forced by $\\lambda$, written as $\\lambda = 1/4 - \\theta^2$.","title":"Selberg gap of an exceptional eigenvalue"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/selberg-family","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.selbergFamily","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#selberg-family","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"The Selberg family is the local-exponent family indexed by the exceptional-eigenvalue witnesses, the pairs $(\\Gamma, \\lambda)$ of a congruence subgroup $\\Gamma$ of $\\mathrm{SL}_2(\\mathbb{Z})$ and a Laplace eigenvalue $\\lambda \\in (0, 1/4)$ of $\\Gamma$, whose local exponent at $(\\Gamma, \\lambda)$ is $\\sqrt{1/4 - \\lambda}$.","title":"Selberg family of archimedean exponents"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/dominates-selberg","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.DominatesSelberg","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#dominates-selberg","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"A local-exponent family $F$ with local exponents $m_i$ ($i \\in \\iota$) dominates the Selberg archimedean slice when, for every congruence subgroup $\\Gamma$ of $\\mathrm{SL}_2(\\mathbb{Z})$ and every eigenvalue $\\lambda$ of the hyperbolic Laplacian on $L^2(\\Gamma \\backslash \\mathbb{H})$ with $0 < \\lambda < 1/4$, there is an index $i$ with $m_i = \\sqrt{1/4 - \\lambda}$.","title":"Families dominating the Selberg exceptional spectrum"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/sqrt-le-ramanujan-exponent-of-of-dominat","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.sqrt_le_ramanujanExponentOf_of_dominatesSelberg","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#sqrt-le-ramanujan-exponent-of-of-dominat","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"Let $F$ be a local-exponent family that dominates the Selberg archimedean slice. If $\\Gamma$ is a congruence subgroup of $\\mathrm{SL}_2(\\mathbb{Z})$ and $\\lambda$ is a real eigenvalue of the hyperbolic Laplacian on $L^2(\\Gamma \\backslash \\mathbb{H})$ with $0 < \\lambda < 1/4$, then $\\sqrt{1/4 - \\lambda}$ is at most the Ramanujan exponent of $F$.","title":"Exceptional eigenvalues bound Selberg-dominating exponents from below"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/selberg-constant-le-of-dominates-selberg","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.selbergConstant_le_of_dominatesSelberg","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#selberg-constant-le-of-dominates-selberg","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"Let $F$ be a local-exponent family that dominates the Selberg archimedean slice, that is, for every congruence subgroup $\\Gamma$ of $\\mathrm{SL}_2(\\mathbb{Z})$ and every Laplace eigenvalue $\\lambda \\in (0, 1/4)$ of $\\Gamma$, the gap $\\sqrt{1/4 - \\lambda}$ is one of the local exponents of $F$. Then Selberg's constant is at most the Ramanujan exponent of $F$.","title":"Selberg's constant is at most every dominating family's exponent"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/ramanujan-exponent-of-selberg-family","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanExponent.ramanujanExponentOf_selbergFamily","parent":10896,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture/index.html#ramanujan-exponent-of-selberg-family","scope":"NumberTheory/modular-forms-q-series/modular-forms/ramanujan-petersson-conjecture","status":"fully_proved","summary":"The Ramanujan exponent of the canonical Selberg family of archimedean exponents is exactly Selberg's congruence spectral-gap constant.","title":"The Selberg family's exponent is Selberg's constant"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching.json","id":"NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching","status":"fully_proved","title":"Taylor\u2013Wiles patching"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching/patching-level","kind":"node","lean":"Algebra.PatchingLevel","parent":10919,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching/index.html#patching-level","scope":"NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching","status":"fully_proved","summary":"One level of a Taylor\u2013Wiles patching datum: a finite module `N` over the power series ring `\ud835\udcaa\u27e6X\u2081, \u2026, X\u1d63\u27e7`, with a surjective algebra map `\u03c8` to `R` killing the `\u03c6`-images of the variables, a surjective `\u03c8`-semilinear map `\u03c0 : N \u2192 M` whose kernel is the span of the `\u03c6 (X\u1d62)`, and a finite family `b` spanning `N` with relations exactly `J`.","title":"Level of a Taylor\u2013Wiles patching datum"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching/patching-datum","kind":"node","lean":"Algebra.PatchingDatum","parent":10919,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching/index.html#patching-datum","scope":"NumberTheory/modular-forms-q-series/modular-forms/taylor-wiles-patching","status":"fully_proved","summary":"A Taylor\u2013Wiles patching datum: a patching level for every `n`, with the relation ideal at level `n` generated by `(1 + X\u2c7c)^(\u2113\u207f) - 1`. This structure records an indexed family of finite-level data.","title":"Taylor\u2013Wiles patching datum"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2.json","id":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","kind":"scope","navigation":"module","parent":10826,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/index.html","scope":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","status":"fully_proved","title":"Weight-two cusp forms on \u0393\u2080(2)"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/norm","kind":"node","lean":"CuspForm.norm","parent":10922,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/index.html#norm","scope":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","status":"fully_proved","summary":"The product norm of a cusp form: for `\ud835\udca2` of finite relative index in `\u210b`, the product of the `\u210b / \ud835\udca2`-translates of `f`, viewed as a cusp form on the larger group `\u210b` at weight `k * [\u210b : \ud835\udca2]`. This lifts Mathlib's `ModularForm.norm` to cusp forms; the only new work is checking vanishing at the cusps, which follows because each translate is a cusp form.","title":"Norm of a cusp form"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/coe-norm-eq-coe-modular-form-norm","kind":"node","lean":"CuspForm.coe_norm_eq_coe_modularFormNorm","parent":10922,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/index.html#coe-norm-eq-coe-modular-form-norm","scope":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","status":"fully_proved","summary":"The coercion of the norm of a cusp form is the norm of its coercion as a modular form.","title":"Cusp form norm agrees with the modular form norm"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/norm-eq-zero-iff","kind":"node","lean":"CuspForm.norm_eq_zero_iff","parent":10922,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/index.html#norm-eq-zero-iff","scope":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","status":"fully_proved","summary":"The norm of a cusp form vanishes if and only if the form itself vanishes: the norm is a product of translates, and a translate of a nonzero form is nonzero.","title":"Norm of a cusp form vanishes only for the zero form"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/gamma0-two-index-eq-three","kind":"node","lean":"CongruenceSubgroup.Gamma0_two_index_eq_three","parent":10922,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/index.html#gamma0-two-index-eq-three","scope":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","status":"fully_proved","summary":"The index of `\u0393\u2080(2)` in `SL(2, \u2124)` is 3: the cosets are classified by the first column mod 2, and there are three nonzero vectors in `(ZMod 2)\u00b2`.","title":"\u0393\u2080(2) has index 3 in SL(2, \u2124)"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/s2-gamma0-one-eq-zero","kind":"node","lean":"ModularForm.S2_Gamma0_one_eq_zero","parent":10922,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/index.html#s2-gamma0-one-eq-zero","scope":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","status":"fully_proved","summary":"Every weight-two cusp form on `\u0393\u2080(1)` is zero: `\u0393\u2080(1)` is the full modular group, and the dimension formula gives rank zero below weight 12.","title":"No nonzero weight-two cusp forms of level one"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/s2-gamma0-2-eq-zero","kind":"node","lean":"ModularForm.S2_Gamma0_2_eq_zero","parent":10922,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/index.html#s2-gamma0-2-eq-zero","scope":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","status":"fully_proved","summary":"Every weight-two cusp form on `\u0393\u2080(2)` is zero. Take the norm of `f` to `\u0393\u2080(1)`: it has weight `2 * [\u0393\u2080(1) : \u0393\u2080(2)] = 6`, hence vanishes by the level-one dimension formula, and the norm vanishes only when `f` does.","title":"No nonzero weight-two cusp forms on \u0393\u2080(2)"},{"id":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/exists-mul-eq-mul-norm-of-forall-slash-m","kind":"node","lean":"ModularForm.exists_mul_eq_mul_norm_of_forall_slash_mul_eq","parent":10922,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2/index.html#exists-mul-eq-mul-norm-of-forall-slash-m","scope":"NumberTheory/modular-forms-q-series/modular-forms/weight-two-cusp-forms-on-2","status":"fully_proved","summary":"The norm trick: two modular forms satisfying a commutation relation under the slash action of `\u210b` give rise to a modular form `\u03a6` of level `\u210b` with `\u03a6 * G = F * norm \u210b G`.","title":"Norm trick for modular forms"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions","kind":"scope","parent":10825,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions","status":"can_prove","summary":"Jacobi and Ramanujan theta functions, q-Pochhammer symbols and Euler products, Rogers\u2013Ramanujan and related continued fractions, mock theta functions, Lambert series and q-series identities.","title":"Theta functions and q-series"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-cube-euler-product-has-sum","kind":"node","lean":"MetaMathlibExt.QPochhammer.jacobi_cube_euler_product_hasSum","parent":10930,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/index.html#jacobi-cube-euler-product-has-sum","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions","status":"fully_proved","summary":"For every complex number $q$ with $|q| < 1$, the series $\\sum_{n=0}^{\\infty} (-1)^n (2n+1) q^{n(n+1)/2}$ converges unconditionally to $(q;q)_\\infty^3$, the cube of the infinite q-Pochhammer symbol $(q;q)_\\infty = \\prod_{n=0}^{\\infty} (1 - q^{n+1})$.","title":"Jacobi's identity for the cube of Euler's product"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","title":"Glaisher's divisor functions and infinite products"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/series-divisor-add-of-mul","kind":"node","lean":"MetaMathlibExt.series_divisor_add_of_mul","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#series-divisor-add-of-mul","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $R$ be a commutative ring and let $A$, $B$ be formal power series over $R$ with coefficients $a(n)$, $b(n)$. Suppose $s_A, s_B$ are maps $\\mathbb{N} \\to R$ with $n\\,a(n) = \\sum_{i=1}^{n} s_A(i)\\,a(n-i)$ and $n\\,b(n) = \\sum_{i=1}^{n} s_B(i)\\,b(n-i)$ for every $n \\ge 0$ (that is, $s_A$ and $s_B$ are series-divisors of $A$ and $B$). Then, writing $c(n)$ for the coefficients of $AB$,\u2026","title":"Series-divisors add under multiplication"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/series-divisor-calculus","kind":"node","lean":"MetaMathlibExt.series_divisor_calculus","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#series-divisor-calculus","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $R$ be a commutative ring, let $A$ be a formal power series over $R$, write $[q^k]F$ for the coefficient of $q^k$ in a power series $F$, and let $s : \\mathbb{N} \\to R$ satisfy $n\\,[q^n]A = \\sum_{i=1}^{n} s(i)\\,[q^{n-i}]A$ for every $n \\ge 0$ (that is, $s$ is a series-divisor of $A$). Then the following hold for every $n \\ge 0$: (a) for every power series $B$ over $R$ with $AB = 1$,\u2026","title":"Calculus of series-divisors"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/series-divisor-pow-recursion","kind":"node","lean":"MetaMathlibExt.series_divisor_pow_recursion","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#series-divisor-pow-recursion","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $R$ be a commutative ring, let $A$ be a formal power series over $R$ with coefficients $a(k)$, and let $s : \\mathbb{N} \\to R$ satisfy $k\\,a(k) = \\sum_{i=1}^{k} s(i)\\,a(k-i)$ for every $k \\ge 0$ (that is, $s$ is a series-divisor of $A$). For natural numbers $m$ and $k$, write $a_m(k)$ for the coefficient of $q^k$ in $A^m$. Then for all natural numbers $m$ and $n$, $n\\,a_m(n) = m \\sum_{j=1}^{n} s(j)\\,a_m(n-j)$.","title":"Coefficient recursion for powers of a power series"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/glaisher-divisors-series-divisor-recurre","kind":"node","lean":"MetaMathlibExt.glaisher_divisors_series_divisor_recurrences","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#glaisher-divisors-series-divisor-recurre","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $n$ be a natural number. For a positive integer $m$, write $\\sigma(m)$ for the sum of the divisors of $m$, $d_B(m)$ and $d_C(m)$ for the sums of the odd and of the even divisors of $m$, $d_D(m)$ and $d_E(m)$ for the sums of the divisors $d$ of $m$ with $m/d$ odd and with $m/d$ even, $d_F(m) = \\sum_{d \\mid m} (-1)^{d-1} d$ and $d_G(m) = \\sum_{d \\mid m} (-1)^{m/d-1} d$, and set all of these to $0$ at $m = 0$. Let\u2026","title":"Glaisher's divisor functions as series-divisors"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/glaisher-partition-recurrences","kind":"node","lean":"MetaMathlibExt.glaisher_partition_recurrences","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#glaisher-partition-recurrences","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $n$ be a natural number. Let $p_e(m)$, $p_o(m)$ and $p_d(m)$ be the numbers of partitions of $m$ into even parts, into odd parts and into distinct parts, and let $\\bar p(m)$ be the number of overpartitions of $m$. For a positive integer $m$, write $\\sigma(m)$ for the sum of the divisors of $m$, $\\sigma(m/2)$ for $\\sigma$ evaluated at $m/2$ when $m$ is even and for $0$ when $m$ is odd, $d_B(m)$ and $d_C(m)$ for\u2026","title":"Recurrences for partitions into even or distinct parts and for overpartitions"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/sum-squares-triangular-power-recurrences","kind":"node","lean":"MetaMathlibExt.sum_squares_triangular_power_recurrences","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#sum-squares-triangular-power-recurrences","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $m \\ge 1$ be an integer. For $n \\ge 0$, let $r_m(n)$ be the number of $x \\in \\mathbb{Z}^m$ with $x_1^2 + \\dots + x_m^2 = n$, and let $t_m(n)$ be the number of $x \\in \\mathbb{Z}_{\\ge 0}^m$ with $\\sum_{i=1}^{m} x_i(x_i+1)/2 = n$. Let $\\sigma$ be the sum-of-divisors function, and set $\\sigma(j/2) = 0$ when $j$ is odd. Then for every $n \\ge 0$,","title":"Recurrences for sums of squares and sums of triangular numbers"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/glaisher-theta-divisor-recursions","kind":"node","lean":"MetaMathlibExt.glaisher_theta_divisor_recursions","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#glaisher-theta-divisor-recursions","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $n \\ge 1$ be a natural number. Write $\\sigma(m)$ for the sum of the positive divisors of $m$, $d_D(m)$ for the sum of the divisors $d$ of $m$ with $m/d$ odd, $d_F(m) = \\sum_{d \\mid m} (-1)^{d-1} d$, $\\bar\\sigma_3(m) = \\sum_{d \\mid m,\\ m/d \\text{ odd}} d^3$, $\\tilde\\sigma_3(m) = \\sum_{d \\mid m} (-1)^{d-1} d^3$ and $S(m) = \\sigma(m) - 4\\sigma(m/4)$, where $\\sigma(m/4) = 0$ unless $4 \\mid m$; all of these are $0$\u2026","title":"Six recursions for Glaisher's divisor functions from the theta functions"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/power-coeff-convolution-zpow","kind":"node","lean":"MetaMathlibExt.power_coeff_convolution_zpow","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#power-coeff-convolution-zpow","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, let $A$ be an invertible formal power series in $K[[X]]$, and let $u$ and $v$ be nonzero integers. Write $a_u(k)$ and $a_v(k)$ for the coefficients of $X^k$ in $A^u$ and $A^v$. Then for every natural number $n$,","title":"Power recursion lemma for nonzero integer exponents"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/power-coeff-convolution-pow","kind":"node","lean":"MetaMathlibExt.power_coeff_convolution_pow","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#power-coeff-convolution-pow","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, let $A \\in K[[X]]$ be a formal power series, and let $u$ and $v$ be positive integers. Write $a_u(k)$ and $a_v(k)$ for the coefficients of $X^k$ in $A^u$ and $A^v$. Then for every natural number $n$,","title":"Power recursion lemma for positive exponents"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/glaisher-power-recursion-divisor-identit","kind":"node","lean":"MetaMathlibExt.glaisher_power_recursion_divisor_identities","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#glaisher-power-recursion-divisor-identit","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $n \\ge 1$ be a natural number. Write $\\sigma(m)$ for the sum of the positive divisors of $m$, $\\bar\\sigma_3(m) = \\sum_{d \\mid m,\\ m/d \\text{ odd}} d^3$, $\\tilde\\sigma_3(m) = \\sum_{d \\mid m} (-1)^{d-1} d^3$ and $S(m) = \\sigma(m) - 4\\sigma(m/4)$, where $\\sigma(m/4) = 0$ unless $4 \\mid m$; all of these are $0$ at every integer $m \\le 0$. Then $2n\\,\\sigma(2n+1) = \\sum_{j=1}^{n} (5j(j+1) - 2n)\\,\\sigma(2n+1-j(j+1))$;\u2026","title":"Six divisor-function identities from the power recursion lemma"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/glaisher-divisor-sum-congruences","kind":"node","lean":"MetaMathlibExt.glaisher_divisor_sum_congruences","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#glaisher-divisor-sum-congruences","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $n \\ge 1$ be a natural number. Write $\\sigma(m)$ for the sum of the positive divisors of $m$ and $\\bar\\sigma_3(m) = \\sum_{d \\mid m,\\ m/d \\text{ odd}} d^3$, and set $\\sigma(m) = \\bar\\sigma_3(m) = 0$ for every integer $m \\le 0$. Then the following hold. If $5 \\nmid n$, then $\\sum_{j=0}^{n} \\sigma(2n+1-j(j+1)) \\equiv 0 \\pmod 5$. The sum $\\sum_{j=0}^{n} \\sigma(n-j(j+1)/2)$ is congruent modulo $4$ to $n$ if\u2026","title":"Glaisher's congruences for divisor sums over squares and triangular numbers"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/sigma-triangular-shift-sum-mod-five","kind":"node","lean":"MetaMathlibExt.sigma_triangular_shift_sum_mod_five","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#sigma-triangular-shift-sum-mod-five","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $\\sigma$ be the sum-of-divisors function, with $\\sigma(r) = 0$ for $r \\le 0$. For every integer $n \\ge 1$, the sum $\\sum_{j=0}^{n} \\sigma\\bigl(2n + 1 - j(j+1)\\bigr)$ is congruent to $1$ modulo $5$ if $n = 5k(k+1)/2$ for some integer $k \\ge 0$, and to $0$ modulo $5$ otherwise.","title":"A congruence modulo 5 for divisor sums over triangular shifts"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/series-divisor-inverse-convolution","kind":"node","lean":"MetaMathlibExt.series_divisor_inverse_convolution","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#series-divisor-inverse-convolution","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $R$ be a commutative ring and let $A$, $B$ be formal power series over $R$ with coefficients $a(k)$, $b(k)$ and $AB = 1$. Let $s : \\mathbb{N} \\to R$ satisfy $s(0) = 0$ and $k\\,a(k) = \\sum_{i=1}^{k} s(i)\\,a(k-i)$ for every $k \\ge 0$ (that is, $s$ is a series-divisor of $A$). Then for every $n \\ge 0$,","title":"Convolution identities for the series-divisor of an invertible series"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/overpartition-glaisher-d-d-formulas","kind":"node","lean":"MetaMathlibExt.overpartition_glaisher_dD_formulas","parent":10932,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#overpartition-glaisher-d-d-formulas","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $n$ be a natural number. Write $\\sigma(m)$ for the sum of the positive divisors of $m$, $\\sigma(m/2)$ for $\\sigma$ evaluated at $m/2$ when $m$ is even and for $0$ when $m$ is odd, and $d_D(m)$ for the sum of the divisors $d$ of $m$ with $m/d$ odd, with $\\sigma(0) = d_D(0) = 0$; let $\\bar p(m)$ be the number of overpartitions of $m$. Then\u2026","title":"Overpartition formulas for a divisor sum with odd cofactors"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/partition-mod-five-iff-sigma-convolution","kind":"node","lean":"MetaMathlibExt.partition_mod_five_iff_sigma_convolution","parent":10932,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's congruence modulo "],["math","\\(5\\)"],["text"," and a convolution of divisor sums"]],"source":"Ramanujan's congruence modulo $5$ and a convolution of divisor sums"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#partition-mod-five-iff-sigma-convolution","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $p(k)$ be the number of partitions of $k$ and $\\sigma(k)$ the sum of the positive divisors of $k$. Then $p(5m+4) \\equiv 0 \\pmod 5$ for every natural number $m$ if and only if $\\sum_{i=1}^{5m} \\sigma(i)\\,\\sigma(5m+1-i) \\equiv 0 \\pmod 5$ for every natural number $m$.","title":"Ramanujan's congruence modulo $5$ and a convolution of divisor sums"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/overpartition-mod-eight-iff-d-d-convolut","kind":"node","lean":"MetaMathlibExt.overpartition_mod_eight_iff_dD_convolution","parent":10932,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overpartitions modulo "],["math","\\(8\\)"],["text"," and a convolution of divisor sums"]],"source":"Overpartitions modulo $8$ and a convolution of divisor sums"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc/index.html#overpartition-mod-eight-iff-d-d-convolut","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/glaisher-s-divisor-functions-and-infinite-produc","status":"can_prove","summary":"Let $\\bar p(k)$ be the number of overpartitions of $k$, and let $d_D(k)$ be the sum of the divisors $d$ of $k$ for which $k/d$ is odd. Then $\\bar p(4m+3) \\equiv 0 \\pmod 8$ for every natural number $m$ if and only if $\\sum_{i=1}^{4m-1} d_D(i)\\,d_D(4m-i) \\equiv 0 \\pmod 4$ for every natural number $m$.","title":"Overpartitions modulo $8$ and a convolution of divisor sums"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials","status":"can_prove","title":"Hyperbolicity of Jensen polynomials"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/integral-exp-div-one-sub-exp-sub-eq-log","kind":"node","lean":"MetaMathlibExt.integral_exp_div_one_sub_exp_sub_eq_log_Gamma","parent":10949,"reading_title":{"format":"inline-parts-v1","parts":[["text","An integral formula for "],["math","\\(\\log \\Gamma(1/N)\\)"]],"source":"An integral formula for $\\log \\Gamma(1/N)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/index.html#integral-exp-div-one-sub-exp-sub-eq-log","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials","status":"can_prove","summary":"For every real number $N > 0$,","title":"An integral formula for $\\log \\Gamma(1/N)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/sum-exp-mul-digamma-div-eq-mul-log-one-s","kind":"node","lean":"MetaMathlibExt.sum_exp_mul_digamma_div_eq_mul_log_one_sub","parent":10949,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/index.html#sum-exp-mul-digamma-div-eq-mul-log-one-s","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials","status":"can_prove","summary":"Let $p$ and $k$ be natural numbers with $1 \\le k < p$, and let $\\psi = \\Gamma'/\\Gamma$ be the digamma function. Then","title":"A twisted sum of digamma values"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/eventually-jensen-hyperbolic-of-log-rati","kind":"node","lean":"MetaMathlibExt.eventually_jensen_hyperbolic_of_log_ratio_expansion","parent":10949,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/index.html#eventually-jensen-hyperbolic-of-log-rati","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials","status":"can_prove","summary":"Let $\\alpha$, $A$ and $\\delta$ be sequences of positive real numbers indexed by $n \\in \\mathbb{N}$ with $\\delta(n) \\to 0$, let $d \\ge 1$ be an integer, and for $3 \\le j \\le d$ let $g_j$ be a real sequence with $g_j(n) = o(\\delta(n)^j)$ as $n \\to \\infty$. Suppose that for every natural number $k \\le d$,","title":"Hyperbolicity of Jensen polynomials from a log-ratio expansion"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/eventually-jensen-hyperbolic-colored-par","kind":"node","lean":"MetaMathlibExt.eventually_jensen_hyperbolic_coloredPartitions","parent":10949,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/index.html#eventually-jensen-hyperbolic-colored-par","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials","status":"can_prove","summary":"Let $r \\ge 1$ and $d \\ge 1$ be integers. For $m \\in \\mathbb{N}$ let","title":"Eventual hyperbolicity for colored partitions"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/eventually-jensen-hyperbolic-partitions","kind":"node","lean":"MetaMathlibExt.eventually_jensen_hyperbolic_partitions_parts_mod","parent":10949,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual hyperbolicity for partitions into parts "],["math","\\(\\equiv a \\pmod{p}\\)"]],"source":"Eventual hyperbolicity for partitions into parts $\\equiv a \\pmod{p}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials/index.html#eventually-jensen-hyperbolic-partitions","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/hyperbolicity-of-jensen-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a$ be an integer with $1 \\le a < p$, and let $d \\ge 1$ be an integer. For $m \\in \\mathbb{N}$ let $b_{a,p}(m)$ be the number of partitions of $m$ all of whose parts are congruent to $a$ modulo $p$. Then for all sufficiently large $n$ the Jensen polynomial $J^{d,n}(X) = \\sum_{j=0}^{d} \\binom{d}{j} b_{a,p}(n + j) X^j$ is hyperbolic: the number of its real roots, counted with multiplicity,\u2026","title":"Eventual hyperbolicity for partitions into parts $\\equiv a \\pmod{p}$"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","title":"Lambert series"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-series-summand","kind":"node","lean":"MetaMathlibExt.lambertSeriesSummand","parent":10955,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-series-summand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and a natural number $k$, the $k$-th summand of the basic Lambert series is $q^{k+1} / (1 - q^{k+1})$, that is, the term $q^n / (1 - q^n)$ with $n = k + 1$.","title":"Lambert series summand"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-series","kind":"node","lean":"MetaMathlibExt.lambertSeries","parent":10955,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-series","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$, the basic Lambert series is $L(q) = \\sum_{n \\ge 1} q^n / (1 - q^n)$.","title":"Lambert series"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-fsummand","kind":"node","lean":"MetaMathlibExt.lambertFSummand","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the weighted Lambert series "],["math","\\(f\\)"]],"source":"Terms of the weighted Lambert series $f$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-fsummand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and a natural number $k$, the term of index $k$ is $n q^n / (1 - q^n)$ with $n = k + 1$. These are the terms of the weighted Lambert series $f(q) = \\sum_{n \\ge 1} n q^n / (1 - q^n)$.","title":"Terms of the weighted Lambert series $f$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-f","kind":"node","lean":"MetaMathlibExt.lambertF","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","The weighted Lambert series "],["math","\\(f\\)"]],"source":"The weighted Lambert series $f$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-f","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$, the weighted Lambert series is $f(q) = \\sum_{n \\ge 1} n q^n / (1 - q^n)$.","title":"The weighted Lambert series $f$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-gsummand","kind":"node","lean":"MetaMathlibExt.lambertGSummand","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the weighted Lambert series "],["math","\\(g\\)"]],"source":"Terms of the weighted Lambert series $g$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-gsummand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and a natural number $k$, the term of index $k$ is $n q^n / (1 + q^n)$ with $n = k + 1$. These are the terms of the weighted Lambert series $g(q) = \\sum_{n \\ge 1} n q^n / (1 + q^n)$.","title":"Terms of the weighted Lambert series $g$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-g","kind":"node","lean":"MetaMathlibExt.lambertG","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","The weighted Lambert series "],["math","\\(g\\)"]],"source":"The weighted Lambert series $g$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-g","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$, the weighted Lambert series is $g(q) = \\sum_{n \\ge 1} n q^n / (1 + q^n)$.","title":"The weighted Lambert series $g$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-fmu-summand","kind":"node","lean":"MetaMathlibExt.lambertFMuSummand","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the power-weighted Lambert series "],["math","\\(f_\\mu\\)"]],"source":"Terms of the power-weighted Lambert series $f_\\mu$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-fmu-summand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$, an integer $\\mu \\ge 1$ and a natural number $k$, the term of index $k$ is $n^\\mu q^n / (1 - q^n)$ with $n = k + 1$. These are the terms of the power-weighted Lambert series $f_\\mu(q) = \\sum_{n \\ge 1} n^\\mu q^n / (1 - q^n)$.","title":"Terms of the power-weighted Lambert series $f_\\mu$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-fmu","kind":"node","lean":"MetaMathlibExt.lambertFMu","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","The power-weighted Lambert series "],["math","\\(f_\\mu\\)"]],"source":"The power-weighted Lambert series $f_\\mu$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-fmu","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and an integer $\\mu \\ge 1$, the power-weighted Lambert series is $f_\\mu(q) = \\sum_{n \\ge 1} n^\\mu q^n / (1 - q^n)$.","title":"The power-weighted Lambert series $f_\\mu$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-gmu-summand","kind":"node","lean":"MetaMathlibExt.lambertGMuSummand","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the power-weighted Lambert series "],["math","\\(g_\\mu\\)"]],"source":"Terms of the power-weighted Lambert series $g_\\mu$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-gmu-summand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$, an integer $\\mu \\ge 1$ and a natural number $k$, the term of index $k$ is $n^\\mu q^n / (1 + q^n)$ with $n = k + 1$. These are the terms of the power-weighted Lambert series $g_\\mu(q) = \\sum_{n \\ge 1} n^\\mu q^n / (1 + q^n)$.","title":"Terms of the power-weighted Lambert series $g_\\mu$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-gmu","kind":"node","lean":"MetaMathlibExt.lambertGMu","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","The power-weighted Lambert series "],["math","\\(g_\\mu\\)"]],"source":"The power-weighted Lambert series $g_\\mu$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-gmu","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and an integer $\\mu \\ge 1$, the power-weighted Lambert series is $g_\\mu(q) = \\sum_{n \\ge 1} n^\\mu q^n / (1 + q^n)$.","title":"The power-weighted Lambert series $g_\\mu$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-lm-summand","kind":"node","lean":"MetaMathlibExt.lambertLmSummand","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the Lambert family "],["math","\\(L_m\\)"]],"source":"Terms of the Lambert family $L_m$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-lm-summand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$, an integer $m \\ge 1$ and a natural number $k$, the term of index $k$ is $q^{mn} / (1 - q^{2n})^m$ with $n = k + 1$. These are the terms of $L_m(q) = \\sum_{n \\ge 1} q^{mn} / (1 - q^{2n})^m$.","title":"Terms of the Lambert family $L_m$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/lambert-lm","kind":"node","lean":"MetaMathlibExt.lambertLm","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lambert family "],["math","\\(L_m\\)"]],"source":"The Lambert family $L_m$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#lambert-lm","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and an integer $m \\ge 1$, the Lambert family is $L_m(q) = \\sum_{n \\ge 1} q^{mn} / (1 - q^{2n})^m$.","title":"The Lambert family $L_m$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/tilde-lmsummand","kind":"node","lean":"MetaMathlibExt.tildeLMSummand","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the Lambert-type series "],["math","\\(\\tilde L_m\\)"]],"source":"Terms of the Lambert-type series $\\tilde L_m$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#tilde-lmsummand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$, an integer $m \\ge 1$ and a natural number $k$, the term of index $k$ is $q^{(2k+1)m} / (1 + q^{2k+1})^{2m}$, that is, $q^{(2n-1)m} / (1 + q^{2n-1})^{2m}$ with $n = k + 1$.","title":"Terms of the Lambert-type series $\\tilde L_m$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/tilde-lmsummable","kind":"node","lean":"MetaMathlibExt.tildeLMSummable","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of the Lambert-type series "],["math","\\(\\tilde L_m\\)"]],"source":"Summability of the Lambert-type series $\\tilde L_m$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#tilde-lmsummable","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and an integer $m \\ge 1$, this predicate holds when the series $\\sum_{n \\ge 1} q^{(2n-1)m} / (1 + q^{2n-1})^{2m}$ is summable in $\\mathbb{C}$.","title":"Summability of the Lambert-type series $\\tilde L_m$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/tilde-lm","kind":"node","lean":"MetaMathlibExt.tildeLM","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lambert-type series "],["math","\\(\\tilde L_m\\)"]],"source":"The Lambert-type series $\\tilde L_m$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#tilde-lm","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and an integer $m \\ge 1$, $\\tilde L_m(q) = \\sum_{n \\ge 1} q^{(2n-1)m} / (1 + q^{2n-1})^{2m}$, where the sum is taken to be $0$ if the series is not summable.","title":"The Lambert-type series $\\tilde L_m$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/hat-lmsummand","kind":"node","lean":"MetaMathlibExt.hatLMSummand","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the Lambert-type series "],["math","\\(\\hat L_m\\)"]],"source":"Terms of the Lambert-type series $\\hat L_m$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#hat-lmsummand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$, an integer $m \\ge 1$ and a natural number $k$, the term of index $k$ is $q^{m(k+1)} / (1 + q^{2(k+1)})^m$, that is, $q^{mn} / (1 + q^{2n})^m$ with $n = k + 1$.","title":"Terms of the Lambert-type series $\\hat L_m$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/hat-lmsummable","kind":"node","lean":"MetaMathlibExt.hatLMSummable","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of the Lambert-type series "],["math","\\(\\hat L_m\\)"]],"source":"Summability of the Lambert-type series $\\hat L_m$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#hat-lmsummable","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and an integer $m \\ge 1$, this predicate holds when the series $\\sum_{n \\ge 1} q^{mn} / (1 + q^{2n})^m$ is summable in $\\mathbb{C}$.","title":"Summability of the Lambert-type series $\\hat L_m$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/hat-lm","kind":"node","lean":"MetaMathlibExt.hatLM","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lambert-type series "],["math","\\(\\hat L_m\\)"]],"source":"The Lambert-type series $\\hat L_m$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#hat-lm","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"For a complex number $q$ with $|q| < 1$ and an integer $m \\ge 1$, $\\hat L_m(q) = \\sum_{n \\ge 1} q^{mn} / (1 + q^{2n})^m$, where the sum is taken to be $0$ if the series is not summable.","title":"The Lambert-type series $\\hat L_m$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/star-lmsummand","kind":"node","lean":"MetaMathlibExt.starLMSummand","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the Lambert-type series "],["math","\\(L_m^*\\)"]],"source":"Terms of the Lambert-type series $L_m^*$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#star-lmsummand","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"Terms of the Lambert-type series $L_m^*$. For a complex number $q$ with $|q| < 1$, an integer $m \\ge 1$ and a natural number $k$, the term of index $k$ is $q^{(2k+1)m} / (1 - q^{2k+1})^{2m}$, that is, $q^{(2n-1)m} / (1 - q^{2n-1})^{2m}$ with $n = k + 1$.","title":"Terms of the Lambert-type series $L_m^*$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/star-lmsummable","kind":"node","lean":"MetaMathlibExt.starLMSummable","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of the Lambert-type series "],["math","\\(L_m^*\\)"]],"source":"Summability of the Lambert-type series $L_m^*$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#star-lmsummable","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"Summability of the Lambert-type series $L_m^*$. For a complex number $q$ with $|q| < 1$ and an integer $m \\ge 1$, this predicate holds when the series $\\sum_{n \\ge 1} q^{(2n-1)m} / (1 - q^{2n-1})^{2m}$ is summable in $\\mathbb{C}$.","title":"Summability of the Lambert-type series $L_m^*$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/star-lm","kind":"node","lean":"MetaMathlibExt.starLM","parent":10955,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lambert-type series "],["math","\\(L_m^*\\)"]],"source":"The Lambert-type series $L_m^*$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series/index.html#star-lm","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/lambert-series","status":"fully_proved","summary":"The Lambert-type series $L_m^*$. For a complex number $q$ with $|q| < 1$ and an integer $m \\ge 1$, $L_m^*(q) = \\sum_{n \\ge 1} q^{(2n-1)m} / (1 - q^{2n-1})^{2m}$, where the sum is taken to be $0$ if the series is not summable.","title":"The Lambert-type series $L_m^*$"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","title":"q-Pochhammer symbol"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/q-pochhammer-inf-raw","kind":"node","lean":"MetaMathlibExt.QPochhammer.qPochhammerInfRaw","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#q-pochhammer-inf-raw","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"For complex numbers $a$ and $q$, the raw infinite q-Pochhammer product is the unconditional infinite product $\\prod_{n=0}^{\\infty} (1 - a q^n)$, defined for every $q$: when this product does not converge unconditionally (in particular for most $q$ with $|q| \\ge 1$), its value is $1$ by convention.","title":"Raw infinite q-Pochhammer product"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/q-pochhammer-inf","kind":"node","lean":"MetaMathlibExt.QPochhammer.qPochhammerInf","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#q-pochhammer-inf","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"For complex numbers $a$ and $q$ with $|q| < 1$, the infinite q-Pochhammer symbol is $(a;q)_\\infty = \\prod_{n=0}^{\\infty} (1 - a q^n)$, the unconditional infinite product of the factors $1 - a q^n$.","title":"Infinite q-Pochhammer symbol"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/q-pochhammer-inf-zero-left","kind":"node","lean":"MetaMathlibExt.QPochhammer.qPochhammerInf_zero_left","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#q-pochhammer-inf-zero-left","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"For every complex number $q$ with $|q| < 1$, $(0;q)_\\infty = 1$, since every factor $1 - 0 \\cdot q^n$ equals $1$.","title":"The q-Pochhammer symbol with a = 0"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/has-prod-q-pochhammer-inf","kind":"node","lean":"MetaMathlibExt.QPochhammer.hasProd_qPochhammerInf","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#has-prod-q-pochhammer-inf","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"For all complex numbers $a$ and $q$ with $|q| < 1$, the infinite product $\\prod_{n=0}^{\\infty} (1 - a q^n)$ converges unconditionally to the infinite q-Pochhammer symbol $(a;q)_\\infty$: the products of the factors $1 - a q^n$ over finite sets of indices $n$ tend to $(a;q)_\\infty$.","title":"Convergence of the infinite q-Pochhammer product"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/q-pochhammer-inf-one-zero","kind":"node","lean":"MetaMathlibExt.QPochhammer.qPochhammerInf_one_zero","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#q-pochhammer-inf-one-zero","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"The infinite q-Pochhammer symbol satisfies $(1;0)_\\infty = 0$, because its factor for $n = 0$ is $1 - 1 \\cdot 0^0 = 0$ (with the convention $0^0 = 1$).","title":"The q-Pochhammer symbol vanishes at a = 1 and q = 0"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/q-pochhammer-inf-raw-two-one","kind":"node","lean":"MetaMathlibExt.QPochhammer.qPochhammerInfRaw_two_one","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#q-pochhammer-inf-raw-two-one","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"The raw infinite q-Pochhammer product at $a = 2$ and $q = 1$ equals $1$.","title":"The raw q-Pochhammer product at a = 2 and q = 1 equals 1"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/q-euler-product","kind":"node","lean":"MetaMathlibExt.qEulerProduct","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#q-euler-product","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"For a natural number $k$, the Euler product $f_k = (q^k; q^k)_\\infty = \\prod_{m \\ge 1} (1 - q^{km})$ is this infinite product in the ring of integer power series in $q$, taken in the coefficientwise topology. It is used for $k \\ge 1$.","title":"The Euler products of partition theory"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/q-euler-product-one","kind":"node","lean":"MetaMathlibExt.qEulerProduct_one","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#q-euler-product-one","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"The Euler product $f_1 = \\prod_{m \\ge 1} (1 - q^m)$ equals the pentagonal series $\\sum_{j \\in \\mathbb{Z}} (-1)^j q^{j(3j-1)/2}$ as an integer power series.","title":"Pentagonal series expansion of the first Euler product"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/constant-coeff-q-euler-product","kind":"node","lean":"MetaMathlibExt.constantCoeff_qEulerProduct","parent":10977,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol/index.html#constant-coeff-q-euler-product","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/q-pochhammer-symbol","status":"fully_proved","summary":"For every natural number $k \\ge 1$, the integer power series $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ has constant coefficient $1$.","title":"Constant term of the Euler products"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","title":"Coefficients of mock theta functions"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-v","kind":"node","lean":"MetaMathlibExt.mockThetaV","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","The eighth-order mock theta function "],["math","\\(v(q)\\)"]],"source":"The eighth-order mock theta function $v(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-v","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"fully_proved","summary":"The mock theta function $v(q) = \\sum_{n \\ge 0} q^{(n+1)^2} (-q; q^2)_n / (q; q^2)_{n+1}$, where $(-q; q^2)_n = \\prod_{k=0}^{n-1} (1 + q^{2k+1})$ and $(q; q^2)_{n+1} = \\prod_{k=0}^{n} (1 - q^{2k+1})$, is taken as a power series in $q$ with integer coefficients.","title":"The eighth-order mock theta function $v(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-sigma","kind":"node","lean":"MetaMathlibExt.mockThetaSigma","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sixth-order mock theta function "],["math","\\(\\sigma(q)\\)"]],"source":"The sixth-order mock theta function $\\sigma(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-sigma","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"fully_proved","summary":"The mock theta function $\\sigma(q) = \\sum_{n \\ge 0} q^{(n+1)(n+2)/2} (-q; q)_n / (q; q^2)_{n+1}$, where $(-q; q)_n = \\prod_{k=1}^{n} (1 + q^k)$ and $(q; q^2)_{n+1} = \\prod_{k=0}^{n} (1 - q^{2k+1})$, is taken as a power series in $q$ with integer coefficients.","title":"The sixth-order mock theta function $\\sigma(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-lambda","kind":"node","lean":"MetaMathlibExt.mockThetaLambda","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sixth-order mock theta function "],["math","\\(\\lambda\\)"]],"source":"The sixth-order mock theta function $\\lambda$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-lambda","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"fully_proved","summary":"With $(a; b)_n = \\prod_{k=0}^{n-1} (1 - a b^k)$, the mock theta function $\\lambda$ is the integer power series","title":"The sixth-order mock theta function $\\lambda$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/coeff-mock-theta-v-zero-one","kind":"node","lean":"MetaMathlibExt.coeff_mockThetaV_zero_one","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","First coefficients of the mock theta function "],["math","\\(v\\)"]],"source":"First coefficients of the mock theta function $v$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#coeff-mock-theta-v-zero-one","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"fully_proved","summary":"Write $v(q) = \\sum_{n \\ge 0} P_v(n) q^n$ for the eighth-order mock theta function $v$. Then $P_v(0) = 0$ and $P_v(1) = 1$.","title":"First coefficients of the mock theta function $v$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/coeff-mock-theta-sigma-zero-one","kind":"node","lean":"MetaMathlibExt.coeff_mockThetaSigma_zero_one","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","First coefficients of the mock theta function "],["math","\\(\\sigma\\)"]],"source":"First coefficients of the mock theta function $\\sigma$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#coeff-mock-theta-sigma-zero-one","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"fully_proved","summary":"Write $\\sigma(q) = \\sum_{n \\ge 0} P_\\sigma(n) q^n$ for the sixth-order mock theta function $\\sigma$. Then $P_\\sigma(0) = 0$ and $P_\\sigma(1) = 1$.","title":"First coefficients of the mock theta function $\\sigma$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/coeff-mock-theta-lambda-zero-one","kind":"node","lean":"MetaMathlibExt.coeff_mockThetaLambda_zero_one","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","First coefficients of the mock theta function "],["math","\\(\\lambda\\)"]],"source":"First coefficients of the mock theta function $\\lambda$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#coeff-mock-theta-lambda-zero-one","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"fully_proved","summary":"Write $\\lambda(q) = \\sum_{n \\ge 0} P_\\lambda(n) q^n$ for the sixth-order mock theta function $\\lambda$. Then $P_\\lambda(0) = 1$ and $P_\\lambda(1) = -1$.","title":"First coefficients of the mock theta function $\\lambda$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/ramanujan-theta-monomial","kind":"node","lean":"MetaMathlibExt.ramanujanThetaMonomial","parent":10987,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#ramanujan-theta-monomial","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"fully_proved","summary":"For an integer $\\varepsilon$ and natural numbers $a$ and $b$, $f(\\varepsilon q^a, \\varepsilon q^b)$ is the integer power series in $q$ whose coefficient of $q^n$ is the sum of $\\varepsilon^{m^2}$ over the integers $m$ with $|m| \\le n + 1$ and $a \\cdot m(m+1)/2 + b \\cdot m(m-1)/2 = n$.","title":"Ramanujan's theta function at monomial arguments"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/coeff-ramanujan-theta-monomial-one-one-t","kind":"node","lean":"MetaMathlibExt.coeff_ramanujanThetaMonomial_one_one_three","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","First coefficients of Ramanujan's theta function "],["math","\\(\\psi\\)"]],"source":"First coefficients of Ramanujan's theta function $\\psi$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#coeff-ramanujan-theta-monomial-one-one-t","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"fully_proved","summary":"Let $\\psi(q) = f(q, q^3) = \\sum_{m \\in \\mathbb{Z}} q^{m(m+1)/2 + 3m(m-1)/2}$, where $f$ is Ramanujan's general theta function. The coefficients of $q^0$, $q^1$, $q^2$, $q^3$ in $\\psi(q)$ are $1$, $1$, $0$, $1$.","title":"First coefficients of Ramanujan's theta function $\\psi$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/ramanujan-psi-p-dissection","kind":"node","lean":"MetaMathlibExt.ramanujanPsi_p_dissection","parent":10987,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#ramanujan-psi-p-dissection","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $p$ be an odd prime, let $f(c, d)$ be Ramanujan's theta function and $\\psi(q) = f(q, q^3)$. Then, as integer power series, $\\psi(q) = \\sum_{m=0}^{(p-3)/2} q^{(m^2+m)/2} f(q^{(p^2+(2m+1)p)/2}, q^{(p^2-(2m+1)p)/2}) + q^{(p^2-1)/8} f(q^{p^2}, q^{3p^2})$, where the last factor is $\\psi(q^{p^2})$. Moreover $(m^2+m)/2 \\not\\equiv (p^2-1)/8 \\pmod p$ for every integer $m$ with $0 \\le m \\le (p-3)/2$.","title":"The p-dissection of Ramanujan's psi function"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/euler-product-p-dissection","kind":"node","lean":"MetaMathlibExt.eulerProduct_p_dissection","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-dissection of "],["math","\\(f_1\\)"]],"source":"The $p$-dissection of $f_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#euler-product-p-dissection","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $p \\ge 5$ be a prime, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for $k \\ge 1$, and let $f(a, b) = \\sum_{m \\in \\mathbb{Z}} a^{m(m+1)/2} b^{m(m-1)/2}$ be Ramanujan's general theta function. Put $c = (p-1)/6$ if $p \\equiv 1 \\pmod 6$ and $c = (-p-1)/6$ otherwise. Then, as integer power series,","title":"The $p$-dissection of $f_1$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/euler-product-cube-p-dissection","kind":"node","lean":"MetaMathlibExt.eulerProduct_cube_p_dissection","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-dissection of "],["math","\\(f_1^3\\)"]],"source":"The $p$-dissection of $f_1^3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#euler-product-cube-p-dissection","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $p$ be an odd prime and, for $k \\ge 1$, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then, as integer power series,","title":"The $p$-dissection of $f_1^3$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-v-odd-generating-function","kind":"node","lean":"MetaMathlibExt.mockThetaV_odd_generatingFunction","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for the odd-indexed coefficients of "],["math","\\(v(q)\\)"]],"source":"Generating function for the odd-indexed coefficients of $v(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-v-odd-generating-function","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $P_v(n)$ be the coefficient of $q^n$ in the eighth-order mock theta function $v(q)$, and for $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then, as integer power series, $f_1 f_2 \\sum_{n \\ge 0} P_v(2n+1) q^n = f_4^3$, that is, $\\sum_{n \\ge 0} P_v(2n+1) q^n = f_4^3 / (f_1 f_2)$.","title":"Generating function for the odd-indexed coefficients of $v(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-v-coeff-six-mul-add-five-mod","kind":"node","lean":"MetaMathlibExt.mockThetaV_coeff_six_mul_add_five_mod_three","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficients of "],["math","\\(q^{6n+5}\\)"],["text"," in "],["math","\\(v(q)\\)"],["text"," are divisible by 3"]],"source":"The coefficients of $q^{6n+5}$ in $v(q)$ are divisible by 3"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-v-coeff-six-mul-add-five-mod","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $P_v(n)$ be the coefficient of $q^n$ in the eighth-order mock theta function $v(q)$. Then $P_v(6n+5) \\equiv 0 \\pmod 3$ for every $n \\ge 0$.","title":"The coefficients of $q^{6n+5}$ in $v(q)$ are divisible by 3"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-v-congruences-mod-two","kind":"node","lean":"MetaMathlibExt.mockThetaV_congruences_mod_two","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinite family of congruences modulo 2 for "],["math","\\(v(q)\\)"]],"source":"Infinite family of congruences modulo 2 for $v(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-v-congruences-mod-two","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $P_v(n)$ be the coefficient of $q^n$ in the eighth-order mock theta function $v(q)$, and for $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Let $p \\ge 3$ be a prime whose Legendre symbol $\\left(\\frac{-2}{p}\\right)$ equals $-1$, and let $\\alpha \\ge 0$ be an integer. Then the integer power series $f_1 \\sum_{n \\ge 0} P_v\\bigl(2 p^{2\\alpha} n + (3 p^{2\\alpha} + 1)/4\\bigr) q^n$ and $f_2 f_4^2$ have congruent\u2026","title":"Infinite family of congruences modulo 2 for $v(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-v-congruences-mod-six","kind":"node","lean":"MetaMathlibExt.mockThetaV_congruences_mod_six","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinite family of congruences modulo 6 for "],["math","\\(v(q)\\)"]],"source":"Infinite family of congruences modulo 6 for $v(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-v-congruences-mod-six","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $P_v(n)$ be the coefficient of $q^n$ in the eighth-order mock theta function $v(q)$, and for $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Let $p \\ge 5$ be a prime whose Legendre symbol $\\left(\\frac{-18}{p}\\right)$ equals $-1$, and let $\\alpha \\ge 0$ be an integer. Then for every $n \\ge 0$, $P_v\\bigl(6 p^{2\\alpha} n + (19 p^{2\\alpha} + 1)/4\\bigr)$ is congruent modulo $6$ to the coefficient of $q^n$ in\u2026","title":"Infinite family of congruences modulo 6 for $v(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-v-odd-eq-sum-regular","kind":"node","lean":"MetaMathlibExt.mockThetaV_odd_eq_sum_regular","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum formula for the odd-indexed coefficients of "],["math","\\(v(q)\\)"]],"source":"Sum formula for the odd-indexed coefficients of $v(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-v-odd-eq-sum-regular","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $P_v(n)$ be the coefficient of $q^n$ in the eighth-order mock theta function $v(q)$, for $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and let $b_4^1(k)$ be the coefficient of $q^k$ in the integer power series $f_4 / f_1$. Then for every $n \\ge 0$, $P_v(2n+1) = \\sum_{m \\ge 0,\\ m(m+1) \\le n} b_4^1\\bigl(n - m(m+1)\\bigr)$.","title":"Sum formula for the odd-indexed coefficients of $v(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-v-six-mul-add-five-eq-sum-reg","kind":"node","lean":"MetaMathlibExt.mockThetaV_six_mul_add_five_eq_sum_regular","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum formula for the coefficients of "],["math","\\(q^{6n+5}\\)"],["text"," in "],["math","\\(v(q)\\)"]],"source":"Sum formula for the coefficients of $q^{6n+5}$ in $v(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-v-six-mul-add-five-eq-sum-reg","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $P_v(n)$ be the coefficient of $q^n$ in the eighth-order mock theta function $v(q)$, for $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and let $b_6^3(k)$ be the coefficient of $q^k$ in the integer power series $(f_6 / f_1)^3$. Then for every $n \\ge 0$,\u2026","title":"Sum formula for the coefficients of $q^{6n+5}$ in $v(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-sigma-odd-generating-function","kind":"node","lean":"MetaMathlibExt.mockThetaSigma_odd_generatingFunction","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for the odd-indexed coefficients of "],["math","\\(\\sigma(q)\\)"]],"source":"Generating function for the odd-indexed coefficients of $\\sigma(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-sigma-odd-generating-function","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $P_\\sigma(n)$ be the coefficient of $q^n$ in the sixth-order mock theta function $\\sigma(q)$, and for $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then, as integer power series, $f_1^2 f_3 \\sum_{n \\ge 0} P_\\sigma(2n+1) q^n = f_2^2 f_6^2$, that is, $\\sum_{n \\ge 0} P_\\sigma(2n+1) q^n = f_2^2 f_6^2 / (f_1^2 f_3)$.","title":"Generating function for the odd-indexed coefficients of $\\sigma(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-sigma-congruences-mod-two","kind":"node","lean":"MetaMathlibExt.mockThetaSigma_congruences_mod_two","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo 2 for the mock theta function "],["math","\\(\\sigma\\)"]],"source":"Congruences modulo 2 for the mock theta function $\\sigma$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-sigma-congruences-mod-two","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Write $\\sigma(q) = \\sum_{n \\ge 0} P_\\sigma(n) q^n$ for the sixth-order mock theta function $\\sigma$ and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-2}{p}\\right) = -1$ and let $\\alpha \\ge 0$ be an integer. Then, coefficientwise,","title":"Congruences modulo 2 for the mock theta function $\\sigma$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-sigma-odd-eq-sum-regular","kind":"node","lean":"MetaMathlibExt.mockThetaSigma_odd_eq_sum_regular","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum formula for the odd-indexed coefficients of "],["math","\\(\\sigma(q)\\)"]],"source":"Sum formula for the odd-indexed coefficients of $\\sigma(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-sigma-odd-eq-sum-regular","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Let $P_\\sigma(n)$ be the coefficient of $q^n$ in the sixth-order mock theta function $\\sigma(q)$, for $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and let $b_2^2(k)$ be the coefficient of $q^k$ in the integer power series $(f_2 / f_1)^2$. Then for every $n \\ge 0$, $P_\\sigma(2n+1) = \\sum_{m \\ge 0,\\ (3m^2+3m)/2 \\le n} b_2^2\\bigl(n - (3m^2+3m)/2\\bigr)$.","title":"Sum formula for the odd-indexed coefficients of $\\sigma(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-lambda-even-eq-sum-regular","kind":"node","lean":"MetaMathlibExt.mockThetaLambda_even_eq_sum_regular","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even-index coefficients of the mock theta function "],["math","\\(\\lambda\\)"]],"source":"Even-index coefficients of the mock theta function $\\lambda$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-lambda-even-eq-sum-regular","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Write $\\lambda(q) = \\sum_{n \\ge 0} P_\\lambda(n) q^n$ for the sixth-order mock theta function $\\lambda$, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and let $b_2^3(m)$ be the coefficient of $q^m$ in $(f_2/f_1)^3$. Then for every $n \\ge 0$,","title":"Even-index coefficients of the mock theta function $\\lambda$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/mock-theta-lambda-six-mul-add-two-eq-sum","kind":"node","lean":"MetaMathlibExt.mockThetaLambda_six_mul_add_two_eq_sum_overpartition","parent":10987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of the mock theta function "],["math","\\(\\lambda\\)"],["text"," at indices "],["math","\\(6n+2\\)"]],"source":"Coefficients of the mock theta function $\\lambda$ at indices $6n+2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions/index.html#mock-theta-lambda-six-mul-add-two-eq-sum","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/coefficients-of-mock-theta-functions","status":"can_prove","summary":"Write $\\lambda(q) = \\sum_{n \\ge 0} P_\\lambda(n) q^n$ for the sixth-order mock theta function $\\lambda$, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and let $\\overline{p}_3(j)$ be the coefficient of $q^j$ in $(f_2/f_1^2)^3$. Then for every $n \\ge 0$,","title":"Coefficients of the mock theta function $\\lambda$ at indices $6n+2$"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","title":"Jacobi triple product"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/jacobi-theta-product","kind":"node","lean":"MetaMathlibExt.QPochhammer.jacobiThetaProduct","parent":11010,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html#jacobi-theta-product","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","summary":"For complex numbers $x$ and $q$ with $|q| < 1$,","title":"Jacobi theta function, product form"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/jacobi-theta-product-symm","kind":"node","lean":"MetaMathlibExt.QPochhammer.jacobiThetaProduct_symm","parent":11010,"reading_title":{"format":"inline-parts-v1","parts":[["text","Symmetry of the Jacobi theta product under "],["math","\\(x \\mapsto q/x\\)"]],"source":"Symmetry of the Jacobi theta product under $x \\mapsto q/x$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html#jacobi-theta-product-symm","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","summary":"Write $(a; q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$ for the infinite $q$-Pochhammer symbol and $j(x; q) = (x; q)_\\infty \\, (q/x; q)_\\infty \\, (q; q)_\\infty$ for the product-form Jacobi theta function. For all complex numbers $x$ and $q$ with $0 < |q| < 1$, $j(q/x; q) = j(x; q)$. This includes $x = 0$, where $q/0$ is read as $0$.","title":"Symmetry of the Jacobi theta product under $x \\mapsto q/x$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/jacobi-theta-product-zero-left","kind":"node","lean":"MetaMathlibExt.QPochhammer.jacobiThetaProduct_zero_left","parent":11010,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html#jacobi-theta-product-zero-left","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","summary":"Write $(a; q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$ for the infinite $q$-Pochhammer symbol and $j(x; q) = (x; q)_\\infty \\, (q/x; q)_\\infty \\, (q; q)_\\infty$ for the product-form Jacobi theta function. For every complex number $q$ with $|q| < 1$, $j(0; q) = (q; q)_\\infty$. Here $q/0$ is read as $0$, so the first two factors of $j(0; q)$ both equal $(0; q)_\\infty = 1$.","title":"Jacobi theta product with vanishing first argument"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/jacobi-theta-product-zero-right","kind":"node","lean":"MetaMathlibExt.QPochhammer.jacobiThetaProduct_zero_right","parent":11010,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html#jacobi-theta-product-zero-right","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","summary":"Write $(a; q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$ for the infinite $q$-Pochhammer symbol and $j(x; q) = (x; q)_\\infty \\, (q/x; q)_\\infty \\, (q; q)_\\infty$ for the product-form Jacobi theta function, where $|q| < 1$. For every complex number $x$, $j(x; 0) = (x; 0)_\\infty$: since $0/x = 0$ (also for $x = 0$, reading $0/0$ as $0$), the last two factors of $j(x; 0)$ both equal $(0; 0)_\\infty = 1$.","title":"Jacobi theta product at nome zero"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/jacobi-triple-product","kind":"node","lean":"MetaMathlibExt.Analysis.SpecialFunctions.JacobiTripleProduct.jacobi_triple_product","parent":11010,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html#jacobi-triple-product","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","summary":"For complex numbers $z \\ne 0$ and $q$ with $|q| < 1$, the bilateral series $\\sum_{n \\in \\mathbb{Z}} z^n q^{n^2}$ is summable, the family of factors $(1 - q^{2n+2})(1 + z q^{2n+1})(1 + z^{-1} q^{2n+1})$, $n \\ge 0$, is multipliable, and","title":"Jacobi triple product identity"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/andrews-jacobi-triple-product","kind":"node","lean":"MetaMathlibExt.andrews_jacobi_triple_product","parent":11010,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html#andrews-jacobi-triple-product","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","summary":"Let $q$ be a complex number with $|q| < 1$, and let $k$ and $i$ be natural numbers with $0 < i \\le k$. Then","title":"Andrews' form of the Jacobi triple product identity"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/andrews-jacobi-triple-product-summable","kind":"node","lean":"MetaMathlibExt.andrews_jacobi_triple_product_summable","parent":11010,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html#andrews-jacobi-triple-product-summable","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","summary":"Let $q$ be a complex number with $|q| < 1$, and let $k$ and $i$ be natural numbers with $i \\le k$. Then the bilateral series","title":"Convergence of the series in Andrews' triple product identity"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/andrews-jacobi-triple-product-multipliab","kind":"node","lean":"MetaMathlibExt.andrews_jacobi_triple_product_multipliable","parent":11010,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product/index.html#andrews-jacobi-triple-product-multipliab","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/jacobi-triple-product","status":"fully_proved","summary":"Let $q$ be a complex number with $|q| < 1$, and let $k$ and $i$ be natural numbers with $0 < i \\le k$. Then the infinite product","title":"Convergence of the product in Andrews' triple product identity"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function","status":"fully_proved","title":"Ramanujan's theta function"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/ramanujan-theta-term","kind":"node","lean":"MetaMathlibExt.ramanujanThetaTerm","parent":11019,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/index.html#ramanujan-theta-term","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function","status":"fully_proved","summary":"For complex numbers $a$, $b$ and an integer $n$, this is the term $a^{n(n+1)/2} b^{n(n-1)/2}$ of index $n$ in Ramanujan's general theta function $f(a,b) = \\sum_{k \\in \\mathbb{Z}} a^{k(k+1)/2} b^{k(k-1)/2}$.","title":"Summand of Ramanujan's general theta function"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/ramanujan-theta-raw","kind":"node","lean":"MetaMathlibExt.ramanujanThetaRaw","parent":11019,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/index.html#ramanujan-theta-raw","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function","status":"fully_proved","summary":"For complex numbers $a$ and $b$, the raw Ramanujan theta sum is $\\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ when this series is summable, and $0$ otherwise.","title":"Raw Ramanujan theta sum"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/ramanujan-theta","kind":"node","lean":"MetaMathlibExt.ramanujanTheta","parent":11019,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/index.html#ramanujan-theta","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function","status":"fully_proved","summary":"For complex numbers $a$ and $b$ such that the series $\\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ is summable, Ramanujan's general theta function is its sum $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$.","title":"Ramanujan's general theta function"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/not-summable-ramanujan-theta-term-one-on","kind":"node","lean":"MetaMathlibExt.not_summable_ramanujanThetaTerm_one_one","parent":11019,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's theta series diverges at "],["math","\\(a = b = 1\\)"]],"source":"Ramanujan's theta series diverges at $a = b = 1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/index.html#not-summable-ramanujan-theta-term-one-on","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function","status":"fully_proved","summary":"The series $\\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ of Ramanujan's general theta function is not summable at $a = b = 1$, since every one of its terms equals $1$.","title":"Ramanujan's theta series diverges at $a = b = 1$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/ramanujan-theta-raw-one-one","kind":"node","lean":"MetaMathlibExt.ramanujanThetaRaw_one_one","parent":11019,"reading_title":{"format":"inline-parts-v1","parts":[["text","The raw Ramanujan theta sum vanishes at "],["math","\\(a = b = 1\\)"]],"source":"The raw Ramanujan theta sum vanishes at $a = b = 1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/index.html#ramanujan-theta-raw-one-one","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function","status":"fully_proved","summary":"The raw Ramanujan theta sum $\\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$, which is assigned the value $0$ whenever the series is not summable, equals $0$ at $a = b = 1$, because there every term equals $1$ and the series is not summable.","title":"The raw Ramanujan theta sum vanishes at $a = b = 1$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/ramanujan-theta-summable","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_summable","parent":11019,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function/index.html#ramanujan-theta-summable","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-theta-function","status":"fully_proved","summary":"For all complex numbers $a$ and $b$ with $|ab| < 1$, the series $\\sum_{n \\in \\mathbb{Z}} |a^{n(n+1)/2} b^{n(n-1)/2}|$ of the absolute values of the terms of Ramanujan's general theta function converges.","title":"Absolute convergence of Ramanujan's general theta series"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"can_prove","title":"Ramanujan's continued fractions of order sixteen"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/lambert-order-sixteen-weight-n","kind":"node","lean":"MetaMathlibExt.lambert_order_sixteen_weight_n","parent":11026,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#lambert-order-sixteen-weight-n","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"can_prove","summary":"Write $(a; q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$, $(a_1, \\dots, a_k; q)_\\infty = (a_1; q)_\\infty \\cdots (a_k; q)_\\infty$ and $f_k = (q^k; q^k)_\\infty$. For every complex number $q$ with $|q| < 1$,","title":"Weighted Lambert series for the continued fractions of order sixteen"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/lambert-order-sixteen-legendre-three","kind":"node","lean":"MetaMathlibExt.lambert_order_sixteen_legendre_three","parent":11026,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lambert series of order sixteen twisted by the Legendre symbol modulo "],["math","\\(3\\)"]],"source":"Lambert series of order sixteen twisted by the Legendre symbol modulo $3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#lambert-order-sixteen-legendre-three","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"can_prove","summary":"Write $(a; q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$, $(a_1, \\dots, a_k; q)_\\infty = (a_1; q)_\\infty \\cdots (a_k; q)_\\infty$ and $f_k = (q^k; q^k)_\\infty$, and let $\\left(\\frac{n}{3}\\right)$ be the Legendre symbol modulo $3$. For every complex number $q$ with $|q| < 1$,","title":"Lambert series of order sixteen twisted by the Legendre symbol modulo $3$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/ramanujan-psi-f-chi-eq-q-pochhammer","kind":"node","lean":"MetaMathlibExt.ramanujan_psi_f_chi_eq_qPochhammer","parent":11026,"reading_title":{"format":"inline-parts-v1","parts":[["text","Product forms of Ramanujan's theta functions "],["math","\\(\\psi\\)"],["text",", "],["math","\\(f\\)"],["text"," and "],["math","\\(\\chi\\)"]],"source":"Product forms of Ramanujan's theta functions $\\psi$, $f$ and $\\chi$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#ramanujan-psi-f-chi-eq-q-pochhammer","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"can_prove","summary":"Write $(a; q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$ and $f_k = (q^k; q^k)_\\infty$, and let $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's general theta function. For every complex number $q$ with $|q| < 1$, $f(q, q^3) = f_2^2 / f_1$, $f(q, -q^2) = f_2^3 / (f_1 f_4)$, $(-q; q^2)_\\infty = f_2^2 / (f_1 f_4)$ and $(q; q^2)_\\infty = f_1 / f_2$.","title":"Product forms of Ramanujan's theta functions $\\psi$, $f$ and $\\chi$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/ramanujan-theta-entry30","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_entry30","parent":11026,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#ramanujan-theta-entry30","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"can_prove","summary":"Let $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's general theta function and $\\psi(q) = f(q, q^3)$. For all complex numbers $a$ and $b$ with $|ab| < 1$, $f(a, ab^2) f(b, a^2 b) = f(a, b) \\psi(ab)$ and $f(a, b) + f(-a, -b) = 2 f(a^3 b, a b^3)$, and if moreover $a \\ne 0$, then $f(a, b) - f(-a, -b) = 2a f(b/a, a^5 b^3)$.","title":"Ramanujan's Entry 30 (i)\u2013(iii) for the general theta function"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/lambert-odd-order-sixteen-eq-continued-f","kind":"node","lean":"MetaMathlibExt.lambert_odd_order_sixteen_eq_continuedFraction","parent":11026,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#lambert-odd-order-sixteen-eq-continued-f","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"can_prove","summary":"Write $(a; q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$ and $(a, b; q)_\\infty = (a; q)_\\infty (b; q)_\\infty$. For a complex number $q$ with $|q| < 1$, put $S(a, b) = \\sum_{k \\ge 0} (q^{a(2k+1)} + q^{b(2k+1)}) / (1 - q^{16(2k+1)})$, a sum over the odd multiples $n = 2k + 1$, and $E = q^4 (q^{32}; q^{32})_\\infty^4 / (q^{16}; q^{16})_\\infty^2$, $I_1 = q (q^6, q^{26}; q^{32})_\\infty / (q^{10}, q^{22}; q^{32})_\\infty$ and\u2026","title":"Odd Lambert series for Ramanujan's continued fractions of order sixteen"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/jacobi-theta-one-sine-raw","kind":"node","lean":"MetaMathlibExt.jacobiThetaOneSineRaw","parent":11026,"reading_title":{"format":"inline-parts-v1","parts":[["text","Raw Jacobi theta function "],["math","\\(\\theta_1\\)"]],"source":"Raw Jacobi theta function $\\theta_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#jacobi-theta-one-sine-raw","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"fully_proved","summary":"For complex numbers $z$ and $\\tau$, the raw Jacobi theta function is","title":"Raw Jacobi theta function $\\theta_1$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/jacobi-theta-one-sine","kind":"node","lean":"MetaMathlibExt.jacobiThetaOneSine","parent":11026,"reading_title":{"format":"inline-parts-v1","parts":[["text","Jacobi theta function "],["math","\\(\\theta_1\\)"]],"source":"Jacobi theta function $\\theta_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#jacobi-theta-one-sine","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"fully_proved","summary":"For complex numbers $z$ and $\\tau$ with $\\operatorname{Im} \\tau > 0$, the Jacobi theta function is","title":"Jacobi theta function $\\theta_1$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/jacobi-theta-one-sine-zero","kind":"node","lean":"MetaMathlibExt.jacobiThetaOneSine_zero","parent":11026,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Jacobi theta function "],["math","\\(\\theta_1\\)"],["text"," vanishes at "],["math","\\(z = 0\\)"]],"source":"The Jacobi theta function $\\theta_1$ vanishes at $z = 0$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#jacobi-theta-one-sine-zero","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"fully_proved","summary":"For every complex number $\\tau$ with $\\operatorname{Im} \\tau > 0$, the Jacobi theta function satisfies $\\theta_1(0 \\mid \\tau) = 0$.","title":"The Jacobi theta function $\\theta_1$ vanishes at $z = 0$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/jacobi-theta-one-sine-neg","kind":"node","lean":"MetaMathlibExt.jacobiThetaOneSine_neg","parent":11026,"reading_title":{"format":"inline-parts-v1","parts":[["text","Oddness of the Jacobi theta function "],["math","\\(\\theta_1\\)"]],"source":"Oddness of the Jacobi theta function $\\theta_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#jacobi-theta-one-sine-neg","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"fully_proved","summary":"For all complex numbers $z$ and $\\tau$ with $\\operatorname{Im} \\tau > 0$, the Jacobi theta function satisfies $\\theta_1(-z \\mid \\tau) = -\\theta_1(z \\mid \\tau)$.","title":"Oddness of the Jacobi theta function $\\theta_1$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/lambert-order-sixteen-sine-eq-jacobi-the","kind":"node","lean":"MetaMathlibExt.lambert_order_sixteen_sine_eq_jacobiThetaOne","parent":11026,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen/index.html#lambert-order-sixteen-sine-eq-jacobi-the","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-continued-fractions-of-order-sixteen","status":"can_prove","summary":"Let $\\tau$ and $z$ be complex numbers with $\\operatorname{Im} \\tau > 0$ and $|\\operatorname{Im} z| < \\pi \\operatorname{Im} \\tau$, let $q = e^{2 \\pi i \\tau}$, write $\\theta(w) = \\theta_1(w \\mid 16\\tau)$ for the Jacobi theta function $\\theta_1$, and let $\\theta'$ be the derivative of $\\theta$. Then","title":"Sine Lambert series of order sixteen as quotients of theta functions"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k","status":"can_prove","title":"Ramanujan's function k"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/ramanujan-k","kind":"node","lean":"MetaMathlibExt.ramanujanK","parent":11037,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's function "],["math","\\(k(q)\\)"]],"source":"Ramanujan's function $k(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/index.html#ramanujan-k","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k","status":"fully_proved","summary":"For a complex number $q$, Ramanujan's function is","title":"Ramanujan's function $k(q)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/ramanujan-k-zero","kind":"node","lean":"MetaMathlibExt.ramanujanK_zero","parent":11037,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's function "],["math","\\(k\\)"],["text"," vanishes at "],["math","\\(0\\)"]],"source":"Ramanujan's function $k$ vanishes at $0$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/index.html#ramanujan-k-zero","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k","status":"fully_proved","summary":"Ramanujan's function satisfies $k(0) = 0$.","title":"Ramanujan's function $k$ vanishes at $0$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/ramanujan-k-eta-quotients","kind":"node","lean":"MetaMathlibExt.ramanujanK_eta_quotients","parent":11037,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eta-quotient expressions for Ramanujan's function "],["math","\\(k\\)"]],"source":"Eta-quotient expressions for Ramanujan's function $k$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/index.html#ramanujan-k-eta-quotients","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k","status":"can_prove","summary":"Let $q$ be a complex number with $0 < |q| < 1$, write $f_m = (q^m;q^m)_\\infty = \\prod_{n \\ge 1} (1 - q^{mn})$ for $m \\ge 1$, and let $k = k(q)$ be the value of Ramanujan's function. Then","title":"Eta-quotient expressions for Ramanujan's function $k$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/ramanujan-k-div-sub-div-sq","kind":"node","lean":"MetaMathlibExt.ramanujanK_div_sub_div_sq","parent":11037,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's function "],["math","\\(k\\)"],["text"," at "],["math","\\(q\\)"],["text"," and "],["math","\\(q^2\\)"],["text"," as an eta quotient"]],"source":"Ramanujan's function $k$ at $q$ and $q^2$ as an eta quotient"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/index.html#ramanujan-k-div-sub-div-sq","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k","status":"can_prove","summary":"For every complex number $q$ with $0 < |q| < 1$, Ramanujan's function $k$ satisfies","title":"Ramanujan's function $k$ at $q$ and $q^2$ as an eta quotient"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/ramanujan-k-sq-relation","kind":"node","lean":"MetaMathlibExt.ramanujanK_sq_relation","parent":11037,"reading_title":{"format":"inline-parts-v1","parts":[["text","Modular equation of degree 2 for Ramanujan's function "],["math","\\(k\\)"]],"source":"Modular equation of degree 2 for Ramanujan's function $k$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/index.html#ramanujan-k-sq-relation","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k","status":"can_prove","summary":"For every complex number $q$ with $|q| < 1$, the values $X = k(q)$ and $Y = k(q^2)$ of Ramanujan's function satisfy $X^2 - Y + 2XY + X^2 Y + Y^2 = 0$.","title":"Modular equation of degree 2 for Ramanujan's function $k$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/ramanujan-theta-level-ten","kind":"node","lean":"MetaMathlibExt.ramanujan_theta_level_ten","parent":11037,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k/index.html#ramanujan-theta-level-ten","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/ramanujan-s-function-k","status":"can_prove","summary":"For complex $x$ with $|x| < 1$, let $\\varphi(x) = \\sum_{n \\in \\mathbb{Z}} x^{n^2}$, $\\psi(x) = \\sum_{n \\ge 0} x^{n(n+1)/2}$ and $f(-x) = \\prod_{n \\ge 1} (1 - x^n)$, so that $f(x) = \\prod_{n \\ge 1} (1 - (-x)^n)$. Then for every complex number $q$ with $|q| < 1$,","title":"Ramanujan's level-10 theta function identities"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction","status":"can_prove","title":"Rogers\u2013Ramanujan continued fraction"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/rogers-ramanujan-cont-frac-real","kind":"node","lean":"MetaMathlibExt.rogersRamanujanContFracReal","parent":11044,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/index.html#rogers-ramanujan-cont-frac-real","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction","status":"fully_proved","summary":"For a real number $x$,","title":"The real Rogers\u2013Ramanujan continued fraction"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/rogers-ramanujan-cont-frac-real-zero","kind":"node","lean":"MetaMathlibExt.rogersRamanujanContFracReal_zero","parent":11044,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/index.html#rogers-ramanujan-cont-frac-real-zero","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction","status":"fully_proved","summary":"The real Rogers\u2013Ramanujan continued fraction $R$ satisfies $R(0) = 0$.","title":"The Rogers\u2013Ramanujan continued fraction vanishes at zero"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/rogers-ramanujan-golden-ratio-theta-rela","kind":"node","lean":"MetaMathlibExt.rogersRamanujan_golden_ratio_theta_relation","parent":11044,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/index.html#rogers-ramanujan-golden-ratio-theta-rela","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction","status":"can_prove","summary":"Let $t > 0$, and put $q = e^{-\\pi t}$ and $q' = e^{-\\pi/t}$. Let $\\psi(x) = \\sum_{n \\ge 0} x^{n(n+1)/2}$ and $E(x) = (x; x)_\\infty = \\prod_{n \\ge 1} (1 - x^n)$, and set $\\tau(q) = \\psi(q) - 5q^3 \\psi(q^{25})$, $\\rho(q) = \\psi(q)^2 - 5q\\,\\psi(q^5)^2$ and","title":"Golden-ratio relation for the Rogers\u2013Ramanujan continued fraction"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/rogers-ramanujan-theta-coeff-eta-quotien","kind":"node","lean":"MetaMathlibExt.rogersRamanujan_theta_coeff_eta_quotient","parent":11044,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction/index.html#rogers-ramanujan-theta-coeff-eta-quotien","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-continued-fraction","status":"can_prove","summary":"In the ring of integer power series in $q$, let $\\psi(q) = \\sum_{m \\ge 0} q^{m(m+1)/2}$ be the triangular theta series and $E_k = (q^k; q^k)_\\infty$ the Euler products, and set $\\tau(q) = \\psi(q) - 5q^3\\psi(q^{25})$ and $\\rho(q) = \\psi(q)^2 - 5q\\,\\psi(q^5)^2$. Then","title":"Eta-quotient form of the Gladun\u2013Chaudhary theta coefficient"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight.json","id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","kind":"scope","navigation":"module","parent":10930,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","title":"Rogers\u2013Ramanujan\u2013Slater functions of order eighteen"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/ramanujan-theta-mul-add-mul-neg","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_mul_add_mul_neg","parent":11049,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's theta function: sum of products when "],["math","\\(ab = cd\\)"]],"source":"Ramanujan's theta function: sum of products when $ab = cd$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html#ramanujan-theta-mul-add-mul-neg","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","summary":"Let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function. For complex numbers $a, b, c, d$ with $ab = cd$ and $|ab| < 1$, $f(a,b) f(c,d) + f(-a,-b) f(-c,-d) = 2 f(ac,bd) f(ad,bc)$.","title":"Ramanujan's theta function: sum of products when $ab = cd$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/ramanujan-theta-mul-sub-mul-neg","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_mul_sub_mul_neg","parent":11049,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's theta function: difference of products when "],["math","\\(ab = cd\\)"]],"source":"Ramanujan's theta function: difference of products when $ab = cd$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html#ramanujan-theta-mul-sub-mul-neg","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","summary":"Let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function. For complex numbers $a, b, c, d$ with $ab = cd$, $|ab| < 1$, $c \\ne 0$ and $d \\ne 0$, $f(a,b) f(c,d) - f(-a,-b) f(-c,-d) = 2a \\, f(b/c, ac^2 d) \\, f(b/d, acd^2)$.","title":"Ramanujan's theta function: difference of products when $ab = cd$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/ramanujan-theta-one-five","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_one_five","parent":11049,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's theta function at "],["math","\\((q, q^5)\\)"]],"source":"Ramanujan's theta function at $(q, q^5)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html#ramanujan-theta-one-five","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","summary":"Let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function and $(a;q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$. For every complex number $q$ with $|q| < 1$, $f(q,q^5) = (-q;q^2)_\\infty \\, f(-q^3,-q^9)$, that is, $f(q,q^5) = \\chi(q) \\psi(-q^3)$ with $\\chi(q) = (-q;q^2)_\\infty$ and $\\psi(q) = f(q,q^3)$.","title":"Ramanujan's theta function at $(q, q^5)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/ramanujan-theta-order-nine-product","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_order_nine_product","parent":11049,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html#ramanujan-theta-order-nine-product","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","summary":"Let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function and write $f(-x) = f(-x,-x^2)$. For every complex number $q$ with $|q| < 1$, $f(-q,-q^8) f(-q^2,-q^7) f(-q^4,-q^5) f(-q^3) = f(-q) f(-q^9)^3$, that is, $f(-q,-q^8) f(-q^2,-q^7) f(-q^4,-q^5) = f(-q) f^3(-q^9)/f(-q^3)$ with the denominator cleared.","title":"Product of the nonic theta functions"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/ramanujan-theta-one-two","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_one_two","parent":11049,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's theta function at "],["math","\\((q, q^2)\\)"]],"source":"Ramanujan's theta function at $(q, q^2)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html#ramanujan-theta-one-two","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","summary":"Let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function and $(a;q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$. For every complex number $q$ with $|q| < 1$, $f(q,q^2) \\, (q;q^2)_\\infty = f(-q^3,-q^3)$, that is, $f(q,q^2) = \\varphi(-q^3)/\\chi(-q)$ with the denominator cleared, where $\\varphi(q) = f(q,q)$ and $\\chi(-q) = (q;q^2)_\\infty$.","title":"Ramanujan's theta function at $(q, q^2)$"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/ramanujan-theta-eta-quotients","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_eta_quotients","parent":11049,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html#ramanujan-theta-eta-quotients","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $(a;q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$ and $f_k = (q^k;q^k)_\\infty$, and let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function. Then $f(q,q) f_1^2 f_4^2 = f_2^5$, $f(q,q^3) f_1 = f_2^2$, $f(-q,-q) f_2 = f_1^2$, $f(-q,-q^3) f_2 = f_1 f_4$, $f(q,-q^2) f_1 f_4 = f_2^3$, $(-q;q^2)_\\infty f_1 f_4 = f_2^2$ and\u2026","title":"Eta-quotients of Ramanujan's theta functions"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/slater-order-eighteen-modular-relations","kind":"node","lean":"MetaMathlibExt.slaterOrderEighteen_modular_relations","parent":11049,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html#slater-order-eighteen-modular-relations","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","summary":"Let $q$ be a complex number with $0 < |q| < 1$. Write $(a;q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$ and $f_k = (q^k;q^k)_\\infty$, let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function, and set $P_n = f(-q^{7n},-q^{11n})/f_{6n}$, $Q_n = f(-q^{5n},-q^{13n})/f_{6n}$ and $R_n = f(-q^n,-q^{17n})/f_{6n}$. Then the following seven relations hold.","title":"Modular relations for the order-eighteen Rogers\u2013Ramanujan\u2013Slater functions"},{"id":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/slater-order-eighteen-nonic-relations","kind":"node","lean":"MetaMathlibExt.slaterOrderEighteen_nonic_relations","parent":11049,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight/index.html#slater-order-eighteen-nonic-relations","scope":"NumberTheory/modular-forms-q-series/theta-q-series-partitions/rogers-ramanujan-slater-functions-of-order-eight","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. Write $(a;q)_\\infty = \\prod_{n \\ge 0} (1 - a q^n)$ and $f_k = (q^k;q^k)_\\infty$, let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function, and set $P_n = f(-q^{7n},-q^{11n})/f_{6n}$, $Q_n = f(-q^{5n},-q^{13n})/f_{6n}$, $R_n = f(-q^n,-q^{17n})/f_{6n}$, $A_n = f(-q^{4n},-q^{5n})/f_{3n}$, $B_n = f(-q^{2n},-q^{7n})/f_{3n}$ and\u2026","title":"Relations between the order-eighteen and the nonic Rogers\u2013Ramanujan functions"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences.json","id":"NumberTheory/modular-forms-q-series/partition-congruences","kind":"scope","parent":10825,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences","status":"can_prove","summary":"Ramanujan-type congruences, dissections and densities for partition functions and their variants (overpartitions, regular, cubic, core, colored, mex, designated-summand and smallest-part partitions), proved with q-series and eta quotients.","title":"Partition congruences"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","title":"(\u2113, k)-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count","kind":"node","lean":"MetaMathlibExt.regularPartitionCount","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\((\\ell, k)\\)"],["text","-regular partitions"]],"source":"Number of $(\\ell, k)$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"fully_proved","summary":"For natural numbers $\\ell$, $k$ and $n$, $b_{\\ell,k}(n)$ is the number of $(\\ell, k)$-regular partitions of $n$, that is, partitions of $n$ none of whose parts is divisible by $\\ell$ or by $k$.","title":"Number of $(\\ell, k)$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-four-nine-zero","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_four_nine_zero","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\((4, 9)\\)"],["text","-regular partition count at zero"]],"source":"The $(4, 9)$-regular partition count at zero"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-four-nine-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"fully_proved","summary":"The number $b_{4,9}(0)$ of $(4, 9)$-regular partitions of $0$, that is, partitions of $0$ none of whose parts is divisible by $4$ or by $9$, equals $1$: the only such partition is the empty partition.","title":"The $(4, 9)$-regular partition count at zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-of-lt","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_of_lt","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","For small "],["math","\\(n\\)"],["text"," every partition is "],["math","\\((\\ell, k)\\)"],["text","-regular"]],"source":"For small $n$ every partition is $(\\ell, k)$-regular"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-of-lt","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"fully_proved","summary":"Let $\\ell$, $k$ and $n$ be natural numbers with $n < \\ell$ and $n < k$, and write $b_{\\ell,k}(n)$ for the number of $(\\ell, k)$-regular partitions of $n$ (partitions none of whose parts is divisible by $\\ell$ or by $k$) and $p(n)$ for the number of all partitions of $n$. Then every partition of $n$ is $(\\ell, k)$-regular, so $b_{\\ell,k}(n) = p(n)$.","title":"For small $n$ every partition is $(\\ell, k)$-regular"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-four-prime-mod-t","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_four_prime_mod_two","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\((4, p)\\)"],["text","-regular partitions in nonresidue progressions"]],"source":"Parity of $(4, p)$-regular partitions in nonresidue progressions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-four-prime-mod-t","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Let $p \\ge 5$ be a prime and let $r$ be a natural number such that $8r + 1$ is not a square modulo $p$ (a quadratic nonresidue). Write $b_{4,p}(m)$ for the number of $(4, p)$-regular partitions of $m$, that is, partitions of $m$ none of whose parts is divisible by $4$ or by $p$. Then $b_{4,p}(pn + r) \\equiv 0 \\pmod 2$ for every natural number $n$.","title":"Parity of $(4, p)$-regular partitions in nonresidue progressions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-two-prime-mod-tw","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_two_prime_mod_two","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\((2, p)\\)"],["text","-regular partitions in nonresidue progressions"]],"source":"Parity of $(2, p)$-regular partitions in nonresidue progressions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-two-prime-mod-tw","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Let $p \\ge 5$ be a prime and let $r$ be a natural number such that $24r + 1$ is not a square modulo $p$ (a quadratic nonresidue). Write $b_{2,p}(m)$ for the number of $(2, p)$-regular partitions of $m$, that is, partitions of $m$ none of whose parts is divisible by $2$ or by $p$. Then $b_{2,p}(pn + r) \\equiv 0 \\pmod 2$ for every natural number $n$.","title":"Parity of $(2, p)$-regular partitions in nonresidue progressions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-four-nine-mod-tw","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_four_nine_mod_twelve","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo "],["math","\\(12\\)"],["text"," for "],["math","\\(b_{4,9}\\)"],["text"," along powers of "],["math","\\(2\\)"]],"source":"Congruences modulo $12$ for $b_{4,9}$ along powers of $2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-four-nine-mod-tw","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Let $b_{4,9}(n)$ be the number of partitions of $n$ with no part divisible by $4$ or by $9$. For all natural numbers $\\alpha \\ge 3$ and $n$, $b_{4,9}(2^\\alpha n + 6 \\cdot 2^{\\alpha - 3} + 1) \\equiv 0 \\pmod{12}$.","title":"Congruences modulo $12$ for $b_{4,9}$ along powers of $2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/f9-div-f1-two-dissection","kind":"node","lean":"MetaMathlibExt.f9_div_f1_two_dissection","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(f_9/f_1\\)"]],"source":"The 2-dissection of $f_9/f_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#f9-div-f1-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Write $f_k = (q^k; q^k)_\\infty$ for the Euler products. In the ring of integer power series in $q$,","title":"The 2-dissection of $f_9/f_1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/f3-div-f1-cube-two-dissection","kind":"node","lean":"MetaMathlibExt.f3_div_f1_cube_two_dissection","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(f_3/f_1^3\\)"]],"source":"The 2-dissection of $f_3/f_1^3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#f3-div-f1-cube-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Write $f_k = (q^k; q^k)_\\infty$ for the Euler products. In the ring of integer power series in $q$,","title":"The 2-dissection of $f_3/f_1^3$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/f3-sq-div-f1-sq-two-dissection","kind":"node","lean":"MetaMathlibExt.f3_sq_div_f1_sq_two_dissection","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(f_3^2/f_1^2\\)"]],"source":"The 2-dissection of $f_3^2/f_1^2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#f3-sq-div-f1-sq-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Write $f_k = (q^k; q^k)_\\infty$ for the Euler products. In the ring of integer power series in $q$,","title":"The 2-dissection of $f_3^2/f_1^2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/one-div-f1-pow-four-eight-twelve-two-dis","kind":"node","lean":"MetaMathlibExt.one_div_f1_pow_four_eight_twelve_two_dissection","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissections of "],["math","\\(1/f_1^4\\)"],["text",", "],["math","\\(1/f_1^8\\)"],["text"," and "],["math","\\(1/f_1^{12}\\)"]],"source":"The 2-dissections of $1/f_1^4$, $1/f_1^8$ and $1/f_1^{12}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#one-div-f1-pow-four-eight-twelve-two-dis","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Write $f_k = (q^k; q^k)_\\infty$ for the Euler products. In the ring of integer power series in $q$, the following three identities hold:","title":"The 2-dissections of $1/f_1^4$, $1/f_1^8$ and $1/f_1^{12}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/one-div-f1-f3-two-dissection","kind":"node","lean":"MetaMathlibExt.one_div_f1_f3_two_dissection","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(1/(f_1 f_3)\\)"]],"source":"The 2-dissection of $1/(f_1 f_3)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#one-div-f1-f3-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Write $f_k = (q^k; q^k)_\\infty$ for the Euler products. In the ring of integer power series in $q$,","title":"The 2-dissection of $1/(f_1 f_3)$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-four-nine-odd-ge","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_four_nine_odd_generatingFunction","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of "],["math","\\(b_{4,9}\\)"],["text"," at odd arguments"]],"source":"Generating function of $b_{4,9}$ at odd arguments"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-four-nine-odd-ge","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Let $b_{4,9}(n)$ be the number of partitions of $n$ with no part divisible by $4$ or by $9$, and write $f_k = (q^k; q^k)_\\infty$ for the Euler products. In the ring of integer power series in $q$,","title":"Generating function of $b_{4,9}$ at odd arguments"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-four-nine-intern","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_four_nine_internal_mod_four","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["text","An internal congruence modulo "],["math","\\(4\\)"],["text"," for "],["math","\\(b_{4,9}\\)"]],"source":"An internal congruence modulo $4$ for $b_{4,9}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-four-nine-intern","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Let $b_{4,9}(n)$ be the number of partitions of $n$ with no part divisible by $4$ or by $9$. For every natural number $n$, $b_{4,9}(16n + 1) \\equiv b_{4,9}(4n + 1) \\pmod 4$.","title":"An internal congruence modulo $4$ for $b_{4,9}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-four-nine-sixtee","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_four_nine_sixteen_mul_add_thirteen","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(b_{4,9}(16n + 13)\\)"],["text"," is divisible by "],["math","\\(4\\)"]],"source":"$b_{4,9}(16n + 13)$ is divisible by $4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-four-nine-sixtee","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Let $b_{4,9}(n)$ be the number of partitions of $n$ with no part divisible by $4$ or by $9$. For every natural number $n$, $b_{4,9}(16n + 13) \\equiv 0 \\pmod 4$.","title":"$b_{4,9}(16n + 13)$ is divisible by $4$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/regular-partition-count-four-nine-thirty","kind":"node","lean":"MetaMathlibExt.regularPartitionCount_four_nine_thirtyTwo_mul_add_twentyFive","parent":11059,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(b_{4,9}(32n + 25)\\)"],["text"," is divisible by "],["math","\\(4\\)"]],"source":"$b_{4,9}(32n + 25)$ is divisible by $4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions/index.html#regular-partition-count-four-nine-thirty","scope":"NumberTheory/modular-forms-q-series/partition-congruences/k-regular-partitions","status":"can_prove","summary":"Let $b_{4,9}(n)$ be the number of partitions of $n$ with no part divisible by $4$ or by $9$. For every natural number $n$, $b_{4,9}(32n + 25) \\equiv 0 \\pmod 4$.","title":"$b_{4,9}(32n + 25)$ is divisible by $4$"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","title":"Analogs of the 5-core partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/ramanujan-theta-pow","kind":"node","lean":"MetaMathlibExt.ramanujanThetaPow","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#ramanujan-theta-pow","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"fully_proved","summary":"For positive integers $i$ and $j$, $f(q^i, q^j)$ is the integer power series $\\sum_{n \\in \\mathbb{Z}} q^{i n(n+1)/2 + j n(n-1)/2}$, that is, Ramanujan's $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ at $a = q^i$ and $b = q^j$.","title":"Ramanujan's general theta function at powers of q"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/five-core-count","kind":"node","lean":"MetaMathlibExt.fiveCoreCount","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#five-core-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"fully_proved","summary":"For a natural number $n$, $c_5(n)$ is the coefficient of $q^n$ in the integer power series $f_5^5 / f_1$, where $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ is the Euler product. The series $f_5^5 / f_1$ is the generating function of 5-core partitions.","title":"The 5-core counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/phi-five-core-analog","kind":"node","lean":"MetaMathlibExt.phiFiveCoreAnalog","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#phi-five-core-analog","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"fully_proved","summary":"Let $\\varphi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n^2}$ be Ramanujan's theta function. For a natural number $n$, $\\bar a_5(n)$ is the coefficient of $q^n$ in the integer power series $\\varphi^5(-q^5)/\\varphi(-q)$ (the denominator has constant term $1$, so the quotient is a power series).","title":"The \u03c6-analog of the 5-core partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/psi-five-core-analog","kind":"node","lean":"MetaMathlibExt.psiFiveCoreAnalog","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#psi-five-core-analog","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"fully_proved","summary":"Let $\\psi(q) = \\sum_{n \\in \\mathbb{Z}} q^{2n^2 - n} = \\sum_{n \\ge 0} q^{n(n+1)/2}$ be Ramanujan's theta function. For a natural number $n$, $\\bar b_5(n)$ is the coefficient of $q^n$ in the integer power series $\\psi^5(-q^5)/\\psi(-q)$ (the denominator has constant term $1$, so the quotient is a power series).","title":"The \u03c8-analog of the 5-core partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/constant-coeff-ramanujan-theta-pow","kind":"node","lean":"MetaMathlibExt.constantCoeff_ramanujanThetaPow","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#constant-coeff-ramanujan-theta-pow","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"fully_proved","summary":"For all positive integers $i$ and $j$, the integer power series $f(q^i, q^j) = \\sum_{n \\in \\mathbb{Z}} q^{i n(n+1)/2 + j n(n-1)/2}$ has constant term $1$: only $n = 0$ contributes to $q^0$.","title":"Constant term of Ramanujan's theta function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/mk-five-core-count-mul","kind":"node","lean":"MetaMathlibExt.mk_fiveCoreCount_mul","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#mk-five-core-count-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"fully_proved","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ and let $c_5(n)$ be the coefficient of $q^n$ in $f_5^5 / f_1$. Then, as integer power series, $\\left(\\sum_{n \\ge 0} c_5(n) q^n\\right) f_1 = f_5^5$, where $f_1$ is written as the pentagonal series $\\sum_{j \\in \\mathbb{Z}} (-1)^j q^{j(3j-1)/2}$.","title":"Generating function of the 5-core counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/mk-phi-five-core-analog-mul","kind":"node","lean":"MetaMathlibExt.mk_phiFiveCoreAnalog_mul","parent":11075,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the "],["math","\\(\\varphi\\)"],["text","-analog of the 5-core counts"]],"source":"Generating function of the $\\varphi$-analog of the 5-core counts"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#mk-phi-five-core-analog-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"fully_proved","summary":"Let $\\varphi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n^2}$ and let $\\bar a_5(n)$ be the coefficient of $q^n$ in $\\varphi(-q^5)^5 / \\varphi(-q)$. Then, as integer power series, $\\left(\\sum_{n \\ge 0} \\bar a_5(n) q^n\\right) \\varphi(-q) = \\varphi(-q^5)^5$.","title":"Generating function of the $\\varphi$-analog of the 5-core counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/mk-psi-five-core-analog-mul","kind":"node","lean":"MetaMathlibExt.mk_psiFiveCoreAnalog_mul","parent":11075,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the "],["math","\\(\\psi\\)"],["text","-analog of the 5-core counts"]],"source":"Generating function of the $\\psi$-analog of the 5-core counts"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#mk-psi-five-core-analog-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"fully_proved","summary":"Let $\\psi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n(2n-1)}$ and let $\\bar b_5(n)$ be the coefficient of $q^n$ in $\\psi(-q^5)^5 / \\psi(-q)$. Then, as integer power series, $\\left(\\sum_{n \\ge 0} \\bar b_5(n) q^n\\right) \\psi(-q) = \\psi(-q^5)^5$.","title":"Generating function of the $\\psi$-analog of the 5-core counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/phi-five-core-analog-eq-five-core-count","kind":"node","lean":"MetaMathlibExt.phiFiveCoreAnalog_eq_fiveCoreCount","parent":11075,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\varphi\\)"],["text","-analog of the 5-core counts in terms of the 5-core counts"]],"source":"The $\\varphi$-analog of the 5-core counts in terms of the 5-core counts"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#phi-five-core-analog-eq-five-core-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, let $c_5(n)$ be the coefficient of $q^n$ in $f_5^5 / f_1$, let $\\varphi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n^2}$ and let $\\bar a_5(n)$ be the coefficient of $q^n$ in $\\varphi(-q^5)^5 / \\varphi(-q)$. For every natural number $n$, $\\bar a_5(5n+2) = 4\\, c_5(5n+1)$, $\\bar a_5(5n+3) = 4\\, c_5(5n+2)$, $\\bar a_5(10n+1) = 2\\, c_5(10n)$, $\\bar a_5(10n+9) = 2\\, c_5(10n+8)$,\u2026","title":"The $\\varphi$-analog of the 5-core counts in terms of the 5-core counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/phi-five-core-analog-five-pow-mul","kind":"node","lean":"MetaMathlibExt.phiFiveCoreAnalog_five_pow_mul","parent":11075,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the "],["math","\\(\\varphi\\)"],["text","-analog of the 5-core counts along powers of 5"]],"source":"Recurrence for the $\\varphi$-analog of the 5-core counts along powers of 5"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#phi-five-core-analog-five-pow-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $\\varphi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n^2}$ and let $\\bar a_5(n)$ be the coefficient of $q^n$ in $\\varphi(-q^5)^5 / \\varphi(-q)$. For every natural number $k \\ge 2$ and every natural number $n$, $\\bar a_5(5^k n) = \\frac{5^k - 1}{4}\\, \\bar a_5(5n) - \\frac{5^k - 5}{4}\\, \\bar a_5(n)$. Both quotients are integers.","title":"Recurrence for the $\\varphi$-analog of the 5-core counts along powers of 5"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/phi-five-core-analog-congruences","kind":"node","lean":"MetaMathlibExt.phiFiveCoreAnalog_congruences","parent":11075,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences for the "],["math","\\(\\varphi\\)"],["text","-analog of the 5-core counts"]],"source":"Congruences for the $\\varphi$-analog of the 5-core counts"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#phi-five-core-analog-congruences","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $\\varphi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n^2}$ and let $\\bar a_5(n)$ be the coefficient of $q^n$ in $\\varphi(-q^5)^5 / \\varphi(-q)$. For every natural number $k \\ge 2$ and every natural number $n$, $\\bar a_5(20n+6) \\equiv 0 \\pmod{10}$, $\\bar a_5(20n+14) \\equiv 0 \\pmod{10}$ and $4\\, \\bar a_5(5^k n) \\equiv 5\\, \\bar a_5(n) - \\bar a_5(5n) \\pmod{5^k}$.","title":"Congruences for the $\\varphi$-analog of the 5-core counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/psi-five-core-analog-four-mul-add-three","kind":"node","lean":"MetaMathlibExt.psiFiveCoreAnalog_four_mul_add_three","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#psi-five-core-analog-four-mul-add-three","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $\\bar b_5(n)$ be the coefficient of $q^n$ in $\\psi^5(-q^5)/\\psi(-q)$, where $\\psi(q) = \\sum_{n \\ge 0} q^{n(n+1)/2}$. For every natural number $n$, $\\bar b_5(4n+3) = 2 \\bar b_5(2n)$.","title":"A doubling identity for the \u03c8-analog of the 5-core partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/psi-five-core-analog-five-pow-mul","kind":"node","lean":"MetaMathlibExt.psiFiveCoreAnalog_five_pow_mul","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#psi-five-core-analog-five-pow-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $\\bar b_5(n)$ be the coefficient of $q^n$ in $\\psi^5(-q^5)/\\psi(-q)$, where $\\psi(q) = \\sum_{n \\ge 0} q^{n(n+1)/2}$. For every integer $k \\ge 2$ and every natural number $n$, $\\bar b_5(5^k(n+3) - 3) = \\frac{5^k - 1}{4} \\bar b_5(5n+12) - \\frac{5^k - 5}{4} \\bar b_5(n)$; both quotients are integers.","title":"A recurrence for the \u03c8-analog of the 5-core partition function along powers of 5"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/psi-five-core-analog-identities","kind":"node","lean":"MetaMathlibExt.psiFiveCoreAnalog_identities","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#psi-five-core-analog-identities","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, $\\varphi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n^2}$ and $\\psi(q) = \\sum_{n \\ge 0} q^{n(n+1)/2}$, and let $c_5(n)$, $\\bar a_5(n)$ and $\\bar b_5(n)$ be the coefficients of $q^n$ in $f_5^5/f_1$, $\\varphi^5(-q^5)/\\varphi(-q)$ and $\\psi^5(-q^5)/\\psi(-q)$. Then for every natural number $n$: $\\bar b_5(4n+1) = c_5(n) - 2 \\bar b_5(2n-1)$, where the last term is read as $0$ for $n = 0$;\u2026","title":"Identities relating the \u03c8-analog to the 5-core counts and the \u03c6-analog"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/psi-five-core-analog-sign-densities","kind":"node","lean":"MetaMathlibExt.psiFiveCoreAnalog_sign_densities","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#psi-five-core-analog-sign-densities","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $\\bar b_5(n)$ be the coefficient of $q^n$ in $\\psi^5(-q^5)/\\psi(-q)$, where $\\psi(q) = \\sum_{n \\ge 0} q^{n(n+1)/2}$, and let the lower density of a set $S$ of natural numbers be $\\liminf_{N \\to \\infty} |S \\cap \\{0, \\dots, N - 1\\}|/N$. The set of positive integers $n$ with $\\bar b_5(n) = 0$ has lower density at least $3/10$, the set of positive $n$ with $\\bar b_5(n) > 0$ has lower density at least $52/100$, and\u2026","title":"Sign densities of the \u03c8-analog of the 5-core partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/phi-five-core-analog-eq-twenty-mul-psi-f","kind":"node","lean":"MetaMathlibExt.phiFiveCoreAnalog_eq_twenty_mul_psiFiveCoreAnalog","parent":11075,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\varphi\\)"],["text","-analog in terms of the "],["math","\\(\\psi\\)"],["text","-analog of the 5-core counts"]],"source":"The $\\varphi$-analog in terms of the $\\psi$-analog of the 5-core counts"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#phi-five-core-analog-eq-twenty-mul-psi-f","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $\\varphi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n^2}$ and $\\psi(q) = \\sum_{n \\in \\mathbb{Z}} q^{n(2n-1)}$, and let $\\bar a_5(n)$ and $\\bar b_5(n)$ be the coefficients of $q^n$ in $\\varphi(-q^5)^5 / \\varphi(-q)$ and $\\psi(-q^5)^5 / \\psi(-q)$ respectively. For every natural number $n$, $\\bar a_5(20n+6) = 20\\, \\bar b_5(10n)$ and $\\bar a_5(20n+14) = 20\\, \\bar b_5(10n+4)$.","title":"The $\\varphi$-analog in terms of the $\\psi$-analog of the 5-core counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/psi-five-core-analog-congruences","kind":"node","lean":"MetaMathlibExt.psiFiveCoreAnalog_congruences","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#psi-five-core-analog-congruences","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $\\bar b_5(n)$ be the coefficient of $q^n$ in $\\psi^5(-q^5)/\\psi(-q)$, where $\\psi(q) = \\sum_{n \\ge 0} q^{n(n+1)/2}$. For every integer $k \\ge 2$ and every natural number $n$, $4 \\bar b_5(5^k(n+3) - 3) \\equiv 5 \\bar b_5(n) - \\bar b_5(5n+12) \\pmod{5^k}$, and both $\\bar b_5(5^k(20n+18) - 3)$ and $\\bar b_5(5^k(20n+22) - 3)$ are divisible by $(5^k - 1)/4$.","title":"Congruences for the \u03c8-analog of the 5-core partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/theta-level-five-identities","kind":"node","lean":"MetaMathlibExt.theta_level_five_identities","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#theta-level-five-identities","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, $f(q) = \\prod_{m \\ge 1} (1 - (-q)^m)$ (so that $f(-q) = f_1$), $\\chi(-q) = \\prod_{m \\ge 0} (1 - q^{2m+1})$ and $\\chi(q) = \\prod_{m \\ge 0} (1 + q^{2m+1})$. Let $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's general theta function, $\\varphi(q) = f(q, q)$ and $\\psi(q) = f(q, q^3)$. Then, as integer power series in $q$ (every denominator below has\u2026","title":"Theta-function identities of level 5"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/theta-five-dissections","kind":"node","lean":"MetaMathlibExt.theta_five_dissections","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#theta-five-dissections","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's general theta function, $\\varphi(q) = f(q, q)$ and $\\psi(q) = f(q, q^3)$. As integer power series in $q$, $\\varphi(q) = \\varphi(q^{25}) + 2q f(q^{15}, q^{35}) + 2q^4 f(q^5, q^{45})$ and $\\psi(q) = f(q^{10}, q^{15}) + q f(q^5, q^{20}) + q^3 \\psi(q^{25})$.","title":"5-dissections of Ramanujan's theta functions \u03c6 and \u03c8"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/five-core-count-identities","kind":"node","lean":"MetaMathlibExt.fiveCoreCount_identities","parent":11075,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function/index.html#five-core-count-identities","scope":"NumberTheory/modular-forms-q-series/partition-congruences/analogs-of-the-5-core-partition-function","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ and let $c_5(n)$ be the coefficient of $q^n$ in $f_5^5 / f_1$. For every natural number $n$, $c_5(4n+1) = c_5(2n)$ and $c_5(5n+4) = 5\\, c_5(n)$.","title":"Two identities for the 5-core counts"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","status":"can_prove","title":"Generalized cubic partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/generalized-cubic-partition-series","kind":"node","lean":"MetaMathlibExt.generalizedCubicPartitionSeries","parent":11096,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/index.html#generalized-cubic-partition-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","status":"fully_proved","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ be the Euler products. For a natural number $c$, this is the integer power series $1/(f_1 f_2^{c-1})$ in $q$, the inverse of $f_1 f_2^{c-1}$ (whose constant term is $1$). For $c \\ge 1$ its coefficient of $q^n$ is the number of partitions of $n$ whose even parts may each come in $c$ colors. For $c = 0$ the exponent $c - 1$ is truncated to $0$, so the series is then the\u2026","title":"Generating function of generalized cubic partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/generalized-cubic-partition-count","kind":"node","lean":"MetaMathlibExt.generalizedCubicPartitionCount","parent":11096,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/index.html#generalized-cubic-partition-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","status":"fully_proved","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ be the Euler products. For natural numbers $c$ and $n$, $a_c(n)$ is the integer coefficient of $q^n$ in the generating function of generalized cubic partitions, the power series $1/(f_1 f_2^{c-1})$. For $c \\ge 1$ it is the number of partitions of $n$ whose even parts may each come in $c$ colors; $c = 1$ gives the partition numbers and $c = 2$ Chan's cubic partition numbers.\u2026","title":"Generalized cubic partition numbers"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/generalized-cubic-partition-count-genera","kind":"node","lean":"MetaMathlibExt.generalizedCubicPartitionCount_generatingFunction","parent":11096,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/index.html#generalized-cubic-partition-count-genera","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","status":"fully_proved","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ be the Euler products and $a_c(n)$ the generalized cubic partition numbers. For every natural number $c$, $f_1 f_2^{c-1} \\sum_{n \\ge 0} a_c(n) q^n = 1$ as integer power series in $q$, where for $c = 0$ the exponent $c - 1$ is truncated to $0$.","title":"Generating function identity for generalized cubic partition numbers"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/generalized-cubic-partition-count-zero","kind":"node","lean":"MetaMathlibExt.generalizedCubicPartitionCount_zero","parent":11096,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/index.html#generalized-cubic-partition-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","status":"fully_proved","summary":"For every natural number $c$, the generalized cubic partition number $a_c(0)$ equals $1$.","title":"Generalized cubic partitions of zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/generalized-cubic-partition-count-seven","kind":"node","lean":"MetaMathlibExt.generalizedCubicPartitionCount_seven_eleven","parent":11096,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/index.html#generalized-cubic-partition-count-seven","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","status":"can_prove","summary":"For every natural number $n$, the generalized cubic partition numbers satisfy $a_3(7n + 4) \\equiv 0 \\pmod{7}$ and $a_5(11n + 10) \\equiv 0 \\pmod{11}$.","title":"Amdeberhan\u2013Sellers\u2013Singh congruences"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/generalized-cubic-partition-count-prime","kind":"node","lean":"MetaMathlibExt.generalizedCubicPartitionCount_prime_five_seven_mod_eight","parent":11096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized cubic partition congruences for primes "],["math","\\(p \\equiv 5, 7 \\pmod 8\\)"]],"source":"Generalized cubic partition congruences for primes $p \\equiv 5, 7 \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/index.html#generalized-cubic-partition-count-prime","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","status":"can_prove","summary":"Let $a_c(n)$ be the generalized cubic partition numbers, let $p$ be a prime with $p \\equiv 5$ or $p \\equiv 7 \\pmod 8$, and let $l$ be a natural number with $p \\mid 8l + 3$. Then $a_{p-4}(pn + l) \\equiv 0 \\pmod p$ for every natural number $n$.","title":"Generalized cubic partition congruences for primes $p \\equiv 5, 7 \\pmod 8$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/generalized-cubic-partition-count-prime-2","kind":"node","lean":"MetaMathlibExt.generalizedCubicPartitionCount_prime_three_seven_mod_eight","parent":11096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized cubic partition congruences modulo primes "],["math","\\(p \\ge 7\\)"],["text",", "],["math","\\(p \\equiv 3, 7 \\pmod 8\\)"]],"source":"Generalized cubic partition congruences modulo primes $p \\ge 7$, $p \\equiv 3, 7 \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions/index.html#generalized-cubic-partition-count-prime-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-cubic-partitions","status":"can_prove","summary":"Let $a_c(n)$ be the generalized cubic partition numbers and let $p \\ge 7$ be a prime with $p \\equiv 3$ or $p \\equiv 7 \\pmod 8$. Then $a_{p-6}\\bigl(pn + 13(p^2 - 1)/24\\bigr) \\equiv 0 \\pmod p$ for every natural number $n$.","title":"Generalized cubic partition congruences modulo primes $p \\ge 7$, $p \\equiv 3, 7 \\pmod 8$"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","title":"Generalized Frobenius partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/frobenius-partition-rep-count","kind":"node","lean":"MetaMathlibExt.frobeniusPartitionRepCount","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Frobenius partitions with at most "],["math","\\(k\\)"],["text"," repetitions"]],"source":"Frobenius partitions with at most $k$ repetitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#frobenius-partition-rep-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $\\varphi_k(n)$ is the number of generalized Frobenius partitions of $n$ in which each integer appears at most $k$ times in each row. Concretely, $\\varphi_k(n)$ is the number of pairs of functions $m_1, m_2 \\colon \\{0, 1, \\dots, n\\} \\to \\{0, 1, \\dots, k\\}$ with $\\sum_i m_1(i) = \\sum_i m_2(i)$ and $\\sum_i (i + 1)\\, m_1(i) + \\sum_i i\\, m_2(i) = n$.","title":"Frobenius partitions with at most $k$ repetitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/frobenius-partition-rep-count-four-zero","kind":"node","lean":"MetaMathlibExt.frobeniusPartitionRepCount_four_zero_one","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial values of "],["math","\\(\\varphi_4\\)"]],"source":"Initial values of $\\varphi_4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#frobenius-partition-rep-count-four-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"fully_proved","summary":"Let $\\varphi_4(n)$ be the number of generalized Frobenius partitions of $n$ in which each integer appears at most $4$ times in each row. Then $\\varphi_4(0) = 1$ (the empty array) and $\\varphi_4(1) = 1$ (the single column with both entries $0$).","title":"Initial values of $\\varphi_4$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/frobenius-partition-rep-count-four-four","kind":"node","lean":"MetaMathlibExt.frobeniusPartitionRepCount_four_four_mul_add_three_mod_two","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\varphi_4(4n+3)\\)"],["text"," is even"]],"source":"$\\varphi_4(4n+3)$ is even"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#frobenius-partition-rep-count-four-four","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $\\varphi_4(n)$ be the number of generalized Frobenius partitions of $n$ in which each integer appears at most $4$ times in each row. For every natural number $n$, $\\varphi_4(4n + 3) \\equiv 0 \\pmod 2$.","title":"$\\varphi_4(4n+3)$ is even"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/frobenius-partition-rep-count-four-two-m","kind":"node","lean":"MetaMathlibExt.frobeniusPartitionRepCount_four_two_mul_add_one_mod_two","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\varphi_4(2N+1)\\)"],["text"," is even when "],["math","\\(5 \\nmid N\\)"]],"source":"$\\varphi_4(2N+1)$ is even when $5 \\nmid N$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#frobenius-partition-rep-count-four-two-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $\\varphi_4(n)$ be the number of generalized Frobenius partitions of $n$ in which each integer appears at most $4$ times in each row. For every positive integer $N$ not divisible by $5$, $\\varphi_4(2N + 1) \\equiv 0 \\pmod 2$.","title":"$\\varphi_4(2N+1)$ is even when $5 \\nmid N$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/frobenius-partition-rep-count-four-four-2","kind":"node","lean":"MetaMathlibExt.frobeniusPartitionRepCount_four_four_mul_add_one_mod_two","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(\\varphi_4(4n+1)\\)"]],"source":"Parity of $\\varphi_4(4n+1)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#frobenius-partition-rep-count-four-four-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $\\varphi_4(n)$ be the number of generalized Frobenius partitions of $n$ in which each integer appears at most $4$ times in each row. For every natural number $n$, $\\varphi_4(4n + 1)$ is odd if and only if $2n = 5k(3k + 1)$ for some integer $k$, that is, $n = 5k(3k \\pm 1)/2$ for some integer $k$.","title":"Parity of $\\varphi_4(4n+1)$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/frobenius-partition-rep-count-four-ten-m","kind":"node","lean":"MetaMathlibExt.frobeniusPartitionRepCount_four_ten_mul_add_six_mod_five","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\varphi_4(10n+6)\\)"],["text"," is divisible by "],["math","\\(5\\)"]],"source":"$\\varphi_4(10n+6)$ is divisible by $5$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#frobenius-partition-rep-count-four-ten-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $\\varphi_4(n)$ be the number of generalized Frobenius partitions of $n$ in which each integer appears at most $4$ times in each row. For every natural number $n$, $\\varphi_4(10n + 6) \\equiv 0 \\pmod 5$.","title":"$\\varphi_4(10n+6)$ is divisible by $5$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/frobenius-partition-rep-count-four-ten-m-2","kind":"node","lean":"MetaMathlibExt.frobeniusPartitionRepCount_four_ten_mul_add_one_mod_five","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\varphi_4(10n+1)\\)"],["text"," modulo "],["math","\\(5\\)"]],"source":"$\\varphi_4(10n+1)$ modulo $5$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#frobenius-partition-rep-count-four-ten-m-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $\\varphi_4(n)$ be the number of generalized Frobenius partitions of $n$ in which each integer appears at most $4$ times in each row, and let $n$ be a natural number. If $n = k(3k + 1)$ for an integer $k$, then $\\varphi_4(10n + 1) \\equiv k + 1 \\pmod 5$; if there is no such integer $k$, then $\\varphi_4(10n + 1) \\equiv 0 \\pmod 5$.","title":"$\\varphi_4(10n+1)$ modulo $5$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/frobenius-partition-rep-count-four-two-d","kind":"node","lean":"MetaMathlibExt.frobeniusPartitionRepCount_four_two_dissection","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of the generating function of "],["math","\\(\\varphi_4\\)"]],"source":"The 2-dissection of the generating function of $\\varphi_4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#frobenius-partition-rep-count-four-two-d","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $\\varphi_4(n)$ be the number of generalized Frobenius partitions of $n$ in which each integer appears at most $4$ times in each row, and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ in the ring of integer power series in $q$. Then","title":"The 2-dissection of the generating function of $\\varphi_4$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/ramanujan-theta-mul-of-mul-eq","kind":"node","lean":"MetaMathlibExt.ramanujanTheta_mul_of_mul_eq","parent":11104,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's theta product identity for "],["math","\\(ab = cd\\)"]],"source":"Ramanujan's theta product identity for $ab = cd$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#ramanujan-theta-mul-of-mul-eq","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $f(a,b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's general theta function. If $a, b, c, d$ are complex numbers with $a \\ne 0$, $|ab| < 1$ and $ab = cd$, then","title":"Ramanujan's theta product identity for $ab = cd$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/one-div-f1-cube-f5-two-dissection","kind":"node","lean":"MetaMathlibExt.one_div_f1_cube_f5_two_dissection","parent":11104,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#one-div-f1-cube-f5-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ in the ring of integer power series in $q$. Then","title":"A 2-dissection of the reciprocal Euler product of levels 1 and 5"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/f2-cube-f5-sq-quotient-identity","kind":"node","lean":"MetaMathlibExt.f2_cube_f5_sq_quotient_identity","parent":11104,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions/index.html#f2-cube-f5-sq-quotient-identity","scope":"NumberTheory/modular-forms-q-series/partition-congruences/generalized-frobenius-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ in the ring of integer power series in $q$. Then $f_2^5 f_5^2 = f_1^5 f_5 f_{10} + 5q f_1^2 f_2 f_{10}^4$.","title":"A level-10 Euler-product identity"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions","status":"can_prove","title":"Mex-related partition functions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/mesh-partition-congruence-lift","kind":"node","lean":"MetaMathlibExt.mesh_partition_congruence_lift","parent":11116,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/index.html#mesh-partition-congruence-lift","scope":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions","status":"fully_proved","summary":"For a partition $\\lambda$, a positive integer $A$ and a natural number $c$, let $\\operatorname{mex}_{A,c}(\\lambda)$ be the smallest positive integer congruent to $c$ modulo $A$ that is not a part of $\\lambda$, and let $p_{A,c}(N)$ be the number of partitions $\\lambda$ of $N$ with $\\operatorname{mex}_{A,c}(\\lambda) \\equiv c \\pmod{2A}$. Let $p(N)$ be the number of partitions of $N$. Let $m$, $a$ and $b$ be natural\u2026","title":"Lifting partition congruences to mex partition functions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/partition-mex","kind":"node","lean":"MetaMathlibExt.partitionMex","parent":11116,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/index.html#partition-mex","scope":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions","status":"fully_proved","summary":"For natural numbers $A$, $a$ and $n$ and a partition $\\lambda$ of $n$, $\\operatorname{mex}_{A,a}(\\lambda)$ is the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $\\lambda$, and $0$ if there is no such integer (which happens only when $A = 0$).","title":"Mex of a partition in a residue class"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/mex-partition-number","kind":"node","lean":"MetaMathlibExt.mexPartitionNumber","parent":11116,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/index.html#mex-partition-number","scope":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions","status":"fully_proved","summary":"For natural numbers $A$, $a$ and $n$, $p_{A,a}(n)$ is the number of partitions $\\lambda$ of $n$ such that $\\operatorname{mex}_{A,a}(\\lambda) \\equiv a \\pmod{2A}$, where $\\operatorname{mex}_{A,a}(\\lambda)$ is the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $\\lambda$, and $0$ if there is no such integer.","title":"Mex-related partition function of Andrews and Newman"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/ramanujan-delta","kind":"node","lean":"MetaMathlibExt.ramanujanDelta","parent":11116,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/index.html#ramanujan-delta","scope":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions","status":"fully_proved","summary":"For natural numbers $p$ and $k$, $\\delta_{p,k}$ is the least natural number $\\delta$ with $24\\delta \\equiv 1 \\pmod{p^k}$, and $0$ if there is no such $\\delta$, which happens exactly when $p^k$ and $24$ have a common factor greater than $1$.","title":"Least inverse of 24 modulo a power"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/ramanujan-mex-partition-congruences","kind":"node","lean":"MetaMathlibExt.ramanujan_mex_partition_congruences","parent":11116,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions/index.html#ramanujan-mex-partition-congruences","scope":"NumberTheory/modular-forms-q-series/partition-congruences/mex-related-partition-functions","status":"can_prove","summary":"For a positive integer $A$, a natural number $a$ and a partition $\\lambda$, let $\\operatorname{mex}_{A,a}(\\lambda)$ be the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $\\lambda$; let $p_{A,a}(N)$ be the number of partitions $\\lambda$ of $N$ with $\\operatorname{mex}_{A,a}(\\lambda) \\equiv a \\pmod{2A}$; and let $\\delta_{p,k}$ be the least natural number $\\delta$ with\u2026","title":"Ramanujan congruences for mex-related partition functions"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","title":"Overcubic partition triples"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-series","kind":"node","lean":"MetaMathlibExt.overcubicTripleSeries","parent":11122,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"fully_proved","summary":"The generating function of overcubic partition triples is the integer power series $f_4^3 / (f_1^6 f_2^3)$ in $q$, where $f_k = \\prod_{j \\ge 1} (1 - q^{kj})$; the inverse of $f_1^6 f_2^3$ exists because its constant coefficient is $1$.","title":"The overcubic partition triple series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount","parent":11122,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{bt}(n)$, the number of overcubic partition triples of $n$, is defined as the integer coefficient of $q^n$ in the power series $f_4^3 / (f_1^6 f_2^3)$, where $f_k = \\prod_{j \\ge 1} (1 - q^{kj})$.","title":"Number of overcubic partition triples"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-generating-functi","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_generatingFunction","parent":11122,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-generating-functi","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"fully_proved","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and let $\\mathrm{bt}(n)$, the number of overcubic partition triples of $n$, be the coefficient of $q^n$ in $f_4^3/(f_1^6 f_2^3)$. Then, as integer power series in $q$,","title":"Generating function for overcubic partition triples"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-zero","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_zero","parent":11122,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overcubic partition triples of "],["math","\\(0\\)"]],"source":"Overcubic partition triples of $0$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"fully_proved","summary":"The number $\\mathrm{bt}(0)$ of overcubic partition triples of $0$ equals $1$.","title":"Overcubic partition triples of $0$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/f1-sq-two-dissection","kind":"node","lean":"MetaMathlibExt.f1_sq_two_dissection","parent":11122,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(f_1^2\\)"]],"source":"The 2-dissection of $f_1^2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#f1-sq-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$. Then","title":"The 2-dissection of $f_1^2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/one-div-f1-sq-two-dissection","kind":"node","lean":"MetaMathlibExt.one_div_f1_sq_two_dissection","parent":11122,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(1/f_1^2\\)"]],"source":"The 2-dissection of $1/f_1^2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#one-div-f1-sq-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$. Then","title":"The 2-dissection of $1/f_1^2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/f1-cube-three-dissection","kind":"node","lean":"MetaMathlibExt.f1_cube_three_dissection","parent":11122,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 3-dissection of "],["math","\\(f_1^3\\)"]],"source":"The 3-dissection of $f_1^3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#f1-cube-three-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$. Then","title":"The 3-dissection of $f_1^3$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-congruences-odd-a","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_congruences_odd_args","parent":11122,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-congruences-odd-a","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"Let $\\mathrm{bt}(n)$ be the number of overcubic partition triples of $n$. For all natural numbers $n$ and $\\alpha$, $\\mathrm{bt}(8n+7)$ is divisible by $32$, and each of $\\mathrm{bt}(8n+5)$, $\\mathrm{bt}(72n+33)$, $\\mathrm{bt}(72n+57)$ and $\\mathrm{bt}(8 \\cdot 9^{\\alpha+2} n + 33 \\cdot 9^{\\alpha+1})$ is divisible by $8$.","title":"Congruences for overcubic partition triples at odd arguments"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-twenty-four-mul-a","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_twentyFour_mul_add_one","parent":11122,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overcubic partition triples of "],["math","\\(24n+1\\)"],["text"," modulo "],["math","\\(4\\)"]],"source":"Overcubic partition triples of $24n+1$ modulo $4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-twenty-four-mul-a","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{j \\ge 1} (1 - q^{kj})$, and let $\\mathrm{bt}(m)$ be the number of overcubic partition triples of $m$, the coefficient of $q^m$ in $f_4^3 / (f_1^6 f_2^3)$. For every natural number $n$, $\\mathrm{bt}(24n+1) \\equiv 2 c_n \\pmod 4$, where $c_n$ is the coefficient of $q^n$ in $f_1$; equivalently, $\\sum_{n \\ge 0} \\mathrm{bt}(24n+1) q^n \\equiv 2 f_1 \\pmod 4$ coefficientwise.","title":"Overcubic partition triples of $24n+1$ modulo $4$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-congruences-two-m","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_congruences_two_mod_four_args","parent":11122,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-congruences-two-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"Let $\\mathrm{bt}(n)$ be the number of overcubic partition triples of $n$. For all natural numbers $n$ and $\\alpha$, both $\\mathrm{bt}(16n+14)$ and $\\mathrm{bt}(16n+10)$ are divisible by $16$; each of $\\mathrm{bt}(144n+66)$, $\\mathrm{bt}(144n+114)$, $\\mathrm{bt}(1296n+594)$, $\\mathrm{bt}(1296n+1026)$ and $\\mathrm{bt}(16 \\cdot 9^{\\alpha+3} n + 66 \\cdot 9^{\\alpha+2})$ is divisible by $8$; and\u2026","title":"Congruences for overcubic partition triples at arguments congruent to 2 modulo 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-congruences-four","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_congruences_four_mul_args","parent":11122,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-congruences-four","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"Let $\\mathrm{bt}(n)$ be the number of overcubic partition triples of $n$. For all natural numbers $n$ and $\\alpha$, both $\\mathrm{bt}(32n+28)$ and $\\mathrm{bt}(32n+20)$ are divisible by $16$, and each of $\\mathrm{bt}(64n+56)$, $\\mathrm{bt}(576n+408)$, $\\mathrm{bt}(5184n+216)$, $\\mathrm{bt}(5184n+3672)$ and $\\mathrm{bt}(64 \\cdot 81^{\\alpha+2} n + 216 \\cdot 81^{\\alpha+1})$ is divisible by $8$.","title":"Congruences for overcubic partition triples at multiples of 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-five-hundred-seve","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_fiveHundredSeventySix_mul_add_twentyFour","parent":11122,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overcubic partition triples of "],["math","\\(576n+24\\)"],["text"," modulo 8"]],"source":"Overcubic partition triples of $576n+24$ modulo 8"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-five-hundred-seve","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"Let $\\mathrm{bt}(n)$ be the number of overcubic partition triples of $n$, and let $f_1 = \\prod_{m \\ge 1} (1 - q^m)$. For every natural number $n$, $\\mathrm{bt}(576n+24) \\equiv 4 c_n \\pmod 8$, where $c_n$ is the coefficient of $q^n$ in $f_1$; that is, $\\sum_{n \\ge 0} \\mathrm{bt}(576n+24) q^n \\equiv 4 f_1 \\pmod 8$.","title":"Overcubic partition triples of $576n+24$ modulo 8"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-congruences-eight","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_congruences_eight_mod_sixteen_args","parent":11122,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-congruences-eight","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"Let $\\mathrm{bt}(n)$ be the number of overcubic partition triples of $n$. For all natural numbers $n$ and $\\alpha$, each of $\\mathrm{bt}(64n+40)$, $\\mathrm{bt}(576n+264)$, $\\mathrm{bt}(576n+456)$ and $\\mathrm{bt}(64 \\cdot 9^{\\alpha+2} n + 264 \\cdot 9^{\\alpha+1})$ is divisible by $8$.","title":"Congruences for overcubic partition triples at arguments congruent to 8 modulo 16"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-one-hundred-ninet","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_oneHundredNinetyTwo_mul_add_eight","parent":11122,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overcubic partition triples of "],["math","\\(192n+8\\)"],["text"," modulo "],["math","\\(4\\)"]],"source":"Overcubic partition triples of $192n+8$ modulo $4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-one-hundred-ninet","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{j \\ge 1} (1 - q^{kj})$, and let $\\mathrm{bt}(m)$ be the number of overcubic partition triples of $m$, the coefficient of $q^m$ in $f_4^3 / (f_1^6 f_2^3)$. For every natural number $n$, $\\mathrm{bt}(192n+8) \\equiv 2 c_n \\pmod 4$, where $c_n$ is the coefficient of $q^n$ in $f_1$; equivalently, $\\sum_{n \\ge 0} \\mathrm{bt}(192n+8) q^n \\equiv 2 f_1 \\pmod 4$ coefficientwise.","title":"Overcubic partition triples of $192n+8$ modulo $4$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/overcubic-triple-count-congruences-sixte","kind":"node","lean":"MetaMathlibExt.overcubicTripleCount_congruences_sixteen_mul_args","parent":11122,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples/index.html#overcubic-triple-count-congruences-sixte","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overcubic-partition-triples","status":"can_prove","summary":"Let $\\mathrm{bt}(n)$ be the number of overcubic partition triples of $n$. For all natural numbers $n$ and $\\alpha$, each of $\\mathrm{bt}(64n+48)$, $\\mathrm{bt}(128n+80)$, $\\mathrm{bt}(128n+96)$, $\\mathrm{bt}(256n+160)$ and $\\mathrm{bt}(256 \\cdot 2^{\\alpha} n + 192)$ is divisible by $4$, and $\\mathrm{bt}(256n+32) \\equiv \\mathrm{bt}(128n+16) \\pmod 4$.","title":"Congruences for overcubic partition triples at multiples of 16"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","status":"can_prove","title":"Overpartition congruences"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/overpartition-mul-five-mod-five","kind":"node","lean":"MetaMathlibExt.overpartition_mul_five_mod_five","parent":11138,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overpartitions of "],["math","\\(5n\\)"],["text"," modulo "],["math","\\(5\\)"],["text",", explicit form"]],"source":"Overpartitions of $5n$ modulo $5$, explicit form"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/index.html#overpartition-mul-five-mod-five","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","status":"fully_proved","summary":"For every integer $n \\ge 1$, the number of pairs $(P, S)$, where $P$ is a finite multiset of positive integers with sum $5n$ and $S$ is a finite set of natural numbers each belonging to $P$, is congruent modulo $5$ to $(-1)^n$ times the number of ordered triples $(x, y, z)$ of integers with $x^2 + y^2 + z^2 = n$.","title":"Overpartitions of $5n$ modulo $5$, explicit form"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/nat-card-eq-card-subtype","kind":"node","lean":"Nat.Overpartition.natCard_eq_card_subtype","parent":11138,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/index.html#nat-card-eq-card-subtype","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","status":"fully_proved","summary":"For every natural number $m$, the number of overpartitions of $m$ equals the number of pairs $(P, S)$ in which $P$ is a finite multiset of positive integers with sum $m$ and $S$ is a finite set of natural numbers each belonging to $P$.","title":"Overpartitions as pairs of a multiset and a set"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/three-square-representation-count-eq-nat","kind":"node","lean":"Nat.threeSquareRepresentationCount_eq_natCard","parent":11138,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/index.html#three-square-representation-count-eq-nat","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","status":"fully_proved","summary":"For every natural number $n$, the three-square representation count $r_3(n)$ equals the number of ordered triples $(x, y, z)$ of integers with $x^2 + y^2 + z^2 = n$.","title":"The three-square count counts all integer solutions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/overpartition-card-mul-five-mod-eq","kind":"node","lean":"MetaMathlibExt.overpartition_card_mul_five_modEq","parent":11138,"reading_title":{"format":"inline-parts-v1","parts":[["text","Overpartitions of "],["math","\\(5n\\)"],["text"," modulo "],["math","\\(5\\)"]],"source":"Overpartitions of $5n$ modulo $5$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/index.html#overpartition-card-mul-five-mod-eq","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","status":"fully_proved","summary":"For every integer $n \\ge 1$, the number of overpartitions of $5n$ is congruent modulo $5$ to $(-1)^n r_3(n)$, where $r_3(n)$ is the number of ordered triples $(x, y, z)$ of integers with $x^2 + y^2 + z^2 = n$.","title":"Overpartitions of $5n$ modulo $5$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/overpartition-three-pow-congr-zero-mod-5","kind":"node","lean":"MetaMathlibExt.overpartition_three_pow_congr_zero_mod_512","parent":11138,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/index.html#overpartition-three-pow-congr-zero-mod-5","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","status":"can_prove","summary":"For all natural numbers $\\alpha$ and $n$, the number $\\overline{p}(m)$ of overpartitions of $m$ satisfies $\\overline{p}(3^{8\\alpha+7}(24n+5)) \\equiv 0 \\pmod{512}$ and $\\overline{p}(3^{8\\alpha+7}(24n+13)) \\equiv 0 \\pmod{512}$.","title":"Liang's congruences modulo 512 for overpartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/euler-function-six-cube-nine-six-div-eq","kind":"node","lean":"MetaMathlibExt.eulerFunction_six_cube_nine_six_div_eq","parent":11138,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eta-quotient identity in the powers "],["math","\\(q^3\\)"],["text",", "],["math","\\(q^6\\)"],["text",", "],["math","\\(q^9\\)"],["text"," and "],["math","\\(q^{18}\\)"]],"source":"Eta-quotient identity in the powers $q^3$, $q^6$, $q^9$ and $q^{18}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/index.html#euler-function-six-cube-nine-six-div-eq","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for each integer $k \\ge 1$ write $E_k = (q^k; q^k)_\\infty$, the infinite q-Pochhammer symbol $\\prod_{n=0}^{\\infty} (1 - q^{k(n+1)})$. Then","title":"Eta-quotient identity in the powers $q^3$, $q^6$, $q^9$ and $q^{18}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/euler-function-one-div-two-sq-eq","kind":"node","lean":"MetaMathlibExt.eulerFunction_one_div_two_sq_eq","parent":11138,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-dissection of "],["math","\\((q;q)_\\infty / (q^2;q^2)_\\infty^2\\)"]],"source":"Three-dissection of $(q;q)_\\infty / (q^2;q^2)_\\infty^2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences/index.html#euler-function-one-div-two-sq-eq","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartition-congruences","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for each integer $k \\ge 1$ write $E_k = (q^k; q^k)_\\infty$, the infinite q-Pochhammer symbol $\\prod_{n=0}^{\\infty} (1 - q^{k(n+1)})$. Then","title":"Three-dissection of $(q;q)_\\infty / (q^2;q^2)_\\infty^2$"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","title":"Overpartitions with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-gf","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinctGF","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of "],["math","\\((j, k)\\)"],["text","-overpartition pairs with even parts distinct"]],"source":"Generating function of $(j, k)$-overpartition pairs with even parts distinct"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-gf","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For natural numbers $j$ and $k$, let $f_a = \\prod_{m \\ge 1} (1 - q^{am})$ be the Euler products. This generating function is the integer power series $f_4^2 \\bigl(\\prod_{i \\ge 0} (1 + q^{j+ki})\\bigr)^2 \\, g$, where $g$ is the inverse of $f_1^2$ (whose constant coefficient is $1$) and the infinite products are taken in the coefficientwise topology. In $q$-Pochhammer notation it is\u2026","title":"Generating function of $(j, k)$-overpartition pairs with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinct","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The number of "],["math","\\((j, k)\\)"],["text","-overpartition pairs with even parts distinct"]],"source":"The number of $(j, k)$-overpartition pairs with even parts distinct"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For natural numbers $j$, $k$ and $n$, $\\mathrm{bped}_{j,k}(n)$ is the integer coefficient of $q^n$ in the generating function of $[j, k]$-overpartition pairs with even parts distinct.","title":"The number of $(j, k)$-overpartition pairs with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-gf-mul","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinctGF_mul_qEulerProduct_sq","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cleared form of the generating function of "],["math","\\((j, k)\\)"],["text","-overpartition pairs with even parts distinct"]],"source":"Cleared form of the generating function of $(j, k)$-overpartition pairs with even parts distinct"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-gf-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"Cleared form of the generating function of $[j, k]$-overpartition pairs with even parts distinct. For natural numbers $j$ and $k$, let $f_a = \\prod_{m \\ge 1} (1 - q^{am})$ be the Euler products and let $G_{j,k}$ be the generating function of $[j, k]$-overpartition pairs with even parts distinct. Then, as integer power series in $q$, $G_{j,k} \\, f_1^2 = f_4^2 \\bigl(\\prod_{i \\ge 0} (1 + q^{j+ki})\\bigr)^2$, where the\u2026","title":"Cleared form of the generating function of $(j, k)$-overpartition pairs with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-zero","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinct_zero","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The count of overpartition pairs of "],["math","\\(0\\)"],["text"," with even parts distinct"]],"source":"The count of overpartition pairs of $0$ with even parts distinct"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For natural numbers $j \\ge 1$ and $k \\ge 1$, the number $\\mathrm{bped}_{j,k}(0)$ of $[j, k]$-overpartition pairs of $0$ with even parts distinct equals $1$ (the pair of empty overpartitions).","title":"The count of overpartition pairs of $0$ with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-three-t","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinct_three_three_congr_neg_six","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Paksok\u2013Saikia congruences for "],["math","\\((3, 3)\\)"],["text","-overpartition pairs (Theorem 1)"]],"source":"Paksok\u2013Saikia congruences for $(3, 3)$-overpartition pairs (Theorem 1)"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-three-t","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{bped}_{j,k}(n)$ be the number of $[j, k]$-overpartition pairs of $n$ with even parts distinct. Let $p > 5$ be a prime with Legendre symbol $\\left(\\frac{-6}{p}\\right) = -1$, and let $k, m, \\alpha, \\beta$ be natural numbers with $p \\nmid m$. Then $\\mathrm{bped}_{3,3}\\bigl(4 \\cdot 3^{4\\alpha+3} \\cdot 5^{2\\beta} p^{2k+1} m + (7 \\cdot 3^{4\\alpha+2} \\cdot 5^{2\\beta} p^{2k+2} - 1)/2\\bigr) \\equiv 0 \\pmod{16}$\u2026","title":"Paksok\u2013Saikia congruences for $(3, 3)$-overpartition pairs (Theorem 1)"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-congr-n","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinct_congr_neg_two_eleven","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Paksok\u2013Saikia congruences for "],["math","\\((3, 3)\\)"],["text","- and "],["math","\\((3, 6)\\)"],["text","-overpartition pairs (Theorem 2)"]],"source":"Paksok\u2013Saikia congruences for $(3, 3)$- and $(3, 6)$-overpartition pairs (Theorem 2)"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-congr-n","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{bped}_{j,k}(n)$ be the number of $[j, k]$-overpartition pairs of $n$ with even parts distinct. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-2}{p}\\right) = -1$, and let $k, m, \\alpha, \\beta$ be natural numbers with $p \\nmid m$. Then\u2026","title":"Paksok\u2013Saikia congruences for $(3, 3)$- and $(3, 6)$-overpartition pairs (Theorem 2)"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-congr-n-2","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinct_congr_neg_two_nineteen","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Paksok\u2013Saikia congruences for "],["math","\\((3, 3)\\)"],["text","- and "],["math","\\((3, 6)\\)"],["text","-overpartition pairs (Theorem 3)"]],"source":"Paksok\u2013Saikia congruences for $(3, 3)$- and $(3, 6)$-overpartition pairs (Theorem 3)"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-congr-n-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{bped}_{j,k}(n)$ be the number of $[j, k]$-overpartition pairs of $n$ with even parts distinct. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-2}{p}\\right) = -1$, and let $k, m, \\alpha, \\beta$ be natural numbers with $p \\nmid m$. Then\u2026","title":"Paksok\u2013Saikia congruences for $(3, 3)$- and $(3, 6)$-overpartition pairs (Theorem 3)"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-congr-n-3","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinct_congr_neg_one_five","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Paksok\u2013Saikia congruences for "],["math","\\((3, 3)\\)"],["text","- and "],["math","\\((3, 6)\\)"],["text","-overpartition pairs (Theorem 4)"]],"source":"Paksok\u2013Saikia congruences for $(3, 3)$- and $(3, 6)$-overpartition pairs (Theorem 4)"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-congr-n-3","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{bped}_{j,k}(n)$ be the number of $[j, k]$-overpartition pairs of $n$ with even parts distinct. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-1}{p}\\right) = -1$, and let $k, m, \\alpha, \\gamma$ be natural numbers with $p \\nmid m$. Then\u2026","title":"Paksok\u2013Saikia congruences for $(3, 3)$- and $(3, 6)$-overpartition pairs (Theorem 4)"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-congr-n-4","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinct_congr_neg_three_thirteen","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Paksok\u2013Saikia congruences for "],["math","\\((3, 3)\\)"],["text","- and "],["math","\\((3, 6)\\)"],["text","-overpartition pairs (Theorem 5)"]],"source":"Paksok\u2013Saikia congruences for $(3, 3)$- and $(3, 6)$-overpartition pairs (Theorem 5)"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-congr-n-4","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{bped}_{j,k}(n)$ be the number of $[j, k]$-overpartition pairs of $n$ with even parts distinct. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-3}{p}\\right) = -1$, and let $k, m, \\alpha, \\beta$ be natural numbers with $p \\nmid m$. Then\u2026","title":"Paksok\u2013Saikia congruences for $(3, 3)$- and $(3, 6)$-overpartition pairs (Theorem 5)"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/overpartition-pair-even-distinct-three-s","kind":"node","lean":"MetaMathlibExt.overpartitionPairEvenDistinct_three_six_congr_neg_one_five","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Paksok\u2013Saikia congruences for "],["math","\\((3, 6)\\)"],["text","-overpartition pairs (Theorem 6)"]],"source":"Paksok\u2013Saikia congruences for $(3, 6)$-overpartition pairs (Theorem 6)"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#overpartition-pair-even-distinct-three-s","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{bped}_{j,k}(n)$ be the number of $[j, k]$-overpartition pairs of $n$ with even parts distinct. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-1}{p}\\right) = -1$, and let $k, m, \\alpha$ be natural numbers with $p \\nmid m$. Then $\\mathrm{bped}_{3,6}\\bigl(3^{4\\alpha+2} p^{2k+1} (12m + 5p)\\bigr) \\equiv 0 \\pmod{16}$ and $\\mathrm{bped}_{3,6}\\bigl(3 p^{2k+1} (12m + 5p)\\bigr) \\equiv 0 \\pmod{16}$.","title":"Paksok\u2013Saikia congruences for $(3, 6)$-overpartition pairs (Theorem 6)"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/q-pochhammer-inf-five-dissection","kind":"node","lean":"MetaMathlibExt.qPochhammerInf_five_dissection","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#q-pochhammer-inf-five-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. For $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{n \\ge 0} (1 - q^k q^{kn})$, and for complex $a$, $b$ with $|a|, |b| < 1$ let $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's general theta function; all these products and series converge, so the raw infinite product and the raw theta sum take these values. For $|x| < 1$ put\u2026","title":"Ramanujan's 5-dissection of Euler's product"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/q-pochhammer-inf-seven-dissection","kind":"node","lean":"MetaMathlibExt.qPochhammerInf_seven_dissection","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#q-pochhammer-inf-seven-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$. For $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{n \\ge 0} (1 - q^k q^{kn})$, and for complex $a$, $b$ with $|a|, |b| < 1$ let $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's general theta function; all these products and series converge, so the raw infinite product and the raw theta sum take these values. For $|x| < 1$ put\u2026","title":"Ramanujan's 7-dissection of Euler's product"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-nine-series","kind":"node","lean":"MetaMathlibExt.pedNineNineSeries","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generating series for "],["math","\\(\\mathrm{ped}_{9,9}\\)"]],"source":"The generating series for $\\mathrm{ped}_{9,9}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-nine-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$ with constant coefficient $1$. This is the integer power series","title":"The generating series for $\\mathrm{ped}_{9,9}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-nine","kind":"node","lean":"MetaMathlibExt.pedNineNine","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partition function "],["math","\\(\\mathrm{ped}_{9,9}\\)"]],"source":"The partition function $\\mathrm{ped}_{9,9}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-nine","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$ with constant coefficient $1$. For a natural number $n$, $\\mathrm{ped}_{9,9}(n)$ is the integer coefficient of $q^n$ in the power series $f_4 f_{18} / (f_1 f_9)$.","title":"The partition function $\\mathrm{ped}_{9,9}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-eighteen-series","kind":"node","lean":"MetaMathlibExt.pedNineEighteenSeries","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generating series for "],["math","\\(\\mathrm{ped}_{9,18}\\)"]],"source":"The generating series for $\\mathrm{ped}_{9,18}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-eighteen-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$ with constant coefficient $1$. This is the integer power series","title":"The generating series for $\\mathrm{ped}_{9,18}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-eighteen","kind":"node","lean":"MetaMathlibExt.pedNineEighteen","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partition function "],["math","\\(\\mathrm{ped}_{9,18}\\)"]],"source":"The partition function $\\mathrm{ped}_{9,18}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-eighteen","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$ with constant coefficient $1$. For a natural number $n$, $\\mathrm{ped}_{9,18}(n)$ is the integer coefficient of $q^n$ in the power series $f_4 f_{18}^2 / (f_1 f_9 f_{36})$.","title":"The partition function $\\mathrm{ped}_{9,18}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-nine-generating-function","kind":"node","lean":"MetaMathlibExt.pedNineNine_generatingFunction","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function identity for "],["math","\\(\\mathrm{ped}_{9,9}\\)"]],"source":"Generating function identity for $\\mathrm{ped}_{9,9}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-nine-generating-function","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$, and let $\\mathrm{ped}_{9,9}(n)$ be the coefficient of $q^n$ in $f_4 f_{18} / (f_1 f_9)$. Then, as integer power series in $q$,","title":"Generating function identity for $\\mathrm{ped}_{9,9}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-eighteen-generating-function","kind":"node","lean":"MetaMathlibExt.pedNineEighteen_generatingFunction","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-eighteen-generating-function","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"Let $\\mathrm{ped}_{9,18}(n)$ be the number of $[9, 18]$-overpartitions of $n$ with even parts distinct, and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then, as integer power series in $q$, $f_1 f_9 f_{36} \\sum_{n \\ge 0} \\mathrm{ped}_{9,18}(n) q^n = f_4 f_{18}^2$.","title":"Product identity for the (9, 18)-overpartition generating function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-nine-zero-and-ped-nine-eighteen","kind":"node","lean":"MetaMathlibExt.pedNineNine_zero_and_pedNineEighteen_zero","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of "],["math","\\(\\mathrm{ped}_{9,9}\\)"],["text"," and "],["math","\\(\\mathrm{ped}_{9,18}\\)"],["text"," at "],["math","\\(0\\)"]],"source":"Values of $\\mathrm{ped}_{9,9}$ and $\\mathrm{ped}_{9,18}$ at $0$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-nine-zero-and-ped-nine-eighteen","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$, and let $\\mathrm{ped}_{9,9}(n)$ and $\\mathrm{ped}_{9,18}(n)$ be the coefficients of $q^n$ in $f_4 f_{18} / (f_1 f_9)$ and in $f_4 f_{18}^2 / (f_1 f_9 f_{36})$. Then $\\mathrm{ped}_{9,9}(0) = \\mathrm{ped}_{9,18}(0) = 1$.","title":"Values of $\\mathrm{ped}_{9,9}$ and $\\mathrm{ped}_{9,18}$ at $0$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/chan-cubic-matrix","kind":"node","lean":"MetaMathlibExt.chanCubicMatrix","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#chan-cubic-matrix","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"Chan's matrix $(m_{i,j})_{i,j \\ge 1}$ is the integer matrix with first column $m_{1,1} = 3$, $m_{2,1} = 2$, $m_{3,1} = 1$ and $m_{i,1} = 0$ for $i \\ge 4$, and with $m_{i,j} = 9 m_{i-1,j-1} + 3 m_{i-2,j-1} + m_{i-3,j-1}$ for $i \\ge 1$ and $j \\ge 2$, where entries with row index $\\le 0$ are taken to be $0$.","title":"Chan's matrix"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/chan-cubic-matrix-first-rows","kind":"node","lean":"MetaMathlibExt.chanCubicMatrix_first_rows","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#chan-cubic-matrix-first-rows","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"The upper left $6 \\times 6$ block of Chan's matrix $(m_{i,j})_{i,j \\ge 1}$ is lower triangular, and its rows up to the diagonal are $(3)$, $(2, 27)$, $(1, 27, 243)$, $(0, 18, 324, 2187)$, $(0, 5, 270, 3645, 19683)$ and $(0, 1, 126, 3645, 39366, 177147)$: for $1 \\le j \\le i \\le 6$, $m_{i,j}$ is the $j$th entry of the $i$th listed row, and $m_{i,j} = 0$ for $1 \\le i < j \\le 6$.","title":"First six rows of Chan's matrix"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/cubic-coeff-vector","kind":"node","lean":"MetaMathlibExt.cubicCoeffVector","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#cubic-coeff-vector","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"Let $(m_{i,j})_{i,j \\ge 1}$ be Chan's matrix, and for $i, j \\ge 1$ let $a_{i,j} = m_{4i-3,i+j-1}$ and $b_{i,j} = m_{4i-1,i+j-1}$ be the entries of the infinite matrices $A$ and $B$. The coefficient vectors $X_\\alpha = (x_{\\alpha,1}, x_{\\alpha,2}, \\dots)$ of integers, for $\\alpha \\ge 0$, are defined by $X_0 = (1, 0, 0, \\dots)$ and, for $j \\ge 1$, $x_{\\alpha+1,j} = \\sum_{i=1}^{3^\\alpha} x_{\\alpha,i} a_{i,j}$ if\u2026","title":"Coefficient vectors from Chan's matrix"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/cubic-coeff-vector-one-two","kind":"node","lean":"MetaMathlibExt.cubicCoeffVector_one_two","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#cubic-coeff-vector-one-two","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"fully_proved","summary":"The coefficient vectors $X_\\alpha = (x_{\\alpha,1}, x_{\\alpha,2}, \\dots)$ built from Chan's matrix satisfy $(x_{1,1}, \\dots, x_{1,5}) = (3, 0, 0, 0, 0)$ and $(x_{2,1}, \\dots, x_{2,5}) = (3, 81, 729, 0, 0)$.","title":"Initial values of the cubic coefficient vectors"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/q-euler-product-fourth-two-dissection","kind":"node","lean":"MetaMathlibExt.qEulerProduct_fourth_two_dissection","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(f_1^4\\)"]],"source":"The 2-dissection of $f_1^4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#q-euler-product-fourth-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$. Then","title":"The 2-dissection of $f_1^4$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/q-euler-product-sq-div-three-dissection","kind":"node","lean":"MetaMathlibExt.qEulerProduct_sq_div_three_dissection","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#q-euler-product-sq-div-three-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"In the ring of integer power series in $q$, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products. Then $f_1^2 f_6 f_9 f_{18} = f_2 f_6 f_9^3 - 2q f_2 f_3 f_{18}^3$. Since every $f_k$ with $k \\ge 1$ has constant term $1$, this is the 3-dissection $f_1^2 / f_2 = f_9^2 / f_{18} - 2q f_3 f_{18}^2 / (f_6 f_9)$ with denominators cleared.","title":"A 3-dissection of a quotient of Euler products"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-nine-congruences-mod-thirty-two","kind":"node","lean":"MetaMathlibExt.pedNineNine_congruences_mod_thirtyTwo","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo 32 for "],["math","\\(\\mathrm{ped}_{9,9}\\)"]],"source":"Congruences modulo 32 for $\\mathrm{ped}_{9,9}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-nine-congruences-mod-thirty-two","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$, and let $\\mathrm{ped}_{9,9}(n)$ be the coefficient of $q^n$ in $f_4 f_{18} / (f_1 f_9)$. Then for all natural numbers $\\alpha$, $\\beta$ and $n$ the following congruences modulo $32$ hold, where every division is exact.","title":"Congruences modulo 32 for $\\mathrm{ped}_{9,9}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-nine-congruences-mod-sixteen","kind":"node","lean":"MetaMathlibExt.pedNineNine_congruences_mod_sixteen","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo 16 for "],["math","\\(\\mathrm{ped}_{9,9}\\)"]],"source":"Congruences modulo 16 for $\\mathrm{ped}_{9,9}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-nine-congruences-mod-sixteen","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$, and let $\\mathrm{ped}_{9,9}(n)$ be the coefficient of $q^n$ in $f_4 f_{18} / (f_1 f_9)$. Then for all natural numbers $\\alpha$, $\\beta$, $\\gamma$ and $n$ the following congruences modulo $16$ hold, where every division is exact.","title":"Congruences modulo 16 for $\\mathrm{ped}_{9,9}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-nine-more-congruences-mod-sixte","kind":"node","lean":"MetaMathlibExt.pedNineNine_more_congruences_mod_sixteen","parent":11146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Further congruences modulo 16 for "],["math","\\(\\mathrm{ped}_{9,9}\\)"]],"source":"Further congruences modulo 16 for $\\mathrm{ped}_{9,9}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-nine-more-congruences-mod-sixte","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, an integer power series in $q$, and let $\\mathrm{ped}_{9,9}(n)$ be the coefficient of $q^n$ in $f_4 f_{18} / (f_1 f_9)$. Then for all natural numbers $\\alpha$, $\\beta$, $\\gamma$ and $n$ the following congruences modulo $16$ hold, where every division is exact.","title":"Further congruences modulo 16 for $\\mathrm{ped}_{9,9}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-eighteen-congruences-mod-sixty","kind":"node","lean":"MetaMathlibExt.pedNineEighteen_congruences_mod_sixtyFour","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-eighteen-congruences-mod-sixty","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{ped}_{9,18}(n)$ be the number of $[9, 18]$-overpartitions of $n$ with even parts distinct, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and read congruences between integer power series in $q$ coefficientwise. For all $\\alpha \\ge 0$ the following hold modulo $64$, where the statements about single values hold for every $n \\ge 0$: (73)\u2026","title":"Congruences modulo 64 for (9, 18)-overpartitions with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-eighteen-congruences-mod-thirty","kind":"node","lean":"MetaMathlibExt.pedNineEighteen_congruences_mod_thirtyTwo","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-eighteen-congruences-mod-thirty","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{ped}_{9,18}(n)$ be the number of $[9, 18]$-overpartitions of $n$ with even parts distinct, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and read congruences between integer power series in $q$ coefficientwise. For all $\\alpha, \\beta \\ge 0$ the following hold modulo $32$, where the statements about single values hold for every $n \\ge 0$: (94)\u2026","title":"Congruences modulo 32 for (9, 18)-overpartitions with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-eighteen-congruences-mod-sixtee","kind":"node","lean":"MetaMathlibExt.pedNineEighteen_congruences_mod_sixteen","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-eighteen-congruences-mod-sixtee","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{ped}_{9,18}(n)$ be the number of $[9, 18]$-overpartitions of $n$ with even parts distinct, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, and read congruences between integer power series in $q$ coefficientwise. For all $\\alpha, \\beta, \\gamma \\ge 0$ the following hold, where the statements about single values hold for every $n \\ge 0$: (105)\u2026","title":"Congruences modulo 16 and 8 for (9, 18)-overpartitions with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-eighteen-congruences-mod-nine-t","kind":"node","lean":"MetaMathlibExt.pedNineEighteen_congruences_mod_nine_twentySeven","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-eighteen-congruences-mod-nine-t","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{ped}_{9,18}(n)$ be the number of $[9, 18]$-overpartitions of $n$ with even parts distinct. For every $n \\ge 0$, $\\mathrm{ped}_{9,18}(27n + 16)$ and $\\mathrm{ped}_{9,18}(27n + 25)$ are divisible by $9$, and $\\mathrm{ped}_{9,18}(243n + 142)$, $\\mathrm{ped}_{9,18}(243n + 223)$ and $\\mathrm{ped}_{9,18}(729n + 547)$ are divisible by $27$.","title":"Congruences modulo 9 and 27 for (9, 18)-overpartitions with even parts distinct"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/ped-nine-eighteen-cubic-coeff-vector-con","kind":"node","lean":"MetaMathlibExt.pedNineEighteen_cubicCoeffVector_congruences","parent":11146,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct/index.html#ped-nine-eighteen-cubic-coeff-vector-con","scope":"NumberTheory/modular-forms-q-series/partition-congruences/overpartitions-with-even-parts-distinct","status":"can_prove","summary":"Let $\\mathrm{ped}_{9,18}(n)$ be the number of $[9, 18]$-overpartitions of $n$ with even parts distinct, let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$, let $X_\\alpha = (x_{\\alpha,1}, x_{\\alpha,2}, \\dots)$ be the coefficient vectors built from Chan's matrix, and read congruences between integer power series in $q$ coefficientwise. For all $\\alpha \\ge 0$, modulo $27$, (124)\u2026","title":"Congruences modulo 27 for (9, 18)-overpartitions from the cubic coefficient vectors"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","title":"Partitions with designated summands"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-series","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsSeries","parent":11180,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"fully_proved","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ be the Euler product. This is the integer power series $f_4 f_6^2 / (f_1 f_3 f_{12})$ in $q$, where the denominator $f_1 f_3 f_{12}$ is invertible because its constant coefficient is $1$. It is the generating function $\\sum_{n \\ge 0} \\mathrm{PD}_2(n) q^n$ of the numbers $\\mathrm{PD}_2(n)$ of partitions of $n$ with designated summands into odd parts.","title":"Generating function of odd designated-summand partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount","parent":11180,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"fully_proved","summary":"For a natural number $n$, the integer $\\mathrm{PD}_2(n)$ is defined as the coefficient of $q^n$ in the integer power series $f_4 f_6^2 / (f_1 f_3 f_{12})$, where $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ is the Euler product (the denominator has constant coefficient $1$, so it is invertible). It counts the partitions of $n$ with designated summands in which all parts are odd.","title":"The odd designated-summand partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-generating","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_generatingFunction","parent":11180,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-generating","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"fully_proved","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ in the ring of integer power series in $q$, and let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd, defined as the coefficient of $q^n$ in $f_4 f_6^2 / (f_1 f_3 f_{12})$. Then $f_1 f_3 f_{12} \\sum_{n \\ge 0} PD_2(n) q^n = f_4 f_6^2$.","title":"Generating function for partitions with designated summands into odd parts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-zero","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_zero","parent":11180,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"fully_proved","summary":"The odd designated-summand partition function satisfies $\\mathrm{PD}_2(0) = 1$.","title":"Odd designated-summand partitions of zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/f1-mul-f3-two-dissection","kind":"node","lean":"MetaMathlibExt.f1_mul_f3_two_dissection","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(f_1 f_3\\)"]],"source":"The 2-dissection of $f_1 f_3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#f1-mul-f3-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products in the ring of integer power series in $q$. Then","title":"The 2-dissection of $f_1 f_3$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-two-pow-mu","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_two_pow_mul_four_mul_add_three","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{PD}_2(2^\\alpha (4n + 3))\\)"],["text"," is divisible by 4"]],"source":"$\\mathrm{PD}_2(2^\\alpha (4n + 3))$ is divisible by 4"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-two-pow-mu","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd. For all natural numbers $\\alpha$ and $n$, $\\mathrm{PD}_2(2^\\alpha (4n + 3)) \\equiv 0 \\pmod 4$.","title":"$\\mathrm{PD}_2(2^\\alpha (4n + 3))$ is divisible by 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-two-pow-mu-2","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_two_pow_mul_eight_mul_add_seven","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{PD}_2(2^\\alpha (8n + 7))\\)"],["text"," is divisible by 8"]],"source":"$\\mathrm{PD}_2(2^\\alpha (8n + 7))$ is divisible by 8"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-two-pow-mu-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd. For all natural numbers $\\alpha$ and $n$, $\\mathrm{PD}_2(2^\\alpha (8n + 7)) \\equiv 0 \\pmod 8$.","title":"$\\mathrm{PD}_2(2^\\alpha (8n + 7))$ is divisible by 8"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-two-dissec","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_two_dissection","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of the generating function of "],["math","\\(\\mathrm{PD}_2\\)"]],"source":"The 2-dissection of the generating function of $\\mathrm{PD}_2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-two-dissec","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd, and write $f_k = \\prod_{j \\ge 1} (1 - q^{kj})$ for the Euler product. In the ring of integer power series in $q$, $f_1^2 f_3^2 f_{12}^2 \\sum_{n \\ge 0} \\mathrm{PD}_2(2n) q^n = f_4^2 f_6^4$ and $f_1^4 f_4^2 f_6^2 \\sum_{n \\ge 0} \\mathrm{PD}_2(2n + 1) q^n = f_2^6 f_{12}^2$.","title":"The 2-dissection of the generating function of $\\mathrm{PD}_2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-four-disse","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_four_dissection","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 4-dissection of the generating function of "],["math","\\(PD_2\\)"]],"source":"The 4-dissection of the generating function of $PD_2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-four-disse","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ in the ring of integer power series in $q$, and let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd. Then $f_1^8 f_3^4 f_4^4 f_6^4 f_{12}^4 \\sum_{n \\ge 0} PD_2(4n) q^n = f_1^4 f_4^8 f_6^{12} + q f_2^{12} f_3^4 f_{12}^8$, $f_1^8 f_3^2 f_4^4 \\sum_{n \\ge 0} PD_2(4n + 1) q^n = f_2^{12} f_6^2$,\u2026","title":"The 4-dissection of the generating function of $PD_2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-two-dissec-2","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_two_dissection_mod_four","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-dissection of "],["math","\\(\\mathrm{PD}_2\\)"],["text"," modulo 4"]],"source":"The 2-dissection of $\\mathrm{PD}_2$ modulo 4"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-two-dissec-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd, and write $f_k = \\prod_{j \\ge 1} (1 - q^{kj})$ for the Euler product, an integer power series in $q$. For every natural number $n$, the $n$th coefficient of $f_1^2 f_3^2 \\sum_{k \\ge 0} \\mathrm{PD}_2(2k) q^k$ is congruent modulo $4$ to the $n$th coefficient of $f_4^2$, and $\\mathrm{PD}_2(2n + 1)$ is\u2026","title":"The 2-dissection of $\\mathrm{PD}_2$ modulo 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-odd-args-m","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_odd_args_mod_four","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo 4 for "],["math","\\(\\mathrm{PD}_2\\)"],["text"," at "],["math","\\(4n + 3\\)"],["text",", "],["math","\\(6n + 3\\)"],["text"," and "],["math","\\(6n + 5\\)"]],"source":"Congruences modulo 4 for $\\mathrm{PD}_2$ at $4n + 3$, $6n + 3$ and $6n + 5$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-odd-args-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd. For every natural number $n$, $\\mathrm{PD}_2(4n + 3)$, $\\mathrm{PD}_2(6n + 3)$ and $\\mathrm{PD}_2(6n + 5)$ are all congruent to $0$ modulo $4$.","title":"Congruences modulo 4 for $\\mathrm{PD}_2$ at $4n + 3$, $6n + 3$ and $6n + 5$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-four-disse-2","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_four_dissection_mod_four","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(PD_2(4n)\\)"],["text"," and "],["math","\\(PD_2(4n+2)\\)"],["text"," modulo 4"]],"source":"$PD_2(4n)$ and $PD_2(4n+2)$ modulo 4"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-four-disse-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ in the ring of integer power series in $q$, and let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd. For every $n \\ge 0$, the coefficient of $q^n$ in $f_6^2 \\sum_{k \\ge 0} PD_2(4k) q^k$ is congruent modulo $4$ to the coefficient of $q^n$ in $f_2^6 + q f_6^2 f_{12}^2$, and $PD_2(4n + 2)$ is congruent modulo $4$ to the\u2026","title":"$PD_2(4n)$ and $PD_2(4n+2)$ modulo 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-eight-mul","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_eight_mul_add_six","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(PD_2(8n+6)\\)"],["text"," is divisible by 4"]],"source":"$PD_2(8n+6)$ is divisible by 4"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-eight-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd. For every $n \\ge 0$, $PD_2(8n + 6) \\equiv 0 \\pmod 4$.","title":"$PD_2(8n+6)$ is divisible by 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-eight-diss","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_eight_dissection_mod_four","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(PD_2(8n)\\)"],["text"," and "],["math","\\(PD_2(8n+4)\\)"],["text"," modulo 4"]],"source":"$PD_2(8n)$ and $PD_2(8n+4)$ modulo 4"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-eight-diss","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ in the ring of integer power series in $q$, and let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd. For every $n \\ge 0$, the coefficient of $q^n$ in $f_3^2 \\sum_{k \\ge 0} PD_2(8k) q^k$ is congruent modulo $4$ to the coefficient of $q^n$ in $f_1^6$, and $PD_2(8n + 4)$ is congruent modulo $4$ to the coefficient of $q^n$ in\u2026","title":"$PD_2(8n)$ and $PD_2(8n+4)$ modulo 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-sixteen-mu","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_sixteen_mul_add_twelve","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{PD}_2(16n + 12)\\)"],["text"," is divisible by 4"]],"source":"$\\mathrm{PD}_2(16n + 12)$ is divisible by 4"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-sixteen-mu","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd. For every natural number $n$, $\\mathrm{PD}_2(16n + 12) \\equiv 0 \\pmod 4$.","title":"$\\mathrm{PD}_2(16n + 12)$ is divisible by 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-four-mul-m","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_four_mul_mod_four","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["text","The internal congruence "],["math","\\(PD_2(4n) \\equiv PD_2(n) \\pmod 4\\)"]],"source":"The internal congruence $PD_2(4n) \\equiv PD_2(n) \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-four-mul-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd. For every $n \\ge 0$, $PD_2(4n) \\equiv PD_2(n) \\pmod 4$.","title":"The internal congruence $PD_2(4n) \\equiv PD_2(n) \\pmod 4$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-four-pow-m","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_four_pow_mul_six_mul_add_five","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(PD_2(4^\\alpha (6n+5))\\)"],["text"," is divisible by 4"]],"source":"$PD_2(4^\\alpha (6n+5))$ is divisible by 4"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-four-pow-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd. For all $\\alpha \\ge 0$ and $n \\ge 0$, $PD_2(4^\\alpha (6n + 5)) \\equiv 0 \\pmod 4$.","title":"$PD_2(4^\\alpha (6n+5))$ is divisible by 4"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-eight-mul-2","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_eight_mul_add_seven","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(PD_2(8n+7)\\)"],["text"," is divisible by 8"]],"source":"$PD_2(8n+7)$ is divisible by 8"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-eight-mul-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd. For every $n \\ge 0$, $PD_2(8n + 7) \\equiv 0 \\pmod 8$.","title":"$PD_2(8n+7)$ is divisible by 8"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-sixteen-mu-2","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_sixteen_mul_add_fourteen","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{PD}_2(16n + 14)\\)"],["text"," is divisible by 8"]],"source":"$\\mathrm{PD}_2(16n + 14)$ is divisible by 8"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-sixteen-mu-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd. For every natural number $n$, $\\mathrm{PD}_2(16n + 14) \\equiv 0 \\pmod 8$.","title":"$\\mathrm{PD}_2(16n + 14)$ is divisible by 8"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-thirty-two","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_thirtyTwo_mul_add_twentyEight","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{PD}_2(32n + 28)\\)"],["text"," is divisible by 8"]],"source":"$\\mathrm{PD}_2(32n + 28)$ is divisible by 8"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-thirty-two","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd. For every natural number $n$, $\\mathrm{PD}_2(32n + 28) \\equiv 0 \\pmod 8$.","title":"$\\mathrm{PD}_2(32n + 28)$ is divisible by 8"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-sixty-four","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_sixtyFour_mul_add_fiftySix","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{PD}_2(64n + 56)\\)"],["text"," is divisible by 8"]],"source":"$\\mathrm{PD}_2(64n + 56)$ is divisible by 8"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-sixty-four","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd. For every natural number $n$, $\\mathrm{PD}_2(64n + 56) \\equiv 0 \\pmod 8$.","title":"$\\mathrm{PD}_2(64n + 56)$ is divisible by 8"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-sixteen-mu-3","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_sixteen_mul_mod_eight","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["text","Internal congruence "],["math","\\(\\mathrm{PD}_2(16n) \\equiv \\mathrm{PD}_2(4n) \\pmod 8\\)"]],"source":"Internal congruence $\\mathrm{PD}_2(16n) \\equiv \\mathrm{PD}_2(4n) \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-sixteen-mu-3","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Write $\\mathrm{PD}_2(m)$ for the number of partitions of $m$ with designated summands in which every part is odd. For every natural number $n$, $\\mathrm{PD}_2(16n) \\equiv \\mathrm{PD}_2(4n) \\pmod 8$.","title":"Internal congruence $\\mathrm{PD}_2(16n) \\equiv \\mathrm{PD}_2(4n) \\pmod 8$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/odd-designated-summands-count-forty-eigh","kind":"node","lean":"MetaMathlibExt.oddDesignatedSummandsCount_fortyEight_mul_add_twentySix","parent":11180,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(PD_2(48n+26)\\)"],["text"," is divisible by 4"]],"source":"$PD_2(48n+26)$ is divisible by 4"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands/index.html#odd-designated-summands-count-forty-eigh","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-designated-summands","status":"can_prove","summary":"Let $PD_2(n)$ be the number of partitions of $n$ with designated summands in which every part is odd. For every $n \\ge 0$, $PD_2(48n + 26) \\equiv 0 \\pmod 4$.","title":"$PD_2(48n+26)$ is divisible by 4"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts","status":"can_prove","title":"Partitions with distinct even parts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts/partitions-even-parts-distinct-mod24-pro","kind":"node","lean":"MetaMathlibExt.partitionsEvenPartsDistinct_mod24_progressions","parent":11204,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts/index.html#partitions-even-parts-distinct-mod24-pro","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts","status":"can_prove","summary":"Write $\\mathrm{ped}(m)$ for the number of partitions of $m$ in which the even parts are distinct (odd parts may repeat). For every natural number $n$, each of $\\mathrm{ped}(225n+43)$, $\\mathrm{ped}(225n+88)$, $\\mathrm{ped}(225n+133)$ and $\\mathrm{ped}(225n+223)$ is congruent to $0$ modulo $24$.","title":"Congruences modulo 24 for partitions with distinct even parts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts/partitions-even-parts-distinct-mod24-fiv","kind":"node","lean":"MetaMathlibExt.partitionsEvenPartsDistinct_mod24_five_power","parent":11204,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts/index.html#partitions-even-parts-distinct-mod24-fiv","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-distinct-even-parts","status":"can_prove","summary":"Write $\\mathrm{ped}(m)$ for the number of partitions of $m$ in which the even parts are distinct (odd parts may repeat). For all natural numbers $k \\ge 1$ and $n$,","title":"Power-of-five congruence modulo 24 for partitions with distinct even parts"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","title":"Partitions with initial repetitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/is-square-mod-two-pow-odd-not-even-iff","kind":"node","lean":"MetaMathlibExt.isSquare_mod_two_pow_odd_not_even_iff","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residues that are squares modulo "],["math","\\(2^{2k-1}\\)"],["text"," but not modulo "],["math","\\(2^{2k}\\)"]],"source":"Residues that are squares modulo $2^{2k-1}$ but not modulo $2^{2k}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#is-square-mod-two-pow-odd-not-even-iff","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $k \\ge 0$ and let $x$ be an integer with $0 \\le x < 2^{2k}$, where the exponents $2k - 1$ and $2k - 2$ are truncated at $0$. Then $x$ is a square modulo $2^{2k-1}$ but not a square modulo $2^{2k}$ if and only if $x = 2^{2k-1}$ or $x = 2^{2k-1} + 2^{2k-2}$.","title":"Residues that are squares modulo $2^{2k-1}$ but not modulo $2^{2k}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/is-square-mod-two-pow-even-not-odd-iff","kind":"node","lean":"MetaMathlibExt.isSquare_mod_two_pow_even_not_odd_iff","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residues that are squares modulo "],["math","\\(2^{2k}\\)"],["text"," but not modulo "],["math","\\(2^{2k+1}\\)"]],"source":"Residues that are squares modulo $2^{2k}$ but not modulo $2^{2k+1}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#is-square-mod-two-pow-even-not-odd-iff","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $k \\ge 0$ and let $x$ be an integer with $0 \\le x < 2^{2k+1}$, where the exponent $2k - 2$ is truncated at $0$. Then $x$ is a square modulo $2^{2k}$ but not a square modulo $2^{2k+1}$ if and only if $x = 2^{2k} + 2^{2k-2}$.","title":"Residues that are squares modulo $2^{2k}$ but not modulo $2^{2k+1}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/card-not-is-square-sixteen-mul-add-one-e","kind":"node","lean":"MetaMathlibExt.card_not_isSquare_sixteen_mul_add_one_eight_mul_add_one","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residues "],["math","\\(r\\)"],["text"," with "],["math","\\(16r + 1\\)"],["text"," and "],["math","\\(8r + 1\\)"],["text"," both non-squares modulo "],["math","\\(p\\)"]],"source":"Residues $r$ with $16r + 1$ and $8r + 1$ both non-squares modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#card-not-is-square-sixteen-mul-add-one-e","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"For every prime $p$, the number of integers $r$ with $1 \\le r \\le p - 1$ such that neither $16r + 1$ nor $8r + 1$ is a square modulo $p$ equals $\\lfloor p/4 \\rfloor$.","title":"Residues $r$ with $16r + 1$ and $8r + 1$ both non-squares modulo $p$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/card-not-is-square-twelve-mul-add-one-ei","kind":"node","lean":"MetaMathlibExt.card_not_isSquare_twelve_mul_add_one_eight_mul_add_one","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residues "],["math","\\(r\\)"],["text"," with "],["math","\\(12r + 1\\)"],["text"," and "],["math","\\(8r + 1\\)"],["text"," both non-squares modulo "],["math","\\(p\\)"]],"source":"Residues $r$ with $12r + 1$ and $8r + 1$ both non-squares modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#card-not-is-square-twelve-mul-add-one-ei","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"For every prime $p$, let $N$ be the number of integers $r$ with $1 \\le r \\le p - 1$ such that neither $12r + 1$ nor $8r + 1$ is a square modulo $p$. Then $N = \\lfloor p/4 \\rfloor - 1$ if $p \\equiv 5 \\pmod{24}$, $N = \\lfloor p/4 \\rfloor + 1$ if $p \\equiv 11 \\pmod{24}$, and $N = \\lfloor p/4 \\rfloor$ otherwise.","title":"Residues $r$ with $12r + 1$ and $8r + 1$ both non-squares modulo $p$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/four-mul-card-legendre-sym-pair-eq-neg-o","kind":"node","lean":"MetaMathlibExt.four_mul_card_legendreSym_pair_eq_neg_one","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(r\\)"],["text"," with "],["math","\\(mr + 1\\)"],["text"," and "],["math","\\(nr + 1\\)"],["text"," both quadratic nonresidues"]],"source":"Counting $r$ with $mr + 1$ and $nr + 1$ both quadratic nonresidues"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#four-mul-card-legendre-sym-pair-eq-neg-o","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $p$ be an odd prime, write $\\left(\\frac{a}{p}\\right)$ for the Legendre symbol of an integer $a$, and let $m$ and $n$ be integers with $p \\nmid mn(n - m)$. If $N$ is the number of integers $r$ with $0 \\le r \\le p - 1$ such that $\\left(\\frac{mr+1}{p}\\right) = \\left(\\frac{nr+1}{p}\\right) = -1$, then","title":"Counting $r$ with $mr + 1$ and $nr + 1$ both quadratic nonresidues"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/card-initial-repetitions-eq-card-not-dvd","kind":"node","lean":"MetaMathlibExt.card_initialRepetitions_eq_card_not_dvd_eq_card_count_le","parent":11207,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#card-initial-repetitions-eq-card-not-dvd","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 0$ be integers. Say that a partition has initial $k$-repetitions if, whenever a part $j$ appears at least $k$ times, every integer $i$ with $1 \\le i < j$ appears at least $k$ times. Then the number of partitions of $n$ with initial $k$-repetitions equals the number of partitions of $n$ into parts not divisible by $2k$, and this number equals the number of partitions of $n$ in which no part\u2026","title":"Andrews' theorem on partitions with initial repetitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c2","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC2","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partition function "],["math","\\(c_2\\)"],["text"," of Nyirenda and Mugwangwavari"]],"source":"The partition function $c_2$ of Nyirenda and Mugwangwavari"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"fully_proved","summary":"For an integer $n \\ge 0$, $c_2(n)$ is the number of partitions of $n$ for which there is an integer $j$ with $1 \\le j \\le n$ such that $j$ appears exactly $j$ times, every part other than $j$ is at least $2j + 1$, and every even part that is at least $2j + 2$ appears only once.","title":"The partition function $c_2$ of Nyirenda and Mugwangwavari"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c3","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC3","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","The initial-repetitions partition function "],["math","\\(c_3\\)"]],"source":"The initial-repetitions partition function $c_3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c3","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"fully_proved","summary":"For $n \\ge 0$, $c_3(n)$ is the number of partitions of $n$ for which there is an integer $j$ with $1 \\le j \\le n$ such that the part $1$ appears exactly $j^2$ or $j^2 + 1$ times, every odd part larger than $1$ appears exactly once, every even part appears exactly once, and every even part greater than $2j$ is at least $4j + 4$ and divisible by $4$.","title":"The initial-repetitions partition function $c_3$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partition function "],["math","\\(c_8\\)"]],"source":"The partition function $c_8$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"fully_proved","summary":"For $n \\ge 0$, $c_8(n)$ is the number of partitions of $n$ such that either (a) all parts are even and distinct, or (b) for some positive integer $j$, the largest odd part is $2j - 1$, every positive odd integer $i \\le j$ appears once or twice, every odd integer $i$ with $j < i \\le 2j - 1$ appears exactly once, all even parts are distinct, and every even part greater than $j$ is at least $2j + 2$.","title":"The partition function $c_8$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8-prime","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8Prime","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","The initial-repetitions partition function "],["math","\\(c_8'\\)"]],"source":"The initial-repetitions partition function $c_8'$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8-prime","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"fully_proved","summary":"For $n \\ge 0$, $c_8'(n)$ is the number of partitions of $n$ into distinct even parts plus the sum, over $m = 1, \\dots, n$, of the number of partitions of $n$ in which every even number $2i$ with $1 \\le i \\le m$ is a part, every even part at most $m$ appears at most twice, every odd part at most $m$ appears exactly once, and every part greater than $m$ is even and appears exactly once.","title":"The initial-repetitions partition function $c_8'$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-values-zero-one","kind":"node","lean":"MetaMathlibExt.initialRepetitions_values_zero_one","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small values of the partition functions "],["math","\\(c_2\\)"],["text",", "],["math","\\(c_3\\)"],["text",", "],["math","\\(c_8\\)"],["text"," and "],["math","\\(c_8'\\)"]],"source":"Small values of the partition functions $c_2$, $c_3$, $c_8$ and $c_8'$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-values-zero-one","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"fully_proved","summary":"We have $c_2(0) = 0$, $c_2(1) = 1$, $c_3(1) = 1$, $c_8(0) = c_8(1) = 1$, $c_8'(0) = 1$ and $c_8'(1) = 0$.","title":"Small values of the partition functions $c_2$, $c_3$, $c_8$ and $c_8'$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c2-five-mul-add-two","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC2_five_mul_add_two_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(c_2(5n + 2)\\)"]],"source":"Parity of $c_2(5n + 2)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c2-five-mul-add-two","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_2(m)$ be the number of partitions of $m$ for which there is an integer $j$ with $1 \\le j \\le m$ such that $j$ appears exactly $j$ times, every part other than $j$ is at least $2j + 1$, and every even part that is at least $2j + 2$ appears only once. Nyirenda and Mugwangwavari: for every integer $n \\ge 0$, the number $c_2(5n + 2)$ is even.","title":"Parity of $c_2(5n + 2)$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c2-prime-mul-add-eve","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC2_prime_mul_add_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(c_2(pn + r)\\)"],["text"," for primes "],["math","\\(p \\ge 5\\)"]],"source":"Parity of $c_2(pn + r)$ for primes $p \\ge 5$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c2-prime-mul-add-eve","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_2(m)$ be the number of partitions of $m$ for which there is an integer $j$ with $1 \\le j \\le m$ such that $j$ appears exactly $j$ times, every part other than $j$ is at least $2j + 1$, and every even part that is at least $2j + 2$ appears only once. Let $p \\ge 5$ be a prime and let $r$ be an integer with $1 \\le r \\le p - 1$ such that neither $16r + 1$ nor $8r + 1$ is a square modulo $p$. Then $c_2(pn + r)$\u2026","title":"Parity of $c_2(pn + r)$ for primes $p \\ge 5$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c3-eleven-mul-add-ev","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC3_eleven_mul_add_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(c_3(11n + 5)\\)"],["text",", "],["math","\\(c_3(11n + 7)\\)"],["text"," and "],["math","\\(c_3(11n + 9)\\)"]],"source":"Parity of $c_3(11n + 5)$, $c_3(11n + 7)$ and $c_3(11n + 9)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c3-eleven-mul-add-ev","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_3(m)$ be the number of partitions of $m$ for which there is an integer $j$ with $1 \\le j \\le m$ such that the part $1$ appears exactly $j^2$ or $j^2 + 1$ times, the odd parts greater than $1$ are distinct, the even parts are distinct, and every even part greater than $2j$ is divisible by $4$ and at least $4j + 4$. Nyirenda and Mugwangwavari: for every integer $n \\ge 0$, the numbers $c_3(11n + 5)$,\u2026","title":"Parity of $c_3(11n + 5)$, $c_3(11n + 7)$ and $c_3(11n + 9)$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c3-prime-mul-add-eve","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC3_prime_mul_add_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(c_3\\)"],["text"," on residue classes modulo a prime"]],"source":"Parity of $c_3$ on residue classes modulo a prime"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c3-prime-mul-add-eve","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_3$ be the initial-repetitions partition function of Nyirenda and Mugwangwavari. Let $p \\ge 5$ be a prime and let $r$ be an integer with $1 \\le r \\le p - 1$ such that neither $12r + 1$ nor $8r + 1$ is a square modulo $p$. Then $c_3(pn + r)$ is even for every integer $n \\ge 0$.","title":"Parity of $c_3$ on residue classes modulo a prime"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8-forty-nine-mul-ad","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8_fortyNine_mul_add_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(c_8\\)"],["text"," is even on six residue classes modulo "],["math","\\(49\\)"]],"source":"$c_8$ is even on six residue classes modulo $49$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8-forty-nine-mul-ad","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_8$ be the initial-repetitions partition function of Nyirenda and Mugwangwavari. For every integer $n \\ge 0$ and every $r \\in \\{6, 20, 27, 34, 41, 48\\}$, the number $c_8(49n + r)$ is even.","title":"$c_8$ is even on six residue classes modulo $49$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/andrews-rogers-ramanujan-first-dissectio","kind":"node","lean":"MetaMathlibExt.andrews_rogersRamanujan_first_dissection","parent":11207,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#andrews-rogers-ramanujan-first-dissectio","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, write $(q;q)_j = \\prod_{i=1}^{j} (1 - q^i)$ for $j \\ge 0$, and let $(q^2;q^2)_\\infty = \\prod_{n=0}^{\\infty} (1 - q^{2n+2})$ be the infinite q-Pochhammer symbol. Then","title":"Andrews' identity for the first Rogers\u2013Ramanujan series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8-even-of-not-is-sq","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8_even_of_not_isSquare","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(c_8(N)\\)"],["text"," is even when "],["math","\\(15N + 1\\)"],["text"," is not a square"]],"source":"$c_8(N)$ is even when $15N + 1$ is not a square"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8-even-of-not-is-sq","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_8$ be the initial-repetitions partition function of Nyirenda and Mugwangwavari. For every integer $N \\ge 0$ such that $15N + 1$ is not a perfect square, $c_8(N)$ is even.","title":"$c_8(N)$ is even when $15N + 1$ is not a square"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/andrews-rogers-ramanujan-second-dissecti","kind":"node","lean":"MetaMathlibExt.andrews_rogersRamanujan_second_dissection","parent":11207,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#andrews-rogers-ramanujan-second-dissecti","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, write $(q;q)_j = \\prod_{i=1}^{j} (1 - q^i)$ for $j \\ge 0$, and let $(q^2;q^2)_\\infty = \\prod_{n=0}^{\\infty} (1 - q^{2n+2})$ be the infinite q-Pochhammer symbol. Then","title":"Andrews' identity for the second Rogers\u2013Ramanujan series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8-prime-even-of-not","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8Prime_even_of_not_isSquare","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(c_8'(N)\\)"],["text"," is even when "],["math","\\(15N + 4\\)"],["text"," is not a square"]],"source":"$c_8'(N)$ is even when $15N + 4$ is not a square"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8-prime-even-of-not","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_8'$ be the partition function that counts partitions with a possibly distinguished part. For every integer $N \\ge 0$ such that $15N + 4$ is not a perfect square, $c_8'(N)$ is even.","title":"$c_8'(N)$ is even when $15N + 4$ is not a square"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8-two-pow-mul-add-e","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8_two_pow_mul_add_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(c_8\\)"],["text"," on residue classes modulo powers of "],["math","\\(2\\)"]],"source":"Parity of $c_8$ on residue classes modulo powers of $2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8-two-pow-mul-add-e","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_8$ be the initial-repetitions partition function of Nyirenda and Mugwangwavari, and let $k \\ge 1$, $n \\ge 0$ and $r \\ge 0$ be integers. If $r < 2^{2k}$ and $15r \\equiv 2^{2k-1} - 1 \\pmod{2^{2k}}$, then $c_8(2^{2k} n + r)$ is even. If $r < 2^{2k}$ and $15r \\equiv 2^{2k-1} + 2^{2k-2} - 1 \\pmod{2^{2k}}$, then $c_8(2^{2k} n + r)$ is even. If $r < 2^{2k+1}$ and $15r \\equiv 2^{2k} + 2^{2k-2} - 1 \\pmod{2^{2k+1}}$,\u2026","title":"Parity of $c_8$ on residue classes modulo powers of $2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8-prime-mul-add-eve","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8_prime_mul_add_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(c_8\\)"],["text"," on residue classes modulo a prime"]],"source":"Parity of $c_8$ on residue classes modulo a prime"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8-prime-mul-add-eve","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_8$ be the initial-repetitions partition function of Nyirenda and Mugwangwavari. Let $p \\ge 7$ be a prime and let $r$ be an integer with $1 \\le r \\le p - 1$ such that $15r + 1$ is not a square modulo $p$. Then $c_8(pn + r)$ is even for every integer $n \\ge 0$.","title":"Parity of $c_8$ on residue classes modulo a prime"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8-prime-two-pow-mul","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8Prime_two_pow_mul_add_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(c_8'\\)"],["text"," on residue classes modulo powers of "],["math","\\(2\\)"]],"source":"Parity of $c_8'$ on residue classes modulo powers of $2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8-prime-two-pow-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_8'$ be the partition function that counts partitions with a possibly distinguished part, and let $k \\ge 1$, $n \\ge 0$ and $r \\ge 0$ be integers. If $r < 2^{2k}$ and $15r \\equiv 2^{2k-1} - 4 \\pmod{2^{2k}}$, then $c_8'(2^{2k} n + r)$ is even. If $r < 2^{2k}$ and $15r \\equiv 2^{2k-1} + 2^{2k-2} - 4 \\pmod{2^{2k}}$, then $c_8'(2^{2k} n + r)$ is even. If $r < 2^{2k+1}$ and\u2026","title":"Parity of $c_8'$ on residue classes modulo powers of $2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/initial-repetitions-c8-prime-prime-mul-a","kind":"node","lean":"MetaMathlibExt.initialRepetitionsC8Prime_prime_mul_add_even","parent":11207,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(c_8'\\)"],["text"," on residue classes modulo a prime"]],"source":"Parity of $c_8'$ on residue classes modulo a prime"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions/index.html#initial-repetitions-c8-prime-prime-mul-a","scope":"NumberTheory/modular-forms-q-series/partition-congruences/partitions-with-initial-repetitions","status":"can_prove","summary":"Let $c_8'$ be the partition function that counts partitions with a possibly distinguished part. Let $p \\ge 7$ be a prime and let $r$ be an integer with $1 \\le r \\le p - 1$ such that $15r + 4$ is not a square modulo $p$. Then $c_8'(pn + r)$ is even for every integer $n \\ge 0$.","title":"Parity of $c_8'$ on residue classes modulo a prime"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","title":"POND and PEND partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/is-pond-parts","kind":"node","lean":"MetaMathlibExt.IsPondParts","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#is-pond-parts","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"fully_proved","summary":"A multiset $s$ of natural numbers (the parts of a partition) satisfies the POND condition when no odd number occurs in $s$ exactly once; equivalently, every odd part that occurs in $s$ has multiplicity at least $2$, so the odd parts are not distinct.","title":"The POND condition"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/is-pend-parts","kind":"node","lean":"MetaMathlibExt.IsPendParts","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#is-pend-parts","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"fully_proved","summary":"A multiset $s$ of natural numbers (the parts of a partition) satisfies the PEND condition when no even number occurs in $s$ exactly once; equivalently, every even part that occurs in $s$ has multiplicity at least $2$, so the even parts are not distinct.","title":"The PEND condition"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pond-count","kind":"node","lean":"MetaMathlibExt.pondCount","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pond-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{pond}(n)$ is the number of partitions of $n$ that are POND, that is, in which every odd part that occurs has multiplicity at least $2$ (OEIS A265256).","title":"Number of POND partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pend-count","kind":"node","lean":"MetaMathlibExt.pendCount","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pend-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{pend}(n)$ is the number of PEND partitions of $n$: partitions of $n$ in which every even part that occurs appears at least twice (OEIS A265254).","title":"The PEND partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pond-count-zero","kind":"node","lean":"MetaMathlibExt.pondCount_zero","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pond-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"fully_proved","summary":"The number $\\mathrm{pond}(0)$ of POND partitions of $0$ equals $1$: the only partition of $0$ is the empty partition, and it is POND.","title":"The number of POND partitions of zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pend-count-zero","kind":"node","lean":"MetaMathlibExt.pendCount_zero","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pend-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"fully_proved","summary":"The number of PEND partitions of $0$ is $1$, that is, $\\mathrm{pend}(0) = 1$.","title":"PEND partitions of zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/is-pond-parts-is-pend-parts-examples","kind":"node","lean":"MetaMathlibExt.isPondParts_isPendParts_examples","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#is-pond-parts-is-pend-parts-examples","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"fully_proved","summary":"The multisets of parts $\\{1, 1\\}$ and $\\{2\\}$ satisfy the POND condition, while $\\{1\\}$ and $\\{2, 1\\}$ do not; the multisets $\\{1\\}$ and $\\{2, 2\\}$ satisfy the PEND condition, while $\\{2\\}$ and $\\{2, 4\\}$ do not.","title":"Examples of POND and PEND parts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pond-count-congruences","kind":"node","lean":"MetaMathlibExt.pondCount_congruences","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pond-count-congruences","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $\\mathrm{pond}(m)$ be the number of POND partitions of $m$. For every natural number $n$, $\\mathrm{pond}(3n + 2) \\equiv 0 \\pmod{2}$, $\\mathrm{pond}(27n + 26) \\equiv 0 \\pmod{3}$ and $\\mathrm{pond}(3n + 1) \\equiv 0 \\pmod{4}$.","title":"Congruences modulo 2, 3 and 4 for POND partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pend-count-twenty-seven-mul-add-nineteen","kind":"node","lean":"MetaMathlibExt.pendCount_twentySeven_mul_add_nineteen","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pend-count-twenty-seven-mul-add-nineteen","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $\\mathrm{pend}(m)$ be the number of PEND partitions of $m$. For every natural number $n$, $\\mathrm{pend}(27n + 19) \\equiv 0 \\pmod{3}$.","title":"Congruence modulo 3 for PEND partitions of 27n + 19"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pond-count-internal-congruence","kind":"node","lean":"MetaMathlibExt.pondCount_internal_congruence","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pond-count-internal-congruence","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $\\mathrm{pond}(m)$ be the number of POND partitions of $m$. For every natural number $n$, $\\mathrm{pond}(27n + 17) \\equiv \\mathrm{pond}(3n + 2) \\pmod{3}$.","title":"Internal congruence modulo 3 for POND partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pend-count-internal-congruence","kind":"node","lean":"MetaMathlibExt.pendCount_internal_congruence","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pend-count-internal-congruence","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $\\mathrm{pend}(m)$ be the number of PEND partitions of $m$. For every natural number $n$, $\\mathrm{pend}(27n + 10) \\equiv \\mathrm{pend}(3n + 1) \\pmod{3}$.","title":"Internal congruence modulo 3 for PEND partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pond-count-infinite-family","kind":"node","lean":"MetaMathlibExt.pondCount_infinite_family","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pond-count-infinite-family","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $\\mathrm{pond}(m)$ be the number of POND partitions of $m$. For all natural numbers $\\alpha \\ge 1$ and $n$, $\\mathrm{pond}\\left(3^{2\\alpha+1} n + (23 \\cdot 3^{2\\alpha} + 1)/8\\right) \\equiv 0 \\pmod{3}$.","title":"Infinite family of congruences modulo 3 for POND partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pend-count-infinite-family","kind":"node","lean":"MetaMathlibExt.pendCount_infinite_family","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pend-count-infinite-family","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $\\mathrm{pend}(m)$ be the number of PEND partitions of $m$. For all natural numbers $\\alpha \\ge 1$ and $n$, $\\mathrm{pend}\\left(3^{2\\alpha+1} n + (17 \\cdot 3^{2\\alpha} - 1)/8\\right) \\equiv 0 \\pmod{3}$.","title":"Infinite family of congruences modulo 3 for PEND partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/f2-div-f1-f4-dissection","kind":"node","lean":"MetaMathlibExt.f2_div_f1_f4_dissection","parent":11232,"reading_title":{"format":"inline-parts-v1","parts":[["text","3-dissection of "],["math","\\(f_2 / (f_1 f_4)\\)"]],"source":"3-dissection of $f_2 / (f_1 f_4)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#f2-div-f1-f4-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products in $\\mathbb{Z}[[q]]$. Then $f_2 f_3^4 f_9^3 f_{12}^4 f_{18}^3 f_{36}^3 = f_1 f_3^2 f_4 f_{12}^2 f_{18}^{12} + q f_1 f_3 f_4 f_6^2 f_9^3 f_{12} f_{18}^6 f_{36}^3 + q^2 f_1 f_4 f_6^4 f_9^6 f_{36}^6$.","title":"3-dissection of $f_2 / (f_1 f_4)$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/f1-f2-dissection","kind":"node","lean":"MetaMathlibExt.f1_f2_dissection","parent":11232,"reading_title":{"format":"inline-parts-v1","parts":[["text","3-dissection of "],["math","\\(f_1 f_2\\)"]],"source":"3-dissection of $f_1 f_2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#f1-f2-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products in $\\mathbb{Z}[[q]]$. Then $f_1 f_2 f_3 f_6 f_9^2 f_{18}^2 = f_6^2 f_9^6 - q f_3 f_6 f_9^3 f_{18}^3 - 2 q^2 f_3^2 f_{18}^6$.","title":"3-dissection of $f_1 f_2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/one-div-f1-f2-dissection","kind":"node","lean":"MetaMathlibExt.one_div_f1_f2_dissection","parent":11232,"reading_title":{"format":"inline-parts-v1","parts":[["text","3-dissection of "],["math","\\(1 / (f_1 f_2)\\)"]],"source":"3-dissection of $1 / (f_1 f_2)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#one-div-f1-f2-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products in $\\mathbb{Z}[[q]]$. Then $f_3^6 f_6^6 f_9^3 f_{18}^3 = f_1 f_2 f_6^4 f_9^{12} + q f_1 f_2 f_3 f_6^3 f_9^9 f_{18}^3 + 3 q^2 f_1 f_2 f_3^2 f_6^2 f_9^6 f_{18}^6 - 2 q^3 f_1 f_2 f_3^3 f_6 f_9^3 f_{18}^9 + 4 q^4 f_1 f_2 f_3^4 f_{18}^{12}$.","title":"3-dissection of $1 / (f_1 f_2)$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/f2-sq-div-f1-dissection","kind":"node","lean":"MetaMathlibExt.f2_sq_div_f1_dissection","parent":11232,"reading_title":{"format":"inline-parts-v1","parts":[["text","3-dissection of "],["math","\\(f_2^2 / f_1\\)"]],"source":"3-dissection of $f_2^2 / f_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#f2-sq-div-f1-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products in $\\mathbb{Z}[[q]]$. Then $f_2^2 f_3 f_9 f_{18} = f_1 f_6 f_9^3 + q f_1 f_3 f_{18}^3$.","title":"3-dissection of $f_2^2 / f_1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/f2-div-f1-sq-dissection","kind":"node","lean":"MetaMathlibExt.f2_div_f1_sq_dissection","parent":11232,"reading_title":{"format":"inline-parts-v1","parts":[["text","3-dissection of "],["math","\\(f_2 / f_1^2\\)"]],"source":"3-dissection of $f_2 / f_1^2$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#f2-div-f1-sq-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products in $\\mathbb{Z}[[q]]$. Then $f_2 f_3^8 f_{18}^3 = f_1^2 f_6^4 f_9^6 + 2 q f_1^2 f_3 f_6^3 f_9^3 f_{18}^3 + 4 q^2 f_1^2 f_3^2 f_6^2 f_{18}^6$.","title":"3-dissection of $f_2 / f_1^2$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/f4-div-f1-dissection","kind":"node","lean":"MetaMathlibExt.f4_div_f1_dissection","parent":11232,"reading_title":{"format":"inline-parts-v1","parts":[["text","3-dissection of "],["math","\\(f_4 / f_1\\)"]],"source":"3-dissection of $f_4 / f_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#f4-div-f1-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products in $\\mathbb{Z}[[q]]$. Then $f_3^4 f_4 f_{18}^2 f_{36}^2 = f_1 f_3 f_{12} f_{18}^6 + q f_1 f_6^2 f_9^3 f_{36}^3 + 2 q^2 f_1 f_3 f_6 f_{18}^3 f_{36}^3$.","title":"3-dissection of $f_4 / f_1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/f3-cube-div-f1-sub","kind":"node","lean":"MetaMathlibExt.f3_cube_div_f1_sub","parent":11232,"reading_title":{"format":"inline-parts-v1","parts":[["text","2-dissection of "],["math","\\(f_3^3 / f_1\\)"]],"source":"2-dissection of $f_3^3 / f_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#f3-cube-div-f1-sub","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ denote the Euler products in $\\mathbb{Z}[[q]]$. Then $f_2^2 f_3^3 f_4 f_{12} - q f_1 f_2^2 f_{12}^4 = f_1 f_4^4 f_6^2$.","title":"2-dissection of $f_3^3 / f_1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/q-euler-product-pow-prime-pow-congruence","kind":"node","lean":"MetaMathlibExt.qEulerProduct_pow_prime_pow_congruence","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#q-euler-product-pow-prime-pow-congruence","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $p$ be a prime, let $j, k, m \\ge 1$ be natural numbers, and let $f_m = \\prod_{i \\ge 1} (1 - q^{mi})$. Then for every natural number $n$, the coefficient of $q^n$ in $f_m^{p^j k}$ is congruent modulo $p^j$ to the coefficient of $q^n$ in $f_{pm}^{p^{j-1} k}$.","title":"Prime-power congruence for powers of Euler products"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pond-count-generating-function","kind":"node","lean":"MetaMathlibExt.pondCount_generatingFunction","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pond-count-generating-function","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $\\mathrm{pond}(n)$ be the number of POND partitions of $n$, and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then, as integer power series in $q$, $f_2^2 f_3 f_{12} \\sum_{n \\ge 0} \\mathrm{pond}(n) q^n = f_4 f_6^2$. This is the identity $\\sum_{n \\ge 0} \\mathrm{pond}(n) q^n = f_4 f_6^2 / (f_2^2 f_3 f_{12})$ with the denominator cleared.","title":"Generating function for POND partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/pend-count-generating-function","kind":"node","lean":"MetaMathlibExt.pendCount_generatingFunction","parent":11232,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions/index.html#pend-count-generating-function","scope":"NumberTheory/modular-forms-q-series/partition-congruences/pond-and-pend-partitions","status":"can_prove","summary":"Let $\\mathrm{pend}(n)$ be the number of PEND partitions of $n$, and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then, as integer power series in $q$, $f_1 f_4 f_6 \\sum_{n \\ge 0} \\mathrm{pend}(n) q^n = f_2 f_{12}$. This is the identity $\\sum_{n \\ge 0} \\mathrm{pend}(n) q^n = f_2 f_{12} / (f_1 f_4 f_6)$ with the denominator cleared.","title":"Generating function for PEND partitions"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","title":"Ramanujan's partition congruences"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/two-mul-choose-two-succ","kind":"node","lean":"MetaMathlibExt.two_mul_choose_two_succ","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#two-mul-choose-two-succ","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"For every natural number $a$, $2\\binom{a+1}{2} = a(a+1)$.","title":"Twice a triangular number"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/jacobi-cube-series","kind":"node","lean":"MetaMathlibExt.jacobiCubeSeries","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#jacobi-cube-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"Jacobi's cube series is the formal power series $J(X) = \\sum_{t \\ge 0} (-1)^t (2t+1) X^{t(t+1)/2}$ in $\\mathbb{Z}[[X]]$: for each natural number $t$ its coefficient at the triangular number $t(t+1)/2$ is $(-1)^t (2t+1)$, and all its other coefficients are $0$.","title":"Jacobi's cube series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/coeff-jacobi-cube-series-triangle","kind":"node","lean":"MetaMathlibExt.coeff_jacobiCubeSeries_triangle","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#coeff-jacobi-cube-series-triangle","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"Let $J(X)$ be Jacobi's cube series. For every natural number $t$, the coefficient of $X^{t(t+1)/2}$ in $J(X)$ is $(-1)^t (2t+1)$.","title":"Coefficients of Jacobi's cube series at triangular numbers"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/coeff-jacobi-cube-series-eq-zero","kind":"node","lean":"MetaMathlibExt.coeff_jacobiCubeSeries_eq_zero","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#coeff-jacobi-cube-series-eq-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"Let $J(X)$ be Jacobi's cube series. If a natural number $b$ is not of the form $t(t+1)/2$ for any natural number $t$, then the coefficient of $X^b$ in $J(X)$ is $0$.","title":"Vanishing coefficients of Jacobi's cube series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/pentagonal-series-pow-three","kind":"node","lean":"MetaMathlibExt.pentagonalSeries_pow_three","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#pentagonal-series-pow-three","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"Let $E(X) = \\sum_{k \\in \\mathbb{Z}} (-1)^k X^{k(3k-1)/2}$ be Euler's pentagonal series and $J(X) = \\sum_{t \\ge 0} (-1)^t (2t+1) X^{t(t+1)/2}$ Jacobi's cube series, both in $\\mathbb{Z}[[X]]$. Then $E(X)^3 = J(X)$.","title":"Jacobi's cube identity"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/dvd-partition-function-of-dvd-coeff-pent","kind":"node","lean":"MetaMathlibExt.dvd_partitionFunction_of_dvd_coeff_pentagonalSeries_pow","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#dvd-partition-function-of-dvd-coeff-pent","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"Let $p$ be a prime, let $r < p$ be a natural number, and let $E(X) = \\sum_{k \\in \\mathbb{Z}} (-1)^k X^{k(3k-1)/2}$ be Euler's pentagonal series in $\\mathbb{Z}[[X]]$. If $p$ divides the coefficient of $X^{pm+r}$ in $E(X)^{p-1}$ for every natural number $m$, then $p$ divides the number of partitions of $pm + r$ for every natural number $m$.","title":"Partition congruences from a power of Euler's series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/ramanujan-partition-congruence-mod-five","kind":"node","lean":"MetaMathlibExt.ramanujan_partition_congruence_mod_five","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#ramanujan-partition-congruence-mod-five","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"For every natural number $m$, the number of partitions of $5m + 4$ is divisible by $5$.","title":"Ramanujan's congruence modulo 5"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/ramanujan-partition-congruence-mod-seven","kind":"node","lean":"MetaMathlibExt.ramanujan_partition_congruence_mod_seven","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#ramanujan-partition-congruence-mod-seven","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"For every natural number $m$, the number of partitions of $7m + 5$ is divisible by $7$.","title":"Ramanujan's congruence modulo 7"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/ramanujan-partition-congruence-mod-eleve","kind":"node","lean":"MetaMathlibExt.ramanujan_partition_congruence_mod_eleven","parent":11256,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences/index.html#ramanujan-partition-congruence-mod-eleve","scope":"NumberTheory/modular-forms-q-series/partition-congruences/ramanujan-s-partition-congruences","status":"fully_proved","summary":"For every natural number $m$, the number of partitions of $11m + 6$ is divisible by $11$.","title":"Ramanujan's congruence modulo 11"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","title":"Regular partitions and a generalized tau function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/constant-coeff-pentagonal-series-int","kind":"node","lean":"MetaMathlibExt.constantCoeff_pentagonalSeries_int","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler's product has constant term "],["math","\\(1\\)"]],"source":"Euler's product has constant term $1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#constant-coeff-pentagonal-series-int","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"fully_proved","summary":"The constant coefficient of the integer power series $\\prod_{m \\ge 1} (1 - X^m)$ in $\\mathbb{Z}[[X]]$ is $1$.","title":"Euler's product has constant term $1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/is-unit-pentagonal-series-int","kind":"node","lean":"MetaMathlibExt.isUnit_pentagonalSeries_int","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler's product is a unit in "],["math","\\(\\mathbb{Z}((X))\\)"]],"source":"Euler's product is a unit in $\\mathbb{Z}((X))$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#is-unit-pentagonal-series-int","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"fully_proved","summary":"Euler's pentagonal series $\\sum_{k \\in \\mathbb{Z}} (-1)^k X^{k(3k - 1)/2}$, which equals $\\prod_{m \\ge 1} (1 - X^m)$, is a unit of the ring $\\mathbb{Z}[[X]]$ of formal power series with integer coefficients, so all its integer powers are defined.","title":"Euler's product is a unit in $\\mathbb{Z}((X))$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau","kind":"node","lean":"MetaMathlibExt.generalizedTau","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"fully_proved","summary":"For an integer $k$ and a natural number $n$, the generalized tau function $\\tau_k(n)$ is the coefficient of $q^n$ in the integer power series $q \\prod_{m \\ge 1} (1 - q^m)^k$. For negative $k$ the power is taken in the group of units of $\\mathbb{Z}[[q]]$, where $\\prod_{m \\ge 1} (1 - q^m)$ is invertible because its constant term is $1$.","title":"Generalized tau function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-zero","kind":"node","lean":"MetaMathlibExt.generalizedTau_zero","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"fully_proved","summary":"For every integer $k$, $\\tau_k(0) = 0$, where $\\tau_k$ is the generalized tau function: the series $q \\prod_{m \\ge 1} (1 - q^m)^k$ has no constant term.","title":"Generalized tau function at zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-one","kind":"node","lean":"MetaMathlibExt.generalizedTau_one","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generalized tau function at "],["math","\\(1\\)"]],"source":"The generalized tau function at $1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-one","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"fully_proved","summary":"For every integer $k$, $\\tau_k(1) = 1$.","title":"The generalized tau function at $1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-nat-cast","kind":"node","lean":"MetaMathlibExt.generalizedTau_natCast","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-nat-cast","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $\\tau_k(n)$ is the coefficient of $q^n$ in the integer power series $q \\bigl(\\prod_{m \\ge 1} (1 - q^m)\\bigr)^k$.","title":"The generalized tau function for natural exponents"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-succ-eq-partition-sum","kind":"node","lean":"MetaMathlibExt.generalizedTau_succ_eq_partition_sum","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-succ-eq-partition-sum","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 0$ be natural numbers. For a partition $\\lambda$ of $n$ and a part $a$ of $\\lambda$, write $f_a$ for the number of times $a$ occurs in $\\lambda$. Then","title":"Partition formulas for the generalized tau function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-succ-mod-eq-card-frequen","kind":"node","lean":"MetaMathlibExt.generalizedTau_succ_modEq_card_frequency_two","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-succ-mod-eq-card-frequen","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 0$ be natural numbers, and let $A$ be the set of integers $a$ with $1 \\le a \\le k$ and $\\binom{k}{a}$ odd. Then $\\tau_k(n + 1)$ is congruent modulo $2$ to the number of partitions of $n$ in which the number of times each part occurs lies in $A$.","title":"Parity of the generalized tau function via part frequencies"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/odd-card-four-regular-iff-triangular","kind":"node","lean":"MetaMathlibExt.odd_card_four_regular_iff_triangular","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of the "],["math","\\(4\\)"],["text","-regular partition function"]],"source":"Parity of the $4$-regular partition function"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#odd-card-four-regular-iff-triangular","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"For a positive integer $n$, let $R_4(n)$ be the number of partitions of $n$ with no part divisible by $4$. Then $R_4(n)$ is odd if and only if $n$ is a triangular number, that is, $n = m(m + 1)/2$ for some integer $m \\ge 0$.","title":"Parity of the $4$-regular partition function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-fourteen-six-odd-index-m","kind":"node","lean":"MetaMathlibExt.generalizedTau_fourteen_six_odd_index_mod_two","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(\\tau_{14}\\)"],["text"," and "],["math","\\(\\tau_6\\)"],["text"," at odd arguments"]],"source":"Parity of $\\tau_{14}$ and $\\tau_6$ at odd arguments"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-fourteen-six-odd-index-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau_k(m)$ be the generalized tau function, the coefficient of $q^m$ in $q \\prod_{j \\ge 1} (1 - q^j)^k$, and let $R_8(n)$ be the number of partitions of $n$ in which no part is divisible by $8$. For every positive integer $n$: (a) $\\tau_{14}(2n + 1) \\equiv R_8(n) \\pmod 2$; (b) $\\tau_6(2n + 1)$ is odd if and only if $n$ is a triangular number, that is, $n = m(m + 1)/2$ for some natural number $m$.","title":"Parity of $\\tau_{14}$ and $\\tau_6$ at odd arguments"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/card-two-pow-regular-mod-eq-generalized","kind":"node","lean":"MetaMathlibExt.card_two_pow_regular_modEq_generalizedTau","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2^s\\)"],["text","-regular partitions and the generalized tau function modulo 2"]],"source":"$2^s$-regular partitions and the generalized tau function modulo 2"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#card-two-pow-regular-mod-eq-generalized","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $s$ be a positive integer and $n$ a natural number. Let $R_{2^s}(n)$ be the number of partitions of $n$ in which no part is divisible by $2^s$, and let $\\tau_k(m)$ be the generalized tau function, the coefficient of $q^m$ in $q \\prod_{j \\ge 1} (1 - q^j)^k$. Then $R_{2^s}(n) \\equiv \\tau_{2^s - 1}(n + 1) \\pmod 2$.","title":"$2^s$-regular partitions and the generalized tau function modulo 2"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-nine-regular","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_nine_regular","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(3\\)"],["text"," via "],["math","\\(9\\)"],["text","-regular partitions"]],"source":"Ramanujan's $\\tau$ modulo $3$ via $9$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-nine-regular","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function, and let $R_9(m)$ be the number of partitions of $m$ with no part divisible by $9$. For every integer $n \\ge 1$, $\\tau(n + 1) \\equiv R_9(n/3) \\pmod 3$ if $3$ divides $n$, and $\\tau(n + 1) \\equiv 0 \\pmod 3$ otherwise.","title":"Ramanujan's $\\tau$ modulo $3$ via $9$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-three-mul-mod-eq-zero","kind":"node","lean":"MetaMathlibExt.ramanujanTau_three_mul_modEq_zero","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's congruence "],["math","\\(\\tau(3n) \\equiv 0 \\pmod 3\\)"]],"source":"Ramanujan's congruence $\\tau(3n) \\equiv 0 \\pmod 3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-three-mul-mod-eq-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function. For every integer $n \\ge 1$, $\\tau(3n) \\equiv 0 \\pmod 3$.","title":"Ramanujan's congruence $\\tau(3n) \\equiv 0 \\pmod 3$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/card-nine-regular-mod-eq-sigma","kind":"node","lean":"MetaMathlibExt.card_nine_regular_modEq_sigma","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#card-nine-regular-mod-eq-sigma","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $R_9(m)$ denote the number of partitions of $m$ in which no part is divisible by $9$, and let $\\sigma(m)$ be the sum of the positive divisors of $m$. For every positive integer $n$, $R_9(n) \\equiv \\sigma(3n + 1) \\pmod 3$.","title":"9-regular partitions modulo 3 and the divisor sum"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-twentyfive-reg","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_twentyfive_regular","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(5\\)"],["text"," via "],["math","\\(25\\)"],["text","-regular partitions"]],"source":"Ramanujan's $\\tau$ modulo $5$ via $25$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-twentyfive-reg","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function, and let $R_{25}(n)$ be the number of partitions of $n$ with no part divisible by $25$. For every integer $n \\ge 1$, $\\tau(n + 1) \\equiv R_{25}(n) \\pmod 5$.","title":"Ramanujan's $\\tau$ modulo $5$ via $25$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/card-twentyfive-regular-mod-eq-mul-sigma","kind":"node","lean":"MetaMathlibExt.card_twentyfive_regular_modEq_mul_sigma","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#card-twentyfive-regular-mod-eq-mul-sigma","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $R_{25}(m)$ denote the number of partitions of $m$ in which no part is divisible by $25$, and let $\\sigma(m)$ be the sum of the positive divisors of $m$. For every positive integer $n$, $R_{25}(n) \\equiv (n + 1) \\, \\sigma(n + 1) \\pmod 5$.","title":"25-regular partitions modulo 5 and the divisor sum"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-triangular-sum","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_triangular_sum_seven","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(7\\)"],["text"," via triangular numbers"]],"source":"Ramanujan's $\\tau$ modulo $7$ via triangular numbers"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-triangular-sum","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function. For every integer $n \\ge 1$, $\\tau(n + 1)$ is congruent modulo $7$ to the sum of $(-1)^{m + r} (2m + 1)(2r + 1)$ over all pairs of integers $m, r \\ge 0$ with $m(m + 1)/2 + 7r(r + 1)/2 = n$.","title":"Ramanujan's $\\tau$ modulo $7$ via triangular numbers"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-pentagonal-sum","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_pentagonal_sum_eleven","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(11\\)"],["text"," via pentagonal numbers"]],"source":"Ramanujan's $\\tau$ modulo $11$ via pentagonal numbers"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-pentagonal-sum","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function, and let $P(r) = r(3r - 1)/2$ for integers $r$. For every integer $n \\ge 1$, $\\tau(n + 1)$ is congruent modulo $11$ to the sum of $(-1)^{l + m + s + r}$ over all quadruples of integers $(l, m, s, r)$ with $P(l) + P(m) + 11P(s) + 11P(r) = n$.","title":"Ramanujan's $\\tau$ modulo $11$ via pentagonal numbers"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-pentagonal-tau","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_pentagonal_tau_eleven","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(13\\)"],["text"," via "],["math","\\(\\tau_{11}\\)"]],"source":"Ramanujan's $\\tau$ modulo $13$ via $\\tau_{11}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-pentagonal-tau","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function, and let $P(r) = r(3r - 1)/2$ for integers $r$. For every integer $n \\ge 1$,","title":"Ramanujan's $\\tau$ modulo $13$ via $\\tau_{11}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-pentagonal-tau-2","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_pentagonal_tau_seven","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(17\\)"],["text"," via "],["math","\\(\\tau_7\\)"]],"source":"Ramanujan's $\\tau$ modulo $17$ via $\\tau_7$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-pentagonal-tau-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function, and let $P(r) = r(3r - 1)/2$ for integers $r$. For every integer $n \\ge 1$,","title":"Ramanujan's $\\tau$ modulo $17$ via $\\tau_7$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-pentagonal-tau-3","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_pentagonal_tau_five","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(19\\)"],["text"," via "],["math","\\(\\tau_5\\)"]],"source":"Ramanujan's $\\tau$ modulo $19$ via $\\tau_5$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-pentagonal-tau-3","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function, and let $P(r) = r(3r - 1)/2$ for integers $r$. For every integer $n \\ge 1$,","title":"Ramanujan's $\\tau$ modulo $19$ via $\\tau_5$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-pentagonal-sum-2","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_pentagonal_sum_twentythree","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(23\\)"],["text"," via pentagonal numbers"]],"source":"Ramanujan's $\\tau$ modulo $23$ via pentagonal numbers"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-pentagonal-sum-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function, and let $P(r) = r(3r - 1)/2$ for integers $r$. For every integer $n \\ge 1$, $\\tau(n + 1)$ is congruent modulo $23$ to the sum of $(-1)^{r + s}$ over all pairs of integers $(r, s)$ with $P(r) + 23P(s) = n$.","title":"Ramanujan's $\\tau$ modulo $23$ via pentagonal numbers"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-succ-mod-eq-five-regular-s","kind":"node","lean":"MetaMathlibExt.ramanujanTau_succ_modEq_five_regular_sum_twentyfive","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\tau\\)"],["text"," modulo "],["math","\\(25\\)"],["text"," via "],["math","\\(5\\)"],["text","-regular partitions"]],"source":"Ramanujan's $\\tau$ modulo $25$ via $5$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-succ-mod-eq-five-regular-s","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau = \\tau_{24}$ be Ramanujan's tau function, where $\\tau_k$ is the generalized tau function. Let $R_5(m)$ be the number of partitions of $m$ with no part divisible by $5$, and let $P(t) = t(3t - 1)/2$ for integers $t$. For every integer $n \\ge 1$,","title":"Ramanujan's $\\tau$ modulo $25$ via $5$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-prime-pow-succ-mod-eq","kind":"node","lean":"MetaMathlibExt.generalizedTau_prime_pow_succ_modEq","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-prime-pow-succ-mod-eq","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $p$ be a prime, let $s \\ge 1$ and $n \\ge 0$ be natural numbers, and for an integer $t$ let $P(t) = t(3t - 1)/2$ be the generalized pentagonal number. Then $\\tau_{p^s}(n + 1)$ is congruent modulo $p$ to the sum of $(-1)^t$ over the integers $t$ with $-n \\le t \\le n$ and $n = p^s P(t)$.","title":"Generalized tau at a prime power modulo the prime"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-two-mul-prime-succ-mod-e","kind":"node","lean":"MetaMathlibExt.generalizedTau_two_mul_prime_succ_modEq","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized tau function "],["math","\\(\\tau_{2p}\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Generalized tau function $\\tau_{2p}$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-two-mul-prime-succ-mod-e","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $p$ be an odd prime, let $\\tau_k$ be the generalized tau function, and for an integer $r$ let $P(r) = r(3r - 1)/2$ be the generalized pentagonal number. For every integer $n \\ge 0$,","title":"Generalized tau function $\\tau_{2p}$ modulo $p$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-two-mul-prime-succ-mod-e-2","kind":"node","lean":"MetaMathlibExt.generalizedTau_two_mul_prime_succ_modEq_zero","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau_{2p}(n + 1)\\)"],["text"," vanishes modulo "],["math","\\(p\\)"],["text"," when "],["math","\\(p \\nmid n\\)"]],"source":"$\\tau_{2p}(n + 1)$ vanishes modulo $p$ when $p \\nmid n$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-two-mul-prime-succ-mod-e-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $p$ be an odd prime and $n$ a natural number not divisible by $p$. Then $\\tau_{2p}(n + 1) \\equiv 0 \\pmod{p}$.","title":"$\\tau_{2p}(n + 1)$ vanishes modulo $p$ when $p \\nmid n$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-two-mul-prime-add-one-su","kind":"node","lean":"MetaMathlibExt.generalizedTau_two_mul_prime_add_one_succ_modEq","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized tau at "],["math","\\(2p + 1\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Generalized tau at $2p + 1$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-two-mul-prime-add-one-su","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $p$ be an odd prime and $n$ a natural number, and for an integer $j$ let $P(j) = j(3j - 1)/2$ be the generalized pentagonal number. Then $\\tau_{2p + 1}(n + 1)$ is congruent modulo $p$ to the sum of $(-1)^{r + s + t}$ over the triples of integers $(r, s, t)$ with $-n \\le r, s, t \\le n$ and $n = P(r) + p(P(s) + P(t))$.","title":"Generalized tau at $2p + 1$ modulo $p$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-prime-sq-add-one-succ-mo","kind":"node","lean":"MetaMathlibExt.generalizedTau_prime_sq_add_one_succ_modEq","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized tau at "],["math","\\(p^2 + 1\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Generalized tau at $p^2 + 1$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-prime-sq-add-one-succ-mo","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $p$ be an odd prime and $n$ a natural number, and for an integer $j$ let $P(j) = j(3j - 1)/2$ be the generalized pentagonal number. Then $\\tau_{p^2 + 1}(n + 1)$ is congruent modulo $p$ to the sum of $(-1)^{r + s}$ over the pairs of integers $(r, s)$ with $-n \\le r, s \\le n$ and $n = P(r) + p^2 P(s)$.","title":"Generalized tau at $p^2 + 1$ modulo $p$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/card-nine-regular-four-mul-add-one-mod-e","kind":"node","lean":"MetaMathlibExt.card_nine_regular_four_mul_add_one_modEq","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","A congruence for 9-regular partitions at "],["math","\\(4n + 1\\)"]],"source":"A congruence for 9-regular partitions at $4n + 1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#card-nine-regular-four-mul-add-one-mod-e","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $R_9(m)$ denote the number of partitions of $m$ in which no part is divisible by $9$. For every positive integer $n$, $R_9(4n + 1) \\equiv R_9(n) \\pmod 3$.","title":"A congruence for 9-regular partitions at $4n + 1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/card-nine-regular-iterate-mod-eq","kind":"node","lean":"MetaMathlibExt.card_nine_regular_iterate_modEq","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#card-nine-regular-iterate-mod-eq","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $R_9(m)$ denote the number of partitions of $m$ in which no part is divisible by $9$, and let $N_s = (4^s - 1)/3$. For all positive integers $r$ and $s$, $R_9((r - 1) 4^{s-1} + N_s) \\equiv R_9(r) \\pmod 3$, $R_9(N_s) \\equiv 1 \\pmod 3$, $R_9(4^{s-1} + N_s) \\equiv 2 \\pmod 3$ and $R_9(2 \\cdot 4^{s-1} + N_s) \\equiv 0 \\pmod 3$.","title":"Iterated congruences for 9-regular partitions modulo 3"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/card-prime-regular-mod-eq-generalized-ta","kind":"node","lean":"MetaMathlibExt.card_prime_regular_modEq_generalizedTau_pred","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#card-prime-regular-mod-eq-generalized-ta","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $p$ be a prime and $n$ a natural number. Let $R_p(n)$ be the number of partitions of $n$ in which no part is divisible by $p$, and let $\\tau_k(m)$ be the generalized tau function, the coefficient of $q^m$ in $q \\prod_{j \\ge 1} (1 - q^j)^k$. Then $R_p(n) \\equiv \\tau_{p-1}(n + 1) \\pmod p$.","title":"Prime-regular partitions and the generalized tau function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/choose-add-mod-eq-prime","kind":"node","lean":"MetaMathlibExt.choose_add_modEq_prime","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial coefficients "],["math","\\(\\binom{n+k}{k}\\)"],["text"," modulo an odd prime"]],"source":"Binomial coefficients $\\binom{n+k}{k}$ modulo an odd prime"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#choose-add-mod-eq-prime","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $l$ be an odd prime, $k$ an integer with $1 \\le k < l$, $n$ a natural number, and $r$ the remainder of $n$ modulo $l$. If $r \\le l - k - 1$, then $\\binom{n+k}{k} \\equiv (-1)^r \\binom{l-k-1}{r} \\pmod l$; otherwise $\\binom{n+k}{k} \\equiv 0 \\pmod l$.","title":"Binomial coefficients $\\binom{n+k}{k}$ modulo an odd prime"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-neg-succ-mod-eq-partitio","kind":"node","lean":"MetaMathlibExt.generalizedTau_neg_succ_modEq_partition_convolution","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau_{-k}\\)"],["text"," and partition convolutions modulo a prime"]],"source":"$\\tau_{-k}$ and partition convolutions modulo a prime"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-neg-succ-mod-eq-partitio","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $l$ be an odd prime, let $k$ be a natural number with $2 \\le k < l$, let $n$ be a natural number, and write $p(s)$ for the number of partitions of $s$. Then","title":"$\\tau_{-k}$ and partition convolutions modulo a prime"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/partition-convolution-two-mod-eq-three","kind":"node","lean":"MetaMathlibExt.partition_convolution_two_modEq_three","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partition convolution modulo "],["math","\\(3\\)"]],"source":"Partition convolution modulo $3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#partition-convolution-two-mod-eq-three","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $p(a)$ be the number of partitions of $a$, and let $\\omega(t)$ be the coefficient of $q^t$ in Euler's product $\\prod_{m \\ge 1} (1 - q^m)$. For every integer $n \\ge 0$,","title":"Partition convolution modulo $3$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/partition-composition-sum-mod-eq-zero","kind":"node","lean":"MetaMathlibExt.partition_composition_sum_modEq_zero","parent":11266,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#partition-composition-sum-mod-eq-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $p(a)$ be the number of partitions of $a$. Let $\\ell \\ge 5$ be a prime and $m \\ge 0$ an integer such that $m \\not\\equiv 0 \\pmod{\\ell}$ and $m \\not\\equiv (-1)^r \\binom{\\ell - 3}{r} \\pmod{\\ell}$ for every integer $r$ with $0 \\le r \\le (\\ell - 3)/2$. Then every integer $n \\ge 0$ with $n \\equiv m \\pmod{\\ell}$ satisfies","title":"Partition composition sums vanishing modulo a prime"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-seven-mul-mod-eq-zero","kind":"node","lean":"MetaMathlibExt.ramanujanTau_seven_mul_modEq_zero","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's congruence modulo "],["math","\\(7\\)"]],"source":"Ramanujan's congruence modulo $7$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-seven-mul-mod-eq-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau(n) = \\tau_{24}(n)$ be Ramanujan's tau function, the generalized tau function with $k = 24$. For every positive integer $n$, $\\tau(7n) \\equiv 0 \\pmod 7$.","title":"Ramanujan's congruence modulo $7$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-mod-eq-zero-of-pentagona","kind":"node","lean":"MetaMathlibExt.generalizedTau_modEq_zero_of_pentagonal","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing of "],["math","\\(\\tau_k(n)\\)"],["text"," modulo the prime "],["math","\\(k - 1\\)"]],"source":"Vanishing of $\\tau_k(n)$ modulo the prime $k - 1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-mod-eq-zero-of-pentagona","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $n \\ge 1$ and let $k$ be a natural number such that $k - 1$ is prime. For an integer $t$ let $P(t) = t(3t - 1)/2$ be the generalized pentagonal number. If $k - 1$ does not divide $n - 1 - P(t)$ for any integer $t$ with $P(t) \\le n - 1$, then the generalized tau function satisfies $\\tau_k(n) \\equiv 0 \\pmod{k - 1}$.","title":"Vanishing of $\\tau_k(n)$ modulo the prime $k - 1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/mul-generalized-tau-succ-mod-eq-zero","kind":"node","lean":"MetaMathlibExt.mul_generalizedTau_succ_modEq_zero","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility of "],["math","\\(n\\,\\tau_k(n + 1)\\)"],["text"," by "],["math","\\(|k|\\)"]],"source":"Divisibility of $n\\,\\tau_k(n + 1)$ by $|k|$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#mul-generalized-tau-succ-mod-eq-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau_k$ be the generalized tau function. For every positive integer $n$ and every integer $k$ with $|k| \\ge 2$, $n\\,\\tau_k(n + 1) \\equiv 0 \\pmod{|k|}$.","title":"Divisibility of $n\\,\\tau_k(n + 1)$ by $|k|$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/generalized-tau-mod-eq-zero-of-coprime","kind":"node","lean":"MetaMathlibExt.generalizedTau_modEq_zero_of_coprime","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing of the generalized tau function modulo "],["math","\\(|k|/d\\)"]],"source":"Vanishing of the generalized tau function modulo $|k|/d$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#generalized-tau-mod-eq-zero-of-coprime","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau_k(n)$ be the generalized tau function, the coefficient of $q^n$ in $q \\prod_{j \\ge 1} (1 - q^j)^k$. Let $k$ be an integer with $|k| \\ge 2$ and $d$ a natural number dividing $|k|$ with $d < |k|$. Then for all natural numbers $m$ and $r$ with $\\gcd(r, |k|/d) = 1$, $\\tau_k(|k| m + d r + 1) \\equiv 0 \\pmod{|k|/d}$.","title":"Vanishing of the generalized tau function modulo $|k|/d$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/ramanujan-tau-mod-eq-divisors-twentyfour","kind":"node","lean":"MetaMathlibExt.ramanujanTau_modEq_divisors_twentyfour","parent":11266,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's tau function modulo divisors of "],["math","\\(24\\)"]],"source":"Ramanujan's tau function modulo divisors of $24$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio/index.html#ramanujan-tau-mod-eq-divisors-twentyfour","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-partitions-and-a-generalized-tau-functio","status":"can_prove","summary":"Let $\\tau(n) = \\tau_{24}(n)$ be Ramanujan's tau function, the generalized tau function with $k = 24$. For every integer $m \\ge 0$: $\\tau(24m + r + 1) \\equiv 0 \\pmod{24}$ for $r \\in \\{1, 5, 7, 11, 13, 17, 19, 23\\}$; $\\tau(24m + r + 1) \\equiv 0 \\pmod 6$ for $r \\in \\{4, 20\\}$; $\\tau(24m + r + 1) \\equiv 0 \\pmod 8$ for $r \\in \\{3, 6, 9, 15\\}$; $\\tau(24m + r + 1) \\equiv 0 \\pmod 3$ for $r \\in \\{8, 16\\}$; and\u2026","title":"Ramanujan's tau function modulo divisors of $24$"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","title":"Smallest parts in overpartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/overpartition-spt2-contribution","kind":"node","lean":"MetaMathlibExt.overpartitionSpt2Contribution","parent":11307,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#overpartition-spt2-contribution","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"fully_proved","summary":"For an overpartition $O$ of $n$, let $s$ be the smallest part of $O$. The contribution of $O$ is the number of copies of $s$ in $O$ if $s$ is even and not overlined, and $0$ otherwise; the empty overpartition, which has no smallest part, contributes $0$.","title":"Smallest-even-part contribution of an overpartition"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/overpartition-spt2","kind":"node","lean":"MetaMathlibExt.overpartitionSpt2","parent":11307,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#overpartition-spt2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"fully_proved","summary":"For a natural number $n$, $\\overline{\\mathrm{spt2}}(n)$ is the number of smallest parts in the overpartitions of $n$ whose smallest part is even and not overlined: the sum, over all overpartitions $O$ of $n$, of the smallest-even-part contribution of $O$, which is the number of copies of the smallest part $s$ of $O$ when $s$ is even and not overlined, and $0$ otherwise (in particular when $O$ is empty). For example\u2026","title":"Number of smallest even parts in overpartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/overpartition-spt2-zero","kind":"node","lean":"MetaMathlibExt.overpartitionSpt2_zero","parent":11307,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\overline{\\mathrm{spt2}}\\)"],["text"," vanishes at zero"]],"source":"$\\overline{\\mathrm{spt2}}$ vanishes at zero"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#overpartition-spt2-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"fully_proved","summary":"Let $\\overline{\\mathrm{spt2}}(n)$ be the number of smallest parts in the overpartitions of $n$ whose smallest part is even and not overlined. Then $\\overline{\\mathrm{spt2}}(0) = 0$: the only overpartition of $0$ is empty and has no smallest part.","title":"$\\overline{\\mathrm{spt2}}$ vanishes at zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/overpartition-spt2-contribution-eq-zero","kind":"node","lean":"MetaMathlibExt.overpartitionSpt2Contribution_eq_zero_of_odd","parent":11307,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#overpartition-spt2-contribution-eq-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"fully_proved","summary":"If $O$ is an overpartition of $n$ with at least one part and the smallest part of $O$ is odd, then the smallest-even-part contribution of $O$ is $0$.","title":"Overpartitions with odd smallest part contribute nothing"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/overpartition-spt2-basic-congruences","kind":"node","lean":"MetaMathlibExt.overpartitionSpt2_basic_congruences","parent":11307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three congruences modulo 4 for "],["math","\\(\\overline{\\mathrm{spt2}}\\)"]],"source":"Three congruences modulo 4 for $\\overline{\\mathrm{spt2}}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#overpartition-spt2-basic-congruences","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $\\overline{\\mathrm{spt2}}(n)$ be the number of smallest parts in the overpartitions of $n$ whose smallest part is even and not overlined. For every natural number $n$, each of $\\overline{\\mathrm{spt2}}(36n + 30)$, $\\overline{\\mathrm{spt2}}(48n + 34)$ and $\\overline{\\mathrm{spt2}}(72n + 42)$ is divisible by $4$.","title":"Three congruences modulo 4 for $\\overline{\\mathrm{spt2}}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/overpartition-spt2-dyadic-congruences","kind":"node","lean":"MetaMathlibExt.overpartitionSpt2_dyadic_congruences","parent":11307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo 4 for "],["math","\\(\\overline{\\mathrm{spt2}}\\)"],["text"," along powers of 2"]],"source":"Congruences modulo 4 for $\\overline{\\mathrm{spt2}}$ along powers of 2"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#overpartition-spt2-dyadic-congruences","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $\\overline{\\mathrm{spt2}}(N)$ be the number of smallest parts in the overpartitions of $N$ whose smallest part is even and not overlined. For all natural numbers $j$ and $n$, $\\overline{\\mathrm{spt2}}(N)$ is divisible by $4$ for each of the six values $N = 2^{j+4} n + 7 \\cdot 2^{j+1}$, $N = 2^{j+2} \\cdot 9 n + 15 \\cdot 2^{j+1}$, $N = 2^{j+4} \\cdot 3 n + 17 \\cdot 2^{j+1}$,\u2026","title":"Congruences modulo 4 for $\\overline{\\mathrm{spt2}}$ along powers of 2"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/overpartition-spt2-prime-congruence","kind":"node","lean":"MetaMathlibExt.overpartitionSpt2_prime_congruence","parent":11307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo 4 for "],["math","\\(\\overline{\\mathrm{spt2}}\\)"],["text"," at primes "],["math","\\(p \\equiv 5, 7 \\pmod 8\\)"]],"source":"Congruences modulo 4 for $\\overline{\\mathrm{spt2}}$ at primes $p \\equiv 5, 7 \\pmod 8$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#overpartition-spt2-prime-congruence","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $\\overline{\\mathrm{spt2}}(N)$ be the number of smallest parts in the overpartitions of $N$ whose smallest part is even and not overlined. Let $p > 3$ be a prime with $p \\equiv 5 \\pmod 8$ or $p \\equiv 7 \\pmod 8$. Then for all natural numbers $k$ and $m$ with $p \\nmid m$, the number $\\overline{\\mathrm{spt2}}(32 p^{2k+1} m + 24 p^{2k+2})$ is divisible by $4$.","title":"Congruences modulo 4 for $\\overline{\\mathrm{spt2}}$ at primes $p \\equiv 5, 7 \\pmod 8$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/q-pochhammer-two-dissection-square","kind":"node","lean":"MetaMathlibExt.qPochhammer_two_dissection_square","parent":11307,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#q-pochhammer-two-dissection-square","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{n=0}^{\\infty} (1 - q^k (q^k)^n)$ be the raw infinite q-Pochhammer product. Then","title":"2-dissection of the square of Euler's product"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/q-pochhammer-two-dissection-inverse-squa","kind":"node","lean":"MetaMathlibExt.qPochhammer_two_dissection_inverse_square","parent":11307,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#q-pochhammer-two-dissection-inverse-squa","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{n=0}^{\\infty} (1 - q^k (q^k)^n)$ be the raw infinite q-Pochhammer product. Then","title":"2-dissection of the reciprocal square of Euler's product"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/q-pochhammer-three-dissection-cube","kind":"node","lean":"MetaMathlibExt.qPochhammer_three_dissection_cube","parent":11307,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 3-dissection of "],["math","\\(f_1^3\\)"],["text"," for complex "],["math","\\(q\\)"]],"source":"The 3-dissection of $f_1^3$ for complex $q$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#q-pochhammer-three-dissection-cube","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{m \\ge 1} (1 - q^{km})$ be the raw infinite q-Pochhammer product with $a = q^k$ and base $q^k$. Then","title":"The 3-dissection of $f_1^3$ for complex $q$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/q-pochhammer-three-dissection-ratio-four","kind":"node","lean":"MetaMathlibExt.qPochhammer_three_dissection_ratio_four_one","parent":11307,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 3-dissection of "],["math","\\(f_4/f_1\\)"],["text"," for complex "],["math","\\(q\\)"]],"source":"The 3-dissection of $f_4/f_1$ for complex $q$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#q-pochhammer-three-dissection-ratio-four","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{m \\ge 1} (1 - q^{km})$ be the raw infinite q-Pochhammer product with $a = q^k$ and base $q^k$. Then","title":"The 3-dissection of $f_4/f_1$ for complex $q$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/q-pochhammer-three-dissection-product-on","kind":"node","lean":"MetaMathlibExt.qPochhammer_three_dissection_product_one_two","parent":11307,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 3-dissection of "],["math","\\(f_1 f_2\\)"],["text"," for complex "],["math","\\(q\\)"]],"source":"The 3-dissection of $f_1 f_2$ for complex $q$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#q-pochhammer-three-dissection-product-on","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{m \\ge 1} (1 - q^{km})$ be the raw infinite q-Pochhammer product with $a = q^k$ and base $q^k$. Then","title":"The 3-dissection of $f_1 f_2$ for complex $q$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/ramanujan-psi-three-dissection","kind":"node","lean":"MetaMathlibExt.ramanujanPsi_three_dissection","parent":11307,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#ramanujan-psi-three-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{n=0}^{\\infty} (1 - q^k (q^k)^n)$ be the raw infinite q-Pochhammer product. Then","title":"3-dissection of the product form of Ramanujan's psi function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/ramanujan-psi-neg-dissection","kind":"node","lean":"MetaMathlibExt.ramanujanPsi_neg_dissection","parent":11307,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#ramanujan-psi-neg-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{n=0}^{\\infty} (1 - q^k (q^k)^n)$ be the raw infinite q-Pochhammer product. Then","title":"3-dissection of the product form of Ramanujan's psi function at negated argument"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/ramanujan-phi-neg-dissection","kind":"node","lean":"MetaMathlibExt.ramanujanPhi_neg_dissection","parent":11307,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions/index.html#ramanujan-phi-neg-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/smallest-parts-in-overpartitions","status":"can_prove","summary":"Let $q$ be a complex number with $|q| < 1$, and for $k \\ge 1$ let $f_k = (q^k; q^k)_\\infty = \\prod_{n=0}^{\\infty} (1 - q^k (q^k)^n)$ be the raw infinite q-Pochhammer product. Then","title":"3-dissection of the product form of Ramanujan's phi function at negated argument"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","title":"Tuples of \u2113-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/l-regular-partition-count","kind":"node","lean":"MetaMathlibExt.lRegularPartitionCount","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(\\ell\\)"],["text","-regular partitions"]],"source":"Number of $\\ell$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#l-regular-partition-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For natural numbers $\\ell$ and $n$, the number of $\\ell$-regular partitions of $n$ is the number of partitions of $n$ none of whose parts is divisible by $\\ell$.","title":"Number of $\\ell$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/k-tuple-regular-partition-count","kind":"node","lean":"MetaMathlibExt.kTupleRegularPartitionCount","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-tuples of "],["math","\\(\\ell\\)"],["text","-regular partitions"]],"source":"Number of $k$-tuples of $\\ell$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#k-tuple-regular-partition-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For natural numbers $\\ell$, $k$ and $n$, the number $T_{\\ell,k}(n)$ is defined by recursion on $k$: $T_{\\ell,0}(n)$ is $1$ if $n = 0$ and $0$ otherwise, and","title":"Number of $k$-tuples of $\\ell$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/k-tuple-regular-partition-count-zero","kind":"node","lean":"MetaMathlibExt.kTupleRegularPartitionCount_zero","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","The empty tuple of "],["math","\\(\\ell\\)"],["text","-regular partitions"]],"source":"The empty tuple of $\\ell$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#k-tuple-regular-partition-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For all natural numbers $\\ell$ and $n$, writing $T_{\\ell,k}(n)$ for the number of $k$-tuples of $\\ell$-regular partitions with total $n$, we have $T_{\\ell,0}(n) = 1$ if $n = 0$ and $T_{\\ell,0}(n) = 0$ otherwise.","title":"The empty tuple of $\\ell$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/k-tuple-regular-partition-count-succ","kind":"node","lean":"MetaMathlibExt.kTupleRegularPartitionCount_succ","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for tuples of "],["math","\\(\\ell\\)"],["text","-regular partitions"]],"source":"Recurrence for tuples of $\\ell$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#k-tuple-regular-partition-count-succ","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For all natural numbers $\\ell$, $k$ and $n$, writing $r_\\ell(i)$ for the number of $\\ell$-regular partitions of $i$ and $T_{\\ell,k}(n)$ for the number of $k$-tuples of $\\ell$-regular partitions with total $n$,","title":"Recurrence for tuples of $\\ell$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/triple-regular-partition-count","kind":"node","lean":"MetaMathlibExt.tripleRegularPartitionCount","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triples of "],["math","\\(\\ell\\)"],["text","-regular partitions"]],"source":"Triples of $\\ell$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#triple-regular-partition-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For natural numbers $\\ell$ and $n$, $T_\\ell(n) = T_{\\ell,3}(n)$ is the number of triples $(\\lambda_1, \\lambda_2, \\lambda_3)$ of $\\ell$-regular partitions (partitions with no part divisible by $\\ell$) whose sizes add up to $n$.","title":"Triples of $\\ell$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/l-regular-partition-count-zero","kind":"node","lean":"MetaMathlibExt.lRegularPartitionCount_zero","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero has one "],["math","\\(\\ell\\)"],["text","-regular partition"]],"source":"Zero has one $\\ell$-regular partition"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#l-regular-partition-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For every natural number $\\ell$, the number of $\\ell$-regular partitions of $0$ is $1$; the only one is the empty partition.","title":"Zero has one $\\ell$-regular partition"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/k-tuple-regular-partition-count-at-zero","kind":"node","lean":"MetaMathlibExt.kTupleRegularPartitionCount_at_zero","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","One tuple of "],["math","\\(\\ell\\)"],["text","-regular partitions of total zero"]],"source":"One tuple of $\\ell$-regular partitions of total zero"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#k-tuple-regular-partition-count-at-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For all natural numbers $\\ell$ and $k$, writing $T_{\\ell,k}(n)$ for the number of $k$-tuples of $\\ell$-regular partitions with total $n$, we have $T_{\\ell,k}(0) = 1$: the only such tuple of total $0$ is the tuple of empty partitions.","title":"One tuple of $\\ell$-regular partitions of total zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/bounded-bilateral-series","kind":"node","lean":"MetaMathlibExt.boundedBilateralSeries","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#bounded-bilateral-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"Given functions $e, w : \\mathbb{Z} \\to \\mathbb{Z}$, the bounded bilateral series of $e$ and $w$ is the power series over $\\mathbb{Z}$ whose coefficient of $q^n$, for each natural number $n$, is the sum of $w(m)$ over the integers $m$ with $-n-1 \\le m \\le n+1$ and $e(m) = n$.","title":"Bounded bilateral series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/coeff-bounded-bilateral-series","kind":"node","lean":"MetaMathlibExt.coeff_boundedBilateralSeries","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#coeff-bounded-bilateral-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For all functions $e, w : \\mathbb{Z} \\to \\mathbb{Z}$ and every natural number $n$, the coefficient of $q^n$ in the bounded bilateral series of $e$ and $w$ equals the sum of $w(m)$ over the integers $m$ with $-n-1 \\le m \\le n+1$ and $e(m) = n$.","title":"Coefficients of the bounded bilateral series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/positive-unilateral-series","kind":"node","lean":"MetaMathlibExt.positiveUnilateralSeries","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#positive-unilateral-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For functions $e : \\mathbb{N} \\to \\mathbb{N}$ and $w : \\mathbb{N} \\to \\mathbb{Z}$, this is the integer power series whose coefficient of $q^n$ is the sum of $w(m)$ over the integers $m$ with $1 \\le m \\le n$ and $e(m) = n$.","title":"Truncated one-sided series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/coeff-positive-unilateral-series","kind":"node","lean":"MetaMathlibExt.coeff_positiveUnilateralSeries","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#coeff-positive-unilateral-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For all functions $e : \\mathbb{N} \\to \\mathbb{N}$ and $w : \\mathbb{N} \\to \\mathbb{Z}$ and every natural number $n$, the coefficient of $q^n$ in the positive unilateral series of $e$ and $w$ equals the sum of $w(m)$ over the natural numbers $m$ with $1 \\le m \\le n$ and $e(m) = n$.","title":"Coefficients of the positive unilateral series"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/triple-two-mod-twenty-four","kind":"node","lean":"MetaMathlibExt.tripleTwo_mod_twentyFour","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#triple-two-mod-twenty-four","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(n)$ be the number of triples of 2-regular partitions (partitions with no even part) whose sizes add up to $n$. If $t$ is a positive integer with $\\gcd(t, 6) = 1$ and $N \\in \\{33, 57\\}$, then for every natural number $n$,","title":"Congruences modulo 24 for triples of 2-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/nath-saikia-sarma-conjecture","kind":"node","lean":"MetaMathlibExt.nathSaikiaSarma_conjecture","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#nath-saikia-sarma-conjecture","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(n)$ be the number of triples of 2-regular partitions (partitions with no even part) whose sizes add up to $n$. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-2}{p}\\right) = -1$, and let $t$ be a positive integer with $\\gcd(t, 6) = 1$ and $p \\mid t$. Then for every natural number $n$ and every $j$ with $1 \\le j \\le p - 1$,","title":"The Nath\u2013Saikia\u2013Sarma conjecture"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/triple-two-prime-power-mod-eight-and-twe","kind":"node","lean":"MetaMathlibExt.tripleTwo_primePower_mod_eight_and_twentyFour","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#triple-two-prime-power-mod-eight-and-twe","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(n)$ be the number of triples of $2$-regular partitions (partitions with no even part) whose sizes add up to $n$. Let $p$ be a prime with $p \\equiv 3$, $5$ or $7 \\pmod 8$, and let $n$ and $\\alpha$ be natural numbers with $p \\nmid n$. Then","title":"Congruences modulo 8 and 24 for triples of 2-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/triple-two-nine-prime-power-mod-twenty-f","kind":"node","lean":"MetaMathlibExt.tripleTwo_nine_primePower_mod_twentyFour","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#triple-two-nine-prime-power-mod-twenty-f","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(n)$ be the number of triples of 2-regular partitions (partitions with no even part) whose sizes add up to $n$. Let $p \\ne 3$ be a prime with $p \\equiv 3$, $5$ or $7 \\pmod 8$. Then for all natural numbers $n$ and $\\alpha$ with $p \\nmid n$,","title":"Prime-power congruences modulo 24 for triples of 2-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/q-euler-product-fifth-power-identity","kind":"node","lean":"MetaMathlibExt.qEulerProduct_fifth_power_identity","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#q-euler-product-fifth-power-identity","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Write $f_k = \\prod_{i \\ge 1} (1 - q^{ki})$ for the Euler products, as integer power series in $q$. Then","title":"A fifth-power Euler product identity"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/rescale-neg-one-q-euler-product-identity","kind":"node","lean":"MetaMathlibExt.rescale_neg_one_qEulerProduct_identity","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#rescale-neg-one-q-euler-product-identity","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Write $f_k = \\prod_{i \\ge 1} (1 - q^{ki})$ for the Euler products, as integer power series in $q$, and let $(-q;-q)_\\infty = \\prod_{i \\ge 1} (1 - (-q)^i)$ be $f_1$ with $q$ replaced by $-q$. Then","title":"A signed Euler product identity"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/q-euler-product-square-theta-identity","kind":"node","lean":"MetaMathlibExt.qEulerProduct_square_theta_identity","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#q-euler-product-square-theta-identity","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Write $f_k = \\prod_{i \\ge 1} (1 - q^{ki})$ for the Euler products, as integer power series in $q$, and let $\\theta = \\sum_{m \\in \\mathbb{Z}} (-1)^m q^{m^2}$. Then $f_1^2 = f_2\\, \\theta$ and $\\theta = 1 + 2 \\sum_{m \\ge 1} (-1)^m q^{m^2}$.","title":"Gauss's theta identity for the Euler products"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/q-euler-product-two-square-identity","kind":"node","lean":"MetaMathlibExt.qEulerProduct_two_square_identity","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#q-euler-product-two-square-identity","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Write $f_k = \\prod_{i \\ge 1} (1 - q^{ki})$ for the Euler products, as integer power series in $q$. Then","title":"The theta identity for the second Euler product"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/q-euler-product-iterated-prime-power-con","kind":"node","lean":"MetaMathlibExt.qEulerProduct_iterated_primePower_congruence","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#q-euler-product-iterated-prime-power-con","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"For a natural number $l$ write $f_l = \\prod_{i \\ge 1} (1 - q^{li})$, an integer power series in $q$. Let $p$ be a prime and let $k, l, s, m$ be natural numbers with $l \\ge 1$, $s \\ge 1$, $m \\ge 1$ and $s \\le k$. Then for every natural number $n$, the coefficients of $q^n$ in $f_l^{p^k m}$ and in $f_{l p^s}^{p^{k-s} m}$ are congruent modulo $p^{k-s+1}$.","title":"Iterated prime-power congruence for Euler products"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/triple-two-odd-iff-triangular","kind":"node","lean":"MetaMathlibExt.tripleTwo_odd_iff_triangular","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#triple-two-odd-iff-triangular","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"For a natural number $n$, let $T_2(n)$ be the number of triples of $2$-regular partitions (partitions with no even part) whose sizes add up to $n$. Then $T_2(n)$ is odd if and only if $n$ is a triangular number, that is, $n = m(m+1)/2$ for some natural number $m$.","title":"Parity of triples of 2-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/k-tuple-regular-partition-prime-power-co","kind":"node","lean":"MetaMathlibExt.kTupleRegularPartition_primePower_congruent","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-power congruences for "],["math","\\(k\\)"],["text","-tuples of "],["math","\\(\\ell\\)"],["text","-regular partitions"]],"source":"Prime-power congruences for $k$-tuples of $\\ell$-regular partitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#k-tuple-regular-partition-prime-power-co","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Write $T_{\\ell,k}(n)$ for the number of $k$-tuples of $\\ell$-regular partitions with total $n$. Let $p$ be a prime, let $m$, $\\ell$ and $s$ be positive integers, let $\\alpha \\ge s$ and $r$ be natural numbers with $1 \\le r \\le p^s - 1$. Then for every natural number $n$,","title":"Prime-power congruences for $k$-tuples of $\\ell$-regular partitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For natural numbers $\\ell$ and $n$, $T_\\ell(n)$ is the number of triples $(\\lambda_1, \\lambda_2, \\lambda_3)$ of integer partitions with $|\\lambda_1| + |\\lambda_2| + |\\lambda_3| = n$ in which no part of any $\\lambda_i$ is divisible by $\\ell$; these triples are the $\\ell$-regular tripartitions of $n$.","title":"Regular tripartition counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-zero","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_zero","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For every natural number $\\ell$, $T_\\ell(0) = 1$, where $T_\\ell(n)$ is the number of $\\ell$-regular tripartitions of $n$: the only tripartition of $0$ is the triple of empty partitions.","title":"The regular tripartition count at zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-one","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_one","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","The number of "],["math","\\(\\ell\\)"],["text","-regular tripartitions of "],["math","\\(1\\)"]],"source":"The number of $\\ell$-regular tripartitions of $1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-one","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"fully_proved","summary":"For every natural number $\\ell \\ne 1$, the number $T_\\ell(1)$ of $\\ell$-regular tripartitions of $1$ is $3$: the single part $1$ lies in one of the three components.","title":"The number of $\\ell$-regular tripartitions of $1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-two-mod-three","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_two_mod_three_six","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-two-mod-three","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(m)$ be the number of $2$-regular tripartitions of $m$. For every natural number $n$, $3 \\mid T_2(3n + 1)$ and $6 \\mid T_2(3n + 2)$.","title":"Congruences modulo 3 and 6 for 2-regular tripartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-three-mod-thr","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_three_mod_three_nine","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-three-mod-thr","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_3(m)$ be the number of $3$-regular tripartitions of $m$. For every natural number $n$, $3 \\mid T_3(3n + 1)$ and $9 \\mid T_3(3n + 2)$.","title":"Congruences modulo 3 and 9 for 3-regular tripartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-three-six-mul","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_three_six_mul_add_four","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-three-six-mul","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_3(m)$ be the number of $3$-regular tripartitions of $m$. For every natural number $n$, $6 \\mid T_3(6n + 4)$.","title":"A congruence modulo 6 for 3-regular tripartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-two-prime-ge","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_two_prime_ge_five_even","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-two-prime-ge","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(n)$ be the number of 2-regular tripartitions of $n$. For every prime $p \\ge 5$, all natural numbers $\\alpha$ and $n$, and every $i$ with $1 \\le i \\le p - 1$, the number $T_2\\bigl(3 p^{2\\alpha+1} (p n + i) + (p^{2\\alpha+2} - 1)/8\\bigr)$ is even (the division is exact).","title":"Even 2-regular tripartition counts for primes at least 5"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-two-odd-prime","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_two_odd_prime_even","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-two-odd-prime","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(n)$ be the number of 2-regular tripartitions of $n$. For every odd prime $p$, all natural numbers $\\alpha$ and $n$, and every $i$ with $1 \\le i \\le p - 1$, the number $T_2\\bigl(9 p^{2\\alpha+1} (p n + i) + (9 p^{2\\alpha+2} - 1)/8\\bigr)$ is even (the division is exact).","title":"Even 2-regular tripartition counts for odd primes"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-three-prime-t","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_three_prime_three_mod_four","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","A congruence modulo 3 for 3-regular tripartitions at primes "],["math","\\(p \\equiv 3 \\pmod 4\\)"]],"source":"A congruence modulo 3 for 3-regular tripartitions at primes $p \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-three-prime-t","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_3(m)$ be the number of $3$-regular tripartitions of $m$, and let $p$ be a prime with $p \\equiv 3 \\pmod 4$. For all natural numbers $\\alpha$ and $n$ and every integer $i$ with $1 \\le i \\le p - 1$,","title":"A congruence modulo 3 for 3-regular tripartitions at primes $p \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/f2-cube-div-f1-cube-three-dissection","kind":"node","lean":"MetaMathlibExt.f2_cube_div_f1_cube_three_dissection","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Toh's 3-dissection of "],["math","\\(f_2^3 / f_1^3\\)"]],"source":"Toh's 3-dissection of $f_2^3 / f_1^3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#f2-cube-div-f1-cube-three-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ be the Euler product, an integer power series in $q$. Then","title":"Toh's 3-dissection of $f_2^3 / f_1^3$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/one-div-f1-cube-three-dissection","kind":"node","lean":"MetaMathlibExt.one_div_f1_cube_three_dissection","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 3-dissection of "],["math","\\(1/f_1^3\\)"]],"source":"The 3-dissection of $1/f_1^3$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#one-div-f1-cube-three-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"For $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ be the Euler product, an integer power series in $q$. Then","title":"The 3-dissection of $1/f_1^3$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/ramanujan-psi-prime-dissection","kind":"node","lean":"MetaMathlibExt.ramanujanPsi_prime_dissection","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cui\u2013Gu "],["math","\\(p\\)"],["text","-dissection of "],["math","\\(\\psi(q)\\)"]],"source":"Cui\u2013Gu $p$-dissection of $\\psi(q)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#ramanujan-psi-prime-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $p$ be an odd prime and $q$ a complex number with $|q| < 1$, let $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function, and let $\\psi(q) = f(q, q^3)$. Then","title":"Cui\u2013Gu $p$-dissection of $\\psi(q)$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/q-pochhammer-prime-dissection","kind":"node","lean":"MetaMathlibExt.qPochhammer_prime_dissection","parent":11323,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cui\u2013Gu "],["math","\\(p\\)"],["text","-dissection of "],["math","\\(f_1\\)"]],"source":"Cui\u2013Gu $p$-dissection of $f_1$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#q-pochhammer-prime-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $p \\ge 5$ be a prime, let $s = (p - 1)/6$ if $p \\equiv 1 \\pmod 6$ and $s = (-p - 1)/6$ otherwise, and let $q$ be a complex number with $|q| < 1$. Write $f_k = (q^k; q^k)_\\infty = \\prod_{m \\ge 1} (1 - q^{km})$, let $f(a, b) = \\sum_{n \\in \\mathbb{Z}} a^{n(n+1)/2} b^{n(n-1)/2}$ be Ramanujan's theta function, and let $K$ be the set of integers $k$ with $(1 - p)/2 \\le k \\le (p - 1)/2$ and $k \\ne s$. Then","title":"Cui\u2013Gu $p$-dissection of $f_1$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-two-three-dis","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_two_three_dissection","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-two-three-dis","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(n)$ be the number of 2-regular tripartitions of $n$ and $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. In the ring $\\mathbb{Z}[[q]]$ of integer power series, $f_1^6 f_3 \\sum_{n \\ge 0} T_2(3n) q^n = f_1^5 f_2 f_3 + 12 q f_2^2 f_6^5$, $f_1^8 f_6 \\sum_{n \\ge 0} T_2(3n + 1) q^n = 3 f_2^4 f_3^5$ and $f_1^7 \\sum_{n \\ge 0} T_2(3n + 2) q^n = 6 f_2^3 f_3^2 f_6^2$.","title":"The 3-dissection of the 2-regular tripartition generating function"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-two-mod-twelv","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_two_mod_twelve","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-two-mod-twelv","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_2(n)$ be the number of 2-regular tripartitions of $n$. For all natural numbers $\\alpha$ and $n$, put $A_\\alpha = \\sum_{i=0}^{2\\alpha+1} 3^{2i}$ and $B_\\alpha = \\sum_{i=0}^{2\\alpha} 3^{2i}$. Then $12$ divides each of $T_2(3^{4\\alpha+4} n + A_\\alpha + 3^{4\\alpha+3})$, $T_2(3^{4\\alpha+4} n + A_\\alpha + 2 \\cdot 3^{4\\alpha+3})$, $T_2(3^{4\\alpha+2} n + B_\\alpha + 3^{4\\alpha+1})$ and\u2026","title":"Mod 12 congruences for 2-regular tripartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/regular-tripartition-count-three-three-d","kind":"node","lean":"MetaMathlibExt.regularTripartitionCount_three_three_dissection","parent":11323,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions/index.html#regular-tripartition-count-three-three-d","scope":"NumberTheory/modular-forms-q-series/partition-congruences/tuples-of-regular-partitions","status":"can_prove","summary":"Let $T_3(n)$ be the number of $3$-regular tripartitions of $n$, and for $k \\ge 1$ let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ be the Euler product. Then, as integer power series in $q$,","title":"The 3-dissection of the generating function of 3-regular tripartitions"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","title":"Two-color partitions modulo powers of 7"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/two-color-multiple-partition-count","kind":"node","lean":"MetaMathlibExt.twoColorMultiplePartitionCount","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#two-color-multiple-partition-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"For natural numbers $\\ell$ and $n$, let $p(m)$ be the number of partitions of $m$, and define","title":"Two-color partitions with one color restricted to multiples"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/two-color-multiple-partition-count-of-lt","kind":"node","lean":"MetaMathlibExt.twoColorMultiplePartitionCount_of_lt","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#two-color-multiple-partition-count-of-lt","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"Let $p_{1,\\ell}(n)$ be the number of 2-color partitions of $n$ in which the parts of the second color are multiples of $\\ell$, and $p(n)$ the number of partitions of $n$. For all natural numbers $\\ell$ and $n$ with $n < \\ell$, the second color is unavailable and $p_{1,\\ell}(n) = p(n)$.","title":"Two-color partitions below the modulus"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/two-color-multiple-partition-count-zero","kind":"node","lean":"MetaMathlibExt.twoColorMultiplePartitionCount_zero","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#two-color-multiple-partition-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"For every natural number $\\ell$, including $\\ell = 0$, the two-color partition count $p_{1,\\ell}$ satisfies $p_{1,\\ell}(0) = 1$.","title":"The two-color partition count at zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/two-color-multiple-partition-count-seven","kind":"node","lean":"MetaMathlibExt.twoColorMultiplePartitionCount_seven_pow_congr","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#two-color-multiple-partition-count-seven","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"For all natural numbers $n$, $\\beta$ and $k \\ge 1$, the two-color partition count $p_{1,\\ell}$ satisfies","title":"Congruences for two-color partitions modulo powers of 7"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-mtable","kind":"node","lean":"MetaMathlibExt.garvanMTable","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-mtable","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"For $1 \\le i \\le 7$ and $j \\ge 1$, the value at $(i, j)$ is the $j$-th entry of row $i$ in the list below, and $0$ if row $i$ has fewer than $j$ entries (row $i$ has $2i$ entries). The value is $0$ if $i = 0$, $j = 0$ or $i > 7$.","title":"Tabulated rows of Garvan's matrix"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-m","kind":"node","lean":"MetaMathlibExt.garvanM","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"Garvan's matrix $(m_{i,j})_{i,j \\ge 1}$ is the array of natural numbers defined as follows. For $1 \\le i \\le 7$, $m_{i,j}$ is the entry in row $i$ and column $j$ of the tabulated rows of Garvan's matrix (Tables 1 and 2 of the source), and $0$ beyond the printed columns. For $i \\ge 8$, $m_{i,1} = m_{i,2} = 0$ and, for $j \\ge 3$,","title":"Garvan's matrix"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-m-one","kind":"node","lean":"MetaMathlibExt.garvanM_one","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-m-one","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"The first two entries of the first row of Garvan's matrix $(m_{i,j})$ are $m_{1,1} = 7$ and $m_{1,2} = 7^2 = 49$.","title":"Initial entries of the first row of Garvan's matrix"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-m-eight-three","kind":"node","lean":"MetaMathlibExt.garvanM_eight_three","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-m-eight-three","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"The entry of Garvan's matrix $(m_{i,j})$ in row $8$ and column $3$, the first entry computed by the recurrence rather than read from the tables, is $m_{8,3} = 2464$.","title":"The first recurrence value of Garvan's matrix"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-m-rec","kind":"node","lean":"MetaMathlibExt.garvanM_rec","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-m-rec","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"For all natural numbers $i \\ge 8$ and $j \\ge 3$, the entries of Garvan's matrix $(m_{i,j})$ satisfy","title":"Recurrence for Garvan's matrix"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-x","kind":"node","lean":"MetaMathlibExt.garvanX","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-x","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"Let $(m_{i,j})$ be Garvan's matrix. Garvan's coefficient vectors $x_k = (x_{k,1}, x_{k,2}, \\dots)$ for $k \\ge 1$ are defined by $x_{1,1} = 7$, $x_{1,2} = 7^2$ and $x_{1,i} = 0$ for $i \\ge 3$, and, for $k \\ge 1$ and $i \\ge 1$, by","title":"Garvan's coefficient vectors"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-x-one","kind":"node","lean":"MetaMathlibExt.garvanX_one","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-x-one","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"The first of Garvan's coefficient vectors, $x_1$, has entries $x_{1,1} = 7$, $x_{1,2} = 7^2 = 49$ and $x_{1,3} = 0$.","title":"Initial entries of Garvan's first coefficient vector"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-yodd","kind":"node","lean":"MetaMathlibExt.garvanYOdd","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-yodd","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"For $k \\ge 1$ and $\\beta \\ge 1$ the vectors $y^{(2k-1)}_\\beta = (y^{(2k-1)}_{\\beta,1}, y^{(2k-1)}_{\\beta,2}, \\dots)$ are built from Garvan's vector $x_{2k-1}$ and Garvan's matrix $m_{i,j}$: $y^{(2k-1)}_{1,j} = x_{2k-1,j}$ and","title":"Coefficient vectors for odd powers of 7"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-yodd-one","kind":"node","lean":"MetaMathlibExt.garvanYOdd_one","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-yodd-one","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"For all natural numbers $k$ and $j$, the first coefficient vector of Theorem 2 agrees with Garvan's coefficient vector $x_{2k-1}$: $y^{(2k-1)}_{1,j} = x_{2k-1,j}$. For $k = 0$ the index $2k - 1$ is read as $0$.","title":"Initial coefficient vector for odd powers of 7"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-yeven","kind":"node","lean":"MetaMathlibExt.garvanYEven","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-yeven","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"Let $(m_{i,j})$ be Garvan's matrix and $x_k$ Garvan's coefficient vectors, and fix $k \\ge 1$. The coefficient vectors $y^{(2k)}_\\beta = (y^{(2k)}_{\\beta,1}, y^{(2k)}_{\\beta,2}, \\dots)$ of Theorem 3, $\\beta \\ge 1$, are defined by $y^{(2k)}_{1,j} = x_{2k-1,j}$ and, for $\\beta \\ge 1$,","title":"Coefficient vectors for even powers of 7"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/garvan-yeven-one","kind":"node","lean":"MetaMathlibExt.garvanYEven_one","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#garvan-yeven-one","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"fully_proved","summary":"For all natural numbers $k$ and $j$, the first coefficient vector of Theorem 3 agrees with Garvan's coefficient vector $x_{2k-1}$: $y^{(2k)}_{1,j} = x_{2k-1,j}$. For $k = 0$ the index $2k - 1$ is read as $0$.","title":"Initial coefficient vector for even powers of 7"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/huffing-xi-inv-pow-eq-sum-garvan-m","kind":"node","lean":"MetaMathlibExt.huffing_xi_inv_pow_eq_sum_garvanM","parent":11362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Garvan's huffing identities for powers of "],["math","\\(\\xi\\)"]],"source":"Garvan's huffing identities for powers of $\\xi$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#huffing-xi-inv-pow-eq-sum-garvan-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"Let $f_k = \\prod_{n \\ge 1} (1 - q^{kn})$ in $\\mathbb{Q}[[q]]$, let $H$ be the huffing operator $H\\bigl(\\sum_n p_n q^n\\bigr) = \\sum_n p_{7n} q^{7n}$, and let $m_{i,j}$ be Garvan's matrix. Write $\\xi^{-a} = q^{2a} f_{49}^a / f_1^a$ and $T^{-b} = q^{7b} f_{49}^{4b} / f_7^{4b}$ for the negative powers of $\\xi = f_1 / (q^2 f_{49})$ and $T = f_7^4 / (q^7 f_{49}^4)$, which are formal power series. For every $i \\ge 1$,","title":"Garvan's huffing identities for powers of $\\xi$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/huffing-eta-quotients-eq-sum-garvan-m","kind":"node","lean":"MetaMathlibExt.huffing_eta_quotients_eq_sum_garvanM","parent":11362,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#huffing-eta-quotients-eq-sum-garvan-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"Let $f_k = \\prod_{n \\ge 1} (1 - q^{kn})$ in $\\mathbb{Q}[[q]]$, let $H$ be the huffing operator $H\\bigl(\\sum_n p_n q^n\\bigr) = \\sum_n p_{7n} q^{7n}$, and let $m_{i,j}$ be Garvan's matrix. For every $i \\ge 1$,","title":"Huffing identities for three eta quotients"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/seven-pow-dvd-garvan-m","kind":"node","lean":"MetaMathlibExt.seven_pow_dvd_garvanM","parent":11362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Garvan's 7-adic bounds for the matrix "],["math","\\(m_{i,j}\\)"]],"source":"Garvan's 7-adic bounds for the matrix $m_{i,j}$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#seven-pow-dvd-garvan-m","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"Let $m_{i,j}$ be Garvan's matrix. For all $i, j \\ge 1$, $7^{\\lfloor (7j - 2i - 1)/4 \\rfloor}$ divides $m_{i,j}$, and in particular $7^{\\lfloor (7j - i - 1)/4 \\rfloor}$ divides $m_{4i,i+j}$, $7^{\\lfloor (7j - i - 3)/4 \\rfloor}$ divides $m_{4i+1,i+j}$ and $7^{\\lfloor (7j - i - 5)/4 \\rfloor}$ divides $m_{4i+2,i+j}$. A negative exponent is replaced by $0$, which gives the same statement.","title":"Garvan's 7-adic bounds for the matrix $m_{i,j}$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/seven-pow-dvd-garvan-x","kind":"node","lean":"MetaMathlibExt.seven_pow_dvd_garvanX","parent":11362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Garvan's 7-adic bounds for the vectors "],["math","\\(x_k\\)"]],"source":"Garvan's 7-adic bounds for the vectors $x_k$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#seven-pow-dvd-garvan-x","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"Let $x_k = (x_{k,1}, x_{k,2}, \\dots)$ be Garvan's coefficient vectors and $\\pi$ the $7$-adic valuation. Then $\\pi(x_{1,1}) = 1$, $\\pi(x_{1,2}) = 2$, and for all $k, j \\ge 1$, $7^{k + 1 + \\lfloor (7j-4)/4 \\rfloor}$ divides $x_{2k+1,j}$ and $7^{k + 1 + \\lfloor (7j-6)/4 \\rfloor}$ divides $x_{2k,j}$.","title":"Garvan's 7-adic bounds for the vectors $x_k$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/seven-pow-dvd-garvan-yeven","kind":"node","lean":"MetaMathlibExt.seven_pow_dvd_garvanYEven","parent":11362,"reading_title":{"format":"inline-parts-v1","parts":[["text","7-adic bounds for the vectors "],["math","\\(y^{(2k)}_\\beta\\)"]],"source":"7-adic bounds for the vectors $y^{(2k)}_\\beta$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#seven-pow-dvd-garvan-yeven","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"Let $y^{(2k)}_\\beta$ be the coefficient vectors for even powers of $7$, built from Garvan's vectors and matrix. For all $k, j \\ge 1$ and $\\beta \\ge 0$, $7^{k + \\beta + \\lfloor (7j-7)/4 \\rfloor}$ divides $y^{(2k)}_{2\\beta+1,j}$ and $7^{k + \\beta + 1 + \\lfloor (7j-7)/4 \\rfloor}$ divides $y^{(2k)}_{2\\beta+2,j}$.","title":"7-adic bounds for the vectors $y^{(2k)}_\\beta$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/seven-pow-dvd-garvan-yodd","kind":"node","lean":"MetaMathlibExt.seven_pow_dvd_garvanYOdd","parent":11362,"reading_title":{"format":"inline-parts-v1","parts":[["text","7-adic bounds for the vectors "],["math","\\(y^{(2k-1)}_\\beta\\)"]],"source":"7-adic bounds for the vectors $y^{(2k-1)}_\\beta$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#seven-pow-dvd-garvan-yodd","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"Let $y^{(2k-1)}_\\beta$ be the coefficient vectors for odd powers of $7$, built from Garvan's vectors and matrix. For all $k, j \\ge 1$ and $\\beta \\ge 0$, $7^{k + \\lfloor (7j-7)/4 \\rfloor}$ divides $y^{(2k-1)}_{\\beta+1,j}$.","title":"7-adic bounds for the vectors $y^{(2k-1)}_\\beta$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/two-color-multiple-partition-count-seven-2","kind":"node","lean":"MetaMathlibExt.twoColorMultiplePartitionCount_seven_pow_odd_generatingFunction","parent":11362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions for "],["math","\\(p_{1,7^{2k-1}}\\)"],["text"," on progressions"]],"source":"Generating functions for $p_{1,7^{2k-1}}$ on progressions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#two-color-multiple-partition-count-seven-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"Let $p_{1,\\ell}(n)$ be the number of 2-color partitions of $n$ in which the parts of the second color are multiples of $\\ell$, let $f_k = \\prod_{n \\ge 1} (1 - q^{kn})$ in $\\mathbb{Q}[[q]]$, and let $y^{(2k-1)}_\\beta$ be the coefficient vectors for odd powers of $7$. For all $k \\ge 1$ and $\\beta \\ge 0$, with $N = 7^{2k+\\beta-1}$,","title":"Generating functions for $p_{1,7^{2k-1}}$ on progressions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/two-color-multiple-partition-count-seven-3","kind":"node","lean":"MetaMathlibExt.twoColorMultiplePartitionCount_seven_pow_even_generatingFunction","parent":11362,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions for "],["math","\\(p_{1,7^{2k}}\\)"],["text"," on progressions"]],"source":"Generating functions for $p_{1,7^{2k}}$ on progressions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7/index.html#two-color-multiple-partition-count-seven-3","scope":"NumberTheory/modular-forms-q-series/partition-congruences/two-color-partitions-modulo-powers-of-7","status":"can_prove","summary":"Let $p_{1,\\ell}(n)$ be the number of 2-color partitions of $n$ in which the parts of the second color are multiples of $\\ell$, let $f_k = \\prod_{n \\ge 1} (1 - q^{kn})$ in $\\mathbb{Q}[[q]]$, and let $y^{(2k)}_\\beta$ be the coefficient vectors for even powers of $7$. For all $k \\ge 1$ and $\\beta \\ge 0$, with $N = 7^{2k+2\\beta-1}$,","title":"Generating functions for $p_{1,7^{2k}}$ on progressions"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","title":"\u2113-regular cubic partitions with overlined odd parts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-series","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedSeries","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-series","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"fully_proved","summary":"For a natural number $\\ell$, this is the integer power series","title":"Generating function of \u2113-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"fully_proved","summary":"For natural numbers $\\ell$ and $n$, $a_\\ell(n)$ is the coefficient of $q^n$ in the integer power series","title":"\u2113-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-generating","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_generatingFunction","parent":11386,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for "],["math","\\(\\ell\\)"],["text","-regular cubic partitions with odd parts overlined"]],"source":"Generating function for $\\ell$-regular cubic partitions with odd parts overlined"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-generating","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"fully_proved","summary":"Let $\\ell \\ge 1$, write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for the Euler products in the ring of integer power series in $q$, and let $a_\\ell(n)$ be the $n$th coefficient of $f_2 f_\\ell^2 f_{4\\ell} / (f_1^2 f_4 f_{2\\ell})$ (the denominator has constant term $1$, so it is invertible). Then","title":"Generating function for $\\ell$-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-zero","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_zero","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"fully_proved","summary":"For every natural number $\\ell \\ge 1$, the coefficient $a_\\ell(0)$ of $q^0$ in the generating function of $\\ell$-regular cubic partitions with odd parts overlined equals $1$.","title":"Constant term of the overlined \u2113-regular cubic partition numbers"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/f3-f5-quotients-two-dissection","kind":"node","lean":"MetaMathlibExt.f3_f5_quotients_two_dissection","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#f3-f5-quotients-two-dissection","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for the Euler products in the ring of integer power series in $q$. Then","title":"Four 2-dissections of quotients of Euler products"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-mod-four","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_mod_four","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-mod-four","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for the Euler products in the ring of integer power series in $q$. For $\\ell \\ge 1$ let $a_\\ell(n)$ be the $n$th coefficient of $f_2 f_\\ell^2 f_{4\\ell} / (f_1^2 f_4 f_{2\\ell})$; for $\\ell > 2$ this is the number of $\\ell$-regular cubic partitions of $n$ with odd parts overlined. Then for all natural numbers $k$ and $n$: (i) if $k > 1$, then $4$ divides $a_{2k}(4n + 3)$;\u2026","title":"Congruences modulo 4 for regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-three-prim","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_three_prime_mod_twelve","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-three-prim","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Let $a_3(n)$ be the number of $3$-regular cubic partitions of $n$ with odd parts overlined, and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-6}{p}\\right) = -1$, and let $\\delta \\ge 0$. Then for every $n \\ge 0$,","title":"Congruences modulo 12 for 3-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-three-mod","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_three_mod_eighteen_twelve","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-three-mod","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Let $a_3(n)$ be the number of $3$-regular cubic partitions of $n$ with odd parts overlined. For every $n \\ge 0$, $a_3(8n + 5)$ is divisible by $18$, and $a_3(16n + 11)$ and $a_3(16n + 15)$ are divisible by $12$.","title":"Congruences modulo 18 and 12 for 3-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-three-prim-2","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_three_prime_mod_six","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-three-prim-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Let $a_3(n)$ be the number of $3$-regular cubic partitions of $n$ with odd parts overlined, and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-3}{p}\\right) = -1$, and let $\\delta \\ge 0$. Then for every $n \\ge 0$,","title":"Congruences modulo 6 for 3-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-four-prime","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_four_prime_mod_four","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-four-prime","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for the Euler products in the ring of integer power series in $q$, and $a_4(n)$ for the $n$th coefficient of $f_2 f_4^2 f_{16} / (f_1^2 f_4 f_8)$, the number of $4$-regular cubic partitions of $n$ with odd parts overlined. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-2}{p}\\right) = -1$ and let $\\delta \\ge 0$. Then for every $n \\ge 0$,\u2026","title":"Congruences modulo 4 for 4-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-four-mod-e","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_four_mod_eight_fiftySix_twentyFour","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-four-mod-e","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Let $a_4(n)$ be the number of $4$-regular cubic partitions of $n$ with odd parts overlined, that is, the $n$th coefficient of $f_2 f_4^2 f_{16} / (f_1^2 f_4 f_8)$, where $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then for every $n \\ge 0$, $8$ divides $a_4(4n + 3)$, $56$ divides $a_4(28n + 4j + 3)$ for each $j = 1, \\ldots, 6$, and $24$ divides $a_4(36n + 19)$.","title":"Congruences modulo 8, 56 and 24 for 4-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-five-prime","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_five_prime_mod_eight","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-five-prime","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for the Euler products in the ring of integer power series in $q$, and $a_5(n)$ for the $n$th coefficient of $f_2 f_5^2 f_{20} / (f_1^2 f_4 f_{10})$, the number of $5$-regular cubic partitions of $n$ with odd parts overlined. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-3}{p}\\right) = -1$, let $s = (p - 1)/6$ if $p \\equiv 1 \\pmod 6$ and $s = (-p - 1)/6$\u2026","title":"Congruences modulo 8 for 5-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-five-mod-s","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_five_mod_sixteen_eight","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-five-mod-s","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Let $a_5(n)$ be the number of $5$-regular cubic partitions of $n$ with odd parts overlined, that is, the $n$th coefficient of $f_2 f_5^2 f_{20} / (f_1^2 f_4 f_{10})$, where $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Then for every $n \\ge 0$, $16$ divides $a_5(8n + 7)$, and $8$ divides $a_5(20n + 2j + 1)$ for each $j = 1, \\ldots, 9$.","title":"Congruences modulo 16 and 8 for 5-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-five-prime-2","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_five_prime_mod_four","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-five-prime-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for the Euler products in the ring of integer power series in $q$, and $a_5(n)$ for the $n$th coefficient of $f_2 f_5^2 f_{20} / (f_1^2 f_4 f_{10})$, the number of $5$-regular cubic partitions of $n$ with odd parts overlined. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-15}{p}\\right) = -1$, let $s = (p - 1)/6$ if $p \\equiv 1 \\pmod 6$ and $s = (-p - 1)/6$\u2026","title":"Congruences modulo 4 for 5-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-eight-prim","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_eight_prime_mod_four_eight","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-eight-prim","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for the Euler products in the ring of integer power series in $q$, and $a_8(n)$ for the $n$th coefficient of $f_2 f_8^2 f_{32} / (f_1^2 f_4 f_{16})$, the number of $8$-regular cubic partitions of $n$ with odd parts overlined. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-3}{p}\\right) = -1$, let $s = (p - 1)/6$ if $p \\equiv 1 \\pmod 6$ and $s = (-p - 1)/6$\u2026","title":"Congruences modulo 4 and 8 for 8-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-eight-prim-2","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_eight_prime_mod_sixteen","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-eight-prim-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Write $f_k = \\prod_{m \\ge 1} (1 - q^{km})$ for the Euler products in the ring of integer power series in $q$, and $a_8(n)$ for the $n$th coefficient of $f_2 f_8^2 f_{32} / (f_1^2 f_4 f_{16})$, the number of $8$-regular cubic partitions of $n$ with odd parts overlined. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-6}{p}\\right) = -1$, let $s = (p - 1)/6$ if $p \\equiv 1 \\pmod 6$ and $s = (-p - 1)/6$\u2026","title":"Congruences modulo 16 for 8-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-nine-prime","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_nine_prime_mod_four","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-nine-prime","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Let $a_9(n)$ be the number of $9$-regular cubic partitions of $n$ with odd parts overlined, and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-2}{p}\\right) = -1$, and let $\\delta \\ge 0$. Then for every $n \\ge 0$,","title":"Congruences modulo 4 for 9-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-nine-four","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_nine_four_mul_add_three","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-nine-four","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"For every $n \\ge 0$, the number $a_9(4n + 3)$ of $9$-regular cubic partitions of $4n + 3$ with odd parts overlined is divisible by $8$.","title":"Congruence modulo 8 for 9-regular cubic partitions with odd parts overlined"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/regular-cubic-overlined-count-sixteen-pr","kind":"node","lean":"MetaMathlibExt.regularCubicOverlinedCount_sixteen_prime_mod_sixteen","parent":11386,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part/index.html#regular-cubic-overlined-count-sixteen-pr","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-cubic-partitions-with-overlined-odd-part","status":"can_prove","summary":"Let $a_{16}(n)$ be the number of $16$-regular cubic partitions of $n$ with odd parts overlined, and let $f_k = \\prod_{m \\ge 1} (1 - q^{km})$. Let $p \\ge 5$ be a prime with Legendre symbol $\\left(\\frac{-2}{p}\\right) = -1$, and let $\\delta \\ge 0$. Then for every $n \\ge 0$,","title":"Congruences modulo 16 for 16-regular cubic partitions with odd parts overlined"},{"data_url":"scopes/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions.json","id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","kind":"scope","navigation":"module","parent":11058,"result":false,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","title":"\u2113-regular overpartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-count","kind":"node","lean":"MetaMathlibExt.regularOverpartitionCount","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(\\ell\\)"],["text","-regular overpartitions"]],"source":"Number of $\\ell$-regular overpartitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-count","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"fully_proved","summary":"For natural numbers $\\ell$ and $n$, $A_\\ell(n)$ is the number of overpartitions of $n$ in which no part is divisible by $\\ell$.","title":"Number of $\\ell$-regular overpartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-count-zero","kind":"node","lean":"MetaMathlibExt.regularOverpartitionCount_zero","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["text","One "],["math","\\(\\ell\\)"],["text","-regular overpartition of zero"]],"source":"One $\\ell$-regular overpartition of zero"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-count-zero","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"fully_proved","summary":"For every natural number $\\ell$, the number $A_\\ell(0)$ of overpartitions of $0$ with no part divisible by $\\ell$ equals $1$: the only one is the empty overpartition.","title":"One $\\ell$-regular overpartition of zero"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-count-one","kind":"node","lean":"MetaMathlibExt.regularOverpartitionCount_one","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two "],["math","\\(\\ell\\)"],["text","-regular overpartitions of one"]],"source":"Two $\\ell$-regular overpartitions of one"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-count-one","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"fully_proved","summary":"For every natural number $\\ell \\ge 2$, the number $A_\\ell(1)$ of overpartitions of $1$ with no part divisible by $\\ell$ equals $2$: they are the single part $1$, without and with an overline.","title":"Two $\\ell$-regular overpartitions of one"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-count-eq-coeff-tru","kind":"node","lean":"MetaMathlibExt.regularOverpartitionCount_eq_coeff_truncatedProduct","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncated generating function for "],["math","\\(\\ell\\)"],["text","-regular overpartitions"]],"source":"Truncated generating function for $\\ell$-regular overpartitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-count-eq-coeff-tru","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Let $\\ell \\ge 1$, $N$ and $n \\le N$ be natural numbers, and let $A_\\ell(n)$ be the number of overpartitions of $n$ with no part divisible by $\\ell$. Then the coefficient of $X^n$ in the polynomial","title":"Truncated generating function for $\\ell$-regular overpartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-prime-power-two-de","kind":"node","lean":"MetaMathlibExt.regularOverpartition_prime_power_two_density","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-regular overpartitions are almost always divisible by "],["math","\\(2^j\\)"]],"source":"$p$-regular overpartitions are almost always divisible by $2^j$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-prime-power-two-de","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Let $p$ be an odd prime, let $j \\ge 1$, and let $A_p(n)$ be the number of overpartitions of $n$ with no part divisible by $p$. Then","title":"$p$-regular overpartitions are almost always divisible by $2^j$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-three-mul-prime-po","kind":"node","lean":"MetaMathlibExt.regularOverpartition_three_mul_prime_power_two_density","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Almost all "],["math","\\(3p\\)"],["text","-regular overpartition counts are divisible by "],["math","\\(2^j\\)"]],"source":"Almost all $3p$-regular overpartition counts are divisible by $2^j$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-three-mul-prime-po","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Write $A_\\ell(m)$ for the number of overpartitions of $m$ none of whose parts is divisible by $\\ell$. Let $p$ be an odd prime and let $j \\ge 1$. Then the number of natural numbers $n$ with $0 \\le n \\le X$ and $2^j \\mid A_{3p}(n)$, divided by $X$, tends to $1$ as the natural number $X$ tends to infinity.","title":"Almost all $3p$-regular overpartition counts are divisible by $2^j$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-three-mul-two-powe","kind":"node","lean":"MetaMathlibExt.regularOverpartition_three_mul_two_power_density","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Almost all "],["math","\\(3 \\cdot 2^\\alpha\\)"],["text","-regular overpartition counts are divisible by "],["math","\\(2^j\\)"]],"source":"Almost all $3 \\cdot 2^\\alpha$-regular overpartition counts are divisible by $2^j$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-three-mul-two-powe","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Write $A_\\ell(m)$ for the number of overpartitions of $m$ none of whose parts is divisible by $\\ell$. Let $\\alpha \\ge 1$ and $j \\ge \\alpha + 1$ be natural numbers. Then the number of natural numbers $n$ with $0 \\le n \\le X$ and $2^j \\mid A_{3 \\cdot 2^\\alpha}(n)$, divided by $X$, tends to $1$ as the natural number $X$ tends to infinity.","title":"Almost all $3 \\cdot 2^\\alpha$-regular overpartition counts are divisible by $2^j$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-three-mul-two-powe-2","kind":"node","lean":"MetaMathlibExt.regularOverpartition_three_mul_two_power_many_primes","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Congruences modulo powers of two for "],["math","\\(3 \\cdot 2^\\alpha\\)"],["text","-regular overpartitions"]],"source":"Congruences modulo powers of two for $3 \\cdot 2^\\alpha$-regular overpartitions"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-three-mul-two-powe-2","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Write $A_\\ell(m)$ for the number of overpartitions of $m$ none of whose parts is divisible by $\\ell$. For every natural number $\\alpha \\ge 1$ there is a natural number $c$ with the following property. Let $d \\ge 1$ be a natural number, let $p_1, \\dots, p_{c+d}$ be distinct primes, each coprime to $6$, and let $n$ be a natural number that is divisible by $24$ and coprime to every $p_i$. Then\u2026","title":"Congruences modulo powers of two for $3 \\cdot 2^\\alpha$-regular overpartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-mod-four-prime-fam","kind":"node","lean":"MetaMathlibExt.regularOverpartition_mod_four_prime_family","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell\\)"],["text","-regular overpartitions vanish modulo "],["math","\\(4\\)"],["text"," along families of primes"]],"source":"$\\ell$-regular overpartitions vanish modulo $4$ along families of primes"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-mod-four-prime-fam","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"For a natural number $\\ell$, let $A_\\ell(m)$ be the number of overpartitions of $m$ with no part divisible by $\\ell$. Let $\\ell \\equiv 3 \\pmod 4$, let $k$, $n$ and $j$ be natural numbers, and let $p_1, \\dots, p_{k+1}$ be primes, not necessarily distinct, each greater than $2$ and congruent to $3$ modulo $4$. If $j \\not\\equiv 0 \\pmod{p_{k+1}}$, then","title":"$\\ell$-regular overpartitions vanish modulo $4$ along families of primes"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-mod-four-single-pr","kind":"node","lean":"MetaMathlibExt.regularOverpartition_mod_four_single_prime","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell\\)"],["text","-regular overpartitions vanish modulo "],["math","\\(4\\)"],["text"," along powers of one prime"]],"source":"$\\ell$-regular overpartitions vanish modulo $4$ along powers of one prime"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-mod-four-single-pr","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"For a natural number $\\ell$, let $A_\\ell(m)$ be the number of overpartitions of $m$ with no part divisible by $\\ell$. Let $\\ell \\equiv 3 \\pmod 4$, and let $p > 2$ be a prime with $p \\not\\equiv 1 \\pmod 6$. Then for all natural numbers $k$, $n$ and $j$ with $j \\not\\equiv 0 \\pmod p$,","title":"$\\ell$-regular overpartitions vanish modulo $4$ along powers of one prime"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-mod-four-progressi","kind":"node","lean":"MetaMathlibExt.regularOverpartition_mod_four_progression","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell\\)"],["text","-regular overpartitions vanish modulo "],["math","\\(4\\)"],["text"," on progressions modulo "],["math","\\(4\\ell\\)"]],"source":"$\\ell$-regular overpartitions vanish modulo $4$ on progressions modulo $4\\ell$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-mod-four-progressi","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"For a natural number $\\ell$, let $A_\\ell(m)$ be the number of overpartitions of $m$ with no part divisible by $\\ell$. Let $\\ell \\equiv 3 \\pmod 4$, and let $n$ and $j$ be natural numbers with $j \\not\\equiv (\\ell - 3)/4 \\pmod{\\ell}$. Then $A_\\ell(4\\ell n + 4j + 3) \\equiv 0 \\pmod 4$.","title":"$\\ell$-regular overpartitions vanish modulo $4$ on progressions modulo $4\\ell$"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-seven-self-similar","kind":"node","lean":"MetaMathlibExt.regularOverpartition_seven_self_similarity","parent":11406,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-seven-self-similar","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Write $A_7(m)$ for the number of overpartitions of $m$ none of whose parts is divisible by $7$. For all natural numbers $n$ and $k$, $A_7(7^{2k}(4n+3)) \\equiv A_7(4n+3) \\pmod 7$.","title":"Self-similarity modulo 7 of 7-regular overpartition counts"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/regular-overpartition-seven-ramanujan-co","kind":"node","lean":"MetaMathlibExt.regularOverpartition_seven_ramanujan_congruences","parent":11406,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#regular-overpartition-seven-ramanujan-co","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Write $A_7(m)$ for the number of overpartitions of $m$ none of whose parts is divisible by $7$. For every natural number $n$, we have $A_7(16n+4) \\equiv 0 \\pmod 7$; $A_7(64n+i) \\equiv 0 \\pmod 7$ for every natural number $i < 64$ with $i \\equiv 4 \\pmod{32}$; $A_7(256n+i) \\equiv 0 \\pmod 7$ for every natural number $i < 256$ with $i \\equiv 4 \\pmod{32}$; and $A_7(1024n+i) \\equiv 0 \\pmod 7$ for every natural number\u2026","title":"Ramanujan-type congruences modulo 7 for 7-regular overpartitions"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/finite-euler-product-prime-power-congrue","kind":"node","lean":"MetaMathlibExt.finiteEulerProduct_primePower_congruence","parent":11406,"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#finite-euler-product-prime-power-congrue","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Let $s$ and $t$ be positive integers, let $p$ be a prime and let $N$ be a natural number. Then every coefficient of the polynomial","title":"Prime-power congruence for truncated Euler products"},{"id":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/exists-gamma0-lower-unipotent-t-zpow-dec","kind":"node","lean":"MetaMathlibExt.exists_gamma0_lowerUnipotent_T_zpow_decomposition","parent":11406,"reading_title":{"format":"inline-parts-v1","parts":[["text","Double coset representatives for "],["math","\\(\\Gamma_0(N)\\)"]],"source":"Double coset representatives for $\\Gamma_0(N)$"},"result":true,"route":"../roadmap/NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions/index.html#exists-gamma0-lower-unipotent-t-zpow-dec","scope":"NumberTheory/modular-forms-q-series/partition-congruences/regular-overpartitions","status":"can_prove","summary":"Let $N$ be a natural number such that either $N$ is squarefree, or $N$ is even and $N/2$ is squarefree. Write $\\Gamma_0(N)$ for the group of matrices in $\\mathrm{SL}_2(\\mathbb{Z})$ whose lower left entry is divisible by $N$, and $T = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}$. Then for every $\\gamma \\in \\mathrm{SL}_2(\\mathbb{Z})$ there is exactly one natural number $\\delta$ dividing $N$ for which there exist\u2026","title":"Double coset representatives for $\\Gamma_0(N)$"},{"data_url":"scopes/NumberTheory/special-numbers.json","id":"NumberTheory/special-numbers","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/special-numbers/index.html","scope":"NumberTheory/special-numbers","status":"can_prove","summary":"Arithmetic of classical number families: Bernoulli, Euler and Genocchi numbers, harmonic numbers, binomial and Catalan-type coefficients.","title":"Special numbers and polynomials"},{"data_url":"scopes/NumberTheory/special-numbers/bernoulli-numbers.json","id":"NumberTheory/special-numbers/bernoulli-numbers","kind":"scope","parent":11422,"result":false,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/index.html","scope":"NumberTheory/special-numbers/bernoulli-numbers","status":"can_prove","summary":"Bernoulli numbers and polynomials, identities, denominators, Kummer congruences, Faulhaber sums, Appell sequences, higher-order and Apostol variants.","title":"Bernoulli numbers and polynomials"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/is-apostol-bernoulli-sequence","kind":"node","lean":"MetaMathlibExt.IsApostolBernoulliSequence","parent":11423,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/index.html#is-apostol-bernoulli-sequence","scope":"NumberTheory/special-numbers/bernoulli-numbers","status":"fully_proved","summary":"Let $\\alpha$ be a complex number and let $\\log$ denote the principal branch of the complex logarithm. A sequence $(\\beta_k)_{k \\ge 0}$ of complex numbers is the sequence of Apostol\u2013Bernoulli numbers $\\beta_k(\\alpha)$ when $\\alpha \\ne 0$, $\\alpha \\ne 1$, and for every complex number $t$ with $|t| < |\\log \\alpha|$ the series $\\sum_{k \\ge 0} \\beta_k t^k / k!$ converges unconditionally to $t / (\\alpha e^t - 1)$.","title":"Apostol\u2013Bernoulli numbers"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/is-appell-sequence","kind":"node","lean":"MetaMathlibExt.IsAppellSequence","parent":11423,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/index.html#is-appell-sequence","scope":"NumberTheory/special-numbers/bernoulli-numbers","status":"fully_proved","summary":"Let $R$ be a field of characteristic zero. A sequence of polynomials $(p_n)_{n \\ge 0}$ over $R$ is an Appell sequence when, for every $n$, the polynomial $p_n$ has degree exactly $n$ (in particular $p_n \\ne 0$) and the formal derivative of $p_{n+1}$ equals $(n + 1)\\, p_n$.","title":"Appell sequence"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/faulhaber-bernoulli-polynomial-sum","kind":"node","lean":"MetaMathlibExt.faulhaber_bernoulli_polynomial_sum","parent":11423,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/index.html#faulhaber-bernoulli-polynomial-sum","scope":"NumberTheory/special-numbers/bernoulli-numbers","status":"fully_proved","summary":"Let $B_i$ denote the $i$-th Bernoulli number, with $B_1 = -1/2$, and let $B_j(X) = \\sum_{i=0}^{j} \\binom{j}{i} B_i X^{j-i} \\in \\mathbb{Q}[X]$ be the $j$-th Bernoulli polynomial. For all natural numbers $k$ and $n$, the following identity holds in $\\mathbb{Q}[X]$:","title":"Faulhaber-type summation formula for Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/kummer-congruence","kind":"node","lean":"MetaMathlibExt.kummer_congruence","parent":11423,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/index.html#kummer-congruence","scope":"NumberTheory/special-numbers/bernoulli-numbers","status":"fully_proved","summary":"Let $p$ be a prime, let $B_j$ denote the $j$-th Bernoulli number, with $B_1 = -1/2$, and let $t$ be a natural number. Let $m$ and $n$ be natural numbers, neither divisible by $p - 1$, with $m \\equiv n \\pmod{(p-1)p^t}$. Then the rational numbers","title":"Kummer's congruence"},{"data_url":"scopes/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n.json","id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","kind":"scope","navigation":"module","parent":11423,"result":false,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","title":"Bernoulli identities via Stirling and harmonic numbers"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/poly-bernoulli-poly","kind":"node","lean":"MetaMathlibExt.polyBernoulliPoly","parent":11428,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#poly-bernoulli-poly","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"fully_proved","summary":"For a natural number $n$, an integer $k$ and a rational number $x$, the poly-Bernoulli polynomial of degree $n$ and order $k$ at $x$ is the rational number","title":"Poly-Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/poly-bernoulli-poly-zero","kind":"node","lean":"MetaMathlibExt.polyBernoulliPoly_zero","parent":11428,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#poly-bernoulli-poly-zero","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"fully_proved","summary":"For every natural number $n$ and every integer $k$, the poly-Bernoulli polynomial of degree $n$ and order $k$ evaluated at $0$ equals the poly-Bernoulli number of index $n$ and order $k$: $B_n^{(k)}(0) = B_n^{(k)}$.","title":"Poly-Bernoulli polynomials at zero"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/poly-bernoulli-poly-zero-index","kind":"node","lean":"MetaMathlibExt.polyBernoulliPoly_zero_index","parent":11428,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#poly-bernoulli-poly-zero-index","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"fully_proved","summary":"For every integer $k$ and every rational number $x$, the poly-Bernoulli polynomial of degree $0$ and order $k$ satisfies $B_0^{(k)}(x) = 1$.","title":"Poly-Bernoulli polynomials of degree zero"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/sum-stirling-harmonic-eq-bernoulli","kind":"node","lean":"MetaMathlibExt.sum_stirling_harmonic_eq_bernoulli","parent":11428,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#sum-stirling-harmonic-eq-bernoulli","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","summary":"Let $j < n$ be natural numbers. Write ${n \\brace k}$ for the Stirling numbers of the second kind, ${k \\brack j}$ for the unsigned Stirling numbers of the first kind (the number of permutations of $k$ elements with exactly $j$ cycles), $H_k = \\sum_{i=1}^{k} 1/i$ for the harmonic numbers (so $H_0 = 0$), and $B_m$ for the Bernoulli numbers, normalized so that $B_1 = -1/2$. Then, in the rational numbers,","title":"Bernoulli numbers from Stirling and harmonic numbers"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/sum-bernoulli-harmonic-power-identity","kind":"node","lean":"MetaMathlibExt.sum_bernoulli_harmonic_power_identity","parent":11428,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#sum-bernoulli-harmonic-power-identity","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","summary":"Let $B_j$ denote the Bernoulli numbers with $B_1 = -\\tfrac12$, and let $H_m = \\sum_{i=1}^{m} 1/i$ denote the harmonic numbers. For all positive integers $n$ and $m$,","title":"A Bernoulli\u2013harmonic power-sum identity"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/sum-bernoulli-one-sub-two-pow","kind":"node","lean":"MetaMathlibExt.sum_bernoulli_one_sub_two_pow","parent":11428,"reading_title":{"format":"inline-parts-v1","parts":[["text","A weighted Bernoulli sum with factors "],["math","\\(1 - 2^j\\)"]],"source":"A weighted Bernoulli sum with factors $1 - 2^j$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#sum-bernoulli-one-sub-two-pow","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","summary":"Let $B_j$ denote the Bernoulli numbers with $B_1 = -\\tfrac12$. For every positive integer $n$,","title":"A weighted Bernoulli sum with factors $1 - 2^j$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/sum-bernoulli-eq-poly-bernoulli","kind":"node","lean":"MetaMathlibExt.sum_bernoulli_eq_polyBernoulli","parent":11428,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#sum-bernoulli-eq-poly-bernoulli","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","summary":"Let $B_j$ denote the Bernoulli numbers with $B_1 = -\\tfrac12$, let $B_n^{(2)}$ denote the poly-Bernoulli numbers of order $2$, and let $B_n^{(2)}(x)$ denote the poly-Bernoulli polynomials of order $2$. For every natural number $n \\ge 2$,","title":"Sums of Bernoulli numbers via poly-Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/sum-prime-mul-bernoulli-mod-prime","kind":"node","lean":"MetaMathlibExt.sum_prime_mul_bernoulli_mod_prime","parent":11428,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence "],["math","\\(\\sum_{j=0}^{p} pB_j \\equiv -1 \\pmod p\\)"]],"source":"The congruence $\\sum_{j=0}^{p} pB_j \\equiv -1 \\pmod p$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#sum-prime-mul-bernoulli-mod-prime","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","summary":"Let $B_j$ denote the Bernoulli numbers with $B_1 = -\\tfrac12$. For every odd prime $p$, the rational number $S = \\sum_{j=0}^{p} p B_j$ has denominator not divisible by $p$, and its residue modulo $p$ (its numerator times the inverse of its denominator in $\\mathbb{Z}/p\\mathbb{Z}$) equals $-1$.","title":"The congruence $\\sum_{j=0}^{p} pB_j \\equiv -1 \\pmod p$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/sum-bernoulli-to-prime-sub-three-mod-pri","kind":"node","lean":"MetaMathlibExt.sum_bernoulli_to_prime_sub_three_mod_prime","parent":11428,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence "],["math","\\(\\sum_{j=0}^{p-3} B_j \\equiv -1 \\pmod p\\)"]],"source":"The congruence $\\sum_{j=0}^{p-3} B_j \\equiv -1 \\pmod p$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#sum-bernoulli-to-prime-sub-three-mod-pri","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","summary":"Let $B_j$ denote the Bernoulli numbers with $B_1 = -\\tfrac12$. For every prime $p \\ge 5$, the rational number $S = B_0 + B_1 + \\dots + B_{p-3}$ has denominator not divisible by $p$, and its residue modulo $p$ (its numerator times the inverse of its denominator in $\\mathbb{Z}/p\\mathbb{Z}$) equals $-1$.","title":"The congruence $\\sum_{j=0}^{p-3} B_j \\equiv -1 \\pmod p$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/sum-euler-eval-zero-mod-prime","kind":"node","lean":"MetaMathlibExt.sum_euler_eval_zero_mod_prime","parent":11428,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence "],["math","\\(\\sum_{j=0}^{p} E_j(0) \\equiv 3/2 \\pmod p\\)"]],"source":"The congruence $\\sum_{j=0}^{p} E_j(0) \\equiv 3/2 \\pmod p$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#sum-euler-eval-zero-mod-prime","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","summary":"Let $E_j(x)$ denote the Euler polynomials. For every odd prime $p$, the rational number $S = E_0(0) + E_1(0) + \\dots + E_p(0)$ has denominator not divisible by $p$, and its residue modulo $p$ (its numerator times the inverse of its denominator in $\\mathbb{Z}/p\\mathbb{Z}$) equals $3 \\cdot 2^{-1}$.","title":"The congruence $\\sum_{j=0}^{p} E_j(0) \\equiv 3/2 \\pmod p$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/prime-mul-sum-bernoulli-div-mod-prime","kind":"node","lean":"MetaMathlibExt.prime_mul_sum_bernoulli_div_mod_prime","parent":11428,"reading_title":{"format":"inline-parts-v1","parts":[["text","The congruence "],["math","\\(p\\sum_{j=0}^{p} B_j/(p-j+1) \\equiv -1 \\pmod p\\)"]],"source":"The congruence $p\\sum_{j=0}^{p} B_j/(p-j+1) \\equiv -1 \\pmod p$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n/index.html#prime-mul-sum-bernoulli-div-mod-prime","scope":"NumberTheory/special-numbers/bernoulli-numbers/bernoulli-identities-via-stirling-and-harmonic-n","status":"can_prove","summary":"Let $B_j$ denote the Bernoulli numbers with $B_1 = -\\tfrac12$. For every odd prime $p$, the rational number","title":"The congruence $p\\sum_{j=0}^{p} B_j/(p-j+1) \\equiv -1 \\pmod p$"},{"data_url":"scopes/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials.json","id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","kind":"scope","navigation":"module","parent":11423,"result":false,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","title":"Denominators of Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/kellner-sondow-bernoulli-denom","kind":"node","lean":"MetaMathlibExt.kellner_sondow_bernoulli_denom","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#kellner-sondow-bernoulli-denom","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"Let $B_k$ denote the $k$-th Bernoulli number, with $B_1 = -1/2$, let $B_n(x) = \\sum_{k=0}^{n} \\binom{n}{k} B_k x^{n-k}$ be the $n$-th Bernoulli polynomial, and let $D_n$ be the least common multiple of the denominators of its nonzero coefficients. For a prime $p$, let $s_p(N)$ denote the sum of the base-$p$ digits of a natural number $N$. Then for every $n \\ge 1$, $D_n$ is the product of three factors: the product\u2026","title":"Kellner\u2013Sondow factorization of the denominators of Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/polynomial-denominator-bernoulli-zero","kind":"node","lean":"MetaMathlibExt.polynomialDenominator_bernoulli_zero","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Bernoulli polynomial "],["math","\\(B_0(x)\\)"],["text"," has denominator one"]],"source":"The Bernoulli polynomial $B_0(x)$ has denominator one"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#polynomial-denominator-bernoulli-zero","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"The Bernoulli polynomial $B_0(x) = 1$ has denominator $\\operatorname{denom}(B_0(x)) = 1$. In the source's notation $\\mathfrak{D}_n = \\operatorname{denom}(B_n(x))$, this is the convention $\\mathfrak{D}_0 = 1$.","title":"The Bernoulli polynomial $B_0(x)$ has denominator one"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominator","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Denominator of "],["math","\\(B_n(x) - B_n\\)"]],"source":"Denominator of $B_n(x) - B_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"For a natural number $n$ and a prime $p$, let $s_p(n)$ be the sum of the base-$p$ digits of $n$. Define $\\mathbb{D}_n = \\prod_{p \\text{ prime},\\ s_p(n) \\ge p} p$, the product of all primes $p$ with $s_p(n) \\ge p$; every such prime satisfies $p \\le n$, so the product is finite.","title":"Denominator of $B_n(x) - B_n$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-small","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominator_small","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","First values of "],["math","\\(\\mathbb{D}_n\\)"]],"source":"First values of $\\mathbb{D}_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-small","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"For a natural number $n$ let $\\mathbb{D}_n$ be the product of the primes $p$ whose base-$p$ digit sum $s_p(n)$ satisfies $s_p(n) \\ge p$. Then $(\\mathbb{D}_1, \\mathbb{D}_2, \\dots, \\mathbb{D}_{10}) = (1, 1, 2, 1, 6, 2, 6, 3, 10, 2)$.","title":"First values of $\\mathbb{D}_n$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-coprime-part","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominatorCoprimePart","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kellner's product "],["math","\\(\\mathbb{D}^\\perp_n\\)"],["text"," over primes not dividing "],["math","\\(n\\)"]],"source":"Kellner's product $\\mathbb{D}^\\perp_n$ over primes not dividing $n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-coprime-part","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, write $s_p(n)$ for the sum of the base-$p$ digits of $n$. The number $\\mathbb{D}^\\perp_n$ is the product of all primes $p$ with $p \\nmid n$ and $s_p(n) \\ge p$; every such prime satisfies $p \\le n$.","title":"Kellner's product $\\mathbb{D}^\\perp_n$ over primes not dividing $n$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-coprime-part-2","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominatorCoprimePart_small","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small values of "],["math","\\(\\mathbb{D}^\\perp_n\\)"]],"source":"Small values of $\\mathbb{D}^\\perp_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-coprime-part-2","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"Write $s_p(n)$ for the sum of the base-$p$ digits of $n$, and let $\\mathbb{D}^\\perp_n$ be the product of the primes $p$ with $p \\nmid n$ and $s_p(n) \\ge p$. Then $\\mathbb{D}^\\perp_1 = 1$, $\\mathbb{D}^\\perp_2 = 1$, $\\mathbb{D}^\\perp_3 = 2$, $\\mathbb{D}^\\perp_5 = 6$ and $\\mathbb{D}^\\perp_9 = 10$.","title":"Small values of $\\mathbb{D}^\\perp_n$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-large-part","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominatorLargePart","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kellner's product "],["math","\\(\\mathbb{D}^+_n\\)"],["text"," over primes above "],["math","\\(\\sqrt{n}\\)"]],"source":"Kellner's product $\\mathbb{D}^+_n$ over primes above $\\sqrt{n}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-large-part","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, write $s_p(n)$ for the sum of the base-$p$ digits of $n$. The number $\\mathbb{D}^+_n$ is the product of all primes $p$ with $p > \\sqrt{n}$ (that is, $p^2 > n$) and $s_p(n) \\ge p$; every such prime satisfies $p \\le n$.","title":"Kellner's product $\\mathbb{D}^+_n$ over primes above $\\sqrt{n}$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-large-part-sm","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominatorLargePart_small","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small values of "],["math","\\(\\mathbb{D}^+_n\\)"]],"source":"Small values of $\\mathbb{D}^+_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-large-part-sm","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"Write $s_p(n)$ for the sum of the base-$p$ digits of $n$, and let $\\mathbb{D}^+_n$ be the product of the primes $p$ with $p^2 > n$ and $s_p(n) \\ge p$. Then $\\mathbb{D}^+_4 = 1$, $\\mathbb{D}^+_5 = 3$ and $\\mathbb{D}^+_9 = 5$.","title":"Small values of $\\mathbb{D}^+_n$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-integral-derivative-indices","kind":"node","lean":"MetaMathlibExt.bernoulliIntegralDerivativeIndices","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Indices of Bernoulli polynomials with integral "],["math","\\(k\\)"],["text","-th derivative"]],"source":"Indices of Bernoulli polynomials with integral $k$-th derivative"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-integral-derivative-indices","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, $\\overline{\\mathcal{S}}_k$ is the set of $n \\ge 1$ such that the $k$-th derivative $B_n^{(k)}(x)$ of the Bernoulli polynomial $B_n(x)$ has only integer coefficients, that is, has denominator equal to $1$. We write $\\overline{\\mathcal{S}}$ for $\\overline{\\mathcal{S}}_1$.","title":"Indices of Bernoulli polynomials with integral $k$-th derivative"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/one-mem-bernoulli-integral-derivative-in","kind":"node","lean":"MetaMathlibExt.one_mem_bernoulliIntegralDerivativeIndices_one","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","The derivative of "],["math","\\(B_1(x)\\)"],["text"," has integer coefficients"]],"source":"The derivative of $B_1(x)$ has integer coefficients"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#one-mem-bernoulli-integral-derivative-in","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"The Bernoulli polynomial $B_1(x) = x - \\tfrac{1}{2}$ has derivative $B_1'(x) = 1 \\in \\mathbb{Z}[x]$, so $1$ belongs to the set $\\overline{\\mathcal{S}}$ of indices $n \\ge 1$ for which $B_n'(x)$ has only integer coefficients.","title":"The derivative of $B_1(x)$ has integer coefficients"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-eq-polynomial","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominator_eq_polynomialDenominator","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Product formula for the denominator of "],["math","\\(B_n(x) - B_n\\)"]],"source":"Product formula for the denominator of $B_n(x) - B_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-eq-polynomial","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, let $B_n(x)$ be the $n$-th Bernoulli polynomial and $B_n = B_n(0)$ the $n$-th Bernoulli number, and write $s_p(n)$ for the sum of the base-$p$ digits of $n$. Then $\\mathbb{D}_n$, the product of the primes $p$ with $s_p(n) \\ge p$, equals the denominator of the polynomial $B_n(x) - B_n$.","title":"Product formula for the denominator of $B_n(x) - B_n$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/kellner-large-part-conjecture","kind":"node","lean":"MetaMathlibExt.KellnerLargePartConjecture","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#kellner-large-part-conjecture","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"For a natural number $n$, let $\\mathbb{D}^+_n$ be the product of the primes $p > \\sqrt{n}$ with $s_p(n) \\ge p$, where $s_p(n)$ is the sum of the base-$p$ digits of $n$. This definition is the proposition that $\\mathbb{D}^+_n > 1$ for every natural number $n > 192$.","title":"Kellner's large-part conjecture"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/kellner-large-part-asymptotic-conjecture","kind":"node","lean":"MetaMathlibExt.KellnerLargePartAsymptoticConjecture","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#kellner-large-part-asymptotic-conjecture","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ let $\\mathbb{D}^{+}_n$ be the product of the primes $p > \\sqrt{n}$ whose base-$p$ digit sum $s_p(n)$ satisfies $s_p(n) \\ge p$, and let $\\omega(m)$ be the number of distinct prime factors of $m$. The conjecture asserts that there is a real number $\\kappa > 1$ such that $\\omega(\\mathbb{D}^{+}_n) \\sim \\kappa \\sqrt{n} / \\log n$ as $n \\to \\infty$.","title":"Kellner's asymptotic conjecture on large prime factors"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/kellner-conjectures-imply-bernoulli-inte","kind":"node","lean":"MetaMathlibExt.kellner_conjectures_imply_bernoulli_integral_derivative_conjectures","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#kellner-conjectures-imply-bernoulli-inte","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $\\mathbb{D}^{+}_n$ be the product of the primes $p > \\sqrt{n}$ whose base-$p$ digit sum $s_p(n)$ satisfies $s_p(n) \\ge p$, let $\\omega(m)$ be the number of distinct prime factors of $m$, and let $\\overline{\\mathcal{S}}$ be the set of $n \\ge 1$ such that the derivative $B_n'(x)$ of the Bernoulli polynomial $B_n(x)$ has integer coefficients. If $\\mathbb{D}^{+}_n > 1$ for every $n > 192$,\u2026","title":"Kellner's large-prime conjectures imply the Bernoulli derivative conjectures"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/kellner-large-part-asymptotic-conjecture-2","kind":"node","lean":"MetaMathlibExt.kellnerLargePartAsymptoticConjecture_and_finite_integral_derivative","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#kellner-large-part-asymptotic-conjecture-2","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $\\mathbb{D}^{+}_n$ be the product of the primes $p > \\sqrt{n}$ whose base-$p$ digit sum $s_p(n)$ satisfies $s_p(n) \\ge p$, and let $\\omega(m)$ be the number of distinct prime factors of $m$. There is a real number $\\kappa > 1$ with $\\omega(\\mathbb{D}^{+}_n) \\sim \\kappa \\sqrt{n} / \\log n$ as $n \\to \\infty$, and the set of $n \\ge 1$ such that the derivative $B_n'(x)$ of the Bernoulli\u2026","title":"Bordell\u00e8s\u2013Luca\u2013Moree\u2013Shparlinski asymptotic and finiteness of integral derivatives"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-integral-derivative-succ-prime","kind":"node","lean":"MetaMathlibExt.bernoulli_integral_derivative_succ_prime_and_large","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-integral-derivative-succ-prime","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"Let $n \\ge 1$ be such that the derivative $B_n'(x)$ of the Bernoulli polynomial $B_n(x)$ has integer coefficients. Then $n + 1$ is prime, and if $n \\notin \\{1, 2, 4, 6, 10, 12, 28, 30, 36, 60\\}$, then $n > 10^7$.","title":"Constraints on Bernoulli polynomials with integral derivative"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-polynomial-denominator-rad-eve","kind":"node","lean":"MetaMathlibExt.bernoulli_polynomial_denominator_rad_even","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-polynomial-denominator-rad-eve","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"Let $n \\ge 1$. Write $\\mathfrak{D}_n$ for the denominator of the Bernoulli polynomial $B_n(x)$, $\\mathcal{D}_n$ for the denominator of $S_n(x) = (B_{n+1}(x) - B_{n+1}) / (n + 1)$, where $B_{n+1}$ is the Bernoulli number, $\\operatorname{rad}(m)$ for the product of the primes dividing $m$, and $\\mathbb{D}^{\\perp}_n$ for the product of the primes $p \\nmid n$ whose base-$p$ digit sum $s_p(n)$ satisfies $s_p(n) \\ge p$.\u2026","title":"Denominators of Bernoulli polynomials are even"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-derivative-integral-iff","kind":"node","lean":"MetaMathlibExt.bernoulli_derivative_integral_iff","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-derivative-integral-iff","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, let $B_m(x)$ denote the $m$-th Bernoulli polynomial, write $s_p(n)$ for the sum of the base-$p$ digits of $n$, and let $\\mathbb{D}^\\perp_n$ be the product of the primes $p$ with $p \\nmid n$ and $s_p(n) \\ge p$. Then $B_n'(x)$ has integer coefficients if and only if $\\mathbb{D}^\\perp_n = 1$, and $\\mathbb{D}^\\perp_n = 1$ if and only if the denominator of $B_{n-1}(x)$ equals the radical\u2026","title":"Integrality of the derivative of a Bernoulli polynomial"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-eq-radical-su","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominator_eq_radical_succ_properties","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Indices with "],["math","\\(\\mathbb{D}_n = \\operatorname{rad}(n + 1)\\)"]],"source":"Indices with $\\mathbb{D}_n = \\operatorname{rad}(n + 1)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-eq-radical-su","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"Write $s_p(m)$ for the sum of the base-$p$ digits of $m$, let $\\mathbb{D}_m$ be the product of the primes $p$ with $s_p(m) \\ge p$, and let $\\mathbb{D}^\\perp_m$ be the product of the primes $p$ with $p \\nmid m$ and $s_p(m) \\ge p$. Let $\\overline{\\mathcal{R}}$ be the set of $n \\ge 1$ with $\\mathbb{D}_n = \\operatorname{rad}(n + 1)$, where $\\operatorname{rad}$ denotes the radical. For every\u2026","title":"Indices with $\\mathbb{D}_n = \\operatorname{rad}(n + 1)$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/odd-radical-conjecture-implies-bernoulli","kind":"node","lean":"MetaMathlibExt.odd_radical_conjecture_implies_bernoulli_integral_derivative_conjecture","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#odd-radical-conjecture-implies-bernoulli","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"For $n \\ge 1$, let $\\mathbb{D}_n$ be the product of the primes $p$ with $s_p(n) \\ge p$, where $s_p(n)$ is the sum of the base-$p$ digits of $n$, and let $\\operatorname{rad}(m)$ be the product of the distinct primes dividing $m$. Let $\\overline{\\mathcal{S}}$ be the set of $n \\ge 1$ such that the derivative $B_n'(x)$ of the Bernoulli polynomial $B_n(x)$ has only integer coefficients. If the odd integers $n \\ge 1$\u2026","title":"The odd radical conjecture implies the integral-derivative conjecture"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-higher-derivative-denominator","kind":"node","lean":"MetaMathlibExt.bernoulli_higher_derivative_denominator","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-higher-derivative-denominator","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"For $m \\ge 0$ let $\\mathfrak{D}_m$ be the denominator of the Bernoulli polynomial $B_m(x)$, and for $k \\ge 1$ let $\\mathfrak{D}_n^{(k)}$ be the denominator of the $k$-th derivative $B_n^{(k)}(x)$ of $B_n(x)$. Write $(n)_k = n(n-1)\\cdots(n-k+1)$ for the falling factorial, $s_p(m)$ for the sum of the base-$p$ digits of $m$, $\\operatorname{rad}(n)$ for the product of the primes dividing $n$, and $\\mathbb{D}^{\\perp}_m$\u2026","title":"Denominators of higher derivatives of Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-integral-derivative-indices-fi","kind":"node","lean":"MetaMathlibExt.bernoulliIntegralDerivativeIndices_finite_and_mono","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finitely many Bernoulli polynomials have an integral "],["math","\\(k\\)"],["text","-th derivative"]],"source":"Finitely many Bernoulli polynomials have an integral $k$-th derivative"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-integral-derivative-indices-fi","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"For $k \\ge 1$ let $\\overline{\\mathcal{S}}_k$ be the set of $n \\ge 1$ such that the $k$-th derivative $B_n^{(k)}(x)$ of the Bernoulli polynomial $B_n(x)$ has integer coefficients. For every $k \\ge 1$ the set $\\overline{\\mathcal{S}}_k$ is finite and $\\overline{\\mathcal{S}}_k \\subseteq \\overline{\\mathcal{S}}_{k+1}$.","title":"Finitely many Bernoulli polynomials have an integral $k$-th derivative"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-parity-radica","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominator_parity_radical_lcm","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity, radical divisibility and an lcm recursion for "],["math","\\(\\mathbb{D}_n\\)"]],"source":"Parity, radical divisibility and an lcm recursion for $\\mathbb{D}_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-parity-radica","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"Write $s_p(m)$ for the sum of the base-$p$ digits of $m$, let $\\mathbb{D}_m$ be the product of the primes $p$ with $s_p(m) \\ge p$, and let $\\mathbb{D}^\\perp_m$ be the product of the primes $p$ with $p \\nmid m$ and $s_p(m) \\ge p$. Let $n \\ge 1$. Then $\\mathbb{D}_n$ is odd if and only if $n = 2^e$ for some $e \\ge 0$. If $n + 1$ is not prime, then the radical $\\operatorname{rad}(n + 1)$ divides both $\\mathbb{D}_n$ and\u2026","title":"Parity, radical divisibility and an lcm recursion for $\\mathbb{D}_n$"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-large-part-dv","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominatorLargePart_dvd_coprimePart","parent":11440,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-large-part-dv","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"Write $s_p(n)$ for the sum of the base-$p$ digits of $n$, let $\\mathbb{D}^+_n$ be the product of the primes $p$ with $p^2 > n$ and $s_p(n) \\ge p$, and let $\\mathbb{D}^\\perp_n$ be the product of the primes $p$ with $p \\nmid n$ and $s_p(n) \\ge p$. Then $\\mathbb{D}^+_n$ divides $\\mathbb{D}^\\perp_n$ for every $n \\ge 1$.","title":"The large-prime part divides the coprime part"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/bernoulli-diff-denominator-large-part-ev","kind":"node","lean":"MetaMathlibExt.bernoulliDiffDenominatorLargePart_eventually_gt_descFactorial","parent":11440,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathbb{D}^+_n\\)"],["text"," eventually exceeds every fixed rising product"]],"source":"$\\mathbb{D}^+_n$ eventually exceeds every fixed rising product"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials/index.html#bernoulli-diff-denominator-large-part-ev","scope":"NumberTheory/special-numbers/bernoulli-numbers/denominators-of-bernoulli-polynomials","status":"can_prove","summary":"Write $s_p(n)$ for the sum of the base-$p$ digits of $n$ and let $\\mathbb{D}^+_n$ be the product of the primes $p$ with $p^2 > n$ and $s_p(n) \\ge p$. For every $k \\ge 1$ there is $N$ such that $\\mathbb{D}^+_n > (n + 1)(n + 2) \\cdots (n + k)$ for all $n > N$.","title":"$\\mathbb{D}^+_n$ eventually exceeds every fixed rising product"},{"data_url":"scopes/NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers.json","id":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers","kind":"scope","navigation":"module","parent":11423,"result":false,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/index.html","scope":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers","status":"fully_proved","title":"Higher-order Bernoulli numbers"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/higher-order-bernoulli-number","kind":"node","lean":"MetaMathlibExt.higherOrderBernoulliNumber","parent":11466,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/index.html#higher-order-bernoulli-number","scope":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers","status":"fully_proved","summary":"For a complex number $\\alpha$ and a natural number $n$, the higher-order Bernoulli number $B_n^{(\\alpha)}$ is $n!$ times the coefficient of $t^n$ in the formal power series $\\left(t / (e^t - 1)\\right)^{\\alpha}$. Here $t / (e^t - 1)$ is the Bernoulli generating series $\\sum_{k \\ge 0} B_k t^k / k!$ (with $B_1 = -1/2$), and its $\\alpha$-th power is the binomial series\u2026","title":"Higher-order Bernoulli numbers"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/higher-order-bernoulli-number-nat","kind":"node","lean":"MetaMathlibExt.higherOrderBernoulliNumber_nat","parent":11466,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/index.html#higher-order-bernoulli-number-nat","scope":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers","status":"fully_proved","summary":"Let $B_k$ denote the Bernoulli numbers, with $B_1 = -1/2$. For natural numbers $d$ and $n$, the higher-order Bernoulli number $B_n^{(d)}$ equals $n!$ times the coefficient of $t^n$ in the $d$-th power of the Bernoulli generating series $\\sum_{k \\ge 0} B_k t^k / k!$.","title":"Higher-order Bernoulli numbers of natural order"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/higher-order-bernoulli-number-one","kind":"node","lean":"MetaMathlibExt.higherOrderBernoulliNumber_one","parent":11466,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/index.html#higher-order-bernoulli-number-one","scope":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers","status":"fully_proved","summary":"For every natural number $n$, the higher-order Bernoulli number $B_n^{(1)}$ equals the Bernoulli number $B_n$, with the convention $B_1 = -1/2$.","title":"Higher-order Bernoulli numbers of order one"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/neto-higher-order-bernoulli-identity","kind":"node","lean":"MetaMathlibExt.neto_higher_order_bernoulli_identity","parent":11466,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers/index.html#neto-higher-order-bernoulli-identity","scope":"NumberTheory/special-numbers/bernoulli-numbers/higher-order-bernoulli-numbers","status":"fully_proved","summary":"For every complex number $x$ and all natural numbers $n$ and $\\ell$,","title":"Neto's identity for higher-order Bernoulli numbers"},{"data_url":"scopes/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers.json","id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","kind":"scope","navigation":"module","parent":11423,"result":false,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","title":"Identities for Bernoulli numbers"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/abel-bernoulli-sum","kind":"node","lean":"MetaMathlibExt.abel_bernoulli_sum","parent":11471,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html#abel-bernoulli-sum","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","summary":"Let $B_k(t)$ denote the $k$-th Bernoulli polynomial. For every natural number $n$ and all complex numbers $x$ and $y$ with $x \\neq 0$,","title":"Abel\u2013Bernoulli sum"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/abel-hurwitz-bernoulli-multinomial","kind":"node","lean":"MetaMathlibExt.abel_hurwitz_bernoulli_multinomial","parent":11471,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html#abel-hurwitz-bernoulli-multinomial","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","summary":"Let $B_k(t)$ denote the $k$-th Bernoulli polynomial. Let $m \\ge 2$ and $n$ be natural numbers, let $x_1, \\dots, x_m$ be rational numbers, and let $r$ be an index with $1 \\le r \\le m$. Then","title":"Abel\u2013Hurwitz\u2013Bernoulli multinomial identity"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/carlitz-bernoulli-sum-symm","kind":"node","lean":"MetaMathlibExt.carlitz_bernoulli_sum_symm","parent":11471,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html#carlitz-bernoulli-sum-symm","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","summary":"Let $B_j$ denote the $j$-th Bernoulli number, with $B_1 = -1/2$. For all natural numbers $m$ and $n$,","title":"Carlitz's Bernoulli identity"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/euler-bernoulli-convolution","kind":"node","lean":"MetaMathlibExt.euler_bernoulli_convolution","parent":11471,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html#euler-bernoulli-convolution","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","summary":"Let $B_j$ denote the $j$-th Bernoulli number, with $B_1 = -1/2$. For every natural number $n \\ge 1$,","title":"Euler's convolution identity for Bernoulli numbers"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/gessel-bernoulli-number-identity","kind":"node","lean":"MetaMathlibExt.gessel_bernoulli_number_identity","parent":11471,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html#gessel-bernoulli-number-identity","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","summary":"Let $B_j$ denote the $j$-th Bernoulli number, with $B_1 = -1/2$. For all natural numbers $n$, $r$ and $m$ with $r$ odd and $m \\ge 1$,","title":"Gessel's Bernoulli-number identity"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/gould-bernoulli-identity","kind":"node","lean":"MetaMathlibExt.gould_bernoulli_identity","parent":11471,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html#gould-bernoulli-identity","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","summary":"Let $B_i$ denote the $i$-th Bernoulli number, with $B_1 = -1/2$, and let $B_j(x) = \\sum_{i=0}^{j} \\binom{j}{i} B_i x^{j-i}$ be the $j$-th Bernoulli polynomial. For every natural number $n$ and every rational number $x$,","title":"Gould's alternating identity for Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/kaneko-bernoulli-stirling-sum","kind":"node","lean":"MetaMathlibExt.kaneko_bernoulli_stirling_sum","parent":11471,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html#kaneko-bernoulli-stirling-sum","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","summary":"Let $B_n$ denote the $n$-th Bernoulli number, with $B_1 = -1/2$, and let $S(n, j)$ denote the Stirling numbers of the second kind. For every natural number $n$,","title":"Kaneko's Stirling-number formula for the Bernoulli numbers"},{"id":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/tuenter-bernoulli-power-sum-identity","kind":"node","lean":"MetaMathlibExt.tuenter_bernoulli_power_sum_identity","parent":11471,"result":true,"route":"../roadmap/NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers/index.html#tuenter-bernoulli-power-sum-identity","scope":"NumberTheory/special-numbers/bernoulli-numbers/identities-for-bernoulli-numbers","status":"fully_proved","summary":"Let $a$ and $b$ be nonzero complex numbers and let $n \\ge 1$ be a natural number with $b^n \\ne a^n$. Let $\\sigma_x(i) \\in \\mathbb{C}$, for $x \\in \\mathbb{C}$ and $i \\in \\mathbb{N}$, satisfy $(i+1)\\,\\sigma_x(i) = B_{i+1}(x+1) - B_{i+1}$ for all $x$ and $i$, where $B_m(t)$ denotes the Bernoulli polynomial and $B_m$ the Bernoulli number, with $B_1 = -\\tfrac{1}{2}$. Then","title":"Tuenter's power-sum identity for Bernoulli numbers"},{"data_url":"scopes/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli.json","id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli","kind":"scope","parent":11422,"result":false,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/index.html","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli","status":"can_prove","summary":"Euler numbers and polynomials, Genocchi numbers, poly-Bernoulli, Bernoulli of the second kind, degenerate variants.","title":"Euler, Genocchi and poly-Bernoulli numbers"},{"data_url":"scopes/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind.json","id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","kind":"scope","navigation":"module","parent":11480,"result":false,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","title":"Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second-power-series","kind":"node","lean":"MetaMathlibExt.bernoulliSecondPowerSeries","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second-power-series","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"fully_proved","summary":"Let $G(t)$ be the inverse, in the ring $\\mathbb{Q}[[t]]$ of formal power series, of the series $\\sum_{n \\ge 0} \\frac{(-1)^n}{n + 1} t^n$, which is invertible because its constant coefficient is $1$.","title":"Generating function of the Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second","kind":"node","lean":"MetaMathlibExt.bernoulliSecond","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"fully_proved","summary":"For a natural number $n$, the Bernoulli number of the second kind $B_n^*$ is the rational number $n!$ times the coefficient of $t^n$ in $G(t) \\in \\mathbb{Q}[[t]]$, the inverse of the formal power series $\\sum_{k \\ge 0} \\frac{(-1)^k}{k + 1} t^k$. Thus $G(t) = \\sum_{n \\ge 0} B_n^* t^n / n!$.","title":"Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second-power-series-eq","kind":"node","lean":"MetaMathlibExt.bernoulliSecondPowerSeries_eq","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second-power-series-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"fully_proved","summary":"The generating function $G(t)$ of the Bernoulli numbers of the second kind equals the inverse, in the ring $\\mathbb{Q}[[t]]$ of formal power series, of the series $\\sum_{n \\ge 0} \\frac{(-1)^n}{n + 1} t^n$.","title":"Generating function of the Bernoulli numbers of the second kind as an inverse"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second-eq","kind":"node","lean":"MetaMathlibExt.bernoulliSecond_eq","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"fully_proved","summary":"For every natural number $n$, the Bernoulli number of the second kind $B_n^*$ equals $n!$ times the coefficient of $t^n$ in the generating function $G(t) \\in \\mathbb{Q}[[t]]$ of the Bernoulli numbers of the second kind.","title":"Bernoulli numbers of the second kind as scaled coefficients"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second-power-series-mul-mk","kind":"node","lean":"MetaMathlibExt.bernoulliSecondPowerSeries_mul_mk","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second-power-series-mul-mk","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"fully_proved","summary":"Defining identity of the generating function of the Bernoulli numbers of the second kind. In the ring $\\mathbb{Q}[[t]]$ of formal power series, the generating function $G(t)$ of the Bernoulli numbers of the second kind satisfies $G(t) \\cdot \\sum_{n \\ge 0} \\frac{(-1)^n}{n + 1} t^n = 1$.","title":"Defining identity of the generating function of the Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/sum-antidiagonal-bernoulli-second-power","kind":"node","lean":"MetaMathlibExt.sum_antidiagonal_bernoulliSecondPowerSeries_coeff","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#sum-antidiagonal-bernoulli-second-power","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"fully_proved","summary":"Coefficient identity for the generating function of the Bernoulli numbers of the second kind. Let $g_i$ be the coefficient of $t^i$ in the generating function $G(t) \\in \\mathbb{Q}[[t]]$ of the Bernoulli numbers of the second kind. For every natural number $n$, the sum of $g_i \\cdot (-1)^j / (j + 1)$ over all pairs $(i, j)$ of natural numbers with $i + j = n$ equals $1$ if $n = 0$ and $0$ if $n \\ge 1$.","title":"Coefficient identity for the generating function of the Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second-div-factorial-eq","kind":"node","lean":"MetaMathlibExt.bernoulliSecond_div_factorial_eq","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second-div-factorial-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the Bernoulli numbers of the second kind $B_k^*$ satisfy $B_n^* / n! = -\\sum_{k=0}^{n-1} (B_k^* / k!) \\cdot (-1)^{n-k} / (n - k + 1)$.","title":"Recurrence for the Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second-kind-asymptotic","kind":"node","lean":"MetaMathlibExt.bernoulli_second_kind_asymptotic","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second-kind-asymptotic","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"fully_proved","summary":"For a natural number $n$ let $b_n = \\frac{1}{n!} \\int_0^1 x (x - 1) \\cdots (x - n + 1) dx$, and for a natural number $k$ let $\\beta_k$ be $(-1)^k$ times the $(k + 1)$-th derivative of $1 / \\Gamma$ at $0$. Then for every natural number $m$, as $n \\to \\infty$,","title":"Asymptotic expansion of the Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-poly-eq","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.bernoulli_poly_eq","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-poly-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $B_m$ be the Bernoulli numbers (with $B_1 = -1/2$), $B_n(X) = \\sum_{i=0}^{n} \\binom{n}{i} B_i X^{n-i}$ the Bernoulli polynomials over $\\mathbb{Q}$, $S(m, j)$ the Stirling numbers of the second kind and $X^{\\underline{k}} = X(X-1)\\cdots(X-k+1)$ the falling factorial. For every natural number $n$, in $\\mathbb{Q}[X]$,","title":"Bernoulli polynomials in the falling-factorial basis"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-eq-neg-sum","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.bernoulli_eq_neg_sum","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-eq-neg-sum","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $B_m$ be the Bernoulli numbers (with $B_1 = -1/2$), $B_m^*$ the Bernoulli numbers of the second kind and $S(m, j)$ the Stirling numbers of the second kind. For every natural number $n \\ge 1$,","title":"Bernoulli numbers in terms of Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/stirling-inversion-iff","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.stirling_inversion_iff","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#stirling-inversion-iff","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $s(m, j)$ be the signed Stirling numbers of the first kind and $S(m, j)$ the Stirling numbers of the second kind. For all sequences $(u_n)_{n \\ge 0}$ and $(v_n)_{n \\ge 0}$ of rational numbers, $u_n = \\sum_{k=0}^{n} s(n, k) \\, v_k$ for every natural number $n$ if and only if $v_n = \\sum_{k=0}^{n} S(n, k) \\, u_k$ for every natural number $n$.","title":"Stirling inversion"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second-eq-neg-sum","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.bernoulliSecond_eq_neg_sum","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second-eq-neg-sum","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $B_m$ be the Bernoulli numbers (with $B_1 = -1/2$), $B_m^*$ the Bernoulli numbers of the second kind and $s(m, j)$ the signed Stirling numbers of the first kind. For every natural number $n \\ge 1$,","title":"Bernoulli numbers of the second kind in terms of Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/desc-pochhammer-eq","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.descPochhammer_eq","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#desc-pochhammer-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $B_m^*$ be the Bernoulli numbers of the second kind, $B_k(X)$ the Bernoulli polynomials over $\\mathbb{Q}$ (so that $B_1(X) = X - 1/2$), $s(m, j)$ the signed Stirling numbers of the first kind and $X^{\\underline{n}} = X(X-1)\\cdots(X-n+1)$ the falling factorial. For every natural number $n$, in $\\mathbb{Q}[X]$,","title":"Falling factorials in the Bernoulli polynomial basis"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/sum-stirling-second-signed-div","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.sum_stirlingSecond_signed_div","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#sum-stirling-second-signed-div","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"For natural numbers $k$ and $n$ with $1 \\le k \\le n$, write $S(m,j)$ for the Stirling numbers of the second kind, $s(m,j)$ for the signed Stirling numbers of the first kind, and $B_m$ for the Bernoulli numbers with $B_1 = -\\tfrac12$. Then, in $\\mathbb{Q}$,","title":"Shifted Stirling convolution formula for Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-eq-sum-stirling-second","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.bernoulli_eq_sum_stirlingSecond","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-eq-sum-stirling-second","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $B_m$ be the Bernoulli numbers (with $B_1 = -1/2$) and $S(m, j)$ the Stirling numbers of the second kind. For every natural number $n$,","title":"Bernoulli numbers via Stirling numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/sum-stirling-second-signed-div-delta","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.sum_stirlingSecond_signed_div_delta","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#sum-stirling-second-signed-div-delta","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"For natural numbers $k$ and $n$ with $1 \\le k \\le n$, write $S(m,j)$ for the Stirling numbers of the second kind, $s(m,j)$ for the signed Stirling numbers of the first kind, and $B_m$ for the Bernoulli numbers with $B_1 = -\\tfrac12$. Then, in $\\mathbb{Q}$,","title":"Unshifted Stirling convolution formula for Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/bernoulli-second-eq-sum","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.bernoulliSecond_eq_sum","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#bernoulli-second-eq-sum","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $B_m^*$ be the Bernoulli numbers of the second kind and $s(m, j)$ the signed Stirling numbers of the first kind. For every natural number $n$,","title":"Bernoulli numbers of the second kind via Stirling numbers of the first kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/sum-signed-stirling-second-div","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.sum_signed_stirlingSecond_div","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#sum-signed-stirling-second-div","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $s(m, j)$ be the signed Stirling numbers of the first kind, $S(m, j)$ the Stirling numbers of the second kind and $B_m^*$ the Bernoulli numbers of the second kind. For all natural numbers $n$ and $k$ with $1 \\le k \\le n$,","title":"Shifted Stirling convolution as a Bernoulli number of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/sum-signed-stirling-second-div-eq","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.sum_signed_stirlingSecond_div_eq","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#sum-signed-stirling-second-div-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $s(m, j)$ be the signed Stirling numbers of the first kind, $S(m, j)$ the Stirling numbers of the second kind and $B_m^*$ the Bernoulli numbers of the second kind. For all natural numbers $n$ and $k$ with $1 \\le k \\le n$,","title":"Stirling convolution as a sum of Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/sum-stirling-second-div-add","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.sum_stirlingSecond_div_add","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#sum-stirling-second-div-add","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"For natural numbers $r \\ge 1$ and $n$, write $S(n,k)$ for the Stirling numbers of the second kind, $c(r,j)$ for the unsigned Stirling numbers of the first kind, and $B_m$ for the Bernoulli numbers with $B_1 = -\\tfrac12$. Then, in $\\mathbb{Q}$,","title":"Weighted alternating sums of Stirling numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/sum-signed-div-add","kind":"node","lean":"MathlibExt.NumberTheory.BernoulliSecondKindWanted.sum_signed_div_add","parent":11481,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind/index.html#sum-signed-div-add","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/bernoulli-numbers-of-the-second-kind","status":"can_prove","summary":"Let $s(m, j)$ be the signed Stirling numbers of the first kind, $S(m, j)$ the Stirling numbers of the second kind and $B_m^*$ the Bernoulli numbers of the second kind. Let $r \\ge 1$ and $n$ be natural numbers, and for $0 \\le k \\le r - 1$ set $\\lambda_{r,n,k} = \\sum_{l=0}^{r-1-k} \\binom{r-1}{l} \\, S(r-1-l, k) \\, n^l$ (with $0^0 = 1$). Then","title":"Sums of signed Stirling numbers of the first kind with shifted reciprocal weights"},{"data_url":"scopes/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials.json","id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","kind":"scope","navigation":"module","parent":11480,"result":false,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"can_prove","title":"Degenerate Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/modified-degenerate-bernoulli","kind":"node","lean":"MetaMathlibExt.modifiedDegenerateBernoulli","parent":11503,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html#modified-degenerate-bernoulli","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"fully_proved","summary":"For a complex number $\\lambda$, a natural number $n$ and a complex number $x$, the modified degenerate Bernoulli polynomial is $\\mathfrak{B}_{n,\\lambda}(x) = c^{n-1} B_n(x)$, where $c = \\log(1+\\lambda)/\\lambda$ with the principal branch of the logarithm, $B_n$ is the $n$-th Bernoulli polynomial (with $B_1(x) = x - 1/2$), and the exponent $n - 1$ is an integer, so that $\\mathfrak{B}_{0,\\lambda}(x) = c^{-1}$. At\u2026","title":"Modified degenerate Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/modified-degenerate-bernoulli-generating","kind":"node","lean":"MetaMathlibExt.modifiedDegenerateBernoulli_generating_function","parent":11503,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html#modified-degenerate-bernoulli-generating","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"fully_proved","summary":"Let $\\lambda$ and $x$ be complex numbers such that $c = \\log(1+\\lambda)/\\lambda$ (principal branch) is nonzero, and write $\\mathfrak{B}_{n,\\lambda}(x)$ for the modified degenerate Bernoulli polynomials. Then, as formal power series in $t$ over $\\mathbb{C}$,","title":"Generating function of the modified degenerate Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/modified-degenerate-bernoulli-one","kind":"node","lean":"MetaMathlibExt.modifiedDegenerateBernoulli_one","parent":11503,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html#modified-degenerate-bernoulli-one","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"fully_proved","summary":"For all complex numbers $\\lambda$ and $x$, the modified degenerate Bernoulli polynomial $\\mathfrak{B}_{1,\\lambda}$ of index $1$ satisfies $\\mathfrak{B}_{1,\\lambda}(x) = x - 1/2$.","title":"The first modified degenerate Bernoulli polynomial"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/modified-degenerate-bernoulli-symmetric","kind":"node","lean":"MetaMathlibExt.modifiedDegenerateBernoulli_symmetric_identity","parent":11503,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html#modified-degenerate-bernoulli-symmetric","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"can_prove","summary":"Let $m \\ge 2$ and $n \\ge 1$ be integers, let $\\lambda$ be a complex number with $\\lambda \\ne 0$ and $\\lambda \\ne -1$, let $r_1, \\ldots, r_m$ and $x_1, \\ldots, x_m$ be complex numbers, write $\\mathfrak{B}_{k,\\lambda}(x)$ for the modified degenerate Bernoulli polynomials, and put $r_0 = n - r_1 - \\cdots - r_m$. Then","title":"Symmetric identity for modified degenerate Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/modified-degenerate-bernoulli-three-term","kind":"node","lean":"MetaMathlibExt.modifiedDegenerateBernoulli_three_term","parent":11503,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html#modified-degenerate-bernoulli-three-term","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, let $\\lambda$ be a complex number with $\\lambda \\ne 0$ and $\\lambda \\ne -1$, write $\\mathfrak{B}_{k,\\lambda}(x)$ for the modified degenerate Bernoulli polynomials, and for complex numbers $s, t, x, y$ put $\\mathfrak{D}_{s,t,\\lambda}(x, y) = \\sum_{k=0}^{n} (-1)^k \\binom{s}{k} \\binom{t}{n-k} \\mathfrak{B}_{n-k,\\lambda}(x) \\mathfrak{B}_{k,\\lambda}(y)$, where $\\binom{w}{k} = w(w-1) \\cdots (w-k+1)/k!$ is\u2026","title":"Three-term identity for modified degenerate Bernoulli polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/modified-degenerate-bernoulli-miki","kind":"node","lean":"MetaMathlibExt.modifiedDegenerateBernoulli_miki","parent":11503,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html#modified-degenerate-bernoulli-miki","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"can_prove","summary":"Let $n \\ge 4$, let $\\lambda$ be a complex number with $\\lambda \\ne 0$ and $\\lambda \\ne -1$, write $\\mathfrak{B}_{k,\\lambda} = \\mathfrak{B}_{k,\\lambda}(0)$ for the value at $0$ of the modified degenerate Bernoulli polynomial $\\mathfrak{B}_{k,\\lambda}(x)$, and let $H_n = \\sum_{k=1}^{n} 1/k$ be the $n$-th harmonic number. Then","title":"Miki-type identity for modified degenerate Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/modified-degenerate-bernoulli-matiyasevi","kind":"node","lean":"MetaMathlibExt.modifiedDegenerateBernoulli_matiyasevich","parent":11503,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html#modified-degenerate-bernoulli-matiyasevi","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"can_prove","summary":"Let $n \\ge 4$, let $\\lambda$ be a complex number with $\\lambda \\ne 0$ and $\\lambda \\ne -1$, and write $\\mathfrak{B}_{k,\\lambda} = \\mathfrak{B}_{k,\\lambda}(0)$ for the value at $0$ of the modified degenerate Bernoulli polynomial $\\mathfrak{B}_{k,\\lambda}(x)$. Then","title":"Matiyasevich-type identity for modified degenerate Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/modified-degenerate-bernoulli-lemma-thre","kind":"node","lean":"MetaMathlibExt.modifiedDegenerateBernoulli_lemma_three","parent":11503,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials/index.html#modified-degenerate-bernoulli-lemma-thre","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/degenerate-bernoulli-polynomials","status":"can_prove","summary":"Let $m \\ge 2$, let $\\lambda$ be a complex number with $\\lambda \\ne 0$ and $\\lambda \\ne -1$, let $s_1, \\ldots, s_m$ be nonnegative integers with $S = s_1 + \\cdots + s_m$, and let $x_1, \\ldots, x_m$ be complex numbers. Writing $\\mathfrak{B}_{n,\\lambda}(x)$ for the modified degenerate Bernoulli polynomials,","title":"A convolution lemma for modified degenerate Bernoulli polynomials"},{"data_url":"scopes/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers.json","id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers","kind":"scope","navigation":"module","parent":11480,"result":false,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/index.html","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers","status":"fully_proved","title":"Euler numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/chapter5-euler-even","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry10bEuleriansymmetry.chapter5EulerEven","parent":11512,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/index.html#chapter5-euler-even","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers","status":"fully_proved","summary":"The integers $E_0, E_2, E_4, \\dots$ defined recursively by $E_0 = 1$ and $E_{2m} = -\\sum_{k=0}^{m-1} \\binom{2m}{2k} E_{2k}$ for $m \\ge 1$; equivalently, $\\sum_{k=0}^{m} \\binom{2m}{2k} E_{2k}$ is $1$ for $m = 0$ and $0$ for $m \\ge 1$. The value at $n$ is $E_{2n}$.","title":"Even-index Euler numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/chapter5-euler-even-zero","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry10bEuleriansymmetry.chapter5EulerEven_zero","parent":11512,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/index.html#chapter5-euler-even-zero","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers","status":"fully_proved","summary":"$E_0 = 1$.","title":"The initial even-index Euler number"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/chapter5-euler-even-step","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry10bEuleriansymmetry.chapter5EulerEven_step","parent":11512,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/index.html#chapter5-euler-even-step","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers","status":"fully_proved","summary":"For every natural number $m \\ge 1$, $E_{2m} = -\\sum_{k=0}^{m-1} \\binom{2m}{2k} E_{2k}$.","title":"Recursion for the even-index Euler numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/chapter5-euler-even-recurrence","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry10bEuleriansymmetry.chapter5EulerEven_recurrence","parent":11512,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/index.html#chapter5-euler-even-recurrence","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers","status":"fully_proved","summary":"For every natural number $m$, $\\sum_{k=0}^{m} \\binom{2m}{2k} E_{2k}$ is $1$ if $m = 0$ and $0$ if $m \\ge 1$.","title":"Binomial recurrence for the even-index Euler numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/ramanujan-part1-ch5-entry10b-euleriansym","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry10bEuleriansymmetry.ramanujan_part1_ch5_entry10b_euleriansymmetry","parent":11512,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers/index.html#ramanujan-part1-ch5-entry10b-euleriansym","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/euler-numbers","status":"fully_proved","summary":"For every natural number $n$, $2^{2n} (2^{2n} - 1) B_{2n} = 2n \\sum_{k=0}^{n-1} \\binom{2n-2}{2k} E_{2k} E_{2n-2k-2}$, where $B_m$ are the Bernoulli numbers and $E_{2k}$ are the even-index Euler numbers.","title":"Bernoulli numbers as a convolution of even-index Euler numbers"},{"data_url":"scopes/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers.json","id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","kind":"scope","navigation":"module","parent":11480,"result":false,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"can_prove","title":"Genocchi numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/genocchi-power-series","kind":"node","lean":"MetaMathlibExt.genocchiPowerSeries","parent":11518,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#genocchi-power-series","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"The Genocchi power series is the formal power series $\\frac{2x}{e^x + 1} = 2x \\cdot (e^x + 1)^{-1}$ in $\\mathbb{Q}[[x]]$, where $e^x = \\sum_{n \\ge 0} x^n / n!$ is the exponential power series and $(e^x + 1)^{-1}$ is the inverse of $e^x + 1$ among formal power series. It is the exponential generating function of the Genocchi numbers $G_n$, that is, $\\frac{2x}{e^x + 1} = \\sum_{n \\ge 0} G_n \\frac{x^n}{n!}$.","title":"Generating function of the Genocchi numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/genocchi-number","kind":"node","lean":"MetaMathlibExt.genocchiNumber","parent":11518,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#genocchi-number","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"For a natural number $n$, the Genocchi number $G_n \\in \\mathbb{Q}$ is $n!$ times the coefficient of $x^n$ in the Genocchi power series $\\frac{2x}{e^x + 1} \\in \\mathbb{Q}[[x]]$, so that $\\frac{2x}{e^x + 1} = \\sum_{n \\ge 0} G_n \\frac{x^n}{n!}$.","title":"Genocchi numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/genocchi-power-series-eq","kind":"node","lean":"MetaMathlibExt.genocchiPowerSeries_eq","parent":11518,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Genocchi power series is "],["math","\\(2x/(e^x + 1)\\)"]],"source":"The Genocchi power series is $2x/(e^x + 1)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#genocchi-power-series-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"The Genocchi power series equals $2x \\cdot (e^x + 1)^{-1}$ in $\\mathbb{Q}[[x]]$, where $e^x = \\sum_{n \\ge 0} x^n / n!$ is the exponential power series and $(e^x + 1)^{-1}$ is the inverse of $e^x + 1$ among formal power series.","title":"The Genocchi power series is $2x/(e^x + 1)$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/genocchi-number-eq","kind":"node","lean":"MetaMathlibExt.genocchiNumber_eq","parent":11518,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#genocchi-number-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"For every natural number $n$, the Genocchi number $G_n$ equals $n!$ times the coefficient of $x^n$ in the Genocchi power series $\\frac{2x}{e^x + 1} \\in \\mathbb{Q}[[x]]$.","title":"Genocchi numbers from their generating function"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/genocchi-number-eq-bernoulli","kind":"node","lean":"MetaMathlibExt.genocchiNumber_eq_bernoulli","parent":11518,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#genocchi-number-eq-bernoulli","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"For every natural number $n$, the Genocchi number $G_n$, defined by $\\frac{2x}{e^x + 1} = \\sum_{m \\ge 0} G_m \\frac{x^m}{m!}$, satisfies $G_n = 2(1 - 2^n) B_n$, where $B_n$ is the $n$-th Bernoulli number with the convention $B_1 = -\\tfrac{1}{2}$.","title":"Genocchi numbers in terms of Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/genocchi-number-via-bernoulli-eq-genocch","kind":"node","lean":"MetaMathlibExt.genocchiNumberViaBernoulli_eq_genocchiNumber","parent":11518,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#genocchi-number-via-bernoulli-eq-genocch","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"For every natural number $n$, the rational number $2(1 - 2^n) B_n$, where $B_n$ is the $n$-th Bernoulli number with the convention $B_1 = -\\tfrac{1}{2}$, equals the Genocchi number $G_n$ defined by the exponential generating function $\\frac{2x}{e^x + 1} = \\sum_{m \\ge 0} G_m \\frac{x^m}{m!}$.","title":"The two definitions of the Genocchi numbers agree"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/diff-pow-eval","kind":"node","lean":"MathlibExt.NumberTheory.GenocchiExplicitWanted.diffPowEval","parent":11518,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterated forward differences of a power on "],["math","\\(\\mathbb{Q}\\)"]],"source":"Iterated forward differences of a power on $\\mathbb{Q}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#diff-pow-eval","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"For a rational number $x$ and natural numbers $k$ and $n$, $\\Delta^k x^n$ denotes the value at $x$ of the $k$-th iterate of the forward difference operator applied to the function $y \\mapsto y^n$ on $\\mathbb{Q}$, where the forward difference of $f \\colon \\mathbb{Q} \\to \\mathbb{Q}$ is the function $y \\mapsto f(y + 1) - f(y)$.","title":"Iterated forward differences of a power on $\\mathbb{Q}$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/zerro-u","kind":"node","lean":"MathlibExt.NumberTheory.GenocchiExplicitWanted.zerroU","parent":11518,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Zerroukhat\u2013Khaldi sum "],["math","\\(U(m, k)\\)"]],"source":"The Zerroukhat\u2013Khaldi sum $U(m, k)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#zerro-u","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"For natural numbers $m$ and $k$, define the rational number","title":"The Zerroukhat\u2013Khaldi sum $U(m, k)$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/zerro-h","kind":"node","lean":"MathlibExt.NumberTheory.GenocchiExplicitWanted.zerroH","parent":11518,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Zerroukhat\u2013Khaldi sum "],["math","\\(H_{n,m}\\)"]],"source":"The Zerroukhat\u2013Khaldi sum $H_{n,m}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#zerro-h","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $m$, define the rational number","title":"The Zerroukhat\u2013Khaldi sum $H_{n,m}$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/genocchi-number-eq-sum","kind":"node","lean":"MathlibExt.NumberTheory.GenocchiExplicitWanted.genocchiNumber_eq_sum","parent":11518,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#genocchi-number-eq-sum","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"can_prove","summary":"Let $n$ and $r$ be positive integers with $n \\ge r + 1$, let $G_n$ be the $n$-th Genocchi number and let $S(a, b)$ denote the Stirling numbers of the second kind. For $k \\ge 1$ write $c_k = \\sum_{j=0}^{r} (-1)^{k+j+r} \\frac{(k+j-1)!}{2^{k+j-1}} S(r, j)$. Then $G_n = (-1)^n \\, n \\sum_{k=1}^{n-r} c_k \\, S(n-r, k)$.","title":"Explicit formula for Genocchi numbers via Stirling numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/neg-one-pow-mul-bernoulli-eq-sum-zero","kind":"node","lean":"MathlibExt.NumberTheory.GenocchiExplicitWanted.neg_one_pow_mul_bernoulli_eq_sum_zero","parent":11518,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli numbers via forward differences at "],["math","\\(0\\)"]],"source":"Bernoulli numbers via forward differences at $0$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#neg-one-pow-mul-bernoulli-eq-sum-zero","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"can_prove","summary":"Let $n$ and $r$ be positive integers with $n \\ge r + 1$, let $B_n$ be the $n$-th Bernoulli number, let $S(a, b)$ denote the Stirling numbers of the second kind and let $\\Delta^k 0^m$ be the $k$-th forward difference of $y \\mapsto y^m$ at $0$. For $k \\ge 1$ write $d_k = \\sum_{j=0}^{r} (-1)^{k+j+r+1} \\, 2^{r-j} \\, \\frac{(k+j-1)!}{k!} \\, S(r, j)$. Then\u2026","title":"Bernoulli numbers via forward differences at $0$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/neg-one-pow-mul-bernoulli-eq-sum-one","kind":"node","lean":"MathlibExt.NumberTheory.GenocchiExplicitWanted.neg_one_pow_mul_bernoulli_eq_sum_one","parent":11518,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli numbers via forward differences at "],["math","\\(1\\)"]],"source":"Bernoulli numbers via forward differences at $1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#neg-one-pow-mul-bernoulli-eq-sum-one","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"can_prove","summary":"Let $n$ and $r$ be positive integers with $n \\ge r + 1$, let $B_n$ be the $n$-th Bernoulli number, let $S(a, b)$ denote the Stirling numbers of the second kind and let $\\Delta^k 1^m$ be the $k$-th forward difference of $y \\mapsto y^m$ at $1$. For $k \\ge 0$ write $e_k = \\sum_{j=0}^{r} (-1)^{k+j+r} \\, 2^{r-j} \\, \\frac{(k+j)!}{k!} \\, S(r, j)$. Then\u2026","title":"Bernoulli numbers via forward differences at $1$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/zerro-u-succ","kind":"node","lean":"MathlibExt.NumberTheory.GenocchiExplicitWanted.zerroU_succ","parent":11518,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the Zerroukhat\u2013Khaldi sums "],["math","\\(U\\)"]],"source":"Recurrence for the Zerroukhat\u2013Khaldi sums $U$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#zerro-u-succ","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"can_prove","summary":"Let $U(m, k)$ be the Zerroukhat\u2013Khaldi sum. For all natural numbers $r$ and $k$ with $k \\ge 1$, $U(r+1, k) = U(r, k+1) + k \\, U(r, k)$.","title":"Recurrence for the Zerroukhat\u2013Khaldi sums $U$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/zerro-h-eq","kind":"node","lean":"MathlibExt.NumberTheory.GenocchiExplicitWanted.zerroH_eq","parent":11518,"reading_title":{"format":"inline-parts-v1","parts":[["text","A step identity for the Zerroukhat\u2013Khaldi sums "],["math","\\(H_{n,m}\\)"]],"source":"A step identity for the Zerroukhat\u2013Khaldi sums $H_{n,m}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers/index.html#zerro-h-eq","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/genocchi-numbers","status":"can_prove","summary":"Let $H_{n,m}$ denote the Zerroukhat\u2013Khaldi sum. For all natural numbers $n$ and $r$ with $r \\ge 1$ and $n \\ge r + 1$, $H_{n,r} = H_{n,r-1}$.","title":"A step identity for the Zerroukhat\u2013Khaldi sums $H_{n,m}$"},{"data_url":"scopes/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials.json","id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","kind":"scope","navigation":"module","parent":11480,"result":false,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"can_prove","title":"Higher-order Euler polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-poly","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerPoly","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-poly","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"fully_proved","summary":"Let $B_m(x)$ denote the $m$-th Bernoulli polynomial. For $n \\ge 0$, the Euler polynomial $E_n(x)$ is the polynomial with rational coefficients $E_n(x) = \\frac{2}{n + 1} \\left(B_{n+1}(x) - 2^{n+1} B_{n+1}(x/2)\\right)$.","title":"Euler polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/gen-stirling-poly","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.genStirlingPoly","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#gen-stirling-poly","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the generalized Stirling polynomial $S(n, k, x)$ is the polynomial in $x$ with rational coefficients","title":"Generalized Stirling polynomial"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-alpha","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerAlpha","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-alpha","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"fully_proved","summary":"The rational numbers $\\alpha_k$, $k \\ge 0$, are defined by $\\alpha_0 = 1$, $\\alpha_1 = 0$ and $2\\alpha_{k+2} + 2\\alpha_{k+1} + \\alpha_k = 0$ for every $k \\ge 0$. They are the Taylor coefficients of $(2 + 2z)/(2 + 2z + z^2)$ at $z = 0$.","title":"Taylor coefficients of (2 + 2z)/(2 + 2z + z\u00b2)"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-num-gen-aux","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerNumGenAux","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-num-gen-aux","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"fully_proved","summary":"Let $E_m(x)$ denote the Euler polynomial. For a complex number $\\alpha$, the complex numbers $a_\\alpha(f, n)$, for natural numbers $f$ (the fuel) and $n$, are defined by $a_\\alpha(0, n) = 1$, $a_\\alpha(f + 1, 0) = 1$ and $a_\\alpha(f + 1, n + 1) = -\\alpha \\, a_\\alpha(f, n) + \\frac{\\alpha}{2} \\sum_{k=0}^{n} \\binom{n}{k} a_\\alpha(f, k) \\, E_{n-k}(0)$.","title":"Fuel-bounded recurrence for the Euler numbers of order \u03b1"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-num-gen","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerNumGen","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-num-gen","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"fully_proved","summary":"Let $E_m(x)$ denote the Euler polynomial. For a complex number $\\alpha$, the Euler numbers of order $\\alpha$ are the complex numbers $E_n^{(\\alpha)}(0)$, $n \\ge 0$, defined by $E_0^{(\\alpha)}(0) = 1$ and $E_{n+1}^{(\\alpha)}(0) = -\\alpha E_n^{(\\alpha)}(0) + \\frac{\\alpha}{2} \\sum_{k=0}^{n} \\binom{n}{k} E_k^{(\\alpha)}(0) \\, E_{n-k}(0)$ for $n \\ge 0$.","title":"Euler numbers of order \u03b1"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-poly-gen","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerPolyGen","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-poly-gen","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"fully_proved","summary":"For a complex number $\\alpha$ and $n \\ge 0$, let $E_k^{(\\alpha)}(0)$ denote the Euler numbers of order $\\alpha$. The Euler polynomial of order $\\alpha$ is the complex polynomial $E_n^{(\\alpha)}(x) = \\sum_{k=0}^{n} \\binom{n}{k} E_k^{(\\alpha)}(0) \\, x^{n-k}$.","title":"Euler polynomials of order \u03b1"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-poly-half-eq-sum","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerPoly_half_eq_sum","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-poly-half-eq-sum","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"can_prove","summary":"Let $E_n(x)$ denote the Euler polynomial, $\\alpha_k$ the Taylor coefficients of $(2 + 2z)/(2 + 2z + z^2)$, and $S(n, k, x) = \\frac{1}{k!} \\sum_{j=0}^{k} \\binom{k}{j} (-1)^{k-j} (x + j)^n$ the generalized Stirling polynomial. For every $n \\ge 0$, as polynomials in $x$ with rational coefficients, $2^n E_n\\left(\\frac{x + 1}{2}\\right) = \\sum_{k=0}^{n} k! \\, \\alpha_k \\, S(n, k, x)$.","title":"Euler polynomials at (x + 1)/2 via generalized Stirling polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-poly-gen-recurrence","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerPolyGen_recurrence","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-poly-gen-recurrence","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"can_prove","summary":"Let $E_m(x)$ denote the Euler polynomial and $E_m^{(\\alpha)}(x)$ the Euler polynomial of order $\\alpha$. For every complex number $\\alpha$ and every integer $n \\ge 1$, as polynomials in $x$, $E_n^{(\\alpha)}(x) = (x - \\alpha/2) \\, E_{n-1}^{(\\alpha)}(x) + \\frac{\\alpha}{2} \\sum_{k=0}^{n-2} \\binom{n-1}{k} (-1)^{n-k} E_{n-1-k}(0) \\, E_k^{(\\alpha)}(x)$, where the sum is empty when $n = 1$.","title":"Recurrence for Euler polynomials of order \u03b1"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-poly-eq-binomial-average","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerPoly_eq_binomial_average","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-poly-eq-binomial-average","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"can_prove","summary":"Let $E_n(x)$ denote the Euler polynomial. For all natural numbers $m \\ge 2$ and $n \\le 2m + 1$, $E_n(x) = \\sum_{k=0}^{m} \\frac{(-1)^k}{2^k} \\binom{m+1}{k+1} \\sum_{j=0}^{k} \\binom{k}{j} \\left(x + j - \\frac{k+1}{2}\\right)^n$ as polynomials in $x$ with rational coefficients.","title":"Euler polynomials as alternating sums of shifted powers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-poly-eq-sum-gen-stirling","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerPoly_eq_sum_genStirling","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-poly-eq-sum-gen-stirling","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"can_prove","summary":"Let $E_n(x)$ denote the Euler polynomial, $\\alpha_k$ the Taylor coefficients of $(2 + 2z)/(2 + 2z + z^2)$, and $S(n, k, x) = \\frac{1}{k!} \\sum_{j=0}^{k} \\binom{k}{j} (-1)^{k-j} (x + j)^n$ the generalized Stirling polynomial. For every integer $n \\ge 1$, as polynomials in $x$ with rational coefficients, $2^n E_n(x) = \\sum_{k=0}^{n} k! \\, \\alpha_k \\, S(n, k, 2x - 1)$.","title":"Euler polynomials via generalized Stirling polynomials"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/euler-poly-eq-double-sum","kind":"node","lean":"MathlibExt.NumberTheory.EulerPolynomialFormulasWanted.eulerPoly_eq_double_sum","parent":11533,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials/index.html#euler-poly-eq-double-sum","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/higher-order-euler-polynomials","status":"can_prove","summary":"Let $E_n(x)$ denote the Euler polynomial and $S(n, k, x) = \\frac{1}{k!} \\sum_{j=0}^{k} \\binom{k}{j} (-1)^{k-j} (x + j)^n$ the generalized Stirling polynomial. For every $n \\ge 0$, as polynomials in $x$ with rational coefficients, $2^n E_n(x) = \\sum_{\\ell=0}^{n} \\frac{(-1)^\\ell}{\\ell + 1} \\sum_{k=0}^{n-\\ell} \\frac{(k + \\ell + 1)!}{2^k} \\binom{\\ell + 1}{k} S(n, k + \\ell, 2x - 1)$.","title":"Double-sum formula for Euler polynomials"},{"data_url":"scopes/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num.json","id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","kind":"scope","navigation":"module","parent":11480,"result":false,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","title":"Landen connection formula and poly-Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/linear-fractional-pole-series","kind":"node","lean":"MetaMathlibExt.linearFractionalPoleSeries","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#linear-fractional-pole-series","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For real numbers $a$ and $c$, this is the power series $F_{a,c}(u)$ of $\\frac{cu}{a - cu}$: it is $\\sum_{n \\ge 1} (c/a)^n u^n$ if $a \\neq 0$, the constant $-1$ if $a = 0$ and $c \\neq 0$, and $0$ (a junk value) if $a = c = 0$. It is $u$ times the summand $\\frac{c}{a - cu}$ of $G_g(u) = \\frac{c}{a - cu} - \\frac{d}{b - du}$ for the matrix $g$ with rows $(a, b)$ and $(c, d)$.","title":"Linear fractional pole series"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/linear-fractional-pole-series-spec","kind":"node","lean":"MetaMathlibExt.linearFractionalPoleSeries_spec","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#linear-fractional-pole-series-spec","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For all real numbers $a$ and $c$, the linear fractional pole series $F_{a,c}(u)$, the power series of $\\frac{cu}{a - cu}$, satisfies $(a - cu) \\cdot F_{a,c}(u) = cu$ in $\\mathbb{R}[[u]]$.","title":"The linear fractional pole series solves its defining equation"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/landen-form-series","kind":"node","lean":"MetaMathlibExt.landenFormSeries","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#landen-form-series","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For real numbers $a, b, c, d$, viewed as the matrix $g$ with rows $(a, b)$ and $(c, d)$, this is the power series $u \\cdot G_g(u) = \\frac{cu}{a - cu} - \\frac{du}{b - du} \\in \\mathbb{R}[[u]]$, where $G_g(t) = \\frac{c}{a - ct} - \\frac{d}{b - dt}$. Here the power series of $\\frac{cu}{a - cu}$ is $\\sum_{n \\ge 1} (c/a)^n u^n$ if $a \\neq 0$, the constant $-1$ if $a = 0$ and $c \\neq 0$, and $0$ if $a = c = 0$; likewise\u2026","title":"Landen form series"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/landen-form-series-identity","kind":"node","lean":"MetaMathlibExt.landenFormSeries_identity","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#landen-form-series-identity","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For the identity matrix $g$, with rows $(1, 0)$ and $(0, 1)$, the Landen form series $u \\cdot G_g(u)$ is the constant power series $1$ in $\\mathbb{R}[[u]]$; that is, $G_g(t) = 1/t$.","title":"Landen form series of the identity matrix"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/integral-div-x","kind":"node","lean":"MetaMathlibExt.integralDivX","parent":11545,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formal integral against "],["math","\\(du/u\\)"]],"source":"Formal integral against $du/u$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#integral-div-x","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For a real power series $h$, this is the power series $\\int_0^z h(u)\\,du/u$ computed coefficientwise: for $n \\ge 1$ its coefficient of $z^n$ is the coefficient of $u^n$ in $h$ divided by $n$, and its constant coefficient is $0$. The constant coefficient of $h$ is ignored, so this is the termwise integral only when $h$ has zero constant term.","title":"Formal integral against $du/u$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/integral-div-x-spec","kind":"node","lean":"MetaMathlibExt.integralDivX_spec","parent":11545,"reading_title":{"format":"inline-parts-v1","parts":[["text","The formal integral against "],["math","\\(du/u\\)"],["text"," is a right inverse of "],["math","\\(z \\, d/dz\\)"]],"source":"The formal integral against $du/u$ is a right inverse of $z \\, d/dz$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#integral-div-x-spec","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $h \\in \\mathbb{R}[[z]]$ be a power series with zero constant term, and let $I(h) = \\sum_{n \\ge 1} \\frac{[z^n] h}{n} z^n$ be its formal integral $\\int_0^z h(u) \\, du / u$, where $[z^n] h$ is the coefficient of $z^n$ in $h$. Then $I(h)$ has zero constant term and $z \\, \\frac{d}{dz} I(h) = h$.","title":"The formal integral against $du/u$ is a right inverse of $z \\, d/dz$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/landen-data","kind":"node","lean":"MetaMathlibExt.LandenData","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#landen-data","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Landen data $g = (g_0, g_1, \\dots, g_s)$ consist of a natural number $s$, real numbers $c_0, d_0$, and real numbers $a_i, b_i, c_i, d_i$ for $1 \\le i \\le s$. They encode the real $2 \\times 2$ matrix $g_0$ with rows $(1, -1)$ and $(c_0, d_0)$ and, for $1 \\le i \\le s$, the matrix $g_i$ with rows $(a_i, b_i)$ and $(c_i, d_i)$. No invertibility is required.","title":"Landen data"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/tilde","kind":"node","lean":"MetaMathlibExt.LandenData.tilde","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#tilde","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For Landen data $g = (g_0, g_1, \\dots, g_s)$ with $g_0 = \\begin{pmatrix} 1 & -1 \\\\ c_0 & d_0 \\end{pmatrix}$ and $g_i = \\begin{pmatrix} a_i & b_i \\\\ c_i & d_i \\end{pmatrix}$ for $1 \\le i \\le s$, the transformed data $\\tilde g = (\\tilde g_0, \\tilde g_1, \\dots, \\tilde g_s)$ has the same $s$, with $\\tilde g_0 = \\begin{pmatrix} 1 & -1 \\\\ -d_0 & -c_0 \\end{pmatrix}$ and, for $1 \\le i \\le s$, the matrix $\\tilde g_i$ has\u2026","title":"Transformed Landen data"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/tilde-tilde","kind":"node","lean":"MetaMathlibExt.LandenData.tilde_tilde","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#tilde-tilde","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For Landen data $g = (g_0, g_1, \\dots, g_s)$ with $g_0 = \\begin{pmatrix} 1 & -1 \\\\ c_0 & d_0 \\end{pmatrix}$, let $\\tilde g$ be the transformed data, given by $\\tilde g_i = \\begin{pmatrix} 1 & 0 \\\\ c_0 - d_0 & -1 \\end{pmatrix} g_i$ for $0 \\le i \\le s$. Then the transformed data of $\\tilde g$ is $g$ itself: $\\tilde{\\tilde g} = g$.","title":"The transformation of Landen data is an involution"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/form-series","kind":"node","lean":"MetaMathlibExt.LandenData.formSeries","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#form-series","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, where $g_i$ has rows $(a_i, b_i)$ and $(c_i, d_i)$ for $1 \\le i \\le s$, and let $w_1, \\dots, w_s$ be real numbers. The weighted form series of $g$ with weights $w$ is the formal power series $u \\sum_{i=1}^{s} w_i G_{g_i}(u) \\in \\mathbb{R}[[u]]$, where $u\\,G_{g_i}(u)$ is the Landen form series of $g_i$: the expansion of\u2026","title":"Weighted form series"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/polylog-rev","kind":"node","lean":"MetaMathlibExt.LandenData.polylogRev","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#polylog-rev","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Fix Landen data $g$, with $g_0 = \\begin{pmatrix} 1 & -1 \\\\ c_0 & d_0 \\end{pmatrix}$ and $g_i = \\begin{pmatrix} a_i & b_i \\\\ c_i & d_i \\end{pmatrix}$ for $1 \\le i \\le s$, and weights $\\beta, \\gamma \\in \\mathbb{R}^s$. For a real matrix $h = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ let $z\\,G_h(z) \\in \\mathbb{R}[[z]]$ be the power series of $\\frac{cz}{a - cz} - \\frac{dz}{b - dz}$, where a term whose numerator and\u2026","title":"Reversed recursion for Kamano's generalized multiple polylogarithm"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-multiple-polylog","kind":"node","lean":"MetaMathlibExt.generalizedMultiplePolylog","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-multiple-polylog","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, where $g_0$ has rows $(1, -1)$ and $(c_0, d_0)$ and $g_i$ has rows $(a_i, b_i)$ and $(c_i, d_i)$, and let $\\beta, \\gamma \\in \\mathbb{R}^s$. For a matrix $h$ with rows $(a, b)$ and $(c, d)$ put $G_h(u) = \\frac{c}{a - c u} - \\frac{d}{b - d u}$, a fraction with zero numerator and zero denominator being read as $0$. For an index $\\mathbf{k} = (k_1, \\dots, k_r)$ of\u2026","title":"Kamano's generalized multiple polylogarithm"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-multiple-polylog-one","kind":"node","lean":"MetaMathlibExt.generalizedMultiplePolylog_one","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-multiple-polylog-one","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, where $g_0$ has rows $(1, -1)$ and $(c_0, d_0)$, and let $\\beta, \\gamma \\in \\mathbb{R}^s$. Then $\\mathrm{Li}_{(1)}(g; (\\beta, \\gamma); z) = \\int_0^z G_{g_0}(u)\\,du$, where $u\\,G_{g_0}(u) = \\frac{c_0 u}{1 - c_0 u} + \\frac{d_0 u}{1 + d_0 u}$ is the Landen form series of $g_0$ and the integral of a power series is taken term by term.","title":"Initial value of the generalized multiple polylogarithm"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-multiple-polylog-up","kind":"node","lean":"MetaMathlibExt.generalizedMultiplePolylog_up","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-multiple-polylog-up","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data and let $\\beta, \\gamma \\in \\mathbb{R}^s$. For every list $(k_1, \\dots, k_{r-1})$ of natural numbers (possibly empty) and every natural number $m$, writing $\\mathbf{k} = (k_1, \\dots, k_{r-1}, m + 1)$, $\\mathrm{Li}_{(k_1, \\dots, k_{r-1}, m + 2)}(g; (\\beta, \\gamma); z) = \\int_0^z \\sum_{i=1}^{s} \\beta_i G_{g_i}(u) \\, \\mathrm{Li}_{\\mathbf{k}}(g; (\\beta, \\gamma); u)\\,du$,\u2026","title":"Recurrence for raising the last entry"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-multiple-polylog-right","kind":"node","lean":"MetaMathlibExt.generalizedMultiplePolylog_right","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-multiple-polylog-right","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data and let $\\beta, \\gamma \\in \\mathbb{R}^s$. For every nonempty list $\\mathbf{k} = (k_1, \\dots, k_r)$ of natural numbers, $\\mathrm{Li}_{(k_1, \\dots, k_r, 1)}(g; (\\beta, \\gamma); z) = \\int_0^z \\sum_{i=1}^{s} \\gamma_i G_{g_i}(u) \\, \\mathrm{Li}_{\\mathbf{k}}(g; (\\beta, \\gamma); u)\\,du$, where $u \\sum_{i} \\gamma_i G_{g_i}(u)$ is the weighted form series of $g$ with weights\u2026","title":"Recurrence for appending the entry 1"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/multiple-polylog-coeff-rev","kind":"node","lean":"MetaMathlibExt.multiplePolylogCoeffRev","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#multiple-polylog-coeff-rev","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For a finite list $\\ell$ of natural numbers and $m \\in \\mathbb{N}$, the real number $c_\\ell(m)$ is defined by recursion on $\\ell$: for the empty list, $c_{()}(m)$ is $1$ if $m = 0$ and $0$ otherwise; for the list $(a, \\ell')$ with first entry $a$ followed by the list $\\ell'$, $c_{(a, \\ell')}(m) = \\big(\\sum_{m' < m} c_{\\ell'}(m')\\big) / m^a$.","title":"Coefficient recursion for multiple polylogarithms"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/multiple-polylog","kind":"node","lean":"MetaMathlibExt.multiplePolylog","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#multiple-polylog","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For a finite sequence $k = (k_1, \\dots, k_r)$ of natural numbers with $r \\ge 1$, the multiple polylogarithm is the formal power series","title":"Multiple polylogarithm"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/coeff-multiple-polylog-singleton","kind":"node","lean":"MetaMathlibExt.coeff_multiplePolylog_singleton","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#coeff-multiple-polylog-singleton","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For every natural number $a$ and every integer $m \\ge 1$, the coefficient of $z^m$ in the multiple polylogarithm $\\mathrm{Li}_{(a)}(z)$ is $1/m^a$.","title":"Depth-one multiple polylogarithm"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/multiple-polylog-star-coeff-rev","kind":"node","lean":"MetaMathlibExt.multiplePolylogStarCoeffRev","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#multiple-polylog-star-coeff-rev","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For a finite list $\\ell$ of natural numbers and $m \\in \\mathbb{N}$, the real number $c_\\ell(m)$ is defined by recursion on $\\ell$: for the empty list, $c_\\ell(m) = 1$ if $m = 0$ and $c_\\ell(m) = 0$ otherwise; for a list $\\ell = (a, \\ell')$ with first entry $a$ followed by the list $\\ell'$,","title":"Coefficient recursion for the non-strict multiple polylogarithm"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/multiple-polylog-star","kind":"node","lean":"MetaMathlibExt.multiplePolylogStar","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#multiple-polylog-star","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For an index $k = (k_1, \\dots, k_r)$ of positive integers with $r \\ge 1$, the non-strict multiple polylogarithm is the formal power series","title":"Non-strict multiple polylogarithm"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/coeff-multiple-polylog-star-singleton","kind":"node","lean":"MetaMathlibExt.coeff_multiplePolylogStar_singleton","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#coeff-multiple-polylog-star-singleton","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For all natural numbers $a$ and $m$, the coefficient of $z^m$ in the non-strict multiple polylogarithm $\\mathrm{Li}^{\\star}_{(a)}(z)$ is $1/m^a$, with the conventions $0^0 = 1$ and $1/0 = 0$. In particular $\\mathrm{Li}^{\\star}_{(a)}(z) = \\sum_{m \\ge 1} z^m / m^a$ for $a \\ge 1$.","title":"Depth-one non-strict multiple polylogarithm"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/hoffman-dual-rev","kind":"node","lean":"MetaMathlibExt.hoffmanDualRev","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#hoffman-dual-rev","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"The auxiliary map $R$ on finite lists $l = (l_1, \\dots, l_m)$ of natural numbers is defined recursively by: $R(()) = ()$; $R(l) = ()$ if $l_1 = 0$; $R((1)) = (1)$; if $l_1 = 1$ and $m \\ge 2$, then $R(l)$ is $R((l_2, \\dots, l_m))$ with its first entry increased by $1$ (unchanged if that list is empty); and if $l_1 \\ge 2$, then $R(l) = (1, r_1, \\dots, r_p)$ where $(r_1, \\dots, r_p) = R((l_1 - 1, l_2, \\dots, l_m))$.","title":"Hoffman dual on reversed lists"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/hoffman-dual","kind":"node","lean":"MetaMathlibExt.hoffmanDual","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#hoffman-dual","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For an index $k = (k_1, \\dots, k_r)$ of positive integers write $k\\uparrow = (k_1, \\dots, k_r + 1)$ and $k\\rightarrow = (k_1, \\dots, k_r, 1)$. The Hoffman dual index $k^\\vee$ is defined recursively by $(1)^\\vee = (1)$, $(k\\uparrow)^\\vee = (k^\\vee)\\rightarrow$ and $(k\\rightarrow)^\\vee = (k^\\vee)\\uparrow$. On other finite lists of natural numbers, the empty list is sent to the empty list and a list with an entry $0$\u2026","title":"Hoffman dual index"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/hoffman-dual-small-values","kind":"node","lean":"MetaMathlibExt.hoffmanDual_small_values","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#hoffman-dual-small-values","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"The Hoffman dual index satisfies $(1)^\\vee = (1)$, $(3)^\\vee = (1, 1, 1)$ and $(1, 2)^\\vee = (2, 1)$.","title":"Small values of the Hoffman dual"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/landen-exp-variable","kind":"node","lean":"MetaMathlibExt.landenExpVariable","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#landen-exp-variable","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"For Landen data $g$ whose matrix $g_0$ has rows $(1, -1)$ and $(c_0, d_0)$, this is the formal power series $z(t) = (e^t - 1)(c_0 e^t + d_0)^{-1}$ in $\\mathbb{R}[[t]]$. The inverse is the power series inverse, which exists exactly when $c_0 + d_0 \\neq 0$; when $c_0 + d_0 = 0$ the inverse is taken to be $0$, so $z(t) = 0$.","title":"Landen exponential variable"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/landen-exp-variable-spec","kind":"node","lean":"MetaMathlibExt.landenExpVariable_spec","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#landen-exp-variable-spec","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g$ be Landen data whose matrix $g_0$ has rows $(1, -1)$ and $(c_0, d_0)$ with $c_0 + d_0 \\neq 0$, and let $z(t) = (e^t - 1)(c_0 e^t + d_0)^{-1} \\in \\mathbb{R}[[t]]$ be its Landen exponential variable. Then $(c_0 e^t + d_0)\\, z(t) = e^t - 1$ in $\\mathbb{R}[[t]]$.","title":"Defining relation of the Landen exponential variable"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-poly-bernoulli-c","kind":"node","lean":"MetaMathlibExt.generalizedPolyBernoulliC","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-poly-bernoulli-c","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, where $(c_0, d_0)$ is the bottom row of the matrix $g_0$, let $\\beta, \\gamma \\in \\mathbb{R}^s$, and let $k = (k_1, \\dots, k_r)$ be a list of natural numbers; write $k\\uparrow = (k_1, \\dots, k_r + 1)$, with $k\\uparrow = ()$ when $k$ is empty. Let $z(t) \\in \\mathbb{R}[[t]]$ be the power series of $(e^t - 1)/(c_0 e^t + d_0)$ (the zero series when $c_0 + d_0 = 0$). The\u2026","title":"Generalized poly-Bernoulli numbers of the first kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-poly-bernoulli-b","kind":"node","lean":"MetaMathlibExt.generalizedPolyBernoulliB","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-poly-bernoulli-b","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, where $(c_0, d_0)$ is the bottom row of the matrix $g_0$, let $\\beta, \\gamma \\in \\mathbb{R}^s$, and let $k = (k_1, \\dots, k_r)$ be a list of natural numbers; write $k\\rightarrow = (k_1, \\dots, k_r, 1)$, with $k\\rightarrow = (1)$ when $k$ is empty. Let $z(t) \\in \\mathbb{R}[[t]]$ be the power series of $(e^t - 1)/(c_0 e^t + d_0)$ (the zero series when $c_0 + d_0 = 0$).\u2026","title":"Generalized poly-Bernoulli numbers of the second kind"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-poly-bernoulli-c-eq-coeff","kind":"node","lean":"MetaMathlibExt.generalizedPolyBernoulliC_eq_coeff","parent":11545,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized poly-Bernoulli numbers "],["math","\\(C_n\\)"],["text"," as Taylor coefficients"]],"source":"Generalized poly-Bernoulli numbers $C_n$ as Taylor coefficients"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-poly-bernoulli-c-eq-coeff","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"fully_proved","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, where $(c_0, d_0)$ is the bottom row of the matrix $g_0$, let $\\beta, \\gamma \\in \\mathbb{R}^s$, let $k = (k_1, \\dots, k_r)$ be a list of natural numbers, and write $k\\uparrow = (k_1, \\dots, k_r + 1)$, with $k\\uparrow = ()$ when $k$ is empty. Let $z(t) \\in \\mathbb{R}[[t]]$ be the power series of $(e^t - 1)/(c_0 e^t + d_0)$ (the zero series when $c_0 + d_0 = 0$). Then\u2026","title":"Generalized poly-Bernoulli numbers $C_n$ as Taylor coefficients"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/multiple-polylog-landen-connection","kind":"node","lean":"MetaMathlibExt.multiplePolylog_landen_connection","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#multiple-polylog-landen-connection","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","summary":"Landen connection formula (Okuda\u2013Ueno). Let $n \\ge 0$ and let $k = (k_1, \\dots, k_r)$ be an index of weight $n$, that is, a finite sequence of positive integers with sum $n$. Then the multiple polylogarithms satisfy the identity","title":"Landen connection formula"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-multiple-polylog-sum-formula","kind":"node","lean":"MetaMathlibExt.generalizedMultiplePolylog_sum_formula","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-multiple-polylog-sum-formula","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, let $\\beta, \\gamma, \\delta \\in \\mathbb{R}^s$, let $\\mu$ be a real number and let $k \\ge 1$ be an integer. Then $\\sum_{\\mathbf{k}} \\mu^{d(\\mathbf{k}) - 1} \\mathrm{Li}_{\\mathbf{k}}(g; (\\beta, \\gamma); z) = \\mathrm{Li}_{(k)}(g; (\\beta + \\mu \\gamma, \\delta); z)$, where the sum runs over all compositions $\\mathbf{k} = (k_1, \\dots, k_r)$ of $k$ (sequences of positive\u2026","title":"Sum formula"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-multiple-polylog-landen-conn","kind":"node","lean":"MetaMathlibExt.generalizedMultiplePolylog_landen_connection","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-multiple-polylog-landen-conn","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, where $g_0$ has rows $(1, -1)$ and $(c_0, d_0)$ and each $g_i$ ($1 \\le i \\le s$) has rows $(a_i, b_i)$ and $(c_i, d_i)$ with $a_i d_i - b_i c_i \\ne 0$. Let $\\tilde g$ be the transformed data: $\\tilde g_0$ has rows $(1, -1)$ and $(-d_0, -c_0)$, and $\\tilde g_i = M g_i$, where $M$ is the matrix with rows $(1, 0)$ and $(c_0 - d_0, -1)$. For all\u2026","title":"Generalized Landen connection formula"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-multiple-polylog-landen-conn-2","kind":"node","lean":"MetaMathlibExt.generalizedMultiplePolylog_landen_connection_classical","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-multiple-polylog-landen-conn-2","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","summary":"Let $g = (g_0, g_1, g_2)$ be the Landen data in which $g_0 = g_2$ has rows $(1, -1)$ and $(1, 0)$ and $g_1$ is the identity matrix, so that $G_{g_1}(t) = 1/t$ and $G_{g_2}(t) = 1/(1 - t)$. For all real numbers $\\beta_1$, $\\beta_2$, $\\gamma_1$, $\\gamma_2$ and every finite (possibly empty) sequence $\\mathbf{k}$ of positive integers,\u2026","title":"Landen formula in the classical case"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/multi-poly-bernoulli-star-duality","kind":"node","lean":"MetaMathlibExt.multiPolyBernoulliStar_duality","parent":11545,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#multi-poly-bernoulli-star-duality","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","summary":"This duality is due to Imatomi. Let $k$ be an index (a composition of a natural number $m$), let $k^\\vee$ be its Hoffman dual, and let $\\mathrm{Li}^\\star_k$ be the non-strict multiple polylogarithm. Let $F, G \\in \\mathbb{R}[[t]]$ satisfy $(e^t - 1) F = \\mathrm{Li}^\\star_k(1 - e^{-t})$ and $(1 - e^{-t}) G = \\mathrm{Li}^\\star_{k^\\vee}(1 - e^{-t})$, and write $F = \\sum_{n \\ge 0} C_{n,\\star}^{(k)} t^n / n!$ and\u2026","title":"Duality of multi-poly-Bernoulli-star numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-poly-bernoulli-c-sum-formula","kind":"node","lean":"MetaMathlibExt.generalizedPolyBernoulliC_sum_formula","parent":11545,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum formula for generalized poly-Bernoulli numbers "],["math","\\(C_n\\)"]],"source":"Sum formula for generalized poly-Bernoulli numbers $C_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-poly-bernoulli-c-sum-formula","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, let $\\beta, \\gamma \\in \\mathbb{R}^s$, let $\\mu$ be a real number, and let $k \\ge 1$ and $n \\ge 0$ be integers. Write $B_n$ and $C_n$ for Kamano's generalized poly-Bernoulli numbers of the second and first kind, $d(\\mathbf{k})$ for the number of entries of an index $\\mathbf{k}$, and $\\{1\\}^k = (1, \\dots, 1)$ for the index of $k$ ones. Then","title":"Sum formula for generalized poly-Bernoulli numbers $C_n$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-poly-bernoulli-b-sum-formula","kind":"node","lean":"MetaMathlibExt.generalizedPolyBernoulliB_sum_formula","parent":11545,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum formula for generalized poly-Bernoulli numbers "],["math","\\(B_n\\)"]],"source":"Sum formula for generalized poly-Bernoulli numbers $B_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-poly-bernoulli-b-sum-formula","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data, let $\\beta, \\gamma \\in \\mathbb{R}^s$, let $\\mu$ be a real number, and let $k \\ge 1$ and $n \\ge 0$ be integers. Write $B_n$ and $C_n$ for Kamano's generalized poly-Bernoulli numbers of the second and first kind, $d(\\mathbf{k})$ for the number of entries of an index $\\mathbf{k}$, and $\\{1\\}^k = (1, \\dots, 1)$ for the index of $k$ ones. Then","title":"Sum formula for generalized poly-Bernoulli numbers $B_n$"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/generalized-poly-bernoulli-c-landen","kind":"node","lean":"MetaMathlibExt.generalizedPolyBernoulliC_landen","parent":11545,"reading_title":{"format":"inline-parts-v1","parts":[["text","Landen formula for generalized poly-Bernoulli numbers "],["math","\\(C_n\\)"]],"source":"Landen formula for generalized poly-Bernoulli numbers $C_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num/index.html#generalized-poly-bernoulli-c-landen","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/landen-connection-formula-and-poly-bernoulli-num","status":"can_prove","summary":"Let $g = (g_0, g_1, \\dots, g_s)$ be Landen data whose matrices $g_1, \\dots, g_s$ are invertible, where $(c_0, d_0)$ is the bottom row of $g_0$. Let $\\tilde g$ be the transformed data: $\\tilde g_0$ has rows $(1, -1)$ and $(-d_0, -c_0)$, and $\\tilde g_i = h g_i$ for $1 \\le i \\le s$, where $h$ is the matrix with rows $(1, 0)$ and $(c_0 - d_0, -1)$. Let $\\beta, \\gamma \\in \\mathbb{R}^s$, let $k$ be a composition of a\u2026","title":"Landen formula for generalized poly-Bernoulli numbers $C_n$"},{"data_url":"scopes/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers.json","id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers","kind":"scope","navigation":"module","parent":11480,"result":false,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers/index.html","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers","status":"fully_proved","title":"Poly-Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers/poly-bernoulli","kind":"node","lean":"MetaMathlibExt.polyBernoulli","parent":11583,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers/index.html#poly-bernoulli","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers","status":"fully_proved","summary":"Let $S(n, r)$ denote the Stirling numbers of the second kind. For a natural number $n$ and an integer $k$, the poly-Bernoulli number $B_n^{(k)}$ is the rational number given by Kaneko's explicit formula","title":"Poly-Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers/arakawa-kaneko-duality","kind":"node","lean":"MetaMathlibExt.arakawa_kaneko_duality","parent":11583,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers/index.html#arakawa-kaneko-duality","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers","status":"fully_proved","summary":"Let $S(n, r)$ denote the Stirling numbers of the second kind. For a natural number $n$ and an integer $k$, let $B_n^{(k)}$ be the poly-Bernoulli number given by Kaneko's explicit formula","title":"Arakawa\u2013Kaneko duality for poly-Bernoulli numbers"},{"id":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers/poly-bernoulli-negative-symmetric-expone","kind":"node","lean":"MetaMathlibExt.polyBernoulli_negative_symmetric_exponential_generating_function","parent":11583,"result":true,"route":"../roadmap/NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers/index.html#poly-bernoulli-negative-symmetric-expone","scope":"NumberTheory/special-numbers/euler-genocchi-poly-bernoulli/poly-bernoulli-numbers","status":"fully_proved","summary":"For natural numbers $n, k$ let $B_{n,k} = \\sum_{j=0}^{\\min(n,k)} (j!)^2 \\, S(n+1, j+1) \\, S(k+1, j+1)$, where $S$ denotes the Stirling numbers of the second kind. Then for all real numbers $x, y$ with $(e^{|x|} - 1)(e^{|y|} - 1) < 1$,","title":"Generating function of negative-index poly-Bernoulli numbers"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers.json","id":"NumberTheory/special-numbers/harmonic-numbers","kind":"scope","parent":11422,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers","status":"can_prove","summary":"Harmonic, hyperharmonic and multiple harmonic sums, binomial-harmonic identities, non-integrality, harmonic congruences, Euler\u2013Mascheroni bounds.","title":"Harmonic numbers and harmonic sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/higher-harmonic","kind":"node","lean":"higherHarmonic","parent":11587,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/index.html#higher-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $r$, the harmonic number of order $r$ is the rational number $H_n^{(r)} = \\sum_{k=1}^{n} 1/k^r$.","title":"Higher-order harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/harmonic-partial-sum-not-int","kind":"node","lean":"MetaMathlibExt.harmonic_partial_sum_not_int","parent":11587,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/index.html#harmonic-partial-sum-not-int","scope":"NumberTheory/special-numbers/harmonic-numbers","status":"fully_proved","summary":"For every natural number $n \\ge 2$, the sum $\\sum_{\\nu=2}^{n} H_\\nu$ is not an integer, where $H_\\nu = \\sum_{j=1}^{\\nu} 1/j$ is the $\\nu$-th harmonic number, viewed as a rational number.","title":"Sums of harmonic numbers starting from the second are not integers"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences.json","id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","title":"2-adic valuations of harmonic number differences"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/weighted-binary-digit-sum","kind":"node","lean":"MetaMathlibExt.weightedBinaryDigitSum","parent":11590,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#weighted-binary-digit-sum","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"fully_proved","summary":"For a natural number $n$, the weighted binary digit sum of $n$ is $a(n) = \\sum_i i \\cdot 2^i$, where $i$ runs over the indices of the $1$ bits in the binary expansion of $n$. Equivalently, if $n = 2^{a_1} + \\dots + 2^{a_t}$ with distinct exponents $a_1, \\dots, a_t$, then $a(n) = a_1 2^{a_1} + \\dots + a_t 2^{a_t}$; in particular $a(0) = 0$.","title":"Weighted binary digit sum"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/weighted-binary-digit-sum-two","kind":"node","lean":"MetaMathlibExt.weightedBinaryDigitSum_two","parent":11590,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#weighted-binary-digit-sum-two","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"fully_proved","summary":"The weighted binary digit sum of $2$ is $a(2) = 2$.","title":"Weighted binary digit sum of 2"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/weighted-binary-digit-sum-four","kind":"node","lean":"MetaMathlibExt.weightedBinaryDigitSum_four","parent":11590,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#weighted-binary-digit-sum-four","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"fully_proved","summary":"The weighted binary digit sum of $4$ is $a(4) = 8$.","title":"Weighted binary digit sum of 4"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/padic-val-rat-two-harmonic-sub-neg","kind":"node","lean":"MetaMathlibExt.padicValRat_two_harmonic_sub_neg","parent":11590,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#padic-val-rat-two-harmonic-sub-neg","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number (so $H_0 = 0$) and let $\\nu_2$ denote the $2$-adic valuation on the rational numbers. For all natural numbers $m$ and $n$ with $m + 2 \\le n$, the $2$-adic valuation $\\nu_2(H_n - H_m)$ is negative; that is, $H_n - H_m$ is not a $2$-adic integer.","title":"K\u00fcrsch\u00e1k's strengthening of Theisinger's theorem"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/card-filter-padic-val-rat-two-harmonic-s","kind":"node","lean":"MetaMathlibExt.card_filter_padicValRat_two_harmonic_sub","parent":11590,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fiber counts of the 2-adic valuations of "],["math","\\(H_n - H_m\\)"]],"source":"Fiber counts of the 2-adic valuations of $H_n - H_m$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#card-filter-padic-val-rat-two-harmonic-s","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number (so $H_0 = 0$) and let $\\nu_2$ denote the $2$-adic valuation on the rational numbers. Let $n$ and $a$ be natural numbers such that the binary digit of $n$ in position $a$ is $1$, that is, $2^a$ occurs in the binary expansion of $n$. Then exactly $2^a$ of the natural numbers $m < n$ satisfy $\\nu_2(H_n - H_m) = -a$.","title":"Fiber counts of the 2-adic valuations of $H_n - H_m$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/padic-val-rat-two-harmonic-two-mul-sub","kind":"node","lean":"MetaMathlibExt.padicValRat_two_harmonic_two_mul_sub","parent":11590,"reading_title":{"format":"inline-parts-v1","parts":[["text","2-adic valuation of "],["math","\\(H_{2n} - H_n\\)"]],"source":"2-adic valuation of $H_{2n} - H_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#padic-val-rat-two-harmonic-two-mul-sub","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number and let $\\nu_2$ denote the $2$-adic valuation on the rational numbers. For every natural number $n \\ge 1$, $\\nu_2(H_{2n} - H_n) = -\\lfloor \\log_2(2n) \\rfloor$.","title":"2-adic valuation of $H_{2n} - H_n$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/padic-val-rat-two-harmonic-pow-sub","kind":"node","lean":"MetaMathlibExt.padicValRat_two_harmonic_pow_sub","parent":11590,"reading_title":{"format":"inline-parts-v1","parts":[["text","2-adic valuation of "],["math","\\(H_{2^a} - H_{2^a - m}\\)"]],"source":"2-adic valuation of $H_{2^a} - H_{2^a - m}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#padic-val-rat-two-harmonic-pow-sub","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number (so $H_0 = 0$) and let $\\nu_2$ denote the $2$-adic valuation on the rational numbers. For all natural numbers $a$ and $m$ with $1 \\le m \\le 2^a$, $\\nu_2(H_{2^a} - H_{2^a - m}) = -a$.","title":"2-adic valuation of $H_{2^a} - H_{2^a - m}$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/sum-padic-val-rat-two-harmonic-sub","kind":"node","lean":"MetaMathlibExt.sum_padicValRat_two_harmonic_sub","parent":11590,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of the 2-adic valuations of "],["math","\\(H_n - H_m\\)"]],"source":"Sum of the 2-adic valuations of $H_n - H_m$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#sum-padic-val-rat-two-harmonic-sub","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number (so $H_0 = 0$), let $\\nu_2$ denote the $2$-adic valuation on the rational numbers, and let $a(n)$ be the weighted binary digit sum of $n$. For every natural number $n$, $\\sum_{m=0}^{n-1} \\nu_2(H_n - H_m) = -a(n)$.","title":"Sum of the 2-adic valuations of $H_n - H_m$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/weighted-binary-digit-sum-recurrence","kind":"node","lean":"MetaMathlibExt.weightedBinaryDigitSum_recurrence","parent":11590,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#weighted-binary-digit-sum-recurrence","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","summary":"Let $a(n)$ denote the weighted binary digit sum of $n$. For every natural number $n$, $a(2n) = 2a(n) + 2n$ and $a(2n + 1) = a(2n)$.","title":"Recurrence for the weighted binary digit sum"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/padic-val-rat-two-harmonic-sub-pow","kind":"node","lean":"MetaMathlibExt.padicValRat_two_harmonic_sub_pow","parent":11590,"reading_title":{"format":"inline-parts-v1","parts":[["text","2-adic valuation of "],["math","\\(H_n - H_{n-2^r}\\)"]],"source":"2-adic valuation of $H_n - H_{n-2^r}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#padic-val-rat-two-harmonic-sub-pow","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number and let $\\nu_2$ denote the $2$-adic valuation on the rational numbers. For all natural numbers $r$ and $n$ with $2^r \\le n$, $\\nu_2(H_n - H_{n-2^r}) = -r - \\nu_2(\\lfloor n / 2^r \\rfloor)$.","title":"2-adic valuation of $H_n - H_{n-2^r}$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/padic-val-rat-two-harmonic-sub-piecewise","kind":"node","lean":"MetaMathlibExt.padicValRat_two_harmonic_sub_piecewise","parent":11590,"reading_title":{"format":"inline-parts-v1","parts":[["text","Piecewise formula for the 2-adic valuation of "],["math","\\(H_n - H_{n-m}\\)"]],"source":"Piecewise formula for the 2-adic valuation of $H_n - H_{n-m}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences/index.html#padic-val-rat-two-harmonic-sub-piecewise","scope":"NumberTheory/special-numbers/harmonic-numbers/2-adic-valuations-of-harmonic-number-differences","status":"can_prove","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number and let $\\nu_2$ denote the $2$-adic valuation on the rational numbers (for a positive integer it is the exponent of $2$ in its factorization). Let $m$ and $n$ be natural numbers with $1 \\le m \\le n$, and put $M = \\lceil \\log_2(m + 1) \\rceil$ and $j = \\lfloor (n - m) / 2^M \\rfloor$. If $2^M j + m \\le n \\le 2^M (j + 1) - 1$, then\u2026","title":"Piecewise formula for the 2-adic valuation of $H_n - H_{n-m}$"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers.json","id":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers","status":"fully_proved","title":"Binomial identities for harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/alternating-binomial-transform-harmonic","kind":"node","lean":"MetaMathlibExt.alternating_binomial_transform_harmonic_numbers","parent":11602,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/index.html#alternating-binomial-transform-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers","status":"fully_proved","summary":"Let $s$ be a complex number that is not a negative integer, let $n$ be a positive integer, and let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number. For complex $z$ and natural $k$ write $\\binom{z}{k} = \\Gamma(z + 1) / (\\Gamma(k + 1)\\,\\Gamma(z - k + 1))$, with $1/\\Gamma$ taken to be $0$ at the poles of $\\Gamma$. Then","title":"Alternating binomial transform of the harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/binomial-harmonic-convolution","kind":"node","lean":"MetaMathlibExt.binomial_harmonic_convolution","parent":11602,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/index.html#binomial-harmonic-convolution","scope":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers","status":"fully_proved","summary":"Let $H_m = \\sum_{i=1}^{m} 1/i$ denote the $m$-th harmonic number. For natural numbers $k$ and $n$ with $1 \\le k \\le n$,","title":"Binomial-harmonic convolution identity"},{"id":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/binomial-harmonic-interchange-identity","kind":"node","lean":"MetaMathlibExt.binomial_harmonic_interchange_identity","parent":11602,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/index.html#binomial-harmonic-interchange-identity","scope":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero. For every natural number $n$ and every sequence $a : \\mathbb{N} \\to K$,","title":"Binomial interchange identity with harmonic weights"},{"id":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/binomial-transform-second-order-harmonic","kind":"node","lean":"MetaMathlibExt.binomial_transform_second_order_harmonic_numbers","parent":11602,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/index.html#binomial-transform-second-order-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers","status":"fully_proved","summary":"For a natural number $m$ let $H_m = \\sum_{j=1}^{m} 1/j$ and $H_m^{(2)} = \\sum_{j=1}^{m} 1/j^2$, and for $z \\in \\mathbb{C}$ and a natural number $r$ let $\\binom{z}{r} = z(z-1)\\cdots(z-r+1)/r!$, the empty product for $r = 0$ being $1$. Let $s \\in \\mathbb{C}$ with $s \\ne -m$ for every positive integer $m$, and let $n \\ge 1$. Then","title":"Alternating binomial transform of second-order harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/binomial-transform-sq-harmonic","kind":"node","lean":"MetaMathlibExt.binomial_transform_sq_harmonic","parent":11602,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers/index.html#binomial-transform-sq-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/binomial-identities-for-harmonic-numbers","status":"fully_proved","summary":"For a complex number $z$ and $k \\in \\mathbb{N}$ write $\\binom{z}{k} = z(z-1)\\cdots(z-k+1)/k!$ (the empty product being $1$), and let $H_k = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{k}$ be the $k$-th harmonic number. Let $s$ be a complex number that is not a negative integer, and let $n \\ge 1$. Then","title":"Alternating binomial transform of the squared harmonic numbers"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct.json","id":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","status":"can_prove","title":"Generalized harmonic numbers and the Gamma function"},{"id":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/generalized-harmonic","kind":"node","lean":"MetaMathlibExt.generalizedHarmonic","parent":11608,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/index.html#generalized-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","status":"fully_proved","summary":"For a real number $s$ and a natural number $n$, the generalized harmonic number of order $s$ is $H_n^{(s)} = \\sum_{k=1}^{n} 1/k^s$, where $k^s$ is the real power.","title":"Generalized harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/generalized-harmonic-zero","kind":"node","lean":"MetaMathlibExt.generalizedHarmonic_zero","parent":11608,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/index.html#generalized-harmonic-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","status":"fully_proved","summary":"For every real $s$, the generalized harmonic number $H_n^{(s)} = \\sum_{k=1}^{n} 1/k^s$ vanishes at $n = 0$: $H_0^{(s)} = 0$.","title":"The zeroth generalized harmonic number"},{"id":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/generalized-harmonic-succ","kind":"node","lean":"MetaMathlibExt.generalizedHarmonic_succ","parent":11608,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/index.html#generalized-harmonic-succ","scope":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","status":"fully_proved","summary":"For every real $s$ and every natural number $n$, the generalized harmonic numbers $H_n^{(s)} = \\sum_{k=1}^{n} 1/k^s$ satisfy $H_{n+1}^{(s)} = H_n^{(s)} + 1/(n+1)^s$.","title":"Recurrence for generalized harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/generalized-harmonic-eq-gamma-expectatio","kind":"node","lean":"MetaMathlibExt.generalizedHarmonic_eq_gamma_expectation","parent":11608,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/index.html#generalized-harmonic-eq-gamma-expectatio","scope":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","status":"can_prove","summary":"Let $s > 0$ be real, let $H_n^{(s)} = \\sum_{k=1}^{n} 1/k^s$ denote the generalized harmonic numbers, and let $X_s$ be a random variable with the Gamma distribution $\\mathrm{Gamma}(s, 1)$ of shape $s$ and rate $1$. Then for every natural number $n$,","title":"Gamma representation of generalized harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/generalized-harmonic-sub-eq-gamma-expect","kind":"node","lean":"MetaMathlibExt.generalizedHarmonic_sub_eq_gamma_expectation","parent":11608,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/index.html#generalized-harmonic-sub-eq-gamma-expect","scope":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","status":"can_prove","summary":"For real $t$ and a natural number $n$, write $H_n^{(t)} = \\sum_{k=1}^{n} 1/k^t$ for the generalized harmonic number. Let $s > 0$ be real and let $X_s$ be a random variable with the Gamma distribution $\\mathrm{Gamma}(s, 1)$ of shape $s$ and rate $1$. Then for every real $u$ and every natural number $n$,","title":"Shifted Gamma representation of generalized harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/sum-choose-mul-generalized-harmonic","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_generalizedHarmonic","parent":11608,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/index.html#sum-choose-mul-generalized-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","status":"can_prove","summary":"Write $[a, b]$ for the unsigned Stirling number of the first kind, the number of permutations of $a$ elements with exactly $b$ cycles, and, for real $t$ and a natural number $n$, write $H_n^{(t)} = \\sum_{k=1}^{n} 1/k^t$ for the generalized harmonic number. For all natural numbers $m$ and $n$ and every real $\\alpha$,","title":"Binomially weighted sums of generalized harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/stirling-first-generalized-harmonic-eq-g","kind":"node","lean":"MetaMathlibExt.stirlingFirst_generalizedHarmonic_eq_gamma_expectation","parent":11608,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct/index.html#stirling-first-generalized-harmonic-eq-g","scope":"NumberTheory/special-numbers/harmonic-numbers/generalized-harmonic-numbers-and-the-gamma-funct","status":"can_prove","summary":"Let $s > 0$ be real and let $X_s$ be a random variable with the Gamma distribution $\\mathrm{Gamma}(s, 1)$ of shape $s$ and rate $1$. Write $[a, b]$ for the unsigned Stirling number of the first kind, the number of permutations of $a$ elements with exactly $b$ cycles, and, for real $t$ and a natural number $n$, write $H_n^{(t)} = \\sum_{k=1}^{n} 1/k^t$ for the generalized harmonic number. Then for all natural numbers\u2026","title":"Gamma representation of the Stirling combination of generalized harmonic numbers"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta.json","id":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta","status":"fully_proved","title":"Harmonic numbers and the Euler\u2013Mascheroni constant"},{"id":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/log-sub-log-floor-le","kind":"node","lean":"Real.log_sub_log_floor_le","parent":11616,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/index.html#log-sub-log-floor-le","scope":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta","status":"fully_proved","summary":"For every real $x \\ge 1$, the floor $\\lfloor x \\rfloor$ of $x$ satisfies $\\log x - \\log \\lfloor x \\rfloor \\le 1/\\lfloor x \\rfloor$.","title":"Logarithm gap between a real number and its floor"},{"id":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/abs-harmonic-sub-log-sub-euler-mascheron","kind":"node","lean":"Real.abs_harmonic_sub_log_sub_eulerMascheroniConstant_le","parent":11616,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/index.html#abs-harmonic-sub-log-sub-euler-mascheron","scope":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta","status":"fully_proved","summary":"Let $H_n = \\sum_{k=1}^{n} 1/k$ be the $n$-th harmonic number and $\\gamma$ the Euler\u2013Mascheroni constant. For every natural number $n \\ge 1$, $|H_n - \\log n - \\gamma| \\le 1/n$.","title":"Explicit harmonic error bound at integers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/abs-harmonic-floor-sub-log-sub-euler-mas","kind":"node","lean":"Real.abs_harmonic_floor_sub_log_sub_eulerMascheroniConstant_le","parent":11616,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/index.html#abs-harmonic-floor-sub-log-sub-euler-mas","scope":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta","status":"fully_proved","summary":"Let $H_n = \\sum_{k=1}^{n} 1/k$ be the $n$-th harmonic number and $\\gamma$ the Euler\u2013Mascheroni constant. For every real $x \\ge 2$, with $\\lfloor x \\rfloor$ the floor of $x$, $|H_{\\lfloor x \\rfloor} - \\log x - \\gamma| \\le 2/x$.","title":"Explicit harmonic error bound at real arguments"},{"id":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/is-big-o-harmonic-floor-sub-log-sub-eule","kind":"node","lean":"Real.isBigO_harmonic_floor_sub_log_sub_eulerMascheroniConstant","parent":11616,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta/index.html#is-big-o-harmonic-floor-sub-log-sub-eule","scope":"NumberTheory/special-numbers/harmonic-numbers/harmonic-numbers-and-the-euler-mascheroni-consta","status":"fully_proved","summary":"Let $H_k$ denote the $k$-th harmonic number, $\\lfloor x \\rfloor$ the floor of a real number $x$, and $\\gamma$ the Euler\u2013Mascheroni constant. Then $H_{\\lfloor x \\rfloor} - \\log x - \\gamma = O(1/x)$ as $x \\to \\infty$.","title":"Harmonic numbers at the floor track the logarithm"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums.json","id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","title":"Hyperharmonic Euler sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/generalized-hyperharmonic","kind":"node","lean":"MetaMathlibExt.generalizedHyperharmonic","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#generalized-hyperharmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and a complex number $r$, the generalized hyperharmonic number $H_n^{[r]}(k)$ is $0$ when $k > n$; when $k \\le n$ it is the elementary symmetric polynomial of degree $n - k$ in $r, r + 1, \\ldots, r + n - 1$ divided by $n!$, that is,","title":"Generalized hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/generalized-hyperharmonic-of-lt","kind":"node","lean":"MetaMathlibExt.generalizedHyperharmonic_of_lt","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized hyperharmonic numbers vanish for "],["math","\\(k > n\\)"]],"source":"Generalized hyperharmonic numbers vanish for $k > n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#generalized-hyperharmonic-of-lt","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For all natural numbers $n$ and $k$ with $n < k$ and every complex number $r$, the generalized hyperharmonic number $H_n^{[r]}(k)$ equals $0$.","title":"Generalized hyperharmonic numbers vanish for $k > n$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/generalized-hyperharmonic-zero-zero","kind":"node","lean":"MetaMathlibExt.generalizedHyperharmonic_zero_zero","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#generalized-hyperharmonic-zero-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For every complex number $r$, the generalized hyperharmonic number $H_0^{[r]}(0)$ equals $1$.","title":"Generalized hyperharmonic numbers with both indices zero"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/generalized-hyperharmonic-one-zero","kind":"node","lean":"MetaMathlibExt.generalizedHyperharmonic_one_zero","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(H_1^{(r)}(0) = r\\)"]],"source":"The value $H_1^{(r)}(0) = r$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#generalized-hyperharmonic-one-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For every complex number $r$, the generalized hyperharmonic number $H_1^{[r]}(0)$ equals $r$.","title":"The value $H_1^{(r)}(0) = r$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/generalized-hyperharmonic-one-one","kind":"node","lean":"MetaMathlibExt.generalizedHyperharmonic_one_one","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(H_1^{(r)}(1) = 1\\)"]],"source":"The value $H_1^{(r)}(1) = 1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#generalized-hyperharmonic-one-one","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For every complex number $r$, the generalized hyperharmonic number $H_1^{[r]}(1)$ equals $1$.","title":"The value $H_1^{(r)}(1) = 1$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For an integer $k \\ge 0$, complex numbers $r$ and $a$ and an integer $p$, the generalized hyperharmonic Euler sum is $S(k, r; p; a) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + a)^p$, where $H_n^{[r]}(k)$ is the generalized hyperharmonic number. A term whose denominator $(n + a)^p$ vanishes counts as $0$, and $S(k, r; p; a) = 0$ when the series is not summable.","title":"Generalized hyperharmonic Euler sum"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-eq-tsum","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_eq_tsum","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-eq-tsum","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For every integer $k \\ge 0$, complex numbers $r$ and $a$, and integer $p$, write $H_n^{[r]}(k)$ for the generalized hyperharmonic number. Then the hyperharmonic Euler sum $S(k, r; p; a) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + a)^p$ equals $\\sum_{n \\ge 0} H_n^{[r]}(k) / (n + a)^p$, because $H_n^{[r]}(k) = 0$ for $n < k$.","title":"Hyperharmonic Euler sums as sums over all indices"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/odd-multiple-harmonic","kind":"node","lean":"MetaMathlibExt.oddMultipleHarmonic","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#odd-multiple-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the depth-$k$ odd harmonic number is the real number","title":"Odd multiple harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/odd-multiple-harmonic-zero","kind":"node","lean":"MetaMathlibExt.oddMultipleHarmonic_zero","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#odd-multiple-harmonic-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For every natural number $n$, the depth-zero odd harmonic number $O_n(0)$ equals $1$.","title":"Odd multiple harmonic numbers of depth zero"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/odd-multiple-harmonic-of-lt","kind":"node","lean":"MetaMathlibExt.oddMultipleHarmonic_of_lt","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#odd-multiple-harmonic-of-lt","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For natural numbers $k$ and $n$ with $k > n$, the depth-$k$ odd harmonic number $O_n(k)$ equals $0$.","title":"Vanishing of odd multiple harmonic numbers of large depth"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/odd-multiple-harmonic-one-one","kind":"node","lean":"MetaMathlibExt.oddMultipleHarmonic_one_one","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#odd-multiple-harmonic-one-one","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"The depth-one odd harmonic number $O_1(1)$ equals $2$.","title":"The first odd harmonic number"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/euler-sum-period-algebra","kind":"node","lean":"MetaMathlibExt.eulerSumPeriodAlgebra","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","The algebra generated by "],["math","\\(\\log 2\\)"],["text",", "],["math","\\(\\pi^2\\)"],["text"," and odd zeta values"]],"source":"The algebra generated by $\\log 2$, $\\pi^2$ and odd zeta values"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#euler-sum-period-algebra","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"This is the $\\mathbb{Q}$-subalgebra of $\\mathbb{C}$ generated by $\\log 2$, $\\pi^2$ and the zeta values $\\zeta(3), \\zeta(5), \\zeta(7), \\dots$ (that is, $\\zeta(2j+1)$ for all integers $j \\ge 1$). It is written $\\mathbb{Q}[\\log 2, \\pi^2, \\zeta(3), \\zeta(5), \\dots]$.","title":"The algebra generated by $\\log 2$, $\\pi^2$ and odd zeta values"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/generators-mem-euler-sum-period-algebra","kind":"node","lean":"MetaMathlibExt.generators_mem_eulerSumPeriodAlgebra","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#generators-mem-euler-sum-period-algebra","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"The numbers $\\log 2$ and $\\pi^2$ and the odd zeta values $\\zeta(2j + 3)$ for all integers $j \\ge 0$ lie in the $\\mathbb{Q}$-subalgebra $\\mathbb{Q}[\\log 2, \\pi^2, \\zeta(2j+1)_{j \\ge 1}]$ of $\\mathbb{C}$ that they generate.","title":"Generators of the Euler sum period algebra"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/euler-sum-beta-period-algebra","kind":"node","lean":"MetaMathlibExt.eulerSumBetaPeriodAlgebra","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","The algebra generated by "],["math","\\(\\log 2\\)"],["text",", "],["math","\\(\\pi\\)"],["text",", odd zeta values and even beta values"]],"source":"The algebra generated by $\\log 2$, $\\pi$, odd zeta values and even beta values"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#euler-sum-beta-period-algebra","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"For integers $s \\ge 2$ let $\\beta(s) = \\sum_{n \\ge 0} (-1)^n / (2n+1)^s$ be the value of the Dirichlet beta function. This is the $\\mathbb{Q}$-subalgebra of $\\mathbb{C}$ generated by $\\log 2$, $\\pi$, the zeta values $\\zeta(2j+1)$ for all integers $j \\ge 1$ and the beta values $\\beta(2j)$ for all integers $j \\ge 1$. It is written $\\mathbb{Q}[\\log 2, \\pi, \\zeta(2j+1)_{j \\ge 1}, \\beta(2j)_{j \\ge 1}]$.","title":"The algebra generated by $\\log 2$, $\\pi$, odd zeta values and even beta values"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/generators-mem-euler-sum-beta-period-alg","kind":"node","lean":"MetaMathlibExt.generators_mem_eulerSumBetaPeriodAlgebra","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#generators-mem-euler-sum-beta-period-alg","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"fully_proved","summary":"The numbers $\\log 2$ and $\\pi$, the zeta values $\\zeta(2j + 3)$ for all integers $j \\ge 0$, and the Dirichlet beta values $\\beta(2j + 2) = \\sum_{n \\ge 0} (-1)^n / (2n + 1)^{2j+2}$ for all integers $j \\ge 0$ lie in the $\\mathbb{Q}$-subalgebra $\\mathbb{Q}[\\log 2, \\pi, \\zeta(2j+1)_{j \\ge 1}, \\beta(2j)_{j \\ge 1}]$ of $\\mathbb{C}$ that they generate.","title":"Generators of the Euler sum beta period algebra"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-eq-iterated-deri","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_eq_iteratedDeriv_beta","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-eq-iterated-deri","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k, p \\ge 0$ be integers and let $r, a$ be complex numbers with $\\operatorname{Re}(r) < p + 1$ and $\\operatorname{Re}(a) > -k$, such that $r$ is not a positive integer and $a$ is not a nonpositive integer. Write $H_n^{[r]}(k)$ for the generalized hyperharmonic number and let $S(k, r; p + 1; a) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + a)^{p+1}$. Then","title":"Beta-function evaluation of hyperharmonic Euler sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-mem-adjoin-digam","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_mem_adjoin_digamma","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-mem-adjoin-digam","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k \\ge 0$, $r$ and $p$ be integers with $1 \\le r \\le p$, and let $a \\in \\mathbb{C} \\setminus \\mathbb{Z}$. Write $H_n^{[r]}(k)$ for the generalized hyperharmonic number. Then the series $S(k, r; p + 1; a) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + a)^{p+1}$ converges absolutely, and its sum lies in the algebra generated over the field $\\mathbb{Q}(a)$ by $\\psi(a) + \\gamma$, the derivatives $\\psi^{(j)}(a)$ for $j \\ge 1$\u2026","title":"Polygamma structure of hyperharmonic Euler sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-rat-mem-pi-mul-a","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_rat_mem_pi_mul_adjoin","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-rat-mem-pi-mul-a","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $a$ and $r$ be rational numbers that are not integers, with $a - r \\in \\mathbb{Z}$, let $N \\ge 1$ be an integer with $N a \\in \\mathbb{Z}$, and let $k \\ge 0$ and $p > r$ be integers. Put $M = \\mathrm{lcm}(2N, 4)$ and $\\omega = e^{2 \\pi i / M}$, a primitive $M$-th root of unity. Write $H_n^{[r]}(k)$ for the generalized hyperharmonic numbers. Then the generalized hyperharmonic Euler sum\u2026","title":"Hyperharmonic Euler sums with rational parameters"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-duality","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_duality","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-duality","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k, p \\ge 0$ be integers and let $r, t$ be complex numbers with $\\operatorname{Re}(r) < p + 1$ and $\\operatorname{Re}(t) < k + 1$. Write $H_n^{[r]}(k)$ for the generalized hyperharmonic number. Then the series $S(k, r; p + 1; 1 - t) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + 1 - t)^{p+1}$ and $S(p, t; k + 1; 1 - r) = \\sum_{n \\ge p} H_n^{[t]}(p) / (n + 1 - r)^{k+1}$ both converge absolutely, and\u2026","title":"Duality for hyperharmonic Euler sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-half-mem-euler-s","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_half_mem_eulerSumPeriodAlgebra","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-half-mem-euler-s","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k \\ge 0$, $r$ and $p$ be integers with $1 \\le r < p$, and let $a \\in \\frac{1}{2} + \\mathbb{Z}$. Write $H_n^{[r]}(k)$ for the generalized hyperharmonic number. Then the series $S(k, r; p; a) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + a)^p$ converges absolutely, and its sum lies in the $\\mathbb{Q}$-subalgebra $\\mathbb{Q}[\\log 2, \\pi^2, \\zeta(2j+1)_{j \\ge 1}]$ of $\\mathbb{C}$ generated by $\\log 2$, $\\pi^2$ and the odd\u2026","title":"Hyperharmonic Euler sums with integer parameter and half-integer shift"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-odd-multiple-harmonic-sum","kind":"node","lean":"MetaMathlibExt.centralBinom_oddMultipleHarmonic_sum_mem_eulerSumPeriodAlgebra","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-odd-multiple-harmonic-sum","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k \\ge 0$ and $p > 0$ be integers, let $a$ be an integer with $a > -k$, and let $O_n(k)$ denote the depth-$k$ odd harmonic number. Then the series of real numbers","title":"Odd harmonic central binomial sums at integer shifts"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-odd-multiple-harmonic-div","kind":"node","lean":"MetaMathlibExt.centralBinom_oddMultipleHarmonic_div_succ_sq_tsum","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial odd harmonic sum with denominator "],["math","\\(4^n (n + 1)^2\\)"]],"source":"Central binomial odd harmonic sum with denominator $4^n (n + 1)^2$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-odd-multiple-harmonic-div","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"For every integer $k \\ge 0$, with $O_n(k)$ the depth-$k$ odd harmonic number,","title":"Central binomial odd harmonic sum with denominator $4^n (n + 1)^2$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/choose-central-binom-odd-multiple-harmon","kind":"node","lean":"MetaMathlibExt.choose_centralBinom_oddMultipleHarmonic_sum_mem_eulerSumPeriodAlgebra","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd harmonic central binomial sums with weight "],["math","\\(\\binom{n}{m}\\)"],["text"," at integer shifts"]],"source":"Odd harmonic central binomial sums with weight $\\binom{n}{m}$ at integer shifts"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#choose-central-binom-odd-multiple-harmon","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k, m \\ge 0$ and $p > m$ be integers, let $a$ be an integer with $a > -k$, and let $O_n(k)$ denote the depth-$k$ odd harmonic number. Then the series of real numbers","title":"Odd harmonic central binomial sums with weight $\\binom{n}{m}$ at integer shifts"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-odd-multiple-harmonic-div-2","kind":"node","lean":"MetaMathlibExt.centralBinom_oddMultipleHarmonic_div_choose_sum_mem_eulerSumPeriodAlgebra","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial odd harmonic sums over "],["math","\\(\\binom{n+m}{m}\\)"],["text"," lie in the period algebra"]],"source":"Central binomial odd harmonic sums over $\\binom{n+m}{m}$ lie in the period algebra"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-odd-multiple-harmonic-div-2","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $O_n(k)$ denote the depth-$k$ odd harmonic number and let $A = \\mathbb{Q}[\\log 2, \\pi^2, \\zeta(3), \\zeta(5), \\ldots] \\subseteq \\mathbb{C}$ be the Euler sum period algebra. For integers $k \\ge 0$, $p \\ge 1$, $m \\ge 1$ and $a > -k$, the series","title":"Central binomial odd harmonic sums over $\\binom{n+m}{m}$ lie in the period algebra"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/catalan-odd-multiple-harmonic-div-tsum","kind":"node","lean":"MetaMathlibExt.catalan_oddMultipleHarmonic_div_tsum","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan\u2013odd harmonic sum with denominator "],["math","\\(4^n n\\)"]],"source":"Catalan\u2013odd harmonic sum with denominator $4^n n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#catalan-odd-multiple-harmonic-div-tsum","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k \\ge 1$ be an integer, let $C_n$ denote the $n$-th Catalan number and $O_n(k)$ the depth-$k$ odd harmonic number. Then","title":"Catalan\u2013odd harmonic sum with denominator $4^n n$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-quarter-mem-eule","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_quarter_mem_eulerSumBetaPeriodAlgebra","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-quarter-mem-eule","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k \\ge 0$ and $0 \\le r < p$ be integers, and let $a$ be a complex number with $a \\in \\tfrac14 + \\mathbb{Z}$ or $a \\in \\tfrac34 + \\mathbb{Z}$. Write $H_n^{[r]}(k)$ for the generalized hyperharmonic numbers. Then the generalized hyperharmonic Euler sum $S(k, r; p; a) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + a)^p$ converges, and its value lies in the $\\mathbb{Q}$-algebra\u2026","title":"Quarter-shift hyperharmonic Euler sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-half-half-mem-pi","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_half_half_mem_pi_mul","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-half-half-mem-pi","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k \\ge 0$ be an integer, let $r, a \\in \\frac{1}{2} + \\mathbb{Z}$, and let $p$ be an integer with $p > r$. Write $H_n^{[r]}(k)$ for the generalized hyperharmonic number. Then the series $S(k, r; p; a) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + a)^p$ converges absolutely, and its sum equals $\\pi x$ for some $x$ in the $\\mathbb{Q}$-subalgebra $\\mathbb{Q}[\\log 2, \\pi^2, \\zeta(2j+1)_{j \\ge 1}]$ of $\\mathbb{C}$ generated\u2026","title":"Hyperharmonic Euler sums with half-integer parameter and shift"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/choose-central-binom-odd-multiple-harmon-2","kind":"node","lean":"MetaMathlibExt.choose_centralBinom_oddMultipleHarmonic_half_sum_mem_pi_mul","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd harmonic central binomial sums with weight "],["math","\\(\\binom{n}{m}\\)"],["text"," at half-integer shifts"]],"source":"Odd harmonic central binomial sums with weight $\\binom{n}{m}$ at half-integer shifts"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#choose-central-binom-odd-multiple-harmon-2","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k, m \\ge 0$ and $p > m$ be integers, let $a = d + 1/2$ with $d$ an integer, and let $O_n(k)$ denote the depth-$k$ odd harmonic number. Then the series of real numbers","title":"Odd harmonic central binomial sums with weight $\\binom{n}{m}$ at half-integer shifts"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-odd-multiple-harmonic-div-3","kind":"node","lean":"MetaMathlibExt.centralBinom_oddMultipleHarmonic_div_choose_half_sum_mem","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial odd harmonic sums over "],["math","\\(\\binom{n+m}{m}\\)"],["text"," at half-integer shifts"]],"source":"Central binomial odd harmonic sums over $\\binom{n+m}{m}$ at half-integer shifts"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-odd-multiple-harmonic-div-3","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $O_n(k)$ denote the depth-$k$ odd harmonic number and let $A = \\mathbb{Q}[\\log 2, \\pi^2, \\zeta(3), \\zeta(5), \\ldots] \\subseteq \\mathbb{C}$ be the Euler sum period algebra. For integers $k \\ge 0$, $p \\ge 1$, $m \\ge 1$ and $d$, put $a = d + 1/2$. Then the series","title":"Central binomial odd harmonic sums over $\\binom{n+m}{m}$ at half-integer shifts"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/catalan-odd-multiple-harmonic-half-sum-s","kind":"node","lean":"MetaMathlibExt.catalan_oddMultipleHarmonic_half_sum_sub_mem_pi_mul","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan\u2013odd harmonic sums at the shift "],["math","\\(1/2\\)"]],"source":"Catalan\u2013odd harmonic sums at the shift $1/2$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#catalan-odd-multiple-harmonic-half-sum-s","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k \\ge 0$ and $p \\ge 1$ be integers, let $C_n$ denote the $n$-th Catalan number and $O_n(k)$ the depth-$k$ odd harmonic number, and let $A = \\mathbb{Q}[\\log 2, \\pi^2, \\zeta(3), \\zeta(5), \\ldots] \\subseteq \\mathbb{C}$ be the Euler sum period algebra. Then the series","title":"Catalan\u2013odd harmonic sums at the shift $1/2$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-div-choose-half-sq-tsum","kind":"node","lean":"MetaMathlibExt.centralBinom_div_choose_half_sq_tsum","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial sum over "],["math","\\(\\binom{n+m}{m} (n + 1/2)^2\\)"]],"source":"Central binomial sum over $\\binom{n+m}{m} (n + 1/2)^2$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-div-choose-half-sq-tsum","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $m \\ge 1$ be an integer and write $O_m = O_m(1) = \\sum_{i=1}^{m} 1/(i - 1/2)$ for the depth-one odd harmonic number. Then","title":"Central binomial sum over $\\binom{n+m}{m} (n + 1/2)^2$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-odd-harmonic-sub-div-choos","kind":"node","lean":"MetaMathlibExt.centralBinom_oddHarmonic_sub_div_choose_half_sq_tsum","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd harmonic central binomial sum over "],["math","\\(\\binom{n+m}{m} (n + 1/2)^2\\)"]],"source":"Odd harmonic central binomial sum over $\\binom{n+m}{m} (n + 1/2)^2$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-odd-harmonic-sub-div-choos","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $m \\ge 1$ be an integer and write $O_j = O_j(1) = \\sum_{i=1}^{j} 1/(i - 1/2)$ for the depth-one odd harmonic number. Then","title":"Odd harmonic central binomial sum over $\\binom{n+m}{m} (n + 1/2)^2$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/hyperharmonic-euler-sum-quarter-mul-mem","kind":"node","lean":"MetaMathlibExt.hyperharmonicEulerSum_quarter_mul_mem_pi_mul","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#hyperharmonic-euler-sum-quarter-mul-mem","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k_1, k_2 \\ge 0$ be integers, let $r_1, r_2 \\in \\tfrac12 + \\mathbb{Z}$, $a \\in \\tfrac14 + \\mathbb{Z}$ and $b \\in \\tfrac34 + \\mathbb{Z}$, and let $p_1, p_2$ be integers with $p_1 > r_1$ and $p_2 > r_2$. Write $H_n^{[r]}(k)$ for the generalized hyperharmonic numbers and $S(k, r; p; a) = \\sum_{n \\ge k} H_n^{[r]}(k) / (n + a)^p$ for the generalized hyperharmonic Euler sum. Then the series $S(k_1, r_1; p_1; a)$ and\u2026","title":"Products of quarter-shift hyperharmonic Euler sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-odd-multiple-harmonic-quar","kind":"node","lean":"MetaMathlibExt.centralBinom_oddMultipleHarmonic_quarter_mul_mem_pi_mul","parent":11621,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-odd-multiple-harmonic-quar","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $O_n(k)$ denote the depth-$k$ odd harmonic number, let $\\beta(s) = \\sum_{n \\ge 0} (-1)^n / (2n+1)^s$ be the Dirichlet beta function, and let $B = \\mathbb{Q}[\\log 2, \\pi, \\zeta(3), \\zeta(5), \\ldots, \\beta(2), \\beta(4), \\ldots] \\subseteq \\mathbb{C}$. For integers $k_1, k_2 \\ge 0$ and $p_1, p_2 \\ge 1$, the series","title":"Products of quarter-shift central binomial odd harmonic sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/choose-central-binom-odd-multiple-harmon-3","kind":"node","lean":"MetaMathlibExt.choose_centralBinom_oddMultipleHarmonic_quarter_sum_mem","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd harmonic central binomial sums with weight "],["math","\\(\\binom{n}{m}\\)"],["text"," at shift "],["math","\\(1/4\\)"]],"source":"Odd harmonic central binomial sums with weight $\\binom{n}{m}$ at shift $1/4$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#choose-central-binom-odd-multiple-harmon-3","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k, m \\ge 0$ and $p > m$ be integers, let $O_n(k)$ denote the depth-$k$ odd harmonic number, let $\\beta$ be the Dirichlet beta function, and let","title":"Odd harmonic central binomial sums with weight $\\binom{n}{m}$ at shift $1/4$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/choose-central-binom-odd-multiple-harmon-4","kind":"node","lean":"MetaMathlibExt.choose_centralBinom_oddMultipleHarmonic_three_quarter_sum_mem","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd harmonic central binomial sums with weight "],["math","\\(\\binom{n}{m}\\)"],["text"," at shift "],["math","\\(3/4\\)"]],"source":"Odd harmonic central binomial sums with weight $\\binom{n}{m}$ at shift $3/4$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#choose-central-binom-odd-multiple-harmon-4","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $k, m \\ge 0$ and $p > m$ be integers, let $O_n(k)$ denote the depth-$k$ odd harmonic number, let $\\beta$ be the Dirichlet beta function, and let","title":"Odd harmonic central binomial sums with weight $\\binom{n}{m}$ at shift $3/4$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-odd-multiple-harmonic-div-4","kind":"node","lean":"MetaMathlibExt.centralBinom_oddMultipleHarmonic_div_choose_quarter_sum_mem","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial odd harmonic sums over "],["math","\\(\\binom{n+m}{m}\\)"],["text"," at the shift "],["math","\\(1/4\\)"]],"source":"Central binomial odd harmonic sums over $\\binom{n+m}{m}$ at the shift $1/4$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-odd-multiple-harmonic-div-4","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $O_n(k)$ denote the depth-$k$ odd harmonic number, let $\\beta(s) = \\sum_{n \\ge 0} (-1)^n / (2n+1)^s$ be the Dirichlet beta function, let $B = \\mathbb{Q}[\\log 2, \\pi, \\zeta(3), \\zeta(5), \\ldots, \\beta(2), \\beta(4), \\ldots] \\subseteq \\mathbb{C}$, and let $\\varpi = 2 \\int_0^1 dt / \\sqrt{1 - t^4}$ be the lemniscate constant. For integers $k \\ge 0$, $p \\ge 1$ and $m \\ge 1$, the series","title":"Central binomial odd harmonic sums over $\\binom{n+m}{m}$ at the shift $1/4$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/central-binom-odd-multiple-harmonic-div-5","kind":"node","lean":"MetaMathlibExt.centralBinom_oddMultipleHarmonic_div_choose_three_quarter_sum_mem","parent":11621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial odd harmonic sums over "],["math","\\(\\binom{n+m}{m}\\)"],["text"," at the shift "],["math","\\(3/4\\)"]],"source":"Central binomial odd harmonic sums over $\\binom{n+m}{m}$ at the shift $3/4$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums/index.html#central-binom-odd-multiple-harmonic-div-5","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-euler-sums","status":"can_prove","summary":"Let $O_n(k)$ denote the depth-$k$ odd harmonic number, let $\\beta(s) = \\sum_{n \\ge 0} (-1)^n / (2n+1)^s$ be the Dirichlet beta function, let $B = \\mathbb{Q}[\\log 2, \\pi, \\zeta(3), \\zeta(5), \\ldots, \\beta(2), \\beta(4), \\ldots] \\subseteq \\mathbb{C}$, and let $\\varpi = 2 \\int_0^1 dt / \\sqrt{1 - t^4}$ be the lemniscate constant. For integers $k \\ge 0$, $p \\ge 1$ and $m \\ge 1$, the series","title":"Central binomial odd harmonic sums over $\\binom{n+m}{m}$ at the shift $3/4$"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes.json","id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","title":"Hyperharmonic numbers and sums over primes"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/hyperharmonic","kind":"node","lean":"MetaMathlibExt.hyperharmonic","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#hyperharmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"The hyperharmonic numbers $h_n^{(r)} \\in \\mathbb{Q}$ of order $r$ and index $n$, for natural numbers $r$ and $n$, are defined by recursion on $r$: $h_n^{(0)} = 1/n$ for $n \\ge 1$, $h_0^{(0)} = 0$, and","title":"Hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/hyperharmonic-succ","kind":"node","lean":"MetaMathlibExt.hyperharmonic_succ","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#hyperharmonic-succ","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For all natural numbers $r$ and $n$, the hyperharmonic numbers $h_m^{(s)}$ (of order $s$ and index $m$) satisfy","title":"Recursion for the hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/hyperharmonic-one","kind":"node","lean":"MetaMathlibExt.hyperharmonic_one","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#hyperharmonic-one","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For every natural number $n$, the hyperharmonic number $h_n^{(1)}$ of order $1$ and index $n$ equals the $n$-th harmonic number $H_n = \\sum_{k=1}^{n} 1/k$ (in particular $h_0^{(1)} = H_0 = 0$).","title":"Hyperharmonic numbers of order one are the harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/hyperharmonic-two-two","kind":"node","lean":"MetaMathlibExt.hyperharmonic_two_two","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","The hyperharmonic number "],["math","\\(h_2^{(2)}\\)"]],"source":"The hyperharmonic number $h_2^{(2)}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#hyperharmonic-two-two","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"The hyperharmonic number of order $2$ and index $2$ is $h_2^{(2)} = 5/2$.","title":"The hyperharmonic number $h_2^{(2)}$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/hyperharmonic-is-equivalent","kind":"node","lean":"MetaMathlibExt.hyperharmonic_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#hyperharmonic-is-equivalent","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"For every natural number $r \\ge 1$, the hyperharmonic numbers $h_n^{(r)}$ of order $r$ are asymptotically equivalent, as $n \\to \\infty$, to $n^{r-1} \\log n / (r-1)!$, where $\\log$ is the natural logarithm:","title":"Asymptotic of the hyperharmonic numbers (Mez\u0151)"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/sum-is-equivalent-of-is-equivalent","kind":"node","lean":"MetaMathlibExt.sum_isEquivalent_of_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#sum-is-equivalent-of-is-equivalent","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"Let $(a_i)_{i \\ge 0}$ and $(b_i)_{i \\ge 0}$ be real sequences with $a_i > 0$ and $b_i > 0$ for every $i \\ge 1$. If $a_i \\sim b_i$ as $i \\to \\infty$ and $\\sum_{i=1}^{n} b_i \\to +\\infty$ as $n \\to \\infty$, then","title":"Jakimczuk's lemma on sums of asymptotically equivalent terms"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/nth-k-prime-factors-is-equivalent","kind":"node","lean":"MetaMathlibExt.nth_kPrimeFactors_isEquivalent","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotics of the "],["math","\\(n\\)"],["text","-th number with "],["math","\\(k\\)"],["text"," prime factors"]],"source":"Asymptotics of the $n$-th number with $k$ prime factors"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#nth-k-prime-factors-is-equivalent","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"Let $k \\ge 1$. For $n \\ge 1$ let $p_{n,k}$ be the $n$-th squarefree natural number with exactly $k$ distinct prime factors, and let $q_{n,k}$ be the $n$-th natural number with exactly $k$ prime factors counted with multiplicity (that is, $\\Omega(q_{n,k}) = k$). Then, as $n \\to \\infty$, $p_{n,k} \\sim q_{n,k} \\sim (k-1)!\\, n \\log n / (\\log \\log n)^{k-1}$ and\u2026","title":"Asymptotics of the $n$-th number with $k$ prime factors"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/sum-pow-mul-log-pow-is-equivalent","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_log_pow_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#sum-pow-mul-log-pow-is-equivalent","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"For all $m, n, k \\in \\mathbb{N}$, as the natural number $x$ tends to infinity,","title":"Asymptotics of power-logarithm sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-generalized-harmonic","kind":"node","lean":"MetaMathlibExt.alternatingGeneralizedHarmonic","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-generalized-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For a real number $s$ and a natural number $n$, the alternating generalized harmonic number is $\\overline{H}_n^{(s)} = \\sum_{j=1}^{n} (-1)^{j-1} / j^s$, which is $0$ for $n = 0$.","title":"Alternating generalized harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-generalized-harmonic-zero","kind":"node","lean":"MetaMathlibExt.alternatingGeneralizedHarmonic_zero","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-generalized-harmonic-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For every real number $s$, the alternating generalized harmonic number $\\overline{H}_0^{(s)}$ is $0$.","title":"Alternating generalized harmonic numbers of index zero"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/signed-iterated-partial-sum","kind":"node","lean":"MetaMathlibExt.signedIteratedPartialSum","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#signed-iterated-partial-sum","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"Let $(b_n)_{n \\ge 0}$ be a real sequence and fix a choice of signs, either alternating, $\\varepsilon_k = (-1)^{k-1}$, or constant, $\\varepsilon_k = 1$. The signed iterated partial sums $S^{(r)}_n$ of $b$, for orders $r \\in \\mathbb{N}$ and $n \\in \\mathbb{N}$, are defined by $S^{(0)}_n = 0$, $S^{(1)}_n = b_n$ and, for $r \\ge 1$,","title":"Signed iterated partial sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/signed-iterated-partial-sum-one","kind":"node","lean":"MetaMathlibExt.signedIteratedPartialSum_one","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#signed-iterated-partial-sum-one","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For every real sequence $(b_n)_{n \\ge 0}$, either choice of signs and every $n \\in \\mathbb{N}$, the signed iterated partial sum of $b$ of order $1$ at $n$ equals $b_n$.","title":"Order one of the signed iterated partial sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/signed-iterated-partial-sum-succ","kind":"node","lean":"MetaMathlibExt.signedIteratedPartialSum_succ","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#signed-iterated-partial-sum-succ","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"Let $(b_n)_{n \\ge 0}$ be a real sequence, fix a choice of signs, either $\\varepsilon_k = (-1)^{k-1}$ for all $k$ or $\\varepsilon_k = 1$ for all $k$, and let $S^{(r)}_n$ denote the signed iterated partial sum of $b$ of order $r$ at $n$ for this choice. Then for every $r \\ge 1$ and every $n \\in \\mathbb{N}$,","title":"Recursion for the signed iterated partial sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-zeta","kind":"node","lean":"MetaMathlibExt.alternatingZeta","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-zeta","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For a natural number $p$, the alternating zeta value $\\bar\\zeta(p)$ is defined by $\\bar\\zeta(1) = \\log 2$, by $\\bar\\zeta(p) = (1 - 2^{1-p})\\,\\zeta(p)$ for $p \\ge 2$, where $\\zeta(p) = \\sum_{n \\ge 1} 1/n^p$, and by $\\bar\\zeta(0) = 0$.","title":"Alternating zeta values"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-zeta-one","kind":"node","lean":"MetaMathlibExt.alternatingZeta_one","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-zeta-one","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"The alternating zeta value at $1$ is $\\bar\\zeta(1) = \\log 2$.","title":"Alternating zeta value at one"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-a","kind":"node","lean":"MetaMathlibExt.liCoeffA","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficients "],["math","\\(a(r, m, j)\\)"]],"source":"Li's coefficients $a(r, m, j)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-a","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"The rational numbers $a(r, m, j)$, for natural numbers $r$, $m$, $j$, are defined by recursion on $r$. For all $m$ and $j$, $a(0, m, j) = 0$; moreover $a(1, 0, 0) = 1$ and $a(1, m, j) = 0$ for $(m, j) \\ne (0, 0)$. For $r \\ge 1$ the values of order $r + 1$ are as follows. First,","title":"Li's coefficients $a(r, m, j)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-a-one-zero-zero","kind":"node","lean":"MetaMathlibExt.liCoeffA_one_zero_zero","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient "],["math","\\(a(1, 0, 0)\\)"]],"source":"The coefficient $a(1, 0, 0)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-a-one-zero-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"The coefficient $a(1, 0, 0)$ of Li's Lemma 3 equals $1$.","title":"The coefficient $a(1, 0, 0)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-a-two-one-zero","kind":"node","lean":"MetaMathlibExt.liCoeffA_two_one_zero","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient "],["math","\\(a(2, 1, 0)\\)"]],"source":"The coefficient $a(2, 1, 0)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-a-two-one-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"The coefficient $a(2, 1, 0)$ of Li's Lemma 3 equals $-1$.","title":"The coefficient $a(2, 1, 0)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-c","kind":"node","lean":"MetaMathlibExt.liCoeffC","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficients "],["math","\\(c(m, j)\\)"]],"source":"Li's coefficients $c(m, j)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-c","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For natural numbers $m$ and $j$, the rational number $c(m, j)$ is $\\frac{1}{m+1} \\binom{m+1}{m+1-j} B_{m+1-j}$ when $j \\le m + 1$ and $0$ otherwise, where $B_i$ denotes the Bernoulli numbers with the convention $B_1 = +\\frac{1}{2}$.","title":"Li's coefficients $c(m, j)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-c-zero-zero","kind":"node","lean":"MetaMathlibExt.liCoeffC_zero_zero","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficient "],["math","\\(c(0, 0)\\)"]],"source":"Li's coefficient $c(0, 0)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-c-zero-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"Li's coefficient $c(0, 0)$ equals $\\frac{1}{2}$ (this is the Bernoulli number $B_1$ in the convention $B_1 = +\\frac{1}{2}$).","title":"Li's coefficient $c(0, 0)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-d","kind":"node","lean":"MetaMathlibExt.liCoeffD","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficients "],["math","\\(d(m, j)\\)"]],"source":"Li's coefficients $d(m, j)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-d","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For natural numbers $m$ and $j$, the rational number $d(m, j)$ is $\\frac{1}{m+1} \\sum_{k} \\binom{m+1}{m-k} B_{m-k} \\binom{1+k}{j} (-1)^{1+k-j}$, the sum running over the natural numbers $k$ with $\\max(j - 1, 0) \\le k \\le m$ (so $d(m, j) = 0$ when $j > m + 1$), where $B_i$ denotes the Bernoulli numbers with the convention $B_1 = +\\frac{1}{2}$.","title":"Li's coefficients $d(m, j)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-d-zero-one","kind":"node","lean":"MetaMathlibExt.liCoeffD_zero_one","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficient "],["math","\\(d(0, 1)\\)"]],"source":"Li's coefficient $d(0, 1)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-d-zero-one","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"Li's coefficient $d(0, 1)$ equals $1$.","title":"Li's coefficient $d(0, 1)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-c1","kind":"node","lean":"MetaMathlibExt.liCoeffC1","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficients "],["math","\\(c_1(m, j)\\)"]],"source":"Li's coefficients $c_1(m, j)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-c1","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For natural numbers $m$ and $j$, the rational number $c_1(m, j)$ is $\\frac{1}{2(m+1)} \\sum_{k=0}^{m-j} \\binom{m+1}{k} B_k 2^k \\binom{m+1-k}{j} (-1)^{m-k-j}$ when $j \\le m$ and $0$ otherwise, where $B_i$ denotes the Bernoulli numbers with the convention $B_1 = +\\frac{1}{2}$.","title":"Li's coefficients $c_1(m, j)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-c1-zero-zero","kind":"node","lean":"MetaMathlibExt.liCoeffC1_zero_zero","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficient "],["math","\\(c_1(0, 0)\\)"]],"source":"Li's coefficient $c_1(0, 0)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-c1-zero-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"Li's coefficient $c_1(0, 0)$ equals $\\frac{1}{2}$.","title":"Li's coefficient $c_1(0, 0)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-d1","kind":"node","lean":"MetaMathlibExt.liCoeffD1","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficients "],["math","\\(d_1(m, j)\\)"]],"source":"Li's coefficients $d_1(m, j)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-d1","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"For natural numbers $m$ and $j$, $d_1(m, j) = \\sum_{k=j}^{m} \\binom{k}{j} (-1)^{k-j} c_1(m, k)$, where $c_1$ denotes Li's coefficients $c_1(m, k)$; in particular $d_1(m, j) = 0$ when $j > m$.","title":"Li's coefficients $d_1(m, j)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-d1-zero-zero","kind":"node","lean":"MetaMathlibExt.liCoeffD1_zero_zero","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficient "],["math","\\(d_1(0, 0)\\)"]],"source":"Li's coefficient $d_1(0, 0)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-d1-zero-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"Li's coefficient $d_1(0, 0)$ equals $\\frac{1}{2}$.","title":"Li's coefficient $d_1(0, 0)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-b1","kind":"node","lean":"MetaMathlibExt.liCoeffB1","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Li's coefficients "],["math","\\(b_1(r, m, j, k)\\)"]],"source":"Li's coefficients $b_1(r, m, j, k)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-b1","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"These rational numbers, indexed by natural numbers $r, m, j, k$, are defined by recursion on $r$ from Li's coefficients $c$, $d$, $c_1$ and $d_1$. All values $b_1(0, m, j, k)$ are $0$, $b_1(1, 0, 0, 2) = 1$, and all other values $b_1(1, m, j, k)$ are $0$. For $s \\ge 1$ let $G = \\lfloor (s-1)/2 \\rfloor$ and $b(m, j, k) = b_1(s, m, j, k)$; then $b_1(s+1, m, j, k)$ is $0$ except in the following cases.","title":"Li's coefficients $b_1(r, m, j, k)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-b1-one-zero-zero-two","kind":"node","lean":"MetaMathlibExt.liCoeffB1_one_zero_zero_two","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient "],["math","\\(b_1(1, 0, 0, 2)\\)"]],"source":"The coefficient $b_1(1, 0, 0, 2)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-b1-one-zero-zero-two","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"The coefficient $b_1(1, 0, 0, 2)$ of Li's Definition 5 equals $1$.","title":"The coefficient $b_1(1, 0, 0, 2)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/li-coeff-b1-one-zero-zero-three","kind":"node","lean":"MetaMathlibExt.liCoeffB1_one_zero_zero_three","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient "],["math","\\(b_1(1, 0, 0, 3)\\)"]],"source":"The coefficient $b_1(1, 0, 0, 3)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#li-coeff-b1-one-zero-zero-three","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"fully_proved","summary":"The coefficient $b_1(1, 0, 0, 3)$ of Li's Definition 5 equals $0$.","title":"The coefficient $b_1(1, 0, 0, 3)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/generalized-hyperharmonic-is-equivalent","kind":"node","lean":"MetaMathlibExt.generalizedHyperharmonic_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#generalized-hyperharmonic-is-equivalent","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"For a natural number $p$ let $H_n^{(p)} = \\sum_{i=1}^{n} 1/i^p$ be the generalized harmonic numbers, and let the generalized hyperharmonic numbers be $H_n^{(p,1)} = H_n^{(p)}$ and $H_n^{(p,r+1)} = \\sum_{k=1}^{n} H_k^{(p,r)}$. Then for all natural numbers $p \\ge 2$ and $r \\ge 1$, $H_n^{(p,r)} \\sim n^{r-1} \\zeta(p) / (r-1)!$ as $n \\to \\infty$, where $\\zeta(p) = \\sum_{i \\ge 1} 1/i^p$.","title":"Asymptotic of generalized hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-hyperharmonic-type-ii-eq-sum","kind":"node","lean":"MetaMathlibExt.alternatingHyperharmonic_typeII_eq_sum","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-hyperharmonic-type-ii-eq-sum","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"For real $s$ write $\\overline{H}_n^{(s)} = \\sum_{i=1}^{n} (-1)^{i-1}/i^s$ for the alternating generalized harmonic numbers. For a natural number $p$, the type II generalized alternating hyperharmonic numbers are $H_n^{(p,1,2)} = \\overline{H}_n^{(p)}$ and $H_n^{(p,r+1,2)} = \\sum_{k=1}^{n} H_k^{(p,r,2)}$. Let $a(r, m, j)$ be Li's rational coefficients. Then for all natural numbers $r, n, p \\ge 1$,","title":"Explicit formula for type II alternating hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-hyperharmonic-type-i-type-ii","kind":"node","lean":"MetaMathlibExt.alternatingHyperharmonic_typeI_typeIII_eq_sum","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-hyperharmonic-type-i-type-ii","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"For real $s$ write $H_n^{(s)} = \\sum_{i=1}^{n} 1/i^s$ and $\\overline{H}_n^{(s)} = \\sum_{i=1}^{n} (-1)^{i-1}/i^s$ for the generalized and the alternating generalized harmonic numbers. For a natural number $p$, the type I and type III generalized alternating hyperharmonic numbers are $H_n^{(p,1,1)} = H_n^{(p)}$, $H_n^{(p,1,3)} = \\overline{H}_n^{(p)}$ and, for $i = 1, 3$,\u2026","title":"Explicit formulas for type I and type III alternating hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-hyperharmonic-type-i-is-equi","kind":"node","lean":"MetaMathlibExt.alternatingHyperharmonic_typeI_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-hyperharmonic-type-i-is-equi","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"For real $s$ let $H_n^{(s)} = \\sum_{i=1}^{n} 1/i^s$ be the generalized harmonic numbers, and let the type I generalized alternating hyperharmonic numbers be $H_n^{(s,1,1)} = H_n^{(s)}$ and $H_n^{(s,r+1,1)} = \\sum_{k=1}^{n} (-1)^{k-1} H_k^{(s,r,1)}$. Let $y \\ge 1$ and $p \\ge 2$ be natural numbers, let $\\zeta(p) = \\sum_{i \\ge 1} 1/i^p$ and let $\\bar\\zeta(p)$ be the alternating zeta value at $p$. Then, as\u2026","title":"Asymptotics of type I alternating hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-hyperharmonic-type-ii-is-equ","kind":"node","lean":"MetaMathlibExt.alternatingHyperharmonic_typeII_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-hyperharmonic-type-ii-is-equ","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"For a natural number $p$ let $\\overline{H}_n^{(p)} = \\sum_{i=1}^{n} (-1)^{i-1}/i^p$ be the alternating generalized harmonic numbers, and let the type II generalized alternating hyperharmonic numbers be $H_n^{(p,1,2)} = \\overline{H}_n^{(p)}$ and $H_n^{(p,r+1,2)} = \\sum_{k=1}^{n} H_k^{(p,r,2)}$. Let $\\bar\\zeta(p)$ be the alternating zeta value at $p$, so that $\\bar\\zeta(1) = \\log 2$. Then for all natural numbers\u2026","title":"Asymptotic of type II alternating hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/alternating-hyperharmonic-type-iii-is-eq","kind":"node","lean":"MetaMathlibExt.alternatingHyperharmonic_typeIII_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#alternating-hyperharmonic-type-iii-is-eq","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"For real $s$ let $\\overline{H}_n^{(s)} = \\sum_{i=1}^{n} (-1)^{i-1}/i^s$ be the alternating generalized harmonic numbers, and let the type III generalized alternating hyperharmonic numbers be $H_n^{(s,1,3)} = \\overline{H}_n^{(s)}$ and $H_n^{(s,r+1,3)} = \\sum_{k=1}^{n} (-1)^{k-1} H_k^{(s,r,3)}$. Let $y \\ge 1$ and $p \\ge 1$ be natural numbers and let $\\bar\\zeta(p)$ be the alternating zeta value at $p$. Then, as\u2026","title":"Asymptotics of type III alternating hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/sum-squarefree-kprime-pow-mul-hyperharmo","kind":"node","lean":"MetaMathlibExt.sum_squarefreeKPrime_pow_mul_hyperharmonic_products_isEquivalent","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mixed hyperharmonic product sums over squarefree numbers with "],["math","\\(k\\)"],["text"," prime factors"]],"source":"Mixed hyperharmonic product sums over squarefree numbers with $k$ prime factors"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#sum-squarefree-kprime-pow-mul-hyperharmo","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"Let $q_1 \\ge 2$ and $q_2, \\alpha, \\beta, m, k, s, n, r_1, r_2 \\ge 1$ be natural numbers. Let $p_{\\ell,k}$ be the $\\ell$th ($\\ell \\ge 1$) squarefree number with exactly $k$ prime factors. Write $H_\\ell^{(q)} = \\sum_{j=1}^{\\ell} j^{-q}$ and $\\bar H_\\ell^{(q)} = \\sum_{j=1}^{\\ell} (-1)^{j-1} j^{-q}$; let $H_\\ell^{(q,1)} = H_\\ell^{(q)}$ and $H_\\ell^{(q,r+1)} = \\sum_{j=1}^{\\ell} H_j^{(q,r)}$, and define\u2026","title":"Mixed hyperharmonic product sums over squarefree numbers with $k$ prime factors"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/sum-primes-pow-mul-alternating-hyperharm","kind":"node","lean":"MetaMathlibExt.sum_primes_pow_mul_alternatingHyperharmonic_typeI_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#sum-primes-pow-mul-alternating-hyperharm","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"Let $\\alpha, m, y \\ge 1$ and $q \\ge 2$ be integers. Let $p_\\ell$ be the $\\ell$-th prime ($p_1 = 2$), let $H_n^{(s,r,1)}$ be the signed iterated partial sum of order $r$, with alternating signs, of the generalized harmonic numbers $H_n^{(s)} = \\sum_{j=1}^{n} j^{-s}$, let $\\zeta(q) = \\sum_{n \\ge 1} n^{-q}$, and let $\\bar\\zeta$ be the alternating zeta function. For a real sequence $f$ write","title":"Prime-weighted power sums of type I alternating hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/sum-squarefree-kprime-pow-mul-alternatin","kind":"node","lean":"MetaMathlibExt.sum_squarefreeKPrime_pow_mul_alternatingHyperharmonic_typeI_isEquivalent","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Type I alternating hyperharmonic power sums weighted by squarefree "],["math","\\(k\\)"],["text","-prime numbers"]],"source":"Type I alternating hyperharmonic power sums weighted by squarefree $k$-prime numbers"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#sum-squarefree-kprime-pow-mul-alternatin","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"Let $\\alpha, m, k, y \\ge 1$ and $q \\ge 2$ be integers. Let $p_{\\ell,k}$ be the $\\ell$-th squarefree number with exactly $k$ prime factors, let $H_n^{(s,r,1)}$ be the signed iterated partial sum of order $r$, with alternating signs, of the generalized harmonic numbers $H_n^{(s)} = \\sum_{j=1}^{n} j^{-s}$, let $\\zeta(q) = \\sum_{n \\ge 1} n^{-q}$, and let $\\bar\\zeta$ be the alternating zeta function. For a real sequence\u2026","title":"Type I alternating hyperharmonic power sums weighted by squarefree $k$-prime numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/sum-pow-mul-alternating-hyperharmonic-ty","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_alternatingHyperharmonic_typeII_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#sum-pow-mul-alternating-hyperharmonic-ty","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"Let $\\alpha, m, q, k, r \\ge 1$ be integers. Let $p_\\ell$ be the $\\ell$-th prime ($p_1 = 2$) and $p_{\\ell,k}$ the $\\ell$-th squarefree number with exactly $k$ prime factors. Let $H_n^{(q,r,2)}$ be the signed iterated partial sum of order $r$, with constant signs, of the alternating generalized harmonic numbers $\\bar H_n^{(q)} = \\sum_{j=1}^{n} (-1)^{j-1} j^{-q}$, and let $\\bar\\zeta$ be the alternating zeta function\u2026","title":"Prime- and squarefree-weighted power sums of type II alternating hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/sum-primes-pow-mul-alternating-hyperharm-2","kind":"node","lean":"MetaMathlibExt.sum_primes_pow_mul_alternatingHyperharmonic_typeIII_isEquivalent","parent":11660,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#sum-primes-pow-mul-alternating-hyperharm-2","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"Let $\\alpha, m, p, y \\ge 1$ and $q \\ge 2$ be integers. Let $p_\\ell$ be the $\\ell$-th prime ($p_1 = 2$), let $H_n^{(s,r,3)}$ be the signed iterated partial sum of order $r$, with alternating signs, of the alternating generalized harmonic numbers $\\bar H_n^{(s)} = \\sum_{j=1}^{n} (-1)^{j-1} j^{-s}$, let $\\zeta(q) = \\sum_{n \\ge 1} n^{-q}$, and let $\\bar\\zeta$ be the alternating zeta function ($\\bar\\zeta(1) = \\log 2$).\u2026","title":"Prime-weighted power sums of type III alternating hyperharmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/sum-squarefree-kprime-pow-mul-alternatin-2","kind":"node","lean":"MetaMathlibExt.sum_squarefreeKPrime_pow_mul_alternatingHyperharmonic_typeIII_isEquivalent","parent":11660,"reading_title":{"format":"inline-parts-v1","parts":[["text","Type III alternating hyperharmonic power sums weighted by squarefree "],["math","\\(k\\)"],["text","-prime numbers"]],"source":"Type III alternating hyperharmonic power sums weighted by squarefree $k$-prime numbers"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes/index.html#sum-squarefree-kprime-pow-mul-alternatin-2","scope":"NumberTheory/special-numbers/harmonic-numbers/hyperharmonic-numbers-and-sums-over-primes","status":"can_prove","summary":"Let $\\alpha, m, k, p, y \\ge 1$ and $q \\ge 2$ be integers. Let $p_{\\ell,k}$ be the $\\ell$-th squarefree number with exactly $k$ prime factors, let $H_n^{(s,r,3)}$ be the signed iterated partial sum of order $r$, with alternating signs, of the alternating generalized harmonic numbers $\\bar H_n^{(s)} = \\sum_{j=1}^{n} (-1)^{j-1} j^{-s}$, let $\\zeta(q) = \\sum_{n \\ge 1} n^{-q}$, and let $\\bar\\zeta$ be the alternating\u2026","title":"Type III alternating hyperharmonic power sums weighted by squarefree $k$-prime numbers"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums.json","id":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums","status":"fully_proved","title":"Multiple harmonic sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/dilcher-finite-multiple-harmonic-identit","kind":"node","lean":"MetaMathlibExt.dilcher_finite_multiple_harmonic_identity_general","parent":11702,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/index.html#dilcher-finite-multiple-harmonic-identit","scope":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums","status":"fully_proved","summary":"For every natural number $m$ (including $m = 0$) and every positive integer $n$,","title":"Dilcher's finite multiple harmonic identity for every exponent"},{"id":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/dilcher-finite-multiple-harmonic-identit-2","kind":"node","lean":"MetaMathlibExt.dilcher_finite_multiple_harmonic_identity","parent":11702,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/index.html#dilcher-finite-multiple-harmonic-identit-2","scope":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums","status":"fully_proved","summary":"For all positive integers $m$ and $n$,","title":"Dilcher's finite multiple harmonic identity"},{"id":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/multiple-harmonic","kind":"node","lean":"MetaMathlibExt.multipleHarmonic","parent":11702,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/index.html#multiple-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums","status":"fully_proved","summary":"For natural numbers $p$, $r$ and $n$, the multiple harmonic sum $H_r^{(n)}$ modulo $p$ is the element","title":"Multiple harmonic sums"},{"id":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/multiple-harmonic-polynomial-congruence","kind":"node","lean":"MetaMathlibExt.multiple_harmonic_polynomial_congruence","parent":11702,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums/index.html#multiple-harmonic-polynomial-congruence","scope":"NumberTheory/special-numbers/harmonic-numbers/multiple-harmonic-sums","status":"fully_proved","summary":"Let $n \\ge 1$ and let $p$ be a prime with $p > n + 1$, write $H_r^{(m)}$ for the multiple harmonic sum modulo $p$, and write $k^{-1}$ for the inverse of $k$ in $\\mathbb{Z}/p\\mathbb{Z}$. Then, as polynomials in $x$ over $\\mathbb{Z}/p\\mathbb{Z}$,","title":"Mattarei\u2013Tauraso polynomial congruence for multiple harmonic sums"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers.json","id":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers","status":"fully_proved","title":"Odd harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/odd-harmonic","kind":"node","lean":"MetaMathlibExt.oddHarmonic","parent":11707,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/index.html#odd-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers","status":"fully_proved","summary":"For a natural number $n$, the $n$-th odd harmonic number is the rational number $O_n = \\sum_{m=1}^{n} \\frac{1}{2m-1}$, the sum of the reciprocals of the first $n$ odd positive integers; in particular $O_0 = 0$.","title":"Odd harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/odd-harmonic-zero","kind":"node","lean":"MetaMathlibExt.oddHarmonic_zero","parent":11707,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/index.html#odd-harmonic-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers","status":"fully_proved","summary":"The odd harmonic number $O_0$ equals $0$.","title":"The zeroth odd harmonic number"},{"id":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/odd-harmonic-succ","kind":"node","lean":"MetaMathlibExt.oddHarmonic_succ","parent":11707,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/index.html#odd-harmonic-succ","scope":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers","status":"fully_proved","summary":"For every natural number $n$, the odd harmonic numbers satisfy $O_{n+1} = O_n + \\frac{1}{2n+1}$.","title":"Recurrence for the odd harmonic numbers"},{"id":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/odd-harmonic-eq-harmonic","kind":"node","lean":"MetaMathlibExt.oddHarmonic_eq_harmonic","parent":11707,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers/index.html#odd-harmonic-eq-harmonic","scope":"NumberTheory/special-numbers/harmonic-numbers/odd-harmonic-numbers","status":"fully_proved","summary":"For every natural number $n$, the odd harmonic number $O_n$ and the harmonic numbers $H_k = \\sum_{j=1}^{k} \\frac{1}{j}$ satisfy $O_n = H_{2n} - \\frac{H_n}{2}$.","title":"Odd harmonic numbers in terms of harmonic numbers"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity.json","id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity","status":"can_prove","title":"Ramanujan's binomial\u2013harmonic identity"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/sum-choose-sub-choose-mul-harmonic-succ","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanHarmonicIdentityWanted.sum_choose_sub_choose_mul_harmonic_succ_eq","parent":11712,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/index.html#sum-choose-sub-choose-mul-harmonic-succ","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity","status":"can_prove","summary":"For natural numbers $k$ and $n$ with $2 \\le k \\le n$,","title":"Shifted binomial\u2013harmonic convolution (Alzer\u2013Kwong, Corollary 1)"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/sum-choose-sub-choose-mul-harmonic-eq-ad","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanHarmonicIdentityWanted.sum_choose_sub_choose_mul_harmonic_eq_add","parent":11712,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/index.html#sum-choose-sub-choose-mul-harmonic-eq-ad","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity","status":"can_prove","summary":"For natural numbers $k$ and $n$ with $1 \\le k \\le n - 1$,","title":"Shifted binomial\u2013harmonic convolution (Alzer\u2013Kwong, Corollary 2)"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/has-sum-ring-choose-mul-sin","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanHarmonicIdentityWanted.hasSum_ringChoose_mul_sin","parent":11712,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/index.html#has-sum-ring-choose-mul-sin","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity","status":"can_prove","summary":"Let $a$, $N$ and $\\theta$ be real numbers with $N \\ge 0$ and $|\\theta| \\le \\pi / 2$. Then the series $\\sum_{\\nu=0}^{\\infty} \\binom{N}{\\nu} \\sin((a + 2\\nu)\\theta)$ converges unconditionally to $2^N (\\cos \\theta)^N \\sin((a + N)\\theta)$. Here $\\binom{N}{\\nu} = N(N-1)\\cdots(N-\\nu+1)/\\nu!$ is the generalized binomial coefficient, and $2^N$ and $(\\cos \\theta)^N$ are real powers.","title":"Ramanujan's binomial sine series"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/sum-two-pow-choose-sub-mul-cos-pow-sin","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanHarmonicIdentityWanted.sum_two_pow_choose_sub_mul_cos_pow_sin","parent":11712,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/index.html#sum-two-pow-choose-sub-mul-cos-pow-sin","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity","status":"can_prove","summary":"Let $k$ and $n$ be natural numbers with $1 \\le k \\le n$, and let $\\theta$ be a real number. Then","title":"The Rung-Obaid trigonometric identity"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/sum-neg-one-pow-div-mul-choose-eq","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanHarmonicIdentityWanted.sum_neg_one_pow_div_mul_choose_eq","parent":11712,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/index.html#sum-neg-one-pow-div-mul-choose-eq","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity","status":"can_prove","summary":"Let $N$ be a natural number and $x$ a real number with $x \\ne \\nu$ for every $\\nu \\in \\{0, 1, \\dots, N\\}$. Then","title":"Gould's reciprocal binomial identity (Alzer\u2013Kwong, Lemma 4)"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/eq-lambda-mul-harmonic-of-sum-choose-sub","kind":"node","lean":"MathlibExt.NumberTheory.RamanujanHarmonicIdentityWanted.eq_lambda_mul_harmonic_of_sum_choose_sub_mul_eq","parent":11712,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity/index.html#eq-lambda-mul-harmonic-of-sum-choose-sub","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-binomial-harmonic-identity","status":"can_prove","summary":"Let $K : \\mathbb{N} \\to \\mathbb{C}$ be a sequence such that for every integer $n \\ge 2$","title":"Uniqueness of the harmonic solution of the diagonal Ramanujan identity"},{"data_url":"scopes/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities.json","id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","kind":"scope","navigation":"module","parent":11587,"result":false,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","title":"Ramanujan's harmonic sum identities"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry1-general","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry1.ramanujan_part1_ch2_entry1_general","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's identity for "],["math","\\(\\sum_{k=1}^{n} 1/(n+k)\\)"],["text",", including "],["math","\\(n = 0\\)"]],"source":"Ramanujan's identity for $\\sum_{k=1}^{n} 1/(n+k)$, including $n = 0$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry1-general","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $n$,","title":"Ramanujan's identity for $\\sum_{k=1}^{n} 1/(n+k)$, including $n = 0$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry1","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry1.ramanujan_part1_ch2_entry1","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's identity for "],["math","\\(\\sum_{k=1}^{n} 1/(n+k)\\)"]],"source":"Ramanujan's identity for $\\sum_{k=1}^{n} 1/(n+k)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry1","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every positive integer $n$,","title":"Ramanujan's identity for $\\sum_{k=1}^{n} 1/(n+k)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry1-corollary","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry1Corollary.ramanujan_part1_ch2_entry1_corollary","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's series "],["math","\\(\\sum_{k \\ge 1} 1/((2k)^3 - 2k) = \\log 2 - 1/2\\)"]],"source":"Ramanujan's series $\\sum_{k \\ge 1} 1/((2k)^3 - 2k) = \\log 2 - 1/2$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry1-corollary","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"The series","title":"Ramanujan's series $\\sum_{k \\ge 1} 1/((2k)^3 - 2k) = \\log 2 - 1/2$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry2-general","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry2.ramanujan_part1_ch2_entry2_general","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's identity for "],["math","\\(\\sum_{k=1}^{2n+1} 1/(n+k)\\)"],["text",", including "],["math","\\(n = 0\\)"]],"source":"Ramanujan's identity for $\\sum_{k=1}^{2n+1} 1/(n+k)$, including $n = 0$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry2-general","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $n$,","title":"Ramanujan's identity for $\\sum_{k=1}^{2n+1} 1/(n+k)$, including $n = 0$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry2.ramanujan_part1_ch2_entry2","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's identity for "],["math","\\(\\sum_{k=1}^{2n+1} 1/(n+k)\\)"]],"source":"Ramanujan's identity for $\\sum_{k=1}^{2n+1} 1/(n+k)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry2","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every positive integer $n$,","title":"Ramanujan's identity for $\\sum_{k=1}^{2n+1} 1/(n+k)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry2-corollary","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry2Corollary.ramanujan_part1_ch2_entry2_corollary","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's series "],["math","\\(\\sum_{k \\ge 1} 1/((3k)^3 - 3k) = (\\log 3 - 1)/2\\)"]],"source":"Ramanujan's series $\\sum_{k \\ge 1} 1/((3k)^3 - 3k) = (\\log 3 - 1)/2$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry2-corollary","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"The series","title":"Ramanujan's series $\\sum_{k \\ge 1} 1/((3k)^3 - 3k) = (\\log 3 - 1)/2$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry4-general","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry4.ramanujan_part1_ch2_entry4_general","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's sum identity for "],["math","\\(1/((4k)^3 - 4k)\\)"],["text",", including "],["math","\\(n = 0\\)"]],"source":"Ramanujan's sum identity for $1/((4k)^3 - 4k)$, including $n = 0$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry4-general","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $n$,","title":"Ramanujan's sum identity for $1/((4k)^3 - 4k)$, including $n = 0$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry4","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry4.ramanujan_part1_ch2_entry4","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's sum identity for "],["math","\\(1/((4k)^3 - 4k)\\)"]],"source":"Ramanujan's sum identity for $1/((4k)^3 - 4k)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry4","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every positive integer $n$,","title":"Ramanujan's sum identity for $1/((4k)^3 - 4k)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry4-alternating-f","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry4AlternatingForm.ramanujan_part1_ch2_entry4_alternating_form_general","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating form of Ramanujan's identity for "],["math","\\(1/((4k)^3 - 4k)\\)"],["text",", including "],["math","\\(n = 0\\)"]],"source":"Alternating form of Ramanujan's identity for $1/((4k)^3 - 4k)$, including $n = 0$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry4-alternating-f","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $n$,","title":"Alternating form of Ramanujan's identity for $1/((4k)^3 - 4k)$, including $n = 0$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry4-alternating-f-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry4AlternatingForm.ramanujan_part1_ch2_entry4_alternating_form","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating form of Ramanujan's sum identity for "],["math","\\(1/((4k)^3 - 4k)\\)"]],"source":"Alternating form of Ramanujan's sum identity for $1/((4k)^3 - 4k)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry4-alternating-f-2","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every positive integer $n$,","title":"Alternating form of Ramanujan's sum identity for $1/((4k)^3 - 4k)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry4-corollary","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry4Corollary.ramanujan_part1_ch2_entry4_corollary","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's series for "],["math","\\(\\frac{3}{2}\\log 2 - 1\\)"]],"source":"Ramanujan's series for $\\frac{3}{2}\\log 2 - 1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry4-corollary","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"The numbers $\\frac{2}{(4k)^3 - 4k}$ for $k = 1, 2, 3, \\dots$ are summable with sum $\\frac{3}{2}\\log 2 - 1$, where $\\log$ is the natural logarithm.","title":"Ramanujan's series for $\\frac{3}{2}\\log 2 - 1$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry5-general","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry5.ramanujan_part1_ch2_entry5_general","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's sum identity for "],["math","\\(1/((6k)^3 - 6k)\\)"],["text",", including "],["math","\\(n = 0\\)"]],"source":"Ramanujan's sum identity for $1/((6k)^3 - 6k)$, including $n = 0$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry5-general","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $n$,","title":"Ramanujan's sum identity for $1/((6k)^3 - 6k)$, including $n = 0$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry5","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry5.ramanujan_part1_ch2_entry5","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's sum identity for "],["math","\\(1/((6k)^3 - 6k)\\)"]],"source":"Ramanujan's sum identity for $1/((6k)^3 - 6k)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry5","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every positive integer $n$,","title":"Ramanujan's sum identity for $1/((6k)^3 - 6k)$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry5-corollary","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry5Corollary.ramanujan_part1_ch2_entry5_corollary","parent":11719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's series for "],["math","\\(\\frac{1}{2}\\log 3 + \\frac{1}{3}\\log 4 - 1\\)"]],"source":"Ramanujan's series for $\\frac{1}{2}\\log 3 + \\frac{1}{3}\\log 4 - 1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry5-corollary","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"The numbers $\\frac{2}{(6k)^3 - 6k}$ for $k = 1, 2, 3, \\dots$ are summable with sum $\\frac{1}{2}\\log 3 + \\frac{1}{3}\\log 4 - 1$, where $\\log$ is the natural logarithm.","title":"Ramanujan's series for $\\frac{1}{2}\\log 3 + \\frac{1}{3}\\log 4 - 1$"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/a","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6.A","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#a","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For natural numbers $n$ and $k$, let $A(n, k) = 3^k n + (3^k - 1)/2$, a natural number. It is the sequence with $A(n, 0) = n$ and $A(n, k + 1) = 3A(n, k) + 1$.","title":"The tripling sequence of Ramanujan's Entry 6"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/a-zero","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6.A_zero","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#a-zero","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $n$, the tripling sequence $A(n, k)$ of Entry 6 satisfies $A(n, 0) = n$.","title":"Starting value of the tripling sequence"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/a-succ","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6.A_succ","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#a-succ","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the tripling sequence $A(n, k)$ of Entry 6 satisfies $A(n, k + 1) = 3A(n, k) + 1$.","title":"Recurrence of the tripling sequence"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/a-ge","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6.A_ge","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#a-ge","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the tripling sequence $A(n, k)$ of Entry 6 satisfies $n \\le A(n, k)$.","title":"The tripling sequence never drops below its starting value"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry6-general","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6.ramanujan_part1_ch2_entry6_general","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry6-general","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For all natural numbers $n$ and $r$, the following identity holds in $\\mathbb{Q}$:","title":"Ramanujan's harmonic sum identity (Entry 6, general form)"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry6","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6.ramanujan_part1_ch2_entry6","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry6","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For all natural numbers $n$ and $r$ with $r > 0$, the following identity holds in $\\mathbb{Q}$:","title":"Ramanujan's harmonic sum identity (Entry 6)"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/a-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6Corollary.A","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#a-2","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For a natural number $k$, let $A(k) = (3^k - 1)/2 = 1 + 3 + \\cdots + 3^{k-1}$, a natural number with $A(0) = 0$.","title":"The tripling sequence starting at zero"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/a-two-mul-add-one","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6Corollary.A_two_mul_add_one","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#a-two-mul-add-one","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $k$, $2A(k) + 1 = 3^k$, where $A(k) = (3^k - 1)/2$; in particular the division by $2$ in the definition of $A(k)$ is exact.","title":"Twice the tripling sequence plus one is a power of three"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry6-corollary-gen","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6Corollary.ramanujan_part1_ch2_entry6_corollary_general","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry6-corollary-gen","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $r$, the following identity holds in $\\mathbb{Q}$:","title":"Ramanujan's harmonic sum identity (corollary to Entry 6, general form)"},{"id":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/ramanujan-part1-ch2-entry6-corollary","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry6Corollary.ramanujan_part1_ch2_entry6_corollary","parent":11719,"result":true,"route":"../roadmap/NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities/index.html#ramanujan-part1-ch2-entry6-corollary","scope":"NumberTheory/special-numbers/harmonic-numbers/ramanujan-s-harmonic-sum-identities","status":"fully_proved","summary":"For every natural number $r > 0$, the following identity holds in $\\mathbb{Q}$:","title":"Ramanujan's harmonic sum identity (corollary to Entry 6)"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients.json","id":"NumberTheory/special-numbers/binomial-coefficients","kind":"scope","parent":11422,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients","status":"can_prove","summary":"Divisibility and valuations of binomial coefficients, Lucas-type theorems, factorial ratios and Fibonomial coefficients; b-nomial, totienomial and other generalized binomial coefficients go to Combinatorics.","title":"Binomial coefficients and factorials"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2.json","id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","title":"Binomial coefficient sums modulo 2"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/bit-run-lengths","kind":"node","lean":"MetaMathlibExt.bitRunLengths","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#bit-run-lengths","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"fully_proved","summary":"Given a list $b_1, \\dots, b_k$ of natural numbers (binary digits) and a natural number $c$, this is the list, in order of occurrence, of the lengths of the maximal runs of $1$s in the sequence consisting of $c$ ones followed by $b_1, \\dots, b_k$, where every digit other than $1$ ends a run.","title":"Run lengths of a digit list"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/binary-run-lengths","kind":"node","lean":"MetaMathlibExt.binaryRunLengths","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#binary-run-lengths","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"fully_proved","summary":"For a natural number $n$, the binary run lengths of $n$ form the list of the lengths of the maximal runs of $1$s in the binary expansion of $n$, in the order in which the runs occur when the digits are read from the least significant one. This is the paper's list $\\mathcal{L}(n)$, whose order does not matter there; for $n = 0$ the list is empty.","title":"Binary run lengths"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform","kind":"node","lean":"MetaMathlibExt.runLengthTransform","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"fully_proved","summary":"For a sequence $S : \\mathbb{N} \\to \\mathbb{Z}$, the run length transform of $S$ is the sequence $T : \\mathbb{N} \\to \\mathbb{Z}$ given by $T(n) = \\prod_{\\ell \\in \\mathcal{L}(n)} S(\\ell)$, where $\\mathcal{L}(n)$ is the list of the lengths of the maximal runs of $1$s in the binary expansion of $n$, so that each run contributes one factor.","title":"Run length transform"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-zero","kind":"node","lean":"MetaMathlibExt.runLengthTransform_zero","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"fully_proved","summary":"For every sequence $S : \\mathbb{N} \\to \\mathbb{Z}$, the run length transform of $S$ takes the value $1$ at $0$, since $0$ has no runs of $1$s in binary.","title":"Run length transform at zero"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-eleven","kind":"node","lean":"MetaMathlibExt.runLengthTransform_eleven","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-eleven","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"fully_proved","summary":"For every sequence $S : \\mathbb{N} \\to \\mathbb{Z}$, the run length transform $T$ of $S$ satisfies $T(11) = S(1) S(2)$: in binary $11$ is $1011$, whose maximal runs of $1$s have lengths $2$ and $1$.","title":"Run length transform at eleven"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/linear-recurrence-eq-companion-pow","kind":"node","lean":"MetaMathlibExt.linearRecurrence_eq_companion_pow","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#linear-recurrence-eq-companion-pow","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $r$ be a natural number, let $d_0, \\dots, d_r$ be integers, and let $S : \\mathbb{N} \\to \\mathbb{Z}$ satisfy $S(0) = 1$ and $S(m + 1) = d_0 S(m) + d_1 S(m - 1) + \\dots + d_r S(m - r)$ for every $m \\ge r$. Let $\\gamma(1)$ be the $(r + 1) \\times (r + 1)$ integer matrix with rows and columns indexed by $0, \\dots, r$ whose row $i < r$ has a $1$ in column $i + 1$ and zeros elsewhere, and whose last row is\u2026","title":"Linear recurrences via powers of the companion matrix"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-linear-recurrence-e","kind":"node","lean":"MetaMathlibExt.runLengthTransform_linearRecurrence_eq_linearRep","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-linear-recurrence-e","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $r \\ge 0$, let $d_0, \\dots, d_r$ be integers, and let $S : \\mathbb{N} \\to \\mathbb{Z}$ satisfy $S(0) = 1$ and $S(m + 1) = d_0 S(m) + d_1 S(m - 1) + \\dots + d_r S(m - r)$ for all $m \\ge r$. Let $T$ be the run length transform of $S$, let $v = (1, 0, \\dots, 0)$ and $w = (S(0), \\dots, S(r))^\\top$, let $\\gamma(0)$ be the $(r + 1) \\times (r + 1)$ matrix with first column $w$ and zeros elsewhere, and let $\\gamma(1)$\u2026","title":"Run length transform of a linear recurrence (Rampersad\u2013Wiebe Theorem 1)"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-sub-two-mul","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_sub_two_mul_fib","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-sub-two-mul","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $(F_j)_{j \\ge 0}$ be the Fibonacci numbers, $F_0 = 0$, $F_1 = 1$, and let $T$ be the run length transform of $S(i) = F_{i+1}$, i.e. of $1, 1, 2, 3, 5, 8, \\dots$. Then for every natural number $n$, $\\sum_{k=0}^{n} \\left(\\binom{n - k}{2k} \\binom{n}{k} \\bmod 2\\right) = T(n)$.","title":"Rampersad\u2013Wiebe Theorem 2"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-three-mul-fi","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_three_mul_fib","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-three-mul-fi","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $(F_j)_{j \\ge 0}$ be the Fibonacci numbers, $F_0 = 0$, $F_1 = 1$, and let $T$ be the run length transform of $S(i) = F_{i+2}$, i.e. of $1, 2, 3, 5, 8, \\dots$. Then for every natural number $n$, $\\sum_{k=0}^{n} \\left(\\binom{3k}{k} \\binom{n}{k} \\bmod 2\\right) = T(n)$.","title":"Rampersad\u2013Wiebe Theorem 3"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-two-mul-pow","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_two_mul_pow_two","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-two-mul-pow","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $T$ be the run length transform of the sequence $S$ with $S(0) = 1$ and $S(i) = 2^{i-1}$ for $i \\ge 1$, i.e. of $1, 1, 2, 4, 8, \\dots$. Then for every natural number $n$, $\\sum_{k=0}^{n} \\left(\\binom{n}{2k} \\binom{n}{k} \\bmod 2\\right) = T(n)$.","title":"Rampersad\u2013Wiebe Theorem 4"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-add-two-mul","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_add_two_mul_one_two","parent":11745,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\binom{n+2k}{2k}\\binom{n}{k} \\bmod 2\\)"],["text"," as a run length transform"]],"source":"Sum of $\\binom{n+2k}{2k}\\binom{n}{k} \\bmod 2$ as a run length transform"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-add-two-mul","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"For every natural number $n$, the sum $\\sum_{k=0}^{n} \\bigl(\\binom{n+2k}{2k}\\binom{n}{k} \\bmod 2\\bigr)$ equals the value at $n$ of the run length transform of the sequence $S$ with $S(0) = 1$ and $S(i) = 2$ for $i \\ge 1$, that is, $1, 2, 2, 2, \\dots$.","title":"Sum of $\\binom{n+2k}{2k}\\binom{n}{k} \\bmod 2$ as a run length transform"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-add-sub-succ","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_add_sub_succ","parent":11745,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\binom{n+k}{n-k}\\binom{n}{k} \\bmod 2\\)"],["text"," as a run length transform"]],"source":"Sum of $\\binom{n+k}{n-k}\\binom{n}{k} \\bmod 2$ as a run length transform"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-add-sub-succ","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"For every natural number $n$, the sum $\\sum_{k=0}^{n} \\bigl(\\binom{n+k}{n-k}\\binom{n}{k} \\bmod 2\\bigr)$ equals the value at $n$ of the run length transform of the sequence $S(i) = i + 1$ of positive integers $1, 2, 3, \\dots$.","title":"Sum of $\\binom{n+k}{n-k}\\binom{n}{k} \\bmod 2$ as a run length transform"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-sub-six-mul","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_sub_six_mul_narayanaCows","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-sub-six-mul","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $(N_j)_{j \\ge 0}$ be Narayana's cows sequence, $N_0 = 0$, $N_1 = N_2 = 1$ and $N_j = N_{j-1} + N_{j-3}$ for $j \\ge 3$, and let $T$ be the run length transform of $S(i) = N_{i+1}$, i.e. of $1, 1, 1, 2, 3, 4, 6, 9, \\dots$. Then for every natural number $n$, $\\sum_{k=0}^{n} \\left(\\binom{n - k}{6k} \\binom{n}{k} \\bmod 2\\right) = T(n)$.","title":"Rampersad\u2013Wiebe Theorem 7"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-add-three-mu","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_add_three_mul_six_mul_doubled","parent":11745,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\binom{n+3k}{6k}\\binom{n}{k} \\bmod 2\\)"],["text"," as a run length transform"]],"source":"Sum of $\\binom{n+3k}{6k}\\binom{n}{k} \\bmod 2$ as a run length transform"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-add-three-mu","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"For every natural number $n$, the sum $\\sum_{k=0}^{n} \\bigl(\\binom{n+3k}{6k}\\binom{n}{k} \\bmod 2\\bigr)$ equals the value at $n$ of the run length transform of the sequence $S(i) = \\lfloor i / 2 \\rfloor + 1$ of doubled positive integers $1, 1, 2, 2, 3, 3, \\dots$.","title":"Sum of $\\binom{n+3k}{6k}\\binom{n}{k} \\bmod 2$ as a run length transform"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-add-two-mul-2","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_add_two_mul_two_mul_sub_lucas","parent":11745,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\binom{n+2k}{2n-k}\\binom{n}{k} \\bmod 2\\)"],["text"," as a run length transform"]],"source":"Sum of $\\binom{n+2k}{2n-k}\\binom{n}{k} \\bmod 2$ as a run length transform"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-add-two-mul-2","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"For every natural number $n$, the sum $\\sum_{k=0}^{n} \\bigl(\\binom{n+2k}{2n-k}\\binom{n}{k} \\bmod 2\\bigr)$ equals the value at $n$ of the run length transform of the sequence $S$ with $S(0) = S(1) = 1$ and $S(i) = L_{i-2}$ for $i \\ge 2$, where $L_0 = 2, L_1 = 1, L_2 = 3, \\dots$ are the Lucas numbers; that is, $S$ is $1, 1, 2, 1, 3, 4, 7, 11, \\dots$.","title":"Sum of $\\binom{n+2k}{2n-k}\\binom{n}{k} \\bmod 2$ as a run length transform"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-seven-mul-su","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_seven_mul_sub_padovan_five","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-seven-mul-su","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $(p_j)_{j \\ge 0}$ be the shifted Padovan sequence, $p_0 = 1$, $p_1 = 0$, $p_2 = 1$ and $p_{j+3} = p_{j+1} + p_j$, and let $T$ be the run length transform of $S(i) = p_{i+2}$, i.e. of $1, 1, 1, 2, 2, 3, 4, 5, \\dots$ (the Padovan numbers, OEIS A000931, from offset $5$). Then for every natural number $n$, $\\sum_{k=0}^{n} \\left(\\binom{7k - n}{n + k} \\binom{n}{k} \\bmod 2\\right) = T(n)$, where a binomial coefficient\u2026","title":"Rampersad\u2013Wiebe Theorem 12"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-add-seven-mu","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_add_seven_mul_padovan_three","parent":11745,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\binom{n+7k}{3n+k}\\binom{n}{k} \\bmod 2\\)"],["text"," as a run length transform"]],"source":"Sum of $\\binom{n+7k}{3n+k}\\binom{n}{k} \\bmod 2$ as a run length transform"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-add-seven-mu","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"For every natural number $n$, the sum $\\sum_{k=0}^{n} \\bigl(\\binom{n+7k}{3n+k}\\binom{n}{k} \\bmod 2\\bigr)$ equals the value at $n$ of the run length transform of the shifted Padovan sequence $P$, given by $P(0) = 1$, $P(1) = 0$, $P(2) = 1$ and $P(i + 3) = P(i + 1) + P(i)$, that is, $1, 0, 1, 1, 1, 2, 2, 3, \\dots$ (the Padovan numbers, OEIS A000931, from offset $3$).","title":"Sum of $\\binom{n+7k}{3n+k}\\binom{n}{k} \\bmod 2$ as a run length transform"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-six-mul-alte","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_six_mul_alternating","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-six-mul-alte","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $T$ be the run length transform of the sequence $S$ with $S(i) = 1$ for even $i$ and $S(i) = (i + 1)/2$ for odd $i$, i.e. of $1, 1, 1, 2, 1, 3, 1, 4, \\dots$. Then for every natural number $n$, $\\sum_{k=0}^{n} \\left(\\binom{6k}{n + 3k} \\binom{n}{k} \\bmod 2\\right) = T(n)$.","title":"Rampersad\u2013Wiebe Theorem 14"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-eight-mul-su","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_eight_mul_sub_period_three","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-eight-mul-su","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $T$ be the run length transform of the periodic sequence $S$ with $S(i) = 0$ if $i \\equiv 2 \\pmod 3$ and $S(i) = 1$ otherwise, i.e. of $1, 1, 0, 1, 1, 0, \\dots$. Then for every natural number $n$, $\\sum_{k=0}^{n} \\left(\\binom{8k - 2n}{n + k} \\binom{n}{k} \\bmod 2\\right) = T(n)$, where a binomial coefficient $\\binom{a}{b}$ with $a < 0$ is $0$.","title":"Rampersad\u2013Wiebe Theorem 15"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-add-five-mul","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_add_five_mul_two_mul_add_two_mul","parent":11745,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\binom{n+5k}{2n+2k}\\binom{n}{k} \\bmod 2\\)"],["text"," as a run length transform"]],"source":"Sum of $\\binom{n+5k}{2n+2k}\\binom{n}{k} \\bmod 2$ as a run length transform"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-add-five-mul","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $S : \\mathbb{N} \\to \\mathbb{Z}$ satisfy $S(0) = S(1) = S(2) = S(3) = 1$ and $S(m) = S(m - 1) + S(m - 2) - S(m - 3) + S(m - 4)$ for every $m \\ge 4$. Then for every natural number $n$, the sum $\\sum_{k=0}^{n} \\bigl(\\binom{n+5k}{2n+2k}\\binom{n}{k} \\bmod 2\\bigr)$ equals the value at $n$ of the run length transform of $S$.","title":"Sum of $\\binom{n+5k}{2n+2k}\\binom{n}{k} \\bmod 2$ as a run length transform"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/run-length-transform-choose-add-five-mul-2","kind":"node","lean":"MetaMathlibExt.runLengthTransform_choose_add_five_mul_two_mul","parent":11745,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\binom{n+5k}{2k}\\binom{n}{k} \\bmod 2\\)"],["text"," as a run length transform"]],"source":"Sum of $\\binom{n+5k}{2k}\\binom{n}{k} \\bmod 2$ as a run length transform"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#run-length-transform-choose-add-five-mul-2","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"Let $S : \\mathbb{N} \\to \\mathbb{Z}$ satisfy $S(0) = 1$, $S(1) = S(2) = 2$, $S(3) = 3$ and $S(m) = S(m - 1) + S(m - 2) - S(m - 3) + S(m - 4)$ for every $m \\ge 4$. Then for every natural number $n$, the sum $\\sum_{k=0}^{n} \\bigl(\\binom{n+5k}{2k}\\binom{n}{k} \\bmod 2\\bigr)$ equals the value at $n$ of the run length transform of $S$.","title":"Sum of $\\binom{n+5k}{2k}\\binom{n}{k} \\bmod 2$ as a run length transform"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/sum-choose-two-pow-mul-mod-two-two-pow-s","kind":"node","lean":"MetaMathlibExt.sum_choose_two_pow_mul_mod_two_two_pow_sub_one","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#sum-choose-two-pow-mul-mod-two-two-pow-s","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"For all natural numbers $m$ and $\\ell$, with $n = 2^\\ell - 1$, the sum","title":"Binomial parity sums at one less than a power of two"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/sum-choose-two-pow-mul-mod-two-eq","kind":"node","lean":"MetaMathlibExt.sum_choose_two_pow_mul_mod_two_eq","parent":11745,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2/index.html#sum-choose-two-pow-mul-mod-two-eq","scope":"NumberTheory/special-numbers/binomial-coefficients/binomial-coefficient-sums-modulo-2","status":"can_prove","summary":"For all natural numbers $m$ and $n$, the sum","title":"Binomial parity sums detect run length divisibility"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic.json","id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","title":"Binomid indices and generalized binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-binomid-index","kind":"node","lean":"MetaMathlibExt.IsBinomidIndex","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-binomid-index","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"An infinite binomid index is a sequence $(e_1, e_2, \\dots)$ of nonnegative integers such that $\\sum_{i=n-k+1}^{n} e_i \\ge \\sum_{i=1}^{k} e_i$ for all $1 \\le k \\le n$.","title":"Binomid index"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-binomid-index-const","kind":"node","lean":"MetaMathlibExt.isBinomidIndex_const","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-binomid-index-const","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For every natural number $c$, the constant sequence $(c, c, c, \\dots)$ is a binomid index.","title":"Constant sequences are binomid indices"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-lex-minimal-extension-of","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-lex-minimal-extension-of","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"Let $e = (e_1, e_2, \\dots)$ be a sequence of nonnegative integers and $m \\ge 0$. Then $e$ is the lex-minimal extension of $(e_1, \\dots, e_m)$ when $e$ is an infinite binomid index and, for every infinite binomid index $g$ with $g_i = e_i$ for all $i \\le m$, either $g = e$ or $e$ is lexicographically smaller than $g$: there is a $j$ with $e_i = g_i$ for all $i < j$ and $e_j < g_j$.","title":"Lex-minimal extension"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-lex-minimal-extension","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtension","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-lex-minimal-extension","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"A sequence $e = (e_1, e_2, \\dots)$ of nonnegative integers is a lex-minimal extension when, for some $m \\ge 0$, it is the lex-minimal extension of $(e_1, \\dots, e_m)$. The set of all lex-minimal extensions is denoted $L$.","title":"Lex-minimal extensions"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-lex-minimal-extension-of-zero","kind":"node","lean":"MetaMathlibExt.isLexMinimalExtensionOf_zero","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-lex-minimal-extension-of-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For every $m \\ge 0$, the zero sequence $(0, 0, \\dots)$ is the lex-minimal extension of its first $m$ entries.","title":"The zero sequence is lex-minimal"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-shift","kind":"node","lean":"MetaMathlibExt.binomidShift","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-shift","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For a natural number $l$, the right shift $\\sigma^l$ sends a sequence $\\eta = (e_1, e_2, \\dots)$ of nonnegative integers to the sequence $(0, \\dots, 0, e_1, e_2, \\dots)$ obtained by prepending $l$ zeros: its first $l$ entries are $0$ and its $(l + i)$-th entry is $e_i$ for every $i \\ge 1$.","title":"Right shift of a sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-shift-zero","kind":"node","lean":"MetaMathlibExt.binomidShift_zero","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-shift-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For every sequence $\\eta$ of nonnegative integers, $\\sigma^0(\\eta) = \\eta$, where $\\sigma^l$ is the right shift that prepends $l$ zeros.","title":"Shifting by zero places is the identity"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-binomid-index-of-monotone","kind":"node","lean":"MetaMathlibExt.isBinomidIndex_of_monotone","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-binomid-index-of-monotone","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Every non-decreasing sequence $e_1 \\le e_2 \\le \\cdots$ of nonnegative integers is an infinite binomid index.","title":"Monotone sequences are binomid indices"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-binomid-index-iff-partial-sum","kind":"node","lean":"MetaMathlibExt.isBinomidIndex_iff_partialSum","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-binomid-index-iff-partial-sum","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $\\eta = (e_1, e_2, \\dots)$ be a sequence of nonnegative integers with partial sums $s_n = e_1 + \\dots + e_n$ (so $s_0 = 0$). Then $\\eta$ is a binomid index if and only if $s_{n-k} + s_k \\le s_n$ for all $k$ and $n$ with $1 \\le k < n$. Moreover, for every $m$, if $\\eta$ is the lex-minimal extension of its first $m$ entries, then for every $n > m$","title":"Partial-sum characterization of binomid indices"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-lex-minimal-extension-of-periodic-of","kind":"node","lean":"MetaMathlibExt.isLexMinimalExtensionOf_periodic_of_monotoneOn","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-lex-minimal-extension-of-periodic-of","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 1$ and let $e_1 \\le e_2 \\le \\cdots \\le e_m$ be nonnegative integers. Then the periodic sequence $(e_1, \\dots, e_m, e_1, \\dots, e_m, \\dots)$ is the lex-minimal extension of $(e_1, \\dots, e_m)$.","title":"Monotone finite binomid indices extend periodically"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/apply-period-eq-apply-zero","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtension.apply_period_eq_apply_zero","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#apply-period-eq-apply-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $\\eta = (e_1, e_2, \\dots)$ be a lex-minimal extension and let $p \\ge 1$. If $e_{i+p} = e_i$ for every $i \\ge 2$, then $e_{1+p} = e_1$, so the whole sequence $\\eta$ is periodic with period $p$.","title":"A lex-minimal extension periodic from its second entry is periodic"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-lex-minimal-extension-of-periodic","kind":"node","lean":"MetaMathlibExt.isLexMinimalExtension_of_periodic","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-lex-minimal-extension-of-periodic","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"If an infinite binomid index $e = (e_1, e_2, \\dots)$ is periodic, that is, there is $p \\ge 1$ with $e_{i+p} = e_i$ for all $i$, then $e$ is a lex-minimal extension: it is the lex-minimal extension of $(e_1, \\dots, e_m)$ for some $m \\ge 0$.","title":"Periodic binomid indices are lex-minimal extensions"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/add","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtension.add","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#add","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"If $\\eta_1$ and $\\eta_2$ are lex-minimal extensions, then their componentwise sum $\\eta_1 + \\eta_2$ is a lex-minimal extension.","title":"Lex-minimal extensions are closed under addition"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-shift-image-inter-eq-zero","kind":"node","lean":"MetaMathlibExt.binomidShift_image_inter_eq_zero","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-shift-image-inter-eq-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $L$ be the set of lex-minimal extensions, and for a natural number $l$ let $\\sigma^l$ be the right shift that prepends $l$ zeros to a sequence. For natural numbers $l \\neq m$, $\\sigma^l(L) \\cap \\sigma^m(L) = \\{(0, 0, 0, \\dots)\\}$: the only sequence in both images is the zero sequence.","title":"Distinct shifts of lex-minimal extensions meet only in zero"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/infinite-set-of-prime-dvd-choose","kind":"node","lean":"MetaMathlibExt.infinite_setOf_prime_dvd_choose","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#infinite-set-of-prime-dvd-choose","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"For every prime $p$ there are infinitely many natural numbers $n$ such that $p$ divides the binomial coefficient $\\binom{n}{k}$ for every $k$ with $1 \\le k < n$.","title":"Primes divide all inner binomial coefficients of infinitely many rows"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/infinite-set-of-prime-dvd-fibonomial","kind":"node","lean":"MetaMathlibExt.infinite_setOf_prime_dvd_fibonomial","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#infinite-set-of-prime-dvd-fibonomial","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers and $F_n! = F_1 F_2 \\cdots F_n$ the Fibonacci factorial. For every prime $p$ there are infinitely many natural numbers $n$ such that for every $k$ with $1 \\le k < n$ the Fibonomial coefficient $F_n! / (F_k! \\, F_{n-k}!)$ is an integer multiple of $p$.","title":"Primes divide all inner Fibonomial coefficients of infinitely many rows"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-lex-minimal-extension-of-of-greedy","kind":"node","lean":"MetaMathlibExt.isLexMinimalExtensionOf_of_greedy","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-lex-minimal-extension-of-of-greedy","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 1$ and let $f = (f_1, f_2, \\dots)$ be a sequence of nonnegative integers whose first $m$ entries form a finite binomid index, that is, $\\sum_{i=n-k+1}^{n} f_i \\ge \\sum_{i=1}^{k} f_i$ for all $1 \\le k \\le n \\le m$. Suppose that for every $n > m$ the entry $f_n$ equals the maximum, over $1 \\le k < n$, of the integers $\\sum_{i=1}^{k} f_i - \\sum_{i=n-k+1}^{n-1} f_i$. Then $f$ is the lex-minimal extension of\u2026","title":"The lex-minimal extension is computed greedily"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/exists-nomid-eq-one","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.exists_nomid_eq_one","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#exists-nomid-eq-one","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 1$, let $\\eta = (e_1, e_2, \\dots)$ be the lex-minimal extension of its first $m$ entries, and let $p$ be a prime. Write $f_i = p^{e_i}$ for $i \\ge 1$ and, for $1 \\le k \\le n$, let $[n, k]_f = \\frac{f_n f_{n-1} \\cdots f_{n-k+1}}{f_k f_{k-1} \\cdots f_1}$ be the $f$-nomid coefficient, the generalized binomial coefficient of $f$. Then for every $n > m$ there is some $k$ with $1 \\le k < n$ and $[n, k]_f = 1$.","title":"Lex-minimal extensions have a nomid coefficient equal to one"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/infinite-set-of-prime-dvd-nomid-of-not-i","kind":"node","lean":"MetaMathlibExt.infinite_setOf_prime_dvd_nomid_of_not_isLexMinimalExtension","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#infinite-set-of-prime-dvd-nomid-of-not-i","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $\\mu = (e_1, e_2, \\dots)$ be a binomid index such that for no $m$ is $\\mu$ the lex-minimal extension of its first $m$ entries, and let $p$ be a prime. Write $f_i = p^{e_i}$ for $i \\ge 1$ and let $[n, k]_f = \\frac{f_n f_{n-1} \\cdots f_{n-k+1}}{f_k f_{k-1} \\cdots f_1}$ be the $f$-nomid coefficient. Then there are infinitely many natural numbers $n$ such that for every $k$ with $1 \\le k < n$ the coefficient\u2026","title":"Nomid coefficients of binomid indices that are not lex-minimal extensions"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-average","kind":"node","lean":"MetaMathlibExt.binomidAverage","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-average","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For a sequence $\\eta = (e_1, e_2, \\dots)$ of nonnegative integers and $i \\ge 1$, the average $A_{i,0}(\\eta)$ is the rational number $s_i / i$, where $s_i = e_1 + \\dots + e_i$ is the sum of the first $i$ entries; for $i = 0$ the value is $A_{0,0}(\\eta) = 0$.","title":"Initial averages of a sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-average-const","kind":"node","lean":"MetaMathlibExt.binomidAverage_const","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-average-const","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For every natural number $c$ and every $i \\ge 1$, the average $A_{i,0}$ of the first $i$ entries of the constant sequence $(c, c, c, \\dots)$ equals $c$.","title":"Averages of a constant sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-upper-average","kind":"node","lean":"MetaMathlibExt.binomidUpperAverage","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-upper-average","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For a sequence $\\eta = (e_1, e_2, \\dots)$ of nonnegative integers with partial sums $s_i = e_1 + \\dots + e_i$ and averages $A_{i,0}(\\eta) = s_i / i$, the upper average of $\\eta$ is $A_0(\\eta) = \\limsup_{i \\to \\infty} A_{i,0}(\\eta)$, taken in the extended real numbers; it is $+\\infty$ when $s_i / i$ is unbounded.","title":"Upper average of a sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-upper-average-const","kind":"node","lean":"MetaMathlibExt.binomidUpperAverage_const","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-upper-average-const","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For every natural number $c$, the upper average $A_0$ of the constant sequence $(c, c, c, \\dots)$ equals $c$.","title":"Upper average of a constant sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-average-condition","kind":"node","lean":"MetaMathlibExt.BinomidAverageCondition","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-average-condition","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"A sequence $\\eta = (e_1, e_2, \\dots)$ of nonnegative integers satisfies the Average Condition when its upper average $A_0(\\eta)$, the limit superior as $n \\to \\infty$ of the averages $A_{n,0}(\\eta) = (e_1 + \\dots + e_n)/n$ taken in the extended real line, equals $A_{i,0}(\\eta)$ for some $i \\ge 1$.","title":"The Average Condition"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-average-condition-const","kind":"node","lean":"MetaMathlibExt.binomidAverageCondition_const","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-average-condition-const","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For every natural number $c$, the constant sequence $(c, c, c, \\dots)$ satisfies the Average Condition.","title":"Constant sequences satisfy the Average Condition"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-average-props","kind":"node","lean":"MetaMathlibExt.IsBinomidIndex.binomidAverage_props","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-average-props","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $\\eta = (e_1, e_2, \\dots)$ be a binomid index, write $s_n = e_1 + \\dots + e_n$, and for $i \\ge 1$ and $l \\ge 0$ let $A_{i,l} = (s_{i+l} - s_l)/i$ be the average of $e_{l+1}, \\dots, e_{l+i}$. Then for all $i, l \\ge 1$: $A_{i,l} \\ge A_{i,0}$; $A_{ki,0} \\ge A_{i,0}$ for every $k \\ge 1$; and $A_{i+l,0} = \\frac{i}{i+l} A_{i,l} + \\frac{l}{l+i} A_{l,0}$.","title":"Averages of a binomid index"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-upper-average-eq-i-sup","kind":"node","lean":"MetaMathlibExt.IsBinomidIndex.binomidUpperAverage_eq_iSup","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-upper-average-eq-i-sup","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $\\eta$ be a binomid index, let $s_i$ be the sum of its first $i$ entries, let $A_{i,0}(\\eta) = s_i / i$ for $i \\ge 1$, and let $A_0(\\eta) = \\limsup_{i \\to \\infty} A_{i,0}(\\eta)$ be its upper average. Then, in the extended real numbers, $A_0(\\eta) = \\sup_{i \\ge 1} A_{i,0}(\\eta)$.","title":"The upper average of a binomid index is the supremum of its averages"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-average-condition-2","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtension.binomidAverageCondition","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-average-condition-2","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Every lex-minimal extension satisfies the Average Condition.","title":"Lex-minimal extensions satisfy the Average Condition"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-lex-minimal-extension-and-eventually","kind":"node","lean":"MetaMathlibExt.IsBinomidIndex.isLexMinimalExtension_and_eventually_periodic","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-lex-minimal-extension-and-eventually","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"If an infinite binomid index $e = (e_1, e_2, \\dots)$ satisfies the Average Condition, then $e$ is a lex-minimal extension, that is, the lex-minimal extension of $(e_1, \\dots, e_m)$ for some $m \\ge 0$, and $e$ is eventually periodic: there are $N$ and $p \\ge 1$ such that $e_{n+p} = e_n$ for all $n \\ge N$.","title":"The Average Condition gives eventually periodic lex-minimal extensions"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-average-one","kind":"node","lean":"MetaMathlibExt.binomidAverage_one","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-average-one","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For every sequence $\\eta = (e_1, e_2, \\dots)$ of nonnegative integers, the initial average $A_{1,0}(\\eta)$ of its first entry equals $e_1$.","title":"The first average is the first entry"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/max-average-indices","kind":"node","lean":"MetaMathlibExt.maxAverageIndices","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#max-average-indices","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"For a sequence $\\eta = (e_1, e_2, \\dots)$ of nonnegative integers and $m \\ge 0$, let $A_i = (e_1 + \\dots + e_i)/i$ be the average of the first $i$ entries of $\\eta$. The set of maximal-average indices of $\\eta$ up to $m$ is the set of all $i$ with $1 \\le i \\le m$ such that $A_j \\le A_i$ for every $j$ with $1 \\le j \\le m$; for $m \\ge 1$ these are the $i$ with $1 \\le i \\le m$ and $A_i = A_{\\max}$, where\u2026","title":"Indices of maximal average"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/max-average-indices-one","kind":"node","lean":"MetaMathlibExt.maxAverageIndices_one","parent":11769,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal-average indices for "],["math","\\(m = 1\\)"]],"source":"Maximal-average indices for $m = 1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#max-average-indices-one","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"Let $\\eta = (e_1, e_2, \\dots)$ be a sequence of nonnegative integers and $A_i = (e_1 + \\dots + e_i)/i$. For $m = 1$, the set of indices $1 \\le i \\le m$ at which $A_i$ is maximal among $A_1, \\dots, A_m$ is $\\{1\\}$.","title":"Maximal-average indices for $m = 1$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-add-monoid-basis","kind":"node","lean":"MetaMathlibExt.IsAddMonoidBasis","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-add-monoid-basis","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"Let $M$ and $B$ be subsets of an additive commutative monoid. Then $B$ is a basis of $M$ when $M$ is the submonoid $\\langle B \\rangle$ generated by $B$ and no element of $B$ is a nonnegative integer combination of elements of $B$ in which its own coefficient is $0$: for all pairwise distinct $\\beta_1, \\dots, \\beta_n \\in B$, all nonnegative integers $c_1, \\dots, c_n$ and every $j$, if\u2026","title":"Basis of a submonoid"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/zero-not-mem","kind":"node","lean":"MetaMathlibExt.IsAddMonoidBasis.zero_notMem","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#zero-not-mem","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid and let $M$ and $B$ be subsets of $\\alpha$. If $B$ is a basis of $M$, then $0 \\notin B$.","title":"A monoid basis does not contain zero"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-add-monoid-basis-empty","kind":"node","lean":"MetaMathlibExt.isAddMonoidBasis_empty","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-add-monoid-basis-empty","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"In every additive commutative monoid, the empty set is a basis of the subset $\\{0\\}$.","title":"The empty set is a basis of the zero submonoid"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-atomic-binomid-index","kind":"node","lean":"MetaMathlibExt.IsAtomicBinomidIndex","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-atomic-binomid-index","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"A sequence $\\eta$ of nonnegative integers is a nonzero atomic binomid index (an element of the paper's set $\\mathrm{Atom}$) when $\\eta$ is a nonzero binomid index that is not the sum of two nonzero binomid indices: whenever $\\eta = \\eta_1 + \\eta_2$ termwise with $\\eta_1$ and $\\eta_2$ binomid indices, $\\eta_1 = 0$ or $\\eta_2 = 0$.","title":"Atomic binomid index"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-atomic-binomid-index-one","kind":"node","lean":"MetaMathlibExt.isAtomicBinomidIndex_one","parent":11769,"reading_title":{"format":"inline-parts-v1","parts":[["text","The constant sequence "],["math","\\(1\\)"],["text"," is atomic"]],"source":"The constant sequence $1$ is atomic"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-atomic-binomid-index-one","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"fully_proved","summary":"The constant sequence $(1, 1, 1, \\dots)$ is a nonzero atomic binomid index: it is a nonzero binomid index that is not the sum of two nonzero binomid indices.","title":"The constant sequence $1$ is atomic"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-lex-minimal-extension-of-iff-forall-l","kind":"node","lean":"MetaMathlibExt.isLexMinimalExtensionOf_iff_forall_lt","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-lex-minimal-extension-of-iff-forall-l","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $k \\ge 1$, let $\\eta = (e_1, e_2, \\dots)$ be a sequence of nonnegative integers and write $S_i = e_1 + \\dots + e_i$. Suppose $S_i + S_j \\le S_{i+j}$ for all $i, j \\ge 1$ with $i + j \\le k$. Then $\\eta$ is the lex-minimal extension of $(e_1, \\dots, e_k)$ if and only if, for every $r > k$, $S_i + S_j \\le S_r$ for all $i, j \\ge 1$ with $i + j = r$, with equality for at least one such pair $(i, j)$.","title":"Partial-sum criterion for lex-minimal extensions"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/exists-knapsack","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.exists_knapsack","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#exists-knapsack","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 1$, let $\\eta = (e_1, e_2, \\dots)$ be the lex-minimal extension of its first $m$ entries, and write $S_i = e_1 + \\dots + e_i$. Then for every $c \\ge 1$ there are nonnegative integers $n_1, \\dots, n_m$ with $\\sum_{i=1}^m n_i i = c$ and $S_c = \\sum_{i=1}^m n_i S_i$ that solve the unbounded knapsack problem with values $S_i$, weights $i$ and capacity $c$: all nonnegative integers $n'_1, \\dots, n'_m$ with\u2026","title":"Partial sums of a lex-minimal extension solve a knapsack problem"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/binomid-average-le","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.binomidAverage_le","parent":11769,"reading_title":{"format":"inline-parts-v1","parts":[["text","The largest of the first "],["math","\\(m\\)"],["text"," averages is the largest average"]],"source":"The largest of the first $m$ averages is the largest average"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#binomid-average-le","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 0$ and let $\\eta = (e_1, e_2, \\dots)$ be the lex-minimal extension of its first $m$ entries, with averages $A_i = (e_1 + \\dots + e_i)/i$ for $i \\ge 1$. If $1 \\le j \\le m$ and $A_j = \\max \\{A_i \\mid 1 \\le i \\le m\\}$, then $A_i \\le A_j$ for every $i \\ge 1$.","title":"The largest of the first $m$ averages is the largest average"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/gcd-max-average-indices-dvd","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.gcd_maxAverageIndices_dvd","parent":11769,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every index with maximal average is a multiple of the gcd of those up to "],["math","\\(m\\)"]],"source":"Every index with maximal average is a multiple of the gcd of those up to $m$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#gcd-max-average-indices-dvd","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 0$, let $\\eta = (e_1, e_2, \\dots)$ be the lex-minimal extension of its first $m$ entries, with averages $A_i = (e_1 + \\dots + e_i)/i$, let $J$ be the set of indices $1 \\le j \\le m$ with $A_j = \\max \\{A_{j'} \\mid 1 \\le j' \\le m\\}$, and let $g = \\gcd J$. Then $g$ divides every $i \\ge 1$ such that $A_i = A_j$ for some $j \\in J$.","title":"Every index with maximal average is a multiple of the gcd of those up to $m$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/gcd-max-average-indices-dvd-minimal-peri","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.gcd_maxAverageIndices_dvd_minimalPeriod","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#gcd-max-average-indices-dvd-minimal-peri","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 1$, let $\\eta = (e_1, e_2, \\dots)$ be the lex-minimal extension of its first $m$ entries, with averages $A_i = (e_1 + \\dots + e_i)/i$, and let $g$ be the greatest common divisor of the indices $1 \\le i \\le m$ with $A_i = \\max \\{A_j \\mid 1 \\le j \\le m\\}$. If $p$ is the least $q \\ge 1$ such that $e_{i+q} = e_i$ for all sufficiently large $i$, then $g$ divides $p$.","title":"The gcd of the maximizing indices divides the minimal period"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-least-period-gcd-max-average-indices","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.isLeast_period_gcd_maxAverageIndices","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-least-period-gcd-max-average-indices","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 1$ and let $\\eta = (e_1, e_2, \\dots)$ be the lex-minimal extension of $(e_1, \\dots, e_m)$. For $1 \\le i \\le m$ let $A_i = (e_1 + \\dots + e_i)/i$, and let $A_{\\max}$ be the largest of $A_1, \\dots, A_m$. Let $g$ be the greatest common divisor of the indices $1 \\le i \\le m$ with $A_i = A_{\\max}$. Then $g$ is the minimal eventual period of $\\eta$: $g \\ge 1$, $e_{i+g} = e_i$ for all sufficiently large $i$,\u2026","title":"Minimal period of a lex-minimal extension"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/exists-period-from-preperiod-bound","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.exists_period_from_preperiod_bound","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#exists-period-from-preperiod-bound","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $m \\ge 0$, let $\\eta = (e_1, e_2, \\dots)$ be the lex-minimal extension of its first $m$ entries, with averages $A_i = (e_1 + \\dots + e_i)/i$, and let $1 \\le k \\le m$ with $A_k = \\max \\{A_i \\mid 1 \\le i \\le m\\}$. Then $\\eta$ is periodic after position $N = k m (m + 1)/2$: there is $p \\ge 1$ with $e_{i+p} = e_i$ for all $i > N$.","title":"A preperiod bound for lex-minimal extensions"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/of-add","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.of_add","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#of-add","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $\\eta_1$ and $\\eta_2$ be infinite binomid indices and let $k \\ge 0$. If the termwise sum $\\eta_1 + \\eta_2$ is the lex-minimal extension of its first $k$ entries, then $\\eta_1$ is the lex-minimal extension of its first $k$ entries and $\\eta_2$ is the lex-minimal extension of its first $k$ entries.","title":"Summands of a lex-minimal extension are lex-minimal"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-add-monoid-basis-iff-eq-atoms","kind":"node","lean":"MetaMathlibExt.isAddMonoidBasis_iff_eq_atoms","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-add-monoid-basis-iff-eq-atoms","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $\\iota$ be a type and let $M$ be a submonoid of $\\mathbb{N}^\\iota$ under componentwise addition. Let $A$ be the set of atoms of $M$: the elements $a \\in M$ with $a \\ne 0$ such that $y = 0$ or $z = 0$ whenever $a = y + z$ with $y, z \\in M$. If every element of $M$ is a finite sum of elements of $A$, then a set $B \\subseteq \\mathbb{N}^\\iota$ is a basis of $M$ if and only if $B = A$; that is, $A$ is the unique\u2026","title":"Atoms are the unique basis of an atom-generated submonoid"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-add-monoid-basis-lex-minimal-extensio","kind":"node","lean":"MetaMathlibExt.isAddMonoidBasis_lexMinimalExtensions_iff","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-add-monoid-basis-lex-minimal-extensio","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $L$ be the set of lex-minimal extensions, viewed inside the sequences of nonnegative integers under componentwise addition. A set $B$ of sequences of nonnegative integers is a basis of $L$ if and only if $B$ is the set of nonzero atomic binomid indices that belong to $L$. Hence that set is the unique basis of $L$.","title":"Basis of the monoid of lex-minimal extensions"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/is-add-monoid-basis-closure-lex-minimal","kind":"node","lean":"MetaMathlibExt.isAddMonoidBasis_closure_lexMinimalExtensionsOf_iff","parent":11769,"reading_title":{"format":"inline-parts-v1","parts":[["text","Basis of the span of the lex-minimal extensions of length "],["math","\\(k\\)"]],"source":"Basis of the span of the lex-minimal extensions of length $k$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#is-add-monoid-basis-closure-lex-minimal","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $k \\ge 0$ and let $L_k$ be the set of sequences $\\eta = (e_1, e_2, \\dots)$ of nonnegative integers that are the lex-minimal extension of $(e_1, \\dots, e_k)$. A set $B$ of sequences of nonnegative integers is a basis of the submonoid generated by $L_k$ under componentwise addition if and only if $B$ is the set of nonzero atomic binomid indices that belong to $L_k$. Hence that set is the unique basis of this\u2026","title":"Basis of the span of the lex-minimal extensions of length $k$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/exists-path","kind":"node","lean":"MetaMathlibExt.IsLexMinimalExtensionOf.exists_path","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#exists-path","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"Let $k \\ge 1$, let $\\eta = (e_1, e_2, \\dots)$ be the lex-minimal extension of its first $k$ entries, and write $S_i = e_1 + \\dots + e_i$. Then for every $l$ there is a map $\\pi \\colon \\mathbb{N} \\to \\mathbb{N}$ such that $1 \\le \\pi(i) < i$ and $S_i = S_{\\pi(i)} + S_{i - \\pi(i)}$ for every $i$ with $k < i \\le l$.","title":"Late partial sums of a lex-minimal extension split"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/fg-closure-lex-minimal-extensions-of","kind":"node","lean":"MetaMathlibExt.fg_closure_lexMinimalExtensionsOf","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#fg-closure-lex-minimal-extensions-of","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"For $k \\ge 0$ let $L_k$ be the set of sequences of nonnegative integers that are the lex-minimal extension of their first $k$ entries. Then the submonoid of $\\mathbb{N}^{\\mathbb{N}}$ (under componentwise addition) generated by $L_k$ is finitely generated, and only finitely many elements of $L_k$ are atomic binomid indices, that is, nonzero binomid indices that are not the sum of two nonzero binomid indices.","title":"Spans of lex-minimal extensions of a fixed length are finitely generated"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/exists-finite-presentation-closure-lex-m","kind":"node","lean":"MetaMathlibExt.exists_finitePresentation_closure_lexMinimalExtensionsOf","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#exists-finite-presentation-closure-lex-m","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"For $k \\ge 0$ let $L_k$ be the set of sequences of nonnegative integers that are the lex-minimal extension of their first $k$ entries, and let $\\langle L_k \\rangle$ be the submonoid of $\\mathbb{N}^{\\mathbb{N}}$ (under componentwise addition) generated by $L_k$. Then there are $n \\ge 0$, a surjective monoid homomorphism $f \\colon \\mathbb{N}^n \\to \\langle L_k \\rangle$ and a finite set $R$ of pairs of elements of\u2026","title":"Spans of lex-minimal extensions of a fixed length are finitely presented"},{"id":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/i-union-closure-lex-minimal-extensions-o","kind":"node","lean":"MetaMathlibExt.iUnion_closure_lexMinimalExtensionsOf","parent":11769,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic/index.html#i-union-closure-lex-minimal-extensions-o","scope":"NumberTheory/special-numbers/binomial-coefficients/binomid-indices-and-generalized-binomial-coeffic","status":"can_prove","summary":"For $k \\ge 0$ let $L_k$ be the set of sequences of nonnegative integers that are the lex-minimal extension of their first $k$ entries, let $\\langle L_k \\rangle$ be the submonoid of $\\mathbb{N}^{\\mathbb{N}}$ (under componentwise addition) generated by $L_k$, and let $\\mathcal{A}_k$ be the set of elements of $L_k$ that are atomic binomid indices, that is, nonzero binomid indices that are not the sum of two nonzero\u2026","title":"The monoid of lex-minimal extensions is a directed union"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients.json","id":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients","status":"can_prove","title":"Cubed central binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/a","kind":"node","lean":"MathlibExt.NumberTheory.OeisA002897Wanted.a","parent":11822,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/index.html#a","scope":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients","status":"fully_proved","summary":"For a natural number $n$, $a(n) = \\binom{2n}{n}^3$.","title":"Cubed central binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/vars","kind":"node","lean":"MathlibExt.NumberTheory.OeisA002897Wanted.Vars","parent":11822,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/index.html#vars","scope":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients","status":"fully_proved","summary":"Three polynomial variables are indexed by the set $\\{0, 1, 2\\}$.","title":"Indices for three variables"},{"id":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/xyz-pow-n","kind":"node","lean":"MathlibExt.NumberTheory.OeisA002897Wanted.xyzPowN","parent":11822,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exponent vector of "],["math","\\(x^n y^n z^n\\)"]],"source":"Exponent vector of $x^n y^n z^n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/index.html#xyz-pow-n","scope":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients","status":"fully_proved","summary":"For a natural number $n$, the exponent vector assigning exponent $n$ to each of the three variables $x, y, z$, that is, the monomial $x^n y^n z^n$.","title":"Exponent vector of $x^n y^n z^n$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/p-poly","kind":"node","lean":"MathlibExt.NumberTheory.OeisA002897Wanted.pPoly","parent":11822,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/index.html#p-poly","scope":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients","status":"fully_proved","summary":"For a natural number $n$, this is the polynomial $(1 + x + y + z)^{2n} (1 + x + y - z)^n (1 + x - y + z)^n$ in three variables $x, y, z$ (indexed by $0, 1, 2$) with integer coefficients.","title":"Product polynomial for the cubed central binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/a-eq-coeff","kind":"node","lean":"MathlibExt.NumberTheory.OeisA002897Wanted.a_eq_coeff","parent":11822,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/index.html#a-eq-coeff","scope":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients","status":"fully_proved","summary":"This is the proposition that for every natural number $n$, the coefficient of $x^n y^n z^n$ in the polynomial $(1 + x + y + z)^{2n} (1 + x + y - z)^n (1 + x - y + z)^n$ in three variables $x, y, z$ with integer coefficients equals $\\binom{2n}{n}^3$.","title":"Diagonal-coefficient property of the cubed central binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/a-eq-coeff-holds","kind":"node","lean":"MathlibExt.NumberTheory.OeisA002897Wanted.a_eq_coeff_holds","parent":11822,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients/index.html#a-eq-coeff-holds","scope":"NumberTheory/special-numbers/binomial-coefficients/cubed-central-binomial-coefficients","status":"can_prove","summary":"For every natural number $n$, the coefficient of $x^n y^n z^n$ in the polynomial $(1 + x + y + z)^{2n} (1 + x + y - z)^n (1 + x - y + z)^n$ in three variables $x, y, z$ with integer coefficients equals $\\binom{2n}{n}^3$.","title":"Cubed central binomial coefficients as diagonal coefficients"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient.json","id":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","status":"can_prove","title":"Divisors of the middle binomial coefficient"},{"id":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/padic-val-nat-factorial-div-factorial-le","kind":"node","lean":"MetaMathlibExt.padicValNat_factorial_div_factorial_le","parent":11829,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/index.html#padic-val-nat-factorial-div-factorial-le","scope":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","status":"can_prove","summary":"Let $m$ and $k$ be natural numbers and $p$ a prime, and write $v_p(n)$ for the exponent of $p$ in a positive integer $n$. Then $v_p\\bigl((m+k)!/m!\\bigr) \\le v_p(k!) + \\max \\{ v_p(m + i) : 1 \\le i \\le k \\}$, where the maximum is $0$ when $k = 0$.","title":"Valuation of a product of consecutive integers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/exists-density-one-padic-val-nat-central","kind":"node","lean":"MetaMathlibExt.exists_density_one_padicValNat_centralBinom_bounds","parent":11829,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/index.html#exists-density-one-padic-val-nat-central","scope":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","status":"can_prove","summary":"Write $v_p(n)$ for the exponent of a prime $p$ in a positive integer $n$, and for $m \\ge 2$ let $\\alpha_p(m) = v_p\\bigl(\\binom{2m}{m}\\bigr) / (\\log m / \\log p)$. There is a set $S$ of natural numbers of natural density $1$ such that: $\\alpha_2(m) \\to 1/2$ as $m \\to \\infty$ within $S$; for every $\\varepsilon > 0$, $\\alpha_3(m) \\ge 34/81 - \\varepsilon$ for all sufficiently large $m \\in S$; and $\\alpha_p(m) \\ge 0.39$\u2026","title":"Valuations of the middle binomial coefficient on a set of density one"},{"id":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/factorial-div-factorial-dvd-central-bino","kind":"node","lean":"MetaMathlibExt.factorial_div_factorial_dvd_centralBinom_density_one","parent":11829,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/index.html#factorial-div-factorial-dvd-central-bino","scope":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","status":"can_prove","summary":"Let $\\eta$ be a real number with $\\eta < 1 / \\log 4$. The set of natural numbers $m$ such that $(m+k)!/m! = (m+1)(m+2) \\cdots (m+k)$ divides $\\binom{2m}{m}$ for every integer $k$ with $1 \\le k \\le \\eta \\log m$ has natural density $1$.","title":"Products of consecutive integers dividing the middle binomial coefficient"},{"id":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/max-padic-val-nat-add-le-density-one","kind":"node","lean":"MetaMathlibExt.max_padicValNat_add_le_density_one","parent":11829,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/index.html#max-padic-val-nat-add-le-density-one","scope":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","status":"can_prove","summary":"Write $v_p(n)$ for the exponent of a prime $p$ in a positive integer $n$. The set of natural numbers $m$ such that $\\max \\{ v_p(m + i) : 1 \\le i \\le k \\} \\le 3 \\log k / \\log p$ for every integer $k$ with $\\frac{1}{2} \\log m < k < \\log m$ and every prime $p \\le 2k$ has natural density $1$.","title":"Short runs of consecutive integers have small valuations almost always"},{"id":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/card-padic-val-nat-central-binom-le-is-l","kind":"node","lean":"MetaMathlibExt.card_padicValNat_centralBinom_le_isLittleO","parent":11829,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/index.html#card-padic-val-nat-central-binom-le-is-l","scope":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","status":"can_prove","summary":"For a natural number $x$ let $K = \\lfloor \\exp(0.8 \\sqrt{\\log x}) \\rfloor$ and, for a prime $p$, let $D_p = 1 + \\lfloor \\log x / \\log p \\rfloor$. Write $v_p(n)$ for the exponent of $p$ in a positive integer $n$. Then the number of integers $m$ with $1 \\le m \\le x$ for which some prime $p \\le 2K$ satisfies $v_p\\bigl(\\binom{2m}{m}\\bigr) \\le D_p / \\log D_p$ is $o(x)$ as $x \\to \\infty$.","title":"Small valuations of the middle binomial coefficient are rare"},{"id":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/card-max-padic-val-nat-add-gt-is-little","kind":"node","lean":"MetaMathlibExt.card_max_padicValNat_add_gt_isLittleO","parent":11829,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/index.html#card-max-padic-val-nat-add-gt-is-little","scope":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","status":"can_prove","summary":"For a natural number $x$ let $K = \\lfloor \\exp(0.8 \\sqrt{\\log x}) \\rfloor$ and, for a prime $p$, let $D_p = 1 + \\lfloor \\log x / \\log p \\rfloor$. Write $v_p(n)$ for the exponent of $p$ in a positive integer $n$. Then the number of integers $m$ with $1 \\le m \\le x$ for which some prime $p < 2K$ satisfies $\\max \\{ v_p(m + i) : 1 \\le i \\le K \\} > D_p / \\log D_p$ is $o(x)$ as $x \\to \\infty$.","title":"Large valuations in short runs of consecutive integers are rare"},{"id":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/choose-dvd-central-binom-density-one","kind":"node","lean":"MetaMathlibExt.choose_dvd_centralBinom_density_one","parent":11829,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient/index.html#choose-dvd-central-binom-density-one","scope":"NumberTheory/special-numbers/binomial-coefficients/divisors-of-the-middle-binomial-coefficient","status":"can_prove","summary":"The set of natural numbers $m$ such that $\\binom{m+k}{k}$ divides $\\binom{2m}{m}$ for every integer $k$ with $1 \\le k \\le \\exp(0.8 \\sqrt{\\log m})$ has natural density $1$.","title":"Binomial coefficients dividing the middle binomial coefficient"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients.json","id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","title":"Equations in central binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-eq-central-binom-mul-c","kind":"node","lean":"MetaMathlibExt.centralBinom_add_eq_centralBinom_mul_choose_iff","parent":11837,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-eq-central-binom-mul-c","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $x$ and $b$ be positive integers and let $a$ be a nonnegative integer. Then $\\binom{2(a+b)}{a+b} = \\binom{2a}{a}\\binom{x+2b}{b}$ if and only if $x = 1$ and $a = 1$.","title":"Allen's theorem on central binomial products"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-le-central-binom-mul-c","kind":"node","lean":"MetaMathlibExt.centralBinom_add_le_centralBinom_mul_choose_of_le","parent":11837,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-le-central-binom-mul-c","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $x$ and $b$ be positive integers and let $a$ be a nonnegative integer with $a \\le x$. Then $\\binom{2(a+b)}{a+b} \\le \\binom{2a}{a}\\binom{x+2b}{b}$, and equality holds if and only if $x = 1$ and $a = 1$.","title":"Allen's inequality for central binomial products"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-eq-central-binom-mul-c-2","kind":"node","lean":"MetaMathlibExt.centralBinom_add_eq_centralBinom_mul_choose_iff_of_le_ten","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial products with "],["math","\\(a + b \\le 10\\)"]],"source":"Central binomial products with $a + b \\le 10$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-eq-central-binom-mul-c-2","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $x$, $a$, $b$ be positive integers with $a + b \\le 10$. Then $\\binom{2(a+b)}{a+b} = \\binom{2a}{a}\\binom{x+2b}{b}$ if and only if $x = 1$ and $a = 1$.","title":"Central binomial products with $a + b \\le 10$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-central-binom-mul-c","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_centralBinom_mul_choose_sub_one","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial products in the case "],["math","\\(x = a - 1\\)"]],"source":"Central binomial products in the case $x = a - 1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-central-binom-mul-c","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $a$ and $b$ be natural numbers with $4 < 2a \\le a + b$. Then $\\binom{2(a+b)}{a+b} \\ne \\binom{2a}{a}\\binom{a+2b-1}{b}$, so the equation $\\binom{2(a+b)}{a+b} = \\binom{2a}{a}\\binom{x+2b}{b}$ has no solution with $x = a - 1$.","title":"Central binomial products in the case $x = a - 1$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-central-binom-mul-c-2","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_centralBinom_mul_choose_sub_two","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial products in the case "],["math","\\(x = a - 2\\)"]],"source":"Central binomial products in the case $x = a - 2$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-central-binom-mul-c-2","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $a$ and $b$ be natural numbers with $4 < 2a \\le a + b$. Then $\\binom{2(a+b)}{a+b} \\ne \\binom{2a}{a}\\binom{a+2b-2}{b}$, so the equation $\\binom{2(a+b)}{a+b} = \\binom{2a}{a}\\binom{x+2b}{b}$ has no solution with $x = a - 2$.","title":"Central binomial products in the case $x = a - 2$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-central-binom-mul-c-3","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_centralBinom_mul_choose_sub_three","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial products in the case "],["math","\\(x = a - 3\\)"]],"source":"Central binomial products in the case $x = a - 3$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-central-binom-mul-c-3","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $a$ and $b$ be natural numbers with $4 < 2a \\le a + b$. Then $\\binom{2(a+b)}{a+b} \\ne \\binom{2a}{a}\\binom{a+2b-3}{b}$, so the equation $\\binom{2(a+b)}{a+b} = \\binom{2a}{a}\\binom{x+2b}{b}$ has no solution with $x = a - 3$.","title":"Central binomial products in the case $x = a - 3$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/exists-prime-dvd-prod-consecutive-gt-thr","kind":"node","lean":"MetaMathlibExt.exists_prime_dvd_prod_consecutive_gt_three_halves_iff","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hanson's theorem on a prime factor above "],["math","\\(3m/2\\)"],["text"," in consecutive integers"]],"source":"Hanson's theorem on a prime factor above $3m/2$ in consecutive integers"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#exists-prime-dvd-prod-consecutive-gt-thr","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $m$ and $n$ be natural numbers with $0 < m < n$. The product $n(n+1)\\cdots(n+m-1)$ has a prime divisor $p$ with $p > 3m/2$ if and only if $(n, m)$ is none of the pairs $(3, 2)$, $(8, 2)$ and $(6, 5)$ (the products $3 \\cdot 4$, $8 \\cdot 9$ and $6 \\cdot 7 \\cdot 8 \\cdot 9 \\cdot 10$).","title":"Hanson's theorem on a prime factor above $3m/2$ in consecutive integers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/choose-mul-mul-bounds","kind":"node","lean":"MetaMathlibExt.choose_mul_mul_bounds","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","St\u0103nic\u0103's bounds for the binomial coefficient "],["math","\\(\\binom{mn}{rn}\\)"]],"source":"St\u0103nic\u0103's bounds for the binomial coefficient $\\binom{mn}{rn}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#choose-mul-mul-bounds","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $m$, $n$ and $r$ be natural numbers with $1 \\le r < m$ and $n \\ge 1$, and put","title":"St\u0103nic\u0103's bounds for the binomial coefficient $\\binom{mn}{rn}$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-of-half-le-of-three","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_of_half_le_of_three_le","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial products with "],["math","\\(a \\ge b \\ge 3\\)"]],"source":"Central binomial products with $a \\ge b \\ge 3$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-of-half-le-of-three","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $x$, $a$, $b$ be natural numbers with $x \\ge 1$ and $3 \\le b \\le a$. Then $\\binom{2(a+b)}{a+b} \\ne \\binom{2a}{a}\\binom{x+2b}{b}$.","title":"Central binomial products with $a \\ge b \\ge 3$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-of-half-le-of-two-l","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_of_half_le_of_two_le","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial products with "],["math","\\(x \\ge 2\\)"],["text"," and "],["math","\\(a \\ge b\\)"]],"source":"Central binomial products with $x \\ge 2$ and $a \\ge b$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-of-half-le-of-two-l","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $x$, $a$, $b$ be natural numbers with $x \\ge 2$ and $1 \\le b \\le a$. Then $\\binom{2(a+b)}{a+b} \\ne \\binom{2a}{a}\\binom{x+2b}{b}$.","title":"Central binomial products with $x \\ge 2$ and $a \\ge b$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/exists-prime-dvd-prod-consecutive-gt-nin","kind":"node","lean":"MetaMathlibExt.exists_prime_dvd_prod_consecutive_gt_nine_fifths","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prime factor above "],["math","\\(9m/5\\)"],["text"," in a product of consecutive integers"]],"source":"A prime factor above $9m/5$ in a product of consecutive integers"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#exists-prime-dvd-prod-consecutive-gt-nin","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $m$ and $n$ be natural numbers with $2 < m < n$ and $n + m \\ge 150$. Then the product $n(n+1)\\cdots(n+m-1)$ has a prime divisor $p$ with $p > 9m/5$.","title":"A prime factor above $9m/5$ in a product of consecutive integers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/exists-prime-dvd-prod-consecutive-gt-fou","kind":"node","lean":"MetaMathlibExt.exists_prime_dvd_prod_consecutive_gt_four_point_four_two","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prime factor above "],["math","\\(4.42m\\)"],["text"," in a product of consecutive integers"]],"source":"A prime factor above $4.42m$ in a product of consecutive integers"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#exists-prime-dvd-prod-consecutive-gt-fou","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $m$ and $n$ be natural numbers with $m > 3$, $n > 4m$ and $n + m \\ge 150$. Then the product $n(n+1)\\cdots(n+m-1)$ has a prime divisor $p$ with $p > 4.42\\,m$.","title":"A prime factor above $4.42m$ in a product of consecutive integers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-of-balanced","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_of_balanced","parent":11837,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-of-balanced","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $a$ and $b$ be positive integers, let $x \\ge 2$ be an integer, and put $k = a + b$. If $k \\ge 75$, $a \\le \\tfrac{9}{10}k$ and $b \\le \\tfrac{4}{5}k$, then $\\binom{2k}{k} \\ne \\binom{2a}{a}\\binom{x+2b}{b}$.","title":"Central binomial products in the balanced range"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-of-large-a","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_of_large_a","parent":11837,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-of-large-a","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"For natural numbers $a$, $b$, $x$ with $b \\ge 1$, $x \\ge 2$, $a + b \\ge 150$ and $a \\ge \\frac{171}{121} x$, we have $\\binom{2(a+b)}{a+b} \\ne \\binom{2a}{a}\\binom{x+2b}{b}$.","title":"No large solutions of the central binomial product equation with a dominant central factor"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-eq-central-binom-mul-c-3","kind":"node","lean":"MetaMathlibExt.centralBinom_add_eq_centralBinom_mul_choose_one_iff","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial products with "],["math","\\(x = 1\\)"]],"source":"Central binomial products with $x = 1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-eq-central-binom-mul-c-3","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $a$ and $b$ be positive integers. Then $\\binom{2(a+b)}{a+b} = \\binom{2a}{a}\\binom{1+2b}{b}$ if and only if $a = 1$.","title":"Central binomial products with $x = 1$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-of-two-le-of-large","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_of_two_le_of_large","parent":11837,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-of-two-le-of-large","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"For natural numbers $a$, $b$, $x$ with $a \\ge 1$, $b \\ge 1$, $x \\ge 2$ and $a + b \\ge 150$, we have $\\binom{2(a+b)}{a+b} \\ne \\binom{2a}{a}\\binom{x+2b}{b}$.","title":"No large solutions of the central binomial product equation with shift at least two"},{"id":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/central-binom-add-ne-of-two-le-of-small","kind":"node","lean":"MetaMathlibExt.centralBinom_add_ne_of_two_le_of_small","parent":11837,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial products with "],["math","\\(x \\ge 2\\)"],["text"," and "],["math","\\(a + b < 150\\)"]],"source":"Central binomial products with $x \\ge 2$ and $a + b < 150$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients/index.html#central-binom-add-ne-of-two-le-of-small","scope":"NumberTheory/special-numbers/binomial-coefficients/equations-in-central-binomial-coefficients","status":"can_prove","summary":"Let $x$, $a$, $b$ be natural numbers with $a \\ge 1$, $b \\ge 1$, $x \\ge 2$ and $x < a + b < 150$. Then $\\binom{2(a+b)}{a+b} \\ne \\binom{2a}{a}\\binom{x+2b}{b}$.","title":"Central binomial products with $x \\ge 2$ and $a + b < 150$"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio.json","id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","title":"Factorials bounded by powers of the golden ratio"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/golden-factorial-exponent","kind":"node","lean":"MetaMathlibExt.goldenFactorialExponent","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#golden-factorial-exponent","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"For a natural number $n$, the golden factorial exponent $\\alpha(n)$ is the least positive integer $m$ with $n! \\le \\varphi^m$, where $\\varphi$ is the golden ratio. This is the sequence OEIS A214048.","title":"Golden factorial exponent"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/exists-golden-ratio-pow-ge-factorial","kind":"node","lean":"MetaMathlibExt.exists_goldenRatio_pow_ge_factorial","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#exists-golden-ratio-pow-ge-factorial","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. For every natural number $n$ there is an integer $m \\ge 1$ with $n! \\le \\varphi^m$.","title":"Every factorial is at most a positive power of the golden ratio"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/golden-factorial-exponent-mem","kind":"node","lean":"MetaMathlibExt.goldenFactorialExponent_mem","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#golden-factorial-exponent-mem","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio and, for a natural number $n$, let $\\alpha(n)$ be the least positive integer $m$ with $n! \\le \\varphi^m$. For every natural number $n$, $\\alpha(n) \\ge 1$ and $n! \\le \\varphi^{\\alpha(n)}$.","title":"Defining property of the golden factorial exponent"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/golden-factorial-exponent-le","kind":"node","lean":"MetaMathlibExt.goldenFactorialExponent_le","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#golden-factorial-exponent-le","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio and, for a natural number $n$, let $\\alpha(n)$ be the least positive integer $m$ with $n! \\le \\varphi^m$. If $n$ and $m$ are natural numbers with $m \\ge 1$ and $n! \\le \\varphi^m$, then $\\alpha(n) \\le m$.","title":"Minimality of the golden factorial exponent"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/golden-factorial-exponent-one","kind":"node","lean":"MetaMathlibExt.goldenFactorialExponent_one","parent":11855,"reading_title":{"format":"inline-parts-v1","parts":[["text","Golden factorial exponent of "],["math","\\(1\\)"]],"source":"Golden factorial exponent of $1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#golden-factorial-exponent-one","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio and, for a natural number $n$, let $\\alpha(n)$ be the least positive integer $m$ with $n! \\le \\varphi^m$. Then $\\alpha(1) = 1$.","title":"Golden factorial exponent of $1$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/golden-factorial-exponent-two","kind":"node","lean":"MetaMathlibExt.goldenFactorialExponent_two","parent":11855,"reading_title":{"format":"inline-parts-v1","parts":[["text","Golden factorial exponent of "],["math","\\(2\\)"]],"source":"Golden factorial exponent of $2$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#golden-factorial-exponent-two","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio and, for a natural number $n$, let $\\alpha(n)$ be the least positive integer $m$ with $n! \\le \\varphi^m$. Then $\\alpha(2) = 2$.","title":"Golden factorial exponent of $2$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/alternating-sign-fibonacci","kind":"node","lean":"MetaMathlibExt.alternatingSignFibonacci","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#alternating-sign-fibonacci","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"The alternating-sign Fibonacci sequence (OEIS A181716) is the integer sequence $(d_m)_{m \\ge 0}$ defined by $d_0 = 0$, $d_1 = 1$ and $d_{m+2} = d_{m+1} + d_m + (-1)^{m+2}$ for every natural number $m$.","title":"Alternating-sign Fibonacci sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/alternating-sign-fibonacci-add-two","kind":"node","lean":"MetaMathlibExt.alternatingSignFibonacci_add_two","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#alternating-sign-fibonacci-add-two","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"For every natural number $m$, the alternating-sign Fibonacci sequence $d$ satisfies $d_{m+2} = d_{m+1} + d_m + (-1)^{m+2}$.","title":"Recurrence of the alternating-sign Fibonacci sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/alternating-sign-fibonacci-small-values","kind":"node","lean":"MetaMathlibExt.alternatingSignFibonacci_small_values","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#alternating-sign-fibonacci-small-values","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"The alternating-sign Fibonacci sequence $d$ satisfies $d_0 = 0$, $d_1 = 1$, $d_2 = 2$, $d_3 = 2$ and $d_4 = 5$.","title":"First values of the alternating-sign Fibonacci sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/kurepa-partial-integral","kind":"node","lean":"MetaMathlibExt.kurepaPartialIntegral","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#kurepa-partial-integral","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"For a natural number $n$ and a real number $x$, Kurepa's partial integral is","title":"Kurepa's partial integral"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/kurepa-partial-integral-zero","kind":"node","lean":"MetaMathlibExt.kurepaPartialIntegral_zero","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#kurepa-partial-integral-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"For every natural number $n$, Kurepa's partial integral $K_n(x) = \\int_0^x e^{-t} \\sum_{k=0}^{n-1} t^k \\, dt$ satisfies $K_n(0) = 0$.","title":"Kurepa's partial integral vanishes at zero"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/golden-factorial-exponent-succ-bounds","kind":"node","lean":"MetaMathlibExt.goldenFactorialExponent_succ_bounds","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#golden-factorial-exponent-succ-bounds","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio and, for a natural number $n$, let $\\alpha(n)$ be the least positive integer $m$ with $n! \\le \\varphi^m$. For every natural number $n > 1$,","title":"Increment bounds for the golden factorial exponent"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/ncard-floor-logb-golden-ratio-eq-alterna","kind":"node","lean":"MetaMathlibExt.ncard_floor_logb_goldenRatio_eq_alternatingSignFibonacci","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#ncard-floor-logb-golden-ratio-eq-alterna","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"Let $d$ be the integer sequence with $d(0) = 0$, $d(1) = 1$ and $d(m) = d(m-1) + d(m-2) + (-1)^m$ for $m \\ge 2$, and let $\\varphi$ be the golden ratio. For every integer $m \\ge 1$, the number of positive integers $n$ with $\\lfloor \\log_\\varphi n \\rfloor = m$ equals $d(m)$.","title":"Counting integers by their golden-ratio logarithm"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/exists-lambert-sequence","kind":"node","lean":"MetaMathlibExt.exists_lambertSequence","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#exists-lambert-sequence","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"fully_proved","summary":"Let $h(y) = y e^y$ be the Lambert map. For every natural number $n \\ge 4$ and every real number $c$ with $0 < c \\le 1$ there is a real number $a \\ge n - 1$ with","title":"Existence of the Lambert sequences"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/lambert-sequence-lt-golden-factorial-exp","kind":"node","lean":"MetaMathlibExt.lambertSequence_lt_goldenFactorialExponent","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#lambert-sequence-lt-golden-factorial-exp","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"Let $n > 4$ be an integer, let $h(y) = y e^y$ be the Lambert map, and let $a$ be a real number with $a \\ge n - 1$ and","title":"The golden factorial exponent exceeds the Lambert sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/two-mul-sub-four-lt-lambert-sequence","kind":"node","lean":"MetaMathlibExt.two_mul_sub_four_lt_lambertSequence","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#two-mul-sub-four-lt-lambert-sequence","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"Let $n \\ge 4$ be a natural number, write $h(y) = y e^y$ for the Lambert map, and let $a$ be a real number with $a \\ge n - 1$ such that","title":"A linear lower bound for the Lambert sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/sum-pow-div-factorial-lt-lambert-sequenc","kind":"node","lean":"MetaMathlibExt.sum_pow_div_factorial_lt_lambertSequence","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#sum-pow-div-factorial-lt-lambert-sequenc","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"Let $n \\ge 5$ and $k$ be natural numbers with $k < n - 2$, let $h(y) = y e^y$ be the Lambert map, and let $a$ be a real number with $a \\ge n - 1$ and","title":"Partial exponential sums at the Lambert sequence"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/kurepa-partial-integral-lambert-sequence","kind":"node","lean":"MetaMathlibExt.kurepaPartialIntegral_lambertSequence_bounds","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#kurepa-partial-integral-lambert-sequence","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"Let $n \\ge 5$ be an integer, let $!n = \\sum_{k=0}^{n-1} k!$ be the left factorial, let $K_n(x) = \\int_0^x e^{-t} \\sum_{k=0}^{n-1} t^k \\, dt$ be Kurepa's partial integral, let $h(y) = y e^y$ be the Lambert map, and let $a$ be a real number with $a \\ge n - 1$ and","title":"Lambert bounds for the left factorial"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/kurepa-partial-integral-golden-factorial","kind":"node","lean":"MetaMathlibExt.kurepaPartialIntegral_goldenFactorialExponent_bounds","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#kurepa-partial-integral-golden-factorial","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"For every integer $n \\ge 1$, let $!n = \\sum_{k=0}^{n-1} k!$ be the left factorial, let $K_n(x) = \\int_0^x e^{-t} \\sum_{k=0}^{n-1} t^k \\, dt$ be Kurepa's partial integral, and let $\\alpha(n)$ be the least positive integer $m$ with $n! \\le \\varphi^m$, where $\\varphi$ is the golden ratio. Then $\\beta(n) = K_n(\\alpha(n))$ satisfies $\\beta(n) < {!n} < \\beta(n) + 1$.","title":"Left factorial bounds at the golden factorial exponent"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/kurepa-partial-integral-lambert-sequence-2","kind":"node","lean":"MetaMathlibExt.kurepaPartialIntegral_lambertSequence_two_bounds","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#kurepa-partial-integral-lambert-sequence-2","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"Let $n \\ge 5$ and $m \\ge 1$ be integers, let $!n = \\sum_{k=0}^{n-1} k!$ be the left factorial, let $K_n(x) = \\int_0^x e^{-t} \\sum_{k=0}^{n-1} t^k \\, dt$ be Kurepa's partial integral, let $h(y) = y e^y$ be the Lambert map, and let $A$ be a real number with $A \\ge n - 1$ and","title":"Two-parameter Lambert bounds for the left factorial"},{"id":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/left-factorial-eq-floor-kurepa-partial-i","kind":"node","lean":"MetaMathlibExt.leftFactorial_eq_floor_kurepaPartialIntegral","parent":11855,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio/index.html#left-factorial-eq-floor-kurepa-partial-i","scope":"NumberTheory/special-numbers/binomial-coefficients/factorials-bounded-by-powers-of-the-golden-ratio","status":"can_prove","summary":"Let $!n = \\sum_{k=0}^{n-1} k!$ be the left factorial, let $K_n(x) = \\int_0^x e^{-t} \\sum_{k=0}^{n-1} t^k \\, dt$ be Kurepa's partial integral, let $h(y) = y e^y$ be the Lambert map, and let $\\alpha(n)$ be the least positive integer $m$ with $n! \\le \\varphi^m$, where $\\varphi$ is the golden ratio. Then the following hold. (i) For every integer $n \\ge 1$, $\\lfloor K_n(\\alpha(n)) \\rfloor + 1 = {!n}$. (ii) For every\u2026","title":"Floor formulas for the left factorial"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients.json","id":"NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients","status":"fully_proved","title":"Fibonomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients/padic-val-nat-fib-mul","kind":"node","lean":"MetaMathlibExt.padicValNat_fib_mul","parent":11877,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients/index.html#padic-val-nat-fib-mul","scope":"NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients","status":"fully_proved","summary":"Let $p$ be an odd prime and $m \\ge 1$ an integer such that $p$ divides the Fibonacci number $F_m$, and write $v_p$ for the $p$-adic valuation. Then $v_p(F_{mj}) = v_p(F_m) + v_p(j)$ for every integer $j \\ge 1$.","title":"Lifting the exponent for Fibonacci numbers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients/fibonomial-prime-power-padic-val-eq-zero","kind":"node","lean":"MetaMathlibExt.fibonomial_primePower_padicVal_eq_zero","parent":11877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing "],["math","\\(p\\)"],["text","-adic valuation of Fibonomial coefficients at prime powers"]],"source":"Vanishing $p$-adic valuation of Fibonomial coefficients at prime powers"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients/index.html#fibonomial-prime-power-padic-val-eq-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients","status":"fully_proved","summary":"Let $p$ be a prime with $p \\ne 2$ and $p \\ne 5$, and let $a, b$ be integers with $0 < a \\le b$ such that $p \\equiv \\pm 1 \\pmod 5$ or $a \\equiv b \\pmod 2$. Write $F_n$ for the $n$-th Fibonacci number and $F_n! = F_1 F_2 \\cdots F_n$ for the Fibonacci factorial, with $F_0! = 1$. Then the $p$-adic valuation of the Fibonomial coefficient","title":"Vanishing $p$-adic valuation of Fibonomial coefficients at prime powers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients/fibonomial-prime-cube-congr","kind":"node","lean":"MetaMathlibExt.fibonomial_prime_cube_congr","parent":11877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kimball\u2013Webb congruence for Fibonomial coefficients modulo "],["math","\\(p^3\\)"]],"source":"Kimball\u2013Webb congruence for Fibonomial coefficients modulo $p^3$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients/index.html#fibonomial-prime-cube-congr","scope":"NumberTheory/special-numbers/binomial-coefficients/fibonomial-coefficients","status":"fully_proved","summary":"Let $F_n$ be the Fibonacci numbers, with $F_1 = F_2 = 1$, and write $F_n! = F_1 F_2 \\cdots F_n$, with $F_0! = 1$. Let $p \\ge 7$ be a prime and $\\varepsilon \\in \\{1, -1\\}$, and suppose that $\\rho = p - \\varepsilon$ is the rank of appearance of $p$, that is, $p \\mid F_\\rho$ and $p \\nmid F_m$ for $0 < m < \\rho$. Then for every natural number $k$ the Fibonomial coefficient","title":"Kimball\u2013Webb congruence for Fibonomial coefficients modulo $p^3$"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/franel-numbers.json","id":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/franel-numbers/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers","status":"can_prove","title":"Franel numbers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers/a265233","kind":"node","lean":"MathlibExt.NumberTheory.OeisA265233Wanted.a265233","parent":11881,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/franel-numbers/index.html#a265233","scope":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers","status":"fully_proved","summary":"The sequence $(a_n)_{n \\ge 0}$ of rational numbers is given by $a_0 = 1$, $a_1 = 1$, $a_2 = 7$, $a_3 = 56$ and, for every integer $N \\ge 4$,","title":"The sequence A265233"},{"id":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers/a265233-eq-sum-franel","kind":"node","lean":"MathlibExt.NumberTheory.OeisA265233Wanted.a265233_eq_sum_franel","parent":11881,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/franel-numbers/index.html#a265233-eq-sum-franel","scope":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be the sequence A265233 and let $f_m = \\sum_{k=0}^{m} \\binom{m}{k}^3$ be the $m$-th Franel number. For every integer $n \\ge 1$,","title":"Explicit formula for A265233 in terms of Franel numbers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers/coeff-pow-eq-franel","kind":"node","lean":"MathlibExt.NumberTheory.OeisA265233Wanted.coeff_pow_eq_franel","parent":11881,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/franel-numbers/index.html#coeff-pow-eq-franel","scope":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers","status":"can_prove","summary":"For every natural number $m$, the coefficient of $x^m y^m z^m$ in the polynomial $\\bigl((x + y)(y + z)(z + x)\\bigr)^m \\in \\mathbb{Q}[x, y, z]$ is the Franel number $f_m = \\sum_{k=0}^{m} \\binom{m}{k}^3$.","title":"Franel numbers as diagonal coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers/tsum-a265233-eq","kind":"node","lean":"MathlibExt.NumberTheory.OeisA265233Wanted.tsum_a265233_eq","parent":11881,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/franel-numbers/index.html#tsum-a265233-eq","scope":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be the sequence A265233 and let ${}_2F_1$ be the Gauss hypergeometric function. For every real number $x$ with $|x| < 1/12$ and $|27x(1 - 4x)^2| < 1$,","title":"Generating function of A265233"},{"id":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers/a265233-is-equivalent","kind":"node","lean":"MathlibExt.NumberTheory.OeisA265233Wanted.a265233_isEquivalent","parent":11881,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/franel-numbers/index.html#a265233-is-equivalent","scope":"NumberTheory/special-numbers/binomial-coefficients/franel-numbers","status":"can_prove","summary":"The sequence $(a_n)_{n \\ge 0}$ A265233 satisfies","title":"Asymptotics of A265233"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co.json","id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","title":"Greatest common divisors of radical-expansion coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/exists-unique-radical-expansion-coeffs","kind":"node","lean":"MetaMathlibExt.existsUnique_radical_expansion_coeffs","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unique integer coordinates of powers of "],["math","\\(u + v^{1/N}\\)"]],"source":"Unique integer coordinates of powers of $u + v^{1/N}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#exists-unique-radical-expansion-coeffs","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $N$ be a natural number with at least two distinct prime divisors, let $u, v$ be coprime positive integers such that $x^N - v$ is irreducible over $\\mathbb{Q}$, and let $\\theta = v^{1/N}$ be the positive real $N$-th root of $v$. Then for every natural number $n$ there is exactly one tuple of integers $(a_0, \\dots, a_{N-1})$ with $(u + \\theta)^n = \\sum_{i=0}^{N-1} a_i \\theta^i$.","title":"Unique integer coordinates of powers of $u + v^{1/N}$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/gcd-radical-expansion-coeffs-eq-one","kind":"node","lean":"MetaMathlibExt.gcd_radical_expansion_coeffs_eq_one","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coprime coefficients of "],["math","\\((u + v^{1/N})^n\\)"],["text"," when "],["math","\\(N\\)"],["text"," has two distinct prime factors"]],"source":"Coprime coefficients of $(u + v^{1/N})^n$ when $N$ has two distinct prime factors"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#gcd-radical-expansion-coeffs-eq-one","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $N$ be a natural number with at least two distinct prime divisors, let $u, v$ be coprime positive integers such that $x^N - v$ is irreducible over $\\mathbb{Q}$, let $\\theta = v^{1/N}$ be the positive real $N$-th root of $v$, and let $a_{n,i}$ (for natural $n$ and $0 \\le i \\le N - 1$) be integers with $(u + \\theta)^n = \\sum_{i=0}^{N-1} a_{n,i} \\theta^i$ for every $n \\ge 1$. Then for every $n \\ge 1$ the greatest\u2026","title":"Coprime coefficients of $(u + v^{1/N})^n$ when $N$ has two distinct prime factors"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/gcd-pow-add-neg-one-pow-mul-eq-one","kind":"node","lean":"MetaMathlibExt.gcd_pow_add_neg_one_pow_mul_eq_one","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(u^N + (-1)^{N+1} v\\)"],["text"," is coprime to "],["math","\\(u\\)"],["text"," and to "],["math","\\(v\\)"]],"source":"$u^N + (-1)^{N+1} v$ is coprime to $u$ and to $v$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#gcd-pow-add-neg-one-pow-mul-eq-one","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $u, v$ be coprime positive integers and let $N \\ge 1$. Then $\\gcd(u^N + (-1)^{N+1} v, u) = 1$ and $\\gcd(u^N + (-1)^{N+1} v, v) = 1$.","title":"$u^N + (-1)^{N+1} v$ is coprime to $u$ and to $v$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/prime-dvd-choose-pow-mul-pow-mul","kind":"node","lean":"MetaMathlibExt.prime_dvd_choose_pow_mul_pow_mul","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text"," divides "],["math","\\(\\binom{p^r a}{p^s b}\\)"],["text"," when "],["math","\\(r > s\\)"]],"source":"$p$ divides $\\binom{p^r a}{p^s b}$ when $r > s$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#prime-dvd-choose-pow-mul-pow-mul","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $p$ be a prime, let $r > s$ be natural numbers, and let $a, b$ be natural numbers not divisible by $p$. Then $p \\mid \\binom{p^r a}{p^s b}$.","title":"$p$ divides $\\binom{p^r a}{p^s b}$ when $r > s$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/choose-pow-mul-mod-eq","kind":"node","lean":"MetaMathlibExt.choose_pow_mul_modEq","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial coefficients "],["math","\\(\\binom{p^r a}{p^r b}\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Binomial coefficients $\\binom{p^r a}{p^r b}$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#choose-pow-mul-mod-eq","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $p$ be a prime, let $r \\ge 1$, and let $a, b$ be natural numbers not divisible by $p$. Then $\\binom{p^r a}{p^r b} \\equiv \\binom{a}{b} \\pmod{p}$.","title":"Binomial coefficients $\\binom{p^r a}{p^r b}$ modulo $p$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/gcd-sqrt-expansion-coeffs","kind":"node","lean":"MetaMathlibExt.gcd_sqrt_expansion_coeffs","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gcd of the coefficients of "],["math","\\((u + \\sqrt{v})^n\\)"]],"source":"Gcd of the coefficients of $(u + \\sqrt{v})^n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#gcd-sqrt-expansion-coeffs","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $u, v$ be coprime positive integers with $\\sqrt{v}$ irrational, let $a_n, b_n$ (for natural $n$) be integers with $(u + \\sqrt{v})^n = a_n + b_n \\sqrt{v}$ for every $n \\ge 1$, and let $D_n \\ge 0$ be the greatest common divisor of $a_n$ and $b_n$. Then for every $n \\ge 1$: (a) if $u^2 - v$ is odd, then $D_n = 1$; (b) if $u^2 - v \\equiv 2 \\pmod 4$, then $D_{2n-1} = 2^{n-1}$ and $D_{2n} = 2^n$; (c) if\u2026","title":"Gcd of the coefficients of $(u + \\sqrt{v})^n$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/gcd-prime-root-expansion-coeffs","kind":"node","lean":"MetaMathlibExt.gcd_prime_root_expansion_coeffs","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gcd of the coefficients of "],["math","\\((u + v^{1/p})^n\\)"],["text"," for an odd prime "],["math","\\(p\\)"]],"source":"Gcd of the coefficients of $(u + v^{1/p})^n$ for an odd prime $p$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#gcd-prime-root-expansion-coeffs","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $p$ be an odd prime, let $u, v$ be coprime positive integers such that $\\theta = v^{1/p}$, the positive real $p$-th root of $v$, is irrational, and let $a_{n,i}$ (for natural $n$ and $0 \\le i \\le p - 1$) be integers with $(u + \\theta)^n = \\sum_{i=0}^{p-1} a_{n,i} \\theta^i$ for every $n \\ge 1$. Write $D_n$ for the nonnegative greatest common divisor of $a_{n,0}, \\dots, a_{n,p-1}$. Then: (a) if $p \\nmid u^p + v$,\u2026","title":"Gcd of the coefficients of $(u + v^{1/p})^n$ for an odd prime $p$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/gcd-radical-expansion-coeffs-dvd-succ","kind":"node","lean":"MetaMathlibExt.gcd_radical_expansion_coeffs_dvd_succ","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient gcd of "],["math","\\((u + v^{1/N})^n\\)"],["text"," divides that of "],["math","\\((u + v^{1/N})^{n+1}\\)"]],"source":"The coefficient gcd of $(u + v^{1/N})^n$ divides that of $(u + v^{1/N})^{n+1}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#gcd-radical-expansion-coeffs-dvd-succ","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $N, u, v$ be natural numbers such that $x^N - v$ is irreducible over $\\mathbb{Q}$, let $\\theta = v^{1/N}$ be the nonnegative real $N$-th root of $v$, and let $a_{n,i}$ (for natural $n$ and $0 \\le i \\le N - 1$) be integers with $(u + \\theta)^n = \\sum_{i=0}^{N-1} a_{n,i} \\theta^i$ for every $n \\ge 1$. Write $D_n$ for the nonnegative greatest common divisor of $a_{n,0}, \\dots, a_{n,N-1}$. Then $D_n \\mid D_{n+1}$\u2026","title":"The coefficient gcd of $(u + v^{1/N})^n$ divides that of $(u + v^{1/N})^{n+1}$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/prime-dvd-gcd-pow-add-radical-coeffs","kind":"node","lean":"MetaMathlibExt.prime_dvd_gcd_pow_add_radical_coeffs","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Common prime factors of "],["math","\\(u^N + (-1)^{N+1} v\\)"],["text"," and the coefficient gcd divide "],["math","\\(N\\)"]],"source":"Common prime factors of $u^N + (-1)^{N+1} v$ and the coefficient gcd divide $N$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#prime-dvd-gcd-pow-add-radical-coeffs","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $N \\ge 2$, let $u, v$ be coprime natural numbers such that $x^N - v$ is irreducible over $\\mathbb{Q}$, let $\\theta = v^{1/N}$ be the nonnegative real $N$-th root of $v$, and let $a_{n,i}$ (for natural $n$ and $0 \\le i \\le N - 1$) be integers with $(u + \\theta)^n = \\sum_{i=0}^{N-1} a_{n,i} \\theta^i$ for every $n \\ge 1$. Write $D_n$ for the nonnegative greatest common divisor of $a_{n,0}, \\dots, a_{n,N-1}$. Then\u2026","title":"Common prime factors of $u^N + (-1)^{N+1} v$ and the coefficient gcd divide $N$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/prime-dvd-pow-add-of-dvd-radical-coeffs","kind":"node","lean":"MetaMathlibExt.prime_dvd_pow_add_of_dvd_radical_coeffs_gcd","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes dividing the coefficient gcd of "],["math","\\((u + v^{1/N})^n\\)"],["text"," divide the norm"]],"source":"Primes dividing the coefficient gcd of $(u + v^{1/N})^n$ divide the norm"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#prime-dvd-pow-add-of-dvd-radical-coeffs","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $u, v, N$ be natural numbers such that $x^N - v$ is irreducible over $\\mathbb{Q}$. Write $\\theta = v^{1/N}$ for the nonnegative real $N$-th root of $v$, and let $a_{n,i}$ be integers ($n \\ge 1$, $0 \\le i < N$) with $(u + \\theta)^n = \\sum_{i=0}^{N-1} a_{n,i} \\theta^i$ for every $n \\ge 1$. If $n \\ge 1$ and $p$ is a prime dividing $D_n = \\gcd(a_{n,0}, \\dots, a_{n,N-1})$, then $p$ divides $u^N + (-1)^{N+1} v$, the\u2026","title":"Primes dividing the coefficient gcd of $(u + v^{1/N})^n$ divide the norm"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/prime-dvd-of-dvd-radical-coeffs-gcd","kind":"node","lean":"MetaMathlibExt.prime_dvd_of_dvd_radical_coeffs_gcd","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes dividing the coefficient gcd of "],["math","\\((u + v^{1/N})^n\\)"],["text"," divide "],["math","\\(N\\)"]],"source":"Primes dividing the coefficient gcd of $(u + v^{1/N})^n$ divide $N$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#prime-dvd-of-dvd-radical-coeffs-gcd","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $N \\ge 2$ and let $u, v$ be coprime natural numbers such that $x^N - v$ is irreducible over $\\mathbb{Q}$. Write $\\theta = v^{1/N}$ for the nonnegative real $N$-th root of $v$, and let $a_{n,i}$ be integers ($n \\ge 1$, $0 \\le i < N$) with $(u + \\theta)^n = \\sum_{i=0}^{N-1} a_{n,i} \\theta^i$ for every $n \\ge 1$. If $n \\ge 1$ and $p$ is a prime dividing $D_n = \\gcd(a_{n,0}, \\dots, a_{n,N-1})$, then $p$ divides $N$.","title":"Primes dividing the coefficient gcd of $(u + v^{1/N})^n$ divide $N$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/gcd-radical-expansion-coeffs-eq-one-of-c","kind":"node","lean":"MetaMathlibExt.gcd_radical_expansion_coeffs_eq_one_of_coprime","parent":11887,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coprime coefficients when "],["math","\\(u^N + (-1)^{N+1} v\\)"],["text"," is coprime to "],["math","\\(N\\)"]],"source":"Coprime coefficients when $u^N + (-1)^{N+1} v$ is coprime to $N$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co/index.html#gcd-radical-expansion-coeffs-eq-one-of-c","scope":"NumberTheory/special-numbers/binomial-coefficients/greatest-common-divisors-of-radical-expansion-co","status":"can_prove","summary":"Let $N \\ge 2$, let $u, v$ be coprime natural numbers such that $x^N - v$ is irreducible over $\\mathbb{Q}$, let $\\theta = v^{1/N}$ be the nonnegative real $N$-th root of $v$, and let $a_{n,i}$ (for natural $n$ and $0 \\le i \\le N - 1$) be integers with $(u + \\theta)^n = \\sum_{i=0}^{N-1} a_{n,i} \\theta^i$ for every $n \\ge 1$. If $u^N + (-1)^{N+1} v$ and $N$ are coprime integers, then for every $n \\ge 1$ the greatest\u2026","title":"Coprime coefficients when $u^N + (-1)^{N+1} v$ is coprime to $N$"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios.json","id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"can_prove","title":"Integral factorial ratios"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/factorial-prod","kind":"node","lean":"factorialProd","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#factorial-prod","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"For a list $L$ of natural numbers and a natural number $n$, the product of scaled factorials of $L$ at $n$ is $\\prod_{a \\in L} (an)!$, the product of the factorials $(an)!$ over the entries $a$ of $L$, counted with multiplicity; for the empty list it is $1$.","title":"Product of scaled factorials"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/is-integral-factorial-ratio","kind":"node","lean":"IsIntegralFactorialRatio","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#is-integral-factorial-ratio","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"For lists $A = (a_1, \\dots, a_k)$ and $B = (b_1, \\dots, b_l)$ of natural numbers, the pair $(A, B)$ is an integral factorial ratio when every $a_i$ and every $b_j$ is positive, $a_1 + \\dots + a_k = b_1 + \\dots + b_l$, and for every natural number $n$ the product $(b_1 n)! \\cdots (b_l n)!$ divides $(a_1 n)! \\cdots (a_k n)!$; equivalently, the quotient $\\frac{(a_1 n)! \\cdots (a_k n)!}{(b_1 n)! \\cdots (b_l n)!}$ is a\u2026","title":"Integral factorial ratio"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/factorial-prod-zero","kind":"node","lean":"factorialProd_zero","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#factorial-prod-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"For every list $L$ of natural numbers, $\\prod_{a \\in L} (a \\cdot 0)! = 1$, since every factor equals $0! = 1$.","title":"Scaled factorial product at index zero"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/factorial-prod-dvd-zero","kind":"node","lean":"factorialProd_dvd_zero","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#factorial-prod-dvd-zero","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"For all lists $A$ and $B$ of natural numbers, $\\prod_{b \\in B} (b \\cdot 0)!$ divides $\\prod_{a \\in A} (a \\cdot 0)!$, since both products equal $1$.","title":"Scaled factorial products at index zero divide"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/is-integral-factorial-ratio-iff","kind":"node","lean":"isIntegralFactorialRatio_iff","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#is-integral-factorial-ratio-iff","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"For lists $A$ and $B$ of natural numbers, $(A, B)$ is an integral factorial ratio if and only if every entry of $A$ is positive, every entry of $B$ is positive, the entries of $A$ and the entries of $B$ have the same sum, and for every natural number $n$ the product $\\prod_{b \\in B} (bn)!$ divides $\\prod_{a \\in A} (an)!$.","title":"Characterization of integral factorial ratios"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/is-integral-factorial-ratio-perm-left","kind":"node","lean":"isIntegralFactorialRatio_perm_left","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#is-integral-factorial-ratio-perm-left","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"Let $A$, $A'$ and $B$ be lists of natural numbers such that $A'$ is a permutation of $A$. If $(A, B)$ is an integral factorial ratio, then so is $(A', B)$.","title":"Permuting the numerator of an integral factorial ratio"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/is-integral-factorial-ratio-perm-right","kind":"node","lean":"isIntegralFactorialRatio_perm_right","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#is-integral-factorial-ratio-perm-right","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"Let $A$, $B$ and $B'$ be lists of natural numbers such that $B'$ is a permutation of $B$. If $(A, B)$ is an integral factorial ratio, then so is $(A, B')$.","title":"Permuting the denominator of an integral factorial ratio"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/is-integral-factorial-ratio-perm","kind":"node","lean":"isIntegralFactorialRatio_perm","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#is-integral-factorial-ratio-perm","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"Let $A$, $A'$, $B$ and $B'$ be lists of natural numbers such that $A'$ is a permutation of $A$ and $B'$ is a permutation of $B$. If $(A, B)$ is an integral factorial ratio, then so is $(A', B')$.","title":"Integral factorial ratios are invariant under permutation"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/is-integral-factorial-ratio-refl","kind":"node","lean":"isIntegralFactorialRatio_refl","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#is-integral-factorial-ratio-refl","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"If every entry of a list $L$ of natural numbers is positive, then $(L, L)$ is an integral factorial ratio.","title":"Equal tuples form an integral factorial ratio"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/is-integral-factorial-ratio-singleton-se","kind":"node","lean":"isIntegralFactorialRatio_singleton_self","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#is-integral-factorial-ratio-singleton-se","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"For every positive natural number $c$, the one-entry tuples $(c)$ and $(c)$ form an integral factorial ratio.","title":"Equal singletons form an integral factorial ratio"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/is-integral-factorial-ratio-central-bino","kind":"node","lean":"isIntegralFactorialRatio_centralBinom","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#is-integral-factorial-ratio-central-bino","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"The tuples $(2)$ and $(1, 1)$ form an integral factorial ratio: their entries are positive, their sums agree ($2 = 1 + 1$), and for every natural number $n$, $n! \\cdot n!$ divides $(2n)!$.","title":"The central binomial factorial ratio is integral"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem728Wanted.conjecture","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#conjecture","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"The proposition that there is a real number $\\delta > 0$ such that for every real number $\\varepsilon$ with $0 < \\varepsilon < \\delta$ and all real numbers $C > 0$ and $C' > C$, there are natural numbers $a$, $b$ and $n \\ge 1$ with $a > \\varepsilon n$, $b > \\varepsilon n$, $a! \\, b! \\mid n! \\, (a + b - n)!$ and $n + C \\log n < a + b < n + C' \\log n$.","title":"Erd\u0151s Problem 728"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem728Wanted.conjecture_holds","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#conjecture-holds","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"can_prove","summary":"There is a real number $\\delta > 0$ such that for every real number $\\varepsilon$ with $0 < \\varepsilon < \\delta$ and all real numbers $C > 0$ and $C' > C$, there are natural numbers $a$, $b$ and $n \\ge 1$ with $a > \\varepsilon n$, $b > \\varepsilon n$, $a! \\, b! \\mid n! \\, (a + b - n)!$ and $n + C \\log n < a + b < n + C' \\log n$.","title":"Erd\u0151s Problem 728 holds"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/conjecture-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem729Wanted.conjecture","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#conjecture-2","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"fully_proved","summary":"The proposition that for every real number $C > 0$ there is a natural number $K \\ge 3$ such that there are infinitely many triples $(a, b, n)$ of positive integers with $a + b > n + C \\log n$ for which no prime $p > K$ divides the denominator of the rational number $n! / (a! \\, b!)$ in lowest terms.","title":"Erd\u0151s Problem 729"},{"id":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/conjecture-holds-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem729Wanted.conjecture_holds","parent":11900,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios/index.html#conjecture-holds-2","scope":"NumberTheory/special-numbers/binomial-coefficients/integral-factorial-ratios","status":"can_prove","summary":"For every real number $C > 0$ there is a natural number $K \\ge 3$ such that there are infinitely many triples $(a, b, n)$ of positive integers with $a + b > n + C \\log n$ for which no prime $p > K$ divides the denominator of the rational number $n! / (a! \\, b!)$ in lowest terms.","title":"Erd\u0151s Problem 729 holds"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients.json","id":"NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients","status":"can_prove","title":"Mid-sized divisors of binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.BinomialMidrangeDivisorWanted.conjecture","parent":11916,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients/index.html#conjecture","scope":"NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients","status":"fully_proved","summary":"The proposition that there is a real number $c > 0$ such that for all natural numbers $n$ and $k$ with $1 \\le k < n$, the binomial coefficient $\\binom{n}{k}$ has a divisor $d$ with $cn < d \\le n$.","title":"Erd\u0151s Problem 387"},{"id":"NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.BinomialMidrangeDivisorWanted.conjecture_refuted","parent":11916,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients/index.html#conjecture-refuted","scope":"NumberTheory/special-numbers/binomial-coefficients/mid-sized-divisors-of-binomial-coefficients","status":"can_prove","summary":"There is no real number $c > 0$ such that for all natural numbers $n$ and $k$ with $1 \\le k < n$ the binomial coefficient $\\binom{n}{k}$ has a divisor $d$ with $cn < d \\le n$.","title":"Erd\u0151s Problem 387 is false"},{"data_url":"scopes/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients.json","id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","kind":"scope","navigation":"module","parent":11744,"result":false,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","title":"Products of factorials and binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/glaisher-kinkelin-constant","kind":"node","lean":"MetaMathlibExt.glaisherKinkelinConstant","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#glaisher-kinkelin-constant","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"The Glaisher\u2013Kinkelin constant is the real number","title":"Glaisher\u2013Kinkelin constant"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/log-glaisher-kinkelin-constant","kind":"node","lean":"MetaMathlibExt.log_glaisherKinkelinConstant","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#log-glaisher-kinkelin-constant","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"The Glaisher\u2013Kinkelin constant $A$ satisfies $\\log A = \\frac{1}{12} - \\operatorname{Re} \\zeta'(-1)$, where $\\zeta'$ is the complex derivative of the Riemann zeta function.","title":"Logarithm of the Glaisher\u2013Kinkelin constant"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/factorial-product-constant","kind":"node","lean":"MetaMathlibExt.factorialProductConstant","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#factorial-product-constant","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"For a natural number $k \\ge 1$, the constant $F_k$ is the positive real number defined by","title":"Factorial-product constants"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/log-factorial-product-constant-one","kind":"node","lean":"MetaMathlibExt.log_factorialProductConstant_one","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#log-factorial-product-constant-one","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"The factorial product constant $F_1$ satisfies $\\log F_1 = \\frac{1}{4} \\log(2\\pi) + \\frac{1}{12} - 2 \\log A$, where $A$ is the Glaisher\u2013Kinkelin constant.","title":"Logarithm of the first factorial product constant"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/factorial-product-constant-pos","kind":"node","lean":"MetaMathlibExt.factorialProductConstant_pos","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#factorial-product-constant-pos","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"For every natural number $k$, the factorial-product constant $F_k$ is positive.","title":"Positivity of the factorial-product constants"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/factorial-product-comparison","kind":"node","lean":"MetaMathlibExt.factorialProductComparison","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#factorial-product-comparison","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"For a natural number $k$, the comparison function $P_k \\colon \\mathbb{R} \\to \\mathbb{R}$ is defined by","title":"Comparison function for products of factorials"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/factorial-product-comparison-one","kind":"node","lean":"MetaMathlibExt.factorialProductComparison_one","parent":11919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of the factorial-product comparison function at "],["math","\\(1\\)"]],"source":"Value of the factorial-product comparison function at $1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#factorial-product-comparison-one","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"For every natural number $k$, the factorial-product comparison function $P_k$ satisfies $P_k(1) = k^k (2\\pi k/e)^{1/2}$.","title":"Value of the factorial-product comparison function at $1$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/binomial-product-comparison","kind":"node","lean":"MetaMathlibExt.binomialProductComparison","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#binomial-product-comparison","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"For natural numbers $a$ and $b$, let $c = a - b$ if $b \\le a$ and $c = 0$ otherwise. The comparison function $P_{a,b} \\colon \\mathbb{R} \\to \\mathbb{R}$ is defined by","title":"Comparison function for products of binomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/binomial-product-comparison-one","kind":"node","lean":"MetaMathlibExt.binomialProductComparison_one","parent":11919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of the binomial-product comparison function at "],["math","\\(1\\)"]],"source":"Value of the binomial-product comparison function at $1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#binomial-product-comparison-one","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"fully_proved","summary":"For natural numbers $a$ and $b$, let $c = a - b$ if $b \\le a$ and $c = 0$ otherwise. Then the binomial-product comparison function $P_{a,b}$ satisfies","title":"Value of the binomial-product comparison function at $1$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/tendsto-prod-factorial-mul-div","kind":"node","lean":"MetaMathlibExt.tendsto_prod_factorial_mul_div","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#tendsto-prod-factorial-mul-div","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $k \\ge 1$ be a natural number, let $F_k$ be the $k$-th factorial product constant and $A$ the Glaisher\u2013Kinkelin constant. As $n \\to \\infty$, the ratio of $\\prod_{\\nu=1}^{n} (k\\nu)!$ to","title":"Asymptotics of products of factorials of multiples"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/factorial-product-constant-strict-anti-t","kind":"node","lean":"MetaMathlibExt.factorialProductConstant_strictAnti_tendsto_one","parent":11919,"reading_title":{"format":"inline-parts-v1","parts":[["text","The factorial-product constants decrease strictly to "],["math","\\(1\\)"]],"source":"The factorial-product constants decrease strictly to $1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#factorial-product-constant-strict-anti-t","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"The sequence $(F_k)_{k \\ge 1}$ of factorial-product constants is strictly decreasing, and $F_k \\to 1$ as $k \\to \\infty$.","title":"The factorial-product constants decrease strictly to $1$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/inv-add-inv-sub-inv-add-bounds","kind":"node","lean":"MetaMathlibExt.inv_add_inv_sub_inv_add_bounds","parent":11919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds for "],["math","\\(1/a + 1/b - 1/(a+b)\\)"]],"source":"Bounds for $1/a + 1/b - 1/(a+b)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#inv-add-inv-sub-inv-add-bounds","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $a$ and $b$ be natural numbers with $a, b \\ge 1$ and $a + b \\ge 3$, and put $g(a,b) = 1/a + 1/b - 1/(a+b)$. Then $0 < g(a,b) \\le 7/6$.","title":"Bounds for $1/a + 1/b - 1/(a+b)$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/factorial-product-constant-ratio-mem-ico","kind":"node","lean":"MetaMathlibExt.factorialProductConstant_ratio_mem_Ico","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#factorial-product-constant-ratio-mem-ico","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $a$ and $b$ be natural numbers with $1 \\le b < a$, and let $F_k$ denote the factorial-product constants. Then $c_{a,b} = F_a / (F_b F_{a-b})$ satisfies","title":"Bounds for the binomial-product constants"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/tendsto-prod-multinomial-div","kind":"node","lean":"MetaMathlibExt.tendsto_prod_multinomial_div","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#tendsto-prod-multinomial-div","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $r \\ge 2$, let $b_1, \\dots, b_r \\ge 1$ be natural numbers with $a = b_1 + \\dots + b_r$, let $F_k$ denote the factorial product constants and $P_k$ the factorial comparison functions, and put $c_{a,b} = F_a / (F_{b_1} \\cdots F_{b_r})$. As $m \\to \\infty$, the ratio of $\\prod_{n=1}^{m} \\binom{an}{b_1 n, \\dots, b_r n}$ to $P_a(m) / (P_{b_1}(m) \\cdots P_{b_r}(m))$ tends to $c_{a,b} (2\\pi)^{(1-r)/4}$; moreover\u2026","title":"Asymptotics of products of multinomial coefficients"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/tendsto-prod-choose-div","kind":"node","lean":"MetaMathlibExt.tendsto_prod_choose_div","parent":11919,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotics of products of binomial coefficients "],["math","\\(\\binom{an}{bn}\\)"]],"source":"Asymptotics of products of binomial coefficients $\\binom{an}{bn}$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#tendsto-prod-choose-div","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $a > b \\ge 1$ be natural numbers, let $F_k$ denote the factorial product constants and $P_{a,b}$ the binomial comparison function, and put $c_{a,b} = F_a / (F_b F_{a-b})$ and $C_{a,b} = c_{a,b} (2\\pi)^{-1/4}$. Then $\\prod_{n=1}^{m} \\binom{an}{bn} / P_{a,b}(m) \\to C_{a,b}$ as $m \\to \\infty$; moreover $C_{2,1} \\le C_{a,b} < (2\\pi)^{-1/4}$ and $c_{2,1} \\le c_{a,b} < 1$, where $c_{2,1} = F_2 / F_1^2$ and\u2026","title":"Asymptotics of products of binomial coefficients $\\binom{an}{bn}$"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/tendsto-prod-central-binom-catalan-div","kind":"node","lean":"MetaMathlibExt.tendsto_prod_centralBinom_catalan_div","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#tendsto-prod-central-binom-catalan-div","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $A$ be the Glaisher\u2013Kinkelin constant, let $\\mathrm{Cat}_n$ denote the $n$-th Catalan number, and put $C_{2,1} = 2^{5/24} A^{3/2} / \\left((2\\pi)^{1/2} e^{1/8}\\right)$. As $m \\to \\infty$, the ratio of $\\prod_{n=1}^{m} \\binom{2n}{n}$ to $2^{m^2} (4e/\\pi)^{m/2} / m^{m/2 + 3/8}$ tends to $C_{2,1}$, and the ratio of $\\prod_{n=1}^{m} \\mathrm{Cat}_n$ to $2^{m^2} (4e^3/\\pi)^{m/2} / m^{3m/2 + 15/8}$ tends to\u2026","title":"Asymptotics of products of central binomial coefficients and Catalan numbers"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/tendsto-prod-choose-row-div","kind":"node","lean":"MetaMathlibExt.tendsto_prod_choose_row_div","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#tendsto-prod-choose-row-div","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $F_1$ be the first factorial product constant and $A$ the Glaisher\u2013Kinkelin constant, and put $C_{\\mathrm{row}} = \\left(F_1 (2\\pi)^{1/4}\\right)^{-1}$. As $n \\to \\infty$, the ratio of $\\prod_{\\nu=0}^{n} \\binom{n}{\\nu}$ to $e^{n^2/2 + n} / \\left((2\\pi)^{n/2} n^{n/2 + 1/3}\\right)$ tends to $C_{\\mathrm{row}}$, and $C_{\\mathrm{row}} = A^2 / \\left((2\\pi)^{1/2} e^{1/12}\\right)$.","title":"Asymptotics of the product of a row of Pascal's triangle"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/tendsto-binomial-product-limit-row","kind":"node","lean":"MetaMathlibExt.tendsto_binomialProduct_limit_row","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#tendsto-binomial-product-limit-row","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $F_k$ denote the factorial product constants and $P_{a,b}$ the binomial comparison functions, and let $n \\ge 1$ be a natural number and $x > 0$ a real number. Then, as $a \\to \\infty$ through the natural numbers, $\\frac{F_a}{F_1 F_{a-1}} (2\\pi)^{-1/4} \\to \\left(F_1 (2\\pi)^{1/4}\\right)^{-1}$, $a^{-n(n+1)/2} \\prod_{\\nu=1}^{n} \\binom{a\\nu}{\\nu} \\to \\prod_{\\nu=1}^{n} \\binom{n}{\\nu}$, and\u2026","title":"Pascal-row asymptotics as a limit of binomial-product asymptotics"},{"id":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/tendsto-hirschhorn-constant","kind":"node","lean":"MetaMathlibExt.tendsto_hirschhorn_constant","parent":11919,"result":true,"route":"../roadmap/NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients/index.html#tendsto-hirschhorn-constant","scope":"NumberTheory/special-numbers/binomial-coefficients/products-of-factorials-and-binomial-coefficients","status":"can_prove","summary":"Let $A$ be the Glaisher\u2013Kinkelin constant and $F_1$ the first factorial product constant. As $n \\to \\infty$,","title":"Hirschhorn's constant"},{"data_url":"scopes/NumberTheory/special-numbers/catalan-numbers.json","id":"NumberTheory/special-numbers/catalan-numbers","kind":"scope","parent":11422,"result":false,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/index.html","scope":"NumberTheory/special-numbers/catalan-numbers","status":"can_prove","summary":"Catalan, Fuss\u2013Catalan, Lucasnomial and FiboCatalan numbers: integrality and prime divisibility.","title":"Catalan and Fuss\u2013Catalan numbers"},{"data_url":"scopes/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers.json","id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","kind":"scope","navigation":"module","parent":11939,"result":false,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","title":"FiboCatalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-factorial","kind":"node","lean":"MetaMathlibExt.fibonacciFactorial","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-factorial","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For a natural number $n$, the Fibonacci factorial $F_n!$ is the rational number $F_n! = F_1 F_2 \\cdots F_n$, the product of the first $n$ Fibonacci numbers (where $F_1 = F_2 = 1$ and $F_{k+2} = F_{k+1} + F_k$); in particular $F_0! = 1$, the empty product.","title":"Fibonacci factorial"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-factorial-zero","kind":"node","lean":"MetaMathlibExt.fibonacciFactorial_zero","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-factorial-zero","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"The Fibonacci factorial $F_0!$ equals $1$, by the empty-product convention.","title":"Fibonacci factorial at zero"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-factorial-pos","kind":"node","lean":"MetaMathlibExt.fibonacciFactorial_pos","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-factorial-pos","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For every natural number $n$, the Fibonacci factorial $F_n!$ is a strictly positive rational number.","title":"Fibonacci factorials are positive"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-factorial-ne","kind":"node","lean":"MetaMathlibExt.fibonacciFactorial_ne","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-factorial-ne","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For every natural number $n$, the Fibonacci factorial $F_n!$ is a nonzero rational number.","title":"Fibonacci factorials are nonzero"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-factorial-one","kind":"node","lean":"MetaMathlibExt.fibonacciFactorial_one","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-factorial-one","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"The Fibonacci factorial $F_1!$ equals $1$, since the first Fibonacci number is $F_1 = 1$.","title":"Fibonacci factorial at one"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-factorial-two","kind":"node","lean":"MetaMathlibExt.fibonacciFactorial_two","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-factorial-two","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"The Fibonacci factorial $F_2! = F_1 F_2$ of the Fibonacci numbers $F_1, F_2$ equals $1$, since $F_1 = F_2 = 1$.","title":"Fibonacci factorial at two"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-factorial-three","kind":"node","lean":"MetaMathlibExt.fibonacciFactorial_three","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-factorial-three","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"The Fibonacci factorial $F_3! = F_1 F_2 F_3$ of the Fibonacci numbers $F_1, F_2, F_3$ equals $2$, since $F_1 = F_2 = 1$ and $F_3 = 2$.","title":"Fibonacci factorial at three"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-jordan-quotient","kind":"node","lean":"MetaMathlibExt.fibonacciJordanQuotient","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-jordan-quotient","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For a natural number $r$, the Jordan-type quotient is the rational number $J_{r,F} = F_{2r+1}! / F_r!$, where $F_m! = F_1 F_2 \\cdots F_m$ is the Fibonacci factorial of the Fibonacci numbers $F_1 = F_2 = 1$, $F_{k+2} = F_{k+1} + F_k$ (so $F_0! = 1$).","title":"Fibonacci Jordan quotient"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-jordan-quotient-pos","kind":"node","lean":"MetaMathlibExt.fibonacciJordanQuotient_pos","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-jordan-quotient-pos","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For every natural number $r$, the Jordan-type quotient $J_{r,F} = F_{2r+1}! / F_r!$ of Fibonacci factorials is a strictly positive rational number.","title":"Fibonacci Jordan quotient is positive"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-jordan-quotient-r0","kind":"node","lean":"MetaMathlibExt.fibonacciJordanQuotient_r0","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-jordan-quotient-r0","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"The Jordan-type quotient $J_{r,F} = F_{2r+1}! / F_r!$ of Fibonacci factorials takes the value $J_{0,F} = F_1! / F_0! = 1$.","title":"Fibonacci Jordan quotient at zero"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonacci-jordan-quotient-r1","kind":"node","lean":"MetaMathlibExt.fibonacciJordanQuotient_r1","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonacci-jordan-quotient-r1","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"The Jordan-type quotient $J_{r,F} = F_{2r+1}! / F_r!$ of Fibonacci factorials takes the value $J_{1,F} = F_3! / F_1! = 2$.","title":"Fibonacci Jordan quotient at one"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/generalized-fibonacci-catalan","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciCatalan","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#generalized-fibonacci-catalan","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"Let $F_m! = F_1 F_2 \\cdots F_m$ denote the Fibonacci factorial of the Fibonacci numbers $F_1 = F_2 = 1$, $F_{k+2} = F_{k+1} + F_k$ (so $F_0! = 1$), and let $J_{r,F} = F_{2r+1}! / F_r!$ be the Fibonacci Jordan quotient. For natural numbers $r$ and $n$, the generalized FiboCatalan number with parameters $r$ and $n$ is the rational number $J_{r,F} \\, F_{2n}! / (F_n! \\, F_{n+r+1}!)$.","title":"Generalized FiboCatalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/generalized-fibonacci-catalan-pos","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciCatalan_pos","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#generalized-fibonacci-catalan-pos","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For all natural numbers $r$ and $n$, the generalized FiboCatalan number with parameters $r$ and $n$ is a strictly positive rational number.","title":"Positivity of generalized FiboCatalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/generalized-fibonacci-catalan-is-pos-int","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciCatalan_isPosInt","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#generalized-fibonacci-catalan-is-pos-int","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For all natural numbers $r$ and $n$, the generalized FiboCatalan number with parameters $r$ and $n$ is a positive integer: it equals $k$ for some natural number $k \\ge 1$.","title":"Generalized FiboCatalan numbers are positive integers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/generalized-fibonacci-catalan-r0","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciCatalan_r0","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#generalized-fibonacci-catalan-r0","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For every natural number $n$, the generalized FiboCatalan number with parameters $r = 0$ and $n$ equals the FiboCatalan number $F_{2n}! / (F_n! \\, F_{n+1}!)$, where $F_i$ is the $i$-th Fibonacci number ($F_1 = F_2 = 1$) and $F_m! = F_1 F_2 \\cdots F_m$ is the Fibonacci factorial (with $F_0! = 1$).","title":"Generalized FiboCatalan numbers at r = 0"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/generalized-fibonacci-catalan-r1","kind":"node","lean":"MetaMathlibExt.generalizedFibonacciCatalan_r1","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#generalized-fibonacci-catalan-r1","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For every natural number $n$, the generalized FiboCatalan number with parameters $r = 1$ and $n$ equals $2 F_{2n}! / (F_{n+2}! \\, F_n!)$, where $F_i$ is the $i$-th Fibonacci number ($F_1 = F_2 = 1$) and $F_m! = F_1 F_2 \\cdots F_m$ is the Fibonacci factorial (with $F_0! = 1$).","title":"Generalized FiboCatalan numbers at r = 1"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibo-catalan","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.fiboCatalan","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibo-catalan","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For a natural number $n$, the FiboCatalan number is the rational number $C_{n,F} = F_{2n}!/(F_n! \\, F_{n+1}!)$, where $F_j$ is the $j$-th Fibonacci number and $F_k! = F_1 F_2 \\cdots F_k$ is the Fibonacci factorial.","title":"FiboCatalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibonomial","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.fibonomial","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibonomial","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the fibonomial coefficient $\\binom{n}{k}_F$ is the rational number $F_n!/(F_k! \\, F_{n-k}!)$ when $k \\le n$, where $F_i$ is the $i$-th Fibonacci number and $F_j! = F_1 F_2 \\cdots F_j$ is the Fibonacci factorial, and $0$ when $k > n$.","title":"Fibonomial coefficients"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/super-fibonacci-catalan-is-pos-int","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.superFibonacciCatalan_isPosInt","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#super-fibonacci-catalan-is-pos-int","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"For all natural numbers $m$ and $n$, the super FiboCatalan number","title":"Super FiboCatalan numbers are positive integers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fib-mul-sub-mul","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.fib_mul_sub_mul","parent":11940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci identity "],["math","\\(F_{2n} F_{n+2} - F_{2n+2} F_n = (-1)^n F_n\\)"]],"source":"Fibonacci identity $F_{2n} F_{n+2} - F_{2n+2} F_n = (-1)^n F_n$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fib-mul-sub-mul","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"For every natural number $n$, the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{k+2} = F_{k+1} + F_k$) satisfy $F_{2n} F_{n+2} - F_{2n+2} F_n = (-1)^n F_n$.","title":"Fibonacci identity $F_{2n} F_{n+2} - F_{2n+2} F_n = (-1)^n F_n$"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fibo-catalan-identity","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.fiboCatalan_identity","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fibo-catalan-identity","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers, $F_k! = F_1 F_2 \\cdots F_k$ the Fibonacci factorial and $C_{k,F} = F_{2k}!/(F_k! \\, F_{k+1}!)$ the FiboCatalan number. For every natural number $n$, in $\\mathbb{Q}$,","title":"An identity between consecutive FiboCatalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/fib-mul-factorial-quotient-eq-and-is-int","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.fib_mul_factorialQuotient_eq_and_isInt","parent":11940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrality of "],["math","\\(F_{2n+1} F_{2n}!/(F_{n+2}! \\, F_n!)\\)"]],"source":"Integrality of $F_{2n+1} F_{2n}!/(F_{n+2}! \\, F_n!)$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#fib-mul-factorial-quotient-eq-and-is-int","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers, $F_k! = F_1 F_2 \\cdots F_k$ the Fibonacci factorial and, for $k \\le m$, $\\binom{m}{k}_F = F_m!/(F_k! \\, F_{m-k}!)$ the fibonomial coefficient. For every natural number $n \\ge 1$,","title":"Integrality of $F_{2n+1} F_{2n}!/(F_{n+2}! \\, F_n!)$"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/lucas-poly","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.lucasPoly","parent":11940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas polynomials in "],["math","\\(s\\)"],["text"," and "],["math","\\(t\\)"]],"source":"Lucas polynomials in $s$ and $t$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#lucas-poly","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"The Lucas polynomials $\\{n\\}$, for natural numbers $n$, are the polynomials in two variables $s$ and $t$ with integer coefficients defined by $\\{0\\} = 0$, $\\{1\\} = 1$ and $\\{n + 2\\} = s\\{n + 1\\} + t\\{n\\}$ for every $n \\ge 0$.","title":"Lucas polynomials in $s$ and $t$"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/lucas-factorial","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.lucasFactorial","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#lucas-factorial","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For a natural number $n$, the Lucas factorial $\\{n\\}!$ is the product $\\{n\\}! = \\{1\\}\\{2\\} \\cdots \\{n\\}$ of the Lucas polynomials $\\{i\\}$ in $s$ and $t$ for $1 \\le i \\le n$, a polynomial with integer coefficients; in particular $\\{0\\}! = 1$, the empty product.","title":"Lucas factorial"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/lucas-kdiv-factorial","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.lucasKDivFactorial","parent":11940,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-divisible Lucas factorial"]],"source":"$k$-divisible Lucas factorial"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#lucas-kdiv-factorial","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the $k$-divisible Lucas factorial $\\{n : k\\}!$ is the product $\\{n : k\\}! = \\{k\\}\\{2k\\} \\cdots \\{nk\\}$ of the Lucas polynomials $\\{ki\\}$ in $s$ and $t$ for $1 \\le i \\le n$; in particular $\\{0 : k\\}! = 1$, the empty product.","title":"$k$-divisible Lucas factorial"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/exists-nonneg-poly-super-lucas-catalan","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.exists_nonneg_poly_superLucasCatalan","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#exists-nonneg-poly-super-lucas-catalan","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"Let $\\{k\\}$ be the Lucas polynomials in $s$ and $t$ ($\\{0\\} = 0$, $\\{1\\} = 1$, $\\{k+2\\} = s\\{k+1\\} + t\\{k\\}$) and let $\\{k\\}! = \\{1\\}\\{2\\} \\cdots \\{k\\}$ be the Lucas factorial. For all natural numbers $m$ and $n$ there is a polynomial $p \\in \\mathbb{Z}[s, t]$ with non-negative coefficients such that $p \\, \\{m\\}! \\, \\{n\\}! \\, \\{m+n\\}! = \\{2m\\}! \\, \\{2n\\}!$; that is, the Lucas analogue\u2026","title":"Lucas super FiboCatalan numbers have non-negative coefficients"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/exists-nonneg-poly-generalized-lucas-cat","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.exists_nonneg_poly_generalizedLucasCatalan","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#exists-nonneg-poly-generalized-lucas-cat","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"Let $\\{k\\}$ be the Lucas polynomials in $s$ and $t$ ($\\{0\\} = 0$, $\\{1\\} = 1$, $\\{k+2\\} = s\\{k+1\\} + t\\{k\\}$) and let $\\{k\\}! = \\{1\\}\\{2\\} \\cdots \\{k\\}$ be the Lucas factorial. For all natural numbers $r$ and $n$ there is a polynomial $p \\in \\mathbb{Z}[s, t]$ with non-negative coefficients such that $p \\, \\{r\\}! \\, \\{n\\}! \\, \\{n+r+1\\}! = \\{2r+1\\}! \\, \\{2n\\}!$; that is, with $J_{\\{r\\}} = \\{2r+1\\}!/\\{r\\}!$, the Lucas\u2026","title":"Lucas generalized FiboCatalan numbers have non-negative coefficients"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/lucas-poly-mul-sub-mul","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.lucasPoly_mul_sub_mul","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#lucas-poly-mul-sub-mul","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"For every natural number $n$, the Lucas polynomials $\\{k\\}$ in $s$ and $t$ satisfy $\\{2n\\}\\{n + 2\\} - \\{2n + 2\\}\\{n\\} = (-1)^n \\{2\\} t^n \\{n\\}$.","title":"Cross-product identity for Lucas polynomials"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/lucas-catalan-identity-mul-factorial","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.lucasCatalan_identity_mul_factorial","parent":11940,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#lucas-catalan-identity-mul-factorial","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"Let $\\{k\\}$ be the Lucas polynomials in $s$ and $t$ ($\\{0\\} = 0$, $\\{1\\} = 1$, $\\{k+2\\} = s\\{k+1\\} + t\\{k\\}$) and let $\\{k\\}! = \\{1\\}\\{2\\} \\cdots \\{k\\}$ be the Lucas factorial. For every natural number $n$, in $\\mathbb{Z}[s, t]$,","title":"Lucas analogue of the FiboCatalan identity, cleared of denominators"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/exists-nonneg-poly-lucas-quotient","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.exists_nonneg_poly_lucasQuotient","parent":11940,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lucas quotient "],["math","\\(\\{2n+1\\} \\{2\\} \\{2n\\}!/(\\{n+2\\}! \\, \\{n\\}!)\\)"],["text"," has non-negative coefficients"]],"source":"The Lucas quotient $\\{2n+1\\} \\{2\\} \\{2n\\}!/(\\{n+2\\}! \\, \\{n\\}!)$ has non-negative coefficients"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#exists-nonneg-poly-lucas-quotient","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"The Lucas quotient $\\{2n+1\\} \\{2\\} \\{2n\\}!/(\\{n+2\\}! \\, \\{n\\}!)$ has non-negative coefficients. Let $\\{k\\}$ be the Lucas polynomials in $s$ and $t$ ($\\{0\\} = 0$, $\\{1\\} = 1$, $\\{k+2\\} = s\\{k+1\\} + t\\{k\\}$) and let $\\{k\\}! = \\{1\\}\\{2\\} \\cdots \\{k\\}$ be the Lucas factorial. For every natural number $n \\ge 1$,","title":"The Lucas quotient $\\{2n+1\\} \\{2\\} \\{2n\\}!/(\\{n+2\\}! \\, \\{n\\}!)$ has non-negative coefficients"},{"id":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/exists-nonneg-poly-k-div-lucas-catalan","kind":"node","lean":"MathlibExt.NumberTheory.SuperFibocatalanWanted.exists_nonneg_poly_kDivLucasCatalan","parent":11940,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-divisible Lucas super Catalan numbers have non-negative coefficients"]],"source":"$k$-divisible Lucas super Catalan numbers have non-negative coefficients"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers/index.html#exists-nonneg-poly-k-div-lucas-catalan","scope":"NumberTheory/special-numbers/catalan-numbers/fibocatalan-numbers","status":"can_prove","summary":"Let $\\{j\\}$ be the Lucas polynomials in $s$ and $t$ ($\\{0\\} = 0$, $\\{1\\} = 1$, $\\{j+2\\} = s\\{j+1\\} + t\\{j\\}$) and let $\\{n : k\\}! = \\{k\\}\\{2k\\} \\cdots \\{nk\\}$ be the $k$-divisible Lucas factorial. For all natural numbers $k$, $m$ and $n$ with $k \\ge 1$ there is a polynomial $p \\in \\mathbb{Z}[s, t]$ with non-negative coefficients such that","title":"$k$-divisible Lucas super Catalan numbers have non-negative coefficients"},{"data_url":"scopes/NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers.json","id":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers","kind":"scope","navigation":"module","parent":11939,"result":false,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/index.html","scope":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers","status":"fully_proved","title":"Fuss\u2013Catalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/fuss-catalan-dvd-choose","kind":"node","lean":"MetaMathlibExt.fuss_catalan_dvd_choose","parent":11972,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/index.html#fuss-catalan-dvd-choose","scope":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers","status":"fully_proved","summary":"For natural numbers $a \\ge 2$ and $n \\ge 1$, the integer $(a-1)n + 1$ divides the binomial coefficient $\\binom{an}{n}$. Hence the Fuss\u2013Catalan number $\\frac{1}{(a-1)n+1}\\binom{an}{n}$ is an integer.","title":"Integrality of the Fuss\u2013Catalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/fuss-catalan-number","kind":"node","lean":"MetaMathlibExt.fussCatalanNumber","parent":11972,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/index.html#fuss-catalan-number","scope":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers","status":"fully_proved","summary":"For natural numbers $k \\ge 1$ (the chord arity) and $m$, the Fuss\u2013Catalan number is the natural-number quotient of $\\binom{km}{m}$ by $(k-1)m+1$, that is $\\lfloor \\binom{km}{m} / ((k-1)m+1) \\rfloor$.","title":"Fuss\u2013Catalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/generalized-catalan-prime-dvd-iff-not-ge","kind":"node","lean":"MetaMathlibExt.generalizedCatalan_prime_dvd_iff_not_geometric_sum","parent":11972,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/index.html#generalized-catalan-prime-dvd-iff-not-ge","scope":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers","status":"fully_proved","summary":"Let $p$ be a prime and $n$ a natural number. Then $p$ divides the generalized Catalan number $\\frac{1}{(p-1)n+1}\\binom{pn}{n}$ (an integer) if and only if $n$ is not of the form $\\frac{p^k-1}{p-1}$ for any natural number $k$.","title":"Prime divisibility of generalized Catalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/generalized-fuss-catalan-number","kind":"node","lean":"MetaMathlibExt.generalizedFussCatalanNumber","parent":11972,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/index.html#generalized-fuss-catalan-number","scope":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers","status":"fully_proved","summary":"For natural numbers $m$, $n$ and $k$, the generalized Fuss\u2013Catalan number $F_m(n, k)$ is the quotient of $k \\binom{mn + k}{n}$ by $mn + k$ in natural-number division, rounded down, with division by $0$ giving $0$.","title":"Generalized Fuss\u2013Catalan number"},{"id":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/generalized-fuss-catalan-number-eq","kind":"node","lean":"MetaMathlibExt.generalizedFussCatalanNumber_eq","parent":11972,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/index.html#generalized-fuss-catalan-number-eq","scope":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers","status":"fully_proved","summary":"For all natural numbers $m$, $n$ and $k$, the generalized Fuss\u2013Catalan number $F_m(n, k)$ equals the quotient of $k \\binom{mn + k}{n}$ by $mn + k$ in natural-number division, rounded down, with division by $0$ giving $0$.","title":"Formula for the generalized Fuss\u2013Catalan number"},{"id":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/generalized-fuss-catalan-number-one","kind":"node","lean":"MetaMathlibExt.generalizedFussCatalanNumber_one","parent":11972,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Fuss\u2013Catalan numbers at "],["math","\\(n = 1\\)"]],"source":"Generalized Fuss\u2013Catalan numbers at $n = 1$"},"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers/index.html#generalized-fuss-catalan-number-one","scope":"NumberTheory/special-numbers/catalan-numbers/fuss-catalan-numbers","status":"fully_proved","summary":"For all natural numbers $m$ and $k$, the generalized Fuss\u2013Catalan number $F_m(1, k)$ equals $k$.","title":"Generalized Fuss\u2013Catalan numbers at $n = 1$"},{"data_url":"scopes/NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers.json","id":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers","kind":"scope","navigation":"module","parent":11939,"result":false,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/index.html","scope":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers","status":"fully_proved","title":"Lucasnomial Fuss\u2013Catalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/generalized-lucasnomial-fuss-catalan-int","kind":"node","lean":"MetaMathlibExt.generalized_lucasnomial_fuss_catalan_integral_general","parent":11979,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/index.html#generalized-lucasnomial-fuss-catalan-int","scope":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers","status":"fully_proved","summary":"Let $P$ and $Q$ be integers with $\\gcd(P, Q) = 1$, let $(U_m)_{m \\ge 0}$ be the Lucas sequence of the first kind with parameters $P$ and $Q$, and suppose $U_m \\ne 0$ for every $m \\ge 1$. Then for all natural numbers $r \\ge 1$, $a \\ge 2$ and $n \\ge 1$, the rational number","title":"Integrality of generalized Lucasnomial Fuss\u2013Catalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/generalized-lucasnomial-fuss-catalan-int-2","kind":"node","lean":"MetaMathlibExt.generalized_lucasnomial_fuss_catalan_integral","parent":11979,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/index.html#generalized-lucasnomial-fuss-catalan-int-2","scope":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers","status":"fully_proved","summary":"Let $P$ and $Q$ be nonzero integers with $\\gcd(P, Q) = 1$, and let $U : \\mathbb{N} \\to \\mathbb{Z}$ satisfy $U_0 = 0$, $U_1 = 1$, $U_{m+2} = P U_{m+1} - Q U_m$ for every natural number $m$, and $U_m \\ne 0$ for every $m \\ge 1$. Then for all natural numbers $r \\ge 1$, $a \\ge 2$ and $n \\ge 1$, the rational number","title":"Generalized Lucasnomial Fuss\u2013Catalan numbers are integers (Ballot)"},{"id":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/generalized-lucasnomial-fuss-catalan","kind":"node","lean":"MetaMathlibExt.generalizedLucasnomialFussCatalan","parent":11979,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/index.html#generalized-lucasnomial-fuss-catalan","scope":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers","status":"fully_proved","summary":"Let $P$ be Lucas parameters, that is, nonzero reals $p$ and $q$ whose Lucas bracket sequence $U_0 = 0$, $U_1 = 1$, $U_{m+2} = pU_{m+1} + qU_m$ satisfies $U_m \\ne 0$ for every $m \\ge 1$. Write $U_m! = U_1 U_2 \\cdots U_m$, and let the Lucasnomial coefficient $\\binom{m}{k}_U$ be $\\frac{U_m!}{U_k! \\, U_{m-k}!}$ for $k \\le m$ and $0$ for $k > m$. For natural numbers $a$, $r$ and $n$, the generalized Lucasnomial\u2026","title":"Generalized Lucasnomial Fuss\u2013Catalan numbers"},{"id":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/generalized-lucasnomial-fuss-catalan-rs","kind":"node","lean":"MetaMathlibExt.generalizedLucasnomialFussCatalanRS","parent":11979,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/index.html#generalized-lucasnomial-fuss-catalan-rs","scope":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers","status":"fully_proved","summary":"Let $P$ be Lucas parameters, that is, nonzero reals $p$ and $q$ whose Lucas bracket sequence $U_0 = 0$, $U_1 = 1$, $U_{m+2} = pU_{m+1} + qU_m$ satisfies $U_m \\ne 0$ for every $m \\ge 1$. Write $U_m! = U_1 U_2 \\cdots U_m$, and let the Lucasnomial coefficient $\\binom{m}{k}_U$ be $\\frac{U_m!}{U_k! \\, U_{m-k}!}$ for $k \\le m$ and $0$ for $k > m$. For natural numbers $r$, $s$ and $t$, define the real number","title":"Generalized Lucasnomial Fuss\u2013Catalan numbers in the r, s, t notation"},{"id":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/generalized-lucasnomial-fuss-catalan-vol","kind":"node","lean":"MetaMathlibExt.generalizedLucasnomialFussCatalan_vol24_eq","parent":11979,"result":true,"route":"../roadmap/NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers/index.html#generalized-lucasnomial-fuss-catalan-vol","scope":"NumberTheory/special-numbers/catalan-numbers/lucasnomial-fuss-catalan-numbers","status":"fully_proved","summary":"For all Lucas parameters $P$, with Lucas bracket sequence $U$, and all natural numbers $a$, $r$ and $n$, the generalized Lucasnomial Fuss\u2013Catalan number $C_{U,a,r}(n)$ equals $C'_{U,a,r}(n)$, where $C'_{U,r,s}(t)$ denotes the generalized Lucasnomial Fuss\u2013Catalan number in the r, s, t notation. The two definitions are the same function.","title":"The two indexings of the generalized Lucasnomial Fuss\u2013Catalan numbers agree"},{"data_url":"scopes/NumberTheory/digits-numeration.json","id":"NumberTheory/digits-numeration","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/index.html","scope":"NumberTheory/digits-numeration","status":"can_state","summary":"Base-b digits of integers, repdigits and palindromes, Kaprekar and happy numbers, Zeckendorf and base-\u03c6 systems, automatic sequences.","title":"Digits and numeration systems"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties.json","id":"NumberTheory/digits-numeration/digit-properties","kind":"scope","parent":11985,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/index.html","scope":"NumberTheory/digits-numeration/digit-properties","status":"can_prove","summary":"Digit sums and products, Niven, happy, Kaprekar, Keith numbers, comma sequences, decimal expansions of fractions (Midy, repetends).","title":"Digit properties of integers"},{"id":"NumberTheory/digits-numeration/digit-properties/grundman-consecutive-niven-bound","kind":"node","lean":"MathlibExt.NumberTheory.Digits.GrundmanNivenBoundWanted.grundman_consecutive_niven_bound","parent":11986,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/index.html#grundman-consecutive-niven-bound","scope":"NumberTheory/digits-numeration/digit-properties","status":"fully_proved","summary":"Let $b \\ge 2$ and $a \\ge 1$ be integers. Then for some $i$ with $0 \\le i \\le 2b$, the integer $a + i$ is not divisible by the sum of its base-$b$ digits. Equivalently, the $2b + 1$ consecutive integers $a, a + 1, \\dots, a + 2b$ are not all $b$-Niven numbers, where a $b$-Niven number is a positive integer divisible by the sum of its base-$b$ digits.","title":"Grundman's bound on consecutive Niven numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/tu-deng-conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.TuDengConjectureWanted.tuDengConjecture_holds","parent":11986,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/index.html#tu-deng-conjecture-holds","scope":"NumberTheory/digits-numeration/digit-properties","status":"can_prove","summary":"For every integer $k \\ge 2$ and every nonzero residue $t$ modulo $2^k - 1$, the number of pairs $(a, b)$ of residues modulo $2^k - 1$ with $a + b = t$ and $w(a) + w(b) \\le k - 1$ is at most $2^{k-1}$, where $w(a)$ is the binary digit sum of the representative of $a$ in $\\{0, 1, \\dots, 2^k - 2\\}$.","title":"Tu\u2013Deng conjecture"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu.json","id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"can_prove","title":"Binary digit sums and the Trollope\u2013Delange formula"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/q-weighted-digit-sum","kind":"node","lean":"MetaMathlibExt.qWeightedDigitSum","parent":11989,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-weighted binary digit sum"]],"source":"The $q$-weighted binary digit sum"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#q-weighted-digit-sum","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"Let $q$ be an element of a commutative ring and let $k$ be a natural number with binary expansion $k = \\sum_{i \\ge 0} \\omega_i 2^i$, where $\\omega_i \\in \\{0, 1\\}$. The $q$-weighted binary digit sum of $k$ is $s_q(k) = \\sum_{i \\ge 0} \\omega_i q^{i+1}$, a finite sum because only finitely many $\\omega_i$ are nonzero.","title":"The $q$-weighted binary digit sum"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/q-weighted-digit-partial-sum","kind":"node","lean":"MetaMathlibExt.qWeightedDigitPartialSum","parent":11989,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sums of the "],["math","\\(q\\)"],["text","-weighted binary digit sum"]],"source":"Partial sums of the $q$-weighted binary digit sum"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#q-weighted-digit-partial-sum","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"For an element $q$ of a commutative ring and a natural number $n$, the partial sum is $S_q(n) = \\sum_{k=0}^{n-1} s_q(k)$, where $s_q$ is the $q$-weighted binary digit sum.","title":"Partial sums of the $q$-weighted binary digit sum"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/takagi-landsberg","kind":"node","lean":"MetaMathlibExt.takagiLandsberg","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#takagi-landsberg","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"For a real or complex parameter $a$ and a real number $x$, let $\\tau(y)$ denote the distance from a real number $y$ to the nearest integer. The Takagi\u2013Landsberg function is $\\mathcal{T}_a(x) = \\sum_{n \\ge 0} a^n \\tau(2^n x)$, where the value is taken to be $0$ when the series does not converge absolutely. The series converges for $|a| < 1$, and at dyadic rationals $x$ it is a finite sum.","title":"The Takagi\u2013Landsberg function"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/q-weighted-digit-sum-zero","kind":"node","lean":"MetaMathlibExt.qWeightedDigitSum_zero","parent":11989,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-weighted digit sum of zero"]],"source":"The $q$-weighted digit sum of zero"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#q-weighted-digit-sum-zero","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"For every element $q$ of a commutative ring, the $q$-weighted binary digit sum of $0$ is $s_q(0) = 0$.","title":"The $q$-weighted digit sum of zero"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/q-weighted-digit-sum-of-pos","kind":"node","lean":"MetaMathlibExt.qWeightedDigitSum_of_pos","parent":11989,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binary recursion for the "],["math","\\(q\\)"],["text","-weighted digit sum"]],"source":"Binary recursion for the $q$-weighted digit sum"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#q-weighted-digit-sum-of-pos","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"Let $q$ be an element of a commutative ring and let $k > 0$ be a natural number. Then the $q$-weighted binary digit sum satisfies $s_q(k) = q\\,(k \\bmod 2) + q\\,s_q(\\lfloor k/2 \\rfloor)$.","title":"Binary recursion for the $q$-weighted digit sum"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/takagi-landsberg-zero","kind":"node","lean":"MetaMathlibExt.takagiLandsberg_zero","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#takagi-landsberg-zero","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"For every real or complex parameter $a$, the Takagi\u2013Landsberg function satisfies $\\mathcal{T}_a(0) = 0$.","title":"The Takagi\u2013Landsberg function vanishes at zero"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/q-weighted-digit-partial-sum-dyadic-fluc","kind":"node","lean":"MetaMathlibExt.qWeightedDigitPartialSum_dyadic_fluctuation","parent":11989,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dyadic values of the normalized "],["math","\\(q\\)"],["text","-weighted fluctuation"]],"source":"Dyadic values of the normalized $q$-weighted fluctuation"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#q-weighted-digit-partial-sum-dyadic-fluc","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"can_prove","summary":"Let $q$ be a complex number with $|q| > 1/2$, let $N > 0$ be a natural number, and let $j \\le 2^{N-1}$ be a natural number. Put $t_j = j / 2^{N-1}$ and $a = 1/(2q)$. Then","title":"Dyadic values of the normalized $q$-weighted fluctuation"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/q-weighted-trollope-delange","kind":"node","lean":"MetaMathlibExt.qWeighted_trollopeDelange","parent":11989,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-weighted Trollope\u2013Delange formula"]],"source":"The $q$-weighted Trollope\u2013Delange formula"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#q-weighted-trollope-delange","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"can_prove","summary":"Let $q$ be a real number with $|q| > 1/2$ and $q \\ne 1$, and put $a = 1/(2q)$. For a positive integer $n$, let $k = \\lfloor \\log_2 n \\rfloor$ and let $u = \\log_2 n - k$ be the fractional part of $\\log_2 n$. Write $F(u) = 2^{1-u}\\,\\mathcal{T}_a(2^{u-1})$. Then","title":"The $q$-weighted Trollope\u2013Delange formula"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/dyadic-mantissa-well-defined-iff-doublin","kind":"node","lean":"MetaMathlibExt.dyadicMantissa_wellDefined_iff_doubling_invariant","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#dyadic-mantissa-well-defined-iff-doublin","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"can_prove","summary":"Let $G$ be a function from the natural numbers to the real numbers, and for a positive integer $n$ let $p_n = 2^{\\lfloor \\log_2 n \\rfloor}$, so that $(n - p_n)/p_n$ is a dyadic rational in $[0, 1)$. The following are equivalent: (i) for all positive integers $m$ and $n$ with $(m - p_m)/p_m = (n - p_n)/p_n$ we have $G(m) = G(n)$, that is, $F((n - p_n)/p_n) = G(n)$ well-defines a function $F$ of these dyadic\u2026","title":"Girgensohn's lemma"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/dyadic-add-nat-carry","kind":"node","lean":"MetaMathlibExt.dyadicAddNatCarry","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#dyadic-add-nat-carry","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"Let $\\omega$ be a dyadic integer, written as its sequence $(\\omega_0, \\omega_1, \\dots)$ of binary digits $\\omega_i \\in \\{0, 1\\}$ (least significant digit first), and let $j$ be a natural number with binary digits $j_0, j_1, \\dots$. The carry $c_i$ into digit $i$ in the binary addition of $j$ to $\\omega$ is defined by $c_0 = 0$ and $c_{i+1} = \\lfloor (\\omega_i + j_i + c_i)/2 \\rfloor$.","title":"Carry in adding a natural number to a dyadic integer"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/dyadic-add-nat","kind":"node","lean":"MetaMathlibExt.dyadicAddNat","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#dyadic-add-nat","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"Let $\\omega$ be a dyadic integer, written as its sequence $(\\omega_0, \\omega_1, \\dots)$ of binary digits $\\omega_i \\in \\{0, 1\\}$ (least significant digit first), and let $j$ be a natural number with binary digits $j_0, j_1, \\dots$. The dyadic integer $\\omega + j$ has $i$-th digit $(\\omega_i + j_i + c_i) \\bmod 2$, where the carries are given by $c_0 = 0$ and $c_{i+1} = \\lfloor (\\omega_i + j_i + c_i)/2 \\rfloor$.","title":"Adding a natural number to a dyadic integer"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/dyadic-odometer","kind":"node","lean":"MetaMathlibExt.dyadicOdometer","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#dyadic-odometer","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"The dyadic odometer is the map $T$ on the dyadic integers $\\mathbb{Z}_2$, viewed as binary digit sequences $\\omega = (\\omega_0, \\omega_1, \\dots)$ (least significant digit first), given by $T\\omega = \\omega + 1$, binary addition of $1$ with carries.","title":"Dyadic odometer"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/dyadic-odometer-zero-apply","kind":"node","lean":"MetaMathlibExt.dyadicOdometer_zero_apply","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#dyadic-odometer-zero-apply","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"Let $T$ be the dyadic odometer $T\\omega = \\omega + 1$ on dyadic integers $\\omega = (\\omega_0, \\omega_1, \\dots)$, written as binary digit sequences. The image $T(0)$ of the zero sequence $(0, 0, 0, \\dots)$ has digits $T(0)_0 = 1$, $T(0)_1 = 0$ and $T(0)_2 = 0$.","title":"First digits of the odometer at zero"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/dyadic-haar","kind":"node","lean":"MetaMathlibExt.dyadicHaar","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#dyadic-haar","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"The Haar measure on the dyadic integers $\\mathbb{Z}_2$, viewed as the space $\\{0, 1\\}^{\\mathbb{N}}$ of binary digit sequences: the infinite product of the uniform probability measures on $\\{0, 1\\}$, one for each digit.","title":"Haar measure on the dyadic integers"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/odometer-digit-partial-sum","kind":"node","lean":"MetaMathlibExt.odometerDigitPartialSum","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#odometer-digit-partial-sum","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"Let $q \\in \\mathbb{C}$, let $\\omega = (\\omega_0, \\omega_1, \\dots)$ be a dyadic integer written as its binary digit sequence, let $T$ be the dyadic odometer $T\\omega = \\omega + 1$, and let $s_q(\\omega) = \\sum_{i \\ge 0} \\omega_i q^{i+1}$, taken to be $0$ when the series is not absolutely summable. For real $x$ the odometer partial sum is","title":"Odometer partial sums"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/odometer-digit-partial-sum-zero","kind":"node","lean":"MetaMathlibExt.odometerDigitPartialSum_zero","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#odometer-digit-partial-sum-zero","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"For every $q \\in \\mathbb{C}$ and every dyadic integer $\\omega$, the interpolated odometer partial sum satisfies $S_{q,\\omega}(0) = 0$.","title":"The odometer partial sum vanishes at zero"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/odometer-max-fluctuation","kind":"node","lean":"MetaMathlibExt.odometerMaxFluctuation","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#odometer-max-fluctuation","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"Let $q$ be a complex number, let $\\omega = (\\omega_0, \\omega_1, \\dots)$ be a dyadic integer (a sequence of binary digits), let $l$ be a natural number, and let $S_{q,\\omega}$ denote the interpolated odometer partial sums. The maximal fluctuation is the supremum over $t \\in [0, 1]$ of $|S_{q,\\omega}(t l) - t\\,S_{q,\\omega}(l)|$.","title":"Maximal fluctuation of the odometer partial sums"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/is-stabilizing-seq","kind":"node","lean":"MetaMathlibExt.IsStabilizingSeq","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#is-stabilizing-seq","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"fully_proved","summary":"Let $q \\in \\mathbb{C}$, let $\\omega$ be a dyadic integer, let $S_{q,\\omega}$ be the interpolated odometer partial sum, and for a natural number $l$ let $M_{q,\\omega}(l) = \\sup_{t \\in [0, 1]} |S_{q,\\omega}(tl) - t S_{q,\\omega}(l)|$ be the maximal fluctuation. A sequence $(l_n)$ of natural numbers is a stabilizing sequence for $(q, \\omega)$ with normalizing sequence $(R_n)$ of complex numbers and limiting curve\u2026","title":"Stabilizing sequence for the dyadic odometer"},{"id":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/odometer-limiting-curve-eq-neg-takagi","kind":"node","lean":"MetaMathlibExt.odometer_limitingCurve_eq_neg_takagi","parent":11989,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu/index.html#odometer-limiting-curve-eq-neg-takagi","scope":"NumberTheory/digits-numeration/digit-properties/binary-digit-sums-and-the-trollope-delange-formu","status":"can_prove","summary":"Let $q \\in \\mathbb{C}$ with $1/2 < |q| < 1$ and put $a = 1/(2q)$. Then for almost every dyadic integer $\\omega$ with respect to the Haar measure on $\\mathbb{Z}_2$ there are a sequence $(l_n)$ of natural numbers and a sequence $(R_n)$ of complex numbers such that $(l_n)$ is a stabilizing sequence for $(q, \\omega)$ with normalizing sequence $(R_n)$ and limiting curve $t \\mapsto -\\mathcal{T}_a(t)$, where\u2026","title":"Odometer limiting curve is the negative Takagi\u2013Landsberg function"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/comma-sequences.json","id":"NumberTheory/digits-numeration/digit-properties/comma-sequences","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","title":"Comma sequences"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/comma-msd","kind":"node","lean":"MetaMathlibExt.commaMsd","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#comma-msd","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"fully_proved","summary":"For natural numbers $b$ and $m$, let $k$ be the base-$b$ logarithm of $m$ rounded down, that is, the largest $k$ with $b^k \\le m$ when $b \\ge 2$ and $m \\ge 1$, and $k = 0$ otherwise. The most significant digit of $m$ in base $b$ is $\\lfloor m / b^k \\rfloor$. When $b \\ge 2$ and $m \\ge 1$ this is the leading digit of the base-$b$ expansion of $m$.","title":"Most significant digit"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/is-comma-candidate","kind":"node","lean":"MetaMathlibExt.IsCommaCandidate","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#is-comma-candidate","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"fully_proved","summary":"Let $b$ and $n$ be natural numbers and let $y$ be a digit with $0 \\le y < b$. Put $m = n + b\\,(n \\bmod b) + y$, where $n \\bmod b$ is the remainder of $n$ on division by $b$. Then $y$ is a comma candidate for $n$ in base $b$ if $m > 0$ and the most significant base-$b$ digit of $m$ equals $y$.","title":"Comma candidate digits"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/is-comma-succ","kind":"node","lean":"MetaMathlibExt.IsCommaSucc","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#is-comma-succ","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"fully_proved","summary":"Let $b$, $n$ and $m$ be natural numbers. Then $m$ is the comma successor of $n$ in base $b$ if there is a comma candidate digit $y < b$ for $n$ in base $b$ such that $m = n + b\\,(n \\bmod b) + y$ and no digit $y' < y$ is a comma candidate for $n$ in base $b$.","title":"Comma successor"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/is-comma-terminal","kind":"node","lean":"MetaMathlibExt.IsCommaTerminal","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#is-comma-terminal","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"fully_proved","summary":"Let $b$ and $n$ be natural numbers. Then $n$ is comma-terminal in base $b$ if no digit $y$ with $0 \\le y < b$ is a comma candidate for $n$ in base $b$, so $n$ has no comma successor in base $b$.","title":"Comma-terminal numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/is-infinite-comma-seq","kind":"node","lean":"MetaMathlibExt.IsInfiniteCommaSeq","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#is-infinite-comma-seq","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"fully_proved","summary":"Let $b$ and $v$ be natural numbers and let $a : \\mathbb{N} \\to \\mathbb{N}$ be a sequence. Then $a$ is an infinite comma sequence in base $b$ starting at $v$ if $b \\ge 2$, $v > 0$, $a_1 = v$, and for every $n \\ge 1$ the term $a_{n+1}$ is the comma successor of $a_n$ in base $b$.","title":"Infinite comma sequences"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/is-comma-terminating","kind":"node","lean":"MetaMathlibExt.IsCommaTerminating","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#is-comma-terminating","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"fully_proved","summary":"Let $b$ and $v$ be natural numbers. The comma sequence in base $b$ starting at $v$ terminates if $b \\ge 2$, $v > 0$, and there is no infinite comma sequence in base $b$ starting at $v$.","title":"Terminating comma sequences"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/comma-sequence-terminates-of-base-le-633","kind":"node","lean":"MetaMathlibExt.commaSequence_terminates_of_base_le_633","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#comma-sequence-terminates-of-base-le-633","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","summary":"For every base $b$ with $3 \\le b \\le 633$ and every positive integer $v$, the comma sequence in base $b$ starting at $v$ terminates, that is, there is no infinite comma sequence in base $b$ starting at $v$.","title":"Comma sequences terminate in bases 3 to 633"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/is-comma-terminal-iff-digits","kind":"node","lean":"MetaMathlibExt.isCommaTerminal_iff_digits","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#is-comma-terminal-iff-digits","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","summary":"Let $b \\ge 2$ and let $n$ be a positive integer. Then no digit $y < b$ is a comma candidate for $n$ in base $b$ (so $n$ has no comma successor) if and only if the base-$b$ representation of $n$, most significant digit first, is $(b-1) \\cdots (b-1)\\, x\\, (b-1-x)$ with some number $i \\ge 0$ of leading digits $b - 1$ and $1 \\le x \\le b - 2$.","title":"Landmines of the comma sequence"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/two-comma-children-iff","kind":"node","lean":"MetaMathlibExt.two_commaChildren_iff","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#two-comma-children-iff","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","summary":"Fix a base $b \\ge 2$, and say that $n$ has at least two comma-children if there are distinct digits $y_1 \\ne y_2$ below $b$ that are both comma candidates for $n$ in base $b$ (distinct candidates $y$ give distinct children $n + b\\,(n \\bmod b) + y$). Then a positive integer $n < b^2$ has at least two comma-children if and only if $n = (b - 1 - 2x)\\, b + x$ for some $x$ with $1 \\le x$ and $2x \\le b - 1$; and an\u2026","title":"Numbers with two comma-children"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/comma-child-parent-eq","kind":"node","lean":"MetaMathlibExt.commaChild_parent_eq","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#comma-child-parent-eq","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","summary":"Fix a base $b \\ge 2$ and a natural number $n$. Call $n$ a comma-child of $k$ if $n = k + b\\,(k \\bmod b) + y$ for some comma candidate digit $y$ of $k$ in base $b$. Let $f$ be the most significant base-$b$ digit of $n$ and let $x = (n - f) \\bmod b$. If $n$ is a comma-child of a positive integer $k$, then $k = n - x b - f$; and if the integer $n - x b - f$ is negative, then $n$ is not a comma-child of any positive\u2026","title":"Recovering the parent of a comma-child"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/not-comma-succ-iff","kind":"node","lean":"MetaMathlibExt.not_commaSucc_iff","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#not-comma-succ-iff","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","summary":"Let $b \\ge 3$ and $n \\ge b^2 - 1$. Then $n$ is not the comma successor in base $b$ of any positive integer if and only if $n = c\\, b^i$ for some $i \\ge 2$ and some $c$ with $2 \\le c \\le b - 1$.","title":"Numbers that are not comma successors"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/exists-comma-parent-of-sq-le","kind":"node","lean":"MetaMathlibExt.exists_commaParent_of_sq_le","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#exists-comma-parent-of-sq-le","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","summary":"Let $b \\ge 2$ and $n \\ge b^2$. Then there are a positive integer $k$ and a comma candidate digit $y$ of $k$ in base $b$ with $n = k + b\\,(k \\bmod b) + y$, that is, $n$ is a comma-child of $k$.","title":"Every number at least the square of the base is a comma-child"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/exists-infinite-comma-child-path","kind":"node","lean":"MetaMathlibExt.exists_infinite_commaChild_path","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#exists-infinite-comma-child-path","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","summary":"For every base $b \\ge 2$ there is a sequence $a_0, a_1, \\dots$ of positive integers such that for every $i$ there is a comma candidate digit $y$ of $a_i$ in base $b$ with $a_{i+1} = a_i + b\\,(a_i \\bmod b) + y$, that is, each $a_{i+1}$ is a comma-child of $a_i$.","title":"The comma-child graph has an infinite path"},{"id":"NumberTheory/digits-numeration/digit-properties/comma-sequences/exists-unique-comma-child-path-base-thre","kind":"node","lean":"MetaMathlibExt.existsUnique_commaChild_path_base_three","parent":12009,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/comma-sequences/index.html#exists-unique-comma-child-path-base-thre","scope":"NumberTheory/digits-numeration/digit-properties/comma-sequences","status":"can_prove","summary":"There is exactly one sequence $a_0, a_1, \\dots$ of natural numbers with $a_0 = 1$ such that for every $i$ there is a comma candidate digit $y$ of $a_i$ in base $3$ with $a_{i+1} = a_i + 3\\,(a_i \\bmod 3) + y$, that is, each $a_{i+1}$ is a base-3 comma-child of $a_i$.","title":"There is a unique infinite comma-child path from 1 in base 3"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri.json","id":"NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri/index.html","scope":"NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri","status":"can_prove","title":"Decimal expansion of reciprocals of Mersenne primes"},{"id":"NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri/ten-pow-block-length-mod-eq-two-pow","kind":"node","lean":"MetaMathlibExt.ten_pow_blockLength_modEq_two_pow","parent":12024,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri/index.html#ten-pow-block-length-mod-eq-two-pow","scope":"NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri","status":"can_prove","summary":"Let $p$ be a natural number such that $M = 2^p - 1$ is prime, let $L$ be the multiplicative order of $10$ modulo $M$ (the period of the decimal expansion of $1/M$), and suppose that $p$ divides $L$. Then there is a natural number $r$ with $1 \\le r \\le p - 1$ such that $10^{L/p} \\equiv 2^r \\pmod{M}$.","title":"The block-length power of ten is a power of two modulo the Mersenne prime"},{"id":"NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri/mersenne-reciprocal-block-recurrence","kind":"node","lean":"MetaMathlibExt.mersenne_reciprocal_block_recurrence","parent":12024,"reading_title":{"format":"inline-parts-v1","parts":[["text","Doubling recurrence for the blocks of "],["math","\\(1/(2^p - 1)\\)"]],"source":"Doubling recurrence for the blocks of $1/(2^p - 1)$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri/index.html#mersenne-reciprocal-block-recurrence","scope":"NumberTheory/digits-numeration/digit-properties/decimal-expansion-of-reciprocals-of-mersenne-pri","status":"can_prove","summary":"Let $p$ be a natural number such that $M = 2^p - 1$ is prime, let $L$ be the multiplicative order of $10$ modulo $M$ (the period of the decimal expansion of $1/M$), and suppose that $p$ divides $L$. Put $\\ell = L/p$ and let $N = (10^L - 1)/M$ be the repeating block of $1/M$, read as an $L$-digit integer with leading zeros. For $1 \\le k \\le p$ let $B_k = \\lfloor N / 10^{\\ell (p - k)} \\rfloor \\bmod 10^\\ell$ be the\u2026","title":"Doubling recurrence for the blocks of $1/(2^p - 1)$"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers.json","id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"can_prove","title":"Digitally robust composite numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/replace-digit","kind":"node","lean":"MetaMathlibExt.replaceDigit","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#replace-digit","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"For natural numbers $N$, $i$ and $d$, the result of replacing the digit of $N$ at position $i$ by $d$ is the natural number","title":"Replacing a base-ten digit"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/delete-digit","kind":"node","lean":"MetaMathlibExt.deleteDigit","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#delete-digit","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"For natural numbers $N$ and $i$, deleting the base-ten digit of $N$ in position $i$, where positions are counted from $0$ at the units digit, gives the number","title":"Deleting a digit"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/insert-digit","kind":"node","lean":"MetaMathlibExt.insertDigit","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#insert-digit","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"For natural numbers $N$, $i$ and $d$, inserting $d$ into the base-ten expansion of $N$ in position $i$, where positions are counted from $0$ at the units digit, gives the number","title":"Inserting a digit"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/is-digitally-robust-composite","kind":"node","lean":"MetaMathlibExt.IsDigitallyRobustComposite","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#is-digitally-robust-composite","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"A natural number $N$ is a digitally robust composite if it is composite (greater than $1$ and not prime), coprime to $10$, and each of the following numbers is greater than $1$ and not prime: the result of replacing the base-ten digit of $N$ in any position $i$ by any digit $d < 10$ other than $\\lfloor N / 10^{i} \\rfloor \\bmod 10$; the result of deleting the single digit in any position $i$ smaller than the number\u2026","title":"Digitally robust composite numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/infinitely-many-digitally-robust-composi","kind":"node","lean":"MetaMathlibExt.infinitelyMany_digitallyRobustComposites","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#infinitely-many-digitally-robust-composi","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"can_prove","summary":"There are infinitely many natural numbers that are digitally robust composites: composite numbers coprime to $10$ that stay composite after replacing any one base-ten digit (including a leading zero) by a different digit, after deleting any one of their digits, and after inserting any digit in any position.","title":"Infinitely many digitally robust composites"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/cyclotomic-prime-divisor-order-and-large","kind":"node","lean":"MetaMathlibExt.cyclotomicPrimeDivisor_order_and_largestPrime","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#cyclotomic-prime-divisor-order-and-large","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"can_prove","summary":"Let $b$ be a positive integer, let $\\Phi_b$ be the $b$-th cyclotomic polynomial with integer coefficients, and let $p$ be a prime that divides the integer $\\Phi_b(10)$. If $b < p$, then the multiplicative order of $10$ modulo $p$ is $b$. If $p \\le b$ and $b > 2$, then $p$ divides $b$, every prime divisor $q$ of $b$ satisfies $q \\le p$ (so $p$ is the largest prime divisor of $b$), and $p^2$ does not divide\u2026","title":"Prime divisors of cyclotomic values at 10"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/replace-digit-25011-units","kind":"node","lean":"MetaMathlibExt.replaceDigit_25011_units","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#replace-digit-25011-units","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"Replacing the base-ten digit of $25011$ at position $0$, its units digit, by $3$ gives $25013$.","title":"Replacing the units digit of 25011 by 3"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/delete-digit-25011-ten-thousands","kind":"node","lean":"MetaMathlibExt.deleteDigit_25011_tenThousands","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#delete-digit-25011-ten-thousands","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"Deleting the base-ten digit in position $4$ of $25011$, where positions are counted from $0$ at the units digit (so position $4$ is the ten-thousands digit $2$), gives $5011$.","title":"Deleting the ten-thousands digit of 25011"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/insert-digit-25011-leading-zeros","kind":"node","lean":"MetaMathlibExt.insertDigit_25011_leadingZeros","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#insert-digit-25011-leading-zeros","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"Inserting the digit $1$ in position $7$ of $25011$, where positions are counted from $0$ at the units digit (so the new digit goes between the second and third leading zeros), gives $10025011$.","title":"Inserting a digit among the leading zeros of 25011"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/replace-digit-25011-leading-zero","kind":"node","lean":"MetaMathlibExt.replaceDigit_25011_leadingZero","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#replace-digit-25011-leading-zero","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"Replacing the base-ten digit in position $6$ of $25011$, where positions are counted from $0$ at the units digit (so this digit is a leading zero), by $0$ gives $25011$ again.","title":"Replacing a leading zero of 25011 by zero"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/digit-value-lt-ten","kind":"node","lean":"MetaMathlibExt.digitValue_lt_ten","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#digit-value-lt-ten","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"For all natural numbers $N$ and $i$, the base-ten digit of $N$ in position $i$ (counted from $0$ at the units digit), namely $\\lfloor N / 10^{i} \\rfloor \\bmod 10$, is less than $10$.","title":"Base-ten digits are less than ten"},{"id":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/not-is-digitally-robust-composite-25011","kind":"node","lean":"MetaMathlibExt.not_isDigitallyRobustComposite_25011","parent":12027,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers/index.html#not-is-digitally-robust-composite-25011","scope":"NumberTheory/digits-numeration/digit-properties/digitally-robust-composite-numbers","status":"fully_proved","summary":"The number $25011$ is not a digitally robust composite: replacing its units digit $1$ by $3$ gives the prime $25013$.","title":"25011 is not a digitally robust composite"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums.json","id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","title":"Dirichlet convolutions of digit sums"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-arith","kind":"node","lean":"MetaMathlibExt.digitSumArith","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-arith","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For a natural number $q$, $s_q$ is the arithmetic function that sends $n$ to $s_q(n)$, the sum of the digits of $n$ in base $q$. Its $k$-th Dirichlet power is the $k$-fold Dirichlet convolution $s_q^{(k)}$, so $s_q^{(1)} = s_q$ and $s_q^{(k)} = s_q * s_q^{(k-1)}$ for $k \\ge 2$.","title":"Digit-sum arithmetic function"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-arith-apply","kind":"node","lean":"MetaMathlibExt.digitSumArith_apply","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-arith-apply","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For natural numbers $q$ and $n$, the base-$q$ digit-sum arithmetic function $s_q$ takes at $n$ the value $s_q(n)$, the sum of the digits of $n$ in base $q$.","title":"Value of the digit-sum arithmetic function"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-arith-sq-apply","kind":"node","lean":"MetaMathlibExt.digitSumArith_sq_apply","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-arith-sq-apply","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For natural numbers $q$ and $n$, the Dirichlet square $s_q^{(2)} = s_q * s_q$ of the base-$q$ digit-sum function $s_q$ satisfies $s_q^{(2)}(n) = \\sum_{de = n} s_q(d) \\, s_q(e)$, where the sum runs over the pairs $(d, e)$ of positive integers with $de = n$ (there are none for $n = 0$, and then both sides are $0$).","title":"The second digit-sum convolution"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/trollope-delange-integral","kind":"node","lean":"MetaMathlibExt.trollopeDelangeIntegral","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#trollope-delange-integral","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For a natural number $q$, Delange's function in base $q$ is the function of a real number $x$ given by $g(x) = \\int_0^x \\left(\\lfloor q t \\rfloor - q \\lfloor t \\rfloor - \\frac{q - 1}{2}\\right) dt$, where $\\lfloor \\cdot \\rfloor$ is the floor function.","title":"Delange's function"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/trollope-delange-integral-zero","kind":"node","lean":"MetaMathlibExt.trollopeDelangeIntegral_zero","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#trollope-delange-integral-zero","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$, Delange's function $g$ in base $q$ satisfies $g(0) = 0$.","title":"Delange's function vanishes at zero"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/trollope-delange-series","kind":"node","lean":"MetaMathlibExt.trollopeDelangeSeries","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#trollope-delange-series","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For a natural number $q$, let $g$ be Delange's function in base $q$. Delange's series in base $q$ is the function of a real number $x$ given by $h(x) = \\sum_{r \\ge 0} g(q^r x)/q^r$, where the sum is taken to be $0$ when the series is not summable.","title":"Delange's series"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/trollope-delange-series-zero","kind":"node","lean":"MetaMathlibExt.trollopeDelangeSeries_zero","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#trollope-delange-series-zero","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$, Delange's series $h$ in base $q$ satisfies $h(0) = 0$.","title":"Delange's series vanishes at zero"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/trollope-delange-fluctuation","kind":"node","lean":"MetaMathlibExt.trollopeDelangeFluctuation","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#trollope-delange-fluctuation","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"Let $q$ be a natural number and let $h$ be Delange's series in base $q$. The Trollope\u2013Delange fluctuation in base $q$ is the function of a real number $x$ given by $F(x) = \\frac{q - 1}{2}(1 + \\lfloor x \\rfloor - x) + q^{1 + \\lfloor x \\rfloor - x}\\, h(q^{x - \\lfloor x \\rfloor - 1})$, where $\\lfloor x \\rfloor$ is the floor of $x$.","title":"Trollope\u2013Delange fluctuation"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/trollope-delange-fluctuation-add-one","kind":"node","lean":"MetaMathlibExt.trollopeDelangeFluctuation_add_one","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#trollope-delange-fluctuation-add-one","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$ and every real number $x$, the Trollope\u2013Delange fluctuation $F$ in base $q$ satisfies $F(x + 1) = F(x)$.","title":"Periodicity of the Trollope\u2013Delange fluctuation"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-remainder","kind":"node","lean":"MetaMathlibExt.digitSumRemainder","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-remainder","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"Let $q$ be a natural number, write $s_q(n)$ for the sum of the base-$q$ digits of a natural number $n$ (with $s_q(n) = n$ when $q \\le 1$), and let $h$ be Delange's series in base $q$. For a real number $x$, let $n_x$ be the floor of $x$ when $x \\ge 0$ and $n_x = 0$ when $x < 0$. The remainder of the Trollope\u2013Delange formula is $J(x) = (1 + n_x - x)\\, s_q(n_x) - h(x)$.","title":"Trollope\u2013Delange remainder"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-remainder-zero","kind":"node","lean":"MetaMathlibExt.digitSumRemainder_zero","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-remainder-zero","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$, the remainder $J$ of the Trollope\u2013Delange formula in base $q$ satisfies $J(0) = 0$.","title":"The Trollope\u2013Delange remainder vanishes at zero"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-fluctuation-integral","kind":"node","lean":"MetaMathlibExt.digitSumFluctuationIntegral","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-fluctuation-integral","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For a natural number $q$ and a real number $x$, let $F$ be the Trollope\u2013Delange fluctuation in base $q$. Then $G(x)$ is the real number","title":"Integral of the Trollope\u2013Delange fluctuation"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-fluctuation-integral-one","kind":"node","lean":"MetaMathlibExt.digitSumFluctuationIntegral_one","parent":12040,"reading_title":{"format":"inline-parts-v1","parts":[["text","The fluctuation integral vanishes at "],["math","\\(1\\)"]],"source":"The fluctuation integral vanishes at $1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-fluctuation-integral-one","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$, let $F$ be the Trollope\u2013Delange fluctuation in base $q$ and $G(x) = \\int_1^x F(\\log u / \\log q) \\, du / u$. Then $G(1) = 0$.","title":"The fluctuation integral vanishes at $1$"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-fluctuation-double-integral","kind":"node","lean":"MetaMathlibExt.digitSumFluctuationDoubleIntegral","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-fluctuation-double-integral","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For a natural number $q$ and a real number $x$, let $F$ be the Trollope\u2013Delange fluctuation in base $q$. Then $H(x)$ is the real number","title":"Double integral of the Trollope\u2013Delange fluctuation"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-fluctuation-double-integral-on","kind":"node","lean":"MetaMathlibExt.digitSumFluctuationDoubleIntegral_one","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-fluctuation-double-integral-on","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$, the double integral $H(x) = \\int_1^x \\int_1^u F(\\log v / \\log q) \\, dv/v \\, du/u$ of the Trollope\u2013Delange fluctuation $F$ for base $q$ satisfies $H(1) = 0$.","title":"The double fluctuation integral vanishes at 1"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-remainder-integral","kind":"node","lean":"MetaMathlibExt.digitSumRemainderIntegral","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-remainder-integral","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For a natural number $q$, let $J$ be the remainder of the Trollope\u2013Delange formula in base $q$. For a real number $x$, define $K(x) = \\int_1^x J(u)/u^2 \\, du$, an oriented interval integral (so $K(x) = -\\int_x^1 J(u)/u^2 \\, du$ when $x < 1$) that is taken to be $0$ when $J(u)/u^2$ is not integrable on the interval between $1$ and $x$.","title":"Weighted integral of the Trollope\u2013Delange remainder"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-remainder-integral-one","kind":"node","lean":"MetaMathlibExt.digitSumRemainderIntegral_one","parent":12040,"reading_title":{"format":"inline-parts-v1","parts":[["text","The remainder integral vanishes at "],["math","\\(1\\)"]],"source":"The remainder integral vanishes at $1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-remainder-integral-one","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$, let $J$ be the remainder of the Trollope\u2013Delange formula in base $q$ and $K(x) = \\int_1^x J(u) \\, du / u^2$. Then $K(1) = 0$.","title":"The remainder integral vanishes at $1$"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-remainder-double-integral","kind":"node","lean":"MetaMathlibExt.digitSumRemainderDoubleIntegral","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-remainder-double-integral","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For a natural number $q$ and a real number $x$, let $J$ be the remainder of the Trollope\u2013Delange formula in base $q$. Then $L(x)$ is the real number","title":"Double integral of the Trollope\u2013Delange remainder"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-remainder-double-integral-one","kind":"node","lean":"MetaMathlibExt.digitSumRemainderDoubleIntegral_one","parent":12040,"reading_title":{"format":"inline-parts-v1","parts":[["text","The double remainder integral vanishes at "],["math","\\(1\\)"]],"source":"The double remainder integral vanishes at $1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-remainder-double-integral-one","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$, let $J$ be the remainder of the Trollope\u2013Delange formula in base $q$ and $L(x) = \\int_1^x \\frac{1}{u} \\int_1^u \\frac{J(v)}{v^2} \\, dv \\, du$. Then $L(1) = 0$.","title":"The double remainder integral vanishes at $1$"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-convolution-coeff","kind":"node","lean":"MetaMathlibExt.digitConvolutionCoeff","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-convolution-coeff","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For a natural number $q$, the base-$q$ digit-convolution coefficients are the real numbers $a(k)$, $k \\ge 0$, given by $a(0) = a(1) = 0$, $a(2) = \\frac{(q - 1)^2}{24 \\log^2 q}$ and, for every $k \\ge 3$,","title":"Digit-convolution coefficients"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-convolution-coeff-two","kind":"node","lean":"MetaMathlibExt.digitConvolutionCoeff_two","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-convolution-coeff-two","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$, the base-$q$ digit-convolution coefficient $a(2)$ equals $\\frac{(q - 1)^2}{24 \\log^2 q}$, with division by $0$ read as $0$ (this matters only for $q \\le 1$).","title":"Second digit-convolution coefficient"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-convolution-coeff-succ","kind":"node","lean":"MetaMathlibExt.digitConvolutionCoeff_succ","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-convolution-coeff-succ","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"fully_proved","summary":"For every natural number $q$ and every integer $k \\ge 2$, the base-$q$ digit-convolution coefficients $a(k)$ satisfy","title":"Recurrence for the digit-convolution coefficients"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-partial-sum-trollope-delange","kind":"node","lean":"MetaMathlibExt.digitSum_partialSum_trollopeDelange","parent":12040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trollope\u2013Delange formula for real "],["math","\\(x\\)"]],"source":"Trollope\u2013Delange formula for real $x$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-partial-sum-trollope-delange","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","summary":"Let $q \\ge 2$ be an integer, let $s_q(n)$ denote the sum of the base-$q$ digits of $n$, and let $F$ be the Trollope\u2013Delange fluctuation and $h$ Delange's series in base $q$. For every real number $x \\ge 1$,","title":"Trollope\u2013Delange formula for real $x$"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/trollope-delange-integral-is-big-o-one","kind":"node","lean":"MetaMathlibExt.trollopeDelangeIntegral_isBigO_one","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#trollope-delange-integral-is-big-o-one","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","summary":"For every natural number $q \\ge 2$, Delange's function $g$ in base $q$ satisfies $g(x) = O(1)$ as $x \\to +\\infty$.","title":"Delange's function is eventually bounded"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-reciprocal-sums-eq","kind":"node","lean":"MetaMathlibExt.digitSum_reciprocal_sums_eq","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-reciprocal-sums-eq","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","summary":"Let $q \\ge 2$ be an integer and let $s_q(n)$ denote the sum of the base-$q$ digits of $n$. Let $F$ be the Trollope\u2013Delange fluctuation and $J$ the remainder of the Trollope\u2013Delange formula in base $q$, and for real $x$ put $G(x) = \\int_1^x F(\\log u / \\log q) \\, du / u$, $H(x) = \\int_1^x G(u) \\, du / u$, $K(x) = \\int_1^x J(u) \\, du / u^2$ and $L(x) = \\int_1^x K(u) \\, du / u$. Then for every real number $x \\ge 1$,","title":"Exact formulas for the reciprocal digit sums"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-auxiliary-is-big-o","kind":"node","lean":"MetaMathlibExt.digitSum_auxiliary_isBigO","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-auxiliary-is-big-o","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","summary":"Let $q \\ge 2$ be an integer. For base $q$, let $h$ be Delange's series, $F$ the Trollope\u2013Delange fluctuation, $J$ the remainder of the Trollope\u2013Delange formula, and $G(x) = \\int_1^x F(\\log u / \\log q) \\, du/u$, $H(x) = \\int_1^x \\int_1^u F(\\log v / \\log q) \\, dv/v \\, du/u$, $K(x) = \\int_1^x J(u) \\, du/u^2$ and $L(x) = \\int_1^x \\int_1^u J(v) \\, dv/v^2 \\, du/u$. Then $F$ is bounded on $\\mathbb{R}$, and as\u2026","title":"Size bounds for the Trollope\u2013Delange auxiliary functions"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-reciprocal-fluctuation-sum-le","kind":"node","lean":"MetaMathlibExt.digitSum_reciprocal_fluctuation_sum_le","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-reciprocal-fluctuation-sum-le","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","summary":"Let $q \\ge 2$ be an integer, let $s_q(n)$ denote the sum of the base-$q$ digits of $n$, let $F$ be the Trollope\u2013Delange fluctuation in base $q$, and let $F_M = \\sup_{t \\in \\mathbb{R}} |F(t)|$ (taken to be $0$ if $|F|$ is unbounded). For all real numbers $x > 0$ and $y$,","title":"Bound for a weighted sum of the fluctuation"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-reciprocal-log-pow-sum-is-big","kind":"node","lean":"MetaMathlibExt.digitSum_reciprocal_log_pow_sum_isBigO","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-reciprocal-log-pow-sum-is-big","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","summary":"Let $q \\ge 2$ and $j \\ge 1$ be integers, and let $s_q(n)$ denote the sum of the base-$q$ digits of $n$. As the real number $x$ tends to infinity,","title":"Reciprocal digit sums weighted by powers of the logarithm"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-arith-sq-partial-sum-is-big-o","kind":"node","lean":"MetaMathlibExt.digitSumArith_sq_partialSum_isBigO","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-arith-sq-partial-sum-is-big-o","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","summary":"Let $q \\ge 2$ be an integer, let $s_q(n)$ be the sum of the base-$q$ digits of $n$, and let $s_q^{(2)} = s_q * s_q$ be its Dirichlet square, so $s_q^{(2)}(n) = \\sum_{de = n} s_q(d) \\, s_q(e)$. Then","title":"Partial sums of the second digit-sum convolution"},{"id":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/digit-sum-arith-pow-partial-sum-is-big-o","kind":"node","lean":"MetaMathlibExt.digitSumArith_pow_partialSum_isBigO","parent":12040,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums/index.html#digit-sum-arith-pow-partial-sum-is-big-o","scope":"NumberTheory/digits-numeration/digit-properties/dirichlet-convolutions-of-digit-sums","status":"can_prove","summary":"Let $q \\ge 2$ be an integer, let $s_q(n)$ be the sum of the base-$q$ digits of $n$, let $s_q^{(k)}$ be the $k$-fold Dirichlet convolution $s_q * \\cdots * s_q$, and let $a(k)$ be the base-$q$ digit-convolution coefficients. For every integer $k \\ge 2$,","title":"Partial sums of the iterated digit-sum convolutions"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/happy-numbers.json","id":"NumberTheory/digits-numeration/digit-properties/happy-numbers","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","title":"Happy numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/happy-six-min","kind":"node","lean":"MetaMathlibExt.happy_six_min","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#happy-six-min","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"A natural number is happy when repeatedly replacing it by the sum of the squares of its decimal digits reaches $1$ after finitely many steps. Let $N_0 = 7899999999999959999999996$. The six consecutive integers $N_0, N_0 + 1, \\dots, N_0 + 5$ are all happy, and every natural number $m$ such that $m, m + 1, \\dots, m + 5$ are all happy satisfies $m \\ge N_0$.","title":"Smallest run of six consecutive happy numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/neg-base-sq-digit-sum-aux","kind":"node","lean":"MetaMathlibExt.negBaseSqDigitSumAux","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#neg-base-sq-digit-sum-aux","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"For an integer $b$, define $T_b(f, a)$ for a natural number $f$ and an integer $a$ by recursion on $f$: $T_b(0, a) = 0$, $T_b(f + 1, 0) = 0$, and for $a \\ne 0$, $T_b(f + 1, a) = r^2 + T_b(f, (a - r)/b)$, where $r = a \\bmod b$ is the remainder of $a$ on division by $b$, with $0 \\le r < |b|$ (for $b = 0$ one takes $r = a$ and $(a - r)/b = 0$). For $b \\le -2$, $T_b(f, a)$ is thus the sum of the squares of the first\u2026","title":"Truncated digit-square sum in a negative base"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/neg-base-sq-digit-sum","kind":"node","lean":"MetaMathlibExt.negBaseSqDigitSum","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#neg-base-sq-digit-sum","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"Let $b \\le -2$ be an integer. Every integer $a$ has an expansion $a = \\sum_{i=0}^{n} a_i b^i$ with digits $0 \\le a_i \\le |b| - 1$, unique up to leading zero digits, and $S_{2,b}(a) = \\sum_{i=0}^{n} a_i^2$ is the sum of the squares of these digits. The digits are produced from the least significant one by taking $a_0 = a \\bmod b$ and passing to $(a - a_0)/b$, until $0$ is reached.","title":"Sum of squared digits in a negative base"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/neg-base-sq-digit-sum-zero","kind":"node","lean":"MetaMathlibExt.negBaseSqDigitSum_zero","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#neg-base-sq-digit-sum-zero","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"For every integer $b$, the base-$b$ digit-square sum $S_{2,b}$ satisfies $S_{2,b}(0) = 0$.","title":"The digit-square sum of zero"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/neg-base-sq-digit-sum-of-digit","kind":"node","lean":"MetaMathlibExt.negBaseSqDigitSum_of_digit","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#neg-base-sq-digit-sum-of-digit","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"Let $b$ and $d$ be integers with $0 < d < -b$, so that $b \\le -2$ and $d$ is a nonzero base-$b$ digit. Then the base-$b$ digit-square sum $S_{2,b}$ satisfies $S_{2,b}(d) = d^2$.","title":"The digit-square sum of a single digit"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/neg-base-sq-digit-sum-samples","kind":"node","lean":"MetaMathlibExt.negBaseSqDigitSum_samples","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#neg-base-sq-digit-sum-samples","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"Write $S_{2,b}$ for the base-$b$ digit-square sum, which sends an integer to the sum of the squares of its base-$b$ digits. Then $S_{2,-10}(-1) = 82$, $S_{2,-10}(10) = 82$ and $S_{2,-4}(5) = 11$. Indeed $-1 = 1 \\cdot (-10) + 9$ and $10 = 1 \\cdot (-10)^2 + 9 \\cdot (-10) + 0$ have base-$(-10)$ digits $1, 9$ and $1, 9, 0$, and $5 = 1 \\cdot (-4)^2 + 3 \\cdot (-4) + 1$ has base-$(-4)$ digits $1, 3, 1$ (most significant\u2026","title":"Sample values of the negative-base digit-square sum"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-happy","kind":"node","lean":"MetaMathlibExt.IsNegBaseHappy","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-happy","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"Let $b$ be an integer and let $S_{2,b}$ be the base-$b$ digit-square sum, which for $b \\le -2$ sends an integer to the sum of the squares of its base-$b$ digits. An integer $a$ is $b$-happy if $S_{2,b}^k(a) = 1$ for some $k \\ge 1$, where $S_{2,b}^k$ denotes the $k$-th iterate of $S_{2,b}$.","title":"Happy integers in a negative base"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-happy-one","kind":"node","lean":"MetaMathlibExt.isNegBaseHappy_one","parent":12071,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(1\\)"],["text"," is happy in every negative base"]],"source":"$1$ is happy in every negative base"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-happy-one","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"For every integer $b \\le -2$, the integer $1$ is $b$-happy.","title":"$1$ is happy in every negative base"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/neg-base-sq-digit-cycles","kind":"node","lean":"MetaMathlibExt.negBaseSqDigitCycles","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#neg-base-sq-digit-cycles","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"For an integer $b$, let $S_{2,b}$ be the base-$b$ digit-square sum, which for $b \\le -2$ sends an integer to the sum of the squares of its base-$b$ digits. Then $U_{2,b}$ is the set of positive integers $u$ such that $S_{2,b}^m(u) = u$ for some $m \\ge 1$, that is, the positive integers lying on a cycle of $S_{2,b}$.","title":"Cycles of the negative-base digit-square map"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/one-mem-neg-base-sq-digit-cycles","kind":"node","lean":"MetaMathlibExt.one_mem_negBaseSqDigitCycles","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#one-mem-neg-base-sq-digit-cycles","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"For every integer $b \\le -2$, $1 \\in U_{2,b}$, where $U_{2,b}$ is the set of positive integers lying on a cycle of the base-$b$ digit-square sum $S_{2,b}$.","title":"One lies on a cycle of the negative-base digit-square map"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-good","kind":"node","lean":"MetaMathlibExt.IsNegBaseGood","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-good","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"fully_proved","summary":"For an integer $b$, let $S_{2,b}$ be the base-$b$ digit-square map, which for $b \\le -2$ sends an integer to the sum of the squares of its base-$b$ digits; write $S_{2,b}^k$ for its $k$-fold iterate, and let $U_{2,b}$ be the set of positive integers $u$ with $S_{2,b}^m(u) = u$ for some $m \\ge 1$. A finite set $T$ of integers is $(2, b)$-good if for each $u \\in U_{2,b}$ there are integers $n \\ge 1$ and $k \\ge 1$\u2026","title":"Good sets for the digit-square map"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-happy-neg-two-iff","kind":"node","lean":"MetaMathlibExt.isNegBaseHappy_neg_two_iff","parent":12071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Happy numbers in base "],["math","\\(-2\\)"]],"source":"Happy numbers in base $-2$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-happy-neg-two-iff","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"There is an integer $n$ such that $n + 3i$ is $(-2)$-happy for every natural number $i$, and an integer $a$ is $(-2)$-happy if and only if $a \\equiv 1 \\pmod 3$.","title":"Happy numbers in base $-2$"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-happy-neg-three-iff","kind":"node","lean":"MetaMathlibExt.isNegBaseHappy_neg_three_iff","parent":12071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Happy numbers in base "],["math","\\(-3\\)"]],"source":"Happy numbers in base $-3$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-happy-neg-three-iff","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"There is an integer $n$ such that $n + 2i$ is $(-3)$-happy for every natural number $i$, and an integer $a$ is $(-3)$-happy if and only if $a \\equiv 1 \\pmod 2$.","title":"Happy numbers in base $-3$"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-happy-consecutive-runs-small","kind":"node","lean":"MetaMathlibExt.isNegBaseHappy_consecutive_runs_small","parent":12071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arbitrarily long runs of consecutive happy numbers in bases "],["math","\\(-4\\)"],["text",", "],["math","\\(-6\\)"],["text",", "],["math","\\(-8\\)"],["text",", "],["math","\\(-10\\)"]],"source":"Arbitrarily long runs of consecutive happy numbers in bases $-4$, $-6$, $-8$, $-10$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-happy-consecutive-runs-small","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"Let $b \\in \\{-4, -6, -8, -10\\}$. For every natural number $N$ there is an integer $n$ such that $n + i$ is $b$-happy for every $0 \\le i < N$.","title":"Arbitrarily long runs of consecutive happy numbers in bases $-4$, $-6$, $-8$, $-10$"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-happy-two-consecutive-runs-o","kind":"node","lean":"MetaMathlibExt.isNegBaseHappy_two_consecutive_runs_odd","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-happy-two-consecutive-runs-o","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"Let $b \\le -3$ be an odd integer. For every natural number $N$ there is an integer $n$ such that $n + 2i$ is $b$-happy for every natural number $i < N$. Thus there are arbitrarily long finite arithmetic progressions with common difference $2$ of $b$-happy integers.","title":"Arithmetic progressions of happy numbers in odd negative bases"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-happy-consecutive-runs-even","kind":"node","lean":"MetaMathlibExt.isNegBaseHappy_consecutive_runs_even","parent":12071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arbitrarily long runs of consecutive happy numbers in even bases at most "],["math","\\(-4\\)"]],"source":"Arbitrarily long runs of consecutive happy numbers in even bases at most $-4$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-happy-consecutive-runs-even","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"Let $b \\le -4$ be an even integer. For every natural number $N$ there is an integer $n$ such that $n + i$ is $b$-happy for every $0 \\le i < N$.","title":"Arbitrarily long runs of consecutive happy numbers in even bases at most $-4$"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-good-of-image-comp","kind":"node","lean":"MetaMathlibExt.isNegBaseGood_of_image_comp","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-good-of-image-comp","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"Let $b \\le -2$ be an integer, let $S_{2,b}$ be the map sending an integer to the sum of the squares of its base-$b$ digits, and let $I(t) = t + 1$. Let $T$ be a finite set of positive integers and let $F$ be a composition $F = f_1 \\circ \\dots \\circ f_m$ with $m \\ge 0$ (the empty composition being the identity), where each $f_i$ is $S_{2,b}$ or $I$. If the image $F(T)$ is $(2, b)$-good, then $T$ is $(2, b)$-good.","title":"Goodness pulls back along compositions of the digit-square map and the successor map"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/exists-iterate-neg-base-sq-digit-sum-lt","kind":"node","lean":"MetaMathlibExt.exists_iterate_negBaseSqDigitSum_lt","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#exists-iterate-neg-base-sq-digit-sum-lt","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"Let $b \\le -7$ be an integer, let $S_{2,b}$ be the map sending an integer to the sum of the squares of its base-$b$ digits, and write $S_{2,b}^k$ for its $k$-fold iterate. For every finite set $A$ of positive integers there is an integer $k \\ge 1$ such that $S_{2,b}^k(a) < 2b^2$ for every $a \\in A$.","title":"A common iterate of the negative-base digit-square map falls below twice the squared base"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/exists-neg-base-sq-digit-sum-congr","kind":"node","lean":"MetaMathlibExt.exists_negBaseSqDigitSum_congr","parent":12071,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#exists-neg-base-sq-digit-sum-congr","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"Let $b \\le -12$ be an even integer, let $S_{2,b}$ be the map sending an integer to the sum of the squares of its base-$b$ digits, and let $v$ be an integer with $0 < v < 2b^2$. Then there are integers $w$ and $c$ with $0 \\le w < v$ and $0 \\le c \\le |b| - 2$ such that","title":"A congruence for the negative-base digit-square map"},{"id":"NumberTheory/digits-numeration/digit-properties/happy-numbers/is-neg-base-good-of-even","kind":"node","lean":"MetaMathlibExt.isNegBaseGood_of_even","parent":12071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even bases at most "],["math","\\(-4\\)"],["text"," make every finite set of positive integers good"]],"source":"Even bases at most $-4$ make every finite set of positive integers good"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/happy-numbers/index.html#is-neg-base-good-of-even","scope":"NumberTheory/digits-numeration/digit-properties/happy-numbers","status":"can_prove","summary":"Let $b \\le -4$ be an even integer. Then every finite set of positive integers is $(2, b)$-good.","title":"Even bases at most $-4$ make every finite set of positive integers good"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/irwin-series.json","id":"NumberTheory/digits-numeration/digit-properties/irwin-series","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","title":"Irwin series"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/digit-count","kind":"node","lean":"MetaMathlibExt.digitCount","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#digit-count","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $d$ and $n$, the digit count of $d$ in $n$ is the number of occurrences of the digit $d$ in the minimal base-$b$ representation of $n$ (its base-$b$ digits without leading zeros). The representation of $0$ is empty, so $0$ has digit count $0$.","title":"Digit count"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/integer-length","kind":"node","lean":"MetaMathlibExt.integerLength","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#integer-length","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$ and $n$, the integer length of $n$ in base $b$ is the number of digits in the minimal base-$b$ representation of $n$ (its base-$b$ digits without leading zeros); in particular $0$ has length $0$.","title":"Integer length"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-sum","kind":"node","lean":"MetaMathlibExt.irwinSum","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-sum","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $d$ and $k$, the Irwin sum $H^{(k)}_{b,d}$ is the sum of $1/n$ over the positive integers $n$ whose base-$b$ expansion contains exactly $k$ occurrences of the digit $d$. It is taken as an unconditional sum of real numbers, with the value $0$ if this family is not summable; positivity and convergence are results, not part of the definition.","title":"Irwin sum"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/string-value","kind":"node","lean":"MetaMathlibExt.stringValue","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#string-value","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For a natural number $b$ and a base-$b$ digit string $X = x_1 x_2 \\cdots x_\\ell$ with digits $x_i \\in \\{0, \\dots, b-1\\}$, written with the most significant digit first, the integer value of $X$ is","title":"Value of a digit string"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/normalized-string-value","kind":"node","lean":"MetaMathlibExt.normalizedStringValue","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#normalized-string-value","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For a base $b$ and a string $X$ of base-$b$ digits, written most significant digit first, with integer value $n(X)$ and length $|X|$, the normalized value of $X$ is the real number $n(X) / b^{|X|}$.","title":"Normalized value of a digit string"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/string-weight","kind":"node","lean":"MetaMathlibExt.stringWeight","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#string-weight","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For a natural number $b$ and a base-$b$ digit string $X$ of length $|X|$, the weight of $X$ is $b^{-|X|}$, regarded as an element of $[0, \\infty]$.","title":"Weight of a digit string"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/string-atom","kind":"node","lean":"MetaMathlibExt.stringAtom","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#string-atom","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For a natural number $b$ and a base-$b$ digit string $X$ of length $|X|$ and integer value $n(X)$, the atom of $X$ is the measure $b^{-|X|}\\,\\delta_{n(X)/b^{|X|}}$ on $\\mathbb{R}$: the Dirac measure at the point $n(X)/b^{|X|}$, scaled by the weight $b^{-|X|}$.","title":"Atom of a digit string"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-measure","kind":"node","lean":"MetaMathlibExt.irwinMeasure","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-measure","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"Let $b$, $d$ and $k$ be natural numbers with $d < b$. For a finite string $X$ of base-$b$ digits (of any length, including the empty string, with leading zeros allowed), read most significant digit first, let $|X|$ be its length and $n(X)$ the integer it represents. The Irwin measure $\\mu_k$ of base $b$ and digit $d$ with index $k$ is the measure on $\\mathbb{R}$ obtained by summing, over all strings $X$ containing\u2026","title":"Irwin measure"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/leading-digits","kind":"node","lean":"MetaMathlibExt.leadingDigits","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#leading-digits","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $l$ and $m$, let $L$ be the number of digits of $m$ in base $b$. The leading-digits integer of $m$ is the natural-number quotient of $m$ by $b^{\\max(L - l,\\, 0)}$, rounded down (with quotient $0$ when the divisor is $0$). For $b \\ge 2$ and $L \\ge l$ this is the integer formed by the first $l$ base-$b$ digits of $m$; when $L < l$ it is $m$ itself, and statements using this notion separately\u2026","title":"Leading digits"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/leading-reciprocal-term","kind":"node","lean":"MetaMathlibExt.leadingReciprocalTerm","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#leading-reciprocal-term","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $d$, $k$, $n$ and $m$, write $L(x)$ for the number of base-$b$ digits of $x$. The term is $1/m$ when $m$ is positive, $L(n) \\le L(m)$, the leading-digits integer of $m$ keeping its first $L(n)$ digits equals $n$, and $m$ has exactly $k$ occurrences of $d$ in base $b$; otherwise the term is $0$. For $b \\ge 2$ the leading-digits condition says that the base-$b$ expansion of $m$ begins with\u2026","title":"Prefix-restricted reciprocal terms"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/leading-irwin-sum","kind":"node","lean":"MetaMathlibExt.leadingIrwinSum","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#leading-irwin-sum","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $d$, $k$ and $n$, this is the sum of the prefix-restricted reciprocal terms over all natural numbers $m$: the sum of $1/m$ over the positive integers $m$ with at least as many base-$b$ digits as $n$, whose leading-digits integer keeping as many digits as $n$ has equals $n$, and with exactly $k$ occurrences of $d$ in base $b$. For $b \\ge 2$ the middle condition says that the base-$b$\u2026","title":"Prefix-restricted Irwin sum"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/ordinary-moment","kind":"node","lean":"MetaMathlibExt.ordinaryMoment","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#ordinary-moment","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $d$, $k$, $m$ with $d < b$, the ordinary moment $u_{k;m}$ is the integral of $x^m$ over $[0,1)$ with respect to the Irwin measure $\\mu_k$ of base $b$ and digit $d$:","title":"Ordinary moments of the Irwin measures"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/complementary-moment","kind":"node","lean":"MetaMathlibExt.complementaryMoment","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#complementary-moment","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $d$, $k$ and $m$ with $d < b$, let $\\mu_k$ be the Irwin measure for base $b$, digit $d$ and digit count $k$. The complementary moment is","title":"Complementary moment"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/digit-power-sum","kind":"node","lean":"MetaMathlibExt.digitPowerSum","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#digit-power-sum","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $d$ and $j$, the digit power sum $\\gamma_j$ is the real number $\\sum_{0 \\le a < b,\\ a \\ne d} a^j$, the sum of the $j$-th powers of the base-$b$ digits other than $d$, with the convention $0^0 = 1$.","title":"Digit power sum"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/complementary-digit-power-sum","kind":"node","lean":"MetaMathlibExt.complementaryDigitPowerSum","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#complementary-digit-power-sum","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$, $d$ and $j$, let $d' = b - 1 - d$, computed by natural-number subtraction (so $d' = 0$ when $d \\ge b - 1$). The complementary digit power sum $\\gamma'_j$ is the real number $\\sum_{0 \\le a < b,\\ a \\ne d'} a^j$, the sum of the $j$-th powers of the base-$b$ digits other than $d'$, with the convention $0^0 = 1$.","title":"Complementary digit power sum"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/ordinary-moment-recurrence","kind":"node","lean":"MetaMathlibExt.OrdinaryMomentRecurrence","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#ordinary-moment-recurrence","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"Let $b$ and $d$ be natural numbers, and for $i \\ge 1$ let $\\gamma_i$ be the sum of the $i$-th powers of the base-$b$ digits $a \\in \\{0, \\dots, b-1\\}$ with $a \\ne d$. A family of real numbers $(u_{j;m})_{j, m \\in \\mathbb{N}}$ satisfies the ordinary moment recurrence for $b$ and $d$ when $u_{0;0} = b$, when for every $m \\ge 1$","title":"Ordinary moment recurrence"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/complementary-moment-recurrence","kind":"node","lean":"MetaMathlibExt.ComplementaryMomentRecurrence","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#complementary-moment-recurrence","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"Let $b$ and $d$ be natural numbers, let $d' = b - 1 - d$ (natural-number subtraction), and let $\\gamma'_i$ be the complementary digit power sum for $b$ and $d$. A family $(v_{j,m})_{j, m \\ge 0}$ of real numbers satisfies the complementary moment recurrence when, for every $m \\ge 0$,","title":"Complementary moment recurrence"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/has-base-depth","kind":"node","lean":"MetaMathlibExt.HasBaseDepth","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#has-base-depth","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For a natural number $b$, a real number $x$ and a natural number $l$, the number $x$ has base-$b$ depth $l$ when $x = z / b^l$ for some integer $z$, and for no $q < l$ is there an integer $z$ with $x = z / b^q$. Thus $l$ is the least exponent for which $x$ can be written as an integer divided by $b^l$.","title":"Base depth of a real number"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/affine-scale-set","kind":"node","lean":"MetaMathlibExt.affineScaleSet","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#affine-scale-set","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For natural numbers $b$ and $l$, a real number $x$ and a set $U \\subseteq \\mathbb{R}$, the affine scale set $b^l U - b^l x$ is the image of $U$ under the map $y \\mapsto b^l y - b^l x$, that is, $\\{\\, b^l y - b^l x : y \\in U \\,\\}$.","title":"Scaled translate of a set"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/digit-count-zero","kind":"node","lean":"MetaMathlibExt.digitCount_zero","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#digit-count-zero","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For all natural numbers $b$ and $d$, the digit $d$ occurs $0$ times in the minimal base-$b$ representation of $0$, which is empty.","title":"Zero has no digits"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/string-value-nil","kind":"node","lean":"MetaMathlibExt.stringValue_nil","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#string-value-nil","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For every natural number $b$, the integer value of the empty base-$b$ digit string is $0$.","title":"The empty digit string has value zero"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/string-atom-nil","kind":"node","lean":"MetaMathlibExt.stringAtom_nil","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#string-atom-nil","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"fully_proved","summary":"For every natural number $b$, the atom $b^{-|X|}\\,\\delta_{n(X)/b^{|X|}}$ of a base-$b$ digit string $X$ is the Dirac measure $\\delta_0$ when $X$ is the empty string.","title":"The atom of the empty string is the unit Dirac mass at the origin"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-sum-has-sum-alternating","kind":"node","lean":"MetaMathlibExt.irwinSum_hasSum_alternating","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-sum-has-sum-alternating","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, let $(u_{j;m})_{j, m \\in \\mathbb{N}}$ be a family of real numbers satisfying the ordinary moment recurrence for $b$ and $d$, and let $k$ and $l \\ge 1$ be natural numbers. For a natural number $n$ write $c(n)$ for the number of occurrences of the digit $d$ in the base-$b$ expansion of $n$, and let $H^{(k)}_{b,d}$ be the Irwin sum, the sum of $1/n$ over the positive\u2026","title":"Alternating series for the Irwin sum"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/ordinary-moment-recurrence-properties","kind":"node","lean":"MetaMathlibExt.ordinaryMomentRecurrence_properties","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#ordinary-moment-recurrence-properties","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, and let $(u_{j;m})_{j, m \\in \\mathbb{N}}$ be a family of real numbers that satisfies the ordinary moment recurrence for $b$ and $d$. Then $u_{j;0} = b$ for every $j$, and for all $j$ and all $m \\ge 1$: $u_{j;m} \\ge 0$, with $u_{j;m} = 0$ if and only if $j = 0$, $b = 2$ and $d = 1$; $u_{j;m+1} \\le u_{j;m}$, with strict inequality when $u_{j;m} \\ne 0$;\u2026","title":"Properties of solutions of the ordinary moment recurrence"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-sum-has-sum-positive","kind":"node","lean":"MetaMathlibExt.irwinSum_hasSum_positive","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-sum-has-sum-positive","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, let $k$ and $l \\ge 1$ be natural numbers, and let $(v_{j,m})_{j, m \\ge 0}$ be a family of real numbers satisfying the complementary moment recurrence for $b$ and $d$. For a natural number $n$ let $k(n)$ be the number of occurrences of $d$ in the base-$b$ representation of $n$, and let $H^{(k)}$ be the Irwin sum, the sum of $1/n$ over the positive integers $n$ with\u2026","title":"Complementary-moment series for the Irwin sums"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/complementary-moment-recurrence-properti","kind":"node","lean":"MetaMathlibExt.complementaryMomentRecurrence_properties","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#complementary-moment-recurrence-properti","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, and let $(v_{j,m})_{j, m \\ge 0}$ be any family of real numbers satisfying the complementary moment recurrence for $b$ and $d$. Then $v_{j,0} = b$ for every $j$. For every $j$ and every $m \\ge 1$: $0 < v_{j,m} \\le b$; if $(b, d, j) = (2, 1, 0)$ then $v_{j,m} = 2$, and otherwise $v_{j,m+1} < v_{j,m}$; and $v_{j+1,m} < v_{j,m}$. Finally, for every $m \\ge 1$,\u2026","title":"Properties of the complementary moment recurrence"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-sum-strict-anti-and-tendsto","kind":"node","lean":"MetaMathlibExt.irwinSum_strictAnti_and_tendsto","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-sum-strict-anti-and-tendsto","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ be a base and $d < b$ a digit, and for $k \\ge 0$ let $H^{(k)}$ be the Irwin sum, the sum of $1/n$ over the positive integers $n$ whose base-$b$ expansion contains exactly $k$ occurrences of $d$. If $d \\ne 0$, then $k \\mapsto H^{(k)}$ is strictly decreasing on $k \\ge 1$; if $d = 0$, it is strictly decreasing on all $k \\ge 0$. In both cases $H^{(k)} \\to b \\log b$ as $k \\to \\infty$.","title":"Monotonicity and limit of the Irwin sums"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-sum-zero-lower-bounds","kind":"node","lean":"MetaMathlibExt.irwinSum_zero_lowerBounds","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-sum-zero-lower-bounds","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 2$ be a base and let $d$ be a nonzero digit, $0 < d < b$. Let $H^{(0)}$ be the Irwin sum of $1/n$ over the positive integers $n$ whose base-$b$ expansion does not contain $d$. Then $b \\log b - b \\log(1 + 1/d) < H^{(0)}$ and $b \\log(b/2) < H^{(0)}$.","title":"Lower bounds for the Irwin sum without the digit"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/integral-eq-leading-reciprocal-has-sum","kind":"node","lean":"MetaMathlibExt.integral_eq_leadingReciprocal_hasSum","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#integral-eq-leading-reciprocal-has-sum","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, let $n > 0$ and $k$ be natural numbers, let $k(n)$ be the number of occurrences of the digit $d$ in the base-$b$ representation of $n$, and let $\\mu_k$ be the Irwin measure of base $b$ and digit $d$ with index $k$. Then the family that assigns $1/m$ to each positive integer $m$ whose base-$b$ representation begins with that of $n$ and contains exactly $k(n) + k$\u2026","title":"Prefix reciprocal sums as integrals against the Irwin measure"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/leading-reciprocal-has-sum-ordinary-mome","kind":"node","lean":"MetaMathlibExt.leadingReciprocal_hasSum_ordinaryMoments","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#leading-reciprocal-has-sum-ordinary-mome","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$, let $d < b$ be a digit, let $n$ be a positive integer, let $k(n)$ be the number of occurrences of $d$ in the base-$b$ expansion of $n$, and let $k \\ge k(n)$. Write $u_{j;m} = \\int_{[0,1)} x^m \\, d\\mu_j(x)$ for the ordinary moments of the Irwin measure $\\mu_j$. Then there is a real number $S$ such that the reciprocals $1/N$ of the positive integers $N$ whose base-$b$ expansion begins with that of $n$\u2026","title":"Prefix sums as alternating ordinary-moment series"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/leading-reciprocal-has-sum-complementary","kind":"node","lean":"MetaMathlibExt.leadingReciprocal_hasSum_complementaryMoments","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#leading-reciprocal-has-sum-complementary","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$, let $d < b$ be a digit, let $n$ be a positive integer, let $k(n)$ be the number of occurrences of $d$ in the base-$b$ expansion of $n$, and let $k \\ge k(n)$. Write $v_{j;m} = \\int_{[0,1)} (1 - x)^m \\, d\\mu_j(x)$ for the complementary moments of the Irwin measure $\\mu_j$. Then there is a real number $S$ such that the reciprocals $1/N$ of the positive integers $N$ whose base-$b$ expansion begins with\u2026","title":"Prefix sums as complementary-moment series"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/integral-irwin-measure-succ","kind":"node","lean":"MetaMathlibExt.integral_irwinMeasure_succ","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#integral-irwin-measure-succ","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, and for each natural number $j$ let $\\mu_j$ be the Irwin measure of base $b$ and digit $d$ with index $j$. Let $f : \\mathbb{R} \\to \\mathbb{R}$ be measurable and bounded on $[0, b)$, and let $k \\ge 1$. Then","title":"Digit decomposition of the Irwin measures, positive index"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/integral-irwin-measure-zero","kind":"node","lean":"MetaMathlibExt.integral_irwinMeasure_zero","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#integral-irwin-measure-zero","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, and let $\\mu_0$ be the Irwin measure of base $b$ and digit $d$ with index $0$. For every measurable function $f : \\mathbb{R} \\to \\mathbb{R}$ that is bounded on $[0, b)$,","title":"Digit decomposition of the Irwin measure, zero index"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-measure-open-local-mass","kind":"node","lean":"MetaMathlibExt.irwinMeasure_open_local_mass","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-measure-open-local-mass","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, and for each natural number $k$ let $\\mu_k$ be the Irwin measure of base $b$ and digit $d$ with index $k$. Let $x \\in [0, 1)$ have base-$b$ depth $l$, that is, $l$ is the least natural number such that $b^l x$ is an integer; let $X$ be a string of $l$ base-$b$ digits, read most significant digit first, whose value $n(X)$ satisfies $n(X)/b^l = x$; and let $j$ be the number\u2026","title":"Local structure of the Irwin measures on open intervals"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-measure-half-open-deeper-local-mas","kind":"node","lean":"MetaMathlibExt.irwinMeasure_halfOpen_deeper_local_mass","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-measure-half-open-deeper-local-mas","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, and for each natural number $k$ let $\\mu_k$ be the Irwin measure of base $b$ and digit $d$ with index $k$. Let $x \\in [0, 1)$ have base-$b$ depth $l$, that is, $l$ is the least natural number such that $b^l x$ is an integer; let $X$ be a string of $l$ base-$b$ digits, read most significant digit first, whose value $n(X)$ satisfies $n(X)/b^l = x$; and let $j$ be the number\u2026","title":"Local structure of the Irwin measures on half-open intervals"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-measure-half-open-local-mass","kind":"node","lean":"MetaMathlibExt.irwinMeasure_halfOpen_local_mass","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-measure-half-open-local-mass","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, and for each natural number $k$ let $\\mu_k$ be the Irwin measure of base $b$ and digit $d$ with index $k$. Let $x \\in [0, 1)$ have base-$b$ depth $l$, that is, $l$ is the least natural number such that $b^l x$ is an integer; let $X$ be a string of $l$ base-$b$ digits, read most significant digit first, whose value $n(X)$ satisfies $n(X)/b^l = x$; and let $j$ be the number\u2026","title":"Local structure of the Irwin measures on cylinders"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/irwin-measure-interval-convergence","kind":"node","lean":"MetaMathlibExt.irwinMeasure_interval_convergence","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#irwin-measure-interval-convergence","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$ and $d < b$ be natural numbers, and for each natural number $k$ let $\\mu_k$ be the Irwin measure of base $b$ and digit $d$ with index $k$; masses below are taken as real numbers. First, for all reals $0 \\le t < u \\le 1$ and every natural number $l'$ such that $b^{l'} t$ and $b^{l'} u$ are integers, $\\mu_k([t, u)) = bu - bt$ for every $k \\ge l'$. Second, for every nonempty interval $I \\subseteq [0, 1)$,\u2026","title":"Masses of intervals under the Irwin measures"},{"id":"NumberTheory/digits-numeration/digit-properties/irwin-series/leading-irwin-sum-tendsto","kind":"node","lean":"MetaMathlibExt.leadingIrwinSum_tendsto","parent":12092,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/irwin-series/index.html#leading-irwin-sum-tendsto","scope":"NumberTheory/digits-numeration/digit-properties/irwin-series","status":"can_prove","summary":"Let $b > 1$, let $d < b$ be a digit and let $n$ be a positive integer. For every $k \\ge 0$, the family of reciprocals $1/m$ of the positive integers $m$ whose base-$b$ expansion begins with the base-$b$ expansion of $n$ and contains exactly $k$ occurrences of $d$ is summable, and its sum tends to $b \\log(1 + 1/n)$ as $k \\to \\infty$.","title":"Limit of the prefix-restricted Irwin sums"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine.json","id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","title":"Kaprekar's routine"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/four-digit-kaprekar-structural-classific","kind":"node","lean":"MetaMathlibExt.four_digit_kaprekar_structural_classification","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#four-digit-kaprekar-structural-classific","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"Let $B > 3$ be an odd integer. For a pair of natural numbers $d = (d_1, d_2)$, let $K(d) = (a - e, b - c)$, where $a \\ge b \\ge c \\ge e$ are the four numbers $d_1 - 1, B - 1, B - 1, B - d_1$ if $d_2 = 0$, and $d_1, d_2 - 1, B - d_2 - 1, B - d_1$ if $d_2 \\ne 0$, sorted in decreasing order (subtraction is truncated at $0$). Let $X_B$ be the set of pairs with $1 \\le d_1 < B$ and $d_2 \\le d_1$, and let $T_B$ be the set\u2026","title":"Structure of four-digit Kaprekar dynamics in odd bases"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-diff","kind":"node","lean":"MetaMathlibExt.kaprekarDiff","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-diff","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"Let $l$ be a list of $k$ natural numbers and let $a_0 \\le a_1 \\le \\dots \\le a_{k-1}$ be its entries in increasing order. The Kaprekar difference of $l$ is $\\alpha_M - \\alpha_L$, where $\\alpha_M = \\sum_{i\\lt k} a_i 10^i$ and $\\alpha_L = \\sum_{i\\lt k} a_{k-1-i} 10^i$. When the entries are decimal digits, $\\alpha_M$ is the number written with the digits of $l$ in descending order and $\\alpha_L$ is the number written\u2026","title":"Kaprekar difference"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-map","kind":"node","lean":"MetaMathlibExt.kaprekarMap","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-map","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"For natural numbers $n$ and $\\alpha$, take the decimal digits of $\\alpha$ (without leading zeros, and no digits at all for $\\alpha = 0$) and, if there are fewer than $n$ of them, pad them with leading zeros to a string of exactly $n$ digits. The Kaprekar map sends $\\alpha$ to the Kaprekar difference $\\alpha_M - \\alpha_L$ of this digit string: the number with these digits in descending order minus the number with\u2026","title":"Kaprekar map"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-map-four-6174","kind":"node","lean":"MetaMathlibExt.kaprekarMap_four_6174","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-map-four-6174","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"The Kaprekar map with $n = 4$ sends $6174$ to itself: $7641 - 1467 = 6174$.","title":"6174 is fixed by the four-digit Kaprekar map"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-map-three-495","kind":"node","lean":"MetaMathlibExt.kaprekarMap_three_495","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-map-three-495","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"The Kaprekar map with $n = 3$ sends $495$ to itself: $954 - 459 = 495$.","title":"495 is fixed by the three-digit Kaprekar map"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-map-four-495","kind":"node","lean":"MetaMathlibExt.kaprekarMap_four_495","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-map-four-495","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"With $n = 4$, the number $495$ is padded to the digit string $0495$, and the Kaprekar map sends it to $9540 - 0459 = 9081$.","title":"The four-digit Kaprekar map sends 495 to 9081"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-map-fixed-iff","kind":"node","lean":"MetaMathlibExt.kaprekarMap_fixed_iff","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-map-fixed-iff","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"Let $n \\ge 3$ and $0 < \\alpha < 10^n$. Then the Kaprekar map on $n$-digit strings fixes $\\alpha$ if and only if $\\alpha$ is the Kaprekar difference of some string $s$ of exactly $n$ decimal digits of one of the following five forms, where juxtaposition is concatenation, $(w)^k$ is the block $w$ repeated $k$ times, and in each form the exponents are chosen independently with $x, x_1, x_2 \\ge 1$ and\u2026","title":"Classification of decimal Kaprekar numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-four-digit-step","kind":"node","lean":"MetaMathlibExt.kaprekarFourDigitStep","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-four-digit-step","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"For a base $B$ and a pair of natural numbers $(d_1, d_2)$, the map $K_B$ forms the list of four digits $[d_1 - 1, B - 1, B - 1, B - d_1]$ if $d_2 = 0$ and $[d_1, d_2 - 1, B - d_2 - 1, B - d_1]$ if $d_2 > 0$, sorts it decreasingly as $a_1 \\ge a_2 \\ge a_3 \\ge a_4$, and sets $K_B(d_1, d_2) = (a_1 - a_4, a_2 - a_3)$. Subtraction of natural numbers is truncated at $0$; for $1 \\le d_1 < B$ and $0 \\le d_2 \\le d_1$ no\u2026","title":"The four-digit Kaprekar map in difference coordinates"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-four-digit-step-examples","kind":"node","lean":"MetaMathlibExt.kaprekarFourDigitStep_examples","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-four-digit-step-examples","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"The four-digit Kaprekar map satisfies $K_{11}(5, 1) = (9, 1)$, $K_{11}(6, 0) = (5, 5)$ and $K_{11}(5, 5) = (2, 0)$ in base $11$, and $K_7(3, 1) = (5, 1)$, $K_7(5, 1) = (5, 3)$ and $K_7(5, 3) = (3, 1)$ in base $7$.","title":"Sample values of the four-digit Kaprekar map"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-four-digit-states","kind":"node","lean":"MetaMathlibExt.kaprekarFourDigitStates","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-four-digit-states","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"For a base $B$, the state space $X_B$ is the finite set of pairs of natural numbers $(d_1, d_2)$ with $1 \\le d_1 < B$ and $0 \\le d_2 \\le d_1$.","title":"The four-digit Kaprekar state space"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-four-digit-states-card-examples","kind":"node","lean":"MetaMathlibExt.kaprekarFourDigitStates_card_examples","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-four-digit-states-card-examples","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"The four-digit Kaprekar state spaces satisfy $|X_3| = 5$ and $|X_7| = 27$.","title":"Sizes of the Kaprekar state spaces in bases 3 and 7"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/projective-order-two","kind":"node","lean":"MetaMathlibExt.projectiveOrderTwo","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#projective-order-two","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"For a natural number $n$, the projective order of $2$ modulo $n$ is the least $m > 0$ with $2^m \\equiv 1$ or $2^m \\equiv -1 \\pmod{n}$,","title":"Projective order of 2"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/projective-order-two-eq-of","kind":"node","lean":"MetaMathlibExt.projectiveOrderTwo_eq_of","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#projective-order-two-eq-of","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"Let $n$ and $m$ be natural numbers with $m > 0$. Suppose that $2^m \\equiv 1$ or $2^m \\equiv -1 \\pmod{n}$, and that for every $k$ with $0 < k < m$ neither $2^k \\equiv 1$ nor $2^k \\equiv -1 \\pmod{n}$. Then the projective order of $2$ modulo $n$ is $\\lambda(n) = m$.","title":"Characterization of the projective order of 2"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/projective-order-two-examples","kind":"node","lean":"MetaMathlibExt.projectiveOrderTwo_examples","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#projective-order-two-examples","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"fully_proved","summary":"Writing $\\lambda(n)$ for the projective order of $2$ modulo $n$, we have $\\lambda(7) = 3$, $\\lambda(9) = 3$, $\\lambda(11) = 5$ and $\\lambda(17) = 4$.","title":"Values of the projective order of 2"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/projective-doubling-minimal-period-le-pr","kind":"node","lean":"MetaMathlibExt.projective_doubling_minimalPeriod_le_projectiveOrderTwo","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#projective-doubling-minimal-period-le-pr","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"Let $B > 3$ be odd. For $x \\in \\mathbb{Z}/B\\mathbb{Z}$ write $[x] = \\{x, -x\\}$ and $2C = \\{2c : c \\in C\\}$ for a subset $C$ of $\\mathbb{Z}/B\\mathbb{Z}$, and let $D$ be the doubling map sending a set $T$ of subsets of $\\mathbb{Z}/B\\mathbb{Z}$ to $\\{2C : C \\in T\\}$. If $S = \\{[r], [s]\\}$ for nonzero $r, s \\in \\mathbb{Z}/B\\mathbb{Z}$ with $[r] \\ne [s]$, then $S$ is a periodic point of $D$ and its minimal period is at\u2026","title":"Cycle lengths of doubling on pairs of projective classes"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/projective-order-two-le-totient-div-two","kind":"node","lean":"MetaMathlibExt.projectiveOrderTwo_le_totient_div_two","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#projective-order-two-le-totient-div-two","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"For every odd natural number $B > 3$, the projective order of $2$ modulo $B$ satisfies $\\lambda(B) \\le \\lfloor \\varphi(B)/2 \\rfloor$, where $\\varphi$ is Euler's totient function.","title":"Projective order of 2 is at most half the totient"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-minimal-period-le-and-prime-of","kind":"node","lean":"MetaMathlibExt.kaprekar_minimalPeriod_le_and_prime_of_eq","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-minimal-period-le-and-prime-of","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"Let $B > 3$ be odd, let $K_B$ be the four-digit Kaprekar map and $X_B$ its state space. Every point of $X_B$ that is periodic for $K_B$ has minimal period at most $(B - 1)/2$, and if some point of $X_B$ has minimal period exactly $(B - 1)/2$ under $K_B$, then $B$ is prime.","title":"Long Kaprekar cycles force a prime base"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-minimal-period-le-and-eq-iff","kind":"node","lean":"MetaMathlibExt.kaprekar_minimalPeriod_le_and_eq_iff","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-minimal-period-le-and-eq-iff","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"Let $B > 3$ be odd, let $K_B$ be the four-digit Kaprekar map and $X_B$ its state space, and let $\\lambda(B)$ be the projective order of $2$ modulo $B$. Every point of $X_B$ that is periodic for $K_B$ has minimal period at most $(B - 1)/2$. Moreover, some point of $X_B$ has minimal period exactly $(B - 1)/2$ under $K_B$ if and only if $B \\ge 7$, $B$ is prime and $\\lambda(B) = (B - 1)/2$.","title":"Maximal Kaprekar cycle length in odd bases"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-card-maximal-cycles","kind":"node","lean":"MetaMathlibExt.kaprekar_card_maximal_cycles","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-card-maximal-cycles","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"Let $B > 3$ be odd, let $K_B$ be the four-digit Kaprekar map and $X_B$ its state space, and put $q = (B - 1)/2$. If some point of $X_B$ has minimal period $q$ under $K_B$, then exactly $q \\lfloor (q - 1)/2 \\rfloor$ points of $X_B$ have minimal period $q$ under $K_B$.","title":"Number of Kaprekar cycles of maximal length"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-card-minimal-period-eq","kind":"node","lean":"MetaMathlibExt.kaprekar_card_minimalPeriod_eq","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-card-minimal-period-eq","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"Let $B > 3$ be odd and $L \\ge 1$, let $K_B$ be the four-digit Kaprekar map and $X_B$ its state space, let $\\varphi$ be Euler's totient function, and let $\\lambda(n)$ denote the projective order of $2$ modulo $n$. For $a \\ge 1$ put","title":"Explicit count of Kaprekar cycles of each length"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/two-mul-projective-order-two-le-card-kap","kind":"node","lean":"MetaMathlibExt.two_mul_projectiveOrderTwo_le_card_kaprekar_cycles","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#two-mul-projective-order-two-le-card-kap","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"Let $\\lambda(B)$ denote the projective order of $2$ modulo $B$. For every odd base $B > 17$, at least $2\\lambda(B)$ points of the state space $X_B$ have minimal period exactly $\\lambda(B)$ under the four-digit Kaprekar map $K_B$.","title":"At least two Kaprekar cycles whose length is the projective order of 2"},{"id":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/kaprekar-unique-cycle-iff","kind":"node","lean":"MetaMathlibExt.kaprekar_unique_cycle_iff","parent":12131,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine/index.html#kaprekar-unique-cycle-iff","scope":"NumberTheory/digits-numeration/digit-properties/kaprekar-s-routine","status":"can_prove","summary":"Let $B \\ge 3$ be odd, let $K_B$ be the four-digit Kaprekar map and $X_B$ its state space. There is a point $c \\in X_B$, periodic for $K_B$, whose forward orbit $\\{K_B^k(c) : k \\ge 0\\}$ contains every point of $X_B$ that is periodic for $K_B$, if and only if $B \\in \\{3, 5, 7\\}$.","title":"Odd bases with a unique Kaprekar cycle"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/keith-numbers.json","id":"NumberTheory/digits-numeration/digit-properties/keith-numbers","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/keith-numbers/index.html","scope":"NumberTheory/digits-numeration/digit-properties/keith-numbers","status":"fully_proved","title":"Keith numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/keith-numbers/keith-sequence-prefix","kind":"node","lean":"MetaMathlibExt.keithSequencePrefix","parent":12154,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/keith-numbers/index.html#keith-sequence-prefix","scope":"NumberTheory/digits-numeration/digit-properties/keith-numbers","status":"fully_proved","summary":"Given a list $d = (d_0, \\dots, d_{k-1})$ of natural numbers and $j \\in \\mathbb{N}$, the Keith sequence prefix of $d$ at step $j$ is the list of length $k + j$ obtained from $d$ by appending, $j$ times, the sum of the last $k$ entries of the current list. This is the recurrence used for Keith numbers.","title":"Keith sequence prefix"},{"id":"NumberTheory/digits-numeration/digit-properties/keith-numbers/keith-sequence","kind":"node","lean":"MetaMathlibExt.keithSequence","parent":12154,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/keith-numbers/index.html#keith-sequence","scope":"NumberTheory/digits-numeration/digit-properties/keith-numbers","status":"fully_proved","summary":"Given a list $d = (d_0, \\dots, d_{k-1})$ of natural numbers, the Keith sequence of $d$ is the sequence $(a_m)_{m \\ge 0}$ with $a_m = d_m$ for $m < k$ and $a_m = \\sum_{i=m-k}^{m-1} a_i$ for $m \\ge k$: after the initial entries, each term is the sum of the preceding $k$ terms.","title":"Keith sequence"},{"id":"NumberTheory/digits-numeration/digit-properties/keith-numbers/is-keith-number-with-digits","kind":"node","lean":"MetaMathlibExt.IsKeithNumberWithDigits","parent":12154,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/keith-numbers/index.html#is-keith-number-with-digits","scope":"NumberTheory/digits-numeration/digit-properties/keith-numbers","status":"fully_proved","summary":"A natural number $N$ is a Keith number with digit list $d = (d_0, \\dots, d_{k-1})$ when $k \\ge 2$, $d_i < 10$ for every $i$, $d_0 \\ne 0$, $N = \\sum_{i=0}^{k-1} d_i \\cdot 10^{k-1-i}$ (so $d$ is the decimal representation of $N$, with at least two digits), and $N = a_m$ for some $m \\ge k$, where $(a_m)_{m \\ge 0}$ is the Keith sequence of $d$ indexed from $0$ (so $N$ occurs after the initial digits).","title":"Keith number with given digits"},{"id":"NumberTheory/digits-numeration/digit-properties/keith-numbers/is-keith-number","kind":"node","lean":"MetaMathlibExt.IsKeithNumber","parent":12154,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/keith-numbers/index.html#is-keith-number","scope":"NumberTheory/digits-numeration/digit-properties/keith-numbers","status":"fully_proved","summary":"A natural number $N$ is a Keith number when it is a Keith number with some digit list: that is, $N$ has a decimal representation $d_0 d_1 \\cdots d_{k-1}$ with $k \\ge 2$ digits and leading digit $d_0 \\ne 0$, and $N = a_j$ for some $j \\ge k$, where $(a_m)_{m \\ge 0}$ is the Keith sequence of $(d_0, \\dots, d_{k-1})$: $a_m = d_m$ for $m < k$, and each later term is the sum of the preceding $k$ terms.","title":"Keith number"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/midy-s-theorem.json","id":"NumberTheory/digits-numeration/digit-properties/midy-s-theorem","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/midy-s-theorem/index.html","scope":"NumberTheory/digits-numeration/digit-properties/midy-s-theorem","status":"fully_proved","title":"Midy's theorem"},{"id":"NumberTheory/digits-numeration/digit-properties/midy-s-theorem/midy-theorem-general","kind":"node","lean":"MetaMathlibExt.midy_theorem_general","parent":12159,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/midy-s-theorem/index.html#midy-theorem-general","scope":"NumberTheory/digits-numeration/digit-properties/midy-s-theorem","status":"fully_proved","summary":"Let $p$ be a prime and $n$ a natural number such that $p$ divides $10^{2n} - 1$ but does not divide $10^n - 1$. Let $Q = (10^{2n} - 1)/p$ and write $Q = 10^n H + L$ with $0 \\le L < 10^n$, so that $L$ is the number formed by the last $n$ decimal digits of $Q$ and $H$ the number formed by the remaining digits. Then $H + L = 10^n - 1$.","title":"Midy's theorem, general form"},{"id":"NumberTheory/digits-numeration/digit-properties/midy-s-theorem/midy-theorem","kind":"node","lean":"MetaMathlibExt.midy_theorem","parent":12159,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/midy-s-theorem/index.html#midy-theorem","scope":"NumberTheory/digits-numeration/digit-properties/midy-s-theorem","status":"fully_proved","summary":"Let $p \\ne 2, 5$ be a prime and $n \\ge 1$ an integer such that $p$ divides $10^{2n} - 1$ and does not divide $10^m - 1$ for any $0 < m < 2n$, that is, the decimal period of $1/p$ is $2n$. Let $Q = (10^{2n} - 1)/p$, whose decimal expansion padded with leading zeros to $2n$ digits is the repetend of $1/p$, and write $Q = 10^n H + L$ with $0 \\le L < 10^n$, so that $H$ and $L$ are the two $n$-digit halves of the\u2026","title":"Midy's theorem"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/prodigious-numbers.json","id":"NumberTheory/digits-numeration/digit-properties/prodigious-numbers","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/prodigious-numbers/index.html","scope":"NumberTheory/digits-numeration/digit-properties/prodigious-numbers","status":"fully_proved","title":"Prodigious numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/prodigious-numbers/nonzero-digit-prod","kind":"node","lean":"MetaMathlibExt.nonzeroDigitProd","parent":12162,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/prodigious-numbers/index.html#nonzero-digit-prod","scope":"NumberTheory/digits-numeration/digit-properties/prodigious-numbers","status":"fully_proved","summary":"For natural numbers $b$ and $n$, this is the product of the nonzero digits of $n$ in base $b$; it equals $1$ when $n = 0$. In the degenerate bases, the digits of $n > 0$ are the single digit $n$ when $b = 0$, and $n$ copies of $1$ when $b = 1$.","title":"Product of the nonzero digits"},{"id":"NumberTheory/digits-numeration/digit-properties/prodigious-numbers/is-prodigious","kind":"node","lean":"MetaMathlibExt.IsProdigious","parent":12162,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/prodigious-numbers/index.html#is-prodigious","scope":"NumberTheory/digits-numeration/digit-properties/prodigious-numbers","status":"fully_proved","summary":"For natural numbers $b$ and $n$, the number $n$ is $b$-prodigious when $b \\ge 2$, $n > 0$, and $n$ is divisible by the product of the nonzero digits of $n$ in base $b$.","title":"Prodigious numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/prodigious-numbers/exists-positive-crt-solution-nonzero-dig","kind":"node","lean":"MetaMathlibExt.exists_positive_crt_solution_nonzero_digit_product_one","parent":12162,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/prodigious-numbers/index.html#exists-positive-crt-solution-nonzero-dig","scope":"NumberTheory/digits-numeration/digit-properties/prodigious-numbers","status":"fully_proved","summary":"Let $b \\ge 3$ and $w \\ge 1$ be integers, and for $v = 1, \\dots, w$ let $m_v$ be a positive integer and $r_v$ an integer. Suppose the system of congruences $x \\equiv r_v \\pmod{m_v}$, $v = 1, \\dots, w$, has an integer solution, and $r_v = 0$ whenever $\\gcd(m_v, b) \\ne 1$. Then the system has a positive integer solution $c$ with $p_b(c) = 1$, where $p_b(c)$ is the product of the nonzero base-$b$ digits of $c$.","title":"Chinese remainder solutions with nonzero-digit product one"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/repetend-tables.json","id":"NumberTheory/digits-numeration/digit-properties/repetend-tables","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","title":"Repetend tables"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/digit-add-mul-modulus-eq","kind":"node","lean":"MetaMathlibExt.digit_add_mul_modulus_eq","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Raising the base by multiples of "],["math","\\(n\\)"],["text"," shifts repetend digits linearly"]],"source":"Raising the base by multiples of $n$ shifts repetend digits linearly"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#digit-add-mul-modulus-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $m$ and $j$ be natural numbers, $n \\ge 1$ and $b \\ge 2$. For a base $B$ write $d_j(B) = \\lfloor m B^{j+1} / n \\rfloor \\bmod B$ for the $j$-th base-$B$ digit of $m/n$. Then there is a natural number $c < n$ such that $d_j(b + tn) = d_j(b) + tc$ for every natural number $t$.","title":"Raising the base by multiples of $n$ shifts repetend digits linearly"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/totient-dvd-card-order-of-eq","kind":"node","lean":"MetaMathlibExt.totient_dvd_card_orderOf_eq","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","The number of units of order "],["math","\\(L\\)"],["text"," is a multiple of "],["math","\\(\\varphi(L)\\)"]],"source":"The number of units of order $L$ is a multiple of $\\varphi(L)$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#totient-dvd-card-order-of-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"For natural numbers $n \\ge 1$ and $L$, the number $E(L, n)$ of elements of order $L$ in the unit group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ is divisible by Euler's totient $\\varphi(L)$.","title":"The number of units of order $L$ is a multiple of $\\varphi(L)$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/card-order-of-eq-totient-classification","kind":"node","lean":"MetaMathlibExt.card_orderOf_eq_totient_classification","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moduli with exactly "],["math","\\(\\varphi(L)\\)"],["text"," units of even order "],["math","\\(L\\)"]],"source":"Moduli with exactly $\\varphi(L)$ units of even order $L$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#card-order-of-eq-totient-classification","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n \\ge 1$ and let $L \\ge 2$ be even. If the number of elements of multiplicative order $L$ in $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ equals $\\varphi(L)$, then either $n = 4$ and $L = 2$, or $n = tq$ for some $t \\in \\{1, 2\\}$ and some odd prime power $q$ with $L \\mid \\varphi(q)$.","title":"Moduli with exactly $\\varphi(L)$ units of even order $L$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/card-order-of-eq-two-eq-two-pow-sub-one","kind":"node","lean":"MetaMathlibExt.card_orderOf_eq_two_eq_two_pow_sub_one","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of order two modulo "],["math","\\(n\\)"]],"source":"Elements of order two modulo $n$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#card-order-of-eq-two-eq-two-pow-sub-one","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"For every $n \\ge 1$, the number of elements of multiplicative order $2$ in $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ is $2^k - 1$ for some natural number $k$.","title":"Elements of order two modulo $n$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/order-of-eq-six-ne-small-residues","kind":"node","lean":"MetaMathlibExt.orderOf_eq_six_ne_small_residues","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Order-six elements modulo a power of a prime congruent to "],["math","\\(1\\)"],["text"," mod "],["math","\\(6\\)"]],"source":"Order-six elements modulo a power of a prime congruent to $1$ mod $6$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#order-of-eq-six-ne-small-residues","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $k \\ge 1$ and put $n = p^k$. If $b \\in \\mathbb{Z}/n\\mathbb{Z}$ has multiplicative order $6$, then $b \\ne 0$, $2b \\ne 1$, $b \\ne 1$, $b \\ne -1$ and $b \\ne 2$ in $\\mathbb{Z}/n\\mathbb{Z}$.","title":"Order-six elements modulo a power of a prime congruent to $1$ mod $6$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/sign-balance-shifted-linear-residues","kind":"node","lean":"MetaMathlibExt.sign_balance_shifted_linear_residues","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#sign-balance-shifted-linear-residues","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be prime, $j \\ge 1$ and $Q = p^j$. Let $g_1, g_2, f_1, f_2$ be natural numbers with $g_1, g_2 < Q$ and $f_1 \\le f_2 < Q$ such that $p$ divides none of $g_1$, $g_2$, $g_2 - g_1$, and let $e_1, e_2$ be reals with $0 < e_1 < e_2 < 1$. For natural $z$ put $c_i(z) = ((g_i z + f_i) \\bmod Q) + e_i$ and $d(z) = c_2(z) - c_1(z)$. Then (a) among $1 \\le z \\le Q - 1$, the number of $z$ with $d(z) > 0$ equals the number\u2026","title":"Sign balance for differences of shifted linear residues"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-block-sum-eq","kind":"node","lean":"MetaMathlibExt.repetend_block_sum_eq","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-block-sum-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $B \\ge 2$, $N$, $x < N$, $d \\ge 1$ and $k \\ge 1$ be natural numbers, and let $a_i = \\lfloor xB^{i+1}/N \\rfloor \\bmod B$ be the $i$-th base-$B$ digit of $x/N$. Suppose $a_{i+dk} = a_i$ for every natural number $i$. Let $S = \\sum_{j\\lt d} \\sum_{i\\lt k} a_{jk+i} B^{k-1-i}$ be the sum of the first $d$ blocks of $k$ digits, each read as a base-$B$ number, and let $R = \\sum_{j\\lt d} (xB^{jk} \\bmod N)$. Then\u2026","title":"Block sums of a repetend"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/long-division-remainder-expansion","kind":"node","lean":"MetaMathlibExt.long_division_remainder_expansion","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#long-division-remainder-expansion","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $m < n$ be natural numbers and $b \\ge 1$. For $j \\ge 0$ let $c_j = m b^j \\bmod n$ (the remainders of the long division of $m/n$ in base $b$) and $r_j = \\lfloor b c_j / n \\rfloor$ (its digits). Then for every $k \\ge 0$","title":"Long division of a proper fraction"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-last-digit-ne-zero","kind":"node","lean":"MetaMathlibExt.repetend_last_digit_ne_zero","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-last-digit-ne-zero","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n \\ge 2$ and let $b \\ge 2$ be prime to $n$, with multiplicative order $L$ modulo $n$ (the repetend length of $1/n$ in base $b$). Then the digit $\\lfloor b^{(L-1)+1}/n \\rfloor \\bmod b$ of $1/n$ at position $L - 1$ is not $0$.","title":"The last repetend digit is nonzero"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/ne-add-one-of-sq-add-mul-mem","kind":"node","lean":"MetaMathlibExt.ne_add_one_of_sq_add_mul_mem","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","A constraint on "],["math","\\(b^2 + un\\)"],["text"," between "],["math","\\(2\\)"],["text"," and "],["math","\\(n - 2\\)"]],"source":"A constraint on $b^2 + un$ between $2$ and $n - 2$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#ne-add-one-of-sq-add-mul-mem","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n$ and $b$ be natural numbers and $u$ an integer, and put $b_2 = b^2 + un$. If $2 \\le b_2 \\le n - 2$, then $b \\ne u + 1$.","title":"A constraint on $b^2 + un$ between $2$ and $n - 2$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/long-division-remainders-erase-one-eq-sm","kind":"node","lean":"MetaMathlibExt.long_division_remainders_erase_one_eq_small_bases","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#long-division-remainders-erase-one-eq-sm","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n$ and $L$ be primes and let $b$ be a natural number of multiplicative order $L$ modulo $n$. Suppose $m = 1$, or $2 \\le m \\le n - 1$ and $m$ has multiplicative order $L$ modulo $n$. Then removing $1$ from the set of remainders $\\{m b^j \\bmod n : 0 \\le j < L\\}$ leaves exactly the set of integers $a$ with $2 \\le a \\le n - 1$ whose multiplicative order modulo $n$ is $L$.","title":"Long-division remainders and small bases of prime order"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/even-card-small-bases-order-of","kind":"node","lean":"MetaMathlibExt.even_card_small_bases_orderOf","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#even-card-small-bases-order-of","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"For every natural number $n$ and every $L > 2$, the number of integers $b$ with $1 \\le b < n$ and $\\gcd(b, n) = 1$ whose multiplicative order modulo $n$ is $L$ is even.","title":"Small bases of order greater than two come in pairs"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-digit-pow-mul-eq-of-mul-mod-eq","kind":"node","lean":"MetaMathlibExt.repetend_digit_pow_mul_eq_of_mul_modEq_one","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-digit-pow-mul-eq-of-mul-mod-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n, b, x, j, j'$ be natural numbers and $L \\ge 1$, and suppose $b$ has multiplicative order $L$ modulo $n$. For $y, c \\ge 0$ put $b_y = b^y \\bmod n$ and $r_{y,c} = \\lfloor b_y^{c+1}/n \\rfloor \\bmod b_y$, the $c$-th digit of $1/n$ in base $b_y$. If $\\gcd(j, L) = 1$ and $j j' \\equiv 1 \\pmod L$, then $r_{xj, j'} = r_{x, j}$.","title":"Repetend digits are symmetric under inverse indices"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-column-eq-column-of-mul-mod-eq","kind":"node","lean":"MetaMathlibExt.repetend_column_eq_column_of_mul_modEq_one","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Columns "],["math","\\(j\\)"],["text"," and "],["math","\\(j^{-1}\\)"],["text"," of a repetend table match"]],"source":"Columns $j$ and $j^{-1}$ of a repetend table match"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-column-eq-column-of-mul-mod-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n$ and $L \\ge 1$ be natural numbers, and consider the repetend table of $1/n$ whose rows are the natural numbers $1 \\le b < n$ coprime to $n$ of multiplicative order $L$ modulo $n$, and whose column $j$ holds the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$. If $j$ and $j'$ are natural numbers with $j$ coprime to $L$, $j^2 \\not\\equiv 1 \\pmod L$ and $jj' \\equiv 1 \\pmod L$, then columns $j$ and $j'$ are equal as\u2026","title":"Columns $j$ and $j^{-1}$ of a repetend table match"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/card-coprime-not-sq-mod-eq-one","kind":"node","lean":"MetaMathlibExt.card_coprime_not_sq_modEq_one","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Units modulo "],["math","\\(L\\)"],["text"," whose square is not "],["math","\\(1\\)"]],"source":"Units modulo $L$ whose square is not $1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#card-coprime-not-sq-mod-eq-one","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $L \\ge 1$, let $e$ be the exponent of $2$ in $L$ and let $f$ be the number of distinct odd prime factors of $L$. Then the number of integers $0 \\le j < L$ with $\\gcd(j, L) = 1$ and $j^2 \\not\\equiv 1 \\pmod{L}$ is $\\varphi(L) - 2^{f + g(e)}$, where $g(e) = \\max(0, \\min(2, e - 1))$.","title":"Units modulo $L$ whose square is not $1$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-last-column-inv-eq","kind":"node","lean":"MetaMathlibExt.repetend_last_column_inv_eq","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-last-column-inv-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n$ and $L \\ge 1$ be natural numbers and let $b < n$ have multiplicative order $L$ modulo $n$. Put $b' = b^{L-1} \\bmod n$, the inverse of $b$ modulo $n$. Then $b$ and $b'$ have the same digit in column $L - 1$: $\\lfloor b'^{L}/n \\rfloor \\bmod b' = \\lfloor b^{L}/n \\rfloor \\bmod b$.","title":"The last column pairs a base with its inverse"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-match-iff-triple-sum-eq","kind":"node","lean":"MetaMathlibExt.repetend_columns_match_iff_triple_sum_eq","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-match-iff-triple-sum-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $a \\ge 1$ and $L = 12a + 6$, let $p$ be a prime with $p \\equiv 1 \\pmod L$, let $k \\ge 1$ and $n = p^k$, and let $b$ be a natural number of multiplicative order $L$ modulo $n$; write $b_i = b^i \\bmod n$. Let the rows be the integers $c$ with $1 \\le c < n$, $\\gcd(c, n) = 1$ and multiplicative order $L$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor c^{j+1}/n \\rfloor \\bmod c$ over these\u2026","title":"Class B matched columns of repetend tables"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-digit-sub-eq","kind":"node","lean":"MetaMathlibExt.repetend_digit_sub_eq","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-digit-sub-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n \\ge 1$ and $b$ be natural numbers with $\\gcd(b, n) = 1$. For $x, j \\ge 0$ put $b_x = b^x \\bmod n$ and $r_{x,j} = \\lfloor b_x^{j+1}/n \\rfloor \\bmod b_x$. Then for all natural numbers $y, m, d, e$,","title":"Differences of repetend digits"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/padic-val-nat-sq-sub-add-one-of-order-of","kind":"node","lean":"MetaMathlibExt.padicValNat_sq_sub_add_one_of_orderOf_eq","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-adic valuation of "],["math","\\(B^2 - B + 1\\)"],["text"," for a base of order "],["math","\\(6p^j\\)"]],"source":"The $p$-adic valuation of $B^2 - B + 1$ for a base of order $6p^j$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#padic-val-nat-sq-sub-add-one-of-order-of","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $i$ and $j$ be natural numbers with $j \\ge 1$ and $i \\ge 2j$, and let $B$ be a natural number whose multiplicative order modulo $p^i$ is $6p^j$. Then the $p$-adic valuation of $B^2 - B + 1$ is exactly $i - j$.","title":"The $p$-adic valuation of $B^2 - B + 1$ for a base of order $6p^j$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/pow-mod-eq-sub-one-of-order-of-eq","kind":"node","lean":"MetaMathlibExt.pow_mod_eq_sub_one_of_orderOf_eq","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","A power of "],["math","\\(B\\)"],["text"," equal to "],["math","\\(B - 1\\)"],["text"," modulo "],["math","\\(p^i\\)"]],"source":"A power of $B$ equal to $B - 1$ modulo $p^i$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#pow-mod-eq-sub-one-of-order-of-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $i$ and $j$ be natural numbers with $j \\ge 1$ and $i \\ge 2j$, put $n = p^i$ and $Q = p^j$, and let $B < n$ be a natural number whose multiplicative order modulo $n$ is $6Q$. Let $z < 6Q$ be a natural number with $z \\equiv 2 \\pmod 6$ such that $zB \\equiv 1 \\pmod Q$ or $z \\equiv 1 - B \\pmod Q$. Then $B^z \\bmod n = B - 1$.","title":"A power of $B$ equal to $B - 1$ modulo $p^i$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/pow-mod-eq-pow-of-mod-eq-six","kind":"node","lean":"MetaMathlibExt.pow_modEq_pow_of_modEq_six","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of a base of order "],["math","\\(6p^j\\)"],["text"," modulo "],["math","\\(p^j\\)"],["text"," depend only on the exponent modulo "],["math","\\(6\\)"]],"source":"Powers of a base of order $6p^j$ modulo $p^j$ depend only on the exponent modulo $6$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#pow-mod-eq-pow-of-mod-eq-six","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime, let $i$ and $j$ be natural numbers with $j \\ge 1$ and $i \\ge 2j$, and let $b$ be a natural number whose multiplicative order modulo $p^i$ is $6p^j$. If $y$ and $s$ are natural numbers with $y \\equiv s \\pmod 6$, then $b^y \\equiv b^s \\pmod{p^j}$.","title":"Powers of a base of order $6p^j$ modulo $p^j$ depend only on the exponent modulo $6$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/pow-mod-eq-pow-mod-add-one-of-crt","kind":"node","lean":"MetaMathlibExt.pow_mod_eq_pow_mod_add_one_of_crt","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of "],["math","\\(b^y \\bmod p^i\\)"],["text"," that differ by one"]],"source":"Powers of $b^y \\bmod p^i$ that differ by one"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#pow-mod-eq-pow-mod-add-one-of-crt","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $i$ and $j$ be natural numbers with $j \\ge 1$ and $i \\ge 2j$, put $n = p^i$ and $Q = p^j$, and let $b$ be a natural number whose multiplicative order modulo $n$ is $6Q$. Let $f \\in \\{1, -1\\}$, let $y$ be a natural number with $y \\equiv s \\pmod 6$ for some $s \\in \\{1, -1\\}$, and put $e = 1$ if $sf = 1$ and $e = 5$ if $sf = -1$. Let $d$ be a natural number with\u2026","title":"Powers of $b^y \\bmod p^i$ that differ by one"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/pow-mod-relations-of-order-of-eq-six-mul","kind":"node","lean":"MetaMathlibExt.pow_mod_relations_of_orderOf_eq_six_mul","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relations among the powers "],["math","\\(B^{zp^j} \\bmod p^i\\)"],["text"," for a base of order "],["math","\\(6p^j\\)"]],"source":"Relations among the powers $B^{zp^j} \\bmod p^i$ for a base of order $6p^j$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#pow-mod-relations-of-order-of-eq-six-mul","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $i$ and $j$ be natural numbers, put $n = p^i$ and $Q = p^j$, and let $B$ be a natural number whose multiplicative order modulo $n$ is $6Q$. Write $B_z = B^z \\bmod n$. Then $B_{2Q} = B_Q - 1$, $B_{4Q} = B_{5Q} - 1$, $B_Q + B_{5Q} = n + 1$ and $B_{3Q} = n - 1$; moreover $Q^2 \\equiv Q \\pmod{6Q}$.","title":"Relations among the powers $B^{zp^j} \\bmod p^i$ for a base of order $6p^j$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-digit-sub-mem-zero-n","kind":"node","lean":"MetaMathlibExt.repetend_digit_sub_mem_zero_n","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digit differences for a base of order "],["math","\\(6p^j\\)"],["text"," modulo "],["math","\\(p^i\\)"]],"source":"Digit differences for a base of order $6p^j$ modulo $p^i$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-digit-sub-mem-zero-n","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $i$ and $j$ be natural numbers, and put $n = p^i$ and $Q = p^j$. Let $B < n$ be a natural number whose multiplicative order modulo $n$ is $6Q$, let $f \\in \\{1, -1\\}$, and let $d$ and $e$ be natural numbers with $d \\equiv 2fQ \\pmod{6Q}$ and $e \\equiv fQ \\pmod{6Q}$. For natural $s$ write $r_s = \\lfloor B^{s+1}/n \\rfloor \\bmod B$ for the $s$-th base-$B$ digit of $1/n$,\u2026","title":"Digit differences for a base of order $6p^j$ modulo $p^i$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-eq-of-crt-sq","kind":"node","lean":"MetaMathlibExt.repetend_columns_eq_of_crt_sq","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-eq-of-crt-sq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $j \\ge 1$ and $i \\ge 2j$, and put $n = p^i$, $Q = p^j$ and $L = 6Q$. Let $b$ be a natural number of multiplicative order $L$ modulo $n$, and let $d, e < L$ be natural numbers with $d \\equiv b^2 \\pmod Q$, $e \\equiv b^4 \\pmod Q$ and $d \\equiv e \\equiv 4 \\pmod 6$. Let the rows be the integers $c$ with $1 \\le c < n$, $\\gcd(c, n) = 1$ and multiplicative order $L$ modulo\u2026","title":"Class C matched columns of repetend tables"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-ne-of-le-two","kind":"node","lean":"MetaMathlibExt.repetend_columns_ne_of_le_two","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Repetend tables of length at most "],["math","\\(2\\)"],["text"," have no matched columns"]],"source":"Repetend tables of length at most $2$ have no matched columns"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-ne-of-le-two","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n$ and $L \\le 2$ be natural numbers, let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $L$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. If there is at least one row, then for all distinct $i, j < L$ columns $i$ and $j$ are different.","title":"Repetend tables of length at most $2$ have no matched columns"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-ne-of-card-eq-one-or-th","kind":"node","lean":"MetaMathlibExt.repetend_columns_ne_of_card_eq_one_or_three","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-ne-of-card-eq-one-or-th","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"For natural numbers $n$ and $L$, let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $L$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. If there are exactly one or exactly three rows, then for all distinct $i, j < L$ columns $i$ and $j$ are different.","title":"Repetend tables with one or three rows have no matched columns"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-digit-diff-of-columns-eq","kind":"node","lean":"MetaMathlibExt.repetend_digit_diff_of_columns_eq","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-digit-diff-of-columns-eq","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n$ and $L$ be natural numbers and let $B$ be the set of integers $c$ with $1 \\le c < n$, $\\gcd(c, n) = 1$ and multiplicative order $L$ modulo $n$. For $c \\in B$ and $k \\ge 0$ let $r_k(c) = \\lfloor c^{k+1}/n \\rfloor \\bmod c$, and let $C_k$ be the multiset of the $r_k(c)$ over $c \\in B$. Suppose $B$ has exactly two elements, $b$ and $b' \\ne b$, and put $D_k = r_k(b') - r_k(b)$. Then for all $i, j \\ge 0$ with\u2026","title":"Matching columns of a two-row repetend table"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/pow-add-half-mod-eq-and-digit-add","kind":"node","lean":"MetaMathlibExt.pow_add_half_mod_eq_and_digit_add","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Half-period symmetry of the repetend of "],["math","\\(1/n\\)"]],"source":"Half-period symmetry of the repetend of $1/n$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#pow-add-half-mod-eq-and-digit-add","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n$ be a natural number whose unit group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ is cyclic, let $L > 0$ be even, and let $b < n$ be a natural number whose multiplicative order modulo $n$ is $L$. For natural numbers $m$ write $b_m = b^m \\bmod n$ and $r_m = \\lfloor b^{m+1}/n \\rfloor \\bmod b$, the $m$-th base-$b$ digit of $1/n$. Then for every natural number $j$ we have $b_{j+L/2} = n - b_j$ and\u2026","title":"Half-period symmetry of the repetend of $1/n$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-ne-of-card-two-of-three","kind":"node","lean":"MetaMathlibExt.repetend_columns_ne_of_card_two_of_three","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-row repetend tables of length "],["math","\\(3\\)"],["text"," have no matched columns"]],"source":"Two-row repetend tables of length $3$ have no matched columns"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-ne-of-card-two-of-three","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"For a natural number $n$, let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $3$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. If there are exactly two rows, then for all distinct $i, j < 3$ columns $i$ and $j$ are different.","title":"Two-row repetend tables of length $3$ have no matched columns"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-ne-of-card-two-of-four","kind":"node","lean":"MetaMathlibExt.repetend_columns_ne_of_card_two_of_four","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-row repetend tables of length "],["math","\\(4\\)"],["text"," have no matched columns"]],"source":"Two-row repetend tables of length $4$ have no matched columns"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-ne-of-card-two-of-four","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"For a natural number $n$, let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $4$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. If there are exactly two rows, then for all distinct $i, j < 4$ columns $i$ and $j$ are different.","title":"Two-row repetend tables of length $4$ have no matched columns"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/eq-of-card-small-bases-order-of-six-eq-t","kind":"node","lean":"MetaMathlibExt.eq_of_card_small_bases_orderOf_six_eq_two","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#eq-of-card-small-bases-order-of-six-eq-t","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $n$ be a natural number. If exactly two integers $b$ with $1 \\le b < n$ and $\\gcd(b, n) = 1$ have multiplicative order $6$ modulo $n$, then either $n = 3^k t \\ge 9$ for some natural number $k$ and some $t \\in \\{1, 2\\}$, or $n = p^k t$ for some prime $p \\equiv 1 \\pmod 6$, some $k \\ge 1$ and some $t \\in \\{1, 2\\}$.","title":"Moduli with exactly two small bases of order six"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-ne-of-three-pow-of-six","kind":"node","lean":"MetaMathlibExt.repetend_columns_ne_of_three_pow_of_six","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Order-"],["math","\\(6\\)"],["text"," repetend tables of "],["math","\\(3^k t\\)"],["text"," have no matched columns"]],"source":"Order-$6$ repetend tables of $3^k t$ have no matched columns"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-ne-of-three-pow-of-six","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $k \\ge 2$ and $t \\in \\{1, 2\\}$, and put $n = 3^k t$. Let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $6$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. Then for all distinct $i, j < 6$ columns $i$ and $j$ are different.","title":"Order-$6$ repetend tables of $3^k t$ have no matched columns"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/pow-mod-eqs-of-order-of-eq-six","kind":"node","lean":"MetaMathlibExt.pow_mod_eqs_of_orderOf_eq_six","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of a base of order "],["math","\\(6\\)"],["text"," modulo "],["math","\\(tp^k\\)"]],"source":"Powers of a base of order $6$ modulo $tp^k$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#pow-mod-eqs-of-order-of-eq-six","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $k \\ge 1$ and $t \\in \\{1, 2\\}$ be natural numbers, and put $n = tp^k$. Let $b_1 < n$ be a natural number whose multiplicative order modulo $n$ is $6$, write $b_m = b_1^m \\bmod n$, and put $H = 0$ if $t = 1$, $H = p^k$ if $t = 2$ and $b_1 < p^k$, and $H = -p^k$ if $t = 2$ and $b_1 \\ge p^k$. Then $b_2 = b_1 - 1 + H$, $b_3 = n - 1$, $b_4 = n - b_1$ and\u2026","title":"Powers of a base of order $6$ modulo $tp^k$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-digit-diffs-of-order-of-eq-six","kind":"node","lean":"MetaMathlibExt.repetend_digit_diffs_of_orderOf_eq_six","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digit differences between the rows "],["math","\\(b_1\\)"],["text"," and "],["math","\\(b_1^5\\)"],["text"," of the order-"],["math","\\(6\\)"],["text"," table"]],"source":"Digit differences between the rows $b_1$ and $b_1^5$ of the order-$6$ table"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-digit-diffs-of-order-of-eq-six","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p \\equiv 1 \\pmod 6$ be prime, $k \\ge 1$, $t \\in \\{1, 2\\}$ and $n = t p^k$, and let $b_1 < n$ have multiplicative order $6$ modulo $n$. Put $H = 0$ if $t = 1$, $H = p^k$ if $t = 2$ and $b_1 < p^k$, and $H = -p^k$ if $t = 2$ and $b_1 \\ge p^k$. Let $b_5 = b_1^5 \\bmod n$ and, for $i \\ge 0$, $D_i = r_i(b_5) - r_i(b_1)$, where $r_i(c) = \\lfloor c^{i+1}/n \\rfloor \\bmod c$. Then $D_0 = D_5 = 0$,\u2026","title":"Digit differences between the rows $b_1$ and $b_1^5$ of the order-$6$ table"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-column-zero-ne-five","kind":"node","lean":"MetaMathlibExt.repetend_column_zero_ne_five","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Columns "],["math","\\(0\\)"],["text"," and "],["math","\\(5\\)"],["text"," of the order-"],["math","\\(6\\)"],["text"," repetend table of "],["math","\\(t p^k\\)"],["text"," differ"]],"source":"Columns $0$ and $5$ of the order-$6$ repetend table of $t p^k$ differ"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-column-zero-ne-five","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $k \\ge 1$ and $t \\in \\{1, 2\\}$, and put $n = t p^k$. Let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $6$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. Then columns $0$ and $5$ are different.","title":"Columns $0$ and $5$ of the order-$6$ repetend table of $t p^k$ differ"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-one-to-three-ne-zero-fi","kind":"node","lean":"MetaMathlibExt.repetend_columns_one_to_three_ne_zero_five","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Columns "],["math","\\(1\\)"],["text",", "],["math","\\(2\\)"],["text",", "],["math","\\(3\\)"],["text"," of the order-"],["math","\\(6\\)"],["text"," repetend table differ from columns "],["math","\\(0\\)"],["text"," and "],["math","\\(5\\)"]],"source":"Columns $1$, $2$, $3$ of the order-$6$ repetend table differ from columns $0$ and $5$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-one-to-three-ne-zero-fi","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p \\equiv 1 \\pmod 6$ be prime, $k \\ge 1$, $t \\in \\{1, 2\\}$ and $n = t p^k$, and let $B$ be the set of integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $6$ modulo $n$. For $c \\ge 0$ let column $c$ be the multiset of the digits $\\lfloor b^{c+1}/n \\rfloor \\bmod b$ over $b \\in B$. Then for every $i \\in \\{1, 2, 3\\}$ and $j \\in \\{0, 5\\}$, columns $i$ and $j$ are different multisets.","title":"Columns $1$, $2$, $3$ of the order-$6$ repetend table differ from columns $0$ and $5$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-column-four-ne-zero-five","kind":"node","lean":"MetaMathlibExt.repetend_column_four_ne_zero_five","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Column "],["math","\\(4\\)"],["text"," of the order-"],["math","\\(6\\)"],["text"," repetend table of "],["math","\\(t p^k\\)"],["text"," differs from columns "],["math","\\(0\\)"],["text"," and "],["math","\\(5\\)"]],"source":"Column $4$ of the order-$6$ repetend table of $t p^k$ differs from columns $0$ and $5$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-column-four-ne-zero-five","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $k \\ge 1$ and $t \\in \\{1, 2\\}$, and put $n = t p^k$. Let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $6$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. Then column $4$ is different from column $0$ and from column $5$.","title":"Column $4$ of the order-$6$ repetend table of $t p^k$ differs from columns $0$ and $5$"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-one-to-four-pairwise-ne","kind":"node","lean":"MetaMathlibExt.repetend_columns_one_to_four_pairwise_ne","parent":12166,"reading_title":{"format":"inline-parts-v1","parts":[["text","Columns "],["math","\\(1\\)"],["text"," to "],["math","\\(4\\)"],["text"," of the order-"],["math","\\(6\\)"],["text"," repetend table of "],["math","\\(t p^k\\)"],["text"," are pairwise different"]],"source":"Columns $1$ to $4$ of the order-$6$ repetend table of $t p^k$ are pairwise different"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-one-to-four-pairwise-ne","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 6$, let $k \\ge 1$ and $t \\in \\{1, 2\\}$, and put $n = t p^k$. Let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $6$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. Then for all distinct $i, j \\in \\{1, 2, 3, 4\\}$ columns $i$ and $j$ are different.","title":"Columns $1$ to $4$ of the order-$6$ repetend table of $t p^k$ are pairwise different"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/repetend-columns-ne-of-card-le-three","kind":"node","lean":"MetaMathlibExt.repetend_columns_ne_of_card_le_three","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#repetend-columns-ne-of-card-le-three","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"For natural numbers $n$ and $L$, let the rows be the integers $b$ with $1 \\le b < n$, $\\gcd(b, n) = 1$ and multiplicative order $L$ modulo $n$, and let column $j$ be the multiset of the digits $\\lfloor b^{j+1}/n \\rfloor \\bmod b$ over these rows. If there are at least one and at most three rows, then for all distinct $i, j < L$ columns $i$ and $j$ are different.","title":"Repetend tables with at most three rows have no matched columns"},{"id":"NumberTheory/digits-numeration/digit-properties/repetend-tables/tsum-inv-pow-succ-lt","kind":"node","lean":"MetaMathlibExt.tsum_inv_pow_succ_lt","parent":12166,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/repetend-tables/index.html#tsum-inv-pow-succ-lt","scope":"NumberTheory/digits-numeration/digit-properties/repetend-tables","status":"can_prove","summary":"For natural numbers $m \\ge 2$ and $s \\ge 1$, $\\sum_{n = s+1}^{\\infty} n^{-m} < \\dfrac{1}{(m-1)\\, s^{m-1}}$.","title":"Tail bound for sums of inverse powers"},{"data_url":"scopes/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes.json","id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","kind":"scope","navigation":"module","parent":11986,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"can_prove","title":"Walking to infinity on primes"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/extended-cunningham-chain","kind":"node","lean":"MetaMathlibExt.extendedCunninghamChain","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#extended-cunningham-chain","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"fully_proved","summary":"For a natural number $p$, the extended Cunningham chain generated by $p$ is the sequence $e_0, e_1, e_2, \\ldots$ of natural numbers given by $e_0 = p$ and $e_k = 2 e_{k-1} + 1$ for $k \\ge 1$; in binary, $e_k$ is obtained from $p$ by appending $k$ digits $1$ on the right.","title":"Extended Cunningham chain"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/extended-cunningham-chain-zero","kind":"node","lean":"MetaMathlibExt.extendedCunninghamChain_zero","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#extended-cunningham-chain-zero","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"fully_proved","summary":"For every natural number $p$, the extended Cunningham chain generated by $p$ has initial term $e_0 = p$.","title":"Initial term of the extended Cunningham chain"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/extended-cunningham-chain-succ","kind":"node","lean":"MetaMathlibExt.extendedCunninghamChain_succ","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#extended-cunningham-chain-succ","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"fully_proved","summary":"For all natural numbers $p$ and $k$, the terms of the extended Cunningham chain generated by $p$ satisfy $e_{k+1} = 2 e_k + 1$.","title":"Recurrence of the extended Cunningham chain"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/extended-cunningham-chain-add-one","kind":"node","lean":"MetaMathlibExt.extendedCunninghamChain_add_one","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#extended-cunningham-chain-add-one","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"fully_proved","summary":"For all natural numbers $p$ and $k$, the $k$-th term $e_k$ of the extended Cunningham chain generated by $p$ satisfies $e_k + 1 = 2^k (p + 1)$, that is, $e_k = 2^k (p + 1) - 1$.","title":"Closed form of the extended Cunningham chain"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/extended-cunningham-chain-two-small","kind":"node","lean":"MetaMathlibExt.extendedCunninghamChain_two_small","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#extended-cunningham-chain-two-small","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"fully_proved","summary":"The extended Cunningham chain $e_0, e_1, e_2, \\ldots$ generated by $2$ has $e_1 = 5$, $e_2 = 11$ and $e_3 = 23$.","title":"First terms of the extended Cunningham chain generated by 2"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/exists-prime-walk-appending-decimal-digi","kind":"node","lean":"MetaMathlibExt.exists_prime_walk_appending_decimal_digits","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#exists-prime-walk-appending-decimal-digi","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"can_prove","summary":"For every prime $p_0$ there is a sequence of primes $p_0, p_1, p_2, \\ldots$ starting at $p_0$ such that for every $i \\ge 1$ there are integers $n_i \\ge 1$ and $k_i$ with $0 < k_i < 10^{n_i}$ and $p_i = 10^{n_i} p_{i-1} + k_i$; that is, each term is obtained from the previous one by appending $n_i$ decimal digits on the right.","title":"Walking to infinity on primes by appending decimal digits"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/two-pow-sub-succ-ne-two-pow","kind":"node","lean":"MetaMathlibExt.two_pow_sub_succ_ne_two_pow","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#two-pow-sub-succ-ne-two-pow","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"can_prove","summary":"Let $p$ and $k$ be natural numbers with $k \\ge \\lceil \\log_2 (p + 1) \\rceil + 2$. Then $2^k - (p + 1)$ is not a power of $2$: there is no natural number $m$ with $2^k - (p + 1) = 2^m$.","title":"A power of two minus a much smaller positive number is not a power of two"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/extended-cunningham-chain-exists-composi","kind":"node","lean":"MetaMathlibExt.extendedCunninghamChain_exists_composite_run","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#extended-cunningham-chain-exists-composi","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"can_prove","summary":"Let $p$ be a prime and let $e_0 = p, e_1, e_2, \\ldots$ be the extended Cunningham chain generated by $p$. For every $n \\ge 1$ there is $i \\ge 1$ such that none of $e_i, e_{i+1}, \\ldots, e_{i+n-1}$ is prime.","title":"Extended Cunningham chains generated by primes contain arbitrarily long composite runs"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/not-exists-binary-prime-walk-appending-a","kind":"node","lean":"MetaMathlibExt.not_exists_binary_prime_walk_appending_at_most_two_digits","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#not-exists-binary-prime-walk-appending-a","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"can_prove","summary":"There is no sequence of primes $p_0, p_1, p_2, \\ldots$ such that for every $i \\ge 0$ there are $n \\in \\{1, 2\\}$ and a natural number $k < 2^n$ with $p_{i+1} = 2^n p_i + k$. That is, starting from any prime, one cannot walk to infinity on primes by appending one or two binary digits at a time on the right.","title":"No binary prime walk to infinity in steps of one or two digits"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/right-trunc-squarefree-count","kind":"node","lean":"MetaMathlibExt.rightTruncSquarefreeCount","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#right-trunc-squarefree-count","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"fully_proved","summary":"For natural numbers $b$ and $k$, $L_{b,k}$ is the number of right-truncatable square-free numbers with exactly $k$ digits in base $b$: the number of natural numbers $x$ with $b^{k-1} \\le x < b^k$ such that $\\lfloor x / b^{\\ell} \\rfloor$ is square-free for every $\\ell \\in \\{0, \\dots, k-1\\}$. When $k = 0$ the range is empty, so $L_{b,0} = 0$.","title":"Counting right-truncatable square-free numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/right-trunc-squarefree-count-small","kind":"node","lean":"MetaMathlibExt.rightTruncSquarefreeCount_small","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#right-trunc-squarefree-count-small","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"fully_proved","summary":"Let $L_{b,k}$ denote the number of right-truncatable square-free numbers with exactly $k$ digits in base $b$. Then $L_{2,1} = 1$, $L_{2,2} = 2$, $L_{2,3} = 3$, $L_{2,4} = 5$ and $L_{10,1} = 6$.","title":"Small counts of right-truncatable square-free numbers"},{"id":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/binary-squarefree-walk-expectation-bound","kind":"node","lean":"MetaMathlibExt.binary_squarefree_walk_expectation_bounds","parent":12208,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes/index.html#binary-squarefree-walk-expectation-bound","scope":"NumberTheory/digits-numeration/digit-properties/walking-to-infinity-on-primes","status":"can_prove","summary":"Let $L_{2,k}$ denote the number of right-truncatable square-free numbers with exactly $k$ digits in base $2$. The real series $E_2 = \\sum_{k \\ge 1} L_{2,k} / 2^k$ satisfies $636163720502 / 2^{38} \\le E_2 \\le 636163930777 / 2^{38}$.","title":"Bounds on the binary expected square-free walk length"},{"data_url":"scopes/NumberTheory/digits-numeration/repdigits-reverse-multiples.json","id":"NumberTheory/digits-numeration/repdigits-reverse-multiples","kind":"scope","parent":11985,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/index.html","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples","status":"can_state","summary":"Repdigits, palindromes, reverse multiples and Young graphs, permutiples, digit reversal, special squares by digits.","title":"Repdigits, palindromes and reverse multiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-digits","kind":"node","lean":"Nat.reverseDigits","parent":12221,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/index.html#reverse-digits","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples","status":"fully_proved","summary":"For natural numbers $b$ and $n$ with $b \\ge 2$, write $n = \\sum_{i\\lt k} d_i b^i$ with digits $0 \\le d_i < b$ and leading digit $d_{k-1} \\ne 0$ (so $k = 0$ when $n = 0$). The base-$b$ reversal of $n$ is the natural number $\\sum_{i\\lt k} d_{k-1-i} b^i$. For the bases $b = 0$ and $b = 1$ the reversal of $n$ is $n$ itself.","title":"Digit reversal"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/finite-near-repdigit-perfect-powers","kind":"node","lean":"MetaMathlibExt.finite_near_repdigit_perfect_powers","parent":12221,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/index.html#finite-near-repdigit-perfect-powers","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples","status":"can_prove","summary":"Fix a natural number $l \\ge 3$. There are only finitely many perfect $l$-th powers $y = x^l$ ($x \\in \\mathbb{N}$) whose decimal digits are all equal but one, that is, some digit $c$ occurs in the decimal representation of $y$ and exactly one digit position of $y$ holds a digit other than $c$, apart from the powers $10^{ln}$ ($n \\in \\mathbb{N}$) and, when $l = 3$, the cubes $8 \\cdot 10^{3n}$ ($n \\in \\mathbb{N}$).","title":"Perfect powers with all decimal digits equal but one (Kihel\u2013Luca)"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/squares-aaabb-classification","kind":"node","lean":"MetaMathlibExt.squares_aaabb_classification","parent":12221,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/index.html#squares-aaabb-classification","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples","status":"can_prove","summary":"Let $n$ be a perfect square whose decimal representation, read from the most significant digit, consists of $r \\ge 1$ copies of a digit $a$ with $1 \\le a \\le 9$ followed by $s \\ge 1$ copies of a digit $b$ with $0 \\le b \\le 9$. Then $n$ is one of $16$, $25$, $36$, $49$, $64$, $81$, $144$, $225$, $441$, $1444$ and $7744$, or there is a natural number $i$ such that $n = 10^{2i}$, $n = 4 \\cdot 10^{2i}$ or\u2026","title":"Squares with two blocks of repeated digits"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/triangular-numbers-single-repeated-digit","kind":"node","lean":"MathlibExt.NumberTheory.Digits.TriangularRepdigitsWanted.triangular_numbers_single_repeated_digit","parent":12221,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/index.html#triangular-numbers-single-repeated-digit","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples","status":"can_prove","summary":"The positive triangular numbers $n(n+1)/2$ with $n \\ge 1$ whose decimal representation consists of a single repeated digit are exactly $1$, $3$, $6$, $55$, $66$ and $666$.","title":"Triangular repdigits"},{"data_url":"scopes/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions.json","id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","kind":"scope","navigation":"module","parent":12221,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"can_prove","title":"Palindromes in arithmetic progressions"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/floor-add-floor-neg-and-floor-add-eq","kind":"node","lean":"MetaMathlibExt.floor_add_floor_neg_and_floor_add_eq","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#floor-add-floor-neg-and-floor-add-eq","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"can_prove","summary":"For every real number $x$, the sum $\\lfloor x \\rfloor + \\lfloor -x \\rfloor$ equals $0$ if $x$ is an integer and $-1$ otherwise. For all real numbers $x$ and $y$, $\\lfloor x + y \\rfloor = \\lfloor x \\rfloor + \\lfloor y \\rfloor$ if $(x - \\lfloor x \\rfloor) + (y - \\lfloor y \\rfloor) < 1$, and $\\lfloor x + y \\rfloor = \\lfloor x \\rfloor + \\lfloor y \\rfloor + 1$ otherwise.","title":"Two floor-function identities"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/palindrome-apcount","kind":"node","lean":"MetaMathlibExt.palindromeAPCount","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#palindrome-apcount","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"fully_proved","summary":"For natural numbers $b$, $m$, $q$ and $r$, $A_b(m, q, r)$ is the number of base-$b$ palindromes $n$ with $1 \\le n \\le m$ and $n \\equiv r \\pmod q$, where a natural number is a base-$b$ palindrome when its string of base-$b$ digits reads the same in both directions.","title":"Palindromes in an arithmetic progression"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/palindromic-half-copy","kind":"node","lean":"MetaMathlibExt.palindromicHalfCopy","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#palindromic-half-copy","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"fully_proved","summary":"For natural numbers $b$ and $m \\ge 1$, write $m = (a_k a_{k-1} \\cdots a_0)_b$ for the base-$b$ digits of $m$. The palindromic half copy of $m$ is $C_b(m) = \\sum_{j=0}^{k} a_{\\max(j, k - j)} b^j$: the $b$-adic palindrome with the same number of digits as $m$ whose upper digits $a_k, \\dots, a_{\\lceil k/2 \\rceil}$ are those of $m$ and whose lower digits mirror them. For $m = 0$, which has no digits, $C_b(0) = 0$.","title":"Palindromic half copy"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/digit-sum-congruence-count","kind":"node","lean":"MetaMathlibExt.digitSumCongruenceCount","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#digit-sum-congruence-count","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"fully_proved","summary":"For natural numbers $b$, $k$, $q$, $x$, $y$ and an integer $r$, $N_b(k, q, r, x, y)$ is the number of tuples $(x_1, \\dots, x_k)$ of natural numbers with $x \\le x_1 < y$ and $x_i < b$ for $2 \\le i \\le k$ such that $x_1 + \\dots + x_k \\equiv r \\pmod q$. The count is meant for $k \\ge 1$; for $k = 0$ it is defined to equal its value for $k = 1$, the number of natural numbers $x_1$ with $x \\le x_1 < y$ and\u2026","title":"Digit-sum congruence count"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/palindrome-apcount-small","kind":"node","lean":"MetaMathlibExt.palindromeAPCount_small","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#palindrome-apcount-small","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"fully_proved","summary":"Write $A_b(m, q, r)$ for the number of base-$b$ palindromes $n$ with $1 \\le n \\le m$ and $n \\equiv r \\pmod q$. Then $A_{10}(100, 1, 0) = 18$, the palindromes being $1, \\dots, 9$ and $11, 22, \\dots, 99$, and $A_2(9, 2, 1) = 5$, the odd base-$2$ palindromes up to $9$ being $1$, $3 = (11)_2$, $5 = (101)_2$, $7 = (111)_2$ and $9 = (1001)_2$.","title":"Two small palindrome counts"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/palindromic-half-copy-example","kind":"node","lean":"MetaMathlibExt.palindromicHalfCopy_example","parent":12226,"reading_title":{"format":"inline-parts-v1","parts":[["text","Palindromic half copies of "],["math","\\(5570\\)"],["text"," in bases "],["math","\\(5\\)"],["text"," and "],["math","\\(8\\)"]],"source":"Palindromic half copies of $5570$ in bases $5$ and $8$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#palindromic-half-copy-example","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"fully_proved","summary":"Write $C_b(m)$ for the palindromic half copy of $m$ in base $b$. For $m = 5570 = (134240)_5 = (12702)_8$, $C_5(m) = (134431)_5 = 5616$ and $C_8(m) = (12721)_8 = 5585$.","title":"Palindromic half copies of $5570$ in bases $5$ and $8$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/digit-sum-congruence-count-small","kind":"node","lean":"MetaMathlibExt.digitSumCongruenceCount_small","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#digit-sum-congruence-count-small","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"fully_proved","summary":"In the notation $N_b(k, q, r, x, y)$ of the digit-sum congruence count, $N_3(2, 2, 0, 0, 3) = 5$: exactly five pairs $(x_1, x_2)$ of natural numbers with $x_1, x_2 < 3$ have even sum.","title":"A small value of the digit-sum congruence count"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/digit-sum-congruence-count-of-mod-eq-one","kind":"node","lean":"MetaMathlibExt.digitSumCongruenceCount_of_modEq_one","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#digit-sum-congruence-count-of-mod-eq-one","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"can_prove","summary":"Let $a$, $b$, $k$ and $q$ be natural numbers, all at least $1$, with $b \\equiv 1 \\pmod q$, and let $r$ be an integer. For natural numbers $x$ and $y$ write $N_b(k, q, r, x, y)$ for the number of tuples $(x_1, \\dots, x_k)$ of natural numbers with $x \\le x_1 < y$, $x_i < b$ for $2 \\le i \\le k$, and $x_1 + \\dots + x_k \\equiv r \\pmod q$. Write $[P]$ for $1$ if $P$ holds and $0$ otherwise, $\\lfloor a/q \\rfloor$ for the\u2026","title":"Counting digit tuples by their sum modulo a divisor of the base minus one"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/palindrome-apcount-of-dvd-base","kind":"node","lean":"MetaMathlibExt.palindromeAPCount_of_dvd_base","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#palindrome-apcount-of-dvd-base","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"can_prove","summary":"Let $b \\ge 2$, $m \\ge 1$ and $q \\ge 1$ be natural numbers with $q \\mid b$. Write $m = (a_k a_{k-1} \\cdots a_0)_b$ in base $b$, with leading digit $a_k$. Write $A_b(m, q, r)$ for the number of base-$b$ palindromes $n$ with $1 \\le n \\le m$ and $n \\equiv r \\pmod q$, $C_b(m)$ for the palindromic half copy of $m$, and $[P]$ for $1$ if $P$ holds and $0$ otherwise. Put","title":"Palindromes in residue classes modulo a divisor of the base"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/palindrome-count-closed-form","kind":"node","lean":"MetaMathlibExt.palindromeCount_closed_form","parent":12226,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting palindromes up to "],["math","\\(m\\)"],["text"," and by parity"]],"source":"Counting palindromes up to $m$ and by parity"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#palindrome-count-closed-form","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"can_prove","summary":"Let $b \\ge 2$ and $m \\ge 1$, and write $m = (a_k a_{k-1} \\cdots a_0)_b$ in base $b$, with leading digit $a_k \\ne 0$. For natural numbers $q$ and $r$, let $A_b(m, q, r)$ be the number of $b$-adic palindromes $n$ with $1 \\le n \\le m$ and $n \\equiv r \\pmod q$, and let $C_b(m)$ be the palindromic half copy of $m$. Put $\\delta = 1$ if $C_b(m) \\le m$ and $\\delta = 0$ otherwise. Then\u2026","title":"Counting palindromes up to $m$ and by parity"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/palindrome-apcount-of-mod-eq-one","kind":"node","lean":"MetaMathlibExt.palindromeAPCount_of_modEq_one","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#palindrome-apcount-of-mod-eq-one","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"can_prove","summary":"Let $b \\ge 2$, $m \\ge 1$ and $q \\ge 1$ be natural numbers with $b \\equiv 1 \\pmod q$, and let $r$ be a natural number with $r < q$. Write $m = (a_k a_{k-1} \\cdots a_0)_b$ in base $b$. Write $A_b(m, q, r)$ for the number of base-$b$ palindromes $n$ with $1 \\le n \\le m$ and $n \\equiv r \\pmod q$, $C_b(m)$ for the palindromic half copy of $m$, $[P]$ for $1$ if $P$ holds and $0$ otherwise, and $t \\bmod d$ for the least\u2026","title":"Palindromes in residue classes modulo a divisor of the base minus one"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/palindrome-apcount-base-pow","kind":"node","lean":"MetaMathlibExt.palindromeAPCount_base_pow","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#palindrome-apcount-base-pow","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"can_prove","summary":"Let $b \\ge 2$, $k \\ge 1$ and $q \\ge 1$ be natural numbers and let $r$ be a natural number with $r < q$. Write $A_b(m, q, r)$ for the number of base-$b$ palindromes $n$ with $1 \\le n \\le m$ and $n \\equiv r \\pmod q$, and $[r = 0]$ for $1$ if $r = 0$ and $0$ otherwise. If $q \\mid b$, then","title":"Palindromes up to a power of the base in residue classes"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/abs-palindrome-apcount-sub-le","kind":"node","lean":"MetaMathlibExt.abs_palindromeAPCount_sub_le","parent":12226,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions/index.html#abs-palindrome-apcount-sub-le","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromes-in-arithmetic-progressions","status":"can_prove","summary":"Let $b \\ge 2$ be a natural number, let $q$ be an odd natural number with $b \\equiv 1 \\pmod q$, and let $r$ be a natural number with $r < q$. For natural numbers $m$, $q'$ and $r'$, write $A_b(m, q', r')$ for the number of base-$b$ palindromes $n$ with $1 \\le n \\le m$ and $n \\equiv r' \\pmod{q'}$; thus $A_b(m, 1, 0)$ is the number of all base-$b$ palindromes in $[1, m]$. Then for every natural number $m \\ge 1$,","title":"Palindromes in residue classes: a logarithmic error bound"},{"data_url":"scopes/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two.json","id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","kind":"scope","navigation":"module","parent":12221,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","title":"Palindromic representations of powers of two"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/min-palindromic-base","kind":"node","lean":"MetaMathlibExt.minPalindromicBase","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#min-palindromic-base","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"fully_proved","summary":"For a natural number $N$, $b(N)$ is the smallest base $b \\ge 2$ in which $N$ is a palindrome, that is, in which the base-$b$ digits of $N$ read the same forwards and backwards. Such a base exists: $0$ has no digits and is a palindrome in every base, and for $N \\ge 1$ the number $N$ is written with the single digit $N$ in base $N + 1$.","title":"Minimum palindromic base"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/has-binomial-form","kind":"node","lean":"MetaMathlibExt.HasBinomialForm","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#has-binomial-form","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"fully_proved","summary":"A natural number $N$ has a base-$b$ representation in binomial form if there are natural numbers $\\alpha \\ge 1$ and $k \\ge 1$ such that the base-$b$ digits of $N$, from the most significant to the least significant, are $\\alpha\\binom{k}{k}, \\alpha\\binom{k}{k-1}, \\dots, \\alpha\\binom{k}{0}$.","title":"Binomial form"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/min-palindromic-base-two","kind":"node","lean":"MetaMathlibExt.minPalindromicBase_two","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#min-palindromic-base-two","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"fully_proved","summary":"The smallest base $b \\ge 2$ in which the number $2$ is a palindrome is $b(2) = 3$: in base $2$ the number $2$ is written $(1, 0)_2$, which is not a palindrome, while in base $3$ it is the one-digit palindrome $(2)_3$.","title":"The minimum palindromic base of two"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/has-binomial-form-16-2023","kind":"node","lean":"MetaMathlibExt.hasBinomialForm_16_2023","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#has-binomial-form-16-2023","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"fully_proved","summary":"The number $2023$ has a base-$16$ representation in binomial form: its base-$16$ digits are $7, 14, 7$, that is $7\\binom{2}{2}, 7\\binom{2}{1}, 7\\binom{2}{0}$.","title":"The binomial form of 2023 in base 16"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/min-palindromic-base-eq-pred","kind":"node","lean":"MetaMathlibExt.minPalindromicBase_eq_pred","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#min-palindromic-base-eq-pred","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $N \\ge 1$ be a natural number and let $b(N)$ be the smallest base $b \\ge 2$ in which $N$ is a palindrome. If $b(N) = N - 1$, then $N \\in \\{3, 4, 6\\}$, or $N > 6$ and $N$ is prime.","title":"Brown's theorem on minimal palindromic base one less than the number"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/digits-mul-one-add-pow","kind":"node","lean":"MetaMathlibExt.digits_mul_one_add_pow","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#digits-mul-one-add-pow","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $\\alpha \\ge 1$, $k$ and $b$ be natural numbers with $\\alpha\\binom{k}{\\lceil k/2 \\rceil} < b$. Then the base-$b$ representation of $\\alpha(1 + b)^k$ has exactly the $k + 1$ digits $\\alpha\\binom{k}{k}, \\alpha\\binom{k}{k-1}, \\dots, \\alpha\\binom{k}{0}$, from the most significant to the least significant.","title":"Digits of a multiple of a power of one more than the base"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/has-binomial-form-two-pow","kind":"node","lean":"MetaMathlibExt.hasBinomialForm_two_pow","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#has-binomial-form-two-pow","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $n$, $x$, $k$ and $r$ be natural numbers with $k \\ge 1$, $n = xk + r$ and $2^r\\binom{k}{\\lceil k/2 \\rceil} < 2^x - 1$. Then $2^n$ has a base-$(2^x - 1)$ representation in binomial form.","title":"Sufficient condition for a binomial form of a power of 2"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/binomial-form-two-pow-base","kind":"node","lean":"MetaMathlibExt.binomialForm_two_pow_base","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#binomial-form-two-pow-base","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $n$, $b \\ge 2$, $\\alpha \\ge 1$ and $k \\ge 1$ be natural numbers. If the base-$b$ digits of $2^n$, from the most significant to the least significant, are $\\alpha\\binom{k}{k}, \\alpha\\binom{k}{k-1}, \\dots, \\alpha\\binom{k}{0}$, then $b = 2^x - 1$ for some natural number $x \\le n$, and $\\alpha = 2^i$ for some natural number $i$.","title":"Bases of binomial forms of powers of 2"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/succ-dvd-of-palindrome-even-length","kind":"node","lean":"MetaMathlibExt.succ_dvd_of_palindrome_even_length","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#succ-dvd-of-palindrome-even-length","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $b \\ge 2$ and let $N$ be a natural number whose base-$b$ digits form a palindrome with an even number of digits. Then $b + 1$ divides $N$.","title":"Even-length palindromes are divisible by the base plus one"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/base-eq-two-pow-sub-one-of-palindrome","kind":"node","lean":"MetaMathlibExt.base_eq_two_pow_sub_one_of_palindrome","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#base-eq-two-pow-sub-one-of-palindrome","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $n$ and $b \\ge 2$ be natural numbers. If the base-$b$ representation of $2^n$ is a palindrome with an even number of digits, then $b = 2^x - 1$ for some natural number $x$.","title":"Even-length palindromic powers of 2 have base one less than a power of 2"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/three-digit-palindrome-two-pow-iff","kind":"node","lean":"MetaMathlibExt.three_digit_palindrome_two_pow_iff","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#three-digit-palindrome-two-pow-iff","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $n$ be a natural number, let $b \\ge 2$ be a base, and let $c = 2^n \\bmod b$. The base-$b$ representation of $2^n$ is a palindrome with exactly three digits if and only if $b^2 + 1 \\le 2^n \\le b^3 - 1$, $c \\ne 0$, and $\\frac{1}{b} \\lfloor 2^n / b \\rfloor - \\frac{b - 1}{b} \\le c \\le \\frac{1}{b} \\lfloor 2^n / b \\rfloor$.","title":"Three-digit palindromic powers of two"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/three-digit-binomial-form-two-pow-iff","kind":"node","lean":"MetaMathlibExt.three_digit_binomialForm_two_pow_iff","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#three-digit-binomial-form-two-pow-iff","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $n$ and $\\alpha \\ge 1$ be natural numbers and let $b \\ge 2$ be a base. The base-$b$ representation of $2^n$ is $(\\alpha, 2\\alpha, \\alpha)_b$ if and only if there is a natural number $i$ with $3i + 2 < n$, $n \\equiv i \\pmod 2$, $\\alpha = 2^i$ and $b = 2^{(n - i)/2} - 1$.","title":"Three-digit binomial forms of powers of two"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/one-c-one-two-pow-iff","kind":"node","lean":"MetaMathlibExt.one_c_one_two_pow_iff","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#one-c-one-two-pow-iff","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $n$ be a natural number and $b \\ge 2$ a base. There is a digit $c \\ne 2$ such that the base-$b$ representation of $2^n$ is $(1, c, 1)_b$, that is, $2^n = 1 + cb + b^2$ with $0 \\le c < b$, if and only if (i) $2^n - 1 = kb$ for some natural number $k$ with $b \\le k \\le 2b - 1$, and (ii) it is not the case that $n$ is even and $b = 2^{n/2} - 1$.","title":"Three-digit representations of powers of two with outer digits one"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/palindrome-two-pow-base-two-pow-sub-one","kind":"node","lean":"MetaMathlibExt.palindrome_two_pow_base_two_pow_sub_one","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#palindrome-two-pow-base-two-pow-sub-one","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $x \\ge 2$ and $n \\ge 1$ be natural numbers, let $b = 2^x - 1$, and suppose that the base-$b$ representation $2^n = (c_k, \\dots, c_0)_b$ is a palindrome with $k + 1$ digits. Then (1) $n > k(x - 1)$; and (2) if $n = kx + r$ for a natural number $r$, and either $k \\le x - r$, or $3 \\le k \\le x - r + 1$ and $x \\ge 3$, then $c_i = 2^r \\binom{k}{i}$ for $i = 0, \\dots, k$, that is, the representation is the binomial\u2026","title":"Palindromic powers of two in a base one less than a power of two"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/exists-small-base-binomial-form-two-pow","kind":"node","lean":"MetaMathlibExt.exists_small_base_binomialForm_two_pow","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#exists-small-base-binomial-form-two-pow","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"For every natural number $n \\ge 2$ there are natural numbers $y$ and $b$ with $y \\le \\lceil \\sqrt{2n} \\rceil + 1$ and $2 \\le b \\le 2^y - 1$ such that $2^n$ has a base-$b$ representation in binomial form.","title":"Powers of 2 have binomial forms in small bases"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/base-eq-prime-pow-sub-one-of-palindrome","kind":"node","lean":"MetaMathlibExt.base_eq_prime_pow_sub_one_of_palindrome","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#base-eq-prime-pow-sub-one-of-palindrome","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $p$ be a prime and let $n$ and $b \\ge 2$ be natural numbers. If the base-$b$ representation of $p^n$ is a palindrome with an even number of digits, then $b = p^x - 1$ for some natural number $x$.","title":"Even-length palindromic prime powers have base one less than a power of the prime"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/digits-pow-and-pow-add-one","kind":"node","lean":"MetaMathlibExt.digits_pow_and_pow_add_one","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#digits-pow-and-pow-add-one","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $z \\ge 2$ and $n$ be natural numbers. Then the base-$z$ digits of $z^n$ are a $1$ followed by $n$ zeros, and, if $n \\ge 1$, the base-$z$ digits of $z^n + 1$ are a $1$, then $n - 1$ zeros, then a $1$.","title":"Digits of a power of the base and of its successor"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/nagell-ljunggren-solutions","kind":"node","lean":"MetaMathlibExt.nagell_ljunggren_solutions","parent":12240,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two/index.html#nagell-ljunggren-solutions","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/palindromic-representations-of-powers-of-two","status":"can_prove","summary":"Let $x, y, n, q$ be natural numbers with $x \\ge 2$, $y \\ge 2$, $n \\ge 3$ and $q \\ge 2$ such that $(x^n - 1)/(x - 1) = y^q$. If $q = 2$, or $3 \\mid n$, or $4 \\mid n$, or $q = 3$ and $n \\not\\equiv 5 \\pmod 6$, then $(x, y, n, q)$ is one of $(3, 11, 5, 2)$, $(7, 20, 4, 2)$ and $(18, 7, 3, 3)$; these are $11^2 = (1,1,1,1,1)_3$, $20^2 = (1,1,1,1)_7$ and $7^3 = (1,1,1)_{18}$.","title":"The Nagell\u2013Ljunggren equation in natural numbers (Bugeaud\u2013Mih\u0103ilescu)"},{"data_url":"scopes/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples.json","id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","kind":"scope","navigation":"module","parent":12221,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","title":"Permutiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-permutiple","kind":"node","lean":"MetaMathlibExt.IsPermutiple","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-permutiple","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$, $b$ and $d_0, \\dots, d_k$ be natural numbers and let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$. The numeral $(d_k, \\dots, d_0)_b$ is an $(n, b, \\sigma)$-permutiple when $d_j < b$ for every $j$ and","title":"Permutiples with a given digit permutation"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-carry","kind":"node","lean":"MetaMathlibExt.permutipleCarry","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-carry","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$, $b$ and $d_0, \\dots, d_k$ be natural numbers and let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$. The carries of the single-digit multiplication $n \\cdot (d_{\\sigma(k)}, \\dots, d_{\\sigma(0)})_b$, where $(e_k, \\dots, e_0)_b$ denotes the base-$b$ numeral $\\sum_{j=0}^{k} e_j b^j$, are the natural numbers $c_0, c_1, c_2, \\dots$ defined by $c_0 = 0$, $c_{j+1} = \\lfloor (n d_{\\sigma(j)} + c_j) / b \\rfloor$\u2026","title":"Carries of a permutiple multiplication"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-carry-zero-one","kind":"node","lean":"MetaMathlibExt.permutipleCarry_zero_one","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-carry-zero-one","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$, $b$ and $d_0, \\dots, d_k$ be natural numbers, let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$, and write $(e_k, \\dots, e_0)_b$ for the base-$b$ numeral $\\sum_{j=0}^{k} e_j b^j$. The carries $c_0, c_1, \\dots$ of the single-digit multiplication $n \\cdot (d_{\\sigma(k)}, \\dots, d_{\\sigma(0)})_b$, defined by $c_0 = 0$ and $c_{j+1} = \\lfloor (n d_{\\sigma(j)} + c_j) / b \\rfloor$ for $0 \\le j \\le k$, satisfy\u2026","title":"First two carries of a permutiple multiplication"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-permutiple-87912","kind":"node","lean":"MetaMathlibExt.isPermutiple_87912","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(87912\\)"],["text"," is a "],["math","\\((4, 10)\\)"],["text","-permutiple by digit reversal"]],"source":"$87912$ is a $(4, 10)$-permutiple by digit reversal"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-permutiple-87912","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"The numeral $(d_4, d_3, d_2, d_1, d_0)_{10} = 87912$ is a $(4, 10, \\sigma)$-permutiple, where $\\sigma$ is the reversal $j \\mapsto 4 - j$ of $\\{0, \\dots, 4\\}$; that is, $87912 = 4 \\cdot 21978$, and $21978$ has the digits of $87912$ in reverse order.","title":"$87912$ is a $(4, 10)$-permutiple by digit reversal"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-carry-relation","kind":"node","lean":"MetaMathlibExt.permutipleCarry_relation","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-carry-relation","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n$ and $b \\ge 2$ be natural numbers, let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$, and let $(d_k, \\dots, d_0)_b$ be an $(n, b, \\sigma)$-permutiple, where $(e_k, \\dots, e_0)_b$ denotes the base-$b$ numeral $\\sum_{j=0}^{k} e_j b^j$. Let $c_0, c_1, \\dots$ be the carries of the multiplication $n \\cdot (d_{\\sigma(k)}, \\dots, d_{\\sigma(0)})_b$. Then for every $0 \\le j \\le k$,","title":"Carry relation for permutiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-carry-le","kind":"node","lean":"MetaMathlibExt.permutipleCarry_le","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-carry-le","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n$ and $b \\ge 2$ be natural numbers, let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$, and let $(d_k, \\dots, d_0)_b$ be an $(n, b, \\sigma)$-permutiple, where $(e_k, \\dots, e_0)_b$ denotes the base-$b$ numeral $\\sum_{j=0}^{k} e_j b^j$. Let $c_0, c_1, \\dots$ be the carries of the multiplication $n \\cdot (d_{\\sigma(k)}, \\dots, d_{\\sigma(0)})_b$. Then for every $0 \\le j \\le k$ we have $c_j \\le n - 1$ if\u2026","title":"Carry bound for permutiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-base-perm-residue-le","kind":"node","lean":"MetaMathlibExt.permutiple_basePerm_residue_le","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-base-perm-residue-le","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n \\ge 1$ and $b \\ge 2$ be natural numbers, let $d_0, \\dots, d_k$ be natural numbers, and write $(e_k, \\dots, e_0)_b$ for the base-$b$ numeral $\\sum_{j=0}^{k} e_j b^j$. Let $\\sigma$, $\\pi$ and $\\tau$ be permutations of $\\{0, \\dots, k\\}$. If $(d_k, \\dots, d_0)_b$ is an $(n, b, \\sigma)$-permutiple and $(d_{\\pi(k)}, \\dots, d_{\\pi(0)})_b$ is an $(n, b, \\tau)$-permutiple, then for every $0 \\le j \\le k$","title":"Holt's residue condition for rearranged permutiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-graph-edges","kind":"node","lean":"MetaMathlibExt.permutipleGraphEdges","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-graph-edges","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $b$ be a base, let $d_0, \\dots, d_k$ be natural numbers, the digits of the base-$b$ numeral $(d_k, \\dots, d_0)_b = \\sum_{j=0}^{k} d_j b^j$, and let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$. The graph $G$ of these digits with digit permutation $\\sigma$ has edge set","title":"Edge set of the graph of a numeral"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-graph-edges-87912","kind":"node","lean":"MetaMathlibExt.permutipleGraphEdges_87912","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-graph-edges-87912","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $(d_0, d_1, d_2, d_3, d_4) = (2, 1, 9, 7, 8)$ be the decimal digits of $87912$, from the units digit up, and let $\\sigma$ be the reversal $\\sigma(j) = 4 - j$ of $\\{0, \\dots, 4\\}$. The edge set $\\{(d_j, d_{\\sigma(j)}) : 0 \\le j \\le 4\\}$ of the graph of these digits with digit permutation $\\sigma$ is exactly $\\{(2, 8), (1, 7), (9, 9), (7, 1), (8, 2)\\}$: the edges $8 \\rightleftarrows 2$ and $7 \\rightleftarrows 1$\u2026","title":"Graph of the permutiple 87912"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-mother-graph-edge","kind":"node","lean":"MetaMathlibExt.IsMotherGraphEdge","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-mother-graph-edge","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$ and $b$ be natural numbers. For natural numbers $x$ and $y$, the pair $(x, y)$ is an edge of the $(n, b)$-mother graph $M$ when $x < b$, $y < b$ and","title":"Edges of the mother graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-mother-graph-edge-two-four","kind":"node","lean":"MetaMathlibExt.isMotherGraphEdge_two_four","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["text","An edge and a non-edge of the "],["math","\\((2, 4)\\)"],["text","-mother graph"]],"source":"An edge and a non-edge of the $(2, 4)$-mother graph"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-mother-graph-edge-two-four","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"In the $(2, 4)$-mother graph, the pair $(0, 2)$ is an edge, since $\\lambda(0 + (4 - 2) \\cdot 2) = \\lambda(4) = 0 \\le 1$, and the pair $(0, 1)$ is not an edge, since $\\lambda(0 + (4 - 2) \\cdot 1) = \\lambda(2) = 2 > 1$, where $\\lambda$ is the least non-negative residue modulo $4$.","title":"An edge and a non-edge of the $(2, 4)$-mother graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/hoey-sloane-carry","kind":"node","lean":"MetaMathlibExt.hoeySloaneCarry","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#hoey-sloane-carry","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$ and $b$ be natural numbers and let $(d_0, \\hat d_0)(d_1, \\hat d_1) \\cdots (d_k, \\hat d_k)$ be an input string of pairs of natural numbers. The carries $c_0, c_1, c_2, \\dots$ of the $(n, b)$-Hoey\u2013Sloane machine on this string are the integers given by $c_0 = 0$,","title":"Carries of the Hoey\u2013Sloane machine"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/hoey-sloane-carry-two-four-example","kind":"node","lean":"MetaMathlibExt.hoeySloaneCarry_two_four_example","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["text","Carries of the "],["math","\\((2, 4)\\)"],["text","-Hoey\u2013Sloane machine on "],["math","\\((0,0)(2,3)(1,2)(3,1)\\)"]],"source":"Carries of the $(2, 4)$-Hoey\u2013Sloane machine on $(0,0)(2,3)(1,2)(3,1)$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#hoey-sloane-carry-two-four-example","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"On the input string $(d_0, \\hat d_0)(d_1, \\hat d_1)(d_2, \\hat d_2)(d_3, \\hat d_3) = (0,0)(2,3)(1,2)(3,1)$, the carries $c_0, c_1, c_2, c_3, c_4$ of the $(2, 4)$-Hoey\u2013Sloane machine are $0, 0, 1, 1, 0$.","title":"Carries of the $(2, 4)$-Hoey\u2013Sloane machine on $(0,0)(2,3)(1,2)(3,1)$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-hoey-sloane-word","kind":"node","lean":"MetaMathlibExt.IsHoeySloaneWord","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-hoey-sloane-word","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$ and $b$ be natural numbers. An input string $(d_0, \\hat d_0) \\cdots (d_k, \\hat d_k)$ of pairs of natural numbers belongs to the language $L$ of the $(n, b)$-Hoey\u2013Sloane machine when, with $c_0, \\dots, c_{k+1}$ the carries of the machine on this string, every input $(d_j, \\hat d_j)$ is an edge of the $(n, b)$-mother graph, $b$ divides $n \\hat d_j - d_j + c_j$ for every $0 \\le j \\le k$, and $c_{k+1} = 0$.","title":"Hoey\u2013Sloane words"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-hoey-sloane-word-two-four-example","kind":"node","lean":"MetaMathlibExt.isHoeySloaneWord_two_four_example","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["text","The string "],["math","\\((0,0)(2,3)(1,2)(3,1)\\)"],["text"," lies in the "],["math","\\((2, 4)\\)"],["text","-Hoey\u2013Sloane language"]],"source":"The string $(0,0)(2,3)(1,2)(3,1)$ lies in the $(2, 4)$-Hoey\u2013Sloane language"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-hoey-sloane-word-two-four-example","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"The input string $(d_0, \\hat d_0) \\cdots (d_3, \\hat d_3) = (0,0)(2,3)(1,2)(3,1)$ belongs to the language $L$ of the $(2, 4)$-Hoey\u2013Sloane machine.","title":"The string $(0,0)(2,3)(1,2)(3,1)$ lies in the $(2, 4)$-Hoey\u2013Sloane language"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-graph-edges-eq-of-conj","kind":"node","lean":"MetaMathlibExt.permutipleGraphEdges_eq_of_conj","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-graph-edges-eq-of-conj","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $d_0, \\dots, d_k$ be natural numbers and let $\\pi_1, \\pi_2, \\tau_1, \\tau_2$ be permutations of $\\{0, \\dots, k\\}$ with $\\pi_1 \\tau_1 \\pi_1^{-1} = \\pi_2 \\tau_2 \\pi_2^{-1}$. Then the graph of the digits $d_{\\pi_1(0)}, \\dots, d_{\\pi_1(k)}$ with digit permutation $\\tau_1$ and the graph of the digits $d_{\\pi_2(0)}, \\dots, d_{\\pi_2(k)}$ with digit permutation $\\tau_2$ have the same edge set:","title":"Conjugate rearrangements have the same graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/exists-cycle-preserving-permutiple","kind":"node","lean":"MetaMathlibExt.exists_cyclePreserving_permutiple","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#exists-cycle-preserving-permutiple","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n$, $b$ and $d_0, \\dots, d_k$ be natural numbers and let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$ such that $(d_k, \\dots, d_0)_b$ is an $(n, b, \\sigma)$-permutiple. Then there is a permutation $\\hat\\sigma$ of $\\{0, \\dots, k\\}$ such that $(d_k, \\dots, d_0)_b$ is also an $(n, b, \\hat\\sigma)$-permutiple and the digits along each cycle of $\\hat\\sigma$ are distinct: for all $j \\in \\{0, \\dots, k\\}$ and all\u2026","title":"Every permutiple has a cycle-preserving digit permutation"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-graph-edges-union-cycles","kind":"node","lean":"MetaMathlibExt.permutipleGraphEdges_union_cycles","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-graph-edges-union-cycles","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n \\ge 1$ and $b \\ge 2$ be natural numbers, let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$, and let $(d_k, \\dots, d_0)_b$ be an $(n, b, \\sigma)$-permutiple, where $(e_k, \\dots, e_0)_b$ denotes the base-$b$ numeral $\\sum_{j=0}^{k} e_j b^j$. Let $E = \\{(d_j, d_{\\sigma(j)}) : 0 \\le j \\le k\\}$ be the edge set of its graph. Then every pair in $E$ is an edge of the $(n, b)$-mother graph, and every $e \\in E$ lies\u2026","title":"Permutiple graphs are unions of simple cycles of the mother graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/hoey-sloane-word-inputs-union-cycles","kind":"node","lean":"MetaMathlibExt.hoeySloaneWord_inputs_union_cycles","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#hoey-sloane-word-inputs-union-cycles","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n \\ge 1$ and $b \\ge 2$ be natural numbers, and let $s = (d_0, \\hat d_0) \\cdots (d_k, \\hat d_k)$ be an input string in the language $L$ of the $(n, b)$-Hoey\u2013Sloane machine such that $\\hat d_j = d_{\\sigma(j)}$ for all $j$, for some permutation $\\sigma$ of $\\{0, \\dots, k\\}$. Let $E = \\{(d_j, \\hat d_j) : 0 \\le j \\le k\\}$ be the set of inputs of $s$. Then every $e \\in E$ lies on a directed cycle with edges in $E$:\u2026","title":"Hoey\u2013Sloane words with permuted digits are unions of cycles"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-mother-graph-edge-four-ten","kind":"node","lean":"MetaMathlibExt.isMotherGraphEdge_four_ten","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sample edges of the "],["math","\\((4, 10)\\)"],["text","-mother graph"]],"source":"Sample edges of the $(4, 10)$-mother graph"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-mother-graph-edge-four-ten","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"In the $(4, 10)$-mother graph, $(2, 8)$ and $(9, 9)$ are edges and $(2, 1)$ is not. With $\\lambda$ the least non-negative residue modulo ten, $\\lambda(2 + 6 \\cdot 8) = 0 \\le 3$ and $\\lambda(9 + 6 \\cdot 9) = 3 \\le 3$, while $\\lambda(2 + 6 \\cdot 1) = 8 > 3$.","title":"Sample edges of the $(4, 10)$-mother graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-hoey-sloane-step","kind":"node","lean":"MetaMathlibExt.IsHoeySloaneStep","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-hoey-sloane-step","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$, $b$, $c_1$, $c_2$, $d_1$ and $d_2$ be natural numbers. The input $e = (d_1, d_2)$ moves state $c_1$ to state $c_2$ in one transition of the $(n, b)$-Hoey-Sloane machine $\\Gamma$ when $c_1 < n$, $c_2 < n$, $(d_1, d_2)$ is an edge of the $(n, b)$-mother graph, and $b c_2 - c_1 = n d_2 - d_1$ in the integers (Equation (1)).","title":"Transitions of the Hoey-Sloane machine"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-hoey-sloane-run","kind":"node","lean":"MetaMathlibExt.IsHoeySloaneRun","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-hoey-sloane-run","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$, $b$, $c$ and $c'$ be natural numbers and let $s = e_1 e_2 \\cdots e_m$ be a finite string of inputs, each a pair of natural numbers. The $(n, b)$-Hoey-Sloane machine can go from state $c$ to state $c'$ reading $s$ from left to right when there are natural numbers $c_0, c_1, \\dots, c_m$ with $c_0 = c$ and $c_m = c'$ such that, for each $1 \\le i \\le m$, the input $e_i$ moves state $c_{i-1}$ to state $c_i$ in\u2026","title":"Runs of the Hoey-Sloane machine"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/hoey-sloane-language","kind":"node","lean":"MetaMathlibExt.hoeySloaneLanguage","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#hoey-sloane-language","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"For natural numbers $n$ and $b$, the language $L$ of the $(n, b)$-Hoey\u2013Sloane machine is the set of strings $s$ of pairs of natural numbers such that the machine, reading $s$ from left to right, can go from state $0$ to state $0$. In particular the empty string belongs to $L$.","title":"Language of the Hoey\u2013Sloane machine"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/mem-hoey-sloane-language-87912","kind":"node","lean":"MetaMathlibExt.mem_hoeySloaneLanguage_87912","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["text","A word in the "],["math","\\((4, 10)\\)"],["text","-Hoey-Sloane language"]],"source":"A word in the $(4, 10)$-Hoey-Sloane language"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#mem-hoey-sloane-language-87912","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"The input string $(2, 8)(1, 7)(9, 9)(7, 1)(8, 2)$ belongs to the language of the $(4, 10)$-Hoey-Sloane machine: reading it from left to right, the machine can go from state $0$ to state $0$.","title":"A word in the $(4, 10)$-Hoey-Sloane language"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-permutiple-string","kind":"node","lean":"MetaMathlibExt.IsPermutipleString","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-permutiple-string","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$ and $b$ be natural numbers, and for natural numbers $a_0, \\dots, a_k$ write $(a_k, \\dots, a_0)_b = \\sum_{j=0}^{k} a_j b^j$. A string $s$ of inputs (pairs of natural numbers) is an $(n, b)$-permutiple string when it belongs to the language $L$ of the $(n, b)$-Hoey-Sloane machine and there are a natural number $k$, natural numbers $d_0, \\dots, d_k$ and a permutation $\\sigma$ of $\\{0, \\dots, k\\}$ such that\u2026","title":"Permutiple strings"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-permutiple-string-87912","kind":"node","lean":"MetaMathlibExt.isPermutipleString_87912","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["text","The permutiple string of "],["math","\\(87912 = 4 \\cdot 21978\\)"]],"source":"The permutiple string of $87912 = 4 \\cdot 21978$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-permutiple-string-87912","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"The string of inputs $(2, 8)(1, 7)(9, 9)(7, 1)(8, 2)$ is a $(4, 10)$-permutiple string. It is produced by the $(4, 10)$-permutiple $87912 = 4 \\cdot 21978$, whose multiplication reverses the digits.","title":"The permutiple string of $87912 = 4 \\cdot 21978$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/cycle-edges","kind":"node","lean":"MetaMathlibExt.cycleEdges","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#cycle-edges","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"For a list $[v_0, \\dots, v_{r-1}]$ of natural numbers, read as the vertex list of a closed walk, the edge list is $[(v_0, v_1), \\dots, (v_{r-2}, v_{r-1}), (v_{r-1}, v_0)]$: each vertex is paired with the next one, and the last vertex with the first. A single vertex $[v_0]$ gives the loop $[(v_0, v_0)]$, and the empty list gives the empty edge list.","title":"Edge list of a closed walk"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/cycle-edges-examples","kind":"node","lean":"MetaMathlibExt.cycleEdges_examples","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edges of the closed walks "],["math","\\(2 \\to 8 \\to 2\\)"],["text"," and "],["math","\\(9 \\to 9\\)"]],"source":"Edges of the closed walks $2 \\to 8 \\to 2$ and $9 \\to 9$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#cycle-edges-examples","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"The closed walk with vertex list $[2, 8]$ has edge list $[(2, 8), (8, 2)]$, and the closed walk with vertex list $[9]$ (a loop at $9$) has edge list $[(9, 9)]$.","title":"Edges of the closed walks $2 \\to 8 \\to 2$ and $9 \\to 9$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-mother-graph-cycle","kind":"node","lean":"MetaMathlibExt.IsMotherGraphCycle","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-mother-graph-cycle","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$ and $b$ be natural numbers. A list $[v_0, \\dots, v_{r-1}]$ of natural numbers is a (directed) cycle of the $(n, b)$-mother graph when it is nonempty, has no repeated entry, and each of its edges $(v_0, v_1), \\dots, (v_{r-2}, v_{r-1}), (v_{r-1}, v_0)$ is an edge of the $(n, b)$-mother graph. In particular a one-element list $[v]$ is a cycle when $(v, v)$ is an edge of the mother graph.","title":"Cycles of the mother graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/cycle-union","kind":"node","lean":"MetaMathlibExt.cycleUnion","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#cycle-union","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"For a multiset $I$ of lists of natural numbers, each read as the vertex list of a closed walk, write $C_j$ for the edge list of $j \\in I$. The multiset union $C_I = \\biguplus_{j \\in I} C_j$ is the multiset of all these edges, counted with multiplicity: each list $j$ contributes its edges as often as $j$ occurs in $I$.","title":"Multiset union of cycle edges"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/hoey-sloane-transition","kind":"node","lean":"MetaMathlibExt.hoeySloaneTransition","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#hoey-sloane-transition","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"For natural numbers $n$ and $b$ and a pair $e = (d_1, d_2)$ of natural numbers, the multi-edge of the $(n, b)$-Hoey\u2013Sloane multigraph carrying the label $e$ is $(c_1, c_2)$, where $c_1 = \\lambda(d_1 + (b - n) d_2)$ with $\\lambda$ the least non-negative residue modulo $b$ (computed in the integers), and $c_2 = \\lfloor (n d_2 - d_1 + c_1) / b \\rfloor$, replaced by $0$ when negative. For $b = 0$ the conventions are\u2026","title":"Hoey\u2013Sloane transition of a digit pair"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/hoey-sloane-transition-four-ten","kind":"node","lean":"MetaMathlibExt.hoeySloaneTransition_four_ten","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two multi-edges of the "],["math","\\((4, 10)\\)"],["text","-Hoey\u2013Sloane multigraph"]],"source":"Two multi-edges of the $(4, 10)$-Hoey\u2013Sloane multigraph"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#hoey-sloane-transition-four-ten","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"In the $(4, 10)$-Hoey\u2013Sloane multigraph the label $(2, 8)$ sits on the multi-edge from state $0$ to state $3$, and the label $(1, 7)$ sits on the loop at state $3$: the Hoey\u2013Sloane transition sends $(2, 8)$ to $(0, 3)$ and $(1, 7)$ to $(3, 3)$.","title":"Two multi-edges of the $(4, 10)$-Hoey\u2013Sloane multigraph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/multi-image-union","kind":"node","lean":"MetaMathlibExt.multiImageUnion","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#multi-image-union","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $n$ and $b$ be natural numbers and let $I$ be a multiset of vertex lists. Let $C_I$ be the multiset cycle union of $I$: the multiset sum, over the lists in $I$, of their edges, where a nonempty list $[v_0, \\dots, v_{r-1}]$ has the edges $(v_0, v_1), \\dots, (v_{r-2}, v_{r-1}), (v_{r-1}, v_0)$ and the empty list has none. The multi-image union $\\Delta_I$ is the multiset obtained by replacing each edge\u2026","title":"Multi-image union of a multiset of cycles"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/has-eulerian-circuit-from-zero","kind":"node","lean":"MetaMathlibExt.HasEulerianCircuitFromZero","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#has-eulerian-circuit-from-zero","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Read a multiset $D$ of pairs of natural numbers as the multi-edges $(c_1, c_2)$ of a directed multigraph. Then $D$ has an Eulerian circuit beginning and ending with the zero state when there is a nonempty list of pairs whose multiset of entries is $D$ (so it uses every multi-edge exactly once), in which each multi-edge ends where the next one begins, the first one starts at $0$ and the last one ends at $0$.","title":"Eulerian circuit through the zero state"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-leulerian","kind":"node","lean":"MetaMathlibExt.IsLEulerian","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L\\)"],["text","-Eulerian multigraphs"]],"source":"$L$-Eulerian multigraphs"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-leulerian","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"fully_proved","summary":"Let $D$ be a finite multiset of multi-edges $(x, y)$ between natural numbers, and call a number a vertex of $D$ when it is an endpoint of some multi-edge in $D$. The multigraph $D$ is $L$-Eulerian when $0$ is a vertex of $D$, every vertex $u$ of $D$ reaches every vertex $v$ of $D$ along a (possibly empty) directed path of multi-edges in $D$, and for every natural number $c$ the number of multi-edges in $D$ ending\u2026","title":"$L$-Eulerian multigraphs"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-edge-residue-le","kind":"node","lean":"MetaMathlibExt.permutiple_edge_residue_le","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-edge-residue-le","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n \\ge 1$, $b$ and $k$ be natural numbers, let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$ and let $(d_k, \\dots, d_0)_b$ be an $(n, b, \\sigma)$-permutiple, that is, $d_i < b$ for all $i$ and $(d_k, \\dots, d_0)_b = n \\cdot (d_{\\sigma(k)}, \\dots, d_{\\sigma(0)})_b$. Then for every $0 \\le j \\le k$, $\\lambda(d_j + (b - n) d_{\\sigma(j)}) \\le n - 1$, where $\\lambda$ is the least non-negative residue modulo $b$ and\u2026","title":"Edge condition for permutiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/permutiple-graph-is-union-mother-graph-c","kind":"node","lean":"MetaMathlibExt.permutiple_graph_isUnion_motherGraphCycles","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#permutiple-graph-is-union-mother-graph-c","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n \\ge 1$, $b$ and $k$ be natural numbers, let $\\sigma$ be a permutation of $\\{0, \\dots, k\\}$ and let $(d_k, \\dots, d_0)_b$ be an $(n, b, \\sigma)$-permutiple, that is, $d_i < b$ for all $i$ and $(d_k, \\dots, d_0)_b = n \\cdot (d_{\\sigma(k)}, \\dots, d_{\\sigma(0)})_b$. Its graph has the edges $(d_i, d_{\\sigma(i)})$ for $0 \\le i \\le k$. Then for every $0 \\le j \\le k$, the pair $(d_j, d_{\\sigma(j)})$ is an edge of\u2026","title":"Edges of a permutiple's graph lie on cycles of the mother graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-hoey-sloane-step-unique","kind":"node","lean":"MetaMathlibExt.isHoeySloaneStep_unique","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-hoey-sloane-step-unique","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n \\le b$. If an input $e$ moves state $c_1$ to state $c_2$ and also moves state $\\hat c_1$ to state $\\hat c_2$, each in one transition of the $(n, b)$-Hoey-Sloane machine, then $c_1 = \\hat c_1$ and $c_2 = \\hat c_2$.","title":"Uniqueness of Hoey-Sloane transitions"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/exists-carries-of-is-mother-graph-edge","kind":"node","lean":"MetaMathlibExt.exists_carries_of_isMotherGraphEdge","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#exists-carries-of-is-mother-graph-edge","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $n$ and $b$ be natural numbers. If $(d_1, d_2)$ is an edge of the $(n, b)$-mother graph, then there are natural numbers $c_1$ and $c_2$ with $c_1 \\le n - 1$ and $c_2 \\le n - 1$ such that $b c_2 - c_1 = n d_2 - d_1$ as integers.","title":"Every mother-graph edge induces a carry transition"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-hoey-sloane-step-hoey-sloane-transiti","kind":"node","lean":"MetaMathlibExt.isHoeySloaneStep_hoeySloaneTransition","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-hoey-sloane-step-hoey-sloane-transiti","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $1 < n < b$, let $e = (d_1, d_2)$ be an edge of the $(n, b)$-mother graph, and let $(c_1, c_2)$ be the Hoey-Sloane transition of $e$, that is, $c_1 = \\lambda(d_1 + (b - n) d_2)$ and $c_2 = (n d_2 - d_1 + c_1) / b$, where $\\lambda$ is the least non-negative residue modulo $b$. Then the input $e$ moves state $c_1$ to state $c_2$ in one transition of the $(n, b)$-Hoey-Sloane machine.","title":"Hoey-Sloane transitions are transitions of the machine"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/is-permutiple-string-inputs-is-union-mot","kind":"node","lean":"MetaMathlibExt.isPermutipleString_inputs_isUnion_motherGraphCycles","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#is-permutiple-string-inputs-is-union-mot","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $1 < n < b$ and let $s$ be an $(n, b)$-permutiple string. Then every input $(d, \\hat d)$ of $s$ is an edge of the $(n, b)$-mother graph, and $\\hat d$ reaches $d$ along a (possibly empty) directed path in the graph whose edges are the inputs of $s$, so $(d, \\hat d)$ lies on a directed cycle of that graph. In this sense the inputs of $s$ form a union of cycles of the mother graph.","title":"Inputs of a permutiple string lie on mother-graph cycles"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/exists-is-permutiple-string-of-has-euler","kind":"node","lean":"MetaMathlibExt.exists_isPermutipleString_of_hasEulerianCircuitFromZero","parent":12259,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#exists-is-permutiple-string-of-has-euler","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $1 < n < b$ and let $I$ be a multiset of cycles of the $(n, b)$-mother graph, with multiset union of cycle edges $C_I$ and multi-image union $\\Delta_I$. If $\\Delta_I$ has an Eulerian circuit beginning and ending at the zero state, then there is an $(n, b)$-permutiple string $s$ whose multiset of entries is $C_I$.","title":"Eulerian circuits through zero yield permutiple strings"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/exists-is-permutiple-string-iff-is-leule","kind":"node","lean":"MetaMathlibExt.exists_isPermutipleString_iff_isLEulerian","parent":12259,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L\\)"],["text","-Eulerian criterion for permutiple strings"]],"source":"$L$-Eulerian criterion for permutiple strings"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples/index.html#exists-is-permutiple-string-iff-is-leule","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/permutiples","status":"can_prove","summary":"Let $1 < n < b$ and let $I$ be a multiset of cycles of the $(n, b)$-mother graph, with multiset union of cycle edges $C_I$ and multi-image union $\\Delta_I$. Then $\\Delta_I$ is $L$-Eulerian if and only if there is an $(n, b)$-permutiple string $s$ whose multiset of entries is $C_I$.","title":"$L$-Eulerian criterion for permutiple strings"},{"data_url":"scopes/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples.json","id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","kind":"scope","navigation":"module","parent":12221,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_state","title":"Reverse multiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/sloane-reverse-multiple-endpoint","kind":"node","lean":"MetaMathlibExt.sloane_reverse_multiple_endpoint","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#sloane-reverse-multiple-endpoint","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $g$ and $k$ be integers with $2 \\le k < g$, and let $N = (a_n a_{n-1} \\cdots a_0)_g$ have base-$g$ digits $a_0, \\dots, a_n \\in \\{0, \\dots, g - 1\\}$, with units digit $a_0$ and leading digit $a_n \\ne 0$. Suppose $N$ is a $(g, k)$ reverse multiple, that is, $k N = (a_0 a_1 \\cdots a_n)_g$, and that $a_0 + a_n = g$. Then $(k + 1) a_n = g$; in particular $k + 1$ divides $g$ and $a_n = g / (k + 1)$.","title":"Sloane's endpoint formula for reverse multiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/sutcliffe-two-to-three-digit-reverse-mul","kind":"node","lean":"MetaMathlibExt.sutcliffe_two_to_three_digit_reverse_multiple","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#sutcliffe-two-to-three-digit-reverse-mul","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $g$ and $k$ be integers with $2 \\le k < g$, and let $a$ and $b$ be base-$g$ digits, $0 \\le a, b < g$, with $a \\ne 0$, such that $(a\\,b)_g$ is a $(g, k)$ reverse multiple, that is, $k (a g + b) = b g + a$. Then $a + b < g$, so $a + b$ is a base-$g$ digit, and $(a\\,(a + b)\\,b)_g$ is again a $(g, k)$ reverse multiple, that is, $k (a g^2 + (a + b) g + b) = b g^2 + (a + b) g + a$.","title":"Sutcliffe's theorem on two- and three-digit reverse multiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/is-young-retained","kind":"node","lean":"MetaMathlibExt.isYoungRetained","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#is-young-retained","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $V$ be a set, let $P$ be a predicate on $V$ (the pivot nodes) and let $\\prec$ be a binary relation on $V$. A node $v \\in V$ is retained with respect to $P$ and $\\prec$ if $P(v)$ holds or there is a node $w \\in V$ with $P(w)$ and $v \\prec w$.","title":"Retained node"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/is-young-retained-iff","kind":"node","lean":"MetaMathlibExt.isYoungRetained_iff","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#is-young-retained-iff","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $V$ be a set, let $P$ be a predicate on $V$ and let $\\prec$ be a binary relation on $V$. A node $v \\in V$ is retained with respect to $P$ and $\\prec$ if and only if $P(v)$ holds or there is a node $w \\in V$ with $P(w)$ and $v \\prec w$.","title":"Characterization of retained nodes"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/young-graph","kind":"node","lean":"MetaMathlibExt.youngGraph","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#young-graph","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $G$ be a labeled directed graph with vertex set $V$, edge set $E$ and label set $L$, let $P$ be a predicate on $V$ and let $\\prec$ be a binary relation on $V$. The Young graph of $G$ with respect to $P$ and $\\prec$ is the labeled directed graph with label set $L$ whose vertices are the vertices of $G$ that are retained with respect to $P$ and $\\prec$, whose edges are the edges of $G$ whose source and target are\u2026","title":"Young graph of a labeled directed graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/young-graph-src","kind":"node","lean":"MetaMathlibExt.youngGraph_src","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#young-graph-src","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $G$ be a labeled directed graph, let $P$ be a predicate on its vertices, let $\\prec$ be a binary relation on its vertices, and let $e$ be an edge of $G$ whose source and target are both retained with respect to $P$ and $\\prec$, viewed as an edge of the Young graph of $G$ with respect to $P$ and $\\prec$. The source of $e$ in the Young graph, viewed as a vertex of $G$, is the source of $e$ in $G$.","title":"Source of an edge in the Young graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/young-graph-tgt","kind":"node","lean":"MetaMathlibExt.youngGraph_tgt","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#young-graph-tgt","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $G$ be a labeled directed graph, let $P$ be a predicate on its vertices, let $\\prec$ be a binary relation on its vertices, and let $e$ be an edge of $G$ whose source and target are both retained with respect to $P$ and $\\prec$, viewed as an edge of the Young graph of $G$ with respect to $P$ and $\\prec$. The target of $e$ in the Young graph, viewed as a vertex of $G$, is the target of $e$ in $G$.","title":"Target of an edge in the Young graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/young-graph-label","kind":"node","lean":"MetaMathlibExt.youngGraph_label","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#young-graph-label","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $G$ be a labeled directed graph, let $P$ be a predicate on its vertices, let $\\prec$ be a binary relation on its vertices, and let $e$ be an edge of $G$ whose source and target are both retained with respect to $P$ and $\\prec$, viewed as an edge of the Young graph of $G$ with respect to $P$ and $\\prec$. The label of $e$ in the Young graph is its label in $G$.","title":"Label of an edge in the Young graph"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/young-reverse-multiple-carry-path-iff-ge","kind":"node","lean":"MetaMathlibExt.young_reverse_multiple_carry_path_iff_general","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#young-reverse-multiple-carry-path-iff-ge","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $g, k, n$ be natural numbers with $2 \\le k < g$, and let $a_0, \\dots, a_{n-1}$ be natural numbers with $a_i < g$ for every $i$. Then $k \\sum_{i\\lt n} a_i g^i = \\sum_{i\\lt n} a_{n-1-i} g^i$ if and only if there is a unique sequence of natural numbers $r_0, \\dots, r_n$ with $r_0 = 0$, $r_n = 0$ and, for every $i < n$, $r_{i+1} < k$ and $k a_i + r_i = a_{n-1-i} + r_{i+1} g$.","title":"Carry equations for reverse multiples, general form"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/young-reverse-multiple-carry-path-iff","kind":"node","lean":"MetaMathlibExt.young_reverse_multiple_carry_path_iff","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#young-reverse-multiple-carry-path-iff","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $g, k, n$ be natural numbers with $2 \\le k < g$ and $n \\ge 1$, and let $a_0, \\dots, a_{n-1}$ be natural numbers with $a_i < g$ for every $i$, $a_{n-1} \\ne 0$ and $a_0 \\ne 0$. Then $k \\sum_{i\\lt n} a_i g^i = \\sum_{i\\lt n} a_{n-1-i} g^i$, that is, $k$ times the number with base-$g$ digits $a_{n-1}, \\dots, a_0$ (most significant first) equals the number with these digits reversed, if and only if there is a unique\u2026","title":"Carry equations for reverse multiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/young-reverse-multiples-equiv-pivot-path","kind":"node","lean":"MetaMathlibExt.young_reverse_multiples_equiv_pivot_paths","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#young-reverse-multiples-equiv-pivot-path","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Let $g$ and $k$ be natural numbers with $2 \\le k < g$. A $(g, k)$ reverse multiple is a list of base-$g$ digits $d_1, d_2, \\dots, d_n$, most significant first, with $n \\ge 2$, $d_1 \\ne 0$, $d_n \\ne 0$ and $d_i < g$ for every $i$, such that $k$ times the number $\\sum_{i=1}^{n} d_i g^{n-i}$ equals the number $\\sum_{i=1}^{n} d_{n+1-i} g^{n-i}$ whose digits are those of the first in reverse order. Consider the graph\u2026","title":"Young's theorem on reverse multiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/is-reverse-divisor","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.IsReverseDivisor","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#is-reverse-divisor","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"A natural number $n$ is a reverse divisor if $n > 0$ and there is a natural number $k \\ge 2$ such that the decimal reversal of $n$ equals $kn$. Since $n > 0$, such a $k$ is unique; it is called the quotient of $n$.","title":"Reverse divisor"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/is-reverse-divisor-iff","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.isReverseDivisor_iff","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#is-reverse-divisor-iff","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"A natural number $n$ is a reverse divisor if and only if $n > 0$ and there is a natural number $k \\ge 2$ such that the decimal reversal of $n$ equals $kn$.","title":"Characterization of reverse divisors"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/has-property-pstar","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.HasPropertyPStar","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Property "],["math","\\(P^*\\)"]],"source":"Property $P^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#has-property-pstar","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Property $P^*$. A natural number $n$ has property $P^*$ if $n$ divides the decimal reversal of every multiple of $n$.","title":"Property $P^*$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/has-property-pstar-iff","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.hasPropertyPStar_iff","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unfolding property "],["math","\\(P^*\\)"]],"source":"Unfolding property $P^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#has-property-pstar-iff","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"Unfolding property $P^*$. A natural number $n$ has property $P^*$ if and only if $n$ divides the decimal reversal of every multiple of $n$.","title":"Unfolding property $P^*$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/has-property-pstar-iff-dvd","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.hasPropertyPStar_iff_dvd","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","The positive integers with property "],["math","\\(P^*\\)"],["text"," are the divisors of "],["math","\\(99\\)"]],"source":"The positive integers with property $P^*$ are the divisors of $99$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#has-property-pstar-iff-dvd","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"The positive integers with property $P^*$ are the divisors of $99$. A natural number $n \\ge 1$ has property $P^*$ if and only if $n$ divides $99$.","title":"The positive integers with property $P^*$ are the divisors of $99$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/has-property-pstar-of-dvd","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.hasPropertyPStar_of_dvd","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisors of "],["math","\\(99\\)"],["text"," have property "],["math","\\(P^*\\)"]],"source":"Divisors of $99$ have property $P^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#has-property-pstar-of-dvd","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Divisors of $99$ have property $P^*$. Every divisor of $99$ has property $P^*$.","title":"Divisors of $99$ have property $P^*$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/not-has-property-pstar-of-not-coprime","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.not_hasPropertyPStar_of_not_coprime","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positive integers sharing a factor with "],["math","\\(10\\)"],["text"," lack property "],["math","\\(P^*\\)"]],"source":"Positive integers sharing a factor with $10$ lack property $P^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#not-has-property-pstar-of-not-coprime","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Positive integers sharing a factor with $10$ lack property $P^*$. If $n \\ge 1$ is a natural number with $\\gcd(n, 10) \\ne 1$, then $n$ does not have the property $P^*$.","title":"Positive integers sharing a factor with $10$ lack property $P^*$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/palindrome-has-property-pstar","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.palindrome_hasPropertyPStar","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Palindromes with property "],["math","\\(P^*\\)"]],"source":"Palindromes with property $P^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#palindrome-has-property-pstar","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Palindromes with property $P^*$. Let $n \\ge 1$ be a natural number that is a decimal palindrome, that is, whose decimal reversal equals $n$. If $n$ has the property $P^*$, then $n$ is one of $1$, $3$, $9$, $11$, $33$ and $99$.","title":"Palindromes with property $P^*$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/reverse-digits-eq-four-or-nine-mul","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.reverseDigits_eq_four_or_nine_mul","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reverse divisors have quotient "],["math","\\(4\\)"],["text"," or "],["math","\\(9\\)"]],"source":"Reverse divisors have quotient $4$ or $9$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#reverse-digits-eq-four-or-nine-mul","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"If $n$ is a reverse divisor, then the decimal reversal of $n$ equals $4n$ or $9n$.","title":"Reverse divisors have quotient $4$ or $9$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/basic-reverse-divisors","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.basic_reverseDivisors","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#basic-reverse-divisors","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"For every natural number $k \\ge 2$, the numbers $11(10^k - 1)$ and $22(10^k - 1)$ both have exactly $k + 2$ decimal digits, the decimal reversal of $11(10^k - 1)$ equals $9 \\cdot 11(10^k - 1)$, and the decimal reversal of $22(10^k - 1)$ equals $4 \\cdot 22(10^k - 1)$.","title":"Basic reverse divisors"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/four-le-length-digits","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.four_le_length_digits","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#four-le-length-digits","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Every reverse divisor has at least four decimal digits.","title":"Reverse divisors have at least four digits"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/eq-basic-of-length-le-seven","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.eq_basic_of_length_le_seven","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#eq-basic-of-length-le-seven","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Let $n$ be a reverse divisor with $L$ decimal digits, where $L \\le 7$. Then $n = 11(10^{L-2} - 1)$ or $n = 22(10^{L-2} - 1)$, where $L - 2$ is read as $0$ when $L < 2$.","title":"Reverse divisors with at most seven digits"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/not-has-property-pstar-of-quotient-four","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.not_hasPropertyPStar_of_quotient_four","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reverse divisors with quotient "],["math","\\(4\\)"],["text"," lack property "],["math","\\(P^*\\)"]],"source":"Reverse divisors with quotient $4$ lack property $P^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#not-has-property-pstar-of-quotient-four","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Reverse divisors with quotient $4$ lack property $P^*$. If $n > 0$ is a natural number whose decimal reversal equals $4n$, then $n$ does not have the property $P^*$.","title":"Reverse divisors with quotient $4$ lack property $P^*$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/not-has-property-pstar-basic-nine","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.not_hasPropertyPStar_basic_nine","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","For "],["math","\\(k \\ge 2\\)"],["text",", the number "],["math","\\(11 \\cdot (10^k - 1)\\)"],["text"," lacks property "],["math","\\(P^*\\)"]],"source":"For $k \\ge 2$, the number $11 \\cdot (10^k - 1)$ lacks property $P^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#not-has-property-pstar-basic-nine","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"For $k \\ge 2$, the number $11 \\cdot (10^k - 1)$ lacks property $P^*$. For every natural number $k \\ge 2$, the number $11 \\cdot (10^k - 1)$ does not have the property $P^*$.","title":"For $k \\ge 2$, the number $11 \\cdot (10^k - 1)$ lacks property $P^*$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/not-has-property-pstar-of-is-reverse-div","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.not_hasPropertyPStar_of_isReverseDivisor","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","No reverse divisor has property "],["math","\\(P^*\\)"]],"source":"No reverse divisor has property $P^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#not-has-property-pstar-of-is-reverse-div","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"No reverse divisor has property $P^*$. If $n$ is a reverse divisor, then $n$ does not have the property $P^*$.","title":"No reverse divisor has property $P^*$"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/mul-nines-mod-eq-eleven","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.mul_nines_modEq_eleven","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","A product congruent to "],["math","\\(11\\)"],["text"," modulo a power of ten"]],"source":"A product congruent to $11$ modulo a power of ten"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#mul-nines-mod-eq-eleven","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"fully_proved","summary":"For every natural number $r$, let $a = (10^r - 1) \\cdot 100 + 89$, the integer whose decimal digits are $r$ nines followed by $89$, and let $b = 10^{r+2} - 1$, the integer whose decimal digits are $r + 2$ nines. Then the remainder of $ab$ modulo $10^{r+2}$ is $11$; that is, the last $r + 2$ decimal digits of $ab$ are $r$ zeros followed by $11$.","title":"A product congruent to $11$ modulo a power of ten"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/dvd-reverse-digits-of-dvd","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.dvd_reverseDigits_of_dvd","parent":12303,"reading_title":{"format":"inline-parts-v1","parts":[["text","In base "],["math","\\(B\\)"],["text",", divisors of "],["math","\\(B^2 - 1\\)"],["text"," divide reversals of their multiples"]],"source":"In base $B$, divisors of $B^2 - 1$ divide reversals of their multiples"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#dvd-reverse-digits-of-dvd","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Let $B \\ge 2$ be a natural number and let $n$ be a natural number dividing $B^2 - 1$. Then $n$ divides the base-$B$ reversal of every multiple of $n$.","title":"In base $B$, divisors of $B^2 - 1$ divide reversals of their multiples"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/infinite-dvd-reverse-digits-mul","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.infinite_dvd_reverseDigits_mul","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#infinite-dvd-reverse-digits-mul","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Let $n$ be a natural number not divisible by $10$, and suppose that $n$ divides the decimal reversal of $k_0 n$ for some natural number $k_0 \\ge 1$. Then there are infinitely many natural numbers $k \\ge 1$ such that $n$ divides the decimal reversal of $k n$.","title":"Infinitely many multiples with divisible reversal"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/exists-dvd-reverse-digits-mul","kind":"node","lean":"MathlibExt.NumberTheory.Digits.ReverseDivisorWanted.exists_dvd_reverseDigits_mul","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#exists-dvd-reverse-digits-mul","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_prove","summary":"Let $n$ be a natural number such that $\\gcd(n, 10) = 1$, or $n$ equals its decimal reversal, or $n$ is a reverse divisor. Then there is a natural number $k_0 \\ge 1$ such that $n$ divides the decimal reversal of $k_0 n$.","title":"Sufficient conditions for a multiple with divisible reversal"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/is-complete-young-graph","kind":"node","lean":"MetaMathlibExt.IsCompleteYoungGraph","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#is-complete-young-graph","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"can_state","summary":"For natural numbers $g$, $k$ and $m$, the Young graph $Y(g, k)$ of the $k$-reverse multiples in base $g$ is complete on $m$ nodes: the nodes other than its distinguished starting node form the complete directed graph on $m$ nodes, so $Y(g, k)$ has $m + 1$ nodes in all; the node $[0, 0]$ is one of these $m$ nodes, and the starting node has an edge to every node except $[0, 0]$.","title":"Complete Young graphs"},{"id":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/sloane-complete-young-graph-characteriza","kind":"node","lean":"MetaMathlibExt.sloane_complete_young_graph_characterization","parent":12303,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples/index.html#sloane-complete-young-graph-characteriza","scope":"NumberTheory/digits-numeration/repdigits-reverse-multiples/reverse-multiples","status":"stated","summary":"Let $g$, $k$ and $m$ be natural numbers with $2 \\le k < g$ and $m \\ge 2$. Then the Young graph $Y(g, k)$ is complete on $m$ nodes if and only if","title":"Characterization of complete Young graphs"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic.json","id":"NumberTheory/digits-numeration/numeration-automatic","kind":"scope","parent":11985,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic","status":"can_prove","summary":"Zeckendorf, base-\u03c6, negative and signed bases, Cobham's theorem, base-recognizable sets, Mahler substitution, regular sequences.","title":"Numeration systems and automatic sequences"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets.json","id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","title":"Cobham's theorems on recognizable sets"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/base-word-value","kind":"node","lean":"MetaMathlibExt.baseWordValue","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#base-word-value","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"The value of a word $w = d_1 d_2 \\cdots d_m$ of base-$k$ digits $d_i \\in \\{0, \\dots, k - 1\\}$, read with the most significant digit first, is $\\sum_{i=1}^{m} d_i k^{m-i}$. The empty word has value $0$.","title":"Value of a base-k digit word"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/is-canonical-base-word","kind":"node","lean":"MetaMathlibExt.IsCanonicalBaseWord","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#is-canonical-base-word","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"A word of base-$k$ digits is canonical when it is empty or its first (most significant) digit is nonzero.","title":"Canonical base-k words"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/is-base-recognizable","kind":"node","lean":"MetaMathlibExt.IsBaseRecognizable","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#is-base-recognizable","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"For a natural number $k$, a set $S$ of natural numbers is recognizable in base $k$ when $k \\ge 2$ and some deterministic finite automaton with a nonempty finite set of states, reading words of base-$k$ digits from the most significant digit, accepts a word $w = d_1 d_2 \\cdots d_m$ exactly when $w$ is canonical (empty, or with $d_1 \\neq 0$) and its value $\\sum_{i=1}^{m} d_i k^{m-i}$ lies in $S$.","title":"Sets recognizable in base k"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/base-digits","kind":"node","lean":"MetaMathlibExt.baseDigits","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#base-digits","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"For a natural number $k \\ge 2$ and a natural number $n$, the base-$k$ expansion of $n$ is the word $d_1 d_2 \\cdots d_m$ of base-$k$ digits, most significant digit first, with $n = \\sum_{i=1}^{m} d_i k^{m-i}$ and $d_1 \\neq 0$; for $n = 0$ it is the empty word.","title":"Base-k expansion"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/bases-multiplicatively-independent","kind":"node","lean":"MetaMathlibExt.BasesMultiplicativelyIndependent","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#bases-multiplicatively-independent","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"Two natural numbers $k$ and $l$ are multiplicatively independent when $k^m \\ne l^n$ for all positive natural numbers $m$ and $n$.","title":"Multiplicatively independent bases"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/is-ultimately-periodic-set","kind":"node","lean":"MetaMathlibExt.IsUltimatelyPeriodicSet","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#is-ultimately-periodic-set","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"A set $S$ of natural numbers is ultimately periodic when there are natural numbers $N$ and $p > 0$ such that, for every $n \\ge N$, $n \\in S$ if and only if $n + p \\in S$.","title":"Ultimately periodic sets of natural numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/cobham-base-dependence","kind":"node","lean":"MetaMathlibExt.cobham_base_dependence","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#cobham-base-dependence","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"Let $k$ and $l$ be multiplicatively independent natural numbers with $l \\ge 2$, and let $S$ be a set of natural numbers that is recognizable in base $k$ and is not ultimately periodic. Then $S$ is not recognizable in base $l$.","title":"Cobham's theorem"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/is-ultimately-periodic-set-iff-is-ultima","kind":"node","lean":"MetaMathlibExt.isUltimatelyPeriodicSet_iff_isUltimatelyPeriodic","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#is-ultimately-periodic-set-iff-is-ultima","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"A set $S$ of natural numbers is ultimately periodic if and only if its characteristic sequence $n \\mapsto (n \\in S)$, a sequence of truth values, is ultimately periodic.","title":"Ultimately periodic sets and their characteristic sequences"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/cobham-base-dependence-is-ultimately-per","kind":"node","lean":"MetaMathlibExt.cobham_base_dependence_isUltimatelyPeriodic","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#cobham-base-dependence-is-ultimately-per","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"Let $k$ and $l$ be multiplicatively independent natural numbers with $l \\ge 2$, and let $S$ be a set of natural numbers that is recognizable in base $k$ and whose characteristic sequence $n \\mapsto (n \\in S)$, a sequence of truth values, is not ultimately periodic. Then $S$ is not recognizable in base $l$.","title":"Cobham's theorem for characteristic sequences"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/cobham-gap-theorem-general","kind":"node","lean":"MetaMathlibExt.cobham_gap_theorem_general","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#cobham-gap-theorem-general","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"Let $k$ be a natural number and let $x_0 < x_1 < x_2 < \\cdots$ be a strictly increasing sequence of natural numbers whose set of values is recognizable in base $k$. Then either the limit superior, as $i \\to \\infty$, of the ratios $x_{i+1} / x_i$ computed in the extended nonnegative reals $[0, \\infty]$ (where a ratio with denominator $0$ is $\\infty$) is greater than $1$, or there is a natural number $C$ with\u2026","title":"Cobham's gap theorem without a separate base hypothesis"},{"id":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/cobham-gap-theorem","kind":"node","lean":"MetaMathlibExt.cobham_gap_theorem","parent":12337,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets/index.html#cobham-gap-theorem","scope":"NumberTheory/digits-numeration/numeration-automatic/cobham-s-theorems-on-recognizable-sets","status":"fully_proved","summary":"Let $k \\ge 2$ be a natural number and let $x_0 < x_1 < x_2 < \\cdots$ be a strictly increasing sequence of natural numbers whose set of values is recognizable in base $k$. Then either the limit superior, as $i \\to \\infty$, of the ratios $x_{i+1} / x_i$ computed in the extended nonnegative reals $[0, \\infty]$ (so an unbounded limit superior is $\\infty$, and a ratio with denominator $0$ is $\\infty$) is greater than\u2026","title":"Cobham's gap theorem"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials.json","id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","title":"Constant terms of powers of Laurent polynomials"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/laurent-cartier","kind":"node","lean":"MetaMathlibExt.laurentCartier","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cartier operator "],["math","\\(\\Lambda_k\\)"],["text"," on Laurent polynomials"]],"source":"Cartier operator $\\Lambda_k$ on Laurent polynomials"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#laurent-cartier","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"Let $R$ be a semiring and $f$ a Laurent polynomial in $r$ variables with coefficients in $R$. For a natural number $k > 0$, $\\Lambda_k f$ keeps the terms of $f$ whose exponents are all multiples of $k$ and substitutes $x_s^k \\mapsto x_s$ for each variable, so that the coefficient of $x^j$ in $\\Lambda_k f$ is the coefficient of $x^{kj}$ in $f$ for every $j \\in \\mathbb{Z}^r$. For $k = 0$ it is set to\u2026","title":"Cartier operator $\\Lambda_k$ on Laurent polynomials"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/coeff-laurent-cartier","kind":"node","lean":"MetaMathlibExt.coeff_laurentCartier","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of Kohen's operator "],["math","\\(\\Lambda_k\\)"]],"source":"Coefficients of Kohen's operator $\\Lambda_k$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#coeff-laurent-cartier","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"Let $R$ be a semiring and let $r$ and $k \\ne 0$ be natural numbers. For every Laurent polynomial $f$ in $r$ variables with coefficients in $R$ and every $j \\in \\mathbb{Z}^r$, the coefficient of $x^j$ in $\\Lambda_k f$ is the coefficient of $x^{kj}$ in $f$.","title":"Coefficients of Kohen's operator $\\Lambda_k$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/laurent-deg","kind":"node","lean":"MetaMathlibExt.laurentDeg","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#laurent-deg","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"Let $R$ be a semiring and let $f$ be a Laurent polynomial in $r$ variables $x_1, \\dots, x_r$ with coefficients in $R$. Kohen's degree $\\deg f$ of $f$ is the largest absolute value $|j_i|$ of a single exponent, taken over all $i$ and all exponent vectors $j \\in \\mathbb{Z}^r$ whose monomial $x^j$ has nonzero coefficient in $f$, and is $0$ if there is no such exponent (in particular for the zero polynomial). For\u2026","title":"Degree of a Laurent polynomial"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/laurent-deg-le-iff","kind":"node","lean":"MetaMathlibExt.laurentDeg_le_iff","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#laurent-deg-le-iff","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"Let $R$ be a semiring, let $r$ and $m$ be natural numbers, and let $f$ be a Laurent polynomial in $r$ variables $x_1, \\dots, x_r$ with coefficients in $R$. Then Kohen's degree of $f$ (the largest absolute value of a single exponent occurring in $f$, or $0$ if there is none) is at most $m$ if and only if every exponent vector $j \\in \\mathbb{Z}^r$ whose monomial $x^j$ has nonzero coefficient in $f$ satisfies\u2026","title":"Exponent criterion for the degree of a Laurent polynomial"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/seq-factor-occurs-at","kind":"node","lean":"MetaMathlibExt.SeqFactorOccursAt","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#seq-factor-occurs-at","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"Let $s = (s_n)_{n \\ge 0}$ be a sequence and let $i$, $\\ell$ and $i'$ be natural numbers. The factor of $s$ of length $\\ell$ starting at position $i$ occurs at position $i'$ when $s_{i'+t} = s_{i+t}$ for every natural number $t < \\ell$.","title":"Occurrence of a sequence factor at a position"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-recurrent-seq","kind":"node","lean":"MetaMathlibExt.IsRecurrentSeq","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-recurrent-seq","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"A sequence $s : \\mathbb{N} \\to \\alpha$ is recurrent if every occurrence of a factor is followed by a later occurrence: for all $i, \\ell \\in \\mathbb{N}$ there is $j > 0$ such that $s(i + j + t) = s(i + t)$ for every $t < \\ell$.","title":"Recurrent sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-uniformly-recurrent-seq","kind":"node","lean":"MetaMathlibExt.IsUniformlyRecurrentSeq","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-uniformly-recurrent-seq","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"A sequence $s : \\mathbb{N} \\to \\alpha$ is uniformly recurrent if for every factor $w$ there is a constant $C_w$ such that every occurrence of $w$ is followed by another occurrence at distance at most $C_w$: for all $i, \\ell \\in \\mathbb{N}$ there is $C \\in \\mathbb{N}$ such that for every $i'$ with $s(i' + t) = s(i + t)$ for all $t < \\ell$, there is $j$ with $0 < j \\le C$ and $s(i' + j + t) = s(i + t)$ for all\u2026","title":"Uniformly recurrent sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-linearly-recurrent-seq","kind":"node","lean":"MetaMathlibExt.IsLinearlyRecurrentSeq","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-linearly-recurrent-seq","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"A sequence $s = (s_n)_{n \\in \\mathbb{N}}$ with values in a set $\\alpha$ is linearly recurrent if there is $K \\in \\mathbb{N}$ such that for all $i, i' \\in \\mathbb{N}$ and $\\ell \\ge 1$ with $s_{i'+t} = s_{i+t}$ for all $t < \\ell$, there is $j$ with $0 < j \\le K \\ell$ and $s_{i'+j+t} = s_{i+t}$ for all $t < \\ell$. That is, each occurrence of a nonempty factor of length $\\ell$ is followed by another occurrence starting\u2026","title":"Linearly recurrent sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-uniformly-recurrent-seq-2","kind":"node","lean":"MetaMathlibExt.IsLinearlyRecurrentSeq.isUniformlyRecurrentSeq","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-uniformly-recurrent-seq-2","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"Every linearly recurrent sequence $s : \\mathbb{N} \\to \\alpha$, with values in any type $\\alpha$, is uniformly recurrent.","title":"Linearly recurrent sequences are uniformly recurrent"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-recurrent-seq-2","kind":"node","lean":"MetaMathlibExt.IsUniformlyRecurrentSeq.isRecurrentSeq","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-recurrent-seq-2","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"Every uniformly recurrent sequence $s : \\mathbb{N} \\to \\alpha$, with values in any type $\\alpha$, is recurrent.","title":"Uniformly recurrent sequences are recurrent"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-linearly-recurrent-seq-const","kind":"node","lean":"MetaMathlibExt.isLinearlyRecurrentSeq_const","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-linearly-recurrent-seq-const","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"fully_proved","summary":"For every set $\\alpha$ and every $c \\in \\alpha$, the constant sequence $n \\mapsto c$ on $\\mathbb{N}$ is linearly recurrent.","title":"Constant sequences are linearly recurrent"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/constant-term-pow-mul-mod-eq-laurent-car","kind":"node","lean":"MetaMathlibExt.constantTerm_pow_mul_modEq_laurentCartier","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#constant-term-pow-mul-mod-eq-laurent-car","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime and let $P$ and $Q$ be Laurent polynomials in $r$ variables with integer coefficients. Write $\\operatorname{ct}[F]$ for the constant term of a Laurent polynomial $F$ and $\\Lambda_p$ for Kohen's operator, which keeps the terms whose exponents are all divisible by $p$ and replaces $x^{pj}$ by $x^j$. Then for all natural numbers $n$ and $k$,\u2026","title":"Rowland\u2013Zeilberger congruence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/laurent-deg-foldl-laurent-cartier-le","kind":"node","lean":"MetaMathlibExt.laurentDeg_foldl_laurentCartier_le","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#laurent-deg-foldl-laurent-cartier-le","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime and let $P$ and $Q$ be Laurent polynomials in $r$ variables with integer coefficients. Write $\\deg F$ for the largest absolute value of an exponent of a variable occurring in $F$, and $\\Lambda_p$ for the operator that keeps the terms whose exponents are all multiples of $p$ and substitutes $x_s^p \\mapsto x_s$. For every finite list $k_1, \\dots, k_n$ of natural numbers with each $k_t < p$,\u2026","title":"Degree bound for the Rowland\u2013Zeilberger states"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/laurent-cartier-pow-mul-single-eq-sum","kind":"node","lean":"MetaMathlibExt.laurentCartier_pow_mul_single_eq_sum","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#laurent-cartier-pow-mul-single-eq-sum","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime and $P$ a Laurent polynomial in $r$ variables with integer coefficients. Write $\\mathrm{coef}_e[F]$ for the coefficient of $x^e$ in a Laurent polynomial $F$, $\\deg F$ for the largest absolute value of an exponent of a variable occurring in $F$, and $\\Lambda_p$ for the Cartier operator with $\\mathrm{coef}_j[\\Lambda_p F] = \\mathrm{coef}_{pj}[F]$. Let $m \\in \\mathbb{N}$ with $\\deg P \\le m + 1$, put\u2026","title":"Cartier operator applied to a power times a monomial"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/coeff-pow-mod-eq-sum-gamma-mul-v","kind":"node","lean":"MetaMathlibExt.coeff_pow_modEq_sum_gamma_mul_V","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kohen's linear representation step modulo "],["math","\\(p\\)"]],"source":"Kohen's linear representation step modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#coeff-pow-mod-eq-sum-gamma-mul-v","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $P$ be a Laurent polynomial in $r$ variables with integer coefficients, and write $\\deg P$ for the largest absolute value of an exponent of a single variable in a monomial occurring in $P$ ($0$ if $P = 0$). Let $m$ be a natural number with $\\deg P \\le m + 1$. Write $T = [-m, m]^r \\subseteq \\mathbb{Z}^r$ and $[x^e]F$ for the coefficient of $x^e$ in a Laurent polynomial $F$. Then for every\u2026","title":"Kohen's linear representation step modulo $p$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/exists-gamma-zero-pow-eq-single","kind":"node","lean":"MetaMathlibExt.exists_gammaZero_pow_eq_single","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of "],["math","\\(\\gamma(0)\\)"],["text"," modulo "],["math","\\(p\\)"],["text"," stabilize at a matrix unit"]],"source":"Powers of $\\gamma(0)$ modulo $p$ stabilize at a matrix unit"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#exists-gamma-zero-pow-eq-single","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $r, m \\ge 0$ be natural numbers, let $T = \\{-m, \\dots, m\\}^r \\subseteq \\mathbb{Z}^r$, and let $\\gamma(0)$ be the $T \\times T$ matrix over $\\mathbb{Z}/p\\mathbb{Z}$ whose $(i, j)$ entry is $1$ if $i = p j$ and $0$ otherwise. Then there is $s \\in \\mathbb{N}$ such that for every $s' \\ge s$, $\\gamma(0)^{s'}$ is the matrix whose $(0, 0)$ entry is $1$ and whose other entries are $0$.","title":"Powers of $\\gamma(0)$ modulo $p$ stabilize at a matrix unit"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/exists-bound-constant-term-pow-dvd","kind":"node","lean":"MetaMathlibExt.exists_bound_constantTerm_pow_dvd","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#exists-bound-constant-term-pow-dvd","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $r$ and $d$ be natural numbers, and write $\\operatorname{ct}[F]$ for the constant term of a Laurent polynomial $F$ and $\\deg F$ for the largest absolute value of an exponent of a single variable in a monomial occurring in $F$ ($0$ if $F = 0$). There is a natural number $B$ such that for every Laurent polynomial $P$ in $r$ variables with integer coefficients and $\\deg P \\le d$, if $p$ divides\u2026","title":"Divisibility of constant terms of powers is decided on a bounded prefix"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/constant-term-zero-density-mod-prime","kind":"node","lean":"MetaMathlibExt.constantTerm_zero_density_mod_prime","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Density-one zeros of constant terms modulo "],["math","\\(p\\)"]],"source":"Density-one zeros of constant terms modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#constant-term-zero-density-mod-prime","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $P$ be a Laurent polynomial in $r$ variables with integer coefficients, write $\\operatorname{ct}[F]$ for the constant term of a Laurent polynomial $F$, and suppose that $p$ divides $\\operatorname{ct}[P^n]$ for some natural number $n$. Then for every Laurent polynomial $Q$ in $r$ variables with integer coefficients, the set of natural numbers $n$ with $p \\mid \\operatorname{ct}[P^n Q]$ has\u2026","title":"Density-one zeros of constant terms modulo $p$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/exists-sigma-pow-v-contains-v-zero","kind":"node","lean":"MetaMathlibExt.exists_sigma_pow_V_contains_V_zero","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Return to the initial coefficient vector modulo "],["math","\\(p\\)"]],"source":"Return to the initial coefficient vector modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#exists-sigma-pow-v-contains-v-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $m \\in \\mathbb{N}$, and let $P$ be a Laurent polynomial with integer coefficients in variables $x_1, \\dots, x_r$ with $\\deg P \\le m + 1$, where $\\deg P$ is the largest absolute value of an exponent of a single variable in a monomial occurring in $P$ (and $0$ if $P = 0$). Let $T = \\{-m, \\dots, m\\}^r$. Then there is $t_0 \\in \\mathbb{N}$ such that for every $n \\in \\mathbb{N}$ for which $p$ does\u2026","title":"Return to the initial coefficient vector modulo $p$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-linearly-recurrent-seq-constant-term","kind":"node","lean":"MetaMathlibExt.isLinearlyRecurrentSeq_constantTerm_mod_prime_iff","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear recurrence of constant terms modulo "],["math","\\(p\\)"]],"source":"Linear recurrence of constant terms modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-linearly-recurrent-seq-constant-term","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime and let $P, Q$ be Laurent polynomials with integer coefficients in $r$ variables; write $\\operatorname{ct}[f]$ for the constant term of $f$. Suppose $p \\nmid \\operatorname{ct}[P^n Q]$ for some $n \\in \\mathbb{N}$. Then the sequence $u_n = \\operatorname{ct}[P^n Q] \\bmod p$ is linearly recurrent if and only if $p \\nmid \\operatorname{ct}[P^n]$ for all $n \\in \\mathbb{N}$, and if $u$ is not linearly\u2026","title":"Linear recurrence of constant terms modulo $p$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-linearly-recurrent-seq-combination-mo","kind":"node","lean":"MetaMathlibExt.isLinearlyRecurrentSeq_combination_mod_prime_iff","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear recurrence of combinations of constant terms modulo "],["math","\\(p\\)"]],"source":"Linear recurrence of combinations of constant terms modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-linearly-recurrent-seq-combination-mo","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $P, Q_1, \\dots, Q_\\ell$ be Laurent polynomials with integer coefficients in $r$ variables, let $k_1, \\dots, k_\\ell \\in \\mathbb{N}$, and let $F \\colon (\\mathbb{Z}/p\\mathbb{Z})^\\ell \\to X$ be a map to a set $X$. Write $\\operatorname{ct}[f]$ for the constant term of $f$ and, for $n \\in \\mathbb{N}$, let\u2026","title":"Linear recurrence of combinations of constant terms modulo $p$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-recurrent-seq-constant-term-mod-prime","kind":"node","lean":"MetaMathlibExt.isRecurrentSeq_constantTerm_mod_prime","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-recurrent-seq-constant-term-mod-prime","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime and let $P$ and $Q$ be Laurent polynomials in $r$ variables with integer coefficients; write $\\mathrm{ct}[F]$ for the constant term of a Laurent polynomial $F$. If $p \\nmid \\mathrm{ct}[P^{n_0}]$ for some $n_0 > 0$, then the sequence of residues $(\\mathrm{ct}[P^n Q] \\bmod p)_{n \\ge 0}$ in $\\mathbb{Z}/p\\mathbb{Z}$ is recurrent.","title":"Recurrence of constant-term sequences modulo a prime"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/exists-constant-term-pow-dvd-iff-tilde","kind":"node","lean":"MetaMathlibExt.exists_constantTerm_pow_dvd_iff_tilde","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility of constant terms of powers of "],["math","\\(P\\)"],["text"," and "],["math","\\(\\tilde P\\)"]],"source":"Divisibility of constant terms of powers of $P$ and $\\tilde P$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#exists-constant-term-pow-dvd-iff-tilde","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a \\ge 1$ and $\\ell$ be natural numbers, and let $P$ and $\\tilde P$ be Laurent polynomials in $r$ variables with integer coefficients such that $P^{p^{a-1}} \\equiv \\tilde P(x^{p^\\ell}) \\pmod{p^a}$ coefficientwise, where $\\tilde P(x^q)$ is obtained from $\\tilde P$ by replacing each monomial $x^e$ with $x^{qe}$. Write $\\operatorname{ct}[F]$ for the constant term of a Laurent polynomial $F$.\u2026","title":"Divisibility of constant terms of powers of $P$ and $\\tilde P$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/constant-term-pow-mul-mod-eq-tilde","kind":"node","lean":"MetaMathlibExt.constantTerm_pow_mul_modEq_tilde","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Passing from "],["math","\\(P\\)"],["text"," to "],["math","\\(\\tilde P\\)"],["text"," modulo "],["math","\\(p^a\\)"]],"source":"Passing from $P$ to $\\tilde P$ modulo $p^a$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#constant-term-pow-mul-mod-eq-tilde","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a \\ge 1$ and $\\ell$ be natural numbers, and let $P$ and $\\tilde P$ be Laurent polynomials in $r$ variables with integer coefficients such that $P^{p^{a-1}} \\equiv \\tilde P(x^{p^\\ell}) \\pmod{p^a}$ coefficientwise, where $\\tilde P(x^q)$ is obtained from $\\tilde P$ by replacing each monomial $x^e$ with $x^{qe}$. Write $\\operatorname{ct}[F]$ for the constant term of a Laurent polynomial $F$ and\u2026","title":"Passing from $P$ to $\\tilde P$ modulo $p^a$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/constant-term-tilde-pow-mul-mod-eq-laure","kind":"node","lean":"MetaMathlibExt.constantTerm_tilde_pow_mul_modEq_laurentCartier","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rowland\u2013Zeilberger congruence modulo "],["math","\\(p^a\\)"]],"source":"Rowland\u2013Zeilberger congruence modulo $p^a$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#constant-term-tilde-pow-mul-mod-eq-laure","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a$ be a natural number, and let $\\tilde P$ be a Laurent polynomial in $r$ variables with integer coefficients such that $\\tilde P^p \\equiv \\tilde P(x^p) \\pmod{p^a}$ coefficientwise, where $\\tilde P(x^p)$ is obtained from $\\tilde P$ by replacing each monomial $x^e$ with $x^{pe}$. Write $\\operatorname{ct}[F]$ for the constant term of a Laurent polynomial $F$ and $\\Lambda_p$ for Kohen's\u2026","title":"Rowland\u2013Zeilberger congruence modulo $p^a$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/laurent-deg-foldl-laurent-cartier-le-zmo","kind":"node","lean":"MetaMathlibExt.laurentDeg_foldl_laurentCartier_le_zmod","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Degree bound for the Rowland\u2013Zeilberger states mod "],["math","\\(p^a\\)"]],"source":"Degree bound for the Rowland\u2013Zeilberger states mod $p^a$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#laurent-deg-foldl-laurent-cartier-le-zmo","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a$ be a natural number, and let $\\tilde P$ and $Q_0$ be Laurent polynomials in $r$ variables with coefficients in $\\mathbb{Z}/p^a\\mathbb{Z}$. Write $\\deg F$ for the largest absolute value of an exponent of a variable occurring in $F$, and $\\Lambda_p$ for the operator that keeps the terms whose exponents are all multiples of $p$ and substitutes $x_s^p \\mapsto x_s$. For every finite list\u2026","title":"Degree bound for the Rowland\u2013Zeilberger states mod $p^a$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/coeff-tilde-pow-mod-eq-sum-gamma-mul-v","kind":"node","lean":"MetaMathlibExt.coeff_tilde_pow_modEq_sum_gamma_mul_V","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kohen's linear representation step modulo "],["math","\\(p^a\\)"]],"source":"Kohen's linear representation step modulo $p^a$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#coeff-tilde-pow-mod-eq-sum-gamma-mul-v","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a$ be a natural number, and let $\\tilde P$ be a Laurent polynomial in $r$ variables with integer coefficients such that $\\tilde P^p \\equiv \\tilde P(x^p) \\pmod{p^a}$ coefficientwise, where $\\tilde P(x^p)$ is obtained from $\\tilde P$ by replacing each monomial $x^e$ with $x^{pe}$. Write $\\deg \\tilde P$ for the largest absolute value of an exponent of a single variable in a monomial occurring\u2026","title":"Kohen's linear representation step modulo $p^a$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/exists-gamma-zero-pow-eq-single-zmod","kind":"node","lean":"MetaMathlibExt.exists_gammaZero_pow_eq_single_zmod","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of "],["math","\\(\\tilde\\gamma(0)\\)"],["text"," modulo "],["math","\\(p^a\\)"],["text"," stabilize at a matrix unit"]],"source":"Powers of $\\tilde\\gamma(0)$ modulo $p^a$ stabilize at a matrix unit"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#exists-gamma-zero-pow-eq-single-zmod","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a \\ge 1$, $r$ and $m$ be natural numbers, let $T = \\{-m, \\dots, m\\}^r \\subseteq \\mathbb{Z}^r$, and let $\\tilde\\gamma(0)$ be the $T \\times T$ matrix over $\\mathbb{Z}/p^a\\mathbb{Z}$ whose $(i, j)$ entry is $1$ if $i = p j$ and $0$ otherwise. Then there is $s \\in \\mathbb{N}$ such that for every $s' \\ge s$, $\\tilde\\gamma(0)^{s'}$ is the matrix whose $(0, 0)$ entry is $1$ and whose other entries\u2026","title":"Powers of $\\tilde\\gamma(0)$ modulo $p^a$ stabilize at a matrix unit"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/constant-term-tilde-zero-density","kind":"node","lean":"MetaMathlibExt.constantTerm_tilde_zero_density","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Density-one zeros of constant terms modulo "],["math","\\(p^a\\)"]],"source":"Density-one zeros of constant terms modulo $p^a$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#constant-term-tilde-zero-density","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a$ be a natural number, and let $\\tilde P$ be a Laurent polynomial in $r$ variables with integer coefficients such that $\\tilde P^p \\equiv \\tilde P(x^p) \\pmod{p^a}$ coefficientwise, where $\\tilde P(x^p)$ is obtained from $\\tilde P$ by replacing each monomial $x^e$ with $x^{pe}$. Write $\\operatorname{ct}[F]$ for the constant term of a Laurent polynomial $F$, and suppose that $p$ divides\u2026","title":"Density-one zeros of constant terms modulo $p^a$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/exists-sigma-pow-v-contains-v-zero-tilde","kind":"node","lean":"MetaMathlibExt.exists_sigma_pow_V_contains_V_zero_tilde","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Return to the initial coefficient vector modulo "],["math","\\(p^a\\)"]],"source":"Return to the initial coefficient vector modulo $p^a$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#exists-sigma-pow-v-contains-v-zero-tilde","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a, m \\in \\mathbb{N}$, and let $\\tilde P$ be a Laurent polynomial with integer coefficients in variables $x_1, \\dots, x_r$ such that for every $j \\in \\mathbb{Z}^r$ the coefficients of $x^j$ in $\\tilde P(x_1, \\dots, x_r)^p$ and in $\\tilde P(x_1^p, \\dots, x_r^p)$ are congruent modulo $p^a$. Suppose $\\deg \\tilde P \\le m + 1$, where $\\deg \\tilde P$ is the largest absolute value of an exponent of\u2026","title":"Return to the initial coefficient vector modulo $p^a$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-linearly-recurrent-seq-constant-term-2","kind":"node","lean":"MetaMathlibExt.isLinearlyRecurrentSeq_constantTerm_mod_primePow_iff","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear recurrence of constant terms modulo "],["math","\\(p^a\\)"]],"source":"Linear recurrence of constant terms modulo $p^a$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-linearly-recurrent-seq-constant-term-2","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a \\ge 1$, and let $P, Q$ be Laurent polynomials with integer coefficients in $r$ variables; write $\\operatorname{ct}[f]$ for the constant term of $f$. Suppose $p^a \\nmid \\operatorname{ct}[P^n Q]$ for some $n \\in \\mathbb{N}$. Then the sequence $u_n = \\operatorname{ct}[P^n Q] \\bmod p^a$ is linearly recurrent if and only if $p \\nmid \\operatorname{ct}[P^n]$ for all $n \\in \\mathbb{N}$, and if\u2026","title":"Linear recurrence of constant terms modulo $p^a$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-linearly-recurrent-seq-combination-mo-2","kind":"node","lean":"MetaMathlibExt.isLinearlyRecurrentSeq_combination_mod_primePow_iff","parent":12349,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear recurrence of combinations of constant terms modulo "],["math","\\(p^a\\)"]],"source":"Linear recurrence of combinations of constant terms modulo $p^a$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-linearly-recurrent-seq-combination-mo-2","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a \\ge 1$, let $P, Q_1, \\dots, Q_\\ell$ be Laurent polynomials with integer coefficients in $r$ variables, let $k_1, \\dots, k_\\ell \\in \\mathbb{N}$, and let $F \\colon (\\mathbb{Z}/p^a\\mathbb{Z})^\\ell \\to X$ be a map to a set $X$. Write $\\operatorname{ct}[f]$ for the constant term of $f$ and, for $n \\in \\mathbb{N}$, let\u2026","title":"Linear recurrence of combinations of constant terms modulo $p^a$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/is-recurrent-seq-constant-term-mod-prime-2","kind":"node","lean":"MetaMathlibExt.isRecurrentSeq_constantTerm_mod_primePow","parent":12349,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials/index.html#is-recurrent-seq-constant-term-mod-prime-2","scope":"NumberTheory/digits-numeration/numeration-automatic/constant-terms-of-powers-of-laurent-polynomials","status":"can_prove","summary":"Let $p$ be a prime, let $a \\ge 1$, and let $P$ and $Q$ be Laurent polynomials in $r$ variables with integer coefficients; write $\\mathrm{ct}[F]$ for the constant term of a Laurent polynomial $F$. If $p \\nmid \\mathrm{ct}[P^{n_0}]$ for some $n_0 > 0$, then the sequence of residues $(\\mathrm{ct}[P^n Q] \\bmod p^a)_{n \\ge 0}$ in $\\mathbb{Z}/p^a\\mathbb{Z}$ is recurrent.","title":"Recurrence of constant-term sequences modulo a prime power"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions.json","id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","title":"Distribution of Zeckendorf expressions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/bijective-finset-sum-iff-eq-two-pow","kind":"node","lean":"MetaMathlibExt.bijective_finset_sum_iff_eq_two_pow","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#bijective-finset-sum-iff-eq-two-pow","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $H \\colon \\mathbb{N} \\to \\mathbb{N}$ be a strictly increasing sequence of positive integers. Then every natural number (including $0$) equals $\\sum_{k \\in s} H_k$ for exactly one finite set $s \\subseteq \\mathbb{N}$ if and only if $H_k = 2^k$ for all $k$.","title":"Powers of two are the only binary fundamental sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/bijective-nonadjacent-finset-sum-iff-eq","kind":"node","lean":"MetaMathlibExt.bijective_nonadjacent_finset_sum_iff_eq_fib","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#bijective-nonadjacent-finset-sum-iff-eq","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $H \\colon \\mathbb{N} \\to \\mathbb{N}$ be a strictly increasing sequence of positive integers, and call a finite set $s \\subseteq \\mathbb{N}$ nonadjacent if $k \\in s$ implies $k + 1 \\notin s$. Then every natural number (including $0$) equals $\\sum_{k \\in s} H_k$ for exactly one nonadjacent finite set $s$ if and only if $H_k = F_{k+2}$ for all $k$, where $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ are the\u2026","title":"The Fibonacci numbers are the only Zeckendorf fundamental sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/card-zeckendorf-binary-lt-sum-two-pow","kind":"node","lean":"MetaMathlibExt.card_zeckendorfBinary_lt_sum_two_pow","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#card-zeckendorf-binary-lt-sum-two-pow","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"For $x \\in \\mathbb{N}$ let $z(x)$ be the number of $n < x$ whose binary expansion has no two adjacent digits equal to $1$. For a finite set $A \\subseteq \\mathbb{N}$, put $\\bar A = A$ if there is no $j$ with $j, j + 1 \\in A$, and otherwise $\\bar A = \\{k \\in A : k \\ge j\\}$, where $j$ is the largest index with $j, j + 1 \\in A$. Then $z\\bigl(\\sum_{k \\in A} 2^k\\bigr) = \\sum_{k \\in \\bar A} F_{k+2}$, where $F_0 = 0$,\u2026","title":"Duality formula for binary expansions without adjacent ones"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/asc-lex-lt","kind":"node","lean":"MetaMathlibExt.AscLexLT","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#asc-lex-lt","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For coefficient functions $\\mu, \\nu \\colon \\mathbb{N} \\to \\mathbb{N}$ with finite support, $\\mu \\lt _a \\nu$ means that there is an index $k$ with $\\mu_k < \\nu_k$ and $\\mu_j = \\nu_j$ for all $j > k$: at the largest index where they differ, $\\mu$ is smaller.","title":"Ascending lexicographic order"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/asc-lex-lt-example","kind":"node","lean":"MetaMathlibExt.ascLexLT_example","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#asc-lex-lt-example","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Write $(a_0, a_1, a_2, a_3, a_4)$ for the coefficient function with finite support whose entry at index $i \\le 4$ is $a_i$ and whose other entries are $0$. Then $(1, 2, 10, 3, 7) \\lt _a (1, 3, 1, 4, 7)$ in the ascending lexicographic order: the largest index where they differ is $3$, where $3 < 4$.","title":"Example of the ascending lexicographic order"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-immediate-predecessor","kind":"node","lean":"MetaMathlibExt.IsImmediatePredecessor","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-immediate-predecessor","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $E$ be a collection of finitely supported coefficient functions $\\mathbb N \\to \\mathbb N$, and let $\\lt _a$ be the ascending lexicographic order. A finitely supported coefficient function $\\mu$ is the immediate predecessor of $\\nu$ in $E$ if $\\mu \\in E$, $\\mu \\lt _a \\nu$, and every $\\xi \\in E$ with $\\xi \\lt _a \\nu$ satisfies $\\xi = \\mu$ or $\\xi \\lt _a \\mu$. That is, $\\mu$ is the largest element of $E$ that is\u2026","title":"Immediate predecessor"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-immediate-successor","kind":"node","lean":"MetaMathlibExt.IsImmediateSuccessor","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-immediate-successor","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $E$ be a collection of finitely supported coefficient functions $\\mathbb N \\to \\mathbb N$, and let $\\lt _a$ be the ascending lexicographic order. A finitely supported coefficient function $\\nu$ is the immediate successor of $\\mu$ in $E$ if $\\nu \\in E$, $\\mu \\lt _a \\nu$, and every $\\xi \\in E$ with $\\mu \\lt _a \\xi$ satisfies $\\xi = \\nu$ or $\\nu \\lt _a \\xi$. That is, $\\nu$ is the smallest element of $E$ that is\u2026","title":"Immediate successor"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-zeckendorf-collection","kind":"node","lean":"MetaMathlibExt.IsZeckendorfCollection","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-zeckendorf-collection","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Index coefficient functions from $1$, let $\\beta^n$ be the basis coefficient function with entry $1$ at index $n$, and for a coefficient function $\\mu$ let $\\operatorname{res}_n(\\mu)$ keep the entries of $\\mu$ at indices $\\le n$ and $\\operatorname{res}^n(\\mu)$ keep those at indices $\\ge n$, the other entries being $0$. A set $E$ of finitely supported coefficient functions $\\mathbb N \\to \\mathbb N$ is a Zeckendorf\u2026","title":"Zeckendorf collections"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zero-mem","kind":"node","lean":"MetaMathlibExt.IsZeckendorfCollection.zero_mem","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zero-mem","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Every Zeckendorf collection $E$ of finitely supported coefficient functions $\\mathbb N \\to \\mathbb N$ contains the zero coefficient function.","title":"Zeckendorf collections contain zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/periodic-theta","kind":"node","lean":"MetaMathlibExt.periodicTheta","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The maximal "],["math","\\(L\\)"],["text","-block "],["math","\\(\\theta^{m+1}\\)"]],"source":"The maximal $L$-block $\\theta^{m+1}$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#periodic-theta","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L = (e_1, \\dots, e_N)$ be a list of natural numbers and $m \\in \\mathbb N$, with indices of coefficient functions starting at $0$. The maximal $L$-block $\\theta^{m+1}$ is the finitely supported coefficient function whose entry at each index $i < m$ is $e_j$, where $j - 1 = (m - 1 - i) \\bmod N$, and whose entries at the indices $i \\ge m$ are $0$ (every entry is $0$ when $L$ is empty).","title":"The maximal $L$-block $\\theta^{m+1}$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/periodic-theta-one-zero-three","kind":"node","lean":"MetaMathlibExt.periodicTheta_one_zero_three","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The maximal block "],["math","\\(\\theta^4\\)"],["text"," in the Fibonacci case"]],"source":"The maximal block $\\theta^4$ in the Fibonacci case"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#periodic-theta-one-zero-three","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For $L = (1, 0)$, the maximal $L$-block $\\theta^4$ is $(1, 0, 1)$: its entries at indices $0$, $1$, $2$ are $1$, $0$, $1$, and all its entries from index $3$ on are $0$.","title":"The maximal block $\\theta^4$ in the Fibonacci case"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-lblock","kind":"node","lean":"MetaMathlibExt.IsLBlock","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-lblock","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L$ be a list of natural numbers, and for each $m$ let $\\theta^{m+1}$ be the maximal $L$-block, which vanishes from index $m$ on (indices are zero-based). A finitely supported coefficient function $\\zeta$ is an $L$-block with support interval $[s, m - 1]$ if $1 \\le m$, $s < m$, and either $s = 0$ and $\\zeta = \\theta^{m+1}$, or $\\zeta_i = \\theta^{m+1}_i$ for every $i > s$, $\\zeta_s < \\theta^{m+1}_s$ and\u2026","title":"L-blocks"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/periodic-zeckendorf-collection","kind":"node","lean":"MetaMathlibExt.periodicZeckendorfCollection","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#periodic-zeckendorf-collection","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For a list $L$ of natural numbers, the periodic Zeckendorf collection determined by $L$ is the set of finitely supported coefficient functions $\\mu$ that can be written as a sum $\\mu = \\zeta_1 + \\dots + \\zeta_r$ of finitely many $L$-blocks $\\zeta_1, \\dots, \\zeta_r$ (possibly none, which gives $\\mu = 0$) whose support intervals are pairwise disjoint.","title":"Periodic Zeckendorf collection"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-eval","kind":"node","lean":"MetaMathlibExt.zeckendorfEval","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-eval","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For a sequence $H$ of natural numbers and a finitely supported coefficient function $\\varepsilon \\colon \\mathbb{N} \\to \\mathbb{N}$, the evaluation of $\\varepsilon$ at $H$ is the natural number $\\mathrm{eval}_H(\\varepsilon) = \\sum_{k \\ge 0} \\varepsilon_k H_k$.","title":"Evaluation of a coefficient function at an integer sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/exists-unique-zeckendorf-collection","kind":"node","lean":"MetaMathlibExt.existsUnique_zeckendorfCollection","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#exists-unique-zeckendorf-collection","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Index coefficient functions from $0$, and for $m \\in \\mathbb N$ let $\\beta^{(m)}$ be the coefficient function with entry $1$ at index $m$ and $0$ elsewhere. Suppose that for each $m \\ge 1$ a finitely supported coefficient function $\\theta^{(m)}$ is given whose entry at index $m - 1$ is nonzero and whose entries at all indices $j \\ge m$ are $0$. Then there is a unique Zeckendorf collection $E$ such that, for every\u2026","title":"Zeckendorf collections are determined by their immediate predecessors"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-zeckendorf-collection-periodic","kind":"node","lean":"MetaMathlibExt.isZeckendorfCollection_periodic","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-zeckendorf-collection-periodic","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L = (e_1, \\dots, e_N)$ be a list of natural numbers with $N \\ge 2$ and $e_1 \\ne 0$, and let $E$ be the periodic Zeckendorf collection determined by $L$ (the sums of finitely many $L$-blocks with pairwise disjoint support intervals). Index coefficient functions from $1$ and let $\\beta^n$ be the basis coefficient function with entry $1$ at index $n$. Then $E$ is a Zeckendorf collection, and for every $n \\ge 2$\u2026","title":"Periodic collections are Zeckendorf collections"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/exists-unique-zeckendorf-fundamental-seq","kind":"node","lean":"MetaMathlibExt.existsUnique_zeckendorf_fundamentalSequence","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#exists-unique-zeckendorf-fundamental-seq","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $E$ be a Zeckendorf collection and index coefficient functions and sequences from $0$. Suppose that, for each $m \\ge 1$, $\\theta^{(m)}$ is the immediate predecessor in $E$ of the coefficient function with entry $1$ at index $m$ and $0$ elsewhere. For a sequence $H$ write $\\mathrm{eval}_H(\\mu) = \\sum_k \\mu_k H_k$. Then there is a unique strictly increasing sequence $H$ of positive integers such that\u2026","title":"Fundamental sequences of Zeckendorf collections"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-collection-subset-iff","kind":"node","lean":"MetaMathlibExt.zeckendorfCollection_subset_iff","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-collection-subset-iff","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $E$ and $\\tilde E$ be Zeckendorf collections. For $n \\ge 2$ let $\\beta^n$ be the basis coefficient function with entry $1$ at position $n$ (positions counted from $1$) and $0$ elsewhere, and let $\\hat\\beta^n$ be the immediate predecessor of $\\beta^n$ in $E$ (the largest element of $E$ that is $\\lt _a \\beta^n$ in the ascending lexicographic order). Then $E \\subseteq \\tilde E$ if and only if\u2026","title":"Criterion for one Zeckendorf collection to contain another"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/subcollection-count","kind":"node","lean":"MetaMathlibExt.subcollectionCount","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#subcollection-count","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For a set $E$ of finitely supported coefficient functions $\\mu : \\mathbb{N} \\to \\mathbb{N}$, a sequence $H : \\mathbb{N} \\to \\mathbb{N}$ and a real number $x$, $z(x)$ is the number of natural numbers $n < x$ of the form $n = \\sum_k \\mu_k H_k$ for some $\\mu \\in E$.","title":"Counting function of a subcollection"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/subcollection-count-of-nonpos","kind":"node","lean":"MetaMathlibExt.subcollectionCount_of_nonpos","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The counting function "],["math","\\(z\\)"],["text"," vanishes for "],["math","\\(x \\le 0\\)"]],"source":"The counting function $z$ vanishes for $x \\le 0$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#subcollection-count-of-nonpos","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $E$ be a set of finitely supported coefficient functions and $H$ a sequence of natural numbers, and for real $x$ let $z(x)$ be the number of natural numbers $n < x$ of the form $n = \\sum_k \\mu_k H_k$ with $\\mu \\in E$. Then $z(x) = 0$ for every real $x \\le 0$.","title":"The counting function $z$ vanishes for $x \\le 0$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/exists-rpow-bounds-subcollection-count","kind":"node","lean":"MetaMathlibExt.exists_rpow_bounds_subcollectionCount","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#exists-rpow-bounds-subcollection-count","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries, and let $E$ and $\\tilde E$ be the periodic Zeckendorf collections they determine, with $E$ a proper subset of $\\tilde E$. Let $\\tilde H$ be a strictly increasing sequence of positive integers such that $\\mu \\mapsto \\sum_k \\mu_k \\tilde H_k$ is a bijection from $\\tilde E$ onto $\\mathbb N$, and let $z(x)$ be the\u2026","title":"Power-law bounds for the subcollection count"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/limsup-zeckendorf-binary-count-div-rpow","kind":"node","lean":"MetaMathlibExt.limsup_zeckendorfBinaryCount_div_rpow","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#limsup-zeckendorf-binary-count-div-rpow","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"For real $x$ let $z(x)$ be the number of natural numbers $n < x$ (including $0$) whose binary expansion has no two adjacent $1$s, let $\\varphi$ be the golden ratio and let $\\gamma = \\log_2 \\varphi$. Then $\\limsup_{x \\to \\infty} z(x) / x^{\\gamma} = 3^{\\gamma} (\\varphi + 2) / 5$.","title":"Limsup of the count of binary expansions without adjacent ones"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/liminf-zeckendorf-binary-count-div-rpow","kind":"node","lean":"MetaMathlibExt.liminf_zeckendorfBinaryCount_div_rpow","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#liminf-zeckendorf-binary-count-div-rpow","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"For real $x$ let $z(x)$ be the number of natural numbers $n < x$ (including $0$) whose binary expansion has no two adjacent $1$s, let $\\varphi$ be the golden ratio and let $\\gamma = \\log_2 \\varphi$. Then $\\liminf_{x \\to \\infty} z(x) / x^{\\gamma} = (3\\varphi + 1) / 5$.","title":"Liminf of the count of binary expansions without adjacent ones"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/exists-is-equivalent-subcollection-count","kind":"node","lean":"MetaMathlibExt.exists_isEquivalent_subcollectionCount","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#exists-is-equivalent-subcollection-count","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries, and let $E$ and $\\tilde E$ be the periodic Zeckendorf collections they determine, with $E$ a proper subset of $\\tilde E$. Let $\\tilde H$ be a strictly increasing sequence of positive integers such that $\\mu \\mapsto \\sum_k \\mu_k \\tilde H_k$ is a bijection from $\\tilde E$ onto $\\mathbb N$, and let $z(x)$ be the\u2026","title":"Asymptotic form of the subcollection count"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/fundamental-sequence-div-rpow-tendsto","kind":"node","lean":"MetaMathlibExt.fundamentalSequence_div_rpow_tendsto","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#fundamental-sequence-div-rpow-tendsto","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries, and let $E$ and $\\tilde E$ be the periodic Zeckendorf collections they determine, with $E$ a proper subset of $\\tilde E$. Let $H$ and $\\tilde H$ be strictly increasing sequences of positive integers, indexed from $0$, such that $\\mu \\mapsto \\sum_k \\mu_k H_k$ is a bijection from $E$ onto $\\mathbb N$ and\u2026","title":"Growth of fundamental sequences"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/subcollection-lt-eval-eq-eval-bar","kind":"node","lean":"MetaMathlibExt.subcollectionLT_eval_eq_eval_bar","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#subcollection-lt-eval-eq-eval-bar","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $E \\subseteq \\tilde E$ be Zeckendorf collections, and let $\\tilde H$ be a strictly increasing sequence of positive integers such that $\\varepsilon \\mapsto \\sum_k \\varepsilon_k \\tilde H_k$ is a bijection from $\\tilde E$ onto $\\mathbb{N}$. For a natural number $x$ let $R(x)$ be the set of natural numbers $n < x$ of the form $n = \\sum_k \\varepsilon_k \\tilde H_k$ with $\\varepsilon \\in E$. Let $\\mu \\in \\tilde E$ be\u2026","title":"Reduction lemma for counting a subcollection"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/subcollection-count-eval-eq-eval-bar","kind":"node","lean":"MetaMathlibExt.subcollectionCount_eval_eq_eval_bar","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#subcollection-count-eval-eq-eval-bar","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $E \\subseteq \\tilde E$ be Zeckendorf collections. Let $H$ and $\\tilde H$ be strictly increasing sequences of positive integers such that $\\mu \\mapsto \\sum_k \\mu_k H_k$ is a bijection from $E$ onto $\\mathbb N$ and $\\mu \\mapsto \\sum_k \\mu_k \\tilde H_k$ is a bijection from $\\tilde E$ onto $\\mathbb N$. For real $x$, let $z(x)$ be the number of natural numbers $n < x$ of the form $n = \\sum_k \\nu_k \\tilde H_k$ with\u2026","title":"Duality formula"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/desc-lex-lt","kind":"node","lean":"MetaMathlibExt.DescLexLT","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#desc-lex-lt","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For coefficient functions $\\varepsilon, \\varepsilon' \\colon \\mathbb{N} \\to \\mathbb{N}$, $\\varepsilon \\lt _d \\varepsilon'$ means that there is an index $k$ with $\\varepsilon_k < \\varepsilon'_k$ and $\\varepsilon_j = \\varepsilon'_j$ for all $j < k$: at the smallest index where they differ, $\\varepsilon$ is smaller.","title":"Descending lexicographic order"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/max-lstar-block","kind":"node","lean":"MetaMathlibExt.maxLStarBlock","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The maximal "],["math","\\(L^*\\)"],["text","-block"]],"source":"The maximal $L^*$-block"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#max-lstar-block","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"The maximal $L^*$-block. Let $L = (e_1, \\dots, e_N)$ be a list of natural numbers and $s$ an index, with indices starting at $0$. The maximal $L^*$-block starting at $s$ is the coefficient function whose entry at index $i$ is $0$ for $i < s$, and $e_j$ for $i \\ge s$, where $j - 1 = (i - s) \\bmod N$ (every entry is $0$ when $L$ is empty). It is the paper's $\\bar\\beta^{s+1}$.","title":"The maximal $L^*$-block"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/max-lstar-block-example","kind":"node","lean":"MetaMathlibExt.maxLStarBlock_example","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The maximal block "],["math","\\(\\bar\\beta^2\\)"],["text"," for "],["math","\\(L = (2, 3, 0)\\)"]],"source":"The maximal block $\\bar\\beta^2$ for $L = (2, 3, 0)$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#max-lstar-block-example","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For $L = (2, 3, 0)$, the maximal $L^*$-block starting at index $1$ (indices start at $0$; the paper's $\\bar\\beta^2$) has entries $0, 2, 3, 0, 2$ at the indices $0, 1, 2, 3, 4$.","title":"The maximal block $\\bar\\beta^2$ for $L = (2, 3, 0)$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-proper-lstar-block","kind":"node","lean":"MetaMathlibExt.IsProperLStarBlock","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Proper "],["math","\\(L^*\\)"],["text","-blocks"]],"source":"Proper $L^*$-blocks"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-proper-lstar-block","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Proper $L^*$-blocks. Let $L$ be a list of natural numbers, index coefficient functions from $1$, and let $\\bar\\beta^s$ be the maximal $L^*$-block starting at index $s$. For $s \\ge 1$, a function $\\zeta : \\mathbb N \\to \\mathbb N$ is a proper $L^*$-block with support interval $[s, k]$ when $s \\le k$, $\\zeta_i = \\bar\\beta^s_i$ for all $i < k$, $\\zeta_k < \\bar\\beta^s_k$, and $\\zeta_i = 0$ for all $i > k$. Since\u2026","title":"Proper $L^*$-blocks"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-lstar-block-sum","kind":"node","lean":"MetaMathlibExt.IsLStarBlockSum","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of disjoint proper "],["math","\\(L^\\ast\\)"],["text","-blocks"]],"source":"Sums of disjoint proper $L^\\ast$-blocks"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-lstar-block-sum","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L$ be a list of natural numbers, and for each $s$ let $\\bar\\beta^{s+1}$ be the maximal $L^*$-block starting at (zero-based) index $s$. Let $I$ be a finite or infinite set of pairs $(s, k)$ of natural numbers. A coefficient function $\\varepsilon : \\mathbb N \\to \\mathbb N$ is the sum of the disjoint proper $L^*$-blocks with support intervals $[s, k]$, $(s, k) \\in I$, if $s \\le k$ for every $(s, k) \\in I$, the\u2026","title":"Sums of disjoint proper $L^\\ast$-blocks"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/l-star-collection","kind":"node","lean":"MetaMathlibExt.lStarCollection","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The periodic collection "],["math","\\(E^*\\)"]],"source":"The periodic collection $E^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#l-star-collection","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"The periodic collection $E^*$. For a list $L$ of natural numbers, $E^*$ is the set of nonzero coefficient functions $\\varepsilon : \\mathbb{N} \\to \\mathbb{N}$ that are sums of finitely or infinitely many disjoint proper $L^*$-blocks, that is, the $\\varepsilon \\ne 0$ that are $L^*$-block sums over some set of support intervals. It is the collection $E^*$ for the interval $(0, 1)$ determined by $L$.","title":"The periodic collection $E^*$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zero-not-mem-l-star-collection","kind":"node","lean":"MetaMathlibExt.zero_notMem_lStarCollection","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zero function is not in the periodic Zeckendorf collection for "],["math","\\((0, 1)\\)"]],"source":"The zero function is not in the periodic Zeckendorf collection for $(0, 1)$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zero-not-mem-l-star-collection","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For every list $L$ of natural numbers, the zero coefficient function $\\mathbb{N} \\to \\mathbb{N}$ does not belong to the periodic Zeckendorf collection $E^*$ for the interval $(0, 1)$ determined by $L$.","title":"The zero function is not in the periodic Zeckendorf collection for $(0, 1)$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/exists-unique-l-star-fundamental-sequenc","kind":"node","lean":"MetaMathlibExt.existsUnique_lStar_fundamentalSequence","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#exists-unique-l-star-fundamental-sequenc","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L = (e_1, \\dots, e_N)$ be a list of natural numbers with $N \\ge 2$ and $e_1 \\ne 0$, and let $E^*$ be its $L^*$-collection; indices of coefficient functions and sequences start at $0$. There is a unique strictly decreasing sequence $(Q_m)_{m \\ge 0}$ in $(0, 1)$ such that $\\varepsilon \\mapsto \\sum_k \\varepsilon_k Q_k$ is a bijection from $E^*$ onto $(0, 1)$. Moreover, every such sequence $Q$ has the following\u2026","title":"Real Zeckendorf expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/l-star-inf","kind":"node","lean":"MetaMathlibExt.lStarInf","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient function "],["math","\\(\\inf(\\varepsilon)\\)"]],"source":"The coefficient function $\\inf(\\varepsilon)$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#l-star-inf","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L$ be a list of natural numbers and $\\varepsilon : \\mathbb{N} \\to \\mathbb{N}$ a coefficient function, with indices starting at $0$. If $\\varepsilon$ has a last nonzero entry, at index $c$, then $\\inf(\\varepsilon)$ agrees with $\\varepsilon$ at the indices $i < c$, equals $\\varepsilon_c - 1$ at $c$, and agrees with the maximal $L^*$-block starting at index $c + 1$ at the indices $i > c$. Otherwise, that is, when\u2026","title":"The coefficient function $\\inf(\\varepsilon)$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/l-star-inf-of-infinite-support","kind":"node","lean":"MetaMathlibExt.lStarInf_of_infinite_support","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\inf(\\varepsilon) = \\varepsilon\\)"],["text"," for infinite support"]],"source":"$\\inf(\\varepsilon) = \\varepsilon$ for infinite support"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#l-star-inf-of-infinite-support","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L$ be a list of natural numbers and $\\varepsilon : \\mathbb{N} \\to \\mathbb{N}$ a coefficient function with infinitely many nonzero entries (for every $c$ there is $i > c$ with $\\varepsilon_i \\ne 0$). Then $\\inf(\\varepsilon) = \\varepsilon$, where $\\inf$ is taken with respect to $L$.","title":"$\\inf(\\varepsilon) = \\varepsilon$ for infinite support"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/tsum-l-star-inf-mul-eq","kind":"node","lean":"MetaMathlibExt.tsum_lStarInf_mul_eq","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\varepsilon\\)"],["text"," and "],["math","\\(\\inf(\\varepsilon)\\)"],["text"," have the same value"]],"source":"$\\varepsilon$ and $\\inf(\\varepsilon)$ have the same value"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#tsum-l-star-inf-mul-eq","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ be a list of natural numbers of length at least $2$ with nonzero first entry, let $E^*$ be the set of nonzero coefficient functions $\\mathbb N \\to \\mathbb N$ that are sums of finitely or infinitely many disjoint proper $L^*$-blocks, and let $Q$ be a strictly decreasing sequence in $(0, 1)$ such that $\\varepsilon \\mapsto \\sum_k \\varepsilon_k Q_k$ is a bijection from $E^*$ onto $(0, 1)$; indices start at $0$.\u2026","title":"$\\varepsilon$ and $\\inf(\\varepsilon)$ have the same value"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/exists-prefix-eq-of-l-star-near","kind":"node","lean":"MetaMathlibExt.exists_prefix_eq_of_lStar_near","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#exists-prefix-eq-of-l-star-near","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ be a list of natural numbers of length at least $2$ with nonzero first entry, let $E^*$ be its $L^*$-collection, and let $Q$ be a strictly decreasing sequence in $(0, 1)$ such that $\\varepsilon \\mapsto \\sum_k \\varepsilon_k Q_k$ is a bijection from $E^*$ onto $(0, 1)$; indices start at $0$. Let $\\varepsilon \\in E^*$, put $x = \\sum_k \\varepsilon_k Q_k$, and let $n \\in \\mathbb N$. Define $\\inf(\\varepsilon)$ as\u2026","title":"Nearby expansions share long prefixes"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/desc-lex-lt-max-lstar-block-iff","kind":"node","lean":"MetaMathlibExt.descLexLT_maxLStarBlock_iff","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#desc-lex-lt-max-lstar-block-iff","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L = (e_1, \\dots, e_N)$ with $N \\ge 2$ and $e_1 \\ne 0$, let $\\varepsilon \\colon \\mathbb{N} \\to \\mathbb{N}$ and $b \\in \\mathbb{N}$ with $\\varepsilon_i = 0$ for all $i < b$, and let $\\bar\\beta$ be the maximal $L^*$-block starting at index $b$. Then $\\varepsilon \\lt _d \\bar\\beta$ in the descending lexicographic order if and only if there is $k \\ge b$ such that the function equal to $\\varepsilon_i$ for $i \\le k$\u2026","title":"Proper blocks below a maximal block"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/exists-l-star-block-sum-of-proper-lstar","kind":"node","lean":"MetaMathlibExt.exists_lStarBlockSum_of_properLStarBlock","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Decomposing "],["math","\\(L^*\\)"],["text","-blocks into "],["math","\\(\\tilde L^*\\)"],["text","-blocks"]],"source":"Decomposing $L^*$-blocks into $\\tilde L^*$-blocks"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#exists-l-star-block-sum-of-proper-lstar","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Decomposing $L^*$-blocks into $\\tilde L^*$-blocks. Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries whose periodic Zeckendorf collections satisfy $E \\subsetneq \\tilde E$. If $\\zeta$ is a nonzero proper $L^*$-block with support interval $[i, b]$, then $\\zeta$ is a sum of disjoint proper $\\tilde L^*$-blocks whose support intervals all lie in $[0, b]$, one of which\u2026","title":"Decomposing $L^*$-blocks into $\\tilde L^*$-blocks"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/mem-l-star-collection-of-l-star-bar-deco","kind":"node","lean":"MetaMathlibExt.mem_lStarCollection_of_lStarBar_decomposition","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The tail of an "],["math","\\(L^\\ast\\)"],["text","-block decomposition lies in "],["math","\\(\\tilde E^\\ast\\)"]],"source":"The tail of an $L^\\ast$-block decomposition lies in $\\tilde E^\\ast$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#mem-l-star-collection-of-l-star-bar-deco","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries, such that the periodic Zeckendorf collection $E$ of $L$ is a proper subset of the periodic Zeckendorf collection $\\tilde E$ of $\\tilde L$, and let $\\tilde E^*$ be the set of nonzero coefficient functions $\\mathbb N \\to \\mathbb N$ that are sums of finitely or infinitely many disjoint proper $\\tilde L^*$-blocks.\u2026","title":"The tail of an $L^\\ast$-block decomposition lies in $\\tilde E^\\ast$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/tsum-max-lstar-block-mul-eq","kind":"node","lean":"MetaMathlibExt.tsum_maxLStarBlock_mul_eq","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#tsum-max-lstar-block-mul-eq","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L = (e_1, \\dots, e_N)$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries, and let $\\tilde Q$ be a strictly decreasing sequence with values in $(0, 1)$ such that $\\varepsilon \\mapsto \\sum_k \\varepsilon_k \\tilde Q_k$ is a bijection from $\\tilde E^*$, the periodic Zeckendorf collection for $(0, 1)$ determined by $\\tilde L$, onto $(0, 1)$. For $s \\in \\mathbb{N}$ let\u2026","title":"Shifted maximal blocks scale geometrically"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/ncard-proper-lstar-block","kind":"node","lean":"MetaMathlibExt.ncard_properLStarBlock","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of proper "],["math","\\(L^*\\)"],["text","-blocks with a given support interval"]],"source":"Number of proper $L^*$-blocks with a given support interval"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#ncard-proper-lstar-block","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Number of proper $L^*$-blocks with a given support interval. Let $L = (e_1, \\dots, e_N)$ be a list of natural numbers with $N \\ge 2$ and $e_1 \\ne 0$, let $s \\le k$ be natural numbers, and let $j$ be the integer with $1 \\le j \\le N$ and $k - s + 1 \\equiv j \\pmod N$. Then there are exactly $e_j$ proper $L^*$-blocks with support interval $[s, k]$.","title":"Number of proper $L^*$-blocks with a given support interval"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/l-star-block-sum-count","kind":"node","lean":"MetaMathlibExt.lStarBlockSumCount","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(L^*\\)"],["text","-block sums of length "],["math","\\(n\\)"]],"source":"Number of $L^*$-block sums of length $n$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#l-star-block-sum-count","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Number of $L^*$-block sums of length $n$. For a list $L$ of natural numbers and $n \\ge 0$, with coefficient functions indexed from $1$, $C_n$ is the number of functions $\\varepsilon : \\mathbb N \\to \\mathbb N$ that are sums of disjoint proper $L^*$-blocks with support intervals inside $[1, n]$. The zero function (the empty sum) is counted, so $C_0 = 1$.","title":"Number of $L^*$-block sums of length $n$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/l-star-block-sum-count-zero","kind":"node","lean":"MetaMathlibExt.lStarBlockSumCount_zero","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#l-star-block-sum-count-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For every list $L$ of natural numbers, $C_0 = 1$, where $C_n$ is the number of sums of disjoint proper $L^*$-blocks with support intervals inside $[1, n]$: for $n = 0$ the zero function is the only such sum.","title":"The empty block-sum count"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/l-star-block-sum-count-recurrence","kind":"node","lean":"MetaMathlibExt.lStarBlockSumCount_recurrence","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the number of "],["math","\\(L^*\\)"],["text","-block sums"]],"source":"Recurrence for the number of $L^*$-block sums"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#l-star-block-sum-count-recurrence","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Recurrence for the number of $L^*$-block sums. Let $L = (e_1, \\dots, e_N)$ be a list of natural numbers with $N \\ge 2$ and $e_1 \\ne 0$, index coefficient functions from $1$, and let $C_n$ be the number of functions $\\mathbb N \\to \\mathbb N$ that are sums of disjoint proper $L^*$-blocks with support intervals inside $[1, n]$. Then $C_k = \\sum_{j=1}^{k} e_j C_{k-j}$ for $1 \\le k \\le N$, and\u2026","title":"Recurrence for the number of $L^*$-block sums"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/has-sum-l-star-block-sum-count","kind":"node","lean":"MetaMathlibExt.hasSum_lStarBlockSumCount","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the "],["math","\\(L^\\ast\\)"],["text","-block-sum counts"]],"source":"Generating function of the $L^\\ast$-block-sum counts"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#has-sum-l-star-block-sum-count","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L = (e_1, \\dots, e_N)$ be a list of natural numbers with $N \\ge 2$ and $e_1 \\ne 0$. For each $k$ let $C_k$ be the number of coefficient functions that are sums of disjoint proper $L^*$-blocks with support intervals inside $[1, k]$ (the zero function included), and put $f(x) = 1 - \\sum_{k=1}^{N-1} e_k x^k - (1 + e_N) x^N$. Then there is a real $r > 0$ such that for every real $x$ with $-r < x < r$ the series\u2026","title":"Generating function of the $L^\\ast$-block-sum counts"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-periodic-zeckendorf-pair","kind":"node","lean":"MetaMathlibExt.IsPeriodicZeckendorfPair","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-periodic-zeckendorf-pair","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Two lists $L$ and $\\tilde L$ of natural numbers form a periodic Zeckendorf pair when both have length at least $2$ and a nonzero first entry, and the periodic Zeckendorf collection $E$ determined by $L$ is a proper subset of the periodic Zeckendorf collection $\\tilde E$ determined by $\\tilde L$.","title":"Periodic Zeckendorf pairs"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/subset","kind":"node","lean":"MetaMathlibExt.IsPeriodicZeckendorfPair.subset","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#subset","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"If the lists $L$ and $\\tilde L$ form a periodic Zeckendorf pair, then the periodic Zeckendorf collection determined by $L$ is contained in the periodic Zeckendorf collection determined by $\\tilde L$.","title":"Periodic Zeckendorf pairs are nested"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-real-eval","kind":"node","lean":"MetaMathlibExt.zeckendorfRealEval","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-real-eval","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For a real sequence $Q$ and a coefficient function $\\varepsilon \\colon \\mathbb{N} \\to \\mathbb{N}$, not necessarily finitely supported, the evaluation $\\sum \\varepsilon Q$ is the sum of the series $\\sum_{k \\ge 0} \\varepsilon_k Q_k$, taken to be $0$ when the series is not absolutely summable.","title":"Evaluation of a coefficient function at a real sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-real-eval-zero","kind":"node","lean":"MetaMathlibExt.zeckendorfRealEval_zero","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-real-eval-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For every real sequence $Q$, the evaluation of the zero coefficient function at $Q$ is $0$.","title":"Evaluating the zero coefficient function"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-lstar-fundamental-sequence","kind":"node","lean":"MetaMathlibExt.IsLStarFundamentalSequence","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fundamental sequence of an "],["math","\\(L^\\ast\\)"],["text","-collection"]],"source":"Fundamental sequence of an $L^\\ast$-collection"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-lstar-fundamental-sequence","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L$ be a list of natural numbers with $L^*$-collection $E^*$. A real sequence $(Q_n)_{n \\ge 0}$ is a fundamental sequence of $E^*$ if it is strictly decreasing, $0 < Q_n < 1$ for every $n$, and the map $\\varepsilon \\mapsto \\sum_k \\varepsilon_k Q_k$ is a bijection from $E^*$ onto the open interval $(0, 1)$.","title":"Fundamental sequence of an $L^\\ast$-collection"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/pos","kind":"node","lean":"MetaMathlibExt.IsLStarFundamentalSequence.pos","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fundamental sequences of "],["math","\\(E^*\\)"],["text"," are positive"]],"source":"Fundamental sequences of $E^*$ are positive"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#pos","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Fundamental sequences of $E^*$ are positive. Let $L$ be a list of natural numbers and let $Q$ be a fundamental sequence of the collection $E^*$ determined by $L$. Then $Q_n > 0$ for every $n$.","title":"Fundamental sequences of $E^*$ are positive"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-rev","kind":"node","lean":"MetaMathlibExt.zeckendorfRev","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reversal of the first "],["math","\\(n\\)"],["text"," entries"]],"source":"Reversal of the first $n$ entries"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-rev","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For $n \\in \\mathbb N$ and a coefficient function $\\varepsilon : \\mathbb N \\to \\mathbb N$, $\\mathrm{rev}_n(\\varepsilon)$ is the finitely supported coefficient function whose entry at index $j$ is $\\varepsilon(n - 1 - j)$ for $j < n$ and $0$ for $j \\ge n$ (indices start at $0$). In the paper's one-based indexing this is $\\mathrm{rev}_n(\\varepsilon) = (\\varepsilon_n, \\dots, \\varepsilon_1)$.","title":"Reversal of the first $n$ entries"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-rev-three","kind":"node","lean":"MetaMathlibExt.zeckendorfRev_three","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-rev-three","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"For every coefficient function $\\varepsilon : \\mathbb N \\to \\mathbb N$, $\\mathrm{rev}_3(\\varepsilon)$ is the finitely supported coefficient function with entries $\\varepsilon(2)$, $\\varepsilon(1)$, $\\varepsilon(0)$ at indices $0$, $1$, $2$ and $0$ at every other index. In the paper's one-based indexing, $\\mathrm{rev}_3(\\varepsilon) = (\\varepsilon_3, \\varepsilon_2, \\varepsilon_1)$.","title":"Reversal of the first three entries"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-bar","kind":"node","lean":"MetaMathlibExt.zeckendorfBar","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficient function "],["math","\\(\\bar\\mu\\)"]],"source":"The coefficient function $\\bar\\mu$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-bar","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $E$ be a set of finitely supported coefficient functions $\\mathbb{N} \\to \\mathbb{N}$ and $\\mu$ a finitely supported coefficient function. If $\\mu$ has an immediate predecessor $\\check\\mu$ in $E$ (the largest element of $E$ that is $\\lt _a \\mu$ in the ascending lexicographic order) and $\\check\\mu$ has an immediate successor in $E$ (the smallest element of $E$ that is $>_a \\check\\mu$), then $\\bar\\mu$ is that\u2026","title":"The coefficient function $\\bar\\mu$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-bar-spec","kind":"node","lean":"MetaMathlibExt.zeckendorfBar_spec","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-bar-spec","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $E$ be a set of finitely supported coefficient functions $\\mathbb{N} \\to \\mathbb{N}$ and $\\mu$ a finitely supported coefficient function. Call $\\nu$ an immediate predecessor of $\\mu$ in $E$ if it is a largest element of $E$ that is $\\lt _a \\mu$ in the ascending lexicographic order, and call $\\xi$ an immediate successor of $\\nu$ in $E$ if it is a smallest element of $E$ that is $>_a \\nu$. Let $\\bar\\mu$ be an\u2026","title":"The bar of a coefficient function is a successor of its predecessor"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/l-star-closure","kind":"node","lean":"MetaMathlibExt.lStarClosure","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The closure "],["math","\\(\\bar E^*\\)"],["text"," of "],["math","\\(E^*\\)"]],"source":"The closure $\\bar E^*$ of $E^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#l-star-closure","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"The closure $\\bar E^*$ of $E^*$. Let $L$ be a list of natural numbers, $E^*$ the collection determined by $L$, and index coefficient functions from $1$. The collection $\\bar E^*$ consists of the elements of $E^*$ together with every function $\\varepsilon : \\mathbb N \\to \\mathbb N$ for which there is an index $b \\ge 0$ such that the function agreeing with $\\varepsilon$ at the indices $\\le b$ and vanishing elsewhere\u2026","title":"The closure $\\bar E^*$ of $E^*$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/max-lstar-block-zero-mem-l-star-closure","kind":"node","lean":"MetaMathlibExt.maxLStarBlock_zero_mem_lStarClosure","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\bar\\beta^1\\)"],["text"," lies in "],["math","\\(\\bar E^*\\)"]],"source":"$\\bar\\beta^1$ lies in $\\bar E^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#max-lstar-block-zero-mem-l-star-closure","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"$\\bar\\beta^1$ lies in $\\bar E^*$. For every list $L$ of natural numbers, the maximal $L^*$-block starting at index $0$ (the paper's $\\bar\\beta^1$) belongs to the collection $\\bar E^*$ determined by $L$. This is the case of the definition of $\\bar E^*$ with no proper blocks, in which the maximal block starts at index $0$.","title":"$\\bar\\beta^1$ lies in $\\bar E^*$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-rev-bar","kind":"node","lean":"MetaMathlibExt.zeckendorfRevBar","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(\\overline{\\mathrm{rev}}\\)"]],"source":"The map $\\overline{\\mathrm{rev}}$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-rev-bar","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Fix a list $L$, and let $E$ be its periodic Zeckendorf collection, $\\bar\\beta^1$ the maximal $L^*$-block starting at index $0$, $\\lt _d$ the descending lexicographic order, $\\mathrm{rev}_n$ the reversal of the first $n$ entries, and $\\bar\\mu$ the bar of a finitely supported $\\mu$ relative to $E$. For a coefficient function $\\varepsilon : \\mathbb N \\to \\mathbb N$, if $\\varepsilon \\lt _d \\bar\\beta^1$ fails then\u2026","title":"The map $\\overline{\\mathrm{rev}}$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-rev-bar-of-not-desc-lex-lt","kind":"node","lean":"MetaMathlibExt.zeckendorfRevBar_of_not_descLexLT","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\overline{\\mathrm{rev}}\\)"],["text"," of a function not below "],["math","\\(\\bar\\beta^1\\)"]],"source":"$\\overline{\\mathrm{rev}}$ of a function not below $\\bar\\beta^1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-rev-bar-of-not-desc-lex-lt","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L$ be a list of natural numbers, let $\\bar\\beta^1$ be the maximal $L^*$-block starting at index $0$, let $\\lt _d$ be the descending lexicographic order, and let $\\overline{\\mathrm{rev}}$ be the map determined by $L$. For every coefficient function $\\varepsilon : \\mathbb N \\to \\mathbb N$ such that $\\varepsilon \\lt _d \\bar\\beta^1$ fails, $\\overline{\\mathrm{rev}}(\\varepsilon) = \\bar\\beta^1$.","title":"$\\overline{\\mathrm{rev}}$ of a function not below $\\bar\\beta^1$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-delta-star-one","kind":"node","lean":"MetaMathlibExt.zeckendorfDeltaStarOne","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function "],["math","\\(\\delta_1^\\ast\\)"]],"source":"The function $\\delta_1^\\ast$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-delta-star-one","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L$ and $\\tilde L$ be lists of natural numbers and $Q$ and $\\tilde Q$ real sequences with first terms $\\omega = Q_0$ and $\\tilde\\omega = \\tilde Q_0$ (indices start at $0$), and put $\\gamma = \\log_{\\tilde\\omega} \\omega$. Let $\\tilde E^*$ be the collection $E^*$ determined by $\\tilde L$, let $\\overline{\\mathrm{rev}}$ be the map determined by $L$, and write $\\sum \\varepsilon R = \\sum_k \\varepsilon_k R_k$ for the\u2026","title":"The function $\\delta_1^\\ast$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-delta-star","kind":"node","lean":"MetaMathlibExt.zeckendorfDeltaStar","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function "],["math","\\(\\delta^*\\)"]],"source":"The function $\\delta^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-delta-star","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"The function $\\delta^*$. Let $L$ and $\\tilde L$ be lists and $Q$ and $\\tilde Q$ real sequences, let $\\tilde\\omega$ be the first term of $\\tilde Q$, and let $\\delta_1^*$ be the function $\\delta_1^*$ of $L$, $\\tilde L$, $Q$, $\\tilde Q$. For real $y$: if some natural number $m$ satisfies $\\tilde\\omega^{m+1} \\le y < \\tilde\\omega^m$, then $\\delta^*(y) = \\delta_1^*(y / \\tilde\\omega^m)$ for a chosen such $m$; otherwise\u2026","title":"The function $\\delta^*$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-delta-star-of-mem-ico","kind":"node","lean":"MetaMathlibExt.zeckendorfDeltaStar_of_mem_Ico","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\delta^*\\)"],["text"," agrees with "],["math","\\(\\delta_1^*\\)"],["text"," on "],["math","\\((\\tilde\\omega, 1)\\)"]],"source":"$\\delta^*$ agrees with $\\delta_1^*$ on $(\\tilde\\omega, 1)$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-delta-star-of-mem-ico","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"$\\delta^*$ agrees with $\\delta_1^*$ on $[\\tilde\\omega, 1)$. Let $L$ and $\\tilde L$ be lists and $Q$ and $\\tilde Q$ real sequences such that the first term $\\tilde\\omega$ of $\\tilde Q$ satisfies $0 < \\tilde\\omega < 1$. Then for every real $y$ with $\\tilde\\omega \\le y < 1$, the functions $\\delta^*$ and $\\delta_1^*$ of $L$, $\\tilde L$, $Q$, $\\tilde Q$ agree at $y$: $\\delta^*(y) = \\delta_1^*(y)$.","title":"$\\delta^*$ agrees with $\\delta_1^*$ on $(\\tilde\\omega, 1)$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-generic-exponent","kind":"node","lean":"MetaMathlibExt.zeckendorfGenericExponent","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-generic-exponent","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $Q$ and $\\tilde Q$ be real sequences with first terms $\\omega$ and $\\tilde\\omega$, and put $\\gamma = \\log_{\\tilde\\omega} \\omega$. The generic exponent $p$ of $Q$ and $\\tilde Q$ is the smallest positive integer $p$ with $\\gamma \\tilde\\omega^{p-1} < \\omega^p$, and $p = 0$ if no positive integer satisfies this inequality.","title":"The generic exponent"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-generic-exponent-spec","kind":"node","lean":"MetaMathlibExt.zeckendorfGenericExponent_spec","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-generic-exponent-spec","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $Q$ and $\\tilde Q$ be real sequences with first terms $\\omega$ and $\\tilde\\omega$, put $\\gamma = \\log_{\\tilde\\omega} \\omega$, and let $p$ be the generic exponent of $Q$ and $\\tilde Q$. If some positive integer $q$ satisfies $\\gamma \\tilde\\omega^{q-1} < \\omega^q$, then $p$ is positive and $\\gamma \\tilde\\omega^{p-1} < \\omega^p$.","title":"The generic exponent satisfies its defining inequality"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-smooth-set","kind":"node","lean":"MetaMathlibExt.zeckendorfSmoothSet","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(U\\)"],["text"," of evaluations of "],["math","\\(\\tilde E^* \\setminus \\bar E^*\\)"]],"source":"The set $U$ of evaluations of $\\tilde E^* \\setminus \\bar E^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-smooth-set","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"The set $U$ of evaluations of $\\tilde E^* \\setminus \\bar E^*$. For lists $L$ and $\\tilde L$ and a real sequence $\\tilde Q$, let $\\tilde E^*$ be the $\\tilde L^*$-collection and $\\bar E^*$ the closure collection of $L$. The set $U \\subseteq \\mathbb R$ is the image of $\\tilde E^* \\setminus \\bar E^*$ under the evaluation $\\varepsilon \\mapsto \\sum_k \\varepsilon_k \\tilde Q_k$.","title":"The set $U$ of evaluations of $\\tilde E^* \\setminus \\bar E^*$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/mem-zeckendorf-smooth-set-iff","kind":"node","lean":"MetaMathlibExt.mem_zeckendorfSmoothSet_iff","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in the set "],["math","\\(U\\)"]],"source":"Membership in the set $U$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#mem-zeckendorf-smooth-set-iff","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"fully_proved","summary":"Let $L$ and $\\tilde L$ be lists of natural numbers and let $\\tilde Q = (\\tilde Q_k)$ be a sequence of real numbers. Let $\\tilde E^*$ be the collection $E^*$ determined by $\\tilde L$, let $\\bar E^*$ be the closure $\\bar E^*$ determined by $L$, and let $U$ be the image of $\\tilde E^* \\setminus \\bar E^*$ under the map $\\varepsilon \\mapsto \\sum_k \\varepsilon_k \\tilde Q_k$. A real number $x$ lies in $U$ if and only if\u2026","title":"Membership in the set $U$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/forall-zeckendorf-rev-mem-iff-mem-l-star","kind":"node","lean":"MetaMathlibExt.forall_zeckendorfRev_mem_iff_mem_lStarClosure","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reversals characterize the nonzero elements of "],["math","\\(\\bar E^\\ast\\)"]],"source":"Reversals characterize the nonzero elements of $\\bar E^\\ast$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#forall-zeckendorf-rev-mem-iff-mem-l-star","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ be a list of natural numbers of length at least $2$ with nonzero first entry, let $E$ be the periodic Zeckendorf collection determined by $L$, and let $\\bar E^*$ be the closure of the collection $E^*$ determined by $L$. For $n \\in \\mathbb N$ and a coefficient function $\\varepsilon : \\mathbb N \\to \\mathbb N$, let $\\mathrm{rev}_n(\\varepsilon)$ be the finitely supported coefficient function whose first $n$\u2026","title":"Reversals characterize the nonzero elements of $\\bar E^\\ast$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/l-star-bar-decomposition","kind":"node","lean":"MetaMathlibExt.lStarBar_decomposition","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Decomposition along "],["math","\\(L^*\\)"],["text","-blocks (Corollary 1)"]],"source":"Decomposition along $L^*$-blocks (Corollary 1)"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#l-star-bar-decomposition","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Decomposition along $L^*$-blocks (Corollary 1). Let $L$ and $\\tilde L$ be a periodic Zeckendorf pair, let $E^*$ and $\\tilde E^*$ be the collections determined by $L$ and $\\tilde L$, and let $\\varepsilon \\in \\tilde E^*$. Index coefficient functions from $1$. For $b \\ge 0$, let $\\eta_b$ agree with $\\varepsilon$ at the indices $\\le b$ and vanish elsewhere, and let $\\mu_b = \\varepsilon - \\eta_b$. Call $b$ a\u2026","title":"Decomposition along $L^*$-blocks (Corollary 1)"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-rev-bar-zeckendorf-rev-eq","kind":"node","lean":"MetaMathlibExt.zeckendorfRev_bar_zeckendorfRev_eq","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lemma 10: "],["math","\\(\\mathrm{rev}_n\\)"],["text"," of the bar of "],["math","\\(\\mathrm{rev}_n(\\varepsilon)\\)"]],"source":"Lemma 10: $\\mathrm{rev}_n$ of the bar of $\\mathrm{rev}_n(\\varepsilon)$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-rev-bar-zeckendorf-rev-eq","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $(L, \\tilde L)$ be a periodic Zeckendorf pair, $E$ the periodic Zeckendorf collection of $L$, and $E^*$ and $\\tilde E^*$ the $L^*$- and $\\tilde L^*$-collections. Indices start at $0$. Write $\\lt _d$ for the descending lexicographic order, $\\bar\\beta^{s+1}$ for the maximal $L^*$-block starting at index $s$, $\\mathrm{rev}_n$ for the reversal of the first $n$ entries, $\\bar\\mu$ for the bar of a finitely supported\u2026","title":"Lemma 10: $\\mathrm{rev}_n$ of the bar of $\\mathrm{rev}_n(\\varepsilon)$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/mem-l-star-closure-iff-zeckendorf-rev-ba","kind":"node","lean":"MetaMathlibExt.mem_lStarClosure_iff_zeckendorfRevBar_eq","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed points of "],["math","\\(\\overline{\\mathrm{rev}}\\)"],["text"," in "],["math","\\(\\tilde E^*\\)"]],"source":"Fixed points of $\\overline{\\mathrm{rev}}$ in $\\tilde E^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#mem-l-star-closure-iff-zeckendorf-rev-ba","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Fixed points of $\\overline{\\mathrm{rev}}$ in $\\tilde E^*$. Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries, such that the periodic Zeckendorf collection $E$ of $L$ is a proper subset of the periodic Zeckendorf collection $\\tilde E$ of $\\tilde L$. Let $\\tilde E^*$ be the collection $E^*$ determined by $\\tilde L$, and let $\\bar E^*$ and $\\overline{\\mathrm{rev}}$ be\u2026","title":"Fixed points of $\\overline{\\mathrm{rev}}$ in $\\tilde E^*$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-rev-bar-zeckendorf-rev-of-ord","kind":"node","lean":"MetaMathlibExt.zeckendorfRevBar_zeckendorfRev_of_order","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\overline{\\mathrm{rev}}\\)"],["text"," of the reversal of an element of "],["math","\\(\\tilde E\\)"],["text"," (Lemma 12)"]],"source":"$\\overline{\\mathrm{rev}}$ of the reversal of an element of $\\tilde E$ (Lemma 12)"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-rev-bar-zeckendorf-rev-of-ord","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, let $E$ and $\\tilde E$ be the periodic Zeckendorf collections determined by $L$ and $\\tilde L$, and index coefficient functions from $0$. Let $\\mu \\in \\tilde E$ and $t \\ge 1$ be such that $\\mu_{t-1} \\ne 0$ and $\\mu_j = 0$ for all $j \\ge t$. Let $\\overline{\\mathrm{rev}}$ be the map determined by $L$, $\\bar\\beta^1$ the maximal $L^*$-block starting at index $0$,\u2026","title":"$\\overline{\\mathrm{rev}}$ of the reversal of an element of $\\tilde E$ (Lemma 12)"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-equivalent-subcollection-count-div-rp","kind":"node","lean":"MetaMathlibExt.isEquivalent_subcollectionCount_div_rpow_deltaStar","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Transition from the subcollection count to "],["math","\\(\\delta^\\ast\\)"]],"source":"Transition from the subcollection count to $\\delta^\\ast$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-equivalent-subcollection-count-div-rp","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, with periodic Zeckendorf collections $E \\subsetneq \\tilde E$, and let $Q$ and $\\tilde Q$ be fundamental sequences of the $L^*$-collection $E^*$ and the $\\tilde L^*$-collection $\\tilde E^*$. Put $\\omega = Q_0$, $\\tilde\\omega = \\tilde Q_0$ and $\\gamma = \\log_{\\tilde\\omega} \\omega$, and let $\\delta^*$ be the function determined by $L$, $\\tilde L$, $Q$ and\u2026","title":"Transition from the subcollection count to $\\delta^\\ast$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-rev-bar-eventually-near","kind":"node","lean":"MetaMathlibExt.zeckendorfRevBar_eventually_near","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lemma 13: behaviour of "],["math","\\(\\overline{\\mathrm{rev}}\\)"],["text"," near a point"]],"source":"Lemma 13: behaviour of $\\overline{\\mathrm{rev}}$ near a point"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-rev-bar-eventually-near","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $(L, \\tilde L)$ be a periodic Zeckendorf pair, let $E^*$ and $\\tilde E^*$ be the $L^*$- and $\\tilde L^*$-collections, $\\bar E^*$ the closure collection of $L$, $\\overline{\\mathrm{rev}}$ the map determined by $L$, $\\inf$ the $L^*$-infimum map, and $\\tilde Q$ a fundamental sequence of $\\tilde E^*$. For $\\eta \\in \\tilde E^*$ write $x(\\eta) = \\sum_k \\eta_k \\tilde Q_k$, and let $\\varepsilon \\in \\tilde E^*$. If\u2026","title":"Lemma 13: behaviour of $\\overline{\\mathrm{rev}}$ near a point"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/infinite-set-of-sub-single-mem-l-star-cl","kind":"node","lean":"MetaMathlibExt.infinite_setOf_sub_single_mem_lStarClosure","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many unit reductions stay in the "],["math","\\(L^\\ast\\)"],["text","-closure"]],"source":"Infinitely many unit reductions stay in the $L^\\ast$-closure"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#infinite-set-of-sub-single-mem-l-star-cl","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L = (e_1, \\dots, e_N)$ be a list of natural numbers with $N \\ge 2$ and $e_1 \\ne 0$, and let $E^* \\subseteq \\bar E^*$ be the $L^*$-collection and the $L^*$-closure determined by $L$. Let $\\varepsilon \\in \\bar E^*$, and suppose it is not the case that $\\varepsilon \\in E^*$ and $\\varepsilon$ has finite support. Then there are infinitely many indices $r$ such that $\\varepsilon_r \\ge 1$ and the coefficient function\u2026","title":"Infinitely many unit reductions stay in the $L^\\ast$-closure"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-smooth-set-eq-i-union","kind":"node","lean":"MetaMathlibExt.zeckendorfSmoothSet_eq_iUnion","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Proposition 3: "],["math","\\(U\\)"],["text"," is a disjoint union of open intervals"]],"source":"Proposition 3: $U$ is a disjoint union of open intervals"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-smooth-set-eq-i-union","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $(L, \\tilde L)$ be a periodic Zeckendorf pair, $E^*$ and $\\tilde E^*$ the $L^*$- and $\\tilde L^*$-collections, $\\bar E^*$ the closure collection of $L$, and $\\tilde Q$ a fundamental sequence of $\\tilde E^*$. Write $x(\\eta) = \\sum_k \\eta_k \\tilde Q_k$, $\\tilde\\omega = \\tilde Q_0$, and $\\bar\\beta^{b+1}$ for the maximal $L^*$-block starting at index $b$ (indices start at $0$), and let\u2026","title":"Proposition 3: $U$ is a disjoint union of open intervals"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/continuous-on-zeckendorf-delta-star","kind":"node","lean":"MetaMathlibExt.continuousOn_zeckendorfDeltaStar","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuity and local monotonicity of "],["math","\\(\\delta^{\\ast}\\)"]],"source":"Continuity and local monotonicity of $\\delta^{\\ast}$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#continuous-on-zeckendorf-delta-star","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, let $Q$ be a fundamental sequence of the collection $E^*$ determined by $L$, and let $\\tilde Q$ be a fundamental sequence of the collection $\\tilde E^*$ determined by $\\tilde L$. Then the function $\\delta^*$ built from $L, \\tilde L, Q, \\tilde Q$ is continuous on $(0, 1)$, and at each point $x$ of the smooth set $U$ of $L, \\tilde L, \\tilde Q$ it is\u2026","title":"Continuity and local monotonicity of $\\delta^{\\ast}$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/not-differentiable-at-zeckendorf-delta-s","kind":"node","lean":"MetaMathlibExt.not_differentiableAt_zeckendorfDeltaStar","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\delta^*\\)"],["text"," is not differentiable at the evaluations of "],["math","\\(\\bar E^*\\)"]],"source":"$\\delta^*$ is not differentiable at the evaluations of $\\bar E^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#not-differentiable-at-zeckendorf-delta-s","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"$\\delta^*$ is not differentiable at the evaluations of $\\bar E^*$. Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries whose periodic Zeckendorf collections $E$ and $\\tilde E$ satisfy $E \\subsetneq \\tilde E$. Let $Q$ be a fundamental sequence of the collection $E^*$ determined by $L$, let $\\tilde Q$ be a fundamental sequence of the collection $\\tilde E^*$ determined\u2026","title":"$\\delta^*$ is not differentiable at the evaluations of $\\bar E^*$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/volume-zeckendorf-smooth-set","kind":"node","lean":"MetaMathlibExt.volume_zeckendorfSmoothSet","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","The smooth set "],["math","\\(U\\)"],["text"," has Lebesgue measure "],["math","\\(1\\)"]],"source":"The smooth set $U$ has Lebesgue measure $1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#volume-zeckendorf-smooth-set","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, let $\\tilde E^*$ be the collection $E^*$ determined by $\\tilde L$, let $\\tilde Q$ be a fundamental sequence of $\\tilde E^*$, and let $\\bar E^*$ be the closure determined by $L$. Let $U$ be the image of $\\tilde E^* \\setminus \\bar E^*$ under the evaluation $\\varepsilon \\mapsto \\sum_k \\varepsilon_k \\tilde Q_k$. Then $U$ has Lebesgue measure $1$.","title":"The smooth set $U$ has Lebesgue measure $1$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-local-min-zeckendorf-delta-star-of-fi","kind":"node","lean":"MetaMathlibExt.isLocalMin_zeckendorfDeltaStar_of_finite","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Local minima of "],["math","\\(\\delta^\\ast\\)"],["text"," at finitely supported points"]],"source":"Local minima of $\\delta^\\ast$ at finitely supported points"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-local-min-zeckendorf-delta-star-of-fi","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, let $Q$ and $\\tilde Q$ be fundamental sequences of the $L^*$-collection $E^*$ and the $\\tilde L^*$-collection $\\tilde E^*$, and let $\\delta^*$ be the function determined by $L$, $\\tilde L$, $Q$ and $\\tilde Q$. If $\\varepsilon \\in E^*$ has finite support, then $\\delta^*$ has a local minimum at $\\sum_k \\varepsilon_k \\tilde Q_k$.","title":"Local minima of $\\delta^\\ast$ at finitely supported points"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/not-is-local-extr-zeckendorf-delta-star","kind":"node","lean":"MetaMathlibExt.not_isLocalExtr_zeckendorfDeltaStar_of_infinite","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\delta^*\\)"],["text"," has no local extremum at infinite expansions in "],["math","\\(E^*\\)"]],"source":"$\\delta^*$ has no local extremum at infinite expansions in $E^*$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#not-is-local-extr-zeckendorf-delta-star","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"$\\delta^*$ has no local extremum at infinite expansions in $E^*$. Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries whose periodic Zeckendorf collections satisfy $E \\subsetneq \\tilde E$. Let $Q$ be a fundamental sequence of the collection $E^*$ determined by $L$, let $\\tilde Q$ be a fundamental sequence of the collection $\\tilde E^*$ determined by $\\tilde L$, and\u2026","title":"$\\delta^*$ has no local extremum at infinite expansions in $E^*$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/eventually-zeckendorf-delta-star-lt-of-m","kind":"node","lean":"MetaMathlibExt.eventually_zeckendorfDeltaStar_lt_of_mem_lStarClosure","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Strict local maxima of "],["math","\\(\\delta^{\\ast}\\)"],["text"," at the closure points"]],"source":"Strict local maxima of $\\delta^{\\ast}$ at the closure points"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#eventually-zeckendorf-delta-star-lt-of-m","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, let $Q$ be a fundamental sequence of the collection $E^*$ determined by $L$, and let $\\tilde Q$ be a fundamental sequence of the collection $\\tilde E^*$ determined by $\\tilde L$. Let $\\varepsilon \\in \\bar E^* \\setminus E^*$, where $\\bar E^*$ is the closure collection of $L$, and put $x = \\sum_k \\varepsilon_k \\tilde Q_k$. Then, for the function $\\delta^*$ built\u2026","title":"Strict local maxima of $\\delta^{\\ast}$ at the closure points"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-local-extr-zeckendorf-delta-star-clas","kind":"node","lean":"MetaMathlibExt.isLocalExtr_zeckendorfDeltaStar_classification","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classification of the local extrema of "],["math","\\(\\delta^\\ast\\)"]],"source":"Classification of the local extrema of $\\delta^\\ast$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-local-extr-zeckendorf-delta-star-clas","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, let $Q$ and $\\tilde Q$ be fundamental sequences of the $L^*$-collection $E^*$ and the $\\tilde L^*$-collection $\\tilde E^*$, let $\\bar E^*$ be the $L^*$-closure, and let $\\delta^*$ be the function determined by $L$, $\\tilde L$, $Q$ and $\\tilde Q$. For a coefficient function $\\varepsilon$ write $x_\\varepsilon = \\sum_k \\varepsilon_k \\tilde Q_k$. Then (i) for\u2026","title":"Classification of the local extrema of $\\delta^\\ast$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-delta-star-lt-generic-bound","kind":"node","lean":"MetaMathlibExt.zeckendorfDeltaStar_lt_genericBound","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generic upper bound for "],["math","\\(\\delta^*\\)"],["text"," (Lemma 17)"]],"source":"Generic upper bound for $\\delta^*$ (Lemma 17)"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-delta-star-lt-generic-bound","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Generic upper bound for $\\delta^*$ (Lemma 17). Let $L$ and $\\tilde L$ be lists of length at least $2$ with nonzero first entries whose periodic Zeckendorf collections satisfy $E \\subsetneq \\tilde E$, and let $Q$ and $\\tilde Q$ be fundamental sequences of the collections $E^*$ and $\\tilde E^*$ of $L$ and $\\tilde L$. Write $\\omega$ and $\\tilde\\omega$ for the first terms of $Q$ and $\\tilde Q$,\u2026","title":"Generic upper bound for $\\delta^*$ (Lemma 17)"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/div-rpow-lt-add-div-add-rpow-of-generic","kind":"node","lean":"MetaMathlibExt.div_rpow_lt_add_div_add_rpow_of_genericBound","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#div-rpow-lt-add-div-add-rpow-of-generic","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, let $Q$ be a fundamental sequence of the collection $E^*$ determined by $L$, and let $\\tilde Q$ be a fundamental sequence of the collection $\\tilde E^*$ determined by $\\tilde L$. Put $\\omega = Q_0$, $\\tilde\\omega = \\tilde Q_0$ and $\\gamma = \\log_{\\tilde\\omega} \\omega$, and let $p$ be the generic exponent of $Q, \\tilde Q$. Let $\\varepsilon \\in \\tilde E^*$ with\u2026","title":"Monotonicity of the ratio under steep increments"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-delta-star-lt-add-fundamental","kind":"node","lean":"MetaMathlibExt.zeckendorfDeltaStar_lt_add_fundamental","parent":12383,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-delta-star-lt-add-fundamental","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ form a periodic Zeckendorf pair, let $Q$ be a fundamental sequence of the collection $E^*$ determined by $L$, and let $\\tilde Q$ be a fundamental sequence of the collection $\\tilde E^*$ determined by $\\tilde L$; indices start at $0$. Put $\\omega = Q_0$, $\\tilde\\omega = \\tilde Q_0$ and $\\gamma = \\log_{\\tilde\\omega} \\omega$, let $p$ be the generic exponent of $Q$ and $\\tilde Q$, let $\\delta^*$\u2026","title":"Monotonicity of the ratio under fundamental increments (Corollary 3)"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/is-max-on-zeckendorf-delta-star-bound","kind":"node","lean":"MetaMathlibExt.isMaxOn_zeckendorfDeltaStar_bound","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Where "],["math","\\(\\delta^\\ast\\)"],["text"," can attain its maximum"]],"source":"Where $\\delta^\\ast$ can attain its maximum"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#is-max-on-zeckendorf-delta-star-bound","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Let $L$ and $\\tilde L$ be lists of natural numbers forming a periodic Zeckendorf pair, let $N$ be the length of $L$, let $E^*$ and $\\tilde E^*$ be the collections determined by $L$ and $\\tilde L$, let $\\bar E^*$ be the closure of $E^*$, and let $Q$ and $\\tilde Q$ be fundamental sequences of $E^*$ and $\\tilde E^*$. Let $\\delta^*$ be the function $\\delta^*$ of $L$, $\\tilde L$, $Q$, $\\tilde Q$, and let $p$ be the\u2026","title":"Where $\\delta^\\ast$ can attain its maximum"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/exists-is-min-on-zeckendorf-delta-star","kind":"node","lean":"MetaMathlibExt.exists_isMinOn_zeckendorfDeltaStar","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Where "],["math","\\(\\delta^*\\)"],["text"," attains its minimum"]],"source":"Where $\\delta^*$ attains its minimum"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#exists-is-min-on-zeckendorf-delta-star","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Where $\\delta^*$ attains its minimum. Let $L$ and $\\tilde L$ be lists of natural numbers of length at least $2$ with nonzero first entries whose periodic Zeckendorf collections satisfy $E \\subsetneq \\tilde E$. Let $Q$ and $\\tilde Q$ be fundamental sequences of the $L^*$-collections $E^*$ of $L$ and $\\tilde E^*$ of $\\tilde L$, let $\\delta^*$ be the function on $(0, 1)$ built from $L$, $\\tilde L$, $Q$ and $\\tilde Q$,\u2026","title":"Where $\\delta^*$ attains its minimum"},{"id":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/zeckendorf-delta-star-lt-shift-base-n","kind":"node","lean":"MetaMathlibExt.zeckendorfDeltaStar_lt_shift_baseN","parent":12383,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shifting a unit one entry down increases "],["math","\\(\\delta^*\\)"],["text"," (Lemma 19)"]],"source":"Shifting a unit one entry down increases $\\delta^*$ (Lemma 19)"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions/index.html#zeckendorf-delta-star-lt-shift-base-n","scope":"NumberTheory/digits-numeration/numeration-automatic/distribution-of-zeckendorf-expressions","status":"can_prove","summary":"Shifting a unit one entry down increases $\\delta^*$ (Lemma 19). Let $N \\ge 3$, let $L = (1, \\dots, 1, 0)$ ($N - 1$ ones followed by $0$, the $N$th order Zeckendorf collection) and $\\tilde L = (N - 1, \\dots, N - 1)$ ($N$ entries, the base-$N$ collection), and let $Q$ and $\\tilde Q$ be fundamental sequences of the collections $E^*$ and $\\tilde E^*$ of $L$ and $\\tilde L$. Write $\\varepsilon_k$ for the $k$th entry of a\u2026","title":"Shifting a unit one entry down increases $\\delta^*$ (Lemma 19)"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base.json","id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","title":"Golden ratio base"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-value","kind":"node","lean":"MetaMathlibExt.basePhiValue","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-value","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For a finite set $S$ of integers, the base phi value of $S$ is the real number $\\sum_{i \\in S} \\varphi^i$, where $\\varphi = (1 + \\sqrt 5)/2$ is the golden ratio. It is the number represented by the base phi expansion with digit $1$ at each position in $S$ and digit $0$ at every other position.","title":"Value of a base phi expansion"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-bergman-expansion","kind":"node","lean":"MetaMathlibExt.IsBergmanExpansion","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-bergman-expansion","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Encode a finite base phi expansion with digits $0$ and $1$ by the finite set $S \\subseteq \\mathbb{Z}$ of positions of its digits $1$. The expansion is a Bergman expansion when no two digits $1$ are adjacent, that is, $i + 1 \\notin S$ for every $i \\in S$.","title":"Bergman expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-golden-mean-flip","kind":"node","lean":"MetaMathlibExt.IsGoldenMeanFlip","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-golden-mean-flip","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Encode a finite base phi expansion with digits $0$ and $1$ by the finite set of positions of its digits $1$. An expansion $T$ arises from an expansion $S$ by a golden mean flip $100 \\to 011$ when there is a position $i \\in S$ with $i - 1 \\notin S$ and $i - 2 \\notin S$ such that $T = (S \\setminus \\{i\\}) \\cup \\{i - 1, i - 2\\}$: the digit $1$ at position $i$ is replaced by digits $1$ at positions $i - 1$ and $i - 2$,\u2026","title":"Golden mean flips"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/satisfies-knott-condition","kind":"node","lean":"MetaMathlibExt.SatisfiesKnottCondition","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#satisfies-knott-condition","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. Encode a finite base $\\varphi$ expansion with digits $0$ and $1$ by the finite set $S \\subseteq \\mathbb{Z}$ of the positions of its digits $1$, and write $a_i$ for its digit at position $i$. The expansion satisfies Knott's condition when it does not end in the digits $0, 1, 1$: if $S$ is nonempty with least element $R$, then $a_{R+2} a_{R+1} a_R \\ne 011$, that is, it is not the\u2026","title":"Knott's truncation rule"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/knott-expansions","kind":"node","lean":"MetaMathlibExt.knottExpansions","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#knott-expansions","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio and $N$ a natural number. Encode a finite base phi expansion with digits $0$ and $1$ by the finite set $S \\subseteq \\mathbb{Z}$ of the positions of its digits $1$. The Knott expansions of $N$ are the expansions of value $N$ that satisfy Knott's truncation rule: the set of all such $S$ with $\\sum_{i \\in S} \\varphi^i = N$ that satisfy Knott's condition.","title":"Knott expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/knott-expansion-count","kind":"node","lean":"MetaMathlibExt.knottExpansionCount","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#knott-expansion-count","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For a natural number $N$, $\\mathrm{Tot}_\\kappa(N)$ is the number of Knott expansions of $N$ (OEIS A289749).","title":"Number of Knott expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/natural-expansions","kind":"node","lean":"MetaMathlibExt.naturalExpansions","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#natural-expansions","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio and $N$ a natural number. Encode a finite base phi expansion with digits $0$ and $1$ by the finite set $S \\subseteq \\mathbb{Z}$ of the positions of its digits $1$. The natural expansions of $N$ are the Knott expansions $S$ of $N$ for which there is a Bergman expansion $T$ with $\\sum_{i \\in T} \\varphi^i = N$ whose lowest digit $1$ is at the same position as the lowest digit $1$ of\u2026","title":"Natural base phi expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/natural-expansion-count","kind":"node","lean":"MetaMathlibExt.naturalExpansionCount","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#natural-expansion-count","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For a natural number $N$, $\\mathrm{Tot}_\\nu(N)$ is the number of natural base phi expansions of $N$.","title":"Number of natural expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/fibonacci-representation-count","kind":"node","lean":"MetaMathlibExt.fibonacciRepresentationCount","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#fibonacci-representation-count","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For a natural number $m$, $\\mathrm{Tot}_{\\mathrm{FIB}}(m)$ is the number of finite sets $T$ of indices $i \\ge 2$ with $\\sum_{i \\in T} F_i = m$, where $F_i$ are the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$); that is, the number of ways to write $m$ as a sum of Fibonacci numbers with distinct indices at least $2$ (OEIS A000119). Indices above $m + 1$ never occur because $F_{m+2} > m$.","title":"Number of Fibonacci representations"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-value-empty","kind":"node","lean":"MetaMathlibExt.basePhiValue_empty","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-value-empty","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The base phi value of the empty set of positions is $0$.","title":"The empty base phi expansion has value zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-value-singleton-zero","kind":"node","lean":"MetaMathlibExt.basePhiValue_singleton_zero","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","The single digit "],["math","\\(1\\)"],["text"," at position "],["math","\\(0\\)"],["text"," has value one"]],"source":"The single digit $1$ at position $0$ has value one"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-value-singleton-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The base phi value of the set $\\{0\\}$ is $1$, that is, $\\varphi^0 = 1$, where $\\varphi$ is the golden ratio.","title":"The single digit $1$ at position $0$ has value one"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-value-one-neg-two","kind":"node","lean":"MetaMathlibExt.basePhiValue_one_neg_two","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","The base phi expansion "],["math","\\(10.01\\)"],["text"," has value two"]],"source":"The base phi expansion $10.01$ has value two"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-value-one-neg-two","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The base phi value of the set $\\{1, -2\\}$ is $2$, that is, $\\varphi + \\varphi^{-2} = 2$, where $\\varphi$ is the golden ratio.","title":"The base phi expansion $10.01$ has value two"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-value-zero-neg-one-neg-two","kind":"node","lean":"MetaMathlibExt.basePhiValue_zero_neg_one_neg_two","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","The base phi expansion "],["math","\\(1.11\\)"],["text"," has value two"]],"source":"The base phi expansion $1.11$ has value two"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-value-zero-neg-one-neg-two","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The base phi value of the set $\\{0, -1, -2\\}$ is $2$, that is, $1 + \\varphi^{-1} + \\varphi^{-2} = 2$, where $\\varphi$ is the golden ratio.","title":"The base phi expansion $1.11$ has value two"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-bergman-expansion-one-neg-two","kind":"node","lean":"MetaMathlibExt.isBergmanExpansion_one_neg_two","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-bergman-expansion-one-neg-two","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The base phi expansion $10.01$, with digits $1$ at positions $1$ and $-2$ and digits $0$ elsewhere, has no two adjacent digits $1$, so it is a Bergman expansion.","title":"The expansion 10.01 is a Bergman expansion"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-golden-mean-flip-one-neg-two","kind":"node","lean":"MetaMathlibExt.isGoldenMeanFlip_one_neg_two","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-golden-mean-flip-one-neg-two","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Applying the golden mean flip $100 \\to 011$ at the leading digit of $10.01$ (the digit $1$ at position $1$, followed by digits $0$ at positions $0$ and $-1$) gives $1.11$, the expansion with digits $1$ at positions $0$, $-1$ and $-2$.","title":"A golden mean flip from 10.01 to 1.11"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/zero-neg-one-neg-two-mem-knott-expansion","kind":"node","lean":"MetaMathlibExt.zero_neg_one_neg_two_mem_knottExpansions","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#zero-neg-one-neg-two-mem-knott-expansion","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. The base $\\varphi$ expansion $1.11$, whose digits $1$ sit at the positions $0$, $-1$ and $-2$, is a Knott expansion of $2$: its value is $1 + \\varphi^{-1} + \\varphi^{-2} = 2$ and it satisfies Knott's truncation rule.","title":"A Knott expansion of two"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/not-satisfies-knott-condition-neg-one-ne","kind":"node","lean":"MetaMathlibExt.not_satisfiesKnottCondition_neg_one_neg_two","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#not-satisfies-knott-condition-neg-one-ne","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The finite base phi expansion $0.11$, with digits $1$ at positions $-1$ and $-2$ and no others, ends in the digits $011$, so it does not satisfy Knott's truncation rule.","title":"The expansion 0.11 violates Knott's truncation rule"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/one-neg-two-mem-natural-expansions","kind":"node","lean":"MetaMathlibExt.one_neg_two_mem_naturalExpansions","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#one-neg-two-mem-natural-expansions","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The finite base phi expansion $10.01$, with digits $1$ exactly at positions $1$ and $-2$, is a natural base phi expansion of $2$.","title":"The expansion 10.01 is a natural expansion of 2"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/fibonacci-representation-count-small","kind":"node","lean":"MetaMathlibExt.fibonacciRepresentationCount_small","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#fibonacci-representation-count-small","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $F_i$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$), and let $\\mathrm{Tot}_{\\mathrm{FIB}}(m)$ be the number of ways to write $m$ as a sum of Fibonacci numbers $F_i$ with distinct indices $i \\ge 2$. Then $\\mathrm{Tot}_{\\mathrm{FIB}}(0), \\dots, \\mathrm{Tot}_{\\mathrm{FIB}}(8)$ are $1, 1, 1, 2, 1, 2, 2, 1, 3$.","title":"Small values of the Fibonacci representation count"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/refl-trans-gen-is-golden-mean-flip-of-is","kind":"node","lean":"MetaMathlibExt.reflTransGen_isGoldenMeanFlip_of_isBergmanExpansion","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#refl-trans-gen-is-golden-mean-flip-of-is","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $N$ be a natural number, let $S$ be a Bergman expansion of $N$ and let $T$ be any finite base phi expansion of $N$ with digits $0$ and $1$. Then $T$ is obtained from $S$ by finitely many (possibly zero) golden mean flips $100 \\to 011$.","title":"Golden mean flips connect all expansions to the Bergman expansion"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/knott-expansion-count-eq-of-recursion","kind":"node","lean":"MetaMathlibExt.knottExpansionCount_eq_of_recursion","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#knott-expansion-count-eq-of-recursion","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio, let $N \\ge 2$ and $n$ be natural numbers, and let $p_0, \\dots, p_n$ be integers with $p_{k+1} \\ge p_k + 2$ for all $k < n$ and $\\sum_{k=0}^{n} \\varphi^{p_k} = N$. Thus $N$ has the base phi expansion $10^{s_n} \\cdots 10^{s_1} 1$ with digits $1$ at positions $p_0 < \\dots < p_n$, where $s_k = p_k - p_{k-1} - 1$ for $1 \\le k \\le n$. Let $r_0, r_1, \\dots$ be integers with $r_0 = 1$; if\u2026","title":"Recursion for the number of Knott expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/bergman-word-fib","kind":"node","lean":"MetaMathlibExt.bergmanWord_fib","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#bergman-word-fib","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$ and write $F_m$ for the $m$-th Fibonacci number, with $F_1 = F_2 = 1$. Let $S$ be a finite set of integers with no two consecutive elements (a Bergman expansion). If the base phi value of $S$ is $F_{2n}$, then $S = \\{t + 4j : 0 \\le j < n\\}$ for some integer $t$. If the base phi value of $S$ is $F_{2n+1}$, then $S = \\{t\\} \\cup \\{t + 3 + 4j : 0 \\le j < n\\}$ for some integer $t$.","title":"Bergman expansions of the Fibonacci numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/knott-expansion-count-fib","kind":"node","lean":"MetaMathlibExt.knottExpansionCount_fib","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#knott-expansion-count-fib","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). For every $n \\ge 1$, the number of Knott expansions of $F_n$ is $F_n$.","title":"Knott expansions of Fibonacci numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/knott-expansion-count-lucas-number","kind":"node","lean":"MetaMathlibExt.knottExpansionCount_lucasNumber","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#knott-expansion-count-lucas-number","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"For every $n \\ge 1$, each of the Lucas numbers $L_{2n}$ and $L_{2n+1}$ (with $L_0 = 2$, $L_1 = 1$) has exactly $2n + 1$ Knott expansions.","title":"Knott expansions of Lucas numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/natural-expansion-count-eq-of-recursion","kind":"node","lean":"MetaMathlibExt.naturalExpansionCount_eq_of_recursion","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#natural-expansion-count-eq-of-recursion","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $N$ and $n$ be natural numbers and let $p_0, \\dots, p_n$ be integers with $p_k + 2 \\le p_{k+1}$ for $k < n$ such that the base phi expansion with digits $1$ at positions $p_0, \\dots, p_n$ represents $N$. Put $s_k = p_k - p_{k-1} - 1$ for $1 \\le k \\le n$. Let $r_0, r_1, \\dots$ be integers with $r_0 = 1$ such that, if $n \\ge 1$, $2 r_1 = s_1 + 2$ when $s_1$ is even and $2 r_1 = s_1 + 1$ when $s_1$ is odd, and for\u2026","title":"Recursion for the number of natural expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/fib-mem-lucas-intervals","kind":"node","lean":"MetaMathlibExt.fib_mem_lucasIntervals","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#fib-mem-lucas-intervals","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$) and $L_k$ the Lucas numbers ($L_0 = 2$, $L_1 = 1$). Call $\\Lambda_{2n} = [L_{2n}, L_{2n+1}]$ the even Lucas interval and $\\Lambda_{2n+1} = [L_{2n+1} + 1, L_{2n+2} - 1]$ the odd Lucas interval. For every $n \\ge 1$, $F_{2n+2} \\in \\Lambda_{2n}$ and $F_{2n+3} \\in \\Lambda_{2n+1}$.","title":"Fibonacci numbers in the Lucas intervals"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/natural-expansion-count-fib","kind":"node","lean":"MetaMathlibExt.naturalExpansionCount_fib","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#natural-expansion-count-fib","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Write $\\mathrm{Tot}_\\nu(N)$ for the number of natural base phi expansions of a natural number $N$ and $F_k$ for the $k$-th Fibonacci number. Then for every $n \\ge 0$, $\\mathrm{Tot}_\\nu(F_{2n+2}) = F_{2n+1}$ and $\\mathrm{Tot}_\\nu(F_{2n+3}) = F_{2n+3}$.","title":"Natural expansions of Fibonacci numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/natural-expansion-count-eq-fibonacci-rep","kind":"node","lean":"MetaMathlibExt.naturalExpansionCount_eq_fibonacciRepresentationCount","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#natural-expansion-count-eq-fibonacci-rep","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Write $F_k$ for the $k$-th Fibonacci number. Let $N > 3$ be a natural number, let $S$ be a Bergman expansion of $N$ and let $R$ be the lowest position of a digit $1$ in $S$. Then the number $\\mathrm{Tot}_\\nu(N)$ of natural base phi expansions of $N$ equals the number $\\mathrm{Tot}_{\\mathrm{FIB}}(F_{2-R} N)$ of representations of $F_{2-R} N$ as a sum of Fibonacci numbers with distinct indices at least $2$.","title":"Natural expansions are counted by Fibonacci representations"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/bergman-tail-parity-iff-lucas-interval","kind":"node","lean":"MetaMathlibExt.bergman_tail_parity_iff_lucasInterval","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lucas interval of "],["math","\\(N\\)"],["text"," from the gap between its two lowest Bergman digits"]],"source":"Lucas interval of $N$ from the gap between its two lowest Bergman digits"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#bergman-tail-parity-iff-lucas-interval","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $N \\ge 2$ be a natural number, let $S$ be a finite set of integers with no two consecutive elements (a Bergman expansion) whose base phi value is $N$, and let $a$ be an integer and $s$ a natural number such that the elements of $S$ that are at most $a + s + 1$ are exactly $a$ and $a + s + 1$ (so the expansion ends in the digits $10^s1$). Write $L_m$ for the $m$-th Lucas number. Then $L_{2n} \\le N \\le L_{2n+1}$\u2026","title":"Lucas interval of $N$ from the gap between its two lowest Bergman digits"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/bergman-lowest-position-lucas-interval","kind":"node","lean":"MetaMathlibExt.bergman_lowest_position_lucasInterval","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lowest Bergman digit of "],["math","\\(N\\)"],["text"," from its Lucas interval"]],"source":"Lowest Bergman digit of $N$ from its Lucas interval"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#bergman-lowest-position-lucas-interval","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $N$ and $n$ be natural numbers, let $S$ be a finite set of integers with no two consecutive elements (a Bergman expansion) whose base phi value is $N$, and let $R$ be the least element of $S$. Write $L_m$ for the $m$-th Lucas number. If $L_{2n} \\le N \\le L_{2n+1}$, then $-R = 2n$. If $L_{2n+1} + 1 \\le N \\le L_{2n+2} - 1$, then $-R = 2n + 2$.","title":"Lowest Bergman digit of $N$ from its Lucas interval"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/bergman-lucas-even-add","kind":"node","lean":"MetaMathlibExt.bergman_lucas_even_add","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bergman expansion of "],["math","\\(L_{2n} + k\\)"]],"source":"Bergman expansion of $L_{2n} + k$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#bergman-lucas-even-add","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$ and let $k$ be a natural number with $k \\le L_{2n-1}$, where $L_m$ is the $m$-th Lucas number. Let $S$ and $T$ be finite sets of integers, each with no two consecutive elements (Bergman expansions), whose base phi values are $k$ and $L_{2n} + k$ respectively. Then $2n \\notin S$, $-2n \\notin S$, and $T = S \\cup \\{2n, -2n\\}$.","title":"Bergman expansion of $L_{2n} + k$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/bergman-lucas-odd-add","kind":"node","lean":"MetaMathlibExt.bergman_lucas_odd_add","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#bergman-lucas-odd-add","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 2$, let $k$ be a natural number, and write $L_m$ for the $m$-th Lucas number. Call a finite set of integers with no two consecutive elements a Bergman expansion. Let $S$ and $T$ be Bergman expansions, let $a$ and $t$ be the least and the greatest element of $S$, let $S' = S \\setminus \\{a, t\\}$, and for an integer $c$ write $S' + c = \\{x + c : x \\in S'\\}$.","title":"Recursive structure of Bergman expansions, odd Lucas case"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-base-phi-expansion","kind":"node","lean":"MetaMathlibExt.IsBasePhiExpansion","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-base-phi-expansion","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. A finite set $S$ of integers, read as the positions of the digits $1$ of a finite expansion with digits $0$ and $1$, is a base phi expansion of the natural number $N$ when it is a Bergman expansion (no two digits $1$ are adjacent: $i \\in S$ implies $i + 1 \\notin S$) and its value $\\sum_{i \\in S} \\varphi^i$ equals $N$.","title":"Base phi expansion"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-base-phi-expansion-zero","kind":"node","lean":"MetaMathlibExt.isBasePhiExpansion_zero","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-base-phi-expansion-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The empty set of positions, the expansion with no digit $1$, is a base phi expansion of $0$: it has no two adjacent digits $1$, and its value, the empty sum, is equal to $0$.","title":"Base phi expansion of zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-base-phi-expansion-one","kind":"node","lean":"MetaMathlibExt.isBasePhiExpansion_one","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-base-phi-expansion-one","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The set $\\{0\\}$, read as the positions of the digits $1$, is a base phi expansion of $1$: the expansion with a single digit $1$ at position $0$ has no two adjacent digits $1$, and its value $\\varphi^0$ equals $1$, where $\\varphi$ is the golden ratio.","title":"Base phi expansion of one"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/has-digit-block","kind":"node","lean":"MetaMathlibExt.HasDigitBlock","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#has-digit-block","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $S$ be a finite set of integers, read as the positions of the digits $1$ of an expansion with digits $d_i$, so $d_i = 1$ if $i \\in S$ and $d_i = 0$ otherwise. For an integer $k$ and a finite word $w = w_0 w_1 \\cdots w_{m-1}$ over the natural numbers, $w$ occurs as a digit block of $S$ at position $k$ when the digits $d_k d_{k-1} \\cdots d_{k-m+1}$, read from the highest position down, are the letters of $w$: for\u2026","title":"Digit block of an expansion"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/has-digit-block-nil","kind":"node","lean":"MetaMathlibExt.hasDigitBlock_nil","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#has-digit-block-nil","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For every finite set $S$ of integers, read as the positions of the digits $1$ of an expansion, and every integer $k$, the empty word occurs as a digit block of $S$ at position $k$.","title":"The empty digit block occurs everywhere"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/has-digit-block-cons","kind":"node","lean":"MetaMathlibExt.hasDigitBlock_cons","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#has-digit-block-cons","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $S$ be a finite set of integers, read as the positions of the digits $1$ of an expansion. For every integer $k$, every natural number $b$ and every finite word $w$ over the natural numbers, the word $bw$ (the letter $b$ followed by $w$) occurs as a digit block of $S$ at position $k$ if and only if both $k \\in S$ holds exactly when $b = 1$, and $w$ occurs as a digit block of $S$ at position $k - 1$.","title":"Peeling off the leading letter of a digit block"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-occurrences","kind":"node","lean":"MetaMathlibExt.basePhiOccurrences","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-occurrences","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For an integer $k$ and a word $w = w_1 w_2 \\cdots w_\\ell$ of natural numbers, the occurrence set of $w$ at position $k$ is the set of natural numbers $N$ that have a base phi expansion $S$ (the set of positions of its digits $1$) whose digits $d_k d_{k-1} \\cdots d_{k-\\ell+1}$ spell $w$: for each $j = 1, \\dots, \\ell$, the position $k - j + 1$ lies in $S$ if and only if $w_j = 1$.","title":"Occurrence set of a digit block"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/one-mem-base-phi-occurrences-one","kind":"node","lean":"MetaMathlibExt.one_mem_basePhiOccurrences_one","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#one-mem-base-phi-occurrences-one","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The number $1$ lies in the occurrence set of the one-letter digit block $1$ placed at position $0$ (the paper's $R_1$): some base phi expansion of $1$ has the digit $1$ at position $0$.","title":"A base phi expansion of 1 ends in the digit 1"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/generalized-beatty","kind":"node","lean":"MetaMathlibExt.generalizedBeatty","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#generalized-beatty","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. For integers $p, q, r$, the generalized Beatty sequence $V(p, q, r)$ is the integer sequence defined for every natural number $n$, including $n = 0$, by $V(p, q, r)(n) = p\\lfloor n\\varphi \\rfloor + qn + r$.","title":"Generalized Beatty sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/generalized-beatty-zero","kind":"node","lean":"MetaMathlibExt.generalizedBeatty_zero","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#generalized-beatty-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. For all integers $p, q, r$, the generalized Beatty sequence $V(p, q, r)(n) = p\\lfloor n\\varphi \\rfloor + qn + r$ takes the value $V(p, q, r)(0) = r$ at $n = 0$.","title":"Generalized Beatty sequence at zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/generalized-beatty-one","kind":"node","lean":"MetaMathlibExt.generalizedBeatty_one","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#generalized-beatty-one","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. For all integers $p, q, r$, the generalized Beatty sequence $V(p, q, r)(n) = p\\lfloor n\\varphi \\rfloor + qn + r$ takes the value $V(p, q, r)(1) = p + q + r$ at $n = 1$, since $\\lfloor \\varphi \\rfloor = 1$.","title":"Generalized Beatty sequence at one"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/beatty-set","kind":"node","lean":"MetaMathlibExt.beattySet","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#beatty-set","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For integers $p$, $q$ and $r$, this is the set of integers $p \\lfloor n\\varphi \\rfloor + qn + r$ for natural numbers $n \\ge 1$, where $\\varphi$ is the golden ratio: the set of terms $V(p, q, r)(n)$, $n \\ge 1$, of the generalized Beatty sequence $V(p, q, r)$.","title":"Terms of a generalized Beatty sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/beatty-set-2","kind":"node","lean":"MetaMathlibExt.beattySet\u2080","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#beatty-set-2","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For integers $p$, $q$ and $r$, this is the set of integers $p \\lfloor n\\varphi \\rfloor + qn + r$ for all natural numbers $n \\ge 0$, where $\\varphi$ is the golden ratio: the set of values $V(p, q, r)(n)$, $n \\ge 0$, of the generalized Beatty sequence $V(p, q, r)$, which the paper denotes $V_0(p, q, r)$.","title":"Terms of a generalized Beatty sequence from index zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/add-add-mem-beatty-set","kind":"node","lean":"MetaMathlibExt.add_add_mem_beattySet","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#add-add-mem-beatty-set","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. For all integers $p, q, r$, the number $p + q + r$ is a term $p \\lfloor n \\varphi \\rfloor + q n + r$, $n \\ge 1$, of the generalized Beatty sequence $V(p, q, r)$; it is the term at $n = 1$, since $\\lfloor \\varphi \\rfloor = 1$.","title":"The value at one is a term of a generalized Beatty sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/mem-beatty-set-self","kind":"node","lean":"MetaMathlibExt.mem_beattySet\u2080_self","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#mem-beatty-set-self","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For all integers $p$, $q$ and $r$, the integer $r$ belongs to the set $V_0(p, q, r)$ of terms $p \\lfloor n\\varphi \\rfloor + qn + r$, $n \\ge 0$, of the generalized Beatty sequence, where $\\varphi$ is the golden ratio; indeed $r$ is its term at $n = 0$.","title":"The initial term of a generalized Beatty set"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-no-block-one-zeros-dot","kind":"node","lean":"MetaMathlibExt.basePhi_no_block_one_zeros_dot","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","The blocks "],["math","\\(10^{2m} \\cdot 1\\)"],["text"," and "],["math","\\(10^{2m+1} \\cdot 0\\)"],["text"," never occur"]],"source":"The blocks $10^{2m} \\cdot 1$ and $10^{2m+1} \\cdot 0$ never occur"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-no-block-one-zeros-dot","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"For every $m \\ge 1$, no base phi expansion of a natural number has digits $d_{2m} \\cdots d_0 \\cdot d_{-1} = 10^{2m} \\cdot 1$, and none has digits $d_{2m+1} \\cdots d_0 \\cdot d_{-1} = 10^{2m+1} \\cdot 0$.","title":"The blocks $10^{2m} \\cdot 1$ and $10^{2m+1} \\cdot 0$ never occur"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-propagation-principle","kind":"node","lean":"MetaMathlibExt.basePhi_propagation_principle","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-propagation-principle","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $w = w_1 \\cdots w_8$ be a word of length $8$ over the natural numbers. Say that a base phi expansion $S$ (the set of positions of its digits $1$) contains $w$ centrally if for each $j = 1, \\dots, 8$ the position $4 - j$ lies in $S$ exactly when $w_j = 1$, that is, if its digits $d_3 d_2 d_1 d_0 \\cdot d_{-1} d_{-2} d_{-3} d_{-4}$ spell $w$. If no base phi expansion of any natural number $N \\le 17$ contains $w$\u2026","title":"Propagation principle for base phi expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/generalized-beatty-comp-wythoff","kind":"node","lean":"MetaMathlibExt.generalizedBeatty_comp_wythoff","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#generalized-beatty-comp-wythoff","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio and, for integers $a$, $b$, $c$ and natural numbers $n$, let $V(a, b, c)(n) = a \\lfloor n\\varphi \\rfloor + bn + c$. Let $A(n) = \\lfloor n\\varphi \\rfloor$ and $B(n) = \\lfloor n\\varphi \\rfloor + n$ be the lower and upper Wythoff sequences. For all integers $p$, $q$, $r$ and every natural number $n \\ge 1$, $V(p, q, r)(A(n)) = V(p + q, p, r - p)(n)$ and\u2026","title":"Generalized Beatty sequences composed with the Wythoff sequences"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/beatty-set-complement-of-lower-wythoff","kind":"node","lean":"MetaMathlibExt.beattySet_complement_of_lowerWythoff","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#beatty-set-complement-of-lower-wythoff","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio and, for integers $a$, $b$, $c$, let $V(a, b, c)$ be the set of integers $a \\lfloor n\\varphi \\rfloor + bn + c$ for natural numbers $n \\ge 1$. Let $p$, $q$, $r$ be integers with $p + q > 0$ and $2p + q > 0$. If $U$ and $W$ are disjoint sets of integers with $U \\cup W = V(p, q, r)$ and $U = V(p + q, p, r - p)$, then $W = V(2p + q, p + q, r)$.","title":"Complement of the lower Wythoff part of a generalized Beatty sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/generalized-beatty-diff-eq-fib-inf","kind":"node","lean":"MetaMathlibExt.generalizedBeatty_diff_eq_fibInf","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#generalized-beatty-diff-eq-fib-inf","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio, for integers $p, q, r$ let $V(p, q, r)$ be the sequence $n \\mapsto p\\lfloor n\\varphi \\rfloor + qn + r$ on the natural numbers, and let $x_0 x_1 x_2 \\cdots$ be the infinite Fibonacci word. For all integers $p, q, r$ and every $n \\ge 1$, the difference $V(p, q, r)(n+1) - V(p, q, r)(n)$ equals $p + q$ if $x_{n-1} = 1$ and $2p + q$ if $x_{n-1} = 0$. Conversely, for all integers $a, b$\u2026","title":"First differences of a generalized Beatty sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-no-end-block-one-zeros-one","kind":"node","lean":"MetaMathlibExt.basePhi_no_end_block_one_zeros_one","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","For "],["math","\\(m \\ge 2\\)"],["text",", no base phi expansion has end block "],["math","\\(1\\,0^{2m}\\,1\\)"]],"source":"For $m \\ge 2$, no base phi expansion has end block $1\\,0^{2m}\\,1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-no-end-block-one-zeros-one","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $m \\ge 2$, let $N$ be a natural number and let $S$ be a base phi expansion of $N$, given as the set of positions of its digits $1$. Then the digits $d_{2m+1} d_{2m} \\cdots d_1 d_0$ at positions $2m+1, \\dots, 0$ are not $1\\,0^{2m}\\,1$: it is not the case that $2m+1 \\in S$, $0 \\in S$ and no $i$ with $1 \\le i \\le 2m$ lies in $S$.","title":"For $m \\ge 2$, no base phi expansion has end block $1\\,0^{2m}\\,1$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-end-block-one-iff-pred","kind":"node","lean":"MetaMathlibExt.basePhi_end_block_one_iff_pred","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","End blocks "],["math","\\(u1 \\cdot 0\\)"],["text"," and "],["math","\\(u0 \\cdot 0\\)"],["text"," of consecutive numbers"]],"source":"End blocks $u1 \\cdot 0$ and $u0 \\cdot 0$ of consecutive numbers"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-end-block-one-iff-pred","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $u$ be a nonempty word of length $m - 1$, read as the digits $d_{m-1} \\cdots d_1$ (a letter $1$ is a digit $1$, any other letter a digit $0$), such that the word $u1$ has no two adjacent letters $1$. Then for every $N \\ge 1$, some base phi expansion of $N$ has digits $d_{m-1} \\cdots d_1 d_0 \\cdot d_{-1} = u1 \\cdot 0$ if and only if some base phi expansion of $N - 1$ has digits\u2026","title":"End blocks $u1 \\cdot 0$ and $u0 \\cdot 0$ of consecutive numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/fib-add-three-mul-fib-succ-eq-lucas-numb","kind":"node","lean":"MetaMathlibExt.fib_add_three_mul_fib_succ_eq_lucasNumber","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#fib-add-three-mul-fib-succ-eq-lucas-numb","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"For every natural number $n$, $F_n + 3F_{n+1} = L_{n+2}$, where $F_j$ is the $j$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$) and $L_j$ is the $j$-th Lucas number ($L_0 = 2$, $L_1 = 1$).","title":"Lucas numbers from Fibonacci numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-lucas-border-codes","kind":"node","lean":"MetaMathlibExt.basePhi_lucas_border_codes","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-lucas-border-codes","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$, and write $F_k$ for the Fibonacci numbers and $L_k$ for the Lucas numbers. Let $S_1, S_2, S_3, S_4$ be base phi expansions (sets of positions of the digits $1$) of $L_{2n}$, $L_{2n+1}$, $L_{2n+1} + 1$ and $L_{2n+2} - 1$ respectively, and let $S_k^-$ be the set of negative positions in $S_k$. Then $S_1^- = \\{-2n\\}$, $S_2^- = \\{-2j : 1 \\le j \\le n\\}$, $S_3^- = \\{-2n-2\\} \\cup \\{1 - 2j : 1 \\le j \\le n\\}$\u2026","title":"Border words and codes of Lucas numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-digits-dot-one","kind":"node","lean":"MetaMathlibExt.basePhi_digits_dot_one","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","The digit blocks "],["math","\\(10 \\cdot 1\\)"],["text"," and "],["math","\\(00 \\cdot 1\\)"],["text"," around the radix point"]],"source":"The digit blocks $10 \\cdot 1$ and $00 \\cdot 1$ around the radix point"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-digits-dot-one","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio, let $N \\ge 1$ and let $S$ be a base phi expansion of $N$, with digits $d_i$ ($d_i = 1$ if $i \\in S$, else $d_i = 0$). Then $d_1 d_0 \\cdot d_{-1} \\ne 10 \\cdot 1$, and $d_1 d_0 \\cdot d_{-1} = 00 \\cdot 1$ if and only if $N = 3 \\lfloor n \\varphi \\rfloor + n + 1$ for some integer $n \\ge 1$.","title":"The digit blocks $10 \\cdot 1$ and $00 \\cdot 1$ around the radix point"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-short-end-blocks-zero","kind":"node","lean":"MetaMathlibExt.basePhi_short_end_blocks_zero","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","End blocks of length at most three ending in "],["math","\\(0\\)"]],"source":"End blocks of length at most three ending in $0$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-short-end-blocks-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. For integers $p, q, r$ let $V(p, q, r) = \\{p \\lfloor n\\varphi \\rfloor + qn + r : n \\ge 1\\}$ and $V_0(p, q, r) = \\{p \\lfloor n\\varphi \\rfloor + qn + r : n \\ge 0\\}$. For a word $w$ of length $\\ell$ let $R_w$ be the set of natural numbers having a base phi expansion whose digits $d_{\\ell-1} \\cdots d_0$ at positions $\\ell - 1, \\dots, 0$ spell $w$. Then for every natural number\u2026","title":"End blocks of length at most three ending in $0$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-short-negative-prefixes","kind":"node","lean":"MetaMathlibExt.basePhi_short_negative_prefixes","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-short-negative-prefixes","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. For integers $p, q, r$ let $V(p, q, r) = \\{p \\lfloor n\\varphi \\rfloor + qn + r : n \\ge 1\\}$ and $V_0(p, q, r) = \\{p \\lfloor n\\varphi \\rfloor + qn + r : n \\ge 0\\}$. For a word $w$ of length $\\ell$ let $R_{\\cdot w}$ be the set of natural numbers having a base phi expansion whose digits $d_{-1} d_{-2} \\cdots d_{-\\ell}$ spell $w$, where $d_i = 1$ if position $i$ carries a digit $1$\u2026","title":"Short prefixes of the negative parts of base phi expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-positive-part-eq-zeckendorf","kind":"node","lean":"MetaMathlibExt.basePhi_positive_part_eq_zeckendorf","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-positive-part-eq-zeckendorf","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio and let $F_j$ be the Fibonacci numbers, $F_0 = 0$, $F_1 = F_2 = 1$. For a natural number $M$ let $Z(M)$ be the set of indices $i \\ge 0$ such that $F_{i+2}$ occurs in the Zeckendorf representation of $M$, so that $M = \\sum_{i \\in Z(M)} F_{i+2}$. Let $N$ be a natural number, let $S$ be a base phi expansion of $N$ (the set of positions of its digits $1$), and let $s(N)$ be the number\u2026","title":"Embedding of base phi expansions into Zeckendorf expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-recursive-structure","kind":"node","lean":"MetaMathlibExt.basePhi_recursive_structure","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-recursive-structure","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 2$ and let $L_j$ be the Lucas numbers; $k$ denotes a natural number, and base phi expansions are given as sets of positions of digits $1$. (I) If $1 \\le k \\le L_{2n-2} - 1$, $T$ is a base phi expansion of $L_{2n-1} + k$ and $S$ is a base phi expansion of $L_{2n+1} + k$, then $2n-1 \\in T$, $2n-2 \\notin T$, $1-2n \\notin T$, $-2n \\in T$ and $S = (T \\setminus \\{2n-1, -2n\\}) \\cup \\{2n+1, 1-2n, -2n-2\\}$. (K)\u2026","title":"Recursive structure theorem for base phi expansions, part (A)"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-recursive-structure-lucas-inter","kind":"node","lean":"MetaMathlibExt.basePhi_recursive_structure_lucas_intervals","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-recursive-structure-lucas-inter","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$, let $L_k$ be the Lucas numbers, and for $j \\ge 0$ let $\\Lambda_{2j}$ be the set of natural numbers in $[L_{2j}, L_{2j+1}]$ and $\\Lambda_{2j+1}$ the set of natural numbers in $[L_{2j+1} + 1, L_{2j+2} - 1]$. Let $T$ be a base phi expansion of a natural number $M$ and $S$ a base phi expansion of a natural number $N$, both given as sets of positions of digits $1$. Then: (A)(a) if $M \\in \\Lambda_{2n-1}$\u2026","title":"Recursive structure of base phi expansions on Lucas intervals"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-short-end-blocks","kind":"node","lean":"MetaMathlibExt.basePhi_short_end_blocks","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-short-end-blocks","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. For integers $p, q, r$ let $V(p, q, r)$ be the set of the integers $p \\lfloor n\\varphi \\rfloor + qn + r$ with $n \\ge 1$, and $V_0(p, q, r)$ the same set with $n \\ge 0$. Then for every natural number $N \\ge 1$: some base phi expansion of $N$ has digit $1$ at position $0$ if and only if $N \\in V_0(1, 2, 1)$; some base phi expansion of $N$ has digits $1, 0$ at positions $1, 0$ if and\u2026","title":"Short end blocks"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-end-blocks-lucas-wythoff","kind":"node","lean":"MetaMathlibExt.basePhi_end_blocks_lucasWythoff","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","End blocks ending in "],["math","\\(0\\)"],["text"," are Lucas\u2013Wythoff sequences"]],"source":"End blocks ending in $0$ are Lucas\u2013Wythoff sequences"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-end-blocks-lucas-wythoff","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio and $L_n$ the Lucas numbers. For integers $p, q, r$ let $V(p, q, r)$ be the set of the integers $p \\lfloor n\\varphi \\rfloor + qn + r$ with $n \\ge 1$, and $V_0(p, q, r)$ the same set with $n \\ge 0$. For a word $w = w_{m-1} \\cdots w_0$ of length $m$ over the natural numbers, let $R_w$ be the set of natural numbers $N$ with a base phi expansion whose digit at position $i$ is $1$\u2026","title":"End blocks ending in $0$ are Lucas\u2013Wythoff sequences"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-end-blocks-one-lucas-wythoff","kind":"node","lean":"MetaMathlibExt.basePhi_end_blocks_one_lucasWythoff","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","End blocks ending in "],["math","\\(1\\)"],["text"," are Lucas\u2013Wythoff sequences"]],"source":"End blocks ending in $1$ are Lucas\u2013Wythoff sequences"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-end-blocks-one-lucas-wythoff","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio and $L_n$ the Lucas numbers. For integers $p, q, r$ let $V(p, q, r)$ be the set of the integers $p \\lfloor n\\varphi \\rfloor + qn + r$ with $n \\ge 1$, and $V_0(p, q, r)$ the same set with $n \\ge 0$. For a word $w = w_{m-1} \\cdots w_0$ of length $m$ over the natural numbers, let $R_w$ be the set of natural numbers $N$ with a base phi expansion whose digit at position $i$ is $1$\u2026","title":"End blocks ending in $1$ are Lucas\u2013Wythoff sequences"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-negative-parts-surject","kind":"node","lean":"MetaMathlibExt.basePhi_negative_parts_surject","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every even-length Zeckendorf word ending in "],["math","\\(1\\)"],["text"," is a negative part"]],"source":"Every even-length Zeckendorf word ending in $1$ is a negative part"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-negative-parts-surject","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$ and let $T$ be a finite set of integers with $T \\subseteq [-2n, -1]$ and $-2n \\in T$ that contains no two consecutive integers. Then there are a natural number $N$ and a base phi expansion $S$ of $N$ (the set of positions of its digits $1$) with $\\{i \\in S : i < 0\\} = T$. In words, every Zeckendorf word of even length $2n \\ge 2$ ending in $1$ is the negative part $\\beta^-(N)$ of some $N$.","title":"Every even-length Zeckendorf word ending in $1$ is a negative part"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-recursive-structure-gamma","kind":"node","lean":"MetaMathlibExt.basePhi_recursive_structure_gamma","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-recursive-structure-gamma","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$, let $L_k$ be the Lucas numbers, and for $j \\ge 0$ let $\\Lambda_{2j}$ be the set of natural numbers in $[L_{2j}, L_{2j+1}]$ and $\\Lambda_{2j+1}$ the set of natural numbers in $[L_{2j+1} + 1, L_{2j+2} - 1]$. Let $T$ be a base phi expansion of a natural number $M$ and $S$ a base phi expansion of a natural number $N$, both given as sets of positions of digits $1$. Write $S^- = \\{i \\in S : i < 0\\}$ and,\u2026","title":"Recursive structure of the negative parts of base phi expansions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-negative-part-order-orbit","kind":"node","lean":"MetaMathlibExt.basePhi_negative_part_order_orbit","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-negative-part-order-orbit","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$, write $F_k$ for the Fibonacci numbers and $L_k$ for the Lucas numbers, and let $c$ be a function from the natural numbers to the natural numbers such that $c(N) = \\sum_{i \\in S,\\, i < 0} F_{i+2n}$ (with $F_k$ read as $0$ for $k < 0$) whenever $S$ is a base phi expansion of $N$. List the values $c(N)$ for $N = L_{2n-1} + 1, \\dots, L_{2n+1}$ (the interval $\\Xi_n$) in increasing order of $N$, and merge\u2026","title":"The codes of the negative parts follow an orbit of addition modulo a Fibonacci number"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/lucas-interval-lo","kind":"node","lean":"MetaMathlibExt.lucasIntervalLo","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#lucas-interval-lo","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $L_0 = 2$, $L_1 = 1$, $L_{j+2} = L_{j+1} + L_j$ be the Lucas numbers, and let the Lucas intervals be $\\Lambda_{2n} = [L_{2n}, L_{2n+1}]$ and $\\Lambda_{2n+1} = [L_{2n+1} + 1, L_{2n+2} - 1]$. The left end of $\\Lambda_m$ is $L_m$ when $m$ is even and $L_m + 1$ when $m$ is odd.","title":"Left end of a Lucas interval"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/lucas-interval-hi","kind":"node","lean":"MetaMathlibExt.lucasIntervalHi","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#lucas-interval-hi","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $L_0 = 2$, $L_1 = 1$, $L_{j+2} = L_{j+1} + L_j$ be the Lucas numbers, and let the Lucas intervals be $\\Lambda_{2n} = [L_{2n}, L_{2n+1}]$ and $\\Lambda_{2n+1} = [L_{2n+1} + 1, L_{2n+2} - 1]$. The right end of $\\Lambda_m$ is $L_{m+1}$ when $m$ is even and $L_{m+1} - 1$ when $m$ is odd.","title":"Right end of a Lucas interval"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/lucas-interval-three","kind":"node","lean":"MetaMathlibExt.lucasInterval_three","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Endpoints of the Lucas interval "],["math","\\(\\Lambda_3\\)"]],"source":"Endpoints of the Lucas interval $\\Lambda_3$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#lucas-interval-three","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The Lucas interval $\\Lambda_3$ has left end $5$ and right end $6$, that is, $\\Lambda_3 = [5, 6]$.","title":"Endpoints of the Lucas interval $\\Lambda_3$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/lucas-interval-four","kind":"node","lean":"MetaMathlibExt.lucasInterval_four","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Endpoints of the Lucas interval "],["math","\\(\\Lambda_4\\)"]],"source":"Endpoints of the Lucas interval $\\Lambda_4$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#lucas-interval-four","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The Lucas interval $\\Lambda_4$ has left end $7$ and right end $11$, that is, $\\Lambda_4 = [7, 11]$.","title":"Endpoints of the Lucas interval $\\Lambda_4$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/lucas-interval-five","kind":"node","lean":"MetaMathlibExt.lucasInterval_five","parent":12475,"reading_title":{"format":"inline-parts-v1","parts":[["text","Endpoints of the Lucas interval "],["math","\\(\\Lambda_5\\)"]],"source":"Endpoints of the Lucas interval $\\Lambda_5$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#lucas-interval-five","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The Lucas interval $\\Lambda_5$ has left end $12$ and right end $17$, that is, $\\Lambda_5 = [12, 17]$.","title":"Endpoints of the Lucas interval $\\Lambda_5$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-lucas-tiling","kind":"node","lean":"MetaMathlibExt.IsLucasTiling","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-lucas-tiling","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Write $\\Lambda_k = [\\ell_k, h_k]$ for the Lucas interval with left end $\\ell_k$ and right end $h_k$. For natural numbers $a$, $b$ and a list $P$ of pairs $(k, t)$ of natural numbers, $P$ is a Lucas tiling of $[a, b]$ when the translated Lucas intervals $\\Lambda_k + t$, $(k, t) \\in P$, taken in the order of $P$, are adjacent and their union is $[a, b]$. Precisely, it is defined by recursion on $P$: the empty list is\u2026","title":"Lucas tilings"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-lucas-tiling-nil","kind":"node","lean":"MetaMathlibExt.isLucasTiling_nil","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-lucas-tiling-nil","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For all natural numbers $a$ and $b$, the empty list is a Lucas tiling of $[a, b]$ if and only if $a = b + 1$.","title":"Lucas tilings by the empty list"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/is-lucas-tiling-cons","kind":"node","lean":"MetaMathlibExt.isLucasTiling_cons","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#is-lucas-tiling-cons","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Write $\\Lambda_k = [\\ell_k, h_k]$ for the Lucas interval with left end $\\ell_k$ and right end $h_k$. For all natural numbers $a$, $b$, $k$, $t$ and every list $P$ of pairs of natural numbers, the list consisting of $(k, t)$ followed by $P$ is a Lucas tiling of $[a, b]$ if and only if $\\ell_k + t = a$ and $P$ is a Lucas tiling of $[h_k + t + 1, b]$.","title":"Lucas tilings of a list with a first interval"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/canonical-lucas-splitting","kind":"node","lean":"MetaMathlibExt.canonicalLucasSplitting","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#canonical-lucas-splitting","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"Let $L_j$ be the $j$-th Lucas number ($L_0 = 2$, $L_1 = 1$, $L_{j+2} = L_{j+1} + L_j$). For a natural number $m$, the canonical splitting of the Lucas interval $\\Lambda_m$ is the list of pairs $(k, t)$ of natural numbers defined by recursion on $m$: for $m \\le 5$ it is the one-pair list $[(m, 0)]$, and for $m \\ge 6$ it is the concatenation, in this order, of the canonical splitting of $\\Lambda_{m-2}$ with $L_{m-1}$\u2026","title":"Canonical splitting of a Lucas interval"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/canonical-lucas-splitting-of-lt-six","kind":"node","lean":"MetaMathlibExt.canonicalLucasSplitting_of_lt_six","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#canonical-lucas-splitting-of-lt-six","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For every natural number $m < 6$, the canonical splitting of the Lucas interval $\\Lambda_m$ is the one-pair list $[(m, 0)]$, that is, $\\Lambda_m$ itself.","title":"Canonical splitting of a short Lucas interval"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/canonical-lucas-splitting-add-six","kind":"node","lean":"MetaMathlibExt.canonicalLucasSplitting_add_six","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#canonical-lucas-splitting-add-six","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"For every natural number $m$, the canonical splitting of the Lucas interval $\\Lambda_{m+6}$ is the concatenation, in this order, of the canonical splitting of $\\Lambda_{m+4}$ with $L_{m+5}$ added to the second coordinate of each pair, the canonical splitting of $\\Lambda_{m+3}$ with $L_{m+6}$ added to the second coordinate of each pair, and the canonical splitting of $\\Lambda_{m+4}$ with $L_{m+6}$ added to the\u2026","title":"Recursion for the canonical splitting"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/canonical-lucas-splitting-six","kind":"node","lean":"MetaMathlibExt.canonicalLucasSplitting_six","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#canonical-lucas-splitting-six","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"fully_proved","summary":"The canonical splitting of the Lucas interval $\\Lambda_6$ is the list $[(4, 11), (3, 18), (4, 18)]$.","title":"Canonical splitting of the sixth Lucas interval"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/lucas-interval-canonical-splitting-codin","kind":"node","lean":"MetaMathlibExt.lucasInterval_canonical_splitting_coding","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#lucas-interval-canonical-splitting-codin","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Write $\\Lambda_m = [\\ell_m, h_m]$ for the Lucas interval with left end $\\ell_m$ and right end $h_m$. For every $n \\ge 3$, every pair $(k, t)$ in the canonical splitting of $\\Lambda_n$ has $k \\in \\{3, 4, 5\\}$, and the canonical splitting of $\\Lambda_n$ is a Lucas tiling of $[\\ell_n, h_n]$. Let $\\kappa$ be the map on finite words of natural numbers that replaces each letter $3$ by the word $5$, each letter $4$ by\u2026","title":"Dekking's Theorem 4: the coding of the canonical splittings of Lucas intervals"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/xi-canonical-splitting-coding","kind":"node","lean":"MetaMathlibExt.xi_canonical_splitting_coding","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#xi-canonical-splitting-coding","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $L_m$ denote the $m$-th Lucas number and $\\Lambda_m$ the Lucas interval, so $\\Lambda_{2k} = [L_{2k}, L_{2k+1}]$ and $\\Lambda_{2k+1} = [L_{2k+1} + 1, L_{2k+2} - 1]$. For $n \\ge 2$, let $P_n$ be the canonical splitting of $\\Lambda_{2n-1}$ followed by the canonical splitting of $\\Lambda_{2n}$, a list of translations $\\Lambda_k + t$. Then every index $k$ occurring in $P_n$ is $3$, $4$ or $5$, and the intervals of\u2026","title":"Canonical splitting and coding of unions of two Lucas intervals"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-trident-splitting","kind":"node","lean":"MetaMathlibExt.basePhi_trident_splitting","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-trident-splitting","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$ and let $L_{2n}$ be the $2n$-th Lucas number. If $S_1$, $S_2$ and $S_3$ are base phi expansions of $L_{2n} - 1$, $L_{2n}$ and $L_{2n} + 1$ respectively, each given as the set of positions of its digits $1$, then the negative positions $i < 0$ of $S_1$, of $S_2$ and of $S_3$ are the same.","title":"Trident splitting at an even Lucas number"},{"id":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/base-phi-negative-part-order-rotation","kind":"node","lean":"MetaMathlibExt.basePhi_negative_part_order_rotation","parent":12475,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base/index.html#base-phi-negative-part-order-rotation","scope":"NumberTheory/digits-numeration/numeration-automatic/golden-ratio-base","status":"can_prove","summary":"Let $n \\ge 1$, write $\\varphi$ for the golden ratio, $F_k$ for the Fibonacci numbers and $L_k$ for the Lucas numbers, and let $c$ be a function from the natural numbers to the natural numbers such that $c(N) = \\sum_{i \\in S,\\, i < 0} F_{i+2n}$ (with $F_k$ read as $0$ for $k < 0$) whenever $S$ is a base phi expansion of $N$. List the values $c(N)$ for $N = L_{2n-1} + 1, \\dots, L_{2n+1}$ (the interval $\\Xi_n$) in\u2026","title":"The codes of the negative parts follow the golden rotation"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions.json","id":"NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions","status":"fully_proved","title":"Hyperfactorial partitions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions/hyperfactorial-digits-normalized","kind":"node","lean":"MetaMathlibExt.HyperfactorialDigitsNormalized","parent":12560,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions/index.html#hyperfactorial-digits-normalized","scope":"NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions","status":"fully_proved","summary":"A vector $(h_1, \\dots, h_k)$ of natural numbers is normalized when either $k = 0$ or its last entry $h_k$ is nonzero. In particular the empty vector is normalized; it is the canonical representation of zero.","title":"Normalized hyperfactorial digit vectors"},{"id":"NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions/hyperfactorial-partition","kind":"node","lean":"MetaMathlibExt.HyperfactorialPartition","parent":12560,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions/index.html#hyperfactorial-partition","scope":"NumberTheory/digits-numeration/numeration-automatic/hyperfactorial-partitions","status":"fully_proved","summary":"A hyperfactorial partition of a natural number $n$ consists of a length $k \\in \\mathbb{N}$ and natural numbers $h_1, \\dots, h_k$ such that $h_i \\le 2i$ for every $1 \\le i \\le k$, the digit vector is normalized (either $k = 0$ or $h_k \\neq 0$), and","title":"Hyperfactorial partition"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/mahler-functions.json","id":"NumberTheory/digits-numeration/numeration-automatic/mahler-functions","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/mahler-functions/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/mahler-functions","status":"can_prove","title":"Mahler functions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/mahler-functions/mahler-subst","kind":"node","lean":"MetaMathlibExt.mahlerSubst","parent":12563,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/mahler-functions/index.html#mahler-subst","scope":"NumberTheory/digits-numeration/numeration-automatic/mahler-functions","status":"fully_proved","summary":"For a formal power series $F \\in \\mathbb{C}[[z]]$ and natural numbers $k$ and $i$, let $m = k^i$; the power series $F(z^{k^i})$ is defined coefficientwise. When $m \\ge 1$, its coefficient of $z^n$ is the coefficient of $z^{n/m}$ in $F$ if $m$ divides $n$, and $0$ otherwise; this is the substitution $z \\mapsto z^m$ in $F$. When $m = 0$ (that is, $k = 0$ and $i \\ge 1$), it is the constant series equal to the constant\u2026","title":"Mahler substitution"},{"id":"NumberTheory/digits-numeration/numeration-automatic/mahler-functions/is-kmahler-function","kind":"node","lean":"MetaMathlibExt.IsKMahlerFunction","parent":12563,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-Mahler functions"]],"source":"$k$-Mahler functions"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/mahler-functions/index.html#is-kmahler-function","scope":"NumberTheory/digits-numeration/numeration-automatic/mahler-functions","status":"fully_proved","summary":"Let $k$ be a natural number. A formal power series $F \\in \\mathbb{C}[[z]]$ is a $k$-Mahler function when $k \\ge 2$ and there exist a natural number $d$ and polynomials $a_0(z), \\dots, a_d(z) \\in \\mathbb{C}[z]$ with $a_0 \\ne 0$ and $a_d \\ne 0$ such that the following identity holds in $\\mathbb{C}[[z]]$:","title":"$k$-Mahler functions"},{"id":"NumberTheory/digits-numeration/numeration-automatic/mahler-functions/nishioka-algebraic-k-mahler-is-rational","kind":"node","lean":"MetaMathlibExt.nishioka_algebraic_kMahler_is_rational","parent":12563,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/mahler-functions/index.html#nishioka-algebraic-k-mahler-is-rational","scope":"NumberTheory/digits-numeration/numeration-automatic/mahler-functions","status":"can_prove","summary":"Let $k$ be a natural number and let $F \\in \\mathbb{C}[[z]]$ be a $k$-Mahler function. If $F$ is algebraic over $\\mathbb{C}[z]$, that is, $P(F) = 0$ for some nonzero polynomial $P$ with coefficients in $\\mathbb{C}[z]$, then $F$ is rational: there are polynomials $p, q \\in \\mathbb{C}[z]$ with $q \\ne 0$ and $qF = p$ in $\\mathbb{C}[[z]]$.","title":"Nishioka's theorem: algebraic Mahler functions are rational"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers.json","id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","title":"Numeration by Fibonacci and Pell numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/fib-odd-mul-golden-ratio","kind":"node","lean":"MetaMathlibExt.fib_odd_mul_goldenRatio","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#fib-odd-mul-golden-ratio","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$) and $\\varphi = (1 + \\sqrt 5)/2$ the golden ratio. For every natural number $k$, $F_{2k+1} \\varphi = F_{2k+2} + \\left((\\sqrt 5 - 1)/2\\right)^{2k+1}$ and $F_{2k+1} (1 + \\varphi) = F_{2k+3} + \\left((\\sqrt 5 - 1)/2\\right)^{2k+1}$.","title":"Odd-indexed Fibonacci numbers times the golden ratio"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/exists-odd-fib-representation","kind":"node","lean":"MetaMathlibExt.exists_odd_fib_representation","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#exists-odd-fib-representation","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$). For every positive integer $t$ there are a natural number $n$ and natural numbers $a_0, a_1, a_2, \\dots$ such that $F_{2n+1} \\le t < F_{2n+3}$, $t = \\sum_{k=0}^{n} a_k F_{2k+1}$, $a_0 \\le 1$, $a_k \\le 2$ for every $k \\ge 1$, $a_1 = 2$ and $a_0 = 1$ do not both hold, and for no $k$ do $a_{k+1} = 2$ and $a_k = 2$ both hold.","title":"Numeration by odd-indexed Fibonacci numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/pell-number-odd-mul-one-add-sqrt-two","kind":"node","lean":"MetaMathlibExt.pellNumber_odd_mul_one_add_sqrt_two","parent":12567,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd Pell and even Pell\u2013Lucas numbers times "],["math","\\(1 + \\sqrt 2\\)"]],"source":"Odd Pell and even Pell\u2013Lucas numbers times $1 + \\sqrt 2$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#pell-number-odd-mul-one-add-sqrt-two","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $P_n$ be the Pell numbers and $H_n$ the Pell\u2013Lucas numbers ($H_0 = H_1 = 1$, $H_{n+2} = 2H_{n+1} + H_n$). For every natural number $k$, $P_{2k+1}(1 + \\sqrt 2) = P_{2k+2} + (\\sqrt 2 - 1)^{2k+1}$ and $H_{2k}(1 + \\sqrt 2) = H_{2k+1} + \\sqrt 2\\,(\\sqrt 2 - 1)^{2k}$.","title":"Odd Pell and even Pell\u2013Lucas numbers times $1 + \\sqrt 2$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/exists-two-pell-half-companion-represent","kind":"node","lean":"MetaMathlibExt.exists_two_pell_halfCompanion_representation","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#exists-two-pell-half-companion-represent","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $P_n$ be the Pell numbers ($P_0 = 0$, $P_1 = 1$, $P_{n+2} = 2P_{n+1} + P_n$) and $H_n$ the half-companion Pell numbers ($H_0 = H_1 = 1$, $H_{n+2} = 2H_{n+1} + H_n$), and let $R_k = 2P_k$ for odd $k$ and $R_k = H_k$ for even $k$. Then for every positive integer $t$ there are a natural number $n$ and natural numbers $a_0, a_1, a_2, \\dots$ such that $R_n \\le t < R_{n+1}$, $t = \\sum_{k=0}^{n} a_k R_k$, $a_0 \\le 1$,\u2026","title":"Numeration by doubled Pell and half-companion Pell numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/exists-pell-half-companion-representatio","kind":"node","lean":"MetaMathlibExt.exists_pell_halfCompanion_representation","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#exists-pell-half-companion-representatio","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $P_n$ be the Pell numbers ($P_0 = 0$, $P_1 = 1$, $P_{n+2} = 2P_{n+1} + P_n$) and $H_n$ the half-companion Pell numbers ($H_0 = H_1 = 1$, $H_{n+2} = 2H_{n+1} + H_n$), and let $R_k = P_k$ for odd $k$ and $R_k = H_k$ for even $k$. Then for every positive integer $t$ there are a natural number $n \\ge 1$ and natural numbers $a_0, a_1, a_2, \\dots$ such that $R_n \\le t < R_{n+1}$, $t = \\sum_{k=1}^{n} a_k R_k$,\u2026","title":"Numeration by Pell and half-companion Pell numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/bronze-lucas-params","kind":"node","lean":"MetaMathlibExt.bronzeLucasParams","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#bronze-lucas-params","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"The bronze Lucas parameters are the Lucas sequence parameters $(P, Q) = (3, -1)$. The Lucas sequence $U(3, -1)$ is the sequence of bronze numbers $B_n$, with $B_0 = 0$, $B_1 = 1$ and $B_{n+2} = 3B_{n+1} + B_n$.","title":"Bronze Lucas parameters"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/lucas-u-bronze-lucas-params-add-two","kind":"node","lean":"MetaMathlibExt.lucasU_bronzeLucasParams_add_two","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#lucas-u-bronze-lucas-params-add-two","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"Let $B_n$ be the bronze numbers, the terms of the Lucas sequence $U(3, -1)$. For every natural number $n$, $B_{n+2} = 3B_{n+1} + B_n$.","title":"Recurrence of the bronze numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/lucas-u-bronze-lucas-params-values","kind":"node","lean":"MetaMathlibExt.lucasU_bronzeLucasParams_values","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#lucas-u-bronze-lucas-params-values","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"Let $B_n$ be the bronze numbers, the terms of the Lucas sequence $U(3, -1)$. Then $B_0, B_1, B_2, B_3, B_4, B_5$ are $0, 1, 3, 10, 33, 109$.","title":"The first bronze numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/bronze-companion-first","kind":"node","lean":"MetaMathlibExt.bronzeCompanionFirst","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#bronze-companion-first","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"Let $B_n$ denote the bronze numbers, the Lucas sequence $U(3, -1)$: $B_0 = 0$, $B_1 = 1$ and $B_{n+2} = 3B_{n+1} + B_n$. For every natural number $n$, the $n$-th companion bronze number of the first type is the integer $C_n = B_{n+1} - 2B_n$.","title":"Companion bronze numbers of the first type"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/bronze-companion-second","kind":"node","lean":"MetaMathlibExt.bronzeCompanionSecond","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#bronze-companion-second","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"Let $B_n$ denote the bronze numbers, the Lucas sequence $U(3, -1)$: $B_0 = 0$, $B_1 = 1$ and $B_{n+2} = 3B_{n+1} + B_n$. For every natural number $n$, the $n$-th companion bronze number of the second type is the integer $D_n = B_{n+1} - B_n$.","title":"Companion bronze numbers of the second type"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/bronze-companion-first-add-two","kind":"node","lean":"MetaMathlibExt.bronzeCompanionFirst_add_two","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#bronze-companion-first-add-two","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"For every natural number $n$, the companion bronze numbers $C_n$ of the first type satisfy $C_{n+2} = 3C_{n+1} + C_n$.","title":"Recurrence for the companion bronze numbers of the first type"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/bronze-companion-second-add-two","kind":"node","lean":"MetaMathlibExt.bronzeCompanionSecond_add_two","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#bronze-companion-second-add-two","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"For every natural number $n$, the companion bronze numbers $D_n$ of the second type satisfy $D_{n+2} = 3D_{n+1} + D_n$.","title":"Recurrence for the companion bronze numbers of the second type"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/bronze-companion-values","kind":"node","lean":"MetaMathlibExt.bronzeCompanion_values","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#bronze-companion-values","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"The companion bronze numbers $C_n$ of the first type satisfy $(C_0, C_1, C_2, C_3, C_4) = (1, 1, 4, 13, 43)$, and the companion bronze numbers $D_n$ of the second type satisfy $(D_0, D_1, D_2, D_3, D_4) = (1, 2, 7, 23, 76)$.","title":"Initial companion bronze numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/beatty-octal-move","kind":"node","lean":"MetaMathlibExt.beattyOctalMove","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#beatty-octal-move","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be real numbers. In the Beatty octal game for $\\alpha$ and $\\beta$, a move takes a heap of $n$ tokens to a heap of $m$ tokens exactly when one of the following holds: $n \\ge 1$, $n = \\lfloor k\\alpha \\rfloor$ for some integer $k \\ge 1$, and $m = 0$ (the whole heap is removed); or $m = n - i$ for some natural number $i$ with $1 \\le i < n$ and $i = \\lfloor k\\beta \\rfloor$ for some integer\u2026","title":"Moves of the Beatty octal game"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/beatty-octal-move-lt","kind":"node","lean":"MetaMathlibExt.beattyOctalMove_lt","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#beatty-octal-move-lt","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"fully_proved","summary":"For all real numbers $\\alpha$, $\\beta$ and all natural numbers $n$, $m$: if a move of the Beatty octal game for $\\alpha$ and $\\beta$ takes a heap of $n$ tokens to a heap of $m$ tokens, then $m < n$.","title":"Moves of a Beatty octal game decrease the heap"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/floor-greedy-odd-fib-sum-mul-golden-rati","kind":"node","lean":"MetaMathlibExt.floor_greedy_odd_fib_sum_mul_goldenRatio","parent":12567,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floor of greedy odd-Fibonacci sums times "],["math","\\(\\varphi\\)"],["text"," and "],["math","\\(1 + \\varphi\\)"]],"source":"Floor of greedy odd-Fibonacci sums times $\\varphi$ and $1 + \\varphi$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#floor-greedy-odd-fib-sum-mul-golden-rati","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$) and $\\varphi = (1 + \\sqrt 5)/2$ the golden ratio. Let $n$ be a natural number and $a_0, a_1, \\dots$ natural numbers such that $\\sum_{k=0}^{m-1} a_k F_{2k+1} < F_{2m+1}$ for every $m \\le n + 1$, and let $t = \\sum_{k=0}^{n} a_k F_{2k+1}$. Then $\\lfloor t \\varphi \\rfloor = \\sum_{k=0}^{n} a_k F_{2k+2}$ and\u2026","title":"Floor of greedy odd-Fibonacci sums times $\\varphi$ and $1 + \\varphi$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/gold-game-p-positions","kind":"node","lean":"MetaMathlibExt.goldGame_pPositions","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#gold-game-p-positions","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio and $F_n$ the Fibonacci numbers ($F_1 = F_2 = 1$). In the gold game a heap of $n$ tokens may be removed entirely when $n = \\lfloor k\\varphi \\rfloor$ for some integer $k \\ge 1$, and $i$ tokens may be removed from a heap of $n > i$ tokens, leaving $n - i$, when $i = \\lfloor k(1 + \\varphi) \\rfloor$ for some integer $k \\ge 1$ (the Beatty octal moves for $\\varphi$ and $1 + \\varphi$).\u2026","title":"P-positions of the gold game"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/floor-greedy-pell-half-companion-sum-mul","kind":"node","lean":"MetaMathlibExt.floor_greedy_pell_halfCompanion_sum_mul_one_add_sqrt_two","parent":12567,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floor of greedy Pell sums times "],["math","\\(1 + \\sqrt 2\\)"]],"source":"Floor of greedy Pell sums times $1 + \\sqrt 2$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#floor-greedy-pell-half-companion-sum-mul","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $P_n$ be the Pell numbers ($P_0 = 0$, $P_1 = 1$, $P_{n+2} = 2P_{n+1} + P_n$) and $H_n$ the half-companion Pell numbers ($H_0 = H_1 = 1$, $H_{n+2} = 2H_{n+1} + H_n$). Let $R_k = P_k$ and $R'_k = H_k$ for odd $k$, and $R_k = H_k$ and $R'_k = P_k$ for even $k$. Let $n$ be a natural number and $a_1, a_2, \\dots$ natural numbers such that $\\sum_{k=1}^{m-1} a_k R_k < R_m$ for every $1 \\le m \\le n + 1$, and let\u2026","title":"Floor of greedy Pell sums times $1 + \\sqrt 2$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/floor-greedy-two-pell-half-companion-sum","kind":"node","lean":"MetaMathlibExt.floor_greedy_two_pell_halfCompanion_sum_mul","parent":12567,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floor of greedy sums of "],["math","\\(2P_k\\)"],["text"," and "],["math","\\(H_k\\)"],["text"," times "],["math","\\((2 + \\sqrt 2)/2\\)"]],"source":"Floor of greedy sums of $2P_k$ and $H_k$ times $(2 + \\sqrt 2)/2$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#floor-greedy-two-pell-half-companion-sum","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $P_k$ be the Pell numbers and $H_k$ the Pell\u2013Lucas numbers ($H_0 = H_1 = 1$, $H_{k+2} = 2H_{k+1} + H_k$). Put $R_k = 2P_k$ and $R'_k = P_k$ for odd $k$, and $R_k = R'_k = H_k$ for even $k$. Let $n$ be a natural number and $a_0, a_1, \\dots$ natural numbers such that $\\sum_{k\\lt m} a_k R_k < R_m$ for every $m \\le n + 1$, and let $t = \\sum_{k=0}^{n} a_k R_k$. Then\u2026","title":"Floor of greedy sums of $2P_k$ and $H_k$ times $(2 + \\sqrt 2)/2$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/silver-game-p-positions","kind":"node","lean":"MetaMathlibExt.silverGame_pPositions","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#silver-game-p-positions","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $P_n$ be the Pell numbers and $H_n$ the Pell\u2013Lucas numbers ($H_0 = H_1 = 1$, $H_{n+2} = 2H_{n+1} + H_n$). In the silver game a heap of $n$ tokens may be removed entirely when $n = \\lfloor k(1 + \\sqrt 2) \\rfloor$ for some integer $k \\ge 1$, and $i$ tokens may be removed from a heap of $n > i$ tokens, leaving $n - i$, when $i = \\lfloor k(2 + \\sqrt 2)/2 \\rfloor$ for some integer $k \\ge 1$ (the Beatty octal moves\u2026","title":"P-positions of the silver game"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/bronze-mul-metallic-identities","kind":"node","lean":"MetaMathlibExt.bronze_mul_metallic_identities","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#bronze-mul-metallic-identities","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $\\alpha = (3 + \\sqrt{13})/2$, $\\beta = (5 + \\sqrt{13})/6$ and $\\gamma = (\\sqrt{13} - 3)/2$, let $B_n$ be the bronze numbers, and let $C_n$ and $D_n$ be the companion bronze numbers of the first and second type. For every natural number $k$, the following hold in $\\mathbb{R}$: (1) $B_{2k+1}\\alpha = B_{2k+2} + \\gamma^{2k+1}$; (2) $C_{2k}\\alpha = C_{2k+1} + \\frac{13 + \\sqrt{13}}{2\\sqrt{13}}\\gamma^{2k}$; (3)\u2026","title":"Metallic identities for the bronze numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/exists-mixed-bronze-representation","kind":"node","lean":"MetaMathlibExt.exists_mixed_bronze_representation","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#exists-mixed-bronze-representation","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $B_n$ be the bronze numbers ($B_0 = 0$, $B_1 = 1$, $B_{n+2} = 3B_{n+1} + B_n$, the Lucas sequence $U(3, -1)$), and let $C_n = B_{n+1} - 2B_n$ and $D_n = B_{n+1} - B_n$ be the companion bronze numbers of the first and second type. Let $R$ be the integer sequence with $R_{3m} = D_{2m}$, $R_{3m+1} = 3B_{2m+1}$ and $R_{3m+2} = B_{2m+1} + C_{2m+2}$ for all $m \\ge 0$. Then for every positive integer $t$ there are a\u2026","title":"Mixed bronze numeration"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/exists-bronze-companion-representation","kind":"node","lean":"MetaMathlibExt.exists_bronze_companion_representation","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#exists-bronze-companion-representation","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $B_n$ be the bronze numbers ($B_0 = 0$, $B_1 = 1$, $B_{n+2} = 3B_{n+1} + B_n$, the Lucas sequence $U(3, -1)$), and let $C_n = B_{n+1} - 2B_n$ and $D_n = B_{n+1} - B_n$ be the companion bronze numbers of the first and second type. Let $R$ be the integer sequence with $R_{3m+1} = B_{2m+1}$, $R_{3m+2} = C_{2m+2}$ and $R_{3m} = D_{2m}$ for all $m \\ge 0$. Then for every positive integer $t$ there are a natural\u2026","title":"Numeration by bronze and companion bronze numbers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/floor-greedy-bronze-companion-sum-mul","kind":"node","lean":"MetaMathlibExt.floor_greedy_bronze_companion_sum_mul","parent":12567,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floor of greedy bronze sums times "],["math","\\((3 + \\sqrt{13})/2\\)"]],"source":"Floor of greedy bronze sums times $(3 + \\sqrt{13})/2$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#floor-greedy-bronze-companion-sum-mul","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $B_n$ be the bronze numbers ($B_0 = 0$, $B_1 = 1$, $B_{n+2} = 3B_{n+1} + B_n$, the Lucas sequence $U(3, -1)$), and let $C_n = B_{n+1} - 2B_n$ and $D_n = B_{n+1} - B_n$ be the companion bronze numbers of the first and second type. Let $R$ and $R'$ be the integer sequences with $R_{3m+1} = B_{2m+1}$, $R_{3m+2} = C_{2m+2}$, $R_{3m} = D_{2m}$ and $R'_{3m+1} = D_{2m+1}$, $R'_{3m+2} = B_{2m+2}$, $R'_{3m} = C_{2m+1}$\u2026","title":"Floor of greedy bronze sums times $(3 + \\sqrt{13})/2$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/floor-greedy-mixed-bronze-sum-mul","kind":"node","lean":"MetaMathlibExt.floor_greedy_mixed_bronze_sum_mul","parent":12567,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floor of greedy mixed bronze sums times "],["math","\\((5 + \\sqrt{13})/6\\)"]],"source":"Floor of greedy mixed bronze sums times $(5 + \\sqrt{13})/6$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#floor-greedy-mixed-bronze-sum-mul","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $B_n$ be the bronze numbers ($B_0 = 0$, $B_1 = 1$, $B_{n+2} = 3B_{n+1} + B_n$, the Lucas sequence $U(3, -1)$), and let $C_n = B_{n+1} - 2B_n$ and $D_n = B_{n+1} - B_n$ be the companion bronze numbers of the first and second type. Let $R$ and $R'$ be the integer sequences with $R_{3m} = D_{2m}$, $R_{3m+1} = 3B_{2m+1}$, $R_{3m+2} = B_{2m+1} + C_{2m+2}$ and $R'_{3m} = D_{2m}$, $R'_{3m+1} = B_{2m+1}$,\u2026","title":"Floor of greedy mixed bronze sums times $(5 + \\sqrt{13})/6$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/bronze-game-p-positions","kind":"node","lean":"MetaMathlibExt.bronzeGame_pPositions","parent":12567,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers/index.html#bronze-game-p-positions","scope":"NumberTheory/digits-numeration/numeration-automatic/numeration-by-fibonacci-and-pell-numbers","status":"can_prove","summary":"Let $\\alpha = (3 + \\sqrt{13})/2$ and $\\beta = (5 + \\sqrt{13})/6$, let $B_n$ be the bronze numbers, and let $C_n$ and $D_n$ be the companion bronze numbers of the first and second type. Let $P$ be the set of natural numbers $x$ such that $x = 0$ or $x \\in \\{B_{2k+1}, C_{2k+2}, D_{2k+2}\\}$ for some natural number $k$. Then, in the Beatty octal game for $\\alpha$ and $\\beta$, no move takes a heap whose size lies in $P$\u2026","title":"P-positions of the bronze game"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence.json","id":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","status":"can_prove","title":"Shakov's 2-regular sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/shakov-seq","kind":"node","lean":"MathlibExt.NumberTheory.ShakovTwoRegularWanted.shakovSeq","parent":12594,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shakov's "],["math","\\(2\\)"],["text","-regular sequence"]],"source":"Shakov's $2$-regular sequence"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/index.html#shakov-seq","scope":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","status":"fully_proved","summary":"The integer sequence $s$ with $s(0) = s(1) = 0$, $s(2) = s(3) = 1$ and, for every $k \\ge 1$, $s(4k) = 2s(2k) - s(k)$, $s(4k+1) = 2s(2k) + s(2k+1)$, $s(4k+2) = 2s(2k+1) + s(2k)$ and $s(4k+3) = 2s(2k+1) - s(k)$.","title":"Shakov's $2$-regular sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/shakov-a","kind":"node","lean":"MathlibExt.NumberTheory.ShakovTwoRegularWanted.shakovA","parent":12594,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/index.html#shakov-a","scope":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","status":"fully_proved","summary":"The integer sequence $a$ with $a(0) = 0$, $a(1) = 1$ and, for every $n \\ge 2$, $a(n) = 2a(n-1)$ if $n \\equiv 0 \\pmod 4$, $a(n) = a(n-1) + 1$ if $n \\equiv 1 \\pmod 4$, $a(n) = 2a(n-1) + 1$ if $n \\equiv 2 \\pmod 4$, and $a(n) = a(n-1) - 1$ if $n \\equiv 3 \\pmod 4$.","title":"Shakov's auxiliary sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/fiber-card","kind":"node","lean":"MathlibExt.NumberTheory.ShakovTwoRegularWanted.fiber_card","parent":12594,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shakov's sequence counts the divisors of "],["math","\\(m^2 + 1\\)"]],"source":"Shakov's sequence counts the divisors of $m^2 + 1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/index.html#fiber-card","scope":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","status":"can_prove","summary":"Let $s$ be Shakov's $2$-regular sequence and let $\\tau(N)$ denote the number of positive divisors of $N$. Then $s(n) \\ge 0$ for every $n \\ge 1$, and for every natural number $m$ the set $\\{n \\ge 1 : s(n) = m\\}$ is finite and has exactly $\\tau(m^2 + 1)$ elements.","title":"Shakov's sequence counts the divisors of $m^2 + 1$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/block-average","kind":"node","lean":"MathlibExt.NumberTheory.ShakovTwoRegularWanted.block_average","parent":12594,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/index.html#block-average","scope":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","status":"can_prove","summary":"Let $s$ be Shakov's $2$-regular sequence. For every natural number $n$,","title":"Block averages of Shakov's sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/prime-iff-fiber","kind":"node","lean":"MathlibExt.NumberTheory.ShakovTwoRegularWanted.prime_iff_fiber","parent":12594,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primality of "],["math","\\(n^2 + 1\\)"],["text"," from the fibers of Shakov's sequence"]],"source":"Primality of $n^2 + 1$ from the fibers of Shakov's sequence"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/index.html#prime-iff-fiber","scope":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","status":"can_prove","summary":"Let $s$ be Shakov's $2$-regular sequence. For every natural number $n \\ge 1$, the number $n^2 + 1$ is prime if and only if the set of natural numbers $m$ with $s(m) = n$ is exactly $\\{2^n, 2^{n+1} - 1\\}$.","title":"Primality of $n^2 + 1$ from the fibers of Shakov's sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/shakov-seq-shakov-a","kind":"node","lean":"MathlibExt.NumberTheory.ShakovTwoRegularWanted.shakovSeq_shakovA","parent":12594,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/index.html#shakov-seq-shakov-a","scope":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","status":"can_prove","summary":"Let $s$ be Shakov's $2$-regular sequence, let $a$ be Shakov's auxiliary sequence, and let $F_k$ denote the $k$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For every natural number $n \\ge 1$, $s(\\max(a(n), 0)) = F_{n-1}$.","title":"Shakov's sequence takes Fibonacci values along the auxiliary sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/fib-sq-add-one-composite","kind":"node","lean":"MathlibExt.NumberTheory.ShakovTwoRegularWanted.fib_sq_add_one_composite","parent":12594,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence/index.html#fib-sq-add-one-composite","scope":"NumberTheory/digits-numeration/numeration-automatic/shakov-s-2-regular-sequence","status":"can_prove","summary":"Let $F_k$ denote the $k$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For every natural number $k \\ge 4$, the number $F_k^2 + 1$ is composite.","title":"Fibonacci squares plus one are eventually composite"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations.json","id":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations","status":"can_prove","title":"Signed binary representations"},{"id":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/is-signed-pow-two-sum","kind":"node","lean":"MetaMathlibExt.IsSignedPowTwoSum","parent":12602,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/index.html#is-signed-pow-two-sum","scope":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations","status":"fully_proved","summary":"For natural numbers $m$, $s$ and $t$, an integer $n$ is a signed power-of-two sum with coefficients in $\\{-1, m\\}$ over the range $s \\le j \\le t$ if $n = \\sum_{j=s}^{t} \\lambda_j 2^j$ for some integers $\\lambda_j$ with every $\\lambda_j \\in \\{-1, m\\}$.","title":"Signed power-of-two sums"},{"id":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/is-signed-pow-two-sum-one","kind":"node","lean":"MetaMathlibExt.isSignedPowTwoSum_one","parent":12602,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/index.html#is-signed-pow-two-sum-one","scope":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations","status":"fully_proved","summary":"The integer $1$ can be written as $\\sum_{j=0}^{1} \\lambda_j 2^j$ with every $\\lambda_j \\in \\{-1, 3\\}$, namely $1 = 3 \\cdot 2^0 + (-1) \\cdot 2^1$.","title":"Example of a signed power-of-two sum"},{"id":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/is-signed-pow-two-sum-iff","kind":"node","lean":"MetaMathlibExt.isSignedPowTwoSum_iff","parent":12602,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/index.html#is-signed-pow-two-sum-iff","scope":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations","status":"can_prove","summary":"Let $m \\ge 1$ and $s \\le t$ be natural numbers and put $L = t - s + 1$. An integer $n$ can be written as $\\sum_{j=s}^{t} \\lambda_j 2^j$ with every $\\lambda_j \\in \\{-1, m\\}$ if and only if $-2^s (2^L - 1) \\le n \\le 2^s m (2^L - 1)$ and $n \\equiv -2^s (2^L - 1) \\pmod{2^s (m + 1)}$.","title":"Interval and congruence characterization of signed power-of-two sums"},{"id":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/is-signed-pow-two-sum-extend","kind":"node","lean":"MetaMathlibExt.isSignedPowTwoSum_extend","parent":12602,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/index.html#is-signed-pow-two-sum-extend","scope":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations","status":"can_prove","summary":"Let $m \\ge 1$, $s$, $t$ and $\\alpha$ be natural numbers, let $\\beta = \\nu_2(m + 1)$ be the exponent of $2$ in $m + 1$, and suppose that $s + \\beta \\le t + 1$ and $2^\\alpha \\equiv 1 \\pmod{(m + 1)/2^\\beta}$. If an integer $n$ can be written as $\\sum_{j=s}^{t} \\lambda_j 2^j$ with every $\\lambda_j \\in \\{-1, m\\}$, then $n$ can also be written as $\\sum_{j=s}^{t+\\alpha} \\mu_j 2^j$ with every $\\mu_j \\in \\{-1, m\\}$.","title":"Extending a signed power-of-two sum to a longer range"},{"id":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/one-add-signed-pow-two-sum-iff","kind":"node","lean":"MetaMathlibExt.one_add_signedPowTwoSum_iff","parent":12602,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations/index.html#one-add-signed-pow-two-sum-iff","scope":"NumberTheory/digits-numeration/numeration-automatic/signed-binary-representations","status":"can_prove","summary":"An integer $n$ can be written as $n = 1 + \\sum_{j=1}^{k} \\lambda_j 2^j$ for some $k \\ge 1$ and some $\\lambda_1, \\dots, \\lambda_k \\in \\{-1, 1\\}$ if and only if $n \\equiv 3 \\pmod 4$.","title":"Integers that are one plus a signed sum of powers of two"},{"data_url":"scopes/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits.json","id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","kind":"scope","navigation":"module","parent":12336,"result":false,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","title":"Zeckendorf sum of digits"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeckendorf-variation-density-fib-eq-one","kind":"node","lean":"MetaMathlibExt.zeckendorf_variation_density_fib_eq_one","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeckendorf-variation-density-fib-eq-one","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $s(n)$ be the number of terms in the Zeckendorf representation of $n$, and let $F_\\ell$ be the $\\ell$-th Fibonacci number ($F_1 = F_2 = 1$). Let $\\ell \\ge 2$ and let $\\mu : \\mathbb{Z} \\to \\mathbb{R}$ be such that for every $d \\in \\mathbb{Z}$ the set $\\{n \\in \\mathbb{N} : s(n + 1) - s(n) = d\\}$ has natural density $\\mu(d)$. Then for every $d \\in \\mathbb{Z}$ the set\u2026","title":"Fibonacci shifts have the same Zeckendorf variation densities as shift one"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeckendorf-variation-density-one","kind":"node","lean":"MetaMathlibExt.zeckendorf_variation_density_one","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeckendorf-variation-density-one","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $s(n)$ be the number of terms in the Zeckendorf representation of $n$, and let $\\varphi$ be the golden ratio. For every $d \\in \\mathbb{Z}$ the set $\\{n \\in \\mathbb{N} : s(n + 1) - s(n) = d\\}$ has natural density $0$ if $d \\ge 2$, $1/\\varphi^2$ if $d = 1$, and $1/\\varphi^{2 - 2d}$ if $d \\le 0$.","title":"Distribution of the Zeckendorf digit-sum variation for shift one"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeckendorf-variation-density-exists-mome","kind":"node","lean":"MetaMathlibExt.zeckendorf_variation_density_exists_moments","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeckendorf-variation-density-exists-mome","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $s(n)$ be the number of terms in the Zeckendorf representation of $n$. For every $r \\in \\mathbb{N}$ there is a function $\\mu : \\mathbb{Z} \\to \\mathbb{R}$ such that for every $d \\in \\mathbb{Z}$ the set $\\{n \\in \\mathbb{N} : s(n + r) - s(n) = d\\}$ has natural density $\\mu(d)$, the series $\\sum_{d \\in \\mathbb{Z}} |d|^k \\mu(d)$ converges absolutely for every $k \\in \\mathbb{N}$, and\u2026","title":"Zeckendorf variation densities exist, with finite moments and mean zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeckendorf-variation-density-tail","kind":"node","lean":"MetaMathlibExt.zeckendorf_variation_density_tail","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeckendorf-variation-density-tail","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $s(n)$ be the number of terms in the Zeckendorf representation of $n$, and let $\\varphi$ be the golden ratio. Let $r \\in \\mathbb{N}$ and let $\\mu : \\mathbb{Z} \\to \\mathbb{R}$ be such that for every $d \\in \\mathbb{Z}$ the set $\\{n \\in \\mathbb{N} : s(n + r) - s(n) = d\\}$ has natural density $\\mu(d)$. Then there is $D \\in \\mathbb{Z}$ such that $\\mu(d - 1) = \\mu(d)/\\varphi^2$ for every\u2026","title":"Geometric negative tail of the Zeckendorf variation densities"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic","kind":"node","lean":"MetaMathlibExt.zeckAdic","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"The Zeckendorf-adic integers $\\mathbb{X}$ are the digit sequences $x = (x_k)_{k \\ge 0}$ with $x_k \\in \\{0, 1\\}$ such that $x_0 = x_1 = 0$ and no two consecutive digits are both $1$, that is, $x_k x_{k+1} = 0$ for every $k$.","title":"The Zeckendorf-adic integers"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-of-nat","kind":"node","lean":"MetaMathlibExt.zeckAdicOfNat","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-of-nat","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Let $F_k$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = F_2 = 1$). Every $n \\in \\mathbb{N}$ has a unique Zeckendorf representation $n = F_{k_1} + \\dots + F_{k_m}$ with $k_1 > \\dots > k_m \\ge 2$ and no two consecutive indices. The Zeckendorf digit sequence of $n$ is the binary sequence $(x_k)_{k \\in \\mathbb{N}}$ with $x_k = 1$ if $k$ is one of the indices $k_1, \\dots, k_m$ and $x_k = 0$ otherwise.","title":"Zeckendorf digit sequence of a natural number"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer","kind":"node","lean":"MetaMathlibExt.zeckOdometer","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"The odometer is the map $T$ on digit sequences $x = (x_k)_{k \\in \\mathbb{N}}$ with $x_k \\in \\{0, 1\\}$ defined as follows. If $x_k = x_{k+1} = 0$ for some $k \\ge 2$, let $k$ be the least such index; then $T(x)_j = 0$ for $j < k$, $T(x)_k = 1$ and $T(x)_j = x_j$ for $j > k$. If there is no such $k$, then $T(x)$ is the zero sequence.","title":"The Zeckendorf odometer"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-of-nat-zero","kind":"node","lean":"MetaMathlibExt.zeckAdicOfNat_zero","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-of-nat-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"The Zeckendorf digit sequence of $0$ is the zero sequence: all of its digits are $0$, because the Zeckendorf representation of $0$ is empty.","title":"Zeckendorf digit sequence of zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-eq-update","kind":"node","lean":"MetaMathlibExt.zeckOdometer_eq_update","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odometer on a point ending in "],["math","\\(00\\)"]],"source":"The odometer on a point ending in $00$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-eq-update","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"If a Zeckendorf-adic integer $x = (x_k)_{k \\ge 2}$ has $x_2 = x_3 = 0$, then the odometer $T(x) = x + 1$ sets the unit digit $x_2$ to $1$ and leaves every other digit of $x$ unchanged.","title":"The odometer on a point ending in $00$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-of-forall","kind":"node","lean":"MetaMathlibExt.zeckOdometer_of_forall","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-of-forall","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Let $T$ be the Zeckendorf odometer and let $x = (x_k)_{k \\in \\mathbb{N}}$ be a digit sequence with $x_k \\in \\{0, 1\\}$ such that for every $k \\ge 2$ we have $x_k = 1$ or $x_{k+1} = 1$. Then $T(x)$ is the zero sequence.","title":"The odometer on sequences without two consecutive zeros"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-trunc-variation","kind":"node","lean":"MetaMathlibExt.zeckAdicTruncVariation","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-trunc-variation","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Let $T$ be the Zeckendorf odometer $x \\mapsto x + 1$ on binary sequences, and write $x + r = T^r(x)$ for its $r$-th iterate. For $r, k \\in \\mathbb{N}$ and a binary sequence $x = (x_i)_{i \\in \\mathbb{N}}$, let $s_k(x) = x_2 + \\dots + x_k$. The truncated variation is the integer $\\Delta_k^{(r)}(x) = s_k(x + r) - s_k(x)$.","title":"Truncated variation of the Zeckendorf digit sum"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-trunc-variation-zero","kind":"node","lean":"MetaMathlibExt.zeckAdicTruncVariation_zero","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-trunc-variation-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Let $T$ be the Zeckendorf odometer and, for a binary sequence $x$, let $s_k(x) = x_2 + \\dots + x_k$ and $\\Delta_k^{(r)}(x) = s_k(T^r(x)) - s_k(x)$. Then $\\Delta_k^{(0)}(x) = 0$ for every $k \\in \\mathbb{N}$ and every binary sequence $x$.","title":"The truncated variation vanishes for zero steps"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For $r \\in \\mathbb{N}$ and a digit sequence $x : \\mathbb{N} \\to \\{0, 1\\}$, let $\\Delta_k^{(r)}(x)$ be the truncated variation of order $k$. If the integer sequence $(\\Delta_k^{(r)}(x))_k$ is eventually constant, the variation $\\Delta^{(r)}(x) = \\lim_{k \\to \\infty} \\Delta_k^{(r)}(x)$ is its eventual value; otherwise $\\Delta^{(r)}(x) = 0$.","title":"The variation of the Zeckendorf digit sum"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation-eq-of-eventually","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation_eq_of_eventually","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation-eq-of-eventually","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Let $r \\in \\mathbb{N}$, let $x$ be a binary sequence and let $d \\in \\mathbb{Z}$. If the truncated variation $\\Delta_k^{(r)}(x)$ equals $d$ for all sufficiently large $k$, then the variation $\\Delta^{(r)}(x)$ equals $d$.","title":"The variation is the eventual value of the truncated variations"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation-zero","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation_zero","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","The variation for "],["math","\\(r = 0\\)"],["text"," vanishes"]],"source":"The variation for $r = 0$ vanishes"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For every digit sequence $x : \\mathbb{N} \\to \\{0, 1\\}$, not only for Zeckendorf-adic integers, the variation satisfies $\\Delta^{(0)}(x) = 0$.","title":"The variation for $r = 0$ vanishes"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-iterate-low-digits","kind":"node","lean":"MetaMathlibExt.zeckOdometer_iterate_low_digits","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Adding "],["math","\\(r\\)"],["text"," without low digits leaves the low digits unchanged"]],"source":"Adding $r$ without low digits leaves the low digits unchanged"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-iterate-low-digits","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Write $F_j$ for the Fibonacci numbers ($F_1 = F_2 = 1$). For $r \\in \\mathbb{N}$ let $Z(r)$ be the set of indices in the Zeckendorf representation $r = \\sum_{j \\in Z(r)} F_j$ (all indices $\\ge 2$, no two consecutive), and let $r_j = 1$ if $j \\in Z(r)$ and $r_j = 0$ otherwise. Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive\u2026","title":"Adding $r$ without low digits leaves the low digits unchanged"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-iterate-high-digits","kind":"node","lean":"MetaMathlibExt.zeckOdometer_iterate_high_digits","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Adding a small "],["math","\\(r\\)"],["text"," leaves the high digits unchanged"]],"source":"Adding a small $r$ leaves the high digits unchanged"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-iterate-high-digits","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Write $F_j$ for the Fibonacci numbers ($F_1 = F_2 = 1$). Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive ones), and write $x + r = T^r(x)$ for the $r$-th iterate of the Zeckendorf odometer $T$. Let $x \\in \\mathbb{X}$, $\\ell \\ge 2$ and $r < F_{\\ell+1}$, and suppose that $x_{\\ell+2} = x_{\\ell+3} = 0$. Then $(x + r)_k = x_k$\u2026","title":"Adding a small $r$ leaves the high digits unchanged"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-iterate-block-digits","kind":"node","lean":"MetaMathlibExt.zeckOdometer_iterate_block_digits","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Adding "],["math","\\(r\\)"],["text"," across a window of four zero digits"]],"source":"Adding $r$ across a window of four zero digits"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-iterate-block-digits","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Write $F_j$ for the Fibonacci numbers ($F_1 = F_2 = 1$). For $r \\in \\mathbb{N}$ let $Z(r)$ be the set of indices in the Zeckendorf representation $r = \\sum_{j \\in Z(r)} F_j$ (all indices $\\ge 2$, no two consecutive), and let $r_j = 1$ if $j \\in Z(r)$ and $r_j = 0$ otherwise. Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive\u2026","title":"Adding $r$ across a window of four zero digits"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-stopping-block-one","kind":"node","lean":"MetaMathlibExt.zeckOdometer_stopping_block_one","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-stopping-block-one","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Write $F_j$ for the Fibonacci numbers ($F_1 = F_2 = 1$). For $r \\in \\mathbb{N}$ let $Z(r)$ be the set of indices in the Zeckendorf representation $r = \\sum_{j \\in Z(r)} F_j$ (all indices $\\ge 2$, no two consecutive), and let $r_j = 1$ if $j \\in Z(r)$ and $r_j = 0$ otherwise. Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive\u2026","title":"Stopping condition for a block of size one"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-stopping-block","kind":"node","lean":"MetaMathlibExt.zeckOdometer_stopping_block","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stopping condition for a block of size "],["math","\\(m + 2\\)"]],"source":"Stopping condition for a block of size $m + 2$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-stopping-block","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Write $F_j$ for the Fibonacci numbers ($F_1 = F_2 = 1$). For $r \\in \\mathbb{N}$ let $Z(r)$ be the set of indices in the Zeckendorf representation $r = \\sum_{j \\in Z(r)} F_j$ (all indices $\\ge 2$, no two consecutive), and let $r_j = 1$ if $j \\in Z(r)$ and $r_j = 0$ otherwise. Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive\u2026","title":"Stopping condition for a block of size $m + 2$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-uniquely-ergodic","kind":"node","lean":"MetaMathlibExt.zeckOdometer_uniquely_ergodic","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-uniquely-ergodic","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X}$ be the set of Zeckendorf-adic integers, a subset of the space $\\{0,1\\}^{\\mathbb{N}}$ of digit sequences with its product $\\sigma$-algebra, and let $T$ be the Zeckendorf odometer on $\\{0,1\\}^{\\mathbb{N}}$. There is exactly one probability measure $\\mu$ on $\\{0,1\\}^{\\mathbb{N}}$ such that $\\mu(\\{0,1\\}^{\\mathbb{N}} \\setminus \\mathbb{X}) = 0$ and $T$ preserves $\\mu$, that is, $T$ is measurable and\u2026","title":"Unique ergodicity of the Zeckendorf odometer"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-invariant-measure-cylinder","kind":"node","lean":"MetaMathlibExt.zeckOdometer_invariant_measure_cylinder","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-invariant-measure-cylinder","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive ones), let $T$ be the Zeckendorf odometer, and let $\\varphi$ be the golden ratio. Let $\\mu$ be a probability measure on $\\{0,1\\}^{\\mathbb{N}}$ with $\\mu(\\mathbb{X}^c) = 0$ that is preserved by $T$. Let $\\ell \\ge 2$ and $y \\in \\mathbb{X}$, and let\u2026","title":"Mass of a cylinder under the invariant measure"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-invariant-measure-renewal","kind":"node","lean":"MetaMathlibExt.zeckOdometer_invariant_measure_renewal","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-invariant-measure-renewal","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive ones) and let $T$ be the Zeckendorf odometer. Let $\\mu$ be a probability measure on $\\{0,1\\}^{\\mathbb{N}}$ with $\\mu(\\mathbb{X}^c) = 0$ that is preserved by $T$. Let $k \\ge 2$, $\\ell_0 \\in \\mathbb{N}$ and $y, z \\in \\mathbb{X}$, and let\u2026","title":"Renewal property of the invariant measure"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/alpha-mixing-le-half-phi-mixing","kind":"node","lean":"MetaMathlibExt.alpha_mixing_le_half_phi_mixing","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bradley's inequality between the "],["math","\\(\\alpha\\)"],["text","- and "],["math","\\(\\varphi\\)"],["text","-mixing coefficients"]],"source":"Bradley's inequality between the $\\alpha$- and $\\varphi$-mixing coefficients"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#alpha-mixing-le-half-phi-mixing","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $(X_j)_{j \\ge 1}$ be random variables on a probability space $(\\Omega, \\mathbb{P})$ with values in a measurable space, and let $k \\ge 1$. Suppose a real number $c$ bounds every term defining the $\\varphi$-mixing coefficient $\\varphi(k)$: for every $p \\ge 1$, every event $A \\in \\sigma(X_j : 1 \\le j \\le p)$ with $\\mathbb{P}(A) > 0$ and every event $B \\in \\sigma(X_j : j \\ge k + p)$,\u2026","title":"Bradley's inequality between the $\\alpha$- and $\\varphi$-mixing coefficients"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-invariant-measure-phi-mixi","kind":"node","lean":"MetaMathlibExt.zeckOdometer_invariant_measure_phi_mixing","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-invariant-measure-phi-mixi","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive ones), let $T$ be the Zeckendorf odometer, and let $\\varphi$ be the golden ratio. Let $\\mu$ be a probability measure on $\\{0,1\\}^{\\mathbb{N}}$ with $\\mu(\\mathbb{X}^c) = 0$ that is preserved by $T$, and let $k, p \\ge 1$. Let $A$ be a set that is measurable for the\u2026","title":"Mixing of the Zeckendorf digits under the invariant measure"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-markov-path","kind":"node","lean":"MetaMathlibExt.zeckMarkovPath","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-markov-path","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio, and let $\\psi(0, t) = 1$ if $t < 1/\\varphi^2$, $\\psi(0, t) = 0$ otherwise, and $\\psi(1, t) = 0$. For a real sequence $(u_n)_{n \\ge 0}$, written $u_n = U_{n+2}$, the chain $(y_j)_{j \\in \\mathbb{N}}$ is the sequence in $\\{0, 1\\}$ with $y_0 = y_1 = 0$ and $y_j = \\psi(y_{j-1}, U_j)$ for $j \\ge 2$; that is, $y_{n+2} = 1$ exactly when $y_{n+1} = 0$ and $u_n < 1/\\varphi^2$.","title":"The Zeckendorf Markov chain"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-markov-path-two","kind":"node","lean":"MetaMathlibExt.zeckMarkovPath_two","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-markov-path-two","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For every real sequence $(u_n)_{n \\ge 0}$, the digit $y_2$ of the chain driven by $(u_n)$ is $1$ exactly when $u_0 < 1/\\varphi^2$, where $\\varphi$ is the golden ratio. In the paper's notation, where $u_n = U_{n+2}$, this says that $y_2 = 1$ if and only if $U_2 < 1/\\varphi^2$.","title":"The first step of the Zeckendorf Markov chain"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-markov-path-succ-of-true","kind":"node","lean":"MetaMathlibExt.zeckMarkovPath_succ_of_true","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-markov-path-succ-of-true","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For every real sequence $(u_n)_{n \\ge 0}$ and every $j \\ge 1$, if the digit $y_j$ of the chain driven by $(u_n)$ is $1$, then $y_{j+1} = 0$.","title":"After a one, the Zeckendorf Markov chain emits a zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-markov-path-law","kind":"node","lean":"MetaMathlibExt.zeckMarkovPath_law","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-markov-path-law","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X}$ be the set of Zeckendorf-adic integers: the digit sequences $x \\in \\{0, 1\\}^{\\mathbb{N}}$ with $x_0 = x_1 = 0$ and no two consecutive digits equal to $1$. Let $\\mu$ be a probability measure on $\\{0, 1\\}^{\\mathbb{N}}$ with $\\mu(\\mathbb{X}^c) = 0$ that is preserved by the odometer $T$. Let $(\\Omega, \\nu)$ be a probability space and let $(U_n)_{n \\ge 0}$ be independent measurable real random variables\u2026","title":"The Zeckendorf Markov chain has the invariant law"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-odometer-invariant-measure-cond-zer","kind":"node","lean":"MetaMathlibExt.zeckOdometer_invariant_measure_cond_zeros","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-odometer-invariant-measure-cond-zer","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X} \\subseteq \\{0,1\\}^{\\mathbb{N}}$ be the set of Zeckendorf-adic integers (sequences with $x_0 = x_1 = 0$ and no two consecutive ones), let $T$ be the Zeckendorf odometer, and let $\\varphi$ be the golden ratio. Let $\\mu$ be a probability measure on $\\{0,1\\}^{\\mathbb{N}}$ with $\\mu(\\mathbb{X}^c) = 0$ that is preserved by $T$. Let $\\ell \\in \\mathbb{N}$ and let $k_0, \\dots, k_\\ell$ be natural numbers with\u2026","title":"Conditional probability that prescribed later digits vanish"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation-ergodic-average-of-g","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation_ergodic_average_of_growth","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation-ergodic-average-of-g","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X}$ be the set of Zeckendorf-adic integers, $T$ the Zeckendorf odometer, and $\\mathbb{P}$ a $T$-invariant probability measure on binary sequences with $\\mathbb{P}(\\mathbb{X}^c) = 0$. Let $s(n)$ be the number of terms in the Zeckendorf representation of $n \\in \\mathbb{N}$, let $r \\ge 1$, and let $\\Delta^{(r)}$ and $\\Delta_k^{(r)}$ be the variation and the truncated variation on binary sequences. Let\u2026","title":"Ergodic averages of the variation for functions of polynomial growth"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation-count-le","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation_count_le","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation-count-le","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X}$ be the set of Zeckendorf-adic integers, $T$ the Zeckendorf odometer, and $\\mathbb{P}$ a $T$-invariant probability measure on binary sequences with $\\mathbb{P}(\\mathbb{X}^c) = 0$. Let $s(n)$ be the number of terms in the Zeckendorf representation of $n \\in \\mathbb{N}$, let $\\iota(n)$ be the Zeckendorf digit sequence of $n$, let $\\Delta^{(r)}$ and $\\Delta_k^{(r)}$ be the variation and the truncated\u2026","title":"Counting values of the variation against the invariant measure"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation-one-exists-iff","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation_one_exists_iff","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","When the variation "],["math","\\(\\Delta^{(1)}\\)"],["text"," exists"]],"source":"When the variation $\\Delta^{(1)}$ exists"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation-one-exists-iff","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $x$ be a Zeckendorf-adic integer with digits $x_k$, and let $\\Delta_k^{(1)}(x)$ be its truncated variation of order $k$ for $r = 1$. The integer sequence $(\\Delta_k^{(1)}(x))_k$ is eventually constant (equivalently, the variation $\\Delta^{(1)}(x) = \\lim_{k \\to \\infty} \\Delta_k^{(1)}(x)$ exists) if and only if there is $d \\ge 0$ such that either $x_{2d+3} = x_{2d+2} = 0$ and $x_{2i+1} = 1$, $x_{2i} = 0$ for\u2026","title":"When the variation $\\Delta^{(1)}$ exists"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation-one-of-mem-cylinder","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation_one_of_mem_cylinder_even","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","The variation "],["math","\\(\\Delta^{(1)}\\)"],["text"," on the cylinder "],["math","\\(C_{00(10)^d}\\)"]],"source":"The variation $\\Delta^{(1)}$ on the cylinder $C_{00(10)^d}$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation-one-of-mem-cylinder","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $d \\ge 0$ and let $x$ be a Zeckendorf-adic integer in the cylinder $C_{00(10)^d}$, that is, one whose digits satisfy $x_{2d+3} = x_{2d+2} = 0$ and $x_{2i+1} = 1$, $x_{2i} = 0$ for $1 \\le i \\le d$. Write $\\Delta_k^{(1)}(x)$ for the truncated variation of order $k$ and $\\Delta^{(1)}(x)$ for the variation, both for $r = 1$. Then $\\Delta^{(1)}(x) = 1 - d$, and for every $k \\ge 2$: $\\Delta_k^{(1)}(x) = (2 - k)/2$ if\u2026","title":"The variation $\\Delta^{(1)}$ on the cylinder $C_{00(10)^d}$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation-one-of-mem-cylinder-2","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation_one_of_mem_cylinder_odd","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","The variation "],["math","\\(\\Delta^{(1)}\\)"],["text"," on the cylinder "],["math","\\(C_{001(01)^d}\\)"]],"source":"The variation $\\Delta^{(1)}$ on the cylinder $C_{001(01)^d}$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation-one-of-mem-cylinder-2","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $d \\ge 0$ and let $x$ be a Zeckendorf-adic integer in the cylinder $C_{001(01)^d}$, that is, one whose digits satisfy $x_{2d+4} = x_{2d+3} = 0$, $x_{2d+2} = 1$ and $x_{2i+1} = 0$, $x_{2i} = 1$ for $1 \\le i \\le d$. Write $\\Delta_k^{(1)}(x)$ for the truncated variation of order $k$ and $\\Delta^{(1)}(x)$ for the variation, both for $r = 1$. Then $\\Delta^{(1)}(x) = -d$, and for every $k \\ge 2$:\u2026","title":"The variation $\\Delta^{(1)}$ on the cylinder $C_{001(01)^d}$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-variation-ergodic-average","kind":"node","lean":"MetaMathlibExt.zeckAdicVariation_ergodic_average","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-variation-ergodic-average","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mu$ be a probability measure on the digit sequences $x : \\mathbb{N} \\to \\{0, 1\\}$ that gives the complement of the Zeckendorf-adic integers $\\mathbb{X}$ measure zero and is preserved by the odometer $T$. Let $r \\ge 1$ and let $f : \\mathbb{Z} \\to \\mathbb{C}$ be such that $f \\circ \\Delta^{(r)}$ is $\\mu$-integrable, where $\\Delta^{(r)}$ is the variation on $\\mathbb{X}$. Write $s(n)$ for the number of terms in\u2026","title":"Ergodic theorem for the Zeckendorf digit-sum variation"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-prefix-value","kind":"node","lean":"MetaMathlibExt.zeckAdicPrefixValue","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-prefix-value","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$ and a binary sequence $x = (x_i)_{i \\in \\mathbb{N}}$, the prefix value of order $k$ is the natural number $v_k(x) = \\sum_{i=2}^{k+1} x_i F_i$, where $F_i$ is the $i$-th Fibonacci number ($F_1 = F_2 = 1$). It is the value of the block $x_{k+1} \\cdots x_2$ of the $k$ rightmost digits of $x$, the name of the cylinder of order $k$ that contains $x$.","title":"Prefix value of a digit sequence"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-adic-prefix-value-one","kind":"node","lean":"MetaMathlibExt.zeckAdicPrefixValue_one","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-adic-prefix-value-one","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For every binary sequence $x = (x_i)_{i \\in \\mathbb{N}}$, the prefix value of order $1$ is $v_1(x) = x_2 F_2$, where $F_2 = 1$ is the second Fibonacci number. It equals $x_2$: it is $1$ if $x_2 = 1$ and $0$ otherwise.","title":"Prefix value of order one"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-cst","kind":"node","lean":"MetaMathlibExt.zeckCST","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sets "],["math","\\(\\mathrm{CST}_k\\)"]],"source":"The sets $\\mathrm{CST}_k$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-cst","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Fix $r, k \\in \\mathbb{N}$. For a digit sequence $x = (x_i)_{i \\in \\mathbb{N}}$ in $\\{0, 1\\}$ let $v_k(x) = \\sum_{i=2}^{k+1} x_i F_i$, where $F_i$ is the $i$-th Fibonacci number ($F_1 = F_2 = 1$). The set $\\mathrm{CST}_k$ (for the shift $r$) consists of the digit sequences $x$ such that either $v_k(x) + r < F_{k+1}$, or both $F_{k+1} \\le v_k(x)$ and $v_k(x) + r < F_{k+2}$.","title":"The sets $\\mathrm{CST}_k$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-cst-one","kind":"node","lean":"MetaMathlibExt.zeckCST_one","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{CST}_1\\)"],["text"," is empty for every positive shift"]],"source":"$\\mathrm{CST}_1$ is empty for every positive shift"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-cst-one","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For every $r \\ge 1$, the set $\\mathrm{CST}_1$ for the shift $r$ is empty.","title":"$\\mathrm{CST}_1$ is empty for every positive shift"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz","kind":"node","lean":"MetaMathlibExt.zeckNIZ","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Let $F_1 = F_2 = 1, F_3 = 2, \\dots$ be the Fibonacci numbers, and for a digit sequence $x = (x_i)_{i \\ge 0}$ with $x_i \\in \\{0, 1\\}$ and $j \\in \\mathbb{N}$ let $v_j(x) = \\sum_{i=2}^{j+1} x_i F_i$. For $r \\in \\mathbb{N}$ let $\\mathrm{CST}_j$ be the set of digit sequences $x$ with $v_j(x) + r < F_{j+1}$, or with $F_{j+1} \\le v_j(x)$ and $v_j(x) + r < F_{j+2}$. For $k \\in \\mathbb{N}$, the new-information zone of order\u2026","title":"New-information zone"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-subset-zeck-cst","kind":"node","lean":"MetaMathlibExt.zeckNIZ_subset_zeckCST","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","New-information zones lie in "],["math","\\(\\mathrm{CST}_k\\)"]],"source":"New-information zones lie in $\\mathrm{CST}_k$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-subset-zeck-cst","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For all $r, k \\in \\mathbb{N}$, the new-information zone $\\mathrm{NIZ}_k$ of order $k$ for the shift $r$ is contained in the set $\\mathrm{CST}_k$ for the shift $r$.","title":"New-information zones lie in $\\mathrm{CST}_k$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-levels","kind":"node","lean":"MetaMathlibExt.zeckNIZ_levels","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-levels","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\ell \\ge 2$ and let $r$ satisfy $F_\\ell \\le r < F_{\\ell+1}$, where $F_1 = F_2 = 1, F_3 = 2, \\dots$ are the Fibonacci numbers. For a Zeckendorf-adic integer $x = (x_i)_{i \\ge 2}$ let $v_k(x) = \\sum_{i=2}^{k+1} x_i F_i$, and let $\\mathrm{NIZ}_k$ be the new-information zone of order $k$ for $r$. Then for every Zeckendorf-adic integer $x$: $x \\notin \\mathrm{NIZ}_k$ whenever $2 \\le k \\le \\ell - 1$;\u2026","title":"The new-information zones as tower levels"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-leading-zeros","kind":"node","lean":"MetaMathlibExt.zeckNIZ_leading_zeros","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Points of "],["math","\\(\\mathrm{NIZ}_k\\)"],["text"," have leading digits "],["math","\\(00\\)"]],"source":"Points of $\\mathrm{NIZ}_k$ have leading digits $00$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-leading-zeros","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $r \\ge 1$ and $k \\ge 2$, and let $\\mathrm{NIZ}_k$ be the new-information zone of order $k$ for $r$. Every Zeckendorf-adic integer $x = (x_i)_{i \\ge 2}$ in $\\mathrm{NIZ}_k$ has $x_{k+1} = x_k = 0$, that is, the name $x_{k+1} \\cdots x_2$ of its cylinder of order $k$ starts with $00$.","title":"Points of $\\mathrm{NIZ}_k$ have leading digits $00$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-cylinder-shift","kind":"node","lean":"MetaMathlibExt.zeckNIZ_cylinder_shift","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-cylinder-shift","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mathbb{X}$ be the set of Zeckendorf-adic integers: the digit sequences $x = (x_i)_{i \\in \\mathbb{N}}$ in $\\{0, 1\\}$ with $x_0 = x_1 = 0$ and no two consecutive digits equal to $1$. Let $r \\ge 1$, $k \\ge 2$ and $y \\in \\mathbb{X}$. Suppose that every $x \\in \\mathbb{X}$ with $x_{k+1} = x_k = 0$ and $x_i = y_i$ for $2 \\le i \\le k - 1$ lies in the new-information zone $\\mathrm{NIZ}_k$ (for the shift $r$). Then\u2026","title":"Shifting a new-information cylinder by two orders"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-variation-shift","kind":"node","lean":"MetaMathlibExt.zeckNIZ_variation_shift","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lifting a cylinder of "],["math","\\(\\mathrm{NIZ}_k\\)"],["text"," to order "],["math","\\(k + 2\\)"],["text"," lowers the variation by "],["math","\\(1\\)"]],"source":"Lifting a cylinder of $\\mathrm{NIZ}_k$ to order $k + 2$ lowers the variation by $1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-variation-shift","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\ell \\ge 2$, let $r$ satisfy $F_\\ell \\le r < F_{\\ell+1}$, where $F_1 = F_2 = 1, F_3 = 2, \\dots$ are the Fibonacci numbers, and let $k \\ge \\ell$. Write $\\Delta^{(r)}$ for the variation of the Zeckendorf sum of digits under adding $r$ on the Zeckendorf-adic integers $\\mathbb{X}$, and $\\mathrm{NIZ}_k$ for the new-information zone of order $k$ for $r$. Let $y \\in \\mathbb{X}$ and $d \\in \\mathbb{Z}$. Let $C$ be the\u2026","title":"Lifting a cylinder of $\\mathrm{NIZ}_k$ to order $k + 2$ lowers the variation by $1$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-values-shift","kind":"node","lean":"MetaMathlibExt.zeckNIZ_values_shift","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of the variation on "],["math","\\(\\mathrm{NIZ}_{k+2}\\)"],["text"," are shifted by "],["math","\\(-1\\)"]],"source":"Values of the variation on $\\mathrm{NIZ}_{k+2}$ are shifted by $-1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-values-shift","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\ell \\ge 2$, let $r$ satisfy $F_\\ell \\le r < F_{\\ell+1}$, where $F_1 = F_2 = 1, F_3 = 2, \\dots$ are the Fibonacci numbers, and let $k \\ge \\ell + 4$. Write $\\Delta^{(r)}$ for the variation of the Zeckendorf sum of digits under adding $r$ on the Zeckendorf-adic integers, $\\mathrm{NIZ}_m$ for the new-information zone of order $m$ for $r$, and $i(v)$ for the Zeckendorf-adic integer whose digits are the Zeckendorf\u2026","title":"Values of the variation on $\\mathrm{NIZ}_{k+2}$ are shifted by $-1$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-count-small-values","kind":"node","lean":"MetaMathlibExt.zeckNIZ_count_small_values","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exactly "],["math","\\(2r\\)"],["text"," new-information cylinders carry each small value"]],"source":"Exactly $2r$ new-information cylinders carry each small value"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-count-small-values","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $r \\ge 1$, and let $F_n$ denote the $n$-th Fibonacci number ($F_1 = F_2 = 1$). There is an integer $D$ such that for every integer $d \\le D$ there are exactly $2r$ pairs $(k, v)$ of natural numbers with $v < F_{k+2}$ such that the Zeckendorf digit sequence $i(v)$ of $v$ lies in the new-information zone $\\mathrm{NIZ}_k$ (for the shift $r$) and the variation $\\Delta^{(r)}$ satisfies $\\Delta^{(r)}(i(v)) = d$.","title":"Exactly $2r$ new-information cylinders carry each small value"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-fib-variation-order","kind":"node","lean":"MetaMathlibExt.zeckNIZ_fib_variation_order","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","The variation by "],["math","\\(F_\\ell\\)"],["text"," on "],["math","\\(\\mathrm{NIZ}_\\ell\\)"],["text"," is "],["math","\\(1\\)"]],"source":"The variation by $F_\\ell$ on $\\mathrm{NIZ}_\\ell$ is $1$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-fib-variation-order","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\ell \\ge 3$, let $F_1 = F_2 = 1, F_3 = 2, \\dots$ be the Fibonacci numbers, and let $\\Delta^{(F_\\ell)}$ be the variation of the Zeckendorf sum of digits under adding $F_\\ell$ on the Zeckendorf-adic integers $\\mathbb{X}$. Then $\\Delta^{(F_\\ell)}(x) = 1$ for every $x \\in \\mathbb{X}$ in the new-information zone $\\mathrm{NIZ}_\\ell$ of order $\\ell$ for $r = F_\\ell$.","title":"The variation by $F_\\ell$ on $\\mathrm{NIZ}_\\ell$ is $1$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-fib-variation-order-succ","kind":"node","lean":"MetaMathlibExt.zeckNIZ_fib_variation_order_succ","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of the variation by "],["math","\\(F_\\ell\\)"],["text"," on "],["math","\\(\\mathrm{NIZ}_{\\ell+1}\\)"]],"source":"Values of the variation by $F_\\ell$ on $\\mathrm{NIZ}_{\\ell+1}$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-fib-variation-order-succ","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\ell \\ge 3$, let $F_1 = F_2 = 1, F_3 = 2, \\dots$ be the Fibonacci numbers, and let $\\Delta^{(F_\\ell)}$ be the variation of the Zeckendorf sum of digits under adding $F_\\ell$ on the Zeckendorf-adic integers. For $v \\in \\mathbb{N}$ write $i(v)$ for the Zeckendorf-adic integer whose digits are the Zeckendorf digits of $v$. Among the $v < F_{\\ell+3}$ with $i(v)$ in the new-information zone $\\mathrm{NIZ}_{\\ell+1}$\u2026","title":"Values of the variation by $F_\\ell$ on $\\mathrm{NIZ}_{\\ell+1}$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-niz-fib-variation-order-add-two","kind":"node","lean":"MetaMathlibExt.zeckNIZ_fib_variation_order_add_two","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","The variation by "],["math","\\(F_\\ell\\)"],["text"," on "],["math","\\(\\mathrm{NIZ}_{\\ell+2}\\)"],["text"," is "],["math","\\(0\\)"]],"source":"The variation by $F_\\ell$ on $\\mathrm{NIZ}_{\\ell+2}$ is $0$"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-niz-fib-variation-order-add-two","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\ell \\ge 3$, let $F_1 = F_2 = 1, F_3 = 2, \\dots$ be the Fibonacci numbers, and let $\\Delta^{(F_\\ell)}$ be the variation of the Zeckendorf sum of digits under adding $F_\\ell$ on the Zeckendorf-adic integers $\\mathbb{X}$. Then $\\Delta^{(F_\\ell)}(x) = 0$ for every $x \\in \\mathbb{X}$ in the new-information zone $\\mathrm{NIZ}_{\\ell+2}$ of order $\\ell + 2$ for $r = F_\\ell$.","title":"The variation by $F_\\ell$ on $\\mathrm{NIZ}_{\\ell+2}$ is $0$"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-blocks","kind":"node","lean":"MetaMathlibExt.zeckBlocks","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-blocks","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"Let $r \\in \\mathbb{N}$ have Zeckendorf representation $r = F_{i_1} + \\dots + F_{i_s}$ with $i_1 > \\dots > i_s \\ge 2$ (Fibonacci numbers with $F_1 = F_2 = 1$). The blocks of $r$ are the maximal runs of successive terms of the list $i_1, \\dots, i_s$ in which each index is exactly $2$ less than the one before it (the maximal factors $(10)^m$ of the expansion, a final $1$ at the unit position counting as $10$). The\u2026","title":"Blocks of a Zeckendorf representation"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-block-prefix","kind":"node","lean":"MetaMathlibExt.zeckBlockPrefix","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-block-prefix","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"The blocks of $r \\in \\mathbb{N}$ are the maximal runs of Zeckendorf indices of $r$ in which consecutive indices differ by exactly $2$, listed from the unit digit. For $i \\in \\mathbb{N}$, the block prefix $r[i]$ is the sum of the Fibonacci numbers $F_j$ over the indices $j$ in the first $i$ blocks of $r$, or in all blocks when $i$ exceeds the number $\\rho(r)$ of blocks. Thus $r[0] = 0$ and $r[i] = r$ for\u2026","title":"Block prefixes of a natural number"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-block-prefix-zero","kind":"node","lean":"MetaMathlibExt.zeckBlockPrefix_zero","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-block-prefix-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"For every natural number $r$, the block prefix $r[0]$, formed by the first $0$ blocks of the Zeckendorf representation of $r$, equals $0$.","title":"The empty block prefix is zero"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-blocks-zero","kind":"node","lean":"MetaMathlibExt.zeckBlocks_zero","parent":12608,"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-blocks-zero","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"fully_proved","summary":"The list of blocks of $0$ is empty; that is, the number $\\rho(0)$ of blocks of $0$ is $0$.","title":"Zero has no blocks"},{"id":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/zeck-block-action-alpha-mixing","kind":"node","lean":"MetaMathlibExt.zeckBlockAction_alpha_mixing","parent":12608,"reading_title":{"format":"inline-parts-v1","parts":[["text","An "],["math","\\(\\alpha\\)"],["text","-mixing bound for the block variations"]],"source":"An $\\alpha$-mixing bound for the block variations"},"result":true,"route":"../roadmap/NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits/index.html#zeck-block-action-alpha-mixing","scope":"NumberTheory/digits-numeration/numeration-automatic/zeckendorf-sum-of-digits","status":"can_prove","summary":"Let $\\mu$ be a probability measure on the digit sequences $x : \\mathbb{N} \\to \\{0, 1\\}$ that gives the complement of the Zeckendorf-adic integers measure zero and is preserved by the odometer $T$. Let $r \\ge 1$, let $\\rho(r)$ be the number of blocks of $r$ and $r[i]$ its block prefixes, and for $1 \\le i \\le \\rho(r)$ let $X_i(x) = \\Delta^{(r[i] - r[i-1])}(T^{r[i-1]}(x))$, where $\\Delta^{(m)}$ is the variation and\u2026","title":"An $\\alpha$-mixing bound for the block variations"},{"data_url":"scopes/NumberTheory/integer-sequences.json","id":"NumberTheory/integer-sequences","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/index.html","scope":"NumberTheory/integer-sequences","status":"can_prove","summary":"Named nonlinear integer sequences: Collatz and iterated maps, self-referential sequences, Sylvester and Euclid sequences, realizable and Dold sequences.","title":"Integer sequences"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential.json","id":"NumberTheory/integer-sequences/iterated-self-referential","kind":"scope","parent":12665,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential","status":"can_prove","summary":"Collatz problem, EKG, Stern diatomic, Hofstadter, Golomb, Ulam, Ducci, Josephus and Conway-type sequences.","title":"Iterated maps and self-referential sequences"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/golomb-like-closed-form","kind":"node","lean":"MetaMathlibExt.golomb_like_closed_form","parent":12666,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/index.html#golomb-like-closed-form","scope":"NumberTheory/integer-sequences/iterated-self-referential","status":"fully_proved","summary":"Let $y$ and $z$ be integers of opposite parity with $y \\ge 2$ and $z \\ge 2 - y$, and put $c = (y + z - 1)/2$. There is exactly one strictly increasing function $f$ from the positive integers to the positive integers such that $f(1) = (y + z + 1)/2$ and $f(f(n)) = yn + z$ for every positive integer $n$. Moreover, for every such $f$, every natural number $k$ and every integer $j$ with $-c\\,y^k \\le j < c\\,y^k$, the\u2026","title":"Closed form for Golomb-like sequences"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/hinman-kuca-schlesinger-sheydvasser-weak","kind":"node","lean":"MetaMathlibExt.hinman_kuca_schlesinger_sheydvasser_weak_rigidity","parent":12666,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/index.html#hinman-kuca-schlesinger-sheydvasser-weak","scope":"NumberTheory/integer-sequences/iterated-self-referential","status":"can_prove","summary":"There exist sequences of integers $(a_k)$, $(b_k)$, $(c_k)$, $(d_k)$ indexed by $k \\in \\mathbb{N}$ with the following property. For every real $C > 0$ there is a natural number $N > 0$ such that for every natural number $n \\ge N$ with $n > 1$, and every sequence of natural numbers $u_0, u_1, u_2, \\dots$ with $u_0 = 1$ and $u_1 = n$ that is strictly increasing and in which, for every $t \\ge 2$, $u_t$ is the least\u2026","title":"Hinman\u2013Kuca\u2013Schlesinger\u2013Sheydvasser weak rigidity theorem for Ulam sequences"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/is-ekgsequence","kind":"node","lean":"MetaMathlibExt.IsEKGSequence","parent":12666,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/index.html#is-ekgsequence","scope":"NumberTheory/integer-sequences/iterated-self-referential","status":"fully_proved","summary":"A sequence $a_0, a_1, a_2, \\dots$ of natural numbers is an EKG sequence when $a_0 = 1$, $a_1 = 2$, and for every $n \\ge 0$ the term $a_{n+2}$ is positive, differs from each of $a_0, \\dots, a_{n+1}$, satisfies $\\gcd(a_{n+1}, a_{n+2}) > 1$, and satisfies $a_{n+2} \\le m$ for every positive integer $m$ that differs from each of $a_0, \\dots, a_{n+1}$ and has $\\gcd(a_{n+1}, m) > 1$. Thus every term from $a_2$ on is the\u2026","title":"The EKG sequence"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem.json","id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","title":"Collatz problem"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/collatz-step","kind":"node","lean":"Nat.collatzStep","parent":12670,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#collatz-step","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"The Collatz map $T : \\mathbb{N} \\to \\mathbb{N}$ sends a natural number $n$ to $n / 2$ if $n$ is even and to $3n + 1$ if $n$ is odd.","title":"The Collatz map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/collatz-iterate","kind":"node","lean":"Nat.collatzIterate","parent":12670,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#collatz-iterate","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"Let $T : \\mathbb{N} \\to \\mathbb{N}$ be the Collatz map, which sends $n$ to $n / 2$ if $n$ is even and to $3n + 1$ if $n$ is odd. For natural numbers $k$ and $n$, the $k$-th Collatz iterate of $n$ is $T^k(n)$, the result of applying $T$ to $n$ exactly $k$ times: $T^0(n) = n$ and $T^{k+1}(n) = T(T^k(n))$.","title":"Iterates of the Collatz map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/collatz-stopping-time","kind":"node","lean":"Nat.collatzStoppingTime","parent":12670,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#collatz-stopping-time","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"Let $T : \\mathbb{N} \\to \\mathbb{N}$ be the Collatz map, which sends $n$ to $n / 2$ if $n$ is even and to $3n + 1$ if $n$ is odd, and let $T^k$ denote its $k$-fold iterate, with $T^0(n) = n$. For a natural number $n$, the Collatz stopping time of $n$ is undefined if $n = 0$ or if $T^k(n) \\ne 1$ for every $k \\in \\mathbb{N}$; otherwise it is the least $k \\in \\mathbb{N}$ with $T^k(n) = 1$.","title":"Collatz stopping time"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-shift","kind":"node","lean":"MetaMathlibExt.qShift","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Division by "],["math","\\(q\\)"],["text"," in "],["math","\\(\\mathbb{F}_2((q))\\)"]],"source":"Division by $q$ in $\\mathbb{F}_2((q))$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-shift","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"For a power series $x = \\sum_n x_n q^n \\in \\mathbb{F}_2[[q]]$, its shift is the power series $\\sum_n x_{n+1} q^n$, obtained by discarding the constant term of $x$ and dividing by $q$. In particular, the shift of $qy$ is $y$ for every $y \\in \\mathbb{F}_2[[q]]$.","title":"Division by $q$ in $\\mathbb{F}_2((q))$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-shift-x-mul","kind":"node","lean":"MetaMathlibExt.qShift_X_mul","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dividing a multiple of "],["math","\\(q\\)"],["text"," by "],["math","\\(q\\)"]],"source":"Dividing a multiple of $q$ by $q$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-shift-x-mul","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"For every power series $y \\in \\mathbb{F}_2[[q]]$, applying the shift $\\sum_n x_n q^n \\mapsto \\sum_n x_{n+1} q^n$ to $qy$ gives back $y$.","title":"Dividing a multiple of $q$ by $q$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz","kind":"node","lean":"MetaMathlibExt.qCollatz","parent":12670,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"For a power series $y = \\sum_n y_n q^n \\in \\mathbb{F}_2[[q]]$, write $y/q$ for the power series $\\sum_n y_{n+1} q^n$: the constant term of $y$ is discarded and the rest is divided by $q$. For power series $A, B \\in \\mathbb{F}_2[[q]]$, the map $T_{A,B} : \\mathbb{F}_2[[q]] \\to \\mathbb{F}_2[[q]]$ is defined by $T_{A,B}(x) = x/q$ if the constant term of $x$ is $0$, and $T_{A,B}(x) = (Ax + B)/q$ otherwise.","title":"The q-analog of the 3x+1 map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-x-mul","kind":"node","lean":"MetaMathlibExt.qCollatz_X_mul","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","The q-analog 3x+1 map on multiples of "],["math","\\(q\\)"]],"source":"The q-analog 3x+1 map on multiples of $q$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-x-mul","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"For all power series $A, B, y \\in \\mathbb{F}_2[[q]]$, the $q$-analog $3x+1$ map satisfies $T_{A,B}(qy) = y$.","title":"The q-analog 3x+1 map on multiples of $q$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-zero","kind":"node","lean":"MetaMathlibExt.qCollatz_zero","parent":12670,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-zero","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"For all power series $A, B \\in \\mathbb{F}_2[[q]]$, the $q$-analog $3x+1$ map satisfies $T_{A,B}(0) = 0$.","title":"Zero is a fixed point of the q-analog 3x+1 map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/two-dvd-of-to-zmod-eq-zero","kind":"node","lean":"MetaMathlibExt.two_dvd_of_toZMod_eq_zero","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisibility by "],["math","\\(2\\)"],["text"," from the residue modulo "],["math","\\(2\\)"]],"source":"Divisibility by $2$ from the residue modulo $2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#two-dvd-of-to-zmod-eq-zero","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"If the residue modulo $2$ of a $2$-adic integer $y \\in \\mathbb{Z}_2$ is zero, then $2$ divides $y$ in $\\mathbb{Z}_2$.","title":"Divisibility by $2$ from the residue modulo $2$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/to-zmod-three-mul-add-one-of-odd","kind":"node","lean":"MetaMathlibExt.toZMod_three_mul_add_one_of_odd","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(3x + 1\\)"],["text"," is even for odd "],["math","\\(2\\)"],["text","-adic "],["math","\\(x\\)"]],"source":"$3x + 1$ is even for odd $2$-adic $x$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#to-zmod-three-mul-add-one-of-odd","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"For every $2$-adic integer $x \\in \\mathbb{Z}_2$ whose residue modulo $2$ is nonzero, the residue of $3x + 1$ modulo $2$ is zero.","title":"$3x + 1$ is even for odd $2$-adic $x$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/two-adic-collatz","kind":"node","lean":"MetaMathlibExt.twoAdicCollatz","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(2\\)"],["text","-adic "],["math","\\(3x+1\\)"],["text"," map"]],"source":"The $2$-adic $3x+1$ map"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#two-adic-collatz","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"The map $T : \\mathbb{Z}_2 \\to \\mathbb{Z}_2$ is defined by $T(x) = x/2$ if $x$ is even and $T(x) = (3x+1)/2$ if $x$ is odd, where even and odd refer to the residue of $x$ modulo $2$.","title":"The $2$-adic $3x+1$ map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/two-mul-two-adic-collatz-of-even","kind":"node","lean":"MetaMathlibExt.two_mul_twoAdicCollatz_of_even","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even branch of the "],["math","\\(2\\)"],["text","-adic "],["math","\\(3x+1\\)"],["text"," map"]],"source":"Even branch of the $2$-adic $3x+1$ map"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#two-mul-two-adic-collatz-of-even","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"Let $T : \\mathbb{Z}_2 \\to \\mathbb{Z}_2$ be the $3x+1$ map on the $2$-adic integers. For every $x \\in \\mathbb{Z}_2$ whose residue modulo $2$ is zero, $2\\,T(x) = x$.","title":"Even branch of the $2$-adic $3x+1$ map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/two-mul-two-adic-collatz-of-odd","kind":"node","lean":"MetaMathlibExt.two_mul_twoAdicCollatz_of_odd","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd branch of the "],["math","\\(2\\)"],["text","-adic "],["math","\\(3x+1\\)"],["text"," map"]],"source":"Odd branch of the $2$-adic $3x+1$ map"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#two-mul-two-adic-collatz-of-odd","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"Let $T : \\mathbb{Z}_2 \\to \\mathbb{Z}_2$ be the $3x+1$ map on the $2$-adic integers. For every $x \\in \\mathbb{Z}_2$ whose residue modulo $2$ is nonzero, $2\\,T(x) = 3x + 1$.","title":"Odd branch of the $2$-adic $3x+1$ map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/two-adic-collatz-one-two","kind":"node","lean":"MetaMathlibExt.twoAdicCollatz_one_two","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(2\\)"],["text","-adic "],["math","\\(3x+1\\)"],["text"," map swaps "],["math","\\(1\\)"],["text"," and "],["math","\\(2\\)"]],"source":"The $2$-adic $3x+1$ map swaps $1$ and $2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#two-adic-collatz-one-two","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"The $2$-adic $3x+1$ map $T : \\mathbb{Z}_2 \\to \\mathbb{Z}_2$ satisfies $T(1) = 2$ and $T(2) = 1$.","title":"The $2$-adic $3x+1$ map swaps $1$ and $2$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-parity-vector","kind":"node","lean":"MetaMathlibExt.qCollatzParityVector","parent":12670,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-parity-vector","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"For power series $A, B, x \\in \\mathbb{F}_2[[q]]$, the parity vector of $x$ is the $2$-adic integer $\\mathrm{PV}_{A,B}(x) = \\sum_{k \\ge 0} x_k 2^k \\in \\mathbb{Z}_2$, where $x_k \\in \\{0, 1\\}$ is the constant coefficient of the $k$-th iterate $T_{A,B}^k(x)$ of the $q$-analog $3x+1$ map $T_{A,B}$.","title":"The parity vector of the q-analog 3x+1 map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-parity-vector-zero","kind":"node","lean":"MetaMathlibExt.qCollatzParityVector_zero","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parity vector of "],["math","\\(0\\)"]],"source":"The parity vector of $0$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-parity-vector-zero","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"fully_proved","summary":"For all $A, B \\in \\mathbb{F}_2[[q]]$, the parity vector of $0$ under the $q$-analog $T_{A,B}$ of the $3x+1$ map is $0$, that is, $\\mathrm{PV}_{A,B}(0) = 0$.","title":"The parity vector of $0$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-conjugate-two-adic-collatz","kind":"node","lean":"MetaMathlibExt.qCollatz_conjugate_twoAdicCollatz","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-Collatz map is conjugate to the "],["math","\\(2\\)"],["text","-adic "],["math","\\(3x+1\\)"],["text"," map"]],"source":"The $q$-Collatz map is conjugate to the $2$-adic $3x+1$ map"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-conjugate-two-adic-collatz","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $A$ and $B$ be odd power series in $\\mathbb{F}_2[[q]]$ (constant term $1$), let $T_{A,B}$ be the $q$-analog of the $3x+1$ map on $\\mathbb{F}_2[[q]]$, and let $T$ be the $3x+1$ map on the $2$-adic integers $\\mathbb{Z}_2$. Then there is a bijection $h : \\mathbb{Z}_2 \\to \\mathbb{F}_2[[q]]$ with $h^{-1}(T_{A,B}(h(x))) = T(x)$ for every $x \\in \\mathbb{Z}_2$.","title":"The $q$-Collatz map is conjugate to the $2$-adic $3x+1$ map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-one-add-x-conjugate-two-adic-c","kind":"node","lean":"MetaMathlibExt.qCollatz_one_add_X_conjugate_twoAdicCollatz","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(T_{1+q,1}\\)"],["text"," is conjugate to the "],["math","\\(2\\)"],["text","-adic "],["math","\\(3x+1\\)"],["text"," map"]],"source":"The map $T_{1+q,1}$ is conjugate to the $2$-adic $3x+1$ map"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-one-add-x-conjugate-two-adic-c","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $T_q = T_{1+q,1}$ be the $q$-analog of the $3x+1$ map with $A = 1 + q$ and $B = 1$, so that $T_q(x) = x/q$ if $q$ divides $x$ and $T_q(x) = ((1+q)x + 1)/q$ otherwise, and let $T$ be the $3x+1$ map on the $2$-adic integers $\\mathbb{Z}_2$. Then there is a bijection $h : \\mathbb{Z}_2 \\to \\mathbb{F}_2[[q]]$ with $h^{-1}(T_q(h(x))) = T(x)$ for every $x \\in \\mathbb{Z}_2$.","title":"The map $T_{1+q,1}$ is conjugate to the $2$-adic $3x+1$ map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-one-one-add-x-sq-polynomial","kind":"node","lean":"MetaMathlibExt.qCollatz_one_one_add_X_sq_polynomial","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial orbits of "],["math","\\(T_{1,1+q^2}\\)"]],"source":"Polynomial orbits of $T_{1,1+q^2}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-one-one-add-x-sq-polynomial","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $T = T_{1,1+q^2}$ be the $q$-analog of the $3x+1$ map with $A = 1$ and $B = 1 + q^2$. For every polynomial $p$ over $\\mathbb{F}_2$, some iterate $T^k(p)$ with $k \\ge 0$ lies in $\\{0, 1 + q, 1, q\\}$, the set formed by the fixed points $0$ and $1 + q$ of $T$ and its $2$-cycle $\\{1, q\\}$. Moreover, for every $x \\in \\mathbb{F}_2[[q]]$, $T(x)$ is a polynomial if and only if $x$ is a polynomial.","title":"Polynomial orbits of $T_{1,1+q^2}$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-add-x-pow-mul","kind":"node","lean":"MetaMathlibExt.qCollatz_add_X_pow_mul","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","A perturbation at "],["math","\\(q^{n+1}\\)"],["text"," moves to "],["math","\\(q^n\\)"],["text"," under "],["math","\\(T_{A,B}\\)"]],"source":"A perturbation at $q^{n+1}$ moves to $q^n$ under $T_{A,B}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-add-x-pow-mul","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $A, B, U, V$ be power series in $\\mathbb{F}_2[[q]]$ with $A$, $B$ and $V$ odd (constant term $1$), let $T_{A,B}$ be the $q$-analog of the $3x+1$ map, and let $n \\ge 0$ be an integer. Then there is an odd power series $W \\in \\mathbb{F}_2[[q]]$ with $T_{A,B}(U + q^{n+1} V) = T_{A,B}(U) + q^n W$.","title":"A perturbation at $q^{n+1}$ moves to $q^n$ under $T_{A,B}$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-iterate-add-x-pow-mul","kind":"node","lean":"MetaMathlibExt.qCollatz_iterate_add_X_pow_mul","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterates of "],["math","\\(T_{A,B}\\)"],["text"," move a perturbation at "],["math","\\(q^{n+1}\\)"],["text"," to "],["math","\\(q^{n+1-k}\\)"]],"source":"Iterates of $T_{A,B}$ move a perturbation at $q^{n+1}$ to $q^{n+1-k}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-iterate-add-x-pow-mul","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $A, B, U, V$ be power series in $\\mathbb{F}_2[[q]]$ with $A$, $B$ and $V$ odd (constant term $1$), let $T_{A,B}$ be the $q$-analog of the $3x+1$ map, and let $n$ and $k$ be natural numbers with $k \\le n + 1$. Write $T_{A,B}^k$ for the $k$-th iterate of $T_{A,B}$. Then there is an odd power series $W \\in \\mathbb{F}_2[[q]]$ with $T_{A,B}^k(U + q^{n+1} V) = T_{A,B}^k(U) + q^{n+1-k} W$.","title":"Iterates of $T_{A,B}$ move a perturbation at $q^{n+1}$ to $q^{n+1-k}$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-iterate-parity-flip","kind":"node","lean":"MetaMathlibExt.qCollatz_iterate_parity_flip","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","A perturbation at "],["math","\\(q^n\\)"],["text"," flips the parity of the "],["math","\\(n\\)"],["text","-th iterate"]],"source":"A perturbation at $q^n$ flips the parity of the $n$-th iterate"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-iterate-parity-flip","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $A, B, U, V \\in \\mathbb{F}_2[[q]]$ with $A$, $B$ and $V$ odd (constant term $1$), let $T_{A,B}$ be the $q$-analog of the $3x+1$ map, and let $n \\ge 0$ be an integer. Then the $n$-th iterates $T_{A,B}^n(U + q^n V)$ and $T_{A,B}^n(U)$ have different constant terms.","title":"A perturbation at $q^n$ flips the parity of the $n$-th iterate"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-parity-vector-mod-eq-iff","kind":"node","lean":"MetaMathlibExt.qCollatzParityVector_modEq_iff","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity vectors detect congruence modulo "],["math","\\(q^{n+1}\\)"]],"source":"Parity vectors detect congruence modulo $q^{n+1}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-parity-vector-mod-eq-iff","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $A, B \\in \\mathbb{F}_2[[q]]$ be odd (constant term $1$), let $\\mathrm{PV}_{A,B}$ be the parity vector map of the $q$-analog $T_{A,B}$ of the $3x+1$ map, and let $n \\ge 0$ be an integer. For all $a, b \\in \\mathbb{F}_2[[q]]$, we have $\\mathrm{PV}_{A,B}(a) \\equiv \\mathrm{PV}_{A,B}(b) \\pmod{2^{n+1}}$ in $\\mathbb{Z}_2$ if and only if $q^{n+1}$ divides $a - b$.","title":"Parity vectors detect congruence modulo $q^{n+1}$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-parity-vector-quotient-bijecti","kind":"node","lean":"MetaMathlibExt.qCollatzParityVector_quotient_bijection","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity vectors induce bijections modulo "],["math","\\(q^n\\)"]],"source":"Parity vectors induce bijections modulo $q^n$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-parity-vector-quotient-bijecti","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $A, B \\in \\mathbb{F}_2[[q]]$ be odd (constant term $1$), let $\\mathrm{PV}_{A,B}$ be the parity vector map of the $q$-analog $T_{A,B}$ of the $3x+1$ map, and let $n \\ge 1$ be an integer. Then there is a bijection $e : \\mathbb{F}_2[[q]]/(q^n) \\to \\mathbb{Z}/2^n\\mathbb{Z}$ such that, for every $a \\in \\mathbb{F}_2[[q]]$, $e$ sends the class of $a$ to the residue of $\\mathrm{PV}_{A,B}(a)$ modulo $2^n$.","title":"Parity vectors induce bijections modulo $q^n$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/q-collatz-fixed-points-and-two-cycle","kind":"node","lean":"MetaMathlibExt.qCollatz_fixedPoints_and_twoCycle","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed points and the "],["math","\\(2\\)"],["text","-cycle of the "],["math","\\(q\\)"],["text","-Collatz map"]],"source":"Fixed points and the $2$-cycle of the $q$-Collatz map"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#q-collatz-fixed-points-and-two-cycle","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $A, B \\in \\mathbb{F}_2[[q]]$ be odd (constant term $1$) and let $T_{A,B}$ be the $q$-analog of the $3x+1$ map. For every $x \\in \\mathbb{F}_2[[q]]$, $T_{A,B}(x) = x$ holds exactly when $x = 0$ or $x = B/(A+q)$. Moreover, for every $x \\in \\mathbb{F}_2[[q]]$, $T_{A,B}(T_{A,B}(x)) = x$ and $T_{A,B}(x) \\ne x$ hold together exactly when $x = B/(A+q^2)$ or $x = qB/(A+q^2)$.","title":"Fixed points and the $2$-cycle of the $q$-Collatz map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/no-collatz-m-cycle-le-ninety-one","kind":"node","lean":"MetaMathlibExt.no_collatz_m_cycle_le_ninety_one","parent":12670,"reading_title":{"format":"inline-parts-v1","parts":[["text","No Collatz "],["math","\\(m\\)"],["text","-cycles with "],["math","\\(m \\le 91\\)"],["text"," (Hercher)"]],"source":"No Collatz $m$-cycles with $m \\le 91$ (Hercher)"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/collatz-problem/index.html#no-collatz-m-cycle-le-ninety-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/collatz-problem","status":"can_prove","summary":"Let $T$ be the shortcut Collatz map on the natural numbers, $T(c) = c/2$ for even $c$ and $T(c) = (3c+1)/2$ for odd $c$, and let $m$ be a natural number with $m \\le 91$. There are no natural numbers $n$ and $p > 0$ such that $T^i(n) > 2$ for all $i < p$, $T^p(n) = n$, $T^k(n) \\ne n$ for all $k$ with $0 < k < p$, and exactly $m$ indices $i < p$ satisfy both $T^i(n) < T^{(i+1) \\bmod p}(n)$ and\u2026","title":"No Collatz $m$-cycles with $m \\le 91$ (Hercher)"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map.json","id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","title":"Arithmetic derivative and the Collatz map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/collatz-accelerated","kind":"node","lean":"MetaMathlibExt.collatzAccelerated","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#collatz-accelerated","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"fully_proved","summary":"The accelerated Collatz map $C : \\mathbb{N} \\to \\mathbb{N}$ sends a natural number $n$ to $n/2$ if $n$ is even and to $(3n + 1)/2$ if $n$ is odd.","title":"The accelerated Collatz map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/collatz-accelerated-of-odd","kind":"node","lean":"MetaMathlibExt.collatzAccelerated_of_odd","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#collatz-accelerated-of-odd","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"fully_proved","summary":"Let $C$ be the accelerated Collatz map and $T$ the Collatz map. For every odd natural number $n$, $C(n) = T(T(n))$.","title":"The accelerated Collatz map on odd numbers"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/collatz-accelerated-of-even","kind":"node","lean":"MetaMathlibExt.collatzAccelerated_of_even","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#collatz-accelerated-of-even","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"fully_proved","summary":"Let $C$ be the accelerated Collatz map and $T$ the Collatz map. For every even natural number $n$, $C(n) = T(n)$.","title":"The accelerated Collatz map on even numbers"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/is-compatible-semiprime-pair","kind":"node","lean":"MetaMathlibExt.IsCompatibleSemiprimePair","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#is-compatible-semiprime-pair","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"fully_proved","summary":"Natural numbers $p, q, r, s$ form a compatible semiprime pair, and the semiprimes $pq$ and $rs$ are called compatible, when $p, q, r, s$ are primes with $2(r + s) = p + q$ and $2rs = 3pq + 1$; equivalently, $r$ and $s$ are the roots of $2x^2 - (p + q)x + 3pq + 1 = 0$.","title":"Compatible semiprimes"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/is-compatible-semiprime-pair-71-1609","kind":"node","lean":"MetaMathlibExt.isCompatibleSemiprimePair_71_1609","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#is-compatible-semiprime-pair-71-1609","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"fully_proved","summary":"The semiprimes $71 \\cdot 1609 = 114239$ and $349 \\cdot 491$ are compatible.","title":"An example of compatible semiprimes"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/arithmetic-derivative-bounds","kind":"node","lean":"MetaMathlibExt.arithmeticDerivative_bounds","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#arithmetic-derivative-bounds","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $n \\ge 2$ be a natural number, let $p$ be the least prime factor of $n$, and let $\\Omega(n)$ be the number of prime factors of $n$ counted with multiplicity. Then the arithmetic derivative $D(n)$ satisfies","title":"Bounds for the arithmetic derivative"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/odd-of-arithmetic-derivative-collatz-com","kind":"node","lean":"MetaMathlibExt.odd_of_arithmeticDerivative_collatz_comm","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#odd-of-arithmetic-derivative-collatz-com","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $D$ be the arithmetic derivative and $C$ the accelerated Collatz map. Every positive integer $n$ with $D(C(n)) = C(D(n))$ is odd.","title":"Positive solutions of the Collatz arithmetic differential equation are odd"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/composite-of-arithmetic-derivative-colla","kind":"node","lean":"MetaMathlibExt.composite_of_arithmeticDerivative_collatz_comm","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#composite-of-arithmetic-derivative-colla","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $D$ be the arithmetic derivative and $C$ the accelerated Collatz map. Every positive integer $n$ with $D(C(n)) = C(D(n))$ is composite, that is, $n > 1$ and $n$ is not prime.","title":"Positive solutions of the Collatz arithmetic differential equation are composite"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/arithmetic-derivative-collatz-comm-prime","kind":"node","lean":"MetaMathlibExt.arithmeticDerivative_collatz_comm_prime_tetration_iff","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#arithmetic-derivative-collatz-comm-prime","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $D$ be the arithmetic derivative and $C$ the accelerated Collatz map. For a prime $p$, $D(C(p^p)) = C(D(p^p))$ if and only if $C(p^p) = q^q$ for some prime $q$.","title":"Prime tetrations where the arithmetic derivative and the Collatz map commute"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/prime-gap-bound-of-arithmetic-derivative","kind":"node","lean":"MetaMathlibExt.prime_gap_bound_of_arithmeticDerivative_collatz_comm","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#prime-gap-bound-of-arithmetic-derivative","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $D$ be the arithmetic derivative and let $C$ be the accelerated Collatz map, $C(n) = n/2$ for even $n$ and $C(n) = (3n + 1)/2$ for odd $n$. If $p$ and $q$ are primes with $q < p$ and $D(C(pq)) = C(D(pq))$, then $p - q \\ge 2\\sqrt{30p^2 + 2} - 10p$.","title":"Prime gap bound for semiprime solutions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/not-special-semiprime-of-arithmetic-deri","kind":"node","lean":"MetaMathlibExt.not_special_semiprime_of_arithmeticDerivative_collatz_comm","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#not-special-semiprime-of-arithmetic-deri","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $D$ be the arithmetic derivative and let $C$ be the accelerated Collatz map, $C(n) = n/2$ for even $n$ and $C(n) = (3n + 1)/2$ for odd $n$. Let $n$ be a positive integer with $D(C(n)) = C(D(n))$ and let $p$ be a prime. Then $n \\ne p^2$; if $p + 2$ is prime then $n \\ne p(p + 2)$; if $p + 4$ is prime then $n \\ne p(p + 4)$; and if $2p + 1$ is prime then $n \\ne p(2p + 1)$. Thus no such $n$ is the square of a prime,\u2026","title":"No special semiprime solves the derivative\u2013Collatz equation"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/collatz-accelerated-mul-eq-and-comm-of-r","kind":"node","lean":"MetaMathlibExt.collatzAccelerated_mul_eq_and_comm_of_roots_prime","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#collatz-accelerated-mul-eq-and-comm-of-r","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $D$ be the arithmetic derivative and $C$ the accelerated Collatz map. Let $p$ and $q$ be primes and let $t \\ge 0$ be an integer with $t^2 = p^2 - 22pq + q^2 - 8$. If $r = (p + q + t)/4$ and $s = (p + q - t)/4$ are both primes, then $C(pq) = rs$ and $D(C(pq)) = C(D(pq))$.","title":"Prime roots give semiprime solutions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/mod-three","kind":"node","lean":"MetaMathlibExt.IsCompatibleSemiprimePair.mod_three","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#mod-three","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $p, q, r, s$ be primes with $2(r + s) = p + q$ and $2rs = 3pq + 1$, so that the semiprimes $pq$ and $rs$ are compatible. Then $pq \\equiv rs \\equiv 2 \\pmod 3$; moreover one of $p, q$ is congruent to $1$ and the other to $2$ modulo $3$, and likewise one of $r, s$ is congruent to $1$ and the other to $2$ modulo $3$.","title":"Compatible semiprimes modulo 3"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/odd","kind":"node","lean":"MetaMathlibExt.IsCompatibleSemiprimePair.odd","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#odd","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $p, q, r, s$ be primes with $2(r + s) = p + q$ and $2rs = 3pq + 1$, so that the semiprimes $pq$ and $rs$ are compatible. Then $p$, $q$, $r$ and $s$ are all odd.","title":"Compatible semiprimes have odd prime factors"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/mod-four","kind":"node","lean":"MetaMathlibExt.IsCompatibleSemiprimePair.mod_four","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#mod-four","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $p, q, r, s$ be primes with $2(r + s) = p + q$ and $2rs = 3pq + 1$, so that the semiprime $pq$ is compatible with the semiprime $rs$. Then $pq \\equiv 3 \\pmod 4$, and one of $p, q$ is congruent to $1$ and the other to $3$ modulo $4$.","title":"Compatible semiprimes modulo 4"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/bounds","kind":"node","lean":"MetaMathlibExt.IsCompatibleSemiprimePair.bounds","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#bounds","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Suppose the semiprimes $pq$ and $rs$ are compatible, where $q < p$. Then $r \\ge 241$, $s \\ge 241$, $3q < r < 3p$ and $3q < s < 3p$.","title":"Bounds on the factors of compatible semiprimes"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/mod-five","kind":"node","lean":"MetaMathlibExt.IsCompatibleSemiprimePair.mod_five","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#mod-five","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $p, q, r, s$ be primes with $2(r + s) = p + q$ and $2rs = 3pq + 1$, so that the semiprimes $pq$ and $rs$ are compatible. Then one of $p, q$ is congruent to $1$ and the other to $4$ modulo $5$, and likewise one of $r, s$ is congruent to $1$ and the other to $4$ modulo $5$.","title":"Compatible semiprimes modulo 5"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/mod-ten","kind":"node","lean":"MetaMathlibExt.IsCompatibleSemiprimePair.mod_ten","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#mod-ten","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Let $p, q, r, s$ be primes with $2(r + s) = p + q$ and $2rs = 3pq + 1$, so that the semiprimes $pq$ and $rs$ are compatible. Then one of $p, q$ is congruent to $1$ and the other to $9$ modulo $10$, and likewise one of $r, s$ is congruent to $1$ and the other to $9$ modulo $10$.","title":"Compatible semiprimes modulo 10"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/summable-inv-compatible-semiprimes","kind":"node","lean":"MetaMathlibExt.summable_inv_compatibleSemiprimes","parent":12697,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map/index.html#summable-inv-compatible-semiprimes","scope":"NumberTheory/integer-sequences/iterated-self-referential/arithmetic-derivative-and-the-collatz-map","status":"can_prove","summary":"Call a natural number $n$ a compatible semiprime if $n = pq$ for primes $p, q$ for which there are primes $r, s$ with $2(r + s) = p + q$ and $2rs = 3pq + 1$. The sum of $1/n$ over all compatible semiprimes $n$ converges.","title":"Reciprocals of compatible semiprimes have a finite sum"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences.json","id":"NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences","status":"fully_proved","title":"Conway\u2013Hofstadter sequences"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences/conway-hofstadter-residue","kind":"node","lean":"MetaMathlibExt.conwayHofstadterResidue","parent":12717,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences/index.html#conway-hofstadter-residue","scope":"NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences","status":"fully_proved","summary":"For natural numbers $a$ and $m$, the one-based residue of $a$ modulo $m$ is $a \\bmod m$ if this is nonzero, and $m$ if $a \\bmod m = 0$ (here $a \\bmod 0 = a$). This is the residue convention used in Pelesko's recurrence.","title":"One-based residue"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences/is-conway-hofstadter-like-sequence","kind":"node","lean":"MetaMathlibExt.IsConwayHofstadterLikeSequence","parent":12717,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences/index.html#is-conway-hofstadter-like-sequence","scope":"NumberTheory/integer-sequences/iterated-self-referential/conway-hofstadter-sequences","status":"fully_proved","summary":"Let $k$ be a natural number. A sequence $c : \\mathbb{N} \\to \\mathbb{N}$ is a Conway\u2013Hofstadter-like sequence of order $k$ when $c(1) = c(2) = 1$ and, for every $n \\ge 3$,","title":"Conway\u2013Hofstadter-like sequence"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences.json","id":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences","status":"can_prove","title":"Ducci sequences"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/ducci-coeff","kind":"node","lean":"MathlibExt.NumberTheory.DucciWanted.ducciCoeff","parent":12720,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/index.html#ducci-coeff","scope":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences","status":"fully_proved","summary":"For natural numbers $n$, $r$ and $s$, the Ducci coefficient is $a_{r,s} = \\sum_{0 \\le j \\le r,\\ j \\equiv s - 1 \\ (\\mathrm{mod}\\ n)} \\binom{r}{j}$, where $s - 1$ is read as $0$ when $s = 0$ and congruence modulo $0$ is equality.","title":"Ducci coefficients"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/choose-prime-pow-sub-one-mod","kind":"node","lean":"MathlibExt.NumberTheory.DucciWanted.choose_prime_pow_sub_one_mod","parent":12720,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial coefficients "],["math","\\(\\binom{p^k - 1}{j}\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Binomial coefficients $\\binom{p^k - 1}{j}$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/index.html#choose-prime-pow-sub-one-mod","scope":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences","status":"can_prove","summary":"Let $p$ be a prime, let $k \\ge 1$, and let $j$ be an integer with $0 \\le j \\le p^k - 1$. Then $\\binom{p^k - 1}{j} \\equiv (-1)^j \\pmod{p}$.","title":"Binomial coefficients $\\binom{p^k - 1}{j}$ modulo $p$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/choose-prime-pow-sub-mod","kind":"node","lean":"MathlibExt.NumberTheory.DucciWanted.choose_prime_pow_sub_mod","parent":12720,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial coefficients "],["math","\\(\\binom{p^k - p^{k-1}}{j}\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Binomial coefficients $\\binom{p^k - p^{k-1}}{j}$ modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/index.html#choose-prime-pow-sub-mod","scope":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences","status":"can_prove","summary":"Let $p$ be an odd prime, let $k \\ge 2$, and let $j$ be an integer with $0 \\le j \\le p^k - p^{k-1}$. If $j = c\\,p^{k-1}$ for an integer $c$, then $\\binom{p^k - p^{k-1}}{j} \\equiv (-1)^c \\pmod{p}$; otherwise $\\binom{p^k - p^{k-1}}{j} \\equiv 0 \\pmod{p}$.","title":"Binomial coefficients $\\binom{p^k - p^{k-1}}{j}$ modulo $p$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/ducci-coeff-vanish-mod","kind":"node","lean":"MathlibExt.NumberTheory.DucciWanted.ducciCoeff_vanish_mod","parent":12720,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing of Ducci coefficients modulo "],["math","\\(p\\)"]],"source":"Vanishing of Ducci coefficients modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/index.html#ducci-coeff-vanish-mod","scope":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences","status":"can_prove","summary":"For integers $N \\ge 1$, $r \\ge 0$ and $s \\ge 1$ write $a^{(N)}_{r,s} = \\sum_{0 \\le j \\le r,\\ j \\equiv s - 1 \\ (\\mathrm{mod}\\ N)} \\binom{r}{j}$ for the Ducci coefficients of length $N$. Let $p$ be a prime, let $k \\ge 0$ and $n_1 \\ge 2$ be integers, and let $n = p^k n_1$. Let $c \\ge 1$, and let $s$ be an integer with $1 \\le s \\le n$ that is not of the form $b\\,p^k + 1$ with $0 \\le b < n_1$. Then\u2026","title":"Vanishing of Ducci coefficients modulo $p$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/ducci-coeff-reduce-mod","kind":"node","lean":"MathlibExt.NumberTheory.DucciWanted.ducciCoeff_reduce_mod","parent":12720,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reduction of Ducci coefficients modulo "],["math","\\(p\\)"]],"source":"Reduction of Ducci coefficients modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/index.html#ducci-coeff-reduce-mod","scope":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences","status":"can_prove","summary":"For integers $N \\ge 1$, $r \\ge 0$ and $s \\ge 1$ write $a^{(N)}_{r,s} = \\sum_{0 \\le j \\le r,\\ j \\equiv s - 1 \\ (\\mathrm{mod}\\ N)} \\binom{r}{j}$ for the Ducci coefficients of length $N$. Let $p$ be a prime, let $k \\ge 0$ and $n_1 \\ge 2$ be integers, and let $n = p^k n_1$. Then for every integer $c \\ge 1$ and every integer $b$ with $0 \\le b < n_1$, $a^{(n)}_{c p^k,\\, b p^k + 1} \\equiv a^{(n_1)}_{c,\\, b + 1} \\pmod{p}$.","title":"Reduction of Ducci coefficients modulo $p$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/ducci-pred-iff","kind":"node","lean":"MathlibExt.NumberTheory.DucciWanted.ducci_pred_iff","parent":12720,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences/index.html#ducci-pred-iff","scope":"NumberTheory/integer-sequences/iterated-self-referential/ducci-sequences","status":"can_prove","summary":"Let $m \\ge 0$ be an integer, let $n \\ge 2$ be even, and write $\\mathbb{Z}_m = \\mathbb{Z}/m\\mathbb{Z}$ (so $\\mathbb{Z}_0 = \\mathbb{Z}$). A tuple $x = (x_1, \\dots, x_n) \\in \\mathbb{Z}_m^n$ has a Ducci predecessor, that is, a tuple $y \\in \\mathbb{Z}_m^n$ with $y_i + y_{i+1} = x_i$ for all $i$ (indices taken cyclically, so $y_{n+1} = y_1$), if and only if $x_1 - x_2 + x_3 - \\cdots + x_{n-1} - x_n = 0$ in $\\mathbb{Z}_m$.","title":"Ducci predecessor criterion for even length"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions.json","id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","title":"Generalized Hofstadter functions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-faux","kind":"node","lean":"MetaMathlibExt.hofstadterFAux","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-faux","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For natural numbers $k$, $p$ and $n$, the value $A_k(p, n)$ is defined by recursion on the fuel $p$, which bounds the recursion depth: $A_k(0, n) = 0$ and $A_k(p + 1, n) = n - g^k(n - 1)$, where $g^k$ is the $k$-fold iterate of the function $g(m) = A_k(p, m)$ and both subtractions are truncated at $0$.","title":"Fuel-bounded generalized Hofstadter recursion"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f","kind":"node","lean":"MetaMathlibExt.hofstadterF","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, the function $F_k : \\mathbb{N} \\to \\mathbb{N}$ is given by $F_k(0) = 0$ and $F_k(n) = n - F_k^k(n - 1)$ for $n \\ge 1$, where $F_k^k$ is the $k$-fold iterate of $F_k$ (the identity when $k = 0$).","title":"Generalized Hofstadter function"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-faux-le","kind":"node","lean":"MetaMathlibExt.hofstadterFAux_le","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-faux-le","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For all natural numbers $k$, $p$ and $n$, the fuel-bounded value $A_k(p, n)$ of the generalized Hofstadter recursion satisfies $A_k(p, n) \\le n$.","title":"The fuel-bounded Hofstadter recursion is bounded by its argument"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-zero","kind":"node","lean":"MetaMathlibExt.hofstadterF_zero","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-zero","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For every natural number $k$, the generalized Hofstadter function $F_k$ satisfies $F_k(0) = 0$.","title":"The generalized Hofstadter functions vanish at zero"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-eq","kind":"node","lean":"MetaMathlibExt.hofstadterF_eq","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-eq","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For every $k \\ge 0$ and every $n \\ge 1$, the generalized Hofstadter function $F_k$ satisfies $F_k(n) = n - F_k^k(n - 1)$, where $F_k^k$ is the $k$-fold iterate of $F_k$.","title":"Recursion for the generalized Hofstadter function"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-small-values","kind":"node","lean":"MetaMathlibExt.hofstadterF_small_values","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-small-values","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $F_k$ denote the generalized Hofstadter functions. For $n = 0, 1, \\dots, 10$ the values $F_2(n)$ are $0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6$ and the values $F_3(n)$ are $0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7$.","title":"First values of Hofstadter's G and H"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-tau","kind":"node","lean":"MetaMathlibExt.hofTau","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-tau","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, the substitution $\\tau_k$ sends the letter $k$ to the word $k\\,1$ and every other letter $i \\in \\mathbb{N}$ to the one-letter word $i + 1$.","title":"The Hofstadter substitution on letters"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-tau-list","kind":"node","lean":"MetaMathlibExt.hofTauList","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-tau-list","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, $\\tau_k$ maps a finite word $w$ over $\\mathbb{N}$ to the concatenation, in order, of the words $\\tau_k(a)$ for the letters $a$ of $w$, where $\\tau_k(k) = k\\,1$ and $\\tau_k(i) = i + 1$ for every letter $i \\ne k$.","title":"The Hofstadter substitution on finite words"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/one-le-length-hof-tau","kind":"node","lean":"MetaMathlibExt.one_le_length_hofTau","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#one-le-length-hof-tau","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For every natural number $k$, let $\\tau_k$ be the substitution on letters in $\\mathbb{N}$ sending $k$ to the two-letter word $k\\,1$ and every other letter $m$ to the one-letter word $m + 1$. For every natural number $i$, the image $\\tau_k(i)$ of the letter $i$ has length at least $1$.","title":"Letter images under the generalized Hofstadter substitution are nonempty"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/length-le-length-hof-tau-list","kind":"node","lean":"MetaMathlibExt.length_le_length_hofTauList","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#length-le-length-hof-tau-list","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For every natural number $k$, let $\\tau_k$ be the substitution on letters in $\\mathbb{N}$ sending $k$ to the two-letter word $k\\,1$ and every other letter $i$ to the one-letter word $i + 1$. For every finite word $w$ over $\\mathbb{N}$, the word $\\tau_k(w)$, obtained by replacing each letter of $w$ by its image under $\\tau_k$, is at least as long as $w$: $|w| \\le |\\tau_k(w)|$.","title":"The generalized Hofstadter substitution does not shorten words"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-tau-list-is-prefix","kind":"node","lean":"MetaMathlibExt.hofTauList_isPrefix","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-tau-list-is-prefix","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, let $\\tau_k$ be the morphism of finite words over $\\mathbb{N}$ that replaces the letter $k$ by the word $k\\,1$ and every other letter $i$ by $i + 1$. For every $k$ and all finite words $u$, $v$ over $\\mathbb{N}$, if $u$ is a prefix of $v$, then $\\tau_k(u)$ is a prefix of $\\tau_k(v)$.","title":"The Hofstadter substitution preserves prefixes"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/exists-hof-tau-list-iterate-eq-cons","kind":"node","lean":"MetaMathlibExt.exists_hofTauList_iterate_eq_cons","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every iterate "],["math","\\(\\tau_k^j(k)\\)"],["text"," starts with "],["math","\\(k\\)"]],"source":"Every iterate $\\tau_k^j(k)$ starts with $k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#exists-hof-tau-list-iterate-eq-cons","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $\\tau_k$ be the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, extended letterwise to words. For all natural numbers $k$ and $j$, the word $\\tau_k^j(k)$, obtained by applying $\\tau_k$ to the one-letter word $k$ a total of $j$ times, has the form $k\\,r$ for some word $r$.","title":"Every iterate $\\tau_k^j(k)$ starts with $k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/succ-le-length-hof-tau-list-iterate","kind":"node","lean":"MetaMathlibExt.succ_le_length_hofTauList_iterate","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#succ-le-length-hof-tau-list-iterate","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For natural numbers $k$ and $j$, let $\\tau_k$ be the substitution on letters in $\\mathbb{N}$ sending $k$ to the two-letter word $k\\,1$ and every other letter $i$ to the one-letter word $i + 1$, extended letter by letter to finite words. The word $\\tau_k^j(k)$, obtained by applying $\\tau_k$ $j$ times to the one-letter word $k$, has length at least $j + 1$.","title":"Growth of the iterated generalized Hofstadter substitution"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-tau-list-iterate-is-prefix-succ","kind":"node","lean":"MetaMathlibExt.hofTauList_iterate_isPrefix_succ","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Consecutive iterates of the Hofstadter substitution on "],["math","\\(k\\)"],["text"," are nested"]],"source":"Consecutive iterates of the Hofstadter substitution on $k$ are nested"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-tau-list-iterate-is-prefix-succ","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, let $\\tau_k$ be the morphism of finite words over $\\mathbb{N}$ that replaces the letter $k$ by the word $k\\,1$ and every other letter $i$ by $i + 1$, and write $\\tau_k^j(k)$ for the word obtained by applying $\\tau_k$ $j$ times to the one-letter word $k$. For all natural numbers $k$ and $j$, $\\tau_k^j(k)$ is a prefix of $\\tau_k^{j+1}(k)$.","title":"Consecutive iterates of the Hofstadter substitution on $k$ are nested"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-tau-list-iterate-is-prefix","kind":"node","lean":"MetaMathlibExt.hofTauList_iterate_isPrefix","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The iterates of the Hofstadter substitution on "],["math","\\(k\\)"],["text"," form a chain of prefixes"]],"source":"The iterates of the Hofstadter substitution on $k$ form a chain of prefixes"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-tau-list-iterate-is-prefix","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, let $\\tau_k$ be the morphism of finite words over $\\mathbb{N}$ that replaces the letter $k$ by the word $k\\,1$ and every other letter $i$ by $i + 1$, and write $\\tau_k^j(k)$ for the word obtained by applying $\\tau_k$ $j$ times to the one-letter word $k$. For every natural number $k$ and all natural numbers $i \\le j$, $\\tau_k^i(k)$ is a prefix of $\\tau_k^j(k)$.","title":"The iterates of the Hofstadter substitution on $k$ form a chain of prefixes"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-grow-aux","kind":"node","lean":"MetaMathlibExt.hofGrowAux","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded growth of a word under "],["math","\\(\\tau_k\\)"]],"source":"Bounded growth of a word under $\\tau_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-grow-aux","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $\\tau_k$ be the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, extended letterwise to words. Given $n$, a bound $f$ and a word $w$, the result is $w$ if $f = 0$ or $w$ has more than $n$ letters, and otherwise it is the result for the bound $f - 1$ and the word $\\tau_k(w)$. Thus $\\tau_k$ is applied to $w$ until the word has more than $n$ letters, at most $f$ times.","title":"Bounded growth of a word under $\\tau_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-grow","kind":"node","lean":"MetaMathlibExt.hofGrow","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The shortest long iterate "],["math","\\(\\tau_k^j(k)\\)"]],"source":"The shortest long iterate $\\tau_k^j(k)$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-grow","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $\\tau_k$ be the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, extended letterwise to words. For natural numbers $k$ and $n$, $\\mathrm{grow}_k(n)$ is the word obtained from the one-letter word $k$ by applying $\\tau_k$ until it has more than $n$ letters, at most $n$ times. Since $\\tau_k^j(k)$ has at least $j + 1$ letters, this is the shortest iterate $\\tau_k^j(k)$ with more\u2026","title":"The shortest long iterate $\\tau_k^j(k)$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-grow-aux-iterate","kind":"node","lean":"MetaMathlibExt.hofGrowAux_iterate","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Growing from an iterate of "],["math","\\(\\tau_k\\)"],["text"," gives an iterate of "],["math","\\(\\tau_k\\)"]],"source":"Growing from an iterate of $\\tau_k$ gives an iterate of $\\tau_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-grow-aux-iterate","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $\\tau_k$ be the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, extended letterwise to words, and let $\\mathrm{grow}_{k,n}^f(w)$ be the word obtained from $w$ by applying $\\tau_k$ until it has more than $n$ letters, at most $f$ times. For all $k, n, f, j$ there is an $m$ with $\\mathrm{grow}_{k,n}^f(\\tau_k^j(k)) = \\tau_k^m(k)$ such that $\\tau_k^m(k)$ has more than $n$ letters\u2026","title":"Growing from an iterate of $\\tau_k$ gives an iterate of $\\tau_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/exists-hof-grow-eq","kind":"node","lean":"MetaMathlibExt.exists_hofGrow_eq","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grown prefix is a long iterate of "],["math","\\(\\tau_k\\)"]],"source":"The grown prefix is a long iterate of $\\tau_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#exists-hof-grow-eq","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $\\tau_k$ be the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, extended letterwise to words, and let $\\mathrm{grow}_k(n)$ be the word obtained from the one-letter word $k$ by applying $\\tau_k$ until it has more than $n$ letters, at most $n$ times. For all natural numbers $k$ and $n$ there is an $m$ with $\\mathrm{grow}_k(n) = \\tau_k^m(k)$, and $\\mathrm{grow}_k(n)$ has more\u2026","title":"The grown prefix is a long iterate of $\\tau_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-word","kind":"node","lean":"MetaMathlibExt.hofWord","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-word","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, let $\\tau_k$ be the substitution on letters in $\\mathbb{N}$ that sends $k$ to the word $k\\,1$ and every other letter $i$ to $i + 1$, applied to words letter by letter, and let $\\tau_k^j(k)$ be the word obtained by applying $\\tau_k$ $j$ times to the one-letter word $k$. Then $x_k[n]$ is the letter at position $n$, counting from $0$, of the shortest word $\\tau_k^j(k)$ with $j \\ge 0$\u2026","title":"Letters of the Hofstadter fixed-point word"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-word-eq-get-elem","kind":"node","lean":"MetaMathlibExt.hofWord_eq_getElem","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Letters of "],["math","\\(x_k\\)"],["text"," from the iterates "],["math","\\(\\tau_k^j(k)\\)"]],"source":"Letters of $x_k$ from the iterates $\\tau_k^j(k)$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-word-eq-get-elem","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, let $\\tau_k$ be the Hofstadter substitution on finite words over $\\mathbb{N}$ ($k \\mapsto k\\,1$, $i \\mapsto i + 1$ for $i \\ne k$), let $x_k = x_k[0]\\,x_k[1] \\cdots$ be its fixed point, and write $\\tau_k^j(k)$ for the word obtained by applying $\\tau_k$ $j$ times to the one-letter word $k$. For all natural numbers $k$, $n$ and $j$, if $\\tau_k^j(k)$ has more than $n$ letters, then $x_k[n]$ is\u2026","title":"Letters of $x_k$ from the iterates $\\tau_k^j(k)$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-word-eq-get-elem-self","kind":"node","lean":"MetaMathlibExt.hofWord_eq_getElem_self","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The letter "],["math","\\(x_k(n)\\)"],["text"," can be read off "],["math","\\(\\tau_k^n(k)\\)"]],"source":"The letter $x_k(n)$ can be read off $\\tau_k^n(k)$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-word-eq-get-elem-self","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, let $\\tau_k$ be the Hofstadter substitution on finite words over $\\mathbb{N}$ ($k \\mapsto k\\,1$, $i \\mapsto i + 1$ for $i \\ne k$), let $x_k = x_k[0]\\,x_k[1] \\cdots$ be its fixed point, and write $\\tau_k^n(k)$ for the word obtained by applying $\\tau_k$ $n$ times to the one-letter word $k$. For all natural numbers $k$ and $n$, $x_k[n]$ is the letter at position $n$ of $\\tau_k^n(k)$, counting\u2026","title":"The letter $x_k(n)$ can be read off $\\tau_k^n(k)$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-prefix","kind":"node","lean":"MetaMathlibExt.hofPrefix","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The prefix "],["math","\\(x_k(0:n)\\)"]],"source":"The prefix $x_k(0:n)$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-prefix","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, let $\\tau_k$ be the substitution $k \\mapsto k\\,1$, $i \\mapsto i + 1$ for $i \\ne k$, applied letter by letter to words, and let $\\tau_k^j$ be its $j$-fold iterate. Then $x_k[0:n)$ is the word formed by the first $n$ letters of the shortest of the words $\\tau_k^j(k)$, $j \\ge 0$, that has more than $n$ letters.","title":"The prefix $x_k(0:n)$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-prefix-eq-map","kind":"node","lean":"MetaMathlibExt.hofPrefix_eq_map","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The prefix "],["math","\\(x_k(0:n)\\)"],["text"," letter by letter"]],"source":"The prefix $x_k(0:n)$ letter by letter"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-prefix-eq-map","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $k$ and $n$ be natural numbers, let $x_k$ be the fixed point of the substitution $\\tau_k$ with letters $x_k[0], x_k[1], \\dots$, and let $x_k[0:n)$ be the word formed by the first $n$ letters of the shortest word $\\tau_k^j(k)$ with more than $n$ letters. Then $x_k[0:n) = x_k[0]\\,x_k[1] \\cdots x_k[n-1]$, the empty word when $n = 0$.","title":"The prefix $x_k(0:n)$ letter by letter"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-l","kind":"node","lean":"MetaMathlibExt.hofL","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The image length "],["math","\\(L_k\\)"]],"source":"The image length $L_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-l","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $L_k(n) = |\\tau_k(x_k[0:n))|$ is the length of the word obtained by applying the substitution $\\tau_k$, letter by letter, to the prefix $x_k[0:n)$ formed by the first $n$ letters of the fixed point $x_k$ of $\\tau_k$.","title":"The image length $L_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-count","kind":"node","lean":"MetaMathlibExt.hofCount","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-count","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $x_k$ be the fixed point, starting with $k$, of the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, and write $x_k[j]$ for its letter at index $j$. For a predicate $P$ on letters, $C_k^{(P)}(n)$ is the number of letters satisfying $P$ in the prefix $x_k[0] \\cdots x_k[n-1]$, that is, $C_k^{(P)}(n) = \\#\\{0 \\le j < n : P(x_k[j])\\}$.","title":"Letter counts in the Hofstadter word"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-count-eq-card","kind":"node","lean":"MetaMathlibExt.hofCount_eq_card","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-count-eq-card","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"Let $x_k$ be the fixed point, starting with $k$, of the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, and write $x_k[j]$ for its letter at index $j$. For every predicate $P$ on letters and every $n$, the number $C_k^{(P)}(n)$ of letters satisfying $P$ in the prefix $x_k[0] \\cdots x_k[n-1]$ equals the number of indices $j < n$ with $P(x_k[j])$.","title":"Letter counts of the Hofstadter word as cardinalities"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-word-small-values","kind":"node","lean":"MetaMathlibExt.hofWord_small_values","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","First values of "],["math","\\(x_2\\)"],["text",", "],["math","\\(x_3\\)"],["text"," and "],["math","\\(L_2\\)"]],"source":"First values of $x_2$, $x_3$ and $L_2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-word-small-values","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"fully_proved","summary":"For a natural number $k$, let $x_k$ be the fixed point of the Hofstadter substitution $\\tau_k$ ($k \\mapsto k\\,1$, $i \\mapsto i + 1$ for $i \\ne k$), and let $L_2(n)$ be the length of $\\tau_2(x_2[0] \\cdots x_2[n-1])$. The first eight letters of the Fibonacci word $x_2$ are $2, 1, 2, 2, 1, 2, 1, 2$; the first eight letters of $x_3$ are $3, 1, 2, 3, 3, 1, 3, 1$; and $L_2(0), \\dots, L_2(5)$ are $0, 2, 3, 5, 7, 8$.","title":"First values of $x_2$, $x_3$ and $L_2$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-le-succ","kind":"node","lean":"MetaMathlibExt.hofstadterF_le_succ","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-le-succ","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"For every $k \\ge 1$ and every natural number $n$, the generalized Hofstadter functions satisfy $F_k(n) \\le F_{k+1}(n)$.","title":"Pointwise order of the generalized Hofstadter functions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-iterate-basic","kind":"node","lean":"MetaMathlibExt.hofstadterF_iterate_basic","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-iterate-basic","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $j, k \\ge 1$, let $F_k$ be the generalized Hofstadter function, and write $F_k^i$ for its $i$-fold iterate. Then $F_k^j(1) = F_k^j(2) = 1$; $F_k^j(n) \\ge 1$ for every $n \\ge 1$; $F_k^j(n) < n$ for every $n \\ge 2$; $F_k(n+1) - F_k(n) = 1 - \\bigl(F_k^k(n) - F_k^k(n-1)\\bigr)$ in $\\mathbb{Z}$ for every $n \\ge 1$; $F_k^j(n+1)$ equals $F_k^j(n)$ or $F_k^j(n) + 1$ for every $n \\ge 0$; and $F_k^j$ is monotone and\u2026","title":"Basic properties of the iterated Hofstadter functions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-tau-list-iterate-self","kind":"node","lean":"MetaMathlibExt.hofTauList_iterate_self","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterates of the Hofstadter substitution on the letter "],["math","\\(k\\)"]],"source":"Iterates of the Hofstadter substitution on the letter $k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-tau-list-iterate-self","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$, let $\\tau_k$ be the morphism of finite words over $\\mathbb{N}$ that replaces the letter $k$ by the word $k\\,1$ and every other letter $i$ by $i + 1$, and write $\\tau_k^j(k)$ for the word obtained by applying $\\tau_k$ $j$ times to the one-letter word $k$. For every natural number $j$: if $j \\le k$, then $\\tau_k^j(k) = k\\,1\\,2 \\cdots j$, the letter $k$ followed by $1, 2, \\dots, j$; and if $j \\ge k$,\u2026","title":"Iterates of the Hofstadter substitution on the letter $k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-l-iterate-basic","kind":"node","lean":"MetaMathlibExt.hofL_iterate_basic","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Basic properties of the iterates of "],["math","\\(L_k\\)"]],"source":"Basic properties of the iterates of $L_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-l-iterate-basic","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$, let $x_k$ be the fixed point of the substitution $\\tau_k$, with letters $x_k[0], x_k[1], \\dots$ and prefixes $x_k[0:n) = x_k[0] \\cdots x_k[n-1]$, let $L_k(n) = |\\tau_k(x_k[0:n))|$, and write $f^j$ for the $j$-fold iterate of a map $f$ (so $f^0$ is the identity). Then for every $j \\ge 0$: (1) $L_k^j(n) = |\\tau_k^j(x_k[0:n))|$ for every $n$; (2) $L_k^j(0) = 0$, and\u2026","title":"Basic properties of the iterates of $L_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/exists-unique-hof-l-iterate-bracket","kind":"node","lean":"MetaMathlibExt.existsUnique_hofL_iterate_bracket","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bracketing by the iterates of "],["math","\\(L_k\\)"]],"source":"Bracketing by the iterates of $L_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#exists-unique-hof-l-iterate-bracket","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"For $k \\ge 1$, let $\\tau_k$ be the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $x_k$ be its fixed point starting with $k$, and let $L_k(n)$ be the length of the image under $\\tau_k$ of the first $n$ letters of $x_k$; write $L_k^j$ for the $j$-fold iterate of $L_k$. For all $j \\ge 0$ and $m \\ge 1$ there is a unique $n \\ge 1$ with $L_k^j(n - 1) < m \\le L_k^j(n)$.","title":"Bracketing by the iterates of $L_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-iterate-preimage-eq-ioc","kind":"node","lean":"MetaMathlibExt.hofstadterF_iterate_preimage_eq_Ioc","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-iterate-preimage-eq-ioc","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$ and $j \\ge 0$. Let $F_k$ be the generalized Hofstadter function, let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $L_k(m)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[m-1])$, and write $f^j$ for the $j$-fold iterate of a function $f$. Then for every $n \\ge 1$, the set of $m$ with\u2026","title":"Fibres of the iterated Hofstadter functions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-hof-l-galois-insertion","kind":"node","lean":"MetaMathlibExt.hofstadterF_hofL_galoisInsertion","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-hof-l-galois-insertion","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$ and $j \\ge 0$. Let $F_k$ be the generalized Hofstadter function, let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $L_k(m)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[m-1])$, and write $f^j$ for the $j$-fold iterate of a function $f$. Then for all $n, m \\ge 0$, $F_k^j(n) \\le m$\u2026","title":"Galois insertion between iterated Hofstadter functions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-iterate-eq-hof-count","kind":"node","lean":"MetaMathlibExt.hofstadterF_iterate_eq_hofCount","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-iterate-eq-hof-count","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 0$. Let $F_k$ be the generalized Hofstadter function and $F_k^j$ its $j$-fold iterate, let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, and for a property $P$ of letters let $C_k^{(P)}(m)$ be the number of indices $0 \\le i < m$ with $P(x_k[i])$. Then\u2026","title":"Iterated Hofstadter functions as letter counts"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-l-eq-add-hof-count","kind":"node","lean":"MetaMathlibExt.hofL_eq_add_hofCount","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The image length "],["math","\\(L_k\\)"],["text"," as a letter count"]],"source":"The image length $L_k$ as a letter count"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-l-eq-add-hof-count","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$. Let $F_k$ be the generalized Hofstadter function and $F_k^{k-1}$ its $(k-1)$-fold iterate, let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $L_k(n)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[n-1])$, and let $C_k^{(=k)}(n)$ be the number of indices $0 \\le i < n$ with\u2026","title":"The image length $L_k$ as a letter count"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-eq-hof-count-ne-one","kind":"node","lean":"MetaMathlibExt.hofstadterF_eq_hofCount_ne_one","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-eq-hof-count-ne-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k > 1$ and $n \\ge 0$. Let $F_k$ be the generalized Hofstadter function, defined by $F_k(0) = 0$ and $F_k(m) = m - F_k^k(m - 1)$ for $m \\ge 1$, where $F_k^k$ is the $k$-fold iterate of $F_k$. Let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, and for a property $P$ of letters let $C_k^{(P)}(n)$ be the\u2026","title":"Generalized Hofstadter function as a letter count"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-word-eq-iff-hofstadter-f-iterate-dif","kind":"node","lean":"MetaMathlibExt.hofWord_eq_iff_hofstadterF_iterate_diff","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The letters of "],["math","\\(x_k\\)"],["text"," from the increments of the iterates of "],["math","\\(F_k\\)"]],"source":"The letters of $x_k$ from the increments of the iterates of $F_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-word-eq-iff-hofstadter-f-iterate-dif","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$, let $F_k$ be the generalized Hofstadter function with $j$-th iterate $F_k^j$ ($F_k^0$ the identity), and let $x_k = x_k[0]\\,x_k[1] \\cdots$ be the fixed point of the substitution $\\tau_k$ ($k \\mapsto k\\,1$, $i \\mapsto i + 1$ for $i \\ne k$). For every natural number $n$: for each $j$ with $1 \\le j < k$, $x_k[n] = j$ if and only if $F_k^{j-1}(n+1) = F_k^{j-1}(n) + 1$ and $F_k^j(n+1) = F_k^j(n)$; and\u2026","title":"The letters of $x_k$ from the increments of the iterates of $F_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-roots-monotone-asymptotics","kind":"node","lean":"MetaMathlibExt.hofstadter_roots_monotone_asymptotics","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-roots-monotone-asymptotics","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"For each integer $k \\ge 1$ let $\\alpha_k$ be a positive zero of $X^k + X - 1$ and $\\beta_k$ a positive zero of $X^k - X^{k-1} - 1$. Then for every $k \\ge 1$ we have $\\alpha_k < \\alpha_{k+1}$, $\\beta_{k+1} < \\beta_k$, $1 + 1/k \\le \\beta_k \\le 1 + 1/\\sqrt{k}$ and $\\sqrt{k} \\le \\beta_k^{k-1} \\le k$. Moreover $\\beta_k \\to 1$ and $\\alpha_k \\to 1$ as $k \\to \\infty$, and $\\alpha_k = 1 - \\ln k / k + o(\\ln k / k)$ and\u2026","title":"Monotonicity and asymptotics of the Hofstadter constants"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-limits","kind":"node","lean":"MetaMathlibExt.hofstadter_limits","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-limits","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$ and $j \\ge 0$, let $\\alpha > 0$ satisfy $\\alpha^k + \\alpha - 1 = 0$ and let $\\beta > 0$ satisfy $\\beta^k - \\beta^{k-1} - 1 = 0$. Write $f^j$ for the $j$-fold iterate of a function $f$ and let $F_k$ be the generalized Hofstadter function. Let $\\tau_k$ be the substitution on letters with $\\tau_k(k) = k\\,1$ and $\\tau_k(i) = i + 1$ for $1 \\le i < k$, let $x_k$ be the infinite word fixed by $\\tau_k$ that\u2026","title":"Limits of Hofstadter iterates and letter frequencies"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-iterate-eventually-lt-of-ro","kind":"node","lean":"MetaMathlibExt.hofstadterF_iterate_eventually_lt_of_root_pow_lt","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-iterate-eventually-lt-of-ro","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k, k' \\ge 1$ and $j, j' \\ge 0$, let $F_k$ be the generalized Hofstadter function, write $F_k^i$ for its $i$-fold iterate, let $\\alpha > 0$ be a root of $X^k + X - 1$, and let $\\alpha' > 0$ be a root of $X^{k'} + X - 1$. If $\\alpha^j < \\alpha'^{j'}$, then $F_k^j(n) < F_{k'}^{j'}(n)$ for all sufficiently large $n$.","title":"Eventual order of iterated Hofstadter functions from their growth rates"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-iterate-eventual-order-succ","kind":"node","lean":"MetaMathlibExt.hofstadterF_iterate_eventual_order_succ","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-iterate-eventual-order-succ","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$, let $F_k$ be the generalized Hofstadter function, write $F_k^i$ for its $i$-fold iterate, let $\\alpha > 0$ be a root of $X^k + X - 1$, and let $\\alpha' > 0$ be a root of $X^{k+1} + X - 1$. Then for all sufficiently large $n$, $F_k(n) < F_{k+1}(n)$, $F_{k+1}^{k+1}(n) < F_k^k(n)$ and $F_{k+1}^{k+2}(n) < F_k^{k+1}(n)$. Moreover, for every $j \\ge 0$: if $j/(j+1) \\le \\alpha$ or $j^2 \\le k$, then\u2026","title":"Eventual order of iterates of consecutive Hofstadter functions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-single-antecedent-count","kind":"node","lean":"MetaMathlibExt.hofstadter_single_antecedent_count","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-single-antecedent-count","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $k \\ge 1$, let $\\alpha > 0$ satisfy $\\alpha^k + \\alpha - 1 = 0$ and let $\\beta > 0$ satisfy $\\beta^k - \\beta^{k-1} - 1 = 0$. Write $f^j$ for the $j$-fold iterate of a function $f$ and let $F_k$ be the generalized Hofstadter function. Let $\\tau_k$ be the substitution on letters with $\\tau_k(k) = k\\,1$ and $\\tau_k(i) = i + 1$ for $1 \\le i < k$, let $x_k$ be the infinite word fixed by $\\tau_k$ that is the limit of\u2026","title":"Values with exactly one preimage under a Hofstadter function"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-l-iterate-le-iff-hofstadter-f-iterat","kind":"node","lean":"MetaMathlibExt.hofL_iterate_le_iff_hofstadterF_iterate_le","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Order of iterates of "],["math","\\(L_k\\)"],["text"," against order of iterates of "],["math","\\(F_k\\)"]],"source":"Order of iterates of $L_k$ against order of iterates of $F_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-l-iterate-le-iff-hofstadter-f-iterat","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $F_k$ be the generalized Hofstadter function, let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $L_k(n)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[n-1])$, and write $f^j$ for the $j$-fold iterate of a function $f$. Let $k, k' \\ge 1$ and $j, j' \\ge 0$. Then $L_k^j(n) \\le L_{k'}^{j'}(n)$\u2026","title":"Order of iterates of $L_k$ against order of iterates of $F_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-l-succ-comparison","kind":"node","lean":"MetaMathlibExt.hofL_succ_comparison","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Comparing "],["math","\\(L_k\\)"],["text"," with "],["math","\\(L_{k+1}\\)"]],"source":"Comparing $L_k$ with $L_{k+1}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-l-succ-comparison","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $L_k(n)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[n-1])$, write $L_k^j$ for the $j$-fold iterate of $L_k$, and let $C_k^{(=k)}(n)$ be the number of indices $i < n$ with $x_k[i] = k$. For all $k \\ge 1$, $n \\ge 1$ and $j$ with $0 \\le j \\le k$:\u2026","title":"Comparing $L_k$ with $L_{k+1}$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-l-iterate-succ-le","kind":"node","lean":"MetaMathlibExt.hofL_iterate_succ_le","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterates of "],["math","\\(L_{k+1}\\)"],["text"," lie below iterates of "],["math","\\(L_k\\)"]],"source":"Iterates of $L_{k+1}$ lie below iterates of $L_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-l-iterate-succ-le","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $L_k(n)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[n-1])$, and write $L_k^j$ for the $j$-fold iterate of $L_k$. For every $k \\ge 1$, every $j \\ge 0$ and every natural number $n$, $L_{k+1}^j(n) \\le L_k^j(n)$.","title":"Iterates of $L_{k+1}$ lie below iterates of $L_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-iterate-le-succ","kind":"node","lean":"MetaMathlibExt.hofstadterF_iterate_le_succ","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-iterate-le-succ","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"For every $k \\ge 1$, $j \\ge 0$ and $n \\ge 0$, $F_k^j(n) \\le F_{k+1}^j(n)$, where $F_k$ is the generalized Hofstadter function and $F_k^j$ is its $j$-fold iterate.","title":"Iterates of consecutive Hofstadter functions are ordered"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hofstadter-f-iterate-succ-le-iterate","kind":"node","lean":"MetaMathlibExt.hofstadterF_iterate_succ_le_iterate","parent":12727,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hofstadter-f-iterate-succ-le-iterate","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $F_k$ denote the generalized Hofstadter functions and write $f^j$ for the $j$-fold iterate of a function $f$. For every $k \\ge 1$, every $j$ with $0 \\le j \\le k$ and every natural number $n$, $F_{k+1}^{j+1}(n) \\le F_k^j(n)$.","title":"Shifted order of iterates of consecutive Hofstadter functions"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-l-iterate-one-succ-sub","kind":"node","lean":"MetaMathlibExt.hofL_iterate_one_succ_sub","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Comparing "],["math","\\(L_{k+1}^{j+1}(1)\\)"],["text"," with "],["math","\\(L_k^j(1)\\)"]],"source":"Comparing $L_{k+1}^{j+1}(1)$ with $L_k^j(1)$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-l-iterate-one-succ-sub","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $L_k(n)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[n-1])$, and write $L_k^j$ for the $j$-fold iterate of $L_k$. Let $k \\ge 1$, and take differences in $\\mathbb{Z}$. Then $L_{k+1}^{j+1}(1) - L_k^j(1) = 1$ for every $j \\le 2k$;\u2026","title":"Comparing $L_{k+1}^{j+1}(1)$ with $L_k^j(1)$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-l-iterate-witness-gt","kind":"node","lean":"MetaMathlibExt.hofL_iterate_witness_gt","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Points where "],["math","\\(L_k^j\\)"],["text"," exceeds "],["math","\\(L_{k+1}^{j+1}\\)"]],"source":"Points where $L_k^j$ exceeds $L_{k+1}^{j+1}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-l-iterate-witness-gt","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $x_k[0]\\,x_k[1] \\cdots$ be the fixed point, beginning with $k$, of the substitution $\\tau_k$ that sends $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, let $L_k(n)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[n-1])$, and write $L_k^j$ for the $j$-fold iterate of $L_k$. Let $k \\ge 1$. Then $L_{k+1}^{j+1}(2k + 3 - j) < L_k^j(2k + 3 - j)$ for every $j$ with $k + 2 \\le j \\le 2k + 2$, and\u2026","title":"Points where $L_k^j$ exceeds $L_{k+1}^{j+1}$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-count-one-succ-le","kind":"node","lean":"MetaMathlibExt.hofCount_one_succ_le","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prefixes of "],["math","\\(x_{k+1}\\)"],["text"," contain no more letters "],["math","\\(1\\)"],["text"," than those of "],["math","\\(x_k\\)"]],"source":"Prefixes of $x_{k+1}$ contain no more letters $1$ than those of $x_k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-count-one-succ-le","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"Let $x_k$ be the fixed point, starting with $k$, of the substitution sending the letter $k$ to $k\\,1$ and every other letter $i$ to $i + 1$, and let $C_k^{(=1)}(n)$ be the number of indices $j < n$ with $x_k[j] = 1$. For $k \\ge 1$ and every $n \\ge 0$, $C_{k+1}^{(=1)}(n) \\le C_k^{(=1)}(n)$.","title":"Prefixes of $x_{k+1}$ contain no more letters $1$ than those of $x_k$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/hof-count-eq-witnesses","kind":"node","lean":"MetaMathlibExt.hofCount_eq_witnesses","parent":12727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Witnesses that the letter counts of "],["math","\\(x_k\\)"],["text"," and "],["math","\\(x_{k+1}\\)"],["text"," are not ordered"]],"source":"Witnesses that the letter counts of $x_k$ and $x_{k+1}$ are not ordered"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions/index.html#hof-count-eq-witnesses","scope":"NumberTheory/integer-sequences/iterated-self-referential/generalized-hofstadter-functions","status":"can_prove","summary":"For a natural number $k$, let $\\tau_k$ be the substitution sending the letter $k$ to $k\\,1$ and every other letter $a$ to $a + 1$, let $x_k = x_k[0]\\,x_k[1] \\cdots$ be its fixed point beginning with $k$, let $C_k^{(=i)}(n)$ be the number of indices $j < n$ with $x_k[j] = i$, let $L_k(n)$ be the length of the word $\\tau_k(x_k[0] \\cdots x_k[n-1])$, and write $L_k^j$ for the $j$-fold iterate of $L_k$. For all natural\u2026","title":"Witnesses that the letter counts of $x_k$ and $x_{k+1}$ are not ordered"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes.json","id":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes","status":"fully_proved","title":"Integer polynomial maps modulo primes"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/integer-polynomial-map","kind":"node","lean":"MetaMathlibExt.IntegerPolynomialMap","parent":12781,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/index.html#integer-polynomial-map","scope":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes","status":"fully_proved","summary":"For natural numbers $n$ and $m$, an integer polynomial map from $n$ variables to $m$ components is an $m$-tuple $F = (F_0, \\dots, F_{m-1})$ of polynomials with integer coefficients in the $n$ variables $x_0, \\dots, x_{n-1}$.","title":"Integer polynomial map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/to-fun","kind":"node","lean":"MetaMathlibExt.IntegerPolynomialMap.toFun","parent":12781,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/index.html#to-fun","scope":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes","status":"fully_proved","summary":"Let $F = (F_0, \\dots, F_{m-1})$ be an integer polynomial map from $n$ variables to $m$ components. The value of $F$ at a point $x \\in \\mathbb{Z}^n$ is the point $(F_0(x), \\dots, F_{m-1}(x)) \\in \\mathbb{Z}^m$.","title":"Evaluation of an integer polynomial map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/to-fun-reduce","kind":"node","lean":"MetaMathlibExt.IntegerPolynomialMap.toFunReduce","parent":12781,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/index.html#to-fun-reduce","scope":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes","status":"fully_proved","summary":"Let $F = (F_0, \\dots, F_{m-1})$ be an integer polynomial map from $n$ variables to $m$ components and let $p$ be a natural number. For $x \\in (\\mathbb{Z}/p\\mathbb{Z})^n$, the value of the reduction of $F$ modulo $p$ at $x$ is $(\\bar F_0(x), \\dots, \\bar F_{m-1}(x)) \\in (\\mathbb{Z}/p\\mathbb{Z})^m$, where $\\bar F_i$ is the polynomial obtained from $F_i$ by reducing its coefficients modulo $p$.","title":"Reduction of an integer polynomial map modulo a natural number"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/iterate","kind":"node","lean":"MetaMathlibExt.IntegerPolynomialMap.iterate","parent":12781,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/index.html#iterate","scope":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes","status":"fully_proved","summary":"Let $F$ be an integer polynomial map from $n$ variables to $n$ components, viewed as the map $\\mathbb{Z}^n \\to \\mathbb{Z}^n$ given by evaluation. For a natural number $k$ and $x \\in \\mathbb{Z}^n$, the $k$-th iterate of $F$ at $x$ is $F^k(x)$, the result of applying $F$ to $x$ exactly $k$ times; in particular $F^0(x) = x$.","title":"Iterates of an integer polynomial map"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/iterate-reduce","kind":"node","lean":"MetaMathlibExt.IntegerPolynomialMap.iterateReduce","parent":12781,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes/index.html#iterate-reduce","scope":"NumberTheory/integer-sequences/iterated-self-referential/integer-polynomial-maps-modulo-primes","status":"fully_proved","summary":"Let $F$ be an integer polynomial map from $n$ variables to $n$ components, let $p$ be a natural number, and let $\\bar F \\colon (\\mathbb{Z}/p\\mathbb{Z})^n \\to (\\mathbb{Z}/p\\mathbb{Z})^n$ be the map obtained by reducing the coefficients of $F$ modulo $p$ and evaluating. For a natural number $k$ and $x \\in (\\mathbb{Z}/p\\mathbb{Z})^n$, the $k$-th iterate of $F$ modulo $p$ at $x$ is $\\bar F^k(x)$, the result of applying\u2026","title":"Iterates of an integer polynomial map modulo a natural number"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem.json","id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","title":"Josephus problem"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus","kind":"node","lean":"MetaMathlibExt.josephus","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the Josephus function $J_k(n)$ is defined by recursion on $n$: $J_k(0) = 0$ and $J_k(n+1) = ((J_k(n) + k - 1) \\bmod (n+1)) + 1$, where subtraction of natural numbers is truncated at $0$.","title":"Josephus survivor function"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-one","kind":"node","lean":"MetaMathlibExt.josephus_one","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"For every natural number $k \\ge 2$, the Josephus function $J_k$ for elimination step $k$ satisfies $J_k(1) = 1$.","title":"The Josephus function at one"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-succ","kind":"node","lean":"MetaMathlibExt.josephus_succ","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-succ","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"For all natural numbers $k \\ge 2$ and $n \\ge 1$, the Josephus function $J_k$ for elimination step $k$ satisfies $J_k(n+1) = ((J_k(n) + k - 1) \\bmod (n+1)) + 1$.","title":"Euler recurrence for the Josephus function"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-range","kind":"node","lean":"MetaMathlibExt.josephus_range","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-range","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"For all natural numbers $k \\ge 2$ and $n \\ge 1$, the Josephus function $J_k$ for elimination step $k$ satisfies $1 \\le J_k(n) \\le n$.","title":"Range of the Josephus function"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point","kind":"node","lean":"MetaMathlibExt.josephusThreeFixedPoint","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Enumeration of the positive fixed points of "],["math","\\(J_3\\)"]],"source":"Enumeration of the positive fixed points of $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"Let $J_3$ be the Josephus function with step $3$. For $\\ell \\in \\mathbb{N}$, $n_\\ell$ is the $\\ell$-th element, in increasing order and counting from $\\ell = 0$, of the set of natural numbers $n \\ge 1$ with $J_3(n) = n$. Thus $n_\\ell$ is the paper's $n_p^{(\\ell+1)}$.","title":"Enumeration of the positive fixed points of $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-gap","kind":"node","lean":"MetaMathlibExt.josephusThreeGap","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gaps between consecutive positive fixed points of "],["math","\\(J_3\\)"]],"source":"Gaps between consecutive positive fixed points of $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-gap","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"Let $J_3$ be the Josephus function with step $3$ and let $n_\\ell$ be the $\\ell$-th natural number $n \\ge 1$ with $J_3(n) = n$, in increasing order and counting from $\\ell = 0$. For $\\ell \\in \\mathbb{N}$, the gap $g_\\ell$ is the number of natural numbers $n$ with $n_\\ell < n < n_{\\ell+1}$ and $J_3(n) + 1 = n$, that is, the number of pure high extremal points strictly between $n_\\ell$ and $n_{\\ell+1}$. Thus $g_\\ell$\u2026","title":"Gaps between consecutive positive fixed points of $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/base-three-halves-digits","kind":"node","lean":"MetaMathlibExt.baseThreeHalvesDigits","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#base-three-halves-digits","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"The base-$3/2$ digit list of a natural number $N$, least significant digit first, is defined by recursion on $N$: the list of $0$ is empty, and for $N \\ge 1$ it is the digit $d = 2N \\bmod 3$ followed by the base-$3/2$ digit list of $(2N - d)/3$.","title":"Base-3/2 digits"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/base-three-halves-digits-four","kind":"node","lean":"MetaMathlibExt.baseThreeHalvesDigits_four","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#base-three-halves-digits-four","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"The base-$3/2$ digit list of $4$, written least significant digit first, is $(2, 1, 2)$; written most significant digit first, this is the paper's $(212)_{3/2}$.","title":"Base-3/2 digits of four"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/base-three-halves-digits-thirteen","kind":"node","lean":"MetaMathlibExt.baseThreeHalvesDigits_thirteen","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#base-three-halves-digits-thirteen","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"The base-$3/2$ digit list of $13$, written least significant digit first, is $(2, 1, 1, 0, 1, 2)$; written most significant digit first, this is the paper's $(210112)_{3/2}$.","title":"Base-3/2 digits of thirteen"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/base-three-halves-digits-value","kind":"node","lean":"MetaMathlibExt.baseThreeHalvesDigits_value","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#base-three-halves-digits-value","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"For every natural number $N$, if $(d_0, d_1, \\dots, d_{k-1})$ is the base-$3/2$ digit list of $N$, least significant digit first, then in $\\mathbb{Q}$","title":"Value of the base-3/2 digits"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/base-three-halves-digits-lt-three","kind":"node","lean":"MetaMathlibExt.baseThreeHalvesDigits_lt_three","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#base-three-halves-digits-lt-three","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"For every natural number $N$, every digit $d$ in the base-$3/2$ digit list of $N$ satisfies $d < 3$, that is, $d \\in \\{0, 1, 2\\}$.","title":"Base-3/2 digits are at most two"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point-zero","kind":"node","lean":"MetaMathlibExt.josephusThreeFixedPoint_zero","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first positive fixed point of "],["math","\\(J_3\\)"],["text"," is "],["math","\\(1\\)"]],"source":"The first positive fixed point of $J_3$ is $1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point-zero","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"Let $J_3$ be the Josephus function with step $3$ and let $n_\\ell$ be the $\\ell$-th natural number $n \\ge 1$ with $J_3(n) = n$, in increasing order and counting from $\\ell = 0$. Then $n_0 = 1$; this is the paper's $n_p^{(1)}$.","title":"The first positive fixed point of $J_3$ is $1$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point-one","kind":"node","lean":"MetaMathlibExt.josephusThreeFixedPoint_one","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"Let $J_3$ be the Josephus function with elimination step $3$. The second smallest positive integer $n$ with $J_3(n) = n$ is $2$; in the zero-indexed enumeration $n_0 < n_1 < \\cdots$ of these fixed points, $n_1 = 2$, the paper's $n_p^{(2)}$.","title":"Second positive fixed point of the Josephus function for step three"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point-two","kind":"node","lean":"MetaMathlibExt.josephusThreeFixedPoint_two","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The third positive fixed point of "],["math","\\(J_3\\)"],["text"," is "],["math","\\(13\\)"]],"source":"The third positive fixed point of $J_3$ is $13$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point-two","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"Let $J_3$ be the Josephus function with step $3$ and let $n_\\ell$ be the $\\ell$-th natural number $n \\ge 1$ with $J_3(n) = n$, in increasing order and counting from $\\ell = 0$. Then $n_2 = 13$; this is the paper's $n_p^{(3)}$.","title":"The third positive fixed point of $J_3$ is $13$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-gap-one","kind":"node","lean":"MetaMathlibExt.josephusThreeGap_one","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The second gap between positive fixed points of "],["math","\\(J_3\\)"],["text"," is "],["math","\\(3\\)"]],"source":"The second gap between positive fixed points of $J_3$ is $3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-gap-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"Let $J_3$ be the Josephus function with step $3$, let $n_\\ell$ be the $\\ell$-th natural number $n \\ge 1$ with $J_3(n) = n$ (in increasing order, counting from $\\ell = 0$), and let $g_\\ell$ be the number of natural numbers $n$ with $n_\\ell < n < n_{\\ell+1}$ and $J_3(n) + 1 = n$. Then $g_1 = 3$; this is the paper's $\\bar m_2 = 3$.","title":"The second gap between positive fixed points of $J_3$ is $3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point-digits-of-gap","kind":"node","lean":"MetaMathlibExt.josephusThreeFixedPoint_digits_of_gap_zero","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point-digits-of-gap","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $J_3$ be the Josephus function with elimination step $3$, let $n_\\ell$ be the $\\ell$-th positive fixed point of $J_3$ in increasing order (counting from $\\ell = 0$), and let $m_\\ell$ be the number of integers $n$ with $n_\\ell < n < n_{\\ell+1}$ and $J_3(n) + 1 = n$ (the pure high extremal points between them). For every $\\ell$, if $m_\\ell = 0$, then the base-$3/2$ digit list of $n_{\\ell+1}$, least significant\u2026","title":"Base-3/2 digits of the next Josephus fixed point, gap zero"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point-digits-of-gap-2","kind":"node","lean":"MetaMathlibExt.josephusThreeFixedPoint_digits_of_gap_one","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point-digits-of-gap-2","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $J_3$ be the Josephus function with elimination step $3$, let $n_\\ell$ be the $\\ell$-th positive fixed point of $J_3$ in increasing order (counting from $\\ell = 0$), and let $m_\\ell$ be the number of integers $n$ with $n_\\ell < n < n_{\\ell+1}$ and $J_3(n) + 1 = n$ (the pure high extremal points between them). For every $\\ell$, if $m_\\ell = 1$, then the base-$3/2$ digit list of $n_{\\ell+1}$, least significant\u2026","title":"Base-3/2 digits of the next Josephus fixed point, gap one"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point-digits-of-gap-3","kind":"node","lean":"MetaMathlibExt.josephusThreeFixedPoint_digits_of_gap_at_least_two","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point-digits-of-gap-3","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $J_3$ be the Josephus function with elimination step $3$, let $n_\\ell$ be the $\\ell$-th positive fixed point of $J_3$ in increasing order (counting from $\\ell = 0$), and let $m_\\ell$ be the number of integers $n$ with $n_\\ell < n < n_{\\ell+1}$ and $J_3(n) + 1 = n$ (the pure high extremal points between them). For every $\\ell$, if $m_\\ell \\ge 2$, then the base-$3/2$ digit list of $n_{\\ell+1}$, least significant\u2026","title":"Base-3/2 digits of the next Josephus fixed point, gap at least two"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point-digits-length","kind":"node","lean":"MetaMathlibExt.josephusThreeFixedPoint_digits_length","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point-digits-length","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $J_3$ be the Josephus function with elimination step $3$, let $n_\\ell$ be the $\\ell$-th positive fixed point of $J_3$ in increasing order (counting from $\\ell = 0$), and let $m_\\ell$ be the number of integers $n$ with $n_\\ell < n < n_{\\ell+1}$ and $J_3(n) + 1 = n$ (the pure high extremal points between them). For every $\\ell$, the base-$3/2$ digit list of $n_{\\ell+1}$ has exactly $m_\\ell + 1$ more digits than\u2026","title":"Digit count of consecutive Josephus fixed points"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/is-josephus-three-high-extremal","kind":"node","lean":"MetaMathlibExt.IsJosephusThreeHighExtremal","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","High extremal points of the Josephus function "],["math","\\(J_3\\)"]],"source":"High extremal points of the Josephus function $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#is-josephus-three-high-extremal","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"A natural number $n$ is a high extremal point of the Josephus function $J_3$ (elimination step $3$) when $n \\ge 1$ and either $J_3(n) = n$ or $J_3(n) = n - 1$.","title":"High extremal points of the Josephus function $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/is-josephus-three-fixed-point","kind":"node","lean":"MetaMathlibExt.IsJosephusThreeFixedPoint","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed points of the Josephus function "],["math","\\(J_3\\)"]],"source":"Fixed points of the Josephus function $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#is-josephus-three-fixed-point","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"A natural number $n$ is a fixed point of the Josephus function $J_3$ (elimination step $3$) when $n \\ge 1$ and $J_3(n) = n$.","title":"Fixed points of the Josephus function $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/is-josephus-three-high-extremal-three","kind":"node","lean":"MetaMathlibExt.isJosephusThreeHighExtremal_three","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(3\\)"],["text"," is a high extremal point but not a fixed point of "],["math","\\(J_3\\)"]],"source":"$3$ is a high extremal point but not a fixed point of $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#is-josephus-three-high-extremal-three","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"The number $3$ is a high extremal point of the Josephus function $J_3$ (elimination step $3$) and is not a fixed point of $J_3$; indeed $J_3(3) = 2$.","title":"$3$ is a high extremal point but not a fixed point of $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/not-is-josephus-three-high-extremal-four","kind":"node","lean":"MetaMathlibExt.not_isJosephusThreeHighExtremal_four","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(4\\)"],["text"," is not a high extremal point of "],["math","\\(J_3\\)"]],"source":"$4$ is not a high extremal point of $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#not-is-josephus-three-high-extremal-four","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"The number $4$ is not a high extremal point of the Josephus function $J_3$ (elimination step $3$).","title":"$4$ is not a high extremal point of $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/is-josephus-three-fixed-point-thirteen","kind":"node","lean":"MetaMathlibExt.isJosephusThreeFixedPoint_thirteen","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(13\\)"],["text"," is a fixed point of "],["math","\\(J_3\\)"]],"source":"$13$ is a fixed point of $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#is-josephus-three-fixed-point-thirteen","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"The number $13$ is a fixed point of the Josephus function $J_3$ (elimination step $3$), that is, $J_3(13) = 13$.","title":"$13$ is a fixed point of $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-high-extremal-succ-cases","kind":"node","lean":"MetaMathlibExt.josephusThree_highExtremal_succ_cases","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Successive high extremal points of "],["math","\\(J_3\\)"]],"source":"Successive high extremal points of $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-high-extremal-succ-cases","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"For every natural number $i$, let $a$ and $b$ be the $(i+1)$-th and $(i+2)$-th smallest high extremal points of the Josephus function $J_3$, and put $f = a - J_3(a)$, $f' = b - J_3(b)$ and $r = a \\bmod 2$. If $(f, r) = (1, 0)$, then $2b = 3a + 2$ and $f' = 0$; if $(f, r) = (1, 1)$, then $2b = 3a + 1$ and $f' = 1$; if $(f, r) = (0, 0)$, then $2b = 3a$ and $f' = 1$; if $(f, r) = (0, 1)$, then $2b = 3a + 1$ and\u2026","title":"Successive high extremal points of $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-high-extremal-recurrences","kind":"node","lean":"MetaMathlibExt.josephusThree_highExtremal_recurrences","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unified recurrences for consecutive high extremal points of "],["math","\\(J_3\\)"]],"source":"Unified recurrences for consecutive high extremal points of $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-high-extremal-recurrences","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"For every natural number $i$, let $a$ and $b$ be the $(i+1)$-th and $(i+2)$-th smallest high extremal points of the Josephus function $J_3$. In the integers, put $r = a \\bmod 2$, $f = a - J_3(a)$ and $s = \\lfloor (3a + 2 - r)/2 \\rfloor \\bmod 2$. Then","title":"Unified recurrences for consecutive high extremal points of $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-fixed-point-recurrence","kind":"node","lean":"MetaMathlibExt.josephusThree_fixedPoint_recurrence","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for consecutive fixed points of "],["math","\\(J_3\\)"]],"source":"Recurrence for consecutive fixed points of $J_3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-fixed-point-recurrence","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"For every natural number $\\ell$, let $a$ and $b$ be the $(\\ell+1)$-th and $(\\ell+2)$-th smallest fixed points of the Josephus function $J_3$, and let $m = \\nu_2(3a + 2)$ be the $2$-adic valuation of $3a + 2$. Then exactly $m$ natural numbers $n$ with $a < n < b$ satisfy $J_3(n) = n - 1$ (the pure high extremal points between $a$ and $b$), and $2^{m+1} b + 2^m = 3^m (3a + 2)$, that is,\u2026","title":"Recurrence for consecutive fixed points of $J_3$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-three-eq-of-mem-fixed-point-int","kind":"node","lean":"MetaMathlibExt.josephusThree_eq_of_mem_fixedPoint_interval","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-three-eq-of-mem-fixed-point-int","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"For every natural number $\\ell$, let $a$ and $b$ be the $(\\ell+1)$-th and $(\\ell+2)$-th smallest fixed points of the Josephus function $J_3$, and let $m = \\nu_2(3a + 2)$ be the $2$-adic valuation of $3a + 2$. For every natural number $n$ with $a + 1 \\le n \\le b$, put $\\mathfrak{m} = \\lceil \\log_{3/2}((2n + 1)/(3a + 2)) \\rceil$; then, in the real numbers,","title":"The Josephus function between consecutive fixed points"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/is-low-extremal","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.IsLowExtremal","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#is-low-extremal","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"For natural numbers $k$ and $m$, the point $m$ is a low extremal point of the Josephus function $J_k$ if $1 \\le J_k(m) \\le k - 1$, where $k - 1$ is truncated at $0$.","title":"Low extremal point of the Josephus function"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/is-high-extremal","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.IsHighExtremal","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#is-high-extremal","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"fully_proved","summary":"For natural numbers $k$ and $m$, the point $m$ is a high extremal point of the Josephus function $J_k$ if $m + 1$ is a low extremal point of $J_k$, that is, if $1 \\le J_k(m + 1) \\le k - 1$.","title":"High extremal point of the Josephus function"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-succ-eq-sub","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.josephus_succ_eq_sub","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-succ-eq-sub","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $k \\ge 2$ and $n \\ge 1$ be natural numbers, let $J_k$ be the Josephus function and write $p = J_k(n)$. If $\\ell$ is a natural number with $\\ell(n+1) + 1 \\le p + k \\le (\\ell+1)(n+1)$, then $J_k(n+1) = p + k - \\ell(n+1)$.","title":"Josephus recurrence by number of wrap-arounds"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-fixed-succ-cases","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.josephus_fixed_succ_cases","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-fixed-succ-cases","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $k \\ge 2$ and $n_p \\ge 1$ be natural numbers with $J_k(n_p) = n_p$, where $J_k$ is the Josephus function. Then $J_k(n_p + 1)$ is determined as follows: (i) if $k \\le n_p + 1$, then $J_k(n_p + 1) = k - 1$; (ii) if $k = \\ell(n_p + 1) + 1$ for some natural number $\\ell \\ge 1$, then $J_k(n_p + 1) = n_p + 1$; (iii) if $k = \\ell(n_p + 1) + k_1$ for some natural numbers $\\ell \\ge 1$ and $k_1$ with $2 \\le k_1 \\le n_p$,\u2026","title":"The Josephus function just after a fixed point"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-eq-self-iff","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.josephus_eq_self_iff","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-eq-self-iff","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $k \\ge 2$ and $n_p$ be natural numbers with $n_p \\ge k - 1$, and let $J_k$ be the Josephus function. Then $J_k(n_p) = n_p$ if and only if $J_k(n_p + 1) = k - 1$.","title":"Josephus fixed points detected by the next value"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-add-eq-of-le","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.josephus_add_eq_of_le","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-add-eq-of-le","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $k \\ge 2$, $n \\ge 1$ and $m$ be natural numbers, let $J_k$ be the Josephus function and write $p = J_k(n)$. If $p + km \\le n + m$, then $J_k(n + m) = p + km$.","title":"Linear growth of the Josephus function before a wrap-around"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-add-eq-sub-of-minimal","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.josephus_add_eq_sub_of_minimal","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-add-eq-sub-of-minimal","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $k \\ge 2$, $n \\ge 1$ and $m$ be natural numbers and let $J_k$ be the Josephus function. Suppose that $m$ is the least natural number with $J_k(n) + km > n + m$, and that moreover $J_k(n) + km \\le 2(n + m)$. Then $J_k(n + m) = J_k(n) + km - (n + m)$.","title":"The Josephus function at the first wrap-around"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/extremal-two-mul-sub","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.extremal_two_mul_sub","parent":12787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extremal points of the Josephus function near "],["math","\\(2k - 2\\)"]],"source":"Extremal points of the Josephus function near $2k - 2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#extremal-two-mul-sub","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and let $J_k$ be the Josephus function; call $m$ low extremal if $1 \\le J_k(m) \\le k - 1$, and high extremal if $m + 1$ is low extremal. If $J_k(2k - 2) \\le k - 1$, then $2k - 3$ is high extremal and $2k - 2$ is low extremal. If $J_k(2k - 2) > k - 1$, then $2k - 2$ is high extremal and $2k - 1$ is low extremal. In particular, $2k - 2$ is always a low or a high extremal point.","title":"Extremal points of the Josephus function near $2k - 2$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/shift-le-iff-le-div","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.shift_le_iff_le_div","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#shift-le-iff-le-div","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $k \\ge 2$ and $n$ be natural numbers, let $J_k$ be the Josephus function, write $j = J_k(n+1)$, and suppose $j \\le k - 1$. Then for every natural number $m$, $j + km \\le n + 1 + m$ holds if and only if $m \\le \\lfloor (n - (j - 1)) / (k - 1) \\rfloor$. Moreover, if $k - 2 + j \\le n$, then $\\lfloor (n - (j - 1)) / (k - 1) \\rfloor \\ge 1$.","title":"Linear-step bound at a low extremal point"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/josephus-add-eq-of-high","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.josephus_add_eq_of_high","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#josephus-add-eq-of-high","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"Let $k \\ge 2$ and $n_e \\ge 2k - 3$ be natural numbers such that $n_e$ is a high extremal point of the Josephus function $J_k$, that is, $1 \\le J_k(n_e + 1) \\le k - 1$, and write $j = J_k(n_e + 1)$. Then $J_k(n_e + 1 + m) = j + km$ for every natural number $m \\le \\lfloor (n_e - (j - 1)) / (k - 1) \\rfloor$.","title":"Linear growth of the Josephus function after a high extremal point"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/infinite-fixed-points","kind":"node","lean":"MathlibExt.NumberTheory.JosephusWanted.infinite_fixedPoints","parent":12787,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/josephus-problem/index.html#infinite-fixed-points","scope":"NumberTheory/integer-sequences/iterated-self-referential/josephus-problem","status":"can_prove","summary":"For every natural number $k \\ge 2$, the set of natural numbers $n$ with $J_k(n) = n$ is infinite, where $J_k$ is the Josephus function.","title":"The Josephus function has infinitely many fixed points"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps.json","id":"NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps","status":"can_prove","title":"Klarner's theorem on iterated affine maps"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps/affine-expand","kind":"node","lean":"MathlibExt.Combinatorics.AffineMapIterateRepetitionErdosProblem481Wanted.affineExpand","parent":12827,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps/index.html#affine-expand","scope":"NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps","status":"fully_proved","summary":"For $r \\in \\mathbb{N}$, maps $a, b : \\{0, \\dots, r - 1\\} \\to \\mathbb{N}$ and a finite sequence $A = (x_1, \\dots, x_n)$ of natural numbers, $T(A)$ is the sequence of length $rn$ obtained by listing the blocks $(a_i x_1 + b_i, \\dots, a_i x_n + b_i)$ one after another for $i = 0, 1, \\dots, r - 1$.","title":"Affine expansion of a sequence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps/affine-iterate-has-repetition","kind":"node","lean":"MathlibExt.Combinatorics.AffineMapIterateRepetitionErdosProblem481Wanted.affineIterateHasRepetition","parent":12827,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps/index.html#affine-iterate-has-repetition","scope":"NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps","status":"fully_proved","summary":"The statement that for every $r \\in \\mathbb{N}$ and all maps $a, b : \\{0, \\dots, r - 1\\} \\to \\mathbb{N}$ with $a_i \\ge 1$ for every $i$ and $\\sum_{i < r} 1 / a_i > 1$, if $T$ is the affine expansion of $a$ and $b$, sending a finite sequence $(x_1, \\dots, x_n)$ of natural numbers to the sequence of all $a_i x_j + b_i$, listed in blocks for $i = 0, 1, \\dots, r - 1$ (the order does not affect whether an entry\u2026","title":"Erd\u0151s Problem 481: repetitions under affine expansion"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps/affine-iterate-has-repetition-holds","kind":"node","lean":"MathlibExt.Combinatorics.AffineMapIterateRepetitionErdosProblem481Wanted.affineIterateHasRepetition_holds","parent":12827,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps/index.html#affine-iterate-has-repetition-holds","scope":"NumberTheory/integer-sequences/iterated-self-referential/klarner-s-theorem-on-iterated-affine-maps","status":"can_prove","summary":"For every $r \\in \\mathbb{N}$ and all maps $a, b : \\{0, \\dots, r - 1\\} \\to \\mathbb{N}$ with $a_i \\ge 1$ for every $i$ and $\\sum_{i < r} 1 / a_i > 1$, let $T$ send a finite sequence $(x_1, \\dots, x_n)$ of natural numbers to the sequence of all $a_i x_j + b_i$, listed in blocks for $i = 0, 1, \\dots, r - 1$. Then, starting from $A_1 = (1)$ with $A_{k+1} = T(A_k)$, some $A_k$ has a repeated entry.","title":"Klarner's theorem (Erd\u0151s Problem 481)"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials.json","id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"can_prove","title":"Locally nilpotent polynomials"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/is-weakly-locally-nilpotent-at","kind":"node","lean":"MetaMathlibExt.IsWeaklyLocallyNilpotentAt","parent":12831,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#is-weakly-locally-nilpotent-at","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"fully_proved","summary":"Let $u \\in \\mathbb{Z}[x]$ and $r \\in \\mathbb{Z}$, and let $A$ be a set of natural numbers. Write $u^{(m)}$ for the $m$-fold iterate of $u$ as a map $\\mathbb{Z} \\to \\mathbb{Z}$. Then $u$ is weakly locally nilpotent at $r$ outside $A$ if for every prime $p \\notin A$ there is $m \\ge 1$ such that $p$ divides $u^{(m)}(r)$. For $A = \\emptyset$ this is called locally nilpotent at $r$.","title":"Weakly locally nilpotent polynomial at a point"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/is-nilpotent-at","kind":"node","lean":"MetaMathlibExt.IsNilpotentAt","parent":12831,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#is-nilpotent-at","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"fully_proved","summary":"Let $u \\in \\mathbb{Z}[x]$ and $r \\in \\mathbb{Z}$, and write $u^{(n)}$ for the $n$-fold iterate of $u$ as a map $\\mathbb{Z} \\to \\mathbb{Z}$. Then $u$ is nilpotent at $r$ if $u^{(n)}(r) = 0$ for some $n \\ge 1$.","title":"Nilpotent polynomial at a point"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/infinite-primes-not-dvd-mul-pow-sub","kind":"node","lean":"MetaMathlibExt.infinite_primes_not_dvd_mul_pow_sub","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many primes avoid "],["math","\\(\\gamma \\alpha^n - \\beta\\)"]],"source":"Infinitely many primes avoid $\\gamma \\alpha^n - \\beta$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#infinite-primes-not-dvd-mul-pow-sub","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"can_prove","summary":"Let $\\alpha, \\beta, \\gamma$ be nonzero integers such that $\\beta \\ne \\gamma \\alpha^k$ and $\\beta \\alpha^k \\ne \\gamma$ for every natural number $k$, that is, $\\beta / \\gamma$ is not $\\alpha^k$ for any integer $k$. Then there are infinitely many primes $p$ such that $p$ divides none of the integers $\\gamma \\alpha^n - \\beta$ with $n \\ge 1$.","title":"Infinitely many primes avoid $\\gamma \\alpha^n - \\beta$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/iterate-eval-x-add-one","kind":"node","lean":"MetaMathlibExt.iterate_eval_X_add_one","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterates of "],["math","\\(x + 1\\)"]],"source":"Iterates of $x + 1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#iterate-eval-x-add-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"fully_proved","summary":"Let $u(x) = x + 1 \\in \\mathbb{Z}[x]$ and write $u^{(m)}$ for its $m$-fold iterate as a map $\\mathbb{Z} \\to \\mathbb{Z}$. For every integer $r$ and every natural number $m$, $u^{(m)}(r) = r + m$.","title":"Iterates of $x + 1$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/is-weakly-locally-nilpotent-at-x-add-one","kind":"node","lean":"MetaMathlibExt.isWeaklyLocallyNilpotentAt_X_add_one","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(x + 1\\)"],["text"," is locally nilpotent at "],["math","\\(1\\)"]],"source":"$x + 1$ is locally nilpotent at $1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#is-weakly-locally-nilpotent-at-x-add-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"fully_proved","summary":"Let $u(x) = x + 1 \\in \\mathbb{Z}[x]$ and write $u^{(m)}$ for the $m$-fold iterate of $u$ as a map $\\mathbb{Z} \\to \\mathbb{Z}$. Then for every prime $p$ there is $m \\ge 1$ such that $p$ divides $u^{(m)}(1)$.","title":"$x + 1$ is locally nilpotent at $1$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/not-is-nilpotent-at-x-add-one","kind":"node","lean":"MetaMathlibExt.not_isNilpotentAt_X_add_one","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(x + 1\\)"],["text"," is not nilpotent at "],["math","\\(1\\)"]],"source":"$x + 1$ is not nilpotent at $1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#not-is-nilpotent-at-x-add-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"fully_proved","summary":"The polynomial $x + 1 \\in \\mathbb{Z}[x]$ is not nilpotent at $1$.","title":"$x + 1$ is not nilpotent at $1$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/locally-nilpotent-at-one-classification","kind":"node","lean":"MetaMathlibExt.locallyNilpotentAt_one_classification","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classification of locally nilpotent polynomials at "],["math","\\(1\\)"]],"source":"Classification of locally nilpotent polynomials at $1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#locally-nilpotent-at-one-classification","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"can_prove","summary":"A polynomial $u \\in \\mathbb{Z}[x]$ of positive degree is locally nilpotent at $1$ if and only if it has one of the following forms, for some $p \\in \\mathbb{Z}[x]$: $(x-1)p(x)$ with $p \\ne 0$; $-2x + 4 + p(x)(x-1)(x-2)$; $-2x^2 + 7x - 3 + p(x)(x-1)(x-2)(x-3)$; or $u = x + 1$.","title":"Classification of locally nilpotent polynomials at $1$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/locally-nilpotent-at-neg-one-classificat","kind":"node","lean":"MetaMathlibExt.locallyNilpotentAt_neg_one_classification","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classification of locally nilpotent polynomials at "],["math","\\(-1\\)"]],"source":"Classification of locally nilpotent polynomials at $-1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#locally-nilpotent-at-neg-one-classificat","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"can_prove","summary":"A polynomial $u \\in \\mathbb{Z}[x]$ of positive degree is locally nilpotent at $-1$ if and only if it has one of the following forms, for some $p \\in \\mathbb{Z}[x]$: $(x+1)p(x)$ with $p \\ne 0$; $-2x - 4 + p(x)(x+1)(x+2)$; $2x^2 + 7x + 3 + p(x)(x+1)(x+2)(x+3)$; or $u = x - 1$.","title":"Classification of locally nilpotent polynomials at $-1$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/locally-nilpotent-not-nilpotent-at-one-n","kind":"node","lean":"MetaMathlibExt.locallyNilpotent_not_nilpotentAt_one_neg_one","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["text","Locally nilpotent but not nilpotent polynomials at "],["math","\\(\\pm 1\\)"]],"source":"Locally nilpotent but not nilpotent polynomials at $\\pm 1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#locally-nilpotent-not-nilpotent-at-one-n","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"can_prove","summary":"The only polynomial $u \\in \\mathbb{Z}[x]$ of positive degree that is locally nilpotent at $1$ but not nilpotent at $1$ is $x + 1$, and the only polynomial of positive degree that is locally nilpotent at $-1$ but not nilpotent at $-1$ is $x - 1$.","title":"Locally nilpotent but not nilpotent polynomials at $\\pm 1$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/locally-nilpotent-at-zero-classification","kind":"node","lean":"MetaMathlibExt.locallyNilpotentAt_zero_classification","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classification of locally nilpotent polynomials at "],["math","\\(0\\)"]],"source":"Classification of locally nilpotent polynomials at $0$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#locally-nilpotent-at-zero-classification","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"can_prove","summary":"A polynomial $u \\in \\mathbb{Z}[x]$ of positive degree is locally nilpotent at $0$ if and only if it has one of the following forms: $x + b$ with $b \\in \\mathbb{Z}$, $b \\ne 0$; $ax + b$ with $a, b \\in \\mathbb{Z}$, $b \\ne 0$, $a$ divisible by at least one prime, and every prime divisor of $a$ dividing $b$; $x\\,p(x)$ with $p \\in \\mathbb{Z}[x]$, $p \\ne 0$; or $(x - a)p(x)$ with $a \\in \\mathbb{Z}$, $a \\ne 0$, and\u2026","title":"Classification of locally nilpotent polynomials at $0$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/locally-nilpotent-not-nilpotent-at-zero","kind":"node","lean":"MetaMathlibExt.locallyNilpotent_not_nilpotentAt_zero","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomials locally nilpotent but not nilpotent at "],["math","\\(0\\)"]],"source":"Polynomials locally nilpotent but not nilpotent at $0$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#locally-nilpotent-not-nilpotent-at-zero","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"can_prove","summary":"A polynomial $u \\in \\mathbb{Z}[x]$ of degree at least $1$ is locally nilpotent at $0$ and not nilpotent at $0$ if and only if either $u = x + b$ for some integer $b \\ne 0$, or $u = ax + b$ for some integers $a$ and $b$ with $b \\ne 0$ such that $a$ has at least one prime divisor and every prime divisor of $a$ divides $b$.","title":"Polynomials locally nilpotent but not nilpotent at $0$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/linear-weakly-locally-nilpotent-not-nilp","kind":"node","lean":"MetaMathlibExt.linear_weaklyLocallyNilpotent_not_nilpotentAt_one","parent":12831,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear polynomials weakly locally nilpotent but not nilpotent at "],["math","\\(1\\)"]],"source":"Linear polynomials weakly locally nilpotent but not nilpotent at $1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials/index.html#linear-weakly-locally-nilpotent-not-nilp","scope":"NumberTheory/integer-sequences/iterated-self-referential/locally-nilpotent-polynomials","status":"can_prove","summary":"Let $A$ be a finite set of primes, and let $c$ range over the positive integers all of whose prime factors lie in $A$. A polynomial $u \\in \\mathbb{Z}[x]$ of degree $1$ is weakly locally nilpotent at $1$ outside $A$ and not nilpotent at $1$ if and only if $u$ is one of: $x + c$; $x - c$ with $c \\ne 1$; $cx + 1$ or $-cx + 1$ with $c \\ne 1$; or $-2x - 1$, the last only when $2 \\in A$.","title":"Linear polynomials weakly locally nilpotent but not nilpotent at $1$"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence.json","id":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence","status":"fully_proved","title":"Stern's diatomic sequence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/stern-sequence","kind":"node","lean":"MetaMathlibExt.sternSequence","parent":12844,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/index.html#stern-sequence","scope":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence","status":"fully_proved","summary":"The Stern sequence is the function $s : \\mathbb{N} \\to \\mathbb{N}$ defined by $s(0) = 0$, $s(1) = 1$, and, for every $n \\ge 1$, $s(2n) = s(n)$ and $s(2n + 1) = s(n) + s(n + 1)$.","title":"Stern's diatomic sequence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/stern-zero","kind":"node","lean":"MetaMathlibExt.stern_zero","parent":12844,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/index.html#stern-zero","scope":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence","status":"fully_proved","summary":"The Stern sequence $s$ satisfies $s(0) = 0$.","title":"Stern sequence at zero"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/stern-one","kind":"node","lean":"MetaMathlibExt.stern_one","parent":12844,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/index.html#stern-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence","status":"fully_proved","summary":"The Stern sequence $s$ satisfies $s(1) = 1$.","title":"Stern sequence at one"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/stern-even","kind":"node","lean":"MetaMathlibExt.stern_even","parent":12844,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/index.html#stern-even","scope":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the Stern sequence $s$ satisfies $s(2n) = s(n)$.","title":"Even-index recurrence for the Stern sequence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/stern-odd","kind":"node","lean":"MetaMathlibExt.stern_odd","parent":12844,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence/index.html#stern-odd","scope":"NumberTheory/integer-sequences/iterated-self-referential/stern-s-diatomic-sequence","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the Stern sequence $s$ satisfies $s(2n + 1) = s(n) + s(n + 1)$.","title":"Odd-index recurrence for the Stern sequence"},{"data_url":"scopes/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences.json","id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","kind":"scope","navigation":"module","parent":12666,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","title":"Three-term Hofstadter-like sequences"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-lookup","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterLookup","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-lookup","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"Given computed terms $B(1), \\dots, B(m)$, a choice of whether the sequence is padded, and an integer $j$, the value at index $j$ is $B(j)$ when $1 \\le j \\le m$, is $0$ when $j \\le 0$ and the sequence is padded, and is undefined otherwise (for $j \\le 0$ in the unpadded case, and for $j > m$).","title":"Value of a partially computed sequence at an index"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-next","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterNext","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-next","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"Given computed terms $B(1), \\dots, B(m)$, padded with $0$ at nonpositive indices or not, the next term is $B(m + 1) = B(m + 1 - B(m)) + B(m + 1 - B(m - 1)) + B(m + 1 - B(m - 2))$. It is undefined when one of the values it uses, $B(m)$, $B(m - 1)$, $B(m - 2)$ or one of the three summands, sits at an undefined index: a nonpositive index in the unpadded case, or an index above $m$.","title":"Next term of the three-term Hofstadter-like recurrence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-extend","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterExtend","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-extend","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"Given terms $B(1), \\dots, B(m)$, padded with $0$ at nonpositive indices or not, and a natural number $k$, this is the list $B(1), \\dots, B(m + k)$ obtained by computing $B(m + 1), \\dots, B(m + k)$ one after another from $B(n) = B(n - B(n-1)) + B(n - B(n-2)) + B(n - B(n-3))$. It is undefined if one of these terms cannot be computed because the formula refers to an undefined index (a nonpositive index in the unpadded\u2026","title":"Computing further terms of a three-term Hofstadter-like sequence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-prefix","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterPrefix","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-prefix","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"Let $x_1, \\dots, x_m$ be integers ($m \\ge 0$) and let $B$ be the padded or the unpadded three-term Hofstadter-like sequence with initial conditions $x_1, \\dots, x_m$: $B(i) = x_i$ for $1 \\le i \\le m$, the padded sequence has $B(i) = 0$ for every $i \\le 0$ while the unpadded one is undefined there, and for $n > m$","title":"Initial segment of a three-term Hofstadter-like sequence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-term","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterTerm","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-term","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"Let $x_1, \\dots, x_m$ be integers ($m \\ge 0$), the initial conditions, and choose whether the sequence is padded. The sequence $B$ has $B(i) = x_i$ for $1 \\le i \\le m$; the padded sequence has $B(i) = 0$ for every integer $i \\le 0$, while the unpadded one is undefined at every $i \\le 0$. For $n > m$,","title":"Three-term Hofstadter-like sequence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-ends-after","kind":"node","lean":"MetaMathlibExt.ThreeTermHofstadterEndsAfter","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-ends-after","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"Take integer initial conditions $B(1), \\dots, B(m)$, and either pad the sequence with $B(n) = 0$ for every $n \\le 0$ or leave it unpadded. Continue it by $B(n) = B(n - B(n-1)) + B(n - B(n-2)) + B(n - B(n-3))$ for $n > m$, where a term cannot be computed if this formula refers to an undefined index: a nonpositive index in the unpadded case, or an index not yet computed. For a natural number $T$, the sequence has\u2026","title":"Exact length of a three-term Hofstadter-like sequence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-term-of-le-length","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterTerm_of_le_length","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-term-of-le-length","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"Let $x_1, \\dots, x_m$ be integers and let $j$ be a natural number with $j \\le m$. For the padded or the unpadded three-term Hofstadter-like sequence with initial conditions $x_1, \\dots, x_m$, the term at index $j$ equals the lookup of index $j$ in $x_1, \\dots, x_m$: it is $x_j$ if $j \\ge 1$, and if $j = 0$ it is $0$ for the padded sequence and undefined for the unpadded one.","title":"The initial conditions are the first terms"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-term-zero","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterTerm_zero","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-term-zero","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"For the three-term Hofstadter-like sequence with the single initial condition $5$, the term at index $0$ is defined and equals $0$ when the sequence is padded, and it is undefined when the sequence is unpadded.","title":"Index zero of the sequence started at 5"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-term-b-sequence-si","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterTerm_bSequence_six_seven","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-term-b-sequence-si","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"For the unpadded three-term Hofstadter-like sequence $B$ with initial conditions $1, 2, 3, 4, 5$, the terms $B(6)$ and $B(7)$ are defined, with $B(6) = B(1) + B(2) + B(3) = 6$ and $B(7) = B(1) + B(2) + B(3) = 6$.","title":"Sixth and seventh terms of the sequence started at 1, 2, 3, 4, 5"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-ends-after-one-two","kind":"node","lean":"MetaMathlibExt.threeTermHofstadterEndsAfter_one_two_three","parent":12850,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sequence from "],["math","\\(1, 2, 3\\)"],["text"," dies after four terms"]],"source":"The sequence from $1, 2, 3$ dies after four terms"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-ends-after-one-two","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"fully_proved","summary":"Run the recurrence $B(n) = B(n - B(n-1)) + B(n - B(n-2)) + B(n - B(n-3))$ from the initial conditions $B(1) = 1$, $B(2) = 2$, $B(3) = 3$ without padding, so that a term whose recurrence refers to a nonpositive index, or to an index not yet computed, cannot be computed. Then the sequence has exactly four terms: $B(4)$ can be computed and $B(5)$ cannot.","title":"The sequence from $1, 2, 3$ dies after four terms"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-linear-dies","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_linear_dies","parent":12850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Death of the unpadded sequence from "],["math","\\(1, \\dots, N\\)"],["text"," for "],["math","\\(N = 3\\)"],["text",", "],["math","\\(4\\)"],["text"," or "],["math","\\(N \\ge 10\\)"]],"source":"Death of the unpadded sequence from $1, \\dots, N$ for $N = 3$, $4$ or $N \\ge 10$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-linear-dies","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"Let $N$ be a natural number with $N = 3$, $N = 4$ or $N \\ge 10$, and let $B_N$ be the sequence with initial conditions $B_N(i) = i$ for $1 \\le i \\le N$, continued without padding by the recurrence $B_N(n) = B_N(n - B_N(n-1)) + B_N(n - B_N(n-2)) + B_N(n - B_N(n-3))$. Then $B_N$ dies: there is a natural number $T$ such that $B_N(1), \\dots, B_N(T)$ can be computed and $B_N(T + 1)$ cannot, because the recurrence refers\u2026","title":"Death of the unpadded sequence from $1, \\dots, N$ for $N = 3$, $4$ or $N \\ge 10$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-linear-length","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_linear_length","parent":12850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Length of the unpadded sequence from "],["math","\\(1, \\dots, N\\)"]],"source":"Length of the unpadded sequence from $1, \\dots, N$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-linear-length","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"For every natural number $N \\ge 14$, the sequence $B_N$ with initial conditions $B_N(i) = i$ for $1 \\le i \\le N$, continued without padding by the recurrence $B_N(n) = B_N(n - B_N(n-1)) + B_N(n - B_N(n-2)) + B_N(n - B_N(n-3))$, has exactly $N + 24$ terms: $B_N(1), \\dots, B_N(N + 24)$ can be computed and $B_N(N + 25)$ cannot, because the recurrence refers to a nonpositive index or to an index not yet computed.","title":"Length of the unpadded sequence from $1, \\dots, N$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-padded-linear-peri","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_padded_linear_period_seven","parent":12850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period-seven pattern of the padded sequence from "],["math","\\(1, \\dots, N\\)"]],"source":"Period-seven pattern of the padded sequence from $1, \\dots, N$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-padded-linear-peri","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"Let $N \\ge 72$ be a natural number, and let $B$ be the sequence with $B(n) = 0$ for $n \\le 0$ and $B(i) = i$ for $1 \\le i \\le N$, continued by the recurrence $B(n) = B(n - B(n-1)) + B(n - B(n-2)) + B(n - B(n-3))$. Let $\\nu = -1, -2, -2, -2, 2, 1, 0$ according as $N \\equiv 0, 1, 2, 3, 4, 5, 6 \\pmod 7$. Then for all natural numbers $k \\ge 1$ and $r < 7$ with $N + 67 \\le N + 7k + r \\le 2N + \\nu$, the term\u2026","title":"Period-seven pattern of the padded sequence from $1, \\dots, N$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-padded-linear-ends","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_padded_linear_endsAfter","parent":12850,"reading_title":{"format":"inline-parts-v1","parts":[["text","Length of the padded sequence from "],["math","\\(1, \\dots, N\\)"]],"source":"Length of the padded sequence from $1, \\dots, N$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-padded-linear-ends","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"Let $N$ be a natural number, and let $t = 196, 2087, 3201, 4315, 200, 32478, 118$ and $e = 27, 254, 524, 560, 20, 4547, 9$ according as $N \\equiv 0, 1, 2, 3, 4, 5, 6 \\pmod 7$. Let $B$ be the sequence with $B(n) = 0$ for $n \\le 0$ and $B(i) = i$ for $1 \\le i \\le N$, continued by the recurrence $B(n) = B(n - B(n-1)) + B(n - B(n-2)) + B(n - B(n-3))$. If $N \\ge t$, then $B$ ends after exactly $2N + e$ terms:\u2026","title":"Length of the padded sequence from $1, \\dots, N$"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-seven-cycle","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_sevenCycle","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-seven-cycle","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"Let $K$, $c$ and $\\gamma$ be natural numbers with $K \\ge 7$, $c \\ge 1$, $\\gamma \\le 6$ and $7c + \\gamma \\le K$, and put $L = K - 7c - \\gamma$ and $M = K + L + 5$. For a tuple $(t_0, \\dots, t_6)$ write $(t_0, \\dots, t_6)_\\gamma$ for its entry $t_\\gamma$. Let $\\lambda$ and $\\mu$ be integers with $\\lambda \\ge -2c + (2, 1, 4, 3, 2, 1, 0)_\\gamma$ and $\\mu \\ge -c + (0, 0, 3, 2, 1, 0, -1)_\\gamma$, and put\u2026","title":"Seven-cycle lemma for the three-term Hofstadter-like recurrence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-two-cycle","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_twoCycle","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-two-cycle","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"Let $K \\ge 1$ be a natural number, let $M$ be an integer with $M \\ge K + 5$, let $a_1, \\dots, a_K$ be arbitrary integers, and let $B$ be the padded three-term Hofstadter-like sequence with initial conditions $a_1, \\dots, a_K, 2, M, 2$. Then for every natural number $k$ the term $B(K + 2k + 1)$ is defined and equals $2$, and for every $k \\ge 1$ the term $B(K + 2k)$ is defined and equals $2^{k-1} M$.","title":"Two-cycle lemma for the three-term Hofstadter-like recurrence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-five-cycle","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_fiveCycle","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-five-cycle","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"Let $K \\ge 3$ be a natural number, $\\mu \\ge 1$ an integer, and $a_4, \\dots, a_K$ arbitrary integers. Let $B$ be the sequence given by $B(n) = B(n - B(n-1)) + B(n - B(n-2)) + B(n - B(n-3))$, padded with $B(n) = 0$ for $n \\le 0$, with initial conditions $1, 2, 3, a_4, \\dots, a_K, K + \\mu, 3, K + 3, K + \\mu + 1, 5$ at the indices $1, \\dots, K + 5$. Then for all natural numbers $k$ and $r < 5$ with\u2026","title":"Five-cycle lemma for the three-term Hofstadter-like recurrence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-sixteen-cycle-two","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_sixteenCycle_two","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-sixteen-cycle-two","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"Let $K \\ge 1$ be a natural number and let $\\lambda$, $\\mu_1$, $\\mu_2$, $\\gamma_1$, $\\gamma_2$, $\\gamma_3$ be integers with $\\lambda > 31 + K$ and $\\mu_1, \\mu_2, \\gamma_1, \\gamma_2, \\gamma_3 > \\lambda$. Let $a_1, \\dots, a_K$ be arbitrary integers and let $B$ be the padded three-term Hofstadter-like sequence with initial conditions\u2026","title":"Second sixteen-cycle lemma for the three-term Hofstadter-like recurrence"},{"id":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/three-term-hofstadter-sixteen-cycle-one","kind":"node","lean":"MetaMathlibExt.threeTermHofstadter_sixteenCycle_one","parent":12850,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences/index.html#three-term-hofstadter-sixteen-cycle-one","scope":"NumberTheory/integer-sequences/iterated-self-referential/three-term-hofstadter-like-sequences","status":"can_prove","summary":"Let $K \\ge 1$ be a natural number and let $\\lambda$, $\\mu_1$, $\\mu_2$, $\\gamma$ be integers with $\\lambda > 31 + K$ and $\\mu_1, \\mu_2, \\gamma > \\lambda$. Let $a_1, \\dots, a_K$ be arbitrary integers and let $B$ be the padded three-term Hofstadter-like sequence with initial conditions $a_1, \\dots, a_K, \\lambda, 7, \\mu_2, 16, \\mu_2, 16, \\mu_1, \\lambda, 7, \\mu_2, 16, 2\\mu_2, 16, \\mu_2, 25, \\gamma, \\lambda, 7$. Then for\u2026","title":"First sixteen-cycle lemma for the three-term Hofstadter-like recurrence"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences","kind":"scope","parent":12665,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences","status":"can_prove","summary":"Dold, Newton and realizable sequences, Sylvester, Euclid and G\u00f6bel sequences, gcd and divisibility sequences, OEIS-specific sequences.","title":"Arithmetic properties of sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/multiple-counting-jacobsthal-sequence","kind":"node","lean":"MetaMathlibExt.multipleCountingJacobsthalSequence","parent":12870,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/index.html#multiple-counting-jacobsthal-sequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences","status":"fully_proved","summary":"For natural numbers $x$, $\\rho$ and $n$, the multiple-counting Jacobsthal sequence $J_{x,\\rho}(n)$ is the number of multiples of $\\rho$ strictly between $x^n$ and $x^{n+1}$, that is, the number of natural numbers $k$ with $x^n < k < x^{n+1}$ and $\\rho \\mid k$.","title":"Multiple-counting Jacobsthal sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/odd-lichtenberg","kind":"node","lean":"MetaMathlibExt.oddLichtenberg","parent":12870,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/index.html#odd-lichtenberg","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences","status":"fully_proved","summary":"The odd Lichtenberg numbers form the sequence $(a_n)_{n \\ge 0}$ of natural numbers defined by $a_0 = 1$ and $a_{n+1} = 4 a_n + 1$, so the sequence begins $1, 5, 21, 85, \\dots$; these are the positive terms of OEIS A002450.","title":"Odd Lichtenberg numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/generalized-euclid-sequence-contains-eve","kind":"node","lean":"MetaMathlibExt.generalized_euclid_sequence_contains_every_prime","parent":12870,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/index.html#generalized-euclid-sequence-contains-eve","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences","status":"fully_proved","summary":"Let $P$ be a finite set of primes and let $k$ be the number of elements of $P$. Then there is an injective sequence $a_0, a_1, a_2, \\dots$ of primes such that $a_0, \\dots, a_{k-1}$ are exactly the elements of $P$, every prime occurs in the sequence, and for every $n \\ge k$ there is a set $I \\subseteq \\{0, \\dots, n-1\\}$ such that $a_n$ divides","title":"Some generalized Euclid sequence contains every prime"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"can_prove","title":"Arithmetic terms for the greatest common divisor"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/gcd-term-seq","kind":"node","lean":"MetaMathlibExt.gcdTermSeq","parent":12874,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html#gcd-term-seq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"fully_proved","summary":"For natural numbers $a$, $b$ and $n$, $s_{a,b}(n)$ is the rational number that is the coefficient of $z^n$ in the inverse of the formal power series $(z^a - 1)(z^b - 1)$ over $\\mathbb{Q}$. For $a, b \\ge 1$ this series has constant coefficient $1$, so $s_{a,b}$ is the coefficient sequence of $f_{a,b}(z) = 1 / ((z^a - 1)(z^b - 1))$. If $a = 0$ or $b = 0$ the series is $0$, whose inverse is taken to be $0$, so\u2026","title":"The gcd-term sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/gcd-term-seq-zero","kind":"node","lean":"MetaMathlibExt.gcdTermSeq_zero","parent":12874,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html#gcd-term-seq-zero","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"fully_proved","summary":"For natural numbers $a, b \\ge 1$, the gcd-term sequence satisfies $s_{a,b}(0) = 1$.","title":"Initial value of the gcd-term sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/mul-mk-gcd-term-seq","kind":"node","lean":"MetaMathlibExt.mul_mk_gcdTermSeq","parent":12874,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html#mul-mk-gcd-term-seq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"fully_proved","summary":"For natural numbers $a, b \\ge 1$, in the ring $\\mathbb{Q}[[z]]$ of formal power series, the gcd-term sequence $s_{a,b}$ satisfies","title":"Generating function of the gcd-term sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/gcd-term-seq-natural-and-eq-gcd-add-one","kind":"node","lean":"MetaMathlibExt.gcdTermSeq_natural_and_eq_gcd_add_one","parent":12874,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html#gcd-term-seq-natural-and-eq-gcd-add-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"can_prove","summary":"Let $a, b \\ge 1$ be natural numbers and let $s_{a,b}$ be the gcd-term sequence. Every value $s_{a,b}(n)$ is a natural number, and $s_{a,b}(ab) = \\gcd(a, b) + 1$.","title":"The gcd-term sequence is integral and recovers the gcd"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/gcd-term-seq-le-gcd-term-seq-one-one","kind":"node","lean":"MetaMathlibExt.gcdTermSeq_le_gcdTermSeq_one_one","parent":12874,"reading_title":{"format":"inline-parts-v1","parts":[["text","The gcd-term sequence is bounded by "],["math","\\(n + 1\\)"]],"source":"The gcd-term sequence is bounded by $n + 1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html#gcd-term-seq-le-gcd-term-seq-one-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"can_prove","summary":"Let $a, b \\ge 1$ be natural numbers, at least one of them greater than $1$, and let $s_{a,b}$ be the gcd-term sequence. For every natural number $n$, $s_{a,b}(n) \\le s_{1,1}(n) = n + 1$, and $s_{a,b}(n) = s_{1,1}(n)$ only if $n = 0$.","title":"The gcd-term sequence is bounded by $n + 1$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/gcd-add-one-eq-floor-div-mod-five","kind":"node","lean":"MetaMathlibExt.gcd_add_one_eq_floor_div_mod_five","parent":12874,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arithmetic term for the gcd in base "],["math","\\(5\\)"]],"source":"Arithmetic term for the gcd in base $5$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html#gcd-add-one-eq-floor-div-mod-five","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"can_prove","summary":"For all natural numbers $a, b \\ge 1$,","title":"Arithmetic term for the gcd in base $5$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/gcd-eq-neg-emod-emod-five-sub-two","kind":"node","lean":"MetaMathlibExt.gcd_eq_neg_emod_emod_five_sub_two","parent":12874,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second arithmetic term for the gcd in base "],["math","\\(5\\)"]],"source":"Second arithmetic term for the gcd in base $5$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html#gcd-eq-neg-emod-emod-five-sub-two","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"can_prove","summary":"For all natural numbers $a, b \\ge 1$,","title":"Second arithmetic term for the gcd in base $5$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/gcd-arithmetic-terms-small-bases","kind":"node","lean":"MetaMathlibExt.gcd_arithmetic_terms_small_bases","parent":12874,"reading_title":{"format":"inline-parts-v1","parts":[["text","The gcd terms in bases "],["math","\\(2\\)"],["text",", "],["math","\\(3\\)"],["text"," and "],["math","\\(4\\)"]],"source":"The gcd terms in bases $2$, $3$ and $4$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor/index.html#gcd-arithmetic-terms-small-bases","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetic-terms-for-the-greatest-common-divisor","status":"can_prove","summary":"Let $c \\in \\{2, 3, 4\\}$, let $a, b \\ge 1$ be natural numbers, and put $E = ab(ab + a + b)$ and $D = (c^{a^2 b} - 1)(c^{a b^2} - 1)$. Then the identity","title":"The gcd terms in bases $2$, $3$ and $4$"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","title":"Binomial sum parities and cotangent floor sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/fried-alternating-binomial-identity","kind":"node","lean":"MetaMathlibExt.fried_alternating_binomial_identity","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#fried-alternating-binomial-identity","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For integers $a$ and $b$ let $\\binom{a}{b}$ be the binomial coefficient when $a, b \\ge 0$ (so $0$ when $b > a$), and $0$ when $a < 0$ or $b < 0$. For natural numbers $n$, $r$, $s$ with $r \\le n$ and $s \\le 2r$, with all arguments computed in the integers,","title":"Fried's alternating binomial identity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/bala-somos-partition-parity","kind":"node","lean":"MetaMathlibExt.bala_somos_partition_parity","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#bala-somos-partition-parity","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For a natural number $k$ let $p(k)$ be the number of partitions of $k$. For natural numbers $m$ and $N$ let $b_m(N) = \\sum_{k=1}^{N} k^{2m+1} p(k) p(N-k)$ and $a(N) = \\sum_{j=0}^{N} c(j)\\, p(N-j)$, where $c(0) = 0$ and $c(j) = \\sum_{d \\mid j,\\ d \\text{ odd}} (-1)^{j/d - 1}$ for $j \\ge 1$. Then for all $m \\ge 1$ and $n \\ge 1$: (a) if $n$ is odd, then $n$ divides $b_m(n)$; (b) $n$ divides $b_m(2n)$, and $b_m(2n)/n$\u2026","title":"The Bala\u2013Somos parity conjectures"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/coeff-diag-trinomial-product-eq-franel","kind":"node","lean":"MetaMathlibExt.coeff_diag_trinomial_product_eq_franel","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#coeff-diag-trinomial-product-eq-franel","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For every $n \\ge 0$, the coefficient of $x^n y^n z^n$ in the integer polynomial $(1 + x + y - z)^n (1 + x - y + z)^n (1 - x + y + z)^n$ equals $(-1)^n \\sum_{k=0}^{n} \\binom{n}{k}^3$ (A000172).","title":"A diagonal coefficient equal to a signed Franel number"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/exists-sq-near-triangular-beatty-sqrt-tw","kind":"node","lean":"MetaMathlibExt.exists_sq_near_triangular_beatty_sqrt_two","parent":12883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triangular numbers near squares along "],["math","\\(\\lfloor n\\sqrt{2} \\rfloor\\)"]],"source":"Triangular numbers near squares along $\\lfloor n\\sqrt{2} \\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#exists-sq-near-triangular-beatty-sqrt-tw","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For a natural number $n$ let $a = \\lfloor n\\sqrt{2} \\rfloor$ (A001951) and $T_a = a(a+1)/2$. Then some integer $k$ satisfies $|T_a - k^2| \\le \\lfloor a/2 \\rfloor$; that is, the distance from $T_a$ to the nearest perfect square is at most $\\lfloor a/2 \\rfloor$.","title":"Triangular numbers near squares along $\\lfloor n\\sqrt{2} \\rfloor$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/ncard-arith-prog-eq-sum-divisor-summator","kind":"node","lean":"MetaMathlibExt.ncard_arithProg_eq_sum_divisorSummatory","parent":12883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting arithmetic progressions in "],["math","\\(\\{1, \\dots, n\\}\\)"]],"source":"Counting arithmetic progressions in $\\{1, \\dots, n\\}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#ncard-arith-prog-eq-sum-divisor-summator","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For every integer $n \\ge 1$, the number of subsets $S \\subseteq \\{1, \\dots, n\\}$ with $|S| \\ge 2$ that are arithmetic progressions, that is, $S = \\{a, a + d, \\dots, a + (|S| - 1)d\\}$ for some natural numbers $a$ and $d \\ge 1$, equals $\\sum_{m=0}^{n-1} D(m)$, where $D(m) = \\sum_{k=1}^{m} \\lfloor m/k \\rfloor$ (A078567, A006218).","title":"Counting arithmetic progressions in $\\{1, \\dots, n\\}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/card-digit-insertion-le-ten-mul-eq-digit","kind":"node","lean":"MetaMathlibExt.card_digitInsertion_le_ten_mul_eq_digitSum","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#card-digit-insertion-le-ten-mul-eq-digit","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"Let $n \\ge 1$ have the $\\ell$ decimal digits $d_{\\ell-1} \\dots d_1 d_0$, so that $n = \\sum_{j\\lt \\ell} d_j 10^j$. Let $S(n)$ be the set of the numbers $\\sum_{j\\lt i} d_j 10^j + x \\cdot 10^i + \\sum_{i \\le j < \\ell} d_j 10^{j+1}$ with $0 \\le i \\le \\ell$ and $x \\in \\{0, \\dots, 9\\}$, excluding $i = \\ell$ with $x = 0$: these are the numbers obtained by inserting one decimal digit $x$ anywhere in the decimal expansion of\u2026","title":"Inserting a digit and the digit sum"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/bala-binomial-sum-odd-imp-mod-four","kind":"node","lean":"MetaMathlibExt.bala_binomial_sum_odd_imp_mod_four","parent":12883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd values of Bala's binomial sum are "],["math","\\(1\\)"],["text"," modulo "],["math","\\(4\\)"]],"source":"Odd values of Bala's binomial sum are $1$ modulo $4$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#bala-binomial-sum-odd-imp-mod-four","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For a natural number $n$, let $a(n) = \\sum_{k=0}^{n} \\binom{2n+1}{n+k+1} (-2)^k$. If $a(n)$ is odd, then $a(n) \\equiv 1 \\pmod 4$.","title":"Odd values of Bala's binomial sum are $1$ modulo $4$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/ceil-cot-pi-div-two-mul-steps","kind":"node","lean":"MetaMathlibExt.ceil_cot_pi_div_two_mul_steps","parent":12883,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ceiling of "],["math","\\(\\cot(\\pi/(2m))\\)"],["text"," rises in steps of "],["math","\\(0\\)"],["text"," or "],["math","\\(1\\)"]],"source":"The ceiling of $\\cot(\\pi/(2m))$ rises in steps of $0$ or $1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#ceil-cot-pi-div-two-mul-steps","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For $m \\ge 1$ let $g(m) = \\lceil \\cot(\\pi/(2m)) \\rceil$. Then for every $m \\ge 1$, $g(m) \\le g(m+1)$ and $g(m+1) - g(m) \\in \\{0, 1\\}$; and for every integer $n \\ge 1$, the number of $m \\ge 1$ with $g(m) = n$ is $1$ or $2$.","title":"The ceiling of $\\cot(\\pi/(2m))$ rises in steps of $0$ or $1$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/ncard-ceil-cot-le-eq-two-mul-sub-singlet","kind":"node","lean":"MetaMathlibExt.ncard_ceil_cot_le_eq_two_mul_sub_singletons","parent":12883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(m\\)"],["text"," with "],["math","\\(\\lceil \\cot(\\pi/(2m)) \\rceil \\le r\\)"]],"source":"Counting $m$ with $\\lceil \\cot(\\pi/(2m)) \\rceil \\le r$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#ncard-ceil-cot-le-eq-two-mul-sub-singlet","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"Let $g(m) = \\lceil \\cot(\\pi / (2m)) \\rceil$ for integers $m \\ge 1$, and call an integer $j$ a singleton if exactly one integer $m \\ge 1$ satisfies $g(m) = j$. For an integer $r \\ge 1$, let $s(r)$ be the number of singletons in $\\{1, \\dots, r\\}$. Then the number of integers $m \\ge 2$ with $g(m) \\le r$ equals $2r - s(r)$.","title":"Counting $m$ with $\\lceil \\cot(\\pi/(2m)) \\rceil \\le r$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/ceil-cot-singleton-iff","kind":"node","lean":"MetaMathlibExt.ceil_cot_singleton_iff","parent":12883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values taken once by the ceiling of "],["math","\\(\\cot(\\pi/(2m))\\)"]],"source":"Values taken once by the ceiling of $\\cot(\\pi/(2m))$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#ceil-cot-singleton-iff","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For $m \\ge 1$ let $g(m) = \\lceil \\cot(\\pi/(2m)) \\rceil$. Let $r \\ge 1$ and $m \\ge 2$ be integers. Then $m$ is the only integer $m' \\ge 1$ with $g(m') = r$ if and only if $g(m-1) = r - 1$, $g(m) = r$ and $g(m+1) = r + 1$.","title":"Values taken once by the ceiling of $\\cot(\\pi/(2m))$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/jeffery-floor-cot-singletons","kind":"node","lean":"MetaMathlibExt.jeffery_floor_cot_singletons","parent":12883,"reading_title":{"format":"inline-parts-v1","parts":[["text","Jeffery's conjecture on singletons of "],["math","\\(\\lfloor \\cot(\\pi/(2m)) \\rfloor\\)"]],"source":"Jeffery's conjecture on singletons of $\\lfloor \\cot(\\pi/(2m)) \\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#jeffery-floor-cot-singletons","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"Let $f(m) = \\lfloor \\cot(\\pi / (2m)) \\rfloor$ for nonzero integers $m$. Let $a_1 < a_2 < \\cdots$ list the integers $k \\ge 1$ for which exactly one integer $m \\ge 2$ satisfies $f(-m) = -k$, and let $b_1 < b_2 < \\cdots$ list the integers $k \\ge 2$ with $f(-k-1) < f(-k) < f(-k+1)$. Then for every integer $n \\ge 1$, $2a_n = b_n + n - 1$ (A223577, A223578).","title":"Jeffery's conjecture on singletons of $\\lfloor \\cot(\\pi/(2m)) \\rfloor$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/ncard-repeating-decimal-ratio-eq-two-pow","kind":"node","lean":"MetaMathlibExt.ncard_repeating_decimal_ratio_eq_two_pow_omega","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#ncard-repeating-decimal-ratio-eq-two-pow","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"For every natural number $n \\ge 1$, the number of natural numbers $x$ with $10^n \\le x < 10^{n+1}$ for which there is a natural number $k$ with $x \\le k < 2x$ and $k / x = 1 + (x - 10^n) / (10^n - 1)$, computed in the rationals, (OEIS A285273) equals $2^{\\omega(10^n - 1)}$, where $\\omega(m)$ is the number of distinct prime factors of $m$.","title":"Counting solutions of a repeating-decimal ratio equation"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/cevian-max-regions","kind":"node","lean":"MetaMathlibExt.cevian_max_regions","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#cevian-max-regions","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"Let $A$, $B$, $C$ be affinely independent points of $\\mathbb{R}^2$ and $T$ the triangle they span; a cevian is a segment joining a vertex of $T$ to a point of the open side opposite it. The regions cut out by cevians $c_1, \\dots, c_n$ are the connected components of the interior of $T$ minus $c_1 \\cup \\dots \\cup c_n$. Then for every $n \\ge 0$, the maximum number of regions cut out by $n$ cevians (not necessarily\u2026","title":"Zakharov's conjecture on regions cut by cevians"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/cevian-max-regions-intersection-count","kind":"node","lean":"MetaMathlibExt.cevian_max_regions_intersection_count","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#cevian-max-regions-intersection-count","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"Let $A$, $B$, $C$ be affinely independent points of $\\mathbb{R}^2$ and $T$ the triangle they span; a cevian is a segment joining a vertex of $T$ to a point of the open side opposite it. Let $n \\ge 0$ and let $c_1, \\dots, c_n$ be cevians such that the interior of $T$ minus $c_1 \\cup \\dots \\cup c_n$ has exactly $\\lceil (n+1)(n+2)/3 \\rceil$ connected components. Then the number of points of the interior of $T$ that\u2026","title":"Zakharov's conjecture on intersection points of cevians"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/two-fan-ray-intersection-count","kind":"node","lean":"MetaMathlibExt.two_fan_ray_intersection_count","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#two-fan-ray-intersection-count","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"Let $m, n \\ge 1$, $P = (-1, 0)$ and $Q = (1, 0)$. From $P$ draw the $m$ closed rays at angles $\\pi - 2\\pi s/m$ for $0 \\le s < m$, and from $Q$ the $n$ closed rays at angles $2\\pi t/n$ for $0 \\le t < n$. Let $h = \\lfloor (m - 1)/2 \\rfloor$. Then the number of points that are $P$, $Q$, or a point off the $x$-axis lying on a ray from $P$ and on a ray from $Q$ is $2 \\sum_{s=1}^{h} \\lceil ns/m - 1/2 \\rceil + 2$ for odd\u2026","title":"Intersection points of two fans of rays"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/two-fan-ray-intersection-count-diag","kind":"node","lean":"MetaMathlibExt.two_fan_ray_intersection_count_diag","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#two-fan-ray-intersection-count-diag","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"Let $n \\ge 1$, $P = (-1, 0)$ and $Q = (1, 0)$. From $P$ draw the $n$ closed rays at angles $\\pi - 2\\pi s/n$ for $0 \\le s < n$, and from $Q$ the $n$ closed rays at angles $2\\pi t/n$ for $0 \\le t < n$. Then the number of points that are $P$, $Q$, or a point off the $x$-axis lying on a ray from $P$ and on a ray from $Q$ is $(n^2 + 7)/4$ for odd $n$ and $(n^2 - 6n + 16)/4$ for even $n$ (A338042).","title":"Blomberg's conjecture on intersection points of two equal fans"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/two-fan-region-count-diag","kind":"node","lean":"MetaMathlibExt.two_fan_region_count_diag","parent":12883,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen/index.html#two-fan-region-count-diag","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/binomial-sum-parities-and-cotangent-floor-sequen","status":"can_prove","summary":"Let $n \\ge 3$, $P = (-1, 0)$ and $Q = (1, 0)$. From $P$ draw the $n$ closed rays at angles $\\pi - 2\\pi s/n$ for $0 \\le s < n$, and from $Q$ the $n$ closed rays at angles $2\\pi t/n$ for $0 \\le t < n$, and let $U$ be the union of these rays. Then there is a real $R_0$ such that for every real $R \\ge R_0$, if $K_R$ is the convex hull of the points of $U$ on the circle $x^2 + y^2 = R^2$, the interior of $K_R$ minus $U$\u2026","title":"Blomberg's conjecture on regions cut out by two equal fans"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","title":"Complementary sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/is-complementary-pair","kind":"node","lean":"MetaMathlibExt.IsComplementaryPair","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#is-complementary-pair","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"Two sequences $a, b : \\mathbb{N} \\to \\mathbb{N}$, indexed from $0$, form a complementary pair when all their terms are positive, both are strictly increasing, no term of $a$ equals a term of $b$, and every positive integer is a term of $a$ or a term of $b$. In particular, the ranges of $a$ and $b$ partition the positive integers.","title":"Complementary pair of sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/satisfies-parametrized-sum-equation","kind":"node","lean":"MetaMathlibExt.SatisfiesParametrizedSumEquation","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subsequence-sum equation "],["math","\\(a_n = b_{hn} + b_{khn}\\)"]],"source":"The subsequence-sum equation $a_n = b_{hn} + b_{khn}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#satisfies-parametrized-sum-equation","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"Sequences $a, b : \\mathbb{N} \\to \\mathbb{N}$ and natural numbers $h$ and $k$ satisfy the parametrized subsequence-sum equation when $h \\ge 2$, $k \\ge 2$, $b_0 = 1$, and $a_n = b_{hn} + b_{khn}$ for every $n \\in \\mathbb{N}$.","title":"The subsequence-sum equation $a_n = b_{hn} + b_{khn}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/satisfies-base-sum-equation","kind":"node","lean":"MetaMathlibExt.SatisfiesBaseSumEquation","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subsequence-sum equation "],["math","\\(a_n = b_{2n} + b_{4n}\\)"]],"source":"The subsequence-sum equation $a_n = b_{2n} + b_{4n}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#satisfies-base-sum-equation","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"Sequences $a, b : \\mathbb{N} \\to \\mathbb{N}$ satisfy this equation when $b_0 = 1$ and $a_n = b_{2n} + b_{4n}$ for every $n \\in \\mathbb{N}$.","title":"The subsequence-sum equation $a_n = b_{2n} + b_{4n}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/satisfies-shifted-sum-equation","kind":"node","lean":"MetaMathlibExt.SatisfiesShiftedSumEquation","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subsequence-sum equation "],["math","\\(a_n = b_{2n} + b_{4n} + 1\\)"]],"source":"The subsequence-sum equation $a_n = b_{2n} + b_{4n} + 1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#satisfies-shifted-sum-equation","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"Sequences $a, b : \\mathbb{N} \\to \\mathbb{N}$ satisfy this equation when $b_0 = 1$ and $a_n = b_{2n} + b_{4n} + 1$ for every $n \\in \\mathbb{N}$.","title":"The subsequence-sum equation $a_n = b_{2n} + b_{4n} + 1$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/is-complementary-determined-by-parametri","kind":"node","lean":"MetaMathlibExt.IsComplementaryDeterminedByParametrizedSumEquation","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Complementary pair determined by "],["math","\\(a_n = b_{hn} + b_{khn}\\)"]],"source":"Complementary pair determined by $a_n = b_{hn} + b_{khn}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#is-complementary-determined-by-parametri","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"Sequences $a, b : \\mathbb{N} \\to \\mathbb{N}$ form a complementary pair determined by this equation with natural-number parameters $h$ and $k$ when they form a complementary pair (both strictly increasing with positive terms, and every positive integer is a term of exactly one of them), $h \\ge 2$, $k \\ge 2$, $b_0 = 1$, and $a_n = b_{hn} + b_{khn}$ for every $n \\in \\mathbb{N}$.","title":"Complementary pair determined by $a_n = b_{hn} + b_{khn}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/is-complementary-determined-by-base-sum","kind":"node","lean":"MetaMathlibExt.IsComplementaryDeterminedByBaseSumEquation","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Complementary pair determined by "],["math","\\(a_n = b_{2n} + b_{4n}\\)"]],"source":"Complementary pair determined by $a_n = b_{2n} + b_{4n}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#is-complementary-determined-by-base-sum","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"Sequences $a, b : \\mathbb{N} \\to \\mathbb{N}$ form a complementary pair determined by this equation when they form a complementary pair (both strictly increasing with positive terms, and every positive integer is a term of exactly one of them) and they satisfy $b_0 = 1$ and $a_n = b_{2n} + b_{4n}$ for every $n \\in \\mathbb{N}$.","title":"Complementary pair determined by $a_n = b_{2n} + b_{4n}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/is-complementary-determined-by-shifted-s","kind":"node","lean":"MetaMathlibExt.IsComplementaryDeterminedByShiftedSumEquation","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Complementary pair determined by "],["math","\\(a_n = b_{2n} + b_{4n} + 1\\)"]],"source":"Complementary pair determined by $a_n = b_{2n} + b_{4n} + 1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#is-complementary-determined-by-shifted-s","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"Sequences $a, b : \\mathbb{N} \\to \\mathbb{N}$ form a complementary pair determined by this equation when they form a complementary pair (both strictly increasing with positive terms, and every positive integer is a term of exactly one of them) and they satisfy $b_0 = 1$ and $a_n = b_{2n} + b_{4n} + 1$ for every $n \\in \\mathbb{N}$.","title":"Complementary pair determined by $a_n = b_{2n} + b_{4n} + 1$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value","kind":"node","lean":"MetaMathlibExt.nthNonValue","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text","-th non-value of a sequence"]],"source":"The $n$-th non-value of a sequence"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"For $f : \\mathbb{N} \\to \\mathbb{N}$ and $n \\ge 1$, the $n$-th non-value of $f$ is the $n$-th smallest positive integer that is not of the form $f(m)$ with $m \\ge 1$, or $0$ if fewer than $n$ such integers exist. At $n = 0$ it takes the same value as at $n = 1$.","title":"The $n$-th non-value of a sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-sq-one","kind":"node","lean":"MetaMathlibExt.nthNonValue_sq_one","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-sq-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"The smallest positive integer that is not the square of a positive integer is $2$; that is, the first non-value of $m \\mapsto m^2$ is $2$.","title":"The first non-square"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-sq-two","kind":"node","lean":"MetaMathlibExt.nthNonValue_sq_two","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-sq-two","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"fully_proved","summary":"The second smallest positive integer that is not the square of a positive integer is $3$; that is, the second non-value of $m \\mapsto m^2$ is $3$.","title":"The second non-square"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-eq-two-step","kind":"node","lean":"MetaMathlibExt.nthNonValue_eq_two_step","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-eq-two-step","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(1) \\ge 1$ and $f(m + 1) \\ge f(m) + m$ for all $m \\ge 1$, let $C(x)$ be the number of $m$ with $1 \\le m \\le x$ and $f(m) \\le x$, and for $n \\ge 1$ let $f_P(n)$ be the $n$-th positive integer that is not a value $f(m)$ with $m \\ge 1$. Then $f_P(n) = n + C(n + C(n))$ for every $n \\ge 1$.","title":"Complementary sequences in two Lambek\u2013Moser steps"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-eq-one-step","kind":"node","lean":"MetaMathlibExt.nthNonValue_eq_one_step","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-eq-one-step","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(1) \\ge 1$ and $f(m + 1) \\ge f(m) + m$ for all $m \\ge 1$, let $C(x)$ be the number of $m$ with $1 \\le m \\le x$ and $f(m) \\le x$, and for $n \\ge 1$ let $f_P(n)$ be the $n$-th positive integer that is not a value $f(m)$ with $m \\ge 1$. Then for every $n \\ge 1$, $f_P(n) = n + C(n) + 1$ if $f(C(n) + 1) \\le n + C(n)$, and $f_P(n) = n + C(n)$ otherwise.","title":"Complementary sequences from one evaluation of the counting function"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-eq-interleave","kind":"node","lean":"MetaMathlibExt.nthNonValue_eq_interleave","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-eq-interleave","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(1) \\ge 1$ and $f(m + 1) \\ge f(m) + m$ for all $m \\ge 1$, and for $n \\ge 1$ let $f_P(n)$ be the $n$-th positive integer that is not a value $f(m)$ with $m \\ge 1$. Let $\\alpha : \\mathbb{N} \\to \\mathbb{R}$ satisfy $f(m - 1) \\le \\alpha(m) \\le f(m)$ for all $m \\ge 2$, and let $h : \\mathbb{N} \\to \\mathbb{N}$ satisfy $h(n) \\ge 1$ and $\\alpha(h(n)) \\le n < \\alpha(h(n) + 1)$\u2026","title":"Complementary sequences from an interleaving function"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-eq-interleave-of-sub-lt","kind":"node","lean":"MetaMathlibExt.nthNonValue_eq_interleave_of_sub_lt","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Interleaving formula when "],["math","\\(f(m) - \\alpha(m) < m\\)"]],"source":"Interleaving formula when $f(m) - \\alpha(m) < m$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-eq-interleave-of-sub-lt","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(1) \\ge 1$ and $f(m + 1) \\ge f(m) + m$ for all $m \\ge 1$, and for $n \\ge 1$ let $f_P(n)$ be the $n$-th positive integer that is not a value $f(m)$ with $m \\ge 1$. Let $\\alpha : \\mathbb{N} \\to \\mathbb{R}$ satisfy $f(m - 1) \\le \\alpha(m) \\le f(m)$ for all $m \\ge 2$, and let $h : \\mathbb{N} \\to \\mathbb{N}$ satisfy $h(n) \\ge 1$ and $\\alpha(h(n)) \\le n < \\alpha(h(n) + 1)$\u2026","title":"Interleaving formula when $f(m) - \\alpha(m) < m$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-eq-interleave-of-le-sub","kind":"node","lean":"MetaMathlibExt.nthNonValue_eq_interleave_of_le_sub","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Interleaving formula when "],["math","\\(f(m) - \\alpha(m) \\ge m\\)"]],"source":"Interleaving formula when $f(m) - \\alpha(m) \\ge m$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-eq-interleave-of-le-sub","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(1) \\ge 1$ and $f(m + 1) \\ge f(m) + m$ for all $m \\ge 1$, and for $n \\ge 1$ let $f_P(n)$ be the $n$-th positive integer that is not a value $f(m)$ with $m \\ge 1$. Let $\\alpha : \\mathbb{N} \\to \\mathbb{R}$ satisfy $f(m - 1) \\le \\alpha(m) \\le f(m)$ for all $m \\ge 2$, and let $h : \\mathbb{N} \\to \\mathbb{N}$ satisfy $h(n) \\ge 1$ and $\\alpha(h(n)) \\le n < \\alpha(h(n) + 1)$\u2026","title":"Interleaving formula when $f(m) - \\alpha(m) \\ge m$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-triangular-eq-round","kind":"node","lean":"MetaMathlibExt.nthNonValue_triangular_eq_round","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formula for the "],["math","\\(n\\)"],["text","-th non-triangular number"]],"source":"Formula for the $n$-th non-triangular number"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-triangular-eq-round","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"For every integer $n \\ge 1$, the $n$-th smallest positive integer that is not a triangular number $j(j+1)/2$ with $j \\ge 1$ is $n + \\lfloor \\sqrt{2n} + 1/2 \\rfloor$.","title":"Formula for the $n$-th non-triangular number"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-sq-eq-round","kind":"node","lean":"MetaMathlibExt.nthNonValue_sq_eq_round","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-sq-eq-round","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"For every integer $n \\ge 1$, the $n$-th positive integer that is not a perfect square is $n + \\lfloor \\sqrt{n} + 1/2 \\rfloor$.","title":"Formula for the non-square numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-polygonal","kind":"node","lean":"MetaMathlibExt.nthNonValue_polygonal","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-polygonal","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k \\ge 3$, and for $j \\ge 1$ let $T(k, j) = (k - 2) j (j - 1)/2 + j$ be the $k$-gonal numbers. For every integer $n \\ge 1$, put $s = \\lfloor \\sqrt{2n/(k - 2)} \\rfloor$. Then the $n$-th positive integer that is not of the form $T(k, j)$ with $j \\ge 1$ equals $n + s + 1$ if $(k - 2) s (s + 1) < 2n$, and $n + s$ otherwise.","title":"Formula for the non-polygonal numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-polygonal-eq-round","kind":"node","lean":"MetaMathlibExt.nthNonValue_polygonal_eq_round","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rounding formula for the non-"],["math","\\(k\\)"],["text","-gonal numbers, "],["math","\\(3 \\le k \\le 10\\)"]],"source":"Rounding formula for the non-$k$-gonal numbers, $3 \\le k \\le 10$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-polygonal-eq-round","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $3 \\le k \\le 10$; the $k$-gonal numbers are $(k - 2) j (j - 1)/2 + j$ for $j \\ge 1$. For every $n \\ge 1$, the $n$-th positive integer that is not a $k$-gonal number equals $n + \\lfloor \\sqrt{2n/(k - 2)} + 1/2 \\rfloor$.","title":"Rounding formula for the non-$k$-gonal numbers, $3 \\le k \\le 10$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-polygonal-shift","kind":"node","lean":"MetaMathlibExt.nthNonValue_polygonal_shift","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-polygonal-shift","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k \\ge 3$, let $r$ be a real number with $-1 \\le r \\le k - 4$, and for $j \\ge 1$ let $T(k, j) = (k - 2) j (j - 1)/2 + j$ be the $k$-gonal numbers. For every integer $n \\ge 1$, put $s = \\lfloor \\sqrt{(2n + r)/(k - 2)} \\rfloor$. Then the $n$-th positive integer that is not of the form $T(k, j)$ with $j \\ge 1$ equals $n + s + 1$ if $(k - 2) s (s + 1) < 2n$, and $n + s$ otherwise.","title":"Shifted formula for the non-polygonal numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-polygonal-eq-round-shift","kind":"node","lean":"MetaMathlibExt.nthNonValue_polygonal_eq_round_shift","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rounding formula for the non-"],["math","\\(k\\)"],["text","-gonal numbers, every "],["math","\\(k \\ge 3\\)"]],"source":"Rounding formula for the non-$k$-gonal numbers, every $k \\ge 3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-polygonal-eq-round-shift","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k \\ge 3$; the $k$-gonal numbers are $(k - 2) j (j - 1)/2 + j$ for $j \\ge 1$. For every $n \\ge 1$, the $n$-th positive integer that is not a $k$-gonal number equals $n + \\lfloor \\sqrt{(2n - 2 + \\lfloor (k + 1)/4 \\rfloor)/(k - 2)} + 1/2 \\rfloor$.","title":"Rounding formula for the non-$k$-gonal numbers, every $k \\ge 3$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-second-pentagonal","kind":"node","lean":"MetaMathlibExt.nthNonValue_secondPentagonal","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-second-pentagonal","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"For every integer $n \\ge 1$, put $s = \\lfloor \\sqrt{2n/3} \\rfloor$. Then the $n$-th positive integer that is not a second pentagonal number $(3j^2 + j)/2$ with $j \\ge 1$ equals $n + s + 1$ if $s (3s + 5) + 4 \\le 2n$, and $n + s$ otherwise.","title":"Formula for the non-second-pentagonal numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-second-polygonal","kind":"node","lean":"MetaMathlibExt.nthNonValue_secondPolygonal","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-second-polygonal","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k > 5$, and for $j \\ge 1$ let $S(k, j) = ((k - 2) j^2 + (k - 4) j)/2$ be the second $k$-gonal numbers. For every integer $n \\ge 1$, put $s = \\lfloor \\sqrt{2n/(k - 2)} \\rfloor$. Then the $n$-th positive integer that is not of the form $S(k, j)$ with $j \\ge 1$ equals $n + s$ if $(k - 2) s (s + 1) < 2n + 4s$, and $n + s - 1$ otherwise.","title":"Formula for the non-second-polygonal numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-centered-polygonal","kind":"node","lean":"MetaMathlibExt.nth_nonCenteredPolygonal","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-centered-polygonal","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k \\ge 3$; the centered $k$-gonal numbers are $k j (j + 1)/2 + 1$ for $j \\ge 0$. For $n \\ge 1$ let $a(n)$ be the $n$-th nonnegative integer that is not a centered $k$-gonal number (so $a(1) = 0$), and let $s = \\lfloor \\sqrt{2n/k} \\rfloor$. Then $a(n) = n + s$ if $k s (s + 1) < 2(n + s - 1)$, and $a(n) = n + s - 1$ otherwise.","title":"Non-centered polygonal numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-pyramidal","kind":"node","lean":"MetaMathlibExt.nthNonValue_pyramidal","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-pyramidal","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $3 \\le k \\le 8$, and for $j \\ge 1$ let $P(k, j) = j (j + 1) ((k - 2) j - (k - 5))/6$ be the $k$-gonal pyramidal numbers. For every integer $n \\ge 1$, put $s = \\lfloor \\sqrt[3]{6n/(k - 2)} \\rfloor$. Then the $n$-th positive integer that is not of the form $P(k, j)$ with $j \\ge 1$ equals $n + s + 1$ if $6n \\ge s^3 (k - 2) + 3 s^2 (k - 1) + s (2k - 1) + 6$; otherwise it equals $n + s - 1$ if\u2026","title":"Formula for the non-pyramidal numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-pyramidal-two-case","kind":"node","lean":"MetaMathlibExt.nthNonValue_pyramidal_two_case","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-case formula for the non-pyramidal numbers when "],["math","\\(k \\le 5\\)"]],"source":"Two-case formula for the non-pyramidal numbers when $k \\le 5$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-pyramidal-two-case","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $3 \\le k \\le 5$, and for $j \\ge 1$ let $P(k, j) = j (j + 1) ((k - 2) j - (k - 5))/6$ be the $k$-gonal pyramidal numbers. For every integer $n \\ge 1$, put $s = \\lfloor \\sqrt[3]{6n/(k - 2)} \\rfloor$. Then the $n$-th positive integer that is not of the form $P(k, j)$ with $j \\ge 1$ equals $n + s - 1$ if $6n \\le s (s - 1) (s (k - 2) + k + 1)$, and $n + s$ otherwise.","title":"Two-case formula for the non-pyramidal numbers when $k \\le 5$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-pyramidal-two-case-le-eigh","kind":"node","lean":"MetaMathlibExt.nthNonValue_pyramidal_two_case_le_eight","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-case formula for the non-pyramidal numbers when "],["math","\\(k \\le 8\\)"]],"source":"Two-case formula for the non-pyramidal numbers when $k \\le 8$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-pyramidal-two-case-le-eigh","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $3 \\le k \\le 8$, and for $j \\ge 1$ let $P(k, j) = j (j + 1) ((k - 2) j - (k - 5))/6$ be the $k$-gonal pyramidal numbers. For every integer $n \\ge 1$, put $s = \\lfloor \\sqrt[3]{6n/(k - 2)} \\rfloor$. Then the $n$-th positive integer that is not of the form $P(k, j)$ with $j \\ge 1$ equals $n + s - 1$ if $6n \\le s (s - 1) (s (k - 2) + k + 1)$, and $n + s$ otherwise.","title":"Two-case formula for the non-pyramidal numbers when $k \\le 8$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-sum-pow","kind":"node","lean":"MetaMathlibExt.nthNonValue_sumPow","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integers missed by the sums "],["math","\\(1^k + \\dots + j^k\\)"]],"source":"Integers missed by the sums $1^k + \\dots + j^k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-sum-pow","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k > 1$ and $n \\ge 1$ be integers, and let $h = \\lfloor ((k+1)n)^{1/(k+1)} \\rfloor$. Then the $n$-th smallest positive integer that is not of the form $\\sum_{i=1}^{j} i^k$ with $j \\ge 1$ is $n + h - 1$ if $n \\le \\sum_{i=1}^{h} (i^k - 1)$, and $n + h$ otherwise.","title":"Integers missed by the sums $1^k + \\dots + j^k$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-add-sum-pow","kind":"node","lean":"MetaMathlibExt.nthNonValue_add_sumPow","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integers not of the form "],["math","\\(m + 1^k + \\cdots + m^k\\)"]],"source":"Integers not of the form $m + 1^k + \\cdots + m^k$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-add-sum-pow","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k > 1$. For $n \\ge 1$ let $a(n)$ be the $n$-th positive integer that is not of the form $m + \\sum_{i=1}^m i^k$ with $m \\ge 1$, and let $h = \\lfloor ((k + 1) n)^{1/(k+1)} \\rfloor$. Then $a(n) = n + h - 1$ if $n \\le \\sum_{i=1}^h i^k$, and $a(n) = n + h$ otherwise.","title":"Integers not of the form $m + 1^k + \\cdots + m^k$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/asc-factorial-bounds-half","kind":"node","lean":"MetaMathlibExt.ascFactorial_bounds_half","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rising factorial between consecutive "],["math","\\(k\\)"],["text","-th powers"]],"source":"Rising factorial between consecutive $k$-th powers"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#asc-factorial-bounds-half","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k$ and $n$ be natural numbers, write $n^{\\overline{k}} = n (n + 1) \\cdots (n + k - 1)$ for the rising factorial, and let $r = \\lfloor (k^2 - 4k + 6)/4 \\rfloor$ if $k$ is even and $r = \\lfloor (k^2 - 6k + 11)/4 \\rfloor$ if $k$ is odd. If $n \\ge r$, then $(n + \\lfloor k/2 \\rfloor - 1)^k \\le n^{\\overline{k}} \\le (n + \\lfloor k/2 \\rfloor)^k$.","title":"Rising factorial between consecutive $k$-th powers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-simplex","kind":"node","lean":"MetaMathlibExt.nthNonValue_simplex","parent":12901,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-simplex","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $k$ satisfy $1 < k \\le 5$, $k = 7$ or $k = 9$. For every integer $n \\ge 1$, put $R = \\lfloor (k!\\, n)^{1/k} \\rfloor$ and let $t = n + R - \\lfloor k/2 \\rfloor$, an integer. Then the $n$-th positive integer that is not a $k$-simplex number $\\binom{j + k - 1}{k}$ with $j \\ge 1$ equals $t + 2$ if $\\binom{R + \\lceil k/2 \\rceil + 1}{k} \\le t + 1$; otherwise it equals $t$ if $t < \\binom{R + \\lceil k/2 \\rceil}{k}$, and\u2026","title":"Formula for the non-simplex numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/nth-non-value-simplex-le-fifteen","kind":"node","lean":"MetaMathlibExt.nthNonValue_simplex_le_fifteen","parent":12901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formula for the non-simplex numbers when "],["math","\\(k \\le 15\\)"]],"source":"Formula for the non-simplex numbers when $k \\le 15$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences/index.html#nth-non-value-simplex-le-fifteen","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/complementary-sequences","status":"can_prove","summary":"Let $1 < k \\le 15$. For every integer $n \\ge 1$, put $R = \\lfloor (k!\\, n)^{1/k} \\rfloor$ and let $t = n + R - \\lfloor k/2 \\rfloor$, an integer. Then the $n$-th positive integer that is not a $k$-simplex number $\\binom{j + k - 1}{k}$ with $j \\ge 1$ equals $t + 2$ if $\\binom{R + \\lceil k/2 \\rceil + 1}{k} \\le t + 1$; otherwise it equals $t$ if $t < \\binom{R + \\lceil k/2 \\rceil}{k}$, and $t + 1$ if not.","title":"Formula for the non-simplex numbers when $k \\le 15$"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","title":"Dold and realizable sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/binomial-power-seq","kind":"node","lean":"MetaMathlibExt.binomialPowerSeq","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#binomial-power-seq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"For natural numbers $r$, $s$ and $n$, the binomial-power number $A_{r,s}(n)$ is the natural number","title":"Binomial-power sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/dold-realizable","kind":"node","lean":"MetaMathlibExt.doldRealizable","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#dold-realizable","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"An integer sequence $(a(n))_{n \\ge 0}$ is Dold realizable when, for every integer $n \\ge 1$, the M\u00f6bius transform $\\sum_{d \\mid n} \\mu(d)\\, a(n/d)$ is divisible by $n$ and non-negative, where $\\mu$ is the M\u00f6bius function and $d$ runs over the positive divisors of $n$.","title":"Dold realizable sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/dold-realizable-dvd","kind":"node","lean":"MetaMathlibExt.doldRealizable_dvd","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#dold-realizable-dvd","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"If an integer sequence $(a(n))_{n \\ge 0}$ is Dold realizable, then for every integer $n \\ge 1$, $n$ divides $\\sum_{d \\mid n} \\mu(d)\\, a(n/d)$, where $\\mu$ is the M\u00f6bius function and $d$ runs over the positive divisors of $n$.","title":"Dold realizable sequences satisfy the Dold congruences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/dold-realizable-nonneg","kind":"node","lean":"MetaMathlibExt.doldRealizable_nonneg","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#dold-realizable-nonneg","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"If an integer sequence $(a(n))_{n \\ge 0}$ is Dold realizable, then for every integer $n \\ge 1$, $\\sum_{d \\mid n} \\mu(d)\\, a(n/d) \\ge 0$, where $\\mu$ is the M\u00f6bius function and $d$ runs over the positive divisors of $n$.","title":"The M\u00f6bius transform of a Dold realizable sequence is non-negative"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/dold-realizable-of-dvd-and-nonneg","kind":"node","lean":"MetaMathlibExt.doldRealizable_of_dvd_and_nonneg","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#dold-realizable-of-dvd-and-nonneg","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"Let $(a(n))_{n \\ge 0}$ be an integer sequence and let $\\mu$ be the M\u00f6bius function. If for every integer $n \\ge 1$ the sum $\\sum_{d \\mid n} \\mu(d)\\, a(n/d)$ over the positive divisors $d$ of $n$ is divisible by $n$, and for every integer $n \\ge 1$ this sum is non-negative, then $a$ is Dold realizable.","title":"Criterion for Dold realizability"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/binomial-power-sequence-dold-realizable","kind":"node","lean":"MetaMathlibExt.binomial_power_sequence_dold_realizable","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#binomial-power-sequence-dold-realizable","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"Let $r \\ge 1$ and $s \\ge 0$ be integers and let $A_{r,s}(n) = \\sum_{k=0}^{n} \\binom{n}{k}^r \\binom{n+k}{k}^s$. Then the integer sequence $(A_{r,s}(n))_{n \\ge 0}$ is Dold realizable: for every integer $n \\ge 1$, the sum $\\sum_{d \\mid n} \\mu(d)\\, A_{r,s}(n/d)$ over the positive divisors $d$ of $n$, where $\\mu$ is the M\u00f6bius function, is non-negative and divisible by $n$.","title":"Binomial-power sequences are Dold realizable"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/binomial-power-seq-dold-congruences","kind":"node","lean":"MetaMathlibExt.binomialPowerSeq_dold_congruences","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#binomial-power-seq-dold-congruences","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"Let $r \\ge 1$ and $s \\ge 0$ be integers and let $A_{r,s}(m) = \\sum_{k=0}^{m} \\binom{m}{k}^r \\binom{m+k}{k}^s$. Then for every integer $n \\ge 1$, $n$ divides $\\sum_{d \\mid n} \\mu(d)\\, A_{r,s}(n/d)$, where $\\mu$ is the M\u00f6bius function and $d$ runs over the positive divisors of $n$.","title":"Dold congruences for binomial-power sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/binomial-power-seq-moebius-sum-nonneg","kind":"node","lean":"MetaMathlibExt.binomialPowerSeq_moebius_sum_nonneg","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#binomial-power-seq-moebius-sum-nonneg","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"Let $r \\ge 1$ and $s \\ge 0$ be integers and let $A_{r,s}(m) = \\sum_{k=0}^{m} \\binom{m}{k}^r \\binom{m+k}{k}^s$. Then for every integer $n \\ge 1$, $\\sum_{d \\mid n} \\mu(d)\\, A_{r,s}(n/d) \\ge 0$, where $\\mu$ is the M\u00f6bius function and $d$ runs over the positive divisors of $n$.","title":"Non-negativity of the M\u00f6bius transform of binomial-power sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/is-dold-sequence","kind":"node","lean":"MetaMathlibExt.IsDoldSequence","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#is-dold-sequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"A sequence $a_0, a_1, a_2, \\dots$ of integers is a Dold sequence when for every integer $m \\ge 1$ there is an $m \\times m$ matrix $A$ with integer entries such that $a_n = \\operatorname{tr}(A^n)$ for every $n$ with $1 \\le n \\le m$. The term $a_0$ is not constrained.","title":"Dold sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/fermat-sequence","kind":"node","lean":"MetaMathlibExt.fermatSequence","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#fermat-sequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"Given a sequence $b : \\mathbb{N} \\to \\mathbb{C}$, the Fermat sequence generated by $b$ is the sequence $a : \\mathbb{N} \\to \\mathbb{C}$ given by $a_n = \\sum_{m \\mid n} m \\, b_m$ for $n \\ge 1$, the sum running over the positive divisors $m$ of $n$, and $a_0 = 0$.","title":"Fermat sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/is-almost-realizable-seq","kind":"node","lean":"MetaMathlibExt.IsAlmostRealizableSeq","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#is-almost-realizable-seq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"A sequence $a$ of natural numbers $a(0), a(1), \\dots$ is almost realizable when there is an integer $m \\ge 1$ such that the integer sequence $(m\\, a(n))_{n \\ge 0}$ is Dold realizable: for every $n \\ge 1$, the sum $\\sum_{d \\mid n} \\mu(d)\\, m\\, a(n/d)$ over the positive divisors $d$ of $n$, where $\\mu$ is the M\u00f6bius function, is non-negative and divisible by $n$.","title":"Almost realizable sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/is-almost-realizable-seq-zero","kind":"node","lean":"MetaMathlibExt.isAlmostRealizableSeq_zero","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#is-almost-realizable-seq-zero","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"The constant sequence $a(n) = 0$ is almost realizable.","title":"The zero sequence is almost realizable"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/is-almost-realizable-seq-binomial-power","kind":"node","lean":"MetaMathlibExt.isAlmostRealizableSeq_binomialPowerSeq","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#is-almost-realizable-seq-binomial-power","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"fully_proved","summary":"For all natural numbers $r \\ge 1$ and $s$, the binomial-power sequence $n \\mapsto \\sum_{k=0}^{n} \\binom{n}{k}^r \\binom{n+k}{k}^s$ is almost realizable.","title":"Binomial-power sequences are almost realizable"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/dold-realizable-domb-type-sum","kind":"node","lean":"MetaMathlibExt.doldRealizable_dombTypeSum","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#dold-realizable-domb-type-sum","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"For all natural numbers $r \\ge 1$, $s$ and $t$, the integer sequence $(D(n, r, s, t))_{n \\ge 0}$ of Domb-type sums $D(n, r, s, t) = \\sum_{k=0}^{n} \\binom{n}{k}^r \\binom{2k}{k}^s \\binom{2(n-k)}{n-k}^t$ is Dold realizable.","title":"Domb-type sums are Dold realizable"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/not-is-almost-realizable-seq-catalan-mot","kind":"node","lean":"MetaMathlibExt.not_isAlmostRealizableSeq_catalan_motzkin_schroder","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#not-is-almost-realizable-seq-catalan-mot","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"None of the following three sequences of non-negative integers is almost realizable: the Catalan numbers $C_n = \\frac{1}{n+1}\\binom{2n}{n}$, the Motzkin numbers $M(n)$, and the large Schr\u00f6der numbers $S(n)$ (OEIS A006318), given by $S(0) = 1$ and $S(n+1) = S(n) + \\sum_{i=0}^{n} S(i) S(n-i)$.","title":"The Catalan, Motzkin and large Schr\u00f6der numbers are not almost realizable"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/choose-mul-prime-mod-eq-of-five-le","kind":"node","lean":"MetaMathlibExt.choose_mul_prime_modEq_of_five_le","parent":12934,"reading_title":{"format":"inline-parts-v1","parts":[["text","Helou\u2013Terjanian congruence for primes "],["math","\\(p \\ge 5\\)"]],"source":"Helou\u2013Terjanian congruence for primes $p \\ge 5$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#choose-mul-prime-mod-eq-of-five-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"Let $p \\ge 5$ be a prime and let $m$ and $n$ be integers with $0 < m < n$. Then $\\binom{np}{mp} \\equiv \\binom{n}{m} \\pmod{p^e}$, where $e = 3 + \\max(\\nu_p(m), \\nu_p(n - m)) + \\nu_p\\big(\\binom{n}{m}\\big)$ and $\\nu_p$ is the $p$-adic valuation.","title":"Helou\u2013Terjanian congruence for primes $p \\ge 5$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/choose-three-mul-mod-eq","kind":"node","lean":"MetaMathlibExt.choose_three_mul_modEq","parent":12934,"reading_title":{"format":"inline-parts-v1","parts":[["text","Helou\u2013Terjanian congruence for "],["math","\\(p = 3\\)"]],"source":"Helou\u2013Terjanian congruence for $p = 3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#choose-three-mul-mod-eq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"Let $m$ and $n$ be integers with $0 < m < n$. Then $\\binom{3n}{3m} \\equiv \\binom{n}{m} \\pmod{3^e}$, where $e = 2 + \\max(\\nu_3(m), \\nu_3(n - m)) + \\nu_3\\big(\\binom{n}{m}\\big)$ and $\\nu_3$ is the $3$-adic valuation.","title":"Helou\u2013Terjanian congruence for $p = 3$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/choose-two-mul-mod-eq","kind":"node","lean":"MetaMathlibExt.choose_two_mul_modEq","parent":12934,"reading_title":{"format":"inline-parts-v1","parts":[["text","Helou\u2013Terjanian congruence for "],["math","\\(p = 2\\)"]],"source":"Helou\u2013Terjanian congruence for $p = 2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#choose-two-mul-mod-eq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"Let $m$ and $n$ be integers with $0 < m < n$. Then $\\binom{2n}{2m} \\equiv \\binom{n}{m} \\pmod{2^e}$, where $e = 1 + \\max(\\nu_2(m), \\nu_2(n - m)) + \\nu_2\\big(\\binom{n}{m}\\big)$ and $\\nu_2$ is the $2$-adic valuation.","title":"Helou\u2013Terjanian congruence for $p = 2$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/choose-mul-prime-mod-eq-choose-div","kind":"node","lean":"MetaMathlibExt.choose_mul_prime_modEq_choose_div","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#choose-mul-prime-mod-eq-choose-div","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"Let $p$ be a prime and let $n \\ge 1$ be an integer divisible by $p$. Then for every integer $\\lambda \\ge 0$ with $\\lambda p \\ne n$,","title":"Binomial coefficients at multiples of a prime divisor"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/moebius-conv-nonneg-of-geometric-growth","kind":"node","lean":"MetaMathlibExt.moebius_conv_nonneg_of_geometric_growth","parent":12934,"reading_title":{"format":"inline-parts-v1","parts":[["text","M\u00f6bius transforms of sequences growing by a factor of at least "],["math","\\(1.221\\)"],["text"," are nonnegative"]],"source":"M\u00f6bius transforms of sequences growing by a factor of at least $1.221$ are nonnegative"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#moebius-conv-nonneg-of-geometric-growth","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"Let $b$ be a sequence of real numbers and $C \\ge 1.221$ a real number such that $b(n) \\ge 0$ and $b(n+1) \\ge C\\, b(n)$ for every $n \\ge 1$. Then for every $n \\ge 1$, with $\\mu$ the M\u00f6bius function,","title":"M\u00f6bius transforms of sequences growing by a factor of at least $1.221$ are nonnegative"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/prime-power-congruence-iff-dold","kind":"node","lean":"MetaMathlibExt.prime_power_congruence_iff_dold","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#prime-power-congruence-iff-dold","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"Let $(V(n))_{n \\ge 1}$ be a sequence of integers and let $\\mu$ denote the M\u00f6bius function. Then $V(n p^m) \\equiv V(n p^{m-1}) \\pmod{p^m}$ holds for all integers $n, m \\ge 1$ and all primes $p$ if and only if $n$ divides $\\sum_{d \\mid n} \\mu(n/d) V(d)$ for every integer $n \\ge 1$.","title":"Dold congruences via prime-power congruences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/dold-realizable-even-binomial-type-sum","kind":"node","lean":"MetaMathlibExt.doldRealizable_evenBinomialTypeSum","parent":12934,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences/index.html#dold-realizable-even-binomial-type-sum","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/dold-and-realizable-sequences","status":"can_prove","summary":"Let $r \\ge 1$, $s$, $t$ and $u$ be natural numbers with $t + u \\ge 1$, and let","title":"Even-binomial-type sums are Dold realizable"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","title":"G\u00f6bel sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence","kind":"node","lean":"MetaMathlibExt.gobelSequence","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For natural numbers $k$ and $l$, the $(k, l)$-G\u00f6bel sequence is the sequence of rational numbers $g_{k,l}(n)$, $n \\ge 0$, given by $g_{k,l}(0) = 0$, $g_{k,l}(1) = l$ and, for $n \\ge 1$,","title":"The (k, l)-G\u00f6bel sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-one","kind":"node","lean":"MetaMathlibExt.gobelSequence_one","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial value of the "],["math","\\((k, l)\\)"],["text","-G\u00f6bel sequence"]],"source":"Initial value of the $(k, l)$-G\u00f6bel sequence"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For all natural numbers $k$ and $l$, the $(k, l)$-G\u00f6bel sequence $g_{k,l}$ satisfies $g_{k,l}(1) = l$.","title":"Initial value of the $(k, l)$-G\u00f6bel sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-succ","kind":"node","lean":"MetaMathlibExt.gobelSequence_succ","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-succ","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For all natural numbers $k$, $l$ and $n \\ge 1$, the $(k, l)$-G\u00f6bel sequence $g_{k,l}$ satisfies, in $\\mathbb{Q}$,","title":"Recursion for the (k, l)-G\u00f6bel sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-two-two-small","kind":"node","lean":"MetaMathlibExt.gobelSequence_two_two_small","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-two-two-small","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"The $(2, 2)$-G\u00f6bel sequence $g_{2,2}$ satisfies $g_{2,2}(2) = 3$, $g_{2,2}(3) = 5$, $g_{2,2}(4) = 10$ and $g_{2,2}(5) = 28$.","title":"G\u00f6bel's original sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-integral-le-six-and-seven","kind":"node","lean":"MetaMathlibExt.gobelSequence_integral_le_six_and_seven_iff","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrality of the early terms of "],["math","\\((k, l)\\)"],["text","-G\u00f6bel sequences"]],"source":"Integrality of the early terms of $(k, l)$-G\u00f6bel sequences"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-integral-le-six-and-seven","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $g_{k,l}$ denote the $(k, l)$-G\u00f6bel sequence. For all natural numbers $k, l \\ge 2$ and $1 \\le n \\le 6$, the value $g_{k,l}(n)$ is an integer. Moreover, for all natural numbers $k, l \\ge 2$, the value $g_{k,l}(7)$ is not an integer if and only if $k \\equiv 2 \\pmod 6$ and $l \\equiv 3 \\pmod 7$.","title":"Integrality of the early terms of $(k, l)$-G\u00f6bel sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-asymptotic-expansion","kind":"node","lean":"MetaMathlibExt.gobelSequence_asymptotic_expansion","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic expansion of the "],["math","\\((k, l)\\)"],["text","-G\u00f6bel sequence"]],"source":"Asymptotic expansion of the $(k, l)$-G\u00f6bel sequence"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-asymptotic-expansion","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k, l \\ge 2$ be natural numbers and let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence. There are a real number $C > 1$ and real numbers $a_1, a_2, \\dots$ such that for every natural number $R$, as $n \\to \\infty$,","title":"Asymptotic expansion of the $(k, l)$-G\u00f6bel sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-padic-key-properties","kind":"node","lean":"MetaMathlibExt.gobelSequence_padic_key_properties","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Key "],["math","\\(p\\)"],["text","-adic properties of the "],["math","\\((k, l)\\)"],["text","-G\u00f6bel sequences"]],"source":"Key $p$-adic properties of the $(k, l)$-G\u00f6bel sequences"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-padic-key-properties","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime, let $N \\ge 1$ be a natural number, let $r = \\nu_p(N!)$, where $\\nu_p$ is the $p$-adic valuation on $\\mathbb{Q}$ (with $\\nu_p(0) = 0$), and write $g_{k,l}$ for the $(k, l)$-G\u00f6bel sequence. Write $\\mathbb{Z}_{(p)}$ for the set of rational numbers $x$ with $\\nu_p(x) \\ge 0$, and for an integer $s$ write $x \\in p^s \\mathbb{Z}_{(p)}$ when $x = 0$ or $\\nu_p(x) \\ge s$. Then: (1) for all natural numbers\u2026","title":"Key $p$-adic properties of the $(k, l)$-G\u00f6bel sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-seven-iff-padic-claims","kind":"node","lean":"MetaMathlibExt.gobelSequence_seven_iff_padic_claims","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrality of G\u00f6bel sequences up to index 7 via finite "],["math","\\(p\\)"],["text","-adic checks"]],"source":"Integrality of G\u00f6bel sequences up to index 7 via finite $p$-adic checks"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-seven-iff-padic-claims","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence. For a prime $p$, call a rational number $x$ $p$-integral if $x \\in \\mathbb{Z}_{(p)}$, that is, if $x = 0$ or its $p$-adic valuation $\\nu_p(x)$ is nonnegative. The following two statements are equivalent.","title":"Integrality of G\u00f6bel sequences up to index 7 via finite $p$-adic checks"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-root-strict-anti","kind":"node","lean":"MetaMathlibExt.gobelSequence_root_strictAnti","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-root-strict-anti","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k, l \\ge 2$ be natural numbers, let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence, and let $C_{k,l}(n) = g_{k,l}(n)^{1/k^n}$, a real number. Then $C_{k,l}(n) > C_{k,l}(n + 1) > 1$ for every natural number $n \\ge 1$.","title":"Strict decrease of the normalized G\u00f6bel roots"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/somos-like-root-antitone-tendsto","kind":"node","lean":"MetaMathlibExt.somosLike_root_antitone_tendsto","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#somos-like-root-antitone-tendsto","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number, let $\\sigma_k = \\exp\\bigl(\\sum_{m \\ge 1} \\log m / k^m\\bigr)$ be the $k$-Somos constant, and let $t : \\mathbb{N} \\to \\mathbb{R}$ satisfy $t(n + 1) = t(n)^k / (n + 1)$ for all $n \\ge 0$. If $t(0) > \\sigma_k$, then the sequence $n \\mapsto t(n)^{1/k^n}$ is nonincreasing and converges to some real number $c > 1$.","title":"Root asymptotics of Somos-type sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/somos-like-le-gobel-sequence","kind":"node","lean":"MetaMathlibExt.somosLike_le_gobelSequence","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#somos-like-le-gobel-sequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k, l \\ge 2$ be natural numbers, let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence, and let $t : \\mathbb{N} \\to \\mathbb{R}$ satisfy $t(n + 1) = t(n)^k / (n + 1)$ for all $n \\ge 0$. If $t(0)^k \\le l$, then $t(n) \\le g_{k,l}(n)$ for every $n \\ge 1$.","title":"Somos-type sequences lie below G\u00f6bel sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-root-tendsto-gt-one","kind":"node","lean":"MetaMathlibExt.gobelSequence_root_tendsto_gt_one","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-root-tendsto-gt-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k, l \\ge 2$ be natural numbers and let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence. There is a real number $C > 1$ such that the real numbers $g_{k,l}(n)^{1/k^n}$ tend to $C$ as $n \\to \\infty$.","title":"The normalized G\u00f6bel roots converge to a limit greater than 1"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-error-is-big-o","kind":"node","lean":"MetaMathlibExt.gobelSequence_error_isBigO","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The G\u00f6bel error term decays faster than every power of "],["math","\\(n\\)"]],"source":"The G\u00f6bel error term decays faster than every power of $n$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-error-is-big-o","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k, l \\ge 2$ be natural numbers, let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence, and let $C$ be a real number such that the real numbers $C_{k,l}(n) = g_{k,l}(n)^{1/k^n}$ tend to $C$ as $n \\to \\infty$. Then for every real number $R > 0$, as $n \\to \\infty$,","title":"The G\u00f6bel error term decays faster than every power of $n$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-error-eq-tsum-le","kind":"node","lean":"MetaMathlibExt.gobelSequence_error_eq_tsum_le","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-error-eq-tsum-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k, l \\ge 2$ be natural numbers, let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence, and let $C$ be a real number such that the real numbers $C_{k,l}(n) = g_{k,l}(n)^{1/k^n}$ tend to $C$ as $n \\to \\infty$. Then for every natural number $n \\ge 1$ the error","title":"Series formula and bound for the G\u00f6bel error"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/somos-tsum-asymptotic-expansion","kind":"node","lean":"MetaMathlibExt.somos_tsum_asymptotic_expansion","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#somos-tsum-asymptotic-expansion","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number. For $j \\ge 1$ put $S_j = \\sum_{i \\ge 1} i^j / k^i$ and $b_j = (-1)^{j-1} S_j / j$, and for $r \\ge 0$ let","title":"Sondow\u2013Hadjicostas asymptotic expansion"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/rat-residue","kind":"node","lean":"MetaMathlibExt.ratResidue","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#rat-residue","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For a natural number $m$ and a rational number $x = a/b$ in lowest terms with $b > 0$, the residue of $x$ modulo $m$ is the element $a \\cdot b^{-1}$ of $\\mathbb{Z}/m\\mathbb{Z}$. When $b$ is prime to $m$, $b^{-1}$ is the inverse of $b$ modulo $m$, so the residue is the class $c$ with $b c \\equiv a \\pmod m$; for other $x$ the inverse, and hence the residue, is a conventional value with no arithmetic meaning.","title":"Residue of a rational number modulo an integer"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/rat-residue-int-cast","kind":"node","lean":"MetaMathlibExt.ratResidue_intCast","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#rat-residue-int-cast","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For every natural number $m$ and every integer $z$, the residue modulo $m$ of $z$, viewed as a rational number, is the image of $z$ in $\\mathbb{Z}/m\\mathbb{Z}$.","title":"Residue of an integer"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-residues-singleton-or-inj","kind":"node","lean":"MetaMathlibExt.gobelSequence_residues_singleton_or_injective","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residues of the "],["math","\\((k, l)\\)"],["text","-G\u00f6bel sequence modulo "],["math","\\(p^r\\)"]],"source":"Residues of the $(k, l)$-G\u00f6bel sequence modulo $p^r$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-residues-singleton-or-inj","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime, let $k, l, r \\ge 2$ be natural numbers, and let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence. Let $I$ be the set of natural numbers $n$ with $1 \\le n < p$ such that $g_{k,l}(n) = 0$ or $\\nu_p(g_{k,l}(n)) \\ge r - 1$, where $\\nu_p$ is the $p$-adic valuation on $\\mathbb{Q}$; that is, $g_{k,l}(n) \\equiv 0 \\pmod{p^{r-1}}$ in $\\mathbb{Z}_{(p)}$. Let $G$ be the set of residues of the numbers $g_{k,l}(n)$,\u2026","title":"Residues of the $(k, l)$-G\u00f6bel sequence modulo $p^r$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-mul-index-congr","kind":"node","lean":"MetaMathlibExt.gobelSequence_mul_index_congr","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Propagation of a congruence along the "],["math","\\((k, l)\\)"],["text","-G\u00f6bel sequence"]],"source":"Propagation of a congruence along the $(k, l)$-G\u00f6bel sequence"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-mul-index-congr","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime, let $k, l, r \\ge 2$ be natural numbers, let $g_{k,l}$ be the $(k, l)$-G\u00f6bel sequence, and let $a, b$ be natural numbers with $1 \\le a < p$ and $b < p$. For rational numbers $x, y$ write $x \\equiv y \\pmod{p^r}$ when $x = y$ or $\\nu_p(x - y) \\ge r$, where $\\nu_p$ is the $p$-adic valuation on $\\mathbb{Q}$. If $g_{k,l}(a) \\equiv b p^{r-1} \\pmod{p^r}$, then\u2026","title":"Propagation of a congruence along the $(k, l)$-G\u00f6bel sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-walk","kind":"node","lean":"MetaMathlibExt.gobelResidueWalk","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The residue walk "],["math","\\(\\tilde g_l\\)"]],"source":"The residue walk $\\tilde g_l$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-walk","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For natural numbers $p$ and $l$, write $\\left(\\frac{a}{p}\\right)$ for the Jacobi symbol. The residue walk $\\tilde g_l \\colon \\mathbb{N} \\to \\mathbb{N}$ of $p$ starting at $l$ is defined by $\\tilde g_l(0) = 0$, $\\tilde g_l(1) = l$ and, for every $n \\ge 1$, $\\tilde g_l(n+1) = \\tilde g_l(n) + \\left(\\frac{n}{p}\\right)\\left(\\frac{\\tilde g_l(n)}{p}\\right)$ if $0 < \\tilde g_l(n) < p$, and $\\tilde g_l(n+1) = \\tilde g_l(n)$\u2026","title":"The residue walk $\\tilde g_l$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-walk-one","kind":"node","lean":"MetaMathlibExt.gobelResidueWalk_one","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-walk-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For all natural numbers $p$ and $l$, the residue walk $\\tilde g_l$ of $p$ starting at $l$ satisfies $\\tilde g_l(1) = l$.","title":"Initial value of the residue walk"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-walk-succ-of-between","kind":"node","lean":"MetaMathlibExt.gobelResidueWalk_succ_of_between","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-walk-succ-of-between","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"Let $p$, $l$ and $n$ be natural numbers with $n \\ge 1$, write $\\tilde g_l$ for the residue walk of $p$ starting at $l$, and write $\\left(\\frac{a}{p}\\right)$ for the Jacobi symbol. If $0 < \\tilde g_l(n) < p$, then","title":"Recurrence of the residue walk at interior values"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-walk-succ-of-eq-zero","kind":"node","lean":"MetaMathlibExt.gobelResidueWalk_succ_of_eq_zero","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The residue walk stays at "],["math","\\(0\\)"]],"source":"The residue walk stays at $0$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-walk-succ-of-eq-zero","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"Let $p$, $l$ and $n$ be natural numbers with $n \\ge 1$, and write $\\tilde g_l$ for the residue walk of $p$ starting at $l$. If $\\tilde g_l(n) = 0$, then $\\tilde g_l(n+1) = 0$.","title":"The residue walk stays at $0$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-walk-succ-of-eq-prime","kind":"node","lean":"MetaMathlibExt.gobelResidueWalk_succ_of_eq_prime","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The residue walk stays at "],["math","\\(p\\)"]],"source":"The residue walk stays at $p$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-walk-succ-of-eq-prime","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"Let $p$, $l$ and $n$ be natural numbers with $n \\ge 1$, and write $\\tilde g_l$ for the residue walk of $p$ starting at $l$. If $\\tilde g_l(n) = p$, then $\\tilde g_l(n+1) = p$.","title":"The residue walk stays at $p$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/is-zigzag-below","kind":"node","lean":"MetaMathlibExt.IsZigzagBelow","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#is-zigzag-below","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For a natural number $p$ and functions $a, b : \\mathbb{N} \\to \\mathbb{Z}$, $a$ lies zigzag-below $b$ up to $p$, written $a \\preceq b$, when there is a natural number $m \\le p$ such that $a(n) < b(n)$ for all $1 \\le n \\le m$ and $a(n) = b(n)$ for all $m < n \\le p$.","title":"Zigzag order on integer sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/is-gobel-asequence","kind":"node","lean":"MetaMathlibExt.IsGobelASequence","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sign sequence "],["math","\\(a_{p,l}\\)"]],"source":"The sign sequence $a_{p,l}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#is-gobel-asequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For natural numbers $p$ and $l$, a function $a : \\mathbb{N} \\to \\mathbb{Z}$ is an $a_{p,l}$-sequence when all of the following hold: $a(n) \\in \\{1, -1\\}$ for $1 \\le n \\le p-1$; $a(n) = 0$ for $n = 0$ and for every $n \\ge p$; $a(1) = 1$; $a(n) = a(p-n)$ for $1 \\le n \\le \\lfloor (p-1)/2 \\rfloor$; $a(n) = -a(l+1-n)$ for $1 \\le n \\le \\lfloor l/2 \\rfloor$; and $a(n) = a(l+1+n)$ for $1 \\le n \\le p-l-2$.","title":"The sign sequence $a_{p,l}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/is-gobel-bsequence","kind":"node","lean":"MetaMathlibExt.IsGobelBSequence","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The periodic sign sequence "],["math","\\(b_{l,s}\\)"]],"source":"The periodic sign sequence $b_{l,s}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#is-gobel-bsequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For natural numbers $l$ and $s$, a function $b : \\mathbb{Z} \\to \\mathbb{Z}$ is a $b_{l,s}$-sequence when all of the following hold: $b(n) \\in \\{1, -1\\}$ for every integer $n$; $b(n + l + 1) = b(n)$ for every integer $n$; $b(1) = 1$; $b(n) = -b(l+1-n)$ for every integer $n$ not divisible by $l+1$; and $b(s-n) = b(s+1+n)$ for every integer $n$.","title":"The periodic sign sequence $b_{l,s}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/is-gobel-left-boundary","kind":"node","lean":"MetaMathlibExt.IsGobelLeftBoundary","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#is-gobel-left-boundary","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"Fix a natural number $p$ and, for each natural number $l$, let $\\tilde g_l$ denote the G\u00f6bel residue walk for $p$ starting at $l$. A natural number $l_L$ is the left boundary for $p$ when there is an even natural number $l_0 < p$ with $\\tilde g_{l_0}(p) = 0$ and $l_L = l_0 + 2$ such that every even natural number $l < p$ with $\\tilde g_l(p) = 0$ satisfies $l \\le l_0$. In other words, the set of even $l < p$ with\u2026","title":"Left boundary of the even G\u00f6bel residue walks"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/is-gobel-right-boundary","kind":"node","lean":"MetaMathlibExt.IsGobelRightBoundary","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#is-gobel-right-boundary","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"Fix a natural number $p$ and, for each natural number $l$, let $\\tilde g_l$ denote the G\u00f6bel residue walk for $p$ starting at $l$. A natural number $l_R$ is the right boundary for $p$ when $l_R$ is even, $l_R < p$, $\\tilde g_{l_R}(p) = p$, and every even natural number $l < p$ with $\\tilde g_l(p) = p$ satisfies $l_R \\le l$. In other words, $l_R$ is the least even $l < p$ with $\\tilde g_l(p) = p$.","title":"Right boundary of the even G\u00f6bel residue walks"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-half-pred-mem-localizatio","kind":"node","lean":"MetaMathlibExt.gobelSequence_half_pred_mem_localization_of_three_mod_four","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","G\u00f6bel sequences at index "],["math","\\(p\\)"],["text"," for primes "],["math","\\(p \\equiv 3 \\pmod 4\\)"]],"source":"G\u00f6bel sequences at index $p$ for primes $p \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-half-pred-mem-localizatio","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $g_{k,l}$ denote the $(k, l)$-G\u00f6bel sequence and let $p$ be a prime with $p \\equiv 3 \\pmod 4$. Then for every natural number $l < p$, the $p$-adic valuation of $g_{(p-1)/2,\\,l}(p)$ is nonnegative (the valuation of $0$ is taken to be $0$); that is, $g_{(p-1)/2,\\,l}(p)$ lies in the localization $\\mathbb{Z}_{(p)}$.","title":"G\u00f6bel sequences at index $p$ for primes $p \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-half-pred-mem-localizatio-2","kind":"node","lean":"MetaMathlibExt.gobelSequence_half_pred_mem_localization_of_one_mod_four_of_odd","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd starting values give a "],["math","\\(p\\)"],["text","-integral "],["math","\\(p\\)"],["text","-th term when "],["math","\\(p \\equiv 1 \\pmod 4\\)"]],"source":"Odd starting values give a $p$-integral $p$-th term when $p \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-half-pred-mem-localizatio-2","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, write $k = (p-1)/2$, and let $g_{k,l}$ denote the $(k,l)$-G\u00f6bel sequence. Then for every odd natural number $l < p$, the term $g_{k,l}(p)$ lies in the localization $\\mathbb{Z}_{(p)}$, that is, its $p$-adic valuation is nonnegative.","title":"Odd starting values give a $p$-integral $p$-th term when $p \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-half-pred-unique-even-loc","kind":"node","lean":"MetaMathlibExt.gobelSequence_half_pred_unique_even_localization_decomposition","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unique boundary pair for G\u00f6bel sequences when "],["math","\\(p \\equiv 1 \\pmod 4\\)"]],"source":"Unique boundary pair for G\u00f6bel sequences when $p \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-half-pred-unique-even-loc","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $g_{k,l}$ denote the $(k, l)$-G\u00f6bel sequence and let $p \\ge 13$ be a prime with $p \\equiv 1 \\pmod 4$. Then there is exactly one pair $(l_L, l_R)$ of even natural numbers with $2 \\le l_L < l_R \\le p-1$ such that, for every even natural number $l < p$, the $p$-adic valuation of $g_{(p-1)/2,\\,l}(p)$ is nonnegative (the valuation of $0$ is taken to be $0$) if and only if $l < l_L$ or $l_R \\le l$.","title":"Unique boundary pair for G\u00f6bel sequences when $p \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-nonintegral-congruence-cl","kind":"node","lean":"MetaMathlibExt.gobelSequence_nonintegral_congruence_class_of_odd_multiple","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-nonintegral-congruence-cl","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $g_{k,l}$ denote the $(k, l)$-G\u00f6bel sequence, let $p \\ge 13$ be a prime with $p \\equiv 1 \\pmod 4$, and let $k = \\frac{p-1}{2} q$ for a positive odd natural number $q$. Then there is an even natural number $l$ with $2 \\le l \\le p-3$ such that for every positive natural number $l'$ with $l' \\equiv l \\pmod p$, the $p$-adic valuation of $g_{k,l'}(p)$ is negative, and some $g_{k,l'}(n)$ with $1 \\le n \\le p$ is not\u2026","title":"A congruence class of non-integral G\u00f6bel sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/odd-multiple-prime-half-sets-not-finitel","kind":"node","lean":"MetaMathlibExt.odd_multiple_prime_half_sets_not_finitely_coverable","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odd multiples of "],["math","\\((p-1)/2\\)"],["text"," admit no finite subcover"]],"source":"The odd multiples of $(p-1)/2$ admit no finite subcover"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#odd-multiple-prime-half-sets-not-finitel","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"For a prime $p$ write $A_p = \\{\\tfrac{p-1}{2}(2m+1) : m \\in \\mathbb{N}\\}$ for the set of odd multiples of $(p-1)/2$. For every finite set $S$ of primes $q$ with $q \\ge 13$ and $q \\equiv 1 \\pmod 4$, there is a natural number $k$ that lies in $A_p$ for some prime $p \\ge 13$ with $p \\equiv 1 \\pmod 4$ but lies in no $A_q$ with $q \\in S$.","title":"The odd multiples of $(p-1)/2$ admit no finite subcover"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-half-pred-congruent-resid","kind":"node","lean":"MetaMathlibExt.gobelSequence_half_pred_congruent_residue_walk","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Index-scaled G\u00f6bel terms with "],["math","\\(k = (p-1)/2\\)"],["text"," are congruent to the residue walk"]],"source":"Index-scaled G\u00f6bel terms with $k = (p-1)/2$ are congruent to the residue walk"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-half-pred-congruent-resid","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be an odd prime and $l \\le p - 1$ a natural number. Write $k = (p-1)/2$, let $g_{k,l}$ be the $(k,l)$-G\u00f6bel sequence, and let $\\tilde g_l$ be the residue walk of $p$ starting at $l$. Then (a) for every $1 \\le n \\le p$, the rational number $n\\,g_{k,l}(n) - \\tilde g_l(n)$ is either $0$ or has $p$-adic valuation at least $1$, that is, $n\\,g_{k,l}(n) \\equiv \\tilde g_l(n) \\pmod{p}$ in $\\mathbb{Z}_{(p)}$; and (b)\u2026","title":"Index-scaled G\u00f6bel terms with $k = (p-1)/2$ are congruent to the residue walk"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-walk-eq-index-of-eq-index","kind":"node","lean":"MetaMathlibExt.gobelResidueWalk_eq_index_of_eq_index","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-walk-eq-index-of-eq-index","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be an odd prime, let $l \\le p - 1$ be a natural number, and let $\\tilde g_l$ be the residue walk of $l$ for $p$. If $1 \\le m \\le p$ and $\\tilde g_l(m) = m$, then $\\tilde g_l(n) = n$ for every $n$ with $m \\le n \\le p$.","title":"The residue walk stays on the diagonal"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/diagonal-is-zigzag-below-gobel-residue-w","kind":"node","lean":"MetaMathlibExt.diagonal_isZigzagBelow_gobelResidueWalk_of_odd","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#diagonal-is-zigzag-below-gobel-residue-w","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be an odd prime, let $l$ be an odd natural number with $l < p$, and let $\\tilde g_l$ be the residue walk of $l$ for $p$. Then the diagonal $n \\mapsto n$ is zigzag-below $\\tilde g_l$ up to $p$: there is $m \\le p$ such that $n < \\tilde g_l(n)$ for all $1 \\le n \\le m$ and $\\tilde g_l(n) = n$ for all $m < n \\le p$.","title":"The diagonal is zigzag-below every odd residue walk"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-walk-is-zigzag-below-antid","kind":"node","lean":"MetaMathlibExt.gobelResidueWalk_isZigzagBelow_antidiagonal_of_three_mod_four","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even residue walks lie zigzag-below the antidiagonal when "],["math","\\(p \\equiv 3 \\pmod 4\\)"]],"source":"Even residue walks lie zigzag-below the antidiagonal when $p \\equiv 3 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-walk-is-zigzag-below-antid","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 3 \\pmod 4$, let $l < p$ be an even natural number, and write $\\tilde g_l$ for the residue walk of $p$ starting at $l$. Then $\\tilde g_l$ is zigzag-below the function $n \\mapsto p - n$ up to $p$: there is an $m \\le p$ such that $\\tilde g_l(n) < p - n$ for $1 \\le n \\le m$ and $\\tilde g_l(n) = p - n$ for $m < n \\le p$.","title":"Even residue walks lie zigzag-below the antidiagonal when $p \\equiv 3 \\pmod 4$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-walk-even-chain-of-one-mod","kind":"node","lean":"MetaMathlibExt.gobelResidueWalk_even_chain_of_one_mod_four","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Consecutive even residue walks form a zigzag chain when "],["math","\\(p \\equiv 1 \\pmod 4\\)"]],"source":"Consecutive even residue walks form a zigzag chain when $p \\equiv 1 \\pmod 4$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-walk-even-chain-of-one-mod","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$, and for a natural number $l$ write $\\tilde g_l$ for the residue walk of $p$ starting at $l$. Then for every even natural number $l$ with $l + 2 \\le p - 1$, the walk $\\tilde g_l$ is zigzag-below $\\tilde g_{l+2}$ up to $p$: there is an $m \\le p$ such that $\\tilde g_l(n) < \\tilde g_{l+2}(n)$ for $1 \\le n \\le m$ and $\\tilde g_l(n) = \\tilde g_{l+2}(n)$ for $m < n \\le p$.\u2026","title":"Consecutive even residue walks form a zigzag chain when $p \\equiv 1 \\pmod 4$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-boundaries-eq-iff-jacobi-symmetrie","kind":"node","lean":"MetaMathlibExt.gobel_boundaries_eq_iff_jacobi_symmetries","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-boundaries-eq-iff-jacobi-symmetrie","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$ and, for each $l$, let $\\tilde g_l$ be the residue walk of $l$ for $p$. Suppose that $l_0$ is the largest even $l < p$ with $\\tilde g_l(p) = 0$, put $l_L = l_0 + 2$, and suppose that $l_R$ is the smallest even $l < p$ with $\\tilde g_l(p) = p$. Then $l_L = l_R$ if and only if there is an even natural number $l \\le p - 3$ such that\u2026","title":"Equality of the left and right boundaries via Legendre symbols"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/arithmetic-billiards-rotated-rectangle","kind":"node","lean":"MetaMathlibExt.arithmetic_billiards_rotated_rectangle","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#arithmetic-billiards-rotated-rectangle","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$ and let $l$ be an even natural number with $l + 3 \\le p$. Put $m = l + 1$ if $2l + 5 \\le p$ and $m = p - (l + 1)$ otherwise, and let $R \\subseteq \\mathbb{R}^2$ be the convex hull of the four points $(0, m)$, $(m, 0)$, $(p/2, p/2 - m)$ and $(p/2 - m, p/2)$. Let $L$ be the set of integer points on the boundary of $R$ other than $(0, m)$ and $(m, 0)$. Call a sequence\u2026","title":"Arithmetic billiards in a tilted rectangle"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/exists-unique-gobel-asequence","kind":"node","lean":"MetaMathlibExt.existsUnique_gobelASequence","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Existence and uniqueness of the sequence "],["math","\\(a_{p,l}\\)"]],"source":"Existence and uniqueness of the sequence $a_{p,l}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#exists-unique-gobel-asequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod 4$ and let $l$ be an even natural number with $l \\le p - 3$. Then exactly one function $a : \\mathbb{N} \\to \\mathbb{Z}$ satisfies the defining conditions of $a_{p,l}$: $a(n) = \\pm 1$ for $1 \\le n \\le p - 1$, $a(n) = 0$ for $n = 0$ and for $n \\ge p$, $a(1) = 1$, $a(n) = a(p - n)$ for $1 \\le n \\le (p - 1)/2$, $a(n) = -a(l + 1 - n)$ for $1 \\le n \\le l/2$, and\u2026","title":"Existence and uniqueness of the sequence $a_{p,l}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/exists-unique-gobel-bsequence","kind":"node","lean":"MetaMathlibExt.existsUnique_gobelBSequence","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Existence and uniqueness of the sequence "],["math","\\(b_{l,s}\\)"]],"source":"Existence and uniqueness of the sequence $b_{l,s}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#exists-unique-gobel-bsequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $l$ be an even natural number and $s$ a natural number with $s \\le l$ such that $2s + 1$ and $l + 1$ are coprime. Then exactly one function $b : \\mathbb{Z} \\to \\mathbb{Z}$ satisfies the defining conditions of $b_{l,s}$: $b(n) = \\pm 1$ for every integer $n$, $b(n + l + 1) = b(n)$ for every $n$, $b(1) = 1$, $b(n) = -b(l + 1 - n)$ for every $n$ not divisible by $l + 1$, and $b(s - n) = b(s + 1 + n)$ for every $n$.","title":"Existence and uniqueness of the sequence $b_{l,s}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-bsequence-complement","kind":"node","lean":"MetaMathlibExt.gobelBSequence_complement","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Complementary sequences "],["math","\\(b_{l,s}\\)"],["text"," and "],["math","\\(b_{l,l-s}\\)"]],"source":"Complementary sequences $b_{l,s}$ and $b_{l,l-s}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-bsequence-complement","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $l \\ge 2$ be an even natural number and $s \\le l$ a natural number such that $2s + 1$ and $l + 1$ are coprime. Let $b_1, b_2 : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy the defining conditions of $b_{l,s}$ and of $b_{l,l-s}$ respectively. Then for every integer $n$, $b_1(n) = b_2(n)$ if $l + 1$ does not divide $n$, and $b_1(n) = -b_2(n)$ if $l + 1$ divides $n$.","title":"Complementary sequences $b_{l,s}$ and $b_{l,l-s}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-bsequence-translation-identities","kind":"node","lean":"MetaMathlibExt.gobelBSequence_translation_identities","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation and reflection identities for "],["math","\\(b_{l,s}\\)"]],"source":"Translation and reflection identities for $b_{l,s}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-bsequence-translation-identities","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $l \\ge 2$ be an even natural number and $s \\le l$ a natural number such that $2s + 1$ and $l + 1$ are coprime, let $b : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy the defining conditions of $b_{l,s}$, and write $t = l/2 - s$. Then the following hold. (a) For every integer $n$ such that neither $n$ nor $n + 2s + 1$ is divisible by $l + 1$, $b(n) = -b(n + 2s + 1)$. (b) If $s < l/2$, then for every integer $n$ such that\u2026","title":"Translation and reflection identities for $b_{l,s}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-asequence-not-multiplicative-at-ev","kind":"node","lean":"MetaMathlibExt.gobelASequence_not_multiplicative_at_even_index","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sequence "],["math","\\(a_{p,l}\\)"],["text"," is not multiplicative at an even index"]],"source":"The sequence $a_{p,l}$ is not multiplicative at an even index"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-asequence-not-multiplicative-at-ev","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p \\ge 13$ be a prime with $p \\equiv 1 \\pmod 4$, let $l$ be an even natural number with $2 \\le l \\le p - 3$, and let $a : \\mathbb{N} \\to \\mathbb{Z}$ satisfy the defining conditions of $a_{p,l}$. Then there is a positive integer $m$ with $2m \\le p - 3$ such that $a(2m) \\ne a(2)\\,a(m)$.","title":"The sequence $a_{p,l}$ is not multiplicative at an even index"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-first-nonintegral-2600725","kind":"node","lean":"MetaMathlibExt.gobelSequence_first_nonintegral_2600725","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first non-integral term of "],["math","\\(g_{2600725,2}\\)"],["text"," is at index "],["math","\\(9011\\)"]],"source":"The first non-integral term of $g_{2600725,2}$ is at index $9011$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-first-nonintegral-2600725","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $g_{k,l}$ denote the $(k,l)$-G\u00f6bel sequence. The rational number $g_{2600725,2}(9011)$ is not an integer, while $g_{2600725,2}(n)$ is an integer for every natural number $n$ with $1 \\le n < 9011$. Writing $N_k$ for the first index at which $g_{k,2}$ is not an integer, as in the paper, this says $N_{2600725} = 9011$.","title":"The first non-integral term of $g_{2600725,2}$ is at index $9011$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-nonintegral-index-le-9011","kind":"node","lean":"MetaMathlibExt.gobelSequence_nonintegral_index_le_9011","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","A non-integral term of "],["math","\\(g_{k,2}\\)"],["text"," by index "],["math","\\(9011\\)"],["text"," for "],["math","\\(2 \\le k \\le 10^7\\)"]],"source":"A non-integral term of $g_{k,2}$ by index $9011$ for $2 \\le k \\le 10^7$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-nonintegral-index-le-9011","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $g_{k,l}$ denote the $(k, l)$-G\u00f6bel sequence. For every natural number $k$ with $2 \\le k \\le 10^7$, there is a natural number $n$ with $1 \\le n \\le 9011$ such that the rational number $g_{k,2}(n)$ is not an integer.","title":"A non-integral term of $g_{k,2}$ by index $9011$ for $2 \\le k \\le 10^7$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/card-gobel-sequence-prime-first-noninteg","kind":"node","lean":"MetaMathlibExt.card_gobelSequence_prime_first_nonintegral_index","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting prime first nonintegral indices of "],["math","\\(g_{k,2}\\)"],["text"," for "],["math","\\(k \\le 10^7\\)"]],"source":"Counting prime first nonintegral indices of $g_{k,2}$ for $k \\le 10^7$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#card-gobel-sequence-prime-first-noninteg","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Exactly $8649270$ natural numbers $k$ with $2 \\le k \\le 10^7$ have the following property: for the $(k, 2)$-G\u00f6bel sequence $g_{k,2}$ there is a prime $q$ such that $g_{k,2}(q)$ is not an integer and $g_{k,2}(n)$ is an integer for every $1 \\le n < q$.","title":"Counting prime first nonintegral indices of $g_{k,2}$ for $k \\le 10^7$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-sequence-nonintegral-index-le-2936","kind":"node","lean":"MetaMathlibExt.gobelSequence_nonintegral_index_le_29363","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","A non-integral term of "],["math","\\(g_{k,2}\\)"],["text"," by index "],["math","\\(29363\\)"],["text"," for "],["math","\\(2 \\le k \\le 10^{14}\\)"]],"source":"A non-integral term of $g_{k,2}$ by index $29363$ for $2 \\le k \\le 10^{14}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-sequence-nonintegral-index-le-2936","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $g_{k,l}$ denote the $(k, l)$-G\u00f6bel sequence. For every natural number $k$ with $2 \\le k \\le 10^{14}$, there is a natural number $n$ with $1 \\le n \\le 29363$ such that the rational number $g_{k,2}(n)$ is not an integer.","title":"A non-integral term of $g_{k,2}$ by index $29363$ for $2 \\le k \\le 10^{14}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue","kind":"node","lean":"MetaMathlibExt.gobelResidue","parent":12957,"reading_title":{"format":"inline-parts-v1","parts":[["text","Residue sequence of the "],["math","\\(k\\)"],["text","-G\u00f6bel sequence"]],"source":"Residue sequence of the $k$-G\u00f6bel sequence"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"Fix natural numbers $k$, $p$, $r$ and write $\\nu_p$ for the $p$-adic valuation. The residue sequence $g_{k,p,r}(n)$, meant for $n \\ge 1$ with $\\nu_p(n!) \\le r$, takes values in $\\mathbb{Z}/p^{r - \\nu_p(n!)}\\mathbb{Z} \\cup \\{F\\}$, where $F$ is a failure value; a residue class is represented by its least nonnegative representative. It is defined by $g_{k,p,r}(0) = F$, $g_{k,p,r}(1) = 2 \\bmod p^r$ and, for $n \\ge 2$,\u2026","title":"Residue sequence of the $k$-G\u00f6bel sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-one","kind":"node","lean":"MetaMathlibExt.gobelResidue_one","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For all natural numbers $k$, $p$, $r$, the residue sequence satisfies $g_{k,p,r}(1) = 2 \\bmod p^r$, the residue class of $2$ modulo $p^r$ (represented by the remainder of $2$ on division by $p^r$).","title":"Initial value of the G\u00f6bel residue sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-succ-of-eq-none","kind":"node","lean":"MetaMathlibExt.gobelResidue_succ_of_eq_none","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-succ-of-eq-none","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For all natural numbers $k$, $p$, $r$ and every $n \\ge 1$, if the residue sequence takes the failure value at $n$, that is $g_{k,p,r}(n) = F$, then $g_{k,p,r}(n + 1) = F$.","title":"Failure of the G\u00f6bel residue sequence persists"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-two-three-one-two","kind":"node","lean":"MetaMathlibExt.gobelResidue_two_three_one_two","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-two-three-one-two","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"fully_proved","summary":"For $k = 2$, $p = 3$ and $r = 1$, the residue sequence satisfies $g_{2,3,1}(2) = 0 \\bmod 3$; in particular it is not the failure value $F$. Indeed $g_{2,3,1}(1) = 2$ and $(1 \\cdot 2 + 2^2) \\cdot 2^{-1} = 6 \\cdot 2^{-1} \\equiv 0 \\pmod 3$.","title":"Example value of the G\u00f6bel residue sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-eq-none-iff","kind":"node","lean":"MetaMathlibExt.gobelResidue_eq_none_iff","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-eq-none-iff","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $k \\ge 2$ and $r \\ge 1$ be integers, let $p$ be a prime, write $\\nu_p$ for the $p$-adic valuation, and let $n \\ge 1$ be an integer with $\\nu_p(n!) \\le r$. Then the residue sequence $g_{k,p,r}$ takes the failure value $F$ at $n$, that is $g_{k,p,r}(n) = F$, if and only if the value $g_{k,2}(n)$ of the $(k, 2)$-G\u00f6bel sequence has negative $p$-adic valuation, that is $g_{k,2}(n) \\notin \\mathbb{Z}_{(p)}$.","title":"Failure of the residue sequence detects non-integrality"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/gobel-residue-eq-of-mod-eq-totient","kind":"node","lean":"MetaMathlibExt.gobelResidue_eq_of_modEq_totient","parent":12957,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences/index.html#gobel-residue-eq-of-mod-eq-totient","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/g-bel-sequences","status":"can_prove","summary":"Let $p$ be a prime, $r \\ge 1$, and $k$, $l$ integers with $r + 1 \\le k \\le l$ and $k \\equiv l \\pmod{\\varphi(p^r)}$, where $\\varphi$ is Euler's totient function. Write $\\nu_p$ for the $p$-adic valuation. Then every integer $n \\ge 1$ with $\\nu_p(n!) \\le r$ satisfies $g_{k,p,r}(n) = g_{l,p,r}(n)$, where both sides are values of the residue sequence (so either both are the failure value $F$ or both are the same residue\u2026","title":"Periodicity of the G\u00f6bel residue sequence in the exponent"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","title":"Integer complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/expr","kind":"node","lean":"IntegerComplexity.Expr","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#expr","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"An expression is either the constant $1$, or a sum $e_1 + e_2$ of two expressions, or a product $e_1 \\times e_2$ of two expressions. Expressions are formal syntax trees, so the tree structure plays the role of parentheses.","title":"Expressions built from one, addition and multiplication"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/eval","kind":"node","lean":"IntegerComplexity.Expr.eval","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#eval","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"The value of an expression is the natural number obtained by evaluating it: the constant $1$ has value $1$, the value of $e_1 + e_2$ is the sum of the values of $e_1$ and $e_2$, and the value of $e_1 \\times e_2$ is the product of their values.","title":"Value of an expression"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/count","kind":"node","lean":"IntegerComplexity.Expr.count","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#count","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"The number of ones of an expression is the number of occurrences of the constant $1$ in it: the constant $1$ has one, and both $e_1 + e_2$ and $e_1 \\times e_2$ have as many as $e_1$ and $e_2$ together.","title":"Number of ones in an expression"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/exists-expr-of-pos","kind":"node","lean":"IntegerComplexity.exists_expr_of_pos","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#exists-expr-of-pos","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For every natural number $n > 0$ there is an expression whose value is $n$.","title":"Every positive integer is the value of an expression"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/pred","kind":"node","lean":"IntegerComplexity.Pred","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#pred","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For a positive integer $n$ and a natural number $c$, the predicate holds when some expression has value $n$ and exactly $c$ ones.","title":"Achievable numbers of ones"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity","kind":"node","lean":"IntegerComplexity.complexity","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"The integer complexity of a positive integer $n$ is the least number of ones in an expression whose value is $n$, that is, the least number of ones needed to write $n$ using $1$, $+$, $\\times$ and parentheses.","title":"Integer complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-le","kind":"node","lean":"IntegerComplexity.complexity_le","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"Let $n$ be a positive integer. If an expression $e$ has value $n$, then the integer complexity of $n$ is at most the number of ones of $e$.","title":"Expressions bound the integer complexity from above"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-add-le","kind":"node","lean":"IntegerComplexity.complexity_add_le","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-add-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For all positive integers $a$ and $b$, the integer complexity of $a + b$ is at most the sum of the integer complexities of $a$ and $b$.","title":"Integer complexity of a sum"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-mul-le","kind":"node","lean":"IntegerComplexity.complexity_mul_le","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-mul-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For all positive integers $a$ and $b$, the integer complexity of $ab$ is at most the sum of the integer complexities of $a$ and $b$.","title":"Integer complexity of a product"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/cx","kind":"node","lean":"MathlibExt.NumberTheory.IntegerComplexityWanted.cx","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#cx","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For a natural number $n$, the value $\\lVert n \\rVert$ is the integer complexity of $n$ when $n$ is positive, that is, the least number of ones needed to write $n$ using $1$, $+$, $\\times$ and parentheses, and $\\lVert 0 \\rVert = 0$.","title":"Integer complexity extended to all natural numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-lower-bound-g","kind":"node","lean":"MathlibExt.NumberTheory.IntegerComplexityWanted.complexityLowerBoundG","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-lower-bound-g","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For a natural number $n$, let $a$ be the largest natural number with $3^a \\le n$ when $n \\ge 1$, and let $a = 0$ when $n = 0$. The function $g$ on the natural numbers is given by $g(n) = 3a$ if $n < \\frac{4}{3} \\cdot 3^a$, $g(n) = 3a + 1$ if $\\frac{4}{3} \\cdot 3^a \\le n < 2 \\cdot 3^a$, and $g(n) = 3a + 2$ if $n \\ge 2 \\cdot 3^a$.","title":"Lower-bound function for integer complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-eq-min","kind":"node","lean":"MathlibExt.NumberTheory.IntegerComplexityWanted.complexity_eq_min","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-eq-min","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\lVert m \\rVert$ for the integer complexity of a positive integer $m$, the least number of ones needed to write $m$ using $1$, $+$, $\\times$ and parentheses. For every integer $n \\ge 2$, $\\lVert n \\rVert$ is the least element of the set consisting of the numbers $\\lVert d \\rVert + \\lVert n/d \\rVert$ for the divisors $d \\ge 2$ of $n$ and the numbers $\\lVert j \\rVert + \\lVert n - j \\rVert$ for the integers $j$\u2026","title":"Integer complexity as a minimum over factorizations and sums"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/le-complexity-of-subadditive","kind":"node","lean":"MathlibExt.NumberTheory.IntegerComplexityWanted.le_complexity_of_subadditive","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#le-complexity-of-subadditive","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\lVert n \\rVert$ for the integer complexity of a positive integer $n$, the least number of ones needed to write $n$ using $1$, $+$, $\\times$ and parentheses. Let $P$ be a real-valued function on the natural numbers with $P(1) = 1$ such that $P(n + m) \\le P(n) + P(m)$ and $P(nm) \\le P(n) + P(m)$ for all positive integers $n$ and $m$. Then $P(n) \\le \\lVert n \\rVert$ for every positive integer $n$.","title":"A criterion for lower bounds on integer complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/logb-two-add-one-le","kind":"node","lean":"MathlibExt.NumberTheory.IntegerComplexityWanted.logb_two_add_one_le","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#logb-two-add-one-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\lVert n \\rVert$ for the integer complexity of a positive integer $n$, the least number of ones needed to write $n$ using $1$, $+$, $\\times$ and parentheses. For every positive integer $n$, $\\log_2(1 + n) \\le \\lVert n \\rVert$.","title":"Logarithmic lower bound for integer complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-two-pow-mul-three-pow","kind":"node","lean":"MathlibExt.NumberTheory.IntegerComplexityWanted.complexity_two_pow_mul_three_pow","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer complexity of "],["math","\\(2^a 3^b\\)"],["text"," for "],["math","\\(a \\le 2\\)"]],"source":"Integer complexity of $2^a 3^b$ for $a \\le 2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-two-pow-mul-three-pow","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\lVert n \\rVert$ for the integer complexity of a positive integer $n$, the least number of ones needed to write $n$ using $1$, $+$, $\\times$ and parentheses. For all natural numbers $a \\le 2$ and $b$, not both zero, $\\lVert 2^a 3^b \\rVert = 2a + 3b$.","title":"Integer complexity of $2^a 3^b$ for $a \\le 2$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-lt-of-gt-extremal","kind":"node","lean":"MathlibExt.NumberTheory.IntegerComplexityWanted.complexity_lt_of_gt_extremal","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integers above "],["math","\\(2^a 3^b\\)"],["text"," have complexity above "],["math","\\(2a + 3b\\)"],["text"," when "],["math","\\(a \\le 2\\)"],["text"," and "],["math","\\(2a + 3b \\ge 2\\)"]],"source":"Integers above $2^a 3^b$ have complexity above $2a + 3b$ when $a \\le 2$ and $2a + 3b \\ge 2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-lt-of-gt-extremal","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\lVert m \\rVert$ for the integer complexity of a positive integer $m$, the least number of ones needed to write $m$ using $1$, $+$, $\\times$ and parentheses. Let $a \\le 2$ and $b$ be natural numbers with $2a + 3b \\ge 2$. Then every natural number $m > 2^a 3^b$ satisfies $\\lVert m \\rVert > 2a + 3b$.","title":"Integers above $2^a 3^b$ have complexity above $2a + 3b$ when $a \\le 2$ and $2a + 3b \\ge 2$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-lower-bound-g-le","kind":"node","lean":"MathlibExt.NumberTheory.IntegerComplexityWanted.complexityLowerBoundG_le","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function "],["math","\\(g\\)"],["text"," bounds integer complexity from below"]],"source":"The function $g$ bounds integer complexity from below"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-lower-bound-g-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\lVert n \\rVert$ for the integer complexity of a positive integer $n$, the least number of ones needed to write $n$ using $1$, $+$, $\\times$ and parentheses. For a positive integer $n$, let $a$ be the largest natural number with $3^a \\le n$, and let $g(n) = 3a$ if $n < \\frac{4}{3} \\cdot 3^a$, $g(n) = 3a + 1$ if $\\frac{4}{3} \\cdot 3^a \\le n < 2 \\cdot 3^a$, and $g(n) = 3a + 2$ if $n \\ge 2 \\cdot 3^a$. Then\u2026","title":"The function $g$ bounds integer complexity from below"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-expr","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.MahlerPopkenExpr","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-expr","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For a natural number $m$, the $m$-ary Mahler\u2013Popken expressions are the formal expressions built from leaves using the binary operations $+$ and $\\times$; the tree structure plays the role of brackets. Each leaf carries a label $i \\in \\{0, \\dots, m - 1\\}$ (so there are $m$ kinds of leaf), and a leaf with label $i$ stands for the number $i + 1 \\in \\{1, \\dots, m\\}$.","title":"Mahler\u2013Popken expressions"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/eval-2","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.MahlerPopkenExpr.eval","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of an "],["math","\\(m\\)"],["text","-ary expression"]],"source":"Value of an $m$-ary expression"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#eval-2","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"The natural number expressed by an $m$-ary Mahler\u2013Popken expression: the leaf with index $i \\in \\{0, \\dots, m-1\\}$ has value $i + 1$, and the value of a sum or of a product of two expressions is the sum or the product of their values.","title":"Value of an $m$-ary expression"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/count-2","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.MahlerPopkenExpr.count","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of leaves of an "],["math","\\(m\\)"],["text","-ary expression"]],"source":"Number of leaves of an $m$-ary expression"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#count-2","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"The number of leaf selections of an $m$-ary Mahler\u2013Popken expression, counting repetitions: a leaf counts $1$, and the count of a sum or of a product of two expressions is the sum of their counts.","title":"Number of leaves of an $m$-ary expression"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $m$ and $n$, the $m$-ary Mahler\u2013Popken complexity $\\lVert n \\rVert_m$ is the smallest number of leaves (counted with repetition) of an $m$-ary Mahler\u2013Popken expression whose value is $n$: the smallest number of possibly repeated selections from $\\{1, \\dots, m\\}$ needed to express $n$ with addition, multiplication and legal bracketing. If no expression has value $n$ (which happens exactly when\u2026","title":"Mahler\u2013Popken complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/exists-mahler-popken-expr-eval","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.exists_mahlerPopkenExpr_eval","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every positive integer has an "],["math","\\(m\\)"],["text","-ary expression"]],"source":"Every positive integer has an $m$-ary expression"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#exists-mahler-popken-expr-eval","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $m \\ge 1$ and $n \\ge 1$, there is an $m$-ary Mahler\u2013Popken expression whose value is $n$ (for instance a sum of ones).","title":"Every positive integer has an $m$-ary expression"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-spec","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_spec","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-spec","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For all natural numbers $m \\ge 1$ and $n \\ge 1$, there is an $m$-ary Mahler\u2013Popken expression whose value is $n$ and whose number of leaves (counted with repetition) equals the $m$-ary Mahler\u2013Popken complexity $\\lVert n \\rVert_m$.","title":"The Mahler\u2013Popken complexity is attained"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-min-le","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_min_le","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Expressions bound the "],["math","\\(m\\)"],["text","-ary complexity from above"]],"source":"Expressions bound the $m$-ary complexity from above"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-min-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $m$ and $n$ and every $m$-ary Mahler\u2013Popken expression $e$ whose value is $n$, the $m$-ary Mahler\u2013Popken complexity $\\|n\\|_m$ is at most the number of leaves of $e$.","title":"Expressions bound the $m$-ary complexity from above"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-pos","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_pos","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(m\\)"],["text","-ary complexity is positive"]],"source":"The $m$-ary complexity is positive"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-pos","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $m \\ge 1$ and $n \\ge 1$, the $m$-ary Mahler\u2013Popken complexity $\\|n\\|_m$ is positive.","title":"The $m$-ary complexity is positive"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-add-le","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_add_le","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subadditivity of the "],["math","\\(m\\)"],["text","-ary complexity under addition"]],"source":"Subadditivity of the $m$-ary complexity under addition"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-add-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $m \\ge 1$, $a \\ge 1$ and $b \\ge 1$, the $m$-ary Mahler\u2013Popken complexity $\\|\\cdot\\|_m$ satisfies $\\|a + b\\|_m \\le \\|a\\|_m + \\|b\\|_m$.","title":"Subadditivity of the $m$-ary complexity under addition"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-mul-le","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_mul_le","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subadditivity of the "],["math","\\(m\\)"],["text","-ary complexity under multiplication"]],"source":"Subadditivity of the $m$-ary complexity under multiplication"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-mul-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $m \\ge 1$, $a \\ge 1$ and $b \\ge 1$, the $m$-ary Mahler\u2013Popken complexity $\\|\\cdot\\|_m$ satisfies $\\|a b\\|_m \\le \\|a\\|_m + \\|b\\|_m$.","title":"Subadditivity of the $m$-ary complexity under multiplication"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/of-integer-complexity-expr","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.MahlerPopkenExpr.ofIntegerComplexityExpr","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integer-complexity expressions as "],["math","\\(1\\)"],["text","-ary Mahler\u2013Popken expressions"]],"source":"Integer-complexity expressions as $1$-ary Mahler\u2013Popken expressions"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#of-integer-complexity-expr","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"The translation of an expression built from one, addition and multiplication into a $1$-ary Mahler\u2013Popken expression replaces each occurrence of $1$ by the leaf standing for $1$ and keeps every addition and multiplication, so the expression tree is unchanged.","title":"Integer-complexity expressions as $1$-ary Mahler\u2013Popken expressions"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/to-integer-complexity-expr","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.MahlerPopkenExpr.toIntegerComplexityExpr","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(1\\)"],["text","-ary Mahler\u2013Popken expressions as integer-complexity expressions"]],"source":"$1$-ary Mahler\u2013Popken expressions as integer-complexity expressions"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#to-integer-complexity-expr","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"The translation of a $1$-ary Mahler\u2013Popken expression into an expression built from one, addition and multiplication replaces each leaf (the only leaf stands for $1$) by $1$ and keeps every addition and multiplication, so the expression tree is unchanged.","title":"$1$-ary Mahler\u2013Popken expressions as integer-complexity expressions"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-one-eq","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_one_eq","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(1\\)"],["text","-ary complexity is the integer complexity"]],"source":"The $1$-ary complexity is the integer complexity"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-one-eq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the $1$-ary Mahler\u2013Popken complexity $\\|n\\|_1$ equals the classical integer complexity of $n$, the least number of ones needed to write $n$ using $1$, $+$, $\\times$ and parentheses.","title":"The $1$-ary complexity is the integer complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-recursion-value","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenRecursionValue","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-recursion-value","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $m$ and $n$, write $\\lVert k \\rVert_m$ for the $m$-ary Mahler\u2013Popken complexity of $k$. The recursion value of $n$ is the smallest of the sums $\\lVert d \\rVert_m + \\lVert n/d \\rVert_m$ over the divisors $d$ of $n$ with $d \\ne 1$ and $d \\ne n$, and of the sums $\\lVert \\eta \\rVert_m + \\lVert n - \\eta \\rVert_m$ over $0 < \\eta < n$; it is $0$ when there are no such sums. This is the right-hand side\u2026","title":"Right-hand side of the Mahler\u2013Popken recursion"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-eq-recursion-va","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_eq_recursionValue_eventually","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(m\\)"],["text","-ary complexity eventually satisfies its recursion"]],"source":"The $m$-ary complexity eventually satisfies its recursion"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-eq-recursion-va","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"For every natural number $m \\ge 1$ there is $N$ such that for every $n \\ge N$, the $m$-ary Mahler\u2013Popken complexity $\\|n\\|_m$ equals the right-hand side of the recursion (9), that is, the least of the values $\\|d\\|_m + \\|n/d\\|_m$ over divisors $d$ of $n$ with $d \\ne 1$ and $d \\ne n$, and the values $\\|\\eta\\|_m + \\|n - \\eta\\|_m$ over $0 < \\eta < n$.","title":"The $m$-ary complexity eventually satisfies its recursion"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-two-eq-recursio","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_two_eq_recursionValue","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-two-eq-recursio","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"For every natural number $n \\ge 3$, the binary ($2$-ary) Mahler\u2013Popken complexity $\\lVert n \\rVert_2$ equals the minimum of the sums $\\lVert d \\rVert_2 + \\lVert n/d \\rVert_2$ over the divisors $d$ of $n$ with $d \\ne 1$ and $d \\ne n$, and of the sums $\\lVert \\eta \\rVert_2 + \\lVert n - \\eta \\rVert_2$ over $0 < \\eta < n$. This is the recursion (9) for $m = 2$.","title":"Recursion for the binary Mahler\u2013Popken complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/logb-two-le-mahler-popken-complexity-two","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.logb_two_le_mahlerPopkenComplexity_two","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#logb-two-le-mahler-popken-complexity-two","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"For every natural number $n \\ge 1$, the $2$-ary (binary) Mahler\u2013Popken complexity $\\|n\\|_2$ satisfies $\\log_2 n \\le \\|n\\|_2$.","title":"Logarithmic lower bound for the binary Mahler\u2013Popken complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/mahler-popken-complexity-two-le-log-add","kind":"node","lean":"MathlibExt.NumberTheory.MahlerPopkenComplexityWanted.mahlerPopkenComplexity_two_le_log_add_digitCount_sub_one","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#mahler-popken-complexity-two-le-log-add","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"For every natural number $n \\ge 2$, the binary ($2$-ary) Mahler\u2013Popken complexity $\\lVert n \\rVert_2$ is at most $\\lfloor \\log_2 n \\rfloor + \\nu_2(n) - 1$, where $\\nu_2(n)$ is the number of ones in the base-$2$ expansion of $n$.","title":"Upper bound for the binary Mahler\u2013Popken complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/can-build","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.CanBuild","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Numbers built from copies of "],["math","\\(l\\)"]],"source":"Numbers built from copies of $l$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#can-build","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $l$, $k$ and $n$, the number $n$ can be built from $k$ copies of $l$ when $n$ is the value of an expression that uses exactly $k$ copies of $l$, combined by addition and multiplication with arbitrary parentheses. Inductively, $l$ is built from $1$ copy, and if $x$ is built from $a$ copies and $y$ from $b$ copies, then $x + y$ and $x y$ are built from $a + b$ copies.","title":"Numbers built from copies of $l$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(l\\)"],["text","-complexity of a natural number"]],"source":"The $l$-complexity of a natural number"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $l$ and $n$, the $l$-complexity $\\|n\\|_l$ of $n$ is the least number $k$ such that $n$ can be built from $k$ copies of $l$ using addition and multiplication (with arbitrary parentheses), and $\\|n\\|_l = 0$ if $n$ cannot be built from copies of $l$ at all.","title":"The $l$-complexity of a natural number"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/one-le","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.CanBuild.one_le","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#one-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"If $n$ can be obtained from $k$ copies of $l$ using addition and multiplication, then $k \\ge 1$.","title":"Every build uses at least one copy"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-spec","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_spec","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","A buildable number is built from as many copies as its "],["math","\\(l\\)"],["text","-complexity"]],"source":"A buildable number is built from as many copies as its $l$-complexity"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-spec","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $l$ and $n$, if $n$ can be built from some number of copies of $l$ using addition and multiplication, then $n$ can be built in this way from $c$ copies of $l$, where $c$ is the $l$-complexity of $n$.","title":"A buildable number is built from as many copies as its $l$-complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-min-le","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_min_le","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(l\\)"],["text","-complexity is at most the size of any build"]],"source":"The $l$-complexity is at most the size of any build"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-min-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $l$, $k$ and $n$, if $n$ can be built from $k$ copies of $l$ using addition and multiplication, then the $l$-complexity of $n$ is at most $k$.","title":"The $l$-complexity is at most the size of any build"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-eq-zero-iff","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_eq_zero_iff","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(l\\)"],["text","-complexity vanishes exactly on numbers that cannot be built"]],"source":"The $l$-complexity vanishes exactly on numbers that cannot be built"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-eq-zero-iff","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $l$ and $n$, the $l$-complexity of $n$ is $0$ if and only if there is no $k$ such that $n$ can be built from $k$ copies of $l$ using addition and multiplication.","title":"The $l$-complexity vanishes exactly on numbers that cannot be built"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/two-complexity-add-two-form","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.twoComplexityAddTwoForm","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#two-complexity-add-two-form","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"fully_proved","summary":"For natural numbers $m$ and $n$, $n$ has one of the four exceptional forms for $m$ when one of the following holds: (a) $n = 2^m + 2^{m_2} + 2^{m_3}$ for some natural numbers $m > m_2 > m_3 \\ge 1$; (b) $n = 2^m + 2^{m_2} + 2^{m_3} + 2^{m_4}$ for some natural numbers $m > m_2 > m_3 > m_4 \\ge 2$ with $m + m_4 = m_2 + m_3$; (c) $n = 2^m + 2^{m-3} + 2^{m_2} + 2^{m_2 - 1}$ for some natural number $m_2$ with\u2026","title":"Zhang's four exceptional forms"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/can-build-max","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.canBuild_max","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The largest number built from "],["math","\\(m\\)"],["text"," copies of "],["math","\\(l\\)"]],"source":"The largest number built from $m$ copies of $l$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#can-build-max","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Let $l > 1$ and $m \\ge 1$ be natural numbers. Then $l^m$ can be built from exactly $m$ copies of $l$ using addition and multiplication, and every natural number that can be built from $m$ copies of $l$ in this way is at most $l^m$.","title":"The largest number built from $m$ copies of $l$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/logb-le-complexity-l-le","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.logb_le_complexityL_le","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logarithmic bounds for the "],["math","\\(l\\)"],["text","-complexity"]],"source":"Logarithmic bounds for the $l$-complexity"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#logb-le-complexity-l-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_l$ for the $l$-complexity of $n$. Let $l > 1$ be a natural number and let $n$ be a positive multiple of $l$. Then $\\log_l n \\le \\|n\\|_l \\le l \\log_l n - 1$, where $\\log_l$ is the real logarithm to base $l$.","title":"Logarithmic bounds for the $l$-complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-pow-add-pow","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_pow_add_pow","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(l\\)"],["text","-complexity of "],["math","\\(l^m + l^{m'}\\)"]],"source":"The $l$-complexity of $l^m + l^{m'}$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-pow-add-pow","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_l$ for the $l$-complexity of $n$. For natural numbers $l > 1$ and $m \\ge m' \\ge 1$, $\\|l^m + l^{m'}\\|_l = m + 1$.","title":"The $l$-complexity of $l^m + l^{m'}$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-two-pow-and-add","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_two_pow_and_add","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-two-pow-and-add","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the $2$-complexity of $n$. For all natural numbers $m_1$ and $m_2$ with $m_1 > m_2 \\ge 1$, one has $\\|2^{m_1}\\|_2 = m_1$ and $\\|2^{m_1} + 2^{m_2}\\|_2 = m_1 + 1$.","title":"The 2-complexity of a power of two and of a sum of two powers of two"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-sum-two-pow-le","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_sum_two_pow_le","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-sum-two-pow-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the $2$-complexity of $n$. Let $S$ be a nonempty finite set of positive integers with $k$ elements and largest element $m_1$. Then $\\|\\sum_{e \\in S} 2^e\\|_2 \\le m_1 + k - 1$.","title":"An upper bound for the 2-complexity of a sum of distinct powers of two"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/logb-two-bounds","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.logb_two_bounds","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#logb-two-bounds","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the 2-complexity of $n$: the least number of copies of $2$ from which $n$ can be built using addition and multiplication, and $0$ if $n$ cannot be built. For every even natural number $n \\ge 1$, $\\log_2 n \\le \\|n\\|_2 \\le 2 \\log_2 n - 1$, where $\\log_2$ is the real base-2 logarithm.","title":"Logarithmic bounds for the 2-complexity (Zhang's Lemma 3(3))"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/logb-two-add-one-le-2","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.logb_two_add_one_le","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#logb-two-add-one-le-2","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the $2$-complexity of $n$. Let $n$ be a positive even integer that cannot be written as $2^a + 2^b$ with natural numbers $a \\le b$. Then $\\log_2 n + 1 \\le \\|n\\|_2$.","title":"A lower bound for the 2-complexity outside sums of two powers of two"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-decomp","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_decomp","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Adding a small multiple of "],["math","\\(l\\)"],["text"," to a multiple of "],["math","\\(l^2\\)"]],"source":"Adding a small multiple of $l$ to a multiple of $l^2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-decomp","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_l$ for the $l$-complexity of $n$. Let $l > 1$, let $b \\ge 1$ be a multiple of $l^2$, let $a$ be a natural number with $a \\le l - 1$, and let $n = b + a l$. Then $\\|n\\|_l = \\|b\\|_l + a$.","title":"Adding a small multiple of $l$ to a multiple of $l^2$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-eq-add-one-iff","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_eq_add_one_iff","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even "],["math","\\(n\\)"],["text"," with "],["math","\\(2^m < n \\le 2^{m+1}\\)"],["text"," and 2-complexity "],["math","\\(m + 1\\)"]],"source":"Even $n$ with $2^m < n \\le 2^{m+1}$ and 2-complexity $m + 1$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-eq-add-one-iff","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the 2-complexity of $n$. Let $m \\ge 1$ and let $n$ be an even natural number with $2^m < n \\le 2^{m+1}$. Then $\\|n\\|_2 = m + 1$ if and only if $n = 2^m + 2^t$ for some natural number $t$ with $1 \\le t \\le m$.","title":"Even $n$ with $2^m < n \\le 2^{m+1}$ and 2-complexity $m + 1$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-eq-add-two-of-form","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_eq_add_two_of_form","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Four forms of even numbers with 2-complexity "],["math","\\(m + 2\\)"]],"source":"Four forms of even numbers with 2-complexity $m + 2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-eq-add-two-of-form","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the 2-complexity of $n$. Let $m \\ge 3$ and let $n$ be an even natural number with $2^m < n \\le 2^{m+1}$. Suppose that $n$ has one of the following forms, where all exponents are natural numbers: (a) $n = 2^m + 2^{m_2} + 2^{m_3}$ with $m > m_2 > m_3 \\ge 1$; (b) $n = 2^m + 2^{m_2} + 2^{m_3} + 2^{m_4}$ with $m > m_2 > m_3 > m_4 \\ge 2$ and $m + m_4 = m_2 + m_3$; (c)\u2026","title":"Four forms of even numbers with 2-complexity $m + 2$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-eq-add-two-iff","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_eq_add_two_iff","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even numbers in "],["math","\\((2^m, 2^{m+1})\\)"],["text"," with 2-complexity "],["math","\\(m + 2\\)"]],"source":"Even numbers in $(2^m, 2^{m+1})$ with 2-complexity $m + 2$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-eq-add-two-iff","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the 2-complexity of $n$. Let $m \\ge 3$ and let $n$ be an even natural number with $2^m < n \\le 2^{m+1}$. Then $\\|n\\|_2 = m + 2$ if and only if $n$ has one of the following forms, where all exponents are natural numbers: (a) $n = 2^m + 2^{m_2} + 2^{m_3}$ with $m > m_2 > m_3 \\ge 1$; (b) $n = 2^m + 2^{m_2} + 2^{m_3} + 2^{m_4}$ with $m > m_2 > m_3 > m_4 \\ge 2$ and $m + m_4 = m_2 + m_3$; (c)\u2026","title":"Even numbers in $(2^m, 2^{m+1})$ with 2-complexity $m + 2$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/logb-two-add-two-le","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.logb_two_add_two_le","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound "],["math","\\(\\log_2 n + 2\\)"],["text"," for the 2-complexity (Zhang's Corollary 4)"]],"source":"Lower bound $\\log_2 n + 2$ for the 2-complexity (Zhang's Corollary 4)"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#logb-two-add-two-le","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the 2-complexity of $n$: the least number of copies of $2$ from which $n$ can be built using addition and multiplication, and $0$ if $n$ cannot be built. Let $m$ and $n$ be natural numbers with $n$ even, $n \\ge 1$ and $2^m < n \\le 2^{m+1}$. Suppose that $n \\ne 2^a + 2^b$ for all natural numbers $a \\le b$, and that $n$ has none of Zhang's four exceptional forms for $m$. Then\u2026","title":"Lower bound $\\log_2 n + 2$ for the 2-complexity (Zhang's Corollary 4)"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/logb-two-add-two-le-of-five-powers","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.logb_two_add_two_le_of_five_powers","parent":13014,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#logb-two-add-two-le-of-five-powers","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the 2-complexity of $n$. Let $n \\ge 1$ be an even natural number and let $S$ be a finite set of natural numbers with at least five elements such that $n = \\sum_{e \\in S} 2^e$. Then $\\log_2 n + 2 \\le \\|n\\|_2$, where $\\log_2$ is the real base-2 logarithm.","title":"Lower bound for the 2-complexity of even sums of five or more distinct powers of two"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-two-pow-mul-six-pow-eq","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_two_pow_mul_six_pow_eq","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-complexity of "],["math","\\(2^m \\cdot 6^r\\)"],["text"," for "],["math","\\(1 \\le r \\le 7\\)"]],"source":"The 2-complexity of $2^m \\cdot 6^r$ for $1 \\le r \\le 7$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-two-pow-mul-six-pow-eq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the $2$-complexity of $n$. For all natural numbers $m$ and $r$ with $1 \\le r \\le 7$, one has $\\|2^m \\cdot 6^r\\|_2 = m + 3r$.","title":"The 2-complexity of $2^m \\cdot 6^r$ for $1 \\le r \\le 7$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-two-pow-mul-six-pow-bounds","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_two_pow_mul_six_pow_bounds","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds for the 2-complexity of "],["math","\\(2^m \\cdot 6^r\\)"],["text"," when "],["math","\\(8 \\le r \\le 9\\)"]],"source":"Bounds for the 2-complexity of $2^m \\cdot 6^r$ when $8 \\le r \\le 9$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-two-pow-mul-six-pow-bounds","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the $2$-complexity of $n$. For all natural numbers $m$ and $r$ with $8 \\le r \\le 9$, one has $m + 3r - 1 \\le \\|2^m \\cdot 6^r\\|_2 \\le m + 3r$.","title":"Bounds for the 2-complexity of $2^m \\cdot 6^r$ when $8 \\le r \\le 9$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-two-pow-mul-ten-pow-eq","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_two_pow_mul_ten_pow_eq","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","The 2-complexity of "],["math","\\(2^m \\cdot 10^r\\)"],["text"," for "],["math","\\(1 \\le r \\le 4\\)"]],"source":"The 2-complexity of $2^m \\cdot 10^r$ for $1 \\le r \\le 4$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-two-pow-mul-ten-pow-eq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the $2$-complexity of $n$. For all natural numbers $m$ and $r$ with $1 \\le r \\le 4$, one has $\\|2^m \\cdot 10^r\\|_2 = m + 4r$.","title":"The 2-complexity of $2^m \\cdot 10^r$ for $1 \\le r \\le 4$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/complexity-l-two-pow-mul-ten-pow-five-bo","kind":"node","lean":"MathlibExt.NumberTheory.TwoComplexityWanted.complexityL_two_pow_mul_ten_pow_five_bounds","parent":13014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds for the 2-complexity of "],["math","\\(2^m \\cdot 10^5\\)"]],"source":"Bounds for the 2-complexity of $2^m \\cdot 10^5$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity/index.html#complexity-l-two-pow-mul-ten-pow-five-bo","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/integer-complexity","status":"can_prove","summary":"Write $\\|n\\|_2$ for the $2$-complexity of $n$. For every natural number $m$, one has $m + 19 \\le \\|2^m \\cdot 10^5\\|_2 \\le m + 20$.","title":"Bounds for the 2-complexity of $2^m \\cdot 10^5$"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","title":"Arithmetical self-similar compact sets"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set","kind":"node","lean":"MetaMathlibExt.IsSelfSimilarCompactSet","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"Let $\\mu$ be a natural number and $\\rho$ a real number. A set $K \\subseteq \\mathbb{R}$ is a self-similar compact set of module $\\mu$ and ratio $\\rho$ when $\\rho > 1$, $K$ is nonempty and compact, $K \\subseteq [0, \\infty)$, $K$ is well ordered by the reverse order $x \\preceq y \\iff x \\ge y$ (every nonempty subset of $K$ has a largest element), and $\\rho K^{(\\mu)} = K$, where $K^{(\\mu)}$ is the $\\mu$-th derived set\u2026","title":"Self-similar compact set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/smul-derived-set-iterate","kind":"node","lean":"MetaMathlibExt.IsSelfSimilarCompactSet.smul_derivedSet_iterate","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#smul-derived-set-iterate","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"If $K \\subseteq \\mathbb{R}$ is a self-similar compact set of module $\\mu$ and ratio $\\rho$, then $\\rho K^{(\\mu)} = K$, where $K^{(\\mu)}$ is the $\\mu$-th derived set of $K$ (the set of accumulation points, iterated $\\mu$ times) and $\\rho K^{(\\mu)}$ is its image under multiplication by $\\rho$.","title":"Self-similarity equation"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/self-similar-layer","kind":"node","lean":"MetaMathlibExt.selfSimilarLayer","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#self-similar-layer","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"For a set $K \\subseteq \\mathbb{R}$ and a natural number $u$, let $K^{(u)}$ be the $u$-th derived set of $K$ (the derived set, i.e. the set of accumulation points, iterated $u$ times, with $K^{(0)} = K$). The layer of $K$ of index $u$ is $T_u = K^{(u)} \\setminus K^{(u+1)}$, the set of points of $K^{(u)}$ that are isolated in $K^{(u)}$.","title":"Layers of the iterated derived sets"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/self-similar-layer-zero","kind":"node","lean":"MetaMathlibExt.selfSimilarLayer_zero","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#self-similar-layer-zero","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"For every set $K \\subseteq \\mathbb{R}$, the layer $T_0$ of $K$ equals $K \\setminus K'$, where $K'$ is the derived set of $K$ (its set of accumulation points); that is, $T_0$ is the set of isolated points of $K$.","title":"The zeroth layer is the set of isolated points"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/self-similar-layer-succ","kind":"node","lean":"MetaMathlibExt.selfSimilarLayer_succ","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#self-similar-layer-succ","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"For a set $A \\subseteq \\mathbb{R}$ and $u \\ge 0$, write $T_u(A) = A^{(u)} \\setminus A^{(u+1)}$ for the $u$-th layer of $A$, where $A^{(u)}$ is the $u$-th derived set of $A$. Then for every set $K \\subseteq \\mathbb{R}$ and every $u \\ge 0$, $T_{u+1}(K) = T_u(K')$, where $K'$ is the derived set of $K$.","title":"Layers of the derived set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set-enumeration","kind":"node","lean":"MetaMathlibExt.isSelfSimilarCompactSet_enumeration","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set-enumeration","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K$ be a self-similar compact set of module $\\mu \\ge 1$ and ratio $\\rho$. Then there is a strictly decreasing map $\\alpha \\mapsto K[\\alpha]$ from the ordinals $\\alpha \\le \\omega^\\omega$ onto $K$ (so $K$ has order type $\\omega^\\omega + 1$ under the reverse order) with $K[\\omega^\\omega] = 0$, $K[0] > 0$ and $K \\subseteq [0, K[0]]$.","title":"Order type of a self-similar compact set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set-eq-union-lay","kind":"node","lean":"MetaMathlibExt.isSelfSimilarCompactSet_eq_union_layers","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set-eq-union-lay","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K$ be a self-similar compact set of module $\\mu$ and ratio $\\rho$, and for $u \\ge 0$ let $T_u = K^{(u)} \\setminus K^{(u+1)}$, where $K^{(u)}$ is the $u$-th derived set of $K$. Then","title":"Decomposition of a self-similar compact set into layers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set-derived-set","kind":"node","lean":"MetaMathlibExt.isSelfSimilarCompactSet_derivedSet_enum","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set-derived-set","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K$ be a self-similar compact set of module $\\mu$ and ratio $\\rho$, let $n \\ge 1$, and write $K^{(n)}$ for the $n$-th derived set of $K$. Let $\\alpha \\mapsto K[\\alpha]$ and $\\alpha \\mapsto K^{(n)}[\\alpha]$ be strictly decreasing maps from the ordinals $\\alpha \\le \\omega^\\omega$ onto $K$ and onto $K^{(n)}$, respectively. Then $K^{(n)}[\\alpha] = K[\\omega^n (1 + \\alpha)]$ (ordinal arithmetic) for every ordinal\u2026","title":"Enumerating the derived sets of a self-similar compact set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set-layer-zero-e","kind":"node","lean":"MetaMathlibExt.isSelfSimilarCompactSet_layer_zero_enum","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set-layer-zero-e","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K$ be a self-similar compact set of module $\\mu$ and ratio $\\rho$, and let $T_0 = K \\setminus K'$ be the set of isolated points of $K$, where $K'$ is the derived set of $K$. Let $\\alpha \\mapsto K[\\alpha]$ be a strictly decreasing map from the ordinals $\\alpha \\le \\omega^\\omega$ onto $K$, and $\\alpha \\mapsto T_0[\\alpha]$ a strictly decreasing map from the ordinals $\\alpha < \\omega^\\omega$ onto $T_0$. Then for\u2026","title":"Enumerating the isolated points of a self-similar compact set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set-layer-enum","kind":"node","lean":"MetaMathlibExt.isSelfSimilarCompactSet_layer_enum","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set-layer-enum","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K$ be a self-similar compact set of module $\\mu$ and ratio $\\rho$, let $u \\ge 1$, and let $T_u = K^{(u)} \\setminus K^{(u+1)}$, where $K^{(u)}$ is the $u$-th derived set of $K$. Let $\\alpha \\mapsto K[\\alpha]$ be a strictly decreasing map from the ordinals $\\alpha \\le \\omega^\\omega$ onto $K$, and $\\alpha \\mapsto T_u[\\alpha]$ a strictly decreasing map from the ordinals $\\alpha < \\omega^\\omega$ onto $T_u$. Then\u2026","title":"Enumerating the higher layers of a self-similar compact set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set-layer-enum-t","kind":"node","lean":"MetaMathlibExt.isSelfSimilarCompactSet_layer_enum_tendsto","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set-layer-enum-t","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K$ be a self-similar compact set of module $\\mu$ and ratio $\\rho$, with layers $T_u = K^{(u)} \\setminus K^{(u+1)}$ for $u \\ge 0$, where $K^{(u)}$ is the $u$-th derived set of $K$. Suppose that for every $u$, $\\alpha \\mapsto T_u[\\alpha]$ is a strictly decreasing map from the ordinals $\\alpha < \\omega^\\omega$ onto $T_u$. Then for every $u \\ge 0$ and every ordinal $\\alpha < \\omega^\\omega$,\u2026","title":"Limits between consecutive layers of a self-similar compact set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set-layer-enum-t-2","kind":"node","lean":"MetaMathlibExt.isSelfSimilarCompactSet_layer_enum_tendsto_last","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set-layer-enum-t-2","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K$ be a self-similar compact set of module $\\mu$ and ratio $\\rho$, with layers $T_u = K^{(u)} \\setminus K^{(u+1)}$ for $u \\ge 0$, where $K^{(u)}$ is the $u$-th derived set of $K$. Suppose that for every $u$, $\\alpha \\mapsto T_u[\\alpha]$ is a strictly decreasing map from the ordinals $\\alpha < \\omega^\\omega$ onto $T_u$. Then for every ordinal $\\alpha < \\omega^\\omega$,\u2026","title":"Limits across the last layer of a self-similar compact set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-quasi-arithmetical-compact-set","kind":"node","lean":"MetaMathlibExt.IsQuasiArithmeticalCompactSet","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-quasi-arithmetical-compact-set","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}$ and let $\\mu$ and $\\rho$ be natural numbers. Then $K$ is a quasi-arithmetical compact set of module $\\mu$ and ratio $\\rho$ when $\\rho \\ge 2$, $K$ is a self-similar compact set of module $\\mu$ and ratio $\\rho$, and there is a function $\\kappa : \\mathbb{N} \\to \\mathbb{N}$ such that $K \\setminus \\{0\\} = \\{n / \\rho^k : n, k \\in \\mathbb{N},\\ \\rho \\nmid n,\\ k \\ge \\kappa(n)\\}$.","title":"Quasi-arithmetical compact sets"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-self-similar-compact-set-2","kind":"node","lean":"MetaMathlibExt.IsQuasiArithmeticalCompactSet.isSelfSimilarCompactSet","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-self-similar-compact-set-2","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"If $K$ is a quasi-arithmetical compact set of module $\\mu$ and natural-number ratio $\\rho$, then $K$ is a self-similar compact set of module $\\mu$ and ratio $\\rho$.","title":"Quasi-arithmetical compact sets are self-similar"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-quasi-arithmetical-compact-set-layer","kind":"node","lean":"MetaMathlibExt.isQuasiArithmeticalCompactSet_layer_index","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-quasi-arithmetical-compact-set-layer","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K \\subseteq \\mathbb{R}$ be a quasi-arithmetical compact set of module $\\mu$ and ratio $\\rho$. For each natural number $j$ let $K^{(j)}$ be the $j$-th iterated derived set of $K$ and $T_j = K^{(j)} \\setminus K^{(j+1)}$ its $j$-th layer. If $n, p, q, v, w$ are natural numbers with $n / \\rho^p \\in T_v$ and $n / \\rho^q \\in T_w$, then $p \\mu - v = q \\mu - w$ as integers.","title":"The layer index of a quasi-arithmetical compact set is well defined (Proposition 5)"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-quasi-arithmetical-compact-set-gauge","kind":"node","lean":"MetaMathlibExt.isQuasiArithmeticalCompactSet_gauge_ratio_mul","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-quasi-arithmetical-compact-set-gauge","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K \\subseteq \\mathbb{R}$ be a quasi-arithmetical compact set of module $\\mu$ and ratio $\\rho$. For each natural number $j$ let $K^{(j)}$ be the $j$-th iterated derived set of $K$ and $T_j = K^{(j)} \\setminus K^{(j+1)}$ its $j$-th layer. If $n, \\ell, u, \\ell', u'$ are natural numbers with $n / \\rho^\\ell \\in T_u$ and $\\rho n / \\rho^{\\ell'} \\in T_{u'}$, then $\\ell' \\mu - u' = \\ell \\mu - u + \\mu$ as integers.","title":"Multiplying by the ratio raises the gauge index by the module (Proposition 6)"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/stability-gauge-index","kind":"node","lean":"MetaMathlibExt.stabilityGaugeIndex","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#stability-gauge-index","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}$, let $\\mu$, $\\rho$ and $n$ be natural numbers, and let $T_u$ denote the layer of $K$ of index $u$. Let $\\ell$ be the least natural number such that $n / \\rho^\\ell$ lies in some layer $T_u$ (division by $0$ gives $0$), and let $u$ be the least index with $n / \\rho^\\ell \\in T_u$. The un-normalized stability gauge of $n$ is the integer $\\ell \\mu - u$. If $n / \\rho^\\ell$ lies in no layer\u2026","title":"Un-normalized stability gauge"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/stability-gauge","kind":"node","lean":"MetaMathlibExt.stabilityGauge","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#stability-gauge","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}$, let $\\mu$ and $\\rho$ be natural numbers, and for a natural number $n$ let $g(n)$ be the un-normalized stability gauge of $n$ for $K$, $\\mu$ and $\\rho$: the integer $\\ell \\mu - u$, where $\\ell$ is least such that $n / \\rho^\\ell$ lies in some layer $T_u$ of $K$ and $u$ is the least such index (and $g(n) = 0$ if there is no such $\\ell$). The stability gauge of $n$ is the integer\u2026","title":"Stability gauge"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/stability-gauge-one","kind":"node","lean":"MetaMathlibExt.stabilityGauge_one","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#stability-gauge-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"For every set $K \\subseteq \\mathbb{R}$ and all natural numbers $\\mu$ and $\\rho$, the stability gauge $n \\mapsto \\lVert n \\rVert_K$ of $K$ with module $\\mu$ and ratio $\\rho$ takes the value $0$ at $n = 1$, that is, $\\lVert 1 \\rVert_K = 0$.","title":"The stability gauge vanishes at one"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-arithmetical-compact-set","kind":"node","lean":"MetaMathlibExt.IsArithmeticalCompactSet","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-arithmetical-compact-set","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}$ and let $\\mu$ and $\\rho$ be natural numbers. For each natural number $j$ let $K^{(j)}$ be the $j$-th iterated derived set (the set of accumulation points, iterated $j$ times) of $K$ and $T_j = K^{(j)} \\setminus K^{(j+1)}$ its $j$-th layer, and for a natural number $m$ let $\\lVert m \\rVert_K$ be the stability gauge of $K$ with module $\\mu$ and ratio $\\rho$ at $m$. Then $K$ is an\u2026","title":"Arithmetical compact sets"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-quasi-arithmetical-compact-set-2","kind":"node","lean":"MetaMathlibExt.IsArithmeticalCompactSet.isQuasiArithmeticalCompactSet","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-quasi-arithmetical-compact-set-2","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"If $K \\subseteq \\mathbb{R}$ is an arithmetical compact set of module $\\mu$ and ratio $\\rho$, then $K$ is a quasi-arithmetical compact set of module $\\mu$ and ratio $\\rho$.","title":"Arithmetical compact sets are quasi-arithmetical"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-arithmetical-compact-set-derived-set","kind":"node","lean":"MetaMathlibExt.isArithmeticalCompactSet_derivedSet_iterate","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-arithmetical-compact-set-derived-set","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $K$ be an arithmetical compact set of module $\\mu$ and ratio $\\rho$. For every natural number $n$, the $n$-th derived set $K^{(n)}$ of $K$ (the set of accumulation points, iterated $n$ times, so that $K^{(0)} = K$) is an arithmetical compact set of module $\\mu$ and ratio $\\rho$.","title":"Derived sets of an arithmetical compact set are arithmetical (Proposition 7)"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-complexity-stable","kind":"node","lean":"MetaMathlibExt.IsComplexityStable","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-complexity-stable","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"A natural number $n$ is stable when $n > 0$ and $\\lVert 3^k n \\rVert = \\lVert n \\rVert + 3k$ for every natural number $k$, where $\\lVert m \\rVert$ is the integer complexity of $m$.","title":"Stable numbers for integer complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/stable-complexity","kind":"node","lean":"MetaMathlibExt.stableComplexity","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#stable-complexity","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"For a natural number $n$, let $\\lVert m \\rVert$ denote the integer complexity of a positive integer $m$, with $\\lVert 0 \\rVert = 0$, and let $a$ be the least natural number such that $3^a n$ is stable, with $a = 0$ if there is none. The stable complexity of $n$ is the integer $\\lVert n \\rVert_{\\mathrm{st}} = \\lVert 3^a n \\rVert - 3a$.","title":"Stable complexity"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/stable-complexity-of-is-complexity-stabl","kind":"node","lean":"MetaMathlibExt.stableComplexity_of_isComplexityStable","parent":13074,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#stable-complexity-of-is-complexity-stabl","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"If $n$ is a stable natural number, then its stable complexity equals its integer complexity: $\\lVert n \\rVert_{\\mathrm{st}} = \\lVert n \\rVert$.","title":"Stable complexity of a stable number"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/complexity-layer-zero","kind":"node","lean":"MetaMathlibExt.complexityLayerZero","parent":13074,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(T_0\\)"],["text"," built from stable numbers"]],"source":"The set $T_0$ built from stable numbers"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#complexity-layer-zero","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"Write $\\lVert n \\rVert$ for the integer complexity of a natural number $n$, and call $n$ stable when $n > 0$ and $\\lVert 3^k n \\rVert = \\lVert n \\rVert + 3k$ for every natural number $k$. The set $T_0 \\subseteq \\mathbb{R}$ consists of the reals $n / 3^k$, where $n$ and $k$ are natural numbers, $n$ is stable and $\\lVert n \\rVert = 3k + 2$.","title":"The set $T_0$ built from stable numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/complexity-layer-zero-subset-ici","kind":"node","lean":"MetaMathlibExt.complexityLayerZero_subset_Ici","parent":13074,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(T_0\\)"],["text"," is nonnegative"]],"source":"The set $T_0$ is nonnegative"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#complexity-layer-zero-subset-ici","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"fully_proved","summary":"Every element of the set $T_0$ of reals $n / 3^k$, where $n$ and $k$ are natural numbers, $n$ is stable and $\\lVert n \\rVert = 3k + 2$ (with $\\lVert n \\rVert$ the integer complexity of $n$), is a nonnegative real number.","title":"The set $T_0$ is nonnegative"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/is-arithmetical-compact-set-closure-comp","kind":"node","lean":"MetaMathlibExt.isArithmeticalCompactSet_closure_complexityLayerZero","parent":13074,"reading_title":{"format":"inline-parts-v1","parts":[["text","The closure of "],["math","\\(T_0\\)"],["text"," is an arithmetical compact set (Theorem 2)"]],"source":"The closure of $T_0$ is an arithmetical compact set (Theorem 2)"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets/index.html#is-arithmetical-compact-set-closure-comp","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/arithmetical-self-similar-compact-sets","status":"can_prove","summary":"Let $T_0$ be the set of reals $n / 3^k$, where $n$ and $k$ are natural numbers, $n$ is stable and $\\lVert n \\rVert = 3k + 2$ (with $\\lVert n \\rVert$ the integer complexity of $n$), and let $K$ be the closure of $T_0$ in $\\mathbb{R}$. Then $K$ is an arithmetical compact set of module $3$ and ratio $3$, and for every natural number $n \\ge 1$ the stability gauge $\\lVert n \\rVert_K$ of $K$ with module $3$ and ratio $3$\u2026","title":"The closure of $T_0$ is an arithmetical compact set (Theorem 2)"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences","status":"fully_proved","title":"Newton sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences/is-newton-sequence","kind":"node","lean":"MetaMathlibExt.IsNewtonSequence","parent":13103,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences/index.html#is-newton-sequence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences","status":"fully_proved","summary":"Let $R$ be a commutative semiring and let $a, c \\colon \\mathbb{N} \\to R$. The sequence $a$ is a Newton sequence generated by $c$ if for every natural number $n$, $a_{n+1} = \\sum_{j=0}^{n-1} a_{n-j} c_{j+1} + (n+1) c_{n+1}$.","title":"Newton sequences"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences/eq-recurrence","kind":"node","lean":"MetaMathlibExt.IsNewtonSequence.eq_recurrence","parent":13103,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences/index.html#eq-recurrence","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/newton-sequences","status":"fully_proved","summary":"Let $R$ be a commutative semiring and let $a, c \\colon \\mathbb{N} \\to R$ be such that $a$ is a Newton sequence generated by $c$. Then for every natural number $n$ we have $a_{n+1} = \\sum_{j=0}^{n-1} a_{n-j} c_{j+1} + (n+1) c_{n+1}$.","title":"The recurrence of a Newton sequence"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"can_prove","title":"Sets of remainders"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/remainder-set","kind":"node","lean":"MetaMathlibExt.remainderSet","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#remainder-set","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"fully_proved","summary":"For a natural number $n$, the remainder set is $S(n) = \\{\\, n \\bmod k : 1 \\le k \\le \\lfloor n/2 \\rfloor \\,\\}$, the set of remainders of $n$ modulo the integers from $1$ to $\\lfloor n/2 \\rfloor$.","title":"Remainder set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/remainder-count","kind":"node","lean":"MetaMathlibExt.remainderCount","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#remainder-count","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"fully_proved","summary":"For a natural number $n$, $s(n)$ is the number of elements of the remainder set $S(n)$, that is, the number of distinct values of $n \\bmod k$ for $1 \\le k \\le \\lfloor n/2 \\rfloor$ (OEIS A283190).","title":"Remainder count"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/remainder-count-eleven","kind":"node","lean":"MetaMathlibExt.remainderCount_eleven","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#remainder-count-eleven","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"fully_proved","summary":"The number $s(11)$ of distinct remainders $11 \\bmod k$ for $1 \\le k \\le 5$ equals $4$.","title":"The remainder count of 11"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/remainder-count-twelve","kind":"node","lean":"MetaMathlibExt.remainderCount_twelve","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#remainder-count-twelve","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"fully_proved","summary":"The number $s(12)$ of distinct remainders $12 \\bmod k$ for $1 \\le k \\le 6$ equals $2$.","title":"The remainder count of 12"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/remainder-constant","kind":"node","lean":"MetaMathlibExt.remainderConstant","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#remainder-constant","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"fully_proved","summary":"The remainder-set constant is $c = \\sum_{p} \\frac{1}{p(p+1)} \\prod_{q < p} \\left(1 - \\frac{1}{q}\\right)$, where $p$ runs over all primes and, for each $p$, $q$ runs over the primes less than $p$.","title":"The remainder-set constant"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/remainder-count-asymptotic","kind":"node","lean":"MetaMathlibExt.remainderCount_asymptotic","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#remainder-count-asymptotic","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"can_prove","summary":"Let $s(n)$ be the number of distinct remainders $n \\bmod k$ with $1 \\le k \\le \\lfloor n/2 \\rfloor$ and let $c$ be the remainder-set constant. Then, as $n \\to \\infty$ over the natural numbers, $s(n) - cn = O\\bigl(n / (\\log n \\cdot \\log \\log n)\\bigr)$, and $s(n)/n \\to c$.","title":"Asymptotic formula for the number of remainders"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/remainder-count-drops-unbounded","kind":"node","lean":"MetaMathlibExt.remainderCount_drops_unbounded","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#remainder-count-drops-unbounded","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"can_prove","summary":"Let $s(n)$ be the number of distinct remainders $n \\bmod k$ with $1 \\le k \\le \\lfloor n/2 \\rfloor$. For every natural number $M$ there are infinitely many $n$ with $s(n+1) + M \\le s(n)$; equivalently, $\\liminf_{n \\to \\infty} (s(n+1) - s(n)) = -\\infty$.","title":"Unbounded drops in the number of remainders"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/remainder-count-consecutive-differences","kind":"node","lean":"MetaMathlibExt.remainderCount_consecutive_differences","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#remainder-count-consecutive-differences","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"can_prove","summary":"Let $s(n)$ be the number of distinct remainders $n \\bmod k$ with $1 \\le k \\le \\lfloor n/2 \\rfloor$. Then $s(n+1) \\le s(n) + 1$ for every $n \\ge 1$; there is a real constant $C$ with $s(n) - C \\log \\log n \\le s(n+1)$ for all sufficiently large $n$; and for every natural number $M$ there are infinitely many $n$ with $s(n+1) + M \\le s(n)$, that is, $\\liminf_{n \\to \\infty} (s(n+1) - s(n)) = -\\infty$.","title":"Consecutive differences of the number of remainders"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/card-least-prime-factor-interval-error","kind":"node","lean":"MetaMathlibExt.card_leastPrimeFactor_interval_error","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#card-least-prime-factor-interval-error","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"can_prove","summary":"Let $a$ and $t$ be positive integers and let $p$ be a prime. Let $N$ be the number of integers $m$ with $a + 1 \\le m \\le a + t$ that are divisible by $p$ and by no prime $q < p$. Then","title":"Counting integers with a given least prime factor in an interval"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/prime-counting-le-thirteen-tenths","kind":"node","lean":"MetaMathlibExt.primeCounting_le_thirteen_tenths","parent":13106,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound "],["math","\\(1.3\\, n / \\log n\\)"],["text"," for the prime-counting function"]],"source":"Upper bound $1.3\\, n / \\log n$ for the prime-counting function"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#prime-counting-le-thirteen-tenths","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"can_prove","summary":"For every natural number $n \\ge 2$, the number $\\pi(n)$ of primes at most $n$ satisfies $\\pi(n) \\le 1.3\\, n / \\log n$, where $\\log$ is the natural logarithm.","title":"Upper bound $1.3\\, n / \\log n$ for the prime-counting function"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/tsum-inv-sq-primes-gt-is-big-o","kind":"node","lean":"MetaMathlibExt.tsum_inv_sq_primes_gt_isBigO","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#tsum-inv-sq-primes-gt-is-big-o","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"can_prove","summary":"As the natural number $n$ tends to infinity, the sum $\\sum_{p > n,\\ p \\text{ prime}} 1/p^2$ of the inverse squares of the primes greater than $n$ is $O(1/(n \\log n))$.","title":"Tail of the sum of inverse prime squares"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/iterated-remainder-set","kind":"node","lean":"MetaMathlibExt.iteratedRemainderSet","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#iterated-remainder-set","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"fully_proved","summary":"For a natural number $n$, the iterated remainder sets $S_j(n)$, $j \\ge 0$, are defined by $S_0(n) = \\{1, \\dots, \\lfloor n/2 \\rfloor\\}$ and, for $j \\ge 0$, $S_{j+1}(n) = \\{n \\bmod k : k \\in S_j(n),\\ k \\ne 0\\}$.","title":"Iterated remainder sets"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/iterated-remainder-set-one","kind":"node","lean":"MetaMathlibExt.iteratedRemainderSet_one","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#iterated-remainder-set-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"fully_proved","summary":"For every natural number $n$, the iterated remainder set $S_1(n)$ equals the remainder set $S(n) = \\{n \\bmod k : 1 \\le k \\le \\lfloor n/2 \\rfloor\\}$.","title":"The first iterated remainder set is the remainder set"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/iterated-remainder-set-thirty-five","kind":"node","lean":"MetaMathlibExt.iteratedRemainderSet_thirtyFive","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#iterated-remainder-set-thirty-five","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"fully_proved","summary":"The iterated remainder sets of $35$ satisfy $S_6(35) = \\{0\\}$ and $S_7(35) = \\emptyset$.","title":"Iterated remainder sets of 35"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/pierce-max-length-eq-succ-iff","kind":"node","lean":"MetaMathlibExt.pierce_maxLength_eq_succ_iff","parent":13106,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders/index.html#pierce-max-length-eq-succ-iff","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sets-of-remainders","status":"can_prove","summary":"Let $n \\ge 3$ and $t$ be natural numbers, let $P(n)$ be the maximal Pierce length of $n$ and let $S_j(n)$ be the iterated remainder sets of $n$. Then the following are equivalent: $P(n) = t + 1$; $S_t(n) = \\{0\\}$; $S_t(n)$ has exactly one element and $S_{t+j}(n) = \\emptyset$ for every $j \\ge 1$.","title":"Maximal Pierce length via iterated remainder sets"},{"data_url":"scopes/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence.json","id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","kind":"scope","navigation":"module","parent":12870,"result":false,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","title":"Sylvester's sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/generalized-sylvester","kind":"node","lean":"Nat.generalizedSylvester","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#generalized-sylvester","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"fully_proved","summary":"For a natural number $n$, the generalized Sylvester sequence $(s_k(n))_{k \\ge 0}$ is the sequence of natural numbers defined by $s_0(n) = n + 1$ and $s_{k+1}(n) = s_k(n)^2 - s_k(n) + 1$ for $k \\ge 0$.","title":"Generalized Sylvester sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/generalized-sylvester-zero","kind":"node","lean":"Nat.generalizedSylvester_zero","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#generalized-sylvester-zero","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"fully_proved","summary":"For every natural number $n$, the generalized Sylvester sequence $(s_k(n))_{k \\ge 0}$ satisfies $s_0(n) = n + 1$.","title":"Initial term of the generalized Sylvester sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/generalized-sylvester-succ","kind":"node","lean":"Nat.generalizedSylvester_succ","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#generalized-sylvester-succ","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the generalized Sylvester sequence $(s_k(n))_{k \\ge 0}$ satisfies $s_{k+1}(n) = s_k(n)^2 - s_k(n) + 1$.","title":"Recurrence for the generalized Sylvester sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/generalized-sylvester-pos","kind":"node","lean":"Nat.generalizedSylvester_pos","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#generalized-sylvester-pos","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the term $s_k(n)$ of the generalized Sylvester sequence is positive.","title":"Positivity of the generalized Sylvester sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-number","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvesterNumber","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-number","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"fully_proved","summary":"The Sylvester numbers $S_0, S_1, S_2, \\dots$ are the natural numbers given by $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$ for $n \\ge 0$, so the sequence begins $2, 3, 7, 43, 1807, \\dots$. It is the generalized Sylvester sequence with parameter $1$.","title":"Sylvester numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-sub-dvd","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_sub_dvd","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-sub-dvd","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For all natural numbers $k$, $m$ and $n$ with $k \\ge 1$, $m \\ge k$ and $n \\ge k$, the integer $(S_k - 1)^2$ divides $S_n - S_m$.","title":"Divisibility of differences of Sylvester numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-mod-eq-three-mod-four","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_modEq_three_mod_four","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-mod-eq-three-mod-four","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For every $n \\ge 1$, $S_n \\equiv 3 \\pmod 4$.","title":"Sylvester numbers modulo 4"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-mod-eq-seven-mod-thirty-six","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_modEq_seven_mod_thirty_six","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-mod-eq-seven-mod-thirty-six","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For every $n \\ge 2$, $S_n \\equiv 7 \\pmod{36}$.","title":"Sylvester numbers modulo 36"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-mod-eq-forty-three-mod-1764","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_modEq_forty_three_mod_1764","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-mod-eq-forty-three-mod-1764","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For every $n \\ge 3$, $S_n \\equiv 43 \\pmod{1764}$.","title":"Sylvester numbers modulo 1764"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-sq-add-one-dvd-sub","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_sq_add_one_dvd_sub","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-sq-add-one-dvd-sub","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For all natural numbers $k$ and $n$ with $1 \\le k \\le n$, the integer $S_k^2 + 1$ divides the integer $S_n - (-1)^{n-k} S_k$.","title":"Alternating divisibility for Sylvester numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-even-index-mod-ten","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_even_index_mod_ten","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-even-index-mod-ten","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For every $i \\ge 1$, $S_{2i} \\equiv 7 \\pmod{10}$.","title":"Sylvester numbers of even index modulo 10"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-odd-index-mod-ten","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_odd_index_mod_ten","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-odd-index-mod-ten","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For every $i \\ge 1$, $S_{2i+1} \\equiv 3 \\pmod{10}$.","title":"Sylvester numbers of odd index modulo 10"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/mersenne-sylvester-values-eq","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.mersenne_sylvester_values_eq","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#mersenne-sylvester-values-eq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. The natural numbers $x$ such that $x = S_n$ for some $n$ and $x = 2^r - 1$ for some $r \\ge 1$ are exactly $3$ and $7$.","title":"Mersenne numbers in the Sylvester sequence"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/triangular-sylvester-values-eq","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.triangular_sylvester_values_eq","parent":13122,"reading_title":{"format":"inline-parts-v1","parts":[["text","The only triangular Sylvester number is "],["math","\\(3\\)"]],"source":"The only triangular Sylvester number is $3$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#triangular-sylvester-values-eq","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $S_n$ denote the $n$-th Sylvester number. A natural number $x$ equals $S_n$ for some $n$ and satisfies $2x = m(m + 1)$ for some natural number $m \\ge 1$ if and only if $x = 3$.","title":"The only triangular Sylvester number is $3$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-not-perfect","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_not_perfect","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-not-perfect","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"For every natural number $n$, the $n$-th Sylvester number $S_n$ is not a perfect number.","title":"No Sylvester number is perfect"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-exists-prime-divisor-mod-four","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_exists_prime_divisor_mod_four_eq_three","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-exists-prime-divisor-mod-four","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For every $n \\ge 1$, $S_n$ has a prime divisor $p$ with $p \\equiv 3 \\pmod 4$.","title":"Prime divisors congruent to 3 modulo 4 of Sylvester numbers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-not-perfect-power","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_not_perfect_power","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-not-perfect-power","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For every $n$ there are no natural numbers $a \\ge 2$ and $m \\ge 2$ with $S_n = a^m$.","title":"Sylvester numbers are not perfect powers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/sylvester-ne-pow-add-or-sub-one","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.sylvester_ne_pow_add_or_sub_one","parent":13122,"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#sylvester-ne-pow-add-or-sub-one","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"Let $(S_n)_{n \\ge 0}$ be the Sylvester sequence, $S_0 = 2$ and $S_{n+1} = S_n^2 - S_n + 1$. For every $n$ other than $1$ and $2$ there are no natural numbers $a \\ge 2$ and $m \\ge 2$ with $S_n = a^m + 1$ or $S_n = a^m - 1$.","title":"Sylvester numbers next to perfect powers"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/lebesgue-nagell-three-solutions-iff","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.lebesgue_nagell_three_solutions_iff","parent":13122,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coprime solutions of "],["math","\\(x^2 + 3 = 4y^n\\)"]],"source":"Coprime solutions of $x^2 + 3 = 4y^n$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#lebesgue-nagell-three-solutions-iff","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"For natural numbers $n$, $x$ and $y$, the conditions $n \\ge 3$, $x \\ge 1$, $y \\ge 1$, $\\gcd(x, y) = 1$ and $x^2 + 3 = 4y^n$ hold if and only if $n \\ge 3$ and either $x = y = 1$ or $(n, x, y) = (3, 37, 7)$ (Luca\u2013Tengely\u2013Togb\u00e9).","title":"Coprime solutions of $x^2 + 3 = 4y^n$"},{"id":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/lebesgue-nagell-seven-solutions-iff","kind":"node","lean":"MathlibExt.NumberTheory.Recurrences.SylvesterSequenceWanted.lebesgue_nagell_seven_solutions_iff","parent":13122,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coprime solutions of "],["math","\\(x^2 + 7 = 4y^n\\)"]],"source":"Coprime solutions of $x^2 + 7 = 4y^n$"},"result":true,"route":"../roadmap/NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence/index.html#lebesgue-nagell-seven-solutions-iff","scope":"NumberTheory/integer-sequences/arithmetic-of-sequences/sylvester-s-sequence","status":"can_prove","summary":"For natural numbers $n$, $x$ and $y$, the conditions $n \\ge 3$, $x \\ge 1$, $y \\ge 1$, $\\gcd(x, y) = 1$ and $x^2 + 7 = 4y^n$ hold if and only if $(n, x, y)$ is $(3, 5, 2)$, $(5, 11, 2)$ or $(13, 181, 2)$ (Luca\u2013Tengely\u2013Togb\u00e9).","title":"Coprime solutions of $x^2 + 7 = 4y^n$"},{"data_url":"scopes/NumberTheory/additive-number-theory.json","id":"NumberTheory/additive-number-theory","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/index.html","scope":"NumberTheory/additive-number-theory","status":"can_prove","summary":"Representing integers as sums from special sets, and the density of integer sets: sumsets of integers, subset sums and complete sequences, primitive sets, natural and logarithmic density, Romanoff's theorem; additive bases and complements and zero-sum constants go to additive combinatorics.","title":"Additive number theory"},{"id":"NumberTheory/additive-number-theory/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1Wanted.conjecture_refuted","parent":13143,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/index.html#conjecture-refuted","scope":"NumberTheory/additive-number-theory","status":"can_prove","summary":"There is no real number $C > 0$ such that for every natural number $N \\ne 0$ and every finite set $A \\subseteq \\{1, \\dots, N\\}$ whose distinct subsets have distinct sums, $C \\cdot 2^{|A|} < N$.","title":"Disproof of Erd\u0151s Problem 1 on distinct subset sums"},{"id":"NumberTheory/additive-number-theory/exists-squarefree-add-two-pow-of-odd-lt","kind":"node","lean":"MetaMathlibExt.exists_squarefree_add_two_pow_of_odd_lt_two_pow_fifty","parent":13143,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd numbers "],["math","\\(1 < n < 2^{50}\\)"],["text"," are squarefree plus a power of two"]],"source":"Odd numbers $1 < n < 2^{50}$ are squarefree plus a power of two"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/index.html#exists-squarefree-add-two-pow-of-odd-lt","scope":"NumberTheory/additive-number-theory","status":"can_prove","summary":"Every odd natural number $n$ with $1 < n < 2^{50}$ can be written as $n = s + 2^k$, where $s$ is a squarefree positive integer and $1 \\le k \\le 13$.","title":"Odd numbers $1 < n < 2^{50}$ are squarefree plus a power of two"},{"data_url":"scopes/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval.json","id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"can_prove","title":"Distinct multiples in an interval"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/has-distinct-multiples-in-interval","kind":"node","lean":"MetaMathlibExt.HasDistinctMultiplesInInterval","parent":13146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Distinct multiples of "],["math","\\(1, \\dots, n\\)"],["text"," in an interval"]],"source":"Distinct multiples of $1, \\dots, n$ in an interval"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#has-distinct-multiples-in-interval","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"For natural numbers $n$, $m$ and $L$, the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$ when there are pairwise distinct natural numbers $a_1, \\dots, a_n$ with $m < a_i \\le m + L$ and $i \\mid a_i$ for every $1 \\le i \\le n$.","title":"Distinct multiples of $1, \\dots, n$ in an interval"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/distinct-multiples-interval-length","kind":"node","lean":"MetaMathlibExt.distinctMultiplesIntervalLength","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#distinct-multiples-interval-length","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"For natural numbers $n$ and $m$, $f(n, m)$ is the least natural number $L$ such that the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$.","title":"Least interval length containing distinct multiples"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/mono","kind":"node","lean":"MetaMathlibExt.HasDistinctMultiplesInInterval.mono","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#mono","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"For natural numbers $n$, $m$, $L$ and $L'$ with $L \\le L'$, if the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$, then so does the interval $(m, m + L']$.","title":"Lengthening the interval preserves distinct multiples"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/exists-has-distinct-multiples-in-interva","kind":"node","lean":"MetaMathlibExt.exists_hasDistinctMultiplesInInterval","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#exists-has-distinct-multiples-in-interva","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"For all natural numbers $n$ and $m$ there is a natural number $L$ such that the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$.","title":"Some interval contains distinct multiples"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/has-distinct-multiples-in-interval-disti","kind":"node","lean":"MetaMathlibExt.hasDistinctMultiplesInInterval_distinctMultiplesIntervalLength","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#has-distinct-multiples-in-interval-disti","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"For natural numbers $n$ and $m$, let $f(n, m)$ be the least natural number $L$ such that the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$. Then the interval $(m, m + f(n, m)]$ contains distinct multiples of $1, \\dots, n$.","title":"The least interval length is admissible"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/distinct-multiples-interval-length-le-if","kind":"node","lean":"MetaMathlibExt.distinctMultiplesIntervalLength_le_iff","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#distinct-multiples-interval-length-le-if","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"For natural numbers $n$ and $m$, let $f(n, m)$ be the least natural number $K$ such that the interval $(m, m + K]$ contains distinct multiples of $1, \\dots, n$. Then for every natural number $L$, $f(n, m) \\le L$ if and only if the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$.","title":"Characterization of the least interval length"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/distinct-multiples-interval-length-one","kind":"node","lean":"MetaMathlibExt.distinctMultiplesIntervalLength_one","parent":13146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The least interval length for "],["math","\\(n = 1\\)"]],"source":"The least interval length for $n = 1$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#distinct-multiples-interval-length-one","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"For natural numbers $n$ and $m$, let $f(n, m)$ be the least natural number $L$ such that the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$. Then $f(1, m) = 1$ for every natural number $m$: the interval $(m, m + 1]$ contains the multiple $m + 1$ of $1$, and the empty interval $(m, m]$ contains nothing.","title":"The least interval length for $n = 1$"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/distinct-multiples-interval-length-mul-l","kind":"node","lean":"MetaMathlibExt.distinctMultiplesIntervalLength_mul_le","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#distinct-multiples-interval-length-mul-l","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"can_prove","summary":"For natural numbers $n$ and $m$, let $f(n, m)$ be the least natural number $L$ such that the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$. Then for all positive integers $k$ and $n$, $kn + f(kn, kn) \\le k^2 n + f(n, k^2 n)$.","title":"An inequality between diagonal and shifted interval lengths"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/distinct-multiples-interval-length-diag","kind":"node","lean":"MetaMathlibExt.distinctMultiplesIntervalLength_diag_bounds","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#distinct-multiples-interval-length-diag","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"can_prove","summary":"For natural numbers $n$ and $m$, let $f(n, m)$ be the least natural number $L$ such that the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$. For every real $\\varepsilon > 0$ and all sufficiently large natural numbers $n$,","title":"Erd\u0151s\u2013Pomerance bounds for the diagonal interval length"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/exists-distinct-multiples-interval-lengt","kind":"node","lean":"MetaMathlibExt.exists_distinctMultiplesIntervalLength_sub_gt","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#exists-distinct-multiples-interval-lengt","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"can_prove","summary":"For natural numbers $n$ and $m$, let $f(n, m)$ be the least natural number $L$ such that the interval $(m, m + L]$ contains distinct multiples of $1, \\dots, n$. For all sufficiently large natural numbers $n$ there is a positive integer $m$ with","title":"Shifting the interval can greatly increase the least length"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/f","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem650Wanted.f","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#f","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"For a natural number $m$, $f(m)$ is the largest natural number $r$ such that for every natural number $N$, every set $A \\subseteq \\{1, \\dots, N\\}$ with exactly $m$ elements and every real $x \\ge 1$, there are distinct $a_1, \\dots, a_r \\in A$ and distinct natural numbers $b_1, \\dots, b_r$ in the open interval $(x, x + 2N)$ with $a_i \\mid b_i$ for every $i$.","title":"Erd\u0151s's matching function for multiples in intervals"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem650Wanted.conjecture","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#conjecture","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"fully_proved","summary":"Let $f$ be Erd\u0151s's matching function for multiples in intervals. The statement is the conjunction of four claims: for every natural number $m$, $f(m) = \\min(m, \\lceil 2\\sqrt{m}\\,\\rceil)$; it is not the case that $f(m) \\le \\sqrt{m}$ for every natural number $m$; $\\sqrt{m} \\le f(m)$ for every natural number $m$; and $f(m^2) \\le 2m$ for every natural number $m$.","title":"Statement of Erd\u0151s Problem 650"},{"id":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem650Wanted.conjecture_holds","parent":13146,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/distinct-multiples-in-an-interval/index.html#conjecture-holds","scope":"NumberTheory/additive-number-theory/distinct-multiples-in-an-interval","status":"can_prove","summary":"For a natural number $m$, let $f(m)$ be the largest natural number $r$ such that for every natural number $N$, every set $A \\subseteq \\{1, \\dots, N\\}$ with exactly $m$ elements and every real $x \\ge 1$, there are distinct $a_1, \\dots, a_r \\in A$ and distinct natural numbers $b_1, \\dots, b_r$ in the open interval $(x, x + 2N)$ with $a_i \\mid b_i$ for every $i$. Then $f(m) = \\min(m, \\lceil 2\\sqrt{m}\\,\\rceil)$ for\u2026","title":"Erd\u0151s Problem 650: matching integers to distinct multiples"},{"data_url":"scopes/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem.json","id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","title":"Erd\u0151s minimum overlap problem"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/overlap","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.Overlap","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#overlap","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For finite sets $A, B \\subseteq \\mathbb{Z}$ and an integer $k$, the overlap of $A$ and $B$ at $k$ is the number of pairs $(a, b) \\in A \\times B$ with $a - b = k$.","title":"Overlap count of two finite sets of integers"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/max-overlap","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.MaxOverlap","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#max-overlap","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For finite sets $A, B \\subseteq \\mathbb{Z}$, the maximum overlap of $A$ and $B$ is the natural number obtained as the supremum over all integers $k$ of $\\#\\{(a, b) \\in A \\times B : a - b = k\\}$.","title":"Maximum overlap"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/m","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.M","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#m","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For a natural number $n$, $M(n)$ is the least value of the maximum overlap of $A$ and $B$, the supremum over $k \\in \\mathbb{Z}$ of $\\#\\{(a, b) \\in A \\times B : a - b = k\\}$, taken over all pairs of disjoint finite sets $A, B \\subseteq \\mathbb{Z}$ with $A \\cup B = \\{1, \\dots, 2n\\}$ and $|A| = |B|$.","title":"Erd\u0151s minimum overlap function"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/overlap-le-max-overlap","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.Overlap_le_MaxOverlap","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#overlap-le-max-overlap","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For finite sets $A, B \\subseteq \\mathbb{Z}$ and every integer $k$, the overlap $\\#\\{(a, b) \\in A \\times B : a - b = k\\}$ is at most the maximum overlap of $A$ and $B$.","title":"Overlaps are bounded by the maximum overlap"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/max-overlap-attained","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.MaxOverlap_attained","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#max-overlap-attained","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For finite sets $A, B \\subseteq \\mathbb{Z}$ there is an integer $k$ such that the maximum overlap of $A$ and $B$, the supremum over $j \\in \\mathbb{Z}$ of $\\#\\{(a, b) \\in A \\times B : a - b = j\\}$, equals $\\#\\{(a, b) \\in A \\times B : a - b = k\\}$.","title":"The maximum overlap is attained"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/balanced-partition-nonempty","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.balancedPartition_nonempty","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Balanced partitions of "],["math","\\(\\{1, \\dots, 2n\\}\\)"],["text"," exist"]],"source":"Balanced partitions of $\\{1, \\dots, 2n\\}$ exist"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#balanced-partition-nonempty","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For every natural number $n$, the set of maximum overlaps of $A$ and $B$, taken over all pairs of disjoint finite sets of integers $A$ and $B$ with $A \\cup B = \\{1, \\dots, 2n\\}$ and $|A| = |B|$, is nonempty.","title":"Balanced partitions of $\\{1, \\dots, 2n\\}$ exist"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/m-attained","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.M_attained","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#m-attained","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For every natural number $n$ there are disjoint finite sets $A, B \\subseteq \\mathbb{Z}$ with $A \\cup B = \\{1, \\dots, 2n\\}$ and $|A| = |B|$ such that the minimum overlap $M(n)$ equals the maximum overlap of $A$ and $B$, the supremum over $k \\in \\mathbb{Z}$ of $\\#\\{(a, b) \\in A \\times B : a - b = k\\}$.","title":"The minimum overlap is attained"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/m-le-max-overlap","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.M_le_MaxOverlap","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#m-le-max-overlap","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"Let $n$ be a natural number and let $A, B \\subseteq \\mathbb{Z}$ be disjoint finite sets with $A \\cup B = \\{1, \\dots, 2n\\}$ and $|A| = |B|$. Then the minimum overlap $M(n)$ is at most the maximum overlap of $A$ and $B$, the supremum over $k \\in \\mathbb{Z}$ of $\\#\\{(a, b) \\in A \\times B : a - b = k\\}$.","title":"The minimum overlap bounds every balanced partition"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/min-overlap-quotient","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.MinOverlapQuotient","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#min-overlap-quotient","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For a natural number $N$, the normalized minimum overlap is the real number $M(N) / N$, where $M(N)$ is the Erd\u0151s minimum overlap function; for $N = 0$ the quotient is $0$ by the convention $x / 0 = 0$.","title":"Normalized minimum overlap"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/discrete-universal-lower-bound","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.DiscreteUniversalLowerBound","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#discrete-universal-lower-bound","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"A real number $c$ is a universal asymptotic lower bound when for every $\\varepsilon > 0$ there is a positive integer $N_0$ such that for every $N \\ge N_0$ and all disjoint finite sets $A, B \\subseteq \\mathbb{Z}$ with $A \\cup B = \\{1, \\dots, 2N\\}$ and $|A| = |B|$, some integer $x$ satisfies $(c - \\varepsilon) N \\le \\#\\{(a, b) \\in A \\times B : a - b = x\\}$.","title":"Universal asymptotic lower bound for discrete overlaps"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-support-interval","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousSupportInterval","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-support-interval","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The support interval of the continuous minimum-overlap problem is the closed interval $[-1, 1]$ of real numbers.","title":"Continuous support interval"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-translation-interval","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousTranslationInterval","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-translation-interval","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The translation interval of the continuous minimum-overlap model is the closed interval $[-2, 2] \\subseteq \\mathbb{R}$.","title":"Translation interval of the continuous overlap model"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-overlap-integrand","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousOverlapIntegrand","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-overlap-integrand","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For functions $f, g : \\mathbb{R} \\to \\mathbb{R}$ and real numbers $x$ and $t$, the overlap integrand is $f(t)\\,g(x + t)$.","title":"Continuous overlap integrand"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-overlap","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousOverlap","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-overlap","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For functions $f, g : \\mathbb{R} \\to \\mathbb{R}$ and a translation $x \\in \\mathbb{R}$, the overlap of $f$ and $g$ at $x$ is $\\int_{-1}^{1} f(t)\\,g(x + t)\\,dt$, the integral of the overlap integrand over the support interval $[-1, 1]$.","title":"Continuous overlap"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-overlap-values","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousOverlapValues","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-overlap-values","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For functions $f, g : \\mathbb{R} \\to \\mathbb{R}$, the set of overlap values of $f$ and $g$ is the image of the translation interval $[-2, 2]$ under the overlap $x \\mapsto \\int_{-1}^{1} f(t)\\,g(x + t)\\,dt$.","title":"Continuous overlap values"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-admissible","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.ContinuousAdmissible","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-admissible","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"A pair of functions $f, g : \\mathbb{R} \\to \\mathbb{R}$ is admissible when $f$ and $g$ are almost everywhere measurable with respect to Lebesgue measure, take values in $[0, 1]$ at every point, vanish at every point outside the support interval $[-1, 1]$, satisfy $f(t) + g(t) = 1$ for every $t \\in [-1, 1]$, and satisfy $\\int_{\\mathbb{R}} f(t)\\,dt = 1$.","title":"Admissible pairs for the continuous minimum-overlap problem"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-pair-score","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousPairScore","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-pair-score","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For functions $f, g : \\mathbb{R} \\to \\mathbb{R}$, the pair score of $f$ and $g$ is the supremum of the overlap values $\\int_{-1}^{1} f(t)\\,g(x + t)\\,dt$ over the translations $x \\in [-2, 2]$.","title":"Continuous pair score"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-universal-lower-bound","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.ContinuousUniversalLowerBound","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-universal-lower-bound","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"Call a pair of functions $f, g : \\mathbb{R} \\to \\mathbb{R}$ admissible when $f$ and $g$ are almost everywhere measurable, take values in $[0, 1]$, vanish outside $[-1, 1]$, satisfy $f + g = 1$ on $[-1, 1]$, and $\\int_{\\mathbb{R}} f = 1$, and let the pair score $S(f, g)$ be the supremum over $x \\in [-2, 2]$ of $\\int_{-1}^{1} f(t)\\, g(x + t)\\, dt$. A real number $c$ is a universal lower bound when $c \\le S(f, g)$ for\u2026","title":"Universal lower bound in the continuous overlap model"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-minimum-overlap-constant","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousMinimumOverlapConstant","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-minimum-overlap-constant","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For an admissible pair $(f, g)$ let $S(f, g)$ be its pair score, the supremum over $x \\in [-2, 2]$ of $\\int_{-1}^{1} f(t)\\,g(x + t)\\,dt$. The continuous minimum-overlap constant $C_{1b}$ is the supremum of the set of real numbers $c$ such that $c \\le S(f, g)$ for every admissible pair $(f, g)$.","title":"The continuous Erd\u0151s minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/c","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.C\u2081\u1d66","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Notation "],["math","\\(C_{1b}\\)"]],"source":"Notation $C_{1b}$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#c","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"We write $C_{1b}$ for the continuous Erd\u0151s minimum-overlap constant.","title":"Notation $C_{1b}$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-minimum-overlap-constant-le-h","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousMinimumOverlapConstant_le_half","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","The continuous minimum-overlap constant is at most "],["math","\\(1/2\\)"]],"source":"The continuous minimum-overlap constant is at most $1/2$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-minimum-overlap-constant-le-h","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 1/2$.","title":"The continuous minimum-overlap constant is at most $1/2$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-discrete-equivalence","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousDiscreteEquivalence","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","The proposition that "],["math","\\(M(N)/N\\)"],["text"," tends to "],["math","\\(C_{1b}\\)"]],"source":"The proposition that $M(N)/N$ tends to $C_{1b}$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-discrete-equivalence","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"For a natural number $N$ let $M(N)$ be the Erd\u0151s minimum overlap function: the least, over all pairs of disjoint finite sets $A, B \\subseteq \\mathbb{Z}$ with $A \\cup B = \\{1, \\dots, 2N\\}$ and $|A| = |B|$, of the largest number of pairs $(a, b) \\in A \\times B$ with a given difference $a - b$. This is the proposition that $M(N)/N$ tends to the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ as $N \\to \\infty$.","title":"The proposition that $M(N)/N$ tends to $C_{1b}$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-discrete-asymptotic-character","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousDiscreteAsymptoticCharacterization","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","The proposition that "],["math","\\(C_{1b}\\)"],["text"," is the largest asymptotic overlap constant"]],"source":"The proposition that $C_{1b}$ is the largest asymptotic overlap constant"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-discrete-asymptotic-character","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"Call a real number $c$ a universal asymptotic lower bound when for every $\\varepsilon > 0$ there is a positive integer $N_0$ such that for every $N \\ge N_0$ and all disjoint finite sets $A, B \\subseteq \\mathbb{Z}$ with $A \\cup B = \\{1, \\dots, 2N\\}$ and $|A| = |B|$, some integer $x$ satisfies $(c - \\varepsilon) N \\le \\#\\{(a, b) \\in A \\times B : a - b = x\\}$. This is the proposition that the continuous Erd\u0151s\u2026","title":"The proposition that $C_{1b}$ is the largest asymptotic overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-e1955-half","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_E1955_half","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s's upper bound "],["math","\\(1/2\\)"],["text"," for the minimum-overlap constant (statement)"]],"source":"Erd\u0151s's upper bound $1/2$ for the minimum-overlap constant (statement)"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-e1955-half","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The statement that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 1/2$.","title":"Erd\u0151s's upper bound $1/2$ for the minimum-overlap constant (statement)"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-erdos-unpublished-four-ninth","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_Erdos_unpublished_fourNinths","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s's unpublished upper bound "],["math","\\(4/9\\)"],["text"," for the minimum-overlap constant (statement)"]],"source":"Erd\u0151s's unpublished upper bound $4/9$ for the minimum-overlap constant (statement)"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-erdos-unpublished-four-ninth","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The statement that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 4/9$.","title":"Erd\u0151s's unpublished upper bound $4/9$ for the minimum-overlap constant (statement)"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-mrs1956-five-twelfths","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_MRS1956_fiveTwelfths","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the Motzkin\u2013Ralston\u2013Selfridge upper bound "],["math","\\(5/12\\)"],["text"," for the minimum-overlap constant"]],"source":"Statement of the Motzkin\u2013Ralston\u2013Selfridge upper bound $5/12$ for the minimum-overlap constant"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-mrs1956-five-twelfths","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"Statement of the Motzkin\u2013Ralston\u2013Selfridge upper bound $5/12$ for the minimum-overlap constant. The proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 5/12$.","title":"Statement of the Motzkin\u2013Ralston\u2013Selfridge upper bound $5/12$ for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-mrs1956-point-four","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_MRS1956_pointFour","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the Motzkin\u2013Ralston\u2013Selfridge upper bound "],["math","\\(0.4\\)"],["text"," for the minimum-overlap constant"]],"source":"Statement of the Motzkin\u2013Ralston\u2013Selfridge upper bound $0.4$ for the minimum-overlap constant"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-mrs1956-point-four","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"Statement of the Motzkin\u2013Ralston\u2013Selfridge upper bound $0.4$ for the minimum-overlap constant. The proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.4$.","title":"Statement of the Motzkin\u2013Ralston\u2013Selfridge upper bound $0.4$ for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-haugland-unpublished-1993","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_Haugland_unpublished_1993","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-haugland-unpublished-1993","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.385694$.","title":"Statement of Haugland's unpublished 1993 upper bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-h1996","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_H1996","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-h1996","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.382002$.","title":"Statement of Haugland's 1996 upper bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-h2016","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_H2016","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-h2016","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.380926$.","title":"Statement of Haugland's 2016 upper bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-ggswt2025-alpha-evolve","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_GGSWT2025_AlphaEvolve","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-ggswt2025-alpha-evolve","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The statement that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.380924$.","title":"AlphaEvolve upper bound for the minimum-overlap constant (statement)"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-yklbmwkczgs2026-tttdiscover","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_YKLBMWKCZGS2026_TTTDiscover","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-yklbmwkczgs2026-tttdiscover","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"This is the proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ is at most $0.380876$.","title":"TTT-Discover 2026 upper bound statement for the minimum overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-t2026-together-ai","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_T2026_TogetherAI","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-t2026-together-ai","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"This is the proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ is at most $0.380871$.","title":"Together AI 2026 upper bound statement for the minimum overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-yltlystyllgdhzswzshmelczx202","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_YLTLYSTYLLGDHZSWZSHMELCZX2026_SimpleTES","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-yltlystyllgdhzswzshmelczx202","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"This is the proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ is at most $0.380868$.","title":"SimpleTES 2026 upper bound statement for the minimum overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-e1955-quarter","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_E1955_quarter","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","The proposition "],["math","\\(1/4 \\le C_{1b}\\)"]],"source":"The proposition $1/4 \\le C_{1b}$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-e1955-quarter","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"This is the proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $1/4 \\le C_{1b}$.","title":"The proposition $1/4 \\le C_{1b}$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-scherk-unpublished-1955","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_Scherk_unpublished_1955","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-scherk-unpublished-1955","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The statement that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $1 - 1/\\sqrt{2} \\le C_{1b}$.","title":"Scherk's lower bound for the minimum-overlap constant (statement)"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-s1958","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_S1958","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","The proposition "],["math","\\((4 - \\sqrt{6})/5 \\le C_{1b}\\)"]],"source":"The proposition $(4 - \\sqrt{6})/5 \\le C_{1b}$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-s1958","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"This is the proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $(4 - \\sqrt{6})/5 \\le C_{1b}$.","title":"The proposition $(4 - \\sqrt{6})/5 \\le C_{1b}$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-m1959","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_M1959","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","The proposition "],["math","\\(\\sqrt{4 - \\sqrt{15}} \\le C_{1b}\\)"]],"source":"The proposition $\\sqrt{4 - \\sqrt{15}} \\le C_{1b}$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-m1959","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"This is the proposition that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $\\sqrt{4 - \\sqrt{15}} \\le C_{1b}$.","title":"The proposition $\\sqrt{4 - \\sqrt{15}} \\le C_{1b}$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-w2022","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_W2022","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-w2022","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"fully_proved","summary":"The statement that the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $0.379005 \\le C_{1b}$, where $0.379005$ is the exact rational number $379005/1000000$.","title":"White's lower bound for the minimum-overlap constant (statement)"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-discrete-equivalence-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousDiscreteEquivalence_holds","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","The normalized minimum overlap tends to "],["math","\\(C_{1b}\\)"]],"source":"The normalized minimum overlap tends to $C_{1b}$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-discrete-equivalence-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"For a natural number $N$ let $M(N)$ be the Erd\u0151s minimum overlap function: the least, over all pairs of disjoint finite sets $A, B \\subseteq \\mathbb{Z}$ with $A \\cup B = \\{1, \\dots, 2N\\}$ and $|A| = |B|$, of the largest number of pairs $(a, b) \\in A \\times B$ with a given difference $a - b$. Then $M(N)/N$ tends to the continuous Erd\u0151s minimum-overlap constant $C_{1b}$ as $N \\to \\infty$.","title":"The normalized minimum overlap tends to $C_{1b}$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/continuous-discrete-asymptotic-character-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.continuousDiscreteAsymptoticCharacterization_holds","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","The continuous constant "],["math","\\(C_{1b}\\)"],["text"," is the largest asymptotic overlap constant"]],"source":"The continuous constant $C_{1b}$ is the largest asymptotic overlap constant"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#continuous-discrete-asymptotic-character-2","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"Call a real number $c$ a universal asymptotic lower bound when for every $\\varepsilon > 0$ there is a positive integer $N_0$ such that for every $N \\ge N_0$ and all disjoint finite sets $A, B \\subseteq \\mathbb{Z}$ with $A \\cup B = \\{1, \\dots, 2N\\}$ and $|A| = |B|$, some integer $x$ satisfies $(c - \\varepsilon) N \\le \\#\\{(a, b) \\in A \\times B : a - b = x\\}$. Then the continuous Erd\u0151s minimum-overlap constant\u2026","title":"The continuous constant $C_{1b}$ is the largest asymptotic overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-erdos-unpublished-four-ninth-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_Erdos_unpublished_fourNinths_holds","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s's unpublished upper bound "],["math","\\(4/9\\)"],["text"," for the minimum-overlap constant"]],"source":"Erd\u0151s's unpublished upper bound $4/9$ for the minimum-overlap constant"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-erdos-unpublished-four-ninth-2","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 4/9$.","title":"Erd\u0151s's unpublished upper bound $4/9$ for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-mrs1956-five-twelfths-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_MRS1956_fiveTwelfths_holds","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Motzkin\u2013Ralston\u2013Selfridge upper bound "],["math","\\(5/12\\)"],["text"," for the minimum-overlap constant"]],"source":"Motzkin\u2013Ralston\u2013Selfridge upper bound $5/12$ for the minimum-overlap constant"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-mrs1956-five-twelfths-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 5/12$.","title":"Motzkin\u2013Ralston\u2013Selfridge upper bound $5/12$ for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-mrs1956-point-four-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_MRS1956_pointFour_holds","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Motzkin\u2013Ralston\u2013Selfridge upper bound "],["math","\\(0.4\\)"],["text"," for the minimum-overlap constant"]],"source":"Motzkin\u2013Ralston\u2013Selfridge upper bound $0.4$ for the minimum-overlap constant"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-mrs1956-point-four-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.4$.","title":"Motzkin\u2013Ralston\u2013Selfridge upper bound $0.4$ for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-haugland-unpublished-1993-ho","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_Haugland_unpublished_1993_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-haugland-unpublished-1993-ho","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.385694$.","title":"Haugland's unpublished 1993 upper bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-h1996-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_H1996_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-h1996-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.382002$.","title":"Haugland's 1996 upper bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-h2016-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_H2016_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-h2016-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.380926$.","title":"Haugland's 2016 upper bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-ggswt2025-alpha-evolve-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_GGSWT2025_AlphaEvolve_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-ggswt2025-alpha-evolve-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.380924$.","title":"AlphaEvolve upper bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-yklbmwkczgs2026-tttdiscover-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_YKLBMWKCZGS2026_TTTDiscover_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-yklbmwkczgs2026-tttdiscover-2","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ is at most $0.380876$.","title":"TTT-Discover 2026 upper bound for the minimum overlap constant holds"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-t2026-together-ai-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_T2026_TogetherAI_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-t2026-together-ai-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $C_{1b} \\le 0.380871$.","title":"Together AI's 2026 upper bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/upper-bound-yltlystyllgdhzswzshmelczx202-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.upperBound_YLTLYSTYLLGDHZSWZSHMELCZX2026_SimpleTES_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#upper-bound-yltlystyllgdhzswzshmelczx202-2","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ is at most $0.380868$.","title":"SimpleTES 2026 upper bound for the minimum overlap constant holds"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-e1955-quarter-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_E1955_quarter_holds","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s's lower bound "],["math","\\(C_{1b} \\ge 1/4\\)"]],"source":"Erd\u0151s's lower bound $C_{1b} \\ge 1/4$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-e1955-quarter-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $1/4 \\le C_{1b}$.","title":"Erd\u0151s's lower bound $C_{1b} \\ge 1/4$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-scherk-unpublished-1955-hold","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_Scherk_unpublished_1955_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-scherk-unpublished-1955-hold","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $1 - 1/\\sqrt{2} \\le C_{1b}$.","title":"Scherk's lower bound for the minimum-overlap constant"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-s1958-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_S1958_holds","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","\u015awierczkowski's lower bound "],["math","\\(C_{1b} \\ge (4 - \\sqrt{6})/5\\)"]],"source":"\u015awierczkowski's lower bound $C_{1b} \\ge (4 - \\sqrt{6})/5$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-s1958-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $(4 - \\sqrt{6})/5 \\le C_{1b}$.","title":"\u015awierczkowski's lower bound $C_{1b} \\ge (4 - \\sqrt{6})/5$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-m1959-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_M1959_holds","parent":13160,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moser's lower bound "],["math","\\(C_{1b} \\ge \\sqrt{4 - \\sqrt{15}}\\)"]],"source":"Moser's lower bound $C_{1b} \\ge \\sqrt{4 - \\sqrt{15}}$"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-m1959-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $\\sqrt{4 - \\sqrt{15}} \\le C_{1b}$.","title":"Moser's lower bound $C_{1b} \\ge \\sqrt{4 - \\sqrt{15}}$"},{"id":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/lower-bound-w2022-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosMinimumOverlapConstantWanted.lowerBound_W2022_holds","parent":13160,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem/index.html#lower-bound-w2022-holds","scope":"NumberTheory/additive-number-theory/erd-s-minimum-overlap-problem","status":"can_prove","summary":"The continuous Erd\u0151s minimum-overlap constant $C_{1b}$ satisfies $0.379005 \\le C_{1b}$.","title":"White's lower bound for the minimum-overlap constant"},{"data_url":"scopes/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate.json","id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","title":"Geometric-progression-free sets of Hurwitz quaternions"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/card-hurwitz-norm-eq","kind":"node","lean":"MetaMathlibExt.card_hurwitz_norm_eq","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#card-hurwitz-norm-eq","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion $q = x_0 + x_1 i + x_2 j + x_3 k$ whose rational coordinates $x_0, x_1, x_2, x_3$ are either all integers or all halves of odd integers, and its norm is $\\mathrm{Nm}(q) = x_0^2 + x_1^2 + x_2^2 + x_3^2$. For every positive integer $N$, the number of Hurwitz quaternions of norm $N$ is $24 \\sum_{d \\mid N,\\ d \\text{ odd}} d$, that is, $24$ times the sum of the odd divisors of $N$.","title":"Number of Hurwitz quaternions of a given norm"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/card-hurwitz-norm-le-asymptotic","kind":"node","lean":"MetaMathlibExt.card_hurwitz_norm_le_asymptotic","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#card-hurwitz-norm-le-asymptotic","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; its norm is the sum of the squares of its coordinates. For a natural number $N$ let $S(N)$ be the set of nonzero Hurwitz quaternions of norm at most $N$.","title":"Counting Hurwitz quaternions of bounded norm"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/density-hurwitz-norm-valuations-in","kind":"node","lean":"MetaMathlibExt.density_hurwitz_norm_valuations_in","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#density-hurwitz-norm-valuations-in","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; its norm is the sum of the squares of its coordinates. For a natural number $N$ let $S(N)$ be the set of nonzero Hurwitz quaternions of norm at most $N$. Let $A$ be a set of natural numbers, let $G$ be the set of positive integers $n$ such that for every prime $p$ the exponent $\\nu_p(n)$ of\u2026","title":"Density of Hurwitz quaternions whose norm has prescribed prime exponents"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/density-hurwitz-norm-in-rankin-greedy","kind":"node","lean":"MetaMathlibExt.density_hurwitz_norm_in_rankin_greedy","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#density-hurwitz-norm-in-rankin-greedy","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; its norm is the sum of the squares of its coordinates. For a natural number $N$ let $S(N)$ be the set of nonzero Hurwitz quaternions of norm at most $N$. Let $G$ be the set of positive integers $n$ such that, for every prime $p$, the base-$3$ digits of the exponent $\\nu_p(n)$ of $p$ in $n$\u2026","title":"Density of Hurwitz quaternions with norm in Rankin's greedy set"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/upper-density-hurwitz-norm-in-interval-u","kind":"node","lean":"MetaMathlibExt.upperDensity_hurwitz_norm_in_interval_union","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#upper-density-hurwitz-norm-in-interval-u","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"Let $Q_{\\mathrm{Hur}}$ be the set of Hurwitz quaternions, the rational quaternions whose four coordinates are all integers or all lie in $\\mathbb{Z} + \\tfrac12$, and let $\\mathrm{Nm}(q)$ be the sum of the squares of the coordinates of $q$. For $N \\in \\mathbb{N}$ let $S(N)$ be the set of nonzero $q \\in Q_{\\mathrm{Hur}}$ with $\\mathrm{Nm}(q) \\le N$, and define the upper density of a set $A$ of quaternions as\u2026","title":"Upper density of Hurwitz quaternions with norms in a lacunary union of intervals"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/disjoint-progressions-hurwitz-card","kind":"node","lean":"MetaMathlibExt.disjoint_progressions_hurwitz_card","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#disjoint-progressions-hurwitz-card","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; its norm is the sum of the squares of its coordinates. For a natural number $N$ let $S(N)$ be the set of nonzero Hurwitz quaternions of norm at most $N$. Fix a Hurwitz quaternion $R$ of norm $2$, and for a natural number $N$ let $E(N)$ be the set of triples $(bR^{3n}, bR^{3n+1}, bR^{3n+2})$,\u2026","title":"Many disjoint geometric progressions among Hurwitz quaternions"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/upper-density-le-of-hurwitz-gp-free","kind":"node","lean":"MetaMathlibExt.upperDensity_le_of_hurwitz_gpFree","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#upper-density-le-of-hurwitz-gp-free","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"Let $Q_{\\mathrm{Hur}}$ be the set of Hurwitz quaternions, the rational quaternions whose four coordinates are all integers or all lie in $\\mathbb{Z} + \\tfrac12$, and let $\\mathrm{Nm}(q)$ be the sum of the squares of the coordinates of $q$. For $N \\in \\mathbb{N}$ let $S(N)$ be the set of nonzero $q \\in Q_{\\mathrm{Hur}}$ with $\\mathrm{Nm}(q) \\le N$. Let $A \\subseteq Q_{\\mathrm{Hur}}$ be free of 3-term geometric\u2026","title":"Upper bound for geometric-progression-free Hurwitz sets"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/s-sup-upper-density-hurwitz-gp-free-boun","kind":"node","lean":"MetaMathlibExt.sSup_upperDensity_hurwitz_gpFree_bounds","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#s-sup-upper-density-hurwitz-gp-free-boun","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"Let $Q_{\\mathrm{Hur}}$ be the set of Hurwitz quaternions, the rational quaternions whose four coordinates are all integers or all lie in $\\mathbb{Z} + \\tfrac12$, and let $\\mathrm{Nm}(q)$ be the sum of the squares of the coordinates of $q$. For $N \\in \\mathbb{N}$ let $S(N)$ be the set of nonzero $q \\in Q_{\\mathrm{Hur}}$ with $\\mathrm{Nm}(q) \\le N$, and define the upper density of a set $A$ of quaternions as\u2026","title":"Bounds on the supremum of upper densities of geometric-progression-free Hurwitz sets"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/quaternion-greedy-squarefree-norm-and-lo","kind":"node","lean":"MetaMathlibExt.quaternionGreedy_squarefree_norm_and_lowerDensity","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#quaternion-greedy-squarefree-norm-and-lo","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; the norm $\\mathrm{Nm}(q)$ of a quaternion $q$ is the sum of the squares of its coordinates. For a natural number $N$ let $S(N)$ be the set of nonzero Hurwitz quaternions of norm at most $N$. Let $\\mathcal{G}$ be a set of quaternions such that a quaternion $q$ lies in $\\mathcal{G}$ if and only\u2026","title":"Squarefree norms and the lower density of the quaternion greedy set"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/quaternion-greedy-ne-norm-preimage","kind":"node","lean":"MetaMathlibExt.quaternionGreedy_ne_norm_preimage","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#quaternion-greedy-ne-norm-preimage","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; the norm $\\mathrm{Nm}(q)$ of a quaternion $q$ is the sum of the squares of its coordinates. Let $\\mathcal{G}$ be a set of quaternions such that a quaternion $q$ lies in $\\mathcal{G}$ if and only if $q$ is a nonzero Hurwitz quaternion and there are no $a \\in \\mathcal{G}$ and Hurwitz quaternion\u2026","title":"The quaternion greedy set is not determined by norms"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/quaternion-greedy-diff-rankin-infinite","kind":"node","lean":"MetaMathlibExt.quaternionGreedy_diff_rankin_infinite","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#quaternion-greedy-diff-rankin-infinite","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; the norm $\\mathrm{Nm}(q)$ of a quaternion $q$ is the sum of the squares of its coordinates. Let $\\mathcal{G}$ be a set of quaternions such that a quaternion $q$ lies in $\\mathcal{G}$ if and only if $q$ is a nonzero Hurwitz quaternion and there are no $a \\in \\mathcal{G}$ and Hurwitz quaternion\u2026","title":"The quaternion greedy set and the norm preimage of Rankin's set differ infinitely"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/exists-hurwitz-norm-sq-not-unit-mul-sq","kind":"node","lean":"MetaMathlibExt.exists_hurwitz_norm_sq_not_unit_mul_sq","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#exists-hurwitz-norm-sq-not-unit-mul-sq","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; its norm is the sum of the squares of its coordinates, and a Hurwitz unit is a Hurwitz quaternion of norm $1$. Let $r$ be a positive integer that is not a sum of three squares of integers, or that is divisible by an odd integer greater than $24$. Then some Hurwitz quaternion $Q$ of norm $r^2$\u2026","title":"Hurwitz quaternions of square norm that are not a unit times a square"},{"id":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/quaternion-greedy-mem-of-not-unit-mul-sq","kind":"node","lean":"MetaMathlibExt.quaternionGreedy_mem_of_not_unit_mul_sq","parent":13217,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate/index.html#quaternion-greedy-mem-of-not-unit-mul-sq","scope":"NumberTheory/additive-number-theory/geometric-progression-free-sets-of-hurwitz-quate","status":"can_prove","summary":"A Hurwitz quaternion is a quaternion whose four rational coordinates are either all integers or all halves of odd integers; its norm $\\mathrm{Nm}(q)$ is the sum of the squares of its coordinates, and a Hurwitz unit is a Hurwitz quaternion of norm $1$. Let $\\mathcal{G}$ be a set of quaternions such that a quaternion $q$ lies in $\\mathcal{G}$ if and only if $q$ is a nonzero Hurwitz quaternion and there are no\u2026","title":"Quaternions of prime-square norm separating the greedy set from Rankin's set"},{"data_url":"scopes/NumberTheory/additive-number-theory/natural-and-logarithmic-density.json","id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","title":"Natural and logarithmic density"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/partial-density","kind":"node","lean":"Set.partialDensity","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#partial-density","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"Let $\\beta$ be a preorder in which every set $\\{x \\in \\beta \\mid x \\le b\\}$ is finite, let $S, A \\subseteq \\beta$ and $b \\in \\beta$, and write $A_{\\lt b} = \\{x \\in A \\mid x < b\\}$. The partial density of $S$ relative to $A$ at $b$ is the proportion $\\lvert S \\cap A_{\\lt b} \\rvert / \\lvert A_{\\lt b} \\rvert$ of elements of $A_{\\lt b}$ that lie in $S$, taken to be $0$ when $A_{\\lt b}$ is empty. By default $A = \\beta$.","title":"Partial density"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/upper-density","kind":"node","lean":"Set.upperDensity","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#upper-density","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"Let $\\beta$ be a preorder in which every set $\\{x \\in \\beta \\mid x \\le b\\}$ is finite, and let $S, A \\subseteq \\beta$. The upper density of $S$ relative to $A$ is the limit superior, as $b \\to \\infty$ in $\\beta$, of the partial densities of $S$ relative to $A$ at $b$. By default $A = \\beta$.","title":"Upper density"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/lower-density","kind":"node","lean":"Set.lowerDensity","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#lower-density","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"Let $\\beta$ be a preorder in which every set $\\{x \\in \\beta \\mid x \\le b\\}$ is finite, and let $S, A \\subseteq \\beta$. The lower density of $S$ relative to $A$ is the limit inferior, as $b \\to \\infty$ in $\\beta$, of the partial densities of $S$ relative to $A$ at $b$. By default $A = \\beta$.","title":"Lower density"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/has-density","kind":"node","lean":"Set.HasDensity","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#has-density","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"Let $\\beta$ be a preorder in which every set $\\{x \\in \\beta \\mid x \\le b\\}$ is finite, let $S, A \\subseteq \\beta$ and $\\alpha \\in \\mathbb{R}$, and write $\\beta_{\\lt b} = \\{x \\in \\beta \\mid x < b\\}$. The set $S$ has density $\\alpha$ relative to $A$ if the partial density $\\lvert S \\cap A \\cap \\beta_{\\lt b} \\rvert / \\lvert A \\cap \\beta_{\\lt b} \\rvert$, taken to be $0$ when the denominator is $0$, tends to $\\alpha$ as\u2026","title":"Density of a set"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/has-pos-density","kind":"node","lean":"Set.HasPosDensity","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#has-pos-density","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"Let $\\beta$ be a preorder in which every set $\\{x \\in \\beta \\mid x \\le b\\}$ is finite, and let $S, A \\subseteq \\beta$. The set $S$ has positive density relative to $A$ if there is a real number $\\alpha > 0$ such that $S$ has density $\\alpha$ relative to $A$. By default $A = \\beta$.","title":"Positive density"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/int-partial-density","kind":"node","lean":"Set.intPartialDensity","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#int-partial-density","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"For a set $S$ of integers and a natural number $N$, the two-sided partial natural density of $S$ at $N$ is $\\lvert S \\cap [-N, N] \\rvert / (2N + 1)$, where $[-N, N]$ is the set of integers $m$ with $-N \\le m \\le N$.","title":"Two-sided partial density"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/has-int-density","kind":"node","lean":"Set.HasIntDensity","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#has-int-density","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"A set $S$ of integers has two-sided natural density $\\alpha \\in \\mathbb{R}$ if $\\lvert S \\cap [-N, N] \\rvert / (2N + 1)$ tends to $\\alpha$ as the natural number $N$ tends to infinity, where $[-N, N]$ is the set of integers $m$ with $-N \\le m \\le N$.","title":"Two-sided natural density"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/univ","kind":"node","lean":"Set.HasDensity.univ","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#univ","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"Let $\\beta$ be a partial order with a least element and at least two elements, in which every closed interval $[a, b]$ is finite. Then the set of all elements of $\\beta$ has density $1$ (relative to all of $\\beta$).","title":"The whole order has density one"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/has-density-even","kind":"node","lean":"Nat.hasDensity_even","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#has-density-even","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"The set of even natural numbers has natural density $1/2$.","title":"Natural density of the even numbers"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/has-density-zero-of-finite","kind":"node","lean":"Nat.hasDensity_zero_of_finite","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#has-density-zero-of-finite","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"Every finite set of natural numbers has natural density $0$.","title":"Finite sets have natural density zero"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/infinite-of-has-density-pos","kind":"node","lean":"Nat.infinite_of_hasDensity_pos","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#infinite-of-has-density-pos","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"If a set $S$ of natural numbers has natural density $\\alpha$ with $\\alpha \\neq 0$, then $S$ is infinite.","title":"Sets of nonzero natural density are infinite"},{"id":"NumberTheory/additive-number-theory/natural-and-logarithmic-density/has-log-density","kind":"node","lean":"Set.HasLogDensity","parent":13231,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/natural-and-logarithmic-density/index.html#has-log-density","scope":"NumberTheory/additive-number-theory/natural-and-logarithmic-density","status":"fully_proved","summary":"A set $A$ of natural numbers has logarithmic density $d \\in \\mathbb{R}$ if $\\frac{1}{\\log n} \\sum_{k \\in A,\\, k \\le n} \\frac{1}{k}$ tends to $d$ as $n \\to \\infty$, where $\\log$ is the natural logarithm, the term $1/k$ for $k = 0$ is taken to be $0$, and the whole expression is taken to be $0$ when $\\log n = 0$.","title":"Logarithmic density"},{"data_url":"scopes/NumberTheory/additive-number-theory/primitive-sets.json","id":"NumberTheory/additive-number-theory/primitive-sets","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/primitive-sets/index.html","scope":"NumberTheory/additive-number-theory/primitive-sets","status":"can_prove","title":"Primitive sets"},{"id":"NumberTheory/additive-number-theory/primitive-sets/is-primitive","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1196Wanted.IsPrimitive","parent":13244,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/primitive-sets/index.html#is-primitive","scope":"NumberTheory/additive-number-theory/primitive-sets","status":"fully_proved","summary":"A subset $A$ of a commutative monoid $M$ is primitive if for all $x, y \\in A$ with $x \\mid y$, the elements $x$ and $y$ are associated, that is, $x u = y$ for some unit $u$ of $M$. In $\\mathbb{N}$, where $1$ is the only unit, this says that no element of $A$ divides a different element of $A$.","title":"Primitive set in a commutative monoid"},{"id":"NumberTheory/additive-number-theory/primitive-sets/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1196Wanted.conjecture","parent":13244,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/primitive-sets/index.html#conjecture","scope":"NumberTheory/additive-number-theory/primitive-sets","status":"fully_proved","summary":"The statement that there is a function $o : \\mathbb{N} \\to \\mathbb{R}$ with $o(x) \\to 0$ as $x \\to \\infty$ such that for every natural number $x > 0$ and every primitive set $A$ of natural numbers contained in $[x, \\infty)$, the series $\\sum_{a \\in A} \\frac{1}{a \\log a}$ is summable and $\\sum_{a \\in A} \\frac{1}{a \\log a} < 1 + o(x)$. Here a set of natural numbers is primitive if no element divides a different\u2026","title":"Asymptotic primitive set bound"},{"id":"NumberTheory/additive-number-theory/primitive-sets/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem1196Wanted.conjecture_holds","parent":13244,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/primitive-sets/index.html#conjecture-holds","scope":"NumberTheory/additive-number-theory/primitive-sets","status":"can_prove","summary":"There is a function $o : \\mathbb{N} \\to \\mathbb{R}$ with $o(x) \\to 0$ as $x \\to \\infty$ such that for every natural number $x > 0$ and every primitive set $A$ of natural numbers contained in $[x, \\infty)$, the series $\\sum_{a \\in A} \\frac{1}{a \\log a}$ is summable and $\\sum_{a \\in A} \\frac{1}{a \\log a} < 1 + o(x)$. Here a set of natural numbers is primitive if no element divides a different element, and the term\u2026","title":"Erd\u0151s Problem 1196 on primitive sets"},{"data_url":"scopes/NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers.json","id":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/index.html","scope":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers","status":"can_prove","title":"Quotients a/gcd(a, b) in sets of integers"},{"id":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/is-cofactor-lower-bound","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem539Wanted.IsCofactorLowerBound","parent":13248,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/index.html#is-cofactor-lower-bound","scope":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers","status":"fully_proved","summary":"For natural numbers $n$ and $m$, $m$ is a cofactor lower bound for $n$ when, for every set $A$ of exactly $n$ positive integers, the set $\\{a/\\gcd(a,b) : a, b \\in A\\}$ has at least $m$ elements.","title":"Cofactor lower bound"},{"id":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/cofactor-threshold","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem539Wanted.cofactorThreshold","parent":13248,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/index.html#cofactor-threshold","scope":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers","status":"fully_proved","summary":"For a natural number $n$, the cofactor threshold $h(n)$ is the largest natural number $m$ such that for every set $A$ of exactly $n$ positive integers, the set $\\{a/\\gcd(a,b) : a, b \\in A\\}$ has at least $m$ elements.","title":"Erd\u0151s cofactor threshold"},{"id":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem539Wanted.conjecture","parent":13248,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/index.html#conjecture","scope":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers","status":"fully_proved","summary":"Let $h(n)$ denote the cofactor threshold of $n$. The statement is that $\\sqrt{n} = O(h(n))$ as $n \\to \\infty$, that is, there are a real constant $C$ and an $n_0$ such that $\\sqrt{n} \\le C\\,h(n)$ for all natural numbers $n \\ge n_0$.","title":"Statement of Erd\u0151s Problem 539"},{"id":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem539Wanted.conjecture_holds","parent":13248,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers/index.html#conjecture-holds","scope":"NumberTheory/additive-number-theory/quotients-a-gcd-a-b-in-sets-of-integers","status":"can_prove","summary":"For a natural number $n$, let $h(n)$ be the largest natural number $m$ such that for every set $A$ of exactly $n$ positive integers, the set $\\{a/\\gcd(a,b) : a, b \\in A\\}$ has at least $m$ elements. Then $\\sqrt{n} = O(h(n))$ as $n \\to \\infty$: there are a real constant $C$ and an $n_0$ such that $\\sqrt{n} \\le C\\,h(n)$ for all natural numbers $n \\ge n_0$.","title":"Erd\u0151s Problem 539: square-root lower bound for the cofactor threshold"},{"data_url":"scopes/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets.json","id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"can_prove","title":"Reciprocal lcm sums of integer sets"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/trunc-log","kind":"node","lean":"MetaMathlibExt.truncLog","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#trunc-log","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a real number $x$, the truncated logarithm is $\\operatorname{Log} x = \\max(\\log x, 1)$. Here $\\log$ is extended to all reals by $\\log 0 = 0$ and $\\log x = \\log |x|$ for $x < 0$, so the definition is total; the paper uses it only for $x > 0$.","title":"Truncated logarithm"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/trunc-log2","kind":"node","lean":"MetaMathlibExt.truncLog2","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#trunc-log2","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a real number $x$, the second iterated truncated logarithm is $\\operatorname{Log}_2 x = \\operatorname{Log}(\\operatorname{Log} x)$, where $\\operatorname{Log}$ is the truncated logarithm.","title":"Second iterated truncated logarithm"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/trunc-log3","kind":"node","lean":"MetaMathlibExt.truncLog3","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#trunc-log3","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For real $x$, let $\\operatorname{Log} x = \\max(\\log x, 1)$ be the truncated logarithm, where $\\log$ is the natural logarithm extended by $\\log x = \\log |x|$ for $x < 0$ and $\\log 0 = 0$, and let $\\operatorname{Log}_2 x = \\operatorname{Log}(\\operatorname{Log} x)$. The third iterated truncated logarithm is the function on all of $\\mathbb{R}$ given by\u2026","title":"Third iterated truncated logarithm"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/one-le-trunc-log","kind":"node","lean":"MetaMathlibExt.one_le_truncLog","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#one-le-trunc-log","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For every real number $x$, the truncated logarithm $\\operatorname{Log} x = \\max(\\log x, 1)$ satisfies $\\operatorname{Log} x \\ge 1$.","title":"The truncated logarithm is at least one"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/naturals-in-set-up-to","kind":"node","lean":"MetaMathlibExt.naturalsInSetUpTo","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#naturals-in-set-up-to","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a set $A$ of natural numbers and a real number $x$, this is the finite set of positive integers $n \\in A$ with $n \\le x$; it is empty when $x < 1$.","title":"Elements of a set up to a bound"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/naturals-in-set-up-to-empty","kind":"node","lean":"MetaMathlibExt.naturalsInSetUpTo_empty","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#naturals-in-set-up-to-empty","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For every real number $x$, the finite set of elements $n$ of the empty set with $1 \\le n \\le x$ is empty.","title":"The empty set has no elements up to any bound"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/recip-sum-up-to","kind":"node","lean":"MetaMathlibExt.recipSumUpTo","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#recip-sum-up-to","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a set $A$ of natural numbers and a real number $x$, this is the finite sum $\\sum_{n \\in A,\\, 1 \\le n \\le x} 1/n$ of the reciprocals of the positive integers in $A$ that are at most $x$; it is $0$ when $x < 1$.","title":"Harmonic sum of a set up to a bound"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/recip-sum-up-to-empty","kind":"node","lean":"MetaMathlibExt.recipSumUpTo_empty","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#recip-sum-up-to-empty","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For every real number $x$, $\\sum_{n \\in \\emptyset,\\, n \\le x} 1/n = 0$.","title":"The harmonic sum over the empty set vanishes"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/lcm-reciprocal-sum","kind":"node","lean":"MetaMathlibExt.lcmReciprocalSum","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#lcm-reciprocal-sum","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a set $A$ of natural numbers and a real number $x$, the double sum of reciprocal least common multiples of $A$ up to $x$ is the real number $\\sum 1/\\operatorname{lcm}(n, m)$, summed over all ordered pairs $(n, m)$ with $n, m \\in A$, $1 \\le n \\le x$ and $1 \\le m \\le x$.","title":"Double sum of reciprocal least common multiples"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/lcm-reciprocal-sum-empty","kind":"node","lean":"MetaMathlibExt.lcmReciprocalSum_empty","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#lcm-reciprocal-sum-empty","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For every real number $x$, the double sum of reciprocal least common multiples of the empty set up to $x$ is $0$.","title":"Double reciprocal lcm sum of the empty set"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/gcd-logarithmic-moment","kind":"node","lean":"MetaMathlibExt.gcdLogarithmicMoment","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#gcd-logarithmic-moment","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a set $A$ of natural numbers and a real number $x$, the logarithmic gcd moment of $A$ up to $x$ is the sum of $\\gcd(n, m)/(nm)$ over all ordered pairs $(n, m)$ with $n, m \\in A$, $1 \\le n \\le x$ and $1 \\le m \\le x$.","title":"Logarithmic gcd moment"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/lcm-defect","kind":"node","lean":"MetaMathlibExt.lcmDefect","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#lcm-defect","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a set $A$ of natural numbers and a real number $x$, let $S_A(x) = \\sum_{n \\in A,\\ 1 \\le n \\le x} 1/n$ be the harmonic sum of $A$ up to $x$, and let $G_A(x)$ be the sum of $\\gcd(n, m)/(nm)$ over all ordered pairs $(n, m)$ with $n, m \\in A$, $1 \\le n \\le x$ and $1 \\le m \\le x$. The gcd defect of $A$ up to $x$ is $\\mathcal{E}_A(x) = G_A(x)/S_A(x)^2 - 1$, where division by $0$ gives $0$; in particular\u2026","title":"Gcd defect"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/lcm-defect-empty","kind":"node","lean":"MetaMathlibExt.lcmDefect_empty","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#lcm-defect-empty","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For every real number $x$, the gcd defect of the empty set up to $x$ equals $-1$.","title":"The gcd defect of the empty set"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/prime-product-layer","kind":"node","lean":"MetaMathlibExt.primeProductLayer","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#prime-product-layer","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a set $\\mathcal{P}$ of natural numbers and $k \\in \\mathbb{N}$, $\\mathcal{P}^{[k]}$ is the set of squarefree natural numbers $n$ whose prime divisors all lie in $\\mathcal{P}$ and which have exactly $k$ distinct prime divisors; that is, the products of exactly $k$ distinct primes from $\\mathcal{P}$. In particular, $\\mathcal{P}^{[0]} = \\{1\\}$.","title":"Products of a fixed number of distinct primes"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/prime-products-at-most","kind":"node","lean":"MetaMathlibExt.primeProductsAtMost","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#prime-products-at-most","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For a set $\\mathcal{P}$ of natural numbers and $k \\in \\mathbb{N}$, $\\mathcal{P}^{[\\le k]}$ is the set of natural numbers that lie in $\\mathcal{P}^{[j]}$ for some $j \\le k$, where $\\mathcal{P}^{[j]}$ is the set of products of exactly $j$ distinct primes from $\\mathcal{P}$; that is, $\\mathcal{P}^{[\\le k]} = \\bigcup_{j \\le k} \\mathcal{P}^{[j]}$.","title":"Products of at most a fixed number of distinct primes"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/prime-products-at-most-zero","kind":"node","lean":"MetaMathlibExt.primeProductsAtMost_zero","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#prime-products-at-most-zero","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"For every set $\\mathcal{P}$ of natural numbers, $\\mathcal{P}^{[\\le 0]} = \\mathcal{P}^{[0]}$, where $\\mathcal{P}^{[\\le k]}$ and $\\mathcal{P}^{[k]}$ are the products of at most, respectively exactly, $k$ distinct primes from $\\mathcal{P}$.","title":"At most zero primes means exactly zero primes"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/lcm-dense-growth-f","kind":"node","lean":"MetaMathlibExt.lcmDenseGrowthF","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#lcm-dense-growth-f","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"fully_proved","summary":"Let $\\operatorname{Log}_2$ be the second iterated truncated logarithm. For real numbers $C_0$ and $x$, let $k$ be the natural number $\\max(\\lceil C_0 \\sqrt{\\operatorname{Log}_2 x}\\,\\rceil - 1, 0)$, and set $h = \\operatorname{Log}_2 x - k^2/C_0^2 + 1$, $x' = \\exp(\\exp((k + 1)^2/C_0^2))$, $h' = \\operatorname{Log}_2 x' - k^2/C_0^2 + 1$ and $\\psi = 1 + h^2/h'$. The growth function at $(C_0, x)$ is\u2026","title":"Growth function of the dense-set construction"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/recip-sum-prime-product-layer-asymp","kind":"node","lean":"MetaMathlibExt.recipSum_primeProductLayer_asymp","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#recip-sum-prime-product-layer-asymp","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"can_prove","summary":"There are constants $c > 0$ and $C > 0$ such that the following holds. Let $\\mathcal{P}$ be a set of primes, let $k$ be a natural number, let $x \\ge 1$ be real, and put $S = \\sum_{p \\in \\mathcal{P},\\, p \\le x} 1/p$, $T_k = \\sum_{n \\in \\mathcal{P}^{[k]},\\, n \\le x} 1/n$ and $T_{\\le k} = \\sum_{n \\in \\mathcal{P}^{[\\le k]},\\, n \\le x} 1/n$, where $\\mathcal{P}^{[k]}$ and $\\mathcal{P}^{[\\le k]}$ are the sets of products\u2026","title":"Harmonic sums over products of distinct primes"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/exists-dense-set-large-lcm","kind":"node","lean":"MetaMathlibExt.exists_dense_set_large_lcm","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#exists-dense-set-large-lcm","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"can_prove","summary":"Let $\\operatorname{Log}_2$ and $\\operatorname{Log}_3$ be the second and third iterated truncated logarithms. For a set $A$ of natural numbers and real $x$, write $S_A(x) = \\sum_{n \\in A,\\ 1 \\le n \\le x} 1/n$, and write $L_A(x)$ for the sum of $1/\\operatorname{lcm}(n, m)$ over all ordered pairs $(n, m)$ with $n, m \\in A$, $1 \\le n \\le x$ and $1 \\le m \\le x$. Then for every real $C_0 > 0$ there is a set $A$ of\u2026","title":"Dense sets with unusually large least common multiples"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/exists-dense-set-controlled-defect","kind":"node","lean":"MetaMathlibExt.exists_dense_set_controlled_defect","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#exists-dense-set-controlled-defect","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"can_prove","summary":"Let $e$ be Euler's number. There is an absolute constant $C > 0$ with the following property. Let $C_0 > 0$ be real, and let $F_{C_0}$ be the growth function of the dense-set construction. Then there is a set $A$ of natural numbers such that, writing $S_A(x) = \\sum_{n \\in A,\\ 1 \\le n \\le x} 1/n$ for its harmonic sum up to $x$ and $\\mathcal{E}_A(x)$ for its gcd defect up to $x$, the following two statements hold.\u2026","title":"Dense sets with controlled gcd defect"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/exists-dense-sets-baby","kind":"node","lean":"MetaMathlibExt.exists_dense_sets_baby","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#exists-dense-sets-baby","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"can_prove","summary":"Let $e$ be Euler's number, let $\\operatorname{Log}_2$ and $\\operatorname{Log}_3$ be the second and third iterated truncated logarithms, and for a set $B$ of natural numbers and real $x$ write $S_B(x) = \\sum_{n \\in B,\\ 1 \\le n \\le x} 1/n$ and $\\mathcal{E}_B(x)$ for the gcd defect of $B$ up to $x$. There is an absolute constant $C > 0$ such that for every real $C_0 > 0$ there are sets $A_0, A_1, A_2, \\dots$ of\u2026","title":"Baby version of the dense-set construction"},{"id":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/uniform-harmonic-bound-of-defect","kind":"node","lean":"MetaMathlibExt.uniform_harmonic_bound_of_defect","parent":13253,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets/index.html#uniform-harmonic-bound-of-defect","scope":"NumberTheory/additive-number-theory/reciprocal-lcm-sums-of-integer-sets","status":"can_prove","summary":"For a set $A$ of natural numbers and a real number $x$, let $S_A(x)$ be the sum of $1/n$ over $n \\in A$ with $1 \\le n \\le x$, and let the defect of $A$ at $x$ be the sum of $\\gcd(n, m)/(nm)$ over $n, m \\in A$ with $1 \\le n, m \\le x$, divided by $S_A(x)^2$, minus $1$ (the defect is $-1$ when $S_A(x) = 0$). Let $\\operatorname{Log} x = \\max(\\log x, 1)$ be the truncated logarithm, let\u2026","title":"Harmonic sums of sets with small gcd defect"},{"data_url":"scopes/NumberTheory/additive-number-theory/romanoff-s-theorem.json","id":"NumberTheory/additive-number-theory/romanoff-s-theorem","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/romanoff-s-theorem/index.html","scope":"NumberTheory/additive-number-theory/romanoff-s-theorem","status":"can_prove","title":"Romanoff's theorem"},{"id":"NumberTheory/additive-number-theory/romanoff-s-theorem/has-at-most-rprime-factors","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem851Wanted.HasAtMostRPrimeFactors","parent":13276,"reading_title":{"format":"inline-parts-v1","parts":[["text","Natural numbers with at most "],["math","\\(r\\)"],["text"," distinct prime factors"]],"source":"Natural numbers with at most $r$ distinct prime factors"},"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/romanoff-s-theorem/index.html#has-at-most-rprime-factors","scope":"NumberTheory/additive-number-theory/romanoff-s-theorem","status":"fully_proved","summary":"For natural numbers $n$ and $r$, $n$ has at most $r$ distinct prime factors when $n = 0$ or at most $r$ distinct primes divide $n$; the number $0$ is counted as having no prime factors.","title":"Natural numbers with at most $r$ distinct prime factors"},{"id":"NumberTheory/additive-number-theory/romanoff-s-theorem/representable","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem851Wanted.Representable","parent":13276,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/romanoff-s-theorem/index.html#representable","scope":"NumberTheory/additive-number-theory/romanoff-s-theorem","status":"fully_proved","summary":"For a natural number $r$, a natural number $m$ is of this form when $m = 2^k + n$ for some natural numbers $k$ and $n$ such that $n = 0$ or at most $r$ distinct primes divide $n$.","title":"Sums of a power of two and a number with few prime factors"},{"id":"NumberTheory/additive-number-theory/romanoff-s-theorem/prime-representable","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem851Wanted.PrimeRepresentable","parent":13276,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/romanoff-s-theorem/index.html#prime-representable","scope":"NumberTheory/additive-number-theory/romanoff-s-theorem","status":"fully_proved","summary":"A natural number $m$ is of this form when $m = 2^k + p$ for some natural number $k$ and some prime $p$.","title":"Sums of a power of two and a prime"},{"id":"NumberTheory/additive-number-theory/romanoff-s-theorem/statement","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem851Wanted.Statement","parent":13276,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/romanoff-s-theorem/index.html#statement","scope":"NumberTheory/additive-number-theory/romanoff-s-theorem","status":"fully_proved","summary":"The proposition asserts two things. First, the set of natural numbers of the form $2^k + p$, with $k$ a natural number and $p$ prime, has positive lower density (Romanoff's theorem). Second, for every real number $\\varepsilon > 0$ there is a natural number $r$ such that the set of natural numbers of the form $2^k + n$, with $k$ and $n$ natural numbers and with $n = 0$ or at most $r$ distinct primes dividing $n$,\u2026","title":"Erd\u0151s Problem 851"},{"id":"NumberTheory/additive-number-theory/romanoff-s-theorem/statement-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem851Wanted.Statement_holds","parent":13276,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/romanoff-s-theorem/index.html#statement-holds","scope":"NumberTheory/additive-number-theory/romanoff-s-theorem","status":"can_prove","summary":"The set of natural numbers of the form $2^k + p$, with $k$ a natural number and $p$ prime, has positive lower density. Moreover, for every real number $\\varepsilon > 0$ there is a natural number $r$ such that the set of natural numbers of the form $2^k + n$, with $k$ and $n$ natural numbers and with $n = 0$ or at most $r$ distinct primes dividing $n$, has lower density at least $1 - \\varepsilon$.","title":"Erd\u0151s Problem 851 is true"},{"data_url":"scopes/NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o.json","id":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/index.html","scope":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o","status":"can_prove","title":"Sets with no element dividing a product of two others"},{"id":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/good","kind":"node","lean":"MathlibExt.NumberTheory.ErdosDivisibilityFamilyGrowthWanted.Good","parent":13282,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/index.html#good","scope":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o","status":"fully_proved","summary":"A finite set $A$ of natural numbers is good when every element of $A$ is positive and no $a \\in A$ divides $bc$ for any $b, c \\in A$ with $a \\ne b$ and $a \\ne c$; here $b = c$ is allowed.","title":"Sets in which no element divides the product of two others"},{"id":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/f","kind":"node","lean":"MathlibExt.NumberTheory.ErdosDivisibilityFamilyGrowthWanted.F","parent":13282,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/index.html#f","scope":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o","status":"fully_proved","summary":"For a natural number $n$, $F(n)$ is the largest size of a good subset of $\\{1, \\dots, n\\}$, defined as the supremum of the sizes of the good sets $A \\subseteq \\{1, \\dots, n\\}$.","title":"Maximum size of a good subset"},{"id":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosDivisibilityFamilyGrowthWanted.conjecture","parent":13282,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/index.html#conjecture","scope":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o","status":"fully_proved","summary":"For a natural number $n$, let $F(n)$ be the largest size of a set $A \\subseteq \\{1, \\dots, n\\}$ with $a \\nmid bc$ for all $a, b, c \\in A$ such that $a \\ne b$ and $a \\ne c$, and let $\\pi(n)$ be the number of primes at most $n$. This is the proposition that there is a real constant $C > 0$ such that $(F(n) - \\pi(n)) / (n^{2/3} / (\\log n)^2) \\to C$ as $n \\to \\infty$. Equivalently,\u2026","title":"Statement of Erd\u0151s Problem 793"},{"id":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosDivisibilityFamilyGrowthWanted.conjecture_holds","parent":13282,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o/index.html#conjecture-holds","scope":"NumberTheory/additive-number-theory/sets-with-no-element-dividing-a-product-of-two-o","status":"can_prove","summary":"Let $F(n)$ be the largest size of a set $A \\subseteq \\{1, \\dots, n\\}$ such that $a \\nmid bc$ whenever $a, b, c \\in A$ with $a \\ne b$ and $a \\ne c$ ($b = c$ is allowed), and let $\\pi(n)$ be the number of primes at most $n$. Then there is a real constant $C > 0$ such that $(F(n) - \\pi(n)) / (n^{2/3} / (\\log n)^2) \\to C$ as $n \\to \\infty$.","title":"Erd\u0151s Problem 793 has a positive answer"},{"data_url":"scopes/NumberTheory/additive-number-theory/strongly-complete-sequences.json","id":"NumberTheory/additive-number-theory/strongly-complete-sequences","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/strongly-complete-sequences/index.html","scope":"NumberTheory/additive-number-theory/strongly-complete-sequences","status":"can_prove","title":"Strongly complete sequences"},{"id":"NumberTheory/additive-number-theory/strongly-complete-sequences/strongly-complete","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem351Wanted.EP351.StronglyComplete","parent":13287,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/strongly-complete-sequences/index.html#strongly-complete","scope":"NumberTheory/additive-number-theory/strongly-complete-sequences","status":"fully_proved","summary":"A set $A$ of rational numbers is strongly complete when, for every finite set $B$ of rational numbers, there is an integer $N_0$ such that every integer $m \\ge N_0$ is the sum of the elements of some finite subset of $A \\setminus B$.","title":"Strongly complete sets of rationals"},{"id":"NumberTheory/additive-number-theory/strongly-complete-sequences/erdos-set351","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem351Wanted.EP351.ErdosSet351","parent":13287,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/strongly-complete-sequences/index.html#erdos-set351","scope":"NumberTheory/additive-number-theory/strongly-complete-sequences","status":"fully_proved","summary":"For a polynomial $p$ with rational coefficients, $A_p$ is the set of rational numbers $p(n) + 1/n$, where $n$ ranges over the positive integers.","title":"Polynomial values plus reciprocals"},{"id":"NumberTheory/additive-number-theory/strongly-complete-sequences/strong-completeness-question","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem351Wanted.EP351.StrongCompletenessQuestion","parent":13287,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/strongly-complete-sequences/index.html#strong-completeness-question","scope":"NumberTheory/additive-number-theory/strongly-complete-sequences","status":"fully_proved","summary":"This is the proposition that for every polynomial $p$ with rational coefficients and positive leading coefficient, the set $A_p$ of rational numbers $p(n) + 1/n$, where $n$ ranges over the positive integers, is strongly complete.","title":"The strong completeness statement of Erd\u0151s Problem 351"},{"id":"NumberTheory/additive-number-theory/strongly-complete-sequences/strong-completeness-question-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem351Wanted.StrongCompletenessQuestion_holds","parent":13287,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/strongly-complete-sequences/index.html#strong-completeness-question-holds","scope":"NumberTheory/additive-number-theory/strongly-complete-sequences","status":"can_prove","summary":"For every polynomial $p$ with rational coefficients and positive leading coefficient, let $A_p$ be the set of rational numbers $p(n) + 1/n$, where $n$ ranges over the positive integers. Then $A_p$ is strongly complete: for every finite set $B$ of rational numbers there is an integer $N_0$ such that every integer $m \\ge N_0$ is the sum of the elements of some finite subset of $A_p \\setminus B$.","title":"Strong completeness of polynomial values plus reciprocals (Erd\u0151s Problem 351)"},{"data_url":"scopes/NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence.json","id":"NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence/index.html","scope":"NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence","status":"can_prove","title":"Sums of consecutive terms of a sequence"},{"id":"NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence/rep-count","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem358IntervalSumsWanted.RepCount","parent":13292,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence/index.html#rep-count","scope":"NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence","status":"fully_proved","summary":"For a sequence $a_0, a_1, a_2, \\ldots$ of natural numbers and a natural number $n$, $f_a(n)$ is the number of pairs $(u, v)$ of natural numbers with $u \\le v$ and $a_u + a_{u+1} + \\cdots + a_v = n$, and $f_a(n) = 0$ if there are infinitely many such pairs.","title":"Interval-sum representation count"},{"id":"NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem358IntervalSumsWanted.conjecture","parent":13292,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence/index.html#conjecture","scope":"NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence","status":"fully_proved","summary":"This is the proposition that there is a strictly increasing sequence $a_0 < a_1 < a_2 < \\cdots$ of positive integers whose interval-sum representation count $f(n)$, the number of pairs $(u, v)$ of natural numbers with $u \\le v$ and $a_u + a_{u+1} + \\cdots + a_v = n$, tends to infinity as $n \\to \\infty$.","title":"The interval-sum conjecture of Erd\u0151s Problem 358"},{"id":"NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem358IntervalSumsWanted.conjecture_holds","parent":13292,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence/index.html#conjecture-holds","scope":"NumberTheory/additive-number-theory/sums-of-consecutive-terms-of-a-sequence","status":"can_prove","summary":"There is a strictly increasing sequence $a_0 < a_1 < a_2 < \\cdots$ of positive integers such that the number $f(n)$ of pairs $(u, v)$ of natural numbers with $u \\le v$ and $a_u + a_{u+1} + \\cdots + a_v = n$ tends to infinity as $n \\to \\infty$.","title":"Interval sums with unbounded representation counts (Erd\u0151s Problem 358)"},{"data_url":"scopes/NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l.json","id":"NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l/index.html","scope":"NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l","status":"can_prove","title":"Sums of numbers of the form 2^k 3^l"},{"id":"NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem845Wanted.conjecture","parent":13296,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l/index.html#conjecture","scope":"NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l","status":"fully_proved","summary":"The proposition asserts that for every real number $C > 0$, the set of all sums $\\sum_{(k, l) \\in B} 2^k 3^l$, where $B$ is a nonempty finite set of pairs of natural numbers such that $\\max_{(k, l) \\in B} 2^k 3^l \\le C \\min_{(k, l) \\in B} 2^k 3^l$, has natural density $0$.","title":"Erd\u0151s Problem 845"},{"id":"NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem845Wanted.conjecture_refuted","parent":13296,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l/index.html#conjecture-refuted","scope":"NumberTheory/additive-number-theory/sums-of-numbers-of-the-form-2-k-3-l","status":"can_prove","summary":"There is a real number $C > 0$ such that the set of all sums $\\sum_{(k, l) \\in B} 2^k 3^l$, where $B$ is a nonempty finite set of pairs of natural numbers such that $\\max_{(k, l) \\in B} 2^k 3^l \\le C \\min_{(k, l) \\in B} 2^k 3^l$, does not have natural density $0$.","title":"Erd\u0151s Problem 845 is false"},{"data_url":"scopes/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes.json","id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","title":"Sums of powers of consecutive primes"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/card-consecutive-prime-power-sums-length","kind":"node","lean":"MetaMathlibExt.card_consecutivePrimePowerSums_length_le","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#card-consecutive-prime-power-sums-length","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 < p_2 < \\cdots$ be the primes. For natural numbers $k > 1$, $m > 0$ and $x$, the number of integers $n \\le x$ of the form $n = p_{\\ell+1}^k + p_{\\ell+2}^k + \\cdots + p_{\\ell+m}^k$ for some $\\ell \\ge 0$ is at most $\\pi(\\lfloor (x/m)^{1/k} \\rfloor)$, the number of primes up to $(x/m)^{1/k}$.","title":"Counting sums of a fixed number of consecutive prime powers"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/max-consecutive-prime-power-sum-length-i","kind":"node","lean":"MetaMathlibExt.maxConsecutivePrimePowerSumLength_isEquivalent","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#max-consecutive-prime-power-sum-length-i","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 < p_2 < \\cdots$ be the primes, let $k > 1$ be a natural number, and for a natural number $x$ let $M(x, k)$ be the largest $m \\le x$ with $p_1^k + p_2^k + \\cdots + p_m^k \\le x$. Then $M(x, k) \\sim (k + 1)\\, x^{1/(k+1)} / (\\log x)^{k/(k+1)}$ (asymptotic equivalence) as $x \\to \\infty$ through the natural numbers.","title":"Length of the longest sum of consecutive prime powers"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/card-consecutive-prime-power-sums-le","kind":"node","lean":"MetaMathlibExt.card_consecutivePrimePowerSums_le","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#card-consecutive-prime-power-sums-le","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 < p_2 < \\cdots$ be the primes and let $k > 1$ be a natural number. For a natural number $x$ let $s_k(x)$ be the number of pairs $(\\ell, m)$ of integers with $0 \\le \\ell < x$ and $1 \\le m \\le x$ such that $p_{\\ell+1}^k + \\cdots + p_{\\ell+m}^k \\le x$. Then for every real $\\varepsilon > 0$, for all sufficiently large $x$, $s_k(x) \\le (1 + \\varepsilon)\\, c_k\\, x^{2/(k+1)} / (\\log x)^{2k/(k+1)}$, where\u2026","title":"Upper bound for the number of sums of consecutive prime powers"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/card-consecutive-prime-power-sums-ge","kind":"node","lean":"MetaMathlibExt.card_consecutivePrimePowerSums_ge","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#card-consecutive-prime-power-sums-ge","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 < p_2 < \\cdots$ be the primes and let $k > 1$ be a natural number. For a natural number $x$ let $s_k(x)$ be the number of pairs $(\\ell, m)$ of integers with $0 \\le \\ell < x$ and $1 \\le m \\le x$ such that $p_{\\ell+1}^k + \\cdots + p_{\\ell+m}^k \\le x$, and let $M(x, k)$ be the largest $m \\le x$ with $p_1^k + \\cdots + p_m^k \\le x$. Then $M(x, k)^2 / 2 \\le s_k(x)$ for every $x$, and for every real\u2026","title":"Lower bound for the number of sums of consecutive prime powers"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/max-chain-length-lt","kind":"node","lean":"MetaMathlibExt.maxChainLength_lt","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#max-chain-length-lt","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 = 2, p_2 = 3, \\dots$ be the primes in increasing order. Let $k > 1$, $M_0 \\ge 6$, $M \\ge M_0$ and $x$ be natural numbers with $p_1^k + \\cdots + p_M^k \\le x < p_1^k + \\cdots + p_{M+1}^k$. Put $y = M_0$,","title":"Upper bound for the maximal gleeful chain length"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/le-max-chain-length","kind":"node","lean":"MetaMathlibExt.le_maxChainLength","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#le-max-chain-length","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 = 2, p_2 = 3, \\dots$ be the primes in increasing order. Let $k > 1$, $M_0 \\ge 6$, $M \\ge M_0$ and $x$ be natural numbers with $p_1^k + \\cdots + p_M^k \\le x < p_1^k + \\cdots + p_{M+1}^k$. Put $y = M_0$ and","title":"Lower bound for the maximal gleeful chain length"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/log-max-chain-length-bounds","kind":"node","lean":"MetaMathlibExt.log_maxChainLength_bounds","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#log-max-chain-length-bounds","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 = 2, p_2 = 3, \\dots$ be the primes in increasing order. Let $k > 1$, $M_0 \\ge 6$, $M \\ge M_0$ and $x$ be natural numbers with $p_1^k + \\cdots + p_M^k \\le x < p_1^k + \\cdots + p_{M+1}^k$. Put $y = M_0$, let $\\log$ be the natural logarithm, and set","title":"Bounds for the logarithm of the maximal gleeful chain length"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/gleeful-rep-count-le","kind":"node","lean":"MetaMathlibExt.gleefulRepCount_le","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#gleeful-rep-count-le","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 = 2, p_2 = 3, \\dots$ be the primes in increasing order, and for natural numbers $k$ and $x$ let $s_k(x)$ be the number of pairs $(n, m)$ of natural numbers with $m \\ge 1$ and $p_{n+1}^k + \\cdots + p_{n+m}^k \\le x$. Let $k > 1$, $M_0 \\ge 6$ and $x$ be natural numbers with $p_1^k + \\cdots + p_{M_0}^k \\le x$. Put $y = M_0$, let $\\log$ be the natural logarithm, and set","title":"Upper bound for the number of gleeful representations"},{"id":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/le-gleeful-rep-count","kind":"node","lean":"MetaMathlibExt.le_gleefulRepCount","parent":13299,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes/index.html#le-gleeful-rep-count","scope":"NumberTheory/additive-number-theory/sums-of-powers-of-consecutive-primes","status":"can_prove","summary":"Let $p_1 = 2, p_2 = 3, \\dots$ be the primes in increasing order, and for natural numbers $k$ and $x$ let $s_k(x)$ be the number of pairs $(n, m)$ of natural numbers with $m \\ge 1$ and $p_{n+1}^k + \\cdots + p_{n+m}^k \\le x$. Let $k > 1$, $M_0 \\ge 6$ and $x$ be natural numbers with $p_1^k + \\cdots + p_{M_0}^k \\le x$. Put $y = M_0$, let $\\log$ be the natural logarithm, and set","title":"Lower bound for the number of gleeful representations"},{"data_url":"scopes/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers.json","id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","kind":"scope","navigation":"module","parent":13143,"result":false,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","title":"Sums of practical and polygonal numbers"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/exists-practical-add-triangular","kind":"node","lean":"MetaMathlibExt.exists_practical_add_triangular","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#exists-practical-add-triangular","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"For every natural number $n > 0$ there are a practical number $q$ and a natural number $k$ with $n = q + k(k+1)/2$; the case $k = 0$, where the triangular number is $0$, is allowed.","title":"Every positive integer is a practical number plus a triangular number"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/eventually-practical-add-two-polygonal","kind":"node","lean":"MetaMathlibExt.eventually_practical_add_two_polygonal","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#eventually-practical-add-two-polygonal","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"Let $s > 3$ be a natural number and, for a natural number $x$, let $P_s(x) = (s-2)\\binom{x}{2} + x$ be the $x$-th $s$-gonal number, so that $P_s(0) = 0$. There is a natural number $N$ such that every natural number $n > N$ can be written as $n = q + P_s(x) + P_s(y)$ with $q$ a practical number and $x, y$ natural numbers.","title":"Large integers as a practical number plus two polygonal numbers"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/exists-sq-mod-eq-eight-mul-add-one","kind":"node","lean":"MetaMathlibExt.exists_sq_modEq_eight_mul_add_one","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#exists-sq-mod-eq-eight-mul-add-one","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"For all natural numbers $m > 0$ and $n > 0$ there is a natural number $x$ with $1 \\le x \\le 2^m - 1$ and $x^2 \\equiv 8n + 1 \\pmod{2^{m+2}}$.","title":"Square roots modulo powers of two"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/is-practical-number-mul-of-le-sigma-add","kind":"node","lean":"MetaMathlibExt.isPracticalNumber_mul_of_le_sigma_add_one","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#is-practical-number-mul-of-le-sigma-add","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"Let $m$ be a practical number and let $\\sigma(m)$ denote the sum of the positive divisors of $m$. Then for every natural number $n$ with $1 \\le n \\le \\sigma(m) + 1$, the product $mn$ is a practical number.","title":"Products of a practical number with small factors"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/sq-sub-one-div-eight-eq-triangular","kind":"node","lean":"MetaMathlibExt.sq_sub_one_div_eight_eq_triangular","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#sq-sub-one-div-eight-eq-triangular","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"For every odd natural number $x$, the number $(x^2 - 1)/8$ is a triangular number: there is a natural number $k$ with $(x^2 - 1)/8 = p_3(k)$, where $p_3(k) = k(k+1)/2$ is the $k$-th $3$-gonal number.","title":"Odd squares and triangular numbers"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/exists-polygonal-add-polygonal-mod-eq-pr","kind":"node","lean":"MetaMathlibExt.exists_polygonal_add_polygonal_modEq_prime","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#exists-polygonal-add-polygonal-mod-eq-pr","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"For a natural number $s > 3$ and a natural number $x$, let $P_s(x) = (s-2)\\binom{x}{2} + x$ be the $x$-th $s$-gonal number. For every odd prime $p$ and every natural number $n > 0$ there are natural numbers $x, y > 0$ with $P_s(x) + P_s(y) \\equiv n \\pmod{p}$.","title":"Sums of two polygonal numbers modulo an odd prime"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/exists-polygonal-add-polygonal-mod-eq-tw","kind":"node","lean":"MetaMathlibExt.exists_polygonal_add_polygonal_modEq_two","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#exists-polygonal-add-polygonal-mod-eq-tw","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"For a natural number $s > 3$ and a natural number $x$, let $P_s(x) = (s-2)\\binom{x}{2} + x$ be the $x$-th $s$-gonal number. For every natural number $n > 0$ there are natural numbers $x, y > 0$ such that $P_s(x_1) + P_s(y_1) \\equiv n \\pmod{2}$ for all natural numbers $x_1, y_1 > 0$ with $x_1 \\equiv x \\pmod{4}$ and $y_1 \\equiv y \\pmod{4}$.","title":"Parity of sums of two polygonal numbers"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/exists-sq-add-sq-mod-eq-prime-pow","kind":"node","lean":"MetaMathlibExt.exists_sq_add_sq_modEq_prime_pow","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#exists-sq-add-sq-mod-eq-prime-pow","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"Let $p$ be a prime with $p \\equiv 1 \\pmod{4}$. For all natural numbers $n > 0$ and $k > 0$ there are natural numbers $x, y > 0$ with $x^2 + y^2 \\equiv n \\pmod{p^k}$ and $p \\nmid y$.","title":"Sums of two squares modulo prime powers"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/exists-prime-polygonal-add-polygonal-mod","kind":"node","lean":"MetaMathlibExt.exists_prime_polygonal_add_polygonal_modEq_prime_pow","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#exists-prime-polygonal-add-polygonal-mod","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"For a natural number $s > 3$ and a natural number $x$, let $P_s(x) = (s-2)\\binom{x}{2} + x$ be the $x$-th $s$-gonal number. There is a prime $p$ with $p \\equiv 1 \\pmod{4}$ and $p \\nmid s - 2$ such that for all natural numbers $k > 0$ and $n > 0$ there are natural numbers $x, y > 0$ with $P_s(x) + P_s(y) \\equiv n \\pmod{p^k}$.","title":"Sums of two polygonal numbers modulo prime powers"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/exists-two-polygonal-two-mul-le-mul-sq","kind":"node","lean":"MetaMathlibExt.exists_two_polygonal_two_mul_le_mul_sq","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#exists-two-polygonal-two-mul-le-mul-sq","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"For a natural number $s > 3$ and a natural number $x$, let $P_s(x) = (s-2)\\binom{x}{2} + x$ be the $x$-th $s$-gonal number. For every natural number $c$ there is a natural number $A$ such that $2 P_s(2cx) \\le A x^2$ for every natural number $x \\ge 1$.","title":"Quadratic upper bound for polygonal numbers"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/is-practical-number-eq-one-or-two","kind":"node","lean":"MetaMathlibExt.isPracticalNumber_eq_one_or_two","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#is-practical-number-eq-one-or-two","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"If $q$ is a practical number that is divisible by neither $3$ nor $4$, then $q = 1$ or $q = 2$.","title":"Practical numbers divisible by neither 3 nor 4"},{"id":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/practical-add-polygonal-exceptions","kind":"node","lean":"MetaMathlibExt.practical_add_polygonal_exceptions","parent":13309,"result":true,"route":"../roadmap/NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers/index.html#practical-add-polygonal-exceptions","scope":"NumberTheory/additive-number-theory/sums-of-practical-and-polygonal-numbers","status":"can_prove","summary":"(a) Let $s \\ge 3$ with $s \\equiv 0$ or $s \\equiv 4 \\pmod{12}$. Then there are infinitely many positive integers $n$ that cannot be written as $n = q + p_s(m)$ with $q$ a practical number and $m \\ge 0$ an integer, where $p_s(m)$ is the $m$-th $s$-gonal number. (b) For every natural number $M$ there is a natural number $S$ such that for every $t \\ge S$ there are at least $M$ positive integers $n$ that cannot be\u2026","title":"Exceptions to sums of a practical number and polygonal numbers"},{"data_url":"scopes/NumberTheory/numerical-semigroups.json","id":"NumberTheory/numerical-semigroups","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/index.html","scope":"NumberTheory/numerical-semigroups","status":"can_prove","summary":"Sylvester's coin theorem, Frobenius numbers, pseudo-Frobenius sets and numerical semigroups generated by special sequences.","title":"Numerical semigroups and the Frobenius problem"},{"data_url":"scopes/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences.json","id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"can_prove","title":"Frobenius numbers of compound sequences"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/compound-sequence","kind":"node","lean":"MetaMathlibExt.compoundSequence","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#compound-sequence","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"fully_proved","summary":"Let $n$ be a natural number and let $a = (a_i)$ and $b = (b_i)$ be sequences of natural numbers. For a natural number $j$, the $j$-th term of the compound sequence of $a$ and $b$ is","title":"Compound sequence"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/compound-sequence-zero","kind":"node","lean":"MetaMathlibExt.compoundSequence_zero","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#compound-sequence-zero","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"fully_proved","summary":"For every natural number $n$ and all sequences $a$, $b$ of natural numbers, the compound sequence $c$ of $a$ and $b$ satisfies $c_0 = a_1 a_2 \\cdots a_n$.","title":"First term of a compound sequence"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/compound-sequence-self","kind":"node","lean":"MetaMathlibExt.compoundSequence_self","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#compound-sequence-self","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"fully_proved","summary":"For every natural number $n$ and all sequences $a$, $b$ of natural numbers, the compound sequence $c$ of $a$ and $b$ satisfies $c_n = b_1 b_2 \\cdots b_n$.","title":"Last term of a compound sequence"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/frobenius-eq-max-apery-sub","kind":"node","lean":"MetaMathlibExt.frobenius_eq_max_apery_sub","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#frobenius-eq-max-apery-sub","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"can_prove","summary":"Let $A_1, \\dots, A_k$ be positive integers with $\\gcd(A_1, \\dots, A_k) = 1$, let $S$ be the additive submonoid of $\\mathbb{N}$ they generate, and let $a = A_i$ be one of them with $a \\ge 2$. For $1 \\le x \\le a - 1$ let $m_x$ be the least element of $S$ congruent to $x$ modulo $a$. If $M$ is the largest of the numbers $m_x$, then $M - a$ is the largest integer not in $S$, that is, the Frobenius number of $S$.","title":"Brauer\u2013Shockley formula for the Frobenius number"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/compound-sequence-apery-and-frobenius","kind":"node","lean":"MetaMathlibExt.compoundSequence_apery_and_frobenius","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#compound-sequence-apery-and-frobenius","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"can_prove","summary":"Let $n$ be a natural number and let $a_1, \\dots, a_n$ and $b_1, \\dots, b_n$ be natural numbers with $0 < a_i < b_i$ for $1 \\le i \\le n$ and $\\gcd(a_i, b_j) = 1$ whenever $1 \\le j \\le i \\le n$. Let $c_0, \\dots, c_n$ be their compound sequence and $S$ the additive submonoid of $\\mathbb{N}$ generated by $c_0, \\dots, c_n$. Then (Kiers, O'Neill and Ponomarenko) the Ap\u00e9ry set of $S$ with respect to $c_0$, the set of\u2026","title":"Ap\u00e9ry set and Frobenius number of a compound sequence"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/compound-sequence-erase-cofinite-iff","kind":"node","lean":"MetaMathlibExt.compoundSequence_erase_cofinite_iff","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#compound-sequence-erase-cofinite-iff","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"can_prove","summary":"Let $n \\ge 1$ and let $a_1, \\dots, a_n$ and $b_1, \\dots, b_n$ be natural numbers with $0 < a_i < b_i$ for $1 \\le i \\le n$ and $\\gcd(a_i, b_j) = 1$ whenever $1 \\le j \\le i \\le n$; let $c_0, \\dots, c_n$ be their compound sequence. For $0 \\le j \\le n$, the additive submonoid of $\\mathbb{N}$ generated by the terms $c_i$ with $0 \\le i \\le n$ and $i \\ne j$ has finite complement in $\\mathbb{N}$ (so it is a numerical\u2026","title":"When deleting a term of a compound sequence leaves a numerical semigroup"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/compound-sequence-erase-frobenius-eq-iff","kind":"node","lean":"MetaMathlibExt.compoundSequence_erase_frobenius_eq_iff","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#compound-sequence-erase-frobenius-eq-iff","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"can_prove","summary":"Let $n$ be a natural number and let $(a_i)_{i \\ge 0}$ and $(b_i)_{i \\ge 0}$ be sequences of natural numbers with $0 < a_i < b_i$ for $1 \\le i \\le n$ and $\\gcd(a_i, b_j) = 1$ whenever $1 \\le j \\le i \\le n$; let $c_0, \\dots, c_n$ be their compound sequence. Let $0 \\le j \\le n$ and suppose that the terms $c_i$ with $0 \\le i \\le n$ and $i \\ne j$ have greatest common divisor $1$. Let $S$ be the additive submonoid of\u2026","title":"Deletions from a compound sequence that keep the Frobenius number"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/compound-sequence-embedding-dimension","kind":"node","lean":"MetaMathlibExt.compoundSequence_embeddingDimension","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#compound-sequence-embedding-dimension","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"can_prove","summary":"Let $n$ be a natural number and let $a_1, \\dots, a_n$ and $b_1, \\dots, b_n$ be natural numbers with $0 < a_i < b_i$ for $1 \\le i \\le n$ and $\\gcd(a_i, b_j) = 1$ whenever $1 \\le j \\le i \\le n$; let $c_0, \\dots, c_n$ be their compound sequence and $S$ the additive submonoid of $\\mathbb{N}$ generated by $c_0, \\dots, c_n$. Let $k$ be the number of indices $1 \\le i \\le n$ with $a_i = 1$. Then the minimal generating set\u2026","title":"Embedding dimension of the semigroup of a compound sequence"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/compound-sequence-minimal-generating-set","kind":"node","lean":"MetaMathlibExt.compoundSequence_minimal_generating_set_least","parent":13323,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences/index.html#compound-sequence-minimal-generating-set","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-compound-sequences","status":"can_prove","summary":"Let $n$ be a natural number and let $a_1, \\dots, a_n$ and $b_1, \\dots, b_n$ be natural numbers with $0 < a_i < b_i$ for $1 \\le i \\le n$ and $\\gcd(a_i, b_j) = 1$ whenever $1 \\le j \\le i \\le n$; let $C = \\{c_0, \\dots, c_n\\}$ be the set of terms of their compound sequence and $S$ the additive submonoid of $\\mathbb{N}$ generated by $C$. Say that a set $G$ of natural numbers has the same Frobenius number as $S$ if some\u2026","title":"The minimal generating set as the least Frobenius-preserving subset"},{"data_url":"scopes/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers.json","id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"can_prove","title":"Frobenius numbers of consecutive star numbers"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/star","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.star","parent":13333,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#star","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"fully_proved","summary":"For a natural number $n$, the $n$-th star number is $S_n = 6n(n-1)+1$.","title":"Star numbers"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/star-rep-count","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.starRepCount","parent":13333,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#star-rep-count","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"fully_proved","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For natural numbers $n$ and $N$, the representation count of $N$ is the number of triples $(x, y, z)$ of natural numbers with $x S_n + y S_{n+1} + z S_{n+2} = N$.","title":"Representation count for three consecutive star numbers"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/is-pfrobenius-star-number","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.IsPFrobeniusStarNumber","parent":13333,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-Frobenius number of three consecutive star numbers"]],"source":"$p$-Frobenius number of three consecutive star numbers"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#is-pfrobenius-star-number","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"fully_proved","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For natural numbers $p$, $n$ and $g$, $g$ is the $p$-Frobenius number of $S_n, S_{n+1}, S_{n+2}$ when at most $p$ triples $(x, y, z)$ of natural numbers satisfy $x S_n + y S_{n+1} + z S_{n+2} = g$, and every natural number $N > g$ has more than $p$ triples $(x, y, z)$ of natural numbers with $x S_n + y S_{n+1} + z S_{n+2} = N$.","title":"$p$-Frobenius number of three consecutive star numbers"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/p-sylvester-star-count","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.pSylvesterStarCount","parent":13333,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-Sylvester number of three consecutive star numbers"]],"source":"$p$-Sylvester number of three consecutive star numbers"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#p-sylvester-star-count","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"fully_proved","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For natural numbers $p$ and $n$, the $p$-Sylvester number of $S_n, S_{n+1}, S_{n+2}$ is the number of natural numbers $N$ for which at most $p$ triples $(x, y, z)$ of natural numbers satisfy $x S_n + y S_{n+1} + z S_{n+2} = N$.","title":"$p$-Sylvester number of three consecutive star numbers"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/gcd-star-succ-succ","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.gcd_star_succ_succ","parent":13333,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#gcd-star-succ-succ","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"can_prove","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For every natural number $n \\ge 1$, $\\gcd(S_n, S_{n+1}, S_{n+2}) = 1$.","title":"Three consecutive star numbers are relatively prime"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/exists-is-pfrobenius-number-star-of-le-n","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.exists_isPFrobeniusNumber_star_of_le_nine","parent":13333,"reading_title":{"format":"inline-parts-v1","parts":[["text","Frobenius numbers of three consecutive star numbers for "],["math","\\(6 \\le n \\le 9\\)"]],"source":"Frobenius numbers of three consecutive star numbers for $6 \\le n \\le 9$"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#exists-is-pfrobenius-number-star-of-le-n","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"can_prove","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For natural numbers $n$ and $p$ with $6 \\le n \\le 9$ and $p \\in \\{0, 1, 2\\}$, there is a natural number $g$ such that at most $p$ triples $(x, y, z)$ of natural numbers satisfy $x S_n + y S_{n+1} + z S_{n+2} = g$, every natural number larger than $g$ has more than $p$ such triples, and $g = 2n S_{n+1} + (p + 2) n S_{n+2} - S_n$. That is, the $p$-Frobenius number\u2026","title":"Frobenius numbers of three consecutive star numbers for $6 \\le n \\le 9$"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/exists-is-pfrobenius-number-star-of-ten","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.exists_isPFrobeniusNumber_star_of_ten_le","parent":13333,"reading_title":{"format":"inline-parts-v1","parts":[["text","Frobenius numbers of three consecutive star numbers for "],["math","\\(n \\ge 10\\)"]],"source":"Frobenius numbers of three consecutive star numbers for $n \\ge 10$"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#exists-is-pfrobenius-number-star-of-ten","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"can_prove","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For natural numbers $n$ and $p$ with $n \\ge 10$ and $p \\in \\{0, 1, 2\\}$, there is a natural number $g$ such that at most $p$ triples $(x, y, z)$ of natural numbers satisfy $x S_n + y S_{n+1} + z S_{n+2} = g$, every natural number larger than $g$ has more than $p$ such triples, and $g = (2n - 11) S_{n+1} + (p + 3) n S_{n+2} - S_n$. That is, the $p$-Frobenius number\u2026","title":"Frobenius numbers of three consecutive star numbers for $n \\ge 10$"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/p-sylvester-count-zero-star","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.pSylvesterCount_zero_star","parent":13333,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(0\\)"],["text","-Sylvester number of three consecutive star numbers"]],"source":"$0$-Sylvester number of three consecutive star numbers"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#p-sylvester-count-zero-star","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"can_prove","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For every natural number $n \\ge 6$, the number of natural numbers that cannot be written as $x S_n + y S_{n+1} + z S_{n+2}$ with $x, y, z$ natural numbers is $n(15n^2 + 8n + 7)$.","title":"$0$-Sylvester number of three consecutive star numbers"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/p-sylvester-count-one-star","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.pSylvesterCount_one_star","parent":13333,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(1\\)"],["text","-Sylvester number of three consecutive star numbers"]],"source":"$1$-Sylvester number of three consecutive star numbers"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#p-sylvester-count-one-star","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"can_prove","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For every natural number $n \\ge 6$, the number of natural numbers $N$ for which at most one triple $(x, y, z)$ of natural numbers satisfies $x S_n + y S_{n+1} + z S_{n+2} = N$ is $25n^3 + 23n^2 + 14n + 1$.","title":"$1$-Sylvester number of three consecutive star numbers"},{"id":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/p-sylvester-count-two-star","kind":"node","lean":"MathlibExt.NumberTheory.PFrobeniusStarNumbersWanted.pSylvesterCount_two_star","parent":13333,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-Sylvester number of three consecutive star numbers"]],"source":"$2$-Sylvester number of three consecutive star numbers"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers/index.html#p-sylvester-count-two-star","scope":"NumberTheory/numerical-semigroups/frobenius-numbers-of-consecutive-star-numbers","status":"can_prove","summary":"Write $S_m$ for the $m$-th star number $6m(m-1)+1$. For every natural number $n \\ge 6$, the number of natural numbers $N$ for which at most two triples $(x, y, z)$ of natural numbers satisfy $x S_n + y S_{n+1} + z S_{n+2} = N$ is $33n^3 + 35n^2 + 20n + 2$.","title":"$2$-Sylvester number of three consecutive star numbers"},{"data_url":"scopes/NumberTheory/numerical-semigroups/generalized-frobenius-numbers.json","id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers/index.html","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers","status":"fully_proved","title":"Generalized Frobenius numbers"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers/frobenius-rep-count","kind":"node","lean":"MetaMathlibExt.frobeniusRepCount","parent":13344,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers/index.html#frobenius-rep-count","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers","status":"fully_proved","summary":"For natural numbers $a_1, \\dots, a_k$ and an integer $m$, the representation count $d(m; a_1, \\dots, a_k)$ is the number of tuples $(x_1, \\dots, x_k)$ of natural numbers with $x_1 a_1 + \\dots + x_k a_k = m$, counted as an extended natural number, so that it is $\\infty$ when there are infinitely many such tuples.","title":"Number of representations by a vector of generators"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers/frobenius-rep-count-eq-zero-of-neg","kind":"node","lean":"MetaMathlibExt.frobeniusRepCount_eq_zero_of_neg","parent":13344,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers/index.html#frobenius-rep-count-eq-zero-of-neg","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers","status":"fully_proved","summary":"For all natural numbers $a_1, \\dots, a_k$ and every integer $m < 0$, the representation count $d(m; a_1, \\dots, a_k)$ is $0$.","title":"Negative integers have no representations"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers/frobenius-rep-count-ne-top","kind":"node","lean":"MetaMathlibExt.frobeniusRepCount_ne_top","parent":13344,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers/index.html#frobenius-rep-count-ne-top","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers","status":"fully_proved","summary":"If $a_1, \\dots, a_k$ are positive natural numbers, then for every integer $m$ the representation count $d(m; a_1, \\dots, a_k)$ is finite.","title":"Finitely many representations for positive generators"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers/is-gen-frobenius-number","kind":"node","lean":"MetaMathlibExt.IsGenFrobeniusNumber","parent":13344,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers/index.html#is-gen-frobenius-number","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers","status":"fully_proved","summary":"For natural numbers $a_1, \\dots, a_k$ and $s$, write $d(m; a)$ for the number of representations of an integer $m$ by $a = (a_1, \\dots, a_k)$. An integer $g$ is the generalized Frobenius number $g(a; s) = \\max \\{m \\in \\mathbb{Z} \\mid d(m; a) \\le s\\}$ when $d(g; a) \\le s$ and $d(m; a) > s$ for every integer $m > g$.","title":"Generalized Frobenius number"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers/unique","kind":"node","lean":"MetaMathlibExt.IsGenFrobeniusNumber.unique","parent":13344,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers/index.html#unique","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers","status":"fully_proved","summary":"For natural numbers $a_1, \\dots, a_k$ and $s$, if the integers $g$ and $g'$ are both the generalized Frobenius number $g(a_1, \\dots, a_k; s)$, then $g = g'$.","title":"Uniqueness of the generalized Frobenius number"},{"data_url":"scopes/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb.json","id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","title":"Generalized Frobenius numbers of triangular numbers"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/triangular-triple","kind":"node","lean":"MetaMathlibExt.triangularTriple","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#triangular-triple","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"fully_proved","summary":"For a natural number $n$, the triangular triple of $n$ is $(t_n, t_{n+1}, t_{n+2})$, where $t_m = m(m+1)/2$ is the $m$-th triangular number.","title":"Three consecutive triangular numbers"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/triangular-triple-one","kind":"node","lean":"MetaMathlibExt.triangularTriple_one","parent":13350,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triangular triple at "],["math","\\(n = 1\\)"]],"source":"Triangular triple at $n = 1$"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#triangular-triple-one","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"fully_proved","summary":"The three consecutive triangular numbers $(t_1, t_2, t_3)$, where $t_m = m(m+1)/2$, are $(1, 3, 6)$.","title":"Triangular triple at $n = 1$"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/triangular-frobenius-bound","kind":"node","lean":"MetaMathlibExt.triangularFrobeniusBound","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#triangular-frobenius-bound","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"fully_proved","summary":"For natural numbers $n$ and $s$, the threshold $N_s$ is the integer $N_s^{\\mathrm{even}} = 6\\lfloor\\sqrt{s+1}\\rfloor - 6$ when $n$ is even and $N_s^{\\mathrm{odd}} = 6\\lfloor(\\sqrt{4s+5} - 1)/2\\rfloor - 3$ when $n$ is odd.","title":"Triangular Frobenius threshold"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/triangular-frobenius-bound-zero","kind":"node","lean":"MetaMathlibExt.triangularFrobeniusBound_zero","parent":13350,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triangular Frobenius threshold at "],["math","\\(s = 0\\)"]],"source":"Triangular Frobenius threshold at $s = 0$"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#triangular-frobenius-bound-zero","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"fully_proved","summary":"At $s = 0$ the triangular Frobenius threshold $N_s$ equals $0$ for $n = 0$ (the even case, $N_0^{\\mathrm{even}} = 6\\lfloor\\sqrt{1}\\rfloor - 6$) and $-3$ for $n = 1$ (the odd case, $N_0^{\\mathrm{odd}} = 6\\lfloor(\\sqrt{5} - 1)/2\\rfloor - 3$).","title":"Triangular Frobenius threshold at $s = 0$"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/triangular-frobenius-x","kind":"node","lean":"MetaMathlibExt.triangularFrobeniusX","parent":13350,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parameter "],["math","\\(x_s\\)"],["text"," of the triangular Frobenius formula"]],"source":"The parameter $x_s$ of the triangular Frobenius formula"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#triangular-frobenius-x","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"fully_proved","summary":"For natural numbers $n$, $k$ and $i$, let $x$ be $i$ if $i \\le k$ and $i - k - 1$ if $i > k$ when $n$ is even, and $2i$ if $i \\le k$ and $2(i - k) - 1$ if $i > k$ when $n$ is odd. For $s = k(k+1) + i$ with $0 \\le i \\le 2k+1$ this is the integer $x_s$ (written $x_s^{\\mathrm{even}}$ or $x_s^{\\mathrm{odd}}$ by the parity of $n$).","title":"The parameter $x_s$ of the triangular Frobenius formula"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/triangular-frobenius-y","kind":"node","lean":"MetaMathlibExt.triangularFrobeniusY","parent":13350,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parameter "],["math","\\(y_s\\)"],["text"," of the triangular Frobenius formula"]],"source":"The parameter $y_s$ of the triangular Frobenius formula"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#triangular-frobenius-y","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"fully_proved","summary":"For natural numbers $n$, $k$ and $i$, let $y$ be $2(k - i)$ if $i \\le k$ and $4k + 3 - 2i$ if $i > k$ when $n$ is even, and $k - i$ if $i \\le k$ and $2k + 1 - i$ if $i > k$ when $n$ is odd, where a difference that would be negative is replaced by $0$ (this happens only for $i > 2k + 1$). For $s = k(k+1) + i$ with $0 \\le i \\le 2k+1$ this is the integer $y_s$ (written $y_s^{\\mathrm{even}}$ or $y_s^{\\mathrm{odd}}$ by\u2026","title":"The parameter $y_s$ of the triangular Frobenius formula"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/triangular-frobenius-xy-five","kind":"node","lean":"MetaMathlibExt.triangularFrobeniusXY_five","parent":13350,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triangular Frobenius parameters at "],["math","\\(s = 5\\)"]],"source":"Triangular Frobenius parameters at $s = 5$"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#triangular-frobenius-xy-five","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"fully_proved","summary":"Take $k = 1$ and $i = 3$, so that $s = k(k+1) + i = 5$. Then $(x_s, y_s) = (1, 1)$ for $n = 0$ (the even case) and $(x_s, y_s) = (3, 0)$ for $n = 1$ (the odd case).","title":"Triangular Frobenius parameters at $s = 5$"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/is-gen-frobenius-number-cons-mul-of-gcd","kind":"node","lean":"MetaMathlibExt.isGenFrobeniusNumber_cons_mul_of_gcd","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#is-gen-frobenius-number-cons-mul-of-gcd","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"Let $K \\ge 2$, let $a_1$ and $\\ell$ be positive integers, and let $a'_2, \\dots, a'_K$ be positive integers with $\\gcd(a'_2, \\dots, a'_K) = 1$ such that $\\gcd(a_1, \\ell a'_2, \\dots, \\ell a'_K) = 1$. Let $s$ be a natural number. If $g'$ is the generalized Frobenius number $g(a_1, a'_2, \\dots, a'_K; s)$, then $\\ell g' + a_1(\\ell - 1)$ is the generalized Frobenius number $g(a_1, \\ell a'_2, \\dots, \\ell a'_K; s)$.","title":"Beck\u2013Kifer reduction by a common factor of all generators but the first"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/frobenius-rep-count-triangular-reduced-e","kind":"node","lean":"MetaMathlibExt.frobeniusRepCount_triangular_reduced_eq_sum","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#frobenius-rep-count-triangular-reduced-e","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"For a natural number $r$ write $t_r = r(r+1)/2$, and for an integer $m$ and natural numbers $a_1, \\dots, a_k$ let $d(m; a_1, \\dots, a_k)$ be the number of representations of $m$ by $(a_1, \\dots, a_k)$. Let $n \\ge 1$, $d_1 = \\gcd(t_{n+1}, t_{n+2})$, $a = t_{n+1}/d_1$ and $b = t_{n+2}/d_1$. Then for every natural number $m$,","title":"Splitting off the first generator of the reduced triangular triple"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/frobenius-rep-count-pair-eq-iff","kind":"node","lean":"MetaMathlibExt.frobeniusRepCount_pair_eq_iff","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#frobenius-rep-count-pair-eq-iff","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"Let $a$ and $b$ be coprime positive integers. For a natural number $r$ let $g(a, b; r)$ be the generalized Frobenius number of $(a, b)$ at $r$, and for an integer $m$ let $d(m; a, b)$ be the number of representations of $m$ by $(a, b)$. Let $i$ and $s$ be natural numbers, let $c$ be a positive integer divisible by $a$ or by $b$, and let $j$ be an integer. Then $d(g(a, b; s) + jc; a, b) = i$ if and only if\u2026","title":"Representation counts between consecutive two-generator Frobenius numbers"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/frobenius-rep-count-pair-sub-ge","kind":"node","lean":"MetaMathlibExt.frobeniusRepCount_pair_sub_ge","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#frobenius-rep-count-pair-sub-ge","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"Let $a < b$ be coprime positive integers, let $s$ and $k$ be natural numbers, let $g$ be the generalized Frobenius number $g(a, b; s)$, and for an integer $m$ let $d(m; a, b)$ be the number of representations of $m$ by $(a, b)$. If $m$ is an integer with $m > g + ka$, then for every natural number $j$, $d(m - ja; a, b) \\ge d(g + (k - j)a; a, b)$.","title":"Representation counts above a shifted two-generator Frobenius number"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/frobenius-rep-count-triangular-reduced-a","kind":"node","lean":"MetaMathlibExt.frobeniusRepCount_triangular_reduced_at_pair_frobenius","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#frobenius-rep-count-triangular-reduced-a","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"For a natural number $r$ write $t_r = r(r+1)/2$. Let $n$, $k$, $i$ be natural numbers with $i \\le 2k + 1$ and put $s = k(k+1) + i$. Let $N_s = 6\\lfloor\\sqrt{s+1}\\rfloor - 6$ if $n$ is even and $N_s = 6\\lfloor(\\sqrt{4s+5} - 1)/2\\rfloor - 3$ if $n$ is odd, and suppose $n > N_s$. For even $n$ let $(x_s, y_s) = (i, 2(k - i))$ if $i \\le k$ and $(x_s, y_s) = (i - k - 1, 4k - 2i + 3)$ if $i > k$; for odd $n$ let\u2026","title":"Representation count of the reduced triangular triple at a shifted Frobenius number"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/is-gen-frobenius-number-triangular-tripl","kind":"node","lean":"MetaMathlibExt.isGenFrobeniusNumber_triangularTriple","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#is-gen-frobenius-number-triangular-tripl","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"For a natural number $j$ write $t_j = j(j+1)/2$. Let $n$ and $s$ be natural numbers, let $r = \\lfloor\\sqrt{s}\\rfloor$, let $\\delta_s = 1$ if $s \\ge r^2 + r$ and $\\delta_s = 0$ otherwise, and put $q_s = 2r + 2 + \\delta_s$ and $c_s = s - r^2 - \\delta_s r$. Suppose that $n > 6\\lfloor\\sqrt{s+1}\\rfloor - 6$ if $n$ is even and $n > 6\\lfloor(\\sqrt{4s+5} - 1)/2\\rfloor - 3$ if $n$ is odd. Then the generalized Frobenius\u2026","title":"Generalized Frobenius numbers of three consecutive triangular numbers"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/is-gen-frobenius-number-triangular-tripl-2","kind":"node","lean":"MetaMathlibExt.isGenFrobeniusNumber_triangularTriple_succ_sub","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#is-gen-frobenius-number-triangular-tripl-2","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"For a natural number $r$ write $t_r = r(r+1)/2$. Let $n$ and $s$ be natural numbers, let $N_{s+1} = 6\\lfloor\\sqrt{s+2}\\rfloor - 6$ if $n$ is even and $N_{s+1} = 6\\lfloor(\\sqrt{4s+9} - 1)/2\\rfloor - 3$ if $n$ is odd, and suppose $n > N_{s+1}$. Let $g_s$ and $g_{s+1}$ be the generalized Frobenius numbers of $(t_n, t_{n+1}, t_{n+2})$ at $s$ and at $s + 1$, and put $P = (n+1)(n+2)$. Then: (i) if $s + 1$ is neither\u2026","title":"Komatsu's conjecture on generalized Frobenius numbers of triangular triples"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/is-gen-frobenius-number-triangular-tripl-3","kind":"node","lean":"MetaMathlibExt.isGenFrobeniusNumber_triangularTriple_of_xy","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#is-gen-frobenius-number-triangular-tripl-3","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"For a natural number $r$ write $t_r = r(r+1)/2$. Let $n$, $k$, $i$ be natural numbers with $i \\le 2k + 1$ and put $s = k(k+1) + i$. Suppose that $s \\le 2$ or that $s$ is neither $m^2$ nor $m(m+1)$ for any integer $m \\ge 1$. Let $N_s = 6\\lfloor\\sqrt{s+1}\\rfloor - 6$ if $n$ is even and $N_s = 6\\lfloor(\\sqrt{4s+5} - 1)/2\\rfloor - 3$ if $n$ is odd, and suppose $n \\ge 2$ and $n > N_s$. For even $n$ let\u2026","title":"Generalized Frobenius numbers of triangular triples in parametrized form"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/is-gen-frobenius-number-triangular-reduc","kind":"node","lean":"MetaMathlibExt.isGenFrobeniusNumber_triangular_reduced","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#is-gen-frobenius-number-triangular-reduc","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"For a natural number $r$ write $t_r = r(r+1)/2$. Let $n$, $k$, $i$ be natural numbers with $i \\le 2k + 1$ and put $s = k(k+1) + i$. Let $N_s = 6\\lfloor\\sqrt{s+1}\\rfloor - 6$ if $n$ is even and $N_s = 6\\lfloor(\\sqrt{4s+5} - 1)/2\\rfloor - 3$ if $n$ is odd, and suppose $n > N_s$. For even $n$ let $(x_s, y_s) = (i, 2(k - i))$ if $i \\le k$ and $(x_s, y_s) = (i - k - 1, 4k - 2i + 3)$ if $i > k$; for odd $n$ let\u2026","title":"Generalized Frobenius numbers of the reduced triangular triple"},{"id":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/is-gen-frobenius-number-triangular-tripl-4","kind":"node","lean":"MetaMathlibExt.isGenFrobeniusNumber_triangularTriple_eleven_to_twenty","parent":13350,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb/index.html#is-gen-frobenius-number-triangular-tripl-4","scope":"NumberTheory/numerical-semigroups/generalized-frobenius-numbers-of-triangular-numb","status":"can_prove","summary":"For a natural number $r$ write $t_r = r(r+1)/2$. For a natural number $n$, a natural number $s$ and an integer $L$, say that $s$ has value $L$ when the generalized Frobenius number $g$ of $(t_n, t_{n+1}, t_{n+2})$ at $s$ exists and satisfies $4(g + 1) = (n+1)(n+2)L$. Then for every natural number $n$: $11$ has value $8n + 12$ if $n = 12$ or $n \\ge 14$; $12$ has value $9n$ for even $n \\ge 14$ and $9n - 3$ for odd\u2026","title":"Explicit generalized Frobenius numbers of triangular triples at levels 11 to 20"},{"data_url":"scopes/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro.json","id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro/index.html","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro","status":"fully_proved","title":"Numerical semigroups generated by arithmetic progressions"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro/tripathi-pseudo-frobenius-arithmetic-pro","kind":"node","lean":"MetaMathlibExt.tripathi_pseudo_frobenius_arithmetic_progression_general","parent":13368,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tripathi's pseudo-Frobenius formula without assuming "],["math","\\(d > 0\\)"]],"source":"Tripathi's pseudo-Frobenius formula without assuming $d > 0$"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro/index.html#tripathi-pseudo-frobenius-arithmetic-pro","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro","status":"fully_proved","summary":"Let $a$, $h$ and $k$ be positive integers and $d$ a natural number with $\\gcd(a, d) = 1$, and let $q$ and $r$ be natural numbers with $1 \\le r \\le k$ and $a - 1 = qk + r$. Let $S$ be the set of all sums $c_0 a + \\sum_{i=1}^{k} c_i (ha + id)$ with natural numbers $c_0, \\dots, c_k$. Then the natural numbers $y \\notin S$ such that $y + s \\in S$ for every nonzero $s \\in S$ are exactly the numbers","title":"Tripathi's pseudo-Frobenius formula without assuming $d > 0$"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro/tripathi-pseudo-frobenius-arithmetic-pro-2","kind":"node","lean":"MetaMathlibExt.tripathi_pseudo_frobenius_arithmetic_progression","parent":13368,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro/index.html#tripathi-pseudo-frobenius-arithmetic-pro-2","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-arithmetic-pro","status":"fully_proved","summary":"Let $a$, $d$, $h$ and $k$ be positive integers with $\\gcd(a, d) = 1$, and let $q$ and $r$ be natural numbers with $1 \\le r \\le k$ and $a - 1 = qk + r$. Let $S$ be the set of all sums $c_0 a + \\sum_{i=1}^{k} c_i (ha + id)$ with natural numbers $c_0, \\dots, c_k$, that is, the additive monoid generated by $a, ha + d, \\dots, ha + kd$. Then the natural numbers $y \\notin S$ such that $y + s \\in S$ for every nonzero\u2026","title":"Tripathi's pseudo-Frobenius formula"},{"data_url":"scopes/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers.json","id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/index.html","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","status":"can_prove","title":"Numerical semigroups generated by powers"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/mem-minimal-generators-iff-first-appeara","kind":"node","lean":"MetaMathlibExt.mem_minimalGenerators_iff_firstAppearance_le","parent":13371,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/index.html#mem-minimal-generators-iff-first-appeara","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","status":"can_prove","summary":"Let $x_0 < x_1 < x_2 < \\cdots$ be a strictly increasing sequence of natural numbers. For $m \\ge 1$ let $S_m$ be the additive submonoid of $\\mathbb{N}$ generated by $x_m, x_{m+1}, \\ldots$, and let $\\beta(S_m)$ be its set of minimal generators, the nonzero elements of $S_m$ that are not the sum of two nonzero elements of $S_m$. For $k \\ge 1$ let $r_k$ be the least $m \\ge 1$ with $x_k \\in \\beta(S_m)$. Then for all\u2026","title":"Minimal generators of the tail semigroups of an increasing sequence"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/embedding-dimension-le-of-first-appearan","kind":"node","lean":"MetaMathlibExt.embeddingDimension_le_of_firstAppearance_ge","parent":13371,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/index.html#embedding-dimension-le-of-first-appearan","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","status":"can_prove","summary":"Let $x_0 < x_1 < x_2 < \\cdots$ be a strictly increasing sequence of natural numbers. For $m \\ge 1$ let $S_m$ be the additive submonoid of $\\mathbb{N}$ generated by $x_m, x_{m+1}, \\ldots$; call a nonzero element of $S_m$ that is not the sum of two nonzero elements of $S_m$ a minimal generator of $S_m$, and let the embedding dimension $e(S_m)$ be the number of minimal generators of $S_m$. For $k \\ge 1$ let $r_k$ be\u2026","title":"Embedding dimension bound from linear growth of first appearances"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/exists-four-squares-large-parts","kind":"node","lean":"MetaMathlibExt.exists_four_squares_large_parts","parent":13371,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/index.html#exists-four-squares-large-parts","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","status":"can_prove","summary":"There is a real constant $C > 0$ such that every positive integer $n$ can be written as $n = a_1^2 + a_2^2 + a_3^2 + a_4^2$ with integers $a_1, a_2, a_3, a_4$, each of which is either $0$ or at least $\\sqrt{n} / C$.","title":"Four squares with large nonzero parts"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/conductor-union-le","kind":"node","lean":"MetaMathlibExt.conductor_union_le","parent":13371,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/index.html#conductor-union-le","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","status":"can_prove","summary":"For a set $X$ of natural numbers, let $\\tau(X)$ be the conductor of the additive submonoid $\\langle X \\rangle$ of $\\mathbb{N}$ generated by $X$, the least natural number $c$ such that every $j \\ge c$ lies in $\\langle X \\rangle$. Let $A$ and $B$ be nonempty sets of natural numbers such that $1$ is the only natural number dividing every element of $A \\cup B$, let $t$ be a natural number dividing every element of $A$,\u2026","title":"Dutch\u2013Rickett conductor inequality for a union of generating sets"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/conductor-powers-le","kind":"node","lean":"MetaMathlibExt.conductor_powers_le","parent":13371,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dutch\u2013Rickett conductor recursion for semigroups of "],["math","\\(k\\)"],["text","-th powers"]],"source":"Dutch\u2013Rickett conductor recursion for semigroups of $k$-th powers"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/index.html#conductor-powers-le","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","status":"can_prove","summary":"Fix an integer $k \\ge 1$. For $m \\ge 1$ let $S_m$ be the additive submonoid of $\\mathbb{N}$ generated by $m^k, (m+1)^k, (m+2)^k, \\ldots$, and let $\\tau(S_m)$ be its conductor, the least natural number $c$ such that every $j \\ge c$ lies in $S_m$. Then for every integer $n \\ge 1$,","title":"Dutch\u2013Rickett conductor recursion for semigroups of $k$-th powers"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/embedding-dimension-squares-is-big-o","kind":"node","lean":"MetaMathlibExt.embeddingDimension_squares_isBigO","parent":13371,"reading_title":{"format":"inline-parts-v1","parts":[["text","Semigroups of squares have embedding dimension "],["math","\\(O(n)\\)"]],"source":"Semigroups of squares have embedding dimension $O(n)$"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/index.html#embedding-dimension-squares-is-big-o","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","status":"can_prove","summary":"For $n \\ge 1$ let $S_n$ be the additive submonoid of $\\mathbb{N}$ generated by $n^2, (n+1)^2, (n+2)^2, \\ldots$, and let $e(S_n)$ be its embedding dimension, the number of nonzero elements of $S_n$ that are not the sum of two nonzero elements of $S_n$. There is a natural number $C$ with $e(S_n) \\le C n$ for all $n \\ge 1$.","title":"Semigroups of squares have embedding dimension $O(n)$"},{"id":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/frobenius-number-powers-is-big-o","kind":"node","lean":"MetaMathlibExt.frobeniusNumber_powers_isBigO","parent":13371,"reading_title":{"format":"inline-parts-v1","parts":[["text","Frobenius numbers of semigroups of "],["math","\\(k\\)"],["text","-th powers are "],["math","\\(O(n^{k+\\varepsilon})\\)"]],"source":"Frobenius numbers of semigroups of $k$-th powers are $O(n^{k+\\varepsilon})$"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers/index.html#frobenius-number-powers-is-big-o","scope":"NumberTheory/numerical-semigroups/numerical-semigroups-generated-by-powers","status":"can_prove","summary":"Let $k \\ge 2$ be an integer and $\\varepsilon > 0$ a real number. For $n \\ge 1$ let $S_n$ be the additive submonoid of $\\mathbb{N}$ generated by $n^k, (n+1)^k, (n+2)^k, \\ldots$. Then there is a real constant $K$ such that for every $n \\ge 1$, if $S_n$ has a Frobenius number $F$ (the largest natural number not in $S_n$), then $F \\le K n^{k + \\varepsilon}$.","title":"Frobenius numbers of semigroups of $k$-th powers are $O(n^{k+\\varepsilon})$"},{"data_url":"scopes/NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato.json","id":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/index.html","scope":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato","status":"can_prove","title":"Primes in numerical semigroups with two generators"},{"id":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/two-generator-dvd-sub-of-eq","kind":"node","lean":"MetaMathlibExt.twoGenerator_dvd_sub_of_eq","parent":13379,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/index.html#two-generator-dvd-sub-of-eq","scope":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato","status":"can_prove","summary":"Let $c$ and $d$ be coprime natural numbers. If natural numbers $x, y, x', y'$ satisfy $cx + dy = cx' + dy'$, then $c$ divides $y - y'$ and $d$ divides $x - x'$ in the integers.","title":"Two representations by coprime generators"},{"id":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/exists-eq-of-ncard-two-generator-reps","kind":"node","lean":"MetaMathlibExt.exists_eq_of_ncard_twoGeneratorReps","parent":13379,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normal form of a number with exactly "],["math","\\(\\ell\\)"],["text"," representations"]],"source":"Normal form of a number with exactly $\\ell$ representations"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/index.html#exists-eq-of-ncard-two-generator-reps","scope":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato","status":"can_prove","summary":"Let $c$ and $d$ be coprime positive integers, and let $n$ and $\\ell$ be natural numbers such that the equation $n = cx + dy$ has exactly $\\ell$ solutions $(x, y)$ in natural numbers. Then there are natural numbers $x_0 < d$ and $y_0 < c$ with $n = (\\ell - 1)cd + cx_0 + dy_0$ in the integers.","title":"Normal form of a number with exactly $\\ell$ representations"},{"id":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/card-primes-ell-numerical-semigroup-eq","kind":"node","lean":"MetaMathlibExt.card_primes_ellNumericalSemigroup_eq","parent":13379,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting primes with more than "],["math","\\(\\ell\\)"],["text"," representations"]],"source":"Counting primes with more than $\\ell$ representations"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/index.html#card-primes-ell-numerical-semigroup-eq","scope":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato","status":"can_prove","summary":"Let $c$ and $d$ be coprime natural numbers with $1 < c < d$, let $\\ell$ be a natural number, and put $g_\\ell(c, d) = (\\ell + 1)cd - c - d$. Write $\\pi_{\\ell,c,d}$ for the number of primes $n \\le g_\\ell(c, d)$ such that the equation $n = cx + dy$ has more than $\\ell$ solutions $(x, y)$ in natural numbers. Then $\\pi_{\\ell,c,d}$ equals the number of primes $n$ with $\\ell cd < n \\le g_\\ell(c, d)$ of the form\u2026","title":"Counting primes with more than $\\ell$ representations"},{"id":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/von-mangoldt-sum-ell-numerical-semigroup","kind":"node","lean":"MetaMathlibExt.vonMangoldt_sum_ellNumericalSemigroup_asymptotic","parent":13379,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted count of primes in the "],["math","\\(\\ell\\)"],["text","-numerical semigroup"]],"source":"Weighted count of primes in the $\\ell$-numerical semigroup"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/index.html#von-mangoldt-sum-ell-numerical-semigroup","scope":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato","status":"can_prove","summary":"For every natural number $\\ell$ and every $\\varepsilon > 0$ there is a natural number $C$ such that the following holds for all coprime natural numbers $c$ and $d$ with $C \\le c < d$. For a natural number $k$ put $g_k(c, d) = (k + 1)cd - c - d$, so that $g_0(c, d) = cd - c - d$, and let $\\psi_{\\ell,c,d}$ be the sum of the von Mangoldt function $\\Lambda(n)$ over the natural numbers $n$ with\u2026","title":"Weighted count of primes in the $\\ell$-numerical semigroup"},{"id":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/card-primes-ell-numerical-semigroup-asym","kind":"node","lean":"MetaMathlibExt.card_primes_ellNumericalSemigroup_asymptotic","parent":13379,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic count of primes with more than "],["math","\\(\\ell\\)"],["text"," representations"]],"source":"Asymptotic count of primes with more than $\\ell$ representations"},"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato/index.html#card-primes-ell-numerical-semigroup-asym","scope":"NumberTheory/numerical-semigroups/primes-in-numerical-semigroups-with-two-generato","status":"can_prove","summary":"For every natural number $\\ell$ and every $\\varepsilon > 0$ there is a natural number $C$ such that the following holds for all coprime natural numbers $c$ and $d$ with $C \\le c < d$. Put $g_\\ell(c, d) = (\\ell + 1)cd - c - d$, and let $\\pi_{\\ell,c,d}$ be the number of primes $n \\le g_\\ell(c, d)$ such that the equation $n = cx + dy$ has more than $\\ell$ solutions $(x, y)$ in natural numbers. Then, with $\\pi$ the\u2026","title":"Asymptotic count of primes with more than $\\ell$ representations"},{"data_url":"scopes/NumberTheory/numerical-semigroups/sylvester-s-coin-theorem.json","id":"NumberTheory/numerical-semigroups/sylvester-s-coin-theorem","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/sylvester-s-coin-theorem/index.html","scope":"NumberTheory/numerical-semigroups/sylvester-s-coin-theorem","status":"fully_proved","title":"Sylvester's coin theorem"},{"id":"NumberTheory/numerical-semigroups/sylvester-s-coin-theorem/sylvester-coin-distinct-odd-primes","kind":"node","lean":"MetaMathlibExt.sylvester_coin_distinct_odd_primes","parent":13385,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/sylvester-s-coin-theorem/index.html#sylvester-coin-distinct-odd-primes","scope":"NumberTheory/numerical-semigroups/sylvester-s-coin-theorem","status":"fully_proved","summary":"Let $p$ and $q$ be distinct odd primes. Then the number of natural numbers $n$ that cannot be written as $n = px + qy$ with natural numbers $x, y$ is $(p-1)(q-1)/2$.","title":"Sylvester's coin theorem for two distinct odd primes"},{"id":"NumberTheory/numerical-semigroups/sylvester-s-coin-theorem/sylvester-coin-theorem","kind":"node","lean":"MetaMathlibExt.sylvester_coin_theorem","parent":13385,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/sylvester-s-coin-theorem/index.html#sylvester-coin-theorem","scope":"NumberTheory/numerical-semigroups/sylvester-s-coin-theorem","status":"fully_proved","summary":"Let $a$ and $b$ be coprime natural numbers. Then the number of natural numbers $n$ that cannot be written as $n = ax + by$ with natural numbers $x, y$ is $(a - 1)(b - 1)/2$.","title":"Sylvester's coin theorem"},{"data_url":"scopes/NumberTheory/numerical-semigroups/telescopic-numerical-semigroups.json","id":"NumberTheory/numerical-semigroups/telescopic-numerical-semigroups","kind":"scope","navigation":"module","parent":13322,"result":false,"route":"../roadmap/NumberTheory/numerical-semigroups/telescopic-numerical-semigroups/index.html","scope":"NumberTheory/numerical-semigroups/telescopic-numerical-semigroups","status":"fully_proved","title":"Telescopic numerical semigroups"},{"id":"NumberTheory/numerical-semigroups/telescopic-numerical-semigroups/telescopic-gcd","kind":"node","lean":"MetaMathlibExt.telescopicGcd","parent":13388,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/telescopic-numerical-semigroups/index.html#telescopic-gcd","scope":"NumberTheory/numerical-semigroups/telescopic-numerical-semigroups","status":"fully_proved","summary":"For a finite sequence $G = (g_1, \\dots, g_k)$ of natural numbers and a natural number $j$, $d_j$ is the greatest common divisor of the first $j$ entries of $G$ (of all of $G$ when $j > k$), with $d_0 = 0$.","title":"Prefix gcds of a sequence"},{"id":"NumberTheory/numerical-semigroups/telescopic-numerical-semigroups/telescopic-c","kind":"node","lean":"MetaMathlibExt.telescopicC","parent":13388,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/telescopic-numerical-semigroups/index.html#telescopic-c","scope":"NumberTheory/numerical-semigroups/telescopic-numerical-semigroups","status":"fully_proved","summary":"For a finite sequence $G = (g_1, \\dots, g_k)$ of natural numbers, let $d_i$ be the greatest common divisor of the first $i$ entries of $G$ (of all of $G$ when $i > k$), with $d_0 = 0$. For $j \\ge 1$ the successive gcd quotient is $c_j = d_{j-1} / d_j$, taken to be $0$ when $d_j = 0$, that is, when the first $j$ entries are all zero; and $c_0 = 0$.","title":"Successive gcd quotients"},{"id":"NumberTheory/numerical-semigroups/telescopic-numerical-semigroups/is-telescopic","kind":"node","lean":"MetaMathlibExt.IsTelescopic","parent":13388,"result":true,"route":"../roadmap/NumberTheory/numerical-semigroups/telescopic-numerical-semigroups/index.html#is-telescopic","scope":"NumberTheory/numerical-semigroups/telescopic-numerical-semigroups","status":"fully_proved","summary":"Let $G = (g_1, \\dots, g_k)$ be a finite sequence of natural numbers. For $0 \\le i \\le k$ let $d_i$ be the greatest common divisor of $g_1, \\dots, g_i$, with $d_0 = 0$, and for $1 \\le j \\le k$ let $c_j = d_{j-1} / d_j$ be the successive gcd quotient, taken to be $0$ when $d_j = 0$. The sequence $G$ is telescopic when $k \\ge 2$, $g_1 + g_2 > 0$, and for every $j$ with $2 \\le j \\le k$ the number $c_j g_j$ lies in the\u2026","title":"Telescopic sequences"},{"data_url":"scopes/NumberTheory/function-fields.json","id":"NumberTheory/function-fields","kind":"scope","parent":4977,"result":false,"route":"../roadmap/NumberTheory/function-fields/index.html","scope":"NumberTheory/function-fields","status":"can_prove","summary":"Number-theoretic analogues over F_q[T] and finite fields: polynomial primes, h-free polynomials, function-field adeles, positive-characteristic zeta values.","title":"Arithmetic of function fields"},{"id":"NumberTheory/function-fields/odd-perfect-polynomial-is-square","kind":"node","lean":"MetaMathlibExt.odd_perfect_polynomial_is_square","parent":13392,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd perfect polynomials over "],["math","\\(\\mathbb{F}_2\\)"],["text"," are squares (Canaday)"]],"source":"Odd perfect polynomials over $\\mathbb{F}_2$ are squares (Canaday)"},"result":true,"route":"../roadmap/NumberTheory/function-fields/index.html#odd-perfect-polynomial-is-square","scope":"NumberTheory/function-fields","status":"fully_proved","summary":"Let $A$ be a polynomial over $\\mathbb{F}_2$ that is coprime to both $X$ and $X + 1$ (that is, odd: $A$ has no linear factor) and that equals the sum of its monic divisors (that is, perfect). Then $A$ is a square: $A = B^2$ for some polynomial $B$ over $\\mathbb{F}_2$.","title":"Odd perfect polynomials over $\\mathbb{F}_2$ are squares (Canaday)"},{"data_url":"scopes/NumberTheory/function-fields/adeles-of-function-fields.json","id":"NumberTheory/function-fields/adeles-of-function-fields","kind":"scope","navigation":"module","parent":13392,"result":false,"route":"../roadmap/NumberTheory/function-fields/adeles-of-function-fields/index.html","scope":"NumberTheory/function-fields/adeles-of-function-fields","status":"fully_proved","title":"Adeles of function fields"},{"id":"NumberTheory/function-fields/adeles-of-function-fields/function-field-adele-ring","kind":"node","lean":"FunctionFieldAdeleRing","parent":13394,"result":true,"route":"../roadmap/NumberTheory/function-fields/adeles-of-function-fields/index.html#function-field-adele-ring","scope":"NumberTheory/function-fields/adeles-of-function-fields","status":"fully_proved","summary":"Let $F$ be a field, $P$ a type (of places) and, for each $p \\in P$, let $O_p$ be a valuation subring of $F$. The ad\u00e8le ring of $F$ with respect to $(O_p)_{p \\in P}$ is the restricted product of the copies of $F$ with respect to the subrings $O_p$: its elements are the families $(a_p)_{p \\in P}$ with $a_p \\in F$ for every $p$ and $a_p \\in O_p$ for all but finitely many $p$.","title":"Ad\u00e8le ring of a field with respect to valuation subrings"},{"id":"NumberTheory/function-fields/adeles-of-function-fields/ext","kind":"node","lean":"FunctionFieldAdeleRing.ext","parent":13394,"result":true,"route":"../roadmap/NumberTheory/function-fields/adeles-of-function-fields/index.html#ext","scope":"NumberTheory/function-fields/adeles-of-function-fields","status":"fully_proved","summary":"Let $F$ be a field, $P$ a type and, for each $p \\in P$, let $O_p$ be a valuation subring of $F$. If two ad\u00e8les $a$ and $b$ of the ad\u00e8le ring of $F$ with respect to $(O_p)_{p \\in P}$ satisfy $a_p = b_p$ for every $p \\in P$, then $a = b$.","title":"Extensionality for restricted-product ad\u00e8les"},{"id":"NumberTheory/function-fields/adeles-of-function-fields/mem-valuation-subring-cofinitely","kind":"node","lean":"FunctionFieldAdeleRing.mem_valuationSubring_cofinitely","parent":13394,"result":true,"route":"../roadmap/NumberTheory/function-fields/adeles-of-function-fields/index.html#mem-valuation-subring-cofinitely","scope":"NumberTheory/function-fields/adeles-of-function-fields","status":"fully_proved","summary":"Let $F$ be a field, $P$ a type and, for each $p \\in P$, let $O_p$ be a valuation subring of $F$. For every ad\u00e8le $a$ of the ad\u00e8le ring of $F$ with respect to $(O_p)_{p \\in P}$, the component $a_p$ lies in $O_p$ for all but finitely many $p \\in P$.","title":"Ad\u00e8les are integral at almost all places"},{"data_url":"scopes/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials.json","id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","kind":"scope","navigation":"module","parent":13392,"result":false,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","title":"Divisor problems for B-free polynomials"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-dirichlet-conv","kind":"node","lean":"MetaMathlibExt.monicDirichletConv","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-dirichlet-conv","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field. The Dirichlet convolution of functions $\\alpha$ and $\\beta$ from $F[T]$ to $\\mathbb{C}$ is the function $\\alpha * \\beta$ given at $f \\in F[T]$ by","title":"Dirichlet convolution over monic divisors"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-dirichlet-conv-one","kind":"node","lean":"MetaMathlibExt.monicDirichletConv_one","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dirichlet convolution at "],["math","\\(1\\)"]],"source":"Dirichlet convolution at $1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-dirichlet-conv-one","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\alpha$ and $\\beta$ be functions from $F[T]$ to $\\mathbb{C}$. Their Dirichlet convolution over monic divisors satisfies $(\\alpha * \\beta)(1) = \\alpha(1) \\beta(1)$, since the only monic divisor of $1$ is $1$.","title":"Dirichlet convolution at $1$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-perron-formula","kind":"node","lean":"MetaMathlibExt.monic_perron_formula","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-perron-formula","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field, let $a$ be a complex-valued function on $F[T]$, and let $\\rho$ be a real number with $0 < \\rho < 1$ such that the series $\\sum_f |a(f)| \\rho^{\\deg f}$ over all monic $f \\in F[T]$ converges. Put $A(u) = \\sum_f a(f) u^{\\deg f}$, the sum again over all monic $f$. Then for every natural number $n$, with the circle $|u| = \\rho$ traversed counterclockwise,","title":"Perron's formula for monic polynomials over a finite field"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-dirichlet-conv-mul-of-completely-m","kind":"node","lean":"MetaMathlibExt.monicDirichletConv_mul_of_completelyMultiplicative","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-dirichlet-conv-mul-of-completely-m","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a field and let $\\alpha, \\beta : F[T] \\to \\mathbb{C}$ be completely multiplicative, that is, monoid homomorphisms from the multiplicative monoid of $F[T]$ to $\\mathbb{C}$. Write $\\gamma = \\alpha * \\beta$ for their Dirichlet convolution over monic divisors and $\\mu$ for the M\u00f6bius function. Then for all monic $f, g \\in F[T]$,","title":"Dirichlet convolution of completely multiplicative functions at a product"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-sum-moebius-div-norm","kind":"node","lean":"MetaMathlibExt.monic_sum_moebius_div_norm","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-sum-moebius-div-norm","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and let $\\mu$ be the M\u00f6bius function of $F[T]$. For every monic $f \\in F[T]$,","title":"M\u00f6bius sum over the monic divisors of a polynomial over a finite field"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-sum-moebius-mul-nat-degree-div-nor","kind":"node","lean":"MetaMathlibExt.monic_sum_moebius_mul_natDegree_div_norm","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-sum-moebius-mul-nat-degree-div-nor","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and let $\\mu$ be the M\u00f6bius function of $F[T]$. For every monic $f \\in F[T]$,","title":"Degree-weighted M\u00f6bius sum over the monic divisors of a polynomial over a finite field"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-count-nat-degree-le","kind":"node","lean":"MetaMathlibExt.monic_count_natDegree_le","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-count-nat-degree-le","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and let $N$ be a natural number. The number of monic polynomials in $F[T]$ of degree at most $N$ is $(q^{N+1} - 1)/(q - 1)$.","title":"Number of monic polynomials of bounded degree"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-sum-divisor-count-nat-degree-le","kind":"node","lean":"MetaMathlibExt.monic_sum_divisorCount_natDegree_le","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-sum-divisor-count-nat-degree-le","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements. For monic $f \\in F[T]$, the Dirichlet convolution $(1 * 1)(f)$ of the constant function $1$ with itself is the number of monic divisors of $f$. Then for every natural number $N$,","title":"Summatory divisor function for monic polynomials over a finite field"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/fq-norm","kind":"node","lean":"MetaMathlibExt.fqNorm","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#fq-norm","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements. The norm of a nonzero $f \\in F[T]$ is the real number $|f| = q^{\\deg f}$, and $|0| = 0$.","title":"Norm of a polynomial over a finite field"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/fq-norm-of-ne-zero","kind":"node","lean":"MetaMathlibExt.fqNorm_of_ne_zero","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#fq-norm-of-ne-zero","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements. For every nonzero $f \\in F[T]$, the norm of $f$ is $|f| = q^{\\deg f}$.","title":"The norm of a nonzero polynomial"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/fq-norm-mul","kind":"node","lean":"MetaMathlibExt.fqNorm_mul","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#fq-norm-mul","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, and let $|f| = q^{\\deg f}$ be the norm of a nonzero $f \\in F[T]$. For all nonzero $f, g \\in F[T]$, $|fg| = |f| \\, |g|$.","title":"Multiplicativity of the norm"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-nat-degree-le-zero-iff","kind":"node","lean":"MetaMathlibExt.monic_natDegree_le_zero_iff","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-nat-degree-le-zero-iff","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field. A polynomial $f \\in F[T]$ is monic of degree at most $0$ if and only if $f = 1$.","title":"Monic polynomials of degree zero"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/is-bfree","kind":"node","lean":"MetaMathlibExt.IsBFree","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(B_\\kappa\\)"],["text","-free polynomials"]],"source":"$B_\\kappa$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#is-bfree","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\kappa$ assign a natural number to each polynomial in $F[T]$. A polynomial $f \\in F[T]$ is $B_\\kappa$-free when $P^{\\kappa(P)}$ does not divide $f$ for any monic irreducible $P \\in F[T]$.","title":"$B_\\kappa$-free polynomials"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/is-bfree-one","kind":"node","lean":"MetaMathlibExt.isBFree_one","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomial "],["math","\\(1\\)"],["text"," is "],["math","\\(B_\\kappa\\)"],["text","-free"]],"source":"The polynomial $1$ is $B_\\kappa$-free"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#is-bfree-one","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\kappa$ assign to each polynomial in $F[T]$ a natural number, with $\\kappa(P) \\ne 0$ for every monic irreducible $P$. Then $1$ is $B_\\kappa$-free: $P^{\\kappa(P)}$ does not divide $1$ for any monic irreducible $P$.","title":"The polynomial $1$ is $B_\\kappa$-free"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/kappa-min","kind":"node","lean":"MetaMathlibExt.kappaMin","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The minimal exponent "],["math","\\(\\kappa_{\\min}\\)"]],"source":"The minimal exponent $\\kappa_{\\min}$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#kappa-min","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\kappa$ be a function from $F[T]$ to the natural numbers. Then $\\kappa_{\\min}$ is the minimum of $\\kappa(P)$ over all monic irreducible polynomials $P \\in F[T]$.","title":"The minimal exponent $\\kappa_{\\min}$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/kappa-min-le","kind":"node","lean":"MetaMathlibExt.kappaMin_le","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\kappa_{\\min}\\)"],["text"," is a lower bound for "],["math","\\(\\kappa\\)"]],"source":"$\\kappa_{\\min}$ is a lower bound for $\\kappa$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#kappa-min-le","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field, let $\\kappa$ assign a natural number to each polynomial in $F[T]$, and let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic irreducible $P \\in F[T]$. Then $\\kappa_{\\min} \\le \\kappa(P)$ for every monic irreducible $P \\in F[T]$.","title":"$\\kappa_{\\min}$ is a lower bound for $\\kappa$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/kappa-pow","kind":"node","lean":"MetaMathlibExt.kappaPow","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\kappa\\)"],["text","-power of a polynomial"]],"source":"The $\\kappa$-power of a polynomial"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#kappa-pow","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\kappa$ be a function from $F[T]$ to the natural numbers. For nonzero $f \\in F[T]$ and a monic irreducible $P$, write $\\nu_P(f)$ for the multiplicity of $P$ in $f$. The $\\kappa$-power of $f$ is $f^\\kappa = \\prod_P P^{\\nu_P(f)\\, \\kappa(P)}$, the product over the monic irreducible $P$ dividing $f$; it does not depend on the leading coefficient of $f$. For $f = 0$ the product is empty and\u2026","title":"The $\\kappa$-power of a polynomial"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/kappa-pow-one","kind":"node","lean":"MetaMathlibExt.kappaPow_one","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\kappa\\)"],["text","-power of "],["math","\\(1\\)"]],"source":"The $\\kappa$-power of $1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#kappa-pow-one","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\kappa$ be a function from $F[T]$ to the natural numbers. Then the $\\kappa$-power of the constant polynomial $1$ is $1^\\kappa = 1$.","title":"The $\\kappa$-power of $1$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/kappa-pow-of-irreducible","kind":"node","lean":"MetaMathlibExt.kappaPow_of_irreducible","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\kappa\\)"],["text","-power of a monic irreducible polynomial"]],"source":"The $\\kappa$-power of a monic irreducible polynomial"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#kappa-pow-of-irreducible","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\kappa$ be a function from $F[T]$ to the natural numbers. For every monic irreducible $P \\in F[T]$, the $\\kappa$-power of $P$ is $P^\\kappa = P^{\\kappa(P)}$.","title":"The $\\kappa$-power of a monic irreducible polynomial"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-divisor-count","kind":"node","lean":"MetaMathlibExt.monicDivisorCount","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h\\)"],["text","-fold divisor function"]],"source":"The $h$-fold divisor function"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-divisor-count","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and $h$ a natural number. For $f \\in F[T]$, $d_h(f)$ is the number of $h$-tuples $(g_1, \\dots, g_h)$ of monic polynomials in $F[T]$ with $g_1 \\cdots g_h = f$.","title":"The $h$-fold divisor function"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-divisor-count-one","kind":"node","lean":"MetaMathlibExt.monicDivisorCount_one","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor functions at "],["math","\\(1\\)"]],"source":"The divisor functions at $1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-divisor-count-one","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and $h$ a natural number. Then $d_h(1) = 1$, where $d_h(f)$ is the number of $h$-tuples of monic polynomials in $F[T]$ whose product is $f$: the only such tuple for $f = 1$ has every entry equal to $1$.","title":"The divisor functions at $1$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-divisor-count-zero-of-ne-one","kind":"node","lean":"MetaMathlibExt.monicDivisorCount_zero_of_ne_one","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor function "],["math","\\(d_0\\)"],["text"," vanishes away from "],["math","\\(1\\)"]],"source":"The divisor function $d_0$ vanishes away from $1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-divisor-count-zero-of-ne-one","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $f \\in F[T]$ with $f \\ne 1$. Then $d_0(f) = 0$, where $d_h(f)$ is the number of $h$-tuples of monic polynomials in $F[T]$ whose product is $f$: the only $0$-tuple is the empty one, whose product is $1$.","title":"The divisor function $d_0$ vanishes away from $1$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-divisor-count-one-left","kind":"node","lean":"MetaMathlibExt.monicDivisorCount_one_left","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The divisor function "],["math","\\(d_1\\)"],["text"," on monic polynomials"]],"source":"The divisor function $d_1$ on monic polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-divisor-count-one-left","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field. For every monic $f \\in F[T]$, $d_1(f) = 1$, where $d_h(f)$ is the number of $h$-tuples of monic polynomials in $F[T]$ whose product is $f$.","title":"The divisor function $d_1$ on monic polynomials"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-conv-shifted-sum","kind":"node","lean":"MetaMathlibExt.monicConvShiftedSum","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-conv-shifted-sum","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha, \\beta \\colon F[T] \\to \\mathbb{C}$, let $\\alpha * \\beta$ be their Dirichlet convolution over monic divisors, let $f \\in F[T]$ and let $n$ be a natural number. Then $S_f\\alpha, \\beta = \\sum_g (\\alpha * \\beta)(fg)$, the sum over the monic $g \\in F[T]$ with $\\deg f + \\deg g \\le n$, where $\\deg 0 = 0$. This is the paper's sum over $g$ of degree at most $n - \\deg f$.","title":"Shifted partial sums of a Dirichlet convolution"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-conv-shifted-sum-of-lt","kind":"node","lean":"MetaMathlibExt.monicConvShiftedSum_of_lt","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shifted partial sums vanish below "],["math","\\(\\deg f\\)"]],"source":"Shifted partial sums vanish below $\\deg f$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-conv-shifted-sum-of-lt","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha, \\beta \\colon F[T] \\to \\mathbb{C}$ and let $f \\in F[T]$. For every natural number $n < \\deg f$, the shifted partial sum of the Dirichlet convolution satisfies $S_f\\alpha, \\beta = 0$.","title":"Shifted partial sums vanish below $\\deg f$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-conv-partial-sum","kind":"node","lean":"MetaMathlibExt.monicConvPartialSum","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-conv-partial-sum","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha, \\beta \\colon F[T] \\to \\mathbb{C}$, and let $\\alpha * \\beta$ be their Dirichlet convolution over monic divisors. For an integer $n$, $D\\alpha, \\beta = \\sum_g (\\alpha * \\beta)(g)$, the sum over the monic $g \\in F[T]$ of degree at most $n$. For $n < 0$ the sum is empty, so $D\\alpha, \\beta = 0$; this is how the paper's Lemma 3 uses $D$ at $n - \\deg(f \\ell)$.","title":"Partial sums of a Dirichlet convolution over monic polynomials"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-conv-partial-sum-of-neg","kind":"node","lean":"MetaMathlibExt.monicConvPartialSum_of_neg","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(D(\\alpha, \\beta)\\)"],["text"," vanishes at negative arguments"]],"source":"$D(\\alpha, \\beta)$ vanishes at negative arguments"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-conv-partial-sum-of-neg","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\alpha, \\beta \\colon F[T] \\to \\mathbb{C}$. For every integer $n < 0$, the partial sum of the Dirichlet convolution of $\\alpha$ and $\\beta$ satisfies $D\\alpha, \\beta = 0$.","title":"$D(\\alpha, \\beta)$ vanishes at negative arguments"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-conv-partial-sum-zero","kind":"node","lean":"MetaMathlibExt.monicConvPartialSum_zero","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partial sum "],["math","\\(D(\\alpha, \\beta)(0)\\)"]],"source":"The partial sum $D(\\alpha, \\beta)(0)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-conv-partial-sum-zero","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\alpha, \\beta \\colon F[T] \\to \\mathbb{C}$. The partial sum of the Dirichlet convolution of $\\alpha$ and $\\beta$ at $0$ is $D\\alpha, \\beta = \\alpha(1)\\, \\beta(1)$.","title":"The partial sum $D(\\alpha, \\beta)(0)$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/weighted-monic-conv-sum","kind":"node","lean":"MetaMathlibExt.weightedMonicConvSum","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#weighted-monic-conv-sum","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field, let $\\gamma, \\alpha, \\beta$ be functions from $F[T]$ to $\\mathbb{C}$, and write $\\alpha * \\beta$ for the monic Dirichlet convolution of $\\alpha$ and $\\beta$. For a natural number $n$, let $M_{\\le n}$ be the set of monic polynomials in $F[T]$ of degree at most $n$. The weighted convolution sum is","title":"Weighted convolution sum"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/weighted-monic-conv-sum-zero","kind":"node","lean":"MetaMathlibExt.weightedMonicConvSum_zero","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#weighted-monic-conv-sum-zero","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\gamma, \\alpha, \\beta$ be functions from $F[T]$ to $\\mathbb{C}$, and write $\\alpha * \\beta$ for the monic Dirichlet convolution. The weighted convolution sum $D(\\gamma, \\alpha, \\beta; 0)$, the sum of $\\gamma(f)\\,(\\alpha * \\beta)(f)$ over the monic $f \\in F[T]$ of degree at most $0$, equals $\\gamma(1)\\,\\alpha(1)\\,\\beta(1)$.","title":"The weighted convolution sum at degree zero"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-conv-shifted-sum-eq-of-completely","kind":"node","lean":"MetaMathlibExt.monicConvShiftedSum_eq_of_completelyMultiplicative","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-conv-shifted-sum-eq-of-completely","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field and let $\\alpha, \\beta \\colon F[T] \\to \\mathbb{C}$ be completely multiplicative, that is, monoid homomorphisms from the multiplicative monoid of $F[T]$ to $\\mathbb{C}$. Write $\\mu$ for the M\u00f6bius function of $F[T]$. For $f \\in F[T]$, a natural number $n$ and an integer $k$, write $S_f\\alpha, \\beta$ and $D\\alpha, \\beta$ for the shifted partial sums and the partial sums of the Dirichlet\u2026","title":"Shifted partial sums for completely multiplicative functions"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/weighted-monic-conv-sum-eq-sum-shifted","kind":"node","lean":"MetaMathlibExt.weightedMonicConvSum_eq_sum_shifted","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#weighted-monic-conv-sum-eq-sum-shifted","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field and let $\\gamma, \\alpha, \\beta$ be functions from $F[T]$ to $\\mathbb{C}$. Write $\\alpha * \\beta$ for the monic Dirichlet convolution, $\\mu$ for the M\u00f6bius function of $F[T]$, and $\\lambda = \\gamma * \\mu$. For a natural number $n$ let $M_{\\le n}$ be the set of monic polynomials in $F[T]$ of degree at most $n$, let\u2026","title":"Weighted convolution sums as sums of shifted sums"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/weighted-monic-conv-sum-eq-of-completely","kind":"node","lean":"MetaMathlibExt.weightedMonicConvSum_eq_of_completelyMultiplicative","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Camargo's identity over "],["math","\\(F(T)\\)"]],"source":"Camargo's identity over $F(T)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#weighted-monic-conv-sum-eq-of-completely","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field and let $\\mu$ be the M\u00f6bius function of $F[T]$. For functions $\\alpha, \\beta \\colon F[T] \\to \\mathbb{C}$ let $(\\alpha * \\beta)(f) = \\sum_{g \\mid f} \\alpha(g) \\beta(f/g)$, over monic divisors $g$, be their Dirichlet convolution, and let $D\\alpha, \\beta$ be the sum of $(\\alpha * \\beta)(g)$ over the monic $g$ of degree at most $k$, which is $0$ for $k < 0$. For a further function $\\gamma$ let\u2026","title":"Camargo's identity over $F(T)$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/monic-dirichlet-conv-b-free-indicator-mo","kind":"node","lean":"MetaMathlibExt.monicDirichletConv_bFreeIndicator_moebius","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution of the "],["math","\\(B_\\kappa\\)"],["text","-free indicator with the M\u00f6bius function"]],"source":"Convolution of the $B_\\kappa$-free indicator with the M\u00f6bius function"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#monic-dirichlet-conv-b-free-indicator-mo","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a field and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ne 0$ for every monic irreducible $P \\in F[T]$. Let $\\mathbf{1}_{B_\\kappa} \\colon F[T] \\to \\mathbb{C}$ be the indicator function of the $B_\\kappa$-free polynomials, let $\\mu$ be the M\u00f6bius function of $F[T]$, and write $r^\\kappa$ for the $\\kappa$-power of a polynomial $r$. Then for every monic $f \\in F[T]$,\u2026","title":"Convolution of the $B_\\kappa$-free indicator with the M\u00f6bius function"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/abs-sum-moebius-kappa-pow-le","kind":"node","lean":"MetaMathlibExt.abs_sum_moebius_kappaPow_le","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#abs-sum-moebius-kappa-pow-le","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. Let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic irreducible $P$, and for monic $r = \\prod_P P^{\\nu_P}$ write $r^\\kappa = \\prod_P P^{\\nu_P \\kappa(P)}$. Then for every natural number $N \\ge 1$,","title":"Bound on the M\u00f6bius sum over \u03ba-th powers"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/sum-divisors-kappa-pow-is-big-o","kind":"node","lean":"MetaMathlibExt.sum_divisors_kappaPow_isBigO","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisor-count bound over "],["math","\\(\\kappa\\)"],["text","-th powers of monic polynomials"]],"source":"Divisor-count bound over $\\kappa$-th powers of monic polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#sum-divisors-kappa-pow-is-big-o","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. For monic $r = \\prod_P P^{\\nu_P}$ write $r^\\kappa = \\prod_P P^{\\nu_P \\kappa(P)}$, and let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic irreducible $P$. Then for every $\\varepsilon > 0$, as $N \\to \\infty$,","title":"Divisor-count bound over $\\kappa$-th powers of monic polynomials"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/sum-moebius-kappa-pow-divisor-count-is-b","kind":"node","lean":"MetaMathlibExt.sum_moebius_kappaPow_divisorCount_isBigO","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler-product asymptotic with the "],["math","\\((h-1)\\)"],["text","-divisor function over "],["math","\\(\\kappa\\)"],["text","-th powers"]],"source":"Euler-product asymptotic with the $(h-1)$-divisor function over $\\kappa$-th powers"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#sum-moebius-kappa-pow-divisor-count-is-b","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\mu$ be the M\u00f6bius function of $F[T]$. Let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. For monic $r = \\prod_P P^{\\nu_P}$ write $r^\\kappa = \\prod_P P^{\\nu_P \\kappa(P)}$, and let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic\u2026","title":"Euler-product asymptotic with the $(h-1)$-divisor function over $\\kappa$-th powers"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/sum-moebius-kappa-pow-moebius-nat-degree","kind":"node","lean":"MetaMathlibExt.sum_moebius_kappaPow_moebius_natDegree_isBigO","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler-product asymptotic for a degree-weighted M\u00f6bius sum over "],["math","\\(\\kappa\\)"],["text","-th powers"]],"source":"Euler-product asymptotic for a degree-weighted M\u00f6bius sum over $\\kappa$-th powers"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#sum-moebius-kappa-pow-moebius-nat-degree","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\mu$ be the M\u00f6bius function of $F[T]$. Let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. For monic $r = \\prod_P P^{\\nu_P}$ write $r^\\kappa = \\prod_P P^{\\nu_P \\kappa(P)}$, and let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic\u2026","title":"Euler-product asymptotic for a degree-weighted M\u00f6bius sum over $\\kappa$-th powers"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/sum-moebius-nat-degree-kappa-pow-is-big","kind":"node","lean":"MetaMathlibExt.sum_moebius_natDegree_kappaPow_isBigO","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler-product asymptotic for a M\u00f6bius sum weighted by "],["math","\\(\\deg r^\\kappa\\)"]],"source":"Euler-product asymptotic for a M\u00f6bius sum weighted by $\\deg r^\\kappa$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#sum-moebius-nat-degree-kappa-pow-is-big","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\mu$ be the M\u00f6bius function of $F[T]$. Let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. For monic $r = \\prod_P P^{\\nu_P}$ write $r^\\kappa = \\prod_P P^{\\nu_P \\kappa(P)}$, and let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic\u2026","title":"Euler-product asymptotic for a M\u00f6bius sum weighted by $\\deg r^\\kappa$"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/norm-sum-mul-char-divisor-count-le","kind":"node","lean":"MetaMathlibExt.norm_sum_mulChar_divisorCount_le","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Character sums of the "],["math","\\(h\\)"],["text","-divisor function over monic polynomials"]],"source":"Character sums of the $h$-divisor function over monic polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#norm-sum-mul-char-divisor-count-le","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $m \\in F[T]$ be monic, and let $\\chi$ be a non-principal Dirichlet character modulo $m$: a multiplicative character of $F[T]/(m)$, vanishing off the units, other than the principal one, evaluated at $f \\in F[T]$ through the class of $f$ modulo $m$. Let $h \\ge 1$ be a natural number and let $d_h$ be the $h$-divisor function, so that $d_h(f)$ is the number of\u2026","title":"Character sums of the $h$-divisor function over monic polynomials"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSum","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(B_\\kappa\\)"],["text","-free divisor sum"]],"source":"The $B_\\kappa$-free divisor sum"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field, let $\\kappa$ assign a natural number to each polynomial in $F[T]$, and let $h$ and $N$ be natural numbers. Then $\\tilde D_{B_\\kappa,h}(N)$ is the sum of $d_h(f)$ over the monic $B_\\kappa$-free polynomials $f \\in F[T]$ of degree at most $N$, where $d_h(f)$ is the number of $h$-tuples of monic polynomials with product $f$, and $f$ is $B_\\kappa$-free when $P^{\\kappa(P)}$ does not divide $f$ for any\u2026","title":"The $B_\\kappa$-free divisor sum"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-zero","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSum_zero","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(B_\\kappa\\)"],["text","-free divisor sum at degree zero"]],"source":"The $B_\\kappa$-free divisor sum at degree zero"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-zero","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field and let $\\kappa$ assign to each polynomial in $F[T]$ a natural number, with $\\kappa(P) \\ne 0$ for every monic irreducible $P$. Call $f \\in F[T]$ $B_\\kappa$-free when $P^{\\kappa(P)}$ does not divide $f$ for any monic irreducible $P$, let $d_h(f)$ be the number of $h$-tuples of monic polynomials with product $f$, and let $\\tilde D_{B_\\kappa,h}(N)$ be the sum of $d_h(f)$ over the $B_\\kappa$-free\u2026","title":"The $B_\\kappa$-free divisor sum at degree zero"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-mod","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSumMod","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-mod","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"For a field $F$, a function $\\kappa : F[T] \\to \\mathbb{N}$, natural numbers $h$ and $N$, and $a, m \\in F[T]$, the natural number $\\tilde D_{\\kappa,h}(N, a, m)$ is the sum of $d_h(f)$ over the monic $B_\\kappa$-free $f \\in F[T]$ of degree at most $N$ with $f \\equiv a \\pmod m$, that is, with $m$ dividing $f - a$. Here $d_h(f)$ is the number of $h$-tuples of monic polynomials with product $f$, and $f$ is\u2026","title":"B-free divisor sum in an arithmetic progression"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-mod-one","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSumMod_one","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-mod-one","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"fully_proved","summary":"Let $F$ be a field, let $\\kappa : F[T] \\to \\mathbb{N}$, and let $h$ and $N$ be natural numbers. Write $\\tilde D_{\\kappa,h}(N)$ for the sum of $d_h(f)$ over the monic $B_\\kappa$-free $f \\in F[T]$ of degree at most $N$, and for $a, m \\in F[T]$ write $\\tilde D_{\\kappa,h}(N, a, m)$ for the same sum restricted to the $f$ with $f \\equiv a \\pmod m$. Here $d_h(f)$ is the number of $h$-tuples of monic polynomials with\u2026","title":"The progression sum modulo 1 is the full sum"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-one-is-big-o","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSum_one_isBigO","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-one-is-big-o","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$; let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic irreducible $P$. Let $\\tilde D_{\\kappa,1}(N)$ be the sum of $d_1(f)$ (which is $1$ for monic $f$) over the monic $B_\\kappa$-free\u2026","title":"Asymptotic count of B-free polynomials"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-two-is-big-o","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSum_two_isBigO","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic for the "],["math","\\(B_\\kappa\\)"],["text","-free sum of the divisor function"]],"source":"Asymptotic for the $B_\\kappa$-free sum of the divisor function"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-two-is-big-o","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and let $\\kappa$ assign to each polynomial in $F[T]$ a natural number, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. A polynomial $f$ is $B_\\kappa$-free when $P^{\\kappa(P)}$ does not divide $f$ for any monic irreducible $P$; $d_2(f)$ is the number of ordered pairs of monic polynomials with product $f$; $\\tilde D_{B_\\kappa,2}(N)$ is the sum of $d_2(f)$ over the\u2026","title":"Asymptotic for the $B_\\kappa$-free sum of the divisor function"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-is-big-o-of-le-two","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSum_isBigO_of_le_two","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-is-big-o-of-le-two","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. Let $h \\in \\{1, 2\\}$ and let $\\tilde D_{\\kappa,h}(N)$ be the sum of $d_h(f)$ over the monic $B_\\kappa$-free $f$ of degree at most $N$, where $d_h(f)$ is the number of $h$-tuples of monic\u2026","title":"Asymptotic for B-free divisor sums when h is at most 2"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-mod-one-is-big-o","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSumMod_one_isBigO","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-mod-one-is-big-o","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$; let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic irreducible $P$. Let $m \\ne 0$ and $a$ be polynomials in $F[T]$ with $a$ coprime to $m$, and let $\\tilde D_{\\kappa,1}(N, a, m)$ be\u2026","title":"Asymptotic count of B-free polynomials in an arithmetic progression"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-mod-two-is-big-o","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSumMod_two_isBigO","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-mod-two-is-big-o","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$; let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic irreducible $P$. Let $m \\ne 0$ and $a$ be polynomials in $F[T]$ with $a$ coprime to $m$, and let $\\tilde D_{\\kappa,2}(N, a, m)$ be\u2026","title":"Asymptotic for the B-free divisor function in an arithmetic progression"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-mod-is-big-o-of-le-tw","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSumMod_isBigO_of_le_two","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-mod-is-big-o-of-le-tw","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. Let $m \\ne 0$ and $a$ be polynomials in $F[T]$ with $a$ coprime to $m$, let $h \\in \\{1, 2\\}$, and let $\\tilde D_{\\kappa,h}(N, a, m)$ be the sum of $d_h(f)$ over the monic $B_\\kappa$-free $f$ of\u2026","title":"Asymptotic for B-free divisor sums in progressions when h is at most 2"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-short-interval","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSum_shortInterval","parent":13398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisor sums over "],["math","\\(B_\\kappa\\)"],["text","-free polynomials in short intervals"]],"source":"Divisor sums over $B_\\kappa$-free polynomials in short intervals"},"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-short-interval","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and let $\\kappa$ assign to each polynomial in $F[T]$ a natural number, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$. Suppose that $\\kappa$ is symmetric: $\\kappa(P^*) = \\kappa(P)$ for every monic irreducible $P$ with $P(0) \\ne 0$, where $P^*$ is the monic scalar multiple of $T^{\\deg P} P(1/T)$. Let $h \\ge 1$ be an integer, let $c$ be a real number with\u2026","title":"Divisor sums over $B_\\kappa$-free polynomials in short intervals"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-short-interval-one","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSum_shortInterval_one","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-short-interval-one","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$; let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic irreducible $P$. Assume $\\kappa$ is symmetric: $\\kappa(P^*) = \\kappa(P)$ for every monic irreducible $P$ with $P(0) \\ne 0$, where\u2026","title":"B-free polynomials in short intervals"},{"id":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/b-free-divisor-sum-short-interval-two","kind":"node","lean":"MetaMathlibExt.bFreeDivisorSum_shortInterval_two","parent":13398,"result":true,"route":"../roadmap/NumberTheory/function-fields/divisor-problems-for-b-free-polynomials/index.html#b-free-divisor-sum-short-interval-two","scope":"NumberTheory/function-fields/divisor-problems-for-b-free-polynomials","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, write $|f| = q^{\\deg f}$ for nonzero $f \\in F[T]$, and let $\\kappa$ assign a natural number to each polynomial in $F[T]$, with $\\kappa(P) \\ge 2$ for every monic irreducible $P$; let $\\kappa_{\\min}$ be the minimum of $\\kappa(P)$ over the monic irreducible $P$. Assume $\\kappa$ is symmetric: $\\kappa(P^*) = \\kappa(P)$ for every monic irreducible $P$ with $P(0) \\ne 0$, where\u2026","title":"The B-free divisor function in short intervals"},{"data_url":"scopes/NumberTheory/function-fields/higher-a-numbers-in-p-towers.json","id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","kind":"scope","navigation":"module","parent":13392,"result":false,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","title":"Higher a-numbers in \u2124_p-towers"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/sum-icc-of-periodic","kind":"node","lean":"MetaMathlibExt.sum_Icc_of_periodic","parent":13452,"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#sum-icc-of-periodic","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, let $f : \\mathbb{N} \\to K$ and let $L \\ge 1$ be a natural number such that $f(i + L) = f(i)$ for all $i \\ge 1$. Let $\\bar f = \\frac{1}{L} \\sum_{k=1}^{L} f(k)$. Then for every natural number $N$,","title":"Sum of a periodic sequence"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/sum-icc-of-eventually-periodic","kind":"node","lean":"MetaMathlibExt.sum_Icc_of_eventually_periodic","parent":13452,"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#sum-icc-of-eventually-periodic","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, let $f : \\mathbb{N} \\to K$, and let $L \\ge 1$ and $D \\ge 1$ be natural numbers such that $f(i + L) = f(i)$ for all $i \\ge D$. Let $\\bar f = \\frac{1}{L} \\sum_{k=D}^{D+L-1} f(k)$ be the average over one period. Then for every natural number $N \\ge D - 1$,","title":"Sum of an eventually periodic sequence"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pdigit","kind":"node","lean":"MetaMathlibExt.basePDigit","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Base-"],["math","\\(p\\)"],["text"," digits of a rational number"]],"source":"Base-$p$ digits of a rational number"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pdigit","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For a natural number $p$, a rational number $x$ and an integer $k$, the $k$-th base-$p$ digit of $x$ is $p^k = \\lfloor x / p^k \\rfloor \\bmod p$; for $p \\ge 1$ this is the residue of $\\lfloor x / p^k \\rfloor$ modulo $p$ in $\\{0, \\dots, p - 1\\}$.","title":"Base-$p$ digits of a rational number"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pdigit-five-two-thirds","kind":"node","lean":"MetaMathlibExt.basePDigit_five_two_thirds","parent":13452,"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pdigit-five-two-thirds","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"Writing $5^j$ for the $j$-th base-$5$ digit of a rational number $x$, we have $5^{-1} = 3$, $5^{-2} = 1$ and $5^0 = 0$.","title":"Base-5 digits of 2/3"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/prime-to-pden","kind":"node","lean":"MetaMathlibExt.primeToPDen","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-to-"],["math","\\(p\\)"],["text"," part of the denominator"]],"source":"Prime-to-$p$ part of the denominator"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#prime-to-pden","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For a natural number $p$ and a rational number $x$ whose denominator in lowest terms is $b \\ge 1$, $x_D$ is the natural number $b / p^k$, where $p^k$ is the largest power of $p$ dividing $b$ (with $k = 0$ when $p = 1$).","title":"Prime-to-$p$ part of the denominator"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/prime-to-pnum","kind":"node","lean":"MetaMathlibExt.primeToPNum","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prime-to-"],["math","\\(p\\)"],["text"," part of the numerator"]],"source":"Prime-to-$p$ part of the numerator"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#prime-to-pnum","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For a natural number $p$ and a rational number $x$ whose numerator in lowest terms is $a$, $x_N$ is the natural number $|a| / p^k$, where $p^k$ is the largest power of $p$ dividing $|a|$, with $k = 0$ when $p = 1$ or $a = 0$. In particular $x_N = 0$ when $x = 0$.","title":"Prime-to-$p$ part of the numerator"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/prime-to-pden-one-prime-to-pnum-one","kind":"node","lean":"MetaMathlibExt.primeToPDen_one_primeToPNum_one","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The prime-to-"],["math","\\(p\\)"],["text"," parts of "],["math","\\(1\\)"]],"source":"The prime-to-$p$ parts of $1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#prime-to-pden-one-prime-to-pnum-one","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For every natural number $p$, the prime-to-$p$ part $x_D$ of the denominator and the prime-to-$p$ part $x_N$ of the numerator of $x = 1$ are both $1$.","title":"The prime-to-$p$ parts of $1$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pperiod","kind":"node","lean":"MetaMathlibExt.basePPeriod","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period of the base-"],["math","\\(p\\)"],["text"," expansion"]],"source":"Period of the base-$p$ expansion"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pperiod","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For a natural number $p$ and a rational number $x$, let $x_D$ be the denominator of $x$ in lowest terms divided by the largest power of $p$ dividing it. The period $L_x$ of the base-$p$ expansion of $x$ is the multiplicative order of $p$ modulo $x_D$: the least $n \\ge 1$ with $p^n \\equiv 1 \\pmod{x_D}$, or $0$ if there is no such $n$. For prime $p$ this is the order of $p$ in $(\\mathbb{Z}/x_D\\mathbb{Z})^\\times$.","title":"Period of the base-$p$ expansion"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pperiod-one","kind":"node","lean":"MetaMathlibExt.basePPeriod_one","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The base-"],["math","\\(p\\)"],["text"," expansion of "],["math","\\(1\\)"],["text"," has period "],["math","\\(1\\)"]],"source":"The base-$p$ expansion of $1$ has period $1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pperiod-one","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For every natural number $p$, the period $L_1$ of the base-$p$ expansion of $1$ equals $1$.","title":"The base-$p$ expansion of $1$ has period $1$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pdelay","kind":"node","lean":"MetaMathlibExt.basePDelay","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Delay of the base-"],["math","\\(p\\)"],["text"," expansion"]],"source":"Delay of the base-$p$ expansion"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pdelay","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For a natural number $p$ and a rational number $x$, the delay of the base-$p$ expansion of $x$ is $D_x = -v_p(x)$, where $v_p$ is the $p$-adic valuation (with $v_p(0) = 0$).","title":"Delay of the base-$p$ expansion"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pdelay-inv-self","kind":"node","lean":"MetaMathlibExt.basePDelay_inv_self","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The delay of "],["math","\\(1/p\\)"],["text"," is "],["math","\\(1\\)"]],"source":"The delay of $1/p$ is $1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pdelay-inv-self","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For every prime $p$, the delay of the base-$p$ expansion of $1/p$ is $D_{1/p} = -v_p(1/p) = 1$, where $v_p$ is the $p$-adic valuation.","title":"The delay of $1/p$ is $1$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pdigit-average","kind":"node","lean":"MetaMathlibExt.basePDigitAverage","parent":13452,"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pdigit-average","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For a natural number $p$ and a rational number $x$, let $D_x$ be the delay and $L_x$ the period of the base-$p$ expansion of $x$, and let $p^k$ be its $k$-th base-$p$ digit. The average repeating digit of $x$ is","title":"Average repeating digit"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/tower-tau","kind":"node","lean":"MetaMathlibExt.towerTau","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ratio "],["math","\\(\\tau = (p + 1)/d\\)"]],"source":"The ratio $\\tau = (p + 1)/d$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#tower-tau","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For natural numbers $p$ and $d$, $\\tau$ is the rational number $(p + 1)/d$, which is $0$ when $d = 0$.","title":"The ratio $\\tau = (p + 1)/d$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/tower-gamma","kind":"node","lean":"MetaMathlibExt.towerGamma","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The constant "],["math","\\(\\gamma\\)"]],"source":"The constant $\\gamma$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#tower-gamma","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For natural numbers $p$, $d$ and $r$, $\\gamma = ((p - 1) r + (p + 1))/d$, a rational number (taken to be $0$ when $d = 0$).","title":"The constant $\\gamma$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/tower-tau-tower-gamma-five-four","kind":"node","lean":"MetaMathlibExt.towerTau_towerGamma_five_four","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of "],["math","\\(\\tau\\)"],["text"," and "],["math","\\(\\gamma\\)"],["text"," for "],["math","\\(p = 5\\)"],["text",", "],["math","\\(d = 4\\)"],["text",", "],["math","\\(r = 2\\)"]],"source":"Values of $\\tau$ and $\\gamma$ for $p = 5$, $d = 4$, $r = 2$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#tower-tau-tower-gamma-five-four","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"With $\\tau = (p + 1)/d$ and $\\gamma = ((p - 1)r + (p + 1))/d$, taking $p = 5$, $d = 4$ and $r = 2$ gives $\\tau = 3/2$ and $\\gamma = 7/2$.","title":"Values of $\\tau$ and $\\gamma$ for $p = 5$, $d = 4$, $r = 2$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/seq-lex-gt","kind":"node","lean":"MetaMathlibExt.SeqLexGt","parent":13452,"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#seq-lex-gt","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For sequences $a, b \\colon \\mathbb{N} \\to \\mathbb{Z}$, $a$ is lexicographically larger than $b$ when there is an index $m$ with $a_j = b_j$ for all $j < m$ and $a_m > b_m$. This is the order used in Definition 1.","title":"Lexicographic order on integer sequences"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-delta","kind":"node","lean":"MetaMathlibExt.higherANumberDelta","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The rounding indicator "],["math","\\(\\delta\\)"]],"source":"The rounding indicator $\\delta$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-delta","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"Let $p$, $d$ and $i$ be natural numbers and let $\\tau = (p + 1)/d$. For a rational $x$ and an integer $k$, write $p^k$ for the residue of $\\lfloor x / p^k \\rfloor$ modulo $p$, the $k$-th base-$p$ digit of $x$. Then $\\delta(i) = 0$ if the integer-part digits $(p^n)_{n \\ge 0}$ are lexicographically larger than the fractional digits $(p^{-1-n})_{n \\ge 0}$, that is, the two sequences agree at every index below some $m$\u2026","title":"The rounding indicator $\\delta$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-delta-five-four","kind":"node","lean":"MetaMathlibExt.higherANumberDelta_five_four","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two values of "],["math","\\(\\delta\\)"],["text"," for "],["math","\\(p = 5\\)"],["text"," and "],["math","\\(d = 4\\)"]],"source":"Two values of $\\delta$ for $p = 5$ and $d = 4$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-delta-five-four","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For $p = 5$ and $d = 4$, so that $\\tau = (p + 1)/d = 3/2$, the rounding indicator satisfies $\\delta(1) = 1$ and $\\delta(2) = 0$.","title":"Two values of $\\delta$ for $p = 5$ and $d = 4$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/tau-den","kind":"node","lean":"MetaMathlibExt.tauDen","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The denominator "],["math","\\(\\tau_D\\)"]],"source":"The denominator $\\tau_D$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#tau-den","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For natural numbers $p$ and $d$, $\\tau_D$ is the denominator in lowest terms of the rational number $\\tau = (p + 1)/d$ (when $d = 0$, $\\tau$ is taken to be $0$, so $\\tau_D = 1$).","title":"The denominator $\\tau_D$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/tau-den-one","kind":"node","lean":"MetaMathlibExt.tauDen_one","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tau_D = 1\\)"],["text"," when "],["math","\\(d = 1\\)"]],"source":"$\\tau_D = 1$ when $d = 1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#tau-den-one","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For every natural number $p$, the denominator $\\tau_D$ of $\\tau = (p + 1)/d$ in lowest terms equals $1$ when $d = 1$.","title":"$\\tau_D = 1$ when $d = 1$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-delta-zero","kind":"node","lean":"MetaMathlibExt.higherANumberDeltaZero","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The variant "],["math","\\(\\delta_0\\)"],["text"," of the rounding indicator"]],"source":"The variant $\\delta_0$ of the rounding indicator"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-delta-zero","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For natural numbers $p$, $d$ and $i$, with $\\delta$ the rounding indicator for $p$ and $d$, the value $\\delta_0(i)$ is $0$ if $p \\mid i$, and $\\delta(i)$ otherwise.","title":"The variant $\\delta_0$ of the rounding indicator"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-delta-zero-zero","kind":"node","lean":"MetaMathlibExt.higherANumberDeltaZero_zero","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The indicator "],["math","\\(\\delta_0\\)"],["text"," vanishes at "],["math","\\(0\\)"]],"source":"The indicator $\\delta_0$ vanishes at $0$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-delta-zero-zero","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For all natural numbers $p$ and $d$, the variant $\\delta_0$ of the rounding indicator for $p$ and $d$ satisfies $\\delta_0(0) = 0$.","title":"The indicator $\\delta_0$ vanishes at $0$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/tower-mu","kind":"node","lean":"MetaMathlibExt.towerMu","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The lattice boundary "],["math","\\(\\mu\\)"]],"source":"The lattice boundary $\\mu$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#tower-mu","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"Let $p$, $d$ and $i$ be natural numbers and let $\\tau = (p + 1)/d$, taken to be $0$ when $d = 0$. For a rational $x$ and an integer $k$, write $p^k$ for the residue of $\\lfloor x/p^k \\rfloor$ modulo $p$, the $k$-th base-$p$ digit of $x$. Then $\\mu(i)$ is $\\lfloor \\tau i \\rfloor$ if the integer-part digit sequence $(p^n)_{n \\ge 0}$ is lexicographically larger than the fractional digit sequence\u2026","title":"The lattice boundary $\\mu$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/tower-mu-five-four-two","kind":"node","lean":"MetaMathlibExt.towerMu_five_four_two","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mu(2) = 3\\)"],["text"," for "],["math","\\(p = 5\\)"],["text"," and "],["math","\\(d = 4\\)"]],"source":"$\\mu(2) = 3$ for $p = 5$ and $d = 4$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#tower-mu-five-four-two","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For $p = 5$ and $d = 4$ the lattice boundary $\\mu(2)$ equals $3$; here $\\tau = (p + 1)/d = 3/2$, so $\\tau \\cdot 2 = 3$ is an integer and its floor and ceiling agree.","title":"$\\mu(2) = 3$ for $p = 5$ and $d = 4$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount","parent":13452,"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For natural numbers $p$, $d$, $r$ and $n$, put $\\tau = (p + 1)/d$, $\\gamma = ((p - 1) r + (p + 1))/d$ and $t_n = \\lfloor \\gamma^{-1} p^n \\rfloor$ (a floor in $\\mathbb{N}$, so $0$ for negative arguments), with the convention that division by $0$ gives $0$; thus $\\tau = \\gamma = 0$ when $d = 0$, and $t_n = 0$ when $\\gamma = 0$. For a rational $x$ and an integer $k$ let $p^k = \\lfloor x / p^k \\rfloor \\bmod p$, the\u2026","title":"The lattice-point count of Theorem 1"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count-three-two-o","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount_three_two_one_zero","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The lattice-point count at "],["math","\\(p = 3\\)"],["text",", "],["math","\\(d = 2\\)"],["text",", "],["math","\\(r = 1\\)"],["text",", "],["math","\\(n = 0\\)"]],"source":"The lattice-point count at $p = 3$, $d = 2$, $r = 1$, $n = 0$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count-three-two-o","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"Let $\\gamma = ((p - 1) r + (p + 1))/d$ and $t_n = \\lfloor \\gamma^{-1} p^n \\rfloor$, and let $\\Delta_n$ be the set of pairs of integers $(i, j)$ with $t_n < i \\le p^n$, $1 \\le j \\le p^n - 1$ and $\\mu(i) \\le j$, where $\\mu$ is the lattice boundary of Definition 1. For $p = 3$, $d = 2$, $r = 1$ and $n = 0$ the lattice-point count $r (p - 1) t_n (t_n + 1)/(2d) + |\\Delta_n|$ of Theorem 1 equals $0$: here $\\gamma = 3$\u2026","title":"The lattice-point count at $p = 3$, $d = 2$, $r = 1$, $n = 0$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/floor-sum-correction","kind":"node","lean":"MetaMathlibExt.floorSumCorrection","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The correction term "],["math","\\(A_{x^{-1}}\\)"],["text"," for sums of floors"]],"source":"The correction term $A_{x^{-1}}$ for sums of floors"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#floor-sum-correction","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For a natural number $p$, a rational $x$ and a natural number $n$, let $x_D$ be the prime-to-$p$ part of the denominator of $x$, write $\\lbrace t \\rbrace = t - \\lfloor t \\rfloor$ for the fractional part of $t$, and put $y = x^{-1} p^n$ (with $y = 0$ when $x = 0$). Define","title":"The correction term $A_{x^{-1}}$ for sums of floors"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/floor-sum-correction-one","kind":"node","lean":"MetaMathlibExt.floorSumCorrection_one","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The correction "],["math","\\(A_{x^{-1}}\\)"],["text"," vanishes at "],["math","\\(x = 1\\)"]],"source":"The correction $A_{x^{-1}}$ vanishes at $x = 1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#floor-sum-correction-one","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For all natural numbers $p$ and $n$, the periodic correction $A_{x^{-1}}(n)$ of Proposition 4 satisfies $A_1(n) = 0$.","title":"The correction $A_{x^{-1}}$ vanishes at $x = 1$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/delta-zero-sum-correction","kind":"node","lean":"MetaMathlibExt.deltaZeroSumCorrection","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The correction term "],["math","\\(F_y\\)"],["text"," for sums of "],["math","\\(\\delta_0\\)"]],"source":"The correction term $F_y$ for sums of $\\delta_0$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#delta-zero-sum-correction","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For natural numbers $p$, $d$ and $e$ and a rational number $y$, let $\\delta_0$ be the variant of the rounding indicator for $p$ and $d$, let $\\tau_D$ be the denominator in lowest terms of $\\tau = (p + 1)/d$ (with $\\tau = 0$ when $d = 0$), and let $m = \\lfloor y p^e \\rfloor \\bmod \\tau_D p$, where $\\lfloor y p^e \\rfloor$ is read as $0$ when $y p^e < 0$. Define","title":"The correction term $F_y$ for sums of $\\delta_0$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/delta-zero-sum-correction-zero","kind":"node","lean":"MetaMathlibExt.deltaZeroSumCorrection_zero","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The correction "],["math","\\(F_y\\)"],["text"," vanishes at "],["math","\\(y = 0\\)"]],"source":"The correction $F_y$ vanishes at $y = 0$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#delta-zero-sum-correction-zero","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"fully_proved","summary":"For all natural numbers $p$, $d$ and $e$, the correction $F_y(e)$ of Lemma 9, taken at $y = 0$, is $0$.","title":"The correction $F_y$ vanishes at $y = 0$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pdigit-div-periodic-one","kind":"node","lean":"MetaMathlibExt.basePDigit_div_periodic_one","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digits of "],["math","\\(m/d\\)"],["text"," after the radix point are constant"]],"source":"Digits of $m/d$ after the radix point are constant"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pdigit-div-periodic-one","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, $d$ a divisor of $p - 1$, and write $p^j$ for the $j$-th base-$p$ digit of a rational number $x$. For every positive integer $m$ and every integer $k \\ge 1$, $p^{-(k+1)} = p^{-k}$; that is, the base-$p$ expansion of $m/d$ is periodic with period $1$ right after the radix point.","title":"Digits of $m/d$ after the radix point are constant"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pdigit-eventually-periodic-iff","kind":"node","lean":"MetaMathlibExt.basePDigit_eventually_periodic_iff","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimal period and delay of base-"],["math","\\(p\\)"],["text"," digits"]],"source":"Minimal period and delay of base-$p$ digits"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pdigit-eventually-periodic-iff","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $x > 0$ a rational number, write $p^j$ for the $j$-th base-$p$ digit of $x$, and let $L_x$ be the period and $D_x = -v_p(x)$ the delay of the base-$p$ expansion of $x$. For every integer $D$ and every natural number $L > 0$, the identity $p^{-(k+L)} = p^{-k}$ holds for all integers $k \\ge D + 1$ if and only if $L_x$ divides $L$ and $D_x \\le D$. Thus $L_x$ is the minimal period and $D_x$ the\u2026","title":"Minimal period and delay of base-$p$ digits"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/base-pdigit-tower-tau-neg-one-eq-zero-if","kind":"node","lean":"MetaMathlibExt.basePDigit_towerTau_neg_one_eq_zero_iff","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digits of "],["math","\\(\\tau i\\)"],["text"," next to the radix point"]],"source":"Digits of $\\tau i$ next to the radix point"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#base-pdigit-tower-tau-neg-one-eq-zero-if","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, $d$ a divisor of $p - 1$ and $\\tau = (p + 1)/d$, and write $p^j$ for the $j$-th base-$p$ digit of a rational number $x$. For every natural number $i$, $p^{-1} = p^0$ if and only if $p$ divides $i$.","title":"Digits of $\\tau i$ next to the radix point"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-delta-eq-one-iff","kind":"node","lean":"MetaMathlibExt.higherANumberDelta_eq_one_iff","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","A digit criterion for "],["math","\\(\\delta(i) = 1\\)"]],"source":"A digit criterion for $\\delta(i) = 1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-delta-eq-one-iff","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, $d$ a divisor of $p - 1$ and $\\tau = (p + 1)/d$. For a rational $t$ and an integer $k$ write $p^k$ for the $k$-th base-$p$ digit of $t$, the residue of $\\lfloor t / p^k \\rfloor$ modulo $p$ in $\\lbrace 0, \\dots, p - 1 \\rbrace$, and let $\\delta$ be the rounding indicator. If $i$ is a natural number not divisible by $p$, then $p^0 < p^{-1}$ if and only if $\\delta(i) = 1$.","title":"A digit criterion for $\\delta(i) = 1$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-delta-reflection","kind":"node","lean":"MetaMathlibExt.higherANumberDelta_reflection","parent":13452,"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-delta-reflection","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, let $d$ be a natural number dividing $p - 1$, let $\\tau_D$ be the denominator of $(p + 1)/d$ in lowest terms, and let $\\delta$ be the rounding indicator for $p$ and $d$. For every natural number $i < \\tau_D p$ with $p \\nmid i$: (i) if $\\tau_D \\nmid i$, then $\\delta(i) + \\delta(\\tau_D p - i) = 1$; (ii) if $\\tau_D \\mid i$, then $\\delta(i) = \\delta(\\tau_D p - i) = 0$. For every natural number\u2026","title":"Reflection symmetry of the rounding indicator"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-delta-periodicity","kind":"node","lean":"MetaMathlibExt.higherANumberDelta_periodicity","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periodicity and reflection of "],["math","\\(\\delta\\)"],["text"," and "],["math","\\(\\delta_0\\)"]],"source":"Periodicity and reflection of $\\delta$ and $\\delta_0$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-delta-periodicity","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $d$ a divisor of $p - 1$, let $\\tau_D$ be the denominator of $\\tau = (p + 1)/d$ in lowest terms, let $\\delta$ be the rounding indicator, and let $\\delta_0(i)$ be $0$ if $p \\mid i$ and $\\delta(i)$ otherwise. Then for all natural numbers $i$ and $n$: (i) $\\delta(p^n i) = \\delta(i)$; (ii) $\\delta_0(i) = \\delta_0(i + \\tau_D p)$; (iii) if $i \\le \\tau_D p$, $\\tau_D \\nmid i$ and $p \\nmid i$, then\u2026","title":"Periodicity and reflection of $\\delta$ and $\\delta_0$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-delta-zero-average","kind":"node","lean":"MetaMathlibExt.higherANumberDeltaZero_average","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Average of the indicator "],["math","\\(\\delta_0\\)"]],"source":"Average of the indicator $\\delta_0$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-delta-zero-average","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, let $d$ be a natural number dividing $p - 1$, let $\\tau_D$ be the denominator of $(p + 1)/d$ in lowest terms, and let $\\delta_0$ be the variant of the rounding indicator for $p$ and $d$. Then","title":"Average of the indicator $\\delta_0$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/fract-mul-pow-periodic","kind":"node","lean":"MetaMathlibExt.fract_mul_pow_periodic","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periodicity and average of the fractional parts of "],["math","\\(x p^n\\)"]],"source":"Periodicity and average of the fractional parts of $x p^n$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#fract-mul-pow-periodic","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $x$ a positive rational. Write $\\lbrace t \\rbrace$ for the fractional part of $t$, $D_x = -v_p(x)$ for the delay and $L_x$ for the period of the base-$p$ expansion of $x$ (the order of $p$ modulo the prime-to-$p$ part of the denominator of $x$), and $\\bar x$ for the average of its repeating base-$p$ digits. Then $\\lbrace x p^{n + L_x} \\rbrace = \\lbrace x p^n \\rbrace$ for every integer\u2026","title":"Periodicity and average of the fractional parts of $x p^n$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/floor-mul-pow-mod-eq","kind":"node","lean":"MetaMathlibExt.floor_mul_pow_modEq","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periodicity of "],["math","\\(\\lfloor x p^n \\rfloor\\)"],["text"," modulo "],["math","\\(x_N\\)"],["text"," and modulo "],["math","\\(p\\)"]],"source":"Periodicity of $\\lfloor x p^n \\rfloor$ modulo $x_N$ and modulo $p$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#floor-mul-pow-mod-eq","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $x$ a positive rational. Write $x_N$ for the prime-to-$p$ part of the numerator of $x$, $D_x = -v_p(x)$ for the delay of its base-$p$ expansion, and $L_x$ for its period, the order of $p$ modulo the prime-to-$p$ part of the denominator of $x$. Then for every integer $n$: (i) if $n \\ge D_x$, then $\\lfloor x p^n \\rfloor \\equiv \\lfloor x p^{n + L_x} \\rfloor \\pmod{x_N}$; (ii) if $n \\ge D_x + 1$,\u2026","title":"Periodicity of $\\lfloor x p^n \\rfloor$ modulo $x_N$ and modulo $p$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/sum-higher-anumber-delta-by-exponent","kind":"node","lean":"MetaMathlibExt.sum_higherANumberDelta_by_exponent","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Splitting the sum of "],["math","\\(\\delta\\)"],["text"," by powers of "],["math","\\(p\\)"]],"source":"Splitting the sum of $\\delta$ by powers of $p$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#sum-higher-anumber-delta-by-exponent","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, $d$ a natural number dividing $p - 1$, $x \\ge 1$ a rational number and $n$ a natural number. With $\\tau = (p + 1)/d$, let $\\delta(i)$ be $0$ if the sequence of base-$p$ digits of $\\tau i$ at $p^0, p^1, p^2, \\ldots$ is lexicographically larger than its sequence of digits at $p^{-1}, p^{-2}, \\ldots$, and $1$ otherwise, and let $\\delta_0(i)$ be $0$ if $p \\mid i$ and $\\delta(i)$ otherwise. Then","title":"Splitting the sum of $\\delta$ by powers of $p$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/sum-pow-sub-floor","kind":"node","lean":"MetaMathlibExt.sum_pow_sub_floor","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(p^n - \\lfloor x i \\rfloor\\)"]],"source":"Sum of $p^n - \\lfloor x i \\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#sum-pow-sub-floor","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $x$ a positive rational with $v_p(x) \\ge 0$. Let $x_D$ be the prime-to-$p$ part of the denominator of $x$, let $D_{x^{-1}}$ and $L_{x^{-1}}$ be the delay and the period of the base-$p$ expansion of $x^{-1}$, and let $A_{x^{-1}}(n)$ be the periodic correction of Proposition 4. Then for every natural number $n$,","title":"Sum of $p^n - \\lfloor x i \\rfloor$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/sum-floor-difference","kind":"node","lean":"MetaMathlibExt.sum_floor_difference","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Difference of the floor sums for "],["math","\\(\\tau\\)"],["text"," and "],["math","\\(\\gamma\\)"]],"source":"Difference of the floor sums for $\\tau$ and $\\gamma$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#sum-floor-difference","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, $d$ a natural number dividing $p - 1$, and $r$ and $n$ natural numbers. Put $\\tau = (p + 1)/d$ and $\\gamma = ((p - 1) r + (p + 1))/d$, let $\\tau_D$ be the denominator of $\\tau$ in lowest terms, and for a rational $x$ let $A_{x^{-1}}(n)$ be the periodic correction of Proposition 4. Then","title":"Difference of the floor sums for $\\tau$ and $\\gamma$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count-eq-sums","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount_eq_sums","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Higher "],["math","\\(a\\)"],["text","-numbers as sums of floors and of "],["math","\\(\\delta\\)"]],"source":"Higher $a$-numbers as sums of floors and of $\\delta$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count-eq-sums","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, let $d$ be a natural number dividing $p - 1$, let $r$ and $n$ be natural numbers, let $\\tau = (p + 1)/d$ and $\\gamma = ((p - 1) r + (p + 1))/d$, let $\\delta$ be the rounding indicator for $p$ and $d$, and let $A_r(n)$ be the lattice-point count for the $r$-th higher $a$-number at level $n$. Put $t_n = \\lfloor \\gamma^{-1} p^n \\rfloor$ and $s_n = \\lfloor \\tau^{-1} p^n \\rfloor$. Then","title":"Higher $a$-numbers as sums of floors and of $\\delta$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/sum-higher-anumber-delta-eq","kind":"node","lean":"MetaMathlibExt.sum_higherANumberDelta_eq","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\delta\\)"],["text"," up to "],["math","\\(\\lfloor p^n/x \\rfloor\\)"]],"source":"Sum of $\\delta$ up to $\\lfloor p^n/x \\rfloor$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#sum-higher-anumber-delta-eq","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $d$ a divisor of $p - 1$; let $\\delta$ be the rounding indicator, $\\tau_D$ the denominator of $\\tau = (p + 1)/d$ in lowest terms, and $F_y(e)$ the $\\delta_0$ correction sum of Lemma 9. For every rational $x \\ge 1$ and every natural number $n$,","title":"Sum of $\\delta$ up to $\\lfloor p^n/x \\rfloor$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/sum-higher-anumber-delta-periodic","kind":"node","lean":"MetaMathlibExt.sum_higherANumberDelta_periodic","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual periodicity of the "],["math","\\(\\delta\\)"],["text"," sums"]],"source":"Eventual periodicity of the $\\delta$ sums"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#sum-higher-anumber-delta-periodic","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $d$ a divisor of $p - 1$; let $\\delta$ be the rounding indicator, $\\tau_D$ the denominator of $\\tau = (p + 1)/d$ in lowest terms, and $F_y(e)$ the $\\delta_0$ correction sum of Lemma 9. Let $x$ be a positive rational such that the prime-to-$p$ part $x_D$ of the denominator of $x$ equals $\\tau_D$, and write $D = D_{x^{-1}}$, $L = L_{x^{-1}}$ and $\\overline{x^{-1}}$ for the delay, the period and\u2026","title":"Eventual periodicity of the $\\delta$ sums"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/delta-zero-sum-correction-average-tower","kind":"node","lean":"MetaMathlibExt.deltaZeroSumCorrection_average_towerTau","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The linear coefficient of Proposition 5 vanishes at "],["math","\\(\\tau\\)"]],"source":"The linear coefficient of Proposition 5 vanishes at $\\tau$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#delta-zero-sum-correction-average-tower","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $d$ a divisor of $p - 1$, let $\\tau = (p + 1)/d$, and let $\\tau_D$ be the denominator of $\\tau$ in lowest terms. Let $L$ be the order of $p$ modulo the prime-to-$p$ part of the denominator of $\\tau^{-1}$ (the period of the base-$p$ expansion of $\\tau^{-1}$), and let $\\overline{\\tau^{-1}}$ be the average of the repeating base-$p$ digits of $\\tau^{-1}$. For $e \\ge 0$ let\u2026","title":"The linear coefficient of Proposition 5 vanishes at $\\tau$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/sum-higher-anumber-delta-difference","kind":"node","lean":"MetaMathlibExt.sum_higherANumberDelta_difference","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(\\delta\\)"],["text"," between the "],["math","\\(\\gamma\\)"],["text","- and "],["math","\\(\\tau\\)"],["text","-cutoffs"]],"source":"Sum of $\\delta$ between the $\\gamma$- and $\\tau$-cutoffs"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#sum-higher-anumber-delta-difference","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, let $d$ be a divisor of $p - 1$ and let $r \\ge 1$ be an integer. Write $\\tau = (p + 1)/d$, $\\gamma = ((p - 1) r + (p + 1))/d$ and $\\tau_D$ for the denominator of $\\tau$ in lowest terms; let $\\delta$ be the rounding indicator, and for a rational $y$ let $L_y$ be the period, $D_y$ the delay and $\\bar y$ the average repeating digit of the base-$p$ expansion of $y$, and let $F_y(e)$ be the\u2026","title":"Sum of $\\delta$ between the $\\gamma$- and $\\tau$-cutoffs"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count-asymptotic","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount_asymptotic","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic formula for higher "],["math","\\(a\\)"],["text","-numbers"]],"source":"Asymptotic formula for higher $a$-numbers"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count-asymptotic","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, let $d$ be a natural number dividing $p - 1$, let $r \\ge 1$, and for each natural number $n$ let $A_r(n)$ be the lattice-point count for the $r$-th higher $a$-number at level $n$. Let $D_r$ be the prime-to-$p$ part of the denominator of $d/((p - 1) r + (p + 1))$, and let $M_r = \\operatorname{lcm}(2, \\operatorname{ord}_{D_r} p)$, where $\\operatorname{ord}_{D_r} p$ is the multiplicative order of\u2026","title":"Asymptotic formula for higher $a$-numbers"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count-eq-explicit","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount_eq_explicit_linear_add_periodic","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Explicit linear term and period for higher "],["math","\\(a\\)"],["text","-numbers"]],"source":"Explicit linear term and period for higher $a$-numbers"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count-eq-explicit","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, let $d$ be a natural number dividing $p - 1$, let $r \\ge 1$, let $\\tau = (p + 1)/d$ and $\\gamma = ((p - 1) r + (p + 1))/d$, let $\\tau_D$ be the denominator of $\\tau$ in lowest terms, let $\\delta_0$ be the variant of the rounding indicator for $p$ and $d$, and for each natural number $n$ let $A_r(n)$ be the lattice-point count for the $r$-th higher $a$-number at level $n$. Put $x = \\gamma^{-1}$.\u2026","title":"Explicit linear term and period for higher $a$-numbers"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count-period-smal","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount_period_small_d","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Period of higher "],["math","\\(a\\)"],["text","-numbers for "],["math","\\(d = 1\\)"],["text"," or "],["math","\\(d = 2\\)"]],"source":"Period of higher $a$-numbers for $d = 1$ or $d = 2$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count-period-smal","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, let $d$ be $1$ or $2$ with $d \\mid p - 1$, let $r \\ge 1$, let $\\tau = (p + 1)/d$ and $\\gamma = ((p - 1) r + (p + 1))/d$, and for each natural number $n$ let $A_r(n)$ be the lattice-point count for the $r$-th higher $a$-number at level $n$ and $R(n) = A_r(n) - \\frac{1}{2} (\\tau^{-1} - \\gamma^{-1}) p^{2n}$. Let $L$ be the period and $D$ the delay of the base-$p$ expansion of $\\gamma^{-1}$. Then\u2026","title":"Period of higher $a$-numbers for $d = 1$ or $d = 2$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count-r-eq-succ","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount_r_eq_succ","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","The lattice-point count for "],["math","\\(r = p + 1\\)"]],"source":"The lattice-point count for $r = p + 1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count-r-eq-succ","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime and $d$ a natural number dividing $p - 1$. Put $\\tau = (p + 1)/d$ and $\\gamma = ((p - 1)(p + 1) + (p + 1))/d$, the value at $r = p + 1$ of $\\gamma = ((p - 1) r + (p + 1))/d$, and write $a^{p+1}(X_n)$ for the lattice-point expression of Theorem 1 with $r = p + 1$. Then for every $n \\ge 1$,","title":"The lattice-point count for $r = p + 1$"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count-linear-coef","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount_linear_coeff_mul_period","parent":13452,"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count-linear-coef","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, let $d$ be a natural number dividing $p - 1$, let $r \\ge 1$, let $\\tau = (p + 1)/d$ and $\\gamma = ((p - 1) r + (p + 1))/d$, and for each natural number $n$ let $A_r(n)$ be the lattice-point count for the $r$-th higher $a$-number at level $n$. Let $\\lambda$ be a rational number, $\\nu$ a function from the natural numbers to the rationals, and $N$ and $L > 0$ natural numbers such that, for every\u2026","title":"The linear coefficient times a period is an integer"},{"id":"NumberTheory/function-fields/higher-a-numbers-in-p-towers/higher-anumber-lattice-count-shift","kind":"node","lean":"MetaMathlibExt.higherANumberLatticeCount_shift","parent":13452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Passing from "],["math","\\(r_0\\)"],["text"," to "],["math","\\((r_0 + 1)p + 1\\)"]],"source":"Passing from $r_0$ to $(r_0 + 1)p + 1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/higher-a-numbers-in-p-towers/index.html#higher-anumber-lattice-count-shift","scope":"NumberTheory/function-fields/higher-a-numbers-in-p-towers","status":"can_prove","summary":"Let $p$ be a prime, $d$ a natural number dividing $p - 1$ and $r_0$ a natural number, and put $r_1 = (r_0 + 1)p + 1$. Write $\\tau = (p + 1)/d$, $\\gamma_k = ((p - 1) r_k + (p + 1))/d$ for $k = 0, 1$, and $a^r(X_n)$ for the lattice-point expression of Theorem 1. For a rational $\\lambda$ and $k = 0, 1$ let $f_k(n) = a^{r_k}(X_n) - \\tfrac12 (\\tau^{-1} - \\gamma_k^{-1}) p^{2n} - \\lambda n$. Then for every rational\u2026","title":"Passing from $r_0$ to $(r_0 + 1)p + 1$"},{"data_url":"scopes/NumberTheory/function-fields/ihara-constant.json","id":"NumberTheory/function-fields/ihara-constant","kind":"scope","navigation":"module","parent":13392,"result":false,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","title":"Ihara constant"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-ratio","kind":"node","lean":"MathlibExt.Analysis.iharaRatio","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-ratio","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"For a sequence $N : \\mathbb{N} \\to \\mathbb{N}$ and $g \\in \\mathbb{N}$, the normalized ratio of $N$ at $g$ is the extended nonnegative real $N(g)/g$ when $g > 0$, and $0$ when $g = 0$.","title":"Normalized Ihara ratio"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-constant-of","kind":"node","lean":"MathlibExt.Analysis.iharaConstantOf","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-constant-of","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"For a sequence $N : \\mathbb{N} \\to \\mathbb{N}$, the abstract Ihara constant of $N$ is $\\limsup_{g \\to \\infty} N(g)/g \\in [0, \\infty]$, the limit superior as $g \\to \\infty$ of the normalized ratios of $N$.","title":"Abstract Ihara constant of a sequence"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-constant-of-le-of-eventually-ratio","kind":"node","lean":"MathlibExt.Analysis.iharaConstantOf_le_of_eventually_ratio_le","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-constant-of-le-of-eventually-ratio","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N : \\mathbb{N} \\to \\mathbb{N}$ and $c \\in [0, \\infty]$. If the normalized ratio of $N$ at $g$ is at most $c$ for all sufficiently large $g$, then the abstract Ihara constant of $N$ is at most $c$.","title":"Eventual ratio bounds bound the abstract Ihara constant"},{"id":"NumberTheory/function-fields/ihara-constant/le-ihara-constant-of-of-frequently-ratio","kind":"node","lean":"MathlibExt.Analysis.le_iharaConstantOf_of_frequently_ratio_le","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#le-ihara-constant-of-of-frequently-ratio","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N : \\mathbb{N} \\to \\mathbb{N}$, let $c \\in [0, \\infty]$, and for each $g \\in \\mathbb{N}$ let $r_N(g) \\in [0, \\infty]$ be the normalized ratio $N(g)/g$, with $r_N(0) = 0$. If $c \\le r_N(g)$ for infinitely many $g$, then $c \\le \\limsup_{g \\to \\infty} r_N(g)$, the Ihara constant of $N$.","title":"Lower bound for the Ihara constant from the normalized ratios"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-ratio-le-of-le-mul","kind":"node","lean":"MathlibExt.Analysis.iharaRatio_le_of_le_mul","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-ratio-le-of-le-mul","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N : \\mathbb{N} \\to \\mathbb{N}$, $g \\in \\mathbb{N}$ and $c \\in [0, \\infty]$. If $N(g) \\le c \\cdot g$ in $[0, \\infty]$, then the normalized ratio of $N$ at $g$ is at most $c$.","title":"Linear upper bound on a normalized ratio"},{"id":"NumberTheory/function-fields/ihara-constant/le-ihara-ratio-of-mul-le","kind":"node","lean":"MathlibExt.Analysis.le_iharaRatio_of_mul_le","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#le-ihara-ratio-of-mul-le","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N : \\mathbb{N} \\to \\mathbb{N}$, $c \\in [0, \\infty]$ and $g > 0$ an integer. If $c \\cdot g \\le N(g)$ in $[0, \\infty]$, then $c \\le N(g)/g$, where the normalized ratio $N(g)/g$ is taken in $[0, \\infty]$.","title":"Lower bound for one normalized Ihara ratio"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-constant-of-le-of-eventually-le","kind":"node","lean":"MathlibExt.Analysis.iharaConstantOf_le_of_eventually_le","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-constant-of-le-of-eventually-le","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N : \\mathbb{N} \\to \\mathbb{N}$ and $c \\in [0, \\infty]$. If $N(g) \\le c \\cdot g$ in $[0, \\infty]$ for all sufficiently large $g$, then the abstract Ihara constant of $N$ is at most $c$.","title":"Eventual linear upper bound on the abstract Ihara constant"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-constant-of-le-of-eventually-le-ad","kind":"node","lean":"MathlibExt.Analysis.iharaConstantOf_le_of_eventually_le_add","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-constant-of-le-of-eventually-le-ad","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N : \\mathbb{N} \\to \\mathbb{N}$, $c \\in [0, \\infty]$ and $d \\in [0, \\infty)$. If $N(g) \\le c \\cdot g + d$ in $[0, \\infty]$ for all sufficiently large $g$, then the abstract Ihara constant of $N$ is at most $c$.","title":"Eventual affine upper bound on the abstract Ihara constant"},{"id":"NumberTheory/function-fields/ihara-constant/le-ihara-constant-of-of-frequently-le","kind":"node","lean":"MathlibExt.Analysis.le_iharaConstantOf_of_frequently_le","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#le-ihara-constant-of-of-frequently-le","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N : \\mathbb{N} \\to \\mathbb{N}$ and $c \\in [0, \\infty]$. If there are infinitely many integers $g > 0$ with $c \\cdot g \\le N(g)$ in $[0, \\infty]$, then $c \\le \\limsup_{g \\to \\infty} N(g)/g$, the Ihara constant of $N$, where the ratio $N(g)/g$ is taken in $[0, \\infty]$ and set to $0$ at $g = 0$.","title":"Lower bound for the Ihara constant from a linear bound"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-ratio-mono","kind":"node","lean":"MathlibExt.Analysis.iharaRatio_mono","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-ratio-mono","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N, M : \\mathbb{N} \\to \\mathbb{N}$ and $g \\in \\mathbb{N}$. If $N(g) \\le M(g)$, then the normalized ratio of $N$ at $g$ is at most the normalized ratio of $M$ at $g$.","title":"Monotonicity of the normalized ratio"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-constant-of-mono-of-eventually-le","kind":"node","lean":"MathlibExt.Analysis.iharaConstantOf_mono_of_eventually_le","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-constant-of-mono-of-eventually-le","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N, M : \\mathbb{N} \\to \\mathbb{N}$. If $N(g) \\le M(g)$ for all sufficiently large $g$, then the abstract Ihara constant of $N$ is at most that of $M$.","title":"Monotonicity of the abstract Ihara constant"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-constant-of-congr","kind":"node","lean":"MathlibExt.Analysis.iharaConstantOf_congr","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-constant-of-congr","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N, M : \\mathbb{N} \\to \\mathbb{N}$. If $N(g) = M(g)$ for all sufficiently large $g$, then $N$ and $M$ have the same abstract Ihara constant. In particular, changing finitely many values of $N$ does not change its abstract Ihara constant.","title":"The abstract Ihara constant ignores finitely many values"},{"id":"NumberTheory/function-fields/ihara-constant/ihara-constant-of-eq-of-eventually-eq","kind":"node","lean":"MathlibExt.Analysis.iharaConstantOf_eq_of_eventually_eq","parent":13506,"result":true,"route":"../roadmap/NumberTheory/function-fields/ihara-constant/index.html#ihara-constant-of-eq-of-eventually-eq","scope":"NumberTheory/function-fields/ihara-constant","status":"fully_proved","summary":"Let $N, M : \\mathbb{N} \\to \\mathbb{N}$ and $G \\in \\mathbb{N}$. If $N(g) = M(g)$ for every $g \\ge G$, then $N$ and $M$ have the same abstract Ihara constant.","title":"Agreement from an index onward preserves the abstract Ihara constant"},{"data_url":"scopes/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic.json","id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","kind":"scope","navigation":"module","parent":13392,"result":false,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"can_prove","title":"Multiple zeta values in positive characteristic"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/monic-strict-anti-nat-degree","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.monicStrictAntiNatDegree","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#monic-strict-anti-nat-degree","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $F$ be a finite field and $r$ a natural number. This is the set of $r$-tuples $(a_1, \\dots, a_r)$ of monic polynomials in $F[t]$ such that $\\deg a_1 > \\deg a_2 > \\dots > \\deg a_r$.","title":"Tuples of monic polynomials with strictly decreasing degrees"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/a","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.\u03b6_A","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#a","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $F$ be a finite field, $A = F[t]$, and $K_\\infty = F((1/t))$ the completion of the rational function field $F(t)$ at the place at infinity. For a natural number $r$ and positive integers $s_1, \\dots, s_r$, the multiple zeta value is","title":"Multiple zeta values in positive characteristic"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/w","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.\ud835\udd16_w","parent":13520,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositions of "],["math","\\(w\\)"],["text"," into "],["math","\\(r\\)"],["text"," parts"]],"source":"Compositions of $w$ into $r$ parts"},"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#w","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"For natural numbers $w$ and $r$, $\\mathfrak{S}_w$ is the set of $r$-tuples $(s_1, \\dots, s_r)$ of positive integers with $s_1 + \\dots + s_r = w$.","title":"Compositions of $w$ into $r$ parts"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/a-weight","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.\u03b6_A_weight","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#a-weight","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $F$ be a finite field and let $w$ and $r$ be natural numbers. The weight of a multiple zeta value $\\zeta_A(s_1, \\dots, s_r)$ over $F$, with $s_1, \\dots, s_r$ positive integers, is $s_1 + \\dots + s_r$. This is the set of multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$ over $F$ of weight $w$, that is, with $s_1, \\dots, s_r$ positive integers and $s_1 + \\dots + s_r = w$.","title":"Multiple zeta values of given depth and weight"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/untitled","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.\ud835\udcb5","parent":13520,"reading_title":{"format":"inline-parts-v1","parts":[["text","The space "],["math","\\(\\mathcal{Z}_w\\)"],["text"," of multiple zeta values of weight "],["math","\\(w\\)"]],"source":"The space $\\mathcal{Z}_w$ of multiple zeta values of weight $w$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#untitled","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $K$ be the field of rational functions in one variable over the finite field $F$, and $K_\\infty$ its completion at infinity. For $w \\in \\mathbb{N}$, $\\mathcal{Z}_w$ is the $K$-linear subspace of $K_\\infty$ spanned by the multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$ for all $r \\ge 0$ and all positive integers $s_1, \\dots, s_r$ with $s_1 + \\dots + s_r = w$.","title":"The space $\\mathcal{Z}_w$ of multiple zeta values of weight $w$"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/aux","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.\ud835\udcaf_aux","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#aux","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, $w$ a natural number and $r$ a positive integer. This is the set of multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$ over $F$ with positive integers $s_1, \\dots, s_r$ such that $s_1 + \\dots + s_r = w$, $s_i \\le q$ for $1 \\le i \\le r - 1$, and $s_r < q$.","title":"Restricted multiple zeta values of given depth"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/untitled-2","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.\ud835\udcaf","parent":13520,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\mathcal{T}_w\\)"],["text"," of restricted multiple zeta values"]],"source":"The set $\\mathcal{T}_w$ of restricted multiple zeta values"},"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#untitled-2","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $q$ be the number of elements of the finite field $F$. For $w \\in \\mathbb{N}$, $\\mathcal{T}_w$ is the set of multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$, over all $r \\ge 1$ and all positive integers $s_1, \\dots, s_r$ with $s_1 + \\dots + s_r = w$, $s_i \\le q$ for $1 \\le i \\le r - 1$, and $s_r < q$.","title":"The set $\\mathcal{T}_w$ of restricted multiple zeta values"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/0-aux","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.\ud835\udcaf0_aux","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#0-aux","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, $w$ a natural number and $r$ a positive integer. This is the set of multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$ over $F$ with positive integers $s_1, \\dots, s_r$ such that $s_1 + \\dots + s_r = w$ and $s_i < q$ for all $i$.","title":"Multiple zeta values of given depth with small entries"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/0","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.\ud835\udcaf0","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#0","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and $w$ a natural number. The set $\\mathcal{T}^0_w$ consists of the multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$ over $F$, for all positive integers $r$ and $s_1, \\dots, s_r$ such that $s_1 + \\dots + s_r = w$ and $s_i < q$ for all $i$.","title":"Multiple zeta values with small entries"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/d","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.d","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#d","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements. The sequence $d(w)$, for natural numbers $w$, is defined by $d(0) = 1$, $d(w) = 2^{w-1}$ for $1 \\le w \\le q - 1$, $d(q) = 2^{q-1} - 1$, and $d(w) = d(w - 1) + d(w - 2) + \\dots + d(w - q)$ for $w > q$.","title":"Dimension sequence for multiple zeta values"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/theorem-a","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.theorem_A","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#theorem-a","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, $K = F(t)$, and $w$ a positive integer. Let $\\mathcal{Z}_w$ be the $K$-vector space spanned by the multiple zeta values over $F$ of weight $w$, and let $\\mathcal{T}_w$ be the set of multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$ with $r \\ge 1$, positive integers $s_1, \\dots, s_r$ with sum $w$, $s_i \\le q$ for $1 \\le i \\le r - 1$, and $s_r < q$. Then $\\mathcal{Z}_w$ is\u2026","title":"Brown's theorem in positive characteristic"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/theorem-b","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.theorem_B","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#theorem-b","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, $K = F(t)$, and $w$ a positive integer. Let $\\mathcal{T}^0_w$ be the set of multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$ over $F$ with $r \\ge 1$, positive integers $s_1, \\dots, s_r$ with sum $w$, and $s_i < q$ for all $i$, and let $\\mathcal{Z}_w$ be the $K$-vector space spanned by all multiple zeta values of weight $w$. Then the elements of $\\mathcal{T}^0_w$ are\u2026","title":"Linear independence of multiple zeta values with small entries"},{"id":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/theorem-d","kind":"node","lean":"MathlibExt.NumberTheory.ZagierHoffmanPositiveCharWanted.ZagierHoffmanPositiveChar.theorem_D","parent":13520,"result":true,"route":"../roadmap/NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic/index.html#theorem-d","scope":"NumberTheory/function-fields/multiple-zeta-values-in-positive-characteristic","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, $K = F(t)$, and $w$ a positive integer with $w \\le 2q - 2$. Let $\\mathcal{T}_w$ be the set of multiple zeta values $\\zeta_A(s_1, \\dots, s_r)$ over $F$ with $r \\ge 1$, positive integers $s_1, \\dots, s_r$ with sum $w$, $s_i \\le q$ for $1 \\le i \\le r - 1$, and $s_r < q$, and let $\\mathcal{Z}_w$ be the $K$-vector space spanned by all multiple zeta values of weight $w$. Then\u2026","title":"Dimension of multiple zeta values in low weight"},{"data_url":"scopes/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po.json","id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","kind":"scope","navigation":"module","parent":13392,"result":false,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","title":"Prime factors of given multiplicity in h-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/fixed-degree-monic","kind":"node","lean":"MetaMathlibExt.fixedDegreeMonic","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#fixed-degree-monic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For a finite field $F$ and a natural number $n$, $M_n$ is the finite set of monic polynomials of degree $n$ over $F$: the polynomials $X^n + c_{n-1} X^{n-1} + \\dots + c_1 X + c_0$ with $c_0, \\dots, c_{n-1} \\in F$.","title":"Monic polynomials of fixed degree"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/mem-fixed-degree-monic-iff","kind":"node","lean":"MetaMathlibExt.mem_fixedDegreeMonic_iff","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in "],["math","\\(M_n\\)"]],"source":"Membership in $M_n$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#mem-fixed-degree-monic-iff","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field and $n \\in \\mathbb{N}$. A polynomial $f$ over $F$ lies in the set $M_n$ of monic polynomials of degree $n$ over $F$ if and only if there are $c_0, \\dots, c_{n-1} \\in F$ with $f = X^n + \\sum_{i=0}^{n-1} c_i X^i$.","title":"Membership in $M_n$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/fixed-degree-monic-zero","kind":"node","lean":"MetaMathlibExt.fixedDegreeMonic_zero","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#fixed-degree-monic-zero","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For every finite field $F$, the set $M_0$ of monic polynomials of degree $0$ over $F$ is $\\{1\\}$.","title":"The only monic polynomial of degree zero"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/monic-irreducible","kind":"node","lean":"MetaMathlibExt.monicIrreducible","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#monic-irreducible","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For a field $F$, $\\mathcal{P}$ is the set of polynomials over $F$ that are monic and irreducible; these are the prime polynomials of the paper.","title":"Monic irreducible polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/mem-monic-irreducible-iff","kind":"node","lean":"MetaMathlibExt.mem_monicIrreducible_iff","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#mem-monic-irreducible-iff","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a field. A polynomial $P$ over $F$ lies in the set $\\mathcal{P}$ of monic irreducible polynomials over $F$ if and only if $P$ is monic and irreducible.","title":"Membership in the set of monic irreducible polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free","kind":"node","lean":"MetaMathlibExt.hFree","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(h\\)"],["text","-free polynomials"]],"source":"$h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a field and $h$ a natural number. The set $S_h$ of $h$-free polynomials over $F$ consists of the monic polynomials $f$ over $F$ such that no monic irreducible polynomial $P$ over $F$ satisfies $P^h \\mid f$.","title":"$h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/one-mem-h-free","kind":"node","lean":"MetaMathlibExt.one_mem_hFree","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomial "],["math","\\(1\\)"],["text"," is "],["math","\\(h\\)"],["text","-free"]],"source":"The polynomial $1$ is $h$-free"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#one-mem-h-free","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For every field $F$ and every natural number $h \\ge 1$, the constant polynomial $1 \\in F[X]$ is $h$-free: it is monic and no $h$-th power of a monic irreducible polynomial divides it.","title":"The polynomial $1$ is $h$-free"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full","kind":"node","lean":"MetaMathlibExt.hFull","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(h\\)"],["text","-full polynomials"]],"source":"$h$-full polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a field and $h \\in \\mathbb{N}$. The set $N_h$ of $h$-full polynomials over $F$ consists of the monic polynomials $f$ such that $P^h$ divides $f$ for every monic irreducible polynomial $P$ that divides $f$.","title":"$h$-full polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/one-mem-h-full","kind":"node","lean":"MetaMathlibExt.one_mem_hFull","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomial "],["math","\\(1\\)"],["text"," is "],["math","\\(h\\)"],["text","-full"]],"source":"The polynomial $1$ is $h$-full"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#one-mem-h-full","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For every field $F$ and every natural number $h$, the constant polynomial $1 \\in F[X]$ is $h$-full: it is monic, and whenever a monic irreducible polynomial $P$ divides it, so does $P^h$.","title":"The polynomial $1$ is $h$-full"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult","kind":"node","lean":"MetaMathlibExt.omegaMult","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a field and $k \\in \\mathbb{N}$. For a nonzero polynomial $f$ over $F$, $\\omega_k(f)$ is the number of distinct monic irreducible polynomials $P$ dividing $f$ whose multiplicity in $f$ (the largest $m$ with $P^m \\mid f$) is exactly $k$. For the zero polynomial, $\\omega_k(0) = 0$ by convention.","title":"Number of prime factors of given multiplicity"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult-zero","kind":"node","lean":"MetaMathlibExt.omegaMult_zero","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\omega_k\\)"],["text"," vanishes at the zero polynomial"]],"source":"$\\omega_k$ vanishes at the zero polynomial"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult-zero","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For every field $F$ and every natural number $k$, $\\omega_k(0) = 0$. Here $\\omega_k(f)$ is the number of distinct monic irreducible polynomials dividing a nonzero polynomial $f$ over $F$ with multiplicity exactly $k$, and $\\omega_k(0) = 0$ by convention, so this records that convention.","title":"$\\omega_k$ vanishes at the zero polynomial"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult-one","kind":"node","lean":"MetaMathlibExt.omegaMult_one","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\omega_k\\)"],["text"," vanishes at the polynomial "],["math","\\(1\\)"]],"source":"$\\omega_k$ vanishes at the polynomial $1$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult-one","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For every field $F$ and every natural number $k$, $\\omega_k(1) = 0$, where $\\omega_k(f)$ is the number of distinct monic irreducible polynomials dividing a nonzero polynomial $f$ over $F$ with multiplicity exactly $k$.","title":"$\\omega_k$ vanishes at the polynomial $1$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/is-degree-nondecreasing","kind":"node","lean":"MetaMathlibExt.IsDegreeNondecreasing","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#is-degree-nondecreasing","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a field, $U$ a set of polynomials over $F$ and $G$ a real-valued function on polynomials over $F$. Then $G$ is degree-nondecreasing on $U$ when for all $f_1, f_2 \\in U$ with $\\deg f_2 \\le \\deg f_1$ we have $G(f_2) \\le G(f_1)$. Here $\\deg$ is the natural degree, with $\\deg 0 = 0$.","title":"Degree-nondecreasing functions"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/is-degree-nondecreasing-const","kind":"node","lean":"MetaMathlibExt.isDegreeNondecreasing_const","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#is-degree-nondecreasing-const","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a field and $U$ a set of polynomials over $F$. For every real number $c$, the constant function with value $c$ is degree-nondecreasing on $U$.","title":"Constant functions are degree-nondecreasing"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/has-normal-order","kind":"node","lean":"MetaMathlibExt.HasNormalOrder","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#has-normal-order","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field, let $U$ be a set of polynomials over $F$, and let $g, G$ be real-valued functions on polynomials over $F$. For $n \\in \\mathbb{N}$ let $M_n$ be the set of monic polynomials of degree $n$ over $F$. Then $g$ has normal order $G$ on $U$ when $g$ and $G$ are nonnegative on $U$, $G$ is degree-nondecreasing on $U$ (for $f_1, f_2 \\in U$, $\\deg f_2 \\le \\deg f_1$ implies $G(f_2) \\le G(f_1)$, with\u2026","title":"Normal order on a set of polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/nonnegative","kind":"node","lean":"MetaMathlibExt.HasNormalOrder.nonnegative","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#nonnegative","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field, $U$ a set of polynomials over $F$, and $g, G$ real-valued functions on polynomials over $F$. If $g$ has normal order $G$ on $U$, then $G(f) \\ge 0$ for every $f \\in U$.","title":"Normal orders are nonnegative"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/erdos-kac-omega-one-h-free","kind":"node","lean":"MetaMathlibExt.erdosKac_omegaOne_hFree","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s\u2013Kac theorem for "],["math","\\(\\omega_1\\)"],["text"," on "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Erd\u0151s\u2013Kac theorem for $\\omega_1$ on $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#erdos-kac-omega-one-h-free","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field, let $h \\ge 2$ and let $\\alpha \\le \\beta$ be real numbers. A monic polynomial $f$ over $F$ is $h$-free if $P^h \\nmid f$ for every monic irreducible $P$, and $\\omega_1(f)$ is the number of distinct monic irreducible $P$ that divide $f$ with multiplicity exactly $1$. For each $n$, let $p_n$ be the number of $h$-free monic polynomials $f$ of degree $n$ over $F$ with","title":"Erd\u0151s\u2013Kac theorem for $\\omega_1$ on $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult-no-normal-order-h-free","kind":"node","lean":"MetaMathlibExt.omegaMult_no_normalOrder_hFree","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\omega_k\\)"],["text"," has no normal order on "],["math","\\(h\\)"],["text","-free polynomials for "],["math","\\(1 < k < h\\)"]],"source":"$\\omega_k$ has no normal order on $h$-free polynomials for $1 < k < h$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult-no-normal-order-h-free","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field, let $h \\ge 2$ and $1 < k < h$, and let $S_h$ be the set of $h$-free monic polynomials over $F$. Let $G$ be a real-valued function on polynomials over $F$ that is nonnegative on $S_h$ and degree-nondecreasing on $S_h$, that is, $G(f_2) \\le G(f_1)$ whenever $f_1, f_2 \\in S_h$ and $\\deg f_2 \\le \\deg f_1$. Then $\\omega_k$, the number of monic irreducible factors of multiplicity exactly $k$,\u2026","title":"$\\omega_k$ has no normal order on $h$-free polynomials for $1 < k < h$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/erdos-kac-omega-h-h-full","kind":"node","lean":"MetaMathlibExt.erdosKac_omegaH_hFull","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s\u2013Kac theorem for "],["math","\\(\\omega_h\\)"],["text"," on "],["math","\\(h\\)"],["text","-full polynomials"]],"source":"Erd\u0151s\u2013Kac theorem for $\\omega_h$ on $h$-full polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#erdos-kac-omega-h-h-full","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field, let $h \\ge 2$ and let $\\alpha \\le \\beta$ be real numbers. A monic polynomial $f$ over $F$ is $h$-full if $P^h \\mid f$ for every monic irreducible $P$ dividing $f$, and $\\omega_h(f)$ is the number of distinct monic irreducible $P$ that divide $f$ with multiplicity exactly $h$. For each $n$, let $p_n$ be the number of $h$-full monic polynomials $f$ of degree $n$ over $F$ with","title":"Erd\u0151s\u2013Kac theorem for $\\omega_h$ on $h$-full polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult-no-normal-order-h-full","kind":"node","lean":"MetaMathlibExt.omegaMult_no_normalOrder_hFull","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\omega_k\\)"],["text"," has no normal order on "],["math","\\(h\\)"],["text","-full polynomials for "],["math","\\(k > h\\)"]],"source":"$\\omega_k$ has no normal order on $h$-full polynomials for $k > h$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult-no-normal-order-h-full","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field, let $2 \\le h < k$, and let $N_h$ be the set of $h$-full monic polynomials over $F$. Let $G$ be a real-valued function on polynomials over $F$ that is nonnegative on $N_h$ and degree-nondecreasing on $N_h$, that is, $G(f_2) \\le G(f_1)$ whenever $f_1, f_2 \\in N_h$ and $\\deg f_2 \\le \\deg f_1$. Then $\\omega_k$, the number of monic irreducible factors of multiplicity exactly $k$, viewed as a\u2026","title":"$\\omega_k$ has no normal order on $h$-full polynomials for $k > h$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/card-h-free-coprime-irreducible-asymptot","kind":"node","lean":"MetaMathlibExt.card_hFree_coprime_irreducible_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(\\ell\\)"],["text","-free polynomials coprime to an irreducible"]],"source":"Counting $\\ell$-free polynomials coprime to an irreducible"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#card-h-free-coprime-irreducible-asymptot","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $\\ell \\ge 2$ and let $d$ be a natural number. Call a monic polynomial over $F$ $\\ell$-free if it is divisible by the $\\ell$-th power of no monic irreducible polynomial. There is a constant $C \\ge 0$ such that for every monic irreducible polynomial $P$ over $F$ of degree $d$, with $|P| = q^{d}$, and every natural number $n$, the number $N$ of $\\ell$-free monic\u2026","title":"Counting $\\ell$-free polynomials coprime to an irreducible"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/perron-formula-polynomial-degree-slice","kind":"node","lean":"MetaMathlibExt.perron_formula_polynomial_degree_slice","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#perron-formula-polynomial-degree-slice","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field, let $M_m$ be the set of monic polynomials of degree $m$ over $F$, let $a \\colon F[X] \\to \\mathbb{C}$, and write $c_m = \\sum_{f \\in M_m} a(f)$. Let $r$ be a real number with $0 < r < 1$ such that the series $\\sum_{m \\ge 0} |c_m| r^m$ converges, and let $A \\colon \\mathbb{C} \\to \\mathbb{C}$ satisfy $A(u) = \\sum_{m \\ge 0} c_m u^m$ for every complex $u$ with $|u| = r$. Then for every $n \\ge 0$,","title":"Perron's formula for degree slices of monic polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/poly-norm","kind":"node","lean":"MetaMathlibExt.polyNorm","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomial norm "],["math","\\(q^{\\deg f}\\)"],["text"," over a finite field"]],"source":"The polynomial norm $q^{\\deg f}$ over a finite field"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#poly-norm","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements. The norm of a polynomial $f \\in F[X]$ is the real number $|f| = q^{\\deg f}$ if $f \\neq 0$, and $|0| = 0$.","title":"The polynomial norm $q^{\\deg f}$ over a finite field"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/poly-norm-zero","kind":"node","lean":"MetaMathlibExt.polyNorm_zero","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zero polynomial has norm "],["math","\\(0\\)"]],"source":"The zero polynomial has norm $0$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#poly-norm-zero","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For every finite field $F$, the norm of the zero polynomial in $F[X]$ is $|0| = 0$.","title":"The zero polynomial has norm $0$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/poly-norm-of-ne-zero","kind":"node","lean":"MetaMathlibExt.polyNorm_of_ne_zero","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","A nonzero polynomial has norm "],["math","\\(q^{\\deg f}\\)"]],"source":"A nonzero polynomial has norm $q^{\\deg f}$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#poly-norm-of-ne-zero","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements. Every nonzero polynomial $f \\in F[X]$ has norm $|f| = q^{\\deg f}$.","title":"A nonzero polynomial has norm $q^{\\deg f}$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/function-field-mertens-constant","kind":"node","lean":"MetaMathlibExt.functionFieldMertensConstant","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#function-field-mertens-constant","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For a finite field $F$ with $q$ elements, the analogue of the first Mertens constant is","title":"Function field Mertens constant"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/function-field-mertens-constant-eq","kind":"node","lean":"MetaMathlibExt.functionFieldMertensConstant_eq","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#function-field-mertens-constant-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"For a finite field $F$ with $q$ elements, the function field Mertens constant $B_1$ of $F$ equals","title":"Series for the function field Mertens constant"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-moment","kind":"node","lean":"MetaMathlibExt.hFreeMoment","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moments of "],["math","\\(\\omega_k\\)"],["text"," over "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Moments of $\\omega_k$ over $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-moment","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field and let $h$, $k$, $j$, $n$ be natural numbers. Let $S_h$ be the set of $h$-free monic polynomials over $F$ (those divisible by no $P^h$ with $P$ monic irreducible), $M_n$ the set of monic polynomials of degree $n$ over $F$, and $\\omega_k(f)$ the number of distinct monic irreducible factors of $f$ whose multiplicity is exactly $k$. The $j$-th moment of $\\omega_k$ over the $h$-free monic\u2026","title":"Moments of $\\omega_k$ over $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-moment-eq","kind":"node","lean":"MetaMathlibExt.hFreeMoment_eq","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-moment-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field and let $h, k, j, n$ be natural numbers. Write $M_n$ for the monic polynomials of degree $n$ over $F$, $S_h$ for the $h$-free monic polynomials (those divisible by the $h$-th power of no monic irreducible polynomial) and $\\omega_k(f)$ for the number of distinct monic irreducible factors of $f$ of multiplicity exactly $k$. Then the $j$-th moment of $\\omega_k$ over $S_h \\cap M_n$ equals","title":"Explicit formula for the moments over h-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-one-sum","kind":"node","lean":"MetaMathlibExt.hFreeOmegaOneSum","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The prime sum "],["math","\\(T_h\\)"]],"source":"The prime sum $T_h$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-one-sum","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ be a natural number, and write $|P| = q^{\\deg P}$ for a monic irreducible polynomial $P$ over $F$. The prime sum $T_h$ is the real number","title":"The prime sum $T_h$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-one-sum-eq","kind":"node","lean":"MetaMathlibExt.hFreeOmegaOneSum_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unfolding the prime sum "],["math","\\(T_h\\)"]],"source":"Unfolding the prime sum $T_h$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-one-sum-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ be a natural number, and write $|P| = q^{\\deg P}$ for a monic irreducible polynomial $P$ over $F$. Then the prime sum $T_h$ equals the real sum over all monic irreducible polynomials $P$ over $F$","title":"Unfolding the prime sum $T_h$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-one-sq-sum","kind":"node","lean":"MetaMathlibExt.hFreeOmegaOneSqSum","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The squared prime sum "],["math","\\(U_h\\)"],["text"," for multiplicity "],["math","\\(1\\)"],["text"," on "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"The squared prime sum $U_h$ for multiplicity $1$ on $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-one-sq-sum","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ be a natural number, and for a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$. Then $U_h$ is the real number","title":"The squared prime sum $U_h$ for multiplicity $1$ on $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-one-sq-sum-eq","kind":"node","lean":"MetaMathlibExt.hFreeOmegaOneSqSum_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Series formula for the squared prime sum "],["math","\\(U_h\\)"]],"source":"Series formula for the squared prime sum $U_h$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-one-sq-sum-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ be a natural number, and for a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$. Then the squared prime sum $U_h$ for multiplicity $1$ on $h$-free polynomials equals","title":"Series formula for the squared prime sum $U_h$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-mult-sum","kind":"node","lean":"MetaMathlibExt.hFreeOmegaMultSum","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The prime sum "],["math","\\(S_{h,k}\\)"],["text"," for multiplicity "],["math","\\(k\\)"],["text"," on "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"The prime sum $S_{h,k}$ for multiplicity $k$ on $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-mult-sum","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $h$, $k$ be natural numbers. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$ and $a_P = |P|^{\\max(h-k-1,\\,0)} (|P| - 1)/(|P|^h - 1)$, read as $0$ when the denominator vanishes (only when $h = 0$). Then $S_{h,k}$ is the real number $\\sum_P a_P$, the sum over all monic irreducible $P$, taken to be $0$ when the series does not converge\u2026","title":"The prime sum $S_{h,k}$ for multiplicity $k$ on $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-mult-sum-eq","kind":"node","lean":"MetaMathlibExt.hFreeOmegaMultSum_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Series formula for the prime sum "],["math","\\(S_{h,k}\\)"]],"source":"Series formula for the prime sum $S_{h,k}$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-mult-sum-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $h$, $k$ be natural numbers. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$ and $a_P = |P|^{\\max(h-k-1,\\,0)} (|P| - 1)/(|P|^h - 1)$, read as $0$ when the denominator vanishes (only when $h = 0$). Then the prime sum $S_{h,k}$ for multiplicity $k$ on $h$-free polynomials equals $\\sum_P a_P$, the sum over all monic irreducible $P$, taken to\u2026","title":"Series formula for the prime sum $S_{h,k}$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-mult-sq-sum","kind":"node","lean":"MetaMathlibExt.hFreeOmegaMultSqSum","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squared prime sum for multiplicity "],["math","\\(k\\)"],["text"," on "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Squared prime sum for multiplicity $k$ on $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-mult-sq-sum","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $h$, $k$ be natural numbers. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$ and $a_P = |P|^{\\max(h-k-1,\\,0)} (|P| - 1)/(|P|^h - 1)$, read as $0$ when the denominator vanishes (only when $h = 0$). The squared prime sum is the real number $\\sum_P a_P^2$, the sum over all monic irreducible $P$, taken to be $0$ when the series does not\u2026","title":"Squared prime sum for multiplicity $k$ on $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-mult-sq-sum-eq","kind":"node","lean":"MetaMathlibExt.hFreeOmegaMultSqSum_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Series formula for the squared prime sum for multiplicity "],["math","\\(k\\)"],["text"," on "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Series formula for the squared prime sum for multiplicity $k$ on $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-mult-sq-sum-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $h$, $k$ be natural numbers. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$ and $a_P = |P|^{\\max(h-k-1,\\,0)} (|P| - 1)/(|P|^h - 1)$, read as $0$ when the denominator vanishes (only when $h = 0$). Then the squared prime sum for multiplicity $k$ on $h$-free polynomials equals $\\sum_P a_P^2$, the sum over all monic irreducible $P$, taken to\u2026","title":"Series formula for the squared prime sum for multiplicity $k$ on $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-one-h-free-mean-asymptotic","kind":"node","lean":"MetaMathlibExt.omegaOne_hFree_mean_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean of "],["math","\\(\\omega_1\\)"],["text"," over "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Mean of $\\omega_1$ over $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-one-h-free-mean-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and let $h \\ge 2$. Let $\\omega_1(f)$ be the number of monic irreducible $P$ dividing $f$ exactly once, write $|P| = q^{\\deg P}$, and let $B_1 = \\gamma + \\sum_P (\\log(1 - 1/|P|) + 1/|P|)$ be the function field Mertens constant, with $\\gamma$ the Euler\u2013Mascheroni constant, and let $T_h = \\sum_P (|P|^{h-1} - 1)/(|P|(|P|^h - 1))$, both sums over monic irreducible $P$. Then\u2026","title":"Mean of $\\omega_1$ over $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-one-h-free-second-moment-asymptoti","kind":"node","lean":"MetaMathlibExt.omegaOne_hFree_secondMoment_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second moment of "],["math","\\(\\omega_1\\)"],["text"," over "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Second moment of $\\omega_1$ over $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-one-h-free-second-moment-asymptoti","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h \\ge 2$ and let $\\varepsilon > 0$. Let $\\omega_1(f)$ be the number of monic irreducible $P$ dividing $f$ exactly once, write $S_h \\cap M_n$ for the set of $h$-free monic polynomials of degree $n$ over $F$ and $|P| = q^{\\deg P}$, and let $B_1 = \\gamma + \\sum_P (\\log(1 - 1/|P|) + 1/|P|)$ be the function field Mertens constant, with $\\gamma$ the Euler\u2013Mascheroni\u2026","title":"Second moment of $\\omega_1$ over $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-one-h-free-variance-asymptotic","kind":"node","lean":"MetaMathlibExt.omegaOne_hFree_variance_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Variance of "],["math","\\(\\omega_1\\)"],["text"," over "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Variance of $\\omega_1$ over $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-one-h-free-variance-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h \\ge 2$ and let $\\varepsilon > 0$. Let $\\omega_1(f)$ be the number of monic irreducible $P$ dividing $f$ exactly once, write $|P| = q^{\\deg P}$, and let $B_1 = \\gamma + \\sum_P (\\log(1 - 1/|P|) + 1/|P|)$ be the function field Mertens constant, with $\\gamma$ the Euler\u2013Mascheroni constant, $T_h = \\sum_P (|P|^{h-1} - 1)/(|P|(|P|^h - 1))$ and\u2026","title":"Variance of $\\omega_1$ over $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult-h-free-mean-asymptotic","kind":"node","lean":"MetaMathlibExt.omegaMult_hFree_mean_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean of "],["math","\\(\\omega_k\\)"],["text"," over "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Mean of $\\omega_k$ over $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult-h-free-mean-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h, k$ be natural numbers with $1 < k < h$, and let $\\varepsilon > 0$. Write $S_h \\cap M_n$ for the set of monic polynomials $f$ of degree $n$ over $F$ that are $h$-free (no $P^h$ with $P$ monic irreducible divides $f$), and $\\omega_k(f)$ for the number of monic irreducible $P$ that divide $f$ exactly $k$ times. For monic irreducible $P$ put $|P| = q^{\\deg P}$ and\u2026","title":"Mean of $\\omega_k$ over $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult-h-free-second-moment-asymptot","kind":"node","lean":"MetaMathlibExt.omegaMult_hFree_secondMoment_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second moment of "],["math","\\(\\omega_k\\)"],["text"," over "],["math","\\(h\\)"],["text","-free polynomials"]],"source":"Second moment of $\\omega_k$ over $h$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult-h-free-second-moment-asymptot","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h, k$ be natural numbers with $1 < k < h$, and let $\\varepsilon > 0$. Write $S_h \\cap M_n$ for the set of monic polynomials $f$ of degree $n$ over $F$ that are $h$-free (no $P^h$ with $P$ monic irreducible divides $f$), and $\\omega_k(f)$ for the number of monic irreducible $P$ that divide $f$ exactly $k$ times. For monic irreducible $P$ put $|P| = q^{\\deg P}$ and\u2026","title":"Second moment of $\\omega_k$ over $h$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/iterated-deriv-sub-is-big-o-of-uniform-a","kind":"node","lean":"MetaMathlibExt.iteratedDeriv_sub_isBigO_of_uniform_approx","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivatives at "],["math","\\(z = 1\\)"],["text"," of uniform approximations"]],"source":"Derivatives at $z = 1$ of uniform approximations"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#iterated-deriv-sub-is-big-o-of-uniform-a","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $r > 0$, let $D$ be the open disk of radius $r$ centred at $1$ in $\\mathbb{C}$, and let $f$ and $G_n$ ($n \\in \\mathbb{N}$) be complex functions analytic on $D$. Suppose there is a real constant $C$ such that $|G_n(z) - f(z)\\, n^{z-1}| \\le C\\, n^{\\operatorname{Re} z - 2}$ for all $n \\ge 1$ and all $z \\in D$. Then for every $k \\in \\mathbb{N}$ and every $\\varepsilon > 0$, as $n \\to \\infty$,","title":"Derivatives at $z = 1$ of uniform approximations"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/selberg-delange-function-field","kind":"node","lean":"MetaMathlibExt.selbergDelange_functionField","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#selberg-delange-function-field","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $q > 1$ and $A > 0$ be real numbers, and let $B_z(n)$ and $C_z(n)$ be complex numbers, for $z \\in \\mathbb{C}$ and integers $n \\ge 0$, such that for all $z$ and $n$","title":"Selberg\u2013Delange theorem for function fields"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/selberg-delange-function-field-pow","kind":"node","lean":"MetaMathlibExt.selbergDelange_functionField_pow","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Selberg\u2013Delange theorem, "],["math","\\(h\\)"],["text","-th power version"]],"source":"Selberg\u2013Delange theorem, $h$-th power version"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#selberg-delange-function-field-pow","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $q > 1$ and $A > 0$ be real numbers, let $h \\ge 1$ be an integer, let $\\xi$ be a primitive $h$-th root of unity in $\\mathbb{C}$, and let $B_z(n)$ and $C_z(n)$ be complex numbers, for $z \\in \\mathbb{C}$ and integers $n \\ge 0$, such that for all $z$ and $n$","title":"Selberg\u2013Delange theorem, $h$-th power version"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-one-b","kind":"node","lean":"MetaMathlibExt.hFreeOmegaOneB","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of "],["math","\\(B_{1,S_h}(u, z)\\)"]],"source":"Coefficients of $B_{1,S_h}(u, z)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-one-b","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ be a natural number and let $z$ be a complex number. Let $S_h$ be the set of $h$-free monic polynomials over $F$ (those divisible by no $P^h$ with $P$ monic irreducible), and $\\omega_1(f)$ the number of distinct monic irreducible factors of $f$ of multiplicity exactly $1$. Write $C_{1,S_h}(u, z) = \\sum_{f \\in S_h} z^{\\omega_1(f)} u^{\\deg f}$ and\u2026","title":"Coefficients of $B_{1,S_h}(u, z)$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-one-b-zero","kind":"node","lean":"MetaMathlibExt.hFreeOmegaOneB_zero","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Constant coefficient of "],["math","\\(B_{1,S_h}\\)"]],"source":"Constant coefficient of $B_{1,S_h}$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-one-b-zero","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $h \\ge 1$ be a natural number. Let $S_h$ be the set of $h$-free monic polynomials over $F$ (those divisible by no $P^h$ with $P$ monic irreducible) and $\\omega_1(f)$ the number of distinct monic irreducible factors of $f$ of multiplicity exactly $1$. For every complex number $z$, the coefficient $B_z(0)$ of $u^0$ in the formal power series","title":"Constant coefficient of $B_{1,S_h}$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-omega-hb","kind":"node","lean":"MetaMathlibExt.hFullOmegaHB","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficients "],["math","\\(B_z(n)\\)"],["text"," of the "],["math","\\(h\\)"],["text","-full series"]],"source":"The coefficients $B_z(n)$ of the $h$-full series"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-omega-hb","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $n$ be natural numbers and let $z$ be a complex number. Write $N_h$ for the set of $h$-full monic polynomials over $F$ (monic $f$ such that $P^h$ divides $f$ for every monic irreducible $P$ dividing $f$), $\\omega_h(f)$ for the number of monic irreducible polynomials that divide $f$ with multiplicity exactly $h$, and $(x)_m = x(x + 1) \\cdots (x + m - 1)$ for\u2026","title":"The coefficients $B_z(n)$ of the $h$-full series"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-omega-hb-zero","kind":"node","lean":"MetaMathlibExt.hFullOmegaHB_zero","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Constant coefficient of "],["math","\\(B_{h,N_h}\\)"]],"source":"Constant coefficient of $B_{h,N_h}$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-omega-hb-zero","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ be a natural number and let $z$ be a complex number. Then the coefficient $B_z(0)$ of the $h$-full series for $F$, $h$ and $z$ equals $1$. For $h \\ge 1$, $B_z(0)$ is the constant coefficient of $B_{h,N_h}(u, z) = (1 - qu^h)^z \\sum_{f \\in N_h} z^{\\omega_h(f)} u^{\\deg f}$, where $N_h$ is the set of $h$-full monic polynomials over $F$ and $\\omega_h(f)$ is the number\u2026","title":"Constant coefficient of $B_{h,N_h}$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-free-omega-one-b-weighted-sum-bounded","kind":"node","lean":"MetaMathlibExt.hFreeOmegaOneB_weighted_sum_bounded","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted coefficient bound for "],["math","\\(B_{1,S_h}\\)"]],"source":"Weighted coefficient bound for $B_{1,S_h}$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-free-omega-one-b-weighted-sum-bounded","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h \\ge 2$ be a natural number, and let $A$ and $\\sigma > 1/2$ be real numbers. Let $S_h$ be the set of $h$-free monic polynomials over $F$ (those divisible by no $P^h$ with $P$ monic irreducible) and $\\omega_1(f)$ the number of distinct monic irreducible factors of $f$ of multiplicity exactly $1$. For a complex number $z$ and a natural number $a$ let $B_z(a)$ be the\u2026","title":"Weighted coefficient bound for $B_{1,S_h}$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-omega-hb-weighted-sum-bounded","kind":"node","lean":"MetaMathlibExt.hFullOmegaHB_weighted_sum_bounded","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniform bound for the weighted coefficients of "],["math","\\(B_{h,N_h}(u, z)\\)"]],"source":"Uniform bound for the weighted coefficients of $B_{h,N_h}(u, z)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-omega-hb-weighted-sum-bounded","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h \\ge 2$ be a natural number, and let $A$ and $\\sigma$ be real numbers with $\\sigma > 1/(h + 1)$. Then there is a real constant $c$ such that for every complex number $z$ with $|z| \\le A$ and every natural number $n \\ge 2$,","title":"Uniform bound for the weighted coefficients of $B_{h,N_h}(u, z)$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-root","kind":"node","lean":"MetaMathlibExt.hFullRoot","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pole parameter "],["math","\\(q^{1/h}\\xi^j\\)"]],"source":"The pole parameter $q^{1/h}\\xi^j$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-root","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite set with $q$ elements, and let $h, j \\in \\mathbb{N}$ and $\\xi \\in \\mathbb{C}$. The pole parameter is the complex number $q^{1/h}\\xi^j$, where $q^{1/h}$ is the nonnegative real $h$-th root of $q$ (for $h = 0$ the exponent $1/h$ is read as $0$, so that $q^{1/h} = 1$).","title":"The pole parameter $q^{1/h}\\xi^j$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-root-eq","kind":"node","lean":"MetaMathlibExt.hFullRoot_eq","parent":13534,"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-root-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite set with $q$ elements, and let $h, j \\in \\mathbb{N}$ and $\\xi \\in \\mathbb{C}$. The pole parameter $q^{1/h}\\xi^j$ equals the real power $q^{1/h}$, whose exponent is the real number $1/h$ (read as $0$ when $h = 0$), viewed as a complex number and multiplied by $\\xi^j$.","title":"Formula for the pole parameter"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-free-euler-h","kind":"node","lean":"MetaMathlibExt.hFullFreeEulerH","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Euler product "],["math","\\(H_{N_h, S_\\ell}\\)"],["text"," for "],["math","\\(h\\)"],["text","-full, "],["math","\\(\\ell\\)"],["text","-free polynomials"]],"source":"The Euler product $H_{N_h, S_\\ell}$ for $h$-full, $\\ell$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-free-euler-h","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field, let $h$ and $\\ell$ be natural numbers and let $u$ be a complex number. Then $H_{N_h, S_\\ell}(u)$ is the complex number","title":"The Euler product $H_{N_h, S_\\ell}$ for $h$-full, $\\ell$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-free-euler-h-eq","kind":"node","lean":"MetaMathlibExt.hFullFreeEulerH_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unfolding the Euler product "],["math","\\(H_{N_h, S_\\ell}\\)"]],"source":"Unfolding the Euler product $H_{N_h, S_\\ell}$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-free-euler-h-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field, let $h$ and $\\ell$ be natural numbers and let $u$ be a complex number. Then $H_{N_h, S_\\ell}(u)$ equals the complex product over all monic irreducible polynomials $P$ over $F$","title":"Unfolding the Euler product $H_{N_h, S_\\ell}$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/card-h-full-h-free-asymptotic","kind":"node","lean":"MetaMathlibExt.card_hFull_hFree_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(h\\)"],["text","-full, "],["math","\\(\\ell\\)"],["text","-free polynomials"]],"source":"Counting $h$-full, $\\ell$-free polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#card-h-full-h-free-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $\\ell$ be natural numbers with $0 < h < \\ell$, let $\\xi \\in \\mathbb{C}$ be a primitive $h$-th root of unity and let $\\varepsilon > 0$. A monic polynomial $f$ over $F$ is $h$-full if $P^h \\mid f$ for every monic irreducible $P$ dividing $f$, and $\\ell$-free if $P^\\ell \\nmid f$ for every monic irreducible $P$. Write $r_j = q^{1/h} \\xi^j$ with $q^{1/h}$ the\u2026","title":"Counting $h$-full, $\\ell$-free polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/card-h-full-h-free-coprime-asymptotic","kind":"node","lean":"MetaMathlibExt.card_hFull_hFree_coprime_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(h\\)"],["text","-full, "],["math","\\(\\ell\\)"],["text","-free polynomials coprime to an irreducible"]],"source":"Counting $h$-full, $\\ell$-free polynomials coprime to an irreducible"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#card-h-full-h-free-coprime-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $\\ell$ be natural numbers with $0 < h < \\ell$, let $\\xi \\in \\mathbb{C}$ be a primitive $h$-th root of unity, let $P_0$ be a monic irreducible polynomial over $F$ and let $\\varepsilon > 0$. A monic polynomial $f$ over $F$ is $h$-full if $P^h \\mid f$ for every monic irreducible $P$ dividing $f$, and $\\ell$-free if $P^\\ell \\nmid f$ for every monic irreducible\u2026","title":"Counting $h$-full, $\\ell$-free polynomials coprime to an irreducible"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-moment","kind":"node","lean":"MetaMathlibExt.hFullMoment","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(j\\)"],["text","-th moment of "],["math","\\(\\omega_k\\)"],["text"," over "],["math","\\(h\\)"],["text","-full polynomials of degree "],["math","\\(n\\)"]],"source":"The $j$-th moment of $\\omega_k$ over $h$-full polynomials of degree $n$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-moment","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field and let $h$, $k$, $j$ and $n$ be natural numbers. Let $N_h \\cap M_n$ be the finite set of $h$-full monic polynomials of degree $n$ over $F$ (monic $f$ such that $P^h$ divides $f$ for every monic irreducible $P$ dividing $f$), and let $\\omega_k(f)$ be the number of monic irreducible polynomials that divide $f$ with multiplicity exactly $k$. The moment is the real number","title":"The $j$-th moment of $\\omega_k$ over $h$-full polynomials of degree $n$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-moment-eq","kind":"node","lean":"MetaMathlibExt.hFullMoment_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h\\)"],["text","-full moment is an average of "],["math","\\(\\omega_k^j\\)"]],"source":"The $h$-full moment is an average of $\\omega_k^j$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-moment-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field and let $h$, $k$, $j$ and $n$ be natural numbers. Let $N_h \\cap M_n$ be the finite set of $h$-full monic polynomials of degree $n$ over $F$ (monic $f$ such that $P^h$ divides $f$ for every monic irreducible $P$ dividing $f$), and let $\\omega_k(f)$ be the number of monic irreducible polynomials that divide $f$ with multiplicity exactly $k$. Then the $j$-th moment of $\\omega_k$ over\u2026","title":"The $h$-full moment is an average of $\\omega_k^j$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-k","kind":"node","lean":"MetaMathlibExt.hFullK","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The local factor "],["math","\\(K(P, h, j)\\)"]],"source":"The local factor $K(P, h, j)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-k","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $j$ be natural numbers, let $\\xi$ be a complex number and let $P$ be a polynomial over $F$. Write $\\rho_j = q^{1/h} \\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ (read as $1$ when $h = 0$), and read $\\rho_j^{-1}$ as $0$ when $\\rho_j = 0$; let $|P| = q^{\\deg P}$ if $P \\ne 0$ and $|0| = 0$; and put $y = (\\rho_j^{-1})^{\\deg P}$, with $\\deg 0$\u2026","title":"The local factor $K(P, h, j)$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-k-eq","kind":"node","lean":"MetaMathlibExt.hFullK_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unfolding the local factor "],["math","\\(K(P, h, j)\\)"]],"source":"Unfolding the local factor $K(P, h, j)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-k-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $j$ be natural numbers, let $\\xi$ be a complex number and let $P$ be a polynomial over $F$. Write $\\rho_j = q^{1/h} \\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ (read as $1$ when $h = 0$), and read $\\rho_j^{-1}$ as $0$ when $\\rho_j = 0$; let $|P| = q^{\\deg P}$ if $P \\ne 0$ and $|0| = 0$; and put $y = (\\rho_j^{-1})^{\\deg P}$, with $\\deg 0$\u2026","title":"Unfolding the local factor $K(P, h, j)$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-euler-product","kind":"node","lean":"MetaMathlibExt.hFullEulerProduct","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Euler product "],["math","\\(\\Pi_j\\)"],["text"," for "],["math","\\(h\\)"],["text","-full polynomials"]],"source":"The Euler product $\\Pi_j$ for $h$-full polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-euler-product","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $j$ be natural numbers and let $\\xi$ be a complex number. Write $\\rho_j = q^{1/h} \\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ (read as $1$ when $h = 0$), read $\\rho_j^{-1}$ as $0$ when $\\rho_j = 0$, and write $|P| = q^{\\deg P}$ for a monic irreducible polynomial $P$ over $F$. Then $\\Pi_j$ is the complex number","title":"The Euler product $\\Pi_j$ for $h$-full polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-euler-product-eq","kind":"node","lean":"MetaMathlibExt.hFullEulerProduct_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unfolding the Euler product "],["math","\\(\\Pi_j\\)"]],"source":"Unfolding the Euler product $\\Pi_j$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-euler-product-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $j$ be natural numbers and let $\\xi$ be a complex number. Write $\\rho_j = q^{1/h} \\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ (read as $1$ when $h = 0$), read $\\rho_j^{-1}$ as $0$ when $\\rho_j = 0$, and write $|P| = q^{\\deg P}$ for a monic irreducible polynomial $P$ over $F$. Then $\\Pi_j$ equals the complex product over all monic\u2026","title":"Unfolding the Euler product $\\Pi_j$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-weighted-mean","kind":"node","lean":"MetaMathlibExt.hFullWeightedMean","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h\\)"],["text","-full weighted mean"]],"source":"The $h$-full weighted mean"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-weighted-mean","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h, n \\in \\mathbb{N}$, $\\xi \\in \\mathbb{C}$ and $X : \\mathbb{N} \\to \\mathbb{C}$. Let $N_h \\cap M_n$ be the set of $h$-full monic polynomials of degree $n$ over $F$. For each $j$ let $\\rho_j = q^{1/h}\\xi^j$ be the pole parameter and let $\\Pi_j = \\prod_P (1 - 1/|P|)(1 + 1/(|P|(1 - \\rho_j^{-\\deg P})))$, the product over monic irreducible $P$ with $|P| = q^{\\deg P}$,\u2026","title":"The $h$-full weighted mean"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-weighted-mean-eq","kind":"node","lean":"MetaMathlibExt.hFullWeightedMean_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formula for the "],["math","\\(h\\)"],["text","-full weighted mean"]],"source":"Formula for the $h$-full weighted mean"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-weighted-mean-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h, n \\in \\mathbb{N}$, $\\xi \\in \\mathbb{C}$ and $X : \\mathbb{N} \\to \\mathbb{C}$. Let $N_h \\cap M_n$ be the set of $h$-full monic polynomials of degree $n$ over $F$. For each $j$ let $\\rho_j = q^{1/h}\\xi^j$ be the pole parameter and let $\\Pi_j = \\prod_P (1 - 1/|P|)(1 + 1/(|P|(1 - \\rho_j^{-\\deg P})))$, the product over monic irreducible $P$ with $|P| = q^{\\deg P}$,\u2026","title":"Formula for the $h$-full weighted mean"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-ksum","kind":"node","lean":"MetaMathlibExt.hFullKSum","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The constant "],["math","\\(\\kappa_j\\)"],["text"," for "],["math","\\(h\\)"],["text","-full polynomials"]],"source":"The constant $\\kappa_j$ for $h$-full polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-ksum","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $j$ be natural numbers and let $\\xi$ be a complex number. Write $\\rho = q^{1/h}\\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ for $h \\ge 1$ and $q^{1/h} = 1$ for $h = 0$, and let $\\rho^{-1}$ be read as $0$ when $\\rho = 0$. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$, $y = (\\rho^{-1})^{\\deg P}$ and\u2026","title":"The constant $\\kappa_j$ for $h$-full polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-ksum-eq","kind":"node","lean":"MetaMathlibExt.hFullKSum_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The series defining "],["math","\\(\\kappa_j\\)"]],"source":"The series defining $\\kappa_j$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-ksum-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $j$ be natural numbers and let $\\xi$ be a complex number. Write $\\rho = q^{1/h}\\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ for $h \\ge 1$ and $q^{1/h} = 1$ for $h = 0$, and let $\\rho^{-1}$ be read as $0$ when $\\rho = 0$. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$, $y = (\\rho^{-1})^{\\deg P}$ and\u2026","title":"The series defining $\\kappa_j$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-ksq-sum","kind":"node","lean":"MetaMathlibExt.hFullKSqSum","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum "],["math","\\(\\sum_P K(P, h, j)^2\\)"],["text"," over monic irreducible polynomials"]],"source":"The sum $\\sum_P K(P, h, j)^2$ over monic irreducible polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-ksq-sum","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$ and $j$ be natural numbers and let $\\xi$ be a complex number. Write $\\rho = q^{1/h}\\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ for $h \\ge 1$ and $q^{1/h} = 1$ for $h = 0$, and let $\\rho^{-1}$ be read as $0$ when $\\rho = 0$. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$, $y = (\\rho^{-1})^{\\deg P}$ and\u2026","title":"The sum $\\sum_P K(P, h, j)^2$ over monic irreducible polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-ksq-sum-eq","kind":"node","lean":"MetaMathlibExt.hFullKSqSum_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unfolding the sum of the squares "],["math","\\(K(P, h, j)^2\\)"]],"source":"Unfolding the sum of the squares $K(P, h, j)^2$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-ksq-sum-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field, let $h$ and $j$ be natural numbers and let $\\xi$ be a complex number. The quantity $\\sum_P K(P, h, j)^2$ of Theorem 5 is the complex sum, over all monic irreducible polynomials $P$ over $F$, of the squares of the local factors $K(P, h, j)$ attached to $h$, $j$ and $\\xi$, taken to be $0$ if it does not converge absolutely.","title":"Unfolding the sum of the squares $K(P, h, j)^2$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-omega-mult-sum","kind":"node","lean":"MetaMathlibExt.hFullOmegaMultSum","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h\\)"],["text","-full prime sum "],["math","\\(\\sum_P b_P\\)"]],"source":"The $h$-full prime sum $\\sum_P b_P$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-omega-mult-sum","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h, k, j \\in \\mathbb{N}$ and $\\xi \\in \\mathbb{C}$, and let $\\rho = q^{1/h}\\xi^j$ be the pole parameter, where $q^{1/h}$ is a real power whose exponent $1/h$ is read as $0$ when $h = 0$. For a monic irreducible polynomial $P$ over $F$ put $|P| = q^{\\deg P}$, $y_P = \\rho^{-\\deg P}$, $K(P, h, j) = (1 - y_P)/(|P|(1 - y_P) + 1)$ and $b_P = |P| \\, y_P^{k} \\, K(P, h, j)$.\u2026","title":"The $h$-full prime sum $\\sum_P b_P$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-omega-mult-sum-eq","kind":"node","lean":"MetaMathlibExt.hFullOmegaMultSum_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formula for the "],["math","\\(h\\)"],["text","-full prime sum "],["math","\\(\\sum_P b_P\\)"]],"source":"Formula for the $h$-full prime sum $\\sum_P b_P$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-omega-mult-sum-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h, k, j \\in \\mathbb{N}$ and $\\xi \\in \\mathbb{C}$, and let $\\rho = q^{1/h}\\xi^j$ be the pole parameter, where $q^{1/h}$ is a real power whose exponent $1/h$ is read as $0$ when $h = 0$. For a monic irreducible polynomial $P$ over $F$ put $|P| = q^{\\deg P}$, $K(P, h, j) = (1 - \\rho^{-\\deg P})/(|P|(1 - \\rho^{-\\deg P}) + 1)$ and\u2026","title":"Formula for the $h$-full prime sum $\\sum_P b_P$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-omega-mult-sq-sum","kind":"node","lean":"MetaMathlibExt.hFullOmegaMultSqSum","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum "],["math","\\(\\sum_P b_P^2\\)"],["text"," over monic irreducible polynomials"]],"source":"The sum $\\sum_P b_P^2$ over monic irreducible polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-omega-mult-sq-sum","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$, $k$ and $j$ be natural numbers and let $\\xi$ be a complex number. Write $\\rho = q^{1/h}\\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ for $h \\ge 1$ and $q^{1/h} = 1$ for $h = 0$, and let $\\rho^{-1}$ be read as $0$ when $\\rho = 0$. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$, $y = (\\rho^{-1})^{\\deg P}$,\u2026","title":"The sum $\\sum_P b_P^2$ over monic irreducible polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/h-full-omega-mult-sq-sum-eq","kind":"node","lean":"MetaMathlibExt.hFullOmegaMultSqSum_eq","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","The series defining "],["math","\\(\\sum_P b_P^2\\)"]],"source":"The series defining $\\sum_P b_P^2$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#h-full-omega-mult-sq-sum-eq","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $h$, $k$ and $j$ be natural numbers and let $\\xi$ be a complex number. Write $\\rho = q^{1/h}\\xi^j$, where $q^{1/h}$ is the positive real $h$-th root of $q$ for $h \\ge 1$ and $q^{1/h} = 1$ for $h = 0$, and let $\\rho^{-1}$ be read as $0$ when $\\rho = 0$. For a monic irreducible polynomial $P$ over $F$ write $|P| = q^{\\deg P}$, $y = (\\rho^{-1})^{\\deg P}$ and\u2026","title":"The series defining $\\sum_P b_P^2$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-h-h-full-mean-asymptotic","kind":"node","lean":"MetaMathlibExt.omegaH_hFull_mean_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean of "],["math","\\(\\omega_h\\)"],["text"," over "],["math","\\(h\\)"],["text","-full polynomials"]],"source":"Mean of $\\omega_h$ over $h$-full polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-h-h-full-mean-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h \\ge 2$, let $\\xi \\in \\mathbb{C}$ be a primitive $h$-th root of unity, and let $\\varepsilon > 0$. Write $N_h \\cap M_n$ for the set of monic polynomials $f$ of degree $n$ over $F$ that are $h$-full (if a monic irreducible $P$ divides $f$, then so does $P^h$), and $\\omega_h(f)$ for the number of monic irreducible $P$ that divide $f$ exactly $h$ times. For monic\u2026","title":"Mean of $\\omega_h$ over $h$-full polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-h-h-full-second-moment-asymptotic","kind":"node","lean":"MetaMathlibExt.omegaH_hFull_secondMoment_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second moment of "],["math","\\(\\omega_h\\)"],["text"," over "],["math","\\(h\\)"],["text","-full polynomials"]],"source":"Second moment of $\\omega_h$ over $h$-full polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-h-h-full-second-moment-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h \\ge 2$, let $\\xi \\in \\mathbb{C}$ be a primitive $h$-th root of unity, and let $\\varepsilon > 0$. Write $N_h \\cap M_n$ for the set of monic polynomials $f$ of degree $n$ over $F$ that are $h$-full (if a monic irreducible $P$ divides $f$, then so does $P^h$), and $\\omega_h(f)$ for the number of monic irreducible $P$ that divide $f$ exactly $h$ times. For monic\u2026","title":"Second moment of $\\omega_h$ over $h$-full polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-h-h-full-variance-asymptotic","kind":"node","lean":"MetaMathlibExt.omegaH_hFull_variance_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Variance of "],["math","\\(\\omega_h\\)"],["text"," over "],["math","\\(h\\)"],["text","-full polynomials"]],"source":"Variance of $\\omega_h$ over $h$-full polynomials"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-h-h-full-variance-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $h \\ge 2$, let $\\xi \\in \\mathbb{C}$ be a primitive $h$-th root of unity, and let $\\varepsilon > 0$. Write $N_h \\cap M_n$ for the set of monic polynomials $f$ of degree $n$ over $F$ that are $h$-full (if a monic irreducible $P$ divides $f$, then so does $P^h$), and $\\omega_h(f)$ for the number of monic irreducible $P$ that divide $f$ exactly $h$ times. For monic\u2026","title":"Variance of $\\omega_h$ over $h$-full polynomials"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult-h-full-mean-asymptotic","kind":"node","lean":"MetaMathlibExt.omegaMult_hFull_mean_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean of "],["math","\\(\\omega_k\\)"],["text"," over "],["math","\\(h\\)"],["text","-full polynomials for "],["math","\\(k > h\\)"]],"source":"Mean of $\\omega_k$ over $h$-full polynomials for $k > h$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult-h-full-mean-asymptotic","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $2 \\le h < k$, let $\\xi \\in \\mathbb{C}$ be a primitive $h$-th root of unity and let $\\varepsilon > 0$. Write $M_n$ for the set of monic polynomials of degree $n$ over $F$, $N_h$ for the set of $h$-full monic polynomials, $|P| = q^{\\deg P}$, and $\\omega_k(f)$ for the number of monic irreducible $P$ dividing $f$ whose multiplicity in $f$ is exactly $k$. For $j < h$ and\u2026","title":"Mean of $\\omega_k$ over $h$-full polynomials for $k > h$"},{"id":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/omega-mult-h-full-second-moment-asymptot","kind":"node","lean":"MetaMathlibExt.omegaMult_hFull_secondMoment_asymptotic","parent":13534,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second moment of "],["math","\\(\\omega_k\\)"],["text"," over "],["math","\\(h\\)"],["text","-full polynomials for "],["math","\\(k > h\\)"]],"source":"Second moment of $\\omega_k$ over $h$-full polynomials for $k > h$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po/index.html#omega-mult-h-full-second-moment-asymptot","scope":"NumberTheory/function-fields/prime-factors-of-given-multiplicity-in-h-free-po","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $2 \\le h < k$, let $\\xi \\in \\mathbb{C}$ be a primitive $h$-th root of unity and let $\\varepsilon > 0$. Write $M_n$ for the set of monic polynomials of degree $n$ over $F$, $N_h$ for the set of $h$-full monic polynomials, $|P| = q^{\\deg P}$, and $\\omega_k(f)$ for the number of monic irreducible $P$ dividing $f$ whose multiplicity in $f$ is exactly $k$. For $j < h$ and\u2026","title":"Second moment of $\\omega_k$ over $h$-full polynomials for $k > h$"},{"data_url":"scopes/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio.json","id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","kind":"scope","navigation":"module","parent":13392,"result":false,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"can_prove","title":"Twin primes and Chowla's conjecture over function fields"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/card-quot-span-eq-pow-nat-degree","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.cardQuot_span_eq_pow_natDegree","parent":13613,"reading_title":{"format":"inline-parts-v1","parts":[["text","Size of the quotient of "],["math","\\(F(T)\\)"],["text"," by a nonzero polynomial"]],"source":"Size of the quotient of $F(T)$ by a nonzero polynomial"},"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#card-quot-span-eq-pow-nat-degree","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"fully_proved","summary":"Let $F$ be a field and let $f \\in F[T]$ be a nonzero polynomial. Then the cardinality of the quotient ring $F[T]/(f)$ is $|F|^{\\deg f}$, where cardinalities are natural numbers and an infinite set has cardinality $0$.","title":"Size of the quotient of $F(T)$ by a nonzero polynomial"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/ring-norm","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.ringNorm","parent":13613,"reading_title":{"format":"inline-parts-v1","parts":[["text","The norm on "],["math","\\(F(T)\\)"]],"source":"The norm on $F(T)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#ring-norm","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"fully_proved","summary":"Let $F$ be a finite field. The ring norm on the polynomial ring $F[T]$ sends $f$ to the number of elements of the quotient ring $F[T]/(f)$, written $|f| = |F[T]/(f)|$; for $f = 0$ the quotient is infinite and $|0| = 0$. It is a ring norm: $|f| = 0$ only for $f = 0$, $|-f| = |f|$, $|f + g| \\le |f| + |g|$ and $|fg| \\le |f|\\,|g|$.","title":"The norm on $F(T)$"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/norm-eq-card-quot","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.norm_eq_cardQuot","parent":13613,"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#norm-eq-card-quot","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"fully_proved","summary":"Let $F$ be a finite field and give $F[T]$ its ring norm. For every $f \\in F[T]$, the norm $|f|$ equals the number of elements of the quotient ring $F[T]/(f)$, where an infinite quotient (the case $f = 0$) counts as $0$.","title":"The norm of a polynomial is the size of its quotient ring"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/norm-eq-pow-nat-degree","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.norm_eq_pow_natDegree","parent":13613,"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#norm-eq-pow-nat-degree","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"fully_proved","summary":"Let $F$ be a finite field and give $F[T]$ its ring norm. For every nonzero $f \\in F[T]$, the norm of $f$ is $|f| = |F|^{\\deg f}$, where $|F|$ is the number of elements of $F$.","title":"Norm of a nonzero polynomial"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/primes","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.primes","parent":13613,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes of "],["math","\\(F(T)\\)"]],"source":"Primes of $F(T)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#primes","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"fully_proved","summary":"Let $F$ be a finite field. The primes of the polynomial ring $F[T]$ are its monic irreducible polynomials.","title":"Primes of $F(T)$"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/indicator","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.indicator","parent":13613,"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#indicator","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"fully_proved","summary":"For a proposition $P$, the indicator $\\mathbf{1}_P$ is the real number $1$ if $P$ holds and $0$ otherwise.","title":"Indicator of a statement"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/q","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.\ud835\udd16_q","parent":13613,"reading_title":{"format":"inline-parts-v1","parts":[["text","Singular series for twin primes in "],["math","\\(F(T)\\)"]],"source":"Singular series for twin primes in $F(T)$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#q","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $h \\in F[T]$. The Hardy\u2013Littlewood constant of the prime pair $\\{f, f + h\\}$ is the infinite product over all primes $P$ of $F[T]$ (the monic irreducible polynomials)","title":"Singular series for twin primes in $F(T)$"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/theorem-1-1","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.theorem_1_1","parent":13613,"reading_title":{"format":"inline-parts-v1","parts":[["text","Twin prime asymptotic over "],["math","\\(\\mathbb{F}_q(T)\\)"],["text"," for large "],["math","\\(q\\)"]],"source":"Twin prime asymptotic over $\\mathbb{F}_q(T)$ for large $q$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#theorem-1-1","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"can_prove","summary":"Let $p$ be an odd prime and let $q$ be a power of $p$ with $q > 685090\\,p^2$; write $\\mathbb{F}_q$ for the finite field with $q$ elements. Then there is a real number $\\varepsilon > 0$ such that for every nonzero $h \\in \\mathbb{F}_q[T]$, as $X \\to \\infty$ through the powers of $q$,","title":"Twin prime asymptotic over $\\mathbb{F}_q(T)$ for large $q$"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/monic-bounded","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.monicBounded","parent":13613,"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#monic-bounded","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"fully_proved","summary":"Let $F$ be a finite field and let $X$ be a real number. This is the set of monic polynomials $f \\in F[T]$ whose norm $|f| = |F[T]/(f)|$, the number of elements of the quotient ring $F[T]/(f)$, satisfies $|f| \\le X$.","title":"Monic polynomials of bounded norm"},{"id":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/theorem-1-3","kind":"node","lean":"MathlibExt.NumberTheory.ChowlaAndTwinPrimeOverFqTWanted.ChowlaAndTwinPrimeOverFqT.theorem_1_3","parent":13613,"reading_title":{"format":"inline-parts-v1","parts":[["text","Chowla's conjecture over "],["math","\\(\\mathbb{F}_q(T)\\)"],["text"," for large "],["math","\\(q\\)"]],"source":"Chowla's conjecture over $\\mathbb{F}_q(T)$ for large $q$"},"result":true,"route":"../roadmap/NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio/index.html#theorem-1-3","scope":"NumberTheory/function-fields/twin-primes-and-chowla-s-conjecture-over-functio","status":"can_prove","summary":"Let $p$ be an odd prime, let $k \\ge 1$ be a natural number, and let $q$ be a power of $p$ with $q > p^2 k^2 e^2$, where $e$ is Euler's number; write $\\mathbb{F}_q$ for the finite field with $q$ elements. Let $h_1, \\dots, h_k \\in \\mathbb{F}_q[T]$ be distinct polynomials. Then, as the real number $X$ tends to $\\infty$,","title":"Chowla's conjecture over $\\mathbb{F}_q(T)$ for large $q$"},{"area":"Algebra & Number Theory","data_url":"chapters/AlgebraicGeometry.json","id":"AlgebraicGeometry","kind":"scope","parent":0,"result":false,"route":"../roadmap/AlgebraicGeometry/index.html","scope":"AlgebraicGeometry","status":"planned","summary":"Algebraic geometry studies spaces cut out by polynomial equations and translates between their geometric and algebraic invariants.","title":"Algebraic Geometry"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry.json","id":"AlgebraicGeometry/varieties-and-projective-geometry","kind":"scope","parent":13624,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry","status":"can_prove","summary":"Classical algebraic geometry of affine and projective varieties: hypersurfaces and their singularities, tangent spaces, plane configurations, intersection multiplicities, linear systems and birational maps.","title":"Varieties and projective geometry"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"can_prove","title":"Birational maps and the Cremona group"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/birational-map","kind":"node","lean":"AlgebraicGeometry.Scheme.BirationalMap","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#birational-map","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$ and $Y$ be irreducible schemes. A birational map from $X$ to $Y$ consists of dominant rational maps $f \\colon X \\dashrightarrow Y$ and $g \\colon Y \\dashrightarrow X$ that are mutually inverse: $g \\circ f$ is the identity rational map of $X$ and $f \\circ g$ is the identity rational map of $Y$.","title":"Birational map"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/refl","kind":"node","lean":"AlgebraicGeometry.Scheme.BirationalMap.refl","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#refl","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"For an irreducible scheme $X$, the identity birational map of $X$ has the identity rational map of $X$ as both its forward and its inverse map.","title":"Identity birational map"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/symm","kind":"node","lean":"AlgebraicGeometry.Scheme.BirationalMap.symm","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#symm","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$ and $Y$ be irreducible schemes. The inverse of a birational map from $X$ to $Y$ is the birational map from $Y$ to $X$ whose forward map is the inverse rational map of the original and whose inverse map is its forward rational map.","title":"Inverse birational map"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/trans","kind":"node","lean":"AlgebraicGeometry.Scheme.BirationalMap.trans","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#trans","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$, $Y$ and $Z$ be irreducible schemes, let $f$ be a birational map from $X$ to $Y$ and let $g$ be a birational map from $Y$ to $Z$. The composite of $f$ and $g$ is the birational map from $X$ to $Z$ whose forward rational map is the forward map of $f$ followed by the forward map of $g$, and whose inverse rational map is the inverse of $g$ followed by the inverse of $f$.","title":"Composition of birational maps"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/is-over","kind":"node","lean":"AlgebraicGeometry.Scheme.BirationalMap.IsOver","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#is-over","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$ and $Y$ be irreducible schemes over a scheme $S$. A birational map from $X$ to $Y$ is over $S$ when its forward rational map $X \\dashrightarrow Y$ is compatible with the structure maps to $S$.","title":"Birational maps over a base"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/birational-aut-over","kind":"node","lean":"AlgebraicGeometry.Scheme.birationalAutOver","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#birational-aut-over","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$ be an irreducible scheme over a scheme $S$. The birational automorphisms of $X$ over $S$ form the subgroup of the group of birational self-maps of $X$ consisting of those birational maps whose forward rational map $X \\dashrightarrow X$ is compatible with the structure maps to $S$.","title":"Birational automorphisms over a base"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/to-birational-map","kind":"node","lean":"AlgebraicGeometry.Scheme.PartialIso.toBirationalMap","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#to-birational-map","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$ and $Y$ be irreducible schemes and let $f$ be a partial isomorphism from $X$ to $Y$, that is, an isomorphism from a dense open subscheme of $X$ onto a dense open subscheme of $Y$. The birational map from $X$ to $Y$ given by $f$ has as forward map the rational map $X \\dashrightarrow Y$ represented by $f$, and as inverse the rational map $Y \\dashrightarrow X$ represented by the inverse of $f$.","title":"Birational map of a partial isomorphism"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/variety","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.Variety","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#variety","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $k$ be a field. A variety over $k$ in this sense is a scheme $X$ together with a morphism $X \\to \\operatorname{Spec} k$ such that $X$ is integral (irreducible and reduced) and the morphism is separated and of finite type (locally of finite type and quasi-compact).","title":"Integral variety over a field"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/is-pseudo-iso","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.IsPseudoIso","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#is-pseudo-iso","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$ and $Y$ be schemes. A partial isomorphism from $X$ to $Y$, that is, an isomorphism $U \\cong V$ between dense open subschemes $U \\subseteq X$ and $V \\subseteq Y$, is a pseudo-isomorphism if it is an isomorphism in codimension one: every point of $X$ outside $U$ and every point of $Y$ outside $V$ has coheight at least $2$ in the specialization order of points.","title":"Pseudo-isomorphism"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/pseudo-regularization","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.PseudoRegularization","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#pseudo-regularization","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$ be a variety over a field $k$ and let $\\varphi$ be a birational automorphism of $X$. A pseudo-regularization of $\\varphi$ consists of a variety $X'$ over $k$, a birational map $\\alpha \\colon X \\dashrightarrow X'$ over $k$, and a partial isomorphism $\\gamma$ from $X'$ to itself over $k$ that is a pseudo-isomorphism, such that $\\varphi = \\alpha^{-1} \\circ \\gamma \\circ \\alpha$, where $\\gamma$ is viewed as a\u2026","title":"Pseudo-regularization of a birational automorphism"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/is-pseudo-regularizable","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.IsPseudoRegularizable","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#is-pseudo-regularizable","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"Let $X$ be a variety over a field $k$. A birational automorphism $\\varphi$ of $X$ is pseudo-regularizable if there exist a variety $X'$ over $k$, a birational map $\\alpha \\colon X \\dashrightarrow X'$ over $k$, and a partial isomorphism $\\gamma$ from $X'$ to itself over $k$ that is a pseudo-isomorphism (an isomorphism in codimension one), such that $\\varphi = \\alpha^{-1} \\circ \\gamma \\circ \\alpha$, where $\\gamma$ is\u2026","title":"Pseudo-regularizable birational automorphism"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/cr","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.Cr","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#cr","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"For a field $k$ and $n \\in \\mathbb{N}$, the Cremona group $\\mathrm{Cr}_n(k)$ is the group of birational automorphisms of affine $n$-space $\\mathbb{A}^n_k$ over $\\operatorname{Spec} k$.","title":"Cremona group"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/gen-by-pseudo-regularizable","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.GenByPseudoRegularizable","parent":13626,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#gen-by-pseudo-regularizable","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"fully_proved","summary":"For a field $k$ and $n \\in \\mathbb{N}$, this is the proposition that the Cremona group $\\mathrm{Cr}_n(k)$ is generated, as a group, by its pseudo-regularizable elements.","title":"Cremona group generated by pseudo-regularizable elements"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/theorem-1-2-1-a","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.theorem_1_2_1_a","parent":13626,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{Cr}_3\\)"],["text"," of a number field is not generated by pseudo-regularizable elements"]],"source":"$\\mathrm{Cr}_3$ of a number field is not generated by pseudo-regularizable elements"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#theorem-1-2-1-a","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"can_prove","summary":"Let $k$ be a number field. The Cremona group $\\mathrm{Cr}_3(k)$, the group of birational automorphisms of affine $3$-space over $k$, is not generated by its pseudo-regularizable elements.","title":"$\\mathrm{Cr}_3$ of a number field is not generated by pseudo-regularizable elements"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/theorem-1-2-1-b","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.theorem_1_2_1_b","parent":13626,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{Cr}_3\\)"],["text"," of a function field over a number field is not generated by pseudo-regularizable elements"]],"source":"$\\mathrm{Cr}_3$ of a function field over a number field is not generated by pseudo-regularizable elements"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#theorem-1-2-1-b","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"can_prove","summary":"$\\mathrm{Cr}_3$ of a function field over a number field is not generated by pseudo-regularizable elements. Let $F$ be a number field and let $X$ be a variety over $F$ of positive dimension, with function field $F(X)$. The Cremona group $\\mathrm{Cr}_3(F(X))$, the group of birational automorphisms of affine $3$-space over $F(X)$, is not generated by its pseudo-regularizable elements.","title":"$\\mathrm{Cr}_3$ of a function field over a number field is not generated by pseudo-regularizable elements"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/theorem-1-2-1-c","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.theorem_1_2_1_c","parent":13626,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{Cr}_3\\)"],["text"," of a function field over a finite field is not generated by pseudo-regularizable elements"]],"source":"$\\mathrm{Cr}_3$ of a function field over a finite field is not generated by pseudo-regularizable elements"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#theorem-1-2-1-c","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"can_prove","summary":"$\\mathrm{Cr}_3$ of a function field over a finite field is not generated by pseudo-regularizable elements. Let $F$ be a finite field and let $X$ be a variety over $F$ of positive dimension, with function field $F(X)$. The Cremona group $\\mathrm{Cr}_3(F(X))$, the group of birational automorphisms of affine $3$-space over $F(X)$, is not generated by its pseudo-regularizable elements.","title":"$\\mathrm{Cr}_3$ of a function field over a finite field is not generated by pseudo-regularizable elements"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/theorem-1-2-1-d","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.theorem_1_2_1_d","parent":13626,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{Cr}_3\\)"],["text"," of a function field over an algebraically closed field is not generated by pseudo-regularizable elements"]],"source":"$\\mathrm{Cr}_3$ of a function field over an algebraically closed field is not generated by pseudo-regularizable elements"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#theorem-1-2-1-d","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"can_prove","summary":"$\\mathrm{Cr}_3$ of a function field over an algebraically closed field is not generated by pseudo-regularizable elements. Let $F$ be an algebraically closed field and let $X$ be a variety over $F$ of positive dimension, with function field $F(X)$. The Cremona group $\\mathrm{Cr}_3(F(X))$, the group of birational automorphisms of affine $3$-space over $F(X)$, is not generated by its pseudo-regularizable elements.","title":"$\\mathrm{Cr}_3$ of a function field over an algebraically closed field is not generated by pseudo-regularizable elements"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/theorem-1-2-2","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.theorem_1_2_2","parent":13626,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{Cr}_n\\)"],["text"," of a subfield of "],["math","\\(\\mathbb{C}\\)"],["text"," is not generated by pseudo-regularizable elements for "],["math","\\(n \\ge 4\\)"]],"source":"$\\mathrm{Cr}_n$ of a subfield of $\\mathbb{C}$ is not generated by pseudo-regularizable elements for $n \\ge 4$"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#theorem-1-2-2","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"can_prove","summary":"$\\mathrm{Cr}_n$ of a subfield of $\\mathbb{C}$ is not generated by pseudo-regularizable elements for $n \\ge 4$. Let $k$ be a subfield of $\\mathbb{C}$ and let $n \\ge 4$. The Cremona group $\\mathrm{Cr}_n(k)$, the group of birational automorphisms of affine $n$-space over $k$, is not generated by its pseudo-regularizable elements.","title":"$\\mathrm{Cr}_n$ of a subfield of $\\mathbb{C}$ is not generated by pseudo-regularizable elements for $n \\ge 4$"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/theorem-1-2-3","kind":"node","lean":"MathlibExt.AlgebraicGeometry.MotivicInvariantsWanted.MotivicInvariants.theorem_1_2_3","parent":13626,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{Cr}_n\\)"],["text"," of an infinite field is not generated by pseudo-regularizable elements for "],["math","\\(n \\ge 5\\)"]],"source":"$\\mathrm{Cr}_n$ of an infinite field is not generated by pseudo-regularizable elements for $n \\ge 5$"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group/index.html#theorem-1-2-3","scope":"AlgebraicGeometry/varieties-and-projective-geometry/birational-maps-and-the-cremona-group","status":"can_prove","summary":"$\\mathrm{Cr}_n$ of an infinite field is not generated by pseudo-regularizable elements for $n \\ge 5$. Let $k$ be an infinite field and let $n \\ge 5$. The Cremona group $\\mathrm{Cr}_n(k)$, the group of birational automorphisms of affine $n$-space over $k$, is not generated by its pseudo-regularizable elements.","title":"$\\mathrm{Cr}_n$ of an infinite field is not generated by pseudo-regularizable elements for $n \\ge 5$"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"can_prove","title":"B\u00e9zout's theorem"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/proj-space","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.ProjSpace","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#proj-space","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"fully_proved","summary":"For a division ring $K$ and $n \\in \\mathbb{N}$, the projective space $\\mathbb{P}^n(K)$ is the projectivization of $K^{n+1}$: the set of nonzero vectors of $K^{n+1}$ modulo multiplication by nonzero scalars, equivalently the lines through the origin.","title":"Projective space"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/vanishing-set","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.vanishingSet","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#vanishing-set","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"fully_proved","summary":"For a field $K$ and a polynomial $f \\in K[X_0, \\dots, X_n]$, the vanishing set of $f$ is the set of points $p \\in \\mathbb{P}^n(K)$ such that $f(v) = 0$, where $v \\in K^{n+1} \\setminus \\{0\\}$ is the chosen representative of $p$.","title":"Projective vanishing set"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/chart-index","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.chartIndex","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#chart-index","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"fully_proved","summary":"For a field $K$ and a point $p \\in \\mathbb{P}^n(K)$ with chosen representative $v = (v_0, \\dots, v_n) \\in K^{n+1} \\setminus \\{0\\}$, the chart index of $p$ is a fixed choice of an index $i \\in \\{0, \\dots, n\\}$ with $v_i \\neq 0$.","title":"Chart index of a projective point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/affine-cone-coord","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.affineConeCoord","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#affine-cone-coord","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"fully_proved","summary":"For a field $K$ and a point $p \\in \\mathbb{P}^n(K)$, let $v = (v_0, \\dots, v_n) \\in K^{n+1} \\setminus \\{0\\}$ be the chosen representative of $p$ and $i$ its chart index (a chosen index with $v_i \\neq 0$). The affine cone coordinates of $p$ are the vector $q \\in K^{n+1}$ with $q_j = v_j / v_i$ for every $j$; in particular $q_i = 1$.","title":"Affine coordinates of a projective point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/eval-at","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.evalAt","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#eval-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"fully_proved","summary":"For a field $K$ and a point $q = (q_0, \\dots, q_n) \\in K^{n+1}$, this is the ring homomorphism $K[X_0, \\dots, X_n] \\to K$ sending each polynomial $g$ to its value $g(q)$; it fixes $K$ and sends $X_j$ to $q_j$.","title":"Evaluation at an affine point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/max-ideal-at","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.maxIdealAt","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#max-ideal-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"fully_proved","summary":"For a field $K$ and a point $q \\in K^{n+1}$, this is the ideal of $K[X_0, \\dots, X_n]$ consisting of the polynomials $g$ with $g(q) = 0$, that is, the kernel of evaluation at $q$.","title":"Maximal ideal of an affine point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/local-ring-at","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.localRingAt","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#local-ring-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"fully_proved","summary":"For a field $K$ and a point $q \\in K^{n+1}$, this is the localization of $K[X_0, \\dots, X_n]$ at the maximal ideal of $q$ (the ideal of polynomials vanishing at $q$), that is, the local ring of affine space $\\mathbb{A}^{n+1}$ at $q$.","title":"Local ring of affine space at a point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/intersection-multiplicity","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.intersectionMultiplicity","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#intersection-multiplicity","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"fully_proved","summary":"Let $K$ be a field and $f_1, \\dots, f_n \\in K[X_0, \\dots, X_n]$. For a point $p \\in \\mathbb{P}^n(K)$, let $i$ be its chart index, $q \\in K^{n+1}$ its affine cone coordinates (so $q_i = 1$), and $\\mathcal{O}_q$ the localization of $K[X_0, \\dots, X_n]$ at the maximal ideal of $q$. The intersection multiplicity of $f_1, \\dots, f_n$ at $p$ is the length, as an $\\mathcal{O}_q$-module, of the quotient of $\\mathcal{O}_q$\u2026","title":"Intersection multiplicity at a projective point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/bezout-multiplicity","kind":"node","lean":"MathlibExt.AlgebraicGeometry.BezoutWanted.AlgebraicGeometry.bezout_multiplicity","parent":13646,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem/index.html#bezout-multiplicity","scope":"AlgebraicGeometry/varieties-and-projective-geometry/b-zout-s-theorem","status":"can_prove","summary":"Let $K$ be an algebraically closed field, let $n \\in \\mathbb{N}$, and let $f_1, \\dots, f_n \\in K[X_0, \\dots, X_n]$ and integers $d_1, \\dots, d_n \\ge 1$ be such that each $f_k$ is homogeneous of degree $d_k$ and has total degree exactly $d_k$. Suppose the common projective zero set $Z = V(f_1) \\cap \\dots \\cap V(f_n) \\subseteq \\mathbb{P}^n(K)$ is finite. Then $\\sum_{p \\in Z} i_p(f_1, \\dots, f_n) = d_1 d_2 \\cdots d_n$\u2026","title":"B\u00e9zout's theorem (with multiplicity)"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","title":"Cubic surfaces"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/cubic-surface","kind":"node","lean":"AlgebraicGeometry.CubicSurface","parent":13656,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#cubic-surface","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"A cubic surface over a field $k$ is a nonzero homogeneous polynomial $f \\in k[x_0, x_1, x_2, x_3]$ of degree $3$ in the four homogeneous coordinates of $\\mathbb{P}^3$, its defining equation. The surface it describes is the projective zero locus $V(f) \\subseteq \\mathbb{P}^3$.","title":"Cubic surface"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/total-degree-defining","kind":"node","lean":"AlgebraicGeometry.CubicSurface.totalDegree_defining","parent":13656,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#total-degree-defining","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"The defining cubic form of a cubic surface over a field $k$ has total degree exactly $3$.","title":"The defining cubic has total degree three"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/is-homogeneous-pderiv","kind":"node","lean":"AlgebraicGeometry.CubicSurface.isHomogeneous_pderiv","parent":13656,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#is-homogeneous-pderiv","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"Let $S$ be a cubic surface over a field $k$ with defining cubic form $f \\in k[x_0, x_1, x_2, x_3]$. For each $i \\in \\{0, 1, 2, 3\\}$, the partial derivative $\\partial f / \\partial x_i$ is homogeneous of degree $2$.","title":"The partial derivatives of a cubic form are quadratic"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/is-singular-point","kind":"node","lean":"AlgebraicGeometry.CubicSurface.IsSingularPoint","parent":13656,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#is-singular-point","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"Let $S$ be a cubic surface over a field $k$ with defining cubic form $f \\in k[x_0, x_1, x_2, x_3]$. A point $x \\in k^4$ is a singular point of the affine cone over $S$ when $f(x) = 0$ and $\\partial f / \\partial x_i$ vanishes at $x$ for every $i \\in \\{0, 1, 2, 3\\}$.","title":"Singular point of a cubic surface"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/is-smooth","kind":"node","lean":"AlgebraicGeometry.CubicSurface.IsSmooth","parent":13656,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#is-smooth","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"Let $k$ be an algebraically closed field and $S$ a cubic surface over $k$ with defining cubic form $f \\in k[x_0, x_1, x_2, x_3]$. The surface $S$ is smooth when every point $x \\in k^4$ at which $f$ and all its partial derivatives $\\partial f / \\partial x_i$ vanish is the origin, that is, when the origin is the only singular point of the affine cone over $S$.","title":"Smooth cubic surface"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/aeval-defining-eq-zero-of-pderiv","kind":"node","lean":"AlgebraicGeometry.CubicSurface.aeval_defining_eq_zero_of_pderiv","parent":13656,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing partial derivatives force the cubic to vanish when "],["math","\\(3 \\neq 0\\)"]],"source":"Vanishing partial derivatives force the cubic to vanish when $3 \\neq 0$"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#aeval-defining-eq-zero-of-pderiv","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"Let $S$ be a cubic surface over a field $k$ in which $3 \\neq 0$, and let $f \\in k[x_0, x_1, x_2, x_3]$ be its defining cubic form. If $x \\in k^4$ is a point at which every partial derivative $\\partial f / \\partial x_i$ vanishes, then $f(x) = 0$.","title":"Vanishing partial derivatives force the cubic to vanish when $3 \\neq 0$"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/is-singular-point-iff","kind":"node","lean":"AlgebraicGeometry.CubicSurface.isSingularPoint_iff","parent":13656,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#is-singular-point-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"Let $S$ be a cubic surface over a field $k$ in which $3 \\neq 0$, and let $f \\in k[x_0, x_1, x_2, x_3]$ be its defining cubic form. A point $x \\in k^4$ is a singular point of the affine cone over $S$ if and only if every partial derivative $\\partial f / \\partial x_i$ vanishes at $x$.","title":"Singular points via partial derivatives"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/fermat-cubic-surface","kind":"node","lean":"AlgebraicGeometry.fermatCubicSurface","parent":13656,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#fermat-cubic-surface","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"Over a field $k$, the Fermat cubic surface is the cubic surface in $\\mathbb{P}^3$ with defining cubic form $x_0^3 + x_1^3 + x_2^3 + x_3^3$.","title":"Fermat cubic surface"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/triple-plane","kind":"node","lean":"AlgebraicGeometry.triplePlane","parent":13656,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces/index.html#triple-plane","scope":"AlgebraicGeometry/varieties-and-projective-geometry/cubic-surfaces","status":"fully_proved","summary":"Over a field $k$, the triple plane is the cubic surface in $\\mathbb{P}^3$ with defining cubic form $x_0^3 \\in k[x_0, x_1, x_2, x_3]$, that is, the plane $x_0 = 0$ counted three times.","title":"Triple plane"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families","status":"can_prove","title":"Equimultiplicity of \u03bc-constant families"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families/milnor-number","kind":"node","lean":"MathlibExt.Geometry.SymplecticMonodromyWanted.SymplecticMonodromy.milnorNumber","parent":13666,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families/index.html#milnor-number","scope":"AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $f$ be a formal power series over $R$ in variables $z_i$ indexed by a set $\\sigma$. The Milnor number $\\mu(f) \\in \\mathbb{N} \\cup \\{\\infty\\}$ is the rank, as an $R$-module, of the quotient of this power series ring by the ideal generated by the partial derivatives $\\partial f / \\partial z_i$ for $i \\in \\sigma$, and $\\infty$ when that rank is infinite. For $R = \\mathbb{C}$ and\u2026","title":"Milnor number"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families/theorem-1-1","kind":"node","lean":"MathlibExt.Geometry.SymplecticMonodromyWanted.SymplecticMonodromy.theorem_1_1","parent":13666,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families/index.html#theorem-1-1","scope":"AlgebraicGeometry/varieties-and-projective-geometry/equimultiplicity-of-constant-families","status":"can_prove","summary":"Let $n \\in \\mathbb{N}$ and let $(f_t)_{t \\in [0,1]}$ be a family of formal power series in $\\mathbb{C}[[z_1, \\dots, z_n]]$ such that each coefficient of $f_t$ depends continuously on $t$ and each $f_t$ has zero constant term. If there is a natural number $\\mu$ such that the Milnor number of $f_t$ equals $\\mu$ for every $t$, then the order of $f_t$ (the least total degree of a monomial with nonzero coefficient,\u2026","title":"Equimultiplicity of \u03bc-constant families"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","status":"fully_proved","title":"Fields of definition of algebraic sets"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/ideal-over-k","kind":"node","lean":"MvPolynomial.idealOverK","parent":13669,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ideal over "],["math","\\(k\\)"],["text"," of a subset of affine space"]],"source":"Ideal over $k$ of a subset of affine space"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/index.html#ideal-over-k","scope":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field with algebraic closure $\\bar k$, and let $Z \\subseteq \\bar k^n$. The ideal $I_k(Z)$ of $k[X_1, \\dots, X_n]$ consists of the polynomials $f$ with coefficients in $k$ whose image in $\\bar k[X_1, \\dots, X_n]$ vanishes at every point of $Z$; that is, $I_k(Z)$ is the preimage of the vanishing ideal of $Z$ in $\\bar k[X_1, \\dots, X_n]$ under the coefficient map\u2026","title":"Ideal over $k$ of a subset of affine space"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/mem-ideal-over-k-iff","kind":"node","lean":"MvPolynomial.mem_idealOverK_iff","parent":13669,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in the ideal over "],["math","\\(k\\)"]],"source":"Membership in the ideal over $k$"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/index.html#mem-ideal-over-k-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field with algebraic closure $\\bar k$, let $Z \\subseteq \\bar k^n$, and let $f \\in k[X_1, \\dots, X_n]$. Then $f$ lies in the ideal $I_k(Z)$ over $k$ of $Z$ if and only if the image of $f$ in $\\bar k[X_1, \\dots, X_n]$ under the coefficient map lies in the vanishing ideal of $Z$, that is, vanishes at every point of $Z$.","title":"Membership in the ideal over $k$"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/is-defined-over","kind":"node","lean":"MvPolynomial.IsDefinedOver","parent":13669,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sets defined over "],["math","\\(k\\)"]],"source":"Sets defined over $k$"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/index.html#is-defined-over","scope":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field with algebraic closure $\\bar k$, let $Z \\subseteq \\bar k^n$, and let $I_k(Z)$ be the ideal of polynomials in $k[X_1, \\dots, X_n]$ whose image in $\\bar k[X_1, \\dots, X_n]$ vanishes at every point of $Z$. The set $Z$ is defined over $k$ if the vanishing ideal of $Z$ in $\\bar k[X_1, \\dots, X_n]$ equals the ideal generated by the image of $I_k(Z)$ under the coefficient map\u2026","title":"Sets defined over $k$"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/is-defined-over-iff","kind":"node","lean":"MvPolynomial.isDefinedOver_iff","parent":13669,"reading_title":{"format":"inline-parts-v1","parts":[["text","Defined over "],["math","\\(k\\)"],["text"," as an equality of ideals"]],"source":"Defined over $k$ as an equality of ideals"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/index.html#is-defined-over-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field with algebraic closure $\\bar k$ and let $Z \\subseteq \\bar k^n$. Then $Z$ is defined over $k$ if and only if the vanishing ideal of $Z$ in $\\bar k[X_1, \\dots, X_n]$ equals the ideal generated by the image of $I_k(Z)$ under the coefficient map $k[X_1, \\dots, X_n] \\to \\bar k[X_1, \\dots, X_n]$.","title":"Defined over $k$ as an equality of ideals"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/eval-of-mem-ideal-over-k","kind":"node","lean":"MvPolynomial.eval_of_mem_idealOverK","parent":13669,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomials in the ideal over "],["math","\\(k\\)"],["text"," vanish on the set"]],"source":"Polynomials in the ideal over $k$ vanish on the set"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/index.html#eval-of-mem-ideal-over-k","scope":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field with algebraic closure $\\bar k$, let $Z \\subseteq \\bar k^n$, and let $f \\in I_k(Z)$. Then for every point $P \\in Z$, the image of $f$ in $\\bar k[X_1, \\dots, X_n]$ evaluated at $P$ is $0$.","title":"Polynomials in the ideal over $k$ vanish on the set"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/ideal-over-k-empty","kind":"node","lean":"MvPolynomial.idealOverK_empty","parent":13669,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ideal over "],["math","\\(k\\)"],["text"," of the empty set"]],"source":"The ideal over $k$ of the empty set"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/index.html#ideal-over-k-empty","scope":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field with algebraic closure $\\bar k$. For every $n$, the ideal $I_k(\\emptyset)$ of the empty subset of $\\bar k^n$ is all of $k[X_1, \\dots, X_n]$.","title":"The ideal over $k$ of the empty set"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/ideal-over-k-anti-mono","kind":"node","lean":"MvPolynomial.idealOverK_anti_mono","parent":13669,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ideal over "],["math","\\(k\\)"],["text"," reverses inclusions"]],"source":"The ideal over $k$ reverses inclusions"},"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets/index.html#ideal-over-k-anti-mono","scope":"AlgebraicGeometry/varieties-and-projective-geometry/fields-of-definition-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field with algebraic closure $\\bar k$. If $Z_1 \\subseteq Z_2 \\subseteq \\bar k^n$, then $I_k(Z_2) \\subseteq I_k(Z_1)$.","title":"The ideal over $k$ reverses inclusions"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","title":"Line arrangements in the projective plane"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/eq-cross-iff","kind":"node","lean":"Projectivization.eq_cross_iff","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#eq-cross-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"Let $K$ be a field and let $v \\ne w$ be two distinct points of the projective plane $\\mathbb{P}(K^3)$. Call two points orthogonal when the dot product of representative vectors in $K^3$ vanishes, and let $v \\times w$ be the point spanned by the cross product of representatives of $v$ and $w$, which is nonzero since $v \\ne w$. Then a point $p$ equals $v \\times w$ if and only if $p$ is orthogonal to both $v$ and $w$.\u2026","title":"Two distinct lines meet in their cross product"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/line-arrangement","kind":"node","lean":"LineArrangement","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#line-arrangement","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"A line arrangement in the projective plane over a field $K$ is a finite set of at least two lines, each line recorded by its coordinate vector, a point of the dual projective plane $\\mathbb{P}(K^3)$.","title":"Line arrangement"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/multiplicity","kind":"node","lean":"LineArrangement.multiplicity","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#multiplicity","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"Let $A$ be a line arrangement in the projective plane over a field. The multiplicity $m_p$ of a point $p$ in $A$ is the number of lines of $A$ through $p$, that is, of lines of $A$ whose coordinate vector has zero dot product with a representative of $p$.","title":"Multiplicity of a point in a line arrangement"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/singular-points","kind":"node","lean":"LineArrangement.singularPoints","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#singular-points","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"The singular points of a line arrangement $A$ in the projective plane over a field form the finite set of intersection points of pairs of distinct lines of $A$, where two distinct lines with coordinate vectors $\\ell$ and $\\ell'$ meet in the point spanned by the cross product $\\ell \\times \\ell'$ of their representatives.","title":"Singular points of a line arrangement"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/mem-singular-points-iff-exists","kind":"node","lean":"LineArrangement.mem_singularPoints_iff_exists","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#mem-singular-points-iff-exists","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"Let $A$ be a line arrangement in the projective plane over a field. A point $p$ is a singular point of $A$ if and only if there are two distinct lines of $A$ that both pass through $p$.","title":"Singular points lie on two distinct lines"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/mem-singular-points-iff","kind":"node","lean":"LineArrangement.mem_singularPoints_iff","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#mem-singular-points-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"Let $A$ be a line arrangement in the projective plane over a field. A point $p$ is a singular point of $A$ if and only if its multiplicity $m_p$ in $A$, the number of lines of $A$ through $p$, satisfies $m_p \\ge 2$.","title":"Singular points are the points of multiplicity at least two"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/multiplicity-le-card","kind":"node","lean":"LineArrangement.multiplicity_le_card","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#multiplicity-le-card","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"Let $A$ be an arrangement of $d$ lines in the projective plane over a field. Every point $p$ has multiplicity $m_p \\le d$ in $A$.","title":"Multiplicity is at most the number of lines"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/singular-points-nonempty","kind":"node","lean":"LineArrangement.singularPoints_nonempty","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#singular-points-nonempty","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"Every line arrangement in the projective plane over a field has at least one singular point, since any two of its distinct lines meet.","title":"Every line arrangement has a singular point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/h-constant","kind":"node","lean":"LineArrangement.hConstant","parent":13677,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane/index.html#h-constant","scope":"AlgebraicGeometry/varieties-and-projective-geometry/line-arrangements-in-the-projective-plane","status":"fully_proved","summary":"Let $A$ be an arrangement of $d$ lines in the projective plane over a field, let $S$ be its set of singular points, with $s$ elements, and let $m_p$ be the multiplicity of a point $p$ in $A$. The Harbourne constant of $A$ is the real number","title":"Harbourne constant of a line arrangement"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","title":"Multiplicity of a hypersurface at a point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/mk-smul-eq","kind":"node","lean":"Projectivization.mk_smul_eq","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#mk-smul-eq","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $V$ be a module over a division ring $K$. For every nonzero scalar $a \\in K$ and every nonzero vector $v \\in V$, the points of the projectivization $\\mathbb{P}(V)$ represented by $a v$ and by $v$ are equal.","title":"Rescaling a representative does not change the projective point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/orthogonal-iff-rep-dot-product-rep","kind":"node","lean":"Projectivization.orthogonal_iff_rep_dotProduct_rep","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#orthogonal-iff-rep-dot-product-rep","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field and $m$ a finite set. Let $p, q$ be points of the projective space $\\mathbb{P}(K^m)$ and let $\\hat p, \\hat q \\in K^m$ be the fixed nonzero representatives chosen for $p$ and $q$. Then $p$ and $q$ are orthogonal (the dot product of a representative of $p$ with a representative of $q$ vanishes, which does not depend on the representatives) if and only if $\\hat p \\cdot \\hat q = 0$.","title":"Orthogonality of projective points via representatives"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/order-at","kind":"node","lean":"MvPolynomial.orderAt","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#order-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $F$ be a polynomial over $R$ in the variables $(X_i)_{i \\in \\sigma}$ and let $v \\in R^\\sigma$. The order of vanishing $\\operatorname{ord}_v F \\in \\mathbb{N} \\cup \\{\\infty\\}$ of $F$ at $v$ is the order at the origin of the Taylor expansion $F(v + X)$, the polynomial obtained from $F$ by substituting $X_i + v_i$ for each $X_i$: the least total degree of a monomial with nonzero\u2026","title":"Order of vanishing at a point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/constant-coeff-aeval-x-add-c","kind":"node","lean":"MvPolynomial.constantCoeff_aeval_X_add_C","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#constant-coeff-aeval-x-add-c","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be a commutative semiring, $F$ a polynomial over $R$ in the variables $(X_i)_{i \\in \\sigma}$, and $v \\in R^\\sigma$. The constant term of the Taylor expansion $F(X + v)$, obtained by substituting $X_i + v_i$ for each $X_i$, is the value $F(v)$.","title":"Constant term of the Taylor expansion"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/pderiv-aeval-x-add-c","kind":"node","lean":"MvPolynomial.pderiv_aeval_X_add_C","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#pderiv-aeval-x-add-c","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $F$ be a polynomial over $R$ in the variables $(X_j)_{j \\in \\sigma}$, let $v \\in R^\\sigma$ and let $i \\in \\sigma$. Writing $G(X + v)$ for the polynomial obtained from a polynomial $G$ by substituting $X_j + v_j$ for each $X_j$, we have $\\frac{\\partial}{\\partial X_i}\\bigl(F(X + v)\\bigr) = \\frac{\\partial F}{\\partial X_i}(X + v)$.","title":"Partial derivatives commute with Taylor expansion"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/one-le-order-at-iff","kind":"node","lean":"MvPolynomial.one_le_orderAt_iff","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#one-le-order-at-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $F$ be a polynomial over $R$ in the variables $(X_i)_{i \\in \\sigma}$ and let $v \\in R^\\sigma$. Then the order of vanishing $\\operatorname{ord}_v F$ of $F$ at $v$ is at least $1$ if and only if $F(v) = 0$.","title":"Vanishing to order at least one"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/order-at-eq-zero-iff","kind":"node","lean":"MvPolynomial.orderAt_eq_zero_iff","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#order-at-eq-zero-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $F$ be a polynomial over $R$ in the variables $(X_i)_{i \\in \\sigma}$ and let $v \\in R^\\sigma$. Then the order of vanishing $\\operatorname{ord}_v F$ of $F$ at $v$ is $0$ if and only if $F(v) \\neq 0$.","title":"Vanishing to order zero"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/order-at-c-of-ne-zero","kind":"node","lean":"MvPolynomial.orderAt_C_of_ne_zero","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#order-at-c-of-ne-zero","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be a commutative semiring and let $a \\in R$ be nonzero. For every point $v \\in R^\\sigma$, the constant polynomial $a$ in the variables $(X_i)_{i \\in \\sigma}$ has order of vanishing $\\operatorname{ord}_v a = 0$ at $v$.","title":"Nonzero constants vanish to order zero"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/order-at-eq-top-iff","kind":"node","lean":"MvPolynomial.orderAt_eq_top_iff","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#order-at-eq-top-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $F$ be a polynomial over $R$ in the variables $(X_i)_{i \\in \\sigma}$ and let $v \\in R^\\sigma$. Then the order of vanishing $\\operatorname{ord}_v F \\in \\mathbb{N} \\cup \\{\\infty\\}$ of $F$ at $v$ equals $\\infty$ if and only if $F = 0$.","title":"Only the zero polynomial vanishes to infinite order"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/nat-cast-add-one-le-order-at-iff","kind":"node","lean":"MvPolynomial.natCast_add_one_le_orderAt_iff","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#nat-cast-add-one-le-order-at-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be an integral domain of characteristic zero, $F$ a polynomial over $R$ in the variables $(X_i)_{i \\in \\sigma}$, $v \\in R^\\sigma$ and $n \\in \\mathbb{N}$. Then $F$ vanishes to order at least $n + 1$ at $v$ if and only if $F(v) = 0$ and every first partial derivative $\\partial F / \\partial X_i$ vanishes to order at least $n$ at $v$.","title":"Order of vanishing via first partial derivatives"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/two-le-order-at-iff","kind":"node","lean":"MvPolynomial.two_le_orderAt_iff","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#two-le-order-at-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $R$ be an integral domain of characteristic zero and let $F$ be a multivariate polynomial over $R$ in the variables $(X_i)_{i \\in \\sigma}$. For every point $v \\in R^\\sigma$, the polynomial $F$ vanishes to order at least $2$ at $v$ if and only if $F(v) = 0$ and $\\frac{\\partial F}{\\partial X_i}(v) = 0$ for every $i \\in \\sigma$.","title":"Vanishing to order at least two via first partial derivatives"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/order-at-smul","kind":"node","lean":"MvPolynomial.orderAt_smul","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#order-at-smul","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field and let $F$ be a polynomial over $K$ in the variables $(X_i)_{i \\in \\sigma}$ that is homogeneous of some degree $d$. For every nonzero $c \\in K$ and every point $v \\in K^\\sigma$, the orders of vanishing agree: $\\operatorname{ord}_{cv} F = \\operatorname{ord}_v F$.","title":"Scaling invariance of the order of vanishing"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/multiplicity-at","kind":"node","lean":"MvPolynomial.multiplicityAt","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#multiplicity-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field and $F$ a polynomial over $K$ in the variables $(X_i)_{i \\in \\sigma}$. The multiplicity of $F$ at a point $p$ of the projective space $\\mathbb{P}(K^\\sigma)$ is the order of vanishing of $F$ at the chosen representative $\\bar p \\in K^\\sigma$ of $p$, as a natural number, with the value $\\infty$ read as $0$.","title":"Multiplicity of a polynomial at a projective point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/multiplicity-at-mk","kind":"node","lean":"MvPolynomial.multiplicityAt_mk","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#multiplicity-at-mk","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field and $F$ a homogeneous polynomial over $K$ in the variables $(X_i)_{i \\in \\sigma}$. For every nonzero $v \\in K^\\sigma$, the multiplicity of $F$ at the point $[v]$ of $\\mathbb{P}(K^\\sigma)$ spanned by $v$ equals the order of vanishing of $F$ at $v$, with the value $\\infty$ read as $0$.","title":"The multiplicity can be computed at any representative"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/nat-cast-multiplicity-at","kind":"node","lean":"MvPolynomial.natCast_multiplicityAt","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#nat-cast-multiplicity-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field and $F \\neq 0$ a polynomial over $K$ in the variables $(X_i)_{i \\in \\sigma}$. For every point $p$ of $\\mathbb{P}(K^\\sigma)$, the multiplicity of $F$ at $p$, viewed in $\\mathbb{N} \\cup \\{\\infty\\}$, equals the order of vanishing of $F$ at the chosen representative $\\bar p \\in K^\\sigma$ of $p$.","title":"The multiplicity of a nonzero polynomial is its order of vanishing"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/multiplicity-at-c","kind":"node","lean":"MvPolynomial.multiplicityAt_C","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#multiplicity-at-c","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field. For every $a \\in K$, the constant polynomial $a$ in the variables $(X_i)_{i \\in \\sigma}$ has multiplicity $0$ at every point $p$ of the projective space $\\mathbb{P}(K^\\sigma)$. For $a = 0$ this is the junk value of the multiplicity of the zero polynomial.","title":"Constants have multiplicity zero"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/multiplicity-at-pos-iff","kind":"node","lean":"MvPolynomial.multiplicityAt_pos_iff","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#multiplicity-at-pos-iff","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field, $F \\neq 0$ a polynomial over $K$ in the variables $(X_i)_{i \\in \\sigma}$, and $p$ a point of $\\mathbb{P}(K^\\sigma)$ with chosen representative $\\bar p \\in K^\\sigma$. The multiplicity of $F$ at $p$ is positive if and only if $F(\\bar p) = 0$.","title":"Positive multiplicity means vanishing"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/order-at-sum-c-mul-x-of-dot-product-ne-z","kind":"node","lean":"MvPolynomial.orderAt_sum_C_mul_X_of_dotProduct_ne_zero","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#order-at-sum-c-mul-x-of-dot-product-ne-z","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field, let $\\sigma$ be a finite set and let $w, v \\in K^\\sigma$ with $w \\cdot v = \\sum_{i \\in \\sigma} w_i v_i \\neq 0$. Then the linear form $L = \\sum_{i \\in \\sigma} w_i X_i$ has order of vanishing $\\operatorname{ord}_v L = 0$ at $v$.","title":"A linear form vanishes to order zero off its hyperplane"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/order-at-sum-c-mul-x-of-dot-product-eq-z","kind":"node","lean":"MvPolynomial.orderAt_sum_C_mul_X_of_dotProduct_eq_zero","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#order-at-sum-c-mul-x-of-dot-product-eq-z","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field, let $\\sigma$ be a finite set and let $w, v \\in K^\\sigma$ with $w \\neq 0$ and $w \\cdot v = \\sum_{i \\in \\sigma} w_i v_i = 0$. Then the linear form $L = \\sum_{i \\in \\sigma} w_i X_i$ has order of vanishing $\\operatorname{ord}_v L = 1$ at $v$.","title":"A nonzero linear form vanishes to order one on its hyperplane"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/multiplicity-at-sum-c-mul-x-of-orthogona","kind":"node","lean":"MvPolynomial.multiplicityAt_sum_C_mul_X_of_orthogonal","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#multiplicity-at-sum-c-mul-x-of-orthogona","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field, $\\sigma$ a finite type and $w \\in K^\\sigma$ nonzero, and let $p$ be a point of $\\mathbb{P}(K^\\sigma)$ orthogonal to the point $[w]$ spanned by $w$, that is, $p$ lies on the hyperplane $\\sum_i w_i x_i = 0$. Then the linear form $\\sum_i w_i X_i$ has multiplicity $1$ at $p$.","title":"A hyperplane has multiplicity one at its points"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/multiplicity-at-sum-c-mul-x-of-not-ortho","kind":"node","lean":"MvPolynomial.multiplicityAt_sum_C_mul_X_of_not_orthogonal","parent":13687,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point/index.html#multiplicity-at-sum-c-mul-x-of-not-ortho","scope":"AlgebraicGeometry/varieties-and-projective-geometry/multiplicity-of-a-hypersurface-at-a-point","status":"fully_proved","summary":"Let $K$ be a field, $\\sigma$ a finite type and $w \\in K^\\sigma$ nonzero, and let $p$ be a point of $\\mathbb{P}(K^\\sigma)$ that is not orthogonal to the point $[w]$ spanned by $w$, that is, $p$ does not lie on the hyperplane $\\sum_i w_i x_i = 0$. Then the linear form $\\sum_i w_i X_i$ has multiplicity $0$ at $p$.","title":"A hyperplane has multiplicity zero off it"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","title":"Regular functions on algebraic sets"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/is-regular-at","kind":"node","lean":"FractionRing.IsRegularAt","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#is-regular-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $A$ and $K$ be commutative rings and $\\mathrm{ev} : A \\to K$ a ring homomorphism. An element $r$ of the total ring of fractions of $A$ (its localization at the non-zero-divisors) is regular at $\\mathrm{ev}$ when there is some $g \\in A$ with $\\mathrm{ev}(g) \\neq 0$ such that $g r$ lies in the image of $A$ in its total ring of fractions.","title":"Regularity of a fraction at an evaluation"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/is-regular-at-algebra-map","kind":"node","lean":"FractionRing.isRegularAt_algebraMap","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#is-regular-at-algebra-map","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $A$ be a commutative ring and $K$ a nontrivial commutative ring. For every ring homomorphism $\\mathrm{ev} : A \\to K$ and every $a \\in A$, the image of $a$ in the total ring of fractions of $A$ is regular at $\\mathrm{ev}$.","title":"Elements of the base ring are regular"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/is-regular-at-zero","kind":"node","lean":"FractionRing.isRegularAt_zero","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#is-regular-at-zero","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $A$ be a commutative ring and $K$ a nontrivial commutative ring. For every ring homomorphism $\\mathrm{ev} : A \\to K$, the element $0$ of the total ring of fractions of $A$ is regular at $\\mathrm{ev}$.","title":"Zero is regular"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/is-regular-at-one","kind":"node","lean":"FractionRing.isRegularAt_one","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#is-regular-at-one","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $A$ be a commutative ring and $K$ a nontrivial commutative ring. For every ring homomorphism $\\mathrm{ev} : A \\to K$, the element $1$ of the total ring of fractions of $A$ is regular at $\\mathrm{ev}$.","title":"One is regular"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/affine-coordinate-ring","kind":"node","lean":"MvPolynomial.AffineCoordinateRing","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#affine-coordinate-ring","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $K$ be a field, $\\sigma$ an index set and $X \\subseteq K^\\sigma$ a set of points. The affine coordinate ring of $X$ is the quotient $K[x_i : i \\in \\sigma] / I(X)$ of the polynomial ring over $K$ in variables indexed by $\\sigma$ by the vanishing ideal $I(X)$, the ideal of all polynomials that vanish at every point of $X$.","title":"Affine coordinate ring"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/eval-at","kind":"node","lean":"MvPolynomial.AffineCoordinateRing.evalAt","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#eval-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $K$ be a field, $\\sigma$ an index set and $X \\subseteq K^\\sigma$ a set of points. For a point $P \\in X$, evaluation at $P$ is the ring homomorphism from the affine coordinate ring of $X$ to $K$ that sends the residue class of a polynomial $f$ to $f(P)$; it is well defined because every polynomial in the vanishing ideal of $X$ vanishes at $P$.","title":"Evaluation at a point of an affine set"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/eval-at-mk","kind":"node","lean":"MvPolynomial.AffineCoordinateRing.evalAt_mk","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#eval-at-mk","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $K$ be a field, $\\sigma$ an index set, $X \\subseteq K^\\sigma$ a set of points and $P \\in X$. For every polynomial $f$ over $K$ in variables indexed by $\\sigma$, evaluating the residue class of $f$ in the affine coordinate ring of $X$ at $P$ gives $f(P)$.","title":"Evaluating a residue class"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/is-regular-at-2","kind":"node","lean":"MvPolynomial.AffineCoordinateRing.IsRegularAt","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#is-regular-at-2","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $K$ be a field, $\\sigma$ an index set and $X \\subseteq K^\\sigma$ a set of points whose affine coordinate ring $A$ is an integral domain. A rational function on $X$ is an element $r$ of the field of fractions of $A$; it is regular at a point $P \\in X$ when there is some $g \\in A$ whose value $g(P)$ under evaluation at $P$ is nonzero such that $g r$ lies in the image of $A$ in its field of fractions.","title":"Regularity of a rational function at a point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/is-regular-at-algebra-map-2","kind":"node","lean":"MvPolynomial.AffineCoordinateRing.isRegularAt_algebraMap","parent":13709,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets/index.html#is-regular-at-algebra-map-2","scope":"AlgebraicGeometry/varieties-and-projective-geometry/regular-functions-on-algebraic-sets","status":"fully_proved","summary":"Let $K$ be a field, $\\sigma$ an index set and $X \\subseteq K^\\sigma$ a set of points whose affine coordinate ring $A$ is an integral domain. For every $a \\in A$ and every point $P \\in X$, the image of $a$ in the field of fractions of $A$ is a rational function on $X$ that is regular at $P$.","title":"Elements of the coordinate ring are regular"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"can_prove","title":"Seshadri constants of points in the plane"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/plane-polynomial","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.PlanePolynomial","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#plane-polynomial","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"A plane polynomial is a polynomial $f \\in \\mathbb{C}[X_0, X_1]$ in two variables $X_0, X_1$ with complex coefficients.","title":"Polynomials in two complex variables"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/affine-point","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.AffinePoint","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#affine-point","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"A point of the complex affine plane is a pair $a = (a_0, a_1)$ of complex numbers, that is, an element of $\\mathbb{C}^2$, with coordinates indexed by $0$ and $1$.","title":"Points of the complex affine plane"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/shifted","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.shifted","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#shifted","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"For a polynomial $f \\in \\mathbb{C}[x_1, x_2]$ and a point $a = (a_1, a_2) \\in \\mathbb{C}^2$, the translate of $f$ by $a$ is the polynomial $f(x_1 + a_1, x_2 + a_2)$, obtained by substituting $x_i + a_i$ for $x_i$. It moves the point $a$ to the origin.","title":"Translate of a plane polynomial"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/monomial-degree","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.monomialDegree","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#monomial-degree","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"The total degree of the exponent vector $m = (m_0, m_1)$ of a monomial $X_0^{m_0} X_1^{m_1}$ in two variables is $m_0 + m_1$.","title":"Total degree of a monomial"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/multiplicity","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.multiplicity","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#multiplicity","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"For a polynomial $f \\in \\mathbb{C}[X_0, X_1]$ and a point $a = (a_0, a_1) \\in \\mathbb{C}^2$, the multiplicity of $f$ at $a$ is the least total degree of a monomial occurring with nonzero coefficient in the translated polynomial $f(X_0 + a_0, X_1 + a_1)$. For the zero polynomial it is $0$.","title":"Multiplicity of a plane polynomial at a point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/multiplicity-x-sub-c","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.multiplicity_X_sub_C","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#multiplicity-x-sub-c","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"For every point $a = (a_0, a_1) \\in \\mathbb{C}^2$, the polynomial $X_0 - a_0 \\in \\mathbb{C}[X_0, X_1]$, which defines the line through $a$ on which the first coordinate is constant, has multiplicity $1$ at $a$.","title":"The first-coordinate line through a point has multiplicity one there"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/seshadri-ratios","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.seshadriRatios","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#seshadri-ratios","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"For a configuration $x = (x_0, \\dots, x_9)$ of ten points of $\\mathbb{C}^2$, not necessarily distinct, the Seshadri ratios of $x$ are the real numbers $\\deg f / \\sum_{i=0}^{9} \\operatorname{mult}_{x_i} f$, where $f$ ranges over the nonzero polynomials in $\\mathbb{C}[X_0, X_1]$ with $\\sum_{i=0}^{9} \\operatorname{mult}_{x_i} f > 0$, $\\deg f$ is the total degree of $f$, and $\\operatorname{mult}_{x_i} f$ is the\u2026","title":"Seshadri ratios of ten points"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/affine-seshadri-constant","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.affineSeshadriConstant","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#affine-seshadri-constant","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"For a configuration $x = (x_0, \\dots, x_9)$ of ten points of $\\mathbb{C}^2$, not necessarily distinct, the affine Seshadri constant of $x$ is the infimum of the ratios $\\deg f / \\sum_{i=0}^{9} \\operatorname{mult}_{x_i} f$ over all nonzero polynomials $f \\in \\mathbb{C}[X_0, X_1]$ with $\\sum_{i=0}^{9} \\operatorname{mult}_{x_i} f > 0$, where $\\deg f$ is the total degree of $f$ and $\\operatorname{mult}_{x_i} f$ is the\u2026","title":"Seshadri constant of ten affine points"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/affine-seshadri-constant-le-one","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.affineSeshadriConstant_le_one","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#affine-seshadri-constant-le-one","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"For every configuration $x = (x_0, \\dots, x_9)$ of ten points of $\\mathbb{C}^2$, not necessarily distinct, the affine Seshadri constant of $x$ is at most $1$.","title":"The affine Seshadri constant is at most one"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/generic-seshadri-constant","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.genericSeshadriConstant","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#generic-seshadri-constant","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"The generic ten-point Seshadri constant is the supremum, over all configurations $x = (x_0, \\dots, x_9)$ of ten pairwise distinct points of $\\mathbb{C}^2$, of the affine Seshadri constant of $x$.","title":"Generic ten-point Seshadri constant"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/seshadri-constant10","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstant.seshadriConstant10","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#seshadri-constant10","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"The ten-point Seshadri constant is the generic ten-point Seshadri constant: the supremum, over all configurations of ten pairwise distinct points of $\\mathbb{C}^2$, of their affine Seshadri constants.","title":"Ten-point Seshadri constant"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/nagata-conjecture10","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstantWanted.nagataConjecture10","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#nagata-conjecture10","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"fully_proved","summary":"The proposition that the generic ten-point Seshadri constant of $\\mathbb{P}^2$, in the affine-chart model, equals $1/\\sqrt{10}$.","title":"Nagata's conjecture for ten points"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/eckl2008-lower-bound-wanted","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstantWanted.eckl2008LowerBound_wanted","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#eckl2008-lower-bound-wanted","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"can_prove","summary":"The generic ten-point Seshadri constant of $\\mathbb{P}^2$, in the affine-chart model, is at least $4/13$.","title":"Eckl's 2008 lower bound for the ten-point Seshadri constant"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/harbourne-roe2009-lower-bound-wanted","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstantWanted.harbourneRoe2009LowerBound_wanted","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#harbourne-roe2009-lower-bound-wanted","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"can_prove","summary":"The generic ten-point Seshadri constant of $\\mathbb{P}^2$, in the affine-chart model, is at least $177/560$.","title":"Harbourne\u2013Ro\u00e9 lower bound for the ten-point Seshadri constant"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/ciliberto-miranda2011-lower-bound-wanted","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstantWanted.cilibertoMiranda2011LowerBound_wanted","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#ciliberto-miranda2011-lower-bound-wanted","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"can_prove","summary":"The generic ten-point Seshadri constant of $\\mathbb{P}^2$, in the affine-chart model, is at least $55/174$.","title":"Ciliberto\u2013Miranda lower bound for the ten-point Seshadri constant"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/eckl2011-lower-bound-wanted","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstantWanted.eckl2011LowerBound_wanted","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#eckl2011-lower-bound-wanted","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"can_prove","summary":"The generic ten-point Seshadri constant of $\\mathbb{P}^2$, in the affine-chart model, is at least $117/370$.","title":"Eckl's 2011 lower bound for the ten-point Seshadri constant"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/submaximal-upper-bound-wanted","kind":"node","lean":"MathlibExt.AlgebraicGeometry.SeshadriConstantWanted.submaximalUpperBound_wanted","parent":13719,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane/index.html#submaximal-upper-bound-wanted","scope":"AlgebraicGeometry/varieties-and-projective-geometry/seshadri-constants-of-points-in-the-plane","status":"can_prove","summary":"The generic ten-point Seshadri constant of $\\mathbb{P}^2$, in the affine-chart model, is at most $1/\\sqrt{10}$.","title":"Submaximal upper bound for the ten-point Seshadri constant"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","status":"fully_proved","title":"Tangent spaces of algebraic sets"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/total-derivative-at","kind":"node","lean":"MvPolynomial.totalDerivativeAt","parent":13737,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/index.html#total-derivative-at","scope":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","status":"fully_proved","summary":"For a field $k$, a polynomial $f \\in k[x_1, \\dots, x_n]$ and $P, v \\in k^n$, the total derivative of $f$ at $P$ along the translated direction $v$ is $\\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i}(P)\\, v_i \\in k$, where $\\frac{\\partial f}{\\partial x_i}$ is the formal partial derivative.","title":"Total derivative at a point"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/total-derivative-at-lin","kind":"node","lean":"MvPolynomial.totalDerivativeAtLin","parent":13737,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/index.html#total-derivative-at-lin","scope":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","status":"fully_proved","summary":"For a field $k$, a polynomial $f \\in k[x_1, \\dots, x_n]$ and a point $P \\in k^n$, this is the $k$-linear map $k^n \\to k$ sending $v$ to the total derivative of $f$ at $P$ along $v$, namely $\\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i}(P)\\, v_i$.","title":"Total derivative as a linear map"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/tangent-space-poly","kind":"node","lean":"MvPolynomial.tangentSpacePoly","parent":13737,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/index.html#tangent-space-poly","scope":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","status":"fully_proved","summary":"For a field $k$, a polynomial $f \\in k[x_1, \\dots, x_n]$ and a point $P \\in k^n$, the tangent space of the hypersurface $f = 0$ at $P$ is the kernel of the total derivative of $f$ at $P$, viewed as a $k$-linear map $k^n \\to k$: the subspace of vectors $v \\in k^n$ with $\\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i}(P)\\, v_i = 0$. The point $P$ is not required to satisfy $f(P) = 0$.","title":"Tangent space of a hypersurface"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/tangent-space-ideal","kind":"node","lean":"MvPolynomial.tangentSpaceIdeal","parent":13737,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/index.html#tangent-space-ideal","scope":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","status":"fully_proved","summary":"For a field $k$, an ideal $I$ of $k[x_1, \\dots, x_n]$ and a point $P \\in k^n$, the tangent space of $I$ at $P$ is the $k$-linear subspace of $k^n$ obtained as the intersection, over all $f \\in I$, of the tangent spaces at $P$ of the hypersurfaces $f = 0$. The point $P$ is not required to be a common zero of $I$.","title":"Tangent space of an ideal"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/total-derivative-at-lin-apply","kind":"node","lean":"MvPolynomial.totalDerivativeAtLin_apply","parent":13737,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/index.html#total-derivative-at-lin-apply","scope":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","status":"fully_proved","summary":"For a field $k$, a polynomial $f \\in k[x_1, \\dots, x_n]$ and $P, v \\in k^n$, the total derivative of $f$ at $P$, viewed as a $k$-linear map $k^n \\to k$, sends $v$ to the total derivative of $f$ at $P$ along $v$.","title":"Evaluating the total derivative linear map"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/mem-tangent-space-poly","kind":"node","lean":"MvPolynomial.mem_tangentSpacePoly","parent":13737,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/index.html#mem-tangent-space-poly","scope":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field, $f \\in k[x_1, \\dots, x_n]$, and $P, v \\in k^n$. Then $v$ lies in the tangent space of the hypersurface $f = 0$ at $P$ if and only if the total derivative of $f$ at $P$ along $v$ is zero.","title":"Membership in the tangent space of a hypersurface"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/mem-tangent-space-ideal","kind":"node","lean":"MvPolynomial.mem_tangentSpaceIdeal","parent":13737,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets/index.html#mem-tangent-space-ideal","scope":"AlgebraicGeometry/varieties-and-projective-geometry/tangent-spaces-of-algebraic-sets","status":"fully_proved","summary":"Let $k$ be a field, $I$ an ideal of $k[x_1, \\dots, x_n]$, and $P, v \\in k^n$. Then $v$ lies in the tangent space of $I$ at $P$ if and only if, for every $f \\in I$, the total derivative of $f$ at $P$ along $v$ is zero.","title":"Membership in the tangent space of an ideal"},{"data_url":"scopes/AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties.json","id":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties","kind":"scope","navigation":"module","parent":13625,"result":false,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/index.html","scope":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties","status":"can_prove","title":"Topological reconstruction of varieties"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/proj-to-spec","kind":"node","lean":"MathlibExt.AlgebraicGeometry.TopologicalReconstructionWanted.AlgebraicGeometry.TopologicalReconstruction.projToSpec","parent":13745,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/index.html#proj-to-spec","scope":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties","status":"fully_proved","summary":"Let $R$ be a commutative ring and $A$ a commutative $R$-algebra graded by $R$-submodules $\\mathcal{A}_0, \\mathcal{A}_1, \\dots$. The structure morphism $\\operatorname{Proj} \\mathcal{A} \\to \\operatorname{Spec} R$ is the canonical morphism $\\operatorname{Proj} \\mathcal{A} \\to \\operatorname{Spec} \\mathcal{A}_0$ followed by the morphism $\\operatorname{Spec} \\mathcal{A}_0 \\to \\operatorname{Spec} R$ induced by the algebra\u2026","title":"Structure morphism of a projective spectrum"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/is-projective-hom","kind":"node","lean":"MathlibExt.AlgebraicGeometry.TopologicalReconstructionWanted.AlgebraicGeometry.TopologicalReconstruction.IsProjectiveHom","parent":13745,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/index.html#is-projective-hom","scope":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties","status":"fully_proved","summary":"Let $R$ be a commutative ring and $X$ a scheme. A morphism $f : X \\to \\operatorname{Spec} R$ is projective if there are $n \\in \\mathbb{N}$ and a closed immersion $i$ of $X$ into $\\operatorname{Proj} R[x_1, \\dots, x_n]$, the polynomial ring graded by total degree, such that $f$ is $i$ followed by the structure morphism $\\operatorname{Proj} R[x_1, \\dots, x_n] \\to \\operatorname{Spec} R$.","title":"Projective morphism to an affine scheme"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/is-normal-scheme","kind":"node","lean":"MathlibExt.AlgebraicGeometry.TopologicalReconstructionWanted.AlgebraicGeometry.TopologicalReconstruction.IsNormalScheme","parent":13745,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/index.html#is-normal-scheme","scope":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties","status":"fully_proved","summary":"A scheme $X$ is normal if for every point $x \\in X$ the local ring $\\mathcal{O}_{X,x}$ is an integrally closed domain.","title":"Normal scheme"},{"id":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/kollar-lieblich-olsson-sawin","kind":"node","lean":"MathlibExt.AlgebraicGeometry.TopologicalReconstructionWanted.AlgebraicGeometry.TopologicalReconstruction.kollar_lieblich_olsson_sawin","parent":13745,"result":true,"route":"../roadmap/AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties/index.html#kollar-lieblich-olsson-sawin","scope":"AlgebraicGeometry/varieties-and-projective-geometry/topological-reconstruction-of-varieties","status":"can_prove","summary":"Let $K$ and $L$ be fields of characteristic zero, and let $X$ and $Y$ be irreducible normal schemes with projective morphisms $f : X \\to \\operatorname{Spec} K$ and $g : Y \\to \\operatorname{Spec} L$. Suppose that the Krull dimension of the underlying topological space of $X$ is at least $4$; or it is at least $3$ and $K$ and $L$ are essentially of finite type over $\\mathbb{Q}$; or it is at least $2$ and $K$ and $L$\u2026","title":"Koll\u00e1r\u2013Lieblich\u2013Olsson\u2013Sawin topological reconstruction theorem"},{"data_url":"scopes/AlgebraicGeometry/elliptic-curves.json","id":"AlgebraicGeometry/elliptic-curves","kind":"scope","parent":13624,"result":false,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/index.html","scope":"AlgebraicGeometry/elliptic-curves","status":"can_prove","summary":"Elliptic curves given by Weierstrass equations, their points over finite fields and number fields, torsion, reduction and Galois representations.","title":"Elliptic curves"},{"id":"AlgebraicGeometry/elliptic-curves/good-reduction-torsion-unramified","kind":"node","lean":"MetaMathlibExt.good_reduction_torsion_unramified","parent":13750,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/index.html#good-reduction-torsion-unramified","scope":"AlgebraicGeometry/elliptic-curves","status":"fully_proved","summary":"Let $R$ be a discrete valuation ring with fraction field $k$, and let $E$ be an elliptic curve over $k$ given by a Weierstrass equation with good reduction at $R$. Let $n$ be a natural number whose image in the residue field of $R$ is nonzero, let $k^{\\mathrm{sep}}$ be a separable closure of $k$, and let $\\mathcal{O}$ be a valuation subring of $k^{\\mathrm{sep}}$ whose intersection with $k$ is $R$. Then every\u2026","title":"N\u00e9ron\u2013Ogg\u2013Shafarevich criterion, easy direction"},{"data_url":"scopes/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points.json","id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","kind":"scope","navigation":"module","parent":13750,"result":false,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","title":"Galois action on torsion points"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-alg-equiv","kind":"node","lean":"MetaMathlibExt.galAlgEquiv","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Galois elements as automorphisms of "],["math","\\(\\overline{\\mathbb{Q}}\\)"]],"source":"Galois elements as automorphisms of $\\overline{\\mathbb{Q}}$"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-alg-equiv","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Each element $\\sigma$ of the absolute Galois group $G_{\\mathbb{Q}}$ of $\\mathbb{Q}$ is regarded as the $\\mathbb{Q}$-algebra automorphism $\\sigma$ of the algebraic closure $\\overline{\\mathbb{Q}}$ of $\\mathbb{Q}$. Since $G_{\\mathbb{Q}}$ is by definition the group of $\\mathbb{Q}$-algebra automorphisms of $\\overline{\\mathbb{Q}}$, this map is the identity.","title":"Galois elements as automorphisms of $\\overline{\\mathbb{Q}}$"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-point-hom","kind":"node","lean":"MetaMathlibExt.galPointHom","parent":13752,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-point-hom","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$ and let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity. For $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$, this is the homomorphism of additive groups $W(\\overline{\\mathbb{Q}}) \\to W(\\overline{\\mathbb{Q}})$ that sends the point\u2026","title":"Galois action on the points of a Weierstrass curve"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-alg-equiv-mul","kind":"node","lean":"MetaMathlibExt.galAlgEquiv_mul","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Galois elements as automorphisms of "],["math","\\(\\overline{\\mathbb{Q}}\\)"],["text"," respect products"]],"source":"Galois elements as automorphisms of $\\overline{\\mathbb{Q}}$ respect products"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-alg-equiv-mul","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"For all $\\sigma, \\tau$ in the absolute Galois group $G_{\\mathbb{Q}}$ of $\\mathbb{Q}$, the $\\mathbb{Q}$-algebra automorphism of the algebraic closure $\\overline{\\mathbb{Q}}$ attached to $\\sigma\\tau$ is the composite of the automorphisms attached to $\\sigma$ and to $\\tau$, with the one attached to $\\tau$ applied first.","title":"Galois elements as automorphisms of $\\overline{\\mathbb{Q}}$ respect products"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-alg-equiv-inv","kind":"node","lean":"MetaMathlibExt.galAlgEquiv_inv","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Galois elements as automorphisms of "],["math","\\(\\overline{\\mathbb{Q}}\\)"],["text"," respect inverses"]],"source":"Galois elements as automorphisms of $\\overline{\\mathbb{Q}}$ respect inverses"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-alg-equiv-inv","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"For every $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ of $\\mathbb{Q}$, the $\\mathbb{Q}$-algebra automorphism of the algebraic closure $\\overline{\\mathbb{Q}}$ attached to $\\sigma^{-1}$ is the inverse of the automorphism attached to $\\sigma$.","title":"Galois elements as automorphisms of $\\overline{\\mathbb{Q}}$ respect inverses"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-to-alg-hom-comp","kind":"node","lean":"MetaMathlibExt.galToAlgHom_comp","parent":13752,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-to-alg-hom-comp","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $\\overline{\\mathbb{Q}}$ be the algebraic closure of $\\mathbb{Q}$. For all $\\sigma, \\tau$ in the absolute Galois group $G_{\\mathbb{Q}}$ of $\\mathbb{Q}$, the $\\mathbb{Q}$-algebra homomorphism $\\overline{\\mathbb{Q}} \\to \\overline{\\mathbb{Q}}$ underlying the product $\\sigma\\tau$ is the composite $\\sigma \\circ \\tau$ of the $\\mathbb{Q}$-algebra homomorphisms underlying $\\sigma$ and $\\tau$.","title":"Galois automorphisms compose as algebra maps"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-point-hom-apply-one","kind":"node","lean":"MetaMathlibExt.galPointHom_apply_one","parent":13752,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-point-hom-apply-one","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$ and let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity. For $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ and $P \\in W(\\overline{\\mathbb{Q}})$, let $\\sigma \\cdot P$ be the point obtained by applying $\\sigma$ to the coordinates of $P$\u2026","title":"The identity acts trivially on points"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-point-hom-apply-mul","kind":"node","lean":"MetaMathlibExt.galPointHom_apply_mul","parent":13752,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-point-hom-apply-mul","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$ and let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity. For $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ and $P \\in W(\\overline{\\mathbb{Q}})$, let $\\sigma \\cdot P$ be the point obtained by applying $\\sigma$ to the coordinates of $P$\u2026","title":"The Galois action on points is compatible with multiplication"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/torsion-pts","kind":"node","lean":"MetaMathlibExt.torsionPts","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p\\)"],["text","-torsion subgroup"]],"source":"The $p$-torsion subgroup"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#torsion-pts","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, let $p$ be a natural number, and let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity. The $p$-torsion subgroup $W[p]$ is the subgroup of points $P \\in W(\\overline{\\mathbb{Q}})$ with $pP = 0$.","title":"The $p$-torsion subgroup"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/mem-torsion-pts-iff","kind":"node","lean":"MetaMathlibExt.mem_torsionPts_iff","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in the "],["math","\\(p\\)"],["text","-torsion subgroup"]],"source":"Membership in the $p$-torsion subgroup"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#mem-torsion-pts-iff","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, let $p$ be a natural number, let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and let $W[p]$ be its $p$-torsion subgroup. A point $x \\in W(\\overline{\\mathbb{Q}})$ lies in $W[p]$ if and only if $p x = 0$, where $p x$ is the sum of $p$ copies\u2026","title":"Membership in the $p$-torsion subgroup"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-point-hom-mem","kind":"node","lean":"MetaMathlibExt.galPointHom_mem","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Galois action preserves "],["math","\\(p\\)"],["text","-torsion points"]],"source":"The Galois action preserves $p$-torsion points"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-point-hom-mem","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, let $p$ be a natural number, let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and let $W[p]$ be its subgroup of points $P$ with $pP = 0$. For $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ and $P \\in W(\\overline{\\mathbb{Q}})$, let\u2026","title":"The Galois action preserves $p$-torsion points"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-torsion-hom","kind":"node","lean":"MetaMathlibExt.galTorsionHom","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Galois action on "],["math","\\(p\\)"],["text","-torsion points"]],"source":"Galois action on $p$-torsion points"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-torsion-hom","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, let $p$ be a natural number, let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and let $W[p]$ be its subgroup of points $P$ with $pP = 0$. For $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$, this is the homomorphism of additive\u2026","title":"Galois action on $p$-torsion points"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-linear-map","kind":"node","lean":"MetaMathlibExt.galLinearMap","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear Galois action on "],["math","\\(W(p)\\)"]],"source":"Linear Galois action on $W(p)$"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-linear-map","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, $p$ a natural number, $W(\\overline{\\mathbb{Q}})$ the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and $W[p]$ its $p$-torsion subgroup; it is a $\\mathbb{Z}/p\\mathbb{Z}$-module because $p$ annihilates it. For $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ of\u2026","title":"Linear Galois action on $W(p)$"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-linear-map-apply","kind":"node","lean":"MetaMathlibExt.galLinearMap_apply","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The linear Galois action on "],["math","\\(W(p)\\)"],["text"," is the action on points"]],"source":"The linear Galois action on $W(p)$ is the action on points"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-linear-map-apply","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, $p$ a natural number, $W(\\overline{\\mathbb{Q}})$ the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and $W[p]$ its $p$-torsion subgroup. For every $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ of $\\mathbb{Q}$ and every $x \\in W[p]$, the image of $x$ under the\u2026","title":"The linear Galois action on $W(p)$ is the action on points"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-linear-map-one-apply","kind":"node","lean":"MetaMathlibExt.galLinearMap_one_apply","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The identity acts trivially on "],["math","\\(W(p)\\)"]],"source":"The identity acts trivially on $W(p)$"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-linear-map-one-apply","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, $p$ a natural number, $W(\\overline{\\mathbb{Q}})$ the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and $W[p]$ its $p$-torsion subgroup. Under the $\\mathbb{Z}/p\\mathbb{Z}$-linear action of the absolute Galois group of $\\mathbb{Q}$ on $W[p]$, the identity element maps every\u2026","title":"The identity acts trivially on $W(p)$"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-linear-map-mul-apply","kind":"node","lean":"MetaMathlibExt.galLinearMap_mul_apply","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The linear Galois action on "],["math","\\(W(p)\\)"],["text"," respects products"]],"source":"The linear Galois action on $W(p)$ respects products"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-linear-map-mul-apply","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, $p$ a natural number, $W(\\overline{\\mathbb{Q}})$ the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and $W[p]$ its $p$-torsion subgroup. For $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ of $\\mathbb{Q}$ and $x \\in W[p]$, write $\\sigma \\cdot x$ for the image of $x$\u2026","title":"The linear Galois action on $W(p)$ respects products"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-linear-map-bijective","kind":"node","lean":"MetaMathlibExt.galLinearMap_bijective","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Galois automorphisms act bijectively on "],["math","\\(W(p)\\)"]],"source":"Galois automorphisms act bijectively on $W(p)$"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-linear-map-bijective","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, $p$ a natural number, $W(\\overline{\\mathbb{Q}})$ the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and $W[p]$ its $p$-torsion subgroup. For every $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ of $\\mathbb{Q}$, the $\\mathbb{Z}/p\\mathbb{Z}$-linear map by which $\\sigma$\u2026","title":"Galois automorphisms act bijectively on $W(p)$"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-linear-equiv","kind":"node","lean":"MetaMathlibExt.galLinearEquiv","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Galois automorphisms as linear automorphisms of "],["math","\\(W(p)\\)"]],"source":"Galois automorphisms as linear automorphisms of $W(p)$"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-linear-equiv","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, $p$ a natural number, $W(\\overline{\\mathbb{Q}})$ the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and $W[p]$ its $p$-torsion subgroup. For $\\sigma$ in the absolute Galois group $G_{\\mathbb{Q}}$ of $\\mathbb{Q}$, this is the $\\mathbb{Z}/p\\mathbb{Z}$-linear map\u2026","title":"Galois automorphisms as linear automorphisms of $W(p)$"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-rep","kind":"node","lean":"MetaMathlibExt.galRep","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","The mod-"],["math","\\(p\\)"],["text"," Galois representation"]],"source":"The mod-$p$ Galois representation"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-rep","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, let $p$ be a natural number, let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and let $W[p]$ be its subgroup of points $P$ with $pP = 0$, viewed as a $\\mathbb{Z}/p\\mathbb{Z}$-module. The mod-$p$ Galois representation is the group\u2026","title":"The mod-$p$ Galois representation"},{"id":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/gal-det-hom","kind":"node","lean":"MetaMathlibExt.galDetHom","parent":13752,"reading_title":{"format":"inline-parts-v1","parts":[["text","Determinant character of the mod-"],["math","\\(p\\)"],["text"," Galois representation"]],"source":"Determinant character of the mod-$p$ Galois representation"},"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points/index.html#gal-det-hom","scope":"AlgebraicGeometry/elliptic-curves/galois-action-on-torsion-points","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over $\\mathbb{Q}$, let $p$ be a natural number, let $W(\\overline{\\mathbb{Q}})$ be the group of nonsingular points of $W$ with coordinates in the algebraic closure $\\overline{\\mathbb{Q}}$, together with the point at infinity, and let $W[p]$ be its $p$-torsion subgroup, viewed as a $\\mathbb{Z}/p\\mathbb{Z}$-module. Let $\\rho$ be the mod-$p$ Galois representation of the absolute Galois\u2026","title":"Determinant character of the mod-$p$ Galois representation"},{"data_url":"scopes/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes.json","id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","kind":"scope","navigation":"module","parent":13750,"result":false,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","title":"Isogeny volcanoes"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/multigraph","kind":"node","lean":"Multigraph","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#multigraph","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"A multigraph on a vertex set $V$ is given by an edge multiplicity function $m$ that assigns to every two vertices $v, w \\in V$ a natural number $m(v, w)$ and is symmetric: $m(v, w) = m(w, v)$ for all $v, w \\in V$. Loops are permitted: $m(v, v)$ may be positive.","title":"Multigraph"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/adj","kind":"node","lean":"Multigraph.Adj","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#adj","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"Two vertices $v$ and $w$ of a multigraph $G$ are adjacent when the edge multiplicity between $v$ and $w$ is positive.","title":"Adjacency in a multigraph"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/degree","kind":"node","lean":"Multigraph.degree","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#degree","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"For a multigraph $G$ on a finite vertex set $V$ with edge multiplicity $m$, the degree of a vertex $v$ is $\\sum_{w \\in V} m(v, w)$. A loop at $v$ contributes its multiplicity once, matching the outgoing-isogeny convention.","title":"Degree of a vertex in a multigraph"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/degree-on","kind":"node","lean":"Multigraph.degreeOn","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#degree-on","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"Let $G$ be a multigraph on a finite vertex set $V$ with edge multiplicity $m$, let $v \\in V$ and let $P$ be a decidable predicate on $V$. The degree of $v$ restricted to $P$ is $\\sum_{w \\in V,\\, P(w)} m(v, w)$.","title":"Degree restricted to a predicate"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/connected","kind":"node","lean":"Multigraph.Connected","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#connected","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"Two vertices $v$ and $w$ of a multigraph $G$ are connected when they are related by the reflexive-transitive closure of adjacency, that is, when there is a finite sequence of vertices $v = u_0, u_1, \\dots, u_k = w$ with $k \\ge 0$ and $u_i$ adjacent to $u_{i+1}$ for each $i < k$.","title":"Connectivity of vertices in a multigraph"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-connected","kind":"node","lean":"Multigraph.IsConnected","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-connected","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"A multigraph $G$ is connected when every two vertices $v$ and $w$ of $G$ are joined by a walk, that is, $w$ is reachable from $v$ by a finite (possibly empty) sequence of steps along edges of positive multiplicity.","title":"Connected multigraph"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/adj-comm","kind":"node","lean":"Multigraph.adj_comm","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#adj-comm","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"In a multigraph $G$, a vertex $v$ is adjacent to a vertex $w$ if and only if $w$ is adjacent to $v$.","title":"Symmetry of adjacency in a multigraph"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/connected-refl","kind":"node","lean":"Multigraph.connected_refl","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#connected-refl","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"In a multigraph $G$, every vertex $v$ is connected to $v$.","title":"Every vertex is connected to itself"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/degree-on-compl","kind":"node","lean":"Multigraph.degreeOn_compl","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#degree-on-compl","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"Let $G$ be a multigraph on a finite vertex set $V$, let $v \\in V$ and let $P$ be a decidable predicate on $V$. Then the degree of $v$ equals the degree of $v$ restricted to $P$ plus the degree of $v$ restricted to $\\neg P$.","title":"Degree splits over a predicate and its complement"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/degree-on-congr","kind":"node","lean":"Multigraph.degreeOn_congr","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#degree-on-congr","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"Let $G$ be a multigraph on a finite vertex set $V$, let $v \\in V$ and let $P$ and $Q$ be decidable predicates on $V$. If $P(w) \\iff Q(w)$ for every vertex $w$ whose edge multiplicity with $v$ is positive, then the degree of $v$ restricted to $P$ equals the degree of $v$ restricted to $Q$.","title":"Restricted degree depends only on the neighbours"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/isogeny-volcano","kind":"node","lean":"IsogenyVolcano","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#isogeny-volcano","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"Let $\\ell$ be a prime. An $\\ell$-isogeny volcano consists of a finite vertex set $V$, a multigraph on $V$ with edge multiplicities $m(v, w)$, a depth $d \\in \\mathbb{N}$, a surjective level map $\\mathrm{lev} : V \\to \\{0, 1, \\dots, d\\}$ and a surface degree $s \\in \\mathbb{N}$ with $s \\le 2$, subject to the following conditions, where the surface is the set of vertices of level $0$. Every surface vertex $v$ satisfies\u2026","title":"Isogeny volcano"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-surface","kind":"node","lean":"IsogenyVolcano.isSurface","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-surface","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"A vertex $v$ of an $\\ell$-isogeny volcano $C$ lies on the surface of $C$ when its level is $0$.","title":"Surface of a volcano"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-floor","kind":"node","lean":"IsogenyVolcano.isFloor","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-floor","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"A vertex $v$ of an $\\ell$-isogeny volcano $C$ of depth $d$ lies on the floor of $C$ when its level is $d$, the last of the levels $0, 1, \\dots, d$.","title":"Floor of a volcano"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/volcano-depth","kind":"node","lean":"IsogenyVolcano.volcanoDepth","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#volcano-depth","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"The depth of an $\\ell$-isogeny volcano $C$ is the natural number $d$ recorded in $C$: the levels of the vertices of $C$ are exactly $0, 1, \\dots, d$, with level $0$ the surface and level $d$ the floor.","title":"Depth of an isogeny volcano"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-horizontal-edge","kind":"node","lean":"IsogenyVolcano.IsHorizontalEdge","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-horizontal-edge","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"In an $\\ell$-isogeny volcano $C$, the pair $(v, w)$ of vertices is a horizontal edge when the edge multiplicity between $v$ and $w$ is positive and both $v$ and $w$ lie on the surface, that is, both have level $0$.","title":"Horizontal edge"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-descending-edge","kind":"node","lean":"IsogenyVolcano.IsDescendingEdge","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-descending-edge","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"In an $\\ell$-isogeny volcano $C$, the pair $(v, w)$ of vertices is a descending edge when the edge multiplicity between $v$ and $w$ is positive and $w$ lies one level below $v$, that is, the level of $v$ plus one equals the level of $w$ (level $0$ being the surface).","title":"Descending edge"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-ascending-edge","kind":"node","lean":"IsogenyVolcano.IsAscendingEdge","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-ascending-edge","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"In an $\\ell$-isogeny volcano $C$, the pair $(v, w)$ of vertices is an ascending edge when the edge multiplicity between $v$ and $w$ is positive and $w$ lies one level above $v$, that is, the level of $w$ plus one equals the level of $v$ (level $0$ being the surface).","title":"Ascending edge"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-vertical-edge","kind":"node","lean":"IsogenyVolcano.IsVerticalEdge","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-vertical-edge","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"In an $\\ell$-isogeny volcano $C$, the pair $(v, w)$ of vertices is a vertical edge when it is a descending edge or an ascending edge, that is, when the edge multiplicity between $v$ and $w$ is positive and the levels of $v$ and $w$ differ by exactly one.","title":"Vertical edge"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/descending-iff-ascending-symm","kind":"node","lean":"IsogenyVolcano.descending_iff_ascending_symm","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#descending-iff-ascending-symm","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"In an $\\ell$-isogeny volcano, for vertices $v$ and $w$, $(v, w)$ is a descending edge if and only if $(w, v)$ is an ascending edge.","title":"Reversing a descending edge gives an ascending edge"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/ascending-iff-descending-symm","kind":"node","lean":"IsogenyVolcano.ascending_iff_descending_symm","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#ascending-iff-descending-symm","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"In an $\\ell$-isogeny volcano, for vertices $v$ and $w$, $(v, w)$ is an ascending edge if and only if $(w, v)$ is a descending edge.","title":"Reversing an ascending edge gives a descending edge"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/edge-direction","kind":"node","lean":"IsogenyVolcano.edge_direction","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#edge-direction","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"Let $C$ be an $\\ell$-isogeny volcano, with levels numbered from $0$ at the surface, and let $v, w$ be vertices of $C$ whose edge multiplicity is positive. Then $(v, w)$ is a horizontal edge (both $v$ and $w$ have level $0$), a descending edge (the level of $w$ is the level of $v$ plus one) or an ascending edge (the level of $v$ is the level of $w$ plus one).","title":"Every edge of an isogeny volcano is horizontal, descending or ascending"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-surface-iff","kind":"node","lean":"IsogenyVolcano.isSurface_iff","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-surface-iff","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"A vertex $v$ of an $\\ell$-isogeny volcano $C$ lies on the surface of $C$ if and only if the level of $v$, as a natural number, is $0$.","title":"Surface vertices are those of level zero"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/is-floor-iff","kind":"node","lean":"IsogenyVolcano.isFloor_iff","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#is-floor-iff","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"A vertex $v$ of an $\\ell$-isogeny volcano $C$ lies on the floor of $C$ if and only if the level of $v$, as a natural number, equals the depth of $C$.","title":"Floor vertices are those whose level equals the depth"},{"id":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/surface-degree-on","kind":"node","lean":"IsogenyVolcano.surface_degreeOn","parent":13772,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/isogeny-volcanoes/index.html#surface-degree-on","scope":"AlgebraicGeometry/elliptic-curves/isogeny-volcanoes","status":"fully_proved","summary":"Let $C$ be an $\\ell$-isogeny volcano with edge multiplicities $m$ and surface degree $s$. For every surface vertex $v$ of $C$, that is, every vertex of level $0$, the sum of $m(v, w)$ over all surface vertices $w$ of $C$ equals $s$.","title":"Surface regularity of an isogeny volcano"},{"data_url":"scopes/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces.json","id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","kind":"scope","navigation":"module","parent":13750,"result":false,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"can_prove","title":"Point counts and Frobenius traces"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/frobenius-trace","kind":"node","lean":"WeierstrassCurve.Affine.frobeniusTrace","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#frobenius-trace","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over a finite field $F$ with $q$ elements, and let $\\#W(F)$ be the number of nonsingular $F$-points of $W$, counting the point at infinity. The Frobenius trace of $W$ is the integer $a_q(W) = q + 1 - \\#W(F)$.","title":"Frobenius trace"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/nat-card-point-eq-nat-card-affine-soluti","kind":"node","lean":"WeierstrassCurve.Affine.natCard_point_eq_natCard_affineSolutions_add_one","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#nat-card-point-eq-nat-card-affine-soluti","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"fully_proved","summary":"Let $W$ be an elliptic curve in Weierstrass form over a finite field $F$. The number of $F$-points of $W$, counting the point at infinity, is one more than the number of pairs $(x, y) \\in F \\times F$ satisfying the Weierstrass equation of $W$.","title":"Point count from affine solutions"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/nat-card-point-le-two-mul-nat-card-add-o","kind":"node","lean":"WeierstrassCurve.Affine.natCard_point_le_two_mul_natCard_add_one","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#nat-card-point-le-two-mul-nat-card-add-o","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over a finite field $F$ with $q$ elements. The number of nonsingular $F$-points of $W$, counting the point at infinity, is at most $2q + 1$.","title":"Upper bound on the point count"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/nat-card-point-eq-nat-card-add-one-sub-f","kind":"node","lean":"WeierstrassCurve.Affine.natCard_point_eq_natCard_add_one_sub_frobeniusTrace","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#nat-card-point-eq-nat-card-add-one-sub-f","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over a finite field $F$ with $q$ elements, and let $\\#W(F)$ be the number of nonsingular $F$-points of $W$, counting the point at infinity. Then $\\#W(F) = q + 1 - a_q(W)$, where $a_q(W)$ is the Frobenius trace of $W$.","title":"Point count from the Frobenius trace"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/nat-card-point-add-frobenius-trace-eq","kind":"node","lean":"WeierstrassCurve.Affine.natCard_point_add_frobeniusTrace_eq","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#nat-card-point-add-frobenius-trace-eq","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over a finite field $F$ with $q$ elements, and let $\\#W(F)$ be the number of nonsingular $F$-points of $W$, counting the point at infinity. Then $\\#W(F) + a_q(W) = q + 1$, where $a_q(W)$ is the Frobenius trace of $W$.","title":"Point count plus Frobenius trace"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/exists-is-elliptic-frobenius-trace-eq-if","kind":"node","lean":"MetaMathlibExt.exists_isElliptic_frobeniusTrace_eq_iff","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#exists-is-elliptic-frobenius-trace-eq-if","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"can_prove","summary":"Let $p$ be a prime, let $a \\ge 1$, and let $F$ be a finite field with $q = p^a$ elements. An integer $\\beta$ is the Frobenius trace $q + 1 - \\#E(F)$ of some elliptic curve $E$ over $F$ if and only if $\\beta^2 \\le 4q$ and one of the following holds: 1. $\\gcd(\\beta, p) = 1$; 2. $a$ is even and $\\beta = \\pm 2p^{a/2}$; 3. $a$ is even, $p \\not\\equiv 1 \\pmod 3$ and $\\beta = \\pm p^{a/2}$; 4. $a$ is odd, $p = 2$ or\u2026","title":"Waterhouse's theorem on Frobenius traces"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/ramanujan-iff-nontrivial-zeros-re-eq-hal","kind":"node","lean":"MetaMathlibExt.ramanujan_iff_nontrivial_zeros_re_eq_half","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#ramanujan-iff-nontrivial-zeros-re-eq-hal","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"can_prove","summary":"Let $q \\ge 2$ be a natural number and let $A$ be a symmetric $n \\times n$ matrix with integer entries. Every complex root $\\mu \\ne q + 1$ of the characteristic polynomial of $A$ satisfies $|\\mu| \\le 2\\sqrt{q}$ if and only if, for every such root $\\mu \\ne q + 1$, every complex number $s$ with $1 - \\mu q^{-s} + q^{1 - 2s} = 0$ has real part $1/2$.","title":"Ihara's criterion for Ramanujan graphs"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/is-weil-number-of-ramanujan","kind":"node","lean":"MetaMathlibExt.isWeilNumber_of_ramanujan","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#is-weil-number-of-ramanujan","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"can_prove","summary":"Let $q$ be a natural number and let $A$ be a symmetric $n \\times n$ matrix with integer entries such that every complex root $\\mu \\ne q + 1$ of its characteristic polynomial satisfies $|\\mu| \\le 2\\sqrt{q}$. Let $\\mu \\ne q + 1$ be a complex root of the characteristic polynomial of $A$ and let $\\alpha \\in \\mathbb{C}$ satisfy $\\alpha^2 - \\mu\\alpha + q = 0$. Then $\\alpha$ is an algebraic integer, and every complex root\u2026","title":"Ramanujan graphs give Weil numbers"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/exists-is-elliptic-card-eq-iff-ramanujan","kind":"node","lean":"MetaMathlibExt.exists_isElliptic_card_eq_iff_ramanujan_two_vertices","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#exists-is-elliptic-card-eq-iff-ramanujan","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"can_prove","summary":"Let $p$ be an odd prime, let $a \\in \\{1, 2\\}$, and let $F$ be a finite field with $q = p^a$ elements. A natural number $N$ is the number $\\#E(F)$ of $F$-points (counting the point at infinity) of some elliptic curve $E$ over $F$ with $4 \\mid N$ if and only if there are natural numbers $l$ and $m$ with $N = 2m$, $2l + m = q + 1$, $m > 0$ and $(2l - m)^2 \\le 4q$.","title":"Elliptic curves with point count divisible by 4 and Ramanujan graphs on two vertices"},{"id":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/exists-ordinary-card-eq-iff-ramanujan-tw","kind":"node","lean":"MetaMathlibExt.exists_ordinary_card_eq_iff_ramanujan_two_vertices","parent":13797,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces/index.html#exists-ordinary-card-eq-iff-ramanujan-tw","scope":"AlgebraicGeometry/elliptic-curves/point-counts-and-frobenius-traces","status":"can_prove","summary":"Ordinary elliptic curves with point count divisible by 4 and Ramanujan graphs on two vertices. Let $p$ be an odd prime, let $a \\ge 1$, and let $F$ be a finite field with $q = p^a$ elements. A natural number $N$ is the number $\\#E(F)$ of $F$-points (counting the point at infinity) of some elliptic curve $E$ over $F$ with $4 \\mid N$ whose Frobenius trace $q + 1 - \\#E(F)$ is coprime to $p$ if and only if there are\u2026","title":"Ordinary elliptic curves with point count divisible by 4 and Ramanujan graphs on two vertices"},{"data_url":"scopes/AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves.json","id":"AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves","kind":"scope","navigation":"module","parent":13750,"result":false,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves/index.html","scope":"AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves","status":"fully_proved","title":"Supersingular elliptic curves"},{"id":"AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves/is-supersingular","kind":"node","lean":"WeierstrassCurve.Affine.IsSupersingular","parent":13808,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves/index.html#is-supersingular","scope":"AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves","status":"fully_proved","summary":"Let $p$ be a prime and let $E$ be an elliptic curve over an algebraic closure $\\overline{\\mathbb{F}}_p$ of $\\mathbb{Z}/p\\mathbb{Z}$. Then $E$ is called supersingular when every point $P \\in E(\\overline{\\mathbb{F}}_p)$ with $pP = 0$ is $P = 0$.","title":"Supersingular elliptic curves"},{"id":"AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves/of-pow-torsion","kind":"node","lean":"WeierstrassCurve.Affine.IsSupersingular.of_pow_torsion","parent":13808,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves/index.html#of-pow-torsion","scope":"AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves","status":"fully_proved","summary":"Let $p$ be a prime and let $E$ be an elliptic curve over an algebraic closure $\\overline{\\mathbb{F}}_p$ of $\\mathbb{Z}/p\\mathbb{Z}$. If for every integer $r \\ge 1$ the only point $P \\in E(\\overline{\\mathbb{F}}_p)$ with $p^r P = 0$ is $P = 0$, then $E$ is supersingular.","title":"Supersingularity from trivial prime-power torsion"},{"id":"AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves/pow-torsion","kind":"node","lean":"WeierstrassCurve.Affine.IsSupersingular.pow_torsion","parent":13808,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves/index.html#pow-torsion","scope":"AlgebraicGeometry/elliptic-curves/supersingular-elliptic-curves","status":"fully_proved","summary":"Let $p$ be a prime and let $E$ be a supersingular elliptic curve over an algebraic closure $\\overline{\\mathbb{F}}_p$ of $\\mathbb{Z}/p\\mathbb{Z}$. Then for every integer $r \\ge 1$, every point $P \\in E(\\overline{\\mathbb{F}}_p)$ with $p^r P = 0$ is $P = 0$.","title":"Supersingular curves have trivial prime-power torsion"},{"data_url":"scopes/AlgebraicGeometry/elliptic-curves/weierstrass-curves.json","id":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","kind":"scope","navigation":"module","parent":13750,"result":false,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/weierstrass-curves/index.html","scope":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","status":"fully_proved","title":"Weierstrass curves"},{"id":"AlgebraicGeometry/elliptic-curves/weierstrass-curves/map-affine-coordinates","kind":"node","lean":"WeierstrassCurve.VariableChange.mapAffineCoordinates","parent":13812,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/weierstrass-curves/index.html#map-affine-coordinates","scope":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","status":"fully_proved","summary":"For a variable change $C = (u, r, s, t)$ over a field $F$, with $u \\in F^\\times$ and $r, s, t \\in F$, the affine coordinate map of $C$ sends $(x, y) \\in F^2$ to $(u^2 x + r,\\ u^3 y + u^2 s x + t)$.","title":"Coordinate map of a Weierstrass variable change"},{"id":"AlgebraicGeometry/elliptic-curves/weierstrass-curves/affine-coordinates-equiv","kind":"node","lean":"WeierstrassCurve.VariableChange.affineCoordinatesEquiv","parent":13812,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/weierstrass-curves/index.html#affine-coordinates-equiv","scope":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","status":"fully_proved","summary":"For a variable change $C = (u, r, s, t)$ over a field $F$, the map $(x, y) \\mapsto (u^2 x + r,\\ u^3 y + u^2 s x + t)$ is a bijection $F^2 \\to F^2$, whose inverse is the corresponding map of the inverse variable change $C^{-1}$.","title":"Variable changes act bijectively on affine coordinates"},{"id":"AlgebraicGeometry/elliptic-curves/weierstrass-curves/equation-map-affine-coordinates-iff","kind":"node","lean":"WeierstrassCurve.VariableChange.equation_mapAffineCoordinates_iff","parent":13812,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/weierstrass-curves/index.html#equation-map-affine-coordinates-iff","scope":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over a field $F$, let $C = (u, r, s, t)$ be a variable change and let $C \\cdot W$ be the transformed curve. A pair $(x, y) \\in F^2$ satisfies the Weierstrass equation of $C \\cdot W$ if and only if $(u^2 x + r,\\ u^3 y + u^2 s x + t)$ satisfies the Weierstrass equation of $W$.","title":"Variable changes transform the Weierstrass equation"},{"id":"AlgebraicGeometry/elliptic-curves/weierstrass-curves/affine-equation-equiv","kind":"node","lean":"WeierstrassCurve.VariableChange.affineEquationEquiv","parent":13812,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/weierstrass-curves/index.html#affine-equation-equiv","scope":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","status":"fully_proved","summary":"Let $W$ be a Weierstrass curve over a field $F$, let $C = (u, r, s, t)$ be a variable change and let $C \\cdot W$ be the transformed curve. The map $(x, y) \\mapsto (u^2 x + r,\\ u^3 y + u^2 s x + t)$ is a bijection from the solutions $(x, y) \\in F^2$ of the Weierstrass equation of $C \\cdot W$ onto the solutions in $F^2$ of the Weierstrass equation of $W$; its inverse is the corresponding map of $C^{-1}$.","title":"Variable changes biject affine solutions"},{"id":"AlgebraicGeometry/elliptic-curves/weierstrass-curves/point-equiv","kind":"node","lean":"WeierstrassCurve.VariableChange.pointEquiv","parent":13812,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/weierstrass-curves/index.html#point-equiv","scope":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","status":"fully_proved","summary":"Let $W$ be an elliptic Weierstrass curve over a field $F$, let $C = (u, r, s, t)$ be a variable change and let $C \\cdot W$ be the transformed curve. There is a bijection from the rational points of $C \\cdot W$ to the rational points of $W$, where in both cases the point at infinity is included; it sends the point at infinity to the point at infinity and an affine point $(x, y)$ to $(u^2 x + r,\\ u^3 y + u^2 s x + t)$.","title":"Variable changes biject rational points"},{"id":"AlgebraicGeometry/elliptic-curves/weierstrass-curves/nat-card-point-eq","kind":"node","lean":"WeierstrassCurve.VariableChange.natCard_point_eq","parent":13812,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/weierstrass-curves/index.html#nat-card-point-eq","scope":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","status":"fully_proved","summary":"Let $W$ be an elliptic Weierstrass curve over a finite field $F$, let $C$ be a variable change and let $C \\cdot W$ be the transformed curve. Then $C \\cdot W$ and $W$ have the same number of rational points.","title":"Variable changes preserve the point count"},{"id":"AlgebraicGeometry/elliptic-curves/weierstrass-curves/xyideal-ne-bot","kind":"node","lean":"WeierstrassCurve.Affine.CoordinateRing.XYIdeal_ne_bot","parent":13812,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/weierstrass-curves/index.html#xyideal-ne-bot","scope":"AlgebraicGeometry/elliptic-curves/weierstrass-curves","status":"fully_proved","summary":"The ideal generated by `X - x` and `Y - y` is nonzero.","title":"The ideal of a point on a Weierstrass curve is nonzero"},{"data_url":"scopes/AlgebraicGeometry/elliptic-curves/hasse-bound.json","id":"AlgebraicGeometry/elliptic-curves/hasse-bound","kind":"scope","navigation":"module","parent":13750,"result":false,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","title":"Hasse bound"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/primitive-triple","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinProjectiveDegree.PrimitiveTriple","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#primitive-triple","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $K$ be a field. Polynomials $a, b, c \\in K[X]$ form a primitive triple when they generate the unit ideal of $K[X]$, that is, when there are $u, v, w \\in K[X]$ with $ua + vb + wc = 1$.","title":"Primitive triple of polynomials"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/primitive-triple-of-no-common-irreducibl","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinProjectiveDegree.primitiveTriple_of_no_common_irreducible","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#primitive-triple-of-no-common-irreducibl","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $K$ be a field and let $a, b, c \\in K[X]$ with $c \\neq 0$. If no irreducible polynomial in $K[X]$ divides all of $a$, $b$ and $c$, then $a, b, c$ form a primitive triple: there are $u, v, w \\in K[X]$ with $ua + vb + wc = 1$.","title":"Polynomials with no common irreducible factor form a primitive triple"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/projective-first-nat-degree","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinProjectiveDegree.projective_first_natDegree","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#projective-first-nat-degree","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $K$ be a field, let $u, v, h \\in K[X]$ form a primitive triple with $h \\neq 0$, and let $x, y \\in K(X)$ be nonzero rational functions with $u/h = xy$ and $v/h = x + y$ in $K(X)$. Then $\\deg u = \\deg \\operatorname{num}(x) + \\deg \\operatorname{num}(y)$, where $\\operatorname{num}$ denotes the numerator of a rational function written in lowest terms.","title":"Numerator degrees from a primitive presentation of product and sum"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/eq-quadratic-of-recurrence","kind":"node","lean":"Int.eq_quadratic_of_recurrence","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#eq-quadratic-of-recurrence","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $d : \\mathbb{Z} \\to \\mathbb{Z}$ satisfy $d(n-1) + d(n+1) = 2d(n) + 2$ for every integer $n$, and let $q, N$ be integers with $d(0) = q$ and $d(-1) = N$. Then $d(n) = n^2 + (q + 1 - N)n + q$ for every integer $n$.","title":"Integer sequences with constant second difference two"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/sq-le-four-mul-of-quadratic-nonnegative","kind":"node","lean":"Int.sq_le_four_mul_of_quadratic_nonnegative","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#sq-le-four-mul-of-quadratic-nonnegative","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $a, q \\in \\mathbb{Z}$. If $n^2 + an + q \\ge 0$ for every integer $n$ and there is no integer $n$ with $n^2 + an + q = 0$ and $(n+1)^2 + a(n+1) + q = 0$, then $a^2 \\le 4q$.","title":"Discriminant bound for a nonnegative integral quadratic without adjacent zeros"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/manin-degree-data","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinDegreeData","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#manin-degree-data","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"For natural numbers $q$ and $N$, Manin degree data for $q$ and $N$ is a sequence $d : \\mathbb{Z} \\to \\mathbb{N}$ such that $d(0) = q$, $d(-1) = N$, $d(n-1) + d(n+1) = 2d(n) + 2$ for every integer $n$, and there is no integer $n$ with $d(n) = d(n+1) = 0$.","title":"Manin degree data"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/of-quadratic","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinDegreeData.ofQuadratic","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#of-quadratic","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $q$ and $N$ be natural numbers and put $a = q + 1 - N \\in \\mathbb{Z}$. Suppose $n^2 + an + q \\ge 0$ for every integer $n$, and there is no integer $n$ at which both $n^2 + an + q$ and $(n+1)^2 + a(n+1) + q$ vanish. Then the sequence $d(n) = n^2 + an + q$ is Manin degree data for $q$ and $N$.","title":"Manin degree data from a quadratic"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/hasse-bound-2","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinDegreeData.hasse_bound","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#hasse-bound-2","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $q$ and $N$ be natural numbers. If there is Manin degree data for $q$ and $N$, then $(q + 1 - N)^2 \\le 4q$, computed in the integers.","title":"Hasse bound from Manin degree data"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/manin-degree-data-of-card-eq-three","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwist.manin_degree_data_of_card_eq_three","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#manin-degree-data-of-card-eq-three","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a field with exactly three elements, let $W$ be any Weierstrass curve over $F$, and let $N$ be the number of nonsingular points of $W$ over $F$, including the point at infinity. Then there is Manin degree data for $(3, N)$: a function $d : \\mathbb{Z} \\to \\mathbb{N}$ with $d(0) = 3$, $d(-1) = N$, $d(n - 1) + d(n + 1) = 2 d(n) + 2$ for every integer $n$, and no integer $n$ with $d(n) = d(n + 1) = 0$.","title":"Manin degree data over the field with three elements"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/manin-degree-data-of-is-char-ne-two-nf","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwist.manin_degree_data_of_isCharNeTwoNF","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#manin-degree-data-of-is-char-ne-two-nf","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $W$ be a Weierstrass curve $y^2 = x^3 + a_2 x^2 + a_4 x + a_6$ over $F$ (that is, $a_1 = a_3 = 0$) with nonzero discriminant, and let $N$ be the number of points of $W$ over $F$, including the point at infinity. Then there is Manin degree data for $(q, N)$: a function $d : \\mathbb{Z} \\to \\mathbb{N}$ with $d(0) = q$, $d(-1) = N$, $d(n - 1) + d(n + 1) = 2 d(n) + 2$ for\u2026","title":"Manin degree data for the odd-characteristic normal form"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/hasse-bound-of-is-char-ne-two-nf","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwist.hasse_bound_of_isCharNeTwoNF","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#hasse-bound-of-is-char-ne-two-nf","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $W$ be a Weierstrass curve $y^2 = x^3 + a_2 x^2 + a_4 x + a_6$ over $F$ (that is, $a_1 = a_3 = 0$) with nonzero discriminant. If $N$ is the number of points of $W$ over $F$, including the point at infinity, then $(q + 1 - N)^2 \\le 4q$.","title":"Hasse bound for the odd-characteristic normal form"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/manin-degree-data-of-card-eq-two","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwistCharTwo.manin_degree_data_of_card_eq_two","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#manin-degree-data-of-card-eq-two","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a field with exactly two elements, let $W$ be any Weierstrass curve over $F$, and let $N$ be the number of nonsingular points of $W$ over $F$, including the point at infinity. Then there is Manin degree data for $(2, N)$: a function $d : \\mathbb{Z} \\to \\mathbb{N}$ with $d(0) = 2$, $d(-1) = N$, $d(n - 1) + d(n + 1) = 2 d(n) + 2$ for every integer $n$, and no integer $n$ with $d(n) = d(n + 1) = 0$.","title":"Manin degree data over the field with two elements"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/manin-degree-data-of-card-eq-four","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwistCharTwo.manin_degree_data_of_card_eq_four","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#manin-degree-data-of-card-eq-four","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a field with exactly four elements, let $W$ be any Weierstrass curve over $F$, and let $N$ be the number of nonsingular points of $W$ over $F$, including the point at infinity. Then there is Manin degree data for $(4, N)$: a function $d : \\mathbb{Z} \\to \\mathbb{N}$ with $d(0) = 4$, $d(-1) = N$, $d(n - 1) + d(n + 1) = 2 d(n) + 2$ for every integer $n$, and no integer $n$ with $d(n) = d(n + 1) = 0$.","title":"Manin degree data over the field with four elements"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/manin-degree-data-of-is-char-two-nf","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwistCharTwo.manin_degree_data_of_isCharTwoNF","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#manin-degree-data-of-is-char-two-nf","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a finite field of characteristic $2$ with $q$ elements, let $W$ be a Weierstrass curve over $F$ with nonzero discriminant that is in one of the two characteristic-two normal forms $y^2 + xy = x^3 + a_2 x^2 + a_6$ (that is, $a_1 = 1$ and $a_3 = a_4 = 0$) or $y^2 + a_3 y = x^3 + a_4 x + a_6$ (that is, $a_1 = a_2 = 0$), and let $N$ be the number of points of $W$ over $F$, including the point at infinity.\u2026","title":"Manin degree data for the characteristic-two normal forms"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/hasse-bound-of-is-char-two-nf","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwistCharTwo.hasse_bound_of_isCharTwoNF","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#hasse-bound-of-is-char-two-nf","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a finite field of characteristic $2$ with $q$ elements and let $W$ be a Weierstrass curve over $F$ with nonzero discriminant that is in one of the two characteristic-two normal forms $y^2 + xy = x^3 + a_2 x^2 + a_6$ (that is, $a_1 = 1$ and $a_3 = a_4 = 0$) or $y^2 + a_3 y = x^3 + a_4 x + a_6$ (that is, $a_1 = a_2 = 0$). If $N$ is the number of points of $W$ over $F$, including the point at infinity, then\u2026","title":"Hasse bound for the characteristic-two normal forms"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/hasse-bound-of-is-char-two-jne-zero-nf","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwistCharTwo.hasse_bound_of_isCharTwoJNeZeroNF","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#hasse-bound-of-is-char-two-jne-zero-nf","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a finite field of characteristic $2$ with $q$ elements and let $W$ be a Weierstrass curve $y^2 + xy = x^3 + a_2 x^2 + a_6$ over $F$ (that is, $a_1 = 1$ and $a_3 = a_4 = 0$) with nonzero discriminant. If $N$ is the number of points of $W$ over $F$, including the point at infinity, then $(q + 1 - N)^2 \\le 4q$.","title":"Hasse bound for the ordinary characteristic-two normal form"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/hasse-bound-of-is-char-two-jeq-zero-nf","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.ManinTwistCharTwo.hasse_bound_of_isCharTwoJEqZeroNF","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#hasse-bound-of-is-char-two-jeq-zero-nf","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a finite field of characteristic $2$ with $q$ elements and let $W$ be a Weierstrass curve $y^2 + a_3 y = x^3 + a_4 x + a_6$ over $F$ (that is, $a_1 = a_2 = 0$) with nonzero discriminant. If $N$ is the number of points of $W$ over $F$, including the point at infinity, then $(q + 1 - N)^2 \\le 4q$.","title":"Hasse bound for the supersingular characteristic-two normal form"},{"id":"AlgebraicGeometry/elliptic-curves/hasse-bound/hasse-bound","kind":"node","lean":"MathlibExt.AlgebraicGeometry.EllipticCurve.HasseBoundWanted.hasse_bound","parent":13820,"result":true,"route":"../roadmap/AlgebraicGeometry/elliptic-curves/hasse-bound/index.html#hasse-bound","scope":"AlgebraicGeometry/elliptic-curves/hasse-bound","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and let $E$ be an elliptic curve over $F$ given by a Weierstrass equation. If $N$ is the number of points of $E$ with coordinates in $F$, including the point at infinity, then $(q + 1 - N)^2 \\le 4q$ in $\\mathbb{Z}$.","title":"Hasse bound"},{"data_url":"scopes/AlgebraicGeometry/algebraic-curves.json","id":"AlgebraicGeometry/algebraic-curves","kind":"scope","parent":13624,"result":false,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/index.html","scope":"AlgebraicGeometry/algebraic-curves","status":"planned","summary":"Algebraic curves and their invariants, including divisors, the genus, Jacobians and the Abel\u2013Jacobi map.","title":"Algebraic curves"},{"data_url":"scopes/AlgebraicGeometry/algebraic-curves/divisors-on-curves.json","id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves","kind":"scope","parent":13839,"result":false,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/index.html","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves","status":"fully_proved","title":"Divisors on curves"},{"data_url":"scopes/AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve.json","id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve","kind":"scope","navigation":"module","parent":13840,"result":false,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve/index.html","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve","status":"fully_proved","title":"Divisors on a curve"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve/curve-divisor","kind":"node","lean":"CurveDivisor","parent":13841,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve/index.html#curve-divisor","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve","status":"fully_proved","summary":"For a set $C$ (the points of a curve), a divisor on $C$ is a finitely supported function $D \\colon C \\to \\mathbb{Z}$. Its support is the finite set of points $P \\in C$ with $D(P) \\neq 0$.","title":"Divisors on a curve"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve/coeff","kind":"node","lean":"CurveDivisor.coeff","parent":13841,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve/index.html#coeff","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/divisors-on-a-curve","status":"fully_proved","summary":"For a divisor $D$ on a set of points $C$ and a point $P \\in C$, the coefficient of $D$ at $P$ is the integer $D(P)$.","title":"Coefficient of a divisor at a point"},{"data_url":"scopes/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors.json","id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","kind":"scope","navigation":"module","parent":13840,"result":false,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/index.html","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","status":"fully_proved","title":"Effective parts of curve divisors"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/div-zeros","kind":"node","lean":"CurveDivisor.divZeros","parent":13844,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/index.html#div-zeros","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","status":"fully_proved","summary":"For a divisor $D$ on a set of points $C$, the zero part $D^+$ of $D$ is the divisor on $C$ whose coefficient at each point $P \\in C$ is $\\max(D(P), 0)$.","title":"Zero part of a divisor"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/div-poles","kind":"node","lean":"CurveDivisor.divPoles","parent":13844,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/index.html#div-poles","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","status":"fully_proved","summary":"For a divisor $D$ on a set of points $C$, the pole part $D^-$ of $D$ is the divisor on $C$ whose coefficient at each point $P \\in C$ is $\\max(-D(P), 0)$.","title":"Pole part of a divisor"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/div-zeros-apply","kind":"node","lean":"CurveDivisor.divZeros_apply","parent":13844,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/index.html#div-zeros-apply","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","status":"fully_proved","summary":"For every divisor $D$ on a set of points $C$ and every point $P \\in C$, the coefficient at $P$ of the zero part $D^+$ of $D$ is $\\max(D(P), 0)$.","title":"Coefficients of the zero part"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/div-poles-apply","kind":"node","lean":"CurveDivisor.divPoles_apply","parent":13844,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/index.html#div-poles-apply","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","status":"fully_proved","summary":"For every divisor $D$ on a set of points $C$ and every point $P \\in C$, the coefficient at $P$ of the pole part $D^-$ of $D$ is $\\max(-D(P), 0)$.","title":"Coefficients of the pole part"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/div-zeros-nonneg","kind":"node","lean":"CurveDivisor.divZeros_nonneg","parent":13844,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/index.html#div-zeros-nonneg","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","status":"fully_proved","summary":"For every divisor $D$ on a set of points $C$, the zero part $D^+$ of $D$ is effective: $D^+(P) \\ge 0$ for every point $P \\in C$.","title":"The zero part of a divisor is effective"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/div-poles-nonneg","kind":"node","lean":"CurveDivisor.divPoles_nonneg","parent":13844,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/index.html#div-poles-nonneg","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","status":"fully_proved","summary":"For every divisor $D$ on a set of points $C$, the pole part $D^-$ of $D$ is effective: $D^-(P) \\ge 0$ for every point $P \\in C$.","title":"The pole part of a divisor is effective"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/div-zeros-sub-div-poles","kind":"node","lean":"CurveDivisor.divZeros_sub_divPoles","parent":13844,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors/index.html#div-zeros-sub-div-poles","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/effective-parts-of-curve-divisors","status":"fully_proved","summary":"For every set of points $C$ and every divisor $D$ on $C$, let $D^+$ be its zero part, with coefficient $\\max(D(P), 0)$ at each point $P$, and $D^-$ its pole part, with coefficient $\\max(-D(P), 0)$ at each point $P$. Then $D^+ - D^- = D$.","title":"A divisor is its zero part minus its pole part"},{"data_url":"scopes/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors.json","id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","kind":"scope","navigation":"module","parent":13840,"result":false,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/index.html","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","status":"fully_proved","title":"Evaluation of curve divisors"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/eval-monoid-hom","kind":"node","lean":"CurveDivisor.evalMonoidHom","parent":13852,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/index.html#eval-monoid-hom","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","status":"fully_proved","summary":"For a commutative group $G$ and a function $f : C \\to G$, this is the monoid homomorphism from the group of divisors on $C$, written multiplicatively, to $G$ that sends a divisor $D$ to the finite product $\\prod_{P \\in \\operatorname{supp} D} f(P)^{D(P)}$.","title":"Evaluation of divisors as a group homomorphism"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/eval","kind":"node","lean":"CurveDivisor.eval","parent":13852,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/index.html#eval","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","status":"fully_proved","summary":"For a commutative group $G$, a function $f : C \\to G$ and a divisor $D$ on $C$, the evaluation of $D$ against $f$ is the element $f(D) = \\prod_{P \\in \\operatorname{supp} D} f(P)^{D(P)}$ of $G$.","title":"Evaluation of a divisor against a function"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/eval-eq-prod","kind":"node","lean":"CurveDivisor.eval_eq_prod","parent":13852,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/index.html#eval-eq-prod","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","status":"fully_proved","summary":"For a commutative group $G$, every function $f : C \\to G$ and every divisor $D$ on $C$, the evaluation of $D$ against $f$ equals $\\prod_{P \\in \\operatorname{supp} D} f(P)^{D(P)}$, the product over the points in the support of $D$.","title":"Evaluation is a product over the support"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/eval-zero","kind":"node","lean":"CurveDivisor.eval_zero","parent":13852,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/index.html#eval-zero","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","status":"fully_proved","summary":"For a commutative group $G$ and every function $f : C \\to G$, the evaluation of the zero divisor on $C$ against $f$ is the identity element $1$ of $G$.","title":"The zero divisor evaluates to one"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/eval-single","kind":"node","lean":"CurveDivisor.eval_single","parent":13852,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/index.html#eval-single","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","status":"fully_proved","summary":"For a commutative group $G$, every function $f : C \\to G$, every point $P \\in C$ and every integer $n$, the divisor $n \\cdot P$, with coefficient $n$ at $P$ and $0$ elsewhere, evaluates against $f$ to $f(P)^n$.","title":"Evaluation of a one-point divisor"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/eval-add","kind":"node","lean":"CurveDivisor.eval_add","parent":13852,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/index.html#eval-add","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","status":"fully_proved","summary":"For a commutative group $G$, every function $f : C \\to G$ and all divisors $D_1, D_2$ on $C$, writing $f(D)$ for the evaluation of a divisor $D$ against $f$, we have $f(D_1 + D_2) = f(D_1) \\, f(D_2)$.","title":"Evaluation is multiplicative"},{"id":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/eval-zsmul","kind":"node","lean":"CurveDivisor.eval_zsmul","parent":13852,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors/index.html#eval-zsmul","scope":"AlgebraicGeometry/algebraic-curves/divisors-on-curves/evaluation-of-curve-divisors","status":"fully_proved","summary":"For a commutative group $G$, every function $f : C \\to G$, every integer $n$ and every divisor $D$ on $C$, writing $f(D)$ for the evaluation of $D$ against $f$, we have $f(nD) = f(D)^n$.","title":"Evaluation of an integer multiple"},{"data_url":"scopes/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces.json","id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","kind":"scope","navigation":"module","parent":13839,"result":false,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"planned","title":"Jacobians of compact Riemann surfaces"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/genus","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.genus","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#genus","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"can_state","summary":"For a compact Riemann surface $X$, that is, a compact connected Hausdorff space with a complex-analytic atlas modelled on $\\mathbb{C}$, the genus of $X$ is a natural number.","title":"Genus of a compact Riemann surface"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/genus-eq-zero-iff-homeo","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.genus_eq_zero_iff_homeo","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#genus-eq-zero-iff-homeo","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ be a compact Riemann surface, that is, a compact connected Hausdorff space with a complex-analytic atlas modelled on $\\mathbb{C}$. The genus of $X$ is $0$ if and only if $X$ is homeomorphic to the unit sphere in Euclidean space $\\mathbb{R}^3$.","title":"A compact Riemann surface has genus zero iff it is a sphere"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/jacobian","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#jacobian","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"can_state","summary":"For a compact Riemann surface $X$, that is, a compact connected Hausdorff space with a complex-analytic atlas modelled on $\\mathbb{C}$, the Jacobian $J(X)$ of $X$ is a type in the same universe as $X$.","title":"Jacobian of a compact Riemann surface"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/of-curve","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.ofCurve","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#of-curve","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"planned","summary":"Let $X$ be a compact Riemann surface with Jacobian $J(X)$. Each point $P \\in X$ gives a map $X \\to J(X)$, the Abel\u2013Jacobi map based at $P$.","title":"Abel\u2013Jacobi map"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/of-curve-cont-mdiff","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.ofCurve_contMDiff","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#of-curve-cont-mdiff","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ be a compact Riemann surface of genus $g$, and let $J(X)$ be its Jacobian, a complex manifold modelled on $\\mathbb{C}^g$. For every point $P \\in X$, the Abel\u2013Jacobi map $X \\to J(X)$ based at $P$ is holomorphic.","title":"The Abel\u2013Jacobi map is holomorphic"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/of-curve-self","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.ofCurve_self","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#of-curve-self","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ be a compact Riemann surface with Jacobian $J(X)$, an abelian group. For every point $P \\in X$, the Abel\u2013Jacobi map $X \\to J(X)$ based at $P$ sends $P$ to $0$.","title":"The Abel\u2013Jacobi map sends the base point to zero"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/of-curve-inj","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.ofCurve_inj","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#of-curve-inj","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ be a compact Riemann surface with Jacobian $J(X)$, and let $P \\in X$. If the genus of $X$ is positive, then the Abel\u2013Jacobi map $X \\to J(X)$ based at $P$ is injective.","title":"The Abel\u2013Jacobi map is injective in positive genus"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pushforward","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pushforward","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pushforward","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"planned","summary":"Let $X$ and $Y$ be compact Riemann surfaces, that is, compact, connected, Hausdorff complex-analytic manifolds of dimension one, and write $J(X)$ for the Jacobian of $X$, an abelian group with a topology. For a holomorphic map $f : X \\to Y$, the pushforward $f_*$ is a continuous group homomorphism from $J(X)$ to $J(Y)$. It is meant to be the homomorphism induced by $f$; its properties are stated separately.","title":"Pushforward on Jacobians"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pushforward-cont-mdiff","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pushforward_contMDiff","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pushforward-cont-mdiff","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ and $Y$ be compact Riemann surfaces, that is, compact, connected, Hausdorff complex-analytic manifolds of dimension one, of genera $g_X$ and $g_Y$. The Jacobian $J(X)$ of $X$ is a complex-analytic manifold modeled on $\\mathbb{C}^{g_X}$, and likewise $J(Y)$ on $\\mathbb{C}^{g_Y}$. For every holomorphic map $f : X \\to Y$, the pushforward $f_* : J(X) \\to J(Y)$ is holomorphic.","title":"The pushforward on Jacobians is holomorphic"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pushforward-id-apply","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pushforward_id_apply","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pushforward-id-apply","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ be a compact Riemann surface, that is, a compact, connected, Hausdorff complex-analytic manifold of dimension one, and write $J(X)$ for its Jacobian. For every $P \\in J(X)$, the pushforward of $P$ along the identity map of $X$ is $P$: $(\\mathrm{id}_X)_*(P) = P$.","title":"Pushforward along the identity"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pushforward-comp-apply","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pushforward_comp_apply","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pushforward-comp-apply","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$, $Y$ and $Z$ be compact Riemann surfaces, that is, compact, connected, Hausdorff complex-analytic manifolds of dimension one, and write $J(X)$ for the Jacobian of $X$. For all holomorphic maps $f : X \\to Y$ and $g : Y \\to Z$ and every $P \\in J(X)$, the pushforwards satisfy $(g \\circ f)_*(P) = g_*(f_*(P))$.","title":"Pushforward respects composition"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pullback","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pullback","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pullback","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"planned","summary":"Let $X$ and $Y$ be compact Riemann surfaces, that is, compact, connected, Hausdorff complex-analytic manifolds of dimension one, and write $J(X)$ for the Jacobian of $X$, an abelian group with a topology. For a holomorphic map $f : X \\to Y$, the pullback $f^*$ is a continuous group homomorphism from $J(Y)$ to $J(X)$. It is meant to be the zero map when $f$ is constant and the usual pullback along $f$ otherwise; its\u2026","title":"Pullback on Jacobians"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pullback-cont-mdiff","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pullback_contMDiff","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pullback-cont-mdiff","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ and $Y$ be compact Riemann surfaces of genus $g_X$ and $g_Y$, with Jacobians $J(X)$ and $J(Y)$, complex manifolds modelled on $\\mathbb{C}^{g_X}$ and $\\mathbb{C}^{g_Y}$. For every holomorphic map $f : X \\to Y$, the induced pullback map $f^* : J(Y) \\to J(X)$ is holomorphic.","title":"The pullback map on Jacobians is holomorphic"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pullback-id-apply","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pullback_id_apply","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pullback-id-apply","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ be a compact Riemann surface, that is, a compact, connected, Hausdorff complex-analytic manifold of dimension one, and write $J(X)$ for its Jacobian. For every $P \\in J(X)$, the pullback of $P$ along the identity map of $X$ is $P$: $(\\mathrm{id}_X)^*(P) = P$.","title":"Pullback along the identity"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pullback-comp-apply","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pullback_comp_apply","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pullback-comp-apply","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$, $Y$ and $Z$ be compact Riemann surfaces, and for a holomorphic map $h$ between compact Riemann surfaces write $h^*$ for the induced pullback map on Jacobians. For all holomorphic maps $f : X \\to Y$ and $g : Y \\to Z$ and every point $P$ of the Jacobian $J(Z)$, we have $(g \\circ f)^*(P) = f^*(g^*(P))$.","title":"Pullback on Jacobians is compatible with composition"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/degree","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.degree","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#degree","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"can_state","summary":"Let $X$ and $Y$ be compact Riemann surfaces, that is, compact connected Hausdorff spaces with a complex-analytic atlas modelled on $\\mathbb{C}$. Every holomorphic map $f : X \\to Y$ has a degree, which is a natural number.","title":"Degree of a holomorphic map"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/pushforward-pullback","kind":"node","lean":"MathlibExt.Geometry.JacobianChallengeWanted.JacobianChallenge.Jacobian.pushforward_pullback","parent":13860,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces/index.html#pushforward-pullback","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-compact-riemann-surfaces","status":"stated","summary":"Let $X$ and $Y$ be compact Riemann surfaces, that is, compact, connected, Hausdorff complex-analytic manifolds of dimension one, and write $J(Y)$ for the Jacobian of $Y$, an abelian group. For every holomorphic map $f : X \\to Y$ and every $P \\in J(Y)$, the pushforward along $f$ of the pullback of $P$ along $f$ is $\\deg(f) \\cdot P$, where the degree $\\deg(f)$ is a natural number: $f_*(f^*(P)) = \\deg(f) \\cdot P$.","title":"Pushforward of a pullback is multiplication by the degree"},{"data_url":"scopes/AlgebraicGeometry/algebraic-curves/jacobians-of-curves.json","id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","kind":"scope","navigation":"module","parent":13839,"result":false,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"planned","title":"Jacobians of curves"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/genus","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.AlgebraicGeometry.JacobianChallenge.genus","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#genus","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"can_state","summary":"For a field $k$ and a curve $C$ over $k$, that is, a $k$-scheme that is proper, smooth of relative dimension $1$ and geometrically irreducible, the genus of $C$ is a natural number.","title":"Genus of a smooth proper curve"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/jacobian","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.AlgebraicGeometry.JacobianChallenge.Jacobian","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#jacobian","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"can_state","summary":"For a field $k$ and a curve $C$ over $k$, that is, a $k$-scheme that is proper, smooth of relative dimension $1$ and geometrically irreducible, the Jacobian $J(C)$ of $C$ is a $k$-scheme.","title":"Jacobian of a smooth proper curve"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/smooth-of-relative-dimension-genus","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.AlgebraicGeometry.JacobianChallenge.Jacobian.smoothOfRelativeDimension_genus","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#smooth-of-relative-dimension-genus","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"stated","summary":"Let $C$ be a proper, smooth, geometrically irreducible curve over a field $k$. The Jacobian $J(C)$ is smooth of relative dimension $g$ over $k$, where $g$ is the genus of $C$.","title":"The Jacobian has dimension equal to the genus"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/of-curve","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.AlgebraicGeometry.JacobianChallenge.Jacobian.ofCurve","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#of-curve","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"planned","summary":"Let $C$ be a proper, smooth, geometrically irreducible curve over a field $k$. Each $k$-rational point $P$ of $C$ gives a morphism of $k$-schemes $C \\to J(C)$ from $C$ to its Jacobian, the Abel\u2013Jacobi map based at $P$.","title":"Abel\u2013Jacobi map"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/comp-of-curve","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.AlgebraicGeometry.JacobianChallenge.Jacobian.comp_ofCurve","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#comp-of-curve","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"stated","summary":"Let $C$ be a proper, smooth, geometrically irreducible curve over a field $k$, let $J(C)$ be its Jacobian, and let $P$ be a $k$-rational point of $C$. The Abel\u2013Jacobi map $C \\to J(C)$ based at $P$ sends $P$ to the identity point of $J(C)$ for its group scheme structure.","title":"The Abel\u2013Jacobi map sends the base point to zero"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/exists-unique-of-curve-comp","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.AlgebraicGeometry.JacobianChallenge.Jacobian.exists_unique_ofCurve_comp","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#exists-unique-of-curve-comp","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"stated","summary":"Let $C$ be a proper, smooth, geometrically irreducible curve over a field $k$, let $J(C)$ be its Jacobian, let $P$ be a $k$-rational point of $C$, and let $\\iota_P : C \\to J(C)$ be the Abel\u2013Jacobi map based at $P$. Let $A$ be a group scheme over $k$ that is smooth, proper and geometrically irreducible. For every morphism of $k$-schemes $f : C \\to A$ sending $P$ to the identity point of $A$, there is a unique\u2026","title":"Universal property of the Jacobian"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/inst-comm-group-eq","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.AlgebraicGeometry.JacobianChallenge.instCommGroup_eq","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#inst-comm-group-eq","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"stated","summary":"Let $C$ be a proper, smooth, geometrically irreducible curve over a field $k$. The commutative group structure on the set of $k$-rational points of the Jacobian $J(C)$ is the one induced by its commutative group scheme structure: the product of two $k$-points $x$ and $y$ is $\\mu \\circ (x, y)$, where $\\mu$ is the multiplication morphism of $J(C)$.","title":"Group law on the rational points of the Jacobian"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/c","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.c","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#c","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"can_state","summary":"For natural numbers $g$ and $d$, $c(g, d)$ is a natural number: the constant in the uniform bound on rational points of curves in Theorem 1.1 of Dimitrov, Gao and Habegger.","title":"Uniform Mordell\u2013Lang constant"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/c-pos","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.c_pos","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#c-pos","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"stated","summary":"For all natural numbers $g$ and $d$, the uniform constant satisfies $c(g, d) \\ge 1$.","title":"The uniform Mordell\u2013Lang constant is at least one"},{"id":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves/theorem-1-1","kind":"node","lean":"MathlibExt.AlgebraicGeometry.JacobianChallengeWanted.theorem_1_1","parent":13878,"result":true,"route":"../roadmap/AlgebraicGeometry/algebraic-curves/jacobians-of-curves/index.html#theorem-1-1","scope":"AlgebraicGeometry/algebraic-curves/jacobians-of-curves","status":"stated","summary":"Let $g \\ge 2$ and $d \\ge 1$ be natural numbers, let $F$ be a number field with $[F : \\mathbb{Q}] \\le d$, and let $C$ be a proper, smooth, geometrically irreducible curve over $F$. If $C$ has genus $g$, then the number of $F$-rational points of $C$, possibly infinite, is at most $c(g, d)^{1 + \\rho}$, where $c$ is the uniform constant and $\\rho$ is the free rank of the group of $F$-rational points of the Jacobian of\u2026","title":"Uniform Mordell\u2013Lang for curves"},{"data_url":"scopes/AlgebraicGeometry/arithmetic-geometry.json","id":"AlgebraicGeometry/arithmetic-geometry","kind":"scope","parent":13624,"result":false,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/index.html","scope":"AlgebraicGeometry/arithmetic-geometry","status":"can_state","summary":"Rational points and point counts of varieties over number fields and finite fields, including the Mordell conjecture and the Weil conjectures.","title":"Arithmetic geometry"},{"id":"AlgebraicGeometry/arithmetic-geometry/weil-conjectures","kind":"node","lean":"MathlibExt.AlgebraicGeometry.WeilConjecturesWanted.AlgebraicGeometry.WeilConjectures.weil_conjectures","parent":13889,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/index.html#weil-conjectures","scope":"AlgebraicGeometry/arithmetic-geometry","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, and let $X$ be a scheme over $\\operatorname{Spec} F$ whose structure morphism is proper, geometrically irreducible and smooth of relative dimension $d$. Then there is a sequence $A_0, A_1, A_2, \\dots$ of finite multisets of complex numbers with $A_0 = \\{1\\}$ such that, for every $0 \\le i \\le 2d$, the multiset $A_{2d-i}$ is the image of $A_i$ under\u2026","title":"The Weil conjectures"},{"data_url":"scopes/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem.json","id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","kind":"scope","navigation":"module","parent":13889,"result":false,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"can_state","title":"Faltings's theorem"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/place","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.Place","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#place","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"fully_proved","summary":"Let $F$ be a field and $K$ a field with $F$ a $K$-algebra. A place of $F/K$ is a valuation ring of $F$ containing $K$ and different from $F$: a $K$-subalgebra $\\mathcal{O}$ of $F$ with $\\mathcal{O} \\neq F$ such that $z \\in \\mathcal{O}$ or $z^{-1} \\in \\mathcal{O}$ for every $z \\in F$. See [Stichtenoth, Definition 1.1.4 and 1.1.8].","title":"Place"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/finite-set-of-place-not-mem","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.finite_setOf_place_notMem","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#finite-set-of-place-not-mem","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"can_prove","summary":"Let $F$ be a function field over a field $K$, that is, a finite extension of the rational function field $K(t)$. For every $x \\in F$, there are only finitely many places $P$ of $F/K$ whose valuation ring $\\mathcal{O}_P$ does not contain $x$, that is, $x$ has only finitely many poles. See [Stichtenoth, Corollary 1.3.4].","title":"Finitely many poles"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/adele","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.Adele","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#adele","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"fully_proved","summary":"Let $F$ be a field and $K$ a field with $F$ a $K$-algebra. The adele ring $\\mathcal{A}_F$ of $F/K$ is the restricted product of copies of $F$ indexed by the places $P$ of $F/K$, with respect to their valuation rings $\\mathcal{O}_P$ and the cofinite filter: its elements are the families $\\alpha = (\\alpha_P)_P$ with $\\alpha_P \\in F$ for every place $P$ and $\\alpha_P \\in \\mathcal{O}_P$ for all but finitely many $P$.\u2026","title":"Adele ring"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/principal","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.Adele.principal","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#principal","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"can_state","summary":"Let $F$ be a function field over a field $K$, that is, a finite extension of the rational function field $K(t)$. The principal adele map is a ring homomorphism from $F$ to the adele ring $\\mathcal{A}_F$ of $F/K$. It is meant to be the diagonal embedding sending $x \\in F$ to the adele whose coordinate at every place $P$ of $F/K$ is $x$, which is well defined since $x$ has only finitely many poles; this coordinate\u2026","title":"Principal adele map"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/principal-apply","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.Adele.principal_apply","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#principal-apply","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"stated","summary":"Let $F$ be a function field over a field $K$, that is, a finite extension of the rational function field $K(t)$. For every $x \\in F$ and every place $P$ of $F/K$, the coordinate at $P$ of the principal adele of $x$ is $x$.","title":"Coordinates of a principal adele"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/inst-algebra-adele-algebra-map","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.instAlgebraAdele_algebraMap","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#inst-algebra-adele-algebra-map","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"stated","summary":"Let $F$ be a function field over a field $K$, that is, a finite extension of the rational function field $K(t)$. The algebra map $F \\to \\mathcal{A}_F$ of the $F$-algebra structure on the adele ring $\\mathcal{A}_F$ equals the principal adele map $F \\to \\mathcal{A}_F$.","title":"The adele algebra map is the principal embedding"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/integral-adele","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.integralAdele","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#integral-adele","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"fully_proved","summary":"Let $F$ be a field and $K$ a field with $F$ a $K$-algebra. The integral adeles $\\mathcal{A}_F(0)$ form the $K$-subspace of the adele ring $\\mathcal{A}_F$ of $F/K$ consisting of the adeles $\\alpha = (\\alpha_P)_P$ whose coordinate $\\alpha_P$ lies in the valuation ring $\\mathcal{O}_P$ of $P$ for every place $P$ of $F/K$. See [Stichtenoth, Definition 1.5.3].","title":"Integral adeles"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/genus","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.genus","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#genus","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"can_state","summary":"For a function field $F$ over a field $K$, that is, a finite extension of the rational function field $K(t)$, the genus of $F/K$ is a natural number $g$, specified by the separately stated formula $g = \\dim_K \\mathcal{A}_F / (\\mathcal{A}_F(0) + F)$, where $\\mathcal{A}_F$ is the adele ring and $\\mathcal{A}_F(0)$ the integral adeles. See [Stichtenoth, Corollary 1.5.5].","title":"Genus of a function field"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/genus-eq","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.genus_eq","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#genus-eq","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"stated","summary":"Let $F$ be a function field over a field $K$, that is, a finite extension of the rational function field $K(t)$. The genus $g$ of $F/K$ equals the dimension over $K$ of the quotient $\\mathcal{A}_F / (\\mathcal{A}_F(0) + F)$, where $\\mathcal{A}_F$ is the adele ring, $\\mathcal{A}_F(0)$ the integral adeles, and $F$ is embedded in $\\mathcal{A}_F$ by the algebra map of the $F$-algebra structure on the adeles (stated\u2026","title":"Genus as the dimension of the adele class space"},{"id":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/faltings","kind":"node","lean":"MathlibExt.AlgebraicGeometry.FaltingsWanted.AlgebraicGeometry.Faltings.faltings","parent":13891,"result":true,"route":"../roadmap/AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem/index.html#faltings","scope":"AlgebraicGeometry/arithmetic-geometry/faltings-s-theorem","status":"stated","summary":"Let $K$ be a number field and $F$ a function field over $K$, that is, a finite extension of the rational function field $K(t)$. If the genus of $F/K$ is at least $2$, then $F/K$ has only finitely many places of degree one, that is, places $P$ whose residue field (the quotient of the valuation ring of $P$ by its maximal ideal) has dimension $1$ over $K$.","title":"Faltings' theorem (Mordell conjecture)"},{"data_url":"scopes/AlgebraicGeometry/schemes-and-morphisms.json","id":"AlgebraicGeometry/schemes-and-morphisms","kind":"scope","parent":13624,"result":false,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/index.html","scope":"AlgebraicGeometry/schemes-and-morphisms","status":"fully_proved","summary":"The scheme-theoretic foundations of algebraic geometry: points, immersions, smooth morphisms, varieties over a field and properties that persist under base change.","title":"Schemes and morphisms"},{"data_url":"scopes/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions.json","id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","kind":"scope","navigation":"module","parent":13902,"result":false,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","title":"Closed and open immersions"},{"id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/is-iso-of-is-closed-immersion-of-finrank","kind":"node","lean":"AlgebraicGeometry.isIso_of_isClosedImmersion_of_finrank_comp_eq","parent":13903,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html#is-iso-of-is-closed-immersion-of-finrank","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","summary":"A closed immersion between maps of constant fibre rank is an isomorphism.","title":"Closed immersions of constant fibre rank are isomorphisms"},{"id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/exists-iso-hom-comp-eq-of-range-eq","kind":"node","lean":"AlgebraicGeometry.IsClosedImmersion.exists_iso_hom_comp_eq_of_range_eq","parent":13903,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html#exists-iso-hom-comp-eq-of-range-eq","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","summary":"Two reduced closed immersions with the same range are isomorphic over the target.","title":"Reduced closed immersions with equal range are isomorphic"},{"id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/of-is-closed-immersion-of-flat-comp-of-e","kind":"node","lean":"AlgebraicGeometry.IsOpenImmersion.of_isClosedImmersion_of_flat_comp_of_etale","parent":13903,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html#of-is-closed-immersion-of-flat-comp-of-e","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","summary":"A closed immersion whose composite with an \u00e9tale map is flat and locally of finite presentation is open.","title":"Closed immersions flat over the base of an \u00e9tale map are open"},{"id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/of-is-pullback","kind":"node","lean":"AlgebraicGeometry.IsClosedImmersion.of_isPullback","parent":13903,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html#of-is-pullback","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","summary":"Closed immersions transfer along pullback squares.","title":"Closed immersions are stable under base change"},{"id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/of-is-pullback-2","kind":"node","lean":"AlgebraicGeometry.Surjective.of_isPullback","parent":13903,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html#of-is-pullback-2","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","summary":"Surjectivity transfers along pullback squares.","title":"Surjectivity is stable under base change"},{"id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/of-is-pullback-3","kind":"node","lean":"AlgebraicGeometry.UniversallyClosed.of_isPullback","parent":13903,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html#of-is-pullback-3","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","summary":"Universal closedness transfers along a pullback square whose base map is a surjective closed immersion.","title":"Universal closedness descends along surjective closed immersions"},{"id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/of-is-pullback-4","kind":"node","lean":"AlgebraicGeometry.IsSeparated.of_isPullback","parent":13903,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html#of-is-pullback-4","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","summary":"Separatedness transfers along a pullback square whose base map is a surjective closed immersion.","title":"Separatedness descends along surjective closed immersions"},{"id":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/of-is-pullback-5","kind":"node","lean":"AlgebraicGeometry.IsProper.of_isPullback","parent":13903,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions/index.html#of-is-pullback-5","scope":"AlgebraicGeometry/schemes-and-morphisms/closed-and-open-immersions","status":"fully_proved","summary":"Properness transfers along a pullback square whose base map is a surjective closed immersion.","title":"Properness descends along surjective closed immersions"},{"data_url":"scopes/AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field.json","id":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field","kind":"scope","navigation":"module","parent":13902,"result":false,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/index.html","scope":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field","status":"fully_proved","title":"Geometric properties of schemes over a field"},{"id":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/geometrically-connected","kind":"node","lean":"AlgebraicGeometry.GeometricallyIrreducible.geometricallyConnected","parent":13912,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/index.html#geometrically-connected","scope":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field","status":"fully_proved","summary":"A geometrically irreducible morphism is geometrically connected.","title":"Geometrically irreducible morphisms are geometrically connected"},{"id":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/of-irreducible-space-of-is-alg-closed","kind":"node","lean":"AlgebraicGeometry.GeometricallyIrreducible.of_irreducibleSpace_of_isAlgClosed","parent":13912,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/index.html#of-irreducible-space-of-is-alg-closed","scope":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field","status":"fully_proved","summary":"An irreducible scheme over an algebraically closed field is geometrically irreducible.","title":"Geometric irreducibility over an algebraically closed field"},{"id":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/of-is-reduced-of-perfect-field","kind":"node","lean":"AlgebraicGeometry.GeometricallyReduced.of_isReduced_of_perfectField","parent":13912,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/index.html#of-is-reduced-of-perfect-field","scope":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field","status":"fully_proved","summary":"A reduced scheme over a perfect field is geometrically reduced.","title":"Reduced schemes over a perfect field are geometrically reduced"},{"id":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/of-is-integral-of-is-alg-closed","kind":"node","lean":"AlgebraicGeometry.GeometricallyIntegral.of_isIntegral_of_isAlgClosed","parent":13912,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field/index.html#of-is-integral-of-is-alg-closed","scope":"AlgebraicGeometry/schemes-and-morphisms/geometric-properties-of-schemes-over-a-field","status":"fully_proved","summary":"An integral scheme over an algebraically closed field is geometrically integral.","title":"Integral schemes over algebraically closed fields: geometric integrality"},{"data_url":"scopes/AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes.json","id":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes","kind":"scope","navigation":"module","parent":13902,"result":false,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/index.html","scope":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes","status":"fully_proved","title":"Points and sections of schemes"},{"id":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/exists-over-hom-base-closed-point-eq-of","kind":"node","lean":"AlgebraicGeometry.exists_over_hom_base_closedPoint_eq_of_isClosed_singleton","parent":13917,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/index.html#exists-over-hom-base-closed-point-eq-of","scope":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes","status":"fully_proved","summary":"A closed point of a scheme of finite type over an algebraically closed field lifts to a point over the base.","title":"Closed points of finite-type schemes over algebraically closed fields"},{"id":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/eq-of-fst-eq-of-snd-eq-of-is-iso-residue","kind":"node","lean":"AlgebraicGeometry.Scheme.Pullback.eq_of_fst_eq_of_snd_eq_of_isIso_residueFieldMap","parent":13917,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/index.html#eq-of-fst-eq-of-snd-eq-of-is-iso-residue","scope":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes","status":"fully_proved","summary":"Two points of a pullback with the same first and second projections are equal, when the residue field map at the second point's first projection is an isomorphism.","title":"Pullback points determined by projections given a residue isomorphism"},{"id":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/exists-section-comp-eq-iff-factors-of-un","kind":"node","lean":"AlgebraicGeometry.exists_section_comp_eq_iff_factors_of_universallyClosed_of_valuationRing","parent":13917,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/index.html#exists-section-comp-eq-iff-factors-of-un","scope":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes","status":"fully_proved","summary":"A section over a valuation ring factors through a universally closed open subscheme exactly when the generic point does.","title":"Valuative criterion for factoring sections through an open subscheme"},{"id":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/dense-set-of-exists-section-of-is-alg-cl","kind":"node","lean":"AlgebraicGeometry.dense_setOf_exists_section_of_isAlgClosed","parent":13917,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/index.html#dense-set-of-exists-section-of-is-alg-cl","scope":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes","status":"fully_proved","summary":"Rational points over an algebraically closed field are dense.","title":"Density of rational points over an algebraically closed field"},{"id":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/ext-of-forall-comp-eq-of-is-alg-closed","kind":"node","lean":"AlgebraicGeometry.ext_of_forall_comp_eq_of_isAlgClosed","parent":13917,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/index.html#ext-of-forall-comp-eq-of-is-alg-closed","scope":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes","status":"fully_proved","summary":"Two morphisms over an algebraically closed field agreeing on all rational points are equal.","title":"Morphisms are determined by their values on rational points"},{"id":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/exists-comp-eq-of-surjective-of-locally","kind":"node","lean":"AlgebraicGeometry.exists_comp_eq_of_surjective_of_locallyOfFiniteType_of_isAlgClosed","parent":13917,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes/index.html#exists-comp-eq-of-surjective-of-locally","scope":"AlgebraicGeometry/schemes-and-morphisms/points-and-sections-of-schemes","status":"fully_proved","summary":"A surjective morphism locally of finite type lifts rational points over an algebraically closed field.","title":"Surjective morphisms of finite type lift rational points"},{"data_url":"scopes/AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes.json","id":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes","kind":"scope","navigation":"module","parent":13902,"result":false,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/index.html","scope":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes","status":"fully_proved","title":"Reduced induced closed subschemes"},{"id":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/is-reduced-subscheme-vanishing-ideal","kind":"node","lean":"AlgebraicGeometry.Scheme.IdealSheafData.isReduced_subscheme_vanishingIdeal","parent":13924,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/index.html#is-reduced-subscheme-vanishing-ideal","scope":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes","status":"fully_proved","summary":"The subscheme cut out by a vanishing ideal is reduced.","title":"The subscheme cut out by a vanishing ideal is reduced"},{"id":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/vanishing-ideal-le-ker","kind":"node","lean":"AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_le_ker","parent":13924,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/index.html#vanishing-ideal-le-ker","scope":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes","status":"fully_proved","summary":"The vanishing ideal of a closed subset is contained in the kernel of any map from a reduced scheme landing in the subset.","title":"Vanishing ideal lies in the kernel of maps from reduced schemes"},{"id":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/exists-lift-subscheme-vanishing-ideal","kind":"node","lean":"AlgebraicGeometry.Scheme.IdealSheafData.exists_lift_subscheme\u03b9_vanishingIdeal","parent":13924,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/index.html#exists-lift-subscheme-vanishing-ideal","scope":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes","status":"fully_proved","summary":"A map from a reduced scheme landing in a closed subset lifts uniquely to the subscheme cut out by the vanishing ideal.","title":"Universal property of the reduced subscheme on a closed subset"},{"id":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/is-reduced-subscheme-vanishing-ideal-and","kind":"node","lean":"AlgebraicGeometry.Scheme.IdealSheafData.isReduced_subscheme_vanishingIdeal_and_le_ker","parent":13924,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/index.html#is-reduced-subscheme-vanishing-ideal-and","scope":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes","status":"fully_proved","summary":"The subscheme cut out by a vanishing ideal is reduced, and its ideal sheaf is contained in the kernel of any map from a reduced scheme landing in the subset.","title":"Reducedness and kernel bound for the vanishing-ideal subscheme"},{"id":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/exists-unique-comp-eq-of-range-subset-of","kind":"node","lean":"AlgebraicGeometry.IsClosedImmersion.existsUnique_comp_eq_of_range_subset_of_isReduced","parent":13924,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes/index.html#exists-unique-comp-eq-of-range-subset-of","scope":"AlgebraicGeometry/schemes-and-morphisms/reduced-induced-closed-subschemes","status":"fully_proved","summary":"A map from a reduced scheme into a closed immersion whose range contains the image lifts uniquely.","title":"Unique lifting of maps from reduced schemes through closed immersions"},{"data_url":"scopes/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms.json","id":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","kind":"scope","navigation":"module","parent":13902,"result":false,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/index.html","scope":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","status":"fully_proved","title":"Smooth morphisms"},{"id":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/ring-krull-dim-le-of-faithfully-flat","kind":"node","lean":"ringKrullDim_le_of_faithfullyFlat","parent":13930,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/index.html#ring-krull-dim-le-of-faithfully-flat","scope":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","status":"fully_proved","summary":"Krull dimension does not increase along faithfully flat ring maps.","title":"Krull dimension does not increase along faithfully flat maps"},{"id":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/ring-krull-dim-le-of-is-standard-smooth","kind":"node","lean":"ringKrullDim_le_of_isStandardSmoothOfRelativeDimension","parent":13930,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/index.html#ring-krull-dim-le-of-is-standard-smooth","scope":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","status":"fully_proved","summary":"A standard smooth algebra of relative dimension `n` has Krull dimension at most `n`.","title":"Krull dimension of a standard smooth algebra"},{"id":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/ring-krull-dim-le-of-ring-hom","kind":"node","lean":"SmoothKrullDimAux.ringKrullDim_le_of_ringHom","parent":13930,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/index.html#ring-krull-dim-le-of-ring-hom","scope":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","status":"fully_proved","summary":"A standard smooth algebra map from a field has Krull dimension at most the relative dimension.","title":"Krull dimension of a standard smooth algebra over a field"},{"id":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/topological-krull-dim-le","kind":"node","lean":"AlgebraicGeometry.SmoothOfRelativeDimension.topologicalKrullDim_le","parent":13930,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dimension of a scheme smooth of relative dimension "],["math","\\(n\\)"]],"source":"Dimension of a scheme smooth of relative dimension $n$"},"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/index.html#topological-krull-dim-le","scope":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","status":"fully_proved","summary":"A scheme smooth of relative dimension `n` over a field has topological Krull dimension at most `n`.","title":"Dimension of a scheme smooth of relative dimension $n$"},{"id":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/is-reduced-of-smooth-over-field","kind":"node","lean":"AlgebraicGeometry.isReduced_of_smooth_over_field","parent":13930,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/index.html#is-reduced-of-smooth-over-field","scope":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","status":"fully_proved","summary":"A scheme smooth over a field is reduced.","title":"Smooth schemes over a field are reduced"},{"id":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/is-reduced-of-is-reduced-of-is-locally-n","kind":"node","lean":"AlgebraicGeometry.Smooth.isReduced_of_isReduced_of_isLocallyNoetherian","parent":13930,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/index.html#is-reduced-of-is-reduced-of-is-locally-n","scope":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","status":"fully_proved","summary":"A smooth morphism with reduced locally Noetherian target has reduced source.","title":"Smooth morphisms preserve reducedness"},{"id":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/exists-smooth-of-relative-dimension-of-p","kind":"node","lean":"AlgebraicGeometry.Smooth.exists_smoothOfRelativeDimension_of_preconnectedSpace","parent":13930,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms/index.html#exists-smooth-of-relative-dimension-of-p","scope":"AlgebraicGeometry/schemes-and-morphisms/smooth-morphisms","status":"fully_proved","summary":"A smooth morphism from a preconnected scheme is smooth of some fixed relative dimension.","title":"Smooth maps from a preconnected scheme have constant relative dimension"},{"data_url":"scopes/AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field.json","id":"AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field","kind":"scope","navigation":"module","parent":13902,"result":false,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field/index.html","scope":"AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field","status":"fully_proved","title":"Varieties over a field"},{"id":"AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field/is-variety","kind":"node","lean":"AlgebraicGeometry.IsVariety","parent":13938,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field/index.html#is-variety","scope":"AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field","status":"fully_proved","summary":"Let $k$ be a field. A scheme $X$ with a chosen structure morphism $X \\to \\operatorname{Spec} k$ is a *variety* over $k$ if this morphism is separated, of finite type (quasi-compact and locally of finite type), and geometrically reduced.","title":"Geometrically reduced variety over a field"},{"id":"AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field/is-variety-iff","kind":"node","lean":"AlgebraicGeometry.IsVariety.isVariety_iff","parent":13938,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field/index.html#is-variety-iff","scope":"AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field","status":"fully_proved","summary":"Let $k$ be a field and let $X$ be a scheme with a structure morphism $X \\to \\operatorname{Spec} k$. Then $X$ is a variety over $k$ if and only if this structure morphism is separated, locally of finite type, quasi-compact and geometrically reduced.","title":"Characterization of varieties"},{"id":"AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field/quasi-separated","kind":"node","lean":"AlgebraicGeometry.IsVariety.quasiSeparated","parent":13938,"result":true,"route":"../roadmap/AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field/index.html#quasi-separated","scope":"AlgebraicGeometry/schemes-and-morphisms/varieties-over-a-field","status":"fully_proved","summary":"If $X$ is a variety over a field $k$, then its structure morphism $X \\to \\operatorname{Spec} k$ is quasi-separated, since it is separated.","title":"Varieties are quasi-separated"},{"area":"Geometry & Topology","data_url":"chapters/Geometry.json","id":"Geometry","kind":"scope","parent":0,"result":false,"route":"../roadmap/Geometry/index.html","scope":"Geometry","status":"can_prove","summary":"Geometry studies shapes, spaces, transformations, curvature, and the invariants preserved by geometric equivalence.","title":"Geometry"},{"data_url":"scopes/Geometry/discrete-geometry.json","id":"Geometry/discrete-geometry","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/discrete-geometry/index.html","scope":"Geometry/discrete-geometry","status":"can_prove","summary":"Finite point configurations and their distances, polytopes and polyhedra, dissections and tilings, lattice points and extremal shape problems.","title":"Discrete geometry"},{"data_url":"scopes/Geometry/discrete-geometry/part-01.json","id":"Geometry/discrete-geometry/part-01","kind":"scope","navigation":"transparent","parent":13943,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/index.html","scope":"Geometry/discrete-geometry/part-01","status":"can_prove","title":"Part 1 of 2"},{"id":"Geometry/discrete-geometry/part-01/discrete-full-rank-iff-is-zlattice","kind":"node","lean":"Submodule.discrete_full_rank_iff_isZLattice","parent":13944,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/index.html#discrete-full-rank-iff-is-zlattice","scope":"Geometry/discrete-geometry/part-01","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space and let $L \\subseteq E$ be a $\\mathbb{Z}$-submodule (an additive subgroup) that is discrete in the subspace topology. Then the $\\mathbb{Z}$-rank of $L$ equals $\\dim_{\\mathbb{R}} E$ if and only if $L$ is a $\\mathbb{Z}$-lattice in $E$, that is, $L$ spans $E$ over $\\mathbb{R}$.","title":"A discrete subgroup is a lattice iff it has full rank"},{"id":"Geometry/discrete-geometry/part-01/erdos-anning","kind":"node","lean":"MetaMathlibExt.erdos_anning","parent":13944,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/index.html#erdos-anning","scope":"Geometry/discrete-geometry/part-01","status":"fully_proved","summary":"An infinite set of points in the plane with pairwise integer distances is collinear.","title":"Erd\u0151s\u2013Anning theorem"},{"id":"Geometry/discrete-geometry/part-01/euler-polyhedron-formula","kind":"node","lean":"MetaMathlibExt.euler_polyhedron_formula","parent":13944,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/index.html#euler-polyhedron-formula","scope":"Geometry/discrete-geometry/part-01","status":"can_prove","summary":"Let $V$, $E$ and $F$ be finite sets, let $\\iota : V \\to \\mathbb{R}^3$ be an injective map into Euclidean space, and let $P = \\operatorname{conv} \\iota(V)$ be the convex hull of its image; assume $P$ has nonempty interior $\\operatorname{int} P$, and write $\\partial P$ for its frontier. Each edge $e \\in E$ has an ordered pair of endpoints $(s(e), t(e)) \\in V \\times V$, with segment $[e] = [\\iota(s(e)), \\iota(t(e))]$;\u2026","title":"Euler's polyhedron formula"},{"id":"Geometry/discrete-geometry/part-01/monsky-theorem","kind":"node","lean":"MetaMathlibExt.monsky_theorem","parent":13944,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/index.html#monsky-theorem","scope":"Geometry/discrete-geometry/part-01","status":"fully_proved","summary":"Let $T$ be a finite set of subsets of the Euclidean plane $\\mathbb{R}^2$ such that every member of $T$ is the convex hull of three points $a, b, c$ of the plane (not necessarily distinct), the union of the members of $T$ is the closed unit square $[0,1]^2$, any two distinct members of $T$ have disjoint interiors, and all members of $T$ have the same Lebesgue measure. Then the number of members of $T$ is even.","title":"Monsky's theorem"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem.json","id":"Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem/index.html","scope":"Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem","status":"can_prove","title":"Blumenthal's minimax angle problem"},{"id":"Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem/minimax-angle","kind":"node","lean":"MathlibExt.Combinatorics.BlumenthalMinimaxAngleErdosProblem504Wanted.minimaxAngle","parent":13949,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem/index.html#minimax-angle","scope":"Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem","status":"fully_proved","summary":"For a natural number $N$, the minimax angle $\\alpha_N$ is the supremum of the $\\alpha \\in [0, \\pi]$ such that every set of $N$ points in the Euclidean plane contains three distinct points $x, y, z$ with $\\angle xyz \\ge \\alpha$, where $\\angle xyz$ is the unoriented angle at $y$.","title":"Minimax angle"},{"id":"Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem/minimax-angle-values","kind":"node","lean":"MathlibExt.Combinatorics.BlumenthalMinimaxAngleErdosProblem504Wanted.minimaxAngleValues","parent":13949,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem/index.html#minimax-angle-values","scope":"Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem","status":"fully_proved","summary":"For a natural number $N$, let $\\alpha_N$ be the minimax angle of $N$ points in the plane: the supremum of the $\\alpha \\in [0, \\pi]$ such that every set of $N$ points in the plane contains three distinct points $x, y, z$ with $\\angle xyz \\ge \\alpha$ (the angle at $y$). The statement is Sendov's answer: $\\alpha_3 = \\pi/3$, $\\alpha_4 = \\pi/2$, and for all natural numbers $n \\ge 2$ and $N$, if\u2026","title":"Blumenthal's minimax angle values (Erd\u0151s Problem 504)"},{"id":"Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem/minimax-angle-values-holds","kind":"node","lean":"MathlibExt.Combinatorics.BlumenthalMinimaxAngleErdosProblem504Wanted.minimaxAngleValues_holds","parent":13949,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem/index.html#minimax-angle-values-holds","scope":"Geometry/discrete-geometry/part-01/blumenthal-s-minimax-angle-problem","status":"can_prove","summary":"For a natural number $N$, let $\\alpha_N$ be the minimax angle of $N$ points in the plane: the supremum of the $\\alpha \\in [0, \\pi]$ such that every set of $N$ points in the plane contains three distinct points $x, y, z$ with $\\angle xyz \\ge \\alpha$ (the angle at $y$). Then $\\alpha_3 = \\pi/3$, $\\alpha_4 = \\pi/2$, and for all natural numbers $n \\ge 2$ and $N$: if $2^n < N \\le 2^n + 2^{n-2}$, then\u2026","title":"Sendov's solution of Blumenthal's minimax angle problem (Erd\u0151s Problem 504)"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns.json","id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","title":"Continuous Tur\u00e1n numbers of planar patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/point-set-contains-pattern","kind":"node","lean":"MetaMathlibExt.PointSetContainsPattern","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#point-set-contains-pattern","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"A set $S \\subseteq \\mathbb{R}^2$ contains a set $P \\subseteq \\mathbb{R}^2$ when there are maps $f_X, f_Y : \\mathbb{R} \\to \\mathbb{R}$ such that $f_X(x_0) - f_X(x_1) \\ge x_0 - x_1$ whenever $x_0 > x_1$ are $x$-coordinates of points of $P$, $f_Y(y_0) - f_Y(y_1) \\ge y_0 - y_1$ whenever $y_0 > y_1$ are $y$-coordinates of points of $P$, and $(f_X(x), f_Y(y)) \\in S$ for every $(x, y) \\in P$.","title":"Containment of planar point sets"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/point-set-contains-pattern-empty","kind":"node","lean":"MetaMathlibExt.pointSetContainsPattern_empty","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#point-set-contains-pattern-empty","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"Every set $S \\subseteq \\mathbb{R}^2$ contains the empty set $\\emptyset$ in the sense of containment of planar point sets.","title":"Every planar set contains the empty pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx","kind":"node","lean":"MetaMathlibExt.opx","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For a real number $n$ and a set $P \\subseteq \\mathbb{R}^2$, the continuous Tur\u00e1n number $\\operatorname{opx}(n, P)$ is the supremum of the Lebesgue measures $\\mu_2(S)$ over all open sets $S \\subseteq [0, n]^2$ that do not contain $P$ in the sense of point-set pattern containment; it is $0$ when there is no such set.","title":"Continuous Tur\u00e1n number in the plane"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-nonneg","kind":"node","lean":"MetaMathlibExt.opx_nonneg","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-nonneg","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every real number $n$ and every set $P \\subseteq \\mathbb{R}^2$, $\\operatorname{opx}(n, P) \\ge 0$.","title":"Nonnegativity of the continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-empty","kind":"node","lean":"MetaMathlibExt.opx_empty","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-empty","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every real number $n$, $\\mathrm{opx}(n, \\emptyset) = 0$: every set contains the empty pattern, so no set avoids it.","title":"Continuous Tur\u00e1n number of the empty pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-rotate-reflect","kind":"node","lean":"MetaMathlibExt.opx_rotate_reflect","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-rotate-reflect","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every real number $n$ and every set $P \\subseteq \\mathbb{R}^2$, the value $\\operatorname{opx}(n, P)$ does not change when $P$ is replaced by its image under the rotation by $90^\\circ$, $(x, y) \\mapsto (-y, x)$, under the reflection $(x, y) \\mapsto (-x, y)$, or under the reflection $(x, y) \\mapsto (x, -y)$.","title":"Rotation and reflection invariance of the continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-translate","kind":"node","lean":"MetaMathlibExt.opx_translate","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-translate","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $n$ be a real number, $P \\subseteq \\mathbb{R}^2$ a set and $v \\in \\mathbb{R}^2$ a vector, and let $P + v = \\{p + v : p \\in P\\}$ be the translate of $P$ by $v$. Then $\\operatorname{opx}(n, P + v) = \\operatorname{opx}(n, P)$.","title":"Translation invariance of the continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-mono-of-contains-pattern","kind":"node","lean":"MetaMathlibExt.opx_mono_of_containsPattern","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-mono-of-contains-pattern","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $n$ be a real number and let $P, Q \\subseteq \\mathbb{R}^2$. If $Q$ contains $P$ in the sense of containment of planar point sets, then $\\operatorname{opx}(n, P) \\le \\operatorname{opx}(n, Q)$.","title":"Monotonicity of the continuous Tur\u00e1n number under containment"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-mono-left","kind":"node","lean":"MetaMathlibExt.opx_mono_left","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-mono-left","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every set $P \\subseteq \\mathbb{R}^2$ and all real numbers $m \\le n$, the continuous Tur\u00e1n numbers satisfy $\\operatorname{opx}(m, P) \\le \\operatorname{opx}(n, P)$.","title":"Monotonicity of the continuous Tur\u00e1n number in the side length"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-superadditive","kind":"node","lean":"MetaMathlibExt.opx_superadditive","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-superadditive","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every set $P \\subseteq \\mathbb{R}^2$ and all real numbers $m, n > 0$, $\\operatorname{opx}(m, P) + \\operatorname{opx}(n, P) \\le \\operatorname{opx}(m + n, P)$.","title":"Superadditivity of the continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-add-lt","kind":"node","lean":"MetaMathlibExt.opx_add_lt","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-add-lt","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ and let $n, \\delta$ be real numbers with $n > 0$ and $0 < \\delta < n$. Then the continuous Tur\u00e1n number $\\mathrm{opx}$ satisfies $\\mathrm{opx}(n, P) \\le \\mathrm{opx}(n + \\delta, P)$ and $\\mathrm{opx}(n + \\delta, P) < \\mathrm{opx}(n, P) + 3 n \\delta$.","title":"Increment bound for the continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-continuous-on","kind":"node","lean":"MetaMathlibExt.opx_continuousOn","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-continuous-on","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every $P \\subseteq \\mathbb{R}^2$, the continuous Tur\u00e1n number $\\mathrm{opx}(n, P)$ is a continuous function of $n$ on $(0, \\infty)$.","title":"Continuity of the continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/volume-section-le-of-segment-free","kind":"node","lean":"MetaMathlibExt.volume_section_le_of_segment_free","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#volume-section-le-of-segment-free","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $a, b, c, n$ be real numbers with $c \\ge 0$, and let $S \\subseteq [0, n]^2$ be an open set that does not contain the closed horizontal segment $[a, a + c] \\times \\{b\\}$ in the sense of containment of planar point sets. Then for every $y \\in [0, n]$ the horizontal section $\\{x \\in \\mathbb{R} : (x, y) \\in S\\}$ has one-dimensional Lebesgue measure at most $c$.","title":"Sections of open sets avoiding a horizontal segment"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-union-segment-le","kind":"node","lean":"MetaMathlibExt.opx_union_segment_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-union-segment-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ and $(x_0, y_0) \\in P$ with $x \\le x_0$ for every $(x, y) \\in P$, and let $c \\ge 0$ and $n > 0$ be real numbers. Then","title":"Extending a pattern by a horizontal segment at its right edge"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-insert-point-le","kind":"node","lean":"MetaMathlibExt.opx_insert_point_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-insert-point-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ and let $(x_0, y_0) \\in P$ be such that every point of $P$ has $x$-coordinate at most $x_0$. Then $\\operatorname{opx}(n, P \\cup \\{(x_0 + c, y_0)\\}) \\le \\operatorname{opx}(n, P) + cn$ for all real numbers $c > 0$ and $n > 0$.","title":"Adding a point to the right of a pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-dilate-le","kind":"node","lean":"MetaMathlibExt.opx_dilate_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-dilate-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ be finite, and let $c > 0$ and $d$ be real numbers such that any two consecutive distinct $x$-coordinates of points of $P$ differ by at least $c$ and at most $d$, and likewise for $y$-coordinates. Let $q > 1$ and $qP = \\{qp : p \\in P\\}$. Then for every real $n > 0$, $\\mathrm{opx}(n, qP) \\le (\\lceil qd/c \\rceil + 1)^2 \\, \\mathrm{opx}(m, P)$, where\u2026","title":"Continuous Tur\u00e1n number of a dilated finite pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-horizontal-segment","kind":"node","lean":"MetaMathlibExt.opx_horizontal_segment","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-horizontal-segment","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $a, b, c, n$ be real numbers with $0 \\le c \\le n$, and let $P = [a, a + c] \\times \\{b\\}$ be the closed horizontal segment of length $c$ at height $b$. Then $\\operatorname{opx}(n, P) = cn$.","title":"Continuous Tur\u00e1n number of a horizontal segment"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-diagonal-segment","kind":"node","lean":"MetaMathlibExt.opx_diagonal_segment","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-diagonal-segment","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $a, b, n$ be real numbers with $0 < a \\le n$ and $0 < b \\le n$, and let $P$ be the closed segment from $(0, 0)$ to $(a, b)$. Then the continuous Tur\u00e1n number satisfies $\\operatorname{opx}(n, P) = (a + b)n - ab$.","title":"Continuous Tur\u00e1n number of a diagonal segment"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-open-is-theta","kind":"node","lean":"MetaMathlibExt.opx_open_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-open-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every nonempty open set $P \\subseteq \\mathbb{R}^2$, $\\operatorname{opx}(n, P) = \\Theta(n^2)$ as the real number $n$ tends to infinity.","title":"Open patterns have quadratic continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-rational-points-is-theta","kind":"node","lean":"MetaMathlibExt.opx_rational_points_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-rational-points-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ be a nonempty open set and let $Q_P$ be the set of points of $P$ both of whose coordinates are rational. Then $\\operatorname{opx}(n, Q_P) = \\Theta(n^2)$ as the real number $n$ tends to infinity.","title":"Rational points of open patterns have quadratic continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-increasing-graph-is-theta","kind":"node","lean":"MetaMathlibExt.opx_increasing_graph_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-increasing-graph-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $a > 0$ and $b$ be real numbers and let $f : \\mathbb{R} \\to \\mathbb{R}$ be strictly increasing on $[0, a]$ with $(f(t) - f(s)) / (t - s) \\le b$ for all $s < t$ in $[0, a]$. Then the graph $P = \\{(t, f(t)) : t \\in [0, a]\\}$ satisfies $\\operatorname{opx}(n, P) = \\Theta(n)$ as the real number $n$ tends to infinity.","title":"Continuous Tur\u00e1n number of the graph of an increasing function"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/exists-large-jrr-free-cells","kind":"node","lean":"MetaMathlibExt.exists_large_Jrr_free_cells","parent":13953,"reading_title":{"format":"inline-parts-v1","parts":[["text","Large sets of grid cells with no "],["math","\\(r \\times r\\)"],["text"," all-ones submatrix"]],"source":"Large sets of grid cells with no $r \\times r$ all-ones submatrix"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#exists-large-jrr-free-cells","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For all natural numbers $r \\ge 2$ and $n \\ge 1$ there is a set $S$ of cells of the $n \\times n$ grid $\\{0, \\dots, n - 1\\}^2$ with $|S| \\ge \\frac{1}{2} n^{2 - 2/(r+1)}$ that contains no $r \\times r$ all-ones submatrix: there are no sets $R, C \\subseteq \\{0, \\dots, n - 1\\}$ with $|R| = |C| = r$ and $R \\times C \\subseteq S$.","title":"Large sets of grid cells with no $r \\times r$ all-ones submatrix"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-equal-sign-is-theta","kind":"node","lean":"MetaMathlibExt.opx_equal_sign_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-equal-sign-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $s > 0$ and $c > 0$ be real numbers and let $P_{s,2,c} = [0, s] \\times \\{c, 2c\\}$ be the equal-sign pattern, two horizontal segments of length $s$ at heights $c$ and $2c$. Then, as the real number $n$ tends to infinity with $s$ and $c$ fixed, $\\operatorname{opx}(n, P_{s,2,c}) = \\Theta(s^{1/2} n^{3/2})$.","title":"Continuous Tur\u00e1n number of the equal-sign pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/volume-staircase-simplex-ge","kind":"node","lean":"MetaMathlibExt.volume_staircase_simplex_ge","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#volume-staircase-simplex-ge","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $t$ be a natural number and let $c, n$ be real numbers with $c \\ge 0$ and $ct \\le n$. Let $X \\subseteq \\mathbb{R}^t$ be the set of points $y = (y_0, \\dots, y_{t-1})$ such that $y_i < n - ic$ for every $i$, $y_{i+1} + c < y_i$ for every $i < t - 1$, and $y_{t-1} > 0$ if $t \\ge 1$. Then the $t$-dimensional Lebesgue measure of $X$ is at least $(n/t - c)^t$, where $n/t$ is read as $0$ when $t = 0$.","title":"Lower bound on the volume of a staircase region"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-stacked-segments-is-big-o","kind":"node","lean":"MetaMathlibExt.opx_stacked_segments_isBigO","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-stacked-segments-is-big-o","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $t \\ge 2$ be a natural number, let $s, c > 0$ be real numbers, and let $P_{s,t,c} = [0, s] \\times \\{c, 2c, \\dots, tc\\}$ be the union of the $t$ horizontal segments of length $s$ at heights $c, 2c, \\dots, tc$. Then $\\operatorname{opx}(n, P_{s,t,c}) = O(s^{1/t} n^{2 - 1/t})$ as the real number $n$ tends to infinity, with $s$, $t$ and $c$ fixed.","title":"Upper bound for stacked horizontal segments"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-horizontal-segments-is-big-o","kind":"node","lean":"MetaMathlibExt.opx_horizontal_segments_isBigO","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-horizontal-segments-is-big-o","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $k$ be a natural number and let $a_i \\le b_i$ and $y_i$ be real numbers for $i = 1, \\dots, k$, such that the intervals $[a_i, b_i]$ are pairwise disjoint and the heights $y_i$ are pairwise distinct, and let $P = \\bigcup_{i=1}^k [a_i, b_i] \\times \\{y_i\\}$. Then the continuous Tur\u00e1n number satisfies $\\operatorname{opx}(n, P) = O(n)$ as the real number $n$ tends to infinity.","title":"Linear bound for separated horizontal segments"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/cells-contain-pattern","kind":"node","lean":"MetaMathlibExt.CellsContainPattern","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#cells-contain-pattern","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"A set of cells $A \\subseteq \\mathbb{N} \\times \\mathbb{N}$, each cell written (column, row), contains a set $P \\subseteq \\mathbb{R}^2$ when there are maps $f_X, f_Y : \\mathbb{R} \\to \\mathbb{N}$ such that $f_X$ is strictly increasing on the set of $x$-coordinates of points of $P$, $f_Y$ is strictly increasing on the set of $y$-coordinates of points of $P$, and $(f_X(x), f_Y(y)) \\in A$ for every $(x, y) \\in P$.","title":"Containment of a point pattern in a set of grid cells"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/pattern-ex","kind":"node","lean":"MetaMathlibExt.patternEx","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#pattern-ex","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For a natural number $n$ and a set $P \\subseteq \\mathbb{R}^2$, $\\mathrm{ex}(n, M_P)$ is the largest number of cells of the $n \\times n$ grid $\\{0, \\dots, n - 1\\}^2$ forming a set that does not contain $P$ (cells contain $P$ when there are strictly increasing maps from the $x$-coordinates of $P$ to columns and from the $y$-coordinates of $P$ to rows that send every point of $P$ to a cell of the set), and $0$ if\u2026","title":"0-1 matrix extremal function of a point pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/pattern-ex-empty","kind":"node","lean":"MetaMathlibExt.patternEx_empty","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#pattern-ex-empty","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every natural number $n$, $\\operatorname{ex}(n, M_\\emptyset) = 0$, since every set of cells contains the empty pattern.","title":"The extremal function of the empty pattern vanishes"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/pattern-ex-singleton","kind":"node","lean":"MetaMathlibExt.patternEx_singleton","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#pattern-ex-singleton","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every natural number $n$ and every point $p \\in \\mathbb{R}^2$, the 0-1 matrix extremal function of the one-point pattern $\\{p\\}$ on the $n \\times n$ grid is zero: $\\mathrm{ex}(n, M_{\\{p\\}}) = 0$. Indeed, a single point is contained in every nonempty set of cells.","title":"The extremal function of a single point vanishes"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/pattern-ex-le-sq","kind":"node","lean":"MetaMathlibExt.patternEx_le_sq","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#pattern-ex-le-sq","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"Let $n$ be a natural number and $P \\subseteq \\mathbb{R}^2$ a set, and let $\\mathrm{ex}(n, M_P)$ be the largest number of cells of the $n \\times n$ grid $\\{0, \\dots, n - 1\\}^2$ forming a set that does not contain $P$ ($0$ if there is no such set). Then $\\mathrm{ex}(n, M_P) \\le n^2$.","title":"Trivial upper bound for the 0-1 matrix extremal function"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/inverse-ackermann","kind":"node","lean":"MetaMathlibExt.inverseAckermann","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#inverse-ackermann","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For a natural number $n$, the inverse Ackermann function $\\alpha(n)$ is the least natural number $k$ with $n \\le A(k, k)$, where $A$ is the two-argument Ackermann function, given for all natural numbers $m$ and $j$ by $A(0, j) = j + 1$, $A(m + 1, 0) = A(m, 1)$ and $A(m + 1, j + 1) = A(m, A(m + 1, j))$.","title":"Inverse Ackermann function"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/le-ack-inverse-ackermann","kind":"node","lean":"MetaMathlibExt.le_ack_inverseAckermann","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#le-ack-inverse-ackermann","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every natural number $n$, $n \\le A(\\alpha(n), \\alpha(n))$, where $\\alpha$ is the inverse Ackermann function and $A$ is the two-argument Ackermann function.","title":"Defining inequality of the inverse Ackermann function"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/inverse-ackermann-le","kind":"node","lean":"MetaMathlibExt.inverseAckermann_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#inverse-ackermann-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, if $n \\le A(k, k)$, where $A$ is the two-argument Ackermann function, then $\\alpha(n) \\le k$, where $\\alpha$ is the inverse Ackermann function.","title":"Upper bound for the inverse Ackermann function"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/inverse-ackermann-two","kind":"node","lean":"MetaMathlibExt.inverseAckermann_two","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#inverse-ackermann-two","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"The inverse Ackermann function $\\alpha$ satisfies $\\alpha(2) = 1$, since $A(0, 0) = 1 < 2 \\le 3 = A(1, 1)$, where $A$ is the two-argument Ackermann function.","title":"The inverse Ackermann function at 2"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/inverse-ackermann-eight","kind":"node","lean":"MetaMathlibExt.inverseAckermann_eight","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#inverse-ackermann-eight","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"The inverse Ackermann function $\\alpha$ satisfies $\\alpha(8) = 3$, since $A(2, 2) = 7 < 8 \\le 61 = A(3, 3)$, where $A$ is the two-argument Ackermann function.","title":"The inverse Ackermann function at 8"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-is-theta-pattern-ex","kind":"node","lean":"MetaMathlibExt.opx_isTheta_patternEx","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-is-theta-pattern-ex","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P$ be a finite subset of $\\mathbb{R}^2$. Then the continuous Tur\u00e1n number $\\mathrm{opx}(n, P)$ and the 0-1 matrix extremal function $\\mathrm{ex}(n, M_P)$ of $P$ on the $n \\times n$ grid satisfy $\\mathrm{opx}(n, P) = \\Theta(\\mathrm{ex}(n, M_P))$ as $n \\to \\infty$ through the natural numbers, with constants depending on $P$.","title":"Continuous Tur\u00e1n numbers versus 0-1 matrix extremal functions"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/pattern-ex-is-big-o-opx","kind":"node","lean":"MetaMathlibExt.patternEx_isBigO_opx","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#pattern-ex-is-big-o-opx","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P$ be a finite subset of $\\mathbb{R}^2$. For a natural number $n$, write $\\mathrm{opx}(n, P)$ for the continuous Tur\u00e1n number of $P$ (the supremum of the Lebesgue measures of the open subsets of $[0, n]^2$ that do not contain $P$, or $0$ if there are none) and $\\mathrm{ex}(n, M_P)$ for the 0-1 matrix extremal function of $P$ on the $n \\times n$ grid. Then $\\mathrm{ex}(n, M_P) = O(\\mathrm{opx}(n, P))$ as\u2026","title":"Lower bound for continuous Tur\u00e1n numbers of finite patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-is-big-o-pattern-ex","kind":"node","lean":"MetaMathlibExt.opx_isBigO_patternEx","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-is-big-o-pattern-ex","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P$ be a finite subset of $\\mathbb{R}^2$. Then the continuous Tur\u00e1n number $\\mathrm{opx}(n, P)$ and the 0-1 matrix extremal function $\\mathrm{ex}(n, M_P)$ of $P$ on the $n \\times n$ grid satisfy $\\mathrm{opx}(n, P) = O(\\mathrm{ex}(n, M_P))$ as $n \\to \\infty$ through the natural numbers.","title":"Upper bound for continuous Tur\u00e1n numbers of finite patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-permutation-pattern-le","kind":"node","lean":"MetaMathlibExt.opx_permutation_pattern_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-permutation-pattern-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"There is a real constant $C$ such that for every real $c > 0$ there is a real constant $K$ with the following property. Let $P \\subseteq \\mathbb{R}^2$ be a finite set of $k$ points whose $x$-coordinates are pairwise distinct and whose $y$-coordinates are pairwise distinct, such that any two consecutive $x$-coordinates of points of $P$ differ by at most $c$, and so do any two consecutive $y$-coordinates. Then\u2026","title":"Linear bound for permutation-like point patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-grid-is-big-o","kind":"node","lean":"MetaMathlibExt.opx_grid_isBigO","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-grid-is-big-o","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $X$ and $Y$ be finite sets of real numbers, let $P = X \\times Y$ be the grid they span and let $t = |X|$ be its number of columns. Then $\\operatorname{opx}(n, P) = O(n^{2 - 1/t})$ as the natural number $n$ tends to infinity, where for $t = 0$ (so that $P$ is empty) the exponent $2 - 1/t$ is read as $2$.","title":"Upper bound for the continuous Tur\u00e1n number of a grid"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/exists-opx-is-big-o-rpow","kind":"node","lean":"MetaMathlibExt.exists_opx_isBigO_rpow","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#exists-opx-is-big-o-rpow","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ be finite. Then there is a real number $\\varepsilon > 0$ such that the continuous Tur\u00e1n number satisfies $\\operatorname{opx}(n, P) = O(n^{2 - \\varepsilon})$ as the natural number $n$ tends to infinity.","title":"Power saving for continuous Tur\u00e1n numbers of finite planar patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-grid-two-columns-is-theta","kind":"node","lean":"MetaMathlibExt.opx_grid_two_columns_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-grid-two-columns-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $X$ and $Y$ be finite sets of real numbers with $|X| = 2$ and $|Y| \\ge 2$, and let $P = X \\times Y$ be the grid with $2$ columns and $|Y|$ rows. Then $\\operatorname{opx}(n, P) = \\Theta(n^{3/2})$ as the natural number $n$ tends to infinity.","title":"Continuous Tur\u00e1n number of a grid with two columns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-grid-three-columns-is-theta","kind":"node","lean":"MetaMathlibExt.opx_grid_three_columns_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-grid-three-columns-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $X$ and $Y$ be finite sets of real numbers with $|X| = 3$ and $|Y| \\ge 3$, and let $P = X \\times Y$ be the grid with $3$ columns and $|Y|$ rows. Then $\\operatorname{opx}(n, P) = \\Theta(n^{5/3})$ as the natural number $n$ tends to infinity.","title":"Continuous Tur\u00e1n number of a grid with three columns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/exists-opx-is-big-o-inverse-ackermann","kind":"node","lean":"MetaMathlibExt.exists_opx_isBigO_inverseAckermann","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#exists-opx-is-big-o-inverse-ackermann","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ be a finite set whose points have pairwise distinct $y$-coordinates. Then there is a natural number $t$ such that $\\operatorname{opx}(n, P) = O(n \\cdot 2^{\\alpha(n)^t})$ as the natural number $n$ tends to infinity, where $\\operatorname{opx}$ is the continuous Tur\u00e1n number and $\\alpha$ is the inverse Ackermann function.","title":"Almost linear bound for patterns with at most one point in each row"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-two-row-growing-iff","kind":"node","lean":"MetaMathlibExt.opx_two_row_growing_iff","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-two-row-growing-iff","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For each real $n$ let $Q_n \\subseteq \\mathbb{R}^2$ be a set all of whose points have $y$-coordinate $0$ or $1$, whose least $x$-coordinate is $0$ and whose greatest $x$-coordinate is $s(n)$. Then the following are equivalent: for every $\\varepsilon > 0$, $\\operatorname{opx}(n, Q_n) \\ge n^{2 - \\varepsilon}$ for all sufficiently large real $n$; and for every $\\varepsilon > 0$, $s(n) \\ge n^{1 - \\varepsilon}$ for all\u2026","title":"Two-row patterns of growing width"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/is-opx-saturating","kind":"node","lean":"MetaMathlibExt.IsOpxSaturating","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#is-opx-saturating","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"Let $n$ be a real number and let $P$ and $S$ be subsets of $\\mathbb{R}^2$. Then $S$ is saturating for $P$ in $[0, n]^2$ when $S$ is an open subset of $[0, n]^2$ that does not contain $P$, while every open set $S'$ with $S \\subsetneq S' \\subseteq [0, n]^2$ contains $P$. Here a set $T$ contains $P$ when there are maps $f_X, f_Y : \\mathbb{R} \\to \\mathbb{R}$ such that $f_X(x) - f_X(x') \\ge x - x'$ whenever $x > x'$ are\u2026","title":"Saturating sets for a point pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/osat","kind":"node","lean":"MetaMathlibExt.osat","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#osat","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For a real number $n$ and a set $P \\subseteq \\mathbb{R}^2$, call an open set $S \\subseteq [0, n]^2$ saturating for $P$ if $S$ does not contain $P$ but every open set $S' \\subseteq [0, n]^2$ properly containing $S$ contains $P$ (containment in the sense of point-set pattern containment). The saturation function $\\operatorname{osat}(n, P)$ is the infimum of the Lebesgue measures $\\mu_2(S)$ over all sets $S$\u2026","title":"Saturation function for continuous patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/osat-nonneg","kind":"node","lean":"MetaMathlibExt.osat_nonneg","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#osat-nonneg","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every real number $n$ and every set $P \\subseteq \\mathbb{R}^2$, $\\operatorname{osat}(n, P) \\ge 0$.","title":"The saturation function is nonnegative"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/osat-singleton-and-opx-singleton","kind":"node","lean":"MetaMathlibExt.osat_singleton_and_opx_singleton","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#osat-singleton-and-opx-singleton","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every real number $n$ and every point $p \\in \\mathbb{R}^2$, $\\operatorname{osat}(n, \\{p\\}) = 0$ and $\\operatorname{opx}(n, \\{p\\}) = 0$.","title":"Saturation function and continuous Tur\u00e1n number of a single point"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/contains-pattern-d","kind":"node","lean":"MetaMathlibExt.ContainsPatternD","parent":13953,"reading_title":{"format":"inline-parts-v1","parts":[["text","Containment of point sets in "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Containment of point sets in $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#contains-pattern-d","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"Let $d$ be a natural number and let $S, P \\subseteq \\mathbb{R}^d$. Then $S$ contains $P$ when there are maps $f_1, \\dots, f_d : \\mathbb{R} \\to \\mathbb{R}$ that expand distances between coordinates of points of $P$, that is, $f_i(p_i) - f_i(q_i) \\ge p_i - q_i$ for every $i$ and all $p, q \\in P$ with $p_i > q_i$, and such that $(f_1(p_1), \\dots, f_d(p_d)) \\in S$ for every $p \\in P$.","title":"Containment of point sets in $\\mathbb{R}^d$"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/contains-pattern-d-empty","kind":"node","lean":"MetaMathlibExt.containsPatternD_empty","parent":13953,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every set in "],["math","\\(\\mathbb{R}^d\\)"],["text"," contains the empty pattern"]],"source":"Every set in $\\mathbb{R}^d$ contains the empty pattern"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#contains-pattern-d-empty","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every natural number $d$ and every set $S \\subseteq \\mathbb{R}^d$, $S$ contains the empty set $\\emptyset \\subseteq \\mathbb{R}^d$ in the sense of containment of point sets in $\\mathbb{R}^d$.","title":"Every set in $\\mathbb{R}^d$ contains the empty pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d","kind":"node","lean":"MetaMathlibExt.opxD","parent":13953,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuous Tur\u00e1n number in "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Continuous Tur\u00e1n number in $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For a natural number $d$, a real number $n$ and a set $P \\subseteq \\mathbb{R}^d$, the continuous Tur\u00e1n number $\\mathrm{opx}(n, P, d)$ is the supremum of the $d$-dimensional Lebesgue measures $\\mu_d(S)$ over all open sets $S \\subseteq [0, n]^d$ that do not contain $P$ in the sense of containment of point sets in $\\mathbb{R}^d$; it is $0$ when there is no such set.","title":"Continuous Tur\u00e1n number in $\\mathbb{R}^d$"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-nonneg","kind":"node","lean":"MetaMathlibExt.opxD_nonneg","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-nonneg","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every dimension $d$, real number $n$ and set $P \\subseteq \\mathbb{R}^d$, $\\mathrm{opx}(n, P, d) \\ge 0$.","title":"Higher-dimensional continuous Tur\u00e1n numbers are nonnegative"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-empty","kind":"node","lean":"MetaMathlibExt.opxD_empty","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-empty","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For every dimension $d$ and every real number $n$, $\\mathrm{opx}(n, \\emptyset, d) = 0$: every set contains the empty pattern, so no set is free of it.","title":"The higher-dimensional continuous Tur\u00e1n number of the empty pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/cells-contain-pattern-d","kind":"node","lean":"MetaMathlibExt.CellsContainPatternD","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#cells-contain-pattern-d","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"Let $d$ be a natural number, let $A \\subseteq \\mathbb{N}^d$ be a set of cells and let $P \\subseteq \\mathbb{R}^d$. Then $A$ contains $P$ when there are maps $f_1, \\dots, f_d : \\mathbb{R} \\to \\mathbb{N}$ such that each $f_i$ is strictly increasing on the set $\\{p_i : p \\in P\\}$ of $i$-th coordinates of points of $P$, and $(f_1(p_1), \\dots, f_d(p_d)) \\in A$ for every $p \\in P$.","title":"Containment of a point pattern in d-dimensional grid cells"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/pattern-ex-d","kind":"node","lean":"MetaMathlibExt.patternExD","parent":13953,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extremal function of a "],["math","\\(d\\)"],["text","-dimensional pattern"]],"source":"Extremal function of a $d$-dimensional pattern"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#pattern-ex-d","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For natural numbers $d$ and $n$ and a set $P \\subseteq \\mathbb{R}^d$, say that a set $A$ of cells of $\\mathbb{N}^d$ contains $P$ when there are maps $f_1, \\dots, f_d$ to $\\mathbb{N}$, each $f_i$ strictly increasing on the $i$-th coordinates of the points of $P$, with $(f_1(p_1), \\dots, f_d(p_d)) \\in A$ for every $p \\in P$. Then $\\operatorname{ex}(n, M_P, d)$ is the largest number of cells of a set of cells of the\u2026","title":"Extremal function of a $d$-dimensional pattern"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/pattern-ex-d-empty","kind":"node","lean":"MetaMathlibExt.patternExD_empty","parent":13953,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(d\\)"],["text","-dimensional extremal function of the empty pattern vanishes"]],"source":"The $d$-dimensional extremal function of the empty pattern vanishes"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#pattern-ex-d-empty","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For all natural numbers $d$ and $n$, $\\operatorname{ex}(n, M_\\emptyset, d) = 0$.","title":"The $d$-dimensional extremal function of the empty pattern vanishes"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/pattern-ex-d-singleton","kind":"node","lean":"MetaMathlibExt.patternExD_singleton","parent":13953,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(d\\)"],["text","-dimensional extremal function of a single point vanishes"]],"source":"The $d$-dimensional extremal function of a single point vanishes"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#pattern-ex-d-singleton","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"fully_proved","summary":"For all natural numbers $d$ and $n$ and every point $p \\in \\mathbb{R}^d$, $\\operatorname{ex}(n, M_{\\{p\\}}, d) = 0$, since a single point is contained in every nonempty set of cells.","title":"The $d$-dimensional extremal function of a single point vanishes"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-mono-of-contains-pattern-d","kind":"node","lean":"MetaMathlibExt.opxD_mono_of_containsPatternD","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-mono-of-contains-pattern-d","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P, Q \\subseteq \\mathbb{R}^d$. If $Q$ contains $P$, then $\\mathrm{opx}(n, P, d) \\le \\mathrm{opx}(n, Q, d)$ for every real number $n$.","title":"Higher-dimensional continuous Tur\u00e1n numbers are monotone under containment"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-mono-left","kind":"node","lean":"MetaMathlibExt.opxD_mono_left","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-mono-left","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every set $P \\subseteq \\mathbb{R}^d$ and all real numbers $m \\le n$, $\\mathrm{opx}(m, P, d) \\le \\mathrm{opx}(n, P, d)$.","title":"Higher-dimensional continuous Tur\u00e1n numbers are monotone in the side length"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-add-lt","kind":"node","lean":"MetaMathlibExt.opxD_add_lt","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-add-lt","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $d \\ge 2$ be a natural number, let $P \\subseteq \\mathbb{R}^d$, and let $n, \\varepsilon$ be real numbers with $n > 0$ and $0 < \\varepsilon < n$. Then the $d$-dimensional continuous Tur\u00e1n number $\\mathrm{opx}$ satisfies $\\mathrm{opx}(n, P, d) \\le \\mathrm{opx}(n + \\varepsilon, P, d)$ and $\\mathrm{opx}(n + \\varepsilon, P, d) < \\mathrm{opx}(n, P, d) + d (2n)^{d-1} \\varepsilon$.","title":"Increment bound for the d-dimensional continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-continuous-on","kind":"node","lean":"MetaMathlibExt.opxD_continuousOn","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-continuous-on","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every natural number $d$ and every $P \\subseteq \\mathbb{R}^d$, the $d$-dimensional continuous Tur\u00e1n number $\\mathrm{opx}(n, P, d)$ is a continuous function of $n$ on $(0, \\infty)$.","title":"Continuity of the d-dimensional continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/mul-opx-d-init-le","kind":"node","lean":"MetaMathlibExt.mul_opxD_init_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#mul-opx-d-init-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $d$ be a natural number, $n$ a real number and $P \\subseteq \\mathbb{R}^{d+1}$, and let $\\Pr(P)$ be the subset of $\\mathbb{R}^d$ obtained by dropping the last coordinate of every point of $P$. Writing $\\mathrm{opx}(n, Q, k)$ for the $k$-dimensional continuous Tur\u00e1n number, we have $n \\cdot \\mathrm{opx}(n, \\Pr(P), d) \\le \\mathrm{opx}(n, P, d + 1)$.","title":"Projection bound for the d-dimensional continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-eq-mul-opx-d-init","kind":"node","lean":"MetaMathlibExt.opxD_eq_mul_opxD_init","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-eq-mul-opx-d-init","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^{d+1}$ be a set whose points all have the same last coordinate, and let $\\mathrm{Pr}(P) \\subseteq \\mathbb{R}^d$ be the set obtained by deleting the last coordinate of each point of $P$. Then for every real number $n$, $\\mathrm{opx}(n, P, d + 1) = n \\cdot \\mathrm{opx}(n, \\mathrm{Pr}(P), d)$.","title":"Patterns with a constant last coordinate"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-rational-points-embed-is-theta","kind":"node","lean":"MetaMathlibExt.opxD_rational_points_embed_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-rational-points-embed-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $m$ be a natural number and let $P \\subseteq \\mathbb{R}^2$ be a nonempty open set. Let $P' = P \\times \\{0\\}^{m+1} \\subseteq \\mathbb{R}^{m+3}$, and let $Q_{P'}$ be the set of points of $P'$ all of whose coordinates are rational. Then $\\mathrm{opx}(n, Q_{P'}, m + 3) = \\Theta(n^{m+3})$ as the real number $n$ tends to infinity.","title":"Rational points of an embedded open planar set"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-open-is-theta","kind":"node","lean":"MetaMathlibExt.opxD_open_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-open-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $d \\ge 2$, let $P \\subseteq \\mathbb{R}^d$ be a nonempty open set, and let $Q_P$ be the set of points of $P$ all of whose coordinates are rational. Then, as the real number $n \\to \\infty$, $\\mathrm{opx}(n, P, d) = \\Theta(n^d)$ and $\\mathrm{opx}(n, Q_P, d) = \\Theta(n^d)$.","title":"Nonempty open patterns in dimension at least two"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-dilate-le","kind":"node","lean":"MetaMathlibExt.opxD_dilate_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-dilate-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $d$ be a natural number, let $P \\subseteq \\mathbb{R}^d$ be finite and let $c_1, c_2, q, n$ be real numbers with $c_1 > 0$, $q > 1$ and $n > 0$. Suppose that for every coordinate $i$ and all $p, p' \\in P$ with $p_i < p'_i$ such that no $r \\in P$ has $p_i < r_i < p'_i$, we have $c_1 \\le p'_i - p_i \\le c_2$. Let $qP = \\{q p : p \\in P\\}$ and $m = \\lceil q c_2 / c_1 \\rceil + 1$. Then the $d$-dimensional continuous\u2026","title":"Dilation bound for the d-dimensional continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-is-theta-pattern-ex-d","kind":"node","lean":"MetaMathlibExt.opxD_isTheta_patternExD","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-is-theta-pattern-ex-d","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $d$ be a natural number and let $P \\subseteq \\mathbb{R}^d$ be finite. Then the $d$-dimensional continuous Tur\u00e1n number $\\mathrm{opx}(n, P, d)$ and the $d$-dimensional 0-1 matrix extremal function $\\mathrm{ex}(n, M_P, d)$ of $P$ satisfy $\\mathrm{opx}(n, P, d) = \\Theta(\\mathrm{ex}(n, M_P, d))$ as the natural number $n$ tends to infinity, with constants depending on $P$.","title":"Higher-dimensional continuous Tur\u00e1n numbers versus extremal functions"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-permutation-pattern-le","kind":"node","lean":"MetaMathlibExt.opxD_permutation_pattern_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-permutation-pattern-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"For every $d$ there is a real number $C$ such that for every real $c > 0$ there is a real number $K$ with the following property. Let $P \\subseteq \\mathbb{R}^d$ be a finite set of $k$ points such that, for each coordinate $i$, distinct points of $P$ have distinct $i$-th coordinates and consecutive values of the $i$-th coordinate on $P$ differ by at most $c$. Then\u2026","title":"Upper bound for higher-dimensional permutation-type patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-grid-is-big-o","kind":"node","lean":"MetaMathlibExt.opxD_grid_isBigO","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-grid-is-big-o","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $X_1, \\dots, X_d$ be finite sets of real numbers, $k_i = |X_i|$, and $P = X_1 \\times \\dots \\times X_d \\subseteq \\mathbb{R}^d$. Put $\\alpha = \\max_i k_i / \\prod_i k_i$ if $d \\ge 1$ and every $k_i$ is positive, and $\\alpha = 0$ otherwise. Then $\\mathrm{opx}(n, P, d) = O(n^{d - \\alpha})$ as the natural number $n \\to \\infty$.","title":"Upper bound for higher-dimensional grid patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/exists-opx-d-is-big-o-rpow","kind":"node","lean":"MetaMathlibExt.exists_opxD_isBigO_rpow","parent":13953,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power saving for continuous Tur\u00e1n numbers of finite patterns in "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Power saving for continuous Tur\u00e1n numbers of finite patterns in $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#exists-opx-d-is-big-o-rpow","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $d$ be a natural number and let $P \\subseteq \\mathbb{R}^d$ be finite. Then there is a real number $\\varepsilon > 0$ such that the $d$-dimensional continuous Tur\u00e1n number satisfies $\\mathrm{opx}(n, P, d) = O(n^{d - \\varepsilon})$ as the natural number $n$ tends to infinity.","title":"Power saving for continuous Tur\u00e1n numbers of finite patterns in $\\mathbb{R}^d$"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-permutation-stack-is-theta","kind":"node","lean":"MetaMathlibExt.opxD_permutation_stack_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-permutation-stack-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ be a finite set of at least two points with pairwise distinct $x$-coordinates and pairwise distinct $y$-coordinates, and for $i \\in \\{2, 3\\}$ let $P_i = \\{(x, y, j) : (x, y) \\in P,\\ j \\in \\{1, \\dots, i\\}\\} \\subseteq \\mathbb{R}^3$. Then, as $n$ tends to infinity through the natural numbers, $\\mathrm{opx}(n, P_2, 3) = \\Theta(n^{5/2})$ and $\\mathrm{opx}(n, P_3, 3) = \\Theta(n^{8/3})$.","title":"Stacked permutation patterns in three dimensions"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-stack-le","kind":"node","lean":"MetaMathlibExt.opxD_stack_le","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-stack-le","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $d \\ge 2$ and $t \\ge 2$ be natural numbers and let $c > 0$. For $P \\subseteq \\mathbb{R}^d$ let $Q_{P,t,c} = \\{(x, ic) : x \\in P,\\ i \\in \\{1, \\dots, t\\}\\} \\subseteq \\mathbb{R}^{d+1}$. Then there is a real constant $K$, depending only on $d$, $t$ and $c$, such that for every $P \\subseteq \\mathbb{R}^d$ and every real $n > 0$,\u2026","title":"Upper bound for continuous Tur\u00e1n numbers of stacked patterns"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-stack-two-is-theta","kind":"node","lean":"MetaMathlibExt.opxD_stack_two_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-stack-two-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ have at least two points, and suppose that $\\mathrm{opx}(n, P) = O(n)$ as the real number $n$ tends to infinity. Let $c > 0$ and $Q_{P,2,c} = \\{(x, y, z) : (x, y) \\in P,\\ z \\in \\{c, 2c\\}\\} \\subseteq \\mathbb{R}^3$. Then $\\mathrm{opx}(n, Q_{P,2,c}, 3) = \\Theta(n^{5/2})$ as the real number $n$ tends to infinity.","title":"Two-layer stacks of patterns with linear continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-stack-three-is-theta","kind":"node","lean":"MetaMathlibExt.opxD_stack_three_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-stack-three-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^2$ contain three points with pairwise distinct $x$-coordinates, and suppose that $\\mathrm{opx}(n, P) = O(n)$ as the real number $n$ tends to infinity. Let $c > 0$ and $Q_{P,3,c} = \\{(x, y, z) : (x, y) \\in P,\\ z \\in \\{c, 2c, 3c\\}\\} \\subseteq \\mathbb{R}^3$. Then $\\mathrm{opx}(n, Q_{P,3,c}, 3) = \\Theta(n^{8/3})$ as the real number $n$ tends to infinity.","title":"Three-layer stacks of patterns with linear continuous Tur\u00e1n number"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-graph-stack-is-theta","kind":"node","lean":"MetaMathlibExt.opxD_graph_stack_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-graph-stack-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $a > 0$, $b$ and $c > 0$ be real numbers and let $f : \\mathbb{R} \\to \\mathbb{R}$ be strictly increasing on $[0, a]$ with $(f(t) - f(s)) / (t - s) \\le b$ whenever $0 \\le s < t \\le a$. Let $P = \\{(t, f(t)) : t \\in [0, a]\\}$ be the graph of $f$ over $[0, a]$ and, for $k \\ge 1$, let $Q_{P,k,c} = \\{(x, y, ic) : (x, y) \\in P,\\ i \\in \\{1, \\dots, k\\}\\} \\subseteq \\mathbb{R}^3$. Then, as the real number $n \\to \\infty$,\u2026","title":"Stacked graphs of increasing functions in three dimensions"},{"id":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/opx-d-plus-stack-is-theta","kind":"node","lean":"MetaMathlibExt.opxD_plus_stack_isTheta","parent":13953,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns/index.html#opx-d-plus-stack-is-theta","scope":"Geometry/discrete-geometry/part-01/continuous-tur-n-numbers-of-planar-patterns","status":"can_prove","summary":"Let $r > 0$ and $c > 0$, let $P = \\{(x, y) \\in \\mathbb{R}^2 : |x| \\le r,\\ |y| \\le r,\\ x = 0 \\text{ or } y = 0\\}$ be the plus sign of size $r$, and for $t \\in \\{2, 3\\}$ let $Q_{P,t,c} = \\{(x, y, ic) : (x, y) \\in P,\\ i \\in \\{1, \\dots, t\\}\\} \\subseteq \\mathbb{R}^3$. Then, as the real number $n$ tends to infinity, $\\mathrm{opx}(n, Q_{P,2,c}, 3) = \\Theta(n^{5/2})$ and $\\mathrm{opx}(n, Q_{P,3,c}, 3) = \\Theta(n^{8/3})$.","title":"Stacked plus signs in three dimensions"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/dehn-invariant.json","id":"Geometry/discrete-geometry/part-01/dehn-invariant","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","title":"Dehn invariant"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-group","kind":"node","lean":"DehnInvariant.DehnGroup","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-group","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"The Dehn group is the tensor product $\\mathbb{R} \\otimes_{\\mathbb{Z}} (\\mathbb{R} / 2\\pi\\mathbb{Z})$ of abelian groups; it is the target of the Dehn invariant.","title":"Dehn group"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/edge","kind":"node","lean":"DehnInvariant.Edge","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#edge","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"An edge, as recorded for the Dehn invariant, is a pair of real numbers: its length and its dihedral angle in radians. No positivity or range condition is imposed on either.","title":"Edge of a polyhedron"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-invariant","kind":"node","lean":"DehnInvariant.dehnInvariant","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-invariant","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"The Dehn invariant of a list of edges is the sum, over the edges, of $\\ell \\otimes \\theta$ in the Dehn group, where $\\ell$ is the length of the edge and $\\theta$ its dihedral angle taken modulo $2\\pi$.","title":"Dehn invariant"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/tetrahedral-angle","kind":"node","lean":"DehnInvariant.tetrahedralAngle","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#tetrahedral-angle","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"The real number $\\arccos(1/3)$, the dihedral angle of a regular tetrahedron in radians.","title":"Dihedral angle of the regular tetrahedron"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/cube-edges","kind":"node","lean":"DehnInvariant.cubeEdges","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#cube-edges","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"For a real number $\\ell$, the edge list of a cube of side $\\ell$ consists of $12$ edges, each of length $\\ell$ with dihedral angle $\\pi / 2$.","title":"Edge list of a cube"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/tetrahedron-edges","kind":"node","lean":"DehnInvariant.tetrahedronEdges","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#tetrahedron-edges","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"For a real number $\\ell$, the edge list of a regular tetrahedron with edge length $\\ell$ is the list of $6$ edges, each of length $\\ell$ and dihedral angle $\\arccos(1/3)$.","title":"Edge list of the regular tetrahedron"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-invariant-nil","kind":"node","lean":"DehnInvariant.dehnInvariant_nil","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-invariant-nil","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"The empty list of edges has Dehn invariant $0$.","title":"Dehn invariant of the empty edge list"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-invariant-cons","kind":"node","lean":"DehnInvariant.dehnInvariant_cons","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-invariant-cons","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"Let $e$ be an edge of length $\\ell$ and dihedral angle $\\theta$, and let $L$ be a list of edges. Then the Dehn invariant of the list obtained by putting $e$ in front of $L$ equals $\\ell \\otimes \\theta$ plus the Dehn invariant of $L$, where $\\theta$ is taken modulo $2\\pi$.","title":"The Dehn invariant splits off the first edge"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-invariant-append","kind":"node","lean":"DehnInvariant.dehnInvariant_append","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-invariant-append","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"For lists of edges $L_1$ and $L_2$, the Dehn invariant of the concatenation of $L_1$ and $L_2$ is the sum of the Dehn invariants of $L_1$ and $L_2$.","title":"Additivity of the Dehn invariant under concatenation"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-invariant-replicate","kind":"node","lean":"DehnInvariant.dehnInvariant_replicate","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-invariant-replicate","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"For a natural number $n$ and an edge of length $\\ell$ and dihedral angle $\\theta$, the Dehn invariant of the list consisting of $n$ copies of that edge is $n \\cdot (\\ell \\otimes \\theta)$ in the Dehn group, where $\\theta$ is taken modulo $2\\pi$.","title":"Dehn invariant of a repeated edge"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/exists-linear-map-of-irrational","kind":"node","lean":"DehnInvariant.exists_linearMap_of_irrational","parent":14032,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(\\mathbb{Q}\\)"],["text","-linear functional separating "],["math","\\(\\alpha\\)"],["text"," from "],["math","\\(\\pi\\)"]],"source":"A $\\mathbb{Q}$-linear functional separating $\\alpha$ from $\\pi$"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#exists-linear-map-of-irrational","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"Let $\\alpha$ be a real number such that $\\alpha / \\pi$ is irrational. Then there is a $\\mathbb{Q}$-linear map $f : \\mathbb{R} \\to \\mathbb{R}$ with $f(\\pi) = 0$ and $f(\\alpha) \\neq 0$.","title":"A $\\mathbb{Q}$-linear functional separating $\\alpha$ from $\\pi$"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/tmul-ne-zero-of-irrational","kind":"node","lean":"DehnInvariant.tmul_ne_zero_of_irrational","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#tmul-ne-zero-of-irrational","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"Let $\\ell$ and $\\alpha$ be real numbers with $\\ell \\neq 0$ and $\\alpha / \\pi$ irrational. Then the pure tensor $\\ell \\otimes [\\alpha]$ is nonzero in the Dehn group $\\mathbb{R} \\otimes_{\\mathbb{Z}} (\\mathbb{R} / 2\\pi\\mathbb{Z})$, where $[\\alpha]$ is the class of $\\alpha$ modulo $2\\pi$.","title":"Nonvanishing of an edge term in the Dehn group"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-invariant-cube-edges","kind":"node","lean":"DehnInvariant.dehnInvariant_cubeEdges","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-invariant-cube-edges","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"For every real number $\\ell$, the edge list of a cube of side $\\ell$ (twelve edges of length $\\ell$, each with dihedral angle $\\pi / 2$) has Dehn invariant $0$.","title":"The Dehn invariant of the cube is zero"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-invariant-tetrahedron-edges-ne-zero","kind":"node","lean":"DehnInvariant.dehnInvariant_tetrahedronEdges_ne_zero","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-invariant-tetrahedron-edges-ne-zero","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"Suppose that $\\arccos(1/3) / \\pi$ is irrational. Then for every nonzero real number $\\ell$, the edge list of a regular tetrahedron of side $\\ell$ (six edges of length $\\ell$, each with dihedral angle $\\arccos(1/3)$) has nonzero Dehn invariant.","title":"Nonzero Dehn invariant of the regular tetrahedron, conditional on irrationality"},{"id":"Geometry/discrete-geometry/part-01/dehn-invariant/dehn-invariant-cube-ne-dehn-invariant-te","kind":"node","lean":"DehnInvariant.dehnInvariant_cube_ne_dehnInvariant_tetrahedron","parent":14032,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/dehn-invariant/index.html#dehn-invariant-cube-ne-dehn-invariant-te","scope":"Geometry/discrete-geometry/part-01/dehn-invariant","status":"fully_proved","summary":"Suppose that $\\arccos(1/3) / \\pi$ is irrational. Then for all real numbers $\\ell$ and $\\ell'$ with $\\ell' \\ne 0$, the edge list of a cube of side $\\ell$ and the edge list of a regular tetrahedron of side $\\ell'$ have different Dehn invariants.","title":"Conditionally, the cube and the regular tetrahedron have different Dehn invariants"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/distinct-and-unit-distances.json","id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"can_prove","title":"Distinct and unit distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/distance-multiplicity","kind":"node","lean":"distanceMultiplicity","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#distance-multiplicity","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"Let $P$ be a finite set of points in a metric space and let $d$ be a real number. The multiplicity of $d$ in $P$ is the number of unordered pairs $\\{p, q\\}$ of distinct points of $P$ with $\\operatorname{dist}(p, q) = d$.","title":"Multiplicity of a distance"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/unit-dist-num","kind":"node","lean":"unitDistNum","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#unit-dist-num","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $S$ of points in a metric space, the number of unit distances of $S$ is the number of unordered pairs $\\{p, q\\}$ of points of $S$ with $\\operatorname{dist}(p, q) = 1$, that is, the multiplicity of the distance $1$ in $S$.","title":"Number of unit distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/distance-set","kind":"node","lean":"distanceSet","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#distance-set","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $P$ of points in a metric space, the distance set of $P$ is the finite set of real numbers $\\operatorname{dist}(p, q)$ with $p$ and $q$ distinct points of $P$.","title":"Distance set of a finite point set"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/distinct-distances","kind":"node","lean":"distinctDistances","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#distinct-distances","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $P$ of points in a metric space, the number of distinct distances of $P$ is the number of elements of its distance set, that is, the number of distinct values $\\operatorname{dist}(p, q)$ with $p$ and $q$ distinct points of $P$.","title":"Number of distinct distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/minimal-distinct-distances","kind":"node","lean":"minimalDistinctDistances","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#minimal-distinct-distances","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"Let $X$ be a metric space and $n$ a natural number such that $X$ contains a set of exactly $n$ points. The minimum number of distinct distances for $n$ points in $X$ is the least number of distinct distances determined by a set of exactly $n$ points of $X$.","title":"Minimum number of distinct distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/distinct-distances-from","kind":"node","lean":"distinctDistancesFrom","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#distinct-distances-from","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $P$ of points in a metric space and a point $x$ of the space, this is the number of distinct values $\\operatorname{dist}(y, x)$ with $y$ a point of $P$ other than $x$.","title":"Number of distinct distances from a point"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/unit-dist-num-le-choose-two","kind":"node","lean":"unitDistNum_le_choose_two","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#unit-dist-num-le-choose-two","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set of $n$ points in a metric space, the number of unit distances is at most $\\binom{n}{2}$, the total number of unordered pairs of distinct points.","title":"Trivial upper bound on unit distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/favourite-distance-count-half-plus-const","kind":"node","lean":"MathlibExt.Combinatorics.FavouriteDistancesFourSpaceErdosProblem754Wanted.favouriteDistanceCountHalfPlusConstant","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#favourite-distance-count-half-plus-const","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"This is the statement that there is a real constant $C$ such that for every nonempty finite set $A$ of points of the Euclidean space $\\mathbb{R}^4$ and every natural number $f$, if for every $x \\in A$ there is a real number $r > 0$ such that at least $f$ points of $A$ lie at distance $r$ from $x$, then $f \\le |A|/2 + C$. Equivalently, $f(n) \\le n/2 + O(1)$, where $f(n)$ is the largest $f$ such that some set $A$ of\u2026","title":"Statement of Erd\u0151s Problem 754"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/favourite-distance-count-half-plus-const-2","kind":"node","lean":"MathlibExt.Combinatorics.FavouriteDistancesFourSpaceErdosProblem754Wanted.favouriteDistanceCountHalfPlusConstant_holds","parent":14048,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s Problem 754: favourite distances in "],["math","\\(\\mathbb{R}^4\\)"]],"source":"Erd\u0151s Problem 754: favourite distances in $\\mathbb{R}^4$"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#favourite-distance-count-half-plus-const-2","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"can_prove","summary":"There is a real constant $C$ such that for every nonempty finite set $A$ of points of the Euclidean space $\\mathbb{R}^4$ and every natural number $f$, if for every $x \\in A$ there is a real number $r > 0$ such that at least $f$ points of $A$ lie at distance $r$ from $x$, then $f \\le |A|/2 + C$.","title":"Erd\u0151s Problem 754: favourite distances in $\\mathbb{R}^4$"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/few-distances-with-four-point-condition","kind":"node","lean":"MathlibExt.Combinatorics.FewDistancesFourPointErdosProblem659Wanted.fewDistancesWithFourPointCondition","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#few-distances-with-four-point-condition","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"This is the statement that there is a real constant $C$ such that for every sufficiently large natural number $n$ there is a set $P$ of $n$ points of the Euclidean plane $\\mathbb{R}^2$ such that every $4$-element subset of $P$ determines at least $3$ distinct distances and $P$ determines at most $C n / \\sqrt{\\log n}$ distinct distances.","title":"Statement of Erd\u0151s Problem 659"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/few-distances-with-four-point-condition-2","kind":"node","lean":"MathlibExt.Combinatorics.FewDistancesFourPointErdosProblem659Wanted.fewDistancesWithFourPointCondition_holds","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#few-distances-with-four-point-condition-2","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"can_prove","summary":"There is a real constant $C$ such that for every sufficiently large natural number $n$ there is a set $P$ of $n$ points of the Euclidean plane $\\mathbb{R}^2$ such that every $4$-element subset of $P$ determines at least $3$ distinct distances and $P$ determines at most $C n / \\sqrt{\\log n}$ distinct distances.","title":"Erd\u0151s Problem 659: few distances with every four points spanning three"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/pairs-at-distance","kind":"node","lean":"MathlibExt.Combinatorics.SphereRepeatedDistancesErdosProblem605Wanted.pairsAtDistance","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#pairs-at-distance","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $P$ of points in $\\mathbb{R}^3$ and a real number $\\delta$, this is the number of unordered pairs $\\{p, q\\}$ of distinct points $p, q \\in P$ with Euclidean distance $|p - q| = \\delta$, computed as half the number of ordered pairs $(p, q)$ of distinct points of $P$ at distance $\\delta$ and taken as a real number.","title":"Number of pairs at a given distance"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/sphere-repeated-distance-superlinear","kind":"node","lean":"MathlibExt.Combinatorics.SphereRepeatedDistancesErdosProblem605Wanted.sphereRepeatedDistanceSuperlinear","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#sphere-repeated-distance-superlinear","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"The proposition that there is a function $f : \\mathbb{N} \\to \\mathbb{R}$ with $f(n) \\to \\infty$ as $n \\to \\infty$ such that for every $n \\in \\mathbb{N}$ there are a set $P$ of $n$ distinct points on the unit sphere $\\{x \\in \\mathbb{R}^3 : \\|x\\| = 1\\}$ and a real number $\\delta$ for which the number of unordered pairs of distinct points of $P$ at distance $\\delta$ is at least $f(n) \\cdot n$.","title":"Erd\u0151s Problem 605 (repeated distances on the sphere)"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/sphere-repeated-distance-superlinear-hol","kind":"node","lean":"MathlibExt.Combinatorics.SphereRepeatedDistancesErdosProblem605Wanted.sphereRepeatedDistanceSuperlinear_holds","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#sphere-repeated-distance-superlinear-hol","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"can_prove","summary":"There is a function $f : \\mathbb{N} \\to \\mathbb{R}$ with $f(n) \\to \\infty$ as $n \\to \\infty$ such that for every $n \\in \\mathbb{N}$ there are a set $P$ of $n$ distinct points on the unit sphere $\\{x \\in \\mathbb{R}^3 : \\|x\\| = 1\\}$ and a real number $\\delta$ for which the number of unordered pairs of distinct points of $P$ at distance $\\delta$ is at least $f(n) \\cdot n$.","title":"Erd\u0151s Problem 605 is true"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/unit-dist","kind":"node","lean":"MathlibExt.Combinatorics.UnitDistanceConjectureFalseWanted.Combinatorics.unitDist","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#unit-dist","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $P$ of points in the Euclidean plane $\\mathbb{R}^2$, the unit distance count of $P$ is the number of unordered pairs of points of $P$ at distance exactly $1$.","title":"Unit distance count of a planar point set"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/erdos-unit-distance-conjecture-false","kind":"node","lean":"MathlibExt.Combinatorics.UnitDistanceConjectureFalseWanted.Combinatorics.erdos_unit_distance_conjecture_false","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#erdos-unit-distance-conjecture-false","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"can_prove","summary":"There is a real number $\\delta > 0$ such that for every natural number $N$ there are an integer $n \\ge N$ and a set $P$ of exactly $n$ points in the Euclidean plane $\\mathbb{R}^2$ with at least $n^{1+\\delta}$ unordered pairs of points of $P$ at distance exactly $1$.","title":"Erd\u0151s's unit distance conjecture is false"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/unit-dist-2","kind":"node","lean":"MathlibExt.Combinatorics.UnitDistanceUpperBoundWanted.Combinatorics.unitDist","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#unit-dist-2","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $P$ of points in the Euclidean plane $\\mathbb{R}^2$, the number of unit-distance pairs of $P$ is the number of unordered pairs of points of $P$ at distance exactly $1$.","title":"Number of unit-distance pairs in a planar point set"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/unit-distance-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.UnitDistanceUpperBoundWanted.Combinatorics.unit_distance_upper_bound","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#unit-distance-upper-bound","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"can_prove","summary":"There is a constant $C > 0$ such that every finite set of $n$ points in the Euclidean plane $\\mathbb{R}^2$ has at most $C n^{4/3}$ unordered pairs of points at distance exactly $1$.","title":"Spencer\u2013Szemer\u00e9di\u2013Trotter unit distance bound"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/plane","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem652Wanted.Plane","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#plane","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"The real Euclidean plane is the space $\\mathbb{R}^2$ of pairs of real numbers with the Euclidean inner product and norm.","title":"Euclidean plane for Erd\u0151s Problem 652"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/count-small-points","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem652Wanted.countSmallPoints","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#count-small-points","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $X$ of points in the Euclidean plane and a real number $t$, the number of points $x \\in X$ such that the number of distinct distances from $x$ to the other points of $X$ is less than $t$.","title":"Points with few distinct distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/has-small-kth","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem652Wanted.HasSmallKth","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#has-small-kth","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For natural numbers $n$, $k$ and a real number $c$, the property that some set $X$ of exactly $n$ points in the Euclidean plane contains at least $k$ points $x$ such that the number of distinct distances from $x$ to the other points of $X$ is less than $c\\sqrt{n}$.","title":"Many points with few distinct distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/is-admissible-bound","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem652Wanted.isAdmissibleBound","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#is-admissible-bound","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"A real number $c$ is admissible for $k \\in \\mathbb{N}$ if there is $N$ such that for every $n \\ge N$ some set $X$ of exactly $n$ points in the Euclidean plane contains at least $k$ points $x$ such that the number of distinct distances from $x$ to the other points of $X$ is less than $c\\sqrt{n}$.","title":"Admissible constants in Erd\u0151s Problem 652"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/erdos-alpha","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem652Wanted.erdosAlpha","parent":14048,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Erd\u0151s constant "],["math","\\(\\alpha_k\\)"]],"source":"The Erd\u0151s constant $\\alpha_k$"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#erdos-alpha","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, $\\alpha_k$ is the infimum of the set of real numbers that are admissible for $k$: the reals $c$ for which there is $N$ such that for every $n \\ge N$ some set $X$ of exactly $n$ points in the Euclidean plane contains at least $k$ points $x$ such that the number of distinct distances from $x$ to the other points of $X$ is less than $c\\sqrt{n}$. If this set is empty or unbounded below,\u2026","title":"The Erd\u0151s constant $\\alpha_k$"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/conjecture","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem652Wanted.conjecture","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#conjecture","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $X$ of points in the Euclidean plane and $x \\in X$, let $R_X(x)$ be the number of distinct distances from $x$ to the other points of $X$. For $k \\in \\mathbb{N}$ let $\\alpha_k$ be the infimum of the reals $c$ for which there is $N$ such that for every $n \\ge N$ some set $X$ of exactly $n$ points in the plane has at least $k$ points $x$ with $R_X(x) < c\\sqrt{n}$ (an empty set or a set unbounded below\u2026","title":"Erd\u0151s Problem 652"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/conjecture-holds","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem652Wanted.conjecture_holds","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#conjecture-holds","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"can_prove","summary":"For a finite set $X$ of points in the Euclidean plane and $x \\in X$, let $R_X(x)$ be the number of distinct distances from $x$ to the other points of $X$. For $k \\in \\mathbb{N}$ let $\\alpha_k$ be the infimum of the reals $c$ for which there is $N$ such that for every $n \\ge N$ some set $X$ of exactly $n$ points in the plane has at least $k$ points $x$ with $R_X(x) < c\\sqrt{n}$ (an empty set or a set unbounded below\u2026","title":"Erd\u0151s Problem 652 is true"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/rich-distances","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem756Wanted.richDistances","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#rich-distances","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $A$ of points in the Euclidean plane and a natural number $k$, the rich distances of $A$ at level $k$ are the distances $d$ between two distinct points of $A$ such that at least $k$ unordered pairs of distinct points of $A$ are at distance $d$.","title":"Rich distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/max-rich-distances","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem756Wanted.maxRichDistances","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#max-rich-distances","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a natural number $n$, this is the largest, over all sets $A$ of $n$ points in the Euclidean plane, of the number of rich distances of $A$ at level $n + 1$, that is, of distances that occur for at least $n + 1$ unordered pairs of distinct points of $A$.","title":"Maximal number of rich distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/conjecture-2","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem756Wanted.conjecture","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#conjecture-2","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"fully_proved","summary":"For a finite set $A$ of points in the Euclidean plane, call a distance $k$-rich for $A$ if at least $k$ unordered pairs of distinct points of $A$ lie at that distance, and let $f(n)$ be the largest number of $(n+1)$-rich distances of a set of $n$ points. This is the assertion that there is a constant $C$ with $n \\le C f(n)$ for all sufficiently large $n$; that for every $n$ there is a set of $n$ points with at\u2026","title":"Statement of Erd\u0151s Problem 756 on rich distances"},{"id":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances/conjecture-holds-2","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem756Wanted.conjecture_holds","parent":14048,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/distinct-and-unit-distances/index.html#conjecture-holds-2","scope":"Geometry/discrete-geometry/part-01/distinct-and-unit-distances","status":"can_prove","summary":"For a finite set $A$ of points in the Euclidean plane, call a distance $k$-rich for $A$ if at least $k$ unordered pairs of distinct points of $A$ lie at that distance, and let $f(n)$ be the largest number of $(n+1)$-rich distances of a set of $n$ points. Then there is a constant $C$ with $n \\le C f(n)$ for all sufficiently large $n$; for every $n$ there is a set of $n$ points with at least $\\lfloor n/4 \\rfloor$\u2026","title":"Bhowmick's theorem on rich distances"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets.json","id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"can_prove","title":"Convex and general position of point sets"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/triplewise","kind":"node","lean":"Set.Triplewise","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#triplewise","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Let $r$ be a ternary relation on a type $\\alpha$ and let $s \\subseteq \\alpha$. The relation $r$ holds triplewise on $s$ if $r(x, y, z)$ holds for all pairwise distinct $x, y, z \\in s$.","title":"Relation holding triplewise on a set"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/non-trilinear","kind":"node","lean":"EuclideanGeometry.NonTrilinear","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#non-trilinear","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"A set $A$ of points in the plane is non-trilinear if no three distinct points of $A$ lie on a common line.","title":"Sets with no three collinear points"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/non-collinear-for","kind":"node","lean":"EuclideanGeometry.NonCollinearFor","parent":14078,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sets with no "],["math","\\(n\\)"],["text"," collinear points"]],"source":"Sets with no $n$ collinear points"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#non-collinear-for","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"A set $S$ of points in a Euclidean affine space has no $n$ collinear points if no subset of $S$ consisting of exactly $n$ points is collinear.","title":"Sets with no $n$ collinear points"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/has-convex-ngon","kind":"node","lean":"EuclideanGeometry.HasConvexNGon","parent":14078,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sets containing a convex "],["math","\\(n\\)"],["text","-gon"]],"source":"Sets containing a convex $n$-gon"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#has-convex-ngon","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"A set $P \\subseteq \\mathbb{R}^2$ contains a convex $n$-gon if there is a set $S \\subseteq P$ of exactly $n$ points that is convex independent: no point of $S$ lies in the convex hull of the other points of $S$.","title":"Sets containing a convex $n$-gon"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-ccw-convex-polygon","kind":"node","lean":"EuclideanGeometry.IsCcwConvexPolygon","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-ccw-convex-polygon","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Let $P$ be an oriented two-dimensional Euclidean affine space. Points $p_0, \\dots, p_{n-1} \\in P$ form a counterclockwise convex polygon if for all indices $i < j < k$ the oriented angle $\\measuredangle p_i p_j p_k$ at $p_j$ has sign $-1$, that is, $p_i, p_j, p_k$ are not collinear and lie in counterclockwise order with respect to the orientation of $P$. Every list of at most two points qualifies.","title":"Counterclockwise convex polygons"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-convex-polygon","kind":"node","lean":"EuclideanGeometry.IsConvexPolygon","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-convex-polygon","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Points $p_0, \\dots, p_{n-1}$ of an oriented two-dimensional Euclidean affine space form a convex polygon if one of the lists $p_0, p_1, \\dots, p_{n-1}$ and $p_0, p_{n-1}, p_{n-2}, \\dots, p_1$ is a counterclockwise convex polygon: for any three of its entries $q, r, s$, in the order in which they occur in that list, the oriented angle $\\measuredangle q r s$ at $r$ has sign $-1$. Thus for $n \\ge 3$ the points are the\u2026","title":"Convex polygons"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-convex-polygon-three-of-affine-indepe","kind":"node","lean":"EuclideanGeometry.isConvexPolygon_three_of_affineIndependent","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-convex-polygon-three-of-affine-indepe","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"If three points $A, B, C$ of an oriented two-dimensional Euclidean affine space are affinely independent, then $A, B, C$ form a convex polygon.","title":"Affinely independent triples are convex polygons"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/oriented-triangle-area","kind":"node","lean":"EuclideanGeometry.orientedTriangleArea","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#oriented-triangle-area","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Let $\\omega$ be the area form of the given orientation of the Euclidean plane, so that $\\omega(u, v)$ is the signed area of the parallelogram spanned by the vectors $u$ and $v$. For points $a$, $b$, $c$ of the plane, the oriented area of the triangle with vertices $a$, $b$, $c$ is the real number $\\omega(a - c, b - c) / 2$.","title":"Oriented triangle area"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/maximal-distinct-distances-from","kind":"node","lean":"EuclideanGeometry.maximalDistinctDistancesFrom","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#maximal-distinct-distances-from","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"For a finite set $X \\subseteq \\mathbb{R}^2$ and a point $x \\in X$, let $R_X(x)$ be the number of distinct distances $\\lvert y - x \\rvert$ from $x$ to the points $y \\in X$ other than $x$. For $n \\in \\mathbb{N}$, $g(n)$ is the maximum, over all sets $X \\subseteq \\mathbb{R}^2$ of exactly $n$ points, of the number of distinct values among the numbers $R_X(x)$ with $x \\in X$.","title":"Maximum number of distinct distance counts"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/in-general-position","kind":"node","lean":"EuclideanGeometry.InGeneralPosition","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#in-general-position","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"A set $X \\subseteq \\mathbb{R}^2$, finite or infinite, is in _general position_ if no three distinct points of $X$ are collinear and no four distinct points of $X$ lie on a common circle.","title":"Points in general position"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-right-angled","kind":"node","lean":"EuclideanGeometry.IsRightAngled","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-right-angled","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Points $a, b, c$ of a Euclidean affine space are the vertices of a right-angled triangle if they are pairwise distinct and the unoriented angle at one of the three vertices equals $\\pi / 2$: $\\angle bac = \\pi / 2$, $\\angle abc = \\pi / 2$ or $\\angle bca = \\pi / 2$.","title":"Right-angled triangles"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-isosceles-trapezoid","kind":"node","lean":"EuclideanGeometry.IsIsoscelesTrapezoid","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-isosceles-trapezoid","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Points $a, b, c, d \\in \\mathbb{R}^2$ are the vertices, in counterclockwise order, of an isosceles trapezoid if they are in strictly convex position and listed counterclockwise (for any three of them $p, q, r$, in the order in which they occur in the list $a, b, c, d$, the oriented angle $\\measuredangle p q r$ at $q$ has sign $-1$), the line through $a$ and $b$ is parallel to the line through $c$ and $d$ (the two\u2026","title":"Isosceles trapezoids"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-isosceles-free","kind":"node","lean":"Set.IsIsoscelesFree","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-isosceles-free","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"A set $A$ in a space with a distance function $d$ is isosceles-free if there are no three distinct points $x, y, z \\in A$ such that two of the distances $d(x, y)$, $d(y, z)$, $d(z, x)$ are equal.","title":"Isosceles-free sets"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-general-position","kind":"node","lean":"IsGeneralPosition","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-general-position","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Let $k$ be a division ring and $P$ an affine space over $k$. A family $(p_i)_{i \\in \\iota}$ of points of $P$ is in general position through dimension $d \\in \\mathbb{N}$ if for every finite set $S \\subseteq \\iota$ of at most $d + 1$ indices, the subfamily $(p_i)_{i \\in S}$ is affinely independent over $k$.","title":"General position of affine points"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-general-position-2","kind":"node","lean":"AffineIndependent.isGeneralPosition","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-general-position-2","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"If a family $(p_i)_{i \\in \\iota}$ of points of an affine space over a division ring $k$ is affinely independent over $k$, then it is in general position through dimension $d$ for every $d \\in \\mathbb{N}$.","title":"Affinely independent families are in general position"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/mono","kind":"node","lean":"IsGeneralPosition.mono","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#mono","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Let $d \\le e$ be natural numbers. If a family of points of an affine space over a division ring $k$ is in general position through dimension $e$, then it is in general position through dimension $d$.","title":"General position in affine space is monotone in the dimension"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-general-position-3","kind":"node","lean":"Projectivization.IsGeneralPosition","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-general-position-3","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Let $k$ be a division ring and $V$ a vector space over $k$, with projective space $\\mathbb{P}(V)$. A family $(p_i)_{i \\in \\iota}$ of points of $\\mathbb{P}(V)$ is in general position through dimension $d \\in \\mathbb{N}$ if for every finite set $S \\subseteq \\iota$ of at most $d + 1$ indices, the subfamily $(p_i)_{i \\in S}$ is projectively independent.","title":"General position of projective points"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/is-general-position-4","kind":"node","lean":"Projectivization.Independent.isGeneralPosition","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#is-general-position-4","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"If a family $(p_i)_{i \\in \\iota}$ of points of the projective space $\\mathbb{P}(V)$ of a vector space $V$ over a division ring $k$ is projectively independent, then it is in general position through dimension $d$ for every $d \\in \\mathbb{N}$.","title":"Projectively independent families are in general position"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/mono-2","kind":"node","lean":"Projectivization.IsGeneralPosition.mono","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#mono-2","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"Let $d \\le e$ be natural numbers. If a family of points of the projective space $\\mathbb{P}(V)$ of a vector space $V$ over a division ring $k$ is in general position through dimension $e$, then it is in general position through dimension $d$.","title":"General position in projective space is monotone in the dimension"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/convex-position-threshold","kind":"node","lean":"MathlibExt.Combinatorics.ConvexPositionHigherDimensionsErdosProblem651Wanted.convexPositionThreshold","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#convex-position-threshold","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $f_k(n)$ is the least natural number $m$ such that every set of $m$ points of $\\mathbb{R}^k$ in general position (any at most $k + 1$ of them affinely independent) contains $n$ points in convex position (none of them in the convex hull of the others). If there is no such $m$, then $f_k(n) = 0$.","title":"Convex position threshold"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/convex-position-threshold-exponential","kind":"node","lean":"MathlibExt.Combinatorics.ConvexPositionHigherDimensionsErdosProblem651Wanted.convexPositionThresholdExponential","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#convex-position-threshold-exponential","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"fully_proved","summary":"For natural numbers $k$ and $n$ let $f_k(n)$ be the least $m$ such that every set of $m$ points of $\\mathbb{R}^k$ in general position (any at most $k + 1$ of them affinely independent) contains $n$ points in convex position (none of them in the convex hull of the others), with $f_k(n) = 0$ if there is no such $m$. The claim is that for every $k \\ge 2$ there is a real $c > 0$ with $(1 + c)^n < f_k(n)$ for all\u2026","title":"Erd\u0151s Problem 651: exponential growth of convex position thresholds"},{"id":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/convex-position-threshold-exponential-re","kind":"node","lean":"MathlibExt.Combinatorics.ConvexPositionHigherDimensionsErdosProblem651Wanted.convexPositionThresholdExponential_refuted","parent":14078,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets/index.html#convex-position-threshold-exponential-re","scope":"Geometry/discrete-geometry/part-01/convex-and-general-position-of-point-sets","status":"can_prove","summary":"For natural numbers $k$ and $n$ let $f_k(n)$ be the least $m$ such that every set of $m$ points of $\\mathbb{R}^k$ in general position (any at most $k + 1$ of them affinely independent) contains $n$ points in convex position (none of them in the convex hull of the others), with $f_k(n) = 0$ if there is no such $m$. It is not the case that for every $k \\ge 2$ there is a real $c > 0$ with $(1 + c)^n < f_k(n)$ for all\u2026","title":"Erd\u0151s Problem 651 has a negative answer"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/equiangular-lines.json","id":"Geometry/discrete-geometry/part-01/equiangular-lines","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/equiangular-lines/index.html","scope":"Geometry/discrete-geometry/part-01/equiangular-lines","status":"can_prove","title":"Equiangular lines"},{"id":"Geometry/discrete-geometry/part-01/equiangular-lines/equiangular-lines","kind":"node","lean":"MathlibExt.Combinatorics.EquiangularLinesFixedAngleWanted.EquiangularLinesFixedAngle.EquiangularLines","parent":14101,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equiangular-lines/index.html#equiangular-lines","scope":"Geometry/discrete-geometry/part-01/equiangular-lines","status":"fully_proved","summary":"For a real number $\\alpha$ and a natural number $d$, an equiangular system with inner product $\\pm\\alpha$ in $\\mathbb{R}^d$ is a set of unit vectors in $d$-dimensional Euclidean space such that any two distinct vectors $x$ and $y$ in it satisfy $|\\langle x, y \\rangle| = \\alpha$.","title":"Equiangular systems of unit vectors"},{"id":"Geometry/discrete-geometry/part-01/equiangular-lines/n","kind":"node","lean":"MathlibExt.Combinatorics.EquiangularLinesFixedAngleWanted.EquiangularLinesFixedAngle.N","parent":14101,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equiangular-lines/index.html#n","scope":"Geometry/discrete-geometry/part-01/equiangular-lines","status":"fully_proved","summary":"For a real number $\\alpha$ and a natural number $d$, $N_\\alpha(d)$ is the supremum, taken in $\\mathbb{N}$, of the cardinalities of the equiangular systems of unit vectors in $\\mathbb{R}^d$ with inner product $\\pm\\alpha$. Here an infinite set counts as having cardinality $0$, and a set of natural numbers that is not bounded above has supremum $0$.","title":"Maximum number of equiangular lines"},{"id":"Geometry/discrete-geometry/part-01/equiangular-lines/spectral-radius-order","kind":"node","lean":"MathlibExt.Combinatorics.EquiangularLinesFixedAngleWanted.EquiangularLinesFixedAngle.spectralRadiusOrder","parent":14101,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equiangular-lines/index.html#spectral-radius-order","scope":"Geometry/discrete-geometry/part-01/equiangular-lines","status":"fully_proved","summary":"For a real number $\\lambda$, the spectral radius order $k(\\lambda) \\in \\mathbb{N} \\cup \\{\\infty\\}$ is the least natural number $k$ such that some simple graph on $k$ vertices has an adjacency matrix with spectral radius $\\max(\\lambda, 0)$, and $k(\\lambda) = \\infty$ if there is no such graph.","title":"Spectral radius order"},{"id":"Geometry/discrete-geometry/part-01/equiangular-lines/theorem-1-2","kind":"node","lean":"MathlibExt.Combinatorics.EquiangularLinesFixedAngleWanted.EquiangularLinesFixedAngle.theorem_1_2","parent":14101,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equiangular-lines/index.html#theorem-1-2","scope":"Geometry/discrete-geometry/part-01/equiangular-lines","status":"can_prove","summary":"Fix a real number $\\alpha$ with $0 < \\alpha < 1$, let $\\lambda = (1 - \\alpha)/(2\\alpha)$ and let $k = k(\\lambda)$ be the spectral radius order of $\\lambda$. Let $N_\\alpha(d)$ be the maximum number of equiangular lines in $\\mathbb{R}^d$ with common angle $\\arccos \\alpha$. Then: (a) if $k < \\infty$, then $N_\\alpha(d) = \\lfloor k(d - 1)/(k - 1) \\rfloor$ for all sufficiently large $d$; (b) if $k = \\infty$, then\u2026","title":"Equiangular lines with a fixed angle"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets.json","id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"can_prove","title":"Equilateral triangles in planar point sets"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/is-equilateral-triangle","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.IsEquilateralTriangle","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#is-equilateral-triangle","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"fully_proved","summary":"For a real number $s$, a finite set $T$ of points in the Euclidean space $\\mathbb{R}^d$ is an equilateral triangle of side $s$ when $T$ has exactly three elements and any two distinct points of $T$ are at distance $s$.","title":"Equilateral triangle with given side"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/is-unit-equilateral-triangle","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.IsUnitEquilateralTriangle","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#is-unit-equilateral-triangle","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"fully_proved","summary":"A finite set $T$ of points in the Euclidean space $\\mathbb{R}^d$ is a unit equilateral triangle when it is an equilateral triangle of side $1$: $T$ has exactly three elements and any two distinct points of $T$ are at distance $1$.","title":"Unit equilateral triangle"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/is-any-size-equilateral-triangle","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.IsAnySizeEquilateralTriangle","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#is-any-size-equilateral-triangle","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"fully_proved","summary":"A finite set $T$ of points in the Euclidean space $\\mathbb{R}^d$ is an equilateral triangle of any size when there is a real number $s > 0$ such that $T$ is an equilateral triangle of side $s$: $T$ has exactly three elements and any two distinct points of $T$ are at distance $s$.","title":"Equilateral triangle of any size"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/unit-equilateral-triangle-count","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.unitEquilateralTriangleCount","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#unit-equilateral-triangle-count","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"fully_proved","summary":"For a finite set $P$ of points in the Euclidean space $\\mathbb{R}^d$, this is the number of three-element subsets of $P$ that are unit equilateral triangles, that is, whose three points are pairwise at distance $1$.","title":"Number of unit equilateral triangles in a point set"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/any-size-equilateral-triangle-count","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.anySizeEquilateralTriangleCount","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#any-size-equilateral-triangle-count","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"fully_proved","summary":"For a finite set $P$ of points in the Euclidean space $\\mathbb{R}^d$, this is the number of three-element subsets of $P$ that are equilateral triangles of some positive side length $s$, that is, whose three points are pairwise at distance $s$.","title":"Number of equilateral triangles in a point set"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/tunit","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.TUnit","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#tunit","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"fully_proved","summary":"For natural numbers $d$ and $n$, $T_1(d, n)$ is the supremum in $\\mathbb{N}$, over all sets $P$ of exactly $n$ points in the Euclidean space $\\mathbb{R}^d$, of the number of unit equilateral triangles spanned by $P$ (the three-element subsets of $P$ whose points are pairwise at distance $1$). When such sets exist this is the largest such number; when none exist (only for $d = 0$ and $n \\ge 2$) it is $0$.","title":"Maximum number of unit equilateral triangles"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/tany-size","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.TAnySize","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#tany-size","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"fully_proved","summary":"For natural numbers $d$ and $n$, $T(d, n)$ is the supremum in $\\mathbb{N}$, over all sets $P$ of exactly $n$ points in the Euclidean space $\\mathbb{R}^d$, of the number of equilateral triangles of positive side length spanned by $P$ (the three-element subsets of $P$ whose points are pairwise at the same positive distance). When such sets exist this is the largest such number; when none exist (only for $d = 0$ and\u2026","title":"Maximum number of equilateral triangles"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/conjecture","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.conjecture","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#conjecture","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"fully_proved","summary":"Let $T_1(n)$ be the largest number of unit equilateral triangles spanned by a set of $n$ points in $\\mathbb{R}^6$, and let $T(n)$ be the largest number of equilateral triangles of positive side length spanned by a set of $n$ points in $\\mathbb{R}^6$; a triangle spanned by a set is a three-element subset of it whose points are pairwise at the same distance. The statement asserts both of the following: there is a\u2026","title":"Statement of Erd\u0151s Problem 755"},{"id":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/conjecture-holds","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem755Wanted.conjecture_holds","parent":14106,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets/index.html#conjecture-holds","scope":"Geometry/discrete-geometry/part-01/equilateral-triangles-in-planar-point-sets","status":"can_prove","summary":"Let $T_1(n)$ be the largest number of unit equilateral triangles spanned by a set of $n$ points in $\\mathbb{R}^6$, and let $T(n)$ be the largest number of equilateral triangles of positive side length spanned by a set of $n$ points in $\\mathbb{R}^6$; a triangle spanned by a set is a three-element subset of it whose points are pairwise at the same distance. Then there is a function $o : \\mathbb{N} \\to \\mathbb{R}$\u2026","title":"Erd\u0151s Problem 755 holds (Clemen\u2013Dumitrescu\u2013Liu)"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields.json","id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"can_prove","title":"Kakeya sets in finite fields"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/space","kind":"node","lean":"MathlibExt.Combinatorics.FraserKakeyaWanted.Combinatorics.FraserKakeyaProblem.Space","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#space","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"fully_proved","summary":"For a type $F$ and a natural number $d$, $F^d$ denotes the set of $d$-tuples $(x_1, \\dots, x_d)$ of elements of $F$, that is, functions from a $d$-element index set to $F$.","title":"Coordinate space"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/dot","kind":"node","lean":"MathlibExt.Combinatorics.FraserKakeyaWanted.Combinatorics.FraserKakeyaProblem.dot","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#dot","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field and $d$ a natural number. For $x, y \\in F^d$, the dot product is $x \\cdot y = \\sum_{i} x_i y_i \\in F$, the sum running over the $d$ coordinates.","title":"Dot product"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/affine-line","kind":"node","lean":"MathlibExt.Combinatorics.FraserKakeyaWanted.Combinatorics.FraserKakeyaProblem.affineLine","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#affine-line","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field and $d$ a natural number. For $y, x \\in F^d$, the affine line with base point $y$ and direction $x$ is the set $\\{y + a x : a \\in F\\} \\subseteq F^d$. When $x = 0$ this set is the single point $\\{y\\}$.","title":"Affine line"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/is-kakeya","kind":"node","lean":"MathlibExt.Combinatorics.FraserKakeyaWanted.Combinatorics.FraserKakeyaProblem.IsKakeya","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#is-kakeya","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field and $d$ a natural number. A set $K \\subseteq F^d$ is a Kakeya set if for every $x \\in F^d$ there is $y \\in F^d$ such that the affine line $\\{y + a x : a \\in F\\}$ is contained in $K$; that is, $K$ contains a line in every direction.","title":"Kakeya set"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/is-probability-measure-on","kind":"node","lean":"MathlibExt.Combinatorics.FraserKakeyaWanted.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#is-probability-measure-on","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field, $d$ a natural number and $K \\subseteq F^d$. A function $\\mu : F^d \\to \\mathbb{R}$ is a probability measure on $K$ if $\\mu(x) \\ge 0$ for every $x \\in F^d$, $\\sum_{x \\in F^d} \\mu(x) = 1$, and $\\mu(x) \\ne 0$ implies $x \\in K$.","title":"Probability measure supported on a set"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/fourier","kind":"node","lean":"MathlibExt.Combinatorics.FraserKakeyaWanted.Combinatorics.FraserKakeyaProblem.fourier","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#fourier","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"fully_proved","summary":"Let $F$ be a finite field, $d$ a natural number and $\\chi$ an additive character of $F$ with values in $\\mathbb{C}$. For $\\mu : F^d \\to \\mathbb{R}$ and $\\xi \\in F^d$, the Fourier transform of $\\mu$ at $\\xi$ is $\\hat{\\mu}(\\xi) = \\sum_{x \\in F^d} \\chi(-\\xi \\cdot x)\\, \\mu(x) \\in \\mathbb{C}$, where $\\xi \\cdot x$ is the dot product.","title":"Finite-field Fourier transform"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/fraser-kakeya-fourier-decay-and-sharp","kind":"node","lean":"MathlibExt.Combinatorics.FraserKakeyaWanted.Combinatorics.FraserKakeyaProblem.fraser_kakeya_fourier_decay_and_sharp","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#fraser-kakeya-fourier-decay-and-sharp","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements, let $d \\ge 2$ be a natural number, let $K \\subseteq F^d$ be a Kakeya set and let $\\chi$ be a nontrivial additive character of $F$ with values in $\\mathbb{C}$. Then, first, there is a probability measure $\\mu$ on $K$ whose Fourier transform $\\hat{\\mu}$ with respect to $\\chi$ satisfies $|\\hat{\\mu}(\\xi)| \\le q^{-1}$ for every nonzero $\\xi \\in F^d$. Second, this bound is\u2026","title":"Fraser's Fourier decay theorem for finite-field Kakeya sets"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/finite-field-kakeya-cube-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.FiniteFieldKakeyaThreeAIMCombinatoricsProblem245Wanted.finiteFieldKakeyaCubeLowerBound","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#finite-field-kakeya-cube-lower-bound","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"fully_proved","summary":"This is the statement that there is a real number $c > 0$ such that for every finite field $F$ and every Kakeya set $E \\subseteq F^3$, that is, a set such that for every $x \\in F^3$ there is $y \\in F^3$ with $\\{y + a x : a \\in F\\} \\subseteq E$, we have $|E| \\ge c\\,|F|^3$.","title":"Statement of the finite field Kakeya bound in dimension three"},{"id":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/finite-field-kakeya-cube-lower-bound-hol","kind":"node","lean":"MathlibExt.Combinatorics.FiniteFieldKakeyaThreeAIMCombinatoricsProblem245Wanted.finiteFieldKakeyaCubeLowerBound_holds","parent":14116,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields/index.html#finite-field-kakeya-cube-lower-bound-hol","scope":"Geometry/discrete-geometry/part-01/kakeya-sets-in-finite-fields","status":"can_prove","summary":"There is a real number $c > 0$ such that for every finite field $F$ and every Kakeya set $E \\subseteq F^3$, that is, a set such that for every $x \\in F^3$ there is $y \\in F^3$ with $\\{y + a x : a \\in F\\} \\subseteq E$, we have $|E| \\ge c\\,|F|^3$.","title":"Finite field Kakeya sets in dimension three have cubic size"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions.json","id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","title":"Lattice points in dilated regions"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/integer-unit-cube","kind":"node","lean":"LatticePointCubeSandwich.integerUnitCube","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#integer-unit-cube","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For $v \\in \\mathbb{Z}^n$, the integer unit cube at $v$ is the half-open cube $Q_v = \\prod_i [v_i, v_i + 1) \\subseteq \\mathbb{R}^n$, the set of $x \\in \\mathbb{R}^n$ with $v_i \\le x_i < v_i + 1$ for every coordinate $i$.","title":"Integer unit cube"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/mem-integer-unit-cube-iff","kind":"node","lean":"LatticePointCubeSandwich.mem_integerUnitCube_iff","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#mem-integer-unit-cube-iff","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For $v \\in \\mathbb{Z}^n$ and $x \\in \\mathbb{R}^n$, $x$ lies in the integer unit cube at $v$ if and only if $v_i \\le x_i < v_i + 1$ for every coordinate $i$.","title":"Membership in an integer unit cube"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/outer-cube-set","kind":"node","lean":"LatticePointCubeSandwich.outerCubeSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#outer-cube-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a set $S \\subseteq \\mathbb{R}^n$ and a real number $t$, the outer cube set of $S$ at scale $t$ is the set of $v \\in \\mathbb{Z}^n$ such that $(x_1/t, \\dots, x_n/t) \\in S$ for some $x$ in the integer unit cube $Q_v = \\prod_i [v_i, v_i + 1)$. For $t > 0$ these are the $v$ with $Q_v$ meeting $tS$; for $t = 0$ the division is read as $x_i/0 = 0$.","title":"Outer cube set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/mem-outer-cube-set-iff","kind":"node","lean":"LatticePointCubeSandwich.mem_outerCubeSet_iff","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#mem-outer-cube-set-iff","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For $S \\subseteq \\mathbb{R}^n$, $t \\in \\mathbb{R}$ and $v \\in \\mathbb{Z}^n$, $v$ belongs to the outer cube set of $S$ at scale $t$ if and only if $(x_1/t, \\dots, x_n/t) \\in S$ for some $x$ in the integer unit cube at $v$.","title":"Membership in the outer cube set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/inner-cube-set","kind":"node","lean":"LatticePointCubeSandwich.innerCubeSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#inner-cube-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a set $S \\subseteq \\mathbb{R}^n$ and a real number $t$, the inner cube set of $S$ at scale $t$ is the set of $v \\in \\mathbb{Z}^n$ such that $(x_1/t, \\dots, x_n/t) \\in S$ for every $x$ in the integer unit cube $Q_v = \\prod_i [v_i, v_i + 1)$. For $t > 0$ these are the $v$ with $Q_v \\subseteq tS$; for $t = 0$ the division is read as $x_i/0 = 0$.","title":"Inner cube set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/mem-inner-cube-set-iff","kind":"node","lean":"LatticePointCubeSandwich.mem_innerCubeSet_iff","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#mem-inner-cube-set-iff","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For $S \\subseteq \\mathbb{R}^n$, $t \\in \\mathbb{R}$ and $v \\in \\mathbb{Z}^n$, $v$ belongs to the inner cube set of $S$ at scale $t$ if and only if $(x_1/t, \\dots, x_n/t) \\in S$ for every $x$ in the integer unit cube at $v$.","title":"Membership in the inner cube set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/inner-cube-set-subset-outer-cube-set","kind":"node","lean":"LatticePointCubeSandwich.innerCubeSet_subset_outerCubeSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#inner-cube-set-subset-outer-cube-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For every set $S \\subseteq \\mathbb{R}^n$ and every real $t$, the inner cube set of $S$ at scale $t$ is contained in the outer cube set of $S$ at scale $t$.","title":"Inner cube indices are outer cube indices"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/finite-outer-cube-set","kind":"node","lean":"LatticePointCubeSandwich.finite_outerCubeSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#finite-outer-cube-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"If $S \\subseteq \\mathbb{R}^n$ is bounded and $t > 0$, then the outer cube set of $S$ at scale $t$, the set of $v \\in \\mathbb{Z}^n$ whose integer unit cube $\\prod_i [v_i, v_i + 1)$ meets $tS$, is finite.","title":"Finiteness of the outer cube set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/cube-measure-sandwich","kind":"node","lean":"LatticePointCubeSandwich.cube_measure_sandwich","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#cube-measure-sandwich","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{R}^n$ be bounded, let $t > 0$, and let $\\mathrm{vol}(S)$ be the Lebesgue (outer) measure of $S$, which is finite. For $v \\in \\mathbb{Z}^n$ let $Q_v = \\prod_i [v_i, v_i + 1)$. Let $I$ be the inner cube set of $S$ at scale $t$, the $v$ with $Q_v \\subseteq tS$, and $O$ the outer cube set, the $v$ with $Q_v$ meeting $tS$. Then $|I| \\le \\mathrm{vol}(S)\\, t^n \\le |O|$.","title":"Cubical measure sandwich"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/std-lattice","kind":"node","lean":"LatticePointEnumerator.stdLattice","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#std-lattice","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, the standard integer lattice $\\mathbb{Z}^d$ is the $\\mathbb{Z}$-submodule of $\\mathbb{R}^d$ (the real-valued functions on $\\{0, \\dots, d-1\\}$) spanned by the standard basis vectors $e_0, \\dots, e_{d-1}$.","title":"The standard integer lattice"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/mem-std-lattice-iff","kind":"node","lean":"LatticePointEnumerator.mem_stdLattice_iff","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#mem-std-lattice-iff","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a vector $x \\in \\mathbb{R}^d$ lies in the standard lattice $\\mathbb{Z}^d$, the integer span of the standard basis vectors of $\\mathbb{R}^d$, exactly when every coordinate $x_i$ of $x$ is an integer.","title":"Membership in the standard lattice"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/lattice-point-set","kind":"node","lean":"LatticePointEnumerator.latticePointSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#lattice-point-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a set $P \\subseteq \\mathbb{R}^d$ and a real number $t$, this is the set $(tP) \\cap \\mathbb{Z}^d$ of points of the integer lattice $\\mathbb{Z}^d$ in the dilate $tP = \\{tp : p \\in P\\}$.","title":"Lattice points in a dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/lattice-enumerator","kind":"node","lean":"LatticePointEnumerator.latticeEnumerator","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#lattice-enumerator","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a set $P \\subseteq \\mathbb{R}^d$ and a real number $t$, this is the number of elements of the set $(tP) \\cap \\mathbb{Z}^d$ of points of the integer lattice $\\mathbb{Z}^d$ in the dilate $tP = \\{tp : p \\in P\\}$, where an infinite set is assigned the value $0$.","title":"Lattice-point count of a real dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/translated-lattice-point-set","kind":"node","lean":"LatticePointEnumerator.translatedLatticePointSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#translated-lattice-point-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, a set $P \\subseteq \\mathbb{R}^d$, a real $t$ and $x \\in \\mathbb{R}^d$, the translated lattice-point set is $((t \\cdot P) + x) \\cap \\mathbb{Z}^d$, the set of points of the standard integer lattice $\\mathbb{Z}^d$ in the dilate $t \\cdot P$ translated by $x$.","title":"Translated lattice-point set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/translated-lattice-enumerator","kind":"node","lean":"LatticePointEnumerator.translatedLatticeEnumerator","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#translated-lattice-enumerator","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, a set $P \\subseteq \\mathbb{R}^d$, a real $t$ and $x \\in \\mathbb{R}^d$, the translated lattice-point count is the number of elements of the translated lattice-point set $((t \\cdot P) + x) \\cap \\mathbb{Z}^d$, taken to be $0$ when this set is infinite.","title":"Translated lattice-point count for arbitrary sets"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/lattice-point-set-pos","kind":"node","lean":"LatticePointEnumerator.latticePointSetPos","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#lattice-point-set-pos","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a set $P \\subseteq \\mathbb{R}^d$ and a real number $t > 0$, this is the set $(tP) \\cap \\mathbb{Z}^d$ of points of the integer lattice $\\mathbb{Z}^d$ in the dilate $tP = \\{tp : p \\in P\\}$.","title":"Lattice points in a positive dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/lattice-enumerator-pos","kind":"node","lean":"LatticePointEnumerator.latticeEnumeratorPos","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#lattice-enumerator-pos","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a bounded set $P \\subseteq \\mathbb{R}^d$ and a real number $t > 0$, the lattice-point enumerator $L_P(t)$ is the number of points of the integer lattice $\\mathbb{Z}^d$ in the dilate $tP = \\{tp : p \\in P\\}$, that is, $L_P(t) = |(tP) \\cap \\mathbb{Z}^d|$.","title":"Lattice-point enumerator"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/translated-lattice-point-set-pos","kind":"node","lean":"LatticePointEnumerator.translatedLatticePointSetPos","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#translated-lattice-point-set-pos","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, a set $P \\subseteq \\mathbb{R}^d$, a real $t > 0$ and $x \\in \\mathbb{R}^d$, this is the set $((t \\cdot P) + x) \\cap \\mathbb{Z}^d$ of points of the standard integer lattice $\\mathbb{Z}^d$ in the dilate $t \\cdot P$ translated by $x$. No boundedness of $P$ is assumed.","title":"Translated lattice-point set at a positive dilation"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/translated-lattice-enumerator-pos","kind":"node","lean":"LatticePointEnumerator.translatedLatticeEnumeratorPos","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#translated-lattice-enumerator-pos","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$, let $P \\subseteq \\mathbb{R}^d$ be a bounded set, let $t > 0$ be real and let $x \\in \\mathbb{R}^d$. The translated lattice-point enumerator is $F_{t,P}(x) = |((t \\cdot P) + x) \\cap \\mathbb{Z}^d|$, the number of points of the standard integer lattice $\\mathbb{Z}^d$ in the dilate $t \\cdot P$ translated by $x$.","title":"Translated lattice-point enumerator of a bounded set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/zero-mem-std-lattice","kind":"node","lean":"LatticePointEnumerator.zero_mem_stdLattice","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#zero-mem-std-lattice","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For every $d \\in \\mathbb{N}$, the origin of $\\mathbb{R}^d$ lies in the standard integer lattice $\\mathbb{Z}^d$.","title":"The origin is a lattice point"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/lattice-enumerator-empty","kind":"node","lean":"LatticePointEnumerator.latticeEnumerator_empty","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#lattice-enumerator-empty","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For every natural number $d$ and every real number $t$, the lattice-point count of the empty set $\\emptyset \\subseteq \\mathbb{R}^d$ at the dilate $t$ is $0$.","title":"The lattice-point count of the empty set vanishes"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/translated-lattice-enumerator-empty","kind":"node","lean":"LatticePointEnumerator.translatedLatticeEnumerator_empty","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#translated-lattice-enumerator-empty","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For every $d \\in \\mathbb{N}$, every real $t$ and every $x \\in \\mathbb{R}^d$, the translated lattice-point count $|((t \\cdot \\emptyset) + x) \\cap \\mathbb{Z}^d|$ is $0$.","title":"The translated lattice-point count of the empty set is zero"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/finite-lattice-point-set","kind":"node","lean":"LatticePointEnumerator.finite_latticePointSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#finite-lattice-point-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a bounded set $P \\subseteq \\mathbb{R}^d$ and any real number $t$, the set $(tP) \\cap \\mathbb{Z}^d$ of points of the integer lattice $\\mathbb{Z}^d$ in the dilate $tP = \\{tp : p \\in P\\}$ is finite.","title":"Finiteness of the lattice points in a dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/finite-lattice-point-set-pos","kind":"node","lean":"LatticePointEnumerator.finite_latticePointSetPos","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#finite-lattice-point-set-pos","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a bounded set $P \\subseteq \\mathbb{R}^d$ and a real number $t > 0$, the set $(tP) \\cap \\mathbb{Z}^d$ of points of the integer lattice $\\mathbb{Z}^d$ in the dilate $tP = \\{tp : p \\in P\\}$ is finite.","title":"Finiteness of the lattice points in a positive dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/finite-translated-lattice-point-set","kind":"node","lean":"LatticePointEnumerator.finite_translatedLatticePointSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#finite-translated-lattice-point-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a bounded set $P \\subseteq \\mathbb{R}^d$, any real number $t$ and a vector $x \\in \\mathbb{R}^d$, the set $(tP + x) \\cap \\mathbb{Z}^d$ of points of the integer lattice $\\mathbb{Z}^d$ in the translated dilate $tP + x = \\{tp + x : p \\in P\\}$ is finite.","title":"Finiteness of the lattice points in a translated dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/finite-translated-lattice-point-set-pos","kind":"node","lean":"LatticePointEnumerator.finite_translatedLatticePointSetPos","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#finite-translated-lattice-point-set-pos","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, a bounded set $P \\subseteq \\mathbb{R}^d$, a real number $t > 0$ and a vector $x \\in \\mathbb{R}^d$, the set $(tP + x) \\cap \\mathbb{Z}^d$ of points of the integer lattice $\\mathbb{Z}^d$ in the translated dilate $tP + x = \\{tp + x : p \\in P\\}$ is finite.","title":"Finiteness of the lattice points in a translated positive dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/unit-cube","kind":"node","lean":"unitCube","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#unit-cube","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For a natural number $d$, the closed unit cube $[0, 1]^d$ is the set of points $x \\in \\mathbb{R}^d$ with $0 \\le x_i \\le 1$ for every coordinate $i$.","title":"The unit cube"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/is-lipschitz-parametrizable","kind":"node","lean":"IsLipschitzParametrizable","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#is-lipschitz-parametrizable","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $X$ be a metric space and $d$ a natural number. A set $B \\subseteq X$ is $d$-Lipschitz parametrizable if there are a natural number $n$ (possibly $0$) and Lipschitz maps $f_1, \\dots, f_n : [0, 1]^d \\to X$ out of the closed unit cube $[0, 1]^d \\subseteq \\mathbb{R}^d$ such that $B$ equals the union of their images, $B = f_1([0, 1]^d) \\cup \\dots \\cup f_n([0, 1]^d)$.","title":"Lipschitz parametrizable sets"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/unit-cube-zero-mem","kind":"node","lean":"unitCube_zero_mem","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#unit-cube-zero-mem","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For every natural number $d$, the origin of $\\mathbb{R}^d$ lies in the closed unit cube $[0, 1]^d$.","title":"The origin lies in the unit cube"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/is-lipschitz-parametrizable-empty","kind":"node","lean":"isLipschitzParametrizable_empty","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#is-lipschitz-parametrizable-empty","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $X$ be a metric space. For every natural number $d$, the empty subset of $X$ is $d$-Lipschitz parametrizable, via the empty family of maps.","title":"The empty set is Lipschitz parametrizable"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/is-lipschitz-parametrizable-singleton","kind":"node","lean":"isLipschitzParametrizable_singleton","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#is-lipschitz-parametrizable-singleton","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $X$ be a metric space. For every natural number $d$ and every point $x \\in X$, the set $\\lbrace x \\rbrace$ is $d$-Lipschitz parametrizable, via a single constant map.","title":"Singletons are Lipschitz parametrizable"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/is-lipschitz-parametrizable-coe","kind":"node","lean":"Finset.isLipschitzParametrizable_coe","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#is-lipschitz-parametrizable-coe","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $X$ be a metric space and $d$ a natural number. Every finite set $s$ of points of $X$, given as a finset, is $d$-Lipschitz parametrizable.","title":"Finsets are Lipschitz parametrizable"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/is-lipschitz-parametrizable-2","kind":"node","lean":"Set.Finite.isLipschitzParametrizable","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#is-lipschitz-parametrizable-2","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $X$ be a metric space. Every finite set $B \\subseteq X$ is $d$-Lipschitz parametrizable for every natural number $d$.","title":"Finite sets are Lipschitz parametrizable"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/image-lipschitz-with","kind":"node","lean":"IsLipschitzParametrizable.image_lipschitzWith","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#image-lipschitz-with","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $X$ and $Y$ be metric spaces, let $d$ be a natural number and let $f : X \\to Y$ be a Lipschitz map. If a set $B \\subseteq X$ is $d$-Lipschitz parametrizable, then its image $f(B) \\subseteq Y$ is $d$-Lipschitz parametrizable.","title":"Lipschitz images of Lipschitz parametrizable sets"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/integer-cube-set","kind":"node","lean":"LipschitzImageCubeCount.integerCubeSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#integer-cube-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $f$ be a map from the unit cube $[0, 1]^d$ to $\\mathbb{R}^n$ and let $t$ be a real number. The integer cube set of $f$ at scale $t$ is the set of integer vectors $v \\in \\mathbb{Z}^n$ for which some point $y$ of the unit cube satisfies $v_i \\le t f(y)_i < v_i + 1$ for every coordinate $i$, that is, the set of integer vectors whose half-open unit cube $\\prod_i [v_i, v_i + 1)$ meets the dilate $t \\cdot f([0, 1]^d)$.","title":"Integer cube set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/mem-integer-cube-set-iff","kind":"node","lean":"LipschitzImageCubeCount.mem_integerCubeSet_iff","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#mem-integer-cube-set-iff","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $f$ be a map from the unit cube $[0, 1]^d$ to $\\mathbb{R}^n$, let $t$ be a real number and let $v \\in \\mathbb{Z}^n$. Then $v$ belongs to the integer cube set of $f$ at scale $t$ if and only if some point $y$ of the unit cube satisfies $v_i \\le t f(y)_i < v_i + 1$ for every coordinate $i$.","title":"Membership in the integer cube set"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/abs-floor-sub-le-ceil","kind":"node","lean":"LipschitzImageCubeCount.abs_floor_sub_le_ceil","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#abs-floor-sub-le-ceil","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"For real numbers $a$, $b$ and $D$ with $|a - b| \\le D$, the integers $\\lfloor a \\rfloor$ and $\\lfloor b \\rfloor$ satisfy $|\\lfloor a \\rfloor - \\lfloor b \\rfloor| \\le \\lceil D \\rceil$.","title":"Floor differences are bounded by a ceiling"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/scaled-coord-abs-le","kind":"node","lean":"LipschitzImageCubeCount.scaled_coord_abs_le","parent":14126,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dilated Lipschitz maps vary by at most "],["math","\\(K\\)"],["text"," on cells of size "],["math","\\(1/T\\)"]],"source":"Dilated Lipschitz maps vary by at most $K$ on cells of size $1/T$"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#scaled-coord-abs-le","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $K \\ge 0$, let $f$ be a $K$-Lipschitz map from the unit cube $[0, 1]^d$ to $\\mathbb{R}^n$, both carrying the sup distance, and let $t$ and $T$ be real numbers with $0 < t \\le T$. If $y$ and $y'$ are points of the unit cube at distance at most $1/T$, then $|t f(y)_i - t f(y')_i| \\le K$ for every coordinate $i$.","title":"Dilated Lipschitz maps vary by at most $K$ on cells of size $1/T$"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/card-integer-cube-set-le","kind":"node","lean":"LipschitzImageCubeCount.card_integerCubeSet_le","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#card-integer-cube-set-le","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $d$ and $n$ be natural numbers, let $K \\ge 0$, and let $f$ be a $K$-Lipschitz map from the unit cube $[0, 1]^d$ to $\\mathbb{R}^n$, both carrying the sup distance. For every real $t \\ge 1$, the cardinality of the integer cube set of $f$ at scale $t$, taken as a natural number, is at most $\\lceil t \\rceil^d \\, (2 \\lceil K \\sqrt{d} \\rceil + 3)^n$.","title":"Counting integer cubes met by a dilated Lipschitz image"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/finite-integer-cube-set","kind":"node","lean":"LipschitzImageCubeCount.finite_integerCubeSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#finite-integer-cube-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $d$ and $n$ be natural numbers, let $K \\ge 0$, and let $f$ be a $K$-Lipschitz map from the unit cube $[0, 1]^d$ to $\\mathbb{R}^n$, both carrying the sup distance. For every real $t \\ge 1$, the integer cube set of $f$ at scale $t$ is finite.","title":"Dilated Lipschitz images meet finitely many integer cubes"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/exists-cube-count-le-const-mul-pow","kind":"node","lean":"LipschitzImageCubeCount.exists_cubeCount_le_const_mul_pow","parent":14126,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dilated Lipschitz images meet "],["math","\\(O(t^d)\\)"],["text"," integer cubes"]],"source":"Dilated Lipschitz images meet $O(t^d)$ integer cubes"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#exists-cube-count-le-const-mul-pow","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $d$ and $n$ be natural numbers, let $K \\ge 0$, and let $f$ be a $K$-Lipschitz map from the unit cube $[0, 1]^d$ to $\\mathbb{R}^n$, both carrying the sup distance. Then there is a real number $C > 0$ such that for every real $t \\ge 1$, the cardinality of the integer cube set of $f$ at scale $t$, taken as a natural number, is at most $C t^d$.","title":"Dilated Lipschitz images meet $O(t^d)$ integer cubes"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/outer-diff-inner-subset-i-union-integer","kind":"node","lean":"LatticePointBoundaryCubes.outer_diff_inner_subset_iUnion_integerCubeSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#outer-diff-inner-subset-i-union-integer","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{R}^n$, let $f_1, \\dots, f_m : [0,1]^d \\to \\mathbb{R}^n$ be maps whose ranges together cover the frontier of $S$, and let $t$ be a real number. For $v \\in \\mathbb{Z}^n$ write $Q_v$ for the half-open unit cube $\\prod_i [v_i, v_i + 1)$. If $v$ lies in the outer cube set of $S$ at scale $t$ (some $x \\in Q_v$ has $x / t \\in S$, coordinatewise, with $x / 0 = 0$) but not in the inner cube set (not\u2026","title":"Boundary cubes meet the dilated frontier charts"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/card-outer-le-card-inner-add-sum","kind":"node","lean":"LatticePointBoundaryCubes.card_outer_le_card_inner_add_sum","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#card-outer-le-card-inner-add-sum","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{R}^n$, let $f_1, \\dots, f_m : [0,1]^d \\to \\mathbb{R}^n$ be maps whose ranges together cover the frontier of $S$, and let $t$ be a real number. Let $O$ and $I$ be the outer and inner cube sets of $S$ at scale $t$: the $v \\in \\mathbb{Z}^n$ for which some, respectively every, point $x$ of the half-open unit cube $\\prod_i [v_i, v_i + 1)$ has $x / t \\in S$ (coordinatewise, with $x / 0 = 0$). Let\u2026","title":"Counting outer cubes by inner cubes and boundary charts"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/raw-subtype-equiv-canonical-point-set","kind":"node","lean":"LatticePointAsymptotics.rawSubtypeEquivCanonicalPointSet","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#raw-subtype-equiv-canonical-point-set","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{R}^n$ and $t > 0$. Viewing an integer vector as a real vector gives a bijection from the set of $x \\in \\mathbb{Z}^n$ with $x / t \\in S$ onto the lattice point set $tS \\cap \\mathbb{Z}^n$.","title":"Integer points of a dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/card-raw-eq-lattice-enumerator","kind":"node","lean":"LatticePointAsymptotics.card_raw_eq_latticeEnumerator","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#card-raw-eq-lattice-enumerator","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{R}^n$ and $t > 0$. The number of integer vectors $x \\in \\mathbb{Z}^n$ with $x / t \\in S$ equals the lattice enumerator of $S$ at $t$, the number of points of $\\mathbb{Z}^n$ in the dilate $tS$; here an infinite set is counted as $0$.","title":"Integer count of a dilate"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/lattice-point-count-error-bound","kind":"node","lean":"LatticePointAsymptotics.lattice_point_count_error_bound","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#lattice-point-count-error-bound","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{R}^n$ be bounded, let $f_1, \\dots, f_m$ be maps from the unit cube $[0,1]^d$ to $\\mathbb{R}^n$ whose ranges cover the boundary $\\partial S$, and let $t > 0$. For each $i$, let $N_i(t)$ be the number of $v \\in \\mathbb{Z}^n$ such that $t f_i(y)$ lies in the half-open cube $\\prod_j [v_j, v_j + 1)$ for some $y \\in [0,1]^d$, and assume each of these sets of $v$ is finite. Then the lattice\u2026","title":"Lattice-point error bound from boundary charts"},{"id":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/lattice-point-count-asymptotics","kind":"node","lean":"LatticePointAsymptotics.lattice_point_count_asymptotics","parent":14126,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions/index.html#lattice-point-count-asymptotics","scope":"Geometry/discrete-geometry/part-01/lattice-points-in-dilated-regions","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{R}^n$ be bounded, and suppose its boundary $\\partial S$ is $(n-1)$-Lipschitz parametrizable: it is the union of the ranges of finitely many Lipschitz maps from the unit cube $[0,1]^{n-1}$, where $n - 1$ is read as $0$ when $n = 0$. Then there is a real constant $C$ such that for all sufficiently large real $t$, the lattice enumerator $L_S(t)$, the number of points of $\\mathbb{Z}^n$ in $tS$,\u2026","title":"Lattice-point count of a dilate with boundary error"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/moving-sofa-problem.json","id":"Geometry/discrete-geometry/part-01/moving-sofa-problem","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","title":"Moving sofa problem"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/corridor","kind":"node","lean":"MovingSofa.corridor","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#corridor","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The corridor is the L-shaped set of points $z \\in \\mathbb{C}$ with $\\operatorname{Re} z \\le 1$ and $0 \\le \\operatorname{Im} z \\le 1$ (the horizontal arm, extending to the left) together with the points $z$ with $0 \\le \\operatorname{Re} z \\le 1$ and $\\operatorname{Im} z \\ge 0$ (the vertical arm, extending upward): two perpendicular semi-infinite arms of width one.","title":"Corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/entrance","kind":"node","lean":"MovingSofa.entrance","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#entrance","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The entrance region of the unit right-angle corridor is the closed half-strip $\\{z \\in \\mathbb{C} : \\operatorname{Re} z \\le 0,\\ 0 \\le \\operatorname{Im} z \\le 1\\}$, the part of its horizontal arm to the left of the corner square.","title":"Entrance of the corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/exit","kind":"node","lean":"MovingSofa.exit","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#exit","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The exit region of the unit right-angle corridor is the closed half-strip $\\{z \\in \\mathbb{C} : 0 \\le \\operatorname{Re} z \\le 1,\\ \\operatorname{Im} z \\ge 1\\}$, the part of its vertical arm above the corner square.","title":"Exit of the corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/motion","kind":"node","lean":"MovingSofa.Motion","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#motion","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"A motion consists of a continuous map $t \\mapsto u_t$ from $[0, 1]$ to the unit circle $\\{u \\in \\mathbb{C} : |u| = 1\\}$ (the rotation) and a continuous map $t \\mapsto c_t$ from $[0, 1]$ to $\\mathbb{C}$ (the translation).","title":"Planar rigid motion"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/position","kind":"node","lean":"MovingSofa.Motion.position","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#position","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"Let $m$ be a motion with rotation $u(t)$ in the unit circle and translation $c(t) \\in \\mathbb{C}$ at each time $t \\in [0, 1]$. The position at time $t$ of a point $z \\in \\mathbb{C}$ is $u(t)\\,z + c(t)$.","title":"Position under a rigid motion"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/can-pass","kind":"node","lean":"MovingSofa.CanPass","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#can-pass","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{C}$ can pass the corridor if there are continuous maps $u$ from $[0, 1]$ to the unit circle $\\{w \\in \\mathbb{C} : |w| = 1\\}$ and $c$ from $[0, 1]$ to $\\mathbb{C}$ such that, writing $K_t = \\{u(t) z + c(t) : z \\in K\\}$ for the position of $K$ at time $t$, $K_0$ lies in the entrance arm, $K_1$ lies in the exit arm, and $K_t$ lies in the corridor for every $t \\in [0, 1]$.","title":"Passing the corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/is-sofa","kind":"node","lean":"MovingSofa.IsSofa","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#is-sofa","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{C}$ is a sofa when it is Lebesgue measurable, nonempty and connected, and can be moved through the unit right-angle corridor: there is a motion (a rotation by a unit complex number and a translation, both varying continuously with time $t \\in [0, 1]$) that places $K$ inside the corridor's entrance region at time $0$, inside its exit region at time $1$, and inside the corridor at every\u2026","title":"Sofa shapes"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/optimal-area","kind":"node","lean":"MovingSofa.optimalArea","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#optimal-area","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"Let the corridor be $L = \\{z \\in \\mathbb{C} : \\operatorname{Re} z \\le 1,\\ 0 \\le \\operatorname{Im} z \\le 1\\} \\cup \\{z \\in \\mathbb{C} : 0 \\le \\operatorname{Re} z \\le 1,\\ 0 \\le \\operatorname{Im} z\\}$, with entrance $E = \\{z \\in \\mathbb{C} : \\operatorname{Re} z \\le 0,\\ 0 \\le \\operatorname{Im} z \\le 1\\}$ and exit $X = \\{z \\in \\mathbb{C} : 0 \\le \\operatorname{Re} z \\le 1,\\ 1 \\le \\operatorname{Im} z\\}$. A sofa is a\u2026","title":"Optimal sofa area"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/unit-square","kind":"node","lean":"MovingSofa.unitSquare","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#unit-square","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The unit square is the set of complex numbers $z$ with $0 \\le \\operatorname{Re} z \\le 1$ and $0 \\le \\operatorname{Im} z \\le 1$, that is, the square $[0,1] \\times [0,1]$ in the complex plane.","title":"Unit square"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-motion","kind":"node","lean":"MovingSofa.squareMotion","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-motion","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The motion with no rotation that sends a point $z$ at time $t \\in [0,1]$ to $z + \\tau(t)$, where $\\tau(t) = 2t - 1$ for $t \\le 1/2$ and $\\tau(t) = (2t - 1)i$ for $t \\ge 1/2$: it moves a shape one unit to the right during the first half of the time interval and then one unit upward.","title":"The square's translation through the corner"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/motion-preserves-distance","kind":"node","lean":"MovingSofa.motion_preserves_distance","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#motion-preserves-distance","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"Let $m$ be a motion, $t \\in [0, 1]$ a time, and write $m_t(z)$ for the position of a point $z \\in \\mathbb{C}$ under $m$ at time $t$. Then $|m_t(z) - m_t(w)| = |z - w|$ for all $z, w \\in \\mathbb{C}$.","title":"Motions preserve distance"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-measurable","kind":"node","lean":"MovingSofa.square_measurable","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-measurable","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The unit square $[0,1] \\times [0,1]$ is a measurable set for the Borel $\\sigma$-algebra of the complex plane.","title":"The unit square is Borel measurable"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-connected","kind":"node","lean":"MovingSofa.square_connected","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-connected","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The unit square $\\{z \\in \\mathbb{C} : 0 \\le \\operatorname{Re} z \\le 1,\\ 0 \\le \\operatorname{Im} z \\le 1\\}$ is a nonempty connected subset of $\\mathbb{C}$.","title":"The unit square is connected"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-area","kind":"node","lean":"MovingSofa.square_area","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-area","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The unit square $\\{z \\in \\mathbb{C} : 0 \\le \\operatorname{Re} z \\le 1,\\ 0 \\le \\operatorname{Im} z \\le 1\\}$ has Lebesgue measure $1$.","title":"The unit square has area one"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-can-pass","kind":"node","lean":"MovingSofa.square_canPass","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-can-pass","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"Let the corridor be $L = \\{z \\in \\mathbb{C} : \\operatorname{Re} z \\le 1,\\ 0 \\le \\operatorname{Im} z \\le 1\\} \\cup \\{z \\in \\mathbb{C} : 0 \\le \\operatorname{Re} z \\le 1,\\ 0 \\le \\operatorname{Im} z\\}$, with entrance $E = \\{z \\in \\mathbb{C} : \\operatorname{Re} z \\le 0,\\ 0 \\le \\operatorname{Im} z \\le 1\\}$ and exit $X = \\{z \\in \\mathbb{C} : 0 \\le \\operatorname{Re} z \\le 1,\\ 1 \\le \\operatorname{Im} z\\}$. For the unit\u2026","title":"The unit square passes the corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-is-sofa","kind":"node","lean":"MovingSofa.square_isSofa","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-is-sofa","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The unit square $[0,1] \\times [0,1]$ in the complex plane is a sofa: it is null-measurable for Lebesgue measure, connected, and there are continuous maps $u \\colon [0,1] \\to \\{w \\in \\mathbb{C} : |w| = 1\\}$ and $v \\colon [0,1] \\to \\mathbb{C}$ such that the motion $z \\mapsto u(t) z + v(t)$ carries the square into the entrance $\\{z : \\operatorname{Re} z \\le 0,\\ 0 \\le \\operatorname{Im} z \\le 1\\}$ at $t = 0$, into the\u2026","title":"The unit square is a sofa"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/one-le-optimal-area","kind":"node","lean":"MovingSofa.one_le_optimalArea","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#one-le-optimal-area","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The optimal sofa area, that is, the supremum in $[0, \\infty]$ of the Lebesgue areas of all sofas, is at least $1$.","title":"The optimal sofa area is at least one"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/corridor-other","kind":"node","lean":"MovingSofa.corridorOther","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#corridor-other","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The mirrored corridor is the set of points $z \\in \\mathbb{C}$ with $\\operatorname{Re} z \\ge -1$ and $0 \\le \\operatorname{Im} z \\le 1$ (the horizontal arm, extending to the right) together with the points $z$ with $-1 \\le \\operatorname{Re} z \\le 0$ and $\\operatorname{Im} z \\ge 0$ (the vertical arm, extending upward). It is the reflection of the corridor in the imaginary axis, so passing it requires the opposite turn.","title":"Mirrored corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/entrance-other","kind":"node","lean":"MovingSofa.entranceOther","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#entrance-other","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The entrance arm of the mirrored corridor is the set of points $z \\in \\mathbb{C}$ with $\\operatorname{Re} z \\ge 0$ and $0 \\le \\operatorname{Im} z \\le 1$: the part of its horizontal arm beyond the corner square $-1 \\le \\operatorname{Re} z \\le 0$, $0 \\le \\operatorname{Im} z \\le 1$.","title":"Entrance arm of the mirrored corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/exit-other","kind":"node","lean":"MovingSofa.exitOther","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#exit-other","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The exit region of the mirrored unit corridor is the closed half-strip $\\{z \\in \\mathbb{C} : -1 \\le \\operatorname{Re} z \\le 0,\\ \\operatorname{Im} z \\ge 1\\}$, the part of its vertical arm above its corner square.","title":"Exit of the mirrored corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/can-pass-other","kind":"node","lean":"MovingSofa.CanPassOther","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#can-pass-other","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{C}$ can pass the mirrored corridor if there are continuous maps $u$ from $[0, 1]$ to the unit circle $\\{w \\in \\mathbb{C} : |w| = 1\\}$ and $c$ from $[0, 1]$ to $\\mathbb{C}$ such that, writing $K_t = \\{u(t) z + c(t) : z \\in K\\}$ for the position of $K$ at time $t$, $K_0$ lies in the entrance arm of the mirrored corridor, $K_1$ lies in the exit arm of the mirrored corridor, and $K_t$ lies in\u2026","title":"Passing the mirrored corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/corridor-z","kind":"node","lean":"MovingSofa.corridorZ","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#corridor-z","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"For a real number $d$, the Z-shaped corridor with corner separation $d$ is the set of points $z \\in \\mathbb{C}$ lying in at least one of three pieces: the lower horizontal arm, where $\\operatorname{Re} z \\le 1$ and $0 \\le \\operatorname{Im} z \\le 1$; the middle channel, where $0 \\le \\operatorname{Re} z \\le 1$ and $0 \\le \\operatorname{Im} z \\le 2 + d$; and the upper horizontal arm, where $\\operatorname{Re} z \\ge 0$\u2026","title":"Z-shaped corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/entrance-z","kind":"node","lean":"MovingSofa.entranceZ","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#entrance-z","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The entrance region of the Z corridor is the closed half-strip $\\{z \\in \\mathbb{C} : \\operatorname{Re} z \\le 0,\\ 0 \\le \\operatorname{Im} z \\le 1\\}$, the part of its horizontal entrance arm to the left of the middle channel.","title":"Entrance arm of the Z corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/exit-z","kind":"node","lean":"MovingSofa.exitZ","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#exit-z","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"For a real number $d$ (the corner separation), the exit region of the Z corridor with separation $d$ is the closed half-strip $\\{z \\in \\mathbb{C} : \\operatorname{Re} z \\ge 1,\\ 1 + d \\le \\operatorname{Im} z \\le 2 + d\\}$, the part of its horizontal exit arm to the right of the middle channel.","title":"Exit arm of the Z corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/can-pass-zat","kind":"node","lean":"MovingSofa.CanPassZAt","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#can-pass-zat","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"For a real number $d$, a set $K \\subseteq \\mathbb{C}$ can pass the Z corridor with corner separation $d$ if there are continuous maps $u$ from $[0, 1]$ to the unit circle $\\{w \\in \\mathbb{C} : |w| = 1\\}$ and $c$ from $[0, 1]$ to $\\mathbb{C}$ such that, writing $K_t = \\{u(t) z + c(t) : z \\in K\\}$ for the position of $K$ at time $t$, $K_0$ lies in the entrance arm of the Z corridor, $K_1$ lies in the exit arm of the\u2026","title":"Passing the Z corridor at a given separation"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/can-pass-z","kind":"node","lean":"MovingSofa.CanPassZ","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#can-pass-z","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{C}$ can pass the Z corridor if there is a real number $D$ such that, for every real $d \\ge D$, $K$ can pass the Z corridor with corner separation $d$; that is, $K$ passes whenever the corners are sufficiently separated.","title":"Passing the Z corridor for all large separations"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-motion-zat","kind":"node","lean":"MovingSofa.squareMotionZAt","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-motion-zat","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"For a real number $d$, the motion with no rotation that sends a point $z$ at time $t \\in [0,1]$ to $z + \\tau(t)$, where $\\tau(t) = 3t - 1$ for $t \\le 1/3$, $\\tau(t) = (1 + d)(3t - 1)i$ for $1/3 \\le t \\le 2/3$, and $\\tau(t) = (3t - 2) + (1 + d)i$ for $t \\ge 2/3$: it translates a shape by $1$, then by $(1 + d)i$, then by $1$, each segment taking a third of the time interval.","title":"The square's translation through the Z corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-can-pass-zat","kind":"node","lean":"MovingSofa.square_canPassZAt","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-can-pass-zat","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"For every real $d \\ge 0$, the unit square passes the Z corridor with corner separation $d$.","title":"Z-corridor passage of the unit square"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-motion-z","kind":"node","lean":"MovingSofa.squareMotionZ","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-motion-z","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The square's translation through the Z corridor at separation $d = 1$: the motion with no rotation that sends a point $z$ at time $t \\in [0,1]$ to $z + \\tau(t)$, where $\\tau(t) = 3t - 1$ for $t \\le 1/3$, $\\tau(t) = 2(3t - 1)i$ for $1/3 \\le t \\le 2/3$, and $\\tau(t) = (3t - 2) + 2i$ for $t \\ge 2/3$.","title":"The square's translation through the Z corridor at separation one"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-can-pass-zat-one","kind":"node","lean":"MovingSofa.square_canPassZAt_one","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-can-pass-zat-one","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The unit square passes the Z corridor with corner separation $1$.","title":"Z-corridor passage of the unit square at separation one"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-can-pass-z","kind":"node","lean":"MovingSofa.square_canPassZ","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-can-pass-z","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"There is a real $D$ such that for every $d \\ge D$ the unit square passes the Z corridor with corner separation $d$.","title":"Z-corridor passage of the unit square for large separations"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/is-ambidextrous-sofa","kind":"node","lean":"MovingSofa.IsAmbidextrousSofa","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#is-ambidextrous-sofa","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{C}$ is an ambidextrous sofa when it is a sofa and there is a real number $D$ such that, for every corner separation $d \\ge D$, a single motion places $K$ inside the entrance arm of the Z corridor with separation $d$ at time $0$, inside its exit arm at time $1$, and inside that corridor at every time; such a passage turns once in each direction.","title":"Ambidextrous sofa shapes"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/ambidextrous-optimal-area","kind":"node","lean":"MovingSofa.ambidextrousOptimalArea","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#ambidextrous-optimal-area","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The ambidextrous optimal area is the supremum, taken in $[0, \\infty]$, of the Lebesgue areas of all ambidextrous sofas $K \\subseteq \\mathbb{C}$.","title":"Optimal area of ambidextrous sofas"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/ambidextrous-optimal-area-le-optimal-are","kind":"node","lean":"MovingSofa.ambidextrousOptimalArea_le_optimalArea","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#ambidextrous-optimal-area-le-optimal-are","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The supremum, in $[0, \\infty]$, of the Lebesgue areas of all ambidextrous sofas is at most the supremum of the Lebesgue areas of all sofas. This holds because every ambidextrous sofa is a sofa.","title":"The ambidextrous optimal area is at most the optimal area"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-motion-other","kind":"node","lean":"MovingSofa.squareMotionOther","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-motion-other","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The motion with no rotation that sends a point $z$ at time $t \\in [0,1]$ to $z + \\tau(t)$, where $\\tau(t) = -2t$ for $t \\le 1/2$ and $\\tau(t) = -1 + (2t - 1)i$ for $t \\ge 1/2$: it moves a shape one unit to the left during the first half of the time interval and then one unit upward.","title":"The square's translation through the mirrored corner"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-can-pass-other","kind":"node","lean":"MovingSofa.square_canPassOther","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-can-pass-other","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"Let the mirrored corridor be $L' = \\{z \\in \\mathbb{C} : -1 \\le \\operatorname{Re} z,\\ 0 \\le \\operatorname{Im} z \\le 1\\} \\cup \\{z \\in \\mathbb{C} : -1 \\le \\operatorname{Re} z \\le 0,\\ 0 \\le \\operatorname{Im} z\\}$, with entrance $E' = \\{z \\in \\mathbb{C} : 0 \\le \\operatorname{Re} z,\\ 0 \\le \\operatorname{Im} z \\le 1\\}$ and exit $X' = \\{z \\in \\mathbb{C} : -1 \\le \\operatorname{Re} z \\le 0,\\ 1 \\le \\operatorname{Im} z\\}$. For\u2026","title":"The unit square passes the mirrored corridor"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/square-is-ambidextrous-sofa","kind":"node","lean":"MovingSofa.square_isAmbidextrousSofa","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#square-is-ambidextrous-sofa","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The unit square $[0,1] \\times [0,1]$ in the complex plane is an ambidextrous sofa.","title":"The unit square is an ambidextrous sofa"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/one-le-ambidextrous-optimal-area","kind":"node","lean":"MovingSofa.one_le_ambidextrousOptimalArea","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#one-le-ambidextrous-optimal-area","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"fully_proved","summary":"The supremum of the Lebesgue areas of all ambidextrous sofas, taken in $[0, \\infty]$, is at least $1$.","title":"The ambidextrous sofa constant is at least one"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/hammersley-upper","kind":"node","lean":"AmbidextrousSofaWanted.hammersley_upper","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#hammersley-upper","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","summary":"The supremum of the areas of ambidextrous sofas, taken in $[0, \\infty]$, is at most $2\\sqrt{2}$.","title":"Hammersley upper bound for the ambidextrous sofa"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/baek-upper","kind":"node","lean":"AmbidextrousSofaWanted.baek_upper","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#baek-upper","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","summary":"The supremum of the areas of ambidextrous sofas, taken in $[0, \\infty]$, is at most $22195/10000 = 2.2195$.","title":"Baek upper bound for the ambidextrous sofa"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/gibbs-lower","kind":"node","lean":"AmbidextrousSofaWanted.gibbs_lower","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#gibbs-lower","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","summary":"The supremum of the areas of ambidextrous sofas, taken in $[0, \\infty]$, is at least $164495/100000 = 1.64495$.","title":"Gibbs lower bound for the ambidextrous sofa"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/romik-lower","kind":"node","lean":"AmbidextrousSofaWanted.romik_lower","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#romik-lower","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","summary":"The supremum of the areas of ambidextrous sofas, taken in $[0, \\infty]$, is at least $164495521/100000000 = 1.64495521$.","title":"Romik lower bound for the ambidextrous sofa"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/hammersley-upper-2","kind":"node","lean":"MovingSofaWanted.hammersley_upper","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#hammersley-upper-2","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","summary":"The moving sofa constant, the supremum of the areas of connected Lebesgue-measurable planar shapes that a continuous rigid motion can carry through the right-angled corridor of width $1$, is at most $2\\sqrt{2}$.","title":"Hammersley's upper bound for the moving sofa constant"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/kallus-romik-upper","kind":"node","lean":"MovingSofaWanted.kallus_romik_upper","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#kallus-romik-upper","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","summary":"The moving sofa constant, the supremum of the areas of connected Lebesgue-measurable planar shapes that a continuous rigid motion can carry through the right-angled corridor of width $1$, is at most $2.37$.","title":"Kallus\u2013Romik upper bound for the moving sofa constant"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/hammersley-lower","kind":"node","lean":"MovingSofaWanted.hammersley_lower","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#hammersley-lower","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","summary":"The moving sofa constant, the supremum of the areas of connected Lebesgue-measurable planar shapes that a continuous rigid motion can carry through the right-angled corridor of width $1$, is at least $\\pi/2 + 2/\\pi$.","title":"Hammersley's lower bound for the moving sofa constant"},{"id":"Geometry/discrete-geometry/part-01/moving-sofa-problem/gerver-printed-lower","kind":"node","lean":"MovingSofaWanted.gerver_printed_lower","parent":14174,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/moving-sofa-problem/index.html#gerver-printed-lower","scope":"Geometry/discrete-geometry/part-01/moving-sofa-problem","status":"can_prove","summary":"The moving sofa constant, the supremum of the areas of connected Lebesgue-measurable planar shapes that a continuous rigid motion can carry through the right-angled corridor of width $1$, is at least $2.2195$.","title":"Gerver's lower bound for the moving sofa constant"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives.json","id":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/index.html","scope":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","status":"can_prove","title":"Pancake cutting with exotic knives"},{"id":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/plane-region-count","kind":"node","lean":"MetaMathlibExt.planeRegionCount","parent":14221,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/index.html#plane-region-count","scope":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","status":"fully_proved","summary":"For a set $K \\subseteq \\mathbb{R}^2$ of cuts, the number of regions into which $K$ divides the plane is the number of connected components of the complement $\\mathbb{R}^2 \\setminus K$, or $0$ if there are infinitely many components.","title":"Number of regions cut out of the plane"},{"id":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/plane-region-count-empty","kind":"node","lean":"MetaMathlibExt.planeRegionCount_empty","parent":14221,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/index.html#plane-region-count-empty","scope":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","status":"fully_proved","summary":"The empty set of cuts divides the plane into exactly one region.","title":"No cuts, one region"},{"id":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/is-greatest-plane-region-count-three-arm","kind":"node","lean":"MetaMathlibExt.isGreatest_planeRegionCount_threeArmedV","parent":14221,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/index.html#is-greatest-plane-region-count-three-arm","scope":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","status":"can_prove","summary":"A long-legged three-armed V is the union of three rays $\\{c + s u_j : s \\ge 0\\}$, $j = 1, 2, 3$, from a common tip $c \\in \\mathbb{R}^2$, with nonzero direction vectors $u_1, u_2, u_3 \\in \\mathbb{R}^2$ in arbitrary directions. For every natural number $n$, the largest number of regions into which the union of $n$ long-legged three-armed V's can divide the plane is $(9n^2 - 5n + 2)/2$.","title":"Maximum region count for three-armed V's"},{"id":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/is-greatest-plane-region-count-three-cha","kind":"node","lean":"MetaMathlibExt.isGreatest_planeRegionCount_threeChain","parent":14221,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/index.html#is-greatest-plane-region-count-three-cha","scope":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","status":"can_prove","summary":"For points $p, q \\in \\mathbb{R}^2$ and nonzero vectors $u, v \\in \\mathbb{R}^2$, the 3-chain is the union of the ray $\\{p + s u : s \\ge 0\\}$, the segment $[p, q]$ and the ray $\\{q + s v : s \\ge 0\\}$; the directions are arbitrary and the 3-chain may intersect itself. For every natural number $n$, the largest number of regions into which the union of $n$ 3-chains can divide the plane is $(9n^2 - 5n + 2)/2$.","title":"Maximum region count for 3-chains"},{"id":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/is-greatest-plane-region-count-long-legg","kind":"node","lean":"MetaMathlibExt.isGreatest_planeRegionCount_longLeggedA","parent":14221,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/index.html#is-greatest-plane-region-count-long-legg","scope":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","status":"can_prove","summary":"For a tip $c \\in \\mathbb{R}^2$, nonzero vectors $u_1, u_2 \\in \\mathbb{R}^2$ and reals $a, b > 0$, the long-legged A is the union of the rays $\\{c + s u_1 : s \\ge 0\\}$ and $\\{c + s u_2 : s \\ge 0\\}$ and the crossbar segment $[c + a u_1, c + b u_2]$. For every natural number $n$, the largest number of regions into which the union of $n$ long-legged A's can divide the plane is $(9n^2 - 5n + 2)/2$.","title":"Maximum region count for long-legged A's"},{"id":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/cos-sub-neg-of-tbar-inter-ncard-four","kind":"node","lean":"MetaMathlibExt.cos_sub_neg_of_tbar_inter_ncard_four","parent":14221,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/index.html#cos-sub-neg-of-tbar-inter-ncard-four","scope":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","status":"can_prove","summary":"For a point $c \\in \\mathbb{R}^2$ and an angle $\\theta \\in \\mathbb{R}$, the constrained long-legged T $\\bar T(c, \\theta)$ is the union of the crossbar, the line $\\{c + s(-\\sin\\theta, \\cos\\theta) : s \\in \\mathbb{R}\\}$, and the stem, the ray $\\{c + s(\\cos\\theta, \\sin\\theta) : s \\ge 0\\}$. For all $c_1, c_2 \\in \\mathbb{R}^2$ and $\\theta_1, \\theta_2 \\in \\mathbb{R}$, if $\\bar T(c_1, \\theta_1) \\cap \\bar T(c_2, \\theta_2)$\u2026","title":"Four-point intersections of constrained T's"},{"id":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/cos-sub-neg-of-lollipop-good-pair","kind":"node","lean":"MetaMathlibExt.cos_sub_neg_of_lollipop_good_pair","parent":14221,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives/index.html#cos-sub-neg-of-lollipop-good-pair","scope":"Geometry/discrete-geometry/part-01/pancake-cutting-with-exotic-knives","status":"can_prove","summary":"For a point $c = (a, b) \\in \\mathbb{R}^2$, a radius $r > 0$ and an angle $\\theta \\in \\mathbb{R}$, the lollipop with center $c$, radius $r$ and stem direction $(\\cos\\theta, \\sin\\theta)$ consists of the circle $\\{(x, y) \\in \\mathbb{R}^2 : (x - a)^2 + (y - b)^2 = r^2\\}$ and the stem $\\{c + s(\\cos\\theta, \\sin\\theta) : s \\ge r\\}$, the part of the ray from $c$ in that direction outside the open disc. For all\u2026","title":"Stem\u2013circle intersections of two lollipops"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/pick-s-theorem.json","id":"Geometry/discrete-geometry/part-01/pick-s-theorem","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"can_prove","title":"Pick's theorem"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/to-real-volume-convex-hull-lattice-trian","kind":"node","lean":"MeasureTheory.toReal_volume_convexHull_lattice_triangle","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#to-real-volume-convex-hull-lattice-trian","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"Let $u, v \\in \\mathbb{Z}^2$ with $u_0 v_1 - u_1 v_0 \\neq 0$, and let $T \\subset \\mathbb{R}^2$ be the convex hull of $0$, $u$ and $v$. Let $I$ be the number of points of $\\mathbb{Z}^2$ in the interior of $T$ and $B$ the number of points of $\\mathbb{Z}^2$ on the boundary of $T$. Then the area (two-dimensional Lebesgue measure) of $T$ equals $I + B/2 - 1$.","title":"Pick's theorem for lattice triangles with a vertex at the origin"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/pick-convex-lattice-polygon","kind":"node","lean":"MetaMathlibExt.pick_convex_lattice_polygon","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#pick-convex-lattice-polygon","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"Let $V \\subset \\mathbb{Z}^2$ be a finite set of lattice points, let $P \\subset \\mathbb{R}^2$ be the convex hull of $V$, and suppose that $P$ has nonempty interior. Let $I$ be the number of points of $\\mathbb{Z}^2$ in the interior of $P$ and $B$ the number of points of $\\mathbb{Z}^2$ on the boundary of $P$. Then the area (two-dimensional Lebesgue measure) of $P$ equals $I + B/2 - 1$.","title":"Pick's theorem for convex lattice polygons"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/to-plane","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.toPlane","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#to-plane","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"The map $\\mathbb{Z}^2 \\to \\mathbb{R}^2$ sending a lattice point $(a, b)$ to the point $(a, b)$ of the real plane.","title":"Lattice points in the plane"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/lat-poly","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.latPoly","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#lat-poly","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"Given integer points $v_0, \\dots, v_{n-1} \\in \\mathbb{Z}^2$, the lattice polygon on $v$ is the polygon in $\\mathbb{R}^2$ with $n$ vertices whose $i$-th vertex is $v_i$, viewed as a point of $\\mathbb{R}^2$.","title":"Lattice polygon"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/inside","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.inside","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#inside","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"For a set $S \\subseteq \\mathbb{R}^2$, the region enclosed by $S$ is the set of points $x \\notin S$ whose connected component in the complement $\\mathbb{R}^2 \\setminus S$ is bounded. For a simple closed curve $S$ this is the Jordan interior.","title":"Region enclosed by a planar set"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/area","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.area","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#area","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"The area enclosed by a set $S \\subseteq \\mathbb{R}^2$ is the Lebesgue measure, as a real number, of the region enclosed by $S$ (the points $x \\notin S$ whose connected component in $\\mathbb{R}^2 \\setminus S$ is bounded); it is taken to be $0$ when that measure is infinite.","title":"Enclosed area"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/adjacent","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.Adjacent","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#adjacent","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"Two indices $i, j \\in \\{0, \\dots, n-1\\}$ are adjacent when $j \\equiv i + 1 \\pmod n$ or $i \\equiv j + 1 \\pmod n$.","title":"Cyclically adjacent indices"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/is-simple","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.IsSimple","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#is-simple","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"A polygon in $\\mathbb{R}^2$ with vertices $p_0, \\dots, p_{n-1}$ (indices taken mod $n$) and edges the segments $e_i = [p_i, p_{i+1}]$ is simple when consecutive vertices are distinct ($p_i \\ne p_{i+1}$ for all $i$), any two edges $e_i$ and $e_j$ with $i \\ne j$, $j \\not\\equiv i + 1$ and $i \\not\\equiv j + 1 \\pmod n$ are disjoint, and every two consecutive edges meet only in their common vertex:\u2026","title":"Simple polygon"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/boundary-pts","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.boundaryPts","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#boundary-pts","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"For integer points $v_0, \\dots, v_{n-1} \\in \\mathbb{Z}^2$, the number of lattice points $z \\in \\mathbb{Z}^2$ that lie on the boundary (the union of the edges) of the lattice polygon on $v$.","title":"Number of boundary lattice points"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/interior-pts","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.interiorPts","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#interior-pts","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"fully_proved","summary":"For integer points $v_0, \\dots, v_{n-1} \\in \\mathbb{Z}^2$, the number of lattice points $z \\in \\mathbb{Z}^2$ that lie in the region enclosed by the boundary of the lattice polygon on $v$, that is, the points off the boundary whose connected component in its complement is bounded.","title":"Number of interior lattice points"},{"id":"Geometry/discrete-geometry/part-01/pick-s-theorem/pick","kind":"node","lean":"MathlibExt.Geometry.PicksTheoremWanted.Geometry.PicksTheorem.pick","parent":14229,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/pick-s-theorem/index.html#pick","scope":"Geometry/discrete-geometry/part-01/pick-s-theorem","status":"can_prove","summary":"Let $n \\ge 3$ and let $v_0, \\dots, v_{n-1} \\in \\mathbb{Z}^2$ be integer points such that the lattice polygon $P$ in $\\mathbb{R}^2$ with vertices $v_0, \\dots, v_{n-1}$ is simple. Then the area of the region enclosed by the boundary of $P$ equals $I + B/2 - 1$, where $I$ is the number of lattice points in that region and $B$ is the number of lattice points on the boundary of $P$.","title":"Pick's theorem"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes.json","id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"can_prove","title":"Platonic solids and regular polytopes"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/platonic-solids-classification","kind":"node","lean":"MetaMathlibExt.platonic_solids_classification","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#platonic-solids-classification","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"Let $V, E, F, p, q$ be natural numbers with $V, E, F > 0$ and $p, q \\ge 3$, and let $l > 0$ and $d > 0$ be real numbers. Let $x_0, \\dots, x_{V-1}$ be points of the Euclidean space $\\mathbb{R}^3$, let $K$ be their convex hull and let $\\partial K$ be its topological boundary. Let each edge $e < E$ be an ordered pair $(a_e, b_e)$ of point indices, with segment $[x_{a_e}, x_{b_e}]$, and let each face $f < F$ be a list\u2026","title":"Classification of the Platonic solids"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/e","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.E","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#e","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"For a natural number $n$, the real Euclidean space $\\mathbb{R}^n$: $n$-tuples of reals with the standard inner product and norm.","title":"Ambient Euclidean space of the Platonic classification"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/convex-polytope","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#convex-polytope","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"A convex polytope in $\\mathbb{R}^n$ is given by its vertex set: a finite nonempty set $V \\subseteq \\mathbb{R}^n$ that equals the set of extreme points of its convex hull $\\operatorname{conv}(V)$. The polytope itself is the convex set $\\operatorname{conv}(V)$.","title":"Convex polytope"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/to-set","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.toSet","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#to-set","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"The underlying set of a convex polytope $P$ in $\\mathbb{R}^n$ is the convex hull of its vertex set.","title":"Underlying set of a convex polytope"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/dim","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.dim","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#dim","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"The dimension of a convex polytope $P \\subseteq \\mathbb{R}^n$ is its affine dimension: the dimension of the real vector space spanned by the differences $x - y$ of points $x, y \\in P$.","title":"Dimension of a convex polytope"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/is-full-dim","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.IsFullDim","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#is-full-dim","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"A convex polytope $P \\subseteq \\mathbb{R}^n$ is full-dimensional if its dimension equals $n$.","title":"Full-dimensional convex polytope"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/is-face","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.IsFace","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#is-face","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"A face of a convex polytope $P \\subseteq \\mathbb{R}^n$ is a nonempty set of the form $\\{x \\in P : \\ell(y) \\le \\ell(x) \\text{ for all } y \\in P\\}$ for some linear functional $\\ell$ on $\\mathbb{R}^n$, that is, the set of points where $\\ell$ attains its maximum over $P$. Taking $\\ell = 0$ shows that $P$ itself is a face.","title":"Face of a convex polytope"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/face-dim","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.faceDim","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#face-dim","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"For a subset $F \\subseteq \\mathbb{R}^n$, its affine dimension is the dimension of the real vector space spanned by the differences $x - y$ with $x, y \\in F$. The empty set and every single point get dimension $0$.","title":"Affine dimension of a set"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/flag","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.Flag","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#flag","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"A flag of a convex polytope $P \\subseteq \\mathbb{R}^n$ is a sequence $F_0, F_1, \\dots, F_{n-1}$ of faces of $P$ such that each $F_k$ has affine dimension $k$ and $F_i \\subsetneq F_j$ whenever $i < j$.","title":"Flag of a convex polytope"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/isom","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.Isom","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#isom","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"The rigid motions of $\\mathbb{R}^n$ are the affine isometric bijections $\\mathbb{R}^n \\to \\mathbb{R}^n$.","title":"Rigid motions"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/is-symmetry","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.isSymmetry","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#is-symmetry","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"An affine isometry $\\varphi$ of $\\mathbb{R}^n$ is a symmetry of a convex polytope $P \\subseteq \\mathbb{R}^n$ if $\\varphi(P) = P$.","title":"Symmetry of a convex polytope"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/is-regular","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.IsRegular","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#is-regular","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"A convex polytope $P \\subseteq \\mathbb{R}^n$ is regular (Platonic) if it is full-dimensional and for any two flags $(F_0, \\dots, F_{n-1})$ and $(G_0, \\dots, G_{n-1})$ of $P$ there is an affine isometry $\\varphi$ of $\\mathbb{R}^n$ with $\\varphi(P) = P$ and $\\varphi(F_k) = G_k$ for every $k < n$.","title":"Regular convex polytope"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/similar","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.ConvexPolytope.Similar","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#similar","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"Two convex polytopes $P$ and $Q$ in $\\mathbb{R}^n$ are similar when there are a real number $a > 0$ and an affine isometry $\\varphi$ of $\\mathbb{R}^n$ such that the underlying set of $Q$ is $\\{a\\,\\varphi(x) : x \\in P\\}$.","title":"Similar polytopes"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/regular-polytopes","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.regularPolytopes","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#regular-polytopes","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, this is the set of regular (Platonic) convex polytopes in $\\mathbb{R}^d$: the convex polytopes in $\\mathbb{R}^d$ that are full-dimensional and whose symmetry group acts transitively on their flags.","title":"Regular polytopes"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/regular-similar","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.regularSimilar","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#regular-similar","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"For regular convex polytopes $P$ and $Q$ in $\\mathbb{R}^d$, $P$ is related to $Q$ when they are similar as convex polytopes, that is, when there are a real number $a > 0$ and an affine isometry $\\varphi$ of $\\mathbb{R}^d$ such that the underlying set of $Q$ is $\\{a\\,\\varphi(x) : x \\in P\\}$.","title":"Similarity of regular polytopes"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/platonic-count","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.platonicCount","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#platonic-count","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, $p_d \\in \\mathbb{N} \\cup \\{\\infty\\}$ is the number of similarity classes of regular convex polytopes in $\\mathbb{R}^d$, with $p_d = \\infty$ when there are infinitely many classes.","title":"Number of Platonic polytopes"},{"id":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/platonic-classification","kind":"node","lean":"MathlibExt.Geometry.PlatonicClassificationWanted.Geometry.PlatonicClassification.platonic_classification","parent":14241,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes/index.html#platonic-classification","scope":"Geometry/discrete-geometry/part-01/platonic-solids-and-regular-polytopes","status":"can_prove","summary":"For $d \\in \\mathbb{N}$ let $p_d \\in \\mathbb{N} \\cup \\{\\infty\\}$ be the number of regular convex polytopes in $\\mathbb{R}^d$ up to similarity. Then $p_2 = \\infty$, $p_3 = 5$, $p_4 = 6$, and $p_d = 3$ for every $d \\ge 5$.","title":"Classification of the Platonic polytopes"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes.json","id":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/index.html","scope":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes","status":"can_prove","title":"Points and spanned hyperplanes"},{"id":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/is-affine-hyperplane","kind":"node","lean":"MathlibExt.Geometry.InfiniteTransversalsSpannedHyperplanesWanted.ITSP.IsAffineHyperplane","parent":14259,"reading_title":{"format":"inline-parts-v1","parts":[["text","Affine hyperplanes in "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Affine hyperplanes in $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/index.html#is-affine-hyperplane","scope":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes","status":"fully_proved","summary":"An affine subspace $A$ of $\\mathbb{R}^d$ is an affine hyperplane when there are a nonzero linear functional $\\ell \\colon \\mathbb{R}^d \\to \\mathbb{R}$ and a real number $c$ such that $A = \\{x \\in \\mathbb{R}^d : \\ell(x) = c\\}$.","title":"Affine hyperplanes in $\\mathbb{R}^d$"},{"id":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/spanned-hyperplane-set","kind":"node","lean":"MathlibExt.Geometry.InfiniteTransversalsSpannedHyperplanesWanted.ITSP.SpannedHyperplaneSet","parent":14259,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/index.html#spanned-hyperplane-set","scope":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes","status":"fully_proved","summary":"For a set $P \\subseteq \\mathbb{R}^d$, the hyperplanes spanned by $P$ are the affine hyperplanes of $\\mathbb{R}^d$ that equal the affine span of some subset of $P$.","title":"Hyperplanes spanned by a point set"},{"id":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/owr1183-question","kind":"node","lean":"MathlibExt.Geometry.InfiniteTransversalsSpannedHyperplanesWanted.ITSP.OWR1183Question","parent":14259,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/index.html#owr1183-question","scope":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes","status":"fully_proved","summary":"The statement that for every $d \\ge 1$ and every set $P \\subseteq \\mathbb{R}^d$ whose affine span is all of $\\mathbb{R}^d$, writing $\\mathcal{H}$ for the set of affine hyperplanes of $\\mathbb{R}^d$ that equal the affine span of some subset of $P$, there is an injective map $f \\colon P \\to \\mathcal{H}$ with $p \\in f(p)$ for every $p \\in P$.","title":"Injective point-to-hyperplane matching question"},{"id":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/owr1183-question-holds","kind":"node","lean":"MathlibExt.Geometry.InfiniteTransversalsSpannedHyperplanesWanted.OWR1183Question_holds","parent":14259,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes/index.html#owr1183-question-holds","scope":"Geometry/discrete-geometry/part-01/points-and-spanned-hyperplanes","status":"can_prove","summary":"For every $d \\ge 1$ and every set $P \\subseteq \\mathbb{R}^d$ whose affine span is all of $\\mathbb{R}^d$, let $\\mathcal{H}$ be the set of affine hyperplanes of $\\mathbb{R}^d$ that equal the affine span of some subset of $P$. Then there is an injective map $f \\colon P \\to \\mathcal{H}$ with $p \\in f(p)$ for every $p \\in P$.","title":"Spanning point sets have injective point-to-hyperplane matchings"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/point-line-incidences.json","id":"Geometry/discrete-geometry/part-01/point-line-incidences","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"can_prove","title":"Point\u2013line incidences"},{"id":"Geometry/discrete-geometry/part-01/point-line-incidences/line-sizes","kind":"node","lean":"MathlibExt.Combinatorics.LineCompatibleSequencesErdosProblem733Wanted.lineSizes","parent":14264,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html#line-sizes","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"fully_proved","summary":"For a finite set $P \\subseteq \\mathbb{R}^2$, this is the multiset of the numbers $|\\ell \\cap P|$, where $\\ell$ ranges over the lines determined by $P$, that is, the distinct lines through two distinct points of $P$, each counted once.","title":"Line sizes of a planar point set"},{"id":"Geometry/discrete-geometry/part-01/point-line-incidences/line-compatible-count","kind":"node","lean":"MathlibExt.Combinatorics.LineCompatibleSequencesErdosProblem733Wanted.lineCompatibleCount","parent":14264,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html#line-compatible-count","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"fully_proved","summary":"For a natural number $n$, $f(n)$ is the number of distinct multisets that arise as the line sizes of some $n$-point set $P \\subseteq \\mathbb{R}^2$, that is, as the multiset of the numbers $|\\ell \\cap P|$ over the distinct lines $\\ell$ through two distinct points of $P$, each line counted once. These multisets are the line-compatible sequences for $n$; there are finitely many, since each has at most $n^2$ entries,\u2026","title":"Number of line-compatible sequences"},{"id":"Geometry/discrete-geometry/part-01/point-line-incidences/line-compatible-count-subexponential","kind":"node","lean":"MathlibExt.Combinatorics.LineCompatibleSequencesErdosProblem733Wanted.lineCompatibleCountSubexponential","parent":14264,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html#line-compatible-count-subexponential","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"fully_proved","summary":"The proposition that there is a real constant $C$ such that for every natural number $n$, the number $f(n)$ of line-compatible sequences for $n$ is at most $e^{C \\sqrt{n}}$; that is, there are at most $\\exp(O(\\sqrt{n}))$ line-compatible sequences. This is Erd\u0151s Problem 733.","title":"Subexponential count of line-compatible sequences (statement)"},{"id":"Geometry/discrete-geometry/part-01/point-line-incidences/line-compatible-count-subexponential-hol","kind":"node","lean":"MathlibExt.Combinatorics.LineCompatibleSequencesErdosProblem733Wanted.lineCompatibleCountSubexponential_holds","parent":14264,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html#line-compatible-count-subexponential-hol","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"can_prove","summary":"There is a real constant $C$ such that for every natural number $n$, the number of line-compatible sequences for $n$, that is, of distinct multisets of the numbers $|\\ell \\cap P|$ over the distinct lines $\\ell$ through two distinct points of an $n$-point set $P \\subseteq \\mathbb{R}^2$, each line counted once, is at most $e^{C \\sqrt{n}}$. This answers Erd\u0151s Problem 733.","title":"Szemer\u00e9di\u2013Trotter bound on line-compatible sequences"},{"id":"Geometry/discrete-geometry/part-01/point-line-incidences/line-incidence-set","kind":"node","lean":"MathlibExt.Combinatorics.LineIncidenceSetsCountErdosProblem607Wanted.lineIncidenceSet","parent":14264,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html#line-incidence-set","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"fully_proved","summary":"For a finite set $P \\subseteq \\mathbb{R}^2$, $A(P)$ is the set of the numbers $|\\ell \\cap P|$ over the lines $\\ell$ determined by $P$, that is, the lines through at least two distinct points of $P$.","title":"Line-incidence set of a finite planar point set"},{"id":"Geometry/discrete-geometry/part-01/point-line-incidences/line-incidence-set-count","kind":"node","lean":"MathlibExt.Combinatorics.LineIncidenceSetsCountErdosProblem607Wanted.lineIncidenceSetCount","parent":14264,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html#line-incidence-set-count","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"fully_proved","summary":"For a finite set $P \\subseteq \\mathbb{R}^2$, let $A(P)$ be the set of the numbers $|\\ell \\cap P|$ over the lines $\\ell$ through at least two distinct points of $P$. For a natural number $n$, $F(n)$ is the number of distinct sets $A(P)$ over all $n$-point sets $P \\subseteq \\mathbb{R}^2$. Every such $A(P)$ is a subset of $\\{0, \\dots, n\\}$.","title":"Number of line-incidence sets"},{"id":"Geometry/discrete-geometry/part-01/point-line-incidences/line-incidence-set-count-subexponential","kind":"node","lean":"MathlibExt.Combinatorics.LineIncidenceSetsCountErdosProblem607Wanted.lineIncidenceSetCountSubexponential","parent":14264,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html#line-incidence-set-count-subexponential","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"fully_proved","summary":"For a finite set $P \\subseteq \\mathbb{R}^2$, let $A(P)$ be the set of the numbers $|\\ell \\cap P|$ over the lines $\\ell$ through at least two distinct points of $P$, and let $F(n)$ be the number of distinct sets $A(P)$ over all $n$-point sets $P \\subseteq \\mathbb{R}^2$. This is the statement that there is a real constant $C$ such that $F(n) \\le e^{C\\sqrt{n}}$ for every natural number $n$, that is,\u2026","title":"Erd\u0151s Problem 607: subexponentially many line-incidence sets"},{"id":"Geometry/discrete-geometry/part-01/point-line-incidences/line-incidence-set-count-subexponential-2","kind":"node","lean":"MathlibExt.Combinatorics.LineIncidenceSetsCountErdosProblem607Wanted.lineIncidenceSetCountSubexponential_holds","parent":14264,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/point-line-incidences/index.html#line-incidence-set-count-subexponential-2","scope":"Geometry/discrete-geometry/part-01/point-line-incidences","status":"can_prove","summary":"For a finite set $P \\subseteq \\mathbb{R}^2$, let $A(P)$ be the set of the numbers $|\\ell \\cap P|$ over the lines $\\ell$ through at least two distinct points of $P$, and let $F(n)$ be the number of distinct sets $A(P)$ over all $n$-point sets $P \\subseteq \\mathbb{R}^2$. There is a real constant $C$ such that $F(n) \\le e^{C\\sqrt{n}}$ for every natural number $n$.","title":"Szemer\u00e9di\u2013Trotter bound for line-incidence sets (Erd\u0151s Problem 607)"},{"data_url":"scopes/Geometry/discrete-geometry/part-01/steiner-ratio.json","id":"Geometry/discrete-geometry/part-01/steiner-ratio","kind":"scope","navigation":"module","parent":13944,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"can_prove","title":"Steiner ratio"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/plane","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.Plane","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#plane","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"The space $\\mathbb{R}^2$ with the Euclidean distance, in which the point sets of the Steiner ratio problem lie.","title":"The Euclidean plane of the Steiner ratio problem"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/edge-length","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.edgeLength","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#edge-length","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"For points $x$ and $y$ of the Euclidean plane, the length of the unordered pair $\\{x, y\\}$ is the Euclidean distance between $x$ and $y$.","title":"Length of an edge"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/total-length","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.totalLength","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#total-length","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"For a finite set $V$ of points in the Euclidean plane and a simple graph $G$ with vertex set $V$, the total length of $G$ is the sum, over all edges $\\{x, y\\}$ of $G$, of the Euclidean distance between $x$ and $y$.","title":"Total length of a graph on points of the plane"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/spanning-tree-lengths","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.spanningTreeLengths","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#spanning-tree-lengths","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"For a finite set $V$ of points in the Euclidean plane, this is the set of real numbers $\\ell$ such that some tree with vertex set $V$ has total length $\\ell$, the total length being the sum of the Euclidean lengths of its edges.","title":"Lengths of spanning trees"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/steiner-tree-lengths","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.steinerTreeLengths","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#steiner-tree-lengths","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"For a finite set $V$ of points in the Euclidean plane, this is the set of real numbers $\\ell$ such that, for some finite set $W$ of points of the plane containing $V$, some tree with vertex set $W$ has total length $\\ell$. The points of $W$ outside $V$ are the Steiner points.","title":"Lengths of Steiner trees"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/mst-length","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.mstLength","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#mst-length","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"For a finite set $V$ of points in the Euclidean plane, the minimum spanning tree length of $V$ is the infimum of the total lengths of the trees with vertex set $V$. For nonempty $V$ this set of lengths is finite and nonempty, so the infimum is a minimum; for empty $V$ the value is $0$ by convention.","title":"Minimum spanning tree length"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/steiner-length","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.steinerLength","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#steiner-length","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"For a finite set $V$ of points in the Euclidean plane, the Steiner length of $V$ is the infimum of the total lengths of the trees whose vertex set is a finite set of points of the plane containing $V$. This set of lengths is nonempty and bounded below by $0$, so the infimum exists.","title":"Steiner minimal tree length"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/steiner-length-le-mst-length","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.steinerLength_le_mstLength","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#steiner-length-le-mst-length","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"For every nonempty finite set $V$ of points in the Euclidean plane, the Steiner minimal tree length of $V$ is at most the minimum spanning tree length of $V$: allowing Steiner points never increases the optimal length.","title":"The Steiner length is at most the minimum spanning tree length"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/steiner-ratio-constant","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.steinerRatioConstant","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#steiner-ratio-constant","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"The Steiner ratio constant is the infimum, over all finite sets $V$ of at least two points in the Euclidean plane $\\mathbb{R}^2$, of the ratio of the Steiner minimal tree length of $V$ to the minimum spanning tree length of $V$.","title":"The Steiner ratio of the Euclidean plane"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/gilbert-pollak-conjecture","kind":"node","lean":"MathlibExt.Geometry.SteinerRatio.GilbertPollakConjecture","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#gilbert-pollak-conjecture","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"fully_proved","summary":"The Steiner ratio of the Euclidean plane, that is, the infimum of the ratio of the Steiner minimal tree length to the minimum spanning tree length over all finite sets of at least two points in the plane, equals $\\sqrt{3}/2$.","title":"Gilbert\u2013Pollak conjecture"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/gilbert-pollak-upper-bound-wanted","kind":"node","lean":"MathlibExt.Geometry.SteinerRatioWanted.gilbertPollakUpperBound_wanted","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#gilbert-pollak-upper-bound-wanted","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"can_prove","summary":"The Steiner ratio constant, the infimum over finite sets $V$ of at least two points in the Euclidean plane of the length of a Steiner minimal tree for $V$ divided by the length of a minimum spanning tree of $V$, is at most $\\sqrt{3}/2$.","title":"Gilbert\u2013Pollak upper bound for the Steiner ratio"},{"id":"Geometry/discrete-geometry/part-01/steiner-ratio/chung-graham-lower-bound-wanted","kind":"node","lean":"MathlibExt.Geometry.SteinerRatioWanted.chungGrahamLowerBound_wanted","parent":14273,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-01/steiner-ratio/index.html#chung-graham-lower-bound-wanted","scope":"Geometry/discrete-geometry/part-01/steiner-ratio","status":"can_prove","summary":"The Steiner ratio constant, the infimum over finite sets $V$ of at least two points in the Euclidean plane of the length of a Steiner minimal tree for $V$ divided by the length of a minimum spanning tree of $V$, is at least $0.824$.","title":"Chung\u2013Graham lower bound for the Steiner ratio"},{"data_url":"scopes/Geometry/discrete-geometry/part-02.json","id":"Geometry/discrete-geometry/part-02","kind":"scope","navigation":"transparent","parent":13943,"result":false,"route":"../roadmap/Geometry/discrete-geometry/part-02/index.html","scope":"Geometry/discrete-geometry/part-02","status":"can_prove","title":"Part 2 of 2"},{"id":"Geometry/discrete-geometry/part-02/squaring-the-square-distinct-squares","kind":"node","lean":"MathlibExt.Geometry.ScottishBook59Wanted.squaringTheSquareDistinctSquares","parent":14286,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-02/index.html#squaring-the-square-distinct-squares","scope":"Geometry/discrete-geometry/part-02","status":"can_prove","summary":"There are an integer $n > 1$ and real numbers $x_i$, $y_i$ and $s_i > 0$ for $i = 1, \\dots, n$, with the $s_i$ pairwise distinct, such that the union of the closed squares $[x_i, x_i + s_i] \\times [y_i, y_i + s_i]$ is the unit square $[0, 1] \\times [0, 1]$ and any two of these squares with different indices have disjoint interiors.","title":"Squaring the square with distinct squares"},{"id":"Geometry/discrete-geometry/part-02/sylvester-gallai","kind":"node","lean":"MathlibExt.Geometry.Euclidean.SylvesterGallaiWanted.sylvester_gallai","parent":14286,"result":true,"route":"../roadmap/Geometry/discrete-geometry/part-02/index.html#sylvester-gallai","scope":"Geometry/discrete-geometry/part-02","status":"fully_proved","summary":"Every finite non-collinear subset of the real Euclidean plane determines an ordinary line: two points of the set such that no third point of the set is collinear with them.","title":"Sylvester\u2013Gallai theorem"},{"data_url":"scopes/Geometry/packings-and-coverings.json","id":"Geometry/packings-and-coverings","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/packings-and-coverings/index.html","scope":"Geometry/packings-and-coverings","status":"can_prove","summary":"Optimal arrangements of congruent bodies in space and the smallest sets or fewest translates needed to cover given families of shapes.","title":"Packings and coverings"},{"id":"Geometry/packings-and-coverings/klartag-packing","kind":"node","lean":"MathlibExt.Geometry.KlartagPackingWanted.Geometry.KlartagPacking.klartag_packing","parent":14289,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/index.html#klartag-packing","scope":"Geometry/packings-and-coverings","status":"can_prove","summary":"There is a real constant $c > 0$ such that for every integer $n \\ge 1$ there is a linear map $\\varphi : \\mathbb{R}^n \\to \\mathbb{R}^n$ for which the ellipsoid $E = \\varphi(B)$, the image of the open Euclidean unit ball $B$ about the origin, has volume exactly $c n^2$ and the origin is the only point of $E$ whose coordinates are all integers.","title":"Klartag's lattice packing theorem"},{"data_url":"scopes/Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem.json","id":"Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem","kind":"scope","navigation":"module","parent":14289,"result":false,"route":"../roadmap/Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem/index.html","scope":"Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem","status":"can_prove","title":"Goodman\u2013Goodman circle covering theorem"},{"id":"Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem/conjecture","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem1121Wanted.conjecture","parent":14291,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem/index.html#conjecture","scope":"Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem","status":"fully_proved","summary":"Write $B(c, r)$ for the closed ball with centre $c$ and radius $r$ in Euclidean space $\\mathbb{R}^d$. Call balls $B(c_1, r_1), \\dots, B(c_n, r_n)$ in $\\mathbb{R}^d$ *nonseparable* if for every nonzero vector $v$ and every real $t$ such that no point $p$ of any of the balls satisfies $\\langle v, p \\rangle = t$, either $\\langle v, c_i \\rangle < t$ for all $i$ or $\\langle v, c_i \\rangle > t$ for all $i$. The statement\u2026","title":"Erd\u0151s Problem 1121 (circle covering)"},{"id":"Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem/conjecture-holds","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem1121Wanted.conjecture_holds","parent":14291,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem/index.html#conjecture-holds","scope":"Geometry/packings-and-coverings/goodman-goodman-circle-covering-theorem","status":"can_prove","summary":"Both parts of Erd\u0151s Problem 1121 hold. For every natural number $d$, and in particular in the plane ($d = 2$): if $B(c_1, r_1), \\dots, B(c_n, r_n)$ are closed balls in Euclidean space $\\mathbb{R}^d$ with centres $c_i$ and radii $r_i > 0$ such that every hyperplane meeting none of the balls has all the centres $c_i$ strictly on the same side, then some closed ball of radius $r_1 + \\dots + r_n$ contains all of the\u2026","title":"Circle covering theorem (Erd\u0151s Problem 1121)"},{"data_url":"scopes/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem.json","id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","kind":"scope","navigation":"module","parent":14289,"result":false,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","title":"Hadwiger covering and illumination problem"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/is-convex-body","kind":"node","lean":"ConvexCovering.IsConvexBody","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#is-convex-body","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"A set $K$ in the Euclidean space $\\mathbb{R}^n$ is a convex body when it is compact and convex and its interior in $\\mathbb{R}^n$ is nonempty.","title":"Convex bodies in Euclidean space"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/covered-by-translates","kind":"node","lean":"ConvexCovering.CoveredByTranslates","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#covered-by-translates","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For sets $K$ and $L$ in the Euclidean space $\\mathbb{R}^n$ and a natural number $q$, $K$ is covered by $q$ translates of $L$ when there are vectors $v_1, \\dots, v_q \\in \\mathbb{R}^n$ with $K \\subseteq \\bigcup_{i=1}^{q} (v_i + L)$, where $v + L = \\{v + y : y \\in L\\}$.","title":"Covering by translates"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/covered-by-interior-translates","kind":"node","lean":"ConvexCovering.CoveredByInteriorTranslates","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#covered-by-interior-translates","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"A set $K$ in the Euclidean space $\\mathbb{R}^n$ is covered by $q$ interior translates, for a natural number $q$, when there are vectors $v_1, \\dots, v_q \\in \\mathbb{R}^n$ such that $K \\subseteq \\bigcup_{i=1}^{q} (v_i + \\operatorname{int} K)$, where $\\operatorname{int} K$ is the interior of $K$ in $\\mathbb{R}^n$ and $v + S = \\{v + y : y \\in S\\}$.","title":"Covering by interior translates"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/uniform-interior-translate-bound","kind":"node","lean":"ConvexCovering.UniformInteriorTranslateBound","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#uniform-interior-translate-bound","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For $n, q \\in \\mathbb{N}$, the uniform interior translate bound holds when every convex body $K \\subseteq \\mathbb{R}^n$ (a compact convex set with nonempty interior) is covered by $q$ translates of its interior $\\operatorname{int} K$, that is, $K \\subseteq (v_1 + \\operatorname{int} K) \\cup \\dots \\cup (v_q + \\operatorname{int} K)$ for some $v_1, \\dots, v_q \\in \\mathbb{R}^n$.","title":"Uniform bound for covering convex bodies by interior translates"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/is-centrally-symmetric","kind":"node","lean":"ConvexCovering.IsCentrallySymmetric","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#is-centrally-symmetric","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"A set $K$ in the Euclidean space $\\mathbb{R}^n$ is centrally symmetric when there is a point $c \\in \\mathbb{R}^n$ such that for every $x \\in \\mathbb{R}^n$, $x \\in K$ if and only if $2c - x \\in K$.","title":"Central symmetry about some center"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/uniform-symmetric-interior-translate-bou","kind":"node","lean":"ConvexCovering.UniformSymmetricInteriorTranslateBound","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#uniform-symmetric-interior-translate-bou","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For $n, q \\in \\mathbb{N}$, the uniform symmetric interior translate bound holds when every convex body $K \\subseteq \\mathbb{R}^n$ (a compact convex set with nonempty interior) that is centrally symmetric, meaning that there is $c \\in \\mathbb{R}^n$ such that for all $x \\in \\mathbb{R}^n$, $x \\in K$ if and only if $2c - x \\in K$, is covered by $q$ translates of its interior, that is,\u2026","title":"Uniform bound for covering centrally symmetric convex bodies by interior translates"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/covered-by-smaller-positive-homothets","kind":"node","lean":"ConvexCovering.CoveredBySmallerPositiveHomothets","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#covered-by-smaller-positive-homothets","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"A set $K$ in the Euclidean space $\\mathbb{R}^n$ is covered by $q$ smaller positive homothets, for a natural number $q$, when there are real numbers $r_1, \\dots, r_q$ with $0 < r_i < 1$ for every $i$ and vectors $v_1, \\dots, v_q \\in \\mathbb{R}^n$ such that every point of $K$ lies in $v_i + r_i K = \\{v_i + r_i y : y \\in K\\}$ for some $i$.","title":"Covering by smaller positive homothets"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/translate-covering-number","kind":"node","lean":"ConvexCovering.translateCoveringNumber","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#translate-covering-number","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For sets $K, L \\subseteq \\mathbb{R}^n$, the translate covering number of $K$ by $L$ is the least natural number $q$ for which there are vectors $v_1, \\dots, v_q \\in \\mathbb{R}^n$ with $K \\subseteq (v_1 + L) \\cup \\dots \\cup (v_q + L)$, and it is $\\infty$ if no such $q$ exists.","title":"Translate covering number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/hadwiger-covering-number","kind":"node","lean":"ConvexCovering.hadwigerCoveringNumber","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#hadwiger-covering-number","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"The Hadwiger covering number of dimension $n$, an element of $\\mathbb{N} \\cup \\{\\infty\\}$, is the least natural number $q$ such that every convex body $K$ in the Euclidean space $\\mathbb{R}^n$ is covered by $q$ translates of its interior $\\operatorname{int} K$, and it is $\\infty$ if no such $q$ exists.","title":"Hadwiger covering number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/symmetric-hadwiger-covering-number","kind":"node","lean":"ConvexCovering.symmetricHadwigerCoveringNumber","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#symmetric-hadwiger-covering-number","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the symmetric Hadwiger covering number in dimension $n$ is the least natural number $q$ such that every centrally symmetric convex body $K \\subseteq \\mathbb{R}^n$ (a compact convex set with nonempty interior that is symmetric about some point) is covered by $q$ translates of its interior, and it is $\\infty$ if no such $q$ exists.","title":"Symmetric Hadwiger covering number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/covered-by-translates-iff","kind":"node","lean":"ConvexCovering.coveredByTranslates_iff","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#covered-by-translates-iff","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For sets $K$ and $L$ in the Euclidean space $\\mathbb{R}^n$ and a natural number $q$, $K$ is covered by $q$ translates of $L$ if and only if there are vectors $v_1, \\dots, v_q \\in \\mathbb{R}^n$ such that every $x \\in K$ lies in $v_i + L = \\{v_i + y : y \\in L\\}$ for some $i$.","title":"Pointwise form of covering by translates"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/covered-by-interior-translates-iff","kind":"node","lean":"ConvexCovering.coveredByInteriorTranslates_iff","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#covered-by-interior-translates-iff","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For every set $K$ in the Euclidean space $\\mathbb{R}^n$ and every natural number $q$, $K$ has a cover by $q$ interior translates if and only if $K$ is covered by $q$ translates of the set $L = \\operatorname{int} K$, the interior of $K$ in $\\mathbb{R}^n$.","title":"Interior-translate covers unfold to translate covers"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/translate-covering-number-eq-of-is-least","kind":"node","lean":"ConvexCovering.translateCoveringNumber_eq_of_isLeast","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#translate-covering-number-eq-of-is-least","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $K, L \\subseteq \\mathbb{R}^n$ and let $q \\in \\mathbb{N}$. If $q$ is the least natural number $r$ such that $K$ is covered by $r$ translates of $L$, then the translate covering number of $K$ by $L$ equals $q$.","title":"The translate covering number is the least covering count"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/hadwiger-covering-number-eq-of-is-least","kind":"node","lean":"ConvexCovering.hadwigerCoveringNumber_eq_of_isLeast","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#hadwiger-covering-number-eq-of-is-least","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"If $q$ is the least natural number such that every convex body $K$ in the Euclidean space $\\mathbb{R}^n$ is covered by $q$ translates of its interior, then the Hadwiger covering number of dimension $n$ equals $q$.","title":"A least uniform bound is the Hadwiger covering number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/symmetric-hadwiger-covering-number-eq-of","kind":"node","lean":"ConvexCovering.symmetricHadwigerCoveringNumber_eq_of_isLeast","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#symmetric-hadwiger-covering-number-eq-of","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"If $q$ is the least natural number such that every centrally symmetric convex body $K$ in the Euclidean space $\\mathbb{R}^n$ is covered by $q$ translates of its interior, then the symmetric Hadwiger covering number of dimension $n$ equals $q$.","title":"A least uniform bound is the symmetric Hadwiger covering number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube","kind":"node","lean":"ConvexCovering.coordinateCube","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the coordinate cube is the set $[-1, 1]^n$ of points $x$ of the Euclidean space $\\mathbb{R}^n$ with $-1 \\le x_i \\le 1$ for every coordinate $i$.","title":"Coordinate cube"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube-eq-abs-set","kind":"node","lean":"ConvexCovering.coordinateCube_eq_absSet","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube-eq-abs-set","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For every $n$, the coordinate cube in $\\mathbb{R}^n$ equals the set of points $x \\in \\mathbb{R}^n$ with $|x_i| \\le 1$ for every coordinate $i$.","title":"The coordinate cube as an absolute-value box"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/is-affine-cube","kind":"node","lean":"ConvexCovering.IsAffineCube","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#is-affine-cube","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{R}^n$ is an affine cube if $K = e([-1, 1]^n)$ for some affine bijection $e : \\mathbb{R}^n \\to \\mathbb{R}^n$, where $[-1, 1]^n$ is the coordinate cube $\\{x \\in \\mathbb{R}^n : -1 \\le x_i \\le 1 \\text{ for all } i\\}$.","title":"Affine cube"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube-interior-iff","kind":"node","lean":"ConvexCovering.coordinateCube_interior_iff","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube-interior-iff","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"A point $x \\in \\mathbb{R}^n$ lies in the interior of the coordinate cube $[-1, 1]^n$ if and only if $-1 < x_i < 1$ for every coordinate $i$.","title":"Interior of the coordinate cube"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube-is-convex-body","kind":"node","lean":"ConvexCovering.coordinateCube_isConvexBody","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube-is-convex-body","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For every $n$, the coordinate cube $[-1, 1]^n$ in $\\mathbb{R}^n$ is a convex body: it is compact and convex, and its interior in $\\mathbb{R}^n$ is nonempty.","title":"The coordinate cube is a convex body"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube-is-centrally-symmetric","kind":"node","lean":"ConvexCovering.coordinateCube_isCentrallySymmetric","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube-is-centrally-symmetric","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"For every $n$, the coordinate cube $[-1, 1]^n$ in $\\mathbb{R}^n$ is centrally symmetric: there is a point $c \\in \\mathbb{R}^n$ such that for every $x \\in \\mathbb{R}^n$, $x$ lies in the cube if and only if $2c - x$ does.","title":"The coordinate cube is centrally symmetric"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube-covering-count-ge-two-po","kind":"node","lean":"ConvexCovering.coordinateCube_coveringCount_ge_two_pow","parent":14294,"reading_title":{"format":"inline-parts-v1","parts":[["text","Covering the cube by translates of its interior needs "],["math","\\(2^n\\)"],["text"," translates"]],"source":"Covering the cube by translates of its interior needs $2^n$ translates"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube-covering-count-ge-two-po","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $C_n = [-1, 1]^n$ be the coordinate cube in $\\mathbb{R}^n$. If $C_n$ is contained in the union of $q$ translates $v_1 + \\operatorname{int} C_n, \\dots, v_q + \\operatorname{int} C_n$ of its interior, then $2^n \\le q$.","title":"Covering the cube by translates of its interior needs $2^n$ translates"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/uniform-interior-translate-bound-ge-two","kind":"node","lean":"ConvexCovering.uniformInteriorTranslateBound_ge_two_pow","parent":14294,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound "],["math","\\(2^n\\)"],["text"," for uniform interior-translate covering"]],"source":"Lower bound $2^n$ for uniform interior-translate covering"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#uniform-interior-translate-bound-ge-two","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $n$ and $q$ be natural numbers. If every convex body $K \\subseteq \\mathbb{R}^n$ (a compact convex set with nonempty interior) can be covered by $q$ not necessarily distinct translates of its interior, then $2^n \\le q$.","title":"Lower bound $2^n$ for uniform interior-translate covering"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/cube-lower-bound","kind":"node","lean":"ConvexCovering.cubeLowerBound","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#cube-lower-bound","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"If $q$ is a natural number such that every convex body $K \\subseteq \\mathbb{R}^3$ is contained in the union of $q$ translates of its interior $\\operatorname{int} K$, then $8 \\le q$.","title":"Uniform interior-translate bounds in dimension three are at least eight"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/symmetric-interior-translate-bound-ge-tw","kind":"node","lean":"ConvexCovering.symmetricInteriorTranslateBound_ge_two_pow","parent":14294,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound "],["math","\\(2^n\\)"],["text"," for centrally symmetric interior-translate covering"]],"source":"Lower bound $2^n$ for centrally symmetric interior-translate covering"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#symmetric-interior-translate-bound-ge-tw","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $n$ and $q$ be natural numbers. If every centrally symmetric convex body $K \\subseteq \\mathbb{R}^n$ (a compact convex set with nonempty interior that is symmetric about some point) can be covered by $q$ not necessarily distinct translates of its interior, then $2^n \\le q$.","title":"Lower bound $2^n$ for centrally symmetric interior-translate covering"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube-three-covered-by-interio","kind":"node","lean":"ConvexCovering.coordinateCube_three_coveredByInteriorTranslates","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube-three-covered-by-interio","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $C_3 = [-1, 1]^3$ be the coordinate cube in $\\mathbb{R}^3$. There are vectors $v_1, \\dots, v_8 \\in \\mathbb{R}^3$ such that $C_3$ is contained in the union of the translates $v_1 + \\operatorname{int} C_3, \\dots, v_8 + \\operatorname{int} C_3$ of its interior.","title":"Eight translates of the interior cover the three-dimensional cube"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube-three-least-covering-cou","kind":"node","lean":"ConvexCovering.coordinateCube_three_least_coveringCount","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube-three-least-covering-cou","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $C_3 = [-1, 1]^3$ be the coordinate cube in $\\mathbb{R}^3$. Then $8$ is the least natural number $q$ such that $C_3$ is contained in the union of $q$ translates of its interior $\\operatorname{int} C_3$.","title":"Eight is the least number of interior translates covering the three-dimensional cube"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/coordinate-cube-three-translate-covering","kind":"node","lean":"ConvexCovering.coordinateCube_three_translateCoveringNumber","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#coordinate-cube-three-translate-covering","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $C_3 = [-1, 1]^3$ be the coordinate cube in $\\mathbb{R}^3$. The translate covering number of $C_3$ by its interior $\\operatorname{int} C_3$, the least number of translates of $\\operatorname{int} C_3$ whose union contains $C_3$, equals $8$.","title":"Translate covering number of the three-dimensional cube by its interior"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/direction-illuminates","kind":"node","lean":"ConvexCovering.DirectionIlluminates","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#direction-illuminates","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}^n$ be a set and let $v, x \\in \\mathbb{R}^n$. The direction $v$ illuminates $x$ with respect to $K$ when $x$ lies on the boundary of $K$, $\\|v\\| = 1$, and $x + tv$ lies in the interior of $K$ for some real $t > 0$.","title":"Illumination by a direction"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/illuminated-by-directions","kind":"node","lean":"ConvexCovering.IlluminatedByDirections","parent":14294,"reading_title":{"format":"inline-parts-v1","parts":[["text","Illumination by "],["math","\\(q\\)"],["text"," directions"]],"source":"Illumination by $q$ directions"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#illuminated-by-directions","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{R}^n$ is illuminated by $q$ directions, for a natural number $q$, if there are unit vectors $v_1, \\dots, v_q \\in \\mathbb{R}^n$ (not necessarily distinct) such that for every boundary point $x$ of $K$ there are an index $i$ and a real $t > 0$ with $x + tv_i$ in the interior of $K$.","title":"Illumination by $q$ directions"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/external-point-illuminates","kind":"node","lean":"ConvexCovering.ExternalPointIlluminates","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#external-point-illuminates","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}^n$ be a set and let $p, x \\in \\mathbb{R}^n$. The point $p$ illuminates $x$ with respect to $K$ when $p \\notin K$, $x$ lies on the boundary of $K$, and $p + t(x - p)$ lies in the interior of $K$ for some real $t > 1$; that is, the ray from $p$ through $x$ meets the interior of $K$ at a point beyond $x$.","title":"Illumination by an external point"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/illuminated-by-external-points","kind":"node","lean":"ConvexCovering.IlluminatedByExternalPoints","parent":14294,"reading_title":{"format":"inline-parts-v1","parts":[["text","Illumination by "],["math","\\(q\\)"],["text"," external points"]],"source":"Illumination by $q$ external points"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#illuminated-by-external-points","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{R}^n$ is illuminated by $q$ external points, for a natural number $q$, if there are points $p_1, \\dots, p_q \\in \\mathbb{R}^n \\setminus K$ (not necessarily distinct) such that for every boundary point $x$ of $K$ there are an index $i$ and a real $t > 1$ with $p_i + t(x - p_i)$ in the interior of $K$.","title":"Illumination by $q$ external points"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/directional-illumination-number","kind":"node","lean":"ConvexCovering.directionalIlluminationNumber","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#directional-illumination-number","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"The directional illumination number of a set $K \\subseteq \\mathbb{R}^n$ is the least natural number $q$ such that $K$ is illuminated by $q$ directions, and $\\infty$ if there is no such $q$.","title":"Directional illumination number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/external-point-illumination-number","kind":"node","lean":"ConvexCovering.externalPointIlluminationNumber","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#external-point-illumination-number","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"fully_proved","summary":"The external-point illumination number of a set $K \\subseteq \\mathbb{R}^n$ is the least natural number $q$ such that $K$ is illuminated by $q$ external points, and $\\infty$ if there is no such $q$.","title":"External-point illumination number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/homothet-interior-translate-formulations","kind":"node","lean":"ConvexCovering.homothetInteriorTranslateFormulationsEquivalent","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#homothet-interior-translate-formulations","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"The following statements are equivalent: for every $n \\ge 1$, every convex body $K \\subseteq \\mathbb{R}^n$ is covered by at most $2^n$ homothets $v_i + r_i K$ with $v_i \\in \\mathbb{R}^n$ and $0 < r_i < 1$; and for every $n \\ge 1$, every convex body $K \\subseteq \\mathbb{R}^n$ is covered by $2^n$ translates of its interior $\\operatorname{int} K$, repetitions allowed.","title":"Two equivalent formulations of Hadwiger's covering conjecture"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/papadoperakis1999-upper-bound","kind":"node","lean":"ConvexCovering.papadoperakis1999UpperBound","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#papadoperakis1999-upper-bound","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"Every convex body $K \\subseteq \\mathbb{R}^3$ is covered by $16$ translates of its interior $\\operatorname{int} K$, repetitions allowed.","title":"Papadoperakis's bound for Hadwiger's covering problem in dimension 3"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/prymak2023-upper-bound","kind":"node","lean":"ConvexCovering.prymak2023UpperBound","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#prymak2023-upper-bound","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"Every convex body $K \\subseteq \\mathbb{R}^3$ is covered by $14$ translates of its interior $\\operatorname{int} K$, repetitions allowed.","title":"Prymak's bound for Hadwiger's covering problem in dimension 3"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/lassak1984-symmetric-exact-value","kind":"node","lean":"ConvexCovering.lassak1984SymmetricExactValue","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#lassak1984-symmetric-exact-value","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"The number $8$ is the least natural number $q$ such that every centrally symmetric convex body $K \\subseteq \\mathbb{R}^3$ is covered by $q$ translates of its interior $\\operatorname{int} K$, repetitions allowed. Here $K$ is centrally symmetric when there is a point $c$ with $x \\in K$ if and only if $2c - x \\in K$.","title":"Lassak's theorem for centrally symmetric bodies in dimension 3"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/directional-external-formulations-equiva","kind":"node","lean":"ConvexCovering.directionalExternalFormulationsEquivalent","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#directional-external-formulations-equiva","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"Let $n \\ge 1$, let $K \\subseteq \\mathbb{R}^n$ be a convex body and let $q$ be a natural number. Then some $q$ unit vectors $v_1, \\dots, v_q$ illuminate $K$ if and only if some $q$ points $p_1, \\dots, p_q$ outside $K$ illuminate $K$. Here the vectors illuminate $K$ when every boundary point $x$ of $K$ has an $i$ and a $t > 0$ with $x + t v_i$ in the interior of $K$, and the points illuminate $K$ when every boundary\u2026","title":"Directional and exterior-point illumination are equivalent"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/directional-homothet-formulations-equiva","kind":"node","lean":"ConvexCovering.directionalHomothetFormulationsEquivalent","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#directional-homothet-formulations-equiva","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"Let $n \\ge 1$, let $K \\subseteq \\mathbb{R}^n$ be a convex body and let $q$ be a natural number. Then some $q$ unit vectors illuminate $K$ if and only if $K$ is covered by $q$ homothets $v_1 + r_1 K, \\dots, v_q + r_q K$ with $v_i \\in \\mathbb{R}^n$ and $0 < r_i < 1$, repetitions allowed.","title":"Directional illumination is equivalent to covering by smaller homothets"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/directional-interior-formulations-equiva","kind":"node","lean":"ConvexCovering.directionalInteriorFormulationsEquivalent","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#directional-interior-formulations-equiva","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"Let $n \\ge 1$, let $K \\subseteq \\mathbb{R}^n$ be a convex body and let $q$ be a natural number. Then some $q$ unit vectors illuminate $K$ if and only if $K$ is covered by $q$ translates $v_1 + \\operatorname{int} K, \\dots, v_q + \\operatorname{int} K$ of its interior, repetitions allowed.","title":"Directional illumination is equivalent to covering by interior translates"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/directional-external-illumination-number","kind":"node","lean":"ConvexCovering.directionalExternalIlluminationNumbersEqual","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#directional-external-illumination-number","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"For every $n \\ge 1$ and every convex body $K \\subseteq \\mathbb{R}^n$, the directional illumination number of $K$ equals its exterior-point illumination number. These are the least $q$ such that some $q$ unit vectors, respectively some $q$ points outside $K$, illuminate $K$, with the value $\\infty$ when no finite family does.","title":"Directional and exterior-point illumination numbers are equal"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/directional-interior-covering-numbers-eq","kind":"node","lean":"ConvexCovering.directionalInteriorCoveringNumbersEqual","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#directional-interior-covering-numbers-eq","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"For every $n \\ge 1$ and every convex body $K \\subseteq \\mathbb{R}^n$, the directional illumination number of $K$ equals the least $q$ such that $K$ is covered by $q$ translates of its interior $\\operatorname{int} K$. Both sides take the value $\\infty$ when no finite family exists.","title":"Directional illumination number equals the interior-translate covering number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/illumination-finite-minimum","kind":"node","lean":"ConvexCovering.illuminationFiniteMinimum","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#illumination-finite-minimum","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"For every $n \\ge 1$ and every convex body $K \\subseteq \\mathbb{R}^n$ there is a least natural number $q$ such that some $q$ unit vectors $v_1, \\dots, v_q$ illuminate $K$: every boundary point $x$ of $K$ has an $i$ and a $t > 0$ with $x + t v_i$ in the interior of $K$.","title":"Every convex body has a finite illumination number"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/uniform-directional-illumination-interpr","kind":"node","lean":"ConvexCovering.uniformDirectionalIlluminationInterpretation","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#uniform-directional-illumination-interpr","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"For integers $n \\ge 1$ and $q \\ge 0$, every convex body in $\\mathbb{R}^n$ (a compact convex set with nonempty interior) can be covered by $q$ translates of its interior if and only if every convex body $K$ in $\\mathbb{R}^n$ is illuminated by $q$ directions: there are unit vectors $v_1, \\dots, v_q$ such that each boundary point $x$ of $K$ has some $i$ and some $t > 0$ with $x + t v_i$ in the interior of $K$.","title":"Uniform covering by interior translates and uniform directional illumination"},{"id":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/uniform-symmetric-external-illumination","kind":"node","lean":"ConvexCovering.uniformSymmetricExternalIlluminationInterpretation","parent":14294,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem/index.html#uniform-symmetric-external-illumination","scope":"Geometry/packings-and-coverings/hadwiger-covering-and-illumination-problem","status":"can_prove","summary":"For integers $n \\ge 1$ and $q \\ge 0$, every centrally symmetric convex body in $\\mathbb{R}^n$ (a compact convex set with nonempty interior that is symmetric about some point) can be covered by $q$ translates of its interior if and only if every centrally symmetric convex body $K$ in $\\mathbb{R}^n$ is illuminated by $q$ external points: there are points $p_1, \\dots, p_q$ outside $K$ such that each boundary point $x$\u2026","title":"Uniform covering of symmetric bodies and uniform illumination by external points"},{"data_url":"scopes/Geometry/packings-and-coverings/kepler-conjecture.json","id":"Geometry/packings-and-coverings/kepler-conjecture","kind":"scope","navigation":"module","parent":14289,"result":false,"route":"../roadmap/Geometry/packings-and-coverings/kepler-conjecture/index.html","scope":"Geometry/packings-and-coverings/kepler-conjecture","status":"can_prove","title":"Kepler conjecture"},{"id":"Geometry/packings-and-coverings/kepler-conjecture/e","kind":"node","lean":"MathlibExt.Geometry.KeplerConjectureWanted.Geometry.KeplerConjectureProblem.E","parent":14341,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/kepler-conjecture/index.html#e","scope":"Geometry/packings-and-coverings/kepler-conjecture","status":"fully_proved","summary":"For a natural number $d$, $\\mathbb{R}^d$ denotes the $d$-dimensional real Euclidean space: $d$-tuples of real numbers with the Euclidean norm.","title":"Euclidean space for sphere packings"},{"id":"Geometry/packings-and-coverings/kepler-conjecture/is-packing","kind":"node","lean":"MathlibExt.Geometry.KeplerConjectureWanted.Geometry.KeplerConjectureProblem.IsPacking","parent":14341,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/kepler-conjecture/index.html#is-packing","scope":"Geometry/packings-and-coverings/kepler-conjecture","status":"fully_proved","summary":"For a natural number $d$, a set $X \\subseteq \\mathbb{R}^d$ is the centre set of a unit sphere packing if the open balls of radius $1$ centred at any two distinct points of $X$ are disjoint.","title":"Unit sphere packing"},{"id":"Geometry/packings-and-coverings/kepler-conjecture/covered-fraction","kind":"node","lean":"MathlibExt.Geometry.KeplerConjectureWanted.Geometry.KeplerConjectureProblem.coveredFraction","parent":14341,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/kepler-conjecture/index.html#covered-fraction","scope":"Geometry/packings-and-coverings/kepler-conjecture","status":"fully_proved","summary":"For a natural number $d$, a set $X \\subseteq \\mathbb{R}^d$ and a real number $R$, the covered fraction of $X$ at radius $R$ is","title":"Covered fraction of a ball"},{"id":"Geometry/packings-and-coverings/kepler-conjecture/density","kind":"node","lean":"MathlibExt.Geometry.KeplerConjectureWanted.Geometry.KeplerConjectureProblem.density","parent":14341,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/kepler-conjecture/index.html#density","scope":"Geometry/packings-and-coverings/kepler-conjecture","status":"fully_proved","summary":"For a natural number $d$, the density of a set $X \\subseteq \\mathbb{R}^d$ is the limit superior, as $R \\to \\infty$, of the covered fraction of $X$ at radius $R$: the fraction of the open ball of radius $R$ about the origin covered by the open unit balls centred at points of $X$.","title":"Density of a sphere packing"},{"id":"Geometry/packings-and-coverings/kepler-conjecture/untitled","kind":"node","lean":"MathlibExt.Geometry.KeplerConjectureWanted.Geometry.KeplerConjectureProblem.\u0394","parent":14341,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/kepler-conjecture/index.html#untitled","scope":"Geometry/packings-and-coverings/kepler-conjecture","status":"fully_proved","summary":"For a natural number $d$, $\\Delta(d)$ is the supremum of the densities of all unit sphere packings in $\\mathbb{R}^d$.","title":"Optimal sphere packing density"},{"id":"Geometry/packings-and-coverings/kepler-conjecture/kepler-conjecture","kind":"node","lean":"MathlibExt.Geometry.KeplerConjectureWanted.Geometry.KeplerConjectureProblem.kepler_conjecture","parent":14341,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/kepler-conjecture/index.html#kepler-conjecture","scope":"Geometry/packings-and-coverings/kepler-conjecture","status":"can_prove","summary":"Let $\\Delta(3)$ be the supremum of the densities of all unit sphere packings in $\\mathbb{R}^3$. Then $\\Delta(3) = \\pi / \\sqrt{18}$.","title":"Kepler conjecture"},{"data_url":"scopes/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem.json","id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","kind":"scope","navigation":"module","parent":14289,"result":false,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","title":"Lebesgue's universal covering problem"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/plane","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.Plane","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#plane","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The ambient space is the Euclidean plane $\\mathbb{R}^2$, with the Euclidean distance.","title":"The Euclidean plane of Lebesgue's covering problem"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/is-unit-diameter-convex-set","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.IsUnitDiameterConvexSet","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#is-unit-diameter-convex-set","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"A set $K$ in the Euclidean plane $\\mathbb{R}^2$ is a unit-diameter convex set when it is nonempty and convex and its diameter $\\sup_{x, y \\in K} \\lVert x - y \\rVert$ is exactly $1$.","title":"Convex sets of diameter one"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/covers-congruent-copy","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.CoversCongruentCopy","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#covers-congruent-copy","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"A planar set $\\Omega$ covers a congruent copy of a planar set $K$ when some isometry $e$ of the plane onto itself satisfies $e(K) \\subseteq \\Omega$.","title":"Covering a congruent copy"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/is-universal-convex-cover","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.IsUniversalConvexCover","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#is-universal-convex-cover","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"A set $\\Omega$ in the Euclidean plane $\\mathbb{R}^2$ is a universal convex cover when it is nonempty and convex and, for every nonempty convex set $K \\subseteq \\mathbb{R}^2$ of diameter $1$, some isometry $e$ of the plane onto itself satisfies $e(K) \\subseteq \\Omega$.","title":"Universal convex covers"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/planar-area","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.planarArea","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#planar-area","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The area of a set $S$ in the Euclidean plane $\\mathbb{R}^2$ is its Lebesgue measure (outer measure if $S$ is not measurable), a value in $[0, \\infty]$, so infinite area is allowed.","title":"Planar area"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/admissible-cover-areas","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.admissibleCoverAreas","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#admissible-cover-areas","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The set of admissible cover areas is the set of all $a \\in [0, \\infty]$ for which there is a universal convex cover $\\Omega$ in the plane (a nonempty convex set containing a congruent copy of every nonempty convex planar set of diameter $1$) whose Lebesgue area is $a$.","title":"Areas of universal convex covers"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/lebesgue-universal-covering-constant","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.lebesgueUniversalCoveringConstant","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#lebesgue-universal-covering-constant","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"Call a set $\\Omega$ in the Euclidean plane $\\mathbb{R}^2$ a universal convex cover if it is nonempty and convex and, for every nonempty convex set $K \\subseteq \\mathbb{R}^2$ of diameter exactly $1$, some isometry of the plane onto itself maps $K$ into $\\Omega$. The Lebesgue universal covering constant is the infimum, taken in $[0, \\infty]$, of the Lebesgue areas of all universal convex covers; the infimum of an\u2026","title":"Lebesgue universal covering constant"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/directional-width","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.directionalWidth","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#directional-width","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"For a planar set $K$ and a vector $u$ in the plane, the width of $K$ in direction $u$ is $\\sup_{x \\in K} \\langle u, x \\rangle - \\inf_{x \\in K} \\langle u, x \\rangle$, where the supremum and infimum are taken in $\\mathbb{R}$, with the convention that each is $0$ when the set of values $\\langle u, x \\rangle$ is empty or unbounded on that side.","title":"Directional width"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/is-constant-width-one","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.IsConstantWidthOne","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#is-constant-width-one","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"A set $K$ in the Euclidean plane $\\mathbb{R}^2$ has constant width $1$ when it is nonempty, compact and convex and, for every unit vector $u$, its width in direction $u$, namely $\\sup_{x \\in K} \\langle u, x \\rangle - \\inf_{x \\in K} \\langle u, x \\rangle$, equals $1$.","title":"Sets of constant width one"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/elekes1994-minimal-cover-exists","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.elekes1994MinimalCoverExists","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#elekes1994-minimal-cover-exists","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"This proposition states that some nonempty convex set $\\Omega$ in the Euclidean plane $\\mathbb{R}^2$ is a universal convex cover, that is, for every nonempty convex set $K \\subseteq \\mathbb{R}^2$ of diameter $1$ some isometry $e$ of the plane onto itself satisfies $e(K) \\subseteq \\Omega$, and that the area (planar Lebesgue measure) of $\\Omega$ equals the Lebesgue universal covering constant, the infimum of the\u2026","title":"Statement of the existence of a minimal universal convex cover"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/vrecica1981-constant-width-one-suffices","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.vrecica1981ConstantWidthOneSuffices","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#vrecica1981-constant-width-one-suffices","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The proposition that every nonempty convex subset $\\Omega$ of the Euclidean plane that contains a congruent copy of every nonempty compact convex planar set of constant width $1$ is a universal convex cover, that is, contains a congruent copy of every nonempty convex planar set of diameter $1$.","title":"Covering sets of constant width one suffices (statement)"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/trivial-jung-elekes1994-upper-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.trivialJungElekes1994UpperBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#trivial-jung-elekes1994-upper-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The proposition that Lebesgue's universal covering constant is at most $\\pi/3$.","title":"Jung's upper bound for Lebesgue's universal covering constant (statement)"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/pal1920-hexagon-upper-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.pal1920HexagonUpperBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#pal1920-hexagon-upper-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The proposition that the Lebesgue universal covering constant is at most $\\sqrt{3}/2$, the bound given by P\u00e1l's regular hexagon.","title":"Statement of P\u00e1l's hexagon bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/pal1920-truncated-hexagon-upper-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.pal1920TruncatedHexagonUpperBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#pal1920-truncated-hexagon-upper-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The proposition that the Lebesgue universal covering constant is at most $2 - 2/\\sqrt{3}$, the bound given by P\u00e1l's truncated hexagon.","title":"Statement of P\u00e1l's truncated hexagon bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/sprague1936-upper-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.sprague1936UpperBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#sprague1936-upper-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The proposition that the Lebesgue universal covering constant is at most $0.844137708436$.","title":"Statement of Sprague's bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/hansen1992-corrected-bbg2015-upper-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.hansen1992CorrectedBBG2015UpperBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#hansen1992-corrected-bbg2015-upper-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"This proposition states that the Lebesgue universal covering constant, the infimum of the areas of the nonempty convex sets in the Euclidean plane that contain a congruent copy of every nonempty convex planar set of diameter $1$, is at most $0.844137708398$.","title":"Statement of Hansen's corrected upper bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/bbg2015-approximate-upper-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.bbg2015ApproximateUpperBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#bbg2015-approximate-upper-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"This is the proposition that there is a real number $b$ with $0.844115297128419059 \\le b < 0.844115297128419060$ such that the Lebesgue universal covering constant (the infimum of the areas of universal convex covers) is at most $b$.","title":"Statement of the Baez\u2013Bagdasaryan\u2013Gibbs upper bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/gibbs2018-upper-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.gibbs2018UpperBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#gibbs2018-upper-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"This proposition states that the Lebesgue universal covering constant, the infimum of the areas of the nonempty convex sets in the Euclidean plane that contain a congruent copy of every nonempty convex planar set of diameter $1$, is at most $0.8440935944$.","title":"Statement of Gibbs's upper bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/trivial-unit-disk-lower-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.trivialUnitDiskLowerBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#trivial-unit-disk-lower-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"The proposition that Lebesgue's universal covering constant is at least $\\pi/4$.","title":"Disk lower bound for Lebesgue's universal covering constant (statement)"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/elekes1994-disk-triangle-lower-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.elekes1994DiskTriangleLowerBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#elekes1994-disk-triangle-lower-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"This is the proposition that the Lebesgue universal covering constant (the infimum of the areas of universal convex covers) is at least $0.8257$.","title":"Statement of Elekes's disk-and-triangle lower bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/elekes1994-regular-polygon-lower-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.elekes1994RegularPolygonLowerBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#elekes1994-regular-polygon-lower-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"This proposition states that the Lebesgue universal covering constant, the infimum of the areas of the nonempty convex sets in the Euclidean plane that contain a congruent copy of every nonempty convex planar set of diameter $1$, is at least $0.8271$.","title":"Statement of Elekes's regular-polygon lower bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/brass-sharifi2005-lower-bound","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.brassSharifi2005LowerBound","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#brass-sharifi2005-lower-bound","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"fully_proved","summary":"This is the proposition that the Lebesgue universal covering constant (the infimum of the areas of universal convex covers) is at least $0.832$.","title":"Statement of the Brass\u2013Sharifi lower bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/elekes1994-minimal-cover-exists-holds","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.elekes1994MinimalCoverExists_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#elekes1994-minimal-cover-exists-holds","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"There is a universal convex cover $\\Omega$ (a nonempty convex planar set containing a congruent copy of every nonempty convex planar set of diameter $1$) whose Lebesgue area equals the Lebesgue universal covering constant, the infimum of the areas of all universal convex covers.","title":"Existence of a minimal universal cover"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/vrecica1981-constant-width-one-suffices-2","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.vrecica1981ConstantWidthOneSuffices_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#vrecica1981-constant-width-one-suffices-2","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"Let $\\Omega$ be a nonempty convex subset of the Euclidean plane that contains a congruent copy (an image under an isometry of the plane, reflections allowed) of every nonempty compact convex planar set of constant width $1$. Then $\\Omega$ contains a congruent copy of every nonempty convex planar set of diameter $1$, that is, $\\Omega$ is a universal convex cover.","title":"Covering sets of constant width one suffices"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/trivial-jung-elekes1994-upper-bound-hold","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.trivialJungElekes1994UpperBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#trivial-jung-elekes1994-upper-bound-hold","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant is at most $\\pi/3$.","title":"Jung's bound for the Lebesgue covering problem"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/pal1920-hexagon-upper-bound-holds","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.pal1920HexagonUpperBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#pal1920-hexagon-upper-bound-holds","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant is at most $\\sqrt{3}/2$.","title":"P\u00e1l's hexagon bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/pal1920-truncated-hexagon-upper-bound-ho","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.pal1920TruncatedHexagonUpperBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#pal1920-truncated-hexagon-upper-bound-ho","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant is at most $2 - 2/\\sqrt{3}$.","title":"P\u00e1l's truncated hexagon bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/sprague1936-upper-bound-holds","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.sprague1936UpperBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#sprague1936-upper-bound-holds","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant is at most $0.844137708436$.","title":"Sprague's bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/hansen1992-corrected-bbg2015-upper-bound-2","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.hansen1992CorrectedBBG2015UpperBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#hansen1992-corrected-bbg2015-upper-bound-2","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant, the infimum of the areas of the nonempty convex sets in the Euclidean plane that contain a congruent copy of every nonempty convex planar set of diameter $1$, is at most $0.844137708398$.","title":"Hansen's upper bound for Lebesgue's covering problem, as corrected"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/bbg2015-approximate-upper-bound-holds","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.bbg2015ApproximateUpperBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#bbg2015-approximate-upper-bound-holds","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"There is a real number $b$ with $0.844115297128419059 \\le b < 0.844115297128419060$ such that the Lebesgue universal covering constant (the infimum of the areas of universal convex covers) is at most $b$.","title":"Baez\u2013Bagdasaryan\u2013Gibbs upper bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/gibbs2018-upper-bound-holds","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.gibbs2018UpperBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#gibbs2018-upper-bound-holds","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant, the infimum of the areas of the nonempty convex sets in the Euclidean plane that contain a congruent copy of every nonempty convex planar set of diameter $1$, is at most $0.8440935944$.","title":"Gibbs's upper bound for Lebesgue's covering problem"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/trivial-unit-disk-lower-bound-holds","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.trivialUnitDiskLowerBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#trivial-unit-disk-lower-bound-holds","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"Lebesgue's universal covering constant, the infimum of the areas of the nonempty convex planar sets that contain a congruent copy of every nonempty convex planar set of diameter $1$, is at least $\\pi/4$, the area of a disk of radius $1/2$.","title":"Disk lower bound for Lebesgue's universal covering constant"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/elekes1994-disk-triangle-lower-bound-hol","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.elekes1994DiskTriangleLowerBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#elekes1994-disk-triangle-lower-bound-hol","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant (the infimum of the areas of universal convex covers) is at least $0.8257$.","title":"Elekes's disk-and-triangle lower bound"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/elekes1994-regular-polygon-lower-bound-h","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.elekes1994RegularPolygonLowerBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#elekes1994-regular-polygon-lower-bound-h","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant, the infimum of the areas of the nonempty convex sets in the Euclidean plane that contain a congruent copy of every nonempty convex planar set of diameter $1$, is at least $0.8271$.","title":"Elekes's regular-polygon lower bound for Lebesgue's covering problem"},{"id":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/brass-sharifi2005-lower-bound-holds","kind":"node","lean":"MathlibExt.Geometry.LebesgueUniversalCoveringConstantWanted.brassSharifi2005LowerBound_holds","parent":14348,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem/index.html#brass-sharifi2005-lower-bound-holds","scope":"Geometry/packings-and-coverings/lebesgue-s-universal-covering-problem","status":"can_prove","summary":"The Lebesgue universal covering constant (the infimum of the areas of universal convex covers) is at least $0.832$.","title":"Brass\u2013Sharifi lower bound"},{"data_url":"scopes/Geometry/packings-and-coverings/moser-s-worm-problem.json","id":"Geometry/packings-and-coverings/moser-s-worm-problem","kind":"scope","navigation":"module","parent":14289,"result":false,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","title":"Moser's worm problem"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/arc-parameter","kind":"node","lean":"MoserConvexWormCoverConstant.ArcParameter","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#arc-parameter","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"Planar arcs are parametrized by the closed unit interval $[0, 1]$ of real numbers.","title":"Parameter interval of unit arcs"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/unit-arc","kind":"node","lean":"MoserConvexWormCoverConstant.UnitArc","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#unit-arc","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A unit arc is a continuous map $\\gamma$ from the closed interval $[0, 1]$ to the complex plane whose length, the total variation of $\\gamma$ on $[0, 1]$, equals $1$.","title":"Unit arc"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/move","kind":"node","lean":"MoserConvexWormCoverConstant.move","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#move","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"For a real angle $\\theta$ and a complex number $\\tau$, the motion with angle $\\theta$ and translation $\\tau$ sends $z \\in \\mathbb{C}$ to $e^{i\\theta} z + \\tau$: a rotation about the origin by $\\theta$ followed by translation by $\\tau$, with neither reflection nor rescaling.","title":"Direct rigid motion of the plane"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/move-isometry","kind":"node","lean":"MoserConvexWormCoverConstant.move_isometry","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#move-isometry","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"For every real angle $\\theta$ and every complex number $\\tau$, the map $z \\mapsto e^{i\\theta} z + \\tau$ of the complex plane is an isometry: it preserves the distance $|z - w|$ between any two points $z, w \\in \\mathbb{C}$.","title":"Direct rigid motions of the plane are isometries"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/straight-path","kind":"node","lean":"MoserConvexWormCoverConstant.straightPath","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#straight-path","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The straight path is the continuous map from the interval $[0, 1]$ to the complex plane that sends each $t \\in [0, 1]$ to $t$ viewed as a complex number; it traces the segment from $0$ to $1$.","title":"Straight path"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/parameter-variation","kind":"node","lean":"MoserConvexWormCoverConstant.parameter_variation","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#parameter-variation","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The total variation of the inclusion $[0, 1] \\to \\mathbb{R}$, $t \\mapsto t$, over the whole interval $[0, 1]$ equals $1$.","title":"The identity on the unit interval has total variation one"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/straight-path-length","kind":"node","lean":"MoserConvexWormCoverConstant.straightPath_length","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#straight-path-length","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The path $[0, 1] \\to \\mathbb{C}$, $t \\mapsto t$, which traces the segment from $0$ to $1$ on the real axis, has total variation (length) exactly $1$ over $[0, 1]$.","title":"The straight segment has length one"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/straight-unit-arc","kind":"node","lean":"MoserConvexWormCoverConstant.straightUnitArc","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#straight-unit-arc","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The straight unit arc is the unit arc whose curve is the straight path $t \\mapsto t$ from $[0, 1]$ to the complex plane, which traces the segment from $0$ to $1$ and has length $1$.","title":"Straight unit arc"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/covers-every-unit-arc","kind":"node","lean":"MoserConvexWormCoverConstant.CoversEveryUnitArc","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#covers-every-unit-arc","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A set $\\Omega \\subseteq \\mathbb{C}$ covers every unit arc when for every unit arc $\\gamma : [0, 1] \\to \\mathbb{C}$, that is, every continuous map whose total variation (length) is exactly $1$, there are an angle $\\theta \\in \\mathbb{R}$ and a translation $\\tau \\in \\mathbb{C}$ such that $e^{i\\theta} \\gamma(t) + \\tau \\in \\Omega$ for every $t \\in [0, 1]$.","title":"Sets that cover every unit arc"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonempty","kind":"node","lean":"MoserConvexWormCoverConstant.CoversEveryUnitArc.nonempty","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonempty","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"If a set $\\Omega \\subseteq \\mathbb{C}$ covers every unit arc, that is, contains a rotated and translated copy of every continuous curve $[0, 1] \\to \\mathbb{C}$ of length exactly one, then $\\Omega$ is nonempty.","title":"Sets covering every unit arc are nonempty"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/empty-not-covers","kind":"node","lean":"MoserConvexWormCoverConstant.empty_not_covers","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#empty-not-covers","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The empty set $\\emptyset \\subseteq \\mathbb{C}$ does not cover every unit arc: not every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be mapped into $\\emptyset$ by a rotation followed by a translation.","title":"The empty set does not cover every unit arc"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/is-convex-universal-cover","kind":"node","lean":"MoserConvexWormCoverConstant.IsConvexUniversalCover","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#is-convex-universal-cover","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A set $\\Omega \\subseteq \\mathbb{C}$ is a convex universal cover when it is nonempty, convex (as a subset of the real plane $\\mathbb{C} \\cong \\mathbb{R}^2$), and covers every unit arc: for every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ there are $\\theta \\in \\mathbb{R}$ and $\\tau \\in \\mathbb{C}$ with $e^{i\\theta} \\gamma(t) + \\tau \\in \\Omega$ for all $t \\in [0, 1]$.","title":"Convex universal covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/is-universal-cover","kind":"node","lean":"MoserConvexWormCoverConstant.IsUniversalCover","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#is-universal-cover","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A set $\\Omega \\subseteq \\mathbb{C}$ is a universal cover when it is nonempty and covers every unit arc: for every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ there are $\\theta \\in \\mathbb{R}$ and $\\tau \\in \\mathbb{C}$ with $e^{i\\theta} \\gamma(t) + \\tau \\in \\Omega$ for all $t \\in [0, 1]$. Neither convexity nor measurability of $\\Omega$ is required.","title":"Universal covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/null-measurable-set","kind":"node","lean":"MoserConvexWormCoverConstant.IsConvexUniversalCover.nullMeasurableSet","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#null-measurable-set","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"If $\\Omega \\subseteq \\mathbb{C}$ is a convex universal cover of unit arcs, then $\\Omega$ is a null-measurable set for Lebesgue measure on $\\mathbb{C}$.","title":"Convex universal covers are null-measurable"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/is-universal-cover-2","kind":"node","lean":"MoserConvexWormCoverConstant.IsConvexUniversalCover.isUniversalCover","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#is-universal-cover-2","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"Every convex universal cover $\\Omega \\subseteq \\mathbb{C}$ is a universal cover.","title":"Convex universal covers are universal covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/convex-cover-areas","kind":"node","lean":"MoserConvexWormCoverConstant.convexCoverAreas","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#convex-cover-areas","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The set of convex cover areas is the set of all values $\\operatorname{vol}(\\Omega) \\in [0, \\infty]$, where $\\Omega \\subseteq \\mathbb{C}$ ranges over the convex universal covers and $\\operatorname{vol}$ is Lebesgue measure on $\\mathbb{C} \\cong \\mathbb{R}^2$.","title":"Areas of convex universal covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/cover-areas","kind":"node","lean":"MoserConvexWormCoverConstant.coverAreas","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#cover-areas","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The set of values $\\lambda(\\Omega) \\in [0, \\infty]$, where $\\Omega \\subseteq \\mathbb{C}$ ranges over the universal covers and $\\lambda$ is Lebesgue measure on $\\mathbb{C} \\cong \\mathbb{R}^2$, applied as an outer measure since $\\Omega$ need not be measurable.","title":"Areas of universal covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/convex-cover-area-infimum","kind":"node","lean":"MoserConvexWormCoverConstant.convexCoverAreaInfimum","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#convex-cover-area-infimum","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The convex cover area infimum is the infimum in $[0, \\infty]$ of the set of convex cover areas, that is, the infimum of the Lebesgue areas of all convex universal covers $\\Omega \\subseteq \\mathbb{C}$.","title":"Infimal area of convex universal covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/cover-area-infimum","kind":"node","lean":"MoserConvexWormCoverConstant.coverAreaInfimum","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#cover-area-infimum","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The infimum in $[0, \\infty]$ of the areas $\\lambda(\\Omega)$ of the universal covers $\\Omega \\subseteq \\mathbb{C}$, where $\\lambda$ is Lebesgue measure on $\\mathbb{C} \\cong \\mathbb{R}^2$, applied as an outer measure since $\\Omega$ need not be measurable.","title":"Infimal area of a universal cover"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/closed-ball-one-covers","kind":"node","lean":"MoserConvexWormCoverConstant.closedBall_one_covers","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#closed-ball-one-covers","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"For every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ there are an angle $\\theta \\in \\mathbb{R}$ and a translation $c \\in \\mathbb{C}$ such that $|e^{i\\theta}\\gamma(t) + c| \\le 1$ for every $t \\in [0, 1]$; that is, the closed disk of radius $1$ centred at $0$ covers every unit arc.","title":"The closed unit disk covers every unit arc"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/closed-ball-one-is-convex-universal-cove","kind":"node","lean":"MoserConvexWormCoverConstant.closedBall_one_isConvexUniversalCover","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#closed-ball-one-is-convex-universal-cove","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The closed disk $\\{z \\in \\mathbb{C} : |z| \\le 1\\}$ is a convex universal cover: it is nonempty, convex, and covers every unit arc.","title":"The closed unit disk is a convex universal cover"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/convex-cover-areas-nonempty","kind":"node","lean":"MoserConvexWormCoverConstant.convexCoverAreas_nonempty","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#convex-cover-areas-nonempty","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The set of convex cover areas, the Lebesgue areas of all convex universal covers in $\\mathbb{C}$, is nonempty; that is, some convex universal cover exists.","title":"The set of convex cover areas is nonempty"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/cover-areas-nonempty","kind":"node","lean":"MoserConvexWormCoverConstant.coverAreas_nonempty","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#cover-areas-nonempty","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The set of values $\\lambda(\\Omega) \\in [0, \\infty]$, where $\\Omega \\subseteq \\mathbb{C}$ ranges over the universal covers and $\\lambda$ is Lebesgue outer measure on $\\mathbb{C} \\cong \\mathbb{R}^2$, is nonempty.","title":"The set of universal cover areas is nonempty"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/convex-cover-area-infimum-ne-top","kind":"node","lean":"MoserConvexWormCoverConstant.convexCoverAreaInfimum_ne_top","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#convex-cover-area-infimum-ne-top","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The convex cover area infimum, the infimum in $[0, \\infty]$ of the Lebesgue areas of all convex universal covers in $\\mathbb{C}$, is not $\\infty$.","title":"The infimal area of convex universal covers is finite"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/cover-area-infimum-ne-top","kind":"node","lean":"MoserConvexWormCoverConstant.coverAreaInfimum_ne_top","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#cover-area-infimum-ne-top","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The cover area infimum, the infimum in $[0, \\infty]$ of the Lebesgue areas of all universal covers in $\\mathbb{C}$ (nonempty sets that cover every unit arc, not necessarily convex), is not $\\infty$.","title":"The infimal area of universal covers is finite"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/convex-worm-cover-constant","kind":"node","lean":"MoserConvexWormCoverConstant.convexWormCoverConstant","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#convex-worm-cover-constant","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The convex worm cover constant is the real number obtained from the convex cover area infimum, the infimum in $[0, \\infty]$ of the Lebesgue areas of all convex universal covers in $\\mathbb{C}$, by the conversion $[0, \\infty] \\to \\mathbb{R}$ that is the identity on finite values and sends $\\infty$ to $0$.","title":"Moser's convex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-worm-cover-constant","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexWormCoverConstant","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-worm-cover-constant","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The nonconvex worm cover constant is the real number obtained from the cover area infimum, the infimum in $[0, \\infty]$ of the Lebesgue outer measures of all universal covers of unit arcs (nonempty sets $\\Omega \\subseteq \\mathbb{C}$ containing a rotated and translated copy of every continuous curve of length one), by converting it to a real number. The conversion would send $\\infty$ to $0$, but this infimum is\u2026","title":"Nonconvex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/convex-worm-cover-constant-nonneg","kind":"node","lean":"MoserConvexWormCoverConstant.convexWormCoverConstant_nonneg","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#convex-worm-cover-constant-nonneg","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The convex worm cover constant is at least $0$.","title":"The convex worm cover constant is nonnegative"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-worm-cover-constant-nonneg","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexWormCoverConstant_nonneg","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-worm-cover-constant-nonneg","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The nonconvex worm cover constant is greater than or equal to $0$.","title":"Nonnegativity of the nonconvex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-le-convex","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvex_le_convex","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-le-convex","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The nonconvex worm cover constant (the infimal area of universal covers of unit arcs) is less than or equal to the convex worm cover constant (the infimal area of convex universal covers of unit arcs).","title":"The nonconvex worm cover constant is at most the convex one"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/half-disk","kind":"node","lean":"MoserConvexWormCoverConstant.halfDisk","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#half-disk","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The closed disk $\\{z \\in \\mathbb{C} : |z| \\le 1/2\\}$ of radius $1/2$ centred at $0$.","title":"Disk of radius one half"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/half-semidisk","kind":"node","lean":"MoserConvexWormCoverConstant.halfSemidisk","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#half-semidisk","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The closed upper half-disk $\\{z \\in \\mathbb{C} : |z| \\le 1/2,\\ \\operatorname{Im} z \\ge 0\\}$ of radius $1/2$ centred at $0$.","title":"Semidisk of radius one half"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/wetzel-sector","kind":"node","lean":"MoserConvexWormCoverConstant.wetzelSector","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#wetzel-sector","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"This is the closed circular sector of radius $1$ and opening angle $30^\\circ$ with apex at the origin, placed symmetrically about the positive real axis: the set of complex numbers $z$ with $|z| \\le 1$, $\\operatorname{Re} z \\ge 0$ and $|\\operatorname{Im} z| \\le \\tan(\\pi/12) \\operatorname{Re} z$.","title":"Wetzel's sector"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-upper-bound-of-cover","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexUpperBound_of_cover","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-upper-bound-of-cover","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"Let $r \\ge 0$ be a real number. If some set $\\Omega \\subseteq \\mathbb{C}$ is a universal cover of unit arcs whose Lebesgue outer measure is at most $r$, then the nonconvex worm cover constant is at most $r$.","title":"Covers give upper bounds on the nonconvex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-pi-over-four","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundPiOverFour","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-pi-over-four","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant is the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation. This proposition states that the convex worm cover constant is at most $\\pi/4$.","title":"Proposition: the convex worm cover constant is at most \u03c0/4"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-pi-over-eight","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundPiOverEight","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-pi-over-eight","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant is the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation. This proposition states that the convex worm cover constant is at most $\\pi/8$.","title":"Proposition: the convex worm cover constant is at most \u03c0/8"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-wetzel1973","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundWetzel1973","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-wetzel1973","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant is the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation. This proposition states that the convex worm cover constant is at most $0.34501$.","title":"Proposition: the convex worm cover constant is at most 0.34501"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-gerriets1972","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundGerriets1972","parent":14384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of Gerriets' upper bound "],["math","\\(0.3214\\)"],["text"," for convex worm covers"]],"source":"Statement of Gerriets' upper bound $0.3214$ for convex worm covers"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-gerriets1972","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The proposition that Moser's convex worm cover constant, the infimum of the areas of convex planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.3214$.","title":"Statement of Gerriets' upper bound $0.3214$ for convex worm covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-gerriets-poole1974","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundGerrietsPoole1974","parent":14384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the Gerriets\u2013Poole upper bound "],["math","\\(0.2861\\)"],["text"," for convex worm covers"]],"source":"Statement of the Gerriets\u2013Poole upper bound $0.2861$ for convex worm covers"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-gerriets-poole1974","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The proposition that Moser's convex worm cover constant, the infimum of the areas of convex planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.2861$.","title":"Statement of the Gerriets\u2013Poole upper bound $0.2861$ for convex worm covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-norwood-poole-laidacker1992","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundNorwoodPooleLaidacker1992","parent":14384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the Norwood\u2013Poole\u2013Laidacker upper bound "],["math","\\(0.27524\\)"],["text"," for convex worm covers"]],"source":"Statement of the Norwood\u2013Poole\u2013Laidacker upper bound $0.27524$ for convex worm covers"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-norwood-poole-laidacker1992","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The proposition that Moser's convex worm cover constant, the infimum of the areas of convex planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.27524$.","title":"Statement of the Norwood\u2013Poole\u2013Laidacker upper bound $0.27524$ for convex worm covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-wang2006","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundWang2006","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-wang2006","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant is the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation. This proposition states that the convex worm cover constant is at most $0.270911861$.","title":"Proposition: the convex worm cover constant is at most 0.270911861"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-pi-over-twelve","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundPiOverTwelve","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-pi-over-twelve","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant is the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation. This proposition states that the convex worm cover constant is at most $\\pi/12$.","title":"Proposition: the convex worm cover constant is at most \u03c0/12"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/lower-bound-wetzel1973","kind":"node","lean":"MoserConvexWormCoverConstant.lowerBoundWetzel1973","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#lower-bound-wetzel1973","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"This proposition asserts that $0.2194 \\le \\mu$, where $\\mu$ is the convex worm cover constant: the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ such that every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by a rotation followed by a translation.","title":"Wetzel's 1973 lower bound for the convex worm cover constant (claim)"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/lower-bound-khandhawit-sriswasdi2009","kind":"node","lean":"MoserConvexWormCoverConstant.lowerBoundKhandhawitSriswasdi2009","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#lower-bound-khandhawit-sriswasdi2009","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"This proposition asserts that $0.227498 \\le \\mu$, where $\\mu$ is the convex worm cover constant: the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ such that every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by a rotation followed by a translation.","title":"Khandhawit\u2013Sriswasdi 2009 lower bound for the convex worm cover constant (claim)"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/lower-bound-khandhawit-pagonakis-sriswas","kind":"node","lean":"MoserConvexWormCoverConstant.lowerBoundKhandhawitPagonakisSriswasdi2013","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#lower-bound-khandhawit-pagonakis-sriswas","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"This proposition asserts that $0.232239 \\le \\mu$, where $\\mu$ is the convex worm cover constant: the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ such that every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by a rotation followed by a translation.","title":"Khandhawit\u2013Pagonakis\u2013Sriswasdi 2013 lower bound on the convex worm cover constant (claim)"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/minimal-convex-cover-exists","kind":"node","lean":"MoserConvexWormCoverConstant.minimalConvexCoverExists","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#minimal-convex-cover-exists","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"Call a nonempty convex set $\\Omega \\subseteq \\mathbb{C}$ a convex universal cover if every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation, and let the convex worm cover constant be the infimum of the areas (Lebesgue measures) of all convex universal covers, which is a finite real number. This is the proposition that some\u2026","title":"The minimal-convex-cover claim"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-cover-norwood-poole2003","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexCoverNorwoodPoole2003","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-cover-norwood-poole2003","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"Call a nonempty set $\\Omega \\subseteq \\mathbb{C}$ a universal cover if every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation. This is the proposition that some universal cover has area (Lebesgue outer measure) at most $0.260437$.","title":"The Norwood\u2013Poole cover claim"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-cover-ploymaklam-wichiramala20","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexCoverPloymaklamWichiramala2018","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-cover-ploymaklam-wichiramala20","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"Call a nonempty set $\\Omega \\subseteq \\mathbb{C}$ a universal cover if every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation. This is the proposition that some universal cover has area (Lebesgue outer measure) at most $0.26007$.","title":"The Ploymaklam\u2013Wichiramala cover claim"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-upper-bound-norwood-poole2003","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexUpperBoundNorwoodPoole2003","parent":14384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the Norwood\u2013Poole upper bound "],["math","\\(0.260437\\)"],["text"," for worm covers"]],"source":"Statement of the Norwood\u2013Poole upper bound $0.260437$ for worm covers"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-upper-bound-norwood-poole2003","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The proposition that the nonconvex worm cover constant, the infimum of the areas (Lebesgue outer measures) of planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.260437$.","title":"Statement of the Norwood\u2013Poole upper bound $0.260437$ for worm covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-upper-bound-ploymaklam-wichira","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexUpperBoundPloymaklamWichiramala2018","parent":14384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the Ploymaklam\u2013Wichiramala upper bound "],["math","\\(0.26007\\)"],["text"," for worm covers"]],"source":"Statement of the Ploymaklam\u2013Wichiramala upper bound $0.26007$ for worm covers"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-upper-bound-ploymaklam-wichira","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The proposition that the nonconvex worm cover constant, the infimum of the areas (Lebesgue outer measures) of planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.26007$.","title":"Statement of the Ploymaklam\u2013Wichiramala upper bound $0.26007$ for worm covers"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-strict-positivity","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexStrictPositivity","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-strict-positivity","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"Call a nonempty set $\\Omega \\subseteq \\mathbb{C}$ a universal cover if every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation, and let the nonconvex worm cover constant be the infimum of the areas (Lebesgue outer measures) of all universal covers, which is a finite real number. This is the proposition that the nonconvex\u2026","title":"The positivity claim for the nonconvex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/half-disk-covers","kind":"node","lean":"MoserConvexWormCoverConstant.halfDiskCovers","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#half-disk-covers","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"This proposition asserts that the closed disk $D = \\{z \\in \\mathbb{C} : |z| \\le 1/2\\}$ covers every unit arc: for every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ there are an angle $\\theta \\in \\mathbb{R}$ and a translation $c \\in \\mathbb{C}$ such that $e^{i\\theta}\\gamma(t) + c \\in D$ for every $t \\in [0, 1]$.","title":"The disk of diameter one covers every unit arc (claim)"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/half-semidisk-covers","kind":"node","lean":"MoserConvexWormCoverConstant.halfSemidiskCovers","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#half-semidisk-covers","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"This proposition asserts that the closed half-disk $H = \\{z \\in \\mathbb{C} : |z| \\le 1/2,\\ \\operatorname{Im} z \\ge 0\\}$ covers every unit arc: for every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ there are an angle $\\theta \\in \\mathbb{R}$ and a translation $c \\in \\mathbb{C}$ such that $e^{i\\theta}\\gamma(t) + c \\in H$ for every $t \\in [0, 1]$.","title":"The half-disk of diameter one covers every unit arc (claim)"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/wetzel-sector-covers","kind":"node","lean":"MoserConvexWormCoverConstant.wetzelSectorCovers","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#wetzel-sector-covers","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"The proposition that the closed circular sector $S$ of radius $1$ and angle $30^\\circ$ with apex $0$ and symmetric about the positive real axis, that is, the set of complex numbers $z$ with $|z| \\le 1$, $\\operatorname{Re} z \\ge 0$ and $|\\operatorname{Im} z| \\le \\tan(\\pi/12)\\operatorname{Re} z$, covers every unit arc: for every continuous curve $\\gamma : [0,1] \\to \\mathbb{C}$ of length (total variation) exactly $1$\u2026","title":"Wetzel's sector covering claim"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-upper-bound-norwood-poole2003-2","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexUpperBoundNorwoodPoole2003_of_cover","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-upper-bound-norwood-poole2003-2","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"Call a nonempty set $\\Omega \\subseteq \\mathbb{C}$ a universal cover if every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation, and let the nonconvex worm cover constant be the infimum of the areas (Lebesgue outer measures) of all universal covers, which is a finite real number. If some universal cover has area at most\u2026","title":"A Norwood\u2013Poole cover bounds the nonconvex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-upper-bound-ploymaklam-wichira-2","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexUpperBoundPloymaklamWichiramala2018_of_cover","parent":14384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Covers of area at most "],["math","\\(0.26007\\)"],["text"," bound the nonconvex worm cover constant"]],"source":"Covers of area at most $0.26007$ bound the nonconvex worm cover constant"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-upper-bound-ploymaklam-wichira-2","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"fully_proved","summary":"If some nonempty planar set of area (Lebesgue outer measure) at most $0.26007$ contains a rotated and translated copy of every planar curve of length $1$, then the nonconvex worm cover constant is at most $0.26007$.","title":"Covers of area at most $0.26007$ bound the nonconvex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-pi-over-four-holds","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundPiOverFour_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-pi-over-four-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant, the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation, is at most $\\pi/4$.","title":"The convex worm cover constant is at most \u03c0/4"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-pi-over-eight-holds","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundPiOverEight_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-pi-over-eight-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant, the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation, is at most $\\pi/8$.","title":"The convex worm cover constant is at most \u03c0/8"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-wetzel1973-holds","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundWetzel1973_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-wetzel1973-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant, the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation, is at most $0.34501$.","title":"The convex worm cover constant is at most 0.34501"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-gerriets1972-holds","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundGerriets1972_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-gerriets1972-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Moser's convex worm cover constant, the infimum of the areas of convex planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.3214$.","title":"Gerriets' upper bound for the convex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-gerriets-poole1974-holds","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundGerrietsPoole1974_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-gerriets-poole1974-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Moser's convex worm cover constant, the infimum of the areas of convex planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.2861$.","title":"Gerriets\u2013Poole upper bound for the convex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-norwood-poole-laidacker1992-2","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundNorwoodPooleLaidacker1992_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-norwood-poole-laidacker1992-2","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Moser's convex worm cover constant, the infimum of the areas of convex planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.27524$.","title":"Norwood\u2013Poole\u2013Laidacker upper bound for the convex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-wang2006-holds","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundWang2006_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-wang2006-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant, the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation, is at most $0.270911861$.","title":"The convex worm cover constant is at most 0.270911861"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/upper-bound-pi-over-twelve-holds","kind":"node","lean":"MoserConvexWormCoverConstant.upperBoundPiOverTwelve_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#upper-bound-pi-over-twelve-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"A unit arc is a continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$. The convex worm cover constant, the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ that contain, for every unit arc $\\gamma$, the image of $\\gamma$ under some rotation followed by a translation, is at most $\\pi/12$.","title":"The convex worm cover constant is at most \u03c0/12"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/lower-bound-wetzel1973-holds","kind":"node","lean":"MoserConvexWormCoverConstant.lowerBoundWetzel1973_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#lower-bound-wetzel1973-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Let $\\mu$ be the convex worm cover constant, the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ such that every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by a rotation followed by a translation. Then $0.2194 \\le \\mu$.","title":"Wetzel's 1973 lower bound for the convex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/lower-bound-khandhawit-sriswasdi2009-hol","kind":"node","lean":"MoserConvexWormCoverConstant.lowerBoundKhandhawitSriswasdi2009_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#lower-bound-khandhawit-sriswasdi2009-hol","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Let $\\mu$ be the convex worm cover constant, the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ such that every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by a rotation followed by a translation. Then $0.227498 \\le \\mu$.","title":"Khandhawit\u2013Sriswasdi 2009 lower bound for the convex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/lower-bound-khandhawit-pagonakis-sriswas-2","kind":"node","lean":"MoserConvexWormCoverConstant.lowerBoundKhandhawitPagonakisSriswasdi2013_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#lower-bound-khandhawit-pagonakis-sriswas-2","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Let $\\mu$ be the convex worm cover constant, the infimum of the areas of the nonempty convex sets $\\Omega \\subseteq \\mathbb{C}$ such that every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by a rotation followed by a translation. Then $0.232239 \\le \\mu$.","title":"Khandhawit\u2013Pagonakis\u2013Sriswasdi 2013 lower bound on the convex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/minimal-convex-cover-exists-holds","kind":"node","lean":"MoserConvexWormCoverConstant.minimalConvexCoverExists_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#minimal-convex-cover-exists-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Call a nonempty convex set $\\Omega \\subseteq \\mathbb{C}$ a convex universal cover if every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation, and let the convex worm cover constant be the infimum of the areas (Lebesgue measures) of all convex universal covers, which is a finite real number. Then some convex universal cover\u2026","title":"A minimal convex cover for unit arcs exists"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-cover-norwood-poole2003-holds","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexCoverNorwoodPoole2003_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-cover-norwood-poole2003-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"There is a nonempty set $\\Omega \\subseteq \\mathbb{C}$ of area (Lebesgue outer measure) at most $0.260437$ such that every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation.","title":"Norwood\u2013Poole universal cover for unit arcs"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-cover-ploymaklam-wichiramala20-2","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexCoverPloymaklamWichiramala2018_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-cover-ploymaklam-wichiramala20-2","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"There is a nonempty set $\\Omega \\subseteq \\mathbb{C}$ of area (Lebesgue outer measure) at most $0.26007$ such that every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation.","title":"Ploymaklam\u2013Wichiramala universal cover for unit arcs"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-upper-bound-norwood-poole2003-3","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexUpperBoundNorwoodPoole2003_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-upper-bound-norwood-poole2003-3","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Call a nonempty set $\\Omega \\subseteq \\mathbb{C}$ a universal cover if every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation, and let the nonconvex worm cover constant be the infimum of the areas (Lebesgue outer measures) of all universal covers, which is a finite real number. Then the nonconvex worm cover constant is at\u2026","title":"Norwood\u2013Poole upper bound for the nonconvex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-upper-bound-ploymaklam-wichira-3","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexUpperBoundPloymaklamWichiramala2018_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-upper-bound-ploymaklam-wichira-3","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"The nonconvex worm cover constant, the infimum of the areas (Lebesgue outer measures) of planar sets containing a rotated and translated copy of every planar curve of length $1$, is at most $0.26007$.","title":"Ploymaklam\u2013Wichiramala upper bound for the nonconvex worm cover constant"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/nonconvex-strict-positivity-holds","kind":"node","lean":"MoserConvexWormCoverConstant.nonconvexStrictPositivity_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#nonconvex-strict-positivity-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Call a nonempty set $\\Omega \\subseteq \\mathbb{C}$ a universal cover if every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ can be moved into $\\Omega$ by some rotation followed by a translation, and let the nonconvex worm cover constant be the infimum of the areas (Lebesgue outer measures) of all universal covers, which is a finite real number. Then the nonconvex worm cover constant is\u2026","title":"The nonconvex worm cover constant is positive"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/half-disk-covers-holds","kind":"node","lean":"MoserConvexWormCoverConstant.halfDiskCovers_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#half-disk-covers-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"For every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ there are an angle $\\theta \\in \\mathbb{R}$ and a translation $c \\in \\mathbb{C}$ such that $|e^{i\\theta}\\gamma(t) + c| \\le 1/2$ for every $t \\in [0, 1]$.","title":"The disk of diameter one covers every unit arc"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/half-semidisk-covers-holds","kind":"node","lean":"MoserConvexWormCoverConstant.halfSemidiskCovers_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#half-semidisk-covers-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Let $H = \\{z \\in \\mathbb{C} : |z| \\le 1/2,\\ \\operatorname{Im} z \\ge 0\\}$. For every continuous curve $\\gamma : [0, 1] \\to \\mathbb{C}$ of length exactly $1$ there are an angle $\\theta \\in \\mathbb{R}$ and a translation $c \\in \\mathbb{C}$ such that $e^{i\\theta}\\gamma(t) + c \\in H$ for every $t \\in [0, 1]$.","title":"The half-disk of diameter one covers every unit arc"},{"id":"Geometry/packings-and-coverings/moser-s-worm-problem/wetzel-sector-covers-holds","kind":"node","lean":"MoserConvexWormCoverConstant.wetzelSectorCovers_holds","parent":14384,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/moser-s-worm-problem/index.html#wetzel-sector-covers-holds","scope":"Geometry/packings-and-coverings/moser-s-worm-problem","status":"can_prove","summary":"Let $S$ be the closed circular sector of radius $1$ and angle $30^\\circ$ with apex $0$ and symmetric about the positive real axis, that is, the set of complex numbers $z$ with $|z| \\le 1$, $\\operatorname{Re} z \\ge 0$ and $|\\operatorname{Im} z| \\le \\tan(\\pi/12)\\operatorname{Re} z$. Then for every continuous curve $\\gamma : [0,1] \\to \\mathbb{C}$ of length (total variation) exactly $1$ there are an angle\u2026","title":"Wetzel's sector covers every unit arc"},{"data_url":"scopes/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions.json","id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","kind":"scope","navigation":"module","parent":14289,"result":false,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"can_prove","title":"Sphere packing in four dimensions"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/space","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.Space","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#space","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The ambient space is $\\mathbb{R}^4$, the space of real $4$-tuples with the standard Euclidean inner product and the distance it induces.","title":"Four-dimensional Euclidean space"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/positive-radius","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.PositiveRadius","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#positive-radius","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"A positive radius is a real number $R$ with $R > 0$; the positive radii form the set $\\{R \\in \\mathbb{R} : R > 0\\}$.","title":"Positive radii"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/packing","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.Packing","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","Packings of congruent balls in "],["math","\\(\\mathbb{R}^4\\)"]],"source":"Packings of congruent balls in $\\mathbb{R}^4$"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#packing","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"A packing in $\\mathbb{R}^4$ consists of a real radius $r > 0$ and a set $C \\subseteq \\mathbb{R}^4$ of centres such that for any two distinct centres $c, c' \\in C$ the open balls of radius $r$ about $c$ and $c'$ are disjoint.","title":"Packings of congruent balls in $\\mathbb{R}^4$"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/packed-ball","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.packedBall","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#packed-ball","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For a packing $P$ in $\\mathbb{R}^4$ with radius $r$ and a point $c \\in \\mathbb{R}^4$, the packed ball of $P$ at $c$ is the closed ball $\\{x \\in \\mathbb{R}^4 : \\|x - c\\| \\le r\\}$, boundary included.","title":"Packed ball"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/packing-region","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.packingRegion","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#packing-region","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For a packing $P$ in $\\mathbb{R}^4$ with radius $r$ and set of centres $C$, the region covered by $P$ is the union, over all centres $c \\in C$, of the packed closed balls $\\{x \\in \\mathbb{R}^4 : \\|x - c\\| \\le r\\}$.","title":"Region covered by a packing"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/finite-radius-density","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.finiteRadiusDensity","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#finite-radius-density","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For a packing $P$ of congruent balls in $\\mathbb{R}^4$, a point $p \\in \\mathbb{R}^4$ and a radius $R > 0$, the density of $P$ in the open ball $B(p, R)$ of radius $R$ about $p$ is $\\operatorname{vol}(U \\cap B(p, R)) / \\operatorname{vol}(B(p, R))$, where $U$ is the union of the closed balls of $P$ and $\\operatorname{vol}$ is Lebesgue measure; the quotient is taken in $[0, \\infty]$.","title":"Density of a packing in a ball"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/centered-radius-density","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.centeredRadiusDensity","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#centered-radius-density","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For a packing $P$ of congruent balls in $\\mathbb{R}^4$ and a radius $R > 0$, the density of $P$ at radius $R$ is the supremum, over all centers $p \\in \\mathbb{R}^4$, of the density of $P$ in the open ball of radius $R$ about $p$; it is a value in $[0, \\infty]$.","title":"Density of a packing at a fixed radius"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-density","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperDensity","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-density","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For a packing $P$ of congruent balls in $\\mathbb{R}^4$, let $U_P$ be the union of the closed balls of the packing, and for $p \\in \\mathbb{R}^4$ and $R > 0$ let $B(p, R)$ be the open ball of radius $R$ about $p$. The upper density of $P$ is the extended nonnegative real number","title":"Upper density of a packing"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/sphere-packing-density4","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.spherePackingDensity4","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sphere-packing density of "],["math","\\(\\mathbb{R}^4\\)"]],"source":"Sphere-packing density of $\\mathbb{R}^4$"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#sphere-packing-density4","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The sphere-packing density $\\Delta_4 \\in [0, \\infty]$ is the supremum, over all packings $P$ of congruent balls in $\\mathbb{R}^4$, of the upper density of $P$. Here the upper density of $P$ is the limit superior, as the real radius $R > 0$ tends to infinity, of the supremum over all points $p \\in \\mathbb{R}^4$ of the ratio of the volume of the part of the open ball $B(p, R)$ covered by $P$ to the volume of $B(p, R)$.","title":"Sphere-packing density of $\\mathbb{R}^4$"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/unit-ball-volume4","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.unitBallVolume4","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","Volume of the unit ball in "],["math","\\(\\mathbb{R}^4\\)"]],"source":"Volume of the unit ball in $\\mathbb{R}^4$"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#unit-ball-volume4","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The volume of the unit ball in $\\mathbb{R}^4$ is the Lebesgue measure, as an extended nonnegative real number, of the open ball of radius $1$ about the origin in four-dimensional Euclidean space $\\mathbb{R}^4$.","title":"Volume of the unit ball in $\\mathbb{R}^4$"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/center-density4","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.centerDensity4","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#center-density4","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The center density of sphere packings in $\\mathbb{R}^4$ is $\\delta_4 = \\Delta_4 / \\operatorname{vol}(B)$, where $\\Delta_4$ is the sphere-packing density of $\\mathbb{R}^4$ and $B$ is the open unit ball of $\\mathbb{R}^4$ centered at the origin; the quotient is taken in $[0, \\infty]$.","title":"Center density in four dimensions"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/is-periodic","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.Packing.IsPeriodic","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#is-periodic","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"A packing $P$ of congruent balls in $\\mathbb{R}^4$ is periodic if there are a basis $b_1, b_2, b_3, b_4$ of $\\mathbb{R}^4$ and a finite set $F \\subseteq \\mathbb{R}^4$ such that the set of centers of $P$ equals $\\{c + w : c \\in F, w \\in \\Lambda\\}$, where $\\Lambda$ is the set of integer linear combinations of $b_1, b_2, b_3, b_4$.","title":"Periodic packings"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-bound-trivial","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperBound_trivial","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trivial upper bound for the sphere-packing density of "],["math","\\(\\mathbb{R}^4\\)"]],"source":"Trivial upper bound for the sphere-packing density of $\\mathbb{R}^4$"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-bound-trivial","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The trivial upper bound is the statement that the sphere-packing density of $\\mathbb{R}^4$ is at most $1$.","title":"Trivial upper bound for the sphere-packing density of $\\mathbb{R}^4$"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/lower-bound-trivial","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.lowerBound_trivial","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#lower-bound-trivial","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"This is the statement that the sphere-packing density $\\Delta_4$ of $\\mathbb{R}^4$ satisfies $0 \\le \\Delta_4$.","title":"The trivial lower bound for the four-dimensional packing density"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/centers-countable","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.centers_countable","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#centers-countable","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For every packing of congruent balls in $\\mathbb{R}^4$, the set of centers of its balls is countable.","title":"A packing has countably many centers"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/packing-region-measurable","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.packingRegion_measurable","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#packing-region-measurable","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For every packing $P$ in $\\mathbb{R}^4$, the region covered by $P$, the union of its packed closed balls, is a Borel set; in particular it is Lebesgue measurable.","title":"The region covered by a packing is measurable"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/observation-volume-ne-zero","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.observationVolume_ne_zero","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","Balls of positive radius in "],["math","\\(\\mathbb{R}^4\\)"],["text"," have nonzero volume"]],"source":"Balls of positive radius in $\\mathbb{R}^4$ have nonzero volume"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#observation-volume-ne-zero","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For every point $p \\in \\mathbb{R}^4$ and every real $R > 0$, the open ball of radius $R$ centred at $p$ has nonzero Lebesgue measure.","title":"Balls of positive radius in $\\mathbb{R}^4$ have nonzero volume"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/observation-volume-ne-top","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.observationVolume_ne_top","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","Balls of positive radius in "],["math","\\(\\mathbb{R}^4\\)"],["text"," have finite volume"]],"source":"Balls of positive radius in $\\mathbb{R}^4$ have finite volume"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#observation-volume-ne-top","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For every point $p \\in \\mathbb{R}^4$ and every real $R > 0$, the open ball of radius $R$ centred at $p$ has finite Lebesgue measure.","title":"Balls of positive radius in $\\mathbb{R}^4$ have finite volume"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/finite-radius-density-le-one","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.finiteRadiusDensity_le_one","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#finite-radius-density-le-one","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For every packing $P$ of congruent balls in $\\mathbb{R}^4$, every point $p \\in \\mathbb{R}^4$ and every radius $R > 0$, the density of $P$ in the open ball of radius $R$ about $p$ is at most $1$.","title":"The density in a ball is at most one"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/centered-radius-density-le-one","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.centeredRadiusDensity_le_one","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#centered-radius-density-le-one","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For every packing $P$ of congruent balls in $\\mathbb{R}^4$ and every radius $R > 0$, the density of $P$ at radius $R$ (the supremum over all centers $p \\in \\mathbb{R}^4$ of the density of $P$ in the open ball of radius $R$ about $p$) is at most $1$.","title":"The density at a fixed radius is at most one"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-density-le-one","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperDensity_le_one","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-density-le-one","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For every packing $P$ of congruent balls in $\\mathbb{R}^4$, the upper density of $P$ is at most $1$.","title":"The upper density of a packing is at most one"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-bound-trivial-proved","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperBound_trivial_proved","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sphere-packing density of "],["math","\\(\\mathbb{R}^4\\)"],["text"," is at most one"]],"source":"The sphere-packing density of $\\mathbb{R}^4$ is at most one"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-bound-trivial-proved","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The trivial upper bound holds: the sphere-packing density of $\\mathbb{R}^4$ is at most $1$.","title":"The sphere-packing density of $\\mathbb{R}^4$ is at most one"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/lower-bound-trivial-proved","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.lowerBound_trivial_proved","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#lower-bound-trivial-proved","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The trivial lower bound holds: the sphere-packing density $\\Delta_4$ of $\\mathbb{R}^4$ satisfies $0 \\le \\Delta_4$.","title":"The four-dimensional packing density is nonnegative"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/unit-ball-volume4-eq","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.unitBallVolume4_eq","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","The unit ball in "],["math","\\(\\mathbb{R}^4\\)"],["text"," has volume "],["math","\\(\\pi^2/2\\)"]],"source":"The unit ball in $\\mathbb{R}^4$ has volume $\\pi^2/2$"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#unit-ball-volume4-eq","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The Lebesgue measure of the open unit ball centred at the origin of $\\mathbb{R}^4$ equals $\\pi^2/2$.","title":"The unit ball in $\\mathbb{R}^4$ has volume $\\pi^2/2$"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/unit-ball-volume4-ne-zero","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.unitBallVolume4_ne_zero","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","The unit ball in "],["math","\\(\\mathbb{R}^4\\)"],["text"," has nonzero volume"]],"source":"The unit ball in $\\mathbb{R}^4$ has nonzero volume"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#unit-ball-volume4-ne-zero","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The Lebesgue volume of the open unit ball of radius $1$ about the origin in $\\mathbb{R}^4$ is not $0$.","title":"The unit ball in $\\mathbb{R}^4$ has nonzero volume"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/unit-ball-volume4-ne-top","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.unitBallVolume4_ne_top","parent":14461,"reading_title":{"format":"inline-parts-v1","parts":[["text","The unit ball in "],["math","\\(\\mathbb{R}^4\\)"],["text"," has finite volume"]],"source":"The unit ball in $\\mathbb{R}^4$ has finite volume"},"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#unit-ball-volume4-ne-top","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The Lebesgue volume of the open unit ball of radius $1$ about the origin in $\\mathbb{R}^4$ is not $\\infty$.","title":"The unit ball in $\\mathbb{R}^4$ has finite volume"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/center-density-upper-iff","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.centerDensity_upper_iff","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#center-density-upper-iff","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For every real number $a \\ge 0$, the center density $\\delta_4$ of sphere packings in $\\mathbb{R}^4$ is at most $a$ if and only if the sphere-packing density $\\Delta_4$ of $\\mathbb{R}^4$ is at most $\\pi^2 a / 2$.","title":"Center-density upper bounds as packing-density upper bounds"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/periodic-approximation","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.periodicApproximation","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#periodic-approximation","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"For a packing $Q$ of congruent balls in $\\mathbb{R}^4$, let its upper density $\\overline{\\delta}(Q)$ be the limit superior, as $R \\to \\infty$, of the supremum over all points $p$ of the fraction of the volume of the open ball $B(p, R)$ covered by the closed balls of $Q$, and let the sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$ be the supremum of $\\overline{\\delta}(Q)$ over all such packings $Q$. This is the\u2026","title":"Statement that periodic packings approach the optimal density in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-bound-ce2003-rogers","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperBound_CE2003_Rogers","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-bound-ce2003-rogers","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The proposition that the sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$, the supremum of the upper densities of all packings of congruent balls in $\\mathbb{R}^4$, satisfies $\\Delta_4 \\le \\frac{\\pi^2}{2} \\cdot \\frac{13127}{100000}$.","title":"Statement of the Rogers upper bound in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-bound-ce2003-new","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperBound_CE2003_new","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-bound-ce2003-new","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The proposition that the sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$, the supremum of the upper densities of all packings of congruent balls in $\\mathbb{R}^4$, satisfies $\\Delta_4 \\le \\frac{\\pi^2}{2} \\cdot \\frac{13126}{100000}$.","title":"Statement of the Cohn\u2013Elkies upper bound in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-bound-d-lov2014","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperBound_dLOV2014","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-bound-d-lov2014","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The proposition that the sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$, the supremum of the upper densities of all packings of congruent balls in $\\mathbb{R}^4$, satisfies $\\Delta_4 \\le \\frac{\\pi^2}{2} \\cdot \\frac{130587}{1000000}$.","title":"Statement of the de Laat\u2013Oliveira\u2013Vallentin upper bound in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/lower-bound-d-lov2014","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.lowerBound_dLOV2014","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#lower-bound-d-lov2014","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"fully_proved","summary":"The proposition that the sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$, the supremum of the upper densities of all packings of congruent balls in $\\mathbb{R}^4$, satisfies $\\frac{\\pi^2}{16} \\le \\Delta_4$.","title":"Statement of the lower bound for sphere packing in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-bound-ce2003-rogers-wanted","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperBound_CE2003_Rogers_wanted","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-bound-ce2003-rogers-wanted","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"can_prove","summary":"The sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$, the supremum of the upper densities of all packings of congruent balls in $\\mathbb{R}^4$, satisfies $\\Delta_4 \\le \\frac{\\pi^2}{2} \\cdot \\frac{13127}{100000}$.","title":"Rogers upper bound for sphere packing in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-bound-ce2003-new-wanted","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperBound_CE2003_new_wanted","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-bound-ce2003-new-wanted","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"can_prove","summary":"The sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$, the supremum of the upper densities of all packings of congruent balls in $\\mathbb{R}^4$, satisfies $\\Delta_4 \\le \\frac{\\pi^2}{2} \\cdot \\frac{13126}{100000}$.","title":"Cohn\u2013Elkies upper bound for sphere packing in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/upper-bound-d-lov2014-wanted","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.upperBound_dLOV2014_wanted","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#upper-bound-d-lov2014-wanted","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"can_prove","summary":"The sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$, the supremum of the upper densities of all packings of congruent balls in $\\mathbb{R}^4$, satisfies $\\Delta_4 \\le \\frac{\\pi^2}{2} \\cdot \\frac{130587}{1000000}$.","title":"de Laat\u2013Oliveira\u2013Vallentin upper bound for sphere packing in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/lower-bound-d-lov2014-wanted","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.lowerBound_dLOV2014_wanted","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#lower-bound-d-lov2014-wanted","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"can_prove","summary":"The sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$, the supremum of the upper densities of all packings of congruent balls in $\\mathbb{R}^4$, satisfies $\\frac{\\pi^2}{16} \\le \\Delta_4$.","title":"Lower bound for sphere packing in dimension four"},{"id":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/periodic-approximation-wanted","kind":"node","lean":"MathlibExt.Geometry.SpherePackingDensityFourWanted.periodicApproximation_wanted","parent":14461,"result":true,"route":"../roadmap/Geometry/packings-and-coverings/sphere-packing-in-four-dimensions/index.html#periodic-approximation-wanted","scope":"Geometry/packings-and-coverings/sphere-packing-in-four-dimensions","status":"can_prove","summary":"For a packing $Q$ of congruent balls in $\\mathbb{R}^4$, let its upper density $\\overline{\\delta}(Q)$ be the limit superior, as $R \\to \\infty$, of the supremum over all points $p$ of the fraction of the volume of the open ball $B(p, R)$ covered by the closed balls of $Q$, and let the sphere packing density $\\Delta_4$ of $\\mathbb{R}^4$ be the supremum of $\\overline{\\delta}(Q)$ over all such packings $Q$. Then for\u2026","title":"Periodic packings approach the optimal density in dimension four"},{"data_url":"scopes/Geometry/convex-geometry.json","id":"Geometry/convex-geometry","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/convex-geometry/index.html","scope":"Geometry/convex-geometry","status":"can_prove","summary":"Convex sets and convex bodies in Euclidean space: Helly-type and Radon-type theorems, partitions, sections, valuations and linear optimization over polyhedra.","title":"Convex geometry"},{"id":"Geometry/convex-geometry/isoperimetric-inequality","kind":"node","lean":"MetaMathlibExt.isoperimetric_inequality","parent":14499,"result":true,"route":"../roadmap/Geometry/convex-geometry/index.html#isoperimetric-inequality","scope":"Geometry/convex-geometry","status":"can_prove","summary":"Let $g$ be a continuous, $1$-periodic, Lipschitz map from $\\mathbb{R}$ to the Euclidean plane $\\mathbb{R}^2$ that is injective on $[0, 1)$, so that its image $C = g(\\mathbb{R})$ is a simple closed curve, and suppose $C$ has finite one-dimensional Hausdorff measure. Let $U \\subseteq \\mathbb{R}^2$ be a bounded, connected open set whose boundary is $C$. Let $L$ be the one-dimensional Hausdorff measure of $C$ (its\u2026","title":"Isoperimetric inequality"},{"id":"Geometry/convex-geometry/jung","kind":"node","lean":"MathlibExt.Geometry.Euclidean.JungWanted.jung","parent":14499,"result":true,"route":"../roadmap/Geometry/convex-geometry/index.html#jung","scope":"Geometry/convex-geometry","status":"fully_proved","summary":"Let $n \\ge 1$ and let $s$ be a bounded subset of the Euclidean space $\\mathbb{R}^n$ with diameter $\\operatorname{diam} s$. Then there is a point $c \\in \\mathbb{R}^n$ such that $s$ is contained in the closed ball with center $c$ and radius $\\sqrt{n / (2(n + 1))} \\cdot \\operatorname{diam} s$.","title":"Jung's theorem"},{"id":"Geometry/convex-geometry/kirchberger","kind":"node","lean":"MetaMathlibExt.kirchberger","parent":14499,"result":true,"route":"../roadmap/Geometry/convex-geometry/index.html#kirchberger","scope":"Geometry/convex-geometry","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets of points in $\\mathbb{R}^d$. Then $A$ and $B$ are strictly separated by a hyperplane, that is, there are a nonzero vector $w \\in \\mathbb{R}^d$ and a real number $\\beta$ with $\\langle w, a \\rangle < \\beta$ for all $a \\in A$ and $\\beta < \\langle w, c \\rangle$ for all $c \\in B$, if and only if for every set $C \\subseteq A \\cup B$ of at most $d + 2$ points, the sets $A \\cap C$ and\u2026","title":"Kirchberger's theorem"},{"id":"Geometry/convex-geometry/rado-centerpoint","kind":"node","lean":"MathlibExt.Geometry.Convex.RadoCenterpointWanted.rado_centerpoint","parent":14499,"result":true,"route":"../roadmap/Geometry/convex-geometry/index.html#rado-centerpoint","scope":"Geometry/convex-geometry","status":"fully_proved","summary":"Let $n \\ge 1$ and let $p_1, \\dots, p_n$ be points of $\\mathbb{R}^d$, not necessarily distinct. Then some point $x$ in the convex hull of $\\{p_1, \\dots, p_n\\}$ has at least $n/(d+1)$ of the points in every closed halfspace containing it: for all $v \\in \\mathbb{R}^d$ and $t \\in \\mathbb{R}$ with $\\langle v, x \\rangle \\le t$, the number of indices $i$ with $\\langle v, p_i \\rangle \\le t$ is at least $n/(d+1)$.","title":"Rado's centerpoint theorem"},{"id":"Geometry/convex-geometry/tverberg","kind":"node","lean":"MathlibExt.Geometry.Convex.TverbergWanted.tverberg","parent":14499,"result":true,"route":"../roadmap/Geometry/convex-geometry/index.html#tverberg","scope":"Geometry/convex-geometry","status":"fully_proved","summary":"Let $d, r$ be natural numbers with $r \\ge 1$ and let $N = (d+1)(r-1)+1$. For any points $p_1, \\dots, p_N$ of $\\mathbb{R}^d$, not necessarily distinct, there are pairwise disjoint sets $P_1, \\dots, P_r \\subseteq \\{1, \\dots, N\\}$ whose union is $\\{1, \\dots, N\\}$ and a point $x \\in \\mathbb{R}^d$ that lies in the convex hull of $\\{p_i : i \\in P_j\\}$ for every $j$.","title":"Tverberg's theorem"},{"id":"Geometry/convex-geometry/ulam-flat-sections-convex","kind":"node","lean":"MathlibExt.Geometry.ScottishBook68Wanted.ulamFlatSectionsConvex","parent":14499,"result":true,"route":"../roadmap/Geometry/convex-geometry/index.html#ulam-flat-sections-convex","scope":"Geometry/convex-geometry","status":"can_prove","summary":"Let $K \\subseteq \\mathbb{R}^3$ be a compact set homeomorphic to the closed unit ball. Suppose that for every nonzero continuous linear functional $F : \\mathbb{R}^3 \\to \\mathbb{R}$ and every real $c$ such that the plane $\\{x : F(x) = c\\}$ meets the interior of $K$, the intersection of this plane with the boundary (frontier) of $K$ is homeomorphic to the unit circle. Then $K$ is convex.","title":"Ulam's flat-sections problem in dimension three"},{"data_url":"scopes/Geometry/convex-geometry/borsuk-s-conjecture.json","id":"Geometry/convex-geometry/borsuk-s-conjecture","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","title":"Borsuk's conjecture"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/has-borsuk-cover","kind":"node","lean":"Metric.HasBorsukCover","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#has-borsuk-cover","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"Let $s$ be a set in an extended pseudometric space and let $k$ be a natural number. The set $s$ has a $k$-piece Borsuk cover if there are $k$ sets whose union contains $s$, each of extended diameter strictly smaller than that of $s$.","title":"Borsuk covers"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/set-borsuk-number","kind":"node","lean":"Metric.setBorsukNumber","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#set-borsuk-number","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"The Borsuk number of a set $s$ in the Euclidean space $\\mathbb{R}^n$ is the least natural number $k$ such that $s$ has a $k$-piece Borsuk cover, that is, $s$ is covered by $k$ sets each of strictly smaller extended diameter than $s$; it is $\\infty$ if no such $k$ exists.","title":"Borsuk number of a set"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/dimension-borsuk-number","kind":"node","lean":"Metric.dimensionBorsukNumber","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#dimension-borsuk-number","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"For a natural number $n$, the Borsuk number of the Euclidean space $\\mathbb{R}^n$ is the supremum, in $\\mathbb{N} \\cup \\{\\infty\\}$, of the Borsuk numbers of the bounded subsets of $\\mathbb{R}^n$ with at least two points.","title":"Borsuk number of a Euclidean space"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/first-borsuk-failure-dimension","kind":"node","lean":"Metric.firstBorsukFailureDimension","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#first-borsuk-failure-dimension","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"The first Borsuk failure dimension is the least positive integer $n$ such that the Borsuk number of the Euclidean space $\\mathbb{R}^n$ is greater than $n + 1$, and $\\infty$ if there is no such $n$.","title":"First Borsuk failure dimension"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/set-borsuk-number-le-of-cover","kind":"node","lean":"Metric.setBorsukNumber_le_of_cover","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#set-borsuk-number-le-of-cover","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"If a set $s$ in the Euclidean space $\\mathbb{R}^n$ has a $k$-piece Borsuk cover for a natural number $k$, then the Borsuk number of $s$ is at most $k$.","title":"A Borsuk cover bounds the Borsuk number"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/has-borsuk-cover-iff-exact-union","kind":"node","lean":"Metric.hasBorsukCover_iff_exactUnion","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#has-borsuk-cover-iff-exact-union","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"For natural numbers $n$ and $k$, a set $s$ in the Euclidean space $\\mathbb{R}^n$ has a $k$-piece Borsuk cover if and only if there are subsets $c_1, \\dots, c_k$ of $s$ with $s = c_1 \\cup \\dots \\cup c_k$, each of extended diameter strictly smaller than that of $s$.","title":"Borsuk covers by subsets"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/has-borsuk-cover-iff-closed-cover","kind":"node","lean":"Metric.hasBorsukCover_iff_closedCover","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#has-borsuk-cover-iff-closed-cover","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"For natural numbers $n$ and $k$, a set $s$ in the Euclidean space $\\mathbb{R}^n$ has a $k$-piece Borsuk cover if and only if there are $k$ closed subsets of $\\mathbb{R}^n$ whose union contains $s$, each of extended diameter strictly smaller than that of $s$.","title":"Borsuk covers by closed sets"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/has-borsuk-cover-pos","kind":"node","lean":"Metric.hasBorsukCover_pos","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#has-borsuk-cover-pos","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers. If a nonempty set $s$ in the Euclidean space $\\mathbb{R}^n$ has a $k$-piece Borsuk cover, then $0 < k$.","title":"Borsuk covers of nonempty sets have at least one piece"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/has-borsuk-cover-mono","kind":"node","lean":"Metric.hasBorsukCover_mono","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#has-borsuk-cover-mono","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"Let $n$, $k$ and $l$ be natural numbers with $k \\le l$. If a nonempty set $s$ in the Euclidean space $\\mathbb{R}^n$ has a $k$-piece Borsuk cover, then $s$ has an $l$-piece Borsuk cover.","title":"Enlarging a Borsuk cover"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/set-borsuk-number-le-iff","kind":"node","lean":"Metric.setBorsukNumber_le_iff","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#set-borsuk-number-le-iff","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"For a nonempty set $s$ in the Euclidean space $\\mathbb{R}^n$ and a natural number $k$, the Borsuk number of $s$ is at most $k$ if and only if $s$ has a $k$-piece Borsuk cover.","title":"Bounds on the Borsuk number of a set"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/dimension-borsuk-number-le-iff","kind":"node","lean":"Metric.dimensionBorsukNumber_le_iff","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#dimension-borsuk-number-le-iff","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the Borsuk number of the Euclidean space $\\mathbb{R}^n$ is at most $k$ if and only if every bounded subset $s$ of $\\mathbb{R}^n$ with at least two points has a $k$-piece Borsuk cover, that is, $s$ is covered by $k$ sets, each of extended diameter strictly smaller than that of $s$.","title":"Upper bounds on the Borsuk number of a Euclidean space"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/borsuk-conjecture-in-dimension","kind":"node","lean":"Metric.borsukConjectureInDimension","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#borsuk-conjecture-in-dimension","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"For a natural number $n$, Borsuk's conjecture in dimension $n$ is the statement that the Borsuk number of the Euclidean space $\\mathbb{R}^n$ is at most $n + 1$.","title":"Borsuk's conjecture in a given dimension"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/first-borsuk-failure-dimension-eq-top-if","kind":"node","lean":"Metric.firstBorsukFailureDimension_eq_top_iff","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#first-borsuk-failure-dimension-eq-top-if","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"The first Borsuk failure dimension is $\\infty$ if and only if, for every positive integer $n$, the Borsuk number of the Euclidean space $\\mathbb{R}^n$ is at most $n + 1$.","title":"When the first Borsuk failure dimension is infinite"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/is-two-distance-set","kind":"node","lean":"Metric.IsTwoDistanceSet","parent":14506,"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#is-two-distance-set","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"A set $s$ in the Euclidean space $\\mathbb{R}^n$ is a two-distance set if there are reals $0 < a < b$ such that the distance between any two distinct points of $s$ is $a$ or $b$, and both $a$ and $b$ occur as the distance between some two points of $s$.","title":"Two-distance sets"},{"id":"Geometry/convex-geometry/borsuk-s-conjecture/two-distance-borsuk-number","kind":"node","lean":"Metric.twoDistanceBorsukNumber","parent":14506,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-distance Borsuk number in dimension "],["math","\\(n\\)"]],"source":"Two-distance Borsuk number in dimension $n$"},"result":true,"route":"../roadmap/Geometry/convex-geometry/borsuk-s-conjecture/index.html#two-distance-borsuk-number","scope":"Geometry/convex-geometry/borsuk-s-conjecture","status":"fully_proved","summary":"The two-distance Borsuk number in dimension $n$ is the supremum, taken in $\\mathbb{N} \\cup \\{\\infty\\}$, of the Borsuk numbers of all bounded two-distance sets with at least two points in the Euclidean space $\\mathbb{R}^n$.","title":"Two-distance Borsuk number in dimension $n$"},{"data_url":"scopes/Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie.json","id":"Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie/index.html","scope":"Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie","status":"fully_proved","title":"Brunn\u2013Minkowski and Pr\u00e9kopa\u2013Leindler inequalities"},{"id":"Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie/prekopa-leindler","kind":"node","lean":"MathlibExt.MeasureTheory.Integral.PrekopaLeindlerWanted.prekopa_leindler","parent":14522,"result":true,"route":"../roadmap/Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie/index.html#prekopa-leindler","scope":"Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie","status":"fully_proved","summary":"Let $n \\ge 0$ and $0 < t < 1$, and let $f, g, h : \\mathbb{R}^n \\to [0, \\infty]$ be Borel measurable functions on Euclidean space $\\mathbb{R}^n$ such that $f(x)^{1-t} g(y)^t \\le h((1-t)x + ty)$ for all $x, y \\in \\mathbb{R}^n$. Then, with all integrals taken over $\\mathbb{R}^n$ against Lebesgue measure,","title":"Pr\u00e9kopa\u2013Leindler inequality"},{"id":"Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie/brunn-minkowski","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BrunnMinkowskiWanted.brunn_minkowski","parent":14522,"result":true,"route":"../roadmap/Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie/index.html#brunn-minkowski","scope":"Geometry/convex-geometry/brunn-minkowski-and-pr-kopa-leindler-inequalitie","status":"fully_proved","summary":"Let $n \\ge 1$, let $A$ and $B$ be nonempty compact subsets of $\\mathbb{R}^n$, let $A + B = \\{a + b : a \\in A,\\ b \\in B\\}$ be their Minkowski sum, and let $\\mathrm{vol}$ denote Lebesgue measure on $\\mathbb{R}^n$. Then","title":"Brunn\u2013Minkowski inequality"},{"data_url":"scopes/Geometry/convex-geometry/colorful-carath-odory-theorem.json","id":"Geometry/convex-geometry/colorful-carath-odory-theorem","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/colorful-carath-odory-theorem/index.html","scope":"Geometry/convex-geometry/colorful-carath-odory-theorem","status":"fully_proved","title":"Colorful Carath\u00e9odory theorem"},{"id":"Geometry/convex-geometry/colorful-carath-odory-theorem/colorful-caratheodory-finrank","kind":"node","lean":"MathlibExt.Geometry.Convex.ColorfulCaratheodoryWanted.colorful_caratheodory_finrank","parent":14525,"result":true,"route":"../roadmap/Geometry/convex-geometry/colorful-carath-odory-theorem/index.html#colorful-caratheodory-finrank","scope":"Geometry/convex-geometry/colorful-carath-odory-theorem","status":"fully_proved","summary":"Let $V$ be a finite-dimensional real inner product space of dimension $n$, and let $S_0, \\dots, S_n$ be finite subsets of $V$ such that $0$ lies in the convex hull of each $S_l$. Then there are points $t_0 \\in S_0, \\dots, t_n \\in S_n$ such that $0$ lies in the convex hull of $\\{t_0, \\dots, t_n\\}$.","title":"Colorful Carath\u00e9odory theorem in finite-dimensional inner product spaces"},{"id":"Geometry/convex-geometry/colorful-carath-odory-theorem/colorful-caratheodory","kind":"node","lean":"MathlibExt.Geometry.Convex.ColorfulCaratheodoryWanted.colorful_caratheodory","parent":14525,"result":true,"route":"../roadmap/Geometry/convex-geometry/colorful-carath-odory-theorem/index.html#colorful-caratheodory","scope":"Geometry/convex-geometry/colorful-carath-odory-theorem","status":"fully_proved","summary":"Let $d \\ge 0$ and let $S_0, \\dots, S_d$ be finite subsets of the Euclidean space $\\mathbb{R}^d$ such that the origin lies in the convex hull of each $S_i$. Then there are points $c_0 \\in S_0, \\dots, c_d \\in S_d$ such that the origin lies in the convex hull of $\\{c_0, \\dots, c_d\\}$.","title":"Colorful Carath\u00e9odory theorem for finite sets"},{"id":"Geometry/convex-geometry/colorful-carath-odory-theorem/exists-colorful-mem-convex-hull","kind":"node","lean":"MathlibExt.Geometry.Convex.ColorfulCaratheodoryWanted.exists_colorful_mem_convexHull","parent":14525,"result":true,"route":"../roadmap/Geometry/convex-geometry/colorful-carath-odory-theorem/index.html#exists-colorful-mem-convex-hull","scope":"Geometry/convex-geometry/colorful-carath-odory-theorem","status":"fully_proved","summary":"Let $d \\ge 0$ and let $S_0, \\dots, S_d$ be subsets of the Euclidean space $\\mathbb{R}^d$ such that the origin lies in the convex hull of each $S_i$. Then there are points $c_0 \\in S_0, \\dots, c_d \\in S_d$ such that the origin lies in the convex hull of $\\{c_0, \\dots, c_d\\}$.","title":"Colorful Carath\u00e9odory theorem"},{"data_url":"scopes/Geometry/convex-geometry/equichordal-point-problem.json","id":"Geometry/convex-geometry/equichordal-point-problem","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/equichordal-point-problem/index.html","scope":"Geometry/convex-geometry/equichordal-point-problem","status":"can_prove","title":"Equichordal point problem"},{"id":"Geometry/convex-geometry/equichordal-point-problem/plane","kind":"node","lean":"MathlibExt.Geometry.EquichordalWanted.Geometry.Equichordal.Plane","parent":14529,"result":true,"route":"../roadmap/Geometry/convex-geometry/equichordal-point-problem/index.html#plane","scope":"Geometry/convex-geometry/equichordal-point-problem","status":"fully_proved","summary":"The Euclidean plane is the space $\\mathbb{R}^2$ of pairs of real numbers with the Euclidean inner product and norm.","title":"Euclidean plane for the equichordal point problem"},{"id":"Geometry/convex-geometry/equichordal-point-problem/is-equichordal-point","kind":"node","lean":"MathlibExt.Geometry.EquichordalWanted.Geometry.Equichordal.IsEquichordalPoint","parent":14529,"result":true,"route":"../roadmap/Geometry/convex-geometry/equichordal-point-problem/index.html#is-equichordal-point","scope":"Geometry/convex-geometry/equichordal-point-problem","status":"fully_proved","summary":"Let $K$ be a subset of the Euclidean plane. A point $P$ is an equichordal point of $K$ if $P$ lies in the interior of $K$ and there is a real $L > 0$ such that for every unit vector $u$ there are reals $a > 0$ and $b > 0$ for which $P + a u$ and $P - b u$ lie in the boundary (frontier) of $K$, the open segment between $P - b u$ and $P + a u$ lies in the interior of $K$, and the distance between $P + a u$ and\u2026","title":"Equichordal point"},{"id":"Geometry/convex-geometry/equichordal-point-problem/equichordal-point-unique","kind":"node","lean":"MathlibExt.Geometry.EquichordalWanted.Geometry.Equichordal.equichordal_point_unique","parent":14529,"result":true,"route":"../roadmap/Geometry/convex-geometry/equichordal-point-problem/index.html#equichordal-point-unique","scope":"Geometry/convex-geometry/equichordal-point-problem","status":"can_prove","summary":"Let $K$ be a compact convex subset of the Euclidean plane with nonempty interior. Then $K$ does not have two distinct equichordal points: there are no points $P \\ne Q$ of the plane such that both $P$ and $Q$ are equichordal points of $K$.","title":"Equichordal point theorem"},{"data_url":"scopes/Geometry/convex-geometry/hadwiger-s-theorem-on-valuations.json","id":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/index.html","scope":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations","status":"can_prove","title":"Hadwiger's theorem on valuations"},{"id":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/e","kind":"node","lean":"MathlibExt.Geometry.HadwigerWanted.ConvexGeometry.E","parent":14533,"result":true,"route":"../roadmap/Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/index.html#e","scope":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations","status":"fully_proved","summary":"For a natural number $n$, $E_n$ is the real Euclidean space $\\mathbb{R}^n$ with its standard inner product and Euclidean norm.","title":"Model space for convex bodies"},{"id":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/is-valuation","kind":"node","lean":"MathlibExt.Geometry.HadwigerWanted.ConvexGeometry.IsValuation","parent":14533,"result":true,"route":"../roadmap/Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/index.html#is-valuation","scope":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations","status":"fully_proved","summary":"For a natural number $n$, a real-valued function $f$ on the convex bodies (nonempty compact convex subsets) of the Euclidean space $\\mathbb{R}^n$ is a continuous invariant valuation if: $f$ is continuous for the Hausdorff metric; $f(A) + f(B) = f(C) + f(D)$ for all convex bodies $A, B, C, D$ with $A = C \\cup D$ and $B = C \\cap D$; $f(e(A)) = f(A)$ for every convex body $A$ and every linear isometry $e$ of\u2026","title":"Continuous invariant valuation"},{"id":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/valuations","kind":"node","lean":"MathlibExt.Geometry.HadwigerWanted.ConvexGeometry.valuations","parent":14533,"result":true,"route":"../roadmap/Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/index.html#valuations","scope":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations","status":"fully_proved","summary":"For a natural number $n$, the continuous invariant valuations on the convex bodies of $\\mathbb{R}^n$ form a real linear subspace of the vector space of all functions from the convex bodies of $\\mathbb{R}^n$ to $\\mathbb{R}$.","title":"The space of continuous invariant valuations"},{"id":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/hadwiger","kind":"node","lean":"MathlibExt.Geometry.HadwigerWanted.ConvexGeometry.hadwiger","parent":14533,"result":true,"route":"../roadmap/Geometry/convex-geometry/hadwiger-s-theorem-on-valuations/index.html#hadwiger","scope":"Geometry/convex-geometry/hadwiger-s-theorem-on-valuations","status":"can_prove","summary":"For every natural number $n$, the real vector space of continuous invariant valuations on the convex bodies of $\\mathbb{R}^n$ (continuous, additive, and invariant under linear isometries and translations) is finite-dimensional of dimension $n + 1$.","title":"Hadwiger's theorem"},{"data_url":"scopes/Geometry/convex-geometry/hanner-polytopes.json","id":"Geometry/convex-geometry/hanner-polytopes","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","title":"Hanner polytopes"},{"id":"Geometry/convex-geometry/hanner-polytopes/hanner-unit-segment","kind":"node","lean":"MetaMathlibExt.hannerUnitSegment","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#hanner-unit-segment","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"The centered unit segment is the closed unit ball $\\{x \\in \\mathbb{R}^1 : \\|x\\| \\le 1\\}$, that is, the segment $[-1, 1]$. It is the one-dimensional leaf in the recursive definition of Hanner polytopes.","title":"Centered unit segment"},{"id":"Geometry/convex-geometry/hanner-polytopes/hanner-infinity-sum","kind":"node","lean":"MetaMathlibExt.hannerInfinitySum","parent":14538,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell_\\infty\\)"],["text"," sum of bodies"]],"source":"$\\ell_\\infty$ sum of bodies"},"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#hanner-infinity-sum","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For sets $K \\subseteq \\mathbb{R}^m$ and $L \\subseteq \\mathbb{R}^n$, the $\\ell_\\infty$ sum of $K$ and $L$ is the set of points of $\\mathbb{R}^{m+n}$ whose first $m$ coordinates form a point of $K$ and whose last $n$ coordinates form a point of $L$; that is, the product $K \\times L$ under the identification $\\mathbb{R}^{m+n} = \\mathbb{R}^m \\times \\mathbb{R}^n$.","title":"$\\ell_\\infty$ sum of bodies"},{"id":"Geometry/convex-geometry/hanner-polytopes/hanner-one-sum","kind":"node","lean":"MetaMathlibExt.hannerOneSum","parent":14538,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell_1\\)"],["text"," sum of bodies"]],"source":"$\\ell_1$ sum of bodies"},"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#hanner-one-sum","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For sets $K \\subseteq \\mathbb{R}^m$ and $L \\subseteq \\mathbb{R}^n$, the $\\ell_1$ sum of $K$ and $L$ is the convex hull in $\\mathbb{R}^{m+n}$ of $(K \\times \\{0\\}) \\cup (\\{0\\} \\times L)$, where $\\mathbb{R}^{m+n}$ is identified with $\\mathbb{R}^m \\times \\mathbb{R}^n$ by splitting off the first $m$ coordinates.","title":"$\\ell_1$ sum of bodies"},{"id":"Geometry/convex-geometry/hanner-polytopes/is-hanner-polytope","kind":"node","lean":"MetaMathlibExt.IsHannerPolytope","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#is-hanner-polytope","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For $K \\subseteq \\mathbb{R}^m$ and $L \\subseteq \\mathbb{R}^n$, identify $\\mathbb{R}^{m+n}$ with $\\mathbb{R}^m \\times \\mathbb{R}^n$ by splitting off the first $m$ coordinates; the $\\ell_\\infty$ sum of $K$ and $L$ is $K \\times L$, and their $\\ell_1$ sum is the convex hull of $(K \\times \\{0\\}) \\cup (\\{0\\} \\times L)$, both viewed as subsets of $\\mathbb{R}^{m+n}$. The Hanner polytopes are the smallest family of subsets\u2026","title":"Hanner polytopes"},{"id":"Geometry/convex-geometry/hanner-polytopes/hanner-cube","kind":"node","lean":"MetaMathlibExt.hannerCube","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#hanner-cube","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the Hanner cube $C_n \\subseteq \\mathbb{R}^{n+1}$ is defined by recursion on $n$: $C_0$ is the centered unit segment $[-1, 1] \\subseteq \\mathbb{R}^1$, and $C_{n+1} \\subseteq \\mathbb{R}^{n+2}$ is the $\\ell_\\infty$ sum of $C_n$ and the centered unit segment.","title":"Hanner cube"},{"id":"Geometry/convex-geometry/hanner-polytopes/hanner-cross-polytope","kind":"node","lean":"MetaMathlibExt.hannerCrossPolytope","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#hanner-cross-polytope","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the Hanner cross-polytope $P_n \\subseteq \\mathbb{R}^{n+1}$ is defined by recursion on $n$: $P_0$ is the centered unit segment $[-1, 1] \\subseteq \\mathbb{R}^1$, and $P_{n+1} \\subseteq \\mathbb{R}^{n+2}$ is the $\\ell_1$ sum of $P_n$ and the centered unit segment.","title":"Hanner cross-polytope"},{"id":"Geometry/convex-geometry/hanner-polytopes/hanner-cube-is-hanner-polytope","kind":"node","lean":"MetaMathlibExt.hannerCube_isHannerPolytope","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#hanner-cube-is-hanner-polytope","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the Hanner cube $C_n \\subseteq \\mathbb{R}^{n+1}$ is a Hanner polytope.","title":"The Hanner cubes are Hanner polytopes"},{"id":"Geometry/convex-geometry/hanner-polytopes/hanner-cross-polytope-is-hanner-polytope","kind":"node","lean":"MetaMathlibExt.hannerCrossPolytope_isHannerPolytope","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#hanner-cross-polytope-is-hanner-polytope","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the Hanner cross-polytope $P_n \\subseteq \\mathbb{R}^{n+1}$ is a Hanner polytope.","title":"The Hanner cross-polytopes are Hanner polytopes"},{"id":"Geometry/convex-geometry/hanner-polytopes/mem-hanner-unit-segment-iff","kind":"node","lean":"MetaMathlibExt.mem_hannerUnitSegment_iff","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#mem-hanner-unit-segment-iff","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"A point $x \\in \\mathbb{R}^1$ lies in the Hanner unit segment if and only if every coordinate $x_i$ of $x$ satisfies $|x_i| \\le 1$.","title":"Membership in the Hanner unit segment"},{"id":"Geometry/convex-geometry/hanner-polytopes/mem-hanner-infinity-sum-iff","kind":"node","lean":"MetaMathlibExt.mem_hannerInfinitySum_iff","parent":14538,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in the "],["math","\\(\\ell_\\infty\\)"],["text"," Hanner sum"]],"source":"Membership in the $\\ell_\\infty$ Hanner sum"},"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#mem-hanner-infinity-sum-iff","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}^m$ and $L \\subseteq \\mathbb{R}^n$. A point $x \\in \\mathbb{R}^{m+n}$ lies in the $\\ell_\\infty$ Hanner sum of $K$ and $L$ if and only if $(x_1, \\dots, x_m) \\in K$ and $(x_{m+1}, \\dots, x_{m+n}) \\in L$.","title":"Membership in the $\\ell_\\infty$ Hanner sum"},{"id":"Geometry/convex-geometry/hanner-polytopes/hanner-cube-eq-coordinate-cube","kind":"node","lean":"MetaMathlibExt.hannerCube_eq_coordinateCube","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#hanner-cube-eq-coordinate-cube","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the Hanner cube $C_n \\subseteq \\mathbb{R}^{n+1}$ equals the coordinate cube $\\{x \\in \\mathbb{R}^{n+1} : -1 \\le x_i \\le 1 \\text{ for every } i\\} = [-1, 1]^{n+1}$.","title":"The Hanner cube is the coordinate cube"},{"id":"Geometry/convex-geometry/hanner-polytopes/dimension-pos","kind":"node","lean":"MetaMathlibExt.IsHannerPolytope.dimension_pos","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#dimension-pos","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"If $K \\subseteq \\mathbb{R}^n$ is a Hanner polytope, then $0 < n$.","title":"Hanner polytopes live in positive dimension"},{"id":"Geometry/convex-geometry/hanner-polytopes/neg-eq-self","kind":"node","lean":"MetaMathlibExt.IsHannerPolytope.neg_eq_self","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#neg-eq-self","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"Every Hanner polytope $K$ is centrally symmetric about the origin: $-K = K$.","title":"Hanner polytopes are centrally symmetric"},{"id":"Geometry/convex-geometry/hanner-polytopes/neg-mem-iff","kind":"node","lean":"MetaMathlibExt.IsHannerPolytope.neg_mem_iff","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#neg-mem-iff","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For every Hanner polytope $K \\subseteq \\mathbb{R}^n$ and every $x \\in \\mathbb{R}^n$, $-x \\in K$ if and only if $x \\in K$.","title":"Negation preserves membership in a Hanner polytope"},{"id":"Geometry/convex-geometry/hanner-polytopes/convex","kind":"node","lean":"MetaMathlibExt.IsHannerPolytope.convex","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#convex","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"Every Hanner polytope $K \\subseteq \\mathbb{R}^n$ is convex.","title":"Hanner polytopes are convex"},{"id":"Geometry/convex-geometry/hanner-polytopes/zero-mem","kind":"node","lean":"MetaMathlibExt.IsHannerPolytope.zero_mem","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#zero-mem","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"The origin belongs to every Hanner polytope.","title":"Hanner polytopes contain the origin"},{"id":"Geometry/convex-geometry/hanner-polytopes/is-compact","kind":"node","lean":"MetaMathlibExt.IsHannerPolytope.isCompact","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#is-compact","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"Every Hanner polytope is compact.","title":"Hanner polytopes are compact"},{"id":"Geometry/convex-geometry/hanner-polytopes/zero-mem-interior","kind":"node","lean":"MetaMathlibExt.IsHannerPolytope.zero_mem_interior","parent":14538,"result":true,"route":"../roadmap/Geometry/convex-geometry/hanner-polytopes/index.html#zero-mem-interior","scope":"Geometry/convex-geometry/hanner-polytopes","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every Hanner polytope $K \\subseteq \\mathbb{R}^n$, the origin is an interior point of $K$.","title":"Hanner polytopes contain the origin in their interior"},{"data_url":"scopes/Geometry/convex-geometry/hyperplane-slicing-problem.json","id":"Geometry/convex-geometry/hyperplane-slicing-problem","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/hyperplane-slicing-problem/index.html","scope":"Geometry/convex-geometry/hyperplane-slicing-problem","status":"can_prove","title":"Hyperplane slicing problem"},{"id":"Geometry/convex-geometry/hyperplane-slicing-problem/is-convex-body","kind":"node","lean":"MathlibExt.Geometry.HyperplaneSlicingUniversalWanted.IsConvexBody","parent":14557,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convex bodies in "],["math","\\(\\mathbb{R}^n\\)"]],"source":"Convex bodies in $\\mathbb{R}^n$"},"result":true,"route":"../roadmap/Geometry/convex-geometry/hyperplane-slicing-problem/index.html#is-convex-body","scope":"Geometry/convex-geometry/hyperplane-slicing-problem","status":"fully_proved","summary":"A set $K$ in the $n$-dimensional Euclidean space $\\mathbb{R}^n$ is a convex body when it is convex, compact and has nonempty interior.","title":"Convex bodies in $\\mathbb{R}^n$"},{"id":"Geometry/convex-geometry/hyperplane-slicing-problem/is-hyperplane","kind":"node","lean":"MathlibExt.Geometry.HyperplaneSlicingUniversalWanted.IsHyperplane","parent":14557,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hyperplanes in "],["math","\\(\\mathbb{R}^n\\)"]],"source":"Hyperplanes in $\\mathbb{R}^n$"},"result":true,"route":"../roadmap/Geometry/convex-geometry/hyperplane-slicing-problem/index.html#is-hyperplane","scope":"Geometry/convex-geometry/hyperplane-slicing-problem","status":"fully_proved","summary":"A set $H$ in the $n$-dimensional Euclidean space $\\mathbb{R}^n$ is a hyperplane when $H = \\{x \\in \\mathbb{R}^n : f(x) = t\\}$ for some nonzero linear functional $f \\colon \\mathbb{R}^n \\to \\mathbb{R}$ and some $t \\in \\mathbb{R}$.","title":"Hyperplanes in $\\mathbb{R}^n$"},{"id":"Geometry/convex-geometry/hyperplane-slicing-problem/conjecture","kind":"node","lean":"MathlibExt.Geometry.HyperplaneSlicingUniversalWanted.conjecture","parent":14557,"result":true,"route":"../roadmap/Geometry/convex-geometry/hyperplane-slicing-problem/index.html#conjecture","scope":"Geometry/convex-geometry/hyperplane-slicing-problem","status":"fully_proved","summary":"The statement that there is a constant $c > 0$ such that for every $n \\ge 1$ and every convex body $K \\subseteq \\mathbb{R}^n$ (a convex compact set with nonempty interior) of Lebesgue volume $1$ there exist a hyperplane $H \\subseteq \\mathbb{R}^n$ (a set $\\{x : f(x) = t\\}$ for a nonzero linear functional $f$ on $\\mathbb{R}^n$ and some $t \\in \\mathbb{R}$) and an isometry\u2026","title":"Hyperplane slicing conjecture"},{"id":"Geometry/convex-geometry/hyperplane-slicing-problem/conjecture-holds","kind":"node","lean":"MathlibExt.Geometry.HyperplaneSlicingUniversalWanted.conjecture_holds","parent":14557,"result":true,"route":"../roadmap/Geometry/convex-geometry/hyperplane-slicing-problem/index.html#conjecture-holds","scope":"Geometry/convex-geometry/hyperplane-slicing-problem","status":"can_prove","summary":"There is a constant $c > 0$ such that for every $n \\ge 1$ and every convex body $K \\subseteq \\mathbb{R}^n$ (a convex compact set with nonempty interior) of Lebesgue volume $1$ there exist a hyperplane $H \\subseteq \\mathbb{R}^n$ (a set $\\{x : f(x) = t\\}$ for a nonzero linear functional $f$ on $\\mathbb{R}^n$ and some $t \\in \\mathbb{R}$) and an isometry $\\varphi \\colon \\mathbb{R}^{n-1} \\to \\mathbb{R}^n$ with image $H$\u2026","title":"Hyperplane slicing theorem"},{"data_url":"scopes/Geometry/convex-geometry/linear-programming.json","id":"Geometry/convex-geometry/linear-programming","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/linear-programming/index.html","scope":"Geometry/convex-geometry/linear-programming","status":"can_prove","title":"Linear programming"},{"id":"Geometry/convex-geometry/linear-programming/strong-linear-programming-duality","kind":"node","lean":"MathlibExt.Analysis.Convex.LPDuality.strong_linear_programming_duality","parent":14562,"result":true,"route":"../roadmap/Geometry/convex-geometry/linear-programming/index.html#strong-linear-programming-duality","scope":"Geometry/convex-geometry/linear-programming","status":"fully_proved","summary":"Let $A$ be a real $m \\times n$ matrix, and let $b \\in \\mathbb{R}^m$ and $c \\in \\mathbb{R}^n$; inequalities between vectors are componentwise. Suppose some $x \\in \\mathbb{R}^n$ satisfies $x \\ge 0$ and $Ax \\le b$, and that $c \\cdot x$ is bounded above over all such $x$. Then there exist $x \\in \\mathbb{R}^n$ and $y \\in \\mathbb{R}^m$ with $x \\ge 0$, $Ax \\le b$, $y \\ge 0$, $A^{\\mathsf{T}} y \\ge c$ and\u2026","title":"Strong linear programming duality"},{"id":"Geometry/convex-geometry/linear-programming/linear-program","kind":"node","lean":"MathlibExt.Geometry.LinearProgrammingWanted.ConvexGeometry.LinearProgram","parent":14562,"result":true,"route":"../roadmap/Geometry/convex-geometry/linear-programming/index.html#linear-program","scope":"Geometry/convex-geometry/linear-programming","status":"fully_proved","summary":"A linear program in standard inequality form with $m$ variables and $n$ constraints is given by an objective vector $c \\in \\mathbb{R}^m$, a right-hand side $b \\in \\mathbb{R}^n$ and an $n \\times m$ real constraint matrix $A$; it stands for the problem of maximising $c \\cdot x$ over $x \\in \\mathbb{R}^m$ subject to $A x \\le b$ and $x \\ge 0$.","title":"Linear program"},{"id":"Geometry/convex-geometry/linear-programming/feasible","kind":"node","lean":"MathlibExt.Geometry.LinearProgrammingWanted.ConvexGeometry.LinearProgram.feasible","parent":14562,"result":true,"route":"../roadmap/Geometry/convex-geometry/linear-programming/index.html#feasible","scope":"Geometry/convex-geometry/linear-programming","status":"fully_proved","summary":"The feasible region of a linear program with $m$ variables, constraint matrix $A$ and right-hand side $b$ is the set of vectors $x \\in \\mathbb{R}^m$ with $A x \\le b$ and $x \\ge 0$, both inequalities holding componentwise. It is a convex polyhedron in $\\mathbb{R}^m$.","title":"Feasible region of a linear program"},{"id":"Geometry/convex-geometry/linear-programming/objective","kind":"node","lean":"MathlibExt.Geometry.LinearProgrammingWanted.ConvexGeometry.LinearProgram.objective","parent":14562,"result":true,"route":"../roadmap/Geometry/convex-geometry/linear-programming/index.html#objective","scope":"Geometry/convex-geometry/linear-programming","status":"fully_proved","summary":"The objective function of a linear program with $m$ variables and objective vector $c \\in \\mathbb{R}^m$ is the map $f \\colon \\mathbb{R}^m \\to \\mathbb{R}$ given by $f(x) = c \\cdot x = \\sum_i c_i x_i$.","title":"Objective function of a linear program"},{"id":"Geometry/convex-geometry/linear-programming/lp-maximum-principle","kind":"node","lean":"MathlibExt.Geometry.LinearProgrammingWanted.ConvexGeometry.lp_maximum_principle","parent":14562,"result":true,"route":"../roadmap/Geometry/convex-geometry/linear-programming/index.html#lp-maximum-principle","scope":"Geometry/convex-geometry/linear-programming","status":"can_prove","summary":"Consider a linear program with $m$ variables and objective vector $c \\in \\mathbb{R}^m$, let $P \\subseteq \\mathbb{R}^m$ be its feasible region and $f(x) = c \\cdot x$ its objective function, and let $x \\in P$ be a local maximiser of $f$ on $P$, that is, $f(y) \\le f(x)$ for all $y \\in P$ in some neighbourhood of $x$. Then $x$ is a global maximiser of $f$ on $P$, and if $c \\ne 0$ then $x$ lies on the topological\u2026","title":"Maximum principle for linear programming"},{"id":"Geometry/convex-geometry/linear-programming/simplex-algorithm","kind":"node","lean":"MathlibExt.Geometry.LinearProgrammingWanted.ConvexGeometry.simplex_algorithm","parent":14562,"result":true,"route":"../roadmap/Geometry/convex-geometry/linear-programming/index.html#simplex-algorithm","scope":"Geometry/convex-geometry/linear-programming","status":"can_prove","summary":"Consider a linear program with $m$ variables and objective vector $c \\in \\mathbb{R}^m$, and let $P \\subseteq \\mathbb{R}^m$ be its feasible region and $f(x) = c \\cdot x$ its objective function. If $P$ is nonempty and bounded, then some $x \\in P$ maximises $f$ on $P$ and is an extreme point of $P$, that is, $x$ does not lie in the open segment between two distinct points of $P$.","title":"Optimal vertices of linear programs"},{"data_url":"scopes/Geometry/convex-geometry/straszewicz-s-theorem.json","id":"Geometry/convex-geometry/straszewicz-s-theorem","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","title":"Straszewicz's theorem"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/convex-hull-of-compact","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.IsCompact.convexHull_of_compact","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#convex-hull-of-compact","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space. If $K \\subseteq E$ is compact, then the convex hull of $K$ is compact.","title":"Convex hull of a compact set is compact"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-quot","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirQuot","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-quot","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space and $E^*$ its continuous dual. For $h : E^* \\to \\mathbb{R}$ and $x, v \\in E^*$, the difference quotients of $h$ at $x$ in direction $v$ form the set $\\{(h(x + t v) - h(x))/t : t > 0\\}$.","title":"Difference quotients"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-deriv","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirDeriv","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-deriv","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space with continuous dual $E^*$. For $h : E^* \\to \\mathbb{R}$ and $x, v \\in E^*$, the directional derivative of $h$ at $x$ in direction $v$ is the infimum of the difference quotients, $h'(x; v) = \\inf_{t > 0} \\frac{h(x + t v) - h(x)}{t}$, which is taken to be $0$ when these quotients are not bounded below.","title":"Directional derivative"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/convex-on-line","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.convexOn_line","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#convex-on-line","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space with continuous dual $E^*$, and let $h : E^* \\to \\mathbb{R}$ be convex. For all $x, v \\in E^*$, the function $s \\mapsto h(x + s v)$ is convex on $\\mathbb{R}$.","title":"Restriction of a convex function to a line"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-quot-bdd-below","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirQuot_bddBelow","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-quot-bdd-below","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space, $E^*$ its continuous dual, and $h : E^* \\to \\mathbb{R}$ a convex function. For all $x, v \\in E^*$, the set of difference quotients $\\{(h(x + t v) - h(x))/t : t > 0\\}$ is bounded below.","title":"Difference quotients of a convex function are bounded below"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-deriv-subadd","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirDeriv_subadd","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-deriv-subadd","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space with continuous dual $E^*$, let $h : E^* \\to \\mathbb{R}$ be convex, and write $h'(x; v)$ for the directional derivative of $h$ at $x$ in direction $v$. For all $x, v, w \\in E^*$, $h'(x; v + w) \\le h'(x; v) + h'(x; w)$.","title":"Subadditivity of the directional derivative"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-deriv-zero","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirDeriv_zero","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-deriv-zero","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space with continuous dual $E^*$ and let $h : E^* \\to \\mathbb{R}$ be any function. For every $x \\in E^*$, the directional derivative of $h$ at $x$ in direction $0$ is $0$.","title":"Directional derivative in direction zero"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-deriv-add-neg-nonneg","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirDeriv_add_neg_nonneg","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-deriv-add-neg-nonneg","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space with continuous dual $E^*$, let $h : E^* \\to \\mathbb{R}$ be convex, and write $h'(x; v)$ for the directional derivative of $h$ at $x$ in direction $v$. For all $x, v \\in E^*$, $h'(x; v) + h'(x; -v) \\ge 0$.","title":"Opposite directional derivatives have nonnegative sum"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/supp-func","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.suppFunc","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#supp-func","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"For a set $K$ in a real normed space $E$, the support function of $K$ sends a continuous linear functional $\\ell$ on $E$ to the supremum $\\sup_{x \\in K} \\ell(x)$ of the values of $\\ell$ on $K$.","title":"Support function"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/supp-func-attain","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.suppFunc_attain","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#supp-func-attain","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty compact subset of a real normed space $E$, and let $h_K(l) = \\sup_{x \\in K} l(x)$ be its support function on the continuous dual $E^*$. For every $l \\in E^*$ there is $x \\in K$ with $l(x) = h_K(l)$.","title":"The support function of a compact set is attained"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/supp-func-convex","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.suppFunc_convex","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#supp-func-convex","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty compact subset of a real normed space $E$. Its support function $h_K(l) = \\sup_{x \\in K} l(x)$ is a convex function on the whole continuous dual $E^*$.","title":"The support function of a compact set is convex"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/supp-func-lipschitz","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.suppFunc_lipschitz","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#supp-func-lipschitz","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty compact subset of a real normed space $E$, and let $h_K(l) = \\sup_{x \\in K} l(x)$ be its support function on the continuous dual $E^*$. There is a constant $C \\ge 0$ such that $h_K(l_1) - h_K(l_2) \\le C \\|l_1 - l_2\\|$ for all $l_1, l_2 \\in E^*$, where $\\|\\cdot\\|$ is the operator norm.","title":"The support function of a compact set is Lipschitz"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/supp-func-continuous","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.suppFunc_continuous","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#supp-func-continuous","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty compact subset of a real normed space $E$. Its support function $h_K(l) = \\sup_{x \\in K} l(x)$ is continuous on the continuous dual $E^*$ with the operator norm.","title":"The support function of a compact set is continuous"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/gateaux-pt","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.GateauxPt","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#gateaux-pt","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space, $E^*$ its continuous dual, and $h : E^* \\to \\mathbb{R}$. For $x, v \\in E^*$ let $h'(x; v)$ be the infimum of the difference quotients $(h(x + t v) - h(x))/t$ over $t > 0$, taken to be $0$ when they are unbounded below. A point $x \\in E^*$ is a Gateaux point of $h$ when $h'(x; v) + h'(x; -v) \\le 0$ for every $v \\in E^*$.","title":"Gateaux points"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-deriv-sublevel-open","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirDeriv_sublevel_open","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-deriv-sublevel-open","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space with continuous dual $E^*$, let $K \\subseteq E$ be a nonempty compact set with support function $h_K$, and write $h_K'(x; v)$ for the directional derivative of $h_K$ at $x$ in direction $v$. For every $v \\in E^*$ and every real $c$, the set of $x \\in E^*$ with $h_K'(x; v) + h_K'(x; -v) < c$ is open.","title":"Strict sublevel sets of the symmetrized directional derivative are open"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-deriv-continuous-in-v","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirDeriv_continuous_in_v","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-deriv-continuous-in-v","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space with continuous dual $E^*$, let $K \\subseteq E$ be a nonempty compact set with support function $h_K$, and write $h_K'(x; w)$ for the directional derivative of $h_K$ at $x$ in direction $w$. For every $x \\in E^*$, the map $w \\mapsto h_K'(x; w)$ is continuous on $E^*$.","title":"Continuity of the directional derivative of a support function in the direction"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dir-deriv-supp-func-bound","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dirDeriv_suppFunc_bound","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dir-deriv-supp-func-bound","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a real normed space with continuous dual $E^*$, let $K \\subseteq E$ be a nonempty compact set with support function $h_K$, and write $h_K'(x; w)$ for the directional derivative of $h_K$ at $x$ in direction $w$. If $C$ is a real number with $h_K(\\ell_1) - h_K(\\ell_2) \\le C \\|\\ell_1 - \\ell_2\\|$ for all $\\ell_1, \\ell_2 \\in E^*$, then $h_K'(x; w) \\le C \\|w\\|$ for all $x, w \\in E^*$.","title":"Lipschitz bound on directional derivatives of a support function"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/dense-gateaux","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.dense_gateaux","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#dense-gateaux","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space with continuous dual $E^*$, let $K \\subseteq E$ be a nonempty compact set with support function $h_K$, and write $h_K'(\\ell; v)$ for the directional derivative of $h_K$ at $\\ell$ in direction $v$. Then the Gateaux points of $h_K$, that is, the functionals $\\ell \\in E^*$ with $h_K'(\\ell; v) + h_K'(\\ell; -v) \\le 0$ for every $v \\in E^*$, form a dense subset of $E^*$.","title":"Mazur's theorem for support functions"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/maximizer-dir-deriv-ge","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.maximizer_dirDeriv_ge","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#maximizer-dir-deriv-ge","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty compact subset of a real normed space $E$, let $h_K(l) = \\sup_{y \\in K} l(y)$ be its support function on the continuous dual $E^*$, and write $h_K'(u; v)$ for the directional derivative of $h_K$ at $u$ in direction $v$. If $u \\in E^*$ and $z \\in K$ satisfy $u(z) = h_K(u)$, then $v(z) \\le h_K'(u; v)$ for every $v \\in E^*$.","title":"A maximizer bounds the directional derivative of the support function from below"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/gateaux-unique-maximizer","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.gateaux_unique_maximizer","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#gateaux-unique-maximizer","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty compact subset of a real normed space $E$, and let $h_K(l) = \\sup_{y \\in K} l(y)$ be its support function on the continuous dual $E^*$. If $l \\in E^*$ is a Gateaux point of $h_K$ and $z_1, z_2 \\in K$ satisfy $l(z_1) = h_K(l)$ and $l(z_2) = h_K(l)$, then $z_1 = z_2$.","title":"Maximizers at a Gateaux point of the support function are unique"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/unique-maximizer-exposed","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.unique_maximizer_exposed","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#unique-maximizer-exposed","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty compact set in a real normed space $E$, let $\\ell$ be a continuous linear functional on $E$, and let $h_K$ be the support function of $K$. If $z \\in K$ satisfies $\\ell(z) = h_K(\\ell)$ and every $y \\in K$ with $\\ell(y) = h_K(\\ell)$ equals $z$, then $z$ is an exposed point of $K$.","title":"A unique maximizer of a linear functional is an exposed point"},{"id":"Geometry/convex-geometry/straszewicz-s-theorem/straszewicz-theorem","kind":"node","lean":"MathlibExt.Analysis.Convex.Straszewicz.straszewicz_theorem","parent":14569,"result":true,"route":"../roadmap/Geometry/convex-geometry/straszewicz-s-theorem/index.html#straszewicz-theorem","scope":"Geometry/convex-geometry/straszewicz-s-theorem","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space and let $A \\subseteq E$ be a compact convex set. Then every extreme point of $A$ lies in the closure of the set of exposed points of $A$.","title":"Straszewicz's theorem"},{"data_url":"scopes/Geometry/convex-geometry/ulam-s-floating-body-problem.json","id":"Geometry/convex-geometry/ulam-s-floating-body-problem","kind":"scope","navigation":"module","parent":14499,"result":false,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"can_prove","title":"Ulam's floating body problem"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/s","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.S","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#s","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"For a natural number $d$, the unit sphere $S^{d-1} = \\{x \\in \\mathbb{R}^d : \\|x\\| = 1\\}$ in $\\mathbb{R}^d$, centered at the origin.","title":"Unit sphere"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/is-origin-symmetric","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.IsOriginSymmetric","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#is-origin-symmetric","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{R}^d$ is origin-symmetric if $K = -K$, where $-K = \\{-x : x \\in K\\}$.","title":"Origin-symmetric set"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/is-centrally-symmetric","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.IsCentrallySymmetric","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#is-centrally-symmetric","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{R}^d$ is centrally symmetric if there is a point $p \\in \\mathbb{R}^d$ such that the translate $K - p = \\{x - p : x \\in K\\}$ is origin-symmetric, that is, $K - p = -(K - p)$.","title":"Centrally symmetric set"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/center-of-mass","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.centerOfMass","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#center-of-mass","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"The center of mass of a set $K \\subseteq \\mathbb{R}^d$ is the average $\\frac{1}{\\operatorname{vol}(K)} \\int_K x \\, dx$, where $dx$ is Lebesgue measure on $\\mathbb{R}^d$. By convention it is $0$ when $\\operatorname{vol}(K)$ is $0$ or infinite, or when $x \\mapsto x$ is not integrable on $K$. For a compact convex body this is the usual center of mass.","title":"Center of mass"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/h-minus","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.H_minus","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#h-minus","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"For a unit vector $\\xi \\in \\mathbb{R}^d$ and $t \\in \\mathbb{R}$, the closed lower half-space is $H^-(\\xi) = \\{p \\in \\mathbb{R}^d : \\xi \\cdot p \\le t\\}$.","title":"Lower half-space"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/cutting-hyperplane-at-level","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.CuttingHyperplaneAtLevel","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#cutting-hyperplane-at-level","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}^d$, let $\\xi \\in \\mathbb{R}^d$ be a unit vector and let $\\delta \\in \\mathbb{R}$. A cutting hyperplane of $K$ in direction $\\xi$ at level $\\delta$ is a real number $t$, standing for the hyperplane $\\{p \\in \\mathbb{R}^d : \\xi \\cdot p = t\\}$, such that the volume of $K \\cap \\{p \\in \\mathbb{R}^d : \\xi \\cdot p \\le t\\}$ equals $\\delta$. Here a set of infinite volume is assigned volume $0$.","title":"Cutting hyperplane at a level"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/line-through-centers-of-mass","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.CuttingHyperplaneAtLevel.lineThroughCentersOfMass","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#line-through-centers-of-mass","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"Let $d$ be a natural number, let $K \\subseteq \\mathbb{R}^d$, let $\\xi$ be a unit vector in $\\mathbb{R}^d$, let $\\delta \\in \\mathbb{R}$, and let $H$ be a cutting hyperplane of $K$ in direction $\\xi$ at level $\\delta$, with translation $t$, so that $H^-(\\xi) = \\{x \\in \\mathbb{R}^d : \\langle x, \\xi \\rangle \\le t\\}$. Writing $c(A)$ for the center of mass of a set $A \\subseteq \\mathbb{R}^d$, the direction vector of the\u2026","title":"Direction of the line through the centers of mass"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/floats-in-equilibrium-in-direction-at-le","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.FloatsInEquilibriumInDirectionAtLevel","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#floats-in-equilibrium-in-direction-at-le","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}^d$, let $\\xi \\in \\mathbb{R}^d$ be a unit vector and let $\\delta \\in \\mathbb{R}$. We say that $K$ floats in equilibrium in direction $\\xi$ at level $\\delta$ if, for every cutting hyperplane $\\{p : \\xi \\cdot p = t\\}$ of $K$ in direction $\\xi$ at level $\\delta$, the center of mass of $K$ minus the center of mass of $K \\cap \\{p : \\xi \\cdot p \\le t\\}$ is a real multiple of $\\xi$ (possibly\u2026","title":"Floating in equilibrium in a direction"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/floats-in-equilibrium-in-every-orientati","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.FloatsInEquilibriumInEveryOrientationAtLevel","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#floats-in-equilibrium-in-every-orientati","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{R}^d$ floats in equilibrium in every orientation at level $\\delta \\in \\mathbb{R}$ if it floats in equilibrium in direction $\\xi$ at level $\\delta$ for every unit vector $\\xi \\in \\mathbb{R}^d$.","title":"Floating in equilibrium in every orientation"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/body-of-revolution","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.BodyOfRevolution","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#body-of-revolution","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"A body of revolution in $\\mathbb{R}^d$ is given by real numbers $R_1, R_2$ and a function $f : \\mathbb{R} \\to \\mathbb{R}$ that vanishes outside the interval $[-R_1, R_2]$, is infinitely differentiable on the open interval $(-R_1, R_2)$ and is concave on $[-R_1, R_2]$. The dimension $d$ is only a parameter; these data do not depend on it.","title":"Body of revolution"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/body","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.BodyOfRevolution.body","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#body","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"Let $d \\ge 1$ and let a body of revolution in $\\mathbb{R}^d$ be given by $R_1$, $R_2$ and $f$. Writing points of $\\mathbb{R}^d$ as $x = (x_0, \\dots, x_{d-1})$, its solid is the set of all $x$ with $x_0 \\in [-R_1, R_2]$ and $x_1^2 + \\dots + x_{d-1}^2 \\le f(x_0)^2$.","title":"Solid of revolution"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/is-strictly-convex-body","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.IsStrictlyConvexBody","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#is-strictly-convex-body","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"fully_proved","summary":"A set $K \\subseteq \\mathbb{R}^d$ is a strictly convex body if it is strictly convex (for distinct $x, y \\in K$, every point of the open segment between $x$ and $y$ lies in the interior of $K$), compact, and has nonempty interior.","title":"Strictly convex body"},{"id":"Geometry/convex-geometry/ulam-s-floating-body-problem/theorem-1","kind":"node","lean":"MathlibExt.Geometry.UlamWanted.Ulam.theorem_1","parent":14592,"result":true,"route":"../roadmap/Geometry/convex-geometry/ulam-s-floating-body-problem/index.html#theorem-1","scope":"Geometry/convex-geometry/ulam-s-floating-body-problem","status":"can_prove","summary":"Let $d \\ge 3$. There exists a body of revolution $K \\subset \\mathbb{R}^d$ that is a strictly convex body, is not centrally symmetric, and floats in equilibrium in every orientation at the level $\\operatorname{vol}(K)/2$.","title":"Ryabogin's negative answer to Ulam's floating body problem"},{"data_url":"scopes/Geometry/euclidean-geometry.json","id":"Geometry/euclidean-geometry","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/index.html","scope":"Geometry/euclidean-geometry","status":"can_prove","summary":"Classical synthetic and metric geometry of the Euclidean plane and space: triangles, circles, conics, polygons, isometries and the foundations of geometry.","title":"Euclidean geometry"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry.json","id":"Geometry/euclidean-geometry/triangle-geometry","kind":"scope","parent":14606,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry","status":"can_prove","summary":"Theorems about triangles, their centers, cevians, inscribed and circumscribed circles, and triangle inequalities.","title":"Triangle geometry"},{"id":"Geometry/euclidean-geometry/triangle-geometry/conway-circle-theorem","kind":"node","lean":"MetaMathlibExt.conway_circle_theorem","parent":14607,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/index.html#conway-circle-theorem","scope":"Geometry/euclidean-geometry/triangle-geometry","status":"fully_proved","summary":"Let $A$, $B$, $C$ be affinely independent points of the Euclidean plane, and write $a = d(B, C)$, $b = d(A, C)$, $c = d(A, B)$ for the side lengths, where $d$ is the Euclidean distance. Extend each side beyond each of its endpoints by the length of the side opposite that endpoint; this gives the six points","title":"Conway circle theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/euler-oi","kind":"node","lean":"MetaMathlibExt.euler_oi","parent":14607,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/index.html#euler-oi","scope":"Geometry/euclidean-geometry/triangle-geometry","status":"fully_proved","summary":"Let $P_0, P_1, P_2$ be affinely independent points of the Euclidean plane, $d$ the Euclidean distance and $\\langle x, y \\rangle$ the inner product. Let $O$ be a point and $R \\ge 0$ with $d(O, P_i) = R$ for each $i$, so that $O$ is the circumcenter and $R$ the circumradius. Let $I = w_0 P_0 + w_1 P_1 + w_2 P_2$ with $w_0, w_1, w_2 > 0$ and $w_0 + w_1 + w_2 = 1$, and let $r \\ge 0$. Suppose there are points $F_0$ on\u2026","title":"Euler's theorem on the circumcenter\u2013incenter distance"},{"id":"Geometry/euclidean-geometry/triangle-geometry/feuerbach-theorem","kind":"node","lean":"MetaMathlibExt.Feuerbach_theorem","parent":14607,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/index.html#feuerbach-theorem","scope":"Geometry/euclidean-geometry/triangle-geometry","status":"fully_proved","summary":"Let $T$ be a triangle in a two-dimensional Euclidean affine space, with incircle $\\omega$, nine-point circle $N$, and excircle $\\omega_i$ opposite its $i$-th vertex for each $i$. If $T$ is not equilateral, then there is a point $F$ at which $\\omega$ is internally tangent to $N$ (that is, $F$ lies on both circles and the center of $\\omega$ lies on the segment from the center of $N$ to $F$), and $F$ is the only\u2026","title":"Feuerbach's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/intercept-theorem","kind":"node","lean":"MetaMathlibExt.intercept_theorem","parent":14607,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/index.html#intercept-theorem","scope":"Geometry/euclidean-geometry/triangle-geometry","status":"fully_proved","summary":"Let $A, B, C, D, E$ be points of the Euclidean plane $\\mathbb{R}^2$, and write $XY$ for the distance between points $X$ and $Y$. Suppose that $C - A$ is not a real multiple of $B - A$; that $AD + DB = AB$ with $AD > 0$ and $DB > 0$, so $D$ lies strictly between $A$ and $B$; that $AE + EC = AC$ with $AE > 0$ and $EC > 0$, so $E$ lies strictly between $A$ and $C$; and that $E - D = r(C - B)$ for some real $r \\ne 0$,\u2026","title":"Intercept theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/marden","kind":"node","lean":"MetaMathlibExt.marden","parent":14607,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/index.html#marden","scope":"Geometry/euclidean-geometry/triangle-geometry","status":"fully_proved","summary":"Let $a, b, c$ be complex numbers such that $(b - a)/(c - a)$ has nonzero imaginary part, so that they are the vertices of a nondegenerate triangle, and let $p(z) = (z - a)(z - b)(z - c)$. Then there are complex numbers $f_1, f_2$ and a real number $L$ such that the roots of $p'$, counted with multiplicity, are exactly $f_1$ and $f_2$, and such that the function $\\sigma(z) = |z - f_1| + |z - f_2|$ satisfies\u2026","title":"Marden's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/routh","kind":"node","lean":"MetaMathlibExt.routh","parent":14607,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/index.html#routh","scope":"Geometry/euclidean-geometry/triangle-geometry","status":"fully_proved","summary":"Let $A, B, C$ be points of the Euclidean plane $\\mathbb{R}^2$ whose triangle (the convex hull of $\\{A, B, C\\}$) has nonzero area, and let $x, y, z$ be positive reals with $xyz \\ne 1$. Let $D = \\frac{x}{x+1} B + \\frac{1}{x+1} C$, $E = \\frac{1}{y+1} A + \\frac{y}{y+1} C$ and $F = \\frac{z}{z+1} A + \\frac{1}{z+1} B$, the points of the sides $BC$, $CA$, $AB$ with $DC/DB = x$, $EA/EC = y$ and $FB/FA = z$. If $P$ lies on\u2026","title":"Routh's theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem","status":"fully_proved","title":"Angle bisector theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/angle-bisector-theorem-general","kind":"node","lean":"MetaMathlibExt.angle_bisector_theorem_general","parent":14614,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/index.html#angle-bisector-theorem-general","scope":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $D$ be points of a Euclidean affine space (a metric affine space over a real inner product space) with $A$, $B$, $C$ affinely independent, so that $ABC$ is a nondegenerate triangle. If $D$ lies strictly between $B$ and $C$ and $\\angle BAD = \\angle DAC$ (unoriented angles), then $BD / DC = AB / AC$.","title":"Angle bisector theorem without distinctness hypotheses"},{"id":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/angle-bisector-theorem","kind":"node","lean":"MetaMathlibExt.angle_bisector_theorem","parent":14614,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/index.html#angle-bisector-theorem","scope":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $D$ be points of a Euclidean affine space (a metric affine space over a real inner product space) with $A$, $B$, $C$ affinely independent, so that $ABC$ is a nondegenerate triangle, and with $A \\ne B$, $A \\ne C$ and $A \\ne D$. If $D$ lies strictly between $B$ and $C$ and $\\angle BAD = \\angle DAC$ (unoriented angles), then $BD / DC = AB / AC$.","title":"Angle bisector theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/angle-bisector-converse-general","kind":"node","lean":"MetaMathlibExt.angle_bisector_converse_general","parent":14614,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/index.html#angle-bisector-converse-general","scope":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $D$ be points of a Euclidean affine space (a metric affine space over a real inner product space) with $A$, $B$, $C$ affinely independent, so that $ABC$ is a nondegenerate triangle. If $D$ lies on the closed segment $BC$ and $BD / DC = AB / AC$, then $\\angle BAD = \\angle DAC$ (unoriented angles), that is, $D$ lies on the internal bisector of the angle at $A$.","title":"Converse of the angle bisector theorem without distinctness hypotheses"},{"id":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/angle-bisector-converse","kind":"node","lean":"MetaMathlibExt.angle_bisector_converse","parent":14614,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem/index.html#angle-bisector-converse","scope":"Geometry/euclidean-geometry/triangle-geometry/angle-bisector-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $D$ be points of a Euclidean affine space (a metric affine space over a real inner product space) with $A$, $B$, $C$ affinely independent, so that $ABC$ is a nondegenerate triangle, and with $A \\ne B$, $A \\ne C$ and $A \\ne D$. If $D$ lies on the closed segment $BC$ and $BD / DC = AB / AC$, then $\\angle BAD = \\angle DAC$ (unoriented angles), that is, $D$ lies on the internal bisector of the angle\u2026","title":"Converse of the angle bisector theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem","status":"fully_proved","title":"Carnot's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem/carnot-inradius-circumradius-general","kind":"node","lean":"MetaMathlibExt.carnot_inradius_circumradius_general","parent":14619,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem/index.html#carnot-inradius-circumradius-general","scope":"Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem","status":"fully_proved","summary":"Let $A, B, C$ be non-collinear points of a Euclidean affine plane lying on the circle with center $O$ and radius $R > 0$. Take unit vectors $n_a$, $n_b$, $n_c$ normal to the sides and pointing toward the opposite vertices: $n_a \\perp C - B$ with $\\langle A - B, n_a \\rangle > 0$, $n_b \\perp A - C$ with $\\langle B - C, n_b \\rangle > 0$, and $n_c \\perp B - A$ with $\\langle C - A, n_c \\rangle > 0$. Let\u2026","title":"Carnot's theorem (inradius and circumradius), general form"},{"id":"Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem/carnot-inradius-circumradius","kind":"node","lean":"MetaMathlibExt.carnot_inradius_circumradius","parent":14619,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem/index.html#carnot-inradius-circumradius","scope":"Geometry/euclidean-geometry/triangle-geometry/carnot-s-theorem","status":"fully_proved","summary":"Let $A, B, C$ be non-collinear points of a Euclidean affine plane lying on the circle with center $O$ and radius $R > 0$, and suppose $O$ lies in the affine span of $A, B, C$. Take unit vectors $n_a$, $n_b$, $n_c$ normal to the sides and pointing toward the opposite vertices: $n_a \\perp C - B$ with $\\langle A - B, n_a \\rangle > 0$, $n_b \\perp A - C$ with $\\langle B - C, n_b \\rangle > 0$, and $n_c \\perp B - A$ with\u2026","title":"Carnot's theorem (inradius and circumradius)"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality.json","id":"Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality","status":"can_prove","title":"Erd\u0151s\u2013Mordell inequality"},{"id":"Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality/conjecture","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem898Wanted.conjecture","parent":14622,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality/index.html#conjecture","scope":"Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality","status":"fully_proved","summary":"Writing $XY$ for the distance between points $X$ and $Y$, this is the assertion that for all affinely independent points $A$, $B$, $C$ of the Euclidean plane, every point $P$ in the interior of the triangle $ABC$, and all points $M$, $N$, $L$ on the lines $BC$, $AB$, $CA$ such that the lines $PM$, $PN$, $PL$ are perpendicular to $BC$, $AB$, $CA$ respectively, we have $PA + PB + PC \\ge 2(PM + PN + PL)$.","title":"Statement of the Erd\u0151s\u2013Mordell inequality"},{"id":"Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality/conjecture-holds","kind":"node","lean":"MathlibExt.Geometry.ErdosProblem898Wanted.conjecture_holds","parent":14622,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality/index.html#conjecture-holds","scope":"Geometry/euclidean-geometry/triangle-geometry/erd-s-mordell-inequality","status":"can_prove","summary":"Let $A$, $B$, $C$ be affinely independent points of the Euclidean plane and $P$ a point in the interior of the triangle $ABC$. Let $M$, $N$, $L$ be points on the lines $BC$, $AB$, $CA$ such that $PM \\perp BC$, $PN \\perp AB$ and $PL \\perp CA$. Then, writing $XY$ for the distance between points $X$ and $Y$, we have $PA + PB + PC \\ge 2(PM + PN + PL)$.","title":"Erd\u0151s\u2013Mordell inequality"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/hinge-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/hinge-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/hinge-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/hinge-theorem","status":"fully_proved","title":"Hinge theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/hinge-theorem/hinge-theorem-of-inner-lt","kind":"node","lean":"MetaMathlibExt.hinge_theorem_of_inner_lt","parent":14625,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/hinge-theorem/index.html#hinge-theorem-of-inner-lt","scope":"Geometry/euclidean-geometry/triangle-geometry/hinge-theorem","status":"fully_proved","summary":"Let $V$ be a real inner product space and let $A, B, C, D, E, F \\in V$. If $\\|B - A\\| = \\|E - D\\|$, $\\|C - A\\| = \\|F - D\\|$ and $\\langle B - A, C - A \\rangle < \\langle E - D, F - D \\rangle$, then $\\|F - E\\| < \\|C - B\\|$.","title":"Hinge theorem in inner product spaces"},{"id":"Geometry/euclidean-geometry/triangle-geometry/hinge-theorem/hinge-theorem","kind":"node","lean":"MetaMathlibExt.hinge_theorem","parent":14625,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/hinge-theorem/index.html#hinge-theorem","scope":"Geometry/euclidean-geometry/triangle-geometry/hinge-theorem","status":"fully_proved","summary":"Let $A, B, C$ and $D, E, F$ be two triples of pairwise distinct points of the Euclidean plane $\\mathbb{R}^2$ forming nondegenerate triangles: $C - A$ is not a real multiple of $B - A$, and $F - D$ is not a real multiple of $E - D$. Write $XY$ for the distance between points $X$ and $Y$. If $AB = DE$, $AC = DF$ and $\\langle B - A, C - A \\rangle < \\langle E - D, F - D \\rangle$, that is, the angle at $A$ is larger\u2026","title":"Hinge theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem","status":"fully_proved","title":"Menelaus's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem/menelaus-general","kind":"node","lean":"MetaMathlibExt.menelaus_general","parent":14628,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem/index.html#menelaus-general","scope":"Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$ be points of the Euclidean plane $\\mathbb{R}^2$ with $A \\ne B$ and $C$ not on the line $AB$. Let $D$, $E$, $F$ be points and let $\\alpha, \\alpha', \\beta, \\beta', \\gamma, \\gamma'$ be real numbers with $F - A = \\alpha (B - A)$, $B - F = \\alpha' (B - A)$, $D - B = \\beta (C - B)$, $C - D = \\beta' (C - B)$, $E - C = \\gamma (A - C)$, $A - E = \\gamma' (A - C)$, and $\\alpha', \\beta', \\gamma' \\ne 0$. Then\u2026","title":"Menelaus's theorem, general form"},{"id":"Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem/menelaus","kind":"node","lean":"MetaMathlibExt.menelaus","parent":14628,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem/index.html#menelaus","scope":"Geometry/euclidean-geometry/triangle-geometry/menelaus-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$ be pairwise distinct points of the Euclidean plane $\\mathbb{R}^2$ with $C$ not on the line $AB$. Let $D$, $E$, $F$ be points on the lines $BC$, $CA$ and $AB$ respectively, with $D \\ne B$, $D \\ne C$, $E \\ne C$, $E \\ne A$, $F \\ne A$ and $F \\ne B$. Let $\\alpha, \\alpha', \\beta, \\beta', \\gamma, \\gamma'$ be real numbers with $F - A = \\alpha (B - A)$, $B - F = \\alpha' (B - A)$, $D - B = \\beta (C - B)$,\u2026","title":"Menelaus's theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem","status":"can_prove","title":"Morley's trisector theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/plane","kind":"node","lean":"MathlibExt.Geometry.MorleyWanted.Geometry.Morley.Plane","parent":14631,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/index.html#plane","scope":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem","status":"fully_proved","summary":"The Euclidean plane $\\mathbb{R}^2$ with its standard Euclidean distance.","title":"Euclidean plane for Morley's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/is-equilateral-triple","kind":"node","lean":"MathlibExt.Geometry.MorleyWanted.Geometry.Morley.IsEquilateralTriple","parent":14631,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/index.html#is-equilateral-triple","scope":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem","status":"fully_proved","summary":"Three points $P$, $Q$, $R$ of the Euclidean plane form an equilateral triple when, writing $d$ for the Euclidean distance, $d(P, Q) = d(Q, R)$ and $d(Q, R) = d(R, P)$. Coincident points are allowed, so $P = Q = R$ qualifies.","title":"Equilateral triple of points"},{"id":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/lies-in-triangle","kind":"node","lean":"MathlibExt.Geometry.MorleyWanted.Geometry.Morley.LiesInTriangle","parent":14631,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/index.html#lies-in-triangle","scope":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem","status":"fully_proved","summary":"A point $X$ of the Euclidean plane lies in the triangle with vertices $A$, $B$, $C$ when $X$ belongs to the convex hull of $\\{A, B, C\\}$, the closed triangle $ABC$ (a segment or a point when $A$, $B$, $C$ are collinear).","title":"Point in a closed triangle"},{"id":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/is-morley-configuration","kind":"node","lean":"MathlibExt.Geometry.MorleyWanted.Geometry.Morley.IsMorleyConfiguration","parent":14631,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/index.html#is-morley-configuration","scope":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem","status":"fully_proved","summary":"Points $A$, $B$, $C$, $P$, $Q$, $R$ of the Euclidean plane form a Morley configuration when $A$, $B$, $C$ are not collinear, each of $P$, $Q$, $R$ lies in the closed triangle $ABC$, and, writing $\\angle XYZ \\in [0, \\pi]$ for the unoriented angle at $Y$, $0 < \\angle CAR = \\angle RAP = \\angle PAB$ and $3 \\angle PAB = \\angle CAB$; $0 < \\angle ABP = \\angle PBQ = \\angle QBC$ and $3 \\angle QBC = \\angle ABC$;\u2026","title":"Morley configuration"},{"id":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/morley-theorem","kind":"node","lean":"MathlibExt.Geometry.MorleyWanted.Geometry.Morley.morley_theorem","parent":14631,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem/index.html#morley-theorem","scope":"Geometry/euclidean-geometry/triangle-geometry/morley-s-trisector-theorem","status":"can_prove","summary":"If points $A$, $B$, $C$, $P$, $Q$, $R$ of the Euclidean plane form a Morley configuration, so that $ABC$ is a nondegenerate triangle and $P$, $Q$, $R$ are the intersections of its adjacent angle trisectors, then $P$, $Q$, $R$ form an equilateral triple: writing $d$ for the Euclidean distance, $d(P, Q) = d(Q, R)$ and $d(Q, R) = d(R, P)$.","title":"Morley's trisector theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem","status":"fully_proved","title":"Napoleon's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem/napoleon-theorem-general","kind":"node","lean":"MetaMathlibExt.napoleon_theorem_general","parent":14637,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem/index.html#napoleon-theorem-general","scope":"Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $A_2$, $B_2$, $C_2$ be points of the Euclidean plane $\\mathbb{R}^2$ with $|BA_2| = |CA_2| = |BC|$, $|CB_2| = |AB_2| = |CA|$ and $|AC_2| = |BC_2| = |AB|$, such that $A_2$ and $A$ lie strictly on opposite sides of the line $BC$, $B_2$ and $B$ lie strictly on opposite sides of the line $CA$, and $C_2$ and $C$ lie strictly on opposite sides of the line $AB$. Let $G_1 = \\frac{1}{3}(B + C + A_2)$,\u2026","title":"Napoleon's theorem, general form"},{"id":"Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem/napoleon-theorem","kind":"node","lean":"MetaMathlibExt.napoleon_theorem","parent":14637,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem/index.html#napoleon-theorem","scope":"Geometry/euclidean-geometry/triangle-geometry/napoleon-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$ be pairwise distinct, non-collinear points of the Euclidean plane $\\mathbb{R}^2$, and let $A_2$, $B_2$, $C_2$ be points with $|BA_2| = |CA_2| = |BC|$, $|CB_2| = |AB_2| = |CA|$ and $|AC_2| = |BC_2| = |AB|$, such that $A_2$ and $A$ lie strictly on opposite sides of the line $BC$, $B_2$ and $B$ lie strictly on opposite sides of the line $CA$, and $C_2$ and $C$ lie strictly on opposite sides of the\u2026","title":"Napoleon's theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem","status":"fully_proved","title":"Pompeiu's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem/pompeiu-general","kind":"node","lean":"MetaMathlibExt.pompeiu_general","parent":14640,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem/index.html#pompeiu-general","scope":"Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem","status":"fully_proved","summary":"Write $XY$ for the Euclidean distance between points $X$ and $Y$ of the Euclidean plane $\\mathbb{R}^2$. Let $A$, $B$, $C$ be points of the plane with $AB = BC = CA$ and $AB > 0$, so that $ABC$ is an equilateral triangle, and let $P$ be any point of the plane. Then $PA \\le PB + PC$, $PB \\le PC + PA$ and $PC \\le PA + PB$; that is, $PA$, $PB$, $PC$ are the side lengths of a (possibly degenerate) triangle.","title":"Pompeiu's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem/pompeiu","kind":"node","lean":"MetaMathlibExt.pompeiu","parent":14640,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem/index.html#pompeiu","scope":"Geometry/euclidean-geometry/triangle-geometry/pompeiu-s-theorem","status":"fully_proved","summary":"Write $XY$ for the Euclidean distance between points $X$ and $Y$ of the Euclidean plane $\\mathbb{R}^2$. Let $A$, $B$, $C$ be points of the plane with $AB = BC = CA$ and $AB > 0$, so that $ABC$ is an equilateral triangle, and let $P$ be a point of the plane with $P \\ne A$, $P \\ne B$ and $P \\ne C$. Then $PA \\le PB + PC$, $PB \\le PC + PA$ and $PC \\le PA + PB$; that is, $PA$, $PB$, $PC$ are the side lengths of a\u2026","title":"Pompeiu's theorem for a point other than a vertex"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem","status":"fully_proved","title":"Van Schooten's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem/van-schooten-general","kind":"node","lean":"MetaMathlibExt.vanSchooten_general","parent":14643,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem/index.html#van-schooten-general","scope":"Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $P$ and $O$ be points of the Euclidean plane $\\mathbb{R}^2$ and let $r > 0$. Suppose that $AB = BC = CA$, that $A$, $B$, $C$ and $P$ lie on the circle of radius $r$ about $O$, and that $CP > AP$ and $CP > BP$. Then $PA + PB = PC$.","title":"Van Schooten's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem/van-schooten","kind":"node","lean":"MetaMathlibExt.vanSchooten","parent":14643,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem/index.html#van-schooten","scope":"Geometry/euclidean-geometry/triangle-geometry/van-schooten-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $P$ and $O$ be points of the Euclidean plane $\\mathbb{R}^2$ and let $r > 0$. Suppose that $AB = BC = CA$, that $A$, $B$, $C$ and $P$ lie on the circle of radius $r$ about $O$, that $P \\ne A$, $P \\ne B$ and $P \\ne C$, and that $CP > AP$ and $CP > BP$. Then $PA + PB = PC$.","title":"Van Schooten's theorem with distinctness hypotheses"},{"data_url":"scopes/Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem.json","id":"Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem","kind":"scope","navigation":"module","parent":14607,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem/index.html","scope":"Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem","status":"fully_proved","title":"Viviani's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem/viviani-general","kind":"node","lean":"MetaMathlibExt.viviani_general","parent":14646,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem/index.html#viviani-general","scope":"Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$ be points of the Euclidean plane $\\mathbb{R}^2$ with $AB = BC = CA$, and let $P$ be a point of the interior of the convex hull of $\\lbrace A, B, C \\rbrace$. Then the sum of the distances from $P$ to the lines $AB$, $BC$ and $CA$ equals the distance from $A$ to the line $BC$, the altitude of the triangle.","title":"Viviani's theorem"},{"id":"Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem/viviani","kind":"node","lean":"MetaMathlibExt.viviani","parent":14646,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem/index.html#viviani","scope":"Geometry/euclidean-geometry/triangle-geometry/viviani-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$ be points of the Euclidean plane $\\mathbb{R}^2$ with $AB = BC = CA$ and $AB > 0$, and let $P$ be a point of the interior of the convex hull of $\\lbrace A, B, C \\rbrace$. Then the sum of the distances from $P$ to the lines $AB$, $BC$ and $CA$ equals the distance from $A$ to the line $BC$, the altitude of the triangle.","title":"Viviani's theorem with a positive side length"},{"data_url":"scopes/Geometry/euclidean-geometry/circles-and-conics.json","id":"Geometry/euclidean-geometry/circles-and-conics","kind":"scope","parent":14606,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/index.html","scope":"Geometry/euclidean-geometry/circles-and-conics","status":"can_prove","summary":"Configurations of circles and conics, tangency, chains of circles, lunes, and mathematical billiards in elliptic and polygonal tables.","title":"Circles and conics"},{"id":"Geometry/euclidean-geometry/circles-and-conics/descartes-circle-theorem","kind":"node","lean":"MetaMathlibExt.descartes_circle_theorem","parent":14649,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/index.html#descartes-circle-theorem","scope":"Geometry/euclidean-geometry/circles-and-conics","status":"fully_proved","summary":"Let four circles in the Euclidean plane have centers $c_i$ and radii $r_i > 0$ for $i = 1, 2, 3, 4$, let $d$ be the Euclidean distance, and let $k_1, k_2, k_3, k_4$ be real numbers. Suppose that either (a) any two of the circles are externally tangent, $d(c_i, c_j) = r_i + r_j$ for $i \\ne j$, and $k_i = 1/r_i$ for every $i$; or (b) for some index $e$, each other circle is internally tangent to circle $e$, that is,\u2026","title":"Descartes' circle theorem"},{"id":"Geometry/euclidean-geometry/circles-and-conics/is-lens-curvature-recurrence","kind":"node","lean":"MetaMathlibExt.IsLensCurvatureRecurrence","parent":14649,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/index.html#is-lens-curvature-recurrence","scope":"Geometry/euclidean-geometry/circles-and-conics","status":"fully_proved","summary":"For real numbers $K$ and $A$, a real sequence $(b_n)_{n \\ge 0}$ satisfies the lens curvature recurrence with parameters $K$ and $A$ when $1 + K \\neq 0$ and, for every $n \\ge 0$,","title":"Lens curvature recurrence"},{"id":"Geometry/euclidean-geometry/circles-and-conics/miquel-theorem","kind":"node","lean":"MetaMathlibExt.miquel_theorem","parent":14649,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/index.html#miquel-theorem","scope":"Geometry/euclidean-geometry/circles-and-conics","status":"fully_proved","summary":"Let $A$, $B$, $C$ be non-collinear points of the Euclidean plane $\\mathbb{R}^2$, and let $D$, $E$, $F$ be points on the lines $BC$, $CA$, $AB$ respectively with $D \\ne B$, $D \\ne C$, $E \\ne C$, $E \\ne A$, $F \\ne A$ and $F \\ne B$. If $A$, $E$, $F$ lie on the circle with center $O_1$ and radius $r_1$, $B$, $F$, $D$ lie on the circle with center $O_2$ and radius $r_2$, and $C$, $D$, $E$ lie on the circle with center\u2026","title":"Miquel's theorem"},{"id":"Geometry/euclidean-geometry/circles-and-conics/pizza-theorem","kind":"node","lean":"MetaMathlibExt.pizza_theorem","parent":14649,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/index.html#pizza-theorem","scope":"Geometry/euclidean-geometry/circles-and-conics","status":"fully_proved","summary":"Let $n \\ge 2$ be a natural number, let $D$ be the open disk of radius $R > 0$ centered at a point $c$ of $\\mathbb{R}^2$, let $p$ be a point of $D$, and let $\\theta_0$ be a real number. Put $\\alpha = \\pi / (2n)$ and, for $k = 0, 1, \\dots, 4n - 1$, let $S_k$ be the set of points of $D$ of the form $p + r(\\cos\\theta, \\sin\\theta)$ with $r \\ge 0$ and $\\theta_0 + k\\alpha \\le \\theta < \\theta_0 + (k + 1)\\alpha$. Then the\u2026","title":"Pizza theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem.json","id":"Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem","kind":"scope","navigation":"module","parent":14649,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem/index.html","scope":"Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem","status":"fully_proved","title":"Casey's theorem"},{"id":"Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem/casey-theorem","kind":"node","lean":"MetaMathlibExt.casey_theorem","parent":14654,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem/index.html#casey-theorem","scope":"Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem","status":"fully_proved","summary":"In the Euclidean plane, let a circle have center $O$ and radius $R > 0$, and let four circles have centers $c_1, c_2, c_3, c_4$ and radii $r_1, r_2, r_3, r_4$ with $0 < r_i < R$ and $|c_i O| + r_i = R$ for each $i$, so that each is internally tangent to the outer circle. Assume that the four circles lie outside each other, $r_i + r_j < |c_i c_j|$ for all $i \\ne j$, and that they appear counterclockwise around $O$\u2026","title":"Casey's theorem"},{"id":"Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem/casey-theorem-cyclic-order-core","kind":"node","lean":"MetaMathlibExt.casey_theorem_cyclic_order_core","parent":14654,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem/index.html#casey-theorem-cyclic-order-core","scope":"Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem","status":"fully_proved","summary":"In the Euclidean plane, write $d(p, q)$ for the distance between points $p$ and $q$. Let $O, c_1, c_2, c_3, c_4$ be points and $R, r_1, r_2, r_3, r_4$ real numbers with $r_i < R$ and $d(c_i, O) + r_i = R$ for each $i$. Suppose $c_i - O = d(c_i, O)(\\cos\\theta_i, \\sin\\theta_i)$ for real numbers $\\theta_1 < \\theta_2 < \\theta_3 < \\theta_4 < \\theta_1 + 2\\pi$. For each pair $(i, j)$ among\u2026","title":"Core identity behind Casey's theorem in cyclic order"},{"id":"Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem/casey-theorem-cyclic-order","kind":"node","lean":"MetaMathlibExt.casey_theorem_cyclic_order","parent":14654,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem/index.html#casey-theorem-cyclic-order","scope":"Geometry/euclidean-geometry/circles-and-conics/casey-s-theorem","status":"fully_proved","summary":"In the Euclidean plane, write $d(p, q)$ for the distance between points $p$ and $q$. Let $O$ be a point and $R > 0$, and let $c_1, c_2, c_3, c_4$ be points and $r_1, r_2, r_3, r_4$ real numbers with $0 < r_i < R$ and $d(c_i, O) + r_i = R$ for each $i$, so that the circle with center $c_i$ and radius $r_i$ is internally tangent to the circle with center $O$ and radius $R$. Suppose any two of the four inner circles\u2026","title":"Casey's theorem for four circles in arbitrary cyclic order"},{"data_url":"scopes/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates.json","id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","kind":"scope","navigation":"module","parent":14649,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"can_prove","title":"Lune of Hippocrates"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/plane","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.Plane","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#plane","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"The coordinate plane is the set $\\mathbb{R} \\times \\mathbb{R}$ of ordered pairs $(x, y)$ of real numbers.","title":"Coordinate plane for Hippocrates' lunes"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/det2","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.det2","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#det2","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For vectors $u = (u_1, u_2)$ and $v = (v_1, v_2)$ of the coordinate plane, $\\det(u, v) = u_1 v_2 - u_2 v_1$, the signed area of the parallelogram spanned by $u$ and $v$.","title":"Planar determinant for Hippocrates' lunes"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/midpoint","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.midpoint","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#midpoint","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of the coordinate plane, their midpoint is $\\left(\\frac{p_1 + q_1}{2}, \\frac{p_2 + q_2}{2}\\right)$.","title":"Midpoint of two points in the plane"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/vec","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.vec","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#vec","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of $\\mathbb{R}^2$, the displacement vector from $p$ to $q$ is $(q_1 - p_1, q_2 - p_2)$.","title":"Displacement vector between two points of the plane"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/euclidean-dist-sq","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.euclideanDistSq","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#euclidean-dist-sq","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of the coordinate plane, the squared Euclidean distance between them is $(p_1 - q_1)^2 + (p_2 - q_2)^2$.","title":"Squared Euclidean distance in the plane"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/closed-half-disk","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.closedHalfDisk","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#closed-half-disk","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"Given points $p$ and $q$ of the coordinate plane and a property $S$ of real numbers, this is the set of points $x$ such that the squared Euclidean distance from $x$ to the midpoint of $p$ and $q$ is at most one quarter of the squared Euclidean distance from $p$ to $q$ (so $x$ lies in the closed disk with diameter $pq$), and $S$ holds of the planar determinant $\\det(q - p, x - p)$. For $p \\ne q$, taking $S(t)$ to be\u2026","title":"Closed half-disk on a diameter"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/a","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.A","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#a","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"The point $A = (0, 0)$, the origin of the coordinate plane; it is the right-angle vertex of the triangle in Hippocrates' theorem on lunes.","title":"Right-angle vertex of the lune triangle"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/b","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.B","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#b","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For a real number $a$, the point $B = (a, 0)$ on the $x$-axis of the coordinate plane; it is the endpoint of the horizontal leg of the right triangle in Hippocrates' theorem on lunes.","title":"Horizontal leg endpoint of the lune triangle"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/c","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.C","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#c","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For a real number $b$, the point $C = (0, b)$ on the $y$-axis of the coordinate plane; it is the endpoint of the vertical leg of the right triangle in Hippocrates' theorem on lunes.","title":"Vertical leg endpoint of the lune triangle"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/hypotenuse-semidisk","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.hypotenuseSemidisk","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#hypotenuse-semidisk","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For real numbers $a$ and $b$, let $B = (a, 0)$ and $C = (0, b)$. The half-disk on the hypotenuse is the set of points $x = (x_1, x_2)$ of the closed disk with diameter $BC$ satisfying $\\det(C - B, x - B) \\ge 0$, that is, $b x_1 + a x_2 \\le ab$. For $a, b > 0$ this is the half of the disk on the side of the line $BC$ containing the right-angle vertex $(0, 0)$.","title":"Half-disk on the hypotenuse"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/horizontal-lune","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.horizontalLune","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#horizontal-lune","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For real numbers $a$ and $b$, let $A = (0, 0)$, $B = (a, 0)$ and $C = (0, b)$. The lune on the horizontal leg is the set of points $x = (x_1, x_2)$ of the closed disk with diameter $AB$ satisfying $a x_2 \\le 0$ that do not lie in the half-disk on the hypotenuse, the set of points of the closed disk with diameter $BC$ satisfying $b x_1 + a x_2 \\le ab$. For $a, b > 0$ this is the set of points of the closed lower\u2026","title":"Lune on the horizontal leg"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/vertical-lune","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.verticalLune","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#vertical-lune","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For real numbers $a$ and $b$, the vertical lune is the set of points $(x, y) \\in \\mathbb{R}^2$ with $x^2 + (y - b/2)^2 \\le b^2/4$ and $bx \\le 0$ that do not satisfy both $(x - a/2)^2 + (y - b/2)^2 \\le (a^2 + b^2)/4$ and $bx + ay \\le ab$.","title":"Hippocrates lune on the vertical leg"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/right-triangle","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.rightTriangle","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#right-triangle","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"fully_proved","summary":"For real numbers $a$ and $b$, the right triangle $T(a, b)$ is the convex hull in the plane $\\mathbb{R}^2$ of the three points $(0, 0)$, $(a, 0)$ and $(0, b)$.","title":"Right triangle with legs on the axes"},{"id":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/hippocrates-lunes","kind":"node","lean":"MathlibExt.Geometry.HippocratesLunesWanted.Geometry.HippocratesLunes.hippocrates_lunes","parent":14658,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates/index.html#hippocrates-lunes","scope":"Geometry/euclidean-geometry/circles-and-conics/lune-of-hippocrates","status":"can_prove","summary":"Let $a, b > 0$ and consider the right triangle with vertices $A = (0, 0)$, $B = (a, 0)$ and $C = (0, b)$. The Lebesgue measure of the lune on the horizontal leg plus the Lebesgue measure of the lune on the vertical leg equals the Lebesgue measure of the triangle, that is, of the convex hull of $A$, $B$ and $C$.","title":"Hippocrates' theorem on lunes"},{"data_url":"scopes/Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards.json","id":"Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards","kind":"scope","navigation":"module","parent":14649,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards/index.html","scope":"Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards","status":"can_prove","title":"Mathematical billiards"},{"id":"Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards/conjecture","kind":"node","lean":"MathlibExt.Geometry.EllipticBilliard602Wanted.conjecture","parent":14673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elliptic billiard invariant "],["math","\\(k_{602}\\)"]],"source":"Elliptic billiard invariant $k_{602}$"},"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards/index.html#conjecture","scope":"Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards","status":"fully_proved","summary":"Write $|XY|$ for the distance between points $X$ and $Y$ of the Euclidean plane. For every natural number $N$, all points $O, f_1, f_2$ of the plane and all reals $a, a_c$ with $O = \\frac{1}{2}(f_1 + f_2)$, $|f_1 f_2| < a$, $|f_1 f_2| < a_c$ and $a_c < a$, there is a real number $C$ with the following property. Let $P_0, P_1, P_2, \\dots$ be points of the plane such that $|P_i f_1| + |P_i f_2| = a$ and\u2026","title":"Elliptic billiard invariant $k_{602}$"},{"id":"Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards/conjecture-holds","kind":"node","lean":"MathlibExt.Geometry.EllipticBilliard602Wanted.conjecture_holds","parent":14673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bialy\u2013Tabachnikov: "],["math","\\(k_{602}\\)"],["text"," is invariant"]],"source":"Bialy\u2013Tabachnikov: $k_{602}$ is invariant"},"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards/index.html#conjecture-holds","scope":"Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards","status":"can_prove","summary":"The elliptic billiard invariant $k_{602}$ conjecture holds. Write $|XY|$ for the distance between points $X$ and $Y$. For every natural number $N$, all points $O, f_1, f_2$ of the Euclidean plane and all reals $a, a_c$ with $O = \\frac{1}{2}(f_1 + f_2)$, $|f_1 f_2| < a$, $|f_1 f_2| < a_c$ and $a_c < a$, there is a real number $C$ with the following property. Let $P_0, P_1, P_2, \\dots$ be points of the plane such\u2026","title":"Bialy\u2013Tabachnikov: $k_{602}$ is invariant"},{"id":"Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards/billiard-unfolding-diagonal-lemma","kind":"node","lean":"MathlibExt.Geometry.ScottishBook147Wanted.billiardUnfoldingDiagonalLemma","parent":14673,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards/index.html#billiard-unfolding-diagonal-lemma","scope":"Geometry/euclidean-geometry/circles-and-conics/mathematical-billiards","status":"can_prove","summary":"For a natural number $j$, let $\\varepsilon_j = 1$ if $j$ is odd and $\\varepsilon_j = 0$ if $j$ is even. Let $a, b > 0$ be real numbers with $a n = b m$ for some positive integers $m, n$. Then there are a real number $t > 0$ and natural numbers $k > 0$, $l > 0$ and $N$ such that $t = k a = l b$, $N + 2 = k + l$, $k$ and $l$ are coprime, and the point $(\\varepsilon_k a, \\varepsilon_l b)$ is one of $(a, 0)$, $(0, b)$,\u2026","title":"Lattice step of the billiard unfolding in a rectangle"},{"data_url":"scopes/Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem.json","id":"Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem","kind":"scope","navigation":"module","parent":14649,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem/index.html","scope":"Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem","status":"fully_proved","title":"Monge's theorem"},{"id":"Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem/monge-three-circles-general","kind":"node","lean":"MetaMathlibExt.monge_three_circles_general","parent":14677,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem/index.html#monge-three-circles-general","scope":"Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem","status":"fully_proved","summary":"Let $V$ be a real vector space, let $c_1, c_2, c_3 \\in V$, and let $r_1, r_2, r_3$ be pairwise distinct real numbers. Then the three points $(r_1 - r_2)^{-1}(r_1 c_2 - r_2 c_1)$, $(r_2 - r_3)^{-1}(r_2 c_3 - r_3 c_2)$ and $(r_1 - r_3)^{-1}(r_1 c_3 - r_3 c_1)$ of $V$ are collinear.","title":"Monge's three-circle theorem, algebraic form"},{"id":"Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem/monge-three-circles","kind":"node","lean":"MetaMathlibExt.monge_three_circles","parent":14677,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem/index.html#monge-three-circles","scope":"Geometry/euclidean-geometry/circles-and-conics/monge-s-theorem","status":"fully_proved","summary":"Let $c_1, c_2, c_3$ be points of the Euclidean plane $\\mathbb{R}^2$ and let $r_1, r_2, r_3$ be pairwise distinct positive real numbers such that the closed disks of radius $r_i$ centered at $c_i$ are pairwise disjoint. Then the external homothety centers $(r_1 - r_2)^{-1}(r_1 c_2 - r_2 c_1)$, $(r_2 - r_3)^{-1}(r_2 c_3 - r_3 c_2)$ and $(r_1 - r_3)^{-1}(r_1 c_3 - r_3 c_1)$ of the three pairs of circles are collinear.","title":"Monge's three-circle theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems.json","id":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems","kind":"scope","navigation":"module","parent":14649,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/index.html","scope":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems","status":"can_prove","title":"Pascal's and Pappus's hexagon theorems"},{"id":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/pascal-theorem","kind":"node","lean":"MetaMathlibExt.pascal_theorem","parent":14680,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/index.html#pascal-theorem","scope":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems","status":"fully_proved","summary":"Let $k$ be a field of characteristic different from $2$ and let $P$ be an affine plane over $k$, that is, an affine space whose space of translations $V$ is a $k$-vector space of dimension $2$. Fix a point $O \\in P$ and a nondegenerate symmetric bilinear form $\\beta$ on $V \\times k$, and call the set of points $x \\in P$ with $\\beta((x - O, 1), (x - O, 1)) = 0$ the conic. Let $v_0, v_1, \\dots, v_5$ be six distinct\u2026","title":"Pascal's theorem in an affine plane"},{"id":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/same-point","kind":"node","lean":"MathlibExt.Geometry.PascalPappusWanted.Geometry.PascalPappus.SamePoint","parent":14680,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/index.html#same-point","scope":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems","status":"fully_proved","summary":"Two vectors $v, w \\in \\mathbb{R}^3$ represent the same point of the real projective plane when $w = c v$ for some real number $c \\ne 0$.","title":"Same projective point"},{"id":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/on-conic","kind":"node","lean":"MathlibExt.Geometry.PascalPappusWanted.Geometry.PascalPappus.OnConic","parent":14680,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/index.html#on-conic","scope":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems","status":"fully_proved","summary":"For a real $3 \\times 3$ matrix $M$, a vector $v \\in \\mathbb{R}^3$ lies on the conic $x^\\top M x = 0$ when $v^\\top M v = 0$.","title":"Points on a conic"},{"id":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/meet","kind":"node","lean":"MathlibExt.Geometry.PascalPappusWanted.Geometry.PascalPappus.meet","parent":14680,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/index.html#meet","scope":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems","status":"fully_proved","summary":"Writing $\\times$ for the cross product on $\\mathbb{R}^3$, the meet of the line through $a$ and $b$ with the line through $c$ and $d$, for $a, b, c, d \\in \\mathbb{R}^3$, is the vector $(a \\times b) \\times (c \\times d)$.","title":"Meet of two lines"},{"id":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/collinear3","kind":"node","lean":"MathlibExt.Geometry.PascalPappusWanted.Geometry.PascalPappus.Collinear3","parent":14680,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/index.html#collinear3","scope":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems","status":"fully_proved","summary":"Writing $\\cdot$ and $\\times$ for the dot and cross products on $\\mathbb{R}^3$, three vectors $p, q, r \\in \\mathbb{R}^3$ represent collinear points of the real projective plane when their triple product vanishes: $p \\cdot (q \\times r) = 0$.","title":"Collinear projective points"},{"id":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/pascal","kind":"node","lean":"MathlibExt.Geometry.PascalPappusWanted.Geometry.PascalPappus.pascal","parent":14680,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems/index.html#pascal","scope":"Geometry/euclidean-geometry/circles-and-conics/pascal-s-and-pappus-s-hexagon-theorems","status":"can_prove","summary":"Let $M$ be a real symmetric $3 \\times 3$ matrix with $\\det M \\ne 0$, and let $a_1, a_2, a_3, b_1, b_2, b_3 \\in \\mathbb{R}^3$ be nonzero vectors on the conic $x^\\top M x = 0$, no two of which represent the same projective point. Writing $\\cdot$ and $\\times$ for the dot and cross products on $\\mathbb{R}^3$, let $p = (a_1 \\times b_2) \\times (a_2 \\times b_1)$, $q = (a_1 \\times b_3) \\times (a_3 \\times b_1)$ and\u2026","title":"Pascal's theorem for real projective conics"},{"data_url":"scopes/Geometry/euclidean-geometry/polygons.json","id":"Geometry/euclidean-geometry/polygons","kind":"scope","parent":14606,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/polygons/index.html","scope":"Geometry/euclidean-geometry/polygons","status":"fully_proved","summary":"Theorems about quadrilaterals, cyclic polygons and squares erected on polygon sides.","title":"Polygons and quadrilaterals"},{"id":"Geometry/euclidean-geometry/polygons/brahmagupta-area","kind":"node","lean":"MetaMathlibExt.brahmagupta_area","parent":14687,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/polygons/index.html#brahmagupta-area","scope":"Geometry/euclidean-geometry/polygons","status":"fully_proved","summary":"Let $P, Q, R, S$ be points of the Euclidean plane with side lengths $|PQ| = a$, $|QR| = b$, $|RS| = c$, $|SP| = d$ and diagonal $|PR| = e$, and let $s = (a + b + c + d)/2$. Suppose that some point $O$ is equidistant from $P, Q, R, S$ (so the quadrilateral $PQRS$ is cyclic) and that some point $T$ satisfies $|PT| + |TR| = |PR|$ and $|QT| + |TS| = |QS|$ with $|PT|, |TR|, |QT|, |TS| > 0$ (so $T$ lies strictly inside\u2026","title":"Brahmagupta's formula"},{"id":"Geometry/euclidean-geometry/polygons/euler-quadrilateral","kind":"node","lean":"MetaMathlibExt.euler_quadrilateral","parent":14687,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/polygons/index.html#euler-quadrilateral","scope":"Geometry/euclidean-geometry/polygons","status":"fully_proved","summary":"For any four points $A, B, C, D$ of the Euclidean plane, write $XY$ for the distance between points $X$ and $Y$, and let $M$ and $N$ be the midpoints of $AC$ and $BD$. Then","title":"Euler's quadrilateral theorem"},{"id":"Geometry/euclidean-geometry/polygons/japanese-theorem-concyclic-quadrilateral","kind":"node","lean":"MetaMathlibExt.japanese_theorem_concyclic_quadrilaterals","parent":14687,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/polygons/index.html#japanese-theorem-concyclic-quadrilateral","scope":"Geometry/euclidean-geometry/polygons","status":"fully_proved","summary":"Let $A, B, C, D$ be pairwise distinct points of the Euclidean plane $\\mathbb{R}^2$ lying on a circle of positive radius, such that the diagonals $AC$ and $BD$, as closed segments, intersect; so $ABCD$ is a convex cyclic quadrilateral. Let $r_1, r_2, r_3, r_4 > 0$, and suppose that for each of the triangles $ABC$, $CDA$, $BCD$ and $DAB$ there is a closed disc of radius $r_1$, $r_2$, $r_3$ and $r_4$ respectively that\u2026","title":"Japanese theorem for concyclic quadrilaterals"},{"id":"Geometry/euclidean-geometry/polygons/varignon","kind":"node","lean":"MetaMathlibExt.varignon","parent":14687,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/polygons/index.html#varignon","scope":"Geometry/euclidean-geometry/polygons","status":"fully_proved","summary":"Let $A$, $B$, $C$, $D$ be any four points of the Euclidean plane $\\mathbb{R}^2$, degenerate and self-crossing configurations included, and let $P$, $Q$, $R$, $S$ be the midpoints of $AB$, $BC$, $CD$, $DA$, that is $2P = A + B$, $2Q = B + C$, $2R = C + D$ and $2S = D + A$. Write $[XYZ]$ for the oriented area of the triangle $XYZ$, positive when $X$, $Y$, $Z$ are listed counterclockwise. Then $P + R = Q + S$, so the\u2026","title":"Varignon's theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem.json","id":"Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem","kind":"scope","navigation":"module","parent":14687,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem/index.html","scope":"Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem","status":"fully_proved","title":"Finsler\u2013Hadwiger theorem"},{"id":"Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem/finsler-hadwiger","kind":"node","lean":"MetaMathlibExt.finsler_hadwiger","parent":14692,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem/index.html#finsler-hadwiger","scope":"Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem","status":"fully_proved","summary":"Let $A, B, C, D, E, F, G$ be points of the Euclidean plane $\\mathbb{R}^2$ with $B \\ne A$ and $E \\ne A$, and let $\\rho(x, y) = (-y, x)$ be the counterclockwise rotation by $90^\\circ$. Suppose that $ABCD$ and $AEFG$ are squares sharing the vertex $A$, in the sense that $D - A = \\rho(B - A)$, $C = B + D - A$, $G - A = \\rho(E - A)$ and $F = E + G - A$. Let $O_1 = (A + C)/2$ and $O_2 = (A + F)/2$ be their centers, and\u2026","title":"Finsler\u2013Hadwiger theorem"},{"id":"Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem/finsler-hadwiger-of-ne","kind":"node","lean":"MetaMathlibExt.finsler_hadwiger_of_ne","parent":14692,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem/index.html#finsler-hadwiger-of-ne","scope":"Geometry/euclidean-geometry/polygons/finsler-hadwiger-theorem","status":"fully_proved","summary":"Let $A, B, C, D, E, F, G$ be points of the Euclidean plane $\\mathbb{R}^2$ with $B \\ne E$, and let $\\rho(x, y) = (-y, x)$ be the counterclockwise rotation by $90^\\circ$. Suppose that $ABCD$ and $AEFG$ are squares sharing the vertex $A$, in the sense that $D - A = \\rho(B - A)$, $C = B + D - A$, $G - A = \\rho(E - A)$ and $F = E + G - A$. Let $O_1 = (A + C)/2$ and $O_2 = (A + F)/2$ be their centers, and let\u2026","title":"Finsler\u2013Hadwiger theorem for distinct squares"},{"data_url":"scopes/Geometry/euclidean-geometry/polygons/van-aubel-s-theorem.json","id":"Geometry/euclidean-geometry/polygons/van-aubel-s-theorem","kind":"scope","navigation":"module","parent":14687,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/polygons/van-aubel-s-theorem/index.html","scope":"Geometry/euclidean-geometry/polygons/van-aubel-s-theorem","status":"fully_proved","title":"Van Aubel's theorem"},{"id":"Geometry/euclidean-geometry/polygons/van-aubel-s-theorem/van-aubel","kind":"node","lean":"MetaMathlibExt.vanAubel","parent":14695,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/polygons/van-aubel-s-theorem/index.html#van-aubel","scope":"Geometry/euclidean-geometry/polygons/van-aubel-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $D$ be any four points of the Euclidean plane $\\mathbb{R}^2$, and let $\\rho(x, y) = (-y, x)$ be the counterclockwise rotation by $90^\\circ$. On each directed side $A \\to B$, $B \\to C$, $C \\to D$, $D \\to A$ erect the square lying to its left; its centre is the side midpoint plus half the side vector rotated by $\\rho$: $O_{AB} = \\tfrac12(A + B) + \\tfrac12\\rho(B - A)$,\u2026","title":"Van Aubel's theorem"},{"id":"Geometry/euclidean-geometry/polygons/van-aubel-s-theorem/van-aubel-right","kind":"node","lean":"MetaMathlibExt.vanAubel_right","parent":14695,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/polygons/van-aubel-s-theorem/index.html#van-aubel-right","scope":"Geometry/euclidean-geometry/polygons/van-aubel-s-theorem","status":"fully_proved","summary":"Let $A$, $B$, $C$, $D$ be any four points of the Euclidean plane $\\mathbb{R}^2$, and let $\\rho(x, y) = (y, -x)$ be the clockwise rotation by $90^\\circ$. On each directed side $A \\to B$, $B \\to C$, $C \\to D$, $D \\to A$ erect the square lying to its right; its centre is the side midpoint plus half the side vector rotated by $\\rho$: $O_{AB} = \\tfrac12(A + B) + \\tfrac12\\rho(B - A)$,\u2026","title":"Van Aubel's theorem with squares on the right"},{"data_url":"scopes/Geometry/euclidean-geometry/solid-and-spherical.json","id":"Geometry/euclidean-geometry/solid-and-spherical","kind":"scope","parent":14606,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/solid-and-spherical/index.html","scope":"Geometry/euclidean-geometry/solid-and-spherical","status":"fully_proved","summary":"Metric relations for tetrahedra and other solids, and trigonometry on the sphere.","title":"Solid and spherical geometry"},{"data_url":"scopes/Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem.json","id":"Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem","kind":"scope","navigation":"module","parent":14698,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem/index.html","scope":"Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem","status":"fully_proved","title":"De Gua's theorem"},{"id":"Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem/de-gua","kind":"node","lean":"MetaMathlibExt.deGua","parent":14699,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem/index.html#de-gua","scope":"Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem","status":"fully_proved","summary":"Let $u, v, w \\in \\mathbb{R}^3$ be pairwise orthogonal vectors, so that the tetrahedron with vertices $0, u, v, w$ has a right angle between any two of the edges at $0$. Write $|x|$ for the Euclidean norm and $\\langle x, y \\rangle$ for the inner product. Let $A_{uv}, A_{uw}, A_{vw}, A$ be real numbers with $A_{uv} = |u||v|/2$, $A_{uw} = |u||w|/2$, $A_{vw} = |v||w|/2$ (the areas of the three right-angled faces) and\u2026","title":"De Gua's theorem"},{"id":"Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem/de-gua-theorem","kind":"node","lean":"MetaMathlibExt.deGua_theorem","parent":14699,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem/index.html#de-gua-theorem","scope":"Geometry/euclidean-geometry/solid-and-spherical/de-gua-s-theorem","status":"fully_proved","summary":"Let $a, b, c > 0$ and let $p_0 = (a, 0, 0)$, $p_1 = (0, b, 0)$, $p_2 = (0, 0, c)$ in $\\mathbb{R}^3$, so that the tetrahedron with vertices $0, p_0, p_1, p_2$ has a right angle between any two of the edges at $0$, with legs of lengths $a$, $b$, $c$. Let $d_{01} = d(p_0, p_1)$, $d_{12} = d(p_1, p_2)$ and $d_{20} = d(p_2, p_0)$ be the Euclidean distances, let $s = (d_{01} + d_{12} + d_{20})/2$, and let\u2026","title":"De Gua's theorem with Heron's formula"},{"data_url":"scopes/Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines.json","id":"Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines","kind":"scope","navigation":"module","parent":14698,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines/index.html","scope":"Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines","status":"fully_proved","title":"Spherical law of cosines"},{"id":"Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines/spherical-law-of-cosines-general","kind":"node","lean":"MetaMathlibExt.spherical_law_of_cosines_general","parent":14702,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines/index.html#spherical-law-of-cosines-general","scope":"Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines","status":"fully_proved","summary":"Write $x \\cdot y = \\sum_{i=1}^{3} x_i y_i$ for the dot product and $\\lVert x \\rVert = \\sqrt{x \\cdot x}$ for the Euclidean norm on $\\mathbb{R}^3$. Let $u, v, w \\in \\mathbb{R}^3$ be unit vectors and let $a, b, c, C$ be real numbers with $0 < a < \\pi$ and $0 < b < \\pi$. Suppose that $v \\cdot w = \\cos a$, $u \\cdot w = \\cos b$, $u \\cdot v = \\cos c$ and","title":"Spherical law of cosines, general form"},{"id":"Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines/spherical-law-of-cosines","kind":"node","lean":"MetaMathlibExt.spherical_law_of_cosines","parent":14702,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines/index.html#spherical-law-of-cosines","scope":"Geometry/euclidean-geometry/solid-and-spherical/spherical-law-of-cosines","status":"fully_proved","summary":"Write $x \\cdot y = \\sum_{i=1}^{3} x_i y_i$ for the dot product and $\\lVert x \\rVert = \\sqrt{x \\cdot x}$ for the Euclidean norm on $\\mathbb{R}^3$. Let $u, v, w \\in \\mathbb{R}^3$ be unit vectors and let $a, b, c, C$ be real numbers, each strictly between $0$ and $\\pi$. Suppose that $v \\cdot w = \\cos a$, $u \\cdot w = \\cos b$, $u \\cdot v = \\cos c$ and","title":"Spherical law of cosines"},{"data_url":"scopes/Geometry/euclidean-geometry/isometries-and-symmetry.json","id":"Geometry/euclidean-geometry/isometries-and-symmetry","kind":"scope","parent":14606,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/index.html","scope":"Geometry/euclidean-geometry/isometries-and-symmetry","status":"can_prove","summary":"Distance-preserving maps of Euclidean space, their classification, and discrete symmetry groups such as wallpaper and space groups.","title":"Isometries and crystallographic groups"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/beckman-quarles","kind":"node","lean":"MathlibExt.Geometry.Euclidean.BeckmanQuarlesWanted.beckman_quarles","parent":14705,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/index.html#beckman-quarles","scope":"Geometry/euclidean-geometry/isometries-and-symmetry","status":"fully_proved","summary":"Let $n \\ge 2$ and let $f : \\mathbb{R}^n \\to \\mathbb{R}^n$ be a map of $n$-dimensional real Euclidean space such that $\\lVert f(x) - f(y) \\rVert = 1$ whenever $\\lVert x - y \\rVert = 1$. Then $f$ is an isometry, that is, $\\lVert f(x) - f(y) \\rVert = \\lVert x - y \\rVert$ for all $x, y \\in \\mathbb{R}^n$.","title":"Beckman\u2013Quarles theorem"},{"data_url":"scopes/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries.json","id":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","kind":"scope","navigation":"module","parent":14705,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/index.html","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","status":"fully_proved","title":"Classification of plane isometries"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/plane","kind":"node","lean":"Isometries.Plane","parent":14707,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/index.html#plane","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","status":"fully_proved","summary":"The Euclidean plane is the real vector space $\\mathbb{R}^2$ with its standard inner product $\\langle x, y \\rangle = x_1 y_1 + x_2 y_2$.","title":"The Euclidean plane"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/linear-det","kind":"node","lean":"Isometries.linearDet","parent":14707,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/index.html#linear-det","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","status":"fully_proved","summary":"The linear determinant of an isometry $f$ of the Euclidean plane is the determinant of the linear part of $f$.","title":"Linear determinant of a plane isometry"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/is-translation","kind":"node","lean":"Isometries.IsTranslation","parent":14707,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/index.html#is-translation","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","status":"fully_proved","summary":"An isometry $f$ of the Euclidean plane is a translation if there is a vector $v$ such that $f(x) = x + v$ for every point $x$.","title":"Translation"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/is-rotation","kind":"node","lean":"Isometries.IsRotation","parent":14707,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/index.html#is-rotation","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","status":"fully_proved","summary":"An isometry $f$ of the Euclidean plane is a rotation if it fixes some point $p$, the determinant of its linear part is $1$, and $f$ is not a translation.","title":"Rotation"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/is-reflection","kind":"node","lean":"Isometries.IsReflection","parent":14707,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/index.html#is-reflection","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","status":"fully_proved","summary":"An isometry $f$ of the Euclidean plane is a reflection if there is a line $L$ such that $f(x) = x$ for every point $x$ of $L$, and the determinant of the linear part of $f$ is $-1$.","title":"Reflection"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/is-glide-reflection","kind":"node","lean":"Isometries.IsGlideReflection","parent":14707,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/index.html#is-glide-reflection","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","status":"fully_proved","summary":"An isometry $f$ of the Euclidean plane is a glide reflection if the determinant of its linear part is $-1$ and there are a line $L$ and a nonzero vector $v$ parallel to $L$ such that $f(x) = x + v$ for every point $x$ of $L$.","title":"Glide reflection"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/plane-isometry-classification","kind":"node","lean":"Isometries.plane_isometry_classification","parent":14707,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries/index.html#plane-isometry-classification","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/classification-of-plane-isometries","status":"fully_proved","summary":"Every isometry of the Euclidean plane is a translation, a rotation, a reflection, or a glide reflection.","title":"Classification of plane isometries"},{"data_url":"scopes/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups.json","id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","kind":"scope","navigation":"module","parent":14705,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"can_prove","title":"Crystallographic groups"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/e","kind":"node","lean":"MathlibExt.Geometry.CrystallographicShared.E","parent":14715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euclidean space "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Euclidean space $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#e","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"fully_proved","summary":"For a natural number $d$, $E(d)$ is the $d$-dimensional Euclidean space $\\mathbb{R}^d$, with coordinates indexed by $0, 1, \\dots, d - 1$ and the standard inner product.","title":"Euclidean space $\\mathbb{R}^d$"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/e-2","kind":"node","lean":"MathlibExt.Geometry.SpaceGroupsWanted.Geometry.SpaceGroupsProblem.E","parent":14715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euclidean space "],["math","\\(\\mathbb{R}^d\\)"],["text"," for the space-group problem"]],"source":"Euclidean space $\\mathbb{R}^d$ for the space-group problem"},"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#e-2","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"fully_proved","summary":"For a natural number $d$, this is the $d$-dimensional real Euclidean space $\\mathbb{R}^d$ with its standard inner product.","title":"Euclidean space $\\mathbb{R}^d$ for the space-group problem"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/is-orientation-preserving-isom","kind":"node","lean":"MathlibExt.Geometry.SpaceGroupsWanted.Geometry.SpaceGroupsProblem.IsOrientationPreservingIsom","parent":14715,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#is-orientation-preserving-isom","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"fully_proved","summary":"An affine isometry of the Euclidean space $\\mathbb{R}^d$ is orientation-preserving if its linear part has positive determinant.","title":"Orientation-preserving isometry"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/aff-opequivalent","kind":"node","lean":"MathlibExt.Geometry.SpaceGroupsWanted.Geometry.SpaceGroupsProblem.AffOPEquivalent","parent":14715,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#aff-opequivalent","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"fully_proved","summary":"Two subgroups $G_1$ and $G_2$ of the group of affine isometries of the Euclidean space $\\mathbb{R}^d$ are orientation-preserving affinely equivalent if there is an affine bijection $\\varphi$ of $\\mathbb{R}^d$ whose linear part has positive determinant such that $\\{\\varphi g \\varphi^{-1} : g \\in G_1\\} = G_2$ as sets of affine maps of $\\mathbb{R}^d$.","title":"Orientation-preserving affine equivalence"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/crystallographic-count-op","kind":"node","lean":"MathlibExt.Geometry.SpaceGroupsWanted.Geometry.SpaceGroupsProblem.crystallographicCountOP","parent":14715,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#crystallographic-count-op","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"fully_proved","summary":"For a natural number $d$, this is the number, in $\\mathbb{N} \\cup \\{\\infty\\}$, of classes of $d$-dimensional crystallographic groups under orientation-preserving affine equivalence. Here a crystallographic group is a subgroup $G$ of the group of affine isometries of $\\mathbb{R}^d$ that is discrete (for every point $x$ and every $\\varepsilon > 0$ only finitely many $g \\in G$ satisfy\u2026","title":"Number of orientation-preserving affine classes of crystallographic groups"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/crystallographic-count-oponly","kind":"node","lean":"MathlibExt.Geometry.SpaceGroupsWanted.Geometry.SpaceGroupsProblem.crystallographicCountOPOnly","parent":14715,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#crystallographic-count-oponly","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"fully_proved","summary":"Number of orientation-preserving affine classes of orientation-preserving crystallographic groups. For a natural number $d$, this is the number, in $\\mathbb{N} \\cup \\{\\infty\\}$, of classes under orientation-preserving affine equivalence of those $d$-dimensional crystallographic groups all of whose elements are orientation-preserving isometries. Here a crystallographic group is a subgroup $G$ of the group of affine\u2026","title":"Number of orientation-preserving affine classes of orientation-preserving crystallographic groups"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/space-groups","kind":"node","lean":"MathlibExt.Geometry.SpaceGroupsWanted.Geometry.SpaceGroupsProblem.space_groups","parent":14715,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#space-groups","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"can_prove","summary":"In dimension $3$ there are exactly $230$ classes of crystallographic groups under orientation-preserving affine equivalence, exactly $219$ classes of crystallographic groups under affine equivalence (conjugation by an arbitrary affine bijection of $\\mathbb{R}^3$), and exactly $65$ classes under orientation-preserving affine equivalence of crystallographic groups all of whose elements are orientation-preserving\u2026","title":"The 230 space groups"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/e-3","kind":"node","lean":"MathlibExt.Geometry.WallpaperGroupsWanted.Geometry.WallpaperGroupsProblem.E","parent":14715,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#e-3","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"fully_proved","summary":"For a natural number $d$, the $d$-dimensional real Euclidean space $\\mathbb{R}^d$ with its standard inner product.","title":"Euclidean space for wallpaper groups"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/there-are-17-wallpaper-groups","kind":"node","lean":"MathlibExt.Geometry.WallpaperGroupsWanted.Geometry.WallpaperGroupsProblem.there_are_17_wallpaper_groups","parent":14715,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups/index.html#there-are-17-wallpaper-groups","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/crystallographic-groups","status":"can_prove","summary":"Call a group $G$ of isometries of the Euclidean plane $\\mathbb{R}^2$ crystallographic if it is discrete, meaning that for every point $x$ and every $\\varepsilon > 0$ only finitely many $g \\in G$ satisfy $\\operatorname{dist}(g x, x) \\le \\varepsilon$, and it contains translations by two linearly independent vectors. Call two such groups $G_1$ and $G_2$ affinely equivalent if some invertible affine map $\\varphi$ of\u2026","title":"There are exactly 17 wallpaper groups"},{"data_url":"scopes/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries.json","id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","kind":"scope","navigation":"module","parent":14705,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","title":"Discrete groups of plane isometries"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/e2","kind":"node","lean":"DiscreteGroups.E2","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#e2","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"$E_2$ is the Euclidean plane: the real vector space $\\mathbb{R}^2$ of pairs of real numbers with the standard Euclidean inner product and norm.","title":"The Euclidean plane for discrete isometry groups"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/m","kind":"node","lean":"DiscreteGroups.M\u2082","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#m","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"$M_2$ is the group, under composition, of affine isometries of the Euclidean plane onto itself.","title":"Isometry group of the Euclidean plane"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/linear-part","kind":"node","lean":"DiscreteGroups.linearPart","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#linear-part","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"The linear part of an affine isometry $f$ of the Euclidean plane is the linear isometry $A$ of the plane such that $f(x + v) = f(x) + A v$ for every point $x$ and every vector $v$.","title":"Linear part of a plane isometry"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/point-group-set","kind":"node","lean":"DiscreteGroups.pointGroupSet","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#point-group-set","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"The point group of a group $G$ of affine isometries of the Euclidean plane is the set of linear parts of the elements of $G$.","title":"Point group of a group of plane isometries"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/translation-lattice","kind":"node","lean":"DiscreteGroups.translationLattice","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#translation-lattice","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"The translation lattice of a subgroup $G$ of the group $M_2$ of affine isometries of the Euclidean plane $E_2$ is the set of vectors $v \\in E_2$ such that the translation $x \\mapsto v + x$ belongs to $G$.","title":"Translation lattice"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/translation-hom","kind":"node","lean":"DiscreteGroups.translationHom","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#translation-hom","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"The group homomorphism from the additive group of the Euclidean plane $\\mathbb{R}^2$ (written multiplicatively) to the group of affine isometries of the plane that sends a vector $v$ to translation by $v$, the isometry $x \\mapsto v + x$.","title":"Translation homomorphism"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/translation-lattice-subgroup","kind":"node","lean":"DiscreteGroups.translationLatticeSubgroup","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#translation-lattice-subgroup","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"For a group $G$ of affine isometries of the Euclidean plane $\\mathbb{R}^2$, the additive subgroup of $\\mathbb{R}^2$ consisting of the vectors $v$ such that translation by $v$ belongs to $G$; it is the preimage of $G$ under the translation homomorphism $v \\mapsto (x \\mapsto v + x)$.","title":"Translation lattice as an additive subgroup"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/mem-translation-lattice-subgroup","kind":"node","lean":"DiscreteGroups.mem_translationLatticeSubgroup","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#mem-translation-lattice-subgroup","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"For every group $G$ of affine isometries of the Euclidean plane and every vector $v$ of the plane, $v$ belongs to the translation lattice subgroup of $G$ if and only if $v$ belongs to the translation lattice of $G$.","title":"Membership in the translation lattice subgroup"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/coe-translation-lattice-subgroup","kind":"node","lean":"DiscreteGroups.coe_translationLatticeSubgroup","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#coe-translation-lattice-subgroup","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"For every group $G$ of affine isometries of the Euclidean plane, the underlying set of the translation lattice subgroup of $G$ is the translation lattice of $G$.","title":"Carrier of the translation lattice subgroup"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/translation-hom-injective","kind":"node","lean":"DiscreteGroups.translationHom_injective","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#translation-hom-injective","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"For vectors $v$ and $w$ of the Euclidean plane $\\mathbb{R}^2$, if translation by $v$ and translation by $w$ are the same affine isometry, then $v = w$. Equivalently, the translation homomorphism $v \\mapsto (x \\mapsto v + x)$ from the additive group of the plane to the group of affine isometries of the plane is injective.","title":"Translations are determined by their vectors"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/is-discrete","kind":"node","lean":"DiscreteGroups.IsDiscrete","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#is-discrete","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"A group $G$ of affine isometries of the Euclidean plane satisfies the separation criterion if there is $\\varepsilon > 0$ such that every element $f \\ne \\mathrm{id}$ of $G$ whose linear part is the identity satisfies $\\|f(0)\\| \\ge \\varepsilon$, and every element $f$ of $G$ whose linear part $A$ is not the identity satisfies $\\|A - I\\| \\ge \\varepsilon$, where $I$ is the identity map and the norm is the operator norm.","title":"Separation criterion for groups of plane isometries"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/translation-lattice-subgroup-discrete-to","kind":"node","lean":"DiscreteGroups.translationLatticeSubgroup_discreteTopology","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#translation-lattice-subgroup-discrete-to","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"Let $G$ be a group of affine isometries of the Euclidean plane $\\mathbb{R}^2$, and suppose there is $\\varepsilon > 0$ such that every non-identity translation in $G$ moves the origin by at least $\\varepsilon$, and every element of $G$ whose linear part $A$ is not the identity satisfies $\\|A - I\\| \\ge \\varepsilon$ in operator norm. Then the translation lattice of $G$, the additive subgroup of vectors $v$ such that\u2026","title":"The translation lattice of a discrete isometry group is discrete"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/conj-translation","kind":"node","lean":"DiscreteGroups.conj_translation","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#conj-translation","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"Let $g$ be an affine isometry of the Euclidean plane with linear part $A$, let $v$ be a vector, and write $\\tau_w$ for translation by a vector $w$. Then $g \\circ \\tau_v \\circ g^{-1} = \\tau_{A v}$.","title":"Conjugating a translation by a plane isometry"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/point-group-preserves-lattice","kind":"node","lean":"DiscreteGroups.point_group_preserves_lattice","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#point-group-preserves-lattice","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"Let $G$ be a group of affine isometries of the Euclidean plane. If $A$ belongs to the point group of $G$, that is, $A$ is the linear part of some element of $G$, and $v$ belongs to the translation lattice of $G$, then $A v$ belongs to the translation lattice of $G$.","title":"The point group preserves the translation lattice"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/discrete-subgroup-r2-classification","kind":"node","lean":"DiscreteGroups.discrete_subgroup_R2_classification","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#discrete-subgroup-r2-classification","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"Every discrete additive subgroup `G` of `\u211d\u00b2` is trivial, cyclic (generated by a single nonzero vector), or a rank-two lattice (generated by two `\u211d`-linearly independent vectors).","title":"Classification of discrete additive subgroups of the plane"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/translation-lattice-classification","kind":"node","lean":"DiscreteGroups.translation_lattice_classification","parent":14725,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries/index.html#translation-lattice-classification","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/discrete-groups-of-plane-isometries","status":"fully_proved","summary":"Let $G$ be a group of affine isometries of the Euclidean plane $\\mathbb{R}^2$, and suppose there is $\\varepsilon > 0$ such that every non-identity translation in $G$ moves the origin by at least $\\varepsilon$, and every element of $G$ whose linear part $A$ is not the identity satisfies $\\|A - I\\| \\ge \\varepsilon$ in operator norm. Let $L$ be the translation lattice of $G$, the additive subgroup of vectors $v$ such\u2026","title":"Classification of translation lattices"},{"data_url":"scopes/Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space.json","id":"Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space","kind":"scope","navigation":"module","parent":14705,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space/index.html","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space","status":"fully_proved","title":"Isometries of Euclidean space"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space/isometry-apply-eq-linear-apply-add-map-z","kind":"node","lean":"isometry_apply_eq_linear_apply_add_map_zero","parent":14742,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space/index.html#isometry-apply-eq-linear-apply-add-map-z","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space","status":"fully_proved","summary":"For every natural number $n$, every isometry $f$ of the Euclidean space $\\mathbb{R}^n$ is affine: there is a linear map $A : \\mathbb{R}^n \\to \\mathbb{R}^n$, the linear part of $f$, such that $f(x) = A x + f(0)$ for every $x \\in \\mathbb{R}^n$.","title":"Isometries of Euclidean space are affine"},{"id":"Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space/isometry-is-linear-map-of-map-zero","kind":"node","lean":"isometry_isLinearMap_of_map_zero","parent":14742,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space/index.html#isometry-is-linear-map-of-map-zero","scope":"Geometry/euclidean-geometry/isometries-and-symmetry/isometries-of-euclidean-space","status":"fully_proved","summary":"For every natural number $n$, if $f$ is an isometry of the Euclidean space $\\mathbb{R}^n$ with $f(0) = 0$, then $f$ is $\\mathbb{R}$-linear: $f(x + y) = f(x) + f(y)$ and $f(c x) = c f(x)$ for all $x, y \\in \\mathbb{R}^n$ and $c \\in \\mathbb{R}$.","title":"Isometries of Euclidean space fixing the origin are linear"},{"data_url":"scopes/Geometry/euclidean-geometry/foundations-and-constructions.json","id":"Geometry/euclidean-geometry/foundations-and-constructions","kind":"scope","parent":14606,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/foundations-and-constructions/index.html","scope":"Geometry/euclidean-geometry/foundations-and-constructions","status":"can_prove","summary":"Axiom systems for plane geometry and the limits of compass-and-straightedge constructions.","title":"Foundations and constructions"},{"id":"Geometry/euclidean-geometry/foundations-and-constructions/angle-trisection-impossible","kind":"node","lean":"MetaMathlibExt.angle_trisection_impossible","parent":14745,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/foundations-and-constructions/index.html#angle-trisection-impossible","scope":"Geometry/euclidean-geometry/foundations-and-constructions","status":"fully_proved","summary":"Let \"constructible\" be any property of real numbers such that a real number $x$ is constructible exactly when there are $n \\ge 0$ and subfields $K_0 \\subseteq K_1 \\subseteq \\dots \\subseteq K_n$ of $\\mathbb{R}$ with $K_0 = \\mathbb{Q}$ (the smallest subfield of $\\mathbb{R}$), $x \\in K_n$, and, for each $i < n$, some $b \\in K_{i+1}$ with $b^2 \\in K_i$ such that every element of $K_{i+1}$ equals $u + bv$ for some\u2026","title":"Impossibility of angle trisection"},{"data_url":"scopes/Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate.json","id":"Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate","kind":"scope","navigation":"module","parent":14745,"result":false,"route":"../roadmap/Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate/index.html","scope":"Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate","status":"can_prove","title":"Independence of the parallel postulate"},{"id":"Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate/tarski-absolute","kind":"node","lean":"MathlibExt.Geometry.ParallelPostulateWanted.Geometry.TarskiAbsolute","parent":14747,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate/index.html#tarski-absolute","scope":"Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate","status":"fully_proved","summary":"A model of Tarski's absolute plane geometry is a type $M$ of points with a betweenness relation $B$ and a congruence relation $C$, where $B(a, b, c)$ means that $b$ lies between $a$ and $c$ (allowing $b = a$ or $b = c$) and $C(a, b, c, d)$ means that the segment $ab$ is congruent to the segment $cd$, satisfying the following axioms, in which unbound variables range over all points of $M$ (axioms A1\u2013A9 and A11 of\u2026","title":"Tarski's absolute plane geometry"},{"id":"Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate/euclidean","kind":"node","lean":"MathlibExt.Geometry.ParallelPostulateWanted.Geometry.Euclidean","parent":14747,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate/index.html#euclidean","scope":"Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate","status":"fully_proved","summary":"For a model $M$ of Tarski's absolute plane geometry with betweenness relation $B$, where $B(a, b, c)$ means that $b$ lies between $a$ and $c$ (allowing $b = a$ or $b = c$), the Euclidean axiom A10 holds in $M$ when for all points $a, b, c, d, t$ of $M$ with $B(a, d, t)$, $B(b, d, c)$ and $a \\ne d$ there are points $x$ and $y$ of $M$ with $B(a, b, x)$, $B(a, c, y)$ and $B(x, t, y)$.","title":"Tarski's Euclidean axiom"},{"id":"Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate/parallel-postulate-independent","kind":"node","lean":"MathlibExt.Geometry.ParallelPostulateWanted.Geometry.parallel_postulate_independent","parent":14747,"result":true,"route":"../roadmap/Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate/index.html#parallel-postulate-independent","scope":"Geometry/euclidean-geometry/foundations-and-constructions/independence-of-the-parallel-postulate","status":"can_prove","summary":"Tarski's Euclidean axiom A10 is independent of Tarski's axioms of absolute plane geometry: there is a model of absolute plane geometry in which the Euclidean axiom holds, and there is a model of absolute plane geometry in which it fails.","title":"Independence of the parallel postulate"},{"data_url":"scopes/Geometry/riemannian-geometry.json","id":"Geometry/riemannian-geometry","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/index.html","scope":"Geometry/riemannian-geometry","status":"can_prove","summary":"Manifolds with smoothly varying inner products or norms on tangent spaces: connections, curvature, geodesics, differential operators and rigidity theorems.","title":"Riemannian and Finsler geometry"},{"id":"Geometry/riemannian-geometry/riemannian-metric-existence","kind":"node","lean":"MathlibExt.Geometry.Manifold.RiemannianMetricExistenceWanted.riemannianMetricExistence","parent":14751,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/index.html#riemannian-metric-existence","scope":"Geometry/riemannian-geometry","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space and let $M$ be a Hausdorff, second-countable $C^\\infty$ manifold modeled on $E$ (possibly with boundary or corners). Then $M$ admits a $C^\\infty$ Riemannian metric on its tangent bundle.","title":"Existence of Riemannian metrics"},{"id":"Geometry/riemannian-geometry/admits-positive-scalar-curvature-metric","kind":"node","lean":"Manifold.AdmitsPositiveScalarCurvatureMetric","parent":14751,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/index.html#admits-positive-scalar-curvature-metric","scope":"Geometry/riemannian-geometry","status":"fully_proved","summary":"A smooth manifold $M$ without boundary modeled on $\\mathbb{R}^n$ admits a metric of positive scalar curvature when some $C^\\infty$ Riemannian metric $g$ on its tangent bundle has scalar curvature, computed for $g$, strictly positive at every point of $M$.","title":"Manifolds admitting a metric of positive scalar curvature"},{"data_url":"scopes/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics.json","id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","title":"Conformal classes of Riemannian metrics"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/of-le","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.ofLE","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#of-le","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $E$ be a real vector bundle over a manifold $B$, let $m \\le n$ be smoothness exponents in $\\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $g$ be a $C^n$ Riemannian metric on $E$. Then $g$, with the same inner product on every fiber, is a $C^m$ Riemannian metric on $E$.","title":"Lowering the smoothness of a Riemannian metric"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/of-le-inner","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.ofLE_inner","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#of-le-inner","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $E$ be a real vector bundle over a manifold $B$, let $m \\le n$ be smoothness exponents in $\\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $g$ be a $C^n$ Riemannian metric on $E$. For every $b \\in B$, the inner product on the fiber $E_b$ of $g$ viewed as a $C^m$ Riemannian metric equals the inner product of $g$ on $E_b$.","title":"Lowering the smoothness keeps the inner product"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/conformal-scale","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.conformalScale","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#conformal-scale","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $E$ be a real vector bundle over a manifold $B$, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, let $g$ be a $C^n$ Riemannian metric on $E$, and let $u : B \\to \\mathbb{R}$ be a $C^n$ function with $u(b) > 0$ for every $b \\in B$. The conformal scaling of $g$ by $u$ is the $C^n$ Riemannian metric $u g$ on $E$ whose inner product on each fiber $E_b$ is $u(b)$ times the inner product of $g$ on $E_b$.","title":"Conformal scaling of a Riemannian metric"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/conformal-scale-inner-apply","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.conformalScale_inner_apply","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#conformal-scale-inner-apply","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $E$ be a real vector bundle over a manifold $B$, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, let $g$ be a $C^n$ Riemannian metric on $E$, with inner product $g_b$ on the fiber $E_b$, and let $u : B \\to \\mathbb{R}$ be a $C^n$ function with $u(b) > 0$ for every $b \\in B$. Then the conformal scaling $u g$ of $g$ by $u$ satisfies $(u g)_b(v, w) = u(b)\\, g_b(v, w)$ for every $b \\in B$ and all $v, w \\in E_b$.","title":"Inner product of a conformally scaled metric"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/pos-of-inner-eq-mul","kind":"node","lean":"Bundle.ContMDiffRiemannianMetric.pos_of_inner_eq_mul","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#pos-of-inner-eq-mul","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $E$ be a real vector bundle over a manifold $B$, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $g$ and $g'$ be $C^n$ Riemannian metrics on $E$, with inner products $g_b$ and $g'_b$ on the fiber $E_b$. Let $u : B \\to \\mathbb{R}$ and $b \\in B$. If $g'_b(v, w) = u(b)\\, g_b(v, w)$ for all $v, w \\in E_b$ and $E_b$ contains a nonzero vector, then $u(b) > 0$.","title":"A conformal factor at a point is positive"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/conformal-class","kind":"node","lean":"Manifold.conformalClass","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#conformal-class","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a $C^\\infty$ manifold modeled on a real normed space, and let $g$ be a $C^\\infty$ Riemannian metric on $M$, with inner product $g_x$ on the tangent space $T_x M$. The conformal class of $g$ is the set of $C^\\infty$ Riemannian metrics $g'$ on $M$ for which there is a $C^\\infty$ function $u : M \\to \\mathbb{R}$ with $u(x) > 0$ for all $x \\in M$ and $g'_x(v, w) = u(x)\\, g_x(v, w)$ for all $x \\in M$ and all\u2026","title":"Conformal class of a Riemannian metric"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/mem-conformal-class-iff-exp","kind":"node","lean":"Manifold.mem_conformalClass_iff_exp","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#mem-conformal-class-iff-exp","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a $C^\\infty$ manifold modeled on a real normed space, and let $g$ and $g'$ be $C^\\infty$ Riemannian metrics on $M$, with inner products $g_x$ and $g'_x$ on the tangent space $T_x M$. Then $g'$ lies in the conformal class of $g$ if and only if there is a $C^\\infty$ function $f : M \\to \\mathbb{R}$ with $g'_x(v, w) = e^{f(x)}\\, g_x(v, w)$ for all $x \\in M$ and all $v, w \\in T_x M$.","title":"Conformal metrics via an exponential factor"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/self-mem-conformal-class","kind":"node","lean":"Manifold.self_mem_conformalClass","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#self-mem-conformal-class","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Every smooth Riemannian metric $g$ on a smooth manifold $M$ lies in the conformal class of $g$.","title":"A metric lies in its own conformal class"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/conformal-scale-mem-conformal-class","kind":"node","lean":"Manifold.conformalScale_mem_conformalClass","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#conformal-scale-mem-conformal-class","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a $C^\\infty$ manifold modeled on a real normed space, let $g$ be a $C^\\infty$ Riemannian metric on $M$, and let $u : M \\to \\mathbb{R}$ be a $C^\\infty$ function with $u(x) > 0$ for all $x \\in M$. Then the conformal scaling $u g$ of $g$ by $u$ lies in the conformal class of $g$.","title":"Conformal scalings lie in the conformal class"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/mem-conformal-class-symm","kind":"node","lean":"Manifold.mem_conformalClass_symm","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#mem-conformal-class-symm","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a $C^\\infty$ manifold modeled on a real normed space, and let $g$ and $g'$ be $C^\\infty$ Riemannian metrics on $M$. If $g'$ lies in the conformal class of $g$, then $g$ lies in the conformal class of $g'$.","title":"Conformal equivalence is symmetric"},{"id":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/mem-conformal-class-trans","kind":"node","lean":"Manifold.mem_conformalClass_trans","parent":14754,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics/index.html#mem-conformal-class-trans","scope":"Geometry/riemannian-geometry/conformal-classes-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a $C^\\infty$ manifold modeled on a real normed space, and let $g$, $g'$ and $g''$ be $C^\\infty$ Riemannian metrics on $M$. If $g'$ lies in the conformal class of $g$ and $g''$ lies in the conformal class of $g'$, then $g''$ lies in the conformal class of $g$.","title":"Conformal equivalence is transitive"},{"data_url":"scopes/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds.json","id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","title":"Curvature of Riemannian manifolds"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/extend-curvature","kind":"node","lean":"CovariantDerivative.extendCurvature","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#extend-curvature","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $\\nabla$ be a covariant derivative on a vector bundle $V$ over a manifold $M$, let $x \\in M$, let $u, v \\in T_x M$ and let $\\sigma \\in V_x$. Let $X$ and $Y$ be the vector fields on $M$ whose coordinates in the preferred local trivialization of $TM$ at $x$ are constant, equal to those of $u$ and $v$, on the base set of that trivialization, with a fixed but unspecified value at each point outside it; let $S$ be\u2026","title":"Curvature expression of a covariant derivative on extensions"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/extend-curvature-antisymm","kind":"node","lean":"CovariantDerivative.extendCurvature_antisymm","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#extend-curvature-antisymm","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"For every covariant derivative $\\nabla$ on a vector bundle $V$ over a manifold $M$, every $x \\in M$, all $u, v \\in T_x M$ and every $\\sigma \\in V_x$, the curvature expression $R$ of $\\nabla$ on extensions satisfies $R(u, v)\\sigma = -R(v, u)\\sigma$.","title":"Antisymmetry of the curvature expression"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/extend-ricci","kind":"node","lean":"CovariantDerivative.extendRicci","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#extend-ricci","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $M$ be a manifold modelled on a finite-dimensional normed space over a nontrivially normed field $\\mathbb{K}$, let $\\nabla$ be a covariant derivative on the tangent bundle of $M$, let $R$ be its curvature expression on extensions, let $x \\in M$ and let $v, w \\in T_x M$. With $e_1, \\dots, e_n$ the fixed basis of $T_x M$ chosen for every finite-dimensional space and $e_1^*, \\dots, e_n^*$ its coordinate\u2026","title":"Ricci contraction of the curvature expression"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/riemann-curvature","kind":"node","lean":"Manifold.riemannCurvature","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#riemann-curvature","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"For a point $x$ of a $C^\\infty$ Riemannian manifold $(M, g)$ with Levi-Civita connection $\\nabla$, and tangent vectors $u, v, w \\in T_x M$,","title":"Riemann curvature endomorphism"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/riemann-curvature-antisymm","kind":"node","lean":"Manifold.riemannCurvature_antisymm","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#riemann-curvature-antisymm","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $(M, g)$ be a $C^\\infty$ Riemannian manifold. Its Riemann curvature endomorphism $R$ is antisymmetric in its first two arguments: $R(u, v) w = -R(v, u) w$ for every point $x \\in M$ and all tangent vectors $u, v, w \\in T_x M$.","title":"Antisymmetry of the Riemann curvature"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/sectional-curvature","kind":"node","lean":"Manifold.sectionalCurvature","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#sectional-curvature","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $(M, g)$ be a $C^\\infty$ Riemannian manifold modeled on a finite-dimensional real space (boundary and corners allowed), with Riemann curvature endomorphism $R$, and let $x \\in M$. The sectional curvature of tangent vectors $u, v \\in T_x M$ is","title":"Sectional curvature"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/sectional-curvature-smul-right","kind":"node","lean":"Manifold.sectionalCurvature_smul_right","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#sectional-curvature-smul-right","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $(M, g)$ be a $C^\\infty$ Riemannian manifold modeled on a finite-dimensional real space (boundary and corners allowed), and let $x \\in M$, $u \\in T_x M$ and $c \\in \\mathbb{R}$. Then the sectional curvature at $x$ satisfies $K(u, cu) = 0$: the denominator $\\|u\\|^2 \\|cu\\|^2 - \\langle u, cu \\rangle^2$ of its defining formula vanishes, and the quotient is then $0$ by definition.","title":"Sectional curvature vanishes on a vector and a multiple of it"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/sectional-curvature-self","kind":"node","lean":"Manifold.sectionalCurvature_self","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#sectional-curvature-self","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"For every point $x$ of a $C^\\infty$ Riemannian manifold $(M, g)$ and every tangent vector $u \\in T_x M$, the sectional curvature formula $K$ gives $K(u, u) = 0$: its denominator $\\|u\\|^2 \\|u\\|^2 - \\langle u, u \\rangle^2$ vanishes, where $\\langle \\cdot, \\cdot \\rangle$ and $\\|\\cdot\\|$ are the inner product and norm on $T_x M$ given by $g$, and a quotient by $0$ is taken to be $0$.","title":"Sectional curvature vanishes on equal vectors"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/ricci","kind":"node","lean":"Manifold.ricci","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#ricci","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"For a point $x$ of a $C^\\infty$ Riemannian manifold $(M, g)$ and tangent vectors $v, w \\in T_x M$, the Ricci curvature is $\\operatorname{Ric}_x(v, w) = \\sum_i e_i^*(R(e_i, v) w)$, where $R$ is the Riemann curvature endomorphism of the Levi-Civita connection of $g$, $(e_i)$ is a fixed basis of $T_x M$ and $(e_i^*)$ is its dual basis; the contraction itself uses no metric. Since $z \\mapsto R(z, v) w$ is linear, this\u2026","title":"Ricci curvature"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/scalar-curvature","kind":"node","lean":"Manifold.scalarCurvature","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#scalar-curvature","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"The scalar curvature of a $C^\\infty$ Riemannian manifold $(M, g)$ at a point $x$ is $\\sum_i \\operatorname{Ric}_x(e_i, e_i)$, the metric trace of the Ricci curvature $\\operatorname{Ric}$, where $(e_i)$ is a fixed orthonormal basis of $T_x M$ ([Gud, Def. 8.32]). The unit $n$-sphere has scalar curvature $n (n - 1)$.","title":"Scalar curvature"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/sectional-curvature-ge","kind":"node","lean":"Manifold.SectionalCurvatureGE","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#sectional-curvature-ge","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"A $C^\\infty$ Riemannian manifold $M$ has sectional curvature at least $k \\in \\mathbb{R}$ when $K(u, v) \\ge k$ for every point $x \\in M$ and all linearly independent tangent vectors $u, v \\in T_x M$, where $K(u, v)$ is the sectional curvature of the plane spanned by $u$ and $v$.","title":"Sectional curvature bounded below"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/sectional-curvature-le","kind":"node","lean":"Manifold.SectionalCurvatureLE","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#sectional-curvature-le","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"A $C^\\infty$ Riemannian manifold $M$ has sectional curvature at most $k \\in \\mathbb{R}$ when $K(u, v) \\le k$ for every point $x \\in M$ and all linearly independent tangent vectors $u, v \\in T_x M$, where $K(u, v)$ is the sectional curvature of the plane spanned by $u$ and $v$.","title":"Sectional curvature bounded above"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/ricci-ge","kind":"node","lean":"Manifold.RicciGE","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#ricci-ge","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"A $C^\\infty$ Riemannian manifold $(M, g)$ has Ricci curvature at least $k \\in \\mathbb{R}$, written $\\operatorname{Ric} \\ge k g$, when $\\operatorname{Ric}_x(v, v) \\ge k \\|v\\|^2$ for every point $x \\in M$ and every tangent vector $v \\in T_x M$, where $\\operatorname{Ric}$ is the Ricci curvature and $\\|v\\|$ is the norm of $v$ given by $g$.","title":"Ricci curvature bounded below"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/scalar-curvature-ge","kind":"node","lean":"Manifold.ScalarCurvatureGE","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#scalar-curvature-ge","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"A $C^\\infty$ Riemannian manifold $M$ has scalar curvature at least $k \\in \\mathbb{R}$ when its scalar curvature at every point $x \\in M$ is at least $k$.","title":"Scalar curvature bounded below"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/is-einstein","kind":"node","lean":"Manifold.IsEinstein","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#is-einstein","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"A $C^\\infty$ Riemannian manifold $(M, g)$ is Einstein when there is a real constant $\\lambda$ such that $\\operatorname{Ric}_x(v, w) = \\lambda \\, g_x(v, w)$ for every point $x \\in M$ and all tangent vectors $v, w \\in T_x M$, where $\\operatorname{Ric}$ is the Ricci curvature ([Lang], Einstein manifolds). Ricci-flat is the case $\\lambda = 0$.","title":"Einstein metric"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/extend-model-space","kind":"node","lean":"FiberBundle.extend_model_space","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#extend-model-space","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $F$ be a normed space over a nontrivially normed field, viewed as a manifold modelled on itself, so that each tangent space of $F$ is $F$. For every point $x \\in F$, tangent vector $u \\in T_xF$ and point $y \\in F$, the extension of $u$ to a vector field on $F$ (built from the trivialization of the tangent bundle at $x$) has value $u$ at $y$.","title":"Extending a tangent vector of a normed space gives a constant vector field"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/mdifferentiable-at-extend-model-space","kind":"node","lean":"FiberBundle.mdifferentiableAt_extend_model_space","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#mdifferentiable-at-extend-model-space","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $F$ be a normed space over a nontrivially normed field, viewed as a manifold modelled on itself. For every point $x \\in F$, tangent vector $u \\in T_xF$ and point $y \\in F$, the extension of $u$ to a vector field on $F$ (built from the trivialization of the tangent bundle at $x$), viewed as a map from $F$ to its tangent bundle, is differentiable at $y$, not only at the base point $x$.","title":"Extensions of tangent vectors of a normed space are differentiable"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/mlie-bracket-extend-model-space","kind":"node","lean":"VectorField.mlieBracket_extend_model_space","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#mlie-bracket-extend-model-space","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $F$ be a normed space over a nontrivially normed field, viewed as a manifold modelled on itself. For points $x, x' \\in F$ and tangent vectors $a \\in T_xF$ and $b \\in T_{x'}F$, let $X$ and $Y$ be the extensions of $a$ and $b$ to vector fields on $F$ (built from the trivializations of the tangent bundle at $x$ and at $x'$). Then the Lie bracket $[X, Y]$ vanishes at every point $y \\in F$.","title":"Lie brackets of extended tangent vectors of a normed space vanish"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/levi-civita-connection-extend-vector-spa","kind":"node","lean":"CovariantDerivative.leviCivitaConnection_extend_vectorSpace","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#levi-civita-connection-extend-vector-spa","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space with the Riemannian metric given by its inner product on every tangent space, and let $\\nabla$ be its Levi-Civita connection. For every point $x \\in F$ and tangent vector $w \\in T_xF$, let $W$ be the extension of $w$ to a vector field on $F$ (built from the trivialization of the tangent bundle at $x$; it is the constant vector field with value $w$). Then the\u2026","title":"The Levi-Civita connection of Euclidean space kills constant vector fields"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/riemann-curvature-vector-space","kind":"node","lean":"Manifold.riemannCurvature_vectorSpace","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#riemann-curvature-vector-space","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space with the Riemannian metric given by its inner product on every tangent space. For every point $x \\in F$ and all tangent vectors $u, v, w \\in T_xF$, the Riemann curvature endomorphism $R$ of $F$ at $x$ satisfies $R(u, v)w = 0$.","title":"Euclidean space is flat"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/sectional-curvature-vector-space","kind":"node","lean":"Manifold.sectionalCurvature_vectorSpace","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#sectional-curvature-vector-space","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space with the Riemannian metric given by its inner product on every tangent space. For every point $x \\in F$ and all tangent vectors $u, v \\in T_xF$, the sectional curvature $K(u, v)$ of $F$ at $x$ is $0$.","title":"Euclidean space has zero sectional curvature"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/ricci-vector-space","kind":"node","lean":"Manifold.ricci_vectorSpace","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#ricci-vector-space","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space with the Riemannian metric given by its inner product on every tangent space. For every point $x \\in F$ and all tangent vectors $v, w \\in T_xF$, the Ricci curvature $\\mathrm{Ric}(v, w)$ of $F$ at $x$ is $0$.","title":"Euclidean space is Ricci-flat"},{"id":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/scalar-curvature-vector-space","kind":"node","lean":"Manifold.scalarCurvature_vectorSpace","parent":14766,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/curvature-of-riemannian-manifolds/index.html#scalar-curvature-vector-space","scope":"Geometry/riemannian-geometry/curvature-of-riemannian-manifolds","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space with the Riemannian metric given by its inner product on every tangent space. For every point $x \\in F$, the scalar curvature of $F$ at $x$ is $0$.","title":"Euclidean space has zero scalar curvature"},{"data_url":"scopes/Geometry/riemannian-geometry/finsler-metrics.json","id":"Geometry/riemannian-geometry/finsler-metrics","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","title":"Finsler metrics"},{"id":"Geometry/riemannian-geometry/finsler-metrics/finsler-metric","kind":"node","lean":"Manifold.FinslerMetric","parent":14790,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html#finsler-metric","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","summary":"Let $M$ be a $C^\\infty$ manifold (possibly with boundary or corners) modelled on a finite-dimensional real normed space $E$, so that each tangent space $T_xM$ is $E$. A $C^\\infty$ Finsler metric on $M$ is a continuous function $F : TM \\to \\mathbb{R}$ that is $C^\\infty$ off the zero section, satisfies $F(x, v) > 0$ for every $x \\in M$ and every nonzero $v \\in T_xM$, is positively homogeneous of degree one on each\u2026","title":"Finsler metric"},{"id":"Geometry/riemannian-geometry/finsler-metrics/is-reversible","kind":"node","lean":"Manifold.FinslerMetric.IsReversible","parent":14790,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html#is-reversible","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","summary":"A Finsler metric $F$ on a manifold $M$ is reversible if $F(x, -v) = F(x, v)$ for every point $x \\in M$ and every tangent vector $v \\in T_xM$.","title":"Reversible Finsler metric"},{"id":"Geometry/riemannian-geometry/finsler-metrics/edist","kind":"node","lean":"Manifold.FinslerMetric.edist","parent":14790,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html#edist","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","summary":"Let $F$ be a Finsler metric on a manifold $M$. The forward Finsler distance from $x \\in M$ to $y \\in M$ is the value in $[0, \\infty]$","title":"Forward Finsler distance"},{"id":"Geometry/riemannian-geometry/finsler-metrics/is-geodesic-on-with-speed","kind":"node","lean":"Manifold.FinslerMetric.IsGeodesicOnWithSpeed","parent":14790,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html#is-geodesic-on-with-speed","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","summary":"Let $F$ be a Finsler metric on a manifold $M$ with forward distance $d_F$, let $\\gamma : \\mathbb{R} \\to M$ be a curve, let $s \\subseteq \\mathbb{R}$ be a set of times and let $c \\ge 0$ be a real number. Then $\\gamma$ is a geodesic of speed $c$ of $F$ on $s$ if for every $t \\in s$ there is $\\varepsilon > 0$ such that $d_F(\\gamma(t_1), \\gamma(t_2)) = c\\,(t_2 - t_1)$ for all $t_1, t_2 \\in s$ with $t_1 \\le t_2$,\u2026","title":"Finsler geodesic of given speed on a set of times"},{"id":"Geometry/riemannian-geometry/finsler-metrics/is-geodesic","kind":"node","lean":"Manifold.FinslerMetric.IsGeodesic","parent":14790,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html#is-geodesic","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","summary":"Let $F$ be a Finsler metric on a manifold $M$. A curve $\\gamma : \\mathbb{R} \\to M$ is a geodesic of $F$ if there is a real number $c \\ge 0$ such that $\\gamma$ is a geodesic of speed $c$ of $F$ on all of $\\mathbb{R}$.","title":"Finsler geodesic"},{"id":"Geometry/riemannian-geometry/finsler-metrics/is-closed-geodesic","kind":"node","lean":"Manifold.FinslerMetric.IsClosedGeodesic","parent":14790,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html#is-closed-geodesic","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","summary":"Let $F$ be a Finsler metric on a manifold $M$. A curve $\\gamma : \\mathbb{R} \\to M$ is a closed geodesic of $F$ if it is a geodesic of $F$, it is periodic with some period $T > 0$, that is $\\gamma(t + T) = \\gamma(t)$ for all real $t$, and it is not constant: $\\gamma(t_1) \\ne \\gamma(t_2)$ for some real $t_1, t_2$.","title":"Closed Finsler geodesic"},{"id":"Geometry/riemannian-geometry/finsler-metrics/euclidean","kind":"node","lean":"Manifold.FinslerMetric.euclidean","parent":14790,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html#euclidean","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","summary":"Let $V$ be a finite-dimensional real inner product space, viewed as a smooth manifold whose tangent spaces are $V$ itself. The Euclidean Finsler metric on $V$ is $F(x, v) = \\|v\\|$ for $x \\in V$ and $v \\in T_xV = V$.","title":"Euclidean Finsler metric"},{"id":"Geometry/riemannian-geometry/finsler-metrics/euclidean-is-reversible","kind":"node","lean":"Manifold.FinslerMetric.euclidean_isReversible","parent":14790,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/finsler-metrics/index.html#euclidean-is-reversible","scope":"Geometry/riemannian-geometry/finsler-metrics","status":"fully_proved","summary":"For every finite-dimensional real inner product space $V$, the Euclidean Finsler metric $F(x, v) = \\|v\\|$ on $V$ is reversible: $F(x, -v) = F(x, v)$ for all $x, v \\in V$.","title":"The Euclidean Finsler metric is reversible"},{"data_url":"scopes/Geometry/riemannian-geometry/hopf-rinow-theorem.json","id":"Geometry/riemannian-geometry/hopf-rinow-theorem","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/hopf-rinow-theorem/index.html","scope":"Geometry/riemannian-geometry/hopf-rinow-theorem","status":"can_prove","title":"Hopf\u2013Rinow theorem"},{"id":"Geometry/riemannian-geometry/hopf-rinow-theorem/is-geodesic","kind":"node","lean":"MathlibExt.Geometry.HopfRinowWanted.Geometry.IsGeodesic","parent":14799,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/hopf-rinow-theorem/index.html#is-geodesic","scope":"Geometry/riemannian-geometry/hopf-rinow-theorem","status":"fully_proved","summary":"Let $M$ be an extended metric space with distance $d$. A path $\\gamma : \\mathbb{R} \\to M$ is a constant-speed geodesic if there is a real number $c \\ge 0$ such that every $t_0 \\in \\mathbb{R}$ has a $\\delta > 0$ with $d(\\gamma(s), \\gamma(t)) = c\\,|t - s|$ for all real $s, t$ with $|s - t_0| < \\delta$ and $|t - t_0| < \\delta$.","title":"Constant-speed geodesic in a metric space"},{"id":"Geometry/riemannian-geometry/hopf-rinow-theorem/is-geodesically-complete","kind":"node","lean":"MathlibExt.Geometry.HopfRinowWanted.Geometry.IsGeodesicallyComplete","parent":14799,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/hopf-rinow-theorem/index.html#is-geodesically-complete","scope":"Geometry/riemannian-geometry/hopf-rinow-theorem","status":"fully_proved","summary":"An extended metric space $M$ with distance $d$ is geodesically complete if the following holds for all real numbers $a < b$ and every path $\\gamma : \\mathbb{R} \\to M$. Suppose there is a real $c \\ge 0$ such that every $t_0 \\in (a, b)$ has a $\\delta > 0$ with $d(\\gamma(s), \\gamma(t)) = c\\,|t - s|$ for all $s, t$ in the interval $(a, b) \\cap (t_0 - \\delta, t_0 + \\delta)$. Then there is a constant-speed geodesic\u2026","title":"Geodesically complete metric space"},{"id":"Geometry/riemannian-geometry/hopf-rinow-theorem/hopf-rinow","kind":"node","lean":"MathlibExt.Geometry.HopfRinowWanted.Geometry.hopf_rinow","parent":14799,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/hopf-rinow-theorem/index.html#hopf-rinow","scope":"Geometry/riemannian-geometry/hopf-rinow-theorem","status":"can_prove","summary":"Let $M$ be a connected, locally compact smooth manifold without boundary, modelled on a finite-dimensional real normed space, with a smooth Riemannian metric whose Riemannian distance is the (extended) distance of $M$. Then $M$ is geodesically complete if and only if $M$ is complete for its distance.","title":"Hopf\u2013Rinow theorem"},{"data_url":"scopes/Geometry/riemannian-geometry/levi-civita-connection.json","id":"Geometry/riemannian-geometry/levi-civita-connection","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/levi-civita-connection/index.html","scope":"Geometry/riemannian-geometry/levi-civita-connection","status":"can_prove","title":"Levi-Civita connection"},{"id":"Geometry/riemannian-geometry/levi-civita-connection/levi-civita-connection-vector-space-appl","kind":"node","lean":"CovariantDerivative.leviCivitaConnection_vectorSpace_apply","parent":14803,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/levi-civita-connection/index.html#levi-civita-connection-vector-space-appl","scope":"Geometry/riemannian-geometry/levi-civita-connection","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space with its standard Riemannian metric, let $V$ be a vector field on $F$ and $x \\in F$, and suppose $V$ is differentiable at $x$ as a section of the tangent bundle. Then for every tangent vector $v \\in T_xF$, the covariant derivative $\\nabla_v V$ of $V$ at $x$ for the Levi-Civita connection $\\nabla$ equals $DV(x)(v)$, the Fr\u00e9chet derivative of $V$ at $x$ applied\u2026","title":"The Levi-Civita connection of a vector space is the Fr\u00e9chet derivative"},{"id":"Geometry/riemannian-geometry/levi-civita-connection/is-metric-compatible","kind":"node","lean":"MathlibExt.Geometry.MetricCompatibilityShared.IsMetricCompatible","parent":14803,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/levi-civita-connection/index.html#is-metric-compatible","scope":"Geometry/riemannian-geometry/levi-civita-connection","status":"fully_proved","summary":"Let $M$ be a smooth manifold (possibly with corners) modelled on a finite-dimensional real normed space, with a smooth Riemannian metric $\\langle \\cdot, \\cdot \\rangle$ on its tangent bundle. A covariant derivative $\\nabla$ on the tangent bundle is compatible with the metric if for all smooth vector fields $Y, Z$ on $M$, every point $x \\in M$ and every tangent vector $v \\in T_x M$,","title":"Metric-compatible covariant derivative"},{"id":"Geometry/riemannian-geometry/levi-civita-connection/koszul-formula","kind":"node","lean":"MathlibExt.Geometry.KoszulFormulaWanted.Geometry.KoszulFormula.koszul_formula","parent":14803,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/levi-civita-connection/index.html#koszul-formula","scope":"Geometry/riemannian-geometry/levi-civita-connection","status":"can_prove","summary":"Let $M$ be a smooth manifold, modelled with corners on a finite-dimensional real normed space, whose tangent bundle carries a smooth Riemannian metric $\\langle \\cdot, \\cdot \\rangle$, and let $\\nabla$ be a smooth covariant derivative on the tangent bundle of $M$ that is torsion-free and compatible with the metric. Then for all smooth vector fields $X$, $Y$, $Z$ on $M$ and every point $x \\in M$, evaluating every term\u2026","title":"Koszul formula"},{"id":"Geometry/riemannian-geometry/levi-civita-connection/same-on-smooth","kind":"node","lean":"MathlibExt.Geometry.LeviCivitaWanted.Geometry.LeviCivita.SameOnSmooth","parent":14803,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/levi-civita-connection/index.html#same-on-smooth","scope":"Geometry/riemannian-geometry/levi-civita-connection","status":"fully_proved","summary":"Let $M$ be a smooth manifold, possibly with boundary or corners, modelled on a finite-dimensional real normed space, whose tangent bundle carries a smooth Riemannian metric. Two covariant derivatives $\\nabla$ and $\\nabla'$ on the tangent bundle of $M$ agree on smooth sections when $\\nabla_v Y = \\nabla'_v Y$ for every smooth vector field $Y$ on $M$, every point $x \\in M$ and every tangent vector $v$ at $x$.","title":"Agreement of covariant derivatives on smooth vector fields"},{"id":"Geometry/riemannian-geometry/levi-civita-connection/levi-civita-exists-unique","kind":"node","lean":"MathlibExt.Geometry.LeviCivitaWanted.Geometry.LeviCivita.levi_civita_exists_unique","parent":14803,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/levi-civita-connection/index.html#levi-civita-exists-unique","scope":"Geometry/riemannian-geometry/levi-civita-connection","status":"can_prove","summary":"Let $M$ be a Hausdorff smooth manifold, possibly with boundary or corners, modelled on a finite-dimensional real normed space, whose tangent bundle carries a smooth Riemannian metric. Then there is a smooth covariant derivative $\\nabla$ on the tangent bundle of $M$ that is torsion-free and compatible with the metric, and every smooth, torsion-free, metric-compatible covariant derivative $\\nabla'$ on the tangent\u2026","title":"Fundamental theorem of Riemannian geometry (Levi-Civita)"},{"data_url":"scopes/Geometry/riemannian-geometry/geodesic-flow.json","id":"Geometry/riemannian-geometry/geodesic-flow","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/geodesic-flow/index.html","scope":"Geometry/riemannian-geometry/geodesic-flow","status":"fully_proved","title":"Geodesic flow"},{"id":"Geometry/riemannian-geometry/geodesic-flow/unit-tangent-bundle","kind":"node","lean":"Manifold.unitTangentBundle","parent":14809,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/geodesic-flow/index.html#unit-tangent-bundle","scope":"Geometry/riemannian-geometry/geodesic-flow","status":"fully_proved","summary":"The unit tangent bundle `SM` of a Riemannian manifold `(M, g)`: the tangent vectors whose norm for the installed metric on their fiber is `1`, as a set of `TangentBundle I M` with the subspace topology. Source: Burns-Matveev, ETDS 41 (2021), \u00a71 (\"the unit tangent bundle ... is denoted by SM\").","title":"Unit tangent bundle of a Riemannian manifold"},{"id":"Geometry/riemannian-geometry/geodesic-flow/is-geodesic-velocity-curve","kind":"node","lean":"Manifold.IsGeodesicVelocityCurve","parent":14809,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/geodesic-flow/index.html#is-geodesic-velocity-curve","scope":"Geometry/riemannian-geometry/geodesic-flow","status":"fully_proved","summary":"Let $M$ be a $C^2$ manifold (boundary and corners allowed) modeled on a finite-dimensional real space, with a $C^1$ Riemannian metric $\\langle \\cdot, \\cdot \\rangle$ and Levi-Civita connection $\\nabla$. A curve $u : \\mathbb{R} \\to TM$, with base curve $\\gamma$ ($\\gamma(t)$ is the base point of $u(t)$), is a geodesic velocity curve when for every $t \\in \\mathbb{R}$: $\\gamma$ is differentiable at $t$ with velocity\u2026","title":"Geodesic velocity curves"},{"id":"Geometry/riemannian-geometry/geodesic-flow/is-geodesic-flow","kind":"node","lean":"Manifold.IsGeodesicFlow","parent":14809,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/geodesic-flow/index.html#is-geodesic-flow","scope":"Geometry/riemannian-geometry/geodesic-flow","status":"fully_proved","summary":"Let $M$ be a $C^2$ manifold (boundary and corners allowed) modeled on a finite-dimensional real space, with a $C^1$ Riemannian metric $g$. A family of maps $\\varphi_t : TM \\to TM$, $t \\in \\mathbb{R}$, is a geodesic flow of $g$ when $\\varphi_0(v) = v$ for every $v \\in TM$, $\\varphi_{t+s}(v) = \\varphi_t(\\varphi_s(v))$ for all $t, s \\in \\mathbb{R}$ and $v \\in TM$, and for every $v \\in TM$ the orbit\u2026","title":"Geodesic flow on the tangent bundle"},{"id":"Geometry/riemannian-geometry/geodesic-flow/is-unit-geodesic-flow","kind":"node","lean":"Manifold.IsUnitGeodesicFlow","parent":14809,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/geodesic-flow/index.html#is-unit-geodesic-flow","scope":"Geometry/riemannian-geometry/geodesic-flow","status":"fully_proved","summary":"Let $M$ be a $C^2$ manifold (boundary and corners allowed) modeled on a finite-dimensional real space, with a $C^1$ Riemannian metric $g$ and unit tangent bundle $SM$. A family of maps $\\varphi_t : SM \\to SM$, $t \\in \\mathbb{R}$, is a geodesic flow on $SM$ when $\\varphi_0(v) = v$ for every $v \\in SM$, $\\varphi_{t+s}(v) = \\varphi_t(\\varphi_s(v))$ for all $t, s \\in \\mathbb{R}$ and $v \\in SM$, and for every $v \\in SM$\u2026","title":"Geodesic flow on the unit tangent bundle"},{"id":"Geometry/riemannian-geometry/geodesic-flow/is-geodesic-velocity-curve-vector-space","kind":"node","lean":"Manifold.isGeodesicVelocityCurve_vectorSpace_iff","parent":14809,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/geodesic-flow/index.html#is-geodesic-velocity-curve-vector-space","scope":"Geometry/riemannian-geometry/geodesic-flow","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product, and identify each tangent space of $F$ with $F$, so that a point of the tangent bundle $TF$ is a pair $(x, v)$ with $x, v \\in F$. A curve $t \\mapsto u(t) = (x(t), v(t))$ in $TF$, defined for all real $t$, is the velocity curve of a geodesic if and only if, for every real $t$,\u2026","title":"Geodesic velocity curves in Euclidean space"},{"id":"Geometry/riemannian-geometry/geodesic-flow/is-geodesic-flow-vector-space-iff","kind":"node","lean":"Manifold.isGeodesicFlow_vectorSpace_iff","parent":14809,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/geodesic-flow/index.html#is-geodesic-flow-vector-space-iff","scope":"Geometry/riemannian-geometry/geodesic-flow","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product, and identify each tangent space of $F$ with $F$, so that a point of the tangent bundle $TF$ is a pair $(x, v)$ with $x, v \\in F$. A family of maps $\\varphi_t \\colon TF \\to TF$, $t \\in \\mathbb{R}$, is a geodesic flow on $TF$ if and only if $\\varphi_t(x, v) = (x + t v, v)$ for every\u2026","title":"The geodesic flow of Euclidean space is the straight-line flow"},{"id":"Geometry/riemannian-geometry/geodesic-flow/is-unit-geodesic-flow-vector-space-iff","kind":"node","lean":"Manifold.isUnitGeodesicFlow_vectorSpace_iff","parent":14809,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/geodesic-flow/index.html#is-unit-geodesic-flow-vector-space-iff","scope":"Geometry/riemannian-geometry/geodesic-flow","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product, and identify each tangent space of $F$ with $F$, so that a point of the tangent bundle is a pair $(x, v)$ with $x, v \\in F$, and the unit tangent bundle $SF$ consists of the pairs with $\\|v\\| = 1$. A family of maps $\\varphi_t \\colon SF \\to SF$, $t \\in \\mathbb{R}$, is a geodesic flow\u2026","title":"The unit geodesic flow of Euclidean space is the straight-line flow"},{"data_url":"scopes/Geometry/riemannian-geometry/killing-vector-fields.json","id":"Geometry/riemannian-geometry/killing-vector-fields","kind":"scope","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/index.html","scope":"Geometry/riemannian-geometry/killing-vector-fields","status":"fully_proved","title":"Killing vector fields"},{"data_url":"scopes/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold.json","id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","kind":"scope","navigation":"module","parent":14817,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","title":"Killing vector fields on a Riemannian manifold"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/is-killing-vector-field","kind":"node","lean":"Manifold.IsKillingVectorField","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#is-killing-vector-field","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"Let $M$ be a finite-dimensional smooth manifold, possibly with boundary or corners, with a smooth Riemannian metric $g = \\langle \\cdot, \\cdot \\rangle$ and its Levi-Civita connection $\\nabla$. A vector field $X$ on $M$ is a Killing vector field if $X$ is smooth and satisfies Killing's equation: for every point $x \\in M$ and all tangent vectors $u, v \\in T_x M$,","title":"Killing vector field"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/inner-levi-civita-connection-self","kind":"node","lean":"Manifold.IsKillingVectorField.inner_leviCivitaConnection_self","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#inner-levi-civita-connection-self","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"Let $X$ be a Killing vector field on a smooth Riemannian manifold $M$ with Levi-Civita connection $\\nabla$. Then $\\langle \\nabla_u X, u \\rangle = 0$ for every point $x \\in M$ and every tangent vector $u \\in T_x M$: a Killing field preserves lengths infinitesimally.","title":"Killing's equation on the diagonal"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/is-killing-vector-field-iff-inner-levi-c","kind":"node","lean":"Manifold.isKillingVectorField_iff_inner_leviCivitaConnection_self","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#is-killing-vector-field-iff-inner-levi-c","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"Let $M$ be a smooth Riemannian manifold with Levi-Civita connection $\\nabla$. A vector field $X$ on $M$ is a Killing vector field if and only if $X$ is smooth and $\\langle \\nabla_u X, u \\rangle = 0$ for every point $x \\in M$ and every tangent vector $u \\in T_x M$.","title":"Killing fields via the diagonal of Killing's equation"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/mvfderiv-inner-sub-inner-mlie-bracket","kind":"node","lean":"Manifold.mvfderiv_inner_sub_inner_mlieBracket","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#mvfderiv-inner-sub-inner-mlie-bracket","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"Let $M$ be a smooth manifold modeled on a finite-dimensional real normed space (possibly with corners), with a smooth Riemannian metric $g = \\langle \\cdot, \\cdot \\rangle$ and Levi-Civita connection $\\nabla$, and let $X$, $Y$, $Z$ be vector fields on $M$ that are differentiable at a point $x$. Write $[\\cdot, \\cdot]$ for the Lie bracket of vector fields and $X_x \\langle Y, Z \\rangle$ for the derivative at $x$ of the\u2026","title":"The Lie derivative of the metric via the Levi-Civita connection"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/is-killing-vector-field-iff-mvfderiv-inn","kind":"node","lean":"Manifold.isKillingVectorField_iff_mvfderiv_inner","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#is-killing-vector-field-iff-mvfderiv-inn","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"Let $M$ be a smooth Riemannian manifold with metric $g = \\langle \\cdot, \\cdot \\rangle$. A vector field $X$ on $M$ is a Killing vector field if and only if $X$ is smooth and, for every point $x \\in M$ and all vector fields $Y$ and $Z$ on $M$ that are differentiable at $x$,","title":"Killing fields are those with vanishing Lie derivative of the metric"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/is-killing-vector-field-zero","kind":"node","lean":"Manifold.isKillingVectorField_zero","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#is-killing-vector-field-zero","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"On a smooth Riemannian manifold, the zero vector field is a Killing vector field.","title":"The zero field is Killing"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/add","kind":"node","lean":"Manifold.IsKillingVectorField.add","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#add","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"On a smooth Riemannian manifold, the sum $X + Y$ of two Killing vector fields $X$ and $Y$ is a Killing vector field.","title":"Sum of Killing fields"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/const-smul","kind":"node","lean":"Manifold.IsKillingVectorField.const_smul","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#const-smul","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"On a smooth Riemannian manifold, if $X$ is a Killing vector field and $c$ is a real number, then $c X$ is a Killing vector field.","title":"Real multiples of Killing fields"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/killing-vector-fields","kind":"node","lean":"Manifold.killingVectorFields","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#killing-vector-fields","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"For a smooth Riemannian manifold $M$, the Killing vector fields on $M$ form a real linear subspace of the real vector space of all (not necessarily continuous) vector fields on $M$.","title":"Space of Killing vector fields"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/mem-killing-vector-fields","kind":"node","lean":"Manifold.mem_killingVectorFields","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#mem-killing-vector-fields","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"A vector field $X$ on a smooth Riemannian manifold $M$ lies in the space of Killing vector fields on $M$ if and only if $X$ is a Killing vector field.","title":"Membership in the space of Killing fields"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/ext","kind":"node","lean":"Manifold.killingVectorFields.ext","parent":14818,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold/index.html#ext","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-a-riemannian-manifold","status":"fully_proved","summary":"Two elements $a$ and $b$ of the space of Killing vector fields on a smooth Riemannian manifold $M$ that agree at every point $x \\in M$ are equal.","title":"Extensionality for Killing fields"},{"data_url":"scopes/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space.json","id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space","kind":"scope","navigation":"module","parent":14817,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space/index.html","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space","status":"fully_proved","title":"Killing vector fields on Euclidean space"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space/is-killing-vector-field-vector-space-iff","kind":"node","lean":"Manifold.isKillingVectorField_vectorSpace_iff","parent":14830,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space/index.html#is-killing-vector-field-vector-space-iff","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product. A vector field $V$ on $F$ is Killing if and only if $V$ is $C^\\infty$ and its derivative is skew-adjoint at every point: for all $x, u, w \\in F$, $\\langle DV(x)\\,u, w \\rangle + \\langle u, DV(x)\\,w \\rangle = 0$.","title":"Killing fields on Euclidean space"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space/is-killing-vector-field-const","kind":"node","lean":"Manifold.isKillingVectorField_const","parent":14830,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space/index.html#is-killing-vector-field-const","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product. For every $c \\in F$, the constant vector field $y \\mapsto c$ on $F$ is Killing.","title":"Translations are Killing"},{"id":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space/is-killing-vector-field-continuous-linea","kind":"node","lean":"Manifold.isKillingVectorField_continuousLinearMap_iff","parent":14830,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space/index.html#is-killing-vector-field-continuous-linea","scope":"Geometry/riemannian-geometry/killing-vector-fields/killing-vector-fields-on-euclidean-space","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product, and let $A \\colon F \\to F$ be a continuous linear map. The linear vector field $y \\mapsto A y$ is Killing if and only if $A$ is skew-adjoint: $\\langle A u, w \\rangle + \\langle u, A w \\rangle = 0$ for all $u, w \\in F$.","title":"Linear Killing fields on Euclidean space"},{"data_url":"scopes/Geometry/riemannian-geometry/mostow-rigidity-theorem.json","id":"Geometry/riemannian-geometry/mostow-rigidity-theorem","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/mostow-rigidity-theorem/index.html","scope":"Geometry/riemannian-geometry/mostow-rigidity-theorem","status":"can_prove","title":"Mostow rigidity theorem"},{"id":"Geometry/riemannian-geometry/mostow-rigidity-theorem/po","kind":"node","lean":"MathlibExt.Geometry.MostowRigidityWanted.Geometry.MostowRigidity.PO","parent":14834,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/mostow-rigidity-theorem/index.html#po","scope":"Geometry/riemannian-geometry/mostow-rigidity-theorem","status":"fully_proved","summary":"For natural numbers $p$ and $q$, let $J$ be the diagonal $(p+q) \\times (p+q)$ real matrix whose first $p$ diagonal entries are $1$ and whose last $q$ are $-1$, and let $\\mathrm{O}(p,q)$ be the group of real $(p+q) \\times (p+q)$ matrices $A$ with $J A^{\\mathsf{T}} J A = A J A^{\\mathsf{T}} J = I$. The group $\\mathrm{PO}(p,q)$ is the quotient of $\\mathrm{O}(p,q)$ by its center.","title":"Projective indefinite orthogonal group"},{"id":"Geometry/riemannian-geometry/mostow-rigidity-theorem/mostow-rigidity","kind":"node","lean":"MathlibExt.Geometry.MostowRigidityWanted.Geometry.MostowRigidity.mostow_rigidity","parent":14834,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/mostow-rigidity-theorem/index.html#mostow-rigidity","scope":"Geometry/riemannian-geometry/mostow-rigidity-theorem","status":"can_prove","summary":"Let $n \\ge 3$, and let $\\Gamma$ and $\\Lambda$ be lattices in the projective orthogonal group $\\mathrm{PO}(n,1)$: discrete subgroups that have a fundamental domain for their action by left multiplication and finite covolume with respect to Haar measure. Then every group isomorphism $f : \\Gamma \\to \\Lambda$ is a conjugation: there is $g \\in \\mathrm{PO}(n,1)$ such that $f(\\gamma) = g \\gamma g^{-1}$ for all\u2026","title":"Mostow rigidity, algebraic form"},{"data_url":"scopes/Geometry/riemannian-geometry/riemannian-gradient-and-hessian.json","id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","title":"Riemannian gradient and Hessian"},{"id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian/riemannian-gradient","kind":"node","lean":"Manifold.riemannianGradient","parent":14837,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html#riemannian-gradient","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","summary":"Let $M$ be a $C^1$ manifold modelled on a complete real normed space, with a Riemannian metric, and let $f : M \\to \\mathbb{R}$ and $x \\in M$. The Riemannian gradient $\\operatorname{grad} f(x) = (df_x)^\\sharp \\in T_xM$ is the image of the differential $df_x$ of $f$ at $x$ under the inverse of the Riesz isomorphism $T_xM \\to (T_xM)^*$, $w \\mapsto \\langle w, \\cdot \\rangle$, given by the metric at $x$. It is the unique\u2026","title":"Riemannian gradient"},{"id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian/inner-riemannian-gradient","kind":"node","lean":"Manifold.inner_riemannianGradient","parent":14837,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html#inner-riemannian-gradient","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","summary":"Let $M$ be a $C^1$ manifold modelled on a complete real normed space, with a Riemannian metric $\\langle \\cdot, \\cdot \\rangle$, and let $f : M \\to \\mathbb{R}$, $x \\in M$ and $v \\in T_xM$. Then $\\langle \\operatorname{grad} f(x), v \\rangle = df_x(v)$, where $\\operatorname{grad} f(x)$ is the Riemannian gradient of $f$ at $x$ and $df_x$ is the differential of $f$ at $x$.","title":"Pairing of the Riemannian gradient with a tangent vector"},{"id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian/riemannian-gradient-eq-iff","kind":"node","lean":"Manifold.riemannianGradient_eq_iff","parent":14837,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html#riemannian-gradient-eq-iff","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","summary":"Let $M$ be a $C^1$ manifold modelled on a complete real normed space, with a Riemannian metric $\\langle \\cdot, \\cdot \\rangle$, and let $f : M \\to \\mathbb{R}$, $x \\in M$ and $w \\in T_xM$. Then the Riemannian gradient $\\operatorname{grad} f(x)$ equals $w$ if and only if $\\langle w, v \\rangle = df_x(v)$ for every $v \\in T_xM$, where $df_x$ is the differential of $f$ at $x$.","title":"Characterization of the Riemannian gradient"},{"id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian/norm-riemannian-gradient","kind":"node","lean":"Manifold.norm_riemannianGradient","parent":14837,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html#norm-riemannian-gradient","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","summary":"Let $M$ be a $C^1$ manifold modelled on a complete real normed space, with a Riemannian metric, and let $f : M \\to \\mathbb{R}$ and $x \\in M$. With $T_xM$ normed by the metric at $x$, the norm of the Riemannian gradient $\\operatorname{grad} f(x)$ equals the operator norm of the differential $df_x : T_xM \\to \\mathbb{R}$ of $f$ at $x$: $\\|\\operatorname{grad} f(x)\\| = \\|df_x\\|$. This is the quantity $|\\nabla f|(x)$.","title":"Norm of the Riemannian gradient"},{"id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian/hessian","kind":"node","lean":"Manifold.hessian","parent":14837,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html#hessian","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","summary":"Let $M$ be a smooth manifold modelled on a finite-dimensional real normed space, with a smooth Riemannian metric and its Levi-Civita connection $\\nabla$. For $f : M \\to \\mathbb{R}$, a point $x \\in M$ and tangent vectors $u, v \\in T_xM$, let $V$ be the extension of $v$ to a vector field with constant coordinates in the preferred chart at $x$, so that $V(x) = v$. The Hessian of $f$ at $x$ is the real number","title":"Covariant Hessian"},{"id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian/riemannian-gradient-vector-space","kind":"node","lean":"Manifold.riemannianGradient_vectorSpace","parent":14837,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html#riemannian-gradient-vector-space","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","summary":"Let $F$ be a real Hilbert space, viewed as a Riemannian manifold with the metric given by its inner product, and let $f : F \\to \\mathbb{R}$ and $x \\in F$. Identifying the tangent space at $x$ with $F$, the Riemannian gradient of $f$ at $x$ is the Hilbert-space gradient $\\nabla f(x)$, the unique vector with $\\langle \\nabla f(x), w \\rangle = Df(x)(w)$ for all $w \\in F$, where $Df(x)$ is the Fr\u00e9chet derivative of $f$\u2026","title":"The Riemannian gradient on a Hilbert space"},{"id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian/hessian-vector-space","kind":"node","lean":"Manifold.hessian_vectorSpace","parent":14837,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html#hessian-vector-space","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product, let $f : F \\to \\mathbb{R}$ and let $x \\in F$. For all tangent vectors $u, v$ at $x$, identified with vectors of $F$, the Hessian of $f$ at $x$ is the derivative at $x$ in direction $u$ of the function $y \\mapsto Df(y)(v)$, where $Df(y)$ is the Fr\u00e9chet derivative of $f$ at $y$:\u2026","title":"The Hessian on Euclidean space as an iterated directional derivative"},{"id":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian/hessian-vector-space-eq-fderiv-fderiv","kind":"node","lean":"Manifold.hessian_vectorSpace_eq_fderiv_fderiv","parent":14837,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-gradient-and-hessian/index.html#hessian-vector-space-eq-fderiv-fderiv","scope":"Geometry/riemannian-geometry/riemannian-gradient-and-hessian","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product, let $f : F \\to \\mathbb{R}$ and let $x \\in F$. If the Fr\u00e9chet derivative $y \\mapsto Df(y)$ of $f$ is differentiable at $x$, then for all tangent vectors $u, v$ at $x$, identified with vectors of $F$, the Hessian of $f$ at $x$ is the second Fr\u00e9chet derivative:\u2026","title":"The Hessian on Euclidean space is the second derivative"},{"data_url":"scopes/Geometry/riemannian-geometry/laplace-beltrami-operator.json","id":"Geometry/riemannian-geometry/laplace-beltrami-operator","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/laplace-beltrami-operator/index.html","scope":"Geometry/riemannian-geometry/laplace-beltrami-operator","status":"fully_proved","title":"Laplace\u2013Beltrami operator"},{"id":"Geometry/riemannian-geometry/laplace-beltrami-operator/laplacian","kind":"node","lean":"Manifold.laplacian","parent":14846,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/laplace-beltrami-operator/index.html#laplacian","scope":"Geometry/riemannian-geometry/laplace-beltrami-operator","status":"fully_proved","summary":"The Laplace-Beltrami operator applied to `f : M \u2192 \u211d` at `x`: `\u0394f(x)` is the metric trace `\u2211 i, hessian I f x (e i) (e i)` over the orthonormal basis `e = stdOrthonormalBasis \u211d (TangentSpace I x)`, the same basis convention as `Manifold.scalarCurvature`. For the textbook Hessian, which is bilinear, this trace does not depend on the orthonormal basis; the regularity caveat of `Manifold.hessian` applies here unchanged.","title":"Laplace\u2013Beltrami operator"},{"id":"Geometry/riemannian-geometry/laplace-beltrami-operator/laplacian-vector-space","kind":"node","lean":"Manifold.laplacian_vectorSpace","parent":14846,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/laplace-beltrami-operator/index.html#laplacian-vector-space","scope":"Geometry/riemannian-geometry/laplace-beltrami-operator","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space of dimension $n$ with its standard Riemannian metric, let $f : F \\to \\mathbb{R}$ and $x \\in F$, and let $e_1, \\dots, e_n$ be the orthonormal basis of the tangent space $T_xF$ used in the definition of the Laplacian, viewed as vectors of $F$. For each $i$ let $g_i(y) = Df(y)(e_i)$, where $D$ denotes the Fr\u00e9chet derivative. Then the Laplacian of $f$ at $x$ is\u2026","title":"Laplacian on a vector space as a sum of second directional derivatives"},{"id":"Geometry/riemannian-geometry/laplace-beltrami-operator/laplacian-vector-space-eq-laplacian","kind":"node","lean":"Manifold.laplacian_vectorSpace_eq_laplacian","parent":14846,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/laplace-beltrami-operator/index.html#laplacian-vector-space-eq-laplacian","scope":"Geometry/riemannian-geometry/laplace-beltrami-operator","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space with its standard Riemannian metric, let $f : F \\to \\mathbb{R}$ and $x \\in F$, and let $Df$ denote the Fr\u00e9chet derivative of $f$, taken to be $0$ at points where $f$ is not differentiable. Suppose $Df$ is differentiable at $x$, with derivative $D^2 f(x)$ there. Then the Laplacian $\\Delta f(x)$ equals $\\sum_i D^2 f(x)(b_i, b_i)$ for any orthonormal basis\u2026","title":"The Laplacian on a vector space is the flat Laplacian at twice-differentiable points"},{"id":"Geometry/riemannian-geometry/laplace-beltrami-operator/laplacian-norm-sq","kind":"node","lean":"Manifold.laplacian_norm_sq","parent":14846,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/laplace-beltrami-operator/index.html#laplacian-norm-sq","scope":"Geometry/riemannian-geometry/laplace-beltrami-operator","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space of dimension $n$ with its standard Riemannian metric. For every $x \\in F$, the Laplacian of the function $y \\mapsto \\|y\\|^2$ at $x$ is $2n$.","title":"Laplacian of the squared norm"},{"id":"Geometry/riemannian-geometry/laplace-beltrami-operator/laplacian-gaussian","kind":"node","lean":"Manifold.laplacian_gaussian","parent":14846,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/laplace-beltrami-operator/index.html#laplacian-gaussian","scope":"Geometry/riemannian-geometry/laplace-beltrami-operator","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space of dimension $n$ with its standard Riemannian metric. For every real number $a$ and every $x \\in F$, the Laplacian of the function $y \\mapsto \\frac{a}{2}\\|y\\|^2$ at $x$ is $a \\cdot n$.","title":"Laplacian of the Gaussian potential"},{"data_url":"scopes/Geometry/riemannian-geometry/ricci-solitons.json","id":"Geometry/riemannian-geometry/ricci-solitons","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/ricci-solitons/index.html","scope":"Geometry/riemannian-geometry/ricci-solitons","status":"fully_proved","title":"Ricci solitons"},{"id":"Geometry/riemannian-geometry/ricci-solitons/is-gradient-ricci-soliton","kind":"node","lean":"Manifold.IsGradientRicciSoliton","parent":14852,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/ricci-solitons/index.html#is-gradient-ricci-soliton","scope":"Geometry/riemannian-geometry/ricci-solitons","status":"fully_proved","summary":"Let $M$ be a smooth manifold (possibly with boundary or corners) modelled on a finite-dimensional real normed space, with a smooth Riemannian metric $g$. A function $f : M \\to \\mathbb{R}$ and a constant $c \\in \\mathbb{R}$ form a gradient Ricci soliton if $f$ is smooth and, for every point $x \\in M$ and all tangent vectors $u, v \\in T_x M$,\u2026","title":"Gradient Ricci soliton"},{"id":"Geometry/riemannian-geometry/ricci-solitons/is-gradient-shrinking-ricci-soliton","kind":"node","lean":"Manifold.IsGradientShrinkingRicciSoliton","parent":14852,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/ricci-solitons/index.html#is-gradient-shrinking-ricci-soliton","scope":"Geometry/riemannian-geometry/ricci-solitons","status":"fully_proved","summary":"On a Riemannian manifold $(M, g)$, a function $f : M \\to \\mathbb{R}$ and a constant $c \\in \\mathbb{R}$ form a gradient shrinking Ricci soliton if they form a gradient Ricci soliton and $c > 0$.","title":"Gradient shrinking Ricci soliton"},{"id":"Geometry/riemannian-geometry/ricci-solitons/is-gradient-ricci-soliton-gaussian","kind":"node","lean":"Manifold.isGradientRicciSoliton_gaussian","parent":14852,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/ricci-solitons/index.html#is-gradient-ricci-soliton-gaussian","scope":"Geometry/riemannian-geometry/ricci-solitons","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product, and let $a \\in \\mathbb{R}$. The function $f(x) = \\frac{a}{2} \\|x\\|^2$ on $F$ is a gradient Ricci soliton with constant $a$.","title":"The Gaussian gradient Ricci soliton"},{"id":"Geometry/riemannian-geometry/ricci-solitons/is-gradient-shrinking-ricci-soliton-gaus","kind":"node","lean":"Manifold.isGradientShrinkingRicciSoliton_gaussian","parent":14852,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/ricci-solitons/index.html#is-gradient-shrinking-ricci-soliton-gaus","scope":"Geometry/riemannian-geometry/ricci-solitons","status":"fully_proved","summary":"Let $F$ be a finite-dimensional real inner product space, viewed as a Riemannian manifold with the metric given by its inner product, and let $a > 0$. The function $f(x) = \\frac{a}{2} \\|x\\|^2$ on $F$ is a gradient shrinking Ricci soliton with constant $a$.","title":"The Gaussian shrinking soliton"},{"id":"Geometry/riemannian-geometry/ricci-solitons/is-gradient-ricci-soliton-gaussian-iff","kind":"node","lean":"Manifold.isGradientRicciSoliton_gaussian_iff","parent":14852,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/ricci-solitons/index.html#is-gradient-ricci-soliton-gaussian-iff","scope":"Geometry/riemannian-geometry/ricci-solitons","status":"fully_proved","summary":"Let $F$ be a real inner product space of positive finite dimension, viewed as a Riemannian manifold with the metric given by its inner product, and let $a, c \\in \\mathbb{R}$. The function $f(x) = \\frac{a}{2} \\|x\\|^2$ on $F$ is a gradient Ricci soliton with constant $c$ if and only if $a = c$.","title":"Gaussian potentials that are gradient Ricci solitons"},{"id":"Geometry/riemannian-geometry/ricci-solitons/is-gradient-shrinking-ricci-soliton-gaus-2","kind":"node","lean":"Manifold.isGradientShrinkingRicciSoliton_gaussian_iff","parent":14852,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/ricci-solitons/index.html#is-gradient-shrinking-ricci-soliton-gaus-2","scope":"Geometry/riemannian-geometry/ricci-solitons","status":"fully_proved","summary":"Let $F$ be a real inner product space of positive finite dimension, viewed as a Riemannian manifold with the metric given by its inner product, and let $a, c \\in \\mathbb{R}$. The function $f(x) = \\frac{a}{2} \\|x\\|^2$ on $F$ is a gradient shrinking Ricci soliton with constant $c$ if and only if $a = c$ and $c > 0$.","title":"Gaussian potentials that are gradient shrinking Ricci solitons"},{"data_url":"scopes/Geometry/riemannian-geometry/riemannian-volume.json","id":"Geometry/riemannian-geometry/riemannian-volume","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-volume/index.html","scope":"Geometry/riemannian-geometry/riemannian-volume","status":"fully_proved","title":"Riemannian volume"},{"id":"Geometry/riemannian-geometry/riemannian-volume/normalized-hausdorff-measure","kind":"node","lean":"Manifold.normalizedHausdorffMeasure","parent":14859,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-volume/index.html#normalized-hausdorff-measure","scope":"Geometry/riemannian-geometry/riemannian-volume","status":"fully_proved","summary":"Let $n$ be a natural number and let $X$ be an extended metric space with its Borel $\\sigma$-algebra. The normalized $n$-dimensional Hausdorff measure of $X$ is the measure $\\frac{\\omega_n}{2^n} \\mathcal{H}^n$, where $\\omega_n$ is the Lebesgue volume of the open unit ball of the Euclidean space $\\mathbb{R}^n$ and $\\mathcal{H}^n$ is the unnormalized $n$-dimensional Hausdorff measure, in which a set of diameter $r$\u2026","title":"Normalized Hausdorff measure"},{"id":"Geometry/riemannian-geometry/riemannian-volume/riemannian-volume","kind":"node","lean":"Manifold.riemannianVolume","parent":14859,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-volume/index.html#riemannian-volume","scope":"Geometry/riemannian-geometry/riemannian-volume","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space, let $I$ be a model with corners on $E$, and let $M$ be a $C^\\infty$ manifold modeled on $I$, so that $M$ has dimension $n = \\dim_{\\mathbb{R}} E$. Suppose that $M$ carries a $C^\\infty$ Riemannian metric and an extended metric equal to the Riemannian distance (the infimum of the lengths of $C^1$ paths), and is given its Borel $\\sigma$-algebra. The Riemannian volume\u2026","title":"Riemannian volume"},{"id":"Geometry/riemannian-geometry/riemannian-volume/riemannian-volume-real","kind":"node","lean":"Manifold.riemannianVolume_real","parent":14859,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/riemannian-volume/index.html#riemannian-volume-real","scope":"Geometry/riemannian-geometry/riemannian-volume","status":"fully_proved","summary":"The Riemannian volume of the real line $\\mathbb{R}$, with its standard Riemannian metric (the usual inner product on each tangent space) and with its usual distance, which is the Riemannian distance, is Lebesgue measure.","title":"The Riemannian volume of the real line is Lebesgue measure"},{"data_url":"scopes/Geometry/riemannian-geometry/spaces-of-riemannian-metrics.json","id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","title":"Spaces of Riemannian metrics"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/metric-in-coordinates","kind":"node","lean":"Manifold.metricInCoordinates","parent":14863,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#metric-in-coordinates","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $g$ be a smooth Riemannian metric on the tangent bundle of a smooth manifold $M$ modeled through a model with corners $I : H \\to E$, and let $e$ be an open partial homeomorphism from $M$ to $H$. Write $\\psi$ for the inverse of the extended chart $I \\circ e$, $T$ for the target of $I \\circ e$, and $D\\psi(x)$ for the derivative of $\\psi$ at $x$ within $T$. The coordinate expression of $g$ in $e$ is the field of\u2026","title":"Coordinate expression of a Riemannian metric"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/metric-in-coordinates-apply","kind":"node","lean":"Manifold.metricInCoordinates_apply","parent":14863,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#metric-in-coordinates-apply","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $g$ be a smooth Riemannian metric on the tangent bundle of a smooth manifold $M$ modeled through a model with corners $I : H \\to E$, and let $e$ be an open partial homeomorphism from $M$ to $H$. Write $g^e$ for the coordinate expression of $g$ in $e$, $\\psi$ for the inverse of the extended chart $I \\circ e$, $T$ for the target of $I \\circ e$, and $D\\psi(x)$ for the derivative of $\\psi$ at $x$ within $T$. Then\u2026","title":"Pointwise formula for a metric in coordinates"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/metric-in-coordinates-conformal-scale","kind":"node","lean":"Manifold.metricInCoordinates_conformalScale","parent":14863,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#metric-in-coordinates-conformal-scale","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $g$ be a smooth Riemannian metric on the tangent bundle of a smooth manifold $M$ modeled through a model with corners $I : H \\to E$, let $u : M \\to \\mathbb{R}$ be a smooth function with $u(p) > 0$ for every $p \\in M$, and let $e$ be an open partial homeomorphism from $M$ to $H$. Write $g^e$ for the coordinate expression of $g$ in $e$ and $\\psi$ for the inverse of the extended chart $I \\circ e$. Then for every\u2026","title":"Conformal scaling of a metric in coordinates"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/cont-diff-on-metric-in-coordinates","kind":"node","lean":"Manifold.contDiffOn_metricInCoordinates","parent":14863,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#cont-diff-on-metric-in-coordinates","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space, let $M$ be a smooth manifold modeled through a model with corners $I : H \\to E$, and let $g$ be a smooth Riemannian metric on the tangent bundle of $M$. For every chart $e$ in the maximal smooth atlas of $M$, the coordinate expression of $g$ in $e$, viewed as a map from $E$ to the continuous bilinear forms on $E$, is $C^\\infty$ on the target of the extended chart\u2026","title":"Smoothness of a metric in coordinates"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/metric-ck-topology","kind":"node","lean":"Manifold.metricCkTopology","parent":14863,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(C^k\\)"],["text"," topology on smooth Riemannian metrics"]],"source":"The $C^k$ topology on smooth Riemannian metrics"},"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#metric-ck-topology","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a smooth manifold modelled on a real normed space $E$ through a model with corners $I$, and let $k \\in \\mathbb{N} \\cup \\{\\infty\\}$. For a chart $\\varphi$ of the maximal smooth atlas of $M$, write $T_\\varphi \\subseteq E$ for the target of the extended chart $I \\circ \\varphi$. The (weak, compact-open) $C^k$ topology on the space of smooth Riemannian metrics on $M$ is the coarsest topology for which, for\u2026","title":"The $C^k$ topology on smooth Riemannian metrics"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/riemannian-metric-with-ck","kind":"node","lean":"Manifold.RiemannianMetricWithCk","parent":14863,"reading_title":{"format":"inline-parts-v1","parts":[["text","The space "],["math","\\(\\mathcal{G}^k(M)\\)"],["text"," of smooth metrics with the "],["math","\\(C^k\\)"],["text"," topology"]],"source":"The space $\\mathcal{G}^k(M)$ of smooth metrics with the $C^k$ topology"},"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#riemannian-metric-with-ck","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"For a smooth manifold $M$ and $k \\in \\mathbb{N} \\cup \\{\\infty\\}$, $\\mathcal{G}^k(M)$ is the space of all smooth Riemannian metrics on $M$, equipped with the $C^k$ topology. It is not the space of $C^k$ metrics: every element is smooth, and $k$ only indexes the topology.","title":"The space $\\mathcal{G}^k(M)$ of smooth metrics with the $C^k$ topology"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/of-metric","kind":"node","lean":"Manifold.RiemannianMetricWithCk.ofMetric","parent":14863,"reading_title":{"format":"inline-parts-v1","parts":[["text","Smooth metrics as elements of "],["math","\\(\\mathcal{G}^k(M)\\)"]],"source":"Smooth metrics as elements of $\\mathcal{G}^k(M)$"},"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#of-metric","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"For a smooth manifold $M$ and $k \\in \\mathbb{N} \\cup \\{\\infty\\}$, the identity map is a bijection from the space of smooth Riemannian metrics on $M$ to $\\mathcal{G}^k(M)$, the same space of metrics equipped with the $C^k$ topology.","title":"Smooth metrics as elements of $\\mathcal{G}^k(M)$"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/tendsto-metric-ck-topology-iff","kind":"node","lean":"Manifold.tendsto_metricCkTopology_iff","parent":14863,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergence in the "],["math","\\(C^k\\)"],["text"," topology on smooth metrics"]],"source":"Convergence in the $C^k$ topology on smooth metrics"},"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#tendsto-metric-ck-topology-iff","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a smooth manifold modelled on a real normed space $E$ through a model with corners $I$, let $k \\in \\mathbb{N} \\cup \\{\\infty\\}$, let $l$ be a filter on a type $\\alpha$, and let $g_a$ ($a \\in \\alpha$) and $g_0$ be smooth Riemannian metrics on $M$. For a chart $\\varphi$ of the maximal smooth atlas of $M$, write $T_\\varphi \\subseteq E$ for the target of the extended chart $I \\circ \\varphi$. Then $g_a$ tends\u2026","title":"Convergence in the $C^k$ topology on smooth metrics"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/tendsto-iff","kind":"node","lean":"Manifold.RiemannianMetricWithCk.tendsto_iff","parent":14863,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergence in "],["math","\\(\\mathcal{G}^k(M)\\)"]],"source":"Convergence in $\\mathcal{G}^k(M)$"},"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#tendsto-iff","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a smooth manifold modelled on a real normed space $E$ through a model with corners $I$, let $k \\in \\mathbb{N} \\cup \\{\\infty\\}$, let $l$ be a filter on a type $\\alpha$, and let $g_a$ ($a \\in \\alpha$) and $g_0$ be smooth Riemannian metrics on $M$. For a chart $\\varphi$ of the maximal smooth atlas of $M$, write $T_\\varphi \\subseteq E$ for the target of the extended chart $I \\circ \\varphi$. Then $g_a$,\u2026","title":"Convergence in $\\mathcal{G}^k(M)$"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/metric-ck-topology-anti","kind":"node","lean":"Manifold.metricCkTopology_anti","parent":14863,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(C^k\\)"],["text"," topologies on smooth metrics get finer as "],["math","\\(k\\)"],["text"," grows"]],"source":"The $C^k$ topologies on smooth metrics get finer as $k$ grows"},"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#metric-ck-topology-anti","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a smooth manifold and let $k \\le k'$ be elements of $\\mathbb{N} \\cup \\{\\infty\\}$. On the space of smooth Riemannian metrics on $M$, the $C^{k'}$ topology is finer than the $C^k$ topology.","title":"The $C^k$ topologies on smooth metrics get finer as $k$ grows"},{"id":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics/tendsto-conformal-scale-const-metric-ck","kind":"node","lean":"Manifold.tendsto_conformalScale_const_metricCkTopology","parent":14863,"reading_title":{"format":"inline-parts-v1","parts":[["text","Constant rescalings converge in every "],["math","\\(C^k\\)"],["text"," topology"]],"source":"Constant rescalings converge in every $C^k$ topology"},"result":true,"route":"../roadmap/Geometry/riemannian-geometry/spaces-of-riemannian-metrics/index.html#tendsto-conformal-scale-const-metric-ck","scope":"Geometry/riemannian-geometry/spaces-of-riemannian-metrics","status":"fully_proved","summary":"Let $M$ be a smooth manifold modelled on a finite-dimensional real normed space, let $g$ be a smooth Riemannian metric on $M$, and let $k \\in \\mathbb{N} \\cup \\{\\infty\\}$. Let $l$ be a filter on a type $\\alpha$ and let $c_a > 0$ ($a \\in \\alpha$) be positive reals with $c_a \\to 1$ along $l$. Then the rescaled metrics $c_a g$ tend to $g$ along $l$ in the $C^k$ topology.","title":"Constant rescalings converge in every $C^k$ topology"},{"data_url":"scopes/Geometry/riemannian-geometry/sphere-theorem.json","id":"Geometry/riemannian-geometry/sphere-theorem","kind":"scope","navigation":"module","parent":14751,"result":false,"route":"../roadmap/Geometry/riemannian-geometry/sphere-theorem/index.html","scope":"Geometry/riemannian-geometry/sphere-theorem","status":"can_prove","title":"Sphere theorem"},{"id":"Geometry/riemannian-geometry/sphere-theorem/curv","kind":"node","lean":"MathlibExt.Geometry.SphereTheoremWanted.Geometry.SphereTheorem.curv","parent":14875,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/sphere-theorem/index.html#curv","scope":"Geometry/riemannian-geometry/sphere-theorem","status":"fully_proved","summary":"Let $\\nabla$ be a covariant derivative on the tangent bundle of a smooth manifold $M$. For vector fields $X, Y, Z$ on $M$, the curvature of $\\nabla$ is the vector field $R(X,Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X,Y]} Z$, where $[X,Y]$ is the Lie bracket of $X$ and $Y$.","title":"Riemann curvature of a covariant derivative"},{"id":"Geometry/riemannian-geometry/sphere-theorem/quarter-pinched","kind":"node","lean":"MathlibExt.Geometry.SphereTheoremWanted.Geometry.SphereTheorem.QuarterPinched","parent":14875,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/sphere-theorem/index.html#quarter-pinched","scope":"Geometry/riemannian-geometry/sphere-theorem","status":"fully_proved","summary":"Let $\\nabla$ be a covariant derivative on the tangent bundle of a Riemannian manifold $M$, with curvature $R$. Then $\\nabla$ is strictly quarter-pinched when, for all smooth vector fields $X, Y$ on $M$ and every point $x \\in M$ at which $X(x)$ and $Y(x)$ are orthonormal, the sectional curvature $K = \\langle R(X,Y)Y(x), X(x) \\rangle$ satisfies $1 < K \\le 4$. The interval $(1, 4]$ is Knill's normalization of strict\u2026","title":"Strictly quarter-pinched curvature"},{"id":"Geometry/riemannian-geometry/sphere-theorem/sphere-theorem","kind":"node","lean":"MathlibExt.Geometry.SphereTheoremWanted.Geometry.SphereTheorem.sphere_theorem","parent":14875,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/sphere-theorem/index.html#sphere-theorem","scope":"Geometry/riemannian-geometry/sphere-theorem","status":"can_prove","summary":"Let $M$ be a compact, Hausdorff, simply connected smooth manifold without boundary, of dimension $d \\ge 2$, with a smooth Riemannian metric. Suppose $M$ carries a smooth covariant derivative $\\nabla$ on its tangent bundle that is torsion-free, compatible with the metric and strictly quarter-pinched (every sectional curvature at an orthonormal pair lies in $(1, 4]$). Then $M$ is homeomorphic to the unit sphere\u2026","title":"Topological sphere theorem"},{"id":"Geometry/riemannian-geometry/sphere-theorem/differentiable-sphere-theorem","kind":"node","lean":"MathlibExt.Geometry.SphereTheoremWanted.Geometry.SphereTheorem.differentiable_sphere_theorem","parent":14875,"result":true,"route":"../roadmap/Geometry/riemannian-geometry/sphere-theorem/index.html#differentiable-sphere-theorem","scope":"Geometry/riemannian-geometry/sphere-theorem","status":"can_prove","summary":"Let $M$ be a compact, Hausdorff, simply connected smooth manifold without boundary, of dimension $d \\ge 2$, with a smooth Riemannian metric. Suppose $M$ carries a smooth covariant derivative $\\nabla$ on its tangent bundle that is torsion-free, compatible with the metric and strictly quarter-pinched (every sectional curvature at an orthonormal pair lies in $(1, 4]$). Then $M$ is diffeomorphic to the unit sphere\u2026","title":"Differentiable sphere theorem"},{"data_url":"scopes/Geometry/curves-and-surfaces.json","id":"Geometry/curves-and-surfaces","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/curves-and-surfaces/index.html","scope":"Geometry/curves-and-surfaces","status":"can_prove","summary":"Curvature, tangents and global properties of curves and surfaces in Euclidean space, including knotted curves and inscribed figures.","title":"Differential geometry of curves and surfaces"},{"data_url":"scopes/Geometry/curves-and-surfaces/curvature-of-hypersurfaces.json","id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","kind":"scope","navigation":"module","parent":14880,"result":false,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","title":"Curvature of hypersurfaces"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/is-local-defining-function","kind":"node","lean":"EuclideanGeometry.IsLocalDefiningFunction","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#is-local-defining-function","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a real inner product space, $S \\subseteq E$, $\\rho : E \\to \\mathbb{R}$ and $p \\in E$. Then $\\rho$ is a local defining function of $S$ at $p$ when $\\rho(p) = 0$, $\\rho$ is of class $C^2$ at $p$, its differential $D\\rho(p)$ is nonzero, and for all $x$ in some neighborhood of $p$, $x \\in S$ if and only if $\\rho(x) = 0$. In particular $p \\in S$.","title":"Local defining function"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/mem","kind":"node","lean":"EuclideanGeometry.IsLocalDefiningFunction.mem","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#mem","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a real inner product space, $S \\subseteq E$, $\\rho : E \\to \\mathbb{R}$ and $p \\in E$. If $\\rho$ is a local defining function of $S$ at $p$, then $p \\in S$.","title":"A point with a local defining function lies in the set"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/level-set-second-fundamental-form","kind":"node","lean":"EuclideanGeometry.levelSetSecondFundamentalForm","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#level-set-second-fundamental-form","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space, $\\rho : E \\to \\mathbb{R}$ and $p \\in E$. Write $\\nabla\\rho(p)$ for the gradient of $\\rho$ at $p$ and $D^2\\rho(p)$ for the second derivative of $\\rho$ at $p$ (a bilinear form on $E$). The second fundamental form at $p$ of the level set of $\\rho$ through $p$, with respect to the unit normal $\\nabla\\rho(p) / \\|\\nabla\\rho(p)\\|$, is defined for all $u, v \\in E$\u2026","title":"Second fundamental form of a level set"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/level-set-mean-curvature","kind":"node","lean":"EuclideanGeometry.levelSetMeanCurvature","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#level-set-mean-curvature","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space, $\\rho : E \\to \\mathbb{R}$ and $p \\in E$. Write $\\nabla\\rho(p)$ for the gradient of $\\rho$ at $p$, $D^2\\rho(p)$ for the second derivative of $\\rho$ at $p$ (a bilinear form on $E$), and $\\Delta\\rho(p) = \\sum_i D^2\\rho(p)(e_i, e_i)$ for a fixed orthonormal basis $(e_i)$ of $E$. The mean curvature at $p$ of the level set of $\\rho$ through $p$, with respect to\u2026","title":"Mean curvature of a level set"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/level-set-second-fundamental-form-smul","kind":"node","lean":"EuclideanGeometry.levelSetSecondFundamentalForm_smul","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#level-set-second-fundamental-form-smul","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space. For every $\\rho : E \\to \\mathbb{R}$, all $p, u, v \\in E$ and every real $c \\neq 0$, the second fundamental form at $p$ of the level set of $c\\rho$ through $p$, evaluated at $(u, v)$, equals $c / |c|$ (the sign of $c$) times the second fundamental form at $p$ of the level set of $\\rho$ through $p$, evaluated at $(u, v)$.","title":"Second fundamental form under rescaling of the defining function"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/level-set-mean-curvature-smul","kind":"node","lean":"EuclideanGeometry.levelSetMeanCurvature_smul","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#level-set-mean-curvature-smul","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space. For every $\\rho : E \\to \\mathbb{R}$, every $p \\in E$ and every real $c \\neq 0$, the mean curvature at $p$ of the level set of $c\\rho$ through $p$ equals $c / |c|$ (the sign of $c$) times the mean curvature at $p$ of the level set of $\\rho$ through $p$.","title":"Mean curvature under rescaling of the defining function"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/level-set-mean-curvature-smul-of-pos","kind":"node","lean":"EuclideanGeometry.levelSetMeanCurvature_smul_of_pos","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#level-set-mean-curvature-smul-of-pos","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space. For every $\\rho : E \\to \\mathbb{R}$, every $p \\in E$ and every real $c > 0$, the mean curvature at $p$ of the level set of $c\\rho$ through $p$ equals the mean curvature at $p$ of the level set of $\\rho$ through $p$.","title":"Positive rescaling preserves the mean curvature"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/level-set-mean-curvature-neg","kind":"node","lean":"EuclideanGeometry.levelSetMeanCurvature_neg","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#level-set-mean-curvature-neg","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space. For every $\\rho : E \\to \\mathbb{R}$ and every $p \\in E$, the mean curvature at $p$ of the level set of $-\\rho$ through $p$ is the negative of the mean curvature at $p$ of the level set of $\\rho$ through $p$: replacing $\\rho$ by $-\\rho$ reverses the unit normal and changes the sign of the mean curvature.","title":"Reversing the normal negates the mean curvature"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/level-set-mean-curvature-sphere","kind":"node","lean":"EuclideanGeometry.levelSetMeanCurvature_sphere","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#level-set-mean-curvature-sphere","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space of dimension $n$. For every $p \\in E$ with $\\|p\\| = 1$, the mean curvature at $p$ of the level set of $\\rho(x) = \\|x\\|^2 - 1$ through $p$ is $n - 1$. That is, the unit sphere of $E$ has mean curvature $n - 1$ with respect to the outward normal, the mean curvature being the sum (not the average) of the principal curvatures.","title":"Mean curvature of the unit sphere"},{"id":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces/level-set-mean-curvature-affine","kind":"node","lean":"EuclideanGeometry.levelSetMeanCurvature_affine","parent":14881,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/curvature-of-hypersurfaces/index.html#level-set-mean-curvature-affine","scope":"Geometry/curves-and-surfaces/curvature-of-hypersurfaces","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space. For all $a \\in E$, $c \\in \\mathbb{R}$ and $p \\in E$, the mean curvature at $p$ of the level set of $\\rho(x) = \\langle a, x \\rangle - c$ through $p$ is $0$. In particular, for $a \\neq 0$ the hyperplane $\\{x \\mid \\langle a, x \\rangle = c\\}$ has mean curvature $0$ at each of its points.","title":"Hyperplanes have zero mean curvature"},{"data_url":"scopes/Geometry/curves-and-surfaces/doubly-ruled-surfaces.json","id":"Geometry/curves-and-surfaces/doubly-ruled-surfaces","kind":"scope","navigation":"module","parent":14880,"result":false,"route":"../roadmap/Geometry/curves-and-surfaces/doubly-ruled-surfaces/index.html","scope":"Geometry/curves-and-surfaces/doubly-ruled-surfaces","status":"can_prove","title":"Doubly ruled surfaces"},{"id":"Geometry/curves-and-surfaces/doubly-ruled-surfaces/line-in-graph","kind":"node","lean":"MathlibExt.Analysis.Calculus.ScottishBook44Wanted.LineInGraph","parent":14892,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/doubly-ruled-surfaces/index.html#line-in-graph","scope":"Geometry/curves-and-surfaces/doubly-ruled-surfaces","status":"fully_proved","summary":"Let $f$ be a real-valued function of two real variables, and let $x_0, y_0, u, v, w$ be real numbers. The line through $(x_0, y_0, f(x_0, y_0))$ with direction $(u, v, w)$ lies in the graph of $f$ when $(u, v) \\ne (0, 0)$ and $f(x_0 + tu, y_0 + tv) = f(x_0, y_0) + tw$ for every real $t$.","title":"Line lying in a graph"},{"id":"Geometry/curves-and-surfaces/doubly-ruled-surfaces/doubly-ruled-graph","kind":"node","lean":"MathlibExt.Analysis.Calculus.ScottishBook44Wanted.DoublyRuledGraph","parent":14892,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/doubly-ruled-surfaces/index.html#doubly-ruled-graph","scope":"Geometry/curves-and-surfaces/doubly-ruled-surfaces","status":"fully_proved","summary":"The graph of a real-valued function $f$ of two real variables is doubly ruled when for every point $(x_0, y_0) \\in \\mathbb{R}^2$ there are $(u_1, v_1, w_1)$ and $(u_2, v_2, w_2)$ in $\\mathbb{R}^3$ such that the lines through $(x_0, y_0, f(x_0, y_0))$ with directions $(u_1, v_1, w_1)$ and $(u_2, v_2, w_2)$ both lie in the graph of $f$, and $(u_2, v_2)$ is not a real multiple of $(u_1, v_1)$.","title":"Doubly ruled graph"},{"id":"Geometry/curves-and-surfaces/doubly-ruled-surfaces/doubly-ruled-graph-is-quadric","kind":"node","lean":"MathlibExt.Analysis.Calculus.ScottishBook44Wanted.doubly_ruled_graph_is_quadric","parent":14892,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/doubly-ruled-surfaces/index.html#doubly-ruled-graph-is-quadric","scope":"Geometry/curves-and-surfaces/doubly-ruled-surfaces","status":"can_prove","summary":"Let $f$ be a real-valued function of two real variables whose graph is doubly ruled; no continuity of $f$ is assumed. Then there are real numbers $a, b, c, d, e, g$ such that $f(x, y) = ax^2 + bxy + cy^2 + dx + ey + g$ for all real $x$ and $y$, and either $a = b = c = 0$ (the graph is a plane) or $b^2 - 4ac > 0$ (the graph is a hyperbolic paraboloid).","title":"Doubly ruled graphs are planes or hyperbolic paraboloids"},{"data_url":"scopes/Geometry/curves-and-surfaces/f-ry-milnor-theorem.json","id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","kind":"scope","navigation":"module","parent":14880,"result":false,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"can_prove","title":"F\u00e1ry\u2013Milnor theorem"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/space","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.Space","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#space","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"Euclidean 3-space is the real vector space $\\mathbb{R}^3$ with the Euclidean inner product and norm.","title":"Euclidean 3-space for the F\u00e1ry\u2013Milnor theorem"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/period","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.period","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#period","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"The period used for parametrised knots is the real number $2\\pi$.","title":"Period of a parametrised knot"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/velocity","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.velocity","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#velocity","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"For a curve $r \\colon \\mathbb{R} \\to \\mathbb{R}^3$ and a real number $t$, the velocity of $r$ at $t$ is its derivative $r'(t)$, taken to be $0$ when $r$ is not differentiable at $t$.","title":"Velocity of a space curve"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/acceleration","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.acceleration","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#acceleration","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"For a curve $r \\colon \\mathbb{R} \\to \\mathbb{R}^3$ and a real number $t$, the acceleration of $r$ at $t$ is the derivative at $t$ of its velocity $s \\mapsto r'(s)$, that is $r''(t)$, where a derivative that does not exist is taken to be $0$.","title":"Acceleration of a space curve"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/curvature","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.curvature","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#curvature","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"For a curve $r \\colon \\mathbb{R} \\to \\mathbb{R}^3$ with velocity $r'$ and acceleration $r''$, the curvature of $r$ at a real number $t$ is $\\kappa(t) = \\lVert r'(t) \\times r''(t) \\rVert / \\lVert r'(t) \\rVert^3$, where $\\times$ is the cross product and $\\lVert \\cdot \\rVert$ is the Euclidean norm; the quotient is $0$ when $r'(t) = 0$.","title":"Curvature of a space curve"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/total-curvature","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.totalCurvature","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#total-curvature","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"For a curve $r \\colon \\mathbb{R} \\to \\mathbb{R}^3$ with velocity $r'$ and curvature $\\kappa$, the total curvature of $r$ is the arc-length integral $\\int_0^{2\\pi} \\kappa(t) \\, \\lVert r'(t) \\rVert \\, dt$ over one period, where the integral is taken to be $0$ if the integrand is not integrable.","title":"Total curvature of a space curve"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/standard-circle","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.standardCircle","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#standard-circle","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"The standard circle is the curve $\\mathbb{R} \\to \\mathbb{R}^3$ given by $t \\mapsto (\\cos t, \\sin t, 0)$: the unit circle centred at the origin in the $xy$-plane.","title":"Standard circle"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/is-smooth-knot","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.IsSmoothKnot","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#is-smooth-knot","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"A curve $r \\colon \\mathbb{R} \\to \\mathbb{R}^3$ is a smooth knot when it is real-analytic, periodic with period $2\\pi$, injective on $[0, 2\\pi)$, and regular, that is, its velocity $r'(t)$ is nonzero for every real number $t$.","title":"Smooth knot"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/is-unknotted","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.IsUnknotted","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#is-unknotted","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"fully_proved","summary":"A curve $r \\colon \\mathbb{R} \\to \\mathbb{R}^3$ is unknotted when there is a real-analytic map $R \\colon \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}^3$ such that $R(t, 0) = r(t)$ and $R(t, 1) = (\\cos t, \\sin t, 0)$, the standard circle, for every real number $t$, and for every $s \\in [0, 1]$ the curve $t \\mapsto R(t, s)$ is a smooth knot.","title":"Unknotted curve"},{"id":"Geometry/curves-and-surfaces/f-ry-milnor-theorem/fary-milnor-total-curvature","kind":"node","lean":"MathlibExt.Geometry.FaryMilnorWanted.Geometry.FaryMilnorProblem.fary_milnor_total_curvature","parent":14896,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/f-ry-milnor-theorem/index.html#fary-milnor-total-curvature","scope":"Geometry/curves-and-surfaces/f-ry-milnor-theorem","status":"can_prove","summary":"Let $r \\colon \\mathbb{R} \\to \\mathbb{R}^3$ be a smooth knot, that is, a real-analytic, $2\\pi$-periodic curve that is injective on $[0, 2\\pi)$ and has nonzero velocity everywhere. If the total curvature of $r$ is at most $4\\pi$, then $r$ is unknotted: there is a real-analytic isotopy from $r$ to the standard circle through smooth knots.","title":"F\u00e1ry\u2013Milnor theorem"},{"data_url":"scopes/Geometry/curves-and-surfaces/hopf-s-umlaufsatz.json","id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","kind":"scope","navigation":"module","parent":14880,"result":false,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"can_prove","title":"Hopf's Umlaufsatz"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/plane","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.Plane","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#plane","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"The Euclidean plane is the space $\\mathbb{R}^2$ of pairs of real numbers with the Euclidean inner product and norm.","title":"Euclidean plane for the Umlaufsatz"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/period","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.period","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#period","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"The parameter period of the closed curves in the Hopf Umlaufsatz is the real number $2\\pi$.","title":"Parameter period for the Umlaufsatz"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/velocity","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.velocity","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#velocity","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"For a curve $r : \\mathbb{R} \\to \\mathbb{R}^2$ in the Euclidean plane and $t \\in \\mathbb{R}$, the velocity of $r$ at $t$ is its derivative $r'(t) \\in \\mathbb{R}^2$, taken to be $0$ where $r$ is not differentiable.","title":"Velocity of a plane curve"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/det2","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.det2","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#det2","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"For vectors $u = (u_0, u_1)$ and $v = (v_0, v_1)$ in the Euclidean plane $\\mathbb{R}^2$, $\\det(u, v) = u_0 v_1 - u_1 v_0$, the signed area of the parallelogram spanned by $u$ and $v$.","title":"Planar determinant for the Umlaufsatz"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/signed-area","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.signedArea","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#signed-area","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"For a curve $r : \\mathbb{R} \\to \\mathbb{R}^2$, the signed area is $\\frac{1}{2}\\int_0^{2\\pi} \\det(r(t), r'(t))\\,dt$, where $r'$ is the velocity of $r$ and $\\det(u, v) = u_0 v_1 - u_1 v_0$ (Green's normalization). Its sign fixes the orientation convention.","title":"Signed area of a closed plane curve"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/is-positive-simple-closed-unit-speed-cur","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.IsPositiveSimpleClosedUnitSpeedCurve","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#is-positive-simple-closed-unit-speed-cur","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"A curve $r : \\mathbb{R} \\to \\mathbb{R}^2$ in the Euclidean plane is a positively oriented simple closed unit-speed curve when it is continuously differentiable, periodic with period $2\\pi$, injective on $[0, 2\\pi)$, has velocity of norm $\\lVert r'(t) \\rVert = 1$ for every $t \\in \\mathbb{R}$, and has positive signed area $\\frac{1}{2}\\int_0^{2\\pi} \\det(r(t), r'(t))\\,dt > 0$, where $\\det(u, v) = u_0 v_1 - u_1 v_0$.","title":"Positively oriented simple closed unit-speed curve"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/is-tangent-angle-lift","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.IsTangentAngleLift","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#is-tangent-angle-lift","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"For a curve $r : \\mathbb{R} \\to \\mathbb{R}^2$ in the Euclidean plane, a function $\\alpha : \\mathbb{R} \\to \\mathbb{R}$ is a tangent-angle lift of $r$ when $\\alpha$ is continuously differentiable and the velocity of $r$ satisfies $r'(t) = (\\cos \\alpha(t), \\sin \\alpha(t))$ for every $t \\in \\mathbb{R}$.","title":"Tangent-angle lift"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/signed-curvature","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.signedCurvature","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#signed-curvature","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"For a function $\\alpha : \\mathbb{R} \\to \\mathbb{R}$ and $t \\in \\mathbb{R}$, the signed curvature of $\\alpha$ at $t$ is the derivative $\\alpha'(t)$, taken to be $0$ where $\\alpha$ is not differentiable at $t$.","title":"Signed curvature"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/total-curvature","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.totalCurvature","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#total-curvature","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"fully_proved","summary":"For a function $\\alpha : \\mathbb{R} \\to \\mathbb{R}$, the total signed curvature is $\\int_0^{2\\pi} \\alpha'(t)\\,dt$, the integral of the signed curvature $\\alpha'$ over one parameter period $[0, 2\\pi]$.","title":"Total signed curvature of a tangent-angle lift"},{"id":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz/hopf-umlaufsatz","kind":"node","lean":"MathlibExt.Geometry.HopfUmlaufsatzWanted.Geometry.HopfUmlaufsatz.hopf_umlaufsatz","parent":14907,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/hopf-s-umlaufsatz/index.html#hopf-umlaufsatz","scope":"Geometry/curves-and-surfaces/hopf-s-umlaufsatz","status":"can_prove","summary":"Let $r : \\mathbb{R} \\to \\mathbb{R}^2$ be a positively oriented simple closed unit-speed plane curve: $r$ is continuously differentiable and $2\\pi$-periodic, injective on $[0, 2\\pi)$, satisfies $\\lVert r'(t) \\rVert = 1$ for all $t$, and has positive signed area $\\frac{1}{2}\\int_0^{2\\pi} \\det(r(t), r'(t))\\,dt > 0$, where $\\det(u, v) = u_0 v_1 - u_1 v_0$. Let $\\alpha : \\mathbb{R} \\to \\mathbb{R}$ be a continuously\u2026","title":"Hopf's Umlaufsatz"},{"data_url":"scopes/Geometry/curves-and-surfaces/inscribed-rectangle-problem.json","id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","kind":"scope","navigation":"module","parent":14880,"result":false,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"can_prove","title":"Inscribed rectangle problem"},{"id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem/axis-aligned-rectangle-corners","kind":"node","lean":"MathlibExt.Geometry.InscribedRectanglesWanted.InscribedRectangles.axisAlignedRectangleCorners","parent":14918,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html#axis-aligned-rectangle-corners","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"fully_proved","summary":"For points $a = (a_1, a_2)$ and $b = (b_1, b_2)$ of $\\mathbb{R}^2$, the axis-aligned (possibly degenerate) rectangle with opposite corners $a$ and $b$ is the list of its four vertices $(a_1, a_2)$, $(b_1, a_2)$, $(b_1, b_2)$, $(a_1, b_2)$, in this order.","title":"Axis-aligned rectangle"},{"id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem/rectangle-corners","kind":"node","lean":"MathlibExt.Geometry.InscribedRectanglesWanted.InscribedRectangles.rectangleCorners","parent":14918,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html#rectangle-corners","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"fully_proved","summary":"For points $a, b \\in \\mathbb{R}^2$ and an angle $\\theta \\in \\mathbb{R}$, the rotated rectangle given by $a$, $b$ and $\\theta$ is the list of four points obtained by applying the rotation $(x, y) \\mapsto (x \\cos\\theta + y \\sin\\theta, -x \\sin\\theta + y \\cos\\theta)$ about the origin to each vertex, in order, of the axis-aligned rectangle with opposite corners $a$ and $b$.","title":"Rotated rectangle"},{"id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem/is-inscribed-rectangle","kind":"node","lean":"MathlibExt.Geometry.InscribedRectanglesWanted.InscribedRectangles.IsInscribedRectangle","parent":14918,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html#is-inscribed-rectangle","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"fully_proved","summary":"Let $\\gamma$ be a map from the unit circle to $\\mathbb{R}^2$. For points $a, b \\in \\mathbb{R}^2$ and an angle $\\theta \\in \\mathbb{R}$, the rotated rectangle given by $a$, $b$ and $\\theta$ is inscribed in $\\gamma$ when its four vertices are pairwise distinct and each lies in the image of $\\gamma$.","title":"Inscribed rectangle"},{"id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem/aspect-ratio","kind":"node","lean":"MathlibExt.Geometry.InscribedRectanglesWanted.InscribedRectangles.aspectRatio","parent":14918,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html#aspect-ratio","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"fully_proved","summary":"For points $a = (a_1, a_2)$ and $b = (b_1, b_2)$ of $\\mathbb{R}^2$, the aspect ratio of the axis-aligned rectangle with opposite corners $a$ and $b$ is $|a_1 - b_1| / |a_2 - b_2|$, the ratio of its horizontal side length to its vertical side length; it is $0$ when $a_2 = b_2$.","title":"Aspect ratio"},{"id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem/x","kind":"node","lean":"MathlibExt.Geometry.InscribedRectanglesWanted.InscribedRectangles.X","parent":14918,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html#x","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"fully_proved","summary":"For a map $\\gamma$ from the unit circle to $\\mathbb{R}^2$, let $X(\\gamma)$ be the set of all $r \\in [0, 1]$ for which there are points $a, b \\in \\mathbb{R}^2$ and an angle $\\theta \\in \\mathbb{R}$ such that the rotated rectangle given by $a$, $b$ and $\\theta$ is inscribed in $\\gamma$ and the axis-aligned rectangle with opposite corners $a$ and $b$ has aspect ratio $\\tan(r\\pi/4)$.","title":"Aspect-ratio parameters of inscribed rectangles"},{"id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem/theorem-1","kind":"node","lean":"MathlibExt.Geometry.InscribedRectanglesWanted.InscribedRectangles.theorem_1","parent":14918,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html#theorem-1","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"can_prove","summary":"Let $\\gamma$ be a smooth Jordan curve, that is, a map from the unit circle to $\\mathbb{R}^2$ that is a $C^\\infty$ immersion and a topological embedding, and let $X$ be the set of all $r \\in [0, 1]$ such that some rectangle inscribed in $\\gamma$ has aspect ratio $\\tan(r\\pi/4)$. Then the Lebesgue measure of $X$ is at least $1/3$.","title":"Hugelmeyer's theorem on aspect ratios of inscribed rectangles"},{"id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem/r","kind":"node","lean":"MathlibExt.Geometry.RectangularPegProblemWanted.RectangularPegProblem.R","parent":14918,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html#r","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"fully_proved","summary":"For $z, w \\in \\mathbb{C}$ and an angle $\\theta$, $R(z, w, \\theta)$ is the list of four points $z e^{i\\theta}$, $(\\operatorname{Re} w + i \\operatorname{Im} z) e^{i\\theta}$, $w e^{i\\theta}$, $(\\operatorname{Re} z + i \\operatorname{Im} w) e^{i\\theta}$: the vertices, in order, of the axis-parallel rectangle with opposite corners $z$ and $w$, rotated about the origin by $\\theta$. The rectangle is degenerate when $z$ and\u2026","title":"Rotated rectangle with given opposite corners"},{"id":"Geometry/curves-and-surfaces/inscribed-rectangle-problem/theorem-1-2","kind":"node","lean":"MathlibExt.Geometry.RectangularPegProblemWanted.RectangularPegProblem.theorem_1","parent":14918,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/inscribed-rectangle-problem/index.html#theorem-1-2","scope":"Geometry/curves-and-surfaces/inscribed-rectangle-problem","status":"can_prove","summary":"Let $\\gamma : S^1 \\to \\mathbb{C}$ be a $C^\\infty$ embedding of the unit circle (a smooth Jordan curve in the plane). For $z, w \\in \\mathbb{C}$ and an angle $\\theta$, write $R(z, w, \\theta)$ for the list of the four points $z e^{i\\theta}$, $(\\operatorname{Re} w + i \\operatorname{Im} z) e^{i\\theta}$, $w e^{i\\theta}$, $(\\operatorname{Re} z + i \\operatorname{Im} w) e^{i\\theta}$, in this order: the vertices of the\u2026","title":"Rectangular peg theorem (Greene\u2013Lobb)"},{"data_url":"scopes/Geometry/curves-and-surfaces/optimal-paper-m-bius-band.json","id":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band","kind":"scope","navigation":"module","parent":14880,"result":false,"route":"../roadmap/Geometry/curves-and-surfaces/optimal-paper-m-bius-band/index.html","scope":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band","status":"can_prove","title":"Optimal paper M\u00f6bius band"},{"id":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band/unit-strip","kind":"node","lean":"MathlibExt.Geometry.OptimalMoebiusWanted.OptimalMoebius.unitStrip","parent":14927,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/optimal-paper-m-bius-band/index.html#unit-strip","scope":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band","status":"fully_proved","summary":"The unit strip is the set of points $(x, y) \\in \\mathbb{R}^2$ with $0 \\le x \\le 1$, that is, $[0, 1] \\times \\mathbb{R}$.","title":"Unit strip"},{"id":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band/period","kind":"node","lean":"MathlibExt.Geometry.OptimalMoebiusWanted.OptimalMoebius.period","parent":14927,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/optimal-paper-m-bius-band/index.html#period","scope":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band","status":"fully_proved","summary":"For a real number $a$, this is the map $\\mathbb{R}^2 \\to \\mathbb{R}^2$ given by $(x, y) \\mapsto (1 - x, y + a)$.","title":"Gluing map of the Moebius band"},{"id":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band/fundamental-domain","kind":"node","lean":"MathlibExt.Geometry.OptimalMoebiusWanted.OptimalMoebius.fundamentalDomain","parent":14927,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/optimal-paper-m-bius-band/index.html#fundamental-domain","scope":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band","status":"fully_proved","summary":"For a real number $a$, this is the set of points $(x, y) \\in \\mathbb{R}^2$ with $0 \\le x \\le 1$ and $0 \\le y < a$.","title":"Fundamental domain of the Moebius band"},{"id":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band/is-moebius-embedding","kind":"node","lean":"MathlibExt.Geometry.OptimalMoebiusWanted.OptimalMoebius.IsMoebiusEmbedding","parent":14927,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/optimal-paper-m-bius-band/index.html#is-moebius-embedding","scope":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band","status":"fully_proved","summary":"Let $a$ be a real number, let $S = [0, 1] \\times \\mathbb{R}$ be the unit strip and let $f : \\mathbb{R}^2 \\to \\mathbb{R}^3$. Then $f$ is a Moebius embedding of aspect ratio $a$ if $f(1 - x, y + a) = f(x, y)$ for every $(x, y) \\in S$, $f$ is $C^\\infty$ on $S$, at every point of $S$ the derivative of $f$ within $S$ is a linear isometry $\\mathbb{R}^2 \\to \\mathbb{R}^3$, and $f$ is injective on $[0, 1] \\times [0, a)$.","title":"Moebius embedding"},{"id":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band/theorem-1-1","kind":"node","lean":"MathlibExt.Geometry.OptimalMoebiusWanted.OptimalMoebius.theorem_1_1","parent":14927,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/optimal-paper-m-bius-band/index.html#theorem-1-1","scope":"Geometry/curves-and-surfaces/optimal-paper-m-bius-band","status":"can_prove","summary":"Let $a > 0$. If $f : \\mathbb{R}^2 \\to \\mathbb{R}^3$ is a Moebius embedding of aspect ratio $a$, then $a > \\sqrt{3}$. That is, a smooth embedded paper Moebius band has aspect ratio greater than $\\sqrt{3}$.","title":"Schwartz's optimal paper Moebius band theorem"},{"data_url":"scopes/Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point.json","id":"Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point","kind":"scope","navigation":"module","parent":14880,"result":false,"route":"../roadmap/Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point/index.html","scope":"Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point","status":"can_prove","title":"Space curves with a tangent line at every point"},{"id":"Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point/has-tangent-line-at","kind":"node","lean":"MathlibExt.Geometry.ScottishBook117Wanted.HasTangentLineAt","parent":14933,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point/index.html#has-tangent-line-at","scope":"Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point","status":"fully_proved","summary":"A `1`-periodic continuous injective-on-`[0, 1)` curve `\u03b3 : \u211d \u2192 \u211d\u00b3` has an (oriented) tangent line at `t` if there is a unit direction `v` such that every nearby secant vector `\u03b3 s - \u03b3 t` is arbitrarily close, relative to its length, to the line spanned by `v`.","title":"Tangent line to a closed space curve"},{"id":"Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point/scottish-book-problem117-claim","kind":"node","lean":"MathlibExt.Geometry.ScottishBook117Wanted.ScottishBookProblem117Claim","parent":14933,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point/index.html#scottish-book-problem117-claim","scope":"Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point","status":"fully_proved","summary":"The claim of Fr\u00e9chet's Problem 117 from the Scottish Book is the statement that for every $1$-periodic continuous map $\\gamma : \\mathbb{R} \\to \\mathbb{R}^3$ that is injective on $[0, 1)$ and has a tangent line at every $t \\in \\mathbb{R}$, there is a $1$-periodic continuous map $p : \\mathbb{R} \\to \\mathbb{R}^3$ that is injective on $[0, 1)$, has the same image as $\\gamma$, and is differentiable at every\u2026","title":"The claim of Fr\u00e9chet's Problem 117"},{"id":"Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point/frechet117-refuted","kind":"node","lean":"MathlibExt.Geometry.ScottishBook117Wanted.frechet117_refuted","parent":14933,"result":true,"route":"../roadmap/Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point/index.html#frechet117-refuted","scope":"Geometry/curves-and-surfaces/space-curves-with-a-tangent-line-at-every-point","status":"can_prove","summary":"There is a $1$-periodic continuous map $\\gamma : \\mathbb{R} \\to \\mathbb{R}^3$, injective on $[0, 1)$, that has a tangent line at every parameter $t$, such that no $1$-periodic continuous map $p : \\mathbb{R} \\to \\mathbb{R}^3$ that is injective on $[0, 1)$ and has the same image as $\\gamma$ is differentiable at every $t$ with $p'(t) \\neq 0$.","title":"Fr\u00e9chet's Problem 117 has a negative answer"},{"data_url":"scopes/Geometry/metric-geometry.json","id":"Geometry/metric-geometry","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/metric-geometry/index.html","scope":"Geometry/metric-geometry","status":"can_prove","summary":"Geometry of general metric spaces: curvature bounds via comparison, length spaces, isometric embeddings and low-distortion embeddings.","title":"Metric geometry"},{"id":"Geometry/metric-geometry/johnson-lindenstrauss","kind":"node","lean":"MathlibExt.Geometry.Euclidean.JohnsonLindenstraussWanted.johnson_lindenstrauss","parent":14937,"result":true,"route":"../roadmap/Geometry/metric-geometry/index.html#johnson-lindenstrauss","scope":"Geometry/metric-geometry","status":"fully_proved","summary":"There is a constant $C > 0$ with the following property. Let $\\varepsilon > 0$, let $m$ and $N \\ge 2$ be natural numbers, let $X$ be a set of exactly $N$ points of $\\mathbb{R}^m$ with the Euclidean norm, and let $d$ be a natural number with $d > C \\varepsilon^{-2} \\log N$. Then there is a map $f : \\mathbb{R}^m \\to \\mathbb{R}^d$ such that\u2026","title":"Johnson\u2013Lindenstrauss lemma"},{"id":"Geometry/metric-geometry/microscopic-weighting-iff-refuted","kind":"node","lean":"MathlibExt.Analysis.MicroscopicWeightingWanted.microscopicWeightingIff_refuted","parent":14937,"result":true,"route":"../roadmap/Geometry/metric-geometry/index.html#microscopic-weighting-iff-refuted","scope":"Geometry/metric-geometry","status":"can_prove","summary":"It is not true that every nonempty finite metric space $X$ has a microscopic weighting if and only if it has finite concentration. Here, for real $t$, $Z_t$ is the $X \\times X$ matrix with entries $e^{-t\\,d(x,y)}$ and $w_t = Z_t^{-1}\\mathbf{1}$ is its weighting, where $\\mathbf{1}$ is the all-ones vector and the inverse of a singular matrix is taken to be $0$. The space $X$ has a microscopic weighting if $w_t$\u2026","title":"Microscopic weighting is not equivalent to finite concentration"},{"id":"Geometry/metric-geometry/no-compact-set-congruent-to-proper-subse","kind":"node","lean":"MathlibExt.Geometry.ScottishBook3Wanted.noCompactSetCongruentToProperSubset","parent":14937,"result":true,"route":"../roadmap/Geometry/metric-geometry/index.html#no-compact-set-congruent-to-proper-subse","scope":"Geometry/metric-geometry","status":"can_prove","summary":"Let $K$ be a compact subset of a metric space $X$. Then there is no map $f : X \\to X$ with $d(f(x), f(y)) = d(x, y)$ for all $x, y \\in X$ such that $f(K) \\subseteq K$ and $f(K) \\neq K$.","title":"No compact set is congruent to a proper subset of itself"},{"data_url":"scopes/Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics.json","id":"Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics","kind":"scope","navigation":"module","parent":14937,"result":false,"route":"../roadmap/Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics/index.html","scope":"Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics","status":"can_prove","title":"Asymptotically equivalent cocompact metrics"},{"id":"Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics/owr1319022-data","kind":"node","lean":"MathlibExt.Geometry.CocompactAsymptoticallyEquivalentMetricsWanted.OWR1319022Data","parent":14941,"result":true,"route":"../roadmap/Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics/index.html#owr1319022-data","scope":"Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics","status":"fully_proved","summary":"Let $\\Gamma$ be a group acting on a complete metric space $(M, d_1)$. A cocompact asymptotically equivalent length-space pair on $M$ is a function $d_2 \\colon M \\times M \\to \\mathbb{R}$ with the following properties. It is a metric: $d_2(x, y) \\ge 0$, $d_2(x, y) = 0$ if and only if $x = y$, $d_2(x, y) = d_2(y, x)$ and $d_2(x, z) \\le d_2(x, y) + d_2(y, z)$. It is complete: every sequence that is Cauchy for $d_2$\u2026","title":"Cocompact asymptotically equivalent length-space pair"},{"id":"Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics/conjecture","kind":"node","lean":"MathlibExt.Geometry.CocompactAsymptoticallyEquivalentMetricsWanted.conjecture","parent":14941,"result":true,"route":"../roadmap/Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics/index.html#conjecture","scope":"Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics","status":"fully_proved","summary":"The statement that for every group $\\Gamma$, every complete metric space $(M, d_1)$ on which $\\Gamma$ acts, and every cocompact asymptotically equivalent length-space pair on $M$ with second metric $d_2$, there is a real number $C$ such that $|d_1(x, y) - d_2(x, y)| \\le C$ for all $x, y \\in M$.","title":"Bounded-difference conjecture for cocompact asymptotically equivalent metrics"},{"id":"Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics/conjecture-refuted","kind":"node","lean":"MathlibExt.Geometry.CocompactAsymptoticallyEquivalentMetricsWanted.conjecture_refuted","parent":14941,"result":true,"route":"../roadmap/Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics/index.html#conjecture-refuted","scope":"Geometry/metric-geometry/asymptotically-equivalent-cocompact-metrics","status":"can_prove","summary":"The conjecture is false: there exist a group $\\Gamma$, a complete metric space $(M, d_1)$ on which $\\Gamma$ acts, and a cocompact asymptotically equivalent length-space pair on $M$ with second metric $d_2$, such that for every real number $C$ there are points $x, y \\in M$ with $|d_1(x, y) - d_2(x, y)| > C$.","title":"Negative answer to the cocompact asymptotic-metrics question"},{"data_url":"scopes/Geometry/metric-geometry/cat-0-spaces.json","id":"Geometry/metric-geometry/cat-0-spaces","kind":"scope","navigation":"module","parent":14937,"result":false,"route":"../roadmap/Geometry/metric-geometry/cat-0-spaces/index.html","scope":"Geometry/metric-geometry/cat-0-spaces","status":"fully_proved","title":"CAT(0) spaces"},{"id":"Geometry/metric-geometry/cat-0-spaces/is-cat0-four-point","kind":"node","lean":"IsCAT0FourPoint","parent":14945,"result":true,"route":"../roadmap/Geometry/metric-geometry/cat-0-spaces/index.html#is-cat0-four-point","scope":"Geometry/metric-geometry/cat-0-spaces","status":"fully_proved","summary":"A metric space $X$ satisfies the CAT(0) four-point comparison condition if for all points $x, y, z, w \\in X$ (not necessarily distinct),","title":"CAT(0) four-point condition"},{"id":"Geometry/metric-geometry/cat-0-spaces/is-cat0-four-point-iff","kind":"node","lean":"isCAT0FourPoint_iff","parent":14945,"result":true,"route":"../roadmap/Geometry/metric-geometry/cat-0-spaces/index.html#is-cat0-four-point-iff","scope":"Geometry/metric-geometry/cat-0-spaces","status":"fully_proved","summary":"A metric space $X$ satisfies the CAT(0) four-point condition if and only if for all points $x, y, z, w \\in X$,","title":"Unfolding the CAT(0) four-point condition"},{"id":"Geometry/metric-geometry/cat-0-spaces/of-isometry","kind":"node","lean":"IsCAT0FourPoint.of_isometry","parent":14945,"result":true,"route":"../roadmap/Geometry/metric-geometry/cat-0-spaces/index.html#of-isometry","scope":"Geometry/metric-geometry/cat-0-spaces","status":"fully_proved","summary":"Let $X$ and $Y$ be metric spaces and let $f : X \\to Y$ be an isometry, that is, $d(f(a), f(b)) = d(a, b)$ for all $a, b \\in X$. If $Y$ satisfies the CAT(0) four-point condition, then so does $X$.","title":"The CAT(0) four-point condition passes to isometrically embedded spaces"},{"id":"Geometry/metric-geometry/cat-0-spaces/iff-of-isometry-equiv","kind":"node","lean":"IsCAT0FourPoint.iff_of_isometryEquiv","parent":14945,"result":true,"route":"../roadmap/Geometry/metric-geometry/cat-0-spaces/index.html#iff-of-isometry-equiv","scope":"Geometry/metric-geometry/cat-0-spaces","status":"fully_proved","summary":"If $X$ and $Y$ are metric spaces and there is an isometric bijection from $X$ onto $Y$, then $X$ satisfies the CAT(0) four-point condition if and only if $Y$ does.","title":"Isometry invariance of the CAT(0) four-point condition"},{"id":"Geometry/metric-geometry/cat-0-spaces/of-subsingleton","kind":"node","lean":"IsCAT0FourPoint.of_subsingleton","parent":14945,"result":true,"route":"../roadmap/Geometry/metric-geometry/cat-0-spaces/index.html#of-subsingleton","scope":"Geometry/metric-geometry/cat-0-spaces","status":"fully_proved","summary":"Every metric space with at most one point (including the empty space) satisfies the CAT(0) four-point condition.","title":"Spaces with at most one point satisfy the CAT(0) four-point condition"},{"data_url":"scopes/Geometry/metric-geometry/geodesics-in-metric-spaces.json","id":"Geometry/metric-geometry/geodesics-in-metric-spaces","kind":"scope","navigation":"module","parent":14937,"result":false,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","title":"Geodesics in metric spaces"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/is-geodesic-on-with-speed","kind":"node","lean":"Metric.IsGeodesicOnWithSpeed","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#is-geodesic-on-with-speed","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $X$ be a pseudo-extended metric space with extended distance $d$, let $\\gamma : \\mathbb{R} \\to X$ be a curve, $s \\subseteq \\mathbb{R}$ a set of times and $c \\ge 0$ a real number. The curve $\\gamma$ is a geodesic of speed $c$ on $s$ if every $t \\in s$ has some $\\varepsilon > 0$ such that $d(\\gamma(t_1), \\gamma(t_2)) = c\\,|t_1 - t_2|$ for all $t_1, t_2 \\in s$ with $|t_1 - t| < \\varepsilon$ and\u2026","title":"Geodesics of a given speed on a set of times"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/is-geodesic-on","kind":"node","lean":"Metric.IsGeodesicOn","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#is-geodesic-on","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $X$ be a pseudo-extended metric space. A curve $\\gamma : \\mathbb{R} \\to X$ is a geodesic on a set of times $s \\subseteq \\mathbb{R}$ if, for some real number $c \\ge 0$, it is a geodesic of speed $c$ on $s$.","title":"Geodesic on a set of times"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/is-geodesic","kind":"node","lean":"Metric.IsGeodesic","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#is-geodesic","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $X$ be a pseudo-extended metric space. A curve $\\gamma : \\mathbb{R} \\to X$ is a geodesic if it is a geodesic on the set of all times $\\mathbb{R}$.","title":"Geodesics defined for all times"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/is-closed-geodesic","kind":"node","lean":"Metric.IsClosedGeodesic","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#is-closed-geodesic","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $X$ be a pseudo-extended metric space. A curve $\\gamma : \\mathbb{R} \\to X$ is a closed geodesic if it is a geodesic defined on all of $\\mathbb{R}$, it is periodic with some period $T > 0$ (that is, $\\gamma(t + T) = \\gamma(t)$ for all $t \\in \\mathbb{R}$), and it is not constant: $\\gamma(t_1) \\ne \\gamma(t_2)$ for some $t_1, t_2 \\in \\mathbb{R}$.","title":"Closed geodesic"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/is-geodesic-const","kind":"node","lean":"Metric.isGeodesic_const","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#is-geodesic-const","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $X$ be a pseudo-extended metric space. For every point $p \\in X$, the constant curve $\\mathbb{R} \\to X$, $t \\mapsto p$, is a geodesic.","title":"Constant curves are geodesics"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/not-is-closed-geodesic-const","kind":"node","lean":"Metric.not_isClosedGeodesic_const","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#not-is-closed-geodesic-const","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $X$ be a pseudo-extended metric space. For every point $p \\in X$, the constant curve $\\mathbb{R} \\to X$, $t \\mapsto p$, is not a closed geodesic.","title":"Constant curves are not closed geodesics"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/is-geodesic-on-with-speed-line-map","kind":"node","lean":"Metric.isGeodesicOnWithSpeed_lineMap","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#is-geodesic-on-with-speed-line-map","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $F$ be a real normed space and $p, v \\in F$. For every set of times $s \\subseteq \\mathbb{R}$, the straight line $t \\mapsto p + t v$ is a geodesic of speed $\\|v\\|$ on $s$.","title":"Straight lines are geodesics of constant speed"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/is-geodesic-line-map","kind":"node","lean":"Metric.isGeodesic_lineMap","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#is-geodesic-line-map","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $F$ be a real normed space. For all $p, v \\in F$, the straight line $\\mathbb{R} \\to F$, $t \\mapsto p + t v$, is a geodesic.","title":"Straight lines are geodesics"},{"id":"Geometry/metric-geometry/geodesics-in-metric-spaces/not-const","kind":"node","lean":"Metric.IsClosedGeodesic.not_const","parent":14951,"result":true,"route":"../roadmap/Geometry/metric-geometry/geodesics-in-metric-spaces/index.html#not-const","scope":"Geometry/metric-geometry/geodesics-in-metric-spaces","status":"fully_proved","summary":"Let $X$ be a pseudo-extended metric space. If $\\gamma : \\mathbb{R} \\to X$ is a closed geodesic, then for every $p \\in X$, $\\gamma$ is not the constant curve $t \\mapsto p$.","title":"Closed geodesics are not constant"},{"data_url":"scopes/Geometry/metric-geometry/gromov-hyperbolic-spaces.json","id":"Geometry/metric-geometry/gromov-hyperbolic-spaces","kind":"scope","navigation":"module","parent":14937,"result":false,"route":"../roadmap/Geometry/metric-geometry/gromov-hyperbolic-spaces/index.html","scope":"Geometry/metric-geometry/gromov-hyperbolic-spaces","status":"fully_proved","title":"Gromov-hyperbolic spaces"},{"id":"Geometry/metric-geometry/gromov-hyperbolic-spaces/is-delta-hyperbolic","kind":"node","lean":"Metric.IsDeltaHyperbolic","parent":14961,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gromov "],["math","\\(\\delta\\)"],["text","-hyperbolic space"]],"source":"Gromov $\\delta$-hyperbolic space"},"result":true,"route":"../roadmap/Geometry/metric-geometry/gromov-hyperbolic-spaces/index.html#is-delta-hyperbolic","scope":"Geometry/metric-geometry/gromov-hyperbolic-spaces","status":"fully_proved","summary":"A pseudometric space `X` is `\u03b4`-hyperbolic (in the sense of Gromov) if `0 \u2264 \u03b4` and the Gromov product satisfies the four-point inequality `(x \u2223 z)_w \u2265 min (x \u2223 y)_w (y \u2223 z)_w - \u03b4` for all points `w x y z`, where `(x \u2223 y)_w = Metric.gromovProduct w x y` (basepoint first, as in Mathlib).","title":"Gromov $\\delta$-hyperbolic space"},{"id":"Geometry/metric-geometry/gromov-hyperbolic-spaces/is-gromov-hyperbolic","kind":"node","lean":"Metric.IsGromovHyperbolic","parent":14961,"result":true,"route":"../roadmap/Geometry/metric-geometry/gromov-hyperbolic-spaces/index.html#is-gromov-hyperbolic","scope":"Geometry/metric-geometry/gromov-hyperbolic-spaces","status":"fully_proved","summary":"A pseudometric space $X$ is Gromov-hyperbolic if there is a real number $\\delta$ such that $X$ is $\\delta$-hyperbolic.","title":"Gromov-hyperbolic spaces"},{"id":"Geometry/metric-geometry/gromov-hyperbolic-spaces/four-point","kind":"node","lean":"Metric.IsDeltaHyperbolic.four_point","parent":14961,"reading_title":{"format":"inline-parts-v1","parts":[["text","Four-point inequality of a "],["math","\\(\\delta\\)"],["text","-hyperbolic space"]],"source":"Four-point inequality of a $\\delta$-hyperbolic space"},"result":true,"route":"../roadmap/Geometry/metric-geometry/gromov-hyperbolic-spaces/index.html#four-point","scope":"Geometry/metric-geometry/gromov-hyperbolic-spaces","status":"fully_proved","summary":"The defining four-point inequality of a `\u03b4`-hyperbolic space.","title":"Four-point inequality of a $\\delta$-hyperbolic space"},{"id":"Geometry/metric-geometry/gromov-hyperbolic-spaces/mono","kind":"node","lean":"Metric.IsDeltaHyperbolic.mono","parent":14961,"result":true,"route":"../roadmap/Geometry/metric-geometry/gromov-hyperbolic-spaces/index.html#mono","scope":"Geometry/metric-geometry/gromov-hyperbolic-spaces","status":"fully_proved","summary":"If a pseudometric space $X$ is $\\delta$-hyperbolic and $\\delta \\le \\delta'$, then $X$ is $\\delta'$-hyperbolic.","title":"Hyperbolicity is monotone in the constant"},{"id":"Geometry/metric-geometry/gromov-hyperbolic-spaces/nonneg","kind":"node","lean":"Metric.IsDeltaHyperbolic.nonneg","parent":14961,"result":true,"route":"../roadmap/Geometry/metric-geometry/gromov-hyperbolic-spaces/index.html#nonneg","scope":"Geometry/metric-geometry/gromov-hyperbolic-spaces","status":"fully_proved","summary":"If a pseudometric space $X$ is $\\delta$-hyperbolic, then $\\delta \\ge 0$.","title":"The hyperbolicity constant is nonnegative"},{"id":"Geometry/metric-geometry/gromov-hyperbolic-spaces/is-delta-hyperbolic-zero-of-subsingleton","kind":"node","lean":"Metric.isDeltaHyperbolic_zero_of_subsingleton","parent":14961,"reading_title":{"format":"inline-parts-v1","parts":[["text","A space with at most one point is "],["math","\\(0\\)"],["text","-hyperbolic"]],"source":"A space with at most one point is $0$-hyperbolic"},"result":true,"route":"../roadmap/Geometry/metric-geometry/gromov-hyperbolic-spaces/index.html#is-delta-hyperbolic-zero-of-subsingleton","scope":"Geometry/metric-geometry/gromov-hyperbolic-spaces","status":"fully_proved","summary":"Every pseudometric space $X$ with at most one point is $0$-hyperbolic.","title":"A space with at most one point is $0$-hyperbolic"},{"data_url":"scopes/Geometry/metric-geometry/pointed-gromov-hausdorff-convergence.json","id":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence","kind":"scope","navigation":"module","parent":14937,"result":false,"route":"../roadmap/Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/index.html","scope":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence","status":"fully_proved","title":"Pointed Gromov\u2013Hausdorff convergence"},{"id":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/pointed-ghconverges","kind":"node","lean":"GromovHausdorff.PointedGHConverges","parent":14968,"result":true,"route":"../roadmap/Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/index.html#pointed-ghconverges","scope":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ and $Y$ be pseudometric spaces with base points $p_n \\in X_n$ and $q \\in Y$. The sequence $(X_n, p_n)$ converges to $(Y, q)$ in the pointed Gromov\u2013Hausdorff sense if for all $R > 0$ and $\\varepsilon > 0$ there is $N$ such that for every $n \\ge N$ there is a map $f$ from the open ball $B(p_n, R)$ to $Y$ with $f(p_n) = q$, with $|d(f(x), f(y)) - d(x, y)| \\le \\varepsilon$ for all\u2026","title":"Pointed Gromov\u2013Hausdorff convergence"},{"id":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/pointed-ghconverges-iff-dis-lt","kind":"node","lean":"GromovHausdorff.pointedGHConverges_iff_dis_lt","parent":14968,"result":true,"route":"../roadmap/Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/index.html#pointed-ghconverges-iff-dis-lt","scope":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ and $Y$ be pseudometric spaces with base points $p_n \\in X_n$ and $q \\in Y$. Then $(X_n, p_n)$ converges to $(Y, q)$ in the pointed Gromov\u2013Hausdorff sense if and only if for all $R > 0$ and $\\varepsilon > 0$ there is $N$ such that for every $n \\ge N$ there are a map $f$ from the open ball $B(p_n, R)$ to $Y$ and a real number $\\delta < \\varepsilon$ with $f(p_n) = q$, with\u2026","title":"Pointed Gromov\u2013Hausdorff convergence with strict distortion"},{"id":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/pointed-ghconverges-const","kind":"node","lean":"GromovHausdorff.pointedGHConverges_const","parent":14968,"result":true,"route":"../roadmap/Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/index.html#pointed-ghconverges-const","scope":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence","status":"fully_proved","summary":"For every pseudometric space $Y$ and every point $q \\in Y$, the constant sequence $(X_n, p_n) = (Y, q)$ converges to $(Y, q)$ in the pointed Gromov\u2013Hausdorff sense.","title":"Constant sequences converge in the pointed Gromov\u2013Hausdorff sense"},{"id":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/pointed-ghconverges-iff-exists-measurabl","kind":"node","lean":"GromovHausdorff.pointedGHConverges_iff_exists_measurable","parent":14968,"result":true,"route":"../roadmap/Geometry/metric-geometry/pointed-gromov-hausdorff-convergence/index.html#pointed-ghconverges-iff-exists-measurabl","scope":"Geometry/metric-geometry/pointed-gromov-hausdorff-convergence","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be separable pseudometric spaces, each with its Borel $\\sigma$-algebra, with base points $p_n \\in X_n$, and let $Y$ be a pseudometric space with any $\\sigma$-algebra and a base point $q \\in Y$. Then $(X_n, p_n)$ converges to $(Y, q)$ in the pointed Gromov\u2013Hausdorff sense if and only if for all $R > 0$ and $\\varepsilon > 0$ there is $N$ such that for every $n \\ge N$ there is a map $f$ from the\u2026","title":"Pointed Gromov\u2013Hausdorff convergence with measurable maps"},{"data_url":"scopes/Geometry/differential-topology.json","id":"Geometry/differential-topology","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/differential-topology/index.html","scope":"Geometry/differential-topology","status":"can_prove","summary":"Smooth manifolds and smooth maps between them: immersions, submersions, transversality, Morse theory, neighbourhood theorems and smooth structures.","title":"Differential topology"},{"id":"Geometry/differential-topology/four-manifold-not-smooth","kind":"node","lean":"MathlibExt.Geometry.FourManifoldNotSmoothWanted.Geometry.four_manifold_not_smooth","parent":14973,"result":true,"route":"../roadmap/Geometry/differential-topology/index.html#four-manifold-not-smooth","scope":"Geometry/differential-topology","status":"can_prove","summary":"There exists a compact, Hausdorff, simply connected topological space $M$ that admits an atlas of charts modelled on $\\mathbb{R}^4$, such that no atlas of charts on $M$ modelled on $\\mathbb{R}^4$ makes $M$ a smooth ($C^\\infty$) manifold.","title":"A closed simply connected topological 4-manifold with no smooth structure"},{"id":"Geometry/differential-topology/topological-collar-neighborhood-of-smoot","kind":"node","lean":"MathlibExt.Geometry.Manifold.CollarNeighborhoodWanted.topological_collar_neighborhood_of_smooth_manifold_with_boundary","parent":14973,"result":true,"route":"../roadmap/Geometry/differential-topology/index.html#topological-collar-neighborhood-of-smoot","scope":"Geometry/differential-topology","status":"fully_proved","summary":"Let $n \\ge 1$ and let $M$ be a Hausdorff, second-countable smooth ($C^\\infty$) manifold with boundary modeled on the closed half-space $\\{x \\in \\mathbb{R}^n : x_1 \\ge 0\\}$, and let $\\partial M$ be its boundary. Then there is an open embedding $c : \\partial M \\times [0, 1) \\to M$ such that $c(b, 0) = b$ for every $b \\in \\partial M$.","title":"Collar neighborhood theorem (topological form)"},{"id":"Geometry/differential-topology/hadamard-caccioppoli-global-inverse","kind":"node","lean":"MathlibExt.Geometry.Manifold.HadamardGlobalInverseWanted.hadamard_caccioppoli_global_inverse","parent":14973,"result":true,"route":"../roadmap/Geometry/differential-topology/index.html#hadamard-caccioppoli-global-inverse","scope":"Geometry/differential-topology","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and let $f : \\mathbb{R}^n \\to \\mathbb{R}^n$ be a proper map that is a $C^\\infty$ local diffeomorphism. Then $f$ is a global $C^\\infty$ diffeomorphism: there is a $C^\\infty$ diffeomorphism $\\Phi : \\mathbb{R}^n \\to \\mathbb{R}^n$ whose underlying map is $f$.","title":"Hadamard\u2013Caccioppoli global inverse theorem"},{"id":"Geometry/differential-topology/hairy-ball-even-sphere-ambient","kind":"node","lean":"MathlibExt.Geometry.Manifold.HairyBallWanted.hairyBall_even_sphere_ambient","parent":14973,"result":true,"route":"../roadmap/Geometry/differential-topology/index.html#hairy-ball-even-sphere-ambient","scope":"Geometry/differential-topology","status":"fully_proved","summary":"Let $k \\in \\mathbb{N}$ and let $E$ be a real inner product space of dimension $2k + 1$, with unit sphere $S = \\{x \\in E : \\|x\\| = 1\\}$. If $f : S \\to E$ is continuous and $\\langle f(x), x \\rangle = 0$ for every $x \\in S$, then $f(x) = 0$ for some $x \\in S$.","title":"Hairy ball theorem"},{"id":"Geometry/differential-topology/tubular-smooth-retraction","kind":"node","lean":"MathlibExt.Geometry.Manifold.TubularNeighborhoodWanted.tubular_smoothRetraction","parent":14973,"reading_title":{"format":"inline-parts-v1","parts":[["text","Smooth retraction onto a closed embedded manifold in "],["math","\\(\\mathbb{R}^n\\)"]],"source":"Smooth retraction onto a closed embedded manifold in $\\mathbb{R}^n$"},"result":true,"route":"../roadmap/Geometry/differential-topology/index.html#tubular-smooth-retraction","scope":"Geometry/differential-topology","status":"fully_proved","summary":"Let $M$ be a Hausdorff, second-countable smooth ($C^\\infty$) manifold without boundary, modelled on a finite-dimensional real normed space, and let $e \\colon M \\to \\mathbb{R}^n$ be a smooth embedding (a $C^\\infty$ immersion that is a topological embedding) whose image $e(M)$ is closed in $\\mathbb{R}^n$. Then there exist an open set $U \\subseteq \\mathbb{R}^n$ containing $e(M)$ and a $C^\\infty$ map $r \\colon U \\to M$\u2026","title":"Smooth retraction onto a closed embedded manifold in $\\mathbb{R}^n$"},{"data_url":"scopes/Geometry/differential-topology/classification-of-one-manifolds.json","id":"Geometry/differential-topology/classification-of-one-manifolds","kind":"scope","navigation":"module","parent":14973,"result":false,"route":"../roadmap/Geometry/differential-topology/classification-of-one-manifolds/index.html","scope":"Geometry/differential-topology/classification-of-one-manifolds","status":"fully_proved","title":"Classification of one-manifolds"},{"id":"Geometry/differential-topology/classification-of-one-manifolds/connected-boundaryless-one-manifold-clas","kind":"node","lean":"MathlibExt.Geometry.Manifold.OneDimensionalWanted.connected_boundaryless_one_manifold_classification","parent":14979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classification of connected "],["math","\\(1\\)"],["text","-manifolds without boundary"]],"source":"Classification of connected $1$-manifolds without boundary"},"result":true,"route":"../roadmap/Geometry/differential-topology/classification-of-one-manifolds/index.html#connected-boundaryless-one-manifold-clas","scope":"Geometry/differential-topology/classification-of-one-manifolds","status":"fully_proved","summary":"Every connected, Hausdorff, second-countable smooth $1$-manifold $M$ without boundary is diffeomorphic either to the circle $S^1$ or to the real line $\\mathbb{R}$.","title":"Classification of connected $1$-manifolds without boundary"},{"id":"Geometry/differential-topology/classification-of-one-manifolds/compact-connected-one-manifold-with-boun","kind":"node","lean":"MathlibExt.Geometry.Manifold.OneDimensionalWanted.compact_connected_one_manifold_with_boundary_classification","parent":14979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classification of compact connected "],["math","\\(1\\)"],["text","-manifolds with boundary"]],"source":"Classification of compact connected $1$-manifolds with boundary"},"result":true,"route":"../roadmap/Geometry/differential-topology/classification-of-one-manifolds/index.html#compact-connected-one-manifold-with-boun","scope":"Geometry/differential-topology/classification-of-one-manifolds","status":"fully_proved","summary":"Every compact, connected, Hausdorff, second-countable smooth $1$-manifold $M$ with (possibly empty) boundary is diffeomorphic either to the circle $S^1$ or to the closed interval $[0, 1]$.","title":"Classification of compact connected $1$-manifolds with boundary"},{"data_url":"scopes/Geometry/differential-topology/immersions.json","id":"Geometry/differential-topology/immersions","kind":"scope","navigation":"module","parent":14973,"result":false,"route":"../roadmap/Geometry/differential-topology/immersions/index.html","scope":"Geometry/differential-topology/immersions","status":"fully_proved","title":"Immersions"},{"id":"Geometry/differential-topology/immersions/comp","kind":"node","lean":"Manifold.IsImmersionAtOfComplement.comp","parent":14982,"result":true,"route":"../roadmap/Geometry/differential-topology/immersions/index.html#comp","scope":"Geometry/differential-topology/immersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be charted spaces modelled through models with corners on normed spaces over $\\mathbb{K}$, where the models of $N$ and $N'$ are boundaryless. Let $F_1$ and $F_2$ be normed spaces over $\\mathbb{K}$. If $f : M \\to N$ is a $C^n$ immersion at a point $x \\in M$ with complement $F_1$ and $g : N \\to N'$\u2026","title":"Composition of immersions at a point with fixed complements"},{"id":"Geometry/differential-topology/immersions/comp-2","kind":"node","lean":"Manifold.IsImmersionAt.comp","parent":14982,"result":true,"route":"../roadmap/Geometry/differential-topology/immersions/index.html#comp-2","scope":"Geometry/differential-topology/immersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be charted spaces modelled through models with corners on normed spaces over $\\mathbb{K}$, where the models of $N$ and $N'$ are boundaryless. If $f : M \\to N$ is a $C^n$ immersion at a point $x \\in M$ and $g : N \\to N'$ is a $C^n$ immersion at $f(x)$, then $g \\circ f$ is a $C^n$ immersion at $x$.","title":"Composition of immersions at a point over boundaryless models"},{"id":"Geometry/differential-topology/immersions/comp-3","kind":"node","lean":"Manifold.IsImmersionOfComplement.comp","parent":14982,"result":true,"route":"../roadmap/Geometry/differential-topology/immersions/index.html#comp-3","scope":"Geometry/differential-topology/immersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be charted spaces modelled through models with corners on normed spaces over $\\mathbb{K}$, where the models of $N$ and $N'$ are boundaryless. Let $F_1$ and $F_2$ be normed spaces over $\\mathbb{K}$. If $f : M \\to N$ is a $C^n$ immersion with complement $F_1$ and $g : N \\to N'$ is a $C^n$ immersion\u2026","title":"Composition of immersions with fixed complements"},{"id":"Geometry/differential-topology/immersions/comp-4","kind":"node","lean":"Manifold.IsImmersion.comp","parent":14982,"result":true,"route":"../roadmap/Geometry/differential-topology/immersions/index.html#comp-4","scope":"Geometry/differential-topology/immersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be charted spaces modelled through models with corners on normed spaces over $\\mathbb{K}$, where the models of $N$ and $N'$ are boundaryless. If $f : M \\to N$ and $g : N \\to N'$ are $C^n$ immersions, then $g \\circ f$ is a $C^n$ immersion.","title":"Composition of immersions over boundaryless models"},{"id":"Geometry/differential-topology/immersions/immersion-at-comp","kind":"node","lean":"MathlibExt.Geometry.Manifold.ImmersionCompWanted.immersionAt_comp","parent":14982,"result":true,"route":"../roadmap/Geometry/differential-topology/immersions/index.html#immersion-at-comp","scope":"Geometry/differential-topology/immersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be $C^n$ manifolds without boundary modeled on Banach spaces over $\\mathbb{K}$. If $f : M \\to N$ is a $C^n$ immersion at a point $x \\in M$ and $g : N \\to N'$ is a $C^n$ immersion at $f(x)$, then $g \\circ f$ is a $C^n$ immersion at $x$.","title":"Composition of immersions at a point"},{"id":"Geometry/differential-topology/immersions/immersion-comp","kind":"node","lean":"MathlibExt.Geometry.Manifold.ImmersionCompWanted.immersion_comp","parent":14982,"result":true,"route":"../roadmap/Geometry/differential-topology/immersions/index.html#immersion-comp","scope":"Geometry/differential-topology/immersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be $C^n$ manifolds without boundary modeled on Banach spaces over $\\mathbb{K}$. If $f : M \\to N$ and $g : N \\to N'$ are $C^n$ immersions, then $g \\circ f$ is a $C^n$ immersion.","title":"Composition of immersions"},{"id":"Geometry/differential-topology/immersions/of-injective-mfderiv-of-closed-complemen","kind":"node","lean":"Manifold.IsImmersionAt.of_injective_mfderiv_of_closedComplemented_range","parent":14982,"result":true,"route":"../roadmap/Geometry/differential-topology/immersions/index.html#of-injective-mfderiv-of-closed-complemen","scope":"Geometry/differential-topology/immersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $n \\ge 1$ be a smoothness degree (possibly $\\infty$ or $\\omega$), and let $M$ and $N$ be $C^n$ manifolds without boundary over $\\mathbb{K}$, modelled on Banach spaces $E$ and $E'$. Let $f \\colon M \\to N$ be $C^n$ at every point of some neighbourhood of a point $x \\in M$, and suppose that its differential $df_x \\colon E \\to E'$ is injective and that there is a\u2026","title":"Injective differential with complemented range gives an immersion"},{"id":"Geometry/differential-topology/immersions/immersion-at-of-injective-mfderiv","kind":"node","lean":"MathlibExt.Geometry.Manifold.ImmersionOfInjectiveMFDerivWanted.immersionAt_of_injective_mfderiv","parent":14982,"result":true,"route":"../roadmap/Geometry/differential-topology/immersions/index.html#immersion-at-of-injective-mfderiv","scope":"Geometry/differential-topology/immersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $n \\ge 1$ be a smoothness degree (possibly $\\infty$ or $\\omega$), and let $M$ and $N$ be $C^n$ manifolds without boundary over $\\mathbb{K}$, modelled on finite-dimensional normed $\\mathbb{K}$-vector spaces. If a map $f \\colon M \\to N$ is $C^n$ at every point of some neighbourhood of a point $x \\in M$ and its differential $df_x$ at $x$ is injective, then $f$ is a\u2026","title":"Injective differential gives an immersion"},{"data_url":"scopes/Geometry/differential-topology/regular-level-sets.json","id":"Geometry/differential-topology/regular-level-sets","kind":"scope","navigation":"module","parent":14973,"result":false,"route":"../roadmap/Geometry/differential-topology/regular-level-sets/index.html","scope":"Geometry/differential-topology/regular-level-sets","status":"fully_proved","title":"Regular level sets"},{"id":"Geometry/differential-topology/regular-level-sets/exists-local-graph-of-surjective-fderiv","kind":"node","lean":"ContDiff.exists_localGraph_of_surjective_fderiv","parent":14991,"result":true,"route":"../roadmap/Geometry/differential-topology/regular-level-sets/index.html#exists-local-graph-of-surjective-fderiv","scope":"Geometry/differential-topology/regular-level-sets","status":"fully_proved","summary":"Let $E$, $G$ and $F$ be finite-dimensional real normed spaces with $\\dim E = \\dim G + \\dim F$, let $f \\colon E \\to F$ be a $C^1$ map and let $x_0 \\in E$ be a point at which the derivative $Df(x_0)$ is surjective. Then there exist a continuous linear isomorphism $e \\colon E \\to G \\times F$, a neighbourhood $U$ in $G$ of the first component of $e(x_0)$, a map $g \\colon G \\to F$ that is $C^1$ on $U$, and a\u2026","title":"Local graph form of a level set near a regular point"},{"id":"Geometry/differential-topology/regular-level-sets/regular-level-set-local-graph","kind":"node","lean":"MathlibExt.Geometry.Manifold.LocalGraphWanted.regularLevelSet_localGraph","parent":14991,"result":true,"route":"../roadmap/Geometry/differential-topology/regular-level-sets/index.html#regular-level-set-local-graph","scope":"Geometry/differential-topology/regular-level-sets","status":"fully_proved","summary":"Let $f \\colon \\mathbb{R}^{k+m} \\to \\mathbb{R}^m$ be a $C^1$ map and let $x_0 \\in \\mathbb{R}^{k+m}$ be a point at which the derivative $Df(x_0)$ is surjective. Then there exist a linear isomorphism $e \\colon \\mathbb{R}^{k+m} \\to \\mathbb{R}^k \\times \\mathbb{R}^m$, a neighbourhood $U$ in $\\mathbb{R}^k$ of the first component of $e(x_0)$, a map $g \\colon \\mathbb{R}^k \\to \\mathbb{R}^m$ that is $C^1$ on $U$, and a\u2026","title":"Regular level sets are locally graphs"},{"id":"Geometry/differential-topology/regular-level-sets/is-immersion-graph","kind":"node","lean":"ContDiff.isImmersion_graph","parent":14991,"result":true,"route":"../roadmap/Geometry/differential-topology/regular-level-sets/index.html#is-immersion-graph","scope":"Geometry/differential-topology/regular-level-sets","status":"fully_proved","summary":"Let $E$ and $F$ be real normed spaces, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$ and let $h \\colon E \\to F$ be a $C^n$ map. Then the graph map $x \\mapsto (x, h(x))$ from $E$ to $E \\times F$ is a $C^n$ immersion, where $E$ and $E \\times F$ are regarded as manifolds modelled on themselves.","title":"Graph maps are immersions"},{"id":"Geometry/differential-topology/regular-level-sets/regular-level-set-local-parametrization","kind":"node","lean":"MathlibExt.Geometry.Manifold.LocalParametrizationWanted.regularLevelSet_localParametrization","parent":14991,"result":true,"route":"../roadmap/Geometry/differential-topology/regular-level-sets/index.html#regular-level-set-local-parametrization","scope":"Geometry/differential-topology/regular-level-sets","status":"fully_proved","summary":"Let $f \\colon \\mathbb{R}^{k+m} \\to \\mathbb{R}^m$ be a $C^1$ map and let $x_0 \\in \\mathbb{R}^{k+m}$ be a point at which the derivative $Df(x_0)$ is surjective. Then there exist an open set $U \\subseteq \\mathbb{R}^k$, a point $u_0 \\in U$, a map $\\varphi \\colon \\mathbb{R}^k \\to \\mathbb{R}^{k+m}$ that is $C^1$ and a $C^1$ immersion on all of $\\mathbb{R}^k$, and a neighbourhood $V$ of $x_0$ such that\u2026","title":"Regular level sets are locally parametrized by immersions"},{"data_url":"scopes/Geometry/differential-topology/sard-s-theorem.json","id":"Geometry/differential-topology/sard-s-theorem","kind":"scope","navigation":"module","parent":14973,"result":false,"route":"../roadmap/Geometry/differential-topology/sard-s-theorem/index.html","scope":"Geometry/differential-topology/sard-s-theorem","status":"fully_proved","title":"Sard's theorem"},{"id":"Geometry/differential-topology/sard-s-theorem/critical-set","kind":"node","lean":"MathlibExt.Geometry.Manifold.SardWanted.criticalSet","parent":14996,"result":true,"route":"../roadmap/Geometry/differential-topology/sard-s-theorem/index.html#critical-set","scope":"Geometry/differential-topology/sard-s-theorem","status":"fully_proved","summary":"Let $M$ be a Hausdorff, second-countable smooth manifold without boundary modelled on a finite-dimensional real normed space, and let $f : M \\to \\mathbb{R}^n$ be a map. The critical set of $f$ is the set of points $x \\in M$ at which the manifold derivative of $f$ at $x$, a linear map $T_xM \\to \\mathbb{R}^n$, is not surjective.","title":"Critical points of a map to Euclidean space"},{"id":"Geometry/differential-topology/sard-s-theorem/critical-values","kind":"node","lean":"MathlibExt.Geometry.Manifold.SardWanted.criticalValues","parent":14996,"result":true,"route":"../roadmap/Geometry/differential-topology/sard-s-theorem/index.html#critical-values","scope":"Geometry/differential-topology/sard-s-theorem","status":"fully_proved","summary":"Let $M$ be a Hausdorff, second-countable smooth manifold without boundary modelled on a finite-dimensional real normed space, and let $f : M \\to \\mathbb{R}^n$ be a map. The set of critical values of $f$ is the image under $f$ of its critical set, the set of points $x \\in M$ at which the manifold derivative of $f$ at $x$ is not surjective.","title":"Critical values of a map to Euclidean space"},{"id":"Geometry/differential-topology/sard-s-theorem/sard-euclidean-open","kind":"node","lean":"MathlibExt.Geometry.Manifold.SardWanted.sard_euclidean_open","parent":14996,"result":true,"route":"../roadmap/Geometry/differential-topology/sard-s-theorem/index.html#sard-euclidean-open","scope":"Geometry/differential-topology/sard-s-theorem","status":"fully_proved","summary":"Let $E$ and $F$ be finite-dimensional real normed spaces, let $\\mu$ be an additive Haar measure on $F$ with its Borel $\\sigma$-algebra, let $U \\subseteq E$ be open, and let $g : E \\to F$ be $C^\\infty$ on $U$. Then the image under $g$ of the set of points $x \\in U$ at which the derivative of $g$ is not surjective has $\\mu$-measure zero.","title":"Sard's theorem on open subsets of finite-dimensional spaces"},{"id":"Geometry/differential-topology/sard-s-theorem/sard-theorem","kind":"node","lean":"MathlibExt.Geometry.Manifold.SardWanted.sard_theorem","parent":14996,"result":true,"route":"../roadmap/Geometry/differential-topology/sard-s-theorem/index.html#sard-theorem","scope":"Geometry/differential-topology/sard-s-theorem","status":"fully_proved","summary":"Let $M$ be a Hausdorff, second-countable smooth manifold without boundary modelled on a finite-dimensional real normed space, and let $f : M \\to \\mathbb{R}^n$ be a $C^\\infty$ map. Then the set of critical values of $f$, the image under $f$ of the points of $M$ at which the manifold derivative of $f$ is not surjective, has Lebesgue measure zero.","title":"Sard's theorem for maps into Euclidean space"},{"id":"Geometry/differential-topology/sard-s-theorem/bad-parameter-set","kind":"node","lean":"MathlibExt.Geometry.Manifold.ParametricSardWanted.badParameterSet","parent":14996,"result":true,"route":"../roadmap/Geometry/differential-topology/sard-s-theorem/index.html#bad-parameter-set","scope":"Geometry/differential-topology/sard-s-theorem","status":"fully_proved","summary":"Let $F \\colon \\mathbb{R}^p \\times \\mathbb{R}^m \\to \\mathbb{R}^n$ be a map and $y \\in \\mathbb{R}^n$. For $a \\in \\mathbb{R}^p$ write $F_a = F(a, \\cdot)$ for the slice $\\mathbb{R}^m \\to \\mathbb{R}^n$. The bad parameter set of $F$ at $y$ is the set of $a \\in \\mathbb{R}^p$ for which there is $x \\in \\mathbb{R}^m$ with $F(a, x) = y$ and the derivative of $F_a$ at $x$ not surjective, that is, the set of $a$ for which $y$\u2026","title":"Bad parameter set"},{"id":"Geometry/differential-topology/sard-s-theorem/parametric-sard","kind":"node","lean":"MathlibExt.Geometry.Manifold.ParametricSardWanted.parametricSard","parent":14996,"result":true,"route":"../roadmap/Geometry/differential-topology/sard-s-theorem/index.html#parametric-sard","scope":"Geometry/differential-topology/sard-s-theorem","status":"fully_proved","summary":"Let $p, m, n$ be natural numbers, let $F \\colon \\mathbb{R}^p \\times \\mathbb{R}^m \\to \\mathbb{R}^n$ be a real analytic ($C^\\omega$) map and let $y \\in \\mathbb{R}^n$. Suppose that the derivative of $F$ is surjective at every point $(a, x)$ with $F(a, x) = y$. Then the bad parameter set of $F$ at $y$, the set of $a \\in \\mathbb{R}^p$ for which some $x \\in \\mathbb{R}^m$ has $F(a, x) = y$ and the derivative of the slice\u2026","title":"Parametric Sard theorem for real analytic families"},{"data_url":"scopes/Geometry/differential-topology/morse-theory.json","id":"Geometry/differential-topology/morse-theory","kind":"scope","navigation":"module","parent":14973,"result":false,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html","scope":"Geometry/differential-topology/morse-theory","status":"can_prove","title":"Morse theory"},{"id":"Geometry/differential-topology/morse-theory/morse-quadratic","kind":"node","lean":"MathlibExt.Geometry.Manifold.MorseTheoryWanted.morseQuadratic","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#morse-quadratic","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the Morse quadratic form of index $k$ on $\\mathbb{R}^n$, with coordinates $v_0, \\dots, v_{n-1}$, is the function $Q_{k,n} \\colon \\mathbb{R}^n \\to \\mathbb{R}$ given by $Q_{k,n}(v) = \\sum_{i=0}^{n-1} \\varepsilon_i v_i^2$, where $\\varepsilon_i = -1$ if $i < k$ and $\\varepsilon_i = 1$ if $i \\ge k$. In particular, if $k \\ge n$ every coefficient is $-1$.","title":"Morse quadratic form"},{"id":"Geometry/differential-topology/morse-theory/morse-quadratic-zero","kind":"node","lean":"MathlibExt.Geometry.Manifold.MorseTheoryWanted.morseQuadratic_zero","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#morse-quadratic-zero","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"For all natural numbers $k$ and $n$, the Morse quadratic form $Q_{k,n}$ of index $k$ on $\\mathbb{R}^n$ satisfies $Q_{k,n}(0) = 0$.","title":"The Morse quadratic form vanishes at the origin"},{"id":"Geometry/differential-topology/morse-theory/cont-diff-morse-quadratic","kind":"node","lean":"MathlibExt.Geometry.Manifold.MorseTheoryWanted.contDiff_morseQuadratic","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#cont-diff-morse-quadratic","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"For all natural numbers $k$ and $n$, the Morse quadratic form $Q_{k,n} \\colon \\mathbb{R}^n \\to \\mathbb{R}$ of index $k$ is $C^\\infty$.","title":"Smoothness of the Morse quadratic form"},{"id":"Geometry/differential-topology/morse-theory/morse-quadratic-zero-index","kind":"node","lean":"MathlibExt.Geometry.Manifold.MorseTheoryWanted.morseQuadratic_zero_index","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#morse-quadratic-zero-index","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"For every natural number $n$, the Morse quadratic form $Q_{0,n}$ of index $0$ on $\\mathbb{R}^n$ satisfies $Q_{0,n}(v) = \\|v\\|^2$ for every $v \\in \\mathbb{R}^n$, where $\\|v\\|$ is the Euclidean norm of $v$.","title":"The Morse quadratic form of index zero"},{"id":"Geometry/differential-topology/morse-theory/morse-quadratic-of-le","kind":"node","lean":"MathlibExt.Geometry.Manifold.MorseTheoryWanted.morseQuadratic_of_le","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#morse-quadratic-of-le","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"If $n \\le k$, then the Morse quadratic form $Q_{k,n}$ of index $k$ on $\\mathbb{R}^n$ satisfies $Q_{k,n}(v) = -\\|v\\|^2$ for every $v \\in \\mathbb{R}^n$, where $\\|v\\|$ is the Euclidean norm of $v$.","title":"The Morse quadratic form when the index is at least the dimension"},{"id":"Geometry/differential-topology/morse-theory/is-morse-function","kind":"node","lean":"MathlibExt.Geometry.Manifold.ReebSphereWanted.IsMorseFunction","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#is-morse-function","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"Let $M$ be a $C^\\infty$ manifold without boundary modelled on a real normed space of finite dimension $d$. A function $f \\colon M \\to \\mathbb{R}$ is a Morse function if it is $C^\\infty$ and, for every critical point $p$ of $f$ (a point where the differential $df_p$ is zero), there exist an integer $0 \\le k \\le d$, a map $\\varphi \\colon M \\to \\mathbb{R}^d$ that is a local $C^\\infty$ diffeomorphism at $p$ with\u2026","title":"Morse function"},{"id":"Geometry/differential-topology/morse-theory/morse-lemma","kind":"node","lean":"MathlibExt.Geometry.Manifold.MorseLemmaWanted.morse_lemma","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#morse-lemma","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"Let $f \\colon \\mathbb{R}^n \\to \\mathbb{R}$ be a $C^\\infty$ function with $Df(0) = 0$ whose Hessian at $0$ is nondegenerate: if $v \\in \\mathbb{R}^n$ satisfies $D^2 f(0)(v, w) = 0$ for all $w \\in \\mathbb{R}^n$, then $v = 0$. Then there exist an integer $0 \\le k \\le n$, a map $\\varphi \\colon \\mathbb{R}^n \\to \\mathbb{R}^n$ that is a local $C^\\infty$ diffeomorphism at $0$ with $\\varphi(0) = 0$, and an open set $U$\u2026","title":"Morse lemma"},{"id":"Geometry/differential-topology/morse-theory/morse-function-exists","kind":"node","lean":"MathlibExt.Geometry.Manifold.MorseExistenceWanted.morse_function_exists","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#morse-function-exists","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"Every Hausdorff, second-countable $C^\\infty$ manifold $M$ without boundary, modelled on a finite-dimensional real normed space, admits a Morse function $f \\colon M \\to \\mathbb{R}$.","title":"Existence of Morse functions"},{"id":"Geometry/differential-topology/morse-theory/alternating-partial-sum","kind":"node","lean":"MathlibExt.Geometry.MorseInequalitiesWanted.Geometry.MorseInequalities.alternatingPartialSum","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#alternating-partial-sum","scope":"Geometry/differential-topology/morse-theory","status":"fully_proved","summary":"For a sequence $a_0, a_1, a_2, \\dots$ of natural numbers and $k \\in \\mathbb{N}$, the alternating partial sum of $a$ up to $k$ is the integer $\\sum_{j=0}^{k} (-1)^{k-j} a_j$.","title":"Alternating partial sum"},{"id":"Geometry/differential-topology/morse-theory/morse-inequality","kind":"node","lean":"MathlibExt.Geometry.MorseInequalitiesWanted.Geometry.MorseInequalities.morse_inequality","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#morse-inequality","scope":"Geometry/differential-topology/morse-theory","status":"can_prove","summary":"Let $M$ be a compact Hausdorff smooth manifold without boundary, modelled on a finite-dimensional real normed space, and let $f : M \\to \\mathbb{R}$ be a Morse function: $f$ is $C^\\infty$, has finitely many critical points, and the Hessian of $f$ at each critical point, computed in the preferred chart, is a nondegenerate bilinear form. The Morse index of a critical point is the largest dimension of a subspace on\u2026","title":"Morse inequalities"},{"id":"Geometry/differential-topology/morse-theory/weak-morse-inequality","kind":"node","lean":"MathlibExt.Geometry.WeakMorseInequalityWanted.Geometry.WeakMorseInequality.weak_morse_inequality","parent":15003,"result":true,"route":"../roadmap/Geometry/differential-topology/morse-theory/index.html#weak-morse-inequality","scope":"Geometry/differential-topology/morse-theory","status":"can_prove","summary":"Let $M$ be a compact Hausdorff smooth ($C^\\infty$) manifold without boundary, modelled on a finite-dimensional real normed space, and let $f \\colon M \\to \\mathbb{R}$ be a Morse function, that is, a smooth function with finitely many critical points, all of them non-degenerate. Then for every $k \\in \\mathbb{N}$ the $k$-th Betti number of $M$, the real dimension of the singular homology $H_k(M; \\mathbb{R})$, is at\u2026","title":"Weak Morse inequalities"},{"data_url":"scopes/Geometry/differential-topology/submersions.json","id":"Geometry/differential-topology/submersions","kind":"scope","navigation":"module","parent":14973,"result":false,"route":"../roadmap/Geometry/differential-topology/submersions/index.html","scope":"Geometry/differential-topology/submersions","status":"fully_proved","title":"Submersions"},{"id":"Geometry/differential-topology/submersions/exists-dom-chart-of-cod-chart","kind":"node","lean":"Manifold.IsSubmersionAtOfComplement.exists_domChart_of_codChart","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#exists-dom-chart-of-cod-chart","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$ and $N$ be charted spaces modelled, through models with corners, on normed spaces $E$ and $E''$ over $\\mathbb{K}$, both models without boundary. Let $f : M \\to N$ be a $C^n$ submersion at $x \\in M$ with complement $F$, and let $e : E \\to E'' \\times F$ be the continuous linear equivalence chosen for it. Then\u2026","title":"Changing the target chart of a submersion at a point"},{"id":"Geometry/differential-topology/submersions/comp","kind":"node","lean":"Manifold.IsSubmersionAtOfComplement.comp","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#comp","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be charted spaces modelled, through models with corners, on normed spaces $E$, $E''$, $E'''$ over $\\mathbb{K}$, where the models of $M$ and $N$ have no boundary. Let $F_1$ and $F_2$ be normed spaces over $\\mathbb{K}$. If $f : M \\to N$ is a $C^n$ submersion at $x \\in M$ with complement $F_1$ and\u2026","title":"Composition of submersions at a point with chosen complements"},{"id":"Geometry/differential-topology/submersions/comp-2","kind":"node","lean":"Manifold.IsSubmersionAt.comp","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#comp-2","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be charted spaces modelled, through models with corners, on normed spaces $E$, $E''$, $E'''$ over $\\mathbb{K}$, where the models of $M$ and $N$ have no boundary. If $f : M \\to N$ is a $C^n$ submersion at $x \\in M$ and $g : N \\to N'$ is a $C^n$ submersion at $f(x)$, then $g \\circ f$ is a $C^n$\u2026","title":"Submersions at a point are closed under composition"},{"id":"Geometry/differential-topology/submersions/comp-3","kind":"node","lean":"Manifold.IsSubmersionOfComplement.comp","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#comp-3","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be charted spaces modelled, through models with corners, on normed spaces $E$, $E''$, $E'''$ over $\\mathbb{K}$, where the models of $M$ and $N$ have no boundary. Let $F_1$ and $F_2$ be normed spaces over $\\mathbb{K}$. If $f : M \\to N$ is a $C^n$ submersion with complement $F_1$ and $g : N \\to N'$\u2026","title":"Composition of submersions with chosen complements"},{"id":"Geometry/differential-topology/submersions/comp-4","kind":"node","lean":"Manifold.IsSubmersion.comp","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#comp-4","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be charted spaces modelled, through models with corners, on normed spaces $E$, $E''$, $E'''$ over $\\mathbb{K}$, where the models of $M$ and $N$ have no boundary. If $f : M \\to N$ and $g : N \\to N'$ are $C^n$ submersions, then so is $g \\circ f$.","title":"Submersions are closed under composition"},{"id":"Geometry/differential-topology/submersions/submersion-at-comp","kind":"node","lean":"MathlibExt.Geometry.Manifold.SubmersionCompWanted.submersionAt_comp","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#submersion-at-comp","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be $C^n$ manifolds without boundary over $\\mathbb{K}$ modelled on Banach spaces. If $f : M \\to N$ is a $C^n$ submersion at $x \\in M$ and $g : N \\to N'$ is a $C^n$ submersion at $f(x)$, then $g \\circ f$ is a $C^n$ submersion at $x$. Here $f$ is a $C^n$ submersion at $x$ when there are charts of the\u2026","title":"Composition of submersions at a point"},{"id":"Geometry/differential-topology/submersions/submersion-comp","kind":"node","lean":"MathlibExt.Geometry.Manifold.SubmersionCompWanted.submersion_comp","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#submersion-comp","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be a nontrivially normed field, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$, and let $M$, $N$, $N'$ be $C^n$ manifolds without boundary over $\\mathbb{K}$ modelled on Banach spaces. If $f : M \\to N$ and $g : N \\to N'$ are $C^n$ submersions, then so is $g \\circ f$. Here $f$ is a $C^n$ submersion when there is one normed space $F$ such that at every $x \\in M$ there are charts of the $C^n$ maximal\u2026","title":"Composition of submersions"},{"id":"Geometry/differential-topology/submersions/of-surjective-mfderiv","kind":"node","lean":"Manifold.IsSubmersionAt.of_surjective_mfderiv","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#of-surjective-mfderiv","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$ with $n \\ge 1$, let $M$ be a $C^n$ manifold without boundary over $\\mathbb{K}$ modelled on a Banach space, and let $N$ be a $C^n$ manifold, possibly with boundary or corners, over $\\mathbb{K}$ modelled on a finite-dimensional space. If $f : M \\to N$ is $C^n$ at every point of some neighbourhood of $x \\in M$ and the\u2026","title":"Surjective differential into a finite-dimensional manifold gives a submersion"},{"id":"Geometry/differential-topology/submersions/submersion-at-of-surjective-mfderiv","kind":"node","lean":"MathlibExt.Geometry.Manifold.SubmersionOfSurjectiveMFDerivWanted.submersionAt_of_surjective_mfderiv","parent":15015,"result":true,"route":"../roadmap/Geometry/differential-topology/submersions/index.html#submersion-at-of-surjective-mfderiv","scope":"Geometry/differential-topology/submersions","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $n \\in \\mathbb{N} \\cup \\{\\infty, \\omega\\}$ with $n \\ge 1$, and let $M$ and $N$ be $C^n$ manifolds without boundary over $\\mathbb{K}$ modelled on finite-dimensional spaces. If $f : M \\to N$ is $C^n$ at every point of some neighbourhood of $x \\in M$ and the manifold derivative of $f$ at $x$ is surjective, then $f$ is a $C^n$ submersion at $x$: there are charts of\u2026","title":"Surjective differential gives a submersion between finite-dimensional manifolds"},{"data_url":"scopes/Geometry/differential-topology/thom-transversality-theorem.json","id":"Geometry/differential-topology/thom-transversality-theorem","kind":"scope","navigation":"module","parent":14973,"result":false,"route":"../roadmap/Geometry/differential-topology/thom-transversality-theorem/index.html","scope":"Geometry/differential-topology/thom-transversality-theorem","status":"fully_proved","title":"Thom transversality theorem"},{"id":"Geometry/differential-topology/thom-transversality-theorem/thom-transversality-uniform-approx-regul","kind":"node","lean":"MathlibExt.Geometry.Manifold.ThomTransversalityWanted.thom_transversality_uniform_approx_regular_value_zero","parent":15025,"result":true,"route":"../roadmap/Geometry/differential-topology/thom-transversality-theorem/index.html#thom-transversality-uniform-approx-regul","scope":"Geometry/differential-topology/thom-transversality-theorem","status":"fully_proved","summary":"Let $M$ be a Hausdorff, second-countable smooth manifold without boundary modelled on a finite-dimensional real normed space, let $f : M \\to \\mathbb{R}^n$ be $C^\\infty$, and let $\\varepsilon > 0$. Then there is a $C^\\infty$ map $g : M \\to \\mathbb{R}^n$ such that $\\lVert f(x) - g(x) \\rVert < \\varepsilon$ for every $x \\in M$ (Euclidean norm) and $0$ is a regular value of $g$, that is, the manifold derivative of $g$\u2026","title":"Approximation by maps having zero as a regular value"},{"id":"Geometry/differential-topology/thom-transversality-theorem/thom-transversality-compact-source-unifo","kind":"node","lean":"MathlibExt.Geometry.Manifold.ThomTransversalityWanted.thom_transversality_compact_source_uniform_approx_regular_value_zero","parent":15025,"result":true,"route":"../roadmap/Geometry/differential-topology/thom-transversality-theorem/index.html#thom-transversality-compact-source-unifo","scope":"Geometry/differential-topology/thom-transversality-theorem","status":"fully_proved","summary":"Let $M$ be a compact, Hausdorff, second-countable smooth manifold without boundary modelled on a finite-dimensional real normed space, let $f : M \\to \\mathbb{R}^n$ be $C^\\infty$, and let $\\varepsilon > 0$. Then there is a $C^\\infty$ map $g : M \\to \\mathbb{R}^n$ such that $\\lVert f(x) - g(x) \\rVert < \\varepsilon$ for every $x \\in M$ (Euclidean norm) and $0$ is a regular value of $g$, that is, the manifold derivative\u2026","title":"Approximation by maps having zero as a regular value, compact case"},{"data_url":"scopes/Geometry/complex-and-symplectic-geometry.json","id":"Geometry/complex-and-symplectic-geometry","kind":"scope","parent":13942,"result":false,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/index.html","scope":"Geometry/complex-and-symplectic-geometry","status":"can_prove","summary":"Manifolds with complex, almost complex, symplectic or contact structures, Riemann surfaces, and integrable Hamiltonian systems.","title":"Complex and symplectic geometry"},{"id":"Geometry/complex-and-symplectic-geometry/rado-riemann-surface","kind":"node","lean":"MathlibExt.Geometry.RadoTheoremWanted.Geometry.rado_riemannSurface","parent":15028,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/index.html#rado-riemann-surface","scope":"Geometry/complex-and-symplectic-geometry","status":"can_prove","summary":"Every connected Riemann surface is second countable: if $X$ is a connected Hausdorff topological space with an atlas of charts to $\\mathbb{C}$ whose transition maps are of class $C^1$ over $\\mathbb{C}$ (that is, holomorphic), then the topology of $X$ is second countable.","title":"Rad\u00f3's theorem"},{"data_url":"scopes/Geometry/complex-and-symplectic-geometry/darboux-s-theorem.json","id":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem","kind":"scope","navigation":"module","parent":15028,"result":false,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/darboux-s-theorem/index.html","scope":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem","status":"can_prove","title":"Darboux's theorem"},{"id":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem/e","kind":"node","lean":"MathlibExt.Geometry.DarbouxWanted.Geometry.Darboux.E","parent":15030,"reading_title":{"format":"inline-parts-v1","parts":[["text","Model space "],["math","\\(\\mathbb{R}^{2n}\\)"],["text"," for Darboux's theorem"]],"source":"Model space $\\mathbb{R}^{2n}$ for Darboux's theorem"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/darboux-s-theorem/index.html#e","scope":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem","status":"fully_proved","summary":"For a natural number $n$, the model space is $\\mathbb{R}^{2n}$, the Euclidean space of dimension $2n$ with coordinates indexed by $0, 1, \\dots, 2n - 1$ and the standard inner product.","title":"Model space $\\mathbb{R}^{2n}$ for Darboux's theorem"},{"id":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem/idx-p","kind":"node","lean":"MathlibExt.Geometry.DarbouxWanted.Geometry.Darboux.idxP","parent":15030,"reading_title":{"format":"inline-parts-v1","parts":[["text","Index of the "],["math","\\(i\\)"],["text","-th "],["math","\\(p\\)"],["text","-coordinate in "],["math","\\(\\mathbb{R}^{2n}\\)"]],"source":"Index of the $i$-th $p$-coordinate in $\\mathbb{R}^{2n}$"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/darboux-s-theorem/index.html#idx-p","scope":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem","status":"fully_proved","summary":"For $i \\in \\{0, \\dots, n - 1\\}$, the index of the $i$-th $p$-coordinate among the coordinates $0, \\dots, 2n - 1$ of $\\mathbb{R}^{2n}$ is $i$ itself.","title":"Index of the $i$-th $p$-coordinate in $\\mathbb{R}^{2n}$"},{"id":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem/idx-q","kind":"node","lean":"MathlibExt.Geometry.DarbouxWanted.Geometry.Darboux.idxQ","parent":15030,"reading_title":{"format":"inline-parts-v1","parts":[["text","Index of the "],["math","\\(i\\)"],["text","-th "],["math","\\(q\\)"],["text","-coordinate in "],["math","\\(\\mathbb{R}^{2n}\\)"]],"source":"Index of the $i$-th $q$-coordinate in $\\mathbb{R}^{2n}$"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/darboux-s-theorem/index.html#idx-q","scope":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem","status":"fully_proved","summary":"For $i \\in \\{0, \\dots, n - 1\\}$, the index of the $i$-th $q$-coordinate among the coordinates $0, \\dots, 2n - 1$ of $\\mathbb{R}^{2n}$ is $i + n$.","title":"Index of the $i$-th $q$-coordinate in $\\mathbb{R}^{2n}$"},{"id":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem/is-darboux-normal","kind":"node","lean":"MathlibExt.Geometry.DarbouxWanted.Geometry.Darboux.IsDarbouxNormal","parent":15030,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/darboux-s-theorem/index.html#is-darboux-normal","scope":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem","status":"fully_proved","summary":"Let $n$ be a natural number and let $e_0, \\dots, e_{2n-1}$ be the standard basis of $\\mathbb{R}^{2n}$. A continuous alternating 2-form $\\omega$ on $\\mathbb{R}^{2n}$ is in Darboux normal form if for all $i, j \\in \\{0, \\dots, n - 1\\}$ we have $\\omega(e_i, e_{n+j}) = 1$ when $i = j$ and $0$ otherwise, $\\omega(e_i, e_j) = 0$, and $\\omega(e_{n+i}, e_{n+j}) = 0$.","title":"Darboux normal form"},{"id":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem/is-symplectic-on","kind":"node","lean":"MathlibExt.Geometry.DarbouxWanted.Geometry.Darboux.IsSymplecticOn","parent":15030,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/darboux-s-theorem/index.html#is-symplectic-on","scope":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem","status":"fully_proved","summary":"Let $n$ be a natural number and $U$ a subset of $\\mathbb{R}^{2n}$. A field $\\alpha$ of continuous alternating 2-forms on $\\mathbb{R}^{2n}$ is symplectic on $U$ if $\\alpha$ is $C^\\infty$ on $U$, its exterior derivative $d\\alpha$ vanishes at every point of $U$, and $\\alpha$ is non-degenerate at every point of $U$: for every $x \\in U$ and every nonzero $v \\in \\mathbb{R}^{2n}$ there is $w \\in \\mathbb{R}^{2n}$ with\u2026","title":"Symplectic 2-form fields"},{"id":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem/darboux","kind":"node","lean":"MathlibExt.Geometry.DarbouxWanted.Geometry.Darboux.darboux","parent":15030,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/darboux-s-theorem/index.html#darboux","scope":"Geometry/complex-and-symplectic-geometry/darboux-s-theorem","status":"can_prove","summary":"Let $n$ be a natural number, let $U$ be an open subset of the model space $\\mathbb{R}^{2n}$, and let $\\alpha$ be a field of continuous alternating 2-forms on $\\mathbb{R}^{2n}$ that is symplectic on $U$. Then for every $x \\in U$ there is an open partial homeomorphism $\\varphi$ of $\\mathbb{R}^{2n}$ whose source contains $x$ and is contained in $U$, such that $\\varphi$ is $C^\\infty$ on its source, $\\varphi^{-1}$ is\u2026","title":"Darboux's theorem"},{"data_url":"scopes/Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph.json","id":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph","kind":"scope","navigation":"module","parent":15028,"result":false,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/index.html","scope":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph","status":"fully_proved","title":"Identity and open mapping theorems for holomorphic maps"},{"id":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/eq-of-eq-on-of-is-open","kind":"node","lean":"MDifferentiable.eq_of_eqOn_of_isOpen","parent":15037,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/index.html#eq-of-eq-on-of-is-open","scope":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph","status":"fully_proved","summary":"Let $M$ be a connected complex manifold without boundary, modeled on a complex normed space with complex $C^1$ (that is, holomorphic) transition maps, and let $F$ be a complex normed space. If two holomorphic (everywhere complex differentiable) maps $f, g : M \\to F$ agree on a nonempty open subset $U \\subseteq M$, then $f = g$.","title":"Holomorphic maps on a connected manifold that agree on a nonempty open set are equal"},{"id":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/holomorphic-identity-on-connected-comple","kind":"node","lean":"MathlibExt.Geometry.Manifold.ComplexRigidityWanted.holomorphic_identity_on_connected_complex_manifold","parent":15037,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/index.html#holomorphic-identity-on-connected-comple","scope":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph","status":"fully_proved","summary":"Let $M$ be a connected complex manifold without boundary, modeled on a complex normed space with complex $C^1$ (that is, holomorphic) transition maps, and let $F$ be a complex normed space. If two holomorphic (everywhere complex differentiable) maps $f, g : M \\to F$ agree on a nonempty open subset $U \\subseteq M$, then $f = g$.","title":"Identity theorem for holomorphic maps"},{"id":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/is-open-map-of-finrank-eq-one","kind":"node","lean":"MDifferentiable.isOpenMap_of_finrank_eq_one","parent":15037,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/index.html#is-open-map-of-finrank-eq-one","scope":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph","status":"fully_proved","summary":"Let $M$ and $N$ be complex manifolds without boundary modeled on complex normed spaces of dimension one, with complex $C^1$ (that is, holomorphic) transition maps, and suppose $M$ is connected. Then every holomorphic (everywhere complex differentiable) map $f : M \\to N$ that is not constant is an open map.","title":"Open mapping theorem for one-dimensional complex manifolds"},{"id":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/holomorphic-open-mapping-one-dim","kind":"node","lean":"MathlibExt.Geometry.Manifold.ComplexRigidityWanted.holomorphic_open_mapping_one_dim","parent":15037,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph/index.html#holomorphic-open-mapping-one-dim","scope":"Geometry/complex-and-symplectic-geometry/identity-and-open-mapping-theorems-for-holomorph","status":"fully_proved","summary":"Let $M$ and $N$ be complex manifolds without boundary modeled on complex normed spaces of dimension one, with complex $C^1$ (that is, holomorphic) transition maps, and suppose $M$ is connected. Then every holomorphic (everywhere complex differentiable) map $f : M \\to N$ that is not constant is an open map.","title":"Open mapping theorem for Riemann surfaces"},{"data_url":"scopes/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem.json","id":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","kind":"scope","navigation":"module","parent":15028,"result":false,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/index.html","scope":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","status":"can_prove","title":"Liouville\u2013Arnold theorem"},{"id":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/e","kind":"node","lean":"MathlibExt.Geometry.LiouvilleArnoldWanted.Geometry.LiouvilleArnold.E","parent":15042,"reading_title":{"format":"inline-parts-v1","parts":[["text","The model space "],["math","\\(\\mathbb{R}^{2n}\\)"],["text"," of the Liouville\u2013Arnold theorem"]],"source":"The model space $\\mathbb{R}^{2n}$ of the Liouville\u2013Arnold theorem"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/index.html#e","scope":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","status":"fully_proved","summary":"For a natural number $n$, $E_n$ is the Euclidean space $\\mathbb{R}^{2n}$, with coordinates indexed by $0, \\dots, 2n-1$ and the standard inner product.","title":"The model space $\\mathbb{R}^{2n}$ of the Liouville\u2013Arnold theorem"},{"id":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/idx-p","kind":"node","lean":"MathlibExt.Geometry.LiouvilleArnoldWanted.Geometry.LiouvilleArnold.idxP","parent":15042,"reading_title":{"format":"inline-parts-v1","parts":[["text","Liouville\u2013Arnold coordinates: index of "],["math","\\(p_i\\)"]],"source":"Liouville\u2013Arnold coordinates: index of $p_i$"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/index.html#idx-p","scope":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","status":"fully_proved","summary":"For $i \\in \\{0, \\dots, n-1\\}$, the coordinate $p_i$ of $\\mathbb{R}^{2n}$ is coordinate number $i$: this map sends $i$ to the same number $i$, regarded as an index in $\\{0, \\dots, 2n-1\\}$.","title":"Liouville\u2013Arnold coordinates: index of $p_i$"},{"id":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/idx-q","kind":"node","lean":"MathlibExt.Geometry.LiouvilleArnoldWanted.Geometry.LiouvilleArnold.idxQ","parent":15042,"reading_title":{"format":"inline-parts-v1","parts":[["text","Liouville\u2013Arnold coordinates: index of "],["math","\\(q_i\\)"]],"source":"Liouville\u2013Arnold coordinates: index of $q_i$"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/index.html#idx-q","scope":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","status":"fully_proved","summary":"For $i \\in \\{0, \\dots, n-1\\}$, the coordinate $q_i$ of $\\mathbb{R}^{2n}$ is coordinate number $n + i$: this map sends $i$ to $n + i \\in \\{0, \\dots, 2n-1\\}$.","title":"Liouville\u2013Arnold coordinates: index of $q_i$"},{"id":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/poisson-bracket","kind":"node","lean":"MathlibExt.Geometry.LiouvilleArnoldWanted.Geometry.LiouvilleArnold.poissonBracket","parent":15042,"reading_title":{"format":"inline-parts-v1","parts":[["text","Standard Poisson bracket on "],["math","\\(\\mathbb{R}^{2n}\\)"]],"source":"Standard Poisson bracket on $\\mathbb{R}^{2n}$"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/index.html#poisson-bracket","scope":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","status":"fully_proved","summary":"Write the coordinates of a point $x \\in \\mathbb{R}^{2n}$ as $p_i = x_i$ and $q_i = x_{n+i}$ for $i = 0, \\dots, n-1$. For functions $F, G \\colon \\mathbb{R}^{2n} \\to \\mathbb{R}$, the Poisson bracket for the symplectic form $\\omega_0 = \\sum_i dp_i \\wedge dq_i$ is the function","title":"Standard Poisson bracket on $\\mathbb{R}^{2n}$"},{"id":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/is-liouville-integrable","kind":"node","lean":"MathlibExt.Geometry.LiouvilleArnoldWanted.Geometry.LiouvilleArnold.IsLiouvilleIntegrable","parent":15042,"reading_title":{"format":"inline-parts-v1","parts":[["text","Liouville integrable system on a subset of "],["math","\\(\\mathbb{R}^{2n}\\)"]],"source":"Liouville integrable system on a subset of $\\mathbb{R}^{2n}$"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/index.html#is-liouville-integrable","scope":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","status":"fully_proved","summary":"Functions $F_0, \\dots, F_{n-1} \\colon \\mathbb{R}^{2n} \\to \\mathbb{R}$ are Liouville integrable on a set $U \\subseteq \\mathbb{R}^{2n}$ if each $F_i$ is $C^\\infty$ on $U$, the Poisson brackets $\\{F_i, F_j\\}$ vanish at every point of $U$ for all $i, j$, and at every $x \\in U$ the Fr\u00e9chet derivatives $dF_0(x), \\dots, dF_{n-1}(x)$ are linearly independent over $\\mathbb{R}$.","title":"Liouville integrable system on a subset of $\\mathbb{R}^{2n}$"},{"id":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/level-set","kind":"node","lean":"MathlibExt.Geometry.LiouvilleArnoldWanted.Geometry.LiouvilleArnold.levelSet","parent":15042,"reading_title":{"format":"inline-parts-v1","parts":[["text","Common level set of "],["math","\\(n\\)"],["text"," functions on "],["math","\\(\\mathbb{R}^{2n}\\)"]],"source":"Common level set of $n$ functions on $\\mathbb{R}^{2n}$"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/index.html#level-set","scope":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","status":"fully_proved","summary":"For functions $F_0, \\dots, F_{n-1} \\colon \\mathbb{R}^{2n} \\to \\mathbb{R}$ and values $c_0, \\dots, c_{n-1} \\in \\mathbb{R}$, the common level set is $\\{x \\in \\mathbb{R}^{2n} : F_i(x) = c_i \\text{ for all } i\\}$.","title":"Common level set of $n$ functions on $\\mathbb{R}^{2n}$"},{"id":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/liouville-arnold","kind":"node","lean":"MathlibExt.Geometry.LiouvilleArnoldWanted.Geometry.LiouvilleArnold.liouville_arnold","parent":15042,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem/index.html#liouville-arnold","scope":"Geometry/complex-and-symplectic-geometry/liouville-arnold-theorem","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{R}^{2n}$ be open and let $F_0, \\dots, F_{n-1} \\colon \\mathbb{R}^{2n} \\to \\mathbb{R}$ be Liouville integrable on $U$. For $c = (c_0, \\dots, c_{n-1}) \\in \\mathbb{R}^n$, let $M_c = \\{x \\in \\mathbb{R}^{2n} : F_i(x) = c_i \\text{ for all } i\\}$ be the common level set. If $M_c \\subseteq U$ and $M_c$ is compact and connected (in particular nonempty), then $M_c$ is homeomorphic to the $n$-torus\u2026","title":"Liouville\u2013Arnold theorem"},{"data_url":"scopes/Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem.json","id":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem","kind":"scope","navigation":"module","parent":15028,"result":false,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/index.html","scope":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem","status":"can_prove","title":"Newlander\u2013Nirenberg theorem"},{"id":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/e","kind":"node","lean":"MathlibExt.Geometry.NewlanderNirenbergWanted.Geometry.NewlanderNirenberg.E","parent":15050,"reading_title":{"format":"inline-parts-v1","parts":[["text","Model space "],["math","\\(\\mathbb{R}^{2n}\\)"],["text"," for the Newlander\u2013Nirenberg theorem"]],"source":"Model space $\\mathbb{R}^{2n}$ for the Newlander\u2013Nirenberg theorem"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/index.html#e","scope":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem","status":"fully_proved","summary":"For a natural number $n$, $E_n$ is the Euclidean space $\\mathbb{R}^{2n}$, the model space carrying the almost complex structure.","title":"Model space $\\mathbb{R}^{2n}$ for the Newlander\u2013Nirenberg theorem"},{"id":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/nijenhuis","kind":"node","lean":"MathlibExt.Geometry.NewlanderNirenbergWanted.Geometry.NewlanderNirenberg.nijenhuis","parent":15050,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/index.html#nijenhuis","scope":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem","status":"fully_proved","summary":"Let $n$ be a natural number, let $J$ assign to each point $y \\in \\mathbb{R}^{2n}$ a real linear map $J(y) : \\mathbb{R}^{2n} \\to \\mathbb{R}^{2n}$, and let $V, W : \\mathbb{R}^{2n} \\to \\mathbb{R}^{2n}$ be vector fields. Write $JV$ for the vector field $y \\mapsto J(y)V(y)$ and $[V, W]$ for the Lie bracket of vector fields. The Nijenhuis tensor of $J$ at a point $x \\in \\mathbb{R}^{2n}$ is","title":"Nijenhuis tensor"},{"id":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/is-almost-complex-on","kind":"node","lean":"MathlibExt.Geometry.NewlanderNirenbergWanted.Geometry.NewlanderNirenberg.IsAlmostComplexOn","parent":15050,"reading_title":{"format":"inline-parts-v1","parts":[["text","Almost complex structure on a subset of "],["math","\\(\\mathbb{R}^{2n}\\)"]],"source":"Almost complex structure on a subset of $\\mathbb{R}^{2n}$"},"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/index.html#is-almost-complex-on","scope":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem","status":"fully_proved","summary":"Let $n$ be a natural number, let $U \\subseteq \\mathbb{R}^{2n}$ and let $J$ assign to each point $y \\in \\mathbb{R}^{2n}$ a real linear map $J(y) : \\mathbb{R}^{2n} \\to \\mathbb{R}^{2n}$. Then $J$ is an almost complex structure on $U$ if $J$ is $C^\\infty$ on $U$ and $J(x)(J(x)v) = -v$ for every $x \\in U$ and every $v \\in \\mathbb{R}^{2n}$.","title":"Almost complex structure on a subset of $\\mathbb{R}^{2n}$"},{"id":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/nijenhuis-vanishes-on","kind":"node","lean":"MathlibExt.Geometry.NewlanderNirenbergWanted.Geometry.NewlanderNirenberg.NijenhuisVanishesOn","parent":15050,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/index.html#nijenhuis-vanishes-on","scope":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem","status":"fully_proved","summary":"Let $n$ be a natural number, let $U \\subseteq \\mathbb{R}^{2n}$, let $J$ assign to each point $y \\in \\mathbb{R}^{2n}$ a real linear map $J(y) : \\mathbb{R}^{2n} \\to \\mathbb{R}^{2n}$, and write $N$ for the Nijenhuis tensor of $J$. The Nijenhuis tensor of $J$ vanishes on $U$ if $N(V, W)(x) = 0$ for all $C^\\infty$ vector fields $V, W : \\mathbb{R}^{2n} \\to \\mathbb{R}^{2n}$ and every $x \\in U$.","title":"Vanishing of the Nijenhuis tensor on a set"},{"id":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/newlander-nirenberg","kind":"node","lean":"MathlibExt.Geometry.NewlanderNirenbergWanted.Geometry.NewlanderNirenberg.newlander_nirenberg","parent":15050,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem/index.html#newlander-nirenberg","scope":"Geometry/complex-and-symplectic-geometry/newlander-nirenberg-theorem","status":"can_prove","summary":"Let $n$ be a natural number, let $U \\subseteq \\mathbb{R}^{2n}$ be open and let $J$ be an almost complex structure on $U$ whose Nijenhuis tensor vanishes on $U$. Then every $x \\in U$ has holomorphic coordinates for $J$: there are open sets $S \\subseteq \\mathbb{R}^{2n}$ and $T \\subseteq \\mathbb{C}^n$ with $x \\in S \\subseteq U$ and a homeomorphism $\\varphi : S \\to T$ such that $\\varphi$ is $C^\\infty$ on $S$,\u2026","title":"Newlander\u2013Nirenberg theorem"},{"data_url":"scopes/Geometry/complex-and-symplectic-geometry/weinstein-conjecture.json","id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","kind":"scope","navigation":"module","parent":15028,"result":false,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"can_prove","title":"Weinstein conjecture"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/one-form","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.OneForm","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#one-form","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"fully_proved","summary":"Let $M$ be a smooth manifold (possibly with boundary or corners). A smooth one-form on $M$ is a smooth section of the cotangent bundle of $M$: a smooth choice of a linear map $\\alpha_x : T_xM \\to \\mathbb{R}$ at each point $x \\in M$.","title":"Smooth one-forms"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/two-form","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.TwoForm","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#two-form","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"fully_proved","summary":"Let $M$ be a smooth manifold (possibly with boundary or corners). A two-form on $M$ is here a smooth section of the bundle whose fiber at $x \\in M$ is the space of bilinear maps $T_xM \\times T_xM \\to \\mathbb{R}$; antisymmetry is not required.","title":"Two-forms as fields of bilinear forms"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/smooth-vector-field","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.SmoothVectorField","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#smooth-vector-field","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"fully_proved","summary":"Let $M$ be a smooth manifold (possibly with boundary or corners). A smooth vector field on $M$ is a smooth section of the tangent bundle of $M$: a smooth choice of a tangent vector $X_x \\in T_xM$ at each point $x \\in M$.","title":"Smooth vector fields"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/is-exterior-derivative","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.IsExteriorDerivative","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#is-exterior-derivative","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"fully_proved","summary":"A two-form $d\\alpha$ on $M$ is the exterior derivative of a one-form $\\alpha$ on $M$ when, for all smooth vector fields $X, Y$ on $M$ and every point $x \\in M$,","title":"Exterior derivative of a one-form"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/is-contact-form3","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.IsContactForm3","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#is-contact-form3","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"fully_proved","summary":"Let $\\alpha$ be a one-form and $d\\alpha$ a two-form on $M$. The pair satisfies the contact condition when $d\\alpha$ is the exterior derivative of $\\alpha$ and, at every point $x \\in M$, the covector $\\alpha_x$ is nonzero and there are tangent vectors $u, v \\in T_xM$ with $\\alpha_x(u) = \\alpha_x(v) = 0$ and $d\\alpha_x(u, v) \\ne 0$.","title":"Contact condition for a one-form"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/is-reeb-vector-field","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.IsReebVectorField","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#is-reeb-vector-field","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"fully_proved","summary":"Let $\\alpha$ be a one-form and $d\\alpha$ a two-form on $M$. An assignment $R$ of a tangent vector $R(x) \\in T_xM$ to each point $x \\in M$ is a Reeb vector field for $(\\alpha, d\\alpha)$ when, at every point $x \\in M$, $\\alpha_x(R(x)) = 1$ and $d\\alpha_x(R(x), v) = 0$ for all $v \\in T_xM$.","title":"Reeb vector fields"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/is-integral-curve","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.IsIntegralCurve","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#is-integral-curve","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"fully_proved","summary":"Let $M$ be a smooth manifold (possibly with boundary or corners) and let $R$ assign to each point $x \\in M$ a tangent vector $R(x) \\in T_xM$. A curve $\\gamma : \\mathbb{R} \\to M$ is an integral curve of $R$ when $\\gamma$ is smooth and $\\gamma'(t) = R(\\gamma(t))$ for every $t \\in \\mathbb{R}$.","title":"Integral curves"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/has-closed-reeb-orbit","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.HasClosedReebOrbit","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#has-closed-reeb-orbit","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"fully_proved","summary":"Let $M$ be a smooth manifold (possibly with boundary or corners) and let $R$ assign to each point $x \\in M$ a tangent vector $R(x) \\in T_xM$. We say that $R$ has a closed orbit when there are a curve $\\gamma : \\mathbb{R} \\to M$ and a real number $T > 0$ such that $\\gamma$ is $T$-periodic, $\\gamma$ is a smooth integral curve of $R$, and $\\gamma$ is not constant.","title":"Vector fields with a closed orbit"},{"id":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture/weinstein-conjecture-dim-three","kind":"node","lean":"MathlibExt.Geometry.WeinsteinConjecture3DWanted.Geometry.WeinsteinConjecture3DProblem.weinstein_conjecture_dim_three","parent":15056,"result":true,"route":"../roadmap/Geometry/complex-and-symplectic-geometry/weinstein-conjecture/index.html#weinstein-conjecture-dim-three","scope":"Geometry/complex-and-symplectic-geometry/weinstein-conjecture","status":"can_prove","summary":"Let $M$ be a nonempty compact Hausdorff smooth manifold without boundary, modelled on a real vector space of dimension $3$. Let $\\alpha$ be a smooth one-form and $d\\alpha$ a smooth two-form on $M$ satisfying the contact condition: $d\\alpha$ is the exterior derivative of $\\alpha$ and, at every point $x$, $\\alpha_x \\ne 0$ and $d\\alpha_x(u, v) \\ne 0$ for some $u, v \\in \\ker \\alpha_x$. Let $R$ be a Reeb vector field\u2026","title":"Weinstein conjecture in dimension three (Taubes)"},{"area":"Geometry & Topology","data_url":"chapters/Topology.json","id":"Topology","kind":"scope","parent":0,"result":false,"route":"../roadmap/Topology/index.html","scope":"Topology","status":"planned","summary":"Topology studies continuity, open sets, compactness, connectedness, and invariants preserved by homeomorphism.","title":"Topology"},{"data_url":"scopes/Topology/knot-theory.json","id":"Topology/knot-theory","kind":"scope","parent":15066,"result":false,"route":"../roadmap/Topology/knot-theory/index.html","scope":"Topology/knot-theory","status":"can_prove","summary":"Embeddings of circles in three-space up to ambient isotopy: knot and link diagrams, chirality, slice knots, and geometric invariants such as ropelength, distortion and quadrisecants.","title":"Knot theory"},{"id":"Topology/knot-theory/exists-chiral-knot","kind":"node","lean":"MathlibExt.Topology.ChiralWanted.KnotTheory.exists_chiral_knot","parent":15067,"result":true,"route":"../roadmap/Topology/knot-theory/index.html#exists-chiral-knot","scope":"Topology/knot-theory","status":"can_prove","summary":"There exists an oriented smooth knot $K$ in $\\mathbb{R}^3$ that is chiral: it is not ambient-isotopic, as an oriented knot, to its mirror image $r \\circ K$, where $r(x, y, z) = (x, y, -z)$ is the reflection through the $xy$-plane.","title":"Existence of a chiral knot"},{"id":"Topology/knot-theory/oesper-weaving-three-determinant","kind":"node","lean":"MetaMathlibExt.oesper_weaving_three_determinant","parent":15067,"result":true,"route":"../roadmap/Topology/knot-theory/index.html#oesper-weaving-three-determinant","scope":"Topology/knot-theory","status":"fully_proved","summary":"Let $k \\ge 1$ be an integer, let $F_i$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$), let $C_j = \\sum_{i=1}^{j} (-1)^{i+1} F_i$ for $j \\ge 0$, and let $M$ be the $3 \\times 3$ integer matrix with rows $(2, -1, 0)$, $(0, 0, 1)$ and $(-1, 0, 2)$. Then the absolute value of the determinant of the $2 \\times 2$ matrix obtained from $M^k - I$ by deleting its first row and first column equals\u2026","title":"Oesper's determinant formula for three-strand weaving knots"},{"data_url":"scopes/Topology/knot-theory/knots-links-and-ambient-isotopy.json","id":"Topology/knot-theory/knots-links-and-ambient-isotopy","kind":"scope","navigation":"module","parent":15067,"result":false,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"can_prove","title":"Knots, links and ambient isotopy"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/parametrized-knot","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedKnot","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#parametrized-knot","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"An oriented parametrized smooth knot in Euclidean 3-space is a $C^\\infty$ curve $\\gamma \\colon \\mathbb{R} \\to \\mathbb{R}^3$ that is $2\\pi$-periodic ($\\gamma(t + 2\\pi) = \\gamma(t)$ for all $t$), injective on $[0, 2\\pi)$, and an immersion ($\\gamma'(t) \\neq 0$ for all $t$).","title":"Parametrized knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/parametrized-link","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#parametrized-link","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"An oriented labeled parametrized smooth link with $n$ components in Euclidean 3-space is a family $(\\gamma_i)_{i < n}$ of parametrized knots whose traces $\\gamma_i(\\mathbb{R})$ are pairwise disjoint.","title":"Parametrized link"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/ambient-isotopy","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.AmbientIsotopy","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#ambient-isotopy","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A smooth ambient isotopy of $\\mathbb{R}^3$ consists of two families of maps $H_t, G_t \\colon \\mathbb{R}^3 \\to \\mathbb{R}^3$, indexed by times $t \\in \\mathbb{R}$, such that $(t, x) \\mapsto H_t(x)$ and $(t, x) \\mapsto G_t(x)$ are $C^\\infty$ on $\\mathbb{R} \\times \\mathbb{R}^3$, $G_t(H_t(x)) = x$ and $H_t(G_t(x)) = x$ for all $t$ and $x$, and $H_0$ is the identity.","title":"Smooth ambient isotopy of Euclidean 3-space"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/id","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.AmbientIsotopy.id","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#id","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The identity ambient isotopy of $\\mathbb{R}^3$ has forward and inverse families $H_t$ and $G_t$ equal to the identity map of $\\mathbb{R}^3$ for every time $t$.","title":"Identity ambient isotopy"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/symm","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.AmbientIsotopy.symm","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#symm","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The inverse of an ambient isotopy with forward family $(H_t)$ and inverse family $(H_t^{-1})$ is the ambient isotopy whose forward family is $(H_t^{-1})$ and whose inverse family is $(H_t)$; at each time $t$ it is the inverse diffeomorphism of $H_t$.","title":"Inverse ambient isotopy"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/comp","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.AmbientIsotopy.comp","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#comp","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"If $\\Phi$ and $\\Psi$ are smooth ambient isotopies of $\\mathbb{R}^3$ with forward families $H_t, H'_t$ and inverse families $G_t, G'_t$, their composition (first $\\Phi$, then $\\Psi$, at the same time parameter) is the ambient isotopy with forward family $H'_t \\circ H_t$ and inverse family $G_t \\circ G'_t$.","title":"Composition of ambient isotopies"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/circle-reparam","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.CircleReparam","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#circle-reparam","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"An orientation-preserving smooth reparametrization of the circle is given by maps $f, g \\colon \\mathbb{R} \\to \\mathbb{R}$, both $C^\\infty$ and inverse to each other ($g(f(t)) = t$ and $f(g(t)) = t$ for all $t$), such that $f(t + 2\\pi) = f(t) + 2\\pi$ and $g(t + 2\\pi) = g(t) + 2\\pi$ for all $t$, and $f$ is strictly increasing. Here $f$ is a lift of the reparametrization and $g$ is a lift of its inverse.","title":"Orientation-preserving reparametrization of the circle"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/id-2","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.CircleReparam.id","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#id-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The identity circle reparametrization has lift and inverse lift both equal to the identity map of $\\mathbb{R}$.","title":"Identity reparametrization of the circle"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/symm-2","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.CircleReparam.symm","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#symm-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The inverse of a circle reparametrization with lift $f$ and inverse lift $f^{-1}$ is the circle reparametrization whose lift is $f^{-1}$ and whose inverse lift is $f$.","title":"Inverse circle reparametrization"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/comp-2","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.CircleReparam.comp","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#comp-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"If $\\sigma$ and $\\tau$ are circle reparametrizations with lifts $f_\\sigma, f_\\tau$ and inverse lifts $g_\\sigma, g_\\tau$, their composition (first $\\sigma$, then $\\tau$) is the circle reparametrization with lift $f_\\tau \\circ f_\\sigma$ and inverse lift $g_\\sigma \\circ g_\\tau$.","title":"Composition of circle reparametrizations"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/isotopic","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedKnot.Isotopic","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#isotopic","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Two parametrized knots $\\gamma_1, \\gamma_2$ in $\\mathbb{R}^3$ are isotopic (oriented ambient isotopy) if there are a smooth ambient isotopy of $\\mathbb{R}^3$ with forward family $H_t$ and an orientation-preserving reparametrization of the circle with lift $f$ such that $H_1(\\gamma_1(t)) = \\gamma_2(f(t))$ for every $t \\in \\mathbb{R}$.","title":"Isotopy of parametrized knots"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/isotopic-2","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.Isotopic","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#isotopic-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Two parametrized links in $\\mathbb{R}^3$ with $n$ components $\\gamma^1_i$ and $\\gamma^2_i$ ($i < n$) are isotopic (oriented, labeled ambient isotopy) if there are a single smooth ambient isotopy of $\\mathbb{R}^3$ with forward family $H_t$ and, for each $i < n$, an orientation-preserving reparametrization of the circle with lift $f_i$, such that $H_1(\\gamma^1_i(t)) = \\gamma^2_i(f_i(t))$ for every $i < n$ and every\u2026","title":"Isotopy of parametrized links"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/trace","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.trace","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#trace","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The trace of a parametrized link $L$ in Euclidean 3-space is the union of the images of its component curves: the set of all points $\\gamma(t)$, where $\\gamma$ is a component of $L$ and $t \\in \\mathbb{R}$.","title":"Trace of a parametrized link"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/total-contour-length","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.totalContourLength","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#total-contour-length","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $L$ be a parametrized link in Euclidean 3-space with components $\\gamma_1, \\dots, \\gamma_n$, each a $2\\pi$-periodic curve. The total contour length of $L$ is the sum over all components $\\gamma_i$ of the contour length of $\\gamma_i$ on one period $[0, 2\\pi]$. It is a value in $[0, \\infty]$.","title":"Total contour length of a parametrized link"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/ropelength-ratio","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.ropelengthRatio","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#ropelength-ratio","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"For a parametrized link $L$, the ropelength ratio of $L$ is its total contour length divided by the thickness of its trace (the union of the images of its components), a value in $[0, \\infty]$ computed with the division of $[0, \\infty]$.","title":"Ropelength ratio of a parametrized link"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/ropelength","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.ropelength","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#ropelength","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"For a parametrized link $L$ with $n$ components, the ropelength of $L$ is the infimum, in $[0, \\infty]$, of the ropelength ratios of all parametrized links $L'$ with $n$ components that are isotopic to $L$. It is the ropelength of the oriented, labeled ambient-isotopy type of $L$.","title":"Ropelength of a parametrized link"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/ropelength-le-ratio-of-isotopic","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.ropelength_le_ratio_of_isotopic","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#ropelength-le-ratio-of-isotopic","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $L$ and $L'$ be parametrized links with $n$ components. If $L'$ is isotopic to $L$, then the ropelength of $L$ is at most the ropelength ratio of $L'$.","title":"Ropelength is at most the ratio of any isotopic link"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/ropelength-le-ratio-self","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.ropelength_le_ratio_self","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#ropelength-le-ratio-self","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"For every parametrized link $L$ with $n$ components, the ropelength of $L$ is at most the ropelength ratio of $L$.","title":"Ropelength is at most the ropelength ratio"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/ropelength-eq-of-isotopic","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.ropelength_eq_of_isotopic","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#ropelength-eq-of-isotopic","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be parametrized links with $n$ components. If $L_1$ is isotopic to $L_2$ (by an oriented, labeled ambient isotopy), then the ropelength of $L_1$ equals the ropelength of $L_2$.","title":"Ropelength is an isotopy invariant"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/to-link","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedKnot.toLink","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#to-link","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A parametrized knot $K$ determines the parametrized link with one component, whose single component is $K$; the pairwise disjointness condition holds trivially.","title":"Parametrized knot as a one-component link"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/to-link-2","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedKnot.Isotopic.toLink","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#to-link-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"If $K_1$ and $K_2$ are parametrized knots and $K_1$ is isotopic to $K_2$, then the one-component parametrized link of $K_1$ is isotopic to the one-component parametrized link of $K_2$.","title":"Isotopic knots give isotopic one-component links"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/ropelength-2","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedKnot.ropelength","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#ropelength-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The ropelength of a parametrized knot $K$ is the ropelength of the one-component parametrized link determined by $K$: the infimum, in $[0, \\infty]$, of the ropelength ratios of all one-component parametrized links that are isotopic to the one-component link of $K$.","title":"Ropelength of a parametrized knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/r3","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.R3","parent":15070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euclidean space "],["math","\\(\\mathbb{R}^3\\)"]],"source":"Euclidean space $\\mathbb{R}^3$"},"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#r3","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The ambient space of the knots here is $\\mathbb{R}^3$ with its standard Euclidean inner product.","title":"Euclidean space $\\mathbb{R}^3$"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/knot","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.Knot","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#knot","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"An oriented smooth knot in $\\mathbb{R}^3$ is a smooth map $\\gamma : \\mathbb{R} \\to \\mathbb{R}^3$ that is $2\\pi$-periodic, injective on $[0, 2\\pi)$, and an immersion, meaning $\\gamma'(t) \\ne 0$ for every real $t$. The orientation is the one induced by the parametrization.","title":"Oriented smooth knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/two-link","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.TwoLink","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#two-link","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"An oriented two-component smooth link in $\\mathbb{R}^3$ is a pair $(K, L)$ of oriented smooth knots whose images in $\\mathbb{R}^3$ are disjoint.","title":"Two-component link"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/ambient-isotopy-2","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.AmbientIsotopy","parent":15070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Smooth ambient isotopy of "],["math","\\(\\mathbb{R}^3\\)"]],"source":"Smooth ambient isotopy of $\\mathbb{R}^3$"},"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#ambient-isotopy-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A smooth ambient isotopy of $\\mathbb{R}^3$ consists of two families of maps $H_t, G_t \\colon \\mathbb{R}^3 \\to \\mathbb{R}^3$ indexed by $t \\in \\mathbb{R}$ such that $(t, x) \\mapsto H_t(x)$ and $(t, x) \\mapsto G_t(x)$ are smooth ($C^\\infty$) maps $\\mathbb{R} \\times \\mathbb{R}^3 \\to \\mathbb{R}^3$, $G_t$ is a two-sided inverse of $H_t$ for every $t$, and $H_0$ is the identity. In particular every $H_t$ is a\u2026","title":"Smooth ambient isotopy of $\\mathbb{R}^3$"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/isotopic-3","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.Knot.Isotopic","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#isotopic-3","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $K_1$ and $K_2$ be oriented knots, each given by its $2\\pi$-periodic parametrization. They are ambient-isotopic if there are a smooth ambient isotopy $(\\Phi_t)_{t \\in \\mathbb{R}}$ of $\\mathbb{R}^3$ and an orientation-preserving smooth reparametrization $\\sigma$ of the circle such that $\\Phi_1(K_1(t)) = K_2(\\sigma(t))$ for every real $t$. Here an orientation-preserving smooth reparametrization of the circle is a\u2026","title":"Ambient isotopy of oriented knots"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/isotopic-4","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.TwoLink.Isotopic","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#isotopic-4","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $L = (K, J)$ and $L' = (K', J')$ be oriented two-component links, each knot given by its $2\\pi$-periodic parametrization. Then $L$ and $L'$ are ambient-isotopic if there are a smooth ambient isotopy $(\\Phi_t)_{t \\in \\mathbb{R}}$ of $\\mathbb{R}^3$ and orientation-preserving smooth reparametrizations $\\sigma$, $\\tau$ of the circle such that $\\Phi_1(K(t)) = K'(\\sigma(t))$ and $\\Phi_1(J(t)) = J'(\\tau(t))$ for every\u2026","title":"Ambient isotopy of oriented two-component links"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/reflect-z","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.reflectZ","parent":15070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reflection in the "],["math","\\(xy\\)"],["text","-plane"]],"source":"Reflection in the $xy$-plane"},"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#reflect-z","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The reflection of $\\mathbb{R}^3$ through the $xy$-plane is the map $(x, y, z) \\mapsto (x, y, -z)$.","title":"Reflection in the $xy$-plane"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/chiral","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.Knot.Chiral","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#chiral","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $K \\colon \\mathbb{R} \\to \\mathbb{R}^3$ be an oriented smooth knot, and let $\\rho(x, y, z) = (x, y, -z)$ be the reflection of $\\mathbb{R}^3$ through the $xy$-plane. Call $\\sigma \\colon \\mathbb{R} \\to \\mathbb{R}$ an orientation-preserving reparametrization of the circle if it is a strictly increasing smooth bijection with smooth inverse satisfying $\\sigma(t + 2\\pi) = \\sigma(t) + 2\\pi$ for all $t \\in \\mathbb{R}$,\u2026","title":"Chiral knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/top-ambient-isotopy","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.TopAmbientIsotopy","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#top-ambient-isotopy","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A topological ambient isotopy of $\\mathbb{R}^3$ is a family of maps $H_t : \\mathbb{R}^3 \\to \\mathbb{R}^3$, indexed by $t \\in \\mathbb{R}$, together with a family $G_t : \\mathbb{R}^3 \\to \\mathbb{R}^3$ such that $(t, x) \\mapsto H_t(x)$ and $(t, x) \\mapsto G_t(x)$ are continuous, $G_t$ is a two-sided inverse of $H_t$ for every $t$ (so each $H_t$ is a homeomorphism of $\\mathbb{R}^3$), and $H_0$ is the identity.","title":"Topological ambient isotopy"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/unoriented-top-amb-isotopic","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.Set.UnorientedTopAmbIsotopic","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#unoriented-top-amb-isotopic","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Two subsets $A$ and $B$ of $\\mathbb{R}^3$ are unoriented topologically ambient isotopic if there is a topological ambient isotopy $(H_t)$ of $\\mathbb{R}^3$ whose time-one map carries $A$ onto $B$, that is, $H_1(A) = B$.","title":"Topological ambient isotopy of subsets"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/disk2","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.disk2","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#disk2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The closed unit Euclidean disk $\\{(x, y) \\in \\mathbb{R} \\times \\mathbb{R} \\mid x^2 + y^2 \\le 1\\}$, the source of the parametrization of a slicing disk.","title":"Closed unit disk in the plane"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/circle1","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.circle1","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#circle1","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The unit Euclidean circle $\\{(x, y) \\in \\mathbb{R} \\times \\mathbb{R} \\mid x^2 + y^2 = 1\\}$, the source of the boundary of a slicing disk.","title":"Unit circle in the plane"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/upper-half","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.upperHalf","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#upper-half","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The closed upper half-space in $\\mathbb{R}^3 \\times \\mathbb{R}$ is the set $\\mathbb{R}^3 \\times [0, \\infty)$ of points $(x, s)$ with $s \\ge 0$.","title":"Closed upper half-space"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/model-plane2","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.modelPlane2","parent":15070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Model interior "],["math","\\(2\\)"],["text","-plane"]],"source":"Model interior $2$-plane"},"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#model-plane2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The model interior $2$-plane in $\\mathbb{R}^3 \\times \\mathbb{R}$ is the set of points $((x_1, x_2, x_3), t)$ with $x_2 = x_3 = 0$, where $x_1$ and $t$ are arbitrary. It is the local model for interior local flatness of an embedded $2$-disk.","title":"Model interior $2$-plane"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/model-half-plane2","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.modelHalfPlane2","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#model-half-plane2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The model half-plane is the set of pairs $((p_1, p_2, p_3), s)$ in $\\mathbb{R}^3 \\times \\mathbb{R}$ with $p_2 = p_3 = 0$ and $s \\ge 0$. It is a copy of the closed half-plane $\\mathbb{R} \\times [0, \\infty)$, with coordinates $p_1$ and $s$, lying in the upper half-space $s \\ge 0$.","title":"Model half-plane"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/smoothly-slice","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.Knot.SmoothlySlice","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#smoothly-slice","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $\\Delta = \\{(x, y) \\in \\mathbb{R}^2 : x^2 + y^2 \\le 1\\}$ be the closed unit disk and $C = \\{(x, y) \\in \\mathbb{R}^2 : x^2 + y^2 = 1\\}$ its boundary circle. A smooth knot $K$ is smoothly slice if there is a smooth ($C^\\infty$) map $f : \\mathbb{R}^2 \\to \\mathbb{R}^3 \\times \\mathbb{R}$ such that $f$ is injective on $\\Delta$; for every $p \\in \\Delta$ the last coordinate of $f(p)$ is nonnegative, and it is $0$\u2026","title":"Smoothly slice (smooth knot version)"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/is-locally-flat-interior","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.IsLocallyFlatInterior","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#is-locally-flat-interior","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $q \\in \\mathbb{R}^3 \\times \\mathbb{R}$ and $D : \\mathbb{R}^2 \\to \\mathbb{R}^3 \\times \\mathbb{R}$. Write $\\Delta = \\{(x, y) : x^2 + y^2 \\le 1\\}$ for the closed unit disk and $P = \\{((p_1, p_2, p_3), s) : p_2 = p_3 = 0\\}$ for the model plane. The image $D(\\Delta)$ is locally flat at $q$ in the interior sense if there is a homeomorphism $h : U \\to V$ between open subsets $U$, $V$ of\u2026","title":"Local flatness at an interior point"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/is-locally-flat-boundary","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.IsLocallyFlatBoundary","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#is-locally-flat-boundary","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $q \\in \\mathbb{R}^3 \\times \\mathbb{R}$ and $D : \\mathbb{R}^2 \\to \\mathbb{R}^3 \\times \\mathbb{R}$. Write $\\Delta = \\{(x, y) : x^2 + y^2 \\le 1\\}$ for the closed unit disk, $H = \\{(p, s) : s \\ge 0\\}$ for the closed upper half-space, and $P = \\{((p_1, p_2, p_3), s) : p_2 = p_3 = 0,\\ s \\ge 0\\}$ for the model half-plane. The image $D(\\Delta)$ is locally flat at $q$ in the boundary sense if there is a homeomorphism\u2026","title":"Local flatness at a boundary point"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/topologically-slice","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.Knot.TopologicallySlice","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#topologically-slice","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Write points of $\\mathbb{R}^3 \\times \\mathbb{R}$ as $(x, s)$ with $x = (x_1, x_2, x_3)$. Let $H = \\{(x, s) : s \\ge 0\\}$ be the closed upper half-space, $P = \\{(x, s) : x_2 = x_3 = 0\\}$ the model plane and $P_+ = P \\cap H$ the model half-plane, and let $D^2 = \\{(a, b) \\in \\mathbb{R}^2 : a^2 + b^2 \\le 1\\}$ and $S^1 = \\{(a, b) \\in \\mathbb{R}^2 : a^2 + b^2 = 1\\}$. A smooth knot $K$ is topologically slice if there is a\u2026","title":"Topologically slice (smooth knot version)"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/pl-curve","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.plCurve","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#pl-curve","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $v_0, \\dots, v_{n-1}$ be a list of $n \\ge 1$ points of $\\mathbb{R}^3$. The polyline curve is the $n$-periodic map $\\gamma : \\mathbb{R} \\to \\mathbb{R}^3$ with $\\gamma(k + \\alpha) = (1 - \\alpha) v_k + \\alpha v_{k+1}$ for integers $0 \\le k < n$ and reals $0 \\le \\alpha < 1$, where indices are taken modulo $n$ (so $v_n = v_0$). For the empty list the curve is constantly $0$.","title":"Parametrization of a closed polyline"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/plknot","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.PLKnot","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#plknot","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A PL knot is a list $v_0, \\dots, v_{n-1}$ of points of $\\mathbb{R}^3$ with $n \\ge 3$, viewed as the vertices, in order, of the closed polyline with edges from $v_i$ to $v_{i+1}$ for $i < n - 1$ and from $v_{n-1}$ back to $v_0$.","title":"PL knot (closed polyline)"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/image","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.PLKnot.image","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#image","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $K$ be a PL knot with vertices $v_0, \\dots, v_{n-1}$. Its image is the subset of $\\mathbb{R}^3$ traced by the closed polyline through these vertices: the image of the half-open interval $[0, n)$ under the parametrization that sends $k + \\alpha$, for an integer $0 \\le k < n$ and a real $0 \\le \\alpha < 1$, to $(1 - \\alpha) v_k + \\alpha v_{k+1}$, with indices taken modulo $n$.","title":"Image of a PL knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/is-simple","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.PLKnot.IsSimple","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#is-simple","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A PL knot $K$ with vertices $v_0, \\dots, v_{n-1}$ is simple if its closed polyline parametrization, which sends $k + \\alpha$, for an integer $0 \\le k < n$ and a real $0 \\le \\alpha < 1$, to $(1 - \\alpha) v_k + \\alpha v_{k+1}$ (indices modulo $n$), is injective on the half-open interval $[0, n)$.","title":"Simple PL knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/has-smooth-representative","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.PLKnot.HasSmoothRepresentative","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#has-smooth-representative","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A smooth knot $J$ is a smooth representative of the piecewise-linear knot $K$ if the image of $J$ and the image of $K$ are unoriented topologically ambient isotopic in $\\mathbb{R}^3$: some topological ambient isotopy $(H_t)$ of $\\mathbb{R}^3$ satisfies $H_1(\\operatorname{im} J) = \\operatorname{im} K$.","title":"Smooth representative of a PL knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/smoothly-slice-2","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.PLKnot.SmoothlySlice","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#smoothly-slice-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A PL knot $K$ is smoothly slice if there is a smooth knot $J$ that is smoothly slice and is a smooth representative of $K$: for some topological ambient isotopy $(H_t)$ of $\\mathbb{R}^3$, the map $H_1$ carries the image of $J$ onto the image (trace) of the polyline $K$.","title":"Smoothly slice (PL knot version)"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/topologically-slice-2","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.PLKnot.TopologicallySlice","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#topologically-slice-2","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A PL knot $K$ is topologically slice if there is a smooth knot $J$ that is a smooth representative of $K$, meaning that the image of $J$ and the image of $K$ are unoriented topologically ambient isotopic subsets of $\\mathbb{R}^3$, and such that $J$ is topologically slice as a smooth knot.","title":"Topologically slice (PL knot version)"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/mk-r3","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.BraidClosure.mkR3","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#mk-r3","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"For real numbers $x$, $y$, $z$, this is the point $(x, y, z)$ of the Euclidean space $\\mathbb{R}^3$.","title":"Point from three coordinates"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/state","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.BraidClosure.State","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#state","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A traversal state for tracing the closure of a braid consists of the current strand position, a natural number, and the list of points of $\\mathbb{R}^3$ accumulated so far, in order, as vertices of the polyline.","title":"Braid closure traversal state"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/apply-crossing","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.BraidClosure.applyCrossing","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#apply-crossing","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let a traversal state consist of a current strand position $p \\in \\mathbb{N}$ and a list of vertices in $\\mathbb{R}^3$, let $w \\in \\mathbb{Z}$ be a braid generator with $i = |w|$, and let $z_{\\mathrm{top}}, z_{\\mathrm{bot}} \\in \\mathbb{R}$. Put $z_{\\mathrm{hi}} = z_{\\mathrm{top}} - (z_{\\mathrm{top}} - z_{\\mathrm{bot}})/4$ and $z_{\\mathrm{lo}} = z_{\\mathrm{bot}} + (z_{\\mathrm{top}} - z_{\\mathrm{bot}})/4$. The new\u2026","title":"Tracing a strand through one crossing"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/apply-word","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.BraidClosure.applyWord","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#apply-word","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Given a traversal state $s$, a braid word $w_0 w_1 \\cdots w_{m-1}$ (a list of integers) and a starting index $k \\in \\mathbb{N}$, the one-crossing step is applied to the state for each generator in turn, starting from $s$: the generator $w_j$ is applied with top height $-(k + j)$ and bottom height $-(k + j + 1)$. The result is the final state; for the empty word it is $s$.","title":"Tracing a strand through a braid word"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/one-pass","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.BraidClosure.onePass","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#one-pass","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $w$ be a braid word (a list of integers) of length $m$. Starting from a traversal state, one pass applies the letters of $w$ in order as crossings, the $k$-th letter (counting from $0$) occupying the slab $-(k+1) \\le z \\le -k$; this moves the state to a strand position $p$ and appends the crossing vertices. It then appends the closure arc: the vertices $(p, 100, -m)$ and $(p, 100, 0)$, followed by $(p, 0, 0)$\u2026","title":"One pass of the braid closure"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/iterate","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.BraidClosure.iterate","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#iterate","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"For a braid word $w$, a number $f$ of remaining passes and a traversal state $S$, the iteration returns $S$ when $f = 0$. When $f > 0$ it applies one pass of $w$ to $S$, obtaining a state $S'$; it returns $S'$ if the strand position of $S'$ is $1$, and otherwise iterates on $S'$ with $f - 1$ passes remaining.","title":"Iterated braid-closure passes"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/braid-closure","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.braidClosure","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#braid-closure","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, and let $w = w_0 w_1 \\cdots w_{m-1}$ be a braid word (a list of integers, where $i > 0$ stands for $\\sigma_i$ and $-i$ for $\\sigma_i^{-1}$). If $m = 0$, the value is the junk list $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$. Otherwise the value is a list of vertices in $\\mathbb{R}^3$ tracing the closure of $w$ as one polyline component. It starts at the vertex $(1, 0, 0)$ on strand position $1$\u2026","title":"Polyline closure of a braid word"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/conway-braid-word","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.conwayBraidWord","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#conway-braid-word","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The braid word $\\sigma_1^{-2} \\sigma_2 \\sigma_1^{-1} \\sigma_2 \\sigma_1^{-1} \\sigma_3 \\sigma_2^{-2} \\sigma_3^{2}$ on $4$ strands, encoded as the list of integers $[-1, -1, 2, -1, 2, -1, 3, -2, -2, 3, 3]$, where an entry $i > 0$ stands for the generator $\\sigma_i$ and $-i$ for $\\sigma_i^{-1}$. It has length $11$ and writhe $-1$, and is a braid word for the Conway knot 11n34.","title":"Braid word for the Conway knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/conway-knot","kind":"node","lean":"MathlibExt.Topology.ConwayKnotShared.KnotTheory.conwayKnot","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#conway-knot","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"The Conway knot 11n34 as a piecewise-linear closed polyline in $\\mathbb{R}^3$: its vertices are those of the polyline closure, on $4$ strands, of the braid word $\\sigma_1^{-2} \\sigma_2 \\sigma_1^{-1} \\sigma_2 \\sigma_1^{-1} \\sigma_3 \\sigma_2^{-2} \\sigma_3^{2}$ for the Conway knot.","title":"The Conway knot"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/to-parametrized-knot","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.toParametrizedKnot","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#to-parametrized-knot","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A knot in $\\mathbb{R}^3$, given by a smooth $2\\pi$-periodic immersion $\\gamma : \\mathbb{R} \\to \\mathbb{R}^3$ that is injective on $[0, 2\\pi)$, is sent to the parametrized knot with the same parametrization $\\gamma$.","title":"Knots as parametrized knots"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/of-parametrized-knot","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.ofParametrizedKnot","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#of-parametrized-knot","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A parametrized knot in $\\mathbb{R}^3$, given by a smooth $2\\pi$-periodic immersion $\\gamma : \\mathbb{R} \\to \\mathbb{R}^3$ that is injective on $[0, 2\\pi)$, is sent to the knot with the same parametrization $\\gamma$.","title":"Parametrized knots as knots"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/knot-equiv","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.knotEquiv","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#knot-equiv","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Sending a knot in $\\mathbb{R}^3$ to the parametrized knot with the same parametrization is a bijection from knots to parametrized knots, whose inverse sends a parametrized knot to the knot with the same parametrization.","title":"Knots are the same as parametrized knots"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/to-euclidean-ambient-isotopy","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.toEuclideanAmbientIsotopy","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#to-euclidean-ambient-isotopy","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"An ambient isotopy of $\\mathbb{R}^3$ in the sense of the Conway knot statements, given by maps $H_t : \\mathbb{R}^3 \\to \\mathbb{R}^3$ for $t \\in \\mathbb{R}$ with $(t, x) \\mapsto H_t(x)$ smooth and $H_0 = \\mathrm{id}$, and maps $G_t : \\mathbb{R}^3 \\to \\mathbb{R}^3$ with $(t, x) \\mapsto G_t(x)$ smooth and $G_t \\circ H_t = H_t \\circ G_t = \\mathrm{id}$ for every $t$, is sent to the smooth ambient isotopy of Euclidean\u2026","title":"Conway-knot ambient isotopies as Euclidean ambient isotopies"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/of-euclidean-ambient-isotopy","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.ofEuclideanAmbientIsotopy","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#of-euclidean-ambient-isotopy","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A smooth ambient isotopy of Euclidean 3-space in the sense of parametrized links, given by maps $H_t : \\mathbb{R}^3 \\to \\mathbb{R}^3$ for $t \\in \\mathbb{R}$ with $(t, x) \\mapsto H_t(x)$ smooth and $H_0 = \\mathrm{id}$, and maps $G_t : \\mathbb{R}^3 \\to \\mathbb{R}^3$ with $(t, x) \\mapsto G_t(x)$ smooth and $G_t \\circ H_t = H_t \\circ G_t = \\mathrm{id}$ for every $t$, is sent to the ambient isotopy of $\\mathbb{R}^3$ in\u2026","title":"Euclidean ambient isotopies as Conway-knot ambient isotopies"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/to-euclidean-circle-reparam","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.toEuclideanCircleReparam","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#to-euclidean-circle-reparam","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"An orientation-preserving smooth reparametrization of the circle in the sense of the Conway knot statements, given by a smooth strictly increasing map $f : \\mathbb{R} \\to \\mathbb{R}$ with smooth two-sided inverse $g$ such that $f(t + 2\\pi) = f(t) + 2\\pi$ and $g(t + 2\\pi) = g(t) + 2\\pi$ for all $t$, is sent to the circle reparametrization, in the sense of parametrized links, with the same lift $f$ and inverse lift\u2026","title":"Conway-knot circle reparametrizations as Euclidean ones"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/of-euclidean-circle-reparam","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.ofEuclideanCircleReparam","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#of-euclidean-circle-reparam","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A circle reparametrization in the sense of parametrized links, given by a smooth strictly increasing map $f : \\mathbb{R} \\to \\mathbb{R}$ with smooth two-sided inverse $g$ such that $f(t + 2\\pi) = f(t) + 2\\pi$ and $g(t + 2\\pi) = g(t) + 2\\pi$ for all $t$, is sent to the orientation-preserving smooth reparametrization of the circle in the sense of the Conway knot statements with the same lift $f$ and inverse lift $g$.","title":"Euclidean circle reparametrizations as Conway-knot ones"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/knot-isotopic-iff","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.knot_isotopic_iff","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#knot-isotopic-iff","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"For all knots $K_1$ and $K_2$ in $\\mathbb{R}^3$, $K_1$ and $K_2$ are ambient isotopic if and only if the parametrized knots with the same parametrizations as $K_1$ and $K_2$ are ambient isotopic as parametrized knots.","title":"Isotopy of knots agrees with isotopy of parametrized knots"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/to-parametrized-link","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.toParametrizedLink","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#to-parametrized-link","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A two-component link in $\\mathbb{R}^3$ with first component $K$ and second component $L$ is sent to the labeled parametrized link with two components whose component $0$ is the parametrized knot with the same parametrization as $K$ and whose component $1$ is the parametrized knot with the same parametrization as $L$.","title":"Two-component links as labeled links"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/of-parametrized-link","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.ofParametrizedLink","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#of-parametrized-link","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"A labeled parametrized link with two components $0$ and $1$ is sent to the two-component link in $\\mathbb{R}^3$ whose first component is the knot with the same parametrization as component $0$ and whose second component is the knot with the same parametrization as component $1$.","title":"Labeled two-component links as two-component links"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/two-link-isotopic-iff","kind":"node","lean":"MathlibExt.Topology.ConwayKnotEuclideanBridge.twoLink_isotopic_iff","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#two-link-isotopic-iff","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be oriented two-component smooth links in $\\mathbb{R}^3$, and regard each as an oriented labeled smooth link with two components, whose component $0$ is its first component and whose component $1$ is its second. Then $L_1$ and $L_2$ are ambient isotopic, meaning that a single smooth ambient isotopy carries each component of $L_1$ onto the corresponding component of $L_2$ up to\u2026","title":"Two-component link isotopy agrees with labeled link isotopy"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/exists-nonisotopic-link","kind":"node","lean":"MathlibExt.Topology.LinkingWanted.KnotTheory.exists_nonisotopic_link","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#exists-nonisotopic-link","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"can_prove","summary":"There exist two oriented smooth two-component links $L_1$ and $L_2$ in $\\mathbb{R}^3$ that are not ambient isotopic as oriented links with ordered components: no ambient isotopy of $\\mathbb{R}^3$ carries the first component of $L_1$ to the first component of $L_2$ and the second component of $L_1$ to the second component of $L_2$, each up to an orientation-preserving reparametrization of the circle.","title":"Non-isotopic two-component links exist"},{"id":"Topology/knot-theory/knots-links-and-ambient-isotopy/exists-nonisotopic-knots","kind":"node","lean":"MathlibExt.Topology.NonIsotopicKnotsWanted.KnotTheory.exists_nonisotopic_knots","parent":15070,"result":true,"route":"../roadmap/Topology/knot-theory/knots-links-and-ambient-isotopy/index.html#exists-nonisotopic-knots","scope":"Topology/knot-theory/knots-links-and-ambient-isotopy","status":"can_prove","summary":"There are two oriented smooth knots $K_1$ and $K_2$ in $\\mathbb{R}^3$ that are not ambient-isotopic.","title":"Existence of non-isotopic knots"},{"data_url":"scopes/Topology/knot-theory/distortion-of-knots.json","id":"Topology/knot-theory/distortion-of-knots","kind":"scope","navigation":"module","parent":15067,"result":false,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html","scope":"Topology/knot-theory/distortion-of-knots","status":"can_prove","title":"Distortion of knots"},{"id":"Topology/knot-theory/distortion-of-knots/speed","kind":"node","lean":"MathlibExt.Topology.PardonDistortionWanted.KnotTheory.PardonDistortion.speed","parent":15141,"result":true,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html#speed","scope":"Topology/knot-theory/distortion-of-knots","status":"fully_proved","summary":"For a knot $K$ with parametrization $\\gamma$ and a real number $t$, the speed of $K$ at $t$ is the Euclidean norm $\\lVert \\gamma'(t) \\rVert$ of the derivative of $\\gamma$ at $t$.","title":"Speed of a knot"},{"id":"Topology/knot-theory/distortion-of-knots/parameter-arc-length","kind":"node","lean":"MathlibExt.Topology.PardonDistortionWanted.KnotTheory.PardonDistortion.parameterArcLength","parent":15141,"result":true,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html#parameter-arc-length","scope":"Topology/knot-theory/distortion-of-knots","status":"fully_proved","summary":"For a knot $K$ with parametrization $\\gamma$ and real numbers $s$ and $t$, the arc length of $K$ between $s$ and $t$ is $\\left| \\int_s^t \\lVert \\gamma'(u) \\rVert \\, du \\right|$, the length of the arc that $\\gamma$ traces between the parameters $s$ and $t$. It does not depend on the order of $s$ and $t$.","title":"Arc length between two parameters"},{"id":"Topology/knot-theory/distortion-of-knots/total-arc-length","kind":"node","lean":"MathlibExt.Topology.PardonDistortionWanted.KnotTheory.PardonDistortion.totalArcLength","parent":15141,"result":true,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html#total-arc-length","scope":"Topology/knot-theory/distortion-of-knots","status":"fully_proved","summary":"For a knot $K$ with parametrization $\\gamma$, the total arc length of $K$ is the arc length of $K$ between the parameters $0$ and $2\\pi$, namely $\\left| \\int_0^{2\\pi} \\lVert \\gamma'(u) \\rVert \\, du \\right|$.","title":"Total arc length of a knot"},{"id":"Topology/knot-theory/distortion-of-knots/intrinsic-distance","kind":"node","lean":"MathlibExt.Topology.PardonDistortionWanted.KnotTheory.PardonDistortion.intrinsicDistance","parent":15141,"result":true,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html#intrinsic-distance","scope":"Topology/knot-theory/distortion-of-knots","status":"fully_proved","summary":"For a knot $K$ and real numbers $s$ and $t$, let $\\ell$ be the arc length of $K$ between the parameters $s$ and $t$, and let $L$ be the total arc length of $K$. The intrinsic distance between $s$ and $t$ is $\\min(\\ell, L - \\ell)$. When $s$ and $t$ lie in one closed interval of length $2\\pi$, this is the length of the shorter of the two arcs of $K$ joining the points with parameters $s$ and $t$.","title":"Intrinsic distance on a knot"},{"id":"Topology/knot-theory/distortion-of-knots/distortion-ratio","kind":"node","lean":"MathlibExt.Topology.PardonDistortionWanted.KnotTheory.PardonDistortion.distortionRatio","parent":15141,"result":true,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html#distortion-ratio","scope":"Topology/knot-theory/distortion-of-knots","status":"fully_proved","summary":"For a knot $K$ with parametrization $\\gamma$ and real numbers $s$ and $t$, the distortion ratio of $K$ at $s$ and $t$ is the intrinsic distance between $s$ and $t$ along $K$ divided by the Euclidean distance $\\lVert \\gamma(s) - \\gamma(t) \\rVert$ in $\\mathbb{R}^3$, where division by $0$ gives $0$.","title":"Distortion ratio"},{"id":"Topology/knot-theory/distortion-of-knots/distortion","kind":"node","lean":"MathlibExt.Topology.PardonDistortionWanted.KnotTheory.PardonDistortion.distortion","parent":15141,"result":true,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html#distortion","scope":"Topology/knot-theory/distortion-of-knots","status":"fully_proved","summary":"For a knot $K$, the distortion of $K$ is the supremum in the extended nonnegative reals $[0, \\infty]$ of the distortion ratios of $K$ at $s$ and $t$, taken over all distinct parameters $s \\ne t$ in the half-open interval $[0, 2\\pi)$.","title":"Distortion of a knot"},{"id":"Topology/knot-theory/distortion-of-knots/standard-torus-curve","kind":"node","lean":"MathlibExt.Topology.PardonDistortionWanted.KnotTheory.PardonDistortion.standardTorusCurve","parent":15141,"result":true,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html#standard-torus-curve","scope":"Topology/knot-theory/distortion-of-knots","status":"fully_proved","summary":"For natural numbers $p$ and $q$, the standard $(p, q)$-torus curve is the map $\\mathbb{R} \\to \\mathbb{R}^3$ sending $t$ to $((2 + \\cos qt) \\cos pt,\\ (2 + \\cos qt) \\sin pt,\\ \\sin qt)$. It lies on the torus of revolution about the $z$-axis with major radius $2$ and minor radius $1$, and as $t$ runs over $[0, 2\\pi]$ it winds $p$ times around the $z$-axis and $q$ times around the core circle of the torus.","title":"Standard torus curve"},{"id":"Topology/knot-theory/distortion-of-knots/pardon-torus-knot-distortion","kind":"node","lean":"MathlibExt.Topology.PardonDistortionWanted.KnotTheory.PardonDistortion.pardon_torus_knot_distortion","parent":15141,"result":true,"route":"../roadmap/Topology/knot-theory/distortion-of-knots/index.html#pardon-torus-knot-distortion","scope":"Topology/knot-theory/distortion-of-knots","status":"can_prove","summary":"Let $p$ and $q$ be coprime positive integers, let $T_{p,q}$ be the standard $(p, q)$-torus curve, and let $K$ be a knot with parametrization $\\gamma$. Suppose there are an ambient isotopy $H$ of $\\mathbb{R}^3$, with time-one map $H_1$, and a strictly increasing diffeomorphism $\\sigma$ of $\\mathbb{R}$ with $\\sigma(t + 2\\pi) = \\sigma(t) + 2\\pi$ for all real $t$, such that $H_1(\\gamma(t)) = T_{p,q}(\\sigma(t))$ for\u2026","title":"Pardon's lower bound for the distortion of torus knots"},{"data_url":"scopes/Topology/knot-theory/quadrisecant-theorem.json","id":"Topology/knot-theory/quadrisecant-theorem","kind":"scope","navigation":"module","parent":15067,"result":false,"route":"../roadmap/Topology/knot-theory/quadrisecant-theorem/index.html","scope":"Topology/knot-theory/quadrisecant-theorem","status":"can_prove","title":"Quadrisecant theorem"},{"id":"Topology/knot-theory/quadrisecant-theorem/period","kind":"node","lean":"MathlibExt.Topology.QuadrisecantWanted.KnotTheory.Quadrisecant.period","parent":15150,"result":true,"route":"../roadmap/Topology/knot-theory/quadrisecant-theorem/index.html#period","scope":"Topology/knot-theory/quadrisecant-theorem","status":"fully_proved","summary":"The period is the real number $2\\pi$.","title":"Period"},{"id":"Topology/knot-theory/quadrisecant-theorem/standard-circle","kind":"node","lean":"MathlibExt.Topology.QuadrisecantWanted.KnotTheory.Quadrisecant.standardCircle","parent":15150,"result":true,"route":"../roadmap/Topology/knot-theory/quadrisecant-theorem/index.html#standard-circle","scope":"Topology/knot-theory/quadrisecant-theorem","status":"fully_proved","summary":"The standard circle is the curve $\\mathbb{R} \\to \\mathbb{R}^3$ given by $t \\mapsto (\\cos t, \\sin t, 0)$, the unit circle in the $xy$-plane.","title":"Standard circle"},{"id":"Topology/knot-theory/quadrisecant-theorem/is-unknotted","kind":"node","lean":"MathlibExt.Topology.QuadrisecantWanted.KnotTheory.Quadrisecant.IsUnknotted","parent":15150,"result":true,"route":"../roadmap/Topology/knot-theory/quadrisecant-theorem/index.html#is-unknotted","scope":"Topology/knot-theory/quadrisecant-theorem","status":"fully_proved","summary":"A knot $K$ is unknotted when there is a knot $J$ whose parametrization is the standard circle $t \\mapsto (\\cos t, \\sin t, 0)$ such that $K$ is ambient-isotopic to $J$ as an oriented knot.","title":"Unknotted knot"},{"id":"Topology/knot-theory/quadrisecant-theorem/has-quadrisecant","kind":"node","lean":"MathlibExt.Topology.QuadrisecantWanted.KnotTheory.Quadrisecant.HasQuadrisecant","parent":15150,"result":true,"route":"../roadmap/Topology/knot-theory/quadrisecant-theorem/index.html#has-quadrisecant","scope":"Topology/knot-theory/quadrisecant-theorem","status":"fully_proved","summary":"A knot $K$ with parametrization $\\gamma \\colon \\mathbb{R} \\to \\mathbb{R}^3$ has a quadrisecant when there are pairwise distinct reals $t_0, t_1, t_2, t_3 \\in [0, 2\\pi)$, a point $p \\in \\mathbb{R}^3$, a nonzero vector $v \\in \\mathbb{R}^3$ and reals $a_0, a_1, a_2, a_3$ such that $\\gamma(t_i) = p + a_i v$ for each $i$, that is, the four points $\\gamma(t_i)$ lie on one line.","title":"Quadrisecant of a knot"},{"id":"Topology/knot-theory/quadrisecant-theorem/smooth-knot-has-quadrisecant","kind":"node","lean":"MathlibExt.Topology.QuadrisecantWanted.KnotTheory.Quadrisecant.smooth_knot_has_quadrisecant","parent":15150,"result":true,"route":"../roadmap/Topology/knot-theory/quadrisecant-theorem/index.html#smooth-knot-has-quadrisecant","scope":"Topology/knot-theory/quadrisecant-theorem","status":"can_prove","summary":"Every smooth knot $K$ in $\\mathbb{R}^3$ that is not unknotted has a quadrisecant.","title":"Quadrisecant theorem (Pannwitz\u2013Kuperberg)"},{"data_url":"scopes/Topology/knot-theory/slice-knots.json","id":"Topology/knot-theory/slice-knots","kind":"scope","navigation":"module","parent":15067,"result":false,"route":"../roadmap/Topology/knot-theory/slice-knots/index.html","scope":"Topology/knot-theory/slice-knots","status":"can_prove","title":"Slice knots"},{"id":"Topology/knot-theory/slice-knots/conway-knot-topologically-slice","kind":"node","lean":"MathlibExt.Topology.ConwayTopologicallySliceWanted.KnotTheory.conway_knot_topologically_slice","parent":15156,"result":true,"route":"../roadmap/Topology/knot-theory/slice-knots/index.html#conway-knot-topologically-slice","scope":"Topology/knot-theory/slice-knots","status":"can_prove","summary":"Let $K$ be the PL knot traced by the braid closure on $4$ strands of the braid word $\\sigma_1^{-2} \\sigma_2 \\sigma_1^{-1} \\sigma_2 \\sigma_1^{-1} \\sigma_3 \\sigma_2^{-2} \\sigma_3^2$ for the Conway knot 11n34. Then $K$ is topologically slice: there is a smooth knot $J$ whose image is unoriented topologically ambient isotopic in $\\mathbb{R}^3$ to the image of $K$, and $J$ is topologically slice.","title":"The Conway knot is topologically slice"},{"id":"Topology/knot-theory/slice-knots/conway-knot-is-simple","kind":"node","lean":"MathlibExt.Topology.PiccirilloWanted.KnotTheory.conwayKnot_isSimple","parent":15156,"result":true,"route":"../roadmap/Topology/knot-theory/slice-knots/index.html#conway-knot-is-simple","scope":"Topology/knot-theory/slice-knots","status":"can_prove","summary":"Let $K$ be the Conway knot $11n34$, realized as a closed polyline in $\\mathbb{R}^3$ (the braid closure of a 4-strand braid word for it), and let $N$ be its number of vertices. The periodic piecewise-linear parametrization of $K$, which traverses each edge in unit time, is injective on the interval $[0, N)$.","title":"The Conway knot polyline is a simple closed curve"},{"id":"Topology/knot-theory/slice-knots/conway-knot-not-smoothly-slice","kind":"node","lean":"MathlibExt.Topology.PiccirilloWanted.KnotTheory.conway_knot_not_smoothly_slice","parent":15156,"result":true,"route":"../roadmap/Topology/knot-theory/slice-knots/index.html#conway-knot-not-smoothly-slice","scope":"Topology/knot-theory/slice-knots","status":"can_prove","summary":"Let $K$ be the Conway knot $11n34$, realized as a closed polyline in $\\mathbb{R}^3$. There is no smooth knot $J$, that is, a smooth $2\\pi$-periodic map $\\mathbb{R} \\to \\mathbb{R}^3$ with nowhere vanishing derivative that is injective on $[0, 2\\pi)$, with both of the following properties: some topological ambient isotopy of $\\mathbb{R}^3$ carries the image of $J$ onto the image of $K$; and there is a smooth map\u2026","title":"The Conway knot is not smoothly slice (Piccirillo)"},{"id":"Topology/knot-theory/slice-knots/exists-topologically-slice-not-smoothly","kind":"node","lean":"MathlibExt.Topology.SliceDichotomyWanted.KnotTheory.exists_topologically_slice_not_smoothly_slice","parent":15156,"result":true,"route":"../roadmap/Topology/knot-theory/slice-knots/index.html#exists-topologically-slice-not-smoothly","scope":"Topology/knot-theory/slice-knots","status":"can_prove","summary":"There is a PL knot $K$ (a closed polyline in $\\mathbb{R}^3$ with at least three vertices, not required to be simple) that is topologically slice but not smoothly slice. Here $K$ is smoothly (respectively topologically) slice when some smooth knot whose image is topologically ambient isotopic to the image of $K$ bounds a smooth (respectively locally flat) properly embedded $2$-disk in\u2026","title":"A topologically slice knot that is not smoothly slice"},{"data_url":"scopes/Topology/knot-theory/knot-diagrams.json","id":"Topology/knot-theory/knot-diagrams","kind":"scope","navigation":"module","parent":15067,"result":false,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","title":"Knot diagrams"},{"id":"Topology/knot-theory/knot-diagrams/dart-diagram","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#dart-diagram","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"A dart diagram consists of a finite set $V$ of crossings, a fixed-point-free involution $\\mathrm{pair}$ of the set $V \\times \\{0, 1, 2, 3\\}$ of darts, and a natural number counting isolated crossingless components.","title":"Dart diagram"},{"id":"Topology/knot-theory/knot-diagrams/dart","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.Dart","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#dart","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"The darts of a dart diagram $D$ with crossing set $V$ are the pairs $(v, i)$ with $v \\in V$ and $i \\in \\{0, 1, 2, 3\\}$; the dart $(v, i)$ is the $i$-th incident half-edge at the crossing $v$.","title":"Darts of a dart diagram"},{"id":"Topology/knot-theory/knot-diagrams/crossing-count","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.crossingCount","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-count","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"The crossing count of a dart diagram $D$ is the number of vertices of $D$, each vertex being one crossing.","title":"Number of crossings of a dart diagram"},{"id":"Topology/knot-theory/knot-diagrams/rot","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.rot","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#rot","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$, the rotation of a dart $(v, i)$ of $D$, with $v$ a crossing and $i \\in \\mathbb{Z}/4$, is the dart $(v, i + 1)$: one step counterclockwise around the crossing $v$.","title":"Rotation of a dart"},{"id":"Topology/knot-theory/knot-diagrams/opposite","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.opposite","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#opposite","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$ and a dart $d = (v, i)$ of $D$, the opposite dart is obtained by rotating $d$ twice counterclockwise around its crossing $v$; it is the dart $(v, i + 2)$, with labels taken modulo $4$.","title":"Opposite dart"},{"id":"Topology/knot-theory/knot-diagrams/is-over","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.isOver","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-over","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"The four darts at a crossing of a dart diagram carry the labels $0, 1, 2, 3$ in cyclic order. A dart lies on the over-strand at its crossing when its label is $0$ or $2$.","title":"Over-strand darts"},{"id":"Topology/knot-theory/knot-diagrams/pair-perm","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.pairPerm","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#pair-perm","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$, the edge-pairing permutation is the edge-pairing involution of $D$ regarded as a permutation of the darts of $D$; it is its own inverse.","title":"Edge-pairing permutation"},{"id":"Topology/knot-theory/knot-diagrams/rot-perm","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.rotPerm","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#rot-perm","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$, the rotation permutation is the permutation of the darts of $D$ sending a dart $(v, i)$, with $v$ a crossing and $i \\in \\mathbb{Z}/4$, to $(v, i + 1)$; its inverse sends $(v, i)$ to $(v, i - 1)$.","title":"Rotation permutation of darts"},{"id":"Topology/knot-theory/knot-diagrams/opposite-perm","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.oppositePerm","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#opposite-perm","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$, the opposite-dart permutation of the darts of $D$ is the square of the quarter-turn rotation permutation: it sends the dart $(v, i)$ at crossing $v$ to the dart $(v, i + 2)$, with labels taken modulo $4$.","title":"Opposite-dart permutation"},{"id":"Topology/knot-theory/knot-diagrams/next","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.next","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#next","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$ and a dart $d$ of $D$, the next dart after $d$ is the dart opposite $\\mathrm{pair}(d)$ at its crossing, where $\\mathrm{pair}$ is the edge-pairing involution of $D$: cross the edge segment at $d$, then go straight through the crossing reached.","title":"Strand traversal step"},{"id":"Topology/knot-theory/knot-diagrams/next-perm","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.nextPerm","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#next-perm","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$, the strand traversal permutation of the darts of $D$ is the edge-pairing permutation followed by the opposite-dart permutation: it sends a dart $d$ to the dart opposite $\\mathrm{pair}(d)$ at its crossing, where $\\mathrm{pair}$ is the edge-pairing involution of $D$.","title":"Strand traversal permutation"},{"id":"Topology/knot-theory/knot-diagrams/same-cycle-next","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.sameCycle_next","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#same-cycle-next","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $D$ be a dart diagram, and for a dart $d$ of $D$ let $\\operatorname{next}(d)$ be the dart reached from $d$ by crossing its edge segment and then going straight through the next crossing. Then for every dart $d$ of $D$, the darts $d$ and $\\operatorname{next}(d)$ lie in the same cycle of the strand traversal permutation of $D$, the permutation of the darts of $D$ that sends each dart $e$ to\u2026","title":"A traversal step stays in its strand cycle"},{"id":"Topology/knot-theory/knot-diagrams/is-alternating","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.IsAlternating","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-alternating","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"A dart diagram is alternating when, for every dart $d$, the dart $\\operatorname{next}(d)$, reached from $d$ by crossing its edge segment and going straight through the next crossing, has the opposite over/under status to $d$. In other words, consecutive crossings on every strand have opposite over/under status.","title":"Alternating dart diagrams"},{"id":"Topology/knot-theory/knot-diagrams/same-component","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.SameComponent","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#same-component","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$, two darts of $D$ lie in the same component when they are related by the equivalence relation on darts generated by $d \\sim \\operatorname{pair}(d)$, crossing an edge segment, and $d \\sim \\operatorname{opposite}(d)$, going straight through a crossing.","title":"Same component relation on darts"},{"id":"Topology/knot-theory/knot-diagrams/component-setoid","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.componentSetoid","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#component-setoid","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a dart diagram $D$, this is the equivalence relation on the darts of $D$ generated by relating each dart $d$ to its partner across an edge segment and to the opposite dart of $d$ at the same crossing. Two darts are related exactly when they lie on the same unoriented strand component.","title":"Strand-connectivity relation on darts"},{"id":"Topology/knot-theory/knot-diagrams/same-component-next","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.sameComponent_next","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#same-component-next","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For every dart $d$ of a dart diagram $D$, the dart $\\operatorname{next}(d)$ reached by one step of strand traversal lies in the same component as $d$.","title":"A traversal step stays in the same component"},{"id":"Topology/knot-theory/knot-diagrams/component-count","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.componentCount","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#component-count","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"The number of components of a dart diagram $D$ is the number of classes of darts of $D$ under unoriented strand connectivity, plus the recorded number of isolated crossingless components of $D$.","title":"Number of components of a dart diagram"},{"id":"Topology/knot-theory/knot-diagrams/isolated-le-component-count","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.isolated_le_componentCount","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#isolated-le-component-count","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For every dart diagram $D$, the number of isolated crossingless components recorded in $D$ is at most the number of components of $D$.","title":"Isolated components are counted in the component count"},{"id":"Topology/knot-theory/knot-diagrams/empty","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.DartDiagram.empty","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#empty","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a natural number $k$, the empty dart diagram with $k$ isolated components has no crossings, hence no darts, and records $k$ isolated crossingless components.","title":"Empty dart diagram"},{"id":"Topology/knot-theory/knot-diagrams/r2","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.R2","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#r2","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"The space $\\mathbb{R}^2$ with its Euclidean inner product and norm, the target plane of the fixed standard projection frame.","title":"Euclidean plane"},{"id":"Topology/knot-theory/knot-diagrams/plane","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.plane","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#plane","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"The standard projection sends a point $x = (x_0, x_1, x_2)$ of Euclidean 3-space to the point $(x_0, x_1)$ of the Euclidean plane, forgetting the third coordinate.","title":"Standard projection to the plane"},{"id":"Topology/knot-theory/knot-diagrams/height","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.height","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#height","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"The height of a point $x = (x_1, x_2, x_3)$ of Euclidean 3-space is its third coordinate $x_3$.","title":"Height in the standard projection frame"},{"id":"Topology/knot-theory/knot-diagrams/param-point","kind":"node","lean":"MathlibExt.Topology.Knot.RegularProjection.ParamPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#param-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a natural number $n$, a parameter point consists of a component index $i \\in \\{0, \\dots, n-1\\}$ and a real parameter $t$ with $0 \\le t < 2\\pi$, the fundamental half-open period. It represents the point at parameter $t$ of component $i$ of a parametrized link with $n$ components.","title":"Parameter point"},{"id":"Topology/knot-theory/knot-diagrams/point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.point","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a parametrized link $L$ with components $\\gamma_0, \\dots, \\gamma_{n-1} \\colon \\mathbb{R} \\to \\mathbb{R}^3$ and a parameter point $a = (i, t)$, with $i < n$ and $t \\in [0, 2\\pi)$, the point of $L$ at $a$ is $\\gamma_i(t) \\in \\mathbb{R}^3$.","title":"Point of a link at a parameter point"},{"id":"Topology/knot-theory/knot-diagrams/proj-curve","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.projCurve","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#proj-curve","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a parametrized link $L$ with components $\\gamma_0, \\dots, \\gamma_{n-1} \\colon \\mathbb{R} \\to \\mathbb{R}^3$ and $i < n$, the projected curve of component $i$ is the map $\\mathbb{R} \\to \\mathbb{R}^2$, $t \\mapsto \\pi(\\gamma_i(t))$, where $\\pi(x_0, x_1, x_2) = (x_0, x_1)$ is the standard projection forgetting the third coordinate.","title":"Projected curve of a link component"},{"id":"Topology/knot-theory/knot-diagrams/proj-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.projPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#proj-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a parametrized link $L$ with components $\\gamma_0, \\dots, \\gamma_{n-1} \\colon \\mathbb{R} \\to \\mathbb{R}^3$ and a parameter point $a = (i, t)$, with $i < n$ and $t \\in [0, 2\\pi)$, the projected point of $L$ at $a$ is $\\pi(\\gamma_i(t)) \\in \\mathbb{R}^2$, where $\\pi(x_0, x_1, x_2) = (x_0, x_1)$ forgets the third coordinate.","title":"Projected point of a link at a parameter point"},{"id":"Topology/knot-theory/knot-diagrams/tangent","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.tangent","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#tangent","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link and let $a$ be a parameter point of $L$, given by a component $i$ and a parameter $t$. The projected tangent vector of $L$ at $a$ is the vector of $\\mathbb{R}^2$ obtained by applying the standard projection $(x, y, z) \\mapsto (x, y)$ to the derivative $\\gamma_i'(t)$ of the curve $\\gamma_i$ of the $i$-th component of $L$ at $t$.","title":"Projected tangent vector"},{"id":"Topology/knot-theory/knot-diagrams/height-at","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.heightAt","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#height-at","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a parametrized link $L$ with $n$ components and a parameter point $a$ of $L$, the height of $L$ at $a$ is the height (the third coordinate) of the point of Euclidean 3-space that $a$ represents on $L$.","title":"Height of a parameter point"},{"id":"Topology/knot-theory/knot-diagrams/double-point-set","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.doublePointSet","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#double-point-set","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For an oriented labeled parametrized link $L$ with $n$ components, the double-point set of $L$ is the set of points $p$ of the plane $\\mathbb{R}^2$ for which there are two distinct parameter points of $L$ (each a component index together with a parameter in $[0, 2\\pi)$) whose images under the standard projection, which forgets the third coordinate, both equal $p$.","title":"Double points of a link projection"},{"id":"Topology/knot-theory/knot-diagrams/is-regular-projection","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.IsRegularProjection","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-regular-projection","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link; a parameter point of $L$ is a component of $L$ together with a parameter in $[0, 2\\pi)$. The standard projection of $L$ to the plane, which forgets the third coordinate, is regular when the following hold: at every parameter point, the projection of the velocity vector of the component curve is nonzero; the double-point set is finite; no three pairwise distinct parameter points have\u2026","title":"Regular projections"},{"id":"Topology/knot-theory/knot-diagrams/height-at-ne-of-proj-point-eq-of-ne","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.heightAt_ne_of_projPoint_eq_of_ne","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#height-at-ne-of-proj-point-eq-of-ne","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link with $n$ components. If two distinct parameter points $a \\ne b$ of $L$ have the same projected point, then the heights of $L$ at $a$ and at $b$ are different.","title":"Points with the same projection have different heights"},{"id":"Topology/knot-theory/knot-diagrams/crossing","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.Crossing","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link with $n$ components. A crossing of $L$ is a point of the double-point set of $L$, that is, a point $p$ of the plane $\\mathbb{R}^2$ such that two distinct parameter points of $L$ both have image $p$ under the standard projection forgetting the third coordinate.","title":"Crossings of a link projection"},{"id":"Topology/knot-theory/knot-diagrams/crossing-count-2","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingCount","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-count-2","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a parametrized link $L$ whose standard projection is regular, the crossing count of $L$ is the number of crossings of the projection, that is, the number of points of its double-point set.","title":"Crossing count of a regular projection"},{"id":"Topology/knot-theory/knot-diagrams/exists-pair-of-mem-double-point-set","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.exists_pair_of_mem_doublePointSet","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#exists-pair-of-mem-double-point-set","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link with $n$ components whose standard projection is regular, and let $p$ be a point of the double-point set of $L$. Then there are parameter points $a \\ne b$ of $L$ whose projected points are both $p$, and every parameter point of $L$ whose projected point is $p$ is $a$ or $b$.","title":"Double points have exactly two preimages"},{"id":"Topology/knot-theory/knot-diagrams/is-over-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.IsOverPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-over-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link and $p$ a crossing of its standard projection to the plane. A parameter point $a$ of $L$ is an over point of $p$ when $a$ projects to $p$ and every other parameter point that projects to $p$ has strictly smaller height than $a$.","title":"Being the over-strand preimage of a crossing"},{"id":"Topology/knot-theory/knot-diagrams/is-under-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.IsUnderPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-under-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link and $p$ a crossing of its standard projection to the plane. A parameter point $a$ of $L$ is an under point of $p$ when $a$ projects to $p$ and every other parameter point that projects to $p$ has strictly greater height than $a$.","title":"Being the under-strand preimage of a crossing"},{"id":"Topology/knot-theory/knot-diagrams/exists-unique-is-over-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.existsUnique_isOverPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#exists-unique-is-over-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link with $n$ components whose standard projection is regular, and let $p$ be a crossing of that projection. There is exactly one parameter point $a$ of $L$ that is an over-strand preimage of $p$: the projected point of $a$ is $p$, and every other parameter point whose projected point is $p$ has height strictly less than the height of $a$.","title":"The over-strand preimage exists and is unique"},{"id":"Topology/knot-theory/knot-diagrams/exists-unique-is-under-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.existsUnique_isUnderPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#exists-unique-is-under-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link with $n$ components whose standard projection is regular, and let $p$ be a crossing of that projection. There is exactly one parameter point $a$ of $L$ that is an under-strand preimage of $p$: the projected point of $a$ is $p$, and every other parameter point whose projected point is $p$ has height strictly greater than the height of $a$.","title":"The under-strand preimage exists and is unique"},{"id":"Topology/knot-theory/knot-diagrams/over-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.overPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#over-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link with $n$ components whose standard projection is regular, and let $p$ be a crossing of that projection. The over-strand preimage of $p$ is the unique parameter point $a$ of $L$ that projects to $p$ and such that every other parameter point projecting to $p$ has strictly smaller height than $a$.","title":"Over-strand preimage of a crossing"},{"id":"Topology/knot-theory/knot-diagrams/under-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.underPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#under-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link whose standard projection is regular, and let $p$ be a crossing of this projection. The under-strand preimage of $p$ is the unique parameter point $a$ of $L$ that projects to $p$ and whose height is strictly less than the height of every other parameter point projecting to $p$; the height of a parameter point is the third coordinate of the corresponding point of $L$ in $\\mathbb{R}^3$.","title":"Under-strand preimage of a crossing"},{"id":"Topology/knot-theory/knot-diagrams/under-point-ne-over-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.underPoint_ne_overPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#under-point-ne-over-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link whose standard projection is regular, and let $p$ be a crossing of this projection. Then the under-strand preimage of $p$ and the over-strand preimage of $p$ are distinct parameter points of $L$.","title":"Over and under preimages differ"},{"id":"Topology/knot-theory/knot-diagrams/height-at-under-point-lt-over-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.heightAt_underPoint_lt_overPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#height-at-under-point-lt-over-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link with $n$ components whose standard projection is regular, and let $p$ be a crossing of that projection. The height of $L$ at the under-strand preimage of $p$ is strictly less than its height at the over-strand preimage of $p$.","title":"The under-strand lies strictly below the over-strand"},{"id":"Topology/knot-theory/knot-diagrams/proj-point-over-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.projPoint_overPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#proj-point-over-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link whose standard projection is regular, and let $p$ be a crossing of $L$, that is, a point of the double-point set of its projection. Then the projected point of the over-strand parameter point at $p$ is $p$.","title":"The over-strand preimage lies over its crossing"},{"id":"Topology/knot-theory/knot-diagrams/proj-point-under-point","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.projPoint_underPoint","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#proj-point-under-point","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link whose standard projection is regular, and let $p$ be a crossing of $L$, that is, a point of the double-point set of its projection. Then the projected point of the under-strand parameter point at $p$ is $p$.","title":"The under-strand preimage lies over its crossing"},{"id":"Topology/knot-theory/knot-diagrams/eq-over-point-or-eq-under-point-of-proj","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.eq_overPoint_or_eq_underPoint_of_projPoint_eq","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#eq-over-point-or-eq-under-point-of-proj","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link with $n$ components whose standard projection is regular, and let $p$ be a crossing of that projection. Every parameter point of $L$ whose projected point is $p$ is either the over-strand preimage or the under-strand preimage of $p$.","title":"The over and under preimages exhaust a crossing"},{"id":"Topology/knot-theory/knot-diagrams/crossing-count-eq-zero-of-double-point-s","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingCount_eq_zero_of_doublePointSet_eq_empty","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-count-eq-zero-of-double-point-s","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"If a parametrized link $L$ has a regular standard projection whose double-point set is empty, then the crossing count of $L$ is $0$.","title":"A crossing-free regular projection has crossing count zero"},{"id":"Topology/knot-theory/knot-diagrams/crossing-count-pos-of-crossing","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingCount_pos_of_crossing","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-count-pos-of-crossing","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"If a parametrized link $L$ has a regular standard projection with at least one crossing, then the crossing count of $L$ is positive.","title":"A regular projection with a crossing has positive crossing count"},{"id":"Topology/knot-theory/knot-diagrams/crossing-number","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingNumber","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-number","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For an oriented labeled parametrized link $L$ with $n$ components, the crossing number of $L$ is the infimum in $\\mathbb{N} \\cup \\{\\infty\\}$, over all parametrized links $L'$ with $n$ components that are ambient isotopic to $L$ and whose standard projection to the plane is regular, of the number of crossings (double points) of that projection of $L'$. It is $\\infty$ when no such $L'$ exists.","title":"Crossing number of a link"},{"id":"Topology/knot-theory/knot-diagrams/crossing-number-le-of-isotopic-of-is-reg","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingNumber_le_of_isotopic_of_isRegularProjection","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-number-le-of-isotopic-of-is-reg","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ and $L'$ be oriented labeled parametrized links with $n$ components, with $L'$ ambient isotopic to $L$, and suppose the standard projection of $L'$ to the plane is regular. Then the crossing number of $L$ is at most the number of crossings of this projection of $L'$.","title":"Regular projections of isotopic links bound the crossing number"},{"id":"Topology/knot-theory/knot-diagrams/crossing-number-le-self-of-is-regular-pr","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingNumber_le_self_of_isRegularProjection","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-number-le-self-of-is-regular-pr","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"If the standard projection of an oriented labeled parametrized link $L$ to the plane is regular, then the crossing number of $L$ is at most the number of crossings of this projection.","title":"A link's own regular projection bounds its crossing number"},{"id":"Topology/knot-theory/knot-diagrams/crossing-number-lt-top-of-isotopic-of-is","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingNumber_lt_top_of_isotopic_of_isRegularProjection","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-number-lt-top-of-isotopic-of-is","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ and $L'$ be oriented labeled parametrized links with $n$ components. If $L'$ is ambient isotopic to $L$ and the standard projection of $L'$ to the plane is regular, then the crossing number of $L$ is finite.","title":"Finiteness of the crossing number"},{"id":"Topology/knot-theory/knot-diagrams/crossing-number-eq-of-isotopic","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingNumber_eq_of_isotopic","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-number-eq-of-isotopic","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"If two parametrized links $L_1$ and $L_2$ with $n$ components are isotopic, then $L_1$ and $L_2$ have the same crossing number.","title":"The crossing number is an isotopy invariant"},{"id":"Topology/knot-theory/knot-diagrams/crossing-number-eq-zero-of-isotopic-of-d","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.crossingNumber_eq_zero_of_isotopic_of_doublePointSet_eq_empty","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#crossing-number-eq-zero-of-isotopic-of-d","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ and $L'$ be parametrized links with $n$ components such that $L'$ is isotopic to $L$. If the standard projection of $L'$ is regular and its double-point set is empty, then the crossing number of $L$ is $0$.","title":"Crossing number zero from a crossing-free representative"},{"id":"Topology/knot-theory/knot-diagrams/is-crossing-preimage","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.IsCrossingPreimage","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-crossing-preimage","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"A parameter point $a$ of a parametrized link $L$ (a component of $L$ together with a parameter in $[0, 2\\pi)$) is a crossing preimage when its image under the standard projection to the plane lies in the double-point set of $L$.","title":"Crossing preimages"},{"id":"Topology/knot-theory/knot-diagrams/is-over-at","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.isOverAt","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-over-at","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link whose standard projection to the plane is regular, and let $a$ be a crossing preimage of $L$. The over/under status at $a$ is true when $a$ is the over-strand preimage of the crossing at the projection of $a$, that is, the preimage of that crossing of greatest height, and false otherwise.","title":"Over/under status at a crossing preimage"},{"id":"Topology/knot-theory/knot-diagrams/is-over-at-ne-of-ne-of-proj-point-eq","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.isOverAt_ne_of_ne_of_projPoint_eq","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-over-at-ne-of-ne-of-proj-point-eq","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link whose standard projection to the plane is regular. If $a \\ne b$ are crossing preimages of $L$ with the same projection, then the over/under status at $a$ differs from the over/under status at $b$.","title":"The two preimages of a crossing have opposite status"},{"id":"Topology/knot-theory/knot-diagrams/in-open-arc","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.inOpenArc","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#in-open-arc","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $a$, $b$, $c$ be parameter points with parameters $s$, $u$, $t \\in [0, 2\\pi)$ respectively. Then $c$ lies in the open counterclockwise arc from $a$ to $b$ when $s < t < u$ in case $s < u$, and when $s < t$ or $t < u$ in case $u \\le s$. Only the parameters are compared; the components of $a$, $b$, $c$ play no role.","title":"Open arc on the parameter circle"},{"id":"Topology/knot-theory/knot-diagrams/cyclically-adjacent","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.CyclicallyAdjacent","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#cyclically-adjacent","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be an oriented labeled parametrized link, with each component parametrized by $t \\in [0, 2\\pi)$. Two distinct parameter points $a$ and $b$ of $L$ on the same component are cyclically adjacent, in this order, when both are crossing preimages (they project to double points of the standard projection) and no crossing preimage on that component lies in the open arc from $a$ to $b$ in the direction of increasing\u2026","title":"Cyclically adjacent crossing preimages"},{"id":"Topology/knot-theory/knot-diagrams/is-alternating-projection","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.IsAlternatingProjection","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-alternating-projection","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link whose standard projection to the plane is regular. The projection is alternating when, for any two cyclically adjacent crossing preimages $a$ and $b$ of $L$ (distinct, on the same component, with no other crossing preimage of that component in the open arc of parameters from $a$ to $b$), the over/under status at $a$ differs from the over/under status at $b$.","title":"Alternating regular projections"},{"id":"Topology/knot-theory/knot-diagrams/is-alternating-projection-of-double-poin","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.isAlternatingProjection_of_doublePointSet_eq_empty","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#is-alternating-projection-of-double-poin","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L$ be a parametrized link whose standard projection to the plane is regular. If the double-point set of this projection is empty, then the projection is alternating.","title":"Crossingless regular projections are alternating"},{"id":"Topology/knot-theory/knot-diagrams/admits-alternating-diagram","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.AdmitsAlternatingDiagram","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#admits-alternating-diagram","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"A parametrized link $L$ with $n$ components admits an alternating diagram when there is a parametrized link $L'$ isotopic to $L$ whose standard projection is regular and alternating.","title":"Links admitting an alternating diagram"},{"id":"Topology/knot-theory/knot-diagrams/of-isotopic","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.AdmitsAlternatingDiagram.of_isotopic","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#of-isotopic","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be parametrized links with $n$ components. If $L_1$ is isotopic to $L_2$ and $L_1$ admits an alternating diagram, then $L_2$ admits an alternating diagram.","title":"Admitting an alternating diagram is an isotopy invariant"},{"id":"Topology/knot-theory/knot-diagrams/admits-alternating-diagram-of-double-poi","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.admitsAlternatingDiagram_of_doublePointSet_eq_empty","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#admits-alternating-diagram-of-double-poi","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"If a parametrized link $L$ has a regular standard projection whose double-point set is empty, then $L$ admits an alternating diagram.","title":"Links with a crossing-free regular projection admit alternating diagrams"},{"id":"Topology/knot-theory/knot-diagrams/i-union-range-proj-curve","kind":"node","lean":"MathlibExt.Topology.Knot.Euclidean.ParametrizedLink.iUnion_range_projCurve","parent":15161,"result":true,"route":"../roadmap/Topology/knot-theory/knot-diagrams/index.html#i-union-range-proj-curve","scope":"Topology/knot-theory/knot-diagrams","status":"fully_proved","summary":"For a parametrized link $L$ with $n$ components, the union over all components $i$ of the images of the projected curves of $L$ equals the image of the trace of $L$ under the standard projection to the plane.","title":"The projected trace is the union of the projected components"},{"data_url":"scopes/Topology/knot-theory/ropelength-and-crossing-number.json","id":"Topology/knot-theory/ropelength-and-crossing-number","kind":"scope","navigation":"module","parent":15067,"result":false,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","title":"Ropelength and crossing number"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/r3","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.R3","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#r3","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The space $\\mathbb{R}^3$ with its Euclidean inner product and norm.","title":"Euclidean 3-space"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/triple-points","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.triplePoints","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#triple-points","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For points $p, q, r$ of Euclidean 3-space, the family indexed by $\\{0, 1, 2\\}$ that sends $0 \\mapsto p$, $1 \\mapsto q$ and $2 \\mapsto r$.","title":"Ordered triple of points"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/triangle-of","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.triangleOf","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#triangle-of","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For points $p, q, r$ of Euclidean 3-space whose ordered triple $(p, q, r)$ is affinely independent, the triangle in Euclidean 3-space with vertices $p$, $q$, $r$, in that order.","title":"Triangle on three affinely independent points"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/circumradius-enn","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.circumradiusENN","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#circumradius-enn","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For points $p, q, r$ of Euclidean 3-space, the extended circumradius of $(p, q, r)$ is the following value in $[0, \\infty]$. If $p, q, r$ are affinely independent, it is the circumradius of the triangle with vertices $p$, $q$ and $r$. Otherwise, that is, when the three points are collinear (in particular when two of them coincide), it is $\\infty$.","title":"Extended circumradius of three points"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/circumradius-enn-eq-of-real-circumradius","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.circumradiusENN_eq_ofReal_circumradius","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#circumradius-enn-eq-of-real-circumradius","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"Let $p, q, r$ be points of Euclidean 3-space and let $T$ be a triangle in Euclidean 3-space whose vertices, in order, are $p$, $q$ and $r$. Then the extended circumradius of $(p, q, r)$ equals the circumradius of $T$.","title":"Extended circumradius of a triangle"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/circumradius-enn-eq-top-of-collinear","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.circumradiusENN_eq_top_of_collinear","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#circumradius-enn-eq-top-of-collinear","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"If $p, q, r$ are points of Euclidean 3-space and the set $\\{p, q, r\\}$ is collinear, then the extended circumradius of $(p, q, r)$ is $\\infty$.","title":"Collinear points have infinite extended circumradius"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/circumradius-enn-eq-top-of-eq-left","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.circumradiusENN_eq_top_of_eq_left","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#circumradius-enn-eq-top-of-eq-left","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"If $p, q, r$ are points of Euclidean 3-space with $p = q$, then the extended circumradius of $(p, q, r)$ is $\\infty$.","title":"Equal first two points give infinite extended circumradius"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/circumradius-enn-swap-left","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.circumradiusENN_swap_left","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#circumradius-enn-swap-left","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For all points $p, q, r$ of Euclidean 3-space, the extended circumradius of $(q, p, r)$ equals the extended circumradius of $(p, q, r)$.","title":"Extended circumradius is symmetric in the first two points"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/circumradius-enn-swap-right","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.circumradiusENN_swap_right","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#circumradius-enn-swap-right","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For all points $p, q, r$ of Euclidean 3-space, the extended circumradius of $(p, r, q)$ equals the extended circumradius of $(p, q, r)$.","title":"Extended circumradius is symmetric in the last two points"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/circumradius-enn-reverse","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.circumradiusENN_reverse","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#circumradius-enn-reverse","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For all points $p, q, r$ of Euclidean 3-space, the extended circumradius of $(r, q, p)$ equals the extended circumradius of $(p, q, r)$.","title":"Extended circumradius is invariant under reversal"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/circumradius-enn-vadd","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.circumradiusENN_vadd","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#circumradius-enn-vadd","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For every vector $v$ and all points $p, q, r$ of Euclidean 3-space, the extended circumradius of $(v + p, v + q, v + r)$ equals the extended circumradius of $(p, q, r)$.","title":"Translation invariance of the extended circumradius"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/thickness","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.thickness","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#thickness","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The thickness of a set $s$ in Euclidean 3-space is the infimum, in $[0, \\infty]$, of the extended circumradii of $(p, q, r)$ over all points $p, q, r \\in s$. Since the extended circumradius of three collinear points (in particular of three points two of which coincide) is $\\infty$, this is also the infimum over the triples of pairwise distinct points of $s$.","title":"Thickness"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/thickness-le","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.thickness_le","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#thickness-le","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For every set $s$ in Euclidean 3-space and all points $p, q, r \\in s$, the thickness of $s$ is at most the extended circumradius of $p, q, r$.","title":"Thickness is at most every circumradius"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/thickness-anti","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.thickness_anti","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#thickness-anti","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"If $s \\subseteq t$ are sets in Euclidean 3-space, then the thickness of $t$ is at most the thickness of $s$: enlarging a set can only add triples and hence can only decrease the infimum.","title":"Thickness is antitone"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/thickness-empty","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.thickness_empty","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#thickness-empty","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The empty subset of Euclidean 3-space has thickness $\\infty$, as an empty infimum in $[0, \\infty]$.","title":"Thickness of the empty set"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/thickness-singleton","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.thickness_singleton","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#thickness-singleton","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For every point $p$ of Euclidean 3-space, the set $\\{p\\}$ has thickness $\\infty$: its only triple $(p, p, p)$ is degenerate.","title":"Thickness of a single point"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/contour-length","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.contourLength","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#contour-length","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"Let $\\gamma$ be a map from $\\mathbb{R}$ to Euclidean 3-space and let $a, b$ be real numbers. The contour length of $\\gamma$ on $[a, b]$ is the total variation of $\\gamma$ on $[a, b]$: the supremum, over all finite sequences $t_0 \\le t_1 \\le \\dots \\le t_n$ of points of $[a, b]$, of $\\sum_{i < n} \\lVert \\gamma(t_{i+1}) - \\gamma(t_i) \\rVert$. It is a value in $[0, \\infty]$.","title":"Contour length"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/contour-length-eq-of-real-integral-norm","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.contourLength_eq_ofReal_integral_norm_deriv","parent":15226,"reading_title":{"format":"inline-parts-v1","parts":[["text","Contour length of a "],["math","\\(C^1\\)"],["text"," curve"]],"source":"Contour length of a $C^1$ curve"},"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#contour-length-eq-of-real-integral-norm","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"Let $a \\le b$ be real numbers and let $\\gamma$ be a map from $\\mathbb{R}$ to Euclidean 3-space that is continuously differentiable on $[a, b]$. Then the contour length of $\\gamma$ on $[a, b]$ equals $\\int_a^b \\lVert \\gamma'(t) \\rVert \\, dt$.","title":"Contour length of a $C^1$ curve"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/ropelength-ratio","kind":"node","lean":"MathlibExt.Geometry.Euclidean.Ropelength.ropelengthRatio","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#ropelength-ratio","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For a curve $\\gamma : \\mathbb{R} \\to \\mathbb{R}^3$ and reals $a, b$, the ropelength ratio of $\\gamma$ on $[a, b]$ is the contour length of $\\gamma$ on $[a, b]$ divided by the thickness of the trace $\\gamma([a, b])$. The quotient is taken in $[0, \\infty]$, where $c / 0 = \\infty$ for $c \\ne 0$, $0 / 0 = 0$ and $c / \\infty = 0$.","title":"Ropelength ratio"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/is-admissible-alternating","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.IsAdmissibleAlternating","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#is-admissible-alternating","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"A real number $c$ is admissible when $c \\ge 0$ and, for every $n$ and every oriented labeled link type with $n$ components that admits an alternating diagram and whose crossing number is a natural number $k$, the ropelength of the type is at least $c k$.","title":"Admissible constants for the tight alternating knot inequality"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/tight-alternating-knot-constant","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.tightAlternatingKnotConstant","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#tight-alternating-knot-constant","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The tight alternating knot constant $C_{22b} = b_0$ is the supremum, taken in the extended reals, of the set of real numbers $c$ admissible for the tight alternating knot inequality: those $c \\ge 0$ such that every link type $K$ that admits an alternating diagram and has finite crossing number $\\mathrm{Cr}(K)$ has ropelength $\\mathrm{L}(K) \\ge c\\,\\mathrm{Cr}(K)$. The supremum is not assumed to be attained.","title":"Tight alternating knot constant"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/is-admissible-alternating-zero","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.isAdmissibleAlternating_zero","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#is-admissible-alternating-zero","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The constant $c = 0$ is admissible for the tight alternating knot inequality.","title":"Zero is an admissible alternating constant"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/of-le","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.IsAdmissibleAlternating.of_le","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#of-le","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"Admissible alternating constants are downward closed among nonnegative reals. If a real number $c$ is admissible for the tight alternating knot inequality and $c'$ is a real number with $0 \\le c' \\le c$, then $c'$ is admissible as well.","title":"Admissible alternating constants are downward closed among nonnegative reals"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/le-tight-alternating-knot-constant","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.le_tightAlternatingKnotConstant","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#le-tight-alternating-knot-constant","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"Admissible constants are at most the tight alternating knot constant. Every real number $c$ that is admissible for the tight alternating knot inequality is at most the tight alternating knot constant.","title":"Admissible constants are at most the tight alternating knot constant"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/zero-le-tight-alternating-knot-constant","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.zero_le_tightAlternatingKnotConstant","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#zero-le-tight-alternating-knot-constant","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The tight alternating knot constant, an extended real number, is at least $0$.","title":"The tight alternating knot constant is nonnegative"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/tight-alternating-knot-constant-le-iff","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.tightAlternatingKnotConstant_le_iff","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#tight-alternating-knot-constant-le-iff","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For every extended real number $b$, the tight alternating knot constant is at most $b$ if and only if every real number $c$ that is admissible for the tight alternating knot inequality satisfies $c \\le b$.","title":"Upper bounds for the tight alternating knot constant"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/crossing-number-lt-top-of-admits-alterna","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.crossingNumber_lt_top_of_admitsAlternatingDiagram","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#crossing-number-lt-top-of-admits-alterna","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"If an oriented labeled link type with $n$ components admits an alternating diagram, then its crossing number is finite: the alternating representative is itself a regular projection.","title":"Alternating link types have finite crossing number"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/exists-crossing-number-eq-nat-cast-of-ad","kind":"node","lean":"MathlibExt.Topology.Knot.TightAlternatingKnot.exists_crossingNumber_eq_natCast_of_admitsAlternatingDiagram","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#exists-crossing-number-eq-nat-cast-of-ad","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"If an oriented labeled link type with $n$ components admits an alternating diagram, then its crossing number equals some natural number $k$.","title":"Alternating link types have a natural-number crossing number"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/is-admissible","kind":"node","lean":"MathlibExt.Topology.Knot.TightKnot.IsAdmissible","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#is-admissible","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"A real number $c$ is admissible for the tight-knot inequality when $c \\ge 0$ and, for every $n \\in \\mathbb{N}$, every parametrized link $L$ with $n$ components and every $k \\in \\mathbb{N}$ such that the crossing number of $L$ equals $k$, the ropelength of $L$ is at least $c \\, k^{3/4}$, the inequality being read in $[0, \\infty]$. Here the crossing number and the ropelength of $L$ are those of its oriented labeled\u2026","title":"Admissible constants for the tight-knot inequality"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/tight-knot-constant","kind":"node","lean":"MathlibExt.Topology.Knot.TightKnot.tightKnotConstant","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#tight-knot-constant","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The tight knot constant $C_{22a}$ is the supremum, taken in the extended reals $[-\\infty, \\infty]$, of the set of real numbers $c$ that are admissible for the tight-knot inequality $\\mathrm{ropelength} \\ge c \\cdot (\\text{crossing number})^{3/4}$ on link types. The supremum is not assumed to be attained.","title":"Tight knot constant"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/is-admissible-zero","kind":"node","lean":"MathlibExt.Topology.Knot.TightKnot.isAdmissible_zero","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#is-admissible-zero","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The real number $0$ is admissible for the tight-knot inequality between the ropelength and the crossing number of link types.","title":"Zero is an admissible tight-knot constant"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/of-le-2","kind":"node","lean":"MathlibExt.Topology.Knot.TightKnot.IsAdmissible.of_le","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#of-le-2","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"If a real number $c$ is admissible for the tight-knot inequality and $c'$ is a real number with $0 \\le c' \\le c$, then $c'$ is admissible for the tight-knot inequality as well.","title":"Admissibility passes to smaller nonnegative constants"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/le-tight-knot-constant","kind":"node","lean":"MathlibExt.Topology.Knot.TightKnot.le_tightKnotConstant","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#le-tight-knot-constant","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"Let $C_{22a}$ be the tight knot constant. Every real number $c$ that is admissible for the tight-knot inequality satisfies $c \\le C_{22a}$ in the extended reals.","title":"Admissible constants are bounded by the tight knot constant"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/zero-le-tight-knot-constant","kind":"node","lean":"MathlibExt.Topology.Knot.TightKnot.zero_le_tightKnotConstant","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#zero-le-tight-knot-constant","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"The tight knot constant $C_{22a}$ satisfies $0 \\le C_{22a}$ in the extended reals.","title":"The tight knot constant is nonnegative"},{"id":"Topology/knot-theory/ropelength-and-crossing-number/tight-knot-constant-le-iff","kind":"node","lean":"MathlibExt.Topology.Knot.TightKnot.tightKnotConstant_le_iff","parent":15226,"result":true,"route":"../roadmap/Topology/knot-theory/ropelength-and-crossing-number/index.html#tight-knot-constant-le-iff","scope":"Topology/knot-theory/ropelength-and-crossing-number","status":"fully_proved","summary":"For every extended real number $b$, the tight knot constant $C_{22a}$ satisfies $C_{22a} \\le b$ if and only if every real number $c$ that is admissible for the tight-knot inequality satisfies $c \\le b$.","title":"Upper bounds for the tight knot constant"},{"data_url":"scopes/Topology/general-topology.json","id":"Topology/general-topology","kind":"scope","parent":15066,"result":false,"route":"../roadmap/Topology/general-topology/index.html","scope":"Topology/general-topology","status":"can_prove","summary":"Point-set topology of topological and metric spaces: connectedness, separation and cardinal invariants, homogeneous spaces, direct and inverse limits of spaces, restricted products, and dimension theory.","title":"General topology"},{"id":"Topology/general-topology/inter-frontier-nonempty-of-inter-compl","kind":"node","lean":"IsPreconnected.inter_frontier_nonempty_of_inter_compl","parent":15262,"result":true,"route":"../roadmap/Topology/general-topology/index.html#inter-frontier-nonempty-of-inter-compl","scope":"Topology/general-topology","status":"fully_proved","summary":"Let $s$ be a preconnected subset of a topological space $X$ and let $A \\subseteq X$. If both $s \\cap A$ and $s \\cap (X \\setminus A)$ are nonempty, then $s$ meets the frontier $\\partial A = \\overline{A} \\setminus \\operatorname{int} A$ of $A$, that is, $s \\cap \\partial A$ is nonempty.","title":"A preconnected set meets the frontier of every set it crosses"},{"id":"Topology/general-topology/dist-div-one-add-dist-balls-equivalent","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BoundedMetricTopologicallyEquivalentWanted.dist_div_one_add_dist_balls_equivalent","parent":15262,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded metric "],["math","\\(d/(1+d)\\)"],["text"," is equivalent to "],["math","\\(d\\)"]],"source":"Bounded metric $d/(1+d)$ is equivalent to $d$"},"result":true,"route":"../roadmap/Topology/general-topology/index.html#dist-div-one-add-dist-balls-equivalent","scope":"Topology/general-topology","status":"can_prove","summary":"Let $(X,d)$ be a metric space and define","title":"Bounded metric $d/(1+d)$ is equivalent to $d$"},{"id":"Topology/general-topology/homogeneous-space","kind":"node","lean":"HomogeneousSpace","parent":15262,"result":true,"route":"../roadmap/Topology/general-topology/index.html#homogeneous-space","scope":"Topology/general-topology","status":"fully_proved","summary":"A topological space is homogeneous if its homeomorphism group acts transitively.","title":"Homogeneous topological space"},{"id":"Topology/general-topology/topological-krull-dim-add-one-le-of-is-c","kind":"node","lean":"IsIrreducible.topologicalKrullDim_add_one_le_of_isClosed_of_ne","parent":15262,"result":true,"route":"../roadmap/Topology/general-topology/index.html#topological-krull-dim-add-one-le-of-is-c","scope":"Topology/general-topology","status":"fully_proved","summary":"A proper closed subset of an irreducible set has Krull dimension at least one less than the ambient set.","title":"Krull dimension drops on proper closed subsets of irreducible sets"},{"id":"Topology/general-topology/kuratowski-closure-complement","kind":"node","lean":"MathlibExt.Topology.KuratowskiClosureComplement.kuratowski_closure_complement","parent":15262,"result":true,"route":"../roadmap/Topology/general-topology/index.html#kuratowski-closure-complement","scope":"Topology/general-topology","status":"fully_proved","summary":"Let $X$ be a topological space and $s \\subseteq X$. The sets that can be obtained from $s$ by applying closure and complement finitely many times (including $s$ itself) form a finite family of at most $14$ distinct sets.","title":"Kuratowski's closure-complement theorem"},{"id":"Topology/general-topology/is-separated","kind":"node","lean":"Metric.IsSeparated'","parent":15262,"result":true,"route":"../roadmap/Topology/general-topology/index.html#is-separated","scope":"Topology/general-topology","status":"fully_proved","summary":"Let $X$ be a pseudo-extended metric space (distances lie in $[0, \\infty]$ and distinct points may be at distance $0$), write $d$ for its extended distance, and let $\\varepsilon \\in [0, \\infty]$. A set $s \\subseteq X$ is $\\varepsilon$-separated if $\\varepsilon \\le d(x, y)$ for all distinct points $x, y \\in s$.","title":"Non-strictly separated sets"},{"data_url":"scopes/Topology/general-topology/covering-and-inductive-dimension.json","id":"Topology/general-topology/covering-and-inductive-dimension","kind":"scope","navigation":"module","parent":15262,"result":false,"route":"../roadmap/Topology/general-topology/covering-and-inductive-dimension/index.html","scope":"Topology/general-topology/covering-and-inductive-dimension","status":"can_prove","title":"Covering and inductive dimension"},{"id":"Topology/general-topology/covering-and-inductive-dimension/cover-le","kind":"node","lean":"MathlibExt.Topology.ScottishBook187bWanted.CoverLE","parent":15269,"result":true,"route":"../roadmap/Topology/general-topology/covering-and-inductive-dimension/index.html#cover-le","scope":"Topology/general-topology/covering-and-inductive-dimension","status":"fully_proved","summary":"A topological space $X$ has covering dimension at most $n \\in \\mathbb{N}$ if every finite open cover $U_1, \\dots, U_m$ of $X$ has a finite open refinement $V_1, \\dots, V_k$ (open sets covering $X$, each contained in some $U_i$) in which any $n + 2$ members with distinct indices have empty intersection, that is, the refinement has order at most $n + 1$.","title":"Covering dimension at most n"},{"id":"Topology/general-topology/covering-and-inductive-dimension/scottish-book187b-refuted","kind":"node","lean":"MathlibExt.Topology.ScottishBook187bWanted.scottishBook187b_refuted","parent":15269,"result":true,"route":"../roadmap/Topology/general-topology/covering-and-inductive-dimension/index.html#scottish-book187b-refuted","scope":"Topology/general-topology/covering-and-inductive-dimension","status":"can_prove","summary":"Scottish Book Problem 187, part (2), has a negative answer: it is not the case that for every compact Hausdorff space $X$ and every $n \\in \\mathbb{N}$, $X$ has small inductive dimension at most $n$ if and only if $X$ has covering dimension at most $n$.","title":"Small inductive and covering dimension can differ for compact Hausdorff spaces"},{"data_url":"scopes/Topology/general-topology/dimension-raising-open-maps.json","id":"Topology/general-topology/dimension-raising-open-maps","kind":"scope","navigation":"module","parent":15262,"result":false,"route":"../roadmap/Topology/general-topology/dimension-raising-open-maps/index.html","scope":"Topology/general-topology/dimension-raising-open-maps","status":"can_prove","title":"Dimension-raising open maps"},{"id":"Topology/general-topology/dimension-raising-open-maps/kolmogorov-interior-map-raises-dimension","kind":"node","lean":"MathlibExt.Topology.ScottishBook136Wanted.kolmogorovInteriorMapRaisesDimension","parent":15272,"result":true,"route":"../roadmap/Topology/general-topology/dimension-raising-open-maps/index.html#kolmogorov-interior-map-raises-dimension","scope":"Topology/general-topology/dimension-raising-open-maps","status":"can_prove","summary":"An interior (open) mapping can raise dimension: there are compact metrizable spaces $X$ and $Y$, where $X$ has small inductive dimension exactly $1$ and $Y$ has small inductive dimension exactly $2$, and a continuous open surjective map $f \\colon X \\to Y$. A space has small inductive dimension exactly $n$ when its small inductive dimension is at most $n$ but not at most $n - 1$.","title":"Kolmogorov's open map raising dimension"},{"id":"Topology/general-topology/dimension-raising-open-maps/cube","kind":"node","lean":"MathlibExt.Topology.ScottishBook187cWanted.Cube","parent":15272,"result":true,"route":"../roadmap/Topology/general-topology/dimension-raising-open-maps/index.html#cube","scope":"Topology/general-topology/dimension-raising-open-maps","status":"fully_proved","summary":"For $p \\in \\mathbb{N}$, the cube $I^p = [0,1]^p$ is the set of points of $\\mathbb{R}^p$ all of whose coordinates lie in $[0,1]$, with the subspace topology.","title":"The p-dimensional cube"},{"id":"Topology/general-topology/dimension-raising-open-maps/scottish-book187c-refuted","kind":"node","lean":"MathlibExt.Topology.ScottishBook187cWanted.scottishBook187c_refuted","parent":15272,"result":true,"route":"../roadmap/Topology/general-topology/dimension-raising-open-maps/index.html#scottish-book187c-refuted","scope":"Topology/general-topology/dimension-raising-open-maps","status":"can_prove","summary":"There are natural numbers $p < q$ and a continuous, open, surjective map $f \\colon I^p \\to I^q$, where $I^p = [0,1]^p$ and $I^q = [0,1]^q$ are the $p$- and $q$-dimensional cubes.","title":"Open maps can raise the dimension of a cube"},{"data_url":"scopes/Topology/general-topology/direct-limits-of-topological-spaces.json","id":"Topology/general-topology/direct-limits-of-topological-spaces","kind":"scope","navigation":"module","parent":15262,"result":false,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","title":"Direct limits of topological spaces"},{"id":"Topology/general-topology/direct-limits-of-topological-spaces/rel","kind":"node","lean":"TopologicalSpace.DirectLimit.Rel","parent":15276,"result":true,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html#rel","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in I}$ be a family of topological spaces indexed by a preorder $I$, with maps $f_{ij} : X_i \\to X_j$ for $i \\le j$ forming a directed system ($f_{ii}$ is the identity and $f_{jk} \\circ f_{ij} = f_{ik}$ for $i \\le j \\le k$). This relation on the disjoint union $\\bigsqcup_{i \\in I} X_i$ relates $(i, x)$ to $(j, f_{ij}(x))$ for all $i \\le j$ and $x \\in X_i$, and relates no other pairs.","title":"Generating relation of a direct limit"},{"id":"Topology/general-topology/direct-limits-of-topological-spaces/direct-limit-setoid","kind":"node","lean":"TopologicalSpace.DirectLimit.directLimitSetoid","parent":15276,"result":true,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html#direct-limit-setoid","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in I}$ be a family of topological spaces indexed by a preorder $I$, with maps $f_{ij} : X_i \\to X_j$ for $i \\le j$ forming a directed system. This is the smallest equivalence relation on the disjoint union $\\bigsqcup_{i \\in I} X_i$ that relates $(i, x)$ to $(j, f_{ij}(x))$ for all $i \\le j$ and $x \\in X_i$.","title":"Direct limit equivalence relation"},{"id":"Topology/general-topology/direct-limits-of-topological-spaces/space","kind":"node","lean":"TopologicalSpace.DirectLimit.Space","parent":15276,"result":true,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html#space","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","summary":"Let $I$ be a preorder, $(X_i)_{i \\in I}$ a family of topological spaces, and $f_{ij} : X_i \\to X_j$ for $i \\le j$ continuous maps forming a directed system ($f_{ii}$ is the identity and $f_{jk} \\circ f_{ij} = f_{ik}$ for $i \\le j \\le k$). The direct limit $\\varinjlim X_i$ is the quotient of the disjoint union $\\bigsqcup_{i \\in I} X_i$ by the equivalence relation generated by $(i, x) \\sim (j, f_{ij}(x))$ for\u2026","title":"Direct limit of topological spaces"},{"id":"Topology/general-topology/direct-limits-of-topological-spaces/of","kind":"node","lean":"TopologicalSpace.DirectLimit.of","parent":15276,"result":true,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html#of","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in I}$ be a directed system of topological spaces over a preorder $I$ with continuous transition maps $f_{ij} : X_i \\to X_j$. For $i \\in I$, the canonical map $\\iota_i : X_i \\to \\varinjlim X_i$ into the direct limit sends $x$ to the class of $(i, x)$.","title":"Canonical map into the direct limit"},{"id":"Topology/general-topology/direct-limits-of-topological-spaces/of-continuous","kind":"node","lean":"TopologicalSpace.DirectLimit.of_continuous","parent":15276,"result":true,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html#of-continuous","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in I}$ be a directed system of topological spaces over a preorder $I$ with continuous transition maps $f_{ij} : X_i \\to X_j$. For every $i \\in I$, the canonical map $\\iota_i : X_i \\to \\varinjlim X_i$ into the direct limit is continuous.","title":"Canonical maps into the direct limit are continuous"},{"id":"Topology/general-topology/direct-limits-of-topological-spaces/of-comp","kind":"node","lean":"TopologicalSpace.DirectLimit.of_comp","parent":15276,"result":true,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html#of-comp","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in I}$ be a directed system of topological spaces over a preorder $I$ with continuous transition maps $f_{ij} : X_i \\to X_j$. The canonical maps $\\iota_i : X_i \\to \\varinjlim X_i$ into the direct limit satisfy $\\iota_j(f_{ij}(x)) = \\iota_i(x)$ for all $i \\le j$ and $x \\in X_i$.","title":"Canonical maps commute with the transition maps"},{"id":"Topology/general-topology/direct-limits-of-topological-spaces/lift","kind":"node","lean":"TopologicalSpace.DirectLimit.lift","parent":15276,"result":true,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html#lift","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in I}$ be a directed system of topological spaces over a preorder $I$ with continuous transition maps $f_{ij} : X_i \\to X_j$, and let $Y$ be a topological space. A family of maps $\\psi_i : X_i \\to Y$ with $\\psi_j(f_{ij}(x)) = \\psi_i(x)$ for all $i \\le j$ and $x \\in X_i$ induces the map $\\varinjlim X_i \\to Y$ from the direct limit that sends the class of $(i, x)$ to $\\psi_i(x)$.","title":"Map out of the direct limit"},{"id":"Topology/general-topology/direct-limits-of-topological-spaces/lift-continuous","kind":"node","lean":"TopologicalSpace.DirectLimit.lift_continuous","parent":15276,"result":true,"route":"../roadmap/Topology/general-topology/direct-limits-of-topological-spaces/index.html#lift-continuous","scope":"Topology/general-topology/direct-limits-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in I}$ be a directed system of topological spaces over a preorder $I$ with continuous transition maps $f_{ij} : X_i \\to X_j$, and let $Y$ be a topological space. If the maps $\\psi_i : X_i \\to Y$ are continuous and satisfy $\\psi_j(f_{ij}(x)) = \\psi_i(x)$ for all $i \\le j$ and $x \\in X_i$, then the induced map $\\varinjlim X_i \\to Y$ from the direct limit, sending the class of $(i, x)$ to $\\psi_i(x)$,\u2026","title":"The induced map out of the direct limit is continuous"},{"data_url":"scopes/Topology/general-topology/inverse-limits-of-topological-spaces.json","id":"Topology/general-topology/inverse-limits-of-topological-spaces","kind":"scope","navigation":"module","parent":15262,"result":false,"route":"../roadmap/Topology/general-topology/inverse-limits-of-topological-spaces/index.html","scope":"Topology/general-topology/inverse-limits-of-topological-spaces","status":"fully_proved","title":"Inverse limits of topological spaces"},{"id":"Topology/general-topology/inverse-limits-of-topological-spaces/carrier","kind":"node","lean":"InverseLimit.carrier","parent":15285,"result":true,"route":"../roadmap/Topology/general-topology/inverse-limits-of-topological-spaces/index.html#carrier","scope":"Topology/general-topology/inverse-limits-of-topological-spaces","status":"fully_proved","summary":"Let $I$ be a preordered set, let $(X_i)_{i \\in I}$ be a family of sets, and for every $i \\le j$ let $f_{ij} \\colon X_j \\to X_i$ be a map. The inverse limit is the set of tuples $x = (x_i)_{i \\in I}$ in the product $\\prod_{i \\in I} X_i$ such that $f_{ij}(x_j) = x_i$ whenever $i \\le j$.","title":"Inverse limit as compatible tuples"},{"id":"Topology/general-topology/inverse-limits-of-topological-spaces/mem-carrier-iff","kind":"node","lean":"InverseLimit.mem_carrier_iff","parent":15285,"result":true,"route":"../roadmap/Topology/general-topology/inverse-limits-of-topological-spaces/index.html#mem-carrier-iff","scope":"Topology/general-topology/inverse-limits-of-topological-spaces","status":"fully_proved","summary":"Let $I$ be a preordered set, let $(X_i)_{i \\in I}$ be a family of sets, and for every $i \\le j$ let $f_{ij} \\colon X_j \\to X_i$ be a map. A tuple $x \\in \\prod_{i \\in I} X_i$ lies in the inverse limit if and only if $f_{ij}(x_j) = x_i$ for all $i \\le j$.","title":"Membership in the inverse limit"},{"id":"Topology/general-topology/inverse-limits-of-topological-spaces/proj-compat","kind":"node","lean":"InverseLimit.proj_compat","parent":15285,"result":true,"route":"../roadmap/Topology/general-topology/inverse-limits-of-topological-spaces/index.html#proj-compat","scope":"Topology/general-topology/inverse-limits-of-topological-spaces","status":"fully_proved","summary":"Let $I$ be a preordered set, let $(X_i)_{i \\in I}$ be a family of sets, and for every $i \\le j$ let $f_{ij} \\colon X_j \\to X_i$ be a map. If $x = (x_i)_{i \\in I}$ lies in the inverse limit, then $f_{ij}(x_j) = x_i$ for all $i \\le j$.","title":"Elements of the inverse limit are compatible"},{"id":"Topology/general-topology/inverse-limits-of-topological-spaces/is-closed-carrier","kind":"node","lean":"InverseLimit.isClosed_carrier","parent":15285,"result":true,"route":"../roadmap/Topology/general-topology/inverse-limits-of-topological-spaces/index.html#is-closed-carrier","scope":"Topology/general-topology/inverse-limits-of-topological-spaces","status":"fully_proved","summary":"Let $I$ be a preordered set, let $(X_i)_{i \\in I}$ be a family of Hausdorff topological spaces, and for every $i \\le j$ let $f_{ij} \\colon X_j \\to X_i$ be a continuous map. Then the inverse limit $\\{x \\in \\prod_{i \\in I} X_i : f_{ij}(x_j) = x_i \\text{ for all } i \\le j\\}$ is closed in the product $\\prod_{i \\in I} X_i$.","title":"The inverse limit is closed"},{"id":"Topology/general-topology/inverse-limits-of-topological-spaces/is-compact-carrier","kind":"node","lean":"InverseLimit.isCompact_carrier","parent":15285,"result":true,"route":"../roadmap/Topology/general-topology/inverse-limits-of-topological-spaces/index.html#is-compact-carrier","scope":"Topology/general-topology/inverse-limits-of-topological-spaces","status":"fully_proved","summary":"Let $I$ be a preordered set, let $(X_i)_{i \\in I}$ be a family of compact Hausdorff topological spaces, and for every $i \\le j$ let $f_{ij} \\colon X_j \\to X_i$ be a continuous map. Then the inverse limit $\\{x \\in \\prod_{i \\in I} X_i : f_{ij}(x_j) = x_i \\text{ for all } i \\le j\\}$ is compact in the product $\\prod_{i \\in I} X_i$.","title":"The inverse limit of compact Hausdorff spaces is compact"},{"data_url":"scopes/Topology/general-topology/networks-and-monolithic-spaces.json","id":"Topology/general-topology/networks-and-monolithic-spaces","kind":"scope","navigation":"module","parent":15262,"result":false,"route":"../roadmap/Topology/general-topology/networks-and-monolithic-spaces/index.html","scope":"Topology/general-topology/networks-and-monolithic-spaces","status":"fully_proved","title":"Networks and monolithic spaces"},{"id":"Topology/general-topology/networks-and-monolithic-spaces/has-countable-network","kind":"node","lean":"HasCountableNetwork","parent":15291,"result":true,"route":"../roadmap/Topology/general-topology/networks-and-monolithic-spaces/index.html#has-countable-network","scope":"Topology/general-topology/networks-and-monolithic-spaces","status":"fully_proved","summary":"A topological space $X$ has a countable network if there is a countable family $\\mathcal{N}$ of subsets of $X$ such that for every open set $U \\subseteq X$ and every point $x \\in U$ there is some $V \\in \\mathcal{N}$ with $x \\in V \\subseteq U$.","title":"Countable network"},{"id":"Topology/general-topology/networks-and-monolithic-spaces/has-countable-network-of-second-countabl","kind":"node","lean":"hasCountableNetwork_of_secondCountable","parent":15291,"result":true,"route":"../roadmap/Topology/general-topology/networks-and-monolithic-spaces/index.html#has-countable-network-of-second-countabl","scope":"Topology/general-topology/networks-and-monolithic-spaces","status":"fully_proved","summary":"Every second-countable topological space has a countable network.","title":"Second-countable spaces have a countable network"},{"id":"Topology/general-topology/networks-and-monolithic-spaces/countably-monolithic-space","kind":"node","lean":"CountablyMonolithicSpace","parent":15291,"result":true,"route":"../roadmap/Topology/general-topology/networks-and-monolithic-spaces/index.html#countably-monolithic-space","scope":"Topology/general-topology/networks-and-monolithic-spaces","status":"fully_proved","summary":"A topological space is countably monolithic (\u03c9-monolithic) if the closure of every countable subspace has a countable network.","title":"Countably monolithic space"},{"data_url":"scopes/Topology/general-topology/restricted-products-of-topological-spaces.json","id":"Topology/general-topology/restricted-products-of-topological-spaces","kind":"scope","navigation":"module","parent":15262,"result":false,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","title":"Restricted products of topological spaces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/xs","kind":"node","lean":"RestrictedProduct.XS","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#xs","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be a family of types with subsets $U_i \\subseteq X_i$. For a finite set $S$ of indices, the piece $X_S$ is the set of tuples $x \\in \\prod_{i \\in \\iota} X_i$ such that $x_i \\in U_i$ for every $i \\notin S$.","title":"Piece of a restricted product supported in a finite set"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/xs-mono","kind":"node","lean":"RestrictedProduct.XS_mono","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#xs-mono","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be a family of types with subsets $U_i \\subseteq X_i$, and for a finite set $S$ of indices let $X_S$ be the set of tuples $x \\in \\prod_{i \\in \\iota} X_i$ with $x_i \\in U_i$ for every $i \\notin S$. If $S \\subseteq T$ are finite sets of indices, then $X_S \\subseteq X_T$.","title":"The pieces grow with the finite set"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/mem-xs-of-restricted","kind":"node","lean":"RestrictedProduct.mem_XS_of_restricted","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#mem-xs-of-restricted","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be sets with subsets $U_i \\subseteq X_i$, and let $x$ be an element of the restricted product of the $X_i$ with respect to the $U_i$, that is, a tuple $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all but finitely many $i$. Then there is a finite set $S \\subseteq \\iota$ such that $x_i \\in U_i$ for all $i \\notin S$, that is, $x$ lies in the piece $X_S$.","title":"Every element of the restricted product lies in a piece"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/support-finset","kind":"node","lean":"RestrictedProduct.supportFinset","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#support-finset","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be sets with subsets $U_i \\subseteq X_i$. The support of an element $x$ of the restricted product of the $X_i$ with respect to the $U_i$ (a tuple with $x_i \\in U_i$ for all but finitely many $i$) is the finite set of indices $i$ with $x_i \\notin U_i$.","title":"Support of an element of a restricted product"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/mem-xs-support","kind":"node","lean":"RestrictedProduct.mem_XS_support","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#mem-xs-support","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be sets with subsets $U_i \\subseteq X_i$, let $x$ be an element of the restricted product of the $X_i$ with respect to the $U_i$ (a tuple with $x_i \\in U_i$ for all but finitely many $i$), and let $\\operatorname{supp}(x)$ be the finite set of indices $i$ with $x_i \\notin U_i$. Then $x_i \\in U_i$ for every $i \\notin \\operatorname{supp}(x)$, that is, $x$ lies in the piece\u2026","title":"An element lies in the piece indexed by its support"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/xs-mem-restricted","kind":"node","lean":"RestrictedProduct.XS_mem_restricted","parent":15295,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tuples in a piece are eventually in the "],["math","\\(U_i\\)"]],"source":"Tuples in a piece are eventually in the $U_i$"},"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#xs-mem-restricted","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be a family of types with subsets $U_i \\subseteq X_i$, and let $S$ be a finite set of indices. If a tuple $x \\in \\prod_{i \\in \\iota} X_i$ lies in the piece $X_S$, that is, $x_i \\in U_i$ for every $i \\notin S$, then $x_i \\in U_i$ for all but finitely many indices $i$.","title":"Tuples in a piece are eventually in the $U_i$"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/support-finset-le","kind":"node","lean":"RestrictedProduct.supportFinset_le","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#support-finset-le","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be sets with subsets $U_i \\subseteq X_i$, and let $x$ be an element of the restricted product of the $X_i$ with respect to the $U_i$ (a tuple with $x_i \\in U_i$ for all but finitely many $i$), with support $\\operatorname{supp}(x) = \\{i : x_i \\notin U_i\\}$. If $T \\subseteq \\iota$ is a finite set with $x_i \\in U_i$ for all $i \\notin T$, then $\\operatorname{supp}(x) \\subseteq T$.","title":"The support lies in the index set of every piece containing the element"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/inclusion-map","kind":"node","lean":"RestrictedProduct.inclusionMap","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#inclusion-map","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$. For finite sets $S \\subseteq T$, the inclusion of pieces is the map $X_S \\to X_T$ sending each tuple $x$ to itself.","title":"Inclusion of pieces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/continuous-inclusion-map","kind":"node","lean":"RestrictedProduct.continuous_inclusionMap","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#continuous-inclusion-map","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$, with the subspace topology of the product topology. For finite sets $S \\subseteq T \\subseteq \\iota$, the inclusion map $X_S \\to X_T$ is continuous.","title":"Continuity of the inclusion of pieces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/transition-maps","kind":"node","lean":"RestrictedProduct.transitionMaps","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#transition-maps","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$. For finite sets $S \\subseteq T$, the transition map $X_S \\to X_T$ of the direct system of pieces sends each tuple $x$ to itself.","title":"Transition maps of the pieces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/continuous-transition-maps","kind":"node","lean":"RestrictedProduct.continuous_transitionMaps","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#continuous-transition-maps","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$, with the subspace topology of the product topology. For all finite sets $S \\subseteq T \\subseteq \\iota$, the transition map $X_S \\to X_T$ of the direct system of the pieces, which is the inclusion, is\u2026","title":"Continuity of the transition maps of the pieces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/direct-limit-xs","kind":"node","lean":"RestrictedProduct.DirectLimitXS","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#direct-limit-xs","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$, with the subspace topology of the product topology. The space $\\varinjlim_S X_S$ is the direct limit of the $X_S$ over the finite subsets $S$ of $\\iota$ ordered by inclusion, with the inclusions\u2026","title":"Direct limit of the finite-support pieces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/cofinite-le-principal-compl","kind":"node","lean":"RestrictedProduct.cofinite_le_principal_compl","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#cofinite-le-principal-compl","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"For every finite set $S$ of indices in $\\iota$, the cofinite filter on $\\iota$ is less than or equal to the principal filter of the complement $\\iota \\setminus S$; that is, $\\iota \\setminus S$ belongs to the cofinite filter.","title":"The cofinite filter lies below the principal filter of the complement of a finite set"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/inclusion-mem-xs-of-principal","kind":"node","lean":"RestrictedProduct.inclusion_mem_XS_of_principal","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#inclusion-mem-xs-of-principal","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be sets with subsets $U_i \\subseteq X_i$, and for a finite set $F \\subseteq \\iota$ let $X_F$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin F$. Let $S \\subseteq \\iota$ have finite complement. Then every tuple $y \\in \\prod_i X_i$ with $y_i \\in U_i$ for all $i \\in S$ lies in $X_{\\iota \\setminus S}$.","title":"Tuples restricted on a cofinite set lie in a piece"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/to-direct-limit","kind":"node","lean":"RestrictedProduct.toDirectLimit","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#to-direct-limit","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and let $L$ be the direct limit, over finite sets $S \\subseteq \\iota$, of the pieces $X_S$ of tuples with $x_i \\in U_i$ for all $i \\notin S$. The forward map sends an element $x$ of the restricted product of the $X_i$ with respect to the $U_i$ (a tuple with $x_i \\in U_i$ for all but finitely many $i$) to the class in $L$ of $x$,\u2026","title":"Forward map to the direct limit of the pieces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/from-direct-limit","kind":"node","lean":"RestrictedProduct.fromDirectLimit","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#from-direct-limit","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$, with the subspace topology of the product topology. The backward map sends the class in the direct limit $\\varinjlim_S X_S$ (over the finite subsets of $\\iota$) of a tuple $x \\in X_S$ to $x$, regarded as\u2026","title":"The map from the direct limit of the pieces to the restricted product"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/from-direct-limit-to-direct-limit","kind":"node","lean":"RestrictedProduct.fromDirectLimit_toDirectLimit","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#from-direct-limit-to-direct-limit","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$. For every $x$ in the restricted product $\\prod'_i (X_i, U_i)$ with respect to the cofinite filter, let $S(x)$ be the finite set of indices $i$ with $x_i \\notin U_i$. Sending $x$ to its class in\u2026","title":"The backward map undoes the forward map"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/to-direct-limit-from-direct-limit","kind":"node","lean":"RestrictedProduct.toDirectLimit_fromDirectLimit","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#to-direct-limit-from-direct-limit","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, let $P$ be their restricted product (the tuples with $x_i \\in U_i$ for all but finitely many $i$), and let $L$ be the direct limit, over finite sets $S \\subseteq \\iota$, of the pieces $X_S$ of tuples with $x_i \\in U_i$ for all $i \\notin S$. Let $f : P \\to L$ be the forward map, sending $x$ to its class in the piece indexed by its\u2026","title":"The forward map undoes the backward map"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/to-chunk","kind":"node","lean":"RestrictedProduct.toChunk","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#to-chunk","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, let $S \\subseteq \\iota$ be finite, and let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$. The chunk map sends each $x \\in X_S$ to the same tuple $x$, viewed as an element of the restricted product of the $X_i$ with respect to the $U_i$ and the principal filter of $\\iota \\setminus S$; this\u2026","title":"Chunk map of a piece"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/continuous-to-chunk","kind":"node","lean":"RestrictedProduct.continuous_toChunk","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#continuous-to-chunk","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, let $S \\subseteq \\iota$ be finite, and let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$, with the subspace topology of the product topology. The map that sends $x \\in X_S$ to $x$, viewed in the restricted product of the $(X_i, U_i)$ with respect to the principal filter of $\\iota \\setminus S$\u2026","title":"Continuity of the chunk map"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/continuous-from-direct-limit","kind":"node","lean":"RestrictedProduct.continuous_fromDirectLimit","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#continuous-from-direct-limit","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$, with the subspace topology of the product topology. The map from the direct limit $\\varinjlim_S X_S$ over the finite subsets of $\\iota$ to the restricted product $\\prod'_i (X_i, U_i)$ (with respect to\u2026","title":"Continuity of the map from the direct limit of the pieces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/continuous-to-direct-limit","kind":"node","lean":"RestrictedProduct.continuous_toDirectLimit","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#continuous-to-direct-limit","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$, with the subspace topology of the product topology. The map from the restricted product $\\prod'_i (X_i, U_i)$ (with respect to the cofinite filter, with its restricted product topology) to the direct\u2026","title":"Continuity of the map to the direct limit of the pieces"},{"id":"Topology/general-topology/restricted-products-of-topological-spaces/homeomorph-direct-limit","kind":"node","lean":"RestrictedProduct.homeomorph_directLimit","parent":15295,"result":true,"route":"../roadmap/Topology/general-topology/restricted-products-of-topological-spaces/index.html#homeomorph-direct-limit","scope":"Topology/general-topology/restricted-products-of-topological-spaces","status":"fully_proved","summary":"Let $(X_i)_{i \\in \\iota}$ be topological spaces with subsets $U_i \\subseteq X_i$, and for a finite set $S \\subseteq \\iota$ let $X_S$ be the set of tuples $x \\in \\prod_i X_i$ with $x_i \\in U_i$ for all $i \\notin S$, with the subspace topology of the product topology. Let $\\prod'_i (X_i, U_i)$ be the restricted product with respect to the cofinite filter, with its restricted product topology, and for $x$ in it let\u2026","title":"The restricted product is the direct limit of its finite-support pieces"},{"data_url":"scopes/Topology/descriptive-set-theory.json","id":"Topology/descriptive-set-theory","kind":"scope","parent":15066,"result":false,"route":"../roadmap/Topology/descriptive-set-theory/index.html","scope":"Topology/descriptive-set-theory","status":"can_prove","summary":"The topology of Polish spaces and their definable subsets: classical characterizations of Cantor space, Baire space and the rationals, Borel complexity, Bernstein sets and topological games.","title":"Descriptive set theory"},{"id":"Topology/descriptive-set-theory/brouwer-cantor-space-characterization","kind":"node","lean":"MathlibExt.Topology.DescriptiveSetTheory.CantorSpaceCharacterizationWanted.brouwer_cantor_space_characterization","parent":15319,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/index.html#brouwer-cantor-space-characterization","scope":"Topology/descriptive-set-theory","status":"fully_proved","summary":"Every nonempty compact metrizable perfect totally disconnected space is homeomorphic to the Cantor space.","title":"Brouwer's characterization of the Cantor space"},{"id":"Topology/descriptive-set-theory/bernstein-set-exists","kind":"node","lean":"MathlibExt.Topology.DescriptiveSetTheory.BernsteinSet.bernstein_set_exists","parent":15319,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/index.html#bernstein-set-exists","scope":"Topology/descriptive-set-theory","status":"fully_proved","summary":"There is a set $B \\subseteq \\mathbb{R}$ such that every closed, uncountable set $F \\subseteq \\mathbb{R}$ meets both $B$ and its complement $\\mathbb{R} \\setminus B$.","title":"Existence of a Bernstein set"},{"id":"Topology/descriptive-set-theory/sierpinski-rationals-characterization","kind":"node","lean":"MathlibExt.Topology.DescriptiveSetTheory.RationalsCharacterizationWanted.sierpinski_rationals_characterization","parent":15319,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/index.html#sierpinski-rationals-characterization","scope":"Topology/descriptive-set-theory","status":"fully_proved","summary":"Every nonempty countable metrizable space with no isolated points is homeomorphic to Q.","title":"Sierpinski's characterization of the rationals"},{"data_url":"scopes/Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa.json","id":"Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa","kind":"scope","navigation":"module","parent":15319,"result":false,"route":"../roadmap/Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa/index.html","scope":"Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa","status":"fully_proved","title":"Alexandrov\u2013Urysohn characterization of Baire space"},{"id":"Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa/is-zero-dimensional-space","kind":"node","lean":"MathlibExt.Topology.DescriptiveSetTheory.BaireSpaceCharacterizationWanted.IsZeroDimensionalSpace","parent":15323,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa/index.html#is-zero-dimensional-space","scope":"Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa","status":"fully_proved","summary":"A topological space $X$ is zero-dimensional if for every point $x \\in X$ and every open set $U \\subseteq X$ containing $x$ there is a clopen set $V$ with $x \\in V \\subseteq U$.","title":"Zero-dimensional space"},{"id":"Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa/baire-space-characterization","kind":"node","lean":"MathlibExt.Topology.DescriptiveSetTheory.BaireSpaceCharacterizationWanted.baire_space_characterization","parent":15323,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa/index.html#baire-space-characterization","scope":"Topology/descriptive-set-theory/alexandrov-urysohn-characterization-of-baire-spa","status":"fully_proved","summary":"Let $X$ be a nonempty Polish space that is zero-dimensional (for every point $x$ and every open set $U$ containing $x$ there is a clopen set $V$ with $x \\in V \\subseteq U$) and in which every compact subset has empty interior. Then $X$ is homeomorphic to the Baire space $\\mathbb{N}^{\\mathbb{N}}$ of all sequences of natural numbers, with the product of the discrete topologies.","title":"Alexandrov\u2013Urysohn theorem on Baire space"},{"data_url":"scopes/Topology/descriptive-set-theory/banach-mazur-game.json","id":"Topology/descriptive-set-theory/banach-mazur-game","kind":"scope","navigation":"module","parent":15319,"result":false,"route":"../roadmap/Topology/descriptive-set-theory/banach-mazur-game/index.html","scope":"Topology/descriptive-set-theory/banach-mazur-game","status":"can_prove","title":"Banach\u2013Mazur game"},{"id":"Topology/descriptive-set-theory/banach-mazur-game/banach-mazur-strategy-valid","kind":"node","lean":"MathlibExt.Topology.ScottishBook43Wanted.BanachMazurStrategyValid","parent":15326,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/banach-mazur-game/index.html#banach-mazur-strategy-valid","scope":"Topology/descriptive-set-theory/banach-mazur-game","status":"fully_proved","summary":"A strategy in the Banach\u2013Mazur game is a map $\\sigma \\colon \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R} \\times \\mathbb{R}$, a pair $(a, b)$ standing for the open interval $(a, b)$. It is valid if for all reals $a < b$ the pair $(c, d) = \\sigma(a, b)$ satisfies $a \\le c < d \\le b$, so that it answers each nonempty open interval with a nonempty open subinterval, possibly the interval itself. Nothing is required of\u2026","title":"Valid strategies in the Banach\u2013Mazur game"},{"id":"Topology/descriptive-set-theory/banach-mazur-game/banach-mazur-play-consistent","kind":"node","lean":"MathlibExt.Topology.ScottishBook43Wanted.BanachMazurPlayConsistent","parent":15326,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/banach-mazur-game/index.html#banach-mazur-play-consistent","scope":"Topology/descriptive-set-theory/banach-mazur-game","status":"fully_proved","summary":"Let $(a, b)$ be a pair of reals and let $\\sigma_A$ and $\\sigma_B$ be strategies for A and B, maps from pairs of reals to pairs of reals. A sequence $(p_n)_{n \\ge 0}$ of pairs of reals is the play that opens with $(a, b)$ and follows $\\sigma_A$ and $\\sigma_B$ if $p_0 = (a, b)$, $p_{n+1} = \\sigma_B(p_n)$ for every even $n$, and $p_{n+1} = \\sigma_A(p_n)$ for every odd $n$: A opens at index $0$ and moves at the other\u2026","title":"Plays of the Banach\u2013Mazur game"},{"id":"Topology/descriptive-set-theory/banach-mazur-game/banach-mazur-ahas-winning-strategy","kind":"node","lean":"MathlibExt.Topology.ScottishBook43Wanted.BanachMazurAHasWinningStrategy","parent":15326,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/banach-mazur-game/index.html#banach-mazur-ahas-winning-strategy","scope":"Topology/descriptive-set-theory/banach-mazur-game","status":"fully_proved","summary":"Let $E \\subseteq \\mathbb{R}$. In this game a move is a pair of reals $(c, d)$, standing for the open interval $(c, d)$, and a valid strategy is a map from pairs to pairs answering each pair $(c, d)$ with $c < d$ by a pair $(c', d')$ with $c \\le c' < d' \\le d$. Player A has a winning strategy for $E$ if there are reals $a < b$ and a valid strategy $\\sigma_A$ such that, for every valid strategy $\\sigma_B$, the play\u2026","title":"Winning strategy for the first player in the Banach\u2013Mazur game"},{"id":"Topology/descriptive-set-theory/banach-mazur-game/banach-mazur-bhas-winning-strategy","kind":"node","lean":"MathlibExt.Topology.ScottishBook43Wanted.BanachMazurBHasWinningStrategy","parent":15326,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/banach-mazur-game/index.html#banach-mazur-bhas-winning-strategy","scope":"Topology/descriptive-set-theory/banach-mazur-game","status":"fully_proved","summary":"Let $E \\subseteq \\mathbb{R}$. In this game a move is a pair of reals $(c, d)$, standing for the open interval $(c, d)$, and a valid strategy is a map from pairs to pairs answering each pair $(c, d)$ with $c < d$ by a pair $(c', d')$ with $c \\le c' < d' \\le d$. Player B has a winning strategy for $E$ if there is a valid strategy $\\sigma_B$ such that, for all reals $a < b$ and every valid strategy $\\sigma_A$, the\u2026","title":"Winning strategy for the second player in the Banach\u2013Mazur game"},{"id":"Topology/descriptive-set-theory/banach-mazur-game/banach-mazur-winning-strategy-characteri","kind":"node","lean":"MathlibExt.Topology.ScottishBook43Wanted.banach_mazur_winning_strategy_characterization","parent":15326,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/banach-mazur-game/index.html#banach-mazur-winning-strategy-characteri","scope":"Topology/descriptive-set-theory/banach-mazur-game","status":"can_prove","summary":"For every set $E \\subseteq \\mathbb{R}$, player A has a winning strategy in the Banach\u2013Mazur game for $E$ if and only if there are reals $a < b$ such that $(a, b) \\setminus E$ is meagre, and player B has a winning strategy in the Banach\u2013Mazur game for $E$ if and only if $E$ is meagre.","title":"Banach\u2013Mazur game theorem"},{"data_url":"scopes/Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions.json","id":"Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions","kind":"scope","navigation":"module","parent":15319,"result":false,"route":"../roadmap/Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions/index.html","scope":"Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions","status":"can_prove","title":"Borel complexity of differentiable functions"},{"id":"Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions/has-derivative-at-every-point","kind":"node","lean":"MathlibExt.Topology.ScottishBook52Wanted.HasDerivativeAtEveryPoint","parent":15332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Functions differentiable at every point of "],["math","\\((0, 1)\\)"]],"source":"Functions differentiable at every point of $(0, 1)$"},"result":true,"route":"../roadmap/Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions/index.html#has-derivative-at-every-point","scope":"Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions","status":"fully_proved","summary":"A continuous function $f \\colon (0, 1) \\to \\mathbb{R}$ has a derivative at every point if for every $x \\in (0, 1)$ the function $\\tilde f \\colon \\mathbb{R} \\to \\mathbb{R}$ equal to $f$ on $(0, 1)$ and to $0$ outside $(0, 1)$ is differentiable at $x$. Since $\\tilde f$ agrees with $f$ near every point of $(0, 1)$, this depends only on $f$: it says that $f$ is differentiable at every point of $(0, 1)$.","title":"Functions differentiable at every point of $(0, 1)$"},{"id":"Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions/everywhere-differentiable-class-not-bore","kind":"node","lean":"MathlibExt.Topology.ScottishBook52Wanted.everywhere_differentiable_class_not_borel","parent":15332,"result":true,"route":"../roadmap/Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions/index.html#everywhere-differentiable-class-not-bore","scope":"Topology/descriptive-set-theory/borel-complexity-of-differentiable-functions","status":"can_prove","summary":"In the space $C((0, 1))$ of continuous real functions on the open interval $(0, 1)$, with the compact-open topology (uniform convergence on compact subsets), the set of functions that have a derivative at every point of $(0, 1)$ is not a Borel set.","title":"Everywhere differentiable functions do not form a Borel set"},{"data_url":"scopes/Topology/algebraic-topology.json","id":"Topology/algebraic-topology","kind":"scope","parent":15066,"result":false,"route":"../roadmap/Topology/algebraic-topology/index.html","scope":"Topology/algebraic-topology","status":"planned","summary":"Homotopy groups, singular homology and other algebraic invariants of spaces, with classical theorems such as Borsuk\u2013Ulam, Hurewicz and Hopf invariant one.","title":"Algebraic topology"},{"id":"Topology/algebraic-topology/h-space-sphere-iff","kind":"node","lean":"MathlibExt.Topology.HopfInvariantOneWanted.Topology.HopfInvariantOne.hSpace_sphere_iff","parent":15335,"result":true,"route":"../roadmap/Topology/algebraic-topology/index.html#h-space-sphere-iff","scope":"Topology/algebraic-topology","status":"can_prove","summary":"For every natural number $n$, the unit sphere $S^n$ of the Euclidean space $\\mathbb{R}^{n+1}$ admits an H-space structure if and only if $n = 0$, $n = 1$, $n = 3$ or $n = 7$. Here an H-space structure on $S^n$ is a continuous map $\\mu \\colon S^n \\times S^n \\to S^n$ together with a point $e \\in S^n$ such that $\\mu(e, e) = e$ and the maps $x \\mapsto \\mu(e, x)$ and $x \\mapsto \\mu(x, e)$ are both homotopic to the\u2026","title":"Adams' theorem on spheres that are H-spaces"},{"id":"Topology/algebraic-topology/borsuk-ulam","kind":"node","lean":"MathlibExt.Topology.Homotopy.BorsukUlamWanted.borsukUlam","parent":15335,"result":true,"route":"../roadmap/Topology/algebraic-topology/index.html#borsuk-ulam","scope":"Topology/algebraic-topology","status":"fully_proved","summary":"Let $n$ be a natural number and let $S^n$ be the unit sphere in the Euclidean space $\\mathbb{R}^{n+1}$. For every continuous map $f \\colon S^n \\to \\mathbb{R}^n$ there is a point $x \\in S^n$ with $f(x) = f(-x)$.","title":"Borsuk\u2013Ulam theorem"},{"id":"Topology/algebraic-topology/ham-sandwich","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.HamSandwichWanted.ham_sandwich","parent":15335,"result":true,"route":"../roadmap/Topology/algebraic-topology/index.html#ham-sandwich","scope":"Topology/algebraic-topology","status":"fully_proved","summary":"Let $n \\ge 1$ and let $\\mu_1, \\dots, \\mu_n$ be finite Borel measures on the Euclidean space $\\mathbb{R}^n$, each absolutely continuous with respect to Lebesgue measure. Then there exist a unit vector $v \\in \\mathbb{R}^n$ and a real number $t$ such that for every $i$, both closed half-spaces $\\{x : \\langle v, x \\rangle \\le t\\}$ and $\\{x : \\langle v, x \\rangle \\ge t\\}$ have $\\mu_i$-measure equal to\u2026","title":"Ham sandwich theorem"},{"data_url":"scopes/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers.json","id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","kind":"scope","navigation":"module","parent":15335,"result":false,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"can_prove","title":"Adapting families of maps to open covers"},{"id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/supported","kind":"node","lean":"MathlibExt.Topology.FamiliesOfMapsB01Wanted.FamiliesOfMapsB01.Supported","parent":15339,"result":true,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html#supported","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"fully_proved","summary":"Let $X$ and $T$ be topological spaces and $P$ a set. A family of maps $f : P \\times X \\to T$ is supported on a set $S \\subseteq X$ when it does not depend on the parameter outside $S$: $f(p, x) = f(p', x)$ for all $p, p' \\in P$ and all $x \\notin S$.","title":"Families of maps supported on a set"},{"id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/adapted-to","kind":"node","lean":"MathlibExt.Topology.FamiliesOfMapsB01Wanted.FamiliesOfMapsB01.AdaptedTo","parent":15339,"result":true,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html#adapted-to","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"fully_proved","summary":"Let $X$ and $T$ be topological spaces, $P$ a set, $U = (U_\\alpha)_{\\alpha \\in \\iota}$ a family of subsets of $X$ and $k$ a natural number. A family of maps $f : P \\times X \\to T$ is adapted to $U$ with bound $k$ when there is a finite set $A$ of at most $k$ indices such that $f$ is supported on $\\bigcup_{\\alpha \\in A} U_\\alpha$.","title":"Adapted families of maps"},{"id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/is-polyhedron","kind":"node","lean":"MathlibExt.Topology.FamiliesOfMapsB01Wanted.FamiliesOfMapsB01.IsPolyhedron","parent":15339,"result":true,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html#is-polyhedron","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"fully_proved","summary":"A subset $P \\subseteq \\mathbb{R}^k$ is a convex linear polyhedron when it is the convex hull of a finite set of points (equivalently, a bounded intersection of finitely many closed half-spaces).","title":"Convex linear polyhedra"},{"id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/subdivision","kind":"node","lean":"MathlibExt.Topology.FamiliesOfMapsB01Wanted.FamiliesOfMapsB01.Subdivision","parent":15339,"result":true,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html#subdivision","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"fully_proved","summary":"A polyhedral subdivision of a set $P \\subseteq \\mathbb{R}^k$ is a geometric simplicial complex in $\\mathbb{R}^k$ with finitely many faces whose underlying space equals $P$.","title":"Polyhedral subdivisions"},{"id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/closed-cell","kind":"node","lean":"MathlibExt.Topology.FamiliesOfMapsB01Wanted.FamiliesOfMapsB01.closedCell","parent":15339,"result":true,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html#closed-cell","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"fully_proved","summary":"Let $P \\subseteq \\mathbb{R}^k$ and let $D \\subseteq \\mathbb{R}^k$ be a finite set (typically the vertex set of a face of a subdivision of $P$). The closed cell of $D$ in $P$ is the set of points of $P$ that lie in the convex hull of $D$, viewed as a subset of $P$.","title":"Closed cells"},{"id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/is-boundary-subpolyhedron","kind":"node","lean":"MathlibExt.Topology.FamiliesOfMapsB01Wanted.FamiliesOfMapsB01.IsBoundarySubpolyhedron","parent":15339,"result":true,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html#is-boundary-subpolyhedron","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"fully_proved","summary":"Let $P \\subseteq \\mathbb{R}^k$. A subset $Q \\subseteq P$ is a boundary subpolyhedron of $P$ when $Q$ is the convex hull of a finite set of points of $\\mathbb{R}^k$, $Q$ lies in the intrinsic frontier (relative boundary) of $P$, and $Q$ is a face of $P$: whenever a point of $Q$ lies in the open segment between two points of $P$, both of those points lie in $Q$.","title":"Boundary subpolyhedra"},{"id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/continuous","kind":"node","lean":"MathlibExt.Topology.FamiliesOfMapsB01Wanted.FamiliesOfMapsB01.continuous","parent":15339,"result":true,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html#continuous","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"can_prove","summary":"Let $X$ be a compact topological space, $T$ a topological space, $P \\subseteq \\mathbb{R}^k$ a convex linear polyhedron, $(U_\\alpha)_{\\alpha \\in \\iota}$ a family of open subsets of $X$, and $\\rho$ a partition of unity on $X$ indexed by $\\iota$ and subordinate to $U$ (the closed support of $\\rho_\\alpha$ lies in $U_\\alpha$). Then for every continuous map $f : P \\times X \\to T$ there is a continuous map\u2026","title":"Morrison\u2013Walker lemma on adapting families of maps to an open cover"},{"id":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/bi-lipschitz","kind":"node","lean":"MathlibExt.Topology.FamiliesOfMapsB01Wanted.FamiliesOfMapsB01.biLipschitz","parent":15339,"result":true,"route":"../roadmap/Topology/algebraic-topology/adapting-families-of-maps-to-open-covers/index.html#bi-lipschitz","scope":"Topology/algebraic-topology/adapting-families-of-maps-to-open-covers","status":"can_prove","summary":"Let $X$ be a compact metric space, $T$ a metric space, $P \\subseteq \\mathbb{R}^k$ a convex linear polyhedron, $(U_\\alpha)_{\\alpha \\in \\iota}$ a family of open subsets of $X$, and $\\rho$ a partition of unity on $X$ indexed by $\\iota$ and subordinate to $U$. Let $f : P \\times X \\to T$ be continuous and suppose that for each $p \\in P$ the map $x \\mapsto f(p, x)$ is a homeomorphism $h_p : X \\to T$, and that for some\u2026","title":"Morrison\u2013Walker lemma on adapting families of maps, bi-Lipschitz case"},{"data_url":"scopes/Topology/algebraic-topology/bing-s-house-with-two-rooms.json","id":"Topology/algebraic-topology/bing-s-house-with-two-rooms","kind":"scope","navigation":"module","parent":15335,"result":false,"route":"../roadmap/Topology/algebraic-topology/bing-s-house-with-two-rooms/index.html","scope":"Topology/algebraic-topology/bing-s-house-with-two-rooms","status":"can_prove","title":"Bing's house with two rooms"},{"id":"Topology/algebraic-topology/bing-s-house-with-two-rooms/house-with-two-rooms","kind":"node","lean":"MathlibExt.Topology.HouseWithTwoRoomsWanted.Topology.HouseWithTwoRooms","parent":15348,"result":true,"route":"../roadmap/Topology/algebraic-topology/bing-s-house-with-two-rooms/index.html#house-with-two-rooms","scope":"Topology/algebraic-topology/bing-s-house-with-two-rooms","status":"fully_proved","summary":"Bing's house with two rooms, in the version drawn in Hatcher's *Algebraic Topology*, is the subset $H$ of $\\mathbb{R} \\times \\mathbb{R} \\times \\mathbb{R}$ consisting of the points of $[0,4] \\times [0,3] \\times \\{0,1,2\\}$ that lie in none of $(2,3) \\times (1,2) \\times \\{0\\}$, $(1,3) \\times (1,2) \\times \\{1\\}$ and $(1,2) \\times (1,2) \\times \\{2\\}$ (three floors with holes), together with the inner walls\u2026","title":"Bing's house with two rooms"},{"id":"Topology/algebraic-topology/bing-s-house-with-two-rooms/contractible-space-house-with-two-rooms","kind":"node","lean":"MathlibExt.Topology.HouseWithTwoRoomsWanted.Topology.contractibleSpace_houseWithTwoRooms","parent":15348,"result":true,"route":"../roadmap/Topology/algebraic-topology/bing-s-house-with-two-rooms/index.html#contractible-space-house-with-two-rooms","scope":"Topology/algebraic-topology/bing-s-house-with-two-rooms","status":"can_prove","summary":"Bing's house with two rooms, regarded as a subspace of $\\mathbb{R} \\times \\mathbb{R} \\times \\mathbb{R}$, is contractible.","title":"Bing's house with two rooms is contractible"},{"data_url":"scopes/Topology/algebraic-topology/homotopy-groups-of-spheres.json","id":"Topology/algebraic-topology/homotopy-groups-of-spheres","kind":"scope","navigation":"module","parent":15335,"result":false,"route":"../roadmap/Topology/algebraic-topology/homotopy-groups-of-spheres/index.html","scope":"Topology/algebraic-topology/homotopy-groups-of-spheres","status":"can_prove","title":"Homotopy groups of spheres"},{"id":"Topology/algebraic-topology/homotopy-groups-of-spheres/pi1-circle-mul-equiv-int","kind":"node","lean":"MathlibExt.Topology.HomotopyGroupsWanted.Topology.pi1_circle_mulEquiv_int","parent":15351,"result":true,"route":"../roadmap/Topology/algebraic-topology/homotopy-groups-of-spheres/index.html#pi1-circle-mul-equiv-int","scope":"Topology/algebraic-topology/homotopy-groups-of-spheres","status":"can_prove","summary":"The fundamental group of the unit circle in $\\mathbb{C}$ based at $1$ is isomorphic, as a group, to the additive group $\\mathbb{Z}$.","title":"The fundamental group of the circle is infinite cyclic"},{"id":"Topology/algebraic-topology/homotopy-groups-of-spheres/pi6-sphere-three-mul-equiv-zmod-twelve","kind":"node","lean":"MathlibExt.Topology.PiSixSphereThreeWanted.Topology.PiSixSphereThree.pi6_sphere_three_mulEquiv_zmod_twelve","parent":15351,"result":true,"route":"../roadmap/Topology/algebraic-topology/homotopy-groups-of-spheres/index.html#pi6-sphere-three-mul-equiv-zmod-twelve","scope":"Topology/algebraic-topology/homotopy-groups-of-spheres","status":"can_prove","summary":"Let $S^3$ be the unit sphere in Euclidean space $\\mathbb{R}^4$. For every point $x \\in S^3$, the homotopy group $\\pi_6(S^3, x)$ is isomorphic as a group to the cyclic group $\\mathbb{Z}/12$.","title":"The sixth homotopy group of the 3-sphere"},{"data_url":"scopes/Topology/algebraic-topology/hurewicz-theorem.json","id":"Topology/algebraic-topology/hurewicz-theorem","kind":"scope","navigation":"module","parent":15335,"result":false,"route":"../roadmap/Topology/algebraic-topology/hurewicz-theorem/index.html","scope":"Topology/algebraic-topology/hurewicz-theorem","status":"can_prove","title":"Hurewicz theorem"},{"id":"Topology/algebraic-topology/hurewicz-theorem/integral-homology","kind":"node","lean":"MathlibExt.Topology.HurewiczWanted.Topology.Hurewicz.IntegralHomology","parent":15354,"result":true,"route":"../roadmap/Topology/algebraic-topology/hurewicz-theorem/index.html#integral-homology","scope":"Topology/algebraic-topology/hurewicz-theorem","status":"fully_proved","summary":"For a natural number $n$ and a topological space $X$, $H_n(X; \\mathbb{Z})$ denotes the $n$-th singular homology group of $X$ with coefficients in $\\mathbb{Z}$, regarded as an abelian group.","title":"Integral singular homology"},{"id":"Topology/algebraic-topology/hurewicz-theorem/hurewicz-h1-abelianization","kind":"node","lean":"MathlibExt.Topology.HurewiczWanted.Topology.Hurewicz.hurewicz_h1_abelianization","parent":15354,"result":true,"route":"../roadmap/Topology/algebraic-topology/hurewicz-theorem/index.html#hurewicz-h1-abelianization","scope":"Topology/algebraic-topology/hurewicz-theorem","status":"can_prove","summary":"Let $X$ be a path-connected topological space and let $x \\in X$. Then the abelianization of the fundamental group $\\pi_1(X, x)$ is isomorphic, as a group, to the first integral singular homology group $H_1(X; \\mathbb{Z})$ of $X$.","title":"First homology is the abelianized fundamental group"},{"data_url":"scopes/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes.json","id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","kind":"scope","navigation":"module","parent":15335,"result":false,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","title":"Solid condensed homology of CW complexes"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/shift","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightProfinite.shift","parent":15357,"reading_title":{"format":"inline-parts-v1","parts":[["text","The shift map on "],["math","\\(\\mathbb{N} \\cup \\{\\infty\\}\\)"]],"source":"The shift map on $\\mathbb{N} \\cup \\{\\infty\\}$"},"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#shift","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{N} \\cup \\{\\infty\\}$ for the one-point compactification of $\\mathbb{N}$, viewed as a light profinite set. The shift map is the continuous map $\\mathbb{N} \\cup \\{\\infty\\} \\to \\mathbb{N} \\cup \\{\\infty\\}$ sending $n$ to $n + 1$ and $\\infty$ to $\\infty$.","title":"The shift map on $\\mathbb{N} \\cup \\{\\infty\\}$"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/untitled","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.\u03b9","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#untitled","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The morphism of light profinite spaces from the one-point space to the one-point compactification $\\mathbb{N} \\cup \\{\\infty\\}$ of $\\mathbb{N}$ that sends the point to $\\infty$.","title":"Inclusion of the point at infinity"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/split","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.\u03b9_split","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#split","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The inclusion of the point at infinity into the one-point compactification $\\mathbb{N} \\cup \\{\\infty\\}$ of $\\mathbb{N}$ is a split monomorphism of light profinite spaces, with retraction the unique map from $\\mathbb{N} \\cup \\{\\infty\\}$ to the one-point space.","title":"Splitting of the inclusion of the point at infinity"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/p-map","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.P_map","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#p-map","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"For a light profinite set $S$, let $\\mathbb{Z}[S]$ be the free light condensed abelian group on $S$, and let $\\iota$ be the inclusion of the point $\\infty$ into $\\mathbb{N} \\cup \\{\\infty\\}$. This is the morphism $\\mathbb{Z}[\\infty] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$ induced by $\\iota$, where $\\mathbb{Z}[\\infty]$ is the free light condensed abelian group on a point.","title":"The map \u2124(\u221e) \u2192 \u2124(\u2115 \u222a {\u221e})"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/p","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.P","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#p","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"For a light profinite set $S$, let $\\mathbb{Z}[S]$ be the free light condensed abelian group on $S$, and let $\\mathbb{N} \\cup \\{\\infty\\}$ be the one-point compactification of $\\mathbb{N}$. Then $P = \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]/\\mathbb{Z}[\\infty]$ is the cokernel of the morphism $\\mathbb{Z}[\\infty] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$ induced by the inclusion of the point $\\infty$.","title":"The light condensed abelian group P"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/p-proj","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.P_proj","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#p-proj","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$, and let $P = \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]/\\mathbb{Z}[\\infty]$. This is the quotient map $\\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}] \\to P$, the cokernel projection of the map $\\mathbb{Z}[\\infty] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$ induced by the inclusion of the point $\\infty$.","title":"The projection onto P"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/psequence","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.PSequence","parent":15357,"reading_title":{"format":"inline-parts-v1","parts":[["text","The short complex defining "],["math","\\(P\\)"]],"source":"The short complex defining $P$"},"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#psequence","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{N} \\cup \\{\\infty\\}$ for the one-point compactification of $\\mathbb{N}$, viewed as a light profinite set, and $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$, with $\\mathbb{Z}[\\infty]$ the one on a point. This is the short complex of light condensed abelian groups $\\mathbb{Z}[\\infty] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}] \\to P$ whose first map is induced by\u2026","title":"The short complex defining $P$"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/psequence-exact","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.PSequence_exact","parent":15357,"reading_title":{"format":"inline-parts-v1","parts":[["text","Short exactness of the sequence defining "],["math","\\(P\\)"]],"source":"Short exactness of the sequence defining $P$"},"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#psequence-exact","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{N} \\cup \\{\\infty\\}$ for the one-point compactification of $\\mathbb{N}$, viewed as a light profinite set, and $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$, with $\\mathbb{Z}[\\infty]$ the one on a point. The short complex of light condensed abelian groups $\\mathbb{Z}[\\infty] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}] \\to P$, whose first map is induced by the\u2026","title":"Short exactness of the sequence defining $P$"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/psequence-split","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.PSequence_split","parent":15357,"reading_title":{"format":"inline-parts-v1","parts":[["text","A splitting of the sequence defining "],["math","\\(P\\)"]],"source":"A splitting of the sequence defining $P$"},"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#psequence-split","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{N} \\cup \\{\\infty\\}$ for the one-point compactification of $\\mathbb{N}$, viewed as a light profinite set, and $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$, with $\\mathbb{Z}[\\infty]$ the one on a point. Let $f \\colon \\mathbb{Z}[\\infty] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$ be induced by the inclusion of the point $\\infty$, and let\u2026","title":"A splitting of the sequence defining $P$"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/p-retract","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.P_retract","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#p-retract","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$. The quotient $P = \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]/\\mathbb{Z}[\\infty]$ by the image of the map induced by the inclusion of the point $\\infty$ is a retract of $\\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$: the short exact sequence $0 \\to \\mathbb{Z}[\\infty] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}] \\to P \\to 0$ splits,\u2026","title":"P is a retract of \u2124(\u2115 \u222a {\u221e})"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/p-hom-mk","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.P_homMk","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#p-hom-mk","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$, and let $P = \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]/\\mathbb{Z}[\\infty]$ be the cokernel of the map $\\mathbb{Z}[\\infty] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$ induced by the inclusion of the point $\\infty$. Let $A$ be a light condensed abelian group and let $f \\colon \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}] \\to A$ be a\u2026","title":"Maps out of P"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/one-minus-shift","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.oneMinusShift'","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#one-minus-shift","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Let $\\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$ be the free light condensed abelian group on the light profinite set $\\mathbb{N} \\cup \\{\\infty\\}$, and let $\\mathrm{shift}$ be its endomorphism induced by the map $n \\mapsto n + 1$, $\\infty \\mapsto \\infty$. This is the endomorphism $1 - \\mathrm{shift}$ of $\\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$.","title":"One minus shift on \u2124(\u2115 \u222a {\u221e})"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/one-minus-shift-2","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.oneMinusShift","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#one-minus-shift-2","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$, let $\\mathbb{N} \\cup \\{\\infty\\}$ be the one-point compactification of $\\mathbb{N}$, and let $P = \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]/\\mathbb{Z}[\\infty]$ be the cokernel of the map induced by the inclusion of the point $\\infty$. Let $\\mathrm{shift}$ be the endomorphism of $\\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$ induced\u2026","title":"One minus shift on P"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/is-solid","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.isSolid","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#is-solid","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$, let $\\mathbb{N} \\cup \\{\\infty\\}$ be the one-point compactification of $\\mathbb{N}$, and let $P = \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]/\\mathbb{Z}[\\infty]$ be the cokernel of the map induced by the inclusion of the point $\\infty$. The shift $n \\mapsto n + 1$, $\\infty \\mapsto \\infty$ induces an endomorphism\u2026","title":"Solid light condensed abelian groups"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/is-solid-2","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.IsSolid","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#is-solid-2","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"For a light condensed abelian group $A$, the class asserting that $A$ is solid.","title":"The solidity class"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/solid","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#solid","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Write $\\mathbb{Z}[S]$ for the free light condensed abelian group on a light profinite set $S$, let $\\mathbb{N} \\cup \\{\\infty\\}$ be the one-point compactification of $\\mathbb{N}$, viewed as a light profinite set, write $\\mathrm{pt}$ for the one-point light profinite set, and let $P$ be the cokernel of the map $\\mathbb{Z}[\\mathrm{pt}] \\to \\mathbb{Z}[\\mathbb{N} \\cup \\{\\infty\\}]$ induced by the inclusion of the point\u2026","title":"The category of light solid abelian groups"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/dlight-cond-ab","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.DLightCondAb","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#dlight-cond-ab","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The derived category of the abelian category of light condensed abelian groups.","title":"Derived category of light condensed abelian groups"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/dsolid","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.DSolid","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#dsolid","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The derived category of the abelian category of light solid abelian groups.","title":"Derived category of light solid abelian groups"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/solidification","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.solidification","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#solidification","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"can_state","summary":"The solidification functor from the category of light condensed abelian groups to the category of light solid abelian groups.","title":"Solidification"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/solidification-adjunction","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.solidificationAdjunction","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#solidification-adjunction","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","summary":"An adjunction exhibiting the solidification functor from light condensed abelian groups to light solid abelian groups as left adjoint to the inclusion of the full subcategory of light solid abelian groups into light condensed abelian groups.","title":"Solidification adjunction"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidification","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"can_state","summary":"A functor from the derived category of light condensed abelian groups to the derived category of light solid abelian groups.","title":"Derived solidification"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-counit","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidificationCounit","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-counit","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","summary":"Let $Q$ denote the localization functors from cochain complexes to derived categories. This is a natural transformation from the functor sending a cochain complex $K$ of light condensed abelian groups to the derived solidification of $Q(K)$ to the functor sending $K$ to $Q(\\mathrm{Sol}(K))$, where $\\mathrm{Sol}(K)$ is the complex of light solid abelian groups obtained by applying solidification to $K$ degreewise.","title":"Comparison map from derived to degreewise solidification"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-is-left-derived-f","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidification_isLeftDerivedFunctor","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-is-left-derived-f","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","summary":"Let $Q$ denote the localization functors from cochain complexes indexed by $\\mathbb{Z}$ to derived categories. Derived solidification, together with the comparison natural transformation from $Q$ followed by derived solidification to degreewise solidification followed by $Q$, is a left derived functor of the functor that applies solidification degreewise to a cochain complex of light condensed abelian groups and\u2026","title":"Derived solidification is the left derived functor of solidification"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/inclusion-complexes","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.inclusionComplexes","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#inclusion-complexes","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The functor from cochain complexes indexed by $\\mathbb{Z}$ of light solid abelian groups to cochain complexes indexed by $\\mathbb{Z}$ of light condensed abelian groups that applies the inclusion of light solid abelian groups into light condensed abelian groups in each degree.","title":"Degreewise inclusion of complexes of light solid abelian groups"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-inclusion","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedInclusion","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-inclusion","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The inclusion $\\iota$ of light solid abelian groups into light condensed abelian groups is exact, so it induces a functor $D(\\iota)$ from the derived category of light solid abelian groups to the derived category of light condensed abelian groups.","title":"Derived inclusion of solid abelian groups"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-inclusion-factors","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedInclusionFactors","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-inclusion-factors","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Let $\\iota$ be the inclusion of light solid abelian groups into light condensed abelian groups, $D(\\iota)$ the induced derived inclusion, and $Q$ the localization functors from cochain complexes to derived categories. This is the natural isomorphism $D(\\iota)(Q K) \\cong Q(\\iota K)$ for cochain complexes $K$ of light solid abelian groups, where $\\iota K$ is $K$ with $\\iota$ applied degreewise.","title":"The derived inclusion is computed degreewise"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-inclusion-comparison","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedInclusionComparison","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-inclusion-comparison","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"Let $\\iota$ be the inclusion of light solid abelian groups into light condensed abelian groups, $D(\\iota)$ the induced derived inclusion, and $Q$ the localization functors from cochain complexes to derived categories. The comparison map is the natural transformation from $K \\mapsto Q(\\iota K)$ to $K \\mapsto D(\\iota)(Q K)$, for cochain complexes $K$ of light solid abelian groups, where $\\iota K$ is $K$ with $\\iota$\u2026","title":"Comparison map for the derived inclusion"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-adjunction","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidificationAdjunction","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-adjunction","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","summary":"An adjunction in which derived solidification, from the derived category of light condensed abelian groups to the derived category of light solid abelian groups, is left adjoint to the derived inclusion $D(\\iota)$ induced by the inclusion $\\iota$ of light solid abelian groups into light condensed abelian groups.","title":"Derived solidification is left adjoint to the derived inclusion"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/free-light-cond-ab-of-top","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.freeLightCondAbOfTop","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#free-light-cond-ab-of-top","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"For a topological space $X$, $\\mathbb{Z}[X]$ is the free light condensed abelian group on the light condensed set associated with $X$.","title":"Free light condensed abelian group on a topological space"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/free-light-cond-ab-of-top-functor","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.freeLightCondAbOfTopFunctor","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#free-light-cond-ab-of-top-functor","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The functor from topological spaces to light condensed abelian groups sending a space $X$ to the free light condensed abelian group $\\mathbb{Z}[X]$ on the light condensed set associated with $X$.","title":"Free light condensed abelian group functor on topological spaces"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/singular-homology-light-cond-ab","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.singularHomologyLightCondAb","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#singular-homology-light-cond-ab","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"For a topological space $X$ and $n \\in \\mathbb{N}$, this is the discrete light condensed abelian group on the $n$-th integral singular homology group $H_n(X; \\mathbb{Z})$.","title":"Singular homology as a light condensed abelian group"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/singular-chains-light-cond-ab-complex-fu","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.singularChainsLightCondAbComplexFunctor","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#singular-chains-light-cond-ab-complex-fu","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"This is the functor from topological spaces to cochain complexes of light condensed abelian groups sending a space $X$ to the complex whose term in degree $-n$, for $n \\ge 0$, is the discrete light condensed abelian group on the integral singular $n$-chains $C_n(X; \\mathbb{Z})$, with the singular boundary maps as differentials and zero in positive degrees.","title":"The singular chain complex as a complex of light condensed abelian groups"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/singular-chains-light-cond-ab-derived-fu","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.singularChainsLightCondAbDerivedFunctor","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#singular-chains-light-cond-ab-derived-fu","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"This is the functor from topological spaces to the derived category of light condensed abelian groups sending a space $X$ to its integral singular chain complex, regarded as a cochain complex of discrete light condensed abelian groups with the $n$-chains in degree $-n$.","title":"The singular chain functor into the derived category"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/singular-chains-light-cond-ab-derived","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.singularChainsLightCondAbDerived","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#singular-chains-light-cond-ab-derived","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"For a topological space $X$, this is the integral singular chain complex of $X$, regarded as a cochain complex of discrete light condensed abelian groups with the $n$-chains in degree $-n$, as an object of the derived category of light condensed abelian groups.","title":"The singular chains of a space in the derived category"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/is-cwtop-cat","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.isCWTopCat","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#is-cwtop-cat","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"A topological space $X$ has this property when it admits a classical CW complex structure whose underlying set is all of $X$.","title":"Spaces admitting a CW complex structure"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/cwtop-cat","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.CWTopCat","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#cwtop-cat","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The full subcategory of the category of topological spaces whose objects are the spaces $X$ admitting a classical CW complex structure with underlying set all of $X$; its morphisms are all continuous maps between such spaces.","title":"The category of spaces admitting a CW structure"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/to-top-cat","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.CWTopCat.toTopCat","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#to-top-cat","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"The inclusion functor from the full subcategory of topological spaces admitting a classical CW complex structure into the category of topological spaces.","title":"Inclusion of CW spaces into topological spaces"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-free-cwfunctor","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidificationFreeCWFunctor","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-free-cwfunctor","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"can_state","summary":"A functor from the category of topological spaces admitting a CW complex structure, with all continuous maps as morphisms, to the derived category of light condensed abelian groups.","title":"Derived solidification on CW complexes"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-free-cwfunctor-sp","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidificationFreeCWFunctorSpec","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-free-cwfunctor-sp","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","summary":"A natural isomorphism between the derived solidification functor on CW complexes and the composite functor that sends a topological space $X$ admitting a CW complex structure to the free light condensed abelian group $\\mathbb{Z}[X]$, views it as a complex concentrated in degree $0$ in the derived category of light condensed abelian groups, applies derived solidification, and then applies the derived inclusion of\u2026","title":"Natural isomorphism specifying derived solidification on CW complexes"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/singular-chains-light-cond-ab-cwderived","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.singularChainsLightCondAbCWDerivedFunctor","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#singular-chains-light-cond-ab-cwderived","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"fully_proved","summary":"This is the functor from the category of topological spaces admitting a CW complex structure, with all continuous maps as morphisms, to the derived category of light condensed abelian groups sending $X$ to its integral singular chain complex, regarded as a cochain complex of discrete light condensed abelian groups with the $n$-chains in degree $-n$.","title":"The singular chain functor on CW spaces"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-free-cw-derived-n","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidification_free_CW_derivedNatIso","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-free-cw-derived-n","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","summary":"A natural isomorphism of functors from the category of spaces admitting a CW structure to the derived category of light condensed abelian groups, from the derived solidification functor $F$ to the integral singular chain complex functor. Here $F(X)$ is the derived inclusion of the derived solidification of the free light condensed abelian group $\\mathbb{Z}[X]$ placed in degree $0$ (up to a specified natural\u2026","title":"Derived solidification of free CW complexes computes singular chains"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-free-cw-derived-i","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidification_free_CW_derivedIso","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-free-cw-derived-i","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","summary":"For a topological space $X$ with a CW complex structure on all of $X$, an isomorphism in the derived category of light condensed abelian groups from the derived inclusion of the derived solidification of the free light condensed abelian group $\\mathbb{Z}[X]$, placed in degree $0$, to the integral singular chain complex of $X$, viewed as a complex of discrete light condensed abelian groups with chains of degree $n$\u2026","title":"Pointwise derived comparison for CW complexes"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-free-cw-derived-i-2","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidification_free_CW_derivedIso_eq","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-free-cw-derived-i-2","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"stated","summary":"Let $X$ be a topological space with a CW complex structure on all of $X$, regarded as an object of the category of spaces admitting a CW structure, and let $\\mathbb{Z}[X]$ be the free light condensed abelian group on $X$. Let $F$ be the derived solidification functor on spaces admitting a CW structure, let $\\sigma$ be the natural isomorphism from $F$ to the functor sending $X$ to the derived inclusion of the\u2026","title":"The pointwise derived comparison is the component of the natural one"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-free-cw-homology","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidification_free_CW_homologyIso","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-free-cw-homology","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"planned","summary":"Let $X$ be a topological space equipped with a CW complex structure, let $n$ be a natural number, and let $\\mathbb{Z}[X]$ be the free light condensed abelian group on $X$, viewed as a complex concentrated in degree $0$ in the derived category of light condensed abelian groups. Apply derived solidification to $\\mathbb{Z}[X]$, take the homology in cohomological degree $-n$, which is a light solid abelian group, and\u2026","title":"Derived solidification computes singular homology"},{"id":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/derived-solidification-free-cw-homology-2","kind":"node","lean":"MathlibExt.Topology.DerivedSolidCWHomologyWanted.LightCondensed.Solid.derivedSolidification_free_CW_homology","parent":15357,"result":true,"route":"../roadmap/Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes/index.html#derived-solidification-free-cw-homology-2","scope":"Topology/algebraic-topology/solid-condensed-homology-of-cw-complexes","status":"stated","summary":"Let $X$ be a topological space with a CW complex structure on all of $X$, let $\\mathbb{Z}[X]$ be the free light condensed abelian group on $X$, and let $n$ be a natural number. Then the cohomology in degree $-n$ of the derived solidification of $\\mathbb{Z}[X]$ placed in degree $0$, a light solid abelian group regarded as a light condensed abelian group, is isomorphic to the integral singular homology group\u2026","title":"Homology of the derived solidification of a free CW complex"},{"data_url":"scopes/Topology/manifolds-geometric-topology.json","id":"Topology/manifolds-geometric-topology","kind":"scope","parent":15066,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/index.html","scope":"Topology/manifolds-geometric-topology","status":"can_prove","summary":"Topological and smooth manifolds in low and high dimensions: surfaces, 3- and 4-manifolds, the Poincar\u00e9 conjecture, exotic spheres, triangulations, diffeomorphism groups and transformation groups.","title":"Manifolds and geometric topology"},{"id":"Topology/manifolds-geometric-topology/budney-gabai-knotted-three-spheres","kind":"node","lean":"MathlibExt.Topology.BudneyGabaiWanted.Topology.budney_gabai_knotted_three_spheres","parent":15403,"reading_title":{"format":"inline-parts-v1","parts":[["text","Budney\u2013Gabai knotted three-spheres in "],["math","\\(S^1 \\times S^3\\)"]],"source":"Budney\u2013Gabai knotted three-spheres in $S^1 \\times S^3$"},"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/index.html#budney-gabai-knotted-three-spheres","scope":"Topology/manifolds-geometric-topology","status":"can_prove","summary":"Let $S^1 \\subset \\mathbb{R}^2$ and $S^3 \\subset \\mathbb{R}^4$ be the unit spheres, and fix a point $x_0 \\in S^1$. There is a sequence $e_0, e_1, e_2, \\dots$ of smooth embeddings $S^3 \\to S^1 \\times S^3$ such that the complement of the image of each $e_n$ is connected, each $e_n$ is homotopic to the standard cross-section $p \\mapsto (x_0, p)$, and for $i \\ne j$ the images of $e_i$ and $e_j$ are not smoothly\u2026","title":"Budney\u2013Gabai knotted three-spheres in $S^1 \\times S^3$"},{"id":"Topology/manifolds-geometric-topology/milnor-exotic-sphere-seven","kind":"node","lean":"MathlibExt.Topology.MilnorExoticSphereSevenWanted.Topology.milnor_exotic_sphere_seven","parent":15403,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/index.html#milnor-exotic-sphere-seven","scope":"Topology/manifolds-geometric-topology","status":"can_prove","summary":"There is a smooth manifold $M$ modelled on $\\mathbb{R}^7$ (a topological space with a $C^\\infty$ atlas of charts into $\\mathbb{R}^7$) that is homeomorphic to the unit sphere $S^7 \\subset \\mathbb{R}^8$ but is not diffeomorphic to $S^7$ with its standard smooth structure.","title":"Milnor's exotic 7-sphere"},{"id":"Topology/manifolds-geometric-topology/hilbert-smith-padic-dimension-three","kind":"node","lean":"MathlibExt.Topology.HilbertSmith3DWanted.Topology.hilbert_smith_padic_dimension_three","parent":15403,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/index.html#hilbert-smith-padic-dimension-three","scope":"Topology/manifolds-geometric-topology","status":"can_prove","summary":"Pardon's theorem: the p-adic integers cannot act faithfully and continuously on a connected 3-manifold. Let $p$ be a prime and $M$ a connected, Hausdorff, second-countable topological space with an atlas of charts modelled on $\\mathbb{R}^3$, that is, a connected topological 3-manifold without boundary. Then there is no faithful action of the additive group $\\mathbb{Z}_p$ of $p$-adic integers on $M$ that is jointly\u2026","title":"Pardon's theorem: the p-adic integers cannot act faithfully and continuously on a connected 3-manifold"},{"data_url":"scopes/Topology/manifolds-geometric-topology/annulus-theorem.json","id":"Topology/manifolds-geometric-topology/annulus-theorem","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"can_prove","title":"Annulus theorem"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/amb","kind":"node","lean":"MathlibExt.Topology.AnnulusShared.Amb","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#amb","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"fully_proved","summary":"For a natural number $n$, the ambient space is the real Euclidean space $\\mathbb{R}^n$ of dimension $n$, with coordinates indexed by $0, \\dots, n-1$ and the Euclidean norm.","title":"Ambient Euclidean space"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/sph","kind":"node","lean":"MathlibExt.Topology.AnnulusShared.Sph","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#sph","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"fully_proved","summary":"For a natural number $n$, the unit sphere $S^{n-1} \\subseteq \\mathbb{R}^n$ is the set of points of $\\mathbb{R}^n$ at Euclidean distance $1$ from the origin.","title":"Unit sphere in Euclidean space"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/model-hyperplane","kind":"node","lean":"MathlibExt.Topology.AnnulusShared.modelHyperplane","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#model-hyperplane","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"fully_proved","summary":"For $n \\ge 1$, the model hyperplane in $\\mathbb{R}^n$ is the set of points $x = (x_0, \\dots, x_{n-1})$ whose first coordinate $x_0$ is $0$. It is the local model for a locally flat embedded sphere.","title":"Model hyperplane"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/locally-flat","kind":"node","lean":"MathlibExt.Topology.AnnulusShared.LocallyFlat","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#locally-flat","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"fully_proved","summary":"Let $n \\ge 1$, let $S^{n-1} \\subseteq \\mathbb{R}^n$ be the unit sphere and let $H = \\{x \\in \\mathbb{R}^n : x_0 = 0\\}$ be the model hyperplane. A map $c : S^{n-1} \\to \\mathbb{R}^n$ is locally flat if for every point $p \\in S^{n-1}$ there are open sets $U, V \\subseteq \\mathbb{R}^n$ with $c(p) \\in U$ and a homeomorphism $h : U \\to V$ such that $h(U \\cap c(S^{n-1})) = V \\cap H$.","title":"Locally flat maps of the sphere"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/flat-sphere","kind":"node","lean":"MathlibExt.Topology.AnnulusShared.FlatSphere","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#flat-sphere","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"fully_proved","summary":"For $n \\ge 1$, a locally flat embedded $(n-1)$-sphere in $\\mathbb{R}^n$ is a map $f : S^{n-1} \\to \\mathbb{R}^n$ from the unit sphere that is continuous, injective and locally flat.","title":"Locally flat embedded sphere"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/inside","kind":"node","lean":"MathlibExt.Topology.AnnulusShared.inside","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#inside","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"fully_proved","summary":"For a set $S \\subseteq \\mathbb{R}^n$, the inside of $S$ is the set of points $x \\notin S$ whose connected component in the complement $\\mathbb{R}^n \\setminus S$ is bounded.","title":"Inside of a set"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/region-between","kind":"node","lean":"MathlibExt.Topology.AnnulusShared.regionBetween","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#region-between","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"fully_proved","summary":"Let $n \\ge 1$ and let $f$ and $g$ be locally flat embedded $(n-1)$-spheres in $\\mathbb{R}^n$. The region between $f$ and $g$ is the closure of the inside of the image of $f$ with the inside of the image of $g$ removed, where the inside of a set $S$ is the set of points $x \\notin S$ whose connected component in $\\mathbb{R}^n \\setminus S$ is bounded.","title":"Region between two flat spheres"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/annulus-theorem-dim-four","kind":"node","lean":"MathlibExt.Topology.AnnulusTheoremDimFourWanted.Topology.annulus_theorem_dim_four","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#annulus-theorem-dim-four","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"can_prove","summary":"Let $f$ and $g$ be locally flat embedded $3$-spheres in $\\mathbb{R}^4$, that is, continuous injective locally flat maps $S^3 \\to \\mathbb{R}^4$ from the unit sphere. Suppose that the images of $f$ and $g$ are disjoint and that the image of $g$ lies in the inside of the image of $f$, the set of points off the image of $f$ whose connected component in its complement is bounded. Then the region between $f$ and $g$, the\u2026","title":"Annulus theorem in dimension four"},{"id":"Topology/manifolds-geometric-topology/annulus-theorem/annulus-theorem-high-dim","kind":"node","lean":"MathlibExt.Topology.AnnulusTheoremHighDimWanted.Topology.annulus_theorem_high_dim","parent":15407,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/annulus-theorem/index.html#annulus-theorem-high-dim","scope":"Topology/manifolds-geometric-topology/annulus-theorem","status":"can_prove","summary":"Let $n \\ge 5$ and let $f$ and $g$ be locally flat embedded $(n-1)$-spheres in $\\mathbb{R}^n$, that is, continuous injective locally flat maps $S^{n-1} \\to \\mathbb{R}^n$ from the unit sphere. Suppose that the images of $f$ and $g$ are disjoint and that the image of $g$ lies in the inside of the image of $f$, the set of points off the image of $f$ whose connected component in its complement is bounded. Then the\u2026","title":"Annulus theorem in dimensions at least five"},{"data_url":"scopes/Topology/manifolds-geometric-topology/aspherical-homology-4-spheres.json","id":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/index.html","scope":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres","status":"can_prove","title":"Aspherical homology 4-spheres"},{"id":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/closed4-manifold","kind":"node","lean":"MathlibExt.Topology.AsphericalHomologySphereWanted.Topology.Closed4Manifold","parent":15417,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/index.html#closed4-manifold","scope":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres","status":"fully_proved","summary":"A closed topological 4-manifold is a nonempty, compact, connected, Hausdorff, second-countable topological space $M$ together with an atlas of charts covering $M$, each a homeomorphism from an open subset of $M$ onto an open subset of $\\mathbb{R}^4$ (charts are modelled on all of $\\mathbb{R}^4$, not on a half-space, so $M$ has no boundary).","title":"Closed topological 4-manifold"},{"id":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/int-homology","kind":"node","lean":"MathlibExt.Topology.AsphericalHomologySphereWanted.Topology.intHomology","parent":15417,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral singular homology as a "],["math","\\(\\mathbb{Z}\\)"],["text","-module"]],"source":"Integral singular homology as a $\\mathbb{Z}$-module"},"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/index.html#int-homology","scope":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres","status":"fully_proved","summary":"For a natural number $k$ and a topological space $X$, $H_k(X; \\mathbb{Z})$ is the $k$-th singular homology of $X$ with coefficients in $\\mathbb{Z}$, viewed as a $\\mathbb{Z}$-module.","title":"Integral singular homology as a $\\mathbb{Z}$-module"},{"id":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/is-aspherical","kind":"node","lean":"MathlibExt.Topology.AsphericalHomologySphereWanted.Topology.Closed4Manifold.IsAspherical","parent":15417,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/index.html#is-aspherical","scope":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres","status":"fully_proved","summary":"A closed topological 4-manifold $M$ is aspherical if for every $k \\ge 2$ and every point $x \\in M$ the homotopy group $\\pi_k(M, x)$ is trivial.","title":"Aspherical closed 4-manifold"},{"id":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/is-integer-homology-sphere","kind":"node","lean":"MathlibExt.Topology.AsphericalHomologySphereWanted.Topology.Closed4Manifold.IsIntegerHomologySphere","parent":15417,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/index.html#is-integer-homology-sphere","scope":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres","status":"fully_proved","summary":"A closed topological 4-manifold $M$ is an integer homology 4-sphere if for every $k \\ge 0$ the integral singular homology $H_k(M; \\mathbb{Z})$ is isomorphic, as a $\\mathbb{Z}$-module, to $H_k(S^4; \\mathbb{Z})$, where $S^4$ is the unit sphere in $\\mathbb{R}^5$.","title":"Integer homology 4-sphere"},{"id":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/aspherical-integer-homology-four-sphere","kind":"node","lean":"MathlibExt.Topology.AsphericalHomologySphereWanted.Topology.aspherical_integer_homology_four_sphere","parent":15417,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/aspherical-homology-4-spheres/index.html#aspherical-integer-homology-four-sphere","scope":"Topology/manifolds-geometric-topology/aspherical-homology-4-spheres","status":"can_prove","summary":"There is a closed connected topological 4-manifold $M$ that is aspherical, meaning that for every $k \\ge 2$ and every point $x \\in M$ the homotopy group $\\pi_k(M, x)$ is trivial, and that is an integer homology 4-sphere, meaning that for every $k \\ge 0$ the integral singular homology $H_k(M; \\mathbb{Z})$ is isomorphic to $H_k(S^4; \\mathbb{Z})$, where $S^4$ is the unit sphere in $\\mathbb{R}^5$.","title":"Existence of an aspherical integer homology 4-sphere"},{"data_url":"scopes/Topology/manifolds-geometric-topology/classification-of-surfaces.json","id":"Topology/manifolds-geometric-topology/classification-of-surfaces","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/classification-of-surfaces/index.html","scope":"Topology/manifolds-geometric-topology/classification-of-surfaces","status":"can_prove","title":"Classification of surfaces"},{"id":"Topology/manifolds-geometric-topology/classification-of-surfaces/closed-unit-disc","kind":"node","lean":"MathlibExt.Topology.ClassificationOfSurfacesWanted.Complex.ClosedUnitDisc","parent":15423,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/classification-of-surfaces/index.html#closed-unit-disc","scope":"Topology/manifolds-geometric-topology/classification-of-surfaces","status":"fully_proved","summary":"The closed unit disc in the complex plane.","title":"Closed unit disc in the complex plane"},{"id":"Topology/manifolds-geometric-topology/classification-of-surfaces/bdy-pt-of-real","kind":"node","lean":"MathlibExt.Topology.ClassificationOfSurfacesWanted.Complex.ClosedUnitDisc.bdyPtOfReal","parent":15423,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/classification-of-surfaces/index.html#bdy-pt-of-real","scope":"Topology/manifolds-geometric-topology/classification-of-surfaces","status":"fully_proved","summary":"For a real number $r$, this is the point $e^{2\\pi i r}$ of the closed unit disc in $\\mathbb{C}$, which lies on its boundary circle.","title":"Boundary point of the unit disc at a real parameter"},{"id":"Topology/manifolds-geometric-topology/classification-of-surfaces/orientable-rel","kind":"node","lean":"MathlibExt.Topology.ClassificationOfSurfacesWanted.Topology.ClassificationOfSurfaces.OrientableRel","parent":15423,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/classification-of-surfaces/index.html#orientable-rel","scope":"Topology/manifolds-geometric-topology/classification-of-surfaces","status":"fully_proved","summary":"For natural numbers $p$ and $n$, put $N = 4p + 3n$ and write $e(t) = e^{2\\pi i t}$ for the boundary point of the closed unit disc at the real parameter $t$. This is the relation on the closed unit disc under which, for every $x \\in [0, 1]$ and every natural number $k < p$, the point $e((4k + x)/N)$ is related to $e((4k + 3 - x)/N)$ and the point $e((4k + 1 + x)/N)$ is related to $e((4k + 4 - x)/N)$, and for every\u2026","title":"Gluing relation for the orientable normal form"},{"id":"Topology/manifolds-geometric-topology/classification-of-surfaces/non-orientable-rel","kind":"node","lean":"MathlibExt.Topology.ClassificationOfSurfacesWanted.Topology.ClassificationOfSurfaces.NonOrientableRel","parent":15423,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/classification-of-surfaces/index.html#non-orientable-rel","scope":"Topology/manifolds-geometric-topology/classification-of-surfaces","status":"fully_proved","summary":"For natural numbers $p$ and $n$, put $N = 2p + 3n$ and write $e(t) = e^{2\\pi i t}$ for the boundary point of the closed unit disc at the real parameter $t$. This is the relation on the closed unit disc under which, for every $x \\in [0, 1]$ and every natural number $k < p$, the point $e((2k + x)/N)$ is related to $e((2k + 1 + x)/N)$, and for every $x \\in [0, 1]$ and every natural number $k < n$, the point\u2026","title":"Gluing relation for the non-orientable normal form"},{"id":"Topology/manifolds-geometric-topology/classification-of-surfaces/classification-of-surfaces","kind":"node","lean":"MathlibExt.Topology.ClassificationOfSurfacesWanted.Topology.ClassificationOfSurfaces.classification_of_surfaces","parent":15423,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/classification-of-surfaces/index.html#classification-of-surfaces","scope":"Topology/manifolds-geometric-topology/classification-of-surfaces","status":"can_prove","summary":"Let $S$ be a nonempty, compact, connected Hausdorff space that is a topological manifold of dimension $2$ with boundary, with charts in the closed half-plane $\\{x \\in \\mathbb{R}^2 : x_1 \\ge 0\\}$. Then $S$ is homeomorphic to the unit sphere in $\\mathbb{R}^3$, or there are natural numbers $p$ and $n$ for which one of the following holds: $p \\ge 1$ or $n \\ge 1$, and $S$ is homeomorphic to the quotient of the closed\u2026","title":"Classification of compact surfaces"},{"data_url":"scopes/Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher.json","id":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/index.html","scope":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher","status":"can_prove","title":"Homeomorphism and diffeomorphism groups of spheres"},{"id":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/cerf-gamma-four","kind":"node","lean":"MathlibExt.Topology.CerfGammaFourWanted.Topology.cerf_gamma_four","parent":15429,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cerf's theorem "],["math","\\(\\Gamma_4 = 0\\)"]],"source":"Cerf's theorem $\\Gamma_4 = 0$"},"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/index.html#cerf-gamma-four","scope":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher","status":"can_prove","summary":"Let $S^3 \\subset \\mathbb{R}^4$ be the unit sphere. Every smooth diffeomorphism $f$ of $S^3$ is smoothly isotopic to the restriction of a linear isometry of $\\mathbb{R}^4$: there are an orthogonal matrix $A \\in O(4)$ and smooth maps $F, F' : [0, 1] \\times S^3 \\to S^3$ such that for every $t \\in [0, 1]$ the maps $F(t, \\cdot)$ and $F'(t, \\cdot)$ are inverse to each other, and $F(0, p) = f(p)$ and $F(1, p) = Ap$ for\u2026","title":"Cerf's theorem $\\Gamma_4 = 0$"},{"id":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/identity-component-of-sphere-homeomorphi","kind":"node","lean":"MathlibExt.Topology.Manifold.ScottishBook29Wanted.identity_component_of_sphere_homeomorphism_group_simple","parent":15429,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/index.html#identity-component-of-sphere-homeomorphi","scope":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher","status":"can_prove","summary":"Let $n \\ne 4$ be a natural number and let $S^n$ be the unit sphere in $\\mathbb{R}^{n+1}$. Give the group $\\mathrm{Homeo}(S^n)$ of homeomorphisms of $S^n$ the topology induced by the embedding $h \\mapsto (h, h^{-1})$ into pairs of continuous self-maps of $S^n$ with the compact-open topology, and let $G_0$ be the connected component of the identity in $\\mathrm{Homeo}(S^n)$. If $N$ is a subgroup of\u2026","title":"Normal subgroups of the identity component of the homeomorphism group of a sphere"},{"id":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/smale-conjecture","kind":"node","lean":"MathlibExt.Topology.SmaleConjectureWanted.Topology.smale_conjecture","parent":15429,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/index.html#smale-conjecture","scope":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher","status":"can_prove","summary":"Let $n \\ge 1$ and let $X$ be a compact, Hausdorff, second-countable smooth manifold with boundary modelled on the $n$-dimensional half-space, with boundary $\\partial X$. Write $S^3$ for the unit sphere of $\\mathbb{R}^4$ and let the orthogonal group $O(4)$ act linearly on it. Let $F, F' : X \\times S^3 \\to S^3$ be smooth maps with $F(x, F'(x, p)) = p$ and $F'(x, F(x, p)) = p$ for all $x \\in X$ and $p \\in S^3$, and\u2026","title":"Smale conjecture (Hatcher), relative parametrized form"},{"id":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/watanabe-four-dim-smale-disproof","kind":"node","lean":"MathlibExt.Topology.WatanabeSmaleDisproofWanted.Topology.watanabe_four_dim_smale_disproof","parent":15429,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher/index.html#watanabe-four-dim-smale-disproof","scope":"Topology/manifolds-geometric-topology/homeomorphism-and-diffeomorphism-groups-of-spher","status":"can_prove","summary":"Write $S^4$ for the unit sphere of $\\mathbb{R}^5$ and let the orthogonal group $O(5)$ act linearly on it. There are an $n \\ge 1$, a compact, Hausdorff, second-countable smooth manifold $X$ with boundary modelled on the $n$-dimensional half-space, with boundary $\\partial X$, smooth maps $F, F' : X \\times S^4 \\to S^4$ with $F(x, F'(x, p)) = p$ and $F'(x, F(x, p)) = p$ for all $x \\in X$ and $p \\in S^4$, and a\u2026","title":"The four-dimensional Smale conjecture is false (Watanabe)"},{"data_url":"scopes/Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups.json","id":"Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups/index.html","scope":"Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups","status":"can_prove","title":"Linearity of 3-manifold groups"},{"id":"Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups/closed3-manifold","kind":"node","lean":"MathlibExt.Topology.NonlinearThreeManifoldGroupWanted.Topology.Closed3Manifold","parent":15434,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups/index.html#closed3-manifold","scope":"Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups","status":"fully_proved","summary":"A closed topological 3-manifold is a nonempty, compact, connected, Hausdorff, second-countable topological space $M$ together with an atlas of charts, homeomorphisms from open subsets of $M$ onto open subsets of $\\mathbb{R}^3$, whose domains cover $M$. Since the charts land in $\\mathbb{R}^3$ itself rather than in a half-space, $M$ has no boundary.","title":"Closed topological 3-manifold"},{"id":"Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups/nonlinear-three-manifold-group","kind":"node","lean":"MathlibExt.Topology.NonlinearThreeManifoldGroupWanted.Topology.nonlinear_three_manifold_group","parent":15434,"reading_title":{"format":"inline-parts-v1","parts":[["text","A closed 3-manifold group with no faithful representation in "],["math","\\(GL_4(\\mathbb{R})\\)"]],"source":"A closed 3-manifold group with no faithful representation in $GL_4(\\mathbb{R})$"},"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups/index.html#nonlinear-three-manifold-group","scope":"Topology/manifolds-geometric-topology/linearity-of-3-manifold-groups","status":"can_prove","summary":"There is a closed connected topological 3-manifold $M$ with a point $x$ such that no group homomorphism from the fundamental group $\\pi_1(M, x)$ to $GL_4(\\mathbb{R})$ is injective.","title":"A closed 3-manifold group with no faithful representation in $GL_4(\\mathbb{R})$"},{"data_url":"scopes/Topology/manifolds-geometric-topology/linearization-of-compact-group-actions.json","id":"Topology/manifolds-geometric-topology/linearization-of-compact-group-actions","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/linearization-of-compact-group-actions/index.html","scope":"Topology/manifolds-geometric-topology/linearization-of-compact-group-actions","status":"can_prove","title":"Linearization of compact group actions"},{"id":"Topology/manifolds-geometric-topology/linearization-of-compact-group-actions/scottish-book-problem183-claim","kind":"node","lean":"MathlibExt.Topology.ScottishBook183Wanted.ScottishBookProblem183Claim","parent":15437,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/linearization-of-compact-group-actions/index.html#scottish-book-problem183-claim","scope":"Topology/manifolds-geometric-topology/linearization-of-compact-group-actions","status":"fully_proved","summary":"The claim that for every $n \\in \\mathbb{N}$ and every compact, connected, locally connected topological group $G$ acting on $\\mathbb{R}^n$ with continuous action map $G \\times \\mathbb{R}^n \\to \\mathbb{R}^n$, there is a homeomorphism $\\varphi$ of $\\mathbb{R}^n$ such that for every $g \\in G$ there is a linear map $L_g \\colon \\mathbb{R}^n \\to \\mathbb{R}^n$ with $\\varphi(g \\cdot x) = L_g(\\varphi(x))$ for all\u2026","title":"Bogolyubov's linearization claim (Scottish Book Problem 183)"},{"id":"Topology/manifolds-geometric-topology/linearization-of-compact-group-actions/bogolyubov183-refuted","kind":"node","lean":"MathlibExt.Topology.ScottishBook183Wanted.bogolyubov183_refuted","parent":15437,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/linearization-of-compact-group-actions/index.html#bogolyubov183-refuted","scope":"Topology/manifolds-geometric-topology/linearization-of-compact-group-actions","status":"can_prove","summary":"Bogolyubov's linearization claim is false: there are $n \\in \\mathbb{N}$ and a compact, connected, locally connected topological group $G$ acting continuously on $\\mathbb{R}^n$ such that for no homeomorphism $\\varphi$ of $\\mathbb{R}^n$ is every map $\\varphi \\circ g \\circ \\varphi^{-1}$, for $g \\in G$, linear.","title":"Bogolyubov's Problem 183 has a negative answer"},{"data_url":"scopes/Topology/manifolds-geometric-topology/poincar-conjecture.json","id":"Topology/manifolds-geometric-topology/poincar-conjecture","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/poincar-conjecture/index.html","scope":"Topology/manifolds-geometric-topology/poincar-conjecture","status":"can_prove","title":"Poincar\u00e9 conjecture"},{"id":"Topology/manifolds-geometric-topology/poincar-conjecture/poincare-3d-smooth","kind":"node","lean":"MathlibExt.Topology.Poincare3DSmoothWanted.Topology.poincare_3d_smooth","parent":15440,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/poincar-conjecture/index.html#poincare-3d-smooth","scope":"Topology/manifolds-geometric-topology/poincar-conjecture","status":"can_prove","summary":"Let $M$ be a nonempty, compact, simply connected Hausdorff space with a smooth ($C^\\infty$) manifold structure modelled on $\\mathbb{R}^3$, without boundary. Then $M$ is $C^\\infty$-diffeomorphic to the unit sphere $S^3 = \\{x \\in \\mathbb{R}^4 : \\|x\\| = 1\\}$ with its standard smooth structure.","title":"Smooth Poincar\u00e9 conjecture in dimension 3"},{"id":"Topology/manifolds-geometric-topology/poincar-conjecture/poincare-3d-topological","kind":"node","lean":"MathlibExt.Topology.Poincare3DTopologicalWanted.Topology.poincare_3d_topological","parent":15440,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/poincar-conjecture/index.html#poincare-3d-topological","scope":"Topology/manifolds-geometric-topology/poincar-conjecture","status":"can_prove","summary":"Let $M$ be a nonempty, compact, simply connected Hausdorff space with an atlas of charts to $\\mathbb{R}^3$, that is, a topological 3-manifold without boundary. Then $M$ is homeomorphic to the unit sphere $S^3 = \\{x \\in \\mathbb{R}^4 : \\|x\\| = 1\\}$.","title":"Topological Poincar\u00e9 conjecture in dimension 3"},{"id":"Topology/manifolds-geometric-topology/poincar-conjecture/poincare-4d-topological","kind":"node","lean":"MathlibExt.Topology.Poincare4DTopologicalWanted.Topology.poincare_4d_topological","parent":15440,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/poincar-conjecture/index.html#poincare-4d-topological","scope":"Topology/manifolds-geometric-topology/poincar-conjecture","status":"can_prove","summary":"Let $M$ be a Hausdorff space with an atlas of charts to $\\mathbb{R}^4$ (a topological 4-manifold without boundary, not assumed second countable). If $M$ is homotopy equivalent to the unit sphere $S^4 = \\{x \\in \\mathbb{R}^5 : \\|x\\| = 1\\}$, then $M$ is homeomorphic to $S^4$.","title":"Topological Poincar\u00e9 conjecture in dimension 4"},{"id":"Topology/manifolds-geometric-topology/poincar-conjecture/poincare-high-dim-topological","kind":"node","lean":"MathlibExt.Topology.PoincareHighDimTopologicalWanted.Topology.poincare_high_dim_topological","parent":15440,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/poincar-conjecture/index.html#poincare-high-dim-topological","scope":"Topology/manifolds-geometric-topology/poincar-conjecture","status":"can_prove","summary":"Let $n \\ge 5$ and let $M$ be a Hausdorff space with an atlas of charts to $\\mathbb{R}^n$ (a topological $n$-manifold without boundary, not assumed second countable). If $M$ is homotopy equivalent to the unit sphere $S^n = \\{x \\in \\mathbb{R}^{n+1} : \\|x\\| = 1\\}$, then $M$ is homeomorphic to $S^n$.","title":"Topological Poincar\u00e9 conjecture in dimensions at least 5"},{"data_url":"scopes/Topology/manifolds-geometric-topology/topological-3-manifolds.json","id":"Topology/manifolds-geometric-topology/topological-3-manifolds","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/topological-3-manifolds/index.html","scope":"Topology/manifolds-geometric-topology/topological-3-manifolds","status":"fully_proved","title":"Topological 3-manifolds"},{"id":"Topology/manifolds-geometric-topology/topological-3-manifolds/is-three-manifold","kind":"node","lean":"IsThreeManifold","parent":15445,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/topological-3-manifolds/index.html#is-three-manifold","scope":"Topology/manifolds-geometric-topology/topological-3-manifolds","status":"fully_proved","summary":"Let $M$ be a topological space equipped with an atlas of charts, each a homeomorphism from an open subset of $M$ onto an open subset of $\\mathbb{R}^3$, whose domains cover $M$. Then $M$ is a topological 3-manifold if it is Hausdorff and second-countable.","title":"Topological 3-manifold"},{"id":"Topology/manifolds-geometric-topology/topological-3-manifolds/is-three-manifold-with-boundary","kind":"node","lean":"IsThreeManifoldWithBoundary","parent":15445,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/topological-3-manifolds/index.html#is-three-manifold-with-boundary","scope":"Topology/manifolds-geometric-topology/topological-3-manifolds","status":"fully_proved","summary":"A topological 3-manifold with boundary: a Hausdorff, second-countable topological space locally homeomorphic to the Euclidean half-space `EuclideanHalfSpace 2`.","title":"Topological 3-manifold with boundary"},{"id":"Topology/manifolds-geometric-topology/topological-3-manifolds/is-three-manifold-iff","kind":"node","lean":"isThreeManifold_iff","parent":15445,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/topological-3-manifolds/index.html#is-three-manifold-iff","scope":"Topology/manifolds-geometric-topology/topological-3-manifolds","status":"fully_proved","summary":"Let $M$ be a topological space equipped with an atlas of charts modelled on $\\mathbb{R}^3$, so that $M$ is locally homeomorphic to $\\mathbb{R}^3$. Then $M$ is a topological 3-manifold if and only if $M$ is Hausdorff and second-countable.","title":"Characterization of topological 3-manifolds"},{"id":"Topology/manifolds-geometric-topology/topological-3-manifolds/locally-compact-space","kind":"node","lean":"IsThreeManifold.locallyCompactSpace","parent":15445,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/topological-3-manifolds/index.html#locally-compact-space","scope":"Topology/manifolds-geometric-topology/topological-3-manifolds","status":"fully_proved","summary":"Every topological 3-manifold $M$ is locally compact, since $\\mathbb{R}^3$ is.","title":"Topological 3-manifolds are locally compact"},{"data_url":"scopes/Topology/manifolds-geometric-topology/triangulation-conjecture.json","id":"Topology/manifolds-geometric-topology/triangulation-conjecture","kind":"scope","navigation":"module","parent":15403,"result":false,"route":"../roadmap/Topology/manifolds-geometric-topology/triangulation-conjecture/index.html","scope":"Topology/manifolds-geometric-topology/triangulation-conjecture","status":"can_prove","title":"Triangulation conjecture"},{"id":"Topology/manifolds-geometric-topology/triangulation-conjecture/closed-top-manifold","kind":"node","lean":"MathlibExt.Topology.ManolescuTriangulationWanted.Topology.ClosedTopManifold","parent":15450,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/triangulation-conjecture/index.html#closed-top-manifold","scope":"Topology/manifolds-geometric-topology/triangulation-conjecture","status":"fully_proved","summary":"For a natural number $n$, a closed topological $n$-manifold is a nonempty, compact, connected, Hausdorff, second-countable topological space $M$ together with an atlas of charts, homeomorphisms from open subsets of $M$ onto open subsets of $\\mathbb{R}^n$, whose domains cover $M$. Since the charts land in $\\mathbb{R}^n$ itself rather than in a half-space, $M$ has no boundary.","title":"Closed topological manifold"},{"id":"Topology/manifolds-geometric-topology/triangulation-conjecture/triangulable","kind":"node","lean":"MathlibExt.Topology.ManolescuTriangulationWanted.Topology.Triangulable","parent":15450,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/triangulation-conjecture/index.html#triangulable","scope":"Topology/manifolds-geometric-topology/triangulation-conjecture","status":"fully_proved","summary":"A closed topological $n$-manifold $M$ is triangulable if, for some natural number $N$, there is a simplicial complex $K$ in $\\mathbb{R}^N$ with finitely many faces such that $M$ is homeomorphic to the geometric realization of $K$, the union of its faces with the subspace topology from $\\mathbb{R}^N$.","title":"Triangulable closed manifold"},{"id":"Topology/manifolds-geometric-topology/triangulation-conjecture/manolescu-triangulation-disproof","kind":"node","lean":"MathlibExt.Topology.ManolescuTriangulationWanted.Topology.manolescu_triangulation_disproof","parent":15450,"result":true,"route":"../roadmap/Topology/manifolds-geometric-topology/triangulation-conjecture/index.html#manolescu-triangulation-disproof","scope":"Topology/manifolds-geometric-topology/triangulation-conjecture","status":"can_prove","summary":"For every natural number $n \\ge 5$ there is a closed connected topological $n$-manifold that is not triangulable.","title":"Manolescu's disproof of the triangulation conjecture"},{"data_url":"scopes/Topology/continua-infinite-dimensional.json","id":"Topology/continua-infinite-dimensional","kind":"scope","parent":15066,"result":false,"route":"../roadmap/Topology/continua-infinite-dimensional/index.html","scope":"Topology/continua-infinite-dimensional","status":"can_prove","summary":"Compact connected spaces, Peano continua and space-filling curves, absolute retracts, hyperspaces, the Hilbert cube and fixed-point properties.","title":"Continuum theory and infinite-dimensional topology"},{"id":"Topology/continua-infinite-dimensional/hahn-mazurkiewicz","kind":"node","lean":"MathlibExt.Topology.Continuum.HahnMazurkiewiczWanted.hahnMazurkiewicz","parent":15454,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/index.html#hahn-mazurkiewicz","scope":"Topology/continua-infinite-dimensional","status":"fully_proved","summary":"Let $X$ be a nonempty topological space that is compact, connected, locally connected, Hausdorff and second countable. Then there is a continuous surjection from the closed unit interval $[0,1]$ onto $X$.","title":"Hahn\u2013Mazurkiewicz theorem"},{"id":"Topology/continua-infinite-dimensional/peano-curve-exists","kind":"node","lean":"MathlibExt.Topology.Continuum.PeanoCurveWanted.peano_curve_exists","parent":15454,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/index.html#peano-curve-exists","scope":"Topology/continua-infinite-dimensional","status":"fully_proved","summary":"There exists a continuous surjection from the closed unit interval to its square.","title":"Peano's space-filling curve"},{"id":"Topology/continua-infinite-dimensional/sphere-in-banach-space-is-unicoherent","kind":"node","lean":"MathlibExt.Topology.ScottishBook46Wanted.sphere_in_banach_space_is_unicoherent","parent":15454,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/index.html#sphere-in-banach-space-is-unicoherent","scope":"Topology/continua-infinite-dimensional","status":"can_prove","summary":"Let $E$ be a real Banach space that is not finite-dimensional, and let $S = \\{x \\in E : \\|x\\| = 1\\}$ be its unit sphere. If $A$ and $B$ are closed connected subsets of $S$ with $A \\cup B = S$, then $A \\cap B$ is connected. Here connected sets are nonempty by definition.","title":"Unit spheres of infinite-dimensional Banach spaces are unicoherent"},{"data_url":"scopes/Topology/continua-infinite-dimensional/absolute-retracts.json","id":"Topology/continua-infinite-dimensional/absolute-retracts","kind":"scope","navigation":"module","parent":15454,"result":false,"route":"../roadmap/Topology/continua-infinite-dimensional/absolute-retracts/index.html","scope":"Topology/continua-infinite-dimensional/absolute-retracts","status":"can_prove","title":"Absolute retracts"},{"id":"Topology/continua-infinite-dimensional/absolute-retracts/is-absolute-retract","kind":"node","lean":"MathlibExt.Topology.ScottishBook138Wanted.IsAbsoluteRetract","parent":15458,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/absolute-retracts/index.html#is-absolute-retract","scope":"Topology/continua-infinite-dimensional/absolute-retracts","status":"fully_proved","summary":"A topological space $K$ is an absolute retract when $K$ is compact and, for every separable metrizable space $X$ and every topological embedding $e \\colon K \\to X$, there is a continuous map $r \\colon X \\to K$ with $r(e(k)) = k$ for all $k \\in K$.","title":"Absolute retract"},{"id":"Topology/continua-infinite-dimensional/absolute-retracts/is-wilson-convex","kind":"node","lean":"MathlibExt.Topology.ScottishBook138Wanted.IsWilsonConvex","parent":15458,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/absolute-retracts/index.html#is-wilson-convex","scope":"Topology/continua-infinite-dimensional/absolute-retracts","status":"fully_proved","summary":"A subset $K$ of a pseudometric space $X$ with distance $d$ is convex in the sense of Wilson when for all $x, y \\in K$ and every real $t$ with $0 \\le t \\le 1$ there is exactly one $z \\in K$ with $d(x, z) = t \\, d(x, y)$ and $d(z, y) = (1 - t) \\, d(x, y)$.","title":"Wilson convexity"},{"id":"Topology/continua-infinite-dimensional/absolute-retracts/eilenberg-compact-convex-absolute-retrac","kind":"node","lean":"MathlibExt.Topology.ScottishBook138Wanted.eilenbergCompactConvexAbsoluteRetract","parent":15458,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/absolute-retracts/index.html#eilenberg-compact-convex-absolute-retrac","scope":"Topology/continua-infinite-dimensional/absolute-retracts","status":"can_prove","summary":"(a) Every nonempty compact convex subset of a complete metrizable locally convex topological vector space over $\\mathbb{R}$ is an absolute retract. (b) Every nonempty compact subset of a pseudometric space that is convex in the sense of Wilson is an absolute retract.","title":"Eilenberg's theorem on compact convex absolute retracts"},{"data_url":"scopes/Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets.json","id":"Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets","kind":"scope","navigation":"module","parent":15454,"result":false,"route":"../roadmap/Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets/index.html","scope":"Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets","status":"can_prove","title":"Fixed points on hyperspaces of convex sets"},{"id":"Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets/hausdorff-continuous-on-convex-compacts","kind":"node","lean":"MathlibExt.Topology.ScottishBook42Wanted.HausdorffContinuousOnConvexCompacts","parent":15462,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets/index.html#hausdorff-continuous-on-convex-compacts","scope":"Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets","status":"fully_proved","summary":"Let $E$ be a real normed space, let $K \\subseteq E$, let $f$ be a map from subsets of $E$ to subsets of $E$, and write $d_H$ for the Hausdorff distance between subsets of $E$. Then $f$ is Hausdorff continuous on the compact convex subsets of $K$ if for every nonempty compact convex set $X \\subseteq K$ and every $\\varepsilon > 0$ there is $\\delta > 0$ such that every nonempty compact convex set $Y \\subseteq K$ with\u2026","title":"Hausdorff continuity on compact convex sets"},{"id":"Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets/fixed-point-for-convex-hyperspace-map","kind":"node","lean":"MathlibExt.Topology.ScottishBook42Wanted.fixed_point_for_convex_hyperspace_map","parent":15462,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets/index.html#fixed-point-for-convex-hyperspace-map","scope":"Topology/continua-infinite-dimensional/fixed-points-on-hyperspaces-of-convex-sets","status":"can_prove","summary":"Let $n \\ge 0$ and let $B$ be the closed unit ball about the origin in the Euclidean space $\\mathbb{R}^n$. Let $f$ be a map from subsets of $\\mathbb{R}^n$ to subsets of $\\mathbb{R}^n$ that sends every nonempty compact convex subset of $B$ to a nonempty compact convex subset of $B$, and that is continuous in the Hausdorff metric on the nonempty compact convex subsets of $B$. Then there is a nonempty compact convex\u2026","title":"Fixed point theorem for maps of compact convex sets"},{"data_url":"scopes/Topology/continua-infinite-dimensional/hilbert-cube.json","id":"Topology/continua-infinite-dimensional/hilbert-cube","kind":"scope","navigation":"module","parent":15454,"result":false,"route":"../roadmap/Topology/continua-infinite-dimensional/hilbert-cube/index.html","scope":"Topology/continua-infinite-dimensional/hilbert-cube","status":"can_prove","title":"Hilbert cube"},{"id":"Topology/continua-infinite-dimensional/hilbert-cube/scottish-book-hilbert-cube","kind":"node","lean":"MathlibExt.Topology.ScottishBook175Wanted.ScottishBookHilbertCube","parent":15465,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/hilbert-cube/index.html#scottish-book-hilbert-cube","scope":"Topology/continua-infinite-dimensional/hilbert-cube","status":"fully_proved","summary":"The Hilbert cube $Q = [0,1]^{\\mathbb{N}}$ is the product of countably many copies of the closed unit interval $[0,1]$, with the product topology.","title":"The Hilbert cube"},{"id":"Topology/continua-infinite-dimensional/hilbert-cube/scottish-book-letter-t","kind":"node","lean":"MathlibExt.Topology.ScottishBook175Wanted.ScottishBookLetterT","parent":15465,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/hilbert-cube/index.html#scottish-book-letter-t","scope":"Topology/continua-infinite-dimensional/hilbert-cube","status":"fully_proved","summary":"The letter-T curve is the subset","title":"The letter T"},{"id":"Topology/continua-infinite-dimensional/hilbert-cube/borsuk175-hilbert-cube-times-letter-t","kind":"node","lean":"MathlibExt.Topology.ScottishBook175Wanted.borsuk175_hilbert_cube_times_letterT","parent":15465,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/hilbert-cube/index.html#borsuk175-hilbert-cube-times-letter-t","scope":"Topology/continua-infinite-dimensional/hilbert-cube","status":"can_prove","summary":"Let $Q = [0,1]^{\\mathbb{N}}$ be the Hilbert cube and let $T \\subset \\mathbb{R}^2$ be the letter-T curve. Both parts of Borsuk's Problem 175 from the Scottish Book have affirmative answers: (a) the product $Q \\times T$ is homeomorphic to $Q$, and (b) the product $T^{\\mathbb{N}}$ of countably many copies of $T$ is homeomorphic to $Q$.","title":"The Hilbert cube absorbs the letter T (Borsuk's Problem 175)"},{"data_url":"scopes/Topology/continua-infinite-dimensional/plane-fixed-point-problem.json","id":"Topology/continua-infinite-dimensional/plane-fixed-point-problem","kind":"scope","navigation":"module","parent":15454,"result":false,"route":"../roadmap/Topology/continua-infinite-dimensional/plane-fixed-point-problem/index.html","scope":"Topology/continua-infinite-dimensional/plane-fixed-point-problem","status":"can_prove","title":"Plane fixed point problem"},{"id":"Topology/continua-infinite-dimensional/plane-fixed-point-problem/has-fixed-point-property","kind":"node","lean":"MathlibExt.Topology.NonseparatingPlanarContinuumWanted.HasFixedPointProperty","parent":15469,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/plane-fixed-point-problem/index.html#has-fixed-point-property","scope":"Topology/continua-infinite-dimensional/plane-fixed-point-problem","status":"fully_proved","summary":"A subset $K$ of the Euclidean plane $\\mathbb{R}^2$ has the fixed point property when every continuous map $f \\colon K \\to K$ has a fixed point, that is, a point $x \\in K$ with $f(x) = x$.","title":"Fixed point property"},{"id":"Topology/continua-infinite-dimensional/plane-fixed-point-problem/conjecture","kind":"node","lean":"MathlibExt.Topology.NonseparatingPlanarContinuumWanted.conjecture","parent":15469,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/plane-fixed-point-problem/index.html#conjecture","scope":"Topology/continua-infinite-dimensional/plane-fixed-point-problem","status":"fully_proved","summary":"The conjecture asserts that every nonempty, compact, path-connected subset $K$ of the Euclidean plane $\\mathbb{R}^2$ whose complement $\\mathbb{R}^2 \\setminus K$ is connected (that is, $K$ does not separate the plane) has the fixed point property: every continuous map $f \\colon K \\to K$ has a point $x \\in K$ with $f(x) = x$.","title":"Nonseparating planar continuum conjecture"},{"id":"Topology/continua-infinite-dimensional/plane-fixed-point-problem/conjecture-holds","kind":"node","lean":"MathlibExt.Topology.NonseparatingPlanarContinuumWanted.conjecture_holds","parent":15469,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/plane-fixed-point-problem/index.html#conjecture-holds","scope":"Topology/continua-infinite-dimensional/plane-fixed-point-problem","status":"can_prove","summary":"Let $K$ be a nonempty, compact, path-connected subset of the Euclidean plane $\\mathbb{R}^2$ whose complement $\\mathbb{R}^2 \\setminus K$ is connected. Then $K$ has the fixed point property: every continuous map $f \\colon K \\to K$ has a point $x \\in K$ with $f(x) = x$.","title":"Path-connected nonseparating plane continua have the fixed point property"},{"data_url":"scopes/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb.json","id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","kind":"scope","navigation":"module","parent":15454,"result":false,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"can_prove","title":"Topology of infinite-dimensional Banach and Hilbert spaces"},{"id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/scottish-book-problem167-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook167Wanted.ScottishBookProblem167Claim","parent":15473,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html#scottish-book-problem167-claim","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"fully_proved","summary":"Ulam's claim that for every infinite-dimensional real Hilbert space $H$ with unit sphere $S = \\{p \\in H : \\|p\\| = 1\\}$, every $n \\in \\mathbb{N}$, all continuous functions $f_1, \\dots, f_n : S \\to \\mathbb{R}$ and every continuous map $T : S \\to S$, there is a point $p_0 \\in S$ with $f_i(T(p_0)) = f_i(p_0)$ for every $i$.","title":"Scottish Book Problem 167"},{"id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/ulam167-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook167Wanted.ulam167_refuted","parent":15473,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html#ulam167-refuted","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"can_prove","summary":"Ulam's claim in Problem 167 is false: there are an infinite-dimensional real Hilbert space $H$ with unit sphere $S$, a number $n \\in \\mathbb{N}$, continuous functions $f_1, \\dots, f_n : S \\to \\mathbb{R}$ and a continuous map $T : S \\to S$ such that no point $p_0 \\in S$ satisfies $f_i(T(p_0)) = f_i(p_0)$ for every $i$.","title":"Negative answer to Scottish Book Problem 167"},{"id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/projective-hilbert-space","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook35Wanted.ProjectiveHilbertSpace","parent":15473,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html#projective-hilbert-space","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"fully_proved","summary":"Let $E$ be a normed additive commutative group, in practice a real inner product space. The projective Hilbert space of $E$ is the quotient of the unit sphere $\\{x \\in E : \\|x\\| = 1\\}$ by the relation identifying each $x$ with $-x$. Its points are the antipodal pairs $\\{x, -x\\}$ of unit vectors, that is, the diameters of the unit sphere.","title":"Projective Hilbert space"},{"id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/projective-hilbert-space-homeomorphic-to","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook35Wanted.ProjectiveHilbertSpaceHomeomorphicToHilbertSpace","parent":15473,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html#projective-hilbert-space-homeomorphic-to","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"fully_proved","summary":"For a real inner product space $E$, the statement that the projective Hilbert space of $E$, the unit sphere $\\{x \\in E : \\|x\\| = 1\\}$ with each $x$ identified with $-x$ and given the quotient topology, is homeomorphic to $E$.","title":"Projective Hilbert space homeomorphic to the space itself"},{"id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/projective-hilbert-space-not-homeomorphi","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook35Wanted.projective_hilbert_space_not_homeomorphic_to_hilbert_space","parent":15473,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html#projective-hilbert-space-not-homeomorphi","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"can_prove","summary":"Let $E$ be an infinite-dimensional real Hilbert space. Then the projective Hilbert space of $E$, the unit sphere $\\{x \\in E : \\|x\\| = 1\\}$ with each $x$ identified with $-x$ and given the quotient topology, is not homeomorphic to $E$.","title":"Projective Hilbert space is not homeomorphic to Hilbert space"},{"id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/hilbert-ball-retracts-onto-sphere","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook36Wanted.hilbertBallRetractsOntoSphere","parent":15473,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html#hilbert-ball-retracts-onto-sphere","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"can_prove","summary":"Let $E$ be an infinite-dimensional real Hilbert space and let $S = \\{x \\in E : \\|x\\| = 1\\}$ be its unit sphere. Then there is a continuous map $f : E \\to E$ such that $f(x) \\in S$ for every $x \\in E$ and $f(x) = x$ for every $x \\in S$.","title":"Retraction onto the unit sphere of Hilbert space"},{"id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/scottish-book-problem7-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook7Wanted.ScottishBookProblem7Claim","parent":15473,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html#scottish-book-problem7-claim","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"fully_proved","summary":"The claim that in every real Banach space $E$, any two closed convex subsets $A, B \\subseteq E$ whose linear spans are infinite-dimensional are homeomorphic, with the subspace topologies.","title":"Scottish Book Problem 7 (Mazur\u2013Banach)"},{"id":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/closed-convex-sets-homeomorphic-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook7Wanted.closedConvexSetsHomeomorphic_refuted","parent":15473,"result":true,"route":"../roadmap/Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb/index.html#closed-convex-sets-homeomorphic-refuted","scope":"Topology/continua-infinite-dimensional/topology-of-infinite-dimensional-banach-and-hilb","status":"can_prove","summary":"The Scottish Book Problem 7 claim is false: there are a real Banach space $E$ and closed convex subsets $A, B \\subseteq E$, each with infinite-dimensional linear span, that are not homeomorphic.","title":"Scottish Book Problem 7 fails: closed convex sets need not be homeomorphic"},{"data_url":"scopes/Topology/topological-groups.json","id":"Topology/topological-groups","kind":"scope","parent":15066,"result":false,"route":"../roadmap/Topology/topological-groups/index.html","scope":"Topology/topological-groups","status":"fully_proved","summary":"Groups carrying a compatible topology: metrizability, profinite groups and profinite completions, and functions on locally compact groups.","title":"Topological groups"},{"id":"Topology/topological-groups/birkhoff-kakutani-metrization","kind":"node","lean":"MathlibExt.Topology.Algebra.BirkhoffKakutani.birkhoff_kakutani_metrization","parent":15482,"result":true,"route":"../roadmap/Topology/topological-groups/index.html#birkhoff-kakutani-metrization","scope":"Topology/topological-groups","status":"fully_proved","summary":"Every Hausdorff first-countable topological group is metrizable.","title":"Birkhoff\u2013Kakutani metrization theorem"},{"id":"Topology/topological-groups/exists-is-open-one-mem-forall-mul-eq-of","kind":"node","lean":"IsLocallyConstant.exists_isOpen_one_mem_forall_mul_eq_of_hasCompactSupport","parent":15482,"result":true,"route":"../roadmap/Topology/topological-groups/index.html#exists-is-open-one-mem-forall-mul-eq-of","scope":"Topology/topological-groups","status":"fully_proved","summary":"A locally constant compactly supported function on a topological group is invariant under left and right multiplication by elements of some open neighborhood of `1`.","title":"Locally constant compactly supported functions have uniform stabilizers"},{"data_url":"scopes/Topology/topological-groups/profinite-completion-of-topological-groups.json","id":"Topology/topological-groups/profinite-completion-of-topological-groups","kind":"scope","navigation":"module","parent":15482,"result":false,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","title":"Profinite completion of topological groups"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/index-2","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.Index","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#index-2","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"For a group $G$ with a topology, this is the set of normal subgroups $H$ of $G$ that have finite index and are open in $G$, ordered by inclusion and viewed as a category: there is a morphism $H \\to K$ exactly when $H \\subseteq K$.","title":"Open finite-index normal subgroups"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/finite-grp-diagram","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.finiteGrpDiagram","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#finite-grp-diagram","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"For a group $G$ with a topology, this is the functor from the category of open finite-index normal subgroups of $G$ to finite groups that sends $H$ to the quotient group $G/H$ and sends an inclusion $H \\subseteq K$ to the projection $G/H \\to G/K$, $gH \\mapsto gK$.","title":"Diagram of finite quotients"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/diagram","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.diagram","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#diagram","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"For a group $G$ with a topology, this is the functor that sends each open finite-index normal subgroup $H$ of $G$ to the finite quotient group $G/H$ with the discrete topology, regarded as a profinite group, and each inclusion $H \\subseteq K$ to the projection $G/H \\to G/K$, $gH \\mapsto gK$. It is the diagram of finite quotients followed by the inclusion of finite groups into profinite groups.","title":"Diagram of finite quotients as profinite groups"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/completion","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.completion","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#completion","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"For a group $G$ with a topology, the open profinite completion of $G$ is the limit, in profinite groups, of the diagram of finite quotients $G/H$, where $H$ runs over the open finite-index normal subgroups of $G$ and the transition maps are the projections $G/H \\to G/K$ for $H \\subseteq K$. Concretely, it is the group of families $(x_H)_H$ with $x_H \\in G/H$ that are compatible under these projections, with the\u2026","title":"Open profinite completion of a group"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/eta-fn","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.etaFn","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#eta-fn","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"For a group $G$ with a topology, this is the map from $G$ to its open profinite completion that sends $x \\in G$ to the family of cosets $(xH)_H$, where $H$ runs over the open finite-index normal subgroups of $G$.","title":"Canonical map into the open profinite completion"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/eta-fn-apply","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.etaFn_apply","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#eta-fn-apply","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"Let $G$ be a group with a topology. For every $x \\in G$ and every open finite-index normal subgroup $H$ of $G$, the $H$-coordinate of the image of $x$ under the canonical map into the open profinite completion of $G$ is the coset $xH \\in G/H$.","title":"Coordinates of the canonical map"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/eta-fn-continuous","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.etaFn_continuous","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#eta-fn-continuous","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"Let $G$ be a group with a topology in which multiplication is separately continuous: $x \\mapsto ax$ and $x \\mapsto xa$ are continuous for every $a \\in G$. Then the canonical map from $G$ to its open profinite completion is continuous.","title":"Continuity of the canonical map"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/eta","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.eta","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#eta","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"Let $G$ be a group with a topology in which multiplication is separately continuous. This is the canonical map from $G$ to its open profinite completion, $x \\mapsto (xH)_H$ with $H$ running over the open finite-index normal subgroups of $G$, regarded as a continuous group homomorphism.","title":"Canonical homomorphism into the open profinite completion"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/dense-range-eta-fn","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.denseRange_etaFn","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#dense-range-eta-fn","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"For every group $G$ with a topology, the image of the canonical map from $G$ to its open profinite completion is dense.","title":"The canonical map has dense image"},{"id":"Topology/topological-groups/profinite-completion-of-topological-groups/dense-range-eta","kind":"node","lean":"ProfiniteGrp.OpenProfiniteCompletion.denseRange_eta","parent":15485,"result":true,"route":"../roadmap/Topology/topological-groups/profinite-completion-of-topological-groups/index.html#dense-range-eta","scope":"Topology/topological-groups/profinite-completion-of-topological-groups","status":"fully_proved","summary":"Let $G$ be a group with a topology in which multiplication is separately continuous. Then the canonical continuous homomorphism from $G$ to its open profinite completion has dense image.","title":"Density of the canonical homomorphism"},{"area":"Analysis & Probability","data_url":"chapters/Analysis.json","id":"Analysis","kind":"scope","parent":0,"result":false,"route":"../roadmap/Analysis/index.html","scope":"Analysis","status":"can_state","summary":"Analysis studies limits, approximation, continuity, and infinite-dimensional spaces of functions and operators.","title":"Analysis"},{"data_url":"scopes/Analysis/operator-theory.json","id":"Analysis/operator-theory","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/operator-theory/index.html","scope":"Analysis/operator-theory","status":"can_state","summary":"Bounded linear operators on Banach and Hilbert spaces: spectra, compact and Fredholm operators, shifts and other concrete operators, and eigenvalue inequalities.","title":"Operator theory"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory.json","id":"Analysis/operator-theory/spectral-theory","kind":"scope","parent":15497,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/index.html","scope":"Analysis/operator-theory/spectral-theory","status":"can_state","summary":"Spectra and spectral theorems: self-adjoint, normal and compact self-adjoint spectral theorems, approximate and essential spectrum, numerical range and radius, Cayley transform, spectra of finite-rank and rank-one operators, almost Mathieu operator.","title":"Spectral theory"},{"id":"Analysis/operator-theory/spectral-theory/krein-rutman","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KreinRutmanWanted.krein_rutman","parent":15498,"reading_title":{"format":"inline-parts-v1","parts":[["text","Krein\u2013Rutman theorem for "],["math","\\(C(K)\\)"]],"source":"Krein\u2013Rutman theorem for $C(K)$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/index.html#krein-rutman","scope":"Analysis/operator-theory/spectral-theory","status":"fully_proved","summary":"Let $K$ be a compact Hausdorff space and let $T$ be a compact bounded linear operator on the real Banach space $C(K, \\mathbb{R})$ of continuous real functions on $K$ that is positive: $Tf \\ge 0$ pointwise whenever $f \\ge 0$ pointwise. Let $r(T)$ be the spectral radius of $T$, the supremum of $|\\lambda|$ over the real spectrum of $T$. If $r(T) > 0$, then there is a nonzero $f \\in C(K, \\mathbb{R})$ with $f \\ge 0$\u2026","title":"Krein\u2013Rutman theorem for $C(K)$"},{"id":"Analysis/operator-theory/spectral-theory/compact-normal-hilbert-basis","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSpectralWanted.compactNormal_hilbertBasis","parent":15498,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/index.html#compact-normal-hilbert-basis","scope":"Analysis/operator-theory/spectral-theory","status":"fully_proved","summary":"Let $E$ be a complex Hilbert space and $T$ a compact normal operator on $E$. Then there are a Hilbert basis $(b_i)_{i \\in I}$ of $E$ and complex numbers $(\\mu_i)_{i \\in I}$ such that $T b_i = \\mu_i b_i$ for every $i$, and the eigenvalues vanish at infinity: for every $\\varepsilon > 0$ the set of $i$ with $|\\mu_i| \\ge \\varepsilon$ is finite.","title":"Spectral theorem for compact normal operators"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/almost-mathieu-operator.json","id":"Analysis/operator-theory/spectral-theory/almost-mathieu-operator","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/almost-mathieu-operator/index.html","scope":"Analysis/operator-theory/spectral-theory/almost-mathieu-operator","status":"can_prove","title":"Almost Mathieu operator"},{"id":"Analysis/operator-theory/spectral-theory/almost-mathieu-operator/ell-two","kind":"node","lean":"MathlibExt.Analysis.TenMartiniWanted.Analysis.TenMartini.EllTwo","parent":15501,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sequence space "],["math","\\(\\ell^2(\\mathbb{Z})\\)"]],"source":"The sequence space $\\ell^2(\\mathbb{Z})$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/almost-mathieu-operator/index.html#ell-two","scope":"Analysis/operator-theory/spectral-theory/almost-mathieu-operator","status":"fully_proved","summary":"We write $\\ell^2(\\mathbb{Z})$ for the Hilbert space of square-summable sequences $(u_n)_{n \\in \\mathbb{Z}}$ of complex numbers.","title":"The sequence space $\\ell^2(\\mathbb{Z})$"},{"id":"Analysis/operator-theory/spectral-theory/almost-mathieu-operator/is-almost-mathieu-operator","kind":"node","lean":"MathlibExt.Analysis.TenMartiniWanted.Analysis.TenMartini.IsAlmostMathieuOperator","parent":15501,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/almost-mathieu-operator/index.html#is-almost-mathieu-operator","scope":"Analysis/operator-theory/spectral-theory/almost-mathieu-operator","status":"fully_proved","summary":"Let $\\alpha$, $\\lambda$ and $\\theta$ be real numbers. A bounded complex-linear operator $H$ on $\\ell^2(\\mathbb{Z})$ is an almost Mathieu operator with frequency $\\alpha$, coupling $\\lambda$ and phase $\\theta$ when, for every $u \\in \\ell^2(\\mathbb{Z})$ and every $n \\in \\mathbb{Z}$,","title":"Almost Mathieu operator"},{"id":"Analysis/operator-theory/spectral-theory/almost-mathieu-operator/ten-martini-problem","kind":"node","lean":"MathlibExt.Analysis.TenMartiniWanted.Analysis.TenMartini.ten_martini_problem","parent":15501,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/almost-mathieu-operator/index.html#ten-martini-problem","scope":"Analysis/operator-theory/spectral-theory/almost-mathieu-operator","status":"can_prove","summary":"Let $\\alpha$, $\\lambda$ and $\\theta$ be real numbers with $\\alpha$ irrational and $\\lambda \\neq 0$. Then there is an almost Mathieu operator $H$ on $\\ell^2(\\mathbb{Z})$ with frequency $\\alpha$, coupling $\\lambda$ and phase $\\theta$, and the spectrum of $H$ is nonempty, compact, perfect and totally disconnected, that is, a Cantor set.","title":"Ten Martini theorem (Avila\u2013Jitomirskaya)"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/approximate-point-spectrum.json","id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","title":"Approximate point spectrum"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/is-approximate-eigenvalue","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.IsApproximateEigenvalue","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#is-approximate-eigenvalue","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"`\u03bc` is an approximate eigenvalue of `T` if there is a sequence of unit vectors `x\u2099` with `(\u03bc \u2022 1 - T) x\u2099 \u2192 0`.","title":"Approximate eigenvalue of an operator"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/is-approximate-eigenvalue-iff-forall-eps","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.isApproximateEigenvalue_iff_forall_eps","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#is-approximate-eigenvalue-iff-forall-eps","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"The `\u2200 \u03b5 > 0` characterization of approximate eigenvalues.","title":"Epsilon characterization of approximate eigenvalues"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/is-star-normal-smul-one-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.isStarNormal_smul_one_sub","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#is-star-normal-smul-one-sub","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"Shifting by a scalar preserves normality.","title":"Scalar shifts of a normal operator are normal"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/is-approximate-eigenvalue-of-mem-spectru","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.isApproximateEigenvalue_of_mem_spectrum","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#is-approximate-eigenvalue-of-mem-spectru","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"Every spectral value of a normal operator is an approximate eigenvalue.","title":"Spectral values of a normal operator are approximate eigenvalues"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/mem-spectrum-of-is-approximate-eigenvalu","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.mem_spectrum_of_isApproximateEigenvalue","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#mem-spectrum-of-is-approximate-eigenvalu","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"Every approximate eigenvalue is a spectral value (no normality needed).","title":"Approximate eigenvalues lie in the spectrum"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/mem-spectrum-iff-is-approximate-eigenval","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.mem_spectrum_iff_isApproximateEigenvalue","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#mem-spectrum-iff-is-approximate-eigenval","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"For a normal operator, the spectrum is exactly the approximate point spectrum.","title":"Spectrum of a normal operator equals its approximate point spectrum"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/mem-spectrum-iff-is-approximate-eigenval-2","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.mem_spectrum_iff_isApproximateEigenvalue_of_isSelfAdjoint","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#mem-spectrum-iff-is-approximate-eigenval-2","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"Weyl's criterion: for a self-adjoint operator, the spectrum is exactly the approximate point spectrum.","title":"Weyl's criterion for self-adjoint operators"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/is-approximate-eigenvalue-of-has-eigenva","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.isApproximateEigenvalue_of_hasEigenvalue","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#is-approximate-eigenvalue-of-has-eigenva","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"Exact eigenvalues are approximate eigenvalues.","title":"Eigenvalues are approximate eigenvalues"},{"id":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum/dense-range-of-is-star-normal-of-ker-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormalApproximateSpectrum.denseRange_of_isStarNormal_of_ker_eq_bot","parent":15505,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/approximate-point-spectrum/index.html#dense-range-of-is-star-normal-of-ker-eq","scope":"Analysis/operator-theory/spectral-theory/approximate-point-spectrum","status":"fully_proved","summary":"A normal operator with trivial kernel has dense range: the residual spectrum of a normal operator is empty.","title":"Normal operator with trivial kernel has dense range"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/cayley-transform.json","id":"Analysis/operator-theory/spectral-theory/cayley-transform","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","title":"Cayley transform"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/is-unit-smul-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.isUnit_smul_one","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#is-unit-smul-one","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"A nonzero scalar operator is a unit.","title":"Nonzero scalar operators are units"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/ring-inverse-smul-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.ringInverse_smul_one","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#ring-inverse-smul-one","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The ring inverse of a scalar operator is the scalar operator of the inverse scalar.","title":"Inverse of a scalar operator"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The Cayley transform of an operator `A`: `(A - i) * (A + i)\u207b\u00b9`. The factor `A + i \u2022 1` is a unit when `A` is self-adjoint on a complete Hilbert space (see `isUnit_add_I_smul_one_of_isSelfAdjoint` below), in which case the ring inverse here is the genuine operator inverse.","title":"Cayley transform of an operator"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/commute-sub-i-add-i","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.commute_sub_I_add_I","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#commute-sub-i-add-i","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The two factors of the Cayley transform commute.","title":"Factors of the Cayley transform commute"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/is-unit-sub-smul-one-of-is-self-adjoint","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.isUnit_sub_smul_one_of_isSelfAdjoint_of_im_ne_zero","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#is-unit-sub-smul-one-of-is-self-adjoint","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"For a self-adjoint operator, `A - w \u2022 1` is a unit whenever `w` has nonzero imaginary part. The proof is the norm estimate `\u2016(A - w \u2022 1) x\u2016 \u2265 |w.im| * \u2016x\u2016`, read off from the imaginary part of `\u27ea(A - w \u2022 1) x, x\u27eb`.","title":"Self-adjoint operator minus a non-real scalar is invertible"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/is-unit-add-i-smul-one-of-is-self-adjoin","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.isUnit_add_I_smul_one_of_isSelfAdjoint","parent":15515,"reading_title":{"format":"inline-parts-v1","parts":[["text","Self-adjoint operator plus "],["math","\\(i\\)"],["text"," is invertible"]],"source":"Self-adjoint operator plus $i$ is invertible"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#is-unit-add-i-smul-one-of-is-self-adjoin","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"For a self-adjoint operator, `A + i \u2022 1` is a unit.","title":"Self-adjoint operator plus $i$ is invertible"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/is-unit-sub-i-smul-one-of-is-self-adjoin","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.isUnit_sub_I_smul_one_of_isSelfAdjoint","parent":15515,"reading_title":{"format":"inline-parts-v1","parts":[["text","Self-adjoint operator minus "],["math","\\(i\\)"],["text"," is invertible"]],"source":"Self-adjoint operator minus $i$ is invertible"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#is-unit-sub-i-smul-one-of-is-self-adjoin","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"For a self-adjoint operator, `A - i \u2022 1` is a unit.","title":"Self-adjoint operator minus $i$ is invertible"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/is-unit-ring-inverse-add-i-of-is-self-ad","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.isUnit_ringInverse_add_I_of_isSelfAdjoint","parent":15515,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inverse of a self-adjoint operator plus "],["math","\\(i\\)"],["text"," is a unit"]],"source":"Inverse of a self-adjoint operator plus $i$ is a unit"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#is-unit-ring-inverse-add-i-of-is-self-ad","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The inverse of `A + i \u2022 1` is a unit for self-adjoint `A`.","title":"Inverse of a self-adjoint operator plus $i$ is a unit"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/star-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.star_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#star-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The adjoint of the Cayley transform.","title":"Adjoint of the Cayley transform"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/star-mul-self-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.star_mul_self_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#star-mul-self-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The Cayley transform of a self-adjoint operator satisfies `U\u2020 * U = 1`.","title":"Adjoint of the Cayley transform is a left inverse"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/mul-star-self-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.mul_star_self_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#mul-star-self-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The Cayley transform of a self-adjoint operator satisfies `U * U\u2020 = 1`.","title":"Adjoint of the Cayley transform is a right inverse"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/is-unit-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.isUnit_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#is-unit-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The Cayley transform of a self-adjoint operator is a unit.","title":"Cayley transform of a self-adjoint operator is invertible"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/mem-unitary-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.mem_unitary_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#mem-unitary-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The Cayley transform of a self-adjoint operator is unitary.","title":"Cayley transform of a self-adjoint operator is unitary"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/norm-cayley-transform-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.norm_cayleyTransform_apply","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#norm-cayley-transform-apply","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The Cayley transform of a self-adjoint operator is an isometry.","title":"Cayley transform preserves norms of vectors"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/norm-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.norm_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#norm-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"On a nontrivial space, the Cayley transform has operator norm `1`.","title":"Operator norm of the Cayley transform is one"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/one-add-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.one_add_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#one-add-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"`1 + U` is `2 \u2022 A` times the inverse of `A + i \u2022 1`.","title":"Formula for one plus the Cayley transform"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/one-sub-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.one_sub_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#one-sub-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"`1 - U` is `2 \u2022 (i \u2022 1)` times the inverse of `A + i \u2022 1`.","title":"Formula for one minus the Cayley transform"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/is-unit-one-sub-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.isUnit_one_sub_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#is-unit-one-sub-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"`1 - U` is a unit for the Cayley transform of a self-adjoint operator: `1` is never in the spectrum of `U`.","title":"One minus the Cayley transform is invertible"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/eq-cayley-transform-inv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.eq_cayleyTransform_inv","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#eq-cayley-transform-inv","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The inverse Cayley formula: `A = i \u2022 (1 + U) * (1 - U)\u207b\u00b9`.","title":"Inverse Cayley formula"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/algebra-map-sub-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.algebraMap_sub_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#algebra-map-sub-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The resolvent-type factorization behind the spectral mapping: for `\u03bc \u2260 1` and `\u03bb = i * (1 + \u03bc) / (1 - \u03bc)`, `\u03bc \u2022 1 - U = ((\u03bc - 1) \u2022 (A - \u03bb \u2022 1)) * (A + i \u2022 1)\u207b\u00b9`.","title":"Resolvent factorization for the Cayley transform"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/one-not-mem-spectrum-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.one_notMem_spectrum_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#one-not-mem-spectrum-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"`1` is never in the spectrum of the Cayley transform.","title":"One is not in the spectrum of the Cayley transform"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/mem-spectrum-cayley-transform-of-mem-spe","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.mem_spectrum_cayleyTransform_of_mem_spectrum","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#mem-spectrum-cayley-transform-of-mem-spe","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The image direction of the spectral mapping: if `w` is in the spectrum of `A`, then `(w - i) / (w + i)` is in the spectrum of `U`.","title":"Spectral mapping for the Cayley transform, image direction"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/exists-mem-spectrum-of-mem-spectrum-cayl","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.exists_mem_spectrum_of_mem_spectrum_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#exists-mem-spectrum-of-mem-spectrum-cayl","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The preimage direction of the spectral mapping: every point of the spectrum of `U` is `(w - i) / (w + i)` for some `w` in the spectrum of `A`.","title":"Spectral mapping for the Cayley transform, preimage direction"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/spectrum-cayley-transform","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.spectrum_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#spectrum-cayley-transform","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The spectrum of the Cayley transform is the image of the spectrum of `A` under the M\u00f6bius map `\u03bb \u21a6 (\u03bb - i) / (\u03bb + i)`.","title":"Spectrum of the Cayley transform"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/norm-eq-one-of-mem-spectrum-cayley-trans","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.norm_eq_one_of_mem_spectrum_cayleyTransform","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#norm-eq-one-of-mem-spectrum-cayley-trans","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"Points of the spectrum of the Cayley transform lie on the unit circle.","title":"Spectrum of the Cayley transform lies on the unit circle"},{"id":"Analysis/operator-theory/spectral-theory/cayley-transform/has-eigenvalue-cayley-transform-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CayleyTransform.ContinuousLinearMap.hasEigenvalue_cayleyTransform_iff","parent":15515,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/cayley-transform/index.html#has-eigenvalue-cayley-transform-iff","scope":"Analysis/operator-theory/spectral-theory/cayley-transform","status":"fully_proved","summary":"The eigenvalue correspondence: for `\u03bc \u2260 1`, `\u03bc` is an eigenvalue of the Cayley transform `U` iff `i * (1 + \u03bc) / (1 - \u03bc)` is an eigenvalue of `A`.","title":"Eigenvalues of the Cayley transform"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases.json","id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"can_state","title":"Ground-state energy of dilute Bose gases"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/box","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.box","parent":15542,"reading_title":{"format":"inline-parts-v1","parts":[["text","The box "],["math","\\(((0,L)^3)^N\\)"]],"source":"The box $((0,L)^3)^N$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#box","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"fully_proved","summary":"For a natural number $N$ and a real number $L$, the box $([0,L]^3)^N$ is the set of configurations $(x_1, \\dots, x_N)$ of $N$ points of $\\mathbb{R}^3$ such that every coordinate of every point $x_i$ lies in the closed interval $[0, L]$.","title":"The box $((0,L)^3)^N$"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/cinfty0-box","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.CInfty0Box","parent":15542,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#cinfty0-box","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"fully_proved","summary":"For a natural number $N$ and a real number $L$, $C_0^\\infty(([0,L]^3)^N) \\setminus \\{0\\}$ consists of the functions $\\Psi : (\\mathbb{R}^3)^N \\to \\mathbb{R}$ that are smooth on the box $([0,L]^3)^N$, whose closed support is contained in the interior of the box, and that are not identically zero.","title":"Smooth functions supported inside the box"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/is-radial","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.IsRadial","parent":15542,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#is-radial","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"fully_proved","summary":"A function $v : \\mathbb{R}^3 \\to \\mathbb{R}$ is radial if $v(x) = v(y)$ for all $x, y \\in \\mathbb{R}^3$ with $\\|x\\| = \\|y\\|$.","title":"Radial functions"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/is-scattering-solution","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.IsScatteringSolution","parent":15542,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#is-scattering-solution","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"fully_proved","summary":"Let $v, w : \\mathbb{R}^3 \\to \\mathbb{R}$. The function $w$ is a scattering solution for the potential $v$ if $w$ and $v\\,(1 - w)$ are locally integrable, $w(x) \\to 0$ as $\\|x\\| \\to \\infty$, and $w$ solves the scattering equation $-\\Delta w = \\frac{1}{2} v\\,(1 - w)$ (Equation (3.4) of Fournais\u2013Solovej) in the weak sense: for every smooth, compactly supported $\\varphi : \\mathbb{R}^3 \\to \\mathbb{R}$,","title":"Scattering solutions"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/a","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.a","parent":15542,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#a","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"fully_proved","summary":"Let $v : \\mathbb{R}^3 \\to \\mathbb{R}$ be a potential. If there is a radial scattering solution $w$ to the scattering equation with potential $v$, the scattering length of $v$ is $a(v) = (8\\pi)^{-1} \\int_{\\mathbb{R}^3} v(x)\\,(1 - w(x))\\,dx$, where $w$ is one such solution, chosen arbitrarily; if there is no radial scattering solution, $a(v) = 0$.","title":"Scattering length"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/untitled","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.\ud835\udce1","parent":15542,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ratio "],["math","\\(\\mathcal{R}(v)\\)"]],"source":"The ratio $\\mathcal{R}(v)$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#untitled","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"fully_proved","summary":"For a potential $v : \\mathbb{R}^3 \\to \\mathbb{R}$ with scattering length $a(v)$, let $\\mathcal{R}(v) = \\int_{\\mathbb{R}^3} v(x)/(8\\pi a(v))\\,dx$. When $a(v) \\neq 0$ this is the ratio of $(8\\pi)^{-1} \\int v$ to $a(v)$.","title":"The ratio $\\mathcal{R}(v)$"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/untitled-2","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.\u03b7","parent":15542,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#untitled-2","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"can_state","summary":"$\\eta$ is a real constant, not given explicitly, that serves as the exponent in the error term $(\\rho a^3)^{1/2 + \\eta}$ of the Lee\u2013Huang\u2013Yang lower bound (Theorem 1.2), where $\\rho$ is the density and $a$ the scattering length of the potential; that theorem asserts $\\eta > 0$.","title":"Lee\u2013Huang\u2013Yang error exponent"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/untitled-3","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.\ud835\udcd2","parent":15542,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#untitled-3","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"can_state","summary":"$\\mathcal{C}$ is a real-valued function of two real variables, not given explicitly. Let $v$ be a potential with scattering length $a$ whose closed support lies in the open ball $B(0, R)$, and let $\\mathcal{R}(v) = \\int v/(8\\pi a)$. In the Lee\u2013Huang\u2013Yang lower bound (Theorem 1.2) for $v$ at density $\\rho$, the error term is $\\mathcal{C}(\\mathcal{R}(v), R/a)\\,(\\rho a^3)^{1/2 + \\eta}$, where $\\eta$ is the\u2026","title":"Lee\u2013Huang\u2013Yang error constant"},{"id":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/theorem-1-2","kind":"node","lean":"MathlibExt.Geometry.BoseGasesWanted.BoseGases.theorem_1_2","parent":15542,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases/index.html#theorem-1-2","scope":"Analysis/operator-theory/spectral-theory/ground-state-energy-of-dilute-bose-gases","status":"stated","summary":"There are a real number $\\eta$ and a function $\\mathcal{C}$ of two real variables such that the following holds. Let $v : \\mathbb{R}^3 \\to \\mathbb{R}$ be a non-negative, radial, integrable potential that is not almost everywhere zero, and let $R > 0$ be such that the closed support of $v$ lies in the open ball $B(0, R)$. Let $a$ be the scattering length of $v$, let $\\mathcal{R}(v) = \\int v/(8\\pi a)$, and let\u2026","title":"Lee\u2013Huang\u2013Yang lower bound for dilute Bose gases"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius.json","id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"can_prove","title":"Numerical range and numerical radius"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/numerical-radius","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#numerical-radius","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"The numerical radius of a bounded linear operator $T$ on a complex Hilbert space $H$ is the real number $w(T) = \\sup \\{ |\\langle T x, x \\rangle| : x \\in H,\\ \\|x\\| = 1 \\}$, the supremum of the moduli of the values $\\langle T x, x \\rangle$ over the unit vectors of $H$ (taken to be $0$ when $H = 0$).","title":"Numerical radius"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/cartesian-real-part","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.cartesianRealPart","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#cartesian-real-part","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For a bounded linear operator $T$ on a complex Hilbert space with adjoint $T^*$, the real Cartesian part of $T$ is the bounded linear operator $(T + T^*)/2$.","title":"Real Cartesian part of an operator"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/cartesian-imag-part","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.cartesianImagPart","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#cartesian-imag-part","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For a bounded linear operator $T$ on a complex Hilbert space with adjoint $T^*$, the imaginary Cartesian part of $T$ is the bounded linear operator $(T - T^*)/(2i)$.","title":"Imaginary Cartesian part of an operator"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/bdd-above-numerical-radius","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.bddAbove_numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#bdd-above-numerical-radius","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded linear operator on a complex inner product space $H$. The set $\\{|\\langle Tx, x\\rangle| : x \\in H,\\ \\|x\\| = 1\\}$ is bounded above.","title":"The numerical-radius family is bounded above"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/numerical-radius-nonneg","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.numericalRadius_nonneg","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#numerical-radius-nonneg","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For every bounded operator $T$ on a complex inner product space, the numerical radius $w(T)$ of $T$ satisfies $w(T) \\ge 0$.","title":"Nonnegativity of the numerical radius"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/le-numerical-radius","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.le_numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#le-numerical-radius","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded linear operator on a complex inner product space $H$, with numerical radius $w(T)$. For every $x \\in H$ with $\\|x\\| = 1$, $|\\langle Tx, x\\rangle| \\le w(T)$.","title":"The numerical radius bounds the quadratic form on unit vectors"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/numerical-radius-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.numericalRadius_le","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#numerical-radius-le","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded operator on a complex inner product space $H$ and let $c \\ge 0$ be a real number. If $|\\langle T x, x\\rangle| \\le c$ for every unit vector $x \\in H$, then the numerical radius of $T$ satisfies $w(T) \\le c$.","title":"Bounding the numerical radius from above"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/numerical-radius-le-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.numericalRadius_le_norm","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#numerical-radius-le-norm","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For every bounded operator $T$ on a complex inner product space, the numerical radius $w(T)$ is at most the operator norm: $w(T) \\le \\|T\\|$.","title":"Numerical radius at most the operator norm"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/quad-form-scale","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.quadForm_scale","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#quad-form-scale","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded operator on a complex inner product space $H$. For every $x \\in H$, $\\langle T(\\|x\\|^{-1} x), \\|x\\|^{-1} x \\rangle = \\|x\\|^{-2} \\langle T x, x \\rangle$, where $0^{-1}$ is read as $0$, so that for $x = 0$ both sides vanish.","title":"Quadratic form at the normalized vector"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/norm-normalize","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.norm_normalize","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#norm-normalize","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $x$ be a nonzero vector in a complex inner product space. Then the vector $\\|x\\|^{-1} x$ has norm $1$.","title":"Normalizing a nonzero vector gives a unit vector"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/abs-re-quad-form-div-le-numerical-radius","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.abs_re_quadForm_div_le_numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#abs-re-quad-form-div-le-numerical-radius","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded linear operator on a complex inner product space $H$, with numerical radius $w(T)$. For every $x \\in H$, $|\\operatorname{Re} \\langle Tx, x\\rangle| / \\|x\\|^2 \\le w(T)$, where division by zero is taken to be $0$, so the case $x = 0$ is trivial.","title":"Numerical radius bounds the normalized real part of the quadratic form"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/abs-im-quad-form-div-le-numerical-radius","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.abs_im_quadForm_div_le_numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#abs-im-quad-form-div-le-numerical-radius","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded linear operator on a complex inner product space $H$, with numerical radius $w(T)$. For every $x \\in H$, $|\\operatorname{Im} \\langle Tx, x\\rangle| / \\|x\\|^2 \\le w(T)$, where division by zero is taken to be $0$, so the case $x = 0$ is trivial.","title":"Numerical radius bounds the normalized imaginary part of the quadratic form"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/abs-rayleigh-quotient-le-numerical-radiu","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.abs_rayleighQuotient_le_numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#abs-rayleigh-quotient-le-numerical-radiu","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded linear operator on a complex inner product space $H$, with numerical radius $w(T)$. For every $x \\in H$, the Rayleigh quotient $R_T(x) = \\operatorname{Re} \\langle Tx, x\\rangle / \\|x\\|^2$, taken to be $0$ at $x = 0$, satisfies $|R_T(x)| \\le w(T)$.","title":"Numerical radius bounds the Rayleigh quotient"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/star-ring-end-half","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.starRingEnd_half","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#star-ring-end-half","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"The complex conjugate of $1/2 \\in \\mathbb{C}$ is $1/2$, that is, $\\overline{1/2} = 1/2$.","title":"Complex conjugate of one half"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/star-ring-end-neg-i-div-two","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.starRingEnd_neg_I_div_two","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#star-ring-end-neg-i-div-two","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"The complex conjugate of $-i/2 \\in \\mathbb{C}$ is $i/2$, that is, $\\overline{-i/2} = i/2$.","title":"Complex conjugate of minus half the imaginary unit"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/is-symmetric-cartesian-real-part","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.isSymmetric_cartesianRealPart","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#is-symmetric-cartesian-real-part","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For every bounded linear operator $T$ on a complex Hilbert space $H$ with adjoint $T^*$, the real Cartesian part $A = (T + T^*)/2$ is symmetric: $\\langle Ax, y\\rangle = \\langle x, Ay\\rangle$ for all $x, y \\in H$.","title":"The real Cartesian part is symmetric"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/is-symmetric-cartesian-imag-part","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.isSymmetric_cartesianImagPart","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#is-symmetric-cartesian-imag-part","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For every bounded linear operator $T$ on a complex Hilbert space $H$ with adjoint $T^*$, the imaginary Cartesian part $B = (T - T^*)/(2i)$ is symmetric: $\\langle Bx, y\\rangle = \\langle x, By\\rangle$ for all $x, y \\in H$.","title":"The imaginary Cartesian part is symmetric"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/quad-form-cartesian-real-part","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.quadForm_cartesianRealPart","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#quad-form-cartesian-real-part","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded operator on a complex Hilbert space $H$ and let $A = (T + T^*)/2$ be the real Cartesian part of $T$. Then for every $x \\in H$ one has $\\langle A x, x \\rangle = \\operatorname{Re} \\langle T x, x \\rangle$.","title":"Quadratic form of the real Cartesian part"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/quad-form-cartesian-imag-part","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.quadForm_cartesianImagPart","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#quad-form-cartesian-imag-part","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded operator on a complex Hilbert space $H$, whose inner product $\\langle \\cdot, \\cdot \\rangle$ is conjugate-linear in its first argument, and let $B = (T - T^*)/(2i)$ be the imaginary Cartesian part of $T$. Then for every $x \\in H$ one has $\\langle B x, x \\rangle = -\\operatorname{Im} \\langle T x, x \\rangle$.","title":"Quadratic form of the imaginary Cartesian part"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/cartesian-decomposition","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.cartesian_decomposition","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#cartesian-decomposition","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Every bounded linear operator $T$ on a complex Hilbert space satisfies $T = A + iB$, where $A = (T + T^*)/2$ and $B = (T - T^*)/(2i)$ are the real and imaginary Cartesian parts of $T$ and $T^*$ is the adjoint of $T$.","title":"Cartesian decomposition"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/norm-cartesian-real-part-le-numerical-ra","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.norm_cartesianRealPart_le_numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#norm-cartesian-real-part-le-numerical-ra","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded operator on a complex Hilbert space, with adjoint $T^*$, and let $A = (T + T^*)/2$ be its real Cartesian part. Then $\\|A\\| \\le w(T)$, where $\\|A\\|$ is the operator norm of $A$ and $w(T)$ is the numerical radius of $T$.","title":"Norm of the real part is at most the numerical radius"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/norm-cartesian-imag-part-le-numerical-ra","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.norm_cartesianImagPart_le_numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#norm-cartesian-imag-part-le-numerical-ra","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded operator on a complex Hilbert space, with adjoint $T^*$, and let $B = (T - T^*)/(2i)$ be its imaginary Cartesian part. Then $\\|B\\| \\le w(T)$, where $\\|B\\|$ is the operator norm of $B$ and $w(T)$ is the numerical radius of $T$.","title":"Norm of the imaginary part is at most the numerical radius"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/norm-le-two-mul-numerical-radius","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.norm_le_two_mul_numericalRadius","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#norm-le-two-mul-numerical-radius","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For every bounded operator $T$ on a complex Hilbert space, $\\|T\\| \\le 2\\,w(T)$, where $\\|T\\|$ is the operator norm and $w(T)$ is the numerical radius of $T$.","title":"Operator norm at most twice the numerical radius"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/numerical-radius-eq-norm-of-is-self-adjo","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.numericalRadius_eq_norm_of_isSelfAdjoint","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#numerical-radius-eq-norm-of-is-self-adjo","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"If $T$ is a self-adjoint bounded operator on a complex Hilbert space, then its numerical radius equals its operator norm: $w(T) = \\|T\\|$.","title":"Numerical radius of a self-adjoint operator"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/norm-quad-form-adjoint","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.norm_quadForm_adjoint","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#norm-quad-form-adjoint","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $T$ be a bounded operator on a complex Hilbert space $H$, with adjoint $T^*$. For every $x \\in H$, $|\\langle T^* x, x\\rangle| = |\\langle T x, x\\rangle|$, since $\\langle T^* x, x\\rangle$ is the complex conjugate of $\\langle T x, x\\rangle$.","title":"The quadratic form of the adjoint has the same modulus"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/numerical-radius-adjoint","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRadius.numericalRadius_adjoint","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#numerical-radius-adjoint","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For every bounded operator $T$ on a complex Hilbert space, the numerical radius of its adjoint $T^*$ equals the numerical radius of $T$: $w(T^*) = w(T)$.","title":"Numerical radius of the adjoint"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/numerical-range","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzHausdorffWanted.numericalRange","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#numerical-range","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"For a bounded linear operator $A$ on a complex inner product space $E$, the numerical range of $A$ is the set $W(A) = \\{\\langle A x, x \\rangle : x \\in E,\\ \\|x\\| = 1\\} \\subseteq \\mathbb{C}$, where the inner product $\\langle \\cdot, \\cdot \\rangle$ of $E$ is conjugate-linear in its first argument and linear in its second.","title":"Numerical range"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/mem-numerical-range-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzHausdorffWanted.mem_numericalRange_iff","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#mem-numerical-range-iff","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $A$ be a bounded linear operator on a complex inner product space $E$. A complex number $c$ lies in the numerical range $W(A)$ if and only if there is a vector $x \\in E$ with $\\|x\\| = 1$ and $\\langle A x, x \\rangle = c$.","title":"Membership in the numerical range"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/inner-mem-numerical-range","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzHausdorffWanted.inner_mem_numericalRange","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#inner-mem-numerical-range","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $A$ be a bounded linear operator on a complex inner product space $E$. For every $x \\in E$ with $\\|x\\| = 1$, the number $\\langle A x, x \\rangle$ lies in the numerical range $W(A)$.","title":"Unit vectors give points of the numerical range"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/toeplitz-hausdorff-general","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzHausdorffWanted.toeplitz_hausdorff_general","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#toeplitz-hausdorff-general","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"The numerical range of a bounded linear operator on a complex inner product space is convex. Completeness of the space is not needed.","title":"Toeplitz\u2013Hausdorff theorem without completeness"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/toeplitz-hausdorff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzHausdorffWanted.toeplitz_hausdorff","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#toeplitz-hausdorff","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"The numerical range of a bounded linear operator on a complex Hilbert space is convex.","title":"Toeplitz\u2013Hausdorff theorem"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/is-unit-of-not-mem-closure-numerical-ran","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRangeSpectrum.isUnit_of_notMem_closure_numericalRange","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#is-unit-of-not-mem-closure-numerical-ran","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $A$ be a bounded linear operator on a complex Hilbert space $E$, whose inner product $\\langle \\cdot, \\cdot \\rangle$ is conjugate-linear in its first argument and linear in its second, let $W(A) = \\{\\langle A x, x \\rangle : x \\in E,\\ \\|x\\| = 1\\}$ be the numerical range of $A$ and let $\\mu \\in \\mathbb{C}$. If $\\mu$ is not in the closure of $W(A)$, then $A - \\overline{\\mu}\\, I$, where $\\overline{\\mu}$ is the\u2026","title":"Conjugate shifts off the closed numerical range are invertible"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/conj-mem-closure-numerical-range-of-mem","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRangeSpectrum.conj_mem_closure_numericalRange_of_mem_spectrum","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#conj-mem-closure-numerical-range-of-mem","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $A$ be a bounded linear operator on a complex Hilbert space $E$, whose inner product $\\langle \\cdot, \\cdot \\rangle$ is conjugate-linear in its first argument and linear in its second, and let $W(A) = \\{\\langle A x, x \\rangle : x \\in E,\\ \\|x\\| = 1\\}$ be the numerical range of $A$. If $z$ is in the spectrum of $A$, then the complex conjugate $\\overline{z}$ lies in the closure of $W(A)$.","title":"Conjugates of spectral values lie in the closed numerical range"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/spectrum-subset-star-image-closure-numer","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRangeSpectrum.spectrum_subset_star_image_closure_numericalRange","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#spectrum-subset-star-image-closure-numer","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $A$ be a bounded linear operator on a complex Hilbert space $E$, whose inner product $\\langle \\cdot, \\cdot \\rangle$ is conjugate-linear in its first argument and linear in its second, and let $W(A) = \\{\\langle A x, x \\rangle : x \\in E,\\ \\|x\\| = 1\\}$ be the numerical range of $A$. Then every point of the spectrum of $A$ is the complex conjugate $\\overline{w}$ of some point $w$ of the closure of $W(A)$.","title":"The spectrum lies in the conjugate of the closed numerical range"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/conj-mem-numerical-range-of-exists-unit","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRangeSpectrum.conj_mem_numericalRange_of_exists_unit_eigenvector","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#conj-mem-numerical-range-of-exists-unit","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $A$ be a bounded linear operator on a complex inner product space $E$, whose inner product $\\langle \\cdot, \\cdot \\rangle$ is conjugate-linear in its first argument and linear in its second, let $W(A) = \\{\\langle A x, x \\rangle : x \\in E,\\ \\|x\\| = 1\\}$ be the numerical range of $A$ and let $z \\in \\mathbb{C}$. If there is a vector $x \\in E$ with $\\|x\\| = 1$ and $A x = z x$, then the complex conjugate\u2026","title":"Conjugates of eigenvalues lie in the numerical range"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/numerical-range-subset-closed-ball-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NumericalRangeSpectrum.numericalRange_subset_closedBall_norm","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#numerical-range-subset-closed-ball-norm","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"fully_proved","summary":"Let $A$ be a bounded linear operator on a complex inner product space $E$. Then every point of the numerical range $W(A)$ has absolute value at most the operator norm $\\|A\\|$: $W(A)$ lies in the closed disc of radius $\\|A\\|$ centred at $0$.","title":"The numerical range is bounded by the operator norm"},{"id":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/inf-dist-numerical-range-mul-norm-le-nor","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.NumericalRangeDistanceLowerBoundWanted.infDist_numericalRange_mul_norm_le_norm_sub_smul_apply","parent":15552,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius/index.html#inf-dist-numerical-range-mul-norm-le-nor","scope":"Analysis/operator-theory/spectral-theory/numerical-range-and-numerical-radius","status":"can_prove","summary":"Let $H$ be a complex Hilbert space with inner product $\\langle \\cdot, \\cdot \\rangle$, conjugate-linear in the first argument and linear in the second, and let $T : H \\to H$ be a bounded linear operator. Let $N(T) = \\{\\langle \\psi, T\\psi \\rangle : \\psi \\in H,\\ \\|\\psi\\| = 1\\} \\subseteq \\mathbb{C}$ be the numerical range of $T$, and for $z \\in \\mathbb{C}$ let $d(z, N(T))$ be the infimum of the distances from $z$ to\u2026","title":"Lower bound from the numerical range"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat.json","id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat/index.html","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat","status":"fully_proved","title":"Spectral theorem for bounded self-adjoint operators"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat/exists-spectral-measure","kind":"node","lean":"IsSelfAdjoint.exists_spectralMeasure","parent":15590,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat/index.html#exists-spectral-measure","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat","status":"fully_proved","summary":"Let $E$ be a complex Hilbert space, $T$ a bounded self-adjoint operator on $E$ and $v \\in E$. There is a finite regular measure $\\mu$ on the spectrum $\\sigma(T) \\subseteq \\mathbb{C}$ of $T$ such that for every continuous function $f : \\sigma(T) \\to \\mathbb{R}$, $\\int_{\\sigma(T)} f \\, d\\mu = \\operatorname{Re} \\langle f(T) v, v \\rangle$, where $f(T)$ is the continuous functional calculus of $T$ applied to $f$ viewed\u2026","title":"Scalar spectral measure of a vector"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat/bounded-self-adjoint-spectral-multiplica","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BoundedSelfAdjointSpectralWanted.boundedSelfAdjoint_spectral_multiplication","parent":15590,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat/index.html#bounded-self-adjoint-spectral-multiplica","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-bounded-self-adjoint-operat","status":"fully_proved","summary":"Let $E$ be a complex Hilbert space and $T$ a bounded self-adjoint operator on $E$. Then there are a measure space $(X, \\mu)$, a measurable function $\\varphi : X \\to \\mathbb{R}$ and a linear isometric isomorphism $U$ of $E$ onto $L^2(X, \\mu; \\mathbb{C})$ such that for every $x \\in E$, $U(Tx)(a) = \\varphi(a)\\,(Ux)(a)$ for $\\mu$-almost every $a \\in X$.","title":"Spectral theorem for bounded self-adjoint operators, multiplication form"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat.json","id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","title":"Spectral theorem for compact self-adjoint operators"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/norm-has-eigenvalue-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointFiniteEigenvalues.norm_hasEigenvalue_le","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#norm-has-eigenvalue-le","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $T$ be a bounded linear operator on an inner product space over $\\mathbb{R}$ or $\\mathbb{C}$. Every eigenvalue $\\mu$ of $T$ satisfies $|\\mu| \\le \\|T\\|$.","title":"Eigenvalues are bounded by the operator norm"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/finite-eigenvalues-of-ge","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointFiniteEigenvalues.finite_eigenvalues_of_ge","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#finite-eigenvalues-of-ge","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $T$ be a compact self-adjoint operator on a Hilbert space over $\\mathbb{R}$ or $\\mathbb{C}$. For every $\\varepsilon > 0$, the set of eigenvalues $\\mu$ of $T$ with $\\varepsilon \\le |\\mu|$ is finite.","title":"Finiteness of large eigenvalues of a compact self-adjoint operator"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/exists-has-eigenvalue-norm-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointNormEigenvalue.exists_hasEigenvalue_norm_eq","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#exists-has-eigenvalue-norm-eq","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $T \\neq 0$ be a compact self-adjoint operator on a Hilbert space $H$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. Then $T$ has an eigenvalue $\\mu \\in \\mathbb{K}$ with $\\mu \\neq 0$ and $|\\mu| = \\|T\\|$, where $\\|T\\|$ is the operator norm of $T$.","title":"A nonzero compact self-adjoint operator has an eigenvalue of modulus equal to its norm"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/exists-unit-eigenvector-real-eigenvalue","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointNormEigenvalue.exists_unit_eigenvector_real_eigenvalue","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#exists-unit-eigenvector-real-eigenvalue","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"A nonzero compact self-adjoint operator has a real eigenvalue of modulus equal to its norm, with a unit eigenvector. Let $T \\neq 0$ be a compact self-adjoint operator on a Hilbert space $H$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. Then there are a real number $r$ and a unit vector $u \\in H$ with $T u = r u$ and $|r| = \\|T\\|$, where $\\|T\\|$ is the operator norm of $T$.","title":"A nonzero compact self-adjoint operator has a real eigenvalue of modulus equal to its norm, with a unit eigenvector"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/exists-next-eigenvector-of-invariant","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointDeflation.exists_next_eigenvector_of_invariant","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#exists-next-eigenvector-of-invariant","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $T$ be a compact self-adjoint operator on a Hilbert space $H$ over $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a linear subspace of $H$ that admits an orthogonal projection $P : H \\to H$ onto $S$ and satisfies $T(S) \\subseteq S$, and let $T_1 = PTP$ be the compression of $T$ to $S$. If $T_1 \\neq 0$, then there are a real number $r$ and a unit vector $v \\in S$ with $Tv = rv$, $|r| = \\|T_1\\|$ and $|r| \\le \\|T\\|$.","title":"Deflation step"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/exists-next-eigenvector-of-finite-eigenv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointDeflation.exists_next_eigenvector_of_finite_eigenvectors","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#exists-next-eigenvector-of-finite-eigenv","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $T$ be a compact self-adjoint operator on a Hilbert space $H$ over $\\mathbb{R}$ or $\\mathbb{C}$, let $u_1, \\dots, u_n \\in H$ and scalars $\\mu_1, \\dots, \\mu_n$ satisfy $T u_i = \\mu_i u_i$ for every $i$, and let $S$ be the orthogonal complement of the span of $u_1, \\dots, u_n$. If $Tz \\neq 0$ for some $z \\in S$, then there are a real number $r$ and a unit vector $v$ with $\\langle v, u_i \\rangle = 0$ for every\u2026","title":"Next eigenvalue after a finite family of eigenvectors"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/exists-finite-rank-approx","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointFiniteRankApprox.exists_finiteRank_approx","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#exists-finite-rank-approx","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $T$ be a compact self-adjoint operator on a Hilbert space $H$ over $\\mathbb{R}$ or $\\mathbb{C}$. For every $\\varepsilon > 0$ there is a bounded self-adjoint operator $F$ on $H$ with finite-dimensional range such that $\\|T - F\\| \\le \\varepsilon$.","title":"Finite-rank approximation of compact self-adjoint operators"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/exists-nonzero-eigenvalue-enumeration","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointMaximumPrinciple.exists_nonzero_eigenvalue_enumeration","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#exists-nonzero-eigenvalue-enumeration","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $T$ be a compact self-adjoint operator on a Hilbert space $H$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. Then one of the following two alternatives holds.","title":"Maximum principle for compact self-adjoint operators"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/exists-finite-eigenvalue-enumeration","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointMaximumPrinciple.exists_finite_eigenvalue_enumeration","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#exists-finite-eigenvalue-enumeration","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $H$ be a finite-dimensional Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $T$ be a compact self-adjoint operator on $H$. Then there are $N \\in \\mathbb{N}$, nonzero real numbers $s_0, \\dots, s_{N-1}$ and an orthonormal family $u_0, \\dots, u_{N-1}$ in $H$ such that $T u_i = s_i u_i$ for every $i < N$, $|s_0| \\ge |s_1| \\ge \\dots \\ge |s_{N-1}|$, and for every $i < N$ every unit vector\u2026","title":"Maximum principle for compact self-adjoint operators in finite dimension"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/exists-ordered-eigenvalue-sequence","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointMaximumPrinciple.exists_ordered_eigenvalue_sequence","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#exists-ordered-eigenvalue-sequence","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $T$ be a compact self-adjoint operator on an infinite-dimensional Hilbert space $H$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. Then there are real numbers $s_0, s_1, s_2, \\dots$ and an orthonormal sequence $u_0, u_1, u_2, \\dots$ in $H$ such that $T u_n = s_n u_n$ and $|s_{n+1}| \\le |s_n|$ for every $n$, $|s_n| \\to 0$ as $n \\to \\infty$, and for every $n$ every unit vector $x \\in H$ orthogonal to\u2026","title":"Maximum principle as a sequence in infinite dimension"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/orthogonal-i-sup-eigenspaces-ne-zero-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointRangeDecomposition.orthogonal_iSup_eigenspaces_ne_zero_eq_ker","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#orthogonal-i-sup-eigenspaces-ne-zero-eq","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Let $H$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $T : H \\to H$ be a compact self-adjoint bounded linear operator. Then the orthogonal complement of the subspace spanned by the eigenspaces $\\ker(T - \\mu I)$ for all nonzero $\\mu \\in \\mathbb{K}$ equals the kernel of $T$.","title":"Orthogonal complement of the nonzero-eigenvalue eigenspaces"},{"id":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/spectral-theorem-range-decomposition","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointRangeDecomposition.spectral_theorem_range_decomposition","parent":15593,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat/index.html#spectral-theorem-range-decomposition","scope":"Analysis/operator-theory/spectral-theory/spectral-theorem-for-compact-self-adjoint-operat","status":"fully_proved","summary":"Spectral theorem for compact self-adjoint operators (range decomposition form). Let $H$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $T : H \\to H$ be a compact self-adjoint bounded linear operator. Then: (1) the closure of the subspace spanned by the eigenspaces $\\ker(T - \\mu I)$ for all nonzero $\\mu \\in \\mathbb{K}$ equals the closure of the range of $T$; (2) the closure of the range of\u2026","title":"Spectral theorem for compact self-adjoint operators (range decomposition form)"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators.json","id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","title":"Spectrum of finite-rank operators"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/eigenvector-mem-range-of-has-eigenvector","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.Module.End.eigenvector_mem_range_of_hasEigenvector","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#eigenvector-mem-range-of-has-eigenvector","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"An eigenvector for a nonzero eigenvalue lies in the range: if `T v = \u03bc \u2022 v` with `\u03bc \u2260 0`, then `v = T (\u03bc\u207b\u00b9 \u2022 v)`.","title":"Eigenvectors for nonzero eigenvalues lie in the range"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/exists-linear-independent-eigenvectors","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.Module.End.exists_linearIndependent_eigenvectors","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#exists-linear-independent-eigenvectors","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"Choose one eigenvector for each nonzero eigenvalue of `T`. The chosen eigenvectors are linearly independent (eigenvectors for distinct eigenvalues) and all lie in the range of `T`.","title":"Independent eigenvectors for distinct nonzero eigenvalues"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/finite-nonzero-has-eigenvalue-of-finite","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.Module.End.finite_nonzero_hasEigenvalue_of_finiteDimensional_range","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#finite-nonzero-has-eigenvalue-of-finite","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"A finite-rank operator has only finitely many distinct nonzero eigenvalues.","title":"Eigenvalue finiteness for finite-rank operators"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/ncard-nonzero-has-eigenvalue-le-finrank","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.Module.End.ncard_nonzero_hasEigenvalue_le_finrank_range","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#ncard-nonzero-has-eigenvalue-le-finrank","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"There are at most `finrank (range T)` distinct nonzero eigenvalues.","title":"Eigenvalue-count bound for finite-rank operators"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/is-compact-operator-of-finite-dimensiona","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.isCompactOperator_of_finiteDimensional_range","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#is-compact-operator-of-finite-dimensiona","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"A finite-rank operator is compact: it factors through its finite-dimensional range, and every continuous linear map out of a finite-dimensional (hence locally compact) space is compact.","title":"Finite-rank operators are compact"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/spectrum-subset-insert-zero-has-eigenval","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.spectrum_subset_insert_zero_hasEigenvalue","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#spectrum-subset-insert-zero-has-eigenval","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"The spectrum of a finite-rank operator is contained in `{0} \u222a {eigenvalues}`: every nonzero spectral value is an eigenvalue, by the Fredholm alternative for compact operators.","title":"Nonzero spectral values of a finite-rank operator are eigenvalues"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/has-eigenvalue-subset-spectrum","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.hasEigenvalue_subset_spectrum","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#has-eigenvalue-subset-spectrum","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"Every eigenvalue of a finite-rank operator lies in its spectrum.","title":"Eigenvalues of a finite-rank operator lie in its spectrum"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/finite-spectrum-of-finite-dimensional-ra","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.finite_spectrum_of_finiteDimensional_range","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#finite-spectrum-of-finite-dimensional-ra","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"The spectrum of a finite-rank operator is finite.","title":"Spectrum of a finite-rank operator is finite"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/ncard-spectrum-le-finrank-range-add-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.ncard_spectrum_le_finrank_range_add_one","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#ncard-spectrum-le-finrank-range-add-one","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"The spectrum of a finite-rank operator has at most `finrank (range T) + 1` points.","title":"Bound on the size of the spectrum of a finite-rank operator"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/spectrum-eq-insert-zero-has-eigenvalue","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.FiniteRankSpectrum.spectrum_eq_insert_zero_hasEigenvalue","parent":15606,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators/index.html#spectrum-eq-insert-zero-has-eigenvalue","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-finite-rank-operators","status":"fully_proved","summary":"The spectrum is exactly `{0} \u222a {eigenvalues}`.","title":"Spectrum of a finite-rank operator on an infinite-dimensional space"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators.json","id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","title":"Spectrum of rank-one operators"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/smul-right-mul-self","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.smulRight_mul_self","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square of the operator "],["math","\\(x \\mapsto f(x)\\,y\\)"]],"source":"Square of the operator $x \\mapsto f(x)\\,y$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#smul-right-mul-self","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ be a normed space over $\\mathbb{K}$, let $f \\colon E \\to \\mathbb{K}$ be a continuous linear functional, let $y \\in E$, and let $T$ be the bounded operator on $E$ given by $T x = f(x)\\,y$. Then $T^2 = f(y)\\,T$.","title":"Square of the operator $x \\mapsto f(x)\\,y$"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/not-mem-spectrum-smul-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.not_mem_spectrum_smulRight","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["text","The spectrum of "],["math","\\(x \\mapsto f(x)\\,y\\)"],["text"," lies in "],["math","\\(\\{0, f(y)\\}\\)"]],"source":"The spectrum of $x \\mapsto f(x)\\,y$ lies in $\\{0, f(y)\\}$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#not-mem-spectrum-smul-right","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ be a normed space over $\\mathbb{K}$, let $f \\colon E \\to \\mathbb{K}$ be a continuous linear functional, let $y \\in E$, and let $T$ be the bounded operator on $E$ given by $T x = f(x)\\,y$. If $\\mu \\in \\mathbb{K}$ satisfies $\\mu \\neq 0$ and $\\mu \\neq f(y)$, then $\\mu$ is not in the spectrum of $T$ in the algebra of bounded operators on $E$.","title":"The spectrum of $x \\mapsto f(x)\\,y$ lies in $\\{0, f(y)\\}$"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/has-eigenvalue-smul-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.hasEigenvalue_smulRight","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(f(y)\\)"],["text"," is an eigenvalue of "],["math","\\(x \\mapsto f(x)\\,y\\)"]],"source":"$f(y)$ is an eigenvalue of $x \\mapsto f(x)\\,y$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#has-eigenvalue-smul-right","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ be a normed space over $\\mathbb{K}$, let $f \\colon E \\to \\mathbb{K}$ be a continuous linear functional, and let $y \\in E$ with $y \\neq 0$. Then $f(y)$ is an eigenvalue of the linear operator $T x = f(x)\\,y$ on $E$.","title":"$f(y)$ is an eigenvalue of $x \\mapsto f(x)\\,y$"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/mem-spectrum-smul-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.mem_spectrum_smulRight","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(f(y)\\)"],["text"," lies in the spectrum of "],["math","\\(x \\mapsto f(x)\\,y\\)"]],"source":"$f(y)$ lies in the spectrum of $x \\mapsto f(x)\\,y$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#mem-spectrum-smul-right","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ be a Banach space over $\\mathbb{K}$, let $f \\colon E \\to \\mathbb{K}$ be a continuous linear functional, and let $y \\in E$ with $y \\neq 0$. Then $f(y)$ lies in the spectrum of the bounded operator $T x = f(x)\\,y$ on $E$.","title":"$f(y)$ lies in the spectrum of $x \\mapsto f(x)\\,y$"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/has-eigenvalue-zero-smul-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.hasEigenvalue_zero_smulRight","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(0\\)"],["text"," is an eigenvalue of "],["math","\\(x \\mapsto f(x)\\,y\\)"],["text"," when "],["math","\\(f\\)"],["text"," has a nontrivial kernel"]],"source":"$0$ is an eigenvalue of $x \\mapsto f(x)\\,y$ when $f$ has a nontrivial kernel"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#has-eigenvalue-zero-smul-right","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ be a normed space over $\\mathbb{K}$, let $f \\colon E \\to \\mathbb{K}$ be a continuous linear functional, and let $y \\in E$. If some $w \\in E$ with $w \\neq 0$ satisfies $f(w) = 0$, then $0$ is an eigenvalue of the linear operator $T x = f(x)\\,y$ on $E$.","title":"$0$ is an eigenvalue of $x \\mapsto f(x)\\,y$ when $f$ has a nontrivial kernel"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/mem-spectrum-smul-right-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.mem_spectrum_smulRight_zero","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(0\\)"],["text"," lies in the spectrum of "],["math","\\(x \\mapsto f(x)\\,y\\)"],["text"," when "],["math","\\(f\\)"],["text"," has a nontrivial kernel"]],"source":"$0$ lies in the spectrum of $x \\mapsto f(x)\\,y$ when $f$ has a nontrivial kernel"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#mem-spectrum-smul-right-zero","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ be a Banach space over $\\mathbb{K}$, let $f \\colon E \\to \\mathbb{K}$ be a continuous linear functional, and let $y \\in E$. If some $w \\in E$ with $w \\neq 0$ satisfies $f(w) = 0$, then $0$ lies in the spectrum of the bounded operator $T x = f(x)\\,y$ on $E$.","title":"$0$ lies in the spectrum of $x \\mapsto f(x)\\,y$ when $f$ has a nontrivial kernel"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/spectrum-smul-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.spectrum_smulRight","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spectrum of the operator "],["math","\\(x \\mapsto f(x)\\,y\\)"]],"source":"Spectrum of the operator $x \\mapsto f(x)\\,y$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#spectrum-smul-right","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ be a Banach space over $\\mathbb{K}$, let $f \\colon E \\to \\mathbb{K}$ be a continuous linear functional, and let $y \\in E$ with $y \\neq 0$. If some $w \\in E$ with $w \\neq 0$ satisfies $f(w) = 0$, then the spectrum of the bounded operator $T x = f(x)\\,y$ on $E$ is exactly $\\{0, f(y)\\}$.","title":"Spectrum of the operator $x \\mapsto f(x)\\,y$"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/spectrum-smul-right-of-apply-eq-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.spectrum_smulRight_of_apply_eq_zero","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spectrum of "],["math","\\(x \\mapsto f(x)\\,y\\)"],["text"," when "],["math","\\(f(y) = 0\\)"]],"source":"Spectrum of $x \\mapsto f(x)\\,y$ when $f(y) = 0$"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#spectrum-smul-right-of-apply-eq-zero","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ be a Banach space over $\\mathbb{K}$, let $f \\colon E \\to \\mathbb{K}$ be a continuous linear functional, and let $y \\in E$ with $y \\neq 0$ and $f(y) = 0$. Then the spectrum of the bounded operator $T x = f(x)\\,y$ on $E$ is $\\{0\\}$.","title":"Spectrum of $x \\mapsto f(x)\\,y$ when $f(y) = 0$"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/spectrum-rank-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RankOneSpectrum.spectrum_rankOne","parent":15617,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spectrum of the operator "],["math","\\(z \\mapsto \\langle y, z \\rangle\\,x\\)"],["text"," on a Hilbert space"]],"source":"Spectrum of the operator $z \\mapsto \\langle y, z \\rangle\\,x$ on a Hilbert space"},"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators/index.html#spectrum-rank-one","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-rank-one-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $H$ be a Hilbert space over $\\mathbb{K}$ whose inner product $\\langle \\cdot, \\cdot \\rangle$ is linear in the second argument, let $x, y \\in H$ with $x \\neq 0$, and let $T$ be the bounded operator on $H$ given by $T z = \\langle y, z \\rangle\\,x$. If some $w \\in H$ with $w \\neq 0$ satisfies $\\langle y, w \\rangle = 0$, then the spectrum of $T$ is\u2026","title":"Spectrum of the operator $z \\mapsto \\langle y, z \\rangle\\,x$ on a Hilbert space"},{"data_url":"scopes/Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators.json","id":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators","kind":"scope","navigation":"module","parent":15498,"result":false,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/index.html","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators","status":"fully_proved","title":"Spectrum of self-adjoint operators"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/spectrum-re-nonneg-of-re-apply-inner-sel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SelfAdjointPositiveSpectrum.spectrum_re_nonneg_of_reApplyInnerSelf_nonneg","parent":15627,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/index.html#spectrum-re-nonneg-of-re-apply-inner-sel","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators","status":"fully_proved","summary":"Let $T$ be a bounded linear operator on a Hilbert space $H$ over $\\mathbb{R}$ or $\\mathbb{C}$. If $\\operatorname{Re}\\langle Tu, u\\rangle \\ge 0$ for every $u \\in H$, then every element $\\mu$ of the spectrum of $T$ satisfies $\\operatorname{Re}\\mu \\ge 0$.","title":"Spectrum of an accretive operator has nonnegative real part"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/positive-iff-spectrum-nonneg","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SelfAdjointPositiveSpectrum.positive_iff_spectrum_nonneg","parent":15627,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/index.html#positive-iff-spectrum-nonneg","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators","status":"fully_proved","summary":"Let $T$ be a self-adjoint bounded linear operator on a nonzero Hilbert space $H$ over $\\mathbb{R}$ or $\\mathbb{C}$. Then $\\operatorname{Re}\\langle Tu, u\\rangle \\ge 0$ for every $u \\in H$ if and only if every element $\\mu$ of the spectrum of $T$ satisfies $\\operatorname{Re}\\mu \\ge 0$.","title":"Positivity iff spectrum nonnegative"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/rayleigh-quotient-algebra-map-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SelfAdjointSpectrumBounds.rayleighQuotient_algebraMap_sub","parent":15627,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/index.html#rayleigh-quotient-algebra-map-sub","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators","status":"fully_proved","summary":"Let $T$ be a bounded linear operator on an inner product space $H$ over $\\mathbb{R}$ or $\\mathbb{C}$, let $I$ be the identity operator on $H$, and write $R_S(x) = \\operatorname{Re}\\langle Sx, x\\rangle / \\|x\\|^2$ for the Rayleigh quotient of an operator $S$. Then for every real number $t$ and every nonzero $x \\in H$, $R_{tI - T}(x) = t - R_T(x)$.","title":"Rayleigh quotient of a shifted operator"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/self-adjoint-spectrum-subset-icc-nonzero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SelfAdjointSpectrumBounds.selfAdjoint_spectrum_subset_Icc_nonzero","parent":15627,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/index.html#self-adjoint-spectrum-subset-icc-nonzero","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators","status":"fully_proved","summary":"Let $T$ be a self-adjoint bounded linear operator on a nonzero Hilbert space $H$ over $\\mathbb{R}$ or $\\mathbb{C}$, with Rayleigh quotient $R_T(x) = \\operatorname{Re}\\langle Tx, x\\rangle / \\|x\\|^2$. Then every $\\mu$ in the spectrum of $T$ satisfies $\\inf_{x \\ne 0} R_T(x) \\le \\operatorname{Re}\\mu \\le \\sup_{x \\ne 0} R_T(x)$, where the infimum and supremum range over the nonzero vectors $x \\in H$.","title":"Spectrum bounds over nonzero vectors"},{"id":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/self-adjoint-spectrum-subset-icc","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SelfAdjointSpectrumBounds.selfAdjoint_spectrum_subset_Icc","parent":15627,"result":true,"route":"../roadmap/Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators/index.html#self-adjoint-spectrum-subset-icc","scope":"Analysis/operator-theory/spectral-theory/spectrum-of-self-adjoint-operators","status":"fully_proved","summary":"Let $T$ be a self-adjoint bounded linear operator on a nonzero Hilbert space $H$ over $\\mathbb{R}$ or $\\mathbb{C}$, with Rayleigh quotient $R_T(x) = \\operatorname{Re}\\langle Tx, x\\rangle / \\|x\\|^2$ for $x \\ne 0$ and $R_T(0) = 0$. Then every $\\mu$ in the spectrum of $T$ satisfies $\\inf_{x \\in H} R_T(x) \\le \\operatorname{Re}\\mu \\le \\sup_{x \\in H} R_T(x)$.","title":"Spectrum bounds over all vectors"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm.json","id":"Analysis/operator-theory/compact-and-fredholm","kind":"scope","parent":15497,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/index.html","scope":"Analysis/operator-theory/compact-and-fredholm","status":"can_prove","summary":"Compact and finite-rank operators, Schauder's theorem, complete continuity, Hilbert\u2013Schmidt operators, Riesz\u2013Schauder theory, Fredholm alternative and index, closed-range theorems.","title":"Compact and Fredholm operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/not-separable-space-ker-of-is-compact-op","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.CompactOperatorKernelNonseparableWanted.not_separableSpace_ker_of_isCompactOperator_of_not_separableSpace","parent":15633,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/index.html#not-separable-space-ker-of-is-compact-op","scope":"Analysis/operator-theory/compact-and-fredholm","status":"can_prove","summary":"Let $H$ be a complex Hilbert space that is not separable and let $A : H \\to H$ be a compact operator. Then the kernel $\\ker A$ is not separable.","title":"The kernel of a compact operator on a nonseparable Hilbert space is nonseparable"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem.json","id":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","status":"can_prove","title":"Closed range theorem"},{"id":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/set-range-eq-coe-linear-map-range","kind":"node","lean":"ContinuousLinearMap.setRange_eq_coe_linearMap_range","parent":15635,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/index.html#set-range-eq-coe-linear-map-range","scope":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","status":"fully_proved","summary":"Let $E'$ and $F'$ be Hilbert spaces over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $S : E' \\to F'$ be a bounded linear operator. The set of values of $S$ equals the underlying set of the range of $S$ regarded as a linear subspace of $F'$.","title":"The range of a bounded operator as a set"},{"id":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/range-comp-subtype-l-orthogonal-compleme","kind":"node","lean":"ContinuousLinearMap.range_comp_subtypeL_orthogonalComplement_ker","parent":15635,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/index.html#range-comp-subtype-l-orthogonal-compleme","scope":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","status":"fully_proved","summary":"Let $E$ be a Hilbert space and $F$ an inner product space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $T : E \\to F$ be a bounded linear operator. Then the restriction of $T$ to the orthogonal complement $(\\ker T)^\\perp$ of its kernel has the same range as $T$: every value $T x$ is attained at the orthogonal projection of $x$ onto $(\\ker T)^\\perp$, since the difference lies in\u2026","title":"Restriction to the orthogonal complement of the kernel keeps the range"},{"id":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/injective-comp-subtype-l-orthogonal-comp","kind":"node","lean":"ContinuousLinearMap.injective_comp_subtypeL_orthogonalComplement_ker","parent":15635,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/index.html#injective-comp-subtype-l-orthogonal-comp","scope":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","status":"fully_proved","summary":"Let $E$ and $F$ be inner product spaces over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $T : E \\to F$ be a bounded linear operator. Then the restriction of $T$ to the orthogonal complement $(\\ker T)^\\perp$ of its kernel is injective: a vector lying in both $\\ker T$ and $(\\ker T)^\\perp$ is zero.","title":"Restriction to the orthogonal complement of the kernel is injective"},{"id":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/is-closed-range-adjoint-of-is-closed-ran","kind":"node","lean":"ContinuousLinearMap.isClosed_range_adjoint_of_isClosed_range","parent":15635,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/index.html#is-closed-range-adjoint-of-is-closed-ran","scope":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","status":"fully_proved","summary":"Let $E$ and $F$ be Hilbert spaces over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $T : E \\to F$ be a bounded linear operator with adjoint $T^\\dagger$. If the range of $T$ is closed in $F$, then the range of $T^\\dagger$ is closed in $E$.","title":"Closed range passes to the adjoint"},{"id":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/is-closed-range-adjoint-iff","kind":"node","lean":"ContinuousLinearMap.isClosed_range_adjoint_iff","parent":15635,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/index.html#is-closed-range-adjoint-iff","scope":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","status":"fully_proved","summary":"Let $E$ and $F$ be Hilbert spaces over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $T : E \\to F$ be a bounded linear operator with adjoint $T^\\dagger$. Then the range of $T$ is closed in $F$ if and only if the range of $T^\\dagger$ is closed in $E$.","title":"Closed range theorem for adjoints"},{"id":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/is-closed-range-of-is-fredholm-adjoint-c","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.AdjointCompFredholmClosedRangeWanted.isClosed_range_of_isFredholm_adjoint_comp","parent":15635,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed range from a Fredholm "],["math","\\(T^*T\\)"]],"source":"Closed range from a Fredholm $T^*T$"},"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/index.html#is-closed-range-of-is-fredholm-adjoint-c","scope":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","status":"can_prove","summary":"Closed range from a Fredholm $T^*T$. Let $X$ and $Y$ be complex Hilbert spaces and let $T : X \\to Y$ be a bounded linear operator with Hilbert space adjoint $T^* : Y \\to X$. If $T^*T : X \\to X$ is a Fredholm operator, then the range of $T$ is closed in $Y$.","title":"Closed range from a Fredholm $T^*T$"},{"id":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/is-closed-range-adjoint-of-is-closed-ran-2","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.HilbertAdjointClosedRangeWanted.isClosed_range_adjoint_of_isClosed_range","parent":15635,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/closed-range-theorem/index.html#is-closed-range-adjoint-of-is-closed-ran-2","scope":"Analysis/operator-theory/compact-and-fredholm/closed-range-theorem","status":"can_prove","summary":"Let $X$ and $Y$ be complex Hilbert spaces and let $A : X \\to Y$ be a bounded linear operator with Hilbert space adjoint $A^* : Y \\to X$. If the range of $A$ is closed in $Y$, then the range of $A^*$ is closed in $X$.","title":"Closed range passes to the Hilbert space adjoint"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators.json","id":"Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators","status":"fully_proved","title":"Complete continuity of compact operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators/tendsto-apply-of-is-compact-operator-of","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.CompleteContinuity.tendsto_apply_of_isCompactOperator_of_forall_tendsto","parent":15643,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators/index.html#tendsto-apply-of-is-compact-operator-of","scope":"Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ and $F$ be normed spaces over $\\mathbb{K}$, and let $T : E \\to F$ be a compact continuous linear operator. If a sequence $(u_n)$ in $E$ converges weakly to $x \\in E$, that is, $f(u_n) \\to f(x)$ for every continuous linear functional $f : E \\to \\mathbb{K}$, then $T u_n \\to T x$ in norm.","title":"Complete continuity of compact operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators/tendsto-apply-of-is-compact-operator-of-2","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.CompleteContinuity.tendsto_apply_of_isCompactOperator_of_tendsto_zero","parent":15643,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators/index.html#tendsto-apply-of-is-compact-operator-of-2","scope":"Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ and $F$ be normed spaces over $\\mathbb{K}$, and let $T : E \\to F$ be a compact continuous linear operator. If a sequence $(u_n)$ in $E$ converges weakly to $0$, that is, $f(u_n) \\to 0$ for every continuous linear functional $f : E \\to \\mathbb{K}$, then $T u_n \\to 0$ in norm.","title":"Compact operators send weakly null sequences to norm null sequences"},{"id":"Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators/tendsto-apply-of-is-compact-operator-of-3","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.CompleteContinuity.tendsto_apply_of_isCompactOperator_of_tendsto_weakSpace","parent":15643,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators/index.html#tendsto-apply-of-is-compact-operator-of-3","scope":"Analysis/operator-theory/compact-and-fredholm/complete-continuity-of-compact-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $E$ and $F$ be normed spaces over $\\mathbb{K}$, and let $T : E \\to F$ be a compact continuous linear operator. If a sequence $(u_n)$ in $E$ converges to $x \\in E$ in the weak topology of $E$, then $T u_n \\to T x$ in norm.","title":"Complete continuity of compact operators, weak-topology form"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators.json","id":"Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators","status":"fully_proved","title":"Finite-rank approximation of compact operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators/exists-finite-rank-approx","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactFiniteRankApprox.exists_finiteRank_approx","parent":15647,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators/index.html#exists-finite-rank-approx","scope":"Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$ ($H$ need not be complete), and let $T$ be a compact bounded linear operator on $H$. For every real $\\varepsilon > 0$ there is a bounded linear operator $F$ on $H$ with finite-dimensional range such that $\\|T - F\\| \\le \\varepsilon$.","title":"Finite-rank approximation of compact operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators/exists-tendsto-finite-rank-approx","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactFiniteRankApprox.exists_tendsto_finiteRank_approx","parent":15647,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators/index.html#exists-tendsto-finite-rank-approx","scope":"Analysis/operator-theory/compact-and-fredholm/finite-rank-approximation-of-compact-operators","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$ ($H$ need not be complete), and let $T$ be a compact bounded linear operator on $H$. Then there is a sequence $(F_n)_{n \\in \\mathbb{N}}$ of bounded linear operators on $H$, each with finite-dimensional range, such that $\\|T - F_n\\| \\to 0$ as $n \\to \\infty$.","title":"Compact operators are norm limits of finite-rank operators"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/fredholm-alternative.json","id":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative","status":"fully_proved","title":"Fredholm alternative"},{"id":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/ker-smul-id-sub-eq-eigenspace","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointFredholmAlternative.ker_smul_id_sub_eq_eigenspace","parent":15650,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/index.html#ker-smul-id-sub-eq-eigenspace","scope":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a bounded linear operator on $H$, let $I$ be the identity of $H$, and let $\\mu \\in \\mathbb{K}$. Then the kernel of $\\mu I - T$ equals the eigenspace $\\{x \\in H : T x = \\mu x\\}$ of $T$ for $\\mu$.","title":"The kernel of a scalar minus an operator is an eigenspace"},{"id":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/fredholm-alternative-of-is-compact-opera","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointFredholmAlternative.fredholm_alternative_of_isCompactOperator_isSelfAdjoint","parent":15650,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/index.html#fredholm-alternative-of-is-compact-opera","scope":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative","status":"fully_proved","summary":"Let $H$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a compact self-adjoint bounded operator on $H$, let $I$ be the identity of $H$, and let $\\mu \\in \\mathbb{K}$ with $\\mu \\neq 0$. Then the range of $\\mu I - T$ equals the orthogonal complement $(\\ker(\\mu I - T))^\\perp$ of its kernel, and either $\\mu I - T$ is bijective, or its kernel is nonzero and finite-dimensional.","title":"Fredholm alternative for compact self-adjoint operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/is-closed-range-smul-id-sub-of-is-compac","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointFredholmAlternative.isClosed_range_smul_id_sub_of_isCompactOperator_isSelfAdjoint","parent":15650,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/index.html#is-closed-range-smul-id-sub-of-is-compac","scope":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative","status":"fully_proved","summary":"Let $H$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a compact self-adjoint bounded operator on $H$, let $I$ be the identity of $H$, and let $\\mu \\in \\mathbb{K}$ with $\\mu \\neq 0$. Then the range of $\\mu I - T$ is a closed subset of $H$.","title":"Closed range for compact self-adjoint perturbations of nonzero scalars"},{"id":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/fredholm-alternative-sub-of-is-compact-o","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactSelfAdjointFredholmAlternative.fredholm_alternative_sub_of_isCompactOperator_isSelfAdjoint","parent":15650,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/fredholm-alternative/index.html#fredholm-alternative-sub-of-is-compact-o","scope":"Analysis/operator-theory/compact-and-fredholm/fredholm-alternative","status":"fully_proved","summary":"Let $H$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a compact self-adjoint bounded operator on $H$, let $I$ be the identity of $H$, and let $\\mu \\in \\mathbb{K}$ with $\\mu \\neq 0$. Then the range of $T - \\mu I$ equals the orthogonal complement $(\\ker(T - \\mu I))^\\perp$ of its kernel, and either $T - \\mu I$ is bijective, or its kernel is nonzero and finite-dimensional.","title":"Fredholm alternative for compact self-adjoint operators, operator-minus-scalar form"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators.json","id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"can_prove","title":"Hilbert\u2013Schmidt operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-norm-sq-2","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.HilbertSchmidtNorm.hsNormSq","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-norm-sq-2","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"Let $H$ be a real inner product space, let $T : H \\to H$ be a continuous linear operator and let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $H$. The squared Hilbert\u2013Schmidt norm of $T$ in the family $(u_n)$ is the extended nonnegative real $\\sum_{n=0}^{\\infty} \\|T u_n\\|^2$, the sum being taken in $[0, \\infty]$. It is $\\infty$ exactly when the real series $\\sum_n \\|T u_n\\|^2$ diverges.","title":"Squared Hilbert\u2013Schmidt norm in a family"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-ennnorm","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.HilbertSchmidtNorm.hsENNNorm","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-ennnorm","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"Let $H$ be a real inner product space, let $T : H \\to H$ be a continuous linear operator and let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $H$. The Hilbert\u2013Schmidt norm of $T$ in the family $(u_n)$ is the square root, taken in $[0, \\infty]$ with $\\sqrt{\\infty} = \\infty$, of the squared Hilbert\u2013Schmidt norm $\\sum_{n=0}^{\\infty} \\|T u_n\\|^2 \\in [0, \\infty]$.","title":"Extended Hilbert\u2013Schmidt norm in a family"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/norm-sq-lp-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.norm_sq_lp_eq_tsum","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#norm-sq-lp-eq-tsum","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"Norm squared of an `\u2113\u00b2(\u2115, \ud835\udd5c)` vector is the sum of the squared coordinate norms.","title":"Squared \u2113\u00b2 norm as a sum of squared coordinates"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/summable-norm-sq-of-mem-p","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.summable_norm_sq_of_mem\u2113p","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#summable-norm-sq-of-mem-p","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"Summability of squared coordinate norms from `Mem\u2113p`.","title":"Square-summability of coordinates of an \u2113\u00b2 vector"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/mem-p-of-summable-norm-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.mem\u2113p_of_summable_norm_sq","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#mem-p-of-summable-norm-sq","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"`Mem\u2113p` at `2` from summability of squared norms.","title":"Square-summable sequences lie in \u2113\u00b2"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.IsHilbertSchmidt","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A matrix is Hilbert\u2013Schmidt if the squared norms of its entries are summable.","title":"Hilbert\u2013Schmidt matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-norm-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsNormSq","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-norm-sq","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The finite squared Hilbert\u2013Schmidt norm: the sum of the squared entry norms. The witness restricts this real quantity to summable matrices, avoiding the zero default of a nonsummable real `tsum`. Divergent operators are handled by the existing extended-valued operator norm.","title":"Squared Hilbert\u2013Schmidt norm of a matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/summable-row-of-is-hilbert-schmidt","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.summable_row_of_isHilbertSchmidt","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#summable-row-of-is-hilbert-schmidt","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"Fiber summability: each row of a Hilbert\u2013Schmidt matrix is square-summable.","title":"Rows of a Hilbert\u2013Schmidt matrix are square-summable"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-row","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsRow","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-row","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The row vector of a Hilbert\u2013Schmidt matrix, as an `\u2113\u00b2` element (conjugated).","title":"Row vector of a Hilbert\u2013Schmidt matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/summable-hs-coord","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.summable_hsCoord","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#summable-hs-coord","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The coordinate series defining the Hilbert\u2013Schmidt operator is summable.","title":"Summability of the Hilbert\u2013Schmidt coordinate series"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-coord-eq-inner","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsCoord_eq_inner","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-coord-eq-inner","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The coordinate series is the inner product with the row vector.","title":"Hilbert\u2013Schmidt coordinates as inner products with rows"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/mem-p-hs-coord","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.mem\u2113p_hsCoord","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#mem-p-hs-coord","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The coordinate series of a Hilbert\u2013Schmidt matrix applied to an `\u2113\u00b2` vector is in `\u2113\u00b2`.","title":"Hilbert\u2013Schmidt matrix maps \u2113\u00b2 vectors into \u2113\u00b2"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsLinMap","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-lin-map","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator of a matrix, as a linear map.","title":"Hilbert\u2013Schmidt operator as a linear map"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/summable-norm-sq-hs-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.summable_norm_sq_hsLinMap","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#summable-norm-sq-hs-lin-map","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The squared norms of the coordinates of a Hilbert\u2013Schmidt image are summable.","title":"Square-summability of a Hilbert\u2013Schmidt image"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsLinMap_apply","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-lin-map-apply","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"Pointwise action of the Hilbert\u2013Schmidt linear map.","title":"Pointwise action of the Hilbert\u2013Schmidt linear map"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/norm-hs-lin-map-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.norm_hsLinMap_le","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#norm-hs-lin-map-le","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator is bounded by the square root of `hsNormSq`.","title":"Norm bound for the Hilbert\u2013Schmidt linear map"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator of a matrix, as a bounded operator on `\u2113\u00b2(\u2115, \ud835\udd5c)`.","title":"Hilbert\u2013Schmidt operator on \u2113\u00b2"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_apply","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-apply","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"Pointwise action of the Hilbert\u2013Schmidt operator.","title":"Pointwise action of the Hilbert\u2013Schmidt operator"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/norm-hs-operator-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.norm_hsOperator_le","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#norm-hs-operator-le","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The operator norm of a Hilbert\u2013Schmidt operator is at most its Hilbert\u2013Schmidt norm.","title":"Operator norm is at most the Hilbert\u2013Schmidt norm"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_zero","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-zero","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The zero matrix is Hilbert\u2013Schmidt.","title":"Zero matrix is Hilbert\u2013Schmidt"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_zero","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-zero","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator of the zero matrix is zero.","title":"Hilbert\u2013Schmidt operator of the zero matrix is zero"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-neg","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_neg","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-neg","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The negative of a Hilbert\u2013Schmidt matrix is Hilbert\u2013Schmidt.","title":"Negation preserves the Hilbert\u2013Schmidt property"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-add","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_add","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-add","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The sum of Hilbert\u2013Schmidt matrices is Hilbert\u2013Schmidt.","title":"Sum of Hilbert\u2013Schmidt matrices is Hilbert\u2013Schmidt"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_sub","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-sub","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The difference of Hilbert\u2013Schmidt matrices is Hilbert\u2013Schmidt.","title":"Difference of Hilbert\u2013Schmidt matrices is Hilbert\u2013Schmidt"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-smul","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_smul","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-smul","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A scalar multiple of a Hilbert\u2013Schmidt matrix is Hilbert\u2013Schmidt.","title":"Scalar multiple of a Hilbert\u2013Schmidt matrix is Hilbert\u2013Schmidt"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-add","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_add","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-add","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator of a sum is the sum of the operators.","title":"Additivity of the Hilbert\u2013Schmidt operator"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_sub","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-sub","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator of a difference is the difference of the operators.","title":"Hilbert\u2013Schmidt operator of a difference of matrices"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-smul","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_smul","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-smul","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator of a scalar multiple is the scalar multiple of the operator.","title":"Hilbert\u2013Schmidt operator of a scalar multiple of a matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-transpose-star","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsTransposeStar","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-transpose-star","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The conjugate transpose of a matrix: `(a\u1d40*) i j = star (a j i)`.","title":"Conjugate transpose of an infinite matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-transpose-star","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_transposeStar","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-transpose-star","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The conjugate transpose of a Hilbert\u2013Schmidt matrix is Hilbert\u2013Schmidt.","title":"Conjugate transpose of a Hilbert\u2013Schmidt matrix is Hilbert\u2013Schmidt"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-matrix","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsMatrix","parent":15655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Matrix of a bounded operator on "],["math","\\(\\ell^2\\)"]],"source":"Matrix of a bounded operator on $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-matrix","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The matrix of a bounded operator on `\u2113\u00b2(\u2115, \ud835\udd5c)`: entry `(i, j)` is the `i`-th coordinate of the image of the `j`-th standard basis vector.","title":"Matrix of a bounded operator on $\\ell^2$"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/eq-hs-operator-of-is-hilbert-schmidt-mat","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.eq_hsOperator_of_isHilbertSchmidt_matrix","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#eq-hs-operator-of-is-hilbert-schmidt-mat","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A bounded operator whose matrix is Hilbert\u2013Schmidt is the Hilbert\u2013Schmidt operator of its matrix.","title":"Operator with Hilbert\u2013Schmidt matrix is the operator of its matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-matrix-hs-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsMatrix_hsOperator","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-matrix-hs-operator","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The matrix of a Hilbert\u2013Schmidt operator is the matrix itself.","title":"Matrix of the Hilbert\u2013Schmidt operator of a matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-congr","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_congr","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-congr","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The Hilbert\u2013Schmidt operator depends only on the matrix, not on the summability proof.","title":"Hilbert\u2013Schmidt operator is independent of the summability proof"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/adjoint-hs-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.adjoint_hsOperator","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#adjoint-hs-operator","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The adjoint of a Hilbert\u2013Schmidt operator is the Hilbert\u2013Schmidt operator of the conjugate-transpose matrix.","title":"Adjoint of a Hilbert\u2013Schmidt operator"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-trunc","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsTrunc","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-trunc","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The truncation of a matrix to the top-left `N \u00d7 N` block.","title":"Truncation of a matrix to its top-left block"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-hs-trunc","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_hsTrunc","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-hs-trunc","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A truncation of any matrix is Hilbert\u2013Schmidt: it is supported on a finite set.","title":"Truncated matrices are Hilbert\u2013Schmidt"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-compact-operator-rank-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isCompactOperator_rankOne","parent":15655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank-one operators on "],["math","\\(\\ell^2\\)"],["text"," are compact"]],"source":"Rank-one operators on $\\ell^2$ are compact"},"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-compact-operator-rank-one","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A rank-one operator on `\u2113\u00b2` is compact: it factors through the one-dimensional space `\ud835\udd5c`.","title":"Rank-one operators on $\\ell^2$ are compact"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/rank-one-single-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.rankOne_single_apply","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#rank-one-single-apply","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The action of a scaled rank-one operator on standard basis vectors, pointwise.","title":"Scaled rank-one operator on standard basis vectors"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-hs-trunc-eq-sum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_hsTrunc_eq_sum","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-hs-trunc-eq-sum","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A truncated Hilbert\u2013Schmidt operator is a finite sum of rank-one operators.","title":"Truncated Hilbert\u2013Schmidt operator as a sum of rank-one operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-compact-operator-hs-operator-hs-trunc","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isCompactOperator_hsOperator_hsTrunc","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-compact-operator-hs-operator-hs-trunc","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A truncated Hilbert\u2013Schmidt operator is compact.","title":"Truncated Hilbert\u2013Schmidt operators are compact"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-norm-sq-sub-hs-trunc","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsNormSq_sub_hsTrunc","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-norm-sq-sub-hs-trunc","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The squared Hilbert\u2013Schmidt norm of the truncation error is the tail sum.","title":"Hilbert\u2013Schmidt norm of the truncation error as a tail sum"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/tendsto-hs-trunc-finset-sum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.tendsto_hsTrunc_finset_sum","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#tendsto-hs-trunc-finset-sum","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The partial sums of the squared norms over expanding squares converge to `hsNormSq`.","title":"Square partial sums converge to the squared Hilbert\u2013Schmidt norm"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/tendsto-hs-norm-sq-sub-hs-trunc","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.tendsto_hsNormSq_sub_hsTrunc","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#tendsto-hs-norm-sq-sub-hs-trunc","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The squared Hilbert\u2013Schmidt norm of the truncation error tends to zero.","title":"Hilbert\u2013Schmidt norm of the truncation error tends to zero"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/tendsto-hs-operator-hs-trunc","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.tendsto_hsOperator_hsTrunc","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#tendsto-hs-operator-hs-trunc","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The truncated operators converge in norm to the Hilbert\u2013Schmidt operator.","title":"Truncated operators converge in norm to the Hilbert\u2013Schmidt operator"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-compact-operator-hs-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isCompactOperator_hsOperator","parent":15655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hilbert\u2013Schmidt operators on "],["math","\\(\\ell^2\\)"],["text"," are compact"]],"source":"Hilbert\u2013Schmidt operators on $\\ell^2$ are compact"},"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-compact-operator-hs-operator","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"Every Hilbert\u2013Schmidt operator on `\u2113\u00b2(\u2115, \ud835\udd5c)` is compact.","title":"Hilbert\u2013Schmidt operators on $\\ell^2$ are compact"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-compact-operator-of-is-hilbert-schmid","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isCompactOperator_of_isHilbertSchmidt_matrix","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-compact-operator-of-is-hilbert-schmid","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A bounded operator whose matrix is Hilbert\u2013Schmidt is compact.","title":"Bounded operator with Hilbert\u2013Schmidt matrix is compact"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/norm-le-of-is-hilbert-schmidt-matrix","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.norm_le_of_isHilbertSchmidt_matrix","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#norm-le-of-is-hilbert-schmidt-matrix","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A bounded operator whose matrix is Hilbert\u2013Schmidt has norm at most its Hilbert\u2013Schmidt norm.","title":"Operator norm bounded by the Hilbert\u2013Schmidt norm"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-diagonal","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsDiagonal","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-diagonal","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The diagonal matrix with diagonal `d`.","title":"Diagonal infinite matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/summable-hs-diagonal-row","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.summable_hsDiagonal_row","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#summable-hs-diagonal-row","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A row of a diagonal matrix is square-summable.","title":"Rows of a diagonal matrix are square-summable"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/tsum-hs-diagonal-row","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.tsum_hsDiagonal_row","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#tsum-hs-diagonal-row","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The squared norms in a row of a diagonal matrix sum to the squared diagonal entry.","title":"Squared row norm of a diagonal matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-hs-diagonal","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_hsDiagonal","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-hs-diagonal","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A diagonal matrix with square-summable diagonal is Hilbert\u2013Schmidt.","title":"Diagonal matrix with square-summable diagonal is Hilbert\u2013Schmidt"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-norm-sq-hs-diagonal","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsNormSq_hsDiagonal","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-norm-sq-hs-diagonal","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The squared Hilbert\u2013Schmidt norm of a diagonal matrix is the sum of the squared diagonal entries.","title":"Hilbert\u2013Schmidt norm of a diagonal matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-hs-diagonal-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_hsDiagonal_apply","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-hs-diagonal-apply","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A diagonal Hilbert\u2013Schmidt operator acts by pointwise multiplication by the diagonal.","title":"Diagonal Hilbert\u2013Schmidt operator acts by pointwise multiplication"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-rank-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsRankOne","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-rank-one","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The rank-one matrix `u i * star (v j)`.","title":"Rank-one infinite matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/summable-hs-rank-one-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.summable_hsRankOne_sq","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#summable-hs-rank-one-sq","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The squared norms of the entries of a rank-one matrix are summable over the product.","title":"Entries of a rank-one matrix are square-summable"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/is-hilbert-schmidt-hs-rank-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.isHilbertSchmidt_hsRankOne","parent":15655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank-one matrix of "],["math","\\(\\ell^2\\)"],["text"," vectors is Hilbert\u2013Schmidt"]],"source":"Rank-one matrix of $\\ell^2$ vectors is Hilbert\u2013Schmidt"},"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#is-hilbert-schmidt-hs-rank-one","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A rank-one matrix with `\u2113\u00b2` vectors is Hilbert\u2013Schmidt.","title":"Rank-one matrix of $\\ell^2$ vectors is Hilbert\u2013Schmidt"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-operator-hs-rank-one-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsOperator_hsRankOne_apply","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-operator-hs-rank-one-apply","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"A rank-one Hilbert\u2013Schmidt operator acts as the inner product with `v` times `u`.","title":"Action of a rank-one Hilbert\u2013Schmidt operator"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/hs-norm-sq-hs-rank-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.hsNormSq_hsRankOne","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#hs-norm-sq-hs-rank-one","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The squared Hilbert\u2013Schmidt norm of a rank-one matrix is `\u2016u\u2016\u00b2 * \u2016v\u2016\u00b2`.","title":"Hilbert\u2013Schmidt norm of a rank-one matrix"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/canonical-hs-norm-sq-eq-of-real-matrix","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.canonical_hsNormSq_eq_ofReal_matrix","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#canonical-hs-norm-sq-eq-of-real-matrix","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"In the standard `\u2113\u00b2` basis, the existing extended operator squared norm equals the finite matrix squared norm. This bridge is specialized to real scalars because the canonical operator norm API is real-only.","title":"Canonical Hilbert\u2013Schmidt norm equals the matrix norm"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/canonical-hs-norm-sq-hs-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertSchmidt.canonical_hsNormSq_hsOperator","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#canonical-hs-norm-sq-hs-operator","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"fully_proved","summary":"The constructed real Hilbert\u2013Schmidt operator has the canonical extended squared norm given by the finite squared norm of its matrix.","title":"Canonical Hilbert\u2013Schmidt norm of the constructed operator"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/exists-hs-norm-le-mul-op-norm-of-finite","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.HilbertSchmidtFiniteRankBoundWanted.exists_hsNorm_le_mul_opNorm_of_finiteDimensional_quotient_ker","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#exists-hs-norm-le-mul-op-norm-of-finite","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"can_prove","summary":"There is a function $C$ from the natural numbers to the positive reals with the following property. Let $H$ be an infinite-dimensional separable real Hilbert space, let $(u_i)_{i \\in \\mathbb{N}}$ be a Hilbert basis of $H$, and let $T : H \\to H$ be a bounded linear operator such that the quotient $H/\\ker T$ is finite-dimensional, of dimension $n$. Then the Hilbert\u2013Schmidt norm of $T$ computed in the basis $(u_i)$,\u2026","title":"The Hilbert\u2013Schmidt norm of a finite-rank operator is controlled by its operator norm"},{"id":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/norm-apply-le-hs-norm-of-norm-eq-one","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.NormApplyLeHilbertSchmidtNormWanted.norm_apply_le_hsNorm_of_norm_eq_one","parent":15655,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators/index.html#norm-apply-le-hs-norm-of-norm-eq-one","scope":"Analysis/operator-theory/compact-and-fredholm/hilbert-schmidt-operators","status":"can_prove","summary":"Let $H$ be a real Hilbert space, let $T : H \\to H$ be a bounded linear operator, and let $(u_n)_{n \\in \\mathbb{N}}$ be an orthonormal basis of $H$ indexed by the natural numbers. Let $\\|T\\|_{\\mathrm{HS}} = \\bigl(\\sum_{n} \\|T u_n\\|^2\\bigr)^{1/2} \\in [0, \\infty]$ be the Hilbert\u2013Schmidt norm of $T$ computed in this basis; it is $\\infty$ when the series diverges. Then $\\|Tx\\| \\le \\|T\\|_{\\mathrm{HS}}$ for every\u2026","title":"The Hilbert\u2013Schmidt norm bounds an operator on unit vectors"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index.json","id":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","status":"fully_proved","title":"Homotopy invariance of the Fredholm index"},{"id":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index-eq-of-preconnected","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.FredholmIndexHomotopy.index_eq_of_preconnected","parent":15719,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index.html#index-eq-of-preconnected","scope":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","status":"fully_proved","summary":"The Fredholm index is constant along a continuous family of Fredholm operators indexed by a preconnected set, because the index is locally constant (Mathlib) and takes values in the discrete space `\u2124`.","title":"Fredholm index is constant on preconnected families"},{"id":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index-eq-of-continuous-on-icc","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.FredholmIndexHomotopy.index_eq_of_continuousOn_Icc","parent":15719,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index.html#index-eq-of-continuous-on-icc","scope":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","status":"fully_proved","summary":"The Fredholm index is constant along a continuous path of Fredholm operators on `[0, 1]`.","title":"Fredholm index is constant along a path continuous on (0, 1)"},{"id":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index-eq-of-continuous-path","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.FredholmIndexHomotopy.index_eq_of_continuous_path","parent":15719,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index.html#index-eq-of-continuous-path","scope":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","status":"fully_proved","summary":"The Fredholm index is constant along a globally continuous path of Fredholm operators on `[0, 1]`.","title":"Fredholm index is constant along a globally continuous path"},{"id":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/has-finite-range-smul-right","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.FredholmIndexHomotopy.hasFiniteRange_smulRight","parent":15719,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index.html#has-finite-range-smul-right","scope":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","status":"fully_proved","summary":"A rank-one operator has finite range.","title":"Rank-one operators have finite range"},{"id":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index-add-has-finite-range","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.FredholmIndexHomotopy.index_add_hasFiniteRange","parent":15719,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index.html#index-add-has-finite-range","scope":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","status":"fully_proved","summary":"Adding a finite-rank operator does not change the Fredholm index. The proof follows the segment `T + t \u2022 F`, `t \u2208 [0, 1]`, which is a continuous path of Fredholm operators.","title":"Finite-rank perturbation preserves the Fredholm index"},{"id":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index-has-finite-range-add","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.FredholmIndexHomotopy.index_hasFiniteRange_add","parent":15719,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index.html#index-has-finite-range-add","scope":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","status":"fully_proved","summary":"Adding a finite-rank operator on the left does not change the Fredholm index.","title":"Left finite-rank perturbation preserves the Fredholm index"},{"id":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index-sub-has-finite-range","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.FredholmIndexHomotopy.index_sub_hasFiniteRange","parent":15719,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index/index.html#index-sub-has-finite-range","scope":"Analysis/operator-theory/compact-and-fredholm/homotopy-invariance-of-the-fredholm-index","status":"fully_proved","summary":"Subtracting a finite-rank operator does not change the Fredholm index.","title":"Subtracting a finite-rank operator preserves the Fredholm index"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory.json","id":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","status":"fully_proved","title":"Riesz\u2013Schauder theory"},{"id":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/finite-dimensional-ker","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.Compact.RieszSchauder.finiteDimensional_ker","parent":15727,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/index.html#finite-dimensional-ker","scope":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","status":"fully_proved","summary":"Let $E$ be a Banach space over $\\mathbb{R}$ or $\\mathbb{C}$, let $T : E \\to E$ be a compact bounded linear operator, let $\\mu \\neq 0$ be a scalar and let $I$ be the identity of $E$. Then the kernel of $\\mu I - T$ is finite-dimensional.","title":"Riesz\u2013Schauder theorem, kernel part"},{"id":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/exists-continuous-projection","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.Compact.RieszSchauder.exists_continuous_projection","parent":15727,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/index.html#exists-continuous-projection","scope":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{R}$ or $\\mathbb{C}$ and let $K$ be a finite-dimensional subspace of $E$. Then there is a continuous linear map $P : E \\to E$ such that $P x = x$ for every $x \\in K$ and $P x \\in K$ for every $x \\in E$.","title":"Continuous projection onto a finite-dimensional subspace"},{"id":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/is-closed-range-smul-one-sub","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.Compact.RieszSchauder.isClosed_range_smul_one_sub","parent":15727,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/index.html#is-closed-range-smul-one-sub","scope":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","status":"fully_proved","summary":"Let $E$ be a Banach space over $\\mathbb{R}$ or $\\mathbb{C}$, let $T : E \\to E$ be a compact bounded linear operator, let $\\mu \\neq 0$ be a scalar and let $I$ be the identity of $E$. Then the range of $\\mu I - T$ is closed in $E$.","title":"Riesz\u2013Schauder theorem, range part"},{"id":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/fredholm-alternative","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.Compact.RieszSchauder.fredholm_alternative","parent":15727,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/index.html#fredholm-alternative","scope":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","status":"fully_proved","summary":"Let $E$ be a Banach space over $\\mathbb{R}$ or $\\mathbb{C}$, let $T : E \\to E$ be a compact bounded linear operator, let $\\mu \\neq 0$ be a scalar and let $I$ be the identity of $E$. Then either $\\mu I - T$ is bijective, or the kernel of $\\mu I - T$ is nonzero and finite-dimensional and the range of $\\mu I - T$ is closed.","title":"Fredholm alternative, Riesz\u2013Schauder form"},{"id":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/range-eq-orthogonal-ker-adjoint","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszSchauder.range_eq_orthogonal_ker_adjoint","parent":15727,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/index.html#range-eq-orthogonal-ker-adjoint","scope":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","status":"fully_proved","summary":"Let $E$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a compact operator on $E$, let $\\mu \\in \\mathbb{K}$ with $\\mu \\ne 0$, and let $I$ be the identity operator on $E$. Then the range of $\\mu I - T$ is the orthogonal complement of the kernel of its adjoint: $\\operatorname{ran}(\\mu I - T) = \\ker\\big((\\mu I - T)^*\\big)^\\perp$.","title":"Riesz\u2013Schauder theorem on a Hilbert space"},{"id":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/adjoint-smul-one-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszSchauder.adjoint_smul_one_sub","parent":15727,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/index.html#adjoint-smul-one-sub","scope":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","status":"fully_proved","summary":"Let $E$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a bounded linear operator on $E$ with adjoint $T^*$, let $\\mu \\in \\mathbb{K}$ and let $I$ be the identity operator on $E$. Then $(\\mu I - T)^* = \\overline{\\mu} I - T^*$, where $\\overline{\\mu}$ is the complex conjugate of $\\mu$ (so $\\overline{\\mu} = \\mu$ when $\\mathbb{K} = \\mathbb{R}$).","title":"Adjoint of a shifted operator"},{"id":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/range-eq-orthogonal-ker-smul-one-sub-adj","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszSchauder.range_eq_orthogonal_ker_smul_one_sub_adjoint","parent":15727,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory/index.html#range-eq-orthogonal-ker-smul-one-sub-adj","scope":"Analysis/operator-theory/compact-and-fredholm/riesz-schauder-theory","status":"fully_proved","summary":"Let $E$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a compact operator on $E$ with adjoint $T^*$, let $\\mu \\in \\mathbb{K}$ with $\\mu \\ne 0$, and let $I$ be the identity operator on $E$. Then the range of $\\mu I - T$ is the orthogonal complement of the kernel of $\\overline{\\mu} I - T^*$, where $\\overline{\\mu}$ is the complex conjugate of $\\mu$:\u2026","title":"Riesz\u2013Schauder theorem on a Hilbert space, adjoint form"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints.json","id":"Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints","status":"fully_proved","title":"Schauder's theorem on compact adjoints"},{"id":"Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints/is-compact-operator-adjoint-of-finite-ra","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactOperatorAdjoint.isCompactOperator_adjoint_of_finiteRank","parent":15735,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints/index.html#is-compact-operator-adjoint-of-finite-ra","scope":"Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints","status":"fully_proved","summary":"Let $E$ and $F$ be Hilbert spaces over $\\mathbb{R}$ or $\\mathbb{C}$. If a bounded linear operator $T : E \\to F$ has finite-dimensional range, then its adjoint $T^* : F \\to E$ is a compact operator.","title":"The adjoint of a finite-rank operator is compact"},{"id":"Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints/is-compact-operator-adjoint","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactOperatorAdjoint.isCompactOperator_adjoint","parent":15735,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints/index.html#is-compact-operator-adjoint","scope":"Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints","status":"fully_proved","summary":"The adjoint of a compact operator between Hilbert spaces is compact.","title":"Schauder's theorem"},{"id":"Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints/is-compact-operator-adjoint-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompactOperatorAdjoint.isCompactOperator_adjoint_iff","parent":15735,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints/index.html#is-compact-operator-adjoint-iff","scope":"Analysis/operator-theory/compact-and-fredholm/schauder-s-theorem-on-compact-adjoints","status":"fully_proved","summary":"A bounded operator between Hilbert spaces is compact if and only if its adjoint is compact.","title":"Schauder's theorem, iff form"},{"data_url":"scopes/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators.json","id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","kind":"scope","navigation":"module","parent":15633,"result":false,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","title":"Spectrum of compact operators"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/eigenvector-not-mem-span-prefix","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorEigenvalues.eigenvector_not_mem_span_prefix","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#eigenvector-not-mem-span-prefix","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a bounded linear operator on $E$, let $(\\lambda_n)_{n \\in \\mathbb{N}}$ be pairwise distinct scalars in $\\mathbb{K}$, and let $(u_n)_{n \\in \\mathbb{N}}$ be nonzero vectors in $E$ with $T u_n = \\lambda_n u_n$ for every $n$. Then for every $n$, the vector $u_n$ does not lie in the linear span of $\\{u_i : i < n\\}$.","title":"Eigenvectors for distinct eigenvalues are not in the span of earlier ones"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/sub-mem-span-prefix","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorEigenvalues.sub_mem_span_prefix","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#sub-mem-span-prefix","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a bounded linear operator on $E$, let $(\\lambda_n)_{n \\in \\mathbb{N}}$ in $\\mathbb{K}$ and $(u_n)_{n \\in \\mathbb{N}}$ in $E$ satisfy $T u_n = \\lambda_n u_n$ for every $n$, and for each $n$ let $V_n$ be the linear span of $\\{u_i : i < n\\}$. Then for every $n$ and every $y \\in V_{n+1}$, we have $T y - \\lambda_n y \\in V_n$.","title":"An operator acts as a scalar on a span of eigenvectors modulo a smaller span"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/map-mem-span-prefix","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorEigenvalues.map_mem_span_prefix","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#map-mem-span-prefix","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a bounded linear operator on $E$, and let $(\\lambda_n)_{n \\in \\mathbb{N}}$ in $\\mathbb{K}$ and $(u_n)_{n \\in \\mathbb{N}}$ in $E$ satisfy $T u_n = \\lambda_n u_n$ for every $n$. Then for every $n$ and every $y$ in the linear span of $\\{u_i : i \\le n\\}$, the vector $T y$ also lies in that span.","title":"Invariance of spans of eigenvectors"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/exists-unit-vector-far","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorEigenvalues.exists_unit_vector_far","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#exists-unit-vector-far","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(u_n)_{n \\in \\mathbb{N}}$ be vectors in $E$, and for each $n$ let $V_n$ be the linear span of $\\{u_i : i < n\\}$. Suppose $u_n \\notin V_n$ for every $n$. Then for every $n$ there is a vector $y \\in V_{n+1}$ with $\\|y\\| = 1$ and $\\|y - z\\| \\ge 1/2$ for every $z \\in V_n$.","title":"Riesz separation for nested spans"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/finite-eigenvalues-of-ge","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorEigenvalues.finite_eigenvalues_of_ge","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#finite-eigenvalues-of-ge","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a compact bounded linear operator on $E$, and let $\\varepsilon > 0$ be real. Then the set of eigenvalues $\\mu \\in \\mathbb{K}$ of $T$ with $|\\mu| \\ge \\varepsilon$ is finite.","title":"Finiteness of eigenvalues of a compact operator bounded away from zero"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/tendsto-eigenvalue-of-injective","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorEigenvalues.tendsto_eigenvalue_of_injective","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#tendsto-eigenvalue-of-injective","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $T$ be a compact bounded linear operator on $E$. If $(\\lambda_n)_{n \\in \\mathbb{N}}$ is a sequence of pairwise distinct eigenvalues of $T$, then $\\lambda_n \\to 0$ as $n \\to \\infty$.","title":"Eigenvalues of a compact operator tend to zero along any injective sequence"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/not-is-unit-of-is-compact-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorNotUnit.not_isUnit_of_isCompactOperator","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#not-is-unit-of-is-compact-operator","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"If `T` is compact and `E` is infinite-dimensional, then `T` has no two-sided inverse among bounded operators. (Completeness of `E` is not needed for this statement.)","title":"A compact operator on an infinite-dimensional space is not a unit"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/not-bijective-of-is-compact-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorNotUnit.not_bijective_of_isCompactOperator","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#not-bijective-of-is-compact-operator","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"Let $E$ be an infinite-dimensional Banach space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $T$ be a compact bounded linear operator on $E$. Then $T$ is not bijective.","title":"A compact operator on an infinite-dimensional Banach space is not bijective"},{"id":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/zero-mem-spectrum-of-is-compact-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CompactOperatorNotUnit.zero_mem_spectrum_of_isCompactOperator","parent":15739,"result":true,"route":"../roadmap/Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators/index.html#zero-mem-spectrum-of-is-compact-operator","scope":"Analysis/operator-theory/compact-and-fredholm/spectrum-of-compact-operators","status":"fully_proved","summary":"Let $E$ be an infinite-dimensional normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $T$ be a compact bounded linear operator on $E$. Then $0$ lies in the spectrum of $T$ in the algebra of bounded linear operators on $E$; that is, $T$ has no bounded two-sided inverse.","title":"Zero lies in the spectrum of a compact operator on an infinite-dimensional space"},{"data_url":"scopes/Analysis/operator-theory/bounded-operators.json","id":"Analysis/operator-theory/bounded-operators","kind":"scope","parent":15497,"result":false,"route":"../roadmap/Analysis/operator-theory/bounded-operators/index.html","scope":"Analysis/operator-theory/bounded-operators","status":"can_prove","summary":"General bounded operators: Douglas lemma, direct sums, strong and weak operator convergence, uniform boundedness, Hellinger\u2013Toeplitz, invariant subspaces (Lomonosov), hypercyclicity criteria, Wold decomposition.","title":"Bounded operators and operator topologies"},{"id":"Analysis/operator-theory/bounded-operators/exists-bound-of-symmetric-linear-map-to","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SymmetricOperatorToDualBoundedWanted.exists_bound_of_symmetric_linearMap_to_strongDual","parent":15749,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/index.html#exists-bound-of-symmetric-linear-map-to","scope":"Analysis/operator-theory/bounded-operators","status":"can_prove","summary":"Let $E$ be a real Banach space with dual space $E^*$ of continuous linear functionals, and let $T : E \\to E^*$ be a linear map that is symmetric: $(Tx)(y) = (Ty)(x)$ for all $x, y \\in E$. Then $T$ is bounded: there is a constant $C \\ge 0$ such that $\\|Tx\\| \\le C \\|x\\|$ for all $x \\in E$, where $\\|Tx\\|$ is the operator norm of the functional $Tx$.","title":"A symmetric linear map from a Banach space to its dual is bounded"},{"id":"Analysis/operator-theory/bounded-operators/bartle-graves","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BartleGravesWanted.bartle_graves","parent":15749,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/index.html#bartle-graves","scope":"Analysis/operator-theory/bounded-operators","status":"fully_proved","summary":"Let $E$ and $F$ be real Banach spaces and let $T : E \\to F$ be a surjective continuous linear map. Then there are a continuous map $g : F \\to E$, in general not linear, and a constant $C \\ge 0$ such that $T(g(y)) = y$ for every $y \\in F$, $g(0) = 0$, $g(t y) = t \\, g(y)$ for every real $t \\ge 0$ and every $y \\in F$, and $\\|g(y)\\| \\le C \\|y\\|$ for every $y \\in F$.","title":"Bartle\u2013Graves theorem"},{"id":"Analysis/operator-theory/bounded-operators/lomonosov-invariant-subspace","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LomonosovInvariantSubspaceWanted.lomonosov_invariantSubspace","parent":15749,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/index.html#lomonosov-invariant-subspace","scope":"Analysis/operator-theory/bounded-operators","status":"fully_proved","summary":"Let $E$ be an infinite-dimensional complex Banach space, let $K$ be a nonzero compact operator on $E$, and let $T$ be a bounded linear operator on $E$ with $TK = KT$. Then there is a closed linear subspace $M$ of $E$ with $M \\ne \\{0\\}$ and $M \\ne E$ that is invariant under $T$: $T x \\in M$ for every $x \\in M$.","title":"Lomonosov's invariant subspace theorem"},{"id":"Analysis/operator-theory/bounded-operators/tendsto-apply-iff-tendsto-inner-and-tend","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.StrongOperatorConvergenceIffWeakAndAdjointWanted.tendsto_apply_iff_tendsto_inner_and_tendsto_inner_adjoint_comp","parent":15749,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/index.html#tendsto-apply-iff-tendsto-inner-and-tend","scope":"Analysis/operator-theory/bounded-operators","status":"can_prove","summary":"Let $H$ be a real or complex Hilbert space with inner product $\\langle \\cdot, \\cdot \\rangle$, let $(T_n)_{n \\in \\mathbb{N}}$ be a sequence of bounded linear operators on $H$, let $T$ be a bounded linear operator on $H$, and write $S^*$ for the adjoint of an operator $S$. Then $T_n \\to T$ in the strong operator topology, that is, $T_n x \\to T x$ for every $x \\in H$, if and only if both $T_n \\to T$ in the weak\u2026","title":"Strong operator convergence characterized by weak operator convergence"},{"data_url":"scopes/Analysis/operator-theory/bounded-operators/density-of-invertible-operators.json","id":"Analysis/operator-theory/bounded-operators/density-of-invertible-operators","kind":"scope","navigation":"module","parent":15749,"result":false,"route":"../roadmap/Analysis/operator-theory/bounded-operators/density-of-invertible-operators/index.html","scope":"Analysis/operator-theory/bounded-operators/density-of-invertible-operators","status":"can_prove","title":"Density of invertible operators"},{"id":"Analysis/operator-theory/bounded-operators/density-of-invertible-operators/scottish-book-problem173-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook173Wanted.ScottishBookProblem173Claim","parent":15754,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/density-of-invertible-operators/index.html#scottish-book-problem173-claim","scope":"Analysis/operator-theory/bounded-operators/density-of-invertible-operators","status":"fully_proved","summary":"Eidelheit's claim that for every real Banach space $E$, the bounded linear operators $E \\to E$ that have a bounded linear two-sided inverse are dense, for the operator norm, in the space of all bounded linear operators on $E$.","title":"Scottish Book Problem 173"},{"id":"Analysis/operator-theory/bounded-operators/density-of-invertible-operators/eidelheit173-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook173Wanted.eidelheit173_refuted","parent":15754,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/density-of-invertible-operators/index.html#eidelheit173-refuted","scope":"Analysis/operator-theory/bounded-operators/density-of-invertible-operators","status":"can_prove","summary":"Eidelheit's claim in Problem 173 is false: there is a real Banach space $E$ in which the bounded linear operators $E \\to E$ that have a bounded linear two-sided inverse are not dense, for the operator norm, in the space of all bounded linear operators on $E$.","title":"Negative answer to Scottish Book Problem 173"},{"data_url":"scopes/Analysis/operator-theory/bounded-operators/direct-sums-of-operators.json","id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","kind":"scope","navigation":"module","parent":15749,"result":false,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","title":"Direct sums of operators"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/prod-map-comp","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.prodMap_comp","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#prod-map-comp","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"Composing direct sums is componentwise.","title":"Composition of direct sum operators is componentwise"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/prod-map-one","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.prodMap_one","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#prod-map-one","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"The identity direct sum is the identity.","title":"Direct sum of identity operators is the identity"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/prod-map-zero","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.prodMap_zero","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#prod-map-zero","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"The zero direct sum is zero.","title":"Direct sum of zero operators is zero"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/prod-map-smul","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.prodMap_smul","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#prod-map-smul","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"Scaling a direct sum scales both summands by the same factor.","title":"Scalar multiplication of a direct sum operator"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/norm-prod-map","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.norm_prodMap","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#norm-prod-map","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"The operator norm of a direct sum is the maximum of the operator norms.","title":"Operator norm of a direct sum is the maximum of the norms"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/ker-prod-map","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.ker_prodMap","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#ker-prod-map","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"The kernel of a direct sum is the product of the kernels.","title":"Kernel of a direct sum operator"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/range-prod-map","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.range_prodMap","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#range-prod-map","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"The range of a direct sum is the product of the ranges.","title":"Range of a direct sum operator"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/is-unit-prod-map-iff","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.isUnit_prodMap_iff","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#is-unit-prod-map-iff","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"A direct sum is a unit iff both summands are units.","title":"Invertibility of a direct sum operator"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/spectrum-prod-map","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.spectrum_prodMap","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#spectrum-prod-map","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"The spectrum of a direct sum is the union of the spectra.","title":"Spectrum of a direct sum operator is the union of spectra"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/resolvent-set-prod-map","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.resolventSet_prodMap","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#resolvent-set-prod-map","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"The resolvent set of a direct sum is the intersection of the resolvent sets.","title":"Resolvent set of a direct sum operator"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/is-compact-operator-prod-map-iff","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.isCompactOperator_prodMap_iff","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#is-compact-operator-prod-map-iff","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"A direct sum is compact iff both summands are compact.","title":"Compactness of a direct sum operator"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/is-fredholm-prod-map-iff","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.isFredholm_prodMap_iff","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#is-fredholm-prod-map-iff","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"A direct sum is Fredholm iff both summands are Fredholm.","title":"Fredholmness of a direct sum operator"},{"id":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index-prod-map","kind":"node","lean":"MathlibExt.Analysis.Normed.Operator.DirectSumOperators.index_prodMap","parent":15757,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/direct-sums-of-operators/index.html#index-prod-map","scope":"Analysis/operator-theory/bounded-operators/direct-sums-of-operators","status":"fully_proved","summary":"The Fredholm index of a direct sum is the sum of the Fredholm indices.","title":"Fredholm index of a direct sum is additive"},{"data_url":"scopes/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma.json","id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","kind":"scope","navigation":"module","parent":15749,"result":false,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","title":"Douglas factorization lemma"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/norm-adjoint","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.norm_adjoint","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#norm-adjoint","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"Adjoints preserve operator norm.","title":"Adjoint preserves the operator norm"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/eq-of-mem-orthogonal-and-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.eq_of_mem_orthogonal_and_apply","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#eq-of-mem-orthogonal-and-apply","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"Two points of `(ker B)\u15ee` with the same image under `B` are equal, since their difference lies in `ker B \u2229 (ker B)\u15ee`.","title":"Injectivity of an operator on the orthogonal complement of its kernel"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/exists-mem-orthogonal-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.exists_mem_orthogonal_apply","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#exists-mem-orthogonal-apply","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"If `A.range \u2264 B.range`, every `A x` has a preimage under `B` in `(ker B)\u15ee`.","title":"Preimages in the orthogonal complement of the kernel under range inclusion"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/douglas-factor-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.douglasFactorFun","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#douglas-factor-fun","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The underlying function of the canonical factor: the chosen preimage of `A x` in `(ker B)\u15ee`.","title":"Underlying function of the Douglas factor"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/douglas-factor-fun-spec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.douglasFactorFun_spec","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#douglas-factor-fun-spec","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The defining properties of the chosen preimage.","title":"Defining properties of the Douglas preimage"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/douglas-factor-lin","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.douglasFactorLin","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#douglas-factor-lin","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The canonical factor as a linear map.","title":"Douglas factor as a linear map"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/continuous-douglas-factor-lin","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.continuous_douglasFactorLin","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#continuous-douglas-factor-lin","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The canonical factor is continuous, by the (sequential) closed graph theorem: if `u n \u2192 x` and `C (u n) \u2192 y`, then `y \u2208 (ker B)\u15ee` (the orthogonal complement is closed) and `B y = A x` (continuity of `A` and `B`), so `y = C x` by uniqueness of the preimage in `(ker B)\u15ee`.","title":"Continuity of the Douglas factor via the closed graph theorem"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/douglas-factor","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.douglasFactor","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#douglas-factor","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The canonical factor in Douglas' lemma: for `x : E`, the unique element of `(ker B)\u15ee` mapped by `B` to `A x`.","title":"Canonical factor in Douglas' lemma"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/douglas-factor-mem-orthogonal","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.douglasFactor_mem_orthogonal","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#douglas-factor-mem-orthogonal","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The canonical factor lands in `(ker B)\u15ee`.","title":"Douglas factor takes values in the kernel's orthogonal complement"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/apply-douglas-factor","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.apply_douglasFactor","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#apply-douglas-factor","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The canonical factor is a preimage: `B (C x) = A x`.","title":"Douglas factor gives a preimage under B"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/eq-comp-douglas-factor","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.eq_comp_douglasFactor","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#eq-comp-douglas-factor","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The canonical factor factors `A` through `B`.","title":"Douglas factor factors A through B"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/douglas-factor-eq-of-mem-orthogonal-and","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.douglasFactor_eq_of_mem_orthogonal_and_apply","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#douglas-factor-eq-of-mem-orthogonal-and","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"The canonical factor is the unique operator with the two defining properties.","title":"Uniqueness of the Douglas factor"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/range-le-iff-exists-factorization","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.range_le_iff_exists_factorization","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#range-le-iff-exists-factorization","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"Douglas' lemma, range form: `A.range \u2264 B.range` iff `A = B \u2218L C` for some bounded `C`.","title":"Douglas' lemma: range inclusion iff factorization"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/majorization-of-factorization","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.majorization_of_factorization","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#majorization-of-factorization","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"If `A = B \u2218L C`, then majorization holds with constant `\u2016C\u2016`.","title":"Factorization implies majorization"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/exists-factorization-of-majorization","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.exists_factorization_of_majorization","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#exists-factorization-of-majorization","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"Douglas' lemma, factorization form with a norm bound: majorization with constant `c` yields a factor `C` with `A = B \u2218L C` and `\u2016C\u2016 \u2264 c`.","title":"Douglas' lemma: majorization gives a norm-bounded factor"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/majorization-iff-exists-factorization","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.majorization_iff_exists_factorization","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#majorization-iff-exists-factorization","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"Douglas' lemma, majorization form: majorization holds iff `A` factors through `B`.","title":"Douglas' lemma: majorization iff factorization"},{"id":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/range-le-iff-majorization","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DouglasLemma.range_le_iff_majorization","parent":15771,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/douglas-factorization-lemma/index.html#range-le-iff-majorization","scope":"Analysis/operator-theory/bounded-operators/douglas-factorization-lemma","status":"fully_proved","summary":"Douglas' lemma, range/majorization form: `A.range \u2264 B.range` iff majorization holds.","title":"Douglas' lemma: range inclusion iff majorization"},{"data_url":"scopes/Analysis/operator-theory/bounded-operators/hypercyclic-operators.json","id":"Analysis/operator-theory/bounded-operators/hypercyclic-operators","kind":"scope","navigation":"module","parent":15749,"result":false,"route":"../roadmap/Analysis/operator-theory/bounded-operators/hypercyclic-operators/index.html","scope":"Analysis/operator-theory/bounded-operators/hypercyclic-operators","status":"fully_proved","title":"Hypercyclic operators"},{"id":"Analysis/operator-theory/bounded-operators/hypercyclic-operators/is-hypercyclic","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Hypercyclicity.IsHypercyclic","parent":15789,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/hypercyclic-operators/index.html#is-hypercyclic","scope":"Analysis/operator-theory/bounded-operators/hypercyclic-operators","status":"fully_proved","summary":"A bounded operator `T` is *hypercyclic* if some vector has a dense orbit under iteration of `T`.","title":"Hypercyclic operator"},{"id":"Analysis/operator-theory/bounded-operators/hypercyclic-operators/is-hypercyclic-of-criterion","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Hypercyclicity.isHypercyclic_of_criterion","parent":15789,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/hypercyclic-operators/index.html#is-hypercyclic-of-criterion","scope":"Analysis/operator-theory/bounded-operators/hypercyclic-operators","status":"fully_proved","summary":"The *hypercyclicity criterion*: along a sequence `n_k \u2192 \u221e`, suppose `T^(n_k) x \u2192 0` on the dense set `X\u2080`, `S k y \u2192 0` on the dense set `Y\u2080`, and `T^(n_k) (S k y) \u2192 y` on `Y\u2080`. Then `T` is hypercyclic. The Baire proof combines all three eventual estimates and uses cofinality to meet each rational ball after every prescribed time threshold. This includes the full-iterate, exact-inverse special case by taking\u2026","title":"Hypercyclicity criterion"},{"data_url":"scopes/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation.json","id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","kind":"scope","navigation":"module","parent":15749,"result":false,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","title":"Robertson\u2013Schr\u00f6dinger uncertainty relation"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/mean-value","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.meanValue","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#mean-value","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The normalized real expectation is Mathlib's canonical Rayleigh quotient. For a nonzero state and self-adjoint observable it is the physical expectation. At zero this total definition is zero; zero is not a physical state.","title":"Normalized expectation value of an observable"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/mean-value-of-norm-eq-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.meanValue_of_norm_eq_one","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#mean-value-of-norm-eq-one","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"On a unit vector, normalized expectation equals the original quadratic mean.","title":"Normalized expectation on a unit vector"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/centered","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.centered","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#centered","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The centered vector subtracts the normalized real expectation times the state. Its squared norm divided by the squared state norm is `variance`.","title":"Centered state vector of an observable"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/variance","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.variance","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#variance","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"Normalized variance is the centered squared norm divided by the squared state norm. For a nonzero state and self-adjoint observable it is the physical variance. The totalized value at zero is zero. Off the unit sphere this corrects the raw quadratic quantities; agreement with the original formulas is asserted only for unit states.","title":"Normalized variance of an observable"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/variance-of-norm-eq-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.variance_of_norm_eq_one","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#variance-of-norm-eq-one","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"On the unit sphere, normalized variance is the centered squared norm.","title":"Normalized variance on the unit sphere"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/variance-smul","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.variance_smul","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#variance-smul","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"Nonzero scalar multiplication does not change normalized variance.","title":"Normalized variance is invariant under nonzero scaling"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/inner-self-of-is-self-adjoint","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.inner_self_of_isSelfAdjoint","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#inner-self-of-is-self-adjoint","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"For a self-adjoint operator, the expectation inner product is the normalized mean multiplied by the squared state norm. The norm factor is required off the unit sphere.","title":"Expectation inner product of a self-adjoint operator"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/inner-right-of-is-self-adjoint","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.inner_right_of_isSelfAdjoint","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#inner-right-of-is-self-adjoint","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The flipped expectation has the same necessary squared-state-norm factor.","title":"Flipped expectation inner product of a self-adjoint operator"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/inner-centered","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.inner_centered","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#inner-centered","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The inner product of two centered vectors, for self-adjoint operators at a unit vector: the uncentered inner product minus the product of the mean values.","title":"Inner product of two centered vectors"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/commutator-inner","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.commutator_inner","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#commutator-inner","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The commutator expectation is the skew part of the inner product of the centered vectors.","title":"Commutator expectation as the skew part of the centered inner product"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/commutator-inner-re","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.commutator_inner_re","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#commutator-inner-re","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The commutator expectation of two self-adjoint operators is purely imaginary: its real part vanishes.","title":"Commutator expectation of self-adjoint operators is purely imaginary"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/norm-sq-commutator-inner","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.norm_sq_commutator_inner","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#norm-sq-commutator-inner","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The squared norm of the commutator expectation is four times the squared imaginary part of the centered inner product.","title":"Squared norm of the commutator expectation"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/norm-commutator-inner-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.norm_commutator_inner_eq","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#norm-commutator-inner-eq","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The commutator expectation has norm twice the absolute imaginary part of the centered inner product.","title":"Norm of the commutator expectation"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/schrodinger","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.schrodinger","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#schrodinger","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The Robertson\u2013Schr\u00f6dinger uncertainty principle: the product of the variances is at least the squared covariance plus a quarter of the squared norm of the commutator expectation.","title":"Robertson\u2013Schr\u00f6dinger uncertainty principle"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/robertson","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.robertson","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#robertson","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The product of the variances is at least a quarter of the squared norm of the commutator expectation.","title":"Robertson's uncertainty principle"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/commutator-eq-zero-of-commute","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.commutator_eq_zero_of_commute","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#commutator-eq-zero-of-commute","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The commutator of commuting operators vanishes.","title":"Commutator of commuting operators vanishes"},{"id":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/variance-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RobertsonUncertainty.variance_eq","parent":15792,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation/index.html#variance-eq","scope":"Analysis/operator-theory/bounded-operators/robertson-schr-dinger-uncertainty-relation","status":"fully_proved","summary":"The variance is the second moment minus the squared mean.","title":"Variance as second moment minus squared mean"},{"data_url":"scopes/Analysis/operator-theory/bounded-operators/uniform-boundedness-principle.json","id":"Analysis/operator-theory/bounded-operators/uniform-boundedness-principle","kind":"scope","navigation":"module","parent":15749,"result":false,"route":"../roadmap/Analysis/operator-theory/bounded-operators/uniform-boundedness-principle/index.html","scope":"Analysis/operator-theory/bounded-operators/uniform-boundedness-principle","status":"can_prove","title":"Uniform boundedness principle"},{"id":"Analysis/operator-theory/bounded-operators/uniform-boundedness-principle/tendsto-apply-of-tendsto-pointwise-of-te","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.PointwiseLimitOperatorsJointContinuityWanted.tendsto_apply_of_tendsto_pointwise_of_tendsto","parent":15810,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/uniform-boundedness-principle/index.html#tendsto-apply-of-tendsto-pointwise-of-te","scope":"Analysis/operator-theory/bounded-operators/uniform-boundedness-principle","status":"can_prove","summary":"Let $E$ and $F$ be real Banach spaces, let $(T_n)$ be a sequence of bounded linear operators from $E$ to $F$, and suppose that for every $x \\in E$ the sequence $(T_n x)$ converges to a limit $T x$. Then for every sequence $x_n \\to x$ in $E$ we have $T_n x_n \\to T x$ in $F$.","title":"Pointwise convergent operators converge along convergent sequences"},{"id":"Analysis/operator-theory/bounded-operators/uniform-boundedness-principle/exists-bound-op-norm-of-forall-inner-bou","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.WeaklyBoundedOperatorFamilyBoundedWanted.exists_bound_opNorm_of_forall_inner_bounded","parent":15810,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/uniform-boundedness-principle/index.html#exists-bound-op-norm-of-forall-inner-bou","scope":"Analysis/operator-theory/bounded-operators/uniform-boundedness-principle","status":"can_prove","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $H$ and $K$ be Hilbert spaces over $\\mathbb{K}$, and let $Q$ be a set of bounded linear operators from $H$ to $K$. Suppose $Q$ is weakly bounded: for every $f \\in H$ and $g \\in K$ there is a real number $\\alpha$ such that $|\\langle A f, g \\rangle| \\le \\alpha$ for all $A \\in Q$. Then $Q$ is bounded in operator norm: there is a real number $M$ such that\u2026","title":"Weakly bounded sets of operators are bounded"},{"data_url":"scopes/Analysis/operator-theory/bounded-operators/wold-decomposition.json","id":"Analysis/operator-theory/bounded-operators/wold-decomposition","kind":"scope","navigation":"module","parent":15749,"result":false,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","title":"Wold decomposition"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/wandering-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.wanderingSub","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#wandering-sub","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The wandering subspace of an isometry `V`: the orthogonal complement of its range. Vectors here are orthogonal to every vector in the range of `V`.","title":"Wandering subspace of an isometry"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/inner-pow-pullback","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.inner_pow_pullback","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#inner-pow-pullback","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"Pulling inner products back along powers of an isometry.","title":"Powers of an isometry preserve inner products"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/pow-mem-range","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.pow_mem_range","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#pow-mem-range","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"A positive power of an isometry lands in the range of the isometry.","title":"Positive powers of an isometry land in its range"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/inner-pow-wandering","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.inner_pow_wandering","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#inner-pow-wandering","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"A wandering vector, pushed forward `k` times, is orthogonal to any vector pushed forward strictly more times.","title":"Orthogonality of pushed-forward wandering vectors"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/inner-pow-wandering-2","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.inner_pow_wandering'","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#inner-pow-wandering-2","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The symmetric form of `inner_pow_wandering`.","title":"Orthogonality of pushed-forward wandering vectors, symmetric form"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/is-compl-wandering","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.isCompl_wandering","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#is-compl-wandering","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The ambient space splits as the wandering subspace plus the range.","title":"Wandering subspace and range are complementary"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/exists-wandering-add-range","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.exists_wandering_add_range","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#exists-wandering-add-range","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"One step of the Wold decomposition: every vector splits as a wandering vector plus a vector in the range.","title":"One-step Wold splitting into wandering vector plus range"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/exists-wold-decomposition","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.exists_wold_decomposition","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#exists-wold-decomposition","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"Finite Wold decomposition: every vector is a finite sum of pushed-forward wandering vectors plus a remainder in the range of `V ^ n`. The wandering coefficients vanish at and beyond `n`.","title":"Finite Wold decomposition"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/wold-norm-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.wold_norm_sq","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#wold-norm-sq","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"Pythagoras for a Wold decomposition: the squared norm of a vector is the sum of the squared norms of its wandering coefficients plus the squared norm of the remainder.","title":"Pythagorean identity for the Wold decomposition"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/wold-decomposition-unique","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.wold_decomposition_unique","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#wold-decomposition-unique","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"Uniqueness of the Wold decomposition at level `n`: the remainder and the finite wandering sum are uniquely determined.","title":"Uniqueness of the finite Wold decomposition"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/residual-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.residualSub","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#residual-sub","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The residual subspace of an isometry: vectors in the range of every power of `V`.","title":"Residual subspace of an isometry"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/maps-to-residual","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.mapsTo_residual","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#maps-to-residual","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The isometry maps its residual subspace into itself.","title":"An isometry maps its residual subspace into itself"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/inner-residual-wandering","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.inner_residual_wandering","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#inner-residual-wandering","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The residual subspace is orthogonal to every pushed-forward wandering vector: the wandering part and the residual part of Wold's decomposition are orthogonal.","title":"Residual subspace is orthogonal to the wandering part"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/restrict-residual-bijective","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.restrict_residual_bijective","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#restrict-residual-bijective","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The restriction of an isometry to its residual subspace is bijective.","title":"Isometry restricted to the residual subspace is bijective"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/inner-restrict-residual","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.inner_restrict_residual","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#inner-restrict-residual","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The restriction of an isometry to its residual subspace preserves inner products.","title":"Isometry restricted to the residual subspace preserves inner products"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/wold-residual-equiv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.woldResidualEquiv","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#wold-residual-equiv","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"The isometry restricts to a unitary operator (a linear isometric equivalence) on its residual subspace.","title":"Unitary part of an isometry on its residual subspace"},{"id":"Analysis/operator-theory/bounded-operators/wold-decomposition/wold-residual-equiv-apply-coe","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WoldDecomposition.woldResidualEquiv_apply_coe","parent":15813,"result":true,"route":"../roadmap/Analysis/operator-theory/bounded-operators/wold-decomposition/index.html#wold-residual-equiv-apply-coe","scope":"Analysis/operator-theory/bounded-operators/wold-decomposition","status":"fully_proved","summary":"Applying the residual unitary is the same as applying `V` and viewing the result in the ambient space.","title":"Action of the residual unitary of an isometry"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel.json","id":"Analysis/operator-theory/shift-toeplitz-hankel","kind":"scope","parent":15497,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel","status":"can_prove","summary":"Unilateral, bilateral and weighted shifts and their spectra, indices and numerical ranges, Rolewicz backward shifts, Toeplitz, Hankel and composition operators on \u2113\u00b2.","title":"Shift, Toeplitz and Hankel operators"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift.json","id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","kind":"scope","navigation":"module","parent":15831,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","title":"Bilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/bilateral-shift-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftOperator.bilateralShiftFun","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#bilateral-shift-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"For a function $x : \\mathbb{Z} \\to K$ with values in any set $K$, the bilateral (forward) shift of $x$ is the function $Sx : \\mathbb{Z} \\to K$ given by $(Sx)(n) = x(n - 1)$.","title":"Bilateral shift of a sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/bilateral-backward-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftOperator.bilateralBackwardFun","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#bilateral-backward-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"For a function $x : \\mathbb{Z} \\to K$ with values in any set $K$, the bilateral backward shift of $x$ is the function $Bx : \\mathbb{Z} \\to K$ given by $(Bx)(n) = x(n + 1)$.","title":"Bilateral backward shift of a sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/bilateral-shift-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftOperator.bilateralShiftLinMap","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#bilateral-shift-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\ell^2(\\mathbb{Z}, \\mathbb{K})$ be the space of square-summable functions $\\mathbb{Z} \\to \\mathbb{K}$. The bilateral shift is the $\\mathbb{K}$-linear map $S : \\ell^2(\\mathbb{Z}, \\mathbb{K}) \\to \\ell^2(\\mathbb{Z}, \\mathbb{K})$ given by $(Sx)(n) = x(n - 1)$ for all $n \\in \\mathbb{Z}$.","title":"Bilateral shift as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/bilateral-backward-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftOperator.bilateralBackwardLinMap","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#bilateral-backward-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\ell^2(\\mathbb{Z}, \\mathbb{K})$ be the space of square-summable functions $\\mathbb{Z} \\to \\mathbb{K}$. The bilateral backward shift is the $\\mathbb{K}$-linear map $B : \\ell^2(\\mathbb{Z}, \\mathbb{K}) \\to \\ell^2(\\mathbb{Z}, \\mathbb{K})$ given by $(Bx)(n) = x(n + 1)$ for all $n \\in \\mathbb{Z}$.","title":"Bilateral backward shift as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/bilateral-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftOperator.bilateralShiftCLM","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#bilateral-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\ell^2(\\mathbb{Z}, \\mathbb{K})$ be the Hilbert space of square-summable functions $\\mathbb{Z} \\to \\mathbb{K}$. The bilateral shift is the bounded $\\mathbb{K}$-linear operator $S$ on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$ given by $(Sx)(n) = x(n - 1)$ for all $n \\in \\mathbb{Z}$.","title":"Bilateral shift operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/bilateral-backward-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftOperator.bilateralBackwardCLM","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#bilateral-backward-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\ell^2(\\mathbb{Z}, \\mathbb{K})$ be the Hilbert space of square-summable functions $\\mathbb{Z} \\to \\mathbb{K}$. The bilateral backward shift is the bounded $\\mathbb{K}$-linear operator $B$ on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$ given by $(Bx)(n) = x(n + 1)$ for all $n \\in \\mathbb{Z}$.","title":"Bilateral backward shift operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/mul-star-eq-one-of-norm-eq-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.mul_star_eq_one_of_norm_eq_one","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#mul-star-eq-one-of-norm-eq-one","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. If $\\mu \\in \\mathbb{K}$ satisfies $|\\mu| = 1$, then $\\mu \\bar{\\mu} = 1$, where $\\bar{\\mu}$ is the complex conjugate of $\\mu$.","title":"A unit scalar times its conjugate is one"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/star-ne-zero-of-norm-eq-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.star_ne_zero_of_norm_eq_one","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#star-ne-zero-of-norm-eq-one","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. If $\\mu \\in \\mathbb{K}$ satisfies $|\\mu| = 1$, then its conjugate $\\bar{\\mu}$ is nonzero.","title":"Conjugates of unimodular scalars are nonzero"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/mul-star-zpow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.mul_star_zpow","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#mul-star-zpow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and write $\\bar{\\mu}$ for the complex conjugate of $\\mu \\in \\mathbb{K}$. If $|\\mu| = 1$, then for every integer $n$, $\\mu \\bar{\\mu}^{\\,n} = \\bar{\\mu}^{\\,n-1}$.","title":"Multiplying a power of the conjugate by a unit scalar"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/approx-eigen-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.approxEigenFun","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#approx-eigen-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\mu \\in \\mathbb{K}$ and $N \\in \\mathbb{N}$, and write $\\bar{\\mu}$ for the complex conjugate of $\\mu$. This is the function $f_{\\mu,N} : \\mathbb{Z} \\to \\mathbb{K}$ with $f_{\\mu,N}(n) = \\bar{\\mu}^{\\,n}$ for $-N \\le n \\le N$ and $f_{\\mu,N}(n) = 0$ otherwise (negative powers are inverses, with the convention $0^{n} = 0$ for $n < 0$).","title":"Truncated approximate eigenfunction of the bilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/approx-eigen-vec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.approxEigenVec","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#approx-eigen-vec","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\mu \\in \\mathbb{K}$ and $N \\in \\mathbb{N}$, and write $\\bar{\\mu}$ for the complex conjugate of $\\mu$. This is the vector $v_{\\mu,N} \\in \\ell^2(\\mathbb{Z}, \\mathbb{K})$ whose $n$-th coordinate is the truncated approximate eigenfunction $f_{\\mu,N}(n)$, that is, $\\bar{\\mu}^{\\,n}$ for $-N \\le n \\le N$ and $0$ otherwise; it lies in $\\ell^2$ because it has finite\u2026","title":"Truncated approximate eigenvector of the bilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/norm-sq-approx-eigen-vec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.norm_sq_approxEigenVec","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#norm-sq-approx-eigen-vec","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\mu \\in \\mathbb{K}$ with $|\\mu| = 1$, and let $N \\in \\mathbb{N}$. The truncated approximate eigenvector $v_{\\mu,N} \\in \\ell^2(\\mathbb{Z}, \\mathbb{K})$, whose $n$-th coordinate is $\\bar{\\mu}^{\\,n}$ for $-N \\le n \\le N$ and $0$ otherwise ($\\bar{\\mu}$ the complex conjugate of $\\mu$), satisfies $\\|v_{\\mu,N}\\|^2 = 2N + 1$.","title":"Norm of the truncated approximate eigenvector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/algebra-map-sub-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.algebraMap_sub_apply","parent":15832,"reading_title":{"format":"inline-parts-v1","parts":[["text","Applying "],["math","\\(\\mu I - T\\)"],["text"," on "],["math","\\(\\ell^2(\\mathbb{Z})\\)"]],"source":"Applying $\\mu I - T$ on $\\ell^2(\\mathbb{Z})$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#algebra-map-sub-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a bounded linear operator on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$, let $\\mu \\in \\mathbb{K}$, and let $\\mu I$ be the image of $\\mu$ under the algebra map from $\\mathbb{K}$ to the bounded operators on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$, that is, $\\mu$ times the identity. Then for every $x \\in \\ell^2(\\mathbb{Z}, \\mathbb{K})$, $(\\mu I - T)x = \\mu x - Tx$.","title":"Applying $\\mu I - T$ on $\\ell^2(\\mathbb{Z})$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/residual-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.residual_apply","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#residual-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be the bilateral shift on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$, and let $\\mu \\in \\mathbb{K}$, $N \\in \\mathbb{N}$ and $n \\in \\mathbb{Z}$. Let $f_{\\mu,N}$ be the truncated approximate eigenfunction and $v_{\\mu,N} \\in \\ell^2(\\mathbb{Z}, \\mathbb{K})$ the corresponding truncated approximate eigenvector. Then the $n$-th coordinate of $(\\mu I - S)v_{\\mu,N}$ is\u2026","title":"Coordinates of the residual of the truncated approximate eigenvector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/norm-sq-residual-approx-eigen-vec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.norm_sq_residual_approxEigenVec","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#norm-sq-residual-approx-eigen-vec","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be the bilateral shift on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$, let $\\mu \\in \\mathbb{K}$ with $|\\mu| = 1$, and let $N \\in \\mathbb{N}$. The truncated approximate eigenvector $v_{\\mu,N}$ satisfies $\\|(\\mu I - S)v_{\\mu,N}\\|^2 = 2$, where $\\mu I$ is $\\mu$ times the identity operator.","title":"Residual of the truncated approximate eigenvector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/mem-spectrum-bilateral-shift-clm-of-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.mem_spectrum_bilateralShiftCLM_of_norm_eq_one","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#mem-spectrum-bilateral-shift-clm-of-norm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $S$ be the bilateral shift on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$. Every $\\mu \\in \\mathbb{K}$ with $|\\mu| = 1$ lies in the spectrum of $S$ in the algebra of bounded operators on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$, that is, $\\mu I - S$ is not invertible there, where $\\mu I$ is $\\mu$ times the identity operator.","title":"Every point of the unit circle lies in the spectrum of the bilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/spectrum-bilateral-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.spectrum_bilateralShiftCLM","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#spectrum-bilateral-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The spectrum of the bilateral shift, as a bounded operator on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$, is the set of scalars $\\mu \\in \\mathbb{K}$ with $|\\mu| = 1$; for $\\mathbb{K} = \\mathbb{C}$ this is the unit circle.","title":"Spectrum of the bilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/spectrum-bilateral-backward-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BilateralShiftSpectrum.spectrum_bilateralBackwardCLM","parent":15832,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift/index.html#spectrum-bilateral-backward-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/bilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $B$ be the bilateral backward shift on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$. The spectrum of $B$ in the algebra of bounded operators on $\\ell^2(\\mathbb{Z}, \\mathbb{K})$ is the unit circle $\\{\\mu \\in \\mathbb{K} : |\\mu| = 1\\}$.","title":"Spectrum of the bilateral backward shift"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on.json","id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","kind":"scope","navigation":"module","parent":15831,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","title":"Composition operators on \u2113\u00b2"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/has-bounded-fibers","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.HasBoundedFibers","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#has-bounded-fibers","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The fibers of `\u03c8` are finite and uniformly bounded by `M`: every value is taken at most `M` times. This is the boundedness hypothesis for the composition operator.","title":"Uniformly bounded fibers of a map on the naturals"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/fiber-card","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.fiberCard","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#fiber-card","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The number of preimages of `m` under `\u03c8`, defined for a finite fiber.","title":"Fiber cardinality of a map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/fiber-card-eq-ncard","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.fiberCard_eq_ncard","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#fiber-card-eq-ncard","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"On its finite domain, the fiber count agrees with the standard natural cardinality.","title":"Fiber count agrees with set cardinality"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/fiber-sup","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.fiberSup","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#fiber-sup","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The real supremum of the fiber counts of `\u03c8`, defined for uniformly bounded finite fibers. The bound in `h\u03c8` ensures that the real-valued family is bounded above.","title":"Supremum of the fiber counts of a map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/fiber-sup-eq-i-sup","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.fiberSup_eq_iSup","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#fiber-sup-eq-i-sup","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"On its bounded finite domain, the fiber supremum agrees with the supremum of natural cardinalities. Its value is independent of the chosen bound and evidence.","title":"Fiber supremum as a supremum of cardinalities"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compFun","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The composition function: precompose a sequence with `\u03c8`.","title":"Precomposition of a sequence with a map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/mem-p-comp-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.mem\u2113p_compFun","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#mem-p-comp-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The composed sequence of an `\u2113\u00b2` sequence is in `\u2113\u00b2`, under the bounded fibers hypothesis.","title":"Precomposition preserves square-summability"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/tsum-comp-fun-norm-sq-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.tsum_compFun_norm_sq_le","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#tsum-comp-fun-norm-sq-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The squared-norm sum of the composed sequence is at most `fiberSup \u03c8 h\u03c8` times the squared-norm sum of the original sequence.","title":"Squared-norm sum bound for a precomposed sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compLinMap","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The composition operator as a linear map.","title":"Composition operator as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compLinMap_apply","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-lin-map-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"Pointwise action of the composition linear map.","title":"Pointwise action of the composition linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/norm-sq-comp-lin-map-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.norm_sq_compLinMap_apply_le","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#norm-sq-comp-lin-map-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The squared norm of `C_\u03c8 x` is at most `fiberSup \u03c8 h\u03c8 * \u2016x\u2016\u00b2`.","title":"Squared-norm bound for the composition linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/fiber-sup-nonneg","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.fiberSup_nonneg","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#fiber-sup-nonneg","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"`fiberSup \u03c8 h\u03c8` is nonnegative.","title":"Fiber supremum is nonnegative"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/norm-comp-lin-map-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.norm_compLinMap_apply_le","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#norm-comp-lin-map-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The composition operator is bounded by `\u221a(fiberSup \u03c8 h\u03c8)`.","title":"Norm bound for the composition linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compCLM","parent":15851,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded composition operator on "],["math","\\(\\ell^2\\)"]],"source":"Bounded composition operator on $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The composition operator as a bounded operator on `\u2113\u00b2(\u2115, \ud835\udd5c)`.","title":"Bounded composition operator on $\\ell^2$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compCLM_apply","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-clm-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"Pointwise action of the composition operator.","title":"Pointwise action of the bounded composition operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/norm-comp-clm-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.norm_compCLM_apply_le","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#norm-comp-clm-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The composition operator is bounded by `\u221a(fiberSup \u03c8 h\u03c8)` on every vector.","title":"Vectorwise norm bound for the composition operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/norm-comp-clm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.norm_compCLM_le","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#norm-comp-clm-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The operator norm of `C_\u03c8` is at most `\u221a(fiberSup \u03c8 h\u03c8)`.","title":"Operator-norm bound for the composition operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/norm-comp-clm-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.norm_compCLM_single","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#norm-comp-clm-single","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The norm of `C_\u03c8` applied to a standard basis vector is the square root of the corresponding fiber count.","title":"Norm of the composition operator on a basis vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/norm-comp-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.norm_compCLM","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#norm-comp-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The operator norm of the composition operator: `\u2016C_\u03c8\u2016 = \u221a(fiberSup \u03c8 h\u03c8)`, the square root of the largest fiber count.","title":"Operator norm of a composition operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/adjoint-comp-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.adjoint_compCLM_apply","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#adjoint-comp-clm-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The adjoint of the composition operator is the fiber-sum operator: `((C_\u03c8)\u2020 y) m` is the sum of `y` over the fiber of `m`.","title":"Adjoint of a composition operator is the fiber-sum operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/has-bounded-fibers-comp","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.hasBoundedFibers_comp","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#has-bounded-fibers-comp","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"Fiber bounds compose: the fibers of `\u03c8 \u2218 \u03c6` are bounded by the product of the bounds.","title":"Bounded fibers are preserved under composition"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-clm-comp","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compCLM_comp","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-clm-comp","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"Composition of composition operators is composition of symbols in the reversed order: `C_\u03c6 \u2218L C_\u03c8 = C_{\u03c8 \u2218 \u03c6}`.","title":"Composition of composition operators"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-clm-eq-of-forall","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compCLM_eq_of_forall","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-clm-eq-of-forall","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"If `\u03c8` is pointwise the identity, `C_\u03c8` is the identity operator.","title":"Composition operator of a pointwise identity symbol"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-clm-id","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compCLM_id","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-clm-id","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"`C_id` is the identity operator.","title":"Composition operator of the identity is the identity"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/has-bounded-fibers-of-injective","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.hasBoundedFibers_of_injective","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#has-bounded-fibers-of-injective","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"An injective symbol has fibers of size at most one.","title":"Injective maps have fibers of size at most one"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/has-bounded-fibers-equiv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.hasBoundedFibers_equiv","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#has-bounded-fibers-equiv","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"A permutation symbol has fibers of size one.","title":"Permutations have fibers of size one"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/adjoint-comp-clm-of-equiv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.adjoint_compCLM_of_equiv","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#adjoint-comp-clm-of-equiv","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"The adjoint of the composition operator for a permutation symbol is the composition operator of the inverse permutation.","title":"Adjoint of a composition operator with a permutation symbol"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/comp-clm-mem-unitary","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.compCLM_mem_unitary","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#comp-clm-mem-unitary","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"A permutation symbol gives a unitary composition operator.","title":"Permutation composition operators are unitary"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/injective-comp-clm-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.injective_compCLM_iff","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#injective-comp-clm-iff","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"`C_\u03c8` is injective iff `\u03c8` is surjective: a missing value gives a basis vector in the kernel, and a surjective symbol lets every coordinate be read off from the image.","title":"Composition operator is injective iff its symbol is surjective"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/surjective-comp-clm-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.surjective_compCLM_iff","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#surjective-comp-clm-iff","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"`C_\u03c8` is surjective iff `\u03c8` is injective: if `\u03c8 a = \u03c8 b` with `a \u2260 b`, the `a`-th and `b`-th coordinates of every vector in the range agree, so `e_a` is not in the range; if `\u03c8` is injective, the preimage of `y` is obtained by applying the adjoint and reading off the fiber sums.","title":"Composition operator is surjective iff its symbol is injective"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/has-bounded-fibers-div-two","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.hasBoundedFibers_div_two","parent":15851,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#has-bounded-fibers-div-two","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"Halving pairs: the map `n \u21a6 n / 2` has fibers of size at most two.","title":"Halving map has fibers of size at most two"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/not-is-compact-operator-comp-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CompositionOperator.not_isCompactOperator_compCLM","parent":15851,"reading_title":{"format":"inline-parts-v1","parts":[["text","Composition operators on "],["math","\\(\\ell^2\\)"],["text"," are never compact"]],"source":"Composition operators on $\\ell^2$ are never compact"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on/index.html#not-is-compact-operator-comp-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/composition-operators-on","status":"fully_proved","summary":"A composition operator on `\u2113\u00b2(\u2115, \ud835\udd5c)` is never compact: the images of the basis vectors indexed by the (necessarily infinite) range of `\u03c8` are pairwise orthogonal and bounded below in norm, so they have no Cauchy subsequence.","title":"Composition operators on $\\ell^2$ are never compact"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators.json","id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","kind":"scope","navigation":"module","parent":15831,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","title":"Hankel operators"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-slice-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelSliceFun","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-slice-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The `k`-th anti-diagonal slice function: reverse the first `k + 1` coordinates, kill the rest.","title":"Anti-diagonal slice of a sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-mem-p-slice-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankel_mem\u2113p_sliceFun","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-mem-p-slice-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The slice of an `\u2113\u00b2` sequence is in `\u2113\u00b2`.","title":"Anti-diagonal slice of an \u2113\u00b2 sequence lies in \u2113\u00b2"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-slice-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelSliceLinMap","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-slice-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The `k`-th anti-diagonal slice as a linear map.","title":"Anti-diagonal slice as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-slice-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelSliceLinMap_apply","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-slice-lin-map-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"Pointwise action of the slice linear map.","title":"Pointwise action of the anti-diagonal slice map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-tsum-norm-sq-slice-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankel_tsum_normSq_slice_le","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-tsum-norm-sq-slice-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The (squared) norms of a sliced sequence sum to at most the squared norms of the original sequence: only the first `k + 1` coordinates survive, in reverse order.","title":"Anti-diagonal slice does not increase the sum of squared norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/norm-hankel-slice-lin-map-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.norm_hankelSliceLinMap_apply_le","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#norm-hankel-slice-lin-map-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"A slice does not increase norms.","title":"Anti-diagonal slice map does not increase norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-slice","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelSlice","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-slice","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The `k`-th anti-diagonal slice as a bounded operator.","title":"Anti-diagonal slice operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-slice-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelSlice_apply","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-slice-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"Pointwise action of the slice operator.","title":"Pointwise action of the anti-diagonal slice operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/norm-hankel-slice-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.norm_hankelSlice_le","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#norm-hankel-slice-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The operator norm of a slice is at most `1`.","title":"Anti-diagonal slice operator has norm at most one"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-slice-inner-symm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelSlice_inner_symm","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-slice-inner-symm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"A slice is symmetric for the inner product: the anti-diagonal symmetry is a finite-sum reflection.","title":"Anti-diagonal slice is symmetric"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/adjoint-hankel-slice","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.adjoint_hankelSlice","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#adjoint-hankel-slice","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"A slice is self-adjoint.","title":"Anti-diagonal slice operator is self-adjoint"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-summable-slices","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankel_summable_slices","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-summable-slices","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The slice operators weighted by `a` are summable in operator norm when `a` is absolutely summable.","title":"Summability of weighted anti-diagonal slices"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelOperator","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-operator","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The Hankel operator with symbol `a`. Defined as the norm-convergent series `\u2211' k, a k \u2022 hankelSlice k` over the anti-diagonal slices.","title":"Hankel operator with absolutely summable symbol"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-has-sum-slices","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankel_hasSum_slices","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-has-sum-slices","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The Hankel operator is the sum of its slices.","title":"Hankel operator as the sum of its slices"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-operator-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelOperator_apply","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-operator-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"Pointwise action of the Hankel operator: `(H\u2090 x) i = \u2211' j, a (i + j) * x j`.","title":"Matrix entries of the Hankel operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-operator-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankelOperator_zero","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-operator-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"The zero symbol gives the zero operator.","title":"Hankel operator with zero symbol is zero"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/norm-hankel-operator-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.norm_hankelOperator_le","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#norm-hankel-operator-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"Operator-norm bound: `\u2016H\u2090\u2016 \u2264 \u2211' k, \u2016a k\u2016`.","title":"Norm bound for the Hankel operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/adjoint-hankel-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.adjoint_hankelOperator","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#adjoint-hankel-operator","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"Adjoint of a Hankel operator: `(H\u2090)\u2020 = H_{star a}`, by passing the adjoint through the finite partial sums and taking the limit.","title":"Adjoint of a Hankel operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/is-self-adjoint-hankel-operator-of-foral","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.isSelfAdjoint_hankelOperator_of_forall_star","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#is-self-adjoint-hankel-operator-of-foral","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"A Hankel operator with a real-valued (`star`-fixed) symbol is self-adjoint.","title":"Hankel operator with real symbol is self-adjoint"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/hankel-is-compact-operator-partial-sum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.hankel_isCompactOperator_partialSum","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#hankel-is-compact-operator-partial-sum","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"A finite partial sum of the slice series has finite-dimensional range, hence is a compact operator.","title":"Partial sums of the Hankel slice series are compact"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/is-compact-operator-hankel-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HankelOperator.isCompactOperator_hankelOperator","parent":15884,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators/index.html#is-compact-operator-hankel-operator","scope":"Analysis/operator-theory/shift-toeplitz-hankel/hankel-operators","status":"fully_proved","summary":"A Hankel operator with an absolutely summable symbol is compact: it is the norm limit of its finite partial sums, each of which has finite-dimensional range.","title":"Hankel operator with absolutely summable symbol is compact"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries.json","id":"Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries","kind":"scope","navigation":"module","parent":15831,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries","status":"can_prove","title":"Summability of Hankel matrix entries"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries/scottish-book-problem88-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook88Wanted.ScottishBookProblem88Claim","parent":15906,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries/index.html#scottish-book-problem88-claim","scope":"Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries","status":"fully_proved","summary":"The claim that every real sequence $(a_n)_{n \\ge 0}$ with the following property satisfies $\\sum_n n |a_n| < \\infty$: for every real sequence $(x_n)_{n \\ge 0}$ with $|x_n| \\le 1$ for all $n$, each series $\\sum_n a_{m+n} x_n$ with $m \\ge 0$ converges, as the limit of its partial sums, to some $L_m$, and $\\sum_m |L_m| < \\infty$.","title":"Scottish Book Problem 88 (Mazur)"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries/mazur88-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook88Wanted.mazur88_refuted","parent":15906,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries/index.html#mazur88-refuted","scope":"Analysis/operator-theory/shift-toeplitz-hankel/summability-of-hankel-matrix-entries","status":"can_prove","summary":"The Scottish Book Problem 88 claim is false: there is a real sequence $(a_n)_{n \\ge 0}$ such that for every real sequence $(x_n)_{n \\ge 0}$ with $|x_n| \\le 1$ for all $n$, each series $\\sum_n a_{m+n} x_n$ with $m \\ge 0$ converges, as the limit of its partial sums, to some $L_m$, and $\\sum_m |L_m| < \\infty$, but $\\sum_n n |a_n|$ diverges.","title":"Mazur's Scottish Book Problem 88 fails"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift.json","id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","kind":"scope","navigation":"module","parent":15831,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"can_prove","title":"Unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.shiftFun","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The forward shift of a sequence $x : \\mathbb{N} \\to \\mathbb{K}$ is the sequence $S x$ with $(S x)_0 = 0$ and $(S x)_{n+1} = x_n$ for every $n \\in \\mathbb{N}$.","title":"Forward shift of a sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.backwardFun","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"For any type $K$, the backward shift of a sequence $x \\colon \\mathbb{N} \\to K$ is the sequence $n \\mapsto x_{n+1}$: it drops the coordinate $x_0$ and shifts every other coordinate down by one.","title":"Backward shift of a sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-p-shift-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.mem\u2113p_shiftFun","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-p-shift-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every square-summable sequence $x = (x_0, x_1, x_2, \\dots)$ in $\\mathbb{K}$, the forward-shifted sequence $(0, x_0, x_1, \\dots)$ is also square-summable, that is, it belongs to $\\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The forward shift of a square-summable sequence is square-summable"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-p-backward-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.mem\u2113p_backwardFun","parent":15909,"reading_title":{"format":"inline-parts-v1","parts":[["text","The backward shift of an "],["math","\\(\\ell^2\\)"],["text"," sequence is in "],["math","\\(\\ell^2\\)"]],"source":"The backward shift of an $\\ell^2$ sequence is in $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-p-backward-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the backward shift $n \\mapsto x_{n+1}$ of the sequence $x$ is square-summable, that is, it lies in $\\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The backward shift of an $\\ell^2$ sequence is in $\\ell^2$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.shiftLinMap","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The forward shift is the $\\mathbb{K}$-linear map $\\ell^2(\\mathbb{N}, \\mathbb{K}) \\to \\ell^2(\\mathbb{N}, \\mathbb{K})$ sending $x$ to the forward shift of the sequence $x$, that is, to $(0, x_0, x_1, \\dots)$, which is again square-summable.","title":"Forward shift as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.backwardLinMap","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The backward-shift linear map is the $\\mathbb{K}$-linear map $B$ from $\\ell^2(\\mathbb{N}, \\mathbb{K})$ to itself given by $(Bx)_n = x_{n+1}$ for all $x$ and all $n \\in \\mathbb{N}$.","title":"Backward shift as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.shiftLinMap_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-lin-map-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $i \\in \\mathbb{N}$, the $i$-th coordinate of the forward-shift linear map applied to $x$ equals the $i$-th term of the forward shift of the sequence $x$, namely $0$ for $i = 0$ and $x_{i-1}$ for $i \\ge 1$.","title":"Coordinates of the forward-shift linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.backwardLinMap_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-lin-map-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $i \\in \\mathbb{N}$, the $i$-th coordinate of the image of $x$ under the backward-shift linear map $B$ is the $i$-th term of the backward shift of the sequence $x$, that is, $(Bx)_i = x_{i+1}$.","title":"Coordinates of the backward-shift linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/tsum-shift-fun-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.tsum_shiftFun_norm","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#tsum-shift-fun-norm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the forward shift $S x = (0, x_0, x_1, \\dots)$ of $x$ satisfies $\\sum_{i=0}^{\\infty} |(S x)_i|^2 = \\sum_{i=0}^{\\infty} |x_i|^2$.","title":"Square sum of the forward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/tsum-backward-fun-norm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.tsum_backwardFun_norm_le","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#tsum-backward-fun-norm-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the backward shift $(x_1, x_2, \\dots)$ of $x$ satisfies $\\sum_{i=0}^{\\infty} |x_{i+1}|^2 \\le \\sum_{i=0}^{\\infty} |x_i|^2$: the left side is a tail of the right side.","title":"Square sum of the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-shift-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.norm_shiftLinMap_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-shift-lin-map-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $S$ be the forward shift on the space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$, viewed as a linear map and given by $S(x_0, x_1, \\dots) = (0, x_0, x_1, \\dots)$. Then $\\|Sx\\| = \\|x\\|$ for every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The forward shift linear map preserves norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-backward-lin-map-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.norm_backwardLinMap_apply_le","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-backward-lin-map-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $B$ be the backward shift on the space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$, viewed as a linear map and given by $(Bx)_n = x_{n+1}$. Then $\\|Bx\\| \\le \\|x\\|$ for every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The backward shift linear map does not increase norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The unilateral shift is the bounded linear operator $S$ on the Hilbert space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$ given by $S(x_0, x_1, x_2, \\dots) = (0, x_0, x_1, \\dots)$. It is bounded because $\\|Sx\\| = \\|x\\|$ for every $x$.","title":"The unilateral shift operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.backwardCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The backward shift is the bounded linear operator $B$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ given by $(Bx)_n = x_{n+1}$ for all $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and $n \\in \\mathbb{N}$.","title":"Backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.shiftCLM_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-clm-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $S$ be the unilateral shift on the Hilbert space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $i \\in \\mathbb{N}$, the coordinate $(Sx)_i$ is the $i$-th term of the forward shift of $x$: $(Sx)_0 = 0$ and $(Sx)_{n+1} = x_n$ for every $n \\in \\mathbb{N}$.","title":"Coordinates of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.backwardCLM_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-clm-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $i \\in \\mathbb{N}$, the $i$-th coordinate of the backward shift $Bx$ is the $i$-th term of the backward shift of the sequence $x$, that is, $(Bx)_i = x_{i+1}$.","title":"Coordinates of the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-shift-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.norm_shiftCLM_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-shift-clm-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $S$ be the unilateral shift on the Hilbert space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$, given by $S(x_0, x_1, \\dots) = (0, x_0, x_1, \\dots)$. Then $\\|Sx\\| = \\|x\\|$ for every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The unilateral shift preserves norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/isometry-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.isometry_shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#isometry-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The unilateral shift $S$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is an isometry: $\\|Sx - Sy\\| = \\|x - y\\|$ for all $x, y \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The unilateral shift preserves distances"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.norm_shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $S$ be the unilateral shift on the Hilbert space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$, given by $S(x_0, x_1, \\dots) = (0, x_0, x_1, \\dots)$. Then the operator norm of $S$ is $\\|S\\| = 1$.","title":"The unilateral shift has operator norm one"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-backward-clm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.norm_backwardCLM_le","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-backward-clm-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $B$ be the backward shift on the Hilbert space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$, given by $(Bx)_n = x_{n+1}$. Then the operator norm of $B$ satisfies $\\|B\\| \\le 1$.","title":"The backward shift has operator norm at most one"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/adjoint-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.adjoint_shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#adjoint-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The adjoint of the unilateral shift $S$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is the backward shift $B$: $S^* = B$.","title":"Adjoint of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-comp-shift","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.backward_comp_shift","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-comp-shift","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The backward shift $B$ is a left inverse of the unilateral shift $S$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$: $BS = I$, where $I$ is the identity operator.","title":"Left inverse of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/injective-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.injective_shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#injective-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The unilateral shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is injective.","title":"The unilateral shift is injective"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-comp-backward-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.shift_comp_backward_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-comp-backward-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and on the Hilbert space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$ let $S$ be the unilateral shift, $S(x_0, x_1, \\dots) = (0, x_0, x_1, \\dots)$, and $B$ the backward shift, $(Bx)_n = x_{n+1}$. Then for every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $i \\in \\mathbb{N}$, the coordinate $(SBx)_i$ is $0$ if $i = 0$ and is $x_i$\u2026","title":"Coordinates of the unilateral shift after the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-comp-backward","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.shift_comp_backward","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-comp-backward","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be the unilateral (forward) shift and $B$ the backward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, and let $e_0$ be the first standard basis vector. Then $S \\circ B = 1 - P$, where $P$ is the rank-one operator $P x = \\langle e_0, x \\rangle e_0 = x_0 e_0$.","title":"The unilateral shift composed with the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/not-surjective-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperator.not_surjective_shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#not-surjective-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $S$ be the unilateral shift on the Hilbert space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences in $\\mathbb{K}$, given by $S(x_0, x_1, \\dots) = (0, x_0, x_1, \\dots)$. Then $S$ is not surjective.","title":"The unilateral shift is not surjective"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-p-geometric","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.mem\u2113p_geometric","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-p-geometric","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $c \\in \\mathbb{K}$ with $\\|c\\| < 1$. Then the sequence $(c^i)_{i \\in \\mathbb{N}}$ lies in $\\ell^2$, that is, $\\sum_{i} \\|c^i\\|^2 < \\infty$.","title":"Geometric sequences with ratio of norm less than one are square-summable"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/geometric-vec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.geometricVec","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#geometric-vec","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $c \\in \\mathbb{K}$ with $\\|c\\| < 1$. The geometric vector of $c$ is the sequence $(1, c, c^2, \\dots)$, that is $i \\mapsto c^i$, regarded as an element of the Hilbert space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences.","title":"Geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/geometric-vec-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.geometricVec_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#geometric-vec-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $c \\in \\mathbb{K}$ with $\\|c\\| < 1$. For every $i \\in \\mathbb{N}$, the $i$-th coordinate of the geometric vector of $c$ in $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is $c^i$.","title":"Coordinates of the geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/geometric-vec-ne-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.geometricVec_ne_zero","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#geometric-vec-ne-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $c \\in \\mathbb{K}$ with $\\|c\\| < 1$. The geometric vector $(1, c, c^2, \\dots)$ in $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is nonzero, since its $0$-th coordinate is $1$.","title":"The geometric vector is nonzero"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-clm-geometric-vec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.backwardCLM_geometricVec","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-clm-geometric-vec","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $c \\in \\mathbb{K}$ with $\\|c\\| < 1$. Let $v = (1, c, c^2, \\dots)$ be the geometric vector of $c$ in $\\ell^2(\\mathbb{N}, \\mathbb{K})$ and let $B$ be the backward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. Then $Bv = c\\,v$.","title":"Backward shift of the geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-one-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.norm_one_le","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-one-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The operator norm of the identity operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is at most $1$.","title":"The identity on \u2113\u00b2(\u2115) has norm at most one"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/algebra-map-sub-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.algebraMap_sub_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#algebra-map-sub-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $T$ be a bounded linear operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, let $\\mu \\in \\mathbb{K}$, and let $\\mu I$ be the image of $\\mu$ under the algebra map from $\\mathbb{K}$ into the bounded operators, i.e. $\\mu$ times the identity $I$. Then $(\\mu I - T)x = \\mu x - Tx$ for every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"Applying a scalar operator minus an operator on \u2113\u00b2(\u2115)"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-spectrum-shift-clm-of-norm-lt","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.mem_spectrum_shiftCLM_of_norm_lt","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-spectrum-shift-clm-of-norm-lt","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $S$ be the unilateral shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. Every $\\mu \\in \\mathbb{K}$ with $\\|\\mu\\| < 1$ lies in the spectrum of $S$ in the algebra of bounded operators on $\\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The open unit ball lies in the spectrum of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-spectrum-backward-clm-of-norm-lt","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.mem_spectrum_backwardCLM_of_norm_lt","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-spectrum-backward-clm-of-norm-lt","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $B$ be the backward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. Every $\\mu \\in \\mathbb{K}$ with $\\|\\mu\\| < 1$ lies in the spectrum of $B$ in the algebra of bounded operators on $\\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The open unit ball lies in the spectrum of the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/spectrum-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.spectrum_shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#spectrum-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The spectrum over $\\mathbb{K}$ of the unilateral shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is the closed unit ball $\\{\\mu \\in \\mathbb{K} : |\\mu| \\le 1\\}$ of $\\mathbb{K}$.","title":"Spectrum of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/spectrum-backward-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpShiftOperatorSpectrum.spectrum_backwardCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#spectrum-backward-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The spectrum of the backward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, in the algebra of bounded operators, is the closed unit ball $\\{\\mu \\in \\mathbb{K} : \\|\\mu\\| \\le 1\\}$: the closed unit disc when $\\mathbb{K} = \\mathbb{C}$ and the interval $[-1, 1]$ when $\\mathbb{K} = \\mathbb{R}$.","title":"Spectrum of the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-fun-iterate-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.shiftFun_iterate_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-fun-iterate-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Iterating the forward-shift function `n` times inserts `n` zeroes in front of the sequence.","title":"Iterated forward shift inserts leading zeroes"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-fun-iterate-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.backwardFun_iterate_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-fun-iterate-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Iterating the backward-shift function `n` times drops the first `n` coordinates.","title":"Iterated backward shift drops leading coordinates"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-clm-pow-coe","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.shiftCLM_pow_coe","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-clm-pow-coe","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The coercion of `S ^ n` is the `n`-fold iterate of `shiftFun`.","title":"Powers of the unilateral shift as iterates of the shift map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-clm-pow-coe","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.backwardCLM_pow_coe","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-clm-pow-coe","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The coercion of `B ^ n` is the `n`-fold iterate of `backwardFun`.","title":"Powers of the backward shift as iterates of the backward map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-clm-pow-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.shiftCLM_pow_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-clm-pow-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Pointwise action of `S ^ n`: it inserts `n` zeroes in front.","title":"Pointwise action of a power of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-clm-pow-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.backwardCLM_pow_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-clm-pow-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Pointwise action of `B ^ n`: it drops the first `n` coordinates.","title":"Pointwise action of a power of the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/backward-clm-pow-comp-shift-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.backwardCLM_pow_comp_shiftCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#backward-clm-pow-comp-shift-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"`B ^ n` is a left inverse of `S ^ n`: shifting up then down recovers the original sequence.","title":"Backward shift power is a left inverse of the shift power"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/injective-shift-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.injective_shiftCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#injective-shift-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The unilateral shift power is injective.","title":"Injectivity of powers of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/surjective-backward-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.surjective_backwardCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#surjective-backward-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The backward shift power is surjective.","title":"Surjectivity of powers of the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-ker-backward-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.mem_ker_backwardCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-ker-backward-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"A vector is in the kernel of `B ^ n` iff its coordinates from `n` onwards all vanish.","title":"Kernel of a power of the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-range-shift-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.mem_range_shiftCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-range-shift-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"A vector is in the range of `S ^ n` iff its first `n` coordinates vanish.","title":"Range of a power of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/first-coords","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.firstCoords","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#first-coords","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The map taking a sequence to its first `n` coordinates.","title":"First-coordinates map on square-summable sequences"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/ext-zero-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.extZeroFun","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#ext-zero-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The extension of a tuple of `n` coordinates by zeroes.","title":"Extension of a finite tuple by zeroes"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-p-ext-zero-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.mem\u2113p_extZeroFun","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-p-ext-zero-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"A finitely supported sequence is in `\u2113\u00b2`.","title":"Zero-extended finite tuples are square-summable"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/ext-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.extZero","parent":15909,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear extension-by-zero map into "],["math","\\(\\ell^2\\)"]],"source":"Linear extension-by-zero map into $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#ext-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The linear extension-by-zero map `(Fin n \u2192 \ud835\udd5c) \u2192\u2097[\ud835\udd5c] \u2113\u00b2(\u2115, \ud835\udd5c)`.","title":"Linear extension-by-zero map into $\\ell^2$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/ext-zero-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.extZero_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#ext-zero-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Pointwise action of the extension-by-zero map.","title":"Pointwise action of the extension-by-zero map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/first-coords-ext-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.firstCoords_extZero","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#first-coords-ext-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Taking first coordinates after extension by zeroes is the identity.","title":"First coordinates of a zero extension recover the tuple"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/ext-zero-first-coords-of-mem-ker","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.extZero_firstCoords_of_mem_ker","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#ext-zero-first-coords-of-mem-ker","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"A vector in the kernel of `B ^ n` is determined by its first `n` coordinates: extending those coordinates by zeroes recovers it.","title":"Kernel vectors of a backward shift power are zero-extended tuples"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/ker-backward-clm-pow-equiv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.kerBackwardCLM_powEquiv","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#ker-backward-clm-pow-equiv","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The kernel of `B ^ n` is linearly equivalent to `Fin n \u2192 \ud835\udd5c`: a vector in the kernel is exactly its first `n` coordinates.","title":"Kernel of the backward shift power as a space of finite tuples"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/finite-dimensional-ker-backward-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.finiteDimensional_ker_backwardCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#finite-dimensional-ker-backward-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The kernel of `B ^ n` is finite-dimensional.","title":"Finite-dimensionality of the kernel of a backward shift power"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/finrank-ker-backward-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.finrank_ker_backwardCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#finrank-ker-backward-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The kernel of `B ^ n` has dimension `n`.","title":"Dimension of the kernel of a backward shift power"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/ker-first-coords-eq-range-shift-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.ker_firstCoords_eq_range_shiftCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#ker-first-coords-eq-range-shift-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The kernel of the first-`n`-coordinates map is the range of `S ^ n`: both consist of the sequences whose first `n` coordinates vanish.","title":"Kernel of the first-coordinates map is the range of the shift power"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/finrank-quotient-range-shift-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.finrank_quotient_range_shiftCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#finrank-quotient-range-shift-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The cokernel of `S ^ n` has dimension `n`: the quotient of `\u2113\u00b2` by the range of `S ^ n` is equivalent to `Fin n \u2192 \ud835\udd5c` via the first `n` coordinates.","title":"Dimension of the cokernel of a unilateral shift power"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/approx-backward-comp-shift","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.approx_backward_comp_shift","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#approx-backward-comp-shift","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The quasi-inverse relation `B ^ n \u2218L S ^ n = 1` holds exactly, hence also modulo finite-rank operators.","title":"Backward shift power inverts the shift power modulo finite rank"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/approx-shift-comp-backward","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.approx_shift_comp_backward","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#approx-shift-comp-backward","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The defect `S ^ n \u2218L B ^ n - 1` has finite rank: its range lies in the finite-dimensional kernel of `B ^ n`, since applying `B ^ n` after it gives `0`.","title":"Shift power inverts the backward shift power modulo finite rank"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-fredholm-shift-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.isFredholm_shiftCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-fredholm-shift-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Every power of the unilateral shift is a Fredholm operator: `B ^ n` is a quasi-inverse.","title":"Powers of the unilateral shift are Fredholm"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-fredholm-backward-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.isFredholm_backwardCLM_pow","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-fredholm-backward-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Every power of the backward shift is a Fredholm operator: `S ^ n` is a quasi-inverse.","title":"Powers of the backward shift are Fredholm"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index-shift-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.index_shiftCLM_pow","parent":15909,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fredholm index of a unilateral shift power is "],["math","\\(-n\\)"]],"source":"Fredholm index of a unilateral shift power is $-n$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#index-shift-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The Fredholm index of `S ^ n` is `-n`: the shift power is injective, and its cokernel has dimension `n`.","title":"Fredholm index of a unilateral shift power is $-n$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index-backward-clm-pow","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.index_backwardCLM_pow","parent":15909,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fredholm index of a backward shift power is "],["math","\\(n\\)"]],"source":"Fredholm index of a backward shift power is $n$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#index-backward-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The Fredholm index of `B ^ n` is `n`: the backward shift power is surjective, and its kernel has dimension `n`.","title":"Fredholm index of a backward shift power is $n$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.index_shiftCLM","parent":15909,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fredholm index of the unilateral shift is "],["math","\\(-1\\)"]],"source":"Fredholm index of the unilateral shift is $-1$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#index-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The Fredholm index of the unilateral shift is `-1`.","title":"Fredholm index of the unilateral shift is $-1$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index-backward-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.index_backwardCLM","parent":15909,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fredholm index of the backward shift is "],["math","\\(1\\)"]],"source":"Fredholm index of the backward shift is $1$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#index-backward-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The Fredholm index of the backward shift is `1`.","title":"Fredholm index of the backward shift is $1$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-fredholm-shift-comp-backward","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.isFredholm_shift_comp_backward","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-fredholm-shift-comp-backward","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"`S \u2218L B` is Fredholm: it is the composite of two Fredholm operators.","title":"Shift composed with backward shift is Fredholm"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index-shift-comp-backward","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftFredholmIndex.index_shift_comp_backward","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#index-shift-comp-backward","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"`S \u2218L B` has Fredholm index `0`, as index additivity under composition demands: `(-1) + 1 = 0`.","title":"Shift composed with backward shift has Fredholm index zero"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/essential-spectrum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.essentialSpectrum","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#essential-spectrum","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The essential spectrum of a bounded operator `T`: the set of scalars `\u03bc` for which `T - \u03bc \u2022 1` is not a Fredholm operator.","title":"Essential spectrum of a bounded operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/one-sub-smul-backward-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.one_sub_smul_backward_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#one-sub-smul-backward-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Pointwise action of `1 - \u03bc \u2022 B`.","title":"Pointwise action of one minus a scaled backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/shift-sub-smul-one-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.shift_sub_smul_one_eq","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#shift-sub-smul-one-eq","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The key factorization: `S - \u03bc = S (1 - \u03bc B) + \u03bc (S B - 1)`.","title":"Factorization of the shifted unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/injective-shift-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.injective_shift_sub","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#injective-shift-sub","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"`S - \u03bc \u2022 1` has trivial kernel for every `\u03bc`: the coordinate equations force every coordinate to vanish.","title":"Shifted unilateral shift has trivial kernel"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-unit-one-sub-smul-backward","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.isUnit_one_sub_smul_backward","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-unit-one-sub-smul-backward","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"`1 - \u03bc \u2022 B` is a unit when `\u2016\u03bc\u2016 < 1`, by the Neumann-series criterion `isUnit_one_sub_of_norm_lt_one`.","title":"One minus a small multiple of the backward shift is invertible"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-fredholm-of-is-unit","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.isFredholm_of_isUnit","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-fredholm-of-is-unit","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"A unit operator is Fredholm, via `IsHomeomorph` and `IsInvertible.isFredholm`.","title":"Invertible operators are Fredholm"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-fredholm-one-sub-smul-backward","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.isFredholm_one_sub_smul_backward","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-fredholm-one-sub-smul-backward","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"`1 - \u03bc \u2022 B` is Fredholm when `\u2016\u03bc\u2016 < 1`.","title":"One minus a small multiple of the backward shift is Fredholm"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-fredholm-shift-sub-smul-one-of-norm-l","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.isFredholm_shift_sub_smul_one_of_norm_lt_one","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-fredholm-shift-sub-smul-one-of-norm-l","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"For `\u2016\u03bc\u2016 < 1`, `S - \u03bc \u2022 1` is Fredholm: it is the Fredholm operator `S (1 - \u03bc B)` plus the finite-rank operator `\u03bc (S B - 1)`.","title":"Shifted unilateral shift is Fredholm inside the unit disk"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index-shift-sub-smul-one-of-norm-lt-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.index_shift_sub_smul_one_of_norm_lt_one","parent":15909,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shifted unilateral shift has index "],["math","\\(-1\\)"],["text"," inside the unit disk"]],"source":"Shifted unilateral shift has index $-1$ inside the unit disk"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#index-shift-sub-smul-one-of-norm-lt-one","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The index of `S - \u03bc \u2022 1` is `-1` for `\u2016\u03bc\u2016 < 1`: the canonical homotopy theorem makes the index constant along `S - (t \u03bc) \u2022 1`, `t \u2208 [0, 1]`, and the canonical shift has index `-1` at `t = 0`.","title":"Shifted unilateral shift has index $-1$ inside the unit disk"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-unit-shift-sub-smul-one-of-one-lt-nor","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.isUnit_shift_sub_smul_one_of_one_lt_norm","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-unit-shift-sub-smul-one-of-one-lt-nor","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"For `\u2016\u03bc\u2016 > 1`, `S - \u03bc \u2022 1` is a unit: it equals `(algebraMap (-\u03bc)) * (1 - \u03bc\u207b\u00b9 \u2022 S)`, a product of units.","title":"Shifted unilateral shift is invertible outside the unit disk"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-fredholm-shift-sub-smul-one-of-one-lt","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.isFredholm_shift_sub_smul_one_of_one_lt_norm","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-fredholm-shift-sub-smul-one-of-one-lt","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"For `\u2016\u03bc\u2016 > 1`, `S - \u03bc \u2022 1` is Fredholm, as it is a unit.","title":"Shifted unilateral shift is Fredholm outside the unit disk"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index-shift-sub-smul-one-of-one-lt-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.index_shift_sub_smul_one_of_one_lt_norm","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#index-shift-sub-smul-one-of-one-lt-norm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"For `\u2016\u03bc\u2016 > 1`, `S - \u03bc \u2022 1` has index `0`: a unit is bijective.","title":"Shifted unilateral shift has index zero outside the unit disk"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/approx-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.approxFun","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#approx-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The truncated geometric sequence `(1, \u03bc\u0304, \u03bc\u0304\u00b2, \u2026, \u03bc\u0304\u1d3a, 0, \u2026)`.","title":"Truncated geometric sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-p-approx-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.mem\u2113p_approxFun","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-p-approx-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The truncated geometric sequence is in `\u2113\u00b2`: it has finite support.","title":"Truncated geometric sequence is square-summable"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/approx-vec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.approxVec","parent":15909,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncated geometric vector in "],["math","\\(\\ell^2\\)"]],"source":"Truncated geometric vector in $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#approx-vec","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The truncated geometric sequence as an `\u2113\u00b2` vector.","title":"Truncated geometric vector in $\\ell^2$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-sq-approx-vec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.norm_sq_approxVec","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-sq-approx-vec","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The squared norm of the truncated geometric vector is `N + 1`, when `\u2016\u03bc\u2016 = 1`.","title":"Squared norm of the truncated geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/sub-smul-one-approx-vec-apply-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.sub_smul_one_approxVec_apply_zero","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#sub-smul-one-approx-vec-apply-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The coordinates of `(S - \u03bc \u2022 1)` applied to the truncated geometric vector: `-\u03bc` at `0`, `(star \u03bc)^N` at `N + 1`, and `0` elsewhere, when `\u2016\u03bc\u2016 = 1`.","title":"Shifted unilateral shift on the truncated geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-sq-sub-smul-one-approx-vec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.norm_sq_sub_smul_one_approxVec","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-sq-sub-smul-one-approx-vec","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The squared norm of `(S - \u03bc \u2022 1)` applied to the truncated geometric vector is `2`, when `\u2016\u03bc\u2016 = 1`.","title":"Squared norm of the shifted shift on the truncated geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/not-is-fredholm-shift-sub-smul-one-of-no","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.not_isFredholm_shift_sub_smul_one_of_norm_eq_one","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#not-is-fredholm-shift-sub-smul-one-of-no","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"For `\u2016\u03bc\u2016 = 1`, `S - \u03bc \u2022 1` is not Fredholm: it has trivial kernel and closed range would make it bounded below, contradicting the approximate eigenvectors.","title":"Shifted unilateral shift is not Fredholm on the unit circle"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/essential-spectrum-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.essentialSpectrum_shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#essential-spectrum-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The essential spectrum of the unilateral shift is the unit circle: `S - \u03bc \u2022 1` is Fredholm exactly off the circle.","title":"Essential spectrum of the unilateral shift is the unit circle"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-essential-spectrum-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftEssentialSpectrum.mem_essentialSpectrum_shiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-essential-spectrum-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Membership form: `\u03bc` is in the essential spectrum of the shift iff `\u2016\u03bc\u2016 = 1`.","title":"Membership in the essential spectrum of the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/quad-form-fwd-shift-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.quadForm_fwdShift_eq_tsum","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#quad-form-fwd-shift-eq-tsum","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The quadratic form of the forward shift as a series over consecutive coordinates: `\u27eaS x, x\u27eb = \u2211' n, conj (x n) * x (n + 1)`.","title":"Quadratic form of the forward shift as a series"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/quad-form-bwd-shift-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.quadForm_bwdShift_eq_tsum","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#quad-form-bwd-shift-eq-tsum","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The quadratic form of the backward shift as a series over consecutive coordinates: `\u27eaB x, x\u27eb = \u2211' n, conj (x (n + 1)) * x n`.","title":"Quadratic form of the backward shift as a series"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/sum-norm-mul-succ-lt-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.sum_norm_mul_succ_lt_one","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#sum-norm-mul-succ-lt-one","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"For a unit vector `x`, the sum of the products of consecutive coordinate norms is strictly less than `1`: termwise AM\u2013GM gives a bound of `(1 + (1 - \u2016x 0\u2016\u00b2)) / 2 \u2264 1`, and the AM\u2013GM inequality is strict at the coordinate supplied by `exists_norm_ne_norm_succ`.","title":"Strict AM-GM bound on consecutive coordinates of a unit vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-p-geom","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.mem\u2113p_geom","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-p-geom","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The geometric sequence attached to `u`, with coordinates `\u221a(1 - \u2016u\u2016\u00b2) * u ^ n`, is in `\u2113\u00b2`. When `\u2016u\u2016 \u2265 1` the square root is `0` and the sequence is identically `0`; the interesting case is `\u2016u\u2016 < 1`.","title":"Square-summability of the scaled geometric sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/geom","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.geom","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#geom","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The geometric vector attached to `u`: the `\u2113\u00b2` sequence with coordinates `\u221a(1 - \u2016u\u2016\u00b2) * u ^ n`.","title":"Geometric unit vector attached to a point of the disc"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/geom-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.geom_apply","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#geom-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Pointwise action of the geometric vector.","title":"Pointwise action of the geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-geom","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.norm_geom","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-geom","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The geometric vector attached to `u` with `\u2016u\u2016 < 1` is a unit vector.","title":"Geometric vector of a point in the open disc is a unit vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/quad-form-fwd-geom","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.quadForm_fwd_geom","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#quad-form-fwd-geom","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The quadratic form of the forward shift at the geometric vector attached to `u` is `u` itself.","title":"Forward shift quadratic form at a geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/quad-form-bwd-geom","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.quadForm_bwd_geom","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#quad-form-bwd-geom","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The quadratic form of the backward shift at the geometric vector attached to `u` is `conj u`.","title":"Backward shift quadratic form at a geometric vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-numerical-range-fwd-shift-of-norm-lt","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.mem_numericalRange_fwdShift_of_norm_lt_one","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-numerical-range-fwd-shift-of-norm-lt","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Every `u` with `\u2016u\u2016 < 1` lies in the numerical range of the forward shift, witnessed by the geometric vector attached to `u`.","title":"Open unit disc lies in the forward shift's numerical range"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/mem-numerical-range-bwd-shift-of-norm-lt","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.mem_numericalRange_bwdShift_of_norm_lt_one","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#mem-numerical-range-bwd-shift-of-norm-lt","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Every `w` with `\u2016w\u2016 < 1` lies in the numerical range of the backward shift, witnessed by the geometric vector attached to `conj w`.","title":"Open unit disc lies in the backward shift's numerical range"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-lt-one-of-mem-numerical-range-fwd-s","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.norm_lt_one_of_mem_numericalRange_fwdShift","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-lt-one-of-mem-numerical-range-fwd-s","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Every point of the numerical range of the forward shift has norm strictly less than `1`, by the AM\u2013GM strict bound.","title":"Forward shift's numerical range lies in the open unit disc"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/norm-lt-one-of-mem-numerical-range-bwd-s","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.norm_lt_one_of_mem_numericalRange_bwdShift","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#norm-lt-one-of-mem-numerical-range-bwd-s","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"Every point of the numerical range of the backward shift has norm strictly less than `1`, by the AM\u2013GM strict bound.","title":"Backward shift's numerical range lies in the open unit disc"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/numerical-range-fwd-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.numericalRange_fwdShiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#numerical-range-fwd-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The numerical range of the unilateral shift on `\u2113\u00b2(\u2115, \u2102)` is the open unit disc.","title":"Numerical range of the unilateral shift is the open unit disc"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/numerical-range-bwd-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ShiftNumericalRange.numericalRange_bwdShiftCLM","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#numerical-range-bwd-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"The numerical range of the backward shift on `\u2113\u00b2(\u2115, \u2102)` is the open unit disc.","title":"Numerical range of the backward shift is the open unit disc"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-shift","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.UnilateralShiftCharacterizationWanted.IsShift","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-shift","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"fully_proved","summary":"A bounded linear operator $T$ on a complex inner product space $H$ is a unilateral shift if there is a Hilbert basis $(b_n)_{n \\in \\mathbb{N}}$ of $H$ such that $T b_n = b_{n+1}$ for every $n \\in \\mathbb{N}$.","title":"Unilateral shift operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/is-shift-iff-isometry-i-inf-range-pow-eq","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.UnilateralShiftCharacterizationWanted.isShift_iff_isometry_iInf_range_pow_eq_bot_finrank_orthogonal_eq_one","parent":15909,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift/index.html#is-shift-iff-isometry-i-inf-range-pow-eq","scope":"Analysis/operator-theory/shift-toeplitz-hankel/unilateral-shift","status":"can_prove","summary":"Let $H$ be a separable complex Hilbert space, and call a bounded linear operator $T : H \\to H$ a shift if there is an orthonormal basis $(e_n)_{n \\in \\mathbb{N}}$ of $H$ with $T e_n = e_{n+1}$ for every $n \\in \\mathbb{N}$. Then a bounded linear operator $T : H \\to H$ is a shift if and only if the following three conditions hold: $T$ is an isometry, that is, $\\langle T x, T y \\rangle = \\langle x, y \\rangle$ for all\u2026","title":"Isometric characterization of the unilateral shift"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts.json","id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","kind":"scope","navigation":"module","parent":15831,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","title":"Rolewicz backward shifts"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/norm-forward-shift-clm-pow-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Rolewicz.norm_forwardShiftCLM_pow_apply","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#norm-forward-shift-clm-pow-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"Powers of the forward shift preserve norms.","title":"Powers of the forward shift preserve norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/norm-backward-shift-clm-pow-apply-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Rolewicz.norm_backwardShiftCLM_pow_apply_le","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#norm-backward-shift-clm-pow-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"Powers of the backward shift do not increase norms.","title":"Powers of the backward shift do not increase norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/smul-backward-shift-clm-pow-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Rolewicz.smul_backwardShiftCLM_pow_apply","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#smul-backward-shift-clm-pow-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"Powers of a scalar multiple of the backward shift, applied pointwise.","title":"Powers of a scalar multiple of the backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/dense-span-single","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Rolewicz.dense_span_single","parent":16018,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finitely supported vectors are dense in "],["math","\\(\\ell^2(\\mathbb{N})\\)"]],"source":"Finitely supported vectors are dense in $\\ell^2(\\mathbb{N})$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#dense-span-single","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The span of the single-coordinate vectors is dense in `\u2113\u00b2(\u2115, \ud835\udd5c)`: every vector is the sum of its canonical series.","title":"Finitely supported vectors are dense in $\\ell^2(\\mathbb{N})$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/exists-forall-eq-zero-of-mem-span-single","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Rolewicz.exists_forall_eq_zero_of_mem_span_single","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#exists-forall-eq-zero-of-mem-span-single","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"Every vector in the span of the single-coordinate vectors is eventually zero: it has finite support.","title":"Vectors in the span of coordinate vectors have finite support"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/is-hypercyclic-smul-backward-shift-clm-i","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Rolewicz.isHypercyclic_smul_backwardShiftCLM_iff","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#is-hypercyclic-smul-backward-shift-clm-i","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The scalar multiple `\u03bc \u2022 B` of the backward shift on `\u2113\u00b2(\u2115, \ud835\udd5c)` is hypercyclic if and only if `\u2016\u03bc\u2016 > 1`.","title":"Rolewicz's theorem"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/smul-backward-clm-pow-apply-coord","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.smul_backwardCLM_pow_apply_coord","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#smul-backward-clm-pow-apply-coord","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"Coordinate form of a power of a scalar multiple of the backward shift.","title":"Coordinates of powers of a scaled backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/smul-backward-clm-pow","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.smul_backwardCLM_pow","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#smul-backward-clm-pow","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"A power of a scalar multiple of the backward shift, as an operator.","title":"Powers of a scaled backward shift as operators"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/is-periodic-point-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.isPeriodicPoint_zero","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#is-periodic-point-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"`0` is a periodic point of every period for every operator.","title":"Zero is a periodic point of every operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/periodic-approx","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.periodicApprox","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#periodic-approx","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The periodic extension of the truncation of `y` to its first `N` coordinates: the geometric series over forward shifts by blocks of length `N`, with coefficients `(\u03bc\u207b\u00b9 ^ N) ^ m`.","title":"Periodic approximant of a truncated sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/norm-inv-pow-lt-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.norm_inv_pow_lt_one","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#norm-inv-pow-lt-one","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The geometric ratio `\u2016\u03bc\u207b\u00b9 ^ N\u2016` is `< 1` when `\u2016\u03bc\u2016 > 1` and `N > 0`.","title":"Geometric ratio of the periodic approximant is below 1"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/summable-geom-trunc","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.summable_geom_trunc","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#summable-geom-trunc","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The geometric series bounding the periodic approximant terms is summable.","title":"Summability of the geometric bound for periodic approximants"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/norm-periodic-approx-term","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.norm_periodicApprox_term","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#norm-periodic-approx-term","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The norm of a term of the periodic approximant.","title":"Norm of a term of the periodic approximant"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/summable-periodic-approx-terms","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.summable_periodicApprox_terms","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#summable-periodic-approx-terms","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"Summability of the terms of the periodic approximant.","title":"Summability of the periodic approximant terms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/periodic-approx-term-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.periodicApprox_term_zero","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#periodic-approx-term-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The `m = 0` term of the periodic approximant is the truncation.","title":"First term of the periodic approximant is the truncation"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/backward-clm-pow-trunc","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.backwardCLM_pow_trunc","parent":16018,"reading_title":{"format":"inline-parts-v1","parts":[["text","Backward shift annihilates truncations after "],["math","\\(N\\)"],["text"," steps"]],"source":"Backward shift annihilates truncations after $N$ steps"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#backward-clm-pow-trunc","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"Applying the backward shift `N` times to a truncation gives `0`.","title":"Backward shift annihilates truncations after $N$ steps"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/periodic-approx-step","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.periodicApprox_step","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#periodic-approx-step","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"How `(\u03bc \u2022 B) ^ N` acts on a term of the periodic approximant: the `m = 0` term is killed, and every later term shifts down by one.","title":"Action of the scaled backward shift power on approximant terms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/summable-periodic-approx-map","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.summable_periodicApprox_map","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#summable-periodic-approx-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"Summability of the terms of the periodic approximant after applying `(\u03bc \u2022 B) ^ N`.","title":"Summability of the shifted periodic approximant terms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/periodic-approx-is-periodic-point","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.periodicApprox_isPeriodicPoint","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#periodic-approx-is-periodic-point","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The periodic approximant is periodic of period `N` when `\u2016\u03bc\u2016 > 1`.","title":"Periodic approximant is a periodic point"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/norm-periodic-approx-sub-trunc","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.norm_periodicApprox_sub_trunc","parent":16018,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#norm-periodic-approx-sub-trunc","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The distance from the periodic approximant to the truncation is at most the geometric tail `(r / (1 - r)) * \u2016y\u2016`, where `r = \u2016\u03bc\u2016\u207b\u00b9 ^ N`.","title":"Distance from the periodic approximant to the truncation"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/dense-periodic-points-of-one-lt-norm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.dense_periodicPoints_of_one_lt_norm","parent":16018,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dense periodic points of the Rolewicz operator for "],["math","\\(|\\mu| > 1\\)"]],"source":"Dense periodic points of the Rolewicz operator for $|\\mu| > 1$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#dense-periodic-points-of-one-lt-norm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The periodic points of `\u03bc \u2022 B` are dense when `\u2016\u03bc\u2016 > 1`.","title":"Dense periodic points of the Rolewicz operator for $|\\mu| > 1$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/periodic-coord-iterate","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.periodic_coord_iterate","parent":16018,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinate recurrence for periodic points of "],["math","\\(\\mu B\\)"]],"source":"Coordinate recurrence for periodic points of $\\mu B$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#periodic-coord-iterate","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"If `x` is periodic for `\u03bc \u2022 B`, its coordinates satisfy `x k = \u03bc ^ (n * m) * x (k + n * m)` for all `m`.","title":"Coordinate recurrence for periodic points of $\\mu B$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/tendsto-norm-coord","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.tendsto_norm_coord","parent":16018,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinates of an "],["math","\\(\\ell^2\\)"],["text"," vector tend to zero"]],"source":"Coordinates of an $\\ell^2$ vector tend to zero"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#tendsto-norm-coord","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The coordinates of an `\u2113\u00b2` vector tend to `0`.","title":"Coordinates of an $\\ell^2$ vector tend to zero"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/eq-zero-of-is-periodic-point-of-norm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.eq_zero_of_isPeriodicPoint_of_norm_le_one","parent":16018,"reading_title":{"format":"inline-parts-v1","parts":[["text","No nonzero periodic points of "],["math","\\(\\mu B\\)"],["text"," when "],["math","\\(|\\mu| \\le 1\\)"]],"source":"No nonzero periodic points of $\\mu B$ when $|\\mu| \\le 1$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#eq-zero-of-is-periodic-point-of-norm-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The only periodic point of `\u03bc \u2022 B` is `0` when `\u2016\u03bc\u2016 \u2264 1`.","title":"No nonzero periodic points of $\\mu B$ when $|\\mu| \\le 1$"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/dense-periodic-points-iff","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RolewiczPeriodic.dense_periodicPoints_iff","parent":16018,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periodic points of "],["math","\\(\\mu B\\)"],["text"," are dense iff "],["math","\\(|\\mu| > 1\\)"]],"source":"Periodic points of $\\mu B$ are dense iff $|\\mu| > 1$"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts/index.html#dense-periodic-points-iff","scope":"Analysis/operator-theory/shift-toeplitz-hankel/rolewicz-backward-shifts","status":"fully_proved","summary":"The periodic points of `\u03bc \u2022 B` are dense if and only if `\u2016\u03bc\u2016 > 1`.","title":"Periodic points of $\\mu B$ are dense iff $|\\mu| > 1$"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators.json","id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","kind":"scope","navigation":"module","parent":15831,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","title":"Toeplitz operators"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-slice-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzSliceFun","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-slice-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The `k`-th diagonal slice function: shift coordinates by `k`, killing the coordinates that fall off the end. For `k \u2265 0` this is the `k`-fold forward shift; for `k < 0` it is the `(-k)`-fold backward shift.","title":"Diagonal slice function of a Toeplitz matrix"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-mem-p-slice-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitz_mem\u2113p_sliceFun","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-mem-p-slice-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The slice of an `\u2113\u00b2` sequence is in `\u2113\u00b2`.","title":"Diagonal slices preserve square-summability"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-slice-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzSliceLinMap","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-slice-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The `k`-th diagonal slice as a linear map.","title":"Diagonal slice as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-slice-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzSliceLinMap_apply","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-slice-lin-map-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"Pointwise action of the slice linear map.","title":"Pointwise action of the diagonal slice linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/norm-toeplitz-slice-lin-map-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.norm_toeplitzSliceLinMap_apply_le","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#norm-toeplitz-slice-lin-map-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"A slice does not increase norms.","title":"Diagonal slices do not increase norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-slice","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzSlice","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-slice","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The `k`-th diagonal slice as a bounded operator.","title":"Diagonal slice as a bounded operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-slice-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzSlice_apply","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-slice-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"Pointwise action of the slice operator.","title":"Pointwise action of the diagonal slice operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-slice-eq-shift-clm-pow-or-backw","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzSlice_eq_shiftCLM_pow_or_backwardCLM_pow","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-slice-eq-shift-clm-pow-or-backw","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"Every nonnegative Toeplitz slice is the corresponding power of the canonical unilateral shift, and every negative slice is the corresponding power of the canonical backward shift.","title":"Diagonal slices are powers of the forward or backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/norm-toeplitz-slice-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.norm_toeplitzSlice_le","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#norm-toeplitz-slice-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The operator norm of a slice is at most `1`.","title":"Operator norm of a diagonal slice is at most one"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-slice-inner-symm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzSlice_inner_symm","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-slice-inner-symm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"A slice is adjoint to the slice at the negated index: the diagonal symmetry exchanges the two sides of the inner product.","title":"Inner product symmetry of opposite diagonal slices"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/adjoint-toeplitz-slice","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.adjoint_toeplitzSlice","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#adjoint-toeplitz-slice","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"A slice is adjoint to the slice at the negated index.","title":"Adjoint of a diagonal slice is the opposite slice"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-summable-slices","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitz_summable_slices","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-summable-slices","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The slice operators weighted by `a` are summable in operator norm when `a` is absolutely summable.","title":"Summability of weighted diagonal slices"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzOperator","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-operator","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The Toeplitz operator with symbol `a`. Defined as the norm-convergent series `\u2211' k, a k \u2022 toeplitzSlice k` over the diagonal slices.","title":"Toeplitz operator with absolutely summable symbol"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-has-sum-slices","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitz_hasSum_slices","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-has-sum-slices","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The Toeplitz operator is the sum of its slices.","title":"Toeplitz operator as the sum of its diagonal slices"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-operator-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzOperator_apply","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-operator-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"Pointwise action of the Toeplitz operator: `(T\u2090 x) i = \u2211' j, a (i - j) * x j`.","title":"Pointwise action of a Toeplitz operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-operator-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzOperator_zero","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-operator-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The zero symbol gives the zero operator.","title":"Toeplitz operator of the zero symbol"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/norm-toeplitz-operator-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.norm_toeplitzOperator_le","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#norm-toeplitz-operator-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"Operator-norm bound: `\u2016T\u2090\u2016 \u2264 \u2211' k, \u2016a k\u2016`.","title":"Operator norm bound for a Toeplitz operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-adjoint-symbol","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzAdjointSymbol","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-adjoint-symbol","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The adjoint symbol: `a* k = star (a (-k))`.","title":"Adjoint symbol of a Toeplitz operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/summable-norm-toeplitz-adjoint-symbol","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.summable_norm_toeplitzAdjointSymbol","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#summable-norm-toeplitz-adjoint-symbol","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The adjoint symbol of an absolutely summable symbol is absolutely summable.","title":"Absolute summability of the adjoint symbol"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/adjoint-toeplitz-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.adjoint_toeplitzOperator","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#adjoint-toeplitz-operator","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"Adjoint of a Toeplitz operator: `(T\u2090)\u2020 = T_{a*}`.","title":"Adjoint of a Toeplitz operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-operator-apply-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzOperator_apply_single","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-operator-apply-single","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The Toeplitz operator applied to a standard basis vector.","title":"Toeplitz operator applied to a standard basis vector"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/is-self-adjoint-toeplitz-operator-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.isSelfAdjoint_toeplitzOperator_iff","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#is-self-adjoint-toeplitz-operator-iff","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"Self-adjointness of a Toeplitz operator iff the symbol satisfies `star (a (-k)) = a k` for all `k`.","title":"Characterization of self-adjoint Toeplitz operators"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/is-self-adjoint-toeplitz-operator-of-for","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.isSelfAdjoint_toeplitzOperator_of_forall_star","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#is-self-adjoint-toeplitz-operator-of-for","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"A Toeplitz operator with a `star`-fixed adjoint symbol is self-adjoint.","title":"Toeplitz operators with self-adjoint symbol are self-adjoint"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/not-is-compact-operator-toeplitz-operato","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.not_isCompactOperator_toeplitzOperator_of_ne","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#not-is-compact-operator-toeplitz-operato","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"A Toeplitz operator with a nonzero symbol is not compact: the images of the standard basis vectors converge coordinatewise to `0` but keep a nonzero coordinate.","title":"Toeplitz operators with nonzero symbol are not compact"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/is-compact-operator-toeplitz-operator-if","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.isCompactOperator_toeplitzOperator_iff","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#is-compact-operator-toeplitz-operator-if","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"A Toeplitz operator is compact iff its symbol is zero.","title":"A Toeplitz operator is compact iff its symbol is zero"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-delta","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzDelta","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-delta","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The delta symbol at `m` with value `c`.","title":"Delta symbol for Toeplitz operators"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/summable-norm-toeplitz-delta","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.summable_norm_toeplitzDelta","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#summable-norm-toeplitz-delta","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"A delta symbol is absolutely summable.","title":"Absolute summability of a delta symbol"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/toeplitz-operator-delta-eq-smul-slice","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.toeplitzOperator_delta_eq_smul_slice","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#toeplitz-operator-delta-eq-smul-slice","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"A Toeplitz operator with a delta symbol is the corresponding scalar multiple of its diagonal slice.","title":"Toeplitz operator of a delta symbol is a scaled slice"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/tsum-norm-toeplitz-delta","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.tsum_norm_toeplitzDelta","parent":16044,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell^1\\)"],["text"," norm of a delta symbol"]],"source":"$\\ell^1$ norm of a delta symbol"},"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#tsum-norm-toeplitz-delta","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The `\u2113\u00b9` norm of a delta symbol is `\u2016c\u2016`.","title":"$\\ell^1$ norm of a delta symbol"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/norm-toeplitz-operator-delta","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ToeplitzOperator.norm_toeplitzOperator_delta","parent":16044,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators/index.html#norm-toeplitz-operator-delta","scope":"Analysis/operator-theory/shift-toeplitz-hankel/toeplitz-operators","status":"fully_proved","summary":"The operator norm of a Toeplitz operator with a delta symbol is `\u2016c\u2016`.","title":"Operator norm of a Toeplitz operator with delta symbol"},{"data_url":"scopes/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts.json","id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","kind":"scope","navigation":"module","parent":15831,"result":false,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","title":"Weighted shifts"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftFun","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For a weight sequence $w : \\mathbb{N} \\to \\mathbb{K}$ and a sequence $x : \\mathbb{N} \\to \\mathbb{K}$, the weighted forward shift of $x$ is the sequence $y$ with $y_0 = 0$ and $y_{n+1} = w_n x_n$ for every $n \\in \\mathbb{N}$.","title":"Weighted forward shift of a sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-backward-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedBackwardFun","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-backward-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For weights $w : \\mathbb{N} \\to \\mathbb{K}$ and a sequence $x : \\mathbb{N} \\to \\mathbb{K}$, the weighted backward shift of $x$ is the sequence whose term at index $i$ is $\\overline{w_i}\\, x_{i+1}$.","title":"Weighted backward shift of a sequence"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-fun-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftFun_zero","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-fun-zero","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For all weights $w : \\mathbb{N} \\to \\mathbb{K}$ and every sequence $x : \\mathbb{N} \\to \\mathbb{K}$, the term at index $0$ of the weighted forward shift of $x$ is $0$.","title":"Weighted forward shift at index zero"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-fun-succ","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftFun_succ","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-fun-succ","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For weights $w : \\mathbb{N} \\to \\mathbb{K}$, a sequence $x : \\mathbb{N} \\to \\mathbb{K}$ and $n \\in \\mathbb{N}$, the term at index $n + 1$ of the weighted forward shift of $x$ is $w_n x_n$.","title":"Weighted forward shift at a successor index"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-backward-fun-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedBackwardFun_apply","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-backward-fun-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For weights $w : \\mathbb{N} \\to \\mathbb{K}$, a sequence $x : \\mathbb{N} \\to \\mathbb{K}$ and $i \\in \\mathbb{N}$, the weighted backward-shift function of $w$ applied to $x$ takes the value $\\overline{w_i}\\, x_{i+1}$ at $i$.","title":"Coordinates of the weighted backward-shift function"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftCLM","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with $\\sup_n |w_n| < \\infty$. The weighted forward shift with weights $w$ is the bounded operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ obtained by first applying the diagonal operator $x \\mapsto (w_n x_n)_{n}$ and then the unilateral forward shift $(x_0, x_1, \\dots) \\mapsto (0, x_0, x_1, \\dots)$.","title":"Weighted forward shift operator"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-backward-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedBackwardCLM","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-backward-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with bounded moduli. The weighted backward shift $B_w$ is the bounded operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ obtained by applying the backward shift $(x_0, x_1, \\dots) \\mapsto (x_1, x_2, \\dots)$ and then multiplying coordinate $i$ by $\\overline{w_i}$; thus $(B_w x)_i = \\overline{w_i}\\, x_{i+1}$.","title":"Weighted backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftLinMap","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $w : \\mathbb{N} \\to \\mathbb{K}$ be a weight sequence whose norms $\\|w_n\\|$ are bounded above. The weighted forward shift linear map is the $\\mathbb{K}$-linear map $\\ell^2(\\mathbb{N}, \\mathbb{K}) \\to \\ell^2(\\mathbb{N}, \\mathbb{K})$ underlying the bounded weighted forward shift operator (multiplication by $w$ followed by the forward shift), which sends $x$ to\u2026","title":"Weighted forward shift as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-backward-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedBackwardLinMap","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-backward-lin-map","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with $\\sup_n |w_n| < \\infty$. The weighted backward shift linear map with weights $w$ is the bounded weighted backward shift operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ with weights $w$, regarded as a $\\mathbb{K}$-linear map from $\\ell^2(\\mathbb{N}, \\mathbb{K})$ to itself (forgetting continuity).","title":"Weighted backward shift as a linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftCLM_apply","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-clm-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with $\\sup_n |w_n| < \\infty$, and let $S_w$ be the weighted forward shift operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ with weights $w$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $i \\in \\mathbb{N}$, the coordinate $(S_w x)_i$ is the term at index $i$ of the weighted forward shift of the sequence $x$;\u2026","title":"Coordinates of the weighted forward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-backward-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedBackwardCLM_apply","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-backward-clm-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with bounded moduli, and let $B_w$ be the weighted backward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $i \\in \\mathbb{N}$, the $i$-th coordinate of $B_w x$ is the weighted backward-shift function of $w$ applied to $x$ at $i$, that is,\u2026","title":"Coordinates of the weighted backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-clm-apply-coe","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftCLM_apply_coe","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-clm-apply-coe","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with $\\sup_n |w_n| < \\infty$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the image of $x$ under the weighted forward shift operator with weights $w$ is as a sequence the weighted forward shift of $x$: its term at index $0$ is $0$ and its term at index $n + 1$ is $w_n x_n$ for every $n$.","title":"The weighted forward shift agrees with the shift of sequences"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-lin-map-apply-coe","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftLinMap_apply_coe","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-lin-map-apply-coe","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $w : \\mathbb{N} \\to \\mathbb{K}$ be a weight sequence whose norms $\\|w_n\\|$ are bounded above. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the image of $x$ under the weighted forward shift linear map, viewed as a sequence, is the weighted forward shift of $x$: its coordinate $0$ is $0$ and its coordinate $n + 1$ is $w_n x_n$ for every $n$.","title":"Coordinates of the weighted forward shift linear map"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-backward-lin-map-apply-coe","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedBackwardLinMap_apply_coe","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-backward-lin-map-apply-coe","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with $\\sup_n |w_n| < \\infty$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the image of $x$ under the weighted backward shift with weights $w$, regarded as a linear map on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, is as a sequence the weighted backward shift of $x$: its term at index $i$ is $\\overline{w_i}\\, x_{i+1}$\u2026","title":"The weighted backward shift linear map agrees with the shift of sequences"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/mem-p-weighted-shift-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.mem\u2113p_weightedShiftFun","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#mem-p-weighted-shift-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{K}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the sequence $y$ with $y_0 = 0$ and $y_{n+1} = w_n x_n$ is square-summable.","title":"Weighted forward shifts preserve square-summability"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/mem-p-weighted-backward-fun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.mem\u2113p_weightedBackwardFun","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#mem-p-weighted-backward-fun","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{K}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the sequence $(\\overline{w_i}\\, x_{i+1})_{i \\in \\mathbb{N}}$ is square-summable.","title":"Weighted backward shifts preserve square-summability"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-clm-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftCLM_single","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-clm-single","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with $\\sup_n |w_n| < \\infty$, and let $(e_n)_{n \\in \\mathbb{N}}$ be the standard basis vectors of $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For every $n$, the weighted forward shift operator with weights $w$ sends $e_n$ to $w_n e_{n+1}$.","title":"The weighted forward shift on basis vectors"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-backward-clm-single-succ","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedBackwardCLM_single_succ","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-backward-clm-single-succ","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with bounded moduli, and write $e_n$ for the $n$-th standard basis vector of $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For every $n$, the weighted backward shift $B_w$ satisfies $B_w e_{n+1} = \\overline{w_n}\\, e_n$.","title":"Weighted backward shift on basis vectors"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/norm-weighted-shift-clm-le-of-forall-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.norm_weightedShiftCLM_le_of_forall_le","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#norm-weighted-shift-clm-le-of-forall-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with bounded moduli, and let $S_w$ be the weighted forward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. If $C$ is a real number with $|w_n| \\le C$ for every $n$, then the operator norm of $S_w$ is at most $C$.","title":"Uniform weight bounds bound the weighted shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/norm-weighted-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.norm_weightedShiftCLM","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#norm-weighted-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with bounded moduli. The weighted forward shift $S_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ has operator norm $\\|S_w\\| = \\sup_n |w_n|$.","title":"Operator norm of a weighted shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/adjoint-weighted-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.adjoint_weightedShiftCLM","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#adjoint-weighted-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{K}$. The adjoint of the weighted forward shift $S_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(S_w x)_0 = 0$ and $(S_w x)_{n+1} = w_n x_n$, is the weighted backward shift $B_w$, given by $(B_w x)_i = \\overline{w_i}\\, x_{i+1}$; that is, $S_w^* = B_w$.","title":"Adjoint of the weighted shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/norm-weighted-backward-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.norm_weightedBackwardCLM","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#norm-weighted-backward-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{K}$. The weighted backward shift $B_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(B_w x)_i = \\overline{w_i}\\, x_{i+1}$, has operator norm $\\|B_w\\| = \\sup_n |w_n|$.","title":"Norm of the weighted backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/norm-weighted-shift-lin-map-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.norm_weightedShiftLinMap_apply_le","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#norm-weighted-shift-lin-map-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with bounded moduli. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the weighted forward shift $S_w$, viewed as a linear map, satisfies $\\|S_w x\\| \\le \\left(\\sup_n |w_n|\\right) \\|x\\|$.","title":"Norm bound for weighted shift images"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/norm-weighted-backward-lin-map-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.norm_weightedBackwardLinMap_apply_le","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#norm-weighted-backward-lin-map-apply-le","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{K}$, and let $B_w$ be the weighted backward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(B_w x)_i = \\overline{w_i}\\, x_{i+1}$. Then $\\|B_w x\\| \\le (\\sup_n |w_n|)\\, \\|x\\|$ for every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"Norm bound for the weighted backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/backward-comp-shift-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.backward_comp_shift_apply","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#backward-comp-shift-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $(w_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{K}$, and on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ let $S_w$ be the weighted forward shift, $(S_w x)_0 = 0$ and $(S_w x)_{n+1} = w_n x_n$, and $B_w$ the weighted backward shift, $(B_w x)_i = \\overline{w_i}\\, x_{i+1}$. Then $(B_w S_w x)_i = |w_i|^2 x_i$ for every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and\u2026","title":"The weighted backward shift after the forward shift is diagonal"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/shift-comp-backward-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.shift_comp_backward_apply","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#shift-comp-backward-apply","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with bounded moduli, and let $S_w$ and $B_w$ be the weighted forward and backward shifts on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the composite $S_w B_w$ satisfies $(S_w B_w x)_0 = 0$ and $(S_w B_w x)_i = |w_{i-1}|^2 x_i$ for every $i \\ge 1$.","title":"Forward-after-backward weighted shift is diagonal"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/backward-comp-shift-eq-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.backward_comp_shift_eq_one","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#backward-comp-shift-eq-one","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a sequence in $\\mathbb{K}$ with $|w_n| = 1$ for every $n$. On $\\ell^2(\\mathbb{N}, \\mathbb{K})$ let $S_w$ be the weighted forward shift, $(S_w x)_0 = 0$ and $(S_w x)_{n+1} = w_n x_n$, and $B_w$ the weighted backward shift, $(B_w x)_i = \\overline{w_i}\\, x_{i+1}$. Then $B_w S_w$ is the identity operator.","title":"The weighted backward shift is a left inverse for unimodular weights"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/injective-weighted-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.injective_weightedShiftCLM","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#injective-weighted-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{K}$ with $w_n \\neq 0$ for every $n$. Then the weighted forward shift $S_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(S_w x)_0 = 0$ and $(S_w x)_{n+1} = w_n x_n$, is injective.","title":"Weighted shifts with nonzero weights are injective"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/not-injective-weighted-shift-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.not_injective_weightedShiftCLM","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#not-injective-weighted-shift-clm","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be weights with bounded moduli. If $w_n = 0$ for some $n$, then the weighted forward shift $S_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is not injective.","title":"A vanishing weight makes the weighted shift non-injective"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/norm-weighted-shift-clm-apply-of-forall","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.norm_weightedShiftCLM_apply_of_forall_norm_eq_one","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#norm-weighted-shift-clm-apply-of-forall","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a sequence in $\\mathbb{K}$ with $|w_n| = 1$ for every $n$. Then the weighted forward shift $S_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(S_w x)_0 = 0$ and $(S_w x)_{n+1} = w_n x_n$, satisfies $\\|S_w x\\| = \\|x\\|$ for every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"Weighted shifts with unimodular weights preserve norms"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/isometry-weighted-shift-clm-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.isometry_weightedShiftCLM_iff","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#isometry-weighted-shift-clm-iff","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $(w_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{K}$. The weighted forward shift $S_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(S_w x)_0 = 0$ and $(S_w x)_{n+1} = w_n x_n$, is an isometry if and only if $|w_n| = 1$ for every $n$.","title":"Isometric weighted shifts"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-clm-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedShiftCLM_one","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-clm-one","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The weighted forward shift operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ whose weights are all equal to $1$ is the unilateral forward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The weighted forward shift with unit weights is the unilateral shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-backward-clm-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftOperator.weightedBackwardCLM_one","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-backward-clm-one","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. With every weight equal to $1$, the weighted backward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ is the canonical backward shift.","title":"Unit weights give the plain backward shift"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/trunc-weights","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.truncWeights","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#trunc-weights","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For a sequence $w : \\mathbb{N} \\to \\mathbb{K}$ and $N \\in \\mathbb{N}$, the truncation of $w$ after $N$ coordinates is the sequence $w^{(N)}$ with $w^{(N)}_n = w_n$ for $n < N$ and $w^{(N)}_n = 0$ for $n \\ge N$.","title":"Truncated weights"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/trunc-weights-apply-lt","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.truncWeights_apply_lt","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#trunc-weights-apply-lt","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$, and let $N, n \\in \\mathbb{N}$ with $n < N$. Then the truncation $w^{(N)}$ of $w$ after $N$ coordinates (equal to $w_k$ for $k < N$ and to $0$ for $k \\ge N$) satisfies $w^{(N)}_n = w_n$.","title":"Truncated weights agree with the weights below the cutoff"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/trunc-weights-apply-ge","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.truncWeights_apply_ge","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#trunc-weights-apply-ge","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$, and let $N, n \\in \\mathbb{N}$ with $n \\ge N$. Then the truncation $w^{(N)}$ of $w$ after $N$ coordinates (equal to $w_k$ for $k < N$ and to $0$ for $k \\ge N$) satisfies $w^{(N)}_n = 0$.","title":"Truncated weights vanish from the cutoff on"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/bdd-above-trunc-weights","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.bddAbove_truncWeights","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#bdd-above-trunc-weights","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$ be bounded, that is, $|w_n| \\le C$ for all $n$ for some constant $C$, and let $N \\in \\mathbb{N}$. Then the truncation $w^{(N)}$ of $w$ after $N$ coordinates, given by $w^{(N)}_n = w_n$ for $n < N$ and $w^{(N)}_n = 0$ for $n \\ge N$, is bounded as well.","title":"Truncated weights are bounded"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/bdd-above-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.bddAbove_sub","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#bdd-above-sub","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $w, v : \\mathbb{N} \\to \\mathbb{K}$ be bounded sequences, that is, there are constants $C$ and $D$ with $|w_n| \\le C$ and $|v_n| \\le D$ for all $n$. Then $w - v$ is bounded: there is a constant $E$ with $|w_n - v_n| \\le E$ for all $n$.","title":"Differences of bounded sequences are bounded"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/weighted-shift-clm-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.weightedShiftCLM_sub","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#weighted-shift-clm-sub","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $w, v : \\mathbb{N} \\to \\mathbb{K}$ be sequences such that $w$, $v$ and $w - v$ are bounded. Writing $S_u$ for the weighted forward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ with bounded weights $u$, sending $x$ to $(0, u_0 x_0, u_1 x_1, \\dots)$, we have $S_{w - v} = S_w - S_v$.","title":"Weighted shift of a difference of weights"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/tendsto-weighted-shift-clm-trunc","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.tendsto_weightedShiftCLM_trunc","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#tendsto-weighted-shift-clm-trunc","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be a bounded sequence with $w_n \\to 0$. For $N \\in \\mathbb{N}$ let $w^{(N)}$ be the truncation of $w$ after $N$ coordinates, with $w^{(N)}_n = w_n$ for $n < N$ and $w^{(N)}_n = 0$ for $n \\ge N$, and for bounded weights $u$ let $S_u$ be the weighted forward shift on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, sending $x$ to\u2026","title":"Weighted shifts with truncated weights converge in operator norm"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/is-compact-operator-weighted-shift-clm-i","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.isCompactOperator_weightedShiftCLM_iff","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#is-compact-operator-weighted-shift-clm-i","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $w : \\mathbb{N} \\to \\mathbb{K}$ be a bounded sequence of weights, that is, $|w_n| \\le C$ for all $n$ for some constant $C$. The weighted forward shift $S_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, which sends $x$ to $(0, w_0 x_0, w_1 x_1, \\dots)$, is a compact operator if and only if $w_n \\to 0$ as $n \\to \\infty$.","title":"Compactness criterion for weighted shifts"},{"id":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/is-compact-operator-weighted-shift-of-fo","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeightedShiftCompact.isCompactOperator_weightedShift_of_forall_eq_zero","parent":16075,"result":true,"route":"../roadmap/Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts/index.html#is-compact-operator-weighted-shift-of-fo","scope":"Analysis/operator-theory/shift-toeplitz-hankel/weighted-shifts","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $w : \\mathbb{N} \\to \\mathbb{K}$ be a bounded sequence of weights, and suppose there is $N \\in \\mathbb{N}$ with $w_n = 0$ for all $n \\ge N$. Then the weighted forward shift $S_w$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, which sends $x$ to $(0, w_0 x_0, w_1 x_1, \\dots)$, is a compact operator.","title":"Weighted shifts with eventually vanishing weights are compact"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication.json","id":"Analysis/operator-theory/integral-and-multiplication","kind":"scope","parent":15497,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/index.html","scope":"Analysis/operator-theory/integral-and-multiplication","status":"can_prove","summary":"Volterra and kernel integral operators, Ces\u00e0ro and Hilbert operators, multiplication operators on C(X) and L\u00b2, diagonal operators, Schur tests, L^p bounds for integral operators.","title":"Integral, multiplication and diagonal operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/e-lp-norm-hilbert-integral-operator-le","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.HilbertIntegralOperatorLpBoundWanted.eLpNorm_hilbertIntegralOperator_le","parent":16118,"reading_title":{"format":"inline-parts-v1","parts":[["text","An "],["math","\\(L^p\\)"],["text"," bound for Hilbert's integral operator"]],"source":"An $L^p$ bound for Hilbert's integral operator"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/index.html#e-lp-norm-hilbert-integral-operator-le","scope":"Analysis/operator-theory/integral-and-multiplication","status":"can_prove","summary":"Let $p$ be a real number with $p > 1$ and let $T$ be the operator on functions $g : \\mathbb{R} \\to \\mathbb{R}$ given by","title":"An $L^p$ bound for Hilbert's integral operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/integrable-kernel-mul-and-exists-clm-int","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.LpKernelIntegralOperatorBoundedWanted.integrable_kernel_mul_and_exists_clm_integral_operator","parent":16118,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral operators with an "],["math","\\(L^p\\)"],["text"," kernel map "],["math","\\(L^q\\)"],["text"," to "],["math","\\(L^p\\)"]],"source":"Integral operators with an $L^p$ kernel map $L^q$ to $L^p$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/index.html#integrable-kernel-mul-and-exists-clm-int","scope":"Analysis/operator-theory/integral-and-multiplication","status":"can_prove","summary":"Let $1 < p < \\infty$ and let $q \\ge 1$ be the dual exponent, $p^{-1} + q^{-1} = 1$. Let $K : \\mathbb{R}^2 \\to \\mathbb{R}$ lie in $L^p((0,1)^2)$, where $L^p((0,1)^2)$, $L^p(0,1)$ and $L^q(0,1)$ are spaces of real functions for Lebesgue measure. Then for every $f \\in L^q(0,1)$ the function $y \\mapsto K(x,y)f(y)$ is integrable on $(0,1)$ for almost every $x \\in (0,1)$, and there is a bounded linear operator\u2026","title":"Integral operators with an $L^p$ kernel map $L^q$ to $L^p$"},{"id":"Analysis/operator-theory/integral-and-multiplication/laplace-transform-bounded-on-lp-iff-eq-t","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.LaplaceTransformLpBoundedIffTwoWanted.laplaceTransform_bounded_on_Lp_iff_eq_two","parent":16118,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Laplace transform is bounded on "],["math","\\(L^p((0,\\infty))\\)"],["text"," exactly when "],["math","\\(p = 2\\)"]],"source":"The Laplace transform is bounded on $L^p((0,\\infty))$ exactly when $p = 2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/index.html#laplace-transform-bounded-on-lp-iff-eq-t","scope":"Analysis/operator-theory/integral-and-multiplication","status":"can_prove","summary":"For a real function $f$ on $\\mathbb{R}$ let","title":"The Laplace transform is bounded on $L^p((0,\\infty))$ exactly when $p = 2$"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator.json","id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","kind":"scope","navigation":"module","parent":16118,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","title":"Ces\u00e0ro operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-space","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.CesaroSpace","parent":16122,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sequence space "],["math","\\(\\ell^2(\\mathbb{N}, \\mathbb{K})\\)"]],"source":"The sequence space $\\ell^2(\\mathbb{N}, \\mathbb{K})$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-space","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"For $\\mathbb{K}$ equal to $\\mathbb{R}$ or $\\mathbb{C}$, this is the space $\\ell^2(\\mathbb{N}, \\mathbb{K})$ of square-summable sequences $x : \\mathbb{N} \\to \\mathbb{K}$, with the norm $\\|x\\| = \\left(\\sum_{n} |x(n)|^2\\right)^{1/2}$.","title":"The sequence space $\\ell^2(\\mathbb{N}, \\mathbb{K})$"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-mean","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroMean","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-mean","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For a sequence $x : \\mathbb{N} \\to \\mathbb{K}$ and $n \\in \\mathbb{N}$, the $n$-th Ces\u00e0ro mean of $x$ is the average of its first $n + 1$ terms, $\\frac{1}{n+1} \\sum_{k=0}^{n} x(k)$.","title":"Ces\u00e0ro means of a sequence"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/norm-cesaro-den","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.norm_cesaro_den","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#norm-cesaro-den","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every natural number $n$, the element $n + 1$ of $\\mathbb{K}$ (the denominator of the $n$-th Ces\u00e0ro mean) has norm equal to the real number $n + 1$.","title":"Norm of the Ces\u00e0ro denominator"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/norm-cesaro-mean-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.norm_cesaroMean_le","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#norm-cesaro-mean-le","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $x : \\mathbb{N} \\to \\mathbb{K}$ be any sequence and let $n$ be a natural number. Then $\\|\\frac{1}{n+1} \\sum_{k=0}^{n} x_k\\| \\le \\frac{1}{n+1} \\sum_{k=0}^{n} \\|x_k\\|$.","title":"A Ces\u00e0ro mean is bounded by the mean of the norms"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/summable-norm-sq-cesaro-mean","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.summable_norm_sq_cesaroMean","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#summable-norm-sq-cesaro-mean","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the Ces\u00e0ro means $\\sigma_n = (x_0 + x_1 + \\dots + x_n)/(n + 1)$ satisfy: the series $\\sum_n |\\sigma_n|^2$ converges.","title":"Ces\u00e0ro means of square-summable sequences are square-summable"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/mem-p-cesaro-mean","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.mem\u2113p_cesaroMean","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#mem-p-cesaro-mean","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. If $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, then the sequence of its Ces\u00e0ro means $n \\mapsto \\frac{1}{n+1} \\sum_{k=0}^{n} x_k$ also lies in $\\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"Ces\u00e0ro means preserve square-summability"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-linear-map","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroLinearMap","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-linear-map","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The Ces\u00e0ro operator is the $\\mathbb{K}$-linear map $C$ from $\\ell^2(\\mathbb{N}, \\mathbb{K})$ to itself that sends a sequence $x$ to its sequence of Ces\u00e0ro means, $(Cx)(n) = \\frac{1}{n+1} \\sum_{k=0}^{n} x(k)$ for $n \\in \\mathbb{N}$.","title":"The Ces\u00e0ro operator as a linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/norm-cesaro-linear-map-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.norm_cesaroLinearMap_le","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#norm-cesaro-linear-map-le","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, viewed as a linear map, given by $(Cx)_n = \\frac{1}{n+1} \\sum_{k=0}^{n} x_k$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, $\\|Cx\\| \\le 2 \\|x\\|$.","title":"The Ces\u00e0ro operator is bounded by 2"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroCLM","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-clm","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The Ces\u00e0ro operator $C$ is the continuous linear map from $\\ell^2(\\mathbb{N}, \\mathbb{K})$ to itself that sends a sequence $x$ to its sequence of averages, $(Cx)_n = (x_0 + x_1 + \\dots + x_n)/(n + 1)$. It is obtained from the Ces\u00e0ro linear map with the bound $\\|Cx\\| \\le 2 \\|x\\|$.","title":"Ces\u00e0ro operator on square-summable sequences"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-clm-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroCLM_apply","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-clm-apply","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $n \\in \\mathbb{N}$, $(Cx)_n = \\bigl(\\sum_{k=0}^{n} x_k\\bigr)/(n + 1)$.","title":"Formula for the Ces\u00e0ro operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-sum-eq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaro_sum_eq","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-sum-eq","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $n \\in \\mathbb{N}$, $\\sum_{k=0}^{n} x(k) = (n+1)\\,(Cx)(n)$.","title":"Partial sums from Ces\u00e0ro means"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-clm-eq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroCLM_eq","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-clm-eq","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, the Ces\u00e0ro operator, as a continuous linear map on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, takes the same value at $x$ as the Ces\u00e0ro linear map, the same averaging map considered only as a linear map.","title":"The Ces\u00e0ro operator agrees with the Ces\u00e0ro linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/norm-cesaro-clm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.norm_cesaroCLM_le","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#norm-cesaro-clm-le","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The Ces\u00e0ro operator $C$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(Cx)_n = \\frac{1}{n+1} \\sum_{k=0}^{n} x_k$, has operator norm $\\|C\\| \\le 2$.","title":"The Ces\u00e0ro operator has norm at most 2"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-e0","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroE0","parent":16122,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first standard basis vector of "],["math","\\(\\ell^2\\)"]],"source":"The first standard basis vector of $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-e0","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The vector $e_0 \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ is the sequence with $e_0(0) = 1$ and $e_0(n) = 0$ for every $n \\ge 1$.","title":"The first standard basis vector of $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-e0-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroE0_apply","parent":16122,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinates of the first standard basis vector of "],["math","\\(\\ell^2\\)"]],"source":"Coordinates of the first standard basis vector of $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-e0-apply","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $e_0 \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ be the first standard basis vector. For every $n \\in \\mathbb{N}$, $e_0(n) = 1$ if $n = 0$ and $e_0(n) = 0$ otherwise.","title":"Coordinates of the first standard basis vector of $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/norm-cesaro-e0","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.norm_cesaroE0","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#norm-cesaro-e0","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The first standard basis vector $e_0 = (1, 0, 0, \\dots)$ of $\\ell^2(\\mathbb{N}, \\mathbb{K})$ has norm $\\|e_0\\| = 1$.","title":"The first basis vector is a unit vector"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/one-le-norm-cesaro-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.one_le_norm_cesaroCLM","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#one-le-norm-cesaro-clm","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The Ces\u00e0ro operator $C$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(Cx)_n = (x_0 + x_1 + \\dots + x_n)/(n + 1)$, has operator norm $\\|C\\| \\ge 1$.","title":"The Ces\u00e0ro operator has norm at least one"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-preimage-succ","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaro_preimage_succ","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-preimage-succ","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For every $x \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $n \\in \\mathbb{N}$, $x(n+1) = (n+2)\\,(Cx)(n+1) - (n+1)\\,(Cx)(n)$.","title":"One step of inverting the Ces\u00e0ro operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-clm-injective","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroCLM_injective","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-clm-injective","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The Ces\u00e0ro operator $C$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(Cx)_n = (x_0 + \\dots + x_n)/(n + 1)$, is injective.","title":"The Ces\u00e0ro operator is injective"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-non-range","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroNonRange","parent":16122,"reading_title":{"format":"inline-parts-v1","parts":[["text","The alternating harmonic sequence in "],["math","\\(\\ell^2\\)"]],"source":"The alternating harmonic sequence in $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-non-range","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The sequence $y$ with $y(n) = (-1)^n / (n + 1)$ for $n \\in \\mathbb{N}$ is square-summable, and it is regarded as an element of $\\ell^2(\\mathbb{N}, \\mathbb{K})$.","title":"The alternating harmonic sequence in $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-non-range-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroNonRange_apply","parent":16122,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinates of the alternating harmonic sequence in "],["math","\\(\\ell^2\\)"]],"source":"Coordinates of the alternating harmonic sequence in $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-non-range-apply","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $y$ be the alternating harmonic sequence, viewed as an element of $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For every $n \\in \\mathbb{N}$, $y(n) = (-1)^n / (n + 1)$.","title":"Coordinates of the alternating harmonic sequence in $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/not-mem-range-cesaro-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.notMem_range_cesaroCLM","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#not-mem-range-cesaro-clm","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The sequence $y \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ with $y_n = (-1)^n/(n + 1)$ is not in the range of the Ces\u00e0ro operator $C$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(Cx)_n = (x_0 + x_1 + \\dots + x_n)/(n + 1)$. In particular $C$ is not surjective.","title":"The Ces\u00e0ro operator is not surjective"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-clm-single-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroCLM_single_apply","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-clm-single-apply","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, and for $n \\in \\mathbb{N}$ let $e_n$ be the standard basis vector with $1$ in coordinate $n$ and $0$ elsewhere. For all $n, k \\in \\mathbb{N}$, $(C e_n)_k = 1/(k + 1)$ if $n \\le k$, and $(C e_n)_k = 0$ if $k < n$.","title":"The Ces\u00e0ro operator on standard basis vectors"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-adjoint-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaro_adjoint_apply","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-adjoint-apply","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(Cx)_n = (x_0 + \\dots + x_n)/(n + 1)$. For every $y \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ and every $n \\in \\mathbb{N}$, the adjoint $C^*$ of $C$ satisfies","title":"Adjoint of the Ces\u00e0ro operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-adjoint-single-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaro_adjoint_single_zero","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-adjoint-single-zero","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, and let $e_0 = (1, 0, 0, \\dots)$ be the first standard basis vector. Then $C^* e_0 = e_0$, where $C^*$ is the adjoint of $C$.","title":"The adjoint of the Ces\u00e0ro operator fixes the first basis vector"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-clm-e0-apply-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroCLM_e0_apply_one","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-clm-e0-apply-one","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, and let $e_0 = (1, 0, 0, \\dots)$ be the first standard basis vector. With coordinates indexed from $0$, $(C e_0)_1 = 1/2$.","title":"Second coordinate of the Ces\u00e0ro operator at the first basis vector"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/not-is-self-adjoint-cesaro-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.not_isSelfAdjoint_cesaroCLM","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#not-is-self-adjoint-cesaro-clm","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The Ces\u00e0ro operator $C$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(Cx)_n = (x_0 + x_1 + \\dots + x_n)/(n + 1)$, is not self-adjoint.","title":"The Ces\u00e0ro operator is not self-adjoint"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-test","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroTest","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-test","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $e_i$ be the $i$-th standard basis vector of $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For a natural number $N$, the test vector $t_N = \\sum_{i < N} \\frac{1}{\\sqrt{N}} e_i$ has coordinate $1/\\sqrt{N}$ at each $j < N$ and $0$ at each $j \\ge N$; in particular $t_0 = 0$.","title":"Ces\u00e0ro test vectors"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-test-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroTest_apply","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-test-apply","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $N$ and $j$ be natural numbers. The $j$-th coordinate of the Ces\u00e0ro test vector $t_N \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$ is $1/\\sqrt{N}$ if $j < N$, and $0$ if $j \\ge N$.","title":"Coordinates of the Ces\u00e0ro test vectors"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/norm-cesaro-test-sqrt","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.norm_cesaroTest_sqrt","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#norm-cesaro-test-sqrt","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every natural number $N$, the norm of $(\\sqrt{N})^{-1}$, computed in $\\mathbb{K}$, equals the real number $(\\sqrt{N})^{-1}$; for $N = 0$ both sides are $0$, with the convention $0^{-1} = 0$.","title":"Norm of the Ces\u00e0ro test-vector coefficient"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/norm-cesaro-test","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.norm_cesaroTest","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#norm-cesaro-test","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every natural number $N \\ge 1$, the test vector $t_N \\in \\ell^2(\\mathbb{N}, \\mathbb{K})$, equal to $1/\\sqrt{N}$ at the coordinates $0, \\dots, N - 1$ and to $0$ elsewhere, has norm $\\|t_N\\| = 1$.","title":"The Ces\u00e0ro test vectors are unit vectors"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-clm-test-apply-lt","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroCLM_test_apply_lt","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-clm-test-apply-lt","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For $N \\in \\mathbb{N}$ let $t_N$ be the test vector whose first $N$ coordinates equal $1/\\sqrt{N}$ and whose other coordinates are $0$. For all $n, N \\in \\mathbb{N}$ with $n < N$, $(C t_N)_n = 1/\\sqrt{N}$.","title":"The Ces\u00e0ro operator on a test vector, within its support"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/cesaro-clm-test-apply-ge","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.cesaroCLM_test_apply_ge","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#cesaro-clm-test-apply-ge","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$. For $N \\in \\mathbb{N}$ let $t_N$ be the test vector whose first $N$ coordinates equal $(\\sqrt{N})^{-1}$ and whose other coordinates are $0$. For all $N, n \\in \\mathbb{N}$ with $N \\le n$, $(C t_N)_n = N (\\sqrt{N})^{-1}/(n + 1)$, where $(\\sqrt{0})^{-1}$ is read as $0$.","title":"The Ces\u00e0ro operator on a test vector, past its support"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/five-div-four-le-norm-sq-cesaro-test","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.five_div_four_le_norm_sq_cesaro_test","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#five-div-four-le-norm-sq-cesaro-test","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$ and let $t_N$ be the $N$-th Ces\u00e0ro test vector. For every natural number $N \\ge 1$, $\\|C t_N\\|^2 \\ge 5/4$.","title":"Lower bound for the Ces\u00e0ro operator on the test vectors"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/tendsto-cesaro-test-coord-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.tendsto_cesaro_test_coord_zero","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#tendsto-cesaro-test-coord-zero","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $C$ be the Ces\u00e0ro operator on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, $(Cx)_n = (x_0 + x_1 + \\dots + x_n)/(n + 1)$, and for $N \\ge 1$ let $t_N$ be the vector equal to $1/\\sqrt{N}$ at the coordinates $0, \\dots, N - 1$ and to $0$ elsewhere. Then for every fixed coordinate $n$, $(C t_{N+1})_n \\to 0$ as $N \\to \\infty$.","title":"The Ces\u00e0ro images of the test vectors tend to zero coordinatewise"},{"id":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/not-is-compact-operator-cesaro-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CesaroOperator.not_isCompactOperator_cesaroCLM","parent":16122,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/ces-ro-operator/index.html#not-is-compact-operator-cesaro-clm","scope":"Analysis/operator-theory/integral-and-multiplication/ces-ro-operator","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. The Ces\u00e0ro operator $C$ on $\\ell^2(\\mathbb{N}, \\mathbb{K})$, given by $(Cx)_n = (x_0 + x_1 + \\dots + x_n)/(n + 1)$, is not a compact operator.","title":"The Ces\u00e0ro operator is not compact"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication/diagonal-operators.json","id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","kind":"scope","navigation":"module","parent":16118,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","title":"Diagonal operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertBasisDiagonal.diagonalOperator","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-operator","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"The bounded diagonal continuous linear map with coefficients `a` in the Hilbert basis `b`. Its uniform coefficient bound is part of the constructor.","title":"Bounded diagonal operator on a Hilbert basis"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/repr-diagonal-operator-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertBasisDiagonal.repr_diagonalOperator_apply","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#repr-diagonal-operator-apply","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $(b_i)_{i \\in \\iota}$ be a Hilbert basis of an inner product space $H$ over $\\mathbb{K}$ ($\\mathbb{R}$ or $\\mathbb{C}$), with inner product $\\langle \\cdot, \\cdot \\rangle$ conjugate-linear in the first argument, let $a : \\iota \\to \\mathbb{K}$ satisfy $|a_i| \\le C$ for all $i$, and let $D$ be the diagonal operator with coefficients $a$. For every $x \\in H$ and $i \\in \\iota$, the $i$-th coordinate of $D x$ in\u2026","title":"Coordinates of a diagonal operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-operator-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertBasisDiagonal.diagonalOperator_eq_tsum","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-operator-eq-tsum","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $(b_i)_{i \\in \\iota}$ be a Hilbert basis of an inner product space $H$ over $\\mathbb{K}$ ($\\mathbb{R}$ or $\\mathbb{C}$), with inner product $\\langle \\cdot, \\cdot \\rangle$ conjugate-linear in the first argument, and let $a : \\iota \\to \\mathbb{K}$ satisfy $|a_i| \\le C$ for all $i$. The diagonal operator $D$ with coefficients $a$ satisfies, for every $x \\in H$,","title":"Series formula for a diagonal operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-operator-apply-basis","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertBasisDiagonal.diagonalOperator_apply_basis","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-operator-apply-basis","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $(b_i)_{i \\in \\iota}$ be a Hilbert basis of an inner product space $H$ over $\\mathbb{K}$ ($\\mathbb{R}$ or $\\mathbb{C}$), let $a : \\iota \\to \\mathbb{K}$ satisfy $|a_i| \\le C$ for all $i$, and let $D$ be the diagonal operator with coefficients $a$. Then $D b_j = a_j b_j$ for every $j \\in \\iota$.","title":"Action of a diagonal operator on the basis"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-operator-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HilbertBasisDiagonal.diagonalOperator_sq","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-operator-sq","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $(b_i)_{i \\in \\iota}$ be a Hilbert basis of an inner product space $H$ over $\\mathbb{K}$ ($\\mathbb{R}$ or $\\mathbb{C}$), and let $a : \\iota \\to \\mathbb{K}$ satisfy $|a_i| \\le C$ for all $i$. Let $D_a$ be the diagonal operator with coefficients $a$, and let $D_{a^2}$ be the diagonal operator with coefficients $a_i^2$, which satisfy $|a_i^2| \\le C^2$. Then $D_a(D_a x) = D_{a^2} x$ for every $x \\in H$.","title":"Square of a diagonal operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/lp-smul-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.lp_smul_apply","parent":16159,"reading_title":{"format":"inline-parts-v1","parts":[["text","Scalar multiplication on "],["math","\\(\\ell^2\\)"],["text"," is pointwise"]],"source":"Scalar multiplication on $\\ell^2$ is pointwise"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#lp-smul-apply","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $c \\in \\mathbb{K}$, every $x \\in \\ell^2(\\iota, \\mathbb{K})$ and every $i \\in \\iota$, the $i$-th coordinate of $c x$ is $(c x)_i = c \\, x_i$.","title":"Scalar multiplication on $\\ell^2$ is pointwise"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/mem-p-diagonal","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.mem\u2113p_diagonal","parent":16159,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded multipliers preserve "],["math","\\(\\ell^2\\)"]],"source":"Bounded multipliers preserve $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#mem-p-diagonal","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. If $a : \\iota \\to \\mathbb{K}$ is bounded, that is, $|a_i| \\le C$ for some real $C$ and all $i$, then for every $x \\in \\ell^2(\\iota, \\mathbb{K})$ the sequence $(a_i x_i)_{i \\in \\iota}$ lies in $\\ell^2(\\iota, \\mathbb{K})$, that is, the family $(|a_i x_i|^2)_{i \\in \\iota}$ is summable.","title":"Bounded multipliers preserve $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.diagonalCLM","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-clm","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type and let $a : \\iota \\to \\mathbb{K}$ be a bounded sequence, that is, $\\|a_i\\| \\le C$ for some real $C$ and all $i \\in \\iota$. The diagonal operator $D_a$ with coefficients $a$ is the bounded linear operator on $\\ell^2(\\iota, \\mathbb{K})$ that sends $x$ to the sequence $(a_i x_i)_{i \\in \\iota}$.","title":"Diagonal operator on \u2113\u00b2"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.diagonalCLM_apply","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-clm-apply","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type, let $a : \\iota \\to \\mathbb{K}$ be a bounded sequence and let $D_a$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ with coefficients $a$. For every $x \\in \\ell^2(\\iota, \\mathbb{K})$ and every $i \\in \\iota$, the $i$-th coordinate of $D_a x$ is $a_i x_i$.","title":"Coordinates of a diagonal operator on \u2113\u00b2"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-lin-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.diagonalLinMap","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-lin-map","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type and let $a : \\iota \\to \\mathbb{K}$ be a bounded sequence. This is the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ with coefficients $a$, regarded as a $\\mathbb{K}$-linear map $\\ell^2(\\iota, \\mathbb{K}) \\to \\ell^2(\\iota, \\mathbb{K})$ by forgetting its continuity.","title":"Diagonal operator on \u2113\u00b2 as a linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.diagonalLinMap_apply","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-lin-map-apply","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type and let $a : \\iota \\to \\mathbb{K}$ be a bounded sequence. For every $x \\in \\ell^2(\\iota, \\mathbb{K})$ and every $i \\in \\iota$, the diagonal operator with coefficients $a$, regarded as a linear map, sends $x$ to a sequence whose $i$-th coordinate is $a_i x_i$.","title":"Coordinates of the diagonal linear map on \u2113\u00b2"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/norm-diagonal-lin-map-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.norm_diagonalLinMap_le","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#norm-diagonal-lin-map-le","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be bounded, and let $D_a$ be the diagonal operator $x \\mapsto (a_i x_i)_{i \\in \\iota}$ on $\\ell^2(\\iota, \\mathbb{K})$, viewed as a linear map. If $C \\ge 0$ and $|a_i| \\le C$ for every $i \\in \\iota$, then $\\|D_a x\\| \\le C \\|x\\|$ for every $x \\in \\ell^2(\\iota, \\mathbb{K})$.","title":"Pointwise bound for diagonal operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-clm-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.diagonalCLM_single","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-clm-single","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type, let $a : \\iota \\to \\mathbb{K}$ be a bounded sequence and let $D_a$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ with coefficients $a$. For $i \\in \\iota$ let $e_i \\in \\ell^2(\\iota, \\mathbb{K})$ be the sequence equal to $1$ at $i$ and $0$ elsewhere. Then for every $i \\in \\iota$ and every $c \\in \\mathbb{K}$ we have\u2026","title":"Diagonal operators on standard basis vectors"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-clm-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.diagonalCLM_one","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-clm-one","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be a type. The diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ whose coefficients all equal $1$ is the identity operator.","title":"The diagonal operator with all coefficients one is the identity"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-clm-comp","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.diagonalCLM_comp","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-clm-comp","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type, and for a bounded sequence $c : \\iota \\to \\mathbb{K}$ let $D_c$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ with coefficients $c$. For bounded sequences $a, b : \\iota \\to \\mathbb{K}$, we have $D_b \\circ D_a = D_{ba}$, where $ba$ is the pointwise product $i \\mapsto b_i a_i$.","title":"Composition of diagonal operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/diagonal-clm-sub","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.diagonalCLM_sub","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#diagonal-clm-sub","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type, and for a bounded sequence $c : \\iota \\to \\mathbb{K}$ let $D_c$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ with coefficients $c$. For bounded sequences $a, b : \\iota \\to \\mathbb{K}$, we have $D_a - D_b = D_{a - b}$, where $a - b$ is the sequence $i \\mapsto a_i - b_i$.","title":"Difference of diagonal operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/norm-diagonal-clm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.norm_diagonalCLM_le","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#norm-diagonal-clm-le","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be bounded, and let $D_a$ be the diagonal operator $x \\mapsto (a_i x_i)_{i \\in \\iota}$ on $\\ell^2(\\iota, \\mathbb{K})$. If $C \\ge 0$ and $|a_i| \\le C$ for every $i \\in \\iota$, then the operator norm satisfies $\\|D_a\\| \\le C$.","title":"Operator norm bound for diagonal operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/norm-diagonal-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.norm_diagonalCLM","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#norm-diagonal-clm","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be bounded, and let $D_a$ be the diagonal operator $x \\mapsto (a_i x_i)_{i \\in \\iota}$ on $\\ell^2(\\iota, \\mathbb{K})$. Then the operator norm of $D_a$ is $\\|D_a\\| = \\sup_{i \\in \\iota} |a_i|$, where the supremum is taken to be $0$ when $\\iota$ is empty.","title":"Operator norm of a diagonal operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/adjoint-diagonal-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.adjoint_diagonalCLM","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#adjoint-diagonal-clm","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type and let $a : \\iota \\to \\mathbb{K}$ be a bounded sequence. The adjoint of the diagonal operator $D_a$ on $\\ell^2(\\iota, \\mathbb{K})$ with coefficients $a$ is the diagonal operator with the conjugate coefficients $i \\mapsto \\overline{a_i}$.","title":"Adjoint of a diagonal operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/is-self-adjoint-diagonal-clm-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.isSelfAdjoint_diagonalCLM_iff","parent":16159,"reading_title":{"format":"inline-parts-v1","parts":[["text","Self-adjoint diagonal operators on "],["math","\\(\\ell^2\\)"]],"source":"Self-adjoint diagonal operators on $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#is-self-adjoint-diagonal-clm-iff","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be bounded, and let $D_a$ be the diagonal operator $x \\mapsto (a_i x_i)_{i \\in \\iota}$ on $\\ell^2(\\iota, \\mathbb{K})$. Then $D_a$ is self-adjoint if and only if $\\overline{a_i} = a_i$, that is, $a_i$ is real, for every $i \\in \\iota$.","title":"Self-adjoint diagonal operators on $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/single-clm-comp-eval-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.singleCLM_comp_evalCLM_apply","parent":16159,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank-one term of the finite-support decomposition on "],["math","\\(\\ell^2\\)"]],"source":"Rank-one term of the finite-support decomposition on $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#single-clm-comp-eval-clm-apply","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"A rank-one term of the finite-support decomposition, evaluated pointwise.","title":"Rank-one term of the finite-support decomposition on $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/is-compact-operator-diagonal-clm-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperator.isCompactOperator_diagonalCLM_iff","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#is-compact-operator-diagonal-clm-iff","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type and let $a : \\iota \\to \\mathbb{K}$ be a bounded sequence. The diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ with coefficients $a$ is compact if and only if $a$ tends to $0$ along the cofinite filter on $\\iota$, that is, for every $\\varepsilon > 0$ only finitely many $i \\in \\iota$ satisfy $\\|a_i\\| \\ge \\varepsilon$.","title":"Compact diagonal operators on \u2113\u00b2"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/smul-one-sub-diagonal-clm-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperatorSpectrum.smul_one_sub_diagonalCLM_eq","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#smul-one-sub-diagonal-clm-eq","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be a bounded family of coefficients, let $D_a$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ given by $(D_a x)_i = a_i x_i$, and let $I$ be the identity operator. For every $\\mu \\in \\mathbb{K}$, $\\mu I - D_a = D_{\\mu - a}$, the diagonal operator with coefficients $\\mu - a_i$.","title":"Shifting a diagonal operator by a scalar"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/is-unit-smul-one-sub-diagonal-clm-of-not","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperatorSpectrum.isUnit_smul_one_sub_diagonalCLM_of_not_mem_closure_range","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#is-unit-smul-one-sub-diagonal-clm-of-not","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be a bounded family of coefficients, let $D_a$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ given by $(D_a x)_i = a_i x_i$, and let $I$ be the identity operator. If $\\mu \\in \\mathbb{K}$ does not lie in the closure of $\\{a_i : i \\in \\iota\\}$, then $\\mu I - D_a$ is invertible in the algebra of bounded operators on\u2026","title":"Invertibility of a shifted diagonal operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/not-mem-spectrum-of-not-mem-closure-rang","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperatorSpectrum.not_mem_spectrum_of_not_mem_closure_range","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#not-mem-spectrum-of-not-mem-closure-rang","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be a bounded family of coefficients, and let $D_a$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ given by $(D_a x)_i = a_i x_i$. If $\\mu \\in \\mathbb{K}$ does not lie in the closure of $\\{a_i : i \\in \\iota\\}$, then $\\mu$ is not in the spectrum of $D_a$ over $\\mathbb{K}$, that is, $\\mu I - D_a$ is invertible as a bounded\u2026","title":"Points off the closure of the coefficients are not in the spectrum"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/range-subset-spectrum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperatorSpectrum.range_subset_spectrum","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#range-subset-spectrum","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be a bounded family of coefficients, and let $D_a$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ given by $(D_a x)_i = a_i x_i$. Then every $a_j$ with $j \\in \\iota$ lies in the spectrum of $D_a$ over $\\mathbb{K}$, the set of $\\mu \\in \\mathbb{K}$ for which $\\mu I - D_a$ is not invertible as a bounded operator, where $I$ is the\u2026","title":"Every coefficient of a diagonal operator lies in its spectrum"},{"id":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators/spectrum-diagonal-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.LpDiagonalOperatorSpectrum.spectrum_diagonalCLM","parent":16159,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/diagonal-operators/index.html#spectrum-diagonal-clm","scope":"Analysis/operator-theory/integral-and-multiplication/diagonal-operators","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $a : \\iota \\to \\mathbb{K}$ be a bounded family of coefficients, and let $D_a$ be the diagonal operator on $\\ell^2(\\iota, \\mathbb{K})$ given by $(D_a x)_i = a_i x_i$. The spectrum of $D_a$ over $\\mathbb{K}$, the set of $\\mu \\in \\mathbb{K}$ for which $\\mu I - D_a$ is not invertible as a bounded operator ($I$ the identity operator), is the closure of\u2026","title":"Spectrum of a diagonal operator"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels.json","id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","kind":"scope","navigation":"module","parent":16118,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","title":"Integral operators with continuous kernels"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-space","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.KernelSpace","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-space","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"$C([0, 1], \\mathbb{R})$ denotes the space of bounded continuous functions $[0, 1] \\to \\mathbb{R}$, with the supremum norm $\\|f\\| = \\sup_{x \\in [0, 1]} |f(x)|$.","title":"Continuous functions on the unit interval"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-type","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.KernelType","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-type","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"A continuous kernel is a bounded continuous function $k : [0, 1] \\times [0, 1] \\to \\mathbb{R}$. These kernels form a normed space under the supremum norm $\\|k\\| = \\sup_{x, y \\in [0, 1]} |k(x, y)|$.","title":"Continuous kernels on the unit square"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-extend","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelExtend","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-extend","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For a bounded continuous function $f$ on $[0, 1]$, its clamped extension is the function $\\tilde f : \\mathbb{R} \\to \\mathbb{R}$ given by $\\tilde f(t) = f(\\max(0, \\min(1, t)))$. Thus $\\tilde f(t) = f(0)$ for $t < 0$, $\\tilde f(t) = f(t)$ for $t \\in [0, 1]$, and $\\tilde f(t) = f(1)$ for $t > 1$.","title":"Clamped extension"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-extend-continuous","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelExtend_continuous","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-extend-continuous","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For every bounded continuous function $f$ on $[0, 1]$, the clamped extension $t \\mapsto f(\\max(0, \\min(1, t)))$ is continuous on $\\mathbb{R}$.","title":"Continuity of the clamped extension"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-extend-apply-of-mem","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelExtend_apply_of_mem","parent":16187,"reading_title":{"format":"inline-parts-v1","parts":[["text","The clamped extension agrees with the function on "],["math","\\((0, 1)\\)"]],"source":"The clamped extension agrees with the function on $(0, 1)$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-extend-apply-of-mem","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $f$ be a bounded continuous function on $[0, 1]$ and let $\\tilde f(s) = f(\\max(0, \\min(1, s)))$ for real $s$ be its clamped extension. Then $\\tilde f(t) = f(t)$ for every $t \\in [0, 1]$.","title":"The clamped extension agrees with the function on $(0, 1)$"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/norm-kernel-extend-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.norm_kernelExtend_le","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#norm-kernel-extend-le","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $\\pi(t) = \\max(0, \\min(1, t))$ be the projection of $\\mathbb{R}$ onto $[0, 1]$. For every $f \\in C([0, 1], \\mathbb{R})$ and every $t \\in \\mathbb{R}$, the clamped extension $f \\circ \\pi$ of $f$ satisfies $|f(\\pi(t))| \\le \\|f\\|$, where $\\|f\\|$ is the supremum norm.","title":"The clamped extension is bounded by the supremum norm"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-integrand","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelIntegrand","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-integrand","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $\\pi(t) = \\max(0, \\min(1, t))$ clamp a real number $t$ to $[0, 1]$. For a continuous kernel $k$ on $[0, 1] \\times [0, 1]$, a bounded continuous function $f$ on $[0, 1]$ and a point $x \\in [0, 1]$, the kernel integrand is the function $\\mathbb{R} \\to \\mathbb{R}$, $t \\mapsto k(x, \\pi(t)) \\, f(\\pi(t))$.","title":"Kernel integrand"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/continuous-kernel-integrand-uncurry","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.continuous_kernelIntegrand_uncurry","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#continuous-kernel-integrand-uncurry","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $\\pi(t) = \\max(0, \\min(1, t))$ clamp a real number $t$ to $[0, 1]$. For every continuous kernel $k$ on $[0, 1] \\times [0, 1]$ and every bounded continuous function $f$ on $[0, 1]$, the kernel integrand $(x, t) \\mapsto k(x, \\pi(t)) \\, f(\\pi(t))$ is jointly continuous on $[0, 1] \\times \\mathbb{R}$.","title":"Joint continuity of the kernel integrand"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-integrand-continuous","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelIntegrand_continuous","parent":16187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuity of the kernel integrand in "],["math","\\(t\\)"]],"source":"Continuity of the kernel integrand in $t$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-integrand-continuous","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $\\pi(t) = \\max(0, \\min(1, t))$ clamp a real number $t$ to $[0, 1]$. For every continuous kernel $k$ on $[0, 1] \\times [0, 1]$, every bounded continuous function $f$ on $[0, 1]$ and every $x \\in [0, 1]$, the function $t \\mapsto k(x, \\pi(t)) \\, f(\\pi(t))$ is continuous on $\\mathbb{R}$.","title":"Continuity of the kernel integrand in $t$"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-value","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelValue","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-value","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $\\pi(t) = \\max(0, \\min(1, t))$ be the projection of $\\mathbb{R}$ onto $[0, 1]$. For a continuous kernel $k$ on $[0, 1] \\times [0, 1]$, a function $f \\in C([0, 1], \\mathbb{R})$ and a point $x \\in [0, 1]$, the kernel value is the real number","title":"Kernel value"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-value-continuous","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelValue_continuous","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-value-continuous","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For every continuous kernel $k$ on $[0, 1] \\times [0, 1]$ and every $f \\in C([0, 1], \\mathbb{R})$, the kernel value of $k$ and $f$, the function $x \\mapsto \\int_0^1 k(x, t)\\, f(t)\\, dt$ on $[0, 1]$, is continuous.","title":"Continuity of the kernel value"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/norm-kernel-value-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.norm_kernelValue_le","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#norm-kernel-value-le","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $k$ be a continuous kernel on $[0, 1] \\times [0, 1]$ and let $f \\in C([0, 1], \\mathbb{R})$. For every $x \\in [0, 1]$, the integral $\\int_0^1 k(x, t) \\, f(t) \\, dt$ has absolute value at most $\\|k\\|_\\infty \\, \\|f\\|_\\infty$.","title":"Pointwise bound for a kernel integral"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-linear-map","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelLinearMap","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-linear-map","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For a continuous kernel $k$ on $[0, 1] \\times [0, 1]$, this is the $\\mathbb{R}$-linear map $K$ from the space of bounded continuous real functions on $[0, 1]$ to itself given by $(Kf)(x) = \\int_0^1 k(x, t) f(t) \\, dt$ for $x \\in [0, 1]$.","title":"The integral operator with continuous kernel, as a linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-linear-map-norm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelLinearMap_norm_le","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-linear-map-norm-le","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Write $\\|\\cdot\\|$ for the sup norm. For every continuous kernel $k$ on $[0, 1] \\times [0, 1]$ and every bounded continuous function $f$ on $[0, 1]$, the function $Kf$ given by $(Kf)(x) = \\int_0^1 k(x, t) f(t) \\, dt$ satisfies $\\|Kf\\| \\le \\|k\\| \\, \\|f\\|$.","title":"Sup-norm bound for the integral operator with continuous kernel"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelOperator","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-operator","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For a continuous kernel $k$ on $[0, 1] \\times [0, 1]$, the integral operator $T_k$ with kernel $k$ is the continuous linear operator on $C([0, 1], \\mathbb{R})$ given, for $f \\in C([0, 1], \\mathbb{R})$ and $x \\in [0, 1]$, by $(T_k f)(x) = \\int_0^1 k(x, t)\\, f(t)\\, dt$.","title":"Integral operator with a continuous kernel"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-operator-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelOperator_apply","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-operator-apply","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $\\pi(t) = \\max(0, \\min(1, t))$ be the projection of $\\mathbb{R}$ onto $[0, 1]$. For a continuous kernel $k$ on $[0, 1] \\times [0, 1]$, write $T_k$ for the integral operator with kernel $k$ on $C([0, 1], \\mathbb{R})$. For every $f \\in C([0, 1], \\mathbb{R})$ and every $x \\in [0, 1]$,","title":"Evaluation formula for the integral operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/norm-kernel-operator-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.norm_kernelOperator_le","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#norm-kernel-operator-le","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For every continuous kernel $k$ on $[0, 1] \\times [0, 1]$, the integral operator $K$ with kernel $k$ on $C([0, 1], \\mathbb{R})$ has operator norm $\\|K\\| \\le \\|k\\|_\\infty$.","title":"Norm bound for a kernel integral operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/norm-kernel-operator-apply-sub-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.norm_kernelOperator_apply_sub_le","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#norm-kernel-operator-apply-sub-le","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $k$ be a continuous kernel on $[0, 1] \\times [0, 1]$, let $K$ be the integral operator with kernel $k$ on $C([0, 1], \\mathbb{R})$, let $f \\in C([0, 1], \\mathbb{R})$, and let $x, x' \\in [0, 1]$ and $\\delta \\in \\mathbb{R}$. If $|k(x, y) - k(x', y)| \\le \\delta$ for every $y \\in [0, 1]$, then $|(Kf)(x) - (Kf)(x')| \\le \\delta \\, \\|f\\|_\\infty$.","title":"Difference bound for the values of a kernel operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/is-compact-operator-kernel-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.isCompactOperator_kernelOperator","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#is-compact-operator-kernel-operator","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For every continuous kernel $k$ on $[0, 1] \\times [0, 1]$, the integral operator $K$ on the space of bounded continuous real functions on $[0, 1]$, given by $(Kf)(x) = \\int_0^1 k(x, t) f(t) \\, dt$, is a compact operator.","title":"Integral operators with continuous kernels are compact"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/finite-dimensional-range-kernel-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.finiteDimensional_range_kernelOperator_of_separable","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#finite-dimensional-range-kernel-operator","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"Let $s$ be a finite set of indices and, for $i \\in s$, let $\\varphi_i$ and $\\psi_i$ be bounded continuous functions on $[0, 1]$. If a continuous kernel $k$ on $[0, 1] \\times [0, 1]$ satisfies $k(x, y) = \\sum_{i \\in s} \\varphi_i(x) \\psi_i(y)$ for all $x, y \\in [0, 1]$, then the range of the integral operator $K$ on the space of bounded continuous real functions on $[0, 1]$, given by\u2026","title":"Separable kernels give finite-rank integral operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-operator-add","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelOperator_add","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-operator-add","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For a continuous kernel $k$ on $[0, 1] \\times [0, 1]$, write $T_k$ for the integral operator with kernel $k$ on $C([0, 1], \\mathbb{R})$. For all continuous kernels $k_1$ and $k_2$, the operator with the pointwise sum kernel is the sum of the operators: $T_{k_1 + k_2} = T_{k_1} + T_{k_2}$.","title":"The integral operator is additive in the kernel"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-operator-smul","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelOperator_smul","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-operator-smul","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"For a continuous kernel $k$ on $[0, 1] \\times [0, 1]$, write $T_k$ for the integral operator with kernel $k$ on $C([0, 1], \\mathbb{R})$. For every real number $c$ and every continuous kernel $k$, $T_{c k} = c\\, T_k$.","title":"The integral operator is homogeneous in the kernel"},{"id":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/kernel-operator-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.KernelOperator.kernelOperator_zero","parent":16187,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels/index.html#kernel-operator-zero","scope":"Analysis/operator-theory/integral-and-multiplication/integral-operators-with-continuous-kernels","status":"fully_proved","summary":"The integral operator on $C([0, 1], \\mathbb{R})$ whose kernel is the zero function on $[0, 1] \\times [0, 1]$ is the zero operator.","title":"The zero kernel gives the zero operator"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l.json","id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","kind":"scope","navigation":"module","parent":16118,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"can_prove","title":"Multiplication operators on L\u00b2"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/essential-range","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.essentialRange","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#essential-range","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $\\varphi : \\alpha \\to \\mathbb{C}$. The essential range of $\\varphi$ with respect to $\\mu$ is the set of $z \\in \\mathbb{C}$ such that $\\mu(\\{x \\in \\alpha : |\\varphi(x) - z| < \\varepsilon\\}) > 0$ for every $\\varepsilon > 0$.","title":"Essential range"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/essential-range-congr-ae","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.essentialRange_congr_ae","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#essential-range-congr-ae","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $\\varphi, \\psi : \\alpha \\to \\mathbb{C}$. If $\\varphi = \\psi$ $\\mu$-almost everywhere, then $\\varphi$ and $\\psi$ have the same essential range with respect to $\\mu$.","title":"The essential range is invariant under almost-everywhere equality"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/not-mem-essential-range-iff","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.not_mem_essentialRange_iff","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#not-mem-essential-range-iff","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $\\varphi : \\alpha \\to \\mathbb{C}$ and let $z \\in \\mathbb{C}$. Then $z$ is not in the essential range of $\\varphi$ with respect to $\\mu$ if and only if there is $\\varepsilon > 0$ such that the preimage under $\\varphi$ of the open ball $B(z, \\varepsilon) = \\{w \\in \\mathbb{C} : |w - z| < \\varepsilon\\}$ has $\\mu$-measure zero.","title":"Points outside the essential range"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/essential-range-const","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.essentialRange_const","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#essential-range-const","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ with $\\mu(\\alpha) \\neq 0$, and let $c \\in \\mathbb{C}$. Then the essential range of the constant function $x \\mapsto c$ with respect to $\\mu$ is $\\{c\\}$.","title":"Essential range of a constant function"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/essential-range-count-eq-range","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.essentialRange_count_eq_range","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#essential-range-count-eq-range","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\alpha$ be a finite type with a measurable structure in which singletons are measurable, and let $\\varphi : \\alpha \\to \\mathbb{C}$. Then the essential range of $\\varphi$ with respect to counting measure on $\\alpha$ is the range $\\varphi(\\alpha)$.","title":"Essential range for counting measure on a finite type"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/mem-lp-mul-of-mem-lp-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.memLp_mul_of_memLp_top","parent":16211,"reading_title":{"format":"inline-parts-v1","parts":[["text","An essentially bounded multiple of an "],["math","\\(L^2\\)"],["text"," function is in "],["math","\\(L^2\\)"]],"source":"An essentially bounded multiple of an $L^2$ function is in $L^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#mem-lp-mul-of-mem-lp-top","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $\\varphi \\colon \\alpha \\to \\mathbb{C}$ be an essentially bounded, almost everywhere strongly measurable function, and let $f \\in L^2(\\mu)$. Then the pointwise product $x \\mapsto \\varphi(x) f(x)$, formed with a representative of $f$, belongs to $\\mathcal{L}^2(\\mu)$: it is almost everywhere strongly measurable and square integrable.","title":"An essentially bounded multiple of an $L^2$ function is in $L^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/lp-mul-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.lpMulCLM","parent":16211,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplication operator on "],["math","\\(L^2\\)"]],"source":"Multiplication operator on $L^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#lp-mul-clm","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $\\varphi \\colon \\alpha \\to \\mathbb{C}$ be an essentially bounded, almost everywhere strongly measurable function. The multiplication operator $M_\\varphi$ is the continuous $\\mathbb{C}$-linear operator on $L^2(\\mu)$ that sends $f$ to the class of the pointwise product $x \\mapsto \\varphi(x) f(x)$.","title":"Multiplication operator on $L^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/lp-mul-clm-congr-ae","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.lpMulCLM_congr_ae","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#lp-mul-clm-congr-ae","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $\\varphi, \\psi \\colon \\alpha \\to \\mathbb{C}$ be essentially bounded, almost everywhere strongly measurable functions. If $\\varphi = \\psi$ $\\mu$-almost everywhere, then the multiplication operators $M_\\varphi$ and $M_\\psi$ on $L^2(\\mu)$ are equal.","title":"Multiplication operators with almost everywhere equal multipliers coincide"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/lp-mul-clm-const-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.lpMulCLM_const_apply","parent":16211,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplication by a constant function on "],["math","\\(L^2\\)"]],"source":"Multiplication by a constant function on $L^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#lp-mul-clm-const-apply","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $c \\in \\mathbb{C}$. The multiplication operator on $L^2(\\mu)$ by the constant function $x \\mapsto c$ sends every $f \\in L^2(\\mu)$ to the scalar multiple $c f$.","title":"Multiplication by a constant function on $L^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/lp-mul-clm-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.lpMulCLM_one","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#lp-mul-clm-one","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$. The multiplication operator on $L^2(\\mu)$ by the constant function $1$ is the identity operator.","title":"Multiplication by one is the identity"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/lp-mul-clm-mul","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.lpMulCLM_mul","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#lp-mul-clm-mul","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and, for an essentially bounded, almost everywhere strongly measurable function $\\chi \\colon \\alpha \\to \\mathbb{C}$, let $M_\\chi$ be the multiplication operator $f \\mapsto \\chi f$ on $L^2(\\mu)$. Let $\\varphi, \\psi \\colon \\alpha \\to \\mathbb{C}$ be essentially bounded, almost everywhere strongly measurable functions such that the pointwise product\u2026","title":"Product of multiplication operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/algebra-map-sub-lp-mul-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.algebraMap_sub_lpMulCLM","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#algebra-map-sub-lp-mul-clm","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $\\varphi : \\alpha \\to \\mathbb{C}$ be in $L^\\infty(\\mu)$, let $z \\in \\mathbb{C}$, and suppose the function $x \\mapsto z - \\varphi(x)$ is also in $L^\\infty(\\mu)$. For $\\psi \\in L^\\infty(\\mu)$ write $M_\\psi$ for the multiplication operator $f \\mapsto \\psi f$ on $L^2(\\mu)$. Then $z I - M_\\varphi = M_{z - \\varphi}$, where $I$ is the identity operator on\u2026","title":"Shifting a multiplication operator by a scalar"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/exists-measurable-mem-lp","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.exists_measurable_memLp","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#exists-measurable-mem-lp","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $\\varphi : \\alpha \\to \\mathbb{C}$ be in $L^\\infty(\\mu)$, that is, $\\mu$-almost everywhere strongly measurable and essentially bounded. Then there is a measurable $\\psi : \\alpha \\to \\mathbb{C}$ such that $\\varphi = \\psi$ $\\mu$-almost everywhere and $\\psi$ is in $L^\\infty(\\mu)$.","title":"Measurable representatives of essentially bounded functions"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/ae-norm-le-e-lp-norm-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.ae_norm_le_eLpNorm_top","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#ae-norm-le-e-lp-norm-top","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $\\varphi : \\alpha \\to \\mathbb{C}$ be in $L^\\infty(\\mu)$, that is, $\\mu$-almost everywhere strongly measurable and essentially bounded. Then $|\\varphi(x)| \\le \\|\\varphi\\|_{L^\\infty(\\mu)}$ for $\\mu$-almost every $x \\in \\alpha$.","title":"Pointwise bound by the essential supremum norm"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/norm-lp-mul-clm-apply-le-of-ae-bound","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.norm_lpMulCLM_apply_le_of_ae_bound","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#norm-lp-mul-clm-apply-le-of-ae-bound","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $\\varphi \\colon \\alpha \\to \\mathbb{C}$ be an essentially bounded, almost everywhere strongly measurable function, and let $M_\\varphi$ be the multiplication operator on $L^2(\\mu)$. If $C$ is a real number with $|\\varphi(x)| \\le C$ for $\\mu$-almost every $x$, then $\\|M_\\varphi f\\| \\le C \\|f\\|$ for every $f \\in L^2(\\mu)$.","title":"Almost everywhere bound for a multiplication operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/norm-lp-mul-clm-apply-le-of-ae-support","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.norm_lpMulCLM_apply_le_of_ae_support","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#norm-lp-mul-clm-apply-le-of-ae-support","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $\\varphi \\colon \\alpha \\to \\mathbb{C}$ be an essentially bounded, almost everywhere strongly measurable function, and let $M_\\varphi$ be the multiplication operator on $L^2(\\mu)$. If $C$ is a real number and $f \\in L^2(\\mu)$ is such that, for $\\mu$-almost every $x$, either $|\\varphi(x)| \\le C$ or $f(x) = 0$, then $\\|M_\\varphi f\\| \\le C \\|f\\|$.","title":"Bound for a multiplication operator on the support of a vector"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/le-norm-lp-mul-clm-of-ae-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.le_norm_lpMulCLM_of_ae_le","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#le-norm-lp-mul-clm-of-ae-le","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $\\varphi \\colon \\alpha \\to \\mathbb{C}$ be an essentially bounded, almost everywhere strongly measurable function, and let $M_\\varphi$ be the multiplication operator $f \\mapsto \\varphi f$ on $L^2(\\mu)$. If $C \\ge 0$ is a real number and $f \\in L^2(\\mu)$ satisfies $\\|f\\| = 1$ and $C\\,|f(x)| \\le |\\varphi(x)|\\,|f(x)|$ for $\\mu$-almost every $x$, then\u2026","title":"Lower bound for the norm of a multiplication operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/norm-lp-mul-clm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.norm_lpMulCLM_le","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#norm-lp-mul-clm-le","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $\\varphi : \\alpha \\to \\mathbb{C}$ be in $L^\\infty(\\mu)$, with essential supremum norm $\\|\\varphi\\|_\\infty$. Then the multiplication operator $M_\\varphi$ on the complex space $L^2(\\mu)$ has operator norm $\\|M_\\varphi\\| \\le \\|\\varphi\\|_\\infty$.","title":"Norm bound for multiplication operators on L\u00b2"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/exists-unit-vector","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.exists_unit_vector","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#exists-unit-vector","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $t \\subseteq \\alpha$ be a measurable set with $0 < \\mu(t) < \\infty$. Then there is $u \\in L^2(\\mu)$ (complex valued) with $\\|u\\| = 1$ such that $u(x) = \\mu(t)^{-1/2}$ for $\\mu$-almost every $x \\in t$ and $u(x) = 0$ for $\\mu$-almost every $x \\notin t$.","title":"Normalized indicator of a set of finite positive measure"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/norm-lp-mul-clm-of-measurable","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.norm_lpMulCLM_of_measurable","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#norm-lp-mul-clm-of-measurable","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a $\\sigma$-finite measure on a measurable space $\\alpha$ and let $\\varphi : \\alpha \\to \\mathbb{C}$ be a measurable function in $L^\\infty(\\mu)$, with essential supremum norm $\\|\\varphi\\|_\\infty$. Then the multiplication operator $M_\\varphi$ on the complex space $L^2(\\mu)$ has operator norm $\\|M_\\varphi\\| = \\|\\varphi\\|_\\infty$.","title":"Norm of an L\u00b2 multiplication operator with measurable multiplier"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/norm-lp-mul-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.norm_lpMulCLM","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#norm-lp-mul-clm","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a $\\sigma$-finite measure on a measurable space $\\alpha$ and let $\\varphi : \\alpha \\to \\mathbb{C}$ be in $L^\\infty(\\mu)$, with essential supremum norm $\\|\\varphi\\|_\\infty$. Then the multiplication operator $M_\\varphi$ on the complex space $L^2(\\mu)$ has operator norm $\\|M_\\varphi\\| = \\|\\varphi\\|_\\infty$.","title":"Norm of a multiplication operator on L\u00b2"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/spectrum-lp-mul-clm-of-measurable","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.spectrum_lpMulCLM_of_measurable","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#spectrum-lp-mul-clm-of-measurable","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a $\\sigma$-finite measure on a measurable space $\\alpha$ and let $\\varphi : \\alpha \\to \\mathbb{C}$ be a measurable function in $L^\\infty(\\mu)$. Then the spectrum of the multiplication operator $M_\\varphi$, as an element of the complex algebra of bounded operators on $L^2(\\mu)$, equals the essential range of $\\varphi$ with respect to $\\mu$.","title":"Spectrum of an L\u00b2 multiplication operator with measurable multiplier"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/spectrum-lp-mul-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.spectrum_lpMulCLM","parent":16211,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#spectrum-lp-mul-clm","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a $\\sigma$-finite measure on a measurable space $\\alpha$ and let $\\varphi : \\alpha \\to \\mathbb{C}$ be in $L^\\infty(\\mu)$. Then the spectrum of the multiplication operator $M_\\varphi$, as an element of the complex algebra of bounded operators on $L^2(\\mu)$, equals the essential range of $\\varphi$ with respect to $\\mu$.","title":"Spectrum of a multiplication operator on L\u00b2"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/has-eigenvalue-lp-mul-clm-iff-of-measura","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.hasEigenvalue_lpMulCLM_iff_of_measurable","parent":16211,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eigenvalues of a multiplication operator on "],["math","\\(L^2\\)"],["text",", measurable multiplier"]],"source":"Eigenvalues of a multiplication operator on $L^2$, measurable multiplier"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#has-eigenvalue-lp-mul-clm-iff-of-measura","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a $\\sigma$-finite measure on a measurable space $\\alpha$, let $\\varphi : \\alpha \\to \\mathbb{C}$ be measurable and in $L^\\infty(\\mu)$, and let $M_\\varphi$ be the multiplication operator $f \\mapsto \\varphi f$ on $L^2(\\mu)$. Then $z \\in \\mathbb{C}$ is an eigenvalue of $M_\\varphi$ if and only if $\\mu(\\{x \\in \\alpha : \\varphi(x) = z\\}) > 0$.","title":"Eigenvalues of a multiplication operator on $L^2$, measurable multiplier"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/has-eigenvalue-lp-mul-clm-iff","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpMultiplicationOperator.hasEigenvalue_lpMulCLM_iff","parent":16211,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eigenvalues of a multiplication operator on "],["math","\\(L^2\\)"]],"source":"Eigenvalues of a multiplication operator on $L^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#has-eigenvalue-lp-mul-clm-iff","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"Let $\\mu$ be a $\\sigma$-finite measure on a measurable space $\\alpha$, let $\\varphi : \\alpha \\to \\mathbb{C}$ be in $L^\\infty(\\mu)$, and let $M_\\varphi$ be the multiplication operator $f \\mapsto \\varphi f$ on $L^2(\\mu)$. Then $z \\in \\mathbb{C}$ is an eigenvalue of $M_\\varphi$ if and only if $\\mu(\\{x \\in \\alpha : \\varphi(x) = z\\}) > 0$.","title":"Eigenvalues of a multiplication operator on $L^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/l2-on-one-two","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.MulByIdOnL2OneTwoSelfAdjointWanted.L2OnOneTwo","parent":16211,"reading_title":{"format":"inline-parts-v1","parts":[["text","The space "],["math","\\(L^2((1,2))\\)"]],"source":"The space $L^2((1,2))$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#l2-on-one-two","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"fully_proved","summary":"$L^2([1,2])$ is the space of real-valued functions on $\\mathbb{R}$ that are square-integrable for Lebesgue measure restricted to $[1,2]$, taken modulo equality almost everywhere on $[1,2]$.","title":"The space $L^2((1,2))$"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/continuous-bijective-self-adjoint-of-mul","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.MulByIdOnL2OneTwoSelfAdjointWanted.continuous_bijective_selfAdjoint_of_mul_id_L2_Icc_one_two","parent":16211,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplication by "],["math","\\(x\\)"],["text"," on "],["math","\\(L^2((1,2))\\)"],["text"," is bounded, bijective and self-adjoint"]],"source":"Multiplication by $x$ on $L^2((1,2))$ is bounded, bijective and self-adjoint"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#continuous-bijective-self-adjoint-of-mul","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"can_prove","summary":"Let $L^2([1,2])$ be the real Hilbert space of square-integrable real functions for Lebesgue measure on $[1,2]$, with inner product $\\langle f, g\\rangle = \\int_1^2 f(x)g(x)\\,dx$. Let $T : L^2([1,2]) \\to L^2([1,2])$ be a linear map such that for every $f$, $(Tf)(x) = x f(x)$ for almost every $x \\in [1,2]$. Then $T$ is continuous and bijective, and $T = T^*$, where $T^*$ is the Hilbert space adjoint of $T$.","title":"Multiplication by $x$ on $L^2((1,2))$ is bounded, bijective and self-adjoint"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/exists-continuous-linear-map-mul-id-l2-i","kind":"node","lean":"MathlibExt.Analysis.OperatorTheory.MulByIdentityBoundedOnL2UnitIntervalWanted.exists_continuousLinearMap_mul_id_L2_Icc","parent":16211,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplication by "],["math","\\(t\\)"],["text"," is bounded on "],["math","\\(L^2((0,1))\\)"]],"source":"Multiplication by $t$ is bounded on $L^2((0,1))$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l/index.html#exists-continuous-linear-map-mul-id-l2-i","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-l","status":"can_prove","summary":"Let $L^2([0,1])$ be the space of real square-integrable functions for Lebesgue measure on $[0,1]$. There is a bounded linear operator $T : L^2([0,1]) \\to L^2([0,1])$ such that for every $f \\in L^2([0,1])$, $(Tf)(t) = t f(t)$ for almost every $t \\in [0,1]$.","title":"Multiplication by $t$ is bounded on $L^2((0,1))$"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous.json","id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous","kind":"scope","navigation":"module","parent":16118,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/index.html","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous","status":"fully_proved","title":"Multiplication operators on spaces of continuous functions"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/mul-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.MultiplicationOperator.mulCLM","parent":16240,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/index.html#mul-clm","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous","status":"fully_proved","summary":"Let $X$ be a compact topological space, let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $C(X, \\mathbb{K})$ be the Banach algebra of continuous functions $X \\to \\mathbb{K}$ with the supremum norm. For $\\varphi \\in C(X, \\mathbb{K})$, the multiplication operator $M_\\varphi$ is the bounded linear operator on $C(X, \\mathbb{K})$ given by $M_\\varphi f = \\varphi f$.","title":"The multiplication operator on continuous functions"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/mul-clm-algebra-map","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.MultiplicationOperator.mulCLM_algebraMap","parent":16240,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/index.html#mul-clm-algebra-map","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous","status":"fully_proved","summary":"Let $X$ be a compact topological space, let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $\\mu \\in \\mathbb{K}$. The multiplication operator on $C(X, \\mathbb{K})$ by the constant function with value $\\mu$ equals $\\mu$ times the identity operator.","title":"Multiplication by a constant on continuous functions"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/spectrum-mul-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.MultiplicationOperator.spectrum_mulCLM","parent":16240,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/index.html#spectrum-mul-clm","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous","status":"fully_proved","summary":"Let $X$ be a compact topological space, let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $\\varphi \\in C(X, \\mathbb{K})$. The spectrum of the multiplication operator $M_\\varphi f = \\varphi f$, as an element of the algebra of bounded linear operators on $C(X, \\mathbb{K})$, is the range $\\varphi(X)$ of $\\varphi$.","title":"Spectrum of a multiplication operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/is-unit-mul-clm-iff","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.MultiplicationOperator.isUnit_mulCLM_iff","parent":16240,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/index.html#is-unit-mul-clm-iff","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous","status":"fully_proved","summary":"Let $X$ be a compact topological space, let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $\\varphi \\in C(X, \\mathbb{K})$. The multiplication operator $M_\\varphi f = \\varphi f$ is invertible in the algebra of bounded linear operators on $C(X, \\mathbb{K})$ if and only if $\\varphi(x) \\neq 0$ for every $x \\in X$.","title":"Invertibility of a multiplication operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/has-eigenvalue-mul-clm-iff","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.MultiplicationOperator.hasEigenvalue_mulCLM_iff","parent":16240,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous/index.html#has-eigenvalue-mul-clm-iff","scope":"Analysis/operator-theory/integral-and-multiplication/multiplication-operators-on-spaces-of-continuous","status":"fully_proved","summary":"Let $X$ be a compact Hausdorff space, let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\varphi \\in C(X, \\mathbb{K})$ and let $\\mu \\in \\mathbb{K}$. Then $\\mu$ is an eigenvalue of the multiplication operator $M_\\varphi f = \\varphi f$ on $C(X, \\mathbb{K})$ if and only if the level set $\\varphi^{-1}(\\{\\mu\\})$ has nonempty interior.","title":"Eigenvalues of a multiplication operator"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication/schur-test.json","id":"Analysis/operator-theory/integral-and-multiplication/schur-test","kind":"scope","navigation":"module","parent":16118,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","title":"Schur test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-pointwise-sq-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTest.schur_pointwise_sq_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-pointwise-sq-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The per-row estimate in Schur's test: the `i`-th coordinate of `A x` satisfies `\u2016(A x) i\u2016\u00b2 \u2264 R * \u2211 j, \u2016A i j\u2016 * \u2016x j\u2016\u00b2` when every row sum of `\u2016A i j\u2016` is at most `R`.","title":"Per-row estimate in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-norm-sq-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTest.schur_norm_sq_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-norm-sq-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The squared-norm estimate in Schur's test: `\u2016A x\u2016\u00b2 \u2264 R * C * \u2016x\u2016\u00b2` when every row sum of `\u2016A i j\u2016` is at most `R` and every column sum is at most `C`.","title":"Squared-norm estimate in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-apply-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTest.schur_apply_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-apply-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The unsquared estimate in Schur's test: `\u2016A x\u2016 \u2264 \u221a(R * C) * \u2016x\u2016`.","title":"Norm estimate in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/norm-to-euclidean-lin-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTest.norm_toEuclideanLin_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#norm-to-euclidean-lin-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Schur's test for the canonical rectangular Euclidean matrix operator.","title":"Schur's test for rectangular Euclidean matrix operators"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/norm-to-euclidean-clm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTest.norm_toEuclideanCLM_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#norm-to-euclidean-clm-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Schur's test for Mathlib's square Euclidean continuous linear map.","title":"Schur's test for square Euclidean continuous linear maps"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/norm-matrix-l2-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTest.norm_matrix_l2_le","parent":16246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Schur's test for the "],["math","\\(L^2\\)"],["text"," operator norm of a matrix"]],"source":"Schur's test for the $L^2$ operator norm of a matrix"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#norm-matrix-l2-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Schur's test for the actual scoped L2 operator norm of a matrix.","title":"Schur's test for the $L^2$ operator norm of a matrix"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-single-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInf_single_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-single-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"A single entry of a Schur matrix is bounded by the row bound.","title":"Entries of a Schur matrix are bounded by the row bound"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-nonneg-of-row","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInf_nonneg_of_row","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-nonneg-of-row","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"A single row bound: the row bound `R` is nonnegative.","title":"Row bound of a Schur matrix is nonnegative"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-finset-cs","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInf_finset_cs","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-finset-cs","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Finite weighted Cauchy\u2013Schwarz for one row.","title":"Weighted Cauchy\u2013Schwarz for one row of a Schur matrix"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/summable-schur-inf-weighted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.summable_schurInf_weighted","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#summable-schur-inf-weighted","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Weighted summability for one row: `\u2211' j, \u2016a i j\u2016 * \u2016x j\u2016\u00b2` converges.","title":"Summability of the weighted row series in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/sum-range-schur-inf-weighted-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.sum_range_schurInf_weighted_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#sum-range-schur-inf-weighted-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"A finite partial sum of the weighted series is bounded by `R * \u2016x\u2016\u00b2`.","title":"Partial sums of the weighted row series in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/summable-schur-inf-norm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.summable_schurInf_norm","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#summable-schur-inf-norm","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Summability of the row series of absolute values.","title":"Absolute summability of rows of a Schur matrix"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/summable-schur-inf-coord","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.summable_schurInf_coord","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#summable-schur-inf-coord","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Summability of the row series defining one coordinate of the operator.","title":"Summability of the row series defining a Schur coordinate"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-pointwise-sq-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInf_pointwise_sq_le","parent":16246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Per-row estimate in Schur's test on "],["math","\\(\\ell^2\\)"]],"source":"Per-row estimate in Schur's test on $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-pointwise-sq-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The per-row estimate: `\u2016(T x) i\u2016\u00b2 \u2264 R * \u2211' j, \u2016a i j\u2016 * \u2016x j\u2016\u00b2`.","title":"Per-row estimate in Schur's test on $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/summable-schur-inf-weighted-tsum","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.summable_schurInf_weighted_tsum","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#summable-schur-inf-weighted-tsum","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Summability of the double weighted series `\u2211' i, \u2211' j, \u2016a i j\u2016 * \u2016x j\u2016\u00b2`.","title":"Summability of the double weighted series in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-tsum-weighted-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInf_tsum_weighted_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-tsum-weighted-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The double weighted series is bounded by `C * \u2016x\u2016\u00b2`, via the `tsum` interchange.","title":"Bound on the double weighted series in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/summable-schur-inf-coord-sq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.summable_schurInf_coordSq","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#summable-schur-inf-coord-sq","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Summability of the squared coordinates of a Schur image.","title":"Squared coordinates of a Schur image are summable"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/mem-p-schur-inf-coord","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.mem\u2113p_schurInfCoord","parent":16246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinates of a Schur image lie in "],["math","\\(\\ell^2\\)"]],"source":"Coordinates of a Schur image lie in $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#mem-p-schur-inf-coord","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The coordinates of a Schur image are in `\u2113\u00b2`.","title":"Coordinates of a Schur image lie in $\\ell^2$"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-lin-map","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInfLinMap","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-lin-map","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The Schur operator of a matrix, as a linear map.","title":"Schur operator of an infinite matrix as a linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-lin-map-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInfLinMap_apply","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-lin-map-apply","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Pointwise action of the Schur linear map.","title":"Pointwise action of the Schur linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/norm-schur-inf-lin-map-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.norm_schurInfLinMap_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#norm-schur-inf-lin-map-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The Schur operator is bounded by `\u221a(R * C)`.","title":"Norm bound for the Schur linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInfOperator","parent":16246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Schur operator on "],["math","\\(\\ell^2(\\mathbb{N})\\)"]],"source":"Schur operator on $\\ell^2(\\mathbb{N})$"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-operator","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The Schur operator of a matrix, as a bounded operator on `\u2113\u00b2(\u2115, \ud835\udd5c)`.","title":"Schur operator on $\\ell^2(\\mathbb{N})$"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-operator-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInfOperator_apply","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-operator-apply","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Pointwise action of the Schur operator.","title":"Pointwise action of the Schur operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/norm-schur-inf-operator-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.norm_schurInfOperator_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#norm-schur-inf-operator-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The operator norm is at most `\u221a(R * C)`.","title":"Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/norm-schur-inf-operator-le-of-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.norm_schurInfOperator_le_of_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#norm-schur-inf-operator-le-of-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The single-bound form: if all row and column sums are at most `B`, then the operator norm is at most `B`.","title":"Schur's test with a single row and column bound"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-operator-congr","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInfOperator_congr","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-operator-congr","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The Schur operator depends only on the matrix, not on the bounds or proofs.","title":"Schur operator depends only on the matrix"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/sum-finset-schur-inf-weighted-fst-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.sum_finset_schurInf_weighted_fst_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#sum-finset-schur-inf-weighted-fst-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"A finite partial sum of the weighted series, grouped by the first index.","title":"Weighted partial sums grouped by the first index"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/sum-finset-schur-inf-weighted-snd-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.sum_finset_schurInf_weighted_snd_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#sum-finset-schur-inf-weighted-snd-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"A finite partial sum of the weighted series, grouped by the second index.","title":"Weighted partial sums grouped by the second index"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/summable-schur-inf-prod","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.summable_schurInf_prod","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#summable-schur-inf-prod","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Product summability for the bilinear form of a Schur matrix.","title":"Product summability for the Schur bilinear form"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/summable-schur-inf-transpose-star-row","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.summable_schurInf_transposeStar_row","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#summable-schur-inf-transpose-star-row","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Row summability for the conjugate-transpose matrix.","title":"Row summability of the conjugate-transpose Schur matrix"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-transpose-star-row-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInf_transposeStar_row_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-transpose-star-row-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Row bounds for the conjugate-transpose matrix.","title":"Row bounds for the conjugate-transpose Schur matrix"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/summable-schur-inf-transpose-star-col","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.summable_schurInf_transposeStar_col","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#summable-schur-inf-transpose-star-col","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Column summability for the conjugate-transpose matrix.","title":"Column summability for the conjugate-transpose matrix in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/schur-inf-transpose-star-col-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.schurInf_transposeStar_col_le","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#schur-inf-transpose-star-col-le","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"Column bounds for the conjugate-transpose matrix.","title":"Column bounds for the conjugate-transpose matrix in Schur's test"},{"id":"Analysis/operator-theory/integral-and-multiplication/schur-test/adjoint-schur-inf-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurTestInfinite.adjoint_schurInfOperator","parent":16246,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/schur-test/index.html#adjoint-schur-inf-operator","scope":"Analysis/operator-theory/integral-and-multiplication/schur-test","status":"fully_proved","summary":"The adjoint of a Schur operator is the Schur operator of the conjugate-transpose matrix.","title":"Schur's test, adjoint form"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem.json","id":"Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem","kind":"scope","navigation":"module","parent":16118,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem/index.html","scope":"Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem","status":"can_prove","title":"Titchmarsh convolution theorem"},{"id":"Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem/scottish-book-problem124-claim","kind":"node","lean":"MathlibExt.Analysis.ScottishBook124Wanted.ScottishBookProblem124Claim","parent":16281,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem/index.html#scottish-book-problem124-claim","scope":"Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem","status":"fully_proved","summary":"The statement that for every continuous function $f : \\mathbb{R} \\to \\mathbb{R}$ with $f(0) \\ne 0$ and every function $y : \\mathbb{R} \\to \\mathbb{R}$ that is Lebesgue integrable on $[0, 1]$, if $\\int_0^x y(t) f(x - t)\\,dt = 0$ for every $x \\in [0, 1]$, then $y = 0$ almost everywhere on $[0, 1]$.","title":"The Volterra-form sub-conjecture of Scottish Book Problem 124"},{"id":"Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem/marcinkiewicz124-holds","kind":"node","lean":"MathlibExt.Analysis.ScottishBook124Wanted.marcinkiewicz124_holds","parent":16281,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem/index.html#marcinkiewicz124-holds","scope":"Analysis/operator-theory/integral-and-multiplication/titchmarsh-convolution-theorem","status":"can_prove","summary":"For every continuous function $f : \\mathbb{R} \\to \\mathbb{R}$ with $f(0) \\ne 0$ and every function $y : \\mathbb{R} \\to \\mathbb{R}$ that is Lebesgue integrable on $[0, 1]$, if $\\int_0^x y(t) f(x - t)\\,dt = 0$ for every $x \\in [0, 1]$, then $y = 0$ almost everywhere on $[0, 1]$.","title":"The Volterra-form sub-conjecture of Scottish Book Problem 124 holds"},{"data_url":"scopes/Analysis/operator-theory/integral-and-multiplication/volterra-operator.json","id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","kind":"scope","navigation":"module","parent":16118,"result":false,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"can_prove","title":"Volterra operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-space","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.VolterraSpace","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-space","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"The space on which the Volterra operator acts is the space of bounded continuous real functions on $[0, 1]$, with the sup norm.","title":"The domain of the Volterra operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/clamp01","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.clamp01","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#clamp01","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For a real number $t$, its clamp to the unit interval is $\\max(0, \\min(t, 1))$, regarded as a point of $[0, 1]$.","title":"Clamping to the unit interval"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-extend","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraExtend","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-extend","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For a bounded continuous real function $f$ on $[0, 1]$, its extension to $\\mathbb{R}$ is the function $t \\mapsto f(\\max(0, \\min(t, 1)))$.","title":"Extension by clamping"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-extend-continuous","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraExtend_continuous","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-extend-continuous","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every bounded continuous real function $f$ on $[0, 1]$, its extension $t \\mapsto f(\\max(0, \\min(t, 1)))$ to $\\mathbb{R}$ by clamping is continuous on $\\mathbb{R}$.","title":"The clamped extension is continuous"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-extend-apply-of-mem","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraExtend_apply_of_mem","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-extend-apply-of-mem","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every bounded continuous real function $f$ on $[0, 1]$ and every real $t$ with $0 \\le t \\le 1$, the extension $s \\mapsto f(\\max(0, \\min(s, 1)))$ of $f$ to $\\mathbb{R}$ takes the value $f(t)$ at $t$.","title":"The clamped extension agrees with the original function on the unit interval"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/norm-volterra-extend-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.norm_volterraExtend_le","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#norm-volterra-extend-le","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every bounded continuous real function $f$ on $[0, 1]$ and every real $t$, the extension of $f$ to $\\mathbb{R}$ by clamping satisfies $|f(\\max(0, \\min(t, 1)))| \\le \\|f\\|_\\infty$, where $\\|f\\|_\\infty = \\sup_{x \\in [0, 1]} |f(x)|$ is the sup norm of $f$.","title":"The clamped extension is bounded by the sup norm"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-primitive-continuous","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraPrimitive_continuous","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-primitive-continuous","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every bounded continuous real function $f$ on $[0, 1]$, the function $x \\mapsto \\int_0^x \\tilde f(t)\\,dt$ on $[0, 1]$ is continuous, where $\\tilde f(t) = f(\\max(0, \\min(t, 1)))$ is the extension of $f$ to $\\mathbb{R}$ by clamping the argument into $[0, 1]$.","title":"Continuity of the Volterra primitive"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-primitive-norm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraPrimitive_norm_le","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-primitive-norm-le","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every bounded continuous real function $f$ on $[0, 1]$ and every $x \\in [0, 1]$, $\\left|\\int_0^x \\tilde f(t)\\,dt\\right| \\le \\|f\\|$, where $\\|f\\|$ is the sup norm of $f$ and $\\tilde f(t) = f(\\max(0, \\min(t, 1)))$ is the extension of $f$ to $\\mathbb{R}$ by clamping the argument into $[0, 1]$.","title":"Sup-norm bound for the Volterra primitive"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-linear-map","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraLinearMap","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-linear-map","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"The real-linear map $V$ on the space of bounded continuous real functions on $[0, 1]$ given by $(Vf)(x) = \\int_0^x f(t)\\,dt$ for $x \\in [0, 1]$.","title":"The Volterra operator as a linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-linear-map-norm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraLinearMap_norm_le","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-linear-map-norm-le","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every bounded continuous real function $f$ on $[0, 1]$, the Volterra linear map $V$ satisfies $\\|Vf\\| \\le 1 \\cdot \\|f\\|$, where $\\|\\cdot\\|$ is the sup norm.","title":"Sup-norm bound for the Volterra linear map"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraOperator","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-operator","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"The bounded real-linear operator $V$ on the space of bounded continuous real functions on $[0, 1]$ (equivalently $C([0, 1], \\mathbb{R})$ with the sup norm) given by $(Vf)(x) = \\int_0^x f(t)\\,dt$ for $x \\in [0, 1]$.","title":"The Volterra operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-operator-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraOperator_apply","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-operator-apply","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every bounded continuous real function $f$ on $[0, 1]$ and every $x \\in [0, 1]$, the Volterra operator $V$ satisfies $(Vf)(x) = \\int_0^x \\tilde f(t)\\,dt$, where $\\tilde f(t) = f(\\max(0, \\min(t, 1)))$ is the extension of $f$ to $\\mathbb{R}$ by clamping the argument into $[0, 1]$.","title":"Evaluation formula for the Volterra operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-operator-apply-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraOperator_apply_zero","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-operator-apply-zero","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every bounded continuous real function $f$ on $[0, 1]$, the Volterra operator $V$ satisfies $(Vf)(0) = 0$.","title":"Volterra integrals vanish at the left endpoint"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/norm-volterra-operator-le-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.norm_volterraOperator_le_one","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#norm-volterra-operator-le-one","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"The Volterra operator $V$, $(Vf)(x) = \\int_0^x f(t)\\,dt$, acting on the space of bounded continuous real functions on $[0, 1]$ with the sup norm, has operator norm $\\|V\\| \\le 1$.","title":"The Volterra operator has norm at most one"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/pow-volterra-operator-apply-norm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.pow_volterraOperator_apply_norm_le","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#pow-volterra-operator-apply-norm-le","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"Let $V$ be the Volterra operator $(Vf)(x) = \\int_0^x f(t)\\,dt$ on the space of bounded continuous real functions on $[0, 1]$. For every natural number $n$, every such function $f$ and every $x \\in [0, 1]$, $|(V^n f)(x)| \\le \\frac{x^n}{n!} \\|f\\|_\\infty$, where $\\|f\\|_\\infty$ is the sup norm of $f$.","title":"Pointwise factorial bound for powers of the Volterra operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/norm-volterra-operator-pow-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.norm_volterraOperator_pow_le","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#norm-volterra-operator-pow-le","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"Let $V$ be the Volterra operator $(Vf)(x) = \\int_0^x f(t)\\,dt$ on the space of bounded continuous real functions on $[0, 1]$ with the sup norm. For every natural number $n$, the operator norm of $V^n$ satisfies $\\|V^n\\| \\le 1/n!$.","title":"Factorial bound for powers of the Volterra operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-operator-mem-resolvent-set-of-n","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraOperator_mem_resolventSet_of_ne_zero","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-operator-mem-resolvent-set-of-n","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"For every nonzero real $z$, $z$ lies in the real resolvent set of the Volterra operator $V$: the bounded operator $z - V$ is invertible.","title":"Nonzero reals lie in the resolvent set of the Volterra operator"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/not-is-unit-volterra-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.not_isUnit_volterraOperator","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#not-is-unit-volterra-operator","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"The Volterra operator $V$, $(Vf)(x) = \\int_0^x f(t)\\,dt$, on the space of bounded continuous real functions on $[0, 1]$ is not a unit in the algebra of bounded linear operators on that space.","title":"The Volterra operator is not invertible"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/volterra-operator-spectrum","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.volterraOperator_spectrum","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#volterra-operator-spectrum","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"The spectrum of the Volterra operator is `{0}`.","title":"The spectrum of the Volterra operator is `{0}`"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/spectral-radius-volterra-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.VolterraOperator.spectralRadius_volterraOperator","parent":16284,"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#spectral-radius-volterra-operator","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"fully_proved","summary":"Let $V$ be the Volterra operator $(Vf)(x) = \\int_0^x f(t)\\,dt$ on the space of bounded continuous real functions on $[0, 1]$. Its spectral radius over the real scalars, the supremum of $|z|$ over real $z$ in the spectrum of $V$, is $0$.","title":"The spectral radius of the Volterra operator is zero"},{"id":"Analysis/operator-theory/integral-and-multiplication/volterra-operator/x-mul-volterra-iterate-eq-sum-bessel","kind":"node","lean":"MetaMathlibExt.xMul_volterra_iterate_eq_sum_bessel","parent":16284,"reading_title":{"format":"inline-parts-v1","parts":[["text","Expansion of "],["math","\\((XV)^n\\)"],["text"," in Bessel numbers"]],"source":"Expansion of $(XV)^n$ in Bessel numbers"},"result":true,"route":"../roadmap/Analysis/operator-theory/integral-and-multiplication/volterra-operator/index.html#x-mul-volterra-iterate-eq-sum-bessel","scope":"Analysis/operator-theory/integral-and-multiplication/volterra-operator","status":"can_prove","summary":"Let $X$ and $V$ be the operators on functions $\\mathbb{R} \\to \\mathbb{R}$ given by $(Xg)(y) = y \\, g(y)$ and $(Vg)(y) = \\int_0^y g(t) \\, dt$, write $(XV)^n$ and $V^m$ for iterated composition, and let $a(m, k)$ be the coefficient of $x^k$ in the Bessel polynomial $y_m$. For every natural number $n \\ge 1$, every continuous function $f : \\mathbb{R} \\to \\mathbb{R}$ and every real number $x$,","title":"Expansion of $(XV)^n$ in Bessel numbers"},{"data_url":"scopes/Analysis/operator-theory/eigenvalue-inequalities.json","id":"Analysis/operator-theory/eigenvalue-inequalities","kind":"scope","parent":15497,"result":false,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/index.html","scope":"Analysis/operator-theory/eigenvalue-inequalities","status":"fully_proved","summary":"Courant\u2013Fischer, Weyl, Ky Fan, Cauchy interlacing, Schur\u2013Horn, Eckart\u2013Young, Bauer\u2013Fike, Peierls\u2013Klein for self-adjoint operators; statements only about finite matrices move to LinearAlgebra.","title":"Eigenvalue inequalities and perturbation"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/schur-diagonal-eigenvalues-doubly-stocha","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SchurHorn.schur_diagonal_eigenvalues_doublyStochastic","parent":16306,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/index.html#schur-diagonal-eigenvalues-doubly-stocha","scope":"Analysis/operator-theory/eigenvalue-inequalities","status":"fully_proved","summary":"Let $E$ be a complex inner product space of finite dimension $n$, let $T : E \\to E$ be a linear operator that is symmetric, that is, $\\langle T x, y \\rangle = \\langle x, T y \\rangle$ for all $x, y \\in E$, let $\\lambda_1, \\dots, \\lambda_n$ be the eigenvalues of $T$ listed with multiplicity, and let $b_1, \\dots, b_n$ be an orthonormal basis of $E$. Then there is a doubly stochastic real $n \\times n$ matrix $D$ such\u2026","title":"Schur's theorem on diagonals of self-adjoint operators"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/eckart-young-best-rank-approx-operator-n","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.EckartYoungWanted.eckartYoung_best_rank_approx_operator_norm","parent":16306,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/index.html#eckart-young-best-rank-approx-operator-n","scope":"Analysis/operator-theory/eigenvalue-inequalities","status":"fully_proved","summary":"Let $E$ and $F$ be finite-dimensional inner product spaces over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T : E \\to F$ be a linear map and let $k \\ge 0$ be an integer. The infimum of the operator norms $\\|T - S\\|$ over all linear maps $S : E \\to F$ whose range has dimension at most $k$ equals $\\sigma_k(T)$, where $\\sigma_0(T) \\ge \\sigma_1(T) \\ge \\cdots$ are the singular values of $T$ listed with multiplicity\u2026","title":"Eckart\u2013Young theorem in operator norm"},{"data_url":"scopes/Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem.json","id":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem","kind":"scope","navigation":"module","parent":16306,"result":false,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/index.html","scope":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem","status":"fully_proved","title":"Bauer\u2013Fike theorem"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/norm-ring-inverse-sub-smul-one-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BauerFike.norm_ringInverse_sub_smul_one_le","parent":16309,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/index.html#norm-ring-inverse-sub-smul-one-le","scope":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem","status":"fully_proved","summary":"The resolvent norm bound for a normal operator: if `z` lies outside the spectrum of a normal `A`, then `\u2016(z \u2022 1 - A)\u207b\u00b9\u2016 \u2264 1 / dist z (spectrum \u2102 A)`.","title":"Resolvent norm bound for a normal operator"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/inf-dist-le-norm-of-mem-spectrum-add","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BauerFike.infDist_le_norm_of_mem_spectrum_add","parent":16309,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/index.html#inf-dist-le-norm-of-mem-spectrum-add","scope":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem","status":"fully_proved","summary":"The Bauer\u2013Fike theorem: every point of the spectrum of `A + E` lies within `\u2016E\u2016` of the spectrum of a normal operator `A`.","title":"Bauer\u2013Fike theorem"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/exists-mem-spectrum-norm-sub-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BauerFike.exists_mem_spectrum_norm_sub_le","parent":16309,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/index.html#exists-mem-spectrum-norm-sub-le","scope":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem","status":"fully_proved","summary":"The attained form of Bauer\u2013Fike: some eigenvalue-like point of the spectrum of `A` is within `\u2016E\u2016` of `\u03bc`.","title":"Attained form of the Bauer\u2013Fike theorem"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/hausdorff-dist-spectrum-le-of-is-self-ad","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BauerFike.hausdorffDist_spectrum_le_of_isSelfAdjoint","parent":16309,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem/index.html#hausdorff-dist-spectrum-le-of-is-self-ad","scope":"Analysis/operator-theory/eigenvalue-inequalities/bauer-fike-theorem","status":"fully_proved","summary":"For self-adjoint `A` and `A + E`, the Hausdorff distance of the two spectra is at most `\u2016E\u2016`.","title":"Hausdorff distance between spectra of self-adjoint perturbations"},{"data_url":"scopes/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem.json","id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","kind":"scope","navigation":"module","parent":16306,"result":false,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","title":"Courant\u2013Fischer min\u2013max theorem"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/rayleigh-form","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.rayleighForm","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#rayleigh-form","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The Rayleigh form of a vector for an operator `T`: `re \u27eaT x, x\u27eb`. On unit vectors this is the Rayleigh quotient.","title":"Rayleigh form of an operator"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/rayleigh-form-smul","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.rayleighForm_smul","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#rayleigh-form-smul","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"Scaling a vector by a real scalar scales the Rayleigh form by the square of the scalar.","title":"Rayleigh form scales quadratically"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/inner-self-apply-eq-expansion","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.inner_self_apply_eq_expansion","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#inner-self-apply-eq-expansion","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"Expansion of `\u27eax, T x\u27eb` in the orthonormal eigenbasis: a weighted sum of the eigenvalues with weights `\u2016\u27eae\u1d62, x\u27eb\u2016\u00b2`.","title":"Eigenbasis expansion of the quadratic form of a self-adjoint operator"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/rayleigh-form-eq-expansion","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.rayleighForm_eq_expansion","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#rayleigh-form-eq-expansion","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The Rayleigh form expanded in the orthonormal eigenbasis.","title":"Eigenbasis expansion of the Rayleigh form"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/inner-eigenvector-basis-eq-zero-of-mem-s","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.inner_eigenvectorBasis_eq_zero_of_mem_span","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#inner-eigenvector-basis-eq-zero-of-mem-s","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"A vector in the span of some of the eigenvectors is orthogonal to the remaining eigenvectors.","title":"Span of some eigenvectors is orthogonal to the remaining ones"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/tail-family","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.tailFamily","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#tail-family","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The tail of the eigenbasis: the eigenvectors with index `\u2265 k`, as a family indexed by `Fin (n - k)`.","title":"Tail of an orthonormal eigenbasis"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/head-family","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.headFamily","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#head-family","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The head of the eigenbasis: the eigenvectors with index `\u2264 k`, as a family indexed by `Fin (k + 1)`.","title":"Head of an orthonormal eigenbasis"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/tail-family-linear-independent","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.tailFamily_linearIndependent","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#tail-family-linear-independent","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The tail family is linearly independent, as a restriction of the orthonormal eigenbasis.","title":"Linear independence of the eigenbasis tail"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/head-family-linear-independent","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.headFamily_linearIndependent","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#head-family-linear-independent","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The head family is linearly independent, as a restriction of the orthonormal eigenbasis.","title":"Linear independence of the eigenbasis head"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/finrank-span-tail-family","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.finrank_span_tailFamily","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#finrank-span-tail-family","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The span of the tail of the eigenbasis has dimension `n - k`.","title":"Dimension of the span of the eigenbasis tail"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/finrank-span-head-family","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.finrank_span_headFamily","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#finrank-span-head-family","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The span of the head of the eigenbasis has dimension `k + 1`.","title":"Dimension of the span of the eigenbasis head"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/rayleigh-form-le-of-mem-tail","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.rayleighForm_le_of_mem_tail","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#rayleigh-form-le-of-mem-tail","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"On the span of the tail of the eigenbasis, the Rayleigh form is at most `\u03bb\u2096 * \u2016x\u2016\u00b2`.","title":"Upper bound for the Rayleigh form on the eigenbasis tail"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/rayleigh-form-ge-of-mem-head","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.rayleighForm_ge_of_mem_head","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#rayleigh-form-ge-of-mem-head","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"On the span of the head of the eigenbasis, the Rayleigh form is at least `\u03bb\u2096 * \u2016x\u2016\u00b2`.","title":"Lower bound for the Rayleigh form on the eigenbasis head"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/exists-eigenvector-mem-tail","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.exists_eigenvector_mem_tail","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#exists-eigenvector-mem-tail","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The `k`-th eigenvector lies in the tail span, is a unit vector, and attains the Rayleigh value `\u03bb\u2096`.","title":"The k-th eigenvector attains the Rayleigh bound on the tail"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/exists-eigenvector-mem-head","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.exists_eigenvector_mem_head","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#exists-eigenvector-mem-head","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"The `k`-th eigenvector lies in the head span, is a unit vector, and attains the Rayleigh value `\u03bb\u2096`.","title":"The k-th eigenvector attains the Rayleigh bound on the head"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/exists-tail-submodule","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.exists_tail_submodule","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#exists-tail-submodule","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"There is a subspace `W` of dimension `n - k` on which every unit vector has Rayleigh quotient at most `\u03bb\u2096`, and the bound is attained.","title":"Courant\u2013Fischer min\u2013max, extremal subspace form"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/exists-unit-ge-of-submodule","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.exists_unit_ge_of_submodule","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#exists-unit-ge-of-submodule","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"Every subspace of dimension `n - k` contains a unit vector with Rayleigh quotient at least `\u03bb\u2096`.","title":"Courant\u2013Fischer min\u2013max, lower-bound form"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/exists-head-submodule","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.exists_head_submodule","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#exists-head-submodule","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"There is a subspace `U` of dimension `k + 1` on which every unit vector has Rayleigh quotient at least `\u03bb\u2096`, and the bound is attained.","title":"Courant\u2013Fischer max\u2013min, extremal subspace form"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/exists-unit-le-of-submodule","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CourantFischer.exists_unit_le_of_submodule","parent":16314,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem/index.html#exists-unit-le-of-submodule","scope":"Analysis/operator-theory/eigenvalue-inequalities/courant-fischer-min-max-theorem","status":"fully_proved","summary":"Every subspace of dimension `k + 1` contains a unit vector with Rayleigh quotient at most `\u03bb\u2096`.","title":"Courant\u2013Fischer max\u2013min, lower-bound form"},{"data_url":"scopes/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem.json","id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","kind":"scope","navigation":"module","parent":16306,"result":false,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","title":"Cauchy interlacing theorem"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/compression","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CauchyInterlacing.compression","parent":16334,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html#compression","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","summary":"The compression of an operator `T` to a subspace `S`: project `T x` orthogonally back onto `S`.","title":"Compression of an operator to a subspace"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/compression-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CauchyInterlacing.compression_apply","parent":16334,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html#compression-apply","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","summary":"The compression acts by projecting the image back onto the subspace.","title":"Action of the compression of an operator"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/compression-is-symmetric","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CauchyInterlacing.compression_isSymmetric","parent":16334,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html#compression-is-symmetric","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","summary":"The compression of a symmetric operator is symmetric.","title":"Compression of a symmetric operator is symmetric"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/compression-rayleigh","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CauchyInterlacing.compression_rayleigh","parent":16334,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html#compression-rayleigh","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","summary":"The Rayleigh form of the compression at `y` equals the Rayleigh form of the original operator at the underlying vector `(y : E)`.","title":"Rayleigh form of a compression"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/exists-submodule-le-finrank-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CauchyInterlacing.exists_submodule_le_finrank_eq","parent":16334,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html#exists-submodule-le-finrank-eq","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","summary":"A subspace of dimension `m` can be thinned to a subspace of any smaller dimension.","title":"Subspaces of every smaller dimension inside a given subspace"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/eigenvalue-compression-le-eigenvalue","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CauchyInterlacing.eigenvalue_compression_le_eigenvalue","parent":16334,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html#eigenvalue-compression-le-eigenvalue","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","summary":"The `k`-th eigenvalue of the compression to a codimension-one subspace is at most the `k`-th eigenvalue of the original operator.","title":"Cauchy interlacing, upper half"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/eigenvalue-le-eigenvalue-compression","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CauchyInterlacing.eigenvalue_le_eigenvalue_compression","parent":16334,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html#eigenvalue-le-eigenvalue-compression","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","summary":"The `k`-th eigenvalue of the compression to a codimension-one subspace is at least the `(k+1)`-st eigenvalue of the original operator.","title":"Cauchy interlacing, lower half"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/cauchy-interlacing","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CauchyInterlacing.cauchy_interlacing","parent":16334,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem/index.html#cauchy-interlacing","scope":"Analysis/operator-theory/eigenvalue-inequalities/cauchy-interlacing-theorem","status":"fully_proved","summary":"For a codimension-one subspace `S`, the eigenvalues `\u03bc\u2096` of the compression of a self-adjoint operator `T` interlace the eigenvalues `\u03bb\u2096` of `T`: `\u03bb\u2096\u208a\u2081 \u2264 \u03bc\u2096 \u2264 \u03bb\u2096`.","title":"Cauchy interlacing"},{"data_url":"scopes/Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality.json","id":"Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality","kind":"scope","navigation":"module","parent":16306,"result":false,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality/index.html","scope":"Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality","status":"fully_proved","title":"Ky Fan eigenvalue inequality"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality/eig-pref-sum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KyFanWanted.eigPrefSum","parent":16343,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality/index.html#eig-pref-sum","scope":"Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality","status":"fully_proved","summary":"For real numbers $\\lambda_0, \\dots, \\lambda_{n-1}$ and a natural number $k \\le n$, the prefix sum of length $k$ is $\\sum_{i < k} \\lambda_i$, the sum of the first $k$ entries. Applied to the eigenvalues of a Hermitian matrix, listed in decreasing order, it is the sum of the $k$ largest ones.","title":"Prefix sum of a finite real sequence"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality/ky-fan-eigenvalue-sum-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KyFanWanted.ky_fan_eigenvalue_sum_le","parent":16343,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality/index.html#ky-fan-eigenvalue-sum-le","scope":"Analysis/operator-theory/eigenvalue-inequalities/ky-fan-eigenvalue-inequality","status":"fully_proved","summary":"Let $A$ and $B$ be Hermitian $n \\times n$ complex matrices and let $k \\le n$. Then the sum of the $k$ largest eigenvalues of $A + B$ is at most the sum of the $k$ largest eigenvalues of $A$ plus the sum of the $k$ largest eigenvalues of $B$, where eigenvalues are counted with multiplicity.","title":"Ky Fan inequality for sums of largest eigenvalues"},{"data_url":"scopes/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities.json","id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","kind":"scope","navigation":"module","parent":16306,"result":false,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","title":"Peierls and Klein inequalities"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/add-deriv-mul-sub-le-of-convex-on","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.add_deriv_mul_sub_le_of_convexOn","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#add-deriv-mul-sub-le-of-convex-on","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"The first-order (tangent-line) characterization of a convex differentiable function: the graph lies above every tangent line.","title":"Convex differentiable functions lie above their tangent lines"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/peierls-inequality","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.peierls_inequality","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#peierls-inequality","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"The diagonal of a self-adjoint operator in any orthonormal basis is majorized by its eigenvalues, as measured by any convex function.","title":"Peierls' inequality"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/klein-inequality","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.klein_inequality","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#klein-inequality","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"Klein's inequality (sum form): for a convex function `f` differentiable at the eigenvalues of `S`, the tangent-line sum at the eigenvalues of `S` evaluated on the diagonal of `T` is at most the sum of `f` over the eigenvalues of `T`.","title":"Klein's inequality"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/peierls-exp","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.peierls_exp","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#peierls-exp","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"Peierls' inequality for the exponential function.","title":"Peierls' inequality for the exponential"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/klein-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.klein_sq","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#klein-sq","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"Klein's inequality for `f x = x ^ 2`.","title":"Klein's inequality for the square function"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/eigenvalues-nonneg-of-forall-nonneg-inne","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.eigenvalues_nonneg_of_forall_nonneg_inner","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#eigenvalues-nonneg-of-forall-nonneg-inne","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"Eigenvalues of a positive semidefinite symmetric operator are nonnegative: an eigenvalue is the Rayleigh quotient at a unit eigenvector.","title":"Eigenvalues of a positive semidefinite operator are nonnegative"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/eigenvalues-pos-of-forall-pos-inner","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.eigenvalues_pos_of_forall_pos_inner","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#eigenvalues-pos-of-forall-pos-inner","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"Eigenvalues of a positive definite symmetric operator are positive.","title":"Eigenvalues of a positive definite operator are positive"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/klein-entropy-tangent","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.klein_entropy_tangent","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#klein-entropy-tangent","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"The entropy tangent bound in an arbitrary orthonormal basis. Its diagonal entries need not align with the eigenvalues of `S`; `klein_entropy` supplies the aligned quantum-relative-entropy form.","title":"Entropy tangent bound in an arbitrary orthonormal basis"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/klein-entropy","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.PeierlsKlein.klein_entropy","parent":16346,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities/index.html#klein-entropy","scope":"Analysis/operator-theory/eigenvalue-inequalities/peierls-and-klein-inequalities","status":"fully_proved","summary":"Klein's entropy inequality in the eigenbasis of `S`. The eigenvalue at index `i` is paired with its eigenvector, so this is the quantum-relative-entropy form up to the trace terms. In particular, its tangent correction vanishes when `T = S`.","title":"Klein's entropy inequality"},{"data_url":"scopes/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities.json","id":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","kind":"scope","navigation":"module","parent":16306,"result":false,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/index.html","scope":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","status":"fully_proved","title":"Weyl's eigenvalue inequalities"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/weyl-eigenvalue-add-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeylEigenvalueSum.weyl_eigenvalue_add_le","parent":16356,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/index.html#weyl-eigenvalue-add-le","scope":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","status":"fully_proved","summary":"For self-adjoint operators `T, S`, `\u03bb_{a+b}(T + S) \u2264 \u03bb_a(T) + \u03bb_b(S)` whenever `a + b < n`.","title":"Weyl's inequality (upper bound)"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/le-weyl-eigenvalue-add","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeylEigenvalueSum.le_weyl_eigenvalue_add","parent":16356,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/index.html#le-weyl-eigenvalue-add","scope":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","status":"fully_proved","summary":"For self-adjoint operators `T, S`, `\u03bb_a(T) + \u03bb_b(S) \u2264 \u03bb_{a+b-(n-1)}(T + S)` whenever `n - 1 \u2264 a + b`.","title":"Weyl's inequality (lower bound)"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/eigenvalues-add-le","kind":"node","lean":"Matrix.IsHermitian.eigenvalues\u2080_add_le","parent":16356,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/index.html#eigenvalues-add-le","scope":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","status":"fully_proved","summary":"For Hermitian matrices `A, B`, `\u03bb_{a+b}(A + B) \u2264 \u03bb_a(A) + \u03bb_b(B)` whenever `a + b < n`.","title":"Weyl's inequality (upper bound), matrix form"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/le-eigenvalues-add","kind":"node","lean":"Matrix.IsHermitian.le_eigenvalues\u2080_add","parent":16356,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/index.html#le-eigenvalues-add","scope":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","status":"fully_proved","summary":"For Hermitian matrices `A, B`, `\u03bb_a(A) + \u03bb_b(B) \u2264 \u03bb_{a+b-(n-1)}(A + B)` whenever `n - 1 \u2264 a + b`.","title":"Weyl's inequality (lower bound), matrix form"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/eigenvalues-congr","kind":"node","lean":"Matrix.IsHermitian.eigenvalues\u2080_congr","parent":16356,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/index.html#eigenvalues-congr","scope":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","status":"fully_proved","summary":"Eigenvalues are unchanged when the matrix is replaced by an equal matrix (useful to transport eigenvalue computations along identities such as `diagonal d\u2081 + diagonal d\u2082 = diagonal (d\u2081 + d\u2082)`).","title":"Eigenvalues of equal Hermitian matrices agree"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/eigenvalues-diagonal-of-antitone","kind":"node","lean":"Matrix.IsHermitian.eigenvalues\u2080_diagonal_of_antitone","parent":16356,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/index.html#eigenvalues-diagonal-of-antitone","scope":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","status":"fully_proved","summary":"The eigenvalues of a diagonal Hermitian matrix whose diagonal entries are already in decreasing order are exactly those entries.","title":"Eigenvalues of a diagonal matrix with decreasing entries"},{"id":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/weyl-eigenvalue-perturbation-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.WeylPerturbationWanted.weyl_eigenvalue_perturbation_le","parent":16356,"result":true,"route":"../roadmap/Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities/index.html#weyl-eigenvalue-perturbation-le","scope":"Analysis/operator-theory/eigenvalue-inequalities/weyl-s-eigenvalue-inequalities","status":"fully_proved","summary":"Let $A$ and $B$ be Hermitian $n \\times n$ complex matrices, and let $\\lambda_1(A) \\ge \\dots \\ge \\lambda_n(A)$ and $\\lambda_1(B) \\ge \\dots \\ge \\lambda_n(B)$ be their eigenvalues, listed with multiplicity in decreasing order. Then $|\\lambda_i(A) - \\lambda_i(B)| \\le \\|A - B\\|$ for every $i$, where $\\|A - B\\|$ is the operator norm of $A - B$ acting on $\\mathbb{C}^n$ with the Euclidean norm.","title":"Weyl's perturbation inequality"},{"data_url":"scopes/Analysis/complex-analysis.json","id":"Analysis/complex-analysis","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/complex-analysis/index.html","scope":"Analysis/complex-analysis","status":"can_prove","summary":"Holomorphic and meromorphic functions of one and several complex variables: Cauchy theory, entire functions, conformal and univalent maps, zeros of polynomials, and Hilbert and Banach spaces of analytic functions.","title":"Complex analysis"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory.json","id":"Analysis/complex-analysis/cauchy-theory","kind":"scope","parent":16364,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html","scope":"Analysis/complex-analysis/cauchy-theory","status":"can_prove","summary":"Contour integrals, Cauchy's theorem, residues, argument principle, Rouch\u00e9, Laurent series, isolated singularities (Casorati\u2013Weierstrass, great Picard), Schwarz reflection, Looman\u2013Menchoff, normal families and local properties of holomorphic functions on discs and annuli.","title":"Complex integration, residues and singularities"},{"id":"Analysis/complex-analysis/cauchy-theory/finite-zeros-in-ball-of-not-identically","kind":"node","lean":"MathlibExt.Analysis.Complex.FiniteZerosInDiscOfHolomorphicWanted.finite_zeros_in_ball_of_not_identically_zero","parent":16365,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html#finite-zeros-in-ball-of-not-identically","scope":"Analysis/complex-analysis/cauchy-theory","status":"can_prove","summary":"Let $P \\in \\mathbb{C}$ and $r > 0$, write $D(P,r)$ for the open disc and $\\overline{D}(P,r)$ for the closed disc of center $P$ and radius $r$, and let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on some open set containing $\\overline{D}(P,r)$. If $f$ does not vanish identically on $D(P,r)$, then $f$ has only finitely many zeros in $D(P,r)$.","title":"A holomorphic function not identically zero has finitely many zeros in a disc"},{"id":"Analysis/complex-analysis/cauchy-theory/sq-s-sup-norm-circle-one-le-mul","kind":"node","lean":"MathlibExt.Analysis.Complex.SupOnTangentCirclesLogConvexWanted.sq_sSup_norm_circle_one_le_mul","parent":16365,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html#sq-s-sup-norm-circle-one-le-mul","scope":"Analysis/complex-analysis/cauchy-theory","status":"can_prove","summary":"Let $f:\\mathbb{C}\\to\\mathbb{C}$ be an entire function and, for $r>0$, let A(r)=\\sup\\{\\lvert f(z)\\rvert:\\lvert z-r\\rvert=r\\}. \\] Then A(1)^{2}\\le A(2)\\,A\\!\\left(\\frac{2}{3}\\right). \\]","title":"A three-circles inequality for circles tangent at the origin"},{"id":"Analysis/complex-analysis/cauchy-theory/casorati-weierstrass","kind":"node","lean":"MathlibExt.Analysis.Complex.CasoratiWeierstrass.casoratiWeierstrass","parent":16365,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html#casorati-weierstrass","scope":"Analysis/complex-analysis/cauchy-theory","status":"fully_proved","summary":"Let $c \\in \\mathbb{C}$, $r > 0$, and let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on the punctured disc $0 < |z - c| < r$ and not meromorphic at $c$ (that is, $c$ is neither a removable singularity nor a pole of $f$). Then the image of the punctured disc $0 < |z - c| < r$ under $f$ is dense in $\\mathbb{C}$.","title":"Casorati\u2013Weierstrass theorem"},{"id":"Analysis/complex-analysis/cauchy-theory/isolated-singularity-trichotomy","kind":"node","lean":"MathlibExt.Analysis.Complex.IsolatedSingularityWanted.isolatedSingularity_trichotomy","parent":16365,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html#isolated-singularity-trichotomy","scope":"Analysis/complex-analysis/cauchy-theory","status":"can_prove","summary":"Let $c \\in \\mathbb{C}$, let $r > 0$, and let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on the punctured disc $\\{z \\in \\mathbb{C} : 0 < |z - c| < r\\}$. Then at least one of the following holds: there is a function $g$ holomorphic on the disc $\\{z \\in \\mathbb{C} : |z - c| < r\\}$ that agrees with $f$ on the punctured disc (removable singularity); or $|f(z)| \\to \\infty$ as $z \\to c$ with $z \\ne c$ (pole); or for\u2026","title":"Classification of isolated singularities"},{"id":"Analysis/complex-analysis/cauchy-theory/laurent-expansion","kind":"node","lean":"MathlibExt.Analysis.Complex.LaurentSeriesWanted.laurentExpansion","parent":16365,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html#laurent-expansion","scope":"Analysis/complex-analysis/cauchy-theory","status":"can_prove","summary":"Let $c \\in \\mathbb{C}$, let $r$ and $R$ be real numbers, and let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on the annulus $A = \\{z \\in \\mathbb{C} : r < |z - c| < R\\}$. Then there are coefficients $a_n \\in \\mathbb{C}$ for $n \\in \\mathbb{Z}$ such that for every $z \\in A$ the series $\\sum_{n \\in \\mathbb{Z}} a_n (z - c)^n$ is unconditionally summable with sum $f(z)$. Here $0^n = 0$ for $n < 0$, which matters only\u2026","title":"Laurent expansion on an annulus"},{"id":"Analysis/complex-analysis/cauchy-theory/looman-menchoff","kind":"node","lean":"MetaMathlibExt.looman_menchoff","parent":16365,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html#looman-menchoff","scope":"Analysis/complex-analysis/cauchy-theory","status":"fully_proved","summary":"Let $s \\subseteq \\mathbb{C}$ be open and let $f : \\mathbb{C} \\to \\mathbb{C}$ be continuous on $s$. Write $u = \\operatorname{Re} f$ and $v = \\operatorname{Im} f$ as functions of $z = x + iy$, and suppose that at every $z \\in s$ the partial derivatives $u_x(z)$, $u_y(z)$, $v_x(z)$, $v_y(z)$ exist and satisfy the Cauchy\u2013Riemann equations $u_x(z) = v_y(z)$ and $u_y(z) = -v_x(z)$. Then $f$ is complex differentiable on\u2026","title":"Looman\u2013Menchoff theorem"},{"id":"Analysis/complex-analysis/cauchy-theory/residue-theorem-closed-ball","kind":"node","lean":"MathlibExt.Analysis.Complex.ResidueTheoremWanted.residueTheorem_closedBall","parent":16365,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html#residue-theorem-closed-ball","scope":"Analysis/complex-analysis/cauchy-theory","status":"can_prove","summary":"Let $c\\in\\mathbb{C}$ and $R>0$, and let $S$ be a finite set of points of the open disc $\\{z:|z-c|\\lt R\\}$. For each $s\\in S$ let $r_s>0$ be such that the closed disc $\\{z:|z-s|\\le r_s\\}$ lies in $\\{z:|z-c|\\lt R\\}$ and contains no point of $S$ other than $s$. Let $f:\\mathbb{C}\\to\\mathbb{C}$ be complex differentiable on $\\{z:|z-c|\\le R\\}\\setminus S$ and integrable on the circle $|z-c|=R$ and on each circle\u2026","title":"Residue theorem on a disc"},{"id":"Analysis/complex-analysis/cauchy-theory/schwarz-reflection","kind":"node","lean":"MetaMathlibExt.schwarz_reflection","parent":16365,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/index.html#schwarz-reflection","scope":"Analysis/complex-analysis/cauchy-theory","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on the open upper half-plane $\\{z : \\operatorname{Im} z > 0\\}$, continuous on the closed upper half-plane $\\{z : \\operatorname{Im} z \\ge 0\\}$, and real-valued on the real axis. Then there is an entire function $g$ that agrees with $f$ on the closed upper half-plane and satisfies $g(\\bar z) = \\overline{g(z)}$ for every $z \\in \\mathbb{C}$.","title":"Schwarz reflection principle"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory/argument-principle.json","id":"Analysis/complex-analysis/cauchy-theory/argument-principle","kind":"scope","navigation":"module","parent":16365,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/argument-principle/index.html","scope":"Analysis/complex-analysis/cauchy-theory/argument-principle","status":"fully_proved","title":"Argument principle"},{"id":"Analysis/complex-analysis/cauchy-theory/argument-principle/argument-principle-closed-ball-general","kind":"node","lean":"MathlibExt.Analysis.Complex.ArgumentPrinciple.argumentPrinciple_closedBall_general","parent":16374,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/argument-principle/index.html#argument-principle-closed-ball-general","scope":"Analysis/complex-analysis/cauchy-theory/argument-principle","status":"fully_proved","summary":"Let $c \\in \\mathbb{C}$, $R > 0$, and let $f : \\mathbb{C} \\to \\mathbb{C}$ be meromorphic on the closed disc $|z - c| \\le R$ with finite order at every point of that disc (so $f$ vanishes identically near no point). Suppose the order of $f$ is zero at every point of the circle $|z - c| = R$. Then the integral of $f'/f$ over the circle $|z - c| = R$, traversed once counterclockwise, equals $2\\pi i$ times the sum of\u2026","title":"Argument principle on a closed disc"},{"id":"Analysis/complex-analysis/cauchy-theory/argument-principle/argument-principle-closed-ball","kind":"node","lean":"MathlibExt.Analysis.Complex.ArgumentPrinciple.argumentPrinciple_closedBall","parent":16374,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/argument-principle/index.html#argument-principle-closed-ball","scope":"Analysis/complex-analysis/cauchy-theory/argument-principle","status":"fully_proved","summary":"Let $c \\in \\mathbb{C}$, $R > 0$, and let $f : \\mathbb{C} \\to \\mathbb{C}$ be meromorphic on the closed disc $|z - c| \\le R$ with finite order at every point of that disc. Suppose the order of $f$ is zero at every point of the circle $|z - c| = R$ and $f'/f$ is integrable on that circle. Then the integral of $f'/f$ over the circle, traversed once counterclockwise, equals $2\\pi i$ times the sum of the orders of $f$ at\u2026","title":"Argument principle on a closed disc, with an integrability hypothesis"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations.json","id":"Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations","kind":"scope","navigation":"module","parent":16365,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations/index.html","scope":"Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations","status":"can_prove","title":"Cauchy\u2013Riemann equations"},{"id":"Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations/not-differentiable-at-conj","kind":"node","lean":"MathlibExt.Analysis.Complex.ConjNowhereDifferentiableWanted.not_differentiableAt_conj","parent":16377,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations/index.html#not-differentiable-at-conj","scope":"Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations","status":"can_prove","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be the function $f(z) = \\bar z$. Then $f$ is not complex-differentiable at any point $z \\in \\mathbb{C}$.","title":"Complex conjugation is nowhere complex-differentiable"},{"id":"Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations/exists-eq-on-const-of-differentiable-on","kind":"node","lean":"MathlibExt.Analysis.Complex.HolomorphicConjConstWanted.exists_eqOn_const_of_differentiableOn_conj","parent":16377,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations/index.html#exists-eq-on-const-of-differentiable-on","scope":"Analysis/complex-analysis/cauchy-theory/cauchy-riemann-equations","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{C}$ be a connected open set and let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on $U$. If the conjugate function $z \\mapsto \\overline{f(z)}$ is also holomorphic on $U$, then $f$ is constant on $U$: there is $c \\in \\mathbb{C}$ with $f(z) = c$ for all $z \\in U$.","title":"A holomorphic function with holomorphic conjugate is constant"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives.json","id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives","kind":"scope","navigation":"module","parent":16365,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives/index.html","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives","status":"can_prove","title":"Contour integrals and antiderivatives"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives/contour-integral-of-has-deriv-at","kind":"node","lean":"MathlibExt.Analysis.Complex.ContourIntegralWanted.contourIntegral_of_hasDerivAt","parent":16380,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives/index.html#contour-integral-of-has-deriv-at","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{C}$ be open and let $F, f : \\mathbb{C} \\to \\mathbb{C}$ be such that $F$ is complex-differentiable at every $z \\in U$ with $F'(z) = f(z)$. Let $\\gamma : [0, 1] \\to \\mathbb{C}$ be a path from $a$ to $b$ whose image lies in $U$ and which is continuously differentiable on $[0, 1]$, with derivative $\\gamma'$ (one-sided at the endpoints). If $t \\mapsto f(\\gamma(t))\\,\\gamma'(t)$ is integrable on\u2026","title":"Fundamental theorem of contour integration"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives/norm-contour-integral-le","kind":"node","lean":"MathlibExt.Analysis.Complex.ContourIntegralWanted.norm_contourIntegral_le","parent":16380,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives/index.html#norm-contour-integral-le","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives","status":"can_prove","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ and let $\\gamma : [0, 1] \\to \\mathbb{C}$ be a path from $a$ to $b$ which is continuously differentiable on $[0, 1]$, with derivative $\\gamma'$ (one-sided at the endpoints). Let $M$ and $L$ be real numbers with $|f(\\gamma(t))| \\le M$ and $|\\gamma'(t)| \\le L$ for every $t \\in [0, 1]$. If $t \\mapsto f(\\gamma(t))\\,\\gamma'(t)$ is integrable on $[0, 1]$, then\u2026","title":"ML estimate for contour integrals"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives/not-exists-has-deriv-at-inv-on-annulus","kind":"node","lean":"MathlibExt.Analysis.Complex.InvNoAntiderivativeAnnulusWanted.not_exists_hasDerivAt_inv_on_annulus","parent":16380,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives/index.html#not-exists-has-deriv-at-inv-on-annulus","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-and-antiderivatives","status":"can_prove","summary":"Let $U = \\{z \\in \\mathbb{C} : 1 < |z| < 2\\}$. There is no function $F : \\mathbb{C} \\to \\mathbb{C}$ that is complex differentiable at every $z \\in U$ with $F'(z) = 1/z$.","title":"The reciprocal function has no antiderivative on an annulus"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles.json","id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","kind":"scope","navigation":"module","parent":16365,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","title":"Contour integrals over rectangles"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/integral-div-add-sq","kind":"node","lean":"Complex.integral_div_add_sq","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#integral-div-add-sq","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For real numbers $a$, $b$ and $d$ with $d \\ne 0$, $\\int_a^b \\frac{d}{x^2 + d^2}\\,dx = \\arctan\\frac{b}{d} - \\arctan\\frac{a}{d}$.","title":"Arctangent integral"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/odd-integral-eq-zero","kind":"node","lean":"Complex.odd_integral_eq_zero","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#odd-integral-eq-zero","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"If $f : \\mathbb{R} \\to \\mathbb{R}$ satisfies $f(-x) = -f(x)$ for every real $x$, then $\\int_{-T}^{T} f(x)\\,dx = 0$ for every real $T$.","title":"Integral of an odd function over a symmetric interval"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/vertical-side-one-div","kind":"node","lean":"Complex.vertical_side_one_div","parent":16384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(1/(c + iy)\\)"],["text"," over a symmetric interval"]],"source":"Integral of $1/(c + iy)$ over a symmetric interval"},"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#vertical-side-one-div","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For real numbers $c \\ne 0$ and $T \\ge 0$, $\\int_{-T}^{T} \\frac{1}{c + iy}\\,dy = 2\\arctan(T/c)$, where the integrand is complex valued and the integral is taken over the real variable $y$.","title":"Integral of $1/(c + iy)$ over a symmetric interval"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/horiz-diff-one-div","kind":"node","lean":"Complex.horiz_diff_one_div","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#horiz-diff-one-div","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $a \\le b$ and $T \\ne 0$ be real numbers. Then $\\int_a^b \\frac{dx}{x - iT} - \\int_a^b \\frac{dx}{x + iT} = 2i\\left(\\arctan\\frac{b}{T} - \\arctan\\frac{a}{T}\\right)$.","title":"Difference of horizontal integrals of the reciprocal"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-rect-one-div","kind":"node","lean":"Complex.boundary_rect_one_div","parent":16384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangle integral of "],["math","\\(1/s\\)"],["text"," around the origin"]],"source":"Rectangle integral of $1/s$ around the origin"},"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-rect-one-div","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $a, b, T$ be real numbers with $a < 0 < b$ and $T > 0$. Then","title":"Rectangle integral of $1/s$ around the origin"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-rect-one-div-outside","kind":"node","lean":"Complex.boundary_rect_one_div_outside","parent":16384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangle integral of "],["math","\\(1/s\\)"],["text"," with the pole outside"]],"source":"Rectangle integral of $1/s$ with the pole outside"},"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-rect-one-div-outside","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $a, b, T$ be real numbers with $0 < a \\le b$ and $T > 0$. Then","title":"Rectangle integral of $1/s$ with the pole outside"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/rectilinear-rectangle","kind":"node","lean":"RectilinearRectangle","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#rectilinear-rectangle","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"A rectilinear rectangle is an axis-parallel rectangle in the complex plane, recorded by an ordered pair of complex numbers $z$ and $w$, its first and second corner, which are opposite corners of the rectangle.","title":"Rectilinear rectangle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/of-bounds","kind":"node","lean":"RectilinearRectangle.ofBounds","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#of-bounds","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For real numbers $x_0, x_1, y_0, y_1$, this is the rectilinear rectangle whose first corner is $x_0 + i y_0$ and whose second corner is $x_1 + i y_1$.","title":"Rectangle with given coordinate bounds"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/carrier","kind":"node","lean":"RectilinearRectangle.carrier","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#carrier","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For an axis-parallel rectangle with opposite corners $z$ and $w$, its carrier is the closed rectangle of all $u \\in \\mathbb{C}$ such that $\\operatorname{Re} u$ lies in the closed interval with endpoints $\\operatorname{Re} z$ and $\\operatorname{Re} w$, and $\\operatorname{Im} u$ lies in the closed interval with endpoints $\\operatorname{Im} z$ and $\\operatorname{Im} w$.","title":"Carrier of an axis-parallel rectangle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-integral","kind":"node","lean":"RectilinearRectangle.boundaryIntegral","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-integral","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space and let $R$ be a rectilinear rectangle with first corner $z$ and second corner $w$. The boundary integral of a function $f : \\mathbb{C} \\to E$ over $R$ is","title":"Boundary integral over a rectangle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-integral-eq-zero","kind":"node","lean":"RectilinearRectangle.boundaryIntegral_eq_zero","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-integral-eq-zero","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space, let $R$ be a rectilinear rectangle and let $f : \\mathbb{C} \\to E$ be complex differentiable on the closed rectangle spanned by the corners of $R$ (the carrier of $R$). Then the boundary integral of $f$ over $R$ is $0$.","title":"Cauchy\u2013Goursat theorem for rectangles"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-integral-of-bounds-horizontal-c","kind":"node","lean":"RectilinearRectangle.boundaryIntegral_ofBounds_horizontal_concat","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-integral-of-bounds-horizontal-c","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space, let $f : \\mathbb{C} \\to E$ and let $x_0, x_1, y_0, y_1, y_2$ be real numbers such that the maps $y \\mapsto f(x_0 + iy)$ and $y \\mapsto f(x_1 + iy)$ are continuous on $\\mathbb{R}$. For real $a, b, c, d$ write $R(a, b, c, d)$ for the rectangle with corners $a + ic$ and $b + id$. Then the boundary integral of $f$ over $R(x_0, x_1, y_0, y_2)$ equals the sum of the boundary integrals\u2026","title":"Gluing rectangles along a horizontal side"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-integral-of-bounds-vertical-con","kind":"node","lean":"RectilinearRectangle.boundaryIntegral_ofBounds_vertical_concat","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-integral-of-bounds-vertical-con","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space, let $f : \\mathbb{C} \\to E$ and let $x_0, x_1, x_2, y_0, y_1$ be real numbers such that the maps $x \\mapsto f(x + iy_0)$ and $x \\mapsto f(x + iy_1)$ are continuous on $\\mathbb{R}$. For real $a, b, c, d$ write $R(a, b, c, d)$ for the rectangle with corners $a + ic$ and $b + id$. Then the boundary integral of $f$ over $R(x_0, x_2, y_0, y_1)$ equals the sum of the boundary integrals\u2026","title":"Gluing rectangles along a vertical side"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-integral-inv-sub-eq-zero-of-not","kind":"node","lean":"RectilinearRectangle.boundaryIntegral_inv_sub_eq_zero_of_not_mem_carrier","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-integral-inv-sub-eq-zero-of-not","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $R$ be a rectilinear rectangle and let $w$ be a complex number that does not lie in the closed rectangle spanned by the corners of $R$ (the carrier of $R$). Then the boundary integral of $z \\mapsto 1/(z - w)$ over $R$ is $0$.","title":"Cauchy kernel with the pole outside a rectangle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-integral-of-bounds-inv-eq-two-p","kind":"node","lean":"RectilinearRectangle.boundaryIntegral_ofBounds_inv_eq_two_pi_mul_I","parent":16384,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangle integral of "],["math","\\(1/z\\)"],["text"," around the origin"]],"source":"Rectangle integral of $1/z$ around the origin"},"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-integral-of-bounds-inv-eq-two-p","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $a, b, T$ be real numbers with $a < 0 < b$ and $T > 0$. Then the boundary integral of $z \\mapsto 1/z$ over the rectangle with corners $a - iT$ and $b + iT$ equals $2\\pi i$.","title":"Rectangle integral of $1/z$ around the origin"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-integral-of-bounds-inv-sub-eq-t","kind":"node","lean":"RectilinearRectangle.boundaryIntegral_ofBounds_inv_sub_eq_translate","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-integral-of-bounds-inv-sub-eq-t","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For all real numbers $x_0, x_1, y_0, y_1$ and every complex number $w$, the boundary integral of $z \\mapsto 1/(z - w)$ over the rectangle with corners $x_0 + iy_0$ and $x_1 + iy_1$ equals the boundary integral of $z \\mapsto 1/z$ over the rectangle with corners $(x_0 - \\operatorname{Re} w) + i(y_0 - \\operatorname{Im} w)$ and $(x_1 - \\operatorname{Re} w) + i(y_1 - \\operatorname{Im} w)$.","title":"Translating the Cauchy kernel to the origin"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/boundary-integral-of-bounds-inv-sub-eq-t-2","kind":"node","lean":"RectilinearRectangle.boundaryIntegral_ofBounds_inv_sub_eq_two_pi_mul_I","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#boundary-integral-of-bounds-inv-sub-eq-t-2","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $x_0, x_1, y_0, y_1$ be real numbers and let $w$ be a complex number with $x_0 < \\operatorname{Re} w < x_1$ and $y_0 < \\operatorname{Im} w < y_1$. Then the boundary integral of $z \\mapsto 1/(z - w)$ over the rectangle with corners $x_0 + iy_0$ and $x_1 + iy_1$ equals $2\\pi i$.","title":"Rectangle integral of the Cauchy kernel at an interior point"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/rectilinear-cycle","kind":"node","lean":"RectilinearCycle","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#rectilinear-cycle","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"A rectilinear cycle is a finitely supported function from rectilinear rectangles to $\\mathbb{Z}$, that is, a finite formal sum $\\sum_r n_r\\, r$ of rectilinear rectangles $r$ with integer coefficients $n_r$, read as the corresponding integer combination of oriented rectangle boundaries.","title":"Rectilinear cycle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/of-rectangle","kind":"node","lean":"RectilinearCycle.ofRectangle","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#of-rectangle","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For a rectilinear rectangle $r$, this is the rectilinear cycle in which $r$ has coefficient $1$ and every other rectangle has coefficient $0$, that is, the boundary of $r$ with the orientation given by its corner order.","title":"The cycle of a single rectangle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/integral","kind":"node","lean":"RectilinearCycle.integral","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#integral","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space and $f : \\mathbb{C} \\to E$. For a rectilinear cycle $\\Gamma$, a finite formal sum $\\sum_r n_r\\, r$ of rectangles $r$ with integer coefficients $n_r$, the contour integral of $f$ over $\\Gamma$ is $\\sum_r n_r \\int_{\\partial r} f$, where $\\int_{\\partial r} f$ is the boundary integral of $f$ over the oriented rectangle $r$.","title":"Contour integral over a rectilinear cycle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/integral-add","kind":"node","lean":"RectilinearCycle.integral_add","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#integral-add","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space and $f : \\mathbb{C} \\to E$. For rectilinear cycles $\\Gamma$ and $\\Lambda$, the contour integral of $f$ over $\\Gamma + \\Lambda$ equals the integral of $f$ over $\\Gamma$ plus the integral of $f$ over $\\Lambda$.","title":"Additivity of the contour integral in the cycle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/integral-neg","kind":"node","lean":"RectilinearCycle.integral_neg","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#integral-neg","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For every rectilinear cycle $\\Gamma$, every complex Banach space $E$ and every function $f : \\mathbb{C} \\to E$, the contour integral of $f$ over $-\\Gamma$ is the negative of the contour integral of $f$ over $\\Gamma$.","title":"Reversing a cycle negates its integral"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/integral-sub","kind":"node","lean":"RectilinearCycle.integral_sub","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#integral-sub","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For rectilinear cycles $\\Gamma$ and $\\Lambda$, a complex Banach space $E$ and a function $f : \\mathbb{C} \\to E$, the contour integral of $f$ over $\\Gamma - \\Lambda$ equals the contour integral of $f$ over $\\Gamma$ minus the contour integral of $f$ over $\\Lambda$.","title":"Integral over a difference of cycles"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/integral-add-hom","kind":"node","lean":"RectilinearCycle.integralAddHom","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#integral-add-hom","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space and $f : \\mathbb{C} \\to E$. The map sending a rectilinear cycle $\\Gamma$ to the contour integral of $f$ over $\\Gamma$ is an additive group homomorphism from rectilinear cycles to $E$.","title":"Contour integration as an additive map"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/filling-supported-in","kind":"node","lean":"RectilinearCycle.FillingSupportedIn","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#filling-supported-in","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $G \\subseteq \\mathbb{C}$. A rectilinear cycle $\\Gamma$, read as a filling by rectangles, is supported in $G$ when, for every rectangle $r$ whose coefficient in $\\Gamma$ is nonzero, the closed carrier of $r$ is contained in $G$.","title":"Rectangle filling supported in a set"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/integral-eq-zero-of-filling-supported-in","kind":"node","lean":"RectilinearCycle.integral_eq_zero_of_fillingSupportedIn","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#integral-eq-zero-of-filling-supported-in","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space, let $G \\subseteq \\mathbb{C}$, and let $f : \\mathbb{C} \\to E$ be complex differentiable on $G$. If $\\Gamma$ is a rectilinear cycle whose rectangle filling is supported in $G$, that is, the closed rectangle of every rectangle occurring in $\\Gamma$ with nonzero coefficient is contained in $G$, then the contour integral of $f$ over $\\Gamma$ is $0$.","title":"Cauchy\u2013Goursat theorem for rectilinear cycles"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/homologous-in","kind":"node","lean":"RectilinearCycle.HomologousIn","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#homologous-in","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $G \\subseteq \\mathbb{C}$. Two rectilinear cycles $\\Gamma$ and $\\Lambda$ are homologous in $G$ when the rectangle filling of $\\Gamma - \\Lambda$ is supported in $G$: every rectangle with nonzero coefficient in $\\Gamma - \\Lambda$ has its closed carrier contained in $G$.","title":"Rectilinear cycles homologous in a set"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/integral-eq-of-homologous-in","kind":"node","lean":"RectilinearCycle.integral_eq_of_homologousIn","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#integral-eq-of-homologous-in","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"Let $E$ be a complex Banach space, let $G \\subseteq \\mathbb{C}$, and let $f : \\mathbb{C} \\to E$ be complex differentiable on $G$. If $\\Gamma$ and $\\Lambda$ are rectilinear cycles that are homologous in $G$, that is, the closed rectangle of every rectangle occurring in $\\Gamma - \\Lambda$ with nonzero coefficient is contained in $G$, then the contour integral of $f$ over $\\Gamma$ equals the contour integral of $f$\u2026","title":"Homologous rectilinear cycles have equal integrals"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index-2","kind":"node","lean":"RectilinearCycle.index","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#index-2","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For a rectilinear cycle $\\Gamma$ and a point $w \\in \\mathbb{C}$, the index of $\\Gamma$ about $w$ is $\\frac{1}{2\\pi i}\\int_\\Gamma \\frac{dz}{z - w}$, the contour integral over $\\Gamma$ of $z \\mapsto (z - w)^{-1}$, multiplied by $(2\\pi i)^{-1}$.","title":"Index of a rectilinear cycle"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index-add-hom","kind":"node","lean":"RectilinearCycle.indexAddHom","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#index-add-hom","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For a fixed point $w \\in \\mathbb{C}$, the map sending a rectilinear cycle $\\Gamma$ to its index about $w$ is an additive group homomorphism from rectilinear cycles to $\\mathbb{C}$.","title":"Winding index as an additive map"},{"id":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index-of-rectangle-eq-zero-of-not-mem-ca","kind":"node","lean":"RectilinearCycle.index_ofRectangle_eq_zero_of_not_mem_carrier","parent":16384,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles/index.html#index-of-rectangle-eq-zero-of-not-mem-ca","scope":"Analysis/complex-analysis/cauchy-theory/contour-integrals-over-rectangles","status":"fully_proved","summary":"For an axis-parallel rectangle $r$ and a point $w \\in \\mathbb{C}$ outside the closed carrier of $r$, the index about $w$ of the cycle consisting of the boundary of $r$ is $0$.","title":"Rectangle index vanishes outside the rectangle"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory/great-picard-theorem.json","id":"Analysis/complex-analysis/cauchy-theory/great-picard-theorem","kind":"scope","navigation":"module","parent":16365,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/great-picard-theorem/index.html","scope":"Analysis/complex-analysis/cauchy-theory/great-picard-theorem","status":"fully_proved","title":"Great Picard theorem"},{"id":"Analysis/complex-analysis/cauchy-theory/great-picard-theorem/is-essential-isolated-singularity-at","kind":"node","lean":"MathlibExt.Analysis.Complex.GreatPicardWanted.IsEssentialIsolatedSingularityAt","parent":16416,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/great-picard-theorem/index.html#is-essential-isolated-singularity-at","scope":"Analysis/complex-analysis/cauchy-theory/great-picard-theorem","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ has an essential isolated singularity at $c \\in \\mathbb{C}$ when $f$ is holomorphic on some punctured disc $0 < |z - c| < r$ with $r > 0$, $f(z)$ has no finite limit as $z \\to c$ with $z \\ne c$, and $|f(z)|$ does not tend to $\\infty$ as $z \\to c$ with $z \\ne c$.","title":"Essential isolated singularity"},{"id":"Analysis/complex-analysis/cauchy-theory/great-picard-theorem/great-picard","kind":"node","lean":"MathlibExt.Analysis.Complex.GreatPicardWanted.greatPicard","parent":16416,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/great-picard-theorem/index.html#great-picard","scope":"Analysis/complex-analysis/cauchy-theory/great-picard-theorem","status":"fully_proved","summary":"Suppose $f : \\mathbb{C} \\to \\mathbb{C}$ has an essential isolated singularity at $c \\in \\mathbb{C}$. Then there is at most one $w \\in \\mathbb{C}$ for which some $r > 0$ makes the set $\\{z : 0 < |z - c| < r,\\ f(z) = w\\}$ finite; that is, with at most one exception, $f$ takes every complex value infinitely often on every punctured disc around $c$.","title":"Great Picard theorem"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory/normal-families.json","id":"Analysis/complex-analysis/cauchy-theory/normal-families","kind":"scope","navigation":"module","parent":16365,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/normal-families/index.html","scope":"Analysis/complex-analysis/cauchy-theory/normal-families","status":"fully_proved","title":"Normal families"},{"id":"Analysis/complex-analysis/cauchy-theory/normal-families/is-locally-uniformly-bounded-on","kind":"node","lean":"MathlibExt.Analysis.Complex.NormalFamilies.IsLocallyUniformlyBoundedOn","parent":16419,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/normal-families/index.html#is-locally-uniformly-bounded-on","scope":"Analysis/complex-analysis/cauchy-theory/normal-families","status":"fully_proved","summary":"Let $U \\subseteq \\mathbb{C}$ and let $(F_n)_{n \\in \\mathbb{N}}$ be a sequence of functions $\\mathbb{C} \\to \\mathbb{C}$. The sequence is locally uniformly bounded on $U$ when every $x \\in U$ has an open neighbourhood $V \\subseteq U$ and a real $C \\ge 0$ such that $|F_n(y)| \\le C$ for all $n \\in \\mathbb{N}$ and all $y \\in V$.","title":"Locally uniformly bounded sequence of functions"},{"id":"Analysis/complex-analysis/cauchy-theory/normal-families/is-locally-uniformly-bounded-on-of-foral","kind":"node","lean":"MathlibExt.Analysis.Complex.NormalFamilies.isLocallyUniformlyBoundedOn_of_forall_norm_le","parent":16419,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/normal-families/index.html#is-locally-uniformly-bounded-on-of-foral","scope":"Analysis/complex-analysis/cauchy-theory/normal-families","status":"fully_proved","summary":"Let $U \\subseteq \\mathbb{C}$ be open, let $(F_n)_{n \\in \\mathbb{N}}$ be a sequence of functions $\\mathbb{C} \\to \\mathbb{C}$ and let $C \\ge 0$ be real. If $|F_n(y)| \\le C$ for all $n \\in \\mathbb{N}$ and all $y \\in U$, then $(F_n)$ is locally uniformly bounded on $U$.","title":"A uniformly bounded sequence on an open set is locally uniformly bounded"},{"id":"Analysis/complex-analysis/cauchy-theory/normal-families/exists-bound-of-is-compact","kind":"node","lean":"MathlibExt.Analysis.Complex.NormalFamilies.IsLocallyUniformlyBoundedOn.exists_bound_of_isCompact","parent":16419,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/normal-families/index.html#exists-bound-of-is-compact","scope":"Analysis/complex-analysis/cauchy-theory/normal-families","status":"fully_proved","summary":"Let $(F_n)_{n \\in \\mathbb{N}}$ be a sequence of functions $\\mathbb{C} \\to \\mathbb{C}$ that is locally uniformly bounded on $U \\subseteq \\mathbb{C}$. Then for every compact set $K \\subseteq U$ there is a real $C \\ge 0$ with $|F_n(y)| \\le C$ for all $n \\in \\mathbb{N}$ and all $y \\in K$.","title":"Locally uniformly bounded sequences are uniformly bounded on compact subsets"},{"id":"Analysis/complex-analysis/cauchy-theory/normal-families/montel","kind":"node","lean":"MathlibExt.Analysis.Complex.NormalFamilies.montel","parent":16419,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/normal-families/index.html#montel","scope":"Analysis/complex-analysis/cauchy-theory/normal-families","status":"fully_proved","summary":"Let $U \\subseteq \\mathbb{C}$ be open and let $(F_n)_{n \\in \\mathbb{N}}$ be a sequence of functions $\\mathbb{C} \\to \\mathbb{C}$, each holomorphic on $U$, that is locally uniformly bounded on $U$. Then there are a strictly increasing map $\\varphi : \\mathbb{N} \\to \\mathbb{N}$ and a function $f : \\mathbb{C} \\to \\mathbb{C}$, holomorphic on $U$, such that $F_{\\varphi(n)} \\to f$ locally uniformly on $U$ as $n \\to \\infty$.","title":"Montel's theorem"},{"id":"Analysis/complex-analysis/cauchy-theory/normal-families/vitali","kind":"node","lean":"MathlibExt.Analysis.Complex.NormalFamilies.vitali","parent":16419,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/normal-families/index.html#vitali","scope":"Analysis/complex-analysis/cauchy-theory/normal-families","status":"fully_proved","summary":"Let $U \\subseteq \\mathbb{C}$ be open and preconnected, and let $(F_n)_{n \\in \\mathbb{N}}$ be a sequence of functions $\\mathbb{C} \\to \\mathbb{C}$, each holomorphic on $U$, that is locally uniformly bounded on $U$. Let $S \\subseteq U$ have an accumulation point $x_0 \\in U$, and let $g : \\mathbb{C} \\to \\mathbb{C}$ satisfy $F_n(z) \\to g(z)$ as $n \\to \\infty$ for every $z \\in S$. Then there is a function\u2026","title":"Vitali's convergence theorem"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions.json","id":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions","kind":"scope","navigation":"module","parent":16365,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/index.html","scope":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions","status":"can_prove","title":"Removable lines for holomorphic functions"},{"id":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/real-axis","kind":"node","lean":"MathlibExt.Analysis.Complex.ContinuousHolomorphicOffAxesEntireWanted.realAxis","parent":16425,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/index.html#real-axis","scope":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions","status":"fully_proved","summary":"The real axis is the set $\\{z \\in \\mathbb{C} : \\operatorname{Im} z = 0\\}$.","title":"Real axis"},{"id":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/imag-axis","kind":"node","lean":"MathlibExt.Analysis.Complex.ContinuousHolomorphicOffAxesEntireWanted.imagAxis","parent":16425,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/index.html#imag-axis","scope":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions","status":"fully_proved","summary":"The imaginary axis is the set $\\{z \\in \\mathbb{C} : \\operatorname{Re} z = 0\\}$.","title":"Imaginary axis"},{"id":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/coord-axes","kind":"node","lean":"MathlibExt.Analysis.Complex.ContinuousHolomorphicOffAxesEntireWanted.coordAxes","parent":16425,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/index.html#coord-axes","scope":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions","status":"fully_proved","summary":"The coordinate axes of $\\mathbb{C}$: the union of the real axis and the imaginary axis.","title":"Coordinate axes"},{"id":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/differentiable-of-continuous-of-differen","kind":"node","lean":"MathlibExt.Analysis.Complex.ContinuousHolomorphicOffAxesEntireWanted.differentiable_of_continuous_of_differentiableOn_compl_coordAxes","parent":16425,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions/index.html#differentiable-of-continuous-of-differen","scope":"Analysis/complex-analysis/cauchy-theory/removable-lines-for-holomorphic-functions","status":"can_prove","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be continuous on all of $\\mathbb{C}$ and holomorphic on the complement of the coordinate axes. Then $f$ is holomorphic on all of $\\mathbb{C}$.","title":"Continuous functions holomorphic off the coordinate axes are entire"},{"data_url":"scopes/Analysis/complex-analysis/cauchy-theory/rouch-s-theorem.json","id":"Analysis/complex-analysis/cauchy-theory/rouch-s-theorem","kind":"scope","navigation":"module","parent":16365,"result":false,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/rouch-s-theorem/index.html","scope":"Analysis/complex-analysis/cauchy-theory/rouch-s-theorem","status":"can_prove","title":"Rouch\u00e9's theorem"},{"id":"Analysis/complex-analysis/cauchy-theory/rouch-s-theorem/rouche-closed-ball","kind":"node","lean":"MathlibExt.Analysis.Complex.Rouche.rouche_closedBall","parent":16430,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/rouch-s-theorem/index.html#rouche-closed-ball","scope":"Analysis/complex-analysis/cauchy-theory/rouch-s-theorem","status":"fully_proved","summary":"Let $c \\in \\mathbb{C}$ and $R > 0$, and let $f$ and $g$ be analytic at every point of the closed disc $|z - c| \\le R$. If $|g(z)| < |f(z)|$ for every $z$ on the circle $|z - c| = R$, then $f$ and $f + g$ have the same number of zeros in the open disc $|z - c| < R$, counted with multiplicity: the divisors of $f$ and of $f + g$ on the open disc have equal sums.","title":"Rouch\u00e9's theorem on a disc"},{"id":"Analysis/complex-analysis/cauchy-theory/rouch-s-theorem/rouche-zero-count-eq","kind":"node","lean":"MathlibExt.Analysis.RoucheWanted.ComplexAnalysis.rouche_zero_count_eq","parent":16430,"result":true,"route":"../roadmap/Analysis/complex-analysis/cauchy-theory/rouch-s-theorem/index.html#rouche-zero-count-eq","scope":"Analysis/complex-analysis/cauchy-theory/rouch-s-theorem","status":"can_prove","summary":"Let $R > 0$, let $f : \\mathbb{C} \\to \\mathbb{C}$ be meromorphic on $\\mathbb{C}$ and in normal form (at every point $f$ is either analytic, or has a pole and takes the value $0$ there), and let $g : \\mathbb{C} \\to \\mathbb{C}$ be analytic on $\\mathbb{C}$. If $|g(z)| < |f(z)|$ for every $z$ with $|z| = R$, then $f + g$ and $f$ have the same number of zeros in the closed disk $|z| \\le R$, counted with multiplicity: the\u2026","title":"Rouch\u00e9's theorem"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions.json","id":"Analysis/complex-analysis/entire-functions","kind":"scope","parent":16364,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/index.html","scope":"Analysis/complex-analysis/entire-functions","status":"can_prove","summary":"Growth and value distribution of entire and meromorphic functions: Wiman\u2013Valiron theory and maximum term, Hayman admissibility, Weierstrass factorization, Mittag-Leffler, Liouville-type rigidity, preimages and zeros/ones on lines.","title":"Entire and meromorphic functions"},{"id":"Analysis/complex-analysis/entire-functions/eq-const-of-periodic-one-of-norm-le-exp","kind":"node","lean":"PeriodicEntire.eq_const_of_periodic_one_of_norm_le_exp_abs_im","parent":16433,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(1\\)"],["text","-periodic entire function bounded by "],["math","\\(C e^{|\\operatorname{Im} s|}\\)"],["text"," is constant"]],"source":"A $1$-periodic entire function bounded by $C e^{|\\operatorname{Im} s|}$ is constant"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/index.html#eq-const-of-periodic-one-of-norm-le-exp","scope":"Analysis/complex-analysis/entire-functions","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be entire with $f(s + 1) = f(s)$ for all $s \\in \\mathbb{C}$, and suppose there is a real constant $C$ such that $|f(s)| \\le C e^{|\\operatorname{Im} s|}$ for every $s$ with $0 \\le \\operatorname{Re} s \\le 1$. Then $f$ is constant.","title":"A $1$-periodic entire function bounded by $C e^{|\\operatorname{Im} s|}$ is constant"},{"id":"Analysis/complex-analysis/entire-functions/exists-entire-power-series-infinite-zero","kind":"node","lean":"MathlibExt.Analysis.Complex.EntireSeriesInfiniteZerosWanted.exists_entire_powerSeries_infinite_zeros","parent":16433,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/index.html#exists-entire-power-series-infinite-zero","scope":"Analysis/complex-analysis/entire-functions","status":"can_prove","summary":"There is a sequence $(a_k)_{k \\ge 0}$ of complex numbers, not all zero, such that the power series $\\sum_{k=0}^{\\infty} a_k z^k$ converges absolutely for every $z \\in \\mathbb{C}$ and its sum vanishes at infinitely many points $z \\in \\mathbb{C}$.","title":"A nonzero entire power series with infinitely many zeros"},{"id":"Analysis/complex-analysis/entire-functions/entire-cube-add-cube-eq-one-const","kind":"node","lean":"MathlibExt.Analysis.Complex.EntireFermatCubicConstantWanted.entire_cube_add_cube_eq_one_const","parent":16433,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/index.html#entire-cube-add-cube-eq-one-const","scope":"Analysis/complex-analysis/entire-functions","status":"can_prove","summary":"Let $f$ and $g$ be entire functions with $f(z)^3 + g(z)^3 = 1$ for every $z \\in \\mathbb{C}$. Then $f$ and $g$ are constant.","title":"Entire solutions of the Fermat cubic are constant"},{"id":"Analysis/complex-analysis/entire-functions/exists-affine-of-differentiable-injectiv","kind":"node","lean":"MathlibExt.Analysis.Complex.InjectiveEntireIsAffineWanted.exists_affine_of_differentiable_injective","parent":16433,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/index.html#exists-affine-of-differentiable-injectiv","scope":"Analysis/complex-analysis/entire-functions","status":"can_prove","summary":"If $f : \\mathbb{C} \\to \\mathbb{C}$ is entire and injective, then there are $a, b \\in \\mathbb{C}$ with $a \\ne 0$ such that $f(z) = az + b$ for all $z \\in \\mathbb{C}$.","title":"Injective entire functions are affine"},{"id":"Analysis/complex-analysis/entire-functions/weierstrass-factorization","kind":"node","lean":"MathlibExt.Analysis.Complex.WeierstrassFactorizationWanted.weierstrass_factorization","parent":16433,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/index.html#weierstrass-factorization","scope":"Analysis/complex-analysis/entire-functions","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{C}$ be a closed set all of whose points are isolated: for each $z \\in S$ there is $\\varepsilon > 0$ such that no point $w \\ne z$ of $S$ satisfies $|w - z| < \\varepsilon$. Then there is an entire function $f$ whose zeros are exactly the points of $S$, all of them simple: $f'(z) \\ne 0$ for every $z \\in S$.","title":"Weierstrass theorem on prescribed simple zeros"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f.json","id":"Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f/index.html","scope":"Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f","status":"can_prove","title":"Countable dense sets as preimages under entire functions"},{"id":"Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f/conjecture","kind":"node","lean":"MathlibExt.Analysis.ErdosEntireFunctionCountableDensePreimageWanted.conjecture","parent":16439,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f/index.html#conjecture","scope":"Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f","status":"fully_proved","summary":"This is the statement that for any two countable dense subsets $A$ and $B$ of $\\mathbb{C}$, there is an entire function $f : \\mathbb{C} \\to \\mathbb{C}$ such that for every $z \\in \\mathbb{C}$, $f(z) \\in B$ if and only if $z \\in A$; that is, $f^{-1}(B) = A$.","title":"Erd\u0151s's problem on countable dense preimages"},{"id":"Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.ErdosEntireFunctionCountableDensePreimageWanted.conjecture_holds","parent":16439,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f/index.html#conjecture-holds","scope":"Analysis/complex-analysis/entire-functions/countable-dense-sets-as-preimages-under-entire-f","status":"can_prove","summary":"For any two countable dense subsets $A$ and $B$ of $\\mathbb{C}$, there is an entire function $f : \\mathbb{C} \\to \\mathbb{C}$ such that for every $z \\in \\mathbb{C}$, $f(z) \\in B$ if and only if $z \\in A$; that is, $f^{-1}(B) = A$.","title":"Every countable dense set is a preimage of any other under an entire function"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/hayman-admissibility.json","id":"Analysis/complex-analysis/entire-functions/hayman-admissibility","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/hayman-admissibility/index.html","scope":"Analysis/complex-analysis/entire-functions/hayman-admissibility","status":"fully_proved","title":"Hayman admissibility"},{"id":"Analysis/complex-analysis/entire-functions/hayman-admissibility/hayman-auxiliary-a","kind":"node","lean":"MetaMathlibExt.haymanAuxiliaryA","parent":16442,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hayman auxiliary function "],["math","\\(A\\)"]],"source":"Hayman auxiliary function $A$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/hayman-admissibility/index.html#hayman-auxiliary-a","scope":"Analysis/complex-analysis/entire-functions/hayman-admissibility","status":"fully_proved","summary":"For a function $G : \\mathbb{C} \\to \\mathbb{C}$ and a real number $r$, $A(r)$ is the real part of $r\\,G'(r)/G(r)$, where $G'$ is the complex derivative of $G$.","title":"Hayman auxiliary function $A$"},{"id":"Analysis/complex-analysis/entire-functions/hayman-admissibility/hayman-auxiliary-b","kind":"node","lean":"MetaMathlibExt.haymanAuxiliaryB","parent":16442,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hayman auxiliary function "],["math","\\(B\\)"]],"source":"Hayman auxiliary function $B$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/hayman-admissibility/index.html#hayman-auxiliary-b","scope":"Analysis/complex-analysis/entire-functions/hayman-admissibility","status":"fully_proved","summary":"For a function $G : \\mathbb{C} \\to \\mathbb{C}$ and a real number $r$, $B(r) = r\\,A'(r)$, where $A$ is the Hayman auxiliary function of $G$, viewed as a real function of $r$, and $A'$ is its derivative.","title":"Hayman auxiliary function $B$"},{"id":"Analysis/complex-analysis/entire-functions/hayman-admissibility/is-hayman-admissible","kind":"node","lean":"MetaMathlibExt.IsHaymanAdmissible","parent":16442,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/hayman-admissibility/index.html#is-hayman-admissible","scope":"Analysis/complex-analysis/entire-functions/hayman-admissibility","status":"fully_proved","summary":"Let $G : \\mathbb{C} \\to \\mathbb{C}$, let $R_0$ and $\\rho$ be real numbers, write $G'$ for the complex derivative of $G$ and $A$, $B$ for the Hayman auxiliary functions of $G$, and say that a property holds for $r$ near $\\rho$ if there is $\\eta > 0$ such that it holds for every $r \\in (R_0, \\rho)$ with $r > \\rho - \\eta$. Then $G$ is Hayman admissible on $(R_0, \\rho)$ when $0 < R_0 < \\rho$; $G$ is analytic on the\u2026","title":"Hayman-admissible function (finite radius)"},{"id":"Analysis/complex-analysis/entire-functions/hayman-admissibility/hayman-coefficient-asymptotic","kind":"node","lean":"MetaMathlibExt.hayman_coefficient_asymptotic","parent":16442,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/hayman-admissibility/index.html#hayman-coefficient-asymptotic","scope":"Analysis/complex-analysis/entire-functions/hayman-admissibility","status":"fully_proved","summary":"Let $G : \\mathbb{C} \\to \\mathbb{C}$ be Hayman admissible on $(R_0, \\rho)$, with Hayman auxiliary functions $A$ and $B$. Let $(c_n)_{n \\ge 0}$ be complex numbers such that $\\sum_{n \\ge 0} c_n z^n$ converges absolutely to $G(z)$ for every complex $z$ with $|z| < \\rho$, and let $(r_n)_{n \\ge 0}$ be real numbers such that for all sufficiently large $n$, $r_n \\in (R_0, \\rho)$, $A(r_n) = n$, and $r_n$ is the only\u2026","title":"Hayman's coefficient asymptotic (finite radius)"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/liouville-type-theorems.json","id":"Analysis/complex-analysis/entire-functions/liouville-type-theorems","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/liouville-type-theorems/index.html","scope":"Analysis/complex-analysis/entire-functions/liouville-type-theorems","status":"can_prove","title":"Liouville-type theorems"},{"id":"Analysis/complex-analysis/entire-functions/liouville-type-theorems/eq-on-const-of-differentiable-on-compl-z","kind":"node","lean":"MathlibExt.Analysis.Complex.BoundedPuncturedPlaneConstWanted.eqOn_const_of_differentiableOn_compl_zero_of_bounded","parent":16447,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/liouville-type-theorems/index.html#eq-on-const-of-differentiable-on-compl-z","scope":"Analysis/complex-analysis/entire-functions/liouville-type-theorems","status":"can_prove","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on $\\mathbb{C} \\setminus \\{0\\}$, and suppose that there is a real number $C$ with $|f(z)| \\le C$ for every $z \\ne 0$. Then there is a constant $c \\in \\mathbb{C}$ with $f(z) = c$ for every $z \\ne 0$.","title":"Bounded holomorphic functions on the punctured plane are constant"},{"id":"Analysis/complex-analysis/entire-functions/liouville-type-theorems/exists-polynomial-of-iterated-deriv-eq-p","kind":"node","lean":"MathlibExt.Analysis.Complex.EntireIteratedDerivPolyWanted.exists_polynomial_of_iteratedDeriv_eq_polynomial","parent":16447,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/liouville-type-theorems/index.html#exists-polynomial-of-iterated-deriv-eq-p","scope":"Analysis/complex-analysis/entire-functions/liouville-type-theorems","status":"can_prove","summary":"Let $k$ be a positive integer and let $f : \\mathbb{C} \\to \\mathbb{C}$ be an entire function. If there is a complex polynomial $p$ such that the $k$th derivative of $f$ satisfies $f^{(k)}(z) = p(z)$ for all $z \\in \\mathbb{C}$, then there is a complex polynomial $q$ such that $f(z) = q(z)$ for all $z \\in \\mathbb{C}$.","title":"An entire function with a polynomial derivative is a polynomial"},{"id":"Analysis/complex-analysis/entire-functions/liouville-type-theorems/exists-const-of-differentiable-of-tendst","kind":"node","lean":"MathlibExt.Analysis.Complex.EntireSublinearGrowthConstantWanted.exists_const_of_differentiable_of_tendsto_div_self_zero","parent":16447,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/liouville-type-theorems/index.html#exists-const-of-differentiable-of-tendst","scope":"Analysis/complex-analysis/entire-functions/liouville-type-theorems","status":"can_prove","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be an entire function such that $f(z)/z \\to 0$ as $|z| \\to \\infty$. Then $f$ is constant: there is $c \\in \\mathbb{C}$ with $f(z) = c$ for all $z \\in \\mathbb{C}$.","title":"An entire function of sublinear growth is constant"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/mittag-leffler-theorem.json","id":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/index.html","scope":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem","status":"fully_proved","title":"Mittag-Leffler theorem"},{"id":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/principal-part","kind":"node","lean":"MathlibExt.Analysis.Complex.MittagLefflerWanted.principalPart","parent":16451,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/index.html#principal-part","scope":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem","status":"fully_proved","summary":"For a point $s \\in \\mathbb{C}$ and a finitely supported sequence $c = (c_k)_{k \\in \\mathbb{N}}$ of complex coefficients, the principal part at $s$ with coefficients $c$ is the function $\\mathbb{C} \\to \\mathbb{C}$ given by $z \\mapsto \\sum_k c_k (z - s)^{-(k+1)}$, the sum running over the finitely many $k$ with $c_k \\neq 0$.","title":"Principal part with finitely many coefficients"},{"id":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/principal-part-analytic-at","kind":"node","lean":"MathlibExt.Analysis.Complex.MittagLefflerWanted.principalPart_analyticAt","parent":16451,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/index.html#principal-part-analytic-at","scope":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem","status":"fully_proved","summary":"For every $s \\in \\mathbb{C}$ and every finitely supported sequence $c$ of complex coefficients, the principal part at $s$ with coefficients $c$ is analytic at every point $z \\neq s$.","title":"The principal part is analytic away from its pole"},{"id":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/principal-part-meromorphic-at","kind":"node","lean":"MathlibExt.Analysis.Complex.MittagLefflerWanted.principalPart_meromorphicAt","parent":16451,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/index.html#principal-part-meromorphic-at","scope":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem","status":"fully_proved","summary":"For every $s \\in \\mathbb{C}$ and every finitely supported sequence $c$ of complex coefficients, the principal part at $s$ with coefficients $c$ is meromorphic at every point of $\\mathbb{C}$.","title":"The principal part is meromorphic"},{"id":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/mittag-leffler-of-is-discrete","kind":"node","lean":"MathlibExt.Analysis.Complex.MittagLefflerWanted.mittag_leffler_of_isDiscrete","parent":16451,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/index.html#mittag-leffler-of-is-discrete","scope":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{C}$ be a closed set that is discrete in the subspace topology, and for each $s \\in S$ let $c_s$ be a finitely supported sequence of complex coefficients. Then there is a function $f : \\mathbb{C} \\to \\mathbb{C}$ that is holomorphic on $\\mathbb{C} \\setminus S$, meromorphic on all of $\\mathbb{C}$, and such that for every $s \\in S$ the difference between $f$ and the principal part at $s$ with\u2026","title":"Mittag-Leffler theorem, discrete-subspace form"},{"id":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/mittag-leffler","kind":"node","lean":"MathlibExt.Analysis.Complex.MittagLefflerWanted.mittag_leffler","parent":16451,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/mittag-leffler-theorem/index.html#mittag-leffler","scope":"Analysis/complex-analysis/entire-functions/mittag-leffler-theorem","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{C}$ be a closed set such that every $z \\in S$ has a radius $\\varepsilon > 0$ for which the punctured disk $\\{w : 0 < |w - z| < \\varepsilon\\}$ contains no point of $S$, and for each $s \\in S$ let $c_s$ be a finitely supported sequence of complex coefficients. Then there is a function $f : \\mathbb{C} \\to \\mathbb{C}$ that is holomorphic on $\\mathbb{C} \\setminus S$, meromorphic on all of\u2026","title":"Mittag-Leffler theorem"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential.json","id":"Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential/index.html","scope":"Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential","status":"can_prove","title":"R\u00e9nyi\u2013Vincze characterization of the exponential"},{"id":"Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential/conjecture","kind":"node","lean":"MathlibExt.Analysis.HaymanProblem232Wanted.conjecture","parent":16457,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential/index.html#conjecture","scope":"Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential","status":"fully_proved","summary":"There exist a function $f : \\mathbb{C} \\to \\mathbb{C}$ and real numbers $a_0, a_1, a_2, \\dots$ such that $f$ is not the function $z \\mapsto c\\,e^z$ for any real $c > 0$; $a_n \\ge 0$ for every $n$; $a_n \\ne 0$ for infinitely many $n$; for every $z \\in \\mathbb{C}$ the series $\\sum_{n \\ge 0} a_n z^n$ has sum $f(z)$; $f$ is entire; and either $\\int_0^\\infty a_n x^n / f(x)\\,dx = 1$ for every $n \\ge 0$ (condition (2.6)),\u2026","title":"Hayman Problem 2.32: a non-exponential solution of the R\u00e9nyi\u2013Vincze conditions"},{"id":"Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential/conjecture-refuted","kind":"node","lean":"MathlibExt.Analysis.HaymanProblem232Wanted.conjecture_refuted","parent":16457,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential/index.html#conjecture-refuted","scope":"Analysis/complex-analysis/entire-functions/r-nyi-vincze-characterization-of-the-exponential","status":"can_prove","summary":"There are no function $f : \\mathbb{C} \\to \\mathbb{C}$ and real numbers $a_0, a_1, a_2, \\dots$ such that $f$ is not the function $z \\mapsto c\\,e^z$ for any real $c > 0$; $a_n \\ge 0$ for every $n$; $a_n \\ne 0$ for infinitely many $n$; for every $z \\in \\mathbb{C}$ the series $\\sum_{n \\ge 0} a_n z^n$ has sum $f(z)$; $f$ is entire; and either $\\int_0^\\infty a_n x^n / f(x)\\,dx = 1$ for every $n \\ge 0$, or\u2026","title":"No non-exponential solution of the R\u00e9nyi\u2013Vincze conditions"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi.json","id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"can_prove","title":"Spherical area and valence of meromorphic functions"},{"id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/root-count","kind":"node","lean":"MathlibExt.Analysis.HaymanSphericalAreaRatioWanted.rootCount","parent":16460,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html#root-count","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"fully_proved","summary":"For $f\\colon \\mathbb{C} \\to \\mathbb{C}$, $a \\in \\mathbb{C}$ and real $r$, the root count of $f$ at $a$ in radius $r$ is the number of roots of $f(z) = a$ in the open disk $|z| < r$, counted with multiplicity: the sum over $z \\in \\mathbb{C}$ of the positive part of the divisor of $f - a$ on that disk at $z$ (the sum is taken to be $0$ if infinitely many terms are nonzero).","title":"Root count"},{"id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/n","kind":"node","lean":"MathlibExt.Analysis.HaymanSphericalAreaRatioWanted.n","parent":16460,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html#n","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"fully_proved","summary":"For $f\\colon \\mathbb{C} \\to \\mathbb{C}$ and real $r$, $n(f, r)$ is the supremum in $\\mathbb{R}$, over all $a \\in \\mathbb{C}$, of the root count of $f$ at $a$ in radius $r$, that is, of the number of roots of $f(z) = a$ in $|z| < r$ counted with multiplicity.","title":"Maximum valence"},{"id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/spherical-density","kind":"node","lean":"MathlibExt.Analysis.HaymanSphericalAreaRatioWanted.sphericalDensity","parent":16460,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html#spherical-density","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"fully_proved","summary":"For $f\\colon \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$, the spherical density of $f$ at $z$ is $|f'(z)|^2 / (1 + |f(z)|^2)^2$, where $f'(z)$ is the complex derivative of $f$ at $z$, taken to be $0$ where $f$ is not differentiable.","title":"Spherical density"},{"id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/a","kind":"node","lean":"MathlibExt.Analysis.HaymanSphericalAreaRatioWanted.A","parent":16460,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html#a","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"fully_proved","summary":"For $f\\colon \\mathbb{C} \\to \\mathbb{C}$ and real $r$, the normalized spherical area $A(f, r)$ of $f$ on the disk $|z| < r$ is $\\frac{1}{\\pi}$ times the Lebesgue integral of the spherical density of $f$ over the open disk $\\{z \\in \\mathbb{C} : |z| < r\\}$ (the integral is taken to be $0$ if the density is not integrable there).","title":"Normalized spherical area"},{"id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/is-nonconstant","kind":"node","lean":"MathlibExt.Analysis.HaymanSphericalAreaRatioWanted.IsNonconstant","parent":16460,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html#is-nonconstant","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"fully_proved","summary":"A function $f\\colon \\mathbb{C} \\to \\mathbb{C}$ is nonconstant if there are points $u, v \\in \\mathbb{C}$ at which $f$ is complex differentiable and $f(u) \\ne f(v)$.","title":"Germ-wise nonconstant function"},{"id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/has-bounded-root-counts","kind":"node","lean":"MathlibExt.Analysis.HaymanSphericalAreaRatioWanted.HasBoundedRootCounts","parent":16460,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html#has-bounded-root-counts","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"fully_proved","summary":"A function $f\\colon \\mathbb{C} \\to \\mathbb{C}$ has bounded root counts if for every real $r$ the root counts of $f$ in radius $r$, that is, the numbers of roots of $f(z) = a$ in $|z| < r$ counted with multiplicity, are bounded above as $a$ ranges over $\\mathbb{C}$.","title":"Bounded root counts"},{"id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/conjecture","kind":"node","lean":"MathlibExt.Analysis.HaymanSphericalAreaRatioWanted.conjecture","parent":16460,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html#conjecture","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"fully_proved","summary":"The statement that every function $f\\colon \\mathbb{C} \\to \\mathbb{C}$ that is meromorphic on all of $\\mathbb{C}$ and nonconstant (complex differentiable at two points where it takes different values) has bounded root counts (for every real $r$, the numbers of roots of $f(z) = a$ in $|z| < r$, counted with multiplicity, are bounded above as $a$ ranges over $\\mathbb{C}$) and satisfies","title":"Hayman's spherical-area ratio conjecture"},{"id":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/conjecture-refuted","kind":"node","lean":"MathlibExt.Analysis.HaymanSphericalAreaRatioWanted.conjecture_refuted","parent":16460,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi/index.html#conjecture-refuted","scope":"Analysis/complex-analysis/entire-functions/spherical-area-and-valence-of-meromorphic-functi","status":"can_prove","summary":"Not every function $f\\colon \\mathbb{C} \\to \\mathbb{C}$ that is meromorphic on all of $\\mathbb{C}$ and nonconstant (complex differentiable at two points where it takes different values) has bounded root counts and satisfies $\\liminf_{r \\to \\infty} n(f, r)/A(f, r) = 1$, where $n(f, r)$ is the maximum valence and $A(f, r)$ the normalized spherical area of $f$ on the disk $|z| < r$.","title":"Hayman's spherical-area ratio conjecture is false"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem.json","id":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/index.html","scope":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem","status":"can_prove","title":"Tur\u00e1n's three-value problem"},{"id":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/root-mult","kind":"node","lean":"MathlibExt.Analysis.TuranThreeValueBoundedDistributionWanted.rootMult","parent":16469,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplicity of a root of "],["math","\\(f = w\\)"]],"source":"Multiplicity of a root of $f = w$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/index.html#root-mult","scope":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ and $w, z \\in \\mathbb{C}$. The multiplicity of $z$ as a root of the equation $f = w$ is an extended natural number: it is $\\infty$ if $f(t) = w$ for every $t \\neq z$ in some neighbourhood of $z$; otherwise it is $n$ if $f - w$ has meromorphic order $n \\ge 0$ at $z$, and it is $0$ if that order is negative or $f - w$ is not meromorphic at $z$.","title":"Multiplicity of a root of $f = w$"},{"id":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/pole-mult","kind":"node","lean":"MathlibExt.Analysis.TuranThreeValueBoundedDistributionWanted.poleMult","parent":16469,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/index.html#pole-mult","scope":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$. The multiplicity of $z$ as a pole of $f$, that is, the multiplicity of the value $\\infty$ at $z$, is $-n$ if $f$ has finite meromorphic order $n < 0$ at $z$, and $0$ otherwise. In particular it is $0$ unless $f$ has a pole at $z$.","title":"Multiplicity of a pole"},{"id":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/root-mult-sph","kind":"node","lean":"MathlibExt.Analysis.TuranThreeValueBoundedDistributionWanted.rootMultSph","parent":16469,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/index.html#root-mult-sph","scope":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$, and let $w$ be a point of the Riemann sphere $\\mathbb{C} \\cup \\{\\infty\\}$. The multiplicity of $w$ as a value of $f$ at $z$ is the multiplicity of $z$ as a root of $f = w$ when $w$ is finite, and the multiplicity of $z$ as a pole of $f$ when $w = \\infty$.","title":"Multiplicity of a value on the Riemann sphere"},{"id":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/is-bounded-value-distribution","kind":"node","lean":"MathlibExt.Analysis.TuranThreeValueBoundedDistributionWanted.IsBoundedValueDistribution","parent":16469,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/index.html#is-bounded-value-distribution","scope":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ has bounded value distribution if for every real $r > 0$ there is a natural number $C$ such that, for every value $w$ of the Riemann sphere $\\mathbb{C} \\cup \\{\\infty\\}$ and every open disc of radius $r$, the multiplicities of $w$ as a value of $f$ at the points of any finite subset of the disc add up to at most $C$.","title":"Bounded value distribution"},{"id":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/conjecture","kind":"node","lean":"MathlibExt.Analysis.TuranThreeValueBoundedDistributionWanted.conjecture","parent":16469,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/index.html#conjecture","scope":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem","status":"fully_proved","summary":"The statement that every function $f : \\mathbb{C} \\to \\mathbb{C}$ with the following properties has bounded value distribution: $f$ is meromorphic on $\\mathbb{C}$; for every $w \\in \\mathbb{C}$ there is a point at which $f - w$ has finite meromorphic order; and there are three distinct values $w_1, w_2, w_3$ of the Riemann sphere $\\mathbb{C} \\cup \\{\\infty\\}$ such that for every real $r > 0$ there is a natural number\u2026","title":"Tur\u00e1n's three-value question"},{"id":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/conjecture-refuted","kind":"node","lean":"MathlibExt.Analysis.TuranThreeValueBoundedDistributionWanted.conjecture_refuted","parent":16469,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem/index.html#conjecture-refuted","scope":"Analysis/complex-analysis/entire-functions/tur-n-s-three-value-problem","status":"can_prove","summary":"There is a function $f : \\mathbb{C} \\to \\mathbb{C}$, meromorphic on $\\mathbb{C}$, such that for every $w \\in \\mathbb{C}$ there is a point at which $f - w$ has finite meromorphic order, together with three distinct values $w_1, w_2, w_3$ of the Riemann sphere $\\mathbb{C} \\cup \\{\\infty\\}$ such that for every real $r > 0$ there is a natural number $C$ for which, for each $i$ and every open disc of radius $r$, the\u2026","title":"Tur\u00e1n's three-value question has a negative answer"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/wiman-valiron-theory.json","id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","title":"Wiman\u2013Valiron theory"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/power-growth-bad-set","kind":"node","lean":"Real.powerGrowthBadSet","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#power-growth-bad-set","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For a function $h : \\mathbb{R} \\to \\mathbb{R}$ and real numbers $p$ and $x_0$, this is the set of real numbers $x \\ge x_0$ such that $h(x)^p \\le h'(x)$, where $h(x)^p$ is the real power and $h'(x)$ is the derivative of $h$ at $x$, taken to be $0$ where $h$ is not differentiable.","title":"Power-growth exceptional set"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/volume-power-growth-bad-set-lt-top","kind":"node","lean":"Real.volume_powerGrowthBadSet_lt_top","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#volume-power-growth-bad-set-lt-top","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $p > 1$ and $x_0$ be real numbers, and let $h : \\mathbb{R} \\to \\mathbb{R}$ be continuously differentiable, with $h(x) > 0$ for all $x \\ge x_0$ and $h$ nondecreasing on $[x_0, \\infty)$. Then the power-growth exceptional set of $h$, the set of $x \\ge x_0$ with $h(x)^p \\le h'(x)$, has finite Lebesgue measure.","title":"The power-growth exceptional set has finite measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-taylor-coefficient","kind":"node","lean":"Complex.wimanTaylorCoefficient","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-taylor-coefficient","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ and $n \\in \\mathbb{N}$, the $n$th Taylor coefficient of $f$ at the origin is $a_n = f^{(n)}(0) / n!$, where $f^{(n)}$ is the $n$th iterated complex derivative of $f$.","title":"Wiman Taylor coefficient"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-term","kind":"node","lean":"Complex.wimanTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$, a real number $r$ and a natural number $n$, the Wiman term of $f$ at radius $r$ and index $n$ is the real number $|a_n| \\, |r|^n$, where $a_n = f^{(n)}(0) / n!$ is the $n$th Taylor coefficient of $f$ at the origin. It is the modulus of the $n$th Taylor term of $f$ on the circle of radius $|r|$.","title":"Wiman term"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-term","kind":"node","lean":"Complex.wimanMaximumTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ and a real number $r$, the Wiman maximum term $\\mu(f, r)$ is the supremum over $n \\in \\mathbb{N}$ of the Wiman terms $|a_n| \\, |r|^n$, where $a_n = f^{(n)}(0) / n!$ is the $n$th Wiman Taylor coefficient of $f$.","title":"Wiman maximum term"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-modulus","kind":"node","lean":"Complex.wimanMaximumModulus","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-modulus","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ and a real number $r$, the Wiman maximum modulus $M(f, r)$ is the supremum of $|f(z)|$ over all $z \\in \\mathbb{C}$ with $|z| = |r|$.","title":"Wiman maximum modulus"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-term-abs","kind":"node","lean":"Complex.wimanTerm_abs","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-term-abs","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every function $f : \\mathbb{C} \\to \\mathbb{C}$, every real number $r$ and every natural number $n$, the Wiman term of $f$ at radius $|r|$ and index $n$ equals the Wiman term of $f$ at radius $r$ and index $n$.","title":"Wiman term at the absolute value of the radius"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-term-abs","kind":"node","lean":"Complex.wimanMaximumTerm_abs","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-term-abs","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every function $f : \\mathbb{C} \\to \\mathbb{C}$ and every real number $r$, the Wiman maximum term of $f$ at radius $|r|$ equals the Wiman maximum term of $f$ at radius $r$.","title":"Wiman maximum term at the absolute value of the radius"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-modulus-abs","kind":"node","lean":"Complex.wimanMaximumModulus_abs","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-modulus-abs","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every function $f : \\mathbb{C} \\to \\mathbb{C}$ and every real number $r$, the Wiman maximum modulus of $f$ at radius $|r|$ equals the Wiman maximum modulus of $f$ at radius $r$.","title":"Wiman maximum modulus at the absolute value of the radius"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-term-neg","kind":"node","lean":"Complex.wimanTerm_neg","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-term-neg","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every function $f : \\mathbb{C} \\to \\mathbb{C}$, every real number $r$ and every natural number $n$, the Wiman term of $f$ at radius $-r$ and index $n$ equals the Wiman term of $f$ at radius $r$ and index $n$.","title":"Wiman term at a negated radius"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-term-neg","kind":"node","lean":"Complex.wimanMaximumTerm_neg","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-term-neg","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every function $f : \\mathbb{C} \\to \\mathbb{C}$ and every real number $r$, the Wiman maximum term of $f$ at radius $-r$ equals the Wiman maximum term of $f$ at radius $r$.","title":"The Wiman maximum term is even in the radius"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-modulus-neg","kind":"node","lean":"Complex.wimanMaximumModulus_neg","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-modulus-neg","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every function $f : \\mathbb{C} \\to \\mathbb{C}$ and every real number $r$, the Wiman maximum modulus of $f$ at radius $-r$ equals the Wiman maximum modulus of $f$ at radius $r$.","title":"The Wiman maximum modulus is even in the radius"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/is-transcendental","kind":"node","lean":"Complex.IsTranscendental","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#is-transcendental","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ is transcendental when there is no complex polynomial $p$ with $f(z) = p(z)$ for all $z \\in \\mathbb{C}$.","title":"Transcendental functions"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/norm-le-wiman-maximum-modulus","kind":"node","lean":"Complex.norm_le_wimanMaximumModulus","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#norm-le-wiman-maximum-modulus","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be continuous, let $r$ be a real number and let $z \\in \\mathbb{C}$ with $|z| = r$. Then $|f(z)| \\le M(f, r)$, where $M(f, r)$ is the Wiman maximum modulus of $f$ at radius $r$.","title":"Values on a circle are bounded by the maximum modulus"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/cube-le-cube-mul-log-sq-of-wiman","kind":"node","lean":"Complex.cube_le_cube_mul_log_sq_of_wiman","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#cube-le-cube-mul-log-sq-of-wiman","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $M$ and $\\mu$ be real numbers with $M \\ge 0$, $\\mu \\ge 0$ and $\\log \\mu \\ge 0$, where $\\log$ is the real logarithm with $\\log 0 = 0$. If $M \\le \\mu (\\log \\mu)^{2/3}$, then $M^3 \\le \\mu^3 (\\log \\mu)^2$.","title":"Cubing a Wiman 2/3 bound"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/eventually-absorb-corrected-wiman-expone","kind":"node","lean":"Complex.eventually_absorb_corrected_wiman_exponent","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#eventually-absorb-corrected-wiman-expone","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every real $C > 0$ there is a real $U > 1$ such that for all reals $\\mu$ and $A$ with $U \\le \\mu \\le A$ and $A \\le C \\mu \\bigl(1 + (\\log A)^{81/128}\\bigr)$, we have $A \\le \\mu (\\log \\mu)^{2/3}$.","title":"Absorbing constants into the exponent gap"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/exists-mem-sphere-eq-wiman-maximum-modul","kind":"node","lean":"Complex.exists_mem_sphere_eq_wimanMaximumModulus","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#exists-mem-sphere-eq-wiman-maximum-modul","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function and let $r \\ge 0$. Then there is a point $z$ with $|z| = r$ such that $|f(w)| \\le |f(z)|$ for every $w$ with $|w| = r$, and the maximum modulus $M(r) = \\sup_{|w| = r} |f(w)|$ equals $|f(z)|$.","title":"The maximum modulus is attained"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-majorant","kind":"node","lean":"Complex.wimanMajorant","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-majorant","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ with Taylor coefficients $a_n = f^{(n)}(0) / n!$ at the origin and a real number $r$, the radial majorant of $f$ at $r$ is $M_f(r) = \\sum_{n \\ge 0} |a_n| \\, |r|^n$, the sum of the radial Taylor terms at radius $|r|$. By convention $M_f(r) = 0$ when this series diverges.","title":"Radial majorant"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-term-nonneg","kind":"node","lean":"Complex.wimanTerm_nonneg","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-term-nonneg","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every function $f : \\mathbb{C} \\to \\mathbb{C}$ with Taylor coefficients $a_n = f^{(n)}(0) / n!$ at the origin, every real $r$ and every $n \\in \\mathbb{N}$, we have $0 \\le |a_n| \\, |r|^n$.","title":"Radial Taylor terms are nonnegative"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/summable-wiman-term","kind":"node","lean":"Complex.summable_wimanTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#summable-wiman-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n = f^{(n)}(0) / n!$ at the origin. For every real $r$, the series $\\sum_{n \\ge 0} |a_n| \\, |r|^n$ converges.","title":"Summability of the radial Taylor terms"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-majorant-nonneg","kind":"node","lean":"Complex.wimanMajorant_nonneg","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-majorant-nonneg","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For every function $f : \\mathbb{C} \\to \\mathbb{C}$ with Taylor coefficients $a_n = f^{(n)}(0) / n!$ at the origin and every real number $r$, the radial majorant $M_f(r) = \\sum_{n \\ge 0} |a_n| \\, |r|^n$ (taken to be $0$ when the series diverges) satisfies $0 \\le M_f(r)$.","title":"The radial majorant is nonnegative"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-term-le-wiman-majorant","kind":"node","lean":"Complex.wimanTerm_le_wimanMajorant","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-term-le-wiman-majorant","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n = f^{(n)}(0) / n!$ at the origin, and let $M_f(r) = \\sum_{k \\ge 0} |a_k| \\, |r|^k$ be its radial majorant. For every real $r$ and every $n \\in \\mathbb{N}$, $|a_n| \\, |r|^n \\le M_f(r)$.","title":"Each radial Taylor term is at most the radial majorant"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-growth","kind":"node","lean":"Complex.wimanGrowth","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-growth","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ have Taylor coefficients $a_n = f^{(n)}(0) / n!$ at the origin, and let $M_f(r) = \\sum_{n \\ge 0} |a_n| \\, |r|^n$ be its radial majorant, taken to be $0$ when the series diverges. For a real number $x$, the Wiman growth of $f$ at $x$ is $\\log M_f(e^x)$, the logarithm of the radial majorant at radius $e^x$, with the convention $\\log 0 = 0$.","title":"Wiman growth function"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-coefficient-measure","kind":"node","lean":"Complex.wimanCoefficientMeasure","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-coefficient-measure","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For $f : \\mathbb{C} \\to \\mathbb{C}$ with Taylor coefficients $a_n = f^{(n)}(0)/n!$ at $0$, the Wiman coefficient measure of $f$ is the measure $\\mu_f = \\sum_{n \\ge 0} |a_n|\\,\\delta_n$ on $\\mathbb{N}$, which gives each $n$ mass $|a_n|$.","title":"Wiman coefficient measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-tilted-measure","kind":"node","lean":"Complex.wimanTiltedMeasure","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-tilted-measure","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ have Taylor coefficients $a_n = f^{(n)}(0)/n!$, and let $\\mu_f$ be its coefficient measure on $\\mathbb{N}$, with mass $|a_n|$ at $n$. For a real number $x$, the tilted coefficient measure $\\mu_{f,x}$ is the exponential tilt of $\\mu_f$ by $n \\mapsto xn$: the measure with density $e^{xn} / \\int e^{xm}\\,d\\mu_f(m)$ with respect to $\\mu_f$.","title":"Tilted coefficient measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-coefficient-measure-singleton","kind":"node","lean":"Complex.wimanCoefficientMeasure_singleton","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-coefficient-measure-singleton","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ and $a_n = f^{(n)}(0)/n!$. For every $n \\in \\mathbb{N}$, the Wiman coefficient measure $\\mu_f$ of $f$ gives the singleton $\\{n\\}$ mass $\\mu_f(\\{n\\}) = |a_n|$.","title":"Point masses of the Wiman coefficient measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/integrable-exp-mul-wiman-coefficient-mea","kind":"node","lean":"Complex.integrable_exp_mul_wimanCoefficientMeasure","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#integrable-exp-mul-wiman-coefficient-mea","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function and let $\\mu_f$ be its Wiman coefficient measure on $\\mathbb{N}$, which gives each $n \\in \\mathbb{N}$ mass $|a_n|$, where $a_n = f^{(n)}(0)/n!$. For every real $x$, the function $n \\mapsto e^{xn}$ is integrable with respect to $\\mu_f$.","title":"Exponential moments of the Wiman coefficient measure are finite"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/integrable-exp-set-wiman-coefficient-mea","kind":"node","lean":"Complex.integrableExpSet_wimanCoefficientMeasure_eq_univ","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#integrable-exp-set-wiman-coefficient-mea","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function and let $\\mu_f$ be its Wiman coefficient measure on $\\mathbb{N}$. The set of real $t$ for which $n \\mapsto e^{tn}$ is integrable with respect to $\\mu_f$ is all of $\\mathbb{R}$.","title":"The exponential-integrability set of the Wiman coefficient measure is the whole line"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/mem-interior-integrable-exp-set-wiman-co","kind":"node","lean":"Complex.mem_interior_integrableExpSet_wimanCoefficientMeasure","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#mem-interior-integrable-exp-set-wiman-co","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function and let $\\mu_f$ be its Wiman coefficient measure on $\\mathbb{N}$. Every real $x$ lies in the interior of the set of real $t$ for which $n \\mapsto e^{tn}$ is integrable with respect to $\\mu_f$.","title":"Every real parameter is interior to the exponential-integrability set"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/integral-exp-mul-wiman-coefficient-measu","kind":"node","lean":"Complex.integral_exp_mul_wimanCoefficientMeasure","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#integral-exp-mul-wiman-coefficient-measu","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n = f^{(n)}(0)/n!$ at $0$, and let $\\mu_f$ be its Wiman coefficient measure on $\\mathbb{N}$. For every real $x$,","title":"Exponential moments of the Wiman coefficient measure equal the Wiman majorant"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-mgf-eq-majorant","kind":"node","lean":"Complex.wiman_mgf_eq_majorant","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-mgf-eq-majorant","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n = f^{(n)}(0)/n!$, and let $\\mu_f$ be its coefficient measure on $\\mathbb{N}$, which has mass $|a_n|$ at $n$. For every real $x$, the moment generating function of $n \\mapsto n$ under $\\mu_f$ at $x$, namely $\\int e^{xn}\\,d\\mu_f(n)$, equals the radial majorant $M_f(e^x)$, where $M_f(r) = \\sum_n |a_n|\\,|r|^n$.","title":"Moment generating function of the coefficient measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-cgf-eq-growth","kind":"node","lean":"Complex.wiman_cgf_eq_growth","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-cgf-eq-growth","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function and let $\\mu_f$ be its Wiman coefficient measure on $\\mathbb{N}$. For every real $x$, the cumulant generating function of $n \\mapsto n$ under $\\mu_f$ at $x$, namely $\\log \\int e^{xn}\\,d\\mu_f(n)$, equals the Wiman growth $G_f(x) = \\log M_f(e^x)$, where $M_f$ is the Wiman majorant of $f$.","title":"The Wiman growth is the cumulant generating function of the coefficient measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-coefficient-measure-ne-zero","kind":"node","lean":"Complex.wimanCoefficientMeasure_ne_zero","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-coefficient-measure-ne-zero","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ and $a_n = f^{(n)}(0)/n!$. If $a_n \\ne 0$ for some $n$, then the Wiman coefficient measure $\\mu_f$ of $f$ is not the zero measure.","title":"A nonzero Taylor coefficient makes the Wiman coefficient measure nonzero"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/is-probability-measure-wiman-tilted-meas","kind":"node","lean":"Complex.isProbabilityMeasure_wimanTiltedMeasure","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#is-probability-measure-wiman-tilted-meas","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function such that $f^{(n)}(0)/n! \\ne 0$ for some $n$, and let $\\mu_f$ be its Wiman coefficient measure on $\\mathbb{N}$. For every real $x$, the Wiman tilted measure of $f$ at $x$, which has density $n \\mapsto e^{xn} / \\int e^{xm}\\,d\\mu_f(m)$ with respect to $\\mu_f$, is a probability measure.","title":"The Wiman tilted measure is a probability measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-tilted-measure-singleton","kind":"node","lean":"Complex.wimanTiltedMeasure_singleton","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-tilted-measure-singleton","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n = f^{(n)}(0)/n!$, radial majorant $M_f(r) = \\sum_n |a_n|\\,|r|^n$ and Taylor terms $T_n(r) = |a_n|\\,|r|^n$. For every real $x$ and every $n \\in \\mathbb{N}$, the tilted coefficient measure $\\mu_{f,x}$ gives the singleton $\\{n\\}$ mass $T_n(e^x) / M_f(e^x)$ (with the convention $t/0 = 0$).","title":"Atoms of the tilted coefficient measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-tilted-measure-eq-sum","kind":"node","lean":"Complex.wimanTiltedMeasure_eq_sum","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-tilted-measure-eq-sum","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n = f^{(n)}(0)/n!$, radial majorant $M_f(r) = \\sum_n |a_n|\\,|r|^n$ and Taylor terms $T_n(r) = |a_n|\\,|r|^n$. For every real $x$, the tilted coefficient measure $\\mu_{f,x}$ equals $\\sum_{n \\in \\mathbb{N}} \\frac{T_n(e^x)}{M_f(e^x)}\\,\\delta_n$, the measure on $\\mathbb{N}$ with mass $T_n(e^x) / M_f(e^x)$ at each $n$ (with the convention $t/0 = 0$).","title":"Atomic form of the tilted coefficient measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/cont-diff-two-wiman-growth","kind":"node","lean":"Complex.contDiff_two_wimanGrowth","parent":16476,"reading_title":{"format":"inline-parts-v1","parts":[["text","The logarithmic majorant is "],["math","\\(C^2\\)"]],"source":"The logarithmic majorant is $C^2$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#cont-diff-two-wiman-growth","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n$ at the origin, and let $\\Phi(x) = \\log \\sum_{n \\ge 0} |a_n| e^{nx}$ be the logarithm of its radial Taylor majorant at radius $e^x$ (with $\\log 0 = 0$). Then $\\Phi$ is twice continuously differentiable on $\\mathbb{R}$.","title":"The logarithmic majorant is $C^2$"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/deriv-wiman-growth-eq-expectation","kind":"node","lean":"Complex.deriv_wimanGrowth_eq_expectation","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#deriv-wiman-growth-eq-expectation","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function whose Taylor coefficients $a_n$ at the origin are not all zero, and let $\\Phi(x) = \\log \\sum_{n \\ge 0} |a_n| e^{nx}$. For real $x$, let $\\nu_x$ be the coefficient measure tilted at $x$, the probability measure on $\\mathbb{N}$ with $\\nu_x(\\{n\\}) = |a_n| e^{nx} / \\sum_{m \\ge 0} |a_m| e^{mx}$. Then $\\Phi'(x) = \\int n \\, d\\nu_x(n)$, the mean of $n$ under $\\nu_x$.","title":"Derivative of the logarithmic majorant as a mean"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/iterated-deriv-two-wiman-growth-eq-varia","kind":"node","lean":"Complex.iteratedDeriv_two_wimanGrowth_eq_variance","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#iterated-deriv-two-wiman-growth-eq-varia","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function whose Taylor coefficients $a_n$ at the origin are not all zero, and let $\\Phi(x) = \\log \\sum_{n \\ge 0} |a_n| e^{nx}$. For real $x$, let $\\nu_x$ be the coefficient measure tilted at $x$, the probability measure on $\\mathbb{N}$ with $\\nu_x(\\{n\\}) = |a_n| e^{nx} / \\sum_{m \\ge 0} |a_m| e^{mx}$. Then $\\Phi''(x)$ is the variance of $n$ under $\\nu_x$.","title":"Second derivative of the logarithmic majorant as a variance"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/iterated-deriv-two-wiman-growth-nonneg","kind":"node","lean":"Complex.iteratedDeriv_two_wimanGrowth_nonneg","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#iterated-deriv-two-wiman-growth-nonneg","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function whose Taylor coefficients $a_n$ at the origin are not all zero, and let $\\Phi(x) = \\log \\sum_{n \\ge 0} |a_n| e^{nx}$. Then $\\Phi''(x) \\ge 0$ for every real $x$.","title":"The logarithmic majorant has nonnegative second derivative"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/bdd-above-range-wiman-term","kind":"node","lean":"Complex.bddAbove_range_wimanTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#bdd-above-range-wiman-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n$ at the origin, and let $r$ be real. Then the set of Taylor terms $\\{|a_n|\\,|r|^n : n \\in \\mathbb{N}\\}$ is bounded above.","title":"The Taylor terms are bounded above"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-term-le-wiman-maximum-term","kind":"node","lean":"Complex.wimanTerm_le_wimanMaximumTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-term-le-wiman-maximum-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n$ at the origin and maximum term $\\mu(r) = \\sup_k |a_k|\\,|r|^k$. For every real $r$ and every $n \\in \\mathbb{N}$, $|a_n|\\,|r|^n \\le \\mu(r)$.","title":"Each Taylor term is at most the maximum term"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-term-le-iff","kind":"node","lean":"Complex.wimanMaximumTerm_le_iff","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-term-le-iff","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n$ at the origin and maximum term $\\mu(r) = \\sup_n |a_n|\\,|r|^n$. For all real $r$ and $c$, $\\mu(r) \\le c$ if and only if $|a_n|\\,|r|^n \\le c$ for every $n \\in \\mathbb{N}$.","title":"Bounding the maximum term"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/exists-wiman-term-eq-wiman-maximum-term","kind":"node","lean":"Complex.exists_wimanTerm_eq_wimanMaximumTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#exists-wiman-term-eq-wiman-maximum-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n$ at the origin and maximum term $\\mu(r) = \\sup_n |a_n|\\,|r|^n$. For every real $r$ there is an $n \\in \\mathbb{N}$ with $|a_n|\\,|r|^n = \\mu(r)$.","title":"The maximum term is attained by a Taylor term"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-majorant-le-eight-mul-maximum-term","kind":"node","lean":"Complex.wimanMajorant_le_eight_mul_maximumTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-majorant-le-eight-mul-maximum-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n = f^{(n)}(0)/n!$, radial majorant $M_f(r) = \\sum_n |a_n|\\,|r|^n$, maximum term $\\mu_f(r) = \\sup_n |a_n|\\,|r|^n$ and logarithmic majorant $g_f(x) = \\log M_f(e^x)$. For every real $x$, $M_f(e^x) \\le 8\\,\\mu_f(e^x)\\,\\bigl(1 + \\sqrt{g_f''(x)}\\bigr)$.","title":"Corrected Rosenbloom estimate"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-term-le-wiman-maximum-modu","kind":"node","lean":"Complex.wimanMaximumTerm_le_wimanMaximumModulus","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-term-le-wiman-maximum-modu","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n$ at the origin, maximum term $\\mu(r) = \\sup_n |a_n|\\,|r|^n$ and maximum modulus $M(r) = \\sup_{|z| = |r|} |f(z)|$. Then $\\mu(r) \\le M(r)$ for every real $r > 0$.","title":"Cauchy's inequality for the maximum term"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/has-sum-wiman-taylor-coefficient-mul-pow","kind":"node","lean":"Complex.hasSum_wimanTaylorCoefficient_mul_pow","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#has-sum-wiman-taylor-coefficient-mul-pow","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n = f^{(n)}(0) / n!$ at the origin. For every $z \\in \\mathbb{C}$, the series $\\sum_{n \\ge 0} a_n z^n$ converges unconditionally to $f(z)$.","title":"Taylor expansion of an entire function"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/infinite-support-wiman-taylor-coefficien","kind":"node","lean":"Complex.infinite_support_wimanTaylorCoefficient","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#infinite-support-wiman-taylor-coefficien","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"A transcendental entire function has infinitely many nonzero Taylor coefficients.","title":"Transcendental entire functions have infinitely many Taylor coefficients"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-maximum-modulus-le-wiman-majorant","kind":"node","lean":"Complex.wimanMaximumModulus_le_wimanMajorant","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-maximum-modulus-le-wiman-majorant","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function with Taylor coefficients $a_n$ at the origin and let $r \\ge 0$. Then the maximum modulus $M(r) = \\sup_{|z| = r} |f(z)|$ satisfies $M(r) \\le \\sum_{n \\ge 0} |a_n| r^n$.","title":"The maximum modulus is at most the radial Taylor majorant"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/tendsto-wiman-maximum-term-at-top","kind":"node","lean":"Complex.tendsto_wimanMaximumTerm_atTop","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#tendsto-wiman-maximum-term-at-top","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function that is not a polynomial function, with Taylor coefficients $a_n$ at the origin and maximum term $\\mu(r) = \\sup_n |a_n|\\,|r|^n$. Then $\\mu(r) \\to \\infty$ as $r \\to \\infty$.","title":"The maximum term of a transcendental entire function tends to infinity"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/eventually-one-lt-wiman-maximum-term","kind":"node","lean":"Complex.eventually_one_lt_wimanMaximumTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#eventually-one-lt-wiman-maximum-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function that is not a polynomial function, with Taylor coefficients $a_n$ at the origin and maximum term $\\mu(r) = \\sup_n |a_n|\\,|r|^n$. Then there is $R$ such that $\\mu(r) > 1$ for all real $r \\ge R$.","title":"The maximum term eventually exceeds one"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/eventually-pos-log-wiman-maximum-term","kind":"node","lean":"Complex.eventually_pos_log_wimanMaximumTerm","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#eventually-pos-log-wiman-maximum-term","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be an entire function that is not a polynomial function, with Taylor coefficients $a_n$ at the origin and maximum term $\\mu(r) = \\sup_n |a_n|\\,|r|^n$. Then there is $R$ such that $\\log \\mu(r) > 0$ for all real $r \\ge R$.","title":"The logarithm of the maximum term is eventually positive"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/eventually-wiman-growth-regular","kind":"node","lean":"Complex.eventually_wimanGrowth_regular","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#eventually-wiman-growth-regular","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be a transcendental entire function with Taylor coefficients $a_n$ at the origin, and let $\\Phi(x) = \\log \\sum_{n \\ge 0} |a_n| e^{nx}$. Then $\\Phi$ is twice continuously differentiable on $\\mathbb{R}$, $\\Phi'(x) \\to \\infty$ as $x \\to \\infty$, and there is a real $x_0$ such that $\\Phi(x) > 0$ for every $x \\ge x_0$ and both $\\Phi$ and $\\Phi'$ are nondecreasing on $[x_0, \\infty)$.","title":"Eventual regularity of the logarithmic majorant"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-exceptional-set","kind":"node","lean":"Complex.wimanExceptionalSet","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-exceptional-set","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ with radial majorant $M_f$, let $g(x) = \\log M_f(e^x)$. For $x_0 \\in \\mathbb{R}$, the Wiman exceptional set $E_f(x_0)$ is the union of the power-growth bad sets of $g$ and of $g'$ with exponent $9/8$ above $x_0$, that is, the set of $x \\ge x_0$ with $g(x)^{9/8} \\le g'(x)$ or $g'(x)^{9/8} \\le g''(x)$.","title":"Wiman exceptional set"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/exists-finite-wiman-exceptional-set","kind":"node","lean":"Complex.exists_finite_wimanExceptionalSet","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#exists-finite-wiman-exceptional-set","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be a transcendental entire function, let $M_f$ be its radial majorant and let $g(x) = \\log M_f(e^x)$. For $x_0 \\in \\mathbb{R}$ let $E_f(x_0)$ be the Wiman exceptional set, the set of $x \\ge x_0$ with $g(x)^{9/8} \\le g'(x)$ or $g'(x)^{9/8} \\le g''(x)$. There is $x_0 \\in \\mathbb{R}$ such that $E_f(x_0)$ has finite Lebesgue measure and $g''(x) \\le g(x)^{81/64}$ for every $x \\ge x_0$ outside $E_f(x_0)$.","title":"The Wiman exceptional set has finite measure"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/wiman-majorant-bound-off-exceptional-set","kind":"node","lean":"Complex.wimanMajorant_bound_off_exceptionalSet","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#wiman-majorant-bound-off-exceptional-set","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be a transcendental entire function with Taylor coefficients $a_n = f^{(n)}(0)/n!$, radial majorant $M_f(r) = \\sum_n |a_n|\\,|r|^n$, maximum term $\\mu_f(r) = \\sup_n |a_n|\\,|r|^n$ and logarithmic majorant $g_f(x) = \\log M_f(e^x)$. There are reals $C > 0$ and $x_0$ such that the Wiman exceptional set $E_f(x_0)$ has finite Lebesgue measure and, for every real $x \\ge x_0$ with $x \\notin E_f(x_0)$,\u2026","title":"Majorant bound off an exceptional set"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/tendsto-wiman-maximum-term-exp-at-top","kind":"node","lean":"Complex.tendsto_wimanMaximumTerm_exp_atTop","parent":16476,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#tendsto-wiman-maximum-term-exp-at-top","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be a transcendental entire function with Taylor coefficients $a_n$ at the origin and maximum term $\\mu(r) = \\sup_{n \\ge 0} |a_n| |r|^n$. Then $\\mu(e^x) \\to \\infty$ as $x \\to \\infty$.","title":"The maximum term tends to infinity along logarithmic radii"},{"id":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory/exists-wiman-radius-two-thirds","kind":"node","lean":"Complex.exists_wiman_radius_two_thirds","parent":16476,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wiman's inequality with exponent "],["math","\\(2/3\\)"],["text"," at arbitrarily large radii"]],"source":"Wiman's inequality with exponent $2/3$ at arbitrarily large radii"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/wiman-valiron-theory/index.html#exists-wiman-radius-two-thirds","scope":"Analysis/complex-analysis/entire-functions/wiman-valiron-theory","status":"fully_proved","summary":"Let $f$ be a transcendental entire function with Taylor coefficients $a_n$ at the origin, maximum modulus $M(r) = \\sup_{|z| = |r|} |f(z)|$ and maximum term $\\mu(r) = \\sup_{n \\ge 0} |a_n| |r|^n$. Then for every real $R$ there is a real $r > R$ with $M(r) \\le \\mu(r) (\\log \\mu(r))^{2/3}$.","title":"Wiman's inequality with exponent $2/3$ at arbitrarily large radii"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua.json","id":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/index.html","scope":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua","status":"can_prove","title":"Growth of entire functions along continua"},{"id":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/entire","kind":"node","lean":"MathlibExt.Analysis.ContinuumLimsupConstantWanted.Entire","parent":16535,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/index.html#entire","scope":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ is entire when it is complex differentiable at every point of $\\mathbb{C}$.","title":"Entire functions"},{"id":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/recedes-to-inf","kind":"node","lean":"MathlibExt.Analysis.ContinuumLimsupConstantWanted.RecedesToInf","parent":16535,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/index.html#recedes-to-inf","scope":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua","status":"fully_proved","summary":"A set $\\Gamma \\subseteq \\mathbb{C}$ is a continuum receding to infinity when it is closed, connected and unbounded.","title":"Continua receding to infinity"},{"id":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/ratio","kind":"node","lean":"MathlibExt.Analysis.ContinuumLimsupConstantWanted.Ratio","parent":16535,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ratio of "],["math","\\(\\log |f|\\)"],["text"," to the logarithm of the maximum modulus"]],"source":"Ratio of $\\log |f|$ to the logarithm of the maximum modulus"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/index.html#ratio","scope":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ and a point $z \\in \\mathbb{C}$, this is the extended real number equal to $-\\infty$ if $f(z) = 0$, and otherwise to the real number $\\log |f(z)| / \\log M(f, |z|)$, where $M(f, r)$ is the Wiman maximum modulus of $f$, $\\log 0$ is taken to be $0$, and division by $0$ gives $0$.","title":"Ratio of $\\log |f|$ to the logarithm of the maximum modulus"},{"id":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/conjecture","kind":"node","lean":"MathlibExt.Analysis.ContinuumLimsupConstantWanted.conjecture","parent":16535,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound "],["math","\\(-1\\)"],["text"," for the log-modulus ratio along continua"]],"source":"Lower bound $-1$ for the log-modulus ratio along continua"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/index.html#conjecture","scope":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua","status":"fully_proved","summary":"Write $M(f, r)$ for the Wiman maximum modulus of a function $f : \\mathbb{C} \\to \\mathbb{C}$ at radius $r$. This is the statement that for every entire, transcendental function $f$ and every continuum $\\Gamma$ receding to infinity (a closed, connected, unbounded subset of $\\mathbb{C}$), the limit superior, as $|z| \\to \\infty$ with $z \\in \\Gamma$, of the extended real ratio $\\log |f(z)| / \\log M(f, |z|)$ is at least\u2026","title":"Lower bound $-1$ for the log-modulus ratio along continua"},{"id":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.ContinuumLimsupConstantWanted.conjecture_holds","parent":16535,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua/index.html#conjecture-holds","scope":"Analysis/complex-analysis/entire-functions/growth-of-entire-functions-along-continua","status":"can_prove","summary":"Write $M(f, r)$ for the Wiman maximum modulus of a function $f : \\mathbb{C} \\to \\mathbb{C}$ at radius $r$. For every entire, transcendental function $f$ and every closed, connected, unbounded set $\\Gamma \\subseteq \\mathbb{C}$, the limit superior, as $|z| \\to \\infty$ with $z \\in \\Gamma$, of the extended real ratio $\\log |f(z)| / \\log M(f, |z|)$ is at least $-1$. Here the ratio is taken to be $-\\infty$ where\u2026","title":"Hayman\u2013Kjellberg theorem on continua receding to infinity"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines.json","id":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/index.html","scope":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines","status":"can_prove","title":"Zeros and ones of entire functions on lines"},{"id":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/is-complex-line","kind":"node","lean":"MathlibExt.Analysis.EntireZerosOnesLinesWanted.IsComplexLine","parent":16541,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/index.html#is-complex-line","scope":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines","status":"fully_proved","summary":"A set $s \\subseteq \\mathbb{C}$ is a straight line if there are $a, v \\in \\mathbb{C}$ with $v \\ne 0$ such that $s = \\{a + tv : t \\in \\mathbb{R}\\}$.","title":"Straight lines in the complex plane"},{"id":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/zeros-on-line","kind":"node","lean":"MathlibExt.Analysis.EntireZerosOnesLinesWanted.ZerosOnLine","parent":16541,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/index.html#zeros-on-line","scope":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ has all its zeros in a set $l \\subseteq \\mathbb{C}$ if every $z \\in \\mathbb{C}$ with $f(z) = 0$ lies in $l$.","title":"Functions whose zeros lie in a given set"},{"id":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/ones-on-line","kind":"node","lean":"MathlibExt.Analysis.EntireZerosOnesLinesWanted.OnesOnLine","parent":16541,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/index.html#ones-on-line","scope":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ has all its ones in a set $m \\subseteq \\mathbb{C}$ if every $z \\in \\mathbb{C}$ with $f(z) = 1$ lies in $m$.","title":"Functions whose ones lie in a given set"},{"id":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/conjecture","kind":"node","lean":"MathlibExt.Analysis.EntireZerosOnesLinesWanted.conjecture","parent":16541,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/index.html#conjecture","scope":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines","status":"fully_proved","summary":"This is the statement that there exist an entire function $f : \\mathbb{C} \\to \\mathbb{C}$ and two distinct straight lines $l \\ne m$ in $\\mathbb{C}$ such that every zero of $f$ lies on $l$, every solution of $f(z) = 1$ lies on $m$, and $f$ has infinitely many zeros and infinitely many solutions of $f(z) = 1$. Here a straight line is a set $\\{a + tv : t \\in \\mathbb{R}\\}$ with $a, v \\in \\mathbb{C}$ and $v \\ne 0$.","title":"Zeros and ones of an entire function on two distinct lines"},{"id":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.EntireZerosOnesLinesWanted.conjecture_holds","parent":16541,"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines/index.html#conjecture-holds","scope":"Analysis/complex-analysis/entire-functions/zeros-and-ones-of-entire-functions-on-lines","status":"can_prove","summary":"There exist an entire function $f : \\mathbb{C} \\to \\mathbb{C}$ and two distinct straight lines $l \\ne m$ in $\\mathbb{C}$ such that every zero of $f$ lies on $l$, every solution of $f(z) = 1$ lies on $m$, and $f$ has infinitely many zeros and infinitely many solutions of $f(z) = 1$. Here a straight line is a set $\\{a + tv : t \\in \\mathbb{R}\\}$ with $a, v \\in \\mathbb{C}$ and $v \\ne 0$.","title":"Entire functions with zeros and ones on two distinct lines exist"},{"data_url":"scopes/Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n.json","id":"Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n","kind":"scope","navigation":"module","parent":16433,"result":false,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n/index.html","scope":"Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n","status":"can_prove","title":"Zeros of the entire function 1 + \u03a3 (z/n)\u207f"},{"id":"Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n/f","kind":"node","lean":"MathlibExt.Analysis.Complex.ZerosOfSumZDivNPowNWanted.F","parent":16547,"reading_title":{"format":"inline-parts-v1","parts":[["text","The entire function "],["math","\\(1+\\sum_{n\\ge 1}(z/n)^n\\)"]],"source":"The entire function $1+\\sum_{n\\ge 1}(z/n)^n$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n/index.html#f","scope":"Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n","status":"fully_proved","summary":"This is the function $F:\\mathbb{C}\\to\\mathbb{C}$ given by","title":"The entire function $1+\\sum_{n\\ge 1}(z/n)^n$"},{"id":"Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n/exists-finset-zeros-sum-analytic-order-n","kind":"node","lean":"MathlibExt.Analysis.Complex.ZerosOfSumZDivNPowNWanted.exists_finset_zeros_sum_analyticOrderNatAt_le","parent":16547,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zeros of "],["math","\\(1+\\sum_{n\\ge 1}(z/n)^n\\)"],["text"," in a disc"]],"source":"Zeros of $1+\\sum_{n\\ge 1}(z/n)^n$ in a disc"},"result":true,"route":"../roadmap/Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n/index.html#exists-finset-zeros-sum-analytic-order-n","scope":"Analysis/complex-analysis/entire-functions/zeros-of-the-entire-function-1-z-n","status":"can_prove","summary":"Let $F$ be the entire function","title":"Zeros of $1+\\sum_{n\\ge 1}(z/n)^n$ in a disc"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory.json","id":"Analysis/complex-analysis/geometric-function-theory","kind":"scope","parent":16364,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/index.html","scope":"Analysis/complex-analysis/geometric-function-theory","status":"can_prove","summary":"Conformal and univalent maps: Riemann mapping, Koebe and Bloch theorems, Schwarz\u2013Pick and half-plane self-maps, starlike and univalent coefficient problems, Loewner equation, holomorphic motions.","title":"Geometric function theory"},{"id":"Analysis/complex-analysis/geometric-function-theory/bij-on-cayley-ball-upper-half-plane","kind":"node","lean":"MathlibExt.Analysis.Complex.DiskToUpperHalfPlaneCayleyWanted.bijOn_cayley_ball_upperHalfPlane","parent":16550,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/index.html#bij-on-cayley-ball-upper-half-plane","scope":"Analysis/complex-analysis/geometric-function-theory","status":"can_prove","summary":"The function $\\phi(z) = i\\,\\frac{1-z}{1+z}$ maps the open unit disk $D = \\{z \\in \\mathbb{C} : |z| < 1\\}$ bijectively onto the upper half-plane $U = \\{z \\in \\mathbb{C} : \\operatorname{Im} z > 0\\}$: it maps $D$ into $U$, it is injective on $D$, and every point of $U$ is $\\phi(z)$ for some $z \\in D$.","title":"A M\u00f6bius map from the unit disk onto the upper half-plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/bloch","kind":"node","lean":"MathlibExt.Analysis.Complex.BlochWanted.bloch","parent":16550,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/index.html#bloch","scope":"Analysis/complex-analysis/geometric-function-theory","status":"fully_proved","summary":"There is a constant $B > 0$ with the following property. Let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on the open unit disc $\\mathbb{D} = \\{z : |z| < 1\\}$ with $f'(0) = 1$. Then there are an open connected set $V \\subseteq \\mathbb{D}$ and a point $w \\in \\mathbb{C}$ such that $f$ is injective on $V$ and $f(V)$ is the open disc $\\{\\zeta : |\\zeta - w| < B\\}$.","title":"Bloch's theorem"},{"id":"Analysis/complex-analysis/geometric-function-theory/two-mul-norm-deriv-le-ediam-image-ball","kind":"node","lean":"MathlibExt.Analysis.Complex.HolomorphicDiskDiameterDerivBoundWanted.two_mul_norm_deriv_le_ediam_image_ball","parent":16550,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/index.html#two-mul-norm-deriv-le-ediam-image-ball","scope":"Analysis/complex-analysis/geometric-function-theory","status":"can_prove","summary":"Let $D$ be the open unit disc in $\\mathbb{C}$ and let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on $D$. Then the diameter $d = \\sup_{z,w \\in D} |f(z) - f(w)|$ of the image $f(D)$, which may be $+\\infty$, satisfies $2|f'(0)| \\le d$.","title":"Diameter bound for the derivative at the center of the disc"},{"id":"Analysis/complex-analysis/geometric-function-theory/not-exists-differentiable-on-ball-is-com","kind":"node","lean":"MathlibExt.Analysis.Complex.NoProperHolomorphicMapUnitDiscWanted.not_exists_differentiableOn_ball_isCompact_preimage","parent":16550,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/index.html#not-exists-differentiable-on-ball-is-com","scope":"Analysis/complex-analysis/geometric-function-theory","status":"can_prove","summary":"Let $\\Delta := \\{z\\in\\mathbb{C}\\mid |z|<1\\}$ be the unit disc in the complex plane. There is no proper holomorphic map $f:\\Delta\\to\\mathbb{C}$, where proper means that the preimage under $f$ of every compact subset of $\\mathbb{C}$ is a compact subset of $\\Delta$.","title":"No proper holomorphic map from the unit disc to the complex plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/ball-subset-image-of-norm-eq-one-on-sphe","kind":"node","lean":"MathlibExt.Analysis.Complex.UnimodularBoundaryImageContainsDiscWanted.ball_subset_image_of_norm_eq_one_on_sphere","parent":16550,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/index.html#ball-subset-image-of-norm-eq-one-on-sphe","scope":"Analysis/complex-analysis/geometric-function-theory","status":"can_prove","summary":"Let $U\\subseteq\\mathbb{C}$ be a connected open set containing the closed unit disc $\\{z:|z|\\le 1\\}$, and let $f:\\mathbb{C}\\to\\mathbb{C}$ be holomorphic on $U$ and non-constant on $U$. If $|f(z)|=1$ whenever $|z|=1$, then every $w$ with $|w|<1$ is $f(z)$ for some $z\\in U$; that is, $f(U)$ contains the open unit disc.","title":"Nonconstant holomorphic functions of modulus one on the unit circle cover the unit disc"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions.json","id":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions","status":"can_prove","title":"Hadamard products of univalent functions"},{"id":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/class-s","kind":"node","lean":"MathlibExt.Analysis.HadamardProductCircleBoundWanted.ClassS","parent":16556,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/index.html#class-s","scope":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions","status":"fully_proved","summary":"An element of the class $S$ is a function $f : \\mathbb{C} \\to \\mathbb{C}$ together with a sequence $(a_m)_{m \\ge 0}$ of complex numbers such that $f$ is holomorphic and injective on the open unit disk $\\mathbb{D}$ of all $z \\in \\mathbb{C}$ with $|z| < 1$, $f(0) = 0$, $f'(0) = 1$, $a_0 = 0$, $a_1 = 1$, and for every $z \\in \\mathbb{D}$ the series $\\sum_{m \\ge 0} a_m z^m$ is unconditionally summable with sum $f(z)$.","title":"Normalized univalent functions with their Taylor coefficients"},{"id":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/hadamard-product","kind":"node","lean":"MathlibExt.Analysis.HadamardProductCircleBoundWanted.hadamardProduct","parent":16556,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/index.html#hadamard-product","scope":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions","status":"fully_proved","summary":"For a sequence $(a_k)_{k \\ge 0}$ of complex numbers, a complex polynomial $P(z) = \\sum_k p_k z^k$ and $n \\in \\mathbb{N}$, this is the polynomial $\\sum_{k=0}^{n} a_k p_k z^k$: the coefficient-wise (Hadamard) product of $P$ with the power series $\\sum_k a_k z^k$, truncated at degree $n$.","title":"Truncated Hadamard product with a polynomial"},{"id":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/circle-max","kind":"node","lean":"MathlibExt.Analysis.HadamardProductCircleBoundWanted.circleMax","parent":16556,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/index.html#circle-max","scope":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions","status":"fully_proved","summary":"For a complex polynomial $Q$, this is the real number $\\sup_{|z| = 1} |Q(z)|$, the supremum of $|Q(z)|$ over all $z \\in \\mathbb{C}$ with $|z| = 1$.","title":"Sup norm of a polynomial on the unit circle"},{"id":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/conjecture","kind":"node","lean":"MathlibExt.Analysis.HadamardProductCircleBoundWanted.conjecture","parent":16556,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/index.html#conjecture","scope":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions","status":"fully_proved","summary":"The proposition that for every $n \\in \\mathbb{N}$, every function $f(z) = \\sum_{m \\ge 0} a_m z^m$ of the class $S$ of normalized univalent functions on the unit disk, and every complex polynomial $P(z) = \\sum_k p_k z^k$ of degree at most $n$,","title":"Statement of the Hadamard product circle bound"},{"id":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.HadamardProductCircleBoundWanted.conjecture_holds","parent":16556,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions/index.html#conjecture-holds","scope":"Analysis/complex-analysis/geometric-function-theory/hadamard-products-of-univalent-functions","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$, every function $f(z) = \\sum_{m \\ge 0} a_m z^m$ of the class $S$ of normalized univalent functions on the unit disk, and every complex polynomial $P(z) = \\sum_k p_k z^k$ of degree at most $n$,","title":"Hadamard product circle bound for univalent functions"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip.json","id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip","status":"can_prove","title":"Holomorphic maps from the disc into a strip"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip/d","kind":"node","lean":"MathlibExt.Analysis.Complex.StripBoundedHolomorphicGrowthWanted.D","parent":16562,"reading_title":{"format":"inline-parts-v1","parts":[["text","The open unit disc "],["math","\\(D\\)"]],"source":"The open unit disc $D$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip/index.html#d","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip","status":"fully_proved","summary":"The set $D=\\{z\\in\\mathbb{C}:|z|<1\\}$ is the open disc of radius $1$ about $0$.","title":"The open unit disc $D$"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip/norm-le-two-div-pi-mul-log-of-abs-re-lt","kind":"node","lean":"MathlibExt.Analysis.Complex.StripBoundedHolomorphicGrowthWanted.norm_le_two_div_pi_mul_log_of_abs_re_lt_one","parent":16562,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip/index.html#norm-le-two-div-pi-mul-log-of-abs-re-lt","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-maps-from-the-disc-into-a-strip","status":"can_prove","summary":"Let $D=\\{z:|z|<1\\}\\subset\\mathbb{C}$ be the unit disk and $g:D\\to\\mathbb{C}$ be a holomorphic function such that $g(0)=0$ and $|\\operatorname{Re} g(z)|<1$ for any $z\\in D$. Then for any $z\\in D$, \\]","title":"Growth bound for holomorphic maps from the unit disc into a vertical strip"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions.json","id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","status":"can_prove","title":"Holomorphic motions"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/unit-disc","kind":"node","lean":"MathlibExt.Analysis.AMR22Problem7082Wanted.UnitDisc","parent":16565,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/index.html#unit-disc","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","status":"fully_proved","summary":"The open unit disc $\\mathbb{D}$ is the set of complex numbers $z$ with $|z| < 1$, the open ball of radius $1$ about $0$ in $\\mathbb{C}$.","title":"Open unit disc"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/is-injective-in-param","kind":"node","lean":"MathlibExt.Analysis.AMR22Problem7082Wanted.IsInjectiveInParam","parent":16565,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/index.html#is-injective-in-param","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","status":"fully_proved","summary":"Let $\\mathbb{D}$ be the open unit disc, $E$ a set of complex numbers and $F$ a function assigning a complex number $F(w, z)$ to each pair of complex numbers $w, z$. Condition (a) holds for $E$ and $F$ when, for every $z \\in \\mathbb{D}$, the map $w \\mapsto F(w, z)$ is injective on $E$.","title":"Injectivity at each parameter of the disc"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/is-analytic-in-disc","kind":"node","lean":"MathlibExt.Analysis.AMR22Problem7082Wanted.IsAnalyticInDisc","parent":16565,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/index.html#is-analytic-in-disc","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","status":"fully_proved","summary":"Let $\\mathbb{D}$ be the open unit disc, $E$ a set of complex numbers and $F$ a function assigning a complex number $F(w, z)$ to each pair of complex numbers $w, z$. Condition (b) holds for $E$ and $F$ when, for every $w \\in E$, the map $z \\mapsto F(w, z)$ is analytic on $\\mathbb{D}$.","title":"Holomorphic dependence of a motion on the parameter"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/is-normalized-at-zero","kind":"node","lean":"MathlibExt.Analysis.AMR22Problem7082Wanted.IsNormalizedAtZero","parent":16565,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/index.html#is-normalized-at-zero","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","status":"fully_proved","summary":"Let $E$ be a set of complex numbers and $F$ a function assigning a complex number $F(w, z)$ to each pair of complex numbers $w, z$. Condition (c) holds for $E$ and $F$ when $F(w, 0) = w$ for every $w \\in E$.","title":"Motion equal to the identity at parameter zero"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/agrees-on-disc","kind":"node","lean":"MathlibExt.Analysis.AMR22Problem7082Wanted.AgreesOnDisc","parent":16565,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/index.html#agrees-on-disc","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","status":"fully_proved","summary":"Let $\\mathbb{D}$ be the open unit disc, $E$ a set of complex numbers, and $F$ and $G$ functions assigning complex numbers $F(w, z)$ and $G(w, z)$ to each pair of complex numbers $w, z$. Then $G$ agrees with $F$ on $E$ when $G(w, z) = F(w, z)$ for every $w \\in E$ and every $z \\in \\mathbb{D}$.","title":"Agreement of two motions on a set"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/conjecture","kind":"node","lean":"MathlibExt.Analysis.AMR22Problem7082Wanted.conjecture","parent":16565,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/index.html#conjecture","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","status":"fully_proved","summary":"Let $\\mathbb{D}$ be the open unit disc. The proposition asserts the following. Let $E$ be a set of complex numbers and $F$ a function assigning a complex number $F(w, z)$ to each pair of complex numbers $w, z$, such that (a) for every $z \\in \\mathbb{D}$ the map $w \\mapsto F(w, z)$ is injective on $E$, (b) for every $w \\in E$ the map $z \\mapsto F(w, z)$ is analytic on $\\mathbb{D}$, and (c) $F(w, 0) = w$ for every\u2026","title":"Extension problem for holomorphic motions"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.AMR22Problem7082Wanted.conjecture_holds","parent":16565,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-motions/index.html#conjecture-holds","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-motions","status":"can_prove","summary":"Let $\\mathbb{D}$ be the open unit disc, $E$ a set of complex numbers and $F$ a function assigning a complex number $F(w, z)$ to each pair of complex numbers $w, z$, such that (a) for every $z \\in \\mathbb{D}$ the map $w \\mapsto F(w, z)$ is injective on $E$, (b) for every $w \\in E$ the map $z \\mapsto F(w, z)$ is analytic on $\\mathbb{D}$, and (c) $F(w, 0) = w$ for every $w \\in E$. Then there is a function $G$ of the\u2026","title":"Holomorphic motions of planar sets extend to the plane"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane.json","id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane","status":"can_prove","title":"Holomorphic self-maps of the upper half-plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/norm-le-norm-cayley-of-upper-half-plane","kind":"node","lean":"MathlibExt.Analysis.Complex.UpperHalfPlaneSchwarzLemmaWanted.norm_le_norm_cayley_of_upperHalfPlane_bounded_vanishing_at_I","parent":16573,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/index.html#norm-le-norm-cayley-of-upper-half-plane","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane","status":"can_prove","summary":"Let $\\mathbb{H}=\\{z\\in\\mathbb{C} : \\mathrm{Im}\\,z>0\\}$ be the upper half-plane, and let $F$ be a complex function that is holomorphic on $\\mathbb{H}$, satisfies $|F(z)|\\le 1$ for every $z\\in\\mathbb{H}$, and vanishes at $i$. Then for every $z\\in\\mathbb{H}$,","title":"Schwarz lemma on the upper half-plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/norm-deriv-i-le-sqrt-three-halves-of-map","kind":"node","lean":"MathlibExt.Analysis.Complex.UpperHalfPlaneSelfMapDerivBoundWanted.norm_deriv_I_le_sqrt_three_halves_of_mapsTo_upperHalfPlane","parent":16573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative bound at "],["math","\\(i\\)"],["text"," for a self-map of the upper half-plane"]],"source":"Derivative bound at $i$ for a self-map of the upper half-plane"},"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/index.html#norm-deriv-i-le-sqrt-three-halves-of-map","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane","status":"can_prove","summary":"Let $\\mathbb{H}=\\{z\\in\\mathbb{C} : \\mathrm{Im}\\,z>0\\}$ be the upper half-plane, and let $f$ be a complex function that is holomorphic on $\\mathbb{H}$, maps $\\mathbb{H}$ into $\\mathbb{H}$, and satisfies $f(i)=e^{2\\pi i/3}$. Then $|f'(i)|\\le\\sqrt{3/2}$.","title":"Derivative bound at $i$ for a self-map of the upper half-plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/tendsto-iterate-two-mul-i-of-fixed-i","kind":"node","lean":"MathlibExt.Analysis.Complex.UpperHalfPlaneSelfMapIterateConvergesWanted.tendsto_iterate_two_mul_I_of_fixed_I","parent":16573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterates of a self-map of the upper half-plane fixing "],["math","\\(i\\)"]],"source":"Iterates of a self-map of the upper half-plane fixing $i$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/index.html#tendsto-iterate-two-mul-i-of-fixed-i","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane","status":"can_prove","summary":"Let $\\mathbb{H}=\\{z\\in\\mathbb{C} : \\mathrm{Im}\\,z>0\\}$ be the upper half-plane, and let $f$ be a complex function that is holomorphic on $\\mathbb{H}$, maps $\\mathbb{H}$ into $\\mathbb{H}$, and satisfies $f(i)=i$ and $f(2i)=\\frac{15}{8}i$. Then the iterates $f^{\\circ n}(2i)$ converge to $i$ as $n\\to\\infty$, where $f^{\\circ n}$ is the $n$-fold composite of $f$ with itself.","title":"Iterates of a self-map of the upper half-plane fixing $i$"},{"id":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/image-upper-half-plane-eq-of-real-on-rea","kind":"node","lean":"MathlibExt.Analysis.Complex.UpperHalfPlaneSelfMapSurjectiveWanted.image_upperHalfPlane_eq_of_real_on_real","parent":16573,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane/index.html#image-upper-half-plane-eq-of-real-on-rea","scope":"Analysis/complex-analysis/geometric-function-theory/holomorphic-self-maps-of-the-upper-half-plane","status":"can_prove","summary":"Let $\\mathbb{C}_{+} = \\{z \\in \\mathbb{C} : \\operatorname{Im} z > 0\\}$ be the open upper half-plane and $\\overline{\\mathbb{C}_{+}} = \\{z \\in \\mathbb{C} : \\operatorname{Im} z \\ge 0\\}$ the closed one. Let $f : \\mathbb{C} \\to \\mathbb{C}$ be continuous on $\\overline{\\mathbb{C}_{+}}$ with $f(\\overline{\\mathbb{C}_{+}}) \\subseteq \\overline{\\mathbb{C}_{+}}$, suppose $f(x)$ is real for every real $x$, and suppose $f$ is\u2026","title":"Surjectivity for half-plane self-maps that are real on the real axis"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem.json","id":"Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem","status":"can_prove","title":"Koebe quarter theorem"},{"id":"Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem/koebe-quarter","kind":"node","lean":"MathlibExt.Analysis.Complex.KoebeQuarterWanted.koebe_quarter","parent":16578,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem/index.html#koebe-quarter","scope":"Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic and injective on the open unit disk $\\mathbb{D} = \\{z : |z| < 1\\}$, with $f(0) = 0$ and $f'(0) = 1$. Then the image $f(\\mathbb{D})$ contains the open disk $\\{w : |w| < 1/4\\}$.","title":"Koebe quarter theorem"},{"id":"Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem/inf-dist-zero-frontier-image-ball-le-one","kind":"node","lean":"MathlibExt.Analysis.Complex.UnivalentImageBoundaryDistLeOneWanted.infDist_zero_frontier_image_ball_le_one","parent":16578,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem/index.html#inf-dist-zero-frontier-image-ball-le-one","scope":"Analysis/complex-analysis/geometric-function-theory/koebe-quarter-theorem","status":"can_prove","summary":"The boundary of a normalized univalent image of the disc is within distance one of the origin. Let $D=\\{z\\in\\mathbb{C}:|z|<1\\}$ be the unit disk and suppose $f:D\\to\\mathbb{C}$ is an injective holomorphic function with $f(0)=0$ and $f'(0)=1$. Let $\\Omega=f(D)$ and let $\\partial\\Omega$ be its boundary. Then $\\mathrm{dist}(0,\\partial\\Omega)=\\inf\\{|z|:z\\in\\partial\\Omega\\}\\le 1$.","title":"The boundary of a normalized univalent image of the disc is within distance one of the origin"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/loewner-equation.json","id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation","kind":"scope","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation","status":"fully_proved","title":"Loewner equation"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/loewner-strip","kind":"node","lean":"StatMech.SLE.loewnerStrip","parent":16581,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/index.html#loewner-strip","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation","status":"fully_proved","summary":"The closed strip `{w \u2208 \u2102 : \u03b4 \u2264 Im w}`.","title":"Closed half-plane above a horizontal line"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation.json","id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","kind":"scope","navigation":"module","parent":16581,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","title":"Basic objects of the chordal Loewner equation"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/closed-upper-half-plane","kind":"node","lean":"StatMech.SLE.closedUpperHalfPlane","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#closed-upper-half-plane","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"The closed upper half-plane `{z : 0 \u2264 Im z}`.","title":"Closed upper half-plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/real-line","kind":"node","lean":"StatMech.SLE.realLine","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#real-line","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"The real axis `{z : Im z = 0}` in `\u2102`.","title":"Real axis in the complex plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/is-compact-hull","kind":"node","lean":"StatMech.SLE.IsCompactHull","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#is-compact-hull","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"A compact `\u210d`-hull: a bounded subset `K` of `\u210d`, relatively closed in `\u210d`, whose complement `\u210d \\ K` is open and simply connected.","title":"Compact hull in the upper half-plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/complement-connected","kind":"node","lean":"StatMech.SLE.IsCompactHull.complement_connected","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#complement-connected","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"The complement of a compact `\u210d`-hull is connected.","title":"Complement of a compact hull is connected"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/is-simply-connected-upper-half-plane","kind":"node","lean":"StatMech.SLE.isSimplyConnected_upperHalfPlane","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#is-simply-connected-upper-half-plane","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"The complex upper half-plane is simply connected.","title":"The upper half-plane is simply connected"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/has-half-plane-capacity","kind":"node","lean":"StatMech.SLE.HasHalfPlaneCapacity","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#has-half-plane-capacity","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"`g` is hydrodynamically normalized at infinity with half-plane capacity `a`: `g(z) - z \u2192 0` and `z (g(z) - z) \u2192 a` as `z \u2192 \u221e`, i.e. `g(z) = z + a/z + o(1/z)`.","title":"Hydrodynamic normalization with half-plane capacity"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/loewner-field","kind":"node","lean":"StatMech.SLE.loewnerField","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#loewner-field","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"The chordal Loewner vector field `w \u21a6 2 / (w - W_t)` at time `t`.","title":"Chordal Loewner vector field"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/is-loewner-solution-at","kind":"node","lean":"StatMech.SLE.IsLoewnerSolutionAt","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#is-loewner-solution-at","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"`t \u21a6 g_t(z)` satisfies the Loewner equation `\u2202_t g_t(z) = 2 / (g_t(z) - W_t)` at time `t`.","title":"Solution of the Loewner equation at a time"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/is-loewner-solution-on","kind":"node","lean":"StatMech.SLE.IsLoewnerSolutionOn","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#is-loewner-solution-on","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"`t \u21a6 g_t(z)` satisfies the Loewner equation on `s` (derivatives within `s`).","title":"Solution of the Loewner equation on a set of times"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/has-loewner-forward-extension","kind":"node","lean":"StatMech.SLE.HasLoewnerForwardExtension","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#has-loewner-forward-extension","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"The Loewner trajectory from `z` extends in `\u210d` beyond time `t`. This is the intrinsic maximal-lifetime condition used to determine the unswallowed domain.","title":"Forward extension of a Loewner trajectory"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/satisfies-loewner-equation","kind":"node","lean":"StatMech.SLE.SatisfiesLoewnerEquation","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#satisfies-loewner-equation","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"`g_0 = id` and `t \u21a6 g_t(z)` solves the Loewner equation for every `z \u2208 \u2102` and every `t \u2265 0`, with no domain restriction (compare `SatisfiesMaximalLoewnerEquation`, where `z` ranges over the unswallowed domain `D_t`).","title":"Loewner equation without domain restriction"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/satisfies-maximal-loewner-equation","kind":"node","lean":"StatMech.SLE.SatisfiesMaximalLoewnerEquation","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#satisfies-maximal-loewner-equation","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"The maximal chordal Loewner evolution with domains `D_t`. The domains are exactly the points whose trajectories extend beyond `t`, and `g_t : D_t \u2192 \u210d` is a conformal bijection.","title":"Maximal chordal Loewner evolution"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/is-standard-brownian-motion","kind":"node","lean":"StatMech.SLE.IsStandardBrownianMotion","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#is-standard-brownian-motion","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"A standard real Brownian motion indexed by `\u211d` (used for `t \u2265 0`): `B_0 = 0` a.s., continuous paths a.s., Gaussian, with independent increments `B_t - B_s ~ N(0, t - s)`.","title":"Standard real Brownian motion"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/sle-driving","kind":"node","lean":"StatMech.SLE.sleDriving","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#sle-driving","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"The SLE`_\u03ba` driving function `W_t = \u221a\u03ba B_t(\u03c9)`.","title":"SLE driving function"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/is-chordal-sle","kind":"node","lean":"StatMech.SLE.IsChordalSLE","parent":16583,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation/index.html#is-chordal-sle","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/basic-objects-of-the-chordal-loewner-equation","status":"fully_proved","summary":"`(D, g)` is chordal SLE`_\u03ba` in `\u210d` from `0` to `\u221e`: `\u03ba \u2265 0`, `B` is a standard Brownian motion and almost surely `g` solves the maximal Loewner equation with driving function `\u221a\u03ba B`. This is a definition only; existence of SLE is not proved.","title":"Chordal Schramm\u2013Loewner evolution"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l.json","id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l","kind":"scope","navigation":"module","parent":16581,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l","status":"fully_proved","title":"Local existence and uniqueness for the chordal Loewner equation"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/picard-exists-ball","kind":"node","lean":"StatMech.SLE.picard_exists_ball","parent":16599,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/index.html#picard-exists-ball","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l","status":"fully_proved","summary":"Picard\u2013Lindel\u00f6f with the solution confined to `closedBall x\u2080 a`: a strengthening of Mathlib's `IsPicardLindelof.exists_eq_forall_mem_Icc_hasDerivWithinAt\u2080` by the bound `\u03b1 t \u2208 closedBall x\u2080 a` for all `t`.","title":"Picard\u2013Lindel\u00f6f theorem with solutions confined to a ball"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/picard-exists-ball-family-lipschitz","kind":"node","lean":"StatMech.SLE.picard_exists_ball_family_lipschitz","parent":16599,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/index.html#picard-exists-ball-family-lipschitz","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l","status":"fully_proved","summary":"Picard\u2013Lindel\u00f6f local flow, Lipschitz in the initial point, with every trajectory in `closedBall x\u2080 a`: a strengthening of Mathlib's `IsPicardLindelof.exists_forall_mem_closedBall_eq_hasDerivWithinAt_lipschitzOnWith` by the ball bound.","title":"Lipschitz Picard\u2013Lindel\u00f6f flow confined to a ball"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/loewner-local-existence","kind":"node","lean":"StatMech.SLE.loewner_local_existence","parent":16599,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/index.html#loewner-local-existence","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l","status":"fully_proved","summary":"For continuous `W` and `Im z > 0` there are `\u03b5 > 0` and `g` with `g 0 = z`, `\u0121(t) = 2 / (g(t) - W_t)` on `[0, \u03b5]` (one-sided derivatives at the endpoints), and `g(t) \u2208 \u210d` on `[0, \u03b5]` (Lawler 2005, Chapter 4).","title":"Local existence for the chordal Loewner equation"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/loewner-local-exists-unique","kind":"node","lean":"StatMech.SLE.loewner_local_existsUnique","parent":16599,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l/index.html#loewner-local-exists-unique","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/local-existence-and-uniqueness-for-the-chordal-l","status":"fully_proved","summary":"Local existence and uniqueness for the chordal Loewner equation: the solution from `loewner_local_existence` is the unique continuous solution on `[0, \u03b5]` with right derivatives on `[0, \u03b5)` that stays in `\u210d`.","title":"Local existence and uniqueness for the chordal Loewner equation"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o.json","id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o","kind":"scope","navigation":"module","parent":16581,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o","status":"fully_proved","title":"Elementary properties of Loewner solutions and of half-plane capacity"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/loewner-reflection","kind":"node","lean":"StatMech.SLE.loewnerReflection","parent":16604,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/index.html#loewner-reflection","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o","status":"fully_proved","summary":"The reflection `z \u21a6 -z\u0304` across the imaginary axis.","title":"Reflection across the imaginary axis"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/has-half-plane-capacity-comp","kind":"node","lean":"StatMech.SLE.hasHalfPlaneCapacity_comp","parent":16604,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/index.html#has-half-plane-capacity-comp","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o","status":"fully_proved","summary":"Half-plane capacity is additive under composition: `hcap(f \u2218 g) = hcap(f) + hcap(g)`.","title":"Additivity of half-plane capacity under composition"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/has-capacity-rate","kind":"node","lean":"StatMech.SLE.HasCapacityRate","parent":16604,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/index.html#has-capacity-rate","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o","status":"fully_proved","summary":"The family `g_t` has half-plane capacity `a t` for every `t \u2265 0`.","title":"Half-plane capacity rate of a Loewner family"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/monotone-hull-family","kind":"node","lean":"StatMech.SLE.MonotoneHullFamily","parent":16604,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o/index.html#monotone-hull-family","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/elementary-properties-of-loewner-solutions-and-o","status":"fully_proved","summary":"An increasing family `(K_t)` of compact `\u210d`-hulls.","title":"Increasing family of compact hulls"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio.json","id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","kind":"scope","navigation":"module","parent":16581,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","title":"Maximal solutions of the chordal Loewner equation and the Loewner hulls"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/positive-on-icc-of-deriv-nonneg-while-po","kind":"node","lean":"StatMech.SLE.positiveOn_Icc_of_deriv_nonneg_while_pos","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#positive-on-icc-of-deriv-nonneg-while-po","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"A continuous function on `[0, T]` that is positive at `0` and has a nonnegative derivative at every interior point where it is positive stays positive on `[0, T]`.","title":"Positivity persists under a nonnegative derivative"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-forward-solution","kind":"node","lean":"StatMech.SLE.LoewnerForwardSolution","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-forward-solution","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"A solution `t \u21a6 g_t(z)` of the Loewner equation on `[0, T]` (one-sided derivatives at the endpoints) with `g_0(z) = z` and `g_t(z) \u2208 \u210d` for `t \u2208 [0, T]`.","title":"Forward solution of the Loewner equation"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/restrict","kind":"node","lean":"StatMech.SLE.LoewnerForwardSolution.restrict","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#restrict","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The restriction of a forward solution to `[0, S]`, `S \u2264 T`.","title":"Restriction of a Loewner forward solution"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/reflect","kind":"node","lean":"StatMech.SLE.LoewnerForwardSolution.reflect","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#reflect","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The reflected solution `-conj g_t(z)`, a forward solution for the driving function `-W`.","title":"Reflection of a Loewner forward solution"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/time-shift","kind":"node","lean":"StatMech.SLE.LoewnerForwardSolution.timeShift","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#time-shift","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The solution restarted at time `s`: `u \u21a6 g_{s+u}(z)`, for the driving function `u \u21a6 W(s + u)`.","title":"Time shift of a Loewner forward solution"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/append-curve","kind":"node","lean":"StatMech.SLE.LoewnerForwardSolution.appendCurve","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#append-curve","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The concatenation of `g` on `[0, S]` with a solution `h` started from `g_S(z)` at time `S`.","title":"Concatenated curve of two Loewner solutions"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/append","kind":"node","lean":"StatMech.SLE.LoewnerForwardSolution.append","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#append","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The concatenation of two forward solutions, a forward solution on `[0, S + T]`.","title":"Concatenation of Loewner forward solutions"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/is-loewner-admissible-duration","kind":"node","lean":"StatMech.SLE.IsLoewnerAdmissibleDuration","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#is-loewner-admissible-duration","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"`T > 0` and there is a forward solution from `z` on `[0, T]`.","title":"Admissible duration for the Loewner equation"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-lifetime","kind":"node","lean":"StatMech.SLE.loewnerLifetime","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-lifetime","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The lifetime `T_z \u2208 [0, \u221e]` of `z`: the supremum of admissible durations.","title":"Lifetime of a point under the Loewner flow"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/is-before-loewner-lifetime","kind":"node","lean":"StatMech.SLE.IsBeforeLoewnerLifetime","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#is-before-loewner-lifetime","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"`0 \u2264 t < T_z`.","title":"Times before the Loewner lifetime"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/selected-loewner-duration","kind":"node","lean":"StatMech.SLE.selectedLoewnerDuration","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#selected-loewner-duration","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"A chosen admissible duration exceeding `t`, for `t` before the lifetime.","title":"Chosen admissible Loewner duration"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/selected-loewner-solution","kind":"node","lean":"StatMech.SLE.selectedLoewnerSolution","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#selected-loewner-solution","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"A chosen forward solution on `[0, selectedLoewnerDuration ht]`.","title":"Chosen Loewner forward solution"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-maximal-curve","kind":"node","lean":"StatMech.SLE.loewnerMaximalCurve","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-maximal-curve","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The maximal solution `t \u21a6 g_t(z)` for `0 \u2264 t < T_z` (independent of choices by uniqueness); it is `z` outside `[0, T_z)`.","title":"Maximal Loewner trajectory"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-maximal-forward-exists","kind":"node","lean":"StatMech.SLE.loewner_maximal_forward_exists","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-maximal-forward-exists","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"For continuous `W` and `z \u2208 \u210d`, the lifetime `T_z` is positive, `g_0(z) = z`, `g_t(z) \u2208 \u210d` for `0 \u2264 t < T_z`, the equation holds at `t = 0` from the right and at every `0 < t < T_z` (Lawler 2005, Chapter 4).","title":"Maximal solution of the chordal Loewner equation"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-unswallowed-domain","kind":"node","lean":"StatMech.SLE.loewnerUnswallowedDomain","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-unswallowed-domain","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The domain `H_t = {z \u2208 \u210d : t < T_z}` of points not swallowed by time `t`.","title":"Unswallowed domain of the Loewner flow"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-maximal-maps","kind":"node","lean":"StatMech.SLE.loewnerMaximalMaps","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-maximal-maps","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The Loewner map `g_t(z)`, the maximal solution at time `t`.","title":"Maximal Loewner maps"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-maximal-maps-inverse","kind":"node","lean":"StatMech.SLE.loewnerMaximalMapsInverse","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-maximal-maps-inverse","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The inverse `g_t\u207b\u00b9 : \u210d \u2192 H_t` (`0` outside `\u210d`).","title":"Inverse of the maximal Loewner map"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-unswallowed-domain-eq-set-of-has","kind":"node","lean":"StatMech.SLE.loewnerUnswallowedDomain_eq_setOf_hasLoewnerForwardExtension","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-unswallowed-domain-eq-set-of-has","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The lifetime-defined unswallowed domain is exactly the intrinsic forward-extension domain.","title":"Unswallowed domain equals the forward-extension domain"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-maximal-maps-homeomorph","kind":"node","lean":"StatMech.SLE.loewnerMaximalMapsHomeomorph","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-maximal-maps-homeomorph","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"`g_t` as a homeomorphism `H_t \u2243\u209c \u210d`.","title":"Loewner map as a homeomorphism onto the half-plane"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/is-simply-connected-loewner-unswallowed","kind":"node","lean":"StatMech.SLE.isSimplyConnected_loewnerUnswallowedDomain","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#is-simply-connected-loewner-unswallowed","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The unswallowed domain is simply connected.","title":"The unswallowed domain is simply connected"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-maximal-maps-satisfies-maximal-l","kind":"node","lean":"StatMech.SLE.loewnerMaximalMaps_satisfiesMaximalLoewnerEquation","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-maximal-maps-satisfies-maximal-l","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The canonical maximal family has the exact lifetime domains and conformal Loewner maps.","title":"Maximal Loewner maps solve the maximal Loewner equation"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-swallowed-set","kind":"node","lean":"StatMech.SLE.loewnerSwallowedSet","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-swallowed-set","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"The Loewner hull `K_t = \u210d \\ H_t` of points swallowed by time `t`.","title":"Loewner hull"},{"id":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/loewner-swallowed-set-is-compact-hull","kind":"node","lean":"StatMech.SLE.loewnerSwallowedSet_isCompactHull","parent":16609,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio/index.html#loewner-swallowed-set-is-compact-hull","scope":"Analysis/complex-analysis/geometric-function-theory/loewner-equation/maximal-solutions-of-the-chordal-loewner-equatio","status":"fully_proved","summary":"For continuous `W` and `t \u2265 0`, the Loewner hull `K_t` is a compact `\u210d`-hull (Lawler 2005, Chapter 4; the capacity normalization is not proved).","title":"Loewner hulls are compact hulls"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions.json","id":"Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions","status":"can_prove","title":"Radius of convexity of starlike functions"},{"id":"Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions/conjecture","kind":"node","lean":"MathlibExt.Analysis.HaymanProblem622Wanted.conjecture","parent":16633,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions/index.html#conjecture","scope":"Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions","status":"fully_proved","summary":"Let $\\mathbb{D}$ be the open unit disc in $\\mathbb{C}$. Call $f : \\mathbb{C} \\to \\mathbb{C}$ a normalized starlike function of order $1/2$ if $f$ is holomorphic and injective on $\\mathbb{D}$, $f(0) = 0$, $f'(0) = 1$, and $\\operatorname{Re}(z f'(z)/f(z)) \\ge 1/2$ for every $z \\in \\mathbb{D}$ with $z \\ne 0$ and $f(z) \\ne 0$. Call $P : \\mathbb{C} \\to \\mathbb{C}$ a Carath\u00e9odory function if $P$ is holomorphic on\u2026","title":"Conjecture: a sharp radius of convexity for starlike functions of order 1/2"},{"id":"Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.HaymanProblem622Wanted.conjecture_holds","parent":16633,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions/index.html#conjecture-holds","scope":"Analysis/complex-analysis/geometric-function-theory/radius-of-convexity-of-starlike-functions","status":"can_prove","summary":"Let $\\mathbb{D}$ be the open unit disc in $\\mathbb{C}$. Call $f : \\mathbb{C} \\to \\mathbb{C}$ a normalized starlike function of order $1/2$ if $f$ is holomorphic and injective on $\\mathbb{D}$, $f(0) = 0$, $f'(0) = 1$, and $\\operatorname{Re}(z f'(z)/f(z)) \\ge 1/2$ for every $z \\in \\mathbb{D}$ with $z \\ne 0$ and $f(z) \\ne 0$. Call $P : \\mathbb{C} \\to \\mathbb{C}$ a Carath\u00e9odory function if $P$ is holomorphic on\u2026","title":"A sharp radius of convexity for starlike functions of order 1/2 exists"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem.json","id":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem","status":"can_prove","title":"Riemann mapping theorem"},{"id":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/riemann-mapping-normalized","kind":"node","lean":"MathlibExt.Analysis.Complex.RiemannMappingWanted.riemannMapping_normalized","parent":16636,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/index.html#riemann-mapping-normalized","scope":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem","status":"fully_proved","summary":"Let $U$ be a simply connected open subset of $\\mathbb{C}$ with $U \\ne \\mathbb{C}$, and let $z_0 \\in U$. Then there is a function $f : \\mathbb{C} \\to \\mathbb{C}$, holomorphic on $U$, that maps $U$ bijectively onto the open unit disc $\\{w : |w| < 1\\}$, with $f(z_0) = 0$ and $f'(z_0)$ a positive real number. It is unique on $U$: every function $g : \\mathbb{C} \\to \\mathbb{C}$ with the same four properties agrees with\u2026","title":"Riemann mapping theorem"},{"id":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/unit-disk","kind":"node","lean":"MathlibExt.Analysis.Complex.InjectiveDiskMapExtremalDerivWanted.unitDisk","parent":16636,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/index.html#unit-disk","scope":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem","status":"fully_proved","summary":"The open unit disc is the set of complex numbers $z$ with $|z| < 1$, the open ball of radius $1$ about $0$ in $\\mathbb{C}$.","title":"Open unit disc"},{"id":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/admissible-family","kind":"node","lean":"MathlibExt.Analysis.Complex.InjectiveDiskMapExtremalDerivWanted.AdmissibleFamily","parent":16636,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/index.html#admissible-family","scope":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem","status":"fully_proved","summary":"For a set $\\Omega \\subseteq \\mathbb{C}$, the admissible family of $\\Omega$ is the set of functions $f : \\mathbb{C} \\to \\mathbb{C}$ that are complex differentiable on $\\Omega$, injective on $\\Omega$, satisfy $f(0) = 0$, and map $\\Omega$ into the open unit disc $\\{z \\in \\mathbb{C} : |z| < 1\\}$.","title":"Admissible maps into the unit disc"},{"id":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/exists-mem-admissible-family-norm-deriv","kind":"node","lean":"MathlibExt.Analysis.Complex.InjectiveDiskMapExtremalDerivWanted.exists_mem_admissibleFamily_norm_deriv_eq_sSup","parent":16636,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem/index.html#exists-mem-admissible-family-norm-deriv","scope":"Analysis/complex-analysis/geometric-function-theory/riemann-mapping-theorem","status":"can_prove","summary":"Let $D$ be the open unit disc $\\{z \\in \\mathbb{C} : |z| < 1\\}$, let $\\Omega \\subseteq D$ be open and connected with $0 \\in \\Omega$, and let $F$ be the family of functions $f$ that are holomorphic and injective on $\\Omega$, map $\\Omega$ into $D$, and satisfy $f(0) = 0$. Then some $g \\in F$ attains the supremum of $|f'(0)|$ over $F$:","title":"Existence of an extremal injective map into the unit disc"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function.json","id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"can_prove","title":"Robertson's conjecture on odd univalent functions"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/hayman-unit-disc","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.haymanUnitDisc","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#hayman-unit-disc","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"The open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$ in the complex plane.","title":"Open unit disc"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/hayman-is-schlicht","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.haymanIsSchlicht","parent":16641,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normalized univalent functions (class "],["math","\\(S\\)"],["text",")"]],"source":"Normalized univalent functions (class $S$)"},"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#hayman-is-schlicht","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ belongs to the class $S$ when it is analytic at every point of the open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$, injective on $\\mathbb{D}$, and satisfies $f(0) = 0$ and $f'(0) = 1$.","title":"Normalized univalent functions (class $S$)"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/hayman-is-starlike","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.haymanIsStarlike","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#hayman-is-starlike","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ is starlike when it belongs to the class $S$ (analytic at every point of the open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$, injective on $\\mathbb{D}$, $f(0) = 0$ and $f'(0) = 1$) and $\\operatorname{Re}\\bigl(z f'(z) / f(z)\\bigr) > 0$ for every $z \\in \\mathbb{D}$ with $z \\ne 0$.","title":"Starlike univalent functions"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/hayman-has-real-coeffs","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.haymanHasRealCoeffs","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#hayman-has-real-coeffs","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ has real Taylor coefficients at $0$ when, for every natural number $n$, the value $f^{(n)}(0)$ of its $n$-th iterated derivative at $0$ is real (equivalently, every Taylor coefficient $f^{(n)}(0)/n!$ is real).","title":"Real Taylor coefficients at the origin"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/hayman-is-close-to-convex","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.haymanIsCloseToConvex","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#hayman-is-close-to-convex","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ is close-to-convex when it belongs to the class $S$ (analytic at every point of the open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$, injective on $\\mathbb{D}$, $f(0) = 0$ and $f'(0) = 1$) and there are a starlike function $g$ and a real number $\\alpha$ such that $\\operatorname{Re}\\bigl(e^{i\\alpha} z f'(z) / g(z)\\bigr) > 0$ for every $z \\in \\mathbb{D}$ with\u2026","title":"Close-to-convex functions"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/hayman-is-odd-sqrt","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.haymanIsOddSqrt","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#hayman-is-odd-sqrt","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"Given $f : \\mathbb{C} \\to \\mathbb{C}$, a function $h : \\mathbb{C} \\to \\mathbb{C}$ is a normalized odd square root $h(z) = f(z^2)^{1/2}$ when, on the open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$, $h$ is analytic at every point and satisfies $h(z)^2 = f(z^2)$ and $h(-z) = -h(z)$ for all $z \\in \\mathbb{D}$, and moreover $h(0) = 0$ and $h'(0) = 1$.","title":"Normalized odd square roots"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/hayman-odd-sqrt-coeff","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.haymanOddSqrtCoeff","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#hayman-odd-sqrt-coeff","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"For a function $h : \\mathbb{C} \\to \\mathbb{C}$ and a natural number $n$, $c_n(h) = h^{(n)}(0) / n!$, where $h^{(n)}$ is the $n$-th iterated derivative of $h$. For $h$ analytic near $0$ this is the $n$-th Taylor coefficient of $h$ at $0$; Robertson's problem uses it at odd $n$.","title":"Taylor coefficients at the origin"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/hayman-robertson-bound","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.haymanRobertsonBound","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#hayman-robertson-bound","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"For a function $h : \\mathbb{C} \\to \\mathbb{C}$ with Taylor coefficients $c_k = h^{(k)}(0)/k!$ at $0$ and a natural number $n$, the Robertson bound for $h$ at $n$ is the inequality $|c_1|^2 + |c_3|^2 + \\dots + |c_{2n-1}|^2 \\le n$ (for $n = 0$ the sum is empty).","title":"Robertson's bound"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/conjecture","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.conjecture","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#conjecture","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"fully_proved","summary":"Let $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$. The statement that for all functions $f, h : \\mathbb{C} \\to \\mathbb{C}$ such that $f$ belongs to the class $S$ (analytic at every point of $\\mathbb{D}$, injective on $\\mathbb{D}$, $f(0) = 0$ and $f'(0) = 1$), $h$ is a normalized odd square root of $f(z^2)$ (analytic at every point of $\\mathbb{D}$, with $h(z)^2 = f(z^2)$ and $h(-z) = -h(z)$ for all\u2026","title":"Robertson's conjecture for real-coefficient and close-to-convex functions"},{"id":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.HaymanRobertson39Wanted.conjecture_holds","parent":16641,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function/index.html#conjecture-holds","scope":"Analysis/complex-analysis/geometric-function-theory/robertson-s-conjecture-on-odd-univalent-function","status":"can_prove","summary":"Let $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$. For all functions $f, h : \\mathbb{C} \\to \\mathbb{C}$ such that $f$ belongs to the class $S$ (analytic at every point of $\\mathbb{D}$, injective on $\\mathbb{D}$, $f(0) = 0$ and $f'(0) = 1$), $h$ is a normalized odd square root of $f(z^2)$ (analytic at every point of $\\mathbb{D}$, with $h(z)^2 = f(z^2)$ and $h(-z) = -h(z)$ for all $z \\in \\mathbb{D}$, $h(0) = 0$ and\u2026","title":"Robertson's conjecture holds for real-coefficient and close-to-convex functions"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma.json","id":"Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma","status":"can_prove","title":"Schwarz\u2013Pick lemma"},{"id":"Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma/schwarz-pick","kind":"node","lean":"MathlibExt.Analysis.Complex.SchwarzPick.schwarzPick","parent":16652,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma/index.html#schwarz-pick","scope":"Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic on the open unit disc $\\mathbb{D} = \\{z : |z| < 1\\}$ and map $\\mathbb{D}$ into itself. Then for all $z, w \\in \\mathbb{D}$, $|f(z) - f(w)| / |1 - \\overline{f(w)}\\, f(z)| \\le |z - w| / |1 - \\bar w z|$.","title":"Schwarz\u2013Pick lemma"},{"id":"Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma/norm-deriv-div-one-sub-sq-le-of-maps-to","kind":"node","lean":"MathlibExt.Analysis.Complex.SchwarzPickDerivBoundWanted.norm_deriv_div_one_sub_sq_le_of_mapsTo_unitBall","parent":16652,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma/index.html#norm-deriv-div-one-sub-sq-le-of-maps-to","scope":"Analysis/complex-analysis/geometric-function-theory/schwarz-pick-lemma","status":"can_prove","summary":"Let $\\mathbb{D}=\\{z\\in\\mathbb{C}:|z|<1\\}$ be the open unit disc. If $f:\\mathbb{D}\\to\\mathbb{D}$ is holomorphic, then","title":"Schwarz\u2013Pick lemma, derivative form"},{"data_url":"scopes/Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions.json","id":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions","kind":"scope","navigation":"module","parent":16550,"result":false,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/index.html","scope":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions","status":"can_prove","title":"Successive coefficients of starlike functions"},{"id":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/is-in-class-s","kind":"node","lean":"MathlibExt.Analysis.StarlikeCoefficientSuccessiveDifferenceWanted.IsInClassS","parent":16655,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normalized univalent functions (the class "],["math","\\(S\\)"],["text",")"]],"source":"Normalized univalent functions (the class $S$)"},"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/index.html#is-in-class-s","scope":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ is in the class $S$ when it is analytic on the open unit disc $\\mathbb{D} = \\{z : |z| < 1\\}$, satisfies $f(0) = 0$ and $f'(0) = 1$, and is injective on $\\mathbb{D}$.","title":"Normalized univalent functions (the class $S$)"},{"id":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/is-starlike","kind":"node","lean":"MathlibExt.Analysis.StarlikeCoefficientSuccessiveDifferenceWanted.IsStarlike","parent":16655,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/index.html#is-starlike","scope":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ is starlike when it is analytic on the open unit disc $\\mathbb{D} = \\{z : |z| < 1\\}$ and, for every $z \\ne 0$ in $\\mathbb{D}$, $f(z) \\ne 0$ and $\\operatorname{Re}\\bigl(z f'(z) / f(z)\\bigr) > 0$.","title":"Starlike functions on the unit disc"},{"id":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/conjecture","kind":"node","lean":"MathlibExt.Analysis.StarlikeCoefficientSuccessiveDifferenceWanted.conjecture","parent":16655,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/index.html#conjecture","scope":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions","status":"fully_proved","summary":"The statement that for every function $f : \\mathbb{C} \\to \\mathbb{C}$ in the class $S$ (analytic on the open unit disc $\\mathbb{D} = \\{z : |z| < 1\\}$, with $f(0) = 0$ and $f'(0) = 1$, and injective on $\\mathbb{D}$) that is moreover starlike ($\\operatorname{Re}\\bigl(z f'(z) / f(z)\\bigr) > 0$ for every $z \\ne 0$ in $\\mathbb{D}$), the Taylor coefficients $a_n = f^{(n)}(0) / n!$ of $f$ at $0$ satisfy\u2026","title":"Hayman's successive coefficient problem for starlike functions"},{"id":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.StarlikeCoefficientSuccessiveDifferenceWanted.conjecture_holds","parent":16655,"result":true,"route":"../roadmap/Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions/index.html#conjecture-holds","scope":"Analysis/complex-analysis/geometric-function-theory/successive-coefficients-of-starlike-functions","status":"can_prove","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be analytic on the open unit disc $\\mathbb{D} = \\{z : |z| < 1\\}$, with $f(0) = 0$ and $f'(0) = 1$, injective on $\\mathbb{D}$, and starlike, meaning that $\\operatorname{Re}\\bigl(z f'(z) / f(z)\\bigr) > 0$ for every $z \\ne 0$ in $\\mathbb{D}$. Let $a_n = f^{(n)}(0) / n!$ be the Taylor coefficients of $f$ at $0$. Then $\\bigl| |a_{n+1}| - |a_n| \\bigr| \\le 1$ for every natural number\u2026","title":"Leung's theorem on successive coefficients of starlike functions"},{"data_url":"scopes/Analysis/complex-analysis/zeros-of-polynomials.json","id":"Analysis/complex-analysis/zeros-of-polynomials","kind":"scope","parent":16364,"result":false,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/index.html","scope":"Analysis/complex-analysis/zeros-of-polynomials","status":"can_prove","summary":"Analytic behaviour of complex polynomials: location of zeros (Enestr\u00f6m\u2013Kakeya, Sendov's problem, Jentzsch), flat and Littlewood polynomials, complementary polynomials and sup norms on the unit circle. Algebraic properties of polynomials go to Algebra; Mahler measure goes to NumberTheory.","title":"Polynomials in the complex plane"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/exists-complementary-polynomial-on-unit","kind":"node","lean":"MathlibExt.Analysis.ComplementaryPolynomialsWanted.ComplexAnalysis.exists_complementary_polynomial_on_unit_circle","parent":16660,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/index.html#exists-complementary-polynomial-on-unit","scope":"Analysis/complex-analysis/zeros-of-polynomials","status":"can_prove","summary":"Let $P$ be a complex polynomial with $|P(z)| \\le 1$ for every complex number $z$ with $|z| = 1$. Then there is a complex polynomial $Q$ with $\\deg Q \\le \\deg P$ such that $|P(z)|^2 + |Q(z)|^2 = 1$ for every $z$ with $|z| = 1$. Here the degree of the zero polynomial is taken to be $0$.","title":"Complementary polynomial on the unit circle"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/enestrom-kakeya-zero-localization","kind":"node","lean":"MathlibExt.Analysis.Complex.EnestromKakeya.enestrom_kakeya_zero_localization","parent":16660,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/index.html#enestrom-kakeya-zero-localization","scope":"Analysis/complex-analysis/zeros-of-polynomials","status":"fully_proved","summary":"Let $n \\ge 1$ and let $a_0, a_1, \\dots, a_n$ be positive real numbers. Then every complex zero $z$ of the polynomial $f(z) = a_0 + a_1 z + \\dots + a_n z^n$ satisfies $\\rho_1 \\le |z| \\le \\rho_2$, where $\\rho_1$ and $\\rho_2$ are the minimum and the maximum of the ratios $a_k / a_{k+1}$ for $0 \\le k \\le n - 1$.","title":"Enestr\u00f6m\u2013Kakeya theorem"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/jentzsch-zeros-of-power-series-sections","kind":"node","lean":"MathlibExt.Analysis.Complex.JentzschPowerSeriesWanted.jentzsch_zeros_of_power_series_sections","parent":16660,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/index.html#jentzsch-zeros-of-power-series-sections","scope":"Analysis/complex-analysis/zeros-of-polynomials","status":"fully_proved","summary":"Let $(a_k)_{k \\ge 0}$ be complex numbers such that the power series $\\sum_{k \\ge 0} a_k z^k$ has radius of convergence $1$, and for $n \\ge 0$ let $s_n(z) = \\sum_{k=0}^{n} a_k z^k$ be its $n$-th partial sum. Then for every complex $z$ with $|z| = 1$, every real $\\varepsilon > 0$ and every natural number $N$, there are $n \\ge N$ and a root $w$ of the polynomial $s_n$ with $|w - z| < \\varepsilon$.","title":"Jentzsch's theorem"},{"data_url":"scopes/Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials.json","id":"Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials","kind":"scope","navigation":"module","parent":16660,"result":false,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials/index.html","scope":"Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials","status":"can_prove","title":"Flat Littlewood polynomials"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials/is-littlewood-polynomial","kind":"node","lean":"MathlibExt.Analysis.FlatLittlewoodPolyWanted.FlatLittlewoodPoly.IsLittlewoodPolynomial","parent":16664,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials/index.html#is-littlewood-polynomial","scope":"Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials","status":"fully_proved","summary":"A polynomial $P$ over a ring $F$, of degree $n$ (with the zero polynomial taken to have degree $0$), is a Littlewood polynomial if each of its coefficients of $X^0, X^1, \\ldots, X^n$ is $1$ or $-1$.","title":"Littlewood polynomials"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials/theorem-1-1","kind":"node","lean":"MathlibExt.Analysis.FlatLittlewoodPolyWanted.FlatLittlewoodPoly.theorem_1_1","parent":16664,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials/index.html#theorem-1-1","scope":"Analysis/complex-analysis/zeros-of-polynomials/flat-littlewood-polynomials","status":"can_prove","summary":"There exist real constants $\\Delta > \\delta > 0$ such that for every natural number $n \\ge 2$ there is a Littlewood polynomial $P$ with complex coefficients and of degree $n$ such that $\\delta \\sqrt{n} \\le |P(z)| \\le \\Delta \\sqrt{n}$ for all $z \\in \\mathbb{C}$ with $|z| = 1$.","title":"Flat Littlewood polynomials exist"},{"data_url":"scopes/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture.json","id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","kind":"scope","navigation":"module","parent":16660,"result":false,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"can_prove","title":"Sendov's conjecture"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/is-admissible-radius","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.IsAdmissibleRadius","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#is-admissible-radius","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"fully_proved","summary":"A real number $R$ is an admissible radius if $R > 0$ and, for every integer $n \\ge 2$ and every complex polynomial $f$ of degree $n$ whose zeros all lie in the closed unit disk $|z| \\le 1$, every zero $\\lambda$ of $f$ has a critical point within distance $R$, that is, a zero $c$ of the derivative $f'$ with $|c - \\lambda| \\le R$.","title":"Admissible radius in Sendov's problem"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/mono","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.IsAdmissibleRadius.mono","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#mono","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"fully_proved","summary":"If $R$ is an admissible radius and $R \\le S$, then $S$ is an admissible radius.","title":"Enlarging an admissible radius"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/sendov-radius","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.sendovRadius","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#sendov-radius","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"fully_proved","summary":"The Sendov radius is the infimum, as a real number, of the set of admissible radii in Sendov's problem.","title":"The Sendov radius"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/bdd-below-admissible-radii","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.bddBelow_admissibleRadii","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#bdd-below-admissible-radii","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"fully_proved","summary":"The set of admissible radii is bounded below.","title":"The admissible radii are bounded below"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/sendov-radius-le-of-is-admissible-radius","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.sendovRadius_le_of_isAdmissibleRadius","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#sendov-radius-le-of-is-admissible-radius","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"fully_proved","summary":"If $R$ is an admissible radius, then the Sendov radius is at most $R$.","title":"Admissible radii bound the Sendov radius"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/sendov-conjecture","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.sendovConjecture","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#sendov-conjecture","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"fully_proved","summary":"Sendov's conjecture is the proposition that the Sendov radius is at most $1$.","title":"Sendov's conjecture"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/lower-bound-one","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.lowerBoundOne","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#lower-bound-one","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"fully_proved","summary":"This is the proposition that the Sendov radius is at least $1$.","title":"Lower bound one for the Sendov radius"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/upper-bound-two","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.upperBoundTwo","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#upper-bound-two","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"fully_proved","summary":"This is the proposition that the Sendov radius is at most $2$.","title":"Upper bound two for the Sendov radius"},{"id":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/sendov-conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.SendovRadiusConstantWanted.sendovConjecture_holds","parent":16667,"result":true,"route":"../roadmap/Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture/index.html#sendov-conjecture-holds","scope":"Analysis/complex-analysis/zeros-of-polynomials/sendov-s-conjecture","status":"can_prove","summary":"Sendov's conjecture is true: the Sendov radius is at most $1$.","title":"Sendov's conjecture holds"},{"data_url":"scopes/Analysis/complex-analysis/several-complex-variables.json","id":"Analysis/complex-analysis/several-complex-variables","kind":"scope","parent":16364,"result":false,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/index.html","scope":"Analysis/complex-analysis/several-complex-variables","status":"fully_proved","summary":"Holomorphic functions on C^n: Hartogs extension and separate analyticity, Weierstrass preparation, complex Hessians.","title":"Several complex variables"},{"id":"Analysis/complex-analysis/several-complex-variables/weierstrass-preparation","kind":"node","lean":"MathlibExt.Analysis.Complex.WeierstrassPreparationWanted.weierstrass_preparation","parent":16677,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/index.html#weierstrass-preparation","scope":"Analysis/complex-analysis/several-complex-variables","status":"fully_proved","summary":"Let $E$ be a finite-dimensional normed vector space over $\\mathbb{C}$ and let $f : E \\times \\mathbb{C} \\to \\mathbb{C}$ be analytic at $0$. Suppose that for some $d \\in \\mathbb{N}$ and some $g : \\mathbb{C} \\to \\mathbb{C}$ analytic at $0$ with $g(0) \\ne 0$, $f(0, w) = w^d g(w)$ for all $w$ in a neighbourhood of $0$. Then there are functions $a_0, \\dots, a_{d-1} : E \\to \\mathbb{C}$, each analytic at $0$ with\u2026","title":"Weierstrass preparation theorem"},{"data_url":"scopes/Analysis/complex-analysis/several-complex-variables/complex-hessian.json","id":"Analysis/complex-analysis/several-complex-variables/complex-hessian","kind":"scope","navigation":"module","parent":16677,"result":false,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/complex-hessian/index.html","scope":"Analysis/complex-analysis/several-complex-variables/complex-hessian","status":"fully_proved","title":"Complex Hessian"},{"id":"Analysis/complex-analysis/several-complex-variables/complex-hessian/real-hessian","kind":"node","lean":"MetaMathlibExt.realHessian","parent":16679,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/complex-hessian/index.html#real-hessian","scope":"Analysis/complex-analysis/several-complex-variables/complex-hessian","status":"fully_proved","summary":"For a function $u : \\mathbb{C}^n \\to \\mathbb{R}$, with $\\mathbb{C}^n$ viewed as a real normed space, write $Du(w)$ for the real Fr\u00e9chet derivative of $u$ at $w$. For points $z, a, b \\in \\mathbb{C}^n$, the real Hessian $D^2u(z)(a, b)$ is the real number obtained by taking the real Fr\u00e9chet derivative at $z$ of the map $w \\mapsto Du(w)$, applying it to $a$, and applying the result to $b$.","title":"Real Hessian"},{"id":"Analysis/complex-analysis/several-complex-variables/complex-hessian/complex-hessian","kind":"node","lean":"MetaMathlibExt.complexHessian","parent":16679,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/complex-hessian/index.html#complex-hessian","scope":"Analysis/complex-analysis/several-complex-variables/complex-hessian","status":"fully_proved","summary":"Let $u : \\mathbb{C}^n \\to \\mathbb{R}$ and $z \\in \\mathbb{C}^n$, write $e_1, \\dots, e_n$ for the standard basis of $\\mathbb{C}^n$, write $i$ for the imaginary unit, and for $a, b \\in \\mathbb{C}^n$ write $D^2u(z)(a, b)$ for the real Hessian of $u$ at $z$. The complex Hessian of $u$ at $z$ is the $n \\times n$ complex matrix whose $(j, k)$ entry is","title":"Complex Hessian"},{"id":"Analysis/complex-analysis/several-complex-variables/complex-hessian/real-hessian-const","kind":"node","lean":"MetaMathlibExt.realHessian_const","parent":16679,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/complex-hessian/index.html#real-hessian-const","scope":"Analysis/complex-analysis/several-complex-variables/complex-hessian","status":"fully_proved","summary":"For every real constant $c$, the function $u : \\mathbb{C}^n \\to \\mathbb{R}$ with constant value $c$ has real Hessian $D^2u(z)(a, b) = 0$ for all $z, a, b \\in \\mathbb{C}^n$.","title":"Constant functions have zero real Hessian"},{"id":"Analysis/complex-analysis/several-complex-variables/complex-hessian/complex-hessian-const","kind":"node","lean":"MetaMathlibExt.complexHessian_const","parent":16679,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/complex-hessian/index.html#complex-hessian-const","scope":"Analysis/complex-analysis/several-complex-variables/complex-hessian","status":"fully_proved","summary":"For every real constant $c$, the function $u : \\mathbb{C}^n \\to \\mathbb{R}$ with constant value $c$ has complex Hessian equal to the zero $n \\times n$ matrix at every point $z \\in \\mathbb{C}^n$.","title":"Constant functions have zero complex Hessian"},{"data_url":"scopes/Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem.json","id":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem","kind":"scope","navigation":"module","parent":16677,"result":false,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/index.html","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem","status":"fully_proved","title":"Hartogs extension theorem"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/continuous-on-fderiv-of-differentiable-o","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.continuousOn_fderiv_of_differentiableOn","parent":16684,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/index.html#continuous-on-fderiv-of-differentiable-o","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $s$ be an open subset of $\\mathbb{C}^n$ and let $f : \\mathbb{C}^n \\to \\mathbb{C}$ be complex differentiable at every point of $s$. Then the complex derivative $x \\mapsto Df(x)$, with values in the complex linear maps $\\mathbb{C}^n \\to \\mathbb{C}$, is continuous on $s$. Here $\\mathbb{C}^n$ carries the Euclidean norm.","title":"Holomorphic functions on open subsets of \u2102\u207f have continuous derivative"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/bar-deriv","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.barDeriv","parent":16684,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/index.html#bar-deriv","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem","status":"fully_proved","summary":"For $f : \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$, the Wirtinger derivative of $f$ at $z$ is $\\frac{\\partial f}{\\partial \\bar z}(z) = \\frac{1}{2}\\bigl(Df(z)(1) + i\\,Df(z)(i)\\bigr)$, where $Df(z)$ is the real derivative of $f$ at $z$, taken to be $0$ when $f$ is not real differentiable at $z$. When $f$ is real differentiable at $z$, this equals\u2026","title":"The Wirtinger derivative with respect to z\u0304"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/bar-deriv-eq-zero-of-differentiable-at","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.barDeriv_eq_zero_of_differentiableAt","parent":16684,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/index.html#bar-deriv-eq-zero-of-differentiable-at","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$. If $f$ is complex differentiable at $z$, then the Wirtinger derivative $\\partial f / \\partial \\bar z$ of $f$ vanishes at $z$.","title":"Vanishing of the Wirtinger derivative at points of complex differentiability"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/differentiable-at-of-bar-deriv-eq-zero","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.differentiableAt_of_barDeriv_eq_zero","parent":16684,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/index.html#differentiable-at-of-bar-deriv-eq-zero","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$. If $f$ is real differentiable at $z$ and its Wirtinger derivative $\\partial f / \\partial \\bar z$ vanishes at $z$, then $f$ is complex differentiable at $z$.","title":"Cauchy\u2013Riemann criterion for complex differentiability"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/exists-differentiable-bar-deriv-eq-of-ha","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.exists_differentiable_barDeriv_eq_of_hasCompactSupport","parent":16684,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/index.html#exists-differentiable-bar-deriv-eq-of-ha","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem","status":"fully_proved","summary":"Let $g : \\mathbb{C} \\to \\mathbb{C}$ be real differentiable on all of $\\mathbb{C}$, with continuous real derivative and compact support. Then there is a function $u : \\mathbb{C} \\to \\mathbb{C}$, real differentiable on all of $\\mathbb{C}$, whose Wirtinger derivative satisfies $\\frac{\\partial u}{\\partial \\bar z}(z) = g(z)$ for every $z \\in \\mathbb{C}$.","title":"Solving the \u2202\u0304-equation for compactly supported C\u00b9 data"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/hartogs-extension","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.hartogs_extension","parent":16684,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem/index.html#hartogs-extension","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-extension-theorem","status":"fully_proved","summary":"Let $n \\ge 2$, let $U \\subseteq \\mathbb{C}^n$ be open and connected, and let $K \\subseteq U$ be compact with $U \\setminus K$ connected. Then for every function $f : \\mathbb{C}^n \\to \\mathbb{C}$ that is holomorphic on $U \\setminus K$ there is a function $F : \\mathbb{C}^n \\to \\mathbb{C}$ that is holomorphic on $U$ and agrees with $f$ on $U \\setminus K$. Here $\\mathbb{C}^n$ carries the Euclidean norm.","title":"Hartogs extension theorem"},{"data_url":"scopes/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity.json","id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","kind":"scope","navigation":"module","parent":16677,"result":false,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","title":"Hartogs' theorem on separate analyticity"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/hartogs-pos-log-norm-le-circle-average","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.hartogs_posLog_norm_le_circleAverage","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#hartogs-pos-log-norm-le-circle-average","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"Let $c \\in \\mathbb{C}$, $R > 0$ and let $f : \\mathbb{C} \\to \\mathbb{C}$ be analytic at every point of the closed disc $|z - c| \\le R$. Then","title":"Sub-mean-value inequality for the positive logarithm of a holomorphic function"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/differentiable-at-uncurry-of-separately","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.differentiableAt_uncurry_of_separately_differentiable_of_bounded","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#differentiable-at-uncurry-of-separately","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"Let $E$ be a complex normed space and let $f(z, w)$ be a complex-valued function of $z \\in E$ and $w \\in \\mathbb{C}$ such that $z \\mapsto f(z, w)$ is complex differentiable on all of $E$ for every $w \\in \\mathbb{C}$ and $w \\mapsto f(z, w)$ is complex differentiable on all of $\\mathbb{C}$ for every $z \\in E$. If there is a real number $M$ with $|f(z, w)| \\le M$ whenever $\\|z\\| \\le 1$ and $|w| \\le 1$, then $f$, as a\u2026","title":"Osgood's lemma at the origin"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/differentiable-at-uncurry-of-separately-2","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.differentiableAt_uncurry_of_separately_differentiable_of_locally_bounded","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#differentiable-at-uncurry-of-separately-2","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"Let $E$ be a complex normed space and let $f(z, w)$ be a complex-valued function of $z \\in E$ and $w \\in \\mathbb{C}$ such that $z \\mapsto f(z, w)$ is complex differentiable on all of $E$ for every $w \\in \\mathbb{C}$ and $w \\mapsto f(z, w)$ is complex differentiable on all of $\\mathbb{C}$ for every $z \\in E$. Let $U \\subseteq E$ and $V \\subseteq \\mathbb{C}$ be open sets and $M$ a real number with $|f(z, w)| \\le M$\u2026","title":"Osgood's lemma for functions bounded on open sets"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/differentiable-on-cauchy-coefficient-of","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.differentiableOn_cauchyCoefficient_of_separately_differentiable_of_bounded","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#differentiable-on-cauchy-coefficient-of","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"Let $E$ be a finite-dimensional complex normed space, let $z_0 \\in E$, $w_0 \\in \\mathbb{C}$, $A > 0$, $B > 0$ and $M \\in \\mathbb{R}$, and let $f(z, w)$ be a complex-valued function of $z \\in E$ and $w \\in \\mathbb{C}$ such that $z \\mapsto f(z, w)$ is complex differentiable on all of $E$ for every $w$ and $w \\mapsto f(z, w)$ is complex differentiable on all of $\\mathbb{C}$ for every $z$. Suppose $|f(z, w)| \\le M$\u2026","title":"Holomorphic dependence of Cauchy coefficients on a parameter"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/norm-cauchy-coefficient-le","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.norm_cauchyCoefficient_le","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#norm-cauchy-coefficient-le","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be complex differentiable on all of $\\mathbb{C}$, let $c \\in \\mathbb{C}$, $R > 0$ and $M \\in \\mathbb{R}$ with $|f(z)| \\le M$ for all $z$ with $|z - c| = R$. Then for every $k \\in \\mathbb{N}$ the $k$-th coefficient of the Cauchy power series of $f$ about $c$ on the circle of radius $R$,","title":"Cauchy's estimate for Taylor coefficients"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/hartogs-polydisc-integral","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.hartogsPolydiscIntegral","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#hartogs-polydisc-integral","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, a function $u : \\mathbb{C}^n \\to \\mathbb{R}$, a centre $c \\in \\mathbb{C}^n$ and a radius $R \\in \\mathbb{R}$, the iterated polydisc integral $I(u, c, R)$ is defined by recursion on $n$. For $n = 0$ it is $u(c)$. For $n = m + 1$, writing $c = (c_0, c_1, \\dots, c_m)$, it is","title":"Iterated integral over a polydisc"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/hartogs-polydisc-submean","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.hartogsPolydisc_submean","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#hartogs-polydisc-submean","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $u : \\mathbb{C}^n \\to \\mathbb{R}$ be continuous, let $c \\in \\mathbb{C}^n$ and $R > 0$. Write $\\overline{P}(c, R) = \\{x \\in \\mathbb{C}^n : |x_j - c_j| \\le R \\text{ for all } j\\}$ for the closed polydisc, and for $x \\in \\mathbb{C}^n$ and $w \\in \\mathbb{C}$ write $x_{j \\leftarrow w}$ for $x$ with its $j$-th coordinate replaced by $w$. Suppose that for every $x \\in \\overline{P}(c, R)$, every\u2026","title":"Sub-mean-value inequality on a polydisc"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/hartogs-eventually-uniform-of-submean","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.hartogs_eventually_uniform_of_submean","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#hartogs-eventually-uniform-of-submean","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, $c \\in \\mathbb{C}^n$, $R > 0$ and $C \\ge 0$. For $\\rho \\ge 0$ write $\\overline{P}(c, \\rho) = \\{x \\in \\mathbb{C}^n : |x_j - c_j| \\le \\rho \\text{ for all } j\\}$ for the closed polydisc, and for $x \\in \\mathbb{C}^n$ and $w \\in \\mathbb{C}$ write $x_{j \\leftarrow w}$ for $x$ with its $j$-th coordinate replaced by $w$. Let $(u_k)_{k \\in \\mathbb{N}}$ be real-valued functions on $\\mathbb{C}^n$ such\u2026","title":"Hartogs' lemma for sub-mean-value functions"},{"id":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/hartogs-separate-analyticity","kind":"node","lean":"MathlibExt.Analysis.Complex.HartogsWanted.hartogs_separate_analyticity","parent":16691,"result":true,"route":"../roadmap/Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity/index.html#hartogs-separate-analyticity","scope":"Analysis/complex-analysis/several-complex-variables/hartogs-theorem-on-separate-analyticity","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and let $f : \\mathbb{C}^n \\to \\mathbb{C}$ be such that for every point $x \\in \\mathbb{C}^n$ and every coordinate $i$, the function $z \\mapsto f(x_1, \\dots, x_{i-1}, z, x_{i+1}, \\dots, x_n)$ is complex differentiable on all of $\\mathbb{C}$. Then $f$ is complex differentiable on all of $\\mathbb{C}^n$.","title":"Hartogs' theorem on separate analyticity"},{"data_url":"scopes/Analysis/complex-analysis/analytic-function-spaces.json","id":"Analysis/complex-analysis/analytic-function-spaces","kind":"scope","parent":16364,"result":false,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/index.html","scope":"Analysis/complex-analysis/analytic-function-spaces","status":"can_prove","summary":"Hardy, Bergman, Dirichlet and Fock spaces and their reproducing kernels, H^\u221e, boundary behaviour (Fatou, F. and M. Riesz), cyclicity and polynomial density.","title":"Spaces of analytic functions"},{"id":"Analysis/complex-analysis/analytic-function-spaces/riesz-brothers-theorem","kind":"node","lean":"MathlibExt.Analysis.RieszBrothersWanted.Analysis.riesz_brothers_theorem","parent":16701,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/index.html#riesz-brothers-theorem","scope":"Analysis/complex-analysis/analytic-function-spaces","status":"can_prove","summary":"Let $\\mu$ be a complex Borel measure on the circle $\\mathbb{T} = \\mathbb{R}/\\mathbb{Z}$. If $\\int_{\\mathbb{T}} e^{2\\pi i n z} \\, d\\mu(z) = 0$ for every integer $n \\ge 1$, then $\\mu$ is absolutely continuous with respect to the Haar probability measure on $\\mathbb{T}$.","title":"F. and M. Riesz theorem"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fatou-radial-limit","kind":"node","lean":"MathlibExt.Analysis.Complex.FatouRadialWanted.fatou_radial_limit","parent":16701,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/index.html#fatou-radial-limit","scope":"Analysis/complex-analysis/analytic-function-spaces","status":"fully_proved","summary":"Let $f : \\mathbb{C} \\to \\mathbb{C}$ be holomorphic and bounded on the open unit disc $|z| < 1$. Then for Lebesgue-almost every $\\theta \\in [0, 2\\pi]$ the radial limit $\\lim_{r \\to 1^-} f(r e^{i\\theta})$ exists in $\\mathbb{C}$.","title":"Fatou's theorem on radial limits"},{"data_url":"scopes/Analysis/complex-analysis/analytic-function-spaces/bergman-space.json","id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","kind":"scope","navigation":"module","parent":16701,"result":false,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"can_prove","title":"Bergman space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/has-sum-succ-mul-pow-complex","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.hasSum_succ_mul_pow_complex","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#has-sum-succ-mul-pow-complex","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The Cauchy product of the geometric series with itself: the series `\u2211' n, (n + 1) * q\u207f` sums to `(1 - q)\u207b\u00b9 ^ 2`.","title":"Square of the complex geometric series"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/has-sum-succ-mul-pow-real","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.hasSum_succ_mul_pow_real","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#has-sum-succ-mul-pow-real","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The real form of the Cauchy product above.","title":"Square of the real geometric series"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/norm-sq-bergman-coeff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.norm_sq_bergman_coeff","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#norm-sq-bergman-coeff","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The squared norms of the Bergman kernel coefficients are `(n + 1) * \u2016z\u2016 ^ (2 * n)`.","title":"Squared norms of the Bergman kernel coefficients"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/mem-p-bergman-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.mem\u2113p_bergmanKernel","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#mem-p-bergman-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The Bergman kernel coefficients are in `\u2113\u00b2` when `z` is in the disc.","title":"Bergman kernel coefficients are square-summable"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanKernel","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The Bergman kernel at `z`: the `\u2113\u00b2(\u2115, \u2102)` sequence with coefficients `\u221a(n + 1) * (conj z)\u207f`.","title":"Bergman kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-kernel-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanKernel_zero","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-kernel-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The Bergman kernel at `0` is the first standard basis vector.","title":"Bergman kernel at the origin"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/norm-sq-bergman-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.norm_sq_bergmanKernel","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#norm-sq-bergman-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The squared norm of the Bergman kernel is `1 / (1 - \u2016z\u2016\u00b2)\u00b2`.","title":"Squared norm of the Bergman kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/norm-bergman-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.norm_bergmanKernel","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#norm-bergman-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The norm of the Bergman kernel is `1 / (1 - \u2016z\u2016\u00b2)`.","title":"Norm of the Bergman kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-eval","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanEval","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-eval","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"Point evaluation on `A\u00b2(\ud835\udd3b)`: the inner product with the Bergman kernel.","title":"Point evaluation on the Bergman space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-eval-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanEval_eq_tsum","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-eval-eq-tsum","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"Point evaluation is the Taylor series `\u2211' n, a n * \u221a(n + 1) * z\u207f`.","title":"Bergman point evaluation as a Taylor series"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/summable-bergman-eval-terms","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.summable_bergmanEval_terms","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#summable-bergman-eval-terms","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The Taylor series of an `A\u00b2` function converges at every point of the disc.","title":"Convergence of the Taylor series of a Bergman space function"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-eval-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanEvalCLM","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-eval-clm","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"Point evaluation as a bounded linear functional on `A\u00b2(\ud835\udd3b)`.","title":"Bounded point-evaluation functional on the Bergman space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-eval-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanEvalCLM_apply","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-eval-clm-apply","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The bounded point-evaluation functional acts by the inner product with the Bergman kernel.","title":"Bergman point evaluation is the inner product with the kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/norm-bergman-eval-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.norm_bergmanEvalCLM","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#norm-bergman-eval-clm","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The operator norm of point evaluation is `1 / (1 - \u2016z\u2016\u00b2)`.","title":"Operator norm of Bergman point evaluation"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/norm-bergman-eval-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.norm_bergmanEval_le","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#norm-bergman-eval-le","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The pointwise bound for `A\u00b2` functions: `\u2016f(z)\u2016 \u2264 \u2016f\u2016 / (1 - \u2016z\u2016\u00b2)`.","title":"Pointwise bound for Bergman space functions"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-eval-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanEval_zero","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-eval-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"Evaluation at `0` is the constant Taylor coefficient.","title":"Bergman evaluation at the origin"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-eval-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanEval_single","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-eval-single","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"Evaluation of a monomial `c * \u221a(n + 1) * z\u207f` (a single coefficient).","title":"Bergman evaluation of a monomial"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/inner-bergman-kernel-right","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.inner_bergmanKernel_right","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#inner-bergman-kernel-right","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The reproducing property in the other order: `\u27eaa, k_z\u27eb = conj (f(z))`.","title":"Reproducing property of the Bergman kernel, conjugate form"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/inner-bergman-kernel-bergman-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.inner_bergmanKernel_bergmanKernel","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#inner-bergman-kernel-bergman-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The inner product of two Bergman kernels is the Bergman kernel function `(1 - z * conj w)\u207b\u00b9 ^ 2`.","title":"Inner product of two Bergman kernels"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-eval-bergman-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanEval_bergmanKernel","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-eval-bergman-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"Evaluation of the kernel at its own point: `f_z(k_z) = (1 - \u2016z\u2016\u00b2)\u207b\u00b9 ^ 2`, so the pointwise bound is sharp.","title":"Bergman kernel evaluated at its own point"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-monomial","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanMonomial","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-monomial","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The rescaled monomial `e\u2099(z) = \u221a(n + 1) z\u207f` for normalized area measure `dA / \u03c0`, represented by the `n`th standard basis vector in the coefficient model.","title":"Rescaled Bergman monomial"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-monomial-orthonormal","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanMonomial_orthonormal","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-monomial-orthonormal","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The coordinate inner products of the rescaled monomials equal the Kronecker delta.","title":"Inner products of rescaled Bergman monomials are Kronecker deltas"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/orthonormal-bergman-monomial","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.orthonormal_bergmanMonomial","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#orthonormal-bergman-monomial","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The rescaled monomial family satisfies Mathlib's standard orthonormal predicate.","title":"Rescaled Bergman monomials are orthonormal"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman-coeff-eq-inner","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BergmanSpace.bergmanCoeff_eq_inner","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman-coeff-eq-inner","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The coefficient of `a` at `n` is the inner product with `e\u2099`.","title":"Bergman coefficients as inner products with monomials"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/disc","kind":"node","lean":"MathlibExt.Analysis.BergmanCyclicityWanted.Disc","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#disc","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"The set $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$.","title":"The open unit disc"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/bergman","kind":"node","lean":"MathlibExt.Analysis.BergmanCyclicityWanted.Bergman","parent":16704,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bergman space "],["math","\\(A^2\\)"],["text"," of the unit disc"]],"source":"Bergman space $A^2$ of the unit disc"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#bergman","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"A function $f : \\mathbb{C} \\to \\mathbb{C}$ belongs to the Bergman space $A^2$ when it is analytic on the open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$ and square-integrable on $\\mathbb{D}$ with respect to Lebesgue area measure $dA$: $f$ is almost everywhere strongly measurable on $\\mathbb{D}$ and $\\int_{\\mathbb{D}} |f|^2 \\, dA < \\infty$.","title":"Bergman space $A^2$ of the unit disc"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/inv","kind":"node","lean":"MathlibExt.Analysis.BergmanCyclicityWanted.Inv","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#inv","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$, the function $z \\mapsto f(z)^{-1}$, with the convention $0^{-1} = 0$ at the zeros of $f$.","title":"Pointwise reciprocal"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/dense","kind":"node","lean":"MathlibExt.Analysis.BergmanCyclicityWanted.Dense","parent":16704,"reading_title":{"format":"inline-parts-v1","parts":[["text","Density of polynomial multiples in "],["math","\\(A^2\\)"]],"source":"Density of polynomial multiples in $A^2$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#dense","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"Write $\\mathbb{D}$ for the open unit disc and $dA$ for Lebesgue area measure on it. For a function $f : \\mathbb{C} \\to \\mathbb{C}$, the statement that the polynomial multiples of $f$ are dense in the Bergman space $A^2$: for every $g \\in A^2$ and every $\\varepsilon > 0$ there is a complex polynomial $p$ with $\\left(\\int_{\\mathbb{D}} |p f - g|^2 \\, dA\\right)^{1/2} < \\varepsilon$.","title":"Density of polynomial multiples in $A^2$"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/conjecture","kind":"node","lean":"MathlibExt.Analysis.BergmanCyclicityWanted.conjecture","parent":16704,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#conjecture","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"fully_proved","summary":"Write $\\mathbb{D}$ for the open unit disc and $dA$ for Lebesgue area measure on it. The statement that for every function $f : \\mathbb{C} \\to \\mathbb{C}$ with no zeros in $\\mathbb{D}$ such that $f$ and its pointwise reciprocal $1/f$ both belong to the Bergman space $A^2$, the polynomial multiples of $f$ are dense in $A^2$: for every $g \\in A^2$ and every $\\varepsilon > 0$ there is a complex polynomial $p$ with\u2026","title":"Bergman cyclicity conjecture"},{"id":"Analysis/complex-analysis/analytic-function-spaces/bergman-space/conjecture-refuted","kind":"node","lean":"MathlibExt.Analysis.BergmanCyclicityWanted.conjecture_refuted","parent":16704,"reading_title":{"format":"inline-parts-v1","parts":[["text","Invertible non-cyclic functions in the Bergman space "],["math","\\(A^2\\)"]],"source":"Invertible non-cyclic functions in the Bergman space $A^2$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/bergman-space/index.html#conjecture-refuted","scope":"Analysis/complex-analysis/analytic-function-spaces/bergman-space","status":"can_prove","summary":"Write $\\mathbb{D}$ for the open unit disc and $dA$ for Lebesgue area measure on it. There is a function $f : \\mathbb{C} \\to \\mathbb{C}$ with no zeros in $\\mathbb{D}$ such that $f$ and its pointwise reciprocal $1/f$ both belong to the Bergman space $A^2$, but the polynomial multiples of $f$ are not dense in $A^2$: there are $g \\in A^2$ and $\\varepsilon > 0$ with\u2026","title":"Invertible non-cyclic functions in the Bergman space $A^2$"},{"data_url":"scopes/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space.json","id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","kind":"scope","navigation":"module","parent":16701,"result":false,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"can_prove","title":"Dirichlet space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/has-sum-div-succ-pow-real","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.hasSum_div_succ_pow_real","parent":16735,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form of "],["math","\\(\\sum r^n/(n+1)\\)"],["text"," for real "],["math","\\(0<r<1\\)"]],"source":"Closed form of $\\sum r^n/(n+1)$ for real $0<r<1$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#has-sum-div-succ-pow-real","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"Closed form of $\\sum r^n/(n+1)$ for real $0\\lt r<1$. The power series `\u2211' n, r\u207f / (n + 1)` sums to `-log (1 - r) / r` for `0 < r < 1`: factor `1 / r` out of the logarithm series.","title":"Closed form of $\\sum r^n/(n+1)$ for real $0<r<1$"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/has-sum-div-succ-pow-complex","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.hasSum_div_succ_pow_complex","parent":16735,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form of "],["math","\\(\\sum u^n/(n+1)\\)"],["text"," for complex "],["math","\\(u\\)"],["text"," in the unit disc"]],"source":"Closed form of $\\sum u^n/(n+1)$ for complex $u$ in the unit disc"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#has-sum-div-succ-pow-complex","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The complex form: `\u2211' n, u\u207f / (n + 1)` sums to `-log (1 - u) / u` for `u \u2260 0` in the unit disc.","title":"Closed form of $\\sum u^n/(n+1)$ for complex $u$ in the unit disc"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-sq-dirichlet-coeff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_sq_dirichlet_coeff","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-sq-dirichlet-coeff","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The squared norms of the Dirichlet kernel coefficients are `\u2016z\u2016 ^ (2 * n) / (n + 1)`.","title":"Squared norms of the Dirichlet kernel coefficients"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/summable-dirichlet-kernel-terms","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.summable_dirichletKernel_terms","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#summable-dirichlet-kernel-terms","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The Dirichlet kernel coefficients are summable in the `\u2113\u00b2` sense: `\u2211' n, (\u2016z\u2016\u00b2)\u207f / (n + 1)` converges, by comparison with the geometric series.","title":"Square-summability of the Dirichlet kernel coefficients"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/mem-p-dirichlet-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.mem\u2113p_dirichletKernel","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#mem-p-dirichlet-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The Dirichlet kernel coefficients are in `\u2113\u00b2` when `z` is in the disc.","title":"Dirichlet kernel coefficients lie in \u2113\u00b2"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletKernel","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The Dirichlet kernel at `z`: the `\u2113\u00b2(\u2115, \u2102)` sequence with coefficients `(conj z)\u207f / \u221a(n + 1)`.","title":"Reproducing kernel of the Dirichlet space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-kernel-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletKernel_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-kernel-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The Dirichlet kernel at `0` is the first standard basis vector.","title":"Dirichlet kernel at the origin is the first basis vector"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-sq-dirichlet-kernel-of-ne-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_sq_dirichletKernel_of_ne_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-sq-dirichlet-kernel-of-ne-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The squared norm of the Dirichlet kernel is `-log (1 - \u2016z\u2016\u00b2) / \u2016z\u2016\u00b2`.","title":"Squared norm of the Dirichlet kernel away from the origin"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-sq-dirichlet-kernel-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_sq_dirichletKernel_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-sq-dirichlet-kernel-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The squared norm of the Dirichlet kernel at `0` is `1`.","title":"Squared norm of the Dirichlet kernel at the origin"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-dirichlet-kernel-of-ne-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_dirichletKernel_of_ne_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-dirichlet-kernel-of-ne-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The norm of the Dirichlet kernel is `\u221a(-log (1 - \u2016z\u2016\u00b2) / \u2016z\u2016\u00b2)`.","title":"Norm of the Dirichlet kernel away from the origin"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-dirichlet-kernel-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_dirichletKernel_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-dirichlet-kernel-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The norm of the Dirichlet kernel at `0` is `1`.","title":"Norm of the Dirichlet kernel at the origin"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-sq-dirichlet-kernel-le-inv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_sq_dirichletKernel_le_inv","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-sq-dirichlet-kernel-le-inv","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The Dirichlet kernel norm is bounded by the Hardy kernel norm: `\u2016k_z\u2016\u00b2 \u2264 1 / (1 - \u2016z\u2016\u00b2)`, via `log x \u2264 x - 1` at `x = (1 - \u2016z\u2016\u00b2)\u207b\u00b9`.","title":"Dirichlet kernel norm is bounded by the Szeg\u0151 kernel norm"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-eval","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletEval","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-eval","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"Point evaluation on `\ud835\udc9f(\ud835\udd3b)`: the inner product with the Dirichlet kernel.","title":"Point evaluation on the Dirichlet space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-eval-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletEval_eq_tsum","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-eval-eq-tsum","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"Point evaluation is the Taylor series `\u2211' n, a n * z\u207f / \u221a(n + 1)`.","title":"Dirichlet-space point evaluation as a Taylor series"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/summable-dirichlet-eval-terms","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.summable_dirichletEval_terms","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#summable-dirichlet-eval-terms","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The Taylor series of a `\ud835\udc9f` function converges absolutely at every point of the disc.","title":"Absolute convergence of Taylor series in the Dirichlet space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-eval-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletEvalCLM","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-eval-clm","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"Point evaluation as a bounded linear functional on `\ud835\udc9f(\ud835\udd3b)`.","title":"Point evaluation as a bounded functional on the Dirichlet space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-eval-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletEvalCLM_apply","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-eval-clm-apply","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The bounded point-evaluation functional acts by the inner product with the Dirichlet kernel.","title":"Reproducing property of the Dirichlet kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-dirichlet-eval-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_dirichletEvalCLM","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-dirichlet-eval-clm","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The operator norm of point evaluation is the norm of the kernel.","title":"Norm of Dirichlet-space point evaluation equals the kernel norm"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-dirichlet-eval-clm-of-ne-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_dirichletEvalCLM_of_ne_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-dirichlet-eval-clm-of-ne-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The operator norm of point evaluation is `\u221a(-log (1 - \u2016z\u2016\u00b2) / \u2016z\u2016\u00b2)`.","title":"Closed-form norm of Dirichlet-space point evaluation"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-dirichlet-eval-clm-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_dirichletEvalCLM_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-dirichlet-eval-clm-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The operator norm of point evaluation at `0` is `1`.","title":"Norm of Dirichlet-space point evaluation at the origin"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/norm-dirichlet-eval-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.norm_dirichletEval_le","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#norm-dirichlet-eval-le","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The pointwise bound for `\ud835\udc9f` functions: `\u2016f(z)\u2016 \u2264 \u2016f\u2016 * \u2016k_z\u2016`.","title":"Pointwise bound for Dirichlet-space functions"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-eval-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletEval_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-eval-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"Evaluation at `0` is the constant Taylor coefficient.","title":"Dirichlet-space evaluation at the origin is the constant coefficient"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-eval-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletEval_single","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-eval-single","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"Evaluation of a monomial `c * z\u207f / \u221a(n + 1)` (a single coefficient).","title":"Dirichlet-space evaluation of a single monomial"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/inner-dirichlet-kernel-right","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.inner_dirichletKernel_right","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#inner-dirichlet-kernel-right","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The reproducing property in the other order: `\u27eaa, k_z\u27eb = conj (f(z))`.","title":"Conjugate reproducing property of the Dirichlet kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/inner-dirichlet-kernel-dirichlet-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.inner_dirichletKernel_dirichletKernel","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#inner-dirichlet-kernel-dirichlet-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The inner product of two Dirichlet kernels is the power series `\u2211' n, (z * conj w)\u207f / (n + 1)`.","title":"Inner product of two Dirichlet kernels as a power series"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/inner-dirichlet-kernel-dirichlet-kernel-2","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.inner_dirichletKernel_dirichletKernel_eq_log","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#inner-dirichlet-kernel-dirichlet-kernel-2","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The inner product of two Dirichlet kernels is the Dirichlet kernel function `-log (1 - z * conj w) / (z * conj w)`.","title":"Logarithmic formula for the inner product of Dirichlet kernels"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-eval-dirichlet-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletEval_dirichletKernel","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-eval-dirichlet-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"Evaluation of the kernel at its own point: `f_z(k_z) = \u2016k_z\u2016\u00b2`.","title":"Dirichlet kernel evaluated at its own point"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-eval-dirichlet-kernel-of-ne-ze","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.DirichletSpace.dirichletEval_dirichletKernel_of_ne_zero","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-eval-dirichlet-kernel-of-ne-ze","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"Evaluation of the kernel at its own point, in closed form: `f_z(k_z) = -log (1 - \u2016z\u2016\u00b2) / \u2016z\u2016\u00b2`, so the pointwise bound is sharp.","title":"Sharpness of the Dirichlet-space pointwise bound"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/dirichlet-space","kind":"node","lean":"MathlibExt.Analysis.DirichletSpacePolynomialDensityWanted.DirichletSpace","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#dirichlet-space","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The Dirichlet space $\\mathcal{D}$ consists of the functions $f : \\mathbb{C} \\to \\mathbb{C}$ that are analytic on the open unit disc $\\{z : |z| < 1\\}$ and have finite Dirichlet integral: $|f'|^2$ is integrable on the disc with respect to Lebesgue (area) measure. The values of $f$ outside the disc are unconstrained.","title":"The Dirichlet space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/poly-times","kind":"node","lean":"MathlibExt.Analysis.DirichletSpacePolynomialDensityWanted.polyTimes","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#poly-times","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"For $g$ in the Dirichlet space $\\mathcal{D}$, the set $Pg$ consists of the functions $h \\in \\mathcal{D}$ for which there is a polynomial $p$ with complex coefficients such that $h(z) = p(z) g(z)$ for all $|z| < 1$; only values on the open unit disc are compared.","title":"Polynomial multiples of a function in the Dirichlet space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/is-dirichlet-dense","kind":"node","lean":"MathlibExt.Analysis.DirichletSpacePolynomialDensityWanted.IsDirichletDense","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#is-dirichlet-dense","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"A set $S$ of elements of the Dirichlet space $\\mathcal{D}$ is dense in the Dirichlet norm when for every $u \\in \\mathcal{D}$ and every real $\\varepsilon > 0$ there is $h \\in S$ with","title":"Density in the Dirichlet norm"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/conjecture","kind":"node","lean":"MathlibExt.Analysis.DirichletSpacePolynomialDensityWanted.conjecture","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#conjecture","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"fully_proved","summary":"The statement that for all $f$ and $g$ in the Dirichlet space $\\mathcal{D}$: if the polynomial multiples $Pg$ are dense in $\\mathcal{D}$ in the Dirichlet norm and $|g(z)| \\le |f(z)|$ for all $|z| < 1$, then the polynomial multiples $Pf$ are also dense in $\\mathcal{D}$ in the Dirichlet norm.","title":"Hayman's Problem 8.11: polynomial density under domination"},{"id":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.DirichletSpacePolynomialDensityWanted.conjecture_holds","parent":16735,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/dirichlet-space/index.html#conjecture-holds","scope":"Analysis/complex-analysis/analytic-function-spaces/dirichlet-space","status":"can_prove","summary":"For all $f$ and $g$ in the Dirichlet space $\\mathcal{D}$: if the polynomial multiples $Pg$ are dense in $\\mathcal{D}$ in the Dirichlet norm and $|g(z)| \\le |f(z)|$ for all $|z| < 1$, then the polynomial multiples $Pf$ are dense in $\\mathcal{D}$ in the Dirichlet norm. This answers Hayman's Problem 8.11 positively.","title":"Polynomial density passes to dominating functions in the Dirichlet space"},{"data_url":"scopes/Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h.json","id":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h","kind":"scope","navigation":"module","parent":16701,"result":false,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/index.html","scope":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h","status":"can_prove","title":"Finitely generated ideals of H^\u221e"},{"id":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/disc","kind":"node","lean":"MathlibExt.Analysis.RubelHInfinityCoherenceWanted.Disc","parent":16769,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/index.html#disc","scope":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h","status":"fully_proved","summary":"The open unit disc $\\mathbb{D} = \\{z \\in \\mathbb{C} : |z| < 1\\}$, the open ball of radius $1$ about $0$ in $\\mathbb{C}$, taken as a space in its own right whose points are the points of the disc.","title":"The open unit disc"},{"id":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/is-hinf","kind":"node","lean":"MathlibExt.Analysis.RubelHInfinityCoherenceWanted.IsHInf","parent":16769,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in "],["math","\\(H^\\infty\\)"],["text"," of the unit disc"]],"source":"Membership in $H^\\infty$ of the unit disc"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/index.html#is-hinf","scope":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h","status":"fully_proved","summary":"A complex-valued function $f$ on the open unit disc $\\mathbb{D}$ belongs to $H^\\infty$ when there is a function $F : \\mathbb{C} \\to \\mathbb{C}$ that is complex differentiable on $\\mathbb{D}$ and satisfies $f(z) = F(z)$ for every $z \\in \\mathbb{D}$, and there is a real number $C \\ge 0$ with $|f(z)| \\le C$ for every $z \\in \\mathbb{D}$.","title":"Membership in $H^\\infty$ of the unit disc"},{"id":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/hinfinity-subring","kind":"node","lean":"MathlibExt.Analysis.RubelHInfinityCoherenceWanted.HInfinitySubring","parent":16769,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ring "],["math","\\(H^\\infty\\)"],["text"," as a subring of functions on the disc"]],"source":"The ring $H^\\infty$ as a subring of functions on the disc"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/index.html#hinfinity-subring","scope":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h","status":"fully_proved","summary":"$H^\\infty$ is the subring of the ring of all complex-valued functions on the open unit disc $\\mathbb{D}$, under pointwise addition and multiplication, consisting of the functions that belong to $H^\\infty$: those that agree on $\\mathbb{D}$ with a function $\\mathbb{C} \\to \\mathbb{C}$ complex differentiable on $\\mathbb{D}$ and are bounded on $\\mathbb{D}$.","title":"The ring $H^\\infty$ as a subring of functions on the disc"},{"id":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/hinfinity","kind":"node","lean":"MathlibExt.Analysis.RubelHInfinityCoherenceWanted.HInfinity","parent":16769,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(H^\\infty\\)"]],"source":"Elements of $H^\\infty$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/index.html#hinfinity","scope":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h","status":"fully_proved","summary":"The space whose points are the members of the subring $H^\\infty$ of complex-valued functions on the open unit disc $\\mathbb{D}$, that is, the bounded functions on $\\mathbb{D}$ that agree on $\\mathbb{D}$ with a function complex differentiable there.","title":"Elements of $H^\\infty$"},{"id":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/rubel-problem524","kind":"node","lean":"MathlibExt.Analysis.RubelHInfinityCoherenceWanted.RubelProblem524","parent":16769,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rubel's problem on intersections of finitely generated ideals in "],["math","\\(H^\\infty\\)"]],"source":"Rubel's problem on intersections of finitely generated ideals in $H^\\infty$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/index.html#rubel-problem524","scope":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h","status":"fully_proved","summary":"The statement that for any two finitely generated ideals $I$ and $J$ of the commutative ring $H^\\infty$ of bounded holomorphic functions on the open unit disc, the intersection $I \\cap J$ is finitely generated.","title":"Rubel's problem on intersections of finitely generated ideals in $H^\\infty$"},{"id":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/rubel-problem524-holds","kind":"node","lean":"MathlibExt.Analysis.RubelHInfinityCoherenceWanted.RubelProblem524_holds","parent":16769,"reading_title":{"format":"inline-parts-v1","parts":[["text","McVoy\u2013Rubel theorem on finitely generated ideals of "],["math","\\(H^\\infty\\)"]],"source":"McVoy\u2013Rubel theorem on finitely generated ideals of $H^\\infty$"},"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h/index.html#rubel-problem524-holds","scope":"Analysis/complex-analysis/analytic-function-spaces/finitely-generated-ideals-of-h","status":"can_prove","summary":"For any two finitely generated ideals $I$ and $J$ of the commutative ring $H^\\infty$ of bounded holomorphic functions on the open unit disc, the intersection $I \\cap J$ is finitely generated. This answers Rubel's Problem 5.24 positively.","title":"McVoy\u2013Rubel theorem on finitely generated ideals of $H^\\infty$"},{"data_url":"scopes/Analysis/complex-analysis/analytic-function-spaces/fock-space.json","id":"Analysis/complex-analysis/analytic-function-spaces/fock-space","kind":"scope","navigation":"module","parent":16701,"result":false,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","title":"Fock space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/norm-sq-fock-coeff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.norm_sq_fock_coeff","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#norm-sq-fock-coeff","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The squared norms of the Fock kernel coefficients are `\u2016z\u2016 ^ (2 * n) / n!`.","title":"Squared norms of the Fock kernel coefficients"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/mem-p-fock-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.mem\u2113p_fockKernel","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#mem-p-fock-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The Fock kernel coefficients are in `\u2113\u00b2` for every `z : \u2102`, since `\u2211' n, \u2016z\u2016 ^ (2 * n) / n! = Real.exp \u2016z\u2016\u00b2`.","title":"Fock kernel coefficients lie in \u2113\u00b2"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockKernel","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The Fock kernel at `z`: the `\u2113\u00b2(\u2115, \u2102)` sequence with coefficients `(conj z)\u207f / \u221a(n!)`.","title":"Reproducing kernel of the Fock space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-kernel-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockKernel_zero","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-kernel-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The Fock kernel at `0` is the first standard basis vector.","title":"Fock kernel at the origin is the first basis vector"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/norm-sq-fock-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.norm_sq_fockKernel","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#norm-sq-fock-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The squared norm of the Fock kernel is `Real.exp \u2016z\u2016\u00b2`.","title":"Squared norm of the Fock kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/norm-fock-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.norm_fockKernel","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#norm-fock-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The norm of the Fock kernel is `Real.exp (\u2016z\u2016\u00b2 / 2)`.","title":"Norm of the Fock kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-eval","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockEval","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-eval","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"Point evaluation on the Fock space: the inner product with the Fock kernel.","title":"Point evaluation on the Fock space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-eval-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockEval_eq_tsum","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-eval-eq-tsum","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"Point evaluation is the Taylor series `\u2211' n, a n * z\u207f / \u221a(n!)`.","title":"Fock-space point evaluation as a Taylor series"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/summable-fock-eval-terms","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.summable_fockEval_terms","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#summable-fock-eval-terms","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The Taylor series of a Fock-space function converges absolutely at every point of `\u2102`.","title":"Absolute convergence of Taylor series in the Fock space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-eval-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockEvalCLM","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-eval-clm","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"Point evaluation as a bounded linear functional on the Fock space.","title":"Point evaluation as a bounded functional on the Fock space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-eval-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockEvalCLM_apply","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-eval-clm-apply","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The bounded point-evaluation functional acts by the inner product with the Fock kernel.","title":"Reproducing property of the Fock kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/norm-fock-eval-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.norm_fockEvalCLM","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#norm-fock-eval-clm","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The operator norm of point evaluation is `Real.exp (\u2016z\u2016\u00b2 / 2)`.","title":"Norm of Fock-space point evaluation"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/norm-fock-eval-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.norm_fockEval_le","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#norm-fock-eval-le","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The pointwise bound for Fock-space functions: `\u2016f(z)\u2016 \u2264 \u2016f\u2016 * Real.exp (\u2016z\u2016\u00b2 / 2)`.","title":"Pointwise bound for Fock-space functions"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-eval-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockEval_zero","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-eval-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"Evaluation at `0` is the constant Taylor coefficient.","title":"Fock-space evaluation at the origin is the constant coefficient"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-eval-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockEval_single","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-eval-single","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"Evaluation of a monomial `c * z\u207f / \u221a(n!)` (a single Taylor coefficient).","title":"Fock-space evaluation of a single monomial"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/inner-fock-kernel-right","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.inner_fockKernel_right","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#inner-fock-kernel-right","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The reproducing property in the other order: `\u27eaa, k_z\u27eb = conj (f(z))`.","title":"Conjugate reproducing property of the Fock kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/inner-fock-kernel-fock-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.inner_fockKernel_fockKernel","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#inner-fock-kernel-fock-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The inner product of two Fock kernels is the Fock kernel function `Complex.exp (z * conj w)`.","title":"Inner product of two Fock kernels is an exponential"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/norm-inner-fock-kernel-fock-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.norm_inner_fockKernel_fockKernel","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#norm-inner-fock-kernel-fock-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The norm of the inner product of two Fock kernels is `Real.exp (z * conj w).re`.","title":"Modulus of the inner product of two Fock kernels"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-eval-fock-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockEval_fockKernel","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-eval-fock-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"Evaluation of the kernel at its own point: `f_z(k_z) = Real.exp \u2016z\u2016\u00b2`, so the pointwise bound is sharp.","title":"Fock kernel evaluated at its own point"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-monomial","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockMonomial","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-monomial","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The normalized monomial `e\u2099(z) = z\u207f / \u221a(n!)`, represented by the `n`th standard basis vector in the Taylor-coefficient model of Fock space.","title":"Normalized monomials in the Fock space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-monomial-orthonormal","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockMonomial_orthonormal","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-monomial-orthonormal","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The rescaled monomials `e\u2099(z) = z\u207f / \u221a(n!)`, i.e. the standard basis vectors of the coefficient model, are orthonormal.","title":"Normalized Fock monomials are orthonormal standard basis vectors"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/orthonormal-fock-monomial","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.orthonormal_fockMonomial","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#orthonormal-fock-monomial","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The normalized monomial family satisfies Mathlib's standard orthonormal predicate.","title":"Normalized Fock monomials form an orthonormal family"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-coeff-eq-inner","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockCoeff_eq_inner","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-coeff-eq-inner","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The coefficient of `a` at `n` is the inner product with `e\u2099`.","title":"Fock coefficients as inner products with normalized monomials"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/fock-coherent","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.fockCoherent","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#fock-coherent","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The coherent state at `z`: the normalized Fock kernel, `\u0109_z = Real.exp (-\u2016z\u2016\u00b2 / 2) \u2022 k_z`.","title":"Coherent state in the Fock space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/norm-fock-coherent","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.norm_fockCoherent","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#norm-fock-coherent","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"Coherent states are unit vectors.","title":"Coherent states are unit vectors"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/inner-fock-coherent-fock-coherent","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.inner_fockCoherent_fockCoherent","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#inner-fock-coherent-fock-coherent","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The inner product of two coherent states.","title":"Inner product of two coherent states"},{"id":"Analysis/complex-analysis/analytic-function-spaces/fock-space/norm-inner-fock-coherent-fock-coherent","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FockSpace.norm_inner_fockCoherent_fockCoherent","parent":16776,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/fock-space/index.html#norm-inner-fock-coherent-fock-coherent","scope":"Analysis/complex-analysis/analytic-function-spaces/fock-space","status":"fully_proved","summary":"The overlap of two coherent states is Gaussian: `\u2016\u27ea\u0109_z, \u0109_w\u27eb\u2016 = Real.exp (-\u2016z - w\u2016\u00b2 / 2)`.","title":"Gaussian overlap of coherent states"},{"data_url":"scopes/Analysis/complex-analysis/analytic-function-spaces/hardy-space.json","id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","kind":"scope","navigation":"module","parent":16701,"result":false,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"can_prove","title":"Hardy space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/norm-sq-szego-coeff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.norm_sq_szego_coeff","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#norm-sq-szego-coeff","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The squared norms of the Szeg\u0151 kernel coefficients form a geometric progression with ratio `\u2016z\u2016\u00b2`.","title":"Squared norms of the Szeg\u0151 kernel coefficients"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/mem-p-szego-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.mem\u2113p_szegoKernel","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#mem-p-szego-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The Szeg\u0151 kernel coefficients are in `\u2113\u00b2` when `z` is in the disc.","title":"Szeg\u0151 kernel coefficients lie in \u2113\u00b2"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/szego-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.szegoKernel","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#szego-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The Szeg\u0151 kernel at `z`: the `\u2113\u00b2(\u2115, \u2102)` sequence with coefficients `(conj z)\u207f`.","title":"Szeg\u0151 kernel of the Hardy space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/szego-kernel-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.szegoKernel_zero","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#szego-kernel-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The Szeg\u0151 kernel at `0` is the first standard basis vector.","title":"Szeg\u0151 kernel at the origin is the first basis vector"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/norm-sq-szego-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.norm_sq_szegoKernel","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#norm-sq-szego-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The squared norm of the Szeg\u0151 kernel is `1 / (1 - \u2016z\u2016\u00b2)`.","title":"Squared norm of the Szeg\u0151 kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/norm-szego-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.norm_szegoKernel","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#norm-szego-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The norm of the Szeg\u0151 kernel is `1 / \u221a(1 - \u2016z\u2016\u00b2)`.","title":"Norm of the Szeg\u0151 kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/hardy-eval","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.hardyEval","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#hardy-eval","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"Point evaluation on `H\u00b2(\ud835\udd3b)`: the inner product with the Szeg\u0151 kernel.","title":"Point evaluation on the Hardy space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/hardy-eval-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.hardyEval_eq_tsum","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#hardy-eval-eq-tsum","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"Point evaluation is the Taylor series `\u2211' n, a n * z\u207f`.","title":"Hardy-space point evaluation as a Taylor series"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/summable-hardy-eval-terms","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.summable_hardyEval_terms","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#summable-hardy-eval-terms","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The Taylor series of an `H\u00b2` function converges absolutely at every point of the disc.","title":"Absolute convergence of Taylor series in the Hardy space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/hardy-eval-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.hardyEvalCLM","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#hardy-eval-clm","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"Point evaluation as a bounded linear functional on `H\u00b2(\ud835\udd3b)`.","title":"Point evaluation as a bounded functional on the Hardy space"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/hardy-eval-clm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.hardyEvalCLM_apply","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#hardy-eval-clm-apply","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The bounded point-evaluation functional acts by the inner product with the Szeg\u0151 kernel.","title":"Reproducing property of the Szeg\u0151 kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/norm-hardy-eval-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.norm_hardyEvalCLM","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#norm-hardy-eval-clm","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The operator norm of point evaluation is `1 / \u221a(1 - \u2016z\u2016\u00b2)`.","title":"Norm of Hardy-space point evaluation"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/norm-hardy-eval-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.norm_hardyEval_le","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#norm-hardy-eval-le","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The pointwise bound for `H\u00b2` functions: `\u2016f(z)\u2016 \u2264 \u2016f\u2016 / \u221a(1 - \u2016z\u2016\u00b2)`.","title":"Pointwise bound for Hardy-space functions"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/hardy-eval-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.hardyEval_zero","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#hardy-eval-zero","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"Evaluation at `0` is the constant Taylor coefficient.","title":"Hardy-space evaluation at the origin is the constant coefficient"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/hardy-eval-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.hardyEval_single","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#hardy-eval-single","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"Evaluation of a monomial `c * z\u207f` (a single Taylor coefficient).","title":"Hardy-space evaluation of a single monomial"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/inner-szego-kernel-right","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.inner_szegoKernel_right","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#inner-szego-kernel-right","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The reproducing property in the other order: `\u27eaa, k_z\u27eb = conj (f(z))`.","title":"Conjugate reproducing property of the Szeg\u0151 kernel"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/inner-szego-kernel-szego-kernel","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.inner_szegoKernel_szegoKernel","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#inner-szego-kernel-szego-kernel","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"The inner product of two Szeg\u0151 kernels is the Szeg\u0151 kernel function `1 / (1 - z * conj w)`.","title":"Inner product of two Szeg\u0151 kernels"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/abs-coeff-le-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.HardySpace.abs_coeff_le_norm","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#abs-coeff-le-norm","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"Taylor coefficients of an `H\u00b2` function are bounded by its norm: a single squared coefficient is at most the sum of all of them.","title":"Taylor coefficients of a Hardy-space function are bounded by its norm"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/h1","kind":"node","lean":"MathlibExt.Analysis.HardySpaceWanted.Analysis.H1","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#h1","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"fully_proved","summary":"Let $\\mathbb{T} = \\mathbb{R}/\\mathbb{Z}$ carry its Haar measure normalized to total mass $1$. The Hardy space $H^1$ is the complex linear subspace of $L^1(\\mathbb{T})$ consisting of those $f$ whose Fourier coefficients $\\hat f(n) = \\int_{\\mathbb{T}} f(t) e^{-2\\pi i n t} dt$ vanish for all integers $n < 0$.","title":"The boundary Hardy space H\u00b9"},{"id":"Analysis/complex-analysis/analytic-function-spaces/hardy-space/h1-not-closed-complemented","kind":"node","lean":"MathlibExt.Analysis.HardySpaceWanted.Analysis.H1_not_closedComplemented","parent":16804,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/hardy-space/index.html#h1-not-closed-complemented","scope":"Analysis/complex-analysis/analytic-function-spaces/hardy-space","status":"can_prove","summary":"Let $L^1(\\mathbb{T})$ be the space of integrable complex functions on the circle $\\mathbb{T} = \\mathbb{R}/\\mathbb{Z}$ with its normalized Haar measure, and let $H^1 \\subseteq L^1(\\mathbb{T})$ be the boundary Hardy space. There is no continuous $\\mathbb{C}$-linear map $\\pi : L^1(\\mathbb{T}) \\to H^1$ with $\\pi(f) = f$ for every $f \\in H^1$.","title":"Newman's theorem: no bounded projection onto the Hardy space"},{"data_url":"scopes/Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc.json","id":"Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc","kind":"scope","navigation":"module","parent":16701,"result":false,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc/index.html","scope":"Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc","status":"can_prove","title":"Lacunary power series on the unit disc"},{"id":"Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc/f","kind":"node","lean":"MathlibExt.Analysis.Complex.LacunarySeriesDiscWanted.f","parent":16825,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc/index.html#f","scope":"Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc","status":"fully_proved","summary":"For $z \\in \\mathbb{C}$, let $f(z) = \\sum_{j=0}^{\\infty} 2^{-j} z^{2^j}$, where $f(z)$ is taken to be $0$ when the series is not summable.","title":"A lacunary power series"},{"id":"Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc/differentiable-on-ball-continuous-on-clo","kind":"node","lean":"MathlibExt.Analysis.Complex.LacunarySeriesDiscWanted.differentiableOn_ball_continuousOn_closedBall_lacunary","parent":16825,"result":true,"route":"../roadmap/Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc/index.html#differentiable-on-ball-continuous-on-clo","scope":"Analysis/complex-analysis/analytic-function-spaces/lacunary-power-series-on-the-unit-disc","status":"can_prove","summary":"Let $f$ be the lacunary series $f(z) = \\sum_{j=0}^{\\infty} 2^{-j} z^{2^j}$. Then $f$ is holomorphic on the open unit disc $\\{z \\in \\mathbb{C} : |z| < 1\\}$ and continuous on its closure, the closed unit disc $\\{z \\in \\mathbb{C} : |z| \\le 1\\}$.","title":"Regularity of a lacunary series on the unit disc"},{"data_url":"scopes/Analysis/harmonic-analysis.json","id":"Analysis/harmonic-analysis","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/index.html","scope":"Analysis/harmonic-analysis","status":"can_prove","summary":"Fourier series, the Fourier transform on R^n, Fourier analysis on finite groups and the Boolean cube, singular integrals, and other integral transforms and moment problems.","title":"Harmonic analysis"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series.json","id":"Analysis/harmonic-analysis/fourier-series","kind":"scope","parent":16828,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/index.html","scope":"Analysis/harmonic-analysis/fourier-series","status":"can_prove","summary":"Fourier series on the circle and torus: Fej\u00e9r and Dirichlet kernels, pointwise and Ces\u00e0ro convergence, Riemann\u2013Lebesgue, uniqueness, absolutely convergent series and Wiener's theorems, periodic convolution, Fourier optimization on the torus.","title":"Fourier series"},{"id":"Analysis/harmonic-analysis/fourier-series/mathieu-fibonacci-fourier-coefficient","kind":"node","lean":"MetaMathlibExt.mathieuFibonacciFourierCoefficient","parent":16829,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/index.html#mathieu-fibonacci-fourier-coefficient","scope":"Analysis/harmonic-analysis/fourier-series","status":"fully_proved","summary":"Let $\\varphi = (1 + \\sqrt{5}) / 2$ be the golden ratio. For each natural number $n \\ge 1$, the $n$-th coefficient is the real number $c_n = (2 / \\log \\varphi) \\, |\\Gamma(1 + 2 n \\pi i / \\log \\varphi)|^2$, where $\\Gamma$ is the complex Gamma function.","title":"Fourier coefficients of the Mathieu\u2013Fibonacci periodic function"},{"id":"Analysis/harmonic-analysis/fourier-series/periodic-convolution","kind":"node","lean":"MathlibExt.Analysis.Fourier.periodic_convolution","parent":16829,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/index.html#periodic-convolution","scope":"Analysis/harmonic-analysis/fourier-series","status":"fully_proved","summary":"Let $A$ be a normed ring that is a normed algebra over $\\mathbb{R}$ (for example $\\mathbb{R}$ or $\\mathbb{C}$), let $T$ be a real number, and let $f, g : \\mathbb{R} \\to A$ with $g$ periodic of period $T$, that is, $g(x + T) = g(x)$ for all real $x$. Then the function $x \\mapsto \\int_0^T f(t)\\, g(x - t)\\, dt$ is periodic of period $T$. The function $f$ need not be periodic or integrable.","title":"Periodic convolution"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona.json","id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona","kind":"scope","parent":16829,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/index.html","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona","status":"fully_proved","title":"Brezis\u2013Gallou\u00ebt inequality on the two-dimensional torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/beta","kind":"node","lean":"TorusSobolev.LogarithmicRemainderWanted.beta","parent":16832,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Bartuccelli\u2013Deane\u2013Zelik constant "],["math","\\(\\beta\\)"]],"source":"The Bartuccelli\u2013Deane\u2013Zelik constant $\\beta$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/index.html#beta","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona","status":"fully_proved","summary":"Let $\\beta = \\pi \\left( 2\\gamma + 2 \\log 2 + 3 \\log \\pi - 4 \\log \\Gamma(\\tfrac14) \\right)$, where $\\gamma$ is the Euler\u2013Mascheroni constant and $\\Gamma$ is the gamma function.","title":"The Bartuccelli\u2013Deane\u2013Zelik constant $\\beta$"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio.json","id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","kind":"scope","navigation":"module","parent":16832,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","title":"Fourier Sobolev optimization on the two-dimensional torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/torus","kind":"node","lean":"TorusSobolev.Torus","parent":16834,"reading_title":{"format":"inline-parts-v1","parts":[["text","The torus of period "],["math","\\(2\\pi\\)"]],"source":"The torus of period $2\\pi$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#torus","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"The two-dimensional torus $\\mathbb{T}^2 = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$, the product of two copies of the circle $\\mathbb{R}/2\\pi\\mathbb{Z}$.","title":"The torus of period $2\\pi$"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/frequency","kind":"node","lean":"TorusSobolev.Frequency","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#frequency","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"The frequencies are the elements of the lattice $\\mathbb{Z}^2$, that is, the pairs $k = (k_1, k_2)$ of integers; they index the Fourier modes of the torus $(\\mathbb{R}/2\\pi\\mathbb{Z})^2$.","title":"Integer frequencies"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/unit-rescaling","kind":"node","lean":"TorusSobolev.unitRescaling","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#unit-rescaling","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"The homeomorphism from the torus $\\mathbb{T}^2 = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$ onto the unit torus $(\\mathbb{R}/\\mathbb{Z})^2$ that sends $(x_1, x_2)$ to $(x_1/(2\\pi), x_2/(2\\pi))$.","title":"Rescaling the torus to unit period"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/unit-representative","kind":"node","lean":"TorusSobolev.unitRepresentative","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#unit-representative","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For a continuous function $u : \\mathbb{T}^2 \\to \\mathbb{R}$ on the torus $\\mathbb{T}^2 = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$, its unit representative is the continuous function $\\tilde u : (\\mathbb{R}/\\mathbb{Z})^2 \\to \\mathbb{C}$ given by $\\tilde u(y) = u(2\\pi y)$, the composition of $u$ with the inverse of the rescaling to unit period, with its real values regarded as complex numbers.","title":"A function on the torus as a function on the unit torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/fourier-coefficient","kind":"node","lean":"TorusSobolev.fourierCoefficient","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#fourier-coefficient","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"Let $u : (\\mathbb{R}/2\\pi\\mathbb{Z})^2 \\to \\mathbb{R}$ be continuous and let $k \\in \\mathbb{Z}^2$ be a frequency. The Fourier coefficient of $u$ at $k$ is $a_k = 2\\pi c_k$, where $c_k = \\int_{[0,1]^2} u(2\\pi y)\\, e^{-2\\pi i\\, k \\cdot y}\\, dy$ is the Fourier coefficient of the rescaled function $y \\mapsto u(2\\pi y)$ on the unit torus with respect to normalized Haar measure. Equivalently,\u2026","title":"Fourier coefficients on the torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/frequency-weight","kind":"node","lean":"TorusSobolev.frequencyWeight","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#frequency-weight","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For a frequency $k = (k_1, k_2) \\in \\mathbb{Z}^2$, the frequency weight is $|k|^2 = k_1^2 + k_2^2$, the squared frequency of the mode $e^{i k \\cdot x}$ on the torus $(\\mathbb{R}/2\\pi\\mathbb{Z})^2$.","title":"Squared frequency"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/gradient-energy","kind":"node","lean":"TorusSobolev.gradientEnergy","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#gradient-energy","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For a continuous function $u : (\\mathbb{R}/2\\pi\\mathbb{Z})^2 \\to \\mathbb{R}$ with Fourier coefficients $a_k$, the gradient energy is $\\sum_{k \\in \\mathbb{Z}^2} |k|^2 \\, |a_k|^2$, where $|k|^2 = k_1^2 + k_2^2$; the sum is taken to be $0$ when the series diverges.","title":"Gradient energy"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/laplace-energy","kind":"node","lean":"TorusSobolev.laplaceEnergy","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#laplace-energy","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For a continuous function $u : (\\mathbb{R}/2\\pi\\mathbb{Z})^2 \\to \\mathbb{R}$ with Fourier coefficients $a_k$, the Laplacian energy is $\\sum_{k \\in \\mathbb{Z}^2} |k|^4 \\, |a_k|^2$, where $|k|^2 = k_1^2 + k_2^2$; the sum is taken to be $0$ when the series diverges.","title":"Laplacian energy"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/is-h2","kind":"node","lean":"TorusSobolev.IsH2","parent":16834,"reading_title":{"format":"inline-parts-v1","parts":[["text","Functions in "],["math","\\(H^2\\)"]],"source":"Functions in $H^2$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#is-h2","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"A continuous function $u : (\\mathbb{R}/2\\pi\\mathbb{Z})^2 \\to \\mathbb{R}$ with Fourier coefficients $a_k$ is in $H^2$ when the series $\\sum_{k \\in \\mathbb{Z}^2} (1 + |k|^2)^2 \\, |a_k|^2$ converges, where $|k|^2 = k_1^2 + k_2^2$.","title":"Functions in $H^2$"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/unit-haar","kind":"node","lean":"TorusSobolev.unitHaar","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#unit-haar","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"The measure on the unit torus $(\\mathbb{R}/\\mathbb{Z})^2$ that is the product of two copies of the Haar probability measure on the circle $\\mathbb{R}/\\mathbb{Z}$.","title":"Normalized Haar measure on the unit torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/mean","kind":"node","lean":"TorusSobolev.mean","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#mean","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For a continuous function $u : \\mathbb{T}^2 \\to \\mathbb{R}$ on the torus $\\mathbb{T}^2 = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$, its mean is $\\int_{(\\mathbb{R}/\\mathbb{Z})^2} u(2\\pi y)\\, d\\mu(y)$, where $\\mu$ is the normalized Haar measure on the unit torus $(\\mathbb{R}/\\mathbb{Z})^2$ and $y \\mapsto 2\\pi y$ is the inverse of the rescaling to unit period.","title":"Mean of a function on the torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/is-zero-mean-h2","kind":"node","lean":"TorusSobolev.IsZeroMeanH2","parent":16834,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero-mean "],["math","\\(H^2\\)"],["text"," functions"]],"source":"Zero-mean $H^2$ functions"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#is-zero-mean-h2","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"A continuous function $u : (\\mathbb{R}/2\\pi\\mathbb{Z})^2 \\to \\mathbb{R}$ is a zero-mean $H^2$ function when it is in $H^2$ and its mean is $0$.","title":"Zero-mean $H^2$ functions"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/conditional-admissible","kind":"node","lean":"TorusSobolev.ConditionalAdmissible","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#conditional-admissible","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"Let $\\delta$ be a real number. A continuous function $u : (\\mathbb{R}/2\\pi\\mathbb{Z})^2 \\to \\mathbb{R}$ is admissible at level $\\delta$ when it is a zero-mean $H^2$ function, its gradient energy equals $1$, and its Laplacian energy equals $\\delta$.","title":"Conditionally admissible functions"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/conditional-point-values","kind":"node","lean":"TorusSobolev.conditionalPointValues","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#conditional-point-values","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For a real number $\\delta$, this is the set of extended real numbers $u(0)^2$, where $u$ ranges over the continuous functions $u : (\\mathbb{R}/2\\pi\\mathbb{Z})^2 \\to \\mathbb{R}$ that are conditionally admissible at level $\\delta$.","title":"Squared values at the origin of admissible functions"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/theta","kind":"node","lean":"TorusSobolev.theta","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#theta","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For real $\\delta$, $\\theta(\\delta)$ is the supremum, taken in the extended reals $[-\\infty, \\infty]$, of the conditional point values at $\\delta$: the numbers $u(0)^2$ over all continuous $u : \\mathbb{T}^2 \\to \\mathbb{R}$ on the torus $\\mathbb{T}^2 = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$ that are conditionally admissible at $\\delta$, that is, of mean zero and in $H^2$ with gradient energy $1$ and Laplace energy $\\delta$.","title":"Conditional supremum of the squared value at the origin"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/frequency-ratio","kind":"node","lean":"TorusSobolev.frequencyRatio","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#frequency-ratio","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For a continuous function $u : (\\mathbb{R}/2\\pi\\mathbb{Z})^2 \\to \\mathbb{R}$, the frequency ratio is the Laplacian energy of $u$ divided by the gradient energy of $u$.","title":"Frequency ratio"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/remainder","kind":"node","lean":"TorusSobolev.remainder","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#remainder","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For real $\\delta$, the remainder is the extended real number $R(\\delta) = 4\\pi\\,\\theta(\\delta) - \\log \\delta - \\log(1 + \\log \\delta)$, where $\\theta(\\delta)$ is the conditional supremum of $u(0)^2$ over the functions conditionally admissible at $\\delta$.","title":"Logarithmic remainder"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/sharp-remainder-values","kind":"node","lean":"TorusSobolev.sharpRemainderValues","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#sharp-remainder-values","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"The set of extended real numbers $R(\\delta)$ for all real $\\delta \\ge 1$, where $R(\\delta) = 4\\pi\\,\\theta(\\delta) - \\log \\delta - \\log(1 + \\log \\delta)$ is the logarithmic remainder and $\\theta(\\delta)$ is the conditional supremum of $u(0)^2$ over the functions conditionally admissible at $\\delta$.","title":"Set of remainder values"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/sharp-remainder-value","kind":"node","lean":"TorusSobolev.sharpRemainderValue","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#sharp-remainder-value","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"The sharp remainder is the supremum, in the extended reals $[-\\infty, \\infty]$, of the remainders $R(\\delta) = 4\\pi\\,\\theta(\\delta) - \\log \\delta - \\log(1 + \\log \\delta)$ over all real $\\delta \\ge 1$, where $\\theta(\\delta)$ is the conditional supremum of $u(0)^2$ over the functions conditionally admissible at $\\delta$.","title":"Sharp logarithmic remainder"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/uniform-remainder-bound","kind":"node","lean":"TorusSobolev.UniformRemainderBound","parent":16834,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio/index.html#uniform-remainder-bound","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-sobolev-optimization-on-the-two-dimensio","status":"fully_proved","summary":"For a real constant $L$, the bound with constant $L$ holds when, for every continuous $u : \\mathbb{T}^2 \\to \\mathbb{R}$ on the torus $\\mathbb{T}^2 = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$ that has mean zero and lies in $H^2$, has positive gradient energy $E(u)$, and, writing $\\Delta(u)$ for its Laplace energy, has frequency ratio $\\Lambda(u) = \\Delta(u)/E(u) \\ge 1$, every point $x$ of the torus satisfies\u2026","title":"Uniform logarithmic remainder bound"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters.json","id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters","kind":"scope","navigation":"module","parent":16832,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/index.html","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters","status":"fully_proved","title":"Explicit extremals and their parameters"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/finite-extremal-parameter","kind":"node","lean":"TorusSobolev.FiniteExtremalParameter","parent":16855,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/index.html#finite-extremal-parameter","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters","status":"fully_proved","summary":"A finite extremal parameter is a real number $\\mu$ with $\\mu < -1$ or $\\mu > 0$.","title":"Finite extremal parameters"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/extremal-parameter","kind":"node","lean":"TorusSobolev.ExtremalParameter","parent":16855,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/index.html#extremal-parameter","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters","status":"fully_proved","summary":"An extremal parameter is either a finite extremal parameter, that is, a real number $\\mu$ with $\\mu < -1$ or $\\mu > 0$, or one of two further parameters: the first-shell parameter, standing for the endpoint $\\mu = -1$, and the biharmonic parameter, standing for the endpoint $\\mu = \\infty$.","title":"Extremal parameters"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/kernel-admissible","kind":"node","lean":"TorusSobolev.KernelAdmissible","parent":16855,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/index.html#kernel-admissible","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters","status":"fully_proved","summary":"Write $|k|^2 = k_1^2 + k_2^2$ for a frequency $k \\in \\mathbb{Z}^2$. A finite extremal parameter $\\mu$ is kernel-admissible when $|k|^2 \\neq 0$ and $1 + \\mu |k|^2 \\neq 0$ for every nonzero frequency $k$; the first-shell parameter is always kernel-admissible; the biharmonic parameter is kernel-admissible when $|k|^2 \\neq 0$ for every nonzero frequency $k$.","title":"Kernel-admissible extremal parameters"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/extremal-coefficient","kind":"node","lean":"TorusSobolev.extremalCoefficient","parent":16855,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/index.html#extremal-coefficient","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters","status":"fully_proved","summary":"Let $p$ be a kernel-admissible extremal parameter and write $|k|^2 = k_1^2 + k_2^2$ for $k \\in \\mathbb{Z}^2$. The extremal coefficients $c_p(k)$, complex numbers indexed by $k \\in \\mathbb{Z}^2$, are as follows. If $p$ is a finite parameter $\\mu$, then $c_p(0) = 0$ and $c_p(k) = 2\\pi / (|k|^2 (1 + \\mu |k|^2))$ for $k \\neq 0$. If $p$ is the first-shell parameter, then $c_p(k) = \\pi$ when $|k|^2 = 1$ and $c_p(k) = 0$\u2026","title":"Extremal coefficients"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/is-explicit-extremal","kind":"node","lean":"TorusSobolev.IsExplicitExtremal","parent":16855,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/index.html#is-explicit-extremal","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters","status":"fully_proved","summary":"A continuous real function $u$ on the torus $\\mathbb{T} = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$ is an explicit extremal when there are a kernel-admissible extremal parameter $p$ and a real number $a \\neq 0$ such that the extremal coefficients $c_p(k)$ are summable over $k \\in \\mathbb{Z}^2$, the Fourier coefficient $\\hat u(k) = \\frac{1}{2\\pi} \\int_{[0, 2\\pi]^2} u(x) e^{-i k \\cdot x} \\, dx$ equals $a \\, c_p(k)$ for every\u2026","title":"Explicit extremals"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/same-extremal-orbit","kind":"node","lean":"TorusSobolev.SameExtremalOrbit","parent":16855,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters/index.html#same-extremal-orbit","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/explicit-extremals-and-their-parameters","status":"fully_proved","summary":"Two continuous real functions $u$ and $v$ on the torus $\\mathbb{T} = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$ lie in the same extremal orbit when there are a real number $a \\neq 0$ and a point $x_0 \\in \\mathbb{T}$ such that $v(x) = a \\, u(x - x_0)$ for every $x \\in \\mathbb{T}$.","title":"Same extremal orbit"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes.json","id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","kind":"scope","navigation":"module","parent":16832,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","title":"Fourier coefficients and the lowest-shell witness"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/m-fourier-coeff-monomial","kind":"node","lean":"TorusSobolev.mFourierCoeff_monomial","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#m-fourier-coeff-monomial","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For frequencies $m, n \\in \\mathbb{Z}^2$, the $n$-th Fourier coefficient, taken with respect to normalized Haar measure on $(\\mathbb{R}/\\mathbb{Z})^2$, of the monomial $e_m(x) = e^{2\\pi i\\, m \\cdot x}$ is $1$ if $n = m$ and $0$ otherwise.","title":"Fourier coefficients of a Fourier monomial on the 2-torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/m-fourier-coeff-integrable","kind":"node","lean":"TorusSobolev.mFourierCoeff_integrable","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#m-fourier-coeff-integrable","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For every continuous function $f : (\\mathbb{R}/\\mathbb{Z})^2 \\to \\mathbb{C}$ and every frequency $n \\in \\mathbb{Z}^2$, the function $x \\mapsto e^{-2\\pi i\\, n \\cdot x} f(x)$ is integrable with respect to normalized Haar measure.","title":"Integrability of Fourier integrands on the 2-torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/m-fourier-coeff-add","kind":"node","lean":"TorusSobolev.mFourierCoeff_add","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#m-fourier-coeff-add","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For continuous functions $f, g : (\\mathbb{R}/\\mathbb{Z})^2 \\to \\mathbb{C}$ and every frequency $n \\in \\mathbb{Z}^2$, the $n$-th Fourier coefficient $\\hat h(n) = \\int e^{-2\\pi i\\, n \\cdot x} h(x)\\,dx$ (normalized Haar measure) satisfies $\\widehat{f + g}(n) = \\hat f(n) + \\hat g(n)$.","title":"Additivity of Fourier coefficients on the 2-torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/m-fourier-coeff-smul","kind":"node","lean":"TorusSobolev.mFourierCoeff_smul","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#m-fourier-coeff-smul","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For every $c \\in \\mathbb{C}$, every continuous function $f : (\\mathbb{R}/\\mathbb{Z})^2 \\to \\mathbb{C}$ and every frequency $n \\in \\mathbb{Z}^2$, the $n$-th Fourier coefficient (normalized Haar measure) satisfies $\\widehat{c f}(n) = c\\, \\hat f(n)$.","title":"Homogeneity of Fourier coefficients on the 2-torus"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/axis-frequency","kind":"node","lean":"TorusSobolev.axisFrequency","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#axis-frequency","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For $i \\in \\{0, 1\\}$, the axis frequency $e_i$ is the frequency in $\\mathbb{Z}^2$ whose $i$-th coordinate is $1$ and whose other coordinate is $0$.","title":"Axis frequency"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/lowest-shell-unit","kind":"node","lean":"TorusSobolev.lowestShellUnit","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#lowest-shell-unit","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For $n \\in \\mathbb{Z}^2$ let $e_n(y) = e^{2\\pi i\\, n \\cdot y}$ be the Fourier monomial on the unit torus $(\\mathbb{R}/\\mathbb{Z})^2$. The lowest-shell unit polynomial is the continuous function $P : (\\mathbb{R}/\\mathbb{Z})^2 \\to \\mathbb{C}$ given by $P = (4\\pi)^{-1}\\bigl(e_{(1,0)} + e_{(-1,0)} + e_{(0,1)} + e_{(0,-1)}\\bigr)$, the normalized sum of the Fourier monomials of the four nonzero frequencies of smallest\u2026","title":"Lowest-shell unit polynomial"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/lowest-shell-witness","kind":"node","lean":"TorusSobolev.lowestShellWitness","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#lowest-shell-witness","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The lowest-shell witness is the continuous real function $u$ on the torus $(\\mathbb{R}/2\\pi\\mathbb{Z})^2$ given by $u(x_1, x_2) = \\frac{1}{4\\pi}(e^{i x_1} + e^{-i x_1} + e^{i x_2} + e^{-i x_2}) = \\frac{\\cos x_1 + \\cos x_2}{2\\pi}$, the normalized sum of the four lowest nonzero Fourier modes.","title":"Lowest-shell witness"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/lowest-shell-unit-im-zero","kind":"node","lean":"TorusSobolev.lowestShellUnit_im_zero","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#lowest-shell-unit-im-zero","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For every point $y$ of the unit torus $(\\mathbb{R}/\\mathbb{Z})^2$, the lowest-shell unit polynomial $P$ satisfies $\\operatorname{Im} P(y) = 0$.","title":"The lowest-shell unit polynomial is real-valued"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/unit-representative-lowest-shell-witness","kind":"node","lean":"TorusSobolev.unitRepresentative_lowestShellWitness","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#unit-representative-lowest-shell-witness","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The unit representative of the lowest-shell witness is the lowest-shell polynomial $(4\\pi)^{-1}\\bigl(e_{(1,0)} + e_{(-1,0)} + e_{(0,1)} + e_{(0,-1)}\\bigr)$ on the unit torus $(\\mathbb{R}/\\mathbb{Z})^2$, where $e_n(y) = e^{2\\pi i\\, n \\cdot y}$ for $n \\in \\mathbb{Z}^2$.","title":"Unit representative of the lowest-shell witness"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/lowest-shell","kind":"node","lean":"TorusSobolev.lowestShell","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#lowest-shell","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The lowest shell is the set $\\{e_1, -e_1, e_2, -e_2\\}$ of frequencies in $\\mathbb{Z}^2$, where $e_1 = (1, 0)$ and $e_2 = (0, 1)$; these are the four nonzero frequencies of smallest length.","title":"Lowest frequency shell"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/fourier-coefficient-lowest-shell-witness","kind":"node","lean":"TorusSobolev.fourierCoefficient_lowestShellWitness","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#fourier-coefficient-lowest-shell-witness","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For every frequency $k \\in \\mathbb{Z}^2$, the Fourier coefficient of the lowest-shell witness at $k$ equals $1/2$ if $k$ lies in the lowest shell, that is, $k$ is one of $(1, 0)$, $(-1, 0)$, $(0, 1)$, $(0, -1)$, and equals $0$ otherwise.","title":"Fourier coefficients of the lowest-shell witness"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/lowest-shell-witness-is-h2","kind":"node","lean":"TorusSobolev.lowestShellWitness_isH2","parent":16862,"reading_title":{"format":"inline-parts-v1","parts":[["text","The lowest-shell witness is in "],["math","\\(H^2\\)"]],"source":"The lowest-shell witness is in $H^2$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#lowest-shell-witness-is-h2","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The lowest-shell witness $u$ belongs to the spectral Sobolev space $H^2$: if $a_k(u)$ denotes its Fourier coefficients, the series $\\sum_{k \\in \\mathbb{Z}^2} (1 + |k|^2)^2 |a_k(u)|^2$ converges.","title":"The lowest-shell witness is in $H^2$"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/gradient-energy-lowest-shell-witness","kind":"node","lean":"TorusSobolev.gradientEnergy_lowestShellWitness","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#gradient-energy-lowest-shell-witness","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The gradient energy of the lowest-shell witness equals $1$.","title":"Gradient energy of the lowest-shell witness"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/laplace-energy-lowest-shell-witness","kind":"node","lean":"TorusSobolev.laplaceEnergy_lowestShellWitness","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#laplace-energy-lowest-shell-witness","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The Laplacian energy of the lowest-shell witness equals $1$.","title":"Laplacian energy of the lowest-shell witness"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/mean-eq-m-fourier-coeff-zero-re","kind":"node","lean":"TorusSobolev.mean_eq_mFourierCoeff_zero_re","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#mean-eq-m-fourier-coeff-zero-re","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For every continuous function $u : \\mathbb{T} \\to \\mathbb{R}$ on the torus $\\mathbb{T} = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$, the mean of $u$ equals the real part of the zeroth Fourier coefficient $\\int_{(\\mathbb{R}/\\mathbb{Z})^2} \\tilde u(y)\\,dy$, taken with respect to normalized Haar measure, of the unit representative $\\tilde u$ of $u$ (the function $u$, viewed as complex-valued and transported to the unit-period\u2026","title":"The mean is the real part of the zeroth Fourier coefficient"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/mean-lowest-shell-witness","kind":"node","lean":"TorusSobolev.mean_lowestShellWitness","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#mean-lowest-shell-witness","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The lowest-shell witness, the function $w(x_1, x_2) = (\\cos x_1 + \\cos x_2)/(2\\pi)$ on the torus $(\\mathbb{R}/2\\pi\\mathbb{Z})^2$, has mean $0$.","title":"The lowest-shell witness has mean zero"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/unit-rescaling-zero","kind":"node","lean":"TorusSobolev.unitRescaling_zero","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#unit-rescaling-zero","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The coordinate rescaling from the torus $(\\mathbb{R}/2\\pi\\mathbb{Z})^2$ to the unit torus $(\\mathbb{R}/\\mathbb{Z})^2$ maps $0$ to $0$.","title":"The unit rescaling fixes the origin"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/lowest-shell-witness-at-zero","kind":"node","lean":"TorusSobolev.lowestShellWitness_at_zero","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#lowest-shell-witness-at-zero","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The lowest-shell witness $u$ takes the value $u(0) = 1/\\pi$ at the origin of the torus.","title":"Value of the lowest-shell witness at the origin"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/lowest-shell-witness-is-zero-mean-h2","kind":"node","lean":"TorusSobolev.lowestShellWitness_isZeroMeanH2","parent":16862,"reading_title":{"format":"inline-parts-v1","parts":[["text","The lowest-shell witness is a zero-mean "],["math","\\(H^2\\)"],["text"," function"]],"source":"The lowest-shell witness is a zero-mean $H^2$ function"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#lowest-shell-witness-is-zero-mean-h2","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The lowest-shell witness $u$ belongs to the spectral Sobolev space $H^2$ and has mean $0$.","title":"The lowest-shell witness is a zero-mean $H^2$ function"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/lowest-shell-witness-admissible","kind":"node","lean":"TorusSobolev.lowestShellWitness_admissible","parent":16862,"reading_title":{"format":"inline-parts-v1","parts":[["text","The lowest-shell witness is admissible at "],["math","\\(\\delta = 1\\)"]],"source":"The lowest-shell witness is admissible at $\\delta = 1$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#lowest-shell-witness-admissible","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"The lowest-shell witness $u$ satisfies the conditional constraints with $\\delta = 1$: it is a zero-mean $H^2$ function on the torus, its gradient energy is $1$, and its Laplacian energy is $1$.","title":"The lowest-shell witness is admissible at $\\delta = 1$"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/theta-le-iff","kind":"node","lean":"TorusSobolev.theta_le_iff","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#theta-le-iff","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"Let $\\mathbb{T} = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$. For every real $\\delta$, let $\\theta(\\delta)$, the conditional supremum, be the supremum in the extended reals of $u(0)^2$ over all continuous functions $u : \\mathbb{T} \\to \\mathbb{R}$ that are conditionally admissible at $\\delta$, that is, zero-mean $H^2$ functions with gradient energy $1$ and Laplacian energy $\\delta$. Then for every extended real number $b$,\u2026","title":"Upper bounds for the conditional supremum"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/sharp-remainder-value-le-iff","kind":"node","lean":"TorusSobolev.sharpRemainderValue_le_iff","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#sharp-remainder-value-le-iff","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"Let $\\mathbb{T} = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$. For real $\\delta$, let $\\theta(\\delta)$ be the supremum in the extended reals of $u(0)^2$ over all continuous functions $u : \\mathbb{T} \\to \\mathbb{R}$ that are zero-mean $H^2$ functions with gradient energy $1$ and Laplacian energy $\\delta$, and let $R(\\delta) = 4\\pi\\,\\theta(\\delta) - \\log\\delta - \\log(1 + \\log\\delta)$, computed in the extended reals. For every\u2026","title":"Upper bounds for the sharp remainder"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/frequency-weight-nonneg","kind":"node","lean":"TorusSobolev.frequencyWeight_nonneg","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#frequency-weight-nonneg","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"For every frequency $k = (k_0, k_1) \\in \\mathbb{Z}^2$, the frequency weight $k_0^2 + k_1^2$ is nonnegative.","title":"Frequency weights are nonnegative"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/gradient-summable","kind":"node","lean":"TorusSobolev.IsH2.gradient_summable","parent":16862,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(H^2\\)"],["text"," functions have a summable gradient energy series"]],"source":"$H^2$ functions have a summable gradient energy series"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#gradient-summable","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"If $u$ is a continuous real function on the torus of period $2\\pi$ that lies in $H^2$, with Fourier coefficients $a_k$, then the family $\\left((k_0^2 + k_1^2) \\, |a_k|^2\\right)_{k \\in \\mathbb{Z}^2}$ is summable.","title":"$H^2$ functions have a summable gradient energy series"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/laplace-summable","kind":"node","lean":"TorusSobolev.IsH2.laplace_summable","parent":16862,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(H^2\\)"],["text"," functions have a summable Laplacian energy series"]],"source":"$H^2$ functions have a summable Laplacian energy series"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#laplace-summable","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"If $u$ is a continuous real function on the torus of period $2\\pi$ that lies in $H^2$, with Fourier coefficients $a_k$, then the family $\\left((k_0^2 + k_1^2)^2 \\, |a_k|^2\\right)_{k \\in \\mathbb{Z}^2}$ is summable.","title":"$H^2$ functions have a summable Laplacian energy series"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/fourier-reconstruction","kind":"node","lean":"TorusSobolev.fourierReconstruction","parent":16862,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes/index.html#fourier-reconstruction","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/fourier-coefficients-and-the-lowest-shell-witnes","status":"fully_proved","summary":"Let $u$ be a continuous real function on the torus of period $2\\pi$ in each coordinate, let $a_k$ ($k \\in \\mathbb{Z}^2$) be its Fourier coefficients, and let $c_k = a_k / (2\\pi)$ be the Fourier coefficients, with respect to normalized Haar measure, of its unit representative on $(\\mathbb{R}/\\mathbb{Z})^2$. If the family $(c_k)_{k \\in \\mathbb{Z}^2}$ is summable, then for every point $x$ of the torus, with unit\u2026","title":"Fourier reconstruction"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu.json","id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu","kind":"scope","navigation":"module","parent":16832,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu/index.html","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu","status":"fully_proved","title":"One-log and double-log bounds on actual torus functions"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu/one-log-bound","kind":"node","lean":"TorusSobolev.OneLogBound","parent":16889,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-log bound with coefficient "],["math","\\(c\\)"],["text"," and constant "],["math","\\(K\\)"]],"source":"One-log bound with coefficient $c$ and constant $K$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu/index.html#one-log-bound","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu","status":"fully_proved","summary":"For real numbers $c$ and $K$, the one-log bound holds when every continuous function $u : \\mathbb{T} \\to \\mathbb{R}$ on the torus $\\mathbb{T} = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$ that is a zero-mean $H^2$ function, has gradient energy $E(u) > 0$ and has frequency ratio $\\delta(u) \\ge 1$ (the Laplacian energy divided by $E(u)$) satisfies, at every point $x \\in \\mathbb{T}$,","title":"One-log bound with coefficient $c$ and constant $K$"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu/double-log-bound","kind":"node","lean":"TorusSobolev.DoubleLogBound","parent":16889,"reading_title":{"format":"inline-parts-v1","parts":[["text","Double-log bound with coefficient "],["math","\\(a\\)"],["text"," and constant "],["math","\\(L\\)"]],"source":"Double-log bound with coefficient $a$ and constant $L$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu/index.html#double-log-bound","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu","status":"fully_proved","summary":"For real numbers $a$ and $L$, the double-log bound holds when every continuous function $u : \\mathbb{T} \\to \\mathbb{R}$ on the torus $\\mathbb{T} = (\\mathbb{R}/2\\pi\\mathbb{Z})^2$ that is a zero-mean $H^2$ function, has gradient energy $E(u) > 0$ and has frequency ratio $\\delta(u) \\ge 1$ (the Laplacian energy divided by $E(u)$) satisfies, at every point $x \\in \\mathbb{T}$,","title":"Double-log bound with coefficient $a$ and constant $L$"},{"id":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu/one-log-leading-coefficient-infimum","kind":"node","lean":"TorusSobolev.oneLogLeadingCoefficientInfimum","parent":16889,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu/index.html#one-log-leading-coefficient-infimum","scope":"Analysis/harmonic-analysis/fourier-series/brezis-gallou-t-inequality-on-the-two-dimensiona/one-log-and-double-log-bounds-on-actual-torus-fu","status":"fully_proved","summary":"The infimum, taken in the extended reals, of all real numbers $c$ for which there is a real constant $K$ such that the one-log bound with coefficient $c$ and constant $K$ holds. It is $+\\infty$ when no such $c$ exists.","title":"Optimal one-log coefficient"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series.json","id":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series","kind":"scope","navigation":"module","parent":16829,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/index.html","scope":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series","status":"can_prove","title":"Convergence of Fourier series"},{"id":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/dirichlet-pointwise-convergence","kind":"node","lean":"MathlibExt.Analysis.Fourier.DirichletTheoremWanted.dirichlet_pointwise_convergence","parent":16893,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/index.html#dirichlet-pointwise-convergence","scope":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{C}$ be $2\\pi$-periodic and of bounded variation on $[-\\pi, \\pi]$, and for $n \\in \\mathbb{Z}$ let $c_n = \\frac{1}{2\\pi} \\int_0^{2\\pi} f(t) e^{-int}\\,dt$ be its Fourier coefficients. Then at every $x \\in \\mathbb{R}$ the one-sided limits $f(x^+) = \\lim_{y \\to x,\\, y > x} f(y)$ and $f(x^-) = \\lim_{y \\to x,\\, y < x} f(y)$ exist, and the symmetric partial sums of the Fourier series\u2026","title":"Dirichlet\u2013Jordan theorem on pointwise convergence of Fourier series"},{"id":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/fourier-partial-sum","kind":"node","lean":"MathlibExt.Analysis.FourierConvergenceWanted.Analysis.fourierPartialSum","parent":16893,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/index.html#fourier-partial-sum","scope":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series","status":"fully_proved","summary":"For $f \\colon \\mathbb{R} \\to \\mathbb{C}$, a natural number $N$ and $x \\in \\mathbb{R}$, the $N$-th symmetric Fourier partial sum of $f$ at $x$ is $S_N(f)(x) = \\sum_{n=-N}^{N} \\hat f_n e^{inx}$, where $\\hat f_n = \\frac{1}{2\\pi} \\int_0^{2\\pi} e^{-int} f(t)\\,dt$ is the $n$-th Fourier coefficient of $f$ on the interval $[0, 2\\pi]$.","title":"Symmetric Fourier partial sums"},{"id":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/fourier-cesaro-mean","kind":"node","lean":"MathlibExt.Analysis.FourierConvergenceWanted.Analysis.fourierCesaroMean","parent":16893,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/index.html#fourier-cesaro-mean","scope":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series","status":"fully_proved","summary":"For $f \\colon \\mathbb{R} \\to \\mathbb{C}$, a natural number $N$ and $x \\in \\mathbb{R}$, the $N$-th Ces\u00e0ro mean of the Fourier series of $f$ at $x$ is the average of the first $N + 1$ symmetric Fourier partial sums, $\\sigma_N(f)(x) = \\frac{1}{N+1} \\sum_{k=0}^{N} S_k(f)(x)$, where $S_k(f)(x) = \\sum_{n=-k}^{k} \\hat f_n e^{inx}$ and $\\hat f_n = \\frac{1}{2\\pi} \\int_0^{2\\pi} e^{-int} f(t)\\,dt$ is the $n$-th Fourier\u2026","title":"Ces\u00e0ro means of Fourier series"},{"id":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/dirichlet-pointwise","kind":"node","lean":"MathlibExt.Analysis.FourierConvergenceWanted.Analysis.dirichlet_pointwise","parent":16893,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/index.html#dirichlet-pointwise","scope":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series","status":"can_prove","summary":"Let $f \\colon \\mathbb{R} \\to \\mathbb{C}$ be $2\\pi$-periodic and continuously differentiable, and for $n \\in \\mathbb{Z}$ let $\\hat f_n = \\frac{1}{2\\pi} \\int_0^{2\\pi} e^{-int} f(t)\\,dt$ be the $n$-th Fourier coefficient of $f$ on $[0, 2\\pi]$. Then for every $x \\in \\mathbb{R}$ the symmetric Fourier partial sums $S_N(f)(x) = \\sum_{n=-N}^{N} \\hat f_n e^{inx}$ converge to $f(x)$ as $N \\to \\infty$.","title":"Dirichlet's theorem on pointwise convergence of Fourier series"},{"id":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/fejer","kind":"node","lean":"MathlibExt.Analysis.FourierConvergenceWanted.Analysis.fejer","parent":16893,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series/index.html#fejer","scope":"Analysis/harmonic-analysis/fourier-series/convergence-of-fourier-series","status":"can_prove","summary":"Let $f \\colon \\mathbb{R} \\to \\mathbb{C}$ be continuous and $2\\pi$-periodic, and write $S_k(f)$ for the $k$-th symmetric Fourier partial sum of $f$. Then the Ces\u00e0ro means $\\sigma_N(f) = (S_0(f) + S_1(f) + \\dots + S_N(f)) / (N + 1)$ converge to $f$ uniformly on $\\mathbb{R}$ as $N \\to \\infty$.","title":"Fej\u00e9r's theorem"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle.json","id":"Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle","kind":"scope","navigation":"module","parent":16829,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle/index.html","scope":"Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle","status":"can_prove","title":"Convolutions and translates on the circle"},{"id":"Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle/t","kind":"node","lean":"MathlibExt.Analysis.Fourier.ConvolutionTranslateClosureCircleWanted.T","parent":16899,"reading_title":{"format":"inline-parts-v1","parts":[["text","The circle "],["math","\\(\\mathbb{R}/2\\pi\\mathbb{Z}\\)"]],"source":"The circle $\\mathbb{R}/2\\pi\\mathbb{Z}$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle/index.html#t","scope":"Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle","status":"fully_proved","summary":"The circle $T = \\mathbb{R}/2\\pi\\mathbb{Z}$, the quotient of the additive group $\\mathbb{R}$ by its subgroup $2\\pi\\mathbb{Z}$.","title":"The circle $\\mathbb{R}/2\\pi\\mathbb{Z}$"},{"id":"Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle/closure-convolution-set-eq-closure-trans","kind":"node","lean":"MathlibExt.Analysis.Fourier.ConvolutionTranslateClosureCircleWanted.closure_convolution_set_eq_closure_translate_span","parent":16899,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle/index.html#closure-convolution-set-eq-closure-trans","scope":"Analysis/harmonic-analysis/fourier-series/convolutions-and-translates-on-the-circle","status":"can_prove","summary":"Let $T = \\mathbb{R}/2\\pi\\mathbb{Z}$ with the measure induced by Lebesgue measure on $[0, 2\\pi)$, and let $L^1(T)$ be the space of real-valued integrable functions on $T$ modulo equality almost everywhere. Fix a real-valued integrable $f$ on $T$. Let $B_f \\subseteq L^1(T)$ be the set of classes of the finite linear combinations of translates $x \\mapsto \\sum_{i=1}^N a_i f(x - \\tau_i)$ with $N \\in \\mathbb{N}$,\u2026","title":"Convolutions and linear combinations of translates have the same closure"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem.json","id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","kind":"scope","navigation":"module","parent":16829,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","title":"Fej\u00e9r kernel and Fej\u00e9r's theorem"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-kernel","kind":"node","lean":"AddCircle.fejerKernel","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-kernel","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and, for $n \\in \\mathbb{Z}$, let $e_n(x) = e^{2\\pi i n x / T}$ be the $n$-th Fourier character on the circle $\\mathbb{R}/T\\mathbb{Z}$. For a natural number $N$, the $N$-th Fej\u00e9r kernel is the function $K_N : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ given by the Ces\u00e0ro mean of the Dirichlet kernels, $K_N(x) = \\frac{1}{N+1} \\sum_{k=0}^{N} \\sum_{n=-k}^{k} e_n(x)$.","title":"The Fej\u00e9r kernel"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fourier-integrable","kind":"node","lean":"AddCircle.fourier_integrable","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fourier-integrable","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$. For every integer $n$, the Fourier character $e_n(x) = e^{2\\pi i n x / T}$ is integrable on the circle $\\mathbb{R}/T\\mathbb{Z}$ with respect to the normalized Haar measure, the Haar measure of total mass $1$.","title":"Integrability of Fourier characters"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/integral-fourier","kind":"node","lean":"AddCircle.integral_fourier","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#integral-fourier","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and, for $n \\in \\mathbb{Z}$, let $e_n(t) = e^{2\\pi i n t / T}$ be the $n$-th Fourier character on the circle $\\mathbb{R}/T\\mathbb{Z}$. With respect to the Haar probability measure on the circle, $\\int e_n = 1$ if $n = 0$ and $\\int e_n = 0$ otherwise. This is the orthogonality of the Fourier basis.","title":"Integral of a Fourier character"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/integral-dirichlet-sum","kind":"node","lean":"AddCircle.integral_dirichlet_sum","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#integral-dirichlet-sum","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$, let $e_n(t) = e^{2\\pi i n t / T}$ be the $n$-th Fourier character on the circle $\\mathbb{R}/T\\mathbb{Z}$, and for $k \\in \\mathbb{N}$ let $D_k = \\sum_{n=-k}^{k} e_n$ be the $k$-th Dirichlet kernel. With respect to the Haar probability measure on the circle, $\\int D_k = 1$; only the term $n = 0$ contributes, by orthogonality.","title":"The Dirichlet kernel has integral one"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-kernel-symmetric","kind":"node","lean":"AddCircle.fejerKernel_symmetric","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-kernel-symmetric","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$. For every natural number $N$ and every $x \\in \\mathbb{R}/T\\mathbb{Z}$, the $N$-th Fej\u00e9r kernel satisfies $K_N(-x) = K_N(x)$.","title":"Symmetry of the Fej\u00e9r kernel"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/integral-fejer-kernel","kind":"node","lean":"AddCircle.integral_fejerKernel","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#integral-fejer-kernel","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and $N \\in \\mathbb{N}$. The $N$-th Fej\u00e9r kernel $K_N$ on the circle $\\mathbb{R}/T\\mathbb{Z}$ satisfies $\\int K_N = 1$ with respect to the Haar probability measure on the circle. This is one of the defining properties of an approximation to the identity.","title":"The Fej\u00e9r kernel has integral one"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/norm-fejer-kernel-le","kind":"node","lean":"AddCircle.norm_fejerKernel_le","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#norm-fejer-kernel-le","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and $N \\in \\mathbb{N}$. For every point $x$ of the circle $\\mathbb{R}/T\\mathbb{Z}$, the $N$-th Fej\u00e9r kernel satisfies $|K_N(x)| \\le N + 1$.","title":"Pointwise bound for the Fej\u00e9r kernel"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-kernel-continuous","kind":"node","lean":"AddCircle.fejerKernel_continuous","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-kernel-continuous","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$. For every natural number $N$, the $N$-th Fej\u00e9r kernel $K_N$ is a continuous function from the circle $\\mathbb{R}/T\\mathbb{Z}$ to $\\mathbb{C}$.","title":"Continuity of the Fej\u00e9r kernel"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-kernel-integrable","kind":"node","lean":"AddCircle.fejerKernel_integrable","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-kernel-integrable","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$. For every natural number $N$, the $N$-th Fej\u00e9r kernel $K_N$ is integrable on the circle $\\mathbb{R}/T\\mathbb{Z}$ with respect to the normalized Haar measure, the Haar measure of total mass $1$.","title":"Integrability of the Fej\u00e9r kernel"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fourier-neg-mul-fourier-eq","kind":"node","lean":"AddCircle.fourier_neg_mul_fourier_eq","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fourier-neg-mul-fourier-eq","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T$ be a real number and, for $n \\in \\mathbb{Z}$, let $e_n$ be the $n$-th Fourier character on $\\mathbb{R}/T\\mathbb{Z}$, given by $e_n(x) = e^{2\\pi i n x / T}$ (the constant $1$ when $T = 0$). For every integer $n$ and all $x, t \\in \\mathbb{R}/T\\mathbb{Z}$, we have $e_{-n}(t)\\, e_n(x) = e_n(x - t)$.","title":"Products of Fourier characters"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/sum-range-fourier-shift-neg","kind":"node","lean":"AddCircle.sum_range_fourier_shift_neg","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#sum-range-fourier-shift-neg","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$, let $e_n(t) = e^{2\\pi i n t / T}$ be the $n$-th Fourier character on the circle $\\mathbb{R}/T\\mathbb{Z}$, and let $N \\in \\mathbb{N}$. For every point $x$ of the circle, $\\sum_{k=0}^{N} e_{k-(N+1)}(x) = \\sum_{n=-(N+1)}^{-1} e_n(x)$.","title":"Reindexing a sum of Fourier characters, negative range"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/sum-range-fourier-shift-pos","kind":"node","lean":"AddCircle.sum_range_fourier_shift_pos","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#sum-range-fourier-shift-pos","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$, let $e_n(t) = e^{2\\pi i n t / T}$ be the $n$-th Fourier character on the circle $\\mathbb{R}/T\\mathbb{Z}$, and let $N \\in \\mathbb{N}$. For every point $x$ of the circle, $\\sum_{k=0}^{N} e_{(N+1)-k}(x) = \\sum_{n=1}^{N+1} e_n(x)$.","title":"Reindexing a sum of Fourier characters, positive range"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-dirichlet-sum-eq-sq","kind":"node","lean":"AddCircle.fejer_dirichlet_sum_eq_sq","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-dirichlet-sum-eq-sq","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and, for $n \\in \\mathbb{Z}$, let $e_n(x) = e^{2\\pi i n x / T}$ be the $n$-th Fourier character on the circle $\\mathbb{R}/T\\mathbb{Z}$. For every natural number $N$ and every $x \\in \\mathbb{R}/T\\mathbb{Z}$, put $G = \\sum_{k=0}^{N} e_k(x)$. Then the iterated sum of Dirichlet kernels satisfies $\\sum_{k=0}^{N} \\sum_{n=-k}^{k} e_n(x) = G\\,\\overline{G} = |G|^2$.","title":"The sum of Dirichlet kernels is a squared modulus"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-kernel-nonneg","kind":"node","lean":"AddCircle.fejerKernel_nonneg","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-kernel-nonneg","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$. For every natural number $N$ and every $x \\in \\mathbb{R}/T\\mathbb{Z}$, the real part of the $N$-th Fej\u00e9r kernel at $x$ is nonnegative: $\\operatorname{Re} K_N(x) \\ge 0$.","title":"Nonnegativity of the Fej\u00e9r kernel"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-kernel-eq-of-real","kind":"node","lean":"AddCircle.fejerKernel_eq_ofReal","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-kernel-eq-of-real","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and, for $k \\in \\mathbb{Z}$, let $e_k(x) = e^{2\\pi i k x / T}$ be the $k$-th Fourier character on the circle $\\mathbb{R}/T\\mathbb{Z}$. For every natural number $N$ and every $x \\in \\mathbb{R}/T\\mathbb{Z}$, the $N$-th Fej\u00e9r kernel satisfies $K_N(x) = \\frac{1}{N+1} \\left|\\sum_{k=0}^{N} e_k(x)\\right|^2$.","title":"The Fej\u00e9r kernel as a squared modulus"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-kernel-properties","kind":"node","lean":"AddCircle.fejer_kernel_properties","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-kernel-properties","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$, let $N$ be a natural number, let $K_N$ be the $N$-th Fej\u00e9r kernel on the circle $\\mathbb{R}/T\\mathbb{Z}$, let $\\mu$ be the normalized Haar measure on $\\mathbb{R}/T\\mathbb{Z}$ (of total mass $1$), and let $e_1(x) = e^{2\\pi i x / T}$. Then $\\operatorname{Re} K_N(x) \\ge 0$ for every $x \\in \\mathbb{R}/T\\mathbb{Z}$; $K_N(-x) = K_N(x)$ for every $x \\in \\mathbb{R}/T\\mathbb{Z}$; the function on $\\mathbb{R}$\u2026","title":"Properties of the Fej\u00e9r kernel"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/partial-fourier-sum","kind":"node","lean":"AddCircle.partialFourierSum","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#partial-fourier-sum","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$. For $N \\in \\mathbb{N}$, the $N$-th partial Fourier sum of $f$ is the function","title":"Partial Fourier sums"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/cesaro-fourier-mean","kind":"node","lean":"AddCircle.cesaroFourierMean","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#cesaro-fourier-mean","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$. For $N \\in \\mathbb{N}$, the $N$-th Ces\u00e0ro\u2013Fourier mean of $f$ is the average of its first $N + 1$ partial Fourier sums,","title":"Ces\u00e0ro\u2013Fourier means"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fourier-sub-eq","kind":"node","lean":"AddCircle.fourier_sub_eq","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fourier-sub-eq","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and for $n \\in \\mathbb{Z}$ let $e_n(x) = e^{2\\pi i n x / T}$ be the $n$-th Fourier character on $\\mathbb{R}/T\\mathbb{Z}$. Then for every $n \\in \\mathbb{Z}$ and all $x, t \\in \\mathbb{R}/T\\mathbb{Z}$, $e_n(x - t) = e_{-n}(t)\\, e_n(x)$.","title":"Fourier characters at a difference"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fourier-smul-integrable","kind":"node","lean":"AddCircle.fourier_smul_integrable","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fourier-smul-integrable","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and let $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ be integrable with respect to the Haar probability measure on the circle. Then for every $n \\in \\mathbb{Z}$ and every $x \\in \\mathbb{R}/T\\mathbb{Z}$, the function $t \\mapsto e^{2\\pi i n t / T} f(x - t)$ is integrable with respect to the same measure.","title":"Integrability of a function twisted by a Fourier character"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fourier-integral-convolution","kind":"node","lean":"AddCircle.fourier_integral_convolution","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fourier-integral-convolution","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$, let $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ be integrable with respect to the Haar probability measure $dt$ on the circle, for $n \\in \\mathbb{Z}$ let $e_n(x) = e^{2\\pi i n x / T}$ be the $n$-th Fourier character, and let $\\hat f(n) = \\int_{\\mathbb{R}/T\\mathbb{Z}} e_{-n}(t)\\, f(t)\\, dt$ be the $n$-th Fourier coefficient of $f$. Then for every $n \\in \\mathbb{Z}$ and every\u2026","title":"Convolution of a function with a Fourier character"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/cesaro-mean-eq-fejer-convolution","kind":"node","lean":"AddCircle.cesaroMean_eq_fejer_convolution","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#cesaro-mean-eq-fejer-convolution","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and let $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ be integrable with respect to the Haar probability measure $dt$ on the circle. Then for every $N \\in \\mathbb{N}$ and every $x \\in \\mathbb{R}/T\\mathbb{Z}$, the $N$-th Ces\u00e0ro\u2013Fourier mean of $f$ satisfies","title":"Ces\u00e0ro\u2013Fourier means are convolutions with the Fej\u00e9r kernel"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/integral-norm-fejer-kernel","kind":"node","lean":"AddCircle.integral_norm_fejerKernel","parent":16902,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Fej\u00e9r kernel has "],["math","\\(L^1\\)"],["text"," norm one"]],"source":"The Fej\u00e9r kernel has $L^1$ norm one"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#integral-norm-fejer-kernel","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$. For every $N \\in \\mathbb{N}$, the $N$-th Fej\u00e9r kernel $K_N$ on $\\mathbb{R}/T\\mathbb{Z}$ satisfies","title":"The Fej\u00e9r kernel has $L^1$ norm one"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/cesaro-mean-uniform-bound","kind":"node","lean":"AddCircle.cesaroMean_uniform_bound","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#cesaro-mean-uniform-bound","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$, let $f$ be a continuous complex-valued function on the circle $\\mathbb{R}/T\\mathbb{Z}$, and write $\\sigma_N f$ for its $N$-th Ces\u00e0ro\u2013Fourier mean. Let $\\varepsilon > 0$ and $\\delta > 0$ be such that $|f(a) - f(b)| < \\varepsilon / 2$ for all points $a, b$ of the circle at distance less than $\\delta$. Then there is $N_0 \\in \\mathbb{N}$ such that $|f(x) - \\sigma_N f(x)| < \\varepsilon$ for every\u2026","title":"Quantitative Fej\u00e9r theorem"},{"id":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/fejer-uniform-convergence","kind":"node","lean":"AddCircle.fejer_uniform_convergence","parent":16902,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem/index.html#fejer-uniform-convergence","scope":"Analysis/harmonic-analysis/fourier-series/fej-r-kernel-and-fej-r-s-theorem","status":"fully_proved","summary":"Let $T > 0$ and let $f$ be a continuous complex-valued function on the circle $\\mathbb{R}/T\\mathbb{Z}$ (equivalently, a continuous $T$-periodic function on $\\mathbb{R}$). Then the Ces\u00e0ro\u2013Fourier means $\\sigma_N f$ converge to $f$ uniformly on the circle as $N \\to \\infty$.","title":"Fej\u00e9r's theorem: uniform convergence of Ces\u00e0ro\u2013Fourier means"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle.json","id":"Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle","kind":"scope","navigation":"module","parent":16829,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle/index.html","scope":"Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle","status":"can_prove","title":"Fourier coefficients of measures on the circle"},{"id":"Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle/fourier-coeff-measure","kind":"node","lean":"MathlibExt.Analysis.WienerAtomWanted.Analysis.fourierCoeffMeasure","parent":16928,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle/index.html#fourier-coeff-measure","scope":"Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle","status":"fully_proved","summary":"For a measure $\\mu$ on the circle $\\mathbb{T} = \\mathbb{R}/2\\pi\\mathbb{Z}$ and an integer $k$, the $k$-th Fourier coefficient of $\\mu$ is $\\hat\\mu_k = \\int_{\\mathbb{T}} e^{ikx} \\, d\\mu(x)$.","title":"Fourier coefficients of a measure on the circle"},{"id":"Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle/wiener-atom-detection","kind":"node","lean":"MathlibExt.Analysis.WienerAtomWanted.Analysis.wiener_atom_detection","parent":16928,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle/index.html#wiener-atom-detection","scope":"Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle","status":"can_prove","summary":"Let $\\mu$ be a probability measure on the circle $\\mathbb{T} = \\mathbb{R}/2\\pi\\mathbb{Z}$, with Fourier coefficients $\\hat\\mu_k = \\int_{\\mathbb{T}} e^{ikx} \\, d\\mu(x)$. Then $\\frac{1}{N} \\sum_{k=1}^{N} \\lvert \\hat\\mu_k \\rvert^2 \\to \\sum_{x \\in \\mathbb{T}} \\mu(\\{x\\})^2$ as $N \\to \\infty$.","title":"Wiener's atom-detection formula"},{"id":"Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle/tendsto-integral-pow-mul-of-tendsto-inte","kind":"node","lean":"MathlibExt.MeasureTheory.CircleMeasureFourierDecayIntegrableWanted.tendsto_integral_pow_mul_of_tendsto_integral_pow","parent":16928,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle/index.html#tendsto-integral-pow-mul-of-tendsto-inte","scope":"Analysis/harmonic-analysis/fourier-series/fourier-coefficients-of-measures-on-the-circle","status":"can_prove","summary":"Let $\\mu$ be a finite Borel measure on $\\mathbb{C}$ concentrated on the unit circle $T = \\{z \\in \\mathbb{C} : |z| = 1\\}$, that is, $\\mu$-almost every $z$ lies in $T$, and suppose that $\\int z^n \\, d\\mu(z) \\to 0$ as $n \\to \\infty$ through $\\mathbb{N}$. Then for every $\\mu$-integrable function $f : \\mathbb{C} \\to \\mathbb{C}$,","title":"Fourier decay of a circle measure passes to integrable densities"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma.json","id":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma","kind":"scope","navigation":"module","parent":16829,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/index.html","scope":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma","status":"fully_proved","title":"Riemann\u2013Lebesgue lemma"},{"id":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/norm-fourier-coeff-le-integral-norm","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.RiemannLebesgue.norm_fourierCoeff_le_integral_norm","parent":16932,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/index.html#norm-fourier-coeff-le-integral-norm","scope":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma","status":"fully_proved","summary":"Let $T > 0$, let $\\mu$ be the Haar probability measure on the circle $\\mathbb{R}/T\\mathbb{Z}$, and let $E$ be a normed space over $\\mathbb{C}$. For every function $f : \\mathbb{R}/T\\mathbb{Z} \\to E$ and every $n \\in \\mathbb{Z}$, the $n$-th Fourier coefficient $\\hat f(n) = \\int e^{-2\\pi i n t / T} f(t)\\,d\\mu(t)$ satisfies $\\|\\hat f(n)\\| \\le \\int \\|f(t)\\|\\,d\\mu(t)$.","title":"A Fourier coefficient is bounded by the L\u00b9 norm"},{"id":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/tendsto-fourier-coeff-cofinite-of-contin","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.RiemannLebesgue.tendsto_fourierCoeff_cofinite_of_continuous","parent":16932,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/index.html#tendsto-fourier-coeff-cofinite-of-contin","scope":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma","status":"fully_proved","summary":"Let $T > 0$ and let $\\mu$ be the Haar probability measure on the circle $\\mathbb{R}/T\\mathbb{Z}$. For every continuous function $g : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$, the Fourier coefficients $\\hat g(n) = \\int e^{-2\\pi i n t / T} g(t)\\,d\\mu(t)$ tend to $0$ as $|n| \\to \\infty$, that is, along the cofinite filter on $\\mathbb{Z}$.","title":"Riemann\u2013Lebesgue for continuous functions"},{"id":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/tendsto-fourier-coeff-cofinite","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.RiemannLebesgue.tendsto_fourierCoeff_cofinite","parent":16932,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/index.html#tendsto-fourier-coeff-cofinite","scope":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma","status":"fully_proved","summary":"Let $T > 0$ and let $\\mu$ be the Haar probability measure on the circle $\\mathbb{R}/T\\mathbb{Z}$. For every $\\mu$-integrable function $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$, the Fourier coefficients $\\hat f(n) = \\int e^{-2\\pi i n t / T} f(t)\\,d\\mu(t)$ tend to $0$ as $|n| \\to \\infty$, that is, along the cofinite filter on $\\mathbb{Z}$.","title":"Riemann\u2013Lebesgue lemma on the circle"},{"id":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/tendsto-fourier-coeff-on-cofinite","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.RiemannLebesgue.tendsto_fourierCoeffOn_cofinite","parent":16932,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma/index.html#tendsto-fourier-coeff-on-cofinite","scope":"Analysis/harmonic-analysis/fourier-series/riemann-lebesgue-lemma","status":"fully_proved","summary":"Let $a < b$ be real numbers and let $f : \\mathbb{R} \\to \\mathbb{C}$ be integrable on $[a, b]$. Then the Fourier coefficients of $f$ on $[a, b]$,","title":"Riemann\u2013Lebesgue lemma on an interval"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series.json","id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","kind":"scope","navigation":"module","parent":16829,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"can_prove","title":"Absolutely convergent Fourier series"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/mem-lp-of-continuous-on-ioc","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.memLp_of_continuousOn_Ioc","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#mem-lp-of-continuous-on-ioc","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"A continuous function on `\u211d` is square-integrable on one period.","title":"Continuous functions are square-integrable on one period"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/lift-periodic-continuous-map","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.liftPeriodicContinuousMap","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#lift-periodic-continuous-map","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"A `T`-periodic continuous function descends to a continuous function on the circle `AddCircle T`, via `AddCircle.liftIoc`.","title":"Lift of a periodic continuous function to the circle"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/lift-periodic-continuous-map-apply","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.liftPeriodicContinuousMap_apply","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#lift-periodic-continuous-map-apply","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"The descended function agrees with `f` on the period `(0, T]`.","title":"Lifted periodic function agrees with the original on a period"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/fourier-coeff-on-zero-add","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.fourierCoeffOn_zero_add","parent":16937,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourier coefficients on "],["math","\\((0, 0+T)\\)"],["text"," equal those on "],["math","\\((0, T)\\)"]],"source":"Fourier coefficients on $(0, 0+T)$ equal those on $(0, T)$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#fourier-coeff-on-zero-add","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"The interval coefficients on `(0, 0 + T]` agree with those on `(0, T]`; this bridges Mathlib's `fourierCoeff_liftIoc_eq`, which is stated on `(a, a + T]`.","title":"Fourier coefficients on $(0, 0+T)$ equal those on $(0, T)$"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/fourier-coeff-lift-periodic-continuous-m","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.fourierCoeff_liftPeriodicContinuousMap","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#fourier-coeff-lift-periodic-continuous-m","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"The Fourier coefficients of the descended function are the interval coefficients `fourierCoeffOn`.","title":"Fourier coefficients of the lifted periodic function"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/norm-fourier-coeff-on-le-of-bound","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.norm_fourierCoeffOn_le_of_bound","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#norm-fourier-coeff-on-le-of-bound","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"A Fourier coefficient bounded by the sup norm on one period.","title":"Fourier coefficient bounded by the sup norm on a period"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/fourier-coeff-on-eq-of-has-deriv-at","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.fourierCoeffOn_eq_of_hasDerivAt","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#fourier-coeff-on-eq-of-has-deriv-at","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"For `n \u2260 0`, `\u0109\u2099(f) = (T / (2\u03c0 i n)) \u00b7 \u0109\u2099(f')`. The boundary term in integration by parts vanishes by periodicity.","title":"Coefficient identity for a `C\u00b9` periodic function"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/norm-fourier-coeff-on-eq-of-has-deriv-at","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.norm_fourierCoeffOn_eq_of_hasDerivAt","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#norm-fourier-coeff-on-eq-of-has-deriv-at","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"Norm form of the coefficient identity.","title":"Norm form of the coefficient identity"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/summable-fourier-coeff-on-of-has-deriv-a","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.summable_fourierCoeffOn_of_hasDerivAt","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#summable-fourier-coeff-on-of-has-deriv-a","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"Fourier coefficients of a `C\u00b9` periodic function are absolutely summable.","title":"Fourier coefficients of a `C\u00b9` periodic function are absolutely summable"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/tendsto-mul-norm-fourier-coeff-on-of-has","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.tendsto_mul_norm_fourierCoeffOn_of_hasDerivAt","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#tendsto-mul-norm-fourier-coeff-on-of-has","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"`n \u00b7 \u2016\u0109\u2099(f)\u2016 \u2192 0` along the cofinite filter.","title":"Decay rate:"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/has-sum-fourier-series-of-has-deriv-at","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.hasSum_fourier_series_of_hasDerivAt","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#has-sum-fourier-series-of-has-deriv-at","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"The Fourier series of a `C\u00b9` periodic function converges uniformly (in `C(AddCircle T, \u2102)`) to `f` on the circle.","title":"Uniform convergence:"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/has-sum-fourier-series-of-has-deriv-at-a","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.C1FourierSummable.hasSum_fourier_series_of_hasDerivAt_apply","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#has-sum-fourier-series-of-has-deriv-at-a","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"Pointwise convergence on one period.","title":"Pointwise convergence on one period"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/in-wiener-algebra","kind":"node","lean":"MathlibExt.Analysis.WienerLevyWanted.Analysis.WienerLevy.InWienerAlgebra","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#in-wiener-algebra","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"Let $T > 0$. A continuous function $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ lies in the Wiener algebra when the family of its Fourier coefficients $\\hat f(n) = \\frac{1}{T} \\int_0^T f(x) e^{-2\\pi i n x / T} \\, dx$, $n \\in \\mathbb{Z}$, is summable in $\\mathbb{C}$.","title":"Wiener algebra on the circle"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/wiener-levy-analytic-calculus","kind":"node","lean":"MathlibExt.Analysis.WienerLevyWanted.Analysis.WienerLevy.wiener_levy_analytic_calculus","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#wiener-levy-analytic-calculus","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"can_prove","summary":"Let $T > 0$, and let $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ be a continuous function in the Wiener algebra. Let $U \\subseteq \\mathbb{C}$ be an open set containing the range of $f$, and let $\\varphi : \\mathbb{C} \\to \\mathbb{C}$ be analytic at every point of $U$. Then there is a continuous function $g : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ with $g(x) = \\varphi(f(x))$ for every $x$, and $g$ lies in the Wiener\u2026","title":"Wiener\u2013L\u00e9vy theorem"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/in-wiener-algebra-2","kind":"node","lean":"MathlibExt.Analysis.WienerOneOverFWanted.Analysis.WienerOneOverF.InWienerAlgebra","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#in-wiener-algebra-2","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"fully_proved","summary":"Let $T > 0$. A continuous function $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ lies in the Wiener algebra when the family of its Fourier coefficients $\\hat f(n) = \\frac{1}{T} \\int_0^T f(x) e^{-2\\pi i n x / T} \\, dx$, $n \\in \\mathbb{Z}$, is summable in $\\mathbb{C}$.","title":"Functions with absolutely convergent Fourier series"},{"id":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/wiener-inverse-closed","kind":"node","lean":"MathlibExt.Analysis.WienerOneOverFWanted.Analysis.WienerOneOverF.wiener_inverse_closed","parent":16937,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series/index.html#wiener-inverse-closed","scope":"Analysis/harmonic-analysis/fourier-series/absolutely-convergent-fourier-series","status":"can_prove","summary":"Let $T > 0$, and let $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ be a continuous function in the Wiener algebra with $f(x) \\ne 0$ for every $x$. Then there is a continuous function $g : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ with $g(x) = f(x)^{-1}$ for every $x$, and $g$ lies in the Wiener algebra.","title":"Wiener's 1/f theorem"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients.json","id":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients","kind":"scope","navigation":"module","parent":16829,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/index.html","scope":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients","status":"fully_proved","title":"Uniqueness of Fourier coefficients"},{"id":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/eq-zero-of-fourier-coeff-eq-zero-lp","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.FourierUniqueness.eq_zero_of_fourierCoeff_eq_zero_Lp","parent":16954,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/index.html#eq-zero-of-fourier-coeff-eq-zero-lp","scope":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients","status":"fully_proved","summary":"Let $T > 0$ and let $\\mu$ be the Haar probability measure on the circle $\\mathbb{R}/T\\mathbb{Z}$. If an element $F$ of $L^2(\\mathbb{R}/T\\mathbb{Z}, \\mu; \\mathbb{C})$ has Fourier coefficients $\\hat F(n) = \\int e^{-2\\pi i n t / T} F(t)\\,d\\mu(t)$ equal to $0$ for every $n \\in \\mathbb{Z}$, then $F = 0$ in $L^2(\\mathbb{R}/T\\mathbb{Z}, \\mu; \\mathbb{C})$.","title":"Uniqueness of Fourier coefficients in L\u00b2"},{"id":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/eq-zero-of-fourier-coeff-eq-zero","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.FourierUniqueness.eq_zero_of_fourierCoeff_eq_zero","parent":16954,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/index.html#eq-zero-of-fourier-coeff-eq-zero","scope":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients","status":"fully_proved","summary":"Let $T > 0$ and let $\\mu$ be the Haar probability measure on the circle $\\mathbb{R}/T\\mathbb{Z}$. If a continuous function $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ has Fourier coefficients $\\hat f(n) = \\int e^{-2\\pi i n t / T} f(t)\\,d\\mu(t)$ equal to $0$ for every $n \\in \\mathbb{Z}$, then $f = 0$.","title":"Uniqueness of Fourier coefficients for continuous functions"},{"id":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/eq-of-fourier-coeff-eq","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.FourierUniqueness.eq_of_fourierCoeff_eq","parent":16954,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/index.html#eq-of-fourier-coeff-eq","scope":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients","status":"fully_proved","summary":"Let $T > 0$ and let $\\mu$ be the Haar probability measure on the circle $\\mathbb{R}/T\\mathbb{Z}$. For a function $h : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ and $n \\in \\mathbb{Z}$, let $\\hat h(n) = \\int e^{-2\\pi i n t / T} h(t)\\,d\\mu(t)$ be the $n$-th Fourier coefficient of $h$. If two continuous functions $f, g : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ satisfy $\\hat f(n) = \\hat g(n)$ for every $n \\in \\mathbb{Z}$,\u2026","title":"Fourier coefficients determine a continuous function"},{"id":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/fourier-coeff-injective","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.FourierUniqueness.fourierCoeff_injective","parent":16954,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/index.html#fourier-coeff-injective","scope":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients","status":"fully_proved","summary":"Let $T > 0$ and let $\\mu$ be the Haar probability measure on the circle $\\mathbb{R}/T\\mathbb{Z}$. The map sending a continuous function $f : \\mathbb{R}/T\\mathbb{Z} \\to \\mathbb{C}$ to its sequence of Fourier coefficients $(\\hat f(n))_{n \\in \\mathbb{Z}}$, where $\\hat f(n) = \\int e^{-2\\pi i n t / T} f(t)\\,d\\mu(t)$, is injective.","title":"Injectivity of the Fourier-coefficient map"},{"id":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/eq-zero-of-fourier-coeff-on-eq-zero","kind":"node","lean":"MathlibExt.Analysis.Fourier.AddCircle.FourierUniqueness.eq_zero_of_fourierCoeffOn_eq_zero","parent":16954,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients/index.html#eq-zero-of-fourier-coeff-on-eq-zero","scope":"Analysis/harmonic-analysis/fourier-series/uniqueness-of-fourier-coefficients","status":"fully_proved","summary":"Let $a, T \\in \\mathbb{R}$ with $T > 0$, and let $f : \\mathbb{R} \\to \\mathbb{C}$ be continuous with $f(a) = f(a + T)$. If the Fourier coefficients of $f$ on $[a, a + T]$,","title":"Uniqueness of Fourier coefficients on an interval"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform.json","id":"Analysis/harmonic-analysis/fourier-transform","kind":"scope","parent":16828,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/index.html","scope":"Analysis/harmonic-analysis/fourier-transform","status":"can_prove","summary":"The Fourier transform on R^n and Schwartz functions: convolution, Bochner, Hausdorff\u2013Young, Heisenberg and fractal uncertainty, band-limited functions and Bernstein inequalities, sampling and Fourier interpolation, Kakeya sets, Beurling\u2013Ahlfors and other singular integrals.","title":"Fourier transform and uncertainty principles"},{"id":"Analysis/harmonic-analysis/fourier-transform/continuous-and-tendsto-cocompact-convolu","kind":"node","lean":"MathlibExt.Analysis.Fourier.L2ConvolutionContinuousVanishingWanted.continuous_and_tendsto_cocompact_convolution_of_memLp_two","parent":16960,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/index.html#continuous-and-tendsto-cocompact-convolu","scope":"Analysis/harmonic-analysis/fourier-transform","status":"can_prove","summary":"Let $n$ be a natural number and let $f, g : \\mathbb{R}^n \\to \\mathbb{C}$ be square-integrable. Then the convolution $(f \\ast g)(x) = \\int_{\\mathbb{R}^n} f(x - y) \\, g(y) \\, dy$ is a continuous function of $x \\in \\mathbb{R}^n$, and $(f \\ast g)(x) \\to 0$ as $|x| \\to \\infty$.","title":"Convolution of two square-integrable functions is continuous and vanishes at infinity"},{"id":"Analysis/harmonic-analysis/fourier-transform/convolution-theorem","kind":"node","lean":"MetaMathlibExt.convolution_theorem","parent":16960,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/index.html#convolution-theorem","scope":"Analysis/harmonic-analysis/fourier-transform","status":"fully_proved","summary":"Let $e$ be a continuous additive character from $\\mathbb{R}$ to the unit circle, and let $\\mu$ be a \u03c3-finite measure on $\\mathbb{R}$ that is preserved by every translation $x \\mapsto a + x$. For $\\varphi : \\mathbb{R} \\to \\mathbb{C}$ write $\\hat\\varphi(w) = \\int e(-xw)\\,\\varphi(x)\\,d\\mu(x)$ for its Fourier transform with respect to $e$ and $\\mu$, and for $f, g : \\mathbb{R} \\to \\mathbb{C}$ write\u2026","title":"Convolution theorem"},{"id":"Analysis/harmonic-analysis/fourier-transform/exists-norm-fourier-to-lp-le-norm-to-lp","kind":"node","lean":"MathlibExt.Analysis.Fourier.HausdorffYoungSchwartzWanted.exists_norm_fourier_toLp_le_norm_toLp_of_holderConjugate","parent":16960,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/index.html#exists-norm-fourier-to-lp-le-norm-to-lp","scope":"Analysis/harmonic-analysis/fourier-transform","status":"can_prove","summary":"Let $n$ be a natural number and let $\\mathcal{F}$ be the Fourier transform on $\\mathbb{R}^n$, $\\mathcal{F}f(\\xi) = \\int_{\\mathbb{R}^n} f(x) e^{-2\\pi i \\langle x, \\xi \\rangle}\\, dx$. There is a constant $C > 0$ such that for all $p, p' \\in [0, \\infty]$ with $1 \\le p \\le 2$ and $\\frac{1}{p} + \\frac{1}{p'} = 1$ (so $p' = \\infty$ when $p = 1$) and every complex-valued Schwartz function $f$ on $\\mathbb{R}^n$,\u2026","title":"Hausdorff\u2013Young inequality for Schwartz functions"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function.json","id":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function","status":"can_prove","title":"Bernstein inequalities for band-limited functions"},{"id":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/dot-hseminorm","kind":"node","lean":"MathlibExt.Analysis.Fourier.BandLimitedDerivSobolevBoundWanted.dotHSeminorm","parent":16964,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/index.html#dot-hseminorm","scope":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function","status":"fully_proved","summary":"For $s \\in \\mathbb{R}$ and a Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ with Fourier transform $\\hat f(\\xi) = \\int_{\\mathbb{R}} f(x) e^{-2\\pi i x\\xi}\\,dx$, the homogeneous Sobolev seminorm of order $s$ is","title":"Homogeneous Sobolev seminorm (band-limited derivative bound)"},{"id":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/norm-deriv-le-rpow-mul-dot-hseminorm-of","kind":"node","lean":"MathlibExt.Analysis.Fourier.BandLimitedDerivSobolevBoundWanted.norm_deriv_le_rpow_mul_dotHSeminorm_of_fourier_support_Icc","parent":16964,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/index.html#norm-deriv-le-rpow-mul-dot-hseminorm-of","scope":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function","status":"can_prove","summary":"Let $s \\in (\\tfrac12, 1]$. For a Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ write $\\hat f(\\xi) = \\int_{\\mathbb{R}} f(x) e^{-2\\pi i x\\xi}\\,dx$ for its Fourier transform and $\\|f\\|_{\\dot H^s} = \\left(\\int_{\\mathbb{R}} |\\xi|^{2s} |\\hat f(\\xi)|^2\\,d\\xi\\right)^{1/2}$ for its homogeneous Sobolev seminorm. There is a constant $C_1 > 0$ such that for every real $A \\ge 0$ and every Schwartz function\u2026","title":"Bernstein-type bound for the derivative of a band-limited Schwartz function"},{"id":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/exists-const-norm-to-lp-le-volume-rpow-m","kind":"node","lean":"MathlibExt.Analysis.Fourier.BernsteinFourierSupportLpLqWanted.exists_const_norm_toLp_le_volume_rpow_mul_norm_toLp_of_tsupport_fourier_subset","parent":16964,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/index.html#exists-const-norm-to-lp-le-volume-rpow-m","scope":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function","status":"can_prove","summary":"Fix $n \\in \\mathbb{N}$. For a Schwartz function $f : \\mathbb{R}^n \\to \\mathbb{C}$ write $\\hat f(\\xi) = \\int_{\\mathbb{R}^n} f(x) e^{-2\\pi i \\langle x, \\xi \\rangle}\\,dx$ for its Fourier transform, and write $|E|$ for the Lebesgue measure of a set $E \\subseteq \\mathbb{R}^n$. There is a constant $C > 0$ such that for every Borel set $E \\subseteq \\mathbb{R}^n$ with $|E| < \\infty$, all $p, q \\in [1, \\infty]$ with\u2026","title":"Bernstein inequality for Schwartz functions with Fourier support of finite measure"},{"id":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/exists-norm-deriv-to-lp-le-of-support-fo","kind":"node","lean":"MathlibExt.Analysis.Fourier.BernsteinInequalityBandLimitedWanted.exists_norm_deriv_toLp_le_of_support_fourier_subset_ball","parent":16964,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function/index.html#exists-norm-deriv-to-lp-le-of-support-fo","scope":"Analysis/harmonic-analysis/fourier-transform/bernstein-inequalities-for-band-limited-function","status":"can_prove","summary":"For a Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ write $\\hat f(\\xi) = \\int_{\\mathbb{R}} f(x) e^{-2\\pi i x\\xi}\\,dx$ for its Fourier transform. There is a constant $C_0 > 0$ such that for every $p \\in [1, \\infty]$ and every Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ with $\\hat f(\\xi) = 0$ whenever $|\\xi| \\ge 1$,","title":"Bernstein inequality for band-limited Schwartz functions"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform.json","id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"can_prove","title":"Beurling\u2013Ahlfors transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/pstar","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlfors.pstar","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#pstar","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"fully_proved","summary":"For a real number $p$, $p^* = \\max(p, p/(p-1))$, with division by zero taken to be $0$ (so $p^* = 1$ at $p = 1$). For $p > 1$ this is the larger of $p$ and its conjugate exponent $p/(p-1)$.","title":"Larger of an exponent and its conjugate"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/beurling-trunc","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlfors.beurlingTrunc","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#beurling-trunc","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"fully_proved","summary":"For a real number $\\varepsilon$, a function $f : \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$, the truncated Beurling\u2013Ahlfors integral at cutoff $\\varepsilon$ is $-\\frac{1}{\\pi} \\int_{|w - z| > \\varepsilon} \\frac{f(w)}{(z - w)^2} \\, dw$, the integral being with respect to Lebesgue measure on $\\mathbb{C}$ and taken to be $0$ when the integrand is not integrable.","title":"Truncated Beurling\u2013Ahlfors integral"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/beurling-transform","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlfors.beurlingTransform","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#beurling-transform","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"fully_proved","summary":"For $f : \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$, $Bf(z)$ is the limit as $\\varepsilon \\to 0^+$ of the truncated Beurling\u2013Ahlfors integrals $-\\frac{1}{\\pi} \\int_{|w - z| > \\varepsilon} \\frac{f(w)}{(z - w)^2} \\, dw$ (each taken to be $0$ when its integrand is not integrable). Where this limit does not exist, $Bf(z)$ is an unspecified complex number.","title":"Beurling\u2013Ahlfors transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/is-bound","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlfors.IsBound","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#is-bound","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"fully_proved","summary":"For real numbers $p$ and $C$, $C$ is an admissible $L^p$ bound constant for the Beurling\u2013Ahlfors transform $B$ when $\\|Bf\\|_q \\le \\max(C, 0) \\, \\|f\\|_q$ for every almost everywhere strongly measurable function $f : \\mathbb{C} \\to \\mathbb{C}$, where $q = \\max(p, 0)$ and $\\|\\cdot\\|_q$ is the extended $L^q$ norm with respect to Lebesgue measure (identically $0$ when $q = 0$), with values in $[0, \\infty]$ and the\u2026","title":"Admissible bounds for the Beurling\u2013Ahlfors transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/beurling-norm","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlfors.beurlingNorm","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#beurling-norm","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"fully_proved","summary":"For a real number $p$, the norm $\\|B\\|_p$ of the Beurling\u2013Ahlfors transform $B$ is the infimum of the real numbers $C \\ge 0$ that are admissible $L^p$ bound constants for $B$, and is $0$ when there is no such $C$. In particular an upper bound on $\\|B\\|_p$ alone does not imply that $B$ is bounded on $L^p$.","title":"Norm of the Beurling\u2013Ahlfors transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/beurling-norm-le-of-is-bound","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlfors.beurlingNorm_le_of_isBound","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#beurling-norm-le-of-is-bound","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"fully_proved","summary":"Let $p$ and $C$ be real numbers with $C \\ge 0$, and let $\\|B\\|_p$ be the infimum of the nonnegative admissible $L^p$ bound constants for the Beurling\u2013Ahlfors transform $B$ ($0$ if there are none). If $C$ is an admissible $L^p$ bound constant for $B$, then $\\|B\\|_p \\le C$.","title":"Nonnegative admissible bounds bound the Beurling\u2013Ahlfors norm"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/beurling-ahlfors-constant","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlfors.beurlingAhlforsConstant","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#beurling-ahlfors-constant","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"fully_proved","summary":"The Beurling\u2013Ahlfors constant is the supremum, over real numbers $p > 1$, of $\\|B\\|_p / (p^* - 1)$, where $B$ is the Beurling\u2013Ahlfors transform, $\\|B\\|_p$ is the infimum of the real numbers $C \\ge 0$ that are admissible $L^p$ bound constants for $B$ (and $0$ if there is no such $C$), and $p^* = \\max(p, p/(p-1))$. The supremum is taken in $\\mathbb{R}$, so it is $0$ when these ratios are not bounded above.","title":"Beurling\u2013Ahlfors constant"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/iwaniec-conjecture","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlfors.IwaniecConjecture","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#iwaniec-conjecture","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"fully_proved","summary":"For every real number $p > 1$, the $L^p$ norm of the Beurling\u2013Ahlfors transform $B$ satisfies $\\|B\\|_p = p^* - 1$, where $p^* = \\max(p, p/(p-1))$.","title":"Iwaniec's conjecture"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/lehto-lower-bound-wanted","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlforsWanted.lehtoLowerBound_wanted","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#lehto-lower-bound-wanted","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"can_prove","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ let $Bf(z)$ be the limit as $\\varepsilon \\to 0^+$ of $-\\frac{1}{\\pi} \\int_{|w - z| > \\varepsilon} \\frac{f(w)}{(z - w)^2} \\, dA(w)$, where $dA$ is Lebesgue area measure (a junk value where the limit does not exist). For real $p > 1$ let $p^* = \\max(p, p/(p - 1))$ and let $\\|B\\|_p$ be the infimum of the constants $C \\ge 0$ such that $\\|Bf\\|_{L^p} \\le C\\,\\|f\\|_{L^p}$ in\u2026","title":"Lehto's lower bound for the Beurling\u2013Ahlfors transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/banuelos-janakiraman-upper-bound-wanted","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlforsWanted.banuelosJanakiramanUpperBound_wanted","parent":16969,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#banuelos-janakiraman-upper-bound-wanted","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"can_prove","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ let $Bf(z)$ be the limit as $\\varepsilon \\to 0^+$ of $-\\frac{1}{\\pi} \\int_{|w - z| > \\varepsilon} \\frac{f(w)}{(z - w)^2} \\, dA(w)$, where $dA$ is Lebesgue area measure (a junk value where the limit does not exist), and for real $p > 1$ let $p^* = \\max(p, p/(p - 1))$. For every real $p > 1$ and every almost everywhere strongly measurable $f : \\mathbb{C} \\to \\mathbb{C}$,\u2026","title":"Ba\u00f1uelos\u2013Janakiraman upper bound for the Beurling\u2013Ahlfors transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/l2-norm-wanted","kind":"node","lean":"MathlibExt.Analysis.BeurlingAhlforsWanted.l2Norm_wanted","parent":16969,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Beurling\u2013Ahlfors transform has "],["math","\\(L^2\\)"],["text"," norm "],["math","\\(1\\)"]],"source":"The Beurling\u2013Ahlfors transform has $L^2$ norm $1$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform/index.html#l2-norm-wanted","scope":"Analysis/harmonic-analysis/fourier-transform/beurling-ahlfors-transform","status":"can_prove","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ let $Bf(z)$ be the limit as $\\varepsilon \\to 0^+$ of $-\\frac{1}{\\pi} \\int_{|w - z| > \\varepsilon} \\frac{f(w)}{(z - w)^2} \\, dA(w)$, where $dA$ is Lebesgue area measure (a junk value where the limit does not exist). The infimum $\\|B\\|_2$ of the constants $C \\ge 0$ such that $\\|Bf\\|_{L^2} \\le C\\,\\|f\\|_{L^2}$ in $[0, \\infty]$ for every almost everywhere strongly\u2026","title":"The Beurling\u2013Ahlfors transform has $L^2$ norm $1$"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24.json","id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"can_prove","title":"Fourier interpolation in dimensions 8 and 24"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/radial","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.Radial","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#radial","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"fully_proved","summary":"A function is radial if it takes the same value at points with the same norm.","title":"Radial function"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/real-incl","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.realIncl","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#real-incl","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"fully_proved","summary":"For a positive integer $n$, this is the map $\\mathbb{R} \\to \\mathbb{R}^n$ sending $r$ to $(r, 0, \\dots, 0)$, whose image is the first coordinate axis.","title":"Embedding of the first coordinate axis"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/coeffs","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.coeffs","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#coeffs","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"fully_proved","summary":"Let $n \\ge 1$ and $k_0 \\ge 0$ be integers, let $f$ be a complex-valued Schwartz function on $\\mathbb{R}^n$ with Fourier transform $\\hat f(\\xi) = \\int f(x) e^{-2\\pi i \\langle x, \\xi \\rangle}\\,dx$, let $e_1$ be the first standard basis vector and let $r_k = \\sqrt{2k}$. The samples of $f$ are the four complex sequences indexed by the integers $k \\ge k_0$ whose $k$-th terms are, in this order, $f(r_k e_1)$,\u2026","title":"Fourier sample sequences"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/rapidly-decreasing","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.RapidlyDecreasing","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#rapidly-decreasing","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"fully_proved","summary":"For a natural number $k_0$, a complex sequence $(a_k)_{k \\ge k_0}$ indexed by the integers $k \\ge k_0$ is rapidly decreasing if $k^m |a_k| \\to 0$ as $k \\to \\infty$ for every natural number $m$.","title":"Rapidly decreasing sequences"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/rapidly-decreasing-samples","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.RapidlyDecreasingSamples","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#rapidly-decreasing-samples","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"fully_proved","summary":"For a natural number $k_0$, a family $(a_{k,j})$ of complex numbers indexed by the integers $k \\ge k_0$ and $j \\in \\{0, 1, 2, 3\\}$ is a rapidly decreasing quadruple if each of its four sequences $(a_{k,j})_{k \\ge k_0}$ is rapidly decreasing, that is, $k^m |a_{k,j}| \\to 0$ as $k \\to \\infty$ for every $j$ and every natural number $m$.","title":"Rapidly decreasing sample quadruples"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/radial-schwartz","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.RadialSchwartz","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#radial-schwartz","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"fully_proved","summary":"For a natural number $n$, this is the set of complex-valued Schwartz functions $f$ on $\\mathbb{R}^n$ that are radial: $f(x) = f(y)$ whenever $|x| = |y|$.","title":"Radial Schwartz functions"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/rapid-sample-data","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.RapidSampleData","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#rapid-sample-data","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"fully_proved","summary":"For a natural number $k_0$, this is the set of families $(a_{k,j})$ of complex numbers indexed by the integers $k \\ge k_0$ and $j \\in \\{0, 1, 2, 3\\}$ whose four sequences $(a_{k,j})_{k \\ge k_0}$ are all rapidly decreasing: $k^m |a_{k,j}| \\to 0$ as $k \\to \\infty$ for every $j$ and every natural number $m$.","title":"Rapidly decreasing sample data"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/ckmrv","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.CKMRV","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#ckmrv","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"fully_proved","summary":"Let $n \\ge 1$ and $k_0 \\ge 0$ be integers. For a complex-valued Schwartz function $f$ on $\\mathbb{R}^n$, let $\\hat f(\\xi) = \\int f(x) e^{-2\\pi i \\langle x, \\xi \\rangle}\\,dx$ be its Fourier transform, let $e_1$ be the first standard basis vector and let $r_k = \\sqrt{2k}$. The sampling map from index $k_0$ sends $f$ to the four complex sequences indexed by the integers $k \\ge k_0$ whose $k$-th terms are $f(r_k e_1)$,\u2026","title":"Radial Fourier interpolation property"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/ckmrv-fourier-interpolation","kind":"node","lean":"MathlibExt.Analysis.CKMRVInterpolationWanted.Analysis.CKMRVInterpolation.ckmrv_fourier_interpolation","parent":16981,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24/index.html#ckmrv-fourier-interpolation","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-interpolation-in-dimensions-8-and-24","status":"can_prove","summary":"Let $(n, k_0)$ be $(8, 1)$ or $(24, 2)$. For a complex-valued Schwartz function $f$ on $\\mathbb{R}^n$, let $\\hat f(\\xi) = \\int f(x) e^{-2\\pi i \\langle x, \\xi \\rangle}\\,dx$ be its Fourier transform, let $e_1$ be the first standard basis vector and let $r_k = \\sqrt{2k}$. The sampling map sends $f$ to the four complex sequences indexed by the integers $k \\ge k_0$ whose $k$-th terms are $f(r_k e_1)$, $\\hat f(r_k e_1)$\u2026","title":"Cohn\u2013Kumar\u2013Miller\u2013Radchenko\u2013Viazovska Fourier interpolation theorem"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations.json","id":"Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations","status":"fully_proved","title":"Fourier transform of dilations"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations/fourier-comp-mul-left","kind":"node","lean":"Real.fourier_comp_mul_left","parent":16991,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations/index.html#fourier-comp-mul-left","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations","status":"fully_proved","summary":"Let $E$ be a complex normed space, and write $\\hat g(y) = \\int_{\\mathbb{R}} e^{-2\\pi i x y} g(x)\\,dx$ for the Fourier transform of a function $g : \\mathbb{R} \\to E$. For every $f : \\mathbb{R} \\to E$, every real $a \\ne 0$ and every real $y$, the Fourier transform of $x \\mapsto f(ax)$ at $y$ equals $|a|^{-1} \\hat f(y/a)$.","title":"Fourier transform of a dilation"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations/fourier-comp-mul-left-of-pos","kind":"node","lean":"Real.fourier_comp_mul_left_of_pos","parent":16991,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations/index.html#fourier-comp-mul-left-of-pos","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations","status":"fully_proved","summary":"Write $\\hat g(y) = \\int_{\\mathbb{R}} e^{-2\\pi i x y} g(x)\\,dx$ for the Fourier transform of a function $g : \\mathbb{R} \\to \\mathbb{C}$. For every $f : \\mathbb{R} \\to \\mathbb{C}$, every real $a > 0$ and every real $y$, the Fourier transform of $x \\mapsto f(ax)$ at $y$ equals $a^{-1} \\hat f(y/a)$.","title":"Fourier transform of a dilation by a positive factor"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations/fourier-coe-comp-mul-left-of-pos","kind":"node","lean":"SchwartzMap.fourier_coe_comp_mul_left_of_pos","parent":16991,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations/index.html#fourier-coe-comp-mul-left-of-pos","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-transform-of-dilations","status":"fully_proved","summary":"Write $\\hat g(y) = \\int_{\\mathbb{R}} e^{-2\\pi i x y} g(x)\\,dx$ for the Fourier transform of a function $g : \\mathbb{R} \\to \\mathbb{C}$. For every Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$, every real $a > 0$ and every real $y$, the Fourier transform of $x \\mapsto f(ax)$ at $y$ equals $a^{-1} \\hat f(y/a)$.","title":"Fourier transform of a dilated Schwartz function"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform.json","id":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform","status":"fully_proved","title":"Fourier\u2013Laplace transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/fourier-laplace","kind":"node","lean":"Fourier.fourierLaplace","parent":16995,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/index.html#fourier-laplace","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform","status":"fully_proved","summary":"For $f : \\mathbb{R} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$, the Fourier\u2013Laplace transform of $f$ at $z$ is the Lebesgue integral $\\int_{\\mathbb{R}} f(x) \\, e^{-2\\pi i x z} \\, dx$, taken to be $0$ when the integrand is not integrable.","title":"Fourier\u2013Laplace transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/fourier-laplace-of-real","kind":"node","lean":"Fourier.fourierLaplace_ofReal","parent":16995,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/index.html#fourier-laplace-of-real","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform","status":"fully_proved","summary":"For every $f : \\mathbb{R} \\to \\mathbb{C}$ and every real $r$, the Fourier\u2013Laplace transform of $f$ at the complex number $r$, namely $\\int_{\\mathbb{R}} f(x) \\, e^{-2\\pi i x r} \\, dx$, equals the Fourier transform of $f$ at $r$ with the $2\\pi$ normalization, $\\hat f(r) = \\int_{\\mathbb{R}} e^{-2\\pi i x r} f(x) \\, dx$.","title":"Fourier\u2013Laplace transform on the real axis"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/has-deriv-at-fourier-laplace","kind":"node","lean":"Fourier.hasDerivAt_fourierLaplace","parent":16995,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/index.html#has-deriv-at-fourier-laplace","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{C}$ be integrable with compact support, and let $F(z) = \\int_{\\mathbb{R}} f(x) \\, e^{-2\\pi i x z} \\, dx$ be its Fourier\u2013Laplace transform. Then at every $z \\in \\mathbb{C}$, $F$ is complex differentiable with derivative $F'(z) = \\int_{\\mathbb{R}} f(x) \\, (-2\\pi i x) \\, e^{-2\\pi i x z} \\, dx$.","title":"Derivative of the Fourier\u2013Laplace transform"},{"id":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/differentiable-fourier-laplace","kind":"node","lean":"Fourier.differentiable_fourierLaplace","parent":16995,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform/index.html#differentiable-fourier-laplace","scope":"Analysis/harmonic-analysis/fourier-transform/fourier-laplace-transform","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{C}$ be integrable with compact support. Then its Fourier\u2013Laplace transform $z \\mapsto \\int_{\\mathbb{R}} f(x) \\, e^{-2\\pi i x z} \\, dx$ is complex differentiable at every $z \\in \\mathbb{C}$, that is, it is entire.","title":"The Fourier\u2013Laplace transform of an integrable compactly supported function is entire"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle.json","id":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle","status":"can_prove","title":"Fractal uncertainty principle"},{"id":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/porous-on-balls","kind":"node","lean":"MathlibExt.Analysis.FractalUncertaintyWanted.FractalUncertainty.PorousOnBalls","parent":17000,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/index.html#porous-on-balls","scope":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle","status":"fully_proved","summary":"Let $d$ be a natural number and let $\\nu$, $\\alpha_0$ and $\\alpha_1$ be real numbers. A set $X$ in the Euclidean space $\\mathbb{R}^d$ is $\\nu$-porous on balls from scales $\\alpha_0$ to $\\alpha_1$ if for every $y \\in \\mathbb{R}^d$ and every $R$ with $\\alpha_0 < R < \\alpha_1$, the closed ball of radius $R/2$ about $y$ contains a point $x$ such that the closed ball of radius $\\nu R$ about $x$ is disjoint from $X$.","title":"Porosity on balls"},{"id":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/porous-on-lines","kind":"node","lean":"MathlibExt.Analysis.FractalUncertaintyWanted.FractalUncertainty.PorousOnLines","parent":17000,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/index.html#porous-on-lines","scope":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle","status":"fully_proved","summary":"Let $d$ be a natural number and let $\\nu$, $\\alpha_0$ and $\\alpha_1$ be real numbers. A set $X$ in the Euclidean space $\\mathbb{R}^d$ is $\\nu$-porous on lines from scales $\\alpha_0$ to $\\alpha_1$ if for every $R$ with $\\alpha_0 < R < \\alpha_1$ and all points $y_1, y_2 \\in \\mathbb{R}^d$ at distance $R$, the closed segment from $y_1$ to $y_2$ contains a point $x$ such that the closed ball of radius $\\nu R$ about $x$\u2026","title":"Porosity on lines"},{"id":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/cube","kind":"node","lean":"MathlibExt.Analysis.FractalUncertaintyWanted.FractalUncertainty.cube","parent":17000,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/index.html#cube","scope":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle","status":"fully_proved","summary":"Let $d$ be a natural number. For real numbers $a$ and $b$, the cube $[a,b]^d$ is the set of points $x$ of the Euclidean space $\\mathbb{R}^d$ with $a \\le x_i \\le b$ for every coordinate $i$.","title":"Closed cubes in Euclidean space"},{"id":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/theorem-1-1","kind":"node","lean":"MathlibExt.Analysis.FractalUncertaintyWanted.FractalUncertainty.theorem_1_1","parent":17000,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle/index.html#theorem-1-1","scope":"Analysis/harmonic-analysis/fourier-transform/fractal-uncertainty-principle","status":"can_prove","summary":"Let $d$ be a natural number and $\\nu > 0$. Then there are real numbers $\\beta > 0$ and $C > 0$ such that for every $h$ with $0 < h < 1/100$ and all measurable sets $X, Y$ in the Euclidean space $\\mathbb{R}^d$ the following holds: if $X \\subseteq [-1,1]^d$ is $\\nu$-porous on balls from scales $h$ to $1$ and $Y \\subseteq [-h^{-1},h^{-1}]^d$ is $\\nu$-porous on lines from scales $1$ to $h^{-1}$, then every\u2026","title":"Cohen's fractal uncertainty principle for line-porous sets"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle.json","id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"can_prove","title":"Heisenberg uncertainty principle"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/schwartz-integral-sq-eq-neg-two-mul","kind":"node","lean":"MathlibExt.Analysis.Fourier.HeisenbergUncertainty.schwartz_integral_sq_eq_neg_two_mul","parent":17005,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#schwartz-integral-sq-eq-neg-two-mul","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"fully_proved","summary":"The integration-by-parts identity underlying Heisenberg's inequality: for real-valued Schwartz `f`, `\u222b (f x)\u00b2 = -2 * \u222b x * f x * f' x`.","title":"Integration-by-parts identity for Heisenberg's inequality"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/heisenberg-uncertainty-real","kind":"node","lean":"MathlibExt.Analysis.Fourier.HeisenbergUncertainty.heisenberg_uncertainty_real","parent":17005,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#heisenberg-uncertainty-real","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"fully_proved","summary":"Heisenberg's uncertainty principle for real-valued Schwartz functions: `(\u222b (f x)\u00b2)\u00b2 \u2264 4 * (\u222b x\u00b2 * (f x)\u00b2) * (\u222b (f' x)\u00b2)`.","title":"Heisenberg's uncertainty principle"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/heisenberg-uncertainty-complex","kind":"node","lean":"MathlibExt.Analysis.Fourier.HeisenbergUncertainty.heisenberg_uncertainty_complex","parent":17005,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#heisenberg-uncertainty-complex","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"fully_proved","summary":"Heisenberg's uncertainty principle for complex-valued Schwartz functions: `(\u222b \u2016f x\u2016\u00b2)\u00b2 \u2264 4 * (\u222b x\u00b2 * \u2016f x\u2016\u00b2) * (\u222b \u2016f' x\u2016\u00b2)`.","title":"Heisenberg's uncertainty principle"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/heisenberg-uncertainty-fourier","kind":"node","lean":"MathlibExt.Analysis.Fourier.HeisenbergUncertainty.heisenberg_uncertainty_fourier","parent":17005,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#heisenberg-uncertainty-fourier","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"fully_proved","summary":"Heisenberg's uncertainty principle in Fourier form: with Mathlib's Fourier normalization, the Fourier second moment about zero is `\u222b \u03be\u00b2 * \u2016\ud835\udcd5 f \u03be\u2016\u00b2`, and `(\u222b \u2016f x\u2016\u00b2)\u00b2 \u2264 16 * \u03c0\u00b2 * (\u222b x\u00b2 * \u2016f x\u2016\u00b2) * (\u222b \u03be\u00b2 * \u2016\ud835\udcd5 f \u03be\u2016\u00b2)`.","title":"Heisenberg's uncertainty principle"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/heisenberg-uncertainty-shifted","kind":"node","lean":"MathlibExt.Analysis.Fourier.HeisenbergUncertainty.heisenberg_uncertainty_shifted","parent":17005,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#heisenberg-uncertainty-shifted","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"fully_proved","summary":"Heisenberg's uncertainty principle about an arbitrary center `a`: the position second moment is measured about `a`; the derivative factor is unchanged.","title":"Heisenberg's uncertainty principle"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/heisenberg-uncertainty-centered","kind":"node","lean":"MathlibExt.Analysis.Fourier.HeisenbergUncertainty.heisenberg_uncertainty_centered","parent":17005,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#heisenberg-uncertainty-centered","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"fully_proved","summary":"Heisenberg's uncertainty principle with the position factor centered: for `\u03bc = (\u222b y * \u2016f y\u2016\u00b2) / (\u222b \u2016f x\u2016\u00b2)`, the position second moment about `\u03bc` satisfies the same bound. For positive L2 mass, `\u03bc` is the mean position. Zero L2 mass occurs only for `f = 0`, where `\u03bc = 0 / 0 = 0` by Lean's convention. Only the position factor is centered.","title":"Heisenberg's uncertainty principle"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/integral-sq-mul-norm-sq-mul-integral-sq","kind":"node","lean":"MathlibExt.Analysis.Fourier.HeisenbergUncertaintyFourierWanted.integral_sq_mul_norm_sq_mul_integral_sq_mul_norm_fourier_sq_ge","parent":17005,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#integral-sq-mul-norm-sq-mul-integral-sq","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"fully_proved","summary":"Denote by $\\mathcal{S}(\\mathbb{R})$ the Schwartz space on $\\mathbb{R}$ consisting of all complex-valued $C^\\infty$ functions $f(x)$ on $\\mathbb{R}$ such that \\sup_{x\\in\\mathbb{R}} |x|^k \\left|\\frac{d^\\ell f}{dx^\\ell}(x)\\right| < \\infty \\quad\\text{for all } k,\\ell\\in\\mathbb{N}\\cup\\{0\\}. \\] Suppose $\\psi(x)\\in\\mathcal{S}(\\mathbb{R})$ satisfies \\int_{-\\infty}^{\\infty}|\\psi(x)|^2\\,dx=1. \\] Denote by $\\hat{\\psi}(\\xi)$\u2026","title":"Heisenberg uncertainty principle"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/exists-const-sq-integral-sq-le-mul-shift","kind":"node","lean":"MathlibExt.Analysis.Fourier.HeisenbergShiftedMomentsWanted.exists_const_sq_integral_sq_le_mul_shifted_moments_fourier","parent":17005,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#exists-const-sq-integral-sq-le-mul-shift","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"can_prove","summary":"For a complex-valued Schwartz function $f$ on $\\mathbb{R}$, write $\\hat{f}(\\xi) = \\int_{\\mathbb{R}} f(x) e^{-2\\pi i x \\xi}\\, dx$ for its Fourier transform. There is a constant $C > 0$ such that for all real numbers $c$ and $d$ and every complex-valued Schwartz function $f$ on $\\mathbb{R}$,","title":"Heisenberg uncertainty inequality with shifted moments"},{"id":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/integral-sq-le-two-mul-sqrt-integral-mul","kind":"node","lean":"MathlibExt.Analysis.Fourier.SchwartzWeightedL2DerivInequalityWanted.integral_sq_le_two_mul_sqrt_integral_mul_sq_mul_sqrt_integral_deriv_sq","parent":17005,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted "],["math","\\(L^2\\)"],["text"," inequality for real Schwartz functions"]],"source":"Weighted $L^2$ inequality for real Schwartz functions"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle/index.html#integral-sq-le-two-mul-sqrt-integral-mul","scope":"Analysis/harmonic-analysis/fourier-transform/heisenberg-uncertainty-principle","status":"can_prove","summary":"For every real-valued Schwartz function $f : \\mathbb{R} \\to \\mathbb{R}$,","title":"Weighted $L^2$ inequality for real Schwartz functions"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/kakeya-sets.json","id":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/kakeya-sets/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets","status":"can_prove","title":"Kakeya sets"},{"id":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets/besicovitch-kakeya-compact-measure-zero","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BesicovitchKakeyaWanted.besicovitch_kakeya_compact_measure_zero","parent":17015,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/kakeya-sets/index.html#besicovitch-kakeya-compact-measure-zero","scope":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets","status":"fully_proved","summary":"There is a compact set $K \\subseteq \\mathbb{R}^2$ of Lebesgue measure zero that contains a unit segment in every direction: $K$ is measurable, and for every unit vector $v \\in \\mathbb{R}^2$ there is a point $a \\in \\mathbb{R}^2$ with $a + t v \\in K$ for all $t \\in [0, 1]$.","title":"Besicovitch's Kakeya set of measure zero"},{"id":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets/space","kind":"node","lean":"MathlibExt.Analysis.WangZahlKakeyaWanted.Analysis.WangZahlKakeya.Space","parent":17015,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/kakeya-sets/index.html#space","scope":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets","status":"fully_proved","summary":"The space $\\mathbb{R}^3$ with the Euclidean inner product.","title":"Three-dimensional Euclidean space"},{"id":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets/is-kakeya","kind":"node","lean":"MathlibExt.Analysis.WangZahlKakeyaWanted.Analysis.WangZahlKakeya.IsKakeya","parent":17015,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/kakeya-sets/index.html#is-kakeya","scope":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets","status":"fully_proved","summary":"A Kakeya set is a compact subset $K$ of three-dimensional Euclidean space that contains a unit line segment in every direction: for every unit vector $v$ there is a point $x$ with $x + t v \\in K$ for all $t \\in [0, 1]$.","title":"Kakeya set"},{"id":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets/wang-zahl-kakeya-dim-h","kind":"node","lean":"MathlibExt.Analysis.WangZahlKakeyaWanted.Analysis.WangZahlKakeya.wang_zahl_kakeya_dimH","parent":17015,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/kakeya-sets/index.html#wang-zahl-kakeya-dim-h","scope":"Analysis/harmonic-analysis/fourier-transform/kakeya-sets","status":"can_prove","summary":"Every Kakeya set in three-dimensional Euclidean space has Hausdorff dimension $3$.","title":"Wang\u2013Zahl theorem: the Kakeya conjecture in three dimensions"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates.json","id":"Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates","status":"can_prove","title":"Linear independence of time\u2013frequency translates"},{"id":"Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates/modulation-one","kind":"node","lean":"MathlibExt.Analysis.FourFunctionHRTWanted.modulationOne","parent":17020,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates/index.html#modulation-one","scope":"Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates","status":"fully_proved","summary":"The function $\\mathbb{R} \\to \\mathbb{C}$ given by $x \\mapsto e^{2\\pi i x}$; multiplying a function by it modulates that function by frequency one.","title":"Frequency-one modulation factor"},{"id":"Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates/conjecture","kind":"node","lean":"MathlibExt.Analysis.FourFunctionHRTWanted.conjecture","parent":17020,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates/index.html#conjecture","scope":"Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates","status":"fully_proved","summary":"For every function $g : \\mathbb{R} \\to \\mathbb{C}$ in $L^2(\\mathbb{R})$ that is not zero almost everywhere, the four functions $g(x)$, $g(x - 1)$, $g(x - \\sqrt{2})$ and $e^{2\\pi i x} g(x)$ are linearly independent: whenever $c_0, c_1, c_2, c_3 \\in \\mathbb{C}$ satisfy $c_0 g(x) + c_1 g(x - 1) + c_2 g(x - \\sqrt{2}) + c_3 e^{2\\pi i x} g(x) = 0$ for almost every $x \\in \\mathbb{R}$, we have $c_0 = c_1 = c_2 = c_3 = 0$.","title":"A four-function case of the HRT conjecture"},{"id":"Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.FourFunctionHRTWanted.conjecture_holds","parent":17020,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates/index.html#conjecture-holds","scope":"Analysis/harmonic-analysis/fourier-transform/linear-independence-of-time-frequency-translates","status":"can_prove","summary":"For every function $g : \\mathbb{R} \\to \\mathbb{C}$ in $L^2(\\mathbb{R})$ that is not zero almost everywhere, the four functions $g(x)$, $g(x - 1)$, $g(x - \\sqrt{2})$ and $e^{2\\pi i x} g(x)$ are linearly independent: whenever $c_0, c_1, c_2, c_3 \\in \\mathbb{C}$ satisfy $c_0 g(x) + c_1 g(x - 1) + c_2 g(x - \\sqrt{2}) + c_3 e^{2\\pi i x} g(x) = 0$ for almost every $x \\in \\mathbb{R}$, then $c_0 = c_1 = c_2 = c_3 = 0$.","title":"A four-function case of the HRT conjecture holds"},{"data_url":"scopes/Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem.json","id":"Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem","kind":"scope","navigation":"module","parent":16960,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem/index.html","scope":"Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem","status":"can_prove","title":"Nyquist\u2013Shannon sampling theorem"},{"id":"Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem/sinc","kind":"node","lean":"MathlibExt.Analysis.NyquistShannonWanted.Analysis.NyquistShannon.sinc","parent":17024,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem/index.html#sinc","scope":"Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem","status":"fully_proved","summary":"The function $\\operatorname{sinc} \\colon \\mathbb{R} \\to \\mathbb{C}$ is defined by $\\operatorname{sinc}(x) = \\frac{\\sin x}{x}$ for $x \\neq 0$ and $\\operatorname{sinc}(0) = 1$, the value that removes the singularity at $0$. Its values are real.","title":"Unnormalised cardinal sine"},{"id":"Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem/fourier-supported-in-nyquist","kind":"node","lean":"MathlibExt.Analysis.NyquistShannonWanted.Analysis.NyquistShannon.FourierSupportedInNyquist","parent":17024,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem/index.html#fourier-supported-in-nyquist","scope":"Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem","status":"fully_proved","summary":"A Schwartz function $f \\colon \\mathbb{R} \\to \\mathbb{C}$ has Fourier support in the Nyquist band when its Fourier transform $\\hat f(\\xi) = \\int_{\\mathbb{R}} f(x)\\, e^{-2\\pi i x \\xi}\\, dx$ vanishes at every real $\\xi$ outside the closed interval $[-\\frac12, \\frac12]$.","title":"Fourier support in the Nyquist band"},{"id":"Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem/nyquist-shannon-sampling","kind":"node","lean":"MathlibExt.Analysis.NyquistShannonWanted.Analysis.NyquistShannon.nyquist_shannon_sampling","parent":17024,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem/index.html#nyquist-shannon-sampling","scope":"Analysis/harmonic-analysis/fourier-transform/nyquist-shannon-sampling-theorem","status":"can_prove","summary":"Let $\\operatorname{sinc}(x) = \\frac{\\sin x}{x}$ for real $x \\neq 0$ and $\\operatorname{sinc}(0) = 1$, and let $f \\colon \\mathbb{R} \\to \\mathbb{C}$ be a Schwartz function whose Fourier transform $\\hat f(\\xi) = \\int_{\\mathbb{R}} f(x)\\, e^{-2\\pi i x \\xi}\\, dx$ vanishes at every real $\\xi$ outside $[-\\frac12, \\frac12]$. Then for every real $t$ the series $\\sum_{n \\in \\mathbb{Z}} f(n) \\operatorname{sinc}(\\pi(n - t))$ is\u2026","title":"Nyquist\u2013Shannon sampling theorem (Whittaker\u2013Shannon interpolation formula)"},{"data_url":"scopes/Analysis/harmonic-analysis/discrete-fourier-analysis.json","id":"Analysis/harmonic-analysis/discrete-fourier-analysis","kind":"scope","parent":16828,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/index.html","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis","status":"fully_proved","summary":"Fourier analysis on finite abelian groups and the Boolean cube: DFT and FFT, Plancherel, finite uncertainty, Young and Hausdorff\u2013Young, Riesz\u2013Thorin on finite sets, influences and noise operators.","title":"Discrete Fourier analysis"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/inv-dft-mul-dft","kind":"node","lean":"ZMod.invDFT_mul_dft","parent":17028,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/index.html#inv-dft-mul-dft","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis","status":"fully_proved","summary":"Let $n \\ge 1$. For $\\varphi : \\mathbb{Z}/n\\mathbb{Z} \\to \\mathbb{C}$ write $\\hat\\varphi(k) = \\sum_{j \\in \\mathbb{Z}/n\\mathbb{Z}} e^{-2\\pi i jk/n} \\varphi(j)$ for its discrete Fourier transform, and let $\\mathcal{F}^{-1}$ be the inverse of the map $\\varphi \\mapsto \\hat\\varphi$. Then for all $f, g : \\mathbb{Z}/n\\mathbb{Z} \\to \\mathbb{C}$ and every $t \\in \\mathbb{Z}/n\\mathbb{Z}$, the value at $t$ of $\\mathcal{F}^{-1}$\u2026","title":"Convolution theorem for the discrete Fourier transform"},{"data_url":"scopes/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube.json","id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","kind":"scope","navigation":"module","parent":17028,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","title":"Fourier analysis on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/bool-cube","kind":"node","lean":"BooleanCube.BoolCube","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#bool-cube","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the Boolean cube of dimension $n$ is the set of functions $x : \\{0, \\dots, n-1\\} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$. It represents $\\{-1, 1\\}^n$, with true standing for $1$ and false for $-1$.","title":"Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-val","kind":"node","lean":"BooleanCube.cubeVal","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-val","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"The sign of a Boolean value $b$ is the real number $\\sigma(b)$ given by $\\sigma(\\text{true}) = 1$ and $\\sigma(\\text{false}) = -1$.","title":"Sign of a Boolean coordinate"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-chi","kind":"node","lean":"BooleanCube.cubeChi","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-chi","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For a set $S$ of coordinates among $n$, the Fourier character $\\chi_S$ is the real-valued function on the Boolean cube in $n$ coordinates given by $\\chi_S(x) = \\prod_{i \\in S} \\sigma(x_i)$, where $\\sigma$ is the sign of a coordinate: $\\sigma(\\text{true}) = 1$ and $\\sigma(\\text{false}) = -1$.","title":"Fourier character on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/bool-fourier-coeff","kind":"node","lean":"BooleanCube.boolFourierCoeff","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#bool-fourier-coeff","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, a function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ and a set $S \\subseteq \\{0, \\dots, n-1\\}$, the normalized Fourier coefficient of $f$ at $S$ is $\\hat f(S) = \\frac{1}{|\\{-1, 1\\}^n|} \\sum_{x \\in \\{-1, 1\\}^n} f(x) \\chi_S(x)$, where $\\chi_S(x) = \\prod_{i \\in S} x_i$. That is, $\\hat f(S)$ is the uniform expectation of $f \\chi_S$.","title":"Fourier coefficient on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/multilinear-degree-le","kind":"node","lean":"BooleanCube.MultilinearDegreeLE","parent":17030,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourier degree at most "],["math","\\(d\\)"]],"source":"Fourier degree at most $d$"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#multilinear-degree-le","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"A real-valued function $f$ on the Boolean cube of dimension $n$ has Fourier degree at most a natural number $d$ when its Fourier coefficient $\\hat f(S)$ vanishes for every set $S$ of coordinates with $|S| > d$.","title":"Fourier degree at most $d$"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-val-true","kind":"node","lean":"BooleanCube.cubeVal_true","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-val-true","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"The sign $\\sigma(\\text{true})$ of the Boolean value true equals $1$.","title":"Sign of true"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-val-false","kind":"node","lean":"BooleanCube.cubeVal_false","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-val-false","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"The sign $\\sigma(\\text{false})$ of the Boolean value false equals $-1$.","title":"Sign of false"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-val-mul-self","kind":"node","lean":"BooleanCube.cubeVal_mul_self","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-val-mul-self","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every Boolean value $b$, its sign $\\sigma(b)$ satisfies $\\sigma(b) \\cdot \\sigma(b) = 1$.","title":"Coordinate signs square to one"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-chi-empty","kind":"node","lean":"BooleanCube.cubeChi_empty","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-chi-empty","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n$ and every point $x$ of the Boolean cube in $n$ coordinates, the Fourier character $\\chi_\\emptyset$ of the empty set of coordinates satisfies $\\chi_\\emptyset(x) = 1$.","title":"The empty Fourier character is constant one"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-chi-singleton","kind":"node","lean":"BooleanCube.cubeChi_singleton","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-chi-singleton","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n$, every coordinate $i$ and every point $x$ of the Boolean cube in $n$ coordinates, the Fourier character $\\chi_{\\{i\\}}$ of the singleton $\\{i\\}$ satisfies $\\chi_{\\{i\\}}(x) = \\sigma(x_i)$, where $\\sigma$ is the sign of a coordinate: $\\sigma(\\text{true}) = 1$ and $\\sigma(\\text{false}) = -1$.","title":"Singleton Fourier characters are coordinate signs"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-chi-mul-self","kind":"node","lean":"BooleanCube.cubeChi_mul_self","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-chi-mul-self","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n$, every set $S$ of the $n$ coordinates and every point $x$ of the Boolean cube in $n$ coordinates, the Fourier character $\\chi_S$ satisfies $\\chi_S(x) \\cdot \\chi_S(x) = 1$.","title":"Fourier characters square to one"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/bool-fourier-coeff-empty","kind":"node","lean":"BooleanCube.boolFourierCoeff_empty","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#bool-fourier-coeff-empty","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$, the normalized Fourier coefficient of $f$ at the empty set is the uniform mean of $f$: $\\hat f(\\emptyset) = \\frac{1}{|\\{-1, 1\\}^n|} \\sum_{x \\in \\{-1, 1\\}^n} f(x)$.","title":"The empty Fourier coefficient is the mean"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/bool-fourier-coeff-zero","kind":"node","lean":"BooleanCube.boolFourierCoeff_zero","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#bool-fourier-coeff-zero","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every $S \\subseteq \\{0, \\dots, n-1\\}$, the normalized Fourier coefficient at $S$ of the identically zero function on $\\{-1, 1\\}^n$ is $0$.","title":"Fourier coefficients of the zero function"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/bool-fourier-coeff-cube-chi-self","kind":"node","lean":"BooleanCube.boolFourierCoeff_cubeChi_self","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#bool-fourier-coeff-cube-chi-self","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every $S \\subseteq \\{0, \\dots, n-1\\}$, the normalized Fourier coefficient at $S$ of the character $\\chi_S(x) = \\prod_{i \\in S} x_i$ on $\\{-1, 1\\}^n$ equals $1$, that is, $\\widehat{\\chi_S}(S) = 1$.","title":"A character's own Fourier coefficient"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/bool-fourier-coeff-const-empty","kind":"node","lean":"BooleanCube.boolFourierCoeff_const_empty","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#bool-fourier-coeff-const-empty","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every real $c$, the normalized Fourier coefficient at the empty set of the constant function $x \\mapsto c$ on the Boolean cube $\\{-1, 1\\}^n$ equals $c$. This includes dimension $n = 0$.","title":"Empty Fourier coefficient of a constant function"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/multilinear-degree-le-zero","kind":"node","lean":"BooleanCube.multilinearDegreeLE_zero","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#multilinear-degree-le-zero","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For all natural numbers $n$ and $d$, the zero function on the Boolean cube of dimension $n$ has Fourier degree at most $d$.","title":"The zero function satisfies every Fourier degree bound"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/mono","kind":"node","lean":"BooleanCube.MultilinearDegreeLE.mono","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#mono","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $f$ be a real-valued function on the Boolean cube of dimension $n$, and let $d \\le e$ be natural numbers. If $f$ has Fourier degree at most $d$, then $f$ has Fourier degree at most $e$.","title":"Fourier degree bounds are monotone"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/multilinear-degree-le-of-dimension-le","kind":"node","lean":"BooleanCube.multilinearDegreeLE_of_dimension_le","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#multilinear-degree-le-of-dimension-le","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $f$ be a real-valued function on the Boolean cube of dimension $n$, and let $d \\ge n$ be a natural number. Then $f$ has Fourier degree at most $d$.","title":"Fourier degree is at most the dimension"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/cube-basis","kind":"node","lean":"BooleanCube.cubeBasis","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#cube-basis","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $x$ be a point of the Boolean cube in $n$ coordinates, indexed by $i \\in \\{0, \\dots, n-1\\}$. Its cube basis polynomial is the real polynomial $\\prod_{i=0}^{n-1} \\ell_i$ in the variables $X_0, \\dots, X_{n-1}$, where $\\ell_i = X_i$ if $x_i$ is true and $\\ell_i = 1 - X_i$ if $x_i$ is false. Reading true as $1$ and false as $0$, it is the Lagrange basis polynomial of $x$ on the grid $\\{0,1\\}^n$.","title":"Lagrange basis polynomial of a cube point"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/multilinear-extension","kind":"node","lean":"BooleanCube.multilinearExtension","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#multilinear-extension","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $f$ be a Boolean-valued function on the Boolean cube of dimension $n$, and read each coordinate and each value as $1$ for true and $0$ for false. The real multilinear extension of $f$ is the polynomial in $\\mathbb{R}[X_0, \\dots, X_{n-1}]$","title":"Multilinear extension"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/eval-cube-basis","kind":"node","lean":"BooleanCube.eval_cubeBasis","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#eval-cube-basis","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $x$ and $y$ be points of the Boolean cube of dimension $n$, and identify $y$ with the point of $\\{0,1\\}^n \\subseteq \\mathbb{R}^n$ whose $i$-th coordinate is $1$ when $y_i$ is true and $0$ when it is false. Then the cube basis polynomial of $x$ evaluated at $y$ equals $1$ if $x = y$ and $0$ otherwise.","title":"Lagrange basis polynomials on the cube are point indicators"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/eval-multilinear-extension","kind":"node","lean":"BooleanCube.eval_multilinearExtension","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#eval-multilinear-extension","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $f$ be a Boolean function on the Boolean cube of dimension $n$, and let $y$ be a point of the cube, identified with the point of $\\{0,1\\}^n \\subseteq \\mathbb{R}^n$ whose $i$-th coordinate is $1$ when $y_i$ is true and $0$ when it is false. Then the real multilinear extension of $f$ evaluated at $y$ equals $1$ if $f(y)$ is true and $0$ if $f(y)$ is false.","title":"The multilinear extension agrees with its function on the cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/multilinear-extension-const","kind":"node","lean":"BooleanCube.multilinearExtension_const","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#multilinear-extension-const","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every natural number $n$ and every Boolean value $b$, the real multilinear extension of the constant function $b$ on the Boolean cube of dimension $n$ is the constant polynomial $1$ if $b$ is true and the constant polynomial $0$ if $b$ is false.","title":"Multilinear extension of a constant function"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/multilinear-extension-projection","kind":"node","lean":"BooleanCube.multilinearExtension_projection","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#multilinear-extension-projection","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every coordinate $i$ of the Boolean cube of dimension $n$, the real multilinear extension of the projection $x \\mapsto x_i$ is the variable $X_i$.","title":"Multilinear extension of a coordinate projection"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/boolean-degree","kind":"node","lean":"BooleanCube.booleanDegree","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#boolean-degree","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $f$ be a Boolean-valued function on the Boolean cube in $n$ coordinates, and read true as $1$ and false as $0$. The Boolean degree of $f$ is the total degree of the multilinear extension of $f$: the real multilinear polynomial in $n$ variables that takes the value $f(x)$ at every point $x \\in \\{0,1\\}^n$.","title":"Boolean degree"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/boolean-degree-const","kind":"node","lean":"BooleanCube.booleanDegree_const","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#boolean-degree-const","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every Boolean value $b$, the constant function $x \\mapsto b$ on the Boolean cube of dimension $n$ has Boolean degree $0$.","title":"Constant functions have Boolean degree zero"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/boolean-degree-projection","kind":"node","lean":"BooleanCube.booleanDegree_projection","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#boolean-degree-projection","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every $i < n$, the coordinate projection $x \\mapsto x_i$ on the Boolean cube of dimension $n$ has Boolean degree $1$.","title":"Coordinate projections have Boolean degree one"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/all-true-function","kind":"node","lean":"BooleanCube.allTrueFunction","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#all-true-function","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the all-true function on the Boolean cube of dimension $n$ is the Boolean AND of all coordinates: it sends $x = (x_0, \\dots, x_{n-1})$ to true exactly when $x_i$ is true for every $i < n$. In dimension $0$ it is constantly true.","title":"All-true function"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/multilinear-extension-all-true","kind":"node","lean":"BooleanCube.multilinearExtension_allTrue","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#multilinear-extension-all-true","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every natural number $n$, the real multilinear extension of the AND function on the Boolean cube of dimension $n$ (true exactly when every coordinate is true) is the monomial $\\prod_{i=0}^{n-1} X_i$.","title":"Multilinear extension of the AND function"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/boolean-degree-all-true-two","kind":"node","lean":"BooleanCube.booleanDegree_allTrue_two","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#boolean-degree-all-true-two","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"The all-true function on the Boolean cube of dimension $2$, $(x_0, x_1) \\mapsto x_0 \\wedge x_1$, has Boolean degree $2$; that is, its real multilinear extension has total degree $2$.","title":"The two-bit AND has Boolean degree two"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/multilinear-degree-le-indicator","kind":"node","lean":"BooleanCube.multilinearDegreeLE_indicator","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#multilinear-degree-le-indicator","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $f$ be a Boolean function on the Boolean cube of dimension $n$, and let $g$ be the real function with $g(x) = 1$ if $f(x)$ is true and $g(x) = 0$ otherwise. Then $g$ has Fourier degree at most the Boolean degree of $f$: every Fourier coefficient $\\hat g(S)$ with $|S|$ greater than the total degree of the multilinear extension of $f$ vanishes.","title":"Fourier degree is at most Boolean degree"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/fourier-char","kind":"node","lean":"BooleanCube.fourierChar","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#fourier-char","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For finite sets $S$ and $T$, the Fourier character of $S$ at $T$ is the real number $\\chi_S(T) = (-1)^{|S \\cap T|}$.","title":"Fourier character on finite sets"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/fourier-coeff","kind":"node","lean":"BooleanCube.fourierCoeff","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#fourier-coeff","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $E$ be a finite subset of a type $\\iota$ and let $f$ be a function from finite subsets of $\\iota$ to $\\mathbb{R}$. For a finite subset $S$ of $\\iota$, the Fourier coefficient of $f$ at $S$ over $E$ is","title":"Fourier coefficient on the subsets of a finite set"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/card-bool-cube","kind":"node","lean":"BooleanCube.card_boolCube","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#card-bool-cube","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the Boolean cube of dimension $n$ has exactly $2^n$ points.","title":"Number of points of the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/powerset-equiv-indicator","kind":"node","lean":"BooleanCube.powersetEquivIndicator","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#powerset-equiv-indicator","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For a finite set $E$, this is the bijection from the subsets $T \\subseteq E$ to the functions $b$ from $E$ to $\\{\\mathrm{true}, \\mathrm{false}\\}$ given by $b(a) = \\mathrm{false}$ if $a \\in T$ and $b(a) = \\mathrm{true}$ if $a \\notin T$. Its inverse sends $b$ to the set $\\{a \\in E : b(a) = \\mathrm{false}\\}$.","title":"Subsets of a finite set as indicator functions"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/powerset-equiv-bool-cube","kind":"node","lean":"BooleanCube.powersetEquivBoolCube","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#powerset-equiv-bool-cube","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For a finite set $E$ with $n = |E|$ elements, fix a bijection $e$ from $E$ to $\\{0, \\dots, n-1\\}$ (an enumeration of $E$, chosen noncanonically). This is the bijection from the subsets $T \\subseteq E$ to the Boolean cube of dimension $n$ that sends $T$ to the point $x$ with $x_{e(a)} = \\mathrm{false}$ if $a \\in T$ and $x_{e(a)} = \\mathrm{true}$ if $a \\notin T$, for every $a \\in E$.","title":"Subsets of a finite set as points of the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/powerset-equiv-bool-cube-apply","kind":"node","lean":"BooleanCube.powersetEquivBoolCube_apply","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#powerset-equiv-bool-cube-apply","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $E$ be a finite set with $m$ elements, listed in the fixed order $e_0, \\dots, e_{m-1}$ used to encode subsets of $E$ as points of the Boolean cube of dimension $m$. For every subset $T \\subseteq E$ and every coordinate $j < m$, the $j$-th coordinate of the point encoding $T$ is true if $e_j \\notin T$ and false if $e_j \\in T$.","title":"Coordinates of an encoded subset"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/reindex-finset","kind":"node","lean":"BooleanCube.reindexFinset","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#reindex-finset","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"For finite sets $S \\subseteq E$ with $n = |E|$, let $e$ be the fixed bijection from $E$ to $\\{0, \\dots, n-1\\}$ (the enumeration of $E$ used to identify subsets of $E$ with points of the Boolean cube of dimension $n$). The reindexed set is the image $e(S) = \\{e(a) : a \\in S\\}$, a subset of $\\{0, \\dots, n-1\\}$.","title":"Reindexing a subset as a set of coordinates"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/decode-bool-cube","kind":"node","lean":"BooleanCube.decodeBoolCube","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#decode-bool-cube","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $E$ be a finite set with $m$ elements, listed in a fixed order as $e_0, \\dots, e_{m-1}$; this order identifies subsets of $E$ with points of the Boolean cube of dimension $m$. A point $x$ of that cube decodes to the subset $\\{e_j : x_j = \\mathrm{false}\\}$ of $E$; this inverts the encoding of subsets of $E$ as cube points.","title":"Decoding a cube point as a subset"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/decode-encode","kind":"node","lean":"BooleanCube.decode_encode","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#decode-encode","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $E$ be a finite set with $m$ elements, listed in a fixed order as $e_0, \\dots, e_{m-1}$. Encode a subset $T \\subseteq E$ as the point $x_T$ of the Boolean cube of dimension $m$ whose $j$-th coordinate is false exactly when $e_j \\in T$, and decode a point $x$ as the subset $\\{e_j : x_j = \\mathrm{false}\\}$. Then for every subset $T \\subseteq E$, decoding $x_T$ gives back $T$.","title":"Decoding inverts encoding"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/fourier-char-eq-cube-chi","kind":"node","lean":"BooleanCube.fourierChar_eq_cubeChi","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#fourier-char-eq-cube-chi","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $E$ be a finite set with $m$ elements, listed in a fixed order as $e_0, \\dots, e_{m-1}$, and encode a subset $T \\subseteq E$ as the point $x_T$ of the Boolean cube of dimension $m$ whose $j$-th coordinate is false exactly when $e_j \\in T$. For all subsets $S, T \\subseteq E$,","title":"Subset characters agree with cube characters"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/fourier-coeff-eq-bool-fourier-coeff","kind":"node","lean":"BooleanCube.fourierCoeff_eq_boolFourierCoeff","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#fourier-coeff-eq-bool-fourier-coeff","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $E$ be a finite subset of a type $\\iota$ with $m$ elements, listed in a fixed order as $e_0, \\dots, e_{m-1}$, and decode a point $x$ of the Boolean cube of dimension $m$ as the subset $D(x) = \\{e_j : x_j = \\mathrm{false}\\}$ of $E$. Let $f$ be a function from finite subsets of $\\iota$ to $\\mathbb{R}$ and let $S \\subseteq E$. Then the Fourier coefficient\u2026","title":"Subset Fourier coefficients agree with cube Fourier coefficients"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/fourier-inversion","kind":"node","lean":"BooleanCube.fourier_inversion","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#fourier-inversion","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $E$ be a finite subset of a type $\\iota$ and let $f$ be a function from finite subsets of $\\iota$ to $\\mathbb{R}$. For every subset $T \\subseteq E$,","title":"Fourier inversion on the subsets of a finite set"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/fourier-plancherel","kind":"node","lean":"BooleanCube.fourier_plancherel","parent":17030,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube/index.html#fourier-plancherel","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-analysis-on-the-boolean-cube","status":"fully_proved","summary":"Let $E$ be a finite subset of a type $\\iota$ and let $f$ be a function from finite subsets of $\\iota$ to $\\mathbb{R}$. Then","title":"Plancherel identity on the subsets of a finite set"},{"data_url":"scopes/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups.json","id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups","kind":"scope","navigation":"module","parent":17028,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/index.html","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups","status":"fully_proved","title":"Fourier transform on finite abelian groups"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/dft-fun","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianDFTWanted.dftFun","parent":17075,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/index.html#dft-fun","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group and $f : G \\to \\mathbb{C}$. The discrete Fourier transform of $f$ is the function on the additive characters $\\psi$ of $G$ with values in $\\mathbb{C}$ (maps with $\\psi(0) = 1$ and $\\psi(a + b) = \\psi(a)\\psi(b)$) given by $\\hat f(\\psi) = \\sum_{x \\in G} \\psi(x) f(x)$.","title":"Discrete Fourier transform on a finite abelian group"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/dft-inversion","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianDFTWanted.dft_inversion","parent":17075,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/index.html#dft-inversion","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group with $|G|$ elements, let $f : G \\to \\mathbb{C}$, and write $\\hat f(\\psi) = \\sum_{y \\in G} \\psi(y) f(y)$ for the discrete Fourier transform of $f$ at an additive character $\\psi$ of $G$ with values in $\\mathbb{C}$. Then for every $x \\in G$, $f(x) = \\frac{1}{|G|} \\sum_{\\psi} \\overline{\\psi(x)} \\, \\hat f(\\psi)$, where $\\psi$ runs over all additive characters of $G$ with values in\u2026","title":"Fourier inversion on a finite abelian group"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/dft-plancherel","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianDFTWanted.dft_plancherel","parent":17075,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/index.html#dft-plancherel","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group with $|G|$ elements, let $f, g : G \\to \\mathbb{C}$, and write $\\hat h(\\psi) = \\sum_{x \\in G} \\psi(x) h(x)$ for the discrete Fourier transform of $h : G \\to \\mathbb{C}$ at an additive character $\\psi$ of $G$ with values in $\\mathbb{C}$. Then $\\sum_{x \\in G} \\overline{f(x)} \\, g(x)$ equals $\\frac{1}{|G|} \\sum_{\\psi} \\overline{\\hat f(\\psi)} \\, \\hat g(\\psi)$, where $\\psi$ runs over all\u2026","title":"Plancherel formula on a finite abelian group"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/dft-fun-translate","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianDFTWanted.dftFun_translate","parent":17075,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups/index.html#dft-fun-translate","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/fourier-transform-on-finite-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group, $f : G \\to \\mathbb{C}$, $t \\in G$, and $\\psi$ an additive character of $G$ with values in $\\mathbb{C}$, and write $\\hat h(\\psi) = \\sum_{x \\in G} \\psi(x) h(x)$ for the discrete Fourier transform of $h : G \\to \\mathbb{C}$. If $g(x) = f(x - t)$ for all $x \\in G$, then $\\hat g(\\psi) = \\psi(t) \\hat f(\\psi)$.","title":"Translation modulates the discrete Fourier transform"},{"data_url":"scopes/Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle.json","id":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle","kind":"scope","navigation":"module","parent":17028,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/index.html","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle","status":"fully_proved","title":"Donoho\u2013Stark uncertainty principle"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/sum-norm-sq-dft-fun","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianUncertainty.sum_norm_sq_dftFun","parent":17080,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/index.html#sum-norm-sq-dft-fun","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle","status":"fully_proved","summary":"Let $G$ be a finite abelian group and $f : G \\to \\mathbb{C}$, with discrete Fourier transform $\\hat f$. Then $\\sum_{\\psi} |\\hat f(\\psi)|^2 = |G| \\sum_{x \\in G} |f(x)|^2$, where $\\psi$ runs over all additive characters of $G$.","title":"Squared-norm Plancherel identity for the finite Fourier transform"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/norm-sq-le-of-dft-fun-eq-zero-of-not-mem","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianUncertainty.norm_sq_le_of_dftFun_eq_zero_of_notMem","parent":17080,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/index.html#norm-sq-le-of-dft-fun-eq-zero-of-not-mem","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle","status":"fully_proved","summary":"Let $G$ be a finite abelian group, let $f : G \\to \\mathbb{C}$, and let $B$ be a set of additive characters of $G$ such that the discrete Fourier transform $\\hat f$ vanishes outside $B$. Then for every $x \\in G$, $|f(x)|^2 \\le \\frac{|B|}{|G|} \\sum_{y \\in G} |f(y)|^2$.","title":"Pointwise bound from a small Fourier support"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/sum-norm-sq-le-of-dft-fun-eq-zero-of-not","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianUncertainty.sum_norm_sq_le_of_dftFun_eq_zero_of_notMem","parent":17080,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/index.html#sum-norm-sq-le-of-dft-fun-eq-zero-of-not","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle","status":"fully_proved","summary":"Let $G$ be a finite abelian group, let $f : G \\to \\mathbb{C}$, and let $B$ be a set of additive characters of $G$ such that the discrete Fourier transform $\\hat f$ vanishes outside $B$. Then for every $A \\subseteq G$, $\\sum_{x \\in A} |f(x)|^2 \\le \\frac{|A| \\cdot |B|}{|G|} \\sum_{x \\in G} |f(x)|^2$.","title":"Concentration bound for a function with small Fourier support"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/sum-norm-sq-dft-fun-le-of-eq-zero-of-not","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianUncertainty.sum_norm_sq_dftFun_le_of_eq_zero_of_notMem","parent":17080,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/index.html#sum-norm-sq-dft-fun-le-of-eq-zero-of-not","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle","status":"fully_proved","summary":"Let $G$ be a finite abelian group, let $A \\subseteq G$, let $B$ be any set of additive characters of $G$, and let $f : G \\to \\mathbb{C}$ vanish outside $A$, with discrete Fourier transform $\\hat f$. Then $\\sum_{\\psi \\in B} |\\hat f(\\psi)|^2 \\le \\frac{|A| \\cdot |B|}{|G|} \\sum_{\\psi} |\\hat f(\\psi)|^2$, where the last sum runs over all additive characters $\\psi$ of $G$.","title":"Concentration bound for the Fourier transform of a function with small support"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/card-le-mul-card-filter-support","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianUncertainty.card_le_mul_card_filter_support","parent":17080,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/index.html#card-le-mul-card-filter-support","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle","status":"fully_proved","summary":"Let $G$ be a finite abelian group and let $f : G \\to \\mathbb{C}$ be a nonzero function with discrete Fourier transform $\\hat f$. Then $|G| \\le |\\{x \\in G : f(x) \\neq 0\\}| \\cdot |\\{\\psi : \\hat f(\\psi) \\neq 0\\}|$, where $\\psi$ ranges over the additive characters of $G$.","title":"Donoho\u2013Stark uncertainty principle"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/eq-zero-of-card-mul-card-lt-card","kind":"node","lean":"MathlibExt.Analysis.Fourier.FiniteAbelianUncertainty.eq_zero_of_card_mul_card_lt_card","parent":17080,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle/index.html#eq-zero-of-card-mul-card-lt-card","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/donoho-stark-uncertainty-principle","status":"fully_proved","summary":"Let $G$ be a finite abelian group, let $A \\subseteq G$, and let $B$ be a set of additive characters of $G$ with $|A| \\cdot |B| < |G|$. If $f : G \\to \\mathbb{C}$ vanishes outside $A$ and its discrete Fourier transform $\\hat f$ vanishes outside $B$, then $f = 0$.","title":"Functions with small support and small Fourier support vanish"},{"data_url":"scopes/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean.json","id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","kind":"scope","navigation":"module","parent":17028,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","title":"Influences and the noise operator on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/flip-coord","kind":"node","lean":"BooleanCube.flipCoord","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#flip-coord","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For a point $x$ of the Boolean cube of dimension $n$ and a coordinate $i \\in \\{0, \\dots, n-1\\}$, the point $x^{\\oplus i}$ is obtained from $x$ by negating its $i$-th coordinate (exchanging true and false) and leaving every other coordinate unchanged.","title":"Coordinate flip on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/flip-coord-apply-self","kind":"node","lean":"BooleanCube.flipCoord_apply_self","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#flip-coord-apply-self","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every point $x$ of the Boolean cube of dimension $n$ and every coordinate $i \\in \\{0, \\dots, n-1\\}$, the $i$-th coordinate of $x^{\\oplus i}$, the point $x$ with its $i$-th coordinate flipped, is the negation $\\lnot x_i$ of $x_i$ (true and false exchanged).","title":"A coordinate flip negates the flipped coordinate"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/flip-coord-apply-of-ne","kind":"node","lean":"BooleanCube.flipCoord_apply_of_ne","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#flip-coord-apply-of-ne","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every point $x$ of the Boolean cube of dimension $n$ and all coordinates $i, j \\in \\{0, \\dots, n-1\\}$ with $j \\ne i$, the $j$-th coordinate of $x^{\\oplus i}$, the point $x$ with its $i$-th coordinate flipped, is $x_j$.","title":"A coordinate flip fixes the other coordinates"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/flip-coord-flip-coord","kind":"node","lean":"BooleanCube.flipCoord_flipCoord","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#flip-coord-flip-coord","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every point $x$ of the Boolean cube of dimension $n$ and every coordinate $i \\in \\{0, \\dots, n-1\\}$, writing $x^{\\oplus i}$ for $x$ with its $i$-th coordinate flipped, we have $(x^{\\oplus i})^{\\oplus i} = x$.","title":"Flipping a coordinate twice"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/flip-coord-involutive","kind":"node","lean":"BooleanCube.flipCoord_involutive","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#flip-coord-involutive","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every coordinate $i \\in \\{0, \\dots, n-1\\}$, the map $x \\mapsto x^{\\oplus i}$ on the Boolean cube of dimension $n$, which flips the $i$-th coordinate, is an involution: applying it twice gives the identity.","title":"Flipping a coordinate is an involution"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/flip-coord-ne-self","kind":"node","lean":"BooleanCube.flipCoord_ne_self","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#flip-coord-ne-self","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every point $x$ of the Boolean cube of dimension $n$ and every coordinate $i \\in \\{0, \\dots, n-1\\}$, the point $x^{\\oplus i}$ obtained from $x$ by flipping its $i$-th coordinate is different from $x$.","title":"A coordinate flip moves every point"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/cube-val-not","kind":"node","lean":"BooleanCube.cubeVal_not","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#cube-val-not","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every Boolean value $b$, the sign of its negation is the negative of its sign: $\\sigma(\\lnot b) = -\\sigma(b)$, where $\\sigma(\\text{true}) = 1$ and $\\sigma(\\text{false}) = -1$.","title":"Sign of a negated Boolean value"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/cube-val-ne-zero","kind":"node","lean":"BooleanCube.cubeVal_ne_zero","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#cube-val-ne-zero","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every Boolean value $b$, its sign $\\sigma(b)$ (equal to $1$ for true and $-1$ for false) is nonzero.","title":"Coordinate signs are nonzero"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/abs-cube-val","kind":"node","lean":"BooleanCube.abs_cubeVal","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#abs-cube-val","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every Boolean value $b$, its sign $\\sigma(b)$, which is $1$ for true and $-1$ for false, satisfies $|\\sigma(b)| = 1$.","title":"Coordinate signs have absolute value one"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/cube-chi-flip-coord","kind":"node","lean":"BooleanCube.cubeChi_flipCoord","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#cube-chi-flip-coord","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"Let $S \\subseteq \\{0, \\dots, n-1\\}$, let $x$ be a point of the Boolean cube in $n$ coordinates, let $i < n$, and write $x^{\\oplus i}$ for $x$ with coordinate $i$ flipped. Then the Fourier character $\\chi_S$ satisfies $\\chi_S(x^{\\oplus i}) = -\\chi_S(x)$ if $i \\in S$, and $\\chi_S(x^{\\oplus i}) = \\chi_S(x)$ if $i \\notin S$.","title":"Flipping a coordinate in a Fourier character"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/sum-comp-flip-coord","kind":"node","lean":"BooleanCube.sum_comp_flipCoord","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#sum-comp-flip-coord","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, $n \\in \\mathbb{N}$, $f$ a function from the Boolean cube of dimension $n$ to $M$, and $i \\in \\{0, \\dots, n-1\\}$. For a point $x$ of the cube write $x^{\\oplus i}$ for the point $x$ with its $i$-th coordinate flipped. Then $\\sum_x f(x^{\\oplus i}) = \\sum_x f(x)$, both sums ranging over all points $x$ of the cube.","title":"Flipping a coordinate preserves sums over the cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/expect-comp-flip-coord","kind":"node","lean":"BooleanCube.expect_comp_flipCoord","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#expect-comp-flip-coord","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid that is a module over the nonnegative rationals, let $f$ be a function from the Boolean cube in $n$ coordinates to $M$, and let $i < n$. Writing $x^{\\oplus i}$ for $x$ with coordinate $i$ flipped, the uniform average of $f(x^{\\oplus i})$ over all points $x$ of the cube equals the uniform average of $f(x)$.","title":"Averages are invariant under flipping a coordinate"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/expect-eq-sum-div-two-pow","kind":"node","lean":"BooleanCube.expect_eq_sum_div_two_pow","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#expect-eq-sum-div-two-pow","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$, the uniform average of $f$ over the Boolean cube $\\{-1, 1\\}^n$ is $\\mathbb{E}_x f(x) = \\frac{1}{2^n} \\sum_{x \\in \\{-1, 1\\}^n} f(x)$.","title":"Uniform average over the Boolean cube as a sum"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-eq-expect","kind":"node","lean":"BooleanCube.boolFourierCoeff_eq_expect","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-eq-expect","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real-valued function $f$ on the Boolean cube in $n$ coordinates and every set $S$ of the $n$ coordinates, the Fourier coefficient $\\hat f(S)$ is the average of $f(x) \\chi_S(x)$ over all points $x$ of the cube, $\\hat f(S) = \\mathbb{E}_x [f(x) \\chi_S(x)]$, where $\\chi_S$ is the Fourier character of $S$.","title":"Fourier coefficients as averages"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-eq-sum-div-two-pow","kind":"node","lean":"BooleanCube.boolFourierCoeff_eq_sum_div_two_pow","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-eq-sum-div-two-pow","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real-valued function $f$ on the Boolean cube in $n$ coordinates and every set $S$ of the $n$ coordinates, the Fourier coefficient of $f$ at $S$ is $\\hat f(S) = \\frac{1}{2^n} \\sum_x f(x) \\chi_S(x)$, where the sum runs over all points $x$ of the cube and $\\chi_S$ is the Fourier character of $S$.","title":"Fourier coefficients as sums over the cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/expect-eq-bool-fourier-coeff-empty","kind":"node","lean":"BooleanCube.expect_eq_boolFourierCoeff_empty","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#expect-eq-bool-fourier-coeff-empty","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$, the uniform average $\\mathbb{E}_x f(x)$ of $f$ over the Boolean cube $\\{-1, 1\\}^n$ equals the Fourier coefficient $\\hat f(\\emptyset)$ of $f$ at the empty set.","title":"Average over the Boolean cube as the empty Fourier coefficient"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/sum-cube-chi-mul-cube-chi-eq-ite","kind":"node","lean":"BooleanCube.sum_cubeChi_mul_cubeChi_eq_ite","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#sum-cube-chi-mul-cube-chi-eq-ite","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $S$ and $T$ be subsets of $\\{0, \\dots, n-1\\}$, and let $\\chi_S$ and $\\chi_T$ be their Fourier characters. Then $\\sum_x \\chi_S(x) \\chi_T(x)$, the sum ranging over all points $x$ of the Boolean cube of dimension $n$, equals $2^n$ if $S = T$ and $0$ otherwise.","title":"Orthogonality of the Boolean cube characters"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/sum-cube-chi-eq-ite","kind":"node","lean":"BooleanCube.sum_cubeChi_eq_ite","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#sum-cube-chi-eq-ite","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every subset $S$ of $\\{0, \\dots, n-1\\}$, the sum of the Fourier character $\\chi_S$ over all points of the Boolean cube of dimension $n$ equals $2^n$ if $S$ is empty and $0$ otherwise.","title":"Sum of a Boolean cube character"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/sum-cube-chi-apply-mul-cube-chi-apply-eq","kind":"node","lean":"BooleanCube.sum_cubeChi_apply_mul_cubeChi_apply_eq_ite","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#sum-cube-chi-apply-mul-cube-chi-apply-eq","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and let $x, y$ be points of the Boolean cube of dimension $n$. For each subset $S$ of $\\{0, \\dots, n-1\\}$ let $\\chi_S$ be the Fourier character of $S$. Then $\\sum_S \\chi_S(x) \\chi_S(y)$, the sum ranging over all subsets $S$ of $\\{0, \\dots, n-1\\}$, equals $2^n$ if $x = y$ and $0$ otherwise.","title":"Dual orthogonality of the Boolean cube characters"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/sum-bool-fourier-coeff-mul-cube-chi","kind":"node","lean":"BooleanCube.sum_boolFourierCoeff_mul_cubeChi","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#sum-bool-fourier-coeff-mul-cube-chi","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $f$ be a real-valued function on the Boolean cube of dimension $n$, and let $x$ be a point of the cube. For each subset $S$ of $\\{0, \\dots, n-1\\}$ let $\\hat f(S)$ be the Fourier coefficient of $f$ at $S$ and $\\chi_S$ the Fourier character of $S$. Then $\\sum_S \\hat f(S) \\chi_S(x) = f(x)$, the sum ranging over all subsets $S$ of $\\{0, \\dots, n-1\\}$.","title":"Fourier inversion on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/expect-mul-eq-sum-bool-fourier-coeff-mul","kind":"node","lean":"BooleanCube.expect_mul_eq_sum_boolFourierCoeff_mul","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#expect-mul-eq-sum-bool-fourier-coeff-mul","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and all functions $f, g : \\{-1, 1\\}^n \\to \\mathbb{R}$, the uniform average of $f g$ over the Boolean cube $\\{-1, 1\\}^n$ is the sum, over all sets $S \\subseteq \\{0, \\dots, n-1\\}$, of the products of the Fourier coefficients of $f$ and $g$ at $S$: $\\mathbb{E}_x f(x) g(x) = \\sum_S \\hat f(S) \\hat g(S)$.","title":"Plancherel's identity on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/expect-sq-eq-sum-bool-fourier-coeff-sq","kind":"node","lean":"BooleanCube.expect_sq_eq_sum_boolFourierCoeff_sq","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#expect-sq-eq-sum-bool-fourier-coeff-sq","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$, the uniform average of $f^2$ over the Boolean cube $\\{-1, 1\\}^n$ is the sum, over all sets $S \\subseteq \\{0, \\dots, n-1\\}$, of the squares of the Fourier coefficients of $f$ at $S$: $\\mathbb{E}_x f(x)^2 = \\sum_S \\hat f(S)^2$.","title":"Parseval's identity on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-injective","kind":"node","lean":"BooleanCube.boolFourierCoeff_injective","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-injective","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"If two real-valued functions $f$ and $g$ on the Boolean cube in $n$ coordinates have the same Fourier coefficient $\\hat f(S) = \\hat g(S)$ at every set $S$ of the $n$ coordinates, then $f = g$.","title":"Fourier coefficients determine the function"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-add","kind":"node","lean":"BooleanCube.boolFourierCoeff_add","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-add","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For all real-valued functions $f$ and $g$ on the Boolean cube in $n$ coordinates and every set $S$ of the $n$ coordinates, the Fourier coefficient of $f + g$ at $S$ is the sum of the Fourier coefficients of $f$ and $g$ at $S$: $\\widehat{f + g}(S) = \\hat f(S) + \\hat g(S)$.","title":"Additivity of Fourier coefficients"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-sub","kind":"node","lean":"BooleanCube.boolFourierCoeff_sub","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-sub","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For real-valued functions $f$ and $g$ on the Boolean cube $\\{-1, 1\\}^n$ and every set $S \\subseteq \\{0, \\dots, n-1\\}$, the Fourier coefficient of $f - g$ at $S$ is $\\widehat{f - g}(S) = \\hat f(S) - \\hat g(S)$.","title":"Fourier coefficients of a difference"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-const-mul","kind":"node","lean":"BooleanCube.boolFourierCoeff_const_mul","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-const-mul","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real $c$, every real-valued function $f$ on the Boolean cube in $n$ coordinates and every set $S$ of the $n$ coordinates, the Fourier coefficient of $c f$ at $S$ is $c$ times the Fourier coefficient of $f$ at $S$: $\\widehat{c f}(S) = c \\hat f(S)$.","title":"Homogeneity of Fourier coefficients"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-const","kind":"node","lean":"BooleanCube.boolFourierCoeff_const","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-const","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real $c$ and every set $S$ of the $n$ coordinates, the Fourier coefficient at $S$ of the constant function $x \\mapsto c$ on the Boolean cube in $n$ coordinates is $c$ if $S = \\emptyset$ and $0$ otherwise.","title":"Fourier coefficients of a constant function"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-cube-chi","kind":"node","lean":"BooleanCube.boolFourierCoeff_cubeChi","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-cube-chi","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For all sets $S$ and $T$ of the $n$ coordinates, the Fourier coefficient at $S$ of the Fourier character $\\chi_T$ on the Boolean cube in $n$ coordinates, $\\chi_T(x) = \\prod_{i \\in T} \\sigma(x_i)$ with $\\sigma(\\text{true}) = 1$ and $\\sigma(\\text{false}) = -1$, is $1$ if $T = S$ and $0$ otherwise.","title":"Fourier coefficients of a character"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-sum-mul-cube-chi","kind":"node","lean":"BooleanCube.boolFourierCoeff_sum_mul_cubeChi","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-sum-mul-cube-chi","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For real numbers $a_T$ indexed by the sets $T \\subseteq \\{0, \\dots, n-1\\}$, let $g(x) = \\sum_T a_T \\chi_T(x)$ on the Boolean cube $\\{-1, 1\\}^n$, where $\\chi_T$ is the Fourier character of $T$. Then for every set $S \\subseteq \\{0, \\dots, n-1\\}$ the Fourier coefficient of $g$ at $S$ is $\\hat g(S) = a_S$.","title":"Uniqueness of the Fourier expansion"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-comp-flip-coord","kind":"node","lean":"BooleanCube.boolFourierCoeff_comp_flipCoord","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-comp-flip-coord","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"Let $f$ be a real-valued function on the Boolean cube in $n$ coordinates, let $i$ be a coordinate, and let $g(x) = f(x^{\\oplus i})$, where $x^{\\oplus i}$ is the point $x$ with its $i$-th coordinate flipped. Write $\\hat f(S)$ and $\\hat g(S)$ for the Fourier coefficients of $f$ and $g$ at a set $S$ of the $n$ coordinates. Then for every such $S$, $\\hat g(S) = -\\hat f(S)$ if $i \\in S$ and $\\hat g(S) = \\hat f(S)$ if\u2026","title":"Flipping a coordinate negates the Fourier coefficients of the sets containing it"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-influence","kind":"node","lean":"BooleanCube.boolInfluence","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-influence","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For a real-valued function $f$ on the Boolean cube $\\{-1, 1\\}^n$ and a coordinate $i \\in \\{0, \\dots, n-1\\}$, the influence of $i$ on $f$ is $\\mathrm{Inf}_i[f] = \\mathbb{E}_x\\big[\\big((f(x) - f(x^{\\oplus i})) / 2\\big)^2\\big]$, where the expectation is over $x$ uniform on the cube and $x^{\\oplus i}$ is $x$ with its $i$-th coordinate flipped.","title":"Influence of a coordinate"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-total-influence","kind":"node","lean":"BooleanCube.boolTotalInfluence","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-total-influence","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For a function $f$ from the Boolean cube $\\{-1, 1\\}^n$ to $\\mathbb{R}$, the total influence of $f$ is the sum of the influences of its $n$ coordinates, $\\mathrm{I}[f] = \\sum_{i=0}^{n-1} \\mathrm{Inf}_i[f]$, where $\\mathrm{Inf}_i[f] = \\mathbb{E}_x \\left[\\left(\\frac{f(x) - f(x^{\\oplus i})}{2}\\right)^2\\right]$, $x^{\\oplus i}$ is $x$ with its $i$-th coordinate flipped and $\\mathbb{E}_x$ is the uniform average over the\u2026","title":"Total influence on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-variance","kind":"node","lean":"BooleanCube.boolVariance","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-variance","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For a function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ on the Boolean cube, its variance under the uniform distribution is $\\mathrm{Var}[f] = \\mathbb{E}_x\\big[(f(x) - \\mathbb{E}_y[f(y)])^2\\big]$, where $\\mathbb{E}_x$ denotes the uniform average over the $2^n$ points $x$ of the cube.","title":"Variance on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-influence-nonneg","kind":"node","lean":"BooleanCube.boolInfluence_nonneg","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-influence-nonneg","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real-valued function $f$ on the Boolean cube $\\{-1, 1\\}^n$ and every coordinate $i \\in \\{0, \\dots, n-1\\}$, the influence $\\mathrm{Inf}_i[f]$ of coordinate $i$ on $f$ satisfies $\\mathrm{Inf}_i[f] \\ge 0$.","title":"Influences are nonnegative"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-total-influence-nonneg","kind":"node","lean":"BooleanCube.boolTotalInfluence_nonneg","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-total-influence-nonneg","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every function $f$ from the Boolean cube $\\{-1, 1\\}^n$ to $\\mathbb{R}$, the total influence of $f$ is nonnegative.","title":"Total influence is nonnegative"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-variance-nonneg","kind":"node","lean":"BooleanCube.boolVariance_nonneg","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-variance-nonneg","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ on the Boolean cube, the variance of $f$ under the uniform distribution is at least $0$.","title":"Variance on the Boolean cube is nonnegative"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-influence-const","kind":"node","lean":"BooleanCube.boolInfluence_const","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-influence-const","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real number $c$ and every coordinate $i \\in \\{0, \\dots, n-1\\}$, the influence of coordinate $i$ on the constant function with value $c$ on the Boolean cube $\\{-1, 1\\}^n$ is $0$.","title":"Constant functions have zero influence"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-total-influence-const","kind":"node","lean":"BooleanCube.boolTotalInfluence_const","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-total-influence-const","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real number $c$, the constant function with value $c$ on the Boolean cube $\\{-1, 1\\}^n$ has total influence $0$.","title":"Constant functions have zero total influence"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-variance-const","kind":"node","lean":"BooleanCube.boolVariance_const","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-variance-const","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real number $c$, the constant function with value $c$ on the Boolean cube $\\{-1, 1\\}^n$ has variance $0$ under the uniform distribution.","title":"Constant functions have zero variance"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-influence-eq-expect-ite-ne","kind":"node","lean":"BooleanCube.boolInfluence_eq_expect_ite_ne","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-influence-eq-expect-ite-ne","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"Let $f$ be a real-valued function on the Boolean cube $\\{-1, 1\\}^n$ that takes only the values $1$ and $-1$, and let $i \\in \\{0, \\dots, n-1\\}$. Then the influence of coordinate $i$ on $f$ equals the probability, for $x$ uniform on the cube, that $f(x^{\\oplus i}) \\ne f(x)$, where $x^{\\oplus i}$ is $x$ with its $i$-th coordinate flipped.","title":"Influence of a Boolean function as a flip probability"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-influence-eq-sum-sq","kind":"node","lean":"BooleanCube.boolInfluence_eq_sum_sq","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-influence-eq-sum-sq","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real-valued function $f$ on the Boolean cube $\\{-1, 1\\}^n$ and every coordinate $i \\in \\{0, \\dots, n-1\\}$, the influence of coordinate $i$ on $f$ is $\\mathrm{Inf}_i[f] = \\sum_{S \\ni i} \\hat f(S)^2$, where $\\hat f(S)$ is the Fourier coefficient of $f$ at $S$ and the sum runs over the sets $S \\subseteq \\{0, \\dots, n-1\\}$ that contain $i$.","title":"Fourier formula for influence"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-variance-eq-sum-sq","kind":"node","lean":"BooleanCube.boolVariance_eq_sum_sq","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-variance-eq-sum-sq","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ on the Boolean cube, the variance of $f$ under the uniform distribution equals $\\sum_{S \\neq \\emptyset} \\hat f(S)^2$, where the sum runs over the nonempty sets $S \\subseteq \\{0, \\dots, n-1\\}$ and $\\hat f(S)$ is the Fourier coefficient of $f$ at $S$.","title":"Variance in terms of Fourier coefficients"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-variance-eq-expect-sq-sub-sq","kind":"node","lean":"BooleanCube.boolVariance_eq_expect_sq_sub_sq","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-variance-eq-expect-sq-sub-sq","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ on the Boolean cube, the variance of $f$ under the uniform distribution equals $\\mathbb{E}_x[f(x)^2] - (\\mathbb{E}_x[f(x)])^2$, where $\\mathbb{E}_x$ denotes the uniform average over the $2^n$ points $x$ of the cube.","title":"Variance as mean square minus squared mean"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-total-influence-eq-sum-card-mul-sq","kind":"node","lean":"BooleanCube.boolTotalInfluence_eq_sum_card_mul_sq","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-total-influence-eq-sum-card-mul-sq","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every function $f$ from the Boolean cube $\\{-1, 1\\}^n$ to $\\mathbb{R}$, the total influence of $f$ is","title":"Total influence in Fourier form"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-variance-le-bool-total-influence","kind":"node","lean":"BooleanCube.boolVariance_le_boolTotalInfluence","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-variance-le-bool-total-influence","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ on the Boolean cube, the variance of $f$ under the uniform distribution is at most the total influence of $f$: $\\mathrm{Var}[f] \\le I[f]$.","title":"Poincar\u00e9 inequality on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-noise-op","kind":"node","lean":"BooleanCube.boolNoiseOp","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-noise-op","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For a real number $\\rho$ and a function $f$ from the Boolean cube $\\{-1, 1\\}^n$ to $\\mathbb{R}$, the noise operator $T_\\rho$ sends $f$ to the function","title":"Noise operator on the Boolean cube"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-noise-op-eq-sum","kind":"node","lean":"BooleanCube.boolNoiseOp_eq_sum","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-noise-op-eq-sum","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real number $\\rho$, every function $f$ from the Boolean cube $\\{-1, 1\\}^n$ to $\\mathbb{R}$ and every point $x$ of the cube,","title":"Fourier expansion of the noise operator"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-fourier-coeff-bool-noise-op","kind":"node","lean":"BooleanCube.boolFourierCoeff_boolNoiseOp","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-fourier-coeff-bool-noise-op","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real $\\rho$, every real-valued function $f$ on the Boolean cube in $n$ coordinates and every set $S$ of the $n$ coordinates, the Fourier coefficient at $S$ of $T_\\rho f$, the noise operator with parameter $\\rho$ applied to $f$, is $\\widehat{T_\\rho f}(S) = \\rho^{|S|} \\hat f(S)$.","title":"Fourier coefficients of the noise operator"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-noise-op-one","kind":"node","lean":"BooleanCube.boolNoiseOp_one","parent":17087,"reading_title":{"format":"inline-parts-v1","parts":[["text","The noise operator "],["math","\\(T_1\\)"],["text"," is the identity"]],"source":"The noise operator $T_1$ is the identity"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-noise-op-one","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every function $f$ from the Boolean cube $\\{-1, 1\\}^n$ to $\\mathbb{R}$, the noise operator with $\\rho = 1$ fixes $f$: $T_1 f = f$.","title":"The noise operator $T_1$ is the identity"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-noise-op-zero","kind":"node","lean":"BooleanCube.boolNoiseOp_zero","parent":17087,"reading_title":{"format":"inline-parts-v1","parts":[["text","The noise operator "],["math","\\(T_0\\)"],["text"," takes the mean"]],"source":"The noise operator $T_0$ takes the mean"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-noise-op-zero","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every function $f$ from the Boolean cube $\\{-1, 1\\}^n$ to $\\mathbb{R}$, the noise operator with $\\rho = 0$ sends $f$ to the constant function whose value is the mean $\\mathbb{E}_x f(x) = 2^{-n} \\sum_x f(x)$ of $f$ under the uniform distribution.","title":"The noise operator $T_0$ takes the mean"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-noise-op-bool-noise-op","kind":"node","lean":"BooleanCube.boolNoiseOp_boolNoiseOp","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-noise-op-bool-noise-op","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For all real numbers $\\rho$ and $\\sigma$ and every real-valued function $f$ on the Boolean cube $\\{-1, 1\\}^n$, applying the noise operator $T_\\sigma$ and then $T_\\rho$ to $f$ gives $T_{\\rho\\sigma} f$: $T_\\rho(T_\\sigma f) = T_{\\rho\\sigma} f$.","title":"Semigroup property of the noise operator"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/expect-bool-noise-op","kind":"node","lean":"BooleanCube.expect_boolNoiseOp","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#expect-bool-noise-op","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real $\\rho$ and every function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ on the Boolean cube, the uniform average over the cube of $T_\\rho f$, the image of $f$ under the noise operator with parameter $\\rho$, equals the uniform average of $f$.","title":"The noise operator preserves the mean"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-noise-op-cube-chi","kind":"node","lean":"BooleanCube.boolNoiseOp_cubeChi","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-noise-op-cube-chi","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real number $\\rho$ and every set $S \\subseteq \\{0, \\dots, n-1\\}$, the noise operator $T_\\rho$ maps the Fourier character $\\chi_S$ on the Boolean cube $\\{-1, 1\\}^n$ to the function $x \\mapsto \\rho^{|S|} \\chi_S(x)$, so $\\chi_S$ is an eigenfunction of $T_\\rho$ with eigenvalue $\\rho^{|S|}$.","title":"Characters are eigenfunctions of the noise operator"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-noise-op-const","kind":"node","lean":"BooleanCube.boolNoiseOp_const","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-noise-op-const","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real number $\\rho$ and every real number $c$, the noise operator $T_\\rho$ maps the constant function with value $c$ on the Boolean cube $\\{-1, 1\\}^n$ to itself.","title":"The noise operator fixes constants"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/bool-noise-op-nonneg","kind":"node","lean":"BooleanCube.boolNoiseOp_nonneg","parent":17087,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean/index.html#bool-noise-op-nonneg","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/influences-and-the-noise-operator-on-the-boolean","status":"fully_proved","summary":"For every real number $\\rho$ with $|\\rho| \\le 1$, the noise operator $T_\\rho$ on the Boolean cube maps nonnegative functions to nonnegative functions: if $f$ is a real function on the cube with $f(y) \\ge 0$ for every point $y$, then $T_\\rho f(x) \\ge 0$ for every point $x$ of the cube.","title":"Positivity of the noise operator"},{"data_url":"scopes/Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem.json","id":"Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem","kind":"scope","navigation":"module","parent":17028,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem/index.html","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem","status":"fully_proved","title":"Riesz\u2013Thorin interpolation theorem"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem/riesz-thorin-finite","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RieszThorinWanted.riesz_thorin_finite","parent":17143,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem/index.html#riesz-thorin-finite","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem","status":"fully_proved","summary":"Let $\\iota$ and $\\kappa$ be finite sets, each with a \u03c3-algebra in which singletons are measurable and with the counting measure, and write $\\|g\\|_r \\in [0, \\infty]$ for the $\\ell^r$ norm of $g$ with respect to the counting measure. Let $p_0, p_1, p, q_0, q_1, q \\in [1, \\infty]$ and $\\theta \\in [0, 1]$ satisfy $1/p = (1-\\theta)/p_0 + \\theta/p_1$ and $1/q = (1-\\theta)/q_0 + \\theta/q_1$, where $1/\\infty = 0$. Let\u2026","title":"Riesz\u2013Thorin interpolation on finite sets, auxiliary form"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem/riesz-thorin","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RieszThorinWanted.riesz_thorin","parent":17143,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem/index.html#riesz-thorin","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/riesz-thorin-interpolation-theorem","status":"fully_proved","summary":"Let $\\iota$ and $\\kappa$ be finite sets, each with a \u03c3-algebra in which singletons are measurable and with the counting measure, and write $\\|g\\|_r \\in [0, \\infty]$ for the $\\ell^r$ norm of $g$ with respect to the counting measure. Let $p_0, p_1, p, q_0, q_1, q \\in [1, \\infty]$ and $\\theta \\in [0, 1]$ satisfy $1/p = (1-\\theta)/p_0 + \\theta/p_1$ and $1/q = (1-\\theta)/q_0 + \\theta/q_1$, where $1/\\infty = 0$. Let\u2026","title":"Riesz\u2013Thorin interpolation theorem on finite sets"},{"data_url":"scopes/Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro.json","id":"Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro","kind":"scope","navigation":"module","parent":17028,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro/index.html","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro","status":"fully_proved","title":"Hausdorff\u2013Young inequality on finite abelian groups"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro/dft-fun-sup-le","kind":"node","lean":"MathlibExt.Analysis.Fourier.HausdorffYoungFiniteAbelian.dftFun_sup_le","parent":17146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p = 1\\)"],["text"," endpoint of Hausdorff\u2013Young"]],"source":"The $p = 1$ endpoint of Hausdorff\u2013Young"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro/index.html#dft-fun-sup-le","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro","status":"fully_proved","summary":"Let $G$ be a finite abelian group and $f : G \\to \\mathbb{C}$, with discrete Fourier transform $\\hat f$. For every additive character $\\psi$ of $G$, $|\\hat f(\\psi)| \\le \\sum_{x \\in G} |f(x)|$.","title":"The $p = 1$ endpoint of Hausdorff\u2013Young"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro/sum-norm-sq-dft-fun","kind":"node","lean":"MathlibExt.Analysis.Fourier.HausdorffYoungFiniteAbelian.sum_norm_sq_dftFun","parent":17146,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p = 2\\)"],["text"," endpoint of Hausdorff\u2013Young (Plancherel)"]],"source":"The $p = 2$ endpoint of Hausdorff\u2013Young (Plancherel)"},"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro/index.html#sum-norm-sq-dft-fun","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro","status":"fully_proved","summary":"Let $G$ be a finite abelian group and let $f : G \\to \\mathbb{C}$ have discrete Fourier transform $\\hat f$. Then, with $\\psi$ running over the additive characters of $G$ and $|G|$ the number of elements of $G$, $\\sum_{\\psi} |\\hat f(\\psi)|^2 = |G| \\sum_{x \\in G} |f(x)|^2$.","title":"The $p = 2$ endpoint of Hausdorff\u2013Young (Plancherel)"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro/hausdorff-young-dft","kind":"node","lean":"MathlibExt.Analysis.Fourier.HausdorffYoungFiniteAbelian.hausdorffYoung_dft","parent":17146,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro/index.html#hausdorff-young-dft","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/hausdorff-young-inequality-on-finite-abelian-gro","status":"fully_proved","summary":"Let $G$ be a finite abelian group, let $f : G \\to \\mathbb{C}$ have discrete Fourier transform $\\hat f$, let $p$ be a real number with $1 < p \\le 2$, and let $q = p/(p - 1)$. Write $|G|$ for the number of elements of $G$ and let $\\psi$ run over the additive characters of $G$. Then $\\left(\\sum_{\\psi} |\\hat f(\\psi)|^q\\right)^{1/q} \\le","title":"Hausdorff\u2013Young inequality on a finite abelian group"},{"data_url":"scopes/Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups.json","id":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups","kind":"scope","navigation":"module","parent":17028,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/index.html","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups","status":"fully_proved","title":"Young's convolution inequality on finite groups"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/fin-group-conv","kind":"node","lean":"MathlibExt.Analysis.Fourier.finGroupConv","parent":17150,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/index.html#fin-group-conv","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group. The convolution of functions $f, g : G \\to \\mathbb{C}$ is the function $f * g : G \\to \\mathbb{C}$ given by $(f * g)(x) = \\sum_{y \\in G} f(y)\\,g(x - y)$, with no normalizing factor $1/|G|$, where $|G|$ is the number of elements of $G$.","title":"Convolution on a finite abelian group"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/fin-group-conv-comm","kind":"node","lean":"MathlibExt.Analysis.Fourier.finGroupConv_comm","parent":17150,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/index.html#fin-group-conv-comm","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups","status":"fully_proved","summary":"For a finite abelian group $G$ and functions $f, g : G \\to \\mathbb{C}$, we have $f * g = g * f$, where the convolution is $(f * g)(x) = \\sum_{y \\in G} f(y)\\,g(x - y)$.","title":"Convolution on a finite abelian group is commutative"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/fin-group-conv-assoc","kind":"node","lean":"MathlibExt.Analysis.Fourier.finGroupConv_assoc","parent":17150,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/index.html#fin-group-conv-assoc","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups","status":"fully_proved","summary":"For a finite abelian group $G$ and functions $f, g, h : G \\to \\mathbb{C}$, we have $(f * g) * h = f * (g * h)$, where the convolution is $(f * g)(x) = \\sum_{y \\in G} f(y)\\,g(x - y)$.","title":"Convolution on a finite abelian group is associative"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/fin-group-conv-delta-left","kind":"node","lean":"MathlibExt.Analysis.Fourier.finGroupConv_delta_left","parent":17150,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/index.html#fin-group-conv-delta-left","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group and let $\\delta_0 : G \\to \\mathbb{C}$ be the function with $\\delta_0(0) = 1$ and $\\delta_0(y) = 0$ for $y \\ne 0$. Then $\\delta_0 * g = g$ for every $g : G \\to \\mathbb{C}$, where the convolution is $(f * g)(x) = \\sum_{y \\in G} f(y)\\,g(x - y)$.","title":"The delta function is a left identity for convolution"},{"id":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/young-convolution-finite-group","kind":"node","lean":"MathlibExt.Analysis.Fourier.young_convolution_finite_group","parent":17150,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups/index.html#young-convolution-finite-group","scope":"Analysis/harmonic-analysis/discrete-fourier-analysis/young-s-convolution-inequality-on-finite-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group and, for $u : G \\to \\mathbb{C}$ and real $s > 0$, write $\\|u\\|_s = \\left(\\sum_{x \\in G} |u(x)|^s\\right)^{1/s}$. Let $f, g : G \\to \\mathbb{C}$ and let $p, q, r$ be real numbers with $p > 1$, $q > 1$, $r > 1$ and $1/p + 1/q = 1 + 1/r$. Then the convolution $(f * g)(x) = \\sum_{y \\in G} f(y)\\,g(x - y)$ satisfies $\\|f * g\\|_r \\le \\|f\\|_p \\, \\|g\\|_q$.","title":"Young's convolution inequality on a finite abelian group"},{"data_url":"scopes/Analysis/harmonic-analysis/integral-transforms.json","id":"Analysis/harmonic-analysis/integral-transforms","kind":"scope","parent":16828,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/index.html","scope":"Analysis/harmonic-analysis/integral-transforms","status":"can_prove","summary":"Laplace, Mellin and Radon transforms (Ramanujan's master theorem, Fourier slice theorem) and Hausdorff moment problems.","title":"Integral transforms and moment problems"},{"id":"Analysis/harmonic-analysis/integral-transforms/ramanujan-master-theorem","kind":"node","lean":"MetaMathlibExt.Ramanujan_master_theorem","parent":17156,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/index.html#ramanujan-master-theorem","scope":"Analysis/harmonic-analysis/integral-transforms","status":"can_prove","summary":"Let $a$ and $b$ be real numbers with $0 < a < b$, and let $s$ be a complex number with $a < \\operatorname{Re} s < b$. Let $\\varphi$ be a function from $\\mathbb{C}$ to $\\mathbb{C}$ that is analytic on the half-plane $\\operatorname{Re} z > -b$, and suppose there are real constants $C > 0$ and $P < \\pi$ such that $|\\varphi(u + iv)| \\le C e^{P|v|}$ for all real $u$ and $v$ with $u > -b$. Let $F$ be a function from\u2026","title":"Ramanujan's master theorem"},{"data_url":"scopes/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem.json","id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","kind":"scope","navigation":"module","parent":17156,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"can_prove","title":"Hausdorff moment problem"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/cube","kind":"node","lean":"MathlibExt.Analysis.HausdorffAbsoluteContinuityWanted.Analysis.HausdorffAbsoluteContinuity.cube","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#cube","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, the closed unit cube $I^d = [0,1]^d$ is the set of points $x \\in \\mathbb{R}^d$ with $0 \\le x_i \\le 1$ for every coordinate $i$.","title":"Closed unit cube"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/monomial","kind":"node","lean":"MathlibExt.Analysis.HausdorffAbsoluteContinuityWanted.Analysis.HausdorffAbsoluteContinuity.monomial","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#monomial","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For a multi-index $n = (n_1, \\dots, n_d) \\in \\mathbb{N}^d$ and a point $x \\in \\mathbb{R}^d$, the monomial $x^n$ is the real number $\\prod_{i=1}^{d} x_i^{n_i}$.","title":"Multivariate monomial"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/moment-of","kind":"node","lean":"MathlibExt.Analysis.HausdorffAbsoluteContinuityWanted.Analysis.HausdorffAbsoluteContinuity.momentOf","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#moment-of","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}^d$ and a multi-index $n \\in \\mathbb{N}^d$, the $n$-th moment of $\\mu$ is the real number $\\int_{I^d} x^n \\, d\\mu(x)$, the integral over the closed unit cube $I^d = [0,1]^d$ of the monomial $x^n = \\prod_{i=1}^{d} x_i^{n_i}$, taken to be $0$ when the monomial is not $\\mu$-integrable on $I^d$.","title":"Moments of a measure on the unit cube"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/multi-choose","kind":"node","lean":"MathlibExt.Analysis.HausdorffAbsoluteContinuityWanted.Analysis.HausdorffAbsoluteContinuity.multiChoose","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#multi-choose","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For multi-indices $n, k \\in \\mathbb{N}^d$, the binomial coefficient $\\binom{n}{k}$ is the natural number $\\prod_{i=1}^{d} \\binom{n_i}{k_i}$.","title":"Multi-index binomial coefficient"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/diff","kind":"node","lean":"MathlibExt.Analysis.HausdorffAbsoluteContinuityWanted.Analysis.HausdorffAbsoluteContinuity.diff","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#diff","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For a function $a : \\mathbb{N}^d \\to \\mathbb{R}$ and multi-indices $k, n \\in \\mathbb{N}^d$,","title":"Iterated backward difference of a multi-sequence"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/uniformly-absolutely-continuous","kind":"node","lean":"MathlibExt.Analysis.HausdorffAbsoluteContinuityWanted.Analysis.HausdorffAbsoluteContinuity.UniformlyAbsolutelyContinuous","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#uniformly-absolutely-continuous","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"A measure $\\mu$ on $\\mathbb{R}^d$ is uniformly absolutely continuous with respect to a measure $\\nu$ on $\\mathbb{R}^d$ if there is a real constant $C \\ge 0$ with $\\mu(S) \\le C \\, \\nu(S)$ for every measurable set $S$.","title":"Uniformly absolutely continuous measures"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/hausdorff-absolute-continuity","kind":"node","lean":"MathlibExt.Analysis.HausdorffAbsoluteContinuityWanted.Analysis.HausdorffAbsoluteContinuity.hausdorff_absolute_continuity","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#hausdorff-absolute-continuity","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"can_prove","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}^d$ with $\\mu(\\mathbb{R}^d \\setminus I^d) = 0$, where $I^d = [0,1]^d$ is the closed unit cube, and let $\\lambda$ be Lebesgue measure restricted to $I^d$. For a measure $\\rho$ write $m_\\rho(n) = \\int_{I^d} x^n \\, d\\rho(x)$ for its moments, indexed by multi-indices $n \\in \\mathbb{N}^d$. Then $\\mu$ is uniformly absolutely continuous with respect to $\\lambda$ if and only\u2026","title":"Hausdorff's absolute continuity criterion"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/cube-2","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.cube","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#cube-2","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, the closed unit cube $I^d = [0,1]^d$ is the set of points $x \\in \\mathbb{R}^d$ with $0 \\le x_i \\le 1$ for every coordinate $i$.","title":"Unit cube of the Hausdorff moment problem"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/monomial-2","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.monomial","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#monomial-2","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For a multi-index $n \\in \\mathbb{N}^d$, the monomial $x^n$ is the real function on $\\mathbb{R}^d$ given by $x^n = \\prod_i x_i^{n_i}$.","title":"Multi-index monomial"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/moment-of-2","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.momentOf","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#moment-of-2","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}^d$ and a multi-index $n \\in \\mathbb{N}^d$, the $n$-th moment of $\\mu$ on the unit cube is the real number $\\int_{[0,1]^d} x^n \\, d\\mu(x)$, where $x^n = \\prod_i x_i^{n_i}$.","title":"Moments on the unit cube"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/is-moment-configuration","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.IsMomentConfiguration","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#is-moment-configuration","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"A real sequence $a$ indexed by $\\mathbb{N}^d$ is a moment configuration if there are finite measures $\\mu$ and $\\nu$ on $\\mathbb{R}^d$ such that for every multi-index $n \\in \\mathbb{N}^d$,","title":"Moment sequences of signed measures on the cube"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/multi-choose-2","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.multiChoose","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#multi-choose-2","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For multi-indices $n, k \\in \\mathbb{N}^d$, the binomial coefficient $\\binom{n}{k}$ is the natural number $\\prod_i \\binom{n_i}{k_i}$.","title":"Multi-index binomial coefficient"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/diff-2","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.diff","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#diff-2","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"For a real sequence $a$ indexed by $\\mathbb{N}^d$ and multi-indices $k, n \\in \\mathbb{N}^d$, the iterated backward difference of $a$ is the real number","title":"Iterated backward difference"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/hausdorff-bounded","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.HausdorffBounded","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#hausdorff-bounded","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"A real sequence $a$ indexed by $\\mathbb{N}^d$ is Hausdorff bounded if there is a real constant $M$ such that for every multi-index $n \\in \\mathbb{N}^d$,","title":"Hausdorff-bounded sequences"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/is-positive-moment-configuration","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.IsPositiveMomentConfiguration","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#is-positive-moment-configuration","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"fully_proved","summary":"A real sequence $a$ indexed by $\\mathbb{N}^d$ is a positive moment configuration if there is a finite measure $\\mu$ on $\\mathbb{R}^d$ such that $a_n = \\int_{[0,1]^d} x^n \\, d\\mu(x)$ for every multi-index $n \\in \\mathbb{N}^d$, where $x^n = \\prod_i x_i^{n_i}$.","title":"Moment sequences of positive measures on the cube"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/hausdorff-hildebrandt-schoenberg","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.hausdorff_hildebrandt_schoenberg","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#hausdorff-hildebrandt-schoenberg","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"can_prove","summary":"A real sequence $a$ indexed by $\\mathbb{N}^d$ is a moment configuration, that is, there are finite measures $\\mu$ and $\\nu$ on $\\mathbb{R}^d$ with $a_n = \\int_{[0,1]^d} x^n \\, d\\mu(x) - \\int_{[0,1]^d} x^n \\, d\\nu(x)$ for every multi-index $n \\in \\mathbb{N}^d$, where $x^n = \\prod_i x_i^{n_i}$, if and only if $a$ is Hausdorff bounded.","title":"Hausdorff\u2013Hildebrandt\u2013Schoenberg moment theorem"},{"id":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/hausdorff-positivity","kind":"node","lean":"MathlibExt.Analysis.HausdorffMomentsWanted.Analysis.hausdorff_positivity","parent":17158,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem/index.html#hausdorff-positivity","scope":"Analysis/harmonic-analysis/integral-transforms/hausdorff-moment-problem","status":"can_prove","summary":"A real sequence $a$ indexed by $\\mathbb{N}^d$ is a positive moment configuration, that is, there is a finite measure $\\mu$ on $\\mathbb{R}^d$ with $a_n = \\int_{[0,1]^d} x^n \\, d\\mu(x)$ for every multi-index $n \\in \\mathbb{N}^d$, where $x^n = \\prod_i x_i^{n_i}$, if and only if $(\\Delta^k a)_n \\ge 0$ for all multi-indices $k, n$ with $k \\le n$ coordinatewise, where $\\Delta^k a$ is the iterated backward difference of\u2026","title":"Hausdorff positivity criterion"},{"data_url":"scopes/Analysis/harmonic-analysis/integral-transforms/laplace-transform.json","id":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","kind":"scope","navigation":"module","parent":17156,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/laplace-transform/index.html","scope":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","status":"fully_proved","title":"Laplace transform"},{"id":"Analysis/harmonic-analysis/integral-transforms/laplace-transform/laplace-convergent","kind":"node","lean":"LaplaceConvergent","parent":17176,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/laplace-transform/index.html#laplace-convergent","scope":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","status":"fully_proved","summary":"Let $E$ be a complex normed space, $f : \\mathbb{R} \\to E$ and $s \\in \\mathbb{C}$. The Laplace transform of $f$ converges at $s$ when the function $t \\mapsto e^{-st} f(t)$ is integrable on $(0, \\infty)$ with respect to Lebesgue measure.","title":"Convergence of the Laplace transform"},{"id":"Analysis/harmonic-analysis/integral-transforms/laplace-transform/laplace","kind":"node","lean":"laplace","parent":17176,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/laplace-transform/index.html#laplace","scope":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","status":"fully_proved","summary":"Let $E$ be a complex normed space, $f : \\mathbb{R} \\to E$ and $s \\in \\mathbb{C}$. The Laplace transform of $f$ at $s$ is the Bochner integral $\\int_{(0, \\infty)} e^{-st} f(t)\\,dt \\in E$.","title":"The one-sided Laplace transform"},{"id":"Analysis/harmonic-analysis/integral-transforms/laplace-transform/has-laplace","kind":"node","lean":"HasLaplace","parent":17176,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/laplace-transform/index.html#has-laplace","scope":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","status":"fully_proved","summary":"Let $E$ be a complex normed space, $f : \\mathbb{R} \\to E$, $s \\in \\mathbb{C}$ and $v \\in E$. Then $f$ has Laplace transform $v$ at $s$ when the Laplace transform of $f$ converges at $s$, that is, $t \\mapsto e^{-st} f(t)$ is integrable on $(0, \\infty)$, and the Bochner integral $\\int_{(0, \\infty)} e^{-st} f(t)\\,dt$ equals $v$; as usual this integral is taken to be $0$ when $E$ is not complete.","title":"Laplace transform with a given value"},{"id":"Analysis/harmonic-analysis/integral-transforms/laplace-transform/laplace-convergent-of-norm-le-exp","kind":"node","lean":"laplaceConvergent_of_norm_le_exp","parent":17176,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/laplace-transform/index.html#laplace-convergent-of-norm-le-exp","scope":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","status":"fully_proved","summary":"Let $E$ be a complex normed space, $f : \\mathbb{R} \\to E$ almost everywhere strongly measurable on $(0, \\infty)$, and $a, C \\in \\mathbb{R}$ with $C \\ge 0$ such that $\\|f(t)\\| \\le C e^{a t}$ for all $t \\ge 0$. Then for every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > a$ the Laplace transform of $f$ converges at $s$, that is, $t \\mapsto e^{-st} f(t)$ is integrable on $(0, \\infty)$.","title":"Convergence of the Laplace transform under an exponential bound"},{"id":"Analysis/harmonic-analysis/integral-transforms/laplace-transform/laplace-convergent-of-is-big-o-exp","kind":"node","lean":"laplaceConvergent_of_isBigO_exp","parent":17176,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/laplace-transform/index.html#laplace-convergent-of-is-big-o-exp","scope":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","status":"fully_proved","summary":"Let $E$ be a complex normed space, $a \\in \\mathbb{R}$, and $f : \\mathbb{R} \\to E$ locally integrable on $[0, \\infty)$ with $\\|f(t)\\| = O(e^{a t})$ as $t \\to \\infty$. Then for every $s \\in \\mathbb{C}$ with $\\operatorname{Re} s > a$ the Laplace transform of $f$ converges at $s$, that is, $t \\mapsto e^{-st} f(t)$ is integrable on $(0, \\infty)$.","title":"Convergence of the Laplace transform under exponential growth"},{"id":"Analysis/harmonic-analysis/integral-transforms/laplace-transform/differentiable-on-laplace-of-norm-le-exp","kind":"node","lean":"differentiableOn_laplace_of_norm_le_exp","parent":17176,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/laplace-transform/index.html#differentiable-on-laplace-of-norm-le-exp","scope":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","status":"fully_proved","summary":"Let $E$ be a complex Banach space, $f : \\mathbb{R} \\to E$ almost everywhere strongly measurable on $(0, \\infty)$, and $a, C \\in \\mathbb{R}$ with $C \\ge 0$ such that $\\|f(t)\\| \\le C e^{a t}$ for all $t \\ge 0$. Then the Laplace transform $s \\mapsto \\int_{(0, \\infty)} e^{-st} f(t)\\,dt$ is complex differentiable at every point of the half-plane $\\{s \\in \\mathbb{C} : \\operatorname{Re} s > a\\}$.","title":"Holomorphy of the Laplace transform under an exponential bound"},{"id":"Analysis/harmonic-analysis/integral-transforms/laplace-transform/differentiable-on-laplace-of-is-big-o-ex","kind":"node","lean":"differentiableOn_laplace_of_isBigO_exp","parent":17176,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/laplace-transform/index.html#differentiable-on-laplace-of-is-big-o-ex","scope":"Analysis/harmonic-analysis/integral-transforms/laplace-transform","status":"fully_proved","summary":"Let $E$ be a complex Banach space, $a \\in \\mathbb{R}$, and $f : \\mathbb{R} \\to E$ locally integrable on $[0, \\infty)$ with $\\|f(t)\\| = O(e^{a t})$ as $t \\to \\infty$. Then the Laplace transform $s \\mapsto \\int_{(0, \\infty)} e^{-st} f(t)\\,dt$ is complex differentiable at every point of the half-plane $\\{s \\in \\mathbb{C} : \\operatorname{Re} s > a\\}$.","title":"Holomorphy of the Laplace transform under exponential growth"},{"data_url":"scopes/Analysis/harmonic-analysis/integral-transforms/radon-transform.json","id":"Analysis/harmonic-analysis/integral-transforms/radon-transform","kind":"scope","navigation":"module","parent":17156,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/radon-transform/index.html","scope":"Analysis/harmonic-analysis/integral-transforms/radon-transform","status":"can_prove","title":"Radon transform"},{"id":"Analysis/harmonic-analysis/integral-transforms/radon-transform/radon","kind":"node","lean":"MathlibExt.Analysis.RadonTransformWanted.Analysis.RadonTransform.radon","parent":17184,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/radon-transform/index.html#radon","scope":"Analysis/harmonic-analysis/integral-transforms/radon-transform","status":"fully_proved","summary":"For $f : \\mathbb{R}^2 \\to \\mathbb{C}$ and $p, \\theta \\in \\mathbb{R}$, the Radon transform of $f$ at $(p, \\theta)$ is the integral of $f$ along the line $x \\cos\\theta + y \\sin\\theta = p$ with respect to arc length,","title":"Radon transform on the plane"},{"id":"Analysis/harmonic-analysis/integral-transforms/radon-transform/fourier1","kind":"node","lean":"MathlibExt.Analysis.RadonTransformWanted.Analysis.RadonTransform.fourier1","parent":17184,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/radon-transform/index.html#fourier1","scope":"Analysis/harmonic-analysis/integral-transforms/radon-transform","status":"fully_proved","summary":"For $g : \\mathbb{R} \\to \\mathbb{C}$ and $k \\in \\mathbb{R}$, the Fourier transform of $g$ at $k$ is","title":"One-dimensional Fourier transform"},{"id":"Analysis/harmonic-analysis/integral-transforms/radon-transform/fourier2","kind":"node","lean":"MathlibExt.Analysis.RadonTransformWanted.Analysis.RadonTransform.fourier2","parent":17184,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/radon-transform/index.html#fourier2","scope":"Analysis/harmonic-analysis/integral-transforms/radon-transform","status":"fully_proved","summary":"For $f : \\mathbb{R}^2 \\to \\mathbb{C}$ and $k = (k_1, k_2) \\in \\mathbb{R}^2$, the Fourier transform of $f$ at $k$ is","title":"Two-dimensional Fourier transform"},{"id":"Analysis/harmonic-analysis/integral-transforms/radon-transform/radon-can-be-diagonalized-and-pseudo-inv","kind":"node","lean":"MathlibExt.Analysis.RadonTransformWanted.Analysis.RadonTransform.radon_can_be_diagonalized_and_pseudo_inverted","parent":17184,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/radon-transform/index.html#radon-can-be-diagonalized-and-pseudo-inv","scope":"Analysis/harmonic-analysis/integral-transforms/radon-transform","status":"can_prove","summary":"Write $R$ for the Radon transform, $\\hat g$ for the one-dimensional Fourier transform of $g : \\mathbb{R} \\to \\mathbb{C}$ and $\\hat f$ for the two-dimensional Fourier transform of $f : \\mathbb{R}^2 \\to \\mathbb{C}$. For every Schwartz function $\\varphi : \\mathbb{R}^2 \\to \\mathbb{C}$ and all $\\theta, k \\in \\mathbb{R}$, the Fourier transform of $p \\mapsto R\\varphi(p, \\theta)$ at $k$ equals\u2026","title":"Fourier slice theorem and injectivity of the Radon transform on Schwartz functions"},{"data_url":"scopes/Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem.json","id":"Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem","kind":"scope","navigation":"module","parent":17156,"result":false,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem/index.html","scope":"Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem","status":"fully_proved","title":"Stieltjes moment problem"},{"id":"Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem/summable-geometric-means","kind":"node","lean":"MetaMathlibExt.summable_geometric_means","parent":17189,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem/index.html#summable-geometric-means","scope":"Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem","status":"fully_proved","summary":"If $(a_n)_{n \\ge 0}$ is a summable sequence of nonnegative real numbers, then the sequence of geometric means $(a_0 a_1 \\cdots a_n)^{1/(n+1)}$, $n \\ge 0$, is summable.","title":"Carleman's inequality, summability form"},{"id":"Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem/stieltjes-moment-determinate","kind":"node","lean":"MetaMathlibExt.StieltjesMomentDeterminate","parent":17189,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem/index.html#stieltjes-moment-determinate","scope":"Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem","status":"fully_proved","summary":"A measure $\\mu$ on $\\mathbb{R}$ is Stieltjes moment determinate when every probability measure $\\nu$ on $\\mathbb{R}$ such that $\\nu((-\\infty, 0)) = 0$, $\\int |x|^k \\, d\\nu(x) < \\infty$ for every integer $k \\ge 1$, and $\\int x^k \\, d\\nu(x) = \\int x^k \\, d\\mu(x)$ for every integer $k \\ge 1$ is equal to $\\mu$.","title":"Stieltjes moment determinacy"},{"id":"Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem/carleman-stieltjes-criterion","kind":"node","lean":"MetaMathlibExt.carleman_stieltjes_criterion","parent":17189,"result":true,"route":"../roadmap/Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem/index.html#carleman-stieltjes-criterion","scope":"Analysis/harmonic-analysis/integral-transforms/stieltjes-moment-problem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with $\\mu((-\\infty, 0)) = 0$ such that $\\int |x|^k \\, d\\mu(x) < \\infty$ for every integer $k \\ge 1$, and write $m_k = \\int x^k \\, d\\mu(x)$. If $\\sum_{k \\ge 1} m_k^{-1/(2k)} = \\infty$, where the terms are taken in $[0, \\infty]$ (so a term with $m_k = 0$ equals $\\infty$), then $\\mu$ is Stieltjes moment determinate.","title":"Carleman's criterion for the Stieltjes moment problem"},{"data_url":"scopes/Analysis/banach-spaces.json","id":"Analysis/banach-spaces","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/banach-spaces/index.html","scope":"Analysis/banach-spaces","status":"can_prove","summary":"Normed, Banach and topological vector spaces: duality, weak topologies, geometry of the unit ball, bases and classical sequence spaces.","title":"Banach space theory"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies.json","id":"Analysis/banach-spaces/duality-and-weak-topologies","kind":"scope","parent":17193,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"can_prove","summary":"Hahn\u2013Banach, annihilators and bipolars, weak and weak-star convergence and topologies, Goldstine, James, Eberlein\u2013\u0160mulian, Krein\u2013\u0160mulian, Banach\u2013Alaoglu, Mazur's lemma, Banach limits, Hamel bases and discontinuous functionals, F-spaces and locally convex spaces, Choquet theory.","title":"Duality, weak topologies and reflexivity"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/goldstine","kind":"node","lean":"MetaMathlibExt.goldstine","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#goldstine","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $E^{**}$ be its bidual and let $J : E \\to E^{**}$ be the canonical embedding. In the weak-star topology of $E^{**}$, the closure of the image under $J$ of the closed unit ball of $E$ is exactly the closed unit ball of $E^{**}$.","title":"Goldstine's theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/continuous-linear-functional-finite-eval","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook10Wanted.continuousLinearFunctionalFiniteEvaluations","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#continuous-linear-functional-finite-eval","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"can_prove","summary":"Let $H$ be a set and give the space $\\mathbb{R}^H$ of all real-valued functions on $H$ the topology of pointwise convergence. Then for every continuous linear functional $f \\colon \\mathbb{R}^H \\to \\mathbb{R}$ there are a natural number $n$, points $t_1, \\dots, t_n \\in H$ and real numbers $\\alpha_1, \\dots, \\alpha_n$ such that $f(x) = \\sum_{i=1}^n \\alpha_i \\, x(t_i)$ for every $x \\in \\mathbb{R}^H$.","title":"Continuous linear functionals on the space of all real functions"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/eberlein-smulian","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BanachSpaceWanted.eberlein_smulian","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#eberlein-smulian","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"fully_proved","summary":"Let $E$ be a Banach space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. A subset of $E$ is compact in the weak topology if and only if it is sequentially compact in the weak topology, that is, every sequence in it has a subsequence that converges weakly to a point of the subset.","title":"Eberlein\u2013\u0160mulian theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/dunford-pettis-compactness","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.DunfordPettisWanted.dunford_pettis_compactness","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#dunford-pettis-compactness","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $\\alpha$ and let $(f_i)_{i \\in \\iota}$ be a family of integrable real functions on $\\alpha$. The family is uniformly integrable, meaning that it is bounded in $L^1(\\mu)$ and that for every $\\varepsilon > 0$ there is $\\delta > 0$ with $\\int_s |f_i| \\, d\\mu \\le \\varepsilon$ for every $i$ and every measurable set $s$ with $\\mu(s) \\le \\delta$, if and only if the\u2026","title":"Dunford\u2013Pettis weak compactness criterion"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/banach-limit","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BanachLimit.banach_limit","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#banach-limit","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"fully_proved","summary":"There is a continuous linear functional $L$ on the real Banach space $\\ell^\\infty$ of bounded real sequences with the supremum norm such that $L(f) \\ge 0$ whenever $f_n \\ge 0$ for every $n$, $L$ maps the constant sequence $1$ to $1$, $L(f) = L(g)$ whenever $g_n = f_{n+1}$ for every $n$, and $L(f) = l$ whenever the sequence $f_n$ converges to $l$.","title":"Existence of Banach limits"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/josefson-nissenzweig","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.JosefsonNissenzweigWanted.josefson_nissenzweig","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#josefson-nissenzweig","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"fully_proved","summary":"Let $E$ be an infinite-dimensional Banach space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. Then there is a sequence $(\\varphi_n)_{n \\in \\mathbb{N}}$ of continuous linear functionals on $E$ with $\\|\\varphi_n\\| = 1$ for every $n$ such that $\\varphi_n(x) \\to 0$ for every $x \\in E$; that is, the unit sphere of the dual of $E$ contains a weak-star null sequence.","title":"Josefson\u2013Nissenzweig theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/krein-smulian","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BanachSpaceWanted.krein_smulian","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#krein-smulian","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"fully_proved","summary":"Let $E$ be a Banach space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, and let $C$ be a subset of its dual $E^*$ that is convex over $\\mathbb{R}$. Then $C$ is weak-star closed if and only if, for every $r > 0$, the set $\\{\\varphi \\in C : \\|\\varphi\\| \\le r\\}$ is weak-star closed.","title":"Krein\u2013Smulian theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/additive-functional-measurable-on-curves","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook24Wanted.additiveFunctionalMeasurableOnCurvesContinuous","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#additive-functional-measurable-on-curves","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"can_prove","summary":"Let $E$ be a real Banach space and let $F \\colon E \\to \\mathbb{R}$ be additive, that is, $F(x + y) = F(x) + F(y)$ for all $x, y \\in E$. If for every continuous map $x \\colon [0, 1] \\to E$ the function $t \\mapsto F(x(t))$ on $[0, 1]$ is Borel measurable, then $F$ is continuous.","title":"Mazur's Problem 24: additive functionals measurable along curves are continuous"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/exists-norm-sub-eq-i-inf-of-reflexive-of","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ReflexiveBanachNearestPointWanted.exists_norm_sub_eq_iInf_of_reflexive_of_convex_of_isClosed","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#exists-norm-sub-eq-i-inf-of-reflexive-of","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"can_prove","summary":"Let $X$ be a reflexive complex Banach space, that is, the canonical embedding of $X$ into its double dual is surjective, and let $C \\subseteq X$ be a nonempty closed convex set. Then for every $x_0 \\in X$ there is $y_0 \\in C$ such that $\\|x_0 - y_0\\| = \\inf_{y \\in C} \\|x_0 - y\\|$.","title":"Nearest points in reflexive Banach spaces"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/principle-of-local-reflexivity","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LocalReflexivityWanted.principle_of_local_reflexivity","parent":17194,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/index.html#principle-of-local-reflexivity","scope":"Analysis/banach-spaces/duality-and-weak-topologies","status":"fully_proved","summary":"Let $E$ be a Banach space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with dual $E^*$, bidual $E^{**}$ and canonical embedding $J \\colon E \\to E^{**}$. Let $M$ and $N$ be finite-dimensional normed spaces over $\\mathbb{K}$ with injective bounded linear maps $i_M \\colon M \\to E^{**}$ and $i_N \\colon N \\to E^*$, and let $\\varepsilon > 0$. Then there is a bounded linear map $T \\colon M \\to E$ such that for all\u2026","title":"Principle of local reflexivity"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem","status":"can_prove","title":"Banach\u2013Alaoglu theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem/weak-star-polar","kind":"node","lean":"MathlibExt.Analysis.BanachAlaogluWanted.Analysis.weakStarPolar","parent":17205,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem/index.html#weak-star-polar","scope":"Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem","status":"fully_proved","summary":"For a real topological vector space $E$ and a subset $s \\subseteq E$, the weak-star polar of $s$ is the set of continuous linear functionals $\\varphi$ on $E$, viewed as elements of the weak dual of $E$, such that $|\\varphi(x)| \\le 1$ for every $x \\in s$.","title":"Weak-star polar"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem/banach-alaoglu-bourbaki","kind":"node","lean":"MathlibExt.Analysis.BanachAlaogluWanted.Analysis.banach_alaoglu_bourbaki","parent":17205,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem/index.html#banach-alaoglu-bourbaki","scope":"Analysis/banach-spaces/duality-and-weak-topologies/banach-alaoglu-theorem","status":"can_prove","summary":"Let $E$ be a real locally convex topological vector space and let $U$ be a neighbourhood of $0$ in $E$. Then the weak-star polar of $U$ is compact in the weak dual of $E$, that is, in the space of continuous linear functionals on $E$ with the weak-star topology.","title":"Banach\u2013Alaoglu\u2013Bourbaki theorem"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory","status":"can_prove","title":"Choquet theory"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/choquet-representation-of-is-compact","kind":"node","lean":"MathlibExt.Analysis.Convex.ChoquetWanted.choquet_representation_of_isCompact","parent":17208,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/index.html#choquet-representation-of-is-compact","scope":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory","status":"fully_proved","summary":"Let $E$ be a second-countable real Banach space with its Borel \u03c3-algebra and let $K \\subseteq E$ be compact. For every $x \\in K$ there is a probability measure $\\mu$ on $E$ such that the set $\\operatorname{ext} K$ of extreme points of $K$ has $\\mu(\\operatorname{ext} K) = 1$, the identity map $y \\mapsto y$ is Bochner integrable for $\\mu$, and $\\int_E y \\, d\\mu(y) = x$.","title":"Choquet representation for compact sets"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/choquet-representation","kind":"node","lean":"MathlibExt.Analysis.Convex.ChoquetWanted.choquet_representation","parent":17208,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/index.html#choquet-representation","scope":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory","status":"fully_proved","summary":"Let $E$ be a second-countable real Banach space with its Borel \u03c3-algebra and let $K \\subseteq E$ be a nonempty compact convex set. For every $x \\in K$ there is a probability measure $\\mu$ on $E$ such that the set $\\operatorname{ext} K$ of extreme points of $K$ has $\\mu(\\operatorname{ext} K) = 1$, the identity map $y \\mapsto y$ is Bochner integrable for $\\mu$, and $\\int_E y \\, d\\mu(y) = x$.","title":"Choquet's theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/choquet","kind":"node","lean":"MathlibExt.Analysis.ChoquetWanted.Analysis.choquet","parent":17208,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/index.html#choquet","scope":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory","status":"can_prove","summary":"Let $X$ be a real Banach space with its Borel \u03c3-algebra and let $K \\subseteq X$ be compact and convex. For every $x \\in K$ there is a probability measure $\\mu$ on $X$ such that the complement of the set of extreme points of $K$ has $\\mu$-measure $0$ and the barycenter $\\int y \\, d\\mu(y)$ (a Bochner integral) equals $x$.","title":"Choquet's representation theorem in Banach spaces"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/bauer-unique","kind":"node","lean":"MathlibExt.Analysis.ChoquetWanted.Analysis.bauer_unique","parent":17208,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory/index.html#bauer-unique","scope":"Analysis/banach-spaces/duality-and-weak-topologies/choquet-theory","status":"can_prove","summary":"Let $X$ be a real Banach space with its Borel \u03c3-algebra, let $K \\subseteq X$ be compact and convex, and let $x$ be an extreme point of $K$. If $\\mu$ is a probability measure on $X$ with $\\mu(X \\setminus K) = 0$ whose barycenter $\\int y \\, d\\mu(y)$ (a Bochner integral) equals $x$, then $\\mu$ is the Dirac measure at $x$.","title":"Bauer's uniqueness theorem at extreme points"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals","status":"fully_proved","title":"Discontinuous linear functionals"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals/exists-discontinuous-linear-map-of-not-f","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.DiscontinuousLinearFunctional.exists_discontinuous_linearMap_of_not_finiteDimensional","parent":17213,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals/index.html#exists-discontinuous-linear-map-of-not-f","scope":"Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. If $E$ is not finite-dimensional, then there exists a linear functional $f \\colon E \\to \\mathbb{K}$ that is not continuous.","title":"A discontinuous linear functional on an infinite-dimensional normed space"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals/forall-linear-map-continuous-iff-finite","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.DiscontinuousLinearFunctional.forall_linearMap_continuous_iff_finiteDimensional","parent":17213,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals/index.html#forall-linear-map-continuous-iff-finite","scope":"Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. Every linear functional $f \\colon E \\to \\mathbb{K}$ is continuous if and only if $E$ is finite-dimensional.","title":"Every linear functional is continuous if and only if the space is finite-dimensional"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals/exists-linear-map-not-continuous-linear","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.DiscontinuousLinearFunctional.exists_linearMap_not_continuousLinearMap","parent":17213,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals/index.html#exists-linear-map-not-continuous-linear","scope":"Analysis/banach-spaces/duality-and-weak-topologies/discontinuous-linear-functionals","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. If $E$ is not finite-dimensional, then there is a linear functional $f \\colon E \\to \\mathbb{K}$ that differs from every continuous linear functional $g \\colon E \\to \\mathbb{K}$.","title":"The algebraic dual is strictly larger than the topological dual"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"can_prove","title":"F-spaces"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/is-type-f","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook15Point1Wanted.IsTypeF","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#is-type-f","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"A real vector space $E$ with a topology is of type (F) when addition $E \\times E \\to E$ and scalar multiplication $\\mathbb{R} \\times E \\to E$ are continuous and there is a metric $d$ on $E$ that is translation invariant, $d(x + z, y + z) = d(x, y)$ for all $x, y, z \\in E$, that induces the topology of $E$, and that is complete: every Cauchy sequence for $d$ converges for $d$.","title":"Spaces of type (F)"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/is-sphere","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook15Point1Wanted.IsSphere","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#is-sphere","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"Let $E$ be a real vector space. A set $K \\subseteq E$ is a sphere when $0 \\notin K$ and for every $x \\ne 0$ in $E$ there is exactly one real $t > 0$ with $t x \\in K$; that is, every open ray from the origin meets $K$ exactly once.","title":"Spheres in the ray sense"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/is-bounded-set","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook15Point1Wanted.IsBoundedSet","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#is-bounded-set","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"Let $E$ be a real vector space with a topology. A set $R \\subseteq E$ is bounded when for every sequence $(x_n)$ of points of $R$ and every sequence $(t_n)$ of real numbers with $t_n \\to 0$, the sequence $(t_n x_n)$ tends to $0$ in $E$.","title":"Sequentially bounded sets"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/admits-banach-norm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook15Point1Wanted.AdmitsBanachNorm","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#admits-banach-norm","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"A real vector space $E$ with a topology is of type (B) when there is a function $\\|\\cdot\\| : E \\to \\mathbb{R}$ with $\\|x\\| \\ge 0$, with $\\|x\\| = 0$ exactly when $x = 0$, with $\\|x + y\\| \\le \\|x\\| + \\|y\\|$ and with $\\|a x\\| = |a| \\, \\|x\\|$ for all $x, y \\in E$ and $a \\in \\mathbb{R}$, such that a set $U \\subseteq E$ is open exactly when every $x \\in U$ has an $\\varepsilon > 0$ with $y \\in U$ whenever\u2026","title":"Spaces of type (B)"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/scottish-book-problem15-point1-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook15Point1Wanted.ScottishBookProblem15Point1Claim","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#scottish-book-problem15-point1-claim","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"The claim of Mazur and Orlicz that every real vector space $E$ with a topology that is of type (F) and contains a bounded sphere is of type (B).","title":"Scottish Book Problem 15.1"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/bounded-sphere-fimplies-b-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook15Point1Wanted.boundedSphereFImpliesB_refuted","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#bounded-sphere-fimplies-b-refuted","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"can_prove","summary":"The Mazur\u2013Orlicz claim of Problem 15.1 is false: there is a real vector space $E$ with a topology that is of type (F) and contains a bounded sphere, but is not of type (B).","title":"Negative answer to Scottish Book Problem 15.1"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/is-type-f-2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook26Wanted.IsTypeF","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#is-type-f-2","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"A real vector space $E$ with a topology is of type (F) if there is a metric $d$ on $E$ that induces its topology, is translation invariant, $d(x + z, y + z) = d(x, y)$ for all $x, y, z \\in E$, and is complete, and if addition $E \\times E \\to E$ and scalar multiplication $\\mathbb{R} \\times E \\to E$ are continuous.","title":"Spaces of type (F) in Scottish Book Problem 26"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/is-bounded-set-2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook26Wanted.IsBoundedSet","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#is-bounded-set-2","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"Let $E$ be a real vector space with a topology. A set $R \\subseteq E$ is bounded if for every sequence $(x_n)_{n \\ge 0}$ of points of $R$ and every real sequence $(t_n)_{n \\ge 0}$ with $t_n \\to 0$, we have $t_n x_n \\to 0$ in $E$.","title":"Bounded sets in Scottish Book Problem 26"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/is-type-fo","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook26Wanted.IsTypeFo","parent":17217,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spaces of type "],["math","\\(\\{F_o\\}\\)"]],"source":"Spaces of type $\\{F_o\\}$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#is-type-fo","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"A real vector space $E$ with a topology is of type $\\{F_o\\}$ if it is of type (F) and, for every sequence $(x_n)_{n \\ge 0}$ in $E$ with $x_n \\to 0$ and every real sequence $(t_n)_{n \\ge 0}$ with $\\sum_n |t_n| < \\infty$, the series $\\sum_n t_n x_n$ converges in $E$.","title":"Spaces of type $\\{F_o\\}$"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/scottish-book-problem26-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook26Wanted.ScottishBookProblem26Claim","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#scottish-book-problem26-claim","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"fully_proved","summary":"The claim that for every real vector space $E$ with a topology that is of type $\\{F_o\\}$ and every sequence $(F_n)_{n \\ge 0}$ of linear functionals $E \\to \\mathbb{R}$, if $F_n \\to 0$ uniformly on every bounded set $R \\subseteq E$, then there is an open set $U \\subseteq E$ containing $0$ on which $F_n \\to 0$ uniformly.","title":"Scottish Book Problem 26 (Mazur and Orlicz's claim)"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/fo-bounded-uniform-neighborhood-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook26Wanted.foBoundedUniformNeighborhood_refuted","parent":17217,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/f-spaces/index.html#fo-bounded-uniform-neighborhood-refuted","scope":"Analysis/banach-spaces/duality-and-weak-topologies/f-spaces","status":"can_prove","summary":"The claim of Scottish Book Problem 26 is false: there are a real vector space $E$ with a topology, of type $\\{F_o\\}$, and a sequence $(F_n)_{n \\ge 0}$ of linear functionals $E \\to \\mathbb{R}$ that converges to $0$ uniformly on every bounded subset of $E$ but does not converge to $0$ uniformly on any open set containing $0$.","title":"Mazur and Orlicz's Problem 26 has a negative answer"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"can_prove","title":"Hahn\u2013Banach theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/hahn-banach-one-step-extension-bound","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.HahnBanachOneStepExtension.hahn_banach_one_step_extension_bound","parent":17229,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html#hahn-banach-one-step-extension-bound","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"fully_proved","summary":"Let $E$ be a seminormed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $M$ be a subspace of $E$, let $C \\ge 0$ be a real number, and let $f \\colon M \\to \\mathbb{K}$ be a continuous linear functional with $\\|f(m)\\| \\le C\\|m\\|$ for all $m \\in M$. If $x_0 \\in E$ does not lie in $M$, then $M$ is strictly contained in $M' = M + \\mathbb{K} x_0$, and there is a continuous linear functional\u2026","title":"One-step Hahn\u2013Banach extension"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/norm-mk-ql-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.DualDistanceFormula.norm_mkQL_apply","parent":17229,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html#norm-mk-ql-apply","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a linear subspace of $E$, and give $E/S$ the quotient seminorm. Then for every $x \\in E$ the class $q(x)$ of $x$ under the quotient map $q \\colon E \\to E/S$ satisfies $\\|q(x)\\| = d(x, S)$, where $d(x, S) = \\inf_{s \\in S} \\|x - s\\|$ is the distance from $x$ to $S$.","title":"Quotient seminorm equals distance to the subspace"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/norm-mk-ql-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.DualDistanceFormula.norm_mkQL_le","parent":17229,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html#norm-mk-ql-le","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a linear subspace of $E$, and give $E/S$ the quotient seminorm. Then the quotient map $q \\colon E \\to E/S$, a continuous linear map, has operator norm $\\|q\\| \\le 1$.","title":"The quotient map has norm at most one"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/exists-annihilating-apply-eq-inf-dist","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.DualDistanceFormula.exists_annihilating_apply_eq_infDist","parent":17229,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html#exists-annihilating-apply-eq-inf-dist","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a linear subspace of $E$, not necessarily closed, and let $x \\in E$. Then there is a continuous linear functional $f \\colon E \\to \\mathbb{K}$ with $\\|f\\| \\le 1$ that vanishes on $S$ and satisfies $f(x) = d(x, S)$, where $d(x, S) = \\inf_{s \\in S} \\|x - s\\|$ is the distance from $x$ to $S$.","title":"Hahn\u2013Banach distance formula, attainment form"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/exists-annihilating-norm-one","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.DualDistanceFormula.exists_annihilating_norm_one","parent":17229,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html#exists-annihilating-norm-one","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a linear subspace of $E$, and let $x \\in E$ have positive distance $d(x, S) = \\inf_{s \\in S} \\|x - s\\| > 0$ from $S$. Then there is a continuous linear functional $f \\colon E \\to \\mathbb{K}$ with $\\|f\\| = 1$ that vanishes on $S$ and satisfies $f(x) = d(x, S)$.","title":"Norm-one annihilating functional at positive distance"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/exists-annihilating-of-not-mem","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.DualDistanceFormula.exists_annihilating_of_notMem","parent":17229,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html#exists-annihilating-of-not-mem","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a closed linear subspace of $E$ and let $x \\in E$ with $x \\notin S$. Then there is a continuous linear functional $f \\colon E \\to \\mathbb{K}$ with $\\|f\\| = 1$ that vanishes on $S$ and satisfies $f(x) = d(x, S)$, where $d(x, S) = \\inf_{s \\in S} \\|x - s\\|$ is the distance from $x$ to $S$.","title":"Norm-one annihilating functional outside a closed subspace"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/inf-dist-eq-s-sup-norm-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.DualDistanceFormula.infDist_eq_sSup_norm_apply","parent":17229,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html#inf-dist-eq-s-sup-norm-apply","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a linear subspace of $E$, not necessarily closed, and let $x \\in E$. Then the distance $d(x, S) = \\inf_{s \\in S} \\|x - s\\|$ from $x$ to $S$ equals the supremum of $|f(x)|$ over all continuous linear functionals $f \\colon E \\to \\mathbb{K}$ with $\\|f\\| \\le 1$ that vanish on $S$.","title":"Hahn\u2013Banach distance formula"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/convex-bodies-separated-by-hyperplane","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook64Wanted.convexBodiesSeparatedByHyperplane","parent":17229,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem/index.html#convex-bodies-separated-by-hyperplane","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hahn-banach-theorem","status":"can_prove","summary":"Let $E$ be a real Banach space and let $A, B \\subseteq E$ be convex bodies, that is, convex, closed and bounded sets with nonempty interior. If there is $\\varepsilon > 0$ with $\\operatorname{dist}(a, b) \\ge \\varepsilon$ for all $a \\in A$ and $b \\in B$, then there are a nonzero continuous linear functional $F : E \\to \\mathbb{R}$ and a constant $c \\in \\mathbb{R}$ with $F(a) \\le c$ for all $a \\in A$ and $c \\le F(b)$\u2026","title":"Separation of convex bodies at positive distance"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","title":"Annihilators and the bipolar theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator","kind":"node","lean":"Submodule.topologicalAnnihilator","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with continuous dual $E^*$. The topological annihilator of a subspace $S$ of $E$ is the subspace of $E^*$ consisting of the continuous linear functionals $f$ with $f(x) = 0$ for all $x \\in S$.","title":"Topological annihilator"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-coannihilator","kind":"node","lean":"Submodule.topologicalCoannihilator","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-coannihilator","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with continuous dual $E^*$. The topological coannihilator of a subspace $\\Phi$ of $E^*$ is the subspace of $E$ consisting of the vectors $x$ with $f(x) = 0$ for all $f \\in \\Phi$.","title":"Topological coannihilator"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/mem-topological-annihilator","kind":"node","lean":"Submodule.mem_topologicalAnnihilator","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#mem-topological-annihilator","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a subspace of $E$, and let $f : E \\to \\mathbb{K}$ be a continuous linear functional. Then $f$ lies in the topological annihilator of $S$ if and only if $f(x) = 0$ for every $x \\in S$.","title":"Membership in the topological annihilator"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/mem-topological-coannihilator","kind":"node","lean":"Submodule.mem_topologicalCoannihilator","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#mem-topological-coannihilator","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $\\Phi$ be a subspace of the continuous dual of $E$, and let $x \\in E$. Then $x$ lies in the topological coannihilator of $\\Phi$ if and only if $f(x) = 0$ for every $f \\in \\Phi$.","title":"Membership in the topological coannihilator"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator-anti","kind":"node","lean":"Submodule.topologicalAnnihilator_anti","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator-anti","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $S \\subseteq T$ be subspaces of $E$. Then the topological annihilator of $T$ is contained in the topological annihilator of $S$.","title":"The topological annihilator reverses inclusion"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-coannihilator-anti","kind":"node","lean":"Submodule.topologicalCoannihilator_anti","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-coannihilator-anti","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with continuous dual $E^*$. If $\\Phi \\subseteq \\Psi$ are subspaces of $E^*$, then the topological coannihilator of $\\Psi$ is contained in the topological coannihilator of $\\Phi$.","title":"The topological coannihilator is antitone"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator-bot","kind":"node","lean":"Submodule.topologicalAnnihilator_bot","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator-bot","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$. The topological annihilator of the zero subspace of $E$ is the whole continuous dual of $E$.","title":"Topological annihilator of the zero subspace"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator-top","kind":"node","lean":"Submodule.topologicalAnnihilator_top","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator-top","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with continuous dual $E^*$. The topological annihilator of $E$, viewed as a subspace of itself, is the zero subspace of $E^*$: a continuous linear functional vanishing everywhere on $E$ is zero.","title":"Topological annihilator of the whole space"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-coannihilator-bot","kind":"node","lean":"Submodule.topologicalCoannihilator_bot","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-coannihilator-bot","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with continuous dual $E^*$. The topological coannihilator of the zero subspace of $E^*$ is all of $E$.","title":"Topological coannihilator of the zero subspace"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-coannihilator-top","kind":"node","lean":"Submodule.topologicalCoannihilator_top","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-coannihilator-top","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with continuous dual $E^*$. The topological coannihilator of $E^*$ itself is the zero subspace of $E$: a vector killed by every continuous linear functional is zero.","title":"Topological coannihilator of the whole dual"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/le-topological-coannihilator-topological","kind":"node","lean":"Submodule.le_topologicalCoannihilator_topologicalAnnihilator","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#le-topological-coannihilator-topological","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$. Every subspace $S$ of $E$ is contained in the topological coannihilator of the topological annihilator of $S$.","title":"A subspace lies in its double annihilator"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator-topological-clos","kind":"node","lean":"Submodule.topologicalAnnihilator_topologicalClosure","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator-topological-clos","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. For every subspace $S$ of $E$, the topological annihilator of the topological closure $\\overline{S}$ equals the topological annihilator of $S$: a continuous linear functional vanishing on $S$ vanishes on $\\overline{S}$.","title":"Topological annihilator of the closure"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/exists-mem-topological-annihilator-apply","kind":"node","lean":"Submodule.exists_mem_topologicalAnnihilator_apply_ne_zero","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#exists-mem-topological-annihilator-apply","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a subspace of $E$, and let $x \\in E$ lie outside the topological closure of $S$. Then some continuous linear functional $f$ in the topological annihilator of $S$ satisfies $f(x) \\neq 0$.","title":"Separation"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/exists-mem-topological-annihilator-apply-2","kind":"node","lean":"Submodule.exists_mem_topologicalAnnihilator_apply_eq_one","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#exists-mem-topological-annihilator-apply-2","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, let $S$ be a subspace of $E$, and let $x \\in E$ lie outside the topological closure of $S$. Then some continuous linear functional $f$ in the topological annihilator of $S$ satisfies $f(x) = 1$.","title":"Separation, normalized"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator-topological-coan","kind":"node","lean":"Submodule.topologicalAnnihilator_topologicalCoannihilator_eq_topologicalClosure","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator-topological-coan","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $S$ be a subspace of $E$. The topological coannihilator of the topological annihilator of $S$, that is, the subspace of vectors of $E$ killed by every continuous linear functional vanishing on $S$, is the topological closure of $S$.","title":"Bipolar theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator-eq-bot-iff-dense","kind":"node","lean":"Submodule.topologicalAnnihilator_eq_bot_iff_dense","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator-eq-bot-iff-dense","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $S$ be a subspace of $E$. The topological annihilator of $S$ is the zero subspace of the continuous dual of $E$ if and only if $S$ is dense in $E$.","title":"Density criterion via the annihilator"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator-eq-iff-topologic","kind":"node","lean":"Submodule.topologicalAnnihilator_eq_iff_topologicalClosure_eq","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator-eq-iff-topologic","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $S$ and $T$ be subspaces of $E$. Then $S$ and $T$ have the same topological annihilator if and only if they have the same topological closure.","title":"The annihilator determines the closure"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-annihilator-sup","kind":"node","lean":"Submodule.topologicalAnnihilator_sup","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-annihilator-sup","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, and let $S$ and $T$ be subspaces of $E$. The topological annihilator of $S + T$ is the intersection of the topological annihilators of $S$ and $T$.","title":"Topological annihilator of a sum of subspaces"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/topological-coannihilator-sup","kind":"node","lean":"Submodule.topologicalCoannihilator_sup","parent":17238,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem/index.html#topological-coannihilator-sup","scope":"Analysis/banach-spaces/duality-and-weak-topologies/annihilators-and-the-bipolar-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with continuous dual $E^*$. For subspaces $\\Phi$ and $\\Psi$ of $E^*$, the topological coannihilator of their supremum $\\Phi + \\Psi$ is the intersection of the topological coannihilators of $\\Phi$ and $\\Psi$.","title":"Topological coannihilator of a sum of subspaces"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces","status":"fully_proved","title":"Hamel bases of Banach spaces"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces/finite-dimensional-of-countable-basis","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CountableBasisFiniteDimensional.finiteDimensional_of_countable_basis","parent":17258,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces/index.html#finite-dimensional-of-countable-basis","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces","status":"fully_proved","summary":"If `E` is a complete normed space over `\ud835\udd5c = \u211d, \u2102` admitting a Hamel basis indexed by a countable type, then `E` is finite-dimensional. Completeness is essential: it is what the Baire category argument uses.","title":"A Banach space with a countable Hamel basis is finite-dimensional"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces/not-countable-basis-of-not-finite-dimens","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.CountableBasisFiniteDimensional.not_countable_basis_of_not_finiteDimensional","parent":17258,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces/index.html#not-countable-basis-of-not-finite-dimens","scope":"Analysis/banach-spaces/duality-and-weak-topologies/hamel-bases-of-banach-spaces","status":"fully_proved","summary":"Let $E$ be a complete normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. If $E$ is not finite-dimensional, then no Hamel basis of $E$ is indexed by a countable set.","title":"An infinite-dimensional Banach space has no countable Hamel basis"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem","status":"fully_proved","title":"James' reflexivity theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem/goldstine-finite","kind":"node","lean":"NormedSpace.goldstine_finite","parent":17261,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem/index.html#goldstine-finite","scope":"Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem","status":"fully_proved","summary":"Let $X$ be a real normed space, write $X^*$ for its dual space of continuous linear functionals and $X^{**}$ for the dual of $X^*$, and let $T \\in X^{**}$ with $\\|T\\| \\le 1$. Then for every finite set $s \\subseteq X^*$ and every real $\\varepsilon > 0$ there is $x \\in X$ with $\\|x\\| \\le 1$ such that $|z(x) - T(z)| < \\varepsilon$ for every $z \\in s$.","title":"Finite-coordinate Goldstine theorem"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem/james-reflexivity","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BanachSpaceWanted.james_reflexivity","parent":17261,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem/index.html#james-reflexivity","scope":"Analysis/banach-spaces/duality-and-weak-topologies/james-reflexivity-theorem","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $E$ be a Banach space over $\\mathbb{K}$. The canonical embedding of $E$ into its bidual $E^{**}$ is surjective if and only if every continuous linear functional $f \\colon E \\to \\mathbb{K}$ attains its norm on the closed unit ball, that is, there is $x \\in E$ with $\\|x\\| \\le 1$ and $|f(x)| = \\|f\\|$.","title":"James' reflexivity theorem"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma","status":"fully_proved","title":"Mazur's lemma"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/mem-closure-convex-hull-of-tendsto-weak","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.MazurLemma.mem_closure_convexHull_of_tendsto_weakSpace","parent":17264,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/index.html#mem-closure-convex-hull-of-tendsto-weak","scope":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, regarded also as a real vector space by restricting scalars. If a sequence $(u_n)_{n \\in \\mathbb{N}}$ in $E$ converges to $x \\in E$ in the weak topology of $E$ (the topology induced by the continuous $\\mathbb{K}$-linear functionals on $E$), then for every $N \\in \\mathbb{N}$ the point $x$ lies in the norm closure of the\u2026","title":"Mazur's lemma, hull form"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/mem-closure-convex-hull-range-of-tendsto","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.MazurLemma.mem_closure_convexHull_range_of_tendsto_weakSpace","parent":17264,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/index.html#mem-closure-convex-hull-range-of-tendsto","scope":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, regarded also as a real vector space by restricting scalars. If a sequence $(u_n)_{n \\in \\mathbb{N}}$ in $E$ converges to $x \\in E$ in the weak topology of $E$ (the topology induced by the continuous $\\mathbb{K}$-linear functionals on $E$), then $x$ lies in the norm closure of the real convex hull of the set\u2026","title":"Mazur's lemma, hull form for the whole sequence"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/exists-finite-convex-combination-norm-su","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.MazurLemma.exists_finite_convex_combination_norm_sub_lt_of_tendsto_weakSpace","parent":17264,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/index.html#exists-finite-convex-combination-norm-su","scope":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, regarded also as a real vector space by restricting scalars. If a sequence $(u_n)_{n \\in \\mathbb{N}}$ in $E$ converges to $x \\in E$ in the weak topology of $E$ (the topology induced by the continuous $\\mathbb{K}$-linear functionals on $E$), then for every $N \\in \\mathbb{N}$ and every real $\\varepsilon > 0$ there are a\u2026","title":"Mazur's lemma, explicit finite convex combinations"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/exists-finite-convex-combination-norm-su-2","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.MazurLemma.exists_finite_convex_combination_norm_sub_lt_of_forall_tendsto","parent":17264,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma/index.html#exists-finite-convex-combination-norm-su-2","scope":"Analysis/banach-spaces/duality-and-weak-topologies/mazur-s-lemma","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K}$, where $\\mathbb{K}$ is $\\mathbb{R}$ or $\\mathbb{C}$, regarded also as a real vector space by restricting scalars. Let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and $x \\in E$ such that $f(u_n) \\to f(x)$ for every continuous $\\mathbb{K}$-linear functional $f \\colon E \\to \\mathbb{K}$. Then for every $N \\in \\mathbb{N}$ and every real $\\varepsilon > 0$ there are a finite\u2026","title":"Mazur's lemma in the textbook phrasing"},{"data_url":"scopes/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence.json","id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","kind":"scope","navigation":"module","parent":17194,"result":false,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","title":"Weak topology and weak convergence"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/bdd-above-norm-range-of-forall-tendsto","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakConvergence.bddAbove_norm_range_of_forall_tendsto","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#bdd-above-norm-range-of-forall-tendsto","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and let $x \\in E$. If $f(u_n) \\to f(x)$ as $n \\to \\infty$ for every continuous linear functional $f \\colon E \\to \\mathbb{K}$, then the norms are bounded above: there is a real number $C$ with $\\|u_n\\| \\le C$ for all $n$.","title":"Weakly convergent sequences are norm-bounded"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/norm-le-liminf-norm-of-forall-tendsto","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakConvergence.norm_le_liminf_norm_of_forall_tendsto","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#norm-le-liminf-norm-of-forall-tendsto","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and let $x \\in E$. If $f(u_n) \\to f(x)$ as $n \\to \\infty$ for every continuous linear functional $f \\colon E \\to \\mathbb{K}$, that is, $u_n$ converges weakly to $x$, then $\\|x\\| \\le \\liminf_{n \\to \\infty} \\|u_n\\|$.","title":"Weak lower semicontinuity of the norm"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/tendsto-apply-of-forall-tendsto","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakConvergence.tendsto_apply_of_forall_tendsto","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#tendsto-apply-of-forall-tendsto","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Let $E$ and $F$ be normed spaces over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T \\colon E \\to F$ be a continuous linear map, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and let $x \\in E$. If $f(u_n) \\to f(x)$ for every continuous linear functional $f \\colon E \\to \\mathbb{K}$, then $g(T u_n) \\to g(T x)$ for every continuous linear functional $g \\colon F \\to \\mathbb{K}$.","title":"Bounded operators preserve weak convergence"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/forall-tendsto-of-tendsto-weak-space","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakConvergence.forall_tendsto_of_tendsto_weakSpace","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#forall-tendsto-of-tendsto-weak-space","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and let $x \\in E$. If $u_n \\to x$ in the weak topology of $E$ (the topology induced by all continuous linear functionals $E \\to \\mathbb{K}$), then $f(u_n) \\to f(x)$ as $n \\to \\infty$ for every continuous linear functional $f \\colon E \\to \\mathbb{K}$.","title":"Weak convergence gives convergence under every functional"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/tendsto-weak-space-of-forall-tendsto","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakConvergence.tendsto_weakSpace_of_forall_tendsto","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#tendsto-weak-space-of-forall-tendsto","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and let $x \\in E$. If $f(u_n) \\to f(x)$ as $n \\to \\infty$ for every continuous linear functional $f \\colon E \\to \\mathbb{K}$, then $u_n \\to x$ in the weak topology of $E$ (the topology induced by all continuous linear functionals $E \\to \\mathbb{K}$).","title":"Convergence under every functional gives weak convergence"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/bdd-above-norm-range-of-tendsto-weak-spa","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakConvergence.bddAbove_norm_range_of_tendsto_weakSpace","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#bdd-above-norm-range-of-tendsto-weak-spa","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and let $x \\in E$. If $u_n \\to x$ in the weak topology of $E$ (the topology induced by all continuous linear functionals $E \\to \\mathbb{K}$), then there is a real number $C$ with $\\|u_n\\| \\le C$ for all $n$.","title":"Weakly convergent sequences are norm-bounded (weak topology form)"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/norm-le-liminf-norm-of-tendsto-weak-spac","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakConvergence.norm_le_liminf_norm_of_tendsto_weakSpace","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#norm-le-liminf-norm-of-tendsto-weak-spac","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and let $x \\in E$. If $u_n \\to x$ in the weak topology of $E$ (the topology induced by all continuous linear functionals $E \\to \\mathbb{K}$), then $\\|x\\| \\le \\liminf_{n \\to \\infty} \\|u_n\\|$.","title":"Weak lower semicontinuity of the norm (weak topology form)"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/tendsto-weak-space-apply-of-tendsto-weak","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakConvergence.tendsto_weakSpace_apply_of_tendsto_weakSpace","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#tendsto-weak-space-apply-of-tendsto-weak","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Let $E$ and $F$ be normed spaces over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $T \\colon E \\to F$ be a continuous linear map, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $E$ and let $x \\in E$. If $u_n \\to x$ in the weak topology of $E$, then $T u_n \\to T x$ in the weak topology of $F$; here the weak topology of a space is the one induced by its continuous linear functionals.","title":"Bounded operators preserve weak convergence (weak topology form)"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/exists-ne-zero-forall-dual-apply-eq-zero","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.exists_ne_zero_forall_dual_apply_eq_zero","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#exists-ne-zero-forall-dual-apply-eq-zero","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Finitely many continuous linear functionals on an infinite-dimensional normed space have a common nonzero kernel vector.","title":"Finitely many functionals have a common nonzero kernel vector"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/exists-line-subset-of-is-open","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.exists_line_subset_of_isOpen","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#exists-line-subset-of-is-open","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Every nonempty weakly open set contains a whole line through each of its points: if `U` is weakly open and `x \u2208 U`, then `x + t \u2022 v \u2208 U` for all scalars `t` and some nonzero `v` (a common kernel vector of the finitely many functionals constraining a basic weak neighbourhood of `x`).","title":"Weakly open sets contain lines through each point"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/exists-norm-gt-of-is-open","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.exists_norm_gt_of_isOpen","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#exists-norm-gt-of-is-open","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"Every nonempty weakly open set in an infinite-dimensional normed space is unbounded in norm.","title":"Weakly open sets are unbounded in infinite dimensions"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/not-is-open-image-ball","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.not_isOpen_image_ball","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#not-is-open-image-ball","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"The open unit ball is not weakly open in infinite dimensions.","title":"Open unit ball is not weakly open in infinite dimensions"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/not-is-open-image-sphere","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.not_isOpen_image_sphere","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#not-is-open-image-sphere","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"The unit sphere is not weakly open in infinite dimensions.","title":"Unit sphere is not weakly open in infinite dimensions"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/closure-image-sphere-eq-image-closed-bal","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.closure_image_sphere_eq_image_closedBall","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#closure-image-sphere-eq-image-closed-bal","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"In an infinite-dimensional normed space, the weak closure of the unit sphere is the closed unit ball.","title":"Weak closure of the unit sphere is the closed unit ball"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/not-is-closed-image-sphere","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.not_isClosed_image_sphere","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#not-is-closed-image-sphere","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"The unit sphere is not weakly closed in infinite dimensions: its weak closure is the whole closed unit ball.","title":"Unit sphere is not weakly closed in infinite dimensions"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/not-is-compact-image-sphere","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.not_isCompact_image_sphere","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#not-is-compact-image-sphere","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"The unit sphere is not weakly compact in infinite dimensions.","title":"Unit sphere is not weakly compact in infinite dimensions"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/is-open-map-to-weak-space-clm-of-finite","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.isOpenMap_toWeakSpaceCLM_of_finiteDimensional","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#is-open-map-to-weak-space-clm-of-finite","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"In finite dimensions the identity map from the norm topology to the weak topology is an open map: the weak topology agrees with the norm topology.","title":"Weak topology equals norm topology in finite dimensions"},{"id":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/not-is-open-map-to-weak-space-clm-of-inf","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.WeakTopologyInfiniteDimensional.not_isOpenMap_toWeakSpaceCLM_of_infiniteDimensional","parent":17269,"result":true,"route":"../roadmap/Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence/index.html#not-is-open-map-to-weak-space-clm-of-inf","scope":"Analysis/banach-spaces/duality-and-weak-topologies/weak-topology-and-weak-convergence","status":"fully_proved","summary":"In infinite dimensions the identity map from the norm topology to the weak topology is not an open map: the open unit ball is norm-open but not weakly open.","title":"Weak topology is strictly coarser in infinite dimensions"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces","kind":"scope","parent":17193,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"can_prove","summary":"Convexity and smoothness of norms (uniform convexity, Kadec\u2013Klee, Birkhoff\u2013James), extreme points of balls, Auerbach and Schauder bases, approximation property, isometries (Banach\u2013Stone, Mazur\u2013Ulam), Banach\u2013Mazur, Dvoretzky\u2013Rogers, Grothendieck, Rademacher type, Daugavet, Bishop\u2013Phelps.","title":"Geometry of Banach spaces"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/banach-mazur-embedding","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.GeometricWanted.banachMazur_embedding","parent":17288,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html#banach-mazur-embedding","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"fully_proved","summary":"Every separable real Banach space $E$ admits a linear isometric embedding $e \\colon E \\to C([0, 1], \\mathbb{R})$ into the space of continuous real functions on $[0, 1]$ with the supremum norm, such that the range of $e$ is closed.","title":"Banach\u2013Mazur embedding theorem"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/banach-stone-real","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BanachStoneWanted.banachStone_real","parent":17288,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html#banach-stone-real","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"fully_proved","summary":"Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X, \\mathbb{R})$ and $C(Y, \\mathbb{R})$ be the real Banach spaces of continuous real functions on them with the supremum norm. If there is a surjective linear isometry from $C(X, \\mathbb{R})$ onto $C(Y, \\mathbb{R})$, then $X$ and $Y$ are homeomorphic.","title":"Banach\u2013Stone theorem for real scalars"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/bishop-phelps","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BanachSpaceWanted.bishop_phelps","parent":17288,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html#bishop-phelps","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"fully_proved","summary":"Let $E$ be a Banach space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with dual space $E^*$. The functionals that attain their norm on the closed unit ball, that is, the $f \\in E^*$ such that $|f(x)| = \\|f\\|$ for some $x \\in E$ with $\\|x\\| \\le 1$, form a dense subset of $E^*$ in the norm topology.","title":"Bishop\u2013Phelps theorem"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/daugavet","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.DaugavetWanted.daugavet","parent":17288,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html#daugavet","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"fully_proved","summary":"Let $C[0,1]$ be the real Banach space of continuous functions $[0,1] \\to \\mathbb{R}$ with the supremum norm, and let $I$ be the identity operator on it. Every compact bounded linear operator $T : C[0,1] \\to C[0,1]$ satisfies $\\|I + T\\| = 1 + \\|T\\|$, where the norms are operator norms.","title":"Daugavet's theorem for compact operators"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/dvoretzky-rogers-unconditional-not-absol","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook122Wanted.dvoretzky_rogers_unconditional_not_absolute","parent":17288,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html#dvoretzky-rogers-unconditional-not-absol","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"can_prove","summary":"Every infinite-dimensional real Banach space $E$ contains a series that converges unconditionally but not absolutely: there is a sequence $(x_n)_{n \\ge 0}$ in $E$ such that the rearranged series $\\sum_n x_{\\sigma(n)}$ converges for every permutation $\\sigma$ of $\\mathbb{N}$, while $\\sum_n \\lVert x_n \\rVert$ diverges.","title":"Dvoretzky\u2013Rogers theorem"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/grothendieck-inequality-real","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.GrothendieckInequalityWanted.grothendieck_inequality_real","parent":17288,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html#grothendieck-inequality-real","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"fully_proved","summary":"There is a constant $K > 0$ such that for all $m, n \\in \\mathbb{N}$, every real $m \\times n$ matrix $A = (a_{ij})$, every real Hilbert space $H$ and all vectors $x_1, \\dots, x_m, y_1, \\dots, y_n \\in H$ of norm at most $1$, there are signs $\\varepsilon_1, \\dots, \\varepsilon_m, \\delta_1, \\dots, \\delta_n \\in \\{1, -1\\}$ such that\u2026","title":"Grothendieck's inequality (real case)"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/milman-pettis","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.MilmanPettisWanted.milman_pettis","parent":17288,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html#milman-pettis","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"fully_proved","summary":"Let $E$ be a uniformly convex Banach space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, and let $E^*$ be the space of continuous linear functionals on $E$. Then every continuous linear functional $\\Phi$ on $E^*$ is evaluation at a point: there is $x \\in E$ with $\\Phi(f) = f(x)$ for every $f \\in E^*$.","title":"Milman\u2013Pettis theorem"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/banach-saks","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BanachSaksWanted.banach_saks","parent":17288,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/index.html#banach-saks","scope":"Analysis/banach-spaces/geometry-of-banach-spaces","status":"fully_proved","summary":"Let $E$ be a uniformly convex real Banach space and let $(x_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $E$. Then there are a strictly increasing map $\\varphi : \\mathbb{N} \\to \\mathbb{N}$ and a point $y \\in E$ such that the Ces\u00e0ro means $\\frac{1}{n} \\sum_{i=0}^{n-1} x_{\\varphi(i)}$ converge to $y$ in norm as $n \\to \\infty$.","title":"Banach\u2013Saks theorem for uniformly convex spaces"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"can_prove","title":"Approximation property"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/has-approximation-property","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ApproximationProperty.HasApproximationProperty","parent":17297,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html#has-approximation-property","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"fully_proved","summary":"A normed space has the approximation property if, on every compact set, the identity operator is a uniform limit of finite-rank continuous linear operators.","title":"Approximation property"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/has-approximation-property-of-finite-dim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ApproximationProperty.hasApproximationProperty_of_finiteDimensional","parent":17297,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html#has-approximation-property-of-finite-dim","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"fully_proved","summary":"Finite-dimensional spaces have the approximation property: the identity operator is itself finite-rank.","title":"Finite-dimensional spaces have the approximation property"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/has-approximation-property-of-schauder-b","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ApproximationProperty.hasApproximationProperty_of_schauderBasis","parent":17297,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html#has-approximation-property-of-schauder-b","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"fully_proved","summary":"A Banach space with a Schauder basis has the approximation property. The partial-sum projections are finite-rank, converge pointwise to the identity, and are uniformly bounded by the Banach\u2013Steinhaus theorem (`SchauderBasis.exists_norm_proj_le`); pointwise convergence plus uniform boundedness gives uniform convergence on compact sets via a finite `\u03b5`-net argument.","title":"Banach spaces with a Schauder basis have the approximation property"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/of-continuous-linear-equiv","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ApproximationProperty.HasApproximationProperty.of_continuousLinearEquiv","parent":17297,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html#of-continuous-linear-equiv","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"fully_proved","summary":"The approximation property is preserved under continuous linear equivalence: conjugate a finite-rank approximant by the equivalence.","title":"Approximation property is invariant under linear isomorphism"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/of-complemented","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ApproximationProperty.HasApproximationProperty.of_complemented","parent":17297,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html#of-complemented","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"fully_proved","summary":"The approximation property passes to complemented subspaces: if `P` is a continuous projection onto the submodule `F`, restrict a finite-rank approximant on the ambient space via `P`. The key point is that `P` fixes every point of `F`, so `P (S x) - x = P (S x - x)` for `x \u2208 F`.","title":"Approximation property passes to complemented subspaces"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/has-approximation-property-of-hilbert-ba","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ApproximationProperty.hasApproximationProperty_of_hilbertBasis","parent":17297,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html#has-approximation-property-of-hilbert-ba","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"fully_proved","summary":"A Hilbert space with a Hilbert basis indexed by `\u2115` has the approximation property, via the associated Schauder basis (`HilbertBasis.toSchauderBasis`).","title":"Separable Hilbert spaces have the approximation property"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/scottish-book-problem153-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook153Wanted.ScottishBookProblem153Claim","parent":17297,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html#scottish-book-problem153-claim","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"fully_proved","summary":"The claim of Mazur's Problem 153 from the Scottish Book, in its original form: for every continuous function $f \\colon [0,1]^2 \\to \\mathbb{R}$ and every real $\\varepsilon > 0$ there are a natural number $n$, points $a_1, \\dots, a_n, b_1, \\dots, b_n \\in [0,1]$ and real numbers $c_1, \\dots, c_n$ such that","title":"Mazur's Problem 153"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/mazur153-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook153Wanted.mazur153_refuted","parent":17297,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property/index.html#mazur153-refuted","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/approximation-property","status":"can_prove","summary":"There are a continuous function $f \\colon [0,1]^2 \\to \\mathbb{R}$ and a real number $\\varepsilon > 0$ such that for every natural number $n$, all points $a_1, \\dots, a_n, b_1, \\dots, b_n \\in [0,1]$ and all real numbers $c_1, \\dots, c_n$ there are $x, y \\in [0,1]$ with","title":"Negative answer to Mazur's Problem 153"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases","status":"fully_proved","title":"Auerbach bases"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/coord-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AuerbachBasis.coordCLM","parent":17306,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/index.html#coord-clm","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases","status":"fully_proved","summary":"The `i`-th coordinate functional of a basis, as a continuous linear map. In finite dimensions every linear map is continuous, so this is just the coordinate functional bundled with its continuity.","title":"Coordinate functional of a basis as a continuous linear map"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/norm-coord-clm-le-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AuerbachBasis.norm_coordCLM_le_one","parent":17306,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/index.html#norm-coord-clm-le-one","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases","status":"fully_proved","summary":"If the `i`-th coordinate functional is bounded by the norm, its operator norm is at most `1`.","title":"Operator norm of a norm-bounded coordinate functional is at most 1"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/norm-coord-clm-eq-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AuerbachBasis.norm_coordCLM_eq_one","parent":17306,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/index.html#norm-coord-clm-eq-one","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases","status":"fully_proved","summary":"If the `i`-th basis vector has norm `1` and the `i`-th coordinate functional is bounded by the norm, its operator norm is exactly `1`: it attains `1` at the `i`-th basis vector.","title":"Coordinate functional of a unit basis vector has operator norm 1"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/is-auerbach-basis","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AuerbachBasis.IsAuerbachBasis","parent":17306,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/index.html#is-auerbach-basis","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases","status":"fully_proved","summary":"An *Auerbach basis* is a basis of unit vectors whose coordinate functionals all have operator norm `1`.","title":"Auerbach basis"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/exists-auerbach-basis","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AuerbachBasis.exists_auerbach_basis","parent":17306,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases/index.html#exists-auerbach-basis","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/auerbach-bases","status":"fully_proved","summary":"Every finite-dimensional normed space over `\ud835\udd5c` has an Auerbach basis, indexed by `Fin (Module.finrank \ud835\udd5c E)`.","title":"Auerbach's theorem"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","title":"Birkhoff\u2013James orthogonality"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.BirkhoffJamesOrthogonal","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"`x` is Birkhoff\u2013James orthogonal to `y` if `x` is a shortest vector on the affine line `x + \ud835\udd5c \u2022 y`.","title":"Birkhoff\u2013James orthogonality"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal-zero-left","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.birkhoffJamesOrthogonal_zero_left","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal-zero-left","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"`0` is Birkhoff\u2013James orthogonal to every vector.","title":"Zero is Birkhoff\u2013James orthogonal to every vector"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal-zero-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.birkhoffJamesOrthogonal_zero_right","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal-zero-right","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"Every vector is Birkhoff\u2013James orthogonal to `0`.","title":"Every vector is Birkhoff\u2013James orthogonal to zero"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal-self-iff","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.birkhoffJamesOrthogonal_self_iff","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal-self-iff","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"A vector is Birkhoff\u2013James orthogonal to itself if and only if it is `0`.","title":"Only zero is Birkhoff\u2013James orthogonal to itself"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal-smul-left-righ","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.birkhoffJamesOrthogonal_smul_left_right","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal-smul-left-righ","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"Birkhoff\u2013James orthogonality is homogeneous in both arguments.","title":"Homogeneity of Birkhoff\u2013James orthogonality"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal-of-exists-norm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.birkhoffJamesOrthogonal_of_exists_norming","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal-of-exists-norm","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"If a norming functional of `x` vanishes at `y`, then `x` is Birkhoff\u2013James orthogonal to `y`.","title":"Annihilating norming functional implies Birkhoff\u2013James orthogonality"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/exists-norming-of-birkhoff-james-orthogo","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.exists_norming_of_birkhoffJamesOrthogonal","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#exists-norming-of-birkhoff-james-orthogo","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"The converse: if `x \u2260 0` is Birkhoff\u2013James orthogonal to `y`, there is a norming functional of `x` vanishing at `y`. Birkhoff\u2013James orthogonality makes the distance from `x` to `\ud835\udd5c \u2219 y` equal to `\u2016x\u2016`, so the dual distance formula supplies an annihilating norming functional.","title":"Birkhoff\u2013James orthogonality gives an annihilating norming functional"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal-iff-exists-nor","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.birkhoffJamesOrthogonal_iff_exists_norming","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal-iff-exists-nor","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"For `x \u2260 0`, Birkhoff\u2013James orthogonality to `y` is equivalent to the existence of a norming functional of `x` vanishing at `y`.","title":"Norming-functional characterization of Birkhoff\u2013James orthogonality"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/duality-mapping","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.dualityMapping","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#duality-mapping","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"The duality mapping (duality set) at `x`: the functionals `f` with `f x = \u2016x\u2016 ^ 2` and `\u2016f\u2016 = \u2016x\u2016`. At `x = 0` this is the singleton `{0}`.","title":"Duality mapping of a normed space"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/duality-mapping-nonempty","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.dualityMapping_nonempty","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#duality-mapping-nonempty","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"The duality mapping is nonempty at every point: take a Hahn\u2013Banach norming functional and scale it by `\u2016x\u2016`.","title":"Duality mapping is nonempty"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal-iff-exists-dua","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.birkhoffJamesOrthogonal_iff_exists_dualityMapping","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal-iff-exists-dua","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"Birkhoff\u2013James orthogonality to `y` is equivalent to the existence of a functional in the duality mapping at `x` that vanishes at `y`.","title":"Duality-mapping characterization of Birkhoff\u2013James orthogonality"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/birkhoff-james-orthogonal-iff-inner-eq-z","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BirkhoffJames.birkhoffJamesOrthogonal_iff_inner_eq_zero","parent":17312,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality/index.html#birkhoff-james-orthogonal-iff-inner-eq-z","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/birkhoff-james-orthogonality","status":"fully_proved","summary":"In an inner product space, Birkhoff\u2013James orthogonality is ordinary inner-product orthogonality.","title":"Birkhoff\u2013James orthogonality in inner product spaces"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls","status":"fully_proved","title":"Extreme points of unit balls"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/extreme-points-closed-ball-continuous-ma","kind":"node","lean":"MathlibExt.Analysis.Convex.ExtremePointsContinuousMapBall.extremePoints_closedBall_continuousMap","parent":17325,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/index.html#extreme-points-closed-ball-continuous-ma","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls","status":"fully_proved","summary":"The extreme points of the closed unit ball of `C(X, \u211d)`, for `X` compact Hausdorff, are exactly the functions with `|f x| = 1` for every `x`.","title":"Extreme points of the unit ball of continuous real functions"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/mem-extreme-points-closed-ball-continuou","kind":"node","lean":"MathlibExt.Analysis.Convex.ExtremePointsContinuousMapBall.mem_extremePoints_closedBall_continuousMap","parent":17325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership criterion for extreme points of the unit ball of "],["math","\\(C(X)\\)"]],"source":"Membership criterion for extreme points of the unit ball of $C(X)$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/index.html#mem-extreme-points-closed-ball-continuou","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls","status":"fully_proved","summary":"Membership form of `extremePoints_closedBall_continuousMap`.","title":"Membership criterion for extreme points of the unit ball of $C(X)$"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/extreme-points-closed-ball-pi-lp-top","kind":"node","lean":"MathlibExt.Analysis.Convex.ExtremePointsLpBalls.extremePoints_closedBall_piLp_top","parent":17325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extreme points of the unit ball of finite "],["math","\\(\\ell^\\infty\\)"]],"source":"Extreme points of the unit ball of finite $\\ell^\\infty$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/index.html#extreme-points-closed-ball-pi-lp-top","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls","status":"fully_proved","summary":"The extreme points of the closed unit ball of `PiLp \u221e (fun _ : \u03b9 => \ud835\udd5c)` are exactly the vectors all of whose coordinates have norm `1`. If some coordinate has norm `< 1`, a small perturbation of that coordinate stays in the ball and exhibits the point as a midpoint; conversely, a vector with all coordinates of norm `1` that is a convex combination of two ball points must agree with both of them coordinatewise, by\u2026","title":"Extreme points of the unit ball of finite $\\ell^\\infty$"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/extreme-points-closed-ball-pi-lp-one","kind":"node","lean":"MathlibExt.Analysis.Convex.ExtremePointsLpBalls.extremePoints_closedBall_piLp_one","parent":17325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extreme points of the unit ball of finite "],["math","\\(\\ell^1\\)"]],"source":"Extreme points of the unit ball of finite $\\ell^1$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls/index.html#extreme-points-closed-ball-pi-lp-one","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/extreme-points-of-unit-balls","status":"fully_proved","summary":"The extreme points of the closed unit ball of `PiLp 1 (fun _ : \u03b9 => \ud835\udd5c)` are exactly the unit multiples of the coordinate vectors. The proof: an extreme point must have norm `1` (otherwise a one-coordinate perturbation stays in the ball and exhibits it as a midpoint), and it cannot have two nonzero coordinates (otherwise mass can be transferred between them, in the direction of the phases of the two coordinates,\u2026","title":"Extreme points of the unit ball of finite $\\ell^1$"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv","status":"can_prove","title":"Injections of a unit sphere into the unit interval"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv/scottish-book-problem144-claim","kind":"node","lean":"MathlibExt.Analysis.ScottishBook144Wanted.ScottishBookProblem144Claim","parent":17330,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv/index.html#scottish-book-problem144-claim","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv","status":"fully_proved","summary":"For every separable real Banach space $E$, with unit sphere $K = \\{x \\in E : \\|x\\| = 1\\}$, there is an injective map $f \\colon K \\to [0, 1]$ such that for every nonempty open subset $U$ of $K$ the image $f(U)$ has positive Lebesgue outer measure.","title":"Scottish Book Problem 144 (Mazur\u2013Ulam), as a claim"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv/mazur-ulam144-refuted","kind":"node","lean":"MathlibExt.Analysis.ScottishBook144Wanted.mazurUlam144_refuted","parent":17330,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv/index.html#mazur-ulam144-refuted","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/injections-of-a-unit-sphere-into-the-unit-interv","status":"can_prove","summary":"The Scottish Book Problem 144 claim fails: some separable real Banach space $E$ admits no injective map $f$ from its unit sphere $K$ to $[0, 1]$ such that the image under $f$ of every nonempty open subset of $K$ has positive Lebesgue outer measure.","title":"Scottish Book Problem 144 is false as stated"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property","status":"fully_proved","title":"Kadec\u2013Klee property"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/tendsto-of-forall-tendsto-and-tendsto-no","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.KadecKlee.tendsto_of_forall_tendsto_and_tendsto_norm","parent":17333,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/index.html#tendsto-of-forall-tendsto-and-tendsto-no","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property","status":"fully_proved","summary":"Let $E$ be a uniformly convex real normed space, let $(u_n)$ be a sequence in $E$ and let $x \\in E$. If $f(u_n) \\to f(x)$ for every continuous linear functional $f \\colon E \\to \\mathbb{R}$ and $\\|u_n\\| \\to \\|x\\|$, then $u_n \\to x$ in norm.","title":"Kadec\u2013Klee property, all-functionals form"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/tendsto-of-tendsto-weak-space-and-tendst","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.KadecKlee.tendsto_of_tendsto_weakSpace_and_tendsto_norm","parent":17333,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/index.html#tendsto-of-tendsto-weak-space-and-tendst","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property","status":"fully_proved","summary":"Let $E$ be a uniformly convex real normed space, let $(u_n)$ be a sequence in $E$ and let $x \\in E$. If $u_n \\to x$ in the weak topology of $E$ (the topology induced by the continuous linear functionals $E \\to \\mathbb{R}$) and $\\|u_n\\| \\to \\|x\\|$, then $u_n \\to x$ in norm.","title":"Kadec\u2013Klee property, weak-topology form"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/tendsto-iff-forall-tendsto-and-tendsto-n","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.KadecKlee.tendsto_iff_forall_tendsto_and_tendsto_norm","parent":17333,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/index.html#tendsto-iff-forall-tendsto-and-tendsto-n","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property","status":"fully_proved","summary":"Let $E$ be a uniformly convex real normed space, let $(u_n)$ be a sequence in $E$ and let $x \\in E$. Then $u_n \\to x$ in norm if and only if $f(u_n) \\to f(x)$ for every continuous linear functional $f \\colon E \\to \\mathbb{R}$ and $\\|u_n\\| \\to \\|x\\|$.","title":"Kadec\u2013Klee iff, all-functionals form"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/tendsto-iff-tendsto-weak-space-and-tends","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.KadecKlee.tendsto_iff_tendsto_weakSpace_and_tendsto_norm","parent":17333,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property/index.html#tendsto-iff-tendsto-weak-space-and-tends","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/kadec-klee-property","status":"fully_proved","summary":"Let $E$ be a uniformly convex real normed space, let $(u_n)$ be a sequence in $E$ and let $x \\in E$. Then $u_n \\to x$ in norm if and only if $u_n \\to x$ in the weak topology of $E$ (the topology induced by the continuous linear functionals $E \\to \\mathbb{R}$) and $\\|u_n\\| \\to \\|x\\|$.","title":"Kadec\u2013Klee iff, weak-topology form"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces","status":"can_prove","title":"Metric characterization of Banach spaces"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/metric-center-sets","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook69Wanted.metricCenterSets","parent":17338,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/index.html#metric-center-sets","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces","status":"fully_proved","summary":"Let $p, q$ be points of a pseudometric space $E$ with distance $d$. The iterated metric center sets $R_n \\subseteq E$ of $(p, q)$, for $n \\in \\mathbb{N}$, are defined by $R_0 = \\{x \\in E : d(p, x) + d(x, q) = d(p, q)\\}$ and $R_{n+1} = \\{r \\in R_n : d(r, x) \\le \\operatorname{diam}(R_n) / 2 \\text{ for all } x \\in R_n\\}$.","title":"Iterated metric center sets"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/is-metric-center","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook69Wanted.IsMetricCenter","parent":17338,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/index.html#is-metric-center","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces","status":"fully_proved","summary":"Let $p, q, c$ be points of a pseudometric space $E$ and let $R_n$, for $n \\in \\mathbb{N}$, be the iterated metric center sets of $(p, q)$. We say that $c$ is the metric center of $(p, q)$ when $c \\in R_n$ for every $n$, and every point $y \\in E$ with $y \\in R_n$ for every $n$ equals $c$.","title":"Metric center of a pair of points"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/scottish-book-problem69-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook69Wanted.ScottishBookProblem69Claim","parent":17338,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/index.html#scottish-book-problem69-claim","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces","status":"fully_proved","summary":"The statement that every complete pseudometric space $E$ with the following two properties is isometric to a real Banach space: (1) for all $p, q \\in E$ there is exactly one point $c \\in E$ that is the metric center of $(p, q)$, and (2) for all $p, q \\in E$ there is exactly one point $x \\in E$ such that $q$ is the metric center of $(p, x)$. Here \"isometric to a real Banach space\" means that there are a complete\u2026","title":"Scottish Book Problem 69 claim"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/mazur-ulam69-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook69Wanted.mazurUlam69_refuted","parent":17338,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces/index.html#mazur-ulam69-refuted","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/metric-characterization-of-banach-spaces","status":"can_prove","summary":"The Scottish Book Problem 69 claim is false: there is a complete pseudometric space $E$ such that (1) for all $p, q \\in E$ there is exactly one point $c \\in E$ that is the metric center of $(p, q)$, and (2) for all $p, q \\in E$ there is exactly one point $x \\in E$ such that $q$ is the metric center of $(p, x)$, but there is no complete real normed space $F$ with an isometric bijection $E \\to F$.","title":"Mazur\u2013Ulam Problem 69 refuted"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals","status":"can_prove","title":"Normed spaces isometric to their duals"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals/scottish-book-problem91-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook91Wanted.ScottishBookProblem91Claim","parent":17343,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals/index.html#scottish-book-problem91-claim","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals","status":"fully_proved","summary":"This is the proposition that every finite-dimensional real normed space $E$ that is linearly isometric to its dual space $E^*$ (the continuous linear functionals on $E$ with the operator norm) is an inner product space, that is, its norm is induced by some inner product on $E$.","title":"Mazur's Problem 91 (Scottish Book), Banach-space form"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals/mazur91-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook91Wanted.mazur91_refuted","parent":17343,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals/index.html#mazur91-refuted","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/normed-spaces-isometric-to-their-duals","status":"can_prove","summary":"There is a finite-dimensional real normed space $E$ that is linearly isometric to its dual space $E^*$ (the continuous linear functionals on $E$ with the operator norm) but whose norm is not induced by any inner product on $E$. Thus the Banach-space form of Mazur's Problem 91 is false.","title":"Mazur's Problem 91 has a negative answer"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"can_prove","title":"Rademacher type and cotype"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/rademacher-sign","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.rademacherSign","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#rademacher-sign","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"The Rademacher sign attached to a coordinate: `1` for `true`, `-1` for `false`.","title":"Rademacher sign of a coordinate"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/rademacher-sign-eq-one-of","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.rademacherSign_eq_one_of","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#rademacher-sign-eq-one-of","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"The sign is `1` on a `true` coordinate.","title":"Rademacher sign on a true coordinate is one"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/rademacher-sign-eq-neg-one-of","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.rademacherSign_eq_neg_one_of","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#rademacher-sign-eq-neg-one-of","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"The sign is `-1` on a `false` coordinate.","title":"Rademacher sign on a false coordinate is minus one"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/norm-rademacher-sign","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.norm_rademacherSign","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#norm-rademacher-sign","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Every Rademacher sign has norm `1`.","title":"Rademacher signs have norm one"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/card-sign-vectors","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.card_sign_vectors","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#card-sign-vectors","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Number of sign vectors: `Fin n \u2192 Bool` has `2 ^ n` elements.","title":"Number of sign vectors of length n"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/sum-rademacher-sq-norm-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.sum_rademacher_sq_norm_eq","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#sum-rademacher-sq-norm-eq","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Exact second moment of a Rademacher sum in a Hilbert space: the sum over all sign vectors of `\u2016\u2211\u1d62 \u03b5\u1d62 \u2022 x\u1d62\u2016\u00b2` equals `2 ^ n * \u2211\u1d62 \u2016x\u1d62\u2016\u00b2`. Cross terms cancel: the induction step pairs the two values of the last sign and applies the parallelogram law.","title":"Second moment of a Rademacher sum in a Hilbert space"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/average-rademacher-sq-norm-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.average_rademacher_sq_norm_eq","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#average-rademacher-sq-norm-eq","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"The average (uniform over sign vectors) of `\u2016\u2211\u1d62 \u03b5\u1d62 \u2022 x\u1d62\u2016\u00b2` is exactly `\u2211\u1d62 \u2016x\u1d62\u2016\u00b2`.","title":"Average squared norm of a Rademacher sum"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/rademacher-type-two","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.rademacher_type_two","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#rademacher-type-two","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Rademacher type 2 with constant `1`: the average of the squared Rademacher sums is at most `\u2211\u1d62 \u2016x\u1d62\u2016\u00b2`.","title":"Hilbert spaces have Rademacher type 2"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/rademacher-cotype-two","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.rademacher_cotype_two","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#rademacher-cotype-two","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Rademacher cotype 2 with constant `1`: `\u2211\u1d62 \u2016x\u1d62\u2016\u00b2` is at most the average of the squared Rademacher sums.","title":"Hilbert spaces have Rademacher cotype 2"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/sqrt-average-rademacher-sq-norm-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.sqrt_average_rademacher_sq_norm_eq","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#sqrt-average-rademacher-sq-norm-eq","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Square-root form of the exact second moment.","title":"Square-root form of the Rademacher second moment"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/average-rademacher-norm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.average_rademacher_norm_le","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#average-rademacher-norm-le","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"The average Rademacher norm is at most `(\u2211\u1d62 \u2016x\u1d62\u2016\u00b2)^{1/2}` (Cauchy\u2013Schwarz for the finite average, plus the exact second moment).","title":"Bound on the average Rademacher norm"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/average-rademacher-sq-norm-eq-of-norm-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.average_rademacher_sq_norm_eq_of_norm_eq_one","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#average-rademacher-sq-norm-eq-of-norm-eq","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"For unit vectors, the average squared Rademacher sum equals `n`.","title":"Average squared Rademacher sum of unit vectors"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/sum-rademacher-sq-real","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RademacherTypeCotype.sum_rademacher_sq_real","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#sum-rademacher-sq-real","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Scalar real form: the second moment of `\u2211\u1d62 \u00b1a\u1d62` over all sign choices is `2 ^ n * \u2211\u1d62 a\u1d62\u00b2`.","title":"Second moment of a real Rademacher sum"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/rademacher-distribution","kind":"node","lean":"MathlibExt.Analysis.RademacherEnfloTypeWanted.RademacherEnfloType.rademacherDistribution","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#rademacher-distribution","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"The Rademacher distribution is the measure on $\\{-1, 1\\}$, viewed as a subset of $\\mathbb{Z}$, that gives mass $1/2$ to each of $1$ and $-1$.","title":"Rademacher distribution"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/rademacher-product-distribution","kind":"node","lean":"MathlibExt.Analysis.RademacherEnfloTypeWanted.RademacherEnfloType.rademacherProductDistribution","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#rademacher-product-distribution","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"For an index type $\\iota$, this is the product over $\\iota$ of copies of the Rademacher distribution: the joint law of a family $(\\varepsilon_i)_{i \\in \\iota}$ of independent Rademacher random variables, as a measure on the functions $\\iota \\to \\{-1, 1\\}$.","title":"Product Rademacher distribution"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/d","kind":"node","lean":"MathlibExt.Analysis.RademacherEnfloTypeWanted.RademacherEnfloType.D","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#d","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Let $X$ be a real Banach space, $n$ a natural number and $f : \\{-1, 1\\}^n \\to X$. For $j \\in \\{1, \\dots, n\\}$ and $\\varepsilon = (\\varepsilon_1, \\dots, \\varepsilon_n) \\in \\{-1, 1\\}^n$, the discrete partial derivative of $f$ in direction $j$ at $\\varepsilon$ is","title":"Discrete partial derivative on the Rademacher cube"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/is-rademacher-type","kind":"node","lean":"MathlibExt.Analysis.RademacherEnfloTypeWanted.RademacherEnfloType.IsRademacherType","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#is-rademacher-type","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Let $X$ be a real Banach space, $C \\in [0, \\infty]$ and $p$ a real number. We say that $X$ has Rademacher type $p$ with constant $C$ if for every $n \\ge 1$ and all $x_1, \\dots, x_n \\in X$,","title":"Rademacher type"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/tr","kind":"node","lean":"MathlibExt.Analysis.RademacherEnfloTypeWanted.RademacherEnfloType.TR","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#tr","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"For a real Banach space $X$ and a real number $p$, $T^R_p(X)$ is the infimum in $[0, \\infty]$ of all $C \\in [0, \\infty]$ such that $X$ has Rademacher type $p$ with constant $C$.","title":"Rademacher type constant"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/is-enflo-type","kind":"node","lean":"MathlibExt.Analysis.RademacherEnfloTypeWanted.RademacherEnfloType.IsEnfloType","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#is-enflo-type","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"Let $X$ be a real Banach space, $C \\in [0, \\infty]$ and $p$ a real number. We say that $X$ has Enflo type $p$ with constant $C$ if for every $n \\ge 1$ and every function $f : \\{-1, 1\\}^n \\to X$,","title":"Enflo type"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/te","kind":"node","lean":"MathlibExt.Analysis.RademacherEnfloTypeWanted.RademacherEnfloType.TE","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#te","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"fully_proved","summary":"For a real Banach space $X$ and a real number $p$, $T^E_p(X)$ is the infimum in $[0, \\infty]$ of all $C \\in [0, \\infty]$ such that $X$ has Enflo type $p$ with constant $C$.","title":"Enflo type constant"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/theorem-1-1","kind":"node","lean":"MathlibExt.Analysis.RademacherEnfloTypeWanted.RademacherEnfloType.theorem_1_1","parent":17346,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype/index.html#theorem-1-1","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/rademacher-type-and-cotype","status":"can_prove","summary":"For every real Banach space $X$ and every real number $p$ with $1 \\le p \\le 2$, the Rademacher type constant and the Enflo type constant of $X$ satisfy $T^R_p(X) \\le T^E_p(X) \\le \\frac{\\pi}{\\sqrt{2}}\\, T^R_p(X)$.","title":"Comparison of Rademacher and Enflo type constants"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases","status":"can_prove","title":"Schauder bases"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases/is-schauder-basis","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook108Wanted.IsSchauderBasis","parent":17368,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases/index.html#is-schauder-basis","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases","status":"fully_proved","summary":"A sequence $(e_n)_{n \\ge 0}$ in a real normed space $E$ is a Schauder basis when for every $x \\in E$ there is exactly one sequence of real numbers $(a_n)_{n \\ge 0}$ such that the partial sums $\\sum_{i < N} a_i e_i$ converge to $x$ as $N \\to \\infty$.","title":"Schauder basis"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases/krein-basis-in-dense-set","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook108Wanted.krein_basis_in_dense_set","parent":17368,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases/index.html#krein-basis-in-dense-set","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/schauder-bases","status":"can_prove","summary":"Let $E$ be a real Banach space that has a Schauder basis. Then for every dense subset $H \\subseteq E$ there is a Schauder basis $(e_n)_{n \\ge 0}$ of $E$ with $e_n \\in H$ for all $n$, and likewise for every dense linear subspace $S \\subseteq E$ there is a Schauder basis $(e_n)_{n \\ge 0}$ of $E$ with $e_n \\in S$ for all $n$.","title":"Krein's theorem on bases in dense sets"},{"data_url":"scopes/Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces.json","id":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces","kind":"scope","navigation":"module","parent":17288,"result":false,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/index.html","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces","status":"fully_proved","title":"Uniformly convex spaces"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/uniform-convex-space-of-finite-dimension","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.UniformConvex.uniformConvexSpace_of_finiteDimensional_strictConvexSpace","parent":17371,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/index.html#uniform-convex-space-of-finite-dimension","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces","status":"fully_proved","summary":"In a finite-dimensional real normed space, strict convexity implies uniform convexity. The proof maximizes `\u2016x + y\u2016` over the compact set of pairs of unit vectors at distance at least `\u03b5`; strict convexity says the maximum is `< 2`, and the gap is the required `\u03b4`.","title":"Strict convexity implies uniform convexity in finite dimensions"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/strict-convex-space-iff-uniform-convex-s","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.UniformConvex.strictConvexSpace_iff_uniformConvexSpace_of_finiteDimensional","parent":17371,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/index.html#strict-convex-space-iff-uniform-convex-s","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces","status":"fully_proved","summary":"In finite dimensions, strict convexity and uniform convexity are equivalent.","title":"Strict and uniform convexity agree in finite dimensions"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/exists-tendsto-of-minimizing-sequence","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.UniformConvex.exists_tendsto_of_minimizingSequence","parent":17371,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/index.html#exists-tendsto-of-minimizing-sequence","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces","status":"fully_proved","summary":"In a complete uniformly convex space, a minimizing sequence for the distance from a point `x` to a closed convex set `K` converges to a point of `K` attaining that distance. The argument is the usual one: if the sequence were not Cauchy, two far-apart almost-minimizing terms would have normalized displacements far apart on the unit sphere, and uniform convexity would force their midpoint to beat the infimum,\u2026","title":"Convergence of minimizing sequences in uniformly convex spaces"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/exists-unique-nearest-point-of-minimizin","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.UniformConvex.exists_unique_nearestPoint_of_minimizingSequence_of_finiteDimensional","parent":17371,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/index.html#exists-unique-nearest-point-of-minimizin","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces","status":"fully_proved","summary":"In a finite-dimensional strictly convex space, a minimizing sequence for the distance to a closed convex set supplies a unique nearest point. The uniqueness clause only assumes that a second point belongs to the set and attains the distance; it does not assume that the minimizing sequence converges to that point.","title":"Unique nearest point from a minimizing sequence in finite dimensions"},{"id":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/exists-unique-tendsto-of-minimizing-sequ","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.UniformConvex.exists_unique_tendsto_of_minimizingSequence_of_finiteDimensional","parent":17371,"result":true,"route":"../roadmap/Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces/index.html#exists-unique-tendsto-of-minimizing-sequ","scope":"Analysis/banach-spaces/geometry-of-banach-spaces/uniformly-convex-spaces","status":"fully_proved","summary":"In a finite-dimensional strictly convex space, a minimizing sequence for the distance to a closed convex set converges to the unique point of the set attaining that distance. This preserves the original combined API as a corollary of nearest-point uniqueness and minimizing-sequence convergence.","title":"Minimizing sequences converge to the unique nearest point"},{"data_url":"scopes/Analysis/banach-spaces/sequence-spaces.json","id":"Analysis/banach-spaces/sequence-spaces","kind":"scope","parent":17193,"result":false,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/index.html","scope":"Analysis/banach-spaces/sequence-spaces","status":"fully_proved","summary":"\u2113^p and c\u2080: duality, Schauder bases, compactness criteria, Schur and Rosenthal \u2113\u00b9 theorems, Pitt's theorem, quotients of \u2113\u00b9, finite \u2113^p geometry, convolution on \u2113^p(\u2124).","title":"Classical sequence spaces"},{"id":"Analysis/banach-spaces/sequence-spaces/pitt","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.PittWanted.pitt","parent":17377,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/index.html#pitt","scope":"Analysis/banach-spaces/sequence-spaces","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $1 \\le q < p < \\infty$. Then every bounded linear operator $T : \\ell^p(\\mathbb{N}, \\mathbb{K}) \\to \\ell^q(\\mathbb{N}, \\mathbb{K})$ is compact.","title":"Pitt's theorem"},{"id":"Analysis/banach-spaces/sequence-spaces/separable-real-banach-quotient-l1","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.L1Quotient.separableRealBanach_quotient_l1","parent":17377,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/index.html#separable-real-banach-quotient-l1","scope":"Analysis/banach-spaces/sequence-spaces","status":"fully_proved","summary":"For every separable real Banach space $E$ there is a surjective bounded linear operator $T \\colon \\ell^1(\\mathbb{N}, \\mathbb{R}) \\to E$.","title":"Separable real Banach spaces are quotients of \u2113\u00b9"},{"data_url":"scopes/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c.json","id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","kind":"scope","navigation":"module","parent":17377,"result":false,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","title":"Duals of \u2113^p and c\u2080"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/lp-dual-pairing","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Dual.lpDualPairing","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Canonical duality pairing between "],["math","\\(\\ell^q\\)"],["text"," and "],["math","\\(\\ell^p\\)"]],"source":"Canonical duality pairing between $\\ell^q$ and $\\ell^p$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#lp-dual-pairing","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"The canonical pairing `\u2113^q \u00d7 \u2113^p \u2192 \ud835\udd5c`, as a continuous linear map `\u2113^q \u2192L[\ud835\udd5c] (\u2113^p \u2192L[\ud835\udd5c] \ud835\udd5c)`.","title":"Canonical duality pairing between $\\ell^q$ and $\\ell^p$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/lp-dual-pairing-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Dual.lpDualPairing_apply","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#lp-dual-pairing-apply","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"The pairing is given by the sum of coordinatewise products.","title":"Duality pairing as a sum of coordinatewise products"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/norm-lp-dual-pairing-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Dual.norm_lpDualPairing_le","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#norm-lp-dual-pairing-le","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"The pairing has operator norm at most one.","title":"Duality pairing has operator norm at most one"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/lp-dual-pairing-surjective","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Dual.lpDualPairing_surjective","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every functional on "],["math","\\(\\ell^p\\)"],["text"," comes from an "],["math","\\(\\ell^q\\)"],["text"," sequence"]],"source":"Every functional on $\\ell^p$ comes from an $\\ell^q$ sequence"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#lp-dual-pairing-surjective","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"The pairing is surjective: every continuous linear functional on `\u2113^p` is given by pairing with some `\u2113^q` sequence.","title":"Every functional on $\\ell^p$ comes from an $\\ell^q$ sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/lp-dual-pairing-norm","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Dual.lpDualPairing_norm","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#lp-dual-pairing-norm","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"The pairing is norm-preserving.","title":"Duality pairing is norm-preserving"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/lp-dual-equiv","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Dual.lpDualEquiv","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dual of "],["math","\\(\\ell^p\\)"],["text"," is isometric to "],["math","\\(\\ell^q\\)"]],"source":"Dual of $\\ell^p$ is isometric to $\\ell^q$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#lp-dual-equiv","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"The dual of `\u2113^p` is isometrically isomorphic to `\u2113^q`.","title":"Dual of $\\ell^p$ is isometric to $\\ell^q$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-submodule","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Submodule","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subspace of null sequences in "],["math","\\(\\ell^\\infty\\)"]],"source":"The subspace of null sequences in $\\ell^\\infty$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-submodule","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. The sequences $x \\in \\ell^\\infty(\\iota, \\mathbb{K})$ whose coordinates $x_i$ tend to $0$ along the cofinite filter on $\\iota$ form a $\\mathbb{K}$-linear subspace of $\\ell^\\infty(\\iota, \\mathbb{K})$.","title":"The subspace of null sequences in $\\ell^\\infty$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sequence space "],["math","\\(c_0\\)"]],"source":"The sequence space $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be a type. The space $c_0(\\iota, \\mathbb{K})$ consists of the functions $x : \\iota \\to \\mathbb{K}$ that tend to $0$ along the cofinite filter, that is, for every $\\varepsilon > 0$ the set $\\{i : |x_i| \\ge \\varepsilon\\}$ is finite, with pointwise operations and the supremum norm $\\|x\\|_\\infty = \\sup_i |x_i|$. It is realized as the space of continuous\u2026","title":"The sequence space $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-tendsto-zero","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0_tendsto_zero","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of "],["math","\\(c_0\\)"],["text"," tend to zero along the cofinite filter"]],"source":"Elements of $c_0$ tend to zero along the cofinite filter"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-tendsto-zero","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be an index type, and let $c_0(\\iota, \\mathbb{K})$ be the space of functions $\\iota \\to \\mathbb{K}$ vanishing at infinity for the discrete topology on $\\iota$. For every $x \\in c_0(\\iota, \\mathbb{K})$, the coordinates $x_i$ tend to $0$ along the cofinite filter on $\\iota$.","title":"Elements of $c_0$ tend to zero along the cofinite filter"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-apply-norm-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0_apply_norm_le","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinates are bounded by the norm in "],["math","\\(c_0\\)"]],"source":"Coordinates are bounded by the norm in $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-apply-norm-le","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $c_0(\\iota, \\mathbb{K})$ be the space of families $x = (x_i)_{i \\in \\iota}$ in $\\mathbb{K}$ that tend to $0$ along the cofinite filter, with the supremum norm. Then $|x_i| \\le \\|x\\|$ for every $x \\in c_0(\\iota, \\mathbb{K})$ and every index $i \\in \\iota$.","title":"Coordinates are bounded by the norm in $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-to-lp","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0ToLp","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inclusion of "],["math","\\(c_0\\)"],["text"," in "],["math","\\(\\ell^\\infty\\)"]],"source":"Inclusion of $c_0$ in $\\ell^\\infty$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-to-lp","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. Every $x \\in c_0(\\iota, \\mathbb{K})$ is a bounded sequence, and this map sends it to the element of $\\ell^\\infty(\\iota, \\mathbb{K})$ with the same coordinates $x_i$.","title":"Inclusion of $c_0$ in $\\ell^\\infty$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-norm","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0_norm","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(c_0\\)"],["text"," norm is the supremum norm"]],"source":"The $c_0$ norm is the supremum norm"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-norm","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type. For every $x \\in c_0(\\iota, \\mathbb{K})$, the norm of $x$ equals the norm $\\sup_{i \\in \\iota} |x_i|$ of $x$ viewed as an element of $\\ell^\\infty(\\iota, \\mathbb{K})$.","title":"The $c_0$ norm is the supremum norm"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-of-lp","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0OfLp","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sequences in "],["math","\\(\\ell^\\infty\\)"],["text"," tending to zero as elements of "],["math","\\(c_0\\)"]],"source":"Sequences in $\\ell^\\infty$ tending to zero as elements of $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-of-lp","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. A bounded sequence $x \\in \\ell^\\infty(\\iota, \\mathbb{K})$ whose coordinates $x_i$ tend to $0$ along the cofinite filter on $\\iota$ defines the element of $c_0(\\iota, \\mathbb{K})$ with the same coordinates.","title":"Sequences in $\\ell^\\infty$ tending to zero as elements of $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-lp-equiv","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0LpEquiv","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(c_0\\)"],["text"," is isometrically isomorphic to the null sequences in "],["math","\\(\\ell^\\infty\\)"]],"source":"$c_0$ is isometrically isomorphic to the null sequences in $\\ell^\\infty$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-lp-equiv","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type. The map that sends $x \\in c_0(\\iota, \\mathbb{K})$ to the same sequence $(x_i)_{i \\in \\iota}$, viewed in the subspace of $\\ell^\\infty(\\iota, \\mathbb{K})$ of sequences tending to $0$ along the cofinite filter, is a linear isometric equivalence onto that subspace.","title":"$c_0$ is isometrically isomorphic to the null sequences in $\\ell^\\infty$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-single","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Single","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-single","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. For an index $i \\in \\iota$ and a scalar $a \\in \\mathbb{K}$, this is the element of $c_0(\\iota, \\mathbb{K})$ whose $i$-th coordinate is $a$ and whose other coordinates are $0$.","title":"Sequence supported at one index"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-single-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Single_apply","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-single-apply","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. For indices $i, j \\in \\iota$ and $a \\in \\mathbb{K}$, the $j$-th coordinate of the element of $c_0(\\iota, \\mathbb{K})$ that is $a$ at $i$ and $0$ elsewhere equals $a$ if $j = i$ and $0$ otherwise.","title":"Coordinates of the sequence supported at one index"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-single-norm","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Single_norm","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-single-norm","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. For $i \\in \\iota$ and $a \\in \\mathbb{K}$, the element of $c_0(\\iota, \\mathbb{K})$ that is $a$ at $i$ and $0$ elsewhere has norm $\\|a\\|$.","title":"Norm of a sequence supported at one index"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-finset-sum","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0FinsetSum","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finitely supported sequences in "],["math","\\(c_0\\)"]],"source":"Finitely supported sequences in $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-finset-sum","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type, and let $e_i \\in c_0(\\iota, \\mathbb{K})$ be the sequence equal to $1$ at $i$ and $0$ elsewhere. For a finite set $s \\subseteq \\iota$ and $c : \\iota \\to \\mathbb{K}$, the finitely supported sequence with coefficients $c$ on $s$ is the element $\\sum_{i \\in s} c_i e_i$ of $c_0(\\iota, \\mathbb{K})$.","title":"Finitely supported sequences in $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-finset-sum-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0FinsetSum_apply","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-finset-sum-apply","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type, and let $e_i \\in c_0(\\iota, \\mathbb{K})$ be the sequence equal to $1$ at $i$ and $0$ elsewhere. For every finite set $s \\subseteq \\iota$, every $c : \\iota \\to \\mathbb{K}$ and every $j \\in \\iota$, the $j$-th coordinate of $\\sum_{i \\in s} c_i e_i$ is $c_j$ if $j \\in s$ and $0$ otherwise.","title":"Coordinates of a finitely supported sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-finset-sum-apply-dual","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0FinsetSum_apply_dual","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-finset-sum-apply-dual","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type, and let $e_i \\in c_0(\\iota, \\mathbb{K})$ be the sequence equal to $1$ at $i$ and $0$ elsewhere. For every continuous linear functional $\\varphi$ on $c_0(\\iota, \\mathbb{K})$, every finite set $s \\subseteq \\iota$ and every $c : \\iota \\to \\mathbb{K}$, we have $\\varphi(\\sum_{i \\in s} c_i e_i) = \\sum_{i \\in s} c_i \\, \\varphi(e_i)$.","title":"A functional on a finitely supported sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-trunc","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Trunc","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite truncation of a "],["math","\\(c_0\\)"],["text"," sequence"]],"source":"Finite truncation of a $c_0$ sequence"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-trunc","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type, let $s \\subseteq \\iota$ be a finite set and let $x \\in c_0(\\iota, \\mathbb{K})$. The truncation of $x$ to $s$ is the element of $c_0(\\iota, \\mathbb{K})$ whose $j$-th coordinate is $x_j$ for $j \\in s$ and $0$ for $j \\notin s$; it is defined as $\\sum_{i \\in s} x_i e_i$, where $e_i$ is the sequence that is $1$ at $i$ and $0$ elsewhere.","title":"Finite truncation of a $c_0$ sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-trunc-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Trunc_apply","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-trunc-apply","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. For a finite set $s \\subseteq \\iota$, an element $x \\in c_0(\\iota, \\mathbb{K})$ and an index $j \\in \\iota$, the $j$-th coordinate of the truncation of $x$ to $s$ equals $x_j$ if $j \\in s$ and $0$ otherwise.","title":"Coordinates of a finite truncation"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/summable-norm-lp-one","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.summable_norm_lp_one","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinate norms of an "],["math","\\(\\ell^1\\)"],["text"," family are summable"]],"source":"Coordinate norms of an $\\ell^1$ family are summable"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#summable-norm-lp-one","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be any index type. For every $a \\in \\ell^1(\\iota, \\mathbb{K})$, the family of norms $(\\|a_i\\|)_{i \\in \\iota}$ is summable.","title":"Coordinate norms of an $\\ell^1$ family are summable"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/summable-c0-pairing","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.summable_c0Pairing","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of the "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(c_0\\)"],["text"," pairing"]],"source":"Summability of the $\\ell^1$\u2013$c_0$ pairing"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#summable-c0-pairing","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be a type. For every $a \\in \\ell^1(\\iota, \\mathbb{K})$ and $x \\in c_0(\\iota, \\mathbb{K})$, the family $(a_i x_i)_{i \\in \\iota}$ is (unconditionally) summable.","title":"Summability of the $\\ell^1$\u2013$c_0$ pairing"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-pairing-lin","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0PairingLin","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pairing with an "],["math","\\(\\ell^1\\)"],["text"," sequence as a linear functional on "],["math","\\(c_0\\)"]],"source":"Pairing with an $\\ell^1$ sequence as a linear functional on $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-pairing-lin","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. For $a \\in \\ell^1(\\iota, \\mathbb{K})$, this is the $\\mathbb{K}$-linear map $c_0(\\iota, \\mathbb{K}) \\to \\mathbb{K}$ sending $x$ to $\\sum_{i \\in \\iota} a_i x_i$.","title":"Pairing with an $\\ell^1$ sequence as a linear functional on $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-pairing-lin-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0PairingLin_apply","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of the functional of an "],["math","\\(\\ell^1\\)"],["text"," sequence on "],["math","\\(c_0\\)"]],"source":"Value of the functional of an $\\ell^1$ sequence on $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-pairing-lin-apply","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be an index type. For $a \\in \\ell^1(\\iota, \\mathbb{K})$ and $x \\in c_0(\\iota, \\mathbb{K})$, the linear functional on $c_0(\\iota, \\mathbb{K})$ given by pairing with $a$ takes the value $\\sum_{i \\in \\iota} a_i x_i$ at $x$.","title":"Value of the functional of an $\\ell^1$ sequence on $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/norm-c0-pairing-lin-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.norm_c0PairingLin_le","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound for the "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(c_0\\)"],["text"," pairing"]],"source":"Bound for the $\\ell^1$\u2013$c_0$ pairing"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#norm-c0-pairing-lin-le","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be a type. For every $a \\in \\ell^1(\\iota, \\mathbb{K})$ and $x \\in c_0(\\iota, \\mathbb{K})$, $|\\sum_{i \\in \\iota} a_i x_i| \\le \\|a\\|_1 \\|x\\|_\\infty$, where $\\|a\\|_1 = \\sum_i |a_i|$ and $\\|x\\|_\\infty = \\sup_i |x_i|$.","title":"Bound for the $\\ell^1$\u2013$c_0$ pairing"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-pairing-lin","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualPairingLin","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(c_0\\)"],["text"," pairing as a linear map"]],"source":"The $\\ell^1$\u2013$c_0$ pairing as a linear map"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-pairing-lin","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, and let $\\ell^1(\\iota, \\mathbb{K})$ be the space of absolutely summable families. This is the linear map from $\\ell^1(\\iota, \\mathbb{K})$ to the space of continuous linear functionals on $c_0(\\iota, \\mathbb{K})$ that sends $a$ to the\u2026","title":"The $\\ell^1$\u2013$c_0$ pairing as a linear map"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-pairing-lin-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualPairingLin_apply","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formula for the "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(c_0\\)"],["text"," pairing as a linear map"]],"source":"Formula for the $\\ell^1$\u2013$c_0$ pairing as a linear map"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-pairing-lin-apply","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, and let $\\ell^1(\\iota, \\mathbb{K})$ be the space of absolutely summable families. For $a \\in \\ell^1(\\iota, \\mathbb{K})$ and $x \\in c_0(\\iota, \\mathbb{K})$, the linear map from $\\ell^1(\\iota, \\mathbb{K})$ to the continuous linear\u2026","title":"Formula for the $\\ell^1$\u2013$c_0$ pairing as a linear map"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/norm-c0-dual-pairing-lin-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.norm_c0DualPairingLin_le","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norm bound for the functional of an "],["math","\\(\\ell^1\\)"],["text"," sequence on "],["math","\\(c_0\\)"]],"source":"Norm bound for the functional of an $\\ell^1$ sequence on $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#norm-c0-dual-pairing-lin-le","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\iota$ be a type. For every $a \\in \\ell^1(\\iota, \\mathbb{K})$, the image of $a$ under the linear pairing map from $\\ell^1(\\iota, \\mathbb{K})$ to the continuous linear functionals on $c_0(\\iota, \\mathbb{K})$, which is the functional $x \\mapsto \\sum_{i \\in \\iota} a_i x_i$, has operator norm at most $\\|a\\|_1 = \\sum_i |a_i|$.","title":"Norm bound for the functional of an $\\ell^1$ sequence on $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-pairing-lin-continuous","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualPairingLin_continuous","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(c_0\\)"],["text"," pairing is continuous in the "],["math","\\(\\ell^1\\)"],["text"," variable"]],"source":"The $\\ell^1$\u2013$c_0$ pairing is continuous in the $\\ell^1$ variable"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-pairing-lin-continuous","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, with the supremum norm, and let $\\ell^1(\\iota, \\mathbb{K})$ be the space of absolutely summable families. The linear map from $\\ell^1(\\iota, \\mathbb{K})$ to the space of continuous linear functionals on $c_0(\\iota, \\mathbb{K})$, with\u2026","title":"The $\\ell^1$\u2013$c_0$ pairing is continuous in the $\\ell^1$ variable"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-pairing","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualPairing","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(c_0\\)"],["text"," duality pairing"]],"source":"The $\\ell^1$\u2013$c_0$ duality pairing"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-pairing","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type. The pairing is the continuous linear map from $\\ell^1(\\iota, \\mathbb{K})$ to the continuous dual of $c_0(\\iota, \\mathbb{K})$ that sends $a$ to the functional $x \\mapsto \\sum_{i \\in \\iota} a_i x_i$.","title":"The $\\ell^1$\u2013$c_0$ duality pairing"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-pairing-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualPairing_apply","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formula for the "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(c_0\\)"],["text"," duality pairing"]],"source":"Formula for the $\\ell^1$\u2013$c_0$ duality pairing"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-pairing-apply","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, and let $\\ell^1(\\iota, \\mathbb{K})$ be the space of absolutely summable families. For $a \\in \\ell^1(\\iota, \\mathbb{K})$ and $x \\in c_0(\\iota, \\mathbb{K})$, the canonical continuous linear map from $\\ell^1(\\iota, \\mathbb{K})$ to the\u2026","title":"Formula for the $\\ell^1$\u2013$c_0$ duality pairing"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/norm-c0-dual-pairing-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.norm_c0DualPairing_le","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pairing map "],["math","\\(\\ell^1 \\to c_0^*\\)"],["text"," does not increase norms"]],"source":"The pairing map $\\ell^1 \\to c_0^*$ does not increase norms"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#norm-c0-dual-pairing-le","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"The pairing map $\\ell^1 \\to c_0^*$ does not increase norms. Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type, and let $c_0(\\iota, \\mathbb{K})^*$ be the space of continuous linear functionals on $c_0(\\iota, \\mathbb{K})$. For every $a \\in \\ell^1(\\iota, \\mathbb{K})$, the image of $a$ under the continuous linear pairing map $\\ell^1(\\iota, \\mathbb{K}) \\to c_0(\\iota, \\mathbb{K})^*$, which is the\u2026","title":"The pairing map $\\ell^1 \\to c_0^*$ does not increase norms"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-coeff","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Coeff","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of a functional on "],["math","\\(c_0\\)"]],"source":"Coefficients of a functional on $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-coeff","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, and for $i \\in \\iota$ let $e_i \\in c_0(\\iota, \\mathbb{K})$ be the family that is $1$ at $i$ and $0$ elsewhere. The coefficients of a continuous linear functional $\\Phi$ on $c_0(\\iota, \\mathbb{K})$ are the family $i \\mapsto \\Phi(e_i)$\u2026","title":"Coefficients of a functional on $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-unit-scalar","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0UnitScalar","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The normalizing scalar "],["math","\\(\\overline{w}/|w|\\)"]],"source":"The normalizing scalar $\\overline{w}/|w|$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-unit-scalar","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For $w \\in \\mathbb{K}$, the normalizing scalar $u(w) \\in \\mathbb{K}$ of $w$ is $0$ if $w = 0$ and $\\overline{w}/|w|$ otherwise, so that $u(w)\\, w = |w|$.","title":"The normalizing scalar $\\overline{w}/|w|$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-unit-scalar-mul","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0UnitScalar_mul","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The normalizing scalar times "],["math","\\(w\\)"],["text"," is "],["math","\\(|w|\\)"]],"source":"The normalizing scalar times $w$ is $|w|$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-unit-scalar-mul","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $w \\in \\mathbb{K}$, the normalizing scalar $u(w)$ of $w$, which is $0$ if $w = 0$ and $\\overline{w}/|w|$ otherwise, satisfies $u(w)\\, w = |w|$, with $|w|$ regarded as an element of $\\mathbb{K}$.","title":"The normalizing scalar times $w$ is $|w|$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/norm-c0-unit-scalar-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.norm_c0UnitScalar_le","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#norm-c0-unit-scalar-le","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $w \\in \\mathbb{K}$, the normalizing scalar $u(w)$ of $w$, which is $0$ if $w = 0$ and $\\overline{w}/|w|$ otherwise, satisfies $|u(w)| \\le 1$.","title":"The normalizing scalar has absolute value at most one"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-coeff-finset-sum-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Coeff_finset_sum_le","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite sums of the coefficients of a functional on "],["math","\\(c_0\\)"]],"source":"Finite sums of the coefficients of a functional on $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-coeff-finset-sum-le","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, with the supremum norm, and for $i \\in \\iota$ let $e_i \\in c_0(\\iota, \\mathbb{K})$ be the family that is $1$ at $i$ and $0$ elsewhere. For every continuous linear functional $\\Phi$ on $c_0(\\iota, \\mathbb{K})$ and every finite set\u2026","title":"Finite sums of the coefficients of a functional on $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-coeff-mem-p","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Coeff_mem\u2113p","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The coefficients of a functional on "],["math","\\(c_0\\)"],["text"," are absolutely summable"]],"source":"The coefficients of a functional on $c_0$ are absolutely summable"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-coeff-mem-p","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, and for $i \\in \\iota$ let $e_i \\in c_0(\\iota, \\mathbb{K})$ be the family that is $1$ at $i$ and $0$ elsewhere. For every continuous linear functional $\\Phi$ on $c_0(\\iota, \\mathbb{K})$, the family $(\\Phi(e_i))_{i \\in \\iota}$ lies in\u2026","title":"The coefficients of a functional on $c_0$ are absolutely summable"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-coeff-lp","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0CoeffLp","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^1\\)"],["text"," coefficient sequence of a functional on "],["math","\\(c_0\\)"]],"source":"The $\\ell^1$ coefficient sequence of a functional on $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-coeff-lp","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, and for $i \\in \\iota$ let $e_i \\in c_0(\\iota, \\mathbb{K})$ be the family that is $1$ at $i$ and $0$ elsewhere. For a continuous linear functional $\\Phi$ on $c_0(\\iota, \\mathbb{K})$, this is the family $(\\Phi(e_i))_{i \\in \\iota}$,\u2026","title":"The $\\ell^1$ coefficient sequence of a functional on $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/norm-c0-coeff-lp-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.norm_c0CoeffLp_le","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell^1\\)"],["text"," norm of the coefficients of a functional on "],["math","\\(c_0\\)"]],"source":"$\\ell^1$ norm of the coefficients of a functional on $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#norm-c0-coeff-lp-le","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be a type and let $\\Phi$ be a continuous linear functional on $c_0(\\iota, \\mathbb{K})$. For $i \\in \\iota$ let $e_i$ be the sequence that is $1$ at $i$ and $0$ elsewhere. The coefficient sequence $(\\Phi(e_i))_{i \\in \\iota}$, as an element of $\\ell^1(\\iota, \\mathbb{K})$, has norm $\\sum_{i \\in \\iota} |\\Phi(e_i)|$ at most the operator norm $\\|\\Phi\\|$.","title":"$\\ell^1$ norm of the coefficients of a functional on $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-pairing-surjective","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualPairing_surjective","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every functional on "],["math","\\(c_0\\)"],["text"," comes from "],["math","\\(\\ell^1\\)"]],"source":"Every functional on $c_0$ comes from $\\ell^1$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-pairing-surjective","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type. For every continuous linear functional $\\varphi$ on $c_0(\\iota, \\mathbb{K})$ there is $a \\in \\ell^1(\\iota, \\mathbb{K})$ with $\\varphi(x) = \\sum_{i \\in \\iota} a_i x_i$ for all $x \\in c_0(\\iota, \\mathbb{K})$; that is, the pairing map from $\\ell^1(\\iota, \\mathbb{K})$ to the continuous dual of $c_0(\\iota, \\mathbb{K})$ is surjective.","title":"Every functional on $c_0$ comes from $\\ell^1$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-pairing-norm","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualPairing_norm","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(c_0\\)"],["text"," pairing is isometric"]],"source":"The $\\ell^1$\u2013$c_0$ pairing is isometric"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-pairing-norm","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type. For every $a \\in \\ell^1(\\iota, \\mathbb{K})$, the continuous linear functional $x \\mapsto \\sum_{i \\in \\iota} a_i x_i$ on $c_0(\\iota, \\mathbb{K})$, the space of sequences $\\iota \\to \\mathbb{K}$ tending to $0$ at infinity with the supremum norm, has operator norm equal to $\\|a\\|_1 = \\sum_{i \\in \\iota} |a_i|$.","title":"The $\\ell^1$\u2013$c_0$ pairing is isometric"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-equiv","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualEquiv","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","The dual of "],["math","\\(c_0\\)"],["text"," is "],["math","\\(\\ell^1\\)"]],"source":"The dual of $c_0$ is $\\ell^1$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-equiv","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $c_0(\\iota, \\mathbb{K})$ be the space of families in $\\mathbb{K}$ indexed by $\\iota$ that tend to $0$ along the cofinite filter, with the supremum norm, and let $\\ell^1(\\iota, \\mathbb{K})$ be the space of absolutely summable families, with the norm $\\|a\\|_1 = \\sum_{i \\in \\iota} |a_i|$. This is the linear isometric isomorphism from $\\ell^1(\\iota, \\mathbb{K})$\u2026","title":"The dual of $c_0$ is $\\ell^1$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-eval","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0Eval","parent":17380,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinate evaluation on "],["math","\\(c_0\\)"]],"source":"Coordinate evaluation on $c_0$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-eval","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type. For $i \\in \\iota$, evaluation at $i$ is the continuous linear functional $x \\mapsto x_i$ on $c_0(\\iota, \\mathbb{K})$.","title":"Coordinate evaluation on $c_0$"},{"id":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/c0-dual-pairing-single-eq-eval","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.DualC0.c0DualPairing_single_eq_eval","parent":17380,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c/index.html#c0-dual-pairing-single-eq-eval","scope":"Analysis/banach-spaces/sequence-spaces/duals-of-p-and-c","status":"fully_proved","summary":"Let $\\mathbb{K}$ be the real or complex field and $\\iota$ an index type, and for $i \\in \\iota$ let $e_i \\in \\ell^1(\\iota, \\mathbb{K})$ be the sequence equal to $1$ at $i$ and $0$ elsewhere. Then the functional $x \\mapsto \\sum_{j \\in \\iota} (e_i)_j x_j$ on $c_0(\\iota, \\mathbb{K})$ given by pairing with $e_i$ is the evaluation $x \\mapsto x_i$ at $i$.","title":"Pairing with a unit vector is evaluation"},{"data_url":"scopes/Analysis/banach-spaces/sequence-spaces/convolution-on-p.json","id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","kind":"scope","navigation":"module","parent":17377,"result":false,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","title":"Convolution on \u2113^p(\u2124)"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/int-conv","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.intConv","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#int-conv","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution of two complex sequences on `\u2124`, as a series. The series is `0` when it does not converge; for `f \u2208 \u2113\u00b9` and `g \u2208 \u2113^p` it always converges (see `summable_intConv_term`).","title":"Convolution of sequences on the integers"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/tsum-norm-l1","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.tsum_norm_l1","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^1\\)"],["text"," norm as a sum of coordinate norms"]],"source":"The $\\ell^1$ norm as a sum of coordinate norms"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#tsum-norm-l1","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The `\u2113\u00b9` norm is the sum of the coordinate norms.","title":"The $\\ell^1$ norm as a sum of coordinate norms"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/summable-int-conv-term","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.summable_intConv_term","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of convolution terms for "],["math","\\(\\ell^1\\)"],["text"," and "],["math","\\(\\ell^p\\)"],["text"," sequences"]],"source":"Summability of convolution terms for $\\ell^1$ and $\\ell^p$ sequences"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#summable-int-conv-term","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The terms of a convolution of an `\u2113\u00b9` sequence with an `\u2113^p` sequence are summable.","title":"Summability of convolution terms for $\\ell^1$ and $\\ell^p$ sequences"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/mem-p-shift","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.mem\u2113p_shift","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation preserves "],["math","\\(\\ell^p\\)"],["text"," membership"]],"source":"Translation preserves $\\ell^p$ membership"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#mem-p-shift","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Translating an `\u2113^p` sequence on `\u2124` preserves `\u2113^p` membership.","title":"Translation preserves $\\ell^p$ membership"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/shift-lp","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.shiftLp","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation of an "],["math","\\(\\ell^p\\)"],["text"," sequence on the integers"]],"source":"Translation of an $\\ell^p$ sequence on the integers"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#shift-lp","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Translation of an `\u2113^p` sequence on `\u2124` by `m`.","title":"Translation of an $\\ell^p$ sequence on the integers"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/shift-lp-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.shiftLp_apply","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pointwise formula for translation of "],["math","\\(\\ell^p\\)"],["text"," sequences"]],"source":"Pointwise formula for translation of $\\ell^p$ sequences"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#shift-lp-apply","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Pointwise action of a translation.","title":"Pointwise formula for translation of $\\ell^p$ sequences"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/norm-shift-lp","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.norm_shiftLp","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation is an isometry of "],["math","\\(\\ell^p(\\mathbb{Z})\\)"]],"source":"Translation is an isometry of $\\ell^p(\\mathbb{Z})$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#norm-shift-lp","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Translation is an isometry of `\u2113^p(\u2124, \u2102)`.","title":"Translation is an isometry of $\\ell^p(\\mathbb{Z})$"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/summable-l1-conv-term","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.summable_l1Conv_term","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of the Banach-space series for "],["math","\\(\\ell^1\\)"],["text"," convolution"]],"source":"Summability of the Banach-space series for $\\ell^1$ convolution"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#summable-l1-conv-term","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The terms of the Banach-space series defining `l1Conv` are summable.","title":"Summability of the Banach-space series for $\\ell^1$ convolution"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution of an "],["math","\\(\\ell^1\\)"],["text"," sequence with an "],["math","\\(\\ell^p\\)"],["text"," sequence"]],"source":"Convolution of an $\\ell^1$ sequence with an $\\ell^p$ sequence"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution of an `\u2113\u00b9` sequence with an `\u2113^p` sequence, as an element of `\u2113^p(\u2124, \u2102)`: the Banach-space sum `\u2211' m, f m \u2022 shift m g`.","title":"Convolution of an $\\ell^1$ sequence with an $\\ell^p$ sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_apply","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pointwise formula for "],["math","\\(\\ell^1\\)"],["text","\u2013"],["math","\\(\\ell^p\\)"],["text"," convolution"]],"source":"Pointwise formula for $\\ell^1$\u2013$\\ell^p$ convolution"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-apply","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Pointwise formula for convolution.","title":"Pointwise formula for $\\ell^1$\u2013$\\ell^p$ convolution"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/norm-l1-conv-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.norm_l1Conv_le","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Young's inequality with one "],["math","\\(\\ell^1\\)"],["text"," factor"]],"source":"Young's inequality with one $\\ell^1$ factor"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#norm-l1-conv-le","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Young's inequality for convolution with one `\u2113\u00b9` factor.","title":"Young's inequality with one $\\ell^1$ factor"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-add-left","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_add_left","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution is additive in the "],["math","\\(\\ell^1\\)"],["text"," argument"]],"source":"Convolution is additive in the $\\ell^1$ argument"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-add-left","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution is additive in the `\u2113\u00b9` argument.","title":"Convolution is additive in the $\\ell^1$ argument"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-add-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_add_right","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution is additive in the "],["math","\\(\\ell^p\\)"],["text"," argument"]],"source":"Convolution is additive in the $\\ell^p$ argument"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-add-right","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution is additive in the `\u2113^p` argument.","title":"Convolution is additive in the $\\ell^p$ argument"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-smul-left","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_smul_left","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution is homogeneous in the "],["math","\\(\\ell^1\\)"],["text"," argument"]],"source":"Convolution is homogeneous in the $\\ell^1$ argument"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-smul-left","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution commutes with scalar multiplication in the `\u2113\u00b9` argument.","title":"Convolution is homogeneous in the $\\ell^1$ argument"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-smul-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_smul_right","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution is homogeneous in the "],["math","\\(\\ell^p\\)"],["text"," argument"]],"source":"Convolution is homogeneous in the $\\ell^p$ argument"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-smul-right","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution commutes with scalar multiplication in the `\u2113^p` argument.","title":"Convolution is homogeneous in the $\\ell^p$ argument"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-zero-left","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_zero_left","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-zero-left","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution with `0` on the left is `0`.","title":"Convolution with zero on the left"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-zero-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_zero_right","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-zero-right","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution with `0` on the right is `0`.","title":"Convolution with zero on the right"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-clmof","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1ConvCLMOf","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution by a fixed "],["math","\\(\\ell^1\\)"],["text"," sequence as a bounded operator"]],"source":"Convolution by a fixed $\\ell^1$ sequence as a bounded operator"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-clmof","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"For fixed `f \u2208 \u2113\u00b9`, convolution by `f` as a bounded operator on `\u2113^p`.","title":"Convolution by a fixed $\\ell^1$ sequence as a bounded operator"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-clmof-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1ConvCLMOf_apply","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-clmof-apply","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Pointwise action of the convolution operator.","title":"Pointwise action of the convolution operator"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/norm-l1-conv-clmof-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.norm_l1ConvCLMOf_le","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Operator norm of convolution by "],["math","\\(f\\)"],["text"," is at most "],["math","\\(\\|f\\|_1\\)"]],"source":"Operator norm of convolution by $f$ is at most $\\|f\\|_1$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#norm-l1-conv-clmof-le","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The operator norm of convolution by `f` is at most `\u2016f\u2016`.","title":"Operator norm of convolution by $f$ is at most $\\|f\\|_1$"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-clm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1ConvCLM","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution action of "],["math","\\(\\ell^1\\)"],["text"," on "],["math","\\(\\ell^p\\)"],["text"," as a bounded bilinear map"]],"source":"Convolution action of $\\ell^1$ on $\\ell^p$ as a bounded bilinear map"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-clm","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The convolution action as a bounded linear map `\u2113\u00b9(\u2124, \u2102) \u2192L[\u2102] (\u2113^p(\u2124, \u2102) \u2192L[\u2102] \u2113^p(\u2124, \u2102))`.","title":"Convolution action of $\\ell^1$ on $\\ell^p$ as a bounded bilinear map"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-clm-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1ConvCLM_apply","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-clm-apply","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Pointwise action of the convolution action.","title":"Pointwise action of the bundled convolution action"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/norm-l1-conv-clm-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.norm_l1ConvCLM_le","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#norm-l1-conv-clm-le","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The convolution action has operator norm at most `1`.","title":"Convolution action has operator norm at most 1"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-comm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_comm","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Commutativity of "],["math","\\(\\ell^1\\)"],["text"," convolution"]],"source":"Commutativity of $\\ell^1$ convolution"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-comm","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution of `\u2113\u00b9` sequences is commutative.","title":"Commutativity of $\\ell^1$ convolution"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-assoc","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_assoc","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Associativity of "],["math","\\(\\ell^1\\)"],["text"," convolution"]],"source":"Associativity of $\\ell^1$ convolution"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-assoc","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution of `\u2113\u00b9` sequences is associative.","title":"Associativity of $\\ell^1$ convolution"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-delta","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Delta","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit sequence in "],["math","\\(\\ell^1(\\mathbb{Z})\\)"]],"source":"Unit sequence in $\\ell^1(\\mathbb{Z})$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-delta","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The unit sequence `\u03b4` at `0` in `\u2113\u00b9(\u2124, \u2102)`.","title":"Unit sequence in $\\ell^1(\\mathbb{Z})$"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-delta-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Delta_apply","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-delta-apply","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Pointwise values of the unit sequence.","title":"Pointwise values of the unit sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/norm-l1-delta","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.norm_l1Delta","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#norm-l1-delta","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The norm of the unit sequence is `1`.","title":"Norm of the unit sequence is 1"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-delta-left","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_delta_left","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-delta-left","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The unit sequence is a left unit for convolution.","title":"Unit sequence is a left identity for convolution"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-delta-right","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_delta_right","parent":17428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit sequence is a right identity for "],["math","\\(\\ell^1\\)"],["text"," convolution"]],"source":"Unit sequence is a right identity for $\\ell^1$ convolution"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-delta-right","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"The unit sequence is a right unit for convolution on `\u2113\u00b9`.","title":"Unit sequence is a right identity for $\\ell^1$ convolution"},{"id":"Analysis/banach-spaces/sequence-spaces/convolution-on-p/l1-conv-single-single","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.IntConvolution.l1Conv_single_single","parent":17428,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/convolution-on-p/index.html#l1-conv-single-single","scope":"Analysis/banach-spaces/sequence-spaces/convolution-on-p","status":"fully_proved","summary":"Convolution of two coordinate vectors: positions add, coefficients multiply.","title":"Convolution of coordinate vectors"},{"data_url":"scopes/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces.json","id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","kind":"scope","navigation":"module","parent":17377,"result":false,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","title":"Geometry of finite-dimensional \u2113^p spaces"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/lp-two-plus","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.lpTwoPlus","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The vector "],["math","\\((1, 1)\\)"],["text"," in "],["math","\\(\\ell^p_2\\)"]],"source":"The vector $(1, 1)$ in $\\ell^p_2$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#lp-two-plus","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For $p \\in [0, \\infty]$, write $\\ell^p_2$ for the space $\\mathbb{R}^2$ regarded as the $\\ell^p$ product of two copies of $\\mathbb{R}$ (normed by the $\\ell^p$ norm when $p \\ge 1$). The vector $u_p \\in \\ell^p_2$ is $(1, 1)$, that is, $e_0 + e_1$, where $e_0 = (1, 0)$ and $e_1 = (0, 1)$ are the standard basis vectors.","title":"The vector $(1, 1)$ in $\\ell^p_2$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/lp-two-minus","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.lpTwoMinus","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The vector "],["math","\\((1, -1)\\)"],["text"," in "],["math","\\(\\ell^p_2\\)"]],"source":"The vector $(1, -1)$ in $\\ell^p_2$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#lp-two-minus","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For $p \\in [0, \\infty]$, write $\\ell^p_2$ for the space $\\mathbb{R}^2$ regarded as the $\\ell^p$ product of two copies of $\\mathbb{R}$ (normed by the $\\ell^p$ norm when $p \\ge 1$). The vector $v_p \\in \\ell^p_2$ is $(1, -1)$, that is, $e_0 - e_1$, where $e_0 = (1, 0)$ and $e_1 = (0, 1)$ are the standard basis vectors.","title":"The vector $(1, -1)$ in $\\ell^p_2$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/lp-two-plus-add-lp-two-minus","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.lpTwoPlus_add_lpTwoMinus","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\((1, 1)\\)"],["text"," and "],["math","\\((1, -1)\\)"],["text"," in "],["math","\\(\\ell^p_2\\)"]],"source":"Sum of $(1, 1)$ and $(1, -1)$ in $\\ell^p_2$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#lp-two-plus-add-lp-two-minus","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For every $p \\in [0, \\infty]$, write $\\ell^p_2$ for the space $\\mathbb{R}^2$ regarded as the $\\ell^p$ product of two copies of $\\mathbb{R}$. The vectors $u_p = (1, 1)$ and $v_p = (1, -1)$ of $\\ell^p_2$ satisfy $u_p + v_p = 2 e_0$, where $e_0 = (1, 0)$ is the first standard basis vector.","title":"Sum of $(1, 1)$ and $(1, -1)$ in $\\ell^p_2$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/lp-two-plus-sub-lp-two-minus","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.lpTwoPlus_sub_lpTwoMinus","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","Difference of "],["math","\\((1, 1)\\)"],["text"," and "],["math","\\((1, -1)\\)"],["text"," in "],["math","\\(\\ell^p_2\\)"]],"source":"Difference of $(1, 1)$ and $(1, -1)$ in $\\ell^p_2$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#lp-two-plus-sub-lp-two-minus","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For every $p \\in [0, \\infty]$, write $\\ell^p_2$ for the space $\\mathbb{R}^2$ regarded as the $\\ell^p$ product of two copies of $\\mathbb{R}$. The vectors $u_p = (1, 1)$ and $v_p = (1, -1)$ of $\\ell^p_2$ satisfy $u_p - v_p = 2 e_1$, where $e_1 = (0, 1)$ is the second standard basis vector.","title":"Difference of $(1, 1)$ and $(1, -1)$ in $\\ell^p_2$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/norm-add-lp-two","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.norm_add_lpTwo","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norm of "],["math","\\((1, 1) + (1, -1)\\)"],["text"," in "],["math","\\(\\ell^p_2\\)"]],"source":"Norm of $(1, 1) + (1, -1)$ in $\\ell^p_2$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#norm-add-lp-two","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For every $p \\in [1, \\infty]$, let $\\ell^p_2$ be $\\mathbb{R}^2$ with the $\\ell^p$ norm, and let $u_p = (1, 1)$ and $v_p = (1, -1)$. Then $\\|u_p + v_p\\| = 2$.","title":"Norm of $(1, 1) + (1, -1)$ in $\\ell^p_2$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/norm-sub-lp-two","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.norm_sub_lpTwo","parent":17460,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#norm-sub-lp-two","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For every extended real $p \\in [1, \\infty]$, in $\\mathbb{R}^2$ with the $\\ell^p$ norm the vectors $u = (1, 1)$ and $v = (1, -1)$ satisfy $\\|u - v\\| = 2$.","title":"The difference of the parallelogram witnesses has norm two"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/norm-lp-two-plus-of-ne-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.norm_lpTwoPlus_of_ne_top","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^p\\)"],["text"," norm of "],["math","\\((1, 1)\\)"],["text"," for finite "],["math","\\(p\\)"]],"source":"The $\\ell^p$ norm of $(1, 1)$ for finite $p$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#norm-lp-two-plus-of-ne-top","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For every real $p$ with $1 \\le p < \\infty$, the vector $(1, 1)$ of $\\mathbb{R}^2$ has $\\ell^p$ norm $\\|(1, 1)\\|_p = 2^{1/p}$, where $\\|(x, y)\\|_p = (|x|^p + |y|^p)^{1/p}$.","title":"The $\\ell^p$ norm of $(1, 1)$ for finite $p$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/norm-lp-two-minus-of-ne-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.norm_lpTwoMinus_of_ne_top","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^p\\)"],["text"," norm of "],["math","\\((1, -1)\\)"],["text"," for finite "],["math","\\(p\\)"]],"source":"The $\\ell^p$ norm of $(1, -1)$ for finite $p$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#norm-lp-two-minus-of-ne-top","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For every real $p$ with $1 \\le p < \\infty$, the vector $(1, -1)$ of $\\mathbb{R}^2$ has $\\ell^p$ norm $\\|(1, -1)\\|_p = 2^{1/p}$, where $\\|(x, y)\\|_p = (|x|^p + |y|^p)^{1/p}$.","title":"The $\\ell^p$ norm of $(1, -1)$ for finite $p$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/norm-lp-two-plus-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.norm_lpTwoPlus_top","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^\\infty\\)"],["text"," norm of "],["math","\\((1, 1)\\)"]],"source":"The $\\ell^\\infty$ norm of $(1, 1)$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#norm-lp-two-plus-top","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"In $\\mathbb{R}^2$ with the $\\ell^\\infty$ norm $\\|(x, y)\\|_\\infty = \\max(|x|, |y|)$, the vector $(1, 1)$ has norm $\\|(1, 1)\\|_\\infty = 1$.","title":"The $\\ell^\\infty$ norm of $(1, 1)$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/norm-lp-two-minus-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.norm_lpTwoMinus_top","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^\\infty\\)"],["text"," norm of "],["math","\\((1, -1)\\)"]],"source":"The $\\ell^\\infty$ norm of $(1, -1)$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#norm-lp-two-minus-top","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"In $\\mathbb{R}^2$ with the $\\ell^\\infty$ norm $\\|(x, y)\\|_\\infty = \\max(|x|, |y|)$, the vector $(1, -1)$ has norm $\\|(1, -1)\\|_\\infty = 1$.","title":"The $\\ell^\\infty$ norm of $(1, -1)$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/norm-sq-lp-two-plus-of-ne-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.norm_sq_lpTwoPlus_of_ne_top","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The squared "],["math","\\(\\ell^p\\)"],["text"," norm of "],["math","\\((1, 1)\\)"],["text"," for finite "],["math","\\(p\\)"]],"source":"The squared $\\ell^p$ norm of $(1, 1)$ for finite $p$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#norm-sq-lp-two-plus-of-ne-top","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For every real $p$ with $1 \\le p < \\infty$, the vector $(1, 1)$ of $\\mathbb{R}^2$ satisfies $\\|(1, 1)\\|_p^2 = 2^{2/p}$, where $\\|(x, y)\\|_p = (|x|^p + |y|^p)^{1/p}$ is the $\\ell^p$ norm.","title":"The squared $\\ell^p$ norm of $(1, 1)$ for finite $p$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/parallelogram-eq-iff-pi-lp-two","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.parallelogram_eq_iff_piLp_two","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parallelogram law for "],["math","\\((1, 1)\\)"],["text"," and "],["math","\\((1, -1)\\)"],["text"," holds only for the "],["math","\\(\\ell^2\\)"],["text"," norm"]],"source":"The parallelogram law for $(1, 1)$ and $(1, -1)$ holds only for the $\\ell^2$ norm"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#parallelogram-eq-iff-pi-lp-two","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"Let $p \\in [1, \\infty]$ and equip $\\mathbb{R}^2$ with the $\\ell^p$ norm. The vectors $u = (1, 1)$ and $v = (1, -1)$ satisfy $\\|u + v\\|^2 + \\|u - v\\|^2 = 2 (\\|u\\|^2 + \\|v\\|^2)$ if and only if $p = 2$.","title":"The parallelogram law for $(1, 1)$ and $(1, -1)$ holds only for the $\\ell^2$ norm"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/parallelogram-fails-pi-lp-two","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.parallelogram_fails_piLp_two","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parallelogram law fails on "],["math","\\(\\mathbb{R}^2\\)"],["text"," with the "],["math","\\(\\ell^p\\)"],["text"," norm for "],["math","\\(p \\neq 2\\)"]],"source":"The parallelogram law fails on $\\mathbb{R}^2$ with the $\\ell^p$ norm for $p \\neq 2$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#parallelogram-fails-pi-lp-two","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"Let $p \\in [1, \\infty]$ with $p \\neq 2$ and equip $\\mathbb{R}^2$ with the $\\ell^p$ norm. Then the vectors $u = (1, 1)$ and $v = (1, -1)$ satisfy $\\|u + v\\|^2 + \\|u - v\\|^2 \\neq 2 (\\|u\\|^2 + \\|v\\|^2)$.","title":"The parallelogram law fails on $\\mathbb{R}^2$ with the $\\ell^p$ norm for $p \\neq 2$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/not-inner-product-space-pi-lp-two","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.not_innerProductSpace_piLp_two","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^p\\)"],["text"," norm on "],["math","\\(\\mathbb{R}^2\\)"],["text"," comes from no inner product unless "],["math","\\(p = 2\\)"]],"source":"The $\\ell^p$ norm on $\\mathbb{R}^2$ comes from no inner product unless $p = 2$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#not-inner-product-space-pi-lp-two","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"Let $p \\in [1, \\infty]$ with $p \\neq 2$. Then the normed space $\\mathbb{R}^2$ with the $\\ell^p$ norm carries no real inner product space structure, that is, there is no real inner product with $\\langle x, x \\rangle = \\|x\\|^2$ for every $x$.","title":"The $\\ell^p$ norm on $\\mathbb{R}^2$ comes from no inner product unless $p = 2$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/pi-lp-two-strict-convex","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.piLp_two_strictConvex","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","Midpoints of distinct "],["math","\\(\\ell^p\\)"],["text"," unit vectors in "],["math","\\(\\mathbb{R}^2\\)"],["text"," have norm less than one"]],"source":"Midpoints of distinct $\\ell^p$ unit vectors in $\\mathbb{R}^2$ have norm less than one"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#pi-lp-two-strict-convex","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"Let $1 < p < \\infty$ and equip $\\mathbb{R}^2$ with the $\\ell^p$ norm. If $x \\neq y$ are vectors with $\\|x\\| = \\|y\\| = 1$, then $\\|\\tfrac{1}{2} (x + y)\\| < 1$.","title":"Midpoints of distinct $\\ell^p$ unit vectors in $\\mathbb{R}^2$ have norm less than one"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/strict-convex-space-pi-lp-two","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.strictConvexSpace_piLp_two","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathbb{R}^2\\)"],["text"," with the "],["math","\\(\\ell^p\\)"],["text"," norm is strictly convex for "],["math","\\(1 < p < \\infty\\)"]],"source":"$\\mathbb{R}^2$ with the $\\ell^p$ norm is strictly convex for $1 < p < \\infty$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#strict-convex-space-pi-lp-two","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"For $1 < p < \\infty$, the real normed space $\\mathbb{R}^2$ with the $\\ell^p$ norm is a strictly convex space: every closed ball of positive radius centred at the origin is a strictly convex set.","title":"$\\mathbb{R}^2$ with the $\\ell^p$ norm is strictly convex for $1 < p < \\infty$"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/not-strict-convex-pi-lp-two-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.not_strictConvex_piLp_two_one","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two distinct "],["math","\\(\\ell^1\\)"],["text"," unit vectors in "],["math","\\(\\mathbb{R}^2\\)"],["text"," with a unit midpoint"]],"source":"Two distinct $\\ell^1$ unit vectors in $\\mathbb{R}^2$ with a unit midpoint"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#not-strict-convex-pi-lp-two-one","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"In $\\mathbb{R}^2$ with the $\\ell^1$ norm there exist vectors $x \\neq y$ with $\\|x\\| = \\|y\\| = 1$ and $\\|\\tfrac{1}{2} (x + y)\\| = 1$.","title":"Two distinct $\\ell^1$ unit vectors in $\\mathbb{R}^2$ with a unit midpoint"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/not-strict-convex-space-pi-lp-two-one","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.not_strictConvexSpace_piLp_two_one","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathbb{R}^2\\)"],["text"," with the "],["math","\\(\\ell^1\\)"],["text"," norm is not strictly convex"]],"source":"$\\mathbb{R}^2$ with the $\\ell^1$ norm is not strictly convex"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#not-strict-convex-space-pi-lp-two-one","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"The real normed space $\\mathbb{R}^2$ with the $\\ell^1$ norm is not a strictly convex space: not every closed ball of positive radius centred at the origin is a strictly convex set.","title":"$\\mathbb{R}^2$ with the $\\ell^1$ norm is not strictly convex"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/not-strict-convex-pi-lp-two-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.not_strictConvex_piLp_two_top","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two distinct "],["math","\\(\\ell^\\infty\\)"],["text"," unit vectors in "],["math","\\(\\mathbb{R}^2\\)"],["text"," with a unit midpoint"]],"source":"Two distinct $\\ell^\\infty$ unit vectors in $\\mathbb{R}^2$ with a unit midpoint"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#not-strict-convex-pi-lp-two-top","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"In $\\mathbb{R}^2$ with the $\\ell^\\infty$ norm there exist vectors $x \\neq y$ with $\\|x\\| = \\|y\\| = 1$ and $\\|\\tfrac{1}{2} (x + y)\\| = 1$.","title":"Two distinct $\\ell^\\infty$ unit vectors in $\\mathbb{R}^2$ with a unit midpoint"},{"id":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/not-strict-convex-space-pi-lp-two-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpGeometry.not_strictConvexSpace_piLp_two_top","parent":17460,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathbb{R}^2\\)"],["text"," with the "],["math","\\(\\ell^\\infty\\)"],["text"," norm is not strictly convex"]],"source":"$\\mathbb{R}^2$ with the $\\ell^\\infty$ norm is not strictly convex"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces/index.html#not-strict-convex-space-pi-lp-two-top","scope":"Analysis/banach-spaces/sequence-spaces/geometry-of-finite-dimensional-p-spaces","status":"fully_proved","summary":"The real normed space $\\mathbb{R}^2$ with the $\\ell^\\infty$ norm is not a strictly convex space: not every closed ball of positive radius centred at the origin is a strictly convex set.","title":"$\\mathbb{R}^2$ with the $\\ell^\\infty$ norm is not strictly convex"},{"data_url":"scopes/Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem.json","id":"Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem","kind":"scope","navigation":"module","parent":17377,"result":false,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem/index.html","scope":"Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem","status":"fully_proved","title":"Rosenthal's \u2113\u00b9 theorem"},{"id":"Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem/has-uniform-l1-lower-bound","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RosenthalL1Wanted.HasUniformL1LowerBound","parent":17481,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniform "],["math","\\(\\ell^1\\)"],["text"," lower bound"]],"source":"Uniform $\\ell^1$ lower bound"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem/index.html#has-uniform-l1-lower-bound","scope":"Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. A sequence $(y_i)_{i \\in \\mathbb{N}}$ in $E$ has a uniform $\\ell^1$ lower bound when there is a real $c > 0$ such that $\\|\\sum_i a_i y_i\\| \\ge c \\sum_i |a_i|$ for every finitely supported sequence $(a_i)_{i \\in \\mathbb{N}}$ of scalars in $\\mathbb{K}$, both sums running over the finitely many $i$ with $a_i \\ne 0$.","title":"Uniform $\\ell^1$ lower bound"},{"id":"Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem/rosenthal-l1","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RosenthalL1Wanted.rosenthal_l1","parent":17481,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rosenthal's "],["math","\\(\\ell^1\\)"],["text"," theorem"]],"source":"Rosenthal's $\\ell^1$ theorem"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem/index.html#rosenthal-l1","scope":"Analysis/banach-spaces/sequence-spaces/rosenthal-s-theorem","status":"fully_proved","summary":"Let $E$ be a Banach space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(x_n)_{n \\in \\mathbb{N}}$ be a bounded sequence in $E$. Then there is a strictly increasing $\\varphi : \\mathbb{N} \\to \\mathbb{N}$ such that either the subsequence $(x_{\\varphi(n)})_{n \\in \\mathbb{N}}$ is weakly Cauchy, that is, for every continuous linear functional $f : E \\to \\mathbb{K}$ and every real $\\varepsilon > 0$ there is $N$\u2026","title":"Rosenthal's $\\ell^1$ theorem"},{"data_url":"scopes/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p.json","id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","kind":"scope","navigation":"module","parent":17377,"result":false,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","title":"Schauder basis of \u2113^p"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/lp-unconditional-schauder-basis","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.lpUnconditionalSchauderBasis","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#lp-unconditional-schauder-basis","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $1 \\le p < \\infty$ and let $\\iota$ be a type. This is the unconditional Schauder basis of $\\ell^p(\\iota, \\mathbb{K})$ whose $i$-th basis vector is the unit vector $e_i$, equal to $1$ at $i$ and to $0$ elsewhere, and whose $i$-th coordinate functional is the evaluation $x \\mapsto x_i$. In particular every $x \\in \\ell^p(\\iota, \\mathbb{K})$ is the limit in norm of\u2026","title":"The unit vectors form an unconditional Schauder basis of \u2113\u1d56"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/lp-schauder-basis","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.lpSchauderBasis","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#lp-schauder-basis","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $1 \\le p < \\infty$. This is the Schauder basis of $\\ell^p(\\mathbb{N}, \\mathbb{K})$ whose $i$-th basis vector is the unit vector $e_i$, equal to $1$ at $i$ and to $0$ elsewhere, and whose $i$-th coordinate functional is the evaluation $x \\mapsto x_i$. In particular every $x \\in \\ell^p(\\mathbb{N}, \\mathbb{K})$ is the limit in norm of its partial sums\u2026","title":"The unit vectors form a Schauder basis of \u2113\u1d56"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/lp-unconditional-schauder-basis-coord-ap","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.lpUnconditionalSchauderBasis_coord_apply","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#lp-unconditional-schauder-basis-coord-ap","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $1 \\le p < \\infty$ and let $\\iota$ be a type. For every $i \\in \\iota$ and every $x \\in \\ell^p(\\iota, \\mathbb{K})$, the $i$-th coordinate functional of the unconditional Schauder basis of $\\ell^p(\\iota, \\mathbb{K})$ by unit vectors sends $x$ to its entry $x_i$.","title":"Coordinate functionals of the unconditional \u2113\u1d56 basis"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/lp-unconditional-schauder-basis-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.lpUnconditionalSchauderBasis_apply","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#lp-unconditional-schauder-basis-apply","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $1 \\le p < \\infty$ and let $\\iota$ be a type. For every $i \\in \\iota$, the $i$-th basis vector of the unconditional Schauder basis of $\\ell^p(\\iota, \\mathbb{K})$ by unit vectors is the unit vector $e_i$, the family equal to $1$ at $i$ and to $0$ elsewhere.","title":"Basis vectors of the unconditional \u2113\u1d56 basis"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/lp-unconditional-schauder-basis-proj-app","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.lpUnconditionalSchauderBasis_proj_apply","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#lp-unconditional-schauder-basis-proj-app","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $1 \\le p < \\infty$ and let $\\iota$ be a type. For every finite set $A \\subseteq \\iota$ and every $x \\in \\ell^p(\\iota, \\mathbb{K})$, the projection $P_A$ of the unconditional Schauder basis of $\\ell^p(\\iota, \\mathbb{K})$ by unit vectors onto the basis vectors indexed by $A$ satisfies $P_A x = \\sum_{i \\in A} x_i e_i$, where $e_i$ is the unit vector equal to $1$ at\u2026","title":"Finite-set projections of the unconditional \u2113\u1d56 basis"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/norm-lp-schauder-basis-proj-apply-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.norm_lpSchauderBasis_proj_apply_le","parent":17484,"reading_title":{"format":"inline-parts-v1","parts":[["text","Basis projections of "],["math","\\(\\ell^p\\)"],["text"," are contractive"]],"source":"Basis projections of $\\ell^p$ are contractive"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#norm-lp-schauder-basis-proj-apply-le","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be an index set and let $1 \\le p < \\infty$. Let $e_i$ ($i \\in \\iota$) be the unit vectors, which form an unconditional Schauder basis of $\\ell^p(\\iota, \\mathbb{K})$. For every finite set $A \\subseteq \\iota$ and every $x \\in \\ell^p(\\iota, \\mathbb{K})$, the basis projection $P_A x = \\sum_{i \\in A} x_i e_i$ satisfies $\\|P_A x\\|_p \\le \\|x\\|_p$.","title":"Basis projections of $\\ell^p$ are contractive"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/norm-lp-schauder-basis-proj-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.norm_lpSchauderBasis_proj_le","parent":17484,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^p\\)"],["text"," basis projections have operator norm at most "],["math","\\(1\\)"]],"source":"The $\\ell^p$ basis projections have operator norm at most $1$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#norm-lp-schauder-basis-proj-le","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $\\iota$ be an index set and let $1 \\le p < \\infty$. Let $e_i$ ($i \\in \\iota$) be the unit vectors, which form an unconditional Schauder basis of $\\ell^p(\\iota, \\mathbb{K})$. For every finite set $A \\subseteq \\iota$, the basis projection $P_A$, the bounded linear operator on $\\ell^p(\\iota, \\mathbb{K})$ given by $P_A x = \\sum_{i \\in A} x_i e_i$, has operator norm\u2026","title":"The $\\ell^p$ basis projections have operator norm at most $1$"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/lp-schauder-basis-coord-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.lpSchauderBasis_coord_apply","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#lp-schauder-basis-coord-apply","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $1 \\le p < \\infty$. For every $i \\in \\mathbb{N}$ and every $x \\in \\ell^p(\\mathbb{N}, \\mathbb{K})$, the $i$-th coordinate functional of the Schauder basis of $\\ell^p(\\mathbb{N}, \\mathbb{K})$ by unit vectors sends $x$ to its $i$-th entry $x_i$.","title":"Coordinate functionals of the \u2113\u1d56 Schauder basis"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/lp-schauder-basis-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.lpSchauderBasis_apply","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#lp-schauder-basis-apply","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $1 \\le p < \\infty$. For every $i \\in \\mathbb{N}$, the $i$-th basis vector of the Schauder basis of $\\ell^p(\\mathbb{N}, \\mathbb{K})$ by unit vectors is the unit vector $e_i$, the sequence equal to $1$ at $i$ and to $0$ elsewhere.","title":"Basis vectors of the \u2113\u1d56 Schauder basis"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/lp-schauder-basis-proj-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.lpSchauderBasis_proj_apply","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#lp-schauder-basis-proj-apply","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $1 \\le p < \\infty$. For every $n \\in \\mathbb{N}$ and every $x \\in \\ell^p(\\mathbb{N}, \\mathbb{K})$, the $n$-th partial-sum projection $P_n$ of the Schauder basis of $\\ell^p(\\mathbb{N}, \\mathbb{K})$ by unit vectors satisfies $P_n x = \\sum_{i < n} x_i e_i$, where $e_i$ is the unit vector equal to $1$ at $i$ and to $0$ elsewhere; that is, $P_n x$ is the\u2026","title":"The \u2113\u1d56 partial-sum projections are truncations"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/norm-lp-schauder-basis-classical-proj-ap","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.norm_lpSchauderBasis_classical_proj_apply_le","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#norm-lp-schauder-basis-classical-proj-ap","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $1 \\le p < \\infty$. For every $n \\in \\mathbb{N}$ and every $x \\in \\ell^p(\\mathbb{N}, \\mathbb{K})$, the $n$-th partial-sum projection $P_n$ of the Schauder basis of $\\ell^p(\\mathbb{N}, \\mathbb{K})$ by unit vectors satisfies $\\|P_n x\\| \\le \\|x\\|$.","title":"The \u2113\u1d56 partial-sum projections are contractions"},{"id":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/norm-lp-schauder-basis-classical-proj-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.SchauderBasis.norm_lpSchauderBasis_classical_proj_le","parent":17484,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p/index.html#norm-lp-schauder-basis-classical-proj-le","scope":"Analysis/banach-spaces/sequence-spaces/schauder-basis-of-p","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $1 \\le p < \\infty$. For every $n \\in \\mathbb{N}$, the $n$-th partial-sum projection of the Schauder basis of $\\ell^p(\\mathbb{N}, \\mathbb{K})$ by unit vectors has operator norm at most $1$.","title":"The \u2113\u1d56 partial-sum projections have norm at most one"},{"data_url":"scopes/Analysis/banach-spaces/sequence-spaces/compactness-in-p.json","id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","kind":"scope","navigation":"module","parent":17377,"result":false,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","title":"Compactness in \u2113^p"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-restrict","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpRestrict","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Restriction of an "],["math","\\(\\ell^p\\)"],["text"," sequence to its first coordinates"]],"source":"Restriction of an $\\ell^p$ sequence to its first coordinates"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-restrict","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The restriction of an `\u2113^p` sequence to its first `N` coordinates.","title":"Restriction of an $\\ell^p$ sequence to its first coordinates"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-restrict-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpRestrict_apply","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pointwise action of the "],["math","\\(\\ell^p\\)"],["text"," restriction"]],"source":"Pointwise action of the $\\ell^p$ restriction"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-restrict-apply","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The restriction acts coordinatewise.","title":"Pointwise action of the $\\ell^p$ restriction"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-extend","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpExtend","parent":17497,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-extend","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The extension of a coordinate vector on `Fin N` by `0` to a total coordinate function on `\u2115`.","title":"Zero extension of a finite coordinate vector"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-extend-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpExtend_apply","parent":17497,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-extend-apply","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The extension acts coordinatewise.","title":"Pointwise action of the zero extension"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-lift","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpLift","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lift of a coordinate function to a truncated "],["math","\\(\\ell^p\\)"],["text"," sequence"]],"source":"Lift of a coordinate function to a truncated $\\ell^p$ sequence"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-lift","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The lift of a coordinate function on `\u2115` to an `\u2113^p` sequence, keeping only its first `N` coordinates.","title":"Lift of a coordinate function to a truncated $\\ell^p$ sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-trunc","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpTrunc","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncation of an "],["math","\\(\\ell^p\\)"],["text"," sequence, zeroing all but its first coordinates"]],"source":"Truncation of an $\\ell^p$ sequence, zeroing all but its first coordinates"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-trunc","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The all-`p` coordinate extension of finite truncation to the first `N` coordinates, also defined at `p = \u221e`. For finite `p \u2265 1`, `lpTrunc_eq_lpSchauderBasis_proj` identifies it with the canonical continuous linear Schauder projection.","title":"Truncation of an $\\ell^p$ sequence, zeroing all but its first coordinates"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-lift-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpLift_apply","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pointwise action of the finite "],["math","\\(\\ell^p\\)"],["text"," lift"]],"source":"Pointwise action of the finite $\\ell^p$ lift"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-lift-apply","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The lift acts coordinatewise.","title":"Pointwise action of the finite $\\ell^p$ lift"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-trunc-apply","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpTrunc_apply","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pointwise action of "],["math","\\(\\ell^p\\)"],["text"," truncation"]],"source":"Pointwise action of $\\ell^p$ truncation"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-trunc-apply","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"Truncation acts coordinatewise.","title":"Pointwise action of $\\ell^p$ truncation"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/lp-trunc-eq-lp-schauder-basis-proj","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.lpTrunc_eq_lpSchauderBasis_proj","parent":17497,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#lp-trunc-eq-lp-schauder-basis-proj","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"For finite `p \u2265 1`, coordinate truncation is the canonical bundled Schauder-basis projection.","title":"Truncation equals the Schauder basis projection"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/norm-lp-trunc-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.norm_lpTrunc_le","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncation is a contraction on "],["math","\\(\\ell^p\\)"]],"source":"Truncation is a contraction on $\\ell^p$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#norm-lp-trunc-le","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"Truncation is a contraction.","title":"Truncation is a contraction on $\\ell^p$"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/norm-sub-lp-trunc-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.norm_sub_lpTrunc_le","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tail of an "],["math","\\(\\ell^p\\)"],["text"," vector is no larger than the vector"]],"source":"Tail of an $\\ell^p$ vector is no larger than the vector"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#norm-sub-lp-trunc-le","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The tail of a vector is no larger than the vector itself.","title":"Tail of an $\\ell^p$ vector is no larger than the vector"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/norm-sub-lp-trunc-mono","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.norm_sub_lpTrunc_mono","parent":17497,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#norm-sub-lp-trunc-mono","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"Tails shrink as the truncation length grows.","title":"Tails shrink as the truncation length grows"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/norm-lp-lift-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.norm_lpLift_le","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norm bound for the finite "],["math","\\(\\ell^p\\)"],["text"," lift"]],"source":"Norm bound for the finite $\\ell^p$ lift"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#norm-lp-lift-le","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The target `\u2113^p` norm of the finite lift is bounded by `N ^ (1 / p)` times the supremum norm of its first `N` coordinates. The norm on the source Pi type `Fin N \u2192 \ud835\udd5c` is the supremum norm.","title":"Norm bound for the finite $\\ell^p$ lift"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/tendsto-lp-trunc","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.tendsto_lpTrunc","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Truncations of an "],["math","\\(\\ell^p\\)"],["text"," vector converge to it"]],"source":"Truncations of an $\\ell^p$ vector converge to it"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#tendsto-lp-trunc","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The truncations of an `\u2113^p` vector converge to the vector.","title":"Truncations of an $\\ell^p$ vector converge to it"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/exists-norm-sub-lp-trunc-lt","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.exists_norm_sub_lpTrunc_lt","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tail of a single "],["math","\\(\\ell^p\\)"],["text"," vector can be made small"]],"source":"Tail of a single $\\ell^p$ vector can be made small"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#exists-norm-sub-lp-trunc-lt","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The tail of a single vector can be made arbitrarily small.","title":"Tail of a single $\\ell^p$ vector can be made small"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/uniform-tail-vanishing","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.UniformTailVanishing","parent":17497,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#uniform-tail-vanishing","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The tails of the elements of a set `K` vanish uniformly: for every `\u03b5 > 0` there is `N` such that every `x \u2208 K` is within `\u03b5` of its truncation to the first `N` coordinates.","title":"Uniform vanishing of tails"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/totally-bounded-iff-bounded-and-uniform","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.totallyBounded_iff_bounded_and_uniformTailVanishing","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion for total boundedness in "],["math","\\(\\ell^p\\)"]],"source":"Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion for total boundedness in $\\ell^p$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#totally-bounded-iff-bounded-and-uniform","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"The Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion for `\u2113^p (\u2115, \ud835\udd5c)`, `1 \u2264 p < \u221e`: a set is totally bounded if and only if it is bounded and its tails vanish uniformly.","title":"Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion for total boundedness in $\\ell^p$"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/is-compact-iff-is-complete-and-bounded-a","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.isCompact_iff_isComplete_and_bounded_and_uniformTailVanishing","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion for compactness in "],["math","\\(\\ell^p\\)"]],"source":"Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion for compactness in $\\ell^p$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#is-compact-iff-is-complete-and-bounded-a","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"Compact form of the Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion.","title":"Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion for compactness in $\\ell^p$"},{"id":"Analysis/banach-spaces/sequence-spaces/compactness-in-p/is-compact-closure-iff-bounded-and-unifo","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.Compactness.isCompact_closure_iff_bounded_and_uniformTailVanishing","parent":17497,"reading_title":{"format":"inline-parts-v1","parts":[["text","Riesz\u2013Fr\u00e9chet\u2013Kolmogorov relative compactness criterion in "],["math","\\(\\ell^p\\)"]],"source":"Riesz\u2013Fr\u00e9chet\u2013Kolmogorov relative compactness criterion in $\\ell^p$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/compactness-in-p/index.html#is-compact-closure-iff-bounded-and-unifo","scope":"Analysis/banach-spaces/sequence-spaces/compactness-in-p","status":"fully_proved","summary":"Relative-compact form of the Riesz\u2013Fr\u00e9chet\u2013Kolmogorov criterion.","title":"Riesz\u2013Fr\u00e9chet\u2013Kolmogorov relative compactness criterion in $\\ell^p$"},{"data_url":"scopes/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in.json","id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","kind":"scope","navigation":"module","parent":17377,"result":false,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","title":"Schur's theorem on weak convergence in \u2113\u00b9"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/l1-norm-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.l1_norm_eq_tsum","parent":17517,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\ell^1\\)"],["text"," norm is the sum of the coordinate norms"]],"source":"The $\\ell^1$ norm is the sum of the coordinate norms"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#l1-norm-eq-tsum","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. Every $x \\in \\ell^1(\\mathbb{N}, \\mathbb{K})$ satisfies $\\|x\\| = \\sum_{i=0}^{\\infty} |x_i|$.","title":"The $\\ell^1$ norm is the sum of the coordinate norms"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/summable-bounded-mul-l1","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.summable_bounded_mul_l1","parent":17517,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pairing of "],["math","\\(\\ell^1\\)"],["text"," with sequences bounded by "],["math","\\(1\\)"],["text"," is summable"]],"source":"The pairing of $\\ell^1$ with sequences bounded by $1$ is summable"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#summable-bounded-mul-l1","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. If $y : \\mathbb{N} \\to \\mathbb{K}$ satisfies $\\|y_i\\| \\le 1$ for every $i$ and $x \\in \\ell^1(\\mathbb{N}, \\mathbb{K})$, then the series $\\sum_i y_i x_i$ is summable.","title":"The pairing of $\\ell^1$ with sequences bounded by $1$ is summable"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/l1-dual-of-bounded","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.l1DualOfBounded","parent":17517,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pairing "],["math","\\(\\ell^1\\)"],["text"," with a sequence bounded by one"]],"source":"Pairing $\\ell^1$ with a sequence bounded by one"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#l1-dual-of-bounded","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $(y_i)_{i \\in \\mathbb{N}}$ be a sequence in $\\mathbb{K}$ with $|y_i| \\le 1$ for every $i$. The pairing with $y$ is the continuous linear functional $\\ell^1(\\mathbb{N}, \\mathbb{K}) \\to \\mathbb{K}$ given by $x \\mapsto \\sum_{i=0}^{\\infty} y_i x_i$.","title":"Pairing $\\ell^1$ with a sequence bounded by one"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/norm-l1-dual-of-bounded-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.norm_l1DualOfBounded_le","parent":17517,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#norm-l1-dual-of-bounded-le","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $(y_i)_{i \\in \\mathbb{N}}$ be a sequence in $\\mathbb{K}$ with $|y_i| \\le 1$ for every $i$. The continuous linear functional $x \\mapsto \\sum_{i=0}^{\\infty} y_i x_i$ on $\\ell^1(\\mathbb{N}, \\mathbb{K})$ has operator norm at most $1$.","title":"Pairing with a sequence bounded by one has norm at most one"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/tsum-split3","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.tsum_split3","parent":17517,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#tsum-split3","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $h : \\mathbb{N} \\to \\mathbb{K}$ be summable and let $a \\le b$ be natural numbers. Then $\\sum_{i=0}^{\\infty} h(i)$ equals $\\sum_{i < a} h(i) + \\sum_{a \\le i < b} h(i) + \\sum_{j=0}^{\\infty} h(j + b)$.","title":"Splitting a series into a head, a middle block and a tail"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/summable-norm-shift","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.summable_norm_shift","parent":17517,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tails of an "],["math","\\(\\ell^1\\)"],["text"," sequence are absolutely summable"]],"source":"Tails of an $\\ell^1$ sequence are absolutely summable"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#summable-norm-shift","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^1(\\mathbb{N}, \\mathbb{K})$ and every $b \\in \\mathbb{N}$, the series $\\sum_{j=0}^{\\infty} \\|x_{j+b}\\|$ is summable.","title":"Tails of an $\\ell^1$ sequence are absolutely summable"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/tail-tsum-norm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.tail_tsum_norm","parent":17517,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tail of the norm series of an "],["math","\\(\\ell^1\\)"],["text"," sequence"]],"source":"Tail of the norm series of an $\\ell^1$ sequence"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#tail-tsum-norm","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in \\ell^1(\\mathbb{N}, \\mathbb{K})$ and every $b \\in \\mathbb{N}$, $\\sum_{j=0}^{\\infty} \\|x_{j+b}\\| = \\|x\\| - \\sum_{i < b} \\|x_i\\|$, where $\\|x\\|$ is the $\\ell^1$ norm of $x$.","title":"Tail of the norm series of an $\\ell^1$ sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/schur-block-of","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.schurBlockOf","parent":17517,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#schur-block-of","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $M : \\mathbb{N} \\to \\mathbb{N}$ be such that every $i \\in \\mathbb{N}$ satisfies $i < M(k + 1)$ for some $k$. The block index of $i \\in \\mathbb{N}$ is the least $k$ with $i < M(k + 1)$.","title":"Block index for a cutoff sequence"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/schur-block-of-mem","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.schurBlockOf_mem","parent":17517,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#schur-block-of-mem","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $M : \\mathbb{N} \\to \\mathbb{N}$ satisfy $M(0) = 0$, and suppose every $i \\in \\mathbb{N}$ satisfies $i < M(j + 1)$ for some $j$. For every $i \\in \\mathbb{N}$, if $k$ is the block index of $i$ (the least $k$ with $i < M(k + 1)$), then $M(k) \\le i < M(k + 1)$.","title":"An index lies in the block it is assigned to"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/schur-block-of-eq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.schurBlockOf_eq","parent":17517,"reading_title":{"format":"inline-parts-v1","parts":[["text","Indices in block "],["math","\\(k\\)"],["text"," have block index "],["math","\\(k\\)"]],"source":"Indices in block $k$ have block index $k$"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#schur-block-of-eq","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $M : \\mathbb{N} \\to \\mathbb{N}$ be monotone, and suppose every $i \\in \\mathbb{N}$ satisfies $i < M(j + 1)$ for some $j$. For all $k, i \\in \\mathbb{N}$ with $M(k) \\le i < M(k + 1)$, the block index of $i$ (the least $j$ with $i < M(j + 1)$) is $k$.","title":"Indices in block $k$ have block index $k$"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/schur-sign-seq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.schurSignSeq","parent":17517,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#schur-sign-seq","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $(x_n)$ be a sequence in $\\ell^1(\\mathbb{N}, \\mathbb{K})$, let $n' : \\mathbb{N} \\to \\mathbb{N}$, and let $M : \\mathbb{N} \\to \\mathbb{N}$ be such that every $i \\in \\mathbb{N}$ satisfies $i < M(j + 1)$ for some $j$. For $i \\in \\mathbb{N}$, let $k$ be the block index of $i$ (the least $k$ with $i < M(k + 1)$) and let $v = x_{n'(k)}(i)$. The $i$-th term $\\sigma_i$\u2026","title":"Conjugate-sign sequence of a block decomposition"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/norm-schur-sign-seq-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.norm_schurSignSeq_le","parent":17517,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#norm-schur-sign-seq-le","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $(x_n)$ be a sequence in $\\ell^1(\\mathbb{N}, \\mathbb{K})$, let $n' : \\mathbb{N} \\to \\mathbb{N}$, and let $M : \\mathbb{N} \\to \\mathbb{N}$ be such that every $i \\in \\mathbb{N}$ satisfies $i < M(j + 1)$ for some $j$. Every term $\\sigma_i$ of the conjugate-sign sequence of $(x_n)$, $n'$ and $M$ satisfies $|\\sigma_i| \\le 1$.","title":"The conjugate-sign sequence is bounded by one"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/schur-sign-seq-mul","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.schurSignSeq_mul","parent":17517,"reading_title":{"format":"inline-parts-v1","parts":[["text","On block "],["math","\\(k\\)"],["text",", the conjugate signs recover absolute values"]],"source":"On block $k$, the conjugate signs recover absolute values"},"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#schur-sign-seq-mul","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $(x_n)$ be a sequence in $\\ell^1(\\mathbb{N}, \\mathbb{K})$, let $n' : \\mathbb{N} \\to \\mathbb{N}$, and let $M : \\mathbb{N} \\to \\mathbb{N}$ be monotone and such that every $i \\in \\mathbb{N}$ satisfies $i < M(j + 1)$ for some $j$. Let $\\sigma$ be the conjugate-sign sequence of $(x_n)$, $n'$ and $M$. For all $k, i \\in \\mathbb{N}$ with $M(k) \\le i < M(k + 1)$, we have\u2026","title":"On block $k$, the conjugate signs recover absolute values"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/schur-tendsto-of-weak-tendsto","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.schur_tendsto_of_weak_tendsto","parent":17517,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#schur-tendsto-of-weak-tendsto","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $(x_n)$ be a sequence in $\\ell^1(\\mathbb{N}, \\mathbb{K})$ and let $x_0 \\in \\ell^1(\\mathbb{N}, \\mathbb{K})$. If $F(x_n) \\to F(x_0)$ for every continuous linear functional $F$ on $\\ell^1(\\mathbb{N}, \\mathbb{K})$, that is, if $(x_n)$ converges weakly to $x_0$, then $x_n \\to x_0$ in the $\\ell^1$ norm.","title":"Schur's theorem"},{"id":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/schur-tendsto-norm-sub-of-weak-tendsto","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SchurL1.schur_tendsto_norm_sub_of_weak_tendsto","parent":17517,"result":true,"route":"../roadmap/Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in/index.html#schur-tendsto-norm-sub-of-weak-tendsto","scope":"Analysis/banach-spaces/sequence-spaces/schur-s-theorem-on-weak-convergence-in","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, let $(x_n)$ be a sequence in $\\ell^1(\\mathbb{N}, \\mathbb{K})$, and let $x \\in \\ell^1(\\mathbb{N}, \\mathbb{K})$. If $F(x_n) \\to F(x)$ as $n \\to \\infty$ for every continuous linear functional $F : \\ell^1(\\mathbb{N}, \\mathbb{K}) \\to \\mathbb{K}$, then $\\|x_n - x\\| \\to 0$ as $n \\to \\infty$.","title":"Schur's theorem, norm form"},{"data_url":"scopes/Analysis/banach-spaces/polynomials-on-banach-spaces.json","id":"Analysis/banach-spaces/polynomials-on-banach-spaces","kind":"scope","navigation":"module","parent":17193,"result":false,"route":"../roadmap/Analysis/banach-spaces/polynomials-on-banach-spaces/index.html","scope":"Analysis/banach-spaces/polynomials-on-banach-spaces","status":"can_prove","title":"Polynomials on Banach spaces"},{"id":"Analysis/banach-spaces/polynomials-on-banach-spaces/is-polynomial-on","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook27Wanted.IsPolynomialOn","parent":17533,"result":true,"route":"../roadmap/Analysis/banach-spaces/polynomials-on-banach-spaces/index.html#is-polynomial-on","scope":"Analysis/banach-spaces/polynomials-on-banach-spaces","status":"fully_proved","summary":"Let $E$ be a complex normed space. A function $F : E \\to \\mathbb{C}$ is a continuous polynomial function if there are a natural number $n$, natural numbers $m_1, \\dots, m_n$ and continuous $\\mathbb{C}$-multilinear maps $A_i : E^{m_i} \\to \\mathbb{C}$, with $A_i$ taking $m_i$ arguments, such that $F(x) = \\sum_{i=1}^{n} A_i(x, \\dots, x)$ for every $x \\in E$.","title":"Continuous polynomial functions on a complex normed space"},{"id":"Analysis/banach-spaces/polynomials-on-banach-spaces/bezout-identity-for-polynomials-without","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook27Wanted.bezout_identity_for_polynomials_without_common_zero","parent":17533,"result":true,"route":"../roadmap/Analysis/banach-spaces/polynomials-on-banach-spaces/index.html#bezout-identity-for-polynomials-without","scope":"Analysis/banach-spaces/polynomials-on-banach-spaces","status":"can_prove","summary":"Let $E$ be a complex Banach space and let $F, G : E \\to \\mathbb{C}$ be continuous polynomial functions with no common zero, that is, $G(x) \\neq 0$ whenever $F(x) = 0$. Then there are continuous polynomial functions $\\varphi, \\psi : E \\to \\mathbb{C}$ such that $F(x) \\varphi(x) + G(x) \\psi(x) = 1$ for every $x \\in E$.","title":"Bezout identity for polynomials without a common zero"},{"id":"Analysis/banach-spaces/polynomials-on-banach-spaces/scottish-book-problem55-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook55Wanted.ScottishBookProblem55Claim","parent":17533,"result":true,"route":"../roadmap/Analysis/banach-spaces/polynomials-on-banach-spaces/index.html#scottish-book-problem55-claim","scope":"Analysis/banach-spaces/polynomials-on-banach-spaces","status":"fully_proved","summary":"The claim that for every real Banach space $E$, every unbounded set $R \\subseteq E$ and every continuous bilinear form $B : E \\times E \\to \\mathbb{R}$, if the quadratic form $W(x) = B(x, x)$ is bounded on the set of points at distance less than $\\varepsilon$ from $R$ for some $\\varepsilon > 0$, then there are a continuous linear functional $L : E \\to \\mathbb{R}$, a constant $c \\in \\mathbb{R}$ and a real polynomial\u2026","title":"Mazur's factorisation claim (Scottish Book Problem 55)"},{"id":"Analysis/banach-spaces/polynomials-on-banach-spaces/mazur55-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook55Wanted.mazur55_refuted","parent":17533,"result":true,"route":"../roadmap/Analysis/banach-spaces/polynomials-on-banach-spaces/index.html#mazur55-refuted","scope":"Analysis/banach-spaces/polynomials-on-banach-spaces","status":"can_prove","summary":"The factorisation claim of Scottish Book Problem 55 is false: there are a real Banach space $E$, an unbounded set $R \\subseteq E$ and a continuous bilinear form $B : E \\times E \\to \\mathbb{R}$ such that for some $\\varepsilon > 0$ the quadratic form $W(x) = B(x, x)$ is bounded on the set of points at distance less than $\\varepsilon$ from $R$, yet there are no continuous linear functional $L : E \\to \\mathbb{R}$,\u2026","title":"Mazur's Problem 55 has a negative answer"},{"id":"Analysis/banach-spaces/polynomials-on-banach-spaces/scottish-book-problem75-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook75Wanted.ScottishBookProblem75Claim","parent":17533,"result":true,"route":"../roadmap/Analysis/banach-spaces/polynomials-on-banach-spaces/index.html#scottish-book-problem75-claim","scope":"Analysis/banach-spaces/polynomials-on-banach-spaces","status":"fully_proved","summary":"The claim that for every real Banach space $E$, every set $R \\subseteq E$, every continuous bilinear form $B$ on $E$ and every real $\\alpha \\ne 0$ the following holds: if for some $\\varepsilon > 0$ the values $|B(x, x)|$ are bounded on the open $\\varepsilon$-neighbourhood of $R$, that is, for the $x \\in E$ with $\\operatorname{dist}(x, r) < \\varepsilon$ for some $r \\in R$, then for some $\\delta > 0$ the values\u2026","title":"Scottish Book Problem 75 (Mazur), general Banach-space form"},{"id":"Analysis/banach-spaces/polynomials-on-banach-spaces/mazur75-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook75Wanted.mazur75_refuted","parent":17533,"result":true,"route":"../roadmap/Analysis/banach-spaces/polynomials-on-banach-spaces/index.html#mazur75-refuted","scope":"Analysis/banach-spaces/polynomials-on-banach-spaces","status":"can_prove","summary":"The Scottish Book Problem 75 claim is false: there are a real Banach space $E$, a set $R \\subseteq E$, a continuous bilinear form $B$ on $E$ and a real $\\alpha \\ne 0$ such that $|B(x, x)|$ is bounded on some open $\\varepsilon$-neighbourhood of $R$ with $\\varepsilon > 0$, but is unbounded on every open $\\delta$-neighbourhood of $\\alpha R = \\{\\alpha r : r \\in R\\}$ with $\\delta > 0$.","title":"Mazur's Scottish Book Problem 75 fails in general Banach spaces"},{"data_url":"scopes/Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets.json","id":"Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets","kind":"scope","navigation":"module","parent":17193,"result":false,"route":"../roadmap/Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets/index.html","scope":"Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets","status":"can_prove","title":"Symmetric hulls of nowhere dense convex sets"},{"id":"Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets/scottish-book-problem65-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook65Wanted.ScottishBookProblem65Claim","parent":17540,"result":true,"route":"../roadmap/Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets/index.html#scottish-book-problem65-claim","scope":"Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets","status":"fully_proved","summary":"The claim that for every real Banach space $E$ and every convex set $W \\subseteq E$ with $0 \\in W$, if the closure of $W$ has empty interior (that is, $W$ is nowhere dense), then the closure of the convex hull of $W \\cup (-W)$ also has empty interior. The convex hull of $W \\cup (-W)$ is the smallest convex set containing $W$ that is symmetric about the origin.","title":"Mazur's symmetric hull claim (Scottish Book Problem 65)"},{"id":"Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets/mazur65-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook65Wanted.mazur65_refuted","parent":17540,"result":true,"route":"../roadmap/Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets/index.html#mazur65-refuted","scope":"Analysis/banach-spaces/symmetric-hulls-of-nowhere-dense-convex-sets","status":"can_prove","summary":"The symmetric hull claim of Scottish Book Problem 65 is false: there are a real Banach space $E$ and a convex set $W \\subseteq E$ with $0 \\in W$ whose closure has empty interior, such that the closure of the convex hull of $W \\cup (-W)$ has nonempty interior.","title":"Mazur's Problem 65 has a negative answer"},{"data_url":"scopes/Analysis/special-functions.json","id":"Analysis/special-functions","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/special-functions/index.html","scope":"Analysis/special-functions","status":"can_prove","summary":"Named transcendental functions and their series, integral representations and identities: gamma and digamma, hypergeometric functions, exponential and logarithmic integrals, Lambert W, Bessel, Dickman, Bernoulli polynomials, polylogarithms and Clausen functions, and elliptic functions (elliptic-function topics sit directly in the field).","title":"Special functions"},{"data_url":"scopes/Analysis/special-functions/gamma-function.json","id":"Analysis/special-functions/gamma-function","kind":"scope","parent":17543,"result":false,"route":"../roadmap/Analysis/special-functions/gamma-function/index.html","scope":"Analysis/special-functions/gamma-function","status":"can_prove","summary":"The gamma, digamma and trigamma functions: series and integral representations, factorial series, growth bounds against exponentials, and Ramanujan's digamma identities.","title":"Gamma and digamma functions"},{"id":"Analysis/special-functions/gamma-function/ramanujan-part1-ch8-entry15-doublegamma","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry15Doublegamma.ramanujan_part1_ch8_entry15_doublegamma","parent":17544,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's asymptotic expansion of "],["math","\\(((x + 1/2)/e^{\\psi(x+1)})^{4n}\\)"]],"source":"Ramanujan's asymptotic expansion of $((x + 1/2)/e^{\\psi(x+1)})^{4n}$"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/index.html#ramanujan-part1-ch8-entry15-doublegamma","scope":"Analysis/special-functions/gamma-function","status":"fully_proved","summary":"For real $x$ let $\\psi(x) = \\Gamma'(x)/\\Gamma(x)$ and $a(x) = e^{\\psi(x+1)}$. Then $a(x) \\to \\infty$ as $x \\to \\infty$, and for every complex number $n$,","title":"Ramanujan's asymptotic expansion of $((x + 1/2)/e^{\\psi(x+1)})^{4n}$"},{"id":"Analysis/special-functions/gamma-function/ramanujan-part1-ch8-entry17-loggammainte","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry17Loggammaintegral.ramanujan_part1_ch8_entry17_loggammaintegral","parent":17544,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's multiplication formula for the series "],["math","\\(\\sum_k (\\log k/k - \\log(k+x)/(k+x))\\)"]],"source":"Ramanujan's multiplication formula for the series $\\sum_k (\\log k/k - \\log(k+x)/(k+x))$"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/index.html#ramanujan-part1-ch8-entry17-loggammainte","scope":"Analysis/special-functions/gamma-function","status":"fully_proved","summary":"For real $x$ let","title":"Ramanujan's multiplication formula for the series $\\sum_k (\\log k/k - \\log(k+x)/(k+x))$"},{"data_url":"scopes/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions.json","id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","kind":"scope","navigation":"module","parent":17544,"result":false,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","title":"Digamma and trigamma functions"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/summable-one-div-add-nat-succ-sq","kind":"node","lean":"Real.summable_one_div_add_nat_succ_sq","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#summable-one-div-add-nat-succ-sq","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"The series of real numbers $\\sum_{j \\ge 0} \\frac{1}{(j+1)^2}$ is summable.","title":"Summability of the Basel series"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/has-sum-digamma-series","kind":"node","lean":"Real.hasSum_digamma_series","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#has-sum-digamma-series","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For every real $x > 0$, the series $\\sum_{j=0}^{\\infty} \\left(\\frac{1}{j+1} - \\frac{1}{x+j+1}\\right)$ has sum $\\psi(x+1) + \\gamma$, where $\\psi = (\\log \\Gamma)'$ with $\\Gamma$ the real Gamma function, and $\\gamma$ is the Euler\u2013Mascheroni constant.","title":"Digamma series on the positive reals"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/has-deriv-at-tsum-digamma-series","kind":"node","lean":"Real.hasDerivAt_tsum_digamma_series","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#has-deriv-at-tsum-digamma-series","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For every real $y > 0$, the real function $z \\mapsto \\sum_{j=0}^{\\infty} \\left(\\frac{1}{j+1} - \\frac{1}{z+j+1}\\right)$ is differentiable at $y$ with derivative $\\sum_{j=0}^{\\infty} \\frac{1}{(y+j+1)^2}$.","title":"Derivative of the digamma series"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/deriv-deriv-log-gamma-add-one","kind":"node","lean":"Real.deriv_deriv_log_Gamma_add_one","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#deriv-deriv-log-gamma-add-one","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For every real $y > 0$, the second derivative of $\\log \\Gamma$ at $y + 1$ equals $\\sum_{j=0}^{\\infty} \\frac{1}{(y + j + 1)^2}$, where $\\Gamma$ is the real Gamma function.","title":"Trigamma series"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/deriv-log-gamma","kind":"node","lean":"Real.deriv_log_Gamma","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#deriv-log-gamma","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For every real $u > 0$, the derivative of $\\log \\Gamma$ at $u$ equals $\\Gamma'(u) / \\Gamma(u)$, where $\\Gamma$ is the real Gamma function.","title":"Logarithmic derivative of the real Gamma function"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/analytic-at-gamma","kind":"node","lean":"Complex.analyticAt_Gamma","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#analytic-at-gamma","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"The complex Gamma function is analytic at every $w \\in \\mathbb{C}$ that is not a nonpositive integer.","title":"Analyticity of the Gamma function"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/analytic-at-digamma","kind":"node","lean":"Complex.analyticAt_digamma","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#analytic-at-digamma","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"The digamma function $\\psi = \\Gamma'/\\Gamma$, where $\\Gamma$ is the complex Gamma function, is complex analytic at every $w \\in \\mathbb{C}$ that is not a nonpositive integer.","title":"Analyticity of the digamma function"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/deriv-gamma-of-real","kind":"node","lean":"Complex.deriv_Gamma_ofReal","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#deriv-gamma-of-real","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For every real $t > 0$, the derivative of the complex Gamma function at $t$ equals the derivative of the real Gamma function at $t$.","title":"Complex and real derivatives of Gamma agree on the positive reals"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/summable-norm-one-div-add-nat-succ-sq","kind":"node","lean":"Complex.summable_norm_one_div_add_nat_succ_sq","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#summable-norm-one-div-add-nat-succ-sq","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For every complex number $z$, the series $\\sum_{k \\ge 0} \\frac{1}{(z + k + 1)^2}$ converges absolutely, that is, the series of absolute values $\\sum_{k \\ge 0} \\left|\\frac{1}{(z + k + 1)^2}\\right|$ is summable, where a term whose denominator vanishes is read as $0$.","title":"Absolute convergence of the trigamma series"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/digamma-series-term","kind":"node","lean":"Complex.digammaSeriesTerm","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#digamma-series-term","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For $j \\in \\mathbb{N}$ and $s \\in \\mathbb{C}$, the $j$-th term of the complex digamma series at $s$ is $\\frac{1}{j+1} - \\frac{1}{j+s}$, where $1/0$ is read as $0$.","title":"Terms of the complex digamma series"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/digamma-series-fun","kind":"node","lean":"Complex.digammaSeriesFun","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#digamma-series-fun","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For $s \\in \\mathbb{C}$, the digamma series function at $s$ is $\\sum_{j=0}^{\\infty} \\left(\\frac{1}{j+1} - \\frac{1}{j+s}\\right)$, the sum of the terms of the complex digamma series at $s$.","title":"Complex digamma series"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/summable-digamma-series-term","kind":"node","lean":"Complex.summable_digammaSeriesTerm","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#summable-digamma-series-term","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the sequence of digamma series terms $\\frac{1}{j+1} - \\frac{1}{j+s}$, for $j = 0, 1, 2, \\dots$, is summable.","title":"Summability of the digamma series terms"},{"id":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/has-sum-digamma-series-2","kind":"node","lean":"Complex.hasSum_digamma_series","parent":17547,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/digamma-and-trigamma-functions/index.html#has-sum-digamma-series-2","scope":"Analysis/special-functions/gamma-function/digamma-and-trigamma-functions","status":"fully_proved","summary":"For every complex number $s$ with $\\operatorname{Re} s > 1$, the series $\\sum_{j=0}^{\\infty} \\left(\\frac{1}{j+1} - \\frac{1}{j+s}\\right)$ of the terms of the complex digamma series at $s$ has sum $\\psi(s) + \\gamma$, where $\\psi = \\Gamma'/\\Gamma$ is the complex digamma function, the logarithmic derivative of the complex Gamma function, and $\\gamma$ is the Euler\u2013Mascheroni constant.","title":"Series expansion of the complex digamma function"},{"data_url":"scopes/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers.json","id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","kind":"scope","navigation":"module","parent":17544,"result":false,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","title":"Iterated factorials and power towers"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/iterate-logb-tetration-strict-mono-and-c","kind":"node","lean":"MetaMathlibExt.iterate_logb_tetration_strictMono_and_converges","parent":17561,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#iterate-logb-tetration-strict-mono-and-c","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"Let $a, b$ be real numbers with $e^{1/e} < a < b$. For $n \\ge 0$ let $b \\uparrow\\uparrow n$ be the $n$-fold iterate of $x \\mapsto b^x$ starting at $1$ (so $b \\uparrow\\uparrow 0 = 1$ and $b \\uparrow\\uparrow 1 = b$), and let $a_n = \\log_a^{(n)}(b \\uparrow\\uparrow n)$ be its $n$-fold iterated base-$a$ logarithm, so that $a_0 = 1$. Then the sequence $(a_n)_{n \\ge 0}$ is strictly increasing and converges to a real number.","title":"Iterated logarithms of power towers increase to a limit"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/div-exp-one-rpow-lt-gamma-add-one","kind":"node","lean":"MetaMathlibExt.div_exp_one_rpow_lt_Gamma_add_one","parent":17561,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound "],["math","\\(\\Gamma(x + 1) > (x/e)^x\\)"],["text"," for the Gamma function"]],"source":"Lower bound $\\Gamma(x + 1) > (x/e)^x$ for the Gamma function"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#div-exp-one-rpow-lt-gamma-add-one","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"For every real number $x \\ge 1$, $\\Gamma(x + 1) > (x/e)^x$.","title":"Lower bound $\\Gamma(x + 1) > (x/e)^x$ for the Gamma function"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/gamma-add-one-mul-log-le-gamma-rpow-self","kind":"node","lean":"MetaMathlibExt.Gamma_add_one_mul_log_le_Gamma_rpow_self","parent":17561,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound "],["math","\\(\\Gamma(x + 1) \\ln \\Gamma(x + 1) \\le \\Gamma(x)^{\\Gamma(x)}\\)"]],"source":"Upper bound $\\Gamma(x + 1) \\ln \\Gamma(x + 1) \\le \\Gamma(x)^{\\Gamma(x)}$"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#gamma-add-one-mul-log-le-gamma-rpow-self","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"For every real number $x > 4$, $\\Gamma(x + 1) \\ln \\Gamma(x + 1) \\le \\Gamma(x)^{\\Gamma(x)}$.","title":"Upper bound $\\Gamma(x + 1) \\ln \\Gamma(x + 1) \\le \\Gamma(x)^{\\Gamma(x)}$"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/exists-tetration-eq-iterate-rpow-eventua","kind":"node","lean":"MetaMathlibExt.exists_tetration_eq_iterate_rpow_eventually","parent":17561,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#exists-tetration-eq-iterate-rpow-eventua","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"Let $a, b > e^{1/e}$ be real numbers with $a \\ne b$. For a real number $c$ and $m \\ge 0$ let $E_m(b, c)$ be the $m$-fold iterate of $x \\mapsto b^x$ starting at $c$ (the tower $b^{b^{\\cdots^{c}}}$ with $m$ copies of $b$), and let $a \\uparrow\\uparrow n$ be the $n$-fold iterate of $x \\mapsto a^x$ starting at $1$. Then there are a natural number $N$, an integer $k$ and a convergent sequence $(c_n)$ of real numbers such\u2026","title":"Normalization of power towers"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/exists-tetration-eq-iterate-rpow","kind":"node","lean":"MetaMathlibExt.exists_tetration_eq_iterate_rpow","parent":17561,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#exists-tetration-eq-iterate-rpow","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"Let $a, b > e^{1/e}$ be real numbers and $n > 0$ a natural number. Let $a \\uparrow\\uparrow n$ be the $n$-fold iterate of $x \\mapsto a^x$ starting at $1$, and $E_p(b, c)$ the $p$-fold iterate of $x \\mapsto b^x$ starting at $c$. Then there are a natural number $p$ and a real number $c$ with $1 < c \\le b$ such that $a \\uparrow\\uparrow n = E_p(b, c)$.","title":"Power towers rewritten in another base"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/iterate-factorial-lt-iterate-factorial","kind":"node","lean":"MetaMathlibExt.iterate_factorial_lt_iterate_factorial","parent":17561,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#iterate-factorial-lt-iterate-factorial","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"For natural numbers $j$ and $m$ let $F_j(m)$ be the $j$-fold iterated factorial $m!\\cdots!$ (with $j$ factorials). Let $n, n' \\ge 3$ and $k > k'$ be natural numbers. If $n' < F_{k-k'}(n)$, then $F_{k'}(n') < F_k(n)$.","title":"Comparison of iterated factorials"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/exists-iterate-factorial-eq-iterate-rpow","kind":"node","lean":"MetaMathlibExt.exists_iterate_factorial_eq_iterate_rpow_ten","parent":17561,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterated factorials as base-"],["math","\\(10\\)"],["text"," power towers"]],"source":"Iterated factorials as base-$10$ power towers"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#exists-iterate-factorial-eq-iterate-rpow","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"Let $k \\ge 3$ be a natural number, let $F_n(k)$ be the $n$-fold iterated factorial of $k$, viewed as a real number, and for a real number $c$ and $m \\ge 0$ let $E_m(10, c)$ be the $m$-fold iterate of $x \\mapsto 10^x$ starting at $c$. Then there are an integer $\\ell$, a natural number $N > 0$, a sequence $(c_n)$ of real numbers and a real number $L$ with the following properties: for every $n \\ge N$ we have\u2026","title":"Iterated factorials as base-$10$ power towers"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/exists-iterate-factorial-eq-tetration","kind":"node","lean":"MetaMathlibExt.exists_iterate_factorial_eq_tetration","parent":17561,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#exists-iterate-factorial-eq-tetration","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"Let $k \\ge 3$ be a natural number and $F_n(k)$ the $n$-fold iterated factorial of $k$; for a real number $\\beta$ let $\\beta \\uparrow\\uparrow n$ be the $n$-fold iterate of $x \\mapsto \\beta^x$ starting at $1$. Then there is a sequence $(b_n)$ of real numbers such that $b_n > e^{1/e}$ and $F_n(k) = b_n \\uparrow\\uparrow n$ for every $n \\ge 1$, and $(b_n)$ converges to a limit $b \\le k!$.","title":"Iterated factorials as tetrations"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/exists-iterate-gamma-eq-iterate-rpow","kind":"node","lean":"MetaMathlibExt.exists_iterate_Gamma_eq_iterate_rpow","parent":17561,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#exists-iterate-gamma-eq-iterate-rpow","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"Let $z > 1$ be a real number with $\\Gamma(z) = z$, let $b > e^{1/e}$ and $x > z$ be real numbers, let $\\Gamma^n$ be the $n$-fold iterate of $\\Gamma$, and for a real number $c$ and $m \\ge 0$ let $E_m(b, c)$ be the $m$-fold iterate of $y \\mapsto b^y$ starting at $c$. Then there are an integer $\\ell$, a natural number $N > 0$, a sequence $(c_n)$ of real numbers and a real number $L$ with the following properties: for\u2026","title":"Iterated Gamma values as power towers"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/iterate-rpow-succ-lt-gamma-iterate-rpow","kind":"node","lean":"MetaMathlibExt.iterate_rpow_succ_lt_Gamma_iterate_rpow","parent":17561,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#iterate-rpow-succ-lt-gamma-iterate-rpow","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"Let $b \\ge e$ and, for $m \\ge 0$, let $E_m(b, b)$ be the $m$-fold iterate of $x \\mapsto b^x$ started at $b$, so that $E_0(b, b) = b$ and $E_{m+1}(b, b) = b^{E_m(b, b)}$. Then for every integer $n > 0$, $E_{n+1}(b, b) < \\Gamma(E_n(b, b))$.","title":"Gamma of an iterated exponential exceeds the next iterate"},{"id":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/exists-tetration-eq-iterate-gamma","kind":"node","lean":"MetaMathlibExt.exists_tetration_eq_iterate_Gamma","parent":17561,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers/index.html#exists-tetration-eq-iterate-gamma","scope":"Analysis/special-functions/gamma-function/iterated-factorials-and-power-towers","status":"can_prove","summary":"Let $z > 1$ be a real number with $\\Gamma(z) = z$, and let $b > z$. Let $b \\uparrow\\uparrow n$ be the $n$-fold iterate of $y \\mapsto b^y$ starting at $1$, and $\\Gamma^n$ the $n$-fold iterate of $\\Gamma$. Then there is a convergent sequence $(x_n)$ of real numbers, strictly decreasing from some index on, such that $b \\uparrow\\uparrow n = \\Gamma^n(x_n)$ for every $n \\ge 1$.","title":"Tetrations as iterated Gamma values"},{"data_url":"scopes/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function.json","id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","kind":"scope","navigation":"module","parent":17544,"result":false,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","title":"Series and integrals for the digamma function"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch6-entry2-stirlingserie","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry2Stirlingseries.ramanujan_part1_ch6_entry2_stirlingseries","parent":17573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Series for the derivative of "],["math","\\(\\Gamma'(z+1)/\\Gamma(z+1)\\)"]],"source":"Series for the derivative of $\\Gamma'(z+1)/\\Gamma(z+1)$"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch6-entry2-stirlingserie","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $\\gamma$ be the Euler\u2013Mascheroni constant and let $\\varphi(w) = \\gamma + \\Gamma'(w+1)/\\Gamma(w+1)$ for complex $w$. For every complex number $z$ other than $-1, -2, -3, \\dots$, the function $\\varphi$ is complex differentiable at $z$, and the series $\\sum_{k=1}^{\\infty} \\frac{1}{(z+k)^2}$ converges absolutely with sum $\\varphi'(z)$.","title":"Series for the derivative of $\\Gamma'(z+1)/\\Gamma(z+1)$"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry10-qgammalimit","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry10Qgammalimit.ramanujan_part1_ch8_entry10_qgammalimit","parent":17573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's integral for "],["math","\\(\\psi(1 - 1/x)\\)"],["text"," (Chapter 8, Entry 10)"]],"source":"Ramanujan's integral for $\\psi(1 - 1/x)$ (Chapter 8, Entry 10)"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry10-qgammalimit","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $x$ be a complex number with $\\operatorname{Re} x < 0$. For real $u$ put $u^x = e^{x \\log |u|}$ when $u \\ne 0$, and define $K_x(0) = 0$, $K_x(1) = -1/x$ and $K_x(u) = \\frac{1 - u}{u\\,(u^x - 1)}$ for every other real $u$, the quotient being taken to be $0$ at $u = -1$, where its denominator vanishes. Then $x \\ne 0$ and $\\Gamma(1 - 1/x) \\ne 0$; for every $u \\in (0, 1)$ the denominator $u\\,(u^x - 1)$ is nonzero\u2026","title":"Ramanujan's integral for $\\psi(1 - 1/x)$ (Chapter 8, Entry 10)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/chapter8-digamma-eq-digamma","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry11Qgammafunctional.chapter8Digamma_eq_digamma","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#chapter8-digamma-eq-digamma","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"For every complex number $z$, $\\Gamma'(z)/\\Gamma(z) = \\psi(z)$, where $\\psi$ is the digamma function.","title":"Digamma as the logarithmic derivative of Gamma (used for Chapter 8, Entry 11)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/chapter8-phi-term-eq-entry12-qgammainteg","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry11Qgammafunctional.chapter8PhiTerm_eq_entry12Qgammaintegers","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#chapter8-phi-term-eq-entry12-qgammainteg","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"For every complex number $x$ and every natural number $j$, with $k = j + 1$, the term $\\frac{2}{(kx)^3 - kx}$ as defined for Entry 11 equals the term given by the same formula for Entry 12.","title":"The Entry 11 and Entry 12 series terms agree"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/chapter8-phi-eq-entry12-qgammaintegers","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry11Qgammafunctional.chapter8Phi_eq_entry12Qgammaintegers","parent":17573,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Entry 11 and Entry 12 definitions of "],["math","\\(\\varphi\\)"],["text"," agree"]],"source":"The Entry 11 and Entry 12 definitions of $\\varphi$ agree"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#chapter8-phi-eq-entry12-qgammaintegers","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"For every complex number $x$, the function $\\varphi(x) = 1 + \\sum_{k \\ge 1} \\frac{2}{(kx)^3 - kx}$ as defined for Entry 11 equals the function given by the same formula for Entry 12.","title":"The Entry 11 and Entry 12 definitions of $\\varphi$ agree"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry11-qgammafuncti","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry11Qgammafunctional.ramanujan_part1_ch8_entry11_qgammafunctional_digamma","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry11-qgammafuncti","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $x$ be a nonzero complex number with $x \\ne 1/k$ and $x \\ne -1/k$ for every positive integer $k$, and put $\\varphi(x) = 1 + \\sum_{k \\ge 1} \\frac{2}{(kx)^3 - kx}$. Then the series $\\sum_{k \\ge 1} \\frac{2}{(kx)^3 - kx}$ converges unconditionally, $\\Gamma(1/x) \\ne 0$, $\\Gamma(1 - 1/x) \\ne 0$, and","title":"Chapter 8, Entry 11 in terms of the digamma function"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry11-qgammafuncti-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry11Qgammafunctional.ramanujan_part1_ch8_entry11_qgammafunctional","parent":17573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's identity for "],["math","\\(\\Gamma'/\\Gamma\\)"],["text"," at "],["math","\\(1/x\\)"],["text"," and "],["math","\\(1 - 1/x\\)"],["text"," (Chapter 8, Entry 11)"]],"source":"Ramanujan's identity for $\\Gamma'/\\Gamma$ at $1/x$ and $1 - 1/x$ (Chapter 8, Entry 11)"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry11-qgammafuncti-2","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $x$ be a nonzero complex number with $x \\ne 1/k$ and $x \\ne -1/k$ for every positive integer $k$, and put $\\varphi(x) = 1 + \\sum_{k \\ge 1} \\frac{2}{(kx)^3 - kx}$. Then the series $\\sum_{k \\ge 1} \\frac{2}{(kx)^3 - kx}$ converges unconditionally, $\\Gamma(1/x) \\ne 0$, $\\Gamma(1 - 1/x) \\ne 0$, and","title":"Ramanujan's identity for $\\Gamma'/\\Gamma$ at $1/x$ and $1 - 1/x$ (Chapter 8, Entry 11)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/chapter8-unit-pow-eq-cpow","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry12Qgammaintegers.chapter8UnitPow_eq_cpow","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#chapter8-unit-pow-eq-cpow","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"For a complex number $x$ and a real number $u$, let $P_x(u) = 0$ if $u = 0$ and $P_x(u) = e^{x \\log |u|}$ otherwise. Then for every complex $x$ and every real $u > 0$, $P_x(u) = u^x$, where $u^x$ is the principal complex power.","title":"Exponential and principal powers agree for positive bases (Chapter 8, Entry 12)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry12-qgammaintege","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry12Qgammaintegers.ramanujan_part1_ch8_entry12_qgammaintegers_cpow","parent":17573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's integral for "],["math","\\(\\varphi(x) - 1\\)"],["text"," with principal powers (Chapter 8, Entry 12)"]],"source":"Ramanujan's integral for $\\varphi(x) - 1$ with principal powers (Chapter 8, Entry 12)"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry12-qgammaintege","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $x$ be a complex number with $\\operatorname{Re} x > 1$. For real $u$ and complex $s$ put $P_s(u) = 0$ if $u = 0$ and $P_s(u) = e^{s \\log |u|}$ otherwise, and define $K_x(0) = 0$, $K_x(1) = 0$ and $K_x(u) = \\frac{P_{x-2}(u)\\,(1 - u)^2}{1 - P_x(u)}$ for every other real $u$, the quotient being taken to be $0$ at $u = -1$, where its denominator vanishes. Let $\\varphi(x) = 1 + \\sum_{k \\ge 1} \\frac{2}{(kx)^3 - kx}$.\u2026","title":"Ramanujan's integral for $\\varphi(x) - 1$ with principal powers (Chapter 8, Entry 12)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry12-qgammaintege-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry12Qgammaintegers.ramanujan_part1_ch8_entry12_qgammaintegers","parent":17573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's integral for "],["math","\\(\\varphi(x) - 1\\)"],["text"," (Chapter 8, Entry 12)"]],"source":"Ramanujan's integral for $\\varphi(x) - 1$ (Chapter 8, Entry 12)"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry12-qgammaintege-2","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $x$ be a complex number with $\\operatorname{Re} x > 1$. For real $u$ and complex $s$ put $P_s(u) = 0$ if $u = 0$ and $P_s(u) = e^{s \\log |u|}$ otherwise, and define $K_x(0) = 0$, $K_x(1) = 0$ and $K_x(u) = \\frac{P_{x-2}(u)\\,(1 - u)^2}{1 - P_x(u)}$ for every other real $u$, the quotient being taken to be $0$ at $u = -1$, where its denominator vanishes. Let $\\varphi(x) = 1 + \\sum_{k \\ge 1} \\frac{2}{(kx)^3 - kx}$.\u2026","title":"Ramanujan's integral for $\\varphi(x) - 1$ (Chapter 8, Entry 12)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/chapter8-digamma-eq-complex-digamma","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry8Digammaseries.chapter8Digamma_eq_Complex_digamma","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#chapter8-digamma-eq-complex-digamma","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"For every complex number $z$, $\\Gamma'(z)/\\Gamma(z) = \\psi(z)$, where $\\psi$ is the digamma function.","title":"Digamma as the logarithmic derivative of Gamma (used for Chapter 8, Entry 8)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry8-digammaseries","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry8Digammaseries.ramanujan_part1_ch8_entry8_digammaseries_complex_digamma","parent":17573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating series "],["math","\\(\\sum_{k \\ge 1} (-1)^{k-1}/(a + kb)\\)"],["text"," via the digamma function"]],"source":"Alternating series $\\sum_{k \\ge 1} (-1)^{k-1}/(a + kb)$ via the digamma function"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry8-digammaseries","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $a$ and $b$ be complex numbers with $b \\ne 0$ and $a + kb \\ne 0$ for every integer $k \\ge 1$, and let $\\psi$ be the digamma function, the logarithmic derivative of $\\Gamma$. Then the partial sums of $\\sum_{k=1}^{\\infty} \\frac{(-1)^{k-1}}{a + kb}$ converge to","title":"Alternating series $\\sum_{k \\ge 1} (-1)^{k-1}/(a + kb)$ via the digamma function"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry8-digammaseries-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry8Digammaseries.ramanujan_part1_ch8_entry8_digammaseries","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry8-digammaseries-2","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $a$ and $b$ be complex numbers with $b \\ne 0$ and $a + kb \\ne 0$ for every integer $k \\ge 1$, suppose that $\\Gamma\\left(\\frac{a+2b}{2b}\\right) \\ne 0$ and $\\Gamma\\left(\\frac{a+b}{2b}\\right) \\ne 0$, and let $\\psi(z) = \\Gamma'(z)/\\Gamma(z)$. Then the partial sums of $\\sum_{k=1}^{\\infty} \\frac{(-1)^{k-1}}{a + kb}$ converge to","title":"Ramanujan's alternating digamma series (Chapter 8, Entry 8)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/chapter8-digamma-eq-digamma-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry9Qgamma.chapter8Digamma_eq_digamma","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#chapter8-digamma-eq-digamma-2","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"For every complex number $z$, $\\Gamma'(z)/\\Gamma(z) = \\psi(z)$, where $\\Gamma'$ is the complex derivative of the Gamma function and $\\psi$ is the digamma function. At the poles of $\\Gamma$ both sides are $0$ by convention.","title":"Digamma as the logarithmic derivative of Gamma (Chapter 8, Entry 9)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/chapter8-unit-pow-eq-cpow-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry9Qgamma.chapter8UnitPow_eq_cpow","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#chapter8-unit-pow-eq-cpow-2","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"For a complex number $x$ and a real number $u$, let $p_x(u) = e^{x \\log\\lvert u\\rvert}$ if $u \\ne 0$ and $p_x(0) = 0$. If $u > 0$, then $p_x(u) = u^x$, the principal complex power of $u$.","title":"The Entry 9 power function on positive reals (Chapter 8)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry9-qgamma-digamm","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry9Qgamma.ramanujan_part1_ch8_entry9_qgamma_digamma","parent":17573,"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry9-qgamma-digamm","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $x$ be a complex number with $\\operatorname{Re} x > 0$, and for real $u$ write $u^x = e^{x \\log u}$ if $u > 0$ and $0^x = 0$. Let $\\psi$ be the digamma function. Then $x \\ne 0$; the function $u \\mapsto u^x/(1 + u^x)$ is integrable on $[0, 1]$; $1 + u^x \\ne 0$ for every $u \\in [0, 1]$; $\\Gamma(1/(2x) + 1) \\ne 0$ and $\\Gamma(1/x + 1) \\ne 0$; and","title":"Ramanujan's integral for the digamma function (Chapter 8, Entry 9)"},{"id":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/ramanujan-part1-ch8-entry9-qgamma","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry9Qgamma.ramanujan_part1_ch8_entry9_qgamma","parent":17573,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's integral for "],["math","\\(\\Gamma'/\\Gamma\\)"],["text"," (Chapter 8, Entry 9)"]],"source":"Ramanujan's integral for $\\Gamma'/\\Gamma$ (Chapter 8, Entry 9)"},"result":true,"route":"../roadmap/Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function/index.html#ramanujan-part1-ch8-entry9-qgamma","scope":"Analysis/special-functions/gamma-function/series-and-integrals-for-the-digamma-function","status":"fully_proved","summary":"Let $x$ be a complex number with $\\operatorname{Re} x > 0$, and for real $u$ write $u^x = e^{x \\log u}$ if $u > 0$ and $0^x = 0$. Write $\\Gamma'$ for the complex derivative of the Gamma function. Then $x \\ne 0$; the function $u \\mapsto u^x/(1 + u^x)$ is integrable on $[0, 1]$; $1 + u^x \\ne 0$ for every $u \\in [0, 1]$; $\\Gamma(1/(2x) + 1) \\ne 0$ and $\\Gamma(1/x + 1) \\ne 0$; and","title":"Ramanujan's integral for $\\Gamma'/\\Gamma$ (Chapter 8, Entry 9)"},{"data_url":"scopes/Analysis/special-functions/hypergeometric-functions.json","id":"Analysis/special-functions/hypergeometric-functions","kind":"scope","parent":17543,"result":false,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/index.html","scope":"Analysis/special-functions/hypergeometric-functions","status":"fully_proved","summary":"Generalized and basic hypergeometric series and their summation theorems (Gauss, Dougall, Dixon, Klamkin, Frisch). Orthogonal polynomial families go to Algebra/orthogonal-polynomials.","title":"Hypergeometric functions"},{"data_url":"scopes/Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series.json","id":"Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series","kind":"scope","navigation":"module","parent":17591,"result":false,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series/index.html","scope":"Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series","status":"fully_proved","title":"Basic hypergeometric series"},{"id":"Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series/basic-hypergeometric-coefficient","kind":"node","lean":"MetaMathlibExt.BasicHypergeometric.basicHypergeometricCoefficient","parent":17592,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series/index.html#basic-hypergeometric-coefficient","scope":"Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series","status":"fully_proved","summary":"Let $r$ and $s$ be natural numbers, let $a_1, \\dots, a_r$, $b_1, \\dots, b_s$ and $q$ be complex numbers, and write $(x;q)_n = \\prod_{j=0}^{n-1} (1 - x q^j)$ for the finite $q$-Pochhammer symbol, so that $(q;q)_n = \\prod_{j=0}^{n-1} (1 - q^{j+1})$. For a natural number $n$, the coefficient of $z^n$ in the basic hypergeometric series ${}_r\\phi_s(a_1, \\dots, a_r; b_1, \\dots, b_s; q, z)$ is","title":"Coefficients of the basic hypergeometric series"},{"id":"Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series/basic-hypergeometric-series","kind":"node","lean":"MetaMathlibExt.BasicHypergeometric.basicHypergeometricSeries","parent":17592,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series/index.html#basic-hypergeometric-series","scope":"Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series","status":"fully_proved","summary":"Let $r$ and $s$ be natural numbers and let $a_1, \\dots, a_r$, $b_1, \\dots, b_s$ and $q$ be complex numbers. The basic hypergeometric series ${}_r\\phi_s(a_1, \\dots, a_r; b_1, \\dots, b_s; q, z)$ is the formal power series in $z$ over $\\mathbb{C}$ whose coefficient of $z^n$, for each natural number $n$, is the $n$-th basic hypergeometric coefficient of these parameters.","title":"Basic hypergeometric series"},{"id":"Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series/basic-hypergeometric-coefficient-phi-suc","kind":"node","lean":"MetaMathlibExt.BasicHypergeometric.basicHypergeometricCoefficient_phi_succ","parent":17592,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of the series "],["math","\\({}_{s+1}\\phi_s\\)"]],"source":"Coefficients of the series ${}_{s+1}\\phi_s$"},"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series/index.html#basic-hypergeometric-coefficient-phi-suc","scope":"Analysis/special-functions/hypergeometric-functions/basic-hypergeometric-series","status":"fully_proved","summary":"Let $s$ be a natural number, let $a_1, \\dots, a_{s+1}$, $b_1, \\dots, b_s$ and $q$ be complex numbers, and write $(x;q)_n = \\prod_{j=0}^{n-1} (1 - x q^j)$ for the finite $q$-Pochhammer symbol, so that $(q;q)_n = \\prod_{j=0}^{n-1} (1 - q^{j+1})$. For every natural number $n$, the coefficient of $z^n$ in the basic hypergeometric series ${}_{s+1}\\phi_s(a_1, \\dots, a_{s+1}; b_1, \\dots, b_s; q, z)$ is","title":"Coefficients of the series ${}_{s+1}\\phi_s$"},{"data_url":"scopes/Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities.json","id":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities","kind":"scope","navigation":"module","parent":17591,"result":false,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/index.html","scope":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities","status":"fully_proved","title":"Frisch and Klamkin binomial identities"},{"id":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/gauss-hypergeometric-summation-frisch","kind":"node","lean":"MetaMathlibExt.gauss_hypergeometric_summation_frisch","parent":17596,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/index.html#gauss-hypergeometric-summation-frisch","scope":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities","status":"fully_proved","summary":"Let $n \\ge 1$ be an integer and let $b$ and $c$ be real numbers with $c > n$ and $b > -1$. Then","title":"Gauss summation at Frisch's specialization"},{"id":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/gauss-hypergeometric-summation-klamkin","kind":"node","lean":"MetaMathlibExt.gauss_hypergeometric_summation_klamkin","parent":17596,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/index.html#gauss-hypergeometric-summation-klamkin","scope":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities","status":"fully_proved","summary":"Let $n \\ge 1$ be an integer and let $a$ and $b$ be real numbers such that $-a$ and $-b$ are not nonnegative integers, $a + 1 + n < 0$, and $a - b + 1$ is not a nonnegative integer. Then","title":"Klamkin's Gauss summation"},{"id":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/has-sum-ordinary-hypergeometric-coeffici","kind":"node","lean":"MetaMathlibExt.hasSum_ordinaryHypergeometricCoefficient_klamkin","parent":17596,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/index.html#has-sum-ordinary-hypergeometric-coeffici","scope":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities","status":"fully_proved","summary":"Let $n \\ge 1$ be an integer and let $a$ and $b$ be real numbers such that $-a$ and $-b$ are not nonnegative integers, $a + 1 + n < 0$, and $a - b + 1$ is not a nonnegative integer. Then the series","title":"Klamkin's Gauss summation as a convergent series"},{"id":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/variant-klamkin","kind":"node","lean":"MetaMathlibExt.variant_Klamkin","parent":17596,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities/index.html#variant-klamkin","scope":"Analysis/special-functions/hypergeometric-functions/frisch-and-klamkin-binomial-identities","status":"fully_proved","summary":"For real numbers $c$ and $d$ write $\\binom{c}{d} = \\Gamma(c+1) / (\\Gamma(d+1) \\, \\Gamma(c-d+1))$; for natural numbers this is the usual binomial coefficient. Let $a$ and $b$ be real numbers such that $-a$ and $-b$ are not in $\\{0, 1, 2, \\dots\\}$ and $a - b$ is not an integer, and let $n \\ge 1$ be a natural number with $n < -a - 1$. Then the series","title":"Abel's variant of Klamkin's identity"},{"data_url":"scopes/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series.json","id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","kind":"scope","navigation":"module","parent":17591,"result":false,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","title":"Generalized hypergeometric series"},{"id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/generalized-hypergeometric-coefficient","kind":"node","lean":"MetaMathlibExt.GeneralizedHypergeometric.generalizedHypergeometricCoefficient","parent":17601,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html#generalized-hypergeometric-coefficient","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","summary":"Let $K$ be a field, let $a$ and $b$ be finite multisets of elements of $K$ (the upper and lower parameters), and let $n$ be a natural number. The $n$-th generalized hypergeometric coefficient is","title":"Generalized hypergeometric coefficient"},{"id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/generalized-hypergeometric-series","kind":"node","lean":"MetaMathlibExt.GeneralizedHypergeometric.generalizedHypergeometricSeries","parent":17601,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html#generalized-hypergeometric-series","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","summary":"Let $K$ be a field and $A$ a topological ring that is a $K$-algebra. For finite multisets $a$ and $b$ of elements of $K$, the generalized hypergeometric series is the formal multilinear series on $A$ whose $n$-th term is the $n$-linear map $(x_1, \\dots, x_n) \\mapsto c_n \\, x_1 x_2 \\cdots x_n$, where $c_n \\in K$ is the $n$-th generalized hypergeometric coefficient with upper parameters $a$ and lower parameters $b$.","title":"Generalized hypergeometric series"},{"id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/lower-parameters-admissible","kind":"node","lean":"MetaMathlibExt.GeneralizedHypergeometric.LowerParametersAdmissible","parent":17601,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html#lower-parameters-admissible","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","summary":"A finite multiset $B$ of complex numbers is admissible as lower parameters if no element of $B$ is $0$ or a negative integer, that is, $-k \\notin B$ for every natural number $k$.","title":"Admissible lower parameters"},{"id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/summable-hypergeometric","kind":"node","lean":"MetaMathlibExt.GeneralizedHypergeometric.SummableHypergeometric","parent":17601,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html#summable-hypergeometric","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","summary":"For finite multisets $a$ and $b$ of complex numbers and $z \\in \\mathbb{C}$, the generalized hypergeometric series with upper parameters $a$ and lower parameters $b$ is summable at $z$ if the family $(c_n z^n)_{n \\ge 0}$ is summable in $\\mathbb{C}$, where $c_n$ is the $n$-th generalized hypergeometric coefficient.","title":"Summability of the generalized hypergeometric series"},{"id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/generalized-hypergeometric-value","kind":"node","lean":"MetaMathlibExt.GeneralizedHypergeometric.generalizedHypergeometricValue","parent":17601,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html#generalized-hypergeometric-value","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","summary":"For finite multisets $a$ and $b$ of complex numbers and $z \\in \\mathbb{C}$, the value ${}_pF_q(a; b; z)$, where $p = |a|$ and $q = |b|$, is the sum $\\sum_{n \\ge 0} c_n z^n$, where $c_n$ is the $n$-th generalized hypergeometric coefficient with upper parameters $a$ and lower parameters $b$.","title":"Generalized hypergeometric function"},{"id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/lower-parameters-admissible-asc-pochhamm","kind":"node","lean":"MetaMathlibExt.GeneralizedHypergeometric.lowerParametersAdmissible_ascPochhammer_ne_zero","parent":17601,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html#lower-parameters-admissible-asc-pochhamm","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","summary":"Let $B$ be a finite multiset of complex numbers that is admissible as lower parameters. Then for every $b \\in B$ and every natural number $n$, the rising factorial $(b)_n = b(b+1)\\cdots(b+n-1)$, with $(b)_0 = 1$, is nonzero.","title":"Rising factorials at admissible lower parameters are nonzero"},{"id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/summable-hypergeometric-of-neg-coe-nat-m","kind":"node","lean":"MetaMathlibExt.GeneralizedHypergeometric.summableHypergeometric_of_neg_coe_nat_mem_upper","parent":17601,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html#summable-hypergeometric-of-neg-coe-nat-m","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","summary":"Let $a$ and $b$ be finite multisets of complex numbers, and let $c_n$ be the $n$-th generalized hypergeometric coefficient with upper parameters $a$ and lower parameters $b$. If $-N \\in a$ for some natural number $N$, then for every $z \\in \\mathbb{C}$ the family $(c_n z^n)_{n \\ge 0}$ is summable, that is, the generalized hypergeometric series with these parameters is summable at $z$.","title":"Terminating generalized hypergeometric series are summable"},{"id":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/generalized-hypergeometric-tsum-eq-finse","kind":"node","lean":"MetaMathlibExt.GeneralizedHypergeometric.generalizedHypergeometric_tsum_eq_finset_sum","parent":17601,"result":true,"route":"../roadmap/Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series/index.html#generalized-hypergeometric-tsum-eq-finse","scope":"Analysis/special-functions/hypergeometric-functions/generalized-hypergeometric-series","status":"fully_proved","summary":"Let $a$ and $b$ be finite multisets of complex numbers, and let $c_n$ be the $n$-th generalized hypergeometric coefficient with upper parameters $a$ and lower parameters $b$. If $-N \\in a$ for some natural number $N$, then for every $z \\in \\mathbb{C}$","title":"Terminating generalized hypergeometric series are finite sums"},{"data_url":"scopes/Analysis/special-functions/elementary-transcendental-functions.json","id":"Analysis/special-functions/elementary-transcendental-functions","kind":"scope","parent":17543,"result":false,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/index.html","scope":"Analysis/special-functions/elementary-transcendental-functions","status":"fully_proved","summary":"Exponential, logarithmic and trigonometric-type functions and their relatives: exponential integral E1 and Ein, logarithmic integral, Lambert W, logistic function, x^y = y^x, exponential sums, Mittag-Leffler expansion of 1/(e^u \u2212 1), modified Bessel functions, Dickman\u2013de Bruijn function, Bell-polynomial functions.","title":"Exponential, logarithmic and related functions"},{"id":"Analysis/special-functions/elementary-transcendental-functions/eq-zero-of-sum-mul-exp-eq-zero","kind":"node","lean":"Real.eq_zero_of_sum_mul_exp_eq_zero","parent":17610,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/index.html#eq-zero-of-sum-mul-exp-eq-zero","scope":"Analysis/special-functions/elementary-transcendental-functions","status":"fully_proved","summary":"If `\u2211_{u \u2208 S} c_u exp(-a u)` vanishes at `|S|` distinct points `a \u2208 P` (the rates `u \u2208 S` are distinct as elements of a `Finset`), every coefficient is zero. Equivalently, the matrix `(exp(-a u))_{a \u2208 P, u \u2208 S}` has full column rank when `|P| \u2265 |S|`.","title":"Exponential sums have full column rank"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exists-parametrization-of-rpow-eq-rpow","kind":"node","lean":"MetaMathlibExt.exists_parametrization_of_rpow_eq_rpow","parent":17610,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parametrization of the solutions of "],["math","\\(x^y = y^x\\)"],["text"," with "],["math","\\(0 < y < x\\)"]],"source":"Parametrization of the solutions of $x^y = y^x$ with $0 < y < x$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/index.html#exists-parametrization-of-rpow-eq-rpow","scope":"Analysis/special-functions/elementary-transcendental-functions","status":"fully_proved","summary":"Let $x$ and $y$ be real numbers with $0 < y < x$ and $x^y = y^x$. Then there is a real number $t > 0$ such that $y = (1 + 1/t)^t$ and $x = (1 + 1/t)^{t+1}$.","title":"Parametrization of the solutions of $x^y = y^x$ with $0 < y < x$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/sigmoid-log-sub-log","kind":"node","lean":"Real.sigmoid_log_sub_log","parent":17610,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/index.html#sigmoid-log-sub-log","scope":"Analysis/special-functions/elementary-transcendental-functions","status":"fully_proved","summary":"For real numbers $a > 0$ and $b > 0$, the logistic function $\\sigma(z) = 1 / (1 + e^{-z})$ satisfies $\\sigma(\\log a - \\log b) = a / (a + b)$.","title":"The logistic function of the log odds"},{"data_url":"scopes/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function.json","id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","kind":"scope","navigation":"module","parent":17610,"result":false,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","title":"Dickman\u2013de Bruijn function"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/is-dickman-de-bruijn","kind":"node","lean":"Real.IsDickmanDeBruijn","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#is-dickman-de-bruijn","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"A function $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function if $\\rho(u) = 0$ for every real $u < 0$, $\\rho$ maps $[0, \\infty)$ into $[0, 1]$, $\\rho$ is continuous on $[0, \\infty)$, $\\rho(u) = 1$ for every $u \\in [0, 1]$, and for every real $u > 1$ the function $\\rho$ is differentiable at $u$ with $u \\rho'(u) + \\rho(u - 1) = 0$.","title":"Dickman\u2013de Bruijn functions"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/eq-zero-of-neg","kind":"node","lean":"Real.IsDickmanDeBruijn.eq_zero_of_neg","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#eq-zero-of-neg","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function, then $\\rho(u) = 0$ for every real $u < 0$.","title":"The Dickman\u2013de Bruijn function vanishes on the negative reals"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/maps-to-ici-icc","kind":"node","lean":"Real.IsDickmanDeBruijn.mapsTo_Ici_Icc","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#maps-to-ici-icc","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function, then $0 \\le \\rho(u) \\le 1$ for every real $u \\ge 0$.","title":"Range of the Dickman\u2013de Bruijn function on the nonnegative reals"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/continuous-on-ici","kind":"node","lean":"Real.IsDickmanDeBruijn.continuousOn_Ici","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#continuous-on-ici","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function, then $\\rho$ is continuous on $[0, \\infty)$.","title":"Continuity of the Dickman\u2013de Bruijn function on the nonnegative reals"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/eq-one-of-mem-icc","kind":"node","lean":"Real.IsDickmanDeBruijn.eq_one_of_mem_Icc","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#eq-one-of-mem-icc","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function, then $\\rho(u) = 1$ for every $u \\in [0, 1]$.","title":"The Dickman\u2013de Bruijn function equals one on the unit interval"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/eq-one-of-nonneg-of-le-one","kind":"node","lean":"Real.IsDickmanDeBruijn.eq_one_of_nonneg_of_le_one","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#eq-one-of-nonneg-of-le-one","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function and $u$ is a real number with $0 \\le u$ and $u \\le 1$, then $\\rho(u) = 1$.","title":"Normalization of the Dickman\u2013de Bruijn function, inequality form"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/differentiable-at-of-one-lt","kind":"node","lean":"Real.IsDickmanDeBruijn.differentiableAt_of_one_lt","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#differentiable-at-of-one-lt","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function, then $\\rho$ is differentiable at every real $u > 1$.","title":"Differentiability of the Dickman\u2013de Bruijn function above one"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/delay-eq","kind":"node","lean":"Real.IsDickmanDeBruijn.delay_eq","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#delay-eq","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function, then $u \\rho'(u) + \\rho(u - 1) = 0$ for every real $u > 1$.","title":"Delay differential equation of the Dickman\u2013de Bruijn function"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/nonneg-of-nonneg","kind":"node","lean":"Real.IsDickmanDeBruijn.nonneg_of_nonneg","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#nonneg-of-nonneg","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function, then $\\rho(u) \\ge 0$ for every real $u \\ge 0$.","title":"Nonnegativity of the Dickman\u2013de Bruijn function"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/le-one-of-nonneg","kind":"node","lean":"Real.IsDickmanDeBruijn.le_one_of_nonneg","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#le-one-of-nonneg","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function, then $\\rho(u) \\le 1$ for every real $u \\ge 0$.","title":"Upper bound one for the Dickman\u2013de Bruijn function on the nonnegative reals"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/mem-icc-of-nonneg","kind":"node","lean":"Real.IsDickmanDeBruijn.mem_Icc_of_nonneg","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#mem-icc-of-nonneg","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"If $\\rho : \\mathbb{R} \\to \\mathbb{R}$ is a Dickman\u2013de Bruijn function and $u \\ge 0$ is real, then $\\rho(u) \\in [0, 1]$.","title":"Pointwise range bound for the Dickman\u2013de Bruijn function"},{"id":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/exists-unique-is-dickman-de-bruijn","kind":"node","lean":"Real.existsUnique_isDickmanDeBruijn","parent":17614,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function/index.html#exists-unique-is-dickman-de-bruijn","scope":"Analysis/special-functions/elementary-transcendental-functions/dickman-de-bruijn-function","status":"fully_proved","summary":"There is exactly one function $\\rho : \\mathbb{R} \\to \\mathbb{R}$ that is a Dickman\u2013de Bruijn function, that is, such that $\\rho(u) = 0$ for $u < 0$, $0 \\le \\rho(u) \\le 1$ for $u \\ge 0$, $\\rho$ is continuous on $[0, \\infty)$, $\\rho(u) = 1$ for $0 \\le u \\le 1$, and for every $u > 1$ the function $\\rho$ is differentiable at $u$ with $u \\rho'(u) + \\rho(u - 1) = 0$.","title":"Existence and uniqueness of the Dickman\u2013de Bruijn function"},{"data_url":"scopes/Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions.json","id":"Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions","kind":"scope","navigation":"module","parent":17610,"result":false,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions/index.html","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions","status":"fully_proved","title":"Exponential autonomous functions"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions/complete-bell-polynomial","kind":"node","lean":"MetaMathlibExt.completeBellPolynomial","parent":17627,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions/index.html#complete-bell-polynomial","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions","status":"fully_proved","summary":"For a sequence $x = (x_0, x_1, x_2, \\dots)$ of complex numbers, the complete Bell polynomials $B_n(x)$ are defined by $B_0(x) = 1$ and, for every $m \\ge 0$, $B_{m+1}(x) = \\sum_{i=0}^{m} \\binom{m}{i} B_{m-i}(x) \\, x_i$.","title":"Complete Bell polynomials"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions/complete-exponential-autonomous-function","kind":"node","lean":"MetaMathlibExt.completeExponentialAutonomousFunction","parent":17627,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions/index.html#complete-exponential-autonomous-function","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-autonomous-functions","status":"fully_proved","summary":"Let $k \\ge 1$, $a \\in \\mathbb{C}$ and $x_0, \\dots, x_{k-1} \\in \\mathbb{C}$. The complete exponential autonomous function of order $k$ with parameter $a$ and variables $x$ is the sequence $f_0, f_1, f_2, \\dots$ of complex numbers given by $f_n = x_n$ for $n < k$, $f_k = a e^{x_0}$, and $f_n = a e^{x_0} B_{n-k}(y)$ for $n > k$, where $B_{n-k}$ is the complete Bell polynomial and $y$ is the sequence with\u2026","title":"Complete exponential autonomous functions"},{"data_url":"scopes/Analysis/special-functions/elementary-transcendental-functions/exponential-integral.json","id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","kind":"scope","navigation":"module","parent":17610,"result":false,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","title":"Exponential integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/ein-term","kind":"node","lean":"Complex.einTerm","parent":17630,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#ein-term","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For a natural number $n$ and a complex number $z$, this is the complex number $\\frac{(-1)^n z^{n+1}}{(n+1) \\cdot (n+1)!}$, the term of index $j = n + 1$ in the series $\\operatorname{Ein}(z) = \\sum_{j \\ge 1} \\frac{(-1)^{j+1} z^j}{j \\cdot j!}$.","title":"Terms of the series for the entire exponential integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/ein","kind":"node","lean":"Complex.ein","parent":17630,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#ein","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"The entire exponential integral is the function $\\operatorname{Ein} : \\mathbb{C} \\to \\mathbb{C}$ defined by $\\operatorname{Ein}(z) = \\sum_{j \\ge 1} \\frac{(-1)^{j+1} z^j}{j \\cdot j!}$.","title":"The entire exponential integral Ein"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/differentiable-ein","kind":"node","lean":"Complex.differentiable_ein","parent":17630,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#differentiable-ein","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"The entire exponential integral $\\operatorname{Ein}$ is complex differentiable at every point of $\\mathbb{C}$.","title":"Differentiability of the entire exponential integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/analytic-on-nhd-ein","kind":"node","lean":"Complex.analyticOnNhd_ein","parent":17630,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#analytic-on-nhd-ein","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"The entire exponential integral $\\operatorname{Ein}$ is complex analytic at every point of $\\mathbb{C}$.","title":"Analyticity of the entire exponential integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/mul-deriv-ein","kind":"node","lean":"Complex.mul_deriv_ein","parent":17630,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#mul-deriv-ein","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For every complex number $z$, the entire exponential integral $\\operatorname{Ein}$ satisfies $z \\operatorname{Ein}'(z) = 1 - e^{-z}$.","title":"Derivative of the entire exponential integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/principal-exponential-integral-e1-raw","kind":"node","lean":"Complex.principalExponentialIntegralE1Raw","parent":17630,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#principal-exponential-integral-e1-raw","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For every complex number $z$, this is $\\operatorname{Ein}(z) - \\operatorname{Log} z - \\gamma$, where $\\operatorname{Ein}(z) = \\sum_{j \\ge 1} (-1)^{j+1} z^j / (j \\cdot j!)$ is the entire exponential integral, $\\operatorname{Log}$ is the principal branch of the complex logarithm (imaginary part in $(-\\pi, \\pi]$) extended by $\\operatorname{Log} 0 = 0$, and $\\gamma$ is the Euler\u2013Mascheroni constant.","title":"Totalized principal exponential integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/principal-exponential-integral-e1","kind":"node","lean":"Complex.principalExponentialIntegralE1","parent":17630,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#principal-exponential-integral-e1","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For a nonzero complex number $z$, the principal exponential integral is $E_1(z) = \\operatorname{Ein}(z) - \\operatorname{Log} z - \\gamma$, where $\\operatorname{Ein}(z) = \\sum_{j \\ge 1} (-1)^{j+1} z^j / (j \\cdot j!)$ is the entire exponential integral, $\\operatorname{Log}$ is the principal branch of the complex logarithm (imaginary part in $(-\\pi, \\pi]$) and $\\gamma$ is the Euler\u2013Mascheroni constant.","title":"The principal exponential integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-term-raw","kind":"node","lean":"Complex.e1AsymptoticTermRaw","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Totalized asymptotic term of "],["math","\\(E_1\\)"]],"source":"Totalized asymptotic term of $E_1$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-term-raw","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and every complex number $z$, the $n$-th term of the asymptotic expansion of $E_1$ is $\\frac{e^{-z}}{z} \\cdot \\frac{(-1)^n \\, n!}{z^n}$, with division by zero read as $0$.","title":"Totalized asymptotic term of $E_1$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-term","kind":"node","lean":"Complex.e1AsymptoticTerm","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic term of "],["math","\\(E_1\\)"]],"source":"Asymptotic term of $E_1$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-term","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and a nonzero complex number $z$, the $n$-th term of the asymptotic expansion of the exponential integral $E_1$ is $\\frac{e^{-z}}{z} \\cdot \\frac{(-1)^n \\, n!}{z^n}$.","title":"Asymptotic term of $E_1$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-partial-raw","kind":"node","lean":"Complex.e1AsymptoticPartialRaw","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Totalized partial sum of the "],["math","\\(E_1\\)"],["text"," asymptotic expansion"]],"source":"Totalized partial sum of the $E_1$ asymptotic expansion"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-partial-raw","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and every complex number $z$, this is $\\sum_{j < n} t_j(z)$, where $t_j(z) = \\frac{e^{-z}}{z} \\cdot \\frac{(-1)^j \\, j!}{z^j}$ with division by zero read as $0$.","title":"Totalized partial sum of the $E_1$ asymptotic expansion"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-partial","kind":"node","lean":"Complex.e1AsymptoticPartial","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sum of the "],["math","\\(E_1\\)"],["text"," asymptotic expansion"]],"source":"Partial sum of the $E_1$ asymptotic expansion"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-partial","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and a nonzero complex number $z$, the sum of the first $n$ terms of the asymptotic expansion of the exponential integral $E_1$ is $\\sum_{j < n} \\frac{e^{-z}}{z} \\cdot \\frac{(-1)^j \\, j!}{z^j}$.","title":"Partial sum of the $E_1$ asymptotic expansion"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-remainder-raw","kind":"node","lean":"Complex.e1AsymptoticRemainderRaw","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Totalized remainder of the "],["math","\\(E_1\\)"],["text"," asymptotic expansion"]],"source":"Totalized remainder of the $E_1$ asymptotic expansion"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-remainder-raw","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and every complex number $z$, this is $\\operatorname{Ein}(z) - \\operatorname{Log} z - \\gamma - S_n(z)$, where $\\operatorname{Ein}$ is the entire exponential integral, $\\operatorname{Log}$ is the principal logarithm extended by $\\operatorname{Log} 0 = 0$, $\\gamma$ is the Euler\u2013Mascheroni constant and $S_n$ is the totalized sum of the first $n$ terms of the asymptotic expansion of $E_1$.","title":"Totalized remainder of the $E_1$ asymptotic expansion"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-remainder","kind":"node","lean":"Complex.e1AsymptoticRemainder","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Remainder of the "],["math","\\(E_1\\)"],["text"," asymptotic expansion"]],"source":"Remainder of the $E_1$ asymptotic expansion"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-remainder","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and a nonzero complex number $z$, the remainder after $n$ terms is $E_1(z) - \\sum_{j < n} \\frac{e^{-z}}{z} \\cdot \\frac{(-1)^j \\, j!}{z^j}$, where $E_1$ is the principal exponential integral.","title":"Remainder of the $E_1$ asymptotic expansion"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/principal-exponential-integral-e1-raw-ze","kind":"node","lean":"Complex.principalExponentialIntegralE1Raw_zero","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Totalized principal "],["math","\\(E_1\\)"],["text"," kernel at zero"]],"source":"Totalized principal $E_1$ kernel at zero"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#principal-exponential-integral-e1-raw-ze","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"The totalized kernel $\\operatorname{Ein}(z) - \\operatorname{Log} z - \\gamma$, where $\\operatorname{Ein}$ is the entire exponential integral, $\\operatorname{Log}$ is the principal logarithm extended by $\\operatorname{Log} 0 = 0$ and $\\gamma$ is the Euler\u2013Mascheroni constant, equals $-\\gamma$ at $z = 0$.","title":"Totalized principal $E_1$ kernel at zero"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-term-raw-zero","kind":"node","lean":"Complex.e1AsymptoticTermRaw_zero","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Totalized "],["math","\\(E_1\\)"],["text"," asymptotic terms vanish at zero"]],"source":"Totalized $E_1$ asymptotic terms vanish at zero"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-term-raw-zero","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the totalized $n$-th term $\\frac{e^{-z}}{z} \\cdot \\frac{(-1)^n \\, n!}{z^n}$ of the asymptotic expansion of $E_1$, with division by zero read as $0$, equals $0$ at $z = 0$.","title":"Totalized $E_1$ asymptotic terms vanish at zero"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-partial-raw-zero","kind":"node","lean":"Complex.e1AsymptoticPartialRaw_zero","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Totalized partial sums of the "],["math","\\(E_1\\)"],["text"," expansion vanish at zero"]],"source":"Totalized partial sums of the $E_1$ expansion vanish at zero"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-partial-raw-zero","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the totalized sum of the first $n$ terms of the asymptotic expansion of $E_1$ equals $0$ at $z = 0$.","title":"Totalized partial sums of the $E_1$ expansion vanish at zero"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/e1-asymptotic-remainder-raw-zero","kind":"node","lean":"Complex.e1AsymptoticRemainderRaw_zero","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Totalized "],["math","\\(E_1\\)"],["text"," remainder at zero"]],"source":"Totalized $E_1$ remainder at zero"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#e1-asymptotic-remainder-raw-zero","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the totalized remainder of the asymptotic expansion of $E_1$ after $n$ terms equals $-\\gamma$ at $z = 0$, where $\\gamma$ is the Euler\u2013Mascheroni constant.","title":"Totalized $E_1$ remainder at zero"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/two-term-kernel-identity","kind":"node","lean":"Complex.two_term_kernel_identity","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-term expansion of "],["math","\\(1/(t + z)\\)"]],"source":"Two-term expansion of $1/(t + z)$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#two-term-kernel-identity","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"Let $z \\ne 0$ be a complex number and $t$ a real number with $t + z \\ne 0$. Then $\\frac{1}{t + z} = \\frac{1}{z} - \\frac{t}{z^2} + \\frac{t^2}{z^2 (t + z)}$.","title":"Two-term expansion of $1/(t + z)$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/integrable-on-exp-neg-mul-pow","kind":"node","lean":"Complex.integrableOn_exp_neg_mul_pow","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrability of "],["math","\\(e^{-t} t^n\\)"],["text"," on "],["math","\\((0, \\infty)\\)"]],"source":"Integrability of $e^{-t} t^n$ on $(0, \\infty)$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#integrable-on-exp-neg-mul-pow","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For every natural number $n$, the function $t \\mapsto e^{-t} t^n$ is Lebesgue integrable on the open interval $(0, \\infty)$.","title":"Integrability of $e^{-t} t^n$ on $(0, \\infty)$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/integral-exp-neg-mul-pow-ioi","kind":"node","lean":"Complex.integral_exp_neg_mul_pow_Ioi","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","The moments of "],["math","\\(e^{-t}\\)"],["text"," are factorials"]],"source":"The moments of $e^{-t}$ are factorials"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#integral-exp-neg-mul-pow-ioi","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"For every natural number $n$, the integral of $e^{-t} t^n$ over the open interval $(0, \\infty)$ equals $n!$.","title":"The moments of $e^{-t}$ are factorials"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/outer-sector-sin-abs-arg-pos","kind":"node","lean":"Complex.outer_sector_sin_abs_arg_pos","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positivity of "],["math","\\(\\sin |\\arg z|\\)"],["text"," in the outer sector"]],"source":"Positivity of $\\sin |\\arg z|$ in the outer sector"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#outer-sector-sin-abs-arg-pos","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"Let $z \\ne 0$ be a complex number with $\\pi/2 \\le |\\arg z| < \\pi$, where $\\arg z \\in (-\\pi, \\pi]$ is the principal argument. Then $\\sin |\\arg z| > 0$.","title":"Positivity of $\\sin |\\arg z|$ in the outer sector"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/outer-sector-distance","kind":"node","lean":"Complex.outer_sector_distance","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for "],["math","\\(|t + z|\\)"],["text"," in the outer sector"]],"source":"Lower bound for $|t + z|$ in the outer sector"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#outer-sector-distance","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"Let $z \\ne 0$ be a complex number with $\\pi/2 \\le |\\arg z| < \\pi$, where $\\arg z \\in (-\\pi, \\pi]$ is the principal argument, and let $t \\ge 0$ be a real number. Then $|z| \\sin |\\arg z| \\le |t + z|$.","title":"Lower bound for $|t + z|$ in the outer sector"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/integral-remainder-bound","kind":"node","lean":"Complex.integral_remainder_bound","parent":17630,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#integral-remainder-bound","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"Let $z \\ne 0$ be a complex number with $\\pi/2 \\le |\\arg z| < \\pi$, where $\\arg z \\in (-\\pi, \\pi]$ is the principal argument, and let $T_2(z) = \\frac{e^{-z}}{z} \\cdot \\frac{(-1)^2 \\, 2!}{z^2}$ be the $n = 2$ term of the asymptotic expansion of $E_1$. Then","title":"Outer-sector bound for the quadratic integral remainder"},{"id":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral/principal-exponential-integral-e1-two-te","kind":"node","lean":"Complex.principal_exponential_integral_e1_two_term_remainder_dlmf","parent":17630,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-term remainder bound for "],["math","\\(E_1\\)"],["text"," in the outer sector"]],"source":"Two-term remainder bound for $E_1$ in the outer sector"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/exponential-integral/index.html#principal-exponential-integral-e1-two-te","scope":"Analysis/special-functions/elementary-transcendental-functions/exponential-integral","status":"fully_proved","summary":"Let $z \\ne 0$ be a complex number with $\\pi/2 \\le |\\arg z| < \\pi$, where $\\arg z \\in (-\\pi, \\pi]$ is the principal argument. Let $E_1(z)$ be the principal exponential integral, let $T_n(z) = \\frac{e^{-z}}{z} \\cdot \\frac{(-1)^n n!}{z^n}$ be the $n$-th term of its asymptotic expansion, and let $R_2(z) = E_1(z) - T_0(z) - T_1(z)$ be the remainder after the first two terms. Then\u2026","title":"Two-term remainder bound for $E_1$ in the outer sector"},{"data_url":"scopes/Analysis/special-functions/elementary-transcendental-functions/lambert-w-function.json","id":"Analysis/special-functions/elementary-transcendental-functions/lambert-w-function","kind":"scope","navigation":"module","parent":17610,"result":false,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/lambert-w-function/index.html","scope":"Analysis/special-functions/elementary-transcendental-functions/lambert-w-function","status":"fully_proved","title":"Lambert W function"},{"id":"Analysis/special-functions/elementary-transcendental-functions/lambert-w-function/lambert-map","kind":"node","lean":"MetaMathlibExt.lambertMap","parent":17655,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/lambert-w-function/index.html#lambert-map","scope":"Analysis/special-functions/elementary-transcendental-functions/lambert-w-function","status":"fully_proved","summary":"The Lambert map is the function $h(y) = y e^y$ of a real variable $y$.","title":"Lambert map"},{"id":"Analysis/special-functions/elementary-transcendental-functions/lambert-w-function/exists-unique-lambert-map-preimage","kind":"node","lean":"MetaMathlibExt.existsUnique_lambertMap_preimage","parent":17655,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/lambert-w-function/index.html#exists-unique-lambert-map-preimage","scope":"Analysis/special-functions/elementary-transcendental-functions/lambert-w-function","status":"fully_proved","summary":"Let $h(y) = y e^y$ be the Lambert map on the real numbers. Every real $x > -e^{-1}$ has exactly one preimage $y > -1$ under $h$, that is, there is exactly one real number $y > -1$ with $y e^y = x$.","title":"Principal-branch preimages of the Lambert map"},{"id":"Analysis/special-functions/elementary-transcendental-functions/lambert-w-function/exists-unique-lambert-map-lower-preimage","kind":"node","lean":"MetaMathlibExt.existsUnique_lambertMap_lower_preimage","parent":17655,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/lambert-w-function/index.html#exists-unique-lambert-map-lower-preimage","scope":"Analysis/special-functions/elementary-transcendental-functions/lambert-w-function","status":"fully_proved","summary":"Let $h(y) = y e^y$ be the Lambert map on the real numbers. Every real $x$ with $-e^{-1} \\le x < 0$ has exactly one preimage $y \\le -1$ under $h$, that is, there is exactly one real number $y \\le -1$ with $y e^y = x$.","title":"Lower-branch preimages of the Lambert map"},{"data_url":"scopes/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral.json","id":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","kind":"scope","navigation":"module","parent":17610,"result":false,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/index.html","scope":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","status":"fully_proved","title":"Logarithmic integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/logarithmic-integral-domain","kind":"node","lean":"Real.LogarithmicIntegralDomain","parent":17659,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/index.html#logarithmic-integral-domain","scope":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","status":"fully_proved","summary":"The domain of the logarithmic integral is the set of real numbers $x$ with $x \\ge 0$ and $x \\ne 1$.","title":"Domain of the logarithmic integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/logarithmic-integral","kind":"node","lean":"Real.logarithmicIntegral","parent":17659,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/index.html#logarithmic-integral","scope":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","status":"fully_proved","summary":"For a real number $x$ in the domain of the logarithmic integral (that is, $x \\ge 0$ and $x \\ne 1$), the logarithmic integral $\\mathrm{li}(x)$ is $\\mathrm{li}(x) = \\int_0^x \\frac{dt}{\\log t}$ if $x < 1$, and if $x > 1$ it is","title":"Logarithmic integral"},{"id":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/logarithmic-integral-zero","kind":"node","lean":"Real.logarithmicIntegral_zero","parent":17659,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/index.html#logarithmic-integral-zero","scope":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","status":"fully_proved","summary":"The logarithmic integral $\\mathrm{li}$ satisfies $\\mathrm{li}(0) = 0$; here $0$ lies in its domain since $0 \\ge 0$ and $0 \\ne 1$.","title":"The logarithmic integral vanishes at zero"},{"id":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/logarithmic-integral-of-lt-one","kind":"node","lean":"Real.logarithmicIntegral_of_lt_one","parent":17659,"reading_title":{"format":"inline-parts-v1","parts":[["text","The logarithmic integral below "],["math","\\(1\\)"]],"source":"The logarithmic integral below $1$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/index.html#logarithmic-integral-of-lt-one","scope":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","status":"fully_proved","summary":"For every real number $x$ with $0 \\le x < 1$, the logarithmic integral $\\mathrm{li}$ satisfies $\\mathrm{li}(x) = \\int_0^x \\frac{dt}{\\log t}$.","title":"The logarithmic integral below $1$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/logarithmic-integral-of-one-lt","kind":"node","lean":"Real.logarithmicIntegral_of_one_lt","parent":17659,"reading_title":{"format":"inline-parts-v1","parts":[["text","The logarithmic integral above "],["math","\\(1\\)"]],"source":"The logarithmic integral above $1$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/index.html#logarithmic-integral-of-one-lt","scope":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","status":"fully_proved","summary":"For every real number $x > 1$, the logarithmic integral $\\mathrm{li}$ satisfies","title":"The logarithmic integral above $1$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/logarithmic-integral-real","kind":"node","lean":"Real.logarithmicIntegralReal","parent":17659,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/index.html#logarithmic-integral-real","scope":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","status":"fully_proved","summary":"This is the function $\\mathbb{R} \\to \\mathbb{R}$ whose value at $x$ is the logarithmic integral $\\operatorname{li}(x)$ when $0 \\le x$ and $x \\ne 1$, and $0$ when $x < 0$ or $x = 1$.","title":"Real logarithmic integral extended by zero"},{"id":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/has-deriv-at-logarithmic-integral-real","kind":"node","lean":"Real.hasDerivAt_logarithmicIntegralReal","parent":17659,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral/index.html#has-deriv-at-logarithmic-integral-real","scope":"Analysis/special-functions/elementary-transcendental-functions/logarithmic-integral","status":"fully_proved","summary":"For every real $x$ with $0 < x$ and $x \\ne 1$, the real logarithmic integral extended by zero has derivative $1/\\log x$ at $x$.","title":"Derivative of the logarithmic integral"},{"data_url":"scopes/Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1.json","id":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1","kind":"scope","navigation":"module","parent":17610,"result":false,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/index.html","scope":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1","status":"fully_proved","title":"Mittag-Leffler expansion of 1/(e^u \u2212 1)"},{"id":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/one-div-exp-sub-one-eq-tsum","kind":"node","lean":"MetaMathlibExt.one_div_exp_sub_one_eq_tsum","parent":17667,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mittag-Leffler expansion of "],["math","\\(1/(e^u - 1)\\)"]],"source":"Mittag-Leffler expansion of $1/(e^u - 1)$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/index.html#one-div-exp-sub-one-eq-tsum","scope":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1","status":"fully_proved","summary":"For every real number $u \\neq 0$,","title":"Mittag-Leffler expansion of $1/(e^u - 1)$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/one-div-exp-sub-one-laurent-remainder","kind":"node","lean":"MetaMathlibExt.oneDivExpSubOneLaurentRemainder","parent":17667,"reading_title":{"format":"inline-parts-v1","parts":[["text","Remainder of the Laurent expansion of "],["math","\\(1/(e^v - 1)\\)"]],"source":"Remainder of the Laurent expansion of $1/(e^v - 1)$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/index.html#one-div-exp-sub-one-laurent-remainder","scope":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1","status":"fully_proved","summary":"For a natural number $N$, the function $R_N : \\mathbb{R} \\to \\mathbb{R}$ is defined by $R_N(0) = 0$ and, for $v \\neq 0$,","title":"Remainder of the Laurent expansion of $1/(e^v - 1)$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/differentiable-one-div-exp-sub-one-laure","kind":"node","lean":"MetaMathlibExt.differentiable_oneDivExpSubOneLaurentRemainder","parent":17667,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Laurent remainder of "],["math","\\(1/(e^v - 1)\\)"],["text"," is differentiable"]],"source":"The Laurent remainder of $1/(e^v - 1)$ is differentiable"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/index.html#differentiable-one-div-exp-sub-one-laure","scope":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1","status":"fully_proved","summary":"For every natural number $N$, the function $R_N : \\mathbb{R} \\to \\mathbb{R}$ given by $R_N(0) = 0$ and, for $v \\neq 0$,","title":"The Laurent remainder of $1/(e^v - 1)$ is differentiable"},{"id":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/one-div-exp-sub-one-laurent-remainder-ab","kind":"node","lean":"MetaMathlibExt.oneDivExpSubOneLaurentRemainder_abs_le","parent":17667,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on the Laurent remainder of "],["math","\\(1/(e^v - 1)\\)"]],"source":"Bound on the Laurent remainder of $1/(e^v - 1)$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/index.html#one-div-exp-sub-one-laurent-remainder-ab","scope":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1","status":"fully_proved","summary":"For a natural number $N$, let $R_N : \\mathbb{R} \\to \\mathbb{R}$ be given by $R_N(0) = 0$ and, for $v \\neq 0$,","title":"Bound on the Laurent remainder of $1/(e^v - 1)$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/abs-deriv-one-div-exp-sub-one-laurent-re","kind":"node","lean":"MetaMathlibExt.abs_deriv_oneDivExpSubOneLaurentRemainder_le","parent":17667,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on the derivative of the Laurent remainder of "],["math","\\(1/(e^v - 1)\\)"]],"source":"Bound on the derivative of the Laurent remainder of $1/(e^v - 1)$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/index.html#abs-deriv-one-div-exp-sub-one-laurent-re","scope":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1","status":"fully_proved","summary":"For a natural number $N$, let $R_N : \\mathbb{R} \\to \\mathbb{R}$ be given by $R_N(0) = 0$ and, for $v \\neq 0$,","title":"Bound on the derivative of the Laurent remainder of $1/(e^v - 1)$"},{"id":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/abs-one-div-exp-sub-one-sub-laurent-le","kind":"node","lean":"MetaMathlibExt.abs_one_div_exp_sub_one_sub_laurent_le","parent":17667,"reading_title":{"format":"inline-parts-v1","parts":[["text","Global remainder bound for the Laurent expansion of "],["math","\\(1/(e^v - 1)\\)"]],"source":"Global remainder bound for the Laurent expansion of $1/(e^v - 1)$"},"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1/index.html#abs-one-div-exp-sub-one-sub-laurent-le","scope":"Analysis/special-functions/elementary-transcendental-functions/mittag-leffler-expansion-of-1-e-u-1","status":"fully_proved","summary":"For every natural number $N$ there is a constant $C \\ge 0$ such that for every real number $v \\neq 0$,","title":"Global remainder bound for the Laurent expansion of $1/(e^v - 1)$"},{"data_url":"scopes/Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind.json","id":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind","kind":"scope","navigation":"module","parent":17610,"result":false,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/index.html","scope":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind","status":"fully_proved","title":"Modified Bessel functions of the first kind"},{"id":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/modified-bessel-i-term","kind":"node","lean":"MetaMathlibExt.modifiedBesselI_term","parent":17674,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/index.html#modified-bessel-i-term","scope":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind","status":"fully_proved","summary":"For natural numbers $m, k$ and a real number $x$, the $k$-th term of the series for the modified Bessel function of the first kind of order $m$ is $x^{2k+m} / (2^{2k+m} \\, k! \\, (k+m)!)$.","title":"Terms of the modified Bessel series"},{"id":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/modified-bessel-i","kind":"node","lean":"MetaMathlibExt.modifiedBesselI","parent":17674,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/index.html#modified-bessel-i","scope":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind","status":"fully_proved","summary":"For a natural number $m$ and real $x$, the modified Bessel function of the first kind of order $m$ is $I_m(x) = \\sum_{k \\ge 0} x^{2k+m} / (2^{2k+m} \\, k! \\, (k+m)!)$.","title":"Modified Bessel function of the first kind"},{"id":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/modified-bessel-i-zero","kind":"node","lean":"MetaMathlibExt.modifiedBesselI_zero","parent":17674,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/index.html#modified-bessel-i-zero","scope":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind","status":"fully_proved","summary":"For a real number $x$, $I_0(x)$ is the modified Bessel function of the first kind of order $0$ at $x$, that is, $I_0(x) = \\sum_{k \\ge 0} x^{2k} / (2^{2k} \\, (k!)^2)$.","title":"Modified Bessel function of the first kind of order zero"},{"id":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/modified-bessel-i-one","kind":"node","lean":"MetaMathlibExt.modifiedBesselI_one","parent":17674,"result":true,"route":"../roadmap/Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind/index.html#modified-bessel-i-one","scope":"Analysis/special-functions/elementary-transcendental-functions/modified-bessel-functions-of-the-first-kind","status":"fully_proved","summary":"For a real number $x$, $I_1(x)$ is the modified Bessel function of the first kind of order $1$ at $x$, that is, $I_1(x) = \\sum_{k \\ge 0} x^{2k+1} / (2^{2k+1} \\, k! \\, (k+1)!)$.","title":"Modified Bessel function of the first kind of order one"},{"data_url":"scopes/Analysis/special-functions/bernoulli-polylog-clausen.json","id":"Analysis/special-functions/bernoulli-polylog-clausen","kind":"scope","parent":17543,"result":false,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/index.html","scope":"Analysis/special-functions/bernoulli-polylog-clausen","status":"can_prove","summary":"Bernoulli and Euler polynomials and their generating-function identities, Berndt's \u03c6_r and power-difference series, Clausen functions, the inverse tangent integral and dilogarithm identities; results whose content is values of \u03b6, Dirichlet L or \u03b2 functions belong to NumberTheory.","title":"Bernoulli polynomials, polylogarithms and Clausen functions"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/ramanujan-part1-ch9-entry17-sec","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry17Sec.ramanujan_part1_ch9_entry17_sec","parent":17679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The inverse tangent integral at "],["math","\\(\\tan x\\)"]],"source":"The inverse tangent integral at $\\tan x$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/index.html#ramanujan-part1-ch9-entry17-sec","scope":"Analysis/special-functions/bernoulli-polylog-clausen","status":"fully_proved","summary":"Let $x$ be a real number with $|x| \\le \\pi/4$, and let $P(x) = x \\log|\\tan x|$ for $x \\ne 0$ and $P(0) = 0$. Then $\\cos x \\ne 0$, the series $T(x) = \\sum_{k=0}^{\\infty} (-1)^k \\frac{\\tan^{2k+1} x}{(2k+1)^2}$ (the inverse tangent integral of $\\tan x$) and $F(x) = \\sum_{j=0}^{\\infty} \\frac{\\sin((4j+2)x)}{(2j+1)^2}$ are both summable, and $T(x) = P(x) + F(x)$.","title":"The inverse tangent integral at $\\tan x$"},{"data_url":"scopes/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function.json","id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","kind":"scope","navigation":"module","parent":17679,"result":false,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","title":"Berndt's power-difference function \u03c6\u1d63"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry10-zeta4","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zeta4.ramanujan_part1_ch7_entry10_zeta4","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry10-zeta4","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Let $n$ be a positive integer, and for $z \\in \\mathbb{C}$ put $B^*(z) = 2\\,\\Gamma(z + 1)\\,\\zeta(z) / (2\\pi)^z$. Then the functions $Z(r) = (1 - n^{r+1})\\,\\zeta(-r)$ and $B(r) = (n^{r+1} - 1) \\sin(\\pi r / 2)\\, B^*(r + 1) / (r + 1)$ are meromorphic on $\\mathbb{C}$, and $Z(r) = B(r)$ for all $r \\in \\mathbb{C}$ outside a closed discrete subset of $\\mathbb{C}$.","title":"The Entry 10 zeta factor in Bernoulli form"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry8","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry8.ramanujan_part1_ch7_entry8","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry8","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For every positive integer $r$ and every $x \\in \\mathbb{C}$, $\\frac{B_{r+1}(x + 1) - B_{r+1}(1)}{r + 1}$ equals $\\frac{1}{r + 1} \\sum_{k=0}^{r} \\binom{r+1}{k} B_k\\, x^{r+1-k} + x^r$, where $B_{r+1}(x)$ is the Bernoulli polynomial and $B_k$ are the Bernoulli numbers with $B_1 = -\\tfrac12$.","title":"Bernoulli-number expansion of a Bernoulli-polynomial difference"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-phi-argument","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Chapter7PhiArgument","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arguments of Berndt's "],["math","\\(\\varphi_r(x)\\)"]],"source":"Arguments of Berndt's $\\varphi_r(x)$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-phi-argument","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"The open interval $(-1, 0)$ of real numbers, that is, the set of real $x$ with $-1 < x < 0$.","title":"Arguments of Berndt's $\\varphi_r(x)$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-phi","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.chapter7Phi","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Berndt's function "],["math","\\(\\varphi_r(x)\\)"],["text"," on "],["math","\\((-1, 0)\\)"]],"source":"Berndt's function $\\varphi_r(x)$ on $(-1, 0)$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-phi","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For a complex number $r$ and a real number $x$ with $-1 < x < 0$, $\\varphi_r(x) = \\zeta(-r) - \\zeta(-r, x + 1)$, where $\\zeta$ is the Riemann zeta function and $\\zeta(s, a)$ is the Hurwitz zeta function, both analytically continued in $s$. At $r = -1$, where both have a pole, $\\zeta(1)$ and $\\zeta(1, x + 1)$ denote the finite values that the formal definitions assign at the pole.","title":"Berndt's function $\\varphi_r(x)$ on $(-1, 0)$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry15-vonstaudtcla","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry15Vonstaudtclausen.ramanujan_part1_ch7_entry15_vonstaudtclausen","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","A sine series for "],["math","\\(\\varphi_r(x-1) - \\varphi_r(-x)\\)"]],"source":"A sine series for $\\varphi_r(x-1) - \\varphi_r(-x)$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry15-vonstaudtcla","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For complex $r$ and real $-1 < y < 0$ let $\\varphi_r(y) = \\zeta(-r) - \\zeta(-r, y + 1)$, where $\\zeta$ is the Riemann zeta function and $\\zeta(s, a)$ is the Hurwitz zeta function. Let $r$ be a complex number with $\\operatorname{Re} r > -1$ and let $0 < x < 1$. Then the partial sums of $\\sum_{k \\ge 1} \\sin(2\\pi k x) / (2\\pi k)^{r+1}$ converge to a limit $L$, and","title":"A sine series for $\\varphi_r(x-1) - \\varphi_r(-x)$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry16-kummercongru","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry16Kummercongruence.ramanujan_part1_ch7_entry16_kummercongruence","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","A cosine series for "],["math","\\(\\varphi_r(x-1) + \\varphi_r(-x)\\)"]],"source":"A cosine series for $\\varphi_r(x-1) + \\varphi_r(-x)$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry16-kummercongru","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For complex $r$ and real $-1 < y < 0$ let $\\varphi_r(y) = \\zeta(-r) - \\zeta(-r, y + 1)$, where $\\zeta$ is the Riemann zeta function and $\\zeta(s, a)$ is the Hurwitz zeta function. Let $r$ be a complex number with $\\operatorname{Re} r > -1$ and let $0 < x < 1$. Then the partial sums of $\\sum_{k \\ge 1} \\cos(2\\pi k x) / (2\\pi k)^{r+1}$ converge to a limit $L$, and","title":"A cosine series for $\\varphi_r(x-1) + \\varphi_r(-x)$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry17-bernoullipol","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry17Bernoullipolysymmetry.ramanujan_part1_ch7_entry17_bernoullipolysymmetry","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\varphi_r(-1/4) - \\varphi_r(-3/4)\\)"],["text"," through the Dirichlet beta function"]],"source":"$\\varphi_r(-1/4) - \\varphi_r(-3/4)$ through the Dirichlet beta function"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry17-bernoullipol","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Let $\\chi$ be the function on $\\mathbb{Z}/4\\mathbb{Z}$ with $\\chi(1) = 1$, $\\chi(3) = -1$ and $\\chi(0) = \\chi(2) = 0$, let $\\beta$ be its Dirichlet $L$-function (analytically continued to $\\mathbb{C}$), and let $E^*(s) = 2\\,\\Gamma(s)\\,\\beta(s) / (\\pi/2)^s$. For complex $r$ and real $-1 < y < 0$ let $\\varphi_r(y) = \\zeta(-r) - \\zeta(-r, y + 1)$, where $\\zeta$ is the Riemann zeta function and $\\zeta(s, a)$ is the\u2026","title":"$\\varphi_r(-1/4) - \\varphi_r(-3/4)$ through the Dirichlet beta function"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry19-zetanegint","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry19Zetanegint.ramanujan_part1_ch7_entry19_zetanegint","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's functional equation for "],["math","\\(\\varphi\\)"],["text"," at rationals with odd denominator"]],"source":"Ramanujan's functional equation for $\\varphi$ at rationals with odd denominator"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry19-zetanegint","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For $-1 < x < 0$ and $r \\in \\mathbb{C}$ let $\\varphi_r(x) = \\zeta(-r) - \\zeta(-r, x + 1)$, where $\\zeta$ is the Riemann zeta function and $\\zeta(s, a)$ the Hurwitz zeta function. Let $p$ and $q$ be natural numbers with $0 < p < q$ and $q$ odd, and for $r \\in \\mathbb{C}$ put $L(r) = \\frac{(2\\pi q)^r}{4\\,\\Gamma(r)} \\bigl(\\varphi_{r-1}(p/q - 1) - \\varphi_{r-1}(-p/q)\\bigr)$ and\u2026","title":"Ramanujan's functional equation for $\\varphi$ at rationals with odd denominator"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry20-generalizedb","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry20Generalizedbernoullisum.ramanujan_part1_ch7_entry20_generalizedbernoullisum","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's functional equation for "],["math","\\(\\varphi\\)"],["text"," at "],["math","\\(-1/3\\)"],["text"," and "],["math","\\(-2/3\\)"]],"source":"Ramanujan's functional equation for $\\varphi$ at $-1/3$ and $-2/3$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry20-generalizedb","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For $-1 < x < 0$ and $r \\in \\mathbb{C}$ let $\\varphi_r(x) = \\zeta(-r) - \\zeta(-r, x + 1)$, where $\\zeta$ is the Riemann zeta function and $\\zeta(s, a)$ the Hurwitz zeta function. Then the functions $L(r) = \\frac{(6\\pi)^r}{2\\sqrt{3}\\,\\Gamma(r)} \\bigl(\\varphi_{r-1}(-1/3) - \\varphi_{r-1}(-2/3)\\bigr)$ and $R(r) = \\sin(\\pi r/2) \\bigl(\\varphi_{-r}(-1/3) - \\varphi_{-r}(-2/3)\\bigr)$ of $r \\in \\mathbb{C}$ are meromorphic on\u2026","title":"Ramanujan's functional equation for $\\varphi$ at $-1/3$ and $-2/3$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-integer-phi","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry8Bernoullibound.chapter7IntegerPhi","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli-polynomial form of Berndt's "],["math","\\(\\varphi_r\\)"]],"source":"Bernoulli-polynomial form of Berndt's $\\varphi_r$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-integer-phi","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For a natural number $r$ and $x \\in \\mathbb{C}$, $P_r(x) = \\frac{B_{r+1}(x + 1) - B_{r+1}(1)}{r + 1}$, where $B_{r+1}$ is the Bernoulli polynomial of degree $r + 1$.","title":"Bernoulli-polynomial form of Berndt's $\\varphi_r$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry8-bernoulliboun","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry8Bernoullibound.ramanujan_part1_ch7_entry8_bernoullibound","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli and zeta expansions of "],["math","\\(P_r\\)"],["text",", and sums of powers"]],"source":"Bernoulli and zeta expansions of $P_r$, and sums of powers"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry8-bernoulliboun","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Let $r$ be a positive integer and $P_r(x) = \\frac{B_{r+1}(x + 1) - B_{r+1}(1)}{r + 1}$, where $B_{r+1}$ is the Bernoulli polynomial of degree $r + 1$. Then for every $x \\in \\mathbb{C}$, $P_r(x) = \\frac{1}{r + 1} \\sum_{k=0}^{r} \\binom{r+1}{k} B_k\\, x^{r+1-k} + x^r$, where $B_k$ are the Bernoulli numbers with $B_1 = -\\tfrac12$, and $P_r(x) = \\frac{x^{r+1}}{r + 1} + \\frac{x^r}{2} - \\frac{2}{r + 1} Z_r(x)$, where\u2026","title":"Bernoulli and zeta expansions of $P_r$, and sums of powers"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-phi-eq-chapter7-integer-phi","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry8Bernoullibound.chapter7Phi_eq_chapter7IntegerPhi","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Berndt's "],["math","\\(\\varphi_r\\)"],["text"," as a Bernoulli-polynomial difference for positive integer "],["math","\\(r\\)"]],"source":"Berndt's $\\varphi_r$ as a Bernoulli-polynomial difference for positive integer $r$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-phi-eq-chapter7-integer-phi","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Let $r$ be a positive integer and $-1 < x < 0$. Then $\\varphi_r(x) = \\zeta(-r) - \\zeta(-r, x + 1)$, where $\\zeta$ is the Riemann zeta function and $\\zeta(s, a)$ the Hurwitz zeta function, equals $P_r(x) = \\frac{B_{r+1}(x + 1) - B_{r+1}(1)}{r + 1}$, where $B_{r+1}$ is the Bernoulli polynomial of degree $r + 1$.","title":"Berndt's $\\varphi_r$ as a Bernoulli-polynomial difference for positive integer $r$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry8-phi","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry8Bernoullibound.ramanujan_part1_ch7_entry8_phi","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli and zeta expansions of Berndt's "],["math","\\(\\varphi_r\\)"],["text"," on "],["math","\\((-1, 0)\\)"]],"source":"Bernoulli and zeta expansions of Berndt's $\\varphi_r$ on $(-1, 0)$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry8-phi","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Let $r$ be a positive integer and, for $-1 < x < 0$, let $\\varphi_r(x) = \\zeta(-r) - \\zeta(-r, x + 1)$, where $\\zeta$ is the Riemann zeta function and $\\zeta(s, a)$ the Hurwitz zeta function. Then for every $-1 < x < 0$, $\\varphi_r(x) = \\frac{1}{r + 1} \\sum_{k=0}^{r} \\binom{r+1}{k} B_k\\, x^{r+1-k} + x^r$, where $B_k$ are the Bernoulli numbers with $B_1 = -\\tfrac12$, and\u2026","title":"Bernoulli and zeta expansions of Berndt's $\\varphi_r$ on $(-1, 0)$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-admissible-3","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.chapter7Admissible","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-admissible-3","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"A complex number $x$ is admissible when $j + 1 + x \\ne 0$ for every natural number $j$, that is, when $x$ is not one of $-1, -2, -3, \\dots$.","title":"Admissible parameters for the power-difference series"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-power-difference-term-4","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.chapter7PowerDifferenceTerm","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-power-difference-term-4","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For complex numbers $r$ and $x$ and a natural number $j$, the $j$-th term is $(j + 1)^r - (j + 1 + x)^r$, where $w^r = e^{r \\log w}$ is the principal power for $w \\ne 0$, while $0^r = 0$ for $r \\ne 0$ and $0^0 = 1$.","title":"Terms of the power-difference series"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-power-difference-sum-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.chapter7PowerDifferenceSum","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-power-difference-sum-2","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For complex numbers $r$ and $x$, $\\varphi_r(x) = \\sum_{j \\ge 0} \\bigl((j + 1)^r - (j + 1 + x)^r\\bigr)$ (principal powers), where the sum is taken to be $0$ when the series is not summable.","title":"The power-difference series"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry10-zeta2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zeta2.ramanujan_part1_ch7_entry10_zeta2","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry10-zeta2","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Let $\\varphi : \\mathbb{C} \\times \\mathbb{C} \\to \\mathbb{C}$, written $(r, x) \\mapsto \\varphi_r(x)$, be such that for every $x \\in \\mathbb{C}$ the function $r \\mapsto \\varphi_r(x)$ is meromorphic on $\\mathbb{C}$, and such that for all $r, x \\in \\mathbb{C}$ with $\\operatorname{Re} r < 0$ the series $\\sum_{j \\ge 0} \\bigl((j + 1)^r - (j + 1 + x)^r\\bigr)$ converges absolutely with sum $\\varphi_r(x)$. Let $n$ be a\u2026","title":"Ramanujan's distribution relation for the power-difference function"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-power-difference-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zeta2.chapter7PowerDifferenceTerm","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-power-difference-term","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"The power-difference term of Ramanujan's Entry 10 is the $j$-th term of the power-difference series, $(j + 1)^r - (j + 1 + x)^r$.","title":"Power difference term in Ramanujan's Entry 10"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-entry10-corollary-left","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7Entry10CorollaryLeft","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-entry10-corollary-left","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For a function $\\varphi : \\mathbb{C} \\to \\mathbb{C} \\to \\mathbb{C}$, a natural number $n$ and $r \\in \\mathbb{C}$, the left side of the corollary to Entry 10 is $\\sum_{k=1}^{n-1} \\varphi(r, -k/n)$, the empty sum $0$ when $n \\le 1$. In Berndt's notation, $\\varphi(r, x)$ is written $\\varphi_r(x)$.","title":"Left side of the corollary to Entry 10"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-entry10-corollary-right","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7Entry10CorollaryRight","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-entry10-corollary-right","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For a natural number $n$ and $r \\in \\mathbb{C}$, the right side of the corollary to Entry 10 is $(n - n^{-r})\\,\\zeta(-r)$, where $n^{-r}$ is the principal complex power and $\\zeta$ is the Riemann zeta function.","title":"Right side of the corollary to Entry 10"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-admissible-neg-div","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7Admissible_neg_div","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","The points "],["math","\\(-k/n\\)"],["text"," are admissible"]],"source":"The points $-k/n$ are admissible"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-admissible-neg-div","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For natural numbers $n$ and $k$ with $1 \\le k \\le n - 1$, the point $x = -k/n$ is admissible, that is, $j + 1 + x \\neq 0$ for every natural number $j$.","title":"The points $-k/n$ are admissible"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-entry10-corollary-left-eq-right","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7Entry10CorollaryLeft_eq_right","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","The corollary to Entry 10 in the half-plane "],["math","\\(\\operatorname{Re} r < -1\\)"]],"source":"The corollary to Entry 10 in the half-plane $\\operatorname{Re} r < -1$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-entry10-corollary-left-eq-right","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Call $x \\in \\mathbb{C}$ admissible if $j + 1 + x \\neq 0$ for every natural number $j$. Let $\\varphi : \\mathbb{C} \\to \\mathbb{C} \\to \\mathbb{C}$ be such that, for every $r \\in \\mathbb{C}$ with $\\operatorname{Re} r < 0$ and every admissible $x$, the series $\\sum_{j \\ge 0} \\bigl((j + 1)^r - (j + 1 + x)^r\\bigr)$ of principal complex powers is unconditionally summable with sum $\\varphi(r, x)$. Then for every natural\u2026","title":"The corollary to Entry 10 in the half-plane $\\operatorname{Re} r < -1$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry10-zetaviaberno","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.ramanujan_part1_ch7_entry10_zetaviabernoulli","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry10-zetaviaberno","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Call $x \\in \\mathbb{C}$ admissible if $j + 1 + x \\neq 0$ for every natural number $j$. Let $\\varphi : \\mathbb{C} \\to \\mathbb{C} \\to \\mathbb{C}$ be such that, for every admissible $x$, the function $r \\mapsto \\varphi(r, x)$ is entire, and, for every $r \\in \\mathbb{C}$ with $\\operatorname{Re} r < 0$ and every admissible $x$, the series $\\sum_{j \\ge 0} \\bigl((j + 1)^r - (j + 1 + x)^r\\bigr)$ of principal complex powers\u2026","title":"Ramanujan's corollary to Entry 10 of Chapter 7"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-admissible","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7Admissible","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-admissible","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Admissibility in the corollary of Ramanujan's Entry 10 is admissibility for the power-difference series: $x$ is not one of $-1, -2, -3, \\dots$.","title":"Admissibility in the corollary of Ramanujan's Entry 10"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-power-difference-term-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7PowerDifferenceTerm","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-power-difference-term-2","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"The power-difference term of the corollary of Ramanujan's Entry 10 is the $j$-th term of the power-difference series, $(j + 1)^r - (j + 1 + x)^r$.","title":"Power difference term in the corollary of Entry 10"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-admissible-zero","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7Admissible_zero","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-admissible-zero","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"The point $x = 0$ is admissible: $j + 1 + x \\neq 0$ for every natural number $j$.","title":"Zero is admissible"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-power-difference-term-zero-left","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7PowerDifferenceTerm_zero_left","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-power-difference-term-zero-left","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For every $x \\in \\mathbb{C}$ and every natural number $j$, the power-difference term $(j + 1)^r - (j + 1 + x)^r$ vanishes at $r = 0$.","title":"Power-difference terms vanish at exponent zero"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-power-difference-term-zero-righ","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry10Zetaviabernoulli.chapter7PowerDifferenceTerm_zero_right","parent":17681,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power-difference terms vanish at "],["math","\\(x = 0\\)"]],"source":"Power-difference terms vanish at $x = 0$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-power-difference-term-zero-righ","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"For every $r \\in \\mathbb{C}$ and every natural number $j$, the power-difference term $(j + 1)^r - (j + 1 + x)^r$ vanishes at $x = 0$.","title":"Power-difference terms vanish at $x = 0$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry9-bernoulligrow","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry9Bernoulligrowth.ramanujan_part1_ch7_entry9_bernoulligrowth","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry9-bernoulligrow","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Let $x$ be an admissible complex number. For every complex $r$ with $\\operatorname{Re} r < 0$ the series $\\sum_{j \\ge 0} \\bigl((j + 1)^r - (j + 1 + x)^r\\bigr)$ (principal powers) is summable, and its sum $\\varphi_r(x)$ is a holomorphic function of $r$ on the half-plane $\\operatorname{Re} r < 0$.","title":"Convergence and holomorphy of the power-difference series"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-admissible-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry9Bernoulligrowth.chapter7Admissible","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-admissible-2","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Admissibility in Ramanujan's Entry 9 is admissibility for the power-difference series: $x$ is not one of $-1, -2, -3, \\dots$.","title":"Admissibility in Ramanujan's Entry 9"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-power-difference-term-3","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry9Bernoulligrowth.chapter7PowerDifferenceTerm","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-power-difference-term-3","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"The power-difference term of Ramanujan's Entry 9 is the $j$-th term of the power-difference series, $(j + 1)^r - (j + 1 + x)^r$.","title":"Power difference term in Ramanujan's Entry 9"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/chapter7-power-difference-sum","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry9Bernoulligrowth.chapter7PowerDifferenceSum","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#chapter7-power-difference-sum","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"The power-difference sum of Ramanujan's Entry 9 is the power-difference series $\\varphi_r(x)$.","title":"Power difference sum in Ramanujan's Entry 9"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/ramanujan-part1-ch7-entry9-zetaeven","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry9Zetaeven.ramanujan_part1_ch7_entry9_zetaeven","parent":17681,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function/index.html#ramanujan-part1-ch7-entry9-zetaeven","scope":"Analysis/special-functions/bernoulli-polylog-clausen/berndt-s-power-difference-function","status":"fully_proved","summary":"Let $f$ and $g$ be complex-valued functions of two complex variables $(r, x)$ such that, for every admissible $x$, both $r \\mapsto f(r, x)$ and $r \\mapsto g(r, x)$ are entire, and for every admissible $x$ and every complex $r$ with $\\operatorname{Re} r < 0$ the series $\\sum_{j \\ge 0} \\bigl((j + 1)^r - (j + 1 + x)^r\\bigr)$ (principal powers) has sum $f(r, x)$ and also has sum $g(r, x)$. Then $f(r, x) = g(r, x)$ for\u2026","title":"Entire continuations of the power-difference series agree and take Bernoulli values"},{"data_url":"scopes/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions.json","id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","kind":"scope","navigation":"module","parent":17679,"result":false,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","title":"Clausen functions"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-clausen-term-norm-bound","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry13Bernoulligen.chapter9ClausenTerm_norm_bound","parent":17715,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-clausen-term-norm-bound","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For natural numbers $m$ and $k$ and a real number $x$, let $t_k = \\sin((k+1)x)/(k+1)^m$ if $m$ is even and $t_k = \\cos((k+1)x)/(k+1)^m$ if $m$ is odd. Then $\\lvert t_k\\rvert \\le 1/(k+1)^m$.","title":"Termwise bound for the Clausen series (Chapter 9, Entry 13)"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-clausen-term-summable-of-two-le","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry13Bernoulligen.chapter9ClausenTerm_summable_of_two_le","parent":17715,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-clausen-term-summable-of-two-le","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For every natural number $m \\ge 2$ and every real $x$, the series $\\sum_{k \\ge 1} \\sin(kx)/k^m$ (if $m$ is even) or $\\sum_{k \\ge 1} \\cos(kx)/k^m$ (if $m$ is odd) is summable.","title":"Summability of the Clausen series (Chapter 9, Entry 13)"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-clausen-zero","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry13Bernoulligen.chapter9Clausen_zero","parent":17715,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-clausen-zero","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"The Clausen-type functions $C_m$ of Entry 13 are defined by $C_1(x) = -\\log\\lvert 2\\sin(x/2)\\rvert$ and, for $m \\ge 2$, $C_m(x) = \\sum_{k \\ge 1} \\sin(kx)/k^m$ for even $m$ and $C_m(x) = \\sum_{k \\ge 1} \\cos(kx)/k^m$ for odd $m$, together with the convention $C_0 = 0$: for every real $x$, $C_0(x) = 0$.","title":"Clausen function of order zero (Chapter 9, Entry 13)"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-clausen-one","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry13Bernoulligen.chapter9Clausen_one","parent":17715,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-clausen-one","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"The order-one Clausen-type function $C_1$ of Entry 13 is the log-sine function: for every real $x$, $C_1(x) = -\\log\\lvert 2\\sin(x/2)\\rvert$, with the convention $\\log 0 = 0$, so that $C_1$ vanishes at the integer multiples of $2\\pi$.","title":"Clausen function of order one (Chapter 9, Entry 13)"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-clausen-of-two-le","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry13Bernoulligen.chapter9Clausen_of_two_le","parent":17715,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-clausen-of-two-le","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"Let $C_m$ be the Clausen-type function of Entry 13, with $C_0 = 0$ and $C_1(x) = -\\log\\lvert 2\\sin(x/2)\\rvert$. For every natural number $m \\ge 2$ and every real $x$, $C_m(x) = \\sum_{k \\ge 1} \\sin(kx)/k^m$ if $m$ is even and $C_m(x) = \\sum_{k \\ge 1} \\cos(kx)/k^m$ if $m$ is odd.","title":"Clausen functions of order at least two as series (Chapter 9, Entry 13)"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/ramanujan-part1-ch9-entry13-bernoulligen","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry13Bernoulligen.ramanujan_part1_ch9_entry13_bernoulligen","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrals of "],["math","\\(u^n \\cot(u/2)\\)"],["text"," via Clausen functions (Chapter 9, Entry 13)"]],"source":"Integrals of $u^n \\cot(u/2)$ via Clausen functions (Chapter 9, Entry 13)"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#ramanujan-part1-ch9-entry13-bernoulligen","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For real $x$ let $C_1(x) = -\\log\\lvert 2\\sin(x/2)\\rvert$ and, for $m \\ge 2$, let $C_m(x) = \\sum_{k \\ge 1} \\sin(kx)/k^m$ if $m$ is even and $C_m(x) = \\sum_{k \\ge 1} \\cos(kx)/k^m$ if $m$ is odd. Let $n \\ge 1$ be a natural number and $x$ a nonzero real number with $\\lvert x\\rvert < 2\\pi$. Then the function $u \\mapsto \\frac{u^n}{2}\\cot\\frac{u}{2}$ is integrable on the interval between $0$ and $x$; the partial sums\u2026","title":"Integrals of $u^n \\cot(u/2)$ via Clausen functions (Chapter 9, Entry 13)"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-odd-clausen-term-norm-bound","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.chapter9OddClausenTerm_norm_bound","parent":17715,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-odd-clausen-term-norm-bound","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For integers $m, k \\ge 0$ let $s_m = \\sin$ if $m$ is even and $s_m = \\cos$ if $m$ is odd. Then for all integers $m, k \\ge 0$ and every real $x$,","title":"Bound on the terms of the odd Clausen series"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-odd-clausen-term-summable-of-tw","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.chapter9OddClausenTerm_summable_of_two_le","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of the odd Clausen series for "],["math","\\(m \\ge 2\\)"]],"source":"Summability of the odd Clausen series for $m \\ge 2$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-odd-clausen-term-summable-of-tw","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"Let $m \\ge 2$ be an integer, and let $s_m = \\sin$ if $m$ is even and $s_m = \\cos$ if $m$ is odd. Then for every real $x$ the series $\\sum_{k \\ge 0} s_m((2k+1)x)/(2k+1)^m$ is summable.","title":"Summability of the odd Clausen series for $m \\ge 2$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-odd-clausen-zero","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.chapter9OddClausen_zero","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odd Clausen function vanishes at "],["math","\\(m = 0\\)"]],"source":"The odd Clausen function vanishes at $m = 0$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-odd-clausen-zero","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For an integer $m \\ge 0$ let $s_m = \\sin$ if $m$ is even and $s_m = \\cos$ if $m$ is odd. The odd Clausen functions $O_m$, for integers $m \\ge 0$, are given by $O_1(x) = -\\tfrac{1}{2} \\log|\\tan(x/2)|$, by $O_m(x) = \\sum_{k \\ge 0} s_m((2k+1)x)/(2k+1)^m$ for $m \\ge 2$, and at $m = 0$ by zero: for every real $x$, $O_0(x) = 0$.","title":"The odd Clausen function vanishes at $m = 0$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-odd-clausen-one","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.chapter9OddClausen_one","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odd Clausen function at "],["math","\\(m = 1\\)"]],"source":"The odd Clausen function at $m = 1$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-odd-clausen-one","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For an integer $m \\ge 0$ let $s_m = \\sin$ if $m$ is even and $s_m = \\cos$ if $m$ is odd. The odd Clausen functions $O_m$, for integers $m \\ge 0$, are given by $O_0 = 0$, by $O_m(x) = \\sum_{k \\ge 0} s_m((2k+1)x)/(2k+1)^m$ for $m \\ge 2$, and at $m = 1$ by a closed form: for every real $x$, $O_1(x) = -\\tfrac{1}{2} \\log|\\tan(x/2)|$.","title":"The odd Clausen function at $m = 1$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-odd-clausen-of-two-le","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.chapter9OddClausen_of_two_le","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","The odd Clausen function as a series for "],["math","\\(m \\ge 2\\)"]],"source":"The odd Clausen function as a series for $m \\ge 2$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-odd-clausen-of-two-le","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For an integer $m \\ge 0$ let $s_m = \\sin$ if $m$ is even and $s_m = \\cos$ if $m$ is odd. For real $x$, the odd Clausen function $O_m(x)$ is $0$ if $m = 0$, is $-\\tfrac{1}{2} \\log|\\tan(x/2)|$ if $m = 1$, and is otherwise given by a series: for every integer $m \\ge 2$ and every real $x$, $O_m(x) = \\sum_{k \\ge 0} s_m((2k+1)x)/(2k+1)^m$.","title":"The odd Clausen function as a series for $m \\ge 2$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-odd-clausen-eq-clausen-sub-of-t","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.chapter9OddClausen_eq_clausen_sub_of_two_le","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd-frequency part of a Clausen series for "],["math","\\(m \\ge 2\\)"]],"source":"Odd-frequency part of a Clausen series for $m \\ge 2$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-odd-clausen-eq-clausen-sub-of-t","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"Let $m \\ge 2$ be an integer, and let $s_m = \\sin$ if $m$ is even and $s_m = \\cos$ if $m$ is odd. For real $x$ write $C_m(x) = \\sum_{k \\ge 1} s_m(kx)/k^m$ and $O_m(x) = \\sum_{k \\ge 0} s_m((2k+1)x)/(2k+1)^m$. Then for every real $x$, $O_m(x) = C_m(x) - 2^{-m} C_m(2x)$.","title":"Odd-frequency part of a Clausen series for $m \\ge 2$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-odd-clausen-eq-clausen-sub","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.chapter9OddClausen_eq_clausen_sub","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd-frequency part of a Clausen function for "],["math","\\(0 < |x| < \\pi\\)"]],"source":"Odd-frequency part of a Clausen function for $0 < |x| < \\pi$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-odd-clausen-eq-clausen-sub","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For real $x$ let $C_1(x) = -\\log|2 \\sin(x/2)|$ and $O_1(x) = -\\tfrac{1}{2} \\log|\\tan(x/2)|$. For an integer $m \\ge 2$, let $s_m = \\sin$ if $m$ is even and $s_m = \\cos$ if $m$ is odd, and let $C_m(x) = \\sum_{k \\ge 1} s_m(kx)/k^m$ and $O_m(x) = \\sum_{k \\ge 0} s_m((2k+1)x)/(2k+1)^m$. Then for every integer $m \\ge 1$ and every real $x$ with $x \\ne 0$ and $|x| < \\pi$, $O_m(x) = C_m(x) - 2^{-m} C_m(2x)$.","title":"Odd-frequency part of a Clausen function for $0 < |x| < \\pi$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/chapter9-chi-at-one-eq-zeta","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.chapter9ChiAtOne_eq_zeta","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd reciprocal power sums in terms of "],["math","\\(\\zeta\\)"]],"source":"Odd reciprocal power sums in terms of $\\zeta$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#chapter9-chi-at-one-eq-zeta","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For every integer $m \\ge 2$,","title":"Odd reciprocal power sums in terms of $\\zeta$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/ramanujan-part1-ch9-entry14-xoversin","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry14Xoversin.ramanujan_part1_ch9_entry14_xoversin","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entry 14 of Chapter 9: the integral of "],["math","\\(u^n / (2 \\sin u)\\)"]],"source":"Entry 14 of Chapter 9: the integral of $u^n / (2 \\sin u)$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#ramanujan-part1-ch9-entry14-xoversin","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For an integer $m \\ge 2$ write $\\lambda(m) = \\sum_{k \\ge 0} (2k+1)^{-m}$. For real $x$ let $C_1(x) = -\\tfrac{1}{2} \\log|\\tan(x/2)|$, and for an integer $m \\ge 2$ let $s_m = \\sin$ if $m$ is even and $s_m = \\cos$ if $m$ is odd, and let $C_m(x) = \\sum_{k \\ge 0} s_m((2k+1)x)/(2k+1)^m$. Let $n \\ge 1$ be an integer and $x$ a real number with $x \\ne 0$ and $|x| < \\pi$. Then $u \\mapsto u^n / (2 \\sin u)$ is integrable on\u2026","title":"Entry 14 of Chapter 9: the integral of $u^n / (2 \\sin u)$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/ramanujan-part1-ch9-entry15-xcot","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry15Xcot.ramanujan_part1_ch9_entry15_xcot","parent":17715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entry 15 of Chapter 9: integrals of "],["math","\\(u^n \\cot u\\)"],["text"," at "],["math","\\(\\pi/2 - x\\)"]],"source":"Entry 15 of Chapter 9: integrals of $u^n \\cot u$ at $\\pi/2 - x$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions/index.html#ramanujan-part1-ch9-entry15-xcot","scope":"Analysis/special-functions/bernoulli-polylog-clausen/clausen-functions","status":"fully_proved","summary":"For an integer $k \\ge 0$ and real $y$ write $F_k(y) = \\int_{\\pi/4}^{y} u^k \\cot u \\, du$. Let $n \\ge 0$ be an integer. Then for every integer $k$ with $0 \\le k \\le n$ and every real $y$ with $0 < y < \\pi$, the function $u \\mapsto u^k \\cot u$ is integrable on the interval between $\\pi/4$ and $y$; and for every real $x$ with $0 < x < \\pi/2$,","title":"Entry 15 of Chapter 9: integrals of $u^n \\cot u$ at $\\pi/2 - x$"},{"data_url":"scopes/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities.json","id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","kind":"scope","navigation":"module","parent":17679,"result":false,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","title":"Dilogarithm identities"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/ramanujan-part1-ch9-entry24-dougall","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry24Dougall.ramanujan_part1_ch9_entry24_dougall","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#ramanujan-part1-ch9-entry24-dougall","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"fully_proved","summary":"Let $x, y \\in [0, 1]$ and $\\theta, \\varphi \\in (-\\pi, \\pi]$ satisfy $x e^{i\\theta} + y e^{i\\varphi} = 1$. Write $\\ell(t) = \\log t$ for $0 < t \\le 1$ and $\\ell(0) = 0$. Then the four series $\\sum_{k \\ge 1} \\frac{x^k \\cos k\\theta}{k^2}$, $\\sum_{k \\ge 1} \\frac{y^k \\cos k\\varphi}{k^2}$, $\\sum_{k \\ge 1} \\frac{x^k \\sin k\\theta}{k^2}$ and $\\sum_{k \\ge 1} \\frac{y^k \\sin k\\varphi}{k^2}$ converge absolutely, and","title":"Entry 24 of Chapter 9 of Ramanujan's notebooks"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/ramanujan-part1-ch9-entry25-dixon","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry25Dixon.ramanujan_part1_ch9_entry25_dixon","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#ramanujan-part1-ch9-entry25-dixon","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"fully_proved","summary":"Let $x, y \\in [0, 1]$ and let $\\theta$ and $\\varphi$ be real numbers with $x e^{i\\theta} + y e^{i\\varphi} = xy \\, e^{i(\\theta + \\varphi)}$. Then $1 - x \\cos\\theta \\ne 0$ and $1 - y \\cos\\varphi \\ne 0$. Put $A_x = 1 - 2x \\cos\\theta + x^2$, $A_y = 1 - 2y \\cos\\varphi + y^2$, $u_x = \\arctan \\frac{x \\sin\\theta}{1 - x \\cos\\theta}$ and $u_y = \\arctan \\frac{y \\sin\\varphi}{1 - y \\cos\\varphi}$. Then $A_x > 0$, $A_y > 0$, the\u2026","title":"Entry 25 of Chapter 9 of Ramanujan's notebooks"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/chapter9-boundary-log-eq-log","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry26Ramanujanpi.chapter9BoundaryLog_eq_log","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#chapter9-boundary-log-eq-log","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"fully_proved","summary":"Let $\\log$ be the real logarithm, extended to all of $\\mathbb{R}$ by $\\log 0 = 0$ and $\\log x = \\log|x|$ for $x < 0$. For every real $x$, the boundary logarithm of $x$, defined as $0$ if $x = 0$ and as $\\log x$ otherwise, equals $\\log x$.","title":"The Entry 26 boundary logarithm is the real logarithm"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/chapter9-dilog-cos-term-eq-entry24","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry26Ramanujanpi.chapter9DilogCosTerm_eq_entry24","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#chapter9-dilog-cos-term-eq-entry24","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"fully_proved","summary":"For all real $\\rho$ and $\\alpha$ and every natural number $j$, the $j$-th term $\\rho^{j+1} \\cos((j+1)\\alpha) / (j+1)^2$ of the dilogarithm cosine series defined for Entry 26 equals the $j$-th term of the cosine series defined for Entry 24, which is given by the same formula.","title":"The Entry 26 and Entry 24 dilogarithm cosine terms agree"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/chapter9-dilog-sin-term-eq-entry24","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry26Ramanujanpi.chapter9DilogSinTerm_eq_entry24","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#chapter9-dilog-sin-term-eq-entry24","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"fully_proved","summary":"For all real $\\rho$ and $\\alpha$ and every natural number $j$, the $j$-th term $\\rho^{j+1} \\sin((j+1)\\alpha) / (j+1)^2$ of the dilogarithm sine series defined for Entry 26 equals the $j$-th term of the sine series defined for Entry 24, which is given by the same formula.","title":"The Entry 26 and Entry 24 dilogarithm sine terms agree"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/ramanujan-part1-ch9-entry26-ramanujanpi","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry26Ramanujanpi.ramanujan_part1_ch9_entry26_ramanujanpi_entry24_terms","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#ramanujan-part1-ch9-entry26-ramanujanpi","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"fully_proved","summary":"For real $\\rho$ and $\\alpha$ and $k \\ge 1$ let $c_k(\\rho, \\alpha) = \\rho^k \\cos(k\\alpha) / k^2$ and $s_k(\\rho, \\alpha) = \\rho^k \\sin(k\\alpha) / k^2$ be the terms of the dilogarithm series of Entry 24, and write $C(\\rho, \\alpha) = \\sum_{j \\ge 0} c_{2j+1}(\\rho, \\alpha)$ and $S(\\rho, \\alpha) = \\sum_{j \\ge 0} s_{2j+1}(\\rho, \\alpha)$. Let $x, y \\in [0, 1]$ and $\\theta, \\varphi \\in (-\\pi, \\pi]$ satisfy\u2026","title":"Ramanujan's Landen-type identity in terms of the Entry 24 series"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/ramanujan-part1-ch9-entry26-ramanujanpi-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry26Ramanujanpi.ramanujan_part1_ch9_entry26_ramanujanpi","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#ramanujan-part1-ch9-entry26-ramanujanpi-2","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"fully_proved","summary":"For real $\\rho$ and $\\alpha$ write $C(\\rho, \\alpha) = \\sum_{j \\ge 0} \\rho^{2j+1} \\cos((2j+1)\\alpha) / (2j+1)^2$ and $S(\\rho, \\alpha) = \\sum_{j \\ge 0} \\rho^{2j+1} \\sin((2j+1)\\alpha) / (2j+1)^2$. Let $x, y \\in [0, 1]$ and $\\theta, \\varphi \\in (-\\pi, \\pi]$ satisfy $x e^{i\\theta} + y e^{i\\varphi} + xy\\, e^{i(\\theta + \\varphi)} = 1$. Then the four series defining $C(x, \\theta)$, $C(y, \\varphi)$, $S(x, \\theta)$ and\u2026","title":"Ramanujan's Landen-type dilogarithm identity (Entry 26)"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/rogers-dilog-hoorfar-qi-series","kind":"node","lean":"MetaMathlibExt.rogersDilog_hoorfarQi_series","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#rogers-dilog-hoorfar-qi-series","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","summary":"Write $L(x) = \\sum_{m \\ge 1} x^m/m^2 + \\frac{1}{2} \\log x \\log(1 - x)$ for the Rogers dilogarithm on $[0, 1]$ (with $\\log 0 = 0$). For positive integers $p, q$ and real numbers $\\theta, \\beta$ with $0 < \\theta < 1$ and $0 < \\beta < 1$,","title":"Hoorfar\u2013Qi dilogarithm series"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/rogers-dilog-pell-positive-series","kind":"node","lean":"MetaMathlibExt.rogersDilog_pell_positive_series","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#rogers-dilog-pell-positive-series","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","summary":"Write $L(x) = \\sum_{m \\ge 1} x^m/m^2 + \\frac{1}{2} \\log x \\log(1 - x)$ for the Rogers dilogarithm on $[0, 1]$ (with $\\log 0 = 0$). Let $n$ be a natural number that is not a perfect square, let $a, b$ be positive rationals with $a^2 - n b^2 = 1$, and put $u = a + b\\sqrt{n}$. Define rationals $a_k, b_k$ by $a_0 = 1$, $b_0 = 0$, $a_{k+1} = a a_k + n b b_k$ and $b_{k+1} = b a_k + a b_k$, so that\u2026","title":"Bridgeman's dilogarithm series for a positive Pell solution"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/rogers-dilog-pell-negative-series","kind":"node","lean":"MetaMathlibExt.rogersDilog_pell_negative_series","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#rogers-dilog-pell-negative-series","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","summary":"Write $L(x) = \\sum_{m \\ge 1} x^m/m^2 + \\frac{1}{2} \\log x \\log(1 - x)$ for the Rogers dilogarithm on $[0, 1]$ (with $\\log 0 = 0$). Let $n$ be a natural number that is not a perfect square, let $a, b$ be positive rationals with $a^2 - n b^2 = -1$, and put $u = a + b\\sqrt{n}$. Define rationals $a_k, b_k$ by $a_0 = 1$, $b_0 = 0$, $a_{k+1} = a a_k + n b b_k$ and $b_{k+1} = b a_k + a b_k$, so that\u2026","title":"Bridgeman's dilogarithm series for a negative Pell solution"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/sanford-coupled-recurrence","kind":"node","lean":"MetaMathlibExt.sanford_coupled_recurrence","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#sanford-coupled-recurrence","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","summary":"Let $a, b$ be real numbers in $(0, 1)$ with $a \\ne b$, and for every natural number $m$ put","title":"Sanford's coupled recurrence"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/rogers-dilog-sanford-principal-series","kind":"node","lean":"MetaMathlibExt.rogersDilog_sanford_principal_series","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#rogers-dilog-sanford-principal-series","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","summary":"Write $L(x) = \\sum_{m \\ge 1} x^m/m^2 + \\frac{1}{2} \\log x \\log(1 - x)$ for the Rogers dilogarithm on $[0, 1]$ (with $\\log 0 = 0$). For real numbers $a, b \\in (0, 1)$ with $a \\ne b$,","title":"Sanford's dilogarithm series identity"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/rogers-dilog-sanford-symmetric-series","kind":"node","lean":"MetaMathlibExt.rogersDilog_sanford_symmetric_series","parent":17732,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#rogers-dilog-sanford-symmetric-series","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","summary":"Write $L(x) = \\sum_{m \\ge 1} x^m/m^2 + \\frac{1}{2} \\log x \\log(1 - x)$ for the Rogers dilogarithm on $[0, 1]$ (with $\\log 0 = 0$). For every real number $a \\in (0, 1)$,","title":"Sanford's symmetric dilogarithm series"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/rogers-dilog-lucas-u-series","kind":"node","lean":"MetaMathlibExt.rogersDilog_lucasU_series","parent":17732,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sanford's Lucas-sequence dilogarithm series for "],["math","\\(Q > 0\\)"]],"source":"Sanford's Lucas-sequence dilogarithm series for $Q > 0$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#rogers-dilog-lucas-u-series","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","summary":"Write $L(x) = \\sum_{m \\ge 1} x^m/m^2 + \\frac{1}{2} \\log x \\log(1 - x)$ for the Rogers dilogarithm on $[0, 1]$ (with $\\log 0 = 0$). Let $(P, Q)$ be Lucas sequence parameters with $P > 0$, $Q > 0$ and $D = P^2 - 4Q > 0$, let $U = U(P, Q)$ be the Lucas sequence, and put $\\alpha = (P + \\sqrt{D})/2$. Then for every integer $k \\ge 1$,","title":"Sanford's Lucas-sequence dilogarithm series for $Q > 0$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/rogers-dilog-lucas-uv-series-of-neg","kind":"node","lean":"MetaMathlibExt.rogersDilog_lucasUV_series_of_neg","parent":17732,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sanford's Lucas-sequence dilogarithm series for "],["math","\\(Q < 0\\)"]],"source":"Sanford's Lucas-sequence dilogarithm series for $Q < 0$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities/index.html#rogers-dilog-lucas-uv-series-of-neg","scope":"Analysis/special-functions/bernoulli-polylog-clausen/dilogarithm-identities","status":"can_prove","summary":"Write $L(x) = \\sum_{m \\ge 1} x^m/m^2 + \\frac{1}{2} \\log x \\log(1 - x)$ for the Rogers dilogarithm on $[0, 1]$ (with $\\log 0 = 0$). Let $(P, Q)$ be Lucas sequence parameters with $P > 0$ and $Q < 0$, let $U = U(P, Q)$ be the Lucas sequence and $V = V(P, Q)$ its companion, and put $D = P^2 - 4Q$ and $\\alpha = (P + \\sqrt{D})/2$. Then for every odd natural number $k$,","title":"Sanford's Lucas-sequence dilogarithm series for $Q < 0$"},{"data_url":"scopes/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials.json","id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","kind":"scope","navigation":"module","parent":17679,"result":false,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","title":"Eulerian polynomials"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/ramanujan-part1-ch5-entry4","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry4.ramanujan_part1_ch5_entry4","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#ramanujan-part1-ch5-entry4","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"Let $A(n,k)$ be the Eulerian numbers, given by $A(0,0) = 1$, $A(0,k) = 0$ for $k \\ge 1$ and $A(n+1,k) = (k+1)A(n,k) + (n+1-k)A(n,k-1)$, where the last term is $0$ when $k = 0$, and let $\\Psi_0(p) = 1$ and $\\Psi_n(p) = \\sum_{k=0}^{n-1} A(n,k)(-p)^k$ for $n \\ge 1$. For every natural number $n$ and every complex number $p$ with $|p| < 1$, we have $p + 1 \\ne 0$, the series $\\sum_{k \\ge 0} (k+1)^n (-p)^k$ is summable,\u2026","title":"Eulerian polynomials and the series of (k+1)^n (-p)^k"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/ramanujan-part1-ch5-entry5-eulerianexpli","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry5Eulerianexplicit.ramanujan_part1_ch5_entry5_eulerianexplicit","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#ramanujan-part1-ch5-entry5-eulerianexpli","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"Let $A(n,k)$ be the Eulerian numbers, given by $A(0,0) = 1$, $A(0,k) = 0$ for $k \\ge 1$ and $A(n+1,k) = (k+1)A(n,k) + (n+1-k)A(n,k-1)$, where the last term is $0$ when $k = 0$, and let $\\Psi_0(p) = 1$ and $\\Psi_n(p) = \\sum_{k=0}^{n-1} A(n,k)(-p)^k$ for $n \\ge 1$. For every integer $n \\ge 1$ and every complex number $p \\ne -1$, we have $p + 1 \\ne 0$, $\\Psi_0(p) = 1$,\u2026","title":"Two finite expansions of the Eulerian polynomials"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/chapter5-entry8-exp-add-one-ne","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry8Eulerianformula.chapter5Entry8_exp_add_one_ne","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#chapter5-entry8-exp-add-one-ne","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"For every complex number $z$ with $|z| < \\pi$, we have $e^z + 1 \\ne 0$.","title":"No zeros of e^z + 1 in the disc of radius pi"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/chapter5-entry8-psi-neg-one","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry8Eulerianformula.chapter5Entry8_psi_neg_one","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#chapter5-entry8-psi-neg-one","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"Let $A(n,k)$ be the Eulerian numbers, given by $A(0,0) = 1$, $A(0,k) = 0$ for $k \\ge 1$ and $A(n+1,k) = (k+1)A(n,k) + (n+1-k)A(n,k-1)$, where the last term is $0$ when $k = 0$, and let $\\Psi_0(p) = 1$ and $\\Psi_n(p) = \\sum_{k=0}^{n-1} A(n,k)(-p)^k$ for $n \\ge 1$. For every natural number $n$, $\\Psi_n(-1) = n!$.","title":"Eulerian polynomials at p = -1"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/chapter5-entry8-psi-one","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry8Eulerianformula.chapter5Entry8_psi_one","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#chapter5-entry8-psi-one","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"Let $A(n,k)$ be the Eulerian numbers, given by $A(0,0) = 1$, $A(0,k) = 0$ for $k \\ge 1$ and $A(n+1,k) = (k+1)A(n,k) + (n+1-k)A(n,k-1)$, where the last term is $0$ when $k = 0$; let $\\Psi_0(p) = 1$ and $\\Psi_n(p) = \\sum_{k=0}^{n-1} A(n,k)(-p)^k$ for $n \\ge 1$, and let $B_m$ be the Bernoulli numbers. For every integer $n \\ge 1$, $\\Psi_n(1) = 2^{n+1}(2^{n+1} - 1)B_{n+1}/(n+1)$.","title":"Eulerian polynomials at p = 1 in terms of Bernoulli numbers"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/chapter5-entry8-has-sum-bernoulli","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry8Eulerianformula.chapter5Entry8_hasSum_bernoulli","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#chapter5-entry8-has-sum-bernoulli","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"Let $B_m$ be the Bernoulli numbers, with $B_1 = -\\tfrac{1}{2}$. For every complex number $z$ with $|z| < \\pi$, the series $\\sum_{j \\ge 0} (1 - 2^{j+1}) B_{j+1} z^j/(j+1)!$ is summable with sum $1/(e^z + 1)$.","title":"Bernoulli-number expansion of 1/(e^z + 1) in the disc of radius pi"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/chapter5-entry8-has-sum-psi","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry8Eulerianformula.chapter5Entry8_hasSum_psi","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#chapter5-entry8-has-sum-psi","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"Let $A(n,k)$ be the Eulerian numbers, given by $A(0,0) = 1$, $A(0,k) = 0$ for $k \\ge 1$ and $A(n+1,k) = (k+1)A(n,k) + (n+1-k)A(n,k-1)$, where the last term is $0$ when $k = 0$, and let $\\Psi_0(p) = 1$ and $\\Psi_n(p) = \\sum_{k=0}^{n-1} A(n,k)(-p)^k$ for $n \\ge 1$. For every complex number $z$ with $|z| < \\pi$, the series $\\sum_{m \\ge 0} (-1)^m \\Psi_m(1) z^m/(2^{m+1} m!)$ is summable with sum $1/(e^z + 1)$.","title":"Eulerian-polynomial expansion of 1/(e^z + 1) in the disc of radius pi"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/ramanujan-part1-ch5-entry8-eulerianformu","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry8Eulerianformula.ramanujan_part1_ch5_entry8_eulerianformula","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#ramanujan-part1-ch5-entry8-eulerianformu","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"Let $A(n,k)$ be the Eulerian numbers, given by $A(0,0) = 1$, $A(0,k) = 0$ for $k \\ge 1$ and $A(n+1,k) = (k+1)A(n,k) + (n+1-k)A(n,k-1)$, where the last term is $0$ when $k = 0$; let $\\Psi_0(p) = 1$ and $\\Psi_n(p) = \\sum_{k=0}^{n-1} A(n,k)(-p)^k$ for $n \\ge 1$, and let $B_m$ be the Bernoulli numbers, with $B_1 = -\\tfrac{1}{2}$. For every integer $n \\ge 1$ and every complex number $z$ with $|z| < \\pi$, we have\u2026","title":"Eulerian polynomials at 1 and -1 and two series for 1/(e^z + 1)"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/ramanujan-part1-ch5-entry3-eulerianformu","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry3Eulerianformula.ramanujan_part1_ch5_entry3_eulerianformula","parent":17748,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials/index.html#ramanujan-part1-ch5-entry3-eulerianformu","scope":"Analysis/special-functions/bernoulli-polylog-clausen/eulerian-polynomials","status":"fully_proved","summary":"Eulerian polynomials at 1 and -1 and two series for 1/(e^z + 1), under the old Entry 3 label. Let $A(n,k)$ be the Eulerian numbers, given by $A(0,0) = 1$, $A(0,k) = 0$ for $k \\ge 1$ and $A(n+1,k) = (k+1)A(n,k) + (n+1-k)A(n,k-1)$, where the last term is $0$ when $k = 0$; let $\\Psi_0(p) = 1$ and $\\Psi_n(p) = \\sum_{k=0}^{n-1} A(n,k)(-p)^k$ for $n \\ge 1$, and let $B_m$ be the Bernoulli numbers, with\u2026","title":"Eulerian polynomials at 1 and -1 and two series for 1/(e^z + 1), under the old Entry 3 label"},{"data_url":"scopes/Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators.json","id":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators","kind":"scope","navigation":"module","parent":17679,"result":false,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/index.html","scope":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators","status":"fully_proved","title":"Inverting shift and difference operators"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/ramanujan-part1-ch5-entry1","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry1.ramanujan_part1_ch5_entry1","parent":17758,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/index.html#ramanujan-part1-ch5-entry1","scope":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators","status":"fully_proved","summary":"For a sequence $a = (a_n)_{n \\ge 0}$ of real numbers, a real number $h$ and a real polynomial $\\varphi$ of degree $d$ (with $d = 0$ for $\\varphi = 0$), let $S_a(h, \\varphi) = \\sum_{n=0}^{d} a_n h^n \\varphi^{(n)}$. Let $b_n = B_n / n!$, where $B_n$ are the Bernoulli numbers with $B_1 = -\\tfrac12$. Then for every real polynomial $\\varphi$ and every real $h$,\u2026","title":"Ramanujan's operator identity for the Bernoulli numbers"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/chapter5-polynomial-shift","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry1Euleriangf.chapter5PolynomialShift","parent":17758,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/index.html#chapter5-polynomial-shift","scope":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators","status":"fully_proved","summary":"For a semiring $R$, an element $h \\in R$ and a polynomial $p \\in R[x]$, the shift of $p$ by $h$ is the polynomial $p(x + h)$, the composition of $p$ with $x + h$.","title":"Polynomial shift"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/ramanujan-part1-ch5-entry1-euleriangf","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry1Euleriangf.ramanujan_part1_ch5_entry1_euleriangf","parent":17758,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's operator identity for "],["math","\\((1 - 2^n) B_n / n!\\)"]],"source":"Ramanujan's operator identity for $(1 - 2^n) B_n / n!$"},"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/index.html#ramanujan-part1-ch5-entry1-euleriangf","scope":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators","status":"fully_proved","summary":"For a sequence $a = (a_n)_{n \\ge 0}$ of real numbers, a real number $h$ and a real polynomial $\\varphi$ of degree $d$ (with $d = 0$ for $\\varphi = 0$), let $S_a(h, \\varphi) = \\sum_{n=0}^{d} a_n h^n \\varphi^{(n)}$. Let $c_n = (1 - 2^n) B_n / n!$, where $B_n$ are the Bernoulli numbers with $B_1 = -\\tfrac12$. Then for every real polynomial $\\varphi$ and every real $h$,\u2026","title":"Ramanujan's operator identity for $(1 - 2^n) B_n / n!$"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/chapter5-polynomial-shift-eq-taylor","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry2iEuleriangf.chapter5PolynomialShift_eq_taylor","parent":17758,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/index.html#chapter5-polynomial-shift-eq-taylor","scope":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators","status":"fully_proved","summary":"For a semiring $R$, an element $h \\in R$ and a polynomial $p \\in R[x]$, the shift $p(x + h)$ (the composition of $p$ with $x + h$) equals the Taylor expansion of $p$ at $h$, the polynomial whose coefficient of $x^k$ is the $k$-th Hasse derivative of $p$ evaluated at $h$.","title":"The polynomial shift is the Taylor shift"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/ramanujan-part1-ch5-entry2i-euleriangf","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry2iEuleriangf.ramanujan_part1_ch5_entry2i_euleriangf","parent":17758,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/index.html#ramanujan-part1-ch5-entry2i-euleriangf","scope":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators","status":"fully_proved","summary":"Let $\\varphi$ be a real polynomial of degree $d$ (with $d = 0$ for $\\varphi = 0$). For $n \\ge 0$ let $F_n(x) = \\varphi(x) + \\sum_{k=1}^{n} \\frac{(-1)^k (n!)^2}{(n+k)!\\,(n-k)!} \\bigl(\\varphi(x+k) + \\varphi(x-k)\\bigr)$, and let $M = \\sum_{n=0}^{\\lfloor d/2 \\rfloor} \\frac{1}{2n+1} F_n$ and $P = \\sum_{n=0}^{\\lfloor d/2 \\rfloor} \\binom{2n}{n} 2^{-n} F_n$. Then $F_n = 0$ for every $n$ with $d < 2n$,\u2026","title":"Ramanujan's solutions of two central difference equations"},{"id":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/ramanujan-part1-ch5-entry3","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch5.Entry3.ramanujan_part1_ch5_entry3","parent":17758,"result":true,"route":"../roadmap/Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators/index.html#ramanujan-part1-ch5-entry3","scope":"Analysis/special-functions/bernoulli-polylog-clausen/inverting-shift-and-difference-operators","status":"fully_proved","summary":"Let $A(n, k)$ be the Eulerian numbers, given by $A(0, 0) = 1$, $A(0, k) = 0$ for $k \\ge 1$ and $A(n + 1, k) = (k + 1) A(n, k) + (n + 1 - k) A(n, k - 1)$, the last term read as $0$ when $k = 0$. For $p \\in \\mathbb{C}$ set $\\psi_0(p) = 1$ and $\\psi_n(p) = \\sum_{k=0}^{n-1} A(n, k) (-p)^k$ for $n \\ge 1$. For a sequence $a_0, a_1, \\ldots$ of complex numbers, $h \\in \\mathbb{C}$ and a polynomial\u2026","title":"Eulerian-polynomial inverse of a shift-plus-multiple operator"},{"data_url":"scopes/Analysis/special-functions/elliptic-functions.json","id":"Analysis/special-functions/elliptic-functions","kind":"scope","parent":17543,"result":false,"route":"../roadmap/Analysis/special-functions/elliptic-functions/index.html","scope":"Analysis/special-functions/elliptic-functions","status":"fully_proved","title":"Elliptic functions"},{"id":"Analysis/special-functions/elliptic-functions/is-lattice-periodic","kind":"node","lean":"PeriodPair.IsLatticePeriodic","parent":17765,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/index.html#is-lattice-periodic","scope":"Analysis/special-functions/elliptic-functions","status":"fully_proved","summary":"Let $L$ be a pair of $\\mathbb{R}$-linearly independent complex numbers $\\omega_1, \\omega_2$ with period lattice $\\Lambda = \\mathbb{Z}\\omega_1 + \\mathbb{Z}\\omega_2$. A function $f : \\mathbb{C} \\to \\mathbb{C}$ is lattice-periodic for $L$ when every element of $\\Lambda$ is a period of $f$, that is, $f(z + \\omega) = f(z)$ for all $\\omega \\in \\Lambda$ and all $z \\in \\mathbb{C}$.","title":"Lattice-periodic functions"},{"id":"Analysis/special-functions/elliptic-functions/is-elliptic-function","kind":"node","lean":"PeriodPair.IsEllipticFunction","parent":17765,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/index.html#is-elliptic-function","scope":"Analysis/special-functions/elliptic-functions","status":"fully_proved","summary":"Let $L$ be a pair of $\\mathbb{R}$-linearly independent complex numbers $\\omega_1, \\omega_2$ with period lattice $\\Lambda = \\mathbb{Z}\\omega_1 + \\mathbb{Z}\\omega_2$. A function $f : \\mathbb{C} \\to \\mathbb{C}$ is an elliptic function for $L$ when it is meromorphic at every point of $\\mathbb{C}$ and lattice-periodic for $L$, that is, $f(z + \\omega) = f(z)$ for all $z \\in \\mathbb{C}$ and all $\\omega \\in \\Lambda$.","title":"Elliptic functions"},{"data_url":"scopes/Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices.json","id":"Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices","kind":"scope","navigation":"module","parent":17765,"result":false,"route":"../roadmap/Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices/index.html","scope":"Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices","status":"fully_proved","title":"Complex multiplication of period lattices"},{"id":"Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices/endomorphism-ring","kind":"node","lean":"PeriodPair.endomorphismRing","parent":17768,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices/index.html#endomorphism-ring","scope":"Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices","status":"fully_proved","summary":"Let $\\omega_1, \\omega_2$ be $\\mathbb{R}$-linearly independent complex numbers with period lattice $\\Lambda = \\mathbb{Z}\\omega_1 + \\mathbb{Z}\\omega_2$. The endomorphism ring of $\\Lambda$ is the subring of $\\mathbb{C}$ consisting of all $a \\in \\mathbb{C}$ such that $a z \\in \\Lambda$ for every $z \\in \\Lambda$.","title":"Endomorphism ring of a period lattice"},{"id":"Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices/is-proper-ideal","kind":"node","lean":"PeriodPair.IsProperIdeal","parent":17768,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices/index.html#is-proper-ideal","scope":"Analysis/special-functions/elliptic-functions/complex-multiplication-of-period-lattices","status":"fully_proved","summary":"Let $\\Lambda$ be the period lattice of a pair of $\\mathbb{R}$-linearly independent complex numbers, and let $\\mathcal{O}$ be a subring of $\\mathbb{C}$. Then $\\Lambda$ is a proper ideal of $\\mathcal{O}$ when the endomorphism ring of $\\Lambda$, the subring of all $a \\in \\mathbb{C}$ with $a z \\in \\Lambda$ for every $z \\in \\Lambda$, equals $\\mathcal{O}$.","title":"Lattices that are proper ideals"},{"data_url":"scopes/Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices.json","id":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices","kind":"scope","navigation":"module","parent":17765,"result":false,"route":"../roadmap/Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/index.html","scope":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices","status":"fully_proved","title":"Fundamental parallelograms of complex lattices"},{"id":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/fundamental-parallelogram","kind":"node","lean":"PeriodPair.fundamentalParallelogram","parent":17771,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/index.html#fundamental-parallelogram","scope":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices","status":"fully_proved","summary":"Let $\\omega_1, \\omega_2$ be $\\mathbb{R}$-linearly independent complex numbers and let $\\alpha \\in \\mathbb{C}$. The fundamental parallelogram with vertex $\\alpha$ is the half-open parallelogram $\\alpha + P$, where $P$ is the set of $z \\in \\mathbb{C}$ whose two coordinates in the $\\mathbb{R}$-basis $\\omega_1, \\omega_2$ of $\\mathbb{C}$ both lie in $[0, 1)$; that is,\u2026","title":"Fundamental parallelogram"},{"id":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/fundamental-parallelogram-zero","kind":"node","lean":"PeriodPair.fundamentalParallelogram_zero","parent":17771,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/index.html#fundamental-parallelogram-zero","scope":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices","status":"fully_proved","summary":"Let $\\omega_1, \\omega_2$ be $\\mathbb{R}$-linearly independent complex numbers. The fundamental parallelogram with vertex $0$ equals the set of $z \\in \\mathbb{C}$ whose two coordinates in the $\\mathbb{R}$-basis $\\omega_1, \\omega_2$ of $\\mathbb{C}$ both lie in $[0, 1)$.","title":"Fundamental parallelogram at the origin"},{"id":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/basis-repr-omega","kind":"node","lean":"PeriodPair.basis_repr_omega","parent":17771,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/index.html#basis-repr-omega","scope":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices","status":"fully_proved","summary":"Let $\\omega_1, \\omega_2$ be $\\mathbb{R}$-linearly independent complex numbers, so that they form a basis of $\\mathbb{C}$ over $\\mathbb{R}$. For all real numbers $t_1, t_2$, the first coordinate of $t_1 \\omega_1 + t_2 \\omega_2$ in this basis is $t_1$ and the second is $t_2$.","title":"Coordinates in the period basis"},{"id":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/mem-fundamental-parallelogram-iff","kind":"node","lean":"PeriodPair.mem_fundamentalParallelogram_iff","parent":17771,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices/index.html#mem-fundamental-parallelogram-iff","scope":"Analysis/special-functions/elliptic-functions/fundamental-parallelograms-of-complex-lattices","status":"fully_proved","summary":"Let $\\omega_1, \\omega_2$ be $\\mathbb{R}$-linearly independent complex numbers and let $\\alpha, z \\in \\mathbb{C}$. Then $z$ lies in the fundamental parallelogram with vertex $\\alpha$ if and only if there are real numbers $t_1, t_2$ with $0 \\le t_1 < 1$ and $0 \\le t_2 < 1$ such that $z = \\alpha + t_1 \\omega_1 + t_2 \\omega_2$.","title":"Membership in a fundamental parallelogram"},{"data_url":"scopes/Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices.json","id":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices","kind":"scope","navigation":"module","parent":17765,"result":false,"route":"../roadmap/Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/index.html","scope":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices","status":"fully_proved","title":"Homothetic complex period lattices"},{"id":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/is-homothetic","kind":"node","lean":"PeriodPair.IsHomothetic","parent":17776,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/index.html#is-homothetic","scope":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices","status":"fully_proved","summary":"A period pair is a pair $(\\omega_1, \\omega_2)$ of complex numbers that are linearly independent over $\\mathbb{R}$, and its lattice is $\\Lambda = \\mathbb{Z}\\omega_1 + \\mathbb{Z}\\omega_2 \\subseteq \\mathbb{C}$. Two period pairs $L$ and $L'$ with lattices $\\Lambda$ and $\\Lambda'$ are homothetic when there is a nonzero complex number $c$ such that $\\Lambda' = c\\Lambda = \\{c\\lambda : \\lambda \\in \\Lambda\\}$.","title":"Homothetic period pairs"},{"id":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/refl","kind":"node","lean":"PeriodPair.IsHomothetic.refl","parent":17776,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/index.html#refl","scope":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices","status":"fully_proved","summary":"Every period pair $L$ is homothetic to itself.","title":"Homothety of period pairs is reflexive"},{"id":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/symm","kind":"node","lean":"PeriodPair.IsHomothetic.symm","parent":17776,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/index.html#symm","scope":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices","status":"fully_proved","summary":"If a period pair $L$ is homothetic to a period pair $L'$, then $L'$ is homothetic to $L$.","title":"Homothety of period pairs is symmetric"},{"id":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/trans","kind":"node","lean":"PeriodPair.IsHomothetic.trans","parent":17776,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/index.html#trans","scope":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices","status":"fully_proved","summary":"For period pairs $L_1$, $L_2$, $L_3$, if $L_1$ is homothetic to $L_2$ and $L_2$ is homothetic to $L_3$, then $L_1$ is homothetic to $L_3$.","title":"Homothety of period pairs is transitive"},{"id":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/is-homothetic-equivalence","kind":"node","lean":"PeriodPair.isHomothetic_equivalence","parent":17776,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices/index.html#is-homothetic-equivalence","scope":"Analysis/special-functions/elliptic-functions/homothetic-complex-period-lattices","status":"fully_proved","summary":"On period pairs, homothety is reflexive, symmetric and transitive.","title":"Homothety of period pairs is an equivalence relation"},{"data_url":"scopes/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli.json","id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","kind":"scope","navigation":"module","parent":17765,"result":false,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","title":"Pole sets and elliptic order of meromorphic elliptic functions"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/pole-multiplicity","kind":"node","lean":"PeriodPair.poleMultiplicity","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#pole-multiplicity","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"For a function $f : \\mathbb{C} \\to \\mathbb{C}$ and a point $z \\in \\mathbb{C}$, let $\\operatorname{ord}_z f \\in \\mathbb{Z} \\cup \\{\\infty\\}$ be the meromorphic order of $f$ at $z$, which by convention is $0$ when $f$ is not meromorphic at $z$. The pole multiplicity of $f$ at $z$ is the natural number $\\max(0, -\\operatorname{ord}_z f)$ when $\\operatorname{ord}_z f$ is finite, and $0$ when\u2026","title":"Pole multiplicity"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/pole-multiplicity-eq-zero-of-not-meromor","kind":"node","lean":"PeriodPair.poleMultiplicity_eq_zero_of_not_meromorphicAt","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#pole-multiplicity-eq-zero-of-not-meromor","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $f \\colon \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$. If $f$ is not meromorphic at $z$, then the pole multiplicity of $f$ at $z$ is $0$.","title":"Pole multiplicity vanishes where the function is not meromorphic"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/pole-multiplicity-eq-zero-of-nonneg","kind":"node","lean":"PeriodPair.poleMultiplicity_eq_zero_of_nonneg","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#pole-multiplicity-eq-zero-of-nonneg","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $f \\colon \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$, and let the order of $f$ at $z$ be its meromorphic order, an element of $\\mathbb{Z} \\cup \\{\\infty\\}$ that is $0$ by convention when $f$ is not meromorphic at $z$. If this order is nonnegative (possibly $\\infty$), then the pole multiplicity of $f$ at $z$ is $0$.","title":"Pole multiplicity vanishes at points of nonnegative order"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/pole-multiplicity-ne-zero-iff","kind":"node","lean":"PeriodPair.poleMultiplicity_ne_zero_iff","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#pole-multiplicity-ne-zero-iff","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $f \\colon \\mathbb{C} \\to \\mathbb{C}$ and $z \\in \\mathbb{C}$, and let the order of $f$ at $z$ be its meromorphic order, an element of $\\mathbb{Z} \\cup \\{\\infty\\}$ that is $0$ by convention when $f$ is not meromorphic at $z$. The pole multiplicity of $f$ at $z$ is nonzero if and only if this order is negative, that is, if and only if $f$ has a pole at $z$.","title":"Pole multiplicity is nonzero exactly at negative order"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/pole-multiplicity-eq-neg-divisor","kind":"node","lean":"PeriodPair.poleMultiplicity_eq_neg_divisor","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#pole-multiplicity-eq-neg-divisor","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $U \\subseteq \\mathbb{C}$, let $f \\colon \\mathbb{C} \\to \\mathbb{C}$ be meromorphic on $U$ and let $z \\in U$. Let $D(z)$ be the value at $z$ of the divisor of $f$ on $U$, that is, the order of $f$ at $z$, taken to be $0$ when that order is infinite. Then the pole multiplicity of $f$ at $z$ equals $\\max(0, -D(z))$.","title":"Pole multiplicity via the divisor"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/pole-set-in-fundamental-parallelogram","kind":"node","lean":"PeriodPair.poleSetInFundamentalParallelogram","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#pole-set-in-fundamental-parallelogram","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"For a period pair $L$, a function $f : \\mathbb{C} \\to \\mathbb{C}$ and a point $\\alpha \\in \\mathbb{C}$, the set of poles of $f$ in the fundamental parallelogram of $L$ with vertex $\\alpha$ is the set of points $z$ of that parallelogram at which the pole multiplicity of $f$ is nonzero.","title":"Poles in a fundamental parallelogram"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/mem-pole-set-in-fundamental-parallelogra","kind":"node","lean":"PeriodPair.mem_poleSetInFundamentalParallelogram_iff","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#mem-pole-set-in-fundamental-parallelogra","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $L$ be a period pair, let $f \\colon \\mathbb{C} \\to \\mathbb{C}$ be any function and let $\\alpha, z \\in \\mathbb{C}$. Then $z$ lies in the pole set of $f$ in the fundamental parallelogram of $L$ with vertex $\\alpha$ if and only if $z$ lies in that parallelogram and the pole multiplicity of $f$ at $z$ is nonzero.","title":"Membership in the pole set of a fundamental parallelogram"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/is-bounded-fundamental-parallelogram","kind":"node","lean":"PeriodPair.isBounded_fundamentalParallelogram","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#is-bounded-fundamental-parallelogram","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $L$ be a period pair. For every $\\alpha \\in \\mathbb{C}$, the fundamental parallelogram of $L$ with vertex $\\alpha$ is a bounded subset of $\\mathbb{C}$.","title":"Fundamental parallelograms are bounded"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/pole-set-in-fundamental-parallelogram-fi","kind":"node","lean":"PeriodPair.poleSetInFundamentalParallelogram_finite","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#pole-set-in-fundamental-parallelogram-fi","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $L$ be a period pair, let $f$ be an elliptic function for $L$, and let $\\alpha \\in \\mathbb{C}$. Then the set of poles of $f$ in the fundamental parallelogram of $L$ with vertex $\\alpha$ is finite.","title":"Finitely many poles in a fundamental parallelogram"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/elliptic-order-in-fundamental-parallelog","kind":"node","lean":"PeriodPair.ellipticOrderInFundamentalParallelogram","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#elliptic-order-in-fundamental-parallelog","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $L$ be a period pair, let $f$ be an elliptic function for $L$ and let $\\alpha \\in \\mathbb{C}$. The elliptic order of $f$ in the fundamental parallelogram of $L$ with vertex $\\alpha$ is the natural number obtained by summing the pole multiplicity of $f$ at $z$ over all points $z$ of the pole set of $f$ in that parallelogram, that is, over the points of the parallelogram where this multiplicity is nonzero.\u2026","title":"Elliptic order in a fundamental parallelogram"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/elliptic-order","kind":"node","lean":"PeriodPair.ellipticOrder","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#elliptic-order","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $L$ be a period pair and let $f$ be an elliptic function for $L$. The elliptic order of $f$ is its elliptic order in the fundamental parallelogram of $L$ with vertex $0$: the sum of the pole multiplicities of $f$ over its poles in that parallelogram.","title":"Elliptic order"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/elliptic-order-eq-order-in-fundamental-p","kind":"node","lean":"PeriodPair.ellipticOrder_eq_orderInFundamentalParallelogram_zero","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#elliptic-order-eq-order-in-fundamental-p","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"Let $L$ be a period pair and let $f$ be an elliptic function for $L$. Then the elliptic order of $f$ equals the elliptic order of $f$ in the fundamental parallelogram of $L$ with vertex $0$. This holds by definition.","title":"The elliptic order is the parallelogram order at vertex zero"},{"id":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/pole-multiplicity-const","kind":"node","lean":"PeriodPair.poleMultiplicity_const","parent":17782,"result":true,"route":"../roadmap/Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli/index.html#pole-multiplicity-const","scope":"Analysis/special-functions/elliptic-functions/pole-sets-and-elliptic-order-of-meromorphic-elli","status":"fully_proved","summary":"For all $c, z \\in \\mathbb{C}$, the pole multiplicity at $z$ of the constant function with value $c$ is $0$. This includes the zero constant, whose meromorphic order at every point is $\\infty$.","title":"Pole multiplicity of a constant function"},{"data_url":"scopes/Analysis/real-functions.json","id":"Analysis/real-functions","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/real-functions/index.html","scope":"Analysis/real-functions","status":"can_prove","summary":"Functions of one or several real variables: differentiation, partial derivatives and multivariable calculus, the Riemann integral and evaluation of definite integrals, continuity and pathological functions, and functional equations on the reals.","title":"Real functions and calculus"},{"data_url":"scopes/Analysis/real-functions/differential-calculus.json","id":"Analysis/real-functions/differential-calculus","kind":"scope","parent":17796,"result":false,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html","scope":"Analysis/real-functions/differential-calculus","status":"can_prove","summary":"Derivatives of real functions of one real variable: mean-value and second-derivative tests, Taylor polynomials and remainders, Borel's lemma, interchange of limits and derivatives, monotone and C^n gluing results; not nowhere-differentiable constructions (see continuity subfield).","title":"Differential calculus of one variable"},{"id":"Analysis/real-functions/differential-calculus/exists-monotone-differentiable-not-conti","kind":"node","lean":"MathlibExt.Analysis.Calculus.MonotoneDiscontinuousDerivWanted.exists_monotone_differentiable_not_continuous_deriv","parent":17797,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html#exists-monotone-differentiable-not-conti","scope":"Analysis/real-functions/differential-calculus","status":"can_prove","summary":"There is a nondecreasing function $f : \\mathbb{R} \\to \\mathbb{R}$ that is differentiable at every point and whose derivative $f'$ is not continuous.","title":"A monotone differentiable function with discontinuous derivative"},{"id":"Analysis/real-functions/differential-calculus/exists-polynomial-eq-of-forall-exists-it","kind":"node","lean":"MathlibExt.Analysis.Calculus.PolynomialOfPointwiseVanishingDerivWanted.exists_polynomial_eq_of_forall_exists_iteratedDerivWithin_eq_zero","parent":17797,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html#exists-polynomial-eq-of-forall-exists-it","scope":"Analysis/real-functions/differential-calculus","status":"can_prove","summary":"A smooth function on the unit interval with some vanishing derivative at each point is a polynomial there. Let $f : \\mathbb{R} \\to \\mathbb{R}$ be infinitely differentiable on $[0, 1]$, and suppose that for every $x \\in [0, 1]$ there is an integer $n \\ge 0$ such that the $n$-th derivative of $f$ within $[0, 1]$ vanishes at $x$ (at the endpoints these derivatives are one-sided). Then there is a polynomial $p$ with\u2026","title":"A smooth function on the unit interval with some vanishing derivative at each point is a polynomial there"},{"id":"Analysis/real-functions/differential-calculus/exists-cont-diff-on-comp-sq-of-even","kind":"node","lean":"MathlibExt.Analysis.Calculus.EvenC2SquareFactorWanted.exists_contDiffOn_comp_sq_of_even","parent":17797,"reading_title":{"format":"inline-parts-v1","parts":[["text","An even "],["math","\\(C^2\\)"],["text"," function is a "],["math","\\(C^1\\)"],["text"," function of "],["math","\\(x^2\\)"]],"source":"An even $C^2$ function is a $C^1$ function of $x^2$"},"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html#exists-cont-diff-on-comp-sq-of-even","scope":"Analysis/real-functions/differential-calculus","status":"can_prove","summary":"Let $a>0$ and let $f : \\mathbb{R} \\to \\mathbb{R}$ be twice continuously differentiable on $[-a,a]$ with $f(x)=f(-x)$ for all $x\\in[-a,a]$. Then there is $g : \\mathbb{R} \\to \\mathbb{R}$, continuously differentiable on $[0,a^2]$, with $f(x)=g(x^2)$ for all $x\\in[-a,a]$.","title":"An even $C^2$ function is a $C^1$ function of $x^2$"},{"id":"Analysis/real-functions/differential-calculus/borel-jet-real","kind":"node","lean":"MathlibExt.Analysis.Calculus.BorelWanted.borel_jet_real","parent":17797,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html#borel-jet-real","scope":"Analysis/real-functions/differential-calculus","status":"fully_proved","summary":"For every sequence $(a_n)_{n \\ge 0}$ of real numbers there is an infinitely differentiable function $f : \\mathbb{R} \\to \\mathbb{R}$ whose derivatives at $0$ are $f^{(n)}(0) = a_n$ for every $n \\ge 0$.","title":"Borel's lemma"},{"id":"Analysis/real-functions/differential-calculus/cont-diff-on-piecewise-icc","kind":"node","lean":"MathlibExt.Analysis.Calculus.PiecewiseContDiffWanted.contDiffOn_piecewise_Icc","parent":17797,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gluing "],["math","\\(C^n\\)"],["text"," functions on adjacent closed intervals"]],"source":"Gluing $C^n$ functions on adjacent closed intervals"},"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html#cont-diff-on-piecewise-icc","scope":"Analysis/real-functions/differential-calculus","status":"can_prove","summary":"Let $E$ be a real normed space, let $n \\in \\mathbb{N} \\cup \\{\\infty\\}$ and let $a \\le b \\le c$ be real numbers. Let $f, g : \\mathbb{R} \\to E$ be such that $f$ is $C^n$ on $[a, b]$ and $g$ is $C^n$ on $[b, c]$, and suppose that for every natural number $k \\le n$ the $k$-th derivative of $f$ within $[a, b]$ and the $k$-th derivative of $g$ within $[b, c]$ agree at $b$. Then the function equal to $f$ on $[a, b]$ and\u2026","title":"Gluing $C^n$ functions on adjacent closed intervals"},{"id":"Analysis/real-functions/differential-calculus/exists-unique-sum-pow-div-factorial-eq-z","kind":"node","lean":"MathlibExt.Analysis.Calculus.ExpTaylorPolyOddUniqueZeroWanted.existsUnique_sum_pow_div_factorial_eq_zero_of_odd","parent":17797,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html#exists-unique-sum-pow-div-factorial-eq-z","scope":"Analysis/real-functions/differential-calculus","status":"can_prove","summary":"For every odd positive integer $n$, the polynomial $1+x+x^2/2!+\\cdots+x^n/n!$ has exactly one real zero.","title":"Odd Taylor polynomials of the exponential have exactly one real zero"},{"id":"Analysis/real-functions/differential-calculus/differentiable-and-not-integrable-on-der","kind":"node","lean":"MathlibExt.Analysis.Calculus.XSqSinInvSqNonintegrableDerivWanted.differentiable_and_not_integrableOn_deriv_sq_mul_sin_inv_sq","parent":17797,"reading_title":{"format":"inline-parts-v1","parts":[["text","The derivative of "],["math","\\(x^2 \\sin(1/x^2)\\)"],["text"," exists everywhere but is not integrable"]],"source":"The derivative of $x^2 \\sin(1/x^2)$ exists everywhere but is not integrable"},"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html#differentiable-and-not-integrable-on-der","scope":"Analysis/real-functions/differential-calculus","status":"can_prove","summary":"Let $F : \\mathbb{R} \\to \\mathbb{R}$ be given by $F(x) = x^2 \\sin(1/x^2)$ for $x \\ne 0$ and $F(0) = 0$. Then $F$ is differentiable at every real $x$, but its derivative $F'$ is not Lebesgue integrable on $[-1,1]$.","title":"The derivative of $x^2 \\sin(1/x^2)$ exists everywhere but is not integrable"},{"id":"Analysis/real-functions/differential-calculus/exists-unique-exp-sub-one-sub-exp-one-su","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ExpSubExpUniqueFixedPointWanted.existsUnique_exp_sub_one_sub_exp_one_sub_eq_self","parent":17797,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(x \\mapsto e^{x-1} - e^{1-x}\\)"],["text"," has a unique fixed point"]],"source":"The map $x \\mapsto e^{x-1} - e^{1-x}$ has a unique fixed point"},"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/index.html#exists-unique-exp-sub-one-sub-exp-one-su","scope":"Analysis/real-functions/differential-calculus","status":"can_prove","summary":"There is exactly one real number $x$ such that $e^{x-1} - e^{1-x} = x$.","title":"The map $x \\mapsto e^{x-1} - e^{1-x}$ has a unique fixed point"},{"data_url":"scopes/Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives.json","id":"Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives","kind":"scope","navigation":"module","parent":17797,"result":false,"route":"../roadmap/Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives/index.html","scope":"Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives","status":"fully_proved","title":"Interchange of limits and derivatives"},{"id":"Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives/has-deriv-within-at-of-tendsto-uniformly","kind":"node","lean":"hasDerivWithinAt_of_tendstoUniformlyOn_Icc","parent":17806,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives/index.html#has-deriv-within-at-of-tendsto-uniformly","scope":"Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives","status":"fully_proved","summary":"Let $a < b$ be real numbers, and let $f_n, f_n' : \\mathbb{R} \\to \\mathbb{R}$ for $n \\in \\mathbb{N}$ and $g, g' : \\mathbb{R} \\to \\mathbb{R}$ be functions. Suppose that for every $n$ and every $x \\in [a, b]$, $f_n$ is differentiable at $x$ (two-sided, also at the endpoints) with derivative $f_n'(x)$; that each $f_n'$ is continuous on $[a, b]$; that $f_n' \\to g'$ uniformly on $[a, b]$; and that $f_n(x) \\to g(x)$ for\u2026","title":"Interchange of limits and derivatives on a closed interval"},{"id":"Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives/has-deriv-at-of-tendsto-uniformly-on-icc","kind":"node","lean":"hasDerivAt_of_tendstoUniformlyOn_Icc","parent":17806,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives/index.html#has-deriv-at-of-tendsto-uniformly-on-icc","scope":"Analysis/real-functions/differential-calculus/interchange-of-limits-and-derivatives","status":"fully_proved","summary":"Let $a < b$ be real numbers, and let $f_n, f_n' : \\mathbb{R} \\to \\mathbb{R}$ for $n \\in \\mathbb{N}$ and $g, g' : \\mathbb{R} \\to \\mathbb{R}$ be functions. Suppose that for every $n$ and every $x \\in [a, b]$, $f_n$ is differentiable at $x$ (two-sided, also at the endpoints) with derivative $f_n'(x)$; that each $f_n'$ is continuous on $[a, b]$; that $f_n' \\to g'$ uniformly on $[a, b]$; and that $f_n(x) \\to g(x)$ for\u2026","title":"Interchange of limits and derivatives at interior points"},{"data_url":"scopes/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test.json","id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","kind":"scope","navigation":"module","parent":17797,"result":false,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","title":"Local extrema and the second-derivative test"},{"id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/is-strict-local-min","kind":"node","lean":"IsStrictLocalMin","parent":17809,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html#is-strict-local-min","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","summary":"Let $f \\colon X \\to Y$ be a function from a topological space $X$ to a set $Y$ equipped with a relation $<$. Then $f$ has a strict local minimum at a point $a \\in X$ if $f(a) < f(x)$ for all $x \\ne a$ in some neighborhood of $a$.","title":"Strict local minimum"},{"id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/is-strict-local-max","kind":"node","lean":"IsStrictLocalMax","parent":17809,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html#is-strict-local-max","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","summary":"Let $f \\colon X \\to Y$ be a function from a topological space $X$ to a set $Y$ equipped with a relation $<$. Then $f$ has a strict local maximum at a point $a \\in X$ if $f(x) < f(a)$ for all $x \\ne a$ in some neighborhood of $a$.","title":"Strict local maximum"},{"id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/is-local-min","kind":"node","lean":"IsStrictLocalMin.isLocalMin","parent":17809,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html#is-local-min","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","summary":"Let $f \\colon X \\to Y$ be a function from a topological space $X$ to a preordered set $Y$. If $f$ has a strict local minimum at $a \\in X$, then $f$ has a local minimum at $a$, that is, $f(a) \\le f(x)$ for all $x$ in some neighborhood of $a$.","title":"A strict local minimum is a local minimum"},{"id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/is-local-max","kind":"node","lean":"IsStrictLocalMax.isLocalMax","parent":17809,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html#is-local-max","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","summary":"Let $f \\colon X \\to Y$ be a function from a topological space $X$ to a preordered set $Y$. If $f$ has a strict local maximum at $a \\in X$, then $f$ has a local maximum at $a$, that is, $f(x) \\le f(a)$ for all $x$ in some neighborhood of $a$.","title":"A strict local maximum is a local maximum"},{"id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/neg","kind":"node","lean":"IsStrictLocalMin.neg","parent":17809,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html#neg","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","summary":"Let $X$ be a topological space and let $Y$ be an additive group with a relation $<$ such that addition on the left and addition on the right are strictly monotone. If $f \\colon X \\to Y$ has a strict local minimum at $a \\in X$, then $x \\mapsto -f(x)$ has a strict local maximum at $a$.","title":"Negation turns a strict local minimum into a strict local maximum"},{"id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/neg-2","kind":"node","lean":"IsStrictLocalMax.neg","parent":17809,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html#neg-2","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","summary":"Let $X$ be a topological space and let $Y$ be an additive group with a relation $<$ such that addition on the left and addition on the right are strictly monotone. If $f \\colon X \\to Y$ has a strict local maximum at $a \\in X$, then $x \\mapsto -f(x)$ has a strict local minimum at $a$.","title":"Negation turns a strict local maximum into a strict local minimum"},{"id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/is-strict-local-min-of-deriv-deriv-pos","kind":"node","lean":"isStrictLocalMin_of_deriv_deriv_pos","parent":17809,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html#is-strict-local-min-of-deriv-deriv-pos","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and $x_0 \\in \\mathbb{R}$. If $f''(x_0) > 0$, $f'(x_0) = 0$, and $f$ is continuous at $x_0$, then $f$ has a strict local minimum at $x_0$.","title":"Strict second-derivative test, minimum version"},{"id":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/is-strict-local-max-of-deriv-deriv-neg","kind":"node","lean":"isStrictLocalMax_of_deriv_deriv_neg","parent":17809,"result":true,"route":"../roadmap/Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test/index.html#is-strict-local-max-of-deriv-deriv-neg","scope":"Analysis/real-functions/differential-calculus/local-extrema-and-the-second-derivative-test","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and $x_0 \\in \\mathbb{R}$. If $f''(x_0) < 0$, $f'(x_0) = 0$, and $f$ is continuous at $x_0$, then $f$ has a strict local maximum at $x_0$.","title":"Strict second-derivative test, maximum version"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus.json","id":"Analysis/real-functions/multivariable-calculus","kind":"scope","parent":17796,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/index.html","scope":"Analysis/real-functions/multivariable-calculus","status":"can_prove","summary":"Functions on R^n: partial derivatives, gradients, Jacobians, Hessians, implicit functions, regular level sets and tangent spaces, curve length, Green's theorem, multivariable Taylor expansions and counterexamples about mixed partials or continuity at the origin.","title":"Multivariable calculus"},{"id":"Analysis/real-functions/multivariable-calculus/inner-deriv-self-eq-zero-of-norm-eq-one","kind":"node","lean":"MathlibExt.Analysis.Calculus.UnitNormCurveDerivOrthogonalWanted.inner_deriv_self_eq_zero_of_norm_eq_one","parent":17818,"reading_title":{"format":"inline-parts-v1","parts":[["text","A unit-norm curve in "],["math","\\(\\mathbb{R}^3\\)"],["text"," is orthogonal to its derivative"]],"source":"A unit-norm curve in $\\mathbb{R}^3$ is orthogonal to its derivative"},"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/index.html#inner-deriv-self-eq-zero-of-norm-eq-one","scope":"Analysis/real-functions/multivariable-calculus","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}^3$ be a differentiable map such that $\\|f(t)\\| = 1$ for every real $t$, where $\\|\\cdot\\|$ is the Euclidean norm. Then $\\langle f'(t), f(t) \\rangle = 0$ for every real $t$, where $\\langle \\cdot, \\cdot \\rangle$ is the standard inner product on $\\mathbb{R}^3$.","title":"A unit-norm curve in $\\mathbb{R}^3$ is orthogonal to its derivative"},{"id":"Analysis/real-functions/multivariable-calculus/exists-implicit-function-sq-mul-add-exp","kind":"node","lean":"MathlibExt.Analysis.Calculus.ImplicitFunctionExpExampleWanted.exists_implicit_function_sq_mul_add_exp_add","parent":17818,"reading_title":{"format":"inline-parts-v1","parts":[["text","An implicit function for "],["math","\\(x^2y_1+e^x+y_2=0\\)"],["text"," near "],["math","\\((1,-1)\\)"]],"source":"An implicit function for $x^2y_1+e^x+y_2=0$ near $(1,-1)$"},"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/index.html#exists-implicit-function-sq-mul-add-exp","scope":"Analysis/real-functions/multivariable-calculus","status":"can_prove","summary":"Let $f(x,y_1,y_2)=x^2y_1+e^x+y_2$ on $\\mathbb{R}^3$. Then there are an open set $U\\subseteq\\mathbb{R}^2$ containing $(1,-1)$ and a function $g : \\mathbb{R}^2 \\to \\mathbb{R}$, differentiable on $U$, with $g(1,-1)=0$ and $f(g(y_1,y_2),y_1,y_2)=0$ for all $(y_1,y_2)\\in U$.","title":"An implicit function for $x^2y_1+e^x+y_2=0$ near $(1,-1)$"},{"id":"Analysis/real-functions/multivariable-calculus/integral-comp-spherical-symm","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.SphericalCoordinatesWanted.integral_comp_spherical_symm","parent":17818,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/index.html#integral-comp-spherical-symm","scope":"Analysis/real-functions/multivariable-calculus","status":"can_prove","summary":"Let $E$ be a real Banach space and let $f \\colon \\mathbb{R}^3 \\to E$ be any function. Then the integral of $f$ over $\\mathbb{R}^3$ equals the integral over the set of $(r, \\theta, \\varphi) \\in \\mathbb{R}^3$ with $r > 0$, $0 < \\theta < 2\\pi$ and $0 < \\varphi < \\pi$ of $r^2 \\sin\\varphi \\cdot f(r \\sin\\varphi \\cos\\theta,\\, r \\sin\\varphi \\sin\\theta,\\, r \\cos\\varphi)$, both integrals being Bochner integrals with respect\u2026","title":"Change of variables to spherical coordinates"},{"id":"Analysis/real-functions/multivariable-calculus/exists-continuous-mul-add-of-fderiv-eq","kind":"node","lean":"MathlibExt.Analysis.Calculus.EqualPartialsFactorSumWanted.exists_continuous_mul_add_of_fderiv_eq","parent":17818,"reading_title":{"format":"inline-parts-v1","parts":[["text","Equal partial derivatives give a factor "],["math","\\(x+y\\)"]],"source":"Equal partial derivatives give a factor $x+y$"},"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/index.html#exists-continuous-mul-add-of-fderiv-eq","scope":"Analysis/real-functions/multivariable-calculus","status":"can_prove","summary":"Let $f : \\mathbb{R}^2 \\to \\mathbb{R}$ be continuously differentiable with $\\partial_1 f = \\partial_2 f$ on $\\mathbb{R}^2$ and $f(0,0)=0$. Then there is a continuous $g : \\mathbb{R}^2 \\to \\mathbb{R}$ with $f(x,y)=g(x,y)\\,(x+y)$ for all $(x,y)\\in\\mathbb{R}^2$.","title":"Equal partial derivatives give a factor $x+y$"},{"id":"Analysis/real-functions/multivariable-calculus/greens-theorem","kind":"node","lean":"MetaMathlibExt.greens_theorem","parent":17818,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/index.html#greens-theorem","scope":"Analysis/real-functions/multivariable-calculus","status":"can_prove","summary":"Let $U \\subseteq \\mathbb{R}^2$ be open, let $D \\subseteq U$ be bounded and measurable, and let $L, M : \\mathbb{R}^2 \\to \\mathbb{R}$ be continuously differentiable on $U$. Let $a\\lt b$ and let $\\gamma=(\\gamma_1,\\gamma_2) : \\mathbb{R} \\to \\mathbb{R}^2$ be continuous on $[a,b]$, with $\\gamma(a)=\\gamma(b)$, injective on $[a,b)$, with $\\gamma(t)\\ne\\gamma(a)$ for $a\\lt t\\lt b$, with image $\\gamma([a,b])$ equal to the\u2026","title":"Green's theorem"},{"id":"Analysis/real-functions/multivariable-calculus/exists-abs-le-mul-abs-mul-abs-of-cont-di","kind":"node","lean":"MathlibExt.Analysis.Calculus.SmoothVanishingOnAxesBoundWanted.exists_abs_le_mul_abs_mul_abs_of_contDiff_of_eq_zero_on_axes","parent":17818,"reading_title":{"format":"inline-parts-v1","parts":[["text","Smooth functions vanishing on the axes are "],["math","\\(O(|x|\\,|y|)\\)"],["text"," near the origin"]],"source":"Smooth functions vanishing on the axes are $O(|x|\\,|y|)$ near the origin"},"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/index.html#exists-abs-le-mul-abs-mul-abs-of-cont-di","scope":"Analysis/real-functions/multivariable-calculus","status":"can_prove","summary":"Let $f : \\mathbb{R}^2 \\to \\mathbb{R}$ be infinitely differentiable with $f(x,y) = 0$ whenever $xy = 0$. Then there is a constant $C > 0$ such that $|f(x,y)| \\le C\\,|x|\\,|y|$ for all real $x$ and $y$ with $|x| < 1$ and $|y| < 1$, that is, on the open unit ball of $\\mathbb{R}^2$ for the maximum norm.","title":"Smooth functions vanishing on the axes are $O(|x|\\,|y|)$ near the origin"},{"id":"Analysis/real-functions/multivariable-calculus/taylor-is-little-o-multivariable","kind":"node","lean":"MathlibExt.Analysis.Calculus.TaylorLittleoRemainderWanted.taylor_isLittleO_multivariable","parent":17818,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/index.html#taylor-is-little-o-multivariable","scope":"Analysis/real-functions/multivariable-calculus","status":"can_prove","summary":"Let $E$ and $F$ be normed vector spaces over $\\mathbb{R}$, let $n$ be a natural number, and let $f : E \\to F$ be $n$ times continuously differentiable on a neighbourhood of a point $x_0 \\in E$. Write $D^k f(x_0)$ for the $k$-th Fr\u00e9chet derivative of $f$ at $x_0$, a continuous $k$-linear map from $E^k$ to $F$. Then, as $x$ tends to $x_0$,","title":"Taylor's theorem with little-o remainder in several variables"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i.json","id":"Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i","kind":"scope","navigation":"module","parent":17818,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i/index.html","scope":"Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i","status":"can_prove","title":"Continuity and differentiability at the origin in the plane"},{"id":"Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i/bounded-partial-derivs-cube-div-sq-add-s","kind":"node","lean":"MathlibExt.Analysis.Calculus.CubicOverSumSquaresPartialsBoundedWanted.bounded_partialDerivs_cube_div_sq_add_sq","parent":17826,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partial derivatives of "],["math","\\(x^3/(x^2+y^2)\\)"],["text"," are bounded"]],"source":"The partial derivatives of $x^3/(x^2+y^2)$ are bounded"},"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i/index.html#bounded-partial-derivs-cube-div-sq-add-s","scope":"Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i","status":"can_prove","summary":"Let $f : \\mathbb{R}^2 \\to \\mathbb{R}$ satisfy $f(0,0)=0$ and $f(x,y)=x^3/(x^2+y^2)$ for $(x,y)\\ne(0,0)$. Then the partial derivatives $D_1 f$ and $D_2 f$ are bounded on $\\mathbb{R}^2$: there is a real $M$ with $|D_1 f(x,y)| \\le M$ for all $x,y\\in\\mathbb{R}$, and a real $M'$ with $|D_2 f(x,y)| \\le M'$ for all $x,y\\in\\mathbb{R}$.","title":"The partial derivatives of $x^3/(x^2+y^2)$ are bounded"},{"id":"Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i/not-differentiable-at-signed-norm-zero","kind":"node","lean":"MathlibExt.Analysis.Calculus.SignedRadialNotDifferentiableWanted.not_differentiableAt_signed_norm_zero","parent":17826,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i/index.html#not-differentiable-at-signed-norm-zero","scope":"Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i","status":"can_prove","summary":"Let $f : \\mathbb{R}^2 \\to \\mathbb{R}$ be the function","title":"The signed radial function is not differentiable at the origin"},{"id":"Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i/continuous-on-compl-and-not-continuous-a","kind":"node","lean":"MathlibExt.Analysis.Calculus.XYOverNormSqDiscontinuityWanted.continuousOn_compl_and_not_continuousAt_mul_div_sq_add_sq","parent":17826,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function "],["math","\\(xy/(x^2+y^2)\\)"],["text"," is discontinuous only at the origin"]],"source":"The function $xy/(x^2+y^2)$ is discontinuous only at the origin"},"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i/index.html#continuous-on-compl-and-not-continuous-a","scope":"Analysis/real-functions/multivariable-calculus/continuity-and-differentiability-at-the-origin-i","status":"can_prove","summary":"Let $f : \\mathbb{R}^2 \\to \\mathbb{R}$ be given by $f(x,y) = \\frac{xy}{x^2+y^2}$ for $(x,y) \\neq (0,0)$ and $f(0,0) = 0$. Then $f$ is continuous on $\\mathbb{R}^2 \\setminus \\{(0,0)\\}$ and is not continuous at $(0,0)$.","title":"The function $xy/(x^2+y^2)$ is discontinuous only at the origin"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives.json","id":"Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives","kind":"scope","navigation":"module","parent":17818,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives/index.html","scope":"Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives","status":"can_prove","title":"Existence of mixed partial derivatives"},{"id":"Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives/scottish-book-problem66-claim","kind":"node","lean":"MathlibExt.Analysis.Calculus.ScottishBook66Wanted.ScottishBookProblem66Claim","parent":17830,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives/index.html#scottish-book-problem66-claim","scope":"Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives","status":"fully_proved","summary":"This is the following statement, the claim of Mazur's Problem 66 from the Scottish Book. Let $Q = (0, 1) \\times (0, 1)$ be the open unit square. For every function $f : \\mathbb{R}^2 \\to \\mathbb{R}$ such that, at every point $(x, y) \\in Q$, the functions $s \\mapsto f(s, y)$ and $s \\mapsto f(x, s)$ are differentiable at $x$ and at $y$ respectively, and their derivatives $\\partial_x f(\\cdot, y)$ and\u2026","title":"Mazur's mixed partial derivatives claim"},{"id":"Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives/mazur66-refuted","kind":"node","lean":"MathlibExt.Analysis.Calculus.ScottishBook66Wanted.mazur66_refuted","parent":17830,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives/index.html#mazur66-refuted","scope":"Analysis/real-functions/multivariable-calculus/existence-of-mixed-partial-derivatives","status":"can_prove","summary":"Mazur's mixed partial derivatives claim is false: there is a function $f : \\mathbb{R}^2 \\to \\mathbb{R}$ such that, at every point $(x, y)$ of the open unit square $Q = (0, 1) \\times (0, 1)$, the functions $s \\mapsto f(s, y)$ and $s \\mapsto f(x, s)$ are differentiable at $x$ and at $y$ respectively, and their derivatives $\\partial_x f(\\cdot, y)$ and $\\partial_y f(x, \\cdot)$ are again differentiable at $x$ and at $y$\u2026","title":"Mazur's Problem 66 has a negative answer"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus/gradient.json","id":"Analysis/real-functions/multivariable-calculus/gradient","kind":"scope","navigation":"module","parent":17818,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/gradient/index.html","scope":"Analysis/real-functions/multivariable-calculus/gradient","status":"fully_proved","title":"Gradient"},{"id":"Analysis/real-functions/multivariable-calculus/gradient/dot-l","kind":"node","lean":"Real.dotL","parent":17833,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/gradient/index.html#dot-l","scope":"Analysis/real-functions/multivariable-calculus/gradient","status":"fully_proved","summary":"For a finite type $\\iota$ and a vector $x \\in \\mathbb{R}^\\iota$, this is the map $v \\mapsto v \\cdot x = \\sum_{i \\in \\iota} v_i x_i$, as a continuous linear functional $\\mathbb{R}^\\iota \\to \\mathbb{R}$.","title":"The dot product with a fixed vector"},{"id":"Analysis/real-functions/multivariable-calculus/gradient/has-gradient-at-of-lp","kind":"node","lean":"Real.hasGradientAt_ofLp","parent":17833,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/gradient/index.html#has-gradient-at-of-lp","scope":"Analysis/real-functions/multivariable-calculus/gradient","status":"fully_proved","summary":"Let $\\iota$ be a finite type, let $f : \\mathbb{R}^\\iota \\to \\mathbb{R}$ and $g, z \\in \\mathbb{R}^\\iota$, and suppose that, for the sup norm on $\\mathbb{R}^\\iota$, $f$ is Fr\u00e9chet differentiable at $z$ with derivative $v \\mapsto v \\cdot g$. Then $f$, viewed as a function on the Euclidean space $\\mathbb{R}^\\iota$ with inner product $\\langle v, w \\rangle = \\sum_{i \\in \\iota} v_i w_i$, has gradient $g$ at $z$.","title":"A Fr\u00e9chet derivative given by a dot product is a gradient"},{"id":"Analysis/real-functions/multivariable-calculus/gradient/has-deriv-at-line-of-has-fderiv-at","kind":"node","lean":"Real.hasDerivAt_line_of_hasFDerivAt","parent":17833,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/gradient/index.html#has-deriv-at-line-of-has-fderiv-at","scope":"Analysis/real-functions/multivariable-calculus/gradient","status":"fully_proved","summary":"Let $\\iota$ be a finite type, let $f : \\mathbb{R}^\\iota \\to \\mathbb{R}$ and $g, z \\in \\mathbb{R}^\\iota$, and suppose $f$ is Fr\u00e9chet differentiable at $z$ with derivative $w \\mapsto w \\cdot g$. Then for every $v \\in \\mathbb{R}^\\iota$, the function $t \\mapsto f(z + t v)$ on $\\mathbb{R}$ has derivative $v \\cdot g$ at $t = 0$.","title":"Directional derivative from the gradient"},{"id":"Analysis/real-functions/multivariable-calculus/gradient/gradient-comp","kind":"node","lean":"MetaMathlibExt.gradient_comp","parent":17833,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/gradient/index.html#gradient-comp","scope":"Analysis/real-functions/multivariable-calculus/gradient","status":"fully_proved","summary":"Let $E$ and $F$ be real Hilbert spaces, let $g : E \\to F$ be differentiable at $x \\in E$, and let $f : F \\to \\mathbb{R}$ be differentiable at $g(x)$. Then $\\nabla (f \\circ g)(x) = Dg(x)^* \\, \\nabla f(g(x))$, where $Dg(x) : E \\to F$ is the Fr\u00e9chet derivative of $g$ at $x$ and $Dg(x)^*$ is its adjoint.","title":"The chain rule for gradients"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus/hessian-matrix.json","id":"Analysis/real-functions/multivariable-calculus/hessian-matrix","kind":"scope","navigation":"module","parent":17818,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/hessian-matrix/index.html","scope":"Analysis/real-functions/multivariable-calculus/hessian-matrix","status":"fully_proved","title":"Hessian matrix"},{"id":"Analysis/real-functions/multivariable-calculus/hessian-matrix/hessian-matrix","kind":"node","lean":"MetaMathlibExt.hessianMatrix","parent":17838,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/hessian-matrix/index.html#hessian-matrix","scope":"Analysis/real-functions/multivariable-calculus/hessian-matrix","status":"fully_proved","summary":"For a function $f : \\mathbb{R}^n \\to \\mathbb{R}$ and a point $x \\in \\mathbb{R}^n$, the Hessian matrix of $f$ at $x$ is the $n \\times n$ real matrix whose $(i, j)$ entry is $\\frac{d}{ds}\\big|_{s=0} \\frac{d}{dt}\\big|_{t=0} f(x + s e_i + t e_j)$, where $e_1, \\dots, e_n$ is the standard basis of $\\mathbb{R}^n$ and a derivative that does not exist is taken to be $0$.","title":"Hessian matrix"},{"id":"Analysis/real-functions/multivariable-calculus/hessian-matrix/hessian-matrix-apply","kind":"node","lean":"MetaMathlibExt.hessianMatrix_apply","parent":17838,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/hessian-matrix/index.html#hessian-matrix-apply","scope":"Analysis/real-functions/multivariable-calculus/hessian-matrix","status":"fully_proved","summary":"For $f : \\mathbb{R}^n \\to \\mathbb{R}$, a point $x \\in \\mathbb{R}^n$ and indices $i, j$, the $(i, j)$ entry of the Hessian matrix of $f$ at $x$ is $\\frac{d}{ds}\\big|_{s=0} \\frac{d}{dt}\\big|_{t=0} f(x + s e_i + t e_j)$, where $e_1, \\dots, e_n$ is the standard basis of $\\mathbb{R}^n$ and a derivative that does not exist is taken to be $0$: differentiate first along $e_j$, then along $e_i$.","title":"Entries of the Hessian matrix"},{"id":"Analysis/real-functions/multivariable-calculus/hessian-matrix/hessian-matrix-eq-fderiv-fderiv","kind":"node","lean":"MetaMathlibExt.hessianMatrix_eq_fderiv_fderiv","parent":17838,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/hessian-matrix/index.html#hessian-matrix-eq-fderiv-fderiv","scope":"Analysis/real-functions/multivariable-calculus/hessian-matrix","status":"fully_proved","summary":"Let $f : \\mathbb{R}^n \\to \\mathbb{R}$ be twice continuously differentiable and let $x \\in \\mathbb{R}^n$. For all indices $i, j$, the $(i, j)$ entry of the Hessian matrix of $f$ at $x$ equals $D(Df)(x)(e_i)(e_j)$, the derivative of the Fr\u00e9chet derivative $Df$ at $x$ in the direction $e_i$, applied to $e_j$, where $e_1, \\dots, e_n$ is the standard basis of $\\mathbb{R}^n$.","title":"Hessian entries as second Fr\u00e9chet derivatives"},{"id":"Analysis/real-functions/multivariable-calculus/hessian-matrix/hessian-matrix-symmetric","kind":"node","lean":"MetaMathlibExt.hessianMatrix_symmetric","parent":17838,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/hessian-matrix/index.html#hessian-matrix-symmetric","scope":"Analysis/real-functions/multivariable-calculus/hessian-matrix","status":"fully_proved","summary":"If $f : \\mathbb{R}^n \\to \\mathbb{R}$ is twice continuously differentiable, then for every $x \\in \\mathbb{R}^n$ the Hessian matrix of $f$ at $x$ is symmetric: it equals its transpose.","title":"Symmetry of the Hessian"},{"id":"Analysis/real-functions/multivariable-calculus/hessian-matrix/deriv-deriv-nonneg","kind":"node","lean":"IsLocalMin.deriv_deriv_nonneg","parent":17838,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/hessian-matrix/index.html#deriv-deriv-nonneg","scope":"Analysis/real-functions/multivariable-calculus/hessian-matrix","status":"fully_proved","summary":"Let $g : \\mathbb{R} \\to \\mathbb{R}$ be twice continuously differentiable. If $g$ has a local minimum at $x$, then $g''(x) \\ge 0$.","title":"The second derivative is nonnegative at a local minimum"},{"id":"Analysis/real-functions/multivariable-calculus/hessian-matrix/is-local-min-hessian-quadratic-nonneg","kind":"node","lean":"MetaMathlibExt.isLocalMin_hessian_quadratic_nonneg","parent":17838,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/hessian-matrix/index.html#is-local-min-hessian-quadratic-nonneg","scope":"Analysis/real-functions/multivariable-calculus/hessian-matrix","status":"fully_proved","summary":"Let $f : \\mathbb{R}^n \\to \\mathbb{R}$ be twice continuously differentiable and let $x$ be a local minimum of $f$. Then the Hessian quadratic form of $f$ at $x$ is nonnegative: writing $H$ for the Hessian matrix of $f$ at $x$, every $v \\in \\mathbb{R}^n$ satisfies $\\sum_{i, j} v_i H_{ij} v_j \\ge 0$.","title":"Second-derivative test, necessary condition"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus/jacobian-matrix.json","id":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","kind":"scope","navigation":"module","parent":17818,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/jacobian-matrix/index.html","scope":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","status":"fully_proved","title":"Jacobian matrix"},{"id":"Analysis/real-functions/multivariable-calculus/jacobian-matrix/jacobian-matrix","kind":"node","lean":"MetaMathlibExt.jacobianMatrix","parent":17845,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/jacobian-matrix/index.html#jacobian-matrix","scope":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","status":"fully_proved","summary":"For $f : \\mathbb{R}^m \\to \\mathbb{R}^n$ and $x \\in \\mathbb{R}^m$, the Jacobian matrix of $f$ at $x$ is the $n \\times m$ real matrix whose $(i, j)$ entry is the partial derivative of the $i$-th component $f_i$ of $f$ in the $j$-th coordinate direction at $x$: the derivative at $t = 0$ of $t \\mapsto f_i(x + t e_j)$, where $e_j$ is the $j$-th standard basis vector of $\\mathbb{R}^m$. A partial derivative that does not\u2026","title":"Jacobian matrix"},{"id":"Analysis/real-functions/multivariable-calculus/jacobian-matrix/jacobian-matrix-apply","kind":"node","lean":"MetaMathlibExt.jacobianMatrix_apply","parent":17845,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/jacobian-matrix/index.html#jacobian-matrix-apply","scope":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","status":"fully_proved","summary":"Let $f : \\mathbb{R}^m \\to \\mathbb{R}^n$ and $x \\in \\mathbb{R}^m$. For every row index $i$ and column index $j$, the $(i, j)$ entry of the Jacobian matrix of $f$ at $x$ is the derivative at $t = 0$ of $t \\mapsto f_i(x + t e_j)$, where $f_i$ is the $i$-th component of $f$ and $e_j$ is the $j$-th standard basis vector of $\\mathbb{R}^m$. This derivative is taken to be $0$ when it does not exist.","title":"Entries of the Jacobian matrix are partial derivatives"},{"id":"Analysis/real-functions/multivariable-calculus/jacobian-matrix/jacobian-matrix-apply-of-differentiable","kind":"node","lean":"MetaMathlibExt.jacobianMatrix_apply_of_differentiableAt","parent":17845,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/jacobian-matrix/index.html#jacobian-matrix-apply-of-differentiable","scope":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","status":"fully_proved","summary":"Let $f : \\mathbb{R}^m \\to \\mathbb{R}^n$ be differentiable at $x \\in \\mathbb{R}^m$. Then for every row index $i$ and column index $j$, the $(i, j)$ entry of the Jacobian matrix of $f$ at $x$ is the $i$-th coordinate of $Df(x)\\, e_j$, where $Df(x)$ is the Fr\u00e9chet derivative of $f$ at $x$ and $e_j$ is the $j$-th standard basis vector of $\\mathbb{R}^m$.","title":"Jacobian entries at a point of differentiability"},{"id":"Analysis/real-functions/multivariable-calculus/jacobian-matrix/jacobian-matrix-eq-to-matrix-fderiv","kind":"node","lean":"MetaMathlibExt.jacobianMatrix_eq_toMatrix'_fderiv","parent":17845,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/jacobian-matrix/index.html#jacobian-matrix-eq-to-matrix-fderiv","scope":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","status":"fully_proved","summary":"Let $f : \\mathbb{R}^m \\to \\mathbb{R}^n$ be differentiable at $x \\in \\mathbb{R}^m$. Then the Jacobian matrix of $f$ at $x$ is the matrix, in the standard bases of $\\mathbb{R}^m$ and $\\mathbb{R}^n$, of the Fr\u00e9chet derivative of $f$ at $x$, viewed as a linear map $\\mathbb{R}^m \\to \\mathbb{R}^n$.","title":"The Jacobian is the matrix of the derivative"},{"id":"Analysis/real-functions/multivariable-calculus/jacobian-matrix/jacobian-matrix-fun-mul-vec","kind":"node","lean":"MetaMathlibExt.jacobianMatrix_fun_mulVec","parent":17845,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/jacobian-matrix/index.html#jacobian-matrix-fun-mul-vec","scope":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","status":"fully_proved","summary":"Let $M$ be a real $n \\times m$ matrix. At every point $x \\in \\mathbb{R}^m$, the Jacobian matrix of the map $y \\mapsto M y$ from $\\mathbb{R}^m$ to $\\mathbb{R}^n$ is $M$.","title":"Jacobian of a linear map given by a matrix"},{"id":"Analysis/real-functions/multivariable-calculus/jacobian-matrix/jacobian-matrix-mul-vec-eq-fderiv","kind":"node","lean":"MetaMathlibExt.jacobianMatrix_mulVec_eq_fderiv","parent":17845,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/jacobian-matrix/index.html#jacobian-matrix-mul-vec-eq-fderiv","scope":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","status":"fully_proved","summary":"Let $f : \\mathbb{R}^m \\to \\mathbb{R}^n$ be differentiable at $x \\in \\mathbb{R}^m$, and let $J$ be the Jacobian matrix of $f$ at $x$. Then for every $v \\in \\mathbb{R}^m$, the product $J v$ equals $Df(x)\\, v$, the Fr\u00e9chet derivative of $f$ at $x$ applied to $v$.","title":"The Jacobian matrix computes the derivative"},{"id":"Analysis/real-functions/multivariable-calculus/jacobian-matrix/jacobian-matrix-comp","kind":"node","lean":"MetaMathlibExt.jacobianMatrix_comp","parent":17845,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/jacobian-matrix/index.html#jacobian-matrix-comp","scope":"Analysis/real-functions/multivariable-calculus/jacobian-matrix","status":"fully_proved","summary":"Let $f : \\mathbb{R}^m \\to \\mathbb{R}^n$, $g : \\mathbb{R}^n \\to \\mathbb{R}^p$ and $x \\in \\mathbb{R}^m$, and write $J_h(y)$ for the Jacobian matrix of $h$ at $y$. If $f$ is differentiable at $x$ and $g$ is differentiable at $f(x)$, then $J_{g \\circ f}(x) = J_g(f(x))\\, J_f(x)$, a product of a $p \\times n$ and an $n \\times m$ matrix.","title":"Chain rule for Jacobian matrices"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus/length-of-a-curve.json","id":"Analysis/real-functions/multivariable-calculus/length-of-a-curve","kind":"scope","navigation":"module","parent":17818,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/length-of-a-curve/index.html","scope":"Analysis/real-functions/multivariable-calculus/length-of-a-curve","status":"fully_proved","title":"Length of a curve"},{"id":"Analysis/real-functions/multivariable-calculus/length-of-a-curve/integral-norm-deriv-eq-integral-norm-der","kind":"node","lean":"intervalIntegral.integral_norm_deriv_eq_integral_norm_derivWithin_Icc","parent":17853,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/length-of-a-curve/index.html#integral-norm-deriv-eq-integral-norm-der","scope":"Analysis/real-functions/multivariable-calculus/length-of-a-curve","status":"fully_proved","summary":"Let $E$ be a real normed space, let $f : \\mathbb{R} \\to E$ be any function and let $a \\le b$ be real numbers. Then $\\int_a^b \\lVert f'(t) \\rVert \\, dt = \\int_a^b \\lVert f'_{[a,b]}(t) \\rVert \\, dt$, where $f'(t)$ is the derivative of $f$ at $t$ and $f'_{[a,b]}(t)$ is its derivative at $t$ within $[a, b]$, each taken to be $0$ where it does not exist.","title":"Integrating the norm of a derivative within a closed interval"},{"id":"Analysis/real-functions/multivariable-calculus/length-of-a-curve/e-variation-on-icc-eq-of-real-integral-n","kind":"node","lean":"ContDiffOn.eVariationOn_Icc_eq_ofReal_integral_norm_derivWithin","parent":17853,"reading_title":{"format":"inline-parts-v1","parts":[["text","Length of a "],["math","\\(C^1\\)"],["text"," curve via its derivative within the interval"]],"source":"Length of a $C^1$ curve via its derivative within the interval"},"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/length-of-a-curve/index.html#e-variation-on-icc-eq-of-real-integral-n","scope":"Analysis/real-functions/multivariable-calculus/length-of-a-curve","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $a \\le b$ be real numbers and let $\\gamma : \\mathbb{R} \\to E$ be continuously differentiable on $[a, b]$. Then the total variation of $\\gamma$ on $[a, b]$, a priori a value in $[0, \\infty]$, equals the real number $\\int_a^b \\lVert \\gamma'_{[a,b]}(t) \\rVert \\, dt$, where $\\gamma'_{[a,b]}(t)$ is the derivative of $\\gamma$ at $t$ within $[a, b]$.","title":"Length of a $C^1$ curve via its derivative within the interval"},{"id":"Analysis/real-functions/multivariable-calculus/length-of-a-curve/curve-length-eq-integral-norm-deriv","kind":"node","lean":"MetaMathlibExt.curve_length_eq_integral_norm_deriv","parent":17853,"reading_title":{"format":"inline-parts-v1","parts":[["text","Length of a "],["math","\\(C^1\\)"],["text"," curve"]],"source":"Length of a $C^1$ curve"},"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/length-of-a-curve/index.html#curve-length-eq-integral-norm-deriv","scope":"Analysis/real-functions/multivariable-calculus/length-of-a-curve","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $a \\le b$ be real numbers, and let $\\gamma : \\mathbb{R} \\to E$ be continuously differentiable on $[a, b]$. Then the total variation of $\\gamma$ on $[a, b]$ (its length, a priori a value in $[0, \\infty]$) equals $\\int_a^b \\lVert \\gamma'(t) \\rVert \\, dt$.","title":"Length of a $C^1$ curve"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia.json","id":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia","kind":"scope","navigation":"module","parent":17818,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/index.html","scope":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia","status":"fully_proved","title":"Partial derivatives and symmetry of mixed partials"},{"id":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/partial-deriv","kind":"node","lean":"MetaMathlibExt.partialDeriv","parent":17857,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/index.html#partial-deriv","scope":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia","status":"fully_proved","summary":"For $f : \\mathbb{R}^n \\to \\mathbb{R}$ and a coordinate index $i$, let $e_i$ be the $i$-th standard basis vector. The partial derivative $\\partial_i f : \\mathbb{R}^n \\to \\mathbb{R}$ sends $x$ to the derivative at $t = 0$ of $t \\mapsto f(x + t e_i)$, taken to be $0$ when this function is not differentiable at $0$.","title":"Partial derivative along a coordinate axis"},{"id":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/partial-deriv-spec","kind":"node","lean":"MetaMathlibExt.partialDeriv_spec","parent":17857,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/index.html#partial-deriv-spec","scope":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia","status":"fully_proved","summary":"For $f : \\mathbb{R}^n \\to \\mathbb{R}$, a coordinate index $i$ and a point $x \\in \\mathbb{R}^n$, let $e_i$ be the $i$-th standard basis vector. The partial derivative of $f$ along the $i$-th coordinate axis at $x$ equals the derivative at $t = 0$ of the one-variable function $t \\mapsto f(x + t e_i)$, taken to be $0$ when that function is not differentiable at $0$.","title":"Partial derivative as the derivative of a coordinate slice"},{"id":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/partial-deriv-eq-fderiv-apply","kind":"node","lean":"MetaMathlibExt.partialDeriv_eq_fderiv_apply","parent":17857,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/index.html#partial-deriv-eq-fderiv-apply","scope":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia","status":"fully_proved","summary":"Let $f : \\mathbb{R}^n \\to \\mathbb{R}$ be differentiable at a point $x$, and let $e_i$ be the $i$-th standard basis vector of $\\mathbb{R}^n$. Then the partial derivative of $f$ along the $i$-th coordinate axis at $x$ equals $Df(x)\\,e_i$, the Fr\u00e9chet derivative of $f$ at $x$ applied to $e_i$.","title":"Partial derivative as a Fr\u00e9chet derivative"},{"id":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/partial-deriv-eq-fderiv-apply-of-differe","kind":"node","lean":"MetaMathlibExt.partialDeriv_eq_fderiv_apply_of_differentiable","parent":17857,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/index.html#partial-deriv-eq-fderiv-apply-of-differe","scope":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia","status":"fully_proved","summary":"Let $f : \\mathbb{R}^n \\to \\mathbb{R}$ be differentiable and let $e_i$ be the $i$-th standard basis vector of $\\mathbb{R}^n$. Then the partial derivative $\\partial_i f$ of $f$ along the $i$-th coordinate axis is the function $x \\mapsto Df(x)\\,e_i$, where $Df(x)$ is the Fr\u00e9chet derivative of $f$ at $x$.","title":"Partial derivative function of a differentiable function"},{"id":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/partial-deriv-partial-deriv-eq-fderiv-fd","kind":"node","lean":"MetaMathlibExt.partialDeriv_partialDeriv_eq_fderiv_fderiv","parent":17857,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/index.html#partial-deriv-partial-deriv-eq-fderiv-fd","scope":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia","status":"fully_proved","summary":"Let $f : \\mathbb{R}^n \\to \\mathbb{R}$ be twice continuously differentiable, write $e_k$ for the $k$-th standard basis vector of $\\mathbb{R}^n$ and $\\partial_k g$ for the partial derivative of a function $g$ along the $k$-th coordinate axis, and let $Df$ be the Fr\u00e9chet derivative of $f$, a map from $\\mathbb{R}^n$ to the linear maps $\\mathbb{R}^n \\to \\mathbb{R}$. Then for all coordinate indices $i, j$ and every\u2026","title":"Mixed partial derivatives via the second Fr\u00e9chet derivative"},{"id":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/iterated-partial-deriv-commute","kind":"node","lean":"MetaMathlibExt.iterated_partialDeriv_commute","parent":17857,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia/index.html#iterated-partial-deriv-commute","scope":"Analysis/real-functions/multivariable-calculus/partial-derivatives-and-symmetry-of-mixed-partia","status":"fully_proved","summary":"For a function $g : \\mathbb{R}^n \\to \\mathbb{R}$, write $\\partial_i g$ for its partial derivative along the $i$-th coordinate axis. If $f : \\mathbb{R}^n \\to \\mathbb{R}$ is twice continuously differentiable, then for all coordinate indices $i, j$ and every point $x \\in \\mathbb{R}^n$, $\\partial_j (\\partial_i f)(x) = \\partial_i (\\partial_j f)(x)$.","title":"Symmetry of mixed partial derivatives (Clairaut's theorem)"},{"data_url":"scopes/Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets.json","id":"Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets","kind":"scope","navigation":"module","parent":17818,"result":false,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets/index.html","scope":"Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets","status":"fully_proved","title":"Tangent spaces of level sets"},{"id":"Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets/level-set-tangent-vel-of-curve","kind":"node","lean":"MetaMathlibExt.levelSet_tangent_vel_of_curve","parent":17864,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets/index.html#level-set-tangent-vel-of-curve","scope":"Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets","status":"fully_proved","summary":"Let $E$ be a real normed space, $c \\in \\mathbb{R}$, and $f : E \\to \\mathbb{R}$ a function differentiable at $x \\in E$. If a curve $\\gamma : \\mathbb{R} \\to E$ satisfies $\\gamma(0) = x$, has derivative $v$ at $0$, and satisfies $f(\\gamma(t)) = c$ for all $t \\in \\mathbb{R}$, then $Df(x)v = 0$.","title":"Velocities of curves in a level set lie in the kernel of the derivative"},{"id":"Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets/exists-level-set-curve","kind":"node","lean":"HasStrictFDerivAt.exists_levelSet_curve","parent":17864,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets/index.html#exists-level-set-curve","scope":"Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets","status":"fully_proved","summary":"Let $E$ be a real Banach space and let $f : E \\to \\mathbb{R}$ have a strict derivative $f'$ at $x \\in E$, a continuous linear map $E \\to \\mathbb{R}$ that is surjective. Then for every $v \\in E$ with $f'(v) = 0$ there is a curve $\\varphi : \\mathbb{R} \\to E$ with $\\varphi(0) = x$ and $\\varphi'(0) = v$ such that $f(\\varphi(t)) = f(x)$ for all $t$ in some neighbourhood of $0$.","title":"Kernel vectors are velocities of local curves in a level set"},{"id":"Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets/level-set-curve-of-tangent-vel","kind":"node","lean":"MetaMathlibExt.levelSet_curve_of_tangent_vel","parent":17864,"result":true,"route":"../roadmap/Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets/index.html#level-set-curve-of-tangent-vel","scope":"Analysis/real-functions/multivariable-calculus/tangent-spaces-of-level-sets","status":"fully_proved","summary":"Let $E$ be a real Banach space, $f : E \\to \\mathbb{R}$ a $C^1$ function, $c \\in \\mathbb{R}$, and $x \\in E$ a point with $f(x) = c$ at which the derivative $Df(x) : E \\to \\mathbb{R}$ is surjective. Then for every $v \\in E$ with $Df(x)v = 0$ there is a curve $\\gamma : \\mathbb{R} \\to E$ with $\\gamma(0) = x$, $\\gamma'(0) = v$ and $f(\\gamma(t)) = c$ for all $t \\in \\mathbb{R}$.","title":"Kernel vectors are velocities of curves in a regular level set"},{"data_url":"scopes/Analysis/real-functions/integration.json","id":"Analysis/real-functions/integration","kind":"scope","parent":17796,"result":false,"route":"../roadmap/Analysis/real-functions/integration/index.html","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"The Riemann integral via tagged partitions, iterated-integral formulas, moments, and closed-form evaluations of specific definite integrals (Serret's integral, integrals of 1/(1+u^n), areas between cosine powers, sinc squared); integrals defining a named special function go to Special functions.","title":"Riemann integration and definite integrals"},{"id":"Analysis/real-functions/integration/eq-on-zero-of-integral-mul-pow-eq-zero","kind":"node","lean":"MathlibExt.Analysis.Calculus.VanishingMomentsImpliesZeroWanted.eqOn_zero_of_integral_mul_pow_eq_zero","parent":17868,"reading_title":{"format":"inline-parts-v1","parts":[["text","A function continuous on "],["math","\\((0,1)\\)"],["text"," with vanishing moments is zero on "],["math","\\((0,1)\\)"]],"source":"A function continuous on $(0,1)$ with vanishing moments is zero on $(0,1)$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#eq-on-zero-of-integral-mul-pow-eq-zero","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[0,1]$, and suppose that $\\int_0^1 f(x)\\,x^n\\,dx = 0$ for every natural number $n = 0, 1, 2, \\dots$. Then $f(x) = 0$ for every $x \\in [0,1]$.","title":"A function continuous on $(0,1)$ with vanishing moments is zero on $(0,1)$"},{"id":"Analysis/real-functions/integration/arcsin-div-sin-integral","kind":"node","lean":"MetaMathlibExt.arcsin_div_sin_integral","parent":17868,"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#arcsin-div-sin-integral","scope":"Analysis/real-functions/integration","status":"fully_proved","summary":"Let $G = \\sum_{n \\ge 0} (-1)^n/(2n+1)^2$ be Catalan's constant, and for $s > 0$ let $S_s = \\sum_{n \\ge 0} 1/(n + s)^2$. Then the integral $\\int_0^{\\pi/2} \\frac{\\arcsin(\\sin^2 x)}{\\sin x}\\,dx$ equals $-2G - \\frac{\\pi^2}{2\\sqrt 2} + \\frac{S_{1/8} + S_{3/8}}{8\\sqrt 2}$.","title":"Bhandari's arcsine-over-sine integral"},{"id":"Analysis/real-functions/integration/interval-integral-interval-integral-eq-i","kind":"node","lean":"MathlibExt.Analysis.Calculus.IteratedIntegralCauchyFormulaWanted.intervalIntegral_intervalIntegral_eq_integral_sub_smul","parent":17868,"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#interval-integral-interval-integral-eq-i","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"Let $E$ be a real Banach space, let $\\alpha \\le \\beta$ be real numbers and let $f : \\mathbb{R} \\to E$ be continuous on $[\\alpha, \\beta]$. Then for every $s \\in [\\alpha, \\beta]$,","title":"Cauchy's formula for the twice-iterated integral"},{"id":"Analysis/real-functions/integration/integral-one-div-add-mul-cos-eq","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.IntegralInvAddMulCosWanted.integral_one_div_add_mul_cos_eq","parent":17868,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(1/(a + b\\cos\\theta)\\)"],["text"," over a period"]],"source":"Integral of $1/(a + b\\cos\\theta)$ over a period"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#integral-one-div-add-mul-cos-eq","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"Let $a$ and $b$ be real numbers with $a > |b|$. Then","title":"Integral of $1/(a + b\\cos\\theta)$ over a period"},{"id":"Analysis/real-functions/integration/integral-sin-div-sq-ioi-eq-pi-div-two","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.IntegralSincSqHalfLineWanted.integral_sin_div_sq_Ioi_eq_pi_div_two","parent":17868,"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#integral-sin-div-sq-ioi-eq-pi-div-two","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"The integral of $(\\sin x / x)^2$ over $(0, \\infty)$ is $\\pi/2$:","title":"Integral of the squared sinc function over the half-line"},{"id":"Analysis/real-functions/integration/integral-log-one-add-div-one-add-sq","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.IntegralLogOnePlusOverOnePlusSqWanted.integral_log_one_add_div_one_add_sq","parent":17868,"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#integral-log-one-add-div-one-add-sq","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"With $\\log$ the natural logarithm,","title":"Serret's integral"},{"id":"Analysis/real-functions/integration/integral-mul-mul-deriv-eq-neg-half","kind":"node","lean":"MathlibExt.Analysis.Calculus.IntegralXFDerivFWanted.integral_mul_mul_deriv_eq_neg_half","parent":17868,"reading_title":{"format":"inline-parts-v1","parts":[["text","The integral of "],["math","\\(x f(x) f'(x)\\)"],["text"," equals "],["math","\\(-1/2\\)"]],"source":"The integral of $x f(x) f'(x)$ equals $-1/2$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#integral-mul-mul-deriv-eq-neg-half","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"Let $a\\lt b$ and let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuously differentiable on $[a,b]$ with $f(a)=f(b)=0$ and","title":"The integral of $x f(x) f'(x)$ equals $-1/2$"},{"id":"Analysis/real-functions/integration/tendsto-cocompact-zero-of-integrable","kind":"node","lean":"MathlibExt.Analysis.Calculus.UniformlyContinuousIntegrableTendstoZeroWanted.UniformContinuous.tendsto_cocompact_zero_of_integrable","parent":17868,"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#tendsto-cocompact-zero-of-integrable","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be uniformly continuous and Lebesgue integrable on $\\mathbb{R}$. Then $f(x) \\to 0$ as $|x| \\to \\infty$.","title":"Uniformly continuous integrable functions vanish at infinity"},{"id":"Analysis/real-functions/integration/abs-integral-sin-sq-lt-inv","kind":"node","lean":"MathlibExt.Analysis.Calculus.IntegralSinSqUnitWindowBoundWanted.abs_integral_sin_sq_lt_inv","parent":17868,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit-window bound for the integral of "],["math","\\(\\sin(t^2)\\)"]],"source":"Unit-window bound for the integral of $\\sin(t^2)$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/index.html#abs-integral-sin-sq-lt-inv","scope":"Analysis/real-functions/integration","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be given by $f(x) = \\int_x^{x+1} \\sin(t^2)\\,dt$. Then $|f(x)| < 1/x$ for every real $x > 0$.","title":"Unit-window bound for the integral of $\\sin(t^2)$"},{"data_url":"scopes/Analysis/real-functions/integration/areas-between-cosine-powers.json","id":"Analysis/real-functions/integration/areas-between-cosine-powers","kind":"scope","navigation":"module","parent":17868,"result":false,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","title":"Areas between cosine powers"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/cos-pow-area","kind":"node","lean":"MetaMathlibExt.cosPowArea","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#cos-pow-area","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the area between the curves $y = \\cos^n x$ and $y = \\cos^n(kx)$ on $[0, \\pi]$ is $A_n(k) = \\int_0^\\pi |\\cos^n x - \\cos^n(kx)|\\,dx$, where $\\cos^n x$ means $(\\cos x)^n$.","title":"Area between powers of cosines"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/cos-pow-area-zero-left","kind":"node","lean":"MetaMathlibExt.cosPowArea_zero_left","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#cos-pow-area-zero-left","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, let $A_n(k) = \\int_0^\\pi |\\cos^n x - \\cos^n(kx)|\\,dx$ be the area between $\\cos^n x$ and $\\cos^n(kx)$ on $[0, \\pi]$. Then $A_0(k) = 0$ for every natural number $k$, since both curves are then the constant $1$.","title":"The area between cosine powers vanishes for exponent zero"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/cos-pow-area-one-right","kind":"node","lean":"MetaMathlibExt.cosPowArea_one_right","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#cos-pow-area-one-right","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, let $A_n(k) = \\int_0^\\pi |\\cos^n x - \\cos^n(kx)|\\,dx$ be the area between $\\cos^n x$ and $\\cos^n(kx)$ on $[0, \\pi]$. Then $A_n(1) = 0$ for every natural number $n$, since the two curves then coincide.","title":"The area between cosine powers vanishes at frequency one"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/tendsto-cos-pow-area-of-odd","kind":"node","lean":"MetaMathlibExt.tendsto_cosPowArea_of_odd","parent":17878,"reading_title":{"format":"inline-parts-v1","parts":[["text","Area between "],["math","\\(\\cos^n x\\)"],["text"," and "],["math","\\(\\cos^n(kx)\\)"],["text"," in the limit, odd "],["math","\\(n\\)"]],"source":"Area between $\\cos^n x$ and $\\cos^n(kx)$ in the limit, odd $n$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#tendsto-cos-pow-area-of-odd","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $n$ be an odd natural number. As $k \\to \\infty$ through the natural numbers, the area $\\int_0^\\pi |\\cos^n x - \\cos^n(kx)|\\,dx$ between $\\cos^n x$ and $\\cos^n(kx)$ on $[0, \\pi]$ converges to","title":"Area between $\\cos^n x$ and $\\cos^n(kx)$ in the limit, odd $n$"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/tendsto-cos-pow-area-of-even","kind":"node","lean":"MetaMathlibExt.tendsto_cosPowArea_of_even","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#tendsto-cos-pow-area-of-even","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"For natural numbers $n$ and $k$, let $A_n(k) = \\int_0^\\pi |\\cos^n x - \\cos^n(kx)|\\,dx$. As $k \\to \\infty$ through the natural numbers: if $n \\equiv 2 \\pmod 4$, then","title":"Limiting area between even cosine powers"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/cos-pow-area-limit-recursion","kind":"node","lean":"MetaMathlibExt.cosPowArea_limit_recursion","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#cos-pow-area-limit-recursion","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"For natural numbers $n$ and $k$, let $A_n(k) = \\int_0^\\pi |\\cos^n x - \\cos^n(kx)|\\,dx$. For every natural number $n \\ge 3$ there are real numbers $L$ and $L'$ such that $A_n(k) \\to L$ and $A_{n-2}(k) \\to L'$ as $k \\to \\infty$ through the natural numbers, and $L = \\frac{n-1}{n} L' + \\frac{c}{n^2 \\pi}$, where $c = 8$ if $n$ is odd and $c = 16$ if $n$ is even.","title":"Recursion for the limiting areas between cosine powers"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/tendsto-cos-pow-area-double-factorial","kind":"node","lean":"MetaMathlibExt.tendsto_cosPowArea_doubleFactorial","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#tendsto-cos-pow-area-double-factorial","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"For natural numbers $n$ and $k$, let $A_n(k) = \\int_0^\\pi |\\cos^n x - \\cos^n(kx)|\\,dx$. As $k \\to \\infty$ through the natural numbers: if $n$ is odd, then","title":"Limiting area between cosine powers via double factorials"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/iterated-deriv-arcsin-div-one-sub-double","kind":"node","lean":"MetaMathlibExt.iteratedDeriv_arcsin_div_one_sub_doubleFactorial","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#iterated-deriv-arcsin-div-one-sub-double","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $n$ be a natural number. If $n$ is odd, the $n$th derivative at $0$ of $x \\mapsto \\arcsin x / (1 - x)$ is","title":"Double-factorial sums as derivatives of arcsine quotients"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/sum-sin-two-pi-mul-div-of-even","kind":"node","lean":"MetaMathlibExt.sum_sin_two_pi_mul_div_of_even","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#sum-sin-two-pi-mul-div-of-even","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $N$ be a positive even natural number. For every natural number $q$, $\\sum_{\\ell=1}^{N/2} \\sin(2\\pi q \\ell / N)$ equals $0$ if $q$ is even and $\\cot(q\\pi / N)$ if $q$ is odd.","title":"Half-period sine sum for even modulus"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/sum-sin-two-pi-mul-div-of-odd","kind":"node","lean":"MetaMathlibExt.sum_sin_two_pi_mul_div_of_odd","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#sum-sin-two-pi-mul-div-of-odd","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $N$ be an odd natural number. For every natural number $q$, $\\sum_{\\ell=1}^{(N-1)/2} \\sin(2\\pi q \\ell / N)$ equals $-\\frac{1}{2} \\tan\\frac{q\\pi}{2N}$ if $q$ is even and $\\frac{1}{2} \\cot\\frac{q\\pi}{2N}$ if $q$ is odd.","title":"Half-period sine sum for odd modulus"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/sum-sin-pi-mul-div-of-even-of-even","kind":"node","lean":"MetaMathlibExt.sum_sin_pi_mul_div_of_even_of_even","parent":17878,"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#sum-sin-pi-mul-div-of-even-of-even","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $N$ be a positive even natural number and $q$ an even natural number. Then $\\sum_{\\ell=1}^{N/2} \\sin(q \\ell \\pi / N)$ equals $0$ if $q \\equiv 0 \\pmod 4$ and $\\cot(q\\pi / (2N))$ if $q \\equiv 2 \\pmod 4$.","title":"Half-step sine sum for even modulus and even multiplier"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/tendsto-cot-div-pred-add-cot-div-succ","kind":"node","lean":"MetaMathlibExt.tendsto_cot_div_pred_add_cot_div_succ","parent":17878,"reading_title":{"format":"inline-parts-v1","parts":[["text","The cotangent limit "],["math","\\(4/x\\)"]],"source":"The cotangent limit $4/x$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#tendsto-cot-div-pred-add-cot-div-succ","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"For every nonzero real number $x$, as $k \\to \\infty$ through the natural numbers,","title":"The cotangent limit $4/x$"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/tendsto-sum-integral-cos-sub-cos-odd-odd","kind":"node","lean":"MetaMathlibExt.tendsto_sum_integral_cos_sub_cos_odd_odd","parent":17878,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral sums of "],["math","\\(\\cos qkx - \\cos qx\\)"],["text"," for odd "],["math","\\(q\\)"],["text"," and odd "],["math","\\(k\\)"]],"source":"Integral sums of $\\cos qkx - \\cos qx$ for odd $q$ and odd $k$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#tendsto-sum-integral-cos-sub-cos-odd-odd","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $q$ be an odd natural number, and for each natural number $m$ put $k = 2m + 1$. Then, as $m \\to \\infty$,","title":"Integral sums of $\\cos qkx - \\cos qx$ for odd $q$ and odd $k$"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/tendsto-sum-integral-cos-sub-cos-odd-eve","kind":"node","lean":"MetaMathlibExt.tendsto_sum_integral_cos_sub_cos_odd_even","parent":17878,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating integral sums of "],["math","\\(\\cos qx - \\cos qkx\\)"],["text"," for even "],["math","\\(q\\)"],["text"," and odd "],["math","\\(k\\)"]],"source":"Alternating integral sums of $\\cos qx - \\cos qkx$ for even $q$ and odd $k$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#tendsto-sum-integral-cos-sub-cos-odd-eve","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $q$ be an even natural number, for each natural number $m$ put $k = 2m + 1$, and let $f(x) = \\cos(qx) - \\cos(qkx)$. Then, as $m \\to \\infty$,","title":"Alternating integral sums of $\\cos qx - \\cos qkx$ for even $q$ and odd $k$"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/tendsto-sum-integral-cos-sub-cos-even-od","kind":"node","lean":"MetaMathlibExt.tendsto_sum_integral_cos_sub_cos_even_odd","parent":17878,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating integral sums of "],["math","\\(\\cos qx - \\cos qkx\\)"],["text"," for odd "],["math","\\(q\\)"],["text"," and even "],["math","\\(k\\)"]],"source":"Alternating integral sums of $\\cos qx - \\cos qkx$ for odd $q$ and even $k$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#tendsto-sum-integral-cos-sub-cos-even-od","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $q$ be an odd natural number, for each natural number $m$ put $k = 2m$, and let $f(x) = \\cos(qx) - \\cos(qkx)$. Then, as $m \\to \\infty$,","title":"Alternating integral sums of $\\cos qx - \\cos qkx$ for odd $q$ and even $k$"},{"id":"Analysis/real-functions/integration/areas-between-cosine-powers/tendsto-sum-integral-cos-sub-cos-even-ev","kind":"node","lean":"MetaMathlibExt.tendsto_sum_integral_cos_sub_cos_even_even","parent":17878,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating integral sums of "],["math","\\(\\cos qx - \\cos qkx\\)"],["text"," for even "],["math","\\(q\\)"],["text"," and even "],["math","\\(k\\)"]],"source":"Alternating integral sums of $\\cos qx - \\cos qkx$ for even $q$ and even $k$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/areas-between-cosine-powers/index.html#tendsto-sum-integral-cos-sub-cos-even-ev","scope":"Analysis/real-functions/integration/areas-between-cosine-powers","status":"can_prove","summary":"Let $q$ be an even natural number, for each natural number $m$ put $k = 2m$, and let $f(x) = \\cos(qx) - \\cos(qkx)$. Then, as $m \\to \\infty$,","title":"Alternating integral sums of $\\cos qx - \\cos qkx$ for even $q$ and even $k$"},{"data_url":"scopes/Analysis/real-functions/integration/improper-integrals.json","id":"Analysis/real-functions/integration/improper-integrals","kind":"scope","navigation":"module","parent":17868,"result":false,"route":"../roadmap/Analysis/real-functions/integration/improper-integrals/index.html","scope":"Analysis/real-functions/integration/improper-integrals","status":"can_prove","title":"Improper integrals"},{"id":"Analysis/real-functions/integration/improper-integrals/dirichlet-integral-conditionally-converg","kind":"node","lean":"MathlibExt.MeasureTheory.Integral.ImproperIntegralAbsoluteConditionalWanted.dirichlet_integral_conditionally_convergent","parent":17895,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional convergence of the Dirichlet integral on "],["math","\\((1, \\infty)\\)"]],"source":"Conditional convergence of the Dirichlet integral on $(1, \\infty)$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/improper-integrals/index.html#dirichlet-integral-conditionally-converg","scope":"Analysis/real-functions/integration/improper-integrals","status":"can_prove","summary":"The function $x \\mapsto \\sin x / x$ is interval integrable on $[1, b]$ for every real $b \\ge 1$, the integrals $\\int_1^b \\frac{\\sin x}{x}\\,dx$ converge to a real limit as $b \\to \\infty$, and $\\sin x / x$ is not Lebesgue integrable on $(1, \\infty)$.","title":"Conditional convergence of the Dirichlet integral on $(1, \\infty)$"},{"id":"Analysis/real-functions/integration/improper-integrals/improper-integral-comparison-test","kind":"node","lean":"MathlibExt.MeasureTheory.Integral.ImproperIntegralComparisonWanted.improper_integral_comparison_test","parent":17895,"result":true,"route":"../roadmap/Analysis/real-functions/integration/improper-integrals/index.html#improper-integral-comparison-test","scope":"Analysis/real-functions/integration/improper-integrals","status":"can_prove","summary":"Let $a \\in \\mathbb{R}$ and let $f, g : \\mathbb{R} \\to \\mathbb{R}$ be interval integrable on $[a, b]$ for every $b \\ge a$, with $0 \\le f(x) \\le g(x)$ for all $x \\ge a$. If $\\int_a^b g(x)\\,dx$ converges to a real limit as $b \\to \\infty$, then $\\int_a^b f(x)\\,dx$ also converges to a real limit as $b \\to \\infty$.","title":"Comparison test for improper integrals"},{"id":"Analysis/real-functions/integration/improper-integrals/improper-integral-limit-comparison-test","kind":"node","lean":"MathlibExt.MeasureTheory.Integral.ImproperIntegralComparisonWanted.improper_integral_limit_comparison_test","parent":17895,"result":true,"route":"../roadmap/Analysis/real-functions/integration/improper-integrals/index.html#improper-integral-limit-comparison-test","scope":"Analysis/real-functions/integration/improper-integrals","status":"can_prove","summary":"Let $a, L \\in \\mathbb{R}$ with $L \\ne 0$, and let $f, g : \\mathbb{R} \\to \\mathbb{R}$ be interval integrable on $[a, b]$ for every $b \\ge a$, with $f(x) > 0$ and $g(x) > 0$ for all $x \\ge a$. If $f(x) / g(x) \\to L$ as $x \\to \\infty$, then $\\int_a^b f(x)\\,dx$ converges to a real limit as $b \\to \\infty$ if and only if $\\int_a^b g(x)\\,dx$ converges to a real limit as $b \\to \\infty$.","title":"Limit comparison test for improper integrals"},{"data_url":"scopes/Analysis/real-functions/integration/integrals-of-1-1-u.json","id":"Analysis/real-functions/integration/integrals-of-1-1-u","kind":"scope","navigation":"module","parent":17868,"result":false,"route":"../roadmap/Analysis/real-functions/integration/integrals-of-1-1-u/index.html","scope":"Analysis/real-functions/integration/integrals-of-1-1-u","status":"fully_proved","title":"Integrals of 1/(1 + u\u207f)"},{"id":"Analysis/real-functions/integration/integrals-of-1-1-u/ramanujan-part1-ch8-entry14-qreflection","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry14Qreflection.ramanujan_part1_ch8_entry14_qreflection","parent":17899,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's closed form for "],["math","\\(\\int_0^x du/(1 + u^{10})\\)"],["text"," (Chapter 8, Entry 14)"]],"source":"Ramanujan's closed form for $\\int_0^x du/(1 + u^{10})$ (Chapter 8, Entry 14)"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/integrals-of-1-1-u/index.html#ramanujan-part1-ch8-entry14-qreflection","scope":"Analysis/real-functions/integration/integrals-of-1-1-u","status":"fully_proved","summary":"Put $a_\\pm = \\sqrt{6 \\pm 2\\sqrt5}$ and $b_\\pm = \\sqrt{10 \\pm 2\\sqrt5}$. For every real $x$ with $0 \\le x < 1$,","title":"Ramanujan's closed form for $\\int_0^x du/(1 + u^{10})$ (Chapter 8, Entry 14)"},{"id":"Analysis/real-functions/integration/integrals-of-1-1-u/ramanujan-part1-ch8-entry14-v-integral5","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry14VIntegral5.ramanujan_part1_ch8_entry14_v_integral5","parent":17899,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's closed form for "],["math","\\(\\int_0^x du/(1 + u^5)\\)"],["text"," (Chapter 8, Entry 14(v))"]],"source":"Ramanujan's closed form for $\\int_0^x du/(1 + u^5)$ (Chapter 8, Entry 14(v))"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/integrals-of-1-1-u/index.html#ramanujan-part1-ch8-entry14-v-integral5","scope":"Analysis/real-functions/integration/integrals-of-1-1-u","status":"fully_proved","summary":"Let $x$ be a real number with $-1 < x < 4/(\\sqrt5 + 1)$, and put $c_\\pm = \\sqrt{10 \\pm 2\\sqrt5}$ and","title":"Ramanujan's closed form for $\\int_0^x du/(1 + u^5)$ (Chapter 8, Entry 14(v))"},{"id":"Analysis/real-functions/integration/integrals-of-1-1-u/ramanujan-part1-ch8-entry14-vi-integral6","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry14ViIntegral6.ramanujan_part1_ch8_entry14_vi_integral6","parent":17899,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's closed form for "],["math","\\(\\int_0^x du/(1 + u^6)\\)"],["text"," (Chapter 8, Entry 14(vi))"]],"source":"Ramanujan's closed form for $\\int_0^x du/(1 + u^6)$ (Chapter 8, Entry 14(vi))"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/integrals-of-1-1-u/index.html#ramanujan-part1-ch8-entry14-vi-integral6","scope":"Analysis/real-functions/integration/integrals-of-1-1-u","status":"fully_proved","summary":"For every real number $x$, the function $u \\mapsto 1/(1 + u^6)$ is integrable on the interval between $0$ and $x$, $1 + \\sqrt3\\,x + x^2 > 0$, $1 - \\sqrt3\\,x + x^2 > 0$, and","title":"Ramanujan's closed form for $\\int_0^x du/(1 + u^6)$ (Chapter 8, Entry 14(vi))"},{"id":"Analysis/real-functions/integration/integrals-of-1-1-u/ramanujan-part1-ch8-entry14-vii-integral","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry14ViiIntegral8.ramanujan_part1_ch8_entry14_vii_integral8","parent":17899,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's closed form for "],["math","\\(\\int_0^x du/(1 + u^8)\\)"],["text"," (Chapter 8, Entry 14(vii))"]],"source":"Ramanujan's closed form for $\\int_0^x du/(1 + u^8)$ (Chapter 8, Entry 14(vii))"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/integrals-of-1-1-u/index.html#ramanujan-part1-ch8-entry14-vii-integral","scope":"Analysis/real-functions/integration/integrals-of-1-1-u","status":"fully_proved","summary":"Put $s_\\pm = \\sqrt{2 \\pm \\sqrt2}$ and, for real $x$, $R_\\pm(x) = \\frac{1 + s_\\pm x + x^2}{1 - s_\\pm x + x^2}$. For every real $x$ with $|x| < 1$, the function $u \\mapsto 1/(1 + u^8)$ is integrable on the interval between $0$ and $x$, $R_+(x) > 0$, $R_-(x) > 0$, $1 - x^2 \\ne 0$, and","title":"Ramanujan's closed form for $\\int_0^x du/(1 + u^8)$ (Chapter 8, Entry 14(vii))"},{"data_url":"scopes/Analysis/real-functions/integration/integrals-of-cos-x-sin-nx.json","id":"Analysis/real-functions/integration/integrals-of-cos-x-sin-nx","kind":"scope","navigation":"module","parent":17868,"result":false,"route":"../roadmap/Analysis/real-functions/integration/integrals-of-cos-x-sin-nx/index.html","scope":"Analysis/real-functions/integration/integrals-of-cos-x-sin-nx","status":"fully_proved","title":"Integrals of cos\u207fx sin nx"},{"id":"Analysis/real-functions/integration/integrals-of-cos-x-sin-nx/ramanujan-part1-ch9-entry33-zeta3integra","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry33Zeta3integral.ramanujan_part1_ch9_entry33_zeta3integral_general","parent":17904,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's integrals of "],["math","\\(\\cos^n x \\sin nx\\)"],["text"," for every "],["math","\\(n \\ge 0\\)"]],"source":"Ramanujan's integrals of $\\cos^n x \\sin nx$ for every $n \\ge 0$"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/integrals-of-cos-x-sin-nx/index.html#ramanujan-part1-ch9-entry33-zeta3integra","scope":"Analysis/real-functions/integration/integrals-of-cos-x-sin-nx","status":"fully_proved","summary":"Let $n$ be a natural number and let $H_n = \\sum_{k=1}^{n} 1/k$ be the $n$-th harmonic number. Then $x \\cos^n x \\sin(nx)$ and $\\cos^n x \\sin(nx)$ are integrable on $[0, \\pi/2]$, $\\int_0^{\\pi/2} x \\cos^n x \\sin(nx)\\,dx = \\frac{\\pi}{2^{n+2}} H_n$ and $\\int_0^{\\pi/2} \\cos^n x \\sin(nx)\\,dx = \\frac{1}{2^{n+1}} \\sum_{k=1}^{n} \\frac{2^k}{k}$.","title":"Ramanujan's integrals of $\\cos^n x \\sin nx$ for every $n \\ge 0$"},{"id":"Analysis/real-functions/integration/integrals-of-cos-x-sin-nx/ramanujan-part1-ch9-entry33-zeta3integra-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry33Zeta3integral.ramanujan_part1_ch9_entry33_zeta3integral","parent":17904,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's integrals of "],["math","\\(\\cos^n x \\sin nx\\)"],["text"," (Entry 33)"]],"source":"Ramanujan's integrals of $\\cos^n x \\sin nx$ (Entry 33)"},"result":true,"route":"../roadmap/Analysis/real-functions/integration/integrals-of-cos-x-sin-nx/index.html#ramanujan-part1-ch9-entry33-zeta3integra-2","scope":"Analysis/real-functions/integration/integrals-of-cos-x-sin-nx","status":"fully_proved","summary":"Let $n$ be a positive integer and let $H_n = \\sum_{k=1}^{n} 1/k$ be the $n$-th harmonic number. Then $x \\cos^n x \\sin(nx)$ and $\\cos^n x \\sin(nx)$ are integrable on $[0, \\pi/2]$, $\\int_0^{\\pi/2} x \\cos^n x \\sin(nx)\\,dx = \\frac{\\pi}{2^{n+2}} H_n$ and $\\int_0^{\\pi/2} \\cos^n x \\sin(nx)\\,dx = \\frac{1}{2^{n+1}} \\sum_{k=1}^{n} \\frac{2^k}{k}$.","title":"Ramanujan's integrals of $\\cos^n x \\sin nx$ (Entry 33)"},{"data_url":"scopes/Analysis/real-functions/integration/riemann-integral.json","id":"Analysis/real-functions/integration/riemann-integral","kind":"scope","navigation":"module","parent":17868,"result":false,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","title":"Riemann integral"},{"id":"Analysis/real-functions/integration/riemann-integral/partition","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#partition","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"For real numbers $a$ and $b$, a partition of the closed interval $[a, b]$ consists of a natural number $n$, the number of subintervals, together with $n + 1$ points forming a strictly increasing sequence $a = x_0 < x_1 < \\dots < x_n = b$.","title":"Partition of an interval"},{"id":"Analysis/real-functions/integration/riemann-integral/tagged-partition","kind":"node","lean":"MathlibExt.Analysis.Integration.TaggedPartition","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#tagged-partition","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"A tagged partition of $[a, b]$ is a partition $a = x_0 < x_1 < \\cdots < x_n = b$ of $[a, b]$ together with a choice of a tag $\\xi_k \\in [x_{k-1}, x_k]$ in each subinterval, for $k = 1, \\dots, n$.","title":"Tagged partition"},{"id":"Analysis/real-functions/integration/riemann-integral/mesh","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.mesh","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#mesh","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"The mesh (also called the norm) of a partition $a = x_0 < x_1 < \\dots < x_n = b$ of $[a, b]$ is the maximum subinterval length $\\max_{1 \\le k \\le n} (x_k - x_{k-1})$ when $n \\ge 1$, and $0$ when $n = 0$.","title":"Mesh of a partition"},{"id":"Analysis/real-functions/integration/riemann-integral/point-set","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.pointSet","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#point-set","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"The point set of a partition $a = x_0 < x_1 < \\dots < x_n = b$ of $[a, b]$ is the set of its points $x_0, x_1, \\dots, x_n$.","title":"Points of a partition"},{"id":"Analysis/real-functions/integration/riemann-integral/is-refinement","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.IsRefinement","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#is-refinement","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"For partitions $P$ and $Q$ of $[a, b]$, $Q$ is a refinement of $P$ when every point of $P$ is a point of $Q$, that is, when the point set of $P$ is contained in the point set of $Q$.","title":"Refinement of a partition"},{"id":"Analysis/real-functions/integration/riemann-integral/riemann-sum","kind":"node","lean":"MathlibExt.Analysis.Integration.riemannSum","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#riemann-sum","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and let $T$ be a tagged partition of $[a, b]$, with points $a = x_0 < x_1 < \\cdots < x_n = b$ and tags $\\xi_k \\in [x_{k-1}, x_k]$ for $k = 1, \\dots, n$. The Riemann sum of $f$ with respect to $T$ is $S_f(x, \\xi) = \\sum_{k=1}^{n} f(\\xi_k)\\,(x_k - x_{k-1})$ (an empty sum, equal to $0$, when $n = 0$).","title":"Riemann sum"},{"id":"Analysis/real-functions/integration/riemann-integral/a-le-b","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.a_le_b","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#a-le-b","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"If $a, b \\in \\mathbb{R}$ and the interval $[a, b]$ has a partition, then $a \\le b$.","title":"Endpoints of a partition are ordered"},{"id":"Analysis/real-functions/integration/riemann-integral/n-pos-of-lt","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.n_pos_of_lt","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#n-pos-of-lt","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"If $a < b$, then every partition $a = x_0 < x_1 < \\dots < x_n = b$ of $[a, b]$ has at least one subinterval, that is, $n \\ge 1$.","title":"A partition of a nondegenerate interval has a subinterval"},{"id":"Analysis/real-functions/integration/riemann-integral/mesh-pos-of-lt","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.mesh_pos_of_lt","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#mesh-pos-of-lt","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"If $a < b$, then every partition of $[a, b]$ has strictly positive mesh.","title":"The mesh is positive on a nondegenerate interval"},{"id":"Analysis/real-functions/integration/riemann-integral/riemann-sum-eq-zero-of-n-eq-zero","kind":"node","lean":"MathlibExt.Analysis.Integration.riemannSum_eq_zero_of_n_eq_zero","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#riemann-sum-eq-zero-of-n-eq-zero","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"For every function $f : \\mathbb{R} \\to \\mathbb{R}$ and every tagged partition of $[a, b]$ with no subintervals, the Riemann sum of $f$ is $0$, since it is an empty sum.","title":"Riemann sum over a trivial partition"},{"id":"Analysis/real-functions/integration/riemann-integral/succ-sub-cast-succ-nonneg","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.succ_sub_castSucc_nonneg","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#succ-sub-cast-succ-nonneg","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"If $a = x_0 < x_1 < \\cdots < x_n = b$ is a partition of $[a, b]$, then $x_{i+1} - x_i \\ge 0$ for every $i$ with $0 \\le i < n$.","title":"Subintervals of a partition have nonnegative length"},{"id":"Analysis/real-functions/integration/riemann-integral/le-mesh","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.le_mesh","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#le-mesh","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $a = x_0 < x_1 < \\dots < x_n = b$ be a partition of $[a, b]$. Then for every $i$ with $0 \\le i < n$, the subinterval length $x_{i+1} - x_i$ is at most the mesh of the partition.","title":"Subinterval lengths are at most the mesh"},{"id":"Analysis/real-functions/integration/riemann-integral/sum-sub-eq","kind":"node","lean":"MathlibExt.Analysis.Integration.Partition.sum_sub_eq","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#sum-sub-eq","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"If $a = x_0 < x_1 < \\cdots < x_n = b$ is a partition of $[a, b]$, then the lengths of its subintervals sum to the length of the interval: $\\sum_{i=0}^{n-1} (x_{i+1} - x_i) = b - a$.","title":"Subinterval lengths of a partition telescope"},{"id":"Analysis/real-functions/integration/riemann-integral/tag-mem-icc","kind":"node","lean":"MathlibExt.Analysis.Integration.TaggedPartition.tag_mem_Icc","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#tag-mem-icc","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Every tag $\\xi_k$ of a tagged partition of $[a, b]$ lies in $[a, b]$.","title":"Tags of a tagged partition lie in the interval"},{"id":"Analysis/real-functions/integration/riemann-integral/riemann-sum-const","kind":"node","lean":"MathlibExt.Analysis.Integration.riemannSum_const","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#riemann-sum-const","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"For every real constant $c$ and every tagged partition of $[a, b]$, the Riemann sum of the constant function $x \\mapsto c$ equals $c(b - a)$.","title":"Riemann sum of a constant function"},{"id":"Analysis/real-functions/integration/riemann-integral/telescope-sum-of-embedding","kind":"node","lean":"MathlibExt.Analysis.Integration.telescope_sum_of_embedding","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#telescope-sum-of-embedding","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $y_0, \\dots, y_N$ be real numbers, let $\\varphi : \\{0, \\dots, m\\} \\to \\{0, \\dots, N\\}$ be strictly increasing, and let $\\sigma : \\{0, \\dots, N-1\\} \\to \\{0, \\dots, m-1\\}$ satisfy $\\varphi(\\sigma(j)) \\le j < \\varphi(\\sigma(j) + 1)$ for every $j < N$. Then for every $k < m$,","title":"Telescoping over the fibers of an assignment"},{"id":"Analysis/real-functions/integration/riemann-integral/is-tagged-refinement","kind":"node","lean":"MathlibExt.Analysis.Integration.TaggedPartition.IsTaggedRefinement","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#is-tagged-refinement","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $T$ and $T'$ be tagged partitions of $[a, b]$ with points $a = x_0 < \\dots < x_n = b$ and $a = x'_0 < \\dots < x'_{n'} = b$ respectively. Tagged-refinement data from $T$ to $T'$ consist of a map $\\sigma$ assigning to each index $j < n'$ an index $\\sigma(j) < n$ such that $x_{\\sigma(j)} \\le x'_j$ and $x'_{j+1} \\le x_{\\sigma(j)+1}$ for every $j < n'$ (so the $j$-th subinterval of $T'$ lies in the $\\sigma(j)$-th\u2026","title":"Tagged-refinement data"},{"id":"Analysis/real-functions/integration/riemann-integral/nonempty-is-tagged-refinement-of-is-refi","kind":"node","lean":"MathlibExt.Analysis.Integration.TaggedPartition.nonempty_isTaggedRefinement_of_isRefinement","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#nonempty-is-tagged-refinement-of-is-refi","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $T$ and $T'$ be tagged partitions of $[a, b]$ such that every point of $T$ is a point of $T'$. Then tagged-refinement data from $T$ to $T'$ exist: an assignment of each subinterval of $T'$ to a subinterval of $T$ that contains it, such that for each subinterval of $T$ the lengths of the subintervals of $T'$ assigned to it add up to its length.","title":"Refinements carry tagged-refinement data"},{"id":"Analysis/real-functions/integration/riemann-integral/riemann-sum-eq-fiber","kind":"node","lean":"MathlibExt.Analysis.Integration.riemannSum_eq_fiber","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#riemann-sum-eq-fiber","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$, let $T$ be a tagged partition of $[a, b]$ with points $x_0 < \\dots < x_n$ and tags $\\xi_0, \\dots, \\xi_{n-1}$, and let $T'$ be a tagged partition of $[a, b]$ with points $x'_0 < \\dots < x'_{n'}$, together with tagged-refinement data from $T$ to $T'$ assigning to each $j < n'$ an index $\\sigma(j) < n$. Then the Riemann sum $S_f(T)$ of $f$ over $T$ satisfies","title":"Riemann sum regrouped over a refinement"},{"id":"Analysis/real-functions/integration/riemann-integral/riemann-sum-sub-le-of-is-tagged-refineme","kind":"node","lean":"MathlibExt.Analysis.Integration.riemannSum_sub_le_of_isTaggedRefinement","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#riemann-sum-sub-le-of-is-tagged-refineme","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$, let $T$ and $T'$ be tagged partitions of $[a, b]$ together with tagged-refinement data from $T$ to $T'$, and let $\\omega \\in \\mathbb{R}$ satisfy $|f(x) - f(y)| \\le \\omega$ for all $x, y \\in [a, b]$ with $|x - y| \\le \\|T\\|$, where $\\|T\\|$ is the mesh of $T$. Writing $S_f(T)$ for the Riemann sum of $f$ over $T$, we have $|S_f(T) - S_f(T')| \\le \\omega (b - a)$.","title":"Riemann sums under refinement, given refinement data"},{"id":"Analysis/real-functions/integration/riemann-integral/riemann-sum-sub-le-of-is-refinement","kind":"node","lean":"MathlibExt.Analysis.Integration.riemannSum_sub_le_of_isRefinement","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#riemann-sum-sub-le-of-is-refinement","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$, let $T$ and $T'$ be tagged partitions of $[a, b]$ such that every point of $T$ is a point of $T'$, and let $\\omega \\in \\mathbb{R}$ satisfy $|f(x) - f(y)| \\le \\omega$ for all $x, y \\in [a, b]$ with $|x - y| \\le \\|T\\|$, where $\\|T\\|$ is the mesh of $T$. Writing $S_f(T)$ for the Riemann sum of $f$ over $T$, we have $|S_f(T) - S_f(T')| \\le \\omega (b - a)$.","title":"Riemann sums under refinement"},{"id":"Analysis/real-functions/integration/riemann-integral/riemann-sum-sub-le-of-common-refinement","kind":"node","lean":"MathlibExt.Analysis.Integration.riemannSum_sub_le_of_commonRefinement","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#riemann-sum-sub-le-of-common-refinement","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and let $T_1$, $T_2$ be tagged partitions of $[a, b]$ with meshes $\\|T_1\\|$ and $\\|T_2\\|$. Suppose $\\omega_1, \\omega_2 \\in \\mathbb{R}$ satisfy, for $i = 1, 2$, $|f(x) - f(y)| \\le \\omega_i$ for all $x, y \\in [a, b]$ with $|x - y| \\le \\|T_i\\|$. Let $T''$ be a tagged partition of $[a, b]$ together with tagged-refinement data from $T_1$ to $T''$ and from $T_2$ to $T''$. Writing\u2026","title":"Riemann sums of two partitions with a common refinement"},{"id":"Analysis/real-functions/integration/riemann-integral/exists-common-refinement","kind":"node","lean":"MathlibExt.Analysis.Integration.TaggedPartition.exists_commonRefinement","parent":17907,"result":true,"route":"../roadmap/Analysis/real-functions/integration/riemann-integral/index.html#exists-common-refinement","scope":"Analysis/real-functions/integration/riemann-integral","status":"fully_proved","summary":"Let $a, b$ be real numbers. For any two tagged partitions $T_1$ and $T_2$ of $[a, b]$, there is a tagged partition $T''$ of $[a, b]$ that is a tagged refinement of both $T_1$ and $T_2$.","title":"Common refinement of two tagged partitions"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations.json","id":"Analysis/real-functions/continuity-and-functional-equations","kind":"scope","parent":17796,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations","status":"can_prove","summary":"Fine structure of real functions: Blumberg's theorem, continuous nowhere-differentiable functions (Weierstrass, Takagi), the rising sun lemma, approximate continuity, midpoint convexity, and the Cauchy, exponential and almost-additive functional equations including Hyers\u2013Ulam stability.","title":"Continuity, regularity and functional equations"},{"id":"Analysis/real-functions/continuity-and-functional-equations/exists-continuous-nowhere-differentiable","kind":"node","lean":"MathlibExt.Analysis.Calculus.NowhereDifferentiable.exists_continuous_nowhere_differentiable","parent":17931,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/index.html#exists-continuous-nowhere-differentiable","scope":"Analysis/real-functions/continuity-and-functional-equations","status":"fully_proved","summary":"There exists a continuous function $f : \\mathbb{R} \\to \\mathbb{R}$ that is differentiable at no point of $\\mathbb{R}$.","title":"A continuous nowhere differentiable function"},{"id":"Analysis/real-functions/continuity-and-functional-equations/countable-set-of-lt-top-of-lt-liminf","kind":"node","lean":"MathlibExt.MeasureTheory.StrictLiminfGapFiniteSetCountableWanted.countable_setOf_lt_top_of_lt_liminf","parent":17931,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/index.html#countable-set-of-lt-top-of-lt-liminf","scope":"Analysis/real-functions/continuity-and-functional-equations","status":"can_prove","summary":"Let $f : [0,1] \\to [0,\\infty]$ be such that for every $y \\in [0,1]$ with $f(y) \\ne \\infty$,","title":"A strict liminf gap at every finite value forces countably many finite values"},{"id":"Analysis/real-functions/continuity-and-functional-equations/blumberg-theorem","kind":"node","lean":"MetaMathlibExt.blumberg_theorem","parent":17931,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/index.html#blumberg-theorem","scope":"Analysis/real-functions/continuity-and-functional-equations","status":"fully_proved","summary":"For every function $f : \\mathbb{R} \\to \\mathbb{R}$ there is a dense set $D \\subseteq \\mathbb{R}$ such that the restriction of $f$ to $D$ is continuous.","title":"Blumberg's theorem"},{"id":"Analysis/real-functions/continuity-and-functional-equations/convex-on-ioo-of-continuous-on-of-midpoi","kind":"node","lean":"MathlibExt.Analysis.Calculus.MidpointConvexContinuousConvexWanted.convexOn_Ioo_of_continuousOn_of_midpoint_le","parent":17931,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/index.html#convex-on-ioo-of-continuous-on-of-midpoi","scope":"Analysis/real-functions/continuity-and-functional-equations","status":"can_prove","summary":"Let $a < b$ be real numbers and let $\\varphi : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $(a, b)$ and satisfy $\\varphi\\bigl(\\frac{x+y}{2}\\bigr) \\le \\frac{\\varphi(x) + \\varphi(y)}{2}$ for all $x, y \\in (a, b)$. Then $\\varphi$ is convex on $(a, b)$.","title":"Continuous midpoint-convex functions are convex"},{"id":"Analysis/real-functions/continuity-and-functional-equations/exists-discontinuous-subadditive-multipl","kind":"node","lean":"MathlibExt.Analysis.Complex.ScottishBook39Wanted.exists_discontinuous_subadditive_multiplicative","parent":17931,"reading_title":{"format":"inline-parts-v1","parts":[["text","Discontinuous subadditive multiplicative functions on "],["math","\\(\\mathbb{C}\\)"]],"source":"Discontinuous subadditive multiplicative functions on $\\mathbb{C}$"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/index.html#exists-discontinuous-subadditive-multipl","scope":"Analysis/real-functions/continuity-and-functional-equations","status":"can_prove","summary":"There is a function $\\varphi : \\mathbb{C} \\to \\mathbb{R}$ that is not continuous, satisfies $\\varphi(x) \\ge 0$ for all $x \\in \\mathbb{C}$, is not identically $0$ and not identically $1$, and satisfies $\\varphi(x + y) \\le \\varphi(x) + \\varphi(y)$ and $\\varphi(xy) = \\varphi(x)\\varphi(y)$ for all $x, y \\in \\mathbb{C}$.","title":"Discontinuous subadditive multiplicative functions on $\\mathbb{C}$"},{"id":"Analysis/real-functions/continuity-and-functional-equations/countable-diff-set-of-exists-seq-tendsto","kind":"node","lean":"MathlibExt.Analysis.Calculus.GraphNonAccumulationPointsCountableWanted.countable_diff_setOf_exists_seq_tendsto_and_tendsto_apply","parent":17931,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/index.html#countable-diff-set-of-exists-seq-tendsto","scope":"Analysis/real-functions/continuity-and-functional-equations","status":"can_prove","summary":"Let $E \\subseteq \\mathbb{R}^n$, let $f : \\mathbb{R}^n \\to \\mathbb{R}^n$, and let $F$ be the set of $x\\in E$ for which there is a sequence $(x_k)$ in $E\\setminus\\{x\\}$ with $x_k \\to x$ and $f(x_k) \\to f(x)$. Then $E \\setminus F$ is countable.","title":"Isolated points of a graph are countable"},{"id":"Analysis/real-functions/continuity-and-functional-equations/steinhaus181-periodization-one","kind":"node","lean":"MathlibExt.Analysis.ScottishBook181Wanted.steinhaus181_periodization_one","parent":17931,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/index.html#steinhaus181-periodization-one","scope":"Analysis/real-functions/continuity-and-functional-equations","status":"can_prove","summary":"There is a continuous function $f : \\mathbb{R} \\to \\mathbb{R}$ with $f(x) > 0$ for every real $x$ such that, for every real $x$, the series $\\sum_{n \\in \\mathbb{Z}} f(x + n)$ converges and its sum is $1$.","title":"Steinhaus's periodization problem (Scottish Book Problem 181)"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions.json","id":"Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions","status":"can_prove","title":"Almost additive functions"},{"id":"Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions/conjecture","kind":"node","lean":"MathlibExt.Analysis.ErdosProblem1126Wanted.conjecture","parent":17939,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions/index.html#conjecture","scope":"Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions","status":"fully_proved","summary":"This is the statement that for every function $f : \\mathbb{R} \\to \\mathbb{R}$ with $f(x + y) = f(x) + f(y)$ for almost every $(x, y) \\in \\mathbb{R}^2$ with respect to Lebesgue measure on $\\mathbb{R}^2$, there is a function $h : \\mathbb{R} \\to \\mathbb{R}$ with $h(x + y) = h(x) + h(y)$ for all $x, y \\in \\mathbb{R}$ and $f(x) = h(x)$ for almost every $x \\in \\mathbb{R}$.","title":"Erd\u0151s Problem 1126 on almost additive functions"},{"id":"Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.ErdosProblem1126Wanted.conjecture_holds","parent":17939,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions/index.html#conjecture-holds","scope":"Analysis/real-functions/continuity-and-functional-equations/almost-additive-functions","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ satisfy $f(x + y) = f(x) + f(y)$ for almost every $(x, y) \\in \\mathbb{R}^2$ with respect to Lebesgue measure on $\\mathbb{R}^2$. Then there is a function $h : \\mathbb{R} \\to \\mathbb{R}$ with $h(x + y) = h(x) + h(y)$ for all $x, y \\in \\mathbb{R}$ and $f(x) = h(x)$ for almost every $x \\in \\mathbb{R}$.","title":"Almost additive functions agree almost everywhere with additive functions"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity.json","id":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity","status":"can_prove","title":"Approximate derivatives and monotonicity"},{"id":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/has-zero-density-at-right","kind":"node","lean":"MathlibExt.Analysis.ScottishBook157Wanted.HasZeroDensityAtRight","parent":17942,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/index.html#has-zero-density-at-right","scope":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity","status":"fully_proved","summary":"A set $S \\subseteq \\mathbb{R}$ has right-hand density $0$ at $0$ if the Lebesgue outer measure of $S \\cap (0, \\delta]$, divided by $\\delta$, tends to $0$ as $\\delta \\to 0^+$.","title":"Right density zero at the origin"},{"id":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/approx-continuous-at","kind":"node","lean":"MathlibExt.Analysis.ScottishBook157Wanted.ApproxContinuousAt","parent":17942,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/index.html#approx-continuous-at","scope":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity","status":"fully_proved","summary":"A function $f \\colon \\mathbb{R} \\to \\mathbb{R}$ is approximately continuous at $x$ if, for every $\\varepsilon > 0$, the set $S_\\varepsilon = \\{y : |f(y) - f(x)| < \\varepsilon\\}$ has density $1$ at $x$: the Lebesgue outer measure of $S_\\varepsilon \\cap (x - \\delta, x + \\delta)$, divided by $2\\delta$, tends to $1$ as $\\delta \\to 0^+$.","title":"Approximate continuity"},{"id":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/has-positive-upper-right-approx-deriv","kind":"node","lean":"MathlibExt.Analysis.ScottishBook157Wanted.HasPositiveUpperRightApproxDeriv","parent":17942,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/index.html#has-positive-upper-right-approx-deriv","scope":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity","status":"fully_proved","summary":"The upper right-hand approximate derivative of $f \\colon \\mathbb{R} \\to \\mathbb{R}$ at $x$ is positive: there are $a > 0$ and $\\varepsilon > 0$ such that for every $\\delta > 0$ there is $\\delta' \\in (0, \\delta]$ for which the Lebesgue outer measure of $\\{h \\in (0, \\delta'] : (f(x + h) - f(x))/h > a\\}$ is at least $\\varepsilon \\delta'$.","title":"Positive upper right approximate derivative"},{"id":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/ward-approximately-continuous-positive-d","kind":"node","lean":"MathlibExt.Analysis.ScottishBook157Wanted.wardApproximatelyContinuousPositiveDerivMonotone","parent":17942,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity/index.html#ward-approximately-continuous-positive-d","scope":"Analysis/real-functions/continuity-and-functional-equations/approximate-derivatives-and-monotonicity","status":"can_prove","summary":"Let $f \\colon \\mathbb{R} \\to \\mathbb{R}$ be approximately continuous at every point, and suppose that at every point $x$ the upper right-hand approximate derivative of $f$ is positive. Then $f$ is monotone nondecreasing.","title":"Approximate derivative test for monotonicity (Scottish Book Problem 157)"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation.json","id":"Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation","status":"can_prove","title":"Exponential functional equation"},{"id":"Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation/exists-eq-exp-mul-of-continuous-add-mul","kind":"node","lean":"MetaMathlibExt.exists_eq_exp_mul_of_continuous_add_mul","parent":17947,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation/index.html#exists-eq-exp-mul-of-continuous-add-mul","scope":"Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation","status":"fully_proved","summary":"If a continuous function $f : \\mathbb{R} \\to \\mathbb{C}$ satisfies $f(0) = 1$ and $f(r + s) = f(r) f(s)$ for all real $r$ and $s$, then there is a complex number $L$ such that $f(r) = e^{rL}$ for every real $r$.","title":"Continuous solutions of the exponential functional equation"},{"id":"Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation/exists-eq-exp-mul-of-map-add-eq-mul","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ContinuousMultiplicativeFunctionalEqExpWanted.exists_eq_exp_mul_of_map_add_eq_mul","parent":17947,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuous nonzero solutions of "],["math","\\(f(x)f(y) = f(x+y)\\)"],["text"," are exponentials"]],"source":"Continuous nonzero solutions of $f(x)f(y) = f(x+y)$ are exponentials"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation/index.html#exists-eq-exp-mul-of-map-add-eq-mul","scope":"Analysis/real-functions/continuity-and-functional-equations/exponential-functional-equation","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a continuous function that is not identically zero and satisfies $f(x)f(y) = f(x+y)$ for all real $x$ and $y$. Then there is a real constant $c$ such that $f(x) = e^{cx}$ for every real $x$.","title":"Continuous nonzero solutions of $f(x)f(y) = f(x+y)$ are exponentials"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences.json","id":"Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences","status":"can_prove","title":"Functions with continuous differences"},{"id":"Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences/conjecture","kind":"node","lean":"MathlibExt.Analysis.ErdosProblem907Wanted.conjecture","parent":17950,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences/index.html#conjecture","scope":"Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences","status":"fully_proved","summary":"The statement that every function $f : \\mathbb{R} \\to \\mathbb{R}$ such that $x \\mapsto f(x + h) - f(x)$ is continuous for every $h > 0$ can be written as $f = g + a$ with $g : \\mathbb{R} \\to \\mathbb{R}$ continuous and $a : \\mathbb{R} \\to \\mathbb{R}$ additive, that is, $a(x + y) = a(x) + a(y)$ for all real $x$ and $y$.","title":"Erd\u0151s Problem 907"},{"id":"Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.ErdosProblem907Wanted.conjecture_holds","parent":17950,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences/index.html#conjecture-holds","scope":"Analysis/real-functions/continuity-and-functional-equations/functions-with-continuous-differences","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that for every $h > 0$ the function $x \\mapsto f(x + h) - f(x)$ is continuous. Then there are functions $g, a : \\mathbb{R} \\to \\mathbb{R}$ with $g$ continuous and $a$ additive, that is, $a(x + y) = a(x) + a(y)$ for all real $x$ and $y$, such that $f = g + a$.","title":"De Bruijn's theorem on continuous differences (Erd\u0151s Problem 907)"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability.json","id":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability","status":"fully_proved","title":"Hyers\u2013Ulam stability"},{"id":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/approx","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.HyersUlam.approx","parent":17953,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/index.html#approx","scope":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability","status":"fully_proved","summary":"The dyadic approximants `2 ^ (-n) \u2022 f (2 ^ n \u2022 x)` used in Hyers' proof.","title":"Dyadic approximants in Hyers' proof"},{"id":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/norm-nsmul-eq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.HyersUlam.norm_nsmul_eq","parent":17953,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/index.html#norm-nsmul-eq","scope":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability","status":"fully_proved","summary":"Norm of a natural-number multiple in a real normed space.","title":"Norm of a natural-number multiple"},{"id":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/exists-add-monoid-hom-of-additive-defect","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.HyersUlam.exists_addMonoidHom_of_additive_defect","parent":17953,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/index.html#exists-add-monoid-hom-of-additive-defect","scope":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability","status":"fully_proved","summary":"If `f : E \u2192 F` between real normed spaces, with `F` complete, satisfies `\u2016f (x + y) - f x - f y\u2016 \u2264 \u03b4` for all `x, y`, then there is an additive map `A : E \u2192+ F` with `\u2016f x - A x\u2016 \u2264 \u03b4` for all `x`. The bound `\u03b4` is sharp: for `f x = x + c` on `\u211d` the defect is `\u2016c\u2016` and the best additive approximant is the identity, at distance exactly `\u2016c\u2016`.","title":"Hyers' theorem (Hyers\u2013Ulam stability of the Cauchy equation)"},{"id":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/eq-of-forall-norm-sub-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.HyersUlam.eq_of_forall_norm_sub_le","parent":17953,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/index.html#eq-of-forall-norm-sub-le","scope":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability","status":"fully_proved","summary":"Two additive maps both within `\u03b4` of the same `f` are equal: their difference is additive, hence `n`-homogeneous, while staying bounded by `2 * \u03b4`.","title":"Uniqueness in Hyers' theorem"},{"id":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/exists-add-monoid-hom-of-jensen-defect","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.HyersUlam.exists_addMonoidHom_of_jensen_defect","parent":17953,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/index.html#exists-add-monoid-hom-of-jensen-defect","scope":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability","status":"fully_proved","summary":"If the midpoint (Jensen) defect `\u2016f ((1/2) \u2022 (x + y)) - (1/2) \u2022 (f x + f y)\u2016` is `\u2264 \u03b4` for all `x, y`, then `f` is within `4 * \u03b4` of an additive map plus the constant `f 0`. The factor `4` comes from doubling the two Jensen estimates at `(x + y, 0)` and `(x, y)`.","title":"Hyers\u2013Ulam stability of Jensen's functional equation"},{"id":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/exists-continuous-linear-map-of-additive","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.HyersUlam.exists_continuousLinearMap_of_additive_defect_of_continuousAt","parent":17953,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability/index.html#exists-continuous-linear-map-of-additive","scope":"Analysis/real-functions/continuity-and-functional-equations/hyers-ulam-stability","status":"fully_proved","summary":"If `f` is continuous at `0` (in particular if `f` itself is continuous), the additive approximant from Hyers' theorem is continuous, hence a continuous real-linear map. The proof: continuity of `f` at `0` bounds `f` on a small ball, so the approximant is bounded on that ball, and an additive map bounded on a ball is continuous at `0` by the dyadic scaling argument.","title":"Continuous Hyers\u2013Ulam stability"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity.json","id":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity","status":"fully_proved","title":"Modulus of continuity"},{"id":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/modulus-of-continuity","kind":"node","lean":"modulusOfContinuity","parent":17960,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/index.html#modulus-of-continuity","scope":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity","status":"fully_proved","summary":"For a function $f : \\mathbb{R} \\to \\mathbb{R}$, real numbers $a$ and $b$, and a scale $\\eta \\in \\mathbb{R}$, the modulus of continuity of $f$ on $[a, b]$ at scale $\\eta$ is","title":"Modulus of continuity"},{"id":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/modulus-of-continuity-bdd-above","kind":"node","lean":"modulusOfContinuity_bddAbove","parent":17960,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/index.html#modulus-of-continuity-bdd-above","scope":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[a, b]$. Then for every $\\eta \\in \\mathbb{R}$ the set $\\{|f(x) - f(y)| : x, y \\in [a, b],\\ |x - y| \\le \\eta\\}$ is bounded above.","title":"The set defining the modulus of continuity is bounded above"},{"id":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/le-modulus-of-continuity","kind":"node","lean":"le_modulusOfContinuity","parent":17960,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/index.html#le-modulus-of-continuity","scope":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[a, b]$, let $\\eta \\in \\mathbb{R}$, and let $\\omega_f(\\eta)$ be the modulus of continuity of $f$ on $[a, b]$ at scale $\\eta$. For all $x, y \\in [a, b]$ with $|x - y| \\le \\eta$, we have $|f(x) - f(y)| \\le \\omega_f(\\eta)$.","title":"The modulus of continuity bounds increments"},{"id":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/modulus-of-continuity-nonneg","kind":"node","lean":"modulusOfContinuity_nonneg","parent":17960,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/index.html#modulus-of-continuity-nonneg","scope":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity","status":"fully_proved","summary":"Let $a \\le b$ and let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[a, b]$. For every $\\eta \\ge 0$, the modulus of continuity $\\omega_f(\\eta)$ of $f$ on $[a, b]$ at scale $\\eta$ satisfies $0 \\le \\omega_f(\\eta)$.","title":"Nonnegativity of the modulus of continuity"},{"id":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/modulus-of-continuity-mono","kind":"node","lean":"modulusOfContinuity_mono","parent":17960,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/index.html#modulus-of-continuity-mono","scope":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity","status":"fully_proved","summary":"Let $a \\le b$, let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[a, b]$, and write $\\omega_f(\\eta)$ for the modulus of continuity of $f$ on $[a, b]$ at scale $\\eta$. If $0 \\le \\eta_1 \\le \\eta_2$, then $\\omega_f(\\eta_1) \\le \\omega_f(\\eta_2)$.","title":"Monotonicity of the modulus of continuity"},{"id":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/modulus-of-continuity-tendsto-zero","kind":"node","lean":"modulusOfContinuity_tendsto_zero","parent":17960,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity/index.html#modulus-of-continuity-tendsto-zero","scope":"Analysis/real-functions/continuity-and-functional-equations/modulus-of-continuity","status":"fully_proved","summary":"Let $a \\le b$ and let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[a, b]$. Then the modulus of continuity $\\omega_f(\\eta)$ of $f$ on $[a, b]$ tends to $0$ as $\\eta \\to 0^+$.","title":"Theorem I (Modulus of continuity vanishes)"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions.json","id":"Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions","status":"can_prove","title":"Pointwise and uniform H\u00f6lder conditions"},{"id":"Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions/scottish-book-problem83-claim","kind":"node","lean":"MathlibExt.Analysis.Calculus.ScottishBook83Wanted.ScottishBookProblem83Claim","parent":17967,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions/index.html#scottish-book-problem83-claim","scope":"Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions","status":"fully_proved","summary":"Auerbach's Problem 83 from the Scottish Book: if a continuous function `f` satisfies, at every point `x`, that the difference quotients `|f (x + h) - f x| / |h| ^ a` stay below a constant `M` for all sufficiently small nonzero `h` \u2014 that is, if the limsup of the quotients as `h \u2192 0` is `< M`, with `0 < a < 1` \u2014 does `f` then satisfy a H\u00f6lder condition of exponent `a`?","title":"Auerbach's H\u00f6lder condition problem (Scottish Book 83)"},{"id":"Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions/auerbach83-refuted","kind":"node","lean":"MathlibExt.Analysis.Calculus.ScottishBook83Wanted.auerbach83_refuted","parent":17967,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions/index.html#auerbach83-refuted","scope":"Analysis/real-functions/continuity-and-functional-equations/pointwise-and-uniform-h-lder-conditions","status":"can_prove","summary":"Auerbach's Problem 83 is false: the pointwise limsup condition does not imply a H\u00f6lder condition. Source: H. Auerbach, Problem 83 of the Scottish Book; typescript Ad 83 records the negative answer, with Marcinkiewicz's counterexample: a triangular function on intervals `1 / n - x\u2099`, `1 / n + x\u2099` with height `1 / n` and suitably small half-widths `x\u2099`, whose difference quotients are small near every point but which\u2026","title":"Pointwise difference-quotient bounds do not imply H\u00f6lder continuity"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma.json","id":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma","status":"can_prove","title":"Rising sun lemma"},{"id":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/rising-sun-set","kind":"node","lean":"MathlibExt.Analysis.RisingSunWanted.Analysis.RisingSun.risingSunSet","parent":17970,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/index.html#rising-sun-set","scope":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma","status":"fully_proved","summary":"For real numbers $a, b$ and a function $f : \\mathbb{R} \\to \\mathbb{R}$, the rising sun set of $f$ on $(a, b)$ is the set of points $x \\in (a, b)$ for which there is some $t$ with $x < t \\le b$ and $f(x) < f(t)$.","title":"Rising sun set"},{"id":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/has-rising-sun-decomposition","kind":"node","lean":"MathlibExt.Analysis.RisingSunWanted.Analysis.RisingSun.HasRisingSunDecomposition","parent":17970,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/index.html#has-rising-sun-decomposition","scope":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma","status":"fully_proved","summary":"A set $E \\subseteq \\mathbb{R}$ has a rising sun decomposition for a function $f : \\mathbb{R} \\to \\mathbb{R}$ if there are sequences $(c_n)_{n \\in \\mathbb{N}}$ and $(d_n)_{n \\in \\mathbb{N}}$ of real numbers such that $E$ is the union of the open intervals $(c_n, d_n)$, these intervals are pairwise disjoint, and $f(c_n) \\le f(d_n)$ for every $n$.","title":"Rising sun decomposition"},{"id":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/has-rising-sun-property","kind":"node","lean":"MathlibExt.Analysis.RisingSunWanted.Analysis.RisingSun.HasRisingSunProperty","parent":17970,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/index.html#has-rising-sun-property","scope":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma","status":"fully_proved","summary":"Let $a, b$ be real numbers, let $f : \\mathbb{R} \\to \\mathbb{R}$, and let $S$ be the rising sun set of $f$ on $(a, b)$. Then $f$ has the rising sun property on $[a, b]$ if $S$ is open, $S$ is empty if and only if $f$ is antitone (non-increasing) on $[a, b]$, and, whenever $S$ is nonempty, $S$ has a rising sun decomposition for $f$.","title":"Rising sun property"},{"id":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/rising-sun-lemma","kind":"node","lean":"MathlibExt.Analysis.RisingSunWanted.Analysis.RisingSun.rising_sun_lemma","parent":17970,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma/index.html#rising-sun-lemma","scope":"Analysis/real-functions/continuity-and-functional-equations/rising-sun-lemma","status":"can_prove","summary":"Let $a < b$ be real numbers and let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[a, b]$. Then $f$ has the rising sun property on $[a, b]$.","title":"Riesz's rising sun lemma"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe.json","id":"Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe","status":"can_prove","title":"Smooth functions and analytic functions on perfect sets"},{"id":"Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe/is-perfect-set","kind":"node","lean":"MathlibExt.Analysis.ScottishBook17Point1Wanted.IsPerfectSet","parent":17975,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe/index.html#is-perfect-set","scope":"Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe","status":"fully_proved","summary":"A set $s \\subseteq \\mathbb{R}$ is perfect if it is closed, nonempty, and has no isolated points: every $x \\in s$ lies in the closure of $s \\setminus \\{x\\}$.","title":"Perfect sets of real numbers"},{"id":"Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe/scottish-book-problem17-point1-claim","kind":"node","lean":"MathlibExt.Analysis.ScottishBook17Point1Wanted.ScottishBookProblem17Point1Claim","parent":17975,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe/index.html#scottish-book-problem17-point1-claim","scope":"Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe","status":"fully_proved","summary":"Ulam's Problem 17.1 from the Scottish Book: for every continuous `f` on `[0, 1]`, do there exist a perfect set `C` in `[0, 1]` and an analytic function `\u03c6` with `f x = \u03c6 x` for every `x \u2208 C`? The claim is stated for `C ^ \u221e` functions, which is the case refuted by Zahorski's construction.","title":"Ulam's problem on analytic agreement on a perfect set"},{"id":"Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe/continuous-function-perfect-analytic-ref","kind":"node","lean":"MathlibExt.Analysis.ScottishBook17Point1Wanted.continuousFunctionPerfectAnalytic_refuted","parent":17975,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe/index.html#continuous-function-perfect-analytic-ref","scope":"Analysis/real-functions/continuity-and-functional-equations/smooth-functions-and-analytic-functions-on-perfe","status":"can_prove","summary":"Problem 17.1 is false: the answer is negative. Source: S. Ulam, Problem 17.1 of the Scottish Book; Z. Zahorski gave a negative answer by constructing a `C ^ \u221e` function that agrees with every real-analytic function on at most finitely many points, so no nonempty perfect set `C` can work for it: Z. Zahorski, \"Sur l'ensemble des points singuliers d'une fonction d'une variable reelle admettant les derivees de tous les\u2026","title":"Zahorski's negative answer to Ulam's perfect set problem"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/takagi-function.json","id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","title":"Takagi function"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-bit","kind":"node","lean":"MetaMathlibExt.takagiBit","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-bit","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For a natural number $n$ and an integer $i$, the binary digit $n_i \\in \\{0, 1\\}$ of $n$ is the coefficient of $2^i$ in the binary expansion of $n$ when $i \\ge 0$, and $n_i = 0$ when $i < 0$.","title":"Binary digits of a natural number"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-ell","kind":"node","lean":"MetaMathlibExt.takagiEll","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-ell","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"Let $n$ be a natural number and $k$ an integer, and write $n_j$ for the binary digit of $n$ at position $j$ (so $n_j = 0$ for $j < 0$). The Takagi coefficients $\\ell_i(n, k)$, for natural numbers $i$, are the natural numbers given by $\\ell_0(n, k) = 0$ and, for every natural number $i$, with $s_i = \\sum_{j=0}^{i-1} n_{k-j}$, by $\\ell_{i+1}(n, k) = s_i$ if $n_{k-i} = 0$ and $\\ell_{i+1}(n, k) = i - s_i$ otherwise.","title":"Takagi coefficients"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-newdef","kind":"node","lean":"MetaMathlibExt.takagiNewdef","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-newdef","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For a natural number $n$ and an integer $k$, the Takagi value $\\tau(n, k)$ of the dyadic presentation $n / 2^{k+1}$ is the sum of the real series $\\sum_{i=0}^{\\infty} \\ell_{i+1}(n, k) / 2^{i+1}$, where $\\ell_i(n, k)$ denotes the $i$-th Takagi coefficient of $n$ and $k$.","title":"Takagi function on dyadic presentations"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-bit-eq","kind":"node","lean":"MetaMathlibExt.takagiBit_eq","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-bit-eq","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every natural number $n$ and every integer $i$, the binary digit $n_i$ of $n$ equals $1$ if $i \\ge 0$ and the coefficient of $2^i$ in the binary expansion of $n$ is $1$, and equals $0$ otherwise.","title":"Formula for the binary digits"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-ell-zero","kind":"node","lean":"MetaMathlibExt.takagiEll_zero","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-ell-zero","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every natural number $n$ and every integer $k$, the Takagi coefficient $\\ell_0(n, k)$ is $0$.","title":"The Takagi coefficient at index zero"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-ell-succ","kind":"node","lean":"MetaMathlibExt.takagiEll_succ","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-ell-succ","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every natural number $n$, every integer $k$ and every natural number $i$, let $s_i = \\sum_{j=0}^{i-1} n_{k-j}$, where $n_j$ denotes the binary digit of $n$ at position $j$. Then the Takagi coefficient $\\ell_{i+1}(n, k)$ equals $s_i$ if $n_{k-i} = 0$, and equals $i - s_i$ otherwise.","title":"Formula for the Takagi coefficients"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-newdef-eq","kind":"node","lean":"MetaMathlibExt.takagiNewdef_eq","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-newdef-eq","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every natural number $n$ and every integer $k$, the Takagi value $\\tau(n, k)$ of the dyadic presentation $n / 2^{k+1}$ satisfies $\\tau(n, k) = \\sum_{i=0}^{\\infty} \\ell_{i+1}(n, k) / 2^{i+1}$, where $\\ell_i(n, k)$ denotes the $i$-th Takagi coefficient of $n$ and $k$.","title":"Series for the Takagi value"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-bit-neg","kind":"node","lean":"MetaMathlibExt.takagiBit_neg","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-bit-neg","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every natural number $n$ and every integer $i < 0$, the binary digit $n_i$ of $n$ is $0$.","title":"Binary digits at negative positions vanish"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-ell-one","kind":"node","lean":"MetaMathlibExt.takagiEll_one","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-ell-one","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every natural number $n$ and every integer $k$, the Takagi coefficient $\\ell_1(n, k)$ is $0$.","title":"The first Takagi coefficient vanishes"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes","kind":"node","lean":"MetaMathlibExt.dcDNodes","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"The number $\\delta(n)$ of D-nodes in the divide-and-conquer tree on $n$ leaves (OEIS A296062) is defined by the recurrence the paper quotes from Coronado et al.: $\\delta(1) = 0$, and for every natural number $m \\ge 1$, $\\delta(2m) = 2\\,\\delta(m)$ and $\\delta(2m+1) = \\delta(m) + \\delta(m+1) + 1$. The value at $0$ is the junk value $\\delta(0) = 0$.","title":"Number of D-nodes of the divide-and-conquer tree"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-one","kind":"node","lean":"MetaMathlibExt.dcDNodes_one","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-one","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"The divide-and-conquer D-node count $\\delta$ satisfies $\\delta(1) = 0$.","title":"The divide-and-conquer D-node count at one leaf"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-two-mul","kind":"node","lean":"MetaMathlibExt.dcDNodes_two_mul","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-two-mul","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every integer $m > 0$, the divide-and-conquer D-node count $\\delta$ satisfies $\\delta(2m) = 2\\delta(m)$.","title":"Doubling recurrence for the divide-and-conquer D-node count"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-two-mul-add-one","kind":"node","lean":"MetaMathlibExt.dcDNodes_two_mul_add_one","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-two-mul-add-one","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every integer $m > 0$, the divide-and-conquer D-node count $\\delta$ satisfies $\\delta(2m+1) = \\delta(m) + \\delta(m+1) + 1$.","title":"Odd-argument recurrence for the divide-and-conquer D-node count"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-small","kind":"node","lean":"MetaMathlibExt.dcDNodes_small","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-small","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"The divide-and-conquer D-node count $\\delta$ satisfies $\\delta(2) = 0$, $\\delta(3) = 1$, $\\delta(4) = 0$, $\\delta(5) = 2$, $\\delta(6) = 2$ and $\\delta(7) = 2$.","title":"Small values of the divide-and-conquer D-node count"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-level-dnodes","kind":"node","lean":"MetaMathlibExt.dcLevelDNodes","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-level-dnodes","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For natural numbers $n$ and $i$, let $n_i$ be the binary digit of $n$ at position $i$ (the coefficient of $2^i$ in the binary expansion of $n$). The level count $\\lambda_i(n)$ is $n \\bmod 2^i$ if $n_i = 0$, and $2^i - (n \\bmod 2^i)$ if $n_i = 1$.","title":"Level counts of D-nodes"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-level-dnodes-of-digit-eq-zero","kind":"node","lean":"MetaMathlibExt.dcLevelDNodes_of_digit_eq_zero","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-level-dnodes-of-digit-eq-zero","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"Let $n$ and $i$ be natural numbers such that the binary digit $n_i$ of $n$ (the coefficient of $2^i$ in the binary expansion of $n$) equals $0$. Then the level count of D-nodes is $\\lambda_i(n) = n \\bmod 2^i$.","title":"Level count of D-nodes when the binary digit is zero"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-level-dnodes-of-digit-eq-one","kind":"node","lean":"MetaMathlibExt.dcLevelDNodes_of_digit_eq_one","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-level-dnodes-of-digit-eq-one","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"Let $n$ and $i$ be natural numbers such that the binary digit $n_i$ of $n$ (the coefficient of $2^i$ in the binary expansion of $n$) equals $1$. Then the level count of D-nodes is $\\lambda_i(n) = 2^i - (n \\bmod 2^i)$.","title":"Level count of D-nodes when the binary digit is one"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-level-dnodes-zero","kind":"node","lean":"MetaMathlibExt.dcLevelDNodes_zero","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-level-dnodes-zero","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"For every natural number $n$, the level count of D-nodes on level $0$ is $\\lambda_0(n) = n \\bmod 2$.","title":"Level count of D-nodes on level zero"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/tree-dnodes","kind":"node","lean":"MetaMathlibExt.treeDNodes","parent":17979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(D\\)"],["text","-nodes of a full binary tree"]],"source":"Number of $D$-nodes of a full binary tree"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#tree-dnodes","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"A $D$-node of a full binary tree is an interior node whose two subtrees have different numbers of leaves. The number of $D$-nodes of a full binary tree is the number of its interior nodes that are $D$-nodes.","title":"Number of $D$-nodes of a full binary tree"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/tree-snodes","kind":"node","lean":"MetaMathlibExt.treeSNodes","parent":17979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(S\\)"],["text","-nodes of a full binary tree"]],"source":"Number of $S$-nodes of a full binary tree"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#tree-snodes","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"An $S$-node of a full binary tree is an interior node whose two subtrees have the same number of leaves. The number of $S$-nodes of a full binary tree is the number of its interior nodes that are $S$-nodes.","title":"Number of $S$-nodes of a full binary tree"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/is-divide-conquer-tree","kind":"node","lean":"MetaMathlibExt.IsDivideConquerTree","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#is-divide-conquer-tree","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"A divide-and-conquer tree is a full binary tree in which, at every interior node, the numbers of leaves of the two subtrees differ by at most $1$. In particular a single leaf is a divide-and-conquer tree.","title":"Divide-and-conquer trees"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/tree-dnodes-node","kind":"node","lean":"MetaMathlibExt.treeDNodes_node","parent":17979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recursion for the number of "],["math","\\(D\\)"],["text","-nodes"]],"source":"Recursion for the number of $D$-nodes"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#tree-dnodes-node","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"Write $D(t)$ for the number of $D$-nodes of a full binary tree $t$. If the root of $t$ has left subtree $l$ and right subtree $r$, then $D(t) = \\varepsilon + D(l) + D(r)$, where $\\varepsilon = 0$ if $l$ and $r$ have the same number of leaves and $\\varepsilon = 1$ otherwise.","title":"Recursion for the number of $D$-nodes"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/tree-snodes-node","kind":"node","lean":"MetaMathlibExt.treeSNodes_node","parent":17979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recursion for the number of "],["math","\\(S\\)"],["text","-nodes"]],"source":"Recursion for the number of $S$-nodes"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#tree-snodes-node","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"Write $S(t)$ for the number of $S$-nodes of a full binary tree $t$. If the root of $t$ has left subtree $l$ and right subtree $r$, then $S(t) = \\varepsilon + S(l) + S(r)$, where $\\varepsilon = 1$ if $l$ and $r$ have the same number of leaves and $\\varepsilon = 0$ otherwise.","title":"Recursion for the number of $S$-nodes"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/tree-nodes-nil","kind":"node","lean":"MetaMathlibExt.treeNodes_nil","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#tree-nodes-nil","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"The full binary tree consisting of a single leaf has no $D$-nodes and no $S$-nodes, and it is a divide-and-conquer tree.","title":"The single-leaf tree"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/is-divide-conquer-tree-node","kind":"node","lean":"MetaMathlibExt.isDivideConquerTree_node","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#is-divide-conquer-tree-node","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"fully_proved","summary":"A full binary tree whose root has left subtree $L$ and right subtree $R$ is a divide-and-conquer tree if and only if the numbers of leaves of $L$ and $R$ differ by at most $1$ and both $L$ and $R$ are divide-and-conquer trees.","title":"The divide-and-conquer condition at the root"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dyadic-mantissa-eq","kind":"node","lean":"MetaMathlibExt.dyadic_mantissa_eq","parent":17979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normalized form of "],["math","\\(n = 2^k + r\\)"]],"source":"Normalized form of $n = 2^k + r$"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dyadic-mantissa-eq","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $n$, $k$ and $r$ be natural numbers with $r < 2^k$ and $n = 2^k + r$, and let $x$ be a rational number with $n = 2^k (1 + x)$. Then $x = r / 2^k$ and $1 + x = n / 2^k$.","title":"Normalized form of $n = 2^k + r$"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/tree-dnodes-eq-num-leaves-sub-one-sub-tr","kind":"node","lean":"MetaMathlibExt.treeDNodes_eq_numLeaves_sub_one_sub_treeSNodes","parent":17979,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(D\\)"],["text","-nodes and "],["math","\\(S\\)"],["text","-nodes of a full binary tree"]],"source":"$D$-nodes and $S$-nodes of a full binary tree"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#tree-dnodes-eq-num-leaves-sub-one-sub-tr","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $t$ be a full binary tree with $n$ leaves and let $a$ be the number of $S$-nodes of $t$. Then the number of $D$-nodes of $t$ is $n - 1 - a$ (subtraction of natural numbers, truncated at $0$).","title":"$D$-nodes and $S$-nodes of a full binary tree"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-dyadic-eq-dc-dnodes-div","kind":"node","lean":"MetaMathlibExt.takagi_dyadic_eq_dcDNodes_div","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-dyadic-eq-dc-dnodes-div","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $k$ and $r$ be natural numbers with $r \\le 2^k$. Write $\\tau(r / 2^k)$ for the Takagi value of the dyadic presentation $r / 2^k$, and $\\delta(n)$ for the number of $D$-nodes of the divide-and-conquer tree on $n$ leaves, given by $\\delta(1) = 0$, $\\delta(2m) = 2 \\delta(m)$ and $\\delta(2m + 1) = \\delta(m) + \\delta(m + 1) + 1$ for $m \\ge 1$. Then $\\tau(r / 2^k) = \\delta(2^k + r) / 2^k$.","title":"The Takagi function at dyadic rationals counts D-nodes (Coronado et al.)"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/tree-dnodes-eq-dc-dnodes-of-is-divide-co","kind":"node","lean":"MetaMathlibExt.treeDNodes_eq_dcDNodes_of_isDivideConquerTree","parent":17979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coronado et al.'s count of "],["math","\\(D\\)"],["text","-nodes in divide-and-conquer trees"]],"source":"Coronado et al.'s count of $D$-nodes in divide-and-conquer trees"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#tree-dnodes-eq-dc-dnodes-of-is-divide-co","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Every divide-and-conquer tree with $n$ leaves has exactly $\\delta(n)$ $D$-nodes, where $\\delta$ is the function on positive integers given by $\\delta(1) = 0$, $\\delta(2m) = 2\\delta(m)$ and $\\delta(2m+1) = \\delta(m) + \\delta(m+1) + 1$ for $m \\ge 1$.","title":"Coronado et al.'s count of $D$-nodes in divide-and-conquer trees"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-eq-midpoint-add-one-sub-lowest","kind":"node","lean":"MetaMathlibExt.dcDNodes_eq_midpoint_add_one_sub_lowestBit","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-eq-midpoint-add-one-sub-lowest","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $\\delta$ be the divide-and-conquer D-node count. Let $k$ and $r$ be natural numbers with $0 < r < 2^k$, put $n = 2^k + r$, and let $\\nu_2(r)$ be the position of the lowest nonzero binary digit of $r$, counted from $0$ (the exponent of the largest power of $2$ dividing $r$). Then $\\delta(n) = \\frac{\\delta(n-1) + \\delta(n+1)}{2} + 1 - \\nu_2(r)$.","title":"Midpoint identity for the divide-and-conquer D-node count"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-eq-sum-dc-level-dnodes","kind":"node","lean":"MetaMathlibExt.dcDNodes_eq_sum_dcLevelDNodes","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-eq-sum-dc-level-dnodes","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $\\delta$ be the divide-and-conquer D-node count. For every integer $n > 0$, let $k = \\lfloor \\log_2 n \\rfloor$, let $\\lambda_i(n)$ be the level-$i$ D-node count, and let $n_i$ be the binary digit of $n$ at $2^i$. Then","title":"Level-sum formula for the divide-and-conquer D-node count"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/hamming-weight-two-pow-add","kind":"node","lean":"MetaMathlibExt.hammingWeight_two_pow_add","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#hamming-weight-two-pow-add","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For a natural number $m$, write $s_1(m)$ for the sum of the binary digits of $m$ (the number of ones in its binary expansion). For all natural numbers $k$ and $r$ with $r < 2^k$, $s_1(2^k + r) = s_1(r) + 1$.","title":"Adding a larger power of two raises the Hamming weight by one"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/hamming-weight-succ-of-even","kind":"node","lean":"MetaMathlibExt.hammingWeight_succ_of_even","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#hamming-weight-succ-of-even","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For a natural number $m$, write $s_1(m)$ for the sum of the binary digits of $m$ (the number of ones in its binary expansion). If $n$ is an even natural number (including $n = 0$), then $s_1(n + 1) = s_1(n) + 1$.","title":"Hamming weight of the successor of an even number"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/hamming-weight-pred-of-odd","kind":"node","lean":"MetaMathlibExt.hammingWeight_pred_of_odd","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#hamming-weight-pred-of-odd","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For a natural number $m$, write $s_1(m)$ for the sum of the binary digits of $m$ (the number of ones in its binary expansion). If $n$ is an odd natural number, then $s_1(n - 1) = s_1(n) - 1$.","title":"Hamming weight of the predecessor of an odd number"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/hamming-weight-two-pow-sub-of-odd","kind":"node","lean":"MetaMathlibExt.hammingWeight_two_pow_sub_of_odd","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#hamming-weight-two-pow-sub-of-odd","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For a natural number $m$, write $s_1(m)$ for the sum of the binary digits of $m$ (the number of ones in its binary expansion). Let $k$ and $r$ be natural numbers with $r$ odd and $2r < 2^k$, that is, $r < 2^{k-1}$. Then $s_1(2^k - r) = 1 + k - s_1(r)$.","title":"Hamming weight of a power of two minus an odd number"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-level-dnodes-sub-pred","kind":"node","lean":"MetaMathlibExt.dcLevelDNodes_sub_pred","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-level-dnodes-sub-pred","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $k$, $r$ and $i$ be natural numbers with $r$ odd, $r < 2^k$ and $0 < i < k$, and let $n = 2^k + r$. Then the level counts of D-nodes satisfy $\\lambda_i(n) - \\lambda_i(n-1) = 1$ if the binary digit $n_i$ of $n$ is $0$, and $\\lambda_i(n) - \\lambda_i(n-1) = -1$ if $n_i = 1$.","title":"Level counts of D-nodes at consecutive integers"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-two-pow-succ-sub","kind":"node","lean":"MetaMathlibExt.dcDNodes_two_pow_succ_sub","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-two-pow-succ-sub","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For all natural numbers $k$ and $r$ with $r \\le 2^k$, $\\delta(2^{k+1} - r) = \\delta(2^k + r)$, where $\\delta(n)$ denotes the number of D-nodes in the divide-and-conquer tree on $n$ leaves.","title":"Symmetry of the D-node counts"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-succ-of-even","kind":"node","lean":"MetaMathlibExt.dcDNodes_succ_of_even","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-succ-of-even","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $\\delta$ be the divide-and-conquer D-node count and $s_1(n)$ the number of ones in the binary expansion of $n$. For every even integer $n > 0$, $\\delta(n+1) = \\delta(n) + \\lfloor \\log_2 n \\rfloor - 2 s_1(n) + 2$.","title":"Successor formula for the D-node count at even arguments"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-succ-of-odd","kind":"node","lean":"MetaMathlibExt.dcDNodes_succ_of_odd","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-succ-of-odd","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $\\delta$ be the divide-and-conquer D-node count and $s_1(n)$ the number of ones in the binary expansion of $n$. For every odd natural number $n$, $\\delta(n+1) = \\delta(n) + \\lfloor \\log_2 n \\rfloor - 2 s_1(n) + 2$.","title":"Successor formula for the D-node count at odd arguments"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-succ","kind":"node","lean":"MetaMathlibExt.dcDNodes_succ","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-succ","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $\\delta$ be the divide-and-conquer D-node count and $s_1(n)$ the number of ones in the binary expansion of $n$. For every integer $n > 0$, $\\delta(n+1) = \\delta(n) + \\lfloor \\log_2 n \\rfloor - 2 s_1(n) + 2$.","title":"Successor formula for the divide-and-conquer D-node count"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/dc-dnodes-eq-sum-digit-weight","kind":"node","lean":"MetaMathlibExt.dcDNodes_eq_sum_digit_weight","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#dc-dnodes-eq-sum-digit-weight","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $\\delta$ be the divide-and-conquer D-node count and let $s_1(m)$ be the number of ones in the binary expansion of $m$. For every integer $n > 0$, let $k = \\lfloor \\log_2 n \\rfloor$ and let $n_j$ be the binary digit of $n$ at $2^j$. Then","title":"Binary-digit formula for the divide-and-conquer D-node count"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/hamming-weight-eq-dc-dnodes-diff","kind":"node","lean":"MetaMathlibExt.hammingWeight_eq_dcDNodes_diff","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#hamming-weight-eq-dc-dnodes-diff","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For every natural number $n > 0$, the sum $s_1(n)$ of the binary digits of $n$ satisfies $s_1(n) = (\\delta(n) - \\delta(n+1) + \\lfloor \\log_2 n \\rfloor)/2 + 1$ in $\\mathbb{Q}$, where $\\delta(m)$ denotes the number of D-nodes in the divide-and-conquer tree on $m$ leaves.","title":"Binary digit sum from consecutive D-node counts"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-dyadic-eq-midpoint-add","kind":"node","lean":"MetaMathlibExt.takagi_dyadic_eq_midpoint_add","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-dyadic-eq-midpoint-add","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $k$ and $r$ be natural numbers with $1 \\le r < 2^k$. For a natural number $m$, write $\\tau(m / 2^k)$ for the Takagi value of the dyadic presentation $m / 2^k$, and let $\\rho_1$ be the $2$-adic valuation of $r$, the position (counted from $0$) of the lowest nonzero binary digit of $r$. Then $\\tau(r / 2^k) = (\\tau((r - 1) / 2^k) + \\tau((r + 1) / 2^k))/2 + (1 - \\rho_1)/2^k$.","title":"Midpoint recurrence for the Takagi function at dyadic rationals"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-dyadic-eq-sum-dc-level-dnodes","kind":"node","lean":"MetaMathlibExt.takagi_dyadic_eq_sum_dcLevelDNodes","parent":17979,"reading_title":{"format":"inline-parts-v1","parts":[["text","Takagi values at dyadic rationals from level "],["math","\\(D\\)"],["text","-node counts"]],"source":"Takagi values at dyadic rationals from level $D$-node counts"},"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-dyadic-eq-sum-dc-level-dnodes","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $k$ and $r$ be natural numbers with $r < 2^k$ and put $n = 2^k + r$. Write $\\tau(r, k-1)$ for the Takagi value of the dyadic presentation $r/2^k$, write $n_i = \\lfloor n/2^i \\rfloor \\bmod 2$ for the binary digit of $n$ at position $i$, and let $\\lambda_i(n)$ be the level-$i$ $D$-node count, which equals $n \\bmod 2^i$ if $n_i = 0$ and $2^i - (n \\bmod 2^i)$ if $n_i = 1$. Then","title":"Takagi values at dyadic rationals from level $D$-node counts"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-dyadic-succ","kind":"node","lean":"MetaMathlibExt.takagi_dyadic_succ","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-dyadic-succ","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $k$ and $r$ be natural numbers with $r < 2^k$ and put $n = 2^k + r$. Write $\\tau(m, k-1)$ for the Takagi value of the dyadic presentation $m/2^k$, and $s_1(n)$ for the number of ones in the binary expansion of $n$. Then $\\tau(r+1, k-1) = \\tau(r, k-1) + (k - 2 s_1(n) + 2)/2^k$.","title":"Consecutive Takagi values at dyadic rationals"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/takagi-dyadic-eq-sum-digit-weight","kind":"node","lean":"MetaMathlibExt.takagi_dyadic_eq_sum_digit_weight","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#takagi-dyadic-eq-sum-digit-weight","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $k$ and $r$ be natural numbers with $r < 2^k$ and put $n = 2^k + r$. Write $\\tau(r, k-1)$ for the Takagi value of the dyadic presentation $r/2^k$, write $n_j = \\lfloor n/2^j \\rfloor \\bmod 2$ for the binary digit of $n$ at position $j$, and write $s_1(m)$ for the number of ones in the binary expansion of $m$. Then","title":"Takagi values at dyadic rationals from binary digits"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/hamming-weight-eq-takagi-diff","kind":"node","lean":"MetaMathlibExt.hammingWeight_eq_takagi_diff","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#hamming-weight-eq-takagi-diff","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"Let $k$ and $r$ be natural numbers with $r < 2^k$, let $n = 2^k + r$, and let $s_1(n)$ be the sum of the binary digits of $n$. Writing $\\tau(m, k - 1)$ for the Takagi value of the dyadic presentation $m / 2^k$, one has $s_1(n) = 2^{k-1} \\bigl(\\tau(r, k - 1) - \\tau(r + 1, k - 1)\\bigr) + (k + 2)/2$.","title":"Binary digit sum from consecutive Takagi values"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/two-mul-sum-hamming-weight-range-two-pow","kind":"node","lean":"MetaMathlibExt.two_mul_sum_hammingWeight_range_two_pow","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#two-mul-sum-hamming-weight-range-two-pow","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For a natural number $i$, let $s_1(i)$ denote the sum of the binary digits of $i$. Then for every natural number $m$, $2 \\sum_{i < 2^m} s_1(i) = 2^m m$; that is, the binary expansions of $0, 1, \\dots, 2^m - 1$ contain $2^m m / 2$ ones in total.","title":"Sum of binary digit sums below a power of two"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/sum-hamming-weight-range-eq-trollope","kind":"node","lean":"MetaMathlibExt.sum_hammingWeight_range_eq_trollope","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#sum-hamming-weight-range-eq-trollope","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For a natural number $m$, write $s_1(m)$ for the sum of the binary digits of $m$ and $S_1(m) = \\sum_{i=0}^{m-1} s_1(i)$. Let $k$ and $r$ be natural numbers with $r < 2^k$, and put $n = 2^k + r$ and $x = r / 2^k$. Write $\\tau(x)$ for the Takagi value of the dyadic presentation $r / 2^k$. Then","title":"Trollope's formula via the Takagi function"},{"id":"Analysis/real-functions/continuity-and-functional-equations/takagi-function/sum-hamming-weight-range-eq-takagi-dc-dn","kind":"node","lean":"MetaMathlibExt.sum_hammingWeight_range_eq_takagi_dcDNodes","parent":17979,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/takagi-function/index.html#sum-hamming-weight-range-eq-takagi-dc-dn","scope":"Analysis/real-functions/continuity-and-functional-equations/takagi-function","status":"can_prove","summary":"For a natural number $m$, write $s_1(m)$ for the sum of the binary digits of $m$ and $S_1(m) = \\sum_{i=0}^{m-1} s_1(i)$. Let $k$ and $r$ be natural numbers with $r < 2^k$, and put $n = 2^k + r$ and $x = r / 2^k$. Write $\\tau(x)$ for the Takagi value of the dyadic presentation $r / 2^k$, and $\\delta(n)$ for the number of $D$-nodes of the divide-and-conquer tree on $n$ leaves, given by $\\delta(1) = 0$,\u2026","title":"Cumulative binary digit sums via the Takagi function"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions.json","id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","title":"Trapezoidal cutoff functions"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff","kind":"node","lean":"Real.intervalCutoff","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For real numbers $a$, $b$ and $\\delta$, the interval cutoff is the function $\\chi_{a,b,\\delta} \\colon \\mathbb{R} \\to \\mathbb{R}$ defined by","title":"Interval cutoff"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-nonneg","kind":"node","lean":"Real.intervalCutoff_nonneg","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-nonneg","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For all real numbers $a$, $b$, $\\delta$ and $x$, the interval cutoff $\\chi_{a,b,\\delta}$ with endpoints $a$, $b$ and width $\\delta$ satisfies $0 \\le \\chi_{a,b,\\delta}(x)$.","title":"The interval cutoff is nonnegative"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-le-one","kind":"node","lean":"Real.intervalCutoff_le_one","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-le-one","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For all real numbers $a$, $b$, $\\delta$ and $x$, the interval cutoff $\\chi_{a,b,\\delta}$ with endpoints $a$, $b$ and width $\\delta$ satisfies $\\chi_{a,b,\\delta}(x) \\le 1$.","title":"The interval cutoff is at most one"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-continuous","kind":"node","lean":"Real.intervalCutoff_continuous","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-continuous","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For all real numbers $a$, $b$ and $\\delta$, including $\\delta = 0$, the interval cutoff $\\chi_{a,b,\\delta} \\colon \\mathbb{R} \\to \\mathbb{R}$ with endpoints $a$, $b$ and width $\\delta$ is continuous.","title":"Continuity of the interval cutoff"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-eq-zero-left","kind":"node","lean":"Real.intervalCutoff_eq_zero_left","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-eq-zero-left","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For all real numbers $a$, $b$ and $\\delta$, the interval cutoff $\\chi_{a,b,\\delta}$ with endpoints $a$, $b$ and width $\\delta$ satisfies $\\chi_{a,b,\\delta}(a) = 0$.","title":"The interval cutoff vanishes at the left endpoint"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-eq-zero-right","kind":"node","lean":"Real.intervalCutoff_eq_zero_right","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-eq-zero-right","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For all real numbers $a$, $b$ and $\\delta$, the interval cutoff $\\chi_{a,b,\\delta}$ with endpoints $a$, $b$ and width $\\delta$ satisfies $\\chi_{a,b,\\delta}(b) = 0$.","title":"The interval cutoff vanishes at the right endpoint"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-zero-width","kind":"node","lean":"Real.intervalCutoff_zero_width","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-zero-width","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For all real numbers $a$, $b$ and $x$, the interval cutoff $\\chi_{a,b,0}$ with endpoints $a$, $b$ and width $0$ satisfies $\\chi_{a,b,0}(x) = 0$.","title":"The interval cutoff of zero width vanishes"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-eq-zero-of-le","kind":"node","lean":"Real.intervalCutoff_eq_zero_of_le","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-eq-zero-of-le","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For real numbers $a$, $b$, $\\delta$ and $x$ with $\\delta > 0$ and $x \\le a$, the interval cutoff $\\chi_{a,b,\\delta}$ with endpoints $a$, $b$ and width $\\delta$ satisfies $\\chi_{a,b,\\delta}(x) = 0$.","title":"The interval cutoff vanishes left of the interval"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-eq-zero-of-ge","kind":"node","lean":"Real.intervalCutoff_eq_zero_of_ge","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-eq-zero-of-ge","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For real numbers $a$, $b$, $\\delta$ and $x$ with $\\delta > 0$ and $b \\le x$, the interval cutoff $\\chi_{a,b,\\delta}$ with endpoints $a$, $b$ and width $\\delta$ satisfies $\\chi_{a,b,\\delta}(x) = 0$.","title":"The interval cutoff vanishes right of the interval"},{"id":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/interval-cutoff-eq-one","kind":"node","lean":"Real.intervalCutoff_eq_one","parent":18030,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions/index.html#interval-cutoff-eq-one","scope":"Analysis/real-functions/continuity-and-functional-equations/trapezoidal-cutoff-functions","status":"fully_proved","summary":"For real numbers $a$, $b$, $\\delta$ and $x$ with $\\delta > 0$ and $a + \\delta \\le x \\le b - \\delta$, the interval cutoff $\\chi_{a,b,\\delta}$ with endpoints $a$, $b$ and width $\\delta$ satisfies $\\chi_{a,b,\\delta}(x) = 1$.","title":"The interval cutoff equals one on its plateau"},{"data_url":"scopes/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function.json","id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","kind":"scope","navigation":"module","parent":17931,"result":false,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","title":"Weierstrass function"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/exists-cos-sub-cos-ge-one","kind":"node","lean":"Real.exists_cos_sub_cos_ge_one","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#exists-cos-sub-cos-ge-one","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"For every real number $\\theta$ there is a real number $\\varphi$ with $\\pi < \\varphi < 3\\pi$ such that $|\\cos\\theta - \\cos(\\theta + \\varphi)| \\ge 1$.","title":"Cosine oscillation (phase-shift form)"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/exists-cos-oscillation","kind":"node","lean":"Real.exists_cos_oscillation","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#exists-cos-oscillation","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"For every real number $c$ and every positive integer $K$, there is a real number $y$ with $c + \\pi / K < y < c + 3\\pi / K$ such that $|\\cos(Kc) - \\cos(Ky)| \\ge 1$.","title":"Cosine oscillation (scaled form)"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/weierstrass-fun","kind":"node","lean":"Real.weierstrassFun","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#weierstrass-fun","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"The Weierstrass function is the function $W : \\mathbb{R} \\to \\mathbb{R}$ given by $W(x) = \\sum_{k \\ge 0} \\cos(160^k x) / 4^k$, the sum running over all natural numbers $k$.","title":"The Weierstrass function"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/norm-weierstrass-fun-term-le","kind":"node","lean":"Real.norm_weierstrassFun_term_le","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#norm-weierstrass-fun-term-le","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"For every natural number $k$ and every real number $x$, $|\\cos(160^k x) / 4^k| \\le (1/4)^k$.","title":"Termwise bound for the Weierstrass series"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/summable-weierstrass-fun-terms","kind":"node","lean":"Real.summable_weierstrassFun_terms","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#summable-weierstrass-fun-terms","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"For every real number $x$, the series $\\sum_{k \\ge 0} \\cos(160^k x) / 4^k$ defining the Weierstrass function is summable, by comparison with the geometric series $\\sum_{k \\ge 0} (1/4)^k$.","title":"Summability of the Weierstrass series"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/summable-abs-weierstrass-fun-terms","kind":"node","lean":"Real.summable_abs_weierstrassFun_terms","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#summable-abs-weierstrass-fun-terms","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"For every real number $x$, the series $\\sum_{k \\ge 0} \\cos(160^k x) / 4^k$ defining the Weierstrass function converges absolutely, that is, $\\sum_{k \\ge 0} |\\cos(160^k x) / 4^k|$ is summable.","title":"Absolute convergence of the Weierstrass series"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/continuous-weierstrass-fun","kind":"node","lean":"Real.continuous_weierstrassFun","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#continuous-weierstrass-fun","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"The Weierstrass function $W(x) = \\sum_{k \\ge 0} \\cos(160^k x) / 4^k$ is continuous on $\\mathbb{R}$.","title":"Continuity of the Weierstrass function"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/weierstrass-fun-abs-le","kind":"node","lean":"Real.weierstrassFun_abs_le","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#weierstrass-fun-abs-le","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"The Weierstrass function $W(x) = \\sum_{k \\ge 0} \\cos(160^k x) / 4^k$ satisfies $|W(x)| \\le 4/3$ for every real number $x$.","title":"Bound on the Weierstrass function"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/reverse-triangle-three","kind":"node","lean":"Real.reverse_triangle_three","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#reverse-triangle-three","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"For all real numbers `a`, `b`, `c`, the absolute value of their sum is at least `|a| - |b| - |c|`.","title":"Reverse triangle inequality, three terms"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/weierstrass-fun-slope-lower-bound","kind":"node","lean":"Real.weierstrassFun_slope_lower_bound","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#weierstrass-fun-slope-lower-bound","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"For every real `c` and every natural number `n`, there exists a point `y \u2260 c` with `|y - c| < 3\u03c0 / 160\u207f` such that the difference quotient of the Weierstrass function between `y` and `c` has absolute value at least `40\u207f / (117 \u03c0)`. Since `40\u207f / (117 \u03c0) \u2192 \u221e`, this witnesses that the function cannot have a finite derivative at `c`.","title":"Slope blow-up lemma"},{"id":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/weierstrass-fun-nowhere-differentiable","kind":"node","lean":"Real.weierstrassFun_nowhere_differentiable","parent":18041,"result":true,"route":"../roadmap/Analysis/real-functions/continuity-and-functional-equations/weierstrass-function/index.html#weierstrass-fun-nowhere-differentiable","scope":"Analysis/real-functions/continuity-and-functional-equations/weierstrass-function","status":"fully_proved","summary":"The Weierstrass function $W(x) = \\sum_{k=0}^{\\infty} \\cos(160^k x) / 4^k$ is differentiable at no point of $\\mathbb{R}$: for every real number $c$, $W$ is not differentiable at $c$.","title":"The Weierstrass function is nowhere differentiable"},{"data_url":"scopes/Analysis/sequences-and-series.json","id":"Analysis/sequences-and-series","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/sequences-and-series/index.html","scope":"Analysis/sequences-and-series","status":"can_prove","summary":"Convergence of numerical sequences and series and of series of functions, summability methods and Tauberian theorems, and closed-form evaluation of infinite series and products.","title":"Sequences, series and summability"},{"data_url":"scopes/Analysis/sequences-and-series/convergence.json","id":"Analysis/sequences-and-series/convergence","kind":"scope","parent":18053,"result":false,"route":"../roadmap/Analysis/sequences-and-series/convergence/index.html","scope":"Analysis/sequences-and-series/convergence","status":"can_prove","summary":"Convergence tests (root test, rearrangement theorem, Stolz\u2013Ces\u00e0ro), normal and uniform convergence of series of functions, and specific limits of sequences and partial sums.","title":"Convergence of sequences and series"},{"id":"Analysis/sequences-and-series/convergence/exists-tendsto-sum-inv-sq-smul-alternati","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.AlternatingPairWeightedSeriesConvergesWanted.exists_tendsto_sum_inv_sq_smul_alternating_pair","parent":18054,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/index.html#exists-tendsto-sum-inv-sq-smul-alternati","scope":"Analysis/sequences-and-series/convergence","status":"can_prove","summary":"Let $\\alpha$ and $\\beta$ be real numbers. For $k \\in \\mathbb{N}$ let $x_k = (\\alpha, \\beta) \\in \\mathbb{R} \\times \\mathbb{R}$ if $k$ is even and $x_k = (\\beta, \\alpha)$ if $k$ is odd. For $n \\in \\mathbb{N}$ let $s_n = \\sum_{k=1}^{n} k^{-2} x_k$ (so $s_0 = 0$). Then the sequence $(s_n)$ converges in $\\mathbb{R} \\times \\mathbb{R}$.","title":"Convergence of an alternating weighted series in the plane"},{"id":"Analysis/sequences-and-series/convergence/monotone-and-tendsto-uniformly-of-monoto","kind":"node","lean":"MathlibExt.Analysis.Calculus.MonotonePointwiseUniformWanted.monotone_and_tendstoUniformly_of_monotone_tendsto_continuous","parent":18054,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/index.html#monotone-and-tendsto-uniformly-of-monoto","scope":"Analysis/sequences-and-series/convergence","status":"can_prove","summary":"Monotone functions on a closed interval with a continuous pointwise limit converge uniformly. Let $a < b$ be real numbers, let $f_n : [a, b] \\to \\mathbb{R}$, $n \\in \\mathbb{N}$, be nondecreasing functions, and let $g : [a, b] \\to \\mathbb{R}$ be continuous with $f_n(x) \\to g(x)$ for every $x \\in [a, b]$. Then $g$ is nondecreasing and $f_n \\to g$ uniformly on $[a, b]$.","title":"Monotone functions on a closed interval with a continuous pointwise limit converge uniformly"},{"id":"Analysis/sequences-and-series/convergence/coeff-eq-of-tsum-eq-on-inv-nat-add-two","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.PowerSeriesCoeffEqOfAgreeOnReciprocalsWanted.coeff_eq_of_tsum_eq_on_inv_nat_add_two","parent":18054,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power series agreeing at "],["math","\\(1/2, 1/3, 1/4, \\dots\\)"],["text"," have equal coefficients"]],"source":"Power series agreeing at $1/2, 1/3, 1/4, \\dots$ have equal coefficients"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/index.html#coeff-eq-of-tsum-eq-on-inv-nat-add-two","scope":"Analysis/sequences-and-series/convergence","status":"can_prove","summary":"Let $(a_k)$ and $(b_k)$ be real sequences such that the series $\\sum_{k=0}^{\\infty} a_k x^k$ and $\\sum_{k=0}^{\\infty} b_k x^k$ converge absolutely for every real $x$ with $|x| < 1$. If $\\sum_{k=0}^{\\infty} a_k x^k = \\sum_{k=0}^{\\infty} b_k x^k$ for $x = 1/2, 1/3, 1/4, \\dots$, then $a_k = b_k$ for all $k \\ge 0$.","title":"Power series agreeing at $1/2, 1/3, 1/4, \\dots$ have equal coefficients"},{"id":"Analysis/sequences-and-series/convergence/stolz-cesaro","kind":"node","lean":"MetaMathlibExt.stolz_cesaro","parent":18054,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/index.html#stolz-cesaro","scope":"Analysis/sequences-and-series/convergence","status":"fully_proved","summary":"Let $(a_n)_{n \\ge 0}$ and $(b_n)_{n \\ge 0}$ be sequences of real numbers such that $(b_n)$ is strictly increasing and $b_n \\to +\\infty$ as $n \\to \\infty$, and let $L$ be a real number. If $\\frac{a_{n+1} - a_n}{b_{n+1} - b_n} \\to L$ as $n \\to \\infty$, then $\\frac{a_n}{b_n} \\to L$ as $n \\to \\infty$.","title":"Stolz\u2013Ces\u00e0ro theorem"},{"id":"Analysis/sequences-and-series/convergence/summable-neg-pow-mul-pow-div-prod-add-mu","kind":"node","lean":"Complex.summable_neg_pow_mul_pow_div_prod_add_mul","parent":18054,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/index.html#summable-neg-pow-mul-pow-div-prod-add-mu","scope":"Analysis/sequences-and-series/convergence","status":"fully_proved","summary":"Let $b$, $c$ and $x$ be complex numbers such that $c + kb \\neq 0$ for every integer $k \\ge 1$. Then the series $\\sum_{j=0}^{\\infty} \\frac{(-b)^j x^{j+1}}{\\prod_{k=1}^{j+1} (c + kb)}$ is summable.","title":"Summability of a rising-product series"},{"id":"Analysis/sequences-and-series/convergence/tendsto-uniformly-on-sum-one-sub-cos-div","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.OneSubCosDivSeriesUniformWanted.tendstoUniformlyOn_sum_one_sub_cos_div","parent":18054,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniform convergence of "],["math","\\(\\sum (1 - \\cos(x/n))\\)"],["text"," on compact intervals"]],"source":"Uniform convergence of $\\sum (1 - \\cos(x/n))$ on compact intervals"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/index.html#tendsto-uniformly-on-sum-one-sub-cos-div","scope":"Analysis/sequences-and-series/convergence","status":"can_prove","summary":"For all real numbers $a < b$ there is a function $g : \\mathbb{R} \\to \\mathbb{R}$ such that the partial sums $\\sum_{n=1}^{N} \\bigl(1 - \\cos(x/n)\\bigr)$ converge to $g(x)$ uniformly for $x \\in [a, b]$ as $N \\to \\infty$.","title":"Uniform convergence of $\\sum (1 - \\cos(x/n))$ on compact intervals"},{"id":"Analysis/sequences-and-series/convergence/cont-fract-tendsto-of-pringsheim","kind":"node","lean":"MetaMathlibExt.contFract_tendsto_of_pringsheim","parent":18054,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/index.html#cont-fract-tendsto-of-pringsheim","scope":"Analysis/sequences-and-series/convergence","status":"can_prove","summary":"Let $a_0, a_1, a_2, \\dots$ and $b_0, b_1, b_2, \\dots$ be complex numbers with $1 + |a_k| \\le |b_k|$ for every $k$. For $n \\ge 0$ let $f_n$ be the $n$-th approximant of the continued fraction $a_0 / (b_0 + a_1 / (b_1 + \\cdots))$, that is $f_0 = 0$ and $f_n = a_0 / (b_0 + a_1 / (b_1 + \\cdots + a_{n-1} / b_{n-1}))$ for $n \\ge 1$. Then there is a complex number $v$ with $|v| \\le 1$ such that $f_n \\to v$ as\u2026","title":"\u015aleszy\u0144ski\u2013Pringsheim convergence criterion"},{"data_url":"scopes/Analysis/sequences-and-series/convergence/alternating-harmonic-series.json","id":"Analysis/sequences-and-series/convergence/alternating-harmonic-series","kind":"scope","navigation":"module","parent":18054,"result":false,"route":"../roadmap/Analysis/sequences-and-series/convergence/alternating-harmonic-series/index.html","scope":"Analysis/sequences-and-series/convergence/alternating-harmonic-series","status":"fully_proved","title":"Alternating harmonic series"},{"id":"Analysis/sequences-and-series/convergence/alternating-harmonic-series/tendsto-sum-alternating-harmonic","kind":"node","lean":"Real.tendsto_sum_alternating_harmonic","parent":18062,"reading_title":{"format":"inline-parts-v1","parts":[["text","The alternating harmonic series sums to "],["math","\\(\\log 2\\)"]],"source":"The alternating harmonic series sums to $\\log 2$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/alternating-harmonic-series/index.html#tendsto-sum-alternating-harmonic","scope":"Analysis/sequences-and-series/convergence/alternating-harmonic-series","status":"fully_proved","summary":"The partial sums $\\sum_{i=0}^{k-1} (-1)^i / (i + 1)$ of real numbers converge to $\\log 2$ as $k \\to \\infty$.","title":"The alternating harmonic series sums to $\\log 2$"},{"id":"Analysis/sequences-and-series/convergence/alternating-harmonic-series/not-summable-abs-alternating-harmonic","kind":"node","lean":"Real.not_summable_abs_alternating_harmonic","parent":18062,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/alternating-harmonic-series/index.html#not-summable-abs-alternating-harmonic","scope":"Analysis/sequences-and-series/convergence/alternating-harmonic-series","status":"fully_proved","summary":"The series of real numbers $\\sum_{n \\ge 0} \\left|(-1)^n / (n + 1)\\right|$ is not summable.","title":"The alternating harmonic series is not absolutely summable"},{"data_url":"scopes/Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences.json","id":"Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences","kind":"scope","navigation":"module","parent":18054,"result":false,"route":"../roadmap/Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences/index.html","scope":"Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences","status":"can_prove","title":"Averaged Cauchy products of convergent sequences"},{"id":"Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences/scottish-book-problem8-claim","kind":"node","lean":"MathlibExt.Analysis.ScottishBook8Wanted.ScottishBookProblem8Claim","parent":18065,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences/index.html#scottish-book-problem8-claim","scope":"Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences","status":"fully_proved","summary":"This is the statement, asked about in form (b) of Mazur's Problem 8 in the Scottish Book typescript, that every convergent sequence $(z_n)_{n \\ge 0}$ of real numbers can be written as $z_n = \\frac{1}{n+1} \\sum_{k=0}^{n} x_k y_{n-k}$ for all $n \\ge 0$, where $(x_n)_{n \\ge 0}$ and $(y_n)_{n \\ge 0}$ are convergent sequences of real numbers.","title":"The claim of Scottish Book Problem 8"},{"id":"Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences/mazur8-cauchy-product-refuted","kind":"node","lean":"MathlibExt.Analysis.ScottishBook8Wanted.mazur8CauchyProduct_refuted","parent":18065,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences/index.html#mazur8-cauchy-product-refuted","scope":"Analysis/sequences-and-series/convergence/averaged-cauchy-products-of-convergent-sequences","status":"can_prove","summary":"There is a convergent sequence $(z_n)_{n \\ge 0}$ of real numbers that cannot be written as $z_n = \\frac{1}{n+1} \\sum_{k=0}^{n} x_k y_{n-k}$ for all $n \\ge 0$ with convergent sequences $(x_n)_{n \\ge 0}$ and $(y_n)_{n \\ge 0}$ of real numbers. That is, the claim of Scottish Book Problem 8 is false.","title":"Mazur's Problem 8 has a negative answer"},{"data_url":"scopes/Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums.json","id":"Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums","kind":"scope","navigation":"module","parent":18054,"result":false,"route":"../roadmap/Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums/index.html","scope":"Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums","status":"can_prove","title":"Limits of specific sequences and partial sums"},{"id":"Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums/x-seq","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.ReciprocalSumAddInvNullWanted.xSeq","parent":18068,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial sums of "],["math","\\(a_k + 1/a_k\\)"]],"source":"Partial sums of $a_k + 1/a_k$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums/index.html#x-seq","scope":"Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums","status":"fully_proved","summary":"For a real sequence $(a_k)_{k \\in \\mathbb{N}}$, the sequence $(x_n)_{n \\in \\mathbb{N}}$ is given by $x_n = \\sum_{k=0}^{n} (a_k + 1/a_k)$ for $n \\in \\mathbb{N}$, with the convention $1/0 = 0$.","title":"Partial sums of $a_k + 1/a_k$"},{"id":"Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums/tendsto-one-div-sum-add-inv-zero","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.ReciprocalSumAddInvNullWanted.tendsto_one_div_sum_add_inv_zero","parent":18068,"reading_title":{"format":"inline-parts-v1","parts":[["text","The reciprocals of "],["math","\\(\\sum_{k=0}^{n} (a_k + 1/a_k)\\)"],["text"," tend to zero"]],"source":"The reciprocals of $\\sum_{k=0}^{n} (a_k + 1/a_k)$ tend to zero"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums/index.html#tendsto-one-div-sum-add-inv-zero","scope":"Analysis/sequences-and-series/convergence/limits-of-specific-sequences-and-partial-sums","status":"can_prove","summary":"Let $(a_n)_{n \\in \\mathbb{N}}$ be a sequence of positive real numbers, and for $n \\in \\mathbb{N}$ let $x_n = \\sum_{k=0}^{n} (a_k + 1/a_k)$. Then $1/x_n \\to 0$ as $n \\to \\infty$.","title":"The reciprocals of $\\sum_{k=0}^{n} (a_k + 1/a_k)$ tend to zero"},{"data_url":"scopes/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions.json","id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","kind":"scope","navigation":"module","parent":18054,"result":false,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","title":"Normal convergence of series of functions"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/sup-norm-on","kind":"node","lean":"supNormOn","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#sup-norm-on","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $X$ be a type and $E$ a seminormed additive commutative group. For a function $g : X \\to E$ and a set $S \\subseteq X$, the supremum norm of $g$ on $S$ is $\\sup_{x \\in S} \\|g(x)\\|$, taken in the extended nonnegative reals $[0, \\infty]$; in particular it is $0$ when $S$ is empty and $\\infty$ when $\\|g\\|$ is unbounded on $S$.","title":"Extended supremum norm on a set"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/converges-normally-on","kind":"node","lean":"ConvergesNormallyOn","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#converges-normally-on","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $X$ be a type and $E$ a seminormed additive commutative group. A series of functions $f_n \\colon X \\to E$ ($n \\in \\mathbb{N}$) converges normally on a set $S \\subseteq X$ when the sum $\\sum_{n} \\sup_{x \\in S} \\|f_n(x)\\|$ is finite, where each supremum norm and the sum are taken in the extended nonnegative reals $[0, \\infty]$.","title":"Normal convergence of a series of functions on a set"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/sup-norm-on-empty","kind":"node","lean":"supNormOn_empty","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#sup-norm-on-empty","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"For every function $g \\colon X \\to E$ into a seminormed additive commutative group $E$, the supremum norm of $g$ on the empty set, valued in $[0, \\infty]$, is $0$.","title":"Supremum norm over the empty set"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/sup-norm-on-zero","kind":"node","lean":"supNormOn_zero","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#sup-norm-on-zero","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"For every set $S \\subseteq X$, the supremum norm on $S$ of the zero function from $X$ to a seminormed additive commutative group $E$ is $0$.","title":"Supremum norm of the zero function"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/sup-norm-on-mono","kind":"node","lean":"supNormOn_mono","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#sup-norm-on-mono","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"If $T \\subseteq S \\subseteq X$, then for every function $g \\colon X \\to E$ into a seminormed additive commutative group $E$, the supremum norm of $g$ on $T$ is at most the supremum norm of $g$ on $S$.","title":"Monotonicity of the supremum norm in the domain"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/converges-normally-on-empty","kind":"node","lean":"convergesNormallyOn_empty","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#converges-normally-on-empty","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Every series of functions $f_n \\colon X \\to E$ ($n \\in \\mathbb{N}$) with values in a seminormed additive commutative group $E$ converges normally on the empty set.","title":"Normal convergence on the empty set"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/converges-normally-on-zero","kind":"node","lean":"convergesNormallyOn_zero","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#converges-normally-on-zero","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"For every set $S \\subseteq X$ and every seminormed additive commutative group $E$, the series of functions $X \\to E$ whose terms are all zero converges normally on $S$.","title":"Normal convergence of the zero series"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/mono","kind":"node","lean":"ConvergesNormallyOn.mono","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#mono","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $f_n \\colon X \\to E$ ($n \\in \\mathbb{N}$) be functions into a seminormed additive commutative group $E$ and let $T \\subseteq S \\subseteq X$. If the series $\\sum_n f_n$ converges normally on $S$, then it converges normally on $T$.","title":"Normal convergence passes to subsets"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/converges-locally-normally-on","kind":"node","lean":"ConvergesLocallyNormallyOn","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#converges-locally-normally-on","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $X$ be a topological space and $E$ a seminormed additive commutative group. A series of functions $f_n \\colon X \\to E$ ($n \\in \\mathbb{N}$) converges locally normally on a set $S \\subseteq X$ when every $x \\in S$ has a neighbourhood $U$ such that the series converges normally on $U \\cap S$.","title":"Local normal convergence of a series of functions on a set"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/exists-nhds-inter","kind":"node","lean":"ConvergesLocallyNormallyOn.exists_nhds_inter","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#exists-nhds-inter","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $X$ be a topological space, $E$ a seminormed additive commutative group, $f_n \\colon X \\to E$ ($n \\in \\mathbb{N}$) functions and $S \\subseteq X$. If the series $\\sum_n f_n$ converges locally normally on $S$, then every $x \\in S$ has a neighbourhood $U$ such that the series converges normally on $U \\cap S$.","title":"Local normal convergence gives a neighbourhood of each point of the set"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/converges-locally-normally-on-empty","kind":"node","lean":"convergesLocallyNormallyOn_empty","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#converges-locally-normally-on-empty","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $X$ be a topological space. Every series of functions $f_n \\colon X \\to E$ ($n \\in \\mathbb{N}$) with values in a seminormed additive commutative group $E$ converges locally normally on the empty set.","title":"Local normal convergence on the empty set"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/converges-locally-normally-on-zero","kind":"node","lean":"convergesLocallyNormallyOn_zero","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#converges-locally-normally-on-zero","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $X$ be a topological space and $E$ a seminormed additive commutative group. For every set $S \\subseteq X$, the series of functions $X \\to E$ whose terms are all zero converges locally normally on $S$.","title":"Local normal convergence of the zero series"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/mono-2","kind":"node","lean":"ConvergesLocallyNormallyOn.mono","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#mono-2","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $X$ be a topological space, let $f_n \\colon X \\to E$ ($n \\in \\mathbb{N}$) be functions into a seminormed additive commutative group $E$ and let $T \\subseteq S \\subseteq X$. If the series $\\sum_n f_n$ converges locally normally on $S$, then it converges locally normally on $T$.","title":"Local normal convergence passes to subsets"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/summable-to-real-sup-norm-on","kind":"node","lean":"ConvergesNormallyOn.summable_toReal_supNormOn","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#summable-to-real-sup-norm-on","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $E$ be a seminormed additive commutative group, $S$ a subset of a type $X$ and $f_n \\colon X \\to E$ for $n \\in \\mathbb{N}$. If the series $\\sum_n f_n$ converges normally on $S$, that is $\\sum_n \\sup_{x \\in S} \\|f_n(x)\\| < \\infty$ with the suprema taken in $[0, \\infty]$, then the sequence of sup norms $\\left(\\sup_{x \\in S} \\|f_n(x)\\|\\right)_n$, viewed as real numbers, is summable.","title":"Normal convergence gives summable real sup norms"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/summable-norm","kind":"node","lean":"ConvergesNormallyOn.summable_norm","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#summable-norm","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $E$ be a seminormed additive commutative group, $S$ a subset of a type $X$ and $f_n \\colon X \\to E$ for $n \\in \\mathbb{N}$. If the series $\\sum_n f_n$ converges normally on $S$, that is $\\sum_n \\sup_{x \\in S} \\|f_n(x)\\| < \\infty$ with the suprema taken in $[0, \\infty]$, then for every $x \\in S$ the series $\\sum_n \\|f_n(x)\\|$ is summable.","title":"Normal convergence implies pointwise absolute convergence"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/summable-norm-2","kind":"node","lean":"ConvergesLocallyNormallyOn.summable_norm","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#summable-norm-2","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $E$ be a seminormed additive commutative group, $S$ a subset of a topological space $X$ and $f_n \\colon X \\to E$ for $n \\in \\mathbb{N}$. If the series $\\sum_n f_n$ converges locally normally on $S$, that is every $x \\in S$ has a neighbourhood $U$ with $\\sum_n \\sup_{y \\in U \\cap S} \\|f_n(y)\\| < \\infty$ (suprema taken in $[0, \\infty]$), then for every $x \\in S$ the series $\\sum_n \\|f_n(x)\\|$ is summable.","title":"Local normal convergence implies pointwise absolute convergence"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/tendsto-uniformly-on-tsum-nat","kind":"node","lean":"ConvergesNormallyOn.tendstoUniformlyOn_tsum_nat","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#tendsto-uniformly-on-tsum-nat","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $E$ be a complete normed additive commutative group, $S$ a subset of a type $X$ and $f_n \\colon X \\to E$ for $n \\in \\mathbb{N}$. If the series $\\sum_n f_n$ converges normally on $S$, that is $\\sum_n \\sup_{x \\in S} \\|f_n(x)\\| < \\infty$ with the suprema taken in $[0, \\infty]$, then the partial sums $x \\mapsto \\sum_{n < N} f_n(x)$ converge uniformly on $S$, as $N \\to \\infty$, to the pointwise sum\u2026","title":"Weierstrass M-test: normal convergence implies uniform convergence"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/deriv","kind":"node","lean":"ConvergesLocallyNormallyOn.deriv","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#deriv","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $E$ be a normed space over $\\mathbb{C}$, $S \\subseteq \\mathbb{C}$ an open set and $f_n \\colon \\mathbb{C} \\to E$ for $n \\in \\mathbb{N}$ functions that are complex differentiable on $S$. If the series $\\sum_n f_n$ converges locally normally on $S$, that is every $z \\in S$ has a neighbourhood $U$ with $\\sum_n \\sup_{w \\in U \\cap S} \\|f_n(w)\\| < \\infty$ (suprema taken in $[0, \\infty]$), then the series of\u2026","title":"Derivative series of a locally normally convergent holomorphic series"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/tendsto-locally-uniformly-on-tsum-nat","kind":"node","lean":"ConvergesLocallyNormallyOn.tendstoLocallyUniformlyOn_tsum_nat","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#tendsto-locally-uniformly-on-tsum-nat","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $E$ be a complete normed additive commutative group, $S$ a subset of a topological space $X$ and $f_n \\colon X \\to E$ for $n \\in \\mathbb{N}$. If the series $\\sum_n f_n$ converges locally normally on $S$, that is every $x \\in S$ has a neighbourhood $U$ with $\\sum_n \\sup_{y \\in U \\cap S} \\|f_n(y)\\| < \\infty$ (suprema taken in $[0, \\infty]$), then the partial sums $x \\mapsto \\sum_{n < N} f_n(x)$ converge locally\u2026","title":"Local normal convergence implies locally uniform convergence"},{"id":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/has-deriv-at-tsum","kind":"node","lean":"ConvergesLocallyNormallyOn.hasDerivAt_tsum","parent":18071,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions/index.html#has-deriv-at-tsum","scope":"Analysis/sequences-and-series/convergence/normal-convergence-of-series-of-functions","status":"fully_proved","summary":"Let $E$ be a complete normed space over $\\mathbb{C}$, $S \\subseteq \\mathbb{C}$ an open set and $f_n \\colon \\mathbb{C} \\to E$ for $n \\in \\mathbb{N}$ functions that are complex differentiable on $S$. If the series $\\sum_n f_n$ converges locally normally on $S$, that is every $z \\in S$ has a neighbourhood $U$ with $\\sum_n \\sup_{w \\in U \\cap S} \\|f_n(w)\\| < \\infty$ (suprema taken in $[0, \\infty]$), then for every\u2026","title":"Termwise differentiation of locally normally convergent holomorphic series"},{"data_url":"scopes/Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem.json","id":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem","kind":"scope","navigation":"module","parent":18054,"result":false,"route":"../roadmap/Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/index.html","scope":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem","status":"fully_proved","title":"Riemann rearrangement theorem"},{"id":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/ordered-has-sum","kind":"node","lean":"MathlibExt.Analysis.Series.RiemannRearrangementWanted.OrderedHasSum","parent":18092,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/index.html#ordered-has-sum","scope":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem","status":"fully_proved","summary":"For a sequence $f : \\mathbb{N} \\to \\mathbb{R}$ and $a \\in \\mathbb{R}$, the series $\\sum_n f(n)$ converges in order to $a$ when its partial sums $\\sum_{n=0}^{N-1} f(n)$ tend to $a$ as $N \\to \\infty$.","title":"Convergence of a series in order"},{"id":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/ordered-has-sum-iff","kind":"node","lean":"MathlibExt.Analysis.Series.RiemannRearrangementWanted.orderedHasSum_iff","parent":18092,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/index.html#ordered-has-sum-iff","scope":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem","status":"fully_proved","summary":"For a sequence $f : \\mathbb{N} \\to \\mathbb{R}$ and $a \\in \\mathbb{R}$, the series $\\sum_n f(n)$ converges in order to $a$ if and only if its partial sums $\\sum_{n=0}^{N-1} f(n)$ tend to $a$ as $N \\to \\infty$.","title":"Convergence in order via partial sums"},{"id":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/ordered-has-sum-tendsto-zero","kind":"node","lean":"MathlibExt.Analysis.Series.RiemannRearrangementWanted.orderedHasSum_tendsto_zero","parent":18092,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/index.html#ordered-has-sum-tendsto-zero","scope":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem","status":"fully_proved","summary":"If $f : \\mathbb{N} \\to \\mathbb{R}$ and the series $\\sum_n f(n)$ converges in order to some $s \\in \\mathbb{R}$, then $f(n) \\to 0$ as $n \\to \\infty$.","title":"Terms of a convergent series tend to zero"},{"id":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/riemann-rearrangement","kind":"node","lean":"MathlibExt.Analysis.Series.RiemannRearrangementWanted.riemann_rearrangement","parent":18092,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem/index.html#riemann-rearrangement","scope":"Analysis/sequences-and-series/convergence/riemann-rearrangement-theorem","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{R}$ be such that the series $\\sum_n f(n)$ converges in order to some real number, while the series $\\sum_n |f(n)|$ is not summable. Then for every $a \\in \\mathbb{R}$ there is a permutation $\\sigma$ of $\\mathbb{N}$ such that the rearranged series $\\sum_n f(\\sigma(n))$ converges in order to $a$.","title":"Riemann rearrangement theorem"},{"data_url":"scopes/Analysis/sequences-and-series/convergence/root-test.json","id":"Analysis/sequences-and-series/convergence/root-test","kind":"scope","navigation":"module","parent":18054,"result":false,"route":"../roadmap/Analysis/sequences-and-series/convergence/root-test/index.html","scope":"Analysis/sequences-and-series/convergence/root-test","status":"fully_proved","title":"Root test"},{"id":"Analysis/sequences-and-series/convergence/root-test/summable-norm-of-root-test-tendsto-lt-on","kind":"node","lean":"summable_norm_of_root_test_tendsto_lt_one","parent":18097,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/root-test/index.html#summable-norm-of-root-test-tendsto-lt-on","scope":"Analysis/sequences-and-series/convergence/root-test","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\alpha$ be a sequence in a normed additive commutative group $\\alpha$. If $\\|f(n)\\|^{1/n}$ tends to a real limit $l$ with $l < 1$ as $n \\to \\infty$, then the series $\\sum_n \\|f(n)\\|$ converges.","title":"Cauchy's root test, absolute convergence case"},{"id":"Analysis/sequences-and-series/convergence/root-test/summable-of-root-test-tendsto-lt-one","kind":"node","lean":"summable_of_root_test_tendsto_lt_one","parent":18097,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/root-test/index.html#summable-of-root-test-tendsto-lt-one","scope":"Analysis/sequences-and-series/convergence/root-test","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\alpha$ be a sequence in a complete normed additive commutative group $\\alpha$. If $\\|f(n)\\|^{1/n}$ tends to a real limit $l$ with $l < 1$ as $n \\to \\infty$, then $f$ is summable, that is, the series $\\sum_n f(n)$ converges unconditionally.","title":"Cauchy's root test, convergence case"},{"id":"Analysis/sequences-and-series/convergence/root-test/not-summable-of-root-test-tendsto-gt-one","kind":"node","lean":"not_summable_of_root_test_tendsto_gt_one","parent":18097,"result":true,"route":"../roadmap/Analysis/sequences-and-series/convergence/root-test/index.html#not-summable-of-root-test-tendsto-gt-one","scope":"Analysis/sequences-and-series/convergence/root-test","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\alpha$ be a sequence in a seminormed additive commutative group $\\alpha$. If $\\|f(n)\\|^{1/n}$ tends to a real limit $l$ with $l > 1$ as $n \\to \\infty$, then $f$ is not summable, that is, the series $\\sum_n f(n)$ does not converge unconditionally.","title":"Cauchy's root test, divergence case"},{"data_url":"scopes/Analysis/sequences-and-series/summability.json","id":"Analysis/sequences-and-series/summability","kind":"scope","parent":18053,"result":false,"route":"../roadmap/Analysis/sequences-and-series/summability/index.html","scope":"Analysis/sequences-and-series/summability","status":"can_prove","summary":"Matrix summability methods (Silverman\u2013Toeplitz, Ces\u00e0ro means, permanent methods) and Tauberian theorems of Tauber, Hardy\u2013Littlewood, Wiener and Newman.","title":"Summability and Tauberian theorems"},{"id":"Analysis/sequences-and-series/summability/is-equivalent-id-of-monotone-on-of-tends","kind":"node","lean":"Asymptotics.isEquivalent_id_of_monotoneOn_of_tendsto_integral_sub_div_sq","parent":18101,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/index.html#is-equivalent-id-of-monotone-on-of-tends","scope":"Analysis/sequences-and-series/summability","status":"fully_proved","summary":"Let $f$ be a real function on $\\mathbb{R}$ that is nondecreasing on $[1, \\infty)$, and suppose that the integral $\\int_1^x (f(t) - t) / t^2 \\, dt$ converges to a finite limit as $x \\to \\infty$. Then $f(x) \\sim x$ as $x \\to \\infty$.","title":"Integral criterion for asymptotic equivalence to the identity"},{"id":"Analysis/sequences-and-series/summability/newman-tauberian","kind":"node","lean":"newman_tauberian","parent":18101,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/index.html#newman-tauberian","scope":"Analysis/sequences-and-series/summability","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{C}$ be almost everywhere strongly measurable on $(0, \\infty)$ and bounded on $[0, \\infty)$: for some real $B \\ge 0$, $|f(t)| \\le B$ for all $t \\ge 0$. Let $G : \\mathbb{C} \\to \\mathbb{C}$ be analytic at every point of the closed half-plane $\\{s : \\operatorname{Re} s \\ge 0\\}$, and suppose that for every $s$ with $\\operatorname{Re} s > 0$ the Laplace transform of $f$ converges at $s$\u2026","title":"Newman's Tauberian theorem"},{"id":"Analysis/sequences-and-series/summability/silverman-toeplitz","kind":"node","lean":"MetaMathlibExt.silverman_toeplitz","parent":18101,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/index.html#silverman-toeplitz","scope":"Analysis/sequences-and-series/summability","status":"fully_proved","summary":"Let $(a_{nk})_{n, k \\ge 0}$ be an infinite matrix of complex numbers. Suppose that there is a real $C$ such that for every $n$ the series $\\sum_k |a_{nk}|$ converges with sum at most $C$, that $a_{nk} \\to 0$ as $n \\to \\infty$ for every $k$, and that $\\sum_k a_{nk} \\to 1$ as $n \\to \\infty$. Then for every sequence $(x_k)$ of complex numbers converging to $L$, the series $\\sum_k a_{nk} x_k$ converges absolutely for\u2026","title":"Silverman\u2013Toeplitz theorem (sufficiency)"},{"id":"Analysis/sequences-and-series/summability/wiener-tauberian-l1","kind":"node","lean":"MathlibExt.Analysis.Fourier.WienerWanted.wiener_tauberian_L1","parent":18101,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wiener's Tauberian theorem on "],["math","\\(L^1(\\mathbb{R})\\)"]],"source":"Wiener's Tauberian theorem on $L^1(\\mathbb{R})$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/index.html#wiener-tauberian-l1","scope":"Analysis/sequences-and-series/summability","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{C}$ be Lebesgue integrable, and suppose that its Fourier transform $\\hat f(w) = \\int_{\\mathbb{R}} e^{-2\\pi i x w} f(x)\\,dx$ vanishes at no real $w$. Then for every Lebesgue integrable $g : \\mathbb{R} \\to \\mathbb{C}$ and every $\\varepsilon > 0$ there are a finite set $S \\subset \\mathbb{R}$ and coefficients $c_t \\in \\mathbb{C}$ for $t \\in S$ such that the function\u2026","title":"Wiener's Tauberian theorem on $L^1(\\mathbb{R})$"},{"data_url":"scopes/Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences.json","id":"Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences","kind":"scope","navigation":"module","parent":18101,"result":false,"route":"../roadmap/Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences/index.html","scope":"Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences","status":"can_prove","title":"Ces\u00e0ro means of random subsequences"},{"id":"Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences/fair-coin-measure","kind":"node","lean":"MathlibExt.Analysis.ScottishBook4Wanted.fairCoinMeasure","parent":18106,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences/index.html#fair-coin-measure","scope":"Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences","status":"fully_proved","summary":"The fair coin measure is the measure on the Booleans $\\{\\mathrm{true}, \\mathrm{false}\\}$ that gives mass $1/2$ to each of $\\mathrm{true}$ and $\\mathrm{false}$, that is, $\\frac{1}{2}\\delta_{\\mathrm{true}} + \\frac{1}{2}\\delta_{\\mathrm{false}}$.","title":"Fair coin measure"},{"id":"Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences/schreier-cesaro-subsequences","kind":"node","lean":"MathlibExt.Analysis.ScottishBook4Wanted.schreierCesaroSubsequences","parent":18106,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences/index.html#schreier-cesaro-subsequences","scope":"Analysis/sequences-and-series/summability/ces-ro-means-of-random-subsequences","status":"can_prove","summary":"Let $(a_n)_{n \\ge 0}$ be a bounded sequence of real numbers whose Ces\u00e0ro means $\\frac{1}{n} \\sum_{i < n} a_i$ converge to a real number $S$ as $n \\to \\infty$. Give the space of Boolean sequences $\\omega = (\\omega_n)_{n \\ge 0}$ the infinite product of fair coin measures. For such $\\omega$ let $n_0 < n_1 < \\cdots$ list, in increasing order, the indices $n$ with $\\omega_n = \\mathrm{true}$, with $n_k = 0$ whenever\u2026","title":"Schreier's theorem on Ces\u00e0ro means of random subsequences"},{"data_url":"scopes/Analysis/sequences-and-series/summability/permanent-summability-methods.json","id":"Analysis/sequences-and-series/summability/permanent-summability-methods","kind":"scope","navigation":"module","parent":18101,"result":false,"route":"../roadmap/Analysis/sequences-and-series/summability/permanent-summability-methods/index.html","scope":"Analysis/sequences-and-series/summability/permanent-summability-methods","status":"can_prove","title":"Permanent summability methods"},{"id":"Analysis/sequences-and-series/summability/permanent-summability-methods/has-density-one","kind":"node","lean":"MathlibExt.Analysis.Series.ScottishBook5BoundedWanted.HasDensityOne","parent":18109,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/permanent-summability-methods/index.html#has-density-one","scope":"Analysis/sequences-and-series/summability/permanent-summability-methods","status":"fully_proved","summary":"A sequence $s \\colon \\mathbb{N} \\to \\mathbb{N}$ has density one when it is strictly increasing and $s(n+1)/(n+1) \\to 1$ as $n \\to \\infty$.","title":"Index sequences of density one"},{"id":"Analysis/sequences-and-series/summability/permanent-summability-methods/asymptotically-convergent","kind":"node","lean":"MathlibExt.Analysis.Series.ScottishBook5BoundedWanted.AsymptoticallyConvergent","parent":18109,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/permanent-summability-methods/index.html#asymptotically-convergent","scope":"Analysis/sequences-and-series/summability/permanent-summability-methods","status":"fully_proved","summary":"A real sequence $(x_n)_{n \\in \\mathbb{N}}$ is asymptotically convergent to a real number $S$ when there is an index sequence $s$ of density one such that $x_{s(n)} \\to S$ as $n \\to \\infty$.","title":"Asymptotically convergent sequences"},{"id":"Analysis/sequences-and-series/summability/permanent-summability-methods/matrix-sums","kind":"node","lean":"MathlibExt.Analysis.Series.ScottishBook5BoundedWanted.MatrixSums","parent":18109,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/permanent-summability-methods/index.html#matrix-sums","scope":"Analysis/sequences-and-series/summability/permanent-summability-methods","status":"fully_proved","summary":"A real matrix $(a_{ik})_{i,k \\in \\mathbb{N}}$ sums a real sequence $(x_k)_{k \\in \\mathbb{N}}$ to a real number $S$ when for every row $i$ the series $\\sum_k a_{ik} x_k$ converges unconditionally (equivalently, absolutely) and the row sums $\\sum_k a_{ik} x_k$ tend to $S$ as $i \\to \\infty$.","title":"Summation by a matrix method"},{"id":"Analysis/sequences-and-series/summability/permanent-summability-methods/is-permanent","kind":"node","lean":"MathlibExt.Analysis.Series.ScottishBook5BoundedWanted.IsPermanent","parent":18109,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/permanent-summability-methods/index.html#is-permanent","scope":"Analysis/sequences-and-series/summability/permanent-summability-methods","status":"fully_proved","summary":"A real matrix $(a_{ik})_{i,k \\in \\mathbb{N}}$ is a permanent (regular, or Toeplitz) method when it sums every convergent real sequence to its limit: for every real sequence $x$ and real number $S$ with $x_n \\to S$, the matrix sums $x$ to $S$.","title":"Permanent matrix methods"},{"id":"Analysis/sequences-and-series/summability/permanent-summability-methods/is-bounded","kind":"node","lean":"MathlibExt.Analysis.Series.ScottishBook5BoundedWanted.IsBounded","parent":18109,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/permanent-summability-methods/index.html#is-bounded","scope":"Analysis/sequences-and-series/summability/permanent-summability-methods","status":"fully_proved","summary":"A real sequence $(x_n)_{n \\in \\mathbb{N}}$ is bounded when there is a real number $B$ with $|x_n| \\le B$ for all $n$.","title":"Bounded sequences"},{"id":"Analysis/sequences-and-series/summability/permanent-summability-methods/scottish-book-problem5-claim","kind":"node","lean":"MathlibExt.Analysis.Series.ScottishBook5BoundedWanted.ScottishBookProblem5Claim","parent":18109,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/permanent-summability-methods/index.html#scottish-book-problem5-claim","scope":"Analysis/sequences-and-series/summability/permanent-summability-methods","status":"fully_proved","summary":"There is a real matrix $(a_{ik})_{i,k \\in \\mathbb{N}}$ that is a permanent matrix method such that for every bounded real sequence $x$ and every real number $S$, the matrix sums $x$ to $S$ if and only if $x$ is asymptotically convergent to $S$.","title":"The bounded case of Scottish Book Problem 5"},{"id":"Analysis/sequences-and-series/summability/permanent-summability-methods/mazur5-bounded-refuted","kind":"node","lean":"MathlibExt.Analysis.Series.ScottishBook5BoundedWanted.mazur5BoundedRefuted","parent":18109,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/permanent-summability-methods/index.html#mazur5-bounded-refuted","scope":"Analysis/sequences-and-series/summability/permanent-summability-methods","status":"can_prove","summary":"There is no real matrix $(a_{ik})_{i,k \\in \\mathbb{N}}$ that is a permanent matrix method such that for every bounded real sequence $x$ and every real number $S$, the matrix sums $x$ to $S$ if and only if $x$ is asymptotically convergent to $S$.","title":"Negative answer to Scottish Book Problem 5 for bounded sequences"},{"data_url":"scopes/Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series.json","id":"Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series","kind":"scope","navigation":"module","parent":18101,"result":false,"route":"../roadmap/Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series/index.html","scope":"Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series","status":"fully_proved","title":"Tauberian theorems for power series"},{"id":"Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series/tauber-tendsto-partial-sums","kind":"node","lean":"MathlibExt.Analysis.Series.Tauberian.tauber_tendsto_partial_sums","parent":18117,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series/index.html#tauber-tendsto-partial-sums","scope":"Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series","status":"fully_proved","summary":"Tauber's theorem (1897). If the power series `\u2211 a\u2099 r\u207f` converges (in the Abel sense) to `A` as `r \u2192 1\u207b` and `n * a\u2099 \u2192 0`, then the partial sums of `a` converge to `A`.","title":"Tauber's theorem"},{"id":"Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series/hardy-littlewood-tendsto-partial-sums","kind":"node","lean":"MathlibExt.Analysis.Series.Tauberian.hardy_littlewood_tendsto_partial_sums","parent":18117,"result":true,"route":"../roadmap/Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series/index.html#hardy-littlewood-tendsto-partial-sums","scope":"Analysis/sequences-and-series/summability/tauberian-theorems-for-power-series","status":"fully_proved","summary":"Hardy\u2013Littlewood Tauberian theorem (1914). If `a\u2099 \u2265 0` and the power series `\u2211 a\u2099 r\u207f` converges for every `r \u2208 (0, 1)` with Abel limit `A` as `r \u2192 1\u207b`, then the partial sums of `a` converge to `A`.","title":"Hardy\u2013Littlewood Tauberian theorem"},{"data_url":"scopes/Analysis/sequences-and-series/series-for-constants.json","id":"Analysis/sequences-and-series/series-for-constants","kind":"scope","parent":18053,"result":false,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/index.html","scope":"Analysis/sequences-and-series/series-for-constants","status":"can_prove","summary":"Series and infinite products evaluating constants: \u03c0, Catalan's constant, Euler's constant, \u221a2, \u03b6(3), reciprocal central binomial sums.","title":"Series for \u03c0, \u03b6(3) and other constants"},{"id":"Analysis/sequences-and-series/series-for-constants/catalan-has-prod-sqrt-two","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.CatalanSqrtTwo.catalan_hasProd_sqrt_two","parent":18120,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan's product for "],["math","\\(\\sqrt{2}\\)"]],"source":"Catalan's product for $\\sqrt{2}$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/index.html#catalan-has-prod-sqrt-two","scope":"Analysis/sequences-and-series/series-for-constants","status":"fully_proved","summary":"The infinite product over all natural numbers $m \\ge 0$ of $\\frac{(4m+2)^2}{(4m+1)(4m+3)}$ converges unconditionally to $\\sqrt{2}$; that is, $\\frac{2 \\cdot 2}{1 \\cdot 3} \\cdot \\frac{6 \\cdot 6}{5 \\cdot 7} \\cdot \\frac{10 \\cdot 10}{9 \\cdot 11} \\cdots = \\sqrt{2}$.","title":"Catalan's product for $\\sqrt{2}$"},{"id":"Analysis/sequences-and-series/series-for-constants/exp-one-sub-sum-inv-factorial-pos-and-lt","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.ExpOneTaylorRemainderBoundWanted.exp_one_sub_sum_inv_factorial_pos_and_lt","parent":18120,"reading_title":{"format":"inline-parts-v1","parts":[["text","Error bound for the partial sums of the series for "],["math","\\(e\\)"]],"source":"Error bound for the partial sums of the series for $e$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/index.html#exp-one-sub-sum-inv-factorial-pos-and-lt","scope":"Analysis/sequences-and-series/series-for-constants","status":"can_prove","summary":"For every integer $n \\ge 1$,","title":"Error bound for the partial sums of the series for $e$"},{"id":"Analysis/sequences-and-series/series-for-constants/ramanujan-part1-ch8-entry16-eulerconstan","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry16Eulerconstantseries.ramanujan_part1_ch8_entry16_eulerconstantseries","parent":18120,"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/index.html#ramanujan-part1-ch8-entry16-eulerconstan","scope":"Analysis/sequences-and-series/series-for-constants","status":"fully_proved","summary":"For $k \\ge 0$ let $A_k = (3^k - 1)/2$. Then the series","title":"Ramanujan's series for Euler's constant"},{"id":"Analysis/sequences-and-series/series-for-constants/reciprocal-central-binomial-sums","kind":"node","lean":"MetaMathlibExt.reciprocal_central_binomial_sums","parent":18120,"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/index.html#reciprocal-central-binomial-sums","scope":"Analysis/sequences-and-series/series-for-constants","status":"fully_proved","summary":"The following three series converge absolutely, with the stated sums:","title":"Sums of reciprocals of central binomial coefficients"},{"data_url":"scopes/Analysis/sequences-and-series/series-for-constants/catalan-s-constant.json","id":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant","kind":"scope","navigation":"module","parent":18120,"result":false,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/catalan-s-constant/index.html","scope":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant","status":"fully_proved","title":"Catalan's constant"},{"id":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant/catalan-constant","kind":"node","lean":"Real.catalanConstant","parent":18125,"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/catalan-s-constant/index.html#catalan-constant","scope":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant","status":"fully_proved","summary":"Catalan's constant is the real number $G = \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n+1)^2}$.","title":"Catalan's constant"},{"id":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant/summable-catalan-constant-terms","kind":"node","lean":"Real.summable_catalanConstant_terms","parent":18125,"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/catalan-s-constant/index.html#summable-catalan-constant-terms","scope":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant","status":"fully_proved","summary":"The series $\\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n+1)^2}$ converges absolutely: its terms are bounded in absolute value by $\\frac{1}{(n+1)^2}$, and $\\sum_{n=0}^{\\infty} \\frac{1}{(n+1)^2}$ converges.","title":"The series for Catalan's constant converges absolutely"},{"id":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant/catalan-constant-eq-tsum","kind":"node","lean":"Real.catalanConstant_eq_tsum","parent":18125,"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/catalan-s-constant/index.html#catalan-constant-eq-tsum","scope":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant","status":"fully_proved","summary":"Catalan's constant $G$ equals $\\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n+1)^2}$.","title":"Catalan's constant as a series"},{"id":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant/has-sum-catalan-constant","kind":"node","lean":"Real.hasSum_catalanConstant","parent":18125,"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/catalan-s-constant/index.html#has-sum-catalan-constant","scope":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant","status":"fully_proved","summary":"The series $\\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n+1)^2}$ converges unconditionally, and its sum is Catalan's constant $G$.","title":"The series for Catalan's constant converges to it"},{"id":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant/catalan-constant-product-formulas","kind":"node","lean":"MetaMathlibExt.catalan_constant_product_formulas","parent":18125,"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-for-constants/catalan-s-constant/index.html#catalan-constant-product-formulas","scope":"Analysis/sequences-and-series/series-for-constants/catalan-s-constant","status":"fully_proved","summary":"Let $G$ be the sum of the series $\\sum_{n \\ge 0} (-1)^n / (2n+1)^2$ (Catalan's constant). Then, as $m \\to \\infty$,","title":"Infinite products for exponentials of Catalan's constant"},{"data_url":"scopes/Analysis/sequences-and-series/trigonometric-and-harmonic-series.json","id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series","kind":"scope","parent":18053,"result":false,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/index.html","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series","status":"can_prove","summary":"Closed forms for series in powers of sin, cos or tan, central binomial series in trigonometric arguments, partial-fraction series of trigonometric functions, and identities with harmonic or odd harmonic numbers, as in Ramanujan's notebooks Chapter 9.","title":"Trigonometric and harmonic-number series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/tsum-sin-sq-mul-div-sq-mul","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.SumSinSqDivSqWanted.tsum_sin_sq_mul_div_sq_mul","parent":18131,"reading_title":{"format":"inline-parts-v1","parts":[["text","A closed form for "],["math","\\(\\sum \\sin^2(n\\delta)/(n^2\\delta)\\)"]],"source":"A closed form for $\\sum \\sin^2(n\\delta)/(n^2\\delta)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/index.html#tsum-sin-sq-mul-div-sq-mul","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series","status":"can_prove","summary":"For every real $\\delta$ with $0 < \\delta < \\pi$,","title":"A closed form for $\\sum \\sin^2(n\\delta)/(n^2\\delta)$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/tsum-one-div-add-int-sq-eq-pi-sq-div-sin","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.TsumInvSqShiftEqPiSqDivSinSqWanted.tsum_one_div_add_int_sq_eq_pi_sq_div_sin_sq","parent":18131,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial fraction expansion of "],["math","\\(\\pi^2/\\sin^2(\\pi u)\\)"]],"source":"Partial fraction expansion of $\\pi^2/\\sin^2(\\pi u)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/index.html#tsum-one-div-add-int-sq-eq-pi-sq-div-sin","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series","status":"can_prove","summary":"For every real number $u$ that is not an integer,","title":"Partial fraction expansion of $\\pi^2/\\sin^2(\\pi u)$"},{"data_url":"scopes/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums.json","id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","kind":"scope","navigation":"module","parent":18131,"result":false,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","title":"Arcsine series and central binomial sums"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/arcsin-central-binomial-reciprocal-serie","kind":"node","lean":"MetaMathlibExt.arcsin_central_binomial_reciprocal_series","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#arcsin-central-binomial-reciprocal-serie","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"For every real number $x$ with $|x| < 1$, the series","title":"Lehmer's central binomial series for arcsin"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/central-binomial-coeff","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.centralBinomialCoeff","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#central-binomial-coeff","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"For a natural number $n$, the normalized central binomial coefficient is the real number $c_n = \\binom{2n}{n}/2^{2n} = \\binom{2n}{n}/4^n$.","title":"Normalized central binomial coefficient"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/central-binomial-coeff-nonneg","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.centralBinomialCoeff_nonneg","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#central-binomial-coeff-nonneg","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"For every natural number $n$, $0 \\le \\binom{2n}{n}/4^n$.","title":"Normalized central binomial coefficients are nonnegative"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/central-binomial-coeff-le-one","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.centralBinomialCoeff_le_one","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#central-binomial-coeff-le-one","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"For every natural number $n$, $\\binom{2n}{n}/4^n \\le 1$.","title":"Normalized central binomial coefficients are at most one"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/arcsin-series","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.arcsinSeries","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#arcsin-series","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"Let $c_n = \\binom{2n}{n}/4^n$. For real $t$, the arcsine Taylor series is","title":"Arcsine Taylor series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/arcsin-integral-series","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.arcsinIntegralSeries","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#arcsin-integral-series","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"Let $c_n = \\binom{2n}{n}/4^n$. For real $t$, the integrated arcsine series is","title":"Integrated arcsine series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/arcsin-series-eq-arcsin","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.arcsinSeries_eq_arcsin","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#arcsin-series-eq-arcsin","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"For every real $t$ with $|t| < 1$,","title":"Taylor series of the arcsine"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/arcsin-integral-series-term-eq","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.arcsinIntegralSeries_term_eq","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#arcsin-integral-series-term-eq","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"Let $c_n = \\binom{2n}{n}/4^n$ be the normalized central binomial coefficient. For every natural number $n$ and every real $t$,","title":"Terms of the integrated arcsine series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/norm-arcsin-integral-series-term-le","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.norm_arcsinIntegralSeries_term_le","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#norm-arcsin-integral-series-term-le","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"Write $c_k = \\binom{2k}{k}/4^k$ for the normalized central binomial coefficient. For every real number $t$ with $|t| \\le 1$ and every natural number $k$, $\\left|\\frac{c_k\\, t^{2k+1}}{(2k+1)^2}\\right| \\le \\frac{1}{(k+1)^2}$.","title":"Bound on the terms of the integrated arcsine series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/summable-arcsin-integral-series-term","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.summable_arcsinIntegralSeries_term","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#summable-arcsin-integral-series-term","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"Write $c_n = \\binom{2n}{n}/4^n$ for the normalized central binomial coefficient. For every real number $t$ with $|t| \\le 1$, the series $\\sum_{n=0}^{\\infty} \\frac{c_n\\, t^{2n+1}}{(2n+1)^2}$ converges absolutely.","title":"Summability of the integrated arcsine series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/arcsin-integral-series-zero","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.arcsinIntegralSeries_zero","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#arcsin-integral-series-zero","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"The integrated arcsine series $F(t) = \\sum_{n \\ge 0} \\binom{2n}{n} 4^{-n}\\, t^{2n+1}/(2n+1)^2$ satisfies $F(0) = 0$.","title":"The integrated arcsine series vanishes at the origin"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/continuous-on-arcsin-integral-series","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.continuousOn_arcsinIntegralSeries","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#continuous-on-arcsin-integral-series","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"The integrated arcsine series $F(t) = \\sum_{n \\ge 0} \\binom{2n}{n} 4^{-n}\\, t^{2n+1}/(2n+1)^2$ is continuous on the closed interval $[-1, 1]$.","title":"Continuity of the integrated arcsine series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/summable-arcsin-series-term","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.summable_arcsinSeries_term","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#summable-arcsin-series-term","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"Write $c_n = \\binom{2n}{n}/4^n$ for the normalized central binomial coefficient. For every real number $y$ with $|y| < 1$, the series $\\sum_{n=0}^{\\infty} \\frac{c_n\\, y^{2n+1}}{2n+1}$ converges absolutely.","title":"Summability of the arcsine series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/has-deriv-at-arcsin-integral-series","kind":"node","lean":"MathlibExt.Analysis.SpecialFunctions.ArcsinTaylorSeries.hasDerivAt_arcsinIntegralSeries","parent":18134,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums/index.html#has-deriv-at-arcsin-integral-series","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arcsine-series-and-central-binomial-sums","status":"fully_proved","summary":"Let $F(t) = \\sum_{n \\ge 0} \\binom{2n}{n} 4^{-n}\\, t^{2n+1}/(2n+1)^2$ be the integrated arcsine series. For every real $y$ with $|y| < 1$ and $y \\neq 0$, $F$ is differentiable at $y$ with $F'(y) = \\arcsin(y)/y$.","title":"Derivative of the integrated arcsine series"},{"data_url":"scopes/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series.json","id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","kind":"scope","navigation":"module","parent":18131,"result":false,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","title":"Arctangent sums and series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry3-general","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry3.ramanujan_part1_ch2_entry3_general","parent":18149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's arctangent identity for "],["math","\\(\\sum_{k=1}^{2n+1} \\arctan(1/(n+k))\\)"],["text",", "],["math","\\(n \\ge 0\\)"]],"source":"Ramanujan's arctangent identity for $\\sum_{k=1}^{2n+1} \\arctan(1/(n+k))$, $n \\ge 0$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry3-general","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"For every natural number $n$,","title":"Ramanujan's arctangent identity for $\\sum_{k=1}^{2n+1} \\arctan(1/(n+k))$, $n \\ge 0$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry3","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry3.ramanujan_part1_ch2_entry3","parent":18149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's arctangent identity for "],["math","\\(\\sum_{k=1}^{2n+1} \\arctan(1/(n+k))\\)"]],"source":"Ramanujan's arctangent identity for $\\sum_{k=1}^{2n+1} \\arctan(1/(n+k))$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry3","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"For every positive integer $n$,","title":"Ramanujan's arctangent identity for $\\sum_{k=1}^{2n+1} \\arctan(1/(n+k))$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry3-infinite-eval","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry3InfiniteEvaluation.ramanujan_part1_ch2_entry3_infinite_evaluation","parent":18149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's arctangent series for "],["math","\\(\\log 3 - \\pi/4\\)"]],"source":"Ramanujan's arctangent series for $\\log 3 - \\pi/4$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry3-infinite-eval","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"The series","title":"Ramanujan's arctangent series for $\\log 3 - \\pi/4$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7.ramanujan_part1_ch2_entry7","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"For every real number $n > 0$ and every natural number $r$,","title":"Ramanujan's finite arctangent sum (Entry 7)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-corollary","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Corollary.ramanujan_part1_ch2_entry7_corollary","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-corollary","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"For every real number $n > 0$, the series","title":"Ramanujan's infinite arctangent series (corollary to Entry 7)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-example1","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Example1.ramanujan_part1_ch2_entry7_example1","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-example1","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"Let $n > 0$ be a real number with $n \\ne (\\sqrt{5} - 1)/2$. Then the series","title":"Arctangent series over consecutive shifts (Entry 7, Example 1)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-example1-boun","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Example1Boundary.ramanujan_part1_ch2_entry7_example1_boundary","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-example1-boun","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"Let $c = (\\sqrt{5} - 1)/2$. The series","title":"Arctangent series at the exceptional point (Entry 7, Example 1)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-example2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Example2.ramanujan_part1_ch2_entry7_example2","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-example2","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"For every real number $n > 0$, the series","title":"Alternating arctangent series (Entry 7, Example 2)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-example3","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Example3.ramanujan_part1_ch2_entry7_example3","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-example3","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"For every real number $n > 0$, the series","title":"Arctangent series with halved squares (Entry 7, Example 3)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-example4","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Example4.ramanujan_part1_ch2_entry7_example4","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-example4","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"The series","title":"Series of arctangents of 2/k\u00b2 (Entry 7, Example 4)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-example5-alte","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Example5Alternating.ramanujan_part1_ch2_entry7_example5_alternating","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-example5-alte","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"The series","title":"Alternating series of arctangents of 2/k\u00b2 (Entry 7, Example 5)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-example5-firs","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Example5First.ramanujan_part1_ch2_entry7_example5_first","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-example5-firs","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"The series","title":"Series of arctangents of 1/(2k\u00b2) (Entry 7, Example 5)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/ramanujan-part1-ch2-entry7-example6","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch2.Entry7Example6.ramanujan_part1_ch2_entry7_example6","parent":18149,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series/index.html#ramanujan-part1-ch2-entry7-example6","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/arctangent-sums-and-series","status":"fully_proved","summary":"The series","title":"Series of arctangents of 1/(1 + k\u221a2)\u00b2 (Entry 7, Example 6)"},{"data_url":"scopes/Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen.json","id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen","kind":"scope","navigation":"module","parent":18131,"result":false,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/index.html","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen","status":"fully_proved","title":"Central binomial series in trigonometric arguments"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/ramanujan-part1-ch9-entry16-tan","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry16Tan.ramanujan_part1_ch9_entry16_tan","parent":18163,"reading_title":{"format":"inline-parts-v1","parts":[["text","The central binomial series in "],["math","\\(\\sin x\\)"],["text"," and "],["math","\\(x \\log|2\\sin x|\\)"]],"source":"The central binomial series in $\\sin x$ and $x \\log|2\\sin x|$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/index.html#ramanujan-part1-ch9-entry16-tan","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen","status":"fully_proved","summary":"Let $x$ be a real number with $|x| \\le \\pi/2$, and let $L(x) = x \\log|2\\sin x|$ for $x \\ne 0$ and $L(0) = 0$. Then $|2\\sin x| > 0$ if $x \\ne 0$, the series $S(x) = \\sum_{k=0}^{\\infty} \\binom{2k}{k} \\frac{\\sin^{2k+1} x}{2^{2k} (2k+1)^2}$ and $F(x) = \\sum_{k=1}^{\\infty} \\frac{\\sin 2kx}{k^2}$ are both summable, and $S(x) = L(x) + \\frac{1}{2} F(x)$.","title":"The central binomial series in $\\sin x$ and $x \\log|2\\sin x|$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/chapter9-entry16-difference-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry16Tanseries.chapter9Entry16DifferenceTerm","parent":18163,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/index.html#chapter9-entry16-difference-term","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen","status":"fully_proved","summary":"For a real number $x$ and a natural number $k$, the $k$-th term is $\\binom{2k}{k} \\frac{\\cos^{2k+1} x - \\sin^{2k+1} x}{2^{2k} (2k+1)^2}$.","title":"Terms of the cosine-minus-sine central binomial series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/ramanujan-part1-ch9-entry16-tanseries","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry16Tanseries.ramanujan_part1_ch9_entry16_tanseries","parent":18163,"reading_title":{"format":"inline-parts-v1","parts":[["text","The central binomial series in "],["math","\\(\\cos x\\)"],["text"," minus that in "],["math","\\(\\sin x\\)"]],"source":"The central binomial series in $\\cos x$ minus that in $\\sin x$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/index.html#ramanujan-part1-ch9-entry16-tanseries","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen","status":"fully_proved","summary":"Let $x$ be a real number with $0 \\le x \\le \\pi/4$, and for real $y$ let $S(y) = \\sum_{k=0}^{\\infty} \\binom{2k}{k} \\frac{\\sin^{2k+1} y}{2^{2k} (2k+1)^2}$. Then $2\\cos x > 0$, the series $D(x) = \\sum_{k=0}^{\\infty} \\binom{2k}{k} \\frac{\\cos^{2k+1} x - \\sin^{2k+1} x} {2^{2k} (2k+1)^2}$ and the series defining $S(2x)$ are both summable, and $D(x) = \\frac{\\pi}{2} \\log(2\\cos x) - \\frac{1}{2} S(2x)$.","title":"The central binomial series in $\\cos x$ minus that in $\\sin x$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/ramanujan-part1-ch9-entry19-catalangf","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry19Catalangf.ramanujan_part1_ch9_entry19_catalangf","parent":18163,"reading_title":{"format":"inline-parts-v1","parts":[["text","The central binomial series in "],["math","\\(\\cos x\\)"],["text"," plus that in "],["math","\\(\\sin x\\)"]],"source":"The central binomial series in $\\cos x$ plus that in $\\sin x$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/index.html#ramanujan-part1-ch9-entry19-catalangf","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen","status":"fully_proved","summary":"Let $x$ be a real number with $0 \\le x \\le \\pi/4$. Then $2\\cos x > 0$, the series $C(x) = \\sum_{k=0}^{\\infty} \\binom{2k}{k} \\frac{\\cos^{2k+1} x + \\sin^{2k+1} x} {2^{2k} (2k+1)^2}$ and $T(x) = \\sum_{k=0}^{\\infty} (-1)^k \\frac{\\tan^{2k+1} x}{(2k+1)^2}$ are both summable, and $C(x) = \\frac{\\pi}{2} \\log(2\\cos x) + T(x)$.","title":"The central binomial series in $\\cos x$ plus that in $\\sin x$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/ramanujan-part1-ch9-entry31-ramanujansum","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry31Ramanujansum.ramanujan_part1_ch9_entry31_ramanujansum","parent":18163,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen/index.html#ramanujan-part1-ch9-entry31-ramanujansum","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/central-binomial-series-in-trigonometric-argumen","status":"fully_proved","summary":"For real $y$ let $\\psi(y) = \\sum_{k \\ge 0} \\binom{2k}{k} \\sin^{2k+1}(\\pi y) / \\bigl(2^{2k} (2k+1)^2\\bigr)$. If $0 \\le x \\le 1/4$, then the series defining $\\psi(1/2 - x)$, $\\psi(2x)$ and $\\psi(x)$ converge absolutely, $2 \\cos(\\pi x) > 0$, and $\\psi(1/2 - x) + \\tfrac12 \\psi(2x) - \\psi(x) = \\tfrac{\\pi}{2} \\log(2 \\cos(\\pi x))$.","title":"Ramanujan's central-binomial sine series identity (Entry 31)"},{"data_url":"scopes/Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers.json","id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers","kind":"scope","navigation":"module","parent":18131,"result":false,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers/index.html","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers","status":"fully_proved","title":"Partial-fraction series over odd integers"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers/ramanujan-part1-ch9-entry1-logratio","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry1Logratio.ramanujan_part1_ch9_entry1_logratio","parent":18169,"reading_title":{"format":"inline-parts-v1","parts":[["text","Difference series over "],["math","\\(2k+1 \\pm a\\)"],["text"," that are polynomials on "],["math","\\((-\\pi, \\pi)\\)"]],"source":"Difference series over $2k+1 \\pm a$ that are polynomials on $(-\\pi, \\pi)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers/index.html#ramanujan-part1-ch9-entry1-logratio","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers","status":"fully_proved","summary":"Let $r$ be a positive integer, $a$ a real number that is not an odd integer, and $x$ a real number with $|x| < \\pi$. For a natural number $s$ let $c_k(s) = (2k+1-a)^{-s} - (2k+1+a)^{-s}$ and $C_s(a) = \\sum_{k=0}^{\\infty} c_k(s)$. Then $\\sum_k c_k(r)$ is summable, and either $r$ is odd, $\\sum_k c_k(r-2j)$ is summable for $0 \\le j \\le (r-1)/2$, and the partial sums of\u2026","title":"Difference series over $2k+1 \\pm a$ that are polynomials on $(-\\pi, \\pi)$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers/ramanujan-part1-ch9-entry2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry2.ramanujan_part1_ch9_entry2","parent":18169,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum series over "],["math","\\(2k+1 \\pm a\\)"],["text"," on "],["math","\\((0, \\pi)\\)"]],"source":"Sum series over $2k+1 \\pm a$ on $(0, \\pi)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers/index.html#ramanujan-part1-ch9-entry2","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/partial-fraction-series-over-odd-integers","status":"fully_proved","summary":"Let $r \\ge 2$ be an integer, $a$ a real number that is not an odd integer, and $x$ a real number with $0 < x < \\pi$. For a natural number $s$ let $d_k(s) = (2k+1-a)^{-s} + (2k+1+a)^{-s}$ and $D_s(a) = \\sum_{k=0}^{\\infty} d_k(s)$. Then $\\sum_k d_k(r)$ is summable, and either $r$ is even, $\\sum_k d_k(r-2j)$ is summable for $0 \\le j < r/2$, and the partial sums of\u2026","title":"Sum series over $2k+1 \\pm a$ on $(0, \\pi)$"},{"data_url":"scopes/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x.json","id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","kind":"scope","navigation":"module","parent":18131,"result":false,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","title":"Power series in sin x and tan x"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/ramanujan-part1-ch9-entry20-piseries1","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry20Piseries1.ramanujan_part1_ch9_entry20_piseries1","parent":18172,"reading_title":{"format":"inline-parts-v1","parts":[["text","A series in even powers of "],["math","\\(\\sin x\\)"],["text"," involving "],["math","\\(\\zeta(3)\\)"]],"source":"A series in even powers of $\\sin x$ involving $\\zeta(3)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html#ramanujan-part1-ch9-entry20-piseries1","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","summary":"Let $x$ be a real number with $|x| \\le \\pi/2$, let $M(x) = \\frac{x^2}{2} \\log|2\\sin x|$ for $x \\ne 0$ and $M(0) = 0$, and let $\\zeta(3) = \\sum_{k=1}^{\\infty} k^{-3}$. Then $|2\\sin x| > 0$ if $x \\ne 0$, the series $\\sum_{k=1}^{\\infty} k^{-3}$, $A(x) = \\sum_{k=0}^{\\infty} \\frac{2^{2k} (k!)^2 \\sin^{2k+2} x}{(2k+1)! \\, (2k+2)^2}$, $B(x) = \\sum_{k=1}^{\\infty} \\frac{\\sin 2kx}{k^2}$ and\u2026","title":"A series in even powers of $\\sin x$ involving $\\zeta(3)$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/ramanujan-part1-ch9-entry21-piseries2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry21Piseries2.ramanujan_part1_ch9_entry21_piseries2","parent":18172,"reading_title":{"format":"inline-parts-v1","parts":[["text","A series in even powers of "],["math","\\(\\tan x\\)"],["text"," with odd harmonic numbers"]],"source":"A series in even powers of $\\tan x$ with odd harmonic numbers"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html#ramanujan-part1-ch9-entry21-piseries2","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","summary":"Let $x$ be a real number with $|x| \\le \\pi/4$. Let $h_k = \\sum_{j=0}^{k-1} \\frac{1}{2j+1}$, let $N(x) = \\frac{x^2}{2} \\log|\\tan x|$ for $x \\ne 0$ and $N(0) = 0$, and let $\\lambda(3) = \\sum_{j=0}^{\\infty} (2j+1)^{-3}$. Then the series $\\sum_{j=0}^{\\infty} (2j+1)^{-3}$, $A(x) = \\sum_{k=1}^{\\infty} (-1)^{k-1} \\frac{h_k \\tan^{2k} x}{(2k)^2}$, $B(x) = \\sum_{j=0}^{\\infty} \\frac{\\sin((4j+2)x)}{(2j+1)^2}$ and\u2026","title":"A series in even powers of $\\tan x$ with odd harmonic numbers"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/chapter9-chi3-term-eq-entry21","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry22Chudnovsky.chapter9Chi3Term_eq_entry21","parent":18172,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html#chapter9-chi3-term-eq-entry21","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","summary":"The sequence $j \\mapsto 1/(2j+1)^3$ ($j = 0, 1, 2, \\dots$) used in the statement of Entry 22 of Chapter 9 of Ramanujan's notebooks is, by definition, the same sequence as the one used in Entry 21.","title":"Odd reciprocal cubes in Entries 21 and 22"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/chapter9-chi3-at-one-eq-entry21","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry22Chudnovsky.chapter9Chi3AtOne_eq_entry21","parent":18172,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html#chapter9-chi3-at-one-eq-entry21","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","summary":"The constant $\\sum_{j \\ge 0} 1/(2j+1)^3$ used in the statement of Entry 22 of Chapter 9 of Ramanujan's notebooks is, by definition, the same constant as the one used in Entry 21.","title":"The sum of the odd reciprocal cubes in Entries 21 and 22"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/chapter9-entry23-factorial-term-eq-entry","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry22Chudnovsky.chapter9Entry23FactorialTerm_eq_entry20","parent":18172,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html#chapter9-entry23-factorial-term-eq-entry","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","summary":"For real $u$ and $k \\ge 0$, the term $2^{2k} (k!)^2 \\sin^{2k+2} u / \\bigl((2k+1)! \\, (2k+2)^2\\bigr)$ used in the statement of Entry 22 of Chapter 9 of Ramanujan's notebooks is, by definition, the same as the one used in Entry 20.","title":"The factorial sine series term in Entries 20 and 22"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/ramanujan-part1-ch9-entry22-chudnovsky","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry22Chudnovsky.ramanujan_part1_ch9_entry22_chudnovsky","parent":18172,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html#ramanujan-part1-ch9-entry22-chudnovsky","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","summary":"For real $u$ and $k \\ge 0$ let $T_k(u) = \\frac{2^{2k} (k!)^2 \\sin^{2k+2} u}{(2k+1)! \\, (2k+2)^2}$. Let $x$ be real with $0 \\le x \\le \\pi/4$. Then $2 \\cos x > 0$, the series $\\sum_{j \\ge 0} \\frac{1}{(2j+1)^3}$, $\\sum_{k \\ge 0} \\bigl(T_k(x) + T_k(\\pi/2 - x)\\bigr)$, $\\sum_{k \\ge 0} \\frac{\\binom{2k}{k} \\cos^{2k+1} x}{2^{2k} (2k+1)^2}$ and $\\sum_{k \\ge 0} T_k(2x)$ converge absolutely, and","title":"Entry 22 of Chapter 9 of Ramanujan's notebooks"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/ramanujan-part1-ch9-entry23-ramanujanpi2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry23Ramanujanpi2.ramanujan_part1_ch9_entry23_ramanujanpi2","parent":18172,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html#ramanujan-part1-ch9-entry23-ramanujanpi2","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","summary":"For real $u$ and $k \\ge 0$ let $T_k(u) = \\frac{2^{2k} (k!)^2 \\sin^{2k+2} u}{(2k+1)! \\, (2k+2)^2}$, and for $k \\ge 1$ let $O_k = 1 + \\frac{1}{3} + \\dots + \\frac{1}{2k-1}$. Let $x$ be real with $|x| \\le \\pi/4$. Then the series $\\sum_{k \\ge 1} \\frac{(-1)^{k-1} O_k \\tan^{2k} x}{(2k)^2}$, $\\sum_{k \\ge 0} T_k(x)$ and $\\sum_{k \\ge 0} T_k(2x)$ converge absolutely, and","title":"Entry 23 of Chapter 9 of Ramanujan's notebooks"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/ramanujan-part1-ch9-entry32-lambertdivis","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry32Lambertdivisor.ramanujan_part1_ch9_entry32_lambertdivisor","parent":18172,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x/index.html#ramanujan-part1-ch9-entry32-lambertdivis","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/power-series-in-sin-x-and-tan-x","status":"fully_proved","summary":"Let $x$ be real with $|x| \\le \\pi/4$. Then $\\cos x > 0$, the series $\\sum_{k \\ge 0} \\frac{2^{2k} (k!)^2}{(2k)!\\,(2k+1)^2} \\sin^{2k+1}(2x)$ and $\\sum_{k \\ge 0} \\frac{(-1)^k \\tan^{2k+1} x}{(2k+1)^2}$ converge absolutely, and the first sum equals twice the second.","title":"Ramanujan's sine series in odd powers of the tangent (Entry 32)"},{"data_url":"scopes/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity.json","id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","kind":"scope","navigation":"module","parent":18131,"result":false,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","title":"Series from N\u00f6rlund's digamma identity"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/sub-two-mul-one-add-tsum-prod-eq-pi-mul","kind":"node","lean":"MetaMathlibExt.sub_two_mul_one_add_tsum_prod_eq_pi_mul_cot","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","A product series for "],["math","\\(\\pi \\cot(\\pi/k)\\)"]],"source":"A product series for $\\pi \\cot(\\pi/k)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#sub-two-mul-one-add-tsum-prod-eq-pi-mul","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"For every integer $k \\ge 2$,","title":"A product series for $\\pi \\cot(\\pi/k)$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-two-pow-prod-eq-neg-pi-mul-tan","kind":"node","lean":"MetaMathlibExt.tsum_two_pow_prod_eq_neg_pi_mul_tan","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-two-pow-prod-eq-neg-pi-mul-tan","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"For every complex number $z$ with $-\\frac12 < \\operatorname{Re} z < \\frac12$,","title":"Adegoke\u2013Frontczak series for the tangent"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/harmonic-nat-sub-half-values","kind":"node","lean":"MetaMathlibExt.harmonic_nat_sub_half_values","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#harmonic-nat-sub-half-values","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $k$ be a natural number. For each integer $r \\ge 1$ let $O^{(r)}_k = \\sum_{j=1}^{k} \\frac{1}{(2j - 1)^r}$, and let $O_k = O^{(1)}_k$. Let $\\psi$ be the digamma function, $\\gamma$ the Euler\u2013Mascheroni constant and $\\zeta$ the Riemann zeta function. Then","title":"Harmonic numbers of half-integer index"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/gamma-add-half-and-gamma-half-sub-nat","kind":"node","lean":"MetaMathlibExt.Gamma_add_half_and_Gamma_half_sub_nat","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Gamma function at "],["math","\\(r + \\tfrac12\\)"],["text"," and at "],["math","\\(\\tfrac12 - r\\)"]],"source":"The Gamma function at $r + \\tfrac12$ and at $\\tfrac12 - r$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#gamma-add-half-and-gamma-half-sub-nat","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"For every complex number $r$ that is not a negative integer,","title":"The Gamma function at $r + \\tfrac12$ and at $\\tfrac12 - r$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-catalan-div-four-pow-mul-binom","kind":"node","lean":"MetaMathlibExt.tsum_catalan_div_four_pow_mul_binom","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-catalan-div-four-pow-mul-binom","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $z$ be a complex number with $\\operatorname{Re} z > 0$, let $\\binom{n+z}{n} = \\prod_{j=1}^{n} \\frac{z + j}{j}$, let $C_n$ be the $n$th Catalan number and $\\psi$ the digamma function. Then","title":"A Catalan series in terms of the digamma function"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-catalan-div-four-pow-special","kind":"node","lean":"MetaMathlibExt.tsum_catalan_div_four_pow_special","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan series summing to "],["math","\\(4(1 - \\log 2)\\)"],["text"," and "],["math","\\(\\frac{10}{3} - 4 \\log 2\\)"]],"source":"Catalan series summing to $4(1 - \\log 2)$ and $\\frac{10}{3} - 4 \\log 2$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-catalan-div-four-pow-special","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"With $C_n$ the $n$th Catalan number,","title":"Catalan series summing to $4(1 - \\log 2)$ and $\\frac{10}{3} - 4 \\log 2$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-odd-harmonic-div-and-catalan-mul-ha","kind":"node","lean":"MetaMathlibExt.tsum_odd_harmonic_div_and_catalan_mul_harmonic_special","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","An odd harmonic series for "],["math","\\(\\pi^2/6\\)"],["text"," and a Catalan\u2013harmonic series for "],["math","\\(8 - 2\\pi^2/3\\)"]],"source":"An odd harmonic series for $\\pi^2/6$ and a Catalan\u2013harmonic series for $8 - 2\\pi^2/3$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-odd-harmonic-div-and-catalan-mul-ha","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $O_k = \\sum_{j=1}^{k} 1/(2j-1)$ be the $k$-th odd harmonic number, $C_n$ the $n$-th Catalan number and $H_k = \\sum_{j=1}^{k} 1/j$ the $k$-th harmonic number. Then","title":"An odd harmonic series for $\\pi^2/6$ and a Catalan\u2013harmonic series for $8 - 2\\pi^2/3$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-catalan-mul-harmonic-div-four-pow-m","kind":"node","lean":"MetaMathlibExt.tsum_catalan_mul_harmonic_div_four_pow_mul_binom","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-catalan-mul-harmonic-div-four-pow-m","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $z$ be a complex number with $\\operatorname{Re} z > 0$, let $\\binom{n+z}{n} = \\prod_{j=1}^{n} \\frac{z + j}{j}$, let $C_n$ be the $n$th Catalan number, $\\psi$ the digamma function and $\\gamma$ the Euler\u2013Mascheroni constant. Then","title":"A Catalan series weighted by harmonic numbers of complex argument"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-odd-mul-catalan-div-four-pow-mul-bi","kind":"node","lean":"MetaMathlibExt.tsum_odd_mul_catalan_div_four_pow_mul_binom","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digamma evaluation of the "],["math","\\((2n+1)\\)"],["text","-weighted Catalan series with binomial denominators"]],"source":"Digamma evaluation of the $(2n+1)$-weighted Catalan series with binomial denominators"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-odd-mul-catalan-div-four-pow-mul-bi","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $C_n$ be the $n$-th Catalan number and $\\psi$ the digamma function, and for complex $z$ write $\\binom{n+z}{n} = \\prod_{j=1}^{n} (z+j)/j$. For every complex number $z$ with $\\operatorname{Re} z > 1/2$,","title":"Digamma evaluation of the $(2n+1)$-weighted Catalan series with binomial denominators"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-odd-mul-catalan-div-four-pow-specia","kind":"node","lean":"MetaMathlibExt.tsum_odd_mul_catalan_div_four_pow_special","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan series over powers of "],["math","\\(4\\)"],["text"," equal to "],["math","\\(4\\ln 2\\)"],["text"," and "],["math","\\(4\\ln 2 - 2\\)"]],"source":"Catalan series over powers of $4$ equal to $4\\ln 2$ and $4\\ln 2 - 2$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-odd-mul-catalan-div-four-pow-specia","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $C_n$ be the $n$-th Catalan number. Then","title":"Catalan series over powers of $4$ equal to $4\\ln 2$ and $4\\ln 2 - 2$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-nat-mul-catalan-div-four-pow-mul-bi","kind":"node","lean":"MetaMathlibExt.tsum_nat_mul_catalan_div_four_pow_mul_binom","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digamma evaluation of the "],["math","\\(n\\)"],["text","-weighted Catalan series with binomial denominators"]],"source":"Digamma evaluation of the $n$-weighted Catalan series with binomial denominators"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-nat-mul-catalan-div-four-pow-mul-bi","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $C_n$ be the $n$-th Catalan number and $\\psi$ the digamma function, and for complex $z$ write $\\binom{n+z}{n} = \\prod_{j=1}^{n} (z+j)/j$. For every complex number $z$ with $\\operatorname{Re} z > 1/2$,","title":"Digamma evaluation of the $n$-weighted Catalan series with binomial denominators"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-odd-mul-catalan-mul-harmonic-div-fo","kind":"node","lean":"MetaMathlibExt.tsum_odd_mul_catalan_mul_harmonic_div_four_pow_mul_binom","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-odd-mul-catalan-mul-harmonic-div-fo","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $z$ be a complex number with $\\operatorname{Re} z > \\frac12$. Write $C_n$ for the $n$-th Catalan number, $\\psi$ for the digamma function, $\\gamma$ for the Euler\u2013Mascheroni constant, $H_\\alpha = \\psi(\\alpha + 1) + \\gamma$ for complex $\\alpha$, and $\\binom{n+z}{n} = \\prod_{j=1}^{n} \\frac{z+j}{j}$. Then","title":"Adegoke\u2013Frontczak series with odd weights, Catalan numbers and harmonic numbers"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-odd-mul-catalan-mul-harmonic-specia","kind":"node","lean":"MetaMathlibExt.tsum_odd_mul_catalan_mul_harmonic_special","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-odd-mul-catalan-mul-harmonic-specia","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $C_n$ be the $n$-th Catalan number, $H_n = \\sum_{k=1}^{n} \\frac1k$ the $n$-th harmonic number and $O_n = \\sum_{k=1}^{n} \\frac{1}{2k-1}$ the $n$-th odd harmonic number. Then","title":"Special values of the odd-weighted Catalan\u2013harmonic series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-nat-mul-catalan-mul-harmonic-div-fo","kind":"node","lean":"MetaMathlibExt.tsum_nat_mul_catalan_mul_harmonic_div_four_pow_mul_binom","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digamma evaluation of the "],["math","\\(n\\)"],["text","-weighted Catalan\u2013harmonic series with binomial denominators"]],"source":"Digamma evaluation of the $n$-weighted Catalan\u2013harmonic series with binomial denominators"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-nat-mul-catalan-mul-harmonic-div-fo","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Digamma evaluation of the $n$-weighted Catalan\u2013harmonic series with binomial denominators. Let $C_n$ be the $n$-th Catalan number, $\\psi$ the digamma function and $\\gamma$ the Euler\u2013Mascheroni constant; for complex $w$ write $H_w = \\psi(w + 1) + \\gamma$, and for complex $z$ write $\\binom{n+z}{n} = \\prod_{j=1}^{n} (z+j)/j$. For every complex number $z$ with $\\operatorname{Re} z > 1/2$,","title":"Digamma evaluation of the $n$-weighted Catalan\u2013harmonic series with binomial denominators"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-nat-mul-catalan-mul-harmonic-specia","kind":"node","lean":"MetaMathlibExt.tsum_nat_mul_catalan_mul_harmonic_special","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Special values of the "],["math","\\(n\\)"],["text","-weighted Catalan\u2013harmonic series"]],"source":"Special values of the $n$-weighted Catalan\u2013harmonic series"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-nat-mul-catalan-mul-harmonic-specia","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $C_n$ be the $n$-th Catalan number, $H_k = \\sum_{j=1}^{k} 1/j$ the $k$-th harmonic number and $O_k = \\sum_{j=1}^{k} 1/(2j-1)$ the $k$-th odd harmonic number. Then","title":"Special values of the $n$-weighted Catalan\u2013harmonic series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-four-pow-div-central-binom-mul-sq-s","kind":"node","lean":"MetaMathlibExt.tsum_four_pow_div_centralBinom_mul_sq_special","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial series for "],["math","\\(\\pi^2/8\\)"],["text"," and a series for "],["math","\\(2\\ln 2\\)"]],"source":"Central binomial series for $\\pi^2/8$ and a series for $2\\ln 2$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-four-pow-div-central-binom-mul-sq-s","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"We have","title":"Central binomial series for $\\pi^2/8$ and a series for $2\\ln 2$"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-four-pow-div-binom-mul-central-bino","kind":"node","lean":"MetaMathlibExt.tsum_four_pow_div_binom_mul_centralBinom_mul_sq","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-four-pow-div-binom-mul-central-bino","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $\\Gamma$ be the Gamma function and $\\psi$ the digamma function, and let $h$ be a complex number that is not a nonnegative integer and has $\\operatorname{Re} h > -1/2$. Writing $\\binom{n}{h}^{-1}$ for $\\Gamma(h+1)\\,\\Gamma(n-h+1)/n!$, we have","title":"Digamma evaluation of a central binomial series with a complex binomial parameter"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-central-binom-div-choose-mul-centra","kind":"node","lean":"MetaMathlibExt.tsum_centralBinom_div_choose_mul_centralBinom_mul_sq","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-central-binom-div-choose-mul-centra","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"For every natural number $h$, with $H_h = \\sum_{j=1}^{h} \\frac{1}{j}$ the $h$th harmonic number,","title":"A shifted central binomial series"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-central-binom-mul-odd-harmonic-div","kind":"node","lean":"MetaMathlibExt.tsum_centralBinom_mul_odd_harmonic_div_choose_mul_centralBinom_mul_sq","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-central-binom-mul-odd-harmonic-div","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"For every natural number $h$, write $H_h = \\sum_{j=1}^{h} \\frac{1}{j}$, $H^{(2)}_h = \\sum_{j=1}^{h} \\frac{1}{j^2}$ and $O_n = \\sum_{j=1}^{n} \\frac{1}{2j - 1}$. Then","title":"A shifted central binomial series with odd harmonic numbers"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-four-pow-div-succ-mul-choose-mul-ce","kind":"node","lean":"MetaMathlibExt.tsum_four_pow_div_succ_mul_choose_mul_centralBinom","parent":18181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Series over "],["math","\\(4^n\\)"],["text"," with binomial and central binomial denominators"]],"source":"Series over $4^n$ with binomial and central binomial denominators"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-four-pow-div-succ-mul-choose-mul-ce","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Write $O^{(2)}_k = \\sum_{j=1}^{k} 1/(2j-1)^2$. For every positive integer $m$,","title":"Series over $4^n$ with binomial and central binomial denominators"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-four-pow-mul-odd-harmonic-div-succ","kind":"node","lean":"MetaMathlibExt.tsum_four_pow_mul_odd_harmonic_div_succ_mul_choose_mul_centralBinom","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-four-pow-mul-odd-harmonic-div-succ","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $\\zeta$ be the Riemann zeta function, let $O^{(r)}_k = \\sum_{j=1}^{k} 1/(2j-1)^r$ be the $k$-th odd harmonic number of order $r$, and write $O_k = O^{(1)}_k$. For every positive integer $m$,","title":"An odd harmonic series with binomial and central binomial denominators"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-catalan-div-choose-mul-central-bino","kind":"node","lean":"MetaMathlibExt.tsum_catalan_div_choose_mul_centralBinom","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-catalan-div-choose-mul-central-bino","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $m$ be a positive integer, $C_n$ the $n$th Catalan number, $H_n = \\sum_{j=1}^{n} \\frac{1}{j}$ the $n$th harmonic number and $O_n = \\sum_{j=1}^{n} \\frac{1}{2j - 1}$. Then","title":"Catalan numbers over binomial and central binomial coefficients"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/tsum-catalan-mul-odd-harmonic-div-choose","kind":"node","lean":"MetaMathlibExt.tsum_catalan_mul_odd_harmonic_div_choose_mul_centralBinom","parent":18181,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity/index.html#tsum-catalan-mul-odd-harmonic-div-choose","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-from-n-rlund-s-digamma-identity","status":"can_prove","summary":"Let $m$ be a positive integer and $C_n$ the $n$th Catalan number, and write $H_n = \\sum_{j=1}^{n} \\frac{1}{j}$, $H^{(2)}_n = \\sum_{j=1}^{n} \\frac{1}{j^2}$, $O_n = \\sum_{j=1}^{n} \\frac{1}{2j - 1}$ and $O^{(2)}_n = \\sum_{j=1}^{n} \\frac{1}{(2j - 1)^2}$. Then","title":"An odd harmonic Catalan series over central binomial coefficients"},{"data_url":"scopes/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers.json","id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers","kind":"scope","navigation":"module","parent":18131,"result":false,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers/index.html","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers","status":"fully_proved","title":"Series with odd harmonic numbers"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers/ramanujan-part1-ch9-entry34-zeta3doubles","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry34Zeta3doublesum.ramanujan_part1_ch9_entry34_zeta3doublesum","parent":18205,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers/index.html#ramanujan-part1-ch9-entry34-zeta3doubles","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers","status":"fully_proved","summary":"For a natural number $m$ let $O_m = \\sum_{j=0}^{m-1} 1/(2j+1)$. Let $x$ be real with $-1/2 < x < 1$. Then $1 + x > 0$, $|x| < 1$ and $|x/(1+x)| < 1$, the series $\\sum_{k \\ge 0} \\frac{1}{(2k+1)^2} \\left(\\frac{x}{1+x}\\right)^{k+1}$ and $\\sum_{k \\ge 0} \\frac{(-1)^k\\, 2^{2k} (k!)^2}{(2k+1)!} O_{k+1} x^{k+1}$ converge absolutely, and their sums are equal.","title":"Ramanujan's series identity with odd harmonic numbers (Entry 34)"},{"id":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers/ramanujan-part1-ch9-entry8-arcsin","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry8Arcsin.ramanujan_part1_ch9_entry8_arcsin","parent":18205,"result":true,"route":"../roadmap/Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers/index.html#ramanujan-part1-ch9-entry8-arcsin","scope":"Analysis/sequences-and-series/trigonometric-and-harmonic-series/series-with-odd-harmonic-numbers","status":"fully_proved","summary":"Let $x$ be a real number with $|x| < 1$. For $n \\ge 0$ put $h_n = 1 + \\frac{1}{3} + \\cdots + \\frac{1}{2n+1}$, for real $y$ let $F(y) = \\sum_{n \\ge 0} \\frac{h_n}{2n+1} y^{2n+1}$, and let $\\operatorname{Li}_2(x) = \\sum_{k \\ge 1} \\frac{x^k}{k^2}$. Then $1 - x > 0$, $2 - x \\ne 0$ and $|x/(2-x)| < 1$; the series defining $F(x/(2-x))$ and the series defining $\\operatorname{Li}_2(x)$ are both summable; and","title":"Ramanujan's odd harmonic series identity"},{"data_url":"scopes/Analysis/sequences-and-series/power-series.json","id":"Analysis/sequences-and-series/power-series","kind":"scope","parent":18053,"result":false,"route":"../roadmap/Analysis/sequences-and-series/power-series/index.html","scope":"Analysis/sequences-and-series/power-series","status":"fully_proved","summary":"Identities and transformations of power series and exponential-type series: Lagrange inversion and the tree function, Abel- and Jensen-type exponential series, coefficients of exp of a power series and Newton's identities for summable sequences, transformations of factorial series and Taylor-shift transformations, and Touchard / generalized Bell series.","title":"Power series and series transformations"},{"id":"Analysis/sequences-and-series/power-series/ramanujan-part1-ch3-entry11","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry11.ramanujan_part1_ch3_entry11","parent":18208,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the coefficients of "],["math","\\(e^{f(x)}\\)"]],"source":"Recurrence for the coefficients of $e^{f(x)}$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/index.html#ramanujan-part1-ch3-entry11","scope":"Analysis/sequences-and-series/power-series","status":"fully_proved","summary":"Let $(A_k)_{k \\ge 1}$ and $(p_n)_{n \\ge 0}$ be sequences of real numbers, let $R > 0$, and let $f : \\mathbb{R} \\to \\mathbb{R}$. Suppose that for every real $x$ with $|x| < R$ the series $\\sum_{k \\ge 1} A_k x^k / k$ is summable with sum $f(x)$ and the series $\\sum_{n \\ge 0} p_n x^n$ is summable with sum $e^{f(x)}$. Then for every integer $n \\ge 1$,","title":"Recurrence for the coefficients of $e^{f(x)}$"},{"data_url":"scopes/Analysis/sequences-and-series/power-series/abel-type-exponential-series.json","id":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","kind":"scope","navigation":"module","parent":18208,"result":false,"route":"../roadmap/Analysis/sequences-and-series/power-series/abel-type-exponential-series/index.html","scope":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","status":"fully_proved","title":"Abel-type exponential series"},{"id":"Analysis/sequences-and-series/power-series/abel-type-exponential-series/ramanujan-part1-ch3-entry12-summable-suc","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry12SummableSuccN.ramanujan_part1_ch3_entry12_summable_succ_n_general","parent":18210,"reading_title":{"format":"inline-parts-v1","parts":[["text","Absolute convergence of the shifted series in Entry 12 for every "],["math","\\(n\\)"]],"source":"Absolute convergence of the shifted series in Entry 12 for every $n$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/abel-type-exponential-series/index.html#ramanujan-part1-ch3-entry12-summable-suc","scope":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","status":"fully_proved","summary":"Let $r$ be an integer and let $n$ and $a$ be complex numbers with $|a| > e$, or with $a = e$ and $r \\le -2$. Then","title":"Absolute convergence of the shifted series in Entry 12 for every $n$"},{"id":"Analysis/sequences-and-series/power-series/abel-type-exponential-series/ramanujan-part1-ch3-entry12-summable-suc-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry12SummableSuccN.ramanujan_part1_ch3_entry12_summable_succ_n","parent":18210,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/abel-type-exponential-series/index.html#ramanujan-part1-ch3-entry12-summable-suc-2","scope":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","status":"fully_proved","summary":"Let $r$ be an integer and let $n$ and $a$ be complex numbers. Suppose that $|a| > e$, or that $a = e$ and $r \\le -2$, and that $n + k \\ne 0$ for every integer $k \\ge 0$ with $r + k < 0$. Then","title":"Absolute convergence of the shifted series in Entry 12"},{"id":"Analysis/sequences-and-series/power-series/abel-type-exponential-series/ramanujan-part1-ch3-entry12","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry12.ramanujan_part1_ch3_entry12","parent":18210,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/abel-type-exponential-series/index.html#ramanujan-part1-ch3-entry12","scope":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","status":"fully_proved","summary":"Fix a complex number $a$. For an integer $r$ and a complex number $n$, let","title":"Ramanujan's recurrence for the series in Entry 12"},{"id":"Analysis/sequences-and-series/power-series/abel-type-exponential-series/ramanujan-part1-ch3-entry12-summable-r-n","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry12SummableRN.ramanujan_part1_ch3_entry12_summable_r_n","parent":18210,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/abel-type-exponential-series/index.html#ramanujan-part1-ch3-entry12-summable-r-n","scope":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","status":"fully_proved","summary":"Let $r$ be an integer and let $n$ and $a$ be complex numbers. Suppose that $|a| > e$, or that $a = e$ and $r \\le -2$, and that $n + k \\ne 0$ for every natural number $k$ with $r + k < 0$. Then","title":"Absolute convergence of the series in Entry 12"},{"id":"Analysis/sequences-and-series/power-series/abel-type-exponential-series/ramanujan-part1-ch3-entry16","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry16.ramanujan_part1_ch3_entry16","parent":18210,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/abel-type-exponential-series/index.html#ramanujan-part1-ch3-entry16","scope":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","status":"fully_proved","summary":"Let $a$, $x$, $n$ be complex numbers and $r$ an integer, where $x \\ne 0$ and $x = a \\log x$ with $\\log$ the principal branch of the complex logarithm. Suppose that $|a| > e$, or that $a = e$ and $r \\le -2$, and that $n + k \\ne 0$ for every natural number $k$ with $r + k < 0$. For an integer $s$ and a complex number $m$ put","title":"Ramanujan's three-term recurrence (Chapter 3, Entry 16(a))"},{"id":"Analysis/sequences-and-series/power-series/abel-type-exponential-series/ramanujan-part1-ch3-entry16-corollary1-j","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry16Corollary1JensenSeries.ramanujan_part1_ch3_entry16_corollary1_jensen_series","parent":18210,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/abel-type-exponential-series/index.html#ramanujan-part1-ch3-entry16-corollary1-j","scope":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","status":"fully_proved","summary":"Let $\\rho$ be a positive real number with $\\rho e^{\\rho + 1} = 1$, let $n$ be a real number with $-\\rho < n < 1$, and let $x$ be a complex number. Then the series","title":"Jensen's series (Chapter 3, Entry 16, Corollary 1)"},{"id":"Analysis/sequences-and-series/power-series/abel-type-exponential-series/ramanujan-part1-ch3-entry16-psi-series","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry16PsiSeries.ramanujan_part1_ch3_entry16_psi_series","parent":18210,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's "],["math","\\(\\psi\\)"],["text","-series (Chapter 3, Entry 16)"]],"source":"Ramanujan's $\\psi$-series (Chapter 3, Entry 16)"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/abel-type-exponential-series/index.html#ramanujan-part1-ch3-entry16-psi-series","scope":"Analysis/sequences-and-series/power-series/abel-type-exponential-series","status":"fully_proved","summary":"For natural numbers $r$, $k$ and a complex number $n$ define $\\psi(r, k, n)$ recursively by $\\psi(0, 1, n) = 1$, $\\psi(0, k, n) = 0$ for $k \\ne 1$, and","title":"Ramanujan's $\\psi$-series (Chapter 3, Entry 16)"},{"data_url":"scopes/Analysis/sequences-and-series/power-series/lagrange-inversion-series.json","id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","kind":"scope","navigation":"module","parent":18208,"result":false,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","title":"Lagrange inversion series"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/ramanujan-part1-ch3-entry13","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry13.ramanujan_part1_ch3_entry13","parent":18218,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's series for powers of a root of "],["math","\\(x = a \\log x\\)"],["text"," (Entry 13)"]],"source":"Ramanujan's series for powers of a root of $x = a \\log x$ (Entry 13)"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#ramanujan-part1-ch3-entry13","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"Let $a$ be a real number with $|a| \\ge e$. Then there is a real number $x > 0$ with $x = a \\log x$ such that for every real number $n$ the series","title":"Ramanujan's series for powers of a root of $x = a \\log x$ (Entry 13)"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/schroder-abel-binomial","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry13CorollarySchroder.schroder_abel_binomial","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#schroder-abel-binomial","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"For a complex number $w$, let $A_0(x) = 1$ and $A_j(x) = x (x + j w)^{j-1}$ for $j \\ge 1$. Then for all complex numbers $x$, $y$ and every natural number $n$,","title":"Abel's binomial identity"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/ramanujan-part1-ch3-entry13-corollary-sc","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry13CorollarySchroder.ramanujan_part1_ch3_entry13_corollary_schroder","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#ramanujan-part1-ch3-entry13-corollary-sc","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"Let $z$ and $w$ be complex numbers with $|w| \\le |e^{w-1}|$ and $|w| \\le 1$. Then the series","title":"Schr\u00f6der's exponential series"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/ramanujan-part1-ch3-entry14-cor1-catalan","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry14Cor1Catalanpowerseries.ramanujan_part1_ch3_entry14_cor1_catalanpowerseries","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#ramanujan-part1-ch3-entry14-cor1-catalan","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"Let $n$ and $a$ be real numbers with $|a| \\le 1/4$. For a natural number $k$ put $t_k = 0$ if $k < 2$, and","title":"Ramanujan's series for powers of the Catalan generating function"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/lagrange-series-coefficient","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry14LagrangeSeries.lagrangeSeriesCoefficient","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#lagrange-series-coefficient","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"For real numbers $p$, $q$, $n$ and a natural number $k$, the Lagrange series coefficient $C_{p,q}(n, k)$ is $1$ if $k = 0$, and for $k \\ge 1$ it is","title":"Lagrange series coefficients"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/lagrange-series-coefficient-zero","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry14LagrangeSeries.lagrangeSeriesCoefficient_zero","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#lagrange-series-coefficient-zero","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"For all real numbers $p$, $q$, $n$, the Lagrange series coefficient $C_{p,q}(n, 0)$ of degree $0$ equals $1$.","title":"Lagrange series coefficient of degree zero"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/lagrange-series-coefficient-of-ne-zero","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry14LagrangeSeries.lagrangeSeriesCoefficient_of_ne_zero","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#lagrange-series-coefficient-of-ne-zero","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"For all real numbers $p$, $q$, $n$ and every natural number $k \\ne 0$, the Lagrange series coefficient of degree $k$ is","title":"Lagrange series coefficients of positive degree"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/lagrange-series-coefficient-add","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry14LagrangeSeries.lagrangeSeriesCoefficient_add","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#lagrange-series-coefficient-add","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"For all real numbers $p$, $q$, $r$, $s$ and every natural number $m$, the Lagrange series coefficients $C_{p,q}$ satisfy","title":"Hagen\u2013Rothe convolution for Lagrange series coefficients"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/ramanujan-part1-ch3-entry14-lagrange-ser","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry14LagrangeSeries.ramanujan_part1_ch3_entry14_lagrange_series","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#ramanujan-part1-ch3-entry14-lagrange-ser","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"Let $p > 0$ and $q > 0$ be real numbers with $p \\ne q$, and let $a$ be a complex number with $|a| \\le p^{-p/q} |p - q|^{(p - q)/q}$. Then there is a complex number $L$ with $a q e^{p L} - e^{q L} + 1 = 0$ such that, for every real number $n$, the series","title":"Ramanujan's Lagrange series"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/ramanujan-part1-ch3-entry15-example1-exp","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry15Example1ExponentialSeries.ramanujan_part1_ch3_entry15_example1_exponential_series","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#ramanujan-part1-ch3-entry15-example1-exp","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"Let $m$ and $n$ be real numbers with $0 < n \\le 2$. Then the series","title":"Ramanujan's exponential series for powers of two"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/ramanujan-part1-ch3-entry15-inverse-seri","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry15InverseSeries.ramanujan_part1_ch3_entry15_inverse_series","parent":18218,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's inverse series for "],["math","\\(u - \\log u\\)"]],"source":"Ramanujan's inverse series for $u - \\log u$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#ramanujan-part1-ch3-entry15-inverse-seri","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"There exist a real number $r > 0$ and real coefficients $b_0, b_1, b_2, \\dots$ with $b_0 = 1$, $b_1 = -1$, $b_2 = 1/3$, $b_3 = -1/36$ and $b_4 = -1/270$ such that the power series $\\sum_k b_k x^k$ is summable for every real $x$ with $|x| < r$, and for every real $x$ with $0 < x < r$ the sum $u$ of this series satisfies $0 < u < 1$ and $u - \\log u = 1 + x^2/2$, and the series $\\sum_{j \\ge 2} (1 - u)^j / j$ converges\u2026","title":"Ramanujan's inverse series for $u - \\log u$"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/ramanujan-part1-ch3-entry15-treeseries","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry15Treeseries.ramanujan_part1_ch3_entry15_treeseries","parent":18218,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#ramanujan-part1-ch3-entry15-treeseries","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"Let $x$ and $u$ be real numbers with $0 < u < 1$ and $u - \\log u = 1 + x^2/2$. Then the series","title":"Ramanujan's tree-function series (Chapter 3, Entry 15)"},{"id":"Analysis/sequences-and-series/power-series/lagrange-inversion-series/ramanujan-part1-ch3-entry17","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry17.ramanujan_part1_ch3_entry17","parent":18218,"reading_title":{"format":"inline-parts-v1","parts":[["text","Taylor coefficients of the solution of "],["math","\\(x^x = a^a e^h\\)"],["text"," (Chapter 3, Entry 17)"]],"source":"Taylor coefficients of the solution of $x^x = a^a e^h$ (Chapter 3, Entry 17)"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/lagrange-inversion-series/index.html#ramanujan-part1-ch3-entry17","scope":"Analysis/sequences-and-series/power-series/lagrange-inversion-series","status":"fully_proved","summary":"For every real number $a > 1/e$ there exist a real number $R > 0$ and a function $x : \\mathbb{R} \\to \\mathbb{R}$, analytic at $0$, with $x(0) = a$, such that $x(h) > 0$ and $x(h)^{x(h)} = a^a e^h$ for all real $h$ with $|h| < R$, and such that the numbers $A_k = (-a)^{k-1} x^{(k)}(0)$ for $k \\ge 1$ satisfy, for every integer $r \\ge 2$,","title":"Taylor coefficients of the solution of $x^x = a^a e^h$ (Chapter 3, Entry 17)"},{"data_url":"scopes/Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences.json","id":"Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences","kind":"scope","navigation":"module","parent":18208,"result":false,"route":"../roadmap/Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences/index.html","scope":"Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences","status":"fully_proved","title":"Newton's identities for summable sequences"},{"id":"Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences/ramanujan-part1-ch3-entry11-corollary-ne","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry11CorollaryNewtonGirard.ramanujan_part1_ch3_entry11_corollary_newton_girard_general","parent":18232,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences/index.html#ramanujan-part1-ch3-entry11-corollary-ne","scope":"Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences","status":"fully_proved","summary":"Let $(a_j)_{j \\ge 0}$ be a sequence of complex numbers, and let $(e_n)_{n \\ge 0}$ and $(p_k)_{k \\ge 1}$ be complex numbers such that for every $n$ the family $\\prod_{j \\in s} a_j$, indexed by the $n$-element finite sets $s$ of natural numbers, is summable with sum $e_n$, and for every $k \\ge 1$ the series $\\sum_{j \\ge 0} a_j^k$ is summable with sum $p_k$. Then for every natural number $n$,","title":"Newton's identities for summable families"},{"id":"Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences/ramanujan-part1-ch3-entry11-corollary-ne-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry11CorollaryNewtonGirard.ramanujan_part1_ch3_entry11_corollary_newton_girard","parent":18232,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences/index.html#ramanujan-part1-ch3-entry11-corollary-ne-2","scope":"Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences","status":"fully_proved","summary":"Let $(a_j)_{j \\ge 0}$ be a sequence of complex numbers with $\\sum_j |a_j| < \\infty$. Let $(e_n)_{n \\ge 0}$ and $(p_k)_{k \\ge 1}$ be complex numbers such that for every $n$ the family $\\prod_{j \\in s} a_j$, indexed by the $n$-element finite sets $s$ of natural numbers, is summable with sum $e_n$, and for every $k \\ge 1$ the series $\\sum_{j \\ge 0} a_j^k$ is summable with sum $p_k$. Then for every $n \\ge 1$,","title":"Newton's identities for absolutely summable sequences"},{"id":"Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences/ramanujan-part1-ch3-entry11-corollary-po","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry11CorollaryPowerSummable.ramanujan_part1_ch3_entry11_corollary_power_summable","parent":18232,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences/index.html#ramanujan-part1-ch3-entry11-corollary-po","scope":"Analysis/sequences-and-series/power-series/newton-s-identities-for-summable-sequences","status":"fully_proved","summary":"Let $(a_j)_{j \\ge 0}$ be a sequence of complex numbers with $\\sum_j |a_j| < \\infty$, and let $n \\ge 1$ be an integer. Then $\\sum_j |a_j^n| < \\infty$.","title":"Powers of an absolutely summable sequence are absolutely summable"},{"data_url":"scopes/Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts.json","id":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts","kind":"scope","navigation":"module","parent":18208,"result":false,"route":"../roadmap/Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/index.html","scope":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts","status":"fully_proved","title":"Series transformations via Taylor shifts"},{"id":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/ramanujan-part1-ch3-entry1-corollary1-bi","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry1Corollary1BinomialSeries.ramanujan_part1_ch3_entry1_corollary1_binomial_series_general","parent":18236,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/index.html#ramanujan-part1-ch3-entry1-corollary1-bi","scope":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts","status":"fully_proved","summary":"Let $n$ and $x$ be complex numbers with $|x| < 1$, write $(n)_k = n(n-1)\\cdots(n-k+1)$ for the falling factorial (with $(n)_0 = 1$), and let $Q_0, Q_1, Q_2, \\ldots$ be complex numbers such that the double series","title":"Ramanujan's binomial series transformation without the radius hypothesis"},{"id":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/ramanujan-part1-ch3-entry1-corollary1-bi-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry1Corollary1BinomialSeries.ramanujan_part1_ch3_entry1_corollary1_binomial_series","parent":18236,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/index.html#ramanujan-part1-ch3-entry1-corollary1-bi-2","scope":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts","status":"fully_proved","summary":"Let $n$ and $x$ be complex numbers with $|x| < 1$, write $(n)_k = n(n-1)\\cdots(n-k+1)$ for the falling factorial (with $(n)_0 = 1$), and let $Q_0, Q_1, Q_2, \\ldots$ be complex numbers such that, for some $R > 0$, the power series $\\sum_k Q_k z^k$ converges absolutely for every complex $z$ with $|z| < R$. Suppose that the double series","title":"Ramanujan's binomial series transformation (Chapter 3, Entry 1, Corollary 1)"},{"id":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/ramanujan-part1-ch3-entry1-series-transf","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry1SeriesTransformation.ramanujan_part1_ch3_entry1_series_transformation_general","parent":18236,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/index.html#ramanujan-part1-ch3-entry1-series-transf","scope":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts","status":"fully_proved","summary":"Let $R_1 > 1$, let $f : \\mathbb{C} \\to \\mathbb{C}$ be analytic on the open disc $|z| < R_1$, and let $Q_0, Q_1, Q_2, \\ldots$ be complex numbers such that the double series","title":"Ramanujan's series transformation without the radius hypothesis"},{"id":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/ramanujan-part1-ch3-entry1-series-transf-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry1SeriesTransformation.ramanujan_part1_ch3_entry1_series_transformation","parent":18236,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts/index.html#ramanujan-part1-ch3-entry1-series-transf-2","scope":"Analysis/sequences-and-series/power-series/series-transformations-via-taylor-shifts","status":"fully_proved","summary":"Let $R_1 > 1$, let $f : \\mathbb{C} \\to \\mathbb{C}$ be analytic on the open disc $|z| < R_1$, and let $Q_0, Q_1, Q_2, \\ldots$ be complex numbers such that, for some $R_2 > 0$, the power series $\\sum_n Q_n z^n$ converges absolutely for every complex $z$ with $|z| < R_2$. Suppose that the double series","title":"Ramanujan's series transformation (Chapter 3, Entry 1)"},{"data_url":"scopes/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series.json","id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","kind":"scope","navigation":"module","parent":18208,"result":false,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","title":"Touchard polynomials and generalized Bell series"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/ramanujan-part1-ch3-entry4-inner-summabl","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry4InnerSummable.ramanujan_part1_ch3_entry4_inner_summable","parent":18241,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#ramanujan-part1-ch3-entry4-inner-summabl","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"For every complex number $x$ and every natural number $n$, the series $\\sum_{j=0}^{\\infty} \\frac{(j+1)^n x^{j+1}}{j!}$ converges absolutely.","title":"Absolute convergence of Ramanujan's Bell-type series (Entry 4)"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/generalized-bell-generating","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry9IiGeneralizedbellgeneratingDefining.generalizedBellGenerating","parent":18241,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#generalized-bell-generating","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"For complex numbers $a$, $b$, $x$ and a natural number $n$, the generalized Bell function is $F(a, b, x; n) = e^{-x} \\sum_{k=0}^{\\infty} \\frac{(a + b(k + 1))^n x^{k+1}}{k!}$.","title":"Ramanujan's generalized Bell function"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/ramanujan-part1-ch3-entry9-ii-generalize-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry9IiGeneralizedbellgeneratingDefining.ramanujan_part1_ch3_entry9_ii_generalizedbellgenerating_defining","parent":18241,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#ramanujan-part1-ch3-entry9-ii-generalize-2","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"Let $F$ denote Ramanujan's generalized Bell function. For complex numbers $a$, $b$, $x$ and every natural number $n$, the series $\\sum_{k=0}^{\\infty} (a + b(k + 1))^n x^{k+1} / k!$ converges absolutely and its sum is $e^{x} F(a, b, x; n)$.","title":"Convergence of the generalized Bell series"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/ramanujan-part1-ch3-entry4","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry4.ramanujan_part1_ch3_entry4","parent":18241,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#ramanujan-part1-ch3-entry4","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"Let $F$ be Ramanujan's generalized Bell function. For all complex numbers $a$ and $x$, let $f(x, n) = F(0, 1, x; n) = e^{-x} \\sum_{j=0}^{\\infty} \\frac{(j+1)^n x^{j+1}}{j!}$ for natural numbers $n$, and set $f(x, -1) = 1$. Then the series $\\sum_{n=0}^{\\infty} \\frac{a^n}{n!} f(x, n-1)$ converges absolutely and its sum is $e^{x(e^a - 1)}$.","title":"Exponential generating function of Ramanujan's Bell-type series (Entry 4)"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/ramanujan-part1-ch3-entry5","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry5.ramanujan_part1_ch3_entry5","parent":18241,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#ramanujan-part1-ch3-entry5","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"Let $F$ be Ramanujan's generalized Bell function. For every natural number $n$ and every complex number $x$, write $f(x, m) = F(0, 1, x; m)$ for natural numbers $m$, and set $f(x, -1) = 1$. Then $f(x, n) = x \\sum_{k=0}^{n} \\binom{n}{k} f(x, k-1)$.","title":"Binomial recurrence for Ramanujan's Bell-type series (Entry 5)"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/ramanujan-part1-ch3-entry9-ii-generalize","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry9IiGeneralizedbellgenerating.ramanujan_part1_ch3_entry9_ii_generalizedbellgenerating","parent":18241,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#ramanujan-part1-ch3-entry9-ii-generalize","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"Let $F(a, b, x; n)$ be Ramanujan's generalized Bell function. For all complex numbers $a$, $b$, $x$ and $y$, the series $\\sum_{n=0}^{\\infty} \\frac{y^n}{n!} F(a, b, x; n)$ converges to $x e^{(a+b)y} e^{x(e^{by} - 1)}$.","title":"Exponential generating function of the generalized Bell function (Entry 9(ii))"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/f","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Section8Example4Integrable.f","parent":18241,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Bell function extended to "],["math","\\(n = -1\\)"]],"source":"Generalized Bell function extended to $n = -1$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#f","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"For an integer $n \\ge -1$ and a complex number $x$, $f(n, x) = 1$ if $n = -1$, and otherwise $f(n, x)$ is the generalized Bell function $F(0, 1, x; n) = e^{-x} \\sum_{k \\ge 0} (k + 1)^n x^{k + 1} / k!$.","title":"Generalized Bell function extended to $n = -1$"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/ramanujan-part1-ch3-section8-example4-in","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Section8Example4Integrable.ramanujan_part1_ch3_section8_example4_integrable","parent":18241,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrability of the scaled generalized Bell integrand on "],["math","\\((0, 1)\\)"]],"source":"Integrability of the scaled generalized Bell integrand on $(0, 1)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#ramanujan-part1-ch3-section8-example4-in","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"Let $f$ be the generalized Bell function extended to $n = -1$. For every natural number $m$ and every complex number $x$, the complex-valued function $t \\mapsto x f(m - 1, t x)$ of a real variable $t$ is interval integrable on $[0, 1]$ with respect to Lebesgue measure.","title":"Integrability of the scaled generalized Bell integrand on $(0, 1)$"},{"id":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/ramanujan-part1-ch3-section8-example4-be","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Section8Example4BernoulliIntegral.ramanujan_part1_ch3_section8_example4_bernoulli_integral","parent":18241,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series/index.html#ramanujan-part1-ch3-section8-example4-be","scope":"Analysis/sequences-and-series/power-series/touchard-polynomials-and-generalized-bell-series","status":"fully_proved","summary":"Let $f(-1, x) = 1$ and, for natural numbers $n$, let $f(n, x) = e^{-x} \\sum_{j=0}^{\\infty} (j + 1)^n x^{j+1} / j!$ for complex $x$. Let $B_n$ be the Bernoulli numbers, with $B_1 = -1/2$. Then for every natural number $m$ and every complex number $x$,","title":"Ramanujan's integral formula with Bernoulli numbers"},{"data_url":"scopes/Analysis/sequences-and-series/power-series/transformations-of-factorial-series.json","id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","kind":"scope","navigation":"module","parent":18208,"result":false,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","title":"Transformations of factorial series"},{"id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series/ramanujan-part1-ch3-entry2-corollary1-de","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry2Corollary1DerivativeSeries.ramanujan_part1_ch3_entry2_corollary1_derivative_series","parent":18251,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html#ramanujan-part1-ch3-entry2-corollary1-de","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","summary":"Let $R > 1$, let $f : \\mathbb{C} \\to \\mathbb{C}$ be analytic on the open disc $|w| < R$, and let $z$ be a complex number with $z + n \\ne 0$ for every natural number $n$. Suppose that the double series $\\sum_{j, m \\ge 0} \\frac{(-1)^j}{z(z+1)\\cdots(z+j)} \\cdot \\frac{f^{(j+m)}(0)}{m!}$ converges absolutely. Then there is a complex number $c$ such that the series $\\sum_{k \\ge 0} \\frac{f^{(k)}(0)}{(z+k)\\, k!}$ and\u2026","title":"Two derivative series with the same sum (Chapter 3, Entry 2, Corollary 1)"},{"id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series/harmonic-eq-mathlib-harmonic","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry2Corollary2HarmonicSeries.harmonic_eq_mathlib_harmonic","parent":18251,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html#harmonic-eq-mathlib-harmonic","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","summary":"For every natural number $n$, the rational number $\\sum_{k=0}^{n-1} \\frac{1}{k+1}$ equals Mathlib's harmonic number $H_n = \\sum_{k=0}^{n-1} (k+1)^{-1}$.","title":"Agreement of two definitions of the harmonic numbers"},{"id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series/ramanujan-part1-ch3-entry2-corollary2-ha","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry2Corollary2HarmonicSeries.ramanujan_part1_ch3_entry2_corollary2_harmonic_series_mathlib","parent":18251,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html#ramanujan-part1-ch3-entry2-corollary2-ha","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","summary":"Let $H_k = \\sum_{i=0}^{k-1} (i+1)^{-1}$ be Mathlib's $k$-th harmonic number. For every complex number $x$, the series $\\sum_{k \\ge 1} \\frac{H_k x^k}{k!}$ and $\\sum_{k \\ge 1} \\frac{(-1)^{k-1} x^k}{k!\\, k}$ converge absolutely, and $\\sum_{k=1}^{\\infty} \\frac{H_k x^k}{k!} = e^x \\sum_{k=1}^{\\infty} \\frac{(-1)^{k-1} x^k}{k!\\, k}$.","title":"Ramanujan's series for the harmonic numbers, with Mathlib's harmonic numbers"},{"id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series/ramanujan-part1-ch3-entry2-corollary2-ha-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry2Corollary2HarmonicSeries.ramanujan_part1_ch3_entry2_corollary2_harmonic_series","parent":18251,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html#ramanujan-part1-ch3-entry2-corollary2-ha-2","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","summary":"Let $H_k = \\sum_{i=1}^{k} \\frac{1}{i}$ be the $k$-th harmonic number. For every complex number $x$, the series $\\sum_{k \\ge 1} \\frac{H_k x^k}{k!}$ and $\\sum_{k \\ge 1} \\frac{(-1)^{k-1} x^k}{k!\\, k}$ converge absolutely, and $\\sum_{k=1}^{\\infty} \\frac{H_k x^k}{k!} = e^x \\sum_{k=1}^{\\infty} \\frac{(-1)^{k-1} x^k}{k!\\, k}$.","title":"Ramanujan's series for the harmonic numbers (Chapter 3, Entry 2, Corollary 2)"},{"id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series/ramanujan-part1-ch3-entry2-rightsummable","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry2Rightsummable.ramanujan_part1_ch3_entry2_rightsummable","parent":18251,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html#ramanujan-part1-ch3-entry2-rightsummable","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","summary":"For all complex numbers $x$ and $z$ such that $z + r \\ne 0$ for every natural number $r$, the series $\\sum_{j \\ge 0} \\frac{(-1)^j x^{j+1}}{z(z+1)\\cdots(z+j)}$ converges absolutely.","title":"Convergence of the right side of Ramanujan's Chapter 3, Entry 2"},{"id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series/ramanujan-part1-ch3-entry2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry2.ramanujan_part1_ch3_entry2","parent":18251,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html#ramanujan-part1-ch3-entry2","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","summary":"For all complex numbers $x$ and $z$ such that $z + r \\ne 0$ for every natural number $r$, the series $\\sum_{j \\ge 0} \\frac{x^{j+1}}{(z+j)\\, j!}$ converges absolutely, and its sum is $e^x \\sum_{j \\ge 0} \\frac{(-1)^j x^{j+1}}{z(z+1)\\cdots(z+j)}$.","title":"Ramanujan's exponential transformation of a factorial series (Chapter 3, Entry 2)"},{"id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series/ramanujan-part1-ch3-entry2-leftsummable","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry2Leftsummable.ramanujan_part1_ch3_entry2_leftsummable","parent":18251,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html#ramanujan-part1-ch3-entry2-leftsummable","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","summary":"For all complex numbers $x$ and $z$ such that $z + r \\ne 0$ for every natural number $r$, the series $\\sum_{j \\ge 0} \\frac{x^{j+1}}{(z+j)\\, j!}$ converges absolutely.","title":"Convergence of the left side of Ramanujan's Chapter 3, Entry 2"},{"id":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series/ramanujan-part1-ch3-entry2-formula23-ser","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry2Formula23SeriesPair.ramanujan_part1_ch3_entry2_formula23_series_pair","parent":18251,"result":true,"route":"../roadmap/Analysis/sequences-and-series/power-series/transformations-of-factorial-series/index.html#ramanujan-part1-ch3-entry2-formula23-ser","scope":"Analysis/sequences-and-series/power-series/transformations-of-factorial-series","status":"fully_proved","summary":"Let $x$ and $z$ be complex numbers with $z \\ne -(n+1)$ for every natural number $n$. Then the series $\\sum_{j \\ge 0} \\frac{(-1)^j x^{j+1}}{(z+j+1)\\, j!}$ converges absolutely, say with sum $S$, and the series $\\sum_{j \\ge 0} \\frac{x^{j+1}}{(z+1)(z+2)\\cdots(z+j+1)}$ converges absolutely with sum $e^x S$.","title":"Ramanujan's companion series transformation (Chapter 3, Entry 2, formula (2.3))"},{"data_url":"scopes/Analysis/sequences-and-series/multisection-of-series-by-index-factorization.json","id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","kind":"scope","navigation":"module","parent":18053,"result":false,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","title":"Multisection of series by index factorization"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-two-adic-parametric-multisection","kind":"node","lean":"MetaMathlibExt.tsum_twoAdic_parametric_multisection","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-two-adic-parametric-multisection","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $q \\in \\mathbb{C}$ and let $b : \\mathbb{N} \\to \\mathbb{C}$ be a sequence (its value at $0$ is not used). Suppose that the families $(q^j b_{(2n-1) 2^j})$ and $(q^j (q - 1) b_{2n \\cdot 2^j})$, indexed by the pairs $(j, n)$ of natural numbers with $n \\ge 1$, are summable. Then","title":"Parametric two-adic multisection"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-three-adic-parametric-multisection","kind":"node","lean":"MetaMathlibExt.tsum_threeAdic_parametric_multisection","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-three-adic-parametric-multisection","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $q$ be a complex number and let $(b_m)_{m \\ge 1}$ be a sequence of complex numbers such that the three families $q^j b_{(3n-1) 3^j}$, $q^j b_{(3n-2) 3^j}$ and $q^j (q - 1) b_{3n \\cdot 3^j}$, each indexed by the pairs $(j, n)$ with $j \\ge 0$ and $n \\ge 1$, are summable. Then","title":"Parametric 3-adic multisection of a series"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-three-adic-multiples-regroup","kind":"node","lean":"MetaMathlibExt.tsum_threeAdic_multiples_regroup","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-three-adic-multiples-regroup","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $(b_m)_{m \\ge 1}$ be a sequence of complex numbers such that the family $(b_{3m})_{m \\ge 1}$ is summable. Then","title":"Regrouping the multiples of 3 by their 3-adic factorization"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-base-adic-parametric-multisection","kind":"node","lean":"MetaMathlibExt.tsum_baseAdic_parametric_multisection","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-base-adic-parametric-multisection","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $B \\ge 2$ be a natural number, let $q$ be a complex number and let $(c_m)_{m \\ge 1}$ be a sequence of complex numbers. Suppose that the family $q^j c_{(Bn-k) B^j}$ indexed by the triples $(j, n, k)$ with $j \\ge 0$, $n \\ge 1$ and $1 \\le k \\le B - 1$ is summable, and that the family $q^j (q - 1) c_{Bn \\cdot B^j}$ indexed by the pairs $(j, n)$ with $j \\ge 0$ and $n \\ge 1$ is summable. Then","title":"Parametric base-B multisection of a series"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-base-adic-multiples-regroup","kind":"node","lean":"MetaMathlibExt.tsum_baseAdic_multiples_regroup","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-base-adic-multiples-regroup","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $B \\ge 2$ be a natural number and let $(c_m)_{m \\ge 1}$ be a sequence of complex numbers such that the family $(c_{Bm})_{m \\ge 1}$ is summable. Then","title":"Regrouping the multiples of B by their B-adic factorization"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-base-adic-weighted-multisection","kind":"node","lean":"MetaMathlibExt.tsum_baseAdic_weighted_multisection","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-base-adic-weighted-multisection","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $B \\ge 2$ be a natural number, let $\\varphi, \\chi : \\mathbb{N} \\to \\mathbb{C}$ be weights and let $(c_m)_{m \\ge 1}$ be a sequence of complex numbers, and for $m \\ge 1$ let $\\nu_B(m)$ be the largest $k$ with $B^k \\mid m$. Suppose that the family $\\varphi(j) c_{(Bn-k) B^j}$ indexed by the triples $(j, n, k)$ with $j \\ge 0$, $n \\ge 1$ and $1 \\le k \\le B - 1$ is summable, and that the family\u2026","title":"Weighted base-B multisection of a series"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-two-adic-truncated-regroup","kind":"node","lean":"MetaMathlibExt.tsum_twoAdic_truncated_regroup","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-two-adic-truncated-regroup","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For every natural number $J$ and all sequences $\\varphi, c : \\mathbb{N} \\to \\mathbb{C}$,","title":"Truncated two-adic regrouping"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-two-adic-high-regroup","kind":"node","lean":"MetaMathlibExt.tsum_twoAdic_high_regroup","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-two-adic-high-regroup","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For every natural number $J$ and every sequence $(c_m)_{m \\ge 1}$ of complex numbers,","title":"Regrouping the multiples of a power of 2 by their 2-adic factorization"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-two-adic-low-regroup","kind":"node","lean":"MetaMathlibExt.tsum_twoAdic_low_regroup","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-two-adic-low-regroup","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For every natural number $J$ and every sequence $c : \\mathbb{N} \\to \\mathbb{C}$,","title":"Two-adic regrouping of the non-multiples of a power of two"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tsum-two-adic-double-multisection","kind":"node","lean":"MetaMathlibExt.tsum_twoAdic_double_multisection","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tsum-two-adic-double-multisection","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $(b_{p,q})_{p, q \\ge 1}$ be a double sequence of complex numbers such that each of the four families $2^{j+k} b_{(2n-1) 2^j, (2m-1) 2^k}$, $2^{j+k} b_{(2n) 2^j, (2m) 2^k}$, $2^{j+k} b_{(2n) 2^j, (2m-1) 2^k}$ and $2^{j+k} b_{(2n-1) 2^j, (2m) 2^k}$, indexed by the quadruples $(j, k, n, m)$ with $j, k \\ge 0$ and $n, m \\ge 1$, is summable. Then","title":"Double 2-adic multisection of a double series"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/lambert-sigma-eq-tsum-two-adic","kind":"node","lean":"MetaMathlibExt.lambert_sigma_eq_tsum_twoAdic","parent":18260,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-adic expansion of the weighted Lambert series "],["math","\\(f\\)"]],"source":"$2$-adic expansion of the weighted Lambert series $f$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#lambert-sigma-eq-tsum-two-adic","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For every complex number $q$ with $|q| < 1$, the weighted Lambert series $f(q) = \\sum_{n \\ge 1} n q^n / (1 - q^n)$ and $g(q) = \\sum_{n \\ge 1} n q^n / (1 + q^n)$ satisfy $f(q) = \\sum_{j, k \\ge 0} 2^{2j+k} \\bigl(g(q^{2^{j+k}}) - 4 g(q^{2^{j+k+1}})\\bigr)$.","title":"$2$-adic expansion of the weighted Lambert series $f$"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tprod-q-pochhammer-two-adic","kind":"node","lean":"MetaMathlibExt.tprod_qPochhammer_twoAdic","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tprod-q-pochhammer-two-adic","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For every complex number $q$ with $|q| < 1$, with the infinite q-Pochhammer symbol $(a; r)_\\infty = \\prod_{k \\ge 0} (1 - a r^k)$, the unconditional infinite product over $j \\ge 0$ satisfies","title":"2-adic dissection of the Euler function"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tprod-q-pochhammer-two-adic-parametric","kind":"node","lean":"MetaMathlibExt.tprod_qPochhammer_twoAdic_parametric","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tprod-q-pochhammer-two-adic-parametric","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $q$ and $p$ be real numbers with $0 < q < 1$, and write $(a; r)_\\infty = \\prod_{k \\ge 0} (1 - a r^k)$ for real $a$ and $r$ (an unconditional infinite product). Then, with real exponents $p - 1$ and $p^j$,","title":"Parametric 2-adic dissection of the Euler function for real q"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tprod-q-pochhammer-three-adic-parametric","kind":"node","lean":"MetaMathlibExt.tprod_qPochhammer_threeAdic_parametric","parent":18260,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parametric "],["math","\\(3\\)"],["text","-adic dissection of the Euler function"]],"source":"Parametric $3$-adic dissection of the Euler function"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tprod-q-pochhammer-three-adic-parametric","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For real numbers $x$ and $y$ with $0 < y < 1$ write $(x; y)_\\infty = \\prod_{k \\ge 0} (1 - x y^k)$. For all real numbers $q$ and $p$ with $0 < q < 1$, $\\prod_{j \\ge 0} \\bigl((q^{3^j}; q^{3^{j+1}})_\\infty\\, (q^{2 \\cdot 3^j}; q^{3^{j+1}})_\\infty / (q^{3^{j+1}}; q^{3^{j+1}})_\\infty^{\\,p - 1}\\bigr)^{p^j} = (q; q)_\\infty$.","title":"Parametric $3$-adic dissection of the Euler function"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tprod-q-pochhammer-two-adic-factorizatio","kind":"node","lean":"MetaMathlibExt.tprod_qPochhammer_twoAdic_factorization","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tprod-q-pochhammer-two-adic-factorizatio","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For every complex number $q$ with $|q| < 1$, with the infinite q-Pochhammer symbol $(a; r)_\\infty = \\prod_{k \\ge 0} (1 - a r^k)$, the unconditional infinite product over $j \\ge 0$ satisfies $\\prod_{j \\ge 0} (q^{2^j}; q^{2^{j+1}})_\\infty = (q; q)_\\infty$.","title":"2-adic product factorization of the Euler function"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tprod-q-pochhammer-three-adic-factorizat","kind":"node","lean":"MetaMathlibExt.tprod_qPochhammer_threeAdic_factorization","parent":18260,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(3\\)"],["text","-adic factorization of the Euler function"]],"source":"$3$-adic factorization of the Euler function"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tprod-q-pochhammer-three-adic-factorizat","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Write $(x; y)_\\infty = \\prod_{k \\ge 0} (1 - x y^k)$ for the infinite $q$-Pochhammer symbol, where $x$ and $y$ are complex and $|y| < 1$. For every complex number $q$ with $|q| < 1$, $\\prod_{j \\ge 0} (q^{3^j}; q^{3^{j+1}})_\\infty\\, (q^{2 \\cdot 3^j}; q^{3^{j+1}})_\\infty = (q; q)_\\infty$.","title":"$3$-adic factorization of the Euler function"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/tprod-q-pochhammer-shifted-two-adic","kind":"node","lean":"MetaMathlibExt.tprod_qPochhammer_shifted_twoAdic","parent":18260,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-adic multisection of the infinite "],["math","\\(q\\)"],["text","-Pochhammer symbol"]],"source":"$2$-adic multisection of the infinite $q$-Pochhammer symbol"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#tprod-q-pochhammer-shifted-two-adic","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Write $(x; y)_\\infty = \\prod_{k \\ge 0} (1 - x y^k)$ for the infinite $q$-Pochhammer symbol, where $x$ and $y$ are complex and $|y| < 1$. For complex numbers $a$ and $q$ with $|q| < 1$, $\\prod_{j \\ge 0} \\bigl((a q^{2^j - 1}; q^{2^{j+1}})_\\infty / (a q^{2^{j+1} - 1}; q^{2^{j+1}})_\\infty\\bigr)^{2^j} = (a; q)_\\infty$.","title":"$2$-adic multisection of the infinite $q$-Pochhammer symbol"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/gamma-ratio-eq-tprod-two-adic","kind":"node","lean":"MetaMathlibExt.gamma_ratio_eq_tprod_twoAdic","parent":18260,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-adic dissection of a ratio of Gamma functions"]],"source":"$2$-adic dissection of a ratio of Gamma functions"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#gamma-ratio-eq-tprod-two-adic","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For complex numbers $a$ and $z$ write $f(a, z) = \\Gamma(a + 1)^2 / \\bigl(\\Gamma(a + 1 - iz)\\, \\Gamma(a + 1 + iz)\\bigr)$. If $a$ is not a negative integer, then for every complex number $z$, $f(a, z) = \\prod_{j \\ge 0} f(a / 2^{j+1} - 1/2,\\ z / 2^{j+1})$.","title":"$2$-adic dissection of a ratio of Gamma functions"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/lambert-mu-sigma-eq-tsum-two-adic","kind":"node","lean":"MetaMathlibExt.lambertMu_sigma_eq_tsum_twoAdic","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#lambert-mu-sigma-eq-tsum-two-adic","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"For every integer $\\mu \\ge 1$ and every complex number $q$ with $|q| < 1$, the power-weighted Lambert series $f_\\mu(q) = \\sum_{n \\ge 1} n^\\mu q^n / (1 - q^n)$ and $g_\\mu(q) = \\sum_{n \\ge 1} n^\\mu q^n / (1 + q^n)$ satisfy","title":"Two-adic multisection of the power-weighted Lambert series"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/eta-mul-lseries-iff-lambert","kind":"node","lean":"MetaMathlibExt.eta_mul_LSeries_iff_lambert","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#eta-mul-lseries-iff-lambert","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $(a_n)$ and $(b_n)$ be sequences of complex numbers and let $C$ and $K$ be real numbers with $|a_n| \\le C n^K$ and $|b_n| \\le C n^K$ for every $n \\ge 1$. Let $\\eta(s) = \\sum_{n \\ge 1} (-1)^{n-1} n^{-s}$ be the Dirichlet eta function. Then $\\eta(s) \\sum_{n \\ge 1} a_n n^{-s} = \\sum_{n \\ge 1} b_n n^{-s}$ for every real number $s > \\max(K + 2, 1)$ if and only if\u2026","title":"Borwein's eta\u2013Lambert correspondence"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/gamma-quadratic-ratio","kind":"node","lean":"MetaMathlibExt.gammaQuadraticRatio","parent":18260,"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#gamma-quadratic-ratio","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"fully_proved","summary":"For complex numbers $a$, $b$ and $z$, the Gamma quadratic ratio is $g(a, b, z) = \\dfrac{b}{b - z} \\cdot \\dfrac{\\Gamma(a - \\sqrt{a^2 - b})\\, \\Gamma(a + \\sqrt{a^2 - b})} {\\Gamma(a - \\sqrt{a^2 - b + z})\\, \\Gamma(a + \\sqrt{a^2 - b + z})}$, where $\\sqrt{\\cdot}$ is the principal complex square root (the value does not depend on the choice of root).","title":"The Gamma quadratic ratio"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/gamma-quadratic-ratio-zero-right","kind":"node","lean":"MetaMathlibExt.gammaQuadraticRatio_zero_right","parent":18260,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Gamma quadratic ratio at "],["math","\\(z = 0\\)"]],"source":"The Gamma quadratic ratio at $z = 0$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#gamma-quadratic-ratio-zero-right","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"fully_proved","summary":"Let $a$ and $b$ be complex numbers with $b \\ne 0$ and $\\Gamma(a - \\sqrt{a^2 - b})\\, \\Gamma(a + \\sqrt{a^2 - b}) \\ne 0$, where $\\sqrt{\\cdot}$ denotes the principal complex square root. Then the Gamma quadratic ratio $g$ satisfies $g(a, b, 0) = 1$.","title":"The Gamma quadratic ratio at $z = 0$"},{"id":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization/gamma-quadratic-ratio-eq-tprod-two-adic","kind":"node","lean":"MetaMathlibExt.gammaQuadraticRatio_eq_tprod_twoAdic","parent":18260,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2\\)"],["text","-adic dissection of the Gamma quadratic ratio"]],"source":"$2$-adic dissection of the Gamma quadratic ratio"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/multisection-of-series-by-index-factorization/index.html#gamma-quadratic-ratio-eq-tprod-two-adic","scope":"Analysis/sequences-and-series/multisection-of-series-by-index-factorization","status":"can_prove","summary":"Let $a, b, z$ be complex numbers such that $m^2 + 2am + b \\ne 0$ and $m^2 + 2am + b - z \\ne 0$ for every integer $m$ that is nonnegative or of the form $m = -2^j$ with $j \\ge 0$ an integer. For $j \\ge 0$ put $a_j = a / 2^{j+1} - 1/2$ and $b_j = 1/4 - a / 2^{j+1} + b / 2^{2j+2}$. Then the Gamma quadratic ratio $g$ satisfies $g(a, b, z) = \\prod_{j \\ge 0} g(a_j, b_j, z / 2^{2j+2})$.","title":"$2$-adic dissection of the Gamma quadratic ratio"},{"data_url":"scopes/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums.json","id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","kind":"scope","navigation":"module","parent":18053,"result":false,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","title":"Ramanujan's extension of partial sums"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/chapter6-shifted-derived-series-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.chapter6ShiftedDerivedSeriesTerm","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#chapter6-shifted-derived-series-term","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"For $f : \\mathbb{R} \\to \\mathbb{R}$, natural numbers $q$ and $k$, and $x \\in \\mathbb{R}$, the $k$-th term of the shifted series of $q$-th derivatives of $f$ is $f^{(q)}(k + 1 + x)$, where $f^{(q)}$ is the $q$-th iterated derivative of $f$ and a derivative that does not exist is taken to be $0$.","title":"Terms of the shifted derivative series"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/chapter6-derived-series-constant","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.chapter6DerivedSeriesConstant","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#chapter6-derived-series-constant","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"For $f : \\mathbb{R} \\to \\mathbb{R}$ and a natural number $q$, the constant $c_q$ is $\\sum_{k \\ge 1} f^{(q)}(k)$, the sum of the values of the $q$-th iterated derivative of $f$ at the positive integers, where a derivative that does not exist is taken to be $0$ and a series that is not summable is given the value $0$.","title":"Ramanujan's derivative-series constants"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/chapter6-convergent-sum-extension","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.chapter6ConvergentSumExtension","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#chapter6-convergent-sum-extension","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"For $f : \\mathbb{R} \\to \\mathbb{R}$, the extension of the partial sums of $f$ to real arguments is the function $F(x) = \\sum_{k \\ge 1} f(k) - \\sum_{k \\ge 1} f(k + x)$ on $\\mathbb{R}$, where each series is an unconditional sum and a series that is not summable is given the value $0$.","title":"Ramanujan's extension of partial sums"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/chapter6-entry6-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.chapter6Entry6Term","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#chapter6-entry6-term","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"For $f : \\mathbb{R} \\to \\mathbb{R}$, $x \\in \\mathbb{R}$ and a natural number $j$, the $j$-th term is $-c_{j+1} x^{j+1} / (j+1)!$, where $c_q = \\sum_{k \\ge 1} f^{(q)}(k)$ with $f^{(q)}$ the $q$-th iterated derivative of $f$; a derivative that does not exist is taken to be $0$ and a series that is not summable is given the value $0$.","title":"Terms of the Taylor series in Entry 6"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/chapter6-partial-sum","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.chapter6PartialSum","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#chapter6-partial-sum","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"For $f : \\mathbb{R} \\to \\mathbb{R}$ and a natural number $n$, the partial sum is $f(1) + f(2) + \\cdots + f(n)$, which is $0$ when $n = 0$.","title":"Partial sums of a real function over the positive integers"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/has-sum-chapter6-derived-series-constant","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.hasSum_chapter6DerivedSeriesConstant","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#has-sum-chapter6-derived-series-constant","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$, let $q$ be a natural number, and let $f^{(q)}$ be the $q$-th iterated derivative of $f$, where a derivative that does not exist is taken to be $0$. If the family $\\bigl(f^{(q)}(k + 1)\\bigr)_{k \\in \\mathbb{N}}$ is summable, then its sum is the constant $c_q = \\sum_{k \\ge 1} f^{(q)}(k)$.","title":"The derivative-series constant is the sum of its series"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/chapter6-convergent-sum-extension-sub-su","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.chapter6ConvergentSumExtension_sub_sub_one","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#chapter6-convergent-sum-extension-sub-su","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and let $F(y) = \\sum_{k \\ge 1} f(k) - \\sum_{k \\ge 1} f(k + y)$, where a series that is not summable is given the value $0$. If $x \\in \\mathbb{R}$ and the series $\\sum_{k \\ge 0} f(k + x)$ is summable, then $F(x) - F(x - 1) = f(x)$.","title":"Difference equation for Ramanujan's extension"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/summable-chapter6-shifted-derived-series","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.summable_chapter6ShiftedDerivedSeriesTerm_natCast","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#summable-chapter6-shifted-derived-series","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and $q \\in \\mathbb{N}$, and write $f^{(q)}$ for the $q$-th iterated derivative of $f$, where a derivative that does not exist is taken to be $0$. If the series $\\sum_{k \\ge 0} f^{(q)}(k + 1)$ converges absolutely, then for every $m \\in \\mathbb{N}$ the series $\\sum_{k \\ge 0} f^{(q)}(k + 1 + m)$ converges absolutely.","title":"Tails of the shifted series of derivatives converge"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/chapter6-convergent-sum-extension-nat-ca","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.chapter6ConvergentSumExtension_natCast","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#chapter6-convergent-sum-extension-nat-ca","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and let $F(x) = \\sum_{k \\ge 1} f(k) - \\sum_{k \\ge 1} f(k + x)$, where a series that is not summable is given the value $0$. If the series $\\sum_{k \\ge 1} f(k)$ is summable, then $F(m) = f(1) + f(2) + \\cdots + f(m)$ for every natural number $m$ (for $m = 0$ the right side is the empty sum $0$).","title":"Ramanujan's extension agrees with the partial sums at natural numbers"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/iterated-deriv-chapter6-convergent-sum-e","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.iteratedDeriv_chapter6ConvergentSumExtension_apply_zero","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#iterated-deriv-chapter6-convergent-sum-e","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be infinitely differentiable and let $\\rho > 0$. Write $c_q = \\sum_{k \\ge 1} f^{(q)}(k)$ and $F(x) = c_0 - \\sum_{k \\ge 1} f(k + x)$, where a series that is not summable is given the value $0$. Suppose that the series $\\sum_{k \\ge 1} f^{(q)}(k)$ is summable for every $q \\ge 0$, and that for every $q \\ge 1$ there is a summable sequence $(b_k)_{k \\ge 0}$ with\u2026","title":"Derivatives of Ramanujan's extension at the origin"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/exists-has-sum-chapter6-entry6-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.exists_hasSum_chapter6Entry6Term","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#exists-has-sum-chapter6-entry6-term","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be infinitely differentiable and let $\\rho > 0$. Write $c_q = \\sum_{k \\ge 1} f^{(q)}(k)$ and $F(x) = c_0 - \\sum_{k \\ge 1} f(k + x)$, where a series that is not summable is given the value $0$. Suppose that the series $\\sum_{k \\ge 1} f^{(q)}(k)$ is summable for every $q \\ge 0$, that for every $q \\ge 1$ there is a summable sequence $(b_k)_{k \\ge 0}$ with\u2026","title":"Taylor expansion of Ramanujan's extension at the origin"},{"id":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/ramanujan-part1-ch6-entry6-harmonic","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry6Harmonic.ramanujan_part1_ch6_entry6_harmonic","parent":18284,"result":true,"route":"../roadmap/Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums/index.html#ramanujan-part1-ch6-entry6-harmonic","scope":"Analysis/sequences-and-series/ramanujan-s-extension-of-partial-sums","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be infinitely differentiable and let $\\rho > 0$. Suppose that for every $q \\in \\mathbb{N}$ and every $x \\in \\mathbb{R}$ the series $\\sum_{k \\ge 0} f^{(q)}(k + 1 + x)$ converges absolutely, and that for every $q \\in \\mathbb{N}$ there is a summable sequence $(b_k)_{k \\ge 0}$ with $|f^{(q)}(k + 1 + x)| \\le b_k$ for all $k \\in \\mathbb{N}$ and all $x \\in [-\\rho, \\rho]$. Put\u2026","title":"Ramanujan's extension of the partial sums of a smooth function"},{"data_url":"scopes/Analysis/sequences-and-series/series-of-the-form-r-1-a-x.json","id":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x","kind":"scope","navigation":"module","parent":18053,"result":false,"route":"../roadmap/Analysis/sequences-and-series/series-of-the-form-r-1-a-x/index.html","scope":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x","status":"fully_proved","title":"Series of the form \u03a3 r\u1d4f/(1 \u2212 a x\u1d4f)"},{"id":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x/ramanujan-part1-ch6-entry13-ramanujanfac","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry13Ramanujanfactorial.ramanujan_part1_ch6_entry13_ramanujanfactorial","parent":18297,"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-of-the-form-r-1-a-x/index.html#ramanujan-part1-ch6-entry13-ramanujanfac","scope":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x","status":"fully_proved","summary":"For every real number $x$ with $|x| > 1$, we have $x^k - 1 \\ne 0$ and $x^{k^2}(x^{2k} - 1) \\ne 0$ for every integer $k \\ge 1$, the series $\\sum_{k \\ge 1} (-1)^{k+1}/(x^k - 1)$ and $\\sum_{k \\ge 1} (-1)^{k+1}(x^{2k} + 1)/(x^{k^2}(x^{2k} - 1))$ are summable, and $\\sum_{k=1}^{\\infty} (-1)^{k+1}/(x^k - 1) = \\sum_{k=1}^{\\infty} (-1)^{k+1}(x^{2k} + 1)/(x^{k^2}(x^{2k} - 1))$.","title":"Ramanujan's transformation of the alternating series of 1/(x^k - 1)"},{"id":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x/ramanujan-part1-ch6-entry16-finite-sum-g","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry16FiniteSum.ramanujan_part1_ch6_entry16_finite_sum_general","parent":18297,"reading_title":{"format":"inline-parts-v1","parts":[["text","A finite identity for "],["math","\\(\\sum_k r^k/(1-ax^k)\\)"],["text"," over a field"]],"source":"A finite identity for $\\sum_k r^k/(1-ax^k)$ over a field"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-of-the-form-r-1-a-x/index.html#ramanujan-part1-ch6-entry16-finite-sum-g","scope":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x","status":"fully_proved","summary":"Let $K$ be a field, let $a, r, x \\in K$ and let $n \\ge 0$ be an integer. Suppose that $1 - a x^k \\ne 0$ and $1 - r x^{k-1} \\ne 0$ for every integer $k$ with $1 \\le k \\le n$. Then $\\sum_{k=1}^{n} \\frac{r^k}{1 - a x^k} = \\sum_{k=1}^{n} \\frac{(a r x^k)^k}{1 - a x^k} + \\sum_{k=1}^{n} \\frac{a^{k-1} \\left( (r x^{k-1})^k - (r x^{k-1})^{n+1} \\right)}{1 - r x^{k-1}}$. For $n = 0$ all three sums are empty.","title":"A finite identity for $\\sum_k r^k/(1-ax^k)$ over a field"},{"id":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x/ramanujan-part1-ch6-entry16-finite-sum","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry16FiniteSum.ramanujan_part1_ch6_entry16_finite_sum","parent":18297,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's finite identity for "],["math","\\(\\sum_k r^k/(1-ax^k)\\)"]],"source":"Ramanujan's finite identity for $\\sum_k r^k/(1-ax^k)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-of-the-form-r-1-a-x/index.html#ramanujan-part1-ch6-entry16-finite-sum","scope":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x","status":"fully_proved","summary":"Let $a, r, x$ be real numbers and let $n$ be a positive integer. Suppose that $1 - a x^k \\ne 0$ and $1 - r x^{k-1} \\ne 0$ for every integer $k$ with $1 \\le k \\le n$. Then $\\sum_{k=1}^{n} \\frac{r^k}{1 - a x^k} = \\sum_{k=1}^{n} \\frac{(a r x^k)^k}{1 - a x^k} + \\sum_{k=1}^{n} \\frac{a^{k-1} \\left( (r x^{k-1})^k - (r x^{k-1})^{n+1} \\right)}{1 - r x^{k-1}}$.","title":"Ramanujan's finite identity for $\\sum_k r^k/(1-ax^k)$"},{"id":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x/ramanujan-part1-ch6-entry16-infinite-sum","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry16InfiniteSum.ramanujan_part1_ch6_entry16_infinite_sum","parent":18297,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's infinite identity for "],["math","\\(\\sum_k r^k/(1-ax^k)\\)"]],"source":"Ramanujan's infinite identity for $\\sum_k r^k/(1-ax^k)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-of-the-form-r-1-a-x/index.html#ramanujan-part1-ch6-entry16-infinite-sum","scope":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x","status":"fully_proved","summary":"Let $a, r, x$ be complex numbers with $|r| < 1$ and $|x| \\le 1$ such that $1 - a x^k \\ne 0$ for every integer $k \\ge 1$, and such that, if $|x| = 1$, then $|a r| < 1$ and there is a real $\\delta > 0$ with $|1 - a x^k| \\ge \\delta$ for every $k \\ge 1$. Then $1 - r x^{k-1} \\ne 0$ for every $k \\ge 1$, the three series $\\sum_{k=1}^{\\infty} \\frac{r^k}{1 - a x^k}$, $\\sum_{k=1}^{\\infty} \\frac{(a r x^k)^k}{1 - a x^k}$ and\u2026","title":"Ramanujan's infinite identity for $\\sum_k r^k/(1-ax^k)$"},{"id":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x/ramanujan-part1-ch6-entry17-stirlingcoef","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry17Stirlingcoeffs.ramanujan_part1_ch6_entry17_stirlingcoeffs","parent":18297,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's identity for "],["math","\\(\\sum_k (a + kb) \\nu^k / (1 - m x^k)\\)"]],"source":"Ramanujan's identity for $\\sum_k (a + kb) \\nu^k / (1 - m x^k)$"},"result":true,"route":"../roadmap/Analysis/sequences-and-series/series-of-the-form-r-1-a-x/index.html#ramanujan-part1-ch6-entry17-stirlingcoef","scope":"Analysis/sequences-and-series/series-of-the-form-r-1-a-x","status":"fully_proved","summary":"Let $a, b, m, \\nu, x$ be real numbers with $|x| \\le 1$, $|\\nu| < 1$ and $m \\ne 0$, such that $|m \\nu| < 1$ if $|x| = 1$, and such that $1 - m x^k \\ne 0$ for every integer $k \\ge 0$ (here $x^0 = 1$). Then $1 - \\nu x^k \\ne 0$ for every $k \\ge 0$, the series $L = \\sum_{k=0}^{\\infty} \\frac{(a + kb) \\, \\nu^k}{1 - m x^k}$,\u2026","title":"Ramanujan's identity for $\\sum_k (a + kb) \\nu^k / (1 - m x^k)$"},{"data_url":"scopes/Analysis/numerical-analysis.json","id":"Analysis/numerical-analysis","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/numerical-analysis/index.html","scope":"Analysis/numerical-analysis","status":"fully_proved","summary":"Algorithms with error analysis: bisection and Newton's method, Euler and Runge\u2013Kutta methods, Galerkin and finite element estimates (C\u00e9a, Aubin\u2013Nitsche, Strang, Banach\u2013Ne\u010das\u2013Babu\u0161ka).","title":"Numerical analysis"},{"data_url":"scopes/Analysis/numerical-analysis/aubin-nitsche-lemma.json","id":"Analysis/numerical-analysis/aubin-nitsche-lemma","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/aubin-nitsche-lemma/index.html","scope":"Analysis/numerical-analysis/aubin-nitsche-lemma","status":"fully_proved","title":"Aubin\u2013Nitsche lemma"},{"id":"Analysis/numerical-analysis/aubin-nitsche-lemma/is-coercive-flip","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.AubinNitsche.isCoercive_flip","parent":18304,"result":true,"route":"../roadmap/Analysis/numerical-analysis/aubin-nitsche-lemma/index.html#is-coercive-flip","scope":"Analysis/numerical-analysis/aubin-nitsche-lemma","status":"fully_proved","summary":"The transpose of a coercive bilinear form is coercive with the same constant, since `B.flip u u = B u u`.","title":"Transpose of a coercive bilinear form is coercive"},{"id":"Analysis/numerical-analysis/aubin-nitsche-lemma/exists-adjoint-solution","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.AubinNitsche.exists_adjointSolution","parent":18304,"result":true,"route":"../roadmap/Analysis/numerical-analysis/aubin-nitsche-lemma/index.html#exists-adjoint-solution","scope":"Analysis/numerical-analysis/aubin-nitsche-lemma","status":"fully_proved","summary":"If `B` is coercive, then for every `e : H` there is `z : H` with `B v z = \u27eaJ e, J v\u27eb_\u211d` for all `v : H`.","title":"Existence for the adjoint problem"},{"id":"Analysis/numerical-analysis/aubin-nitsche-lemma/aubin-nitsche-pointwise","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.AubinNitsche.aubinNitsche_pointwise","parent":18304,"result":true,"route":"../roadmap/Analysis/numerical-analysis/aubin-nitsche-lemma/index.html#aubin-nitsche-pointwise","scope":"Analysis/numerical-analysis/aubin-nitsche-lemma","status":"fully_proved","summary":"If `e` satisfies Galerkin orthogonality against `Vh` and `z` solves the adjoint problem, then for every competitor `zh \u2208 Vh`, `\u2016J e\u2016\u00b2 \u2264 \u2016B\u2016 * \u2016e\u2016 * \u2016z - zh\u2016`.","title":"Aubin\u2013Nitsche lemma, pointwise form"},{"id":"Analysis/numerical-analysis/aubin-nitsche-lemma/aubin-nitsche-core","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.AubinNitsche.aubinNitsche_core","parent":18304,"result":true,"route":"../roadmap/Analysis/numerical-analysis/aubin-nitsche-lemma/index.html#aubin-nitsche-core","scope":"Analysis/numerical-analysis/aubin-nitsche-lemma","status":"fully_proved","summary":"With the same hypotheses, `\u2016J e\u2016\u00b2 \u2264 \u2016B\u2016 * \u2016e\u2016 * Metric.infDist z Vh`.","title":"Aubin\u2013Nitsche lemma, infimum form"},{"id":"Analysis/numerical-analysis/aubin-nitsche-lemma/adjoint-solution-norm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.AubinNitsche.adjointSolution_norm_le","parent":18304,"result":true,"route":"../roadmap/Analysis/numerical-analysis/aubin-nitsche-lemma/index.html#adjoint-solution-norm-le","scope":"Analysis/numerical-analysis/aubin-nitsche-lemma","status":"fully_proved","summary":"If `B` is coercive with constant `\u03b1` and `\u2016J\u2016 \u2264 Cemb`, then the adjoint solution satisfies `\u03b1 * \u2016z\u2016 \u2264 Cemb * \u2016J e\u2016`.","title":"Adjoint stability"},{"id":"Analysis/numerical-analysis/aubin-nitsche-lemma/aubin-nitsche-rate","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.AubinNitsche.aubinNitsche_rate","parent":18304,"result":true,"route":"../roadmap/Analysis/numerical-analysis/aubin-nitsche-lemma/index.html#aubin-nitsche-rate","scope":"Analysis/numerical-analysis/aubin-nitsche-lemma","status":"fully_proved","summary":"If in addition the adjoint solution is approximable in `Vh` with `\u2016z - zh\u2016 \u2264 Capp * h * \u2016z\u2016` for some `zh \u2208 Vh`, then the weak-norm error gains the factor `h`: `\u03b1 * \u2016J e\u2016 \u2264 \u2016B\u2016 * Cemb * Capp * h * \u2016e\u2016`.","title":"Aubin\u2013Nitsche lemma, rate form"},{"data_url":"scopes/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem.json","id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","title":"Banach\u2013Ne\u010das\u2013Babu\u0161ka theorem"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/is-inf-sup","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.IsInfSup","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#is-inf-sup","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The inf\u2013sup condition: there is `\u03b2 > 0` such that `\u03b2 * \u2016x\u2016 \u2264 \u2016B x\u2016` for every `x`. The right-hand side is the dual norm of the functional `B x`, i.e. `sup_{\u2016y\u2016 \u2264 1} |B x y|` (see `norm_apply_eq_iSup`).","title":"Inf-sup condition"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/is-nondegenerate","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.IsNondegenerate","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#is-nondegenerate","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The only `y` annihilated by every `B x` is `y = 0`.","title":"Nondegeneracy in the second variable"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/norm-apply-eq-i-sup","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.norm_apply_eq_iSup","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#norm-apply-eq-i-sup","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The dual norm is the supremum over the closed unit ball.","title":"Dual norm as a supremum over the unit ball"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/is-inf-sup-iff","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.isInfSup_iff","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#is-inf-sup-iff","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The inf\u2013sup condition in classical supremum form.","title":"Inf-sup condition in supremum form"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/bilinear-form-sharp","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.bilinearFormSharp","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#bilinear-form-sharp","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The operator `E \u2192L[\u211d] F` associated to a bounded bilinear form `B : E \u2192L[\u211d] F \u2192L[\u211d] \u211d`, characterized by `\u27eabilinearFormSharp B x, y\u27eb = B x y`. This is the two-space analogue of Mathlib's `continuousLinearMapOfBilin` (which is stated for `E = F`).","title":"Operator associated to a bounded bilinear form"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/sharp-antilipschitz","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.sharp_antilipschitz","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#sharp-antilipschitz","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The inf\u2013sup condition makes `bilinearFormSharp B` antilipschitz.","title":"Inf-sup condition makes the associated operator antilipschitz"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/sharp-ker-eq-bot","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.sharp_ker_eq_bot","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#sharp-ker-eq-bot","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The inf\u2013sup condition makes `bilinearFormSharp B` injective.","title":"Inf-sup condition makes the associated operator injective"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/sharp-is-closed-range","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.sharp_isClosed_range","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#sharp-is-closed-range","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The inf\u2013sup condition makes the range of `bilinearFormSharp B` closed.","title":"Inf-sup condition makes the associated operator's range closed"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/sharp-range-eq-top","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.sharp_range_eq_top","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#sharp-range-eq-top","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The inf\u2013sup condition plus nondegeneracy make `bilinearFormSharp B` surjective: the orthogonal complement of its range is trivial, and a closed subspace of a Hilbert space with trivial orthogonal complement is everything.","title":"Inf-sup and nondegeneracy make the associated operator surjective"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/sharp-equiv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.sharpEquiv","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#sharp-equiv","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The Banach\u2013Ne\u010das\u2013Babu\u0161ka isomorphism: the operator associated to a bilinear form satisfying the inf\u2013sup condition and nondegeneracy is a continuous linear equivalence.","title":"Banach\u2013Ne\u010das\u2013Babu\u0161ka isomorphism"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/sharp-equiv-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.sharpEquiv_apply","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#sharp-equiv-apply","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The equivalence `sharpEquiv` is characterized by the bilinear form.","title":"Banach\u2013Ne\u010das\u2013Babu\u0161ka isomorphism is characterized by the form"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/norm-sharp-equiv-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.norm_sharpEquiv_apply","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#norm-sharp-equiv-apply","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The norm of `sharpEquiv` applied to `x` is the dual norm of `B x`.","title":"Norm of the Banach\u2013Ne\u010das\u2013Babu\u0161ka isomorphism applied to a vector"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/norm-sharp-equiv-symm-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.norm_sharpEquiv_symm_apply_le","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#norm-sharp-equiv-symm-apply-le","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"Pointwise inverse bound for `sharpEquiv`: the inverse operator is bounded by `\u03b2\u207b\u00b9` when `\u03b2` is an inf\u2013sup constant.","title":"Pointwise inverse bound for the Banach\u2013Ne\u010das\u2013Babu\u0161ka isomorphism"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/norm-sharp-equiv-symm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.norm_sharpEquiv_symm_le","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#norm-sharp-equiv-symm-le","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"Operator-norm bound for the inverse: `\u2016(sharpEquiv)\u207b\u00b9\u2016 \u2264 \u03b2\u207b\u00b9`.","title":"Operator-norm bound for the inverse Banach\u2013Ne\u010das\u2013Babu\u0161ka isomorphism"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/continuous-linear-equiv-of-bilinear-form","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.continuousLinearEquivOfBilinearForm","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#continuous-linear-equiv-of-bilinear-form","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The Banach\u2013Ne\u010das\u2013Babu\u0161ka theorem: a bilinear form satisfying the inf\u2013sup condition and nondegeneracy is itself a continuous linear equivalence `E \u2243L[\u211d] (F \u2192L[\u211d] \u211d)`.","title":"Banach\u2013Ne\u010das\u2013Babu\u0161ka theorem"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/continuous-linear-equiv-of-bilinear-form-2","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.continuousLinearEquivOfBilinearForm_apply","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#continuous-linear-equiv-of-bilinear-form-2","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The equivalence `continuousLinearEquivOfBilinearForm` acts as `B`.","title":"Banach\u2013Ne\u010das\u2013Babu\u0161ka equivalence acts as the bilinear form"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/exists-unique-apply-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.exists_unique_apply_eq","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#exists-unique-apply-eq","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The variational problem `B x y = \u03c6 y` for all `y` has a unique solution.","title":"Unique solvability of the inf-sup variational problem"},{"id":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/exists-apply-eq-and-norm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.BanachNecasBabuska.exists_apply_eq_and_norm_le","parent":18311,"result":true,"route":"../roadmap/Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem/index.html#exists-apply-eq-and-norm-le","scope":"Analysis/numerical-analysis/banach-ne-as-babu-ka-theorem","status":"fully_proved","summary":"The variational problem has a solution with the stability bound `\u2016x\u2016 \u2264 \u03b2\u207b\u00b9 * \u2016\u03c6\u2016`.","title":"Stability bound for the inf-sup variational problem"},{"data_url":"scopes/Analysis/numerical-analysis/bisection-method.json","id":"Analysis/numerical-analysis/bisection-method","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","title":"Bisection method"},{"id":"Analysis/numerical-analysis/bisection-method/is-bisection-sequence","kind":"node","lean":"MetaMathlibExt.IsBisectionSequence","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#is-bisection-sequence","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"For a function $f : \\mathbb{R} \\to \\mathbb{R}$ and reals $a$, $b$, two sequences $u, v : \\mathbb{N} \\to \\mathbb{R}$ form a bisection sequence for $f$ from $a$ to $b$ when $u_0 = a$, $v_0 = b$, and for every $n$, writing $m_n = (u_n + v_n)/2$ for the midpoint, either $f(m_n) \\le 0$, $u_{n+1} = m_n$ and $v_{n+1} = v_n$, or $f(m_n) \\ge 0$, $u_{n+1} = u_n$ and $v_{n+1} = m_n$.","title":"Bisection sequence"},{"id":"Analysis/numerical-analysis/bisection-method/le-and-le","kind":"node","lean":"MetaMathlibExt.IsBisectionSequence.le_and_le","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#le-and-le","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$, let $a \\le b$, and let $u, v$ be a bisection sequence for $f$ from $a$ to $b$. Then $a \\le u_n \\le v_n \\le b$ for every $n \\in \\mathbb{N}$.","title":"Bisection intervals stay inside the initial interval"},{"id":"Analysis/numerical-analysis/bisection-method/invariants","kind":"node","lean":"MetaMathlibExt.IsBisectionSequence.invariants","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#invariants","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and $a \\le b$ with $f(a) \\le 0 \\le f(b)$, and let $u, v$ be a bisection sequence for $f$ from $a$ to $b$. Then for every $n \\in \\mathbb{N}$ we have $a \\le u_n \\le v_n \\le b$ and $f(u_n) \\le 0 \\le f(v_n)$.","title":"Bisection invariants"},{"id":"Analysis/numerical-analysis/bisection-method/nested","kind":"node","lean":"MetaMathlibExt.IsBisectionSequence.nested","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#nested","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$, let $a \\le b$, and let $u, v$ be a bisection sequence for $f$ from $a$ to $b$. Then for every $n \\in \\mathbb{N}$, $u_n \\le u_{n+1} \\le v_{n+1} \\le v_n$.","title":"Bisection intervals are nested"},{"id":"Analysis/numerical-analysis/bisection-method/sub-eq","kind":"node","lean":"MetaMathlibExt.IsBisectionSequence.sub_eq","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#sub-eq","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$, let $a, b$ be real, and let $u, v$ be a bisection sequence for $f$ from $a$ to $b$. Then $v_n - u_n = (b - a)/2^n$ for every $n \\in \\mathbb{N}$.","title":"Length of the bisection intervals"},{"id":"Analysis/numerical-analysis/bisection-method/exists-is-bisection-sequence","kind":"node","lean":"MetaMathlibExt.exists_isBisectionSequence","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#exists-is-bisection-sequence","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"For every function $f : \\mathbb{R} \\to \\mathbb{R}$ and all reals $a$, $b$, there are sequences $u, v$ forming a bisection sequence for $f$ from $a$ to $b$.","title":"Existence of bisection sequences"},{"id":"Analysis/numerical-analysis/bisection-method/exists-is-bisection-sequence-nested","kind":"node","lean":"MetaMathlibExt.exists_isBisectionSequence_nested","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#exists-is-bisection-sequence-nested","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and $a \\le b$ with $f(a) \\le 0 \\le f(b)$. Then there is a bisection sequence $u, v$ for $f$ from $a$ to $b$ such that for every $n \\in \\mathbb{N}$: $u_n \\le u_{n+1} \\le v_{n+1} \\le v_n$, $v_n - u_n = (b - a)/2^n$, and $f(u_n)\\,f(v_n) \\le 0$.","title":"Nested bisection intervals from a sign bracket"},{"id":"Analysis/numerical-analysis/bisection-method/bisection-nested-intervals","kind":"node","lean":"MetaMathlibExt.bisection_nested_intervals","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#bisection-nested-intervals","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[a, b]$, where $a \\le b$ and $f(a) \\le 0 \\le f(b)$. Then there is a bisection sequence $u, v$ for $f$ from $a$ to $b$ such that for every $n \\in \\mathbb{N}$: $u_n \\le u_{n+1} \\le v_{n+1} \\le v_n$, $v_n - u_n = (b - a)/2^n$, and $f(u_n)\\,f(v_n) \\le 0$.","title":"Bisection method: nested intervals"},{"id":"Analysis/numerical-analysis/bisection-method/bisection-approximate-root-with-rate","kind":"node","lean":"MetaMathlibExt.bisection_approximate_root_with_rate","parent":18330,"result":true,"route":"../roadmap/Analysis/numerical-analysis/bisection-method/index.html#bisection-approximate-root-with-rate","scope":"Analysis/numerical-analysis/bisection-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[a, b]$, where $a \\le b$ and $f(a) \\le 0 \\le f(b)$, and let $u, v$ be a bisection sequence for $f$ from $a$ to $b$. Then there is $c \\in [a, b]$ with $f(c) = 0$ such that the midpoints $(u_n + v_n)/2$ converge to $c$ and $|(u_n + v_n)/2 - c| \\le (b - a)/2^{n+1}$ for every $n \\in \\mathbb{N}$.","title":"Bisection method converges at a linear rate"},{"data_url":"scopes/Analysis/numerical-analysis/c-a-s-lemma.json","id":"Analysis/numerical-analysis/c-a-s-lemma","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","title":"C\u00e9a's lemma"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/restrict-bilin","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.restrictBilin","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#restrict-bilin","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"Restriction of a bounded bilinear form to a submodule.","title":"Restriction of a bilinear form to a subspace"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/is-coercive-restrict-bilin","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.isCoercive_restrictBilin","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#is-coercive-restrict-bilin","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"Coercivity is inherited by restriction to a submodule.","title":"Coercivity passes to restrictions to subspaces"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/variational-solution","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.variationalSolution","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#variational-solution","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"The solution of the variational problem `B u w = L w` for all `w`, obtained from the Lax\u2013Milgram equivalence and the Riesz representation.","title":"Solution of a coercive variational problem"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/variational-solution-spec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.variationalSolution_spec","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#variational-solution-spec","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"The variational solution satisfies the variational equation.","title":"Variational solution satisfies the variational equation"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/eq-of-forall-bilin","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.eq_of_forall_bilin","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#eq-of-forall-bilin","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"Uniqueness for the variational problem: two vectors satisfying the same variational equation agree, by coercivity.","title":"Uniqueness for coercive variational problems"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/variational-solution-eq-of-forall","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.variationalSolution_eq_of_forall","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#variational-solution-eq-of-forall","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"The variational solution is the unique solution of the variational equation.","title":"Variational solution is the unique solution"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/exists-variational-solution","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.exists_variationalSolution","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#exists-variational-solution","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"Existence for the variational problem, in bare existential form.","title":"Existence of a variational solution"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/galerkin-solution","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.galerkinSolution","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#galerkin-solution","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"The Galerkin solution in `W`: the solution of the variational problem for the restricted form and the restricted functional.","title":"Galerkin solution"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/galerkin-solution-spec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.galerkinSolution_spec","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#galerkin-solution-spec","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"The Galerkin solution satisfies the Galerkin equations.","title":"Galerkin solution satisfies the Galerkin equations"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/galerkin-solution-eq-of-forall","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.galerkinSolution_eq_of_forall","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#galerkin-solution-eq-of-forall","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"The Galerkin solution is the unique element of `W` satisfying the Galerkin equations.","title":"Galerkin solution is the unique solution of the Galerkin equations"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/exists-galerkin-solution","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.exists_galerkinSolution","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#exists-galerkin-solution","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"Existence for the Galerkin problem, in bare existential form.","title":"Existence of a Galerkin solution"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/galerkin-orthogonality","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.galerkin_orthogonality","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#galerkin-orthogonality","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"Galerkin orthogonality: the error `u - u_h` is `B`-orthogonal to `W`.","title":"Galerkin orthogonality"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/norm-sub-galerkin-solution-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.norm_sub_galerkinSolution_le","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#norm-sub-galerkin-solution-le","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"C\u00e9a's lemma (pointwise form): the Galerkin error is bounded by `(\u2016B\u2016 / \u03b1)` times the distance from `u` to any element `v` of `W`.","title":"C\u00e9a's lemma"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/norm-sub-galerkin-solution-le-i-inf","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.norm_sub_galerkinSolution_le_iInf","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#norm-sub-galerkin-solution-le-i-inf","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"C\u00e9a's lemma (infimum form): the Galerkin error is bounded by `(\u2016B\u2016 / \u03b1)` times the best-approximation error `\u2a05 v : W, \u2016u - v\u2016`.","title":"C\u00e9a's lemma"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/norm-variational-solution-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.norm_variationalSolution_le","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#norm-variational-solution-le","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"Stability of the variational solution: `\u2016u\u2016 \u2264 \u2016L\u2016 / \u03b1`.","title":"Stability bound for the variational solution"},{"id":"Analysis/numerical-analysis/c-a-s-lemma/norm-galerkin-solution-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.CeaLemma.norm_galerkinSolution_le","parent":18340,"result":true,"route":"../roadmap/Analysis/numerical-analysis/c-a-s-lemma/index.html#norm-galerkin-solution-le","scope":"Analysis/numerical-analysis/c-a-s-lemma","status":"fully_proved","summary":"Stability of the Galerkin solution: `\u2016u_h\u2016 \u2264 \u2016L\u2016 / \u03b1`.","title":"Stability bound for the Galerkin solution"},{"data_url":"scopes/Analysis/numerical-analysis/euler-method.json","id":"Analysis/numerical-analysis/euler-method","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/euler-method/index.html","scope":"Analysis/numerical-analysis/euler-method","status":"fully_proved","title":"Euler method"},{"id":"Analysis/numerical-analysis/euler-method/exists-explicit-euler-local-truncation-e","kind":"node","lean":"ODE.exists_explicitEuler_localTruncationError_le","parent":18357,"result":true,"route":"../roadmap/Analysis/numerical-analysis/euler-method/index.html#exists-explicit-euler-local-truncation-e","scope":"Analysis/numerical-analysis/euler-method","status":"fully_proved","summary":"Let $E$ be a real normed space, let $t_0 < T$ be real numbers, let $f : \\mathbb{R} \\times E \\to E$ be $C^2$, and let $x : \\mathbb{R} \\to E$ be differentiable at every $t \\in [t_0, T]$ with $x'(t) = f(t, x(t))$. Then there is a constant $C \\ge 0$ such that $\\|x(t + h) - (x(t) + h f(t, x(t)))\\| \\le C h^2$ for every $h \\in (0, T - t_0]$ and every $t \\in [t_0, T]$ with $t + h \\le T$.","title":"Uniform quadratic bound on the explicit Euler local error"},{"id":"Analysis/numerical-analysis/euler-method/explicit-euler-first-order-convergence","kind":"node","lean":"MathlibExt.Analysis.ODE.ExplicitEulerWanted.explicit_euler_first_order_convergence","parent":18357,"result":true,"route":"../roadmap/Analysis/numerical-analysis/euler-method/index.html#explicit-euler-first-order-convergence","scope":"Analysis/numerical-analysis/euler-method","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $t_0 < T$, let $x_0 \\in E$ and $L \\ge 0$, and let $f : \\mathbb{R} \\times E \\to E$ be $C^2$ with $f(t, \\cdot)$ $L$-Lipschitz for every $t \\in [t_0, T]$. Then there are a function $x : \\mathbb{R} \\to E$ with $x(t_0) = x_0$ and $x'(t) = f(t, x(t))$ for every $t \\in [t_0, T]$, and a constant $C$, such that the following holds for every step size $h \\in (0, T - t_0]$: if\u2026","title":"First-order convergence of the explicit Euler method"},{"id":"Analysis/numerical-analysis/euler-method/explicit-euler-local-truncation-error","kind":"node","lean":"MathlibExt.Analysis.ODE.ExplicitEulerWanted.explicit_euler_local_truncation_error","parent":18357,"result":true,"route":"../roadmap/Analysis/numerical-analysis/euler-method/index.html#explicit-euler-local-truncation-error","scope":"Analysis/numerical-analysis/euler-method","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $t_0 < T$, let $f : \\mathbb{R} \\times E \\to E$ be $C^2$, let $x_0 \\in E$, and let $x : \\mathbb{R} \\to E$ satisfy $x(t_0) = x_0$ and $x'(t) = f(t, x(t))$ for every $t \\in [t_0, T]$. Then there is a constant $C$ such that $\\|x(t + h) - (x(t) + h f(t, x(t)))\\| \\le C h^2$ for every $h \\in (0, T - t_0]$ and every $t \\in [t_0, T]$ with $t + h \\le T$.","title":"Local truncation error of the explicit Euler method"},{"data_url":"scopes/Analysis/numerical-analysis/finite-element-method-in-one-dimension.json","id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","title":"Finite element method in one dimension"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/width","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.width","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#width","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The length `h\u1d62 = x\u1d62\u208a\u2081 - x\u1d62` of the `i`-th element of the mesh.","title":"Length of a mesh element"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/element-slope","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.elementSlope","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#element-slope","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The slope `(d\u1d62\u208a\u2081 - d\u1d62) / h\u1d62` on the `i`-th element of the P1 function with nodal values `d`.","title":"Slope of a P1 function on a mesh element"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/ramp","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.ramp","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#ramp","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The clamped ramp of the `i`-th element: `0` left of `x\u1d62`, `x - x\u1d62` on `[x\u1d62, x\u1d62\u208a\u2081]`, and `h\u1d62` right of `x\u1d62\u208a\u2081`.","title":"Clamped ramp function of a mesh element"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/p1","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.p1","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#p1","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The continuous piecewise linear (P1) function with nodal values `d`: `d\u2080 + \u2211\u1d62 slope\u1d62 \u00b7 ramp\u1d62(x)`. On `[0, 1]` it is linear on each element and takes the value `d\u2c7c` at `x\u2c7c`; outside `[0, 1]` it is constant.","title":"Continuous piecewise linear function with given nodal values"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/hat","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.hat","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#hat","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The hat function `\u03c6\u2c7c` of node `j`: the P1 function equal to `1` at `x\u2c7c` and `0` at every other node.","title":"Hat function of a mesh node"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/interp","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.interp","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#interp","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The nodal interpolant `I_h u`: the P1 function with nodal values `u(x\u2c7c)`.","title":"Nodal interpolant of a function"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/p1-map","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.p1Map","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#p1-map","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"`p1` as a linear map from nodal values to functions.","title":"Linear map from nodal values to P1 functions"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/dirichlet-nodal","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.dirichletNodal","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#dirichlet-nodal","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Nodal vectors with zero boundary values `d\u2080 = d\u2099 = 0`.","title":"Nodal vectors with zero boundary values"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/p1-space","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.p1Space","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#p1-space","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The P1 finite element space `V_h` with zero boundary values: the image under `p1Map` of the nodal vectors vanishing at both ends.","title":"P1 finite element space with zero boundary values"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy","parent":18361,"reading_title":{"format":"inline-parts-v1","parts":[["text","Energy form of the equation "],["math","\\(-u'' = f\\)"]],"source":"Energy form of the equation $-u'' = f$"},"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The energy form `a(u, v) = \u222b\u2080\u00b9 u'(x) v'(x) dx` of `-u'' = f`. For a P1 function, `deriv` is junk at the mesh nodes, a null set.","title":"Energy form of the equation $-u'' = f$"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/element-vector","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.elementVector","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#element-vector","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The `i`-th element vector `e\u1d62 = \u03b4_{i+1} - \u03b4\u1d62` of the assembly.","title":"Element vector of the stiffness assembly"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/stiffness","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.stiffness","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#stiffness","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The assembled stiffness matrix `K = \u2211\u1d62 (1 / h\u1d62) e\u1d62 e\u1d62\u1d40` on all nodes: each element contributes `(1 / h\u1d62) [[1, -1], [-1, 1]]` at nodes `i, i + 1`.","title":"Assembled stiffness matrix of the P1 method"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/load","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.load","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#load","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The load vector `F\u2c7c = \u222b\u2080\u00b9 f \u03c6\u2c7c`.","title":"Load vector of the P1 method"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/is-classical-solution","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.IsClassicalSolution","parent":18361,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classical solution of the Dirichlet problem "],["math","\\(-u'' = f\\)"]],"source":"Classical solution of the Dirichlet problem $-u'' = f$"},"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#is-classical-solution","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"`u` is a classical solution of `-u'' = f` on `(0, 1)` with `u(0) = u(1) = 0`. We ask `u` to be `C\u00b2` on all of `\u211d`, which is no loss for the interval problem (a `C\u00b2` function on `[0, 1]` extends), and `f` to be continuous. The equation is stated on the open interval, as in the Dirichlet problem: requiring it at the endpoints without regularity of `f` would be a genuine strengthening. With `f` continuous, endpoint\u2026","title":"Classical solution of the Dirichlet problem $-u'' = f$"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/eq-icc","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.IsClassicalSolution.eq_Icc","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#eq-icc","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Endpoint equality, derived from the open-interval equation by continuity of `-u''` and `f`: the two sides agree on `(0, 1)`, hence on its closure `[0, 1]`.","title":"Classical solutions satisfy the equation on the closed interval"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/is-galerkin-solution","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.IsGalerkinSolution","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#is-galerkin-solution","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The coefficient vector `c` solves the Galerkin system: zero boundary values and `(K c)\u2c7c = F\u2c7c` at every interior node `j`. The Galerkin solution is `u_h = p1 P c`.","title":"Solution of the Galerkin system"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/width-pos","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.width_pos","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#width-pos","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Every element has positive length.","title":"Mesh elements have positive length"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/width-le-mesh","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.width_le_mesh","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#width-le-mesh","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Every element length is at most the mesh size `h`.","title":"Element lengths are bounded by the mesh size"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/p1-eq-of-mem-icc","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.p1_eq_of_mem_Icc","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#p1-eq-of-mem-icc","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"On the `i`-th element, `p1 P d` is the linear function through `(x\u1d62, d\u1d62)` with slope `slope\u1d62 = elementSlope P d i`.","title":"P1 functions are linear on each element"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/p1-points","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.p1_points","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#p1-points","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"`p1 P d` interpolates the nodal values.","title":"P1 functions interpolate their nodal values"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/continuous-p1","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.continuous_p1","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#continuous-p1","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"P1 functions are continuous.","title":"Continuity of P1 functions"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/has-deriv-at-p1","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.hasDerivAt_p1","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#has-deriv-at-p1","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Inside an element, `p1 P d` has derivative `slope\u1d62`.","title":"Derivative of a P1 function inside an element"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/integral-eq-sum","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.integral_eq_sum","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#integral-eq-sum","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"An integral over `[0, 1]` is the sum of the integrals over the elements.","title":"Integral over the unit interval as a sum over elements"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-p1","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_p1","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-p1","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The key identity of the 1D P1 method: for `w` continuous on each element, differentiable inside it, and with integrable derivative there, `a(w, p1 d) = \u2211\u1d62 slope\u1d62(d) (w(x\u1d62\u208a\u2081) - w(x\u1d62))`.","title":"Energy pairing with a P1 function via nodal differences"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/mem-p1-space","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.mem_p1Space","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#mem-p1-space","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Membership in the P1 space: `v = p1 P d` for nodal values `d` with `d\u2080 = d\u2099 = 0`.","title":"Characterization of membership in the P1 space"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/hat-mem-p1-space","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.hat_mem_p1Space","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#hat-mem-p1-space","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The hat function of an interior node lies in the P1 space.","title":"Interior hat functions lie in the P1 space"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-eq-integral-mul-of-cont-diff","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_eq_integral_mul_of_contDiff","parent":18361,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weak form of the classical solution for "],["math","\\(C^1\\)"],["text"," test functions"]],"source":"Weak form of the classical solution for $C^1$ test functions"},"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-eq-integral-mul-of-cont-diff","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The weak form for `C\u00b9` test functions vanishing at `0` and `1`: `\u222b\u2080\u00b9 u' v' = \u222b\u2080\u00b9 f v`.","title":"Weak form of the classical solution for $C^1$ test functions"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-eq-integral-mul-of-mem-p1-space","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_eq_integral_mul_of_mem_p1Space","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-eq-integral-mul-of-mem-p1-space","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The weak form for P1 test functions: `\u222b\u2080\u00b9 u' v_h' = \u222b\u2080\u00b9 f v_h`.","title":"Weak form of the classical solution for P1 test functions"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-p1-p1","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_p1_p1","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-p1-p1","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The energy form of two P1 functions: `a(p1 c, p1 d) = \u2211\u1d62 (c\u1d62\u208a\u2081 - c\u1d62)(d\u1d62\u208a\u2081 - d\u1d62) / h\u1d62`.","title":"Energy form of two P1 functions"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/dot-product-stiffness-mul-vec","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.dotProduct_stiffness_mulVec","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#dot-product-stiffness-mul-vec","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The quadratic form of the stiffness matrix: `d\u1d40 K c = \u2211\u1d62 (c\u1d62\u208a\u2081 - c\u1d62)(d\u1d62\u208a\u2081 - d\u1d62) / h\u1d62`.","title":"Quadratic form of the stiffness matrix"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-hat-hat","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_hat_hat","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-hat-hat","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The stiffness matrix is the Gram matrix of the hat functions: `a(\u03c6\u2c7c, \u03c6\u2096) = K\u2c7c\u2096`.","title":"Stiffness matrix is the Gram matrix of the hat functions"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/stiffness-cast-succ-succ","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.stiffness_castSucc_succ","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#stiffness-cast-succ-succ","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The off-diagonal entry of an element: `K_{i, i+1} = -1 / h\u1d62`.","title":"Off-diagonal stiffness entry of an element"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/stiffness-succ-succ","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.stiffness_succ_succ","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#stiffness-succ-succ","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The diagonal entry at an interior node `x\u2c7c` shared by consecutive elements `i` and `j`: `K = 1 / h\u1d62 + 1 / h\u2c7c`.","title":"Diagonal stiffness entry at an interior node"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/stiffness-eq-zero","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.stiffness_eq_zero","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#stiffness-eq-zero","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Entries between nodes two or more apart vanish (tridiagonality).","title":"Tridiagonality of the stiffness matrix"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/is-galerkin-solution-nodal","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.isGalerkinSolution_nodal","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#is-galerkin-solution-nodal","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The nodal values of the exact solution solve the Galerkin system.","title":"Nodal values of the exact solution solve the Galerkin system"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/is-galerkin-solution-unique","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.isGalerkinSolution_unique","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#is-galerkin-solution-unique","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The Galerkin system has at most one solution (`K` is positive definite on nodal vectors with zero boundary values).","title":"Uniqueness of the Galerkin solution"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/p1-eq-interp-of-is-galerkin-solution","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.p1_eq_interp_of_isGalerkinSolution","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#p1-eq-interp-of-is-galerkin-solution","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"In one dimension the P1 Galerkin solution is the nodal interpolant of the exact solution: `u_h = I_h u`.","title":"In one dimension the Galerkin solution is the nodal interpolant"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/galerkin-orthogonality","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.galerkin_orthogonality","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#galerkin-orthogonality","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"`a(u - u_h, v_h) = 0` for every `v_h \u2208 V_h`.","title":"Galerkin orthogonality"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-sub-p1-self","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_sub_p1_self","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-sub-p1-self","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Element-wise form of the energy of `u - p1 d` for a `C\u00b2` function `u`: `a(u - p1 d, u - p1 d) = \u2211\u1d62 \u222b_{x\u1d62}^{x\u1d62\u208a\u2081} (u' - slope\u1d62(d))\u00b2`.","title":"Element-wise energy of a function minus a P1 function"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-sub-galerkin-le","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_sub_galerkin_le","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-sub-galerkin-le","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Best approximation (C\u00e9a's lemma for the symmetric energy form, with constant `1`): `\u2016u - u_h\u2016_a \u2264 \u2016u - v_h\u2016_a` for every `v_h \u2208 V_h`.","title":"Best approximation"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/integral-sq-le-of-eq-zero","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.integral_sq_le_of_eq_zero","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#integral-sq-le-of-eq-zero","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"Wirtinger's inequality on `[a, b]`: a `C\u00b9` function vanishing at `a` and `b` satisfies `\u222b\u2090\u1d47 g\u00b2 \u2264 ((b - a) / \u03c0)\u00b2 \u222b\u2090\u1d47 (g')\u00b2`.","title":"Wirtinger's inequality on an interval"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/integral-deriv-sq-le-of-eq-zero","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.integral_deriv_sq_le_of_eq_zero","parent":18361,"reading_title":{"format":"inline-parts-v1","parts":[["text","Element estimate for "],["math","\\(C^2\\)"],["text"," functions vanishing at both ends"]],"source":"Element estimate for $C^2$ functions vanishing at both ends"},"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#integral-deriv-sq-le-of-eq-zero","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The element estimate: a `C\u00b2` function vanishing at `a` and `b` satisfies `\u222b\u2090\u1d47 (g')\u00b2 \u2264 ((b - a) / \u03c0)\u00b2 \u222b\u2090\u1d47 (g'')\u00b2`.","title":"Element estimate for $C^2$ functions vanishing at both ends"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-sub-interp-le","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_sub_interp_le","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-sub-interp-le","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"`\u2016u - I_h u\u2016_a\u00b2 \u2264 (h / \u03c0)\u00b2 \u2016u''\u2016\u00b2_{L\u00b2(0,1)}`, with `h` the mesh size.","title":"Interpolation estimate"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/energy-sub-galerkin-le-mesh","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.energy_sub_galerkin_le_mesh","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#energy-sub-galerkin-le-mesh","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"`\u2016u - u_h\u2016_a\u00b2 \u2264 (h / \u03c0)\u00b2 \u2016f\u2016\u00b2_{L\u00b2(0,1)}`, so the energy error is `O(h)`.","title":"Energy error estimate"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/sqrt-energy-sub-galerkin-le-mesh","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.sqrt_energy_sub_galerkin_le_mesh","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#sqrt-energy-sub-galerkin-le-mesh","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"The energy error estimate in norm form: `\u2016u - u_h\u2016_a \u2264 (h / \u03c0) \u2016f\u2016_{L\u00b2(0,1)}`.","title":"Energy error estimate for the P1 Galerkin solution"},{"id":"Analysis/numerical-analysis/finite-element-method-in-one-dimension/integral-sq-sub-galerkin-le-mesh","kind":"node","lean":"MathlibExt.Analysis.ODE.FiniteElement1D.integral_sq_sub_galerkin_le_mesh","parent":18361,"result":true,"route":"../roadmap/Analysis/numerical-analysis/finite-element-method-in-one-dimension/index.html#integral-sq-sub-galerkin-le-mesh","scope":"Analysis/numerical-analysis/finite-element-method-in-one-dimension","status":"fully_proved","summary":"`\u2016u - u_h\u2016\u00b2_{L\u00b2(0,1)} \u2264 (h / \u03c0)\u2074 \u2016f\u2016\u00b2_{L\u00b2(0,1)}`, so the `L\u00b2` error is `O(h\u00b2)`. The proof uses nodal exactness and Wirtinger's inequality on each element, not the Aubin\u2013Nitsche duality argument.","title":"`L\u00b2` error estimate"},{"data_url":"scopes/Analysis/numerical-analysis/newton-s-method.json","id":"Analysis/numerical-analysis/newton-s-method","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/newton-s-method/index.html","scope":"Analysis/numerical-analysis/newton-s-method","status":"fully_proved","title":"Newton's method"},{"id":"Analysis/numerical-analysis/newton-s-method/newton-step-sub-root-abs-le","kind":"node","lean":"MetaMathlibExt.newton_step_sub_root_abs_le","parent":18408,"result":true,"route":"../roadmap/Analysis/numerical-analysis/newton-s-method/index.html#newton-step-sub-root-abs-le","scope":"Analysis/numerical-analysis/newton-s-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be differentiable with $f(c) = 0$, let $x \\in \\mathbb{R}$, let $m > 0$ and let $K \\ge 0$. If $f'$ is Lipschitz with constant $K$ on the closed interval between $c$ and $x$ and $m \\le |f'(x)|$, then the Newton step from $x$ satisfies $\\left| x - f(x) / f'(x) - c \\right| \\le (K / m)\\, |x - c|^2$.","title":"One-step error bound for Newton's method"},{"id":"Analysis/numerical-analysis/newton-s-method/newton-method-local-quadratic-convergenc","kind":"node","lean":"MetaMathlibExt.newton_method_local_quadratic_convergence","parent":18408,"result":true,"route":"../roadmap/Analysis/numerical-analysis/newton-s-method/index.html#newton-method-local-quadratic-convergenc","scope":"Analysis/numerical-analysis/newton-s-method","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be twice continuously differentiable and let $c$ be a simple root of $f$: $f(c) = 0$ and $f'(c) \\ne 0$. Then there exist $\\delta > 0$ and a real constant $C$ with the following property. For every $x_0 \\in (c - \\delta, c + \\delta)$ and every real sequence $(x_n)$ that starts at $x_0$ and satisfies $x_{n+1} = x_n - f(x_n) / f'(x_n)$ for all $n$, every $x_n$ lies in\u2026","title":"Local quadratic convergence of Newton's method"},{"data_url":"scopes/Analysis/numerical-analysis/runge-kutta-methods.json","id":"Analysis/numerical-analysis/runge-kutta-methods","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","title":"Runge\u2013Kutta methods"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/explicit-rk","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.ExplicitRK","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#explicit-rk","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"An explicit Runge\u2013Kutta method with $s$ stages consists of a real $s \\times s$ Butcher tableau $a = (a_{ij})$ that is strictly lower triangular, meaning $a_{ij} = 0$ whenever $i \\le j$, and real weights $b_1, \\dots, b_s$.","title":"Explicit Runge\u2013Kutta method"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/c","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.ExplicitRK.c","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#c","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"For an explicit Runge\u2013Kutta method with $s$ stages and Butcher tableau $a$, the node of stage $i$ is the row sum $c_i = \\sum_j a_{ij}$, taken over all $s$ stages $j$.","title":"Nodes of a Runge\u2013Kutta method"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/stage-row","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.stageRow","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#stage-row","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"Let $E$ be a real normed space, fix an explicit Runge\u2013Kutta method with $s$ stages, strictly lower-triangular tableau $a$ and nodes $c_i = \\sum_j a_{ij}$, a vector field $f$ assigning a vector $f(t', x) \\in E$ to each time $t' \\in \\mathbb{R}$ and state $x \\in E$, a time $t$, a state $y \\in E$ and a step $h \\in \\mathbb{R}$. The rows $K^{(n)} \\in E^s$, indexed by $0 \\le i < s$, are defined by recursion on $n$:\u2026","title":"Stages computed row by row"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/stage","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.stage","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#stage","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"Let $E$ be a real normed space, fix an explicit Runge\u2013Kutta method with $s$ stages, strictly lower-triangular tableau $a$ and nodes $c_i = \\sum_j a_{ij}$, a vector field $f$ assigning a vector $f(t', x) \\in E$ to each time $t' \\in \\mathbb{R}$ and state $x \\in E$, a time $t$, a state $y \\in E$ and a step $h \\in \\mathbb{R}$. Let the rows $K^{(n)} \\in E^s$, indexed by $0 \\le i < s$, be defined by $K^{(0)} = 0$, and\u2026","title":"Runge\u2013Kutta stages"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/stage-row-eq-of-lt","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.stageRow_eq_of_lt","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#stage-row-eq-of-lt","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"Let $E$ be a real normed vector space and $f(t, y)$ a vector field $\\mathbb{R} \\times E \\to E$, and fix an explicit Runge\u2013Kutta method, a time $t$, a state $y \\in E$ and a step size $h \\in \\mathbb{R}$. Index the stages from $0$. For every natural number $m$ and every stage index $i < m$, the $i$-th entry of the $m$-th partial row for $f$, $t$, $y$ and $h$ (the row-by-row construction that fills in the stages with\u2026","title":"Partial stage rows agree with the stages"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/stage-eq","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.stage_eq","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#stage-eq","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"Let $E$ be a real normed vector space and $f(t, y)$ a vector field $\\mathbb{R} \\times E \\to E$. For an explicit Runge\u2013Kutta method with tableau $a$ and nodes $c_i = \\sum_j a_{ij}$, a time $t$, a state $y \\in E$ and a step size $h$, the stages $k_i$ satisfy $k_i = f\\bigl(t + c_i h, \\; y + h \\sum_{j < i} a_{ij} k_j\\bigr)$ for every stage index $i$.","title":"The defining equation of the stages"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/rk-step","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.rkStep","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#rk-step","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"Let $E$ be a real normed vector space and $f(t, y)$ a vector field $\\mathbb{R} \\times E \\to E$. For an explicit Runge\u2013Kutta method with weights $b$, a time $t$, a state $y \\in E$ and a step size $h \\in \\mathbb{R}$, one step of the method is $y + h \\sum_i b_i k_i$, where $k_i$ is the $i$-th stage of the method for $f$, $t$, $y$ and $h$, and the sum runs over the stages.","title":"One Runge\u2013Kutta step"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/order-one","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.OrderOne","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#order-one","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"An explicit Runge\u2013Kutta method with weights $b_i$ satisfies the order-one condition when $\\sum_i b_i = 1$, the sum running over all of its stages $i$.","title":"Order-one condition for Runge\u2013Kutta methods"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/order-two","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.OrderTwo","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#order-two","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"An explicit Runge\u2013Kutta method with tableau $a$, weights $b$ and nodes $c_i = \\sum_j a_{ij}$ satisfies the order-two conditions when it satisfies the order-one condition $\\sum_i b_i = 1$ and moreover $\\sum_i b_i c_i = \\frac{1}{2}$, all sums running over the stages.","title":"Order-two conditions for Runge\u2013Kutta methods"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/order-three","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.OrderThree","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#order-three","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"An explicit Runge\u2013Kutta method with tableau $a$, weights $b$ and nodes $c_i = \\sum_j a_{ij}$ satisfies the order-three conditions when it satisfies the order-two conditions $\\sum_i b_i = 1$ and $\\sum_i b_i c_i = \\frac{1}{2}$, and moreover $\\sum_i b_i c_i^2 = \\frac{1}{3}$ and $\\sum_{i, j} b_i a_{ij} c_j = \\frac{1}{6}$, all sums running over the stages.","title":"Order-three conditions for Runge\u2013Kutta methods"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/order-four","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.OrderFour","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#order-four","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"An explicit Runge\u2013Kutta method with Butcher tableau $a$, weights $b$ and nodes $c$ satisfies the order-four conditions when it satisfies the order-three conditions, namely $\\sum_i b_i = 1$, $\\sum_i b_i c_i = 1/2$, $\\sum_i b_i c_i^2 = 1/3$ and $\\sum_{i,j} b_i a_{ij} c_j = 1/6$, together with $\\sum_i b_i c_i^3 = 1/4$, $\\sum_{i,j} b_i c_i a_{ij} c_j = 1/8$, $\\sum_{i,j} b_i a_{ij} c_j^2 = 1/12$ and\u2026","title":"Order-four conditions"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/euler-rk","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.eulerRK","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#euler-rk","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"The explicit Runge\u2013Kutta method with one stage, Butcher tableau $a = (0)$ and weight $b_1 = 1$.","title":"Euler tableau"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/midpoint-rk","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.midpointRK","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#midpoint-rk","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"The explicit Runge\u2013Kutta method with two stages, Butcher tableau entries $a_{21} = 1/2$ and $a_{11} = a_{12} = a_{22} = 0$, and weights $b_1 = 0$, $b_2 = 1$.","title":"Midpoint tableau"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/heun-rk","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.heunRK","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#heun-rk","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"The explicit Runge\u2013Kutta method with two stages, Butcher tableau entries $a_{21} = 1$ and $a_{11} = a_{12} = a_{22} = 0$, and weights $b_1 = b_2 = 1/2$.","title":"Heun's tableau"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/rk4","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.rk4","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#rk4","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"The classical fourth-order Runge\u2013Kutta method is the explicit four-stage Runge\u2013Kutta method whose tableau, with stages indexed $1, \\dots, 4$, has $a_{21} = \\frac{1}{2}$, $a_{32} = \\frac{1}{2}$, $a_{43} = 1$ and all other $a_{ij} = 0$, and whose weights are $b = (\\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{6})$.","title":"The classical RK4 tableau"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/euler-rk-order-one","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.eulerRK_orderOne","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#euler-rk-order-one","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"The weights of the Euler tableau sum to $1$.","title":"The Euler tableau satisfies the order-one condition"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/midpoint-rk-order-two","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.midpointRK_orderTwo","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#midpoint-rk-order-two","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"For the midpoint tableau, with weights $b_i$ and nodes $c_i$, we have $\\sum_i b_i = 1$ and $\\sum_i b_i c_i = 1/2$.","title":"The midpoint tableau satisfies the order-two conditions"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/heun-rk-order-two","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.heunRK_orderTwo","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#heun-rk-order-two","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"For Heun's tableau, with weights $b_i$ and nodes $c_i$, we have $\\sum_i b_i = 1$ and $\\sum_i b_i c_i = 1/2$.","title":"Heun's tableau satisfies the order-two conditions"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/rk4-order-four","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.rk4_orderFour","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#rk4-order-four","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"The classical RK4 tableau satisfies all eight order-four conditions.","title":"The classical RK4 tableau satisfies all eight order-four conditions"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/euler-step","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.euler_step","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#euler-step","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"Let $E$ be a real normed space and let $f$ assign to each time $t \\in \\mathbb{R}$ and state $y \\in E$ a vector $f(t, y) \\in E$. For all $t, h \\in \\mathbb{R}$ and $y \\in E$, one Runge\u2013Kutta step of size $h$ from the state $y$ at time $t$ with the Euler tableau equals $y + h f(t, y)$.","title":"The Euler tableau gives the explicit Euler step"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/stage-tendsto","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.stage_tendsto","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#stage-tendsto","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"Let $E$ be a real normed space, fix an explicit Runge\u2013Kutta method with $s$ stages, a vector field $f$ assigning a vector $f(t', x) \\in E$ to each time $t' \\in \\mathbb{R}$ and state $x \\in E$, a time $t$ and a state $y \\in E$, and suppose that $(t', x) \\mapsto f(t', x)$ is continuous at $(t, y)$. For a step $h \\in \\mathbb{R}$, let $k_i(h)$ be the $i$-th stage of the method with step $h$. Then for every natural\u2026","title":"Runge\u2013Kutta stages tend to the vector field as the step shrinks"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/rk-step-sub-inv-smul-tendsto","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.rkStep_sub_inv_smul_tendsto","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#rk-step-sub-inv-smul-tendsto","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"Let $E$ be a real normed vector space, let $f(t, y)$ be a vector field $\\mathbb{R} \\times E \\to E$, let $t \\in \\mathbb{R}$ and $y \\in E$, and for an explicit Runge\u2013Kutta method with weights $b$ let $y_1(h)$ be the result of one step of size $h$ from the state $y$ at time $t$. If $\\sum_i b_i = 1$ and $f$ is continuous at $(t, y)$ as a function of both variables, then $h^{-1} (y_1(h) - y) \\to f(t, y)$ as $h \\to 0$\u2026","title":"Consistency of explicit Runge\u2013Kutta methods"},{"id":"Analysis/numerical-analysis/runge-kutta-methods/rk4-stability","kind":"node","lean":"MathlibExt.Analysis.ODE.RungeKutta.rk4_stability","parent":18411,"result":true,"route":"../roadmap/Analysis/numerical-analysis/runge-kutta-methods/index.html#rk4-stability","scope":"Analysis/numerical-analysis/runge-kutta-methods","status":"fully_proved","summary":"For all real numbers $\\lambda$, $y$ and $h$, one step of size $h$ of the classical fourth-order Runge\u2013Kutta method applied to the scalar linear test equation $u' = \\lambda u$ on $\\mathbb{R}$, starting from the state $y$ at time $0$, gives $y \\, R(\\lambda h)$, where $R(z) = 1 + z + \\frac{z^2}{2} + \\frac{z^3}{6} + \\frac{z^4}{24}$.","title":"Stability function of the classical RK4 method"},{"data_url":"scopes/Analysis/numerical-analysis/strang-s-lemma.json","id":"Analysis/numerical-analysis/strang-s-lemma","kind":"scope","navigation":"module","parent":18303,"result":false,"route":"../roadmap/Analysis/numerical-analysis/strang-s-lemma/index.html","scope":"Analysis/numerical-analysis/strang-s-lemma","status":"fully_proved","title":"Strang's lemma"},{"id":"Analysis/numerical-analysis/strang-s-lemma/strang-first-lemma","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.StrangLemma.strang_first_lemma","parent":18435,"result":true,"route":"../roadmap/Analysis/numerical-analysis/strang-s-lemma/index.html#strang-first-lemma","scope":"Analysis/numerical-analysis/strang-s-lemma","status":"fully_proved","summary":"Strang's first lemma (pointwise form): if `u` solves the exact variational problem `B u w = \u2113 w` and `uh` solves the perturbed problem `Bh uh w = \u2113h w` on the discrete space `V` (embedded by the linear isometry `J`), and `Bh` is coercive with constant `\u03b1`, then for every competitor `vh : V` the error `\u2016u - J uh\u2016` is bounded by the best-approximation error of `u` at `vh` plus the consistency errors. Here\u2026","title":"Strang's first lemma"},{"id":"Analysis/numerical-analysis/strang-s-lemma/strang-first-lemma-i-inf","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.StrangLemma.strang_first_lemma_iInf","parent":18435,"result":true,"route":"../roadmap/Analysis/numerical-analysis/strang-s-lemma/index.html#strang-first-lemma-i-inf","scope":"Analysis/numerical-analysis/strang-s-lemma","status":"fully_proved","summary":"Strang's first lemma (infimum form): the error is bounded by the best-approximation error `\u2a05 vh : V, \u2016u - J vh\u2016` plus the consistency errors.","title":"Strang's first lemma"},{"data_url":"scopes/Analysis/inequalities.json","id":"Analysis/inequalities","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/inequalities/index.html","scope":"Analysis/inequalities","status":"can_prove","summary":"Classical inequalities for sums, sequences, means and integrals of real or complex numbers: Hardy, Hilbert, Carleman, Wirtinger, Opial, Landau\u2013Kolmogorov, Hermite\u2013Hadamard, Kantorovich and Popoviciu, power-sum and elementary log/exp bounds; inequalities stated with Sobolev norms go to Function spaces, eigenvalue inequalities to Operator theory.","title":"Inequalities"},{"id":"Analysis/inequalities/one-div-one-add-sub-one-div-one-add-mul","kind":"node","lean":"MathlibExt.Analysis.Calculus.ReciprocalDifferenceSqrtBoundWanted.one_div_one_add_sub_one_div_one_add_mul_le","parent":18438,"result":true,"route":"../roadmap/Analysis/inequalities/index.html#one-div-one-add-sub-one-div-one-add-mul","scope":"Analysis/inequalities","status":"can_prove","summary":"For all real numbers $a > 1$ and $x \\ge 1$,","title":"A bound on a difference of reciprocals"},{"id":"Analysis/inequalities/hardy-inequality","kind":"node","lean":"MetaMathlibExt.hardy_inequality","parent":18438,"result":true,"route":"../roadmap/Analysis/inequalities/index.html#hardy-inequality","scope":"Analysis/inequalities","status":"fully_proved","summary":"Let $p > 1$ be a real number and let $(a_n)_{n \\ge 0}$ be a sequence of nonnegative real numbers such that $\\sum_n a_n^p$ converges. Write $A_n = \\frac{1}{n+1} \\sum_{i=0}^{n} a_i$ for the average of $a_0, \\dots, a_n$. Then $\\sum_n A_n^p$ converges and","title":"Hardy's inequality"},{"id":"Analysis/inequalities/hermite-hadamard","kind":"node","lean":"MetaMathlibExt.hermite_hadamard","parent":18438,"result":true,"route":"../roadmap/Analysis/inequalities/index.html#hermite-hadamard","scope":"Analysis/inequalities","status":"fully_proved","summary":"Let $a < b$ be real numbers and let $f \\colon \\mathbb{R} \\to \\mathbb{R}$ be convex on $[a, b]$. Then $f\\left(\\frac{a+b}{2}\\right) \\le \\frac{1}{b-a} \\int_a^b f(x)\\,dx$ and $\\frac{1}{b-a} \\int_a^b f(x)\\,dx \\le \\frac{f(a) + f(b)}{2}$.","title":"Hermite\u2013Hadamard inequality"},{"id":"Analysis/inequalities/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.SquareSummableComplexInequalityWanted.conjecture_holds","parent":18438,"result":true,"route":"../roadmap/Analysis/inequalities/index.html#conjecture-holds","scope":"Analysis/inequalities","status":"can_prove","summary":"Let $(\\alpha_n)_{n \\ge 1}$ be complex numbers with $\\sum_{n \\ge 1} |\\alpha_n|^2 < \\infty$, and for $k \\ge 0$ put $\\beta_k = \\sum_{l \\ge 0} \\alpha_{2^k(2l+1)} / (l+1)$. Then each series defining $\\beta_k$ converges absolutely, $\\sum_{k \\ge 0} |\\beta_k|^2 < \\infty$, and","title":"Inequality for square-summable complex sequences"},{"id":"Analysis/inequalities/integral-abs-mul-deriv-le-of-eq-zero-lef","kind":"node","lean":"MathlibExt.Analysis.Calculus.OpialTypeIntegralInequalityWanted.integral_abs_mul_deriv_le_of_eq_zero_left","parent":18438,"result":true,"route":"../roadmap/Analysis/inequalities/index.html#integral-abs-mul-deriv-le-of-eq-zero-lef","scope":"Analysis/inequalities","status":"can_prove","summary":"Let $\\alpha \\le \\beta$ be real numbers and let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuously differentiable on $[\\alpha, \\beta]$ with $f(\\alpha) = 0$. Then","title":"Opial's inequality with one boundary condition"},{"id":"Analysis/inequalities/wirtinger-inequality","kind":"node","lean":"MetaMathlibExt.wirtinger_inequality","parent":18438,"result":true,"route":"../roadmap/Analysis/inequalities/index.html#wirtinger-inequality","scope":"Analysis/inequalities","status":"fully_proved","summary":"Let $f, f' : \\mathbb{R} \\to \\mathbb{R}$ be functions such that for every $x \\in [0, \\pi]$ the function $f$ has derivative $f'(x)$ at $x$ within $[0, \\pi]$, $f'$ is continuous on $[0, \\pi]$, and $f(0) = f(\\pi) = 0$. Then $\\int_0^\\pi f(x)^2 \\, dx \\le \\int_0^\\pi f'(x)^2 \\, dx$.","title":"Wirtinger's inequality"},{"data_url":"scopes/Analysis/inequalities/carleman-s-inequality.json","id":"Analysis/inequalities/carleman-s-inequality","kind":"scope","navigation":"module","parent":18438,"result":false,"route":"../roadmap/Analysis/inequalities/carleman-s-inequality/index.html","scope":"Analysis/inequalities/carleman-s-inequality","status":"fully_proved","title":"Carleman's inequality"},{"id":"Analysis/inequalities/carleman-s-inequality/carleman-geom-mean","kind":"node","lean":"MetaMathlibExt.carlemanGeomMean","parent":18445,"result":true,"route":"../roadmap/Analysis/inequalities/carleman-s-inequality/index.html#carleman-geom-mean","scope":"Analysis/inequalities/carleman-s-inequality","status":"fully_proved","summary":"The geometric mean of the first `n + 1` terms of a sequence `a`.","title":"Geometric mean of the initial terms of a sequence"},{"id":"Analysis/inequalities/carleman-s-inequality/carleman-geom-mean-sum-add-mul-le","kind":"node","lean":"MetaMathlibExt.carlemanGeomMean_sum_add_mul_le","parent":18445,"result":true,"route":"../roadmap/Analysis/inequalities/carleman-s-inequality/index.html#carleman-geom-mean-sum-add-mul-le","scope":"Analysis/inequalities/carleman-s-inequality","status":"fully_proved","summary":"Strengthened finite Carleman inequality: the partial sum of the geometric means plus `(M + 1)` times the last geometric mean is bounded by `e` times the partial sum of the terms.","title":"Strengthened finite Carleman inequality"},{"id":"Analysis/inequalities/carleman-s-inequality/carleman-geom-mean-sum-le","kind":"node","lean":"MetaMathlibExt.carlemanGeomMean_sum_le","parent":18445,"result":true,"route":"../roadmap/Analysis/inequalities/carleman-s-inequality/index.html#carleman-geom-mean-sum-le","scope":"Analysis/inequalities/carleman-s-inequality","status":"fully_proved","summary":"Finite Carleman inequality: the partial sum of the geometric means is bounded by `e` times the partial sum of the terms.","title":"Finite Carleman inequality"},{"id":"Analysis/inequalities/carleman-s-inequality/carleman-inequality","kind":"node","lean":"MetaMathlibExt.carleman_inequality","parent":18445,"result":true,"route":"../roadmap/Analysis/inequalities/carleman-s-inequality/index.html#carleman-inequality","scope":"Analysis/inequalities/carleman-s-inequality","status":"fully_proved","summary":"Carleman's inequality (sharp form): for a summable nonnegative sequence, the series of geometric means of the initial products is summable and bounded by `e` times the series of the terms.","title":"Carleman's inequality"},{"data_url":"scopes/Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials.json","id":"Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials","kind":"scope","navigation":"module","parent":18438,"result":false,"route":"../roadmap/Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials/index.html","scope":"Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials","status":"can_prove","title":"Coefficient bounds for homogeneous polynomials"},{"id":"Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials/eval-poly","kind":"node","lean":"MathlibExt.Analysis.ScottishBook74Wanted.EvalPoly","parent":18450,"result":true,"route":"../roadmap/Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials/index.html#eval-poly","scope":"Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials","status":"fully_proved","summary":"Given a natural number $n$, real coefficients $a_k$ indexed by multi-indices $k = (k_1, \\dots, k_n) \\in \\mathbb{N}^n$, and a point $t = (t_1, \\dots, t_n) \\in \\mathbb{R}^n$, the value is","title":"Evaluation of a homogeneous polynomial"},{"id":"Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials/scottish-book-problem74-claim","kind":"node","lean":"MathlibExt.Analysis.ScottishBook74Wanted.ScottishBookProblem74Claim","parent":18450,"result":true,"route":"../roadmap/Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials/index.html#scottish-book-problem74-claim","scope":"Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials","status":"fully_proved","summary":"This is the statement that for every $n \\ge 1$ and all real coefficients $a_k$ indexed by multi-indices $k = (k_1, \\dots, k_n) \\in \\mathbb{N}^n$ the following holds. Let $P(t) = \\sum_{k_1 + \\cdots + k_n = n} a_k \\, t_1^{k_1} \\cdots t_n^{k_n}$ be the corresponding homogeneous polynomial of degree $n$ on $\\mathbb{R}^n$. If $|P(t)| \\le 1$ for every $t \\in \\mathbb{R}^n$ with $|t_1| + \\cdots + |t_n| \\le 1$, then\u2026","title":"The coefficient bound of Scottish Book Problem 74"},{"id":"Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials/scottish-book74-holds","kind":"node","lean":"MathlibExt.Analysis.ScottishBook74Wanted.scottishBook74_holds","parent":18450,"result":true,"route":"../roadmap/Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials/index.html#scottish-book74-holds","scope":"Analysis/inequalities/coefficient-bounds-for-homogeneous-polynomials","status":"can_prove","summary":"Let $n \\ge 1$, let $a_k$ be real coefficients indexed by multi-indices $k = (k_1, \\dots, k_n) \\in \\mathbb{N}^n$, and let $P(t) = \\sum_{k_1 + \\cdots + k_n = n} a_k \\, t_1^{k_1} \\cdots t_n^{k_n}$ be the corresponding homogeneous polynomial of degree $n$ on $\\mathbb{R}^n$. If $|P(t)| \\le 1$ for every $t \\in \\mathbb{R}^n$ with $|t_1| + \\cdots + |t_n| \\le 1$, then $|a_k| \\le n^n / (k_1! \\cdots k_n!)$ for every\u2026","title":"Scottish Book Problem 74: the coefficient bound holds"},{"data_url":"scopes/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm.json","id":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","kind":"scope","navigation":"module","parent":18438,"result":false,"route":"../roadmap/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/index.html","scope":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","status":"fully_proved","title":"Elementary bounds for exponentials and logarithms"},{"id":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/log-one-sub-le-neg","kind":"node","lean":"Real.log_one_sub_le_neg","parent":18454,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear upper bound for "],["math","\\(\\log(1 - x)\\)"]],"source":"Linear upper bound for $\\log(1 - x)$"},"result":true,"route":"../roadmap/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/index.html#log-one-sub-le-neg","scope":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","status":"fully_proved","summary":"For every real number $x$ with $1 - x > 0$, $\\log(1 - x) \\le -x$.","title":"Linear upper bound for $\\log(1 - x)$"},{"id":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/neg-add-sq-le-log-one-sub","kind":"node","lean":"Real.neg_add_sq_le_log_one_sub","parent":18454,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quadratic lower bound for "],["math","\\(\\log(1 - x)\\)"]],"source":"Quadratic lower bound for $\\log(1 - x)$"},"result":true,"route":"../roadmap/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/index.html#neg-add-sq-le-log-one-sub","scope":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","status":"fully_proved","summary":"For every real number $x$ with $0 < x < 1/2$, $-x - x^2 \\le \\log(1 - x)$.","title":"Quadratic lower bound for $\\log(1 - x)$"},{"id":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/log-one-sub-bounds","kind":"node","lean":"Real.log_one_sub_bounds","parent":18454,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-sided bound for "],["math","\\(\\log(1 - x)\\)"]],"source":"Two-sided bound for $\\log(1 - x)$"},"result":true,"route":"../roadmap/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/index.html#log-one-sub-bounds","scope":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","status":"fully_proved","summary":"For every real number $x$ with $0 < x < 1/2$, $-x - x^2 \\le \\log(1 - x) \\le -x$.","title":"Two-sided bound for $\\log(1 - x)$"},{"id":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/exp-neg-le-one-sub-add-sq","kind":"node","lean":"Real.exp_neg_le_one_sub_add_sq","parent":18454,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quadratic upper bound for "],["math","\\(e^{-x}\\)"]],"source":"Quadratic upper bound for $e^{-x}$"},"result":true,"route":"../roadmap/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/index.html#exp-neg-le-one-sub-add-sq","scope":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","status":"fully_proved","summary":"For every real number $x \\ge 0$, $e^{-x} \\le 1 - x + x^2$.","title":"Quadratic upper bound for $e^{-x}$"},{"id":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/pow-le-nat-ceil-mul-exp","kind":"node","lean":"Real.pow_le_natCeil_mul_exp","parent":18454,"result":true,"route":"../roadmap/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/index.html#pow-le-nat-ceil-mul-exp","scope":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","status":"fully_proved","summary":"An explicit natural multiple of a positive-rate exponential bounds a fixed power.","title":"Explicit exponential bound on a fixed power"},{"id":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/exists-nat-pow-le-mul-exp","kind":"node","lean":"Real.exists_nat_pow_le_mul_exp","parent":18454,"result":true,"route":"../roadmap/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/index.html#exists-nat-pow-le-mul-exp","scope":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","status":"fully_proved","summary":"A single natural constant bounds a fixed power by a positive-rate exponential.","title":"Existence of a constant bounding a power by an exponential"},{"id":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/exists-nat-pow-le-mul-two-rpow","kind":"node","lean":"Real.exists_nat_pow_le_mul_two_rpow","parent":18454,"result":true,"route":"../roadmap/Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm/index.html#exists-nat-pow-le-mul-two-rpow","scope":"Analysis/inequalities/elementary-bounds-for-exponentials-and-logarithm","status":"fully_proved","summary":"The source estimate: a natural-input polynomial is bounded by a base-two exponential.","title":"Polynomial bounded by a base-two exponential"},{"data_url":"scopes/Analysis/inequalities/hilbert-s-inequality.json","id":"Analysis/inequalities/hilbert-s-inequality","kind":"scope","navigation":"module","parent":18438,"result":false,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","title":"Hilbert's inequality"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernel","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The Hilbert kernel with `0`-based indices: `1 / ((m + 1) + (n + 1))`.","title":"Hilbert kernel"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel-row","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernelRow","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel-row","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The row integrand `x \u21a6 1 / ((M + x) * \u221ax)` used to bound the row sums of the Hilbert kernel by an integral.","title":"Row integrand for the Hilbert kernel"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel-nonneg","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernel_nonneg","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel-nonneg","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The Hilbert kernel is nonnegative.","title":"Nonnegativity of the Hilbert kernel"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel-row-nonneg","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernelRow_nonneg","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel-row-nonneg","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The row integrand is nonnegative on `(0, \u221e)` when `M > 0`.","title":"Nonnegativity of the Hilbert row integrand"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel-row-continuous-on","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernelRow_continuousOn","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel-row-continuous-on","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The row integrand is continuous on `(0, \u221e)` when `M > 0`.","title":"Continuity of the Hilbert row integrand"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel-row-antitone-on","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernelRow_antitoneOn","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel-row-antitone-on","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The row integrand is antitone on `[1, \u221e)` when `M > 0`.","title":"Hilbert row integrand is decreasing on (1, \u221e)"},{"id":"Analysis/inequalities/hilbert-s-inequality/has-deriv-at-hilbert-antideriv","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hasDerivAt_hilbertAntideriv","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#has-deriv-at-hilbert-antideriv","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The derivative of the antiderivative `(2 / \u221aM) * arctan \u221a(x / M)`.","title":"Derivative of the arctangent antiderivative of the row integrand"},{"id":"Analysis/inequalities/hilbert-s-inequality/interval-integral-hilbert-kernel-row","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.intervalIntegral_hilbertKernelRow","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#interval-integral-hilbert-kernel-row","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The finite integral of the row integrand, via its antiderivative.","title":"Finite integral of the Hilbert row integrand"},{"id":"Analysis/inequalities/hilbert-s-inequality/integrable-on-ioi-hilbert-kernel-row","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.integrableOn_Ioi_hilbertKernelRow","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#integrable-on-ioi-hilbert-kernel-row","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The row integrand is integrable on `(1, \u221e)`, with a uniform bound on the interval integrals.","title":"Integrability of the Hilbert row integrand on (1, \u221e)"},{"id":"Analysis/inequalities/hilbert-s-inequality/integral-ioi-hilbert-kernel-row","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.integral_Ioi_hilbertKernelRow","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#integral-ioi-hilbert-kernel-row","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The improper integral of the row integrand over `(1, \u221e)`.","title":"Improper integral of the Hilbert row integrand"},{"id":"Analysis/inequalities/hilbert-s-inequality/div-one-add-sq-le-arctan","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.div_one_add_sq_le_arctan","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#div-one-add-sq-le-arctan","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"A lower bound for `arctan`: `t / (1 + t ^ 2) \u2264 arctan t` for `t \u2265 0`.","title":"Lower bound for the arctangent"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel-row-seq","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernelRowSeq","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel-row-seq","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The `n`-th term of the row series, `1 / ((M + (n + 1)) * \u221a((n + 1)))`, as a function of `n : \u2115`.","title":"Terms of the Hilbert row series"},{"id":"Analysis/inequalities/hilbert-s-inequality/summable-hilbert-kernel-row-seq","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.summable_hilbertKernelRowSeq","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#summable-hilbert-kernel-row-seq","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"Summability of the row series.","title":"Summability of the Hilbert row series"},{"id":"Analysis/inequalities/hilbert-s-inequality/summable-hilbert-kernel-row","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.summable_hilbertKernelRow","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#summable-hilbert-kernel-row","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"Summability of the row series in unfurled form.","title":"Summability of the Hilbert row series in unfolded form"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel-row-sum-le","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernel_row_sum_le","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel-row-sum-le","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The row sum of the Hilbert kernel is at most `\u03c0 / \u221a(m + 1)`.","title":"Row sum bound for the Hilbert kernel"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-kernel-symm","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbertKernel_symm","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-kernel-symm","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The Hilbert kernel is symmetric.","title":"Symmetry of the Hilbert kernel"},{"id":"Analysis/inequalities/hilbert-s-inequality/schur-test-finite","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.schurTest_finite","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#schur-test-finite","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The Schur test for finite bilinear forms: a weighted row/column bound implies a bound on the bilinear form with constant `\u221a(\u03b1 * \u03b2)`.","title":"Schur test for finite bilinear forms"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-inequality-finite","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbert_inequality_finite","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-inequality-finite","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The finite form of Hilbert's inequality, from the Schur test with the weights `r n = 1 / \u221a(n + 1)` and the row bound `hilbertKernel_row_sum_le`.","title":"Finite Hilbert inequality"},{"id":"Analysis/inequalities/hilbert-s-inequality/summable-hilbert-terms","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.summable_hilbert_terms","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#summable-hilbert-terms","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"The nonnegative Hilbert double series is summable for square-summable inputs.","title":"Summability of the Hilbert double series"},{"id":"Analysis/inequalities/hilbert-s-inequality/hilbert-inequality","kind":"node","lean":"MathlibExt.Analysis.Series.HilbertInequality.hilbert_inequality","parent":18462,"result":true,"route":"../roadmap/Analysis/inequalities/hilbert-s-inequality/index.html#hilbert-inequality","scope":"Analysis/inequalities/hilbert-s-inequality","status":"fully_proved","summary":"Hilbert's inequality for square-summable real sequences.","title":"Hilbert's inequality"},{"data_url":"scopes/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit.json","id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","kind":"scope","navigation":"module","parent":18438,"result":false,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","title":"Kantorovich, Bhatia\u2013Davis and Popoviciu inequalities"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/weighted-variance-eq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.weighted_variance_eq","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#weighted-variance-eq","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"The weighted variance equals the weighted squared deviation from the weighted mean.","title":"Weighted variance as weighted squared deviation from the mean"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/weighted-bhatia-davis","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.weighted_bhatia_davis","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#weighted-bhatia-davis","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"Bhatia\u2013Davis inequality, scalar weighted form: if the weights are nonnegative, sum to `1`, and every `\u03bb i` lies in `[m, M]`, then the weighted variance is at most `(M - \u03bc) * (\u03bc - m)`, where `\u03bc` is the weighted mean.","title":"Bhatia\u2013Davis inequality"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/weighted-popoviciu","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.weighted_popoviciu","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#weighted-popoviciu","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"Popoviciu's inequality, scalar weighted form: the weighted variance is at most `((M - m) / 2)\u00b2`.","title":"Popoviciu's inequality"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/weighted-kantorovich","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.weighted_kantorovich","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#weighted-kantorovich","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"Kantorovich's inequality, scalar weighted form: if the weights are nonnegative, sum to `1`, every `\u03bb i` lies in `[m, M]`, and `0 < m`, then `(\u2211 w i * \u03bb i) * (\u2211 w i / \u03bb i) \u2264 (M + m)\u00b2 / (4 * m * M)`.","title":"Kantorovich's inequality"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/apply-eigenvector-basis-real","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.apply_eigenvectorBasis_real","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#apply-eigenvector-basis-real","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"Real-valued form of `LinearMap.IsSymmetric.apply_eigenvectorBasis`: the eigenvalue is a plain real scalar, with no `\u211d \u2192 \ud835\udd5c` coercion.","title":"Eigenvector basis equation with real eigenvalues"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/eigenvalue-mem-icc-of-rayleigh-bounds","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.eigenvalue_mem_Icc_of_rayleigh_bounds","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#eigenvalue-mem-icc-of-rayleigh-bounds","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"The eigenvalues of a symmetric operator lie in `[m, M]` whenever its Rayleigh quotients do.","title":"Eigenvalues lie within the Rayleigh quotient bounds"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/sum-weights-eq-norm-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.sum_weights_eq_norm_sq","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#sum-weights-eq-norm-sq","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"The eigenbasis weights sum to `\u2016x\u2016\u00b2`.","title":"Eigenbasis weights sum to the squared norm"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/rayleigh-eq-weighted-sum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.rayleigh_eq_weighted_sum","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#rayleigh-eq-weighted-sum","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"The Rayleigh quotient is the weighted average of the eigenvalues, with weights `((eigenvectorBasis).repr x i)\u00b2`.","title":"Rayleigh quotient as a weighted average of eigenvalues"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/second-moment-eq-weighted-sum","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.second_moment_eq_weighted_sum","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#second-moment-eq-weighted-sum","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"The second-moment quadratic form is the weighted average of the squared eigenvalues.","title":"Second moment as a weighted average of squared eigenvalues"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/inv-rayleigh-eq-weighted-sum-div","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.inv_rayleigh_eq_weighted_sum_div","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#inv-rayleigh-eq-weighted-sum-div","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"The quadratic form of a two-sided inverse is the weighted average of the reciprocal eigenvalues.","title":"Inverse quadratic form as a weighted average of reciprocal eigenvalues"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/variance-eq-weighted-sum-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.variance_eq_weighted_sum_sq","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#variance-eq-weighted-sum-sq","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"The variance of `T` at a unit vector is the weighted squared deviation of the eigenvalues from the Rayleigh quotient.","title":"Operator variance as weighted squared deviation of eigenvalues"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/bhatia-davis-inequality","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.bhatia_davis_inequality","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#bhatia-davis-inequality","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"Bhatia\u2013Davis inequality for a symmetric operator: the variance of `T` at a unit vector is at most `(M - \u03bc) * (\u03bc - m)`, where `\u03bc` is the Rayleigh quotient.","title":"Bhatia\u2013Davis inequality"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/popoviciu-inequality","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.popoviciu_inequality","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#popoviciu-inequality","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"Popoviciu's inequality for a symmetric operator: the variance of `T` at a unit vector is at most `((M - m) / 2)\u00b2`.","title":"Popoviciu's inequality"},{"id":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/kantorovich-inequality","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KantorovichInequality.kantorovich_inequality","parent":18483,"result":true,"route":"../roadmap/Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit/index.html#kantorovich-inequality","scope":"Analysis/inequalities/kantorovich-bhatia-davis-and-popoviciu-inequalit","status":"fully_proved","summary":"Kantorovich's inequality for a symmetric operator: for a unit vector `x`, the product of the Rayleigh quotient of `T` and the Rayleigh quotient of its two-sided inverse `S` is at most `(M + m)\u00b2 / (4 * m * M)`.","title":"Kantorovich's inequality"},{"data_url":"scopes/Analysis/inequalities/landau-kolmogorov-inequality.json","id":"Analysis/inequalities/landau-kolmogorov-inequality","kind":"scope","navigation":"module","parent":18438,"result":false,"route":"../roadmap/Analysis/inequalities/landau-kolmogorov-inequality/index.html","scope":"Analysis/inequalities/landau-kolmogorov-inequality","status":"fully_proved","title":"Landau\u2013Kolmogorov inequality"},{"id":"Analysis/inequalities/landau-kolmogorov-inequality/landau-inequality","kind":"node","lean":"MetaMathlibExt.landau_inequality","parent":18498,"result":true,"route":"../roadmap/Analysis/inequalities/landau-kolmogorov-inequality/index.html#landau-inequality","scope":"Analysis/inequalities/landau-kolmogorov-inequality","status":"fully_proved","summary":"Landau\u2013Kolmogorov inequality on `\u211d` (case `n = 2`, `k = 1`): if `f` is twice continuously differentiable with `|f| \u2264 M\u2080` and `|f''| \u2264 M\u2082` everywhere, then `(f' x)\u00b2 \u2264 2 * M\u2080 * M\u2082` for every `x`.","title":"Landau\u2013Kolmogorov inequality"},{"id":"Analysis/inequalities/landau-kolmogorov-inequality/landau-inequality-abs","kind":"node","lean":"MetaMathlibExt.landau_inequality_abs","parent":18498,"result":true,"route":"../roadmap/Analysis/inequalities/landau-kolmogorov-inequality/index.html#landau-inequality-abs","scope":"Analysis/inequalities/landau-kolmogorov-inequality","status":"fully_proved","summary":"Square-root form of the Landau\u2013Kolmogorov inequality.","title":"Landau\u2013Kolmogorov inequality, square-root form"},{"id":"Analysis/inequalities/landau-kolmogorov-inequality/landau-inequality-complex","kind":"node","lean":"MetaMathlibExt.landau_inequality_complex","parent":18498,"result":true,"route":"../roadmap/Analysis/inequalities/landau-kolmogorov-inequality/index.html#landau-inequality-complex","scope":"Analysis/inequalities/landau-kolmogorov-inequality","status":"fully_proved","summary":"Landau\u2013Kolmogorov inequality, complex-valued case (constant `4`): if `f : \u211d \u2192 \u2102` is twice continuously differentiable with `\u2016f\u2016 \u2264 M\u2080` and `\u2016f''\u2016 \u2264 M\u2082` everywhere, then `\u2016f' x\u2016\u00b2 \u2264 4 * M\u2080 * M\u2082` for every `x`. The constant `4` comes from applying the real inequality to the real and imaginary parts and adding.","title":"Landau\u2013Kolmogorov inequality, complex-valued case"},{"id":"Analysis/inequalities/landau-kolmogorov-inequality/bdd-above-abs-deriv-and-i-sup-le-two-mul","kind":"node","lean":"MathlibExt.Analysis.Calculus.LandauKolmogorovSecondOrderWanted.bddAbove_abs_deriv_and_iSup_le_two_mul_sqrt","parent":18498,"reading_title":{"format":"inline-parts-v1","parts":[["text","Landau's inequality "],["math","\\(M_1 \\le 2\\sqrt{M_0 M_2}\\)"]],"source":"Landau's inequality $M_1 \\le 2\\sqrt{M_0 M_2}$"},"result":true,"route":"../roadmap/Analysis/inequalities/landau-kolmogorov-inequality/index.html#bdd-above-abs-deriv-and-i-sup-le-two-mul","scope":"Analysis/inequalities/landau-kolmogorov-inequality","status":"fully_proved","summary":"For $f \\in C^2(\\mathbb{R})$, let $M_k = \\sup_x |f^{(k)}(x)|$. Prove that $M_1 \\le 2\\sqrt{M_0 M_2}$.","title":"Landau's inequality $M_1 \\le 2\\sqrt{M_0 M_2}$"},{"data_url":"scopes/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences.json","id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","kind":"scope","navigation":"module","parent":18438,"result":false,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","title":"Power sum inequalities for monotone sequences"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/power-sum-rpow-one-div-le","kind":"node","lean":"MetaMathlibExt.power_sum_rpow_one_div_le","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#power-sum-rpow-one-div-le","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $n \\ge 0$ and let $x_1, \\dots, x_n$ be positive real numbers. If $\\alpha$ and $\\beta$ are real numbers with $0 < \\beta < \\alpha$, then $(\\sum_{k=1}^{n} x_k^\\alpha)^{1/\\alpha} \\le (\\sum_{k=1}^{n} x_k^\\beta)^{1/\\beta}$.","title":"Normalised power sums decrease in the exponent"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/power-sum-mul-div-max-le","kind":"node","lean":"MetaMathlibExt.power_sum_mul_div_max_le","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#power-sum-mul-div-max-le","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $0 < x_1 \\le \\dots \\le x_N$ be $N \\ge 1$ real numbers, and for real $p$ write $S^{[p]} = \\sum_{k=1}^{N} x_k^p$. For all real $r, s > 0$, $S^{[r]} S^{[s]} / \\max(x_N^{r/2} S^{[s/2]}, x_N^{s/2} S^{[r/2]}) \\le S^{[(r+s)/2]}$.","title":"Lower bound for a power sum at the mean exponent"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/power-sum-le-rpow-of-le","kind":"node","lean":"MetaMathlibExt.power_sum_le_rpow_of_le","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#power-sum-le-rpow-of-le","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $x_1, \\dots, x_N$ be $N \\ge 1$ positive real numbers, and for real $p$ write $S^{[p]} = \\sum_{k=1}^{N} x_k^p$. If $\\alpha$ and $\\beta$ are real numbers with $0 < \\beta \\le \\alpha$, then $S^{[\\alpha]} \\le (S^{[\\beta]})^{\\alpha/\\beta}$.","title":"Power sums under a larger exponent"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/power-sum-le-card-mul-rpow-of-lt","kind":"node","lean":"MetaMathlibExt.power_sum_le_card_mul_rpow_of_lt","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#power-sum-le-card-mul-rpow-of-lt","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $x_1, \\dots, x_N$ be $N \\ge 1$ positive real numbers, and for real $p$ write $S^{[p]} = \\sum_{k=1}^{N} x_k^p$. If $\\alpha$ and $\\beta$ are real numbers with $0 < \\alpha < \\beta$, then $S^{[\\alpha]} \\le N (S^{[\\beta]} / N)^{\\alpha/\\beta}$. If moreover $x_k \\ge 1$ for every $k$, then also $N (S^{[\\beta]} / N)^{\\alpha/\\beta} \\le S^{[\\beta]}$.","title":"Power mean bound for power sums with a smaller exponent"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/sum-fib-rpow-le-of-two-le","kind":"node","lean":"MetaMathlibExt.sum_fib_rpow_le_of_two_le","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#sum-fib-rpow-le-of-two-le","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $F_k$ denote the $k$-th Fibonacci number. For every natural number $n \\ge 1$ and every real $\\alpha \\ge 2$, $\\sum_{k=1}^{n} F_k^\\alpha \\le (F_n F_{n+1})^{\\alpha/2}$.","title":"Upper bound for Fibonacci power sums with exponent at least 2"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/sum-fib-rpow-le-card-mul-rpow","kind":"node","lean":"MetaMathlibExt.sum_fib_rpow_le_card_mul_rpow","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#sum-fib-rpow-le-card-mul-rpow","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $F_k$ denote the $k$-th Fibonacci number. For every natural number $n \\ge 1$ and every real $\\alpha$ with $0 < \\alpha < 2$, $\\sum_{k=1}^{n} F_k^\\alpha \\le n (F_n F_{n+1} / n)^{\\alpha/2} \\le F_n F_{n+1}$.","title":"Upper bounds for Fibonacci power sums with exponent below 2"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/weighted-power-sum-mul-div-max-le","kind":"node","lean":"MetaMathlibExt.weighted_power_sum_mul_div_max_le","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#weighted-power-sum-mul-div-max-le","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $0 < x_1 \\le \\dots \\le x_N$ be $N \\ge 1$ real numbers and $w_1, \\dots, w_N$ positive real weights, and for real $p$ write $S^{[p]} = \\sum_{k=1}^{N} w_k x_k^p$. For all real $r, s > 0$, $S^{[r]} S^{[s]} / \\max(x_N^{r/2} S^{[s/2]}, x_N^{s/2} S^{[r/2]}) \\le S^{[(r+s)/2]}$.","title":"Lower bound for a weighted power sum at the mean exponent"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/weighted-power-sum-le-mul-rpow-of-lt","kind":"node","lean":"MetaMathlibExt.weighted_power_sum_le_mul_rpow_of_lt","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#weighted-power-sum-le-mul-rpow-of-lt","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $x_1, \\dots, x_N$ be $N \\ge 1$ positive real numbers and $w_1, \\dots, w_N$ positive real weights, write $W = \\sum_{k=1}^{N} w_k$, and for real $p$ write $S^{[p]} = \\sum_{k=1}^{N} w_k x_k^p$. If $\\alpha$ and $\\beta$ are real numbers with $0 < \\alpha < \\beta$, then $S^{[\\alpha]} \\le W (S^{[\\beta]} / W)^{\\alpha/\\beta}$. If moreover $x_k \\ge 1$ for every $k$, then also\u2026","title":"Weighted power mean bound for power sums with a smaller exponent"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/weighted-sum-fib-rpow-le-mul-rpow","kind":"node","lean":"MetaMathlibExt.weighted_sum_fib_rpow_le_mul_rpow","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#weighted-sum-fib-rpow-le-mul-rpow","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $F_k$ be the $k$-th Fibonacci number ($F_1 = F_2 = 1$), let $n \\ge 1$, let $w_1, \\dots, w_n$ be positive reals with sum $W = \\sum_{k=1}^n w_k$, and let $\\alpha$ be a real number with $0 < \\alpha < 2$. Then","title":"Weighted power-mean bounds for Fibonacci numbers"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/fib-power-sum-identities","kind":"node","lean":"MetaMathlibExt.fib_power_sum_identities","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#fib-power-sum-identities","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $F_k$ denote the $k$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For every natural number $n \\ge 1$, the following identities hold in the integers: $\\sum_{i=1}^{n} F_{2i-1} = F_{2n}$; $\\sum_{i=1}^{n} F_{2i} = F_{2n+1} - 1$; $\\sum_{i=1}^{n} i F_i = n F_{n+2} - F_{n+3} + 2$; $\\sum_{i=1}^{n} F_i F_{3i} = F_n F_{n+1} F_{2n+1}$; and $\\sum_{i=1}^{n} \\binom{n}{i} F_i = F_{2n}$.","title":"Classical Fibonacci sum identities"},{"id":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/sum-fib-rpow-mul-sum-fib-rpow-ge","kind":"node","lean":"MetaMathlibExt.sum_fib_rpow_mul_sum_fib_rpow_ge","parent":18503,"result":true,"route":"../roadmap/Analysis/inequalities/power-sum-inequalities-for-monotone-sequences/index.html#sum-fib-rpow-mul-sum-fib-rpow-ge","scope":"Analysis/inequalities/power-sum-inequalities-for-monotone-sequences","status":"can_prove","summary":"Let $F_k$ denote the $k$-th Fibonacci number. For every natural number $n \\ge 1$ and all real $r, s$ with $r + s \\ge 4$, $(F_n F_{n+1})^2 \\le (\\sum_{k=1}^{n} F_k^r)(\\sum_{k=1}^{n} F_k^s)$, where $F_k^r$ and $F_k^s$ are real powers, with equality if and only if $n = 1$, or $n = 2$, or $n \\ge 3$ and $r = s = 2$.","title":"Product of two Fibonacci power sums"},{"data_url":"scopes/Analysis/asymptotic-analysis.json","id":"Analysis/asymptotic-analysis","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/index.html","scope":"Analysis/asymptotic-analysis","status":"can_prove","summary":"Asymptotic expansions and methods: Euler\u2013Maclaurin and Watson's lemma, Laplace-type integrals, coefficient asymptotics of power series (Bender's lemma), Stirling-type and inverse-factorial expansions, and asymptotic expansions of power sums, Lambert series and log sums; complex-analytic saddle-point results on entire functions go to Complex analysis.","title":"Asymptotic analysis"},{"id":"Analysis/asymptotic-analysis/euler-maclaurin-formula","kind":"node","lean":"MetaMathlibExt.eulerMaclaurinFormula","parent":18515,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/index.html#euler-maclaurin-formula","scope":"Analysis/asymptotic-analysis","status":"fully_proved","summary":"Let $a < b$ be integers, let $p \\ge 1$ be a natural number, and let $f : \\mathbb{R} \\to \\mathbb{R}$ be $p$ times continuously differentiable on $[a, b]$, its derivatives $f^{(k)}$ being taken relative to $[a, b]$ (one-sided at the endpoints). Let $B_j$ be the Bernoulli numbers with $B_1 = -\\tfrac{1}{2}$. Then there are $1$-periodic functions $P_m : \\mathbb{R} \\to \\mathbb{R}$ for $m \\ge 0$ such that $P_p$ agrees on\u2026","title":"Euler\u2013Maclaurin formula"},{"id":"Analysis/asymptotic-analysis/watson-inverse-factorial-series","kind":"node","lean":"MetaMathlibExt.watson_inverse_factorial_series","parent":18515,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/index.html#watson-inverse-factorial-series","scope":"Analysis/asymptotic-analysis","status":"can_prove","summary":"Let $A,B,\\rho,\\sigma,\\alpha,\\delta>0$, $\\gamma\\ge 0$ and $p$ be real numbers with $\\alpha+3\\delta\\lt \\pi/2$ and $1\\lt p<1+e^{-\\pi\\cot\\alpha}$. Let $S$ be the sector $\\{z\\in\\mathbb{C} : |z|>\\gamma,\\ |\\arg z|\\le\\pi/2+\\alpha+3\\delta\\}$, where $\\arg$ is the principal argument. Let $f$ be a complex function that is analytic at every point of the half-plane $\\{z : \\mathrm{Re}\\,z>0\\}$ and at every point of $S$, let\u2026","title":"Watson's inverse factorial expansion"},{"id":"Analysis/asymptotic-analysis/watson-laplace-integral-asymptotic","kind":"node","lean":"MetaMathlibExt.watson_laplace_integral_asymptotic","parent":18515,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/index.html#watson-laplace-integral-asymptotic","scope":"Analysis/asymptotic-analysis","status":"fully_proved","summary":"Let $g : \\mathbb{R} \\to \\mathbb{R}$ and let $(c_k)_{k \\ge 1}$ be real numbers such that for every $N \\ge 0$, $g(s) - \\sum_{k=1}^{N} c_k s^k = O(s^{N+1})$ as $s \\to 0^+$. Suppose there is $m_0 \\in \\mathbb{R}$ such that $s \\mapsto g(s) e^{-ms}$ is integrable on $(0, \\infty)$ for every $m \\ge m_0$. Then for every $N \\ge 0$, $\\int_0^\\infty g(s) e^{-ms} \\, ds - \\sum_{k=1}^{N} k! \\, c_k / m^{k+1} = O(1 / m^{N+2})$ as\u2026","title":"Watson's lemma for Laplace integrals"},{"data_url":"scopes/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages.json","id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","kind":"scope","navigation":"module","parent":18515,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","title":"Asymptotic expansion of Poisson averages"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-example1-sqr","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10Example1SqrtPoissonAsymptotic.ramanujan_part1_ch3_entry10_example1_sqrt_poisson_asymptotic","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-example1-sqr","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"For real $x$ let $S(x) = \\sum_{k=1}^{\\infty} x^k \\sqrt{k} / k!$. Then, as $x \\to \\infty$,","title":"Logarithmic asymptotics of the square-root Poisson series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-example1-sum-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10Example1SummableEventually.ramanujan_part1_ch3_entry10_example1_summable_eventually","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-example1-sum-2","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"For all sufficiently large real $x$, the series $\\sum_{k=1}^{\\infty} x^k \\sqrt{k} / k!$ is summable.","title":"Eventual summability of the square-root Poisson series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-example1-sum","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10Example1SumPositiveEventually.ramanujan_part1_ch3_entry10_example1_sum_positive_eventually","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-example1-sum","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"For all sufficiently large real $x$, the sum $\\sum_{k=1}^{\\infty} x^k \\sqrt{k} / k!$ is strictly positive.","title":"Eventual positivity of the square-root Poisson series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-example2-log","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10Example2LogPoissonAsymptotic.ramanujan_part1_ch3_entry10_example2_log_poisson_asymptotic","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-example2-log","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"There are real constants $C > 0$ and $x_0$ such that for every real $x \\ge x_0$ the series $\\sum_{k=1}^{\\infty} x^k \\log(k + 1) / k!$ has a sum $S$ satisfying","title":"Asymptotics of the logarithmic Poisson series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-example2-sum","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10Example2Summable.ramanujan_part1_ch3_entry10_example2_summable","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-example2-sum","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"For every real $x$, the series $\\sum_{k=1}^{\\infty} e^{-x} x^k \\log(k + 1) / k!$ is summable.","title":"Summability of the logarithmic Poisson series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-example4-sum","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10Example4Summable.ramanujan_part1_ch3_entry10_example4_summable","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-example4-sum","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number (so $H_0 = 0$). For every real $x$, the series $\\sum_{k=0}^{\\infty} e^{-x} x^k H_k / k!$ is summable.","title":"Summability of the harmonic Poisson series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-example4-har","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10Example4HarmonicPoissonAsymptotic.ramanujan_part1_ch3_entry10_example4_harmonic_poisson_asymptotic","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-example4-har","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"Let $H_k = \\sum_{j=1}^{k} 1/j$ be the $k$-th harmonic number (so $H_0 = 0$) and let $\\gamma$ be the Euler\u2013Mascheroni constant. Then, as $x \\to \\infty$,","title":"Asymptotics of the harmonic Poisson series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/two-assoc-stirling-second","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10PoissonAsymptotic.twoAssocStirlingSecond","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#two-assoc-stirling-second","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $S_2(n, k)$ is the integer $\\sum_{j=0}^{k} (-1)^j \\binom{n}{j} S(n - j, k - j)$, where $S$ denotes the Stirling numbers of the second kind.","title":"2-associated Stirling numbers in Ramanujan's Entry 10"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/two-assoc-stirling-second-eq-canonical","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10PoissonAsymptotic.twoAssocStirlingSecond_eq_canonical","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#two-assoc-stirling-second-eq-canonical","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the integer $\\sum_{j=0}^{k} (-1)^j \\binom{n}{j} S(n - j, k - j)$ used in Ramanujan's Entry 10, where $S$ denotes the Stirling numbers of the second kind, equals the 2-associated Stirling number of the second kind $S_2(n, k)$.","title":"The Entry 10 Stirling numbers are the 2-associated Stirling numbers"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-poisson-asym","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10PoissonAsymptotic.ramanujan_part1_ch3_entry10_poisson_asymptotic","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-poisson-asym","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"Let $M \\ge 1$ be an integer, $\\varphi : \\mathbb{R} \\to \\mathbb{C}$ and $G : \\mathbb{R} \\to \\mathbb{R}$ functions, $A \\ge 1$ a real number and $F$ a finite set of natural numbers. Write $\\varphi^{(n)}$ for the $n$-th iterated derivative of $\\varphi$ and $S_2(n, k)$ for the 2-associated Stirling numbers of the second kind. Assume that there are $C \\ge 0$ and $p \\in \\mathbb{N}$ such that $|\\varphi(x)| \\le C x^p$ and\u2026","title":"Ramanujan's asymptotic expansion of Poisson averages"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/ramanujan-part1-ch3-entry10-poisson-summ","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry10PoissonSummable.ramanujan_part1_ch3_entry10_poisson_summable","parent":18519,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages/index.html#ramanujan-part1-ch3-entry10-poisson-summ","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-poisson-averages","status":"fully_proved","summary":"Let $\\varphi : \\mathbb{R} \\to \\mathbb{C}$ be polynomially bounded at infinity, that is, there are a real number $C \\ge 0$ and a natural number $p$ such that $|\\varphi(x)| \\le C x^p$ for all sufficiently large real $x$. Let $F$ be a finite set of natural numbers. Then for all sufficiently large real $x$, the family indexed by the natural numbers $j$ whose $j$-th term is $0$ for $j \\in F$ and $x^j \\varphi(j) / j!$\u2026","title":"Eventual summability of the series in Entry 10"},{"data_url":"scopes/Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers.json","id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers","kind":"scope","navigation":"module","parent":18515,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/index.html","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers","status":"fully_proved","title":"Asymptotic expansion of the Catalan numbers"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/tendsto-pow-mul-sub-bernoulli-sum-of-log","kind":"node","lean":"Real.tendsto_pow_mul_sub_bernoulli_sum_of_log_diff","parent":18531,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/index.html#tendsto-pow-mul-sub-bernoulli-sum-of-log","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers","status":"fully_proved","summary":"Let $\\alpha$, $\\beta$ and $c$ be real numbers and let $(u_n)$ be a real sequence with $u_n \\to 0$ such that, for all sufficiently large $n$, $u_{n+1} - u_n = \\log(1 + \\alpha/(n + c)) - \\log(1 + \\beta/(n + c)) + (\\beta - \\alpha) \\log(1 + 1/(n + c))$. Let $B_k(x)$ be the $k$-th Bernoulli polynomial and, for $j \\ge 0$, put $a_j = (-1)^j \\left(B_{j+2}(\\alpha) - B_{j+2}(\\beta)\\right) / ((j + 1)(j + 2))$. Then for every\u2026","title":"Bernoulli-polynomial expansion from a logarithmic difference equation"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/tendsto-pow-mul-log-catalan-sub-bernoull","kind":"node","lean":"tendsto_pow_mul_log_catalan_sub_bernoulli_sum","parent":18531,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/index.html#tendsto-pow-mul-log-catalan-sub-bernoull","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers","status":"fully_proved","summary":"Let $c$ be a real number, let $C_n$ be the $n$-th Catalan number and $B_k(x)$ the $k$-th Bernoulli polynomial, and for $j \\ge 0$ put $a_j = (-1)^j \\left(B_{j+2}(1/2 - c) - B_{j+2}(2 - c)\\right) / ((j + 1)(j + 2))$. Then for every natural number $M$, $(n + c)^M \\left(\\log\\left(C_n \\sqrt{\\pi (n + c)^3} / 4^n\\right) - \\sum_{j=0}^{M-1} a_j (n + c)^{-(j+1)}\\right) \\to 0$ as $n \\to \\infty$.","title":"All-orders logarithmic expansion of the Catalan numbers at a real shift"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/catalan-stirling-asymptotic-expansions","kind":"node","lean":"MetaMathlibExt.catalan_stirling_asymptotic_expansions","parent":18531,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/index.html#catalan-stirling-asymptotic-expansions","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers","status":"fully_proved","summary":"Let $C_n$ be the $n$-th Catalan number and $B_k$ the $k$-th Bernoulli number, and let $(E_k)$ be a sequence of integers with $E_0 = 1$ and $\\sum_{j=0}^{\\lfloor n/2 \\rfloor} \\binom{n}{2j} E_{n-2j} = 0$ for every $n > 0$ (these are the Euler numbers $1, 0, -1, 0, 5, \\dots$). For $k \\ge 1$ put $c_k = (-1)^{k+1} \\left((2^{-k} - 2) B_{k+1} - k - 1\\right) / (k (k + 1))$ and $d_k = 2^{-4k-2} (4 - E_{2k}) / k$. Then for\u2026","title":"Stirling-type asymptotic expansions of the Catalan numbers"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/catalan-odd-terms-shift-unique","kind":"node","lean":"MetaMathlibExt.catalan_odd_terms_shift_unique","parent":18531,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers/index.html#catalan-odd-terms-shift-unique","scope":"Analysis/asymptotic-analysis/asymptotic-expansion-of-the-catalan-numbers","status":"fully_proved","summary":"Let $C_n = \\binom{2n}{n} / (n + 1)$ be the $n$-th Catalan number. For a real number $\\alpha$, there is a real sequence $(a_k)_{k \\ge 0}$ with $a_{2k+1} = 0$ for every $k$ such that, for every $N \\ge 0$, the difference $C_n \\sqrt{\\pi} \\, (n + \\alpha)^{3/2} / 4^n - \\sum_{k=0}^{N} a_k (n + \\alpha)^{-k}$ is $o\\left((n + \\alpha)^{-N}\\right)$ as $n \\to \\infty$, if and only if $\\alpha = 3/4$.","title":"Uniqueness of the even-power shift in Catalan asymptotics"},{"data_url":"scopes/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series.json","id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","kind":"scope","navigation":"module","parent":18515,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","title":"Asymptotic expansions of factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/chapter6-section13-example-cterm","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry5Digamma.chapter6Section13ExampleCTerm","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#chapter6-section13-example-cterm","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"For a real number $x$ and a natural number $j$, the $j$-th term of Ramanujan's factorial series is $(-1)^j / ((x + 1)(x + 2) \\cdots (x + j + 1))$, read as $0$ when the denominator vanishes.","title":"Terms of Ramanujan's factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/chapter6-section13-example-csum","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry5Digamma.chapter6Section13ExampleCSum","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#chapter6-section13-example-csum","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"For a real number $x$, $C(x)$ is the sum $\\sum_{j \\ge 0} (-1)^j / ((x + 1)(x + 2) \\cdots (x + j + 1))$ of Ramanujan's factorial series, where a term is read as $0$ when its denominator vanishes and the sum is taken to be $0$ when the series is not summable.","title":"Ramanujan's factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/summable-chapter6-section13-example-cter","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry5Digamma.summable_chapter6Section13ExampleCTerm","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#summable-chapter6-section13-example-cter","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"For every real $x > 0$, the series $\\sum_{j \\ge 0} (-1)^j / ((x + 1)(x + 2) \\cdots (x + j + 1))$ is summable.","title":"Convergence of Ramanujan's factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/chapter6-section13-example-cterm-succ","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry5Digamma.chapter6Section13ExampleCTerm_succ","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#chapter6-section13-example-cterm-succ","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Write $c_j(x) = (-1)^j / ((x + 1)(x + 2) \\cdots (x + j + 1))$ for the $j$-th term of Ramanujan's factorial series, with division by $0$ read as $0$. For every real $x$ and every natural number $j$, $c_{j+1}(x) = -\\frac{1}{x + 1} \\, c_j(x + 1)$.","title":"Recursion for the terms of Ramanujan's factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/chapter6-section13-example-csum-eq","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry5Digamma.chapter6Section13ExampleCSum_eq","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#chapter6-section13-example-csum-eq","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $C(x) = \\sum_{j \\ge 0} (-1)^j / ((x + 1)(x + 2) \\cdots (x + j + 1))$ be Ramanujan's factorial series. For every real $x > 0$, $C(x) = \\frac{1}{x + 1} - \\frac{C(x + 1)}{x + 1}$.","title":"Functional equation of Ramanujan's factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/is-big-o-chapter6-section13-example-csum","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry5Digamma.isBigO_chapter6Section13ExampleCSum_sub_sum_bell","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#is-big-o-chapter6-section13-example-csum","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $C(x) = \\sum_{j \\ge 0} (-1)^j / ((x + 1)(x + 2) \\cdots (x + j + 1))$ be Ramanujan's factorial series and let $B_n$ denote the $n$-th Bell number. For every natural number $N$,","title":"Asymptotic expansion of Ramanujan's factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch6-entry5-digamma","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry5Digamma.ramanujan_part1_ch6_entry5_digamma","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch6-entry5-digamma","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $c_j(x) = (-1)^j / ((x + 1)(x + 2) \\cdots (x + j + 1))$, let $C(x) = \\sum_{j \\ge 0} c_j(x)$, and let $B_n$ denote the $n$-th Bell number. For all sufficiently large real $x$, $x > 0$ and the series $\\sum_{j \\ge 0} c_j(x)$ is summable; and for every natural number $N$,","title":"Convergence and asymptotic expansion of Ramanujan's factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-entry3-psisummable-g","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry3Psisummable.ramanujan_part1_ch3_entry3_psisummable_general","parent":18536,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of Ramanujan's factorial series in a sector, for every "],["math","\\(x\\)"]],"source":"Summability of Ramanujan's factorial series in a sector, for every $x$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-entry3-psisummable-g","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $x$ be a complex number and let $\\varepsilon > 0$ be a real number. Then there is a real number $R$ such that for every complex number $z$ with $|z| \\ge R$ whose principal argument satisfies $-\\pi + \\varepsilon \\le \\arg z \\le \\pi - \\varepsilon$, the series $\\sum_{j=0}^{\\infty} \\frac{(-1)^j x^{j+1}}{(z+1)(z+2)\\cdots(z+j+1)}$ is summable.","title":"Summability of Ramanujan's factorial series in a sector, for every $x$"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-entry3-psisummable","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry3Psisummable.ramanujan_part1_ch3_entry3_psisummable","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-entry3-psisummable","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $x$ be a nonzero complex number and let $\\varepsilon$ be a real number with $0 < \\varepsilon < \\pi$. Then there is a real number $R$ such that for every complex number $z$ with $|z| \\ge R$ whose principal argument satisfies $-\\pi + \\varepsilon \\le \\arg z \\le \\pi - \\varepsilon$, the series $\\sum_{j=0}^{\\infty} \\frac{(-1)^j x^{j+1}}{(z+1)(z+2)\\cdots(z+j+1)}$ is summable.","title":"Summability of Ramanujan's factorial series in a sector"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/entry9-p","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry9Odd.entry9_P","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#entry9-p","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"For complex numbers $a$, $b$, $x$, $z$, Ramanujan's product series of Entry 9(i) is","title":"Ramanujan's product series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/entry9-p-closed","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry9Odd.entry9_P_closed","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#entry9-p-closed","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $a, b, x, z \\in \\mathbb{C}$ with $b \\neq 0$, let $0 < \\varepsilon < \\pi$, and put $w = (z + a)/b$. Suppose that $w \\neq 0$, that $-\\pi + \\varepsilon \\le \\arg w \\le \\pi - \\varepsilon$, and that $w + r + 1 \\neq 0$ for every natural number $r$. Then Ramanujan's product series $P$ satisfies","title":"Closed form of Ramanujan's product series off the poles"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-entry9-odd","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry9Odd.ramanujan_part1_ch3_entry9_odd","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-entry9-odd","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $a, b, x \\in \\mathbb{C}$ with $b \\neq 0$, let $0 < \\varepsilon < \\pi$, and let $n$ be a natural number. Write $F(a, b, x; k)$ for Ramanujan's generalized Bell function and $P(a, b, x; z)$ for Ramanujan's product series. Then","title":"Asymptotic expansion of Ramanujan's product series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/f-coeff-eq-generalized-bell-generating","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry3.fCoeff_eq_generalizedBellGenerating","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#f-coeff-eq-generalized-bell-generating","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $F$ be Ramanujan's generalized Bell function. For every complex number $x$ and every natural number $j$, let $f(x, j) = e^{-x} \\sum_{k=0}^{\\infty} \\frac{(k+1)^j x^{k+1}}{k!}$. Then $f(x, j) = F(0, 1, x; j)$.","title":"Entry 3 coefficients as generalized Bell values"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/psi-eq-entry9-p","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry3.psi_eq_entry9_P","parent":18536,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramanujan's factorial series as the Entry 9 product series at "],["math","\\(a = 0\\)"],["text",", "],["math","\\(b = 1\\)"]],"source":"Ramanujan's factorial series as the Entry 9 product series at $a = 0$, $b = 1$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#psi-eq-entry9-p","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $P$ be Ramanujan's Entry 9(i) product series. For all complex numbers $x$ and $z$, let $\\psi(x, z) = \\sum_{j=0}^{\\infty} t_j$, where $t_j = \\frac{(-1)^j x^{j+1}}{(z+1)(z+2)\\cdots(z+j+1)}$ when the product in the denominator is nonzero and $t_j = 0$ otherwise. Then $\\psi(x, z) = P(0, 1, x; z)$.","title":"Ramanujan's factorial series as the Entry 9 product series at $a = 0$, $b = 1$"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-entry3-general","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry3.ramanujan_part1_ch3_entry3_general","parent":18536,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic expansion of Ramanujan's factorial series, for every "],["math","\\(x\\)"]],"source":"Asymptotic expansion of Ramanujan's factorial series, for every $x$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-entry3-general","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $x$ be a complex number. For natural numbers $j$ let $f(x, j) = e^{-x} \\sum_{k=0}^{\\infty} \\frac{(k+1)^j x^{k+1}}{k!}$, and for complex $z$ let $\\psi(x, z) = \\sum_{j=0}^{\\infty} t_j$, where $t_j = \\frac{(-1)^j x^{j+1}}{(z+1)(z+2)\\cdots(z+j+1)}$ when the denominator is nonzero and $t_j = 0$ otherwise. Let $\\varepsilon$ be a real number with $0 < \\varepsilon < \\pi$, and let $S_\\varepsilon$ be the set of complex\u2026","title":"Asymptotic expansion of Ramanujan's factorial series, for every $x$"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-entry3","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry3.ramanujan_part1_ch3_entry3","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-entry3","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $x$ be a nonzero complex number. For natural numbers $j$ let $f(x, j) = e^{-x} \\sum_{k=0}^{\\infty} \\frac{(k+1)^j x^{k+1}}{k!}$, and for complex $z$ let $\\psi(x, z) = \\sum_{j=0}^{\\infty} t_j$, where $t_j = \\frac{(-1)^j x^{j+1}}{(z+1)(z+2)\\cdots(z+j+1)}$ when the denominator is nonzero and $t_j = 0$ otherwise. Let $\\varepsilon$ be a real number with $0 < \\varepsilon < \\pi$, and let $S_\\varepsilon$ be the set of\u2026","title":"Ramanujan's asymptotic expansion of a factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-entry9-odd-productsu","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Entry9OddProductsummable.ramanujan_part1_ch3_entry9_odd_productsummable","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-entry9-odd-productsu","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $a, b, x \\in \\mathbb{C}$ with $b \\neq 0$ and let $0 < \\varepsilon < \\pi$. Then there is a real number $R$ such that for every complex $z$ with $|(z + a)/b| \\ge R$ and $-\\pi + \\varepsilon \\le \\arg((z + a)/b) \\le \\pi - \\varepsilon$, the series","title":"Convergence of Ramanujan's product series in a sector"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-section8-example1-as","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Section8Example1Asymptotic.ramanujan_part1_ch3_section8_example1_asymptotic","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-section8-example1-as","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $a_0, a_1, \\ldots$ be complex numbers, let $\\psi : \\mathbb{C} \\to \\mathbb{C}$, let $\\lambda$ be either a real number or $-\\infty$, and let $0 < \\varepsilon < \\pi/2$. Suppose that for every complex $z$ that is not a negative integer and satisfies $\\operatorname{Re} z > \\lambda$, the series","title":"Asymptotic expansion of a factorial series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-section8-example1-su","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Section8Example1Summable.ramanujan_part1_ch3_section8_example1_summable_general","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-section8-example1-su","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $a_0, a_1, \\ldots$ be complex numbers, let $\\lambda$ be either a real number or $-\\infty$, and let $\\varepsilon > 0$. Suppose that the series","title":"Convergence of a factorial series in a sector, general form"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/ramanujan-part1-ch3-section8-example1-su-2","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch3.Section8Example1Summable.ramanujan_part1_ch3_section8_example1_summable","parent":18536,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series/index.html#ramanujan-part1-ch3-section8-example1-su-2","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-factorial-series","status":"fully_proved","summary":"Let $a_0, a_1, \\ldots$ be complex numbers, let $\\lambda$ be either a real number or $-\\infty$, and let $\\varepsilon > 0$, with $\\varepsilon < \\pi/2$ when $\\lambda$ is real and $\\varepsilon < \\pi$ when $\\lambda = -\\infty$. Suppose that the series","title":"Convergence of a factorial series in a sector"},{"data_url":"scopes/Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series.json","id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series","kind":"scope","navigation":"module","parent":18515,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/index.html","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series","status":"fully_proved","title":"Asymptotic expansions of Lambert series"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/summable-chapter6-entry14-minus-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry10Centralbinomialfull.summable_chapter6Entry14MinusTerm","parent":18557,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of the Lambert series "],["math","\\(\\sum_k 1/(e^{kx}-1)\\)"]],"source":"Summability of the Lambert series $\\sum_k 1/(e^{kx}-1)$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/index.html#summable-chapter6-entry14-minus-term","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series","status":"fully_proved","summary":"For every real $x > 0$, the series $\\sum_{k=1}^{\\infty} \\frac{1}{e^{kx} - 1}$ is summable.","title":"Summability of the Lambert series $\\sum_k 1/(e^{kx}-1)$"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/chapter6-entry14-minus-sum-error-le","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry10Centralbinomialfull.chapter6Entry14MinusSum_error_le","parent":18557,"reading_title":{"format":"inline-parts-v1","parts":[["text","Remainder bound for the Lambert series "],["math","\\(\\sum_k 1/(e^{kx}-1)\\)"]],"source":"Remainder bound for the Lambert series $\\sum_k 1/(e^{kx}-1)$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/index.html#chapter6-entry14-minus-sum-error-le","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series","status":"fully_proved","summary":"Let $x > 0$ be real and let $n \\ge 1$ be an integer. Let $S(x) = \\sum_{k=1}^{\\infty} \\frac{1}{e^{kx} - 1}$, let $B_m$ be the $m$-th Bernoulli number, let $\\gamma$ be the Euler\u2013Mascheroni constant, and let $C_n(x) = \\sum_{k=1}^{n} \\frac{B_{2k}^2 \\, x^{2k-1}}{2k \\, (2k)!}$. Then\u2026","title":"Remainder bound for the Lambert series $\\sum_k 1/(e^{kx}-1)$"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/ramanujan-part1-ch6-entry10-centralbinom","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry10Centralbinomialfull.ramanujan_part1_ch6_entry10_centralbinomialfull","parent":18557,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic expansion of the Lambert series "],["math","\\(\\sum_k 1/(e^{kx}-1)\\)"]],"source":"Asymptotic expansion of the Lambert series $\\sum_k 1/(e^{kx}-1)$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/index.html#ramanujan-part1-ch6-entry10-centralbinom","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series","status":"fully_proved","summary":"Let $x > 0$ be real and let $n \\ge 1$ be an integer. Let $B_m$ be the $m$-th Bernoulli number, let $\\gamma$ be the Euler\u2013Mascheroni constant, and let $C_n(x) = \\sum_{k=1}^{n} \\frac{B_{2k}^2 \\, x^{2k-1}}{2k \\, (2k)!}$. Then the series $\\sum_{k=1}^{\\infty} \\frac{1}{e^{kx} - 1}$ is summable, and its sum $S(x)$ satisfies\u2026","title":"Asymptotic expansion of the Lambert series $\\sum_k 1/(e^{kx}-1)$"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/summable-chapter6-entry14-plus-term","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry10Centralbinomialexpansion.summable_chapter6Entry14PlusTerm","parent":18557,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summability of the Lambert series "],["math","\\(\\sum_k 1/(e^{kx}+1)\\)"]],"source":"Summability of the Lambert series $\\sum_k 1/(e^{kx}+1)$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/index.html#summable-chapter6-entry14-plus-term","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series","status":"fully_proved","summary":"For every real $x > 0$, the series $\\sum_{k=1}^{\\infty} \\frac{1}{e^{kx} + 1}$ is summable.","title":"Summability of the Lambert series $\\sum_k 1/(e^{kx}+1)$"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/ramanujan-part1-ch6-entry10-centralbinom-2","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch6.Entry10Centralbinomialexpansion.ramanujan_part1_ch6_entry10_centralbinomialexpansion","parent":18557,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic expansion of the Lambert series "],["math","\\(\\sum_k 1/(e^{kx}+1)\\)"]],"source":"Asymptotic expansion of the Lambert series $\\sum_k 1/(e^{kx}+1)$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series/index.html#ramanujan-part1-ch6-entry10-centralbinom-2","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-lambert-series","status":"fully_proved","summary":"Let $x > 0$ be real and let $n \\ge 1$ be an integer. Let $B_m$ be the $m$-th Bernoulli number and let $D_n(x) = \\sum_{k=1}^{n} \\frac{(2^{2k} - 1) \\, B_{2k}^2 \\, x^{2k-1}}{2k \\, (2k)!}$. Then the series $\\sum_{k=1}^{\\infty} \\frac{1}{e^{kx} + 1}$ is summable, and its sum $T(x)$ satisfies\u2026","title":"Asymptotic expansion of the Lambert series $\\sum_k 1/(e^{kx}+1)$"},{"data_url":"scopes/Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums.json","id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums","kind":"scope","navigation":"module","parent":18515,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums/index.html","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums","status":"can_prove","title":"Asymptotic expansions of power sums"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums/ramanujan-part1-ch7-entry1-powersum","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch7.Entry1Powersum.ramanujan_part1_ch7_entry1_powersum","parent":18563,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums/index.html#ramanujan-part1-ch7-entry1-powersum","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums","status":"fully_proved","summary":"Let $r \\in \\mathbb{C}$ with $r \\ne -1$, and let $N$ be a positive integer with $\\mathrm{Re}\\, r < 2N$. For natural $n$ put $A_N(n) = \\zeta(-r) + \\frac{n^{r+1}}{r+1} + \\frac{n^r}{2} + \\sum_{k=1}^{N} T_k(n)$, where $T_k(n) = \\frac{B_{2k}}{(2k)!}\\, r(r-1)\\cdots(r-2k+2)\\, n^{r-2k+1}$ (a product of $2k - 1$ factors), $\\zeta$ is the Riemann zeta function, $B_{2k}$ are the Bernoulli numbers and complex powers are\u2026","title":"Ramanujan's asymptotic expansion of power sums"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums/ramanujan-part1-ch8-entry19-gaussdigamma","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch8.Entry19Gaussdigamma.ramanujan_part1_ch8_entry19_gaussdigamma","parent":18563,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums/index.html#ramanujan-part1-ch8-entry19-gaussdigamma","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums","status":"fully_proved","summary":"For natural numbers $m$ and $x$ let $S_m(x) = \\sum_{k=1}^{x} (\\log k)^m$, let $c_m = (-1)^m \\zeta^{(m)}(0)$, where $\\zeta^{(m)}(0)$ is the $m$-th derivative at $0$ of the real function $s \\mapsto \\operatorname{Re} \\zeta(s)$, and let $\\psi_m(x) = S_m(x) - c_m$. Let $n$ be a natural number. Then for every $m \\le n$,","title":"Ramanujan's asymptotic expansion of a binomial combination of log-power sums"},{"id":"Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums/ramanujan-part1-ch9-entry27-clausen","kind":"node","lean":"MathlibExt.Analysis.Ramanujan.Part1Ch9.Entry27Clausen.ramanujan_part1_ch9_entry27_clausen","parent":18563,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums/index.html#ramanujan-part1-ch9-entry27-clausen","scope":"Analysis/asymptotic-analysis/asymptotic-expansions-of-power-sums","status":"can_prove","summary":"Let $r > -1$ be a real number. For a natural number $n$ write $\\varphi_r(n) = \\sum_{k=1}^{n} k^r \\log k$ and $S_r(n) = \\sum_{k=1}^{n} k^r$, and for a natural number $m$ let $D_m(r) = \\sum_{j=0}^{m} \\prod_{0 \\le q \\le m,\\ q \\ne j} (r - q)$, the derivative of $r(r-1)\\cdots(r-m)$ with respect to $r$. Let $B_j$ be the Bernoulli numbers and $\\zeta$ the Riemann zeta function. Then there is a real constant $C$ such that\u2026","title":"Asymptotic expansion of partial sums of powers times logarithms"},{"data_url":"scopes/Analysis/asymptotic-analysis/bender-s-lemma.json","id":"Analysis/asymptotic-analysis/bender-s-lemma","kind":"scope","navigation":"module","parent":18515,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/bender-s-lemma/index.html","scope":"Analysis/asymptotic-analysis/bender-s-lemma","status":"fully_proved","title":"Bender's lemma"},{"id":"Analysis/asymptotic-analysis/bender-s-lemma/bender-product-coefficient-asymptotic-ge","kind":"node","lean":"MathlibExt.Analysis.Asymptotics.BenderProductCoefficient.bender_product_coefficient_asymptotic_general","parent":18567,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/bender-s-lemma/index.html#bender-product-coefficient-asymptotic-ge","scope":"Analysis/asymptotic-analysis/bender-s-lemma","status":"fully_proved","summary":"Let $(a_n)$ and $(b_n)$ be real sequences, let $A(z) = \\sum_n a_n z^n$ have radius of convergence $\\alpha \\in [0, \\infty]$, and let $\\beta \\ge 0$ be a real number with $\\beta < \\alpha$. Suppose that $b_n \\neq 0$ for all sufficiently large $n$, that $b_{n-1} / b_n \\to \\beta$ as $n \\to \\infty$, and that $A(\\beta) \\neq 0$. Then the coefficients $c_n = \\sum_{k=0}^{n} a_k b_{n-k}$ of the product of the formal power\u2026","title":"Bender's lemma without a radius condition on the second series"},{"id":"Analysis/asymptotic-analysis/bender-s-lemma/bender-product-coefficient-asymptotic","kind":"node","lean":"MathlibExt.Analysis.Asymptotics.BenderProductCoefficient.bender_product_coefficient_asymptotic","parent":18567,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/bender-s-lemma/index.html#bender-product-coefficient-asymptotic","scope":"Analysis/asymptotic-analysis/bender-s-lemma","status":"fully_proved","summary":"Let $(a_n)$ and $(b_n)$ be real sequences such that $A(z) = \\sum_n a_n z^n$ has radius of convergence $\\alpha \\in [0, \\infty]$ and $\\sum_n b_n z^n$ has radius of convergence $\\beta$, where $\\beta \\ge 0$ is a real number with $\\beta < \\alpha$. Suppose that $b_n \\neq 0$ for all sufficiently large $n$, that $b_{n-1} / b_n \\to \\beta$ as $n \\to \\infty$, and that $A(\\beta) \\neq 0$. Then the coefficients\u2026","title":"Bender's lemma on coefficients of a product of power series"},{"data_url":"scopes/Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi.json","id":"Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi","kind":"scope","navigation":"module","parent":18515,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi/index.html","scope":"Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi","status":"can_prove","title":"Convolution asymptotics of subexponential functions"},{"id":"Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi/f","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.ExpNegSqrtNormSelfConvolutionLimitWanted.f","parent":18570,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function "],["math","\\(e^{-\\sqrt{\\|x\\|}}\\)"],["text"," on the plane"]],"source":"The function $e^{-\\sqrt{\\|x\\|}}$ on the plane"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi/index.html#f","scope":"Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi","status":"fully_proved","summary":"Let $f : \\mathbb{R}^2 \\to \\mathbb{R}$ be $f(x) = e^{-\\sqrt{\\|x\\|}}$, where $\\|x\\|$ is the Euclidean norm of $x$.","title":"The function $e^{-\\sqrt{\\|x\\|}}$ on the plane"},{"id":"Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi/tendsto-convolution-self-div-at-top-norm","kind":"node","lean":"MathlibExt.Analysis.SpecificLimits.ExpNegSqrtNormSelfConvolutionLimitWanted.tendsto_convolution_self_div_atTop_norm","parent":18570,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotics of the self-convolution of "],["math","\\(e^{-\\sqrt{\\|x\\|}}\\)"]],"source":"Asymptotics of the self-convolution of $e^{-\\sqrt{\\|x\\|}}$"},"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi/index.html#tendsto-convolution-self-div-at-top-norm","scope":"Analysis/asymptotic-analysis/convolution-asymptotics-of-subexponential-functi","status":"can_prove","summary":"Let $f : \\mathbb{R}^2 \\to \\mathbb{R}$ be $f(x) = e^{-\\sqrt{\\|x\\|}}$, where $\\|x\\|$ is the Euclidean norm, and let $f * f$ be the convolution of $f$ with itself with respect to Lebesgue measure. Then","title":"Asymptotics of the self-convolution of $e^{-\\sqrt{\\|x\\|}}$"},{"data_url":"scopes/Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences.json","id":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences","kind":"scope","navigation":"module","parent":18515,"result":false,"route":"../roadmap/Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/index.html","scope":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences","status":"fully_proved","title":"Taylor's law for integer sequences"},{"id":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/is-taylor-admissible-sequence","kind":"node","lean":"MetaMathlibExt.IsTaylorAdmissibleSequence","parent":18573,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/index.html#is-taylor-admissible-sequence","scope":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences","status":"fully_proved","summary":"A sequence $a : \\mathbb{N} \\to \\mathbb{N}$, indexed from $1$ (its value at $0$ is ignored), is admissible when $a(n) > 0$ for every $n \\ge 1$ and there is $n_0 \\ge 1$ such that $a(n) < a(n + 1)$ for every $n \\ge n_0$.","title":"Admissible sequences for Taylor's law"},{"id":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/initial-segment-mean","kind":"node","lean":"MetaMathlibExt.initialSegmentMean","parent":18573,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/index.html#initial-segment-mean","scope":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences","status":"fully_proved","summary":"For a sequence $a : \\mathbb{N} \\to \\mathbb{N}$ and $n \\in \\mathbb{N}$, the initial-segment mean is the real number $m(a, n) = \\frac{1}{n} \\sum_{j=1}^{n} a(j)$, with the convention $1 / 0 = 0$, so that $m(a, 0) = 0$.","title":"Initial-segment mean"},{"id":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/initial-segment-sample-variance","kind":"node","lean":"MetaMathlibExt.initialSegmentSampleVariance","parent":18573,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/index.html#initial-segment-sample-variance","scope":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences","status":"fully_proved","summary":"For a sequence $a : \\mathbb{N} \\to \\mathbb{N}$ and $n \\in \\mathbb{N}$, the unbiased initial-segment sample variance is the real number $v(a, n) = \\frac{1}{n - 1} \\sum_{j=1}^{n} \\left(a(j) - \\frac{1}{n} \\sum_{k=1}^{n} a(k)\\right)^2$, with the convention $1 / 0 = 0$; in particular $v(a, 0) = v(a, 1) = 0$.","title":"Initial-segment sample variance"},{"id":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/satisfies-taylors-law-asymptotically","kind":"node","lean":"MetaMathlibExt.SatisfiesTaylorsLawAsymptotically","parent":18573,"result":true,"route":"../roadmap/Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences/index.html#satisfies-taylors-law-asymptotically","scope":"Analysis/asymptotic-analysis/taylor-s-law-for-integer-sequences","status":"fully_proved","summary":"A sequence $a : \\mathbb{N} \\to \\mathbb{N}$, indexed from $1$ (its value at $0$ is ignored), satisfies Taylor's law asymptotically when it is admissible for Taylor's law and, writing $m(a, n)$ and $v(a, n)$ for its initial-segment mean and initial-segment sample variance, there are real numbers $c > 0$ and $b$ such that $v(a, n) / m(a, n)^b \\to c$ as $n \\to \\infty$.","title":"Taylor's law, asymptotic form"},{"data_url":"scopes/Analysis/potential-theory.json","id":"Analysis/potential-theory","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/potential-theory/index.html","scope":"Analysis/potential-theory","status":"can_prove","summary":"Harmonic and subharmonic functions in the plane and in R^n, harmonic conjugates, the Newton and logarithmic potentials and fundamental solutions of the Laplacian.","title":"Potential theory"},{"id":"Analysis/potential-theory/eq-zero-of-harmonic-on-nhd-of-vanishing","kind":"node","lean":"MathlibExt.Analysis.Complex.HarmonicVanishingOnPositiveMeasureWanted.eq_zero_of_harmonicOnNhd_of_vanishing_on_pos_volume","parent":18578,"result":true,"route":"../roadmap/Analysis/potential-theory/index.html#eq-zero-of-harmonic-on-nhd-of-vanishing","scope":"Analysis/potential-theory","status":"can_prove","summary":"Let $\\Omega \\subseteq \\mathbb{C}$ be a nonempty connected open set and let $h : \\mathbb{C} \\to \\mathbb{R}$ be harmonic on $\\Omega$. If $h$ vanishes on some subset $s \\subseteq \\Omega$ of positive (outer) Lebesgue measure, then $h(z) = 0$ for all $z \\in \\Omega$.","title":"A harmonic function vanishing on a set of positive measure is zero"},{"id":"Analysis/potential-theory/exists-differentiable-on-re-eq-of-harmon","kind":"node","lean":"MathlibExt.Analysis.Complex.HarmonicConjugateSimplyConnectedWanted.exists_differentiableOn_re_eq_of_harmonicOnNhd_of_isSimplyConnected","parent":18578,"result":true,"route":"../roadmap/Analysis/potential-theory/index.html#exists-differentiable-on-re-eq-of-harmon","scope":"Analysis/potential-theory","status":"can_prove","summary":"Let $\\Omega \\subseteq \\mathbb{C}$ be a simply connected open set and let $u : \\mathbb{C} \\to \\mathbb{R}$ be harmonic on $\\Omega$. Then there is a function $f$ holomorphic on $\\Omega$ with $\\operatorname{Re} f(z) = u(z)$ for all $z \\in \\Omega$, and $f$ is unique up to an additive constant: if $f$ and $g$ are holomorphic on $\\Omega$ with $\\operatorname{Re} f(z) = u(z)$ and $\\operatorname{Re} g(z) = u(z)$ for all\u2026","title":"Harmonic functions on simply connected domains are real parts"},{"id":"Analysis/potential-theory/eq-zero-of-harmonic-on-nhd-upper-half-pl","kind":"node","lean":"MathlibExt.Analysis.Complex.NonnegHarmonicUpperHalfPlaneAxisIntegrableWanted.eq_zero_of_harmonicOnNhd_upperHalfPlane_of_nonneg_of_integrableOn_imagAxis","parent":18578,"result":true,"route":"../roadmap/Analysis/potential-theory/index.html#eq-zero-of-harmonic-on-nhd-upper-half-pl","scope":"Analysis/potential-theory","status":"can_prove","summary":"Nonnegative harmonic functions on the upper half-plane integrable along the imaginary axis vanish. Let $\\mathbb{C}_{+}=\\{z\\in\\mathbb{C}:\\Im z>0\\}$ be the upper half-plane. Suppose $f:\\mathbb{C}_{+}\\to\\mathbb{R}$ is harmonic and non-negative, and satisfies \\int_{0}^{\\infty}\\lvert f(iy)\\rvert\\,dy\\lt \\infty. \\] Then $f(z)=0$ for every $z\\in\\mathbb{C}_{+}$.","title":"Nonnegative harmonic functions on the upper half-plane integrable along the imaginary axis vanish"},{"data_url":"scopes/Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials.json","id":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials","kind":"scope","navigation":"module","parent":18578,"result":false,"route":"../roadmap/Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/index.html","scope":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials","status":"can_prove","title":"Equilibrium points of logarithmic potentials"},{"id":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/charge-seq","kind":"node","lean":"MathlibExt.Analysis.LogarithmicPotentialEquilibriumFreeWanted.ChargeSeq","parent":18582,"result":true,"route":"../roadmap/Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/index.html#charge-seq","scope":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials","status":"fully_proved","summary":"A charge sequence is an arbitrary sequence $(z_n)_{n \\in \\mathbb{N}}$ of complex numbers, read as the positions of the charges.","title":"Charge sequences"},{"id":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/points-distinct","kind":"node","lean":"MathlibExt.Analysis.LogarithmicPotentialEquilibriumFreeWanted.PointsDistinct","parent":18582,"result":true,"route":"../roadmap/Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/index.html#points-distinct","scope":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials","status":"fully_proved","summary":"A charge sequence $(z_n)$ has distinct positions when its terms are pairwise distinct: $z_m \\neq z_n$ whenever $m \\neq n$.","title":"Distinct charge positions"},{"id":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/is-summable","kind":"node","lean":"MathlibExt.Analysis.LogarithmicPotentialEquilibriumFreeWanted.IsSummable","parent":18582,"result":true,"route":"../roadmap/Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/index.html#is-summable","scope":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials","status":"fully_proved","summary":"A charge sequence $(z_n)$ is summable when $z_n \\neq 0$ for every $n$ and $\\sum_n 1/|z_n| < \\infty$.","title":"Summable charge sequences"},{"id":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/field-sum","kind":"node","lean":"MathlibExt.Analysis.LogarithmicPotentialEquilibriumFreeWanted.FieldSum","parent":18582,"result":true,"route":"../roadmap/Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/index.html#field-sum","scope":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials","status":"fully_proved","summary":"For a charge sequence $(z_n)$ and a point $z \\in \\mathbb{C}$, the field at $z$ is the sum $\\sum_n 1/(z - z_n)$, where a term with $z_n = z$ counts as $0$ and the sum is taken to be $0$ when the family $\\bigl(1/(z - z_n)\\bigr)_n$ is not summable.","title":"Field of a charge sequence"},{"id":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/conjecture","kind":"node","lean":"MathlibExt.Analysis.LogarithmicPotentialEquilibriumFreeWanted.conjecture","parent":18582,"result":true,"route":"../roadmap/Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/index.html#conjecture","scope":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials","status":"fully_proved","summary":"The conjecture states that there exists a charge sequence $(z_n)$ whose positions are pairwise distinct, which is summable ($z_n \\neq 0$ for every $n$ and $\\sum_n 1/|z_n| < \\infty$), and whose field $\\sum_n 1/(z - z_n)$ is nonzero at every point $z \\in \\mathbb{C}$ that is not one of the $z_n$.","title":"Equilibrium-free logarithmic potential conjecture"},{"id":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/conjecture-refuted","kind":"node","lean":"MathlibExt.Analysis.LogarithmicPotentialEquilibriumFreeWanted.conjecture_refuted","parent":18582,"result":true,"route":"../roadmap/Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials/index.html#conjecture-refuted","scope":"Analysis/potential-theory/equilibrium-points-of-logarithmic-potentials","status":"can_prove","summary":"There is no charge sequence $(z_n)$ whose positions are pairwise distinct, which is summable ($z_n \\neq 0$ for every $n$ and $\\sum_n 1/|z_n| < \\infty$), and whose field $\\sum_n 1/(z - z_n)$ is nonzero at every point $z \\in \\mathbb{C}$ that is not one of the $z_n$.","title":"The equilibrium-free logarithmic potential conjecture is false"},{"data_url":"scopes/Analysis/potential-theory/fundamental-solutions-of-the-laplacian.json","id":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian","kind":"scope","navigation":"module","parent":18578,"result":false,"route":"../roadmap/Analysis/potential-theory/fundamental-solutions-of-the-laplacian/index.html","scope":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian","status":"can_prove","title":"Fundamental solutions of the Laplacian"},{"id":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian/harmonic-on-nhd-one-div-norm-pow-sub-two","kind":"node","lean":"MathlibExt.Analysis.Calculus.NormPowTwoSubDimHarmonicWanted.harmonicOnNhd_one_div_norm_pow_sub_two","parent":18589,"result":true,"route":"../roadmap/Analysis/potential-theory/fundamental-solutions-of-the-laplacian/index.html#harmonic-on-nhd-one-div-norm-pow-sub-two","scope":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian","status":"can_prove","summary":"Let $m > 2$ be a natural number. The function $x \\mapsto 1/|x|^{m-2}$ on the Euclidean space $\\mathbb{R}^m$, where $|x|$ is the Euclidean norm, is harmonic on $\\mathbb{R}^m \\setminus \\{0\\}$.","title":"The Newtonian kernel is harmonic away from the origin"},{"id":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian/harmonic-on-nhd-log-norm-euclidean-space","kind":"node","lean":"MathlibExt.Analysis.Complex.LogNormHarmonicPlaneWanted.harmonicOnNhd_log_norm_euclideanSpace_two","parent":18589,"result":true,"route":"../roadmap/Analysis/potential-theory/fundamental-solutions-of-the-laplacian/index.html#harmonic-on-nhd-log-norm-euclidean-space","scope":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian","status":"can_prove","summary":"The function $g(x) = \\log|x|$, where $|x|$ is the Euclidean norm of $x \\in \\mathbb{R}^2$, is harmonic on $\\mathbb{R}^2 \\setminus \\{0\\}$: near every point $x \\ne 0$ it is twice continuously differentiable and its Laplacian vanishes.","title":"The logarithm of the norm is harmonic on the punctured plane"},{"id":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian/exists-fundamental-solution-laplacian","kind":"node","lean":"MathlibExt.Analysis.Distribution.NewtonPotentialWanted.exists_fundamental_solution_laplacian","parent":18589,"result":true,"route":"../roadmap/Analysis/potential-theory/fundamental-solutions-of-the-laplacian/index.html#exists-fundamental-solution-laplacian","scope":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian","status":"can_prove","summary":"Let $E$ be a real inner product space of dimension $3$. Then there is a complex-valued tempered distribution $N$ on $E$ such that $\\Delta N = \\delta_0$, where $\\Delta$ is the Laplacian in the sense of distributions and $\\delta_0$ is the Dirac delta at the origin.","title":"Fundamental solution of the Laplacian in dimension three"},{"id":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian/newton-potential-harmonic","kind":"node","lean":"MathlibExt.Analysis.Distribution.NewtonPotentialWanted.newton_potential_harmonic","parent":18589,"result":true,"route":"../roadmap/Analysis/potential-theory/fundamental-solutions-of-the-laplacian/index.html#newton-potential-harmonic","scope":"Analysis/potential-theory/fundamental-solutions-of-the-laplacian","status":"can_prove","summary":"Let $E$ be a real inner product space of dimension $3$, and define $N : E \\to \\mathbb{R}$ by $N(y) = -1/(4\\pi\\|y\\|)$ for $y \\neq 0$ and $N(0) = 0$. Then the Laplacian of $N$ vanishes at every point $x \\neq 0$ of $E$.","title":"The Newton potential is harmonic away from the origin"},{"data_url":"scopes/Analysis/potential-theory/harmonic-extension-to-a-half-space.json","id":"Analysis/potential-theory/harmonic-extension-to-a-half-space","kind":"scope","navigation":"module","parent":18578,"result":false,"route":"../roadmap/Analysis/potential-theory/harmonic-extension-to-a-half-space/index.html","scope":"Analysis/potential-theory/harmonic-extension-to-a-half-space","status":"can_prove","title":"Harmonic extension to a half-space"},{"id":"Analysis/potential-theory/harmonic-extension-to-a-half-space/r2","kind":"node","lean":"MathlibExt.Analysis.Fourier.HalfSpaceLaplaceDecayEstimateWanted.R2","parent":18594,"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-extension-to-a-half-space/index.html#r2","scope":"Analysis/potential-theory/harmonic-extension-to-a-half-space","status":"fully_proved","summary":"The Euclidean plane $\\mathbb{R}^2$ is the space of pairs of real numbers with the Euclidean inner product and norm.","title":"The Euclidean plane"},{"id":"Analysis/potential-theory/harmonic-extension-to-a-half-space/r3","kind":"node","lean":"MathlibExt.Analysis.Fourier.HalfSpaceLaplaceDecayEstimateWanted.R3","parent":18594,"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-extension-to-a-half-space/index.html#r3","scope":"Analysis/potential-theory/harmonic-extension-to-a-half-space","status":"fully_proved","summary":"Euclidean three-space $\\mathbb{R}^3$ is the space of triples of real numbers with the Euclidean inner product and norm.","title":"Euclidean three-space"},{"id":"Analysis/potential-theory/harmonic-extension-to-a-half-space/lower-half-space","kind":"node","lean":"MathlibExt.Analysis.Fourier.HalfSpaceLaplaceDecayEstimateWanted.LowerHalfSpace","parent":18594,"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-extension-to-a-half-space/index.html#lower-half-space","scope":"Analysis/potential-theory/harmonic-extension-to-a-half-space","status":"fully_proved","summary":"The closed lower half-space is the set of points $(p_1, p_2, p_3)$ of Euclidean three-space $\\mathbb{R}^3$ with $p_3 \\le 0$.","title":"Closed lower half-space"},{"id":"Analysis/potential-theory/harmonic-extension-to-a-half-space/open-lower-half-space","kind":"node","lean":"MathlibExt.Analysis.Fourier.HalfSpaceLaplaceDecayEstimateWanted.OpenLowerHalfSpace","parent":18594,"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-extension-to-a-half-space/index.html#open-lower-half-space","scope":"Analysis/potential-theory/harmonic-extension-to-a-half-space","status":"fully_proved","summary":"The open lower half-space is the set of points $(p_1, p_2, p_3)$ of Euclidean three-space $\\mathbb{R}^3$ with $p_3 < 0$.","title":"Open lower half-space"},{"id":"Analysis/potential-theory/harmonic-extension-to-a-half-space/point3","kind":"node","lean":"MathlibExt.Analysis.Fourier.HalfSpaceLaplaceDecayEstimateWanted.point3","parent":18594,"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-extension-to-a-half-space/index.html#point3","scope":"Analysis/potential-theory/harmonic-extension-to-a-half-space","status":"fully_proved","summary":"For a point $x = (x_1, x_2)$ of the Euclidean plane $\\mathbb{R}^2$ and a real number $z$, this is the point $(x_1, x_2, z)$ of Euclidean three-space $\\mathbb{R}^3$.","title":"Appending a third coordinate"},{"id":"Analysis/potential-theory/harmonic-extension-to-a-half-space/exists-abs-le-inv-one-add-abs-of-harmoni","kind":"node","lean":"MathlibExt.Analysis.Fourier.HalfSpaceLaplaceDecayEstimateWanted.exists_abs_le_inv_one_add_abs_of_harmonic_lowerHalfSpace","parent":18594,"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-extension-to-a-half-space/index.html#exists-abs-le-inv-one-add-abs-of-harmoni","scope":"Analysis/potential-theory/harmonic-extension-to-a-half-space","status":"can_prove","summary":"Write a point of $\\mathbb{R}^3$ as $(x, z)$ with $x = (x_1, x_2) \\in \\mathbb{R}^2$ and $z \\in \\mathbb{R}$. Let $\\phi : \\mathbb{R}^2 \\to \\mathbb{R}$ be a Schwartz function and let $u : \\mathbb{R}^3 \\to \\mathbb{R}$ be harmonic at every point of the open lower half-space $\\{(x, z) : z < 0\\}$, continuous on the closed lower half-space $\\{(x, z) : z \\le 0\\}$ and bounded there, with $u(x, 0) = \\phi(x)$ for every\u2026","title":"Decay of bounded harmonic extensions of Schwartz data in a half-space"},{"data_url":"scopes/Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c.json","id":"Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c","kind":"scope","navigation":"module","parent":18578,"result":false,"route":"../roadmap/Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c/index.html","scope":"Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c","status":"can_prove","title":"Harmonic functions vanishing on the edges of a cube"},{"id":"Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c/cube","kind":"node","lean":"MathlibExt.Analysis.ScottishBook151Wanted.Cube","parent":18601,"reading_title":{"format":"inline-parts-v1","parts":[["text","The closed cube "],["math","\\((-1, 1)^3\\)"]],"source":"The closed cube $(-1, 1)^3$"},"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c/index.html#cube","scope":"Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c","status":"fully_proved","summary":"The closed cube $[-1, 1]^3$ in $\\mathbb{R}^3$: the set of points of three-dimensional Euclidean space all of whose coordinates lie in $[-1, 1]$.","title":"The closed cube $(-1, 1)^3$"},{"id":"Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c/cube-edges","kind":"node","lean":"MathlibExt.Analysis.ScottishBook151Wanted.CubeEdges","parent":18601,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edges of the cube "],["math","\\((-1, 1)^3\\)"]],"source":"Edges of the cube $(-1, 1)^3$"},"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c/index.html#cube-edges","scope":"Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c","status":"fully_proved","summary":"The union of the edges of the cube $[-1, 1]^3$ in $\\mathbb{R}^3$: the set of points $x$ with every coordinate in $[-1, 1]$ such that at least two distinct coordinates $x_i$ and $x_j$ are each equal to $1$ or $-1$.","title":"Edges of the cube $(-1, 1)^3$"},{"id":"Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c/harmonic-function-vanishes-on-cube-edges","kind":"node","lean":"MathlibExt.Analysis.ScottishBook151Wanted.harmonicFunctionVanishesOnCubeEdges","parent":18601,"result":true,"route":"../roadmap/Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c/index.html#harmonic-function-vanishes-on-cube-edges","scope":"Analysis/potential-theory/harmonic-functions-vanishing-on-the-edges-of-a-c","status":"can_prove","summary":"There exist an open set $U \\subseteq \\mathbb{R}^3$ whose interior contains the closed cube $[-1, 1]^3$ and a function $f : \\mathbb{R}^3 \\to \\mathbb{R}$ that is harmonic at every point of $U$, vanishes at every point of the edges of the cube $[-1, 1]^3$, and satisfies $f(x) \\neq 0$ for some $x \\in U$.","title":"Scottish Book Problem 151: a harmonic function vanishing on the edges of a cube"},{"data_url":"scopes/Analysis/potential-theory/subharmonic-functions.json","id":"Analysis/potential-theory/subharmonic-functions","kind":"scope","navigation":"module","parent":18578,"result":false,"route":"../roadmap/Analysis/potential-theory/subharmonic-functions/index.html","scope":"Analysis/potential-theory/subharmonic-functions","status":"can_prove","title":"Subharmonic functions"},{"id":"Analysis/potential-theory/subharmonic-functions/is-subharmonic","kind":"node","lean":"MathlibExt.Analysis.ScottishBook184Wanted.IsSubharmonic","parent":18605,"result":true,"route":"../roadmap/Analysis/potential-theory/subharmonic-functions/index.html#is-subharmonic","scope":"Analysis/potential-theory/subharmonic-functions","status":"fully_proved","summary":"A function $\\varphi : \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$ is subharmonic if it is upper semicontinuous and, for every point $(a, b)$ of the plane and every radius $r > 0$,","title":"Subharmonic functions on the plane"},{"id":"Analysis/potential-theory/subharmonic-functions/has-second-partial-derivatives","kind":"node","lean":"MathlibExt.Analysis.ScottishBook184Wanted.HasSecondPartialDerivatives","parent":18605,"result":true,"route":"../roadmap/Analysis/potential-theory/subharmonic-functions/index.html#has-second-partial-derivatives","scope":"Analysis/potential-theory/subharmonic-functions","status":"fully_proved","summary":"A function $\\varphi : \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$ has second partial derivatives if at every point of the plane the partial derivatives $\\varphi_x$ and $\\varphi_y$ exist, and the partial derivatives $\\varphi_{xx} = (\\varphi_x)_x$ and $\\varphi_{yy} = (\\varphi_y)_y$ also exist.","title":"Existence of the pure second partial derivatives"},{"id":"Analysis/potential-theory/subharmonic-functions/saks184-subharmonic-laplacian-nonneg","kind":"node","lean":"MathlibExt.Analysis.ScottishBook184Wanted.saks184_subharmonic_laplacian_nonneg","parent":18605,"result":true,"route":"../roadmap/Analysis/potential-theory/subharmonic-functions/index.html#saks184-subharmonic-laplacian-nonneg","scope":"Analysis/potential-theory/subharmonic-functions","status":"can_prove","summary":"Let $\\varphi : \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$ be subharmonic, and suppose that its partial derivatives $\\varphi_x$, $\\varphi_{xx}$, $\\varphi_y$ and $\\varphi_{yy}$ exist at every point of the plane. Then $\\varphi_{xx}(p) + \\varphi_{yy}(p) \\ge 0$ for every point $p$ of the plane.","title":"Subharmonic functions with second partial derivatives have nonnegative Laplacian"},{"data_url":"scopes/Analysis/hilbert-spaces.json","id":"Analysis/hilbert-spaces","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/index.html","scope":"Analysis/hilbert-spaces","status":"can_prove","summary":"Geometry of inner product spaces: nearest-point projections, orthonormal and Hilbert bases, separability and \u2113\u00b2, frames and Riesz bases, weak convergence (Opial, Radon\u2013Riesz), Kirszbraun extension, derivatives of the norm, representer theorems; operators on Hilbert spaces go to Operator theory, analytic RKHS to Complex analysis.","title":"Hilbert spaces"},{"id":"Analysis/hilbert-spaces/orthogonal-polynomial-hilbert-basis","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.OrthogonalPolynomialHilbertBasisWanted.orthogonalPolynomial_hilbertBasis","parent":18609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hilbert basis of orthogonal polynomials on "],["math","\\((-1, 1)\\)"]],"source":"Hilbert basis of orthogonal polynomials on $(-1, 1)$"},"result":true,"route":"../roadmap/Analysis/hilbert-spaces/index.html#orthogonal-polynomial-hilbert-basis","scope":"Analysis/hilbert-spaces","status":"can_prove","summary":"Let $L^2([-1, 1])$ be the real Hilbert space of square-integrable real-valued functions on $[-1, 1]$ with respect to Lebesgue measure. There is a Hilbert basis $(b_n)_{n \\in \\mathbb{N}}$ of $L^2([-1, 1])$ such that for every $n \\in \\mathbb{N}$ there is a real polynomial $q_n$ of degree exactly $n$ (for $n = 0$, a constant polynomial) with $b_n(x) = q_n(x)$ for almost every $x \\in [-1, 1]$.","title":"Hilbert basis of orthogonal polynomials on $(-1, 1)$"},{"data_url":"scopes/Analysis/hilbert-spaces/derivatives-of-the-norm.json","id":"Analysis/hilbert-spaces/derivatives-of-the-norm","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","title":"Derivatives of the norm"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/re-inner-sl","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.reInnerSL","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#re-inner-sl","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The real-linear functional `y \u21a6 re \u27eax, y\u27eb`, the unnormalized gradient of the norm at `x`. The scalar field `K` is an explicit argument because the type `E \u2192L[\u211d] \u211d` does not determine it.","title":"Real part of the inner product as a real-linear functional"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/has-strict-fderiv-at-norm-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.hasStrictFDerivAt_norm_sq","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#has-strict-fderiv-at-norm-sq","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The derivative of the squared norm is twice the real inner product with `x`.","title":"Strict derivative of the squared norm"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/has-strict-fderiv-at-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.hasStrictFDerivAt_norm","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#has-strict-fderiv-at-norm","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The Fr\u00e9chet derivative of the norm at a nonzero point `x` is the norming functional `y \u21a6 re \u27eax, y\u27eb / \u2016x\u2016`.","title":"Strict Fr\u00e9chet derivative of the norm at a nonzero point"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/has-fderiv-at-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.hasFDerivAt_norm","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#has-fderiv-at-norm","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The non-strict form of `hasStrictFDerivAt_norm`.","title":"Fr\u00e9chet derivative of the norm at a nonzero point"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/fderiv-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.fderiv_norm","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#fderiv-norm","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The `fderiv` of the norm at a nonzero point.","title":"Formula for the derivative of the norm"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/fderiv-norm-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.fderiv_norm_apply","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#fderiv-norm-apply","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The `fderiv` of the norm at a nonzero point, applied to a vector.","title":"Derivative of the norm applied to a vector"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/norm-re-inner-sl","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.norm_reInnerSL","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#norm-re-inner-sl","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The norming functional `reInnerSL K x` has operator norm `\u2016x\u2016` (it attains `\u2016x\u2016` at `\u2016x\u2016\u207b\u00b9 \u2022 x`).","title":"Operator norm of the real inner product functional"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/norm-fderiv-norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.norm_fderiv_norm","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#norm-fderiv-norm","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The Fr\u00e9chet derivative of the norm at a nonzero point has operator norm `1`: the norming functional is a norming functional. This statement needs the inner product structure, so `K` is an explicit argument.","title":"Derivative of the norm has operator norm one"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/re-inner-sl-real","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.reInnerSL_real","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#re-inner-sl-real","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"Over `\u211d`, the norming functional is just `innerSL \u211d x`.","title":"Over the reals the norming functional is the inner product"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/has-strict-fderiv-at-norm-real","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.hasStrictFDerivAt_norm_real","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#has-strict-fderiv-at-norm-real","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The Fr\u00e9chet derivative of the norm in a real inner product space.","title":"Strict derivative of the norm in a real inner product space"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/fderiv-norm-apply-real","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.fderiv_norm_apply_real","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#fderiv-norm-apply-real","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The `fderiv` of the norm in a real inner product space, applied to a vector.","title":"Derivative of the norm in a real space applied to a vector"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/norm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.HasStrictFDerivAt.norm","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#norm","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"Chain rule: the derivative of the norm of a differentiable map.","title":"Chain rule for the strict derivative of a norm"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/norm-2","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.HasFDerivAt.norm","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#norm-2","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"Chain rule: the non-strict derivative of the norm of a differentiable map.","title":"Chain rule for the Fr\u00e9chet derivative of a norm"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/norm-3","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.HasDerivAt.norm","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#norm-3","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The derivative of the norm along a curve is the normalized real inner product of the position with the velocity.","title":"Derivative of the norm along a curve"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/re-inner-self-deriv-eq-zero-of-forall-no","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.re_inner_self_deriv_eq_zero_of_forall_norm_eq_one","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#re-inner-self-deriv-eq-zero-of-forall-no","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"A differentiable curve lying on the unit sphere has velocity orthogonal to its position (in the real-inner-product sense).","title":"Curves on the unit sphere have velocity orthogonal to position"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/has-strict-fderiv-at-dist-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.hasStrictFDerivAt_dist_sq","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#has-strict-fderiv-at-dist-sq","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The gradient of the squared distance to a point `a` is `2 \u2022 reInnerSL K (x - a)`.","title":"Strict derivative of the squared distance to a point"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/fderiv-dist-sq-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.fderiv_dist_sq_apply","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#fderiv-dist-sq-apply","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The `fderiv` of the squared distance to a point, applied to a vector.","title":"Derivative of the squared distance to a point applied to a vector"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/inf-dist-eq-norm-sub-star-projection","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.infDist_eq_norm_sub_starProjection","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#inf-dist-eq-norm-sub-star-projection","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The distance from `y` to a subspace with an orthogonal projection is the norm of the residual `y - S.starProjection y`.","title":"Distance to a subspace as the norm of the projection residual"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/has-strict-fderiv-at-residual-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.hasStrictFDerivAt_residual_sq","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#has-strict-fderiv-at-residual-sq","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The gradient of the squared residual `\u2016y - S.starProjection y\u2016 ^ 2` is `2 \u2022 reInnerSL K (x - S.starProjection x)`, using that the residual operator `1 - S.starProjection` is self-adjoint and idempotent.","title":"Strict derivative of the squared projection residual"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/has-strict-fderiv-at-inf-dist-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.NormDerivative.hasStrictFDerivAt_infDist_sq","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#has-strict-fderiv-at-inf-dist-sq","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"The gradient of the squared distance to a subspace `S` is twice the real inner product with the residual `x - S.starProjection x`.","title":"Strict derivative of the squared distance to a subspace"},{"id":"Analysis/hilbert-spaces/derivatives-of-the-norm/fderiv-fderiv-norm-sq-apply","kind":"node","lean":"fderiv_fderiv_norm_sq_apply","parent":18611,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/derivatives-of-the-norm/index.html#fderiv-fderiv-norm-sq-apply","scope":"Analysis/hilbert-spaces/derivatives-of-the-norm","status":"fully_proved","summary":"Let $E$ be a real inner product space. For all $p, u, v \\in E$, the second Fr\u00e9chet derivative of $x \\mapsto \\|x\\|^2$ at $p$, applied to $u$ and then to $v$, equals $2 \\langle u, v \\rangle$.","title":"Second derivative of the squared norm"},{"data_url":"scopes/Analysis/hilbert-spaces/frames-in-hilbert-spaces.json","id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","title":"Frames in Hilbert spaces"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/is-frame","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.IsFrame","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#is-frame","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. A finite family $(f_i)_{i \\in \\iota}$ in $E$ is a frame with frame bounds $A, B \\in \\mathbb{R}$ if $A > 0$, $B > 0$ and, for every $x \\in E$, $A \\|x\\|^2 \\le \\sum_{i \\in \\iota} |\\langle f_i, x \\rangle|^2 \\le B \\|x\\|^2$.","title":"Finite frame"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/pos-left","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.IsFrame.pos_left","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#pos-left","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. If a finite family $(f_i)_{i \\in \\iota}$ in $E$ is a frame with frame bounds $A$ and $B$, then $A > 0$.","title":"The lower frame bound is positive"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/pos-right","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.IsFrame.pos_right","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#pos-right","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"If a finite family $(f_i)_{i \\in \\iota}$ in an inner product space $E$ over $\\mathbb{K}$ ($\\mathbb{R}$ or $\\mathbb{C}$) is a frame with frame bounds $A$ and $B$, then $0 < B$.","title":"The upper frame bound is positive"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/lower","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.IsFrame.lower","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#lower","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(f_i)_{i \\in \\iota}$ be a finite frame in $E$ with frame bounds $A$ and $B$. Then $A \\|x\\|^2 \\le \\sum_{i \\in \\iota} |\\langle f_i, x \\rangle|^2$ for every $x \\in E$.","title":"Lower frame inequality"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/upper","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.IsFrame.upper","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#upper","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"If a finite family $(f_i)_{i \\in \\iota}$ in an inner product space $E$ over $\\mathbb{K}$ ($\\mathbb{R}$ or $\\mathbb{C}$) is a frame with frame bounds $A$ and $B$, then $\\sum_{i} |\\langle f_i, x \\rangle|^2 \\le B \\|x\\|^2$ for every $x \\in E$.","title":"Upper frame inequality"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/frame-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.frameOperator","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#frame-operator","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"For a finite family $(f_i)_{i \\in \\iota}$ in an inner product space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, the frame operator is the bounded linear operator $S \\colon E \\to E$ given by $S x = \\sum_{i \\in \\iota} \\langle f_i, x \\rangle f_i$, the sum of the rank-one operators $x \\mapsto \\langle f_i, x \\rangle f_i$. The inner product is conjugate-linear in its first argument.","title":"Frame operator"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/frame-operator-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.frameOperator_apply","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#frame-operator-apply","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"For a finite family $(f_i)_{i \\in \\iota}$ in an inner product space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, the frame operator $S$ of the family satisfies $S x = \\sum_{i \\in \\iota} \\langle f_i, x \\rangle f_i$ for every $x \\in E$, where the inner product is conjugate-linear in its first argument.","title":"Frame operator formula"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/inner-frame-operator-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.inner_frameOperator_apply","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#inner-frame-operator-apply","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with inner product conjugate-linear in the first argument, and let $S$ be the frame operator of a finite family $(f_i)_{i \\in \\iota}$ in $E$. Then $\\langle S x, y \\rangle = \\sum_{i \\in \\iota} \\overline{\\langle f_i, x \\rangle} \\, \\langle f_i, y \\rangle$ for all $x, y \\in E$.","title":"Inner product against the frame operator"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/inner-frame-operator-apply-self","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.inner_frameOperator_apply_self","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#inner-frame-operator-apply-self","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $S$ be the frame operator of a finite family $(f_i)_{i \\in \\iota}$ in an inner product space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. For every $x \\in E$, $\\langle S x, x \\rangle = \\sum_{i \\in \\iota} |\\langle f_i, x \\rangle|^2$; the right side is a real number, viewed as an element of $\\mathbb{K}$.","title":"Quadratic form of the frame operator"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/re-inner-frame-operator-apply-self","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.re_inner_frameOperator_apply_self","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#re-inner-frame-operator-apply-self","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$ be a finite family in an inner product space $E$ over $\\mathbb{K}$ ($\\mathbb{R}$ or $\\mathbb{C}$), with frame operator $S$. For every $x \\in E$, $\\operatorname{Re} \\langle S x, x \\rangle = \\sum_{i} |\\langle f_i, x \\rangle|^2$.","title":"Real part of the frame quadratic form"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/sq-norm-inner-frame-operator-apply-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.sq_norm_inner_frameOperator_apply_le","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#sq-norm-inner-frame-operator-apply-le","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$ be a finite family in an inner product space $E$ over $\\mathbb{K}$ ($\\mathbb{R}$ or $\\mathbb{C}$), with frame operator $S$. For all $x, y \\in E$,","title":"Cauchy\u2013Schwarz bound for the frame operator"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/is-self-adjoint-frame-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.isSelfAdjoint_frameOperator","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#is-self-adjoint-frame-operator","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. For every finite family $(f_i)_{i \\in \\iota}$ in $E$, not necessarily a frame, the frame operator of $(f_i)_{i \\in \\iota}$ is self-adjoint.","title":"The frame operator is self-adjoint"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/le-norm-frame-operator-apply","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.le_norm_frameOperator_apply","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#le-norm-frame-operator-apply","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(f_i)_{i \\in \\iota}$ be a finite frame in $E$ with frame bounds $A$ and $B$ and frame operator $S$. Then $A \\|x\\| \\le \\|S x\\|$ for every $x \\in E$.","title":"The frame operator is bounded below"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/is-unit-frame-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.isUnit_frameOperator","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#is-unit-frame-operator","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(f_i)_{i \\in \\iota}$ be a finite frame in $E$ with frame bounds $A$ and $B$. Then its frame operator $S$ is a unit in the algebra of bounded linear operators on $E$, that is, $S$ has a bounded linear two-sided inverse.","title":"The frame operator of a frame is invertible"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/norm-frame-operator-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.norm_frameOperator_le","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#norm-frame-operator-le","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(f_i)_{i \\in \\iota}$ be a finite frame in $E$ with frame bounds $A$ and $B$ and frame operator $S$. Then the operator norm of $S$ satisfies $\\|S\\| \\le B$.","title":"The frame operator has norm at most the upper frame bound"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/frame-operator-inv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.frameOperatorInv","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#frame-operator-inv","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$, with $\\iota$ finite, be a frame with bounds $A$ and $B$ in a Hilbert space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$. Its frame operator $S$ is invertible in the algebra of bounded linear operators on $E$, and the inverse frame operator $S^{-1}$ is its inverse there.","title":"Inverse frame operator"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/frame-operator-comp-frame-operator-inv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.frameOperator_comp_frameOperatorInv","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#frame-operator-comp-frame-operator-inv","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$, with $\\iota$ finite, be a frame with bounds $A$ and $B$ in a Hilbert space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with frame operator $S$ and inverse frame operator $S^{-1}$. Then $S \\circ S^{-1}$ is the identity operator on $E$.","title":"The inverse frame operator is a right inverse"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/frame-operator-inv-comp-frame-operator","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.frameOperatorInv_comp_frameOperator","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#frame-operator-inv-comp-frame-operator","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$, with $\\iota$ finite, be a frame with bounds $A$ and $B$ in a Hilbert space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, with frame operator $S$ and inverse frame operator $S^{-1}$. Then $S^{-1} \\circ S$ is the identity operator on $E$.","title":"The inverse frame operator is a left inverse"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/is-self-adjoint-frame-operator-inv","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.isSelfAdjoint_frameOperatorInv","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#is-self-adjoint-frame-operator-inv","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(f_i)_{i \\in \\iota}$ be a finite frame in $E$ with frame bounds $A$ and $B$. Then the inverse $S^{-1}$ of its frame operator $S$ is self-adjoint.","title":"The inverse frame operator is self-adjoint"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/frame-reconstruction","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.frame_reconstruction","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#frame-reconstruction","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$, with $\\iota$ finite, be a frame with bounds $A$ and $B$ in a Hilbert space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, and let $S^{-1}$ be the inverse of its frame operator. Then every $x \\in E$ satisfies $x = \\sum_{i \\in \\iota} \\langle f_i, S^{-1} x \\rangle f_i$.","title":"Frame reconstruction formula"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/dual-frame","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.dualFrame","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#dual-frame","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$, with $\\iota$ finite, be a frame with bounds $A$ and $B$ in a Hilbert space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, and let $S^{-1}$ be the inverse of its frame operator. The canonical dual frame is the family $(S^{-1} f_i)_{i \\in \\iota}$.","title":"Canonical dual frame"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/frame-operator-dual-frame","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.frameOperator_dualFrame","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#frame-operator-dual-frame","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$, with $\\iota$ finite, be a frame with bounds $A$ and $B$ in a Hilbert space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, and let $S^{-1}$ be the inverse of its frame operator. Then the frame operator of the canonical dual frame $(S^{-1} f_i)_{i \\in \\iota}$ is $S^{-1}$.","title":"Frame operator of the canonical dual frame"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/is-frame-dual-frame","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.isFrame_dualFrame","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#is-frame-dual-frame","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $E$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(f_i)_{i \\in \\iota}$ be a finite frame in $E$ with frame bounds $A$ and $B$ and frame operator $S$. Then the canonical dual frame $(S^{-1} f_i)_{i \\in \\iota}$ is a frame with lower frame bound $1/B$ and upper frame bound $1/A$.","title":"The canonical dual frame is a frame"},{"id":"Analysis/hilbert-spaces/frames-in-hilbert-spaces/frame-operator-eq-smul-one-of-is-tight-f","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.FiniteFrame.frameOperator_eq_smul_one_of_isTightFrame","parent":18633,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/frames-in-hilbert-spaces/index.html#frame-operator-eq-smul-one-of-is-tight-f","scope":"Analysis/hilbert-spaces/frames-in-hilbert-spaces","status":"fully_proved","summary":"Let $(f_i)_{i \\in \\iota}$, with $\\iota$ finite, be a frame in a Hilbert space $E$ over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ whose lower and upper frame bounds are both equal to $A$. Then its frame operator is $S = A \\cdot I$, where $I$ is the identity operator on $E$.","title":"Tight frames have scalar frame operator"},{"data_url":"scopes/Analysis/hilbert-spaces/hilbert-projection-theorem.json","id":"Analysis/hilbert-spaces/hilbert-projection-theorem","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","title":"Hilbert projection theorem"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/exists-unique-nearest-point-of-complete","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.exists_unique_nearest_point_of_complete_convex","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#exists-unique-nearest-point-of-complete","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$. For every $x \\in F$ there is exactly one $p \\in K$ with $\\|x - p\\| = \\inf_{w \\in K} \\|x - w\\|$, that is, exactly one point of $K$ nearest to $x$.","title":"Existence and uniqueness of the nearest point"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/exists-convex-projection-spec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.exists_convexProjection_spec","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#exists-convex-projection-spec","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"For every subset $K$ of a real inner product space $F$ and every $x \\in F$, there is a point $p \\in F$ such that, if $K$ is nonempty, complete and convex, then $p \\in K$ and $\\|x - p\\| = \\inf_{w \\in K} \\|x - w\\|$.","title":"Totalized existence of the convex projection"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"For a subset $K$ of a real inner product space $F$ and a point $x \\in F$, the nearest-point projection $P_K(x)$ is a chosen point of $F$ such that, whenever $K$ is nonempty, complete and convex, $P_K(x) \\in K$ and $\\|x - P_K(x)\\| = \\inf_{w \\in K} \\|x - w\\|$; it is then the unique point of $K$ nearest to $x$. When $K$ fails one of these hypotheses, the value of $P_K(x)$ is unspecified.","title":"Nearest-point projection onto a convex set"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection-spec","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection_spec","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection-spec","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, and let $P_K$ be the nearest-point projection onto $K$. For every $x \\in F$, $P_K(x) \\in K$ and $\\|x - P_K(x)\\| = \\inf_{w \\in K} \\|x - w\\|$.","title":"Defining property of the convex projection"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection-mem","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection_mem","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection-mem","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, and let $P_K$ be the nearest-point projection onto $K$. Then $P_K(x) \\in K$ for every $x \\in F$.","title":"The convex projection lies in the set"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection-norm-eq-i-inf","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection_norm_eq_iInf","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection-norm-eq-i-inf","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, and let $P_K$ be the nearest-point projection onto $K$. Then $\\|x - P_K(x)\\| = \\inf_{w \\in K} \\|x - w\\|$ for every $x \\in F$.","title":"The convex projection minimizes the distance to the set"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection-variational","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection_variational","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection-variational","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, and let $P_K$ be the nearest-point projection onto $K$. For every $x \\in F$ and every $w \\in K$, $\\langle x - P_K(x), w - P_K(x) \\rangle \\le 0$.","title":"Variational inequality for the convex projection"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection-eq-of-is-nearest-point","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection_eq_of_isNearestPoint","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection-eq-of-is-nearest-point","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, let $P_K$ be the nearest-point projection onto $K$, and let $x \\in F$. If $p \\in K$ satisfies $\\|x - p\\| = \\inf_{w \\in K} \\|x - w\\|$, then $P_K(x) = p$.","title":"Nearest points coincide with the convex projection"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection-eq-self-of-mem","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection_eq_self_of_mem","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection-eq-self-of-mem","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, and let $P_K$ be the nearest-point projection onto $K$. Then $P_K(x) = x$ for every $x \\in K$.","title":"The convex projection fixes the points of the set"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection-idem","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection_idem","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection-idem","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, and let $P_K$ be the nearest-point projection onto $K$. Then $P_K(P_K(x)) = P_K(x)$ for every $x \\in F$.","title":"Idempotence of the convex projection"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/norm-sub-le-of-is-nearest-point","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.norm_sub_le_of_isNearestPoint","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#norm-sub-le-of-is-nearest-point","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a convex subset of a real inner product space $F$, let $x, y \\in F$, and let $p, q \\in K$ satisfy $\\|x - p\\| = \\inf_{w \\in K} \\|x - w\\|$ and $\\|y - q\\| = \\inf_{w \\in K} \\|y - w\\|$, so that $p$ is a point of $K$ nearest to $x$ and $q$ is a point of $K$ nearest to $y$. Then $\\|p - q\\| \\le \\|x - y\\|$.","title":"Nonexpansiveness of nearest-point selection"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/norm-sub-sq-add-norm-sub-sq-le-of-is-nea","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.norm_sub_sq_add_norm_sub_sq_le_of_isNearestPoint","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#norm-sub-sq-add-norm-sub-sq-le-of-is-nea","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a convex subset of a real inner product space $F$, let $x, y \\in F$, and let $p, q \\in K$ satisfy $\\|x - p\\| = \\inf_{w \\in K} \\|x - w\\|$ and $\\|y - q\\| = \\inf_{w \\in K} \\|y - w\\|$, so that $p$ is a point of $K$ nearest to $x$ and $q$ is a point of $K$ nearest to $y$. Then $\\|p - q\\|^2 + \\|(x - p) - (y - q)\\|^2 \\le \\|x - y\\|^2$.","title":"Firm nonexpansiveness of nearest-point selection"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/lipschitz-with-convex-projection","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.lipschitzWith_convexProjection","parent":18658,"reading_title":{"format":"inline-parts-v1","parts":[["text","The convex projection is "],["math","\\(1\\)"],["text","-Lipschitz"]],"source":"The convex projection is $1$-Lipschitz"},"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#lipschitz-with-convex-projection","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, and let $P_K$ be the nearest-point projection onto $K$. Then $\\|P_K x - P_K y\\| \\le \\|x - y\\|$ for all $x, y \\in F$, that is, $P_K$ is $1$-Lipschitz.","title":"The convex projection is $1$-Lipschitz"},{"id":"Analysis/hilbert-spaces/hilbert-projection-theorem/convex-projection-firmly-nonexpansive","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.ConvexProjection.convexProjection_firmly_nonexpansive","parent":18658,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/hilbert-projection-theorem/index.html#convex-projection-firmly-nonexpansive","scope":"Analysis/hilbert-spaces/hilbert-projection-theorem","status":"fully_proved","summary":"Let $K$ be a nonempty complete convex subset of a real inner product space $F$, and let $P_K$ be the nearest-point projection onto $K$. For all $x, y \\in F$, $\\|P_K(x) - P_K(y)\\|^2 + \\|(x - P_K(x)) - (y - P_K(y))\\|^2 \\le \\|x - y\\|^2$.","title":"Firm nonexpansiveness of the convex projection"},{"data_url":"scopes/Analysis/hilbert-spaces/kirszbraun-theorem.json","id":"Analysis/hilbert-spaces/kirszbraun-theorem","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/kirszbraun-theorem/index.html","scope":"Analysis/hilbert-spaces/kirszbraun-theorem","status":"fully_proved","title":"Kirszbraun theorem"},{"id":"Analysis/hilbert-spaces/kirszbraun-theorem/kirszbraun-general","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KirszbraunWanted.kirszbraun_general","parent":18673,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/kirszbraun-theorem/index.html#kirszbraun-general","scope":"Analysis/hilbert-spaces/kirszbraun-theorem","status":"fully_proved","summary":"Let $E$ be a real inner product space (not necessarily complete), let $F$ be a complete real inner product space, let $s \\subseteq E$ and let $K \\ge 0$. Every map $f : E \\to F$ that is $K$-Lipschitz on $s$ agrees on $s$ with some map $g : E \\to F$ that is $K$-Lipschitz on all of $E$.","title":"Kirszbraun extension theorem"},{"id":"Analysis/hilbert-spaces/kirszbraun-theorem/kirszbraun","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.KirszbraunWanted.kirszbraun","parent":18673,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/kirszbraun-theorem/index.html#kirszbraun","scope":"Analysis/hilbert-spaces/kirszbraun-theorem","status":"fully_proved","summary":"Let $E$ and $F$ be real Hilbert spaces, let $s \\subseteq E$ and let $K \\ge 0$. Every map $f : E \\to F$ that is $K$-Lipschitz on $s$ agrees on $s$ with some map $g : E \\to F$ that is $K$-Lipschitz on all of $E$.","title":"Kirszbraun extension theorem for Hilbert spaces"},{"data_url":"scopes/Analysis/hilbert-spaces/opial-property.json","id":"Analysis/hilbert-spaces/opial-property","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/opial-property/index.html","scope":"Analysis/hilbert-spaces/opial-property","status":"fully_proved","title":"Opial property"},{"id":"Analysis/hilbert-spaces/opial-property/bdd-above-norm-of-weakly-tendsto","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Opial.bddAbove_norm_of_weakly_tendsto","parent":18676,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/opial-property/index.html#bdd-above-norm-of-weakly-tendsto","scope":"Analysis/hilbert-spaces/opial-property","status":"fully_proved","summary":"A weakly convergent sequence is norm-bounded (Banach\u2013Steinhaus applied to the evaluation maps on the dual space).","title":"Weakly convergent sequences are norm-bounded"},{"id":"Analysis/hilbert-spaces/opial-property/tendsto-re-inner-sub-of-weakly-tendsto","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Opial.tendsto_re_inner_sub_of_weakly_tendsto","parent":18676,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/opial-property/index.html#tendsto-re-inner-sub-of-weakly-tendsto","scope":"Analysis/hilbert-spaces/opial-property","status":"fully_proved","summary":"The cross term in the squared-norm identity tends to `0` under weak convergence, via the Fr\u00e9chet\u2013Riesz representation of the functional `z \u21a6 \u27eax - y, z\u27eb`.","title":"Cross term vanishes under weak convergence"},{"id":"Analysis/hilbert-spaces/opial-property/opial-liminf-sq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Opial.opial_liminf_sq","parent":18676,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/opial-property/index.html#opial-liminf-sq","scope":"Analysis/hilbert-spaces/opial-property","status":"fully_proved","summary":"If `u n \u21c0 x` weakly and `y \u2260 x`, then `liminf \u2016u n - x\u2016\u00b2 < liminf \u2016u n - y\u2016\u00b2`.","title":"Opial's property, squared-norm form"},{"id":"Analysis/hilbert-spaces/opial-property/opial-liminf","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Opial.opial_liminf","parent":18676,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/opial-property/index.html#opial-liminf","scope":"Analysis/hilbert-spaces/opial-property","status":"fully_proved","summary":"If `u n \u21c0 x` weakly and `y \u2260 x`, then `liminf \u2016u n - x\u2016 < liminf \u2016u n - y\u2016`.","title":"Opial's property"},{"data_url":"scopes/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series.json","id":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/index.html","scope":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","status":"can_prove","title":"Orthogonal systems and general orthogonal series"},{"id":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/banach186-orthonormal-system-divergent-a","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook186Wanted.banach186_orthonormal_system_divergent_ae","parent":18681,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/index.html#banach186-orthonormal-system-divergent-a","scope":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","status":"can_prove","summary":"There is a sequence $(\\varphi_i)_{i \\ge 0}$ of square-integrable real functions on $[0, 1]$, with respect to Lebesgue measure, that is orthonormal, $\\int_0^1 \\varphi_i \\varphi_j \\, dx = \\delta_{ij}$ for all $i, j$, and complete, in the sense that every square-integrable $g$ on $[0, 1]$ with $\\int_0^1 g \\varphi_i \\, dx = 0$ for all $i$ vanishes almost everywhere on $[0, 1]$, such that the following holds: for every\u2026","title":"Banach's Problem 186: a complete orthonormal system with divergent Fourier series"},{"id":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/sequence-systems-are-countable","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook51Wanted.SequenceSystemsAreCountable","parent":18681,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/index.html#sequence-systems-are-countable","scope":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","status":"fully_proved","summary":"This is the statement that every set $S$ of real sequences $(f_n)_{n \\in \\mathbb{N}}$ with the following two properties is at most countable: no member of $S$ is the zero sequence; and for any two distinct $f, g \\in S$ the series $\\sum_{n} f_n g_n$ converges absolutely and its sum is $0$.","title":"Countability of orthogonal systems of sequences"},{"id":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/function-systems-are-countable","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook51Wanted.FunctionSystemsAreCountable","parent":18681,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/index.html#function-systems-are-countable","scope":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","status":"fully_proved","summary":"Let $\\lambda$ be Lebesgue measure on $[0, 1]$. This is the statement that every set $S$ of functions $[0, 1] \\to \\mathbb{R}$ with the following three properties is at most countable: no member of $S$ is $\\lambda$-almost everywhere equal to $0$; any two members of $S$ that are equal $\\lambda$-almost everywhere are equal; and any two members $f, g$ of $S$ that are not equal $\\lambda$-almost everywhere are measurable,\u2026","title":"Countability of orthogonal systems of functions"},{"id":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/orthogonal-systems-not-necessarily-count","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook51Wanted.orthogonal_systems_not_necessarily_countable","parent":18681,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/index.html#orthogonal-systems-not-necessarily-count","scope":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","status":"can_prove","summary":"Both of the following statements are false. (a) Every set $S$ of real sequences $(f_n)_{n \\in \\mathbb{N}}$ in which no member is the zero sequence, and in which for any two distinct $f, g \\in S$ the series $\\sum_n f_n g_n$ converges absolutely with sum $0$, is at most countable. (b) Let $\\lambda$ be Lebesgue measure on $[0, 1]$. Every set $S$ of functions $[0, 1] \\to \\mathbb{R}$ in which no member is\u2026","title":"Orthogonal systems need not be countable"},{"id":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/scottish-book86-measure","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook86Wanted.scottishBook86Measure","parent":18681,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/index.html#scottish-book86-measure","scope":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","status":"fully_proved","summary":"Lebesgue measure on $\\mathbb{R}$ restricted to the unit interval $[0, 1]$, the ambient measure for Problem 86.","title":"Lebesgue measure on the unit interval (Scottish Book Problem 86)"},{"id":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/scottish-book-problem86-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook86Wanted.ScottishBookProblem86Claim","parent":18681,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/index.html#scottish-book-problem86-claim","scope":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","status":"fully_proved","summary":"The claim that for every real $M > 0$ and every sequence $(\\varphi_n)_{n \\in \\mathbb{N}}$ of functions $\\mathbb{R} \\to \\mathbb{R}$ that are almost everywhere measurable on $[0, 1]$, orthonormal in $L^2[0, 1]$ (that is, $\\int_0^1 \\varphi_n \\varphi_m \\, dt$ is $1$ if $n = m$ and $0$ otherwise) and satisfy $|\\varphi_n(t)| \\le M$ for all $n$ and all $t \\in [0, 1]$, there is a sequence $(\\psi_n)_{n \\in \\mathbb{N}}$ of\u2026","title":"Scottish Book Problem 86 (Banach)"},{"id":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/banach86-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook86Wanted.banach86_refuted","parent":18681,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series/index.html#banach86-refuted","scope":"Analysis/hilbert-spaces/orthogonal-systems-and-general-orthogonal-series","status":"can_prove","summary":"The Scottish Book Problem 86 claim is false: there are a real $M > 0$ and a sequence $(\\varphi_n)_{n \\in \\mathbb{N}}$ of functions $\\mathbb{R} \\to \\mathbb{R}$, almost everywhere measurable on $[0, 1]$, orthonormal in $L^2[0, 1]$ and with $|\\varphi_n(t)| \\le M$ for all $n$ and all $t \\in [0, 1]$, such that there is no sequence $(\\psi_n)_{n \\in \\mathbb{N}}$ of functions $\\mathbb{R} \\to \\mathbb{R}$, almost everywhere\u2026","title":"Banach's Scottish Book Problem 86 fails"},{"data_url":"scopes/Analysis/hilbert-spaces/radon-riesz-property.json","id":"Analysis/hilbert-spaces/radon-riesz-property","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/radon-riesz-property/index.html","scope":"Analysis/hilbert-spaces/radon-riesz-property","status":"fully_proved","title":"Radon\u2013Riesz property"},{"id":"Analysis/hilbert-spaces/radon-riesz-property/tendsto-of-tendsto-inner-and-tendsto-nor","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RadonRiesz.tendsto_of_tendsto_inner_and_tendsto_norm","parent":18689,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/radon-riesz-property/index.html#tendsto-of-tendsto-inner-and-tendsto-nor","scope":"Analysis/hilbert-spaces/radon-riesz-property","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, not necessarily complete, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $H$ and let $x \\in H$. If $\\langle u_n, y \\rangle \\to \\langle x, y \\rangle$ for every $y \\in H$ and $\\|u_n\\| \\to \\|x\\|$, then $u_n \\to x$ in norm.","title":"Radon\u2013Riesz theorem, inner-product form"},{"id":"Analysis/hilbert-spaces/radon-riesz-property/tendsto-of-forall-tendsto-and-tendsto-no","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RadonRiesz.tendsto_of_forall_tendsto_and_tendsto_norm","parent":18689,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/radon-riesz-property/index.html#tendsto-of-forall-tendsto-and-tendsto-no","scope":"Analysis/hilbert-spaces/radon-riesz-property","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, not necessarily complete, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $H$ and let $x \\in H$. If $f(u_n) \\to f(x)$ for every continuous linear functional $f : H \\to \\mathbb{K}$ and $\\|u_n\\| \\to \\|x\\|$, then $u_n \\to x$ in norm.","title":"Radon\u2013Riesz theorem, dual-functionals form"},{"id":"Analysis/hilbert-spaces/radon-riesz-property/tendsto-of-tendsto-weak-space-and-tendst","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RadonRiesz.tendsto_of_tendsto_weakSpace_and_tendsto_norm","parent":18689,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/radon-riesz-property/index.html#tendsto-of-tendsto-weak-space-and-tendst","scope":"Analysis/hilbert-spaces/radon-riesz-property","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, not necessarily complete, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $H$ and let $x \\in H$. If $u_n \\to x$ in the weak topology of $H$ (the coarsest topology making every continuous linear functional $H \\to \\mathbb{K}$ continuous) and $\\|u_n\\| \\to \\|x\\|$, then $u_n \\to x$ in norm.","title":"Radon\u2013Riesz theorem, weak-topology form"},{"id":"Analysis/hilbert-spaces/radon-riesz-property/tendsto-iff-tendsto-inner-and-tendsto-no","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RadonRiesz.tendsto_iff_tendsto_inner_and_tendsto_norm","parent":18689,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/radon-riesz-property/index.html#tendsto-iff-tendsto-inner-and-tendsto-no","scope":"Analysis/hilbert-spaces/radon-riesz-property","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, not necessarily complete, let $(u_n)_{n \\in \\mathbb{N}}$ be a sequence in $H$ and let $x \\in H$. Then $u_n \\to x$ in norm if and only if $\\langle u_n, y \\rangle \\to \\langle x, y \\rangle$ for every $y \\in H$ and $\\|u_n\\| \\to \\|x\\|$.","title":"Radon\u2013Riesz characterization of norm convergence"},{"data_url":"scopes/Analysis/hilbert-spaces/representer-theorem.json","id":"Analysis/hilbert-spaces/representer-theorem","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","title":"Representer theorem"},{"id":"Analysis/hilbert-spaces/representer-theorem/regularized-risk","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.regularizedRisk","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#regularized-risk","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The regularized risk `L (f x\u2081, \u2026, f x\u2099) + \u03a9 \u2016f\u2016` of `f` in the RKHS `H`.","title":"Regularized risk in a reproducing kernel Hilbert space"},{"id":"Analysis/hilbert-spaces/representer-theorem/kernel-expansion","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.kernelExpansion","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#kernel-expansion","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The kernel expansion map `c \u21a6 \u2211 i, kerFun H (x i) (c i)`. Its range is the span of the kernel functions at the sample points `x i`.","title":"Kernel expansion map centered at the sample points"},{"id":"Analysis/hilbert-spaces/representer-theorem/mem-topological-closure-range-of-is-mini","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.mem_topologicalClosure_range_of_isMinimizer","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#mem-topological-closure-range-of-is-mini","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The representer theorem (closed-span form): if `\u03a9` is strictly increasing on `[0, \u221e)`, every global minimizer of the regularized risk lies in the closure of the span of the kernel functions at the sample points.","title":"The representer theorem"},{"id":"Analysis/hilbert-spaces/representer-theorem/exists-eq-kernel-expansion-of-is-minimiz","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.exists_eq_kernelExpansion_of_isMinimizer","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#exists-eq-kernel-expansion-of-is-minimiz","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The representer theorem (Sch\u00f6lkopf, Herbrich and Smola, Theorem 1): for a finite-dimensional output space `V` (for instance a scalar kernel, `V = \ud835\udd5c`) and `\u03a9` strictly increasing on `[0, \u221e)`, every global minimizer of `L (f x\u2081, \u2026, f x\u2099) + \u03a9 \u2016f\u2016` is a kernel expansion `\u2211 i, kerFun H (x i) (c i)`.","title":"The representer theorem"},{"id":"Analysis/hilbert-spaces/representer-theorem/exists-kernel-expansion-is-minimizer","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.exists_kernelExpansion_isMinimizer","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#exists-kernel-expansion-is-minimizer","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"For finite-dimensional `V`, a continuous loss bounded below, and a continuous regularizer that is monotone on `[0, \u221e)` and tends to `\u221e`, the regularized risk has a global minimizer, and it can be taken to be a kernel expansion.","title":"Existence of a minimizer"},{"id":"Analysis/hilbert-spaces/representer-theorem/is-minimizer-unique","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.isMinimizer_unique","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#is-minimizer-unique","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"Uniqueness of the minimizer (real scalars): for a convex loss `L` and a regularizer `\u03a9` that is strictly convex and monotone on `[0, \u221e)` (for instance `\u03a9 t = \u03bb t\u00b2` with `\u03bb > 0`), the regularized risk has at most one global minimizer.","title":"Uniqueness of the minimizer"},{"id":"Analysis/hilbert-spaces/representer-theorem/kernel-gram","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.kernelGram","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#kernel-gram","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The Gram matrix `K i j = k(x i, x j)` of a real scalar kernel at the sample points, where `k(x, y) = kernel H x y 1`.","title":"Kernel Gram matrix"},{"id":"Analysis/hilbert-spaces/representer-theorem/kernel-ridge-risk","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.kernelRidgeRisk","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#kernel-ridge-risk","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The kernel ridge objective `\u2211 i, (y i - f (x i))\u00b2 + \u03bb \u2016f\u2016\u00b2`.","title":"Kernel ridge regression objective"},{"id":"Analysis/hilbert-spaces/representer-theorem/kernel-ridge-fit","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.kernelRidgeFit","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#kernel-ridge-fit","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The kernel ridge fit: the kernel expansion with coefficients `(K + \u03bb I)\u207b\u00b9 y`.","title":"Kernel ridge regression fit"},{"id":"Analysis/hilbert-spaces/representer-theorem/kernel-ridge-risk-eq-regularized-risk","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.kernelRidgeRisk_eq_regularizedRisk","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#kernel-ridge-risk-eq-regularized-risk","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The kernel ridge objective is the regularized risk with the squared loss and `\u03a9 t = \u03bb t\u00b2`.","title":"Kernel ridge objective as a regularized risk"},{"id":"Analysis/hilbert-spaces/representer-theorem/kernel-expansion-apply-sample","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.kernelExpansion_apply_sample","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#kernel-expansion-apply-sample","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The values of a kernel expansion at the sample points are `K c`.","title":"Values of a kernel expansion at the sample points"},{"id":"Analysis/hilbert-spaces/representer-theorem/pos-semidef-kernel-gram","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.posSemidef_kernelGram","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#pos-semidef-kernel-gram","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The kernel Gram matrix is positive semidefinite.","title":"Kernel Gram matrix is positive semidefinite"},{"id":"Analysis/hilbert-spaces/representer-theorem/kernel-ridge-fit-apply","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.kernelRidgeFit_apply","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#kernel-ridge-fit-apply","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The fitted values of kernel ridge regression are `K (K + \u03bb I)\u207b\u00b9 y`.","title":"Fitted values of kernel ridge regression"},{"id":"Analysis/hilbert-spaces/representer-theorem/kernel-ridge-risk-kernel-ridge-fit-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.kernelRidgeRisk_kernelRidgeFit_le","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#kernel-ridge-risk-kernel-ridge-fit-le","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"Kernel ridge regression (Kimeldorf and Wahba): for `\u03bb > 0` the kernel ridge fit minimizes `\u2211 i, (y i - f (x i))\u00b2 + \u03bb \u2016f\u2016\u00b2` over the whole RKHS.","title":"Kernel ridge regression"},{"id":"Analysis/hilbert-spaces/representer-theorem/eq-kernel-ridge-fit-of-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RepresenterTheorem.eq_kernelRidgeFit_of_le","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#eq-kernel-ridge-fit-of-le","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"The kernel ridge fit is the unique minimizer for `\u03bb > 0`.","title":"Kernel ridge fit is the unique minimizer"},{"id":"Analysis/hilbert-spaces/representer-theorem/exists-mem-span-norm-sq-add","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RepresenterTheorem.exists_mem_span_norm_sq_add","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#exists-mem-span-norm-sq-add","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"Orthogonal projection onto the span of the Riesz representers preserves the values of the functionals, and the Pythagorean identity holds for the residual.","title":"Projection onto Riesz representers with Pythagorean residual"},{"id":"Analysis/hilbert-spaces/representer-theorem/exists-mem-span-values-eq-norm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RepresenterTheorem.exists_mem_span_values_eq_norm_le","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#exists-mem-span-values-eq-norm-le","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"Orthogonal projection onto the span of the Riesz representers does not increase the norm.","title":"Projection onto Riesz representers does not increase the norm"},{"id":"Analysis/hilbert-spaces/representer-theorem/mem-span-of-is-min-on","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RepresenterTheorem.mem_span_of_isMinOn","parent":18694,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/representer-theorem/index.html#mem-span-of-is-min-on","scope":"Analysis/hilbert-spaces/representer-theorem","status":"fully_proved","summary":"Any minimizer of a ridge-regularized problem `F (fun i => L i g) + \u039b * \u2016g\u2016 ^ 2` with `\u039b > 0` lies in the span of the Riesz representers of the functionals `L i`.","title":"Representer theorem"},{"data_url":"scopes/Analysis/hilbert-spaces/riesz-bases.json","id":"Analysis/hilbert-spaces/riesz-bases","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","title":"Riesz bases"},{"id":"Analysis/hilbert-spaces/riesz-bases/is-riesz-basis","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasis.IsRieszBasis","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#is-riesz-basis","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"A family `f : \u03b9 \u2192 E` is a Riesz basis (with respect to the Hilbert basis `b`) if it is the image of `b` under a bounded invertible operator.","title":"Riesz basis of a Hilbert space"},{"id":"Analysis/hilbert-spaces/riesz-bases/linear-independent","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasis.IsRieszBasis.linearIndependent","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#linear-independent","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"A Riesz basis is linearly independent.","title":"Riesz bases are linearly independent"},{"id":"Analysis/hilbert-spaces/riesz-bases/dense-span","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasis.IsRieszBasis.dense_span","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#dense-span","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"A Riesz basis has dense span.","title":"Riesz bases have dense span"},{"id":"Analysis/hilbert-spaces/riesz-bases/exists-dual","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasis.IsRieszBasis.exists_dual","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#exists-dual","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"A Riesz basis admits a biorthogonal dual family with a reconstruction formula: there is `g : \u03b9 \u2192 E` with `\u27eaf i, g j\u27eb = if i = j then 1 else 0` and `x = \u2211' i, \u27eag i, x\u27eb \u2022 f i` for every `x`. The dual is `g j = (T\u207b\u00b9)\u2020 (b j)` for the operator `T` witnessing that `f` is a Riesz basis.","title":"Biorthogonal dual of a Riesz basis"},{"id":"Analysis/hilbert-spaces/riesz-bases/norm-seq","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.normSeq","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#norm-seq","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The sequence of norms of a square-summable family, as an element of `\u2113\u00b2(\u03b9, \u211d)`.","title":"Norm sequence of a square-summable family"},{"id":"Analysis/hilbert-spaces/riesz-bases/norm-seq-lp","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.normSeqLp","parent":18713,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norm sequence of an "],["math","\\(\\ell^2\\)"],["text"," coefficient sequence"]],"source":"Norm sequence of an $\\ell^2$ coefficient sequence"},"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#norm-seq-lp","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The sequence of norms of an `\u2113\u00b2` coefficient sequence, as an element of `\u2113\u00b2(\u03b9, \u211d)`.","title":"Norm sequence of an $\\ell^2$ coefficient sequence"},{"id":"Analysis/hilbert-spaces/riesz-bases/norm-norm-seq-lp","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.norm_normSeqLp","parent":18713,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell^2\\)"],["text"," norm of the norm sequence"]],"source":"$\\ell^2$ norm of the norm sequence"},"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#norm-norm-seq-lp","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The `\u2113\u00b2` norm of the norms of a coefficient sequence is the norm of the sequence.","title":"$\\ell^2$ norm of the norm sequence"},{"id":"Analysis/hilbert-spaces/riesz-bases/summable-norm-synthesis","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.summable_norm_synthesis","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#summable-norm-synthesis","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The norms of the terms of a synthesis series are summable when the coefficient sequence is in `\u2113\u00b2` and the synthesized family is square-summable: this is Cauchy\u2013Schwarz for `\u2113\u00b2`, via `lp.tsum_mul_le_mul_norm` applied to the two real `\u2113\u00b2` sequences of norms.","title":"Absolute summability of a synthesis series"},{"id":"Analysis/hilbert-spaces/riesz-bases/summable-synthesis","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.summable_synthesis","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#summable-synthesis","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The terms of a synthesis series are summable when the coefficient sequence is in `\u2113\u00b2` and the synthesized family is square-summable.","title":"Summability of a synthesis series"},{"id":"Analysis/hilbert-spaces/riesz-bases/norm-tsum-smul-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.norm_tsum_smul_le","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#norm-tsum-smul-le","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"Norm bound for a synthesis series, by Cauchy\u2013Schwarz for `\u2113\u00b2`.","title":"Norm bound for a synthesis series"},{"id":"Analysis/hilbert-spaces/riesz-bases/synthesis-linear-map","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.synthesisLinearMap","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#synthesis-linear-map","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The linear map underlying the synthesis operator.","title":"Linear map underlying the synthesis operator"},{"id":"Analysis/hilbert-spaces/riesz-bases/synthesis-clm","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.synthesisCLM","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#synthesis-clm","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The synthesis operator: for a square-summable family `d : \u03b9 \u2192 E`, the bounded operator `\u2113\u00b2(\u03b9, \ud835\udd5c) \u2192L[\ud835\udd5c] E` sending `c` to `\u2211' i, c i \u2022 d i`.","title":"Synthesis operator of a square-summable family"},{"id":"Analysis/hilbert-spaces/riesz-bases/norm-synthesis-clm-le","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.norm_synthesisCLM_le","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#norm-synthesis-clm-le","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The operator norm of the synthesis operator is at most `\u221a(\u2211' i, \u2016d i\u2016 ^ 2)`.","title":"Operator norm bound for the synthesis operator"},{"id":"Analysis/hilbert-spaces/riesz-bases/synthesis-clm-single","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.synthesisCLM_single","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#synthesis-clm-single","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"The synthesis operator at a basis vector picks out one term of the family.","title":"Synthesis operator on a basis vector"},{"id":"Analysis/hilbert-spaces/riesz-bases/is-riesz-basis-of-tsum-norm-sq-lt-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.RieszBasisStability.isRieszBasis_of_tsum_norm_sq_lt_one","parent":18713,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/riesz-bases/index.html#is-riesz-basis-of-tsum-norm-sq-lt-one","scope":"Analysis/hilbert-spaces/riesz-bases","status":"fully_proved","summary":"If `b` is a Hilbert basis and `\u2211' i, \u2016f i - b i\u2016 ^ 2 < 1`, then `f` is a Riesz basis.","title":"Paley\u2013Wiener stability theorem"},{"data_url":"scopes/Analysis/hilbert-spaces/separable-hilbert-spaces.json","id":"Analysis/hilbert-spaces/separable-hilbert-spaces","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/separable-hilbert-spaces/index.html","scope":"Analysis/hilbert-spaces/separable-hilbert-spaces","status":"fully_proved","title":"Separable Hilbert spaces"},{"id":"Analysis/hilbert-spaces/separable-hilbert-spaces/separable-space-of-countable-hilbert-bas","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SeparableHilbertBasis.separableSpace_of_countable_hilbertBasis","parent":18729,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/separable-hilbert-spaces/index.html#separable-space-of-countable-hilbert-bas","scope":"Analysis/hilbert-spaces/separable-hilbert-spaces","status":"fully_proved","summary":"If `H` admits a Hilbert basis indexed by a countable type, then `H` is separable. No completeness hypothesis is needed: it is the density of the span of the basis, not completeness, that is used.","title":"A Hilbert space with a countable orthonormal basis is separable"},{"id":"Analysis/hilbert-spaces/separable-hilbert-spaces/countable-of-orthonormal","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SeparableHilbertCountableBasis.countable_of_orthonormal","parent":18729,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/separable-hilbert-spaces/index.html#countable-of-orthonormal","scope":"Analysis/hilbert-spaces/separable-hilbert-spaces","status":"fully_proved","summary":"Let $H$ be a separable inner product space over $\\mathbb{R}$ or $\\mathbb{C}$. If $(v_i)_{i \\in \\iota}$ is an orthonormal family in $H$, then the index set $\\iota$ is countable.","title":"An orthonormal family in a separable space is countable"},{"id":"Analysis/hilbert-spaces/separable-hilbert-spaces/exists-countable-hilbert-basis","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SeparableHilbertCountableBasis.exists_countable_hilbertBasis","parent":18729,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/separable-hilbert-spaces/index.html#exists-countable-hilbert-basis","scope":"Analysis/hilbert-spaces/separable-hilbert-spaces","status":"fully_proved","summary":"Let $H$ be a separable Hilbert space over $\\mathbb{R}$ or $\\mathbb{C}$. Then $H$ has a Hilbert basis $(e_i)_{i \\in \\iota}$ (an orthonormal family whose linear span is dense in $H$) indexed by a countable set $\\iota$.","title":"A separable Hilbert space admits a countable Hilbert basis"},{"id":"Analysis/hilbert-spaces/separable-hilbert-spaces/exists-hilbert-basis-nat","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SeparableHilbertIsoL2.exists_hilbertBasis_nat","parent":18729,"reading_title":{"format":"inline-parts-v1","parts":[["text","An infinite-dimensional separable Hilbert space admits a Hilbert basis indexed by "],["math","\\(\\mathbb{N}\\)"]],"source":"An infinite-dimensional separable Hilbert space admits a Hilbert basis indexed by $\\mathbb{N}$"},"result":true,"route":"../roadmap/Analysis/hilbert-spaces/separable-hilbert-spaces/index.html#exists-hilbert-basis-nat","scope":"Analysis/hilbert-spaces/separable-hilbert-spaces","status":"fully_proved","summary":"An infinite-dimensional separable Hilbert space admits a Hilbert basis indexed by $\\mathbb{N}$. Let $H$ be a separable Hilbert space over $\\mathbb{R}$ or $\\mathbb{C}$ that is not finite-dimensional. Then $H$ has a Hilbert basis $(e_n)_{n \\in \\mathbb{N}}$ indexed by the natural numbers.","title":"An infinite-dimensional separable Hilbert space admits a Hilbert basis indexed by $\\mathbb{N}$"},{"id":"Analysis/hilbert-spaces/separable-hilbert-spaces/separable-hilbert-iso-l2","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.SeparableHilbertIsoL2.separable_hilbert_iso_l2","parent":18729,"reading_title":{"format":"inline-parts-v1","parts":[["text","An infinite-dimensional separable Hilbert space is isometrically isomorphic to "],["math","\\(\\ell^2\\)"]],"source":"An infinite-dimensional separable Hilbert space is isometrically isomorphic to $\\ell^2$"},"result":true,"route":"../roadmap/Analysis/hilbert-spaces/separable-hilbert-spaces/index.html#separable-hilbert-iso-l2","scope":"Analysis/hilbert-spaces/separable-hilbert-spaces","status":"fully_proved","summary":"An infinite-dimensional separable Hilbert space is isometrically isomorphic to $\\ell^2$. Let $H$ be a separable Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ that is not finite-dimensional. Then there is a $\\mathbb{K}$-linear isometric isomorphism from $H$ onto $\\ell^2(\\mathbb{N}, \\mathbb{K})$, the space of square-summable sequences in $\\mathbb{K}$.","title":"An infinite-dimensional separable Hilbert space is isometrically isomorphic to $\\ell^2$"},{"data_url":"scopes/Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences.json","id":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences","kind":"scope","navigation":"module","parent":18609,"result":false,"route":"../roadmap/Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/index.html","scope":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences","status":"fully_proved","title":"Weak convergence of orthonormal sequences"},{"id":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/tendsto-inner-cofinite-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.OrthonormalWeakConvergence.tendsto_inner_cofinite_zero","parent":18735,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/index.html#tendsto-inner-cofinite-zero","scope":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(v_i)_{i \\in \\iota}$ be an orthonormal family in $H$ and let $x \\in H$. Then the inner products $\\langle v_i, x \\rangle$ tend to $0$ along the cofinite filter on $\\iota$: for every $\\varepsilon > 0$, $|\\langle v_i, x \\rangle| < \\varepsilon$ for all but finitely many $i$.","title":"Fourier coefficients along an orthonormal family tend to `0`"},{"id":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/tendsto-apply-cofinite-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.OrthonormalWeakConvergence.tendsto_apply_cofinite_zero","parent":18735,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/index.html#tendsto-apply-cofinite-zero","scope":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences","status":"fully_proved","summary":"Let $H$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(v_i)_{i \\in \\iota}$ be an orthonormal family in $H$ and let $f : H \\to \\mathbb{K}$ be a continuous linear functional. Then $f(v_i)$ tends to $0$ along the cofinite filter on $\\iota$: for every $\\varepsilon > 0$, $|f(v_i)| < \\varepsilon$ for all but finitely many $i$.","title":"An orthonormal family tends weakly to `0`"},{"id":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/tendsto-inner-at-top-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.OrthonormalWeakConvergence.tendsto_inner_atTop_zero","parent":18735,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/index.html#tendsto-inner-at-top-zero","scope":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$, let $(v_n)_{n \\in \\mathbb{N}}$ be an orthonormal sequence in $H$ and let $x \\in H$. Then $\\langle v_n, x \\rangle \\to 0$ as $n \\to \\infty$.","title":"Fourier coefficients along an orthonormal sequence tend to `0`"},{"id":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/tendsto-apply-at-top-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.OrthonormalWeakConvergence.tendsto_apply_atTop_zero","parent":18735,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/index.html#tendsto-apply-at-top-zero","scope":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences","status":"fully_proved","summary":"Let $H$ be a Hilbert space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(v_n)_{n \\in \\mathbb{N}}$ be an orthonormal sequence in $H$. Then for every continuous linear functional $f : H \\to \\mathbb{K}$, we have $f(v_n) \\to 0$ as $n \\to \\infty$.","title":"An orthonormal sequence tends weakly to `0`"},{"id":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/not-tendsto-at-top-zero","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.OrthonormalWeakConvergence.not_tendsto_atTop_zero","parent":18735,"result":true,"route":"../roadmap/Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences/index.html#not-tendsto-at-top-zero","scope":"Analysis/hilbert-spaces/weak-convergence-of-orthonormal-sequences","status":"fully_proved","summary":"Let $H$ be an inner product space over $\\mathbb{K} = \\mathbb{R}$ or $\\mathbb{C}$ and let $(v_n)_{n \\in \\mathbb{N}}$ be an orthonormal sequence in $H$. Then $v_n$ does not tend to $0$ in norm as $n \\to \\infty$.","title":"An orthonormal sequence does not converge strongly to `0`"},{"data_url":"scopes/Analysis/operator-algebras.json","id":"Analysis/operator-algebras","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/operator-algebras/index.html","scope":"Analysis/operator-algebras","status":"can_prove","summary":"Banach algebras and C*- and von Neumann algebras: exponentials and product formulas in Banach algebras, GNS, Gelfand\u2013Naimark, bicommutant, Kaplansky density, Russo\u2013Dye, Stinespring, Kadison\u2013Singer and the Marcus\u2013Spielman\u2013Srivastava theorem.","title":"Banach and operator algebras"},{"id":"Analysis/operator-algebras/c-star-algebra-gelfand-naimark","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.LandmarkWanted.cStarAlgebra_gelfandNaimark","parent":18741,"result":true,"route":"../roadmap/Analysis/operator-algebras/index.html#c-star-algebra-gelfand-naimark","scope":"Analysis/operator-algebras","status":"fully_proved","summary":"For every unital C*-algebra $A$ there are a complex Hilbert space $H$ and a unital $*$-homomorphism $\\pi : A \\to B(H)$ into the algebra of bounded operators on $H$ such that $\\pi$ is injective and isometric: $\\|\\pi(a)\\| = \\|a\\|$ for every $a \\in A$.","title":"Gelfand\u2013Naimark theorem"},{"id":"Analysis/operator-algebras/c-star-algebra-russo-dye","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.LandmarkWanted.cStarAlgebra_russoDye","parent":18741,"result":true,"route":"../roadmap/Analysis/operator-algebras/index.html#c-star-algebra-russo-dye","scope":"Analysis/operator-algebras","status":"fully_proved","summary":"In a unital C*-algebra $A$, the norm closure of the convex hull of the unitary elements of $A$ equals the closed unit ball $\\{a \\in A : \\|a\\| \\le 1\\}$.","title":"Russo\u2013Dye theorem"},{"id":"Analysis/operator-algebras/stinespring-dilation","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.StinespringWanted.stinespring_dilation","parent":18741,"result":true,"route":"../roadmap/Analysis/operator-algebras/index.html#stinespring-dilation","scope":"Analysis/operator-algebras","status":"fully_proved","summary":"Let $A$ be a unital C*-algebra with its standard order, let $H$ be a complex Hilbert space, and let $\\varphi : A \\to B(H)$ be a completely positive map. Then there are a complex Hilbert space $K$, a unital $*$-homomorphism $\\pi : A \\to B(K)$ and a bounded linear map $V : H \\to K$ such that $\\varphi(a) = V^* \\pi(a) V$ for every $a \\in A$, where $V^*$ is the adjoint of $V$.","title":"Stinespring dilation theorem"},{"data_url":"scopes/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula.json","id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","kind":"scope","navigation":"module","parent":18741,"result":false,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","title":"Exponential limits and the Lie\u2013Trotter formula"},{"id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/norm-pow-sub-pow-le","kind":"node","lean":"NormedSpace.norm_pow_sub_pow_le","parent":18745,"result":true,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html#norm-pow-sub-pow-le","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","summary":"Telescoping estimate for powers in a normed ring with `\u20161\u2016 = 1`: `\u2016a ^ m - b ^ m\u2016 \u2264 m \u2016a - b\u2016 max(\u2016a\u2016, \u2016b\u2016) ^ (m - 1)`.","title":"Telescoping estimate for differences of powers in a normed ring"},{"id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/tendsto-one-add-inv-smul-pow","kind":"node","lean":"NormedSpace.tendsto_one_add_inv_smul_pow","parent":18745,"result":true,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html#tendsto-one-add-inv-smul-pow","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","summary":"If `Y M \u2192 X` and `m M / M \u2192 t`, then `(1 + M\u207b\u00b9 \u2022 Y M) ^ m M \u2192 exp (t \u2022 X)` as `M \u2192 \u221e`.","title":"Compound-interest limit in a Banach algebra"},{"id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/norm-exp-le-exp-norm","kind":"node","lean":"MathlibExt.Analysis.Normed.Algebra.TrotterProductFormula.norm_exp_le_exp_norm","parent":18745,"result":true,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html#norm-exp-le-exp-norm","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","summary":"The exponential is bounded by the real exponential of the norm.","title":"Norm of the exponential is bounded by the exponential of the norm"},{"id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/norm-exp-sub-one-sub-id-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Algebra.TrotterProductFormula.norm_exp_sub_one_sub_id_le","parent":18745,"result":true,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html#norm-exp-sub-one-sub-id-le","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","summary":"The Taylor remainder of `exp` at order one.","title":"First-order Taylor remainder bound for the exponential"},{"id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/norm-exp-mul-exp-sub-exp-add-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Algebra.TrotterProductFormula.norm_exp_mul_exp_sub_exp_add_le","parent":18745,"result":true,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html#norm-exp-mul-exp-sub-exp-add-le","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","summary":"The one-step Trotter estimate: `exp x * exp y` differs from `exp (x + y)` by at most a quadratic expression in `\u2016x\u2016` and `\u2016y\u2016`.","title":"One-step Trotter estimate"},{"id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/norm-pow-sub-pow-le-2","kind":"node","lean":"MathlibExt.Analysis.Normed.Algebra.TrotterProductFormula.norm_pow_sub_pow_le","parent":18745,"result":true,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html#norm-pow-sub-pow-le-2","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","summary":"Powers differ at most linearly in the difference of the bases. The `1` in the maximum is needed: without it the bound fails when the norms are `< 1`.","title":"Linear bound on differences of powers for the Trotter formula"},{"id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/tendsto-trotter-product","kind":"node","lean":"MathlibExt.Analysis.Normed.Algebra.TrotterProductFormula.tendsto_trotter_product","parent":18745,"result":true,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html#tendsto-trotter-product","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","summary":"The Trotter product formula for bounded elements of a complete normed algebra: `(exp (a / n) * exp (b / n)) ^ n \u2192 exp (a + b)` in norm.","title":"Trotter product formula"},{"id":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/trotter-product-of-commute","kind":"node","lean":"MathlibExt.Analysis.Normed.Algebra.TrotterProductFormula.trotter_product_of_commute","parent":18745,"result":true,"route":"../roadmap/Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula/index.html#trotter-product-of-commute","scope":"Analysis/operator-algebras/exponential-limits-and-the-lie-trotter-formula","status":"fully_proved","summary":"If `a` and `b` commute, the Trotter approximants are exactly `exp (a + b)` for every positive `n`.","title":"Trotter product formula for commuting elements"},{"data_url":"scopes/Analysis/operator-algebras/gns-construction.json","id":"Analysis/operator-algebras/gns-construction","kind":"scope","navigation":"module","parent":18741,"result":false,"route":"../roadmap/Analysis/operator-algebras/gns-construction/index.html","scope":"Analysis/operator-algebras/gns-construction","status":"fully_proved","title":"GNS construction"},{"id":"Analysis/operator-algebras/gns-construction/exists-gns-cyclic-vector","kind":"node","lean":"PositiveLinearMap.exists_gns_cyclic_vector","parent":18754,"result":true,"route":"../roadmap/Analysis/operator-algebras/gns-construction/index.html#exists-gns-cyclic-vector","scope":"Analysis/operator-algebras/gns-construction","status":"fully_proved","summary":"Let $A$ be a unital C*-algebra with its standard order and let $f$ be a positive linear functional on $A$ with $f(1) = 1$. Let $H_f$ be the GNS Hilbert space of $f$ and $\\pi_f : A \\to B(H_f)$ its GNS representation. Then there is a vector $\\xi \\in H_f$ with $\\|\\xi\\| = 1$ such that $f(a) = \\langle \\xi, \\pi_f(a) \\xi \\rangle$ for every $a \\in A$, and the linear span of $\\{\\pi_f(a) \\xi : a \\in A\\}$ is dense in $H_f$.","title":"Existence of a unit cyclic vector in the GNS space"},{"id":"Analysis/operator-algebras/gns-construction/c-star-algebra-gns-cyclic-vector","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.LandmarkWanted.cStarAlgebra_gns_cyclic_vector","parent":18754,"result":true,"route":"../roadmap/Analysis/operator-algebras/gns-construction/index.html#c-star-algebra-gns-cyclic-vector","scope":"Analysis/operator-algebras/gns-construction","status":"fully_proved","summary":"Let $A$ be a unital C*-algebra with its standard order and let $f$ be a positive linear functional on $A$ with $f(1) = 1$. Let $H_f$ be the GNS Hilbert space of $f$ and $\\pi_f : A \\to B(H_f)$ its GNS representation. Then there is a vector $\\xi \\in H_f$ with $\\|\\xi\\| = 1$ such that $f(a) = \\langle \\xi, \\pi_f(a) \\xi \\rangle$ for every $a \\in A$, and the linear span of $\\{\\pi_f(a) \\xi : a \\in A\\}$ is dense in $H_f$.","title":"GNS cyclic vector of a state"},{"data_url":"scopes/Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge.json","id":"Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge","kind":"scope","navigation":"module","parent":18741,"result":false,"route":"../roadmap/Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge/index.html","scope":"Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge","status":"can_prove","title":"Inverses as limits of polynomials in Banach algebras"},{"id":"Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge/scottish-book-problem176-claim","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook176Wanted.ScottishBookProblem176Claim","parent":18757,"result":true,"route":"../roadmap/Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge/index.html#scottish-book-problem176-claim","scope":"Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge","status":"fully_proved","summary":"The claim that in every complete normed ring $A$ (so $\\|xy\\| \\le \\|x\\| \\, \\|y\\|$, and $A$ has a unit) that is a normed algebra over $\\mathbb{R}$, the inverse of every invertible element is a limit of real polynomials in that element: whenever $a, b \\in A$ satisfy $ab = 1$ and $ba = 1$, there is a sequence $(p_n)_{n \\ge 0}$ of real polynomials with $p_n(a) \\to b$ as $n \\to \\infty$.","title":"Scottish Book Problem 176 (Eidelheit's claim)"},{"id":"Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge/eidelheit176-refuted","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook176Wanted.eidelheit176_refuted","parent":18757,"result":true,"route":"../roadmap/Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge/index.html#eidelheit176-refuted","scope":"Analysis/operator-algebras/inverses-as-limits-of-polynomials-in-banach-alge","status":"can_prove","summary":"The claim of Scottish Book Problem 176 is false: there are a complete normed ring $A$ that is a normed algebra over $\\mathbb{R}$ and elements $a, b \\in A$ with $ab = 1$ and $ba = 1$ such that $b$ is not the limit of any sequence $(p_n(a))_{n \\ge 0}$ with $p_n$ real polynomials.","title":"Eidelheit's Problem 176 has a negative answer"},{"data_url":"scopes/Analysis/operator-algebras/kadison-singer-problem.json","id":"Analysis/operator-algebras/kadison-singer-problem","kind":"scope","navigation":"module","parent":18741,"result":false,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","title":"Kadison\u2013Singer problem"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-det-polynomial","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedDetPolynomial","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-det-polynomial","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For square complex matrices $A_1, \\dots, A_m$ whose rows and columns are indexed by a finite set $d$, the mixed determinant polynomial is $\\det(x I + \\sum_{i=1}^m z_i A_i)$, viewed as a polynomial with complex coefficients in the $m + 1$ variables $x, z_1, \\dots, z_m$, where $I$ is the identity matrix.","title":"Mixed determinant polynomial"},{"id":"Analysis/operator-algebras/kadison-singer-problem/eval-mixed-det-polynomial","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.eval_mixedDetPolynomial","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#eval-mixed-det-polynomial","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Let $A_1, \\dots, A_m$ be square complex matrices whose rows and columns are indexed by a finite set $d$. For all complex numbers $x, z_1, \\dots, z_m$, the value of the mixed determinant polynomial of $A_1, \\dots, A_m$ at the point $(x, z_1, \\dots, z_m)$ is $\\det(x I + \\sum_{i=1}^m z_i A_i)$.","title":"Values of the mixed determinant polynomial"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-det-polynomial-pderiv-sq-eq-zero-o","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedDetPolynomial_pderiv_sq_eq_zero_of_eq_vecMulVec","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-det-polynomial-pderiv-sq-eq-zero-o","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Let $A_1, \\dots, A_m$ be square complex matrices whose rows and columns are indexed by a finite set $d$, and suppose that $A_i = u v^{\\mathsf T}$ for some index $i$ and vectors $u, v \\in \\mathbb{C}^d$. Consider the mixed determinant polynomial $\\det(x I + \\sum_{j=1}^m z_j A_j)$ in the variables $x, z_1, \\dots, z_m$, where $I$ is the identity matrix. Then its second partial derivative with respect to $z_i$ is zero;\u2026","title":"Rank-one coefficients make the mixed determinant polynomial affine"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For square complex matrices $A_1, \\dots, A_m$ whose rows and columns are indexed by a finite set $d$, consider the mixed determinant polynomial $\\det(x I + \\sum_{i=1}^m z_i A_i)$ in the variables $x, z_1, \\dots, z_m$, where $I$ is the identity matrix. The mixed characteristic polynomial is the polynomial in $x$ with complex coefficients obtained by applying the operator $\\prod_{i=1}^m (1 - \\partial_{z_i})$ to it\u2026","title":"Mixed characteristic polynomial"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial-update-s","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial_update_sum","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial-update-s","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Let $A_1, \\dots, A_m$ be square complex matrices whose rows and columns are indexed by a finite set $d$, and fix an index $i$. Let $(C_a)_{a \\in \\iota}$ be a family of such matrices indexed by a finite set $\\iota$, with real weights $w_a$, not necessarily nonnegative, such that $\\sum_a w_a = 1$. Then the mixed characteristic polynomial of the family obtained from $A_1, \\dots, A_m$ by replacing $A_i$ with\u2026","title":"The mixed characteristic polynomial is affine in each matrix"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial-update-a","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial_update_affine","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial-update-a","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Let $A_1, \\dots, A_m$ be square complex matrices whose rows and columns are indexed by a finite set $d$, fix an index $i$, let $B$ and $C$ be two such matrices and let $t$ be a real number. Write $\\mu_M$ for the mixed characteristic polynomial of the family obtained from $A_1, \\dots, A_m$ by replacing $A_i$ with $M$. Then $\\mu_{(1 - t) B + t C} = (1 - t)\\, \\mu_B + t\\, \\mu_C$.","title":"The mixed characteristic polynomial along a line"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial-eq-charp","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial_eq_charpoly_sum_of_eq_vecMulVec","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial-eq-charp","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Let $A_1, \\dots, A_m$ be square complex matrices whose rows and columns are indexed by a finite set $d$, and suppose that $A_i = u_i v_i^{\\mathsf T}$ for vectors $u_i, v_i \\in \\mathbb{C}^d$, for every $i$. Then the mixed characteristic polynomial of $A_1, \\dots, A_m$ equals the characteristic polynomial of the sum $\\sum_{i=1}^m A_i$, that is, the polynomial $\\det(x I - \\sum_{i=1}^m A_i)$ in the variable $x$, where\u2026","title":"Mixed characteristic polynomials of outer products"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial-fin-one","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial_fin_one","parent":18760,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mixed characteristic polynomial of a single "],["math","\\(1 \\times 1\\)"],["text"," matrix"]],"source":"Mixed characteristic polynomial of a single $1 \\times 1$ matrix"},"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial-fin-one","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For every complex number $a$, the mixed characteristic polynomial of the family consisting of the single $1 \\times 1$ complex matrix $(a)$ is the polynomial $x - a$ in the variable $x$, which is the characteristic polynomial of $(a)$.","title":"Mixed characteristic polynomial of a single $1 \\times 1$ matrix"},{"id":"Analysis/operator-algebras/kadison-singer-problem/finite-mssbound-2","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.FiniteMSSBound","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#finite-mssbound-2","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"The proposition that the following holds for every natural number $m$ and every finite index set $D$. Let $s_1, \\dots, s_m$ be positive integers, let $p_{i,a} \\ge 0$ be reals and $v_{i,a} \\in \\mathbb{C}^D$ be vectors for $1 \\le i \\le m$ and $1 \\le a \\le s_i$, and let $\\eta \\ge 0$ be real. Suppose that $\\sum_a p_{i,a} = 1$ for each $i$, that $\\sum_i \\sum_a p_{i,a} v_{i,a} v_{i,a}^* = \\mathrm{Id}_D$, and that\u2026","title":"Finite-support Marcus\u2013Spielman\u2013Srivastava bound"},{"id":"Analysis/operator-algebras/kadison-singer-problem/exists-frame-partition-of-finite-mss","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.exists_frame_partition_of_finiteMSS","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#exists-frame-partition-of-finite-mss","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Assume the finite-support Marcus\u2013Spielman\u2013Srivastava bound holds. Let $I$ be a finite index set, let $d$ and $r$ be natural numbers with $r > 0$, let $\\delta \\ge 0$ be real, and let $u_i \\in \\mathbb{C}^d$ for $i \\in I$ be vectors with $\\sum_{i \\in I} u_i u_i^* = \\mathrm{Id}_d$ and $\\|u_i\\|^2 \\le \\delta$ for every $i$. Then there is a map $c : I \\to \\{1, \\dots, r\\}$ such that for every $j$,\u2026","title":"Partitioning a Parseval frame with the finite MSS bound"},{"id":"Analysis/operator-algebras/kadison-singer-problem/anderson-paving-of-finite-mss","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.andersonPaving_of_finiteMSS","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#anderson-paving-of-finite-mss","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Assume the finite-support Marcus\u2013Spielman\u2013Srivastava bound holds. Let $n$ and $r$ be natural numbers with $r > 0$, and let $A$ be a Hermitian complex $n \\times n$ matrix with zero diagonal and operator norm $\\|A\\| \\le 1$. Then there is a coloring $c$ of the indices $\\{1, \\dots, n\\}$ with $r^2$ colors such that for every color $j$, writing $P_j$ for the diagonal matrix with entry $1$ at the indices of color $j$ and\u2026","title":"Anderson paving from the finite MSS bound"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-det-polynomial-is-stable-of-pos-se","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedDetPolynomial_isStable_of_posSemidef","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-det-polynomial-is-stable-of-pos-se","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For every finite family $A_1, \\dots, A_m$ of positive semidefinite complex square matrices whose rows and columns are indexed by a finite set $d$, the polynomial $\\det\\left(x I + \\sum_{i=1}^m z_i A_i\\right)$ in the variables $x, z_1, \\dots, z_m$, where $I$ is the identity matrix, is stable.","title":"Stability of the determinant polynomial"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-det-polynomial-is-real-stable-of-p","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedDetPolynomial_isRealStable_of_posSemidef","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-det-polynomial-is-real-stable-of-p","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For every finite family $A_1, \\dots, A_m$ of positive semidefinite complex square matrices whose rows and columns are indexed by a finite set $d$, the polynomial $\\det\\left(x I + \\sum_{i=1}^m z_i A_i\\right)$ in the variables $x, z_1, \\dots, z_m$, where $I$ is the identity matrix, is real stable.","title":"Real stability of the determinant polynomial"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial-monic","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial_monic","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial-monic","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For every finite family $A_1, \\dots, A_m$ of complex square matrices of a common size, the mixed characteristic polynomial of $A_1, \\dots, A_m$ is monic.","title":"Mixed characteristic polynomials are monic"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial-nat-degr","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial_natDegree","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial-nat-degr","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For every finite family $A_1, \\dots, A_m$ of complex square matrices whose rows and columns are indexed by a finite set $d$, the mixed characteristic polynomial of $A_1, \\dots, A_m$ has degree $|d|$, the number of elements of $d$.","title":"Degree of a mixed characteristic polynomial"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial-is-real","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial_isRealRooted_of_posSemidef","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial-is-real","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For every finite family $A_1, \\dots, A_m$ of positive semidefinite complex square matrices of a common size, the mixed characteristic polynomial of $A_1, \\dots, A_m$ is real-rooted.","title":"Mixed characteristic polynomials of positive semidefinite matrices are real-rooted"},{"id":"Analysis/operator-algebras/kadison-singer-problem/mixed-characteristic-polynomial-max-real","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.mixedCharacteristicPolynomial_maxRealRoot_le","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#mixed-characteristic-polynomial-max-real","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Let $d$ be a nonempty finite index set, let $A_1, \\dots, A_m$ be positive semidefinite complex $d \\times d$ matrices with $\\sum_i A_i = I$, and let $\\eta \\ge 0$ be a real number such that $\\operatorname{Re} \\operatorname{tr} A_i \\le \\eta$ for every $i$. Then the largest real part of a root of the mixed characteristic polynomial of $A_1, \\dots, A_m$ is at most $(1 + \\sqrt{\\eta})^2$.","title":"Marcus--Spielman--Srivastava barrier bound"},{"id":"Analysis/operator-algebras/kadison-singer-problem/exists-outcome-max-real-root-le-mixed-ch","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.exists_outcome_maxRealRoot_le_mixedCharacteristicPolynomial","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#exists-outcome-max-real-root-le-mixed-ch","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Let $d$ be a nonempty finite index set. For $i = 1, \\dots, m$ let $s_i \\ge 1$, let $v_{i,1}, \\dots, v_{i,s_i} \\in \\mathbb{C}^d$, let $p_{i,1}, \\dots, p_{i,s_i} \\ge 0$ be reals with $\\sum_a p_{i,a} = 1$, and put $A_i = \\sum_a p_{i,a} v_{i,a} v_{i,a}^*$. Then there is a choice of indices $\\omega_i \\in \\{1, \\dots, s_i\\}$, one for each $i$, such that the largest real part of a root of the characteristic polynomial of\u2026","title":"An outcome bounded by the mixed characteristic polynomial"},{"id":"Analysis/operator-algebras/kadison-singer-problem/finite-mssbound","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSinger.finiteMSSBound","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#finite-mssbound","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Let $d$ be a finite index set and, for $i = 1, \\dots, m$, let $s_i \\ge 1$, let $v_{i,1}, \\dots, v_{i,s_i} \\in \\mathbb{C}^d$, and let $p_{i,1}, \\dots, p_{i,s_i} \\ge 0$ be reals with $\\sum_a p_{i,a} = 1$. Let $\\eta \\ge 0$ be a real number, and suppose that $\\sum_i \\sum_a p_{i,a} v_{i,a} v_{i,a}^* = I$ and $\\sum_a p_{i,a} \\|v_{i,a}\\|^2 \\le \\eta$ for every $i$, where $\\|\\cdot\\|$ is the Euclidean norm. Then there is a\u2026","title":"Marcus--Spielman--Srivastava bound, finite-support form"},{"id":"Analysis/operator-algebras/kadison-singer-problem/paving-proj","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSingerWanted.pavingProj","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#paving-proj","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"Given a colouring $c$ of the indices $\\{0, \\dots, n-1\\}$ with the colours $\\{0, \\dots, r-1\\}$ and a colour $j$, the paving projection of $c$ onto colour $j$ is the diagonal complex $n \\times n$ matrix whose $i$-th diagonal entry is $1$ if $c(i) = j$ and $0$ otherwise.","title":"Paving projection"},{"id":"Analysis/operator-algebras/kadison-singer-problem/kadison-singer-paving","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KadisonSingerWanted.kadison_singer_paving","parent":18760,"result":true,"route":"../roadmap/Analysis/operator-algebras/kadison-singer-problem/index.html#kadison-singer-paving","scope":"Analysis/operator-algebras/kadison-singer-problem","status":"fully_proved","summary":"For every real $\\varepsilon > 0$ there is a positive integer $r$ such that for every $n \\in \\mathbb{N}$ and every Hermitian complex $n \\times n$ matrix $A$ with zero diagonal and $\\|A\\| \\le 1$, where $\\|\\cdot\\|$ is the operator norm for the Euclidean norm on $\\mathbb{C}^n$, there is a colouring $c$ of the indices $\\{0, \\dots, n-1\\}$ with the $r$ colours $\\{0, \\dots, r-1\\}$ such that $\\|P_j A P_j\\| \\le \\varepsilon$\u2026","title":"Anderson paving theorem"},{"data_url":"scopes/Analysis/operator-algebras/kaplansky-density-theorem.json","id":"Analysis/operator-algebras/kaplansky-density-theorem","kind":"scope","navigation":"module","parent":18741,"result":false,"route":"../roadmap/Analysis/operator-algebras/kaplansky-density-theorem/index.html","scope":"Analysis/operator-algebras/kaplansky-density-theorem","status":"fully_proved","title":"Kaplansky density theorem"},{"id":"Analysis/operator-algebras/kaplansky-density-theorem/to-bounded","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KaplanskyDensityWanted.toBounded","parent":18782,"result":true,"route":"../roadmap/Analysis/operator-algebras/kaplansky-density-theorem/index.html#to-bounded","scope":"Analysis/operator-algebras/kaplansky-density-theorem","status":"fully_proved","summary":"For a complex Hilbert space $H$, this sends a bounded linear operator on $H$, viewed in the space of operators with the strong operator topology (pointwise convergence on $H$), to the same operator viewed in $B(H)$ with the operator norm. It is the identity on underlying maps.","title":"Strong-operator operators as bounded operators"},{"id":"Analysis/operator-algebras/kaplansky-density-theorem/kaplansky-density-theorem","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.KaplanskyDensityWanted.kaplansky_density_theorem","parent":18782,"result":true,"route":"../roadmap/Analysis/operator-algebras/kaplansky-density-theorem/index.html#kaplansky-density-theorem","scope":"Analysis/operator-algebras/kaplansky-density-theorem","status":"fully_proved","summary":"Let $H$ be a complex Hilbert space and let $S$ be a unital $*$-subalgebra of the algebra $B(H)$ of bounded operators on $H$, with operator norm $\\|\\cdot\\|$. Take closures in the strong operator topology on $B(H)$, the topology of pointwise convergence on $H$, and write $\\overline{S}$ for the strong closure of $S$. Then the strong closure of the unit ball $\\{x \\in S : \\|x\\| \\le 1\\}$ of $S$ equals the unit ball\u2026","title":"Kaplansky density theorem"},{"data_url":"scopes/Analysis/operator-algebras/von-neumann-bicommutant-theorem.json","id":"Analysis/operator-algebras/von-neumann-bicommutant-theorem","kind":"scope","navigation":"module","parent":18741,"result":false,"route":"../roadmap/Analysis/operator-algebras/von-neumann-bicommutant-theorem/index.html","scope":"Analysis/operator-algebras/von-neumann-bicommutant-theorem","status":"can_prove","title":"Von Neumann bicommutant theorem"},{"id":"Analysis/operator-algebras/von-neumann-bicommutant-theorem/bicommutant-finite-dimensional-matrix","kind":"node","lean":"MathlibExt.Analysis.CStarAlgebra.Bicommutant.bicommutant_finiteDimensional_matrix","parent":18785,"result":true,"route":"../roadmap/Analysis/operator-algebras/von-neumann-bicommutant-theorem/index.html#bicommutant-finite-dimensional-matrix","scope":"Analysis/operator-algebras/von-neumann-bicommutant-theorem","status":"fully_proved","summary":"Let $n$ be a natural number and let $S$ be a unital $*$-subalgebra of the algebra $M_n(\\mathbb{C})$ of complex $n \\times n$ matrices. Then the commutant of the commutant of $S$ equals $S$: a matrix commutes with every matrix that commutes with all elements of $S$ if and only if it lies in $S$.","title":"Finite-dimensional bicommutant theorem"},{"id":"Analysis/operator-algebras/von-neumann-bicommutant-theorem/von-neumann-double-commutant-tfae","kind":"node","lean":"MathlibExt.Analysis.VonNeumannDoubleCommutantWanted.Analysis.vonNeumann_doubleCommutant_tfae","parent":18785,"result":true,"route":"../roadmap/Analysis/operator-algebras/von-neumann-bicommutant-theorem/index.html#von-neumann-double-commutant-tfae","scope":"Analysis/operator-algebras/von-neumann-bicommutant-theorem","status":"can_prove","summary":"Let $H$ be a complex Hilbert space and let $S$ be a unital $*$-subalgebra of the algebra of bounded linear operators on $H$. Write $S'$ for the commutant of $S$, the set of bounded operators on $H$ that commute with every element of $S$. The following are equivalent: $S$ equals its double commutant $S''$; $S$ is closed in the weak operator topology; $S$ is closed in the strong operator topology, the topology of\u2026","title":"Von Neumann's double commutant theorem"},{"data_url":"scopes/Analysis/function-spaces.json","id":"Analysis/function-spaces","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/function-spaces/index.html","scope":"Analysis/function-spaces","status":"can_prove","summary":"L^p and Sobolev spaces of functions: embeddings (Morrey, Rellich, radial, trace), Ehrling, Agmon, Ladyzhenskaya, Gagliardo\u2013Nirenberg and weighted Hardy\u2013Sobolev inequalities, sup-norm bounds by H\u00b9 energy, L^p limits and weak convergence in L^p.","title":"Function spaces and Sobolev inequalities"},{"id":"Analysis/function-spaces/exists-norm-to-lp-top-le-mul-sqrt-norm-t","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AgmonInequalitySchwartzLineWanted.exists_norm_toLp_top_le_mul_sqrt_norm_toLp_two_mul_sqrt_deriv","parent":18788,"result":true,"route":"../roadmap/Analysis/function-spaces/index.html#exists-norm-to-lp-top-le-mul-sqrt-norm-t","scope":"Analysis/function-spaces","status":"can_prove","summary":"There is a constant $C > 0$ such that every real-valued Schwartz function $f \\colon \\mathbb{R} \\to \\mathbb{R}$ satisfies","title":"Agmon's inequality on the line"},{"id":"Analysis/function-spaces/finite-dimensional-of-is-closed-of-le-ra","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ClosedL2SubspaceOfContinuousFiniteDimWanted.finiteDimensional_of_isClosed_of_le_range_toLp","parent":18788,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed subspaces of "],["math","\\(L^2((0,1))\\)"],["text"," inside "],["math","\\(C((0,1))\\)"],["text"," are finite-dimensional"]],"source":"Closed subspaces of $L^2((0,1))$ inside $C((0,1))$ are finite-dimensional"},"result":true,"route":"../roadmap/Analysis/function-spaces/index.html#finite-dimensional-of-is-closed-of-le-ra","scope":"Analysis/function-spaces","status":"can_prove","summary":"Let $H$ be a closed linear subspace of the real space $L^2([0,1])$ with respect to Lebesgue measure, and suppose that every element of $H$ is the class of a continuous function $[0,1] \\to \\mathbb{R}$. Then $H$ is finite-dimensional.","title":"Closed subspaces of $L^2((0,1))$ inside $C((0,1))$ are finite-dimensional"},{"id":"Analysis/function-spaces/ehrling-inequality","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.EhrlingInequalityWanted.ehrling_inequality","parent":18788,"result":true,"route":"../roadmap/Analysis/function-spaces/index.html#ehrling-inequality","scope":"Analysis/function-spaces","status":"can_prove","summary":"Let $X$, $Y$, $Z$ be real Banach spaces, let $i \\colon X \\to Y$ be an injective compact bounded linear operator, and let $j \\colon Y \\to Z$ be an injective bounded linear operator. Then for every $\\varepsilon > 0$ there is $C_\\varepsilon > 0$ such that $\\|i(u)\\|_Y \\le \\varepsilon \\|u\\|_X + C_\\varepsilon \\|j(i(u))\\|_Z$ for all $u \\in X$.","title":"Ehrling's inequality"},{"id":"Analysis/function-spaces/nonlinear-interpolation-lp","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ScottishBook87Wanted.nonlinearInterpolationLp","parent":18788,"result":true,"route":"../roadmap/Analysis/function-spaces/index.html#nonlinear-interpolation-lp","scope":"Analysis/function-spaces","status":"can_prove","summary":"Let $\\mu$ be Lebesgue measure on $[0, 1]$, write $\\|f\\|_p$ for the $L^p(\\mu)$ norm, let $1 \\le \\beta < \\alpha \\le \\infty$, and let $U$ map functions $\\mathbb{R} \\to \\mathbb{R}$ to functions $\\mathbb{R} \\to \\mathbb{R}$. Suppose that $U$ maps $L^\\beta(\\mu)$ into $L^\\beta(\\mu)$; that there is $L \\ge 0$ with $\\|Ux - Uy\\|_\\beta \\le L \\|x - y\\|_\\beta$ for all $x, y \\in L^\\beta(\\mu)$; and that there is $M_\\alpha \\ge 0$\u2026","title":"Nonlinear interpolation of Lipschitz maps on Lebesgue spaces (Scottish Book Problem 87)"},{"id":"Analysis/function-spaces/exists-bound-integral-sq-trace-of-integr","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.SobolevTraceR3PlaneBoundedWanted.exists_bound_integral_sq_trace_of_integral_sq_add_fderiv_sq_le_one","parent":18788,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sobolev trace bound on the plane "],["math","\\(z = 0\\)"],["text"," in "],["math","\\(\\mathbb{R}^3\\)"]],"source":"Sobolev trace bound on the plane $z = 0$ in $\\mathbb{R}^3$"},"result":true,"route":"../roadmap/Analysis/function-spaces/index.html#exists-bound-integral-sq-trace-of-integr","scope":"Analysis/function-spaces","status":"can_prove","summary":"Let $(f_n)_{n \\in \\mathbb{N}}$ be a sequence of continuously differentiable functions $\\mathbb{R}^3 \\to \\mathbb{R}$, where $\\mathbb{R}^3$ has coordinates $(x, y, z)$ and the max norm. For $p \\in \\mathbb{R}^3$ write $\\|Df_n(p)\\|$ for the operator norm of the derivative of $f_n$ at $p$, which equals $|\\partial_x f_n(p)| + |\\partial_y f_n(p)| + |\\partial_z f_n(p)|$, the sum of the absolute values of the partial\u2026","title":"Sobolev trace bound on the plane $z = 0$ in $\\mathbb{R}^3$"},{"id":"Analysis/function-spaces/exists-e-lp-norm-convolution-gt-one-sub","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.YoungConvolutionSharpWanted.exists_eLpNorm_convolution_gt_one_sub_mul","parent":18788,"reading_title":{"format":"inline-parts-v1","parts":[["text","The constant in Young's "],["math","\\(L^1\\)"],["text","\u2013"],["math","\\(L^p\\)"],["text"," convolution inequality cannot be lowered"]],"source":"The constant in Young's $L^1$\u2013$L^p$ convolution inequality cannot be lowered"},"result":true,"route":"../roadmap/Analysis/function-spaces/index.html#exists-e-lp-norm-convolution-gt-one-sub","scope":"Analysis/function-spaces","status":"can_prove","summary":"Let $1 \\le p \\le \\infty$ and $\\varepsilon > 0$. Then there exist real-valued functions $f \\in L^1(\\mathbb{R})$ and $g \\in L^p(\\mathbb{R})$ such that","title":"The constant in Young's $L^1$\u2013$L^p$ convolution inequality cannot be lowered"},{"id":"Analysis/function-spaces/exists-const-integral-pow-four-mul-weigh","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.WeightedLadyzhenskayaPlaneWanted.exists_const_integral_pow_four_mul_weight_le","parent":18788,"result":true,"route":"../roadmap/Analysis/function-spaces/index.html#exists-const-integral-pow-four-mul-weigh","scope":"Analysis/function-spaces","status":"can_prove","summary":"There is a constant $C > 0$ such that for every $C^1$ function $\\omega : \\mathbb{R}^2 \\to \\mathbb{R}$ with $\\omega(x) > 0$ for all $x$ and $\\|\\nabla\\omega/\\omega\\|_{L^\\infty(\\mathbb{R}^2)} \\le 1$, and every smooth compactly supported function $u : \\mathbb{R}^2 \\to \\mathbb{R}$,","title":"Weighted Ladyzhenskaya inequality in the plane"},{"data_url":"scopes/Analysis/function-spaces/gagliardo-nirenberg-inequality.json","id":"Analysis/function-spaces/gagliardo-nirenberg-inequality","kind":"scope","navigation":"module","parent":18788,"result":false,"route":"../roadmap/Analysis/function-spaces/gagliardo-nirenberg-inequality/index.html","scope":"Analysis/function-spaces/gagliardo-nirenberg-inequality","status":"can_prove","title":"Gagliardo\u2013Nirenberg inequality"},{"id":"Analysis/function-spaces/gagliardo-nirenberg-inequality/exists-integral-pow-six-le-integral-deri","kind":"node","lean":"MathlibExt.Analysis.Calculus.SixthPowerIntegralLeDerivSqWanted.exists_integral_pow_six_le_integral_deriv_sq_of_integral_sq_eq_one","parent":18796,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sixth-power integral bounded by the Dirichlet energy under "],["math","\\(L^2\\)"],["text"," normalization"]],"source":"Sixth-power integral bounded by the Dirichlet energy under $L^2$ normalization"},"result":true,"route":"../roadmap/Analysis/function-spaces/gagliardo-nirenberg-inequality/index.html#exists-integral-pow-six-le-integral-deri","scope":"Analysis/function-spaces/gagliardo-nirenberg-inequality","status":"can_prove","summary":"There is a constant $C_2 > 0$ such that every infinitely differentiable, compactly supported function $u : \\mathbb{R} \\to \\mathbb{R}$ with $\\int_{\\mathbb{R}} u(t)^2 \\, dt = 1$ satisfies","title":"Sixth-power integral bounded by the Dirichlet energy under $L^2$ normalization"},{"id":"Analysis/function-spaces/gagliardo-nirenberg-inequality/exists-integral-pow-six-mul-weight-le","kind":"node","lean":"MathlibExt.Analysis.Calculus.WeightedSixthPowerGagliardoNirenbergWanted.exists_integral_pow_six_mul_weight_le","parent":18796,"reading_title":{"format":"inline-parts-v1","parts":[["text","A weighted "],["math","\\(L^6\\)"],["text"," bound with a slowly varying weight"]],"source":"A weighted $L^6$ bound with a slowly varying weight"},"result":true,"route":"../roadmap/Analysis/function-spaces/gagliardo-nirenberg-inequality/index.html#exists-integral-pow-six-mul-weight-le","scope":"Analysis/function-spaces/gagliardo-nirenberg-inequality","status":"can_prove","summary":"There is a constant $C > 0$ with the following property. Let $\\theta : \\mathbb{R} \\to \\mathbb{R}$ be continuously differentiable, with $\\theta(x) > 0$ for every $x$ and $\\|\\theta'/\\theta\\|_{L^\\infty(\\mathbb{R})} \\le 1$, and let $u : \\mathbb{R} \\to \\mathbb{R}$ be smooth with compact support. Then","title":"A weighted $L^6$ bound with a slowly varying weight"},{"data_url":"scopes/Analysis/function-spaces/radial-sobolev-embedding.json","id":"Analysis/function-spaces/radial-sobolev-embedding","kind":"scope","navigation":"module","parent":18788,"result":false,"route":"../roadmap/Analysis/function-spaces/radial-sobolev-embedding/index.html","scope":"Analysis/function-spaces/radial-sobolev-embedding","status":"can_prove","title":"Radial Sobolev embedding"},{"id":"Analysis/function-spaces/radial-sobolev-embedding/rn","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RadialSobolevExteriorBallWanted.Rn","parent":18799,"reading_title":{"format":"inline-parts-v1","parts":[["text","Real "],["math","\\(n\\)"],["text","-space with the Euclidean norm"]],"source":"Real $n$-space with the Euclidean norm"},"result":true,"route":"../roadmap/Analysis/function-spaces/radial-sobolev-embedding/index.html#rn","scope":"Analysis/function-spaces/radial-sobolev-embedding","status":"fully_proved","summary":"For a natural number $n$, $\\mathbb{R}^n$ denotes the $n$-dimensional real Euclidean space: real $n$-tuples with the Euclidean inner product and norm.","title":"Real $n$-space with the Euclidean norm"},{"id":"Analysis/function-spaces/radial-sobolev-embedding/closed-unit-ball","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RadialSobolevExteriorBallWanted.ClosedUnitBall","parent":18799,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed unit ball of "],["math","\\(\\mathbb{R}^n\\)"]],"source":"Closed unit ball of $\\mathbb{R}^n$"},"result":true,"route":"../roadmap/Analysis/function-spaces/radial-sobolev-embedding/index.html#closed-unit-ball","scope":"Analysis/function-spaces/radial-sobolev-embedding","status":"fully_proved","summary":"For a natural number $n$, the closed unit ball of $\\mathbb{R}^n$ is the set $\\{x \\in \\mathbb{R}^n : \\|x\\| \\le 1\\}$, where $\\|\\cdot\\|$ is the Euclidean norm.","title":"Closed unit ball of $\\mathbb{R}^n$"},{"id":"Analysis/function-spaces/radial-sobolev-embedding/is-radial","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RadialSobolevExteriorBallWanted.IsRadial","parent":18799,"reading_title":{"format":"inline-parts-v1","parts":[["text","Radial functions on "],["math","\\(\\mathbb{R}^n\\)"]],"source":"Radial functions on $\\mathbb{R}^n$"},"result":true,"route":"../roadmap/Analysis/function-spaces/radial-sobolev-embedding/index.html#is-radial","scope":"Analysis/function-spaces/radial-sobolev-embedding","status":"fully_proved","summary":"A function $f \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is radial if $f(x) = f(y)$ for all $x, y \\in \\mathbb{R}^n$ with $\\|x\\| = \\|y\\|$, where $\\|\\cdot\\|$ is the Euclidean norm.","title":"Radial functions on $\\mathbb{R}^n$"},{"id":"Analysis/function-spaces/radial-sobolev-embedding/exists-e-lp-norm-le-sobolev-of-radial-of","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.RadialSobolevExteriorBallWanted.exists_eLpNorm_le_sobolev_of_radial_of_tsupport_subset_compl_closedBall","parent":18799,"result":true,"route":"../roadmap/Analysis/function-spaces/radial-sobolev-embedding/index.html#exists-e-lp-norm-le-sobolev-of-radial-of","scope":"Analysis/function-spaces/radial-sobolev-embedding","status":"can_prove","summary":"Let $n \\ge 2$, let $B$ be the closed unit ball of $\\mathbb{R}^n$, and let $1 < p < 2$ and $p \\le q \\le \\infty$. Then there is a constant $C > 0$ such that every radial function $f \\colon \\mathbb{R}^n \\to \\mathbb{R}$ that has compact support, whose closed support is contained in $\\mathbb{R}^n \\setminus B$, and which is $C^\\infty$ on $\\mathbb{R}^n \\setminus B$, satisfies","title":"Radial Sobolev embedding outside the unit ball"},{"data_url":"scopes/Analysis/function-spaces/rellich-kondrachov-theorem.json","id":"Analysis/function-spaces/rellich-kondrachov-theorem","kind":"scope","navigation":"module","parent":18788,"result":false,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","title":"Rellich\u2013Kondrachov theorem"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/rellich-weight","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.rellichWeight","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#rellich-weight","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The Rellich weight `1 + n ^ 2`.","title":"Rellich weight"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/mem-rellich-ball","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.memRellichBall","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#mem-rellich-ball","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"Membership in the Rellich ball: every finite weighted partial sum is bounded by `M ^ 2`.","title":"Membership in the Rellich ball via partial sums"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/rellich-ball","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.rellichBall","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#rellich-ball","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The Rellich ball of radius `M`: sequences in `\u2113\u00b2(\u2115, \ud835\udd5c)` whose weighted `\u2113\u00b2` energy is at most `M ^ 2`.","title":"Rellich ball of weighted square-summable sequences"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/norm-sq-eq-tsum","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.norm_sq_eq_tsum","parent":18804,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squared "],["math","\\(\\ell^2\\)"],["text"," norm as a sum of squared coordinates"]],"source":"Squared $\\ell^2$ norm as a sum of squared coordinates"},"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#norm-sq-eq-tsum","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The `\u2113\u00b2` norm squared is the sum of the squared coordinate norms.","title":"Squared $\\ell^2$ norm as a sum of squared coordinates"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/mem-rellich-ball-of-support","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.memRellichBall_of_support","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#mem-rellich-ball-of-support","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"If `x` is supported in a finite set `S` and its weighted sum over `S` is at most `M ^ 2`, then `x` lies in the Rellich ball.","title":"Finitely supported sequences in the Rellich ball"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/norm-sub-lp-trunc-sq-eq","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.norm_sub_lpTrunc_sq_eq","parent":18804,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tail norm identity for "],["math","\\(\\ell^2\\)"],["text"," truncation"]],"source":"Tail norm identity for $\\ell^2$ truncation"},"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#norm-sub-lp-trunc-sq-eq","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The tail norm identity: the distance from `x` to its truncation is the `\u2113\u00b2` norm of the tail.","title":"Tail norm identity for $\\ell^2$ truncation"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/norm-sub-lp-trunc-sq-le","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.norm_sub_lpTrunc_sq_le","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#norm-sub-lp-trunc-sq-le","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The tail estimate: the squared distance from `x` to its truncation is at most `M ^ 2 / (1 + N ^ 2)`.","title":"Tail estimate for the Rellich ball"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/lp-trunc-mem-span","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.lpTrunc_mem_span","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#lp-trunc-mem-span","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"Canonical truncations land in the span of the first `N` basis vectors.","title":"Truncations lie in the span of the first basis vectors"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/rellich-ball-abs","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.rellichBall_abs","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#rellich-ball-abs","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The Rellich ball only depends on `|M|`, since the bound is on `M ^ 2`.","title":"Rellich ball depends only on the absolute radius"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/totally-bounded-rellich-ball","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.totallyBounded_rellichBall","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#totally-bounded-rellich-ball","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The Rellich ball is totally bounded in `\u2113\u00b2(\u2115, \ud835\udd5c)`.","title":"Rellich\u2013Kondrachov, total boundedness form"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/is-closed-rellich-ball","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.isClosed_rellichBall","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#is-closed-rellich-ball","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The Rellich ball is closed.","title":"Rellich\u2013Kondrachov, closedness form"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/is-compact-rellich-ball","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.isCompact_rellichBall","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#is-compact-rellich-ball","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"The Rellich ball is compact in `\u2113\u00b2(\u2115, \ud835\udd5c)`.","title":"Rellich\u2013Kondrachov, compactness form"},{"id":"Analysis/function-spaces/rellich-kondrachov-theorem/exists-subseq-tendsto-of-mem-rellich-bal","kind":"node","lean":"MathlibExt.Analysis.Normed.Lp.RellichKondrachov.exists_subseq_tendsto_of_memRellichBall","parent":18804,"result":true,"route":"../roadmap/Analysis/function-spaces/rellich-kondrachov-theorem/index.html#exists-subseq-tendsto-of-mem-rellich-bal","scope":"Analysis/function-spaces/rellich-kondrachov-theorem","status":"fully_proved","summary":"Every sequence in the Rellich ball has a subsequence converging in `\u2113\u00b2(\u2115, \ud835\udd5c)` to a point of the ball.","title":"Rellich\u2013Kondrachov, sequential form"},{"data_url":"scopes/Analysis/function-spaces/sobolev-embedding-theorem.json","id":"Analysis/function-spaces/sobolev-embedding-theorem","kind":"scope","navigation":"module","parent":18788,"result":false,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"can_prove","title":"Sobolev embedding theorem"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/dot-hseminorm","kind":"node","lean":"MathlibExt.Analysis.Fourier.HomogeneousSobolevHolderEmbeddingWanted.dotHSeminorm","parent":18818,"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#dot-hseminorm","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"fully_proved","summary":"For a real number $s$ and a Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ with Fourier transform $\\hat f(\\xi) = \\int_{\\mathbb{R}} e^{-2\\pi i x \\xi} f(x) \\, dx$, the homogeneous Sobolev seminorm of $f$ is","title":"Homogeneous Sobolev seminorm for the H\u00f6lder embedding"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/exists-holder-seminorm-le-mul-dot-hsemin","kind":"node","lean":"MathlibExt.Analysis.Fourier.HomogeneousSobolevHolderEmbeddingWanted.exists_holderSeminorm_le_mul_dotHSeminorm","parent":18818,"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#exists-holder-seminorm-le-mul-dot-hsemin","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"can_prove","summary":"Let $s \\in (\\tfrac12, 1]$, and for a Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ with Fourier transform $\\hat f$ let $\\|f\\|_{\\dot H^s} = \\left( \\int_{\\mathbb{R}} |\\xi|^{2s} |\\hat f(\\xi)|^2 \\, d\\xi \\right)^{1/2}$ be its homogeneous Sobolev seminorm. There is a constant $C_3 > 0$ such that for every Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$,","title":"The homogeneous Sobolev seminorm controls the H\u00f6lder seminorm"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/e","kind":"node","lean":"MathlibExt.Analysis.SobolevMorreyWanted.Analysis.SobolevMorreyProblem.E","parent":18818,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euclidean space "],["math","\\(\\mathbb{R}^n\\)"]],"source":"Euclidean space $\\mathbb{R}^n$"},"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#e","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"fully_proved","summary":"For a natural number $n$, $\\mathbb{R}^n$ denotes the real Euclidean space of dimension $n$: real vectors with coordinates indexed by $0, \\dots, n-1$, with the Euclidean inner product and norm.","title":"Euclidean space $\\mathbb{R}^n$"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/partial-deriv","kind":"node","lean":"MathlibExt.Analysis.SobolevMorreyWanted.Analysis.SobolevMorreyProblem.partialDeriv","parent":18818,"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#partial-deriv","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"fully_proved","summary":"For a coordinate index $i \\in \\{0, \\dots, n-1\\}$ and a function $\\varphi : \\mathbb{R}^n \\to \\mathbb{R}$, the partial derivative $\\partial_i \\varphi$ is the function $\\mathbb{R}^n \\to \\mathbb{R}$ sending $x$ to $D\\varphi(x)(e_i)$, the Fr\u00e9chet derivative of $\\varphi$ at $x$ applied to the $i$-th standard basis vector $e_i$. At points where $\\varphi$ is not differentiable its value is $0$.","title":"Partial derivative"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/mixed-deriv","kind":"node","lean":"MathlibExt.Analysis.SobolevMorreyWanted.Analysis.SobolevMorreyProblem.mixedDeriv","parent":18818,"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#mixed-deriv","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"fully_proved","summary":"For a multi-index $m = (m_0, \\dots, m_{n-1})$ of natural numbers and a function $\\varphi : \\mathbb{R}^n \\to \\mathbb{R}$, the mixed partial derivative is $D^m \\varphi = \\partial_0^{m_0} \\partial_1^{m_1} \\cdots \\partial_{n-1}^{m_{n-1}} \\varphi$, where $\\partial_i^{j}$ is the $j$-fold iterate of the partial derivative $\\partial_i$; thus $\\partial_{n-1}$ is applied first and $\\partial_0$ last.","title":"Mixed partial derivative"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/is-weak-deriv","kind":"node","lean":"MathlibExt.Analysis.SobolevMorreyWanted.Analysis.SobolevMorreyProblem.IsWeakDeriv","parent":18818,"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#is-weak-deriv","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"fully_proved","summary":"Let $f, g : \\mathbb{R}^n \\to \\mathbb{R}$ and let $m = (m_0, \\dots, m_{n-1})$ be a multi-index of order $|m| = m_0 + \\cdots + m_{n-1}$. Then $g$ is a weak derivative $D^m f$ of $f$ when every smooth ($C^\\infty$) compactly supported function $\\varphi : \\mathbb{R}^n \\to \\mathbb{R}$ satisfies","title":"Weak derivative"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/mem-sobolev-wk","kind":"node","lean":"MathlibExt.Analysis.SobolevMorreyWanted.Analysis.SobolevMorreyProblem.MemSobolevWk","parent":18818,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sobolev space "],["math","\\(W^{k,p}(\\mathbb{R}^n)\\)"]],"source":"Sobolev space $W^{k,p}(\\mathbb{R}^n)$"},"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#mem-sobolev-wk","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"fully_proved","summary":"For a natural number $k$ and $p \\in [0, \\infty]$, a function $f : \\mathbb{R}^n \\to \\mathbb{R}$ belongs to $W^{k,p}(\\mathbb{R}^n)$ when $f$ is locally integrable and, for every multi-index $m = (m_0, \\dots, m_{n-1})$ of natural numbers with $m_0 + \\cdots + m_{n-1} \\le k$, $f$ has a weak derivative $D^m f$ that lies in $L^p(\\mathbb{R}^n)$. Local integrability and $L^p$ are with respect to Lebesgue measure.","title":"Sobolev space $W^{k,p}(\\mathbb{R}^n)$"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/mem-holder","kind":"node","lean":"MathlibExt.Analysis.SobolevMorreyWanted.Analysis.SobolevMorreyProblem.MemHolder","parent":18818,"reading_title":{"format":"inline-parts-v1","parts":[["text","H\u00f6lder space "],["math","\\(C^{r,\\alpha}(\\mathbb{R}^n)\\)"]],"source":"H\u00f6lder space $C^{r,\\alpha}(\\mathbb{R}^n)$"},"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#mem-holder","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"fully_proved","summary":"For a natural number $r$ and a real number $\\alpha$, a function $g : \\mathbb{R}^n \\to \\mathbb{R}$ belongs to $C^{r,\\alpha}(\\mathbb{R}^n)$ when $g$ is $r$ times continuously differentiable, its iterated derivatives $D^j g$ for $j \\le r$ are bounded on $\\mathbb{R}^n$, and its $r$-th derivative is $\\alpha$-H\u00f6lder: there is $C \\ge 0$ with $\\|D^r g(x) - D^r g(y)\\| \\le C \\|x - y\\|^{\\alpha}$ for all\u2026","title":"H\u00f6lder space $C^{r,\\alpha}(\\mathbb{R}^n)$"},{"id":"Analysis/function-spaces/sobolev-embedding-theorem/sobolev-embedding","kind":"node","lean":"MathlibExt.Analysis.SobolevMorreyWanted.Analysis.SobolevMorreyProblem.sobolev_embedding","parent":18818,"result":true,"route":"../roadmap/Analysis/function-spaces/sobolev-embedding-theorem/index.html#sobolev-embedding","scope":"Analysis/function-spaces/sobolev-embedding-theorem","status":"can_prove","summary":"Let $n, k, r$ be natural numbers and $\\alpha, p$ real numbers with $n < p$, $0 < \\alpha \\le 1$ and $r + \\alpha < k - n/p$. Then every function $f : \\mathbb{R}^n \\to \\mathbb{R}$ in the Sobolev space $W^{k,p}(\\mathbb{R}^n)$ is equal almost everywhere, for Lebesgue measure, to some function $g$ in the H\u00f6lder space $C^{r,\\alpha}(\\mathbb{R}^n)$.","title":"Sobolev embedding theorem (Morrey regime)"},{"data_url":"scopes/Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval.json","id":"Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval","kind":"scope","navigation":"module","parent":18788,"result":false,"route":"../roadmap/Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval/index.html","scope":"Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval","status":"can_prove","title":"Sup-norm bounds by derivatives on an interval"},{"id":"Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval/exists-const-s-sup-norm-le-mul-s-sup-nor","kind":"node","lean":"MathlibExt.Analysis.Calculus.SupNormEpsDerivPlusIntegralWanted.exists_const_sSup_norm_le_mul_sSup_norm_deriv_add_mul_integral","parent":18828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sup norm on "],["math","\\((0,1)\\)"],["text"," bounded by the derivative and the "],["math","\\(L^1\\)"],["text"," norm"]],"source":"Sup norm on $(0,1)$ bounded by the derivative and the $L^1$ norm"},"result":true,"route":"../roadmap/Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval/index.html#exists-const-s-sup-norm-le-mul-s-sup-nor","scope":"Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval","status":"can_prove","summary":"For every $\\varepsilon > 0$ there is a constant $C_\\varepsilon > 0$ such that every function $f : \\mathbb{R} \\to \\mathbb{R}$ that is continuously differentiable on $[0,1]$ satisfies","title":"Sup norm on $(0,1)$ bounded by the derivative and the $L^1$ norm"},{"id":"Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval/s-sup-norm-sq-le-half-integral-sq-add-de","kind":"node","lean":"MathlibExt.Analysis.Calculus.SupNormSqLeHalfH1NormWanted.sSup_norm_sq_le_half_integral_sq_add_deriv_sq","parent":18828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sup-norm bound by the "],["math","\\(H^1\\)"],["text"," energy for functions vanishing at both ends"]],"source":"Sup-norm bound by the $H^1$ energy for functions vanishing at both ends"},"result":true,"route":"../roadmap/Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval/index.html#s-sup-norm-sq-le-half-integral-sq-add-de","scope":"Analysis/function-spaces/sup-norm-bounds-by-derivatives-on-an-interval","status":"can_prove","summary":"Let $\\alpha \\le \\beta$ be real numbers and let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuously differentiable on $[\\alpha, \\beta]$ with $f(\\alpha) = f(\\beta) = 0$. Then","title":"Sup-norm bound by the $H^1$ energy for functions vanishing at both ends"},{"data_url":"scopes/Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions.json","id":"Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions","kind":"scope","navigation":"module","parent":18788,"result":false,"route":"../roadmap/Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions/index.html","scope":"Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions","status":"can_prove","title":"Sup-norm bounds for high-frequency functions"},{"id":"Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions/dot-hseminorm","kind":"node","lean":"MathlibExt.Analysis.Fourier.HighFrequencySupNormSobolevBoundWanted.dotHSeminorm","parent":18831,"result":true,"route":"../roadmap/Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions/index.html#dot-hseminorm","scope":"Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions","status":"fully_proved","summary":"For a real number $s$ and a Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ with Fourier transform $\\hat f(\\xi) = \\int_{\\mathbb{R}} e^{-2\\pi i x \\xi} f(x) \\, dx$, the homogeneous Sobolev seminorm of $f$ is","title":"Homogeneous Sobolev seminorm for the high-frequency sup-norm bound"},{"id":"Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions/s-sup-norm-le-rpow-mul-dot-hseminorm-of","kind":"node","lean":"MathlibExt.Analysis.Fourier.HighFrequencySupNormSobolevBoundWanted.sSup_norm_le_rpow_mul_dotHSeminorm_of_tsupport_fourier_compl_Icc","parent":18831,"result":true,"route":"../roadmap/Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions/index.html#s-sup-norm-le-rpow-mul-dot-hseminorm-of","scope":"Analysis/function-spaces/sup-norm-bounds-for-high-frequency-functions","status":"can_prove","summary":"Let $s \\in (\\tfrac12, 1]$, and for a Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ with Fourier transform $\\hat f$ let $\\|f\\|_{\\dot H^s} = \\left( \\int_{\\mathbb{R}} |\\xi|^{2s} |\\hat f(\\xi)|^2 \\, d\\xi \\right)^{1/2}$ be its homogeneous Sobolev seminorm. There is a constant $C_2 > 0$ such that for every real $A > 0$ and every Schwartz function $f : \\mathbb{R} \\to \\mathbb{C}$ whose Fourier transform has closed\u2026","title":"Sup bound for Schwartz functions with high-frequency Fourier support"},{"data_url":"scopes/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu.json","id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","kind":"scope","navigation":"module","parent":18788,"result":false,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"can_prove","title":"Weighted Hardy\u2013Sobolev inequalities on an annulus"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/r2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedHardyInequalityWanted.R2","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#r2","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"The Euclidean plane.","title":"Euclidean plane"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/b","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedHardyInequalityWanted.B","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#b","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"For a real number $r$, $B_r = \\{x \\in \\mathbb{R}^2 : |x| < r\\}$ is the open disc of radius $r$ about the origin of the Euclidean plane $\\mathbb{R}^2$, where $|x|$ is the Euclidean norm.","title":"Open discs about the origin for the annulus Hardy inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/omega1","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedHardyInequalityWanted.Omega1","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#omega1","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"$\\Omega_1 = B_3 \\setminus \\overline{B}_1$, where $B_r$ is the open disc of radius $r$ about the origin of the Euclidean plane $\\mathbb{R}^2$ and $\\overline{B}_1$ is the closure of $B_1$; that is, writing $|x|$ for the Euclidean norm, $\\Omega_1$ is the annulus $\\{x \\in \\mathbb{R}^2 : 1 < |x| < 3\\}$.","title":"Full annulus of the weighted Hardy inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/omega2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedHardyInequalityWanted.Omega2","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#omega2","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"$\\Omega_2 = B_3 \\setminus \\overline{B}_2$, where $B_r$ is the open disc of radius $r$ about the origin of the Euclidean plane $\\mathbb{R}^2$ and $\\overline{B}_2$ is the closure of $B_2$; that is, writing $|x|$ for the Euclidean norm, $\\Omega_2$ is the annulus $\\{x \\in \\mathbb{R}^2 : 2 < |x| < 3\\}$.","title":"Outer annulus of the weighted Hardy inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/phi","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedHardyInequalityWanted.phi","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#phi","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"For $x$ in the Euclidean plane $\\mathbb{R}^2$, $\\phi(x) = \\max\\{|x| - 1, 0\\}$, where $|x|$ is the Euclidean norm; this is the distance from $x$ to the closed unit disc.","title":"Weight of the annulus Hardy inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/exists-e-lp-norm-le-weighted-grad-add-ou","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedHardyInequalityWanted.exists_eLpNorm_le_weighted_grad_add_outer_annulus","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#exists-e-lp-norm-le-weighted-grad-add-ou","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"can_prove","summary":"For $r > 0$ let $B_r$ be the open disc of radius $r$ about the origin of $\\mathbb{R}^2$, let $|x|$ be the Euclidean norm, and let $\\phi(x) = \\max\\{|x| - 1, 0\\}$ be the distance from $x$ to the closed unit disc. Let $\\Omega_1 = B_3 \\setminus \\overline{B}_1$ and $\\Omega_2 = B_3 \\setminus \\overline{B}_2$, with Lebesgue measure. For every real number $p \\ge 1$ there is a constant $C > 0$ such that every function\u2026","title":"Weighted Hardy inequality on the annulus"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/r2-2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedSobolevCriticalWanted.R2","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#r2-2","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"$\\mathbb{R}^2$ denotes the Euclidean plane, the space of pairs of real numbers with the Euclidean norm.","title":"The plane of the critical weighted Sobolev inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/b-2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedSobolevCriticalWanted.B","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#b-2","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"For a real number $r$, $B_r$ is the open disc $\\{x \\in \\mathbb{R}^2 : |x| < r\\}$ of radius $r$ about the origin of the Euclidean plane $\\mathbb{R}^2$.","title":"Discs about the origin for the critical weighted Sobolev inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/omega1-2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedSobolevCriticalWanted.Omega1","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#omega1-2","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"Let $B_r$ denote the open disc of radius $r$ about the origin of the Euclidean plane $\\mathbb{R}^2$. The annulus $\\Omega_1$ is $B_3 \\setminus \\overline{B}_1$, the set of points of $B_3$ outside the closure of $B_1$, that is, $\\Omega_1 = \\{x \\in \\mathbb{R}^2 : 1 < |x| < 3\\}$.","title":"The annulus of the critical weighted Sobolev inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/omega2-2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedSobolevCriticalWanted.Omega2","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#omega2-2","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"Let $B_r$ denote the open disc of radius $r$ about the origin of the Euclidean plane $\\mathbb{R}^2$. The outer annulus $\\Omega_2$ is $B_3 \\setminus \\overline{B}_2$, the set of points of $B_3$ outside the closure of $B_2$, that is, $\\Omega_2 = \\{x \\in \\mathbb{R}^2 : 2 < |x| < 3\\}$.","title":"The outer annulus of the critical weighted Sobolev inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/phi-2","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedSobolevCriticalWanted.phi","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#phi-2","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"For $x$ in the Euclidean plane $\\mathbb{R}^2$, the weight is $\\phi(x) = \\max\\{|x| - 1, 0\\}$, the distance from $x$ to the closed unit disc.","title":"The weight of the critical weighted Sobolev inequality"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/p-star","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedSobolevCriticalWanted.pStar","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#p-star","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"fully_proved","summary":"For a real number $p$, the Sobolev conjugate exponent is $p^* = 2p/(2 - p)$ (at $p = 2$, where the denominator vanishes, this is $0$ by the convention that division by zero gives zero).","title":"Sobolev conjugate exponent in dimension two"},{"id":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/exists-e-lp-norm-phi-mul-le-weighted-gra","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.AnnulusWeightedSobolevCriticalWanted.exists_eLpNorm_phi_mul_le_weighted_grad_add_outer_annulus","parent":18834,"result":true,"route":"../roadmap/Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu/index.html#exists-e-lp-norm-phi-mul-le-weighted-gra","scope":"Analysis/function-spaces/weighted-hardy-sobolev-inequalities-on-an-annulu","status":"can_prove","summary":"For $r > 0$ let $B_r$ be the open disc of radius $r$ about the origin in the Euclidean plane $\\mathbb{R}^2$, let $\\Omega_1 = B_3 \\setminus \\overline{B}_1$ and $\\Omega_2 = B_3 \\setminus \\overline{B}_2$, and let $\\phi(x) = \\max\\{|x| - 1, 0\\}$, the distance from $x$ to the closed unit disc. Let $1 \\le p < 2$ and $p^* = 2p/(2-p)$. Then there is a real number $C > 0$ such that every function\u2026","title":"Critical weighted Sobolev inequality on an annulus"},{"data_url":"scopes/Analysis/distributions.json","id":"Analysis/distributions","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/distributions/index.html","scope":"Analysis/distributions","status":"can_prove","summary":"Test functions, Schwartz functions, tempered distributions and distributional derivatives: principal values, Plemelj limits, Dirac comb, partitions of unity.","title":"Distributions"},{"id":"Analysis/distributions/cauchy-principal-value-exists","kind":"node","lean":"MathlibExt.Analysis.Distribution.CauchyPrincipalValueWanted.cauchyPrincipalValue_exists","parent":18848,"reading_title":{"format":"inline-parts-v1","parts":[["text","Existence of the Cauchy principal value of "],["math","\\(\\varphi(x)/x\\)"]],"source":"Existence of the Cauchy principal value of $\\varphi(x)/x$"},"result":true,"route":"../roadmap/Analysis/distributions/index.html#cauchy-principal-value-exists","scope":"Analysis/distributions","status":"can_prove","summary":"For every real-valued Schwartz function $\\varphi$ on $\\mathbb{R}$ there is a real number $L$ such that $\\int_{\\{x \\in \\mathbb{R} \\,:\\, |x| \\ge \\varepsilon\\}} \\varphi(x)/x \\, dx \\to L$ as $\\varepsilon \\to 0^+$, the integrals being taken with respect to Lebesgue measure.","title":"Existence of the Cauchy principal value of $\\varphi(x)/x$"},{"id":"Analysis/distributions/test-function-partition-of-unity","kind":"node","lean":"MathlibExt.Analysis.Distribution.TestFunctionPartitionOfUnityWanted.testFunction_partitionOfUnity","parent":18848,"result":true,"route":"../roadmap/Analysis/distributions/index.html#test-function-partition-of-unity","scope":"Analysis/distributions","status":"can_prove","summary":"Let $E$ be a finite-dimensional real normed space, $K \\subseteq E$ a compact set, and $(U_i)_{i \\in \\iota}$ a finite family of open subsets of $E$ with $K \\subseteq \\bigcup_i U_i$. Then there are smooth compactly supported functions $\\varphi_i : E \\to \\mathbb{R}$, one for each $i \\in \\iota$, such that $\\varphi_i(x) \\ge 0$ for all $i$ and all $x \\in E$, the closed support of each $\\varphi_i$ is contained in $U_i$,\u2026","title":"Smooth partitions of unity subordinate to a finite open cover"},{"id":"Analysis/distributions/dirac-comb-to-tempered-distribution","kind":"node","lean":"MathlibExt.Analysis.Distribution.TemperedExamplesWanted.diracComb_toTemperedDistribution","parent":18848,"result":true,"route":"../roadmap/Analysis/distributions/index.html#dirac-comb-to-tempered-distribution","scope":"Analysis/distributions","status":"can_prove","summary":"There is a tempered distribution $T$ on $\\mathbb{R}$, that is, a continuous $\\mathbb{C}$-linear functional on the space of complex-valued Schwartz functions on $\\mathbb{R}$, such that $T(g)=\\sum_{n\\in\\mathbb{Z}} g(n)$ for every complex-valued Schwartz function $g$ on $\\mathbb{R}$.","title":"The Dirac comb is a tempered distribution"},{"data_url":"scopes/Analysis/distributions/distributional-derivatives.json","id":"Analysis/distributions/distributional-derivatives","kind":"scope","navigation":"module","parent":18848,"result":false,"route":"../roadmap/Analysis/distributions/distributional-derivatives/index.html","scope":"Analysis/distributions/distributional-derivatives","status":"can_prove","title":"Distributional derivatives"},{"id":"Analysis/distributions/distributional-derivatives/exists-fderiv-distribution","kind":"node","lean":"MathlibExt.Analysis.Distribution.DistributionFderivWanted.exists_fderiv_distribution","parent":18852,"result":true,"route":"../roadmap/Analysis/distributions/distributional-derivatives/index.html#exists-fderiv-distribution","scope":"Analysis/distributions/distributional-derivatives","status":"can_prove","summary":"Let $E$ be a real normed space, $\\Omega \\subseteq E$ an open set and $F$ a real topological vector space, and let $\\mathcal{D}'(\\Omega, F)$ be the space of $F$-valued distributions on $\\Omega$ (continuous linear maps from the real-valued test functions on $\\Omega$ to $F$), with the topology of uniform convergence on compact sets of test functions. For $v \\in E$ and $T \\in \\mathcal{D}'(\\Omega, F)$, let\u2026","title":"The full derivative of a distribution"},{"id":"Analysis/distributions/distributional-derivatives/ae-eq-quadratic-of-integral-mul-iterated","kind":"node","lean":"MathlibExt.Analysis.Distribution.WeakThirdDerivativeZeroQuadraticWanted.ae_eq_quadratic_of_integral_mul_iteratedDeriv_three_eq_zero","parent":18852,"reading_title":{"format":"inline-parts-v1","parts":[["text","A function with vanishing weak third derivative on "],["math","\\((0,1)\\)"],["text"," is a.e. a polynomial of degree at most "],["math","\\(2\\)"],["text"," on "],["math","\\((0,1)\\)"]],"source":"A function with vanishing weak third derivative on $(0,1)$ is a.e. a polynomial of degree at most $2$ on $(0,1)$"},"result":true,"route":"../roadmap/Analysis/distributions/distributional-derivatives/index.html#ae-eq-quadratic-of-integral-mul-iterated","scope":"Analysis/distributions/distributional-derivatives","status":"can_prove","summary":"A function with vanishing weak third derivative on $(0,1)$ is a.e. a polynomial of degree at most $2$ on $[0,1]$. Let $f:\\mathbb{R}\\to\\mathbb{R}$ be integrable on $[0,1]$, and suppose that $\\int_{0}^{1} f(x)\\,g'''(x)\\,dx=0$ for every smooth function $g:\\mathbb{R}\\to\\mathbb{R}$ whose support is contained in some closed interval $[s,t] \\subseteq (0,1)$. Then there are real numbers $a$, $b$, $c$ such that\u2026","title":"A function with vanishing weak third derivative on $(0,1)$ is a.e. a polynomial of degree at most $2$ on $(0,1)$"},{"data_url":"scopes/Analysis/distributions/schwartz-space.json","id":"Analysis/distributions/schwartz-space","kind":"scope","navigation":"module","parent":18848,"result":false,"route":"../roadmap/Analysis/distributions/schwartz-space/index.html","scope":"Analysis/distributions/schwartz-space","status":"can_prove","title":"Schwartz space"},{"id":"Analysis/distributions/schwartz-space/gaussian-mem-schwartz","kind":"node","lean":"MathlibExt.Analysis.Distribution.GaussianSchwartzWanted.gaussian_mem_schwartz","parent":18855,"result":true,"route":"../roadmap/Analysis/distributions/schwartz-space/index.html#gaussian-mem-schwartz","scope":"Analysis/distributions/schwartz-space","status":"can_prove","summary":"Let $E$ be a finite-dimensional real inner product space. Then there is a real-valued Schwartz function $g$ on $E$ such that $g(x) = e^{-\\|x\\|^2}$ for every $x \\in E$.","title":"The Gaussian is a Schwartz function"},{"id":"Analysis/distributions/schwartz-space/exists-eq-smul-of-comp-mul-eq-of-comp-de","kind":"node","lean":"MathlibExt.Analysis.Distribution.SchwartzCommutantMulDerivWanted.exists_eq_smul_of_comp_mul_eq_of_comp_deriv_eq","parent":18855,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear maps on Schwartz space commuting with "],["math","\\(x\\)"],["text"," and "],["math","\\(d/dx\\)"],["text"," are scalars"]],"source":"Linear maps on Schwartz space commuting with $x$ and $d/dx$ are scalars"},"result":true,"route":"../roadmap/Analysis/distributions/schwartz-space/index.html#exists-eq-smul-of-comp-mul-eq-of-comp-de","scope":"Analysis/distributions/schwartz-space","status":"can_prove","summary":"Let $\\mathcal{S}(\\mathbb{R})$ be the space of complex-valued Schwartz functions on $\\mathbb{R}$, and let $M:\\mathcal{S}(\\mathbb{R})\\to\\mathcal{S}(\\mathbb{R})$ be a $\\mathbb{C}$-linear map with $(Mf)(x)=x f(x)$ for all $f\\in\\mathcal{S}(\\mathbb{R})$ and $x\\in\\mathbb{R}$. If $T:\\mathcal{S}(\\mathbb{R})\\to\\mathcal{S}(\\mathbb{R})$ is a $\\mathbb{C}$-linear map (not assumed continuous) with $TM=MT$ and\u2026","title":"Linear maps on Schwartz space commuting with $x$ and $d/dx$ are scalars"},{"data_url":"scopes/Analysis/distributions/sokhotski-plemelj-formula.json","id":"Analysis/distributions/sokhotski-plemelj-formula","kind":"scope","navigation":"module","parent":18848,"result":false,"route":"../roadmap/Analysis/distributions/sokhotski-plemelj-formula/index.html","scope":"Analysis/distributions/sokhotski-plemelj-formula","status":"can_prove","title":"Sokhotski\u2013Plemelj formula"},{"id":"Analysis/distributions/sokhotski-plemelj-formula/u-plus-eps","kind":"node","lean":"MathlibExt.Analysis.Distribution.PlemeljLimitTemperedDistributionWanted.uPlusEps","parent":18858,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pairing with "],["math","\\((x + i\\varepsilon)^{-1}\\)"]],"source":"The pairing with $(x + i\\varepsilon)^{-1}$"},"result":true,"route":"../roadmap/Analysis/distributions/sokhotski-plemelj-formula/index.html#u-plus-eps","scope":"Analysis/distributions/sokhotski-plemelj-formula","status":"fully_proved","summary":"For a real number $\\varepsilon$ and a Schwartz function $\\phi : \\mathbb{R} \\to \\mathbb{C}$, $u_{+,\\varepsilon}(\\phi)$ is the complex number $\\int_{\\mathbb{R}} (x + i\\varepsilon)^{-1}\\phi(x)\\, dx$.","title":"The pairing with $(x + i\\varepsilon)^{-1}$"},{"id":"Analysis/distributions/sokhotski-plemelj-formula/u-minus-eps","kind":"node","lean":"MathlibExt.Analysis.Distribution.PlemeljLimitTemperedDistributionWanted.uMinusEps","parent":18858,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pairing with "],["math","\\((x - i\\varepsilon)^{-1}\\)"]],"source":"The pairing with $(x - i\\varepsilon)^{-1}$"},"result":true,"route":"../roadmap/Analysis/distributions/sokhotski-plemelj-formula/index.html#u-minus-eps","scope":"Analysis/distributions/sokhotski-plemelj-formula","status":"fully_proved","summary":"For a real number $\\varepsilon$ and a Schwartz function $\\phi : \\mathbb{R} \\to \\mathbb{C}$, $u_{-,\\varepsilon}(\\phi)$ is the complex number $\\int_{\\mathbb{R}} (x - i\\varepsilon)^{-1}\\phi(x)\\, dx$.","title":"The pairing with $(x - i\\varepsilon)^{-1}$"},{"id":"Analysis/distributions/sokhotski-plemelj-formula/exists-tempered-distribution-tendsto-inv","kind":"node","lean":"MathlibExt.Analysis.Distribution.PlemeljLimitTemperedDistributionWanted.exists_temperedDistribution_tendsto_inv_add_mul_I","parent":18858,"reading_title":{"format":"inline-parts-v1","parts":[["text","The tempered distributions "],["math","\\((x \\pm i\\varepsilon)^{-1}\\)"],["text"," converge as "],["math","\\(\\varepsilon \\to 0^+\\)"]],"source":"The tempered distributions $(x \\pm i\\varepsilon)^{-1}$ converge as $\\varepsilon \\to 0^+$"},"result":true,"route":"../roadmap/Analysis/distributions/sokhotski-plemelj-formula/index.html#exists-tempered-distribution-tendsto-inv","scope":"Analysis/distributions/sokhotski-plemelj-formula","status":"can_prove","summary":"Write $S(\\mathbb{R})$ for the space of complex-valued Schwartz functions on $\\mathbb{R}$, and for $\\varepsilon > 0$ and $\\phi \\in S(\\mathbb{R})$ let $u_{\\pm,\\varepsilon}(\\phi) = \\int_{\\mathbb{R}} (x \\pm i\\varepsilon)^{-1}\\phi(x)\\, dx$. Then for every $\\varepsilon > 0$ both $u_{+,\\varepsilon}$ and $u_{-,\\varepsilon}$ are tempered distributions, that is, continuous $\\mathbb{C}$-linear functionals on $S(\\mathbb{R})$,\u2026","title":"The tempered distributions $(x \\pm i\\varepsilon)^{-1}$ converge as $\\varepsilon \\to 0^+$"},{"data_url":"scopes/Analysis/ordinary-differential-equations.json","id":"Analysis/ordinary-differential-equations","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html","scope":"Analysis/ordinary-differential-equations","status":"can_prove","summary":"Existence and uniqueness (Cauchy\u2013Peano, Picard, maximal solutions), linear systems and fundamental matrices, stability and Lyapunov's indirect method, Sturm\u2013Liouville and boundary value problems with their Green's operators and eigenfunctions, oscillation and comparison theorems.","title":"Ordinary differential equations"},{"id":"Analysis/ordinary-differential-equations/cauchy-peano-exists","kind":"node","lean":"MathlibExt.Analysis.ODE.CauchyPeanoWanted.cauchy_peano_exists","parent":18862,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html#cauchy-peano-exists","scope":"Analysis/ordinary-differential-equations","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space and let $f : \\mathbb{R} \\times E \\to E$ be continuous. Then for every $t_0 \\in \\mathbb{R}$ and $x_0 \\in E$ there are $\\varepsilon > 0$ and a function $x : \\mathbb{R} \\to E$ with $x(t_0) = x_0$ such that, for every $t \\in (t_0 - \\varepsilon, t_0 + \\varepsilon)$, $x$ is differentiable at $t$ with $x'(t) = f(t, x(t))$.","title":"Cauchy\u2013Peano existence theorem"},{"id":"Analysis/ordinary-differential-equations/bvp-comparison","kind":"node","lean":"MathlibExt.Analysis.BVPComparisonWanted.Analysis.ODE.bvp_comparison","parent":18862,"reading_title":{"format":"inline-parts-v1","parts":[["text","Comparison principle for the Dirichlet problem on "],["math","\\((0, 1)\\)"]],"source":"Comparison principle for the Dirichlet problem on $(0, 1)$"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html#bvp-comparison","scope":"Analysis/ordinary-differential-equations","status":"can_prove","summary":"Let $J \\subseteq \\mathbb{R}$ be an open set containing $[0, 1]$, and let $u, v : \\mathbb{R} \\to \\mathbb{R}$ be functions that are differentiable at every point of $J$ and whose derivatives $u'$ and $v'$ are differentiable at every point of $J$. If $-u''(x) \\le -v''(x)$ for every $x \\in (0, 1)$, $u(0) \\le v(0)$ and $u(1) \\le v(1)$, then $u(x) \\le v(x)$ for every $x \\in [0, 1]$.","title":"Comparison principle for the Dirichlet problem on $(0, 1)$"},{"id":"Analysis/ordinary-differential-equations/cubic-decay-asymptotic","kind":"node","lean":"MathlibExt.Analysis.CubicDecayWanted.Analysis.ODE.cubic_decay_asymptotic","parent":18862,"reading_title":{"format":"inline-parts-v1","parts":[["text","Decay rate of solutions of "],["math","\\(y' = -y^3\\)"]],"source":"Decay rate of solutions of $y' = -y^3$"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html#cubic-decay-asymptotic","scope":"Analysis/ordinary-differential-equations","status":"can_prove","summary":"Let $y : \\mathbb{R} \\to \\mathbb{R}$ be differentiable at every $t > 0$ with $y'(t) = -y(t)^3$, continuous at $0$ within $[0, \\infty)$, and such that $y(0) = 1$. Then $y(t) \\sqrt{t} \\to 1/\\sqrt{2}$ as $t \\to \\infty$.","title":"Decay rate of solutions of $y' = -y^3$"},{"id":"Analysis/ordinary-differential-equations/dirichlet-eigenvalues-eq-nat-sq","kind":"node","lean":"MathlibExt.Analysis.DirichletEigenvaluesWanted.Analysis.ODE.dirichlet_eigenvalues_eq_nat_sq","parent":18862,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dirichlet eigenvalues of "],["math","\\(-y'' = \\lambda y\\)"],["text"," on "],["math","\\((0, \\pi)\\)"]],"source":"Dirichlet eigenvalues of $-y'' = \\lambda y$ on $(0, \\pi)$"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html#dirichlet-eigenvalues-eq-nat-sq","scope":"Analysis/ordinary-differential-equations","status":"can_prove","summary":"For every real number $\\lambda$, the following are equivalent: (i) there are a function $y : \\mathbb{R} \\to \\mathbb{R}$ and an open set $J \\subseteq \\mathbb{R}$ containing $[0, \\pi]$ such that $y$ is differentiable at every point of $J$, its derivative $y'$ is differentiable at every $x \\in J$ with $y''(x) = -\\lambda y(x)$, $y(0) = y(\\pi) = 0$, and $y(x) \\ne 0$ for some $x \\in (0, \\pi)$; (ii) $\\lambda = n^2$ for\u2026","title":"Dirichlet eigenvalues of $-y'' = \\lambda y$ on $(0, \\pi)$"},{"id":"Analysis/ordinary-differential-equations/duhamel-variation-of-constants","kind":"node","lean":"MathlibExt.Analysis.ODE.DuhamelWanted.duhamel_variation_of_constants","parent":18862,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html#duhamel-variation-of-constants","scope":"Analysis/ordinary-differential-equations","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $A : E \\to E$ be a bounded linear operator, let $b : \\mathbb{R} \\to E$ be continuous, and let $t_0 \\in \\mathbb{R}$ and $x_0 \\in E$. Write $e^{sA}$ for the operator exponential and define $x : \\mathbb{R} \\to E$ by $x(t) = e^{(t - t_0)A} x_0 + \\int_{t_0}^{t} e^{(t - s)A} b(s)\\, ds$. Then $x(t_0) = x_0$ and $x$ is differentiable with $x'(t) = A x(t) + b(t)$ for every\u2026","title":"Duhamel's formula (variation of constants)"},{"id":"Analysis/ordinary-differential-equations/exists-unique-periodic-solution-and-is-b","kind":"node","lean":"MathlibExt.Analysis.Calculus.NonresonantForcedOscillatorPeriodicWanted.existsUnique_periodic_solution_and_isBounded_of_ne_natCast","parent":18862,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html#exists-unique-periodic-solution-and-is-b","scope":"Analysis/ordinary-differential-equations","status":"can_prove","summary":"Let $\\omega > 0$ be a real number that is not an integer, and let $g : \\mathbb{R} \\to \\mathbb{R}$ be continuous and $2\\pi$-periodic. Then the equation $u'' + \\omega^2 u = g$ has exactly one twice continuously differentiable solution $u : \\mathbb{R} \\to \\mathbb{R}$ that is $2\\pi$-periodic, and every twice continuously differentiable solution $u : \\mathbb{R} \\to \\mathbb{R}$ of this equation is bounded.","title":"Nonresonant forced oscillator"},{"id":"Analysis/ordinary-differential-equations/set-infinite-zero-of-mul-iterated-deriv","kind":"node","lean":"MathlibExt.Analysis.Calculus.BesselTypeOdeInfinitelyManyZerosWanted.setInfinite_zero_of_mul_iteratedDeriv_two_add_two_deriv_add_eq_zero","parent":18862,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(xu''+2u'+u=0\\)"],["text"," have infinitely many zeros"]],"source":"Solutions of $xu''+2u'+u=0$ have infinitely many zeros"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html#set-infinite-zero-of-mul-iterated-deriv","scope":"Analysis/ordinary-differential-equations","status":"can_prove","summary":"Let $u : \\mathbb{R} \\to \\mathbb{R}$ be twice continuously differentiable on $(0,\\infty)$ and satisfy","title":"Solutions of $xu''+2u'+u=0$ have infinitely many zeros"},{"id":"Analysis/ordinary-differential-equations/sturm-separation","kind":"node","lean":"MathlibExt.Analysis.SturmSeparationWanted.Analysis.ODE.sturm_separation","parent":18862,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/index.html#sturm-separation","scope":"Analysis/ordinary-differential-equations","status":"can_prove","summary":"Let $p, q, y_1, y_2 : \\mathbb{R} \\to \\mathbb{R}$ and $a < b$, and let $J \\subseteq \\mathbb{R}$ be an open connected set (an open interval, possibly unbounded) containing $[a, b]$. Suppose that $p$ and $q$ are continuous on $J$, and that $y_1$ and $y_2$ are twice differentiable at every point of $J$ and solve $y'' + p(x)\\, y' + q(x)\\, y = 0$ at every $x \\in J$. Suppose further that the Wronskian\u2026","title":"Sturm separation theorem"},{"data_url":"scopes/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0.json","id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","kind":"scope","navigation":"module","parent":18862,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","title":"Green's operator of the Dirichlet Laplacian on [0, 1]"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/l2-unit-interval","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.L2UnitInterval","parent":18871,"reading_title":{"format":"inline-parts-v1","parts":[["text","The space "],["math","\\(L^2((0, 1))\\)"]],"source":"The space $L^2((0, 1))$"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#l2-unit-interval","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"$L^2([0, 1])$ is the real Hilbert space of square-integrable real-valued functions on the unit interval, up to equality almost everywhere, with Lebesgue measure restricted to $[0, 1]$.","title":"The space $L^2((0, 1))$"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/sine-basis-fun","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.sineBasisFun","parent":18871,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sine basis functions on "],["math","\\((0, 1)\\)"]],"source":"Sine basis functions on $(0, 1)$"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#sine-basis-fun","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"For a positive integer $k$, the $k$-th sine basis function is $u_k : \\mathbb{R} \\to \\mathbb{R}$, $u_k(x) = \\sqrt{2} \\sin(k\\pi x)$.","title":"Sine basis functions on $(0, 1)$"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-eigenvalue","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.greensEigenvalue","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-eigenvalue","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"For a positive integer $k$, the $k$-th eigenvalue of the Green's operator of $-\\frac{d^2}{dx^2}$ on $[0, 1]$ with Dirichlet boundary conditions $u(0) = u(1) = 0$ is $\\lambda_k = \\frac{1}{k^2 \\pi^2}$.","title":"Eigenvalues of the Dirichlet Green's operator"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-eigenvalue-sqrt","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.greensEigenvalueSqrt","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-eigenvalue-sqrt","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"For a positive integer $k$, set $\\mu_k = \\frac{1}{k\\pi}$. This is the square root of the $k$-th eigenvalue $\\frac{1}{k^2\\pi^2}$ of the Green's operator of $-\\frac{d^2}{dx^2}$ on $[0, 1]$ with Dirichlet boundary conditions.","title":"Square roots of the Green's operator eigenvalues"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-operator-sqrt","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.greensOperatorSqrt","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-operator-sqrt","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Let $L^2([0, 1])$ be the real Hilbert space of square-integrable functions on $[0, 1]$, and let $(b_k)_{k \\ge 1}$ be a Hilbert basis of $L^2([0, 1])$ indexed by the positive integers with $b_k(x) = \\sqrt{2} \\sin(k\\pi x)$ for almost every $x \\in [0, 1]$. Define $A^{1/2} : L^2([0, 1]) \\to L^2([0, 1])$ by","title":"Square root of the Dirichlet Green's operator"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-kernel","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.greensKernel","parent":18871,"reading_title":{"format":"inline-parts-v1","parts":[["text","Green's kernel of the Dirichlet Laplacian on "],["math","\\((0, 1)\\)"]],"source":"Green's kernel of the Dirichlet Laplacian on $(0, 1)$"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-kernel","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"For real numbers $x$ and $t$, set $G(x, t) = \\min(x, t)\\,(1 - \\max(x, t))$.","title":"Green's kernel of the Dirichlet Laplacian on $(0, 1)$"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-kernel-integral-sine-eq","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.greens_kernel_integral_sine_eq","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-kernel-integral-sine-eq","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Let $k$ be a positive integer, let $G(x, t) = \\min(x, t)\\,(1 - \\max(x, t))$ be the Green's kernel, and let $u_k(x) = \\sqrt{2} \\sin(k\\pi x)$ and $\\lambda_k = \\frac{1}{k^2\\pi^2}$. Then for every $x \\in [0, 1]$,","title":"Green's kernel eigenfunction identity for sines"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-operator-eigendecomposition","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.greens_operator_eigendecomposition","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-operator-eigendecomposition","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Write $L^2([0, 1])$ for the real Hilbert space of square-integrable functions on $[0, 1]$ with Lebesgue measure, let $G(x, t) = \\min(x, t)\\,(1 - \\max(x, t))$ be the Green's kernel, and for each positive integer $k$ let $u_k(x) = \\sqrt{2} \\sin(k\\pi x)$ and $\\lambda_k = \\frac{1}{k^2\\pi^2}$. Let $A$ be a linear map from $L^2([0, 1])$ to itself that is symmetric, $\\langle Af, g \\rangle = \\langle f, Ag \\rangle$ for all\u2026","title":"Spectral decomposition of the Dirichlet Green's operator"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/sine-product-integral","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.sine_product_integral","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#sine-product-integral","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The canonical sine basis is orthonormal on the unit interval, in product-integral form.","title":"Orthonormality of the sine basis on the unit interval"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/sine-hilbert-basis","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.sineHilbertBasis","parent":18871,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sine Hilbert basis of "],["math","\\(L^2((0, 1))\\)"]],"source":"The sine Hilbert basis of $L^2((0, 1))$"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#sine-hilbert-basis","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The sine Hilbert basis is a Hilbert basis $(b_k)_{k \\ge 1}$ of the real Hilbert space $L^2([0, 1])$, indexed by the positive integers, with $b_k(x) = \\sqrt{2} \\sin(k\\pi x)$ for almost every $x \\in [0, 1]$. Such a basis exists, and since an element of $L^2([0, 1])$ is determined by its values almost everywhere, it is unique.","title":"The sine Hilbert basis of $L^2((0, 1))$"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/sine-hilbert-basis-coe","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperator.sineHilbertBasis_coe","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#sine-hilbert-basis-coe","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"For every positive integer $k$, the $k$-th element $b_k$ of the sine Hilbert basis of $L^2([0, 1])$ satisfies $b_k(x) = \\sqrt{2} \\sin(k\\pi x)$ for almost every $x \\in [0, 1]$.","title":"The sine Hilbert basis consists of sine functions"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/dirichlet-laplacian-eigenvalue","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.dirichletLaplacianEigenvalue","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#dirichlet-laplacian-eigenvalue","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The `k`-th Dirichlet eigenvalue: `\u03bb_k = (k\u03c0)\u00b2`, the eigenvalue of `-d\u00b2/dx\u00b2` for the eigenfunction `sineBasisFun k`.","title":"Dirichlet eigenvalues of the Laplacian on the unit interval"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/dirichlet-sine-has-deriv-at","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.dirichletSine_hasDerivAt","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#dirichlet-sine-has-deriv-at","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"First derivative of the `k`-th sine eigenfunction.","title":"Pointwise derivative of a Dirichlet sine eigenfunction"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/dirichlet-sine-deriv","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.dirichletSine_deriv","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#dirichlet-sine-deriv","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The derivative of the `k`-th sine eigenfunction, as a function.","title":"Derivative function of a Dirichlet sine eigenfunction"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/dirichlet-sine-iterated-deriv-two","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.dirichletSine_iteratedDeriv_two","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#dirichlet-sine-iterated-deriv-two","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Second-derivative (eigenvalue) equation for the `k`-th sine eigenfunction: `u_k'' = -(k\u03c0)\u00b2 u_k`.","title":"Second derivative of a Dirichlet sine eigenfunction"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/neg-iterated-deriv-two-eq","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.neg_iteratedDeriv_two_eq","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#neg-iterated-deriv-two-eq","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The Dirichlet eigenvalue equation in Sturm-Liouville form: `-u_k'' = \u03bb_k u_k` with `\u03bb_k = (k\u03c0)\u00b2`.","title":"Dirichlet eigenvalue equation in Sturm-Liouville form"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/dirichlet-sine-zero","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.dirichletSine_zero","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#dirichlet-sine-zero","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Left Dirichlet boundary value: `u_k(0) = 0`.","title":"Sine eigenfunctions vanish at the left endpoint"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/dirichlet-sine-one","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.dirichletSine_one","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#dirichlet-sine-one","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Right Dirichlet boundary value: `u_k(1) = 0`.","title":"Sine eigenfunctions vanish at the right endpoint"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/dirichlet-sine-cont-diff","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.dirichletSine_contDiff","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#dirichlet-sine-cont-diff","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The sine eigenfunctions are smooth.","title":"Smoothness of the Dirichlet sine eigenfunctions"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/dirichlet-sine-ne-zero","kind":"node","lean":"MathlibExt.Analysis.ODE.DirichletSineEigenfunctions.dirichletSine_ne_zero","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#dirichlet-sine-ne-zero","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The sine eigenfunctions are nonzero (so they are genuine eigenfunctions, not the trivial solution). The witness point is `x = 1 / (2k)`, where `u_k = \u221a2`.","title":"Dirichlet sine eigenfunctions are nonzero"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-operator-sqrt-eq-diagonal","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorSqrtSquared.greensOperatorSqrt_eq_diagonal","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-operator-sqrt-eq-diagonal","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Let $(b_k)_{k \\ge 1}$ be a Hilbert basis of $L^2([0, 1])$ indexed by the positive integers with $b_k(x) = \\sqrt{2} \\sin(k\\pi x)$ for almost every $x \\in [0, 1]$, let $\\mu_k = \\frac{1}{k\\pi}$, and let $C$ be a real number with $|\\mu_k| \\le C$ for every $k$. Then for every $f \\in L^2([0, 1])$, the square root $A^{1/2} f$ of the Green's operator defined from $(b_k)$ equals $D f$, where $D$ is the bounded diagonal\u2026","title":"The square root of the Green's operator is a diagonal operator"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-operator-sqrt-sq","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorSqrtSquared.greensOperatorSqrt_sq","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-operator-sqrt-sq","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Let $G(x, t) = \\min(x, t)\\,(1 - \\max(x, t))$ be the Green's kernel and let $A$ be a linear map from $L^2([0, 1])$ to itself that is symmetric, $\\langle Af, g \\rangle = \\langle f, Ag \\rangle$ for all $f, g$, and is given by the Green's kernel, $(Af)(x) = \\int_{[0, 1]} G(x, t) f(t)\\, dt$ for almost every $x \\in [0, 1]$ and every $f$. Let $(b_k)_{k \\ge 1}$ be a Hilbert basis of $L^2([0, 1])$ indexed by the positive\u2026","title":"The square root of the Green's operator squares to the Green's operator"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/summable-greens-eigenvalues","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorTrace.summable_greensEigenvalues","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#summable-greens-eigenvalues","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The Green's eigenvalues $\\lambda_k = 1/(k^2\\pi^2)$, indexed by the positive integers $k$, form a summable family of real numbers.","title":"The Green's eigenvalues are summable"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/tendsto-cofinite-greens-eigenvalues","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorTrace.tendsto_cofinite_greensEigenvalues","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#tendsto-cofinite-greens-eigenvalues","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The Green's eigenvalues $\\lambda_k = 1/(k^2\\pi^2)$ tend to $0$ along the cofinite filter on the positive integers, that is, $\\lambda_k \\to 0$ as $k \\to \\infty$.","title":"The Green's eigenvalues tend to zero"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/tsum-greens-eigenvalues","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorTrace.tsum_greensEigenvalues","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#tsum-greens-eigenvalues","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"For the Green's eigenvalues $\\lambda_k = 1/(k^2\\pi^2)$, indexed by the positive integers $k$, we have $\\sum_{k \\ge 1} \\lambda_k = 1/6$.","title":"Sum of the Green's eigenvalues"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/tsum-greens-eigenvalues-sq","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorTrace.tsum_greensEigenvalues_sq","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#tsum-greens-eigenvalues-sq","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"For the Green's eigenvalues $\\lambda_k = 1/(k^2\\pi^2)$, indexed by the positive integers $k$, we have $\\sum_{k \\ge 1} \\lambda_k^2 = 1/90$.","title":"Sum of the squared Green's eigenvalues"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/integral-greens-kernel-diag","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorTrace.integral_greensKernel_diag","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#integral-greens-kernel-diag","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The Green's kernel $G$ on $[0, 1]$ satisfies $\\int_0^1 G(x, x)\\,dx = 1/6$, since $G(x, x) = x(1 - x)$.","title":"Diagonal integral of the Green's kernel"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/greens-eigenvalue-sum-eq-kernel-diagonal","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorTrace.greens_eigenvalue_sum_eq_kernel_diagonal_integral","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#greens-eigenvalue-sum-eq-kernel-diagonal","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"The sum of the Green's eigenvalues $\\lambda_k = 1/(k^2\\pi^2)$ over the positive integers $k$ equals the integral of the Green's kernel $G$ along the diagonal: $\\sum_{k \\ge 1} \\lambda_k = \\int_0^1 G(x, x)\\,dx$.","title":"The Green's eigenvalues sum to the diagonal integral of the kernel"},{"id":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/hs-norm-sq-greens-operator","kind":"node","lean":"MathlibExt.Analysis.ODE.GreensOperatorTrace.hsNormSq_greensOperator","parent":18871,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0/index.html#hs-norm-sq-greens-operator","scope":"Analysis/ordinary-differential-equations/green-s-operator-of-the-dirichlet-laplacian-on-0","status":"fully_proved","summary":"Let $G(x, t) = \\min(x, t)\\,(1 - \\max(x, t))$ be the Green's kernel and let $A$ be a continuous linear operator on $L^2([0, 1])$ that is symmetric, $\\langle Af, g \\rangle = \\langle f, Ag \\rangle$ for all $f, g$, and is given by the Green's kernel, $(Af)(x) = \\int_{[0, 1]} G(x, t) f(t)\\, dt$ for almost every $x \\in [0, 1]$ and every $f$. Let $(b_k)_{k \\ge 1}$ be the sine Hilbert basis of $L^2([0, 1])$. Then the\u2026","title":"Squared Hilbert\u2013Schmidt norm of the Green's operator"},{"data_url":"scopes/Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem.json","id":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem","kind":"scope","navigation":"module","parent":18862,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/index.html","scope":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem","status":"can_prove","title":"Hyperbolic motions in the N-body problem"},{"id":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/r","kind":"node","lean":"MathlibExt.Analysis.ViscositySolutionsWanted.ViscositySolutions.r","parent":18901,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/index.html#r","scope":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem","status":"fully_proved","summary":"For a configuration $x = (x_1, \\dots, x_N) \\in E^N$ of $N$ points in a finite-dimensional real inner product space $E$ and indices $i, j$, the mutual distance of the $i$-th and $j$-th points is $r_{ij}(x) = \\lVert x_i - x_j \\rVert$.","title":"Mutual distances"},{"id":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/without-collisions","kind":"node","lean":"MathlibExt.Analysis.ViscositySolutionsWanted.ViscositySolutions.WithoutCollisions","parent":18901,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/index.html#without-collisions","scope":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem","status":"fully_proved","summary":"A configuration $x = (x_1, \\dots, x_N) \\in E^N$ of $N$ points in a finite-dimensional real inner product space $E$ is without collisions if $r_{ij}(x) = \\lVert x_i - x_j \\rVert \\neq 0$ for all $i \\neq j$.","title":"Configurations without collisions"},{"id":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/is-nbody-solution","kind":"node","lean":"MathlibExt.Analysis.ViscositySolutionsWanted.ViscositySolutions.IsNBodySolution","parent":18901,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of the "],["math","\\(N\\)"],["text","-body problem"]],"source":"Solutions of the $N$-body problem"},"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/index.html#is-nbody-solution","scope":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space, let $N$ be a natural number, give $E^N$ the inner product $\\langle u, v \\rangle = \\sum_i \\langle u_i, v_i \\rangle$, and let $m = (m_1, \\dots, m_N) \\in \\mathbb{R}^N$ be masses. Let $U(y) = \\sum_{i < j} m_i m_j \\lVert y_i - y_j \\rVert^{-1}$ be the Newtonian potential on $E^N$, and for $f : E^N \\to \\mathbb{R}$ let $\\nabla_m f(y) \\in E^N$ be the mass-weighted\u2026","title":"Solutions of the $N$-body problem"},{"id":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/theorem-1-1","kind":"node","lean":"MathlibExt.Analysis.ViscositySolutionsWanted.ViscositySolutions.theorem_1_1","parent":18901,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem/index.html#theorem-1-1","scope":"Analysis/ordinary-differential-equations/hyperbolic-motions-in-the-n-body-problem","status":"can_prove","summary":"Let $E$ be a finite-dimensional real inner product space of dimension at least $2$, let $N$ be a natural number, and let $m = (m_1, \\dots, m_N)$ be positive masses. Let $x_0 \\in E^N$, let $a \\in E^N$ be a configuration without collisions with $\\lVert a \\rVert_m = 1$, where $\\lVert a \\rVert_m = \\bigl(\\sum_i m_i \\lVert a_i \\rVert^2\\bigr)^{1/2}$, and let $h > 0$. Then there are a solution $x : \\mathbb{R} \\to E^N$ of\u2026","title":"Maderna\u2013Venturelli theorem on hyperbolic motions"},{"data_url":"scopes/Analysis/ordinary-differential-equations/lyapunov-s-indirect-method.json","id":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method","kind":"scope","navigation":"module","parent":18862,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/index.html","scope":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method","status":"can_prove","title":"Lyapunov's indirect method"},{"id":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/exists-norm-exp-smul-mul-vec-le","kind":"node","lean":"Matrix.IsHurwitz.exists_norm_exp_smul_mulVec_le","parent":18906,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/index.html#exists-norm-exp-smul-mul-vec-le","scope":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method","status":"fully_proved","summary":"If `A` is Hurwitz, there are `M, \u03b1 > 0` with `\u2016exp(t A) v\u2016 \u2264 M e^{-\u03b1 t} \u2016v\u2016` for all `t \u2265 0` and all vectors `v`.","title":"Exponential decay of a Hurwitz semigroup"},{"id":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/global-solution-decay-of-has-fderiv-at","kind":"node","lean":"MathlibExt.Analysis.ODE.LinearizedStability.globalSolution_decay_of_hasFDerivAt","parent":18906,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/index.html#global-solution-decay-of-has-fderiv-at","scope":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method","status":"fully_proved","summary":"Let `f (x\u22c6) = 0` and let `f` be differentiable at `x\u22c6` with Jacobian matrix `A`. If `A` is Hurwitz, then for some `\u03b4, C, \u03b1 > 0`, every solution of `x' = f(x)` on `[0, \u221e)` with `\u2016x(0) - x\u22c6\u2016 < \u03b4` satisfies `\u2016x(t) - x\u22c6\u2016 \u2264 C e^{-\u03b1 t} \u2016x(0) - x\u22c6\u2016` for all `t \u2265 0`. This is conditional on the solution being defined on all of `[0, \u221e)`; it does not assert solution existence, so it is not by itself local exponential\u2026","title":"Lyapunov's indirect method (linearized stability), conditional decay form"},{"id":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/linear-ode-asymptotic-stability","kind":"node","lean":"MathlibExt.Analysis.LinearStabilityWanted.Analysis.ODE.linear_ode_asymptotic_stability","parent":18906,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/index.html#linear-ode-asymptotic-stability","scope":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method","status":"can_prove","summary":"Let $n \\ge 0$ and let $A$ be a real $n \\times n$ matrix such that every complex eigenvalue $\\mu$ of $A$ (an eigenvalue of $A$ viewed as a complex matrix acting on $\\mathbb{C}^n$) satisfies $\\operatorname{Re} \\mu < 0$. Let $x : \\mathbb{R} \\to \\mathbb{R}^n$ be a function that, at every $t > 0$, is differentiable with derivative $x'(t) = A\\,x(t)$. Then $\\|x(t)\\| \\to 0$ as $t \\to \\infty$, where $\\|\\cdot\\|$ is the sup\u2026","title":"Asymptotic stability of linear ODEs with eigenvalues in the open left half-plane"},{"id":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/exists-unstable-of-has-fderiv-at","kind":"node","lean":"MathlibExt.Analysis.ODE.LinearizedInstabilityWanted.exists_unstable_of_hasFDerivAt","parent":18906,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/lyapunov-s-indirect-method/index.html#exists-unstable-of-has-fderiv-at","scope":"Analysis/ordinary-differential-equations/lyapunov-s-indirect-method","status":"can_prove","summary":"Lyapunov's indirect method, instability (Khalil 2002, Theorem 4.7). If `f` is `C\u00b9`, `f x\u22c6 = 0`, and the Jacobian matrix `A` of `f` at `x\u22c6` has an eigenvalue with positive real part, then `x\u22c6` is unstable: for some `\u03b5 > 0`, every neighbourhood of `x\u22c6` contains an initial point `x\u2080` from which a solution segment of `x' = f(x)` on some finite interval `[0, T]` reaches distance `\u2265 \u03b5` from `x\u22c6`. Quantifying over finite\u2026","title":"Lyapunov's indirect method, instability"},{"data_url":"scopes/Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence.json","id":"Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence","kind":"scope","navigation":"module","parent":18862,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence/index.html","scope":"Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence","status":"fully_proved","title":"Maximal solutions and global existence"},{"id":"Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence/has-deriv-at-piecewise-union-of-eq-on","kind":"node","lean":"hasDerivAt_piecewise_union_of_eqOn","parent":18911,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence/index.html#has-deriv-at-piecewise-union-of-eq-on","scope":"Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence","status":"fully_proved","summary":"Let $E$ be a real normed space, let $v : \\mathbb{R} \\times E \\to E$, let $I, J \\subseteq \\mathbb{R}$ be open, and let $x, y : \\mathbb{R} \\to E$ satisfy $x'(t) = v(t, x(t))$ for every $t \\in I$ and $y'(t) = v(t, y(t))$ for every $t \\in J$, with $x = y$ on $I \\cap J$. Then there is $z : \\mathbb{R} \\to E$ with $z = x$ on $I$, $z = y$ on $J$ and $z'(t) = v(t, z(t))$ for every $t \\in I \\cup J$.","title":"Gluing solutions of an ODE"},{"id":"Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence/ode-global-exists-of-global-lipschitz","kind":"node","lean":"MathlibExt.Analysis.ODE.MaximalSolutionWanted.ode_global_exists_of_global_lipschitz","parent":18911,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence/index.html#ode-global-exists-of-global-lipschitz","scope":"Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $f : \\mathbb{R} \\times E \\to E$ be continuous, and let $L \\ge 0$ be such that $f(t, \\cdot)$ is $L$-Lipschitz for every $t \\in \\mathbb{R}$. Then for every $t_0 \\in \\mathbb{R}$ and $x_0 \\in E$ there is exactly one function $x : \\mathbb{R} \\to E$ with $x(t_0) = x_0$ and $x'(t) = f(t, x(t))$ for every $t \\in \\mathbb{R}$.","title":"Global existence and uniqueness for globally Lipschitz ODEs"},{"id":"Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence/ode-maximal-solution-exit-compact","kind":"node","lean":"MathlibExt.Analysis.ODE.MaximalSolutionWanted.ode_maximal_solution_exit_compact","parent":18911,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence/index.html#ode-maximal-solution-exit-compact","scope":"Analysis/ordinary-differential-equations/maximal-solutions-and-global-existence","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $U \\subseteq \\mathbb{R} \\times E$ be open, let $(t_0, x_0) \\in U$, and let $f : \\mathbb{R} \\times E \\to E$ be continuous on $U$ and locally Lipschitz in the state variable, uniformly in time: every $p \\in U$ has a neighbourhood $N$ and a constant $C \\ge 0$ such that $\\|f(t, u) - f(t, w)\\| \\le C \\|u - w\\|$ whenever $(t, u)$ and $(t, w)$ lie in $N \\cap U$. Then there are an open\u2026","title":"Solutions of locally Lipschitz ODEs that leave every compact set"},{"data_url":"scopes/Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices.json","id":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices","kind":"scope","navigation":"module","parent":18862,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/index.html","scope":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices","status":"fully_proved","title":"Linear systems and fundamental matrices"},{"id":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/exists-unique-has-deriv-at-eq-continuous","kind":"node","lean":"ODE.existsUnique_hasDerivAt_eq_continuousLinearMap","parent":18915,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/index.html#exists-unique-has-deriv-at-eq-continuous","scope":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices","status":"fully_proved","summary":"Let $E$ be a real Banach space, write $L(E)$ for the space of continuous linear maps $E \\to E$ with the operator norm, let $B : \\mathbb{R} \\to L(E)$ be continuous, and let $t_0 \\in \\mathbb{R}$ and $x_0 \\in E$. Then there is exactly one function $x : \\mathbb{R} \\to E$ with $x(t_0) = x_0$ such that $x$ is differentiable at every $t \\in \\mathbb{R}$ with $x'(t) = B(t)\\,x(t)$.","title":"Global existence and uniqueness for homogeneous linear ODEs"},{"id":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/det-of-mul","kind":"node","lean":"HasDerivAt.det_of_mul","parent":18915,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/index.html#det-of-mul","scope":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $A$ be a real $n \\times n$ matrix, let $\\Phi : \\mathbb{R} \\to \\mathbb{R}^{n \\times n}$, and let $t \\in \\mathbb{R}$. If $\\Phi$ is differentiable at $t$ with $\\Phi'(t) = A \\Phi(t)$, then $s \\mapsto \\det \\Phi(s)$ is differentiable at $t$ with derivative $\\operatorname{tr}(A) \\det \\Phi(t)$.","title":"Jacobi's formula when the derivative is a left multiple of the matrix"},{"id":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/fundamental-matrix-linear-system","kind":"node","lean":"MathlibExt.Analysis.ODE.LinearSystemFundamentalMatrixWanted.fundamental_matrix_linear_system","parent":18915,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/index.html#fundamental-matrix-linear-system","scope":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $A : \\mathbb{R} \\to \\mathbb{R}^{n \\times n}$ be continuous, let $t_0 \\in \\mathbb{R}$, and write $I$ for the $n \\times n$ identity matrix. Then there is a function $\\Phi : \\mathbb{R} \\to \\mathbb{R}^{n \\times n}$ with $\\Phi(t_0) = I$, $\\Phi'(t) = A(t) \\Phi(t)$ and $\\det \\Phi(t) \\ne 0$ for every $t \\in \\mathbb{R}$, such that for every $x_0 \\in \\mathbb{R}^n$ the function $x(t) = \\Phi(t) x_0$\u2026","title":"Fundamental matrix of a linear system"},{"id":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/fundamental-matrix-liouville-det","kind":"node","lean":"MathlibExt.Analysis.ODE.LinearSystemFundamentalMatrixWanted.fundamental_matrix_liouville_det","parent":18915,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices/index.html#fundamental-matrix-liouville-det","scope":"Analysis/ordinary-differential-equations/linear-systems-and-fundamental-matrices","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $A : \\mathbb{R} \\to \\mathbb{R}^{n \\times n}$ be continuous, let $t_0 \\in \\mathbb{R}$, write $I$ for the $n \\times n$ identity matrix, and let $\\Phi : \\mathbb{R} \\to \\mathbb{R}^{n \\times n}$ satisfy $\\Phi(t_0) = I$ and $\\Phi'(t) = A(t) \\Phi(t)$ for every $t \\in \\mathbb{R}$. Then for every $t \\in \\mathbb{R}$,\u2026","title":"Liouville's formula for the principal matrix solution"},{"data_url":"scopes/Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices.json","id":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices","kind":"scope","navigation":"module","parent":18862,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/index.html","scope":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices","status":"fully_proved","title":"One-parameter groups of matrices"},{"id":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/is-one-parameter-group","kind":"node","lean":"OneParameterGroup.IsOneParameterGroup","parent":18920,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/index.html#is-one-parameter-group","scope":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices","status":"fully_proved","summary":"For $m \\in \\mathbb{N}$, a map $\\varphi \\colon \\mathbb{R} \\to M_m(\\mathbb{C})$ into the complex $m \\times m$ matrices is a one-parameter group when $\\varphi(0)$ is the identity matrix $I$, $\\varphi(s + t) = \\varphi(s)\\,\\varphi(t)$ for all real $s$ and $t$, and $\\varphi$ is differentiable.","title":"One-parameter group of complex matrices"},{"id":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/is-one-parameter-group-in","kind":"node","lean":"OneParameterGroup.IsOneParameterGroupIn","parent":18920,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/index.html#is-one-parameter-group-in","scope":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices","status":"fully_proved","summary":"Let $m \\in \\mathbb{N}$ and let $G$ be a subgroup of $\\mathrm{GL}_m(\\mathbb{C})$. A map $\\varphi \\colon \\mathbb{R} \\to \\mathrm{GL}_m(\\mathbb{C})$ is a one-parameter group in $G$ when the matrix-valued map $t \\mapsto \\varphi(t)$ is a one-parameter group and $\\varphi(t) \\in G$ for every real $t$.","title":"One-parameter group in a subgroup of the general linear group"},{"id":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/to-gl","kind":"node","lean":"OneParameterGroup.IsOneParameterGroup.toGL","parent":18920,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/index.html#to-gl","scope":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices","status":"fully_proved","summary":"Let $m \\in \\mathbb{N}$ and let $\\varphi \\colon \\mathbb{R} \\to M_m(\\mathbb{C})$ be a one-parameter group of complex $m \\times m$ matrices. For every real $t$ the matrix $\\varphi(t)$ is invertible with inverse $\\varphi(-t)$; this gives the element of $\\mathrm{GL}_m(\\mathbb{C})$ with matrix $\\varphi(t)$ and inverse $\\varphi(-t)$.","title":"The lift of a one-parameter group to the general linear group"},{"id":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/coe-to-gl","kind":"node","lean":"OneParameterGroup.IsOneParameterGroup.coe_toGL","parent":18920,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/index.html#coe-to-gl","scope":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices","status":"fully_proved","summary":"Let $m \\in \\mathbb{N}$ and let $\\varphi \\colon \\mathbb{R} \\to M_m(\\mathbb{C})$ be a one-parameter group of complex $m \\times m$ matrices. For every real $t$, the element of $\\mathrm{GL}_m(\\mathbb{C})$ assigned to $t$ by the lift of $\\varphi$ has underlying matrix $\\varphi(t)$.","title":"Underlying matrix of the lift of a one-parameter group"},{"id":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/one-param-subgroup-form","kind":"node","lean":"OneParameterGroup.one_param_subgroup_form","parent":18920,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/index.html#one-param-subgroup-form","scope":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices","status":"fully_proved","summary":"Let $m \\in \\mathbb{N}$. If $\\varphi \\colon \\mathbb{R} \\to M_m(\\mathbb{C})$ is a one-parameter group of complex $m \\times m$ matrices, then there is a unique matrix $A \\in M_m(\\mathbb{C})$ such that $\\varphi(t) = \\exp(tA)$ for every real $t$.","title":"One-parameter groups of matrices are exponentials"},{"id":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/is-one-parameter-group-exp-smul","kind":"node","lean":"OneParameterGroup.isOneParameterGroup_exp_smul","parent":18920,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices/index.html#is-one-parameter-group-exp-smul","scope":"Analysis/ordinary-differential-equations/one-parameter-groups-of-matrices","status":"fully_proved","summary":"For every $m \\in \\mathbb{N}$ and every complex $m \\times m$ matrix $A$, the map $t \\mapsto \\exp(tA)$ from $\\mathbb{R}$ to the complex $m \\times m$ matrices is a one-parameter group.","title":"Exponential curves are one-parameter groups"},{"data_url":"scopes/Analysis/ordinary-differential-equations/picard-iteration.json","id":"Analysis/ordinary-differential-equations/picard-iteration","kind":"scope","navigation":"module","parent":18862,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/picard-iteration/index.html","scope":"Analysis/ordinary-differential-equations/picard-iteration","status":"fully_proved","title":"Picard iteration"},{"id":"Analysis/ordinary-differential-equations/picard-iteration/picard-iterates-converge","kind":"node","lean":"MathlibExt.Analysis.ODE.PicardIterationWanted.picard_iterates_converge","parent":18927,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/picard-iteration/index.html#picard-iterates-converge","scope":"Analysis/ordinary-differential-equations/picard-iteration","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $f : \\mathbb{R} \\times E \\to E$ be continuous, and let $L \\ge 0$ be such that $f(t, \\cdot)$ is $L$-Lipschitz for every $t \\in \\mathbb{R}$. Fix $t_0 \\in \\mathbb{R}$ and $x_0 \\in E$, and let $(u_n)_{n \\ge 0}$ be maps $\\mathbb{R} \\to E$ with $u_0(t) = x_0$ and $u_{n+1}(t) = x_0 + \\int_{t_0}^{t} f(s, u_n(s))\\,ds$ for all $n$ and $t$. Then there is a map $x : \\mathbb{R} \\to E$ with\u2026","title":"Picard iterates converge"},{"id":"Analysis/ordinary-differential-equations/picard-iteration/picard-iterates-tendsto-of-has-deriv-at","kind":"node","lean":"MathlibExt.Analysis.ODE.PicardIterationWanted.picard_iterates_tendsto_of_hasDerivAt","parent":18927,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/picard-iteration/index.html#picard-iterates-tendsto-of-has-deriv-at","scope":"Analysis/ordinary-differential-equations/picard-iteration","status":"fully_proved","summary":"Let $E$ be a real Banach space, let $f : \\mathbb{R} \\times E \\to E$ be continuous, and let $L \\ge 0$ be such that $f(t, \\cdot)$ is $L$-Lipschitz for every $t \\in \\mathbb{R}$. Fix $t_0 \\in \\mathbb{R}$ and $x_0 \\in E$, and let $(u_n)_{n \\ge 0}$ be maps $\\mathbb{R} \\to E$ with $u_0(t) = x_0$ and $u_{n+1}(t) = x_0 + \\int_{t_0}^{t} f(s, u_n(s))\\,ds$ for all $n$ and $t$. If $y : \\mathbb{R} \\to E$ satisfies $y(t_0) = x_0$\u2026","title":"Picard iterates converge to every global solution of the initial value problem"},{"id":"Analysis/ordinary-differential-equations/picard-iteration/picard-iterates-step-estimate","kind":"node","lean":"MathlibExt.Analysis.ODE.PicardIterationWanted.picard_iterates_step_estimate","parent":18927,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/picard-iteration/index.html#picard-iterates-step-estimate","scope":"Analysis/ordinary-differential-equations/picard-iteration","status":"fully_proved","summary":"Let $E$ be a real normed space, let $f : \\mathbb{R} \\times E \\to E$ be continuous, and let $L \\ge 0$ be such that $f(t, \\cdot)$ is $L$-Lipschitz for every $t \\in \\mathbb{R}$. Fix $t_0 \\in \\mathbb{R}$ and $x_0 \\in E$, and let $(u_n)_{n \\ge 0}$ be maps $\\mathbb{R} \\to E$ with $u_0(t) = x_0$ and $u_{n+1}(t) = x_0 + \\int_{t_0}^{t} f(s, u_n(s))\\,ds$ for all $n$ and $t$. If $M$ is a real number with $\\|f(s, x_0)\\| \\le M$\u2026","title":"Step estimate for Picard iterates"},{"data_url":"scopes/Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem.json","id":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem","kind":"scope","navigation":"module","parent":18862,"result":false,"route":"../roadmap/Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/index.html","scope":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem","status":"can_prove","title":"Sturm\u2013Liouville Dirichlet problem"},{"id":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/sturm-liouville-existence","kind":"node","lean":"MathlibExt.Analysis.ODE.SturmLiouvilleExistence.sturm_liouville_existence","parent":18931,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/index.html#sturm-liouville-existence","scope":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem","status":"fully_proved","summary":"Let $V$ and $f$ be continuous real functions on $[0, 1]$ with $V \\ge 0$. Then there is a function $u \\colon \\mathbb{R} \\to \\mathbb{R}$ that is $C^2$ on $[0, 1]$, satisfies $-u''(x) + V(x)\\,u(x) = f(x)$ for every $x \\in [0, 1]$, with the second derivative taken within $[0, 1]$, and has $u(0) = u(1) = 0$.","title":"Existence for the Sturm\u2013Liouville Dirichlet problem on the closed unit interval"},{"id":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/sturm-liouville-uniqueness","kind":"node","lean":"MathlibExt.Analysis.ODE.SturmLiouvilleUniqueness.sturm_liouville_uniqueness","parent":18931,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/index.html#sturm-liouville-uniqueness","scope":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem","status":"fully_proved","summary":"Let $V$ and $f$ be continuous real functions on $[0, 1]$ with $V \\ge 0$, and let $u_1, u_2 \\colon \\mathbb{R} \\to \\mathbb{R}$ be $C^2$ on $[0, 1]$. Suppose that for $i = 1, 2$, $-u_i''(x) + V(x)\\,u_i(x) = f(x)$ for every $x \\in [0, 1]$, with the second derivative taken within $[0, 1]$, and $u_i(0) = u_i(1) = 0$. Then $u_1$ and $u_2$ agree on $[0, 1]$.","title":"Uniqueness for the Sturm\u2013Liouville Dirichlet problem on the closed unit interval"},{"id":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/dirichlet-1d-exists","kind":"node","lean":"MathlibExt.Analysis.Distribution.Dirichlet1DWanted.dirichlet_1d_exists","parent":18931,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/index.html#dirichlet-1d-exists","scope":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be continuous on $[0, 1]$. Then there is a twice continuously differentiable function $u : \\mathbb{R} \\to \\mathbb{R}$ with $u(0) = u(1) = 0$ and $u''(x) = f(x)$ for every $x \\in (0, 1)$.","title":"Existence for the one-dimensional Dirichlet problem"},{"id":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/dirichlet-1d-unique","kind":"node","lean":"MathlibExt.Analysis.Distribution.Dirichlet1DWanted.dirichlet_1d_unique","parent":18931,"result":true,"route":"../roadmap/Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem/index.html#dirichlet-1d-unique","scope":"Analysis/ordinary-differential-equations/sturm-liouville-dirichlet-problem","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be any function, and let $u_1, u_2 : \\mathbb{R} \\to \\mathbb{R}$ be twice continuously differentiable functions such that, for $i = 1, 2$, $u_i(0) = u_i(1) = 0$ and $u_i''(x) = f(x)$ for every $x \\in (0, 1)$. Then $u_1(x) = u_2(x)$ for every $x \\in [0, 1]$.","title":"Uniqueness for the one-dimensional Dirichlet problem"},{"data_url":"scopes/Analysis/partial-differential-equations.json","id":"Analysis/partial-differential-equations","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/partial-differential-equations/index.html","scope":"Analysis/partial-differential-equations","status":"can_prove","summary":"PDE existence, regularity and qualitative results: Cauchy\u2013Kovalevskaya, heat and Airy equations, Euler singularity formation, \u03c3\u2082 Hessian estimates, radial symmetry, viscosity solutions, Dirichlet and Neumann spectra of the Laplacian.","title":"Partial differential equations"},{"data_url":"scopes/Analysis/partial-differential-equations/airy-equation-and-kato-smoothing.json","id":"Analysis/partial-differential-equations/airy-equation-and-kato-smoothing","kind":"scope","navigation":"module","parent":18936,"result":false,"route":"../roadmap/Analysis/partial-differential-equations/airy-equation-and-kato-smoothing/index.html","scope":"Analysis/partial-differential-equations/airy-equation-and-kato-smoothing","status":"can_prove","title":"Airy equation and Kato smoothing"},{"id":"Analysis/partial-differential-equations/airy-equation-and-kato-smoothing/airy-solution","kind":"node","lean":"MathlibExt.Analysis.Fourier.AiryKatoSmoothingIdentityWanted.AirySolution","parent":18937,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/airy-equation-and-kato-smoothing/index.html#airy-solution","scope":"Analysis/partial-differential-equations/airy-equation-and-kato-smoothing","status":"fully_proved","summary":"For $u_0 : \\mathbb{R} \\to \\mathbb{C}$, let $\\widehat{u_0}(\\xi) = (2\\pi)^{-1/2} \\int_{\\mathbb{R}} u_0(y) e^{-iy\\xi}\\,dy$ be its unitary Fourier transform. The Airy solution with initial datum $u_0$ is the function $u : \\mathbb{R}^2 \\to \\mathbb{C}$ given by","title":"Fourier solution of the Airy equation"},{"id":"Analysis/partial-differential-equations/airy-equation-and-kato-smoothing/exists-pos-const-integral-norm-deriv-air","kind":"node","lean":"MathlibExt.Analysis.Fourier.AiryKatoSmoothingIdentityWanted.exists_pos_const_integral_norm_deriv_airySolution_sq_eq","parent":18937,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/airy-equation-and-kato-smoothing/index.html#exists-pos-const-integral-norm-deriv-air","scope":"Analysis/partial-differential-equations/airy-equation-and-kato-smoothing","status":"can_prove","summary":"For $u_0 : \\mathbb{R} \\to \\mathbb{C}$ let $\\widehat{u_0}(\\xi) = (2\\pi)^{-1/2} \\int_{\\mathbb{R}} u_0(y) e^{-iy\\xi}\\,dy$ be its unitary Fourier transform, and let $u(t, x) = (2\\pi)^{-1/2} \\int_{\\mathbb{R}} e^{i(t\\xi^3 + x\\xi)}\\, \\widehat{u_0}(\\xi)\\,d\\xi$ be the Airy solution with initial datum $u_0$. There is a constant $c_0 > 0$ such that for every smooth compactly supported $u_0 : \\mathbb{R} \\to \\mathbb{C}$ and\u2026","title":"Kato smoothing identity for the Airy equation"},{"data_url":"scopes/Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem.json","id":"Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem","kind":"scope","navigation":"module","parent":18936,"result":false,"route":"../roadmap/Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem/index.html","scope":"Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem","status":"can_prove","title":"Cauchy\u2013Kovalevskaya theorem"},{"id":"Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem/e","kind":"node","lean":"MathlibExt.Analysis.CauchyKovalevskayaWanted.Analysis.E","parent":18940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Model space "],["math","\\(\\mathbb{R}^d\\)"],["text"," of the Cauchy\u2013Kovalevskaya problem"]],"source":"Model space $\\mathbb{R}^d$ of the Cauchy\u2013Kovalevskaya problem"},"result":true,"route":"../roadmap/Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem/index.html#e","scope":"Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem","status":"fully_proved","summary":"For a natural number $d$, this is the real Euclidean space $\\mathbb{R}^d$ of dimension $d$, with the Euclidean inner product and norm.","title":"Model space $\\mathbb{R}^d$ of the Cauchy\u2013Kovalevskaya problem"},{"id":"Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem/cauchy-kovalevskaya","kind":"node","lean":"MathlibExt.Analysis.CauchyKovalevskayaWanted.Analysis.cauchy_kovalevskaya","parent":18940,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem/index.html#cauchy-kovalevskaya","scope":"Analysis/partial-differential-equations/cauchy-kovalevskaya-theorem","status":"can_prove","summary":"Let $d$ be a natural number, write $\\mathbb{R}^d$ for real Euclidean $d$-space, and let $F : \\mathbb{R}^d \\times \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}^d$, $f : \\mathbb{R}^d \\times \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$ and $u_0 : \\mathbb{R}^d \\to \\mathbb{R}$ be real analytic at every point. Then for every $x_0 \\in \\mathbb{R}^d$ there are an open set $U \\subseteq \\mathbb{R}^d \\times \\mathbb{R}$\u2026","title":"Cauchy\u2013Kovalevskaya theorem"},{"data_url":"scopes/Analysis/partial-differential-equations/heat-equation.json","id":"Analysis/partial-differential-equations/heat-equation","kind":"scope","navigation":"module","parent":18936,"result":false,"route":"../roadmap/Analysis/partial-differential-equations/heat-equation/index.html","scope":"Analysis/partial-differential-equations/heat-equation","status":"can_prove","title":"Heat equation"},{"id":"Analysis/partial-differential-equations/heat-equation/heat-solution","kind":"node","lean":"MathlibExt.Analysis.HeatKernelWanted.Analysis.ODE.heatSolution","parent":18943,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/heat-equation/index.html#heat-solution","scope":"Analysis/partial-differential-equations/heat-equation","status":"fully_proved","summary":"For $f\\colon \\mathbb{R} \\to \\mathbb{R}$, the heat solution $u(t, x)$ of $f$ is given for $t > 0$ by","title":"Heat-kernel solution"},{"id":"Analysis/partial-differential-equations/heat-equation/heat-kernel-solves-heat-equation","kind":"node","lean":"MathlibExt.Analysis.HeatKernelWanted.Analysis.ODE.heat_kernel_solves_heat_equation","parent":18943,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/heat-equation/index.html#heat-kernel-solves-heat-equation","scope":"Analysis/partial-differential-equations/heat-equation","status":"can_prove","summary":"Let $f\\colon \\mathbb{R} \\to \\mathbb{R}$ be continuous and bounded, and let $u(t, x)$ be the heat solution of $f$. Then for every $t > 0$ and every $x \\in \\mathbb{R}$ there are a function $u_x\\colon \\mathbb{R} \\to \\mathbb{R}$ and a real number $u_{xx}$ such that $u(t, \\cdot)$ has derivative $u_x(y)$ at every $y \\in \\mathbb{R}$, $u_x$ has derivative $u_{xx}$ at $x$, and $u(\\cdot, x)$ has derivative $u_{xx}$ at $t$;\u2026","title":"The Gaussian heat kernel solves the heat equation"},{"data_url":"scopes/Analysis/partial-differential-equations/hessian-estimates-for-the-equation.json","id":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation","kind":"scope","navigation":"module","parent":18936,"result":false,"route":"../roadmap/Analysis/partial-differential-equations/hessian-estimates-for-the-equation/index.html","scope":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation","status":"can_prove","title":"Hessian estimates for the \u03c3\u2082 equation"},{"id":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation/e-c1-norm-on","kind":"node","lean":"MathlibExt.Analysis.HessianEstimatesWanted.HessianEstimates.eC1NormOn","parent":18946,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/hessian-estimates-for-the-equation/index.html#e-c1-norm-on","scope":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation","status":"fully_proved","summary":"Let $E$ and $F$ be real normed spaces and let $\\mu$ be a measure on $E$ (by default the volume measure). For a function $f : E \\to F$ and a set $s \\subseteq E$, write $\\mu_s$ for the restriction of $\\mu$ to $s$. The $C^1$ norm of $f$ on $s$ is the extended nonnegative real number","title":"C\u00b9 norm on a set"},{"id":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation/h","kind":"node","lean":"MathlibExt.Analysis.HessianEstimatesWanted.HessianEstimates.H","parent":18946,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/hessian-estimates-for-the-equation/index.html#h","scope":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation","status":"fully_proved","summary":"Let $E$ be a real Hilbert space, let $f : E \\to \\mathbb{R}$, let $s \\subseteq E$ and let $x \\in E$. Write $D^2 f(x)$ for the second Fr\u00e9chet derivative of $f$ within $s$ at $x$, that is, the derivative within $s$ at $x$ of the map sending $y$ to the derivative of $f$ within $s$ at $y$; it is a continuous bilinear form on $E$. The Hessian operator of $f$ within $s$ at $x$ is the continuous linear map\u2026","title":"Hessian operator within a set"},{"id":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation/frobenius-norm","kind":"node","lean":"MathlibExt.Analysis.HessianEstimatesWanted.HessianEstimates.frobeniusNorm","parent":18946,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/hessian-estimates-for-the-equation/index.html#frobenius-norm","scope":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation","status":"fully_proved","summary":"Let $E$ and $F$ be finite-dimensional real inner product spaces. The Frobenius norm of a continuous linear map $T : E \\to F$ is $\\|T\\|_F = \\sqrt{\\operatorname{tr}(T^* T)}$, where $T^* : F \\to E$ is the adjoint of $T$ and the trace is taken on $E$.","title":"Frobenius norm"},{"id":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation/untitled","kind":"node","lean":"MathlibExt.Analysis.HessianEstimatesWanted.HessianEstimates.\u03c3\u2082","parent":18946,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/hessian-estimates-for-the-equation/index.html#untitled","scope":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space of dimension $n$. If $T : E \\to E$ is a symmetric linear map with eigenvalues $\\lambda_1, \\dots, \\lambda_n$, listed with multiplicity, then $\\sigma_2(T) = \\sum_{1 \\le i < j \\le n} \\lambda_i \\lambda_j$. If $T$ is not symmetric, then $\\sigma_2(T) = 0$ by convention.","title":"Second elementary symmetric function of the eigenvalues"},{"id":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation/theorem-1-1","kind":"node","lean":"MathlibExt.Analysis.HessianEstimatesWanted.HessianEstimates.theorem_1_1","parent":18946,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/hessian-estimates-for-the-equation/index.html#theorem-1-1","scope":"Analysis/partial-differential-equations/hessian-estimates-for-the-equation","status":"can_prove","summary":"There is a function $C : [0, \\infty] \\to \\mathbb{R}$ with the following property. Let $B_1(0)$ be the open unit ball in $\\mathbb{R}^4$, and let $u : \\mathbb{R}^4 \\to \\mathbb{R}$ be smooth ($C^\\infty$) on $B_1(0)$ with finite $C^1$ norm $\\|u\\|_{C^1(B_1(0))}$ with respect to Lebesgue measure. Suppose that for every $x \\in B_1(0)$ the Laplacian satisfies $\\Delta u(x) > 0$ and the Hessian operator $Hu(x)$ of $u$ within\u2026","title":"Shankar\u2013Yuan Hessian estimate for the sigma-2 equation in dimension four"},{"data_url":"scopes/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues.json","id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","kind":"scope","navigation":"module","parent":18936,"result":false,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","title":"P\u00f3lya's conjecture on Laplacian eigenvalues"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/nonnegative-discrete-spectrum","kind":"node","lean":"NonnegativeDiscreteSpectrum","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#nonnegative-discrete-spectrum","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"A nonnegative discrete spectrum is a sequence of real numbers $E_0, E_1, E_2, \\ldots$, the eigenvalues listed with multiplicity, such that $E_j \\ge 0$ for every $j$, $E_i \\le E_j$ whenever $i \\le j$, and $E_j \\to +\\infty$ as $j \\to \\infty$.","title":"Nonnegative discrete spectrum"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/ext","kind":"node","lean":"NonnegativeDiscreteSpectrum.ext","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#ext","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"If two nonnegative discrete spectra $S$ and $T$ have the same $j$-th eigenvalue for every index $j \\in \\mathbb{N}$, then $S = T$.","title":"A nonnegative discrete spectrum is determined by its eigenvalues"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/sublevel-set","kind":"node","lean":"NonnegativeDiscreteSpectrum.sublevelSet","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#sublevel-set","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For a nonnegative discrete spectrum with eigenvalues $E_0 \\le E_1 \\le \\cdots$ and a real number $\\Lambda$, the strict sublevel set at $\\Lambda$ is the set of indices $\\{j \\in \\mathbb{N} : E_j < \\Lambda\\}$.","title":"Strict sublevel set"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/sublevel-set-finite","kind":"node","lean":"NonnegativeDiscreteSpectrum.sublevelSet_finite","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#sublevel-set-finite","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For a nonnegative discrete spectrum with eigenvalues $E_0 \\le E_1 \\le \\cdots$ and every real number $\\Lambda$, the strict sublevel set $\\{j \\in \\mathbb{N} : E_j < \\Lambda\\}$ is finite.","title":"Strict sublevel sets are finite"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/sublevel-finset","kind":"node","lean":"NonnegativeDiscreteSpectrum.sublevelFinset","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#sublevel-finset","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For a nonnegative discrete spectrum with eigenvalues $E_0 \\le E_1 \\le \\cdots$ and a real number $\\Lambda$, the finite strict sublevel set at $\\Lambda$ is the strict sublevel set $\\{j \\in \\mathbb{N} : E_j < \\Lambda\\}$, regarded as a finite set of indices; it is finite because $E_j \\to +\\infty$.","title":"Finite strict sublevel set"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/mem-sublevel-finset","kind":"node","lean":"NonnegativeDiscreteSpectrum.mem_sublevelFinset","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#mem-sublevel-finset","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For a nonnegative discrete spectrum with eigenvalues $E_0 \\le E_1 \\le \\cdots$, a real number $\\Lambda$ and an index $j \\in \\mathbb{N}$, the index $j$ lies in the finite strict sublevel set at $\\Lambda$ if and only if $E_j < \\Lambda$.","title":"Membership in the finite strict sublevel set"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/counting-function","kind":"node","lean":"NonnegativeDiscreteSpectrum.countingFunction","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#counting-function","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For a nonnegative discrete spectrum with eigenvalues $E_0 \\le E_1 \\le \\cdots$ and a real number $\\Lambda$, the counting function $N(\\Lambda)$ is the number of indices $j \\in \\mathbb{N}$ with $E_j < \\Lambda$, so eigenvalues are counted with multiplicity.","title":"Eigenvalue counting function"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/riesz-mean","kind":"node","lean":"NonnegativeDiscreteSpectrum.rieszMean","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#riesz-mean","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For a nonnegative discrete spectrum with eigenvalues $E_0 \\le E_1 \\le \\cdots$ and real numbers $\\sigma$ and $\\Lambda$, the Riesz mean of order $\\sigma$ at $\\Lambda$ is the finite sum $R_\\sigma(\\Lambda) = \\sum_{j \\in \\mathbb{N},\\, E_j < \\Lambda} (\\Lambda - E_j)^\\sigma$ over indices, where $x^\\sigma$ is the real power.","title":"Riesz mean"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/riesz-mean-zero","kind":"node","lean":"NonnegativeDiscreteSpectrum.rieszMean_zero","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#riesz-mean-zero","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For a nonnegative discrete spectrum with eigenvalues $E_0 \\le E_1 \\le \\cdots$ and every real number $\\Lambda$, the Riesz mean of order $0$ at $\\Lambda$ equals the counting function at $\\Lambda$, viewed as a real number: $R_0(\\Lambda) = N(\\Lambda)$.","title":"The Riesz mean of order zero is the counting function"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/smooth-test-space","kind":"node","lean":"DirichletLaplacian.SmoothTestSpace","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#smooth-test-space","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For an open set $\\Omega \\subseteq \\mathbb{R}^n$, the space of smooth test functions on $\\Omega$ is the real vector space of infinitely differentiable functions $f : \\mathbb{R}^n \\to \\mathbb{R}$ whose support is compact and contained in $\\Omega$.","title":"Smooth test functions"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/rayleigh-quotient","kind":"node","lean":"DirichletLaplacian.rayleighQuotient","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#rayleigh-quotient","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For a smooth test function $f$ on an open set $\\Omega \\subseteq \\mathbb{R}^n$, the Rayleigh quotient of $f$ is","title":"Rayleigh quotient"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/variational-eigenvalue","kind":"node","lean":"DirichletLaplacian.variationalEigenvalue","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#variational-eigenvalue","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For an open set $\\Omega \\subseteq \\mathbb{R}^n$ and an index $j \\ge 0$, the variational eigenvalue of $\\Omega$ at index $j$ is the Courant\u2013Fischer min\u2013max value","title":"Variational eigenvalue"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/is-dirichlet-spectrum","kind":"node","lean":"NonnegativeDiscreteSpectrum.IsDirichletSpectrum","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#is-dirichlet-spectrum","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"A nonnegative discrete spectrum $S = (S_j)_{j \\ge 0}$ is the Dirichlet spectrum of an open set $\\Omega \\subseteq \\mathbb{R}^n$ when $S_j = \\lambda_j(\\Omega)$ for every $j \\ge 0$, where $\\lambda_j(\\Omega)$ is the variational eigenvalue of $\\Omega$ at index $j$.","title":"Dirichlet spectrum of an open set"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/unit-ball-volume","kind":"node","lean":"EuclideanLaplacian.unitBallVolume","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#unit-ball-volume","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For $n \\ge 0$, $\\omega_n$ is the Lebesgue measure of the open unit ball $\\{x \\in \\mathbb{R}^n : |x| < 1\\}$ of Euclidean space $\\mathbb{R}^n$, as a real number.","title":"Volume of the unit ball"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/domain-volume","kind":"node","lean":"EuclideanLaplacian.domainVolume","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#domain-volume","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For an open set $\\Omega \\subseteq \\mathbb{R}^n$, the volume $|\\Omega|$ is the Lebesgue measure of $\\Omega$ taken as a real number, with $|\\Omega| = 0$ when the Lebesgue measure of $\\Omega$ is infinite.","title":"Volume of a domain"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/semiclassical-counting-coefficient","kind":"node","lean":"EuclideanLaplacian.semiclassicalCountingCoefficient","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#semiclassical-counting-coefficient","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For an open set $\\Omega \\subseteq \\mathbb{R}^n$, the semiclassical counting coefficient is $\\omega_n |\\Omega| / (2\\pi)^n$, where $\\omega_n$ is the volume of the unit ball in $\\mathbb{R}^n$ and $|\\Omega|$ is the Lebesgue measure of $\\Omega$ taken as a real number, with $|\\Omega| = 0$ when that measure is infinite.","title":"Semiclassical counting coefficient"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/polya-eigenvalue-benchmark","kind":"node","lean":"EuclideanLaplacian.polyaEigenvalueBenchmark","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#polya-eigenvalue-benchmark","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For an open set $\\Omega \\subseteq \\mathbb{R}^n$ and an index $j \\ge 0$, the P\u00f3lya benchmark at index $j$ is","title":"P\u00f3lya eigenvalue benchmark"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/neumann-polya-eigenvalue-benchmark","kind":"node","lean":"EuclideanLaplacian.neumannPolyaEigenvalueBenchmark","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#neumann-polya-eigenvalue-benchmark","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For an open set $\\Omega \\subseteq \\mathbb{R}^n$ and a natural number $j$, the Neumann P\u00f3lya benchmark of $\\Omega$ at index $j$ is the real number $(2\\pi)^2 \\left(j / (\\omega_n |\\Omega|)\\right)^{2/n}$, where $\\omega_n$ is the volume of the unit ball of $\\mathbb{R}^n$, $|\\Omega|$ is the volume of $\\Omega$, the power is the real power with exponent $2/n$, and a quotient by $0$ is read as $0$.","title":"Neumann P\u00f3lya benchmark"},{"id":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/neumann-polya-eigenvalue-benchmark-succ","kind":"node","lean":"EuclideanLaplacian.neumannPolyaEigenvalueBenchmark_succ","parent":18952,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues/index.html#neumann-polya-eigenvalue-benchmark-succ","scope":"Analysis/partial-differential-equations/p-lya-s-conjecture-on-laplacian-eigenvalues","status":"fully_proved","summary":"For every open set $\\Omega \\subseteq \\mathbb{R}^n$ and every natural number $j$, the Neumann P\u00f3lya benchmark of $\\Omega$ at index $j + 1$ equals the P\u00f3lya eigenvalue benchmark of $\\Omega$ at index $j$.","title":"Neumann and Dirichlet P\u00f3lya benchmarks agree after an index shift"},{"data_url":"scopes/Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations.json","id":"Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations","kind":"scope","navigation":"module","parent":18936,"result":false,"route":"../roadmap/Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations/index.html","scope":"Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations","status":"can_prove","title":"Radial symmetry of semilinear elliptic equations"},{"id":"Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations/solves-semilinear-poisson","kind":"node","lean":"MathlibExt.Analysis.RadialSymmetryWanted.Analysis.PDE.SolvesSemilinearPoisson","parent":18972,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations/index.html#solves-semilinear-poisson","scope":"Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations","status":"fully_proved","summary":"Let $n$ be a natural number, let $f : \\mathbb{R} \\to \\mathbb{R}$, and let $u : \\mathbb{R}^n \\to \\mathbb{R}$ be a function on Euclidean space. Then $u$ solves the semilinear Poisson problem for $f$ if $-\\Delta u(x) = f(u(x))$ for every $x$ in the open unit ball and $u(x) = 0$ for every $x$ on the unit sphere.","title":"Semilinear Poisson problem on the unit ball"},{"id":"Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations/semilinear-poisson-radial-symmetry","kind":"node","lean":"MathlibExt.Analysis.RadialSymmetryWanted.Analysis.PDE.semilinear_poisson_radial_symmetry","parent":18972,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations/index.html#semilinear-poisson-radial-symmetry","scope":"Analysis/partial-differential-equations/radial-symmetry-of-semilinear-elliptic-equations","status":"can_prove","summary":"Let $n \\ge 1$, let $f : \\mathbb{R} \\to \\mathbb{R}$ be Lipschitz, and let $u : \\mathbb{R}^n \\to \\mathbb{R}$ be a function on Euclidean space that is $C^2$ on the closed unit ball $\\overline{B}$. Suppose that $u$ solves the semilinear Poisson problem for $f$, that is, $-\\Delta u = f(u)$ on the open unit ball $B$ and $u = 0$ on the unit sphere, and that $u > 0$ on $B$. Then there is a function\u2026","title":"Radial symmetry of positive solutions of the semilinear Poisson problem"},{"data_url":"scopes/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations.json","id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","kind":"scope","navigation":"module","parent":18936,"result":false,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"can_prove","title":"Singularity formation for the Euler equations"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/is-bounded-fun-on","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.IsBoundedFunOn","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#is-bounded-fun-on","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"A function $f : \\beta \\to \\alpha$ into a space $\\alpha$ with a bornology is bounded on a set $s \\subseteq \\beta$ if the image $f(s)$ is a bounded subset of $\\alpha$.","title":"Functions bounded on a set"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/is-bounded-fun","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.IsBoundedFun","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#is-bounded-fun","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"A function $f : \\beta \\to \\alpha$ into a space $\\alpha$ with a bornology is bounded if its range $f(\\beta)$ is a bounded subset of $\\alpha$.","title":"Bounded functions"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/holder","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.Holder","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#holder","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"Let $X$ and $Y$ be pseudo-extended metric spaces (distances take values in $[0, \\infty]$), let $r \\ge 0$ be a real number and $f : X \\to Y$. Then $f$ is $r$-H\u00f6lder if there is a constant $C \\ge 0$ such that $d_Y(f(x_1), f(x_2)) \\le C \\, d_X(x_1, x_2)^r$ for all $x_1, x_2 \\in X$.","title":"H\u00f6lder continuity"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/holder-on","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.HolderOn","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#holder-on","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"Let $X$ and $Y$ be pseudo-extended metric spaces (distances take values in $[0, \\infty]$), let $r \\ge 0$ be a real number, $s \\subseteq X$ and $f : X \\to Y$. Then $f$ is $r$-H\u00f6lder on $s$ if there is a constant $C \\ge 0$ such that $d_Y(f(x_1), f(x_2)) \\le C \\, d_X(x_1, x_2)^r$ for all $x_1, x_2 \\in s$.","title":"H\u00f6lder continuity on a set"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/is-cont-diff-holder","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.IsContDiffHolder","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#is-cont-diff-holder","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"Let $\\mathbb{k}$ be a nontrivially normed field, $E$ and $F$ normed spaces over $\\mathbb{k}$ and $\\alpha \\ge 0$ a real number. A function $f : E \\to F$ is in $C^{1,\\alpha}(E)$ if $f$ is continuously differentiable over $\\mathbb{k}$ and its derivative $Df$, a map from $E$ to the space $L(E, F)$ of continuous $\\mathbb{k}$-linear maps with the operator norm, is bounded and $\\alpha$-H\u00f6lder.","title":"Continuously differentiable functions with H\u00f6lder derivative"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/is-cont-diff-holder-on","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.IsContDiffHolderOn","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#is-cont-diff-holder-on","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"Let $\\mathbb{k}$ be a nontrivially normed field, $E$ and $F$ normed spaces over $\\mathbb{k}$, $s \\subseteq E$ and $\\alpha \\ge 0$ a real number. A function $f : E \\to F$ is in $C^{1,\\alpha}(s)$ if $f$ is continuously differentiable on $s$ over $\\mathbb{k}$ and its derivative within $s$, a map from $E$ to the space $L(E, F)$ of continuous $\\mathbb{k}$-linear maps with the operator norm, is bounded on $s$ and\u2026","title":"Continuously differentiable functions with H\u00f6lder derivative on a set"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/directional-deriv","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.directionalDeriv","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#directional-deriv","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"Given $u, v : \\mathbb{R}^3 \\times \\mathbb{R} \\to \\mathbb{R}^3$, the spatial directional derivative $(u \\cdot \\nabla_x) v$ of $v$ in the direction of $u$ is the function $(x, t) \\mapsto D_x v(x, t)\\,[u(x, t)]$, where $D_x v(x, t)$ is the Fr\u00e9chet derivative of $y \\mapsto v(y, t)$ at $x$ (taken to be $0$ if this map is not differentiable at $x$).","title":"Spatial directional derivative of a time-dependent vector field"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/time-deriv","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.timeDeriv","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#time-deriv","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"For a function $u(x, t) \\in \\mathbb{R}^3$ of a point $x \\in \\mathbb{R}^3$ and a time $t \\in \\mathbb{R}$, the time derivative $\\partial_t u(x, t) \\in \\mathbb{R}^3$ is the derivative at $t$ of the function $s \\mapsto u(x, s)$, taken to be $0$ when that function is not differentiable at $t$.","title":"Time derivative"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/space-grad","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.spaceGrad","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#space-grad","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"For a function $p(x, t)$ of a point $x \\in \\mathbb{R}^3$ and a time $t \\in \\mathbb{R}$, the spatial gradient $\\nabla_x p(x, t) \\in \\mathbb{R}^3$ is the gradient at $x$ of the function $y \\mapsto p(y, t)$, taken to be $0$ when that function is not differentiable at $x$.","title":"Spatial gradient"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/div","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.div","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#div","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"The divergence of a vector field $u : \\mathbb{R}^3 \\to \\mathbb{R}^3$ at a point $x \\in \\mathbb{R}^3$ is the trace of the Fr\u00e9chet derivative $Du(x)$, a linear map from $\\mathbb{R}^3$ to $\\mathbb{R}^3$ (taken to be $0$ if $u$ is not differentiable at $x$).","title":"Divergence"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/e","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.e","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#e","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"For $i \\in \\{0, 1, 2\\}$, $e_i \\in \\mathbb{R}^3$ is the $i$-th standard basis vector: its $i$-th coordinate is $1$ and its other coordinates are $0$.","title":"Standard basis vectors of \u211d\u00b3"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/vorticity","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.vorticity","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#vorticity","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"The vorticity of a vector field $u : \\mathbb{R}^3 \\to \\mathbb{R}^3$ is its curl $(\\nabla \\times u)(x) = \\sum_{i=1}^{3} e_i \\times Du(x)\\,e_i$, where $e_1, e_2, e_3$ is the standard basis of $\\mathbb{R}^3$, $\\times$ is the cross product and $Du(x)\\,e_i$ is the partial derivative of $u$ at $x$ in direction $e_i$ (all taken to be $0$ when $u$ is not differentiable at $x$).","title":"Vorticity"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/is-local-euler-equation-solution","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.IsLocalEulerEquationSolution","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#is-local-euler-equation-solution","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"Let $T \\in \\mathbb{R}$, let $u_0 : \\mathbb{R}^3 \\to \\mathbb{R}^3$, and let $p : \\mathbb{R}^3 \\times \\mathbb{R} \\to \\mathbb{R}$ and $u : \\mathbb{R}^3 \\times \\mathbb{R} \\to \\mathbb{R}^3$. Write $\\partial_t u$ for the time derivative of $u$, $(u \\cdot \\nabla_x) u$ for the spatial directional derivative of $u$ in the direction of $u$, and $\\nabla_x p$ for the spatial gradient of $p$. Then $u$ is a local solution of the\u2026","title":"Local solutions of the incompressible Euler equations"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/is-odd","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.IsOdd","parent":18975,"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#is-odd","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"fully_proved","summary":"A vector field $u : \\mathbb{R}^3 \\to \\mathbb{R}^3$ with components $u_1, u_2, u_3$ is odd if, for each $i \\in \\{1, 2, 3\\}$, the component $u_i$ is an odd function of the $i$-th coordinate and an even function of each of the other two coordinates, when the remaining coordinates are held fixed.","title":"Odd vector fields in three dimensions"},{"id":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/theorem-1","kind":"node","lean":"MathlibExt.Analysis.FiniteTimeSingularityWanted.FiniteTimeSingularity.theorem_1","parent":18975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elgindi's theorem: finite-time singularity of "],["math","\\(C^{1,\\alpha}\\)"],["text"," Euler flows"]],"source":"Elgindi's theorem: finite-time singularity of $C^{1,\\alpha}$ Euler flows"},"result":true,"route":"../roadmap/Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations/index.html#theorem-1","scope":"Analysis/partial-differential-equations/singularity-formation-for-the-euler-equations","status":"can_prove","summary":"There exist a real number $\\alpha > 0$ and a vector field $u_0 : \\mathbb{R}^3 \\to \\mathbb{R}^3$ that is odd, has divergence $0$ at every point and lies in $C^{1,\\alpha}(\\mathbb{R}^3)$, together with a constant $C > 0$ such that its vorticity satisfies $|(\\nabla \\times u_0)(x)| \\le C / (|x|^\\alpha + 1)$ for every $x \\in \\mathbb{R}^3$, and there exist functions $p : \\mathbb{R}^3 \\times \\mathbb{R} \\to \\mathbb{R}$ and\u2026","title":"Elgindi's theorem: finite-time singularity of $C^{1,\\alpha}$ Euler flows"},{"data_url":"scopes/Analysis/convex-analysis-and-variations.json","id":"Analysis/convex-analysis-and-variations","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/index.html","scope":"Analysis/convex-analysis-and-variations","status":"can_prove","summary":"Convex functions and sets in analysis and optimization: Fenchel conjugates, polar sets, Moreau decomposition, convex projections, KKT conditions, gradient descent rates, Stampacchia, Tikhonov regularization, Euler\u2013Lagrange, mountain pass and Ekeland principles; linear programming goes to convex geometry.","title":"Convex analysis and calculus of variations"},{"id":"Analysis/convex-analysis-and-variations/convex-on-iff-forall-le-add-fderiv-withi","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.ConvexOnIffAboveTangentWanted.convexOn_iff_forall_le_add_fderivWithin","parent":18991,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/index.html#convex-on-iff-forall-le-add-fderiv-withi","scope":"Analysis/convex-analysis-and-variations","status":"can_prove","summary":"Let $E$ be a real Banach space, let $X \\subseteq E$ be an open convex set, and let $f \\colon E \\to \\mathbb{R}$ be twice continuously differentiable on $X$. Then $f$ is convex on $X$ if and only if $f(x) \\ge f(a) + Df(a)(x - a)$ for all $a, x \\in X$, where $Df(a)$ is the Fr\u00e9chet derivative of $f$ at $a$ within $X$.","title":"Convexity via the tangent-plane inequality"},{"id":"Analysis/convex-analysis-and-variations/karush-kuhn-tucker-inequality","kind":"node","lean":"MathlibExt.Analysis.Convex.KarushKuhnTucker.karush_kuhn_tucker_inequality","parent":18991,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/index.html#karush-kuhn-tucker-inequality","scope":"Analysis/convex-analysis-and-variations","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real normed space, let $f \\colon E \\to \\mathbb{R}$ and $g_1, \\dots, g_m \\colon E \\to \\mathbb{R}$, and let $x \\in E$ be a point at which $f$ and every $g_i$ are differentiable; write $D$ for the Fr\u00e9chet derivative at $x$. Suppose that $x$ is feasible, that is $g_i(x) \\le 0$ for every $i$; that $x$ is a local minimizer of $f$ on the feasible set\u2026","title":"Karush\u2013Kuhn\u2013Tucker conditions for inequality constraints"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/ekeland-variational-principle.json","id":"Analysis/convex-analysis-and-variations/ekeland-variational-principle","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/ekeland-variational-principle/index.html","scope":"Analysis/convex-analysis-and-variations/ekeland-variational-principle","status":"fully_proved","title":"Ekeland variational principle"},{"id":"Analysis/convex-analysis-and-variations/ekeland-variational-principle/ekeland-variational-principle-general","kind":"node","lean":"MetaMathlibExt.ekeland_variational_principle_general","parent":18994,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/ekeland-variational-principle/index.html#ekeland-variational-principle-general","scope":"Analysis/convex-analysis-and-variations/ekeland-variational-principle","status":"fully_proved","summary":"Let $(X, d)$ be a complete metric space, let $f : X \\to \\mathbb{R}$ be lower semicontinuous and bounded below, and let $\\varepsilon > 0$. If $x_0 \\in X$ satisfies $f(x_0) \\le f(x) + \\varepsilon$ for every $x \\in X$, then there is $x_\\varepsilon \\in X$ with $f(x_\\varepsilon) \\le f(x_0)$, $d(x_\\varepsilon, x_0) \\le 1$, and $f(x_\\varepsilon) < f(y) + \\varepsilon\\, d(x_\\varepsilon, y)$ for every $y \\ne x_\\varepsilon$.","title":"Ekeland variational principle without a nonemptiness assumption"},{"id":"Analysis/convex-analysis-and-variations/ekeland-variational-principle/ekeland-variational-principle","kind":"node","lean":"MetaMathlibExt.ekeland_variational_principle","parent":18994,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/ekeland-variational-principle/index.html#ekeland-variational-principle","scope":"Analysis/convex-analysis-and-variations/ekeland-variational-principle","status":"fully_proved","summary":"Let $(X, d)$ be a nonempty complete metric space, let $f : X \\to \\mathbb{R}$ be lower semicontinuous and bounded below, and let $\\varepsilon > 0$. If $x_0 \\in X$ satisfies $f(x_0) \\le f(x) + \\varepsilon$ for every $x \\in X$, then there is $x_\\varepsilon \\in X$ with $f(x_\\varepsilon) \\le f(x_0)$, $d(x_\\varepsilon, x_0) \\le 1$, and $f(x_\\varepsilon) < f(y) + \\varepsilon\\, d(x_\\varepsilon, y)$ for every\u2026","title":"Ekeland variational principle"},{"id":"Analysis/convex-analysis-and-variations/ekeland-variational-principle/caristi-fixed-point","kind":"node","lean":"MetaMathlibExt.caristi_fixedPoint","parent":18994,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/ekeland-variational-principle/index.html#caristi-fixed-point","scope":"Analysis/convex-analysis-and-variations/ekeland-variational-principle","status":"fully_proved","summary":"Let $(X, d)$ be a nonempty complete metric space and let $\\varphi : X \\to \\mathbb{R}$ be lower semicontinuous and bounded below. If a map $T : X \\to X$ satisfies $d(x, T x) \\le \\varphi(x) - \\varphi(T x)$ for every $x \\in X$, then $T$ has a fixed point: there is $x \\in X$ with $T x = x$.","title":"Caristi fixed-point theorem"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/euler-lagrange-equation.json","id":"Analysis/convex-analysis-and-variations/euler-lagrange-equation","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/euler-lagrange-equation/index.html","scope":"Analysis/convex-analysis-and-variations/euler-lagrange-equation","status":"can_prove","title":"Euler\u2013Lagrange equation"},{"id":"Analysis/convex-analysis-and-variations/euler-lagrange-equation/lagrangian-partial-x","kind":"node","lean":"MathlibExt.Analysis.EulerLagrangeWanted.Analysis.lagrangianPartialX","parent":18998,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/euler-lagrange-equation/index.html#lagrangian-partial-x","scope":"Analysis/convex-analysis-and-variations/euler-lagrange-equation","status":"fully_proved","summary":"For a real function $L(t, y, v)$ of three real variables, a function $x : \\mathbb{R} \\to \\mathbb{R}$ and a real number $t$, this is the value $\\frac{\\partial L}{\\partial y}(t, x(t), x'(t))$, the derivative at $y = x(t)$ of the function $y \\mapsto L(t, y, x'(t))$.","title":"Partial derivative of the Lagrangian in position"},{"id":"Analysis/convex-analysis-and-variations/euler-lagrange-equation/lagrangian-partial-v","kind":"node","lean":"MathlibExt.Analysis.EulerLagrangeWanted.Analysis.lagrangianPartialV","parent":18998,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/euler-lagrange-equation/index.html#lagrangian-partial-v","scope":"Analysis/convex-analysis-and-variations/euler-lagrange-equation","status":"fully_proved","summary":"For a real function $L(t, y, v)$ of three real variables, a function $x : \\mathbb{R} \\to \\mathbb{R}$ and a real number $t$, this is the value $\\frac{\\partial L}{\\partial v}(t, x(t), x'(t))$, the derivative at $v = x'(t)$ of the function $v \\mapsto L(t, x(t), v)$.","title":"Partial derivative of the Lagrangian in velocity"},{"id":"Analysis/convex-analysis-and-variations/euler-lagrange-equation/is-variational-extremum","kind":"node","lean":"MathlibExt.Analysis.EulerLagrangeWanted.Analysis.IsVariationalExtremum","parent":18998,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/euler-lagrange-equation/index.html#is-variational-extremum","scope":"Analysis/convex-analysis-and-variations/euler-lagrange-equation","status":"fully_proved","summary":"Let $a, b$ be real numbers and let $L(t, y, v)$ be a real function of three real variables. A function $x : \\mathbb{R} \\to \\mathbb{R}$ is a variational extremum of the action of $L$ on $(a, b)$ if $x$ is continuously differentiable and, for every infinitely differentiable function $h : \\mathbb{R} \\to \\mathbb{R}$ of compact support whose closed support lies in the open interval $(a, b)$, the derivative at\u2026","title":"Variational extremum"},{"id":"Analysis/convex-analysis-and-variations/euler-lagrange-equation/euler-lagrange-equation","kind":"node","lean":"MathlibExt.Analysis.EulerLagrangeWanted.Analysis.euler_lagrange_equation","parent":18998,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/euler-lagrange-equation/index.html#euler-lagrange-equation","scope":"Analysis/convex-analysis-and-variations/euler-lagrange-equation","status":"can_prove","summary":"Let $a < b$ be real numbers, let $L(t, y, v)$ be a real function of three real variables that is twice continuously differentiable on $\\mathbb{R}^3$, and let $x : \\mathbb{R} \\to \\mathbb{R}$ be twice continuously differentiable. Write $\\partial L / \\partial y$ and $\\partial L / \\partial v$ for the partial derivatives of $L$ in its second and third arguments. If $x$ is a variational extremum of the action of $L$ on\u2026","title":"Euler\u2013Lagrange equation"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/fenchel-moreau-theorem.json","id":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/index.html","scope":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem","status":"fully_proved","title":"Fenchel\u2013Moreau theorem"},{"id":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/fenchel-conj","kind":"node","lean":"MathlibExt.Analysis.Convex.FenchelMoreau.fenchelConj","parent":19003,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/index.html#fenchel-conj","scope":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem","status":"fully_proved","summary":"Let $E$ be a real vector space with a topology and let $f : E \\to [-\\infty, \\infty]$. The Fenchel conjugate of $f$ at a continuous linear functional $L : E \\to \\mathbb{R}$ is the extended real number $f^*(L) = \\sup_{y \\in E} \\big(L(y) - f(y)\\big)$.","title":"Fenchel conjugate"},{"id":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/fenchel-biconj","kind":"node","lean":"MathlibExt.Analysis.Convex.FenchelMoreau.fenchelBiconj","parent":19003,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/index.html#fenchel-biconj","scope":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem","status":"fully_proved","summary":"Let $E$ be a real vector space with a topology and let $f : E \\to [-\\infty, \\infty]$. The biconjugate of $f$ at $x \\in E$ is the extended real number $f^{**}(x) = \\sup_L \\big(L(x) - f^*(L)\\big)$, where $L$ ranges over the continuous linear functionals $E \\to \\mathbb{R}$ and $f^*$ is the Fenchel conjugate of $f$.","title":"Fenchel biconjugate"},{"id":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/is-closed-epi-real","kind":"node","lean":"MathlibExt.Analysis.Convex.FenchelMoreau.isClosed_epi_real","parent":19003,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/index.html#is-closed-epi-real","scope":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem","status":"fully_proved","summary":"Let $E$ be a topological space and let $f : E \\to [-\\infty, \\infty]$ be lower semicontinuous. Then the set $\\{(x, r) \\in E \\times \\mathbb{R} : f(x) \\le r\\}$ is closed in $E \\times \\mathbb{R}$.","title":"Lower semicontinuous functions have closed epigraphs"},{"id":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/biconj-le","kind":"node","lean":"MathlibExt.Analysis.Convex.FenchelMoreau.biconj_le","parent":19003,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/index.html#biconj-le","scope":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem","status":"fully_proved","summary":"Let $E$ be a real vector space with a topology and let $f : E \\to [-\\infty, \\infty]$ never take the value $-\\infty$. Then $f^{**}(x) \\le f(x)$ for every $x \\in E$, where $f^{**}$ is the Fenchel biconjugate of $f$.","title":"The biconjugate lies below the function"},{"id":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/fenchel-moreau","kind":"node","lean":"MathlibExt.Analysis.Convex.FenchelMoreau.fenchel_moreau","parent":19003,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/fenchel-moreau-theorem/index.html#fenchel-moreau","scope":"Analysis/convex-analysis-and-variations/fenchel-moreau-theorem","status":"fully_proved","summary":"Let $E$ be a locally convex real topological vector space, not necessarily Hausdorff. Let $f : E \\to [-\\infty, \\infty]$ be lower semicontinuous, never equal to $-\\infty$, not identically $+\\infty$, and convex in the sense that the set $\\{(x, r) \\in E \\times \\mathbb{R} : f(x) \\le r\\}$ is convex. Then for every $x \\in E$, $f(x) = \\sup_L \\big(L(x) - \\sup_{y \\in E} (L(y) - f(y))\\big)$, where $L$ ranges over the\u2026","title":"Fenchel\u2013Moreau theorem"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/gradient-descent.json","id":"Analysis/convex-analysis-and-variations/gradient-descent","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","title":"Gradient descent"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/gradient-descent-convex-sublinear-rate","kind":"node","lean":"MetaMathlibExt.gradient_descent_convex_sublinear_rate","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#gradient-descent-convex-sublinear-rate","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"Let $E$ be a finite-dimensional real inner product space, let $L > 0$, and let $f : E \\to \\mathbb{R}$ be convex and differentiable with $L$-Lipschitz gradient $\\nabla f$. Let $x^\\star \\in E$ be a global minimizer of $f$, let $0 < \\eta \\le 1/L$, and let $(x_k)_{k \\ge 0}$ be any sequence in $E$ with $x_{k+1} = x_k - \\eta \\nabla f(x_k)$ for all $k$. Then for every $k \\ge 1$,\u2026","title":"Sublinear rate of gradient descent for smooth convex functions"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/gradient-descent-trajectory","kind":"node","lean":"gradientDescentTrajectory","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#gradient-descent-trajectory","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"Let $E$ be a real Hilbert space, $f : E \\to \\mathbb{R}$, $s : \\mathbb{N} \\to \\mathbb{R}$ a sequence of step sizes and $x_0 \\in E$. The gradient descent trajectory of $f$ from $x_0$ with steps $s$ is the sequence $(x_k)_{k \\in \\mathbb{N}}$ in $E$ starting at $x_0$ with $x_{k+1} = x_k - s_k \\nabla f(x_k)$.","title":"Gradient descent with a step schedule"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/is-unit-smooth-convex","kind":"node","lean":"IsUnitSmoothConvex","parent":19009,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convex functions with "],["math","\\(1\\)"],["text","-Lipschitz gradient"]],"source":"Convex functions with $1$-Lipschitz gradient"},"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#is-unit-smooth-convex","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"A function $f : E \\to \\mathbb{R}$ on a real Hilbert space $E$ is unit-smooth convex if it is convex, differentiable, and its gradient is $1$-Lipschitz: $\\lVert \\nabla f(x) - \\nabla f(y) \\rVert \\le \\lVert x - y \\rVert$ for all $x, y \\in E$.","title":"Convex functions with $1$-Lipschitz gradient"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/gradient-descent-uniform-rate","kind":"node","lean":"GradientDescentUniformRate","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#gradient-descent-uniform-rate","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"A step schedule $s : \\mathbb{N} \\to \\mathbb{R}$ achieves a uniform last-iterate rate with exponent $c \\in \\mathbb{R}$ if there are a real number $C > 0$ and an integer $K \\ge 1$ with the following property. For every $d \\in \\mathbb{N}$, every convex differentiable function $f : \\mathbb{R}^d \\to \\mathbb{R}$ whose gradient is $1$-Lipschitz for the Euclidean norm, every $x^\\star \\in \\mathbb{R}^d$ with\u2026","title":"Uniform last-iterate rate of a step schedule"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/gradient-descent-exponent","kind":"node","lean":"gradientDescentExponent","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#gradient-descent-exponent","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"The gradient descent exponent is the supremum, in the extended reals $[-\\infty, +\\infty]$, of the set of real numbers $c$ such that some step schedule $s : \\mathbb{N} \\to \\mathbb{R}$ achieves a uniform last-iterate rate with exponent $c$. In particular, it is $-\\infty$ if there is no such $c$, and $+\\infty$ if these exponents are unbounded above.","title":"The gradient descent exponent"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/gradient-descent-silver-ratio","kind":"node","lean":"gradientDescentSilverRatio","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#gradient-descent-silver-ratio","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"The silver ratio is the real number $\\rho = 1 + \\sqrt{2}$.","title":"Silver ratio"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/silver-gradient-descent-exponent","kind":"node","lean":"silverGradientDescentExponent","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#silver-gradient-descent-exponent","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"The silver exponent is $\\log_2 \\rho = \\log \\rho / \\log 2$, where $\\rho = 1 + \\sqrt{2}$ is the silver ratio.","title":"Silver exponent"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/silver-gradient-descent-schedule","kind":"node","lean":"silverGradientDescentSchedule","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#silver-gradient-descent-schedule","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"The silver step schedule is the sequence $s_t = 1 + \\rho^{\\nu_2(t + 1) - 1}$ for $t \\in \\mathbb{N}$, where $\\rho = 1 + \\sqrt{2}$ is the silver ratio and $\\nu_2(n)$ is the exponent of $2$ in the prime factorization of $n$.","title":"Silver step schedule"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/gradient-descent-uniform-rate-one","kind":"node","lean":"gradientDescentUniformRate_one","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#gradient-descent-uniform-rate-one","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"The step schedule with every step size equal to $1$ achieves a uniform last-iterate rate with exponent $1$.","title":"Constant unit steps achieve exponent one"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/one-le-gradient-descent-exponent","kind":"node","lean":"one_le_gradientDescentExponent","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#one-le-gradient-descent-exponent","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"The gradient descent exponent is at least $1$.","title":"The gradient descent exponent is at least one"},{"id":"Analysis/convex-analysis-and-variations/gradient-descent/gradient-descent-trajectory-zero-schedul","kind":"node","lean":"gradientDescentTrajectory_zeroSchedule","parent":19009,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/gradient-descent/index.html#gradient-descent-trajectory-zero-schedul","scope":"Analysis/convex-analysis-and-variations/gradient-descent","status":"fully_proved","summary":"Let $E$ be a real Hilbert space, $f : E \\to \\mathbb{R}$ and $x_0 \\in E$. If every step size is $0$, then for every $k \\in \\mathbb{N}$ the $k$-th gradient descent iterate of $f$ from $x_0$ equals $x_0$.","title":"Gradient descent with zero step sizes stays put"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution.json","id":"Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution/index.html","scope":"Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution","status":"can_prove","title":"Lipschitz regularization by inf-convolution"},{"id":"Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution/g","kind":"node","lean":"MathlibExt.MeasureTheory.InfConvolutionLipschitzWanted.g","parent":19021,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution/index.html#g","scope":"Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution","status":"fully_proved","summary":"Let $X$ be a metric space with metric $d$, let $f : X \\to [0,\\infty]$, and let $n$ be a natural number. For $x \\in X$,","title":"Inf-convolution with a multiple of the distance"},{"id":"Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution/abs-to-real-g-sub-le-mul-dist","kind":"node","lean":"MathlibExt.MeasureTheory.InfConvolutionLipschitzWanted.abs_toReal_g_sub_le_mul_dist","parent":19021,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution/index.html#abs-to-real-g-sub-le-mul-dist","scope":"Analysis/convex-analysis-and-variations/lipschitz-regularization-by-inf-convolution","status":"can_prove","summary":"Let $X$ be a metric space with metric $d$, and let $f : X \\to [0,\\infty]$ be lower semicontinuous with $f(p) < \\infty$ for at least one $p \\in X$. For a natural number $n$ and $x \\in X$ let","title":"The inf-convolution of a lower semicontinuous function is Lipschitz"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/moreau-decomposition.json","id":"Analysis/convex-analysis-and-variations/moreau-decomposition","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","title":"Moreau decomposition"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/convex-projection-eq-iff","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.convexProjection_eq_iff","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#convex-projection-eq-iff","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"Variational characterization of the projection onto a cone with complete carrier: `p` is the projection of `x` iff the residual pairs nonpositively with the cone and is orthogonal to `p`. The ambient space need not be complete.","title":"Variational characterization of projection onto a convex cone"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/convex-projection-inner-le-zero","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.convexProjection_inner_le_zero","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#convex-projection-inner-le-zero","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"The residual of the cone projection pairs nonpositively with every point of the cone.","title":"Cone projection residual pairs nonpositively with the cone"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/convex-projection-inner-self","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.convexProjection_inner_self","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#convex-projection-inner-self","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"The residual of the cone projection is orthogonal to the projection.","title":"Cone projection residual is orthogonal to the projection"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/convex-projection-eq-of-forall-inner-le","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.convexProjection_eq_of_forall_inner_le_zero","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#convex-projection-eq-of-forall-inner-le","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"A point satisfying the cone variational characterization is the projection.","title":"Variational characterization determines the cone projection"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/polar-cone","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.polarCone","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#polar-cone","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"The negative polar is the standard dual cone of the negated set. Continuity makes its carrier closed without requiring completeness of the ambient space.","title":"Negative polar cone"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/mem-polar-cone","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.mem_polarCone","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#mem-polar-cone","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"Membership in the negative polar uses the nonpositive inner-product convention.","title":"Membership in the negative polar cone"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/sub-convex-projection-mem-polar-cone","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.sub_convexProjection_mem_polarCone","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#sub-convex-projection-mem-polar-cone","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"The residual of a cone projection lies in its negative polar.","title":"Cone projection residual lies in the negative polar"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/polar-cone-eq-inner-dual-neg","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.polarCone_eq_innerDual_neg","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#polar-cone-eq-inner-dual-neg","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"In a complete ambient space, the negative polar agrees with Mathlib's inner dual of the negated carrier.","title":"Negative polar agrees with the inner dual of the negated cone"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/convex-projection-polar-cone","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.convexProjection_polarCone","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#convex-projection-polar-cone","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"The cone projection's residual is the projection onto its negative polar. This accepts a Mathlib proper cone directly.","title":"Moreau decomposition, polar form"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/moreau-decomposition","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.moreau_decomposition","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#moreau-decomposition","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"A point is the sum of its projections onto a proper cone and its negative polar, and the projections are orthogonal.","title":"Moreau decomposition"},{"id":"Analysis/convex-analysis-and-variations/moreau-decomposition/polar-cone-polar-cone","kind":"node","lean":"MathlibExt.Analysis.Convex.MoreauDecomposition.polarCone_polarCone","parent":19024,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/moreau-decomposition/index.html#polar-cone-polar-cone","scope":"Analysis/convex-analysis-and-variations/moreau-decomposition","status":"fully_proved","summary":"The negative bipolar of a proper cone is itself, by Mathlib's inner-dual bipolar theorem.","title":"Bipolar theorem for closed convex cones"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/mountain-pass-theorem.json","id":"Analysis/convex-analysis-and-variations/mountain-pass-theorem","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/mountain-pass-theorem/index.html","scope":"Analysis/convex-analysis-and-variations/mountain-pass-theorem","status":"can_prove","title":"Mountain pass theorem"},{"id":"Analysis/convex-analysis-and-variations/mountain-pass-theorem/is-critical-point","kind":"node","lean":"MathlibExt.Analysis.MountainPassWanted.Analysis.MountainPassProblem.IsCriticalPoint","parent":19036,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/mountain-pass-theorem/index.html#is-critical-point","scope":"Analysis/convex-analysis-and-variations/mountain-pass-theorem","status":"fully_proved","summary":"Let $E$ be a real Banach space and $f : E \\to \\mathbb{R}$. A point $x \\in E$ is a critical point of $f$ when the Fr\u00e9chet derivative of $f$ at $x$ vanishes, $f'(x) = 0$. Here $f'(x)$ is taken to be $0$ when $f$ is not differentiable at $x$, so such points are critical as well.","title":"Critical point"},{"id":"Analysis/convex-analysis-and-variations/mountain-pass-theorem/palais-smale","kind":"node","lean":"MathlibExt.Analysis.MountainPassWanted.Analysis.MountainPassProblem.PalaisSmale","parent":19036,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/mountain-pass-theorem/index.html#palais-smale","scope":"Analysis/convex-analysis-and-variations/mountain-pass-theorem","status":"fully_proved","summary":"Let $E$ be a real Banach space. A function $f : E \\to \\mathbb{R}$ satisfies the Palais\u2013Smale condition when every sequence $(u_k)$ in $E$ such that $(f(u_k))$ is bounded and the Fr\u00e9chet derivatives $f'(u_k)$ tend to $0$ in operator norm has a subsequence converging in $E$. Here $f'(x)$ is taken to be $0$ at points where $f$ is not differentiable.","title":"Palais\u2013Smale condition"},{"id":"Analysis/convex-analysis-and-variations/mountain-pass-theorem/mountain-range","kind":"node","lean":"MathlibExt.Analysis.MountainPassWanted.Analysis.MountainPassProblem.MountainRange","parent":19036,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/mountain-pass-theorem/index.html#mountain-range","scope":"Analysis/convex-analysis-and-variations/mountain-pass-theorem","status":"fully_proved","summary":"Let $E$ be a real Banach space, $f : E \\to \\mathbb{R}$, $a, b \\in E$ and $\\varepsilon, r \\in \\mathbb{R}$. The points $a$ and $b$ are separated by a mountain range of $f$ at height $\\varepsilon$ and radius $r$ when $f(a) = 0$, $\\varepsilon > 0$, $r > 0$, $f(y) \\ge \\varepsilon$ for every $y \\in E$ with $\\|y - a\\| = r$, $\\|b - a\\| > r$ and $f(b) \\le 0$.","title":"Mountain range"},{"id":"Analysis/convex-analysis-and-variations/mountain-pass-theorem/mountain-pass-level","kind":"node","lean":"MathlibExt.Analysis.MountainPassWanted.Analysis.MountainPassProblem.mountainPassLevel","parent":19036,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/mountain-pass-theorem/index.html#mountain-pass-level","scope":"Analysis/convex-analysis-and-variations/mountain-pass-theorem","status":"fully_proved","summary":"Let $E$ be a real Banach space, $f : E \\to \\mathbb{R}$ and $a, b \\in E$. The mountain pass level of $f$ between $a$ and $b$ is","title":"Mountain pass level"},{"id":"Analysis/convex-analysis-and-variations/mountain-pass-theorem/mountain-pass","kind":"node","lean":"MathlibExt.Analysis.MountainPassWanted.Analysis.MountainPassProblem.mountain_pass","parent":19036,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/mountain-pass-theorem/index.html#mountain-pass","scope":"Analysis/convex-analysis-and-variations/mountain-pass-theorem","status":"can_prove","summary":"Let $E$ be a real Banach space and let $f : E \\to \\mathbb{R}$ be continuously differentiable and satisfy the Palais\u2013Smale condition. Let $a, b \\in E$ and let $\\varepsilon, r$ be real numbers such that $a$ and $b$ are separated by a mountain range of $f$ at height $\\varepsilon$ and radius $r$: $f(a) = 0$, $\\varepsilon > 0$, $r > 0$, $f(y) \\ge \\varepsilon$ for every $y$ with $\\|y - a\\| = r$, $\\|b - a\\| > r$ and\u2026","title":"Mountain pass theorem"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/polar-sets.json","id":"Analysis/convex-analysis-and-variations/polar-sets","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","title":"Polar sets"},{"id":"Analysis/convex-analysis-and-variations/polar-sets/polar-set","kind":"node","lean":"MetaMathlibExt.polarSet","parent":19042,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html#polar-set","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","summary":"Let $E$ be a real inner product space. The polar of a set $A \\subseteq E$ is the set $A^\\circ = \\{y \\in E : \\langle x, y \\rangle \\le 1 \\text{ for all } x \\in A\\}$. This uses the one-sided bound $\\langle x, y \\rangle \\le 1$, not $|\\langle x, y \\rangle| \\le 1$.","title":"Polar of a set"},{"id":"Analysis/convex-analysis-and-variations/polar-sets/mem-polar-set","kind":"node","lean":"MetaMathlibExt.mem_polarSet","parent":19042,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html#mem-polar-set","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","summary":"Let $E$ be a real inner product space, $A \\subseteq E$ and $y \\in E$. Then $y$ lies in the polar $A^\\circ$ of $A$ if and only if $\\langle x, y \\rangle \\le 1$ for every $x \\in A$.","title":"Membership in the polar"},{"id":"Analysis/convex-analysis-and-variations/polar-sets/polar-set-empty","kind":"node","lean":"MetaMathlibExt.polarSet_empty","parent":19042,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html#polar-set-empty","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","summary":"In a real inner product space $E$, the polar of the empty set is all of $E$, since the defining condition is vacuous.","title":"Polar of the empty set"},{"id":"Analysis/convex-analysis-and-variations/polar-sets/zero-mem-polar-set","kind":"node","lean":"MetaMathlibExt.zero_mem_polarSet","parent":19042,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html#zero-mem-polar-set","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","summary":"In a real inner product space $E$, for every set $A \\subseteq E$ the origin $0$ lies in the polar $A^\\circ$ of $A$, since $\\langle x, 0 \\rangle = 0 \\le 1$ for every $x \\in A$.","title":"The origin lies in every polar"},{"id":"Analysis/convex-analysis-and-variations/polar-sets/polar-set-subset-polar-set","kind":"node","lean":"MetaMathlibExt.polarSet_subset_polarSet","parent":19042,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html#polar-set-subset-polar-set","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","summary":"Let $A$ and $B$ be subsets of a real inner product space. If $A \\subseteq B$, then the polar $B^\\circ$ of $B$ is contained in the polar $A^\\circ$ of $A$.","title":"The polar reverses inclusion"},{"id":"Analysis/convex-analysis-and-variations/polar-sets/polar-set-closed-unit-ball","kind":"node","lean":"MetaMathlibExt.polarSet_closedUnitBall","parent":19042,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html#polar-set-closed-unit-ball","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","summary":"In a real inner product space $E$, the polar of the closed unit ball $\\{x \\in E : \\|x\\| \\le 1\\}$ is the closed unit ball itself.","title":"The closed unit ball is self-polar"},{"id":"Analysis/convex-analysis-and-variations/polar-sets/polar-set-coordinate-cube","kind":"node","lean":"MetaMathlibExt.polarSet_coordinateCube","parent":19042,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polar of the cube is the "],["math","\\(\\ell^1\\)"],["text"," unit ball"]],"source":"The polar of the cube is the $\\ell^1$ unit ball"},"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html#polar-set-coordinate-cube","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","summary":"For every $n$, the polar of the coordinate cube $[-1, 1]^n$ in the Euclidean space $\\mathbb{R}^n$, that is the set of all $y$ with $\\langle x, y \\rangle \\le 1$ for every $x$ in the cube, is the closed $\\ell^1$ unit ball $\\{y \\in \\mathbb{R}^n : \\sum_i |y_i| \\le 1\\}$.","title":"The polar of the cube is the $\\ell^1$ unit ball"},{"id":"Analysis/convex-analysis-and-variations/polar-sets/polar-set-image-eq-preimage-adjoint","kind":"node","lean":"MetaMathlibExt.polarSet_image_eq_preimage_adjoint","parent":19042,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/polar-sets/index.html#polar-set-image-eq-preimage-adjoint","scope":"Analysis/convex-analysis-and-variations/polar-sets","status":"fully_proved","summary":"Let $E$ and $F$ be finite-dimensional real inner product spaces, let $f : E \\to F$ be a linear map with adjoint $f^* : F \\to E$, and let $K \\subseteq E$. Writing $S^\\circ$ for the polar of a set $S$, we have $(f(K))^\\circ = (f^*)^{-1}(K^\\circ)$, that is, $y \\in F$ lies in the polar of $f(K)$ if and only if $f^*(y)$ lies in the polar of $K$.","title":"Polar of a linear image"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/stampacchia-s-theorem.json","id":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/stampacchia-s-theorem/index.html","scope":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem","status":"fully_proved","title":"Stampacchia's theorem"},{"id":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem/stampacchia-eq-of-univ","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Stampacchia.stampacchia_eq_of_univ","parent":19051,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/stampacchia-s-theorem/index.html#stampacchia-eq-of-univ","scope":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem","status":"fully_proved","summary":"On the whole space, a variational-inequality solution satisfies `A u = f`.","title":"Variational inequality on the whole space reduces to an equation"},{"id":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem/exists-unique-stampacchia","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Stampacchia.exists_unique_stampacchia","parent":19051,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/stampacchia-s-theorem/index.html#exists-unique-stampacchia","scope":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem","status":"fully_proved","summary":"For a coercive bounded linear operator `A` on a real Hilbert space, a nonempty closed convex set `K`, and `f \u2208 E`, there is exactly one `u \u2208 K` with `\u27eaA u - f, v - u\u27eb_\u211d \u2265 0` for all `v \u2208 K`.","title":"Stampacchia's theorem, operator form"},{"id":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem/exists-unique-stampacchia-bilinear","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Stampacchia.exists_unique_stampacchia_bilinear","parent":19051,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/stampacchia-s-theorem/index.html#exists-unique-stampacchia-bilinear","scope":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem","status":"fully_proved","summary":"For a coercive bounded bilinear form `B` on a real Hilbert space, a nonempty closed convex set `K`, and a continuous linear functional `\u03c6`, there is exactly one `u \u2208 K` with `B u (v - u) \u2265 \u03c6 (v - u)` for all `v \u2208 K`.","title":"Stampacchia's theorem, bilinear-form form"},{"id":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem/exists-eq-of-coercive","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Stampacchia.exists_eq_of_coercive","parent":19051,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/stampacchia-s-theorem/index.html#exists-eq-of-coercive","scope":"Analysis/convex-analysis-and-variations/stampacchia-s-theorem","status":"fully_proved","summary":"A coercive bounded linear operator on a real Hilbert space is surjective. This is the whole-space case of Stampacchia's theorem (compare Mathlib's Lax\u2013Milgram theorem, which is the analogous statement for bilinear forms).","title":"Coercive bounded operators on a real Hilbert space are surjective"},{"data_url":"scopes/Analysis/convex-analysis-and-variations/tikhonov-regularization.json","id":"Analysis/convex-analysis-and-variations/tikhonov-regularization","kind":"scope","navigation":"module","parent":18991,"result":false,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","title":"Tikhonov regularization"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/tikhonov-functional","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.tikhonovFunctional","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#tikhonov-functional","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The Tikhonov (ridge) functional `J(x) = \u2016A x - b\u2016 ^ 2 + \u039b * \u2016x\u2016 ^ 2`.","title":"Tikhonov functional"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/tikhonov-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.tikhonovOperator","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#tikhonov-operator","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal operator `A\u2020 \u2218L A + \u039b \u2022 1` of the Tikhonov problem.","title":"Normal operator of the Tikhonov problem"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/tikhonov-operator-inner-self","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.tikhonovOperator_inner_self","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#tikhonov-operator-inner-self","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The inner product of the normal operator applied to `x` with `x` itself.","title":"Inner product of the Tikhonov normal operator with its argument"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/re-inner-tikhonov-operator-self","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.re_inner_tikhonovOperator_self","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#re-inner-tikhonov-operator-self","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The real part of the inner product of the normal operator applied to `x` with `x` itself.","title":"Real part of the Tikhonov normal operator's quadratic form"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/is-self-adjoint-tikhonov-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.isSelfAdjoint_tikhonovOperator","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#is-self-adjoint-tikhonov-operator","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal operator is self-adjoint, for every real `\u039b`.","title":"Tikhonov normal operator is self-adjoint"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/tikhonov-operator-le-norm","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.tikhonovOperator_le_norm","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#tikhonov-operator-le-norm","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal operator is bounded below by `\u039b`, for every real `\u039b`.","title":"Tikhonov normal operator is bounded below by the regularization parameter"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/ker-tikhonov-operator-eq-bot","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.ker_tikhonovOperator_eq_bot","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#ker-tikhonov-operator-eq-bot","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal operator has trivial kernel when `\u039b > 0`.","title":"Tikhonov normal operator has trivial kernel"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/is-closed-range-tikhonov-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.isClosed_range_tikhonovOperator","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#is-closed-range-tikhonov-operator","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal operator has closed range when `\u039b > 0`, since it is bounded below.","title":"Tikhonov normal operator has closed range"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/range-tikhonov-operator-eq-top","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.range_tikhonovOperator_eq_top","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#range-tikhonov-operator-eq-top","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal operator is surjective when `\u039b > 0`: its range is closed and its orthogonal complement is its kernel, which is trivial since the operator is self-adjoint.","title":"Tikhonov normal operator is surjective"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/bijective-tikhonov-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.bijective_tikhonovOperator","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#bijective-tikhonov-operator","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal operator is bijective when `\u039b > 0`.","title":"Tikhonov normal operator is bijective"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/is-unit-tikhonov-operator","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.isUnit_tikhonovOperator","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#is-unit-tikhonov-operator","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal operator is a unit when `\u039b > 0`.","title":"Tikhonov normal operator is invertible"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/exists-tikhonov-normal-equations","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.exists_tikhonov_normal_equations","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#exists-tikhonov-normal-equations","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The normal equations are solvable when `\u039b > 0`.","title":"Solvability of the Tikhonov normal equations"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/tikhonov-functional-add-of-normal-equati","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.tikhonovFunctional_add_of_normal_equations","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#tikhonov-functional-add-of-normal-equati","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The quadratic identity for the Tikhonov functional around a solution of the normal equations: the cross term vanishes by the normal equations.","title":"Quadratic identity for the Tikhonov functional at a normal-equation solution"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/is-min-on-tikhonov-functional-of-normal","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.isMinOn_tikhonovFunctional_of_normal_equations","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#is-min-on-tikhonov-functional-of-normal","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"A solution of the normal equations minimizes the Tikhonov functional.","title":"Solutions of the normal equations minimize the Tikhonov functional"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/eq-of-is-min-on-tikhonov-functional","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.eq_of_isMinOn_tikhonovFunctional","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#eq-of-is-min-on-tikhonov-functional","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"A minimizer of the Tikhonov functional equals a solution of the normal equations, when `\u039b > 0`.","title":"Tikhonov minimizers solve the normal equations"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/exists-unique-tikhonov-minimizer","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.existsUnique_tikhonov_minimizer","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#exists-unique-tikhonov-minimizer","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"The Tikhonov functional has a unique minimizer when `\u039b > 0`.","title":"Existence and uniqueness of the Tikhonov minimizer"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/norm-le-of-tikhonov-normal-equations","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.norm_le_of_tikhonov_normal_equations","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#norm-le-of-tikhonov-normal-equations","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"Stability bound for a solution of the normal equations.","title":"Stability bound for Tikhonov regularized solutions"},{"id":"Analysis/convex-analysis-and-variations/tikhonov-regularization/tendsto-tikhonov-solution-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.TikhonovRegularization.tendsto_tikhonov_solution_zero","parent":19056,"result":true,"route":"../roadmap/Analysis/convex-analysis-and-variations/tikhonov-regularization/index.html#tendsto-tikhonov-solution-zero","scope":"Analysis/convex-analysis-and-variations/tikhonov-regularization","status":"fully_proved","summary":"Solutions of the Tikhonov normal equations tend to `0` as `\u039b \u2192 \u221e`.","title":"Tikhonov solutions tend to zero as the parameter grows"},{"data_url":"scopes/Analysis/fixed-point-theory.json","id":"Analysis/fixed-point-theory","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/fixed-point-theory/index.html","scope":"Analysis/fixed-point-theory","status":"can_prove","summary":"Fixed points of nonexpansive, Lipschitz and affine maps: the Browder\u2013G\u00f6hde\u2013Kirk theorem and Kirk's normal structure, Markov\u2013Kakutani, Browder's demiclosedness principle, maps with contracting iterates.","title":"Fixed-point theory"},{"id":"Analysis/fixed-point-theory/exists-fixed-point-of-iterate-lipschitz","kind":"node","lean":"MathlibExt.Analysis.Calculus.IterateLipschitzNullFixedPointWanted.exists_fixedPoint_of_iterate_lipschitz_tendsto_zero","parent":19075,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/index.html#exists-fixed-point-of-iterate-lipschitz","scope":"Analysis/fixed-point-theory","status":"can_prove","summary":"Let $X$ be a nonempty complete metric space, let $f : X \\to X$, and write $f^n$ for the $n$-th iterate of $f$. Let $(q_n)_{n \\in \\mathbb{N}}$ be a sequence of nonnegative real numbers with $q_n \\to 0$ such that $d(f^n(x), f^n(y)) \\le q_n \\, d(x, y)$ for every $n \\ge 1$ and all $x, y \\in X$. Then $f$ has a fixed point: there is $x \\in X$ with $f(x) = x$.","title":"Fixed points of maps whose iterates have null Lipschitz constants"},{"id":"Analysis/fixed-point-theory/markov-kakutani-common-fixed-point","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.MarkovKakutani.markovKakutani_commonFixedPoint","parent":19075,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/index.html#markov-kakutani-common-fixed-point","scope":"Analysis/fixed-point-theory","status":"fully_proved","summary":"Let $K$ be a nonempty compact convex subset of a real normed space $E$, and let $(F_i)_{i \\in I}$ be a family of continuous affine maps $E \\to E$ such that each $F_i$ maps $K$ into $K$ and $F_i(F_j(x)) = F_j(F_i(x))$ for all $i, j \\in I$ and all $x \\in E$. Then there is $x \\in K$ with $F_i(x) = x$ for every $i \\in I$.","title":"Markov\u2013Kakutani fixed-point theorem"},{"data_url":"scopes/Analysis/fixed-point-theory/browder-s-demiclosedness-principle.json","id":"Analysis/fixed-point-theory/browder-s-demiclosedness-principle","kind":"scope","navigation":"module","parent":19075,"result":false,"route":"../roadmap/Analysis/fixed-point-theory/browder-s-demiclosedness-principle/index.html","scope":"Analysis/fixed-point-theory/browder-s-demiclosedness-principle","status":"fully_proved","title":"Browder's demiclosedness principle"},{"id":"Analysis/fixed-point-theory/browder-s-demiclosedness-principle/demiclosedness-of-lipschitz-with-one","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Demiclosedness.demiclosedness_of_lipschitzWith_one","parent":19078,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-s-demiclosedness-principle/index.html#demiclosedness-of-lipschitz-with-one","scope":"Analysis/fixed-point-theory/browder-s-demiclosedness-principle","status":"fully_proved","summary":"If `T` is nonexpansive, `u n \u21c0 x` weakly, and `u n - T (u n) \u2192 0`, then `T x = x`.","title":"Browder's demiclosedness principle"},{"id":"Analysis/fixed-point-theory/browder-s-demiclosedness-principle/fixed-point-of-weakly-tendsto-of-forall","kind":"node","lean":"MathlibExt.Analysis.InnerProductSpace.Demiclosedness.fixedPoint_of_weakly_tendsto_of_forall_fixedPoint","parent":19078,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-s-demiclosedness-principle/index.html#fixed-point-of-weakly-tendsto-of-forall","scope":"Analysis/fixed-point-theory/browder-s-demiclosedness-principle","status":"fully_proved","summary":"The fixed-point set of a nonexpansive map is weakly sequentially closed: if every `u n` is a fixed point of `T` and `u n \u21c0 x` weakly, then `T x = x`.","title":"Fixed points of a nonexpansive map are weakly sequentially closed"},{"data_url":"scopes/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem.json","id":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","kind":"scope","navigation":"module","parent":19075,"result":false,"route":"../roadmap/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/index.html","scope":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","status":"can_prove","title":"Browder\u2013G\u00f6hde\u2013Kirk fixed-point theorem"},{"id":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/browder-goehde-kirk-fixed-point","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BrowderKirkWanted.browderGoehdeKirk_fixedPoint","parent":19081,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/index.html#browder-goehde-kirk-fixed-point","scope":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","status":"fully_proved","summary":"Let $E$ be a uniformly convex real Banach space and let $K \\subseteq E$ be a nonempty, closed, bounded, convex set. If a map $f : E \\to E$ sends $K$ into $K$ and is nonexpansive on $K$, that is, $\\|f(x) - f(y)\\| \\le \\|x - y\\|$ for all $x, y \\in K$, then $f$ has a fixed point in $K$: there is $y \\in K$ with $f(y) = y$.","title":"Browder\u2013G\u00f6hde\u2013Kirk fixed point theorem"},{"id":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/metric-diameter","kind":"node","lean":"MathlibExt.Topology.KirkNormalStructureWanted.Topology.KirkNormalStructure.metricDiameter","parent":19081,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/index.html#metric-diameter","scope":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","status":"fully_proved","summary":"The diameter of a set $s$ in a real normed space is the supremum in $\\mathbb{R}$ of the distances $\\operatorname{dist}(x, y)$ over all $x, y \\in s$; by the convention for real suprema it is $0$ when $s$ is empty or these distances are unbounded.","title":"Diameter of a set"},{"id":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/point-radius-in","kind":"node","lean":"MathlibExt.Topology.KirkNormalStructureWanted.Topology.KirkNormalStructure.pointRadiusIn","parent":19081,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/index.html#point-radius-in","scope":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","status":"fully_proved","summary":"The radius of a point $x$ in a set $s$ of a real normed space is the supremum in $\\mathbb{R}$ of the distances $\\operatorname{dist}(x, y)$ over all $y \\in s$; by the convention for real suprema it is $0$ when $s$ is empty or these distances are unbounded.","title":"Radius of a point in a set"},{"id":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/is-diametral-point","kind":"node","lean":"MathlibExt.Topology.KirkNormalStructureWanted.Topology.KirkNormalStructure.IsDiametralPoint","parent":19081,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/index.html#is-diametral-point","scope":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","status":"fully_proved","summary":"A point $x$ of a real normed space is diametral in a set $s$ if $x \\in s$ and the radius of $x$ in $s$ equals the diameter of $s$.","title":"Diametral point"},{"id":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/has-normal-structure","kind":"node","lean":"MathlibExt.Topology.KirkNormalStructureWanted.Topology.KirkNormalStructure.HasNormalStructure","parent":19081,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/index.html#has-normal-structure","scope":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","status":"fully_proved","summary":"A set $K$ in a real normed space has normal structure if every convex subset $H \\subseteq K$ with at least two points contains a point that is not diametral in $H$.","title":"Normal structure"},{"id":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/is-nonexpansive-self-map","kind":"node","lean":"MathlibExt.Topology.KirkNormalStructureWanted.Topology.KirkNormalStructure.IsNonexpansiveSelfMap","parent":19081,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/index.html#is-nonexpansive-self-map","scope":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","status":"fully_proved","summary":"A map $T : K \\to K$ from a subset $K$ of a real normed space to itself is nonexpansive if $\\|T(x) - T(y)\\| \\le \\|x - y\\|$ for all $x, y \\in K$, that is, if $T$ is $1$-Lipschitz for the distance inherited from the space.","title":"Nonexpansive self-map"},{"id":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/kirk-normal-structure","kind":"node","lean":"MathlibExt.Topology.KirkNormalStructureWanted.Topology.KirkNormalStructure.kirk_normal_structure","parent":19081,"result":true,"route":"../roadmap/Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem/index.html#kirk-normal-structure","scope":"Analysis/fixed-point-theory/browder-g-hde-kirk-fixed-point-theorem","status":"can_prove","summary":"Let $E$ be a reflexive real Banach space, meaning that the canonical embedding of $E$ into its double dual is surjective, and let $K \\subseteq E$ be a nonempty, closed, bounded, convex set with normal structure. Then every nonexpansive self-map $T : K \\to K$ has a fixed point.","title":"Kirk's fixed-point theorem for sets with normal structure"},{"data_url":"scopes/Analysis/approximation-theory.json","id":"Analysis/approximation-theory","kind":"scope","parent":15496,"result":false,"route":"../roadmap/Analysis/approximation-theory/index.html","scope":"Analysis/approximation-theory","status":"can_prove","summary":"Approximation and interpolation of functions: Chebyshev equioscillation, best approximation, Lagrange interpolation, Bernstein polynomials and Voronovskaya, Korovkin, M\u00fcntz\u2013Sz\u00e1sz, Runge, Kolmogorov superposition, universal approximation.","title":"Approximation theory"},{"id":"Analysis/approximation-theory/muntz-szasz-nat","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.MuntzSzaszWanted.muntz_szasz_nat","parent":19089,"result":true,"route":"../roadmap/Analysis/approximation-theory/index.html#muntz-szasz-nat","scope":"Analysis/approximation-theory","status":"fully_proved","summary":"Let $\\Lambda : \\mathbb{N} \\to \\mathbb{N}$ be strictly increasing with $\\Lambda(0) = 0$. The real linear span of the monomials $x \\mapsto x^{\\Lambda(n)}$, $n \\in \\mathbb{N}$, restricted to $[0, 1]$, is dense in $C([0, 1], \\mathbb{R})$ with the uniform norm if and only if the series $\\sum_{n \\ge 0} 1/\\Lambda(n+1)$ diverges, that is, the sequence $(1/\\Lambda(n+1))_{n \\in \\mathbb{N}}$ is not summable.","title":"M\u00fcntz\u2013Sz\u00e1sz theorem for integer exponents"},{"id":"Analysis/approximation-theory/runge","kind":"node","lean":"MathlibExt.Analysis.RungeWanted.ComplexAnalysis.runge","parent":19089,"result":true,"route":"../roadmap/Analysis/approximation-theory/index.html#runge","scope":"Analysis/approximation-theory","status":"can_prove","summary":"Let $K \\subseteq \\mathbb{C}$ be compact, let $U \\subseteq \\mathbb{C}$ be open with $K \\subseteq U$, and let $f : \\mathbb{C} \\to \\mathbb{C}$ be analytic at every point of $U$. Then for every real $\\varepsilon > 0$ there are complex polynomials $p$ and $q$ such that $q(z) \\ne 0$ for every $z \\in K$ and $|f(z) - p(z)/q(z)| < \\varepsilon$ for every $z \\in K$.","title":"Runge's theorem"},{"id":"Analysis/approximation-theory/universal-approximation","kind":"node","lean":"MetaMathlibExt.universal_approximation","parent":19089,"result":true,"route":"../roadmap/Analysis/approximation-theory/index.html#universal-approximation","scope":"Analysis/approximation-theory","status":"fully_proved","summary":"Let $n$ be a natural number, let $\\mathbb{R}^n$ carry its standard inner product $\\langle \\cdot, \\cdot \\rangle$, let $K \\subseteq \\mathbb{R}^n$ be compact, and let $\\sigma : \\mathbb{R} \\to \\mathbb{R}$ be continuous and not a polynomial, that is, there is no real polynomial $p$ with $\\sigma(x) = p(x)$ for all real $x$. Then for every function $f : \\mathbb{R}^n \\to \\mathbb{R}$ that is continuous on $K$ and every\u2026","title":"Universal approximation theorem"},{"data_url":"scopes/Analysis/approximation-theory/bernstein-polynomials.json","id":"Analysis/approximation-theory/bernstein-polynomials","kind":"scope","navigation":"module","parent":19089,"result":false,"route":"../roadmap/Analysis/approximation-theory/bernstein-polynomials/index.html","scope":"Analysis/approximation-theory/bernstein-polynomials","status":"fully_proved","title":"Bernstein polynomials"},{"id":"Analysis/approximation-theory/bernstein-polynomials/mean","kind":"node","lean":"bernstein.mean","parent":19093,"result":true,"route":"../roadmap/Analysis/approximation-theory/bernstein-polynomials/index.html#mean","scope":"Analysis/approximation-theory/bernstein-polynomials","status":"fully_proved","summary":"For `n \u2260 0`, `\u2211 k : Fin (n + 1), (x - k / n) * bernstein n k x = 0`: the binomial frequency `S / n` has mean `x`.","title":"First central moment of the Bernstein weights"},{"id":"Analysis/approximation-theory/bernstein-polynomials/fourth-moment","kind":"node","lean":"bernstein.fourth_moment","parent":19093,"result":true,"route":"../roadmap/Analysis/approximation-theory/bernstein-polynomials/index.html#fourth-moment","scope":"Analysis/approximation-theory/bernstein-polynomials","status":"fully_proved","summary":"For `n \u2260 0`, `\u2211 k : Fin (n + 1), (x - k / n) ^ 4 * bernstein n k x = x (1 - x) (1 + 3 (n - 2) x (1 - x)) / n\u00b3`, i.e. `E[(S - n x) ^ 4] = n x (1 - x) (1 + 3 (n - 2) x (1 - x))` for `S ~ Bin(n, x)`.","title":"Fourth central moment of the Bernstein weights"},{"id":"Analysis/approximation-theory/bernstein-polynomials/fourth-moment-le","kind":"node","lean":"bernstein.fourth_moment_le","parent":19093,"result":true,"route":"../roadmap/Analysis/approximation-theory/bernstein-polynomials/index.html#fourth-moment-le","scope":"Analysis/approximation-theory/bernstein-polynomials","status":"fully_proved","summary":"The fourth central moment of the Bernstein weights is at most `1 / n ^ 2`, uniformly in `x \u2208 [0, 1]`.","title":"Fourth central moment bound for Bernstein weights"},{"id":"Analysis/approximation-theory/bernstein-polynomials/tendsto-voronovskaya","kind":"node","lean":"bernsteinApproximation.tendsto_voronovskaya","parent":19093,"result":true,"route":"../roadmap/Analysis/approximation-theory/bernstein-polynomials/index.html#tendsto-voronovskaya","scope":"Analysis/approximation-theory/bernstein-polynomials","status":"fully_proved","summary":"For `f : \u211d \u2192 \u211d` of class `C\u00b2` and `x \u2208 [0, 1]`, the Bernstein approximations of `f` on `[0, 1]` satisfy `n \u00b7 (bernsteinApproximation n f x - f x) \u2192 x (1 - x) / 2 \u00b7 f''(x)` as `n \u2192 \u221e`.","title":"Voronovskaya's theorem (1932)"},{"data_url":"scopes/Analysis/approximation-theory/best-approximation-in-normed-spaces.json","id":"Analysis/approximation-theory/best-approximation-in-normed-spaces","kind":"scope","navigation":"module","parent":19089,"result":false,"route":"../roadmap/Analysis/approximation-theory/best-approximation-in-normed-spaces/index.html","scope":"Analysis/approximation-theory/best-approximation-in-normed-spaces","status":"fully_proved","title":"Best approximation in normed spaces"},{"id":"Analysis/approximation-theory/best-approximation-in-normed-spaces/exists-best-approximation-of-finite-dime","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.BestApproximation.exists_bestApproximation_of_finiteDimensional","parent":19098,"result":true,"route":"../roadmap/Analysis/approximation-theory/best-approximation-in-normed-spaces/index.html#exists-best-approximation-of-finite-dime","scope":"Analysis/approximation-theory/best-approximation-in-normed-spaces","status":"fully_proved","summary":"Every point of a normed space has a nearest point in a finite-dimensional subspace (proximinality). No completeness hypothesis on the ambient space is needed: completeness enters only through the finite-dimensional subspace, whose closed balls are compact.","title":"Existence of best approximations in finite-dimensional subspaces"},{"id":"Analysis/approximation-theory/best-approximation-in-normed-spaces/eq-of-is-best-approximation-of-strict-co","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.BestApproximation.eq_of_isBestApproximation_of_strictConvexSpace","parent":19098,"result":true,"route":"../roadmap/Analysis/approximation-theory/best-approximation-in-normed-spaces/index.html#eq-of-is-best-approximation-of-strict-co","scope":"Analysis/approximation-theory/best-approximation-in-normed-spaces","status":"fully_proved","summary":"In a strictly convex real normed space, a convex set contains at most one point nearest to a given point: the midpoint of two distinct nearest points would be strictly closer, by `norm_midpoint_lt_iff`.","title":"Uniqueness of best approximation in strictly convex spaces"},{"id":"Analysis/approximation-theory/best-approximation-in-normed-spaces/exists-unique-best-approximation-of-fini","kind":"node","lean":"MathlibExt.Analysis.Normed.Module.BestApproximation.exists_unique_bestApproximation_of_finiteDimensional","parent":19098,"result":true,"route":"../roadmap/Analysis/approximation-theory/best-approximation-in-normed-spaces/index.html#exists-unique-best-approximation-of-fini","scope":"Analysis/approximation-theory/best-approximation-in-normed-spaces","status":"fully_proved","summary":"Existence plus uniqueness: a finite-dimensional subspace of a strictly convex real normed space admits a unique best approximation to every point.","title":"Unique best approximation in finite-dimensional subspaces"},{"data_url":"scopes/Analysis/approximation-theory/chebyshev-equioscillation-theorem.json","id":"Analysis/approximation-theory/chebyshev-equioscillation-theorem","kind":"scope","navigation":"module","parent":19089,"result":false,"route":"../roadmap/Analysis/approximation-theory/chebyshev-equioscillation-theorem/index.html","scope":"Analysis/approximation-theory/chebyshev-equioscillation-theorem","status":"fully_proved","title":"Chebyshev equioscillation theorem"},{"id":"Analysis/approximation-theory/chebyshev-equioscillation-theorem/equioscillates","kind":"node","lean":"MathlibExt.Analysis.Approximation.Equioscillation.Equioscillates","parent":19102,"result":true,"route":"../roadmap/Analysis/approximation-theory/chebyshev-equioscillation-theorem/index.html#equioscillates","scope":"Analysis/approximation-theory/chebyshev-equioscillation-theorem","status":"fully_proved","summary":"A polynomial `p` of degree at most `n` equioscillates against `f` on `[a, b]` with amplitude `E` and sign `\u03c3 = \u00b11` at the `n + 2` points `x 0 < \u22ef < x (n+1)`, if `f (x i) - p (x i) = \u03c3 * (-1) ^ i * E` for every `i \u2264 n + 1`, and `|f y - p.eval y| \u2264 E` for every `y \u2208 [a, b]`.","title":"Equioscillation of a polynomial approximation"},{"id":"Analysis/approximation-theory/chebyshev-equioscillation-theorem/eq-of-equioscillates","kind":"node","lean":"MathlibExt.Analysis.Approximation.Equioscillation.eq_of_equioscillates","parent":19102,"result":true,"route":"../roadmap/Analysis/approximation-theory/chebyshev-equioscillation-theorem/index.html#eq-of-equioscillates","scope":"Analysis/approximation-theory/chebyshev-equioscillation-theorem","status":"fully_proved","summary":"If `p` equioscillates against `f` on `[a, b]`, then `p` is the unique polynomial of degree at most `n` whose uniform error on `[a, b]` is at most `E`. In particular `p` is the unique best uniform approximation to `f` among polynomials of degree at most `n`.","title":"Uniqueness of the equioscillating best approximation"},{"id":"Analysis/approximation-theory/chebyshev-equioscillation-theorem/not-lt-of-equioscillates","kind":"node","lean":"MathlibExt.Analysis.Approximation.Equioscillation.not_lt_of_equioscillates","parent":19102,"result":true,"route":"../roadmap/Analysis/approximation-theory/chebyshev-equioscillation-theorem/index.html#not-lt-of-equioscillates","scope":"Analysis/approximation-theory/chebyshev-equioscillation-theorem","status":"fully_proved","summary":"If `p` equioscillates against `f`, then no polynomial of degree at most `n` has uniform error strictly less than `E` on `[a, b]`: `p` is a best uniform approximation.","title":"Chebyshev equioscillation theorem (sufficiency)"},{"id":"Analysis/approximation-theory/chebyshev-equioscillation-theorem/de-la-vallee-poussin","kind":"node","lean":"MathlibExt.Analysis.Approximation.Equioscillation.de_la_vallee_poussin","parent":19102,"result":true,"route":"../roadmap/Analysis/approximation-theory/chebyshev-equioscillation-theorem/index.html#de-la-vallee-poussin","scope":"Analysis/approximation-theory/chebyshev-equioscillation-theorem","status":"fully_proved","summary":"If the error `f - p` of a polynomial `p` of degree at most `n` alternates with a one-sided lower bound `\u03b7` at `n + 2` strictly increasing points of `[a, b]`, `\u03b7 \u2264 \u03c3 * (-1) ^ i * (f (x i) - p (x i))`, then every polynomial of degree at most `n` has uniform error at least `\u03b7` on `[a, b]`.","title":"De la Vall\u00e9e Poussin's theorem"},{"data_url":"scopes/Analysis/approximation-theory/kolmogorov-superposition-theorem.json","id":"Analysis/approximation-theory/kolmogorov-superposition-theorem","kind":"scope","navigation":"module","parent":19089,"result":false,"route":"../roadmap/Analysis/approximation-theory/kolmogorov-superposition-theorem/index.html","scope":"Analysis/approximation-theory/kolmogorov-superposition-theorem","status":"can_prove","title":"Kolmogorov superposition theorem"},{"id":"Analysis/approximation-theory/kolmogorov-superposition-theorem/kolmogorov-superposition","kind":"node","lean":"MathlibExt.Analysis.ContinuousMap.KolmogorovSuperpositionWanted.kolmogorov_superposition","parent":19107,"result":true,"route":"../roadmap/Analysis/approximation-theory/kolmogorov-superposition-theorem/index.html#kolmogorov-superposition","scope":"Analysis/approximation-theory/kolmogorov-superposition-theorem","status":"fully_proved","summary":"Let $n \\ge 2$. There are continuous functions $\\psi_{q,p} : [0,1] \\to \\mathbb{R}$ for $0 \\le q \\le 2n$ and $0 \\le p < n$, depending only on $n$, such that for every continuous function $f : [0,1]^n \\to \\mathbb{R}$ there are continuous functions $\\varphi_q : \\mathbb{R} \\to \\mathbb{R}$ for $0 \\le q \\le 2n$ with $f(x) = \\sum_{q=0}^{2n} \\varphi_q\\left(\\sum_{p=0}^{n-1} \\psi_{q,p}(x_p)\\right)$ for every\u2026","title":"Kolmogorov superposition theorem"},{"id":"Analysis/approximation-theory/kolmogorov-superposition-theorem/kolmogorov-arnold","kind":"node","lean":"MathlibExt.Analysis.KolmogorovArnoldWanted.Analysis.kolmogorov_arnold","parent":19107,"result":true,"route":"../roadmap/Analysis/approximation-theory/kolmogorov-superposition-theorem/index.html#kolmogorov-arnold","scope":"Analysis/approximation-theory/kolmogorov-superposition-theorem","status":"can_prove","summary":"Let $n \\ge 1$ and let $f : \\mathbb{R}^n \\to \\mathbb{R}$ be continuous on the unit cube $[0, 1]^n$. Then there exist a continuous function $g : \\mathbb{R} \\to \\mathbb{R}$ and continuous functions $\\varphi_{k,l} : \\mathbb{R} \\to \\mathbb{R}$ for $0 \\le k \\le 2n$ and $0 \\le l < n$ such that for every $x = (x_0, \\dots, x_{n-1}) \\in [0, 1]^n$,","title":"Kolmogorov\u2013Arnold superposition with a single outer function"},{"data_url":"scopes/Analysis/approximation-theory/korovkin-s-theorem.json","id":"Analysis/approximation-theory/korovkin-s-theorem","kind":"scope","navigation":"module","parent":19089,"result":false,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","title":"Korovkin's theorem"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/korovkin-id","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.korovkinId","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#korovkin-id","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"The function $x \\mapsto x$ on $[0, 1]$, as an element of $C([0, 1], \\mathbb{R})$.","title":"The identity test function"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/korovkin-id-sq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.korovkinIdSq","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#korovkin-id-sq","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"The function $x \\mapsto x^2$ on $[0, 1]$, as an element of $C([0, 1], \\mathbb{R})$.","title":"The square test function"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/korovkin-shift-sq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.korovkinShiftSq","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#korovkin-shift-sq","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"For $x \\in [0, 1]$, the function $y \\mapsto (y - x)^2$ on $[0, 1]$, as an element of $C([0, 1], \\mathbb{R})$.","title":"The shifted square"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/korovkin-shift-sq-eq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.korovkinShiftSq_eq","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#korovkin-shift-sq-eq","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"Write $e(y) = y$ and $q(y) = y^2$ for the identity and square functions and $1$ for the constant function one, as elements of $C([0, 1], \\mathbb{R})$. For every $x \\in [0, 1]$, the function $y \\mapsto (y - x)^2$ in $C([0, 1], \\mathbb{R})$ equals $q - 2x \\cdot e + x^2 \\cdot 1$.","title":"Expansion of the shifted square"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/tendsto-apply-of-tendsto","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.tendsto_apply_of_tendsto","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#tendsto-apply-of-tendsto","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"If a sequence $(u_n)$ in $C([0, 1], \\mathbb{R})$ converges to $e$ in the sup norm, then $u_n(x) \\to e(x)$ for every $x \\in [0, 1]$.","title":"Uniform convergence implies pointwise convergence"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/le-apply-of-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.le_apply_of_le","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#le-apply-of-le","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"Let $L$ be a real-linear operator on $C([0, 1], \\mathbb{R})$ that is positive: $(L f)(x) \\ge 0$ for all $x \\in [0, 1]$ whenever $f(x) \\ge 0$ for all $x \\in [0, 1]$. If $g, h \\in C([0, 1], \\mathbb{R})$ satisfy $g(x) \\le h(x)$ for all $x \\in [0, 1]$, then $(L g)(x) \\le (L h)(x)$ for every $x \\in [0, 1]$.","title":"Positive linear operators are monotone"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/abs-apply-le","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.abs_apply_le","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#abs-apply-le","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"Let $L$ be a real-linear operator on $C([0, 1], \\mathbb{R})$ that is positive: $(L f)(x) \\ge 0$ for all $x \\in [0, 1]$ whenever $f(x) \\ge 0$ for all $x \\in [0, 1]$. If $g, h \\in C([0, 1], \\mathbb{R})$ satisfy $|g(y)| \\le h(y)$ for all $y \\in [0, 1]$, then $|(L g)(x)| \\le (L h)(x)$ for every $x \\in [0, 1]$.","title":"Positive linear operators bound absolute values"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/abs-sub-le-add-sq","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.abs_sub_le_add_sq","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#abs-sub-le-add-sq","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"Let $f \\in C([0, 1], \\mathbb{R})$ have sup norm $\\|f\\|$, let $x \\in [0, 1]$, and let $\\delta > 0$, $\\varepsilon \\ge 0$ and $C \\ge 0$ be real numbers with $2\\|f\\| \\le C \\delta^2$. If $|f(y) - f(x)| < \\varepsilon$ for every $y \\in [0, 1]$ with $|y - x| < \\delta$, then $|f(y) - f(x)| \\le \\varepsilon + C (y - x)^2$ for every $y \\in [0, 1]$.","title":"Quadratic bound from a modulus of continuity"},{"id":"Analysis/approximation-theory/korovkin-s-theorem/korovkin","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.Korovkin.korovkin","parent":19110,"result":true,"route":"../roadmap/Analysis/approximation-theory/korovkin-s-theorem/index.html#korovkin","scope":"Analysis/approximation-theory/korovkin-s-theorem","status":"fully_proved","summary":"Write $e_0$, $e_1$ and $e_2$ for the functions $x \\mapsto 1$, $x \\mapsto x$ and $x \\mapsto x^2$ in $C([0, 1], \\mathbb{R})$. Let $(L_n)_{n \\in \\mathbb{N}}$ be real-linear operators on $C([0, 1], \\mathbb{R})$, each positive: $(L_n f)(x) \\ge 0$ for all $x \\in [0, 1]$ whenever $f(x) \\ge 0$ for all $x \\in [0, 1]$. If $L_n e_i \\to e_i$ in the sup norm for $i = 0, 1, 2$, then $L_n f \\to f$ in the sup norm for every\u2026","title":"Korovkin's theorem"},{"data_url":"scopes/Analysis/approximation-theory/lagrange-interpolation.json","id":"Analysis/approximation-theory/lagrange-interpolation","kind":"scope","navigation":"module","parent":19089,"result":false,"route":"../roadmap/Analysis/approximation-theory/lagrange-interpolation/index.html","scope":"Analysis/approximation-theory/lagrange-interpolation","status":"fully_proved","title":"Lagrange interpolation"},{"id":"Analysis/approximation-theory/lagrange-interpolation/eval-basis","kind":"node","lean":"Lagrange.evalBasis","parent":19120,"result":true,"route":"../roadmap/Analysis/approximation-theory/lagrange-interpolation/index.html#eval-basis","scope":"Analysis/approximation-theory/lagrange-interpolation","status":"fully_proved","summary":"For real nodes $x_0, \\dots, x_{n-1}$, an index $k < n$ and a real number $t$, the value at $t$ of the $k$-th Lagrange basis polynomial is $\\ell_k(t) = \\prod_{j \\ne k} (x_k - x_j)^{-1} (t - x_j)$, the product running over the indices $j < n$ with $j \\ne k$.","title":"Evaluated Lagrange basis polynomial"},{"id":"Analysis/approximation-theory/lagrange-interpolation/lebesgue-function","kind":"node","lean":"Lagrange.lebesgueFunction","parent":19120,"result":true,"route":"../roadmap/Analysis/approximation-theory/lagrange-interpolation/index.html#lebesgue-function","scope":"Analysis/approximation-theory/lagrange-interpolation","status":"fully_proved","summary":"For real nodes $x_0, \\dots, x_{n-1}$ and a real number $t$, the Lebesgue function of the nodes at $t$ is $L_n(t) = \\sum_{k < n} |\\ell_k(t)|$, where $\\ell_k$ is the $k$-th evaluated Lagrange basis polynomial for these nodes.","title":"Lebesgue function"},{"id":"Analysis/approximation-theory/lagrange-interpolation/eval-interpolant","kind":"node","lean":"Lagrange.evalInterpolant","parent":19120,"result":true,"route":"../roadmap/Analysis/approximation-theory/lagrange-interpolation/index.html#eval-interpolant","scope":"Analysis/approximation-theory/lagrange-interpolation","status":"fully_proved","summary":"For real nodes $x_0, \\dots, x_{n-1}$, a function $f : \\mathbb{R} \\to \\mathbb{R}$ and a real number $t$, the Lagrange interpolant of $f$ at $t$ is $\\mathcal{L}^n f(t) = \\sum_{k < n} f(x_k)\\, \\ell_k(t)$, where $\\ell_k$ is the $k$-th evaluated Lagrange basis polynomial for these nodes.","title":"Lagrange interpolant"},{"id":"Analysis/approximation-theory/lagrange-interpolation/eval-basis-self","kind":"node","lean":"Lagrange.evalBasis_self","parent":19120,"result":true,"route":"../roadmap/Analysis/approximation-theory/lagrange-interpolation/index.html#eval-basis-self","scope":"Analysis/approximation-theory/lagrange-interpolation","status":"fully_proved","summary":"If the real nodes $x_0, \\dots, x_{n-1}$ are pairwise distinct, then $\\ell_k(x_k) = 1$ for every $k < n$, where $\\ell_k$ is the $k$-th evaluated Lagrange basis polynomial for these nodes.","title":"Lagrange basis polynomials equal one at their own node"},{"id":"Analysis/approximation-theory/lagrange-interpolation/eval-basis-of-ne","kind":"node","lean":"Lagrange.evalBasis_of_ne","parent":19120,"result":true,"route":"../roadmap/Analysis/approximation-theory/lagrange-interpolation/index.html#eval-basis-of-ne","scope":"Analysis/approximation-theory/lagrange-interpolation","status":"fully_proved","summary":"For real nodes $x_0, \\dots, x_{n-1}$ and indices $k \\ne j$ below $n$, the $k$-th evaluated Lagrange basis polynomial $\\ell_k$ for these nodes satisfies $\\ell_k(x_j) = 0$.","title":"Lagrange basis polynomials vanish at the other nodes"},{"id":"Analysis/approximation-theory/lagrange-interpolation/eval-interpolant-at-node","kind":"node","lean":"Lagrange.evalInterpolant_at_node","parent":19120,"result":true,"route":"../roadmap/Analysis/approximation-theory/lagrange-interpolation/index.html#eval-interpolant-at-node","scope":"Analysis/approximation-theory/lagrange-interpolation","status":"fully_proved","summary":"If the real nodes $x_0, \\dots, x_{n-1}$ are pairwise distinct, then for every function $f : \\mathbb{R} \\to \\mathbb{R}$ and every $j < n$, the Lagrange interpolant of $f$ on these nodes satisfies $\\mathcal{L}^n f(x_j) = f(x_j)$.","title":"The Lagrange interpolant matches the function at the nodes"},{"id":"Analysis/approximation-theory/lagrange-interpolation/lebesgue-function-at-node","kind":"node","lean":"Lagrange.lebesgueFunction_at_node","parent":19120,"result":true,"route":"../roadmap/Analysis/approximation-theory/lagrange-interpolation/index.html#lebesgue-function-at-node","scope":"Analysis/approximation-theory/lagrange-interpolation","status":"fully_proved","summary":"If the real nodes $x_0, \\dots, x_{n-1}$ are pairwise distinct, then their Lebesgue function satisfies $L_n(x_j) = 1$ for every $j < n$.","title":"The Lebesgue function equals one at the nodes"},{"data_url":"scopes/Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow.json","id":"Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow","kind":"scope","navigation":"module","parent":19089,"result":false,"route":"../roadmap/Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow/index.html","scope":"Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow","status":"can_prove","title":"Uniform approximation by partial sums of one power series"},{"id":"Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow/scottish-book-problem159-claim","kind":"node","lean":"MathlibExt.Analysis.ScottishBook159Wanted.ScottishBookProblem159Claim","parent":19128,"result":true,"route":"../roadmap/Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow/index.html#scottish-book-problem159-claim","scope":"Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow","status":"fully_proved","summary":"There is a sequence of real numbers $a_0, a_1, a_2, \\dots$ such that for every $\\varepsilon > 0$ there is a natural number $N$ with the following property: for every function $f : \\mathbb{R} \\to \\mathbb{R}$ that is continuous on $[0, 1]$ and satisfies $f(0) = 0$ and $0 \\le f(x) \\le 1$ for all $x \\in [0, 1]$, there is a natural number $n \\le N$ with $|f(x) - P_n(x)| < \\varepsilon$ for all $x \\in [0, 1]$, where\u2026","title":"The claim of Scottish Book Problem 159"},{"id":"Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow/problem159-refuted","kind":"node","lean":"MathlibExt.Analysis.ScottishBook159Wanted.problem159Refuted","parent":19128,"result":true,"route":"../roadmap/Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow/index.html#problem159-refuted","scope":"Analysis/approximation-theory/uniform-approximation-by-partial-sums-of-one-pow","status":"can_prove","summary":"There is no sequence of real numbers $a_0, a_1, a_2, \\dots$ with the following property: for every $\\varepsilon > 0$ there is a natural number $N$ such that for every function $f : \\mathbb{R} \\to \\mathbb{R}$ that is continuous on $[0, 1]$ and satisfies $f(0) = 0$ and $0 \\le f(x) \\le 1$ for all $x \\in [0, 1]$, there is a natural number $n \\le N$ with $\\left|f(x) - \\sum_{k=0}^{n} a_k x^k\\right| < \\varepsilon$ for all\u2026","title":"Scottish Book Problem 159 has a negative answer"},{"area":"Analysis & Probability","data_url":"chapters/MeasureTheory.json","id":"MeasureTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/MeasureTheory/index.html","scope":"MeasureTheory","status":"can_prove","summary":"Measure theory provides the language of size and integration underlying probability and modern analysis.","title":"Measure Theory"},{"data_url":"scopes/MeasureTheory/geometric-measure-theory.json","id":"MeasureTheory/geometric-measure-theory","kind":"scope","parent":19131,"result":false,"route":"../roadmap/MeasureTheory/geometric-measure-theory/index.html","scope":"MeasureTheory/geometric-measure-theory","status":"can_prove","summary":"Hausdorff measures and dimension, dimension functions and codimension, fractal sets such as attractors of iterated function systems, and the measure and dimension of distance sets.","title":"Geometric measure theory"},{"data_url":"scopes/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures.json","id":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","kind":"scope","navigation":"module","parent":19132,"result":false,"route":"../roadmap/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/index.html","scope":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","status":"fully_proved","title":"Dimension functions for Hausdorff measures"},{"id":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/hausdorff-dimension-function","kind":"node","lean":"HausdorffDimensionFunction","parent":19133,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/index.html#hausdorff-dimension-function","scope":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","status":"fully_proved","summary":"A Hausdorff dimension function (gauge) is a continuous, non-decreasing function $\\psi : [0,\\infty) \\to [0,\\infty)$ with $\\psi(0) = 0$ and $\\psi(r) > 0$ for every $r > 0$. In particular $\\psi(r) \\to 0$ as $r \\to 0^+$.","title":"Hausdorff dimension function"},{"id":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/continuous","kind":"node","lean":"HausdorffDimensionFunction.continuous","parent":19133,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/index.html#continuous","scope":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","status":"fully_proved","summary":"Every Hausdorff dimension function $\\psi : [0,\\infty) \\to [0,\\infty)$ is continuous.","title":"Continuity of a Hausdorff dimension function"},{"id":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/monotone","kind":"node","lean":"HausdorffDimensionFunction.monotone","parent":19133,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/index.html#monotone","scope":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","status":"fully_proved","summary":"Every Hausdorff dimension function $\\psi : [0,\\infty) \\to [0,\\infty)$ is non-decreasing: if $r \\le s$ then $\\psi(r) \\le \\psi(s)$.","title":"Monotonicity of a Hausdorff dimension function"},{"id":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/map-zero","kind":"node","lean":"HausdorffDimensionFunction.map_zero","parent":19133,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/index.html#map-zero","scope":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","status":"fully_proved","summary":"Every Hausdorff dimension function $\\psi$ satisfies $\\psi(0) = 0$.","title":"A Hausdorff dimension function vanishes at zero"},{"id":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/map-pos","kind":"node","lean":"HausdorffDimensionFunction.map_pos","parent":19133,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/index.html#map-pos","scope":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","status":"fully_proved","summary":"If $\\psi$ is a Hausdorff dimension function and $r > 0$ is a real number, then $\\psi(r) > 0$.","title":"A Hausdorff dimension function is positive at positive radii"},{"id":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/map-eq-zero-iff","kind":"node","lean":"HausdorffDimensionFunction.map_eq_zero_iff","parent":19133,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/index.html#map-eq-zero-iff","scope":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","status":"fully_proved","summary":"If $\\psi$ is a Hausdorff dimension function and $r \\ge 0$, then $\\psi(r) = 0$ if and only if $r = 0$.","title":"A Hausdorff dimension function vanishes only at zero"},{"id":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/tendsto-zero","kind":"node","lean":"HausdorffDimensionFunction.tendsto_zero","parent":19133,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures/index.html#tendsto-zero","scope":"MeasureTheory/geometric-measure-theory/dimension-functions-for-hausdorff-measures","status":"fully_proved","summary":"Every Hausdorff dimension function $\\psi$ satisfies $\\psi(r) \\to 0$ as $r \\to 0^+$, that is, as $r$ tends to $0$ through positive values.","title":"A Hausdorff dimension function tends to zero at zero"},{"data_url":"scopes/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem.json","id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","kind":"scope","navigation":"module","parent":19132,"result":false,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","title":"Falconer distance set problem"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/distance-set","kind":"node","lean":"FalconerDistanceSet.distanceSet","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#distance-set","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"For a set $E \\subseteq \\mathbb{R}^d$, the distance set of $E$ is $\\Delta(E) = \\{\\lvert x - y \\rvert : x, y \\in E\\}$, the set of Euclidean distances between pairs of points of $E$.","title":"Euclidean distance set"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-threshold-property","kind":"node","lean":"FalconerDistanceSet.planarThresholdProperty","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-threshold-property","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"An element $s \\in [0, \\infty]$ has the planar threshold property when every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension $\\dim_H E > s$ has a distance set $\\{|x - y| : x, y \\in E\\}$ of positive one-dimensional Lebesgue measure.","title":"Planar Falconer threshold property"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/admissible-planar-threshold","kind":"node","lean":"FalconerDistanceSet.admissiblePlanarThreshold","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#admissible-planar-threshold","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"An extended nonnegative real $s$ is an admissible planar threshold when $0 \\le s \\le 2$ and $s$ has the planar threshold property: every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension $\\dim_H E > s$ has a distance set of positive one-dimensional Lebesgue measure.","title":"Admissible planar threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-distance-threshold","kind":"node","lean":"FalconerDistanceSet.planarDistanceThreshold","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-distance-threshold","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The planar distance threshold is the infimum, taken in $[0, \\infty]$, of the admissible planar thresholds, that is, of all $s \\in [0, 2]$ such that every compact subset $E$ of the Euclidean plane $\\mathbb{R}^2$ with Hausdorff dimension greater than $s$ has a distance set of positive Lebesgue measure.","title":"Planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/distance-set-subset-nonnegative","kind":"node","lean":"FalconerDistanceSet.distanceSet_subset_nonnegative","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#distance-set-subset-nonnegative","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"For every set $E \\subseteq \\mathbb{R}^d$, the distance set of $E$ is contained in $[0, \\infty)$.","title":"Distances are nonnegative"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/distance-set-empty","kind":"node","lean":"FalconerDistanceSet.distanceSet_empty","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#distance-set-empty","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"For every $d$, the distance set of the empty subset of $\\mathbb{R}^d$ is empty.","title":"Distance set of the empty set"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/distance-set-singleton","kind":"node","lean":"FalconerDistanceSet.distanceSet_singleton","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#distance-set-singleton","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"For every point $x \\in \\mathbb{R}^d$, the distance set of $\\{x\\}$ is $\\{0\\}$.","title":"Distance set of a singleton"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/distance-set-is-compact","kind":"node","lean":"FalconerDistanceSet.distanceSet_isCompact","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#distance-set-is-compact","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"If $E \\subseteq \\mathbb{R}^d$ is compact, then the distance set of $E$ is a compact subset of $\\mathbb{R}$.","title":"Distance sets of compact sets are compact"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/distance-set-measurable-set","kind":"node","lean":"FalconerDistanceSet.distanceSet_measurableSet","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#distance-set-measurable-set","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"If $E \\subseteq \\mathbb{R}^d$ is compact, then the distance set of $E$ is a Borel measurable subset of $\\mathbb{R}$.","title":"Distance sets of compact sets are Borel"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-threshold-property-mono","kind":"node","lean":"FalconerDistanceSet.planarThresholdProperty_mono","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-threshold-property-mono","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"Let $s \\le t$ be elements of $[0, \\infty]$. If $s$ has the planar threshold property, that is, every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension $\\dim_H E > s$ has a distance set of positive one-dimensional Lebesgue measure, then $t$ has the planar threshold property.","title":"The planar threshold property is upward closed"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/dim-h-plane-le-two","kind":"node","lean":"FalconerDistanceSet.dimH_plane_le_two","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#dim-h-plane-le-two","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"Every subset $E$ of the Euclidean plane $\\mathbb{R}^2$ satisfies $\\dim_H E \\le 2$.","title":"Subsets of the plane have Hausdorff dimension at most two"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/two-admissible-planar-threshold","kind":"node","lean":"FalconerDistanceSet.two_admissiblePlanarThreshold","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#two-admissible-planar-threshold","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The number $2$ is an admissible planar threshold: $2 \\in [0, 2]$, and every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension $\\dim_H E > 2$ has a distance set of positive Lebesgue measure. This holds vacuously, since no subset of the plane has Hausdorff dimension greater than $2$.","title":"Two is an admissible planar threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/admissible-planar-thresholds-nonempty","kind":"node","lean":"FalconerDistanceSet.admissiblePlanarThresholds_nonempty","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#admissible-planar-thresholds-nonempty","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"There is an admissible planar threshold, that is, the set of admissible planar thresholds is nonempty.","title":"Admissible planar thresholds exist"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-distance-threshold-le-of-admissib","kind":"node","lean":"FalconerDistanceSet.planarDistanceThreshold_le_of_admissible","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-distance-threshold-le-of-admissib","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"For every $s \\in [0, \\infty]$ that is an admissible planar threshold, the planar distance threshold is at most $s$.","title":"Admissible thresholds bound the planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-distance-threshold-le-two","kind":"node","lean":"FalconerDistanceSet.planarDistanceThreshold_le_two","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-distance-threshold-le-two","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The planar distance threshold is at most $2$.","title":"The planar distance threshold is at most two"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-distance-threshold-ne-top","kind":"node","lean":"FalconerDistanceSet.planarDistanceThreshold_ne_top","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-distance-threshold-ne-top","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The planar distance threshold, an element of $[0, \\infty]$, is not $\\infty$.","title":"The planar distance threshold is finite"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-distance-threshold-of-real-to-rea","kind":"node","lean":"FalconerDistanceSet.planarDistanceThreshold_ofReal_toReal","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-distance-threshold-of-real-to-rea","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"Let $\\tau \\in [0, \\infty]$ be the planar distance threshold and let $\\tau_{\\mathbb{R}}$ be its value as a real number (with the convention that $\\infty$ goes to $0$). Viewing $\\tau_{\\mathbb{R}}$ again as an element of $[0, \\infty]$ gives back $\\tau$.","title":"Real round trip of the planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/trivial-lower-bound","kind":"node","lean":"FalconerDistanceSet.trivialLowerBound","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#trivial-lower-bound","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The planar distance threshold is at least $0$.","title":"Trivial lower bound for the planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-distance-threshold-le-of-property","kind":"node","lean":"FalconerDistanceSet.planarDistanceThreshold_le_of_property","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-distance-threshold-le-of-property","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"Thresholds up to two with the planar threshold property bound the planar distance threshold. For every $s \\in [0, \\infty]$ with $s \\le 2$ that has the planar threshold property, the planar distance threshold is at most $s$.","title":"Thresholds up to two with the planar threshold property bound the planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/le-planar-distance-threshold-of-zero-mea","kind":"node","lean":"FalconerDistanceSet.le_planarDistanceThreshold_of_zeroMeasureWitness","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#le-planar-distance-threshold-of-zero-mea","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"Let $t \\in [0, \\infty]$. If $E$ is a compact subset of the Euclidean plane $\\mathbb{R}^2$ whose Hausdorff dimension $\\dim_H E$ satisfies $t \\le \\dim_H E$ and whose distance set has Lebesgue measure zero, then $t$ is at most the planar distance threshold.","title":"Zero-measure distance sets give lower bounds for the planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/of-real-admissible-planar-threshold-iff","kind":"node","lean":"FalconerDistanceSet.ofReal_admissiblePlanarThreshold_iff","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#of-real-admissible-planar-threshold-iff","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"For every real number $s \\ge 0$, $s$ is an admissible planar threshold (as an element of $[0, \\infty]$) if and only if $s \\in [0, 2]$ and $s$ has the planar threshold property.","title":"Real form of admissible planar thresholds"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-threshold-property-planar-distanc","kind":"node","lean":"FalconerDistanceSet.planarThresholdProperty_planarDistanceThreshold","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-threshold-property-planar-distanc","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"Let $s_0 \\in [0, \\infty]$ be the planar distance threshold, the infimum of the admissible planar thresholds. Every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension $\\dim_H E > s_0$ has a distance set of positive one-dimensional Lebesgue measure.","title":"The planar distance threshold is itself a threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-distance-threshold-admissible","kind":"node","lean":"FalconerDistanceSet.planarDistanceThreshold_admissible","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-distance-threshold-admissible","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The planar distance threshold is an admissible planar threshold: it lies in $[0, 2]$ and has the planar threshold property.","title":"The planar distance threshold is admissible"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/is-least-admissible-planar-thresholds","kind":"node","lean":"FalconerDistanceSet.isLeast_admissiblePlanarThresholds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#is-least-admissible-planar-thresholds","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The planar distance threshold is the least admissible planar threshold: it is itself an admissible planar threshold, and it is at most every admissible planar threshold.","title":"The planar distance threshold is attained"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/real-threshold-family-eq-to-real-image","kind":"node","lean":"FalconerDistanceSet.realThresholdFamily_eq_toReal_image","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#real-threshold-family-eq-to-real-image","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The set of real numbers $r \\in [0, 2]$ such that $r$, viewed as an element of $[0, \\infty]$, has the planar threshold property equals the image of the set of admissible planar thresholds under the map $[0, \\infty] \\to \\mathbb{R}$ that is the identity on finite values and sends $\\infty$ to $0$.","title":"Real form of the admissible threshold family"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/planar-distance-threshold-to-real-eq-rea","kind":"node","lean":"FalconerDistanceSet.planarDistanceThreshold_toReal_eq_real_sInf","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#planar-distance-threshold-to-real-eq-rea","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The value of the planar distance threshold as a real number (with the convention that $\\infty$ goes to $0$) equals the infimum in $\\mathbb{R}$ of the set of real numbers $r \\in [0, 2]$ that, as elements of $[0, \\infty]$, have the planar threshold property.","title":"The planar distance threshold as a real infimum"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/falconer1985-general-sufficiency","kind":"node","lean":"FalconerDistanceSetWanted.falconer1985GeneralSufficiency","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#falconer1985-general-sufficiency","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"This is the proposition that for every integer $d \\ge 2$ and every compact set $E \\subseteq \\mathbb{R}^d$ whose Hausdorff dimension is strictly greater than $(d+1)/2$, the distance set $\\{\\|x - y\\| : x, y \\in E\\}$ has positive one-dimensional Lebesgue measure.","title":"Statement of Falconer's distance set theorem"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/wolff1999-planar-sufficiency","kind":"node","lean":"FalconerDistanceSetWanted.wolff1999PlanarSufficiency","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#wolff1999-planar-sufficiency","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The proposition that every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension greater than $4/3$ has a distance set $\\{|x - y| : x, y \\in E\\}$ of positive Lebesgue measure.","title":"Wolff's planar distance set condition, as a proposition"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/guth-iosevich-ou-wang2018-planar-suffici","kind":"node","lean":"FalconerDistanceSetWanted.guthIosevichOuWang2018PlanarSufficiency","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#guth-iosevich-ou-wang2018-planar-suffici","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"This is the proposition that $5/4$ has the planar threshold property: every compact set $E \\subseteq \\mathbb{R}^2$ whose Hausdorff dimension is strictly greater than $5/4$ has a distance set $\\{\\|x - y\\| : x, y \\in E\\}$ of positive one-dimensional Lebesgue measure.","title":"Statement of the Guth\u2013Iosevich\u2013Ou\u2013Wang planar distance set theorem"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/falconer1985-lattice-examples","kind":"node","lean":"FalconerDistanceSetWanted.falconer1985LatticeExamples","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#falconer1985-lattice-examples","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"This is the proposition that for every integer $d \\ge 2$ and every $s$ with $0 \\le s \\le d/2$, there is a compact set $E \\subseteq \\mathbb{R}^d$ of Hausdorff dimension exactly $s$ whose distance set $\\{\\|x - y\\| : x, y \\in E\\}$ has one-dimensional Lebesgue measure zero.","title":"Statement of Falconer's measure-zero distance set examples"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/upper-bound-falconer1985","kind":"node","lean":"FalconerDistanceSetWanted.upperBound_Falconer1985","parent":19141,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the upper bound "],["math","\\(3/2\\)"],["text"," for the planar distance threshold"]],"source":"Statement of the upper bound $3/2$ for the planar distance threshold"},"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#upper-bound-falconer1985","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"This is the proposition that the planar distance threshold, the infimum of all $s \\in [0, 2]$ such that every compact set $E \\subseteq \\mathbb{R}^2$ of Hausdorff dimension strictly greater than $s$ has a distance set $\\{\\|x - y\\| : x, y \\in E\\}$ of positive one-dimensional Lebesgue measure, is at most $3/2$.","title":"Statement of the upper bound $3/2$ for the planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/upper-bound-wolff1999","kind":"node","lean":"FalconerDistanceSetWanted.upperBound_Wolff1999","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#upper-bound-wolff1999","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The proposition that the planar Falconer distance threshold is at most $4/3$. Here the planar Falconer distance threshold is the infimum of the $s \\in [0, 2]$ such that every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension greater than $s$ has a distance set $\\{|x - y| : x, y \\in E\\}$ of positive Lebesgue measure.","title":"The Wolff bound, as a proposition"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/upper-bound-giow2018","kind":"node","lean":"FalconerDistanceSetWanted.upperBound_GIOW2018","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#upper-bound-giow2018","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"The proposition that the planar Falconer distance threshold is at most $5/4$. Here the planar Falconer distance threshold is the infimum of the $s \\in [0, 2]$ such that every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension greater than $s$ has a distance set $\\{|x - y| : x, y \\in E\\}$ of positive Lebesgue measure.","title":"The Guth\u2013Iosevich\u2013Ou\u2013Wang bound, as a proposition"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/lower-bound-falconer1985","kind":"node","lean":"FalconerDistanceSetWanted.lowerBound_Falconer1985","parent":19141,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the lower bound "],["math","\\(1\\)"],["text"," for the planar distance threshold"]],"source":"Statement of the lower bound $1$ for the planar distance threshold"},"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#lower-bound-falconer1985","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"fully_proved","summary":"This is the proposition that the planar distance threshold, the infimum of all $s \\in [0, 2]$ such that every compact set $E \\subseteq \\mathbb{R}^2$ of Hausdorff dimension strictly greater than $s$ has a distance set $\\{\\|x - y\\| : x, y \\in E\\}$ of positive one-dimensional Lebesgue measure, is at least $1$.","title":"Statement of the lower bound $1$ for the planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/falconer1985-general-sufficiency-holds","kind":"node","lean":"FalconerDistanceSetWanted.falconer1985GeneralSufficiency_holds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#falconer1985-general-sufficiency-holds","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","summary":"For every integer $d \\ge 2$ and every compact set $E \\subseteq \\mathbb{R}^d$ whose Hausdorff dimension is strictly greater than $(d+1)/2$, the distance set $\\{\\|x - y\\| : x, y \\in E\\}$ has positive one-dimensional Lebesgue measure.","title":"Falconer's distance set theorem"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/wolff1999-planar-sufficiency-holds","kind":"node","lean":"FalconerDistanceSetWanted.wolff1999PlanarSufficiency_holds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#wolff1999-planar-sufficiency-holds","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","summary":"Every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension greater than $4/3$ has a distance set $\\{|x - y| : x, y \\in E\\}$ of positive Lebesgue measure.","title":"Wolff's planar distance set theorem"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/guth-iosevich-ou-wang2018-planar-suffici-2","kind":"node","lean":"FalconerDistanceSetWanted.guthIosevichOuWang2018PlanarSufficiency_holds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#guth-iosevich-ou-wang2018-planar-suffici-2","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","summary":"Every compact set $E \\subseteq \\mathbb{R}^2$ whose Hausdorff dimension is strictly greater than $5/4$ has a distance set $\\{\\|x - y\\| : x, y \\in E\\}$ of positive one-dimensional Lebesgue measure.","title":"Guth\u2013Iosevich\u2013Ou\u2013Wang planar distance set theorem"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/falconer1985-lattice-examples-holds","kind":"node","lean":"FalconerDistanceSetWanted.falconer1985LatticeExamples_holds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#falconer1985-lattice-examples-holds","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","summary":"For every integer $d \\ge 2$ and every $s$ with $0 \\le s \\le d/2$, there is a compact set $E \\subseteq \\mathbb{R}^d$ of Hausdorff dimension exactly $s$ whose distance set $\\{\\|x - y\\| : x, y \\in E\\}$ has one-dimensional Lebesgue measure zero.","title":"Falconer's measure-zero distance set examples"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/upper-bound-falconer1985-holds","kind":"node","lean":"FalconerDistanceSetWanted.upperBound_Falconer1985_holds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#upper-bound-falconer1985-holds","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","summary":"The planar distance threshold, the infimum of all $s \\in [0, 2]$ such that every compact set $E \\subseteq \\mathbb{R}^2$ of Hausdorff dimension strictly greater than $s$ has a distance set $\\{\\|x - y\\| : x, y \\in E\\}$ of positive one-dimensional Lebesgue measure, is at most $3/2$.","title":"Falconer's upper bound for the planar distance threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/upper-bound-wolff1999-holds","kind":"node","lean":"FalconerDistanceSetWanted.upperBound_Wolff1999_holds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#upper-bound-wolff1999-holds","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","summary":"The planar Falconer distance threshold, the infimum of the $s \\in [0, 2]$ such that every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension greater than $s$ has a distance set $\\{|x - y| : x, y \\in E\\}$ of positive Lebesgue measure, is at most $4/3$.","title":"Wolff's upper bound for the planar Falconer threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/upper-bound-giow2018-holds","kind":"node","lean":"FalconerDistanceSetWanted.upperBound_GIOW2018_holds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#upper-bound-giow2018-holds","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","summary":"The planar Falconer distance threshold, the infimum of the $s \\in [0, 2]$ such that every compact set $E \\subseteq \\mathbb{R}^2$ with Hausdorff dimension greater than $s$ has a distance set $\\{|x - y| : x, y \\in E\\}$ of positive Lebesgue measure, is at most $5/4$.","title":"Guth\u2013Iosevich\u2013Ou\u2013Wang upper bound for the planar Falconer threshold"},{"id":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/lower-bound-falconer1985-holds","kind":"node","lean":"FalconerDistanceSetWanted.lowerBound_Falconer1985_holds","parent":19141,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/falconer-distance-set-problem/index.html#lower-bound-falconer1985-holds","scope":"MeasureTheory/geometric-measure-theory/falconer-distance-set-problem","status":"can_prove","summary":"The planar distance threshold, the infimum of all $s \\in [0, 2]$ such that every compact set $E \\subseteq \\mathbb{R}^2$ of Hausdorff dimension strictly greater than $s$ has a distance set $\\{\\|x - y\\| : x, y \\in E\\}$ of positive one-dimensional Lebesgue measure, is at least $1$.","title":"Falconer's lower bound for the planar distance threshold"},{"data_url":"scopes/MeasureTheory/geometric-measure-theory/hausdorff-codimension.json","id":"MeasureTheory/geometric-measure-theory/hausdorff-codimension","kind":"scope","navigation":"module","parent":19132,"result":false,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-codimension/index.html","scope":"MeasureTheory/geometric-measure-theory/hausdorff-codimension","status":"fully_proved","title":"Hausdorff codimension"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-codimension/has-hausdorff-codim-at-least","kind":"node","lean":"Set.HasHausdorffCodimAtLeast","parent":19184,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-codimension/index.html#has-hausdorff-codim-at-least","scope":"MeasureTheory/geometric-measure-theory/hausdorff-codimension","status":"fully_proved","summary":"Let $S$ be a subset of an extended metric space and let $n, c$ be natural numbers. The set $S$ has Hausdorff codimension at least $c$ in dimension $n$ if $S$ is empty or its Hausdorff dimension satisfies $\\dim_H S + c \\le n$, the inequality being taken in $[0, \\infty]$.","title":"Lower bound on Hausdorff codimension"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-codimension/has-hausdorff-codim-at-least-empty","kind":"node","lean":"Set.hasHausdorffCodimAtLeast_empty","parent":19184,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-codimension/index.html#has-hausdorff-codim-at-least-empty","scope":"MeasureTheory/geometric-measure-theory/hausdorff-codimension","status":"fully_proved","summary":"In an extended metric space, the empty set has Hausdorff codimension at least $c$ in dimension $n$ for all natural numbers $n$ and $c$.","title":"The empty set has every codimension"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-codimension/mono","kind":"node","lean":"Set.HasHausdorffCodimAtLeast.mono","parent":19184,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-codimension/index.html#mono","scope":"MeasureTheory/geometric-measure-theory/hausdorff-codimension","status":"fully_proved","summary":"Let $T \\subseteq S$ be subsets of an extended metric space and let $n, c$ be natural numbers. If $S$ has Hausdorff codimension at least $c$ in dimension $n$, then so does $T$.","title":"Subsets inherit a codimension bound"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-codimension/eq-empty-of-lt","kind":"node","lean":"Set.HasHausdorffCodimAtLeast.eq_empty_of_lt","parent":19184,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-codimension/index.html#eq-empty-of-lt","scope":"MeasureTheory/geometric-measure-theory/hausdorff-codimension","status":"fully_proved","summary":"Let $S$ be a subset of an extended metric space and let $n < c$ be natural numbers. If $S$ has Hausdorff codimension at least $c$ in dimension $n$, then $S$ is empty.","title":"Sets of codimension larger than the dimension are empty"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-codimension/has-hausdorff-codim-at-least-of-dim-h-le","kind":"node","lean":"Set.hasHausdorffCodimAtLeast_of_dimH_le","parent":19184,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-codimension/index.html#has-hausdorff-codim-at-least-of-dim-h-le","scope":"MeasureTheory/geometric-measure-theory/hausdorff-codimension","status":"fully_proved","summary":"Let $S$ be a subset of an extended metric space and let $c \\le n$ be natural numbers. If the Hausdorff dimension of $S$ satisfies $\\dim_H S \\le n - c$, then $S$ has Hausdorff codimension at least $c$ in dimension $n$.","title":"A dimension bound gives a codimension bound"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-codimension/has-hausdorff-codim-at-least-iff","kind":"node","lean":"Set.hasHausdorffCodimAtLeast_iff","parent":19184,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-codimension/index.html#has-hausdorff-codim-at-least-iff","scope":"MeasureTheory/geometric-measure-theory/hausdorff-codimension","status":"fully_proved","summary":"Let $S$ be a subset of an extended metric space and let $n, c$ be natural numbers. Then $S$ has Hausdorff codimension at least $c$ in dimension $n$ if and only if either $S$ is empty, or both $c \\le n$ and the Hausdorff dimension of $S$ satisfies $\\dim_H S \\le n - c$.","title":"Characterization of Hausdorff codimension"},{"data_url":"scopes/MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets.json","id":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets","kind":"scope","navigation":"module","parent":19132,"result":false,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/index.html","scope":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets","status":"can_prove","title":"Hausdorff dimension of self-similar sets"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/is-attractor","kind":"node","lean":"MathlibExt.Dynamics.MoranDimensionWanted.Dynamics.IsAttractor","parent":19191,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/index.html#is-attractor","scope":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets","status":"fully_proved","summary":"Let $X$ be a topological space and let $(f_i)_{i < n}$ be a finite family of maps $X \\to X$. A set $S \\subseteq X$ is an attractor of the family if it is nonempty and compact and is fixed by the Hutchinson operator, that is, $S = \\bigcup_{i < n} f_i(S)$.","title":"Attractor of an iterated function system"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/is-affine-symmetric-ifs","kind":"node","lean":"MathlibExt.Dynamics.MoranDimensionWanted.Dynamics.IsAffineSymmetricIFS","parent":19191,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/index.html#is-affine-symmetric-ifs","scope":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets","status":"fully_proved","summary":"Let $(f_i)_{i < n}$ be a finite family of maps $f_i \\colon \\mathbb{R}^d \\to \\mathbb{R}^d$ on Euclidean space. The family is affine-symmetric with common contraction factor $\\lambda$ if $0 < \\lambda < 1$ and there are linear isometries $A_i \\colon \\mathbb{R}^d \\to \\mathbb{R}^d$ (orthogonal transformations) and vectors $\\beta_i \\in \\mathbb{R}^d$ such that $f_i(x) = \\lambda A_i(x) + \\beta_i$ for every $i < n$ and\u2026","title":"Affine-symmetric iterated function system"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/open-set-condition","kind":"node","lean":"MathlibExt.Dynamics.MoranDimensionWanted.Dynamics.OpenSetCondition","parent":19191,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/index.html#open-set-condition","scope":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets","status":"fully_proved","summary":"A finite family of maps $f_i \\colon \\mathbb{R}^d \\to \\mathbb{R}^d$, $i < n$, satisfies the open set condition if there is a nonempty open set $G \\subseteq \\mathbb{R}^d$ such that $f_i(G) \\subseteq G$ for every $i < n$ and the images $f_i(G)$ and $f_j(G)$ are disjoint whenever $i \\neq j$.","title":"Open set condition"},{"id":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/moran-equality-affine","kind":"node","lean":"MathlibExt.Dynamics.MoranDimensionWanted.Dynamics.moran_equality_affine","parent":19191,"result":true,"route":"../roadmap/MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets/index.html#moran-equality-affine","scope":"MeasureTheory/geometric-measure-theory/hausdorff-dimension-of-self-similar-sets","status":"can_prove","summary":"Let $n \\ge 1$ and let $(f_i)_{i < n}$ be an affine-symmetric iterated function system on $\\mathbb{R}^d$ with common contraction factor $\\lambda \\in (0,1)$ that satisfies the open set condition. Then every attractor $S \\subseteq \\mathbb{R}^d$ of the system has Hausdorff dimension","title":"Moran's equality for affine-symmetric iterated function systems"},{"data_url":"scopes/MeasureTheory/measures.json","id":"MeasureTheory/measures","kind":"scope","parent":19131,"result":false,"route":"../roadmap/MeasureTheory/measures/index.html","scope":"MeasureTheory/measures","status":"can_prove","summary":"Construction and uniqueness of measures, atomless and vector measures, measures on Boolean algebras, and the limits of measurability shown by non-measurable sets and paradoxical decompositions.","title":"Measures and measurable sets"},{"id":"MeasureTheory/measures/szpilrajn170-homeomorphic-images-measura","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook170Wanted.szpilrajn170_homeomorphic_images_measurable_imp_absolute","parent":19196,"result":true,"route":"../roadmap/MeasureTheory/measures/index.html#szpilrajn170-homeomorphic-images-measura","scope":"MeasureTheory/measures","status":"can_prove","summary":"Let $S$ be a subset of the plane $\\mathbb{R}^2$ such that every subset $T$ of $\\mathbb{R}^2$ homeomorphic to $S$ (as a subspace) is Lebesgue measurable, that is, Carath\u00e9odory measurable with respect to Lebesgue outer measure on $\\mathbb{R}^2$. Then $S$ is absolutely measurable: for every measure $\\mu$ on the Borel $\\sigma$-algebra of $\\mathbb{R}^2$, the set $S$ is Carath\u00e9odory measurable with respect to the outer\u2026","title":"Szpilrajn's Problem 170: plane sets with measurable homeomorphic copies"},{"id":"MeasureTheory/measures/vitali-nonmeasurable","kind":"node","lean":"MeasureTheory.vitali_nonmeasurable","parent":19196,"result":true,"route":"../roadmap/MeasureTheory/measures/index.html#vitali-nonmeasurable","scope":"MeasureTheory/measures","status":"fully_proved","summary":"There exists a subset of $\\mathbb{R}$ that is not Lebesgue measurable.","title":"Vitali's theorem"},{"data_url":"scopes/MeasureTheory/measures/atomless-measures.json","id":"MeasureTheory/measures/atomless-measures","kind":"scope","navigation":"module","parent":19196,"result":false,"route":"../roadmap/MeasureTheory/measures/atomless-measures/index.html","scope":"MeasureTheory/measures/atomless-measures","status":"fully_proved","title":"Atomless measures"},{"id":"MeasureTheory/measures/atomless-measures/is-atomless","kind":"node","lean":"MeasureTheory.IsAtomless","parent":19199,"result":true,"route":"../roadmap/MeasureTheory/measures/atomless-measures/index.html#is-atomless","scope":"MeasureTheory/measures/atomless-measures","status":"fully_proved","summary":"A measure $\\mu$ on a measurable space is atomless when every measurable set $s$ with $\\mu(s) > 0$ contains a measurable subset $t \\subseteq s$ with $0 < \\mu(t) < \\mu(s)$.","title":"Atomless measure"},{"id":"MeasureTheory/measures/atomless-measures/is-atomless-restrict","kind":"node","lean":"MeasureTheory.isAtomless_restrict","parent":19199,"result":true,"route":"../roadmap/MeasureTheory/measures/atomless-measures/index.html#is-atomless-restrict","scope":"MeasureTheory/measures/atomless-measures","status":"fully_proved","summary":"The restriction of an atomless measure to a measurable set is atomless.","title":"Restriction of an atomless measure"},{"id":"MeasureTheory/measures/atomless-measures/is-atomless-exists-subset-measure-eq","kind":"node","lean":"MeasureTheory.isAtomless_exists_subset_measure_eq","parent":19199,"result":true,"route":"../roadmap/MeasureTheory/measures/atomless-measures/index.html#is-atomless-exists-subset-measure-eq","scope":"MeasureTheory/measures/atomless-measures","status":"fully_proved","summary":"Let $\\mu$ be a finite atomless measure. For every measurable set $s$ and every $t \\in [0, \\infty]$ with $t \\le \\mu(s)$, there is a measurable set $u \\subseteq s$ with $\\mu(u) = t$.","title":"Sierpi\u0144ski's theorem on atomless measures"},{"data_url":"scopes/MeasureTheory/measures/banach-tarski-paradox.json","id":"MeasureTheory/measures/banach-tarski-paradox","kind":"scope","navigation":"module","parent":19196,"result":false,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","title":"Banach\u2013Tarski paradox"},{"id":"MeasureTheory/measures/banach-tarski-paradox/finitely-equidecomposable-by-isometries","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BanachTarskiWanted.FinitelyEquidecomposableByIsometries","parent":19203,"result":true,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html#finitely-equidecomposable-by-isometries","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","summary":"Let $X$ be a metric space. Subsets $A$ and $B$ of $X$ are finitely equidecomposable by isometries if there are a natural number $n$, sets $A_1, \\dots, A_n$ and $B_1, \\dots, B_n$ in $X$, and isometric bijections $g_1, \\dots, g_n$ of $X$ onto itself such that the sets $A_i$ are pairwise disjoint, the sets $B_i$ are pairwise disjoint, $A = A_1 \\cup \\dots \\cup A_n$, $B = B_1 \\cup \\dots \\cup B_n$, and $g_i(A_i) = B_i$\u2026","title":"Finite equidecomposability by isometries"},{"id":"MeasureTheory/measures/banach-tarski-paradox/finitely-equidecomposable-by-isometries-2","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BanachTarskiWanted.finitelyEquidecomposableByIsometries_iff_bijOn","parent":19203,"result":true,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html#finitely-equidecomposable-by-isometries-2","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","summary":"Let $A$ and $B$ be subsets of a metric space $X$. Then $A$ is finitely equidecomposable by isometries with $B$ if and only if there are a map $f : X \\to X$ and a finite set $S$ of isometric bijections of $X$ onto itself such that $f$ restricts to a bijection from $A$ onto $B$ and for every $x \\in A$ there is $g \\in S$ with $f(x) = g(x)$.","title":"Equidecomposability via piecewise-isometric bijections"},{"id":"MeasureTheory/measures/banach-tarski-paradox/refl","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BanachTarskiWanted.FinitelyEquidecomposableByIsometries.refl","parent":19203,"result":true,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html#refl","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","summary":"Every subset $A$ of a metric space $X$ is finitely equidecomposable by isometries with itself.","title":"Finite equidecomposability by isometries is reflexive"},{"id":"MeasureTheory/measures/banach-tarski-paradox/symm","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BanachTarskiWanted.FinitelyEquidecomposableByIsometries.symm","parent":19203,"result":true,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html#symm","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","summary":"Let $A$ and $B$ be subsets of a metric space $X$. If $A$ is finitely equidecomposable by isometries with $B$, then $B$ is finitely equidecomposable by isometries with $A$.","title":"Finite equidecomposability by isometries is symmetric"},{"id":"MeasureTheory/measures/banach-tarski-paradox/trans","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BanachTarskiWanted.FinitelyEquidecomposableByIsometries.trans","parent":19203,"result":true,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html#trans","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","summary":"Let $A$, $B$ and $C$ be subsets of a metric space $X$. If $A$ is finitely equidecomposable by isometries with $B$ and $B$ is finitely equidecomposable by isometries with $C$, then $A$ is finitely equidecomposable by isometries with $C$.","title":"Finite equidecomposability by isometries is transitive"},{"id":"MeasureTheory/measures/banach-tarski-paradox/union","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BanachTarskiWanted.FinitelyEquidecomposableByIsometries.union","parent":19203,"result":true,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html#union","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","summary":"Let $A_1$, $A_2$, $B_1$ and $B_2$ be subsets of a metric space $X$ with $A_1 \\cap A_2 = \\emptyset$ and $B_1 \\cap B_2 = \\emptyset$. If $A_1$ is finitely equidecomposable by isometries with $B_1$ and $A_2$ is finitely equidecomposable by isometries with $B_2$, then $A_1 \\cup A_2$ is finitely equidecomposable by isometries with $B_1 \\cup B_2$.","title":"Finite equidecomposability by isometries respects disjoint unions"},{"id":"MeasureTheory/measures/banach-tarski-paradox/image-isometry-equiv","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BanachTarskiWanted.FinitelyEquidecomposableByIsometries.image_isometryEquiv","parent":19203,"result":true,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html#image-isometry-equiv","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","summary":"Let $g$ be an isometric bijection of a metric space $X$ onto itself. Then every subset $A$ of $X$ is finitely equidecomposable by isometries with its image $g(A)$.","title":"Equidecomposability of a set with its isometric image"},{"id":"MeasureTheory/measures/banach-tarski-paradox/banach-tarski-closed-ball-unit","kind":"node","lean":"MathlibExt.MeasureTheory.Geometry.BanachTarskiWanted.banach_tarski_closedBall_unit","parent":19203,"result":true,"route":"../roadmap/MeasureTheory/measures/banach-tarski-paradox/index.html#banach-tarski-closed-ball-unit","scope":"MeasureTheory/measures/banach-tarski-paradox","status":"fully_proved","summary":"In three-dimensional Euclidean space $\\mathbb{R}^3$, write $\\bar B(c, r)$ for the closed ball of radius $r$ about $c$. There are points $c_1, c_2 \\in \\mathbb{R}^3$ such that $\\bar B(c_1, 1)$ and $\\bar B(c_2, 1)$ are disjoint and the closed unit ball $\\bar B(0, 1)$ is finitely equidecomposable by isometries with $\\bar B(c_1, 1) \\cup \\bar B(c_2, 1)$.","title":"Banach\u2013Tarski paradox for the closed unit ball"},{"data_url":"scopes/MeasureTheory/measures/carath-odory-s-extension-theorem.json","id":"MeasureTheory/measures/carath-odory-s-extension-theorem","kind":"scope","navigation":"module","parent":19196,"result":false,"route":"../roadmap/MeasureTheory/measures/carath-odory-s-extension-theorem/index.html","scope":"MeasureTheory/measures/carath-odory-s-extension-theorem","status":"fully_proved","title":"Carath\u00e9odory's extension theorem"},{"id":"MeasureTheory/measures/carath-odory-s-extension-theorem/is-set-semiring-of-is-set-algebra","kind":"node","lean":"MeasureTheory.isSetSemiring_of_isSetAlgebra","parent":19212,"result":true,"route":"../roadmap/MeasureTheory/measures/carath-odory-s-extension-theorem/index.html#is-set-semiring-of-is-set-algebra","scope":"MeasureTheory/measures/carath-odory-s-extension-theorem","status":"fully_proved","summary":"If $\\mathcal{A}$ is an algebra of sets on $\\alpha$, then $\\mathcal{A}$ is a semiring of sets: $\\emptyset \\in \\mathcal{A}$, $s \\cap t \\in \\mathcal{A}$ for all $s, t \\in \\mathcal{A}$, and for all $s, t \\in \\mathcal{A}$ the difference $s \\setminus t$ is the union of finitely many pairwise disjoint members of $\\mathcal{A}$.","title":"Every algebra of sets is a semiring of sets"},{"id":"MeasureTheory/measures/carath-odory-s-extension-theorem/is-set-algebra-is-pi-system","kind":"node","lean":"MeasureTheory.isSetAlgebra_isPiSystem","parent":19212,"result":true,"route":"../roadmap/MeasureTheory/measures/carath-odory-s-extension-theorem/index.html#is-set-algebra-is-pi-system","scope":"MeasureTheory/measures/carath-odory-s-extension-theorem","status":"fully_proved","summary":"If $\\mathcal{A}$ is an algebra of sets on $\\alpha$, then $\\mathcal{A}$ is a $\\pi$-system: for all $s, t \\in \\mathcal{A}$ with $s \\cap t$ nonempty, we have $s \\cap t \\in \\mathcal{A}$.","title":"Every algebra of sets is a \u03c0-system"},{"id":"MeasureTheory/measures/carath-odory-s-extension-theorem/exists-measure-extension","kind":"node","lean":"MeasureTheory.exists_measure_extension","parent":19212,"result":true,"route":"../roadmap/MeasureTheory/measures/carath-odory-s-extension-theorem/index.html#exists-measure-extension","scope":"MeasureTheory/measures/carath-odory-s-extension-theorem","status":"fully_proved","summary":"Let $\\mathcal{A}$ be an algebra of sets on $\\alpha$ and let $\\mu_0$ be an additive content on $\\mathcal{A}$ with values in $[0, \\infty]$ that is $\\sigma$-subadditive: whenever $s_0, s_1, \\dots \\in \\mathcal{A}$ and $\\bigcup_i s_i \\in \\mathcal{A}$, we have $\\mu_0(\\bigcup_i s_i) \\le \\sum_i \\mu_0(s_i)$. Then there is a measure $\\mu$ on the $\\sigma$-algebra $\\sigma(\\mathcal{A})$ generated by $\\mathcal{A}$ such that\u2026","title":"Existence of the Carath\u00e9odory extension"},{"id":"MeasureTheory/measures/carath-odory-s-extension-theorem/caratheodory-extension-theorem","kind":"node","lean":"MeasureTheory.caratheodory_extension_theorem","parent":19212,"result":true,"route":"../roadmap/MeasureTheory/measures/carath-odory-s-extension-theorem/index.html#caratheodory-extension-theorem","scope":"MeasureTheory/measures/carath-odory-s-extension-theorem","status":"fully_proved","summary":"Let $\\mathcal{A}$ be an algebra of sets on $\\alpha$ and let $\\mu_0$ be an additive content on $\\mathcal{A}$ with values in $[0, \\infty]$ that is $\\sigma$-subadditive: whenever $s_0, s_1, \\dots \\in \\mathcal{A}$ and $\\bigcup_i s_i \\in \\mathcal{A}$, we have $\\mu_0(\\bigcup_i s_i) \\le \\sum_i \\mu_0(s_i)$. Suppose there are sets $B_0, B_1, \\dots \\in \\mathcal{A}$ with $\\bigcup_i B_i = \\alpha$ and $\\mu_0(B_i) < \\infty$ for\u2026","title":"Carath\u00e9odory extension theorem"},{"data_url":"scopes/MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas.json","id":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas","kind":"scope","navigation":"module","parent":19196,"result":false,"route":"../roadmap/MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/index.html","scope":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas","status":"can_prove","title":"Congruence-invariant extensions of Lebesgue measure"},{"id":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/are-congruent","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook169Wanted.AreCongruent","parent":19217,"result":true,"route":"../roadmap/MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/index.html#are-congruent","scope":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas","status":"fully_proved","summary":"Two sets $E$ and $F$ in the Euclidean plane $\\mathbb{R}^2$ are congruent when some isometry of the plane onto itself maps $E$ onto $F$.","title":"Congruent plane sets"},{"id":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/is-congruence-invariant","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook169Wanted.IsCongruenceInvariant","parent":19217,"result":true,"route":"../roadmap/MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/index.html#is-congruence-invariant","scope":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas","status":"fully_proved","summary":"A function $\\mu$ that assigns to every subset of the Euclidean plane $\\mathbb{R}^2$ a value in $[0, \\infty]$ is congruence invariant when $\\mu(E) = \\mu(F)$ for all congruent plane sets $E$ and $F$.","title":"Congruence-invariant set functions on the plane"},{"id":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/is-countably-additive","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook169Wanted.IsCountablyAdditive","parent":19217,"result":true,"route":"../roadmap/MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/index.html#is-countably-additive","scope":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas","status":"fully_proved","summary":"`\u03bc` is countably additive on pairwise-disjoint families. This is the countably-additive reading of \"additive\" in Problem 169; the typescript does not say whether finite or countable additivity is meant.","title":"Countably additive set function"},{"id":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/countably-additive-congruence-claim","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook169Wanted.CountablyAdditiveCongruenceClaim","parent":19217,"result":true,"route":"../roadmap/MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/index.html#countably-additive-congruence-claim","scope":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas","status":"fully_proved","summary":"The countably additive reading of Problem 169: there is a function `\u03bc`, defined for all plane sets, equal on congruent sets, countably additive, extending the Caratheodory linear measure (read as Lebesgue volume on measurable sets), with values in `[0, +\u221e]` (automatic for `ENNReal`).","title":"Problem of a congruence-invariant countably additive plane measure"},{"id":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/no-countably-additive-congruence-measure","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook169Wanted.no_countably_additive_congruence_measure","parent":19217,"result":true,"route":"../roadmap/MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas/index.html#no-countably-additive-congruence-measure","scope":"MeasureTheory/measures/congruence-invariant-extensions-of-lebesgue-meas","status":"can_prove","summary":"The countably additive reading is false in ZFC. Sketch: let `V\u2080 \u2286 [0,1]` meet each `\u211a`-coset of `\u211d` exactly once (Vitali selector) and put `V = V\u2080 \u00d7 [0,1]`. Enumerating `\u211a \u2229 [-1,1]` as `(q\u2099)`, the translates `T\u2099 = V + (q\u2099, 0)` are pairwise disjoint, each congruent to `V`, so all have the same mass `c = \u03bc V`. Their union contains the unit square `Q = [0,1] \u00d7 [0,1]` and lies in the strip `[-1,2] \u00d7 [0,1]` of volume 3,\u2026","title":"No countably additive congruence-invariant measure on all plane sets"},{"data_url":"scopes/MeasureTheory/measures/independent-bases-of-atomless-measures.json","id":"MeasureTheory/measures/independent-bases-of-atomless-measures","kind":"scope","navigation":"module","parent":19196,"result":false,"route":"../roadmap/MeasureTheory/measures/independent-bases-of-atomless-measures/index.html","scope":"MeasureTheory/measures/independent-bases-of-atomless-measures","status":"can_prove","title":"Independent bases of atomless measures"},{"id":"MeasureTheory/measures/independent-bases-of-atomless-measures/is-convex-measure","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook191Wanted.IsConvexMeasure","parent":19223,"result":true,"route":"../roadmap/MeasureTheory/measures/independent-bases-of-atomless-measures/index.html#is-convex-measure","scope":"MeasureTheory/measures/independent-bases-of-atomless-measures","status":"fully_proved","summary":"A measure $\\mu$ on a measurable space $X$ is convex (in Fr\u00e9chet's sense, \"sans singularit\u00e9s\") if $\\mu(X) = 1$ and every measurable set $A$ with $\\mu(A) > 0$ contains a measurable subset $B$ with $0 < \\mu(B) < \\mu(A)$.","title":"Convex measure"},{"id":"MeasureTheory/measures/independent-bases-of-atomless-measures/is-independent-class","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook191Wanted.IsIndependentClass","parent":19223,"result":true,"route":"../roadmap/MeasureTheory/measures/independent-bases-of-atomless-measures/index.html#is-independent-class","scope":"MeasureTheory/measures/independent-bases-of-atomless-measures","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $X$. A class $B$ of subsets of $X$ is stochastically independent with respect to $\\mu$ if for every finite collection of distinct sets $A_1, \\dots, A_n$ in $B$, including the empty one, $\\mu(A_1 \\cap \\dots \\cap A_n) = \\mu(A_1) \\cdots \\mu(A_n)$. The empty intersection is $X$ and the empty product is $1$, so this forces $\\mu(X) = 1$. The sets in $B$ need not be measurable;\u2026","title":"Stochastically independent class of sets"},{"id":"MeasureTheory/measures/independent-bases-of-atomless-measures/is-base","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook191Wanted.IsBase","parent":19223,"result":true,"route":"../roadmap/MeasureTheory/measures/independent-bases-of-atomless-measures/index.html#is-base","scope":"MeasureTheory/measures/independent-bases-of-atomless-measures","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $X$. A class $B$ of subsets of $X$ is a base of $\\mu$ if every set in $B$ is measurable, $B$ is stochastically independent with respect to $\\mu$, and every measurable set $A \\subseteq X$ agrees up to a null set with a set $C$ in the $\\sigma$-algebra generated by $B$, that is, $\\mu(A \\triangle C) = 0$.","title":"Base of a measure"},{"id":"MeasureTheory/measures/independent-bases-of-atomless-measures/scottish-book-problem191-claim","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook191Wanted.ScottishBookProblem191Claim","parent":19223,"result":true,"route":"../roadmap/MeasureTheory/measures/independent-bases-of-atomless-measures/index.html#scottish-book-problem191-claim","scope":"MeasureTheory/measures/independent-bases-of-atomless-measures","status":"fully_proved","summary":"The proposition, Szpilrajn's Problem 191 from the Scottish Book in its original form, that for every measurable space $X$, every convex measure $\\mu$ on $X$ has a base, that is, some class $B$ of subsets of $X$ is a base of $\\mu$.","title":"The claim of Szpilrajn's Problem 191"},{"id":"MeasureTheory/measures/independent-bases-of-atomless-measures/szpilrajn191-refuted","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook191Wanted.szpilrajn191_refuted","parent":19223,"result":true,"route":"../roadmap/MeasureTheory/measures/independent-bases-of-atomless-measures/index.html#szpilrajn191-refuted","scope":"MeasureTheory/measures/independent-bases-of-atomless-measures","status":"can_prove","summary":"The claim of Szpilrajn's Problem 191 is false: not every convex measure possesses a base. That is, there are a measurable space $X$ and a convex measure $\\mu$ on $X$ such that no class of subsets of $X$ is a base of $\\mu$.","title":"Szpilrajn's Problem 191 has a negative answer"},{"data_url":"scopes/MeasureTheory/measures/lyapunov-convexity-theorem.json","id":"MeasureTheory/measures/lyapunov-convexity-theorem","kind":"scope","navigation":"module","parent":19196,"result":false,"route":"../roadmap/MeasureTheory/measures/lyapunov-convexity-theorem/index.html","scope":"MeasureTheory/measures/lyapunov-convexity-theorem","status":"fully_proved","title":"Lyapunov convexity theorem"},{"id":"MeasureTheory/measures/lyapunov-convexity-theorem/lyapunov-range","kind":"node","lean":"MathlibExt.MeasureTheory.Integral.LyapunovConvexityWanted.lyapunovRange","parent":19229,"result":true,"route":"../roadmap/MeasureTheory/measures/lyapunov-convexity-theorem/index.html#lyapunov-range","scope":"MeasureTheory/measures/lyapunov-convexity-theorem","status":"fully_proved","summary":"For a measure $\\mu$ on a measurable space $\\alpha$ and a function $f : \\alpha \\to E$ into a real normed space $E$, the Lyapunov range of $f$ with respect to $\\mu$ is the set $\\lbrace \\int_s f \\, d\\mu : s \\subseteq \\alpha \\text{ measurable} \\rbrace$ of integrals of $f$ over measurable subsets of $\\alpha$.","title":"Lyapunov range"},{"id":"MeasureTheory/measures/lyapunov-convexity-theorem/lyapunov-convexity","kind":"node","lean":"MathlibExt.MeasureTheory.Integral.LyapunovConvexityWanted.lyapunov_convexity","parent":19229,"result":true,"route":"../roadmap/MeasureTheory/measures/lyapunov-convexity-theorem/index.html#lyapunov-convexity","scope":"MeasureTheory/measures/lyapunov-convexity-theorem","status":"fully_proved","summary":"Let $\\mu$ be a finite atomless measure on a measurable space $\\alpha$, let $E$ be a finite-dimensional real normed space, and let $f : \\alpha \\to E$ be Bochner integrable with respect to $\\mu$. Then the Lyapunov range of $f$ with respect to $\\mu$, the set of integrals $\\int_s f \\, d\\mu$ over measurable sets $s \\subseteq \\alpha$, is compact and convex.","title":"Lyapunov convexity theorem"},{"data_url":"scopes/MeasureTheory/measures/measures-on-complete-boolean-algebras.json","id":"MeasureTheory/measures/measures-on-complete-boolean-algebras","kind":"scope","navigation":"module","parent":19196,"result":false,"route":"../roadmap/MeasureTheory/measures/measures-on-complete-boolean-algebras/index.html","scope":"MeasureTheory/measures/measures-on-complete-boolean-algebras","status":"can_prove","title":"Measures on complete Boolean algebras"},{"id":"MeasureTheory/measures/measures-on-complete-boolean-algebras/is-von-neumann-measure","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook163Wanted.IsVonNeumannMeasure","parent":19232,"result":true,"route":"../roadmap/MeasureTheory/measures/measures-on-complete-boolean-algebras/index.html#is-von-neumann-measure","scope":"MeasureTheory/measures/measures-on-complete-boolean-algebras","status":"fully_proved","summary":"A real-valued function $\\mu$ on a complete Boolean algebra $B$ is a measure in von Neumann's sense when $\\mu(\\bot) = 0$, $\\mu(a) > 0$ for every $a \\ne \\bot$, and $\\mu$ is countably additive: for every sequence $(a_i)_{i \\in \\mathbb{N}}$ of pairwise disjoint elements of $B$ (that is, $a_i \\wedge a_j = \\bot$ whenever $i \\ne j$), the series $\\sum_i \\mu(a_i)$ converges and $\\mu(\\bigvee_i a_i) = \\sum_i \\mu(a_i)$.","title":"Von Neumann measures on a complete Boolean algebra"},{"id":"MeasureTheory/measures/measures-on-complete-boolean-algebras/satisfies-countable-chain-condition","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook163Wanted.SatisfiesCountableChainCondition","parent":19232,"result":true,"route":"../roadmap/MeasureTheory/measures/measures-on-complete-boolean-algebras/index.html#satisfies-countable-chain-condition","scope":"MeasureTheory/measures/measures-on-complete-boolean-algebras","status":"fully_proved","summary":"A complete Boolean algebra $B$ satisfies the countable chain condition (von Neumann's condition (5)) when every subset $S \\subseteq B$ whose distinct elements are pairwise disjoint (that is, $a \\wedge b = \\bot$ for all distinct $a, b \\in S$) is at most countable.","title":"The countable chain condition for complete Boolean algebras"},{"id":"MeasureTheory/measures/measures-on-complete-boolean-algebras/satisfies-generalized-distributivity","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook163Wanted.SatisfiesGeneralizedDistributivity","parent":19232,"result":true,"route":"../roadmap/MeasureTheory/measures/measures-on-complete-boolean-algebras/index.html#satisfies-generalized-distributivity","scope":"MeasureTheory/measures/measures-on-complete-boolean-algebras","status":"fully_proved","summary":"A complete Boolean algebra $B$ satisfies the generalized distributivity law (von Neumann's condition (6)) when, for every double sequence $(a_{i,j})_{i, j \\in \\mathbb{N}}$ of elements of $B$ that is increasing in $j$ for each fixed $i$ (that is, $a_{i,0} \\le a_{i,1} \\le \\cdots$ for every $i$),","title":"Von Neumann's generalized distributivity law"},{"id":"MeasureTheory/measures/measures-on-complete-boolean-algebras/von-neumann-sufficiency-claim","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook163Wanted.VonNeumannSufficiencyClaim","parent":19232,"result":true,"route":"../roadmap/MeasureTheory/measures/measures-on-complete-boolean-algebras/index.html#von-neumann-sufficiency-claim","scope":"MeasureTheory/measures/measures-on-complete-boolean-algebras","status":"fully_proved","summary":"The statement that every complete Boolean algebra satisfying the countable chain condition and the generalized distributivity law carries a measure in von Neumann's sense.","title":"Von Neumann's sufficiency claim"},{"id":"MeasureTheory/measures/measures-on-complete-boolean-algebras/von-neumann-conditions-not-sufficient","kind":"node","lean":"MathlibExt.MeasureTheory.ScottishBook163Wanted.vonNeumannConditionsNotSufficient","parent":19232,"result":true,"route":"../roadmap/MeasureTheory/measures/measures-on-complete-boolean-algebras/index.html#von-neumann-conditions-not-sufficient","scope":"MeasureTheory/measures/measures-on-complete-boolean-algebras","status":"can_prove","summary":"There is a complete Boolean algebra that satisfies the countable chain condition and the generalized distributivity law but carries no measure in von Neumann's sense; that is, von Neumann's sufficiency claim is false. This answers Problem 163 of the Scottish Book in the negative, in its stated form.","title":"Von Neumann's conditions do not guarantee a measure"},{"data_url":"scopes/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space.json","id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","kind":"scope","navigation":"module","parent":19196,"result":false,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","title":"Uniqueness of measures on Euclidean space"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/rectangle-vertex","kind":"node","lean":"MeasureTheory.rectangleVertex","parent":19238,"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#rectangle-vertex","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"Let $d$ be a natural number, let $a, b \\in \\mathbb{R}^d$, and let $S$ be a finite set of coordinate indices in $\\{0, \\dots, d-1\\}$. The vertex of the rectangle $(a, b]$ selected by $S$ is the point $v \\in \\mathbb{R}^d$ with $v_i = a_i$ for $i \\in S$ and $v_i = b_i$ for $i \\notin S$.","title":"Vertex of a rectangle"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/rectangle-value","kind":"node","lean":"MeasureTheory.rectangleValue","parent":19238,"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#rectangle-value","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"For $F : \\mathbb{R}^d \\to \\mathbb{R}$ and arbitrary $a, b \\in \\mathbb{R}^d$, the rectangle value of $F$ at $a, b$ is","title":"Rectangle value of a function"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/is-pi-system-pi-iic","kind":"node","lean":"MeasureTheory.isPiSystem_piIic","parent":19238,"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#is-pi-system-pi-iic","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"For $a \\in \\mathbb{R}^d$ let $L_a = \\prod_{i=1}^d (-\\infty, a_i]$. If two such boxes $L_a$ and $L_b$ intersect, then their intersection is $L_c$ for some $c \\in \\mathbb{R}^d$.","title":"Lower orthants form a \u03c0-system"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/generate-from-pi-iic","kind":"node","lean":"MeasureTheory.generateFrom_piIic","parent":19238,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower orthants generate the product \u03c3-algebra on "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Lower orthants generate the product \u03c3-algebra on $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#generate-from-pi-iic","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"The product \u03c3-algebra on $\\mathbb{R}^d$, built from the Borel \u03c3-algebra on each coordinate, is generated by the boxes $\\prod_{i=1}^d (-\\infty, a_i]$ for $a \\in \\mathbb{R}^d$.","title":"Lower orthants generate the product \u03c3-algebra on $\\mathbb{R}^d$"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/ext-of-iic-pi","kind":"node","lean":"MeasureTheory.Measure.ext_of_Iic_pi","parent":19238,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quadrant uniqueness for measures on "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Quadrant uniqueness for measures on $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#ext-of-iic-pi","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"Let $\\mu$ and $\\nu$ be finite measures on $\\mathbb{R}^d$. If $\\mu(\\{x : x \\le a\\}) = \\nu(\\{x : x \\le a\\})$ for every $a \\in \\mathbb{R}^d$, where $x \\le a$ means $x_i \\le a_i$ for every $i$, then $\\mu = \\nu$.","title":"Quadrant uniqueness for measures on $\\mathbb{R}^d$"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/c-ioc-pi","kind":"node","lean":"MeasureTheory.C_Ioc_pi","parent":19238,"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#c-ioc-pi","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"The family of all intervals $(l, u] \\subseteq \\mathbb{R}$ with real endpoints $l < u$. It is used as a generating \u03c0-system for the Borel \u03c3-algebra on $\\mathbb{R}$.","title":"Nonempty bounded half-open intervals"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/is-pi-system-c-ioc-pi","kind":"node","lean":"MeasureTheory.isPiSystem_C_Ioc_pi","parent":19238,"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#is-pi-system-c-ioc-pi","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"If two intervals $(l, u]$ and $(l', u']$ in $\\mathbb{R}$ with $l < u$ and $l' < u'$ intersect, then their intersection is again an interval $(l'', u'']$ with $l'' < u''$.","title":"Nonempty bounded half-open intervals form a \u03c0-system"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/generate-from-c-ioc-pi","kind":"node","lean":"MeasureTheory.generateFrom_C_Ioc_pi","parent":19238,"reading_title":{"format":"inline-parts-v1","parts":[["text","Half-open intervals generate the Borel \u03c3-algebra on "],["math","\\(\\mathbb{R}\\)"]],"source":"Half-open intervals generate the Borel \u03c3-algebra on $\\mathbb{R}$"},"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#generate-from-c-ioc-pi","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"The Borel \u03c3-algebra on $\\mathbb{R}$ is the \u03c3-algebra generated by the nonempty bounded half-open intervals $(l, u]$ with $l < u$.","title":"Half-open intervals generate the Borel \u03c3-algebra on $\\mathbb{R}$"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/is-countably-spanning-c-ioc-pi","kind":"node","lean":"MeasureTheory.isCountablySpanning_C_Ioc_pi","parent":19238,"reading_title":{"format":"inline-parts-v1","parts":[["text","Half-open intervals countably span "],["math","\\(\\mathbb{R}\\)"]],"source":"Half-open intervals countably span $\\mathbb{R}$"},"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#is-countably-spanning-c-ioc-pi","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"There is a sequence of nonempty bounded half-open intervals $(l, u]$ with $l < u$ whose union is $\\mathbb{R}$; for example, the intervals $(-n - 1, n + 1]$ for $n \\in \\mathbb{N}$.","title":"Half-open intervals countably span $\\mathbb{R}$"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/eq-of-forall-ioc-eq","kind":"node","lean":"MeasureTheory.Measure.eq_of_forall_Ioc_eq","parent":19238,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangle uniqueness for measures on "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Rectangle uniqueness for measures on $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#eq-of-forall-ioc-eq","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"Let $\\mu_1$ and $\\mu_2$ be finite measures on $\\mathbb{R}^d$. If $\\mu_1((a, b]) = \\mu_2((a, b])$ for all $a, b \\in \\mathbb{R}^d$ with $a_i \\le b_i$ for every $i$, where $(a, b] = \\prod_{i=1}^d (a_i, b_i]$, then $\\mu_1 = \\mu_2$.","title":"Rectangle uniqueness for measures on $\\mathbb{R}^d$"},{"id":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/eq-of-forall-rectangle-value-eq","kind":"node","lean":"MeasureTheory.Measure.eq_of_forall_rectangleValue_eq","parent":19238,"result":true,"route":"../roadmap/MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space/index.html#eq-of-forall-rectangle-value-eq","scope":"MeasureTheory/measures/uniqueness-of-measures-on-euclidean-space","status":"fully_proved","summary":"Let $F : \\mathbb{R}^d \\to \\mathbb{R}$, and let $\\mu_1$ and $\\mu_2$ be measures on $\\mathbb{R}^d$. Suppose that for all $a, b \\in \\mathbb{R}^d$ with $a_i \\le b_i$ for every $i$, both $\\mu_1$ and $\\mu_2$ assign the rectangle $(a, b] = \\prod_{i=1}^d (a_i, b_i]$ the mass $\\max(\\Delta_{a,b} F, 0)$, where $\\Delta_{a,b} F$ is the rectangle value of $F$ at $a, b$. Then $\\mu_1 = \\mu_2$.","title":"Uniqueness from rectangle values"},{"data_url":"scopes/MeasureTheory/lebesgue-measure.json","id":"MeasureTheory/lebesgue-measure","kind":"scope","parent":19131,"result":false,"route":"../roadmap/MeasureTheory/lebesgue-measure/index.html","scope":"MeasureTheory/lebesgue-measure","status":"can_prove","summary":"Lebesgue measure on the real line and Euclidean space: volumes of concrete bodies, null sets and density phenomena.","title":"Lebesgue measure"},{"id":"MeasureTheory/lebesgue-measure/exists-gt-not-mem-of-volume-lt-top","kind":"node","lean":"MeasureTheory.exists_gt_not_mem_of_volume_lt_top","parent":19250,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/index.html#exists-gt-not-mem-of-volume-lt-top","scope":"MeasureTheory/lebesgue-measure","status":"fully_proved","summary":"Let $E \\subseteq \\mathbb{R}$ have finite Lebesgue outer measure ($E$ need not be measurable). Then for every real $X$ there is a real $x > X$ with $x \\notin E$; that is, $E$ contains no half-line $(X, \\infty)$.","title":"Sets of finite Lebesgue measure contain no right half-line"},{"id":"MeasureTheory/lebesgue-measure/exists-add-mem-and-sub-mem-of-volume-pos","kind":"node","lean":"MathlibExt.MeasureTheory.PositiveMeasureThreeTermProgressionWanted.exists_add_mem_and_sub_mem_of_volume_pos","parent":19250,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/index.html#exists-add-mem-and-sub-mem-of-volume-pos","scope":"MeasureTheory/lebesgue-measure","status":"can_prove","summary":"For every Borel set $A \\subseteq \\mathbb{R}$ of positive Lebesgue measure there exist $x \\in A$ and $y > 0$ such that $x + y \\in A$ and $x - y \\in A$.","title":"Sets of positive measure contain three-term arithmetic progressions"},{"data_url":"scopes/MeasureTheory/lebesgue-measure/null-sets.json","id":"MeasureTheory/lebesgue-measure/null-sets","kind":"scope","navigation":"module","parent":19250,"result":false,"route":"../roadmap/MeasureTheory/lebesgue-measure/null-sets/index.html","scope":"MeasureTheory/lebesgue-measure/null-sets","status":"can_prove","title":"Null sets"},{"id":"MeasureTheory/lebesgue-measure/null-sets/measure-image-null-of-locally-null","kind":"node","lean":"MathlibExt.MeasureTheory.measure_image_null_of_locally_null","parent":19253,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/null-sets/index.html#measure-image-null-of-locally-null","scope":"MeasureTheory/lebesgue-measure/null-sets","status":"fully_proved","summary":"Let $X$ be a second-countable topological space, $Y$ a measurable space with a measure $\\mu$, $f : X \\to Y$ a function and $s \\subseteq X$. If every point $x \\in s$ has a neighbourhood $V$ with $\\mu(f(V \\cap s)) = 0$, then $\\mu(f(s)) = 0$.","title":"Sets with locally null image have null image"},{"id":"MeasureTheory/lebesgue-measure/null-sets/measure-set-of-eval-eq-zero-of-ne-zero","kind":"node","lean":"MvPolynomial.measure_setOf_eval_eq_zero_of_ne_zero","parent":19253,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/null-sets/index.html#measure-set-of-eval-eq-zero-of-ne-zero","scope":"MeasureTheory/lebesgue-measure/null-sets","status":"fully_proved","summary":"A nonzero multivariate polynomial over a nondiscrete locally compact second-countable Hausdorff topological field has an additive Haar-null zero set. This includes the empty variable type.","title":"Nonzero polynomials over nondiscrete fields have Haar-null zero sets"},{"id":"MeasureTheory/lebesgue-measure/null-sets/measurable-set-and-volume-eq-zero-graph","kind":"node","lean":"MathlibExt.MeasureTheory.GraphMeasurableNullWanted.measurableSet_and_volume_eq_zero_graph","parent":19253,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/null-sets/index.html#measurable-set-and-volume-eq-zero-graph","scope":"MeasureTheory/lebesgue-measure/null-sets","status":"can_prove","summary":"Let $d \\ge 0$ be an integer and let $f : \\mathbb{R}^d \\to \\mathbb{R}$ be Borel measurable. Identify $\\mathbb{R}^d \\times \\mathbb{R}$ with $\\mathbb{R}^{d+1}$ by taking the last coordinate as $y$. Then the graph $\\Gamma = \\{(x, y) \\in \\mathbb{R}^d \\times \\mathbb{R} : y = f(x)\\}$ is a Borel measurable subset of $\\mathbb{R}^{d+1}$ and has Lebesgue measure $0$.","title":"The graph of a measurable function is a null set"},{"data_url":"scopes/MeasureTheory/lebesgue-measure/volumes-of-polytopes.json","id":"MeasureTheory/lebesgue-measure/volumes-of-polytopes","kind":"scope","navigation":"module","parent":19250,"result":false,"route":"../roadmap/MeasureTheory/lebesgue-measure/volumes-of-polytopes/index.html","scope":"MeasureTheory/lebesgue-measure/volumes-of-polytopes","status":"fully_proved","title":"Volumes of polytopes"},{"id":"MeasureTheory/lebesgue-measure/volumes-of-polytopes/volume-coordinate-cross-polytope","kind":"node","lean":"MetaMathlibExt.volume_coordinateCrossPolytope","parent":19257,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/volumes-of-polytopes/index.html#volume-coordinate-cross-polytope","scope":"MeasureTheory/lebesgue-measure/volumes-of-polytopes","status":"fully_proved","summary":"For every integer $n \\ge 1$, the cross-polytope $\\{x \\in \\mathbb{R}^n : |x_1| + \\dots + |x_n| \\le 1\\}$ in the Euclidean space $\\mathbb{R}^n$ has volume $2^n / n!$.","title":"Volume of the cross-polytope"},{"id":"MeasureTheory/lebesgue-measure/volumes-of-polytopes/volume-coordinate-cube","kind":"node","lean":"MetaMathlibExt.volume_coordinateCube","parent":19257,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/volumes-of-polytopes/index.html#volume-coordinate-cube","scope":"MeasureTheory/lebesgue-measure/volumes-of-polytopes","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the coordinate cube $[-1, 1]^n$ in the Euclidean space $\\mathbb{R}^n$ has volume $2^n$.","title":"Volume of the coordinate cube"},{"id":"MeasureTheory/lebesgue-measure/volumes-of-polytopes/to-real-volume-convex-hull-triangle","kind":"node","lean":"MeasureTheory.toReal_volume_convexHull_triangle","parent":19257,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/volumes-of-polytopes/index.html#to-real-volume-convex-hull-triangle","scope":"MeasureTheory/lebesgue-measure/volumes-of-polytopes","status":"fully_proved","summary":"Write $x = (x_1, x_2)$ for the coordinates of a point $x$ of the Euclidean plane $\\mathbb{R}^2$. For all points $a, b, c$ in $\\mathbb{R}^2$, the Lebesgue measure (area) of the convex hull of $\\{a, b, c\\}$, as a real number, equals $\\frac{1}{2}\\,\\lvert (b_1 - a_1)(c_2 - a_2) - (b_2 - a_2)(c_1 - a_1) \\rvert$.","title":"Determinant formula for the area of a triangle"},{"id":"MeasureTheory/lebesgue-measure/volumes-of-polytopes/volume-unit-square","kind":"node","lean":"MeasureTheory.volume_unitSquare","parent":19257,"result":true,"route":"../roadmap/MeasureTheory/lebesgue-measure/volumes-of-polytopes/index.html#volume-unit-square","scope":"MeasureTheory/lebesgue-measure/volumes-of-polytopes","status":"fully_proved","summary":"The closed unit square $\\{x \\in \\mathbb{R}^2 : 0 \\le x_1 \\le 1 \\text{ and } 0 \\le x_2 \\le 1\\}$ in the Euclidean plane has Lebesgue measure (area) $1$.","title":"Area of the unit square"},{"data_url":"scopes/MeasureTheory/haar-measure.json","id":"MeasureTheory/haar-measure","kind":"scope","parent":19131,"result":false,"route":"../roadmap/MeasureTheory/haar-measure/index.html","scope":"MeasureTheory/haar-measure","status":"fully_proved","summary":"Invariant measures on locally compact groups, unimodularity, and fundamental domains for the actions of discrete subgroups.","title":"Haar measure and fundamental domains"},{"data_url":"scopes/MeasureTheory/haar-measure/bruhat-functions.json","id":"MeasureTheory/haar-measure/bruhat-functions","kind":"scope","navigation":"module","parent":19262,"result":false,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","title":"Bruhat functions"},{"id":"MeasureTheory/haar-measure/bruhat-functions/lintegral-mul-eq-of-normalized-smul","kind":"node","lean":"MeasureTheory.lintegral_mul_eq_of_normalized_smul","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#lintegral-mul-eq-of-normalized-smul","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"Nonnegative integrals of an invariant function agree for orbit-normalized weights. Normalization is needed only where the function is nonzero, and the common integral is allowed to be infinite.","title":"Orbit-normalized weights give equal nonnegative integrals"},{"id":"MeasureTheory/haar-measure/bruhat-functions/integrable-mul-iff-of-normalized-smul","kind":"node","lean":"MeasureTheory.integrable_mul_iff_of_normalized_smul","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#integrable-mul-iff-of-normalized-smul","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"Integrability of an invariant complex function times a nonnegative real weight is independent of the weight when both orbit averages are one on the function's support. Neither weighted function is required to be integrable.","title":"Integrability is independent of the orbit-normalized weight"},{"id":"MeasureTheory/haar-measure/bruhat-functions/integral-mul-eq-of-normalized-smul","kind":"node","lean":"MeasureTheory.integral_mul_eq_of_normalized_smul","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#integral-mul-eq-of-normalized-smul","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"Weighted integrals of an invariant complex function agree for nonnegative orbit-normalized cutoffs. If one weighted function is not integrable, neither is the other, and both Bochner integrals are zero.","title":"Orbit-normalized cutoffs give equal complex integrals"},{"id":"MeasureTheory/haar-measure/bruhat-functions/integral-mul-eq-of-normalized-subgroup","kind":"node","lean":"MeasureTheory.integral_mul_eq_of_normalized_subgroup","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#integral-mul-eq-of-normalized-subgroup","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"Subgroup-normalized nonnegative weights give the same integral of an invariant complex function. This measure-theoretic statement requires neither a closed subgroup nor compact supports nor a bounded function.","title":"Subgroup-normalized weights give equal integrals"},{"id":"MeasureTheory/haar-measure/bruhat-functions/exists-continuous-cutoff-of-is-closed-em","kind":"node","lean":"MeasureTheory.exists_continuous_cutoff_of_isClosedEmbedding","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#exists-continuous-cutoff-of-is-closed-em","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"A continuous nonnegative compactly supported cutoff normalized on every left orbit meeting `K`, for a closed-embedding homomorphism `\u03b9`. The measure may have any scale. No Hausdorff or countability hypotheses are required.","title":"Continuous cutoff for a closed-embedding homomorphism"},{"id":"MeasureTheory/haar-measure/bruhat-functions/exists-continuous-cutoff-of-subset-mul","kind":"node","lean":"MeasureTheory.exists_continuous_cutoff_of_subset_mul","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#exists-continuous-cutoff-of-subset-mul","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"A continuous nonnegative compactly supported Bruhat cutoff for a closed subgroup, normalized on any subset of the saturation of a compact set.","title":"Continuous Bruhat cutoff for a closed subgroup"},{"id":"MeasureTheory/haar-measure/bruhat-functions/exists-has-compact-support-integral-subg","kind":"node","lean":"MeasureTheory.exists_hasCompactSupport_integral_subgroup_translate_eq_one_of_subset_mul","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#exists-has-compact-support-integral-subg","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"A Bruhat section: given a closed subgroup `T` of a locally compact group and a set `E` contained in `T * C` for compact `C`, there is a compactly supported nonnegative measurable function `w` whose integral over each `T`-translate meeting `E` equals `1`.","title":"Existence of a Bruhat section for a closed subgroup"},{"id":"MeasureTheory/haar-measure/bruhat-functions/integrable-mul-of-integral-subgroup-tran","kind":"node","lean":"MeasureTheory.integrable_mul_of_integral_subgroup_translate_eq_one","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#integrable-mul-of-integral-subgroup-tran","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"A bounded measurable `T`-invariant complex function is integrable against any nonnegative measurable compactly supported weight whose `T`-average is one wherever the function is nonzero. The weight need not be bounded.","title":"Bounded invariant functions are integrable against normalized cutoffs"},{"id":"MeasureTheory/haar-measure/bruhat-functions/integral-mul-eq-integral-mul-of-integral","kind":"node","lean":"MeasureTheory.integral_mul_eq_integral_mul_of_integral_subgroup_translate_eq_one","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#integral-mul-eq-integral-mul-of-integral","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"The integral of a bounded measurable `T`-invariant complex function against a nonnegative measurable compactly supported cutoff does not depend on the cutoff, when each cutoff has subgroup integral one on the support of that function. Normalization is not required outside its support.","title":"Cutoff independence of integrals of bounded invariant functions"},{"id":"MeasureTheory/haar-measure/bruhat-functions/exists-pos-integral-cutoff-of-cocompact","kind":"node","lean":"MeasureTheory.exists_pos_integral_cutoff_of_cocompact","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#exists-pos-integral-cutoff-of-cocompact","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"A compact cocompactness witness gives a continuous normalized cutoff and a positive common integral for all nonnegative measurable compactly supported normalized cutoffs. Every such cutoff is genuinely integrable, even when it is not bounded. Neither the ambient group nor the subgroup must be Hausdorff.","title":"Positive common integral of cocompact subgroup cutoffs"},{"id":"MeasureTheory/haar-measure/bruhat-functions/exists-is-locally-constant-integral-subg","kind":"node","lean":"MeasureTheory.exists_isLocallyConstant_integral_subgroup_mul_eq_one","parent":19263,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/bruhat-functions/index.html#exists-is-locally-constant-integral-subg","scope":"MeasureTheory/haar-measure/bruhat-functions","status":"fully_proved","summary":"A compact set in a topological group with a compact open subgroup has a nonnegative locally constant cutoff whose integral over every left `T`-orbit meeting that set is one, for a closed commutative subgroup `T`. The same compactly supported function works simultaneously on all of `T * \u03a9`. The Haar measure need not have any prescribed normalization.","title":"Existence of a locally constant normalized subgroup cutoff"},{"data_url":"scopes/MeasureTheory/haar-measure/fundamental-domains.json","id":"MeasureTheory/haar-measure/fundamental-domains","kind":"scope","navigation":"module","parent":19262,"result":false,"route":"../roadmap/MeasureTheory/haar-measure/fundamental-domains/index.html","scope":"MeasureTheory/haar-measure/fundamental-domains","status":"fully_proved","title":"Fundamental domains"},{"id":"MeasureTheory/haar-measure/fundamental-domains/i-union-inv-smul-of-left-coset-represent","kind":"node","lean":"MeasureTheory.IsFundamentalDomain.iUnion_inv_smul_of_leftCosetRepresentatives","parent":19275,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/fundamental-domains/index.html#i-union-inv-smul-of-left-coset-represent","scope":"MeasureTheory/haar-measure/fundamental-domains","status":"fully_proved","summary":"From a fundamental domain for `\u0393\u2081`, a countable union of inverse coset-representative translates gives a fundamental domain for a subgroup `\u0393\u2082 \u2264 \u0393\u2081` of countable index.","title":"Fundamental domain for a subgroup of countable index"},{"id":"MeasureTheory/haar-measure/fundamental-domains/exists-measurable-set-is-fundamental-dom","kind":"node","lean":"MeasureTheory.exists_measurableSet_isFundamentalDomain_op_of_discreteTopology","parent":19275,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/fundamental-domains/index.html#exists-measurable-set-is-fundamental-dom","scope":"MeasureTheory/haar-measure/fundamental-domains","status":"fully_proved","summary":"Every discrete subgroup of a second-countable Hausdorff topological group admits a measurable fundamental domain for the opposite action, simultaneously for every measure.","title":"Measurable fundamental domains for discrete subgroups"},{"id":"MeasureTheory/haar-measure/fundamental-domains/set-lintegral-i-union-inv-smul-eq-and-se","kind":"node","lean":"MeasureTheory.IsFundamentalDomain.setLIntegral_iUnion_inv_smul_eq_and_setIntegral_eq_of_leftCosetRepresentatives","parent":19275,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/fundamental-domains/index.html#set-lintegral-i-union-inv-smul-eq-and-se","scope":"MeasureTheory/haar-measure/fundamental-domains","status":"fully_proved","summary":"If `\ud835\udcd5` is a fundamental domain for `\u0393\u2081` and `R` is a system of representatives for the left cosets of `\u0393\u2082` in `\u0393\u2081`, then integrals over the union of the translates `((R i)\u207b\u00b9) \u2022 \ud835\udcd5` unfold into sums over `i` of integrals over `\ud835\udcd5`, for both `\u211d\u22650\u221e`-valued and Bochner integrals.","title":"Integrals over a subgroup fundamental domain as sums over cosets"},{"data_url":"scopes/MeasureTheory/haar-measure/unimodular-groups.json","id":"MeasureTheory/haar-measure/unimodular-groups","kind":"scope","navigation":"module","parent":19262,"result":false,"route":"../roadmap/MeasureTheory/haar-measure/unimodular-groups/index.html","scope":"MeasureTheory/haar-measure/unimodular-groups","status":"fully_proved","title":"Unimodular groups"},{"id":"MeasureTheory/haar-measure/unimodular-groups/is-inv-invariant-of-is-mul-right-invaria","kind":"node","lean":"MeasureTheory.Measure.isInvInvariant_of_isMulRightInvariant","parent":19279,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/unimodular-groups/index.html#is-inv-invariant-of-is-mul-right-invaria","scope":"MeasureTheory/haar-measure/unimodular-groups","status":"fully_proved","summary":"A right-invariant Haar measure is invariant under inversion.","title":"Right-invariant Haar measures are inversion-invariant"},{"id":"MeasureTheory/haar-measure/unimodular-groups/modular-character-fin-two-eq-one","kind":"node","lean":"Matrix.GeneralLinearGroup.modularCharacter_fin_two_eq_one","parent":19279,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/unimodular-groups/index.html#modular-character-fin-two-eq-one","scope":"MeasureTheory/haar-measure/unimodular-groups","status":"fully_proved","summary":"The modular character of `GL (2, F)` is trivial: `GL (2, F)` is unimodular.","title":"GL(2, F) is unimodular"},{"id":"MeasureTheory/haar-measure/unimodular-groups/is-mul-right-invariant-and-is-inv-invari","kind":"node","lean":"Matrix.GeneralLinearGroup.isMulRightInvariant_and_isInvInvariant_of_isHaarMeasure_fin_two","parent":19279,"result":true,"route":"../roadmap/MeasureTheory/haar-measure/unimodular-groups/index.html#is-mul-right-invariant-and-is-inv-invari","scope":"MeasureTheory/haar-measure/unimodular-groups","status":"fully_proved","summary":"Every Haar measure on `GL (2, F)` is right-invariant and inversion-invariant.","title":"Haar measures on GL(2, F) are right- and inversion-invariant"},{"data_url":"scopes/MeasureTheory/integration.json","id":"MeasureTheory/integration","kind":"scope","parent":19131,"result":false,"route":"../roadmap/MeasureTheory/integration/index.html","scope":"MeasureTheory/integration","status":"can_prove","summary":"The Lebesgue integral and its convergence theorems, modes of convergence of measurable functions, approximation by simple functions, product measures, convolution, differentiation under the integral sign and integral inequalities.","title":"Integration and convergence"},{"id":"MeasureTheory/integration/exists-measure-lt-tendsto-uniformly-on-c","kind":"node","lean":"MathlibExt.MeasureTheory.EgorovDominatedRealLineWanted.exists_measure_lt_tendstoUniformlyOn_compl_of_ae_tendsto_of_abs_le","parent":19283,"result":true,"route":"../roadmap/MeasureTheory/integration/index.html#exists-measure-lt-tendsto-uniformly-on-c","scope":"MeasureTheory/integration","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be any function and let $f_n : \\mathbb{R} \\to \\mathbb{R}$, $n \\in \\mathbb{N}$, be Borel measurable functions with $f_n(x) \\to f(x)$ for almost every $x \\in \\mathbb{R}$. Let $g : \\mathbb{R} \\to \\mathbb{R}$ be Lebesgue integrable with $|f_n(x)| \\le g(x)$ for all $n$ and all $x \\in \\mathbb{R}$. Then for every $\\varepsilon > 0$ there is a Borel set $E \\subseteq \\mathbb{R}$ of\u2026","title":"Egorov's theorem on the real line under integrable domination"},{"id":"MeasureTheory/integration/lusin","kind":"node","lean":"MathlibExt.MeasureTheory.Function.Lusin.lusin","parent":19283,"result":true,"route":"../roadmap/MeasureTheory/integration/index.html#lusin","scope":"MeasureTheory/integration","status":"fully_proved","summary":"Let $X$ be a locally compact Hausdorff topological space with its Borel $\\sigma$-algebra, and let $\\mu$ be a finite measure on $X$ that is regular: finite on compact sets, outer regular, and inner regular on open sets by compact sets. Let $Y$ be a second-countable topological space with its Borel $\\sigma$-algebra, and let $f : X \\to Y$ be measurable. Then for every real $\\varepsilon > 0$ there is a compact set\u2026","title":"Lusin's theorem"},{"data_url":"scopes/MeasureTheory/integration/almost-everywhere-convergence.json","id":"MeasureTheory/integration/almost-everywhere-convergence","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/almost-everywhere-convergence/index.html","scope":"MeasureTheory/integration/almost-everywhere-convergence","status":"can_prove","title":"Almost everywhere convergence"},{"id":"MeasureTheory/integration/almost-everywhere-convergence/exists-tendsto-abs-at-top-and-ae-tendsto","kind":"node","lean":"MathlibExt.MeasureTheory.AeNullSequenceUnboundedMultipliersWanted.exists_tendsto_abs_atTop_and_ae_tendsto_mul_zero","parent":19286,"result":true,"route":"../roadmap/MeasureTheory/integration/almost-everywhere-convergence/index.html#exists-tendsto-abs-at-top-and-ae-tendsto","scope":"MeasureTheory/integration/almost-everywhere-convergence","status":"can_prove","summary":"Let $(f_n)_{n \\in \\mathbb{N}}$ be a sequence of Borel measurable functions $f_n : \\mathbb{R} \\to \\mathbb{R}$ such that $f_n(x) \\to 0$ for almost every $x \\in [0, 1]$. Then there is a sequence of real numbers $(c_n)_{n \\in \\mathbb{N}}$ with $|c_n| \\to \\infty$ such that $c_n f_n(x) \\to 0$ for almost every $x \\in [0, 1]$.","title":"Unbounded multipliers preserving almost-everywhere convergence to zero"},{"id":"MeasureTheory/integration/almost-everywhere-convergence/ae-tendsto-rpow-neg-mul-comp-mul-zero","kind":"node","lean":"MathlibExt.MeasureTheory.DilatedL1FunctionDecayAeWanted.ae_tendsto_rpow_neg_mul_comp_mul_zero","parent":19286,"result":true,"route":"../roadmap/MeasureTheory/integration/almost-everywhere-convergence/index.html#ae-tendsto-rpow-neg-mul-comp-mul-zero","scope":"MeasureTheory/integration/almost-everywhere-convergence","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be Lebesgue integrable and let $\\lambda > 0$. Then for almost every $x \\in \\mathbb{R}$, $n^{-\\lambda} f(nx) \\to 0$ as $n \\to \\infty$ through the positive integers.","title":"Decay of dilates of an integrable function"},{"data_url":"scopes/MeasureTheory/integration/approximation-by-simple-functions.json","id":"MeasureTheory/integration/approximation-by-simple-functions","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/approximation-by-simple-functions/index.html","scope":"MeasureTheory/integration/approximation-by-simple-functions","status":"fully_proved","title":"Approximation by simple functions"},{"id":"MeasureTheory/integration/approximation-by-simple-functions/eapprox-sub-lt-of-le","kind":"node","lean":"MathlibExt.MeasureTheory.Function.SimpleFuncUniformApprox.eapprox_sub_lt_of_le","parent":19289,"result":true,"route":"../roadmap/MeasureTheory/integration/approximation-by-simple-functions/index.html#eapprox-sub-lt-of-le","scope":"MeasureTheory/integration/approximation-by-simple-functions","status":"fully_proved","summary":"Let $\\alpha$ be a measurable space and $f : \\alpha \\to [0, \\infty]$ a measurable function. Let $r_0, r_1, \\ldots$ be the fixed enumeration of $\\mathbb{Q}$ by natural-number codes ($r_k = 0$ when $k$ codes no rational), put $q_k = \\max(r_k, 0)$, and let $\\varphi_n(a) = \\max(\\{0\\} \\cup \\{q_k : k < n,\\ q_k \\le f(a)\\})$ be the $n$-th standard simple approximation of $f$. Then for every $B \\in [0, \\infty)$ and every\u2026","title":"The standard simple approximations converge uniformly where the function is bounded"},{"id":"MeasureTheory/integration/approximation-by-simple-functions/exists-monotone-tendsto","kind":"node","lean":"MathlibExt.MeasureTheory.Function.SimpleFuncUniformApprox.exists_monotone_tendsto","parent":19289,"result":true,"route":"../roadmap/MeasureTheory/integration/approximation-by-simple-functions/index.html#exists-monotone-tendsto","scope":"MeasureTheory/integration/approximation-by-simple-functions","status":"fully_proved","summary":"Let $\\alpha$ be a measurable space and $f : \\alpha \\to [0, \\infty]$ a measurable function. There is a sequence of simple functions $\\varphi_n : \\alpha \\to [0, \\infty]$ ($n \\in \\mathbb{N}$) such that: (a) $\\varphi_0 \\le \\varphi_1 \\le \\cdots \\le f$ pointwise; (b) $\\varphi_n(a) \\to f(a)$ for every $a \\in \\alpha$; and (c) for every $B \\in [0, \\infty)$ and every $\\varepsilon \\in (0, \\infty]$ there is $N \\in \\mathbb{N}$\u2026","title":"Approximation of nonnegative measurable functions by simple functions"},{"data_url":"scopes/MeasureTheory/integration/bounded-convergence-theorem.json","id":"MeasureTheory/integration/bounded-convergence-theorem","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/bounded-convergence-theorem/index.html","scope":"MeasureTheory/integration/bounded-convergence-theorem","status":"fully_proved","title":"Bounded convergence theorem"},{"id":"MeasureTheory/integration/bounded-convergence-theorem/tendsto-integral-norm-sub-of-bounded-con","kind":"node","lean":"MeasureTheory.tendsto_integral_norm_sub_of_bounded_convergence","parent":19292,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded convergence theorem, "],["math","\\(L^1\\)"],["text"," form"]],"source":"Bounded convergence theorem, $L^1$ form"},"result":true,"route":"../roadmap/MeasureTheory/integration/bounded-convergence-theorem/index.html#tendsto-integral-norm-sub-of-bounded-con","scope":"MeasureTheory/integration/bounded-convergence-theorem","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $\\alpha$, let $G$ be a real normed space, let $C$ be a real number, and let $F_n : \\alpha \\to G$ for $n \\in \\mathbb{N}$ and $f : \\alpha \\to G$ be functions. Suppose that each $F_n$ is almost everywhere strongly measurable, that for every $n$ we have $\\|F_n(a)\\| \\le C$ for $\\mu$-almost every $a$, and that $F_n \\to f$ in $\\mu$-measure. Then\u2026","title":"Bounded convergence theorem, $L^1$ form"},{"id":"MeasureTheory/integration/bounded-convergence-theorem/tendsto-integral-of-bounded-convergence","kind":"node","lean":"MeasureTheory.tendsto_integral_of_bounded_convergence","parent":19292,"result":true,"route":"../roadmap/MeasureTheory/integration/bounded-convergence-theorem/index.html#tendsto-integral-of-bounded-convergence","scope":"MeasureTheory/integration/bounded-convergence-theorem","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $\\alpha$, let $G$ be a real normed space, let $C$ be a real number, and let $F_n : \\alpha \\to G$ for $n \\in \\mathbb{N}$ and $f : \\alpha \\to G$ be functions. Suppose that each $F_n$ is almost everywhere strongly measurable, that for every $n$ we have $\\|F_n(a)\\| \\le C$ for $\\mu$-almost every $a$, and that $F_n \\to f$ in $\\mu$-measure (in probability). Then\u2026","title":"Bounded convergence theorem"},{"data_url":"scopes/MeasureTheory/integration/convergence-in-measure.json","id":"MeasureTheory/integration/convergence-in-measure","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/convergence-in-measure/index.html","scope":"MeasureTheory/integration/convergence-in-measure","status":"can_prove","title":"Convergence in measure"},{"id":"MeasureTheory/integration/convergence-in-measure/l0-unit-interval","kind":"node","lean":"MathlibExt.MeasureTheory.L0MetricTendstoInMeasureWanted.L0UnitInterval","parent":19295,"reading_title":{"format":"inline-parts-v1","parts":[["text","Measurable functions representing "],["math","\\(L^0((0,1))\\)"]],"source":"Measurable functions representing $L^0((0,1))$"},"result":true,"route":"../roadmap/MeasureTheory/integration/convergence-in-measure/index.html#l0-unit-interval","scope":"MeasureTheory/integration/convergence-in-measure","status":"fully_proved","summary":"The type of Borel measurable functions $f : \\mathbb{R} \\to \\mathbb{R}$, used as representatives of the elements of $L^0([0,1])$.","title":"Measurable functions representing $L^0((0,1))$"},{"id":"MeasureTheory/integration/convergence-in-measure/l0-pseudo-metric","kind":"node","lean":"MathlibExt.MeasureTheory.L0MetricTendstoInMeasureWanted.L0PseudoMetric","parent":19295,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(L^0\\)"],["text"," pseudometric on "],["math","\\((0,1)\\)"]],"source":"The $L^0$ pseudometric on $(0,1)$"},"result":true,"route":"../roadmap/MeasureTheory/integration/convergence-in-measure/index.html#l0-pseudo-metric","scope":"MeasureTheory/integration/convergence-in-measure","status":"fully_proved","summary":"For Borel measurable functions $f, g : \\mathbb{R} \\to \\mathbb{R}$,","title":"The $L^0$ pseudometric on $(0,1)$"},{"id":"MeasureTheory/integration/convergence-in-measure/tendsto-l0-pseudo-metric-zero-iff-tendst","kind":"node","lean":"MathlibExt.MeasureTheory.L0MetricTendstoInMeasureWanted.tendsto_L0PseudoMetric_zero_iff_tendstoInMeasure","parent":19295,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergence in the "],["math","\\(L^0\\)"],["text"," pseudometric is convergence in measure"]],"source":"Convergence in the $L^0$ pseudometric is convergence in measure"},"result":true,"route":"../roadmap/MeasureTheory/integration/convergence-in-measure/index.html#tendsto-l0-pseudo-metric-zero-iff-tendst","scope":"MeasureTheory/integration/convergence-in-measure","status":"can_prove","summary":"Let $f$ and $f_0, f_1, f_2, \\ldots$ be Borel measurable functions $\\mathbb{R} \\to \\mathbb{R}$, and let $d$ be the pseudometric","title":"Convergence in the $L^0$ pseudometric is convergence in measure"},{"data_url":"scopes/MeasureTheory/integration/convolution.json","id":"MeasureTheory/integration/convolution","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/convolution/index.html","scope":"MeasureTheory/integration/convolution","status":"can_prove","title":"Convolution"},{"id":"MeasureTheory/integration/convolution/uniform-continuous-convolution-integrabl","kind":"node","lean":"MathlibExt.MeasureTheory.ConvolutionIntegrableBoundedUniformWanted.uniformContinuous_convolution_integrable_bounded","parent":19299,"result":true,"route":"../roadmap/MeasureTheory/integration/convolution/index.html#uniform-continuous-convolution-integrabl","scope":"MeasureTheory/integration/convolution","status":"can_prove","summary":"Let $d \\in \\mathbb{N}$, let $f : \\mathbb{R}^d \\to \\mathbb{R}$ be Lebesgue integrable, and let $g : \\mathbb{R}^d \\to \\mathbb{R}$ be Borel measurable and bounded. Then the convolution","title":"Convolution of an integrable and a bounded function is uniformly continuous"},{"id":"MeasureTheory/integration/convolution/integrable-comp-sub-mul-of-integrable-fi","kind":"node","lean":"MathlibExt.MeasureTheory.ConvolutionIntegrandIntegrableEuclideanWanted.integrable_comp_sub_mul_of_integrable_fin_append","parent":19299,"result":true,"route":"../roadmap/MeasureTheory/integration/convolution/index.html#integrable-comp-sub-mul-of-integrable-fi","scope":"MeasureTheory/integration/convolution","status":"can_prove","summary":"Let $d \\in \\mathbb{N}$ and let $f, g : \\mathbb{R}^d \\to \\mathbb{R}$ be Lebesgue integrable. Identify $\\mathbb{R}^{2d}$ with $\\mathbb{R}^d \\times \\mathbb{R}^d$ by writing a point as $(x, y)$, where $x$ consists of its first $d$ coordinates and $y$ of its last $d$ coordinates. Then the function $(x, y) \\mapsto f(x - y)\\, g(y)$ is Lebesgue integrable on $\\mathbb{R}^{2d}$.","title":"Integrability of the convolution integrand"},{"id":"MeasureTheory/integration/convolution/tendsto-mul-integral-integral-exp-neg-tw","kind":"node","lean":"MathlibExt.MeasureTheory.ExponentialKernelApproxIdentityWanted.tendsto_mul_integral_integral_exp_neg_two_mul_abs_sub","parent":19299,"result":true,"route":"../roadmap/MeasureTheory/integration/convolution/index.html#tendsto-mul-integral-integral-exp-neg-tw","scope":"MeasureTheory/integration/convolution","status":"can_prove","summary":"For every function $f : \\mathbb{R} \\to \\mathbb{R}$ in $L^2(\\mathbb{R})$,","title":"Exponential kernel approximate identity"},{"data_url":"scopes/MeasureTheory/integration/differentiation-under-the-integral-sign.json","id":"MeasureTheory/integration/differentiation-under-the-integral-sign","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/differentiation-under-the-integral-sign/index.html","scope":"MeasureTheory/integration/differentiation-under-the-integral-sign","status":"can_prove","title":"Differentiation under the integral sign"},{"id":"MeasureTheory/integration/differentiation-under-the-integral-sign/cont-diff-integral-smul-comp-of-cont-dif","kind":"node","lean":"MeasureTheory.contDiff_integral_smul_comp_of_contDiff_of_hasCompactSupport","parent":19303,"result":true,"route":"../roadmap/MeasureTheory/integration/differentiation-under-the-integral-sign/index.html#cont-diff-integral-smul-comp-of-cont-dif","scope":"MeasureTheory/integration/differentiation-under-the-integral-sign","status":"fully_proved","summary":"The integral `p \u21a6 \u222b x, w x \u2022 \u03a8 (c x, p) \u2202\u03bc` of a smooth compactly supported function is smooth in the parameter `p`.","title":"Smoothness of compactly supported parameter integrals"},{"id":"MeasureTheory/integration/differentiation-under-the-integral-sign/exists-has-deriv-at-integral-tsum-ae","kind":"node","lean":"MathlibExt.MeasureTheory.DerivIntegralSeriesAbsContWanted.exists_hasDerivAt_integral_tsum_ae","parent":19303,"result":true,"route":"../roadmap/MeasureTheory/integration/differentiation-under-the-integral-sign/index.html#exists-has-deriv-at-integral-tsum-ae","scope":"MeasureTheory/integration/differentiation-under-the-integral-sign","status":"can_prove","summary":"Let $d \\in \\mathbb{N}$, let $\\Omega$ be an open subset of $\\mathbb{R}^d$, let $a < c < b$ be real numbers, and for each $n \\in \\mathbb{N}$ let $f_n : \\mathbb{R}^d \\times \\mathbb{R} \\to \\mathbb{C}$ be almost everywhere measurable on $\\Omega \\times (a, b)$ with respect to Lebesgue measure. Write $\\partial_y f_n(x, y)$ for the derivative of $t \\mapsto f_n(x, t)$ at $y$, taken to be $0$ where this derivative does not\u2026","title":"Differentiating an integral of a series term by term"},{"data_url":"scopes/MeasureTheory/integration/jensen-s-inequality-for-integrals.json","id":"MeasureTheory/integration/jensen-s-inequality-for-integrals","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/jensen-s-inequality-for-integrals/index.html","scope":"MeasureTheory/integration/jensen-s-inequality-for-integrals","status":"can_prove","title":"Jensen's inequality for integrals"},{"id":"MeasureTheory/integration/jensen-s-inequality-for-integrals/set-integral-log-le-measure-mul-log-inv","kind":"node","lean":"MathlibExt.MeasureTheory.SetIntegralLogDensityBoundWanted.setIntegral_log_le_measure_mul_log_inv_measure","parent":19306,"result":true,"route":"../roadmap/MeasureTheory/integration/jensen-s-inequality-for-integrals/index.html#set-integral-log-le-measure-mul-log-inv","scope":"MeasureTheory/integration/jensen-s-inequality-for-integrals","status":"can_prove","summary":"Let $\\mu$ be a measure on a measurable space $X$ and let $f : X \\to \\mathbb{R}$ be integrable with respect to $\\mu$, with $f(x) > 0$ for every $x \\in X$ and $\\int_X f\\,d\\mu = 1$. Then for every measurable set $E \\subseteq X$ with $0 < \\mu(E) < \\infty$ such that $\\log f$ is integrable on $E$,","title":"Upper bound for the integral of the logarithm of a probability density over a set"},{"id":"MeasureTheory/integration/jensen-s-inequality-for-integrals/sqrt-one-add-sq-integral-le-and-le-one-a","kind":"node","lean":"MathlibExt.MeasureTheory.SqrtOnePlusSqIntegralBoundsWanted.sqrt_one_add_sq_integral_le_and_le_one_add","parent":19306,"result":true,"route":"../roadmap/MeasureTheory/integration/jensen-s-inequality-for-integrals/index.html#sqrt-one-add-sq-integral-le-and-le-one-a","scope":"MeasureTheory/integration/jensen-s-inequality-for-integrals","status":"can_prove","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$ and let $h : \\Omega \\to [0,\\infty)$ be a measurable function with $\\int_\\Omega h\\,d\\mu < \\infty$. If $A = \\int_\\Omega h\\,d\\mu$, then","title":"Two-sided bound for the integral of the square root of one plus a square"},{"data_url":"scopes/MeasureTheory/integration/products-of-measures-with-densities.json","id":"MeasureTheory/integration/products-of-measures-with-densities","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/products-of-measures-with-densities/index.html","scope":"MeasureTheory/integration/products-of-measures-with-densities","status":"fully_proved","title":"Products of measures with densities"},{"id":"MeasureTheory/integration/products-of-measures-with-densities/lmarginal-prod","kind":"node","lean":"MeasureTheory.lmarginal_prod","parent":19309,"result":true,"route":"../roadmap/MeasureTheory/integration/products-of-measures-with-densities/index.html#lmarginal-prod","scope":"MeasureTheory/integration/products-of-measures-with-densities","status":"fully_proved","summary":"Let $\\mu_i$ be \u03c3-finite measures on measurable spaces $X_i$, indexed by a finite type $\\iota$, and let $g_i : X_i \\to [0, \\infty]$ be measurable functions. For a finite set $s \\subseteq \\iota$ and a point $x \\in \\prod_i X_i$, integrating $y \\mapsto \\prod_{i \\in \\iota} g_i(y_i)$ over the coordinates $y_i$ with $i \\in s$, each against $\\mu_i$, while the other coordinates are fixed to those of $x$, gives\u2026","title":"Iterated integrals of a product of one-variable functions"},{"id":"MeasureTheory/integration/products-of-measures-with-densities/lintegral-fintype-prod-eq-prod","kind":"node","lean":"MeasureTheory.lintegral_fintype_prod_eq_prod","parent":19309,"result":true,"route":"../roadmap/MeasureTheory/integration/products-of-measures-with-densities/index.html#lintegral-fintype-prod-eq-prod","scope":"MeasureTheory/integration/products-of-measures-with-densities","status":"fully_proved","summary":"Let $\\mu_i$ be \u03c3-finite measures on measurable spaces $X_i$, indexed by a finite type $\\iota$, and let $g_i : X_i \\to [0, \\infty]$ be measurable functions. Then the integral of $y \\mapsto \\prod_{i \\in \\iota} g_i(y_i)$ with respect to the product measure $\\prod_i \\mu_i$ equals $\\prod_{i \\in \\iota} \\int g_i \\, d\\mu_i$.","title":"Tonelli for products"},{"id":"MeasureTheory/integration/products-of-measures-with-densities/pi-with-density","kind":"node","lean":"MeasureTheory.Measure.pi_withDensity","parent":19309,"result":true,"route":"../roadmap/MeasureTheory/integration/products-of-measures-with-densities/index.html#pi-with-density","scope":"MeasureTheory/integration/products-of-measures-with-densities","status":"fully_proved","summary":"Let $\\mu_i$ be \u03c3-finite measures on measurable spaces $X_i$, indexed by a finite type $\\iota$, and let $f_i : X_i \\to [0, \\infty]$ be measurable functions such that, for each $i$, the measure $f_i \\mu_i$ with density $f_i$ with respect to $\\mu_i$ is \u03c3-finite. Then the product measure $\\prod_i f_i \\mu_i$ equals the measure with density $y \\mapsto \\prod_{i \\in \\iota} f_i(y_i)$ with respect to the product measure\u2026","title":"A product of measures with densities"},{"data_url":"scopes/MeasureTheory/integration/scheff-s-lemma.json","id":"MeasureTheory/integration/scheff-s-lemma","kind":"scope","navigation":"module","parent":19283,"result":false,"route":"../roadmap/MeasureTheory/integration/scheff-s-lemma/index.html","scope":"MeasureTheory/integration/scheff-s-lemma","status":"can_prove","title":"Scheff\u00e9's lemma"},{"id":"MeasureTheory/integration/scheff-s-lemma/scheffe-lemma","kind":"node","lean":"MeasureTheory.scheffe_lemma","parent":19313,"result":true,"route":"../roadmap/MeasureTheory/integration/scheff-s-lemma/index.html#scheffe-lemma","scope":"MeasureTheory/integration/scheff-s-lemma","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $f_n : \\alpha \\to \\mathbb{R}$ ($n \\in \\mathbb{N}$) be $\\mu$-almost everywhere strongly measurable, and let $g : \\alpha \\to \\mathbb{R}$ be in $L^1(\\mu)$. Write $\\|h\\|_1 = \\int |h| \\, d\\mu \\in [0, \\infty]$ for the $L^1$ seminorm. If $f_n(x) \\to g(x)$ for $\\mu$-almost every $x$ and $\\|f_n\\|_1 \\to \\|g\\|_1$ in $[0, \\infty]$, then $\\|f_n - g\\|_1 \\to 0$, that is,\u2026","title":"Scheff\u00e9's lemma"},{"id":"MeasureTheory/integration/scheff-s-lemma/tendsto-e-lp-norm-one-of-tendsto-in-meas","kind":"node","lean":"MeasureTheory.tendsto_eLpNorm_one_of_tendstoInMeasure_of_tendsto_eLpNorm","parent":19313,"result":true,"route":"../roadmap/MeasureTheory/integration/scheff-s-lemma/index.html#tendsto-e-lp-norm-one-of-tendsto-in-meas","scope":"MeasureTheory/integration/scheff-s-lemma","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $f_n : \\alpha \\to \\mathbb{R}$ ($n \\in \\mathbb{N}$) and $g : \\alpha \\to \\mathbb{R}$ be in $L^1(\\mu)$, and write $\\|h\\|_1 = \\int |h| \\, d\\mu$. If $f_n \\to g$ in $\\mu$-measure (for every $\\varepsilon > 0$, $\\mu\\{x : |f_n(x) - g(x)| \\ge \\varepsilon\\} \\to 0$) and $\\|f_n\\|_1 \\to \\|g\\|_1$, then $\\|f_n - g\\|_1 \\to 0$, that is, $f_n \\to g$ in $L^1(\\mu)$.","title":"Scheff\u00e9's lemma for convergence in measure"},{"id":"MeasureTheory/integration/scheff-s-lemma/uniform-integrable-of-tendsto-in-measure","kind":"node","lean":"MeasureTheory.uniformIntegrable_of_tendstoInMeasure_of_tendsto_eLpNorm","parent":19313,"result":true,"route":"../roadmap/MeasureTheory/integration/scheff-s-lemma/index.html#uniform-integrable-of-tendsto-in-measure","scope":"MeasureTheory/integration/scheff-s-lemma","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $f_n : \\alpha \\to \\mathbb{R}$ ($n \\in \\mathbb{N}$) and $g : \\alpha \\to \\mathbb{R}$ be in $L^1(\\mu)$, and write $\\|h\\|_1 = \\int |h| \\, d\\mu$. If $f_n \\to g$ in $\\mu$-measure (for every $\\varepsilon > 0$, $\\mu\\{x : |f_n(x) - g(x)| \\ge \\varepsilon\\} \\to 0$) and $\\|f_n\\|_1 \\to \\|g\\|_1$, then the family $(f_n)$ is uniformly integrable:\u2026","title":"Convergence in measure with converging L\u00b9 norms implies uniform integrability"},{"id":"MeasureTheory/integration/scheff-s-lemma/uniform-integrable-tfae-of-tendsto-in-me","kind":"node","lean":"MeasureTheory.uniformIntegrable_tfae_of_tendstoInMeasure","parent":19313,"result":true,"route":"../roadmap/MeasureTheory/integration/scheff-s-lemma/index.html#uniform-integrable-tfae-of-tendsto-in-me","scope":"MeasureTheory/integration/scheff-s-lemma","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $\\alpha$, let $f_n : \\alpha \\to \\mathbb{R}$ ($n \\in \\mathbb{N}$) and $g : \\alpha \\to \\mathbb{R}$ be in $L^1(\\mu)$, write $\\|h\\|_1 = \\int |h| \\, d\\mu$, and suppose that $f_n \\to g$ in $\\mu$-measure (for every $\\varepsilon > 0$, $\\mu\\{x : |f_n(x) - g(x)| \\ge \\varepsilon\\} \\to 0$). Then the following are equivalent: (1) the family $(f_n)$ is uniformly integrable,\u2026","title":"Uniform integrability equivalences"},{"id":"MeasureTheory/integration/scheff-s-lemma/tendsto-integral-abs-sub-of-ae-tendsto-o","kind":"node","lean":"MathlibExt.MeasureTheory.ScheffeL1ConvergenceWanted.tendsto_integral_abs_sub_of_ae_tendsto_of_tendsto_integral_abs","parent":19313,"result":true,"route":"../roadmap/MeasureTheory/integration/scheff-s-lemma/index.html#tendsto-integral-abs-sub-of-ae-tendsto-o","scope":"MeasureTheory/integration/scheff-s-lemma","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be integrable and let $(f_n)_{n \\in \\mathbb{N}}$ be a sequence of integrable real functions $f_n$ on $\\mathbb{R}$ such that $f_n(x) \\to f(x)$ for almost every $x \\in \\mathbb{R}$. If $\\int_{\\mathbb{R}} |f_n| \\to \\int_{\\mathbb{R}} |f|$ as $n \\to \\infty$, then $\\int_{\\mathbb{R}} |f_n - f| \\to 0$.","title":"Scheff\u00e9's lemma on the real line"},{"data_url":"scopes/MeasureTheory/lp-spaces.json","id":"MeasureTheory/lp-spaces","kind":"scope","parent":19131,"result":false,"route":"../roadmap/MeasureTheory/lp-spaces/index.html","scope":"MeasureTheory/lp-spaces","status":"can_prove","summary":"The spaces Lp of p-th power integrable functions: membership criteria, inclusions, dense subspaces, the essential supremum and weak convergence.","title":"Lebesgue spaces"},{"id":"MeasureTheory/lp-spaces/integral-abs-rpow-sub-rpow-le-integral-r","kind":"node","lean":"MathlibExt.MeasureTheory.LpSubadditivePowLtOneWanted.integral_abs_rpow_sub_rpow_le_integral_rpow_abs_sub","parent":19319,"reading_title":{"format":"inline-parts-v1","parts":[["text","An integral inequality for "],["math","\\(p\\)"],["text","-th powers with "],["math","\\(0 < p < 1\\)"]],"source":"An integral inequality for $p$-th powers with $0 < p < 1$"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/index.html#integral-abs-rpow-sub-rpow-le-integral-r","scope":"MeasureTheory/lp-spaces","status":"can_prove","summary":"Let $\\mu$ be a measure on a measurable space and let $0 < p < 1$. Then for all real-valued $f, g \\in L^p(\\mu)$,","title":"An integral inequality for $p$-th powers with $0 < p < 1$"},{"data_url":"scopes/MeasureTheory/lp-spaces/borel-sets-in-lp-spaces.json","id":"MeasureTheory/lp-spaces/borel-sets-in-lp-spaces","kind":"scope","navigation":"module","parent":19319,"result":false,"route":"../roadmap/MeasureTheory/lp-spaces/borel-sets-in-lp-spaces/index.html","scope":"MeasureTheory/lp-spaces/borel-sets-in-lp-spaces","status":"can_prove","title":"Borel sets in Lp spaces"},{"id":"MeasureTheory/lp-spaces/borel-sets-in-lp-spaces/vanishing-integral-set","kind":"node","lean":"MathlibExt.MeasureTheory.LpVanishingHalfLineIntegralBorelWanted.vanishingIntegralSet","parent":19321,"reading_title":{"format":"inline-parts-v1","parts":[["text","Functions with vanishing integrals over "],["math","\\((0, L)\\)"]],"source":"Functions with vanishing integrals over $(0, L)$"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/borel-sets-in-lp-spaces/index.html#vanishing-integral-set","scope":"MeasureTheory/lp-spaces/borel-sets-in-lp-spaces","status":"fully_proved","summary":"For $0 \\le p \\le \\infty$, this is the set of real-valued $f \\in L^p(\\mathbb{R})$ (with respect to Lebesgue measure) such that $\\int_{[0, L]} f(x)\\,dx \\to 0$ as $L \\to +\\infty$, where the integral of a function that is not integrable on $[0, L]$ counts as $0$.","title":"Functions with vanishing integrals over $(0, L)$"},{"id":"MeasureTheory/lp-spaces/borel-sets-in-lp-spaces/measurable-set-vanishing-integral-set","kind":"node","lean":"MathlibExt.MeasureTheory.LpVanishingHalfLineIntegralBorelWanted.measurableSet_vanishingIntegralSet","parent":19321,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^p\\)"],["text"," functions whose integrals over "],["math","\\((0, L)\\)"],["text"," tend to "],["math","\\(0\\)"],["text"," form a Borel set"]],"source":"$L^p$ functions whose integrals over $(0, L)$ tend to $0$ form a Borel set"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/borel-sets-in-lp-spaces/index.html#measurable-set-vanishing-integral-set","scope":"MeasureTheory/lp-spaces/borel-sets-in-lp-spaces","status":"can_prove","summary":"Let $1 \\le p \\le \\infty$. The set of real-valued $f \\in L^p(\\mathbb{R})$ (with respect to Lebesgue measure) such that $\\int_0^L f(x)\\,dx \\to 0$ as $L \\to +\\infty$ is a Borel subset of $L^p(\\mathbb{R})$.","title":"$L^p$ functions whose integrals over $(0, L)$ tend to $0$ form a Borel set"},{"data_url":"scopes/MeasureTheory/lp-spaces/criteria-for-lp-membership.json","id":"MeasureTheory/lp-spaces/criteria-for-lp-membership","kind":"scope","navigation":"module","parent":19319,"result":false,"route":"../roadmap/MeasureTheory/lp-spaces/criteria-for-lp-membership/index.html","scope":"MeasureTheory/lp-spaces/criteria-for-lp-membership","status":"can_prove","title":"Criteria for Lp membership"},{"id":"MeasureTheory/lp-spaces/criteria-for-lp-membership/lintegral-eq-i-sup-lintegral-icc-inter","kind":"node","lean":"MathlibExt.MeasureTheory.Function.MemLpTruncation.lintegral_eq_iSup_lintegral_Icc_inter","parent":19324,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/criteria-for-lp-membership/index.html#lintegral-eq-i-sup-lintegral-icc-inter","scope":"MeasureTheory/lp-spaces/criteria-for-lp-membership","status":"fully_proved","summary":"Let $\\lambda$ be Lebesgue measure on $\\mathbb{R}$, let $S \\subseteq \\mathbb{R}$ be measurable, and let $g : \\mathbb{R} \\to [0, \\infty]$ be measurable. Then","title":"Lebesgue integrals over truncations of a set"},{"id":"MeasureTheory/lp-spaces/criteria-for-lp-membership/mem-lp-iff-i-sup-lintegral-icc-inter-lt","kind":"node","lean":"MathlibExt.MeasureTheory.Function.MemLpTruncation.memLp_iff_iSup_lintegral_Icc_inter_lt_top","parent":19324,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(L^p\\)"],["text"," membership via truncated integrals"]],"source":"Characterization of $L^p$ membership via truncated integrals"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/criteria-for-lp-membership/index.html#mem-lp-iff-i-sup-lintegral-icc-inter-lt","scope":"MeasureTheory/lp-spaces/criteria-for-lp-membership","status":"fully_proved","summary":"Let $\\lambda$ be Lebesgue measure on $\\mathbb{R}$, let $F$ be a second-countable normed additive commutative group with its Borel $\\sigma$-algebra, let $0 < p < \\infty$, let $S \\subseteq \\mathbb{R}$ be measurable, and let $f : \\mathbb{R} \\to F$ be measurable. Then $f$ belongs to $L^p$ of $\\lambda$ restricted to $S$ if and only if","title":"Characterization of $L^p$ membership via truncated integrals"},{"id":"MeasureTheory/lp-spaces/criteria-for-lp-membership/mem-lp-of-polynomial-decay-ae","kind":"node","lean":"MathlibExt.MeasureTheory.Function.PolynomialDecay.memLp_of_polynomial_decay_ae","parent":19324,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial decay implies "],["math","\\(L^p\\)"],["text"," membership"]],"source":"Polynomial decay implies $L^p$ membership"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/criteria-for-lp-membership/index.html#mem-lp-of-polynomial-decay-ae","scope":"MeasureTheory/lp-spaces/criteria-for-lp-membership","status":"fully_proved","summary":"Let $E$ be a normed additive commutative group and let $f : \\mathbb{R} \\to E$ be almost everywhere strongly measurable. Suppose there are real numbers $C \\ge 0$ and $q > 1$ such that $\\|f(x)\\| \\le C (1 + |x|)^{-q}$ for almost every $x \\in \\mathbb{R}$. Then $f \\in L^p(\\mathbb{R})$, with respect to Lebesgue measure, for every $p \\in [1, \\infty]$.","title":"Polynomial decay implies $L^p$ membership"},{"id":"MeasureTheory/lp-spaces/criteria-for-lp-membership/mem-lp-of-polynomial-decay","kind":"node","lean":"MathlibExt.MeasureTheory.Function.PolynomialDecay.memLp_of_polynomial_decay","parent":19324,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pointwise polynomial decay implies "],["math","\\(L^p\\)"],["text"," membership"]],"source":"Pointwise polynomial decay implies $L^p$ membership"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/criteria-for-lp-membership/index.html#mem-lp-of-polynomial-decay","scope":"MeasureTheory/lp-spaces/criteria-for-lp-membership","status":"fully_proved","summary":"Let $E$ be a normed additive commutative group and let $f : \\mathbb{R} \\to E$ be almost everywhere strongly measurable. Suppose there are real numbers $C \\ge 0$ and $q > 1$ such that $\\|f(x)\\| \\le C (1 + |x|)^{-q}$ for every $x \\in \\mathbb{R}$. Then $f \\in L^p(\\mathbb{R})$, with respect to Lebesgue measure, for every $p \\in [1, \\infty]$.","title":"Pointwise polynomial decay implies $L^p$ membership"},{"id":"MeasureTheory/lp-spaces/criteria-for-lp-membership/is-partition01","kind":"node","lean":"MathlibExt.MeasureTheory.LpCharacterizationPartitionAveragesWanted.IsPartition01","parent":19324,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/criteria-for-lp-membership/index.html#is-partition01","scope":"MeasureTheory/lp-spaces/criteria-for-lp-membership","status":"fully_proved","summary":"For a natural number $n$, real numbers $t_0, t_1, \\dots, t_n$ form a partition of $[0, 1]$ when $0 = t_0 < t_1 < \\dots < t_n = 1$.","title":"Partitions of the unit interval"},{"id":"MeasureTheory/lp-spaces/criteria-for-lp-membership/partition-average-sum","kind":"node","lean":"MathlibExt.MeasureTheory.LpCharacterizationPartitionAveragesWanted.partitionAverageSum","parent":19324,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/criteria-for-lp-membership/index.html#partition-average-sum","scope":"MeasureTheory/lp-spaces/criteria-for-lp-membership","status":"fully_proved","summary":"For a function $f : \\mathbb{R} \\to \\mathbb{R}$, a real number $p$ and real numbers $t_0, t_1, \\dots, t_n$, the partition average sum is","title":"Partition average sum"},{"id":"MeasureTheory/lp-spaces/criteria-for-lp-membership/mem-lp-iff-bdd-above-partition-average-s","kind":"node","lean":"MathlibExt.MeasureTheory.LpCharacterizationPartitionAveragesWanted.memLp_iff_bddAbove_partitionAverageSum","parent":19324,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^p\\)"],["text"," on "],["math","\\((0,1)\\)"],["text"," via averages over partitions"]],"source":"$L^p$ on $(0,1)$ via averages over partitions"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/criteria-for-lp-membership/index.html#mem-lp-iff-bdd-above-partition-average-s","scope":"MeasureTheory/lp-spaces/criteria-for-lp-membership","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be in $L^1([0, 1])$ and let $p > 1$ be a real number. Then $f \\in L^p([0, 1])$ if and only if there is a real number $C$ such that","title":"$L^p$ on $(0,1)$ via averages over partitions"},{"data_url":"scopes/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces.json","id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","kind":"scope","navigation":"module","parent":19319,"result":false,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","title":"Dense subspaces of Lp spaces"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/vanishing-cutoff","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.vanishingCutoff","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#vanishing-cutoff","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"For real numbers $a$, $b$, $\\delta$ and $x$, the value at $x$ of the vanishing cutoff factor with endpoints $a$, $b$ and width $\\delta$ is $\\max\\bigl(0, \\min\\bigl(1, \\min((x - a)/\\delta, (b - x)/\\delta)\\bigr)\\bigr)$, where division by $0$ is read as $0$.","title":"Vanishing cutoff factor"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/vanishing-cutoff-nonneg","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.vanishingCutoff_nonneg","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#vanishing-cutoff-nonneg","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"For all real numbers $a$, $b$, $\\delta$ and $x$, the value at $x$ of the vanishing cutoff factor with endpoints $a$, $b$ and width $\\delta$ is at least $0$.","title":"The vanishing cutoff factor is nonnegative"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/vanishing-cutoff-le-one","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.vanishingCutoff_le_one","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#vanishing-cutoff-le-one","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"For all real numbers $a$, $b$, $\\delta$ and $x$, the value at $x$ of the vanishing cutoff factor with endpoints $a$, $b$ and width $\\delta$ is at most $1$.","title":"The vanishing cutoff factor is at most one"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/vanishing-cutoff-continuous","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.vanishingCutoff_continuous","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#vanishing-cutoff-continuous","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"For all $a, b, \\delta \\in \\mathbb{R}$, the vanishing cutoff factor with endpoints $a$, $b$ and width $\\delta$ is a continuous function $\\mathbb{R} \\to \\mathbb{R}$.","title":"Continuity of the vanishing cutoff factor"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/vanishing-cutoff-eq-zero-left","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.vanishingCutoff_eq_zero_left","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#vanishing-cutoff-eq-zero-left","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"For all $a, b, \\delta \\in \\mathbb{R}$, the vanishing cutoff factor with endpoints $a$, $b$ and width $\\delta$ takes the value $0$ at $a$.","title":"The vanishing cutoff factor vanishes at the left endpoint"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/vanishing-cutoff-eq-zero-right","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.vanishingCutoff_eq_zero_right","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#vanishing-cutoff-eq-zero-right","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"For all real numbers $a$, $b$ and $\\delta$, the vanishing cutoff factor with endpoints $a$, $b$ and width $\\delta$ takes the value $0$ at $b$.","title":"The vanishing cutoff factor vanishes at the right endpoint"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/vanishing-cutoff-eq-one","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.vanishingCutoff_eq_one","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#vanishing-cutoff-eq-one","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"Let $a, b \\in \\mathbb{R}$ and $\\delta > 0$. For every real $x$ with $a + \\delta \\le x \\le b - \\delta$, the vanishing cutoff factor with endpoints $a$, $b$ and width $\\delta$ takes the value $1$ at $x$.","title":"The vanishing cutoff factor equals one on its plateau"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/cutoff-approx","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.cutoffApprox","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#cutoff-approx","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"Let $a, b \\in \\mathbb{R}$, let $E$ be a real normed space, let $g_0 : \\mathbb{R} \\to E$ and $n \\in \\mathbb{N}$, and put $\\delta_n = (b - a)/(n + 1)$. The $n$-th cutoff approximant of $g_0$ with endpoints $a$ and $b$ is the function $x \\mapsto \\varphi(x)\\, g_0(x)$, where $\\varphi$ is the vanishing cutoff factor with endpoints $a$, $b$ and width $\\delta_n$.","title":"Cutoff approximants"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/cutoff-approx-continuous","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.cutoffApprox_continuous","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#cutoff-approx-continuous","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"Let $a, b \\in \\mathbb{R}$, let $E$ be a real normed space and let $g_0 : \\mathbb{R} \\to E$ be continuous. Then for every $n \\in \\mathbb{N}$ the $n$-th cutoff approximant of $g_0$ with endpoints $a$ and $b$ is continuous.","title":"Continuity of the cutoff approximants"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/cutoff-approx-eq-zero-left","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.cutoffApprox_eq_zero_left","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#cutoff-approx-eq-zero-left","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"Let $a, b \\in \\mathbb{R}$, let $E$ be a real normed space and let $g_0 : \\mathbb{R} \\to E$ be any function. For every $n \\in \\mathbb{N}$, the $n$-th cutoff approximant of $g_0$ with endpoints $a$ and $b$ takes the value $0$ at $a$.","title":"Cutoff approximants vanish at the left endpoint"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/cutoff-approx-eq-zero-right","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.cutoffApprox_eq_zero_right","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#cutoff-approx-eq-zero-right","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"Let $a, b \\in \\mathbb{R}$, let $E$ be a real normed space and let $g_0 : \\mathbb{R} \\to E$ be any function. For every $n \\in \\mathbb{N}$, the $n$-th cutoff approximant of $g_0$ with endpoints $a$ and $b$ takes the value $0$ at $b$.","title":"Cutoff approximants vanish at the right endpoint"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/cutoff-approx-mem-lp","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.cutoffApprox_memLp","parent":19332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cutoff approximants stay in "],["math","\\(L^p\\)"]],"source":"Cutoff approximants stay in $L^p$"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#cutoff-approx-mem-lp","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"Let $E$ be a real normed space, let $\\mu$ be a measure on $\\mathbb{R}$, let $p \\in [0, \\infty]$, and let $g_0 : \\mathbb{R} \\to E$ be continuous with $g_0 \\in L^p(\\mu)$. Then for all $a, b \\in \\mathbb{R}$ and $n \\in \\mathbb{N}$, the $n$-th cutoff approximant of $g_0$ with endpoints $a$ and $b$ lies in $L^p(\\mu)$.","title":"Cutoff approximants stay in $L^p$"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/norm-sub-cutoff-approx-le","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.norm_sub_cutoffApprox_le","parent":19332,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#norm-sub-cutoff-approx-le","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"Let $a, b \\in \\mathbb{R}$, let $E$ be a real normed space and let $g_0 : \\mathbb{R} \\to E$. If $g_n$ denotes the $n$-th cutoff approximant of $g_0$ with endpoints $a$ and $b$, then $\\|g_0(x) - g_n(x)\\| \\le \\|g_0(x)\\|$ for every $n \\in \\mathbb{N}$ and every $x \\in \\mathbb{R}$.","title":"Pointwise bound on the cutoff error"},{"id":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/exists-continuous-vanishing-endpoints-e","kind":"node","lean":"MathlibExt.MeasureTheory.Function.VanishingEndpointsDenseLp.exists_continuous_vanishing_endpoints_eLpNorm_sub_lt","parent":19332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuous functions vanishing at the endpoints are dense in "],["math","\\(L^p\\)"],["text"," of a compact interval"]],"source":"Continuous functions vanishing at the endpoints are dense in $L^p$ of a compact interval"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces/index.html#exists-continuous-vanishing-endpoints-e","scope":"MeasureTheory/lp-spaces/dense-subspaces-of-lp-spaces","status":"fully_proved","summary":"Let $a < b$ be real numbers, let $1 \\le p < \\infty$ and let $E$ be a real normed space. Write $L^p([a, b])$ for $L^p$ with respect to Lebesgue measure restricted to $[a, b]$. For every $f : \\mathbb{R} \\to E$ in $L^p([a, b])$ and every $\\varepsilon \\in (0, \\infty]$, there is a continuous $g : \\mathbb{R} \\to E$ with $g(a) = g(b) = 0$, $g \\in L^p([a, b])$ and $\\|f - g\\|_{L^p([a, b])} < \\varepsilon$.","title":"Continuous functions vanishing at the endpoints are dense in $L^p$ of a compact interval"},{"data_url":"scopes/MeasureTheory/lp-spaces/essential-supremum.json","id":"MeasureTheory/lp-spaces/essential-supremum","kind":"scope","navigation":"module","parent":19319,"result":false,"route":"../roadmap/MeasureTheory/lp-spaces/essential-supremum/index.html","scope":"MeasureTheory/lp-spaces/essential-supremum","status":"can_prove","title":"Essential supremum"},{"id":"MeasureTheory/lp-spaces/essential-supremum/e-lp-norm-ess-sup-eq-enorm-continuous-ma","kind":"node","lean":"MathlibExt.MeasureTheory.Function.EssSupContinuousMap.eLpNormEssSup_eq_enorm_continuousMap","parent":19347,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/essential-supremum/index.html#e-lp-norm-ess-sup-eq-enorm-continuous-ma","scope":"MeasureTheory/lp-spaces/essential-supremum","status":"fully_proved","summary":"Let $\\alpha$ be a compact topological space with its Borel $\\sigma$-algebra, let $\\mu$ be a measure on $\\alpha$ that is positive on every nonempty open set, let $E$ be a seminormed additive commutative group, and let $f : \\alpha \\to E$ be continuous. Then the essential supremum of $\\|f\\|$ with respect to $\\mu$ equals the supremum norm $\\sup_{x \\in \\alpha} \\|f(x)\\|$ of $f$, as an equality in $[0, \\infty]$.","title":"Essential supremum of a continuous map"},{"id":"MeasureTheory/lp-spaces/essential-supremum/e-lp-norm-top-eq-enorm-continuous-map","kind":"node","lean":"MathlibExt.MeasureTheory.Function.EssSupContinuousMap.eLpNorm_top_eq_enorm_continuousMap","parent":19347,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(L^\\infty\\)"],["text"," norm of a continuous map is its supremum norm"]],"source":"The $L^\\infty$ norm of a continuous map is its supremum norm"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/essential-supremum/index.html#e-lp-norm-top-eq-enorm-continuous-map","scope":"MeasureTheory/lp-spaces/essential-supremum","status":"fully_proved","summary":"Let $\\alpha$ be a compact topological space with its Borel $\\sigma$-algebra, let $\\mu$ be a measure on $\\alpha$ that is positive on every nonempty open set, let $E$ be a seminormed additive commutative group such that $\\alpha$ or $E$ is second-countable, and let $f : \\alpha \\to E$ be continuous. Then $\\|f\\|_{L^\\infty(\\mu)} = \\sup_{x \\in \\alpha} \\|f(x)\\|$, as an equality in $[0, \\infty]$.","title":"The $L^\\infty$ norm of a continuous map is its supremum norm"},{"id":"MeasureTheory/lp-spaces/essential-supremum/tendsto-e-lp-norm-at-top-e-lp-norm-top-o","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.LpNormTendstoEssSupWanted.tendsto_eLpNorm_atTop_eLpNorm_top_of_isFiniteMeasure","parent":19347,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(L^p\\)"],["text"," norm tends to the "],["math","\\(L^\\infty\\)"],["text"," norm"]],"source":"The $L^p$ norm tends to the $L^\\infty$ norm"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/essential-supremum/index.html#tendsto-e-lp-norm-at-top-e-lp-norm-top-o","scope":"MeasureTheory/lp-spaces/essential-supremum","status":"can_prove","summary":"Let $\\mu$ be a finite measure on a measurable space $\\alpha$ and let $f \\colon \\alpha \\to \\mathbb{R}$ be essentially bounded, $f \\in L^\\infty(\\mu)$. Then $\\|f\\|_{L^p(\\mu)} \\to \\|f\\|_{L^\\infty(\\mu)}$ as the real exponent $p$ tends to $\\infty$.","title":"The $L^p$ norm tends to the $L^\\infty$ norm"},{"id":"MeasureTheory/lp-spaces/essential-supremum/e-lp-norm-top-le-of-forall-one-le-e-lp-n","kind":"node","lean":"MathlibExt.MeasureTheory.LinftyNormLeSupLpNormUnitIntervalWanted.eLpNorm_top_le_of_forall_one_le_eLpNorm_le","parent":19347,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniform "],["math","\\(L^p\\)"],["text"," bounds on "],["math","\\((0,1)\\)"],["text"," give the same "],["math","\\(L^\\infty\\)"],["text"," bound"]],"source":"Uniform $L^p$ bounds on $(0,1)$ give the same $L^\\infty$ bound"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/essential-supremum/index.html#e-lp-norm-top-le-of-forall-one-le-e-lp-n","scope":"MeasureTheory/lp-spaces/essential-supremum","status":"can_prove","summary":"Let $M$ be a real number, write $M^+ = \\max(M, 0)$, and let $f_0, f_1, f_2, \\dots : \\mathbb{R} \\to \\mathbb{R}$ be functions that are almost everywhere strongly measurable with respect to Lebesgue measure restricted to $[0,1]$. If $\\|f_n\\|_{L^p[0,1]} \\le M^+$ for every $n$ and every real $p \\ge 1$, then $\\|f_n\\|_{L^\\infty[0,1]} \\le M^+$ for every $n$. All norms are taken with respect to Lebesgue measure restricted\u2026","title":"Uniform $L^p$ bounds on $(0,1)$ give the same $L^\\infty$ bound"},{"data_url":"scopes/MeasureTheory/lp-spaces/inclusions-between-lp-spaces.json","id":"MeasureTheory/lp-spaces/inclusions-between-lp-spaces","kind":"scope","navigation":"module","parent":19319,"result":false,"route":"../roadmap/MeasureTheory/lp-spaces/inclusions-between-lp-spaces/index.html","scope":"MeasureTheory/lp-spaces/inclusions-between-lp-spaces","status":"can_prove","title":"Inclusions between Lp spaces"},{"id":"MeasureTheory/lp-spaces/inclusions-between-lp-spaces/mem-lp-two-and-e-lp-norm-two-le-of-bound","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.BoundedL1MemL2NormBoundWanted.memLp_two_and_eLpNorm_two_le_of_bounded_of_memLp_one","parent":19352,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/inclusions-between-lp-spaces/index.html#mem-lp-two-and-e-lp-norm-two-le-of-bound","scope":"MeasureTheory/lp-spaces/inclusions-between-lp-spaces","status":"can_prove","summary":"Let $d$ be a natural number, let $M \\ge 0$, and let $f \\colon \\mathbb{R}^d \\to \\mathbb{R}$ satisfy $|f(x)| \\le M$ for every $x \\in \\mathbb{R}^d$. If $f \\in L^1(\\mathbb{R}^d)$, then $f \\in L^2(\\mathbb{R}^d)$ and","title":"A bounded integrable function is square integrable"},{"id":"MeasureTheory/lp-spaces/inclusions-between-lp-spaces/not-mem-lp-two-subset-one-and-not-mem-lp","kind":"node","lean":"MathlibExt.MeasureTheory.LpTwoLpOneIncomparableWanted.not_memLp_two_subset_one_and_not_memLp_one_subset_two","parent":19352,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^1(\\mathbb{R}^d)\\)"],["text"," and "],["math","\\(L^2(\\mathbb{R}^d)\\)"],["text"," are incomparable"]],"source":"$L^1(\\mathbb{R}^d)$ and $L^2(\\mathbb{R}^d)$ are incomparable"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/inclusions-between-lp-spaces/index.html#not-mem-lp-two-subset-one-and-not-mem-lp","scope":"MeasureTheory/lp-spaces/inclusions-between-lp-spaces","status":"can_prove","summary":"Let $d \\ge 1$. With respect to Lebesgue measure on $\\mathbb{R}^d$, there is a function $f : \\mathbb{R}^d \\to \\mathbb{R}$ in $L^2(\\mathbb{R}^d)$ that is not in $L^1(\\mathbb{R}^d)$, and there is a function $g : \\mathbb{R}^d \\to \\mathbb{R}$ in $L^1(\\mathbb{R}^d)$ that is not in $L^2(\\mathbb{R}^d)$.","title":"$L^1(\\mathbb{R}^d)$ and $L^2(\\mathbb{R}^d)$ are incomparable"},{"id":"MeasureTheory/lp-spaces/inclusions-between-lp-spaces/mem-lp-of-mem-lp-of-lintegral-rpow-lt-to","kind":"node","lean":"MathlibExt.MeasureTheory.MemLpOfLqIntegrableOnFiniteMeasureSetsWanted.memLp_of_memLp_of_lintegral_rpow_lt_top_of_measure_lt_top","parent":19352,"reading_title":{"format":"inline-parts-v1","parts":[["text","An "],["math","\\(L^p\\)"],["text"," function with finite "],["math","\\(L^q\\)"],["text"," mass on finite-measure sets is in "],["math","\\(L^q\\)"]],"source":"An $L^p$ function with finite $L^q$ mass on finite-measure sets is in $L^q$"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/inclusions-between-lp-spaces/index.html#mem-lp-of-mem-lp-of-lintegral-rpow-lt-to","scope":"MeasureTheory/lp-spaces/inclusions-between-lp-spaces","status":"can_prove","summary":"Let $1 \\le p < q < \\infty$ and let $f : \\mathbb{R} \\to \\mathbb{R}$ belong to $L^p(\\mathbb{R})$. If $\\int_A |f|^q < \\infty$ for every Borel set $A \\subseteq \\mathbb{R}$ of finite Lebesgue measure, then $f \\in L^q(\\mathbb{R})$.","title":"An $L^p$ function with finite $L^q$ mass on finite-measure sets is in $L^q$"},{"data_url":"scopes/MeasureTheory/lp-spaces/reverse-h-lder-inequality.json","id":"MeasureTheory/lp-spaces/reverse-h-lder-inequality","kind":"scope","navigation":"module","parent":19319,"result":false,"route":"../roadmap/MeasureTheory/lp-spaces/reverse-h-lder-inequality/index.html","scope":"MeasureTheory/lp-spaces/reverse-h-lder-inequality","status":"can_prove","title":"Reverse H\u00f6lder inequality"},{"id":"MeasureTheory/lp-spaces/reverse-h-lder-inequality/icc01","kind":"node","lean":"MathlibExt.MeasureTheory.LargeLevelSetReverseHolderWanted.\u03bcIcc01","parent":19356,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/reverse-h-lder-inequality/index.html#icc01","scope":"MeasureTheory/lp-spaces/reverse-h-lder-inequality","status":"fully_proved","summary":"The measure on $\\mathbb{R}$ that assigns to each set $A \\subseteq \\mathbb{R}$ the Lebesgue measure of $A \\cap [0,1]$.","title":"Lebesgue measure on the unit interval"},{"id":"MeasureTheory/lp-spaces/reverse-h-lder-inequality/exists-tendsto-at-top-inf-measure-level","kind":"node","lean":"MathlibExt.MeasureTheory.LargeLevelSetReverseHolderWanted.exists_tendsto_atTop_inf_measure_levelSet_pos","parent":19356,"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/reverse-h-lder-inequality/index.html#exists-tendsto-at-top-inf-measure-level","scope":"MeasureTheory/lp-spaces/reverse-h-lder-inequality","status":"can_prove","summary":"Let $\\mu$ be Lebesgue measure on $[0,1]$, and let $f_0, f_1, f_2, \\ldots : \\mathbb{R} \\to \\mathbb{R}$ lie in $L^2(\\mu)$ with $\\|f_n\\|_{L^2(\\mu)} \\le 2024\\,\\|f_n\\|_{L^1(\\mu)}$ for every $n$ and $\\|f_n\\|_{L^1(\\mu)} \\to \\infty$ as $n \\to \\infty$. Then there is a sequence of real numbers $y_n \\to \\infty$ such that","title":"Uniformly large level sets under a reverse H\u00f6lder bound"},{"data_url":"scopes/MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces.json","id":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces","kind":"scope","navigation":"module","parent":19319,"result":false,"route":"../roadmap/MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/index.html","scope":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces","status":"can_prove","title":"Weak convergence in Lp spaces"},{"id":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/i01","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.WeakConvergenceOfBoundedAeNullLpWanted.\u03bcI01","parent":19359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lebesgue measure on "],["math","\\((0,1)\\)"]],"source":"Lebesgue measure on $(0,1)$"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/index.html#i01","scope":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces","status":"fully_proved","summary":"The measure on $\\mathbb{R}$ obtained by restricting Lebesgue measure to the open interval $(0, 1)$.","title":"Lebesgue measure on $(0,1)$"},{"id":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/tendsto-integral-mul-of-ae-tendsto-zero","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.WeakConvergenceOfBoundedAeNullLpWanted.tendsto_integral_mul_of_ae_tendsto_zero_of_eLpNorm_le_one","parent":19359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded a.e.-null sequences in "],["math","\\(L^p(0,1)\\)"],["text"," converge weakly to zero"]],"source":"Bounded a.e.-null sequences in $L^p(0,1)$ converge weakly to zero"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/index.html#tendsto-integral-mul-of-ae-tendsto-zero","scope":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces","status":"can_prove","summary":"Let $1 < p < \\infty$ and let $q$ be the conjugate exponent, so that $1/p + 1/q = 1$. Write $L^p(0,1)$ and $L^q(0,1)$ for the Lebesgue spaces of real functions on $(0,1)$ with Lebesgue measure. Let $(f_n)_{n \\in \\mathbb{N}}$ be a sequence in $L^p(0,1)$ with $\\|f_n\\|_{L^p} \\le 1$ for every $n$ and $f_n(x) \\to 0$ for almost every $x \\in (0,1)$. Then for every $g \\in L^q(0,1)$,","title":"Bounded a.e.-null sequences in $L^p(0,1)$ converge weakly to zero"},{"id":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/ae-nonneg-of-weak-tendsto-of-ae-nonneg","kind":"node","lean":"MathlibExt.Analysis.FunctionalAnalysis.WeakLimitOfNonnegLpNonnegWanted.ae_nonneg_of_weak_tendsto_of_ae_nonneg","parent":19359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weak "],["math","\\(L^p\\)"],["text"," limits of nonnegative functions are nonnegative"]],"source":"Weak $L^p$ limits of nonnegative functions are nonnegative"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/index.html#ae-nonneg-of-weak-tendsto-of-ae-nonneg","scope":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces","status":"can_prove","summary":"Let $1 < p < \\infty$ and write $L^p([0,1])$ for the Lebesgue space of real functions on $[0,1]$ with Lebesgue measure. Let $(f_n)_{n \\in \\mathbb{N}}$ be a sequence in $L^p([0,1])$ such that for each $n$, $f_n(x) \\ge 0$ for almost every $x \\in [0,1]$, and let $f \\in L^p([0,1])$. If $f_n$ converges weakly to $f$, that is, $\\ell(f_n) \\to \\ell(f)$ for every continuous linear functional\u2026","title":"Weak $L^p$ limits of nonnegative functions are nonnegative"},{"id":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/tendsto-integral-mul-of-ae-tendsto-of-e","kind":"node","lean":"MathlibExt.MeasureTheory.WeakLpConvergenceOfAeConvergenceWanted.tendsto_integral_mul_of_ae_tendsto_of_eLpNorm_bounded","parent":19359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded almost everywhere convergent sequences in "],["math","\\(L^p\\)"],["text",", "],["math","\\(1 < p < \\infty\\)"],["text",", converge weakly"]],"source":"Bounded almost everywhere convergent sequences in $L^p$, $1 < p < \\infty$, converge weakly"},"result":true,"route":"../roadmap/MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces/index.html#tendsto-integral-mul-of-ae-tendsto-of-e","scope":"MeasureTheory/lp-spaces/weak-convergence-in-lp-spaces","status":"can_prove","summary":"Bounded almost everywhere convergent sequences in $L^p$, $1 < p < \\infty$, converge weakly. Let $1 < p < \\infty$ and let $q$ be the conjugate exponent, $1/p + 1/q = 1$. Let $f$ and $f_0, f_1, f_2, \\dots$ be real-valued functions in $L^p(\\mathbb{R})$ such that $f_n \\to f$ almost everywhere and $\\sup_n \\|f_n\\|_{L^p} < \\infty$. Then $f_n$ converges weakly to $f$: for every $g \\in L^q(\\mathbb{R})$,","title":"Bounded almost everywhere convergent sequences in $L^p$, $1 < p < \\infty$, converge weakly"},{"data_url":"scopes/MeasureTheory/real-functions.json","id":"MeasureTheory/real-functions","kind":"scope","parent":19131,"result":false,"route":"../roadmap/MeasureTheory/real-functions/index.html","scope":"MeasureTheory/real-functions","status":"can_prove","summary":"Absolutely continuous, Lipschitz and bounded-variation functions of a real variable, maximal functions and almost-everywhere properties of real functions.","title":"Differentiation of real functions"},{"id":"MeasureTheory/real-functions/exists-continuous-monotone-sub-of-bounde","kind":"node","lean":"MathlibExt.MeasureTheory.ContinuousBVDiffContinuousMonotoneWanted.exists_continuous_monotone_sub_of_boundedVariationOn","parent":19364,"result":true,"route":"../roadmap/MeasureTheory/real-functions/index.html#exists-continuous-monotone-sub-of-bounde","scope":"MeasureTheory/real-functions","status":"can_prove","summary":"Let $F : \\mathbb{R} \\to \\mathbb{R}$ be continuous and of bounded variation on all of $\\mathbb{R}$. Then there are continuous non-decreasing functions $F_1, F_2 : \\mathbb{R} \\to \\mathbb{R}$ such that $F = F_1 - F_2$.","title":"Jordan decomposition of a continuous function of bounded variation"},{"id":"MeasureTheory/real-functions/not-exists-continuous-ae-eq-indicator-ic","kind":"node","lean":"MathlibExt.MeasureTheory.IndicatorIccNotAEContinuousWanted.not_exists_continuous_ae_eq_indicator_Icc","parent":19364,"reading_title":{"format":"inline-parts-v1","parts":[["text","No continuous function agrees a.e. with the indicator of "],["math","\\((0,1)\\)"]],"source":"No continuous function agrees a.e. with the indicator of $(0,1)$"},"result":true,"route":"../roadmap/MeasureTheory/real-functions/index.html#not-exists-continuous-ae-eq-indicator-ic","scope":"MeasureTheory/real-functions","status":"can_prove","summary":"Let $\\chi_{[0,1]}$ be the characteristic function of $[0,1]$. There is no continuous function $f : \\mathbb{R} \\to \\mathbb{R}$ such that $f(x) = \\chi_{[0,1]}(x)$ for almost every $x \\in \\mathbb{R}$ with respect to Lebesgue measure.","title":"No continuous function agrees a.e. with the indicator of $(0,1)$"},{"data_url":"scopes/MeasureTheory/real-functions/absolutely-continuous-functions.json","id":"MeasureTheory/real-functions/absolutely-continuous-functions","kind":"scope","navigation":"module","parent":19364,"result":false,"route":"../roadmap/MeasureTheory/real-functions/absolutely-continuous-functions/index.html","scope":"MeasureTheory/real-functions/absolutely-continuous-functions","status":"can_prove","title":"Absolutely continuous functions"},{"id":"MeasureTheory/real-functions/absolutely-continuous-functions/volume-image-eq-zero-of-absolutely-conti","kind":"node","lean":"MathlibExt.MeasureTheory.AbsContLusinNPropertyWanted.volume_image_eq_zero_of_absolutelyContinuous","parent":19367,"result":true,"route":"../roadmap/MeasureTheory/real-functions/absolutely-continuous-functions/index.html#volume-image-eq-zero-of-absolutely-conti","scope":"MeasureTheory/real-functions/absolutely-continuous-functions","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be absolutely continuous on every closed bounded interval $[a, b]$. Then $f$ maps Lebesgue null sets to null sets: for every set $s \\subseteq \\mathbb{R}$ of Lebesgue outer measure zero, the image $f(s)$ has Lebesgue outer measure zero.","title":"Absolutely continuous functions have the Lusin N property"},{"id":"MeasureTheory/real-functions/absolutely-continuous-functions/lipschitz-with-iff-absolutely-continuous","kind":"node","lean":"MathlibExt.MeasureTheory.LipschitzIffAbsContBoundedDerivWanted.lipschitzWith_iff_absolutelyContinuous_and_ae_norm_deriv_le","parent":19367,"result":true,"route":"../roadmap/MeasureTheory/real-functions/absolutely-continuous-functions/index.html#lipschitz-with-iff-absolutely-continuous","scope":"MeasureTheory/real-functions/absolutely-continuous-functions","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ and let $M \\ge 0$ be a real number. Then $|f(x) - f(y)| \\le M|x - y|$ for all $x, y \\in \\mathbb{R}$ if and only if $f$ is absolutely continuous on every closed bounded interval and $|f'(x)| \\le M$ for almost every $x \\in \\mathbb{R}$ (with respect to Lebesgue measure), where $f'(x)$ is taken to be $0$ at points where $f$ is not differentiable.","title":"Lipschitz functions are the absolutely continuous functions with bounded derivative"},{"id":"MeasureTheory/real-functions/absolutely-continuous-functions/absolutely-continuous-on-interval-and-e","kind":"node","lean":"MathlibExt.MeasureTheory.LipschitzOneAbsContDerivLinftyWanted.absolutelyContinuousOnInterval_and_eLpNorm_deriv_lt_top_of_lipschitzOnWith","parent":19367,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(1\\)"],["text","-Lipschitz function on "],["math","\\((a,b)\\)"],["text"," is absolutely continuous with essentially bounded derivative"]],"source":"A $1$-Lipschitz function on $(a,b)$ is absolutely continuous with essentially bounded derivative"},"result":true,"route":"../roadmap/MeasureTheory/real-functions/absolutely-continuous-functions/index.html#absolutely-continuous-on-interval-and-e","scope":"MeasureTheory/real-functions/absolutely-continuous-functions","status":"can_prove","summary":"A $1$-Lipschitz function on $[a,b]$ is absolutely continuous with essentially bounded derivative. Let $a \\le b$ be real numbers and let $f : \\mathbb{R} \\to \\mathbb{R}$ satisfy $|f(x) - f(y)| \\le |x - y|$ for all $x, y \\in [a, b]$. Then $f$ is absolutely continuous on $[a, b]$, and its derivative $f'$, taken to be $0$ at points where $f$ is not differentiable, has finite norm $\\|f'\\|_{L^\\infty[a,b]}$, the essential\u2026","title":"A $1$-Lipschitz function on $(a,b)$ is absolutely continuous with essentially bounded derivative"},{"data_url":"scopes/MeasureTheory/real-functions/hardy-littlewood-maximal-function.json","id":"MeasureTheory/real-functions/hardy-littlewood-maximal-function","kind":"scope","navigation":"module","parent":19364,"result":false,"route":"../roadmap/MeasureTheory/real-functions/hardy-littlewood-maximal-function/index.html","scope":"MeasureTheory/real-functions/hardy-littlewood-maximal-function","status":"can_prove","title":"Hardy\u2013Littlewood maximal function"},{"id":"MeasureTheory/real-functions/hardy-littlewood-maximal-function/ae-tendsto-maximal-zero-of-ae-tendsto-ze","kind":"node","lean":"MathlibExt.MeasureTheory.MaximalFunctionAeTendstoZeroWanted.ae_tendsto_maximal_zero_of_ae_tendsto_zero_of_bounded","parent":19371,"result":true,"route":"../roadmap/MeasureTheory/real-functions/hardy-littlewood-maximal-function/index.html#ae-tendsto-maximal-zero-of-ae-tendsto-ze","scope":"MeasureTheory/real-functions/hardy-littlewood-maximal-function","status":"can_prove","summary":"Let $(f_n)_{n \\in \\mathbb{N}}$ be a sequence of functions $f_n : \\mathbb{R} \\to \\mathbb{R}$, each Lebesgue measurable on $[0,1]$, such that $f_n(x) \\in [0,1]$ for every $n$ and every $x \\in [0,1]$, and $f_n(x) \\to 0$ as $n \\to \\infty$ for almost every $x \\in [0,1]$. For $x \\in [0,1]$ let","title":"Maximal functions of bounded a.e.-null sequences tend to zero a.e"},{"id":"MeasureTheory/real-functions/hardy-littlewood-maximal-function/measure-set-of-maximal-function-lt-top-e","kind":"node","lean":"MathlibExt.MeasureTheory.SingularMeasureMaximalFunctionWanted.measure_setOf_maximalFunction_lt_top_eq_zero_of_mutuallySingular","parent":19371,"result":true,"route":"../roadmap/MeasureTheory/real-functions/hardy-littlewood-maximal-function/index.html#measure-set-of-maximal-function-lt-top-e","scope":"MeasureTheory/real-functions/hardy-littlewood-maximal-function","status":"can_prove","summary":"The maximal function of a singular measure is infinite almost everywhere for that measure. Let $n$ be a natural number and let $\\mu$ be a Borel measure on $\\mathbb{R}^n$ that is mutually singular with respect to Lebesgue measure $m_n$. Write $B(x, r)$ for the open ball of radius $r$ about $x$ and define $M_\\mu : \\mathbb{R}^n \\to [0, \\infty]$ by","title":"The maximal function of a singular measure is infinite almost everywhere for that measure"},{"data_url":"scopes/MeasureTheory/optimal-transport.json","id":"MeasureTheory/optimal-transport","kind":"scope","parent":19131,"result":false,"route":"../roadmap/MeasureTheory/optimal-transport/index.html","scope":"MeasureTheory/optimal-transport","status":"can_prove","summary":"Couplings of probability measures, transport costs and Kantorovich duality.","title":"Optimal transport"},{"data_url":"scopes/MeasureTheory/optimal-transport/kantorovich-duality.json","id":"MeasureTheory/optimal-transport/kantorovich-duality","kind":"scope","navigation":"module","parent":19374,"result":false,"route":"../roadmap/MeasureTheory/optimal-transport/kantorovich-duality/index.html","scope":"MeasureTheory/optimal-transport/kantorovich-duality","status":"fully_proved","title":"Kantorovich duality"},{"id":"MeasureTheory/optimal-transport/kantorovich-duality/is-coupling","kind":"node","lean":"MathlibExt.MeasureTheory.OptimalTransport.KantorovichWanted.IsCoupling","parent":19375,"result":true,"route":"../roadmap/MeasureTheory/optimal-transport/kantorovich-duality/index.html#is-coupling","scope":"MeasureTheory/optimal-transport/kantorovich-duality","status":"fully_proved","summary":"Let $\\mu$ and $\\nu$ be probability measures on measurable spaces $X$ and $Y$. A probability measure $\\pi$ on $X \\times Y$ is a coupling of $\\mu$ and $\\nu$ when its marginals are $\\mu$ and $\\nu$, that is, when the pushforward of $\\pi$ under the projection $X \\times Y \\to X$ is $\\mu$ and the pushforward of $\\pi$ under the projection $X \\times Y \\to Y$ is $\\nu$.","title":"Coupling of two probability measures"},{"id":"MeasureTheory/optimal-transport/kantorovich-duality/primal-values","kind":"node","lean":"MathlibExt.MeasureTheory.OptimalTransport.KantorovichWanted.primalValues","parent":19375,"result":true,"route":"../roadmap/MeasureTheory/optimal-transport/kantorovich-duality/index.html#primal-values","scope":"MeasureTheory/optimal-transport/kantorovich-duality","status":"fully_proved","summary":"Let $\\mu$ and $\\nu$ be probability measures on measurable spaces $X$ and $Y$, and let $c : X \\times Y \\to \\mathbb{R}$. The primal values of $c$ for $\\mu$ and $\\nu$ form the set of real numbers $\\int c \\, d\\pi$, where $\\pi$ ranges over the couplings of $\\mu$ and $\\nu$.","title":"Values of the transport cost over couplings"},{"id":"MeasureTheory/optimal-transport/kantorovich-duality/dual-feasible","kind":"node","lean":"MathlibExt.MeasureTheory.OptimalTransport.KantorovichWanted.DualFeasible","parent":19375,"result":true,"route":"../roadmap/MeasureTheory/optimal-transport/kantorovich-duality/index.html#dual-feasible","scope":"MeasureTheory/optimal-transport/kantorovich-duality","status":"fully_proved","summary":"Let $X$ and $Y$ be topological spaces and let $c : X \\times Y \\to \\mathbb{R}$. A pair of functions $\\varphi : X \\to \\mathbb{R}$ and $\\psi : Y \\to \\mathbb{R}$ is dual feasible for the cost $c$ when $\\varphi$ and $\\psi$ are continuous and $\\varphi(x) + \\psi(y) \\le c(x, y)$ for all $x \\in X$ and $y \\in Y$.","title":"Dual feasible pair of potentials"},{"id":"MeasureTheory/optimal-transport/kantorovich-duality/kantorovich-duality-compact","kind":"node","lean":"MathlibExt.MeasureTheory.OptimalTransport.KantorovichWanted.kantorovich_duality_compact","parent":19375,"result":true,"route":"../roadmap/MeasureTheory/optimal-transport/kantorovich-duality/index.html#kantorovich-duality-compact","scope":"MeasureTheory/optimal-transport/kantorovich-duality","status":"fully_proved","summary":"Let $X$ and $Y$ be nonempty compact metric spaces, each carrying its Borel $\\sigma$-algebra, let $\\mu$ and $\\nu$ be probability measures on $X$ and $Y$, and let $c : X \\times Y \\to \\mathbb{R}$ be continuous. Then the infimum of $\\int c \\, d\\pi$ over all couplings $\\pi$ of $\\mu$ and $\\nu$ equals the supremum of $\\int \\varphi \\, d\\mu + \\int \\psi \\, d\\nu$ over all pairs of continuous functions\u2026","title":"Kantorovich duality on compact metric spaces"},{"data_url":"scopes/MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem.json","id":"MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem","kind":"scope","navigation":"module","parent":19374,"result":false,"route":"../roadmap/MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem/index.html","scope":"MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem","status":"can_prove","title":"Monge\u2013Kantorovich existence theorem"},{"id":"MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem/couplings","kind":"node","lean":"MathlibExt.Analysis.OptimalTransportWanted.Analysis.Couplings","parent":19380,"result":true,"route":"../roadmap/MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem/index.html#couplings","scope":"MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem","status":"fully_proved","summary":"Let $X$ and $Y$ be measurable spaces, $P$ a measure on $X$ and $Q$ a measure on $Y$. A coupling of $P$ and $Q$ is a probability measure $\\pi$ on $X \\times Y$ whose marginals are $P$ and $Q$: the pushforward of $\\pi$ under the projection $X \\times Y \\to X$ is $P$, and its pushforward under the projection $X \\times Y \\to Y$ is $Q$. The set of couplings is written $\\Pi(P, Q)$.","title":"Couplings"},{"id":"MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem/kantorovich-cost","kind":"node","lean":"MathlibExt.Analysis.OptimalTransportWanted.Analysis.kantorovichCost","parent":19380,"result":true,"route":"../roadmap/MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem/index.html#kantorovich-cost","scope":"MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem","status":"fully_proved","summary":"Let $X$ and $Y$ be measurable spaces and $c \\colon X \\times Y \\to [0, \\infty]$ a cost function. The Kantorovich cost of a measure $\\pi$ on $X \\times Y$ is $I(\\pi) = \\int_{X \\times Y} c \\, d\\pi \\in [0, \\infty]$, the Lebesgue integral of $c$ against $\\pi$ (for a non-measurable $c$, the lower Lebesgue integral).","title":"Kantorovich cost"},{"id":"MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem/monge-kantorovich-exists","kind":"node","lean":"MathlibExt.Analysis.OptimalTransportWanted.Analysis.monge_kantorovich_exists","parent":19380,"result":true,"route":"../roadmap/MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem/index.html#monge-kantorovich-exists","scope":"MeasureTheory/optimal-transport/monge-kantorovich-existence-theorem","status":"can_prove","summary":"Let $X$ and $Y$ be Polish spaces with their Borel $\\sigma$-algebras, let $P$ and $Q$ be probability measures on $X$ and $Y$, and let $c \\colon X \\times Y \\to [0, \\infty]$ be continuous. Then there is a coupling $\\pi$ of $P$ and $Q$ whose Kantorovich cost $\\int c \\, d\\pi$ is at most the Kantorovich cost $\\int c \\, d\\pi'$ of every coupling $\\pi'$ of $P$ and $Q$: the Kantorovich cost attains its minimum on the set of\u2026","title":"Monge\u2013Kantorovich existence theorem"},{"area":"Analysis & Probability","data_url":"chapters/Probability.json","id":"Probability","kind":"scope","parent":0,"result":false,"route":"../roadmap/Probability/index.html","scope":"Probability","status":"can_prove","summary":"Probability studies random phenomena through distributions, expectation, independence, concentration, and stochastic processes.","title":"Probability"},{"data_url":"scopes/Probability/laws-of-large-numbers.json","id":"Probability/laws-of-large-numbers","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/laws-of-large-numbers/index.html","scope":"Probability/laws-of-large-numbers","status":"can_prove","summary":"Almost-sure behaviour of sums of independent variables: strong and weak laws of large numbers, Kolmogorov's three-series theorem and maximal inequality, truncation techniques, the law of the iterated logarithm, and the Monte Carlo variance rate of sample means.","title":"Laws of large numbers and random series"},{"id":"Probability/laws-of-large-numbers/kolmogorov-maximal-inequality","kind":"node","lean":"MathlibExt.Probability.KolmogorovMaximal.kolmogorov_maximal_inequality","parent":19385,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/index.html#kolmogorov-maximal-inequality","scope":"Probability/laws-of-large-numbers","status":"fully_proved","summary":"Let $\\alpha$ be a finite nonempty set and $n \\in \\mathbb{N}$. For each $i \\in \\{0, \\dots, n-1\\}$ let $p_i$ be a probability vector on $\\alpha$, so $p_i(a) \\ge 0$ and $\\sum_{a \\in \\alpha} p_i(a) = 1$, and let $X_i : \\alpha \\to \\mathbb{R}$ satisfy $\\sum_{a \\in \\alpha} p_i(a) X_i(a) = 0$. Give each $\\omega \\in \\alpha^n$ the weight $\\prod_{i=0}^{n-1} p_i(\\omega_i)$, so that the coordinates are independent with laws\u2026","title":"Kolmogorov's maximal inequality for finitely-valued variables"},{"id":"Probability/laws-of-large-numbers/tendsto-volume-abs-sum-ge-mul-sqrt","kind":"node","lean":"MathlibExt.MeasureTheory.MeanZeroSumDeviationMeasureWanted.tendsto_volume_abs_sum_ge_mul_sqrt","parent":19385,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/index.html#tendsto-volume-abs-sum-ge-mul-sqrt","scope":"Probability/laws-of-large-numbers","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be square integrable on $[0,1]$ with $\\int_0^1 f = 0$ and $\\int_0^1 |f|^2 > 0$, and let $(a_n)_{n \\in \\mathbb{N}}$ be a sequence of positive reals with $a_n \\to \\infty$. For each $n$ let","title":"Large deviations of mean-zero sums on the cube have vanishing measure"},{"id":"Probability/laws-of-large-numbers/variance-sample-mean-of-iid","kind":"node","lean":"MathlibExt.Probability.variance_sample_mean_of_iid","parent":19385,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/index.html#variance-sample-mean-of-iid","scope":"Probability/laws-of-large-numbers","status":"fully_proved","summary":"Let $X_0, X_1, X_2, \\dots$ be real-valued random variables on a measure space $(\\Omega, \\mu)$ that are mutually independent and each have the same distribution as $X_0$, and suppose $X_0$ is square-integrable. Then for every integer $n \\ge 1$, with variances taken with respect to $\\mu$,","title":"Variance of the sample mean of i.i.d. samples"},{"data_url":"scopes/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem.json","id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","kind":"scope","navigation":"module","parent":19385,"result":false,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","title":"Kolmogorov's three-series theorem"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/e-lp-norm-one-le-sqrt-integral-sq","kind":"node","lean":"ProbabilityTheory.eLpNorm_one_le_sqrt_integral_sq","parent":19389,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^1\\)"],["text"," norm bounded by the square root of the second moment"]],"source":"$L^1$ norm bounded by the square root of the second moment"},"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#e-lp-norm-one-le-sqrt-integral-sq","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"If `\u222b \u2016f\u2016\u00b2 d\u03bc \u2264 C` for an `L\u00b2` function on a probability space, then `\u2016f\u2016\u2081 \u2264 \u221aC`.","title":"$L^1$ norm bounded by the square root of the second moment"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/ae-tendsto-sum-of-indep-centered-l1bdd","kind":"node","lean":"ProbabilityTheory.ae_tendsto_sum_of_indep_centered_L1bdd","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#ae-tendsto-sum-of-indep-centered-l1bdd","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Let $Z_0, Z_1, \\dots$ be independent integrable real random variables on a probability space with $E[Z_n] = 0$ for every $n$, and let $S_n = \\sum_{k \\le n} Z_k$. If there is a constant $R \\ge 0$ with $E|S_n| \\le R$ for every $n$, then $S_n$ converges almost surely to a finite limit.","title":"Almost-sure convergence of L\u00b9-bounded sums of independent centered variables"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncated-rv","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncatedRV","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncated-rv","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"For a sequence $X_0, X_1, \\dots$ of real-valued functions on $\\Omega$, a threshold $A \\in \\mathbb{R}$ and an index $n$, the truncation of $X_n$ at $A$ is $Y_n = X_n \\mathbf{1}_{\\{|X_n| \\le A\\}}$, that is, $Y_n(\\omega) = X_n(\\omega)$ when $|X_n(\\omega)| \\le A$ and $Y_n(\\omega) = 0$ otherwise.","title":"Truncation of a random variable"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncated-rv-apply","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncatedRV_apply","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncated-rv-apply","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"The canonical truncation is the identity inside the cutoff and zero outside it.","title":"Formula for the truncated random variable"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/measurable-truncated-rv","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.measurable_truncatedRV","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#measurable-truncated-rv","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"The canonical truncation of a measurable random variable is measurable.","title":"Truncation preserves measurability"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/i-indep-fun-truncated-rv","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.iIndepFun_truncatedRV","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#i-indep-fun-truncated-rv","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Truncation preserves independence.","title":"Truncation preserves independence"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/abs-truncated-rv-le","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.abs_truncatedRV_le","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#abs-truncated-rv-le","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"A canonical truncation is bounded by a nonnegative cutoff in absolute value.","title":"Truncation is bounded by a nonnegative cutoff"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/abs-truncated-rv-le-abs","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.abs_truncatedRV_le_abs","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#abs-truncated-rv-le-abs","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"A canonical truncation is bounded by the absolute value of any cutoff.","title":"Truncation is bounded by the absolute cutoff"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncated-rv-eq-of-abs-le","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncatedRV_eq_of_abs_le","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncated-rv-eq-of-abs-le","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Truncation fixes points inside or on the cutoff.","title":"Truncation fixes values within the cutoff"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/ae-tendsto-centered-of-indep-summable-va","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.ae_tendsto_centered_of_indep_summableVar","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#ae-tendsto-centered-of-indep-summable-va","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"If the truncated variables have summable variances, then almost surely the partial sums of the centered truncations converge: `\u2211 (Y\u1d62 - E[Y\u1d62])` converges a.s.","title":"Almost sure convergence of centered truncated sums"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/ae-eventually-truncation-eq","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.ae_eventually_truncation_eq","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#ae-eventually-truncation-eq","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a probability space $(\\Omega, P)$, let $A$ be a real number, and let $Y_n = X_n \\mathbf{1}_{\\{|X_n| \\le A\\}}$ be the truncation of $X_n$ at $A$. If $\\sum_n P(|X_n| > A) < \\infty$, then for almost every $\\omega$ we have $X_n(\\omega) = Y_n(\\omega)$ for all sufficiently large $n$.","title":"Truncations eventually agree with the variables"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/ae-tendsto-of-ae-eventually-eq-and-ae-te","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.ae_tendsto_of_ae_eventually_eq_and_ae_tendsto","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#ae-tendsto-of-ae-eventually-eq-and-ae-te","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$ and let $f_n, g_n : \\Omega \\to \\mathbb{R}$ ($n \\ge 0$) be functions such that for $\\mu$-almost every $\\omega$ we have $f_n(\\omega) = g_n(\\omega)$ for all sufficiently large $n$. If there is a function $S : \\Omega \\to \\mathbb{R}$ with $\\sum_{i < n} f_i(\\omega) \\to S(\\omega)$ as $n \\to \\infty$ for $\\mu$-almost every $\\omega$, then there is a function $T : \\Omega \\to \\mathbb{R}$ with\u2026","title":"Eventually equal series inherit almost-everywhere convergence"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/ae-tendsto-truncated-of-conditions","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.ae_tendsto_truncated_of_conditions","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#ae-tendsto-truncated-of-conditions","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent real random variables on a probability space $(\\Omega, P)$, let $A > 0$, and let $Y_n = X_n \\mathbf{1}_{\\{|X_n| \\le A\\}}$ be the truncation of $X_n$ at $A$. If the series $\\sum_n E[Y_n]$ converges absolutely and $\\sum_n \\operatorname{Var}(Y_n) < \\infty$, then there is a function $S : \\Omega \\to \\mathbb{R}$ with $\\sum_{i < n} Y_i(\\omega) \\to S(\\omega)$ as $n \\to \\infty$ for\u2026","title":"Truncated sums converge almost surely"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/kolmogorov-three-series-sufficiency","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.kolmogorov_three_series_sufficiency","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#kolmogorov-three-series-sufficiency","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent real random variables on a probability space $(\\Omega, P)$, fix $A > 0$, and write $Y_n = X_n \\mathbf{1}_{\\{|X_n| \\le A\\}}$ for the truncations. If $\\sum_n P(|X_n| > A) < \\infty$, the series $\\sum_n E[Y_n]$ converges absolutely, and $\\sum_n \\operatorname{Var}(Y_n) < \\infty$, then there is a function $S : \\Omega \\to \\mathbb{R}$ with $\\sum_{i < n} X_i(\\omega) \\to S(\\omega)$ as\u2026","title":"Kolmogorov's three-series theorem (sufficiency direction)"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/summable-tail-prob-of-ae-tendsto","kind":"node","lean":"ProbabilityTheory.summable_tail_prob_of_ae_tendsto","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#summable-tail-prob-of-ae-tendsto","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Let $(X_n)_{n \\ge 0}$ be a sequence of independent measurable real random variables on a probability space $(\\Omega, \\mu)$ whose partial sums converge almost surely: there is a function $S : \\Omega \\to \\mathbb{R}$ such that $\\sum_{i < n} X_i(\\omega) \\to S(\\omega)$ as $n \\to \\infty$ for $\\mu$-almost every $\\omega$. Then for every $A > 0$ the tail probabilities are summable:\u2026","title":"Kolmogorov three-series theorem, necessity, first series"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/not-ae-tendsto-partial-sum-of-tsum-tail","kind":"node","lean":"ProbabilityTheory.not_ae_tendsto_partial_sum_of_tsum_tail_prob_eq_top","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#not-ae-tendsto-partial-sum-of-tsum-tail","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Let $(X_n)_{n \\ge 0}$ be a sequence of independent measurable real random variables on a probability space $(\\Omega, \\mu)$, and let $A > 0$. If $\\sum_n \\mu(\\{\\omega : |X_n(\\omega)| > A\\}) = \\infty$, then there is no function $S : \\Omega \\to \\mathbb{R}$ such that, for $\\mu$-almost every $\\omega$, the partial sums $\\sum_{i < n} X_i(\\omega)$ converge to $S(\\omega)$ as $n \\to \\infty$.","title":"Divergent tail probabilities prevent almost-sure convergence of partial sums"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/pair-rv","kind":"node","lean":"MathlibExt.Probability.ThreeSeriesNecessity.pairRV","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#pair-rv","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"The pair of the `n`-th coordinates of two independent copies.","title":"Pair of coordinates from two independent copies"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/symm-rv","kind":"node","lean":"MathlibExt.Probability.ThreeSeriesNecessity.symmRV","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#symm-rv","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"The symmetrized variable `Z n (\u03c9, \u03c9') = Y n \u03c9 - Y n \u03c9'`.","title":"Symmetrized random variable"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/integral-mul-of-indep-comp","kind":"node","lean":"MathlibExt.Probability.ThreeSeriesNecessity.integral_mul_of_indep_comp","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#integral-mul-of-indep-comp","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"If `f` and `g` are independent, the expectation of a product of functions of `f` and `g` factors.","title":"Expectation of a product of independent functions factors"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/integral-pow-four-le","kind":"node","lean":"MathlibExt.Probability.ThreeSeriesNecessity.integral_pow_four_le","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#integral-pow-four-le","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Fourth-moment induction for sums of independent centered bounded random variables: `E[S\u2074] \u2264 3 E[S\u00b2]\u00b2 + B\u00b2 E[S\u00b2]`, together with the variance-of-sum identity and `E[S] = 0`.","title":"Fourth-moment bound for sums of bounded independent variables"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/summable-variance-of-bounded-ae-tendsto","kind":"node","lean":"MathlibExt.Probability.ThreeSeriesNecessity.summable_variance_of_bounded_ae_tendsto","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#summable-variance-of-bounded-ae-tendsto","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"For a uniformly bounded independent sequence whose partial sums converge almost surely, the variances are summable. Proved by symmetrization: the symmetrized sums are centered, so Paley\u2013Zygmund gives a uniform lower bound on block-tail probabilities, contradicting convergence in measure unless the variances are summable.","title":"Three-series necessity, second half of the variance step"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/tendsto-sum-integral-of-bounded-ae-tends","kind":"node","lean":"MathlibExt.Probability.ThreeSeriesNecessity.tendsto_sum_integral_of_bounded_ae_tendsto","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#tendsto-sum-integral-of-bounded-ae-tends","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"For a uniformly bounded independent sequence with summable variances whose partial sums converge almost surely, the partial sums of the means converge (to some limit). Proved by centering and applying the canonical centered-sum convergence theorem with an `L\u00b9` bound from the total variance. The mean partial sums are the difference of the two convergent series at a single sample point.","title":"Mean step of three-series necessity"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/three-series-necessity-mean-variance","kind":"node","lean":"MathlibExt.Probability.ThreeSeriesNecessity.three_series_necessity_mean_variance","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#three-series-necessity-mean-variance","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"If the partial sums of an independent sequence `X\u2099` converge almost surely, then for every `A > 0` the truncations `Y\u2099 = X\u2099 \u00b7 1_{|X\u2099| \u2264 A}` have convergent mean partial sums and summable variances. This is the corrected form of v1's `summable_mean_and_variance_of_ae_tendsto_ax`: v1 claimed the means themselves are `Summable`, which is false (see `summable_mean_necessity_false`); only convergence of the mean partial\u2026","title":"Kolmogorov three-series necessity, series (2)\u2013(3)"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/summable-mean-necessity-false","kind":"node","lean":"MathlibExt.Probability.ThreeSeriesNecessity.summable_mean_necessity_false","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#summable-mean-necessity-false","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"The deterministic sequence `X k \u03c9 = (-1)^k / (k + 1)` on the one-point probability space `(Unit, dirac ())` has converging partial sums, is constant (hence measurable and independent), agrees with its truncations at `A = 1`, and has all variances `0` \u2014 yet its means `\u222b truncatedRV X 1 k = (-1)^k / (k + 1)` are not `Summable`, since `Summable.abs` would force summability of the harmonic series. This is why the\u2026","title":"The `Summable`-means necessity claim in v1 is false"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncation-fun","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncationFun","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncation-fun","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Scalar truncation is canonical truncation of the identity sequence.","title":"Scalar truncation function"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncated-rv-eq-zero-of-abs-gt","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncatedRV_eq_zero_of_abs_gt","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncated-rv-eq-zero-of-abs-gt","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Truncation of a value strictly outside the closed band is zero.","title":"Truncation vanishes outside the closed band"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncated-rv-nonneg-of-nonneg","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncatedRV_nonneg_of_nonneg","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncated-rv-nonneg-of-nonneg","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Canonical truncation preserves nonnegativity.","title":"Canonical truncation preserves nonnegativity"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncation-fun-eq-self-of-abs-le","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncationFun_eq_self_of_abs_le","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncation-fun-eq-self-of-abs-le","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Scalar truncation fixes values inside or on the cutoff.","title":"Scalar truncation fixes values within the cutoff"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncation-fun-eq-zero-of-abs-gt","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncationFun_eq_zero_of_abs_gt","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncation-fun-eq-zero-of-abs-gt","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Scalar truncation kills values strictly outside the cutoff.","title":"Scalar truncation vanishes beyond the cutoff"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncation-fun-nonneg-of-nonneg","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncationFun_nonneg_of_nonneg","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncation-fun-nonneg-of-nonneg","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Scalar truncation preserves nonnegativity.","title":"Scalar truncation preserves nonnegativity"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncated-rv-eq-comp","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncatedRV_eq_comp","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncated-rv-eq-comp","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Canonical truncation factors through its scalar specialization.","title":"Canonical truncation factors through scalar truncation"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/measurable-truncation-fun","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.measurable_truncationFun","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#measurable-truncation-fun","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"The scalar specialization of canonical truncation is measurable.","title":"Measurability of scalar truncation"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/measurable-truncated-rv-of-measurable","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.measurable_truncatedRV_of_measurable","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#measurable-truncated-rv-of-measurable","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Canonical truncation needs measurability only of the selected variable.","title":"Measurability of a truncated random variable"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/truncated-rv-eq-of-bdd","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.truncatedRV_eq_of_bdd","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#truncated-rv-eq-of-bdd","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"A family bounded by the cutoff is unchanged by canonical truncation.","title":"Truncation leaves a family bounded by the cutoff unchanged"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/ae-tendsto-truncated-rv-of-bdd","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.ae_tendsto_truncatedRV_of_bdd","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#ae-tendsto-truncated-rv-of-bdd","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"A uniformly bounded family keeps its almost-sure partial-sum limit under canonical truncation, on the same measure.","title":"Truncation preserves almost-sure limits of bounded partial sums"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/mem-lp-truncated-rv","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.memLp_truncatedRV","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#mem-lp-truncated-rv","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Canonical truncation is in every `L\u1d56` on a finite measure space.","title":"Truncated random variables lie in every Lp space"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/integrable-truncated-rv","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.integrable_truncatedRV","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#integrable-truncated-rv","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"Canonical truncation is integrable on a finite measure space.","title":"Integrability of truncated random variables"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/variance-truncated-rv-le","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.variance_truncatedRV_le","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#variance-truncated-rv-le","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"The variance of canonical truncation is at most `A\u00b2`.","title":"Variance bound for a truncated random variable"},{"id":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/abs-integral-truncated-rv-le","kind":"node","lean":"ProbabilityTheory.KolmogorovThreeSeries.abs_integral_truncatedRV_le","parent":19389,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem/index.html#abs-integral-truncated-rv-le","scope":"Probability/laws-of-large-numbers/kolmogorov-s-three-series-theorem","status":"fully_proved","summary":"The absolute mean of canonical truncation is at most `|A|`.","title":"Mean bound for a truncated random variable"},{"data_url":"scopes/Probability/laws-of-large-numbers/law-of-the-iterated-logarithm.json","id":"Probability/laws-of-large-numbers/law-of-the-iterated-logarithm","kind":"scope","navigation":"module","parent":19385,"result":false,"route":"../roadmap/Probability/laws-of-large-numbers/law-of-the-iterated-logarithm/index.html","scope":"Probability/laws-of-large-numbers/law-of-the-iterated-logarithm","status":"fully_proved","title":"Law of the iterated logarithm"},{"id":"Probability/laws-of-large-numbers/law-of-the-iterated-logarithm/law-of-the-iterated-logarithm-of-mem-lp","kind":"node","lean":"MathlibExt.Probability.LawIteratedLogarithmWanted.law_of_the_iterated_logarithm_of_memLp","parent":19429,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/law-of-the-iterated-logarithm/index.html#law-of-the-iterated-logarithm-of-mem-lp","scope":"Probability/laws-of-large-numbers/law-of-the-iterated-logarithm","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent, identically distributed, measurable real random variables on a probability space with $\\mathbb{E}[X_0^2] < \\infty$, $\\mathbb{E}[X_0] = 0$ and $\\operatorname{Var}(X_0) = 1$, and let $S_n = X_0 + \\dots + X_{n-1}$. Then almost surely $\\limsup_{n \\to \\infty} S_n / \\sqrt{2 n \\log \\log n} = 1$ and $\\liminf_{n \\to \\infty} S_n / \\sqrt{2 n \\log \\log n} = -1$.","title":"Law of the iterated logarithm, square-integrable form"},{"id":"Probability/laws-of-large-numbers/law-of-the-iterated-logarithm/law-of-the-iterated-logarithm","kind":"node","lean":"MathlibExt.Probability.LawIteratedLogarithmWanted.law_of_the_iterated_logarithm","parent":19429,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/law-of-the-iterated-logarithm/index.html#law-of-the-iterated-logarithm","scope":"Probability/laws-of-large-numbers/law-of-the-iterated-logarithm","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent, identically distributed, measurable real random variables on a probability space such that $X_0$ is integrable, $\\mathbb{E}[X_0^2] < \\infty$, $\\mathbb{E}[X_0] = 0$ and $\\operatorname{Var}(X_0) = 1$, and let $S_n = X_0 + \\dots + X_{n-1}$. Then almost surely $\\limsup_{n \\to \\infty} S_n / \\sqrt{2 n \\log \\log n} = 1$ and $\\liminf_{n \\to \\infty} S_n / \\sqrt{2 n \\log \\log n} = -1$.","title":"Law of the iterated logarithm"},{"data_url":"scopes/Probability/laws-of-large-numbers/strong-law-of-large-numbers.json","id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","kind":"scope","navigation":"module","parent":19385,"result":false,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","title":"Strong law of large numbers"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/partial-sum-indicators","kind":"node","lean":"ProbabilityTheory.partialSumIndicators","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#partial-sum-indicators","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"For sets $A_0, A_1, \\dots$ in $\\Omega$ and $n \\in \\mathbb{N}$, $S_n = \\sum_{i < n} 1_{A_i}$ is the sum of the indicators of the first $n$ sets, viewed as a real-valued function on $\\Omega$.","title":"Partial sums of indicators"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/partial-sum-indicators-eq","kind":"node","lean":"ProbabilityTheory.partialSumIndicators_eq","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#partial-sum-indicators-eq","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"For sets $A_0, A_1, \\dots$ in $\\Omega$ and every $n$, the partial sum $S_n$, defined pointwise by $S_n(\\omega) = \\sum_{i < n} 1_{A_i}(\\omega)$, equals, as a function on $\\Omega$, the sum $\\sum_{i < n} 1_{A_i}$ of the real-valued indicator functions.","title":"The partial sum as a sum of indicator functions"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/comap-indicator-one-le-generate-from","kind":"node","lean":"ProbabilityTheory.comap_indicator_one_le_generateFrom","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#comap-indicator-one-le-generate-from","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"For a set $s \\subseteq \\Omega$, the \u03c3-algebra on $\\Omega$ generated by the indicator function $1_s : \\Omega \\to \\mathbb{R}$ (the preimages of Borel sets of $\\mathbb{R}$) is contained in the \u03c3-algebra generated by $s$.","title":"The \u03c3-algebra generated by an indicator function"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/indep-fun-indicator-of-indep-set","kind":"node","lean":"ProbabilityTheory.indepFun_indicator_of_indepSet","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#indep-fun-indicator-of-indep-set","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"If two sets $s$ and $t$ are independent with respect to a probability measure $\\mu$, then their real-valued indicator functions $1_s$ and $1_t$ are independent random variables with respect to $\\mu$.","title":"Independent events have independent indicators"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/mem-lp-indicator-one","kind":"node","lean":"ProbabilityTheory.memLp_indicator_one","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#mem-lp-indicator-one","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"If $\\mu$ is a probability measure and $s$ is a measurable set, then the real-valued indicator function $1_s$ lies in $L^2(\\mu)$.","title":"Indicators are square-integrable"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/indicator-one-sq","kind":"node","lean":"ProbabilityTheory.indicator_one_sq","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#indicator-one-sq","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"For every set $s \\subseteq \\Omega$, the real-valued indicator function $1_s$ satisfies $1_s^2 = 1_s$.","title":"The square of an indicator is itself"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/variance-indicator-le","kind":"node","lean":"ProbabilityTheory.variance_indicator_le","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#variance-indicator-le","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$ and let $s \\subseteq \\Omega$ be measurable. Then the indicator function $1_s$ satisfies $\\operatorname{Var}_\\mu(1_s) \\le \\mu(s)$.","title":"Variance of an indicator is at most its probability"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/integral-partial-sum-indicators","kind":"node","lean":"ProbabilityTheory.integral_partialSumIndicators","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#integral-partial-sum-indicators","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$ and let $A_0, A_1, \\dots$ be measurable sets. For every $n$, the partial sum $S_n = \\sum_{i < n} 1_{A_i}$ has expectation $E[S_n] = \\sum_{i < n} \\mu(A_i)$.","title":"Expectation of a sum of indicators"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/variance-partial-sum-indicators-le","kind":"node","lean":"ProbabilityTheory.variance_partialSumIndicators_le","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#variance-partial-sum-indicators-le","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$ and let $A_0, A_1, \\dots$ be pairwise independent measurable subsets of $\\Omega$. Then for every $n$, the partial sum $S_n = \\sum_{i < n} 1_{A_i}$ satisfies $\\operatorname{Var}(S_n) \\le \\mathbb{E}[S_n]$, where variance and expectation are taken with respect to $\\mu$.","title":"Variance of a sum of pairwise independent indicators is at most its mean"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/es-tendsto-at-top","kind":"node","lean":"ProbabilityTheory.ES_tendsto_atTop","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#es-tendsto-at-top","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots \\subseteq \\Omega$ be measurable sets with $\\sum_n \\mu(A_n) = \\infty$, and let $S_n = \\sum_{i\\lt n} 1_{A_i}$. Then $\\mathbb{E}[S_n] \\to \\infty$ as $n \\to \\infty$.","title":"Divergent total probability makes the expected partial sums diverge"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/es-mono","kind":"node","lean":"ProbabilityTheory.ES_mono","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#es-mono","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots \\subseteq \\Omega$ be measurable sets and let $S_n = \\sum_{i\\lt n} 1_{A_i}$. Then $n \\mapsto \\mathbb{E}[S_n]$ is nondecreasing.","title":"Monotonicity of the expected partial sums"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/es-step-le-one","kind":"node","lean":"ProbabilityTheory.ES_step_le_one","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#es-step-le-one","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots \\subseteq \\Omega$ be measurable sets and let $S_n = \\sum_{i\\lt n} 1_{A_i}$. Then $\\mathbb{E}[S_{n+1}] - \\mathbb{E}[S_n] \\le 1$ for every $n \\in \\mathbb{N}$.","title":"The expected partial sums grow by at most one per step"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/slln-subseq","kind":"node","lean":"ProbabilityTheory.sllnSubseq","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#slln-subseq","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots$ be subsets of $\\Omega$, and write $S_n = \\sum_{i < n} 1_{A_i}$. Assuming $\\mathbb{E}[S_n] = \\int S_n \\, d\\mu \\to \\infty$ as $n \\to \\infty$, this is the sequence $\\varphi : \\mathbb{N} \\to \\mathbb{N}$ for which $\\varphi(k)$ is the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$.","title":"Quadratic subsequence for the pairwise-independent strong law"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/slln-subseq-spec","kind":"node","lean":"ProbabilityTheory.sllnSubseq_spec","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#slln-subseq-spec","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots$ be subsets of $\\Omega$, write $S_n = \\sum_{i < n} 1_{A_i}$, and suppose that $\\mathbb{E}[S_n] = \\int S_n \\, d\\mu \\to \\infty$ as $n \\to \\infty$. If $\\varphi(k)$ is the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$, then $\\mathbb{E}[S_{\\varphi(k)}] \\ge (k+1)^2$ for every natural number $k$.","title":"Lower bound on the expected count along the quadratic subsequence"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/slln-subseq-pos","kind":"node","lean":"ProbabilityTheory.sllnSubseq_pos","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#slln-subseq-pos","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots$ be subsets of $\\Omega$, write $S_n = \\sum_{i < n} 1_{A_i}$, and suppose that $\\mathbb{E}[S_n] = \\int S_n \\, d\\mu \\to \\infty$ as $n \\to \\infty$. If $\\varphi(k)$ is the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$, then $\\varphi(k) > 0$ for every natural number $k$.","title":"Indices of the quadratic subsequence are positive"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/slln-subseq-strict-mono","kind":"node","lean":"ProbabilityTheory.sllnSubseq_strictMono","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#slln-subseq-strict-mono","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots$ be measurable subsets of $\\Omega$, write $S_n = \\sum_{i < n} 1_{A_i}$, and suppose that $\\mathbb{E}[S_n] = \\int S_n \\, d\\mu \\to \\infty$ as $n \\to \\infty$. If $\\varphi(k)$ is the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$, then $\\varphi : \\mathbb{N} \\to \\mathbb{N}$ is strictly increasing.","title":"The quadratic subsequence is strictly increasing"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/es-slln-subseq-pos","kind":"node","lean":"ProbabilityTheory.ES_sllnSubseq_pos","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#es-slln-subseq-pos","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots \\subseteq \\Omega$, let $S_n = \\sum_{i\\lt n} 1_{A_i}$, and suppose that $\\mathbb{E}[S_n] \\to \\infty$. For $k \\in \\mathbb{N}$ let $\\varphi(k)$ be the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$. Then $\\mathbb{E}[S_{\\varphi(k)}] > 0$ for every $k \\in \\mathbb{N}$.","title":"The expected partial sums along the subsequence are positive"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/mem-lp-partial-sum-indicators","kind":"node","lean":"ProbabilityTheory.memLp_partialSumIndicators","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#mem-lp-partial-sum-indicators","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure and let $A_0, A_1, \\dots$ be measurable sets. For every $n$, the partial sum $S_n = \\sum_{i < n} 1_{A_i}$ lies in $L^2(\\mu)$.","title":"Partial sums of indicators are square-integrable"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/ae-tendsto-of-ae-eventually-abs-sub-lt","kind":"node","lean":"ProbabilityTheory.ae_tendsto_of_ae_eventually_abs_sub_lt","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#ae-tendsto-of-ae-eventually-abs-sub-lt","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $f_k : \\Omega \\to \\mathbb{R}$ for $k \\in \\mathbb{N}$, and let $a \\in \\mathbb{R}$. Suppose that for every $m \\in \\mathbb{N}$, for $\\mu$-almost every $\\omega$, we have $|f_k(\\omega) - a| < 1/(m+1)$ for all sufficiently large $k$. Then for $\\mu$-almost every $\\omega$, $f_k(\\omega) \\to a$ as $k \\to \\infty$.","title":"Almost sure convergence from shrinking tail bounds"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/chebyshev-subseq-bound","kind":"node","lean":"ProbabilityTheory.chebyshev_subseq_bound","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#chebyshev-subseq-bound","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots \\subseteq \\Omega$ be pairwise independent measurable sets, let $S_n = \\sum_{i\\lt n} 1_{A_i}$, and suppose that $\\mathbb{E}[S_n] \\to \\infty$. For $k \\in \\mathbb{N}$ let $\\varphi(k)$ be the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$. Then for every $\\varepsilon > 0$ and every $k \\in \\mathbb{N}$, the probability of the set of $\\omega$ with\u2026","title":"Chebyshev bound along the subsequence"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/summable-chebyshev-subseq","kind":"node","lean":"ProbabilityTheory.summable_chebyshev_subseq","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#summable-chebyshev-subseq","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots$ be pairwise independent measurable subsets of $\\Omega$, write $S_n = \\sum_{i < n} 1_{A_i}$, and suppose that $\\mathbb{E}[S_n] = \\int S_n \\, d\\mu \\to \\infty$ as $n \\to \\infty$. Let $\\varphi(k)$ be the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$. Then for every real $\\varepsilon > 0$, the sum over $k \\in \\mathbb{N}$ of the probabilities\u2026","title":"Summable deviation probabilities along the quadratic subsequence"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/partial-sum-indicators-mono","kind":"node","lean":"ProbabilityTheory.partialSumIndicators_mono","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#partial-sum-indicators-mono","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"For sets $A_0, A_1, \\dots$ and every $\\omega \\in \\Omega$, the map $n \\mapsto S_n(\\omega)$, where $S_n = \\sum_{i < n} 1_{A_i}$, is nondecreasing.","title":"Partial sums of indicators are nondecreasing"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/partial-sum-indicators-nonneg","kind":"node","lean":"ProbabilityTheory.partialSumIndicators_nonneg","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#partial-sum-indicators-nonneg","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"For sets $A_0, A_1, \\dots$, every $n$ and every $\\omega \\in \\Omega$, the partial sum $S_n(\\omega) = \\sum_{i < n} 1_{A_i}(\\omega)$ satisfies $S_n(\\omega) \\ge 0$.","title":"Partial sums of indicators are nonnegative"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/exists-bracket-of-strict-mono","kind":"node","lean":"ProbabilityTheory.exists_bracket_of_strictMono","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#exists-bracket-of-strict-mono","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\varphi : \\mathbb{N} \\to \\mathbb{N}$ be strictly increasing and let $n \\ge \\varphi(0)$. Then there is $k \\in \\mathbb{N}$ with $\\varphi(k) \\le n < \\varphi(k+1)$.","title":"Bracketing by a strictly increasing sequence"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/es-slln-subseq-upper-bound","kind":"node","lean":"ProbabilityTheory.ES_sllnSubseq_upper_bound","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#es-slln-subseq-upper-bound","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots \\subseteq \\Omega$ be measurable sets, let $S_n = \\sum_{i\\lt n} 1_{A_i}$, and suppose that $\\mathbb{E}[S_n] \\to \\infty$. For $k \\in \\mathbb{N}$ let $\\varphi(k)$ be the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$. Then $\\mathbb{E}[S_{\\varphi(k)}] < (k+1)^2 + 1$ for every $k \\in \\mathbb{N}$.","title":"Upper bound for the expected partial sums along the subsequence"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/es-subseq-ratio-tendsto-one","kind":"node","lean":"ProbabilityTheory.ES_subseq_ratio_tendsto_one","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#es-subseq-ratio-tendsto-one","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $A_0, A_1, \\dots \\subseteq \\Omega$ be measurable sets, let $S_n = \\sum_{i\\lt n} 1_{A_i}$, and suppose that $\\mathbb{E}[S_n] \\to \\infty$. For $k \\in \\mathbb{N}$ let $\\varphi(k)$ be the least $n$ with $\\mathbb{E}[S_n] \\ge (k+1)^2$. Then $\\mathbb{E}[S_{\\varphi(k+1)}] / \\mathbb{E}[S_{\\varphi(k)}] \\to 1$ as $k \\to \\infty$.","title":"Ratios of consecutive subsequence expectations tend to one"},{"id":"Probability/laws-of-large-numbers/strong-law-of-large-numbers/pairwise-indep-slln","kind":"node","lean":"ProbabilityTheory.pairwise_indep_slln","parent":19432,"result":true,"route":"../roadmap/Probability/laws-of-large-numbers/strong-law-of-large-numbers/index.html#pairwise-indep-slln","scope":"Probability/laws-of-large-numbers/strong-law-of-large-numbers","status":"fully_proved","summary":"Let $A_1, A_2, \\dots$ be pairwise independent events in a probability space with $\\sum_n P(A_n) = \\infty$, and write $S_n = \\sum_{i=1}^n 1_{A_i}$. Then almost surely $S_n / E[S_n] \\to 1$ as $n \\to \\infty$, where $E[S_n] = \\sum_{i=1}^n P(A_i)$.","title":"Pairwise-independent strong law for indicator sums"},{"data_url":"scopes/Probability/central-limit-theorems.json","id":"Probability/central-limit-theorems","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/central-limit-theorems/index.html","scope":"Probability/central-limit-theorems","status":"can_prove","summary":"Convergence in law of normalized sums and their error terms: central limit theorems (Lindeberg\u2013Feller, lacunary and infinite-variance variants), Berry\u2013Esseen bounds and constants, global and local de Moivre\u2013Laplace theorems, Edgeworth expansions, Donsker's invariance principle, and Poisson limit theorems for binomial and urn models.","title":"Central limit theorems and Poisson approximation"},{"id":"Probability/central-limit-theorems/urn-converse-poisson","kind":"node","lean":"MathlibExt.Probability.ScottishBook17Wanted.urnConversePoisson","parent":19460,"result":true,"route":"../roadmap/Probability/central-limit-theorems/index.html#urn-converse-poisson","scope":"Probability/central-limit-theorems","status":"can_prove","summary":"Let $(\\Omega, \\mu)$ be a probability space, let $(p_n)_{n \\ge 0}$ be real numbers in $[0, 1]$, and let $(X_n)_{n \\ge 0}$ be pairwise independent measurable real random variables on $\\Omega$ such that each $X_n$ almost surely takes only the values $0$ and $1$, and $\\mu(X_n = 1) = p_n$. Then for every real number $L$ and almost every $\\omega \\in \\Omega$: if $\\frac{1}{n} \\sum_{i=0}^{n-1} X_i(\\omega) \\to L$ as\u2026","title":"Converse of Poisson's theorem for urns"},{"id":"Probability/central-limit-theorems/khintchine-infinite-variance-limit","kind":"node","lean":"MathlibExt.Probability.ScottishBook126Wanted.khintchine_infinite_variance_limit","parent":19460,"result":true,"route":"../roadmap/Probability/central-limit-theorems/index.html#khintchine-infinite-variance-limit","scope":"Probability/central-limit-theorems","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a measurable function that is integrable on $[0, 1]$ and satisfies $\\int_0^1 f(x)\\,dx = 0$ and $\\int_0^1 f(x)^2\\,dx = \\infty$. Then the complex numbers $\\left(\\int_0^1 \\exp\\left(i f(x) / \\sqrt{n}\\right) dx\\right)^n$ tend to $0$ as $n \\to \\infty$ through the natural numbers.","title":"Kac's Scottish Book Problem 126"},{"id":"Probability/central-limit-theorems/stirling-edgeworth-expansion","kind":"node","lean":"MetaMathlibExt.stirling_edgeworth_expansion","parent":19460,"result":true,"route":"../roadmap/Probability/central-limit-theorems/index.html#stirling-edgeworth-expansion","scope":"Probability/central-limit-theorems","status":"can_prove","summary":"Let $r \\ge 0$ be an integer and let $L \\subseteq (0, \\infty)$ be compact. For $\\theta > 0$ and integers $n \\ge 2$ and $1 \\le k \\le n$, let $s(n, k)$ be the unsigned Stirling number of the first kind (the number of permutations of $n$ elements with exactly $k$ cycles), let $p_n(k; \\theta) = \\theta^k s(n, k) / (\\theta (\\theta + 1) \\cdots (\\theta + n - 1))$ be the probability that an Ewens permutation with parameter\u2026","title":"Local Edgeworth expansion for the Ewens distribution"},{"id":"Probability/central-limit-theorems/tendsto-integral-abs-sum-cube-div-sqrt","kind":"node","lean":"MathlibExt.MeasureTheory.CubeAbsSumIntegralAsymptoticWanted.tendsto_integral_abs_sum_cube_div_sqrt","parent":19460,"result":true,"route":"../roadmap/Probability/central-limit-theorems/index.html#tendsto-integral-abs-sum-cube-div-sqrt","scope":"Probability/central-limit-theorems","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let","title":"Mean absolute sum over the unit cube"},{"data_url":"scopes/Probability/central-limit-theorems/berry-esseen-theorem.json","id":"Probability/central-limit-theorems/berry-esseen-theorem","kind":"scope","navigation":"module","parent":19460,"result":false,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","title":"Berry\u2013Esseen theorem"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/berry-esseen-finset-sum-gaussian","kind":"node","lean":"MathlibExt.Probability.BerryEsseenWanted.berry_esseen_finsetSum_gaussian","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#berry-esseen-finset-sum-gaussian","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $(\\xi_j)_{j \\in \\mathbb{N}}$ be measurable, mutually independent real random variables on a probability space, and let $s$ be a finite set of indices such that for each $j \\in s$ the variable $\\xi_j$ has a finite third moment and mean $\\mathbb{E}[\\xi_j] = 0$, and such that $\\sum_{j \\in s} \\mathbb{E}[\\xi_j^2] = 1$. Write $\\Phi$ for the distribution function of the standard normal distribution. If $\\gamma \\ge 0$\u2026","title":"Berry\u2013Esseen bound for finite sums"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/berry-esseen-explicit","kind":"node","lean":"MathlibExt.Probability.BerryEsseenWanted.berry_esseen_explicit","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#berry-esseen-explicit","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $X_0, X_1, X_2, \\dots$ be measurable, mutually independent real random variables on a probability space, each with the same distribution as $X_0$, where $X_0$ has a finite third moment, mean $\\mathbb{E}[X_0] = 0$ and variance $\\sigma^2 = \\operatorname{Var}(X_0) > 0$ with $\\sigma > 0$. Write $\\rho = \\mathbb{E}|X_0|^3$, $S_n = X_0 + \\dots + X_{n-1}$, and $\\Phi$ for the distribution function of the standard normal\u2026","title":"Berry\u2013Esseen theorem with constant 42"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/berry-esseen","kind":"node","lean":"MathlibExt.Probability.BerryEsseenWanted.berry_esseen","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#berry-esseen","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"There is a constant $C > 0$ with the following property. Let $X_0, X_1, X_2, \\dots$ be measurable, mutually independent real random variables on a probability space, each with the same distribution as $X_0$, where $X_0$ is integrable and has a finite third moment, mean $\\mathbb{E}[X_0] = 0$ and variance $\\sigma^2 = \\operatorname{Var}(X_0) > 0$ with $\\sigma > 0$. Write $\\rho = \\mathbb{E}|X_0|^3$,\u2026","title":"Berry\u2013Esseen theorem"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/is-standardized-law","kind":"node","lean":"BerryEsseenModel.IsStandardizedLaw","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#is-standardized-law","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"A measure $\\mu$ on $\\mathbb{R}$ is a standardized law when it is a probability measure with mean $\\int x \\, d\\mu(x) = 0$ and variance $1$, and has finite third absolute moment $\\int |x|^3 \\, d\\mu(x) < \\infty$.","title":"Standardized laws with finite third moment"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/iid-law","kind":"node","lean":"BerryEsseenModel.iidLaw","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#iid-law","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}$, the i.i.d. product law of $\\mu$ is the measure on the sequence space $\\mathbb{R}^{\\mathbb{N}}$ that is the infinite product $\\mu^{\\otimes \\mathbb{N}}$ of countably many copies of $\\mu$ when $\\mu$ is a probability measure, and the zero measure otherwise.","title":"I.i.d. product law"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/normalized-sum","kind":"node","lean":"BerryEsseenModel.normalizedSum","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#normalized-sum","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For a positive integer $n$ and a real sequence $\\omega = (\\omega_0, \\omega_1, \\dots)$, the normalized sum $S_n(\\omega)$ is the sum of the first $n$ coordinates divided by $\\sqrt{n}$: $S_n(\\omega) = (\\omega_0 + \\dots + \\omega_{n-1})/\\sqrt{n}$.","title":"Normalized sum"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/normalized-law","kind":"node","lean":"BerryEsseenModel.normalizedLaw","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#normalized-law","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}$ and a positive integer $n$, the normalized law of $\\mu$ at $n$ is the pushforward of the i.i.d. law of $\\mu$ on $\\mathbb{R}^{\\mathbb{N}}$ (the product measure $\\mu^{\\otimes \\mathbb{N}}$ when $\\mu$ is a probability measure, and the zero measure otherwise) under the normalized sum $\\omega \\mapsto (\\omega_0 + \\dots + \\omega_{n-1})/\\sqrt{n}$.","title":"Law of the normalized sum"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/normal-cdf","kind":"node","lean":"BerryEsseenModel.normalCDF","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#normal-cdf","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For $x \\in \\mathbb{R}$, $\\Phi(x)$ is the value at $x$ of the cumulative distribution function of the standard normal distribution $\\mathcal{N}(0, 1)$, that is, the probability that a standard normal random variable is at most $x$.","title":"Standard normal CDF"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/third-moment","kind":"node","lean":"BerryEsseenModel.thirdMoment","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#third-moment","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"The third absolute moment of a measure $\\mu$ on $\\mathbb{R}$ is the real number $\\int |x|^3 \\, d\\mu(x)$, taken to be $0$ when $x \\mapsto |x|^3$ is not integrable with respect to $\\mu$.","title":"Third absolute moment"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/mem-lp-three-iff-integrable-abs-cube","kind":"node","lean":"BerryEsseenModel.memLp_three_iff_integrable_absCube","parent":19465,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite third absolute moment as an "],["math","\\(L^3\\)"],["text"," condition"]],"source":"Finite third absolute moment as an $L^3$ condition"},"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#mem-lp-three-iff-integrable-abs-cube","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}$, the identity function $x \\mapsto x$ belongs to $L^3(\\mu)$ if and only if the function $x \\mapsto |x|^3$ is $\\mu$-integrable.","title":"Finite third absolute moment as an $L^3$ condition"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/kolmogorov-error","kind":"node","lean":"BerryEsseenModel.kolmogorovError","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#kolmogorov-error","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}$ and a positive integer $n$, the Kolmogorov error of $\\mu$ at $n$ is the real supremum $\\sup_{x \\in \\mathbb{R}} |F_{\\mu,n}(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal cumulative distribution function and $F_{\\mu,n}$ is Mathlib's cumulative distribution function of the normalized law of $\\mu$ at $n$. When $\\mu$ is a probability measure, $F_{\\mu,n}(x)$ is the mass the\u2026","title":"Kolmogorov error"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/iid-law-probability","kind":"node","lean":"BerryEsseenModel.iidLaw_probability","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#iid-law-probability","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"If $\\mu$ is a standardized law on $\\mathbb{R}$, then the i.i.d. product law of $\\mu$ on $\\mathbb{R}^{\\mathbb{N}}$ is a probability measure.","title":"The i.i.d. product of a standardized law is a probability measure"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/normalized-sum-measurable","kind":"node","lean":"BerryEsseenModel.normalizedSum_measurable","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#normalized-sum-measurable","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For every positive integer $n$, the map sending a real sequence $\\omega = (\\omega_0, \\omega_1, \\dots)$ to $(\\omega_0 + \\dots + \\omega_{n-1})/\\sqrt{n}$ is measurable with respect to the product $\\sigma$-algebra on $\\mathbb{R}^{\\mathbb{N}}$.","title":"Measurability of the normalized sum"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/normalized-law-probability","kind":"node","lean":"BerryEsseenModel.normalizedLaw_probability","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#normalized-law-probability","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For every standardized law $\\mu$ on $\\mathbb{R}$ (a probability measure with mean $0$, variance $1$ and finite third absolute moment) and every positive integer $n$, the normalized law of $\\mu$ at $n$ is a probability measure.","title":"The normalized law is a probability measure"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/iid-law-independent","kind":"node","lean":"BerryEsseenModel.iidLaw_independent","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#iid-law-independent","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"If $\\mu$ is a standardized law on $\\mathbb{R}$, then the coordinate maps $\\omega \\mapsto \\omega_i$ on $\\mathbb{R}^{\\mathbb{N}}$, for $i \\in \\mathbb{N}$, are mutually independent under the i.i.d. product law of $\\mu$.","title":"Coordinates of the i.i.d. product are independent"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/iid-law-map-eval","kind":"node","lean":"BerryEsseenModel.iidLaw_map_eval","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#iid-law-map-eval","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"If $\\mu$ is a standardized law on $\\mathbb{R}$ and $i$ is a natural number, then the pushforward of the i.i.d. product law of $\\mu$ under the coordinate map $\\omega \\mapsto \\omega_i$ from $\\mathbb{R}^{\\mathbb{N}}$ to $\\mathbb{R}$ is $\\mu$.","title":"Coordinate laws of the i.i.d. product"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/iid-law-ident-distrib","kind":"node","lean":"BerryEsseenModel.iidLaw_identDistrib","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#iid-law-ident-distrib","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $\\mu$ be a standardized law on $\\mathbb{R}$ and let $i$ be a natural number. The coordinate map $\\omega \\mapsto \\omega_i$ on $\\mathbb{R}^{\\mathbb{N}}$ under the i.i.d. product law of $\\mu$ and the identity map of $\\mathbb{R}$ under $\\mu$ are identically distributed.","title":"Coordinates of the i.i.d. product are distributed like the original law"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/normalized-cdf-eq","kind":"node","lean":"BerryEsseenModel.normalized_cdf_eq","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#normalized-cdf-eq","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $\\mu$ be a standardized law on $\\mathbb{R}$ (a probability measure with mean $0$, variance $1$ and finite third absolute moment), let $n$ be a positive integer and let $x \\in \\mathbb{R}$. Then the cumulative distribution function of the normalized law of $\\mu$ at $n$, evaluated at $x$, equals the probability under the i.i.d. law $\\mu^{\\otimes \\mathbb{N}}$ on $\\mathbb{R}^{\\mathbb{N}}$ of the event\u2026","title":"CDF of the normalized sum"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/third-moment-integrable","kind":"node","lean":"BerryEsseenModel.thirdMoment_integrable","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#third-moment-integrable","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"If $\\mu$ is a standardized law on $\\mathbb{R}$, then the function $x \\mapsto |x|^3$ is integrable with respect to $\\mu$.","title":"Standardized laws have an integrable third absolute moment"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/third-moment-nonnegative","kind":"node","lean":"BerryEsseenModel.thirdMoment_nonnegative","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#third-moment-nonnegative","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}$, the third absolute moment $\\int_{\\mathbb{R}} |x|^3 \\, d\\mu(x)$ is nonnegative.","title":"Nonnegativity of the third absolute moment"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/third-moment-positive","kind":"node","lean":"BerryEsseenModel.thirdMoment_positive","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#third-moment-positive","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"If $\\mu$ is a standardized law on $\\mathbb{R}$, that is, a probability measure with mean $0$, variance $1$ and finite third absolute moment, then its third absolute moment $\\int |x|^3 \\, d\\mu(x)$ is strictly positive.","title":"The third absolute moment of a standardized law is positive"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/pointwise-error-le-one","kind":"node","lean":"BerryEsseenModel.pointwise_error_le_one","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#pointwise-error-le-one","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}$, $n$ a positive integer and $x$ a real number. Let $F_{\\mu,n}$ be the distribution function of the normalized law of $\\mu$ at $n$ and $\\Phi$ the standard normal distribution function. Then $|F_{\\mu,n}(x) - \\Phi(x)| \\le 1$.","title":"The CDF error is at most one"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/error-range-bdd-above","kind":"node","lean":"BerryEsseenModel.errorRange_bddAbove","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#error-range-bdd-above","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}$, let $n$ be a positive integer, let $F$ be the cumulative distribution function of the normalized law of $\\mu$ at $n$, and let $\\Phi$ be the standard normal cumulative distribution function. Then the set $\\{|F(x) - \\Phi(x)| : x \\in \\mathbb{R}\\}$ is bounded above.","title":"The CDF errors are bounded above"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/pointwise-le-kolmogorov-error","kind":"node","lean":"BerryEsseenModel.pointwise_le_kolmogorovError","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#pointwise-le-kolmogorov-error","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}$, $n$ a positive integer, $F_{\\mu,n}$ the distribution function of the normalized law of $\\mu$ at $n$ and $\\Phi$ the standard normal distribution function. Then for every real $x$, $|F_{\\mu,n}(x) - \\Phi(x)| \\le \\sup_{y \\in \\mathbb{R}} |F_{\\mu,n}(y) - \\Phi(y)|$, the right side being the Kolmogorov error of $\\mu$ at $n$.","title":"Each CDF error is at most the Kolmogorov error"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/kolmogorov-error-nonnegative","kind":"node","lean":"BerryEsseenModel.kolmogorovError_nonnegative","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#kolmogorov-error-nonnegative","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}$ and every positive integer $n$, the Kolmogorov error of $\\mu$ at $n$, that is $\\sup_{x \\in \\mathbb{R}} |F_{\\mu,n}(x) - \\Phi(x)|$ where $F_{\\mu,n}$ is Mathlib's cumulative distribution function of the normalized law of $\\mu$ at $n$ and $\\Phi$ is the standard normal cumulative distribution function, is nonnegative.","title":"The Kolmogorov error is nonnegative"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/kolmogorov-error-le-one","kind":"node","lean":"BerryEsseenModel.kolmogorovError_le_one","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#kolmogorov-error-le-one","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}$ and every positive integer $n$, the Kolmogorov error of $\\mu$ at $n$, that is $\\sup_{x \\in \\mathbb{R}} |F_{\\mu,n}(x) - \\Phi(x)|$ where $F_{\\mu,n}$ is Mathlib's cumulative distribution function of the normalized law of $\\mu$ at $n$ and $\\Phi$ is the standard normal cumulative distribution function, is at most $1$.","title":"The Kolmogorov error is at most one"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/error-le-iff","kind":"node","lean":"BerryEsseenModel.error_le_iff","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#error-le-iff","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}$, let $n$ be a positive integer, let $F$ be the cumulative distribution function of the normalized law of $\\mu$ at $n$, and let $\\Phi$ be the standard normal cumulative distribution function. Then for every real $a$, the Kolmogorov error of $\\mu$ at $n$ is at most $a$ if and only if $|F(x) - \\Phi(x)| \\le a$ for every real $x$.","title":"Kolmogorov error bounds are uniform CDF bounds"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/standard-gaussian-is-standardized","kind":"node","lean":"BerryEsseenModel.standardGaussian_isStandardized","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#standard-gaussian-is-standardized","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"The standard normal law $N(0,1)$ on $\\mathbb{R}$ is a standardized law: it is a probability measure with mean $0$, variance $1$ and finite third absolute moment.","title":"The standard Gaussian is a standardized law"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/normalized-law-one","kind":"node","lean":"BerryEsseenModel.normalizedLaw_one","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#normalized-law-one","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"For every standardized law $\\mu$ on $\\mathbb{R}$ (a probability measure with mean $0$, variance $1$ and finite third absolute moment), the normalized law of $\\mu$ at $n = 1$ is $\\mu$ itself.","title":"The normalized law with one summand"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/standard-gaussian-error-one","kind":"node","lean":"BerryEsseenModel.standardGaussian_error_one","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#standard-gaussian-error-one","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $F$ be the distribution function of the normalized law of the standard normal law $N(0,1)$ at $n = 1$, and $\\Phi$ the standard normal distribution function. Then the Kolmogorov error of $N(0,1)$ at $n = 1$, $\\sup_{x \\in \\mathbb{R}} |F(x) - \\Phi(x)|$, equals $0$.","title":"The Kolmogorov error vanishes for one standard Gaussian summand"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/is-iidmodel","kind":"node","lean":"BerryEsseenModel.IsIIDModel","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#is-iidmodel","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $\\Omega$ be a measurable space with a measure $P$, and let $(X_i)_{i \\in \\mathbb{N}}$ be real-valued functions on $\\Omega$. They form a standardized i.i.d. model when $P$ is a probability measure, every $X_i$ is measurable, the $X_i$ are mutually independent under $P$, every $X_i$ has the same law under $P$ as $X_0$, and $X_0$ has mean $\\int_\\Omega X_0 \\, dP = 0$, variance $1$ and finite third absolute moment\u2026","title":"Standardized i.i.d. models"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/standardized-law","kind":"node","lean":"BerryEsseenModel.IsIIDModel.standardizedLaw","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#standardized-law","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $P$ be a measure on a measurable space $\\Omega$ and $X_0, X_1, \\dots$ real-valued functions on $\\Omega$ forming an i.i.d. model: $P$ is a probability measure, the $X_i$ are measurable, independent and identically distributed, and $X_0$ has mean $0$, variance $1$ and finite third absolute moment. Then the law of $X_0$, the pushforward of $P$ under $X_0$, is a standardized law.","title":"The common law of an i.i.d. model is standardized"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/joint-law","kind":"node","lean":"BerryEsseenModel.IsIIDModel.jointLaw","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#joint-law","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"If $(X_i)_{i \\in \\mathbb{N}}$ is a standardized i.i.d. model on a measurable space $\\Omega$ with measure $P$, then the law under $P$ of the sequence map $\\omega \\mapsto (X_i(\\omega))_{i \\in \\mathbb{N}}$ from $\\Omega$ to $\\mathbb{R}^{\\mathbb{N}}$ equals the i.i.d. product law of the law of $X_0$ under $P$.","title":"The joint law of a standardized i.i.d. model"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/normalized-law-eq","kind":"node","lean":"BerryEsseenModel.IsIIDModel.normalizedLaw_eq","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#normalized-law-eq","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $P$ be a probability measure on a measurable space $\\Omega$ and let $X_0, X_1, \\dots$ be real random variables on $\\Omega$ that are measurable, independent and identically distributed under $P$, with $X_0$ of mean $0$, variance $1$ and finite third absolute moment. Then for every positive integer $n$, the law under $P$ of $(X_0 + \\dots + X_{n-1})/\\sqrt{n}$ equals the normalized law at $n$ of the law of $X_0$.","title":"Normalized sums of an i.i.d. model have the canonical law"},{"id":"Probability/central-limit-theorems/berry-esseen-theorem/third-moment-eq","kind":"node","lean":"BerryEsseenModel.IsIIDModel.thirdMoment_eq","parent":19465,"result":true,"route":"../roadmap/Probability/central-limit-theorems/berry-esseen-theorem/index.html#third-moment-eq","scope":"Probability/central-limit-theorems/berry-esseen-theorem","status":"fully_proved","summary":"Let $P$ be a measure on a measurable space $\\Omega$ and $X_0, X_1, \\dots$ real-valued functions on $\\Omega$ forming an i.i.d. model: $P$ is a probability measure, the $X_i$ are measurable, independent and identically distributed, and $X_0$ has mean $0$, variance $1$ and finite third absolute moment. Then the third absolute moment of the law of $X_0$ equals $\\int_\\Omega |X_0(\\omega)|^3 \\, dP(\\omega)$.","title":"Third absolute moment of the common law"},{"data_url":"scopes/Probability/central-limit-theorems/de-moivre-laplace-theorem.json","id":"Probability/central-limit-theorems/de-moivre-laplace-theorem","kind":"scope","navigation":"module","parent":19460,"result":false,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","title":"De Moivre\u2013Laplace theorem"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/standardized-binomial","kind":"node","lean":"MathlibExt.Probability.CharFunStandardizedBinomial.standardizedBinomial","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#standardized-binomial","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"For a natural number $n$ and $p \\in [0, 1]$, the standardized binomial distribution is the measure on $\\mathbb{R}$ given by the law of $(S_n - np)/\\sqrt{np(1-p)}$, where $S_n$ has the binomial distribution with parameters $n$ and $p$. When $np(1-p) = 0$ the division by $0$ is read as $0$, so the law is the point mass at $0$.","title":"Standardized binomial distribution"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/char-fun-map-affine","kind":"node","lean":"MathlibExt.Probability.CharFunStandardizedBinomial.charFun_map_affine","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#char-fun-map-affine","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}$, let $a, s, t \\in \\mathbb{R}$, and let $\\nu$ be the image of $\\mu$ under the map $x \\mapsto (x - a)/s$. Then the characteristic functions satisfy $\\varphi_\\nu(t) = \\varphi_\\mu(t s^{-1}) \\, e^{-i a t s^{-1}}$. Here $s^{-1}$ and division by $s$ are read as $0$ when $s = 0$, so no hypothesis $s \\neq 0$ is needed.","title":"Affine transformation of a characteristic function"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/char-fun-binomial-real","kind":"node","lean":"MathlibExt.Probability.CharFunStandardizedBinomial.charFun_binomial_real","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#char-fun-binomial-real","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"For every natural number $n$, every $p \\in [0, 1]$ and every real $t$, the characteristic function $\\varphi_{\\mathrm{Bin}(n, p)}$ of the binomial distribution $\\mathrm{Bin}(n, p)$, viewed as a measure on $\\mathbb{R}$, satisfies $\\varphi_{\\mathrm{Bin}(n, p)}(t) = \\left((1 - p) + p e^{it}\\right)^n$.","title":"Characteristic function of the real-valued binomial distribution"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/exp-taylor2-err","kind":"node","lean":"MathlibExt.Probability.CharFunStandardizedBinomial.exp_taylor2_err","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#exp-taylor2-err","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"For every complex number $z$ with $|z| \\le 1$, the remainder after the degree-2 Taylor polynomial of the exponential satisfies $|e^z - 1 - z - z^2/2| \\le 2|z|^3$.","title":"Second-order Taylor remainder bound for the exponential"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/char-fun-standardized-binomial","kind":"node","lean":"MathlibExt.Probability.CharFunStandardizedBinomial.charFun_standardizedBinomial","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#char-fun-standardized-binomial","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"For every natural number $n$, every $p \\in [0, 1]$ and every real $t$, put $s = \\sqrt{np(1-p)}$, with $s^{-1}$ read as $0$ when $s = 0$. Let $\\psi$ be the characteristic function of the binomial distribution $\\mathrm{Bin}(n, p)$ viewed as a measure on $\\mathbb{R}$, and $\\varphi$ that of the standardized binomial distribution with parameters $n$ and $p$. Then $\\varphi(t) = \\psi(t s^{-1}) \\, e^{-i n p t s^{-1}}$.","title":"Characteristic function of the standardized binomial"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/tendsto-char-fun-standardized-binomial","kind":"node","lean":"MathlibExt.Probability.CharFunStandardizedBinomial.tendsto_charFun_standardizedBinomial","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#tendsto-char-fun-standardized-binomial","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"Let $0 < p < 1$ and $t \\in \\mathbb{R}$. As $n \\to \\infty$, the characteristic function of the standardized binomial distribution with parameters $n$ and $p$, evaluated at $t$, converges to $e^{-t^2/2}$, the value at $t$ of the characteristic function of the standard normal distribution.","title":"Convergence of characteristic functions in De Moivre\u2013Laplace"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/de-moivre-laplace-standardized","kind":"node","lean":"ProbabilityTheory.deMoivreLaplaceStandardized","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#de-moivre-laplace-standardized","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"For a sequence $X_0, X_1, \\dots$ of real-valued functions on a set $\\Omega$, a real number $p$ and $n \\in \\mathbb{N}$, the standardized partial sum is the function $\\omega \\mapsto (S_n(\\omega) - np)/\\sqrt{np(1-p)}$ on $\\Omega$, where $S_n = X_0 + \\dots + X_{n-1}$.","title":"Standardized partial sums"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/standardized-bernoulli","kind":"node","lean":"ProbabilityTheory.standardizedBernoulli","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#standardized-bernoulli","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"For a real number $p$, the standardization map sends $x \\in \\mathbb{R}$ to $(x - p)/\\sqrt{p(1-p)}$.","title":"Bernoulli standardization map"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/integrable-of-ae-eq-zero-or-one","kind":"node","lean":"ProbabilityTheory.integrable_of_ae_eq_zero_or_one","parent":19501,"reading_title":{"format":"inline-parts-v1","parts":[["text","Almost surely "],["math","\\(\\{0, 1\\}\\)"],["text","-valued random variables are integrable"]],"source":"Almost surely $\\{0, 1\\}$-valued random variables are integrable"},"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#integrable-of-ae-eq-zero-or-one","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"Let $P$ be a probability measure on $\\Omega$ and let $f : \\Omega \\to \\mathbb{R}$ be almost everywhere measurable with $f(\\omega) \\in \\{0, 1\\}$ for $P$-almost every $\\omega$. Then $f$ is integrable with respect to $P$.","title":"Almost surely $\\{0, 1\\}$-valued random variables are integrable"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/integral-sq-eq-integral-of-ae-eq-zero-or","kind":"node","lean":"ProbabilityTheory.integral_sq_eq_integral_of_ae_eq_zero_or_one","parent":19501,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second moment of a "],["math","\\(\\{0, 1\\}\\)"],["text","-valued random variable"]],"source":"Second moment of a $\\{0, 1\\}$-valued random variable"},"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#integral-sq-eq-integral-of-ae-eq-zero-or","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"Let $P$ be a probability measure on $\\Omega$ and let $f : \\Omega \\to \\mathbb{R}$ satisfy $f(\\omega) \\in \\{0, 1\\}$ for $P$-almost every $\\omega$. Then $\\int f^2 \\, dP = \\int f \\, dP$.","title":"Second moment of a $\\{0, 1\\}$-valued random variable"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/de-moivre-laplace-tendsto-in-distributio","kind":"node","lean":"ProbabilityTheory.deMoivreLaplace_tendstoInDistribution","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#de-moivre-laplace-tendsto-in-distributio","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent, identically distributed real random variables on a probability space $(\\Omega, P)$, each almost surely equal to $0$ or $1$, with mean $p$ for some $0 < p < 1$. Write $S_n = X_0 + \\dots + X_{n-1}$. Then the standardized sums $(S_n - np)/\\sqrt{np(1-p)}$ converge in distribution to the standard normal distribution $\\mathcal{N}(0, 1)$ as $n \\to \\infty$.","title":"De Moivre\u2013Laplace limit theorem, convergence in distribution"},{"id":"Probability/central-limit-theorems/de-moivre-laplace-theorem/de-moivre-laplace-interval-probability","kind":"node","lean":"ProbabilityTheory.deMoivreLaplace_interval_probability","parent":19501,"result":true,"route":"../roadmap/Probability/central-limit-theorems/de-moivre-laplace-theorem/index.html#de-moivre-laplace-interval-probability","scope":"Probability/central-limit-theorems/de-moivre-laplace-theorem","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent, identically distributed real random variables on a probability space $(\\Omega, P)$, each almost surely equal to $0$ or $1$, with mean $p$ for some $0 < p < 1$. Write $S_n = X_0 + \\dots + X_{n-1}$. Then for all real numbers $a \\le b$, the probability $P\\left(a \\le (S_n - np)/\\sqrt{np(1-p)} \\le b\\right)$ tends, as $n \\to \\infty$, to the mass that the standard normal distribution\u2026","title":"De Moivre\u2013Laplace limit theorem, interval form"},{"data_url":"scopes/Probability/central-limit-theorems/donsker-s-invariance-principle.json","id":"Probability/central-limit-theorems/donsker-s-invariance-principle","kind":"scope","navigation":"module","parent":19460,"result":false,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","title":"Donsker's invariance principle"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/donsker-clamp","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.donskerClamp","parent":19514,"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#donsker-clamp","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"For a real number $x$, its clamp to $[0,1]$ is $\\min(\\max(x, 0), 1)$.","title":"Clamping to the unit interval"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/donsker-weight","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.donskerWeight","parent":19514,"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#donsker-weight","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"For natural numbers $n$ and $i$ and $t \\in [0,1]$, the weight is $w_{n,i}(t) = \\min(\\max(nt - i, 0), 1)$, the clamp of $nt - i$ to $[0,1]$.","title":"Polygonal interpolation weight"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/donsker-weight-cm","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.donskerWeightCM","parent":19514,"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#donsker-weight-cm","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"For natural numbers $n$ and $i$, this is the function $t \\mapsto \\min(\\max(nt - i, 0), 1)$ on $[0,1]$, regarded as an element of the space $C([0,1], \\mathbb{R})$ of continuous real functions on $[0,1]$.","title":"Polygonal interpolation weight as a continuous function"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/donsker-process","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.donskerProcess","parent":19514,"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#donsker-process","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$. For $n \\in \\mathbb{N}$ and $\\omega \\in \\Omega$, $S_n(\\omega)$ is the element of the space $C([0,1], \\mathbb{R})$ of continuous real functions on $[0,1]$ defined by $S_0(\\omega) = 0$ and, for $n \\ge 1$ and $t \\in [0,1]$,","title":"Polygonal partial-sum process"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/donsker-process-apply-eq","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.donskerProcess_apply_eq","parent":19514,"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#donsker-process-apply-eq","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$ and let $S_n$ be their polygonal partial-sum process. For $n \\ge 1$, $\\omega \\in \\Omega$ and $t \\in [0,1]$, write $m = \\lfloor nt \\rfloor$. Then","title":"Explicit formula for the polygonal partial-sum process"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/icc-to-nnreal","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.iccToNNReal","parent":19514,"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#icc-to-nnreal","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"This map sends $t \\in [0,1]$ to $t$ regarded as a nonnegative real number.","title":"Inclusion of the unit interval into the nonnegative reals"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/continuous-map-measure-ext-of-finset-eva","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.continuousMap_measure_ext_of_finset_eval","parent":19514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite-dimensional distributions determine laws on "],["math","\\(C((0,1))\\)"]],"source":"Finite-dimensional distributions determine laws on $C((0,1))$"},"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#continuous-map-measure-ext-of-finset-eva","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"Probability measures on `C([0,1], \u211d)` are determined by their finite-dimensional distributions. Compatibility wrapper around `MathlibExt.Probability.ContinuousPathConvergence.continuousPath_measure_ext_of_finset_eval` at `T = 1`; the projective-limit argument lives there, not here.","title":"Finite-dimensional distributions determine laws on $C((0,1))$"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/etemadi-maximal-inequality","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.etemadi_maximal_inequality","parent":19514,"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#etemadi-maximal-inequality","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent real random variables on a probability space $(\\Omega, P)$ and write $S_k = X_0 + \\dots + X_{k-1}$, so that $S_0 = 0$. Then for every natural number $n$ and every real number $a$,","title":"Etemadi's maximal inequality"},{"id":"Probability/central-limit-theorems/donsker-s-invariance-principle/donsker-invariance-principle","kind":"node","lean":"MathlibExt.Probability.DonskerWanted.donsker_invariance_principle","parent":19514,"result":true,"route":"../roadmap/Probability/central-limit-theorems/donsker-s-invariance-principle/index.html#donsker-invariance-principle","scope":"Probability/central-limit-theorems/donsker-s-invariance-principle","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent, identically distributed, measurable real random variables on a probability space $(\\Omega, P)$ such that $X_0$ is integrable, $\\mathbb{E}[X_0^2] < \\infty$, $\\mathbb{E}[X_0] = 0$ and $\\operatorname{Var}(X_0) = 1$. Let $(B_t)_{t \\ge 0}$ be a real Brownian motion on a probability space $(\\Omega', P')$: its finite-dimensional laws are those of Brownian motion and its paths are\u2026","title":"Donsker's invariance principle"},{"data_url":"scopes/Probability/central-limit-theorems/lacunary-central-limit-theorem.json","id":"Probability/central-limit-theorems/lacunary-central-limit-theorem","kind":"scope","navigation":"module","parent":19460,"result":false,"route":"../roadmap/Probability/central-limit-theorems/lacunary-central-limit-theorem/index.html","scope":"Probability/central-limit-theorems/lacunary-central-limit-theorem","status":"can_prove","title":"Lacunary central limit theorem"},{"id":"Probability/central-limit-theorems/lacunary-central-limit-theorem/lacunary-clt","kind":"node","lean":"MathlibExt.Probability.ScottishBook161Wanted.LacunaryCLT","parent":19524,"result":true,"route":"../roadmap/Probability/central-limit-theorems/lacunary-central-limit-theorem/index.html#lacunary-clt","scope":"Probability/central-limit-theorems/lacunary-central-limit-theorem","status":"fully_proved","summary":"A sequence of integers $(r_k)_{k \\ge 0}$ satisfies the central limit theorem for lacunary sine sums if for all real numbers $a < b$, the Lebesgue measure of the set of $x \\in [0, 1]$ with $a < \\frac{1}{\\sqrt{n}} \\sum_{k=0}^{n-1} \\sin(2 \\pi r_k x) < b$ tends to $\\frac{1}{\\sqrt{\\pi}} \\int_a^b e^{-y^2} \\, dy$ as $n \\to \\infty$.","title":"Central limit theorem for lacunary sine sums"},{"id":"Probability/central-limit-theorems/lacunary-central-limit-theorem/kac-lacunary-central-limit-theorem","kind":"node","lean":"MathlibExt.Probability.ScottishBook161Wanted.kacLacunaryCentralLimitTheorem","parent":19524,"result":true,"route":"../roadmap/Probability/central-limit-theorems/lacunary-central-limit-theorem/index.html#kac-lacunary-central-limit-theorem","scope":"Probability/central-limit-theorems/lacunary-central-limit-theorem","status":"can_prove","summary":"Let $(r_n)_{n \\ge 0}$ be a sequence of integers such that $r_n - (r_0 + r_1 + \\dots + r_{n-1})$ tends to $+\\infty$ as $n \\to \\infty$. Then $(r_n)$ satisfies the central limit theorem for lacunary sine sums. The sequence $r_n = 2^n$ satisfies it as well.","title":"Kac's lacunary central limit theorem"},{"data_url":"scopes/Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem.json","id":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem","kind":"scope","navigation":"module","parent":19460,"result":false,"route":"../roadmap/Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/index.html","scope":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem","status":"fully_proved","title":"Lindeberg\u2013Feller central limit theorem"},{"id":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/lindeberg-condition-triangular","kind":"node","lean":"ProbabilityTheory.LindebergConditionTriangular","parent":19527,"result":true,"route":"../roadmap/Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/index.html#lindeberg-condition-triangular","scope":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem","status":"fully_proved","summary":"A triangular array of real random variables $X_{n,k}$ ($n \\in \\mathbb{N}$, $0 \\le k < n$) on a measure space $(\\Omega, \\mu)$ satisfies the Lindeberg condition if for every $\\varepsilon > 0$ the truncated second moments","title":"Lindeberg condition for a triangular array"},{"id":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/norm-exp-of-real-mul-i-sub-one-sub-le","kind":"node","lean":"ProbabilityTheory.norm_exp_ofReal_mul_I_sub_one_sub_le","parent":19527,"result":true,"route":"../roadmap/Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/index.html#norm-exp-of-real-mul-i-sub-one-sub-le","scope":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem","status":"fully_proved","summary":"For every real $x$ with $|x| \\le 1$, $|e^{ix} - 1 - ix| \\le x^2$.","title":"Second-order Taylor bound for the imaginary exponential"},{"id":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/prod-one-add-sub-exp-sum-le","kind":"node","lean":"ProbabilityTheory.prod_one_add_sub_exp_sum_le","parent":19527,"result":true,"route":"../roadmap/Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/index.html#prod-one-add-sub-exp-sum-le","scope":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem","status":"fully_proved","summary":"Let $s$ be a finite index set and let $u_i \\in \\mathbb{C}$ for $i \\in s$. If $|u_i| \\le 1/2$ for every $i \\in s$, then","title":"Comparing a finite product with the exponential of a sum"},{"id":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/lindeberg-char-fun-product-tendsto","kind":"node","lean":"ProbabilityTheory.lindeberg_charFun_product_tendsto","parent":19527,"result":true,"route":"../roadmap/Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/index.html#lindeberg-char-fun-product-tendsto","scope":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$ and let $X_{n,k}$ ($n \\in \\mathbb{N}$, $0 \\le k < n$) be real random variables on $\\Omega$ such that each $X_{n,k}$ has mean $0$ and each $X_{n,k}^2$ is integrable (expectations are taken under $\\mu$). Suppose that for some $\\sigma > 0$ the row variances $\\sum_{k < n} \\mathbb{E}[X_{n,k}^2]$ converge to $\\sigma^2$ and that the array satisfies the Lindeberg condition.\u2026","title":"Analytic core of the Lindeberg\u2013Feller theorem"},{"id":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/lindeberg-feller-clt-triangular","kind":"node","lean":"ProbabilityTheory.lindeberg_feller_clt_triangular","parent":19527,"result":true,"route":"../roadmap/Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem/index.html#lindeberg-feller-clt-triangular","scope":"Probability/central-limit-theorems/lindeberg-feller-central-limit-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$ and let $X_{n,k}$ ($n \\in \\mathbb{N}$, $0 \\le k < n$) be a triangular array of real random variables on $\\Omega$ such that for each $n$ the variables $X_{n,0}, \\dots, X_{n,n-1}$ are independent, each $X_{n,k}$ has mean $0$, and each $X_{n,k}^2$ is integrable (expectations are taken under $\\mu$). Suppose that for some $\\sigma > 0$ the row variances\u2026","title":"Lindeberg\u2013Feller central limit theorem"},{"data_url":"scopes/Probability/central-limit-theorems/local-de-moivre-laplace-theorem.json","id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","kind":"scope","navigation":"module","parent":19460,"result":false,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","title":"Local de Moivre\u2013Laplace theorem"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/binom-prob-half","kind":"node","lean":"ProbabilityTheory.binomProbHalf","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#binom-prob-half","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The point mass `P(B_n = k) = C(n,k) / 2\u207f` for a `Binomial(n, 1/2)` success count `B_n`.","title":"Binomial point mass with success probability one half"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/gaussian-approx-half","kind":"node","lean":"ProbabilityTheory.gaussianApproxHalf","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#gaussian-approx-half","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The Gaussian (local CLT) approximation to `binomProbHalf n k`, `\u221a(2/(\u03c0 n)) \u00b7 exp(-(k - n/2)\u00b2 / (n/2))`.","title":"Gaussian approximation to the symmetric binomial point mass"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/local-dmlindex","kind":"node","lean":"ProbabilityTheory.localDMLIndex","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#local-dmlindex","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The integer index `k = \u230an/2 + x\u00b7\u221an/2\u230b` corresponding to the parameter `x` in the local DeMoivre\u2013Laplace statement: it picks the lattice point near `n/2 + x \u221an / 2`, so that `(2k - n)/\u221an \u2248 x`.","title":"Lattice index in the local De Moivre\u2013Laplace theorem"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/local-dmlindex-tendsto-at-top","kind":"node","lean":"ProbabilityTheory.localDMLIndex_tendsto_atTop","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#local-dmlindex-tendsto-at-top","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The index `localDMLIndex x n` tends to infinity as `n \u2192 \u221e`, for any fixed `x : \u211d`.","title":"The De Moivre\u2013Laplace index tends to infinity"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/n-sub-local-dmlindex-tendsto-at-top","kind":"node","lean":"ProbabilityTheory.n_sub_localDMLIndex_tendsto_atTop","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#n-sub-local-dmlindex-tendsto-at-top","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The complementary index `n - localDMLIndex x n` also tends to infinity, so both `k` and `n - k` are eventually arbitrarily large.","title":"The complementary De Moivre\u2013Laplace index tends to infinity"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/local-dmlindex-ratio-half","kind":"node","lean":"ProbabilityTheory.localDMLIndex_ratio_half","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#local-dmlindex-ratio-half","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The ratio `localDMLIndex x n / n` tends to `1/2` as `n \u2192 \u221e`, because the correction `x \u221an / 2` is `o(n)`.","title":"The De Moivre\u2013Laplace index over n tends to one half"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/factorial-eq-stirling-seq-mul","kind":"node","lean":"ProbabilityTheory.factorial_eq_stirlingSeq_mul","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#factorial-eq-stirling-seq-mul","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"Rewriting of `m!` in terms of Mathlib's `Stirling.stirlingSeq`: `m! = stirlingSeq(m) \u00b7 \u221a(2m) \u00b7 (m/e)^m`.","title":"Factorial in terms of the Stirling sequence"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/binom-prob-div-gaussian-approx-eq","kind":"node","lean":"ProbabilityTheory.binomProb_div_gaussianApprox_eq","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#binom-prob-div-gaussian-approx-eq","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"Algebraic rewriting of the ratio `binomProbHalf n k / gaussianApproxHalf n k` using Stirling's formula, separating it into a Stirling-sequence factor, a square-root factor, a power-of-`n` factor, and an exponential correction.","title":"Stirling decomposition of the binomial-to-Gaussian ratio"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/entropy-residual-bound","kind":"node","lean":"ProbabilityTheory.entropy_residual_bound","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#entropy-residual-bound","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"Third-order Taylor bound for the symmetric entropy expansion: `|(1+u) log(1+u) + (1-u) log(1-u) - u\u00b2| \u2264 6 |u|\u00b3` whenever `|u| < 1/2`. Used to control the deviation of the binomial coefficient from its Gaussian approximation.","title":"Cubic bound on the symmetric entropy residual"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/local-dmlindex-deviation-bound","kind":"node","lean":"ProbabilityTheory.localDMLIndex_deviation_bound","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#local-dmlindex-deviation-bound","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"Eventually, the deviation `|2k - n|` for `k = localDMLIndex x n` is at most `|x| \u221an + 2`, since `k = \u230an/2 + x \u221an / 2\u230b`.","title":"Deviation bound for the De Moivre\u2013Laplace index"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/log-power-factor-formula","kind":"node","lean":"ProbabilityTheory.log_power_factor_formula","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#log-power-factor-formula","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"Identity rewriting the logarithm of the power factor `n\u207f / (k\u1d4f (n-k)^{n-k} 2\u207f)` plus the Gaussian exponent `(k - n/2)\u00b2 / (n/2)` as `-(n/2) \u00b7 ((1+u) log(1+u) + (1-u) log(1-u) - u\u00b2)`, where `u = (2k - n)/n`. This combines the log-binomial expansion with the Gaussian correction.","title":"Logarithm of the binomial power factor"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/binom-prob-half-is-equivalent-fixed-gaus","kind":"node","lean":"ProbabilityTheory.binomProbHalf_isEquivalent_fixedGaussian","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#binom-prob-half-is-equivalent-fixed-gaus","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"For any binomial success-index sequence whose standardized displacement converges to `x`, the point mass is equivalent to the Gaussian expression evaluated at the fixed limit `x`. The number of trials may follow any natural-valued sequence tending to infinity.","title":"Binomial point mass is equivalent to a fixed Gaussian value"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/local-de-moivre-laplace-half-of-tendsto","kind":"node","lean":"ProbabilityTheory.local_deMoivreLaplace_half_of_tendsto","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#local-de-moivre-laplace-half-of-tendsto","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The source's random-walk local limit theorem: every integer displacement sequence with `2 k\u2099 / \u221a(2n) \u2192 x` has point mass equivalent to `(\u221a(\u03c0 n))\u207b\u00b9 exp(-x\u00b2/2)`. The scaling hypothesis itself implies that `n + k\u2099` is a valid success index eventually.","title":"Local limit theorem for the simple random walk"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/local-dmlindex-standardized-tendsto","kind":"node","lean":"ProbabilityTheory.localDMLIndex_standardized_tendsto","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#local-dmlindex-standardized-tendsto","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The rounded success index has standardized displacement converging to `x`.","title":"Standardized displacement of the rounded index converges"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/dml-power-factor-tendsto-one","kind":"node","lean":"ProbabilityTheory.dml_power_factor_tendsto_one","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#dml-power-factor-tendsto-one","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The product of the power factor `n\u207f / (k\u1d4f (n-k)^{n-k} 2\u207f)` with the Gaussian correction `exp((k - n/2)\u00b2 / (n/2))`, evaluated at `k = localDMLIndex x n`, tends to `1` as `n \u2192 \u221e`. This is the key analytic estimate behind the local DeMoivre\u2013Laplace limit theorem.","title":"The corrected binomial power factor tends to one"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/n-sub-local-dmlindex-ratio-half","kind":"node","lean":"ProbabilityTheory.n_sub_localDMLIndex_ratio_half","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#n-sub-local-dmlindex-ratio-half","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"Companion to `localDMLIndex_ratio_half`: the ratio `(n - k) / n` for `k = localDMLIndex x n` also tends to `1/2`.","title":"The complementary index over n tends to one half"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/binom-prob-half-is-equivalent-gaussian-a","kind":"node","lean":"ProbabilityTheory.binomProbHalf_isEquivalent_gaussianApproxHalf","parent":19533,"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#binom-prob-half-is-equivalent-gaussian-a","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"The binomial point mass and its Gaussian approximation are asymptotically equivalent along the sequence `k = localDMLIndex x n`: `binomProbHalf n k ~ gaussianApproxHalf n k` as `n \u2192 \u221e`. Combines Stirling's formula with the power-factor and entropy estimates.","title":"Binomial point mass is equivalent to its Gaussian approximation"},{"id":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem/local-de-moivre-laplace-half","kind":"node","lean":"ProbabilityTheory.local_deMoivreLaplace_half","parent":19533,"reading_title":{"format":"inline-parts-v1","parts":[["text","Local De Moivre\u2013Laplace theorem for "],["math","\\(p = 1/2\\)"]],"source":"Local De Moivre\u2013Laplace theorem for $p = 1/2$"},"result":true,"route":"../roadmap/Probability/central-limit-theorems/local-de-moivre-laplace-theorem/index.html#local-de-moivre-laplace-half","scope":"Probability/central-limit-theorems/local-de-moivre-laplace-theorem","status":"fully_proved","summary":"Canonical rounded-sequence corollary of the local De Moivre\u2013Laplace theorem: for any `x \u2208 \u211d`, the binomial point mass divided by its moving Gaussian approximation along `localDMLIndex x` tends to `1`. The arbitrary integer-displacement statement with the fixed Gaussian value at `x` is `local_deMoivreLaplace_half_of_tendsto`.","title":"Local De Moivre\u2013Laplace theorem for $p = 1/2$"},{"data_url":"scopes/Probability/central-limit-theorems/optimal-berry-esseen-constant.json","id":"Probability/central-limit-theorems/optimal-berry-esseen-constant","kind":"scope","navigation":"module","parent":19460,"result":false,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","title":"Optimal Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/is-admissible","kind":"node","lean":"BerryEsseenConstant.IsAdmissible","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#is-admissible","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"A real number $C$ is an admissible Berry\u2013Esseen coefficient when, for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3 \\, d\\mu(x)$, and every positive integer $n$, the Kolmogorov error $\\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$ is at most $C \\rho / \\sqrt{n}$. Here $F_n$ is the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for\u2026","title":"Admissible Berry\u2013Esseen coefficient"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/value","kind":"node","lean":"BerryEsseenConstant.value","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#value","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Call a real number $C$ admissible if for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3 \\, d\\mu$, and every positive integer $n$,","title":"The optimal Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/esseen-constant","kind":"node","lean":"BerryEsseenConstant.esseenConstant","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#esseen-constant","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Esseen's constant is the real number $(\\sqrt{10} + 3)/(6\\sqrt{2\\pi})$.","title":"Esseen's lower-bound constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/esseen-constant-positive","kind":"node","lean":"BerryEsseenConstant.esseenConstant_positive","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#esseen-constant-positive","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The number $(\\sqrt{10} + 3)/(6\\sqrt{2\\pi})$ is positive.","title":"Esseen's lower-bound constant is positive"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/is-admissible-nonnegative","kind":"node","lean":"BerryEsseenConstant.isAdmissible_nonnegative","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#is-admissible-nonnegative","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Every admissible Berry\u2013Esseen coefficient $C \\in \\mathbb{R}$ satisfies $C \\ge 0$.","title":"Admissible coefficients are nonnegative"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/is-admissible-forty-two","kind":"node","lean":"BerryEsseenConstant.isAdmissible_fortyTwo","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#is-admissible-forty-two","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The number $42$ is an admissible Berry\u2013Esseen coefficient: for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3 \\, d\\mu(x)$, and every positive integer $n$, the Kolmogorov error $\\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$ is at most $42 \\rho / \\sqrt{n}$, where $F_n$ is the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for\u2026","title":"The Berry\u2013Esseen bound with coefficient 42"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/admissible-nonempty","kind":"node","lean":"BerryEsseenConstant.admissible_nonempty","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#admissible-nonempty","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The set of real numbers $C$ that are admissible Berry\u2013Esseen coefficients (common Berry\u2013Esseen coefficients for all standardized laws and all positive sample sizes) is nonempty.","title":"Admissible Berry\u2013Esseen coefficients exist"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/admissible-bdd-below","kind":"node","lean":"BerryEsseenConstant.admissible_bddBelow","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#admissible-bdd-below","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The set of real numbers $C$ that are admissible Berry\u2013Esseen coefficients (common Berry\u2013Esseen coefficients for all standardized laws and all positive sample sizes) is bounded below.","title":"The admissible Berry\u2013Esseen coefficients are bounded below"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/value-nonnegative","kind":"node","lean":"BerryEsseenConstant.value_nonnegative","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#value-nonnegative","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The infimum $C^*$ of all admissible Berry\u2013Esseen coefficients satisfies $0 \\le C^*$.","title":"The optimal Berry\u2013Esseen coefficient is nonnegative"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/value-le-of-is-admissible","kind":"node","lean":"BerryEsseenConstant.value_le_of_isAdmissible","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#value-le-of-is-admissible","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"If $C$ is an admissible Berry\u2013Esseen coefficient, then the infimum $C^*$ of all admissible coefficients satisfies $C^* \\le C$.","title":"The optimal Berry\u2013Esseen coefficient is at most every admissible one"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/value-le-forty-two","kind":"node","lean":"BerryEsseenConstant.value_le_fortyTwo","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#value-le-forty-two","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The infimum $C^*$ of all admissible Berry\u2013Esseen coefficients satisfies $C^* \\le 42$.","title":"The optimal Berry\u2013Esseen coefficient is at most 42"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/is-admissible-mono","kind":"node","lean":"BerryEsseenConstant.isAdmissible_mono","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#is-admissible-mono","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"If $C \\le D$ are real numbers and $C$ is an admissible Berry\u2013Esseen coefficient, then $D$ is an admissible Berry\u2013Esseen coefficient.","title":"Admissibility is upward closed"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/value-bound","kind":"node","lean":"BerryEsseenConstant.value_bound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#value-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Let $C^*$ be the infimum of all admissible Berry\u2013Esseen coefficients. For every standardized law $\\mu$ on $\\mathbb{R}$ and every positive integer $n$, the Kolmogorov error of $\\mu$ at sample size $n$ is at most $C^* \\rho(\\mu) / \\sqrt{n}$, where $\\rho(\\mu) = \\int |x|^3 \\, d\\mu(x)$ is the third absolute moment of $\\mu$.","title":"The optimal Berry\u2013Esseen coefficient satisfies the Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/value-is-admissible","kind":"node","lean":"BerryEsseenConstant.value_isAdmissible","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#value-is-admissible","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The infimum $C^*$ of all admissible Berry\u2013Esseen coefficients is itself an admissible coefficient.","title":"The optimal Berry\u2013Esseen coefficient is admissible"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/value-is-least","kind":"node","lean":"BerryEsseenConstant.value_isLeast","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#value-is-least","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The infimum $C^*$ of all admissible Berry\u2013Esseen coefficients is the least element of the set of admissible coefficients: $C^*$ is admissible, and $C^* \\le C$ for every admissible coefficient $C$.","title":"The Berry\u2013Esseen constant is the least admissible coefficient"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/is-admissible-iff-value-le","kind":"node","lean":"BerryEsseenConstant.isAdmissible_iff_value_le","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#is-admissible-iff-value-le","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"A real number $C$ is an admissible Berry\u2013Esseen coefficient if and only if $C_0 \\le C$, where $C_0$ is the optimal Berry\u2013Esseen coefficient, the infimum of the set of admissible coefficients.","title":"Admissible coefficients are exactly those above the optimal one"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/iid-bound","kind":"node","lean":"BerryEsseenConstant.iid_bound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#iid-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Let $C$ be an admissible Berry\u2013Esseen coefficient. Let $X_0, X_1, X_2, \\dots$ be measurable real random variables on a probability space $(\\Omega, P)$ that are independent and identically distributed, with $\\mathbb{E}[X_0] = 0$, $\\operatorname{Var}(X_0) = 1$ and $\\mathbb{E}|X_0|^3 < \\infty$. Then for every positive integer $n$ and every real $x$,","title":"Admissible coefficients bound every standardized i.i.d. sequence"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/is-bernoulli-law","kind":"node","lean":"BerryEsseenConstant.IsBernoulliLaw","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#is-bernoulli-law","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"A measure $\\mu$ on $\\mathbb{R}$ is a standardized two-point law when it is a probability measure with mean $0$, variance $1$ and finite third absolute moment, and there are real numbers $a \\ne b$ with $\\mu(\\mathbb{R} \\setminus \\{a, b\\}) = 0$.","title":"Standardized two-point law"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/symmetric-bernoulli-law","kind":"node","lean":"BerryEsseenConstant.symmetricBernoulliLaw","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#symmetric-bernoulli-law","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The symmetric Bernoulli law is the measure $\\frac12 \\delta_{-1} + \\frac12 \\delta_1$ on $\\mathbb{R}$, where $\\delta_a$ denotes the Dirac mass at $a$; it puts mass $1/2$ at each of the points $-1$ and $1$.","title":"Symmetric Bernoulli law"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/symmetric-bernoulli-mem-lp","kind":"node","lean":"BerryEsseenConstant.symmetricBernoulli_memLp","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#symmetric-bernoulli-mem-lp","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"For every exponent $p \\in [0, \\infty]$, the identity function of $\\mathbb{R}$ belongs to $L^p$ of the symmetric Bernoulli law.","title":"The symmetric Bernoulli law has moments of all orders"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/symmetric-bernoulli-integral","kind":"node","lean":"BerryEsseenConstant.symmetricBernoulli_integral","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#symmetric-bernoulli-integral","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"For every function $f : \\mathbb{R} \\to \\mathbb{R}$, the integral of $f$ against the symmetric Bernoulli law $\\frac12 \\delta_{-1} + \\frac12 \\delta_1$ equals $\\frac12 f(-1) + \\frac12 f(1)$.","title":"Integrals against the symmetric Bernoulli law"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/symmetric-bernoulli-is-standardized","kind":"node","lean":"BerryEsseenConstant.symmetricBernoulli_isStandardized","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#symmetric-bernoulli-is-standardized","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The symmetric Bernoulli law, which puts mass $1/2$ at each of the points $-1$ and $1$, is a standardized law: it is a probability measure on $\\mathbb{R}$ with mean $0$ and variance $1$, and it has a finite third absolute moment.","title":"The symmetric Bernoulli law is standardized"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/symmetric-bernoulli-is-bernoulli","kind":"node","lean":"BerryEsseenConstant.symmetricBernoulli_isBernoulli","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#symmetric-bernoulli-is-bernoulli","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The law $\\frac12 \\delta_{-1} + \\frac12 \\delta_1$ on $\\mathbb{R}$ is a probability measure with mean $0$, variance $1$ and finite third absolute moment, and it is concentrated on the two distinct points $-1$ and $1$.","title":"The symmetric Bernoulli law is a standardized two-point law"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/symmetric-bernoulli-third-moment","kind":"node","lean":"BerryEsseenConstant.symmetricBernoulli_thirdMoment","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#symmetric-bernoulli-third-moment","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The third absolute moment $\\int |x|^3 \\, d\\mu(x)$ of the symmetric Bernoulli law $\\mu$ equals $1$.","title":"Third absolute moment of the symmetric Bernoulli law"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/is-bernoulli-admissible","kind":"node","lean":"BerryEsseenConstant.IsBernoulliAdmissible","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#is-bernoulli-admissible","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"A real number $C$ is an admissible Berry\u2013Esseen coefficient for standardized two-point laws when, for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3 \\, d\\mu(x)$ that is concentrated on two distinct points, and every positive integer $n$, the Kolmogorov error $\\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$ is at most $C \\rho / \\sqrt{n}$.\u2026","title":"Admissible Berry\u2013Esseen coefficient for two-point laws"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/bernoulli-value","kind":"node","lean":"BerryEsseenConstant.bernoulliValue","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#bernoulli-value","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The Bernoulli Berry\u2013Esseen constant is the infimum, in $\\mathbb{R}$, of the set of real numbers $C$ that are admissible Berry\u2013Esseen coefficients for the Bernoulli subclass: for every standardized law $\\mu$ (a probability measure on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment) concentrated on two distinct points, and every positive integer $n$, the supremum over $x \\in \\mathbb{R}$ of\u2026","title":"Bernoulli Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/is-bernoulli-admissible-nonnegative","kind":"node","lean":"BerryEsseenConstant.isBernoulliAdmissible_nonnegative","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#is-bernoulli-admissible-nonnegative","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Every real number $C$ that is an admissible Berry\u2013Esseen coefficient for standardized two-point laws satisfies $C \\ge 0$.","title":"Bernoulli-admissible coefficients are nonnegative"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/bernoulli-admissible-nonempty","kind":"node","lean":"BerryEsseenConstant.bernoulliAdmissible_nonempty","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#bernoulli-admissible-nonempty","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The set of real numbers $C$ that are admissible Berry\u2013Esseen coefficients for the Bernoulli subclass (common Berry\u2013Esseen coefficients for all standardized laws concentrated on two points and all positive sample sizes) is nonempty.","title":"Bernoulli-admissible coefficients exist"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/bernoulli-admissible-bdd-below","kind":"node","lean":"BerryEsseenConstant.bernoulliAdmissible_bddBelow","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#bernoulli-admissible-bdd-below","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The set of real numbers $C$ that are admissible Berry\u2013Esseen coefficients for the Bernoulli subclass (common Berry\u2013Esseen coefficients for all standardized laws concentrated on two points and all positive sample sizes) is bounded below.","title":"The Bernoulli-admissible coefficients are bounded below"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/bernoulli-value-nonnegative","kind":"node","lean":"BerryEsseenConstant.bernoulliValue_nonnegative","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#bernoulli-value-nonnegative","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The infimum of the real numbers that are admissible Berry\u2013Esseen coefficients for the Bernoulli subclass is nonnegative.","title":"The Bernoulli Berry\u2013Esseen constant is nonnegative"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/bernoulli-value-le-value","kind":"node","lean":"BerryEsseenConstant.bernoulliValue_le_value","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#bernoulli-value-le-value","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The infimum of the real numbers that are admissible Berry\u2013Esseen coefficients for the Bernoulli subclass is at most the infimum of the real numbers that are admissible Berry\u2013Esseen coefficients for all standardized laws.","title":"The Bernoulli Berry\u2013Esseen constant is at most the general one"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/zolotarev1967-upper-bound","kind":"node","lean":"BerryEsseenWanted.zolotarev1967UpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#zolotarev1967-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The proposition that the Berry\u2013Esseen constant $C_0$ is at most $0.82$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent\u2026","title":"Statement of Zolotarev's Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/van-beek1972-upper-bound","kind":"node","lean":"BerryEsseenWanted.vanBeek1972UpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#van-beek1972-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The proposition that the Berry\u2013Esseen constant $C_0$ is at most $0.7975$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent\u2026","title":"Statement of van Beek's Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/shiganov1986-upper-bound","kind":"node","lean":"BerryEsseenWanted.shiganov1986UpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#shiganov1986-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The proposition that the Berry\u2013Esseen constant $C_0$ is at most $0.7655$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent\u2026","title":"Statement of Shiganov's Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/shevtsova2006-upper-bound","kind":"node","lean":"BerryEsseenWanted.shevtsova2006UpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#shevtsova2006-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Let $C_0$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$ with law $\\mu$, satisfies $\\sup_x |F_n(x) - \\Phi(x)| \\le C \\rho / \\sqrt{n}$, where $\\Phi$ is the standard normal\u2026","title":"Statement of Shevtsova's 2006 Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/korolev-shevtsova2009-upper-bound","kind":"node","lean":"BerryEsseenWanted.korolevShevtsova2009UpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#korolev-shevtsova2009-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"For a probability measure $\\mu$ on $\\mathbb{R}$ and an integer $n \\ge 1$, let $F_n$ be the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, let $\\Delta_n(\\mu) = \\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal distribution function, and let $\\beta_3(\\mu) = \\int |x|^3 \\, d\\mu$. Let $C_0$ be the optimal\u2026","title":"Statement of the Korolev\u2013Shevtsova upper bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/tyurin2009-upper-bound","kind":"node","lean":"BerryEsseenWanted.tyurin2009UpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#tyurin2009-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The proposition that the Berry\u2013Esseen constant $C_0$ is at most $0.4785$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent\u2026","title":"Statement of Tyurin's Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/shevtsova2011-upper-bound","kind":"node","lean":"BerryEsseenWanted.shevtsova2011UpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#shevtsova2011-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Let $C_0$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$ with law $\\mu$, satisfies $\\sup_x |F_n(x) - \\Phi(x)| \\le C \\rho / \\sqrt{n}$, where $\\Phi$ is the standard normal\u2026","title":"Statement of Shevtsova's 2011 Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/shevtsova2013-upper-bound","kind":"node","lean":"BerryEsseenWanted.shevtsova2013UpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#shevtsova2013-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Let $C_0$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$ with law $\\mu$, satisfies $\\sup_x |F_n(x) - \\Phi(x)| \\le C \\rho / \\sqrt{n}$, where $\\Phi$ is the standard normal\u2026","title":"Statement of Shevtsova's 2013 Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/xiao-li2026-unverified-upper-bound","kind":"node","lean":"BerryEsseenWanted.xiaoLi2026UnverifiedUpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#xiao-li2026-unverified-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"The proposition that the Berry\u2013Esseen constant $C_0$ is at most $0.4395$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent\u2026","title":"Statement of the unverified Berry\u2013Esseen bound 0.4395"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/lin2026-unverified-upper-bound","kind":"node","lean":"BerryEsseenWanted.lin2026UnverifiedUpperBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#lin2026-unverified-upper-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Let $C_0$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$ with law $\\mu$, satisfies $\\sup_x |F_n(x) - \\Phi(x)| \\le C \\rho / \\sqrt{n}$, where $\\Phi$ is the standard normal\u2026","title":"Statement of Lin's 2026 Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/korolev-shevtsova2009-structural","kind":"node","lean":"BerryEsseenWanted.korolevShevtsova2009Structural","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#korolev-shevtsova2009-structural","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"For a probability measure $\\mu$ on $\\mathbb{R}$ and an integer $n \\ge 1$, let $F_n$ be the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, let $\\Delta_n(\\mu) = \\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal distribution function, and let $\\beta_3(\\mu) = \\int |x|^3 \\, d\\mu$. This is the proposition that\u2026","title":"Statement of the Korolev\u2013Shevtsova structural Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/esseen1956-lower-bound","kind":"node","lean":"BerryEsseenWanted.esseen1956LowerBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#esseen1956-lower-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"For a probability measure $\\mu$ on $\\mathbb{R}$ and an integer $n \\ge 1$, let $F_n$ be the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, let $\\Delta_n(\\mu) = \\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal distribution function, and let $\\beta_3(\\mu) = \\int |x|^3 \\, d\\mu$. Let $C_0$ be the optimal\u2026","title":"Statement of Esseen's lower bound for the Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/esseen-bernoulli-asymptotic-lower-bound","kind":"node","lean":"BerryEsseenWanted.esseenBernoulliAsymptoticLowerBound","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#esseen-bernoulli-asymptotic-lower-bound","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"For a probability measure $\\mu$ on $\\mathbb{R}$ and an integer $n \\ge 1$, let $F_n$ be the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, let $\\Delta_n(\\mu) = \\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal distribution function, let $\\beta_3(\\mu) = \\int |x|^3 \\, d\\mu$, and let\u2026","title":"Statement of Esseen's asymptotic lower bound for a two-point law"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/schulz2016-bernoulli-optimum","kind":"node","lean":"BerryEsseenWanted.schulz2016BernoulliOptimum","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#schulz2016-bernoulli-optimum","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"Let $C_B$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$ that is concentrated on two points, that is, $\\mu(\\mathbb{R} \\setminus \\{a, b\\}) = 0$ for some reals $a \\ne b$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$\u2026","title":"Statement of Schulz's Bernoulli Berry\u2013Esseen optimum"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/left-right-cdferror-equivalence","kind":"node","lean":"BerryEsseenWanted.leftRightCDFErrorEquivalence","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#left-right-cdferror-equivalence","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"fully_proved","summary":"This is the proposition that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment and every integer $n \\ge 1$, if $S_n = X_1 + \\dots + X_n$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, then","title":"Statement of the Kolmogorov distance via left-continuous distribution functions"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/zolotarev1967-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.zolotarev1967UpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#zolotarev1967-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"The Berry\u2013Esseen constant $C_0$ is at most $0.82$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent with law $\\mu$,\u2026","title":"Zolotarev's upper bound for the Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/van-beek1972-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.vanBeek1972UpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#van-beek1972-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"The Berry\u2013Esseen constant $C_0$ is at most $0.7975$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent with law $\\mu$,\u2026","title":"Van Beek's upper bound for the Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/shiganov1986-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.shiganov1986UpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#shiganov1986-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"The Berry\u2013Esseen constant $C_0$ is at most $0.7655$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent with law $\\mu$,\u2026","title":"Shiganov's upper bound for the Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/shevtsova2006-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.shevtsova2006UpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#shevtsova2006-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"Let $C_0$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$ with law $\\mu$, satisfies $\\sup_x |F_n(x) - \\Phi(x)| \\le C \\rho / \\sqrt{n}$, where $\\Phi$ is the standard normal\u2026","title":"Shevtsova's 2006 Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/korolev-shevtsova2009-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.korolevShevtsova2009UpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#korolev-shevtsova2009-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"For a probability measure $\\mu$ on $\\mathbb{R}$ and an integer $n \\ge 1$, let $F_n$ be the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, let $\\Delta_n(\\mu) = \\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal distribution function, and let $\\beta_3(\\mu) = \\int |x|^3 \\, d\\mu$. Let $C_0$ be the optimal\u2026","title":"Korolev\u2013Shevtsova upper bound for the Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/tyurin2009-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.tyurin2009UpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#tyurin2009-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"The Berry\u2013Esseen constant $C_0$ is at most $0.4785$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent with law $\\mu$,\u2026","title":"Tyurin's upper bound for the Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/shevtsova2011-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.shevtsova2011UpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#shevtsova2011-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"Let $C_0$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$ with law $\\mu$, satisfies $\\sup_x |F_n(x) - \\Phi(x)| \\le C \\rho / \\sqrt{n}$, where $\\Phi$ is the standard normal\u2026","title":"Shevtsova's 2011 Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/shevtsova2013-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.shevtsova2013UpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#shevtsova2013-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"Let $C_0$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$ with law $\\mu$, satisfies $\\sup_x |F_n(x) - \\Phi(x)| \\le C \\rho / \\sqrt{n}$, where $\\Phi$ is the standard normal\u2026","title":"Shevtsova's 2013 Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/xiao-li2026-unverified-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.xiaoLi2026UnverifiedUpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#xiao-li2026-unverified-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"The Berry\u2013Esseen constant $C_0$ is at most $0.4395$. Here $C_0$ is the infimum of the real numbers $C$ such that for every probability measure $\\mu$ on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment $\\beta_3 = \\int |x|^3 \\, d\\mu(x)$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, where $X_1, \\dots, X_n$ are independent with law $\\mu$,\u2026","title":"The unverified upper bound 0.4395 for the Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/lin2026-unverified-upper-bound-holds","kind":"node","lean":"BerryEsseenWanted.lin2026UnverifiedUpperBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#lin2026-unverified-upper-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"Let $C_0$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$ with law $\\mu$, satisfies $\\sup_x |F_n(x) - \\Phi(x)| \\le C \\rho / \\sqrt{n}$, where $\\Phi$ is the standard normal\u2026","title":"Lin's 2026 Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/korolev-shevtsova2009-structural-holds","kind":"node","lean":"BerryEsseenWanted.korolevShevtsova2009Structural_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#korolev-shevtsova2009-structural-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"For a probability measure $\\mu$ on $\\mathbb{R}$ and an integer $n \\ge 1$, let $F_n$ be the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, let $\\Delta_n(\\mu) = \\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal distribution function, and let $\\beta_3(\\mu) = \\int |x|^3 \\, d\\mu$. Then for every probability\u2026","title":"Korolev\u2013Shevtsova structural Berry\u2013Esseen bound"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/esseen1956-lower-bound-holds","kind":"node","lean":"BerryEsseenWanted.esseen1956LowerBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#esseen1956-lower-bound-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"For a probability measure $\\mu$ on $\\mathbb{R}$ and an integer $n \\ge 1$, let $F_n$ be the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, let $\\Delta_n(\\mu) = \\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal distribution function, and let $\\beta_3(\\mu) = \\int |x|^3 \\, d\\mu$. Let $C_0$ be the optimal\u2026","title":"Esseen's lower bound for the Berry\u2013Esseen constant"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/esseen-bernoulli-asymptotic-lower-bound-2","kind":"node","lean":"BerryEsseenWanted.esseenBernoulliAsymptoticLowerBound_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#esseen-bernoulli-asymptotic-lower-bound-2","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"For a probability measure $\\mu$ on $\\mathbb{R}$ and an integer $n \\ge 1$, let $F_n$ be the distribution function of $(X_1 + \\dots + X_n)/\\sqrt{n}$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$, let $\\Delta_n(\\mu) = \\sup_{x \\in \\mathbb{R}} |F_n(x) - \\Phi(x)|$, where $\\Phi$ is the standard normal distribution function, let $\\beta_3(\\mu) = \\int |x|^3 \\, d\\mu$, and let\u2026","title":"Esseen's asymptotic lower bound for a two-point law"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/schulz2016-bernoulli-optimum-holds","kind":"node","lean":"BerryEsseenWanted.schulz2016BernoulliOptimum_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#schulz2016-bernoulli-optimum-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"Let $C_B$ be the infimum of all real $C$ such that for every real probability law $\\mu$ with mean $0$, variance $1$ and finite third absolute moment $\\rho = \\int |x|^3\\,d\\mu$ that is concentrated on two points, that is, $\\mu(\\mathbb{R} \\setminus \\{a, b\\}) = 0$ for some reals $a \\ne b$, and every integer $n \\ge 1$, the distribution function $F_n$ of $(X_1 + \\dots + X_n)/\\sqrt{n}$, for independent $X_1, \\dots, X_n$\u2026","title":"Schulz's Bernoulli Berry\u2013Esseen optimum"},{"id":"Probability/central-limit-theorems/optimal-berry-esseen-constant/left-right-cdferror-equivalence-holds","kind":"node","lean":"BerryEsseenWanted.leftRightCDFErrorEquivalence_holds","parent":19552,"result":true,"route":"../roadmap/Probability/central-limit-theorems/optimal-berry-esseen-constant/index.html#left-right-cdferror-equivalence-holds","scope":"Probability/central-limit-theorems/optimal-berry-esseen-constant","status":"can_prove","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with mean $0$, variance $1$ and finite third absolute moment, let $n \\ge 1$ be an integer, and let $S_n = X_1 + \\dots + X_n$ for independent random variables $X_1, X_2, \\dots$ each with law $\\mu$. Then","title":"Kolmogorov distance via left-continuous distribution functions"},{"data_url":"scopes/Probability/central-limit-theorems/poisson-limit-theorem.json","id":"Probability/central-limit-theorems/poisson-limit-theorem","kind":"scope","navigation":"module","parent":19460,"result":false,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","title":"Poisson limit theorem"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/elem-symm-poly-tendsto-poisson","kind":"node","lean":"ProbabilityTheory.elem_symm_poly_tendsto_poisson","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#elem-symm-poly-tendsto-poisson","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"Let $\\lambda \\ge 0$ be a real number, let $k : \\mathbb{N} \\to \\mathbb{N}$, and for each $n$ let $p_{n,i}$, $0 \\le i < k(n)$, be real numbers. Suppose that $p_{n,i} \\ge 0$ for all $n$ and $i$, that for every $\\varepsilon > 0$ we have $p_{n,i} < \\varepsilon$ for all $i < k(n)$ once $n$ is large enough, and that $\\sum_{i < k(n)} p_{n,i} \\to \\lambda$ as $n \\to \\infty$. Then for every natural number $j$, as\u2026","title":"Analytic core of Poisson convergence"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/binomial-prob-measure","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.binomialProbMeasure","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#binomial-prob-measure","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"The binomial law `Bin(n, p)` on `\u211d`, bundled as a probability measure.","title":"Binomial law as a probability measure"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/poisson-prob-measure","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.poissonProbMeasure","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#poisson-prob-measure","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"The Poisson law of rate `\u03bb` on `\u211d`, bundled as a probability measure.","title":"Poisson law as a probability measure"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/poisson-approx-prob","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.poissonApproxProb","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#poisson-approx-prob","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"The success probability `\u03bb / (n + \u03bb)` used in the binomial approximation to the Poisson law. It lies in `[0, 1]` for every `n`, and `n * p\u2099 \u2192 \u03bb`.","title":"Success probability of the binomial approximation to Poisson"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/binomial-poisson-approx","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.binomialPoissonApprox","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#binomial-poisson-approx","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"The binomial law `Bin(n, \u03bb / (n + \u03bb))` on `\u211d`.","title":"Binomial approximation to the Poisson law"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/char-fun-binomial-prob-measure","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.charFun_binomialProbMeasure","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#char-fun-binomial-prob-measure","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"`\u03c6_{Bin(n, p)}(t) = ((1 - p) + p * e^{it})\u207f`.","title":"Characteristic function of a binomial law"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/char-fun-poisson-prob-measure","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.charFun_poissonProbMeasure","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#char-fun-poisson-prob-measure","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"`\u03c6_{Poisson(\u03bb)}(t) = exp(\u03bb * (e^{it} - 1))`.","title":"Characteristic function of a Poisson law"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/tendsto-mul-poisson-approx-prob","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.tendsto_mul_poissonApproxProb","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#tendsto-mul-poisson-approx-prob","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"The expected number of successes tends to the rate: `n * (\u03bb / (n + \u03bb)) \u2192 \u03bb`.","title":"Mean of the binomial approximation tends to the Poisson rate"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/tendsto-char-fun-binomial-poisson-approx","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.tendsto_charFun_binomialPoissonApprox","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#tendsto-char-fun-binomial-poisson-approx","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"The characteristic functions of the approximating binomial laws converge pointwise to the characteristic function of the Poisson law.","title":"Convergence of characteristic functions to the Poisson law"},{"id":"Probability/central-limit-theorems/poisson-limit-theorem/tendsto-binomial-poisson-approx","kind":"node","lean":"MathlibExt.Probability.PoissonWeakConvergence.tendsto_binomialPoissonApprox","parent":19614,"result":true,"route":"../roadmap/Probability/central-limit-theorems/poisson-limit-theorem/index.html#tendsto-binomial-poisson-approx","scope":"Probability/central-limit-theorems/poisson-limit-theorem","status":"fully_proved","summary":"The binomial laws `Bin(n, \u03bb / (n + \u03bb))` converge weakly to the Poisson law of rate `\u03bb`.","title":"Poisson convergence in law (binomial approximation)"},{"data_url":"scopes/Probability/markov-chains.json","id":"Probability/markov-chains","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/markov-chains/index.html","scope":"Probability/markov-chains","status":"fully_proved","summary":"Discrete-time Markov chains (construction, path measures, finite-dimensional distributions, Chapman\u2013Kolmogorov, Markov and strong Markov properties), finite chains, ergodicity and convergence to stationarity, mixing, reversibility and Kolmogorov's cycle criterion, and finite continuous-time Markov chains with rate matrices, semigroups and hitting times.","title":"Markov chains"},{"data_url":"scopes/Probability/markov-chains/chapman-kolmogorov-equations.json","id":"Probability/markov-chains/chapman-kolmogorov-equations","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/chapman-kolmogorov-equations/index.html","scope":"Probability/markov-chains/chapman-kolmogorov-equations","status":"fully_proved","title":"Chapman\u2013Kolmogorov equations"},{"id":"Probability/markov-chains/chapman-kolmogorov-equations/is-markov-chain","kind":"node","lean":"MathlibExt.Probability.MarkovChainChapmanKolmogorov.IsMarkovChain","parent":19626,"result":true,"route":"../roadmap/Probability/markov-chains/chapman-kolmogorov-equations/index.html#is-markov-chain","scope":"Probability/markov-chains/chapman-kolmogorov-equations","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on $\\Omega$, and let $\\kappa$ be a Markov kernel from a measurable space $S$ to itself. A sequence of random variables $X_n : \\Omega \\to S$ is a Markov chain with respect to $(\\mathcal{F}_n)$ with transition kernel $\\kappa$ under $\\mu$ if it is adapted to $(\\mathcal{F}_n)$ and, for every $n \\in \\mathbb{N}$ and every\u2026","title":"Markov chain with respect to a filtration"},{"id":"Probability/markov-chains/chapman-kolmogorov-equations/cond-exp-indicator-eq","kind":"node","lean":"MathlibExt.Probability.MarkovChainChapmanKolmogorov.IsMarkovChain.condExp_indicator_eq","parent":19626,"result":true,"route":"../roadmap/Probability/markov-chains/chapman-kolmogorov-equations/index.html#cond-exp-indicator-eq","scope":"Probability/markov-chains/chapman-kolmogorov-equations","status":"fully_proved","summary":"Let $X = (X_n)_{n \\in \\mathbb{N}}$ be a Markov chain with state space $S$, with respect to a filtration $(\\mathcal{F}_n)$, with Markov transition kernel $\\kappa$, under a probability measure $\\mu$. Then for all $m, n \\in \\mathbb{N}$ and every measurable $B \\subseteq S$,","title":"Chapman\u2013Kolmogorov equations, conditional form"},{"id":"Probability/markov-chains/chapman-kolmogorov-equations/map-eq-bind-pow","kind":"node","lean":"MathlibExt.Probability.MarkovChainChapmanKolmogorov.IsMarkovChain.map_eq_bind_pow","parent":19626,"result":true,"route":"../roadmap/Probability/markov-chains/chapman-kolmogorov-equations/index.html#map-eq-bind-pow","scope":"Probability/markov-chains/chapman-kolmogorov-equations","status":"fully_proved","summary":"Let $X = (X_n)_{n \\in \\mathbb{N}}$ be a Markov chain with state space $S$, with respect to a filtration, with Markov transition kernel $\\kappa$, under a probability measure $\\mu$, and let $\\kappa^n$ be the $n$-fold composition of $\\kappa$ (the identity kernel for $n = 0$). Then for all $m, n \\in \\mathbb{N}$ the law of $X_{m+n}$ is the law $\\nu_m$ of $X_m$ pushed forward through $\\kappa^n$: for every measurable\u2026","title":"Chapman\u2013Kolmogorov equations, forward (law) form"},{"id":"Probability/markov-chains/chapman-kolmogorov-equations/cond-exp-comp-eq-integral-pow","kind":"node","lean":"MathlibExt.Probability.MarkovChainFunctionCK.IsMarkovChain.condExp_comp_eq_integral_pow","parent":19626,"result":true,"route":"../roadmap/Probability/markov-chains/chapman-kolmogorov-equations/index.html#cond-exp-comp-eq-integral-pow","scope":"Probability/markov-chains/chapman-kolmogorov-equations","status":"fully_proved","summary":"For a Markov chain `X` with transition kernel `\u03ba`, the conditional expectation of `f (X (m + n))` given the past up to time `m` is the `\u03ba ^ n`-integral of `f` evaluated at `X m`. For `n = 1` this is the one-step function form; the general case follows by integrating the canonical conditional-indicator formula over each past-measurable set.","title":"Chapman\u2013Kolmogorov equations for bounded measurable functions"},{"id":"Probability/markov-chains/chapman-kolmogorov-equations/cond-exp-one-step-eq-integral","kind":"node","lean":"MathlibExt.Probability.MarkovChainFunctionCK.IsMarkovChain.condExp_one_step_eq_integral","parent":19626,"result":true,"route":"../roadmap/Probability/markov-chains/chapman-kolmogorov-equations/index.html#cond-exp-one-step-eq-integral","scope":"Probability/markov-chains/chapman-kolmogorov-equations","status":"fully_proved","summary":"For a bounded measurable `f`, the conditional expectation of `f (X (k+1))` given `\u2131 k` is the `\u03ba`-integral of `f` evaluated at `X k`.","title":"One-step Chapman\u2013Kolmogorov equation for functions"},{"id":"Probability/markov-chains/chapman-kolmogorov-equations/integral-comp-eq","kind":"node","lean":"MathlibExt.Probability.MarkovChainFunctionCK.IsMarkovChain.integral_comp_eq","parent":19626,"result":true,"route":"../roadmap/Probability/markov-chains/chapman-kolmogorov-equations/index.html#integral-comp-eq","scope":"Probability/markov-chains/chapman-kolmogorov-equations","status":"fully_proved","summary":"The unconditional expectation of `f (X (m + n))` is the expectation of the `\u03ba ^ n`-integral of `f` evaluated at `X m`.","title":"Integrated Chapman\u2013Kolmogorov equation for functions"},{"data_url":"scopes/Probability/markov-chains/finite-continuous-time-markov-chains.json","id":"Probability/markov-chains/finite-continuous-time-markov-chains","kind":"scope","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/index.html","scope":"Probability/markov-chains/finite-continuous-time-markov-chains","status":"fully_proved","title":"Finite continuous-time Markov chains"},{"data_url":"scopes/Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c.json","id":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c","kind":"scope","navigation":"module","parent":19633,"result":false,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/index.html","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c","status":"fully_proved","title":"Rate matrices of finite continuous-time Markov chains"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/is-rate-matrix","kind":"node","lean":"ProbabilityTheory.IsRateMatrix","parent":19634,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/index.html#is-rate-matrix","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"A real square matrix `Q` is a rate matrix (a `Q`-matrix in Norris's terminology) if its off-diagonal entries are nonnegative and each of its rows sums to `0` (Norris 1997, \u00a72.1).","title":"Rate matrix"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/diag-eq","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.diag_eq","parent":19634,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/index.html#diag-eq","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"The diagonal entry of a rate matrix is minus the total jump rate out of the state.","title":"Diagonal of a rate matrix is minus the total jump rate"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/diag-nonpos","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.diag_nonpos","parent":19634,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/index.html#diag-nonpos","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"The diagonal entries of a rate matrix are nonpositive.","title":"Diagonal entries of a rate matrix are nonpositive"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/mul-vec-one","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.mulVec_one","parent":19634,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/index.html#mul-vec-one","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"A rate matrix annihilates the constant vector `1`.","title":"Rate matrix annihilates the constant vector"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/add-smul-one-nonneg","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.add_smul_one_nonneg","parent":19634,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/index.html#add-smul-one-nonneg","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"If `c` bounds every exit rate `-Q i i`, then `Q + c \u2022 1` has nonnegative entries (Jensen 1953).","title":"Uniformization"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/exists-add-smul-one-nonneg","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.exists_add_smul_one_nonneg","parent":19634,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c/index.html#exists-add-smul-one-nonneg","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/rate-matrices-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"Every rate matrix admits a uniformization constant `c \u2265 0`.","title":"Existence of a uniformization constant for a rate matrix"},{"data_url":"scopes/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous.json","id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","kind":"scope","navigation":"module","parent":19633,"result":false,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","title":"The transition semigroup of a finite continuous-time Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition","kind":"node","lean":"ProbabilityTheory.ctmcTransition","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"The transition function `P t = exp (t \u2022 Q)` of the finite continuous-time Markov chain with generator `Q` (Norris 1997, Theorem 2.1.1).","title":"Transition semigroup of a continuous-time Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition-zero","kind":"node","lean":"ProbabilityTheory.ctmcTransition_zero","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition-zero","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"At time `0` the transition function is the identity.","title":"Transition function at time zero is the identity"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition-add","kind":"node","lean":"ProbabilityTheory.ctmcTransition_add","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition-add","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"The semigroup property (Chapman\u2013Kolmogorov equation): `P (s + t) = P s * P t`.","title":"Chapman\u2013Kolmogorov equation for the transition semigroup"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition-smul","kind":"node","lean":"ProbabilityTheory.ctmcTransition_smul","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition-smul","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"Rescaling the generator rescales time: `exp (t \u2022 (a \u2022 Q)) = exp ((a * t) \u2022 Q)`.","title":"Rescaling the generator rescales time"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/commute-ctmc-transition","kind":"node","lean":"ProbabilityTheory.commute_ctmcTransition","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#commute-ctmc-transition","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"The generator commutes with the transition function.","title":"Generator commutes with the transition function"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/has-deriv-at-ctmc-transition-backward","kind":"node","lean":"ProbabilityTheory.hasDerivAt_ctmcTransition_backward","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#has-deriv-at-ctmc-transition-backward","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"The Kolmogorov backward equation `P' t = Q * P t` (Norris 1997, Theorem 2.1.1).","title":"Kolmogorov backward equation"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/has-deriv-at-ctmc-transition-forward","kind":"node","lean":"ProbabilityTheory.hasDerivAt_ctmcTransition_forward","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#has-deriv-at-ctmc-transition-forward","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"The Kolmogorov forward equation `P' t = P t * Q` (Norris 1997, Theorem 2.1.1).","title":"Kolmogorov forward equation"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition-eq-exp-smul-uniformized","kind":"node","lean":"ProbabilityTheory.ctmcTransition_eq_exp_smul_uniformized","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition-eq-exp-smul-uniformized","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"For every real `c`, `exp (t \u2022 Q) = e^{-c t} \u2022 exp (t \u2022 (Q + c \u2022 1))` (Jensen 1953).","title":"Uniformization"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition-nonneg-of-nonneg","kind":"node","lean":"ProbabilityTheory.ctmcTransition_nonneg_of_nonneg","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition-nonneg-of-nonneg","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"If `A` has nonnegative entries, so does `exp (t \u2022 A)` for `t \u2265 0`.","title":"Exponential of a nonnegative matrix is nonnegative"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition-mul-vec-one","kind":"node","lean":"ProbabilityTheory.ctmcTransition_mulVec_one","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition-mul-vec-one","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"If `Q` annihilates the constant vector `1`, then `P t` fixes it: `P t *\u1d65 1 = 1`.","title":"Transition function fixes the constant vector"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/sum-ctmc-transition","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.sum_ctmcTransition","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#sum-ctmc-transition","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"The rows of the transition function of a rate matrix sum to `1`, for every real `t`.","title":"Rows of the transition function sum to one"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition-nonneg","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcTransition_nonneg","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition-nonneg","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"The transition function of a rate matrix has nonnegative entries for `t \u2265 0`, by uniformization.","title":"Nonnegativity of the transition function of a rate matrix"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/ctmc-transition-mem-row-stochastic","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcTransition_mem_rowStochastic","parent":19641,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous/index.html#ctmc-transition-mem-row-stochastic","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-transition-semigroup-of-a-finite-continuous","status":"fully_proved","summary":"The transition function of a rate matrix is row-stochastic for `t \u2265 0` (Norris 1997, Theorem 2.1.2).","title":"Transition function of a rate matrix is row-stochastic"},{"data_url":"scopes/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro.json","id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","kind":"scope","navigation":"module","parent":19633,"result":false,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","title":"Finite continuous-time Markov chains as path processes"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-matrix","kind":"node","lean":"ProbabilityTheory.ctmcJumpMatrix","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-matrix","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The jump matrix `\u03a0` of a matrix `Q` (Norris 1997, \u00a72.6): for a state with exit rate `q i = -Q i i \u2260 0`, `\u03a0 i j = Q i j / q i` for `j \u2260 i` and `\u03a0 i i = 0`; a state with `q i = 0` is absorbing, `\u03a0 i j = \u03b4 i j`.","title":"Jump matrix of a rate matrix"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-matrix-nonneg","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcJumpMatrix_nonneg","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-matrix-nonneg","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The entries of the jump matrix of a rate matrix are nonnegative.","title":"Nonnegativity of the jump matrix"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/sum-ctmc-jump-matrix","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.sum_ctmcJumpMatrix","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#sum-ctmc-jump-matrix","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The rows of the jump matrix of a rate matrix sum to `1`.","title":"Rows of the jump matrix sum to one"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-matrix-mem-row-stochastic","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcJumpMatrix_mem_rowStochastic","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-matrix-mem-row-stochastic","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The jump matrix of a rate matrix is row-stochastic.","title":"Jump matrix is row-stochastic"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-pmf","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcJumpPMF","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-pmf","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"Row `i` of the jump matrix of a rate matrix, as a probability mass function on `\u03b9`.","title":"Row of the jump matrix as a probability mass function"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-step-measure","kind":"node","lean":"ProbabilityTheory.ctmcStepMeasure","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-step-measure","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The law of one step `(\u03be, E)` of the construction: `\u03be : \u03b9 \u2192 \u03b9` has independent coordinates with `\u03be i \u223c \u03a0 i`, and `E` is an independent exponential clock of rate `1`.","title":"Law of one step of the chain construction"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-input-measure","kind":"node","lean":"ProbabilityTheory.ctmcInputMeasure","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-input-measure","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The law of the i.i.d. step sequence `\u03c9 : \u2115 \u2192 (\u03b9 \u2192 \u03b9) \u00d7 \u211d` driving the chain.","title":"Law of the driving step sequence of a Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-chain","kind":"node","lean":"ProbabilityTheory.ctmcJumpChain","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-chain","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The jump chain started at `i`: `Y 0 = i` and `Y (n + 1) = \u03be n (Y n)`.","title":"Jump chain of a continuous-time Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-holding-time","kind":"node","lean":"ProbabilityTheory.ctmcHoldingTime","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-holding-time","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The `n`-th holding time `S n = E n / q (Y n)`, where `q k = -Q k k`; it is `\u221e` when `Y n` is absorbing (`q (Y n) = 0`). Negative clock values, a null event, give `S n = 0`.","title":"Holding times of a continuous-time Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-time","kind":"node","lean":"ProbabilityTheory.ctmcJumpTime","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-time","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The `n`-th jump time `T n = S 0 + \u22ef + S (n - 1)`, with `T 0 = 0`.","title":"Jump times of a continuous-time Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-path","kind":"node","lean":"ProbabilityTheory.ctmcPath","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-path","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The path of the chain started at `i`: `X t = Y n` for the `n` with `T n \u2264 t < T (n + 1)`. If `t` is at or after the explosion time `sup T n` (a null event, by `IsRateMatrix.ae_tendsto_ctmcJumpTime`), the value is an arbitrary state that does not depend on `i`, `Q` or `\u03c9`.","title":"Path of a continuous-time Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-shift","kind":"node","lean":"ProbabilityTheory.ctmcShift","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-shift","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The shift of the step sequence by `n`: `(shift n \u03c9) m = \u03c9 (n + m)`.","title":"Shift of the step sequence of a Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-time-mono","kind":"node","lean":"ProbabilityTheory.ctmcJumpTime_mono","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-time-mono","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The jump times are nondecreasing.","title":"Jump times are nondecreasing"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-path-eq-of-mem","kind":"node","lean":"ProbabilityTheory.ctmcPath_eq_of_mem","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-path-eq-of-mem","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"If `T n \u2264 t < T (n + 1)`, then `X t = Y n`.","title":"Path value between consecutive jump times"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-chain-shift","kind":"node","lean":"ProbabilityTheory.ctmcJumpChain_shift","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-chain-shift","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The jump chain of the shifted steps started at `Y n` is the jump chain after `n`.","title":"Jump chain of the shifted steps"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-holding-time-shift","kind":"node","lean":"ProbabilityTheory.ctmcHoldingTime_shift","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-holding-time-shift","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The holding times of the shifted steps started at `Y n` are the holding times after `n`.","title":"Holding times of the shifted steps"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-jump-time-add","kind":"node","lean":"ProbabilityTheory.ctmcJumpTime_add","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-jump-time-add","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"`T (n + m) = T n + T' m`, where `T'` are the jump times of the shifted steps started at `Y n`.","title":"Additivity of jump times under shifting"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-path-jump-time-add","kind":"node","lean":"ProbabilityTheory.ctmcPath_jumpTime_add","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-path-jump-time-add","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"After a finite jump time `T n`, the path is the path started at `Y n` driven by the shifted steps: `X (T n + s) = X' s` with `X'` built from `shift n \u03c9` started at `Y n`.","title":"Path restarts at a jump time"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-path-eq-ite","kind":"node","lean":"ProbabilityTheory.ctmcPath_eq_ite","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-path-eq-ite","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"First-jump decomposition of the path: before the first holding time the chain is at `i`; afterwards it is the chain started at `\u03be 0 i` driven by the shifted steps.","title":"First-jump decomposition"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/measurable-ctmc-jump-chain","kind":"node","lean":"ProbabilityTheory.measurable_ctmcJumpChain","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#measurable-ctmc-jump-chain","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The jump chain `Y n` is a measurable function of the steps.","title":"Measurability of the jump chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/measurable-ctmc-holding-time","kind":"node","lean":"ProbabilityTheory.measurable_ctmcHoldingTime","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#measurable-ctmc-holding-time","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The holding time `S n` is a measurable function of the steps.","title":"Measurability of the holding times"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/measurable-ctmc-jump-time","kind":"node","lean":"ProbabilityTheory.measurable_ctmcJumpTime","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#measurable-ctmc-jump-time","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The jump time `T n` is a measurable function of the steps.","title":"Measurability of the jump times"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/measurable-ctmc-path-uncurry","kind":"node","lean":"ProbabilityTheory.measurable_ctmcPath_uncurry","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#measurable-ctmc-path-uncurry","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The path `(t, \u03c9) \u21a6 X t` is jointly measurable in the time and the steps.","title":"Joint measurability of the chain path in time and steps"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/measurable-ctmc-path","kind":"node","lean":"ProbabilityTheory.measurable_ctmcPath","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#measurable-ctmc-path","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"The state `X t` is a measurable function of the steps.","title":"Measurability of the chain state at a fixed time"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ae-tendsto-ctmc-jump-time","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ae_tendsto_ctmcJumpTime","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ae-tendsto-ctmc-jump-time","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"Non-explosion (Norris 1997, \u00a72.7, for a finite state space): almost surely the jump times of the chain started at `i` tend to `\u221e`, so `X t` is a genuine state of the jump chain for every `t \u2265 0`.","title":"Non-explosion"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-input-measure-ctmc-path-jump-time-a","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcInputMeasure_ctmcPath_jumpTime_add","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-input-measure-ctmc-path-jump-time-a","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"Strong Markov property at jump times (Norris 1997, \u00a72.6\u20132.7): let `A` be a measurable set of histories of the first `n` steps such that `T n < \u221e` whenever the history lies in `A`; these histories determine `Y 0, \u2026, Y n` and `S 0, \u2026, S (n - 1)`. Then for every state `k` and every measurable set `B` of paths, the path after `T n` is independent of the event `{history \u2208 A, Y n = k}` and distributed as the chain\u2026","title":"Strong Markov property at jump times"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-input-measure-real-ctmc-path-eq","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcInputMeasure_real_ctmcPath_eq","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-input-measure-real-ctmc-path-eq","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"Kolmogorov (Norris 1997, \u00a72.8, Theorem 2.8.2 for a finite state space): the chain built from the jump chain and the holding times has transition function `exp (t \u2022 Q)`: `P_i(X t = j) = exp (t Q) i j` for every `t \u2265 0`.","title":"Kolmogorov"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-input-measure-real-ctmc-path-add","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcInputMeasure_real_ctmcPath_add","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-input-measure-real-ctmc-path-add","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"Restart law at a fixed time (cf. the Markov property, Norris 1997, \u00a72.8), for events of `X s` and of the path after `s` only: for the chain started at `i`, the probability that `X s = k` and that the path after `s` lies in a measurable set `B` is `exp (s Q) i k` times the probability that the chain started at `k` lies in `B`. This is weaker than the Markov property with respect to the natural filtration: arbitrary\u2026","title":"Restart law at a fixed time"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/ctmc-input-measure-real-ctmc-path-fdd","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcInputMeasure_real_ctmcPath_fdd","parent":19655,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro/index.html#ctmc-input-measure-real-ctmc-path-fdd","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/finite-continuous-time-markov-chains-as-path-pro","status":"fully_proved","summary":"Finite-dimensional distributions (Norris 1997, \u00a72.8, Theorem 2.8.2 (c)): with increments `s 0, \u2026, s (n - 1) \u2265 0`, times `t k = s 0 + \u22ef + s k`, and states `j 0, \u2026, j (n - 1)`, `P_i(X (t k) = j k for all k) = \u220f\u2096 exp (s k Q) (j (k - 1)) (j k)`, where `j (-1) = i`.","title":"Finite-dimensional distributions"},{"data_url":"scopes/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c.json","id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","kind":"scope","navigation":"module","parent":19633,"result":false,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","title":"Hitting times of finite continuous-time Markov chains"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/ctmc-hitting-time","kind":"node","lean":"ProbabilityTheory.ctmcHittingTime","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#ctmc-hitting-time","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"The hitting time `inf {t \u2265 0 | X t \u2208 A}` of a set of states `A` by the chain `X = ctmcPath Q i \u03c9` started at `i`, valued in `\u211d\u22650\u221e` (it is `\u221e` if `A` is never visited).","title":"Hitting time of a continuous-time Markov chain"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/ctmc-hitting-time-eq-add","kind":"node","lean":"ProbabilityTheory.ctmcHittingTime_eq_add","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#ctmc-hitting-time-eq-add","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"From a state `i \u2209 A`, the hitting time of `A` is the first holding time plus the hitting time of the chain restarted after the first jump.","title":"First-jump decomposition of a hitting time"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/ctmc-hitting-time-eq-zero","kind":"node","lean":"ProbabilityTheory.ctmcHittingTime_eq_zero","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#ctmc-hitting-time-eq-zero","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"From a state `i \u2208 A`, the hitting time of `A` is `0` as soon as the first holding time is positive (which fails only on a null event, see `ctmcHoldingTime`).","title":"Hitting time from inside the target set is zero"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/lintegral-ctmc-hitting-time","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.lintegral_ctmcHittingTime","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#lintegral-ctmc-hitting-time","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"Mean hitting time, first-step equation (compare Norris 1997, \u00a73.3): from `i \u2209 A` with `q i = -Q i i \u2260 0`, `E_i H_A = 1 / q i + \u2211\u2c7c \u03a0 i j E_j H_A` in `\u211d\u22650\u221e`.","title":"Mean hitting time, first-step equation"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/lintegral-ctmc-hitting-time-of-mem","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.lintegral_ctmcHittingTime_of_mem","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#lintegral-ctmc-hitting-time-of-mem","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"From a state of `A`, the mean hitting time of `A` is `0`.","title":"Mean hitting time from inside the target set is zero"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/lintegral-ctmc-hitting-time-le","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.lintegral_ctmcHittingTime_le","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#lintegral-ctmc-hitting-time-le","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"Minimality of the mean hitting time (compare Norris 1997, \u00a73.3): if every state off `A` has a positive exit rate, every `m : \u03b9 \u2192 \u211d\u22650\u221e` with `1 / q i + \u2211\u2c7c \u03a0 i j m j \u2264 m i` off `A` dominates `E_i H_A`.","title":"Minimality of the mean hitting time"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/lintegral-ctmc-hitting-time-mul","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.lintegral_ctmcHittingTime_mul","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#lintegral-ctmc-hitting-time-mul","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"From `i \u2209 A \u222a B` with `q i \u2260 0`, `E_i (H_A H_B) = (E_i H_A + E_i H_B) / q i + \u2211\u2c7c \u03a0 i j E_j (H_A H_B)` in `\u211d\u22650\u221e`.","title":"Product moment of two hitting times, first-step equation"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/lintegral-ctmc-hitting-time-mul-of-mem","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.lintegral_ctmcHittingTime_mul_of_mem","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#lintegral-ctmc-hitting-time-mul-of-mem","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"From a state of `A \u222a B`, the product moment of the hitting times of `A` and `B` is `0`.","title":"Product moment of hitting times from inside their union is zero"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/lintegral-ctmc-hitting-time-mul-le","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.lintegral_ctmcHittingTime_mul_le","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#lintegral-ctmc-hitting-time-mul-le","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"If `mA`, `mB` are supersolutions of the mean equations of `A` and `B` (as in `lintegral_ctmcHittingTime_le`) and `F` satisfies `(mA i + mB i) / q i + \u2211\u2c7c \u03a0 i j F j \u2264 F i` off `A \u222a B`, then `F` dominates `E_i (H_A H_B)`.","title":"Minimality of the product moment"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/ctmc-hitting-time-backward","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcHittingTime_backward","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#ctmc-hitting-time-backward","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"Backward equations for the mean hitting time (compare Norris 1997, \u00a73.3): if `m : \u03b9 \u2192 \u211d` is nonnegative and `(Q m) i = -1` for `i \u2209 A`, then `E_i H_A \u2264 m i` (so it is finite), `E_i H_A = 0` on `A`, and the real vector `E_i H_A` solves the same backward equations. With uniqueness of the solution, this identifies `E_i H_A`.","title":"Backward equations for the mean hitting time"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/ctmc-hitting-time-mul-backward","kind":"node","lean":"ProbabilityTheory.IsRateMatrix.ctmcHittingTime_mul_backward","parent":19685,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c/index.html#ctmc-hitting-time-mul-backward","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/hitting-times-of-finite-continuous-time-markov-c","status":"fully_proved","summary":"Let `mA`, `mB`, `f` be nonnegative with `(Q mA) i = -1` off `A`, `(Q mB) i = -1` off `B` and `(Q f) i = -(mA i + mB i)` off `A \u222a B`. Then `E_i (H_A H_B) \u2264 f i` (so it is finite), it is `0` on `A \u222a B`, and its real part `p` solves `(Q p) i = -(E_i H_A + E_i H_B)` off `A \u222a B`.","title":"Backward equations for the product moment of two hitting times"},{"data_url":"scopes/Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con.json","id":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con","kind":"scope","navigation":"module","parent":19633,"result":false,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/index.html","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con","status":"fully_proved","title":"Stationarity and detailed balance for finite continuous-time Markov chains"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/is-detailed-balance","kind":"node","lean":"ProbabilityTheory.IsDetailedBalance","parent":19697,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/index.html#is-detailed-balance","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con","status":"fully_proved","summary":"The row vector `\u03c0` is in detailed balance with the matrix `Q`: `\u03c0 i * Q i j = \u03c0 j * Q j i` for all `i j` (Norris 1997, \u00a73.7). This is only the algebraic equation: it holds for `\u03c0 = 0` and every `Q`. For a probability vector `\u03c0` and a stochastic matrix it is the reversibility of the chain started from `\u03c0`.","title":"Detailed balance"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/vec-mul-eq-zero-iff-forall-vec-mul-ctmc","kind":"node","lean":"ProbabilityTheory.vecMul_eq_zero_iff_forall_vecMul_ctmcTransition","parent":19697,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/index.html#vec-mul-eq-zero-iff-forall-vec-mul-ctmc","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con","status":"fully_proved","summary":"`\u03c0 Q = 0` if and only if `\u03c0 P t = \u03c0` for every real `t` (Norris 1997, \u00a73.5, finite state space).","title":"Invariance under the semigroup"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/vec-mul-eq-zero","kind":"node","lean":"ProbabilityTheory.IsDetailedBalance.vecMul_eq_zero","parent":19697,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/index.html#vec-mul-eq-zero","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con","status":"fully_proved","summary":"Detailed balance with a matrix whose rows sum to `0` implies invariance `\u03c0 Q = 0` (Norris 1997, \u00a73.7).","title":"Detailed balance implies invariance for the generator"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/ctmc-transition","kind":"node","lean":"ProbabilityTheory.IsDetailedBalance.ctmcTransition","parent":19697,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/index.html#ctmc-transition","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con","status":"fully_proved","summary":"Detailed balance for `Q` implies `\u03c0 i P t i j = \u03c0 j P t j i` for every `t` (Norris 1997, \u00a73.7). For a probability vector `\u03c0` and a rate matrix `Q` this is the reversibility of the chain started from `\u03c0`.","title":"Detailed balance propagates to the transition function"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/vec-mul-ctmc-transition","kind":"node","lean":"ProbabilityTheory.IsDetailedBalance.vecMul_ctmcTransition","parent":19697,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con/index.html#vec-mul-ctmc-transition","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/stationarity-and-detailed-balance-for-finite-con","status":"fully_proved","summary":"Detailed balance with a matrix whose rows sum to `0` (for instance a rate matrix) makes `\u03c0` invariant for every `P t`.","title":"Detailed balance implies invariance for the transition semigroup"},{"data_url":"scopes/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state.json","id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","kind":"scope","navigation":"module","parent":19633,"result":false,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","title":"The sticky (equal-input) kernel on a finite state space"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/equal-input","kind":"node","lean":"ProbabilityTheory.equalInput","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#equal-input","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"The equal-input matrix `\ud835\udfd9 \u03c0\u1d40`: every row equals `\u03c0`.","title":"Equal-input matrix"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel","kind":"node","lean":"ProbabilityTheory.stickyKernel","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"The sticky kernel with persistence `c`: `c \u2022 1 + (1 - c) \u2022 \ud835\udfd9 \u03c0\u1d40`. Started from `i`, it stays at `i` with probability `c` and otherwise redraws the state from `\u03c0`.","title":"Sticky kernel"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/equal-input-rate","kind":"node","lean":"ProbabilityTheory.equalInputRate","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#equal-input-rate","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"The equal-input generator `\u03ba \u2022 (\ud835\udfd9 \u03c0\u1d40 - 1)` with rate `\u03ba` (Felsenstein's F81 model on an arbitrary finite state space): jumps at rate `\u03ba` to a state drawn from `\u03c0`.","title":"Equal-input generator (F81 model)"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/equal-input-apply","kind":"node","lean":"ProbabilityTheory.equalInput_apply","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#equal-input-apply","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"Entrywise form of the equal-input matrix.","title":"Entries of the equal-input matrix"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/equal-input-mul-self","kind":"node","lean":"ProbabilityTheory.equalInput_mul_self","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#equal-input-mul-self","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"With `\u2211 \u03c0 = 1`, `\ud835\udfd9 \u03c0\u1d40` is idempotent.","title":"Equal-input matrix is idempotent"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-apply","kind":"node","lean":"ProbabilityTheory.stickyKernel_apply","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-apply","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"Entrywise form of the sticky kernel.","title":"Entries of the sticky kernel"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-one","kind":"node","lean":"ProbabilityTheory.stickyKernel_one","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-one","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"The sticky kernel at persistence `1` is the identity.","title":"Sticky kernel at persistence one is the identity"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-zero","kind":"node","lean":"ProbabilityTheory.stickyKernel_zero","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-zero","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"The sticky kernel at persistence `0` is `\ud835\udfd9 \u03c0\u1d40`.","title":"Sticky kernel at persistence zero is the equal-input matrix"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-nonneg","kind":"node","lean":"ProbabilityTheory.stickyKernel_nonneg","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-nonneg","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"For `0 \u2264 c \u2264 1` and `\u03c0 \u2265 0` the entries of the sticky kernel are nonnegative.","title":"Nonnegativity of the sticky kernel"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-reversible","kind":"node","lean":"ProbabilityTheory.stickyKernel_reversible","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-reversible","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"`\u03c0 i K i j = \u03c0 j K j i` (detailed balance), for all real `c`, `\u03c0`.","title":"Reversibility"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/continuous-sticky-kernel","kind":"node","lean":"ProbabilityTheory.continuous_stickyKernel","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#continuous-sticky-kernel","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"The sticky kernel depends continuously on the persistence.","title":"Continuity of the sticky kernel in the persistence"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/tendsto-sticky-kernel-at-top","kind":"node","lean":"ProbabilityTheory.tendsto_stickyKernel_atTop","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#tendsto-sticky-kernel-at-top","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"For `\u03ba > 0`, `K (e^{-\u03ba t}) \u2192 \ud835\udfd9 \u03c0\u1d40` as `t \u2192 \u221e`.","title":"Long-distance limit"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/tendsto-sticky-kernel-nhds-zero","kind":"node","lean":"ProbabilityTheory.tendsto_stickyKernel_nhds_zero","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#tendsto-sticky-kernel-nhds-zero","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"`K (e^{-\u03ba t}) \u2192 1` as `t \u2192 0`, for every real `\u03ba`.","title":"Short-distance limit"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sum-sticky-kernel","kind":"node","lean":"ProbabilityTheory.sum_stickyKernel","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sum-sticky-kernel","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"With `\u2211 \u03c0 = 1`, the rows of the sticky kernel sum to `1` (for every real `c`).","title":"Rows of the sticky kernel sum to one"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-mem-row-stochastic","kind":"node","lean":"ProbabilityTheory.stickyKernel_mem_rowStochastic","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-mem-row-stochastic","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"For `0 \u2264 c \u2264 1`, `\u03c0 \u2265 0` and `\u2211 \u03c0 = 1` the sticky kernel is row-stochastic.","title":"Sticky kernel is row-stochastic"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/vec-mul-sticky-kernel","kind":"node","lean":"ProbabilityTheory.vecMul_stickyKernel","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#vec-mul-sticky-kernel","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"With `\u2211 \u03c0 = 1`, `\u03c0` is a stationary vector of the sticky kernel: `\u03c0 K = \u03c0`.","title":"Stationary vector of the sticky kernel"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-mul","kind":"node","lean":"ProbabilityTheory.stickyKernel_mul","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-mul","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"With `\u2211 \u03c0 = 1`, `K c * K c' = K (c c')` for all real `c`, `c'`.","title":"Composition"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-pow","kind":"node","lean":"ProbabilityTheory.stickyKernel_pow","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-pow","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"With `\u2211 \u03c0 = 1`, `K c ^ n = K (c ^ n)`: `n` steps of the sticky kernel are one step with persistence `c ^ n`.","title":"Powers of the sticky kernel"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/is-rate-matrix-equal-input-rate","kind":"node","lean":"ProbabilityTheory.isRateMatrix_equalInputRate","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#is-rate-matrix-equal-input-rate","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"For `\u03ba \u2265 0`, `\u03c0 \u2265 0` and `\u2211 \u03c0 = 1`, the equal-input generator is a rate matrix.","title":"Equal-input generator is a rate matrix"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/ctmc-transition-equal-input-rate","kind":"node","lean":"ProbabilityTheory.ctmcTransition_equalInputRate","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#ctmc-transition-equal-input-rate","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"With `\u2211 \u03c0 = 1`, `exp (t \u2022 \u03ba (\ud835\udfd9 \u03c0\u1d40 - 1)) = K (e^{-\u03ba t})` for all real `\u03ba` and `t`.","title":"The sticky kernel is the equal-input transition function"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/sticky-kernel-exp-add","kind":"node","lean":"ProbabilityTheory.stickyKernel_exp_add","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#sticky-kernel-exp-add","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"Semigroup law of the sticky kernel in time: `K (e^{-\u03ba s}) * K (e^{-\u03ba t}) = K (e^{-\u03ba (s + t)})` (with `\u2211 \u03c0 = 1`).","title":"Semigroup law"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/covariance-sticky-kernel","kind":"node","lean":"ProbabilityTheory.covariance_stickyKernel","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#covariance-sticky-kernel","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"If `(X, Y)` has joint law `P(X = i, Y = j) = \u03c0 i K i j` (stationary start, one step of the kernel), then `Cov(f X, g Y) = c Cov_\u03c0(f, g)`. The algebraic identity holds for all real `\u03c0`, `f`, `g`, `c`; it is a covariance of a probability law when `\u03c0` is a probability vector and `0 \u2264 c \u2264 1`.","title":"Covariance under the sticky kernel"},{"id":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/covariance-indicator-sticky-kernel-two","kind":"node","lean":"ProbabilityTheory.covariance_indicator_stickyKernel_two","parent":19703,"result":true,"route":"../roadmap/Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state/index.html#covariance-indicator-sticky-kernel-two","scope":"Probability/markov-chains/finite-continuous-time-markov-chains/the-sticky-equal-input-kernel-on-a-finite-state","status":"fully_proved","summary":"Two-state ancestry covariance (the local-ancestry chain of a two-way pulse admixture). With ancestry proportions `\u03c0 = (\u03b1, 1 - \u03b1)` and persistence `c = (1 - r) ^ n` (no recombination between the two sites along the `n` meioses since admixture, recombination fraction `r` per meiosis), the indicators `I_x = [X = 0]`, `I_y = [Y = 0]` of ancestry `0` at the two sites have covariance `\u03b1 (1 - \u03b1) (1 - r) ^ n`. The kernel\u2026","title":"Two-state ancestry covariance"},{"data_url":"scopes/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain.json","id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","title":"Finite-dimensional distributions of Markov chains"},{"id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/markov-fdd","kind":"node","lean":"MathlibExt.Probability.MarkovChainFDD.markovFDD","parent":19727,"result":true,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html#markov-fdd","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","summary":"For a measure $\\mu$ on a measurable space $S$, a kernel $\\kappa$ from $S$ to $S$, a natural number $n$ and sets $A_0, \\dots, A_n \\subseteq S$, the finite-dimensional distribution $\\mathrm{FDD}_n(\\mu, \\kappa; A_0, \\dots, A_n) \\in [0, \\infty]$ is defined by recursion on $n$: $\\mathrm{FDD}_0(\\mu, \\kappa; A_0) = \\mu(A_0)$ and\u2026","title":"Finite-dimensional distribution of a Markov chain"},{"id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/markov-fdd-measurable","kind":"node","lean":"MathlibExt.Probability.MarkovChainFDD.markovFDD_measurable","parent":19727,"result":true,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html#markov-fdd-measurable","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","summary":"Let $\\kappa$ be a Markov kernel on a measurable space $S$, let $n$ be a natural number and let $A_0, \\dots, A_n$ be measurable subsets of $S$. Write $\\mathrm{FDD}_n(\\mu, \\kappa; A_0, \\dots, A_n)$ for the finite-dimensional distribution of a Markov chain with initial measure $\\mu$ and transition kernel $\\kappa$. Then the map $x \\mapsto \\mathrm{FDD}_n(\\kappa(x), \\kappa; A_0, \\dots, A_n)$ from $S$ to $[0, \\infty]$ is\u2026","title":"Measurability of the finite-dimensional distributions in the starting state"},{"id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/comap-sigma-mono","kind":"node","lean":"MathlibExt.Probability.MarkovChainFDD.comap_sigma_mono","parent":19727,"result":true,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html#comap-sigma-mono","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be maps from a measurable space $\\Omega$ to a measurable space $S$, and for each $k$ let $\\sigma(X_0, \\dots, X_k)$ be the \u03c3-algebra on $\\Omega$ generated by $\\omega \\mapsto (X_0(\\omega), \\dots, X_k(\\omega))$, with the product \u03c3-algebra on $S^{k+1}$. Then $\\sigma(X_0, \\dots, X_k) \\subseteq \\sigma(X_0, \\dots, X_{k+1})$ for every $k$.","title":"The \u03c3-algebras generated by a sequence increase"},{"id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/comap-inter-preimage","kind":"node","lean":"MathlibExt.Probability.MarkovChainFDD.comap_inter_preimage","parent":19727,"result":true,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html#comap-inter-preimage","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be maps from a measurable space $\\Omega$ to a measurable space $S$, and write $\\sigma(X_0, \\dots, X_k)$ for the \u03c3-algebra on $\\Omega$ generated by $\\omega \\mapsto (X_0(\\omega), \\dots, X_k(\\omega))$, with the product \u03c3-algebra on $S^{k+1}$. If $C$ is measurable with respect to $\\sigma(X_0, \\dots, X_k)$ and $B \\subseteq S$ is measurable, then $C \\cap X_{k+1}^{-1}(B)$ is measurable with respect\u2026","title":"Adding the next step to a past event"},{"id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/lintegral-measurable-extension","kind":"node","lean":"MathlibExt.Probability.MarkovChainFDD.lintegral_measurable_extension","parent":19727,"result":true,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html#lintegral-measurable-extension","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","summary":"Let $P$ be a probability measure on a measurable space $\\Omega$, let $X_0, X_1, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$, and let $\\kappa$ be a Markov kernel on $S$. Write $\\sigma(X_0, \\dots, X_k)$ for the \u03c3-algebra on $\\Omega$ generated by $\\omega \\mapsto (X_0(\\omega), \\dots, X_k(\\omega))$, with the product \u03c3-algebra on $S^{k+1}$. Suppose that for every $k$, every measurable\u2026","title":"The Markov property for measurable functions"},{"id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/set-lintegral-inter-preimage","kind":"node","lean":"MathlibExt.Probability.MarkovChainFDD.set_lintegral_inter_preimage","parent":19727,"result":true,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html#set-lintegral-inter-preimage","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","summary":"Let $P$ be a measure on a measurable space $\\Omega$, let $X : \\Omega \\to S$ be a measurable map to a measurable space $S$, let $A \\subseteq S$ be measurable, let $C \\subseteq \\Omega$ be any set and let $f : S \\to [0, \\infty]$ be any function. Then $\\int_{C \\cap X^{-1}(A)} f(X(\\omega)) \\, dP(\\omega) = \\int_C \\mathbf{1}_A(X(\\omega)) \\, f(X(\\omega)) \\, dP(\\omega)$.","title":"Restricting an integral to a preimage"},{"id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/markov-chain-fdd-aux","kind":"node","lean":"MathlibExt.Probability.MarkovChainFDD.markov_chain_fdd_aux","parent":19727,"result":true,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html#markov-chain-fdd-aux","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","summary":"Let $P$ be a probability measure on a measurable space $\\Omega$, let $X_0, X_1, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$, and let $\\kappa$ be a Markov kernel on $S$. Write $\\sigma(X_0, \\dots, X_k)$ for the \u03c3-algebra on $\\Omega$ generated by $\\omega \\mapsto (X_0(\\omega), \\dots, X_k(\\omega))$, with the product \u03c3-algebra on $S^{k+1}$, and $\\mathrm{FDD}_n(\\mu, \\kappa; A_0, \\dots, A_n)$ for the\u2026","title":"Finite-dimensional distributions given the past"},{"id":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/markov-chain-finite-dimensional-distribu","kind":"node","lean":"MathlibExt.Probability.MarkovChainFDD.markov_chain_finite_dimensional_distributions","parent":19727,"result":true,"route":"../roadmap/Probability/markov-chains/finite-dimensional-distributions-of-markov-chain/index.html#markov-chain-finite-dimensional-distribu","scope":"Probability/markov-chains/finite-dimensional-distributions-of-markov-chain","status":"fully_proved","summary":"Let $P$ be a probability measure on a measurable space $\\Omega$, let $X_0, X_1, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$, and let $\\kappa$ be a Markov kernel on $S$. Write $\\sigma(X_0, \\dots, X_k)$ for the \u03c3-algebra on $\\Omega$ generated by $\\omega \\mapsto (X_0(\\omega), \\dots, X_k(\\omega))$, with the product \u03c3-algebra on $S^{k+1}$. Suppose $X$ is a Markov chain with transition kernel\u2026","title":"Finite-dimensional distributions of a Markov chain"},{"data_url":"scopes/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility.json","id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","title":"Kolmogorov's criterion for reversibility"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/stochastic-matrix","kind":"node","lean":"StochasticMatrix","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#stochastic-matrix","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"For a natural number $M$, a stochastic matrix of size $M + 1$ is a square matrix $P = (P_{ij})$ indexed by the states $\\{0, 1, \\dots, M\\}$, with real entries $P_{ij} \\ge 0$ for all states $i, j$, whose rows sum to $1$: $\\sum_j P_{ij} = 1$ for every $i$. It is the transition matrix of a Markov chain on these $M + 1$ states.","title":"Stochastic matrix"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/mat-pow","kind":"node","lean":"StochasticMatrix.matPow","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#mat-pow","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"For a stochastic matrix $P$ on the finite state space $\\{0, 1, \\dots, M\\}$ and $n \\in \\mathbb{N}$, the $n$-th matrix power $P^n$ is defined by $P^0 = I$, the identity matrix, and $(P^{n+1})_{ij} = \\sum_k P(i, k) (P^n)_{kj}$. The entry $(P^n)_{ij}$ is the probability of moving from state $i$ to state $j$ in exactly $n$ steps.","title":"Matrix powers of a stochastic matrix"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/is-irreducible","kind":"node","lean":"StochasticMatrix.IsIrreducible","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#is-irreducible","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix on the finite state space $\\{0, 1, \\dots, M\\}$ and write $P^n$ for its $n$-th matrix power. Then $P$ is irreducible if for all states $i$ and $j$ there is an $n \\ge 1$ with $(P^n)_{ij} > 0$; that is, every state can be reached from every state, itself included, in a positive number of steps.","title":"Irreducible stochastic matrix"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/distribution","kind":"node","lean":"StochasticMatrix.Distribution","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#distribution","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"A probability distribution on the finite state space $\\{0, 1, \\dots, M\\}$ is a function $\\pi$ from the states to $\\mathbb{R}$ with $\\pi(i) \\ge 0$ for every state $i$ and $\\sum_i \\pi(i) = 1$.","title":"Probability distribution on a finite state space"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/is-stationary","kind":"node","lean":"StochasticMatrix.IsStationary","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#is-stationary","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"A probability distribution $\\pi$ on the finite state space $\\{0, 1, \\dots, M\\}$ is stationary for a stochastic matrix $P$ on the same states if $\\pi P = \\pi$, that is, $\\sum_i \\pi(i) P(i, j) = \\pi(j)$ for every state $j$.","title":"Stationary distribution"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/detailed-balance","kind":"node","lean":"StochasticMatrix.DetailedBalance","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#detailed-balance","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"A probability distribution $\\pi$ and a stochastic matrix $P$ on the finite state space $\\{0, 1, \\dots, M\\}$ satisfy the detailed balance (reversibility) equations when $\\pi(i) P(i, j) = \\pi(j) P(j, i)$ for all states $i$ and $j$.","title":"Detailed balance"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/is-reversible","kind":"node","lean":"StochasticMatrix.IsReversible","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#is-reversible","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"A stochastic matrix $P$ on the finite state space $\\{0, 1, \\dots, M\\}$ is reversible if there is a probability distribution $\\pi$ on the states that is stationary for $P$ and satisfies the detailed balance equations $\\pi(i) P(i, j) = \\pi(j) P(j, i)$ for all states $i, j$.","title":"Reversible stochastic matrix"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/kolmogorov-cycle-condition","kind":"node","lean":"StochasticMatrix.KolmogorovCycleCondition","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#kolmogorov-cycle-condition","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"A stochastic matrix $P$ on the finite state space $\\{0, 1, \\dots, M\\}$ satisfies Kolmogorov's cycle condition when (1) for all states $x, y$, $P(x, y) > 0$ implies $P(y, x) > 0$; and (2) for every $n \\ge 0$ and every loop of states $x_0, x_1, \\dots, x_n$ with $x_n = x_0$ and $\\prod_{i=0}^{n-1} P(x_{i+1}, x_i) > 0$, we have $\\prod_{i=0}^{n-1} P(x_i, x_{i+1}) = \\prod_{i=0}^{n-1} P(x_{i+1}, x_i)$.","title":"Kolmogorov's cycle condition"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/detailed-balance-is-stationary","kind":"node","lean":"StochasticMatrix.detailedBalance_isStationary","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#detailed-balance-is-stationary","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix on the finite state space $\\{0, 1, \\dots, M\\}$ and let $\\pi$ be a probability distribution on the same states. If $\\pi$ and $P$ satisfy the detailed balance equations $\\pi(i) P(i, j) = \\pi(j) P(j, i)$ for all states $i, j$, then $\\pi$ is stationary for $P$: $\\sum_i \\pi(i) P(i, j) = \\pi(j)$ for every state $j$.","title":"Detailed balance implies stationarity"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/mat-pow-nonneg","kind":"node","lean":"StochasticMatrix.matPow_nonneg","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#mat-pow-nonneg","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Every entry of every power of a stochastic matrix is nonnegative.","title":"Powers of a stochastic matrix are nonnegative"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/distribution-pos-of-irreducible-detailed","kind":"node","lean":"StochasticMatrix.distribution_pos_of_irreducible_detailedBalance","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#distribution-pos-of-irreducible-detailed","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be an irreducible stochastic matrix on the finite state space $\\{0, 1, \\dots, M\\}$. If a probability distribution $\\pi$ satisfies the detailed balance equations $\\pi(i) P(i, j) = \\pi(j) P(j, i)$ for all states $i, j$, then $\\pi(i) > 0$ for every state $i$.","title":"Detailed balance distributions of irreducible chains are positive"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/kolmogorov-cycle-necessity","kind":"node","lean":"StochasticMatrix.kolmogorov_cycle_necessity","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#kolmogorov-cycle-necessity","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be an irreducible stochastic matrix on the finite state space $\\{0, 1, \\dots, M\\}$. If $P$ is reversible, that is, some probability distribution $\\pi$ is stationary for $P$ and satisfies the detailed balance equations $\\pi(i) P(i, j) = \\pi(j) P(j, i)$ for all states $i, j$, then $P$ satisfies Kolmogorov's cycle condition: (1) for all states $x, y$, $P(x, y) > 0$ implies $P(y, x) > 0$; and (2) for every\u2026","title":"Kolmogorov's cycle theorem, necessity half"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-path-product","kind":"node","lean":"StochasticMatrix.listPathProduct","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-path-product","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"For a stochastic matrix $P$ and a path of states $x_0, x_1, \\dots, x_k$, the path product is the real number $\\prod_{i=0}^{k-1} P(x_i, x_{i+1})$, the product of the forward transition probabilities along the path. For the empty path and for a path consisting of a single state it is $1$.","title":"Forward path product"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-path-product-nonneg","kind":"node","lean":"StochasticMatrix.listPathProduct_nonneg","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-path-product-nonneg","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix on the states $\\{0, \\dots, M\\}$. For every path (finite list of states), the product of the forward transition probabilities $P(x, y)$ over consecutive states $x, y$ along the path is nonnegative.","title":"Path products are nonnegative"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-ratio-product","kind":"node","lean":"StochasticMatrix.listRatioProduct","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-ratio-product","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"For a stochastic matrix $P$ and a path of states $x_0, x_1, \\dots, x_k$, the ratio product is the real number $\\prod_{i=0}^{k-1} \\frac{P(x_i, x_{i+1})}{P(x_{i+1}, x_i)}$, where a quotient with denominator $0$ is taken to be $0$. For the empty path and for a path consisting of a single state it is $1$.","title":"Ratio product along a path"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-ratio-product-nonneg","kind":"node","lean":"StochasticMatrix.listRatioProduct_nonneg","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-ratio-product-nonneg","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"For a stochastic matrix $P$, the ratio product along every path of states is nonnegative.","title":"The ratio product is nonnegative"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-ratio-product-pos-of-list-path-prod","kind":"node","lean":"StochasticMatrix.listRatioProduct_pos_of_listPathProduct_pos","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-ratio-product-pos-of-list-path-prod","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix such that $P(x, y) > 0$ implies $P(y, x) > 0$ for all states $x$ and $y$. If the forward path product along a path of states is positive, then the ratio product along that path is positive.","title":"Positive paths have positive ratio product"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/exists-path-of-mat-pow-pos","kind":"node","lean":"StochasticMatrix.exists_path_of_matPow_pos","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#exists-path-of-mat-pow-pos","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix on the states $\\{0, \\dots, M\\}$. If $n > 0$ and the $(i, j)$ entry of the $n$-th matrix power $P^n$ is positive, then there is a path from $i$ to $j$ (a finite list of states starting at $i$ and ending at $j$) along which the product of the transition probabilities $P(x, y)$ over consecutive states $x, y$ is positive.","title":"Paths from positive matrix powers"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/exists-path","kind":"node","lean":"StochasticMatrix.IsIrreducible.exists_path","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#exists-path","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be an irreducible stochastic matrix on the states $\\{0, \\dots, M\\}$. For any two states $i, j$ there is a path from $i$ to $j$ (a finite list of states starting at $i$ and ending at $j$) along which the product of the transition probabilities $P(x, y)$ over consecutive states $x, y$ is strictly positive.","title":"Irreducible chains have positive paths"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-ratio-product-cons-of-fn","kind":"node","lean":"StochasticMatrix.listRatioProduct_cons_ofFn","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-ratio-product-cons-of-fn","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix, $n$ a natural number and $c_0, c_1, \\dots, c_{n+1}$ states. Then the ratio product along $c_0, c_1, \\dots, c_{n+1}$ equals $\\frac{P(c_0, c_1)}{P(c_1, c_0)}$ times the ratio product along $c_1, \\dots, c_{n+1}$, where a quotient with denominator $0$ is taken to be $0$.","title":"First step of the ratio product"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-ratio-product-of-fn","kind":"node","lean":"StochasticMatrix.listRatioProduct_ofFn","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-ratio-product-of-fn","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix, $n$ a natural number and $c_0, c_1, \\dots, c_n$ states. Then the ratio product along $c_0, c_1, \\dots, c_n$ equals $\\prod_{i=0}^{n-1} \\frac{P(c_i, c_{i+1})}{P(c_{i+1}, c_i)}$, where a quotient with denominator $0$ is taken to be $0$.","title":"The ratio product as a finite product"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-ratio-product-cycle-eq-one","kind":"node","lean":"StochasticMatrix.listRatioProduct_cycle_eq_one","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-ratio-product-cycle-eq-one","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix satisfying Kolmogorov's cycle condition, and let $x_0, x_1, \\dots, x_n$ be a path of states with $n \\ge 1$ and $x_0 = x_n$ such that $P(x_{i+1}, x_i) > 0$ for every $i < n$. Then the ratio product along $x_0, x_1, \\dots, x_n$ equals $1$.","title":"The ratio product around a loop is one"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/cycle-weight","kind":"node","lean":"StochasticMatrix.cycleWeight","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#cycle-weight","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be an irreducible stochastic matrix on the states $\\{0, \\dots, M\\}$ and fix a reference state $i_0$. The cycle weight of a state $j$ relative to $i_0$ is the product of the ratios $P(x, y) / P(y, x)$ over consecutive states $x, y$ along a path (a finite list of states) from $i_0$ to $j$ whose product of transition probabilities $P(x, y)$ over consecutive states is positive, where a ratio with $P(y, x) = 0$\u2026","title":"Cycle weight"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-ratio-product-append","kind":"node","lean":"StochasticMatrix.listRatioProduct_append","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-ratio-product-append","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix and let $x_0, \\dots, x_k$ and $y_0, \\dots, y_l$ be paths of states with $x_k = y_0$. Then the ratio product along the joined path $x_0, \\dots, x_k, y_1, \\dots, y_l$ equals the ratio product along $x_0, \\dots, x_k$ times the ratio product along $y_0, \\dots, y_l$.","title":"Ratio product of joined paths"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-path-product-pos-implies-transition","kind":"node","lean":"StochasticMatrix.listPathProduct_pos_implies_transitions","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-path-product-pos-implies-transition","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix on the states $\\{0, \\dots, M\\}$, and let $l$ be a path (a finite list of states) whose $i$-th state, counting from $0$, is $l_i$. If the product of the transition probabilities $P(x, y)$ over consecutive states $x, y$ along $l$ is positive, then $P(l_i, l_{i+1}) > 0$ for every $i$ such that $i + 1$ is less than the length of $l$.","title":"Positive path products have positive steps"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-path-product-append-pos","kind":"node","lean":"StochasticMatrix.listPathProduct_append_pos","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-path-product-append-pos","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix on the states $\\{0, \\dots, M\\}$, and let $l_1, l_2$ be nonempty paths (finite lists of states) such that the last state of $l_1$ is the first state of $l_2$. If the products of the transition probabilities $P(x, y)$ over consecutive states $x, y$ along $l_1$ and along $l_2$ are both positive, then so is the product along the path that follows $l_1$ and then $l_2$ with its first state\u2026","title":"Joining positive paths"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/cycle-weight-path-independent","kind":"node","lean":"StochasticMatrix.cycleWeight_path_independent","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#cycle-weight-path-independent","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be an irreducible stochastic matrix on the states $\\{0, \\dots, M\\}$ satisfying Kolmogorov's cycle condition, and let $i_0, j$ be states. If two paths (finite lists of states) both start at $i_0$, end at $j$, and have positive product of the transition probabilities $P(x, y)$ over consecutive states $x, y$, then the products of the ratios $P(x, y) / P(y, x)$ over consecutive states $x, y$ along the two paths\u2026","title":"Path independence of cycle weight"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/list-ratio-product-snoc","kind":"node","lean":"StochasticMatrix.listRatioProduct_snoc","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#list-ratio-product-snoc","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be a stochastic matrix, $x_0, \\dots, x_k$ a path of states and $j$ a state. Then the ratio product along $x_0, \\dots, x_k, j$ equals the ratio product along $x_0, \\dots, x_k$ times $\\frac{P(x_k, j)}{P(j, x_k)}$, where a quotient with denominator $0$ is taken to be $0$.","title":"Ratio product after appending a state"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/cycle-weight-extend","kind":"node","lean":"StochasticMatrix.cycleWeight_extend","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#cycle-weight-extend","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be an irreducible stochastic matrix on the states $\\{0, \\dots, M\\}$ satisfying Kolmogorov's cycle condition, fix a reference state $i_0$, and write $w(j)$ for the cycle weight of a state $j$ relative to $i_0$. If $i, j$ are states with $P(i, j) > 0$, then $w(i) \\cdot P(i, j) / P(j, i) = w(j)$.","title":"Cycle weights along a positive transition"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/cycle-weight-detailed-balance","kind":"node","lean":"StochasticMatrix.cycleWeight_detailed_balance","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#cycle-weight-detailed-balance","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"Let $P$ be an irreducible stochastic matrix on the states $\\{0, \\dots, M\\}$ satisfying Kolmogorov's cycle condition, fix a reference state $i_0$, and write $w(j)$ for the cycle weight of a state $j$ relative to $i_0$. Then for all states $i, j$ we have $w(i) \\cdot P(i, j) = w(j) \\cdot P(j, i)$.","title":"Detailed balance for cycle weights"},{"id":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/kolmogorov-cycle-sufficiency","kind":"node","lean":"StochasticMatrix.kolmogorov_cycle_sufficiency","parent":19736,"result":true,"route":"../roadmap/Probability/markov-chains/kolmogorov-s-criterion-for-reversibility/index.html#kolmogorov-cycle-sufficiency","scope":"Probability/markov-chains/kolmogorov-s-criterion-for-reversibility","status":"fully_proved","summary":"For an irreducible finite Markov chain satisfying the cycle condition, there exists a reversible measure: a distribution satisfying the detailed balance equations (and hence stationary).","title":"Kolmogorov's cycle theorem, sufficiency"},{"data_url":"scopes/Probability/markov-chains/finite-markov-chains.json","id":"Probability/markov-chains/finite-markov-chains","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/finite-markov-chains/index.html","scope":"Probability/markov-chains/finite-markov-chains","status":"fully_proved","title":"Finite Markov chains"},{"id":"Probability/markov-chains/finite-markov-chains/is-finite-markov-chain","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChain.IsFiniteMarkovChain","parent":19768,"result":true,"route":"../roadmap/Probability/markov-chains/finite-markov-chains/index.html#is-finite-markov-chain","scope":"Probability/markov-chains/finite-markov-chains","status":"fully_proved","summary":"`IsFiniteMarkovChain X P \u2131 \u03bc` asserts that the sequence `X : \u2115 \u2192 \u03a9 \u2192 Fin (M+1)` is a Markov chain with transition matrix `P`, adapted to the filtration `\u2131`, under the measure `\u03bc`. Concretely: each `X n` is `\u2131 n`-measurable, and the conditional probability that `X (n+1) = j` given `\u2131 n` equals `P.prob (X n) j`, matching the Markov property `P(X_{n+1} = j | X_n = i, ...) = P_{ij}`. Ported verbatim from\u2026","title":"Finite-state Markov chain with a given transition matrix"},{"id":"Probability/markov-chains/finite-markov-chains/finite-chain-kernel","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChain.finiteChainKernel","parent":19768,"result":true,"route":"../roadmap/Probability/markov-chains/finite-markov-chains/index.html#finite-chain-kernel","scope":"Probability/markov-chains/finite-markov-chains","status":"fully_proved","summary":"The transition kernel of a stochastic matrix: `finiteChainKernel P x` is the probability measure on `Fin (M + 1)` putting mass `P.prob x j` on each state `j`. This is the kernel form of the transition matrix.","title":"Transition kernel of a stochastic matrix"},{"id":"Probability/markov-chains/finite-markov-chains/finite-chain-kernel-apply","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChain.finiteChainKernel_apply","parent":19768,"result":true,"route":"../roadmap/Probability/markov-chains/finite-markov-chains/index.html#finite-chain-kernel-apply","scope":"Probability/markov-chains/finite-markov-chains","status":"fully_proved","summary":"The kernel of a stochastic matrix evaluated at a finite set is the sum of the transition probabilities into that set.","title":"Transition kernel of a stochastic matrix evaluated on a set"},{"id":"Probability/markov-chains/finite-markov-chains/cond-exp-indicator","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChain.IsFiniteMarkovChain.condExp_indicator","parent":19768,"result":true,"route":"../roadmap/Probability/markov-chains/finite-markov-chains/index.html#cond-exp-indicator","scope":"Probability/markov-chains/finite-markov-chains","status":"fully_proved","summary":"Bridge: finite-state Markov chains satisfy the general one-step Markov property. For a finite Markov chain, the conditional expectation of the indicator of `X (n+1) \u2208 B` given `\u2131 n` is the kernel of `P` evaluated at `X n`, almost surely. This is the one-step form of `v1`'s general `IsMarkovChain.markov_prop` specialized to the matrix kernel. No measurability hypothesis on `B` is needed: the indicator is defined for\u2026","title":"Bridge: finite-state Markov chains satisfy the general one-step Markov property"},{"id":"Probability/markov-chains/finite-markov-chains/to-is-markov-chain","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChain.IsFiniteMarkovChain.toIsMarkovChain","parent":19768,"result":true,"route":"../roadmap/Probability/markov-chains/finite-markov-chains/index.html#to-is-markov-chain","scope":"Probability/markov-chains/finite-markov-chains","status":"fully_proved","summary":"Convert the finite singleton-probability specification to the canonical general Markov-chain predicate under the probability-space hypotheses of `condExp_indicator`.","title":"Finite Markov chains satisfy the general Markov property"},{"id":"Probability/markov-chains/finite-markov-chains/map-succ-eq-bind","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChain.IsFiniteMarkovChain.map_succ_eq_bind","parent":19768,"result":true,"route":"../roadmap/Probability/markov-chains/finite-markov-chains/index.html#map-succ-eq-bind","scope":"Probability/markov-chains/finite-markov-chains","status":"fully_proved","summary":"The law of `X (n+1)` is the law of `X n` transported by the transition kernel of `P`.","title":"One-step forward equation"},{"data_url":"scopes/Probability/markov-chains/construction-of-markov-chains.json","id":"Probability/markov-chains/construction-of-markov-chains","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","title":"Construction of Markov chains"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-lift-kernel","kind":"node","lean":"MathlibExt.Probability.MarkovChainConstruction.markovLiftKernel","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-lift-kernel","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"For a kernel $\\kappa$ from a measurable space $S$ to itself and $n \\in \\mathbb{N}$, the lifted kernel from $S^{\\{0, \\dots, n\\}}$ to $S$ sends a history $(x_0, \\dots, x_n)$ to $\\kappa(x_n, \\cdot)$: it reads off the last coordinate and applies $\\kappa$.","title":"Lifting a kernel to histories"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-chain-measure","kind":"node","lean":"MathlibExt.Probability.MarkovChainConstruction.markovChainMeasure","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-chain-measure","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $\\kappa$ be a Markov kernel from a measurable space $S$ to itself and let $\\mu$ be a measure on $S$. The canonical Markov chain measure $P_\\mu$ is the measure on the path space $S^{\\mathbb{N}}$ given by the Ionescu\u2013Tulcea theorem for the initial measure $\\mu$ and, at each time $n$, the kernel sending a history $(x_0, \\dots, x_n)$ to $\\kappa(x_n, \\cdot)$. When $\\mu$ is a probability measure, $P_\\mu$ is the law\u2026","title":"Canonical Markov chain measure"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-chain-measure-initial-distributio","kind":"node","lean":"MathlibExt.Probability.MarkovChainConstruction.markovChainMeasure_initial_distribution","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-chain-measure-initial-distributio","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $\\kappa$ be a Markov kernel from a measurable space $S$ to itself and let $\\mu$ be a measure on $S$. Under the canonical Markov chain measure $P_\\mu$ on $S^{\\mathbb{N}}$ with initial measure $\\mu$ and transition kernel $\\kappa$, the law of the initial coordinate is $\\mu$: the pushforward of $P_\\mu$ along $\\omega \\mapsto \\omega_0$ equals $\\mu$.","title":"Initial distribution of the canonical Markov chain measure"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-chain-measure-cond-distrib","kind":"node","lean":"MathlibExt.Probability.MarkovChainConstruction.markovChainMeasure_condDistrib","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-chain-measure-cond-distrib","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $S$ be a nonempty standard Borel space, let $\\kappa$ be a Markov kernel from $S$ to $S$, let $\\mu$ be a probability measure on $S$, and let $P_\\mu$ be the canonical Markov chain measure on $S^{\\mathbb{N}}$ with initial distribution $\\mu$ and transition kernel $\\kappa$. Write $X_n(\\omega) = \\omega_n$. For every $n \\in \\mathbb{N}$, the regular conditional distribution of $X_{n+1}$ given $(X_0, \\dots, X_n)$ under\u2026","title":"Markov property of the canonical Markov chain measure"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-chain-construction","kind":"node","lean":"MathlibExt.Probability.MarkovChainConstruction.markov_chain_construction","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-chain-construction","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $S$ be a nonempty standard Borel space, let $\\kappa$ be a Markov kernel from $S$ to $S$ and let $\\mu$ be a probability measure on $S$. Then there is a probability measure $P$ on the path space $S^{\\mathbb{N}}$ such that, writing $X_n(\\omega) = \\omega_n$ for the coordinates, the law of $X_0$ under $P$ is $\\mu$, and for every $n \\in \\mathbb{N}$ the regular conditional distribution of $X_{n+1}$ given\u2026","title":"Markov chain construction theorem"},{"id":"Probability/markov-chains/construction-of-markov-chains/ae-eval-zero-markov-chain-measure","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChain.ae_eval_zero_markovChainMeasure","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#ae-eval-zero-markov-chain-measure","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"The canonical Markov law with initial distribution `dirac i` starts at `i` almost surely. Its initial marginal is supplied by the existing construction.","title":"Canonical Markov law started from a point starts there almost surely"},{"id":"Probability/markov-chains/construction-of-markov-chains/is-finite-markov-chain-markov-chain-meas","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChain.isFiniteMarkovChain_markovChainMeasure","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#is-finite-markov-chain-markov-chain-meas","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"The coordinate process under the existing canonical matrix-derived Markov law satisfies the original finite-state Markov-chain specification on `Filtration.piLE`.","title":"Canonical Markov law satisfies the finite Markov chain specification"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-lift-kernel-2","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.markovLiftKernel","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-lift-kernel-2","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"For a kernel $\\kappa$ on $S$ and $n \\in \\mathbb{N}$, this is the kernel from histories $(x_0, \\dots, x_n)$ of states in $S$, indexed by $\\{0, \\dots, n\\}$, to $S$ that sends $(x_0, \\dots, x_n)$ to $\\kappa(x_n, \\cdot)$. It is the Markov lift kernel of $\\kappa$ at $n$.","title":"Lifted transition kernel on histories"},{"id":"Probability/markov-chains/construction-of-markov-chains/partial-traj-succ-map-drop","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.partialTraj_succ_map_drop","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#partial-traj-succ-map-drop","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $\\kappa$ be a Markov kernel on a measurable space $S$, let $a, b \\in \\mathbb{N}$ and let $z = (z_0, \\dots, z_{a+b})$ be a history in $S^{\\{0, \\dots, a+b\\}}$. Let $\\mu_z$ be the one-step partial trajectory of the lifted kernel of $\\kappa$ from time $a + b$ to time $a + b + 1$ started at $z$, that is, the law of $(z_0, \\dots, z_{a+b}, X)$ where $X$ has law $\\kappa(z_{a+b})$. Then the image of $\\mu_z$ under the\u2026","title":"One-step shift of a Markov partial trajectory"},{"id":"Probability/markov-chains/construction-of-markov-chains/last-hist","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.lastHist","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#last-hist","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"For $a \\in \\mathbb{N}$ and a history $x = (x_0, \\dots, x_a)$ of states in $S$, indexed by $\\{0, \\dots, a\\}$, the last-coordinate history of $x$ is the history indexed by $\\{0\\}$ whose only entry is $x_a$.","title":"Last-coordinate history"},{"id":"Probability/markov-chains/construction-of-markov-chains/partial-traj-succ-map-drop-kernel","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.partialTraj_succ_map_drop_kernel","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#partial-traj-succ-map-drop-kernel","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $\\kappa$ be a Markov kernel on $S$ and $a, b \\in \\mathbb{N}$. For a history $z = (z_0, \\dots, z_{a+b})$ of states in $S$, extend $z$ by one step, drawing $z_{a+b+1}$ from $\\kappa(z_{a+b}, \\cdot)$, and then keep the entries at times $a, \\dots, a + b + 1$, reindexed as $0, \\dots, b + 1$. The law of the result equals the law obtained by first keeping the entries of $z$ at times $a, \\dots, a + b$, reindexed as\u2026","title":"One-step shift of the partial trajectory kernel"},{"id":"Probability/markov-chains/construction-of-markov-chains/partial-traj-map-drop","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.partialTraj_map_drop","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#partial-traj-map-drop","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $\\kappa$ be a Markov kernel on $S$ and $a, b \\in \\mathbb{N}$. For a history $x = (x_0, \\dots, x_a)$ of states in $S$, let $Q_x$ be the law of the history $(X_0, \\dots, X_{a+b})$ that equals $x$ at times $0, \\dots, a$ and then draws each next state from $\\kappa$ applied to the current state (the Ionescu\u2013Tulcea partial trajectory from time $a$ to time $a + b$). Then the image of $Q_x$ under\u2026","title":"Shifted finite-dimensional distributions"},{"id":"Probability/markov-chains/construction-of-markov-chains/shift-path","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.shiftPath","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#shift-path","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $S$ be a set and $n \\in \\mathbb{N}$. The time-$n$ shift on the path space $S^{\\mathbb{N}}$ sends a path $\\omega$ to the path $k \\mapsto \\omega_{n+k}$.","title":"Time shift on path space"},{"id":"Probability/markov-chains/construction-of-markov-chains/frestrict-le-comp-shift-path","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.frestrictLe_comp_shiftPath","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#frestrict-le-comp-shift-path","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"For all $n, b \\in \\mathbb{N}$ and every path $\\omega : \\mathbb{N} \\to S$, let $\\theta_n \\omega$ be the shifted path, $(\\theta_n \\omega)_k = \\omega_{n+k}$. The restriction of $\\theta_n \\omega$ to the times $0, \\dots, b$ equals the restriction of $\\omega$ to the times $0, \\dots, n + b$ with its entries at times $n, \\dots, n + b$ kept and reindexed as $0, \\dots, b$; that is, it is the tuple\u2026","title":"Finite restrictions of the shifted path"},{"id":"Probability/markov-chains/construction-of-markov-chains/traj-map-shift-path-eq","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.traj_map_shiftPath_eq","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#traj-map-shift-path-eq","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $\\kappa$ be a Markov kernel on a measurable space $S$, let $n \\in \\mathbb{N}$ and let $x = (x_0, \\dots, x_n) \\in S^{\\{0, \\dots, n\\}}$. Let $P_x$ be the trajectory measure on $S^{\\mathbb{N}}$ of the lifted kernel of $\\kappa$ started at time $n$ from the history $x$, and let $P_{x_n}$ be the trajectory measure of the lifted kernel started at time $0$ from the one-point history $x_n$. Then the image of $P_x$ under\u2026","title":"The shifted trajectory of a Markov chain restarts from its last state"},{"id":"Probability/markov-chains/construction-of-markov-chains/path-filtration","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.pathFiltration","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#path-filtration","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $S$ be a measurable space. The path filtration on the path space $S^{\\mathbb{N}}$ is the filtration indexed by $\\mathbb{N}$, relative to the product \u03c3-algebra, whose \u03c3-algebra at time $n$ is the one generated by the coordinate maps $\\omega \\mapsto \\omega_i$ for $i \\le n$.","title":"Path filtration"},{"id":"Probability/markov-chains/construction-of-markov-chains/path-filtration-le-pi","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.pathFiltration_le_pi","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#path-filtration-le-pi","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $S$ be a measurable space. For every $n \\in \\mathbb{N}$, the $n$-th \u03c3-algebra of the path filtration on $S^{\\mathbb{N}}$, generated by the coordinate maps $\\omega \\mapsto \\omega_i$ for $i \\le n$, is contained in the product \u03c3-algebra on $S^{\\mathbb{N}}$.","title":"The path filtration is contained in the product \u03c3-algebra"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-chain-path-measure","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.MarkovChainPathMeasure","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-chain-path-measure","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"For a measure $\\mu$ on $S$ and a Markov kernel $\\kappa$ on $S$, the path measure $P_\\mu$ is the measure on $S^{\\mathbb{N}}$, with the product $\\sigma$-algebra, obtained by integrating against $\\mu(dx)$ the Ionescu\u2013Tulcea trajectory measure that starts at $x$ at time $0$ and draws each next state from $\\kappa$ applied to the current state. It is the Markov chain measure of $\\kappa$ and $\\mu$; when $\\mu$ is a\u2026","title":"Markov chain path measure"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-expectation","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.markovExpectation","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-expectation","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"For a Markov kernel $\\kappa$ on $S$, a function $Y : S^{\\mathbb{N}} \\to \\mathbb{R}$ of paths and a state $x \\in S$, the expectation $E_x Y$ is the integral of $Y$ against the Markov chain path measure with transition kernel $\\kappa$ and initial distribution the Dirac measure at $x$.","title":"Expectation under the chain started from a point"},{"id":"Probability/markov-chains/construction-of-markov-chains/measurable-markov-expectation","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.measurable_markovExpectation","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#measurable-markov-expectation","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $S$ be a nonempty standard Borel space, $\\kappa$ a Markov kernel on $S$, and $Y : S^{\\mathbb{N}} \\to \\mathbb{R}$ a bounded function, measurable for the product $\\sigma$-algebra. Then the map $x \\mapsto E_x Y$, sending a state $x$ to the expectation of $Y$ under the Markov chain with transition kernel $\\kappa$ started from $x$, is measurable.","title":"Measurability of the Markov expectation"},{"id":"Probability/markov-chains/construction-of-markov-chains/markov-chain-path-measure-dirac-eq-traj","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.MarkovChainPathMeasure_dirac_eq_traj","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#markov-chain-path-measure-dirac-eq-traj","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $\\kappa$ be a Markov kernel on $S$, $n \\in \\mathbb{N}$, and $x = (x_0, \\dots, x_n)$ a history of states in $S$. The law on path space of the Markov chain with transition kernel $\\kappa$ and initial distribution the Dirac measure at $x_n$ equals the Ionescu\u2013Tulcea trajectory measure on $S^{\\mathbb{N}}$ started at time $0$ from the one-entry history $(x_n)$ (the last-coordinate history of $x$), in which each next\u2026","title":"Path measure from a point as a trajectory measure"},{"id":"Probability/markov-chains/construction-of-markov-chains/integral-shift-path-traj-eq-markov-expec","kind":"node","lean":"MathlibExt.Probability.MarkovChainPath.integral_shiftPath_traj_eq_markovExpectation","parent":19775,"result":true,"route":"../roadmap/Probability/markov-chains/construction-of-markov-chains/index.html#integral-shift-path-traj-eq-markov-expec","scope":"Probability/markov-chains/construction-of-markov-chains","status":"fully_proved","summary":"Let $S$ be a nonempty standard Borel space, $\\kappa$ a Markov kernel on $S$, and $Y : S^{\\mathbb{N}} \\to \\mathbb{R}$ a bounded function, measurable for the product $\\sigma$-algebra. Fix $n \\in \\mathbb{N}$ and a history $x = (x_0, \\dots, x_n)$ of states in $S$, and let $P_{n,x}$ be the Ionescu\u2013Tulcea trajectory measure on $S^{\\mathbb{N}}$ that equals $x$ at times $0, \\dots, n$ and afterwards draws each next state\u2026","title":"Integral of a shifted path functional"},{"data_url":"scopes/Probability/markov-chains/markov-and-strong-markov-properties.json","id":"Probability/markov-chains/markov-and-strong-markov-properties","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","title":"Markov and strong Markov properties"},{"id":"Probability/markov-chains/markov-and-strong-markov-properties/markov-set-integral-shift","kind":"node","lean":"MathlibExt.Probability.MarkovPropertyChain.markov_setIntegral_shift","parent":19798,"result":true,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html#markov-set-integral-shift","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","summary":"Let $S$ be a nonempty standard Borel space, $\\mu$ a finite measure on $S$, $\\kappa$ a Markov kernel on $S$, and $P_\\mu$ the path measure on $S^{\\mathbb{N}}$ of the Markov chain with initial measure $\\mu$ and transition kernel $\\kappa$. Let $Y : S^{\\mathbb{N}} \\to \\mathbb{R}$ be bounded and measurable for the product $\\sigma$-algebra, let $n \\in \\mathbb{N}$, let $\\theta_n$ be the shift\u2026","title":"Markov property, set-integral form"},{"id":"Probability/markov-chains/markov-and-strong-markov-properties/markov-property-full","kind":"node","lean":"MathlibExt.Probability.MarkovPropertyChain.markov_property_full","parent":19798,"result":true,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html#markov-property-full","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","summary":"Let $S$ be a nonempty standard Borel space, $\\mu$ a finite measure on $S$, $\\kappa$ a Markov kernel on $S$, and $P_\\mu$ the path measure on $S^{\\mathbb{N}}$ of the Markov chain with initial measure $\\mu$ and transition kernel $\\kappa$. Let $Y : S^{\\mathbb{N}} \\to \\mathbb{R}$ be bounded and measurable for the product $\\sigma$-algebra, let $n \\in \\mathbb{N}$, let $\\theta_n$ be the shift\u2026","title":"Markov property"},{"id":"Probability/markov-chains/markov-and-strong-markov-properties/stopped-path-shift","kind":"node","lean":"MathlibExt.Probability.StrongMarkovChain.stoppedPathShift","parent":19798,"result":true,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html#stopped-path-shift","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","summary":"For a type $S$ and a function $T : (\\mathbb{N} \\to S) \\to \\mathbb{N}$ (typically a stopping time), the shift by $T$ sends a path $\\omega : \\mathbb{N} \\to S$ to the path $k \\mapsto \\omega(T(\\omega) + k)$.","title":"Path shifted by a random time"},{"id":"Probability/markov-chains/markov-and-strong-markov-properties/stopped-path-shift-eq-shift-path-on-eq","kind":"node","lean":"MathlibExt.Probability.StrongMarkovChain.stoppedPathShift_eq_shiftPath_on_eq","parent":19798,"result":true,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html#stopped-path-shift-eq-shift-path-on-eq","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","summary":"Let $S$ be a type, let $T : (\\mathbb{N} \\to S) \\to \\mathbb{N}$ and $n \\in \\mathbb{N}$, and let $\\omega : \\mathbb{N} \\to S$ be a path with $T(\\omega) = n$. Then the path $\\omega$ shifted by $T$, namely $k \\mapsto \\omega(T(\\omega) + k)$, equals the path $\\omega$ shifted by $n$, namely $k \\mapsto \\omega(n + k)$.","title":"The shift by a random time agrees with the fixed shift on its level set"},{"id":"Probability/markov-chains/markov-and-strong-markov-properties/ae-eq-of-ae-eq-restrict-fibers","kind":"node","lean":"MathlibExt.Probability.StrongMarkovChain.ae_eq_of_ae_eq_restrict_fibers","parent":19798,"result":true,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html#ae-eq-of-ae-eq-restrict-fibers","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, let $T : \\Omega \\to \\mathbb{N}$ be a function whose fibers $\\{T = n\\}$, $n \\in \\mathbb{N}$, are all measurable, and let $f, g : \\Omega \\to \\mathbb{R}$. If for every $n \\in \\mathbb{N}$ we have $f = g$ almost everywhere with respect to the restriction of $\\mu$ to $\\{T = n\\}$, then $f = g$ $\\mu$-almost everywhere.","title":"A.e. equality from a.e. equality on the fibers of a natural-valued function"},{"id":"Probability/markov-chains/markov-and-strong-markov-properties/ae-eq-of-ae-eq-restrict-finite-fibers","kind":"node","lean":"MathlibExt.Probability.StrongMarkovChain.ae_eq_of_ae_eq_restrict_finite_fibers","parent":19798,"result":true,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html#ae-eq-of-ae-eq-restrict-finite-fibers","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, let $\\tau : \\Omega \\to \\mathbb{N} \\cup \\{\\infty\\}$ satisfy $\\tau \\ne \\infty$ $\\mu$-almost everywhere and have every fiber $\\{\\tau = n\\}$, $n \\in \\mathbb{N}$, measurable, and let $f, g : \\Omega \\to \\mathbb{R}$. If for every $n \\in \\mathbb{N}$ we have $f = g$ almost everywhere with respect to the restriction of $\\mu$ to $\\{\\tau = n\\}$, then $f = g$ $\\mu$-almost\u2026","title":"A.e. equality from a.e. equality on the finite fibers"},{"id":"Probability/markov-chains/markov-and-strong-markov-properties/strong-markov-chain-ae-finite","kind":"node","lean":"MathlibExt.Probability.StrongMarkovChain.strong_markov_chain_ae_finite","parent":19798,"result":true,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html#strong-markov-chain-ae-finite","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","summary":"Let $S$ be a nonempty standard Borel space, $\\mu$ a finite measure on $S$, $\\kappa$ a Markov kernel on $S$, and $P_\\mu$ the law on the path space $S^{\\mathbb{N}}$ of the Markov chain with initial distribution $\\mu$ and transition kernel $\\kappa$. For $x \\in S$ write $E_x$ for the expectation under the chain started from $x$, and for a path $\\omega$ and $n \\in \\mathbb{N}$ write $\\theta_n \\omega$ for the shifted path\u2026","title":"Strong Markov property for a.s.-finite stopping times"},{"id":"Probability/markov-chains/markov-and-strong-markov-properties/strong-markov-chain","kind":"node","lean":"MathlibExt.Probability.StrongMarkovChain.strong_markov_chain","parent":19798,"result":true,"route":"../roadmap/Probability/markov-chains/markov-and-strong-markov-properties/index.html#strong-markov-chain","scope":"Probability/markov-chains/markov-and-strong-markov-properties","status":"fully_proved","summary":"Let $S$ be a nonempty standard Borel space, $\\mu$ a finite measure on $S$, $\\kappa$ a Markov kernel on $S$, and $P_\\mu$ the law on the path space $S^{\\mathbb{N}}$ of the Markov chain with initial distribution $\\mu$ and transition kernel $\\kappa$; for $x \\in S$ write $E_x$ for the expectation under the chain started from $x$. Let $T : S^{\\mathbb{N}} \\to \\mathbb{N}$ be a stopping time for the path filtration, with\u2026","title":"Strong Markov property for everywhere-finite stopping times"},{"data_url":"scopes/Probability/markov-chains/markov-processes-from-transition-semigroups.json","id":"Probability/markov-chains/markov-processes-from-transition-semigroups","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","title":"Markov processes from transition semigroups"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/finite-chain-kernel","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.finiteChainKernel","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#finite-chain-kernel","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The law, as a kernel in its starting point, at finitely many ordered times. `s` is the time of the input state and `t` lists the requested absolute times.","title":"Finite-dimensional kernel of a Markov semigroup at ordered times"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/finite-chain-law","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.finiteChainLaw","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#finite-chain-law","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The ordered finite-dimensional law started from `x` at time `s`.","title":"Finite-dimensional law of a Markov semigroup from a starting point"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/finite-chain-kernel-comp","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.finiteChainKernel_comp","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#finite-chain-kernel-comp","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Moving the starting time forward and then evolving to the first requested time gives the same finite chain as evolving there directly.","title":"Composition rule for finite-chain kernels"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/erase-coordinate","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.eraseCoordinate","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#erase-coordinate","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Remove one coordinate from a finite vector.","title":"Removing one coordinate from a finite vector"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/finite-chain-kernel-map-erase-coordinate","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.finiteChainKernel_map_eraseCoordinate_zero","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#finite-chain-kernel-map-erase-coordinate","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Deleting the first requested time composes the initial transition with the transition to the next requested time.","title":"Deleting the first time from a finite Markov chain kernel"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/finite-chain-kernel-map-erase-coordinate-2","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.finiteChainKernel_map_eraseCoordinate","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#finite-chain-kernel-map-erase-coordinate-2","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Deleting one requested time from an ordered finite Markov chain composes the two adjacent transition kernels.","title":"Deleting one time from a finite Markov chain kernel"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/finite-chain-kernel-map-strict-mono","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.finiteChainKernel_map_strictMono","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#finite-chain-kernel-map-strict-mono","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Restrict an ordered finite Markov law along any strictly increasing map of coordinate sets.","title":"Restricting a finite Markov kernel along an increasing reindexing"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/finite-chain-law-map-strict-mono","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.finiteChainLaw_map_strictMono","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#finite-chain-law-map-strict-mono","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Measure form of `finiteChainKernel_map_strictMono`.","title":"Restricting a finite Markov law along an increasing reindexing"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/order-measurable-equiv","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.orderMeasurableEquiv","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#order-measurable-equiv","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Reindex an ordered vector by the increasing enumeration of a finset.","title":"Reindexing an ordered vector by a finite set"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/order-restriction","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.orderRestriction","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#order-restriction","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The increasing reindexing of the elements of `J` among those of a larger finset `I`.","title":"Increasing reindexing of a subset within a finite set"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/finite-law","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.finiteLaw","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#finite-law","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The canonical law at an arbitrary finite set of nonnegative times.","title":"Markov law at a finite set of times"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/is-projective-measure-family-finite-law","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.isProjectiveMeasureFamily_finiteLaw","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#is-projective-measure-family-finite-law","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The finite laws obtained by sorting requested times form a projective family; projectivity is a consequence of Chapman--Kolmogorov, not a premise.","title":"Finite Markov laws form a projective family"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/exists-projective-limit","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.exists_projectiveLimit","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#exists-projective-limit","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Kolmogorov extension applied to the proved finite Markov laws.","title":"Kolmogorov extension for Markov semigroup laws"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/path-measure","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.pathMeasure","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#path-measure","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The canonical path-space measure selected from Kolmogorov extension.","title":"Path-space measure of a Markov semigroup"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/is-projective-limit-path-measure","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.isProjectiveLimit_pathMeasure","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#is-projective-limit-path-measure","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The canonical path measure has exactly the constructed finite laws.","title":"Path measure is the projective limit of the finite Markov laws"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/map-restrict-path-measure","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.map_restrict_pathMeasure","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#map-restrict-path-measure","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Every finite coordinate projection of the canonical path measure is its explicit iterated-kernel law.","title":"Finite projections of the Markov path measure"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/map-ordered-path-measure","kind":"node","lean":"MathlibExt.Probability.MarkovSemigroupProjective.map_ordered_pathMeasure","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#map-ordered-path-measure","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The coordinates in increasing finset order have exactly the corresponding iterated-kernel law.","title":"Ordered finite projections of the Markov path measure"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/exists-markov-kernel-of-positive-linear","kind":"node","lean":"MathlibExt.Probability.PositiveOperatorKernel.exists_markovKernelOfPositiveLinearMap","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#exists-markov-kernel-of-positive-linear","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"A positive unital linear map from `C(E, \u211d)` to itself is represented by a Markov kernel. Continuity of the operator is unnecessary: each output already belongs to `C(E, \u211d)`.","title":"Representation of positive unital operators by Markov kernels"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/exists-markov-kernel-of-positive-operato","kind":"node","lean":"MathlibExt.Probability.PositiveOperatorKernel.exists_markovKernelOfPositiveOperator","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#exists-markov-kernel-of-positive-operato","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"On a compact metrizable Borel space, a positive continuous linear operator on `C(E, \u211d)` that fixes `1` is represented by a Markov kernel. The integral identity holds for every continuous function and every starting state.","title":"Positive-operator-to-Markov-kernel bridge"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/markov-kernel-of-positive-operator","kind":"node","lean":"MathlibExt.Probability.PositiveOperatorKernel.markovKernelOfPositiveOperator","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#markov-kernel-of-positive-operator","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The canonical chosen Markov kernel representing `T`. Its characterization theorem is `integral_markovKernelOfPositiveOperator`; clients should use that theorem rather than unfold this definition.","title":"Markov kernel of a positive unital operator"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/integral-markov-kernel-of-positive-opera","kind":"node","lean":"MathlibExt.Probability.PositiveOperatorKernel.integral_markovKernelOfPositiveOperator","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#integral-markov-kernel-of-positive-opera","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The chosen kernel represents `T` on every real-valued continuous function.","title":"Kernel of a positive operator represents it on continuous functions"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/kernel-ext-of-forall-integral-continuous","kind":"node","lean":"MathlibExt.Probability.PositiveOperatorKernel.kernel_ext_of_forall_integral_continuous_eq","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#kernel-ext-of-forall-integral-continuous","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Two Markov kernels on a compact metrizable Borel space are equal when they integrate every real-valued continuous function equally at every starting state.","title":"Markov kernels are determined by integrals of continuous functions"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/markov-kernel-of-positive-operator-id","kind":"node","lean":"MathlibExt.Probability.PositiveOperatorKernel.markovKernelOfPositiveOperator_id","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#markov-kernel-of-positive-operator-id","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"The kernel represented by the identity operator is Mathlib's identity kernel.","title":"Identity operator gives the identity kernel"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/markov-kernel-of-positive-operator-id-ap","kind":"node","lean":"MathlibExt.Probability.PositiveOperatorKernel.markovKernelOfPositiveOperator_id_apply","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#markov-kernel-of-positive-operator-id-ap","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Pointwise form of the identity theorem: every transition measure is Dirac.","title":"Identity operator kernel is pointwise Dirac"},{"id":"Probability/markov-chains/markov-processes-from-transition-semigroups/markov-kernel-of-positive-operator-comp","kind":"node","lean":"MathlibExt.Probability.PositiveOperatorKernel.markovKernelOfPositiveOperator_comp","parent":19807,"result":true,"route":"../roadmap/Probability/markov-chains/markov-processes-from-transition-semigroups/index.html#markov-kernel-of-positive-operator-comp","scope":"Probability/markov-chains/markov-processes-from-transition-semigroups","status":"fully_proved","summary":"Operator composition is represented by canonical kernel composition. The order is reversed: `T.comp U` acts as `f \u21a6 T (U f)`, so the represented kernel first uses `K_T` and then `K_U`.","title":"Operator composition corresponds to kernel composition"},{"data_url":"scopes/Probability/markov-chains/mixing-times-of-markov-chains.json","id":"Probability/markov-chains/mixing-times-of-markov-chains","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","title":"Mixing times of Markov chains"},{"id":"Probability/markov-chains/mixing-times-of-markov-chains/torus","kind":"node","lean":"MathlibExt.Probability.MarkovMixing.Torus","parent":19833,"result":true,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html#torus","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","summary":"For natural numbers $d$ and $n$, the vertices of the discrete torus $\\mathbb{Z}_n^d$ are the functions from $\\{0, \\dots, d-1\\}$ to $\\{0, \\dots, n-1\\}$, that is, the $d$-tuples of residues modulo $n$.","title":"Discrete torus"},{"id":"Probability/markov-chains/mixing-times-of-markov-chains/step-up","kind":"node","lean":"MathlibExt.Probability.MarkovMixing.stepUp","parent":19833,"result":true,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html#step-up","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","summary":"For a vertex $x$ of the discrete torus $\\mathbb{Z}_n^d$ and a coordinate $k \\in \\{0, \\dots, d-1\\}$, the neighbour $x + e_k$ is obtained from $x$ by replacing its $k$-th coordinate $x_k$ by $(x_k + 1) \\bmod n$ and leaving the other coordinates unchanged.","title":"Coordinate step on the discrete torus"},{"id":"Probability/markov-chains/mixing-times-of-markov-chains/sgn","kind":"node","lean":"MathlibExt.Probability.MarkovMixing.sgn","parent":19833,"result":true,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html#sgn","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","summary":"The sign of a Boolean spin is the real number $+1$ if the spin is true and $-1$ if it is false.","title":"Sign of a Boolean spin"},{"id":"Probability/markov-chains/mixing-times-of-markov-chains/tv-dist","kind":"node","lean":"MathlibExt.Probability.MarkovMixing.tvDist","parent":19833,"result":true,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html#tv-dist","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","summary":"For real-valued functions $\\mu$ and $\\nu$ on a finite set $S$, their total variation distance is $\\frac{1}{2} \\sum_{y \\in S} |\\mu(y) - \\nu(y)|$.","title":"Total variation distance of functions on a finite set"},{"id":"Probability/markov-chains/mixing-times-of-markov-chains/tv-row","kind":"node","lean":"MathlibExt.Probability.MarkovMixing.tvRow","parent":19833,"result":true,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html#tv-row","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","summary":"For a real $S \\times S$ matrix $Q$ indexed by a finite set $S$, a function $\\pi : S \\to \\mathbb{R}$ and $x \\in S$, this is the total variation distance $\\frac{1}{2} \\sum_{y \\in S} |Q(x, y) - \\pi(y)|$ of the row $x$ of $Q$ from $\\pi$.","title":"Total variation distance of a matrix row from a function"},{"id":"Probability/markov-chains/mixing-times-of-markov-chains/is-mixed-at","kind":"node","lean":"MathlibExt.Probability.MarkovMixing.IsMixedAt","parent":19833,"reading_title":{"format":"inline-parts-v1","parts":[["text","Being "],["math","\\(1/4\\)"],["text","-mixed at a given time"]],"source":"Being $1/4$-mixed at a given time"},"result":true,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html#is-mixed-at","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","summary":"Let $S$ be a finite set, $P$ a real $S \\times S$ matrix and $\\pi : S \\to \\mathbb{R}$. Then $P$ is mixed at time $t \\in \\mathbb{N}$ with respect to $\\pi$ if for every $x \\in S$ the total variation distance $\\frac{1}{2} \\sum_{y \\in S} |P^t(x, y) - \\pi(y)|$ between the row $x$ of the matrix power $P^t$ and $\\pi$ is at most $1/4$.","title":"Being $1/4$-mixed at a given time"},{"id":"Probability/markov-chains/mixing-times-of-markov-chains/mono-edges","kind":"node","lean":"MathlibExt.Probability.MarkovMixing.monoEdges","parent":19833,"result":true,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html#mono-edges","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","summary":"For a simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ and a colouring $\\sigma$ of its vertices with colours in $\\{0, \\dots, q-1\\}$, this is the number of edges $uv$ of $G$ with $\\sigma_u = \\sigma_v$, as a real number, each edge counted once.","title":"Number of monochromatic edges"},{"id":"Probability/markov-chains/mixing-times-of-markov-chains/has-cutoff","kind":"node","lean":"MathlibExt.Probability.MarkovMixing.HasCutoff","parent":19833,"result":true,"route":"../roadmap/Probability/markov-chains/mixing-times-of-markov-chains/index.html#has-cutoff","scope":"Probability/markov-chains/mixing-times-of-markov-chains","status":"fully_proved","summary":"A family of real numbers $d_n(t)$, indexed by $n \\in \\mathbb{N}$ and times $t \\in \\mathbb{N}$, exhibits cutoff if there are real numbers $t_n$ with $t_n \\to \\infty$ such that for every $\\eta$ with $0 < \\eta < 1$, as $n \\to \\infty$, $d_n(\\lceil (1+\\eta) t_n \\rceil) \\to 0$ and $d_n(\\lfloor (1-\\eta) t_n \\rfloor) \\to 1$. Here the ceiling and floor are taken in $\\mathbb{N}$, so a negative argument gives $0$.","title":"Cutoff for a family of distance profiles"},{"data_url":"scopes/Probability/markov-chains/convergence-theorem-for-finite-markov-chains.json","id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","kind":"scope","navigation":"module","parent":19625,"result":false,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","title":"Convergence theorem for finite Markov chains"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/to-matrix","kind":"node","lean":"StochasticMatrix.toMatrix","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#to-matrix","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Underlying matrix of probabilities of a `StochasticMatrix`.","title":"Underlying matrix of a stochastic matrix"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/is-ergodic","kind":"node","lean":"StochasticMatrix.IsErgodic","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#is-ergodic","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"A finite-state Markov chain (given by its stochastic transition matrix `P`) is *ergodic* if some positive power `P^n` has all strictly positive entries.","title":"Ergodic finite Markov chain"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/mat-pow-eq-pow","kind":"node","lean":"StochasticMatrix.matPow_eq_pow","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#mat-pow-eq-pow","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The recursively defined power `matPow` agrees with the ordinary matrix power.","title":"Recursive matrix power agrees with the matrix power"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/is-ergodic-iff-is-primitive","kind":"node","lean":"StochasticMatrix.isErgodic_iff_isPrimitive","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#is-ergodic-iff-is-primitive","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Ergodicity is exactly primitivity of the transition matrix. This is the bridge to Mathlib: `Matrix.IsPrimitive` is the same condition for a bare matrix, while `IsErgodic` is the stochastic-matrix form, with nonnegativity supplied by the `StochasticMatrix` structure.","title":"Ergodicity is primitivity of the transition matrix"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/mat-pow-row-sum","kind":"node","lean":"StochasticMatrix.matPow_row_sum","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#mat-pow-row-sum","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Every row of every power of a stochastic matrix sums to `1`: powers of a stochastic matrix are stochastic.","title":"Powers of a stochastic matrix are stochastic"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/mat-pow-one","kind":"node","lean":"StochasticMatrix.matPow_one","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#mat-pow-one","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"One step of the power recursion: `matPow 1` is the matrix itself.","title":"First recursive power of a stochastic matrix"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/mat-pow-add","kind":"node","lean":"StochasticMatrix.matPow_add","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#mat-pow-add","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Powers add: `matPow (a + b)` is the matrix product of `matPow a` and `matPow b`.","title":"Additivity of recursive matrix powers"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/mat-pow-pos-of-le-of-pos-pow","kind":"node","lean":"StochasticMatrix.matPow_pos_of_le_of_pos_pow","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#mat-pow-pos-of-le-of-pos-pow","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"If some power `w` of a stochastic matrix is all-positive, then every later power `m \u2265 w` is all-positive: write `m = (m - w) + w`, use `matPow_add`, and note that row `i` of `matPow (m - w)` sums to `1`, so some entry of that row is positive, and that entry times the positive `matPow w` entry gives a positive summand.","title":"Later powers of a positive stochastic matrix power stay positive"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/exists-mat-pow-pos","kind":"node","lean":"StochasticMatrix.IsErgodic.exists_matPow_pos","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#exists-mat-pow-pos","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"An ergodic chain has, for every pair of states, some positive power with a positive `(i, j)`-entry. (This is the per-pair form; it is strictly weaker than ergodicity, which asks for one power working for all pairs at once.)","title":"Pairwise positive powers of an ergodic chain"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/eventually-all-mat-pow-pos","kind":"node","lean":"StochasticMatrix.IsErgodic.eventually_all_matPow_pos","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#eventually-all-mat-pow-pos","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Once a power of an ergodic chain is all-positive, every later power is all-positive: there is a threshold `w` past which every power is all-positive.","title":"Powers of an ergodic chain are eventually all positive"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/pow","kind":"node","lean":"StochasticMatrix.pow","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#pow","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The `p`-th power of a stochastic matrix, as a stochastic matrix.","title":"Power of a stochastic matrix"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/to-matrix-pow","kind":"node","lean":"StochasticMatrix.toMatrix_pow","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#to-matrix-pow","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The underlying matrix of `P.pow p` is the `p`-th matrix power of `P`.","title":"Underlying matrix of a stochastic matrix power"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/mat-pow-pow","kind":"node","lean":"StochasticMatrix.matPow_pow","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#mat-pow-pow","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Powers of a power: `(P.pow p).matPow n = P.matPow (p * n)`.","title":"Recursive powers of a stochastic matrix power"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/pow-2","kind":"node","lean":"StochasticMatrix.IsErgodic.pow","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#pow-2","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Every positive power of an ergodic chain is ergodic.","title":"Powers of an ergodic Markov chain are ergodic"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/uniform","kind":"node","lean":"StochasticMatrix.uniform","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#uniform","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The uniform chain: every transition has probability `1 / (M + 1)`.","title":"Uniform Markov chain on finitely many states"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/prob-uniform","kind":"node","lean":"StochasticMatrix.prob_uniform","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#prob-uniform","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The uniform chain has constant transition probabilities `1 / (M + 1)`.","title":"Transition probabilities of the uniform chain"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/is-ergodic-uniform","kind":"node","lean":"StochasticMatrix.isErgodic_uniform","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#is-ergodic-uniform","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The uniform chain is ergodic, with witness `n = 1`.","title":"The uniform Markov chain is ergodic"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/identity","kind":"node","lean":"StochasticMatrix.identity","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#identity","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The identity chain: `prob i j` is `1` on the diagonal and `0` elsewhere.","title":"Identity Markov chain"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/prob-identity","kind":"node","lean":"StochasticMatrix.prob_identity","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#prob-identity","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The identity chain has the displayed transition probabilities.","title":"Transition probabilities of the identity chain"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/mat-pow-identity","kind":"node","lean":"StochasticMatrix.matPow_identity","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#mat-pow-identity","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Every power of the identity chain is the identity matrix.","title":"Powers of the identity chain are the identity matrix"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/not-is-ergodic-identity","kind":"node","lean":"StochasticMatrix.not_isErgodic_identity","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#not-is-ergodic-identity","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The identity chain is not ergodic (for `M \u2265 1`): every power has a zero off-diagonal entry.","title":"The identity Markov chain is not ergodic"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/tv-dist-pow-le-of-minorization","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChainConvergence.tvDist_pow_le_of_minorization","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#tv-dist-pow-le-of-minorization","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"Doeblin's argument (Levin, Peres and Wilmer, proof of Theorem 4.9): if the `r`-step transition probabilities dominate `\u03b4 \u03c0`, then the total variation distance of `P\u207f(x, \u00b7)` from the stationary distribution `\u03c0` is at most `(1 - \u03b4) ^ \u230an / r\u230b`. The bound is informative when `r > 0` and `0 < \u03b4`; for `r = 0` (`n / 0 = 0`) or `\u03b4 \u2264 0` it reads `d_TV \u2264 1` or weaker.","title":"Doeblin's argument"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/exists-tv-dist-pow-le","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChainConvergence.exists_tvDist_pow_le","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#exists-tv-dist-pow-le","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The convergence theorem (Levin, Peres and Wilmer, Theorem 4.9): a primitive (irreducible and aperiodic) stochastic matrix with stationary distribution `\u03c0` converges to `\u03c0` geometrically fast in total variation, uniformly in the starting state.","title":"The convergence theorem"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/tendsto-pow-apply","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChainConvergence.tendsto_pow_apply","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#tendsto-pow-apply","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"For a primitive stochastic matrix with stationary distribution `\u03c0`, every row of `P\u207f` converges to `\u03c0`: `P\u207f(x, y) \u2192 \u03c0(y)`.","title":"Convergence of matrix powers to the stationary distribution"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/stationary-unique","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChainConvergence.stationary_unique","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#stationary-unique","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"A primitive stochastic matrix has at most one stationary distribution. In fact `\u03c0'` may be any stationary vector of total mass `1`, not necessarily nonnegative.","title":"Uniqueness of the stationary distribution of a primitive chain"},{"id":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains/stationary-pos","kind":"node","lean":"MathlibExt.Probability.FiniteMarkovChainConvergence.stationary_pos","parent":19842,"result":true,"route":"../roadmap/Probability/markov-chains/convergence-theorem-for-finite-markov-chains/index.html#stationary-pos","scope":"Probability/markov-chains/convergence-theorem-for-finite-markov-chains","status":"fully_proved","summary":"The stationary distribution of a primitive stochastic matrix has full support.","title":"Stationary distribution of a primitive chain has full support"},{"data_url":"scopes/Probability/stochastic-calculus.json","id":"Probability/stochastic-calculus","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/stochastic-calculus/index.html","scope":"Probability/stochastic-calculus","status":"fully_proved","summary":"Continuous-time processes with continuous paths or independent increments: Kolmogorov's continuity theorem, Blumenthal-type zero\u2013one laws, Markov property for independent-increment processes, quadratic variation, the It\u00f4 integral and It\u00f4's formula, stochastic differential equations, the Ornstein\u2013Uhlenbeck process, and Fokker\u2013Planck equations for diffusions.","title":"Brownian motion and stochastic calculus"},{"data_url":"scopes/Probability/stochastic-calculus/blumenthal-zero-one-law.json","id":"Probability/stochastic-calculus/blumenthal-zero-one-law","kind":"scope","navigation":"module","parent":19869,"result":false,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","title":"Blumenthal zero\u2013one law"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/germ-sigma-algebra-zero","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.germSigmaAlgebraZero","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#germ-sigma-algebra-zero","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"The *germ \u03c3-algebra at `0`* of a filtration `\ud835\udcd5` indexed by `NNReal`, defined as the intersection `\u2a05_{t > 0} \ud835\udcd5_t`. For Brownian motion this is the `\ud835\udcd5_0^+` \u03c3-algebra of events observable \"immediately after\" time `0`. Ported verbatim from `v1/Atlas/TheoryOfProbability/code/Blumenthal01.lean`.","title":"Germ sigma-algebra at time zero"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/is-continuous-independent-increment-proc","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.IsContinuousIndependentIncrementProcess","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#is-continuous-independent-increment-proc","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"A continuous process `B : NNReal \u2192 \u03a9 \u2192 E` with independent increments, started at `x` and adapted to `\ud835\udcd5`. The measure is a probability measure, `B 0 = x` a.e., each increment after `s` is independent of `\ud835\udcd5 s`, and a.e. paths are continuous. These are the five meaningful fields of the archived v1 specification; its tautological pushforward-law field is omitted. No Gaussian-law or stationary-increment condition is\u2026","title":"Continuous process with independent increments"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/measurable","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.IsContinuousIndependentIncrementProcess.measurable","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#measurable","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"Each coordinate of the adapted continuous process is measurable.","title":"Coordinates of a continuous independent-increment process are measurable"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/natural-filtration","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.naturalFiltration","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#natural-filtration","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"The natural filtration of a process: `\ud835\udcd5^nat_t = \u03c3(B_u : u \u2264 t)`. Unlike Mathlib's `Filtration.natural`, this is built from measurability rather than strong measurability, which is all that is available from adaptedness.","title":"Natural filtration of a process"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/natural-germ","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.naturalGerm","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#natural-germ","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"The germ \u03c3-algebra of the natural filtration of `B`.","title":"Germ sigma-algebra of the natural filtration"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/measurable-natural-filtration","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.measurable_naturalFiltration","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#measurable-natural-filtration","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"Each `B u` is measurable with respect to the natural filtration at any later time `t`.","title":"Process is measurable for its natural filtration at later times"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/natural-filtration-le","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.naturalFiltration_le","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#natural-filtration-le","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"The natural filtration is contained in any filtration to which `B` is adapted.","title":"Natural filtration is the smallest adapted filtration"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/germ-le-of-pos","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.germ_le_of_pos","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#germ-le-of-pos","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"The germ \u03c3-algebra is contained in every positive-time component of the filtration.","title":"Germ sigma-algebra lies in every positive-time level"},{"id":"Probability/stochastic-calculus/blumenthal-zero-one-law/zero-one-natural-germ","kind":"node","lean":"MathlibExt.Probability.BlumenthalZeroOne.zero_one_naturalGerm","parent":19870,"result":true,"route":"../roadmap/Probability/stochastic-calculus/blumenthal-zero-one-law/index.html#zero-one-natural-germ","scope":"Probability/stochastic-calculus/blumenthal-zero-one-law","status":"fully_proved","summary":"Every event in the natural germ of a continuous process with independent increments and a deterministic a.e. initial value has probability `0` or `1`. This general form includes the classical Brownian natural-filtration law.","title":"Blumenthal zero-one law"},{"data_url":"scopes/Probability/stochastic-calculus/fokker-planck-equation.json","id":"Probability/stochastic-calculus/fokker-planck-equation","kind":"scope","navigation":"module","parent":19869,"result":false,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","title":"Fokker\u2013Planck equation"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/free-energy","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.freeEnergy","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#free-energy","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The free energy `F(t) = \u222b\u2090\u1d47 (\u03a8 p + D p log p) dx` of the density `p(t, \u00b7)` on `[a, b]` (potential `\u03a8`, diffusion coefficient `D`).","title":"Free energy of a Fokker\u2013Planck density"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/neg-flux","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.negFlux","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#neg-flux","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The (negative) probability flux `p \u03a8' + D \u2202\u2093 p` of the gradient-drift Fokker\u2013Planck equation at `(t, x)`; the flux itself is `-(p \u03a8' + D \u2202\u2093 p)`.","title":"Negative probability flux of the Fokker\u2013Planck equation"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/is-no-flux-solution-on","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.IsNoFluxSolutionOn","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#is-no-flux-solution-on","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"`p` solves the gradient-drift Fokker\u2013Planck equation `\u2202\u209c p = \u2202\u2093 (p \u03a8' + D \u2202\u2093 p)` on `[a, b]` with zero flux at both ends, for times `t \u2208 I`.","title":"No-flux solution of the gradient-drift Fokker\u2013Planck equation"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/has-deriv-at-free-energy","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.hasDerivAt_freeEnergy","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#has-deriv-at-free-energy","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"Let `p` be `C\u00b2` in `(t, x)`, positive for times in an open set `I`, and solve the no-flux gradient-drift Fokker\u2013Planck equation on `[a, b]` for `t \u2208 I`, with `C\u00b2` potential `\u03a8` (any real `D`). Then for `t \u2208 I`, `F'(t) = -\u222b\u2090\u1d47 p (\u03a8' + D \u2202\u2093 log p)\u00b2 dx`. Only the values of `p` on `[a, b]` enter `F`; positivity is asked of the whole `C\u00b2` function `p t` so that `log (p t)` is `C\u00b2`.","title":"Dissipation identity"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/antitone-on-free-energy","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.antitoneOn_freeEnergy","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#antitone-on-free-energy","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"Under the hypotheses of `hasDerivAt_freeEnergy`, on a convex open time set `I` the free energy is nonincreasing.","title":"Free-energy decay"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/has-deriv-at-mass","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.hasDerivAt_mass","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#has-deriv-at-mass","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"If `p` is `C\u00b2` and solves the no-flux equation on `[a, b]` for times in `I` (positivity is not needed), the total mass `\u222b\u2090\u1d47 p(t, x) dx` has zero time derivative at every `t \u2208 I`.","title":"Mass conservation"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/pderiv","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.pderiv","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#pderiv","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The partial derivative `\u2202\u1d62 f (x) = Df(x) e\u1d62` on `Fin n \u2192 \u211d`.","title":"Partial derivative on Euclidean coordinate space"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/grad","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.grad","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#grad","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The gradient `\u2207f(x) = (\u2202\u1d62 f(x))\u1d62`.","title":"Gradient of a function on Euclidean coordinate space"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/divergence","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.divergence","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#divergence","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The divergence `div F (x) = \u2211\u1d62 \u2202\u1d62 F\u1d62 (x)` of a vector field.","title":"Divergence of a vector field"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/flux","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.flux","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#flux","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The probability flux `J = b p - D \u2207p` of the diffusion `dX = b dt + \u221a(2D) dW` with density `p`.","title":"Probability flux of a diffusion"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/fokker-planck-op","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.fokkerPlanckOp","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#fokker-planck-op","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The Fokker\u2013Planck operator `-div J`: the density evolves by `\u2202\u209c p = fokkerPlanckOp D b p`, and `p` is stationary iff this vanishes.","title":"Fokker\u2013Planck operator"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/gibbs","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.gibbs","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#gibbs","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The (unnormalized) Gibbs density `e^{-\u03a8(x) / D}`.","title":"Gibbs density"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/flux-gibbs-eq-zero","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.flux_gibbs_eq_zero","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#flux-gibbs-eq-zero","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The Gibbs density has zero flux under the gradient drift `-\u2207\u03a8`: for differentiable `\u03a8`, `D \u2260 0` and any constant `c`, `J = -\u2207\u03a8 \u00b7 c e^{-\u03a8/D} - D \u2207(c e^{-\u03a8/D}) = 0`.","title":"The Gibbs density has zero flux"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/fokker-planck-op-gibbs-eq-zero","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.fokkerPlanckOp_gibbs_eq_zero","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#fokker-planck-op-gibbs-eq-zero","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The Gibbs density is stationary for the gradient drift: `-div J = 0`.","title":"The Gibbs density is stationary"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/pderiv-eq-of-flux-eq-zero","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.pderiv_eq_of_flux_eq_zero","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#pderiv-eq-of-flux-eq-zero","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"If `p > 0` is differentiable and the gradient drift `-\u2207\u03a8` has zero flux for `p`, then `\u2202\u1d62 \u03a8 = -D \u2202\u1d62 log p`.","title":"A zero-flux density determines the drift"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/exists-const-of-flux-eq-zero","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.exists_const_of_flux_eq_zero","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#exists-const-of-flux-eq-zero","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"Two differentiable potentials whose gradient drifts both have zero flux for the same positive differentiable density differ by a constant.","title":"Identifiability of the potential"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/flux-rotational-gibbs","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.flux_rotational_gibbs","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#flux-rotational-gibbs","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"The flux of the Gibbs density under the drift `-(\u2207\u03a8 + A \u2207\u03a8)` is the rotational part `-(A \u2207\u03a8) e^{-\u03a8 / D}` (for differentiable `\u03a8`, `D \u2260 0`, any matrix `A`).","title":"Flux of the Gibbs density under a rotational gradient drift"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/divergence-flux-rotational-gibbs","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.divergence_flux_rotational_gibbs","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#divergence-flux-rotational-gibbs","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"Rotational drifts share the Gibbs stationary density (non-identifiability): for a `C\u00b2` potential `\u03a8`, `D \u2260 0` and an antisymmetric matrix `A` (`A\u1d40 = -A`), the drift `-(\u2207\u03a8 + A \u2207\u03a8)` satisfies `div J = 0` for `p = e^{-\u03a8 / D}`.","title":"Rotational drifts share the Gibbs stationary density"},{"id":"Probability/stochastic-calculus/fokker-planck-equation/exists-rotational-flux-ne-zero","kind":"node","lean":"MathlibExt.Probability.FokkerPlanck.exists_rotational_flux_ne_zero","parent":19880,"result":true,"route":"../roadmap/Probability/stochastic-calculus/fokker-planck-equation/index.html#exists-rotational-flux-ne-zero","scope":"Probability/stochastic-calculus/fokker-planck-equation","status":"fully_proved","summary":"An explicit rotational drift with nonzero flux (our derivation): on `\u211d\u00b2` with `\u03a8(x) = (x\u2080\u00b2 + x\u2081\u00b2) / 2`, `D = 1` and the rotation generator `A = [[0, 1], [-1, 0]]`, the drift `-(\u2207\u03a8 + A \u2207\u03a8)` has the Gibbs (standard Gaussian, up to normalization) density as a stationary density, yet its flux at `(1, 0)` is nonzero, so it is not the gradient drift.","title":"An explicit rotational drift with nonzero flux"},{"data_url":"scopes/Probability/stochastic-calculus/kolmogorov-continuity-theorem.json","id":"Probability/stochastic-calculus/kolmogorov-continuity-theorem","kind":"scope","navigation":"module","parent":19869,"result":false,"route":"../roadmap/Probability/stochastic-calculus/kolmogorov-continuity-theorem/index.html","scope":"Probability/stochastic-calculus/kolmogorov-continuity-theorem","status":"fully_proved","title":"Kolmogorov continuity theorem"},{"id":"Probability/stochastic-calculus/kolmogorov-continuity-theorem/dyadic-rationals","kind":"node","lean":"MathlibExt.Probability.KolmogorovContinuity.dyadicRationals","parent":19900,"result":true,"route":"../roadmap/Probability/stochastic-calculus/kolmogorov-continuity-theorem/index.html#dyadic-rationals","scope":"Probability/stochastic-calculus/kolmogorov-continuity-theorem","status":"fully_proved","summary":"The set of dyadic rationals: real numbers of the form `k / 2 ^ n` for some integer `k` and natural number `n`.","title":"Dyadic rationals"},{"id":"Probability/stochastic-calculus/kolmogorov-continuity-theorem/dyadic-holder-of-increment-bound","kind":"node","lean":"MathlibExt.Probability.KolmogorovContinuity.dyadic_holder_of_increment_bound","parent":19900,"result":true,"route":"../roadmap/Probability/stochastic-calculus/kolmogorov-continuity-theorem/index.html#dyadic-holder-of-increment-bound","scope":"Probability/stochastic-calculus/kolmogorov-continuity-theorem","status":"fully_proved","summary":"If every adjacent dyadic increment at level `m` is at most `D * 2^{-m\u03b3}`, then `f` is `\u03b3`-H\u00f6lder on the dyadic rationals of `[0, 1]`, with an explicit constant.","title":"Deterministic chaining"},{"id":"Probability/stochastic-calculus/kolmogorov-continuity-theorem/kolmogorov-ae-holder-on-dyadics","kind":"node","lean":"MathlibExt.Probability.KolmogorovContinuity.kolmogorov_ae_holder_on_dyadics","parent":19900,"result":true,"route":"../roadmap/Probability/stochastic-calculus/kolmogorov-continuity-theorem/index.html#kolmogorov-ae-holder-on-dyadics","scope":"Probability/stochastic-calculus/kolmogorov-continuity-theorem","status":"fully_proved","summary":"A single full-measure set carries a H\u00f6lder bound for the original process on the dyadic rationals of `[0, 1]`. The moment-integrability correction in the module docstring is required for this Bochner formulation.","title":"Almost sure H\u00f6lder bound on the dyadic rationals"},{"id":"Probability/stochastic-calculus/kolmogorov-continuity-theorem/kolmogorov-continuity-theorem","kind":"node","lean":"MathlibExt.Probability.KolmogorovContinuity.kolmogorov_continuity_theorem","parent":19900,"result":true,"route":"../roadmap/Probability/stochastic-calculus/kolmogorov-continuity-theorem/index.html#kolmogorov-continuity-theorem","scope":"Probability/stochastic-calculus/kolmogorov-continuity-theorem","status":"fully_proved","summary":"Kolmogorov continuity theorem, in the original modification formulation. The direct dyadic bound permits choosing `Y = X`. See the module docstring for the necessary moment-integrability correction to the archived v1 statement.","title":"Kolmogorov continuity theorem"},{"id":"Probability/stochastic-calculus/kolmogorov-continuity-theorem/kolmogorov-ae-holder-on-dyadics-of-is-ko","kind":"node","lean":"MathlibExt.Probability.KolmogorovContinuity.kolmogorov_ae_holder_on_dyadics_of_isKolmogorovProcess","parent":19900,"result":true,"route":"../roadmap/Probability/stochastic-calculus/kolmogorov-continuity-theorem/index.html#kolmogorov-ae-holder-on-dyadics-of-is-ko","scope":"Probability/stochastic-calculus/kolmogorov-continuity-theorem","status":"fully_proved","summary":"The canonical Kolmogorov-process condition implies a common a.e. H\u00f6lder bound for `X` on the dyadics of `[0, 1]`. Its measurable-pair and finite lintegral fields supply the Bochner hypotheses internally, including when `M = 0`. No claim is made about the raw paths at every real time.","title":"H\u00f6lder bound on dyadics for a Kolmogorov process"},{"id":"Probability/stochastic-calculus/kolmogorov-continuity-theorem/exists-continuous-modification","kind":"node","lean":"MathlibExt.Probability.KolmogorovContinuousModification.exists_continuous_modification","parent":19900,"result":true,"route":"../roadmap/Probability/stochastic-calculus/kolmogorov-continuity-theorem/index.html#exists-continuous-modification","scope":"Probability/stochastic-calculus/kolmogorov-continuity-theorem","status":"fully_proved","summary":"A real-valued Kolmogorov process on `NNReal` whose time exponent is greater than one has one modification with measurable coordinates and continuous paths on every sample. The same modification agrees exactly at time zero and preserves every closed set containing an original sample path. Fixed-time almost-everywhere equality is asserted separately at each time; no indistinguishability claim is made for the raw\u2026","title":"Kolmogorov continuous-modification theorem for real-valued processes"},{"data_url":"scopes/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in.json","id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","kind":"scope","navigation":"module","parent":19869,"result":false,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","title":"Markov property of processes with independent increments"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-prev","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPrev","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-prev","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"For a finite set $S \\subseteq \\mathbb{R}_{\\ge 0}$ and $t \\in \\mathbb{R}_{\\ge 0}$, the predecessor of $t$ in $S$ is the largest element of $S$ strictly below $t$, or $0$ if there is no such element.","title":"Predecessor of a time in a finite set"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-inc","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmInc","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-inc","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a real-valued process on $\\Omega$, let $s \\ge 0$, and let $S \\subset \\mathbb{R}_{\\ge 0}$ be a finite set. For $t \\ge 0$ let $t^-$ be the largest element of $S$ strictly below $t$, or $0$ if there is none. The increment at $t$ is the random variable $\\omega \\mapsto B_{s+t}(\\omega) - B_{s+t^-}(\\omega)$.","title":"Increment over the predecessor interval"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-shifted","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmShifted","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-shifted","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"For a real-valued process $(B_t)_{t \\ge 0}$ on $\\Omega$ indexed by $t \\in \\mathbb{R}_{\\ge 0}$ and a time $s \\ge 0$, the shifted path is the map sending $\\omega \\in \\Omega$ to the path $t \\mapsto B_{s+t}(\\omega)$ in $\\mathbb{R}^{\\mathbb{R}_{\\ge 0}}$.","title":"Shifted path of a process"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-centered-shift","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmCenteredShift","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-centered-shift","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a real-valued process on $\\Omega$ and let $s \\ge 0$. The centered shifted path at time $s$ sends $\\omega \\in \\Omega$ to the function $t \\mapsto B_{s+t}(\\omega) - B_s(\\omega)$ on $\\mathbb{R}_{\\ge 0}$.","title":"Centered shifted path"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-centered-law","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmCenteredLaw","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-centered-law","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a real-valued process on a measure space $(\\Omega, \\mu)$ and let $s \\ge 0$. This is the pushforward of $\\mu$ under the map sending $\\omega \\in \\Omega$ to the path $t \\mapsto B_{s+t}(\\omega) - B_s(\\omega)$, a measure on the space of functions $\\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$ with its product \u03c3-algebra.","title":"Law of the centered shifted path"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-path-expectation","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPathExpectation","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-path-expectation","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$, let $(B_t)_{t \\ge 0}$ be a real process on $\\Omega$ indexed by $t \\in \\mathbb{R}_{\\ge 0}$, let $s \\ge 0$, and let $\\nu_s$ be the law under $\\mu$ (the pushforward of $\\mu$) of the centered shifted path $\\omega \\mapsto (t \\mapsto B_{s+t}(\\omega) - B_s(\\omega))$. For a real function $Y$ on paths $\\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$ and $x \\in \\mathbb{R}$, the path expectation is the\u2026","title":"Time-dependent path expectation"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-adapted","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.BmAdapted","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-adapted","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"A real-valued process $(B_t)_{t \\ge 0}$ on $\\Omega$, indexed by $t \\in \\mathbb{R}_{\\ge 0}$, is adapted to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ when $B_t$ is $\\mathcal{F}_t$-measurable for every $t \\ge 0$.","title":"Adapted real process"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-indep-increments","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.BmIndepIncrements","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-indep-increments","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"A real-valued process $(B_t)_{t \\ge 0}$ on $\\Omega$ has independent increments with respect to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ and a measure $\\mu$ when, for all $0 \\le s \\le t$, the \u03c3-algebra generated by $B_t - B_s$ is independent of $\\mathcal{F}_s$ under $\\mu$.","title":"Independent increments with respect to a filtration"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-bm-shifted","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_bmShifted","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-bm-shifted","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a measurable space $\\Omega$, adapted to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ of its \u03c3-algebra, and give the space $\\mathbb{R}^{[0, \\infty)}$ of real paths indexed by $t \\ge 0$ the product \u03c3-algebra. For every $s \\ge 0$, the map from $\\Omega$ to $\\mathbb{R}^{[0, \\infty)}$ sending $\\omega$ to the path $t \\mapsto B_{s+t}(\\omega)$ is measurable.","title":"Measurability of the shifted path"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-bm-centered-shift","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_bmCenteredShift","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-bm-centered-shift","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(\\mathcal{F}_t)_{t \\ge 0}$ be a filtration on a measurable space $\\Omega$ indexed by $\\mathbb{R}_{\\ge 0}$, and let $(B_t)_{t \\ge 0}$ be a real-valued process adapted to it (each $B_t$ is $\\mathcal{F}_t$-measurable). Then for every $s \\ge 0$ the map $\\omega \\mapsto (t \\mapsto B_{s+t}(\\omega) - B_s(\\omega))$ from $\\Omega$ to $\\mathbb{R}^{\\mathbb{R}_{\\ge 0}}$, with the product \u03c3-algebra, is measurable.","title":"Measurability of the centered shifted path"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-bm-inc-of-le","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_bmInc_of_le","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-bm-inc-of-le","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a measurable space $\\Omega$, adapted to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ of its \u03c3-algebra, let $S$ be a finite set of nonnegative reals and let $s, t, u \\ge 0$. Write $p_S(t)$ for the largest element of $S$ strictly below $t$, or $0$ if there is none. If $s + t \\le s + u$ and $s + p_S(t) \\le s + u$, then the increment $B_{s+t} - B_{s+p_S(t)}$ is\u2026","title":"Measurability of an increment before a given time"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-prev-insert-of-lt","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPrev_insert_of_lt","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-prev-insert-of-lt","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"For a finite set $T \\subseteq \\mathbb{R}_{\\ge 0}$ and $t \\in \\mathbb{R}_{\\ge 0}$, call the largest element of $T$ strictly below $t$ (or $0$ if there is none) the predecessor of $t$ in $T$. Let $S \\subseteq \\mathbb{R}_{\\ge 0}$ be finite and let $t, a \\in \\mathbb{R}_{\\ge 0}$ with $t < a$. Then the predecessor of $t$ in $S \\cup \\{a\\}$ equals the predecessor of $t$ in $S$.","title":"Inserting a larger element does not change the predecessor"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-prev-insert-self","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPrev_insert_self","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-prev-insert-self","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"For a finite set $T \\subseteq \\mathbb{R}_{\\ge 0}$ and $t \\in \\mathbb{R}_{\\ge 0}$, call the largest element of $T$ strictly below $t$ (or $0$ if there is none) the predecessor of $t$ in $T$. Let $S \\subseteq \\mathbb{R}_{\\ge 0}$ be finite and let $a \\in \\mathbb{R}_{\\ge 0}$ with $x < a$ for every $x \\in S$. Then the predecessor of $a$ in $S \\cup \\{a\\}$ is the supremum of $S$ in $\\mathbb{R}_{\\ge 0}$, that is, its\u2026","title":"Predecessor of a new maximum"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measure-inter-bi-inter-bm-inc","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measure_inter_biInter_bmInc","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measure-inter-bi-inter-bm-inc","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure and let $(B_t)_{t \\ge 0}$ be a real process adapted to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ such that $B_t - B_s$ is independent of $\\mathcal{F}_s$ under $\\mu$ whenever $s \\le t$. Let $s \\in \\mathbb{R}_{\\ge 0}$ and let $S \\subseteq \\mathbb{R}_{\\ge 0}$ be finite. For $t \\in S$ let $\\Delta_t = B_{s+t} - B_{s+t^-}$, where $t^-$ is the largest element of $S$ strictly below $t$, or\u2026","title":"Past events factor against grid increments"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/sum-bm-inc-eq","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.sum_bmInc_eq","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#sum-bm-inc-eq","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a family of real functions on $\\Omega$, let $s \\in \\mathbb{R}_{\\ge 0}$ and let $S \\subseteq \\mathbb{R}_{\\ge 0}$ be a nonempty finite set. For $u \\in S$ let $\\Delta_u = B_{s+u} - B_{s+u^-}$, where $u^-$ is the largest element of $S$ strictly below $u$, or $0$ if there is none. Then for every $\\omega \\in \\Omega$, $\\sum_{u \\in S} \\Delta_u(\\omega) = B_{s + \\max S}(\\omega) - B_s(\\omega)$.","title":"Telescoping sum of grid increments"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-centered-eq-sum-bm-inc","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmCentered_eq_sum_bmInc","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-centered-eq-sum-bm-inc","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a real-valued process on $\\Omega$, let $s \\ge 0$, and let $S \\subset \\mathbb{R}_{\\ge 0}$ be a finite set. For $u \\ge 0$ let $u^-$ be the largest element of $S$ strictly below $u$, or $0$ if there is none. Then for every $t \\in S$ and every $\\omega \\in \\Omega$, $B_{s+t}(\\omega) - B_s(\\omega) = \\sum_{u \\in S : u \\le t} (B_{s+u}(\\omega) - B_{s+u^-}(\\omega))$.","title":"Telescoping the centered shifted path over a finite set"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-bm-inc","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_bmInc","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-bm-inc","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a measurable space $\\Omega$, adapted to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ of its \u03c3-algebra. For every finite set $S$ of nonnegative reals and all $s, t \\ge 0$, the increment $B_{s+t} - B_{s+p_S(t)}$ is measurable, where $p_S(t)$ is the largest element of $S$ strictly below $t$, or $0$ if there is none.","title":"Measurability of the increments"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-inc-rectangles","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmIncRectangles","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-inc-rectangles","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a real-valued process on $\\Omega$, let $s \\ge 0$, and let $S \\subset \\mathbb{R}_{\\ge 0}$ be a finite set. For $t \\ge 0$ let $t^-$ be the largest element of $S$ strictly below $t$, or $0$ if there is none. The increment rectangles are the events $\\bigcap_{t \\in S} \\{\\omega \\in \\Omega : B_{s+t}(\\omega) - B_{s+t^-}(\\omega) \\in C_t\\}$, where each $C_t \\subseteq \\mathbb{R}$ is a Borel set.","title":"Increment rectangles"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-inc-sigma","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmIncSigma","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-inc-sigma","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a real-valued process on $\\Omega$, let $s \\ge 0$, and let $S \\subset \\mathbb{R}_{\\ge 0}$ be a finite set. For $t \\ge 0$ let $t^-$ be the largest element of $S$ strictly below $t$, or $0$ if there is none. This is the \u03c3-algebra on $\\Omega$ generated by the increment rectangles, the events $\\bigcap_{t \\in S} \\{\\omega \\in \\Omega : B_{s+t}(\\omega) - B_{s+t^-}(\\omega) \\in C_t\\}$ with each\u2026","title":"\u03c3-algebra of the increments over a finite set"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-set-bm-inc-preimage","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurableSet_bmInc_preimage","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-set-bm-inc-preimage","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a measurable space $\\Omega$, let $s \\ge 0$ and let $S$ be a finite set of nonnegative reals. For $u \\ge 0$ let $p_S(u)$ be the largest element of $S$ strictly below $u$, or $0$ if there is none, and let $\\Delta_u = B_{s+u} - B_{s+p_S(u)}$. Let $\\mathcal{G}_S$ be the \u03c3-algebra on $\\Omega$ generated by the sets $\\bigcap_{u \\in S} \\{\\omega : \\Delta_u(\\omega) \\in E_u\\}$\u2026","title":"Events of a single increment lie in the increment \u03c3-algebra"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-bm-inc-inc-sigma","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_bmInc_incSigma","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-bm-inc-inc-sigma","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a real-valued process on $\\Omega$, let $s \\ge 0$, and let $S \\subseteq \\mathbb{R}_{\\ge 0}$ be finite; for $t \\in S$ let $t^-$ be the largest element of $S$ strictly below $t$, or $0$ if there is none. Then for every $t \\in S$, the increment $\\omega \\mapsto B_{s+t}(\\omega) - B_{s+t^-}(\\omega)$ is measurable with respect to the \u03c3-algebra generated by the events\u2026","title":"Measurability of an increment in the increment \u03c3-algebra"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/indep-bm-inc-sigma","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.indep_bmIncSigma","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#indep-bm-inc-sigma","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(\\mathcal{F}_t)_{t \\ge 0}$ be a filtration indexed by $\\mathbb{R}_{\\ge 0}$, and let $(B_t)_{t \\ge 0}$ be a real-valued process that is adapted (each $B_t$ is $\\mathcal{F}_t$-measurable) and has independent increments (for $u \\le t$, the increment $B_t - B_u$ is independent of $\\mathcal{F}_u$ under $\\mu$). For a finite set $S \\subseteq \\mathbb{R}_{\\ge 0}$ and\u2026","title":"Independence of future increments from the present"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-centered-cylinders","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmCenteredCylinders","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-centered-cylinders","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $(B_t)_{t \\ge 0}$ be a real-valued process on $\\Omega$ and let $s \\ge 0$. The centered cylinders at time $s$ are the events $\\bigcap_{t \\in S} \\{\\omega \\in \\Omega : B_{s+t}(\\omega) - B_s(\\omega) \\in C_t\\}$, where $S \\subset \\mathbb{R}_{\\ge 0}$ is a finite set and each $C_t \\subseteq \\mathbb{R}$ is a Borel set.","title":"Cylinder events of the centered shifted path"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-set-bm-centered-cylinder","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurableSet_bmCentered_cylinder","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-set-bm-centered-cylinder","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a measurable space $\\Omega$, let $s \\ge 0$ and let $S$ be a finite set of nonnegative reals. For $u \\ge 0$ let $p_S(u)$ be the largest element of $S$ strictly below $u$, or $0$ if there is none, and let $\\Delta_u = B_{s+u} - B_{s+p_S(u)}$. Let $\\mathcal{G}_S$ be the \u03c3-algebra on $\\Omega$ generated by the sets $\\bigcap_{u \\in S} \\{\\omega : \\Delta_u(\\omega) \\in E_u\\}$\u2026","title":"Centered cylinders lie in the increment \u03c3-algebra"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-set-pi-eval-preimage","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurableSet_pi_eval_preimage","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-set-pi-eval-preimage","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"For every $t \\in \\mathbb{R}_{\\ge 0}$ and every Borel set $D \\subseteq \\mathbb{R}$, the set of paths $f : \\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$ with $f(t) \\in D$ is measurable for the product \u03c3-algebra on the space of all such paths.","title":"Evaluation events are measurable"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/indep-bm-centered-shift","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.indep_bmCenteredShift","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#indep-bm-centered-shift","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(\\mathcal{F}_t)_{t \\ge 0}$ be a filtration indexed by $\\mathbb{R}_{\\ge 0}$, and let $(B_t)_{t \\ge 0}$ be a real-valued process that is adapted (each $B_t$ is $\\mathcal{F}_t$-measurable) and has independent increments (for $u \\le t$, the increment $B_t - B_u$ is independent of $\\mathcal{F}_u$ under $\\mu$). Then for every $s \\ge 0$, $\\mathcal{F}_s$ is independent\u2026","title":"Independence of the centered future path from the present"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-path-cylinders","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPathCylinders","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-path-cylinders","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"The collection of sets of paths $f : \\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$ of the form $\\bigcap_{t \\in S} \\{f : f(t) \\in C_t\\}$, where $S \\subseteq \\mathbb{R}_{\\ge 0}$ is a finite set and each $C_t$ is a measurable subset of $\\mathbb{R}$.","title":"Cylinder sets of path space"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/is-pi-system-bm-path-cylinders","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.isPiSystem_bmPathCylinders","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#is-pi-system-bm-path-cylinders","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Consider the subsets of $\\mathbb{R}^{\\mathbb{R}_{\\ge 0}}$ of the form $\\bigcap_{t \\in S} \\{f : f(t) \\in C_t\\}$, where $S \\subseteq \\mathbb{R}_{\\ge 0}$ is finite and every $C_t \\subseteq \\mathbb{R}$ is Borel. They form a \u03c0-system: whenever two of them have nonempty intersection, the intersection is again of this form.","title":"Path cylinders form a \u03c0-system"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/pi-eq-generate-from-bm-path-cylinders","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.pi_eq_generateFrom_bmPathCylinders","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#pi-eq-generate-from-bm-path-cylinders","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"The product \u03c3-algebra on the space of paths $f : \\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$ equals the \u03c3-algebra generated by the cylinders, the sets $\\{f : f(t) \\in C_t \\text{ for all } t \\in S\\}$ with $S$ a finite subset of $\\mathbb{R}_{\\ge 0}$ and $(C_t)_{t \\ge 0}$ a family of Borel subsets of $\\mathbb{R}$.","title":"Cylinders generate the product \u03c3-algebra on path space"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-cyl-rectangles","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmCylRectangles","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-cyl-rectangles","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Write $\\mathbb{R}^{\\mathbb{R}_{\\ge 0}}$ for the space of functions $\\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$. These are the subsets of $\\mathbb{R} \\times \\mathbb{R}^{\\mathbb{R}_{\\ge 0}}$ of the form $U \\times \\{f : f(t) \\in V_t \\text{ for all } t \\in S\\}$, where $U \\subseteq \\mathbb{R}$ is a Borel set, $S \\subset \\mathbb{R}_{\\ge 0}$ is a finite set, and each $V_t \\subseteq \\mathbb{R}$ is a Borel set.","title":"Rectangles of a Borel set and a path cylinder"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/is-pi-system-bm-cyl-rectangles","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.isPiSystem_bmCylRectangles","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#is-pi-system-bm-cyl-rectangles","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Consider the subsets of $\\mathbb{R} \\times \\mathbb{R}^{\\mathbb{R}_{\\ge 0}}$ of the form $U \\times \\bigcap_{t \\in S} \\{f \\in \\mathbb{R}^{\\mathbb{R}_{\\ge 0}} : f(t) \\in V_t\\}$, where $U \\subseteq \\mathbb{R}$ is Borel, $S \\subseteq \\mathbb{R}_{\\ge 0}$ is finite and every $V_t \\subseteq \\mathbb{R}$ is Borel. They form a \u03c0-system: whenever two of them have nonempty intersection, the intersection is again of this form.","title":"Rectangles of a Borel set and a path cylinder form a \u03c0-system"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/mem-generate-from-bm-cyl-rectangles","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.mem_generateFrom_bmCylRectangles","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#mem-generate-from-bm-cyl-rectangles","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $U \\subseteq \\mathbb{R}$ be a Borel set and let $V$ be a set of paths $f : \\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$ that is measurable for the product \u03c3-algebra. Then $U \\times V$ belongs to the \u03c3-algebra on $\\mathbb{R} \\times (\\mathbb{R}_{\\ge 0} \\to \\mathbb{R})$ generated by the cylinder rectangles, the sets $U' \\times \\{f : f(t) \\in C_t \\text{ for all } t \\in S\\}$ with $U'$ Borel, $S$ a finite subset of\u2026","title":"Measurable rectangles lie in the \u03c3-algebra of cylinder rectangles"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/prod-eq-generate-from-bm-cyl-rectangles","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.prod_eq_generateFrom_bmCylRectangles","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#prod-eq-generate-from-bm-cyl-rectangles","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"The product \u03c3-algebra on $\\mathbb{R} \\times (\\mathbb{R}_{\\ge 0} \\to \\mathbb{R})$, where the path space carries its product \u03c3-algebra, equals the \u03c3-algebra generated by the cylinder rectangles, the sets $U \\times \\{f : f(t) \\in V_t \\text{ for all } t \\in S\\}$ with $U$ Borel, $S$ a finite subset of $\\mathbb{R}_{\\ge 0}$ and $(V_t)_{t \\ge 0}$ a family of Borel subsets of $\\mathbb{R}$.","title":"Cylinder rectangles generate the product \u03c3-algebra"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-pair","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPair","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-pair","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"For a real process $(B_t)_{t \\ge 0}$ on $\\Omega$ indexed by $t \\in \\mathbb{R}_{\\ge 0}$ and a time $s \\ge 0$, this is the map $\\Omega \\to \\mathbb{R} \\times \\mathbb{R}^{\\mathbb{R}_{\\ge 0}}$ sending $\\omega$ to the pair consisting of $B_s(\\omega)$ and the path $t \\mapsto B_{s+t}(\\omega) - B_s(\\omega)$.","title":"The process value paired with its centered future path"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-phi","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPhi","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-phi","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"The map sending a pair $(x, f)$, with $x \\in \\mathbb{R}$ and $f : \\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$, to the path $t \\mapsto x + f(t)$.","title":"Translating a path by a constant"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-bm-pair","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_bmPair","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-bm-pair","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a measurable space $\\Omega$, adapted to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ of its \u03c3-algebra, and give the space $\\mathbb{R}^{[0, \\infty)}$ of real paths indexed by $t \\ge 0$ the product \u03c3-algebra. For every $s \\ge 0$, the map from $\\Omega$ to $\\mathbb{R} \\times \\mathbb{R}^{[0, \\infty)}$ sending $\\omega$ to the pair of $B_s(\\omega)$ and the centered shifted path\u2026","title":"Measurability of the pair map"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-bm-phi","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_bmPhi","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-bm-phi","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Give the space $\\mathbb{R}^{[0, \\infty)}$ of real paths indexed by $t \\ge 0$ the product \u03c3-algebra. The map from $\\mathbb{R} \\times \\mathbb{R}^{[0, \\infty)}$ to $\\mathbb{R}^{[0, \\infty)}$ sending $(x, f)$ to the path $t \\mapsto x + f(t)$ is measurable.","title":"Translation of paths is measurable"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-phi-bm-pair","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPhi_bmPair","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-phi-bm-pair","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"For a real process $(B_t)_{t \\ge 0}$ on $\\Omega$ indexed by $t \\in \\mathbb{R}_{\\ge 0}$, a time $s \\ge 0$ and $\\omega \\in \\Omega$, translating the centered shifted path $t \\mapsto B_{s+t}(\\omega) - B_s(\\omega)$ by the constant $B_s(\\omega)$ gives the shifted path $t \\mapsto B_{s+t}(\\omega)$.","title":"Translating the centered future path recovers the shifted path"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-section-bm-centered-law","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_section_bmCenteredLaw","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-section-bm-centered-law","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a probability space $(\\Omega, \\mu)$, let $s \\ge 0$, and let $\\nu_s$ be the law of the centered shifted path $t \\mapsto B_{s+t} - B_s$ on the space $\\mathbb{R}^{[0, \\infty)}$ of real paths indexed by $t \\ge 0$, with its product \u03c3-algebra. For every measurable set $T \\subseteq \\mathbb{R} \\times \\mathbb{R}^{[0, \\infty)}$, the function\u2026","title":"Section measures of the centered law are measurable"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/pair-section-rectangle","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.pair_section_rectangle","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#pair-section-rectangle","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure and let $(B_t)_{t \\ge 0}$ be a real process adapted to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ such that $B_t - B_s$ is independent of $\\mathcal{F}_s$ under $\\mu$ whenever $s \\le t$. Fix $s \\in \\mathbb{R}_{\\ge 0}$, let $W^s$ be the centered shifted path $t \\mapsto B_{s+t} - B_s$ and let $\\nu_s$ be its law under $\\mu$. Let $U \\subseteq \\mathbb{R}$ be Borel, let\u2026","title":"The pair identity on cylinder rectangles"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/pair-section-lintegral","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.pair_section_lintegral","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#pair-section-lintegral","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure and let $(B_t)_{t \\ge 0}$ be a real process adapted to a filtration $(\\mathcal{F}_t)_{t \\ge 0}$ such that $B_t - B_s$ is independent of $\\mathcal{F}_s$ under $\\mu$ whenever $s \\le t$. Fix $s \\in \\mathbb{R}_{\\ge 0}$, let $W^s$ be the centered shifted path $t \\mapsto B_{s+t} - B_s$ and let $\\nu_s$ be its law under $\\mu$. Then for every set\u2026","title":"The pair-section identity for measurable sets"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-lintegral-bm-phi","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_lintegral_bmPhi","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-lintegral-bm-phi","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a probability space $(\\Omega, \\mu)$, let $s \\ge 0$, and let $\\nu_s$ be the law of the centered shifted path $t \\mapsto B_{s+t} - B_s$ on the space $\\mathbb{R}^{[0, \\infty)}$ of real paths indexed by $t \\ge 0$, with its product \u03c3-algebra. For every measurable $Y : \\mathbb{R}^{[0, \\infty)} \\to [0, \\infty]$, the function $x \\mapsto \\int Y(x + g) \\, d\\nu_s(g)$ on\u2026","title":"Nonnegative translated integrals against the centered law are measurable"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/independent-increments-lintegral-shift","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.independent_increments_lintegral_shift","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#independent-increments-lintegral-shift","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(\\mathcal{F}_t)_{t \\ge 0}$ be a filtration indexed by $\\mathbb{R}_{\\ge 0}$, and let $(B_t)_{t \\ge 0}$ be a real-valued process that is adapted (each $B_t$ is $\\mathcal{F}_t$-measurable) and has independent increments (for $u \\le t$, the increment $B_t - B_u$ is independent of $\\mathcal{F}_u$ under $\\mu$). Fix $s \\ge 0$ and let $\\nu_s$ be the law under $\\mu$ of\u2026","title":"Nonnegative integrals of future path functionals over present events"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/measurable-bm-path-expectation","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.measurable_bmPathExpectation","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#measurable-bm-path-expectation","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $B = (B_t)_{t \\ge 0}$ be a real process on a probability space $(\\Omega, \\mu)$, let $s \\ge 0$, and let $\\nu_s$ be the law of the centered shifted path $t \\mapsto B_{s+t} - B_s$ on the space $\\mathbb{R}^{[0, \\infty)}$ of real paths indexed by $t \\ge 0$, with its product \u03c3-algebra. For every measurable $Y : \\mathbb{R}^{[0, \\infty)} \\to \\mathbb{R}$, the function $x \\mapsto \\int Y(x + f) \\, d\\nu_s(f)$ on\u2026","title":"The path expectation is measurable in the starting point"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-path-expectation-bound","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPathExpectation_bound","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-path-expectation-bound","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(B_t)_{t \\ge 0}$ be a real process on $\\Omega$ indexed by $t \\in \\mathbb{R}_{\\ge 0}$, let $s \\ge 0$, and let $\\nu_s$ be the law under $\\mu$ of the path $\\omega \\mapsto (t \\mapsto B_{s+t}(\\omega) - B_s(\\omega))$. If $Y$ is a measurable real function on paths $\\mathbb{R}_{\\ge 0} \\to \\mathbb{R}$ with $|Y(p)| \\le C$ for every path $p$, then for every\u2026","title":"Bound on the path expectation"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/bm-path-expectation-set-integral","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.bmPathExpectation_setIntegral","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#bm-path-expectation-set-integral","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(\\mathcal{F}_t)_{t \\ge 0}$ be a filtration of the \u03c3-algebra of $\\Omega$ indexed by $\\mathbb{R}_{\\ge 0}$, and let $(B_t)_{t \\ge 0}$ be a real process such that each $B_t$ is $\\mathcal{F}_t$-measurable and, for all $u \\le t$, the increment $B_t - B_u$ is independent of $\\mathcal{F}_u$ under $\\mu$. Let $s \\ge 0$, let $Y$ be a measurable real function on paths\u2026","title":"Set integrals of the path expectation"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/independent-increments-set-integral","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.independent_increments_setIntegral","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#independent-increments-set-integral","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(\\mathcal{F}_t)_{t \\ge 0}$ be a filtration indexed by $\\mathbb{R}_{\\ge 0}$, and let $(B_t)_{t \\ge 0}$ be a real-valued process that is adapted (each $B_t$ is $\\mathcal{F}_t$-measurable) and has independent increments (for $u \\le t$, the increment $B_t - B_u$ is independent of $\\mathcal{F}_u$ under $\\mu$). Fix $s \\ge 0$ and let $\\nu_s$ be the law under $\\mu$ of\u2026","title":"Bounded integrals of future path functionals over present events"},{"id":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/independent-increments-cond-exp","kind":"node","lean":"MathlibExt.Probability.IndependentIncrementsMarkovProperty.independent_increments_condExp","parent":19907,"result":true,"route":"../roadmap/Probability/stochastic-calculus/markov-property-of-processes-with-independent-in/index.html#independent-increments-cond-exp","scope":"Probability/stochastic-calculus/markov-property-of-processes-with-independent-in","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(\\mathcal{F}_t)_{t \\ge 0}$ be a filtration indexed by $\\mathbb{R}_{\\ge 0}$, and let $(B_t)_{t \\ge 0}$ be a real-valued process that is adapted (each $B_t$ is $\\mathcal{F}_t$-measurable) and has independent increments (for $u \\le t$, the increment $B_t - B_u$ is independent of $\\mathcal{F}_u$ under $\\mu$). Fix $s \\ge 0$ and let $\\nu_s$ be the law under $\\mu$ of\u2026","title":"Conditional expectation of a future path functional"},{"data_url":"scopes/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion.json","id":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","kind":"scope","navigation":"module","parent":19869,"result":false,"route":"../roadmap/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/index.html","scope":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","status":"fully_proved","title":"Quadratic variation of Brownian motion"},{"id":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/weighted-qverror","kind":"node","lean":"ProbabilityTheory.weightedQVError","parent":19956,"result":true,"route":"../roadmap/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/index.html#weighted-qverror","scope":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","status":"fully_proved","summary":"The weighted quadratic-variation error along the uniform partition of `[0, t]` into `n` cells: `\u2211_{k < n} \u03c6(B (t k / n)) ((B (t (k + 1) / n) - B (t k / n))\u00b2 - t / n)`.","title":"Weighted quadratic-variation error of Brownian motion"},{"id":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/integral-weighted-qverror-sq-le","kind":"node","lean":"ProbabilityTheory.integral_weightedQVError_sq_le","parent":19956,"result":true,"route":"../roadmap/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/index.html#integral-weighted-qverror-sq-le","scope":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","status":"fully_proved","summary":"For a pre-Brownian motion `B` adapted to `F` with increments independent of the past, and a measurable weight `\u03c6` bounded by `C`, `E[(\u2211_{k < n} \u03c6(B_{t_k}) ((\u0394B_k)\u00b2 - \u0394t))\u00b2] \u2264 2 C\u00b2 t\u00b2 / n` on the uniform partition `t_k = t k / n` (`n \u2265 1`).","title":"Second moment of the weighted quadratic-variation error"},{"id":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/tendsto-e-lp-norm-weighted-qverror","kind":"node","lean":"ProbabilityTheory.tendsto_eLpNorm_weightedQVError","parent":19956,"result":true,"route":"../roadmap/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/index.html#tendsto-e-lp-norm-weighted-qverror","scope":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","status":"fully_proved","summary":"Under the hypotheses of `integral_weightedQVError_sq_le`, `\u2211_{k < n} \u03c6(B_{t_k}) ((\u0394B_k)\u00b2 - \u0394t) \u2192 0` in `L\u00b2(P)`.","title":"Weighted quadratic variation of Brownian motion"},{"id":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/indep-natural","kind":"node","lean":"ProbabilityTheory.IsPreBrownianReal.indep_natural","parent":19956,"result":true,"route":"../roadmap/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/index.html#indep-natural","scope":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","status":"fully_proved","summary":"For a pre-Brownian motion with measurable coordinates, each increment `B t - B s` (`s \u2264 t`) is independent of the natural filtration at time `s` (a consequence of `IsPreBrownianReal.indepFun_shift`, the weak Markov property).","title":"Increments are independent of the past"},{"id":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/adapted-natural-of-measurable","kind":"node","lean":"ProbabilityTheory.adapted_natural_of_measurable","parent":19956,"result":true,"route":"../roadmap/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/index.html#adapted-natural-of-measurable","scope":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","status":"fully_proved","summary":"A process with measurable coordinates is adapted to its natural filtration.","title":"Processes with measurable coordinates are adapted to their natural filtration"},{"id":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/tendsto-e-lp-norm-sum-sq-sub","kind":"node","lean":"ProbabilityTheory.IsPreBrownianReal.tendsto_eLpNorm_sum_sq_sub","parent":19956,"result":true,"route":"../roadmap/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/index.html#tendsto-e-lp-norm-sum-sq-sub","scope":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","status":"fully_proved","summary":"Quadratic variation of Brownian motion, `[B]_t = t`: along the uniform partitions of `[0, t]`, `\u2211_{k < n} (B (t (k + 1) / n) - B (t k / n))\u00b2 \u2192 t` in `L\u00b2(P)`, for every pre-Brownian motion `B` (no filtration and no measurability of the coordinates is assumed; \u00d8ksendal 2003, Exercise 2.17; Karatzas\u2013Shreve 1991, Theorem 1.5.8 treats continuous square-integrable martingales in probability).","title":"Quadratic variation of Brownian motion"},{"id":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/tendsto-in-measure-sum-sq","kind":"node","lean":"ProbabilityTheory.IsPreBrownianReal.tendstoInMeasure_sum_sq","parent":19956,"result":true,"route":"../roadmap/Probability/stochastic-calculus/quadratic-variation-of-brownian-motion/index.html#tendsto-in-measure-sum-sq","scope":"Probability/stochastic-calculus/quadratic-variation-of-brownian-motion","status":"fully_proved","summary":"Along the uniform partitions of `[0, t]`, `\u2211_{k < n} (B (t (k + 1) / n) - B (t k / n))\u00b2 \u2192 t` in probability.","title":"Quadratic variation of Brownian motion in probability"},{"data_url":"scopes/Probability/stochastic-calculus/it-integral.json","id":"Probability/stochastic-calculus/it-integral","kind":"scope","navigation":"module","parent":19869,"result":false,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","title":"It\u00f4 integral"},{"id":"Probability/stochastic-calculus/it-integral/elementary-integrand","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#elementary-integrand","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"An elementary integrand for a filtration `F` indexed by `\u211d\u22650`: partition times `time 0 \u2264 \u22ef \u2264 time n` and coefficients `coeff i`, bounded by `bound` and measurable with respect to `F (time i)`, representing the process `\u2211 i, coeff i 1_{(time i, time (i + 1)]}`. Equal consecutive times are allowed (the corresponding interval is empty).","title":"Elementary integrand"},{"id":"Probability/stochastic-calculus/it-integral/integral-sub-sq","kind":"node","lean":"ProbabilityTheory.IsPreBrownianReal.integral_sub_sq","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integral-sub-sq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The second moment of a Brownian increment: `E[(B v - B r)\u00b2] = v - r` for `r \u2264 v`.","title":"Second moment of a Brownian increment"},{"id":"Probability/stochastic-calculus/it-integral/to-process","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.toProcess","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#to-process","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The elementary integrand as a process: `u \u21a6 \u2211 i, coeff i 1_{(time i, time (i + 1)]}(u)`.","title":"Elementary integrand as a process"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.itoIntegral","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral of an elementary integrand against a process `B`: `t \u21a6 \u2211 i, coeff i (B (time (i + 1) \u2227 t) - B (time i \u2227 t))`.","title":"It\u00f4 integral of an elementary integrand"},{"id":"Probability/stochastic-calculus/it-integral/single","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.single","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#single","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The elementary integrand `\u03be 1_{(a, b]}` for `a \u2264 b` and an `F a`-measurable `\u03be` bounded by `C`.","title":"Single-step elementary integrand"},{"id":"Probability/stochastic-calculus/it-integral/to-process-single","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.toProcess_single","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#to-process-single","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The process of `single a b hab \u03be h\u03be C h\u03beC` is `u \u21a6 \u03be 1_{(a, b]}(u)`.","title":"Process of a single-step elementary integrand"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-single","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.itoIntegral_single","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-single","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral of `\u03be 1_{(a, b]}` is `\u03be (B (b \u2227 t) - B (a \u2227 t))` (Karatzas\u2013Shreve 1991, Section 3.2.A); this holds for every process `B` and every `\u03c9`, by definition.","title":"Elementary integrands"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-zero","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.itoIntegral_zero","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-zero","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The elementary It\u00f4 integral vanishes at time `0`.","title":"Elementary It\u00f4 integral vanishes at time zero"},{"id":"Probability/stochastic-calculus/it-integral/integral-ito-integral-sq-eq-sum","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.integral_itoIntegral_sq_eq_sum","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integral-ito-integral-sq-eq-sum","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a pre-Brownian motion `B` adapted to `F` whose increments `B t - B s` are independent of `F s`, `E[(\u222b_0^t H dB)\u00b2] = \u2211 i, E[\u03be_i\u00b2] (t_{i+1} \u2227 t - t_i \u2227 t)`.","title":"It\u00f4 isometry, discrete form"},{"id":"Probability/stochastic-calculus/it-integral/disjoint-ioc-time","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.disjoint_Ioc_time","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#disjoint-ioc-time","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The partition intervals of an elementary integrand are pairwise disjoint.","title":"Partition intervals of an elementary integrand are disjoint"},{"id":"Probability/stochastic-calculus/it-integral/to-process-eq-of-mem","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.toProcess_eq_of_mem","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#to-process-eq-of-mem","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"On its `i`-th interval, an elementary integrand equals its `i`-th coefficient.","title":"Elementary integrand on a partition interval"},{"id":"Probability/stochastic-calculus/it-integral/integral-to-process-sq","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.integral_toProcess_sq","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integral-to-process-sq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The time integral of the squared elementary integrand, path by path: `\u222b_0^t H_u\u00b2 du = \u2211 i, \u03be_i\u00b2 (t_{i+1} \u2227 t - t_i \u2227 t)`.","title":"Time integral of a squared elementary integrand"},{"id":"Probability/stochastic-calculus/it-integral/integral-ito-integral-sq","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.integral_itoIntegral_sq","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integral-ito-integral-sq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"It\u00f4 isometry for elementary integrands: for a pre-Brownian motion `B` adapted to `F` whose increments `B t - B s` are independent of `F s`, `E[(\u222b_0^t H dB)\u00b2] = E \u222b_0^t H_u\u00b2 du` (Karatzas\u2013Shreve 1991, Section 3.2.A).","title":"It\u00f4 isometry"},{"id":"Probability/stochastic-calculus/it-integral/martingale-ito-integral","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.martingale_itoIntegral","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#martingale-ito-integral","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a pre-Brownian motion `B` adapted to `F` whose increments `B t - B s` are independent of `F s`, the elementary It\u00f4 integral `t \u21a6 \u222b_0^t H dB` is an `F`-martingale.","title":"Martingale property"},{"id":"Probability/stochastic-calculus/it-integral/integral-ito-integral-mul","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.integral_itoIntegral_mul","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integral-ito-integral-mul","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"Polarized It\u00f4 isometry for elementary integrands: for a pre-Brownian motion `B` adapted to `F` whose increments `B t - B s` are independent of `F s`, and two elementary integrands `H`, `K` (with possibly different partitions), `E[(\u222b_0^t H dB) (\u222b_0^t K dB)] = E \u222b_0^t H_u K_u du`.","title":"Polarized It\u00f4 isometry"},{"id":"Probability/stochastic-calculus/it-integral/tendsto-sum-mul-sub-uniform","kind":"node","lean":"ProbabilityTheory.tendsto_sum_mul_sub_uniform","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#tendsto-sum-mul-sub-uniform","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a `C\u00b9` function `g`, a continuous path `w` and `t \u2265 0`, the left-point sums `\u2211_{k < n} g (t k / n) (w (t (k + 1) / n) - w (t k / n))` converge, as `n \u2192 \u221e`, to `g t w t - g 0 w 0 - \u222b_0^t g' w`.","title":"Riemann\u2013Stieltjes integration by parts, uniform partitions"},{"id":"Probability/stochastic-calculus/it-integral/riemann","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.riemann","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#riemann","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The left-point step approximation `\u2211_{k < n} g (t k / n) 1_{(t k / n, t (k + 1) / n]}` of a deterministic `g` on the uniform partition of `[0, t]` into `n` cells, as an elementary integrand for any filtration `F` (deterministic coefficients are measurable for every \u03c3-algebra).","title":"Left-point step approximation of a deterministic integrand"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-riemann","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.itoIntegral_riemann","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-riemann","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The elementary It\u00f4 integral of `riemann F g t n` at time `t` is the It\u00f4\u2013Riemann sum `\u2211_{k < n} g (t k / n) (B (t (k + 1) / n) - B (t k / n))`.","title":"It\u00f4 integral of a Riemann step approximation"},{"id":"Probability/stochastic-calculus/it-integral/tendsto-ito-integral-riemann","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.tendsto_itoIntegral_riemann","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#tendsto-ito-integral-riemann","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"Along every continuous path of `B`, the elementary It\u00f4 integrals at time `t` of the left-point approximations of `g` converge to Wiener's integration-by-parts expression `g t B t - g 0 B 0 - \u222b_0^t g'(u) B u du`.","title":"It\u00f4\u2013Riemann sums of a deterministic `C\u00b9` integrand"},{"id":"Probability/stochastic-calculus/it-integral/ou-path-solution-eq-integration-by-parts","kind":"node","lean":"ProbabilityTheory.ouPathSolution_eq_integration_by_parts","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ou-path-solution-eq-integration-by-parts","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The pathwise Ornstein\u2013Uhlenbeck solution in integration-by-parts form: with `g u = e^{\u03b8u}`, `ouPathSolution \u03b8 \u03c3 x\u2080 w t = e^{-\u03b8t} x\u2080 + \u03c3 e^{-\u03b8t} (g t w t - g 0 w 0 - \u222b_0^t g'(u) w u du)`, for any path `w` and any real `\u03b8`, `\u03c3`, `x\u2080`, `t`.","title":"Ornstein\u2013Uhlenbeck path solution by integration by parts"},{"id":"Probability/stochastic-calculus/it-integral/ae-tendsto-ou-ito-integral-riemann","kind":"node","lean":"ProbabilityTheory.IsBrownianReal.ae_tendsto_ou_itoIntegral_riemann","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ae-tendsto-ou-ito-integral-riemann","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a Brownian motion `B`, almost surely, for every `t \u2265 0` and every initial value `X\u2080`, the It\u00f4\u2013Riemann sums `e^{-\u03b8t} X\u2080 + \u03c3 e^{-\u03b8t} \u2211_{k < n} e^{\u03b8 t k / n} (B (t (k + 1) / n) - B (t k / n))` of `e^{-\u03b8t} X\u2080 + \u03c3 \u222b_0^t e^{-\u03b8(t-s)} dB_s` converge to the pathwise solution `ouPathSolution` (any real `\u03b8`, `\u03c3`).","title":"The It\u00f4 form of the Ornstein\u2013Uhlenbeck solution"},{"id":"Probability/stochastic-calculus/it-integral/integrand-dist","kind":"node","lean":"ProbabilityTheory.integrandDist","parent":19964,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squared "],["math","\\(L^2\\)"],["text"," distance between integrands"]],"source":"Squared $L^2$ distance between integrands"},"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integrand-dist","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The squared `L\u00b2(P \u2297 du)` distance on `(0, t] \u00d7 \u03a9` between two processes, in `\u211d\u22650\u221e`: `\u222b\u207b \u03c9, \u222b\u207b u in (0, t], |H u \u03c9 - K u \u03c9|\u00b2 du dP`.","title":"Squared $L^2$ distance between integrands"},{"id":"Probability/stochastic-calculus/it-integral/restrict-ioc","kind":"node","lean":"ProbabilityTheory.restrictIoc","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#restrict-ioc","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The process `H` with its values at times outside `(0, t]` set to `0`. Only the values at times in `(0, t]` enter the It\u00f4 integral at time `t`.","title":"Restriction of a process to the time interval (0, t)"},{"id":"Probability/stochastic-calculus/it-integral/is-ito-approximable","kind":"node","lean":"ProbabilityTheory.IsItoApproximable","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#is-ito-approximable","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"A process `H` is It\u00f4-approximable on `[0, t]` for the filtration `F` if it is jointly measurable on `(0, t] \u00d7 \u03a9` and is the `L\u00b2(P \u2297 du)`-limit on `(0, t] \u00d7 \u03a9` of elementary integrands. Neither condition involves the values of `H` at time `0` or after `t`.","title":"It\u00f4-approximable process"},{"id":"Probability/stochastic-calculus/it-integral/to-l2","kind":"node","lean":"ProbabilityTheory.toL2","parent":19964,"reading_title":{"format":"inline-parts-v1","parts":[["text","Function as an element of "],["math","\\(L^2\\)"]],"source":"Function as an element of $L^2$"},"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#to-l2","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"A function as an element of `L\u00b2(P)`, or `0` if it is not square integrable.","title":"Function as an element of $L^2$"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-l2","kind":"node","lean":"ProbabilityTheory.itoIntegralL2","parent":19964,"reading_title":{"format":"inline-parts-v1","parts":[["text","It\u00f4 integral as an element of "],["math","\\(L^2\\)"]],"source":"It\u00f4 integral as an element of $L^2$"},"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-l2","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral `\u222b_0^t H dB` as an element of `L\u00b2(P)`: the `L\u00b2` limit of the elementary It\u00f4 integrals of an approximating sequence of elementary integrands when `H` is It\u00f4-approximable on `[0, t]`, and `0` otherwise.","title":"It\u00f4 integral as an element of $L^2$"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-2","kind":"node","lean":"ProbabilityTheory.itoIntegral","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-2","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral `t \u21a6 \u222b_0^t H dB` as a real process: an `F t`-strongly measurable version of `itoIntegralL2 P F B H t` when one exists (which is the case under the hypotheses of the theorems below), and a representative of `itoIntegralL2 P F B H t` otherwise.","title":"It\u00f4 integral process"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-ae-eq","kind":"node","lean":"ProbabilityTheory.itoIntegral_ae_eq","parent":19964,"reading_title":{"format":"inline-parts-v1","parts":[["text","It\u00f4 integral process is a version of the "],["math","\\(L^2\\)"],["text"," integral"]],"source":"It\u00f4 integral process is a version of the $L^2$ integral"},"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-ae-eq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral process is almost surely equal to `itoIntegralL2` at each time.","title":"It\u00f4 integral process is a version of the $L^2$ integral"},{"id":"Probability/stochastic-calculus/it-integral/measurable-restrict-ioc","kind":"node","lean":"ProbabilityTheory.measurable_restrictIoc","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#measurable-restrict-ioc","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"Restricting a jointly measurable process to the times `(0, t]` keeps it jointly measurable.","title":"Restriction to (0, t) preserves joint measurability"},{"id":"Probability/stochastic-calculus/it-integral/integrand-dist-restrict-ioc","kind":"node","lean":"ProbabilityTheory.integrandDist_restrictIoc","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integrand-dist-restrict-ioc","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"`integrandDist P X Y t` only depends on `X` and `Y` at times in `(0, t]`.","title":"Integrand distance depends only on times in (0, t)"},{"id":"Probability/stochastic-calculus/it-integral/congr","kind":"node","lean":"ProbabilityTheory.IsItoApproximable.congr","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#congr","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"It\u00f4-approximability on `[0, t]` only depends on the integrand at times in `(0, t]`.","title":"Locality of approximability"},{"id":"Probability/stochastic-calculus/it-integral/measurable-to-process","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.measurable_toProcess","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#measurable-to-process","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"An elementary integrand is jointly measurable in `(u, \u03c9)` (reading `u : \u211d` through `Real.toNNReal`).","title":"Elementary integrands are jointly measurable"},{"id":"Probability/stochastic-calculus/it-integral/abs-to-process-le","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.abs_toProcess_le","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#abs-to-process-le","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"An elementary integrand with `n` intervals is bounded by `n * bound`.","title":"Uniform bound on an elementary integrand"},{"id":"Probability/stochastic-calculus/it-integral/mem-lp-ito-integral-2","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.memLp_itoIntegral","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#mem-lp-ito-integral-2","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The elementary It\u00f4 integral against a pre-Brownian motion is square integrable.","title":"Square integrability of the elementary It\u00f4 integral"},{"id":"Probability/stochastic-calculus/it-integral/integrand-dist-ne-top","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.integrandDist_ne_top","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integrand-dist-ne-top","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The `L\u00b2(P \u2297 du)` distance between two elementary integrands is finite.","title":"Finite distance between elementary integrands"},{"id":"Probability/stochastic-calculus/it-integral/integrand-dist-eq-dist-sq","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.integrandDist_eq_dist_sq","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integrand-dist-eq-dist-sq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"`integrandDist` of two elementary integrands is the squared `L\u00b2(P)` distance of their It\u00f4 integrals.","title":"It\u00f4 isometry for elementary integrands"},{"id":"Probability/stochastic-calculus/it-integral/integrand-dist-le","kind":"node","lean":"ProbabilityTheory.integrandDist_le","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integrand-dist-le","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"Pointwise `|x - y|\u00b2 \u2264 2 |x - z|\u00b2 + 2 |y - z|\u00b2`, integrated: a quasi-triangle inequality for `integrandDist` between jointly measurable processes.","title":"Quasi-triangle inequality for the integrand distance"},{"id":"Probability/stochastic-calculus/it-integral/tendsto-to-l2-ito-integral","kind":"node","lean":"ProbabilityTheory.tendsto_toL2_itoIntegral","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#tendsto-to-l2-ito-integral","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a pre-Brownian motion `B` adapted to `F` with increments independent of the past, an `H` jointly measurable on `(0, t] \u00d7 \u03a9`, and *any* sequence of elementary integrands `Kn` converging to `H` in `L\u00b2(P \u2297 du)` on `(0, t] \u00d7 \u03a9`, the elementary It\u00f4 integrals of `Kn` converge in `L\u00b2(P)` to `itoIntegralL2 P F B H t`.","title":"The It\u00f4 integral is well defined"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-l2-congr","kind":"node","lean":"ProbabilityTheory.itoIntegralL2_congr","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-l2-congr","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral at time `t` only depends on the integrand at times in `(0, t]`.","title":"Locality of the It\u00f4 integral"},{"id":"Probability/stochastic-calculus/it-integral/is-ito-approximable-to-process","kind":"node","lean":"ProbabilityTheory.isItoApproximable_toProcess","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#is-ito-approximable-to-process","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"An elementary integrand is It\u00f4-approximable on every `[0, t]` (by itself).","title":"Elementary integrands are It\u00f4-approximable"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-to-process-ae-eq","kind":"node","lean":"ProbabilityTheory.itoIntegral_toProcess_ae_eq","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-to-process-ae-eq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"On an elementary integrand, the It\u00f4 integral agrees almost surely with the elementary It\u00f4 integral.","title":"It\u00f4 integral agrees with the elementary integral on elementary integrands"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-indicator-ioc","kind":"node","lean":"ProbabilityTheory.itoIntegral_indicator_Ioc","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-indicator-ioc","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For `a \u2264 b` and a bounded `F a`-measurable `\u03be`, the It\u00f4 integral of `\u03be 1_{(a, b]}` against a pre-Brownian motion `B` adapted to `F` with increments independent of the past is `\u03be (B (b \u2227 t) - B (a \u2227 t))` almost surely (Karatzas\u2013Shreve 1991, Section 3.2.A).","title":"Elementary integrands"},{"id":"Probability/stochastic-calculus/it-integral/tendsto-integrand-dist-riemann","kind":"node","lean":"ProbabilityTheory.tendsto_integrandDist_riemann","parent":19964,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^2\\)"],["text"," convergence of left-point approximations of a continuous function"]],"source":"$L^2$ convergence of left-point approximations of a continuous function"},"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#tendsto-integrand-dist-riemann","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The left-point approximations of a continuous deterministic `g` converge to `g` in `L\u00b2(P \u2297 du)` on `(0, t] \u00d7 \u03a9`.","title":"$L^2$ convergence of left-point approximations of a continuous function"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-deterministic-eq","kind":"node","lean":"ProbabilityTheory.itoIntegral_deterministic_eq","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-deterministic-eq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a Brownian motion `B` adapted to `F` with increments independent of the past and a deterministic `C\u00b9` integrand `g`, the It\u00f4 integral is `\u222b_0^t g dB = g t B t - g 0 B 0 - \u222b_0^t g'(u) B u du` almost surely (Karatzas\u2013Shreve 1991, Section 3.3; the right side is a pathwise Riemann integral).","title":"Wiener's integration by parts"},{"id":"Probability/stochastic-calculus/it-integral/ito-ou-solution-eq-ou-path-solution","kind":"node","lean":"ProbabilityTheory.ito_ou_solution_eq_ouPathSolution","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-ou-solution-eq-ou-path-solution","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a Brownian motion `B` adapted to `F` with increments independent of the past, any initial value `X\u2080` and any real `\u03b8`, `\u03c3`, the It\u00f4 form `e^{-\u03b8t} X\u2080 + \u03c3 e^{-\u03b8t} \u222b_0^t e^{\u03b8s} dB_s = e^{-\u03b8t} X\u2080 + \u03c3 \u222b_0^t e^{-\u03b8(t-s)} dB_s` of the solution of `dX = -\u03b8 X dt + \u03c3 dB` (Karatzas\u2013Shreve 1991, Section 5.6) agrees almost surely with the pathwise solution `ouPathSolution`, which solves `X t = X\u2080 - \u03b8 \u222b_0^t X + \u03c3 (B t - B 0)`\u2026","title":"The Ornstein\u2013Uhlenbeck solution in It\u00f4 form"},{"id":"Probability/stochastic-calculus/it-integral/aestrongly-measurable-ito-integral-l2","kind":"node","lean":"ProbabilityTheory.aestronglyMeasurable_itoIntegralL2","parent":19964,"reading_title":{"format":"inline-parts-v1","parts":[["text","Measurability of the "],["math","\\(L^2\\)"],["text"," It\u00f4 integral"]],"source":"Measurability of the $L^2$ It\u00f4 integral"},"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#aestrongly-measurable-ito-integral-l2","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The `L\u00b2` It\u00f4 integral is (almost surely equal to) an `F t`-measurable function.","title":"Measurability of the $L^2$ It\u00f4 integral"},{"id":"Probability/stochastic-calculus/it-integral/martingale-ito-integral-2","kind":"node","lean":"ProbabilityTheory.martingale_itoIntegral","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#martingale-ito-integral-2","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a pre-Brownian motion `B` adapted to `F` with increments independent of the past and an integrand `H` that is It\u00f4-approximable on every `[0, t]`, the It\u00f4 integral `t \u21a6 \u222b_0^t H dB` is an `F`-martingale (Karatzas\u2013Shreve 1991, Proposition 3.2.10).","title":"Martingale property of the It\u00f4 integral"},{"id":"Probability/stochastic-calculus/it-integral/integrand-dist-eq-e-lp-norm-sq","kind":"node","lean":"ProbabilityTheory.integrandDist_eq_eLpNorm_sq","parent":19964,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrand distance as a squared "],["math","\\(L^2\\)"],["text"," seminorm"]],"source":"Integrand distance as a squared $L^2$ seminorm"},"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integrand-dist-eq-e-lp-norm-sq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The integrand distance on `(0, t]` is the square of the `L\u00b2` seminorm on `\u03a9 \u00d7 (0, t]`.","title":"Integrand distance as a squared $L^2$ seminorm"},{"id":"Probability/stochastic-calculus/it-integral/lintegral-ito-integral-sq","kind":"node","lean":"ProbabilityTheory.lintegral_itoIntegral_sq","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#lintegral-ito-integral-sq","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"It\u00f4 isometry for the `L\u00b2` It\u00f4 integral, in `\u211d\u22650\u221e` form: for a pre-Brownian motion `B` adapted to `F` with increments independent of the past and an integrand `H` that is It\u00f4-approximable on `[0, t]`, `E[(\u222b_0^t H dB)\u00b2] = E \u222b_0^t H_u\u00b2 du` (Karatzas\u2013Shreve 1991, Section 3.2).","title":"It\u00f4 isometry"},{"id":"Probability/stochastic-calculus/it-integral/ae-tendsto-average-icc","kind":"node","lean":"MeasureTheory.LocallyIntegrable.ae_tendsto_average_Icc","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ae-tendsto-average-icc","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a locally integrable `f : \u211d \u2192 \u211d`, for almost every `u`, the averages of `f` over intervals `[a n, b n]` converge to `f u` whenever the lengths `b n - a n > 0` tend to `0` and the distance from `u` to the midpoint of `[a n, b n]` is at most twice the length.","title":"Lebesgue differentiation along shrinking off-center intervals"},{"id":"Probability/stochastic-calculus/it-integral/measurable-to-nnreal","kind":"node","lean":"ProbabilityTheory.IsProgressive.measurable_toNNReal","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#measurable-to-nnreal","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"A progressive process is jointly measurable in `(u, \u03c9)` (reading `u : \u211d` through `Real.toNNReal`).","title":"Progressive processes are jointly measurable"},{"id":"Probability/stochastic-calculus/it-integral/truncation","kind":"node","lean":"ProbabilityTheory.IsProgressive.truncation","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#truncation","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"Truncating a progressive process at level `M` (Mathlib's `ProbabilityTheory.truncation`, which keeps the values in `(-M, M]` and sets the others to `0`) gives a progressive process.","title":"Truncation of a progressive process is progressive"},{"id":"Probability/stochastic-calculus/it-integral/measurable-average-icc","kind":"node","lean":"ProbabilityTheory.IsProgressive.measurable_average_Icc","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#measurable-average-icc","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a progressive `H` and `b \u2264 T`, the average of `u \u21a6 H u \u03c9` over `[a, b]` is `F T`-measurable in `\u03c9`.","title":"Measurability of time averages of a progressive process"},{"id":"Probability/stochastic-calculus/it-integral/delayed-average","kind":"node","lean":"ProbabilityTheory.delayedAverage","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#delayed-average","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The delayed-average approximation of a progressive process `H` on `[0, t]`, truncated at level `M`, with `n + 1` intervals: on `(t_k, t_{k+1}]`, with `t_k = k t / (n + 1)`, it equals the average of `truncation (H \u00b7) M` over `[t_{k-1}, t_k]` for `k \u2265 1`, and `0` for `k = 0`. For `M \u2264 0` the truncation, and hence the approximation, is `0`.","title":"Delayed-average approximation of a progressive process"},{"id":"Probability/stochastic-calculus/it-integral/tendsto-integrand-dist-delayed-average","kind":"node","lean":"ProbabilityTheory.tendsto_integrandDist_delayedAverage","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#tendsto-integrand-dist-delayed-average","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a progressive `H` and a finite measure `P`, the delayed-average approximations of the truncation of `H` at level `M` converge to it in `L\u00b2(P \u2297 du)` on `(0, t] \u00d7 \u03a9` as the mesh `t / (n + 1)` tends to `0` (Karatzas\u2013Shreve 1991, proof of Lemma 3.2.4(b)).","title":"Convergence of the delayed averages"},{"id":"Probability/stochastic-calculus/it-integral/tendsto-integrand-dist-truncation","kind":"node","lean":"ProbabilityTheory.tendsto_integrandDist_truncation","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#tendsto-integrand-dist-truncation","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a jointly measurable `H` with `E \u222b_0^t H_u\u00b2 du < \u221e`, the truncations of `H` at levels `M \u2192 \u221e` converge to `H` in `L\u00b2(P \u2297 du)` on `(0, t] \u00d7 \u03a9` (Karatzas\u2013Shreve 1991, proof of Lemma 3.2.4(c)).","title":"Truncation"},{"id":"Probability/stochastic-calculus/it-integral/is-ito-approximable-of-is-progressive","kind":"node","lean":"ProbabilityTheory.isItoApproximable_of_isProgressive","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#is-ito-approximable-of-is-progressive","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"A progressive integrand `H` with `E \u222b_0^t H_u\u00b2 du < \u221e` is the `L\u00b2(P \u2297 du)`-limit on `(0, t] \u00d7 \u03a9` of elementary integrands (Karatzas\u2013Shreve 1991, Lemma 3.2.4 and Proposition 3.2.6).","title":"Density of elementary integrands"},{"id":"Probability/stochastic-calculus/it-integral/mem-lp-ito-integral","kind":"node","lean":"ProbabilityTheory.memLp_itoIntegral","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#mem-lp-ito-integral","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral at each time is square integrable: it is a version of the element `itoIntegralL2 P F B H t` of `L\u00b2(P)` (which is `0` when `H` is not It\u00f4-approximable).","title":"Square integrability of the It\u00f4 integral"},{"id":"Probability/stochastic-calculus/it-integral/lintegral-ito-integral-sq-of-is-progress","kind":"node","lean":"ProbabilityTheory.lintegral_itoIntegral_sq_of_isProgressive","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#lintegral-ito-integral-sq-of-is-progress","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a pre-Brownian motion `B` adapted to `F` with increments independent of the past and an `F`-progressive integrand `H` with `E \u222b_0^t H_u\u00b2 du < \u221e`, `E[(\u222b_0^t H dB)\u00b2] = E \u222b_0^t H_u\u00b2 du` (Karatzas\u2013Shreve 1991, Section 3.2).","title":"It\u00f4 isometry for progressive integrands"},{"id":"Probability/stochastic-calculus/it-integral/martingale-ito-integral-of-is-progressiv","kind":"node","lean":"ProbabilityTheory.martingale_itoIntegral_of_isProgressive","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#martingale-ito-integral-of-is-progressiv","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For a pre-Brownian motion `B` adapted to `F` with increments independent of the past and an `F`-progressive integrand `H` with `E \u222b_0^t H_u\u00b2 du < \u221e` for every `t`, the It\u00f4 integral `t \u21a6 \u222b_0^t H dB` is an `F`-martingale (Karatzas\u2013Shreve 1991, Proposition 3.2.10).","title":"Martingale property for progressive integrands"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-continuous-modification","kind":"node","lean":"ProbabilityTheory.itoIntegral_continuous_modification","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-continuous-modification","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral admits a modification with continuous paths: there is a process `J` with continuous paths such that `J t = \u222b_0^t H dB` almost surely for each `t` (Karatzas\u2013Shreve 1991, Section 3.2). `ProbabilityTheory.itoIntegral` chooses its version separately at each time, so it is only determined up to a null set that may depend on `t`; the statement therefore asks for a modification rather than for a process\u2026","title":"Continuous modification"},{"id":"Probability/stochastic-calculus/it-integral/integrand-dist-eq-dist-sq-ito-integral-l","kind":"node","lean":"ProbabilityTheory.integrandDist_eq_dist_sq_itoIntegralL2","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#integrand-dist-eq-dist-sq-ito-integral-l","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The squared `L\u00b2(P \u2297 du)` distance between two It\u00f4-approximable integrands is the squared `L\u00b2(P)` distance between their It\u00f4 integrals.","title":"It\u00f4 isometry for It\u00f4-approximable integrands"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-add-of-is-pre-brownian-real","kind":"node","lean":"ProbabilityTheory.itoIntegral_add_of_isPreBrownianReal","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-add-of-is-pre-brownian-real","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral is additive in the integrand: for `F`-progressive integrands `H\u2081, H\u2082` with `E \u222b_0^t H_i\u00b2 du < \u221e`, the integral of the sum is almost surely the sum of the integrals. Together with homogeneity and the difference isometry this pins down the integral on the whole `L\u00b2` space of progressive integrands. Only the pre-Brownian finite-dimensional laws are used; path continuity is not required.","title":"Linearity (additivity)"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-add","kind":"node","lean":"ProbabilityTheory.itoIntegral_add","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-add","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral is additive in the integrand: for `F`-progressive integrands `H\u2081, H\u2082` with `E \u222b_0^t H_i\u00b2 du < \u221e`, the integral of the sum is almost surely the sum of the integrals. Together with homogeneity and the difference isometry this pins down the integral on the whole `L\u00b2` space of progressive integrands.","title":"Linearity (additivity)"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-smul-of-is-pre-brownian-rea","kind":"node","lean":"ProbabilityTheory.itoIntegral_smul_of_isPreBrownianReal","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-smul-of-is-pre-brownian-rea","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral is homogeneous in the integrand for a pre-Brownian integrator; path continuity is not required.","title":"Linearity (homogeneity)"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-smul","kind":"node","lean":"ProbabilityTheory.itoIntegral_smul","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-smul","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"The It\u00f4 integral is homogeneous in the integrand.","title":"Linearity (homogeneity)"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-isometry-sub-of-is-pre-brow","kind":"node","lean":"ProbabilityTheory.itoIntegral_isometry_sub_of_isPreBrownianReal","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-isometry-sub-of-is-pre-brow","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For two `F`-progressive integrands with `E \u222b_0^t H_i\u00b2 du < \u221e`, the difference of the integrals is square-integrable and `E[(\u222b H\u2081 dB - \u222b H\u2082 dB)\u00b2] = E \u222b (H\u2081 - H\u2082)\u00b2 du`. With linearity this is the two-integrand isometry that, together with the elementary-integrand formula, determines the construction on the whole `L\u00b2` space. Only the pre-Brownian finite-dimensional laws are used; path continuity is not required.","title":"Difference isometry"},{"id":"Probability/stochastic-calculus/it-integral/ito-integral-isometry-sub","kind":"node","lean":"ProbabilityTheory.itoIntegral_isometry_sub","parent":19964,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-integral/index.html#ito-integral-isometry-sub","scope":"Probability/stochastic-calculus/it-integral","status":"fully_proved","summary":"For two `F`-progressive integrands with `E \u222b_0^t H_i\u00b2 du < \u221e`, the difference of the integrals is square-integrable and `E[(\u222b H\u2081 dB - \u222b H\u2082 dB)\u00b2] = E \u222b (H\u2081 - H\u2082)\u00b2 du`. With linearity this is the two-integrand isometry that, together with the elementary-integrand formula, determines the construction on the whole `L\u00b2` space.","title":"Difference isometry"},{"data_url":"scopes/Probability/stochastic-calculus/it-s-formula.json","id":"Probability/stochastic-calculus/it-s-formula","kind":"scope","navigation":"module","parent":19869,"result":false,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","title":"It\u00f4's formula"},{"id":"Probability/stochastic-calculus/it-s-formula/tendsto-taylor-sum-sub","kind":"node","lean":"ProbabilityTheory.tendsto_taylor_sum_sub","parent":20033,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html#tendsto-taylor-sum-sub","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","summary":"For `f` of class `C\u00b2`, a continuous path `w`, `t \u2265 0`, and a sequence of mesh counts `ns j \u2192 \u221e` along which the quadratic sums `\u2211_{k < ns j} (w (t (k + 1) / ns j) - w (t k / ns j))\u00b2` stay bounded, the telescoped increment `f (w t) - f (w 0)` minus the second-order Taylor sums `\u2211_k (f'(w_k) \u0394w_k + (1/2) f''(w_k) \u0394w_k\u00b2)` tends to `0`.","title":"Second-order Taylor sums along a continuous path"},{"id":"Probability/stochastic-calculus/it-s-formula/tendsto-sum-riemann-uniform","kind":"node","lean":"ProbabilityTheory.tendsto_sum_riemann_uniform","parent":20033,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html#tendsto-sum-riemann-uniform","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","summary":"Left-point Riemann sums of a continuous function on the uniform partition of `[0, t]` converge to the integral: `\u2211_{k < n} g (t k / n) (t / n) \u2192 \u222b_0^t g` (`t \u2265 0`).","title":"Left-point Riemann sums of a continuous function"},{"id":"Probability/stochastic-calculus/it-s-formula/left-point-process","kind":"node","lean":"ProbabilityTheory.leftPointProcess","parent":20033,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html#left-point-process","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","summary":"The left-point step process `\u2211_{k < n} g(B (t k / n)) 1_{(t k / n, t (k + 1) / n]}` of `u \u21a6 g (B u)` on the uniform partition of `[0, t]` into `n` cells.","title":"Left-point step process along a Brownian path"},{"id":"Probability/stochastic-calculus/it-s-formula/tendsto-lintegral-left-point-process-sub","kind":"node","lean":"ProbabilityTheory.tendsto_lintegral_leftPointProcess_sub_sq","parent":20033,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html#tendsto-lintegral-left-point-process-sub","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","summary":"For a pre-Brownian motion `B` that is jointly measurable in `(u, \u03c9)` and an `L`-Lipschitz `g`, the left-point step processes of `u \u21a6 g (B u)` converge to it in `L\u00b2(P \u2297 du)` on `(0, t] \u00d7 \u03a9`: `E \u222b_0^t |leftPointProcess g B t n u - g (B u)|\u00b2 du \u2192 0`.","title":"Left-point approximation of a Lipschitz function of Brownian motion"},{"id":"Probability/stochastic-calculus/it-s-formula/left-point","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.leftPoint","parent":20033,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html#left-point","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","summary":"The left-point elementary integrand `\u2211_{k < n} g(B (t k / n)) 1_{(t k / n, t (k + 1) / n]}` of `u \u21a6 g (B u)` on the uniform partition of `[0, t]`, for a measurable `g` bounded by `C` and an adapted `B`.","title":"Left-point elementary integrand along a Brownian path"},{"id":"Probability/stochastic-calculus/it-s-formula/ito-integral-left-point","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.itoIntegral_leftPoint","parent":20033,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html#ito-integral-left-point","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","summary":"The elementary It\u00f4 integral of `leftPoint` at time `t` is the left-point It\u00f4 sum `\u2211_{k < n} g(B (t k / n)) (B (t (k + 1) / n) - B (t k / n))`.","title":"It\u00f4 integral of the left-point integrand is the left-point sum"},{"id":"Probability/stochastic-calculus/it-s-formula/to-process-left-point","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.toProcess_leftPoint","parent":20033,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html#to-process-left-point","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","summary":"The process of `leftPoint` is `leftPointProcess`.","title":"Process of the left-point integrand is the left-point step process"},{"id":"Probability/stochastic-calculus/it-s-formula/ito-formula","kind":"node","lean":"ProbabilityTheory.ito_formula","parent":20033,"result":true,"route":"../roadmap/Probability/stochastic-calculus/it-s-formula/index.html#ito-formula","scope":"Probability/stochastic-calculus/it-s-formula","status":"fully_proved","summary":"It\u00f4's formula for Brownian motion: for a Brownian motion `B` adapted to `F` with increments independent of the past and jointly measurable in `(t, \u03c9)`, and `f` of class `C\u00b2` with `|f'| \u2264 C` and `|f''| \u2264 C`, `f(B t) = f(B 0) + \u222b_0^t f'(B u) dB u + (1/2) \u222b_0^t f''(B u) du` almost surely, for every `t \u2265 0` (Karatzas\u2013Shreve 1991, Theorem 3.3.3). The last integral is a pathwise Riemann integral; it is the integral of a\u2026","title":"It\u00f4's formula"},{"data_url":"scopes/Probability/stochastic-calculus/ornstein-uhlenbeck-process.json","id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","kind":"scope","navigation":"module","parent":19869,"result":false,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","title":"Ornstein\u2013Uhlenbeck process"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-covariance","kind":"node","lean":"ProbabilityTheory.ouCovariance","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-covariance","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The covariance function of the Ornstein\u2013Uhlenbeck process started at `0`: `(\u03c3\u00b2 / (2\u03b8)) (e^{-\u03b8|t-s|} - e^{-\u03b8(t+s)})` (Karatzas\u2013Shreve 1991, Section 5.6). It is meant for `\u03b8 > 0` and times `s, t \u2265 0`; at `\u03b8 = 0` the division by zero makes it `0`.","title":"Covariance of the Ornstein\u2013Uhlenbeck process started at zero"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-stationary-covariance","kind":"node","lean":"ProbabilityTheory.ouStationaryCovariance","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-stationary-covariance","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The covariance function of the stationary Ornstein\u2013Uhlenbeck process: `(\u03c3\u00b2 / (2\u03b8)) e^{-\u03b8|t-s|}` (Uhlenbeck\u2013Ornstein 1930). It is meant for `\u03b8 > 0`.","title":"Covariance of the stationary Ornstein\u2013Uhlenbeck process"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-covariance-comm","kind":"node","lean":"ProbabilityTheory.ouCovariance_comm","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-covariance-comm","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The OU covariance is symmetric.","title":"Symmetry of the Ornstein\u2013Uhlenbeck covariance"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-covariance-self","kind":"node","lean":"ProbabilityTheory.ouCovariance_self","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-covariance-self","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The variance of the OU process started at `0`: `(\u03c3\u00b2 / (2\u03b8)) (1 - e^{-2\u03b8t})`.","title":"Variance of the Ornstein\u2013Uhlenbeck process started at zero"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-stationary-covariance-self","kind":"node","lean":"ProbabilityTheory.ouStationaryCovariance_self","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-stationary-covariance-self","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The stationary variance `\u03c3\u00b2 / (2\u03b8)`.","title":"Variance of the stationary Ornstein\u2013Uhlenbeck process"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/tendsto-ou-covariance-add","kind":"node","lean":"ProbabilityTheory.tendsto_ouCovariance_add","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#tendsto-ou-covariance-add","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Shifting both times by `h \u2192 \u221e` turns `ouCovariance` into `ouStationaryCovariance`.","title":"Ornstein\u2013Uhlenbeck covariance converges to the stationary one"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/is-ornstein-uhlenbeck-from-zero","kind":"node","lean":"ProbabilityTheory.IsOrnsteinUhlenbeckFromZero","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#is-ornstein-uhlenbeck-from-zero","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"`X` is an Ornstein\u2013Uhlenbeck process started at `0` with rate `\u03b8 > 0` and noise amplitude `\u03c3` under `P`: a centered Gaussian process with covariance `ouCovariance \u03b8 \u03c3`. Only the finite-dimensional laws are fixed (no path regularity).","title":"Ornstein\u2013Uhlenbeck process started at zero"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/is-stationary-ornstein-uhlenbeck","kind":"node","lean":"ProbabilityTheory.IsStationaryOrnsteinUhlenbeck","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#is-stationary-ornstein-uhlenbeck","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"`X` is a stationary Ornstein\u2013Uhlenbeck process with rate `\u03b8 > 0` and noise amplitude `\u03c3` under `P`: a centered Gaussian process with covariance `ouStationaryCovariance \u03b8 \u03c3`. Only the finite-dimensional laws are fixed.","title":"Stationary Ornstein\u2013Uhlenbeck process"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-clock","kind":"node","lean":"ProbabilityTheory.ouClock","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-clock","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Doob's clock `\u03c4 t = (e^{2\u03b8t} - 1)/(2\u03b8)` for the process started at `0`. The value is nonnegative for every real `\u03b8`, so `Real.toNNReal` only changes its type.","title":"Doob's clock for the Ornstein\u2013Uhlenbeck process from zero"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-stationary-clock","kind":"node","lean":"ProbabilityTheory.ouStationaryClock","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-stationary-clock","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Doob's clock `t \u21a6 e^{2\u03b8t}/(2\u03b8)` for the stationary process (positive for `\u03b8 > 0`).","title":"Doob's clock for the stationary Ornstein\u2013Uhlenbeck process"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-of-brownian","kind":"node","lean":"ProbabilityTheory.ouOfBrownian","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-of-brownian","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Doob's time change `t \u21a6 \u03c3 e^{-\u03b8t} B(\u03c4 t)` of a process `B`, with `\u03c4 = ouClock \u03b8`.","title":"Doob's time change of Brownian motion"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-stationary-of-brownian","kind":"node","lean":"ProbabilityTheory.ouStationaryOfBrownian","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-stationary-of-brownian","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Doob's stationary time change `t \u21a6 \u03c3 e^{-\u03b8t} B(e^{2\u03b8t}/(2\u03b8))` of a process `B`.","title":"Doob's stationary time change of Brownian motion"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/is-ornstein-uhlenbeck-from-zero-ou-of-br","kind":"node","lean":"ProbabilityTheory.IsPreBrownianReal.isOrnsteinUhlenbeckFromZero_ouOfBrownian","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#is-ornstein-uhlenbeck-from-zero-ou-of-br","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The time change `\u03c3 e^{-\u03b8t} B((e^{2\u03b8t} - 1)/(2\u03b8))` of a pre-Brownian motion is an Ornstein\u2013Uhlenbeck process started at `0`.","title":"Construction (Doob 1942)"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/is-stationary-ornstein-uhlenbeck-ou-stat","kind":"node","lean":"ProbabilityTheory.IsPreBrownianReal.isStationaryOrnsteinUhlenbeck_ouStationaryOfBrownian","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#is-stationary-ornstein-uhlenbeck-ou-stat","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The time change `\u03c3 e^{-\u03b8t} B(e^{2\u03b8t}/(2\u03b8))` of a pre-Brownian motion is a stationary Ornstein\u2013Uhlenbeck process.","title":"Construction (Doob 1942)"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/exists-is-pre-brownian-real-eval","kind":"node","lean":"ProbabilityTheory.exists_isPreBrownianReal_eval","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#exists-is-pre-brownian-real-eval","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"There is a probability measure on paths `\u211d\u22650 \u2192 \u211d` under which the coordinate process is a pre-Brownian motion (Kolmogorov extension of the Brownian finite-dimensional laws).","title":"Existence of a pre-Brownian motion on path space"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/exists-is-ornstein-uhlenbeck-from-zero","kind":"node","lean":"ProbabilityTheory.exists_isOrnsteinUhlenbeckFromZero","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#exists-is-ornstein-uhlenbeck-from-zero","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"An Ornstein\u2013Uhlenbeck process started at `0` exists for every `\u03b8 > 0` and `\u03c3`.","title":"Existence of the Ornstein\u2013Uhlenbeck process started at zero"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/exists-is-stationary-ornstein-uhlenbeck","kind":"node","lean":"ProbabilityTheory.exists_isStationaryOrnsteinUhlenbeck","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#exists-is-stationary-ornstein-uhlenbeck","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"A stationary Ornstein\u2013Uhlenbeck process exists for every `\u03b8 > 0` and `\u03c3`.","title":"Existence of the stationary Ornstein\u2013Uhlenbeck process"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/has-law-eval","kind":"node","lean":"ProbabilityTheory.IsOrnsteinUhlenbeckFromZero.hasLaw_eval","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#has-law-eval","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The one-time law of the OU process started at `0`: `X t ~ N(0, (\u03c3\u00b2 / (2\u03b8)) (1 - e^{-2\u03b8t}))`.","title":"One-time law of the Ornstein\u2013Uhlenbeck process from zero"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/eval-zero-ae-eq-zero","kind":"node","lean":"ProbabilityTheory.IsOrnsteinUhlenbeckFromZero.eval_zero_ae_eq_zero","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#eval-zero-ae-eq-zero","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The OU process of `IsOrnsteinUhlenbeckFromZero` starts at `0` almost surely.","title":"The Ornstein\u2013Uhlenbeck process from zero starts at zero"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/tendsto-covariance-add","kind":"node","lean":"ProbabilityTheory.IsOrnsteinUhlenbeckFromZero.tendsto_covariance_add","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#tendsto-covariance-add","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For the process started at `0`, `cov(X (s + h), X (t + h)) \u2192 (\u03c3\u00b2 / (2\u03b8)) e^{-\u03b8|t-s|}` as `h \u2192 \u221e`.","title":"Relaxation to stationarity"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/has-law-eval-2","kind":"node","lean":"ProbabilityTheory.IsStationaryOrnsteinUhlenbeck.hasLaw_eval","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#has-law-eval-2","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Each `X t` of a stationary OU process is `N(0, \u03c3\u00b2 / (2\u03b8))`.","title":"Stationary law"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/map-restrict-eq-of-covariance-eq","kind":"node","lean":"ProbabilityTheory.IsGaussianProcess.map_restrict_eq_of_covariance_eq","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#map-restrict-eq-of-covariance-eq","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The finite-dimensional laws of a real Gaussian process are determined by its mean and covariance functions: two Gaussian processes with the same means and covariances (possibly on different probability spaces) have the same finite-dimensional laws.","title":"Gaussian process laws are determined by mean and covariance"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/shift","kind":"node","lean":"ProbabilityTheory.IsStationaryOrnsteinUhlenbeck.shift","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#shift","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"A time shift of a stationary OU process is a stationary OU process.","title":"Time shifts of a stationary Ornstein\u2013Uhlenbeck process"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/map-restrict-shift","kind":"node","lean":"ProbabilityTheory.IsStationaryOrnsteinUhlenbeck.map_restrict_shift","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#map-restrict-shift","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Every finite-dimensional law of a stationary OU process is invariant under time shifts.","title":"Strict stationarity"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/cond-exp-natural-eq-of-covariance","kind":"node","lean":"ProbabilityTheory.IsGaussianProcess.condExp_natural_eq_of_covariance","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#cond-exp-natural-eq-of-covariance","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For a centered Gaussian process with `cov(X u, X t) = c cov(X u, X s)` for all `u \u2264 s`, the conditional expectation of `X t` given the natural filtration at `s` is `c X s`.","title":"Conditional expectation of a Gaussian process given its past"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/cond-distrib-eval-eq-of-covariance","kind":"node","lean":"ProbabilityTheory.IsGaussianProcess.condDistrib_eval_eq_of_covariance","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#cond-distrib-eval-eq-of-covariance","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For a centered Gaussian process with `cov(X s, X t) = c Var(X s)`, the conditional law of `X t` given `X s = x` is `N(c x, Var(X t) - c\u00b2 Var(X s))` for almost every `x`.","title":"Conditional law of a Gaussian process at one time given another"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/cond-exp-natural","kind":"node","lean":"ProbabilityTheory.IsOrnsteinUhlenbeckFromZero.condExp_natural","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#cond-exp-natural","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For the OU process started at `0` and `s \u2264 t`, `E[X t | F s] = e^{-\u03b8(t-s)} X s`, where `F` is the natural filtration of `X`.","title":"Mean reversion"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/cond-exp-natural-2","kind":"node","lean":"ProbabilityTheory.IsStationaryOrnsteinUhlenbeck.condExp_natural","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#cond-exp-natural-2","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For the stationary OU process and `s \u2264 t`, `E[X t | F s] = e^{-\u03b8(t-s)} X s`, where `F` is the natural filtration of `X`.","title":"Mean reversion"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/cond-distrib-eval","kind":"node","lean":"ProbabilityTheory.IsOrnsteinUhlenbeckFromZero.condDistrib_eval","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#cond-distrib-eval","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For the OU process started at `0` and `s \u2264 t`, the conditional law of `X t` given `X s = x` is `N(e^{-\u03b8(t-s)} x, (\u03c3\u00b2 / (2\u03b8)) (1 - e^{-2\u03b8(t-s)}))` for almost every `x`.","title":"Transition law"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/cond-distrib-eval-2","kind":"node","lean":"ProbabilityTheory.IsStationaryOrnsteinUhlenbeck.condDistrib_eval","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#cond-distrib-eval-2","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For the stationary OU process and `s \u2264 t`, the conditional law of `X t` given `X s = x` is `N(e^{-\u03b8(t-s)} x, (\u03c3\u00b2 / (2\u03b8)) (1 - e^{-2\u03b8(t-s)}))` for almost every `x`.","title":"Transition law"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-path-solution","kind":"node","lean":"ProbabilityTheory.ouPathSolution","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-path-solution","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The explicit solution of the pathwise Langevin equation driven by a path `w`: `e^{-\u03b8t} x\u2080 + \u03c3 (w t - e^{-\u03b8t} w 0 - \u03b8 \u222b_0^t e^{-\u03b8(t-u)} w u du)`. This is `e^{-\u03b8t} x\u2080 + \u03c3 \u222b_0^t e^{-\u03b8(t-u)} dw(u)` after integration by parts (the It\u00f4-integral form for a Brownian path is a wanted statement).","title":"Pathwise solution of the Langevin equation"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/continuous-ou-path-solution","kind":"node","lean":"ProbabilityTheory.continuous_ouPathSolution","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#continuous-ou-path-solution","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"The pathwise solution is continuous when the driving path is.","title":"Continuity of the pathwise Langevin solution"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ou-path-solution-integral-eq","kind":"node","lean":"ProbabilityTheory.ouPathSolution_integral_eq","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ou-path-solution-integral-eq","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For a continuous path `w` and every real `t`, `ouPathSolution \u03b8 \u03c3 x\u2080 w` satisfies `x t = x\u2080 - \u03b8 \u222b_0^t x + \u03c3 (w t - w 0)` (any real `\u03b8`, `\u03c3`, `x\u2080`).","title":"Pathwise Langevin equation"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/eq-ou-path-solution-of-integral-eq","kind":"node","lean":"ProbabilityTheory.eq_ouPathSolution_of_integral_eq","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#eq-ou-path-solution-of-integral-eq","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"A continuous solution of `x t = x\u2080 - \u03b8 \u222b_0^t x + \u03c3 (w t - w 0)` on `t \u2265 0` equals `ouPathSolution \u03b8 \u03c3 x\u2080 w` there.","title":"Uniqueness"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/ito-integral-ae-eq-of-ae-eq","kind":"node","lean":"ProbabilityTheory.ElementaryIntegrand.itoIntegral_ae_eq_of_ae_eq","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#ito-integral-ae-eq-of-ae-eq","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Elementary It\u00f4 integrals against two modifications of the same process agree almost surely.","title":"Elementary It\u00f4 integrals agree for modifications"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/option-elim","kind":"node","lean":"ProbabilityTheory.IsGaussianProcess.option_elim","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#option-elim","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"If `X\u2080` is a real Gaussian random variable independent of the real Gaussian process `B`, the process indexed by `Option T` that is `X\u2080` at `none` and `B t` at `some t` is a Gaussian process.","title":"Adjoining an independent Gaussian variable"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/integral-ito-integral-l2-exp-mul","kind":"node","lean":"ProbabilityTheory.integral_itoIntegralL2_exp_mul","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#integral-ito-integral-l2-exp-mul","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Covariance of the Wiener integrals `I_t = \u222b_0^t e^{\u03b8u} dB_u`: for a pre-Brownian motion `B` adapted to `F` with increments independent of the past and `\u03b8 \u2260 0`, `E[I_s I_t] = (e^{2\u03b8 (s \u2227 t)} - 1) / (2\u03b8)`, by the martingale property and the It\u00f4 isometry.","title":"Covariance of the Wiener integrals"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/is-gaussian-process-ou-path-solution","kind":"node","lean":"ProbabilityTheory.IsBrownianReal.isGaussianProcess_ouPathSolution","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#is-gaussian-process-ou-path-solution","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"For a Brownian motion `B`, `\u03b8 \u2260 0`, any real `\u03c3`, and a start `X\u2080 ~ N(0, v)` independent of `B`, the process `X t = ouPathSolution \u03b8 \u03c3 X\u2080 B t` (`t \u2265 0`) is a Gaussian process with `E[X t] = 0` and `cov[X s, X t] = e^{-\u03b8(s+t)} (v + \u03c3\u00b2 (e^{2\u03b8 (s \u2227 t)} - 1) / (2\u03b8))` (Karatzas\u2013Shreve 1991, Section 5.6).","title":"The pathwise Langevin solution driven by Brownian motion is Gaussian"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/is-ornstein-uhlenbeck-from-zero-ou-path","kind":"node","lean":"ProbabilityTheory.IsBrownianReal.isOrnsteinUhlenbeckFromZero_ouPathSolution","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#is-ornstein-uhlenbeck-from-zero-ou-path","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Driven by a Brownian motion and started at `0`, `ouPathSolution` is an Ornstein\u2013Uhlenbeck process started at `0` (Karatzas\u2013Shreve 1991, Section 5.6). No filtration and no measurability of the coordinates of `B` is assumed.","title":"The pathwise Langevin solution is an OU process"},{"id":"Probability/stochastic-calculus/ornstein-uhlenbeck-process/is-stationary-ornstein-uhlenbeck-ou-path","kind":"node","lean":"ProbabilityTheory.IsBrownianReal.isStationaryOrnsteinUhlenbeck_ouPathSolution","parent":20042,"result":true,"route":"../roadmap/Probability/stochastic-calculus/ornstein-uhlenbeck-process/index.html#is-stationary-ornstein-uhlenbeck-ou-path","scope":"Probability/stochastic-calculus/ornstein-uhlenbeck-process","status":"fully_proved","summary":"Driven by a Brownian motion `B` and started from `X\u2080 ~ N(0, \u03c3\u00b2/(2\u03b8))` independent of `B`, `ouPathSolution` is a stationary Ornstein\u2013Uhlenbeck process (Karatzas\u2013Shreve 1991, Section 5.6).","title":"Stationary start"},{"data_url":"scopes/Probability/percolation.json","id":"Probability/percolation","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/percolation/index.html","scope":"Probability/percolation","status":"can_prove","summary":"Bernoulli bond and site percolation: percolation probability, critical points, number and uniqueness of infinite clusters, sharpness and sharp thresholds, and crossing estimates in two dimensions.","title":"Percolation"},{"data_url":"scopes/Probability/percolation/phase-transition.json","id":"Probability/percolation/phase-transition","kind":"scope","parent":20083,"result":false,"route":"../roadmap/Probability/percolation/phase-transition/index.html","scope":"Probability/percolation/phase-transition","status":"can_prove","summary":"The percolation probability theta, bounds 0 < p_c < 1 (Peierls and enclosure arguments), percolation on product graphs, the number and uniqueness of infinite clusters (Burton\u2013Keane), and sharp thresholds from influences (KKL).","title":"Phase transition and critical probability"},{"data_url":"scopes/Probability/percolation/phase-transition/percolation-on-product-graphs.json","id":"Probability/percolation/phase-transition/percolation-on-product-graphs","kind":"scope","navigation":"module","parent":20084,"result":false,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","title":"Percolation on product graphs"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/int-line","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.intLine","parent":20085,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#int-line","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"The integer line is the simple graph on $\\mathbb{Z}$ in which $m$ and $n$ are adjacent exactly when $|m - n| = 1$.","title":"The integer line graph"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/prod-graph","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.prodGraph","parent":20085,"reading_title":{"format":"inline-parts-v1","parts":[["text","The product graph "],["math","\\(G \\times \\mathbb{Z}\\)"]],"source":"The product graph $G \\times \\mathbb{Z}$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#prod-graph","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"For a simple graph $G$ on $V$, the graph $G \\times \\mathbb{Z}$ is the box product of $G$ with the integer line: its vertex set is $V \\times \\mathbb{Z}$, and $(v, m)$ and $(w, n)$ are adjacent exactly when either $v$ and $w$ are adjacent in $G$ and $m = n$, or $v = w$ and $|m - n| = 1$.","title":"The product graph $G \\times \\mathbb{Z}$"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/config-graph","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.configGraph","parent":20085,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#config-graph","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on $V$, let $G \\times \\mathbb{Z}$ denote the box product of $G$ with the integer line, and let $\\omega$ assign true or false to every unordered pair of points of $V \\times \\mathbb{Z}$. The open subgraph of $G \\times \\mathbb{Z}$ determined by $\\omega$ is the simple graph on $V \\times \\mathbb{Z}$ in which $a$ and $b$ are adjacent exactly when they are adjacent in $G \\times \\mathbb{Z}$ and\u2026","title":"Open subgraph of the product graph"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/cluster","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.cluster","parent":20085,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#cluster","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on $V$, let $G \\times \\mathbb{Z}$ denote the box product of $G$ with the integer line, let $\\omega$ assign true or false to every unordered pair of points of $V \\times \\mathbb{Z}$, and let $x \\in V \\times \\mathbb{Z}$. The open cluster of $x$ is the set of points $y \\in V \\times \\mathbb{Z}$ that are reachable from $x$ (including $x$ itself) in the open subgraph of $G \\times \\mathbb{Z}$\u2026","title":"Open cluster in the product graph"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/bernoulli-law","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.bernoulliLaw","parent":20085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli bond law on "],["math","\\(V \\times \\mathbb{Z}\\)"]],"source":"Bernoulli bond law on $V \\times \\mathbb{Z}$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#bernoulli-law","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"For a type $V$ and $0 \\le p \\le 1$, the Bernoulli($p$) bond law on $V \\times \\mathbb{Z}$ is the product measure on configurations $\\omega$, which assign true (open) or false (closed) to every unordered pair $\\{a, b\\}$ of points of $V \\times \\mathbb{Z}$ (with $a = b$ allowed), under which the values $\\omega(e)$ are independent and each is true with probability $p$ and false with probability $1 - p$.","title":"Bernoulli bond law on $V \\times \\mathbb{Z}$"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/almost-surely","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.AlmostSurely","parent":20085,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#almost-surely","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"Let $V$ be a type, let $p$ be a real number with $0 \\le p \\le 1$, and let $P$ be a property of configurations $\\omega$, that is, of functions assigning true or false to every unordered pair $\\{a, b\\}$ of points of $V \\times \\mathbb{Z}$ (with $a = b$ allowed). We say that $P$ holds almost surely if $P(\\omega)$ holds for almost every $\\omega$ with respect to the Bernoulli($p$) bond law on $V \\times \\mathbb{Z}$.","title":"Almost-sure properties under the Bernoulli bond law"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/open-subgraph","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.openSubgraph","parent":20085,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#open-subgraph","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on $V$ and let $\\omega$ assign true or false to every unordered pair of elements of $V$. The open subgraph of $G$ determined by $\\omega$ is the simple graph on $V$ in which $a$ and $b$ are adjacent exactly when they are adjacent in $G$ and $\\omega(\\{a, b\\})$ is true.","title":"Open subgraph of a graph"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/gcluster","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.gcluster","parent":20085,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#gcluster","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on $V$, let $\\omega$ assign true or false to every unordered pair of elements of $V$, and let $x \\in V$. The open cluster of $x$ is the set of vertices $y \\in V$ that are reachable from $x$ (including $x$ itself) in the open subgraph of $G$ determined by $\\omega$, that is, joined to $x$ by a path whose edges $e$ are edges of $G$ with $\\omega(e)$ true.","title":"Open cluster in a graph"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/gbernoulli-law","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.gbernoulliLaw","parent":20085,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli bond law on "],["math","\\(V\\)"]],"source":"Bernoulli bond law on $V$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#gbernoulli-law","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"For a type $V$ and $0 \\le p \\le 1$, the Bernoulli($p$) bond law on $V$ is the product measure on configurations $\\omega$, which assign true (open) or false (closed) to every unordered pair $\\{a, b\\}$ of elements of $V$ (with $a = b$ allowed), under which the values $\\omega(e)$ are independent and each is true with probability $p$ and false with probability $1 - p$.","title":"Bernoulli bond law on $V$"},{"id":"Probability/percolation/phase-transition/percolation-on-product-graphs/critical-prob","kind":"node","lean":"MathlibExt.Probability.ProductBondPercolation.criticalProb","parent":20085,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-on-product-graphs/index.html#critical-prob","scope":"Probability/percolation/phase-transition/percolation-on-product-graphs","status":"fully_proved","summary":"For a simple graph $G$ on a countable vertex set $V$, the critical probability $p_c(G)$ is the infimum of the set consisting of $1$ together with every $p \\in [0, 1]$ for which, under the Bernoulli($p$) bond law on $V$, the event that the open cluster in $G$ of some vertex is infinite has positive probability. In particular $p_c(G) = 1$ when no such $p$ exists.","title":"Critical probability for bond percolation"},{"data_url":"scopes/Probability/percolation/phase-transition/percolation-probability-and-critical-point.json","id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","kind":"scope","navigation":"module","parent":20084,"result":false,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","title":"Percolation probability and critical point"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/is-open-edge","kind":"node","lean":"StatMech.Lattice.IsOpenEdge","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#is-open-edge","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The lattice edge `{x, y}` is open in the configuration `\u03c9`: `x` and `y` are nearest neighbours and `\u03c9 s(x, y) = true`.","title":"Open lattice edge of a bond configuration"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/open-subgraph","kind":"node","lean":"StatMech.Lattice.openSubgraph","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#open-subgraph","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The open subgraph of `\u03c9`: the spanning subgraph of `\u2124^d` consisting of the open edges.","title":"Open subgraph of a bond configuration"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/connected","kind":"node","lean":"StatMech.Lattice.Connected","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#connected","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"`x` and `y` are connected by an open path, that is, reachable in `openSubgraph d \u03c9`.","title":"Connection by an open path"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/cluster","kind":"node","lean":"StatMech.Lattice.cluster","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#cluster","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The open cluster of `x`: the set of sites connected to `x` by an open path.","title":"Open cluster of a site"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/open-subgraph-induce","kind":"node","lean":"StatMech.Lattice.openSubgraphInduce","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#open-subgraph-induce","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The open subgraph of `\u03c9` induced on a set `S` of sites.","title":"Open subgraph induced on a set of sites"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/connected-within","kind":"node","lean":"StatMech.Lattice.ConnectedWithin","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#connected-within","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"`x` and `y` in `S` are connected by an open path that stays inside `S`.","title":"Connection by an open path within a set"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/connected-within-mono-set","kind":"node","lean":"StatMech.Lattice.connectedWithin_mono_set","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#connected-within-mono-set","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"Connection inside `S` implies connection inside any `T \u2287 S`.","title":"Open connection within a set is monotone in the set"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/is-open-circuit","kind":"node","lean":"StatMech.Lattice.IsOpenCircuit","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#is-open-circuit","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"A closed walk in the open subgraph that repeats no edge (an open circuit).","title":"Open circuit in a bond configuration"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/origin","kind":"node","lean":"StatMech.Percolation.origin","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#origin","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The origin `0 \u2208 \u2124^d`.","title":"Origin of the hypercubic lattice"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/percolation-event","kind":"node","lean":"StatMech.Percolation.percolationEvent","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#percolation-event","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The percolation event `{0 \u2194 \u221e}`: the open cluster of the origin is infinite.","title":"Percolation event for the origin"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/theta","kind":"node","lean":"StatMech.Percolation.theta","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#theta","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The percolation probability `\u03b8_d(p) = \u2119_p(0 \u2194 \u221e)` of Bernoulli bond percolation on `\u2124^d` with edge density `p`.","title":"Percolation probability of bond percolation"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/subcritical-set","kind":"node","lean":"StatMech.Percolation.subcriticalSet","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#subcritical-set","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The set of densities `p \u2208 [0, 1]` at which `\u03b8_d(p) = 0`.","title":"Set of densities with no percolation"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/pc","kind":"node","lean":"StatMech.Percolation.pc","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#pc","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The critical probability `p_c(d) = sup {p \u2208 [0, 1] : \u03b8_d(p) = 0}` of bond percolation on `\u2124^d`.","title":"Critical probability of bond percolation on the hypercubic lattice"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/cluster-eq-gcluster","kind":"node","lean":"StatMech.Percolation.cluster_eq_gcluster","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#cluster-eq-gcluster","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The open cluster of `x` in the lattice model is the cluster `MathlibExt.Probability.ProductBondPercolation.gcluster` of bond percolation on the graph `hypercubicLattice d` with the same edge configuration.","title":"Lattice clusters agree with general graph clusters"},{"id":"Probability/percolation/phase-transition/percolation-probability-and-critical-point/box-finset","kind":"node","lean":"StatMech.Percolation.boxFinset","parent":20096,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/percolation-probability-and-critical-point/index.html#box-finset","scope":"Probability/percolation/phase-transition/percolation-probability-and-critical-point","status":"fully_proved","summary":"The box `\u039b_n = [-n, n]^d` as a finset.","title":"Lattice box as a finite set"},{"data_url":"scopes/Probability/percolation/phase-transition/bounds-on-the-critical-probability.json","id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","kind":"scope","navigation":"module","parent":20084,"result":false,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","title":"Bounds on the critical probability"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/vert-of","kind":"node","lean":"StatMech.Percolation.vertOf","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#vert-of","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The `k`-th vertex of the lattice path from the origin that takes step `\u00b1e_i` given by `g j` at time `j`.","title":"Vertices of a lattice path encoded by a step sequence"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/edge-of","kind":"node","lean":"StatMech.Percolation.edgeOf","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#edge-of","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The `k`-th edge `{vertOf g k, vertOf g (k + 1)}` of the path encoded by `g`.","title":"Edges of a lattice path encoded by a step sequence"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/word-event","kind":"node","lean":"StatMech.Percolation.wordEvent","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#word-event","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The event that the first `n` edges of the path encoded by `g` are open.","title":"Event that an encoded lattice path is open"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/good-word","kind":"node","lean":"StatMech.Percolation.goodWord","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#good-word","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The first `n + 1` vertices of the path encoded by `g` are distinct.","title":"Self-avoiding initial segment of an encoded lattice path"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/dir-of","kind":"node","lean":"StatMech.Percolation.dirOf","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#dir-of","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The direction `(i, \u00b1)` of the `k`-th step of an open walk from the origin (arbitrary past its end).","title":"Step directions of an open walk from the origin"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-pos","kind":"node","lean":"StatMech.Percolation.pc_pos","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-pos","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"`p_c(d) > 0` for bond percolation on `\u2124^d`, `d \u2265 1` (Hammersley 1957, by counting paths: `p_c(d) \u2265 1/(2d)`).","title":"Positivity of the critical probability (Hammersley's bound)"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-upper-bound","kind":"node","lean":"StatMech.Percolation.pcUpperBound","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-upper-bound","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The Peierls series `\u2211_\u2113 \u2113 4^\u2113 (1 - p)^\u2113` bounding the probability of a closed dual circuit around the origin.","title":"Peierls series bounding closed dual circuits"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-ratio","kind":"node","lean":"StatMech.Percolation.pcRatio","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-ratio","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The ratio `4(1 - p)` of the geometric part of `pcUpperBound`.","title":"Geometric ratio of the Peierls series"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-contour-bound","kind":"node","lean":"StatMech.Percolation.PcContourBound","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-contour-bound","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The Peierls contour bound `1 - \u03b8_2(p) \u2264 pcUpperBound p` at density `p`.","title":"Peierls contour bound on the percolation probability"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-lt-one","kind":"node","lean":"StatMech.Percolation.pc_lt_one","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(2) < 1\\)"],["text"," from the Peierls contour bound"]],"source":"$p_c(2) < 1$ from the Peierls contour bound"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-lt-one","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"`p_c(2) < 1`, assuming the Peierls contour bound `PcContourBound` for all `p` above some `p\u2080 < 1`.","title":"$p_c(2) < 1$ from the Peierls contour bound"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/primal-edge-finset","kind":"node","lean":"StatMech.Percolation.primalEdgeFinset","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#primal-edge-finset","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The set of primal edges crossed by the darts of a walk `c` in the dual (face) lattice.","title":"Primal edges crossed by a dual walk"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/closed-circuit-event","kind":"node","lean":"StatMech.Percolation.closedCircuitEvent","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#closed-circuit-event","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The event that every primal edge crossed by the dual walk `c` is closed.","title":"Event that a dual circuit is closed"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/anchor-finset","kind":"node","lean":"StatMech.Percolation.anchorFinset","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#anchor-finset","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The `2\u2113` anchor sites `(j, 0)` and `(j, -1)`, `0 \u2264 j < \u2113`.","title":"Anchor sites for dual circuits"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/cycles-at-len","kind":"node","lean":"StatMech.Percolation.cyclesAtLen","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#cycles-at-len","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The finite set of dual cycles of length `\u2113` based at an anchor site of `anchorFinset \u2113`.","title":"Anchored dual cycles of a given length"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-summand-two","kind":"node","lean":"StatMech.Percolation.pcSummandTwo","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-summand-two","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The summand `2\u2113 \u00b7 4^\u2113 \u00b7 (1 - p)^\u2113`.","title":"Summand of the anchored Peierls series"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-anchored-enclosure","kind":"node","lean":"StatMech.Percolation.PcAnchoredEnclosure","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-anchored-enclosure","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"Every finite cluster of the origin is enclosed by a cycle of its face-boundary graph based at an anchor site of `anchorFinset \u2113`, where `\u2113` is the cycle length.","title":"Anchored dual-circuit enclosure property"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-lt-one-of-enclosure","kind":"node","lean":"StatMech.Percolation.pc_lt_one_of_enclosure","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(2) < 1\\)"],["text"," from anchored dual-circuit enclosure"]],"source":"$p_c(2) < 1$ from anchored dual-circuit enclosure"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-lt-one-of-enclosure","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"`p_c(2) < 1`, assuming the anchored enclosure property `PcAnchoredEnclosure`.","title":"$p_c(2) < 1$ from anchored dual-circuit enclosure"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/bernoulli-product-map-injective-restrict","kind":"node","lean":"StatMech.FK.bernoulliProduct_map_injective_restrict","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#bernoulli-product-map-injective-restrict","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"Pulling a Bernoulli product configuration back along an injective map `f : I \u2192 E` gives the Bernoulli product measure on `I`. See also `MeasureTheory.Measure.infinitePi_map_restrict'` for the case of a subset.","title":"Bernoulli product pulled back along an injection is Bernoulli"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/fk-prob-one-pattern","kind":"node","lean":"StatMech.FK.fkProbOne_pattern","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(q = 1\\)"],["text"," probability of an edge pattern is its Bernoulli weight"]],"source":"$q = 1$ probability of an edge pattern is its Bernoulli weight"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#fk-prob-one-pattern","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The `q = 1` random-cluster probability of a given pattern on the graph edges is its Bernoulli weight.","title":"$q = 1$ probability of an edge pattern is its Bernoulli weight"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/fk-prob-one-event-eq-bernoulli","kind":"node","lean":"StatMech.FK.fkProbOne_event_eq_bernoulli","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(q = 1\\)"],["text"," measure equals Bernoulli percolation on graph-edge events"]],"source":"$q = 1$ measure equals Bernoulli percolation on graph-edge events"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#fk-prob-one-event-eq-bernoulli","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The `q = 1` random-cluster measure on a finite graph agrees with Bernoulli percolation on events that depend only on the graph edges.","title":"$q = 1$ measure equals Bernoulli percolation on graph-edge events"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/conn-to-bdry-event-depends-on","kind":"node","lean":"StatMech.FK.connToBdryEvent_dependsOn","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#conn-to-bdry-event-depends-on","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The event that `x` is connected to the boundary depends only on the graph edges.","title":"Connection to the boundary depends only on graph edges"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/bernoulli-product-map-box-restrict","kind":"node","lean":"StatMech.FK.bernoulliProduct_map_boxRestrict","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#bernoulli-product-map-box-restrict","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"Restricting Bernoulli percolation on `\u2124^d` to the box gives Bernoulli percolation on the box.","title":"Bernoulli percolation restricted to a box is Bernoulli on the box"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/bernoulli-product-box-restrict-preimage","kind":"node","lean":"StatMech.FK.bernoulliProduct_boxRestrict_preimage","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#bernoulli-product-box-restrict-preimage","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The Bernoulli probability of a box cylinder event equals its probability on the box.","title":"Bernoulli box cylinder probability equals its probability on the box"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/wired-finite-box-bdry-conn-event-le-bern","kind":"node","lean":"StatMech.FK.wiredFinite_boxBdryConnEvent_le_bernoulli","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," origin-to-boundary probability at most Bernoulli"]],"source":"Wired $q = 2$ origin-to-boundary probability at most Bernoulli"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#wired-finite-box-bdry-conn-event-le-bern","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The wired `q = 2` box measure of \"the origin is connected to the box boundary\" is at most its Bernoulli probability.","title":"Wired $q = 2$ origin-to-boundary probability at most Bernoulli"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/fk-theta-two-le-bernoulli-theta","kind":"node","lean":"StatMech.FK.fkTheta_two_le_bernoulliTheta","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," percolation probability at most the Bernoulli one"]],"source":"Wired $q = 2$ percolation probability at most the Bernoulli one"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#fk-theta-two-le-bernoulli-theta","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The wired `q = 2` percolation probability is at most the Bernoulli one: `\u03b8\u00b9(p, 2) \u2264 \u03b8(p)`.","title":"Wired $q = 2$ percolation probability at most the Bernoulli one"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-site-embed","kind":"node","lean":"StatMech.Percolation.pcv_siteEmbed","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Plane embedding of "],["math","\\(\\mathbb{Z}^2\\)"],["text"," into "],["math","\\(\\mathbb{Z}^{2+k}\\)"]],"source":"Plane embedding of $\\mathbb{Z}^2$ into $\\mathbb{Z}^{2+k}$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-site-embed","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The embedding of `\u2124\u00b2` into `\u2124^{2+k}` as the plane of the first two coordinates.","title":"Plane embedding of $\\mathbb{Z}^2$ into $\\mathbb{Z}^{2+k}$"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-site-embed-cast-add","kind":"node","lean":"StatMech.Percolation.pcv_siteEmbed_castAdd","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-site-embed-cast-add","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The first two coordinates of an embedded site.","title":"First two coordinates of an embedded site are the original ones"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-site-embed-nat-add","kind":"node","lean":"StatMech.Percolation.pcv_siteEmbed_natAdd","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Last "],["math","\\(k\\)"],["text"," coordinates of an embedded site vanish"]],"source":"Last $k$ coordinates of an embedded site vanish"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-site-embed-nat-add","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The last `k` coordinates of an embedded site vanish.","title":"Last $k$ coordinates of an embedded site vanish"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-site-embed-injective","kind":"node","lean":"StatMech.Percolation.pcv_siteEmbed_injective","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-site-embed-injective","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The plane embedding is injective.","title":"Injectivity of the plane embedding"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-site-embed-adj-iff","kind":"node","lean":"StatMech.Percolation.pcv_siteEmbed_adj_iff","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-site-embed-adj-iff","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The plane embedding preserves and reflects lattice adjacency.","title":"Plane embedding preserves and reflects adjacency"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-edge-embed","kind":"node","lean":"StatMech.Percolation.pcv_edgeEmbed","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-edge-embed","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The plane embedding on edges.","title":"Plane embedding on edges"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-edge-embed-injective","kind":"node","lean":"StatMech.Percolation.pcv_edgeEmbed_injective","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-edge-embed-injective","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The plane embedding on edges is injective.","title":"Injectivity of the plane embedding on edges"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-plane-restrict","kind":"node","lean":"StatMech.Percolation.pcv_planeRestrict","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-plane-restrict","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"Restrict a configuration on `\u2124^{2+k}` to the embedded plane.","title":"Restriction of a configuration to the embedded plane"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/bernoulli-product-map-plane-restrict","kind":"node","lean":"StatMech.Percolation.bernoulliProduct_map_planeRestrict","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#bernoulli-product-map-plane-restrict","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"Restricting Bernoulli percolation on `\u2124^{2+k}` to the plane gives Bernoulli percolation on `\u2124\u00b2`.","title":"Bernoulli percolation restricted to the plane is planar Bernoulli"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-site-embed-origin","kind":"node","lean":"StatMech.Percolation.pcv_siteEmbed_origin","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-site-embed-origin","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The plane embedding sends the origin to the origin.","title":"Plane embedding fixes the origin"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-connected-of-plane-restrict","kind":"node","lean":"StatMech.Percolation.pcv_connected_of_planeRestrict","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Plane open connections are open connections in "],["math","\\(\\mathbb{Z}^{2+k}\\)"]],"source":"Plane open connections are open connections in $\\mathbb{Z}^{2+k}$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-connected-of-plane-restrict","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"Open connections in the restricted plane configuration are open connections in `\u2124^{2+k}`.","title":"Plane open connections are open connections in $\\mathbb{Z}^{2+k}$"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pcv-plane-percolation-subset","kind":"node","lean":"StatMech.Percolation.pcv_plane_percolation_subset","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Percolation in the plane implies percolation in "],["math","\\(\\mathbb{Z}^{2+k}\\)"]],"source":"Percolation in the plane implies percolation in $\\mathbb{Z}^{2+k}$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pcv-plane-percolation-subset","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"Percolation in the plane implies percolation in `\u2124^{2+k}`.","title":"Percolation in the plane implies percolation in $\\mathbb{Z}^{2+k}$"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/theta-two-le-theta-add","kind":"node","lean":"StatMech.Percolation.theta_two_le_theta_add","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\theta_2(p) \\le \\theta_{2+k}(p)\\)"],["text"," for Bernoulli percolation"]],"source":"$\\theta_2(p) \\le \\theta_{2+k}(p)$ for Bernoulli percolation"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#theta-two-le-theta-add","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The Bernoulli percolation probability does not decrease with the dimension: `\u03b8_2(p) \u2264 \u03b8_{2+k}(p)`.","title":"$\\theta_2(p) \\le \\theta_{2+k}(p)$ for Bernoulli percolation"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/theta-mono-parameter","kind":"node","lean":"StatMech.Percolation.theta_mono_parameter","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity in "],["math","\\(p\\)"],["text"," of the Bernoulli percolation probability"]],"source":"Monotonicity in $p$ of the Bernoulli percolation probability"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#theta-mono-parameter","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The Bernoulli percolation probability `\u03b8_d(p)` is monotone in `p`.","title":"Monotonicity in $p$ of the Bernoulli percolation probability"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-le-of-theta-pos","kind":"node","lean":"StatMech.Percolation.pc_le_of_theta_pos","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positive percolation at "],["math","\\(p\\)"],["text"," gives "],["math","\\(p_c(d) \\le p\\)"]],"source":"Positive percolation at $p$ gives $p_c(d) \\le p$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-le-of-theta-pos","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"Positive percolation probability at `p` gives `p_c(d) \u2264 p`.","title":"Positive percolation at $p$ gives $p_c(d) \\le p$"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/exists-theta-two-pos-below-one","kind":"node","lean":"StatMech.Percolation.exists_theta_two_pos_below_one","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positive planar percolation at some "],["math","\\(p < 1\\)"],["text"," via the Ising model"]],"source":"Positive planar percolation at some $p < 1$ via the Ising model"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#exists-theta-two-pos-below-one","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"There is `p < 1` with `\u03b8_2(p) > 0`, from spontaneous magnetization of the two-dimensional Ising model.","title":"Positive planar percolation at some $p < 1$ via the Ising model"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-two-lt-one-via-fk","kind":"node","lean":"StatMech.Percolation.pc_two_lt_one_via_fk","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli critical probability on "],["math","\\(\\mathbb{Z}^2\\)"],["text"," is below one"]],"source":"Bernoulli critical probability on $\\mathbb{Z}^2$ is below one"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-two-lt-one-via-fk","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The Bernoulli critical probability on `\u2124\u00b2` is below `1`.","title":"Bernoulli critical probability on $\\mathbb{Z}^2$ is below one"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-two-nontrivial","kind":"node","lean":"StatMech.Percolation.pc_two_nontrivial","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nontrivial Bernoulli critical probability on "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Nontrivial Bernoulli critical probability on $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-two-nontrivial","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"`0 < p_c(2) < 1`.","title":"Nontrivial Bernoulli critical probability on $\\mathbb{Z}^2$"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-lt-one-of-two-le-via-fk","kind":"node","lean":"StatMech.Percolation.pc_lt_one_of_two_le_via_fk","parent":20112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli critical probability below one for "],["math","\\(d \\ge 2\\)"]],"source":"Bernoulli critical probability below one for $d \\ge 2$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-lt-one-of-two-le-via-fk","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"The Bernoulli critical probability on `\u2124^d` is below `1` for `d \u2265 2`.","title":"Bernoulli critical probability below one for $d \\ge 2$"},{"id":"Probability/percolation/phase-transition/bounds-on-the-critical-probability/pc-nontrivial-of-two-le","kind":"node","lean":"StatMech.Percolation.pc_nontrivial_of_two_le","parent":20112,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/bounds-on-the-critical-probability/index.html#pc-nontrivial-of-two-le","scope":"Probability/percolation/phase-transition/bounds-on-the-critical-probability","status":"fully_proved","summary":"For Bernoulli bond percolation on `\u2124^d` with `d \u2265 2`, `0 < p_c(d) < 1`.","title":"Non-triviality of the percolation critical probability"},{"data_url":"scopes/Probability/percolation/phase-transition/number-of-infinite-clusters.json","id":"Probability/percolation/phase-transition/number-of-infinite-clusters","kind":"scope","navigation":"module","parent":20084,"result":false,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","title":"Number of infinite clusters"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/ergodic-ae-const-of-shift-invariant","kind":"node","lean":"StatMech.ConfigSpace.ergodic_ae_const_of_shiftInvariant","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#ergodic-ae-const-of-shift-invariant","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"A measurable function with countable values that is invariant under all shifts is almost surely constant under an ergodic probability measure. See also Mathlib's `ErgodicSMul` and `aeconst_of_forall_preimage_smul_ae_eq`, which state ergodicity of a group action through a.e.-invariant sets.","title":"Shift-invariant countably valued functions are a.s. constant under ergodicity"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/translate-hom","kind":"node","lean":"StatMech.Percolation.translateHom","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#translate-hom","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"Translation by `g` as a graph homomorphism between the open subgraphs of `\u03c9` and of its shift.","title":"Translation of the open subgraph as a graph homomorphism"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/translate-hom-inv","kind":"node","lean":"StatMech.Percolation.translateHomInv","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#translate-hom-inv","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The inverse of `translateHom`.","title":"Inverse translation of the open subgraph"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/infinite-clusters","kind":"node","lean":"StatMech.Percolation.infiniteClusters","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#infinite-clusters","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The set of infinite open clusters of `\u03c9`.","title":"Set of infinite open clusters"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/num-infinite-clusters","kind":"node","lean":"StatMech.Percolation.numInfiniteClusters","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#num-infinite-clusters","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The number `N(\u03c9) \u2208 \u2115\u221e` of infinite open clusters of `\u03c9`.","title":"Number of infinite open clusters"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/measurable-set-crossing-event","kind":"node","lean":"StatMech.Percolation.measurableSet_crossingEvent","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#measurable-set-crossing-event","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The one-arm event `crossingEvent d n` (the origin is connected to a site outside `box d (n - 1)`) is measurable.","title":"Measurability of the one-arm event"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/rep-set","kind":"node","lean":"StatMech.Percolation.repSet","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#rep-set","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The set of sites `x` in an infinite cluster that are the `Encodable`-least site of their cluster (one representative per infinite cluster).","title":"Representative sites of the infinite clusters"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/num-infinite-clusters-ae-const","kind":"node","lean":"StatMech.Percolation.numInfiniteClusters_ae_const","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#num-infinite-clusters-ae-const","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The number of infinite clusters is almost surely constant (Newman\u2013Schulman 1981) for any translation-ergodic probability measure on bond configurations of `\u2124^d`.","title":"Number of infinite clusters is almost surely constant"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/at-least-two-infinite","kind":"node","lean":"StatMech.Percolation.atLeastTwoInfinite","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#at-least-two-infinite","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The event that there are at least two infinite open clusters.","title":"Event of at least two infinite open clusters"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/force-open-finset","kind":"node","lean":"StatMech.Percolation.forceOpenFinset","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#force-open-finset","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The configuration obtained from `\u03c9` by opening every edge of `F`.","title":"Configuration with a finite edge set forced open"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/anchor-sites","kind":"node","lean":"StatMech.Percolation.anchorSites","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#anchor-sites","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"The endpoints of the edges of `F`.","title":"Endpoints of a finite edge set"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/merge-witness","kind":"node","lean":"StatMech.Percolation.MergeWitness","parent":20158,"reading_title":{"format":"inline-parts-v1","parts":[["text","Configurations with "],["math","\\(k\\)"],["text"," infinite clusters, two merged by opening "],["math","\\(F\\)"]],"source":"Configurations with $k$ infinite clusters, two merged by opening $F$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#merge-witness","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"Configurations with exactly `k` infinite clusters in which opening `F` connects two distinct infinite clusters.","title":"Configurations with $k$ infinite clusters, two merged by opening $F$"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/has-finite-energy-merge","kind":"node","lean":"StatMech.Percolation.HasFiniteEnergyMerge","parent":20158,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#has-finite-energy-merge","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"Finite energy for opening edges: for every finite edge set `F`, the law of `forceOpenFinset F \u03c9` is absolutely continuous with respect to `\u03bc`.","title":"Finite energy for opening finite edge sets"},{"id":"Probability/percolation/phase-transition/number-of-infinite-clusters/num-infinite-clusters-zero-or-one","kind":"node","lean":"StatMech.Percolation.numInfiniteClusters_zero_or_one","parent":20158,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero or one infinite cluster, given ergodicity and "],["math","\\(\\mu(N \\ge 2) = 0\\)"]],"source":"Zero or one infinite cluster, given ergodicity and $\\mu(N \\ge 2) = 0$"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/number-of-infinite-clusters/index.html#num-infinite-clusters-zero-or-one","scope":"Probability/percolation/phase-transition/number-of-infinite-clusters","status":"fully_proved","summary":"For an ergodic measure with `\u03bc(N \u2265 2) = 0`, either a.s. `N = 0` or a.s. `N = 1`.","title":"Zero or one infinite cluster, given ergodicity and $\\mu(N \\ge 2) = 0$"},{"data_url":"scopes/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation.json","id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","kind":"scope","navigation":"module","parent":20084,"result":false,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","title":"Peierls argument for bond percolation"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/axis-site","kind":"node","lean":"StatMech.Lattice.axisSite","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#axis-site","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The site `(j, 0)` on the positive horizontal axis.","title":"Site on the positive horizontal axis"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-pred","kind":"node","lean":"StatMech.Lattice.exitPred","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-pred","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The site `(j + 1, 0)` is not in the open cluster of the origin.","title":"Axis site lies outside the origin cluster"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-index","kind":"node","lean":"StatMech.Lattice.exitIndex","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-index","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"For a finite origin cluster, the least `j` such that `(j + 1, 0)` is not in the cluster.","title":"Exit index of a finite origin cluster"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-face-up","kind":"node","lean":"StatMech.Lattice.exitFaceUp","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-face-up","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The unit face directly above the exit edge from `(j, 0)` to `(j + 1, 0)`.","title":"Face above the exit edge"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-face-down","kind":"node","lean":"StatMech.Lattice.exitFaceDown","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-face-down","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The unit face directly below the exit edge from `(j, 0)` to `(j + 1, 0)`.","title":"Face below the exit edge"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/anchored-cycle-exists","kind":"node","lean":"StatMech.Lattice.AnchoredCycleExists","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#anchored-cycle-exists","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"Some cycle of the face-boundary graph of the origin cluster is based at the upper exit face and is longer than the exit index.","title":"Existence of an anchored dual cycle around the origin cluster"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/pc-anchored-enclosure-of-anchored-cycle","kind":"node","lean":"StatMech.Lattice.pcAnchoredEnclosure_of_anchoredCycle","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#pc-anchored-enclosure-of-anchored-cycle","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"If every finite origin cluster has an anchored dual cycle (`AnchoredCycleExists`), then the anchored enclosure property `StatMech.Percolation.PcAnchoredEnclosure` holds. This reduces the Peierls bound to `AnchoredCycleExists`; it does not prove that hypothesis.","title":"Anchored dual cycles give the anchored enclosure property"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/cycle-has-left-face","kind":"node","lean":"StatMech.Lattice.CycleHasLeftFace","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#cycle-has-left-face","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"Some cycle of the face-boundary graph of the origin cluster, based at the upper exit face, visits a face whose first coordinate is at most zero.","title":"Dual cycle reaching the left half-plane"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/bd-edge-set2","kind":"node","lean":"StatMech.Lattice.bdEdgeSet2","parent":20173,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary edges of a set of sites in "],["math","\\(\\mathbb{Z}^2\\)"],["text",", as unordered pairs"]],"source":"Boundary edges of a set of sites in $\\mathbb{Z}^2$, as unordered pairs"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#bd-edge-set2","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The edges of `\u2124\u00b2` with exactly one endpoint in `S`, as unordered pairs.","title":"Boundary edges of a set of sites in $\\mathbb{Z}^2$, as unordered pairs"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/cut-lattice","kind":"node","lean":"StatMech.Lattice.cutLattice","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#cut-lattice","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The square lattice with the boundary edges of `S` deleted.","title":"Cut lattice of a set of sites"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/inside-region","kind":"node","lean":"StatMech.Lattice.insideRegion","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#inside-region","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The sites reachable from `o` in the cut lattice of the open cluster of `o`.","title":"Inside region of an open cluster in the cut lattice"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/cluster-open-to-barrier-hom","kind":"node","lean":"StatMech.Lattice.clusterOpenToBarrierHom","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#cluster-open-to-barrier-hom","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The open subgraph on the cluster of `o` maps into the lattice with the boundary edges of the cluster removed.","title":"Open cluster maps into the lattice minus its boundary edges"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/ext-to-barrier-hom","kind":"node","lean":"StatMech.Lattice.extToBarrierHom","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#ext-to-barrier-hom","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The lattice on the exterior of a box disjoint from `K` maps into the lattice with the boundary edges of `K` removed.","title":"Box exterior maps into the lattice minus boundary edges"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/discrete-jordan-separation","kind":"node","lean":"StatMech.Lattice.discrete_jordan_separation","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#discrete-jordan-separation","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"A discrete Jordan separation for a finite open cluster: removing its boundary edges leaves the cluster connected, the far exterior connected, and no path between the cluster and its complement.","title":"Discrete Jordan separation for finite open clusters"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/left-axis-site","kind":"node","lean":"StatMech.Lattice.leftAxisSite","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#left-axis-site","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The site `(-k, 0)` on the negative horizontal axis.","title":"Site on the negative horizontal axis"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/left-exit-pred","kind":"node","lean":"StatMech.Lattice.leftExitPred","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#left-exit-pred","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The site `(-(k + 1), 0)` is not in the open cluster of the origin.","title":"Left exit condition at an axis site"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/left-exit-index","kind":"node","lean":"StatMech.Lattice.leftExitIndex","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#left-exit-index","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"For a finite origin cluster, the least `k` such that `(-(k + 1), 0)` is not in the cluster.","title":"Left exit index of a finite cluster"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/left-exit-face-up","kind":"node","lean":"StatMech.Lattice.leftExitFaceUp","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#left-exit-face-up","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The unit face directly above the left exit edge.","title":"Face above the left exit edge"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/left-exit-face-down","kind":"node","lean":"StatMech.Lattice.leftExitFaceDown","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#left-exit-face-down","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The unit face directly below the left exit edge.","title":"Face below the left exit edge"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/enclosing-contour-links-exits","kind":"node","lean":"StatMech.Lattice.EnclosingContourLinksExits","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#enclosing-contour-links-exits","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"Some cycle of the face-boundary graph of the origin cluster, based at the upper exit face on the positive axis, passes through the upper exit face on the negative axis.","title":"Origin cluster's enclosing contour links the two axis exits"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/pc-lt-one-via-boundary","kind":"node","lean":"StatMech.Lattice.pc_lt_one_via_boundary","parent":20173,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(\\mathbb{Z}^2) < 1\\)"],["text"," via boundary connectivity, assuming linked exits"]],"source":"$p_c(\\mathbb{Z}^2) < 1$ via boundary connectivity, assuming linked exits"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#pc-lt-one-via-boundary","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"`p_c(\u2124\u00b2) < 1`, under the hypothesis that `EnclosingContourLinksExits` holds for every configuration with a finite origin cluster.","title":"$p_c(\\mathbb{Z}^2) < 1$ via boundary connectivity, assuming linked exits"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-dart","kind":"node","lean":"StatMech.Lattice.exitDart","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-dart","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The boundary dart of the origin cluster across the exit edge on the positive horizontal axis.","title":"Exit dart on the positive horizontal axis"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/left-exit-dart","kind":"node","lean":"StatMech.Lattice.leftExitDart","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#left-exit-dart","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The boundary dart of the origin cluster across the exit edge on the negative horizontal axis.","title":"Exit dart on the negative horizontal axis"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/neg-part","kind":"node","lean":"StatMech.Lattice.negPart","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#neg-part","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The negative part `min z 0` of an integer.","title":"Negative part of an integer"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/dart-face","kind":"node","lean":"StatMech.Lattice.dartFace","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#dart-face","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The unit face to the left of the dart `e`, labelled by its lower left corner.","title":"Unit face to the left of a dart"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-darts-same-orbit","kind":"node","lean":"StatMech.Lattice.ExitDartsSameOrbit","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-darts-same-orbit","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The left exit dart lies in the boundary orbit of the right exit dart.","title":"Left and right exit darts lie in the same boundary orbit"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/pc-lt-one-via-dart-orbit","kind":"node","lean":"StatMech.Lattice.pc_lt_one_via_dartOrbit","parent":20173,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(\\mathbb{Z}^2) < 1\\)"],["text"," via exit dart orbits, assuming linked exits"]],"source":"$p_c(\\mathbb{Z}^2) < 1$ via exit dart orbits, assuming linked exits"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#pc-lt-one-via-dart-orbit","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"`p_c(\u2124\u00b2) < 1`, under the hypothesis that `EnclosingContourLinksExits` holds for every configuration with a finite origin cluster.","title":"$p_c(\\mathbb{Z}^2) < 1$ via exit dart orbits, assuming linked exits"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/dart-primal-edge","kind":"node","lean":"StatMech.Lattice.dartPrimalEdge","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#dart-primal-edge","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The undirected lattice edge underlying a dart.","title":"Undirected lattice edge of a dart"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/canonical-contour","kind":"node","lean":"StatMech.Lattice.canonicalContour","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#canonical-contour","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The canonical contour of a finite origin cluster: the closed walk around the boundary orbit of the exit dart on the positive axis.","title":"Canonical contour of a finite origin cluster"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/canonical-contour-darts","kind":"node","lean":"StatMech.Lattice.canonicalContourDarts","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#canonical-contour-darts","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The darts of one period of the boundary orbit of the exit dart.","title":"Darts of the canonical contour"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/canonical-primal-edge-finset","kind":"node","lean":"StatMech.Lattice.canonicalPrimalEdgeFinset","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#canonical-primal-edge-finset","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The primal edges of the canonical contour.","title":"Primal edges of the canonical contour"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/closed-dart-contour-event","kind":"node","lean":"StatMech.Lattice.closedDartContourEvent","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#closed-dart-contour-event","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The set of configurations in which every primal edge of the canonical contour of `\u03c9` is closed.","title":"Event that every edge of the canonical contour is closed"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/canonical-contour-winds","kind":"node","lean":"StatMech.Lattice.CanonicalContourWinds","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#canonical-contour-winds","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The period of the canonical contour exceeds the exit index, the length condition of an anchored enclosure.","title":"Canonical contour is longer than the exit index"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/canonical-face-walk-aux","kind":"node","lean":"StatMech.Lattice.canonicalFaceWalkAux","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#canonical-face-walk-aux","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The walk of faces to the left of the first `n` successors of a boundary dart.","title":"Face walk along the first successors of a boundary dart"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/canonical-face-walk","kind":"node","lean":"StatMech.Lattice.canonicalFaceWalk","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#canonical-face-walk","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The closed walk of faces to the left of the darts of the canonical contour, based at the upper exit face.","title":"Closed face walk of the canonical contour"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/pc-lt-one-via-canonical-contour","kind":"node","lean":"StatMech.Lattice.pc_lt_one_via_canonicalContour","parent":20173,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(\\mathbb{Z}^2) < 1\\)"],["text"," via the canonical contour, assuming anchored cycles"]],"source":"$p_c(\\mathbb{Z}^2) < 1$ via the canonical contour, assuming anchored cycles"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#pc-lt-one-via-canonical-contour","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"`p_c(\u2124\u00b2) < 1`, under the hypothesis that `AnchoredCycleExists` holds for every configuration with a finite origin cluster.","title":"$p_c(\\mathbb{Z}^2) < 1$ via the canonical contour, assuming anchored cycles"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/dart-orbit-face-walk","kind":"node","lean":"StatMech.Lattice.dartOrbitFaceWalk","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#dart-orbit-face-walk","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The walk in the face-boundary graph of `K` through the faces of `e` and its first `n` successors.","title":"Face walk along a boundary orbit"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/dart-orbit-face-loop","kind":"node","lean":"StatMech.Lattice.dartOrbitFaceLoop","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#dart-orbit-face-loop","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The closed walk of faces along one period of the boundary orbit of `e`, for finite `K`.","title":"Face loop of a boundary orbit in the face-boundary graph"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-orbit-face-loop","kind":"node","lean":"StatMech.Lattice.exitOrbitFaceLoop","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-orbit-face-loop","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The closed face walk of the exit orbit of a finite origin cluster, based at the upper exit face.","title":"Face loop of the exit orbit of a finite cluster"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/canonical-orbit-cycle","kind":"node","lean":"StatMech.Lattice.CanonicalOrbitCycle","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#canonical-orbit-cycle","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The face loop of the exit orbit is a cycle, and the left exit dart lies in the exit orbit.","title":"Canonical orbit cycle condition"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/pc-lt-one-orbit","kind":"node","lean":"StatMech.Lattice.pc_lt_one_orbit","parent":20173,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(\\mathbb{Z}^2) < 1\\)"],["text"," assuming a canonical orbit cycle"]],"source":"$p_c(\\mathbb{Z}^2) < 1$ assuming a canonical orbit cycle"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#pc-lt-one-orbit","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"`p_c(\u2124\u00b2) < 1`, under the hypothesis that `CanonicalOrbitCycle` holds for every configuration with a finite origin cluster.","title":"$p_c(\\mathbb{Z}^2) < 1$ assuming a canonical orbit cycle"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-base-sub","kind":"node","lean":"StatMech.Lattice.exitBaseSub","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-base-sub","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The exit dart of a finite origin cluster, as an element of the subtype of boundary darts.","title":"Exit dart of a finite origin cluster"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/exit-min-face-loop","kind":"node","lean":"StatMech.Lattice.exitMinFaceLoop","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#exit-min-face-loop","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The closed walk of faces along one period of the exit orbit, based at the upper exit face.","title":"Minimal-period face loop of the exit orbit"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/orbit-face-simple","kind":"node","lean":"StatMech.Lattice.OrbitFaceSimple","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#orbit-face-simple","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The exit orbit has period at least three and visits distinct faces during one period.","title":"Simplicity of the exit orbit's face walk"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/pc-lt-one-of-orbit-face-simple","kind":"node","lean":"StatMech.Lattice.pc_lt_one_of_orbitFaceSimple","parent":20173,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(\\mathbb{Z}^2) < 1\\)"],["text"," assuming simple exit orbits"]],"source":"$p_c(\\mathbb{Z}^2) < 1$ assuming simple exit orbits"},"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#pc-lt-one-of-orbit-face-simple","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"`p_c(\u2124\u00b2) < 1`, under the hypothesis that for every configuration with a finite origin cluster the exit orbit is simple (`OrbitFaceSimple`) and contains the left exit dart.","title":"$p_c(\\mathbb{Z}^2) < 1$ assuming simple exit orbits"},{"id":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/pww-bd-edge-match","kind":"node","lean":"StatMech.Lattice.pww_BdEdgeMatch","parent":20173,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/peierls-argument-for-bond-percolation/index.html#pww-bd-edge-match","scope":"Probability/percolation/phase-transition/peierls-argument-for-bond-percolation","status":"fully_proved","summary":"The set `K` and the inside `jec_leftRegion Vc` of the closed walk `Vc` have the same boundary edges.","title":"Set and walk interior share boundary edges"},{"data_url":"scopes/Probability/percolation/phase-transition/sharp-thresholds-from-influences.json","id":"Probability/percolation/phase-transition/sharp-thresholds-from-influences","kind":"scope","navigation":"module","parent":20084,"result":false,"route":"../roadmap/Probability/percolation/phase-transition/sharp-thresholds-from-influences/index.html","scope":"Probability/percolation/phase-transition/sharp-thresholds-from-influences","status":"fully_proved","title":"Sharp thresholds from influences"},{"id":"Probability/percolation/phase-transition/sharp-thresholds-from-influences/open-edge","kind":"node","lean":"StatMech.BeffaraDC.openEdge","parent":20220,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/sharp-thresholds-from-influences/index.html#open-edge","scope":"Probability/percolation/phase-transition/sharp-thresholds-from-influences","status":"fully_proved","summary":"The event `{\u03c9_e = 1}` that the edge `e` is open.","title":"Event that an edge is open"},{"id":"Probability/percolation/phase-transition/sharp-thresholds-from-influences/russo-formula","kind":"node","lean":"StatMech.BeffaraDC.russoFormula","parent":20220,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/sharp-thresholds-from-influences/index.html#russo-formula","scope":"Probability/percolation/phase-transition/sharp-thresholds-from-influences","status":"fully_proved","summary":"Russo's formula, covariance form: for increasing `A` and `p \u2209 {0, 1}`, `d/dp \u2119_p(A) = (p(1 - p))\u207b\u00b9 \u2211_e (\u2119_p(A \u2229 {\u03c9_e = 1}) - \u2119_p(\u03c9_e = 1) \u2119_p(A))`.","title":"Russo's formula"},{"id":"Probability/percolation/phase-transition/sharp-thresholds-from-influences/influence","kind":"node","lean":"StatMech.BeffaraDC.influence","parent":20220,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/sharp-thresholds-from-influences/index.html#influence","scope":"Probability/percolation/phase-transition/sharp-thresholds-from-influences","status":"fully_proved","summary":"The influence `I_e^p(A) = \u2119_p(e is pivotal for A)` of the coordinate `e` on the event `A` under the `p`-biased product measure.","title":"Influence of an edge on an event"},{"id":"Probability/percolation/phase-transition/sharp-thresholds-from-influences/sth-sharp-threshold","kind":"node","lean":"StatMech.TwoDim.sth_sharpThreshold","parent":20220,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/sharp-thresholds-from-influences/index.html#sth-sharp-threshold","scope":"Probability/percolation/phase-transition/sharp-thresholds-from-influences","status":"fully_proved","summary":"If `A` is increasing, `\u2119_{1/2}(A) = 1/2`, and the total influence `\u2211_e I_e^p(A)` is at least `M` for all `p \u2208 (1/2, q)`, then `\u2119_q(A) \u2265 1/2 + M (q - 1/2)`.","title":"Sharp threshold from influences"},{"id":"Probability/percolation/phase-transition/sharp-thresholds-from-influences/ktr-sharp-threshold-via-kkl","kind":"node","lean":"StatMech.TwoDim.ktr_sharpThreshold_via_kkl","parent":20220,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/sharp-thresholds-from-influences/index.html#ktr-sharp-threshold-via-kkl","scope":"Probability/percolation/phase-transition/sharp-thresholds-from-influences","status":"fully_proved","summary":"Sharp threshold from a KKL-type inequality: if `\u2119_{1/2}(A) = 1/2` and for all `p \u2208 (1/2, q)` `\u2211_e I_e^p(A) \u2265 C \u00b7 \u2119_p(A)(1 - \u2119_p(A)) \u00b7 L'(p)`, `\u2119_p(A)(1 - \u2119_p(A)) \u2265 v\u2080` and `L'(p) \u2265 L`, then `\u2119_q(A) \u2265 1/2 + C v\u2080 L (q - 1/2)`. The KKL-type inequality is a hypothesis.","title":"Sharp threshold from a KKL-type influence inequality"},{"data_url":"scopes/Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster.json","id":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster","kind":"scope","navigation":"module","parent":20084,"result":false,"route":"../roadmap/Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/index.html","scope":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster","status":"can_prove","title":"Uniqueness of the infinite cluster"},{"id":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/remove-site","kind":"node","lean":"StatMech.Percolation.removeSite","parent":20226,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/index.html#remove-site","scope":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster","status":"fully_proved","summary":"The configuration obtained from `\u03c9` by closing every edge incident to `x`.","title":"Configuration with all edges at a site closed"},{"id":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/is-trifurcation","kind":"node","lean":"StatMech.Percolation.IsTrifurcation","parent":20226,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/index.html#is-trifurcation","scope":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster","status":"fully_proved","summary":"`x` is connected to three distinct sites with infinite `\u03c9`-clusters that are pairwise disconnected after removing `x`. Unlike the classical definition, the three branches at `x` are not required to be infinite.","title":"Trifurcation point, without requiring infinite branches"},{"id":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/box-finset-bk","kind":"node","lean":"StatMech.Percolation.boxFinsetBK","parent":20226,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/index.html#box-finset-bk","scope":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster","status":"fully_proved","summary":"The sites of the box `\u039b_n` as a finset.","title":"Sites of a box as a finite set"},{"id":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/tcount","kind":"node","lean":"StatMech.Percolation.Tcount","parent":20226,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/index.html#tcount","scope":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster","status":"fully_proved","summary":"The number of trifurcation points of `\u03c9` in the box `\u039b_n`.","title":"Number of trifurcation points in a box"},{"id":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/burton-keane-bernoulli","kind":"node","lean":"MathlibExt.Probability.Percolation.BurtonKeaneWanted.burton_keane_bernoulli","parent":20226,"result":true,"route":"../roadmap/Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster/index.html#burton-keane-bernoulli","scope":"Probability/percolation/phase-transition/uniqueness-of-the-infinite-cluster","status":"can_prove","summary":"Burton\u2013Keane theorem for Bernoulli bond percolation. For every dimension `d` and every `p \u2208 [0, 1]`, almost surely under the Bernoulli(`p`) bond percolation measure on `\u2124^d` there is at most one infinite open cluster.","title":"Burton\u2013Keane theorem"},{"data_url":"scopes/Probability/percolation/sharpness.json","id":"Probability/percolation/sharpness","kind":"scope","parent":20083,"result":false,"route":"../roadmap/Probability/percolation/sharpness/index.html","scope":"Probability/percolation/sharpness","status":"fully_proved","summary":"The Duminil-Copin\u2013Tassion proof of exponential decay below p_c: the quantity phi_p(S), Russo's formula and pivotal edges, exploration and last-exit decompositions, the differential inequality, the mean-field bound and subcritical decay.","title":"Sharpness of the phase transition"},{"data_url":"scopes/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t.json","id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","title":"The Duminil-Copin\u2013Tassion quantity \u03c6_p(S) and the threshold p\u0303_c"},{"id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/coord-shift","kind":"node","lean":"StatMech.Percolation.coordShift","parent":20233,"result":true,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html#coord-shift","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","summary":"The site `x + \u03b4 e_i` obtained by moving `x` by `\u03b4` in coordinate `i`.","title":"Shift of a lattice site in one coordinate"},{"id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/step-sign","kind":"node","lean":"StatMech.Percolation.stepSign","parent":20233,"result":true,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html#step-sign","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","summary":"The unit step `+1` (for `true`) or `-1` (for `false`).","title":"Sign of a unit lattice step"},{"id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/boundary-edges","kind":"node","lean":"StatMech.Percolation.boundaryEdges","parent":20233,"result":true,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html#boundary-edges","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","summary":"The oriented edge boundary `\u2202S = {(x, x \u00b1 e_i) : x \u2208 S, x \u00b1 e_i \u2209 S}` of a finite set `S \u2286 \u2124^d`, as ordered pairs (inner endpoint first).","title":"Oriented edge boundary of a finite lattice set"},{"id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/card-boundary-edges-le","kind":"node","lean":"StatMech.Percolation.card_boundaryEdges_le","parent":20233,"result":true,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html#card-boundary-edges-le","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","summary":"The edge boundary of `S` has at most `2d |S|` elements.","title":"Size bound for the edge boundary"},{"id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/conn-within-prob","kind":"node","lean":"StatMech.Percolation.connWithinProb","parent":20233,"result":true,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html#conn-within-prob","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","summary":"The probability `\u2119_p(0 \u2194 x inside S)` that the origin and `x` are joined by an open path using only vertices of `S`; it is `0` if `0 \u2209 S` or `x \u2209 S`.","title":"Probability of connection inside a set"},{"id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/phi","kind":"node","lean":"StatMech.Percolation.phi","parent":20233,"result":true,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html#phi","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","summary":"The Duminil-Copin\u2013Tassion quantity `\u03c6_p(S) = p \u2211_{(x, y) \u2208 \u2202S} \u2119_p(0 \u2194 x inside S)`.","title":"Duminil-Copin\u2013Tassion quantity"},{"id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/tilde-pc-set","kind":"node","lean":"StatMech.Percolation.tildePcSet","parent":20233,"result":true,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html#tilde-pc-set","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","summary":"The set of densities `p \u2208 [0, 1]` for which some finite `S \u220b 0` has `\u03c6_p(S) < 1`.","title":"Densities with a finite set of small Duminil-Copin\u2013Tassion quantity"},{"id":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/tilde-pc","kind":"node","lean":"StatMech.Percolation.tildePc","parent":20233,"result":true,"route":"../roadmap/Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t/index.html#tilde-pc","scope":"Probability/percolation/sharpness/the-duminil-copin-tassion-quantity-p-s-and-the-t","status":"fully_proved","summary":"The Duminil-Copin\u2013Tassion threshold `p\u0303_c(d) = sup {p : \u2203 S \u220b 0 finite, \u03c6_p(S) < 1}`.","title":"Duminil-Copin\u2013Tassion threshold"},{"data_url":"scopes/Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature.json","id":"Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature/index.html","scope":"Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature","status":"fully_proved","title":"Bernoulli percolation in the inverse-temperature parametrization"},{"id":"Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature/p-beta","kind":"node","lean":"StatMech.Sharpness.pBeta","parent":20242,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bond probability "],["math","\\(p(\\beta) = 1 - e^{-\\beta J}\\)"]],"source":"Bond probability $p(\\beta) = 1 - e^{-\\beta J}$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature/index.html#p-beta","scope":"Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature","status":"fully_proved","summary":"The bond-percolation edge probability `p(\u03b2) = 1 - exp(-\u03b2J)` attached to coupling `J` at inverse temperature `\u03b2`.","title":"Bond probability $p(\\beta) = 1 - e^{-\\beta J}$"},{"id":"Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature/beta-tilde-c","kind":"node","lean":"StatMech.Sharpness.betaTildeC","parent":20242,"reading_title":{"format":"inline-parts-v1","parts":[["text","Auxiliary critical inverse temperature from "],["math","\\(\\tilde p_c\\)"]],"source":"Auxiliary critical inverse temperature from $\\tilde p_c$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature/index.html#beta-tilde-c","scope":"Probability/percolation/sharpness/bernoulli-percolation-in-the-inverse-temperature","status":"fully_proved","summary":"The auxiliary critical inverse temperature `\u03b2\u0303_c = -log(1 - p\u0303_c)/J`, so that `p(\u03b2\u0303_c) = p\u0303_c`.","title":"Auxiliary critical inverse temperature from $\\tilde p_c$"},{"data_url":"scopes/Probability/percolation/sharpness/russo-s-formula-for-percolation.json","id":"Probability/percolation/sharpness/russo-s-formula-for-percolation","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/russo-s-formula-for-percolation/index.html","scope":"Probability/percolation/sharpness/russo-s-formula-for-percolation","status":"fully_proved","title":"Russo's formula for percolation"},{"id":"Probability/percolation/sharpness/russo-s-formula-for-percolation/measure-prob","kind":"node","lean":"StatMech.Percolation.measureProb","parent":20245,"result":true,"route":"../roadmap/Probability/percolation/sharpness/russo-s-formula-for-percolation/index.html#measure-prob","scope":"Probability/percolation/sharpness/russo-s-formula-for-percolation","status":"fully_proved","summary":"The probability `\u2119_p(A)` of `A` under the Bernoulli(`p`) product measure on `{0, 1}^E` for `p \u2208 [0, 1]`; outside `[0, 1]` it is the polynomial `StatMech.prob p A`, so that the function is defined on all of `\u211d`.","title":"Probability of an event under Bernoulli product measure"},{"id":"Probability/percolation/sharpness/russo-s-formula-for-percolation/russo-derivative","kind":"node","lean":"StatMech.Percolation.russo_derivative","parent":20245,"result":true,"route":"../roadmap/Probability/percolation/sharpness/russo-s-formula-for-percolation/index.html#russo-derivative","scope":"Probability/percolation/sharpness/russo-s-formula-for-percolation","status":"fully_proved","summary":"Russo's formula (Margulis\u2013Russo): for an increasing event `A` on a finite edge set and `0 < p < 1`, `p \u21a6 \u2119_p(A)` has derivative `\u2211_e \u2119_p(e is pivotal for A)` at `p`. See also `StatMech.hasDerivAt_prob_eq_sum_pivotalProb` (the polynomial form, for all `p`) and `StatMech.hasDerivAt_realProb_cylinder_pivotal` (cylinder events on an infinite edge set).","title":"Russo's formula"},{"id":"Probability/percolation/sharpness/russo-s-formula-for-percolation/russo-formula","kind":"node","lean":"StatMech.Percolation.russo_formula","parent":20245,"result":true,"route":"../roadmap/Probability/percolation/sharpness/russo-s-formula-for-percolation/index.html#russo-formula","scope":"Probability/percolation/sharpness/russo-s-formula-for-percolation","status":"fully_proved","summary":"Russo's formula, `deriv` form: `d/dp \u2119_p(A) = \u2211_e \u2119_p(e is pivotal for A)` for an increasing event `A` and `0 < p < 1`.","title":"Russo's formula"},{"id":"Probability/percolation/sharpness/russo-s-formula-for-percolation/russo-derivative-measure","kind":"node","lean":"StatMech.Percolation.russo_derivative_measure","parent":20245,"result":true,"route":"../roadmap/Probability/percolation/sharpness/russo-s-formula-for-percolation/index.html#russo-derivative-measure","scope":"Probability/percolation/sharpness/russo-s-formula-for-percolation","status":"fully_proved","summary":"Russo's formula for `p : \u211d\u22650`, packaged with the identification of `measureProb A p` with `(bernoulliProductMeasure p hp).real A`.","title":"Russo's formula for the Bernoulli product measure"},{"id":"Probability/percolation/sharpness/russo-s-formula-for-percolation/open-ind","kind":"node","lean":"StatMech.Sharpness.openInd","parent":20245,"result":true,"route":"../roadmap/Probability/percolation/sharpness/russo-s-formula-for-percolation/index.html#open-ind","scope":"Probability/percolation/sharpness/russo-s-formula-for-percolation","status":"fully_proved","summary":"The indicator `\u03c9_e \u2208 {0, 1}` (as a real number) that the edge `e` is open in `\u03c9`.","title":"Real-valued indicator that an edge is open"},{"id":"Probability/percolation/sharpness/russo-s-formula-for-percolation/has-deriv-at-prob-eq-pivotal-factor","kind":"node","lean":"StatMech.Sharpness.hasDerivAt_prob_eq_pivotalFactor","parent":20245,"result":true,"route":"../roadmap/Probability/percolation/sharpness/russo-s-formula-for-percolation/index.html#has-deriv-at-prob-eq-pivotal-factor","scope":"Probability/percolation/sharpness/russo-s-formula-for-percolation","status":"fully_proved","summary":"Russo's formula in pivotal-factor form: for increasing `A` and `0 < p < 1`, `d/dp \u2119_p(A) = (p(1 - p))\u207b\u00b9 \u2211_e \ud835\udd3c_p[1_A (\u03c9_e - p)]`.","title":"Russo's formula"},{"data_url":"scopes/Probability/percolation/sharpness/exponential-decay-below-the-critical-point.json","id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","title":"Exponential decay below the critical point"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/w1","kind":"node","lean":"StatMech.Percolation.w1","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#w1","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The Bernoulli(`p`) weight of a single edge state: `p` for open, `1 - p` for closed.","title":"Bernoulli weight of a single edge"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/w","kind":"node","lean":"StatMech.Percolation.W","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#w","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The product weight `\u220f_e w1(\u03c9_e)` of a configuration on a finite edge set.","title":"Product weight of an edge configuration"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/wsum","kind":"node","lean":"StatMech.Percolation.wsum","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#wsum","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The expectation `\u2211_\u03c9 f(\u03c9) W(\u03c9)` of `f` under the Bernoulli(`p`) product weights on a finite edge set.","title":"Expectation under Bernoulli product weights"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/cluster-within","kind":"node","lean":"StatMech.Percolation.clusterWithin","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#cluster-within","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The open cluster `C_S(o)` of `o` inside `S`: sites of `S` joined to `o` by an open path in `S`.","title":"Open cluster of a site inside a set"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/incident-within","kind":"node","lean":"StatMech.Percolation.incidentWithin","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#incident-within","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The set of edges `{x, y}` with `x \u2208 K` and `y \u2208 S`.","title":"Edges from a subset into a set"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/cluster-event","kind":"node","lean":"StatMech.Percolation.clusterEvent","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#cluster-event","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The event `{C_S(o) = K}` that the open cluster of `o` inside `S` is exactly `K`.","title":"Event that the cluster of the origin inside a set equals a given set"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/incident-within-finset","kind":"node","lean":"StatMech.Percolation.incidentWithinFinset","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#incident-within-finset","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The finite set of edges `{x, y}` with `x \u2208 K` and `y \u2208 S`.","title":"Edges joining a finite set to a region"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/exploration-decomposition","kind":"node","lean":"StatMech.Percolation.exploration_decomposition","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#exploration-decomposition","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"Exploration independence: for `K \u2286 S` finite and an event `B` depending only on edges in `U`, disjoint from the edges between `K` and `S`, `\u2119_p(C_S(o) = K, B) = \u2119_p(C_S(o) = K) \u2119_p(B)`.","title":"Exploration independence of a cluster inside a set"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/off-cluster-crossing","kind":"node","lean":"StatMech.Percolation.offClusterCrossing","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#off-cluster-crossing","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"`y \u2209 C` is joined by an open path avoiding `C` to a site outside `\u039b_{n-1}`.","title":"Crossing to the box exterior that avoids a cluster"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/boundary-open","kind":"node","lean":"StatMech.Percolation.boundaryOpen","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#boundary-open","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The edge `{x, y}` is open, with `x \u2208 C` and `y \u2209 C`.","title":"Open boundary edge of a cluster"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/off-cluster-box-crossing","kind":"node","lean":"StatMech.Percolation.offClusterBoxCrossing","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#off-cluster-box-crossing","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"`y \u2208 \u039b_n \\ C` is joined by an open path inside `\u039b_n \\ C` to a site outside `\u039b_{n-1}`.","title":"Crossing inside the box that avoids a cluster"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/box-off-cedges","kind":"node","lean":"StatMech.Percolation.boxOffCEdges","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#box-off-cedges","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The edges of `\u039b_n` with no endpoint in `C`.","title":"Box edges with no endpoint in a cluster"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/crossing-event","kind":"node","lean":"StatMech.Percolation.crossingEvent","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#crossing-event","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The event `{0 \u2194 \u2124^d \\ \u039b_{n-1}}` that the origin is joined by an open path to a site outside `\u039b_{n-1}`.","title":"One-arm event from the origin"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/cross-prob","kind":"node","lean":"StatMech.Percolation.crossProb","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#cross-prob","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The one-arm probability `\u2119_p(0 \u2194 \u2124^d \\ \u039b_{n-1})`.","title":"One-arm probability"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/subcritical-decay","kind":"node","lean":"StatMech.Percolation.subcritical_decay","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#subcritical-decay","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"If `\u03c6_p(S) < 1` and `crossProb((k + 1)L) \u2264 \u03c6_p(S) \u00b7 crossProb(kL)` for all `k`, then `crossProb d p n \u2264 C e^{-cn}` for some `c, C > 0`.","title":"Exponential decay from a one-step recursion"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/bernoulli-translation-invariant","kind":"node","lean":"StatMech.Percolation.bernoulli_translationInvariant","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#bernoulli-translation-invariant","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The Bernoulli product measure on `{0, 1}^E` is invariant under the shift action of `G`.","title":"Translation invariance of the Bernoulli product measure"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/crossing-event-from","kind":"node","lean":"StatMech.Percolation.crossingEventFrom","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#crossing-event-from","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"The event that `y` is joined by an open path to a site outside `\u039b_{n-1}`.","title":"One-arm event from a shifted origin"},{"id":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point/cross-prob-exp-decay-of-phi-lt-one","kind":"node","lean":"StatMech.Percolation.crossProb_exp_decay_of_phi_lt_one","parent":20252,"result":true,"route":"../roadmap/Probability/percolation/sharpness/exponential-decay-below-the-critical-point/index.html#cross-prob-exp-decay-of-phi-lt-one","scope":"Probability/percolation/sharpness/exponential-decay-below-the-critical-point","status":"fully_proved","summary":"Exponential decay below `p\u0303_c` (Duminil-Copin\u2013Tassion 2016): if `\u03c6_p(S) < 1` for some finite `S \u220b 0`, then `\u2119_p(0 \u2194 \u2124^d \\ \u039b_{n-1}) \u2264 C e^{-cn}` for some `c, C > 0`.","title":"Exponential decay below `p\u0303_c`"},{"data_url":"scopes/Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t.json","id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t/index.html","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t","status":"fully_proved","title":"Sharpness of the phase transition, assuming its two analytic inputs"},{"id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t/cross-prob-real","kind":"node","lean":"StatMech.Percolation.crossProbReal","parent":20271,"result":true,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t/index.html#cross-prob-real","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t","status":"fully_proved","summary":"The one-arm probability `crossProb d t n` as a function of a real density `t` (`0` outside `[0, 1]`).","title":"One-arm probability as a function of a real density"},{"id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t/tilde-pc-eq-pc","kind":"node","lean":"StatMech.Percolation.tildePc_eq_pc","parent":20271,"reading_title":{"format":"inline-parts-v1","parts":[["text","Equality of the Duminil-Copin\u2013Tassion threshold and "],["math","\\(p_c\\)"]],"source":"Equality of the Duminil-Copin\u2013Tassion threshold and $p_c$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t/index.html#tilde-pc-eq-pc","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t","status":"fully_proved","summary":"`p\u0303_c = p_c`, assuming exponential decay on `tildePcSet`, `\u03b8(1) \u2260 0`, and `\u03b8(q) > 0` for `p\u0303_c < q < 1`.","title":"Equality of the Duminil-Copin\u2013Tassion threshold and $p_c$"},{"id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t/sharpness","kind":"node","lean":"StatMech.Percolation.sharpness","parent":20271,"result":true,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t/index.html#sharpness","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-assuming-its-t","status":"fully_proved","summary":"Sharpness of the phase transition, conditional form: from `p\u0303_c > 0`, `\u03b8(1) \u2260 0`, exponential decay of the one-arm probability on `tildePcSet`, and the Duminil-Copin\u2013Tassion differential inequality on `(p\u0303_c, q)` (with continuity and derivative hypotheses), conclude `p\u0303_c = p_c`, exponential decay on `tildePcSet`, and `\u03b8(q) \u2265 (q - p\u0303_c)/(q(1 - p\u0303_c))` for `p\u0303_c < q < 1`. The hypotheses are discharged in\u2026","title":"Sharpness of the phase transition, conditional form"},{"data_url":"scopes/Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b.json","id":"Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b/index.html","scope":"Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b","status":"fully_proved","title":"Mean-field lower bound and critical points for Bernoulli percolation"},{"id":"Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b/item1-meanfield-lower-bound","kind":"node","lean":"StatMech.Sharpness.item1_meanfield_lower_bound","parent":20275,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean-field lower bound for "],["math","\\(\\theta(p)\\)"],["text"," from a differential inequality"]],"source":"Mean-field lower bound for $\\theta(p)$ from a differential inequality"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b/index.html#item1-meanfield-lower-bound","scope":"Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b","status":"fully_proved","summary":"The mean-field lower bound `\u03b8(p) \u2265 (p - p\u2080)/(p(1 - p\u2080))` from the differential inequality for crossing probabilities. Reference: H. Duminil-Copin, V. Tassion, \"A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model\", Comm. Math. Phys. 343 (2016) 725\u2013745.","title":"Mean-field lower bound for $\\theta(p)$ from a differential inequality"},{"id":"Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b/pc-le-tilde-pc","kind":"node","lean":"StatMech.Sharpness.pc_le_tildePc","parent":20275,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c \\le \\tilde p_c\\)"],["text"," if "],["math","\\(\\theta > 0\\)"],["text"," above "],["math","\\(\\tilde p_c\\)"]],"source":"$p_c \\le \\tilde p_c$ if $\\theta > 0$ above $\\tilde p_c$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b/index.html#pc-le-tilde-pc","scope":"Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b","status":"fully_proved","summary":"`p_c \u2264 p\u0303_c` when `\u03b8` is positive above `p\u0303_c`. See also `Percolation.pc_le_tildePc` (MathlibExt), stated with hypotheses on `\u03b8(1)` and on `(p\u0303_c, 1)`.","title":"$p_c \\le \\tilde p_c$ if $\\theta > 0$ above $\\tilde p_c$"},{"id":"Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b/tilde-pc-le-pc","kind":"node","lean":"StatMech.Sharpness.tildePc_le_pc","parent":20275,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\tilde p_c \\le p_c\\)"],["text"," if "],["math","\\(\\theta = 0\\)"],["text"," on the set defining "],["math","\\(\\tilde p_c\\)"]],"source":"$\\tilde p_c \\le p_c$ if $\\theta = 0$ on the set defining $\\tilde p_c$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b/index.html#tilde-pc-le-pc","scope":"Probability/percolation/sharpness/mean-field-lower-bound-and-critical-points-for-b","status":"fully_proved","summary":"`p\u0303_c \u2264 p_c` when `\u03b8` vanishes on the set defining `p\u0303_c`. See also `Percolation.tildePc_le_pc` (MathlibExt), stated with exponential decay of crossing probabilities.","title":"$\\tilde p_c \\le p_c$ if $\\theta = 0$ on the set defining $\\tilde p_c$"},{"data_url":"scopes/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality.json","id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","title":"Duminil-Copin\u2013Tassion differential inequality"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/within-conn-event","kind":"node","lean":"StatMech.Percolation.withinConnEvent","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#within-conn-event","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The event `{o \u2194 x inside S}`: `o` and `x` lie in `S` and are joined by an open path in `S`.","title":"Event of a connection inside a finite set"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/edges-within-finset","kind":"node","lean":"StatMech.Percolation.edgesWithinFinset","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#edges-within-finset","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"All unordered pairs of sites of the finite set `S`.","title":"Pairs of sites of a finite set"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/box-edge-finset","kind":"node","lean":"StatMech.Percolation.boxEdgeFinset","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#box-edge-finset","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"All unordered pairs of sites of `\u039b_n` (containing the lattice edges of `\u039b_n`).","title":"Pairs of sites of a lattice box"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/box-crossing-event","kind":"node","lean":"StatMech.Percolation.boxCrossingEvent","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#box-crossing-event","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The event that the origin is joined by an open path inside `\u039b_n` to the vertex boundary of `\u039b_n`.","title":"Box-crossing event"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/box-cross-prob","kind":"node","lean":"StatMech.Percolation.boxCrossProb","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#box-cross-prob","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The probability of `boxCrossingEvent d n`.","title":"Box-crossing probability"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/pivotal-event","kind":"node","lean":"StatMech.Percolation.pivotalEvent","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#pivotal-event","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The event that the edge `e` is pivotal for `A` (flipping `e` changes whether `A` occurs).","title":"Pivotal edge event"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/conn-bdry","kind":"node","lean":"StatMech.Percolation.connBdry","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#conn-bdry","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"`x \u2208 \u039b_n` is joined by an open path inside `\u039b_n` to a site of the vertex boundary of `\u039b_n`.","title":"Connection of a site to the box boundary"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/surface-set","kind":"node","lean":"StatMech.Percolation.surfaceSet","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#surface-set","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The random set `\ud835\udcae(\u03c9)` of sites of `\u039b_n` not joined inside `\u039b_n` to the vertex boundary.","title":"Set of sites not connected to the box boundary"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/surface-event","kind":"node","lean":"StatMech.Percolation.surfaceEvent","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#surface-event","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The event `{\ud835\udcae = S}`.","title":"Event that the unconnected set equals a given set"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/box-edges-outside","kind":"node","lean":"StatMech.Percolation.boxEdgesOutside","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#box-edges-outside","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The pairs of sites of `\u039b_n` that are not pairs of sites of `S`.","title":"Pairs of box sites outside a given set"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/dct-differential-inequality","kind":"node","lean":"StatMech.Percolation.dct_differential_inequality","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#dct-differential-inequality","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The Duminil-Copin\u2013Tassion differential inequality: if `\u03c6_p(S) \u2265 m` for every finite `S \u220b 0`, then `d/dp \u2119_p(A_n) \u2265 (p(1 - p))\u207b\u00b9 m (1 - \u2119_p(A_n))` for `0 < p < 1` and `n \u2265 1` (Duminil-Copin\u2013Tassion 2016).","title":"Duminil-Copin\u2013Tassion differential inequality"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/cross-poly","kind":"node","lean":"StatMech.Percolation.crossPoly","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#cross-poly","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The polynomial `t \u21a6 \u2119_t(A_{n+1})` (the box-crossing probability as a polynomial in the density), defined for all real `t`.","title":"Box-crossing probability as a polynomial in the density"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/theta-ge-mean-field-of-tilde-pc-lt","kind":"node","lean":"StatMech.Percolation.theta_ge_meanField_of_tildePc_lt","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#theta-ge-mean-field-of-tilde-pc-lt","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"Mean-field lower bound (Duminil-Copin\u2013Tassion 2016): for `p\u0303_c < q < 1` (with `p\u0303_c > 0`), `\u03b8(q) \u2265 (q - p\u0303_c) / (q(1 - p\u0303_c))`.","title":"Mean-field lower bound"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/internal-edges","kind":"node","lean":"StatMech.Sharpness.internalEdges","parent":20279,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edges of "],["math","\\(\\mathbb{Z}^d\\)"],["text"," inside a set"]],"source":"Edges of $\\mathbb{Z}^d$ inside a set"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#internal-edges","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The edges of `\u2124^d` with both endpoints in `S`.","title":"Edges of $\\mathbb{Z}^d$ inside a set"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/mpi-crossing-event","kind":"node","lean":"StatMech.Sharpness.mpiCrossingEvent","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#mpi-crossing-event","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The event that `o` is connected to `B`.","title":"Event that a vertex is connected to a target set"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/mpi-blocked-set","kind":"node","lean":"StatMech.Sharpness.mpiBlockedSet","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#mpi-blocked-set","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The set of vertices not connected to `B` in `\u03c9`.","title":"Vertices not connected to a target set"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/mpi-surface-event","kind":"node","lean":"StatMech.Sharpness.mpiSurfaceEvent","parent":20279,"reading_title":{"format":"inline-parts-v1","parts":[["text","Event that the set of vertices not connected to "],["math","\\(B\\)"],["text"," is "],["math","\\(S\\)"]],"source":"Event that the set of vertices not connected to $B$ is $S$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#mpi-surface-event","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The event that the set of vertices not connected to `B` is exactly `S`.","title":"Event that the set of vertices not connected to $B$ is $S$"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/mpi-internal-edges","kind":"node","lean":"StatMech.Sharpness.mpiInternalEdges","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#mpi-internal-edges","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The edges with both endpoints in `S`.","title":"Edges with both endpoints in a vertex set"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/mpi-phi","kind":"node","lean":"StatMech.Sharpness.mpiPhi","parent":20279,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inhomogeneous Duminil-Copin\u2013Tassion quantity "],["math","\\(\\varphi(S)\\)"]],"source":"Inhomogeneous Duminil-Copin\u2013Tassion quantity $\\varphi(S)$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#mpi-phi","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The inhomogeneous Duminil-Copin\u2013Tassion quantity `\u03c6(S) = \u2211_{(x,y) \u2208 \u2202S} (1 - e^{-\u03b2J_{xy}}) P(o \u2194 x in S)`.","title":"Inhomogeneous Duminil-Copin\u2013Tassion quantity $\\varphi(S)$"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/mpi-pivotal-closed","kind":"node","lean":"StatMech.Sharpness.mpiPivotalClosed","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#mpi-pivotal-closed","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The event that `e` is closed and pivotal for `A`.","title":"Event that an edge is closed and pivotal"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/mpi-weight-off","kind":"node","lean":"StatMech.Sharpness.mpiWeightOff","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#mpi-weight-off","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The product of the edge laws over all edges other than `e`.","title":"Product of the edge laws over all other edges"},{"id":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/mpi-meanfield-perco","kind":"node","lean":"StatMech.Sharpness.mpi_meanfield_perco","parent":20279,"result":true,"route":"../roadmap/Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality/index.html#mpi-meanfield-perco","scope":"Probability/percolation/sharpness/duminil-copin-tassion-differential-inequality","status":"fully_proved","summary":"The mean-field differential inequality for inhomogeneous percolation: Russo's formula `\u2202_\u03b2 P(o \u2194 B) = \u2211_e J_e P(e closed pivotal)` and, if `\u03c6(S) \u2265 m` for all `S \u220b o`, `(m/\u03b2)(1 - P(o \u2194 B)) \u2264 \u2202_\u03b2 P(o \u2194 B)`. Reference: H. Duminil-Copin, V. Tassion, \"A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model\", Comm. Math. Phys. 343 (2016) 725\u2013745.","title":"Mean-field differential inequality for inhomogeneous percolation"},{"data_url":"scopes/Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli.json","id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/index.html","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli","status":"fully_proved","title":"Sharpness of the phase transition for Bernoulli bond percolation on \u2124^d"},{"id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/tilde-pc-pos","kind":"node","lean":"StatMech.Percolation.tildePc_pos","parent":20302,"result":true,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/index.html#tilde-pc-pos","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli","status":"fully_proved","summary":"`p\u0303_c(d) > 0` for `d \u2265 1`: at `p = 1/(4d + 1)`, `\u03c6_p({0}) \u2264 2dp < 1`.","title":"Positivity of the Duminil-Copin\u2013Tassion threshold"},{"id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/line-word","kind":"node","lean":"StatMech.Percolation.lineWord","parent":20302,"result":true,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/index.html#line-word","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli","status":"fully_proved","summary":"The direction sequence of the straight path from the origin along the first coordinate axis.","title":"Straight path along the first coordinate axis"},{"id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/cross-prob-exp-decay-of-mem-tilde-pc-set","kind":"node","lean":"StatMech.Percolation.crossProb_exp_decay_of_mem_tildePcSet","parent":20302,"result":true,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/index.html#cross-prob-exp-decay-of-mem-tilde-pc-set","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli","status":"fully_proved","summary":"Exponential decay of the one-arm probability for every `p \u2208 tildePcSet d`, i.e. every `p` admitting a finite `S \u220b 0` with `\u03c6_p(S) < 1`.","title":"Exponential decay of the one-arm probability below the threshold"},{"id":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/sharp-phase-transition","kind":"node","lean":"StatMech.Percolation.sharp_phase_transition","parent":20302,"result":true,"route":"../roadmap/Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli/index.html#sharp-phase-transition","scope":"Probability/percolation/sharpness/sharpness-of-the-phase-transition-for-bernoulli","status":"fully_proved","summary":"Sharpness of the phase transition for Bernoulli bond percolation on `\u2124^d`, `d \u2265 1` (Duminil-Copin\u2013Tassion 2016; Aizenman\u2013Barsky 1987; Menshikov 1986):","title":"Sharpness of the phase transition"},{"data_url":"scopes/Probability/percolation/sharpness/aizenman-barsky-inequality.json","id":"Probability/percolation/sharpness/aizenman-barsky-inequality","kind":"scope","navigation":"module","parent":20232,"result":false,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","title":"Aizenman\u2013Barsky inequality"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abcr-endpoint-sum","kind":"node","lean":"StatMech.Sharpness.abcrEndpointSum","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abcr-endpoint-sum","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The function `s(x, y) \u21a6 F(x) + F(y)` on unordered pairs.","title":"Endpoint sum on unordered pairs"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abcr-incident-coupling","kind":"node","lean":"StatMech.Sharpness.abcrIncidentCoupling","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abcr-incident-coupling","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The incident coupling `\u2211_{y \u223c x} J_{xy}` of `x`, as a sum over darts starting at `x`.","title":"Incident coupling of a vertex"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/edge-weight-v","kind":"node","lean":"StatMech.Sharpness.edgeWeightV","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#edge-weight-v","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The probability `p_e` or `1 - p_e` of the state of edge `e` in `\u03c9`.","title":"Edge weight under an inhomogeneous product measure"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/config-weight-v","kind":"node","lean":"StatMech.Sharpness.configWeightV","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#config-weight-v","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The product weight `\u220f_e (p_e or 1 - p_e)` of a configuration.","title":"Product weight of a configuration"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/weight-off-v","kind":"node","lean":"StatMech.Sharpness.weightOffV","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#weight-off-v","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The product weight of a configuration over all edges other than `e`.","title":"Product weight of a configuration off one edge"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/prob-v","kind":"node","lean":"StatMech.Sharpness.probV","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#prob-v","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The probability `\u2211_\u03c9 1_A(\u03c9) \u220f_e (p_e or 1 - p_e)` of `A` under the inhomogeneous product measure.","title":"Probability under an inhomogeneous product measure"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/pivotal-prob-v","kind":"node","lean":"StatMech.Sharpness.pivotalProbV","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#pivotal-prob-v","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The probability that `e` is pivotal for `A`.","title":"Pivotal probability of an edge"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/param-on","kind":"node","lean":"StatMech.Sharpness.paramOn","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#param-on","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The parameters `p` with the value `c` on the edges of `s`.","title":"Edge parameters set to a common value on a subset"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/q-field","kind":"node","lean":"StatMech.Sharpness.qField","parent":20307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ghost-edge probability "],["math","\\(1 - e^{-h}\\)"]],"source":"Ghost-edge probability $1 - e^{-h}$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#q-field","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The ghost-edge probability `1 - e^{-h}`.","title":"Ghost-edge probability $1 - e^{-h}$"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abfs-option-equiv","kind":"node","lean":"StatMech.Sharpness.abfsOptionEquiv","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abfs-option-equiv","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The extension of a vertex permutation to the ghost graph fixing the ghost.","title":"Extension of a vertex permutation to the ghost graph"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abfs-edge-equiv","kind":"node","lean":"StatMech.Sharpness.abfsEdgeEquiv","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abfs-edge-equiv","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The induced permutation of unordered pairs of the ghost graph.","title":"Induced permutation of ghost-graph edges"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abgi-closed-pivotal","kind":"node","lean":"StatMech.Sharpness.abgiClosedPivotal","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abgi-closed-pivotal","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Configurations in which `e` is closed and pivotal for `A`.","title":"Configurations with an edge closed and pivotal"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abpd-beta-field-params","kind":"node","lean":"StatMech.Sharpness.abpdBetaFieldParams","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abpd-beta-field-params","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Edge parameters `1 - e^{-\u03b2J_e}` off `s` and `q` on `s`.","title":"Edge parameters with a field value on a subset"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/sab-induced-edge-set","kind":"node","lean":"StatMech.Sharpness.sab_inducedEdgeSet","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#sab-induced-edge-set","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The edges of `V` used by a walk in the open subgraph induced on `A`.","title":"Edges used by an open walk in an induced subgraph"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/sab-induced-walk-transfer","kind":"node","lean":"StatMech.Sharpness.sab_inducedWalkTransfer","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#sab-induced-walk-transfer","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"An open walk in `A` for `\u03c9`, read as an open walk for `\u03c9'` when `\u03c9` and `\u03c9'` agree on its edges.","title":"Transfer of an open walk between agreeing configurations"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/sab-transfer-avoid","kind":"node","lean":"StatMech.Sharpness.sab_transfer_avoid","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#sab-transfer-avoid","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"An open walk for `\u03c9_S` avoiding the edge `xy`, read as an open walk for `\u03c9` when the two agree off `xy`.","title":"Transfer of an open walk avoiding a fixed edge"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abfa-base-coupling","kind":"node","lean":"StatMech.Sharpness.abfaBaseCoupling","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abfa-base-coupling","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The coupling `J` restricted to the edges of `G`.","title":"Coupling restricted to the edges of a graph"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abfa-lift-coupling","kind":"node","lean":"StatMech.Sharpness.abfaLiftCoupling","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abfa-lift-coupling","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The coupling on the ghost graph: `J` on the edges of `G` and `0` on the ghost edges.","title":"Coupling lifted to the ghost graph"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abfa-params","kind":"node","lean":"StatMech.Sharpness.abfaParams","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abfa-params","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Edge parameters `1 - e^{-\u03b2J_e}` on the edges of `G` and `1 - e^{-h}` on the ghost edges.","title":"Edge parameters with a ghost field"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abfa-mag","kind":"node","lean":"StatMech.Sharpness.abfaMag","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abfa-mag","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The percolation magnetization `M(\u03b2, h) = P(o \u2194 ghost)`.","title":"Finite-volume percolation magnetization"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abfa-aizenman-barsky","kind":"node","lean":"StatMech.Sharpness.abfa_aizenmanBarsky","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abfa-aizenman-barsky","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The Aizenman\u2013Barsky inequality on a finite graph: if all row sums equal `J\u2080` and all sites have the same magnetization, then `\u2202_\u03b2 M \u2264 J\u2080 M \u2202_h M` for `\u03b2, h \u2265 0`. Reference: M. Aizenman, D. J. Barsky, \"Sharpness of the phase transition in percolation models\", Comm. Math. Phys. 108 (1987) 489\u2013526.","title":"Aizenman\u2013Barsky inequality on a finite graph"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abgr-merge","kind":"node","lean":"StatMech.Sharpness.abgrMerge","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abgr-merge","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The configuration equal to `\u03b3` on `s` and to `\u03b7` off `s`.","title":"Merging configurations on and off a set"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abgr-keep-ghost","kind":"node","lean":"StatMech.Sharpness.abgrKeepGhost","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abgr-keep-ghost","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Close all ghost edges except `{x, ghost}`.","title":"Closing all ghost edges but one"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abgr-close-ghost","kind":"node","lean":"StatMech.Sharpness.abgrCloseGhost","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abgr-close-ghost","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Close all ghost edges.","title":"Closing all ghost edges"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abgr-active","kind":"node","lean":"StatMech.Sharpness.abgrActive","parent":20307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sites connected to "],["math","\\(o\\)"],["text"," with ghost edges closed"]],"source":"Sites connected to $o$ with ghost edges closed"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abgr-active","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The sites connected to `o` when all ghost edges are closed.","title":"Sites connected to $o$ with ghost edges closed"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abgr-ghost-edge-equiv","kind":"node","lean":"StatMech.Sharpness.abgrGhostEdgeEquiv","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abgr-ghost-edge-equiv","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Sites correspond to ghost edges.","title":"Correspondence between sites and ghost edges"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abgr-active-ghost","kind":"node","lean":"StatMech.Sharpness.abgrActiveGhost","parent":20307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ghost edges at the physical cluster of "],["math","\\(o\\)"]],"source":"Ghost edges at the physical cluster of $o$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abgr-active-ghost","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The ghost edges at sites of the cluster of `o` in the physical configuration `\u03b7`.","title":"Ghost edges at the physical cluster of $o$"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abgr-off-weight","kind":"node","lean":"StatMech.Sharpness.abgrOffWeight","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abgr-off-weight","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The probability of the physical configuration `\u03b7`.","title":"Probability of a physical configuration"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/aizenman-barsky-inequality","kind":"node","lean":"StatMech.Sharpness.AizenmanBarskyInequality","parent":20307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Aizenman\u2013Barsky inequality for all "],["math","\\(\\beta\\)"],["text"," and "],["math","\\(h\\)"]],"source":"Aizenman\u2013Barsky inequality for all $\\beta$ and $h$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#aizenman-barsky-inequality","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The Aizenman\u2013Barsky inequality `\u2202_\u03b2 M(\u03b2, h) \u2264 J M(\u03b2, h) \u2202_h M(\u03b2, h)` for all `\u03b2, h`.","title":"Aizenman\u2013Barsky inequality for all $\\beta$ and $h$"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/aizenman-barsky-pdi","kind":"node","lean":"StatMech.Sharpness.AizenmanBarskyPDI","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#aizenman-barsky-pdi","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The second Aizenman\u2013Barsky inequality `M \u2264 h \u2202_h M + M\u00b2 + \u03b2 M \u2202_\u03b2 M`.","title":"Second Aizenman\u2013Barsky inequality"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/ab-perco-meanfield-sqrt","kind":"node","lean":"StatMech.Sharpness.ab_perco_meanfield_sqrt","parent":20307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Aizenman\u2013Barsky critical bound "],["math","\\(M(\\beta_c, h) \\ge c\\sqrt{h}\\)"]],"source":"Aizenman\u2013Barsky critical bound $M(\\beta_c, h) \\ge c\\sqrt{h}$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#ab-perco-meanfield-sqrt","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The Aizenman\u2013Barsky critical bound: from `\u2202_\u03b2 M \u2264 J M \u2202_h M` and `1 - M(\u03b2_c, h) \u2264 \u03b2_c \u2202_\u03b2 M(\u03b2_c, h)` with `M(\u03b2_c, 0) = 0`, `M(\u03b2_c, h) \u2265 c \u221ah` for `0 < h \u2264 \u03b2_c J/4`. Reference: M. Aizenman, D. J. Barsky, \"Sharpness of the phase transition in percolation models\", Comm. Math. Phys. 108 (1987) 489\u2013526.","title":"Aizenman\u2013Barsky critical bound $M(\\beta_c, h) \\ge c\\sqrt{h}$"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/aizenman-barsky-inequality-physical","kind":"node","lean":"StatMech.Sharpness.AizenmanBarskyInequalityPhysical","parent":20307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Aizenman\u2013Barsky inequality for "],["math","\\(\\beta \\ge 0\\)"],["text"," and "],["math","\\(h > 0\\)"]],"source":"Aizenman\u2013Barsky inequality for $\\beta \\ge 0$ and $h > 0$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#aizenman-barsky-inequality-physical","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The Aizenman\u2013Barsky inequality `\u2202_\u03b2 M \u2264 J M \u2202_h M` for `\u03b2 \u2265 0` and `h > 0`.","title":"Aizenman\u2013Barsky inequality for $\\beta \\ge 0$ and $h > 0$"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-real-param","kind":"node","lean":"StatMech.Sharpness.abigRealParam","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-real-param","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The real edge parameters: `1 - e^{-\u03b2J_e}` on physical edges and `1 - e^{-h}` on ghost edges.","title":"Real edge parameters with a ghost field"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-param","kind":"node","lean":"StatMech.Sharpness.abigParam","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-param","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The edge parameters as elements of `\u211d\u22650`.","title":"Edge parameters as nonnegative reals"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-measure","kind":"node","lean":"StatMech.Sharpness.abigMeasure","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-measure","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Independent bond percolation on the ghost graph with parameters `abigParam`.","title":"Bond percolation measure on the ghost graph"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-mag","kind":"node","lean":"StatMech.Sharpness.abigMag","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-mag","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The percolation magnetization `M(\u03b2, h) = P(o \u2194 ghost)`.","title":"Infinite-volume percolation magnetization"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-internal-edges","kind":"node","lean":"StatMech.Sharpness.abigInternalEdges","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-internal-edges","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The unordered pairs with both endpoints in `T`.","title":"Internal edges of a finite set"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-internal-edge-equiv","kind":"node","lean":"StatMech.Sharpness.abigInternalEdgeEquiv","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-internal-edge-equiv","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Unordered pairs of `T` correspond to `abigInternalEdges T`.","title":"Unordered pairs of a set as its internal edges"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-ghost-window","kind":"node","lean":"StatMech.Sharpness.abigGhostWindow","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-ghost-window","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The window `S \u222a {ghost}` in the ghost graph.","title":"Ghost window of a finite set"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-ghost-window-equiv","kind":"node","lean":"StatMech.Sharpness.abigGhostWindowEquiv","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-ghost-window-equiv","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"`Option S` corresponds to the window `S \u222a {ghost}`.","title":"Ghost window as a set with one added point"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-window-edge-equiv","kind":"node","lean":"StatMech.Sharpness.abigWindowEdgeEquiv","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-window-edge-equiv","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Unordered pairs of `Option S` correspond to the internal edges of the window.","title":"Unordered pairs of the extended set as window edges"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-window-graph","kind":"node","lean":"StatMech.Sharpness.abigWindowGraph","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-window-graph","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The subgraph of `G` induced on `S`.","title":"Subgraph induced on a window"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-window-coupling","kind":"node","lean":"StatMech.Sharpness.abigWindowCoupling","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-window-coupling","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The coupling restricted to the window.","title":"Coupling restricted to a window"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-dart-fiber-equiv-neighbor","kind":"node","lean":"StatMech.Sharpness.abigDartFiberEquivNeighbor","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-dart-fiber-equiv-neighbor","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"Darts starting at `x` correspond to neighbours of `x`.","title":"Darts at a vertex as its neighbours"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-uniform-finite-row-bound","kind":"node","lean":"StatMech.Sharpness.abigUniformFiniteRowBound","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-uniform-finite-row-bound","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"All row sums of `J` in all finite windows are at most `J\u2080`.","title":"Uniform row-sum bound in finite windows"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-window-envelope","kind":"node","lean":"StatMech.Sharpness.abigWindowEnvelope","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-window-envelope","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The largest site magnetization in the window.","title":"Largest site magnetization in a window"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abig-aizenman-barsky-physical-of-tsum-tr","kind":"node","lean":"StatMech.Sharpness.abig_aizenmanBarskyPhysical_of_tsum_transitive","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abig-aizenman-barsky-physical-of-tsum-tr","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The Aizenman\u2013Barsky inequality on infinite vertex-transitive graphs with translation-invariant couplings supported on edges and row sums at most `J\u2080`: `\u2202_\u03b2 M \u2264 J\u2080 M \u2202_h M` for `\u03b2 \u2265 0`, `h > 0`. Reference: M. Aizenman, D. J. Barsky, \"Sharpness of the phase transition in percolation models\", Comm. Math. Phys. 108 (1987) 489\u2013526.","title":"Aizenman\u2013Barsky inequality on infinite vertex-transitive graphs"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abl-box-finset","kind":"node","lean":"StatMech.Sharpness.ablBoxFinset","parent":20307,"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abl-box-finset","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The box `\u039b_n` as a finite set of sites.","title":"Box as a finite set of sites"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abl-aizenman-barsky-hypercubic-neighbour","kind":"node","lean":"StatMech.Sharpness.abl_aizenmanBarsky_hypercubic_neighbourSum","parent":20307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Aizenman\u2013Barsky inequality on "],["math","\\(\\mathbb{Z}^d\\)"],["text"," with neighbour row sum"]],"source":"Aizenman\u2013Barsky inequality on $\\mathbb{Z}^d$ with neighbour row sum"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abl-aizenman-barsky-hypercubic-neighbour","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The Aizenman\u2013Barsky inequality on `\u2124^d` with the row sum written over lattice neighbours. Reference: M. Aizenman, D. J. Barsky, \"Sharpness of the phase transition in percolation models\", Comm. Math. Phys. 108 (1987) 489\u2013526. Scope and fidelity: this is the first Aizenman\u2013Barsky differential inequality `\u2202_\u03b2 M \u2264 |J| M \u2202_h M` for independent bond percolation on `\u2124^d` with nearest-neighbour, translation-invariant\u2026","title":"Aizenman\u2013Barsky inequality on $\\mathbb{Z}^d$ with neighbour row sum"},{"id":"Probability/percolation/sharpness/aizenman-barsky-inequality/abl-aizenman-barsky-hypercubic","kind":"node","lean":"StatMech.Sharpness.abl_aizenmanBarsky_hypercubic","parent":20307,"reading_title":{"format":"inline-parts-v1","parts":[["text","Aizenman\u2013Barsky inequality on "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Aizenman\u2013Barsky inequality on $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/Probability/percolation/sharpness/aizenman-barsky-inequality/index.html#abl-aizenman-barsky-hypercubic","scope":"Probability/percolation/sharpness/aizenman-barsky-inequality","status":"fully_proved","summary":"The Aizenman\u2013Barsky inequality on `\u2124^d`: `\u2202_\u03b2 M \u2264 |J| M \u2202_h M` with `|J| = \u2211_y J_{0y}`. Reference: M. Aizenman, D. J. Barsky, \"Sharpness of the phase transition in percolation models\", Comm. Math. Phys. 108 (1987) 489\u2013526. Scope and fidelity: this is the first Aizenman\u2013Barsky differential inequality `\u2202_\u03b2 M \u2264 |J| M \u2202_h M` for independent bond percolation on `\u2124^d` with nearest-neighbour, translation-invariant\u2026","title":"Aizenman\u2013Barsky inequality on $\\mathbb{Z}^d$"},{"data_url":"scopes/Probability/percolation/crossings.json","id":"Probability/percolation/crossings","kind":"scope","parent":20083,"result":false,"route":"../roadmap/Probability/percolation/crossings/index.html","scope":"Probability/percolation/crossings","status":"can_prove","summary":"Crossings of rectangles and annuli in bond percolation on Z^2, the box-crossing property, gluing and square-root tricks, Russo\u2013Seymour\u2013Welsh theory and planar separation lemmas.","title":"Crossing probabilities in two dimensions"},{"data_url":"scopes/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice.json","id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","kind":"scope","navigation":"module","parent":20358,"result":false,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","title":"Rectangle crossings on the square lattice"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/rectangle","kind":"node","lean":"StatMech.TwoDim.rectangle","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#rectangle","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"The rectangle `[0, a] \u00d7 [0, b]` in `\u2124\u00b2`.","title":"Rectangle at the origin in the plane lattice"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/left-side","kind":"node","lean":"StatMech.TwoDim.leftSide","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#left-side","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"The left side `{0} \u00d7 [0, b]` of `rectangle a b`.","title":"Left side of a rectangle at the origin"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/right-side","kind":"node","lean":"StatMech.TwoDim.rightSide","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#right-side","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"The right side `{a} \u00d7 [0, b]` of `rectangle a b`.","title":"Right side of a rectangle at the origin"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/bottom-side","kind":"node","lean":"StatMech.TwoDim.bottomSide","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#bottom-side","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"The bottom side `[0, a] \u00d7 {0}` of `rectangle a b`.","title":"Bottom side of a rectangle at the origin"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/top-side","kind":"node","lean":"StatMech.TwoDim.topSide","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#top-side","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"The top side `[0, a] \u00d7 {b}` of `rectangle a b`.","title":"Top side of a rectangle at the origin"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/horizontal-crossing","kind":"node","lean":"StatMech.TwoDim.HorizontalCrossing","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#horizontal-crossing","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"`\u03c9` has an open left\u2013right crossing of `rectangle a b`: an open path inside the rectangle from its left side to its right side.","title":"Open left\u2013right crossing of a rectangle at the origin"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/horizontal-crossing-event","kind":"node","lean":"StatMech.TwoDim.horizontalCrossingEvent","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#horizontal-crossing-event","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"The event that `rectangle a b` has an open left\u2013right crossing.","title":"Left\u2013right crossing event of a rectangle at the origin"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/vertical-crossing","kind":"node","lean":"StatMech.TwoDim.VerticalCrossing","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#vertical-crossing","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"`\u03c9` has an open bottom\u2013top crossing of `rectangle a b`: an open path inside the rectangle from its bottom side to its top side.","title":"Open bottom\u2013top crossing of a rectangle at the origin"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/vertical-crossing-event","kind":"node","lean":"StatMech.TwoDim.verticalCrossingEvent","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#vertical-crossing-event","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"The event that `rectangle a b` has an open bottom\u2013top crossing.","title":"Bottom\u2013top crossing event of a rectangle at the origin"},{"id":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/open-subgraph-mono","kind":"node","lean":"StatMech.TwoDim.openSubgraph_mono","parent":20359,"result":true,"route":"../roadmap/Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice/index.html#open-subgraph-mono","scope":"Probability/percolation/crossings/rectangle-crossings-on-the-square-lattice","status":"fully_proved","summary":"The open subgraph is monotone in the configuration (`d = 2` case of `StatMech.Universality.openSubgraph_mono_dim`).","title":"Monotonicity of the open subgraph in the plane"},{"data_url":"scopes/Probability/percolation/crossings/russo-seymour-welsh-theory.json","id":"Probability/percolation/crossings/russo-seymour-welsh-theory","kind":"scope","parent":20358,"result":false,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/index.html","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory","status":"can_prove","title":"Russo\u2013Seymour\u2013Welsh theory"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/arc-separating-set","kind":"node","lean":"StatMech.Lattice.ArcSeparatingSet","parent":20370,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/index.html#arc-separating-set","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory","status":"fully_proved","summary":"There is a set containing the left side of `[\u03b1, \u03b2] \u00d7 [c, d]` and disjoint from its right side such that every lattice edge of the rectangle crossing its boundary has an endpoint in `P`.","title":"Separating set for a lattice rectangle"},{"data_url":"scopes/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing.json","id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","kind":"scope","navigation":"module","parent":20370,"result":false,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","title":"Rectangles, crossings, annuli and the box-crossing property on \u2124\u00b2"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/rect","kind":"node","lean":"StatMech.RSW.Box.rect","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#rect","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The rectangle `[a, b] \u00d7 [c, d]` in `\u2124\u00b2`.","title":"Integer rectangle in the plane lattice"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/measurable-set-connected-within","kind":"node","lean":"StatMech.RSW.Box.measurableSet_connectedWithin","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#measurable-set-connected-within","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The event that two vertices are connected by an open path inside `S` is measurable.","title":"Measurability of connection inside a set"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/left-side","kind":"node","lean":"StatMech.RSW.Box.leftSide","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#left-side","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The left side `{a} \u00d7 [c, d]` of `rect a b c d`.","title":"Left side of a general lattice rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/right-side","kind":"node","lean":"StatMech.RSW.Box.rightSide","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#right-side","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The right side `{b} \u00d7 [c, d]` of `rect a b c d`.","title":"Right side of a general lattice rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/bottom-side","kind":"node","lean":"StatMech.RSW.Box.bottomSide","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#bottom-side","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The bottom side `[a, b] \u00d7 {c}` of `rect a b c d`.","title":"Bottom side of a general lattice rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/top-side","kind":"node","lean":"StatMech.RSW.Box.topSide","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#top-side","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The top side `[a, b] \u00d7 {d}` of `rect a b c d`.","title":"Top side of a general lattice rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/horizontal-crossing","kind":"node","lean":"StatMech.RSW.Box.HorizontalCrossing","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#horizontal-crossing","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"An open left\u2013right crossing of `rect a b c d` by a path inside the rectangle.","title":"Open left\u2013right crossing of a general rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/vertical-crossing","kind":"node","lean":"StatMech.RSW.Box.VerticalCrossing","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#vertical-crossing","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"An open bottom\u2013top crossing of `rect a b c d` by a path inside the rectangle.","title":"Open bottom\u2013top crossing of a general rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/horizontal-crossing-event","kind":"node","lean":"StatMech.RSW.Box.horizontalCrossingEvent","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#horizontal-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The event that `rect a b c d` has an open left\u2013right crossing.","title":"Left\u2013right crossing event of a general rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/vertical-crossing-event","kind":"node","lean":"StatMech.RSW.Box.verticalCrossingEvent","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#vertical-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The event that `rect a b c d` has an open bottom\u2013top crossing.","title":"Bottom\u2013top crossing event of a general rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/measurable-set-horizontal-crossing-event","kind":"node","lean":"StatMech.RSW.Box.measurableSet_horizontalCrossingEvent","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#measurable-set-horizontal-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"A left\u2013right crossing event of a rectangle is measurable.","title":"Measurability of the left\u2013right crossing event"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/measurable-set-vertical-crossing-event","kind":"node","lean":"StatMech.RSW.Box.measurableSet_verticalCrossingEvent","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#measurable-set-vertical-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"A bottom\u2013top crossing event of a rectangle is measurable.","title":"Measurability of the bottom\u2013top crossing event"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/dual-horizontal-crossing","kind":"node","lean":"StatMech.RSW.Box.DualHorizontalCrossing","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#dual-horizontal-crossing","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"A left\u2013right crossing of `rect a b c d` in the dual configuration `dualConfig \u03c9` (an open crossing of the dual lattice, read through the primal coordinates of `dualConfig`).","title":"Dual left\u2013right crossing of a rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/dual-vertical-crossing","kind":"node","lean":"StatMech.RSW.Box.DualVerticalCrossing","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#dual-vertical-crossing","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"A bottom\u2013top crossing of `rect a b c d` in the dual configuration `dualConfig \u03c9`.","title":"Dual bottom\u2013top crossing of a rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/dual-horizontal-crossing-event","kind":"node","lean":"StatMech.RSW.Box.dualHorizontalCrossingEvent","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#dual-horizontal-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The event `DualHorizontalCrossing`.","title":"Dual left\u2013right crossing event"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/dual-vertical-crossing-event","kind":"node","lean":"StatMech.RSW.Box.dualVerticalCrossingEvent","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#dual-vertical-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The event `DualVerticalCrossing`.","title":"Dual bottom\u2013top crossing event"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/annulus","kind":"node","lean":"StatMech.RSW.Box.annulus","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#annulus","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The annulus `z + (\u039b_{2n} \\ \u039b_n)`, where `\u039b_n = [-n, n]\u00b2`.","title":"Square annulus in the plane lattice"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/is-annulus-circuit","kind":"node","lean":"StatMech.RSW.Box.IsAnnulusCircuit","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#is-annulus-circuit","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The closed walk `w` is an open circuit all of whose vertices lie in `annulus z n`.","title":"Open circuit lying in an annulus"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/annulus-circuit","kind":"node","lean":"StatMech.RSW.Box.AnnulusCircuit","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#annulus-circuit","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"There is an open circuit inside `annulus z n`.","title":"Existence of an open circuit in an annulus"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/annulus-circuit-event","kind":"node","lean":"StatMech.RSW.Box.annulusCircuitEvent","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#annulus-circuit-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The event `AnnulusCircuit`.","title":"Open circuit event in an annulus"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/annulus-circuit0","kind":"node","lean":"StatMech.RSW.Box.AnnulusCircuit0","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#annulus-circuit0","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"There is an open circuit inside the annulus `\u039b_{2n} \\ \u039b_n` centered at the origin.","title":"Open circuit in an annulus centered at the origin"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/box-crossing-event","kind":"node","lean":"StatMech.RSW.Box.boxCrossingEvent","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#box-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The left\u2013right open crossing of the box `[0, \u230a\u03b1n\u230b] \u00d7 [0, n]` (aspect ratio `\u03b1`).","title":"Left\u2013right crossing event of a box with given aspect ratio"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/p5","kind":"node","lean":"StatMech.RSW.Box.P5","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#p5","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The box-crossing property for a sequence of set functions `\u03bc n`: for every aspect ratio `\u03b1 > 0` there is `c > 0` with `\u03bc n (boxCrossingEvent \u03b1 n) \u2265 c` for all `n`.","title":"Box-crossing property for a sequence of measures"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/p5c","kind":"node","lean":"StatMech.RSW.Box.P5c","parent":20372,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing/index.html#p5c","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/rectangles-crossings-annuli-and-the-box-crossing","status":"fully_proved","summary":"The uniform two-sided box-crossing property for a family `\u03bc \u03be n`: for every `\u03b1 > 0` there is `c > 0` with `c \u2264 \u03bc \u03be n (boxCrossingEvent \u03b1 n) \u2264 1 - c` for all `\u03be` and `n`.","title":"Uniform two-sided box-crossing property"},{"data_url":"scopes/Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl.json","id":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl","kind":"scope","navigation":"module","parent":20370,"result":false,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/index.html","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl","status":"fully_proved","title":"The left region of a rectangle cut by an open cluster"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/box-minus-p","kind":"node","lean":"StatMech.Lattice.boxMinusP","parent":20397,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/index.html#box-minus-p","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl","status":"fully_proved","summary":"The square lattice restricted to the rectangle `[\u03b1, \u03b2] \u00d7 [c, d]`, with the open cluster of `x_B` in the overlap rectangle removed.","title":"Rectangle lattice with an open cluster removed"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/flood-from-left","kind":"node","lean":"StatMech.Lattice.floodFromLeft","parent":20397,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/index.html#flood-from-left","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl","status":"fully_proved","summary":"The sites reachable from the left side of `[\u03b1, \u03b2] \u00d7 [c, d]` inside the rectangle without touching the open cluster of `x_B` in the overlap.","title":"Sites reachable from the left side of a rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/arc-s-left-region","kind":"node","lean":"StatMech.Lattice.arcS_leftRegion","parent":20397,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/index.html#arc-s-left-region","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl","status":"fully_proved","summary":"The left region: the left side of `[\u03b1, \u03b2] \u00d7 [c, d]` together with `floodFromLeft`.","title":"Left region of a rectangle cut by an open cluster"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/arc-s-two-paths-cross","kind":"node","lean":"StatMech.Lattice.ArcS_TwoPathsCross","parent":20397,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl/index.html#arc-s-two-paths-cross","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/the-left-region-of-a-rectangle-cut-by-an-open-cl","status":"fully_proved","summary":"Every lattice path inside `[\u03b1, \u03b2] \u00d7 [c, d]` from the left side to the right side meets the open cluster of `x_B` in the overlap `[m, m'] \u00d7 [c, d]`. This fails for some configurations (`StatMech.Lattice.tpc_arcS_twoPathsCross_false`).","title":"Condition that every left-right crossing meets the open cluster"},{"data_url":"scopes/Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles.json","id":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles","kind":"scope","navigation":"module","parent":20370,"result":false,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/index.html","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles","status":"fully_proved","title":"Gluing crossings of adjacent rectangles"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/left-half-crossing","kind":"node","lean":"StatMech.RSW.Strip.LeftHalfCrossing","parent":20402,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/index.html#left-half-crossing","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles","status":"fully_proved","summary":"An open path inside `rect a m c d` from its left side to the point `z` (on its right side).","title":"Open path from the left side of a rectangle to a point"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/right-half-crossing","kind":"node","lean":"StatMech.RSW.Strip.RightHalfCrossing","parent":20402,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/index.html#right-half-crossing","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles","status":"fully_proved","summary":"An open path inside `rect m b c d` from the point `z` (on its left side) to its right side.","title":"Open path from a point to the right side of a rectangle"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/left-half-crossing-event","kind":"node","lean":"StatMech.RSW.Strip.leftHalfCrossingEvent","parent":20402,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/index.html#left-half-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles","status":"fully_proved","summary":"The event `LeftHalfCrossing`.","title":"Left half-crossing event"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/right-half-crossing-event","kind":"node","lean":"StatMech.RSW.Strip.rightHalfCrossingEvent","parent":20402,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/index.html#right-half-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles","status":"fully_proved","summary":"The event `RightHalfCrossing`.","title":"Right half-crossing event"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/measurable-set-left-half-crossing-event","kind":"node","lean":"StatMech.RSW.Strip.measurableSet_leftHalfCrossingEvent","parent":20402,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/index.html#measurable-set-left-half-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles","status":"fully_proved","summary":"A left half-crossing event is measurable.","title":"Measurability of the left half-crossing event"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/measurable-set-right-half-crossing-event","kind":"node","lean":"StatMech.RSW.Strip.measurableSet_rightHalfCrossingEvent","parent":20402,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles/index.html#measurable-set-right-half-crossing-event","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/gluing-crossings-of-adjacent-rectangles","status":"fully_proved","summary":"A right half-crossing event is measurable.","title":"Measurability of the right half-crossing event"},{"data_url":"scopes/Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per.json","id":"Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per","kind":"scope","navigation":"module","parent":20370,"result":false,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per/index.html","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per","status":"can_prove","title":"Russo\u2013Seymour\u2013Welsh bounds for critical bond percolation on \u2124\u00b2"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per/critical-bond-measure","kind":"node","lean":"MathlibExt.Probability.Percolation.RSWCriticalBondWanted.criticalBondMeasure","parent":20409,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per/index.html#critical-bond-measure","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per","status":"fully_proved","summary":"The critical bond percolation measure `\u2119_{1/2}` on `{0, 1}^{Sym2 \u2124\u00b2}`: every coordinate is open independently with probability `1/2`.","title":"Critical bond percolation measure on the square lattice"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per/rsw-box-crossing-lower","kind":"node","lean":"MathlibExt.Probability.Percolation.RSWCriticalBondWanted.rsw_box_crossing_lower","parent":20409,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per/index.html#rsw-box-crossing-lower","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per","status":"can_prove","summary":"Russo\u2013Seymour\u2013Welsh theorem (lower bound) for critical bond percolation on `\u2124\u00b2`: for every `\u03b1 > 0` there is `c > 0` with `\u2119_{1/2}(open left\u2013right crossing of [0, \u230a\u03b1n\u230b] \u00d7 [0, n]) \u2265 c` for every `n`.","title":"Russo\u2013Seymour\u2013Welsh theorem"},{"id":"Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per/rsw-box-crossing-two-sided","kind":"node","lean":"MathlibExt.Probability.Percolation.RSWCriticalBondWanted.rsw_box_crossing_two_sided","parent":20409,"result":true,"route":"../roadmap/Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per/index.html#rsw-box-crossing-two-sided","scope":"Probability/percolation/crossings/russo-seymour-welsh-theory/russo-seymour-welsh-bounds-for-critical-bond-per","status":"can_prove","summary":"Russo\u2013Seymour\u2013Welsh theorem (two-sided form) for critical bond percolation on `\u2124\u00b2`: for every `\u03b1 > 0` there is `c > 0` with `c \u2264 \u2119_{1/2}(open left\u2013right crossing of [0, \u230a\u03b1n\u230b] \u00d7 [0, n]) \u2264 1 - c` for every `n` with `\u03b1n \u2265 1`. The upper bound follows from the lower bound for the dual crossing.","title":"Russo\u2013Seymour\u2013Welsh theorem"},{"data_url":"scopes/Probability/percolation/crossings/box-crossing-property-for-positively-associated.json","id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated","kind":"scope","parent":20358,"result":false,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/index.html","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated","status":"fully_proved","title":"Box-crossing property for positively associated measures"},{"data_url":"scopes/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even.json","id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","kind":"scope","navigation":"module","parent":20413,"result":false,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","title":"Positive association, translates of crossing events, and the box-crossing property"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/origin-cluster","kind":"node","lean":"StatMech.Universality.originCluster","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#origin-cluster","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The open cluster of the origin of `\u2124^d`.","title":"Open cluster of the origin"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/percolation-event","kind":"node","lean":"StatMech.Universality.percolationEvent","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#percolation-event","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The event `{0 \u2194 \u221e}` that the open cluster of the origin is infinite.","title":"Event that the origin's cluster is infinite"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/percolation-probability","kind":"node","lean":"StatMech.Universality.percolationProbability","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#percolation-probability","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The percolation probability `\u03b8_d(p) = \u2119_p(0 \u2194 \u221e)`, as an extended nonnegative real.","title":"Percolation probability as an extended real"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/subcritical-densities","kind":"node","lean":"StatMech.Universality.subcriticalDensities","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#subcritical-densities","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The set of real densities `p \u2208 [0, 1]` with `\u03b8_d(p) = 0`.","title":"Real densities with no percolation"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/critical-probability","kind":"node","lean":"StatMech.Universality.criticalProbability","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#critical-probability","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The critical probability `p_c(d) = sup {p \u2208 [0, 1] : \u03b8_d(p) = 0}` as a real number (see `criticalProbability_eq_pc`).","title":"Critical probability as a real number"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/percolation-event-eq","kind":"node","lean":"StatMech.Universality.percolationEvent_eq","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#percolation-event-eq","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"`percolationEvent` is `StatMech.Percolation.percolationEvent`.","title":"Agreement of the two percolation events"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/critical-probability-eq-pc","kind":"node","lean":"StatMech.Universality.criticalProbability_eq_pc","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#critical-probability-eq-pc","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"`criticalProbability` is the real-valued form of `StatMech.Percolation.pc`.","title":"Agreement of the two critical probabilities"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/positively-associated","kind":"node","lean":"StatMech.Universality.PositivelyAssociated","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#positively-associated","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"A finite measure on configurations is positively associated if `\u03bc(A) \u03bc(B) \u2264 \u03bc(A \u2229 B) \u03bc(univ)` for all measurable increasing events `A`, `B`. The total-mass factor makes the Harris--FKG inequality invariant under finite scaling.","title":"Positively associated measure"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/positively-associated-iff-of-is-probabil","kind":"node","lean":"StatMech.Universality.positivelyAssociated_iff_of_isProbabilityMeasure","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#positively-associated-iff-of-is-probabil","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"For a probability measure, normalized positive association is the usual Harris--FKG inequality.","title":"Positive association for probability measures is the Harris\u2013FKG inequality"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/probability","kind":"node","lean":"StatMech.Universality.PositivelyAssociated.probability","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#probability","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The usual unnormalized Harris--FKG inequality for a positively associated probability measure.","title":"Harris\u2013FKG inequality for a positively associated probability measure"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/translate-config","kind":"node","lean":"StatMech.Universality.translateConfig","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#translate-config","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The configuration `\u03c9` translated by `-v`: the state of `e` is the state of `e + v` in `\u03c9`.","title":"Translate of a bond configuration"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/translated-horizontal-crossing-event","kind":"node","lean":"StatMech.Universality.translatedHorizontalCrossingEvent","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#translated-horizontal-crossing-event","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The event that the rectangle `v + [0, a] \u00d7 [0, b]` has an open left\u2013right crossing.","title":"Left\u2013right crossing event of a translated rectangle"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/translated-vertical-crossing-event","kind":"node","lean":"StatMech.Universality.translatedVerticalCrossingEvent","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#translated-vertical-crossing-event","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The event that the rectangle `v + [0, a] \u00d7 [0, b]` has an open bottom\u2013top crossing.","title":"Bottom\u2013top crossing event of a translated rectangle"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/box-crossing-probabilities","kind":"node","lean":"StatMech.Universality.boxCrossingProbabilities","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#box-crossing-probabilities","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The set of crossing probabilities, for `n \u2265 n\u2080` and all translates, of the long direction of `\u230a\u03c1n\u230b \u00d7 n` and `n \u00d7 \u230a\u03c1n\u230b` rectangles.","title":"Set of long-direction box-crossing probabilities"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/box-crossing-inf","kind":"node","lean":"StatMech.Universality.boxCrossingInf","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#box-crossing-inf","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The infimum of `boxCrossingProbabilities \u03bc \u03c1 n\u2080`.","title":"Infimum of the box-crossing probabilities"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/box-crossing-property","kind":"node","lean":"StatMech.Universality.BoxCrossingProperty","parent":20414,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even/index.html#box-crossing-property","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/positive-association-translates-of-crossing-even","status":"fully_proved","summary":"The box-crossing property with aspect ratio `\u03c1 > 1`: for some `n\u2080`, the probabilities of crossing all translates of `\u230a\u03c1n\u230b \u00d7 n` rectangles in the long direction, `n \u2265 n\u2080`, are bounded below by a positive constant.","title":"Box-crossing property with a given aspect ratio"},{"data_url":"scopes/Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet.json","id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet","kind":"scope","navigation":"module","parent":20413,"result":false,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet/index.html","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet","status":"fully_proved","title":"Horizontal and vertical crossings meet"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet/hvmeet-property","kind":"node","lean":"StatMech.Universality.HVMeetProperty","parent":20431,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet/index.html#hvmeet-property","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet","status":"fully_proved","summary":"For the configuration `\u03c9`: whenever `x_L` (left side of `[\u03b1, \u03b2] \u00d7 [c, d]`) is joined to the right side and `x_B` (bottom of the overlap `[m, m'] \u00d7 [c, d]`) is joined to its top, there is a point `z` of the overlap reached from `x_L` inside `[\u03b1, \u03b2] \u00d7 [c, d]` and from `x_B` inside the overlap.","title":"Horizontal and vertical crossings meet"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet/overlap-comp","kind":"node","lean":"StatMech.Universality.overlapComp","parent":20431,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet/index.html#overlap-comp","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet","status":"fully_proved","summary":"The open cluster of `x_B` inside the overlap rectangle `[m, m'] \u00d7 [c, d]`.","title":"Open cluster inside the overlap rectangle"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet/vseparates-h","kind":"node","lean":"StatMech.Universality.VSeparatesH","parent":20431,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet/index.html#vseparates-h","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/horizontal-and-vertical-crossings-meet","status":"fully_proved","summary":"For the configuration `\u03c9` and every point `x_B` of the overlap `[m, m'] \u00d7 [c, d]`, every open left\u2013right path of `[\u03b1, \u03b2] \u00d7 [c, d]` meets the open cluster of `x_B` inside the overlap. For `c < d` this fails for some `\u03c9`; see `CrossingSeparatesH` for the version that only concerns bottom points `x_B` joined to the top.","title":"Every overlap cluster meets every horizontal crossing"},{"data_url":"scopes/Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu.json","id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu","kind":"scope","navigation":"module","parent":20413,"result":false,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/index.html","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu","status":"fully_proved","title":"The square-root trick and gluing crossings through an overlap"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/rsw-sqrt-trick","kind":"node","lean":"StatMech.Universality.rsw_sqrt_trick","parent":20435,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/index.html#rsw-sqrt-trick","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu","status":"fully_proved","summary":"The square-root trick: for a positively associated probability measure and increasing measurable `A`, `B`, `1 - \u221a(1 - \u03bc(A \u222a B)) \u2264 max(\u03bc(A), \u03bc(B))`.","title":"Square-root trick"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/overlap-connection","kind":"node","lean":"StatMech.Universality.OverlapConnection","parent":20435,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/index.html#overlap-connection","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu","status":"fully_proved","summary":"An open path inside the overlap `rect m m' c d` from `p` to `q`.","title":"Open connection inside an overlap rectangle"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/overlap-connection-event","kind":"node","lean":"StatMech.Universality.overlapConnectionEvent","parent":20435,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/index.html#overlap-connection-event","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu","status":"fully_proved","summary":"The event `OverlapConnection`.","title":"Connection event inside an overlap rectangle"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/measurable-set-overlap-connection-event","kind":"node","lean":"StatMech.Universality.measurableSet_overlapConnectionEvent","parent":20435,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/index.html#measurable-set-overlap-connection-event","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu","status":"fully_proved","summary":"A connection event inside a rectangular overlap is measurable.","title":"Measurability of the overlap connection event"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/hvintersection-property","kind":"node","lean":"StatMech.Universality.HVIntersectionProperty","parent":20435,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu/index.html#hvintersection-property","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/the-square-root-trick-and-gluing-crossings-throu","status":"fully_proved","summary":"For the configuration `\u03c9`: horizontal crossings of `[a, m'] \u00d7 [c, d]` and of `[m, b] \u00d7 [c, d]` together with a vertical crossing of the overlap `[m, m'] \u00d7 [c, d]` imply a horizontal crossing of `[a, b] \u00d7 [c, d]`.","title":"Gluing horizontal crossings through a vertical overlap crossing"},{"data_url":"scopes/Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column.json","id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column","kind":"scope","navigation":"module","parent":20413,"result":false,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column/index.html","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column","status":"fully_proved","title":"Left\u2013right paths cross every column"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column/proj-hom","kind":"node","lean":"StatMech.Universality.projHom","parent":20441,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column/index.html#proj-hom","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column","status":"fully_proved","summary":"The graph homomorphism from the open subgraph induced on `R` to the lattice `\u2124\u00b2` (forgetting `R`).","title":"Projection of an induced open subgraph to the lattice"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column/lift-walk","kind":"node","lean":"StatMech.Universality.liftWalk","parent":20441,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column/index.html#lift-walk","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column","status":"fully_proved","summary":"A walk in the open subgraph induced on `R`, mapped to a walk in the lattice `\u2124\u00b2`.","title":"Walk in an induced open subgraph as a lattice walk"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column/col-leq","kind":"node","lean":"StatMech.Universality.colLeq","parent":20441,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column/index.html#col-leq","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/left-right-paths-cross-every-column","status":"fully_proved","summary":"The half-plane `{z \u2208 \u2124\u00b2 : z_0 \u2264 k}`.","title":"Half-plane left of a column"},{"data_url":"scopes/Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta.json","id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta","kind":"scope","navigation":"module","parent":20413,"result":false,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta/index.html","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta","status":"fully_proved","title":"Vertical crossings separate the sides of a rectangle"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta/crossing-separates-h","kind":"node","lean":"StatMech.Universality.CrossingSeparatesH","parent":20445,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta/index.html#crossing-separates-h","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta","status":"fully_proved","summary":"For the configuration `\u03c9`: whenever `x_B` on the bottom side of the overlap `[m, m'] \u00d7 [c, d]` is joined to the top side inside the overlap, every open left\u2013right path of `[\u03b1, \u03b2] \u00d7 [c, d]` meets the open cluster of `x_B` inside the overlap.","title":"Vertical overlap crossings meet every horizontal crossing"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta/crossing-arc-sides","kind":"node","lean":"StatMech.Universality.CrossingArcSides","parent":20445,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta/index.html#crossing-arc-sides","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta","status":"fully_proved","summary":"For the configuration `\u03c9`: whenever `x_B` on the bottom of the overlap is joined to its top, there is a set `L` containing the left side of `[\u03b1, \u03b2] \u00d7 [c, d]` and disjoint from its right side such that every lattice edge of the rectangle crossing the boundary of `L` has an endpoint in the open cluster of `x_B` inside the overlap.","title":"Vertical overlap crossings separate the sides of a rectangle"},{"id":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta/rsw-strip-glue-of-crossing-arc-sides","kind":"node","lean":"StatMech.Universality.rsw_strip_glue_of_crossingArcSides","parent":20445,"result":true,"route":"../roadmap/Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta/index.html#rsw-strip-glue-of-crossing-arc-sides","scope":"Probability/percolation/crossings/box-crossing-property-for-positively-associated/vertical-crossings-separate-the-sides-of-a-recta","status":"fully_proved","summary":"RSW gluing: for a positively associated probability measure `\u03bc`, if `CrossingArcSides` holds for every configuration, then `\u03bc(H_{[a, m']}) \u03bc(V_{[m, m']}) \u03bc(H_{[m, b]}) \u2264 \u03bc(H_{[a, b]})`.","title":"RSW gluing from side separation"},{"data_url":"scopes/Probability/correlation-inequalities.json","id":"Probability/correlation-inequalities","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/correlation-inequalities/index.html","scope":"Probability/correlation-inequalities","status":"can_prove","summary":"Inequalities comparing probabilities of monotone or disjointly occurring events under product measures, and stochastic orderings and monotone couplings of measures.","title":"Correlation inequalities and stochastic domination"},{"data_url":"scopes/Probability/correlation-inequalities/harris-fkg-bk.json","id":"Probability/correlation-inequalities/harris-fkg-bk","kind":"scope","parent":20449,"result":false,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/index.html","scope":"Probability/correlation-inequalities/harris-fkg-bk","status":"fully_proved","summary":"Increasing events, the Harris and FKG inequalities, Holley's inequality, disjoint occurrence and the van den Berg\u2013Kesten inequality (finite, countable and inhomogeneous), pivotal coordinates and Russo's formula on product spaces.","title":"Harris, FKG and BK inequalities"},{"data_url":"scopes/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition.json","id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","kind":"scope","navigation":"module","parent":20450,"result":false,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","title":"Increasing events and the FKG lattice condition"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/is-increasing","kind":"node","lean":"StatMech.IsIncreasing","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#is-increasing","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"An event `A \u2286 ConfigSpace E` is *increasing* if it is an upper set for the coordinatewise order: `\u03c9 \u2208 A` and `\u03c9 \u2264 \u03c9'` imply `\u03c9' \u2208 A`. This is `IsUpperSet A`.","title":"Increasing event"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/is-decreasing","kind":"node","lean":"StatMech.IsDecreasing","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#is-decreasing","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"An event `A \u2286 ConfigSpace E` is *decreasing* if it is a lower set for the coordinatewise order. This is `IsLowerSet A`.","title":"Decreasing event"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/compl","kind":"node","lean":"StatMech.IsIncreasing.compl","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#compl","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"The complement of an increasing event is decreasing (`IsUpperSet.compl`).","title":"Complement of an increasing event is decreasing"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/compl-2","kind":"node","lean":"StatMech.IsDecreasing.compl","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#compl-2","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"The complement of a decreasing event is increasing (`IsLowerSet.compl`).","title":"Complement of a decreasing event is increasing"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/inter","kind":"node","lean":"StatMech.IsIncreasing.inter","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#inter","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"The intersection of two increasing events is increasing (`IsUpperSet.inter`).","title":"Intersection of two increasing events is increasing"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/union","kind":"node","lean":"StatMech.IsIncreasing.union","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#union","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"The union of two increasing events is increasing (`IsUpperSet.union`).","title":"Union of two increasing events is increasing"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/is-increasing-i-inter","kind":"node","lean":"StatMech.isIncreasing_iInter","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#is-increasing-i-inter","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"An intersection of increasing events is increasing (`isUpperSet_iInter`).","title":"Arbitrary intersections of increasing events are increasing"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/indicator-monotone","kind":"node","lean":"StatMech.IsIncreasing.indicator_monotone","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#indicator-monotone","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"The real indicator function of an increasing event is monotone.","title":"Indicator of an increasing event is monotone"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/is-increasing-iff-indicator-monotone","kind":"node","lean":"StatMech.isIncreasing_iff_indicator_monotone","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#is-increasing-iff-indicator-monotone","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"An event is increasing iff its real indicator function is monotone. See also `Set.monotone_mem`, the analogous statement for the `Prop`-valued membership predicate.","title":"An event is increasing iff its indicator is monotone"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/fkglattice-condition","kind":"node","lean":"StatMech.FKGLatticeCondition","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#fkglattice-condition","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"The *FKG lattice condition* on a weight `\u03c0 : ConfigSpace E \u2192 \u211d`: `\u03c0 \u03c9 * \u03c0 \u03c9' \u2264 \u03c0 (\u03c9 \u2294 \u03c9') * \u03c0 (\u03c9 \u2293 \u03c9')` for all configurations `\u03c9 \u03c9'`. For a positive weight this is log-supermodularity. Mathlib's `fkg` and `holley` state it in the form `\u03c0 a * \u03c0 b \u2264 \u03c0 (a \u2293 b) * \u03c0 (a \u2294 b)`.","title":"FKG lattice condition"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/of-log-modular","kind":"node","lean":"StatMech.FKGLatticeCondition.of_logModular","parent":20451,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition/index.html#of-log-modular","scope":"Probability/correlation-inequalities/harris-fkg-bk/increasing-events-and-the-fkg-lattice-condition","status":"fully_proved","summary":"A log-modular weight, `\u03c0 \u03c9 * \u03c0 \u03c9' = \u03c0 (\u03c9 \u2294 \u03c9') * \u03c0 (\u03c9 \u2293 \u03c9')`, satisfies the FKG lattice condition.","title":"Log-modular weights satisfy the FKG lattice condition"},{"data_url":"scopes/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities.json","id":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","kind":"scope","navigation":"module","parent":20450,"result":false,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/index.html","scope":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","status":"fully_proved","title":"FKG and Holley inequalities"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/fkg-inequality","kind":"node","lean":"StatMech.fkg_inequality","parent":20463,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/index.html#fkg-inequality","scope":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","status":"fully_proved","summary":"The FKG inequality (Fortuin, Kasteleyn, Ginibre 1971). Let `\u03c0` be a probability weight on `ConfigSpace E` (`\u03c0 \u2265 0`, `\u2211 \u03c0 = 1`) satisfying the FKG lattice condition. Then any two monotone functions `f g : ConfigSpace E \u2192 \u211d` are positively correlated: `E_\u03c0[f] * E_\u03c0[g] \u2264 E_\u03c0[f * g]`.","title":"FKG inequality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/fkg-inequality-events","kind":"node","lean":"StatMech.fkg_inequality_events","parent":20463,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/index.html#fkg-inequality-events","scope":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","status":"fully_proved","summary":"The FKG inequality for events: under a probability weight `\u03c0` satisfying the FKG lattice condition, two increasing events are positively correlated, `\u03c0(A) * \u03c0(B) \u2264 \u03c0(A \u2229 B)`.","title":"FKG inequality for increasing events"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/holley-dominates","kind":"node","lean":"StatMech.holley_dominates","parent":20463,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/index.html#holley-dominates","scope":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","status":"fully_proved","summary":"The Holley inequality for increasing events (Holley 1974). If `\u03bc\u2081, \u03bc\u2082 \u2265 0` have equal total mass and satisfy `\u03bc\u2081 a * \u03bc\u2082 b \u2264 \u03bc\u2081 (a \u2293 b) * \u03bc\u2082 (a \u2294 b)`, then `\u03bc\u2081(A) \u2264 \u03bc\u2082(A)` for every increasing event `A`. This is Mathlib's `holley` applied to the indicator of `A`.","title":"Holley inequality for increasing events"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/meas-of-mass","kind":"node","lean":"StatMech.FK.measOfMass","parent":20463,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/index.html#meas-of-mass","scope":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","status":"fully_proved","summary":"The finite measure `\u2211_\u03c9 \u03bc(\u03c9) \u03b4_\u03c9` on `ConfigSpace E` with point masses `\u03bc(\u03c9)` (negative masses are truncated to `0`). For a normalized nonnegative `\u03bc` it is the measure of the `PMF` `PMF.ofFintype \u03bc`; see `measOfMass_eq_toMeasure`.","title":"Finite measure with prescribed point masses"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/meas-of-mass-eq-to-measure","kind":"node","lean":"StatMech.FK.measOfMass_eq_toMeasure","parent":20463,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/index.html#meas-of-mass-eq-to-measure","scope":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","status":"fully_proved","summary":"For a normalized nonnegative mass function, `measOfMass \u03bc` is the measure of the `PMF` `PMF.ofFintype (ENNReal.ofReal \u2218 \u03bc)`.","title":"Normalized point masses give the probability mass function's measure"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/holley-stochastically-dominated","kind":"node","lean":"StatMech.FK.holley_stochasticallyDominated","parent":20463,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/index.html#holley-stochastically-dominated","scope":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","status":"fully_proved","summary":"If `\u03bc\u2081, \u03bc\u2082 \u2265 0` on `{0,1}^E` have equal total mass and satisfy the Holley lattice condition `\u03bc\u2081(a) \u03bc\u2082(b) \u2264 \u03bc\u2081(a \u2227 b) \u03bc\u2082(a \u2228 b)`, then `measOfMass \u03bc\u2081` is stochastically dominated by `measOfMass \u03bc\u2082`. Strict positivity, assumed in Holley's paper, is not needed. See R. Holley, *Remarks on the FKG inequalities*, Comm. Math. Phys. 36 (1974); the real-valued form is `StatMech.holley_dominates`, from Mathlib's `holley`.","title":"Holley's inequality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/holley-exists-monotone-coupling","kind":"node","lean":"StatMech.FK.holley_exists_monotoneCoupling","parent":20463,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities/index.html#holley-exists-monotone-coupling","scope":"Probability/correlation-inequalities/harris-fkg-bk/fkg-and-holley-inequalities","status":"fully_proved","summary":"If `\u03bc\u2081, \u03bc\u2082` are strictly positive probability mass functions on `{0,1}^E` satisfying the Holley lattice condition `\u03bc\u2081(a) \u03bc\u2082(b) \u2264 \u03bc\u2081(a \u2227 b) \u03bc\u2082(a \u2228 b)`, there is a finite measure on pairs with marginals `measOfMass \u03bc\u2081` and `measOfMass \u03bc\u2082`, concentrated on `{(a, b) | a \u2264 b}`. See R. Holley, *Remarks on the FKG inequalities*, Comm. Math. Phys. 36 (1974), and G. Grimmett, *The Random-Cluster Model* (2006), Chapter 2.","title":"Holley's coupling theorem"},{"data_url":"scopes/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates.json","id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","kind":"scope","navigation":"module","parent":20450,"result":false,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","title":"Russo's formula and pivotal coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/set-open","kind":"node","lean":"StatMech.setOpen","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#set-open","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"`setOpen e \u03c9` is `\u03c9` with coordinate `e` set to `true` (open).","title":"Opening a coordinate of a configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/set-closed","kind":"node","lean":"StatMech.setClosed","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#set-closed","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"`setClosed e \u03c9` is `\u03c9` with coordinate `e` set to `false` (closed).","title":"Closing a coordinate of a configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/set-closed-le-set-open","kind":"node","lean":"StatMech.setClosed_le_setOpen","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#set-closed-le-set-open","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Closing a coordinate gives a smaller configuration than opening it.","title":"Closed configuration lies below the opened one"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/set-open-congr","kind":"node","lean":"StatMech.setOpen_congr","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#set-open-congr","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"`setOpen e \u03c9` depends only on the values of `\u03c9` off `e`.","title":"Opening a coordinate depends only on the other coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/set-closed-congr","kind":"node","lean":"StatMech.setClosed_congr","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#set-closed-congr","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"`setClosed e \u03c9` depends only on the values of `\u03c9` off `e`.","title":"Closing a coordinate depends only on the other coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/is-pivotal","kind":"node","lean":"StatMech.IsPivotal","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#is-pivotal","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The coordinate `e` is *pivotal* for the event `A` at `\u03c9` if exactly one of `setOpen e \u03c9` and `setClosed e \u03c9` lies in `A`. The condition does not depend on `\u03c9 e`.","title":"Pivotal coordinate for an event"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/is-pivotal-comm","kind":"node","lean":"StatMech.isPivotal_comm","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#is-pivotal-comm","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Pivotality with the two sides swapped.","title":"Symmetric form of pivotality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/is-pivotal-congr","kind":"node","lean":"StatMech.isPivotal_congr","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#is-pivotal-congr","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Pivotality of `e` depends only on the values of `\u03c9` off `e`.","title":"Pivotality depends only on the other coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/is-pivotal-set-open","kind":"node","lean":"StatMech.isPivotal_setOpen","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#is-pivotal-set-open","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Opening `e` does not change whether `e` is pivotal.","title":"Opening a coordinate preserves its pivotality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/is-pivotal-set-closed","kind":"node","lean":"StatMech.isPivotal_setClosed","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#is-pivotal-set-closed","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Closing `e` does not change whether `e` is pivotal.","title":"Closing a coordinate preserves its pivotality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/is-pivotal-iff-of-is-increasing","kind":"node","lean":"StatMech.isPivotal_iff_of_isIncreasing","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#is-pivotal-iff-of-is-increasing","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"For an increasing event `A`, the coordinate `e` is pivotal at `\u03c9` iff `setClosed e \u03c9 \u2209 A` and `setOpen e \u03c9 \u2208 A`.","title":"Pivotal coordinates of an increasing event"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/edge-weight","kind":"node","lean":"StatMech.edgeWeight","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#edge-weight","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The weight of coordinate `e` in configuration `\u03c9` with parameter `p`: `p` if `\u03c9 e` is open (`true`), `1 - p` otherwise.","title":"Bernoulli weight of a single coordinate"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/weight-off","kind":"node","lean":"StatMech.weightOff","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#weight-off","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The product of the coordinate weights over all coordinates except `e`.","title":"Product of coordinate weights off one coordinate"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/config-weight","kind":"node","lean":"StatMech.configWeight","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#config-weight","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The product weight `\u220f e, edgeWeight p \u03c9 e` of a configuration; for `p \u2208 [0, 1]` the probability of `\u03c9` under independent Bernoulli(`p`) coordinates.","title":"Bernoulli product weight of a configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/prob","kind":"node","lean":"StatMech.prob","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#prob","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The weight `\u2211 \u03c9 \u2208 A, configWeight p \u03c9` of an event `A`; for `p \u2208 [0, 1]` its probability under independent Bernoulli(`p`) coordinates.","title":"Bernoulli weight of an event"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/pivotal-prob","kind":"node","lean":"StatMech.pivotalProb","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#pivotal-prob","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The weight of the event that coordinate `e` is pivotal for `A`.","title":"Weight of the event that a coordinate is pivotal"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/config-weight-eq-pweight","kind":"node","lean":"StatMech.configWeight_eq_pweight","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#config-weight-eq-pweight","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"`configWeight` is the product weight `StatMech.pweight` for Bernoulli coordinate weights.","title":"Configuration weight as a product weight"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/config-weight-nonneg","kind":"node","lean":"StatMech.configWeight_nonneg","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#config-weight-nonneg","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The product weight `configWeight p \u03c9` is nonnegative for `p \u2208 [0, 1]`.","title":"Nonnegativity of the configuration weight"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/pivotal-prob-nonneg","kind":"node","lean":"StatMech.pivotalProb_nonneg","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#pivotal-prob-nonneg","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The pivotal weight `pivotalProb p A e` is nonnegative for `p \u2208 [0, 1]`.","title":"Nonnegativity of the pivotal weight"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/prob-eq-wprob","kind":"node","lean":"StatMech.prob_eq_wprob","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#prob-eq-wprob","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"`prob` is `StatMech.wprob` for Bernoulli coordinate weights.","title":"Event weight as a product-weight probability"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/config-weight-eq","kind":"node","lean":"StatMech.configWeight_eq","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#config-weight-eq","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The product weight splits as the weight of `e` times the weight of the other coordinates.","title":"Configuration weight splits at one coordinate"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/weight-off-congr","kind":"node","lean":"StatMech.weightOff_congr","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#weight-off-congr","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"`weightOff p e \u03c9` depends only on the values of `\u03c9` off `e`.","title":"Off-coordinate weight depends only on the other coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/has-deriv-at-edge-weight","kind":"node","lean":"StatMech.hasDerivAt_edgeWeight","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#has-deriv-at-edge-weight","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The derivative in `p` of a coordinate weight is `1` if the coordinate is open, `-1` otherwise.","title":"Derivative of a single coordinate weight"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/has-deriv-at-config-weight","kind":"node","lean":"StatMech.hasDerivAt_configWeight","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#has-deriv-at-config-weight","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The derivative in `p` of the product weight (product rule).","title":"Derivative of the configuration weight"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/has-deriv-at-prob","kind":"node","lean":"StatMech.hasDerivAt_prob","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#has-deriv-at-prob","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The derivative in `p` of `prob p A`, before grouping by pivotal coordinates.","title":"Derivative of the event weight before grouping"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/set-open-symm","kind":"node","lean":"StatMech.setOpen_symm","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#set-open-symm","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Opening `e` in the configuration with value `b` at `e` and `\u03b7` elsewhere.","title":"Opening a coordinate of a split configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/set-closed-symm","kind":"node","lean":"StatMech.setClosed_symm","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#set-closed-symm","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Closing `e` in the configuration with value `b` at `e` and `\u03b7` elsewhere.","title":"Closing a coordinate of a split configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/russo-per-eta","kind":"node","lean":"StatMech.russo_per_eta","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#russo-per-eta","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Russo's formula on one fiber `{\u03c9t, \u03c9f}` of configurations differing only at `e`.","title":"Russo's formula on a single fiber"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/russo-per-edge","kind":"node","lean":"StatMech.russo_per_edge","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#russo-per-edge","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"The contribution of coordinate `e` to the derivative of `prob p A` is `pivotalProb p A e`, for increasing `A`.","title":"Contribution of one coordinate to Russo's formula"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/has-deriv-at-prob-eq-sum-pivotal-prob","kind":"node","lean":"StatMech.hasDerivAt_prob_eq_sum_pivotalProb","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#has-deriv-at-prob-eq-sum-pivotal-prob","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Russo's formula (Russo 1981; Margulis 1974): for an increasing event `A` on a finite set of coordinates, `p \u21a6 prob p A` has derivative `\u2211 e, pivotalProb p A e` at every `p`.","title":"Russo's formula"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/deriv-prob-eq-sum-pivotal-prob","kind":"node","lean":"StatMech.deriv_prob_eq_sum_pivotalProb","parent":20471,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates/index.html#deriv-prob-eq-sum-pivotal-prob","scope":"Probability/correlation-inequalities/harris-fkg-bk/russo-s-formula-and-pivotal-coordinates","status":"fully_proved","summary":"Russo's formula, `deriv` form: `d/dp prob p A = \u2211 e, pivotalProb p A e` for increasing `A`.","title":"Russo's formula"},{"data_url":"scopes/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality.json","id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","kind":"scope","navigation":"module","parent":20450,"result":false,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","title":"Harris inequality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/harris-inequality","kind":"node","lean":"StatMech.harris_inequality","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#harris-inequality","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"The Harris inequality (Harris 1960): under the product Bernoulli(`p`) measure on a finite configuration space, increasing events are positively correlated, `P_p(A) * P_p(B) \u2264 P_p(A \u2229 B)`. No measurability hypothesis is needed since `E` is finite. See also `IsUpperSet.le_card_inter_finset` (Harris\u2013Kleitman, counting form).","title":"Harris inequality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-event-depends-on-of-indicator","kind":"node","lean":"StatMech.ih_event_dependsOn_of_indicator","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-event-depends-on-of-indicator","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"If the indicator of `A` depends only on the coordinates in `S` (Mathlib's `DependsOn`), so does the event `A` (`StatMech.DependsOn`).","title":"An event depends on the coordinates its indicator depends on"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-indicator-depends-on-of-event","kind":"node","lean":"StatMech.ih_indicator_dependsOn_of_event","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-indicator-depends-on-of-event","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"If the event `A` depends only on `S`, so does its indicator function.","title":"Indicator of an event depends on the coordinates the event depends on"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-depends-on-inter","kind":"node","lean":"StatMech.ih_dependsOn_inter","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-depends-on-inter","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"Events depending only on `S` are closed under intersection.","title":"Events depending on a coordinate set are closed under intersection"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-depends-on-i-union","kind":"node","lean":"StatMech.ih_dependsOn_iUnion","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-depends-on-i-union","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"Events depending only on `S` are closed under arbitrary unions.","title":"Events depending on a coordinate set are closed under unions"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-fill","kind":"node","lean":"StatMech.ih_fill","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-fill","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"The configuration on `E` equal to `\u03b7` on the finite set `F` and closed (`false`) off `F`.","title":"Filling a finite configuration with closed coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-fill-mono","kind":"node","lean":"StatMech.ih_fill_mono","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-fill-mono","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"`ih_fill F` is monotone.","title":"Monotonicity of the closed filling map"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-section","kind":"node","lean":"StatMech.ih_section","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-section","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"The section of an event `A` on `F`: the configurations `\u03b7` on `F` with `ih_fill F \u03b7 \u2208 A`.","title":"Section of an event on a finite coordinate set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-section-is-increasing","kind":"node","lean":"StatMech.ih_section_isIncreasing","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-section-is-increasing","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"The section of an increasing event is increasing.","title":"Section of an increasing event is increasing"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-eq-cylinder-of-depends-on","kind":"node","lean":"StatMech.ih_eq_cylinder_of_dependsOn","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-eq-cylinder-of-depends-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"An event depending only on the coordinates in `F` is the cylinder over its section on `F`.","title":"Finitely dependent event is a cylinder over its section"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-harris-of-finite-depends-on","kind":"node","lean":"StatMech.ih_harris_of_finite_dependsOn","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-harris-of-finite-depends-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"The Harris inequality on a countable configuration space for increasing events that depend on finitely many coordinates: `P_p(A) * P_p(B) \u2264 P_p(A \u2229 B)`.","title":"Harris inequality for finitely dependent increasing events"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-harris-i-inter-of-antitone","kind":"node","lean":"StatMech.ih_harris_iInter_of_antitone","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-harris-i-inter-of-antitone","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"Positive correlation passes to limits of decreasing sequences of measurable events: if `\u03bc(A n) \u03bc(B n) \u2264 \u03bc(A n \u2229 B n)` for all `n`, the same holds for `\u22c2 n, A n` and `\u22c2 n, B n`.","title":"Positive correlation passes to decreasing intersections"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/ih-tendsto-measure-real-i-inter","kind":"node","lean":"StatMech.ih_tendsto_measureReal_iInter","parent":20503,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/harris-inequality/index.html#ih-tendsto-measure-real-i-inter","scope":"Probability/correlation-inequalities/harris-fkg-bk/harris-inequality","status":"fully_proved","summary":"Continuity from above of a finite measure, real-valued form: `\u03bc.real (A n)` tends to `\u03bc.real (\u22c2 n, A n)` for a decreasing sequence of measurable sets.","title":"Continuity from above of a finite measure, real-valued form"},{"data_url":"scopes/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality.json","id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","kind":"scope","navigation":"module","parent":20450,"result":false,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","title":"Van den Berg\u2013Kesten inequality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/agree-on","kind":"node","lean":"StatMech.agreeOn","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#agree-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`agreeOn K \u03c9 \u03c9'`: the configurations `\u03c9` and `\u03c9'` take the same value at every coordinate of `K`. This is `Set.EqOn \u03c9' \u03c9 K` (see `StatMech.agreeOn_iff_eqOn`).","title":"Agreement of configurations on a set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/agree-on-iff-eq-on","kind":"node","lean":"StatMech.agreeOn_iff_eqOn","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#agree-on-iff-eq-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`agreeOn` is the pointwise form of `Set.EqOn`.","title":"Agreement on a set as pointwise equality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/agree-on-refl","kind":"node","lean":"StatMech.agreeOn_refl","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#agree-on-refl","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Every configuration agrees with itself (`Set.eqOn_refl`).","title":"Reflexivity of agreement on a set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/agree-on-symm","kind":"node","lean":"StatMech.agreeOn_symm","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#agree-on-symm","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`agreeOn` is symmetric (`Set.EqOn.symm`).","title":"Symmetry of agreement on a set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/agree-on-trans","kind":"node","lean":"StatMech.agreeOn_trans","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#agree-on-trans","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`agreeOn` is transitive (`Set.EqOn.trans`).","title":"Transitivity of agreement on a set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/agree-on-mono","kind":"node","lean":"StatMech.agreeOn_mono","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#agree-on-mono","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Agreement on a set implies agreement on any subset (`Set.EqOn.mono`).","title":"Agreement on a set restricts to subsets"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/occurs-on","kind":"node","lean":"StatMech.OccursOn","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#occurs-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`OccursOn A K \u03c9`: the event `A` occurs on the coordinate set `K` at `\u03c9`, meaning that every configuration agreeing with `\u03c9` on `K` lies in `A`.","title":"Occurrence of an event on a coordinate set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/mem-self","kind":"node","lean":"StatMech.OccursOn.mem_self","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#mem-self","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"If `A` occurs on some `K` at `\u03c9`, then `\u03c9 \u2208 A`.","title":"An event occurring on a set contains the configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/mono","kind":"node","lean":"StatMech.OccursOn.mono","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#mono","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Occurrence on `K` implies occurrence on any larger coordinate set.","title":"Occurrence is monotone in the coordinate set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/mono-event","kind":"node","lean":"StatMech.OccursOn.mono_event","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#mono-event","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Occurrence of `A` on `K` implies occurrence of any larger event `B \u2287 A` on `K`.","title":"Occurrence is monotone in the event"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/disjoint-occurrence","kind":"node","lean":"StatMech.disjointOccurrence","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#disjoint-occurrence","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The *disjoint occurrence* `A \u25a1 B`: the configurations `\u03c9` for which there are disjoint coordinate sets `K` and `L` such that `A` occurs on `K` at `\u03c9` and `B` occurs on `L` at `\u03c9`.","title":"Disjoint occurrence of events"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/mem-disjoint-occurrence","kind":"node","lean":"StatMech.mem_disjointOccurrence","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#mem-disjoint-occurrence","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Membership in `A \u25a1 B` unfolded.","title":"Membership in a disjoint occurrence"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/disjoint-occurrence-notation","kind":"node","lean":"StatMech.disjointOccurrence_notation","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#disjoint-occurrence-notation","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The notation `A \u25a1 B` unfolds to `disjointOccurrence A B`.","title":"Notation for disjoint occurrence"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/disjoint-occurrence-subset-inter","kind":"node","lean":"StatMech.disjointOccurrence_subset_inter","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#disjoint-occurrence-subset-inter","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Disjoint occurrence implies occurrence of both events: `A \u25a1 B \u2286 A \u2229 B`.","title":"Disjoint occurrence lies in the intersection"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/disjoint-occurrence-comm","kind":"node","lean":"StatMech.disjointOccurrence_comm","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#disjoint-occurrence-comm","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Disjoint occurrence is symmetric: `A \u25a1 B = B \u25a1 A`.","title":"Disjoint occurrence is symmetric"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/disjoint-occurrence-mono","kind":"node","lean":"StatMech.disjointOccurrence_mono","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#disjoint-occurrence-mono","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Disjoint occurrence is monotone in both events.","title":"Disjoint occurrence is monotone in both events"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/mono-config","kind":"node","lean":"StatMech.OccursOn.mono_config","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#mono-config","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"If an increasing event `A` occurs on `K` at `\u03c9` and `\u03c9 \u2264 \u03c9'`, then `A` occurs on `K` at `\u03c9'`.","title":"Occurrence of an increasing event is monotone in the configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/disjoint-occurrence-is-increasing","kind":"node","lean":"StatMech.disjointOccurrence_isIncreasing","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#disjoint-occurrence-is-increasing","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The disjoint occurrence of two increasing events is increasing.","title":"Disjoint occurrence of increasing events is increasing"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/pweight","kind":"node","lean":"StatMech.pweight","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#pweight","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The product weight `\u220f x, \u03c6 x (\u03c9 x)` of a configuration `\u03c9`, where `\u03c6 x b` is the weight of value `b` at coordinate `x`.","title":"Product weight of a configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/pweight-nonneg","kind":"node","lean":"StatMech.pweight_nonneg","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#pweight-nonneg","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"A product of nonnegative coordinate weights is nonnegative.","title":"Nonnegativity of the product weight"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/wprob","kind":"node","lean":"StatMech.wprob","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#wprob","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The `pweight \u03c6`-mass `\u2211 \u03c9 \u2208 S, pweight \u03c6 \u03c9` of an event `S`. When `\u03c6 \u2265 0` and `\u03c6 x false + \u03c6 x true = 1` for all `x`, this is the probability of `S` under the product measure whose coordinate `x` has law `\u03c6 x`.","title":"Weighted probability of an event"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/wprob-mono","kind":"node","lean":"StatMech.wprob_mono","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#wprob-mono","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`wprob \u03c6` is monotone for nonnegative weights.","title":"Monotonicity of the weighted probability"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/sum-pweight-eq-one","kind":"node","lean":"StatMech.sum_pweight_eq_one","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#sum-pweight-eq-one","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"A product probability weight has total mass `1`.","title":"Product probability weight has total mass one"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/wprob-union-inter","kind":"node","lean":"StatMech.wprob_union_inter","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#wprob-union-inter","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Inclusion\u2013exclusion for `wprob`.","title":"Inclusion\u2013exclusion for the weighted probability"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/bk-algebra","kind":"node","lean":"StatMech.bk_algebra","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#bk-algebra","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The scalar inequality behind the induction step of the BK inequality: with `q = 1 - p`, `p, q \u2265 0`, if `c0 \u2264 m \u2264 a0 b0`, `c1 \u2264 a0 b1 + a1 b0 - m` and `c1 \u2264 a1 b1`, then `q c0 + p c1 \u2264 (q a0 + p a1) (q b0 + p b1)`.","title":"Scalar inequality behind the BK induction step"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/slice","kind":"node","lean":"StatMech.slice","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#slice","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The section of an event `C` on `Fin (n + 1)` at first coordinate `x`: the configurations `\u03c9'` on `Fin n` with `Fin.cons x \u03c9' \u2208 C`.","title":"Slice of an event at the first coordinate"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/slice-set","kind":"node","lean":"StatMech.sliceSet","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#slice-set","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The coordinate set `{j | j.succ \u2208 K}`: `K` with the first coordinate removed, shifted down.","title":"Coordinate set with the first coordinate removed"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/indicator-slice","kind":"node","lean":"StatMech.indicator_slice","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#indicator-slice","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The indicator of `S` at `Fin.cons x \u03c9'` is the indicator of the slice at `\u03c9'`.","title":"Indicator of an event at a prepended configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/pweight-cons","kind":"node","lean":"StatMech.pweight_cons","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#pweight-cons","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The product weight factors over the first coordinate.","title":"Product weight factors over the first coordinate"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/wprob-slice","kind":"node","lean":"StatMech.wprob_slice","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#wprob-slice","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Conditioning on the first coordinate: `wprob \u03c6 S = \u03c6 0 false * wprob \u03c8 (slice S false) + \u03c6 0 true * wprob \u03c8 (slice S true)`, where `\u03c8` is `\u03c6` on the remaining coordinates.","title":"Weighted probability by conditioning on the first coordinate"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/slice-is-increasing","kind":"node","lean":"StatMech.slice_isIncreasing","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#slice-is-increasing","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Slices of an increasing event are increasing.","title":"Slices of an increasing event are increasing"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/slice-false-subset-true","kind":"node","lean":"StatMech.slice_false_subset_true","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#slice-false-subset-true","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"For an increasing event, the `false` slice is contained in the `true` slice.","title":"False slice of an increasing event lies in the true slice"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/occurs-on-slice-of-false","kind":"node","lean":"StatMech.occursOn_slice_of_false","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#occurs-on-slice-of-false","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Occurrence on `K` at `Fin.cons false \u03c9'` passes to the `false` slice.","title":"Occurrence with first coordinate false passes to the false slice"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/occurs-on-slice-true","kind":"node","lean":"StatMech.occursOn_slice_true","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#occurs-on-slice-true","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Occurrence on `K` at `Fin.cons true \u03c9'` passes to the `true` slice.","title":"Occurrence with first coordinate true passes to the true slice"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/occurs-on-slice-false-notmem","kind":"node","lean":"StatMech.occursOn_slice_false_notmem","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#occurs-on-slice-false-notmem","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"If the witness set `K` avoids the first coordinate, occurrence on `K` passes to the `false` slice whatever the value of the first coordinate.","title":"Occurrence avoiding the first coordinate passes to the false slice"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/slice-set-disjoint","kind":"node","lean":"StatMech.sliceSet_disjoint","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#slice-set-disjoint","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`sliceSet` preserves disjointness.","title":"Removing the first coordinate preserves disjointness"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/slice-disjoint-occurrence-false","kind":"node","lean":"StatMech.slice_disjointOccurrence_false","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#slice-disjoint-occurrence-false","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The `false` slice of `A \u25a1 B` is contained in the disjoint occurrence of the `false` slices.","title":"False slice of a disjoint occurrence"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/slice-disjoint-occurrence-true","kind":"node","lean":"StatMech.slice_disjointOccurrence_true","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#slice-disjoint-occurrence-true","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The `true` slice of `A \u25a1 B` is contained in `(A\u2080 \u25a1 B\u2081) \u222a (A\u2081 \u25a1 B\u2080)`, where `A\u2080, A\u2081` are the slices of `A` (similarly for `B`): the first coordinate is used by at most one witness.","title":"True slice of a disjoint occurrence"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/occurs-on-subsingleton","kind":"node","lean":"StatMech.occursOn_subsingleton","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#occurs-on-subsingleton","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"On a one-point configuration space, `OccursOn A K \u03c9` reduces to `\u03c9 \u2208 A`.","title":"Occurrence on a one-point configuration space"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/disjoint-occurrence-subsingleton","kind":"node","lean":"StatMech.disjointOccurrence_subsingleton","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#disjoint-occurrence-subsingleton","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"On a one-point configuration space, `A \u25a1 B = A \u2229 B`.","title":"Disjoint occurrence on a one-point space is the intersection"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/bk-wprob","kind":"node","lean":"StatMech.bk_wprob","parent":20517,"reading_title":{"format":"inline-parts-v1","parts":[["text","BK inequality for product weights on "],["math","\\(n\\)"],["text"," ordered coordinates"]],"source":"BK inequality for product weights on $n$ ordered coordinates"},"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#bk-wprob","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The BK inequality on `Fin n` for a product probability weight `\u03c6`: `wprob \u03c6 (A \u25a1 B) \u2264 wprob \u03c6 A * wprob \u03c6 B` for increasing `A, B`.","title":"BK inequality for product weights on $n$ ordered coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/cfg-equiv","kind":"node","lean":"StatMech.cfgEquiv","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#cfg-equiv","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The bijection `\u03c9 \u21a6 \u03c9 \u2218 e` from configurations on `F` to configurations on `E` induced by `e : E \u2243 F`. This is `e.symm.arrowCongr (Equiv.refl Bool)` (`StatMech.cfgEquiv_eq_arrowCongr`).","title":"Relabeling configurations along a bijection"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/cfg-equiv-eq-arrow-congr","kind":"node","lean":"StatMech.cfgEquiv_eq_arrowCongr","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#cfg-equiv-eq-arrow-congr","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`cfgEquiv e` agrees with Mathlib's `Equiv.arrowCongr`.","title":"Configuration relabeling agrees with the library arrow equivalence"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/cfg-equiv-mono","kind":"node","lean":"StatMech.cfgEquiv_mono","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#cfg-equiv-mono","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`cfgEquiv e` is monotone.","title":"Monotonicity of configuration relabeling"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/agree-on-cfg-equiv","kind":"node","lean":"StatMech.agreeOn_cfgEquiv","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#agree-on-cfg-equiv","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Agreement on `e '' K` corresponds to agreement on `K` after transport by `cfgEquiv e`.","title":"Agreement on a set under configuration relabeling"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/occurs-on-cfg-equiv","kind":"node","lean":"StatMech.occursOn_cfgEquiv","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#occurs-on-cfg-equiv","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Occurrence is preserved by relabeling the coordinates.","title":"Occurrence is invariant under relabeling coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/preimage-disjoint-occurrence","kind":"node","lean":"StatMech.preimage_disjointOccurrence","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#preimage-disjoint-occurrence","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Disjoint occurrence commutes with relabeling the coordinates.","title":"Disjoint occurrence commutes with relabeling coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/pweight-transfer","kind":"node","lean":"StatMech.pweight_transfer","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#pweight-transfer","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The product weight is invariant under relabeling the coordinates.","title":"Product weight is invariant under relabeling coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/wprob-transfer","kind":"node","lean":"StatMech.wprob_transfer","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#wprob-transfer","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`wprob` is invariant under relabeling the coordinates.","title":"Weighted probability is invariant under relabeling coordinates"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/bk-wprob-general","kind":"node","lean":"StatMech.bk_wprob_general","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#bk-wprob-general","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The BK inequality for a product probability weight on a finite set `E`: if `\u03c6 \u2265 0` and `\u03c6 x false + \u03c6 x true = 1` for all `x`, then `wprob \u03c6 (A \u25a1 B) \u2264 wprob \u03c6 A * wprob \u03c6 B` for all increasing events `A, B`.","title":"BK inequality for product weights on a finite set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/bernoulli-measure-real-false-add-true","kind":"node","lean":"StatMech.bernoulliMeasure_real_false_add_true","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#bernoulli-measure-real-false-add-true","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The two point masses of the Bernoulli(`p`) measure add up to `1`.","title":"Bernoulli point masses sum to one"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/bernoulli-real-eq-wprob","kind":"node","lean":"StatMech.bernoulli_real_eq_wprob","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#bernoulli-real-eq-wprob","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The product Bernoulli measure of an event is its `wprob` for the Bernoulli marginals.","title":"Bernoulli product measure as a weighted probability"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/bk-inequality","kind":"node","lean":"StatMech.bk_inequality","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#bk-inequality","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The van den Berg\u2013Kesten (BK) inequality (van den Berg and Kesten 1985): under the product Bernoulli(`p`) measure on a finite configuration space, for increasing events `A` and `B`, `P_p(A \u25a1 B) \u2264 P_p(A) * P_p(B)`.","title":"Van den Berg\u2013Kesten inequality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/ibk-occurs-on-inter-finite","kind":"node","lean":"StatMech.ibk_occursOn_inter_finite","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#ibk-occurs-on-inter-finite","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"If `A` depends only on `F` and occurs on `K` at `\u03c9`, then it occurs on `K \u2229 F` at every configuration agreeing with `\u03c9` on `F`.","title":"Occurrence of a finitely dependent event within its window"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/ibk-disjoint-occurrence-depends-on","kind":"node","lean":"StatMech.ibk_disjointOccurrence_dependsOn","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#ibk-disjoint-occurrence-depends-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"If `A` and `B` depend only on the coordinates in `F`, so does `A \u25a1 B`.","title":"Disjoint occurrence preserves finite dependence"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/ibk-section-disjoint-occurrence-subset","kind":"node","lean":"StatMech.ibk_section_disjointOccurrence_subset","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#ibk-section-disjoint-occurrence-subset","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The section on `F` of `A \u25a1 B` is contained in the disjoint occurrence of the sections.","title":"Section of a disjoint occurrence"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/ibk-inequality-of-finite-depends-on","kind":"node","lean":"StatMech.ibk_inequality_of_finite_dependsOn","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#ibk-inequality-of-finite-depends-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The BK inequality on a countable configuration space: for increasing events `A, B` depending only on the coordinates in a finite set `F`, `P_p(A \u25a1 B) \u2264 P_p(A) * P_p(B)`.","title":"BK inequality on countable configuration spaces"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/ibk-triple-of-finite-depends-on","kind":"node","lean":"StatMech.ibk_triple_of_finite_dependsOn","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#ibk-triple-of-finite-depends-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The BK inequality for three increasing events depending on finitely many coordinates: `P_p(P \u25a1 (Q \u25a1 R)) \u2264 P_p(P) * P_p(Q) * P_p(R)`.","title":"BK inequality for three increasing events"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/ibk-criterion-of-finite-depends-on","kind":"node","lean":"StatMech.ibk_criterion_of_finite_dependsOn","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#ibk-criterion-of-finite-depends-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Union bound from the three-event BK inequality: if `C \u2286 \u22c3 i \u2208 s, P i \u25a1 (Q i \u25a1 R i)` with all events increasing and depending on a common finite window, then `P_p(C) \u2264 \u2211 i \u2208 s, P_p(P i) * P_p(Q i) * P_p(R i)`.","title":"Union bound from the three-event BK inequality"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/ibk-measure-mono-parameter","kind":"node","lean":"StatMech.ibk_measure_mono_parameter","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#ibk-measure-mono-parameter","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"Monotonicity in the parameter: for an increasing event `A` depending only on a finite set of coordinates and `p \u2264 q`, `P_p(A) \u2264 P_q(A)`. The proof uses Russo's formula.","title":"Increasing-event probabilities are monotone in the parameter"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/inhom-bernoulli-product-measure","kind":"node","lean":"StatMech.inhomBernoulliProductMeasure","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#inhom-bernoulli-product-measure","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The product measure on `ConfigSpace E` under which coordinate `e` is open (`true`) with probability `p e`, independently.","title":"Inhomogeneous Bernoulli product measure"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/inhom-bernoulli-product-measure-const","kind":"node","lean":"StatMech.inhomBernoulliProductMeasure_const","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#inhom-bernoulli-product-measure-const","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"For a constant parameter, the inhomogeneous product is `bernoulliProductMeasure`.","title":"Inhomogeneous Bernoulli product with constant parameter"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/inhom-bk-inequality","kind":"node","lean":"StatMech.inhom_bk_inequality","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#inhom-bk-inequality","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The BK inequality for an inhomogeneous Bernoulli product measure on a finite set.","title":"BK inequality for inhomogeneous Bernoulli measures on a finite set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/inhom-bk-of-finite-depends-on","kind":"node","lean":"StatMech.inhom_bk_of_finite_dependsOn","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#inhom-bk-of-finite-depends-on","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The BK inequality for an inhomogeneous Bernoulli product measure on a countable set, for increasing events depending on finitely many coordinates.","title":"BK inequality for inhomogeneous measures on a countable set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/bk-criterion","kind":"node","lean":"StatMech.Sharpness.bk_criterion","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#bk-criterion","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The BK criterion: if `C \u2286 \u22c3_i P_i \u2218 Q_i \u2218 R_i` with increasing events, then `P_p(C) \u2264 \u2211_i P_p(P_i) P_p(Q_i) P_p(R_i)` for the Bernoulli product measure. Reference: J. van den Berg, H. Kesten, \"Inequalities with applications to percolation and reliability\", J. Appl. Probab. 22 (1985) 556\u2013569.","title":"BK criterion for Bernoulli product measures"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/open-sub","kind":"node","lean":"StatMech.Sharpness.openSub","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#open-sub","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The subgraph of `G` of edges open in the configuration `\u03c9`.","title":"Open subgraph of a configuration"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/conn-within","kind":"node","lean":"StatMech.Sharpness.ConnWithin","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#conn-within","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`x` and `y` are joined by an open path staying in `A`.","title":"Open connection between two vertices inside a set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/conn-to-set","kind":"node","lean":"StatMech.Sharpness.ConnToSet","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#conn-to-set","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"`u \u2208 A` is joined by an open path in `A` to some vertex of `A \u2229 B`.","title":"Open connection inside a set to a target set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/conn-event","kind":"node","lean":"StatMech.Sharpness.connEvent","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#conn-event","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The event that `u` is connected to `B` by an open path inside `A`.","title":"Connection event inside a set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/edge-open-event","kind":"node","lean":"StatMech.Sharpness.edgeOpenEvent","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#edge-open-event","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The event that the edge `{x, y}` is open.","title":"Event that an edge is open"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/shk-boundary-pairs","kind":"node","lean":"StatMech.Sharpness.shk_boundaryPairs","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#shk-boundary-pairs","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The ordered pairs `(x, y)` with `x \u2208 S`, `y \u2209 S` adjacent in `G`.","title":"Ordered boundary pairs of a set"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/shk-edge-law","kind":"node","lean":"StatMech.Sharpness.shk_edgeLaw","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#shk-edge-law","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The edge law of inhomogeneous bond percolation: open with probability `1 - exp(-\u03b2J_e)`, closed with probability `exp(-\u03b2J_e)`.","title":"Edge law of inhomogeneous bond percolation"},{"id":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/shk-first-exit-support","kind":"node","lean":"StatMech.Sharpness.shk_firstExitSupport","parent":20517,"result":true,"route":"../roadmap/Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality/index.html#shk-first-exit-support","scope":"Probability/correlation-inequalities/harris-fkg-bk/van-den-berg-kesten-inequality","status":"fully_proved","summary":"The edges inside `S`, the boundary edges of `S`, and the edges inside the window `\u039b`: the support of the events in the first-exit decomposition.","title":"Support of the first-exit decomposition events"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality.json","id":"Probability/correlation-inequalities/reimer-inequality","kind":"scope","parent":20449,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/index.html","scope":"Probability/correlation-inequalities/reimer-inequality","status":"can_prove","summary":"Reimer's inequality for arbitrary events and its proof machinery: configurations as finsets, down-compression, doubled ground sets, disjoint-pair counts and weighted pair sums, and special and mixed-monotone cases.","title":"Reimer's inequality"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-inequality","kind":"node","lean":"MathlibExt.Probability.ReimerInequalityWanted.reimer_inequality","parent":20591,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/index.html#reimer-inequality","scope":"Probability/correlation-inequalities/reimer-inequality","status":"can_prove","summary":"Reimer's inequality (van den Berg\u2013Kesten conjecture). For every `n`, every product probability weight `\u03c6` on `{0, 1}^n` (`\u03c6 i b \u2265 0`, `\u03c6 i false + \u03c6 i true = 1`) and all events `A`, `B \u2286 {0, 1}^n`, `P(A \u25a1 B) \u2264 P(A) P(B)`, where `A \u25a1 B` is the event that `A` and `B` occur on disjoint sets of coordinates.","title":"Reimer's inequality"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-wprob-core","kind":"node","lean":"MathlibExt.Probability.ReimerInequalityWanted.reimer_wprob_core","parent":20591,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/index.html#reimer-wprob-core","scope":"Probability/correlation-inequalities/reimer-inequality","status":"can_prove","summary":"Reimer's inequality in the form of the ported hypothesis `StatMech.ReimerWprobCore` (definitionally `reimer_inequality`).","title":"Reimer's inequality, ported hypothesis form"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the.json","id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","kind":"scope","navigation":"module","parent":20591,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","title":"Reimer's inequality: statement, reductions, and the monotone cases"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/reimer-wprob-core","kind":"node","lean":"StatMech.ReimerWprobCore","parent":20594,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html#reimer-wprob-core","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","summary":"Reimer's inequality (the van den Berg\u2013Kesten conjecture, proved by Reimer 2000) for product probability weights on `Fin n`: for every `n`, every weight `\u03c6` with `\u03c6 \u2265 0` and `\u03c6 i false + \u03c6 i true = 1`, and all events `A, B`, `wprob \u03c6 (A \u25a1 B) \u2264 wprob \u03c6 A * wprob \u03c6 B`.","title":"Reimer's inequality"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/reimer-wprob-general-of-core","kind":"node","lean":"StatMech.reimer_wprob_general_of_core","parent":20594,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html#reimer-wprob-general-of-core","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","summary":"Reimer's inequality for product probability weights on any finite set, assuming `ReimerWprobCore`.","title":"Reimer's inequality for product weights, assuming the core case"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/reimer-inequality-of-core","kind":"node","lean":"StatMech.reimer_inequality_of_core","parent":20594,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html#reimer-inequality-of-core","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","summary":"Reimer's inequality `P_p(A \u25a1 B) \u2264 P_p(A) * P_p(B)` for all events under the product Bernoulli measure on a finite set, assuming `ReimerWprobCore`.","title":"Reimer's inequality for Bernoulli measures, assuming the core case"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/reimer-wprob-zero","kind":"node","lean":"StatMech.reimer_wprob_zero","parent":20594,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html#reimer-wprob-zero","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","summary":"Reimer's inequality on the one-point configuration space `ConfigSpace (Fin 0)`.","title":"Reimer's inequality on the one-point configuration space"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/flip-cfg","kind":"node","lean":"StatMech.flipCfg","parent":20594,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html#flip-cfg","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","summary":"The configuration with every coordinate flipped, `x \u21a6 !(\u03c9 x)`.","title":"Flipping every coordinate of a configuration"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/flip-equiv","kind":"node","lean":"StatMech.flipEquiv","parent":20594,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html#flip-equiv","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","summary":"Flipping all coordinates, as an involutive equivalence of `ConfigSpace E`.","title":"Coordinate flip as an involutive equivalence"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/bk-wprob-decreasing","kind":"node","lean":"StatMech.bk_wprob_decreasing","parent":20594,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html#bk-wprob-decreasing","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","summary":"The BK inequality for two decreasing events under a product probability weight, obtained from `StatMech.bk_wprob_general` by flipping all coordinates.","title":"BK inequality for decreasing events under product weights"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/bk-inequality-decreasing","kind":"node","lean":"StatMech.bk_inequality_decreasing","parent":20594,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the/index.html#bk-inequality-decreasing","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-statement-reductions-and-the","status":"fully_proved","summary":"The BK inequality for decreasing events: under the product Bernoulli(`p`) measure on a finite set, `P_p(A \u25a1 B) \u2264 P_p(A) * P_p(B)` for decreasing `A, B`.","title":"BK inequality for decreasing events"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr.json","id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr","kind":"scope","navigation":"module","parent":20591,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/index.html","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr","status":"fully_proved","title":"Reimer's inequality for an increasing and a decreasing event"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/wprob-fkg-pos","kind":"node","lean":"StatMech.wprob_fkg_pos","parent":20603,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/index.html#wprob-fkg-pos","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr","status":"fully_proved","summary":"The Harris\u2013FKG inequality for a product probability weight: increasing events are positively correlated.","title":"Harris\u2013FKG inequality for increasing events"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/wprob-fkg-neg","kind":"node","lean":"StatMech.wprob_fkg_neg","parent":20603,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/index.html#wprob-fkg-neg","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr","status":"fully_proved","summary":"The Harris\u2013FKG inequality for an increasing and a decreasing event: they are negatively correlated.","title":"Harris\u2013FKG inequality for an increasing and a decreasing event"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/disjoint-occurrence-eq-inter-of-mixed","kind":"node","lean":"StatMech.disjointOccurrence_eq_inter_of_mixed","parent":20603,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/index.html#disjoint-occurrence-eq-inter-of-mixed","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr","status":"fully_proved","summary":"For `A` increasing and `B` decreasing, `A \u25a1 B = A \u2229 B`.","title":"Disjoint occurrence of an increasing and a decreasing event"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/reimer-mixed-wprob","kind":"node","lean":"StatMech.reimerMixed_wprob","parent":20603,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/index.html#reimer-mixed-wprob","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr","status":"fully_proved","summary":"Reimer's inequality for an increasing and a decreasing event under a product probability weight.","title":"Reimer's inequality for mixed monotone events, product weights"},{"id":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/reimer-mixed-inequality","kind":"node","lean":"StatMech.reimerMixed_inequality","parent":20603,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr/index.html#reimer-mixed-inequality","scope":"Probability/correlation-inequalities/reimer-inequality/reimer-s-inequality-for-an-increasing-and-a-decr","status":"fully_proved","summary":"Reimer's inequality for an increasing and a decreasing event under the product Bernoulli measure.","title":"Reimer's inequality for mixed monotone events, Bernoulli measure"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality.json","id":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","kind":"scope","navigation":"module","parent":20591,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/index.html","scope":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","status":"fully_proved","title":"Special cases of Reimer's inequality"},{"id":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/negatively-correlated","kind":"node","lean":"StatMech.NegativelyCorrelated","parent":20609,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/index.html#negatively-correlated","scope":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","status":"fully_proved","summary":"`A` and `B` are negatively correlated under the weight `\u03c6`: `wprob \u03c6 (A \u2229 B) \u2264 wprob \u03c6 A * wprob \u03c6 B`.","title":"Negatively correlated events"},{"id":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/reimer-wprob-of-neg-corr","kind":"node","lean":"StatMech.reimer_wprob_of_negCorr","parent":20609,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/index.html#reimer-wprob-of-neg-corr","scope":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","status":"fully_proved","summary":"Reimer's inequality for negatively correlated events (since `A \u25a1 B \u2286 A \u2229 B`).","title":"Reimer's inequality for negatively correlated events"},{"id":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/reimer-inequality-of-inter-empty","kind":"node","lean":"StatMech.reimer_inequality_of_inter_empty","parent":20609,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/index.html#reimer-inequality-of-inter-empty","scope":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","status":"fully_proved","summary":"Reimer's inequality for disjoint events under the product Bernoulli measure: both sides are compared with `P(A \u25a1 B) = 0`.","title":"Reimer's inequality for disjoint events"},{"id":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/ev-xor","kind":"node","lean":"StatMech.evXor","parent":20609,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/index.html#ev-xor","scope":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","status":"fully_proved","summary":"The event `\u03c9 0 \u2260 \u03c9 1` on two coordinates.","title":"Event that two coordinates differ"},{"id":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/ev-xnor","kind":"node","lean":"StatMech.evXnor","parent":20609,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/index.html#ev-xnor","scope":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","status":"fully_proved","summary":"The event `\u03c9 0 = \u03c9 1` on two coordinates.","title":"Event that two coordinates agree"},{"id":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/reimer-xor-xnor-inequality","kind":"node","lean":"StatMech.reimer_xor_xnor_inequality","parent":20609,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/index.html#reimer-xor-xnor-inequality","scope":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","status":"fully_proved","summary":"Reimer's inequality for `evXor` and `evXnor`, which are neither increasing nor decreasing (and are disjoint).","title":"Reimer's inequality for the XOR and XNOR events"},{"id":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/slice-disjoint-occurrence-le-of-is-incre","kind":"node","lean":"StatMech.slice_disjointOccurrence_le_of_isIncreasing","parent":20609,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality/index.html#slice-disjoint-occurrence-le-of-is-incre","scope":"Probability/correlation-inequalities/reimer-inequality/special-cases-of-reimer-s-inequality","status":"fully_proved","summary":"The slice inequalities of the one-coordinate BK induction step, with `A\u2080, A\u2081, B\u2080, B\u2081` the slices at the first coordinate: `P(A\u2080 \u25a1 B\u2080) \u2264 P((A\u2080 \u25a1 B\u2081) \u2229 (A\u2081 \u25a1 B\u2080))` and `P((A\u2080 \u25a1 B\u2081) \u222a (A\u2081 \u25a1 B\u2080)) \u2264 P(A\u2081) P(B\u2081)`, for increasing events `A, B`.","title":"Slice inequalities in the one-coordinate BK induction step"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events.json","id":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events","kind":"scope","navigation":"module","parent":20591,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/index.html","scope":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events","status":"fully_proved","title":"Families of finsets as events"},{"id":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/is-lower-set-iff-is-down-at-all","kind":"node","lean":"StatMech.isLowerSet_iff_isDownAt_all","parent":20617,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/index.html#is-lower-set-iff-is-down-at-all","scope":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events","status":"fully_proved","summary":"A family of finsets is a lower set iff it is closed under erasing any single element.","title":"Lower sets are families closed under erasing an element"},{"id":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/family-event","kind":"node","lean":"StatMech.familyEvent","parent":20617,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/index.html#family-event","scope":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events","status":"fully_proved","summary":"The event that the support of a configuration lies in the family `\ud835\udc9c`.","title":"Event of a family of finsets"},{"id":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/reimer-wprob-family-event-of-is-lower-se","kind":"node","lean":"StatMech.reimer_wprob_familyEvent_of_isLowerSet","parent":20617,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/index.html#reimer-wprob-family-event-of-is-lower-se","scope":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events","status":"fully_proved","summary":"Reimer's inequality for the events of two lower families (the BK inequality for decreasing events).","title":"Reimer's inequality for two lower families"},{"id":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/reimer-wprob-family-event-of-is-upper-se","kind":"node","lean":"StatMech.reimer_wprob_familyEvent_of_isUpperSet","parent":20617,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events/index.html#reimer-wprob-family-event-of-is-upper-se","scope":"Probability/correlation-inequalities/reimer-inequality/families-of-finsets-as-events","status":"fully_proved","summary":"Reimer's inequality for the events of two upper families (the BK inequality).","title":"Reimer's inequality for two upper families"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums.json","id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","kind":"scope","navigation":"module","parent":20591,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","title":"Weighted disjoint-pair sums"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/rby-cex-a","kind":"node","lean":"StatMech.rbyCexA","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#rby-cex-a","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"The family `{\u2205, {0}}` on `Fin 2`, used in the counterexamples of this file.","title":"First family in the weighted-pair counterexamples"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/rby-cex-b","kind":"node","lean":"StatMech.rbyCexB","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#rby-cex-b","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"The family `{\u2205}` on `Fin 2`, used in the counterexamples of this file.","title":"Second family in the weighted-pair counterexamples"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/rby-separate-compression-strictly-decrea","kind":"node","lean":"StatMech.rby_separate_compression_strictly_decreases","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#rby-separate-compression-strictly-decrea","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"Down-compressing `\ud835\udc9c` and up-compressing `\u212c` at the same coordinate can strictly decrease the number of disjoint pairs.","title":"Down- and up-compressing two families can decrease disjoint pairs"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/fwt","kind":"node","lean":"StatMech.fwt","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#fwt","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"The product weight `\u220f x, w x (x \u2208 S)` of the configuration with support `S`.","title":"Product weight of a finset"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/fwt-2","kind":"node","lean":"StatMech.fwt'","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#fwt-2","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"The product weight of `S` over all coordinates except `i`.","title":"Product weight of a finset off one coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/wdpairs","kind":"node","lean":"StatMech.wdpairs","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#wdpairs","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"The weighted number of disjoint pairs: `\u2211 fwt w S * fwt w T` over disjoint `(S, T) \u2208 \ud835\udc9c \u00d7 \u212c`.","title":"Weighted number of disjoint pairs"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/wdpairs-2","kind":"node","lean":"StatMech.wdpairs'","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#wdpairs-2","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"`wdpairs` with coordinate `i` omitted from the weights.","title":"Weighted number of disjoint pairs off one coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/fmarg","kind":"node","lean":"StatMech.fmarg","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#fmarg","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"The weight `\u2211 S \u2208 \ud835\udc9c, fwt w S` of a family.","title":"Weight of a family of finsets"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/fmarg-2","kind":"node","lean":"StatMech.fmarg'","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#fmarg-2","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"`fmarg` with coordinate `i` omitted from the weights.","title":"Weight of a family off one coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/wdpairs-le-mul","kind":"node","lean":"StatMech.wdpairs_le_mul","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#wdpairs-le-mul","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"For nonnegative weights, `wdpairs w \ud835\udc9c \u212c \u2264 fmarg w \ud835\udc9c * fmarg w \u212c` (drop the disjointness constraint).","title":"Weighted disjoint pairs bounded by the product of family weights"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/wbutterfly-deficit-identity","kind":"node","lean":"StatMech.wbutterfly_deficit_identity","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#wbutterfly-deficit-identity","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"The deficit `fmarg w \ud835\udc9c * fmarg w \u212c - wdpairs w \ud835\udc9c \u212c` expanded over coordinate `i`.","title":"Expansion of the Reimer deficit over one coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/wbox-value","kind":"node","lean":"StatMech.wboxValue","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#wbox-value","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"`(\u220f x, w x true)^2 * #\ud835\udcb0`: for a symmetric weight, the weight of a doubled family.","title":"Box value of a doubled family under a symmetric weight"},{"id":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/wdpairs-not-le-wbox-value-general","kind":"node","lean":"StatMech.wdpairs_not_le_wboxValue_general","parent":20622,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums/index.html#wdpairs-not-le-wbox-value-general","scope":"Probability/correlation-inequalities/reimer-inequality/weighted-disjoint-pair-sums","status":"fully_proved","summary":"For a non-symmetric weight, `wdpairs w \ud835\udc9c \u212c \u2264 wboxValue w (boxDoubled \ud835\udc9c \u212c)` can fail.","title":"Box-value bound fails for non-symmetric weights"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof.json","id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","kind":"scope","navigation":"module","parent":20591,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","title":"Doubled ground set in Reimer's proof"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/dbl","kind":"node","lean":"StatMech.dbl","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#dbl","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The finset `inl '' S \u222a inr '' T` of `\u03b1 \u2295 \u03b1`, encoding the pair `(S, T)`.","title":"Encoding a pair of finsets in the doubled ground set"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/box-doubled","kind":"node","lean":"StatMech.boxDoubled","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#box-doubled","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The family of `dbl S T` over disjoint pairs `S \u2208 \ud835\udc9c`, `T \u2208 \u212c`.","title":"Doubled family of disjoint pairs"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/card-box-doubled","kind":"node","lean":"StatMech.card_boxDoubled","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#card-box-doubled","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The size of `boxDoubled \ud835\udc9c \u212c` is the number of disjoint pairs in `\ud835\udc9c \u00d7 \u212c`.","title":"Size of the doubled disjoint-pair family"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/card-box-family-le-box-doubled","kind":"node","lean":"StatMech.card_boxFamily_le_boxDoubled","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#card-box-family-le-box-doubled","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"`#(\ud835\udc9c \u25cb \u212c) \u2264 #(boxDoubled \ud835\udc9c \u212c)`: distinct disjoint pairs can have the same union.","title":"Disjoint-union family is no larger than the doubled family"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/up-comp","kind":"node","lean":"StatMech.upComp","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#up-comp","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"Up-compression of a family on `\u03b1 \u2295 \u03b1` at the point `a` (`UV.compression {a} \u2205`).","title":"Up-compression at a point of the doubled ground set"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/down-left","kind":"node","lean":"StatMech.downLeft","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#down-left","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"Down-compression at the left copy `inl i`.","title":"Down-compression at the left copy of a coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/up-right","kind":"node","lean":"StatMech.upRight","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#up-right","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"Up-compression at the right copy `inr i`.","title":"Up-compression at the right copy of a coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/double-compress","kind":"node","lean":"StatMech.doubleCompress","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#double-compress","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"Down-compression at `inl i` followed by up-compression at `inr i`.","title":"Double compression at a coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/is-down-at-left","kind":"node","lean":"StatMech.IsDownAtLeft","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#is-down-at-left","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"`IsDownAtLeft i \ud835\udc9c`: `\ud835\udc9c` is closed under erasing `inl i`.","title":"Family closed under erasing the left copy"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/is-up-at-right","kind":"node","lean":"StatMech.IsUpAtRight","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#is-up-at-right","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"`IsUpAtRight i \ud835\udc9c`: `\ud835\udc9c` is closed under inserting `inr i`.","title":"Family closed under inserting the right copy"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/left-block","kind":"node","lean":"StatMech.leftBlock","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#left-block","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The left copy `{inl a}` of `\u03b1` in `\u03b1 \u2295 \u03b1`.","title":"Left copy in the doubled ground set"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/right-block","kind":"node","lean":"StatMech.rightBlock","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#right-block","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The right copy `{inr a}` of `\u03b1` in `\u03b1 \u2295 \u03b1`.","title":"Right copy in the doubled ground set"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/left-cfg","kind":"node","lean":"StatMech.leftCfg","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#left-cfg","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The restriction of a configuration on `\u03b1 \u2295 \u03b1` to the left copy.","title":"Restriction of a doubled configuration to the left copy"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/right-cfg","kind":"node","lean":"StatMech.rightCfg","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#right-cfg","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The restriction of a configuration on `\u03b1 \u2295 \u03b1` to the right copy.","title":"Restriction of a doubled configuration to the right copy"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/left-event","kind":"node","lean":"StatMech.leftEvent","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#left-event","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The event on `\u03b1 \u2295 \u03b1` that the left restriction lies in `A`.","title":"Event on the left copy of the doubled ground set"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/right-event","kind":"node","lean":"StatMech.rightEvent","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#right-event","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The event on `\u03b1 \u2295 \u03b1` that the right restriction lies in `B`.","title":"Event on the right copy of the doubled ground set"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/double-weight","kind":"node","lean":"StatMech.doubleWeight","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#double-weight","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"The coordinate weight `w` on both copies of `\u03b1`.","title":"Doubled coordinate weight"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/reimer-wprob-doubled-split","kind":"node","lean":"StatMech.reimer_wprob_doubled_split","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#reimer-wprob-doubled-split","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"Reimer's inequality, with equality, for a left event and a right event.","title":"Reimer equality for a left event and a right event"},{"id":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/wprob-family-event-box-doubled-eq-wdpair","kind":"node","lean":"StatMech.wprob_familyEvent_boxDoubled_eq_wdpairs","parent":20636,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof/index.html#wprob-family-event-box-doubled-eq-wdpair","scope":"Probability/correlation-inequalities/reimer-inequality/doubled-ground-set-in-reimer-s-proof","status":"fully_proved","summary":"`wdpairs w \ud835\udc9c \u212c` is the doubled-weight probability of `familyEvent (boxDoubled \ud835\udc9c \u212c)`.","title":"Weighted disjoint pairs as a doubled-weight probability"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs.json","id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","kind":"scope","navigation":"module","parent":20591,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","title":"Down-compression and disjoint pairs"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/cfg-support","kind":"node","lean":"StatMech.cfgSupport","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#cfg-support","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"The support `{i | \u03c9 i = true}` of a configuration, as a finset.","title":"Support of a configuration as a finset"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/support-cfg","kind":"node","lean":"StatMech.supportCfg","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#support-cfg","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"The configuration whose open coordinates are exactly the elements of `s`.","title":"Configuration with a given support"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/cfg-equiv-finset","kind":"node","lean":"StatMech.cfgEquivFinset","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#cfg-equiv-finset","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Configurations on a finite set are the same as finsets (their supports).","title":"Configurations on a finite set as finsets"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/down-compress","kind":"node","lean":"StatMech.downCompress","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#down-compress","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Down-compression at `i` of a family of configurations, through `cfgEquivFinset`.","title":"Down-compression of a family of configurations"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/reimer-wprob-one","kind":"node","lean":"StatMech.reimer_wprob_one","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#reimer-wprob-one","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Reimer's inequality on one coordinate, for all events.","title":"Reimer's inequality on one coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/depends-on","kind":"node","lean":"StatMech.DependsOn","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#depends-on","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"`DependsOn A S`: whether a configuration lies in `A` depends only on its values on `S`. This is Mathlib's `DependsOn` for the membership predicate of `A`.","title":"Event depending only on a set of coordinates"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/disjoint-occurrence-eq-inter-of-depends","kind":"node","lean":"StatMech.disjointOccurrence_eq_inter_of_dependsOn","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#disjoint-occurrence-eq-inter-of-depends","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"If `A` depends only on `S`, `B` only on `T`, and `S, T` are disjoint, then `A \u25a1 B = A \u2229 B`.","title":"Disjoint occurrence equals intersection for disjoint dependence sets"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/wprob-inter-mul-of-split","kind":"node","lean":"StatMech.wprob_inter_mul_of_split","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#wprob-inter-mul-of-split","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Events depending on complementary coordinate sets are independent under a product probability weight.","title":"Independence of events on complementary coordinate sets"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/reimer-wprob-of-disjoint-support","kind":"node","lean":"StatMech.reimer_wprob_of_disjoint_support","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#reimer-wprob-of-disjoint-support","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Reimer's inequality, with equality, for events depending on disjoint coordinate sets.","title":"Reimer's inequality for events on disjoint coordinate sets"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/dpairs-count","kind":"node","lean":"StatMech.dpairsCount","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#dpairs-count","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"The number of disjoint pairs `(S, T) \u2208 \ud835\udc9c \u00d7 \u212c`.","title":"Number of disjoint pairs from two families"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/dpairs-count-fiber-recursion","kind":"node","lean":"StatMech.dpairsCount_fiber_recursion","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#dpairs-count-fiber-recursion","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Splitting disjoint pairs by membership of `i`: pairs where neither, only the first, or only the second set contains `i`.","title":"Recursion for disjoint pairs by membership of a coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/reimer-butterfly-compress-step","kind":"node","lean":"StatMech.reimer_butterfly_compress_step","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#reimer-butterfly-compress-step","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Bookkeeping for one compression step at coordinate `i` on `boxDoubled \ud835\udc9c \u212c`: `doubleCompress i` preserves its size and makes it down-closed at `inl i` and up-closed at `inr i`, and sizes split over `memberSubfamily i` and `nonMemberSubfamily i`.","title":"One compression step on the doubled disjoint-pair family"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/dpairs-count-down-down-mono","kind":"node","lean":"StatMech.dpairsCount_downDown_mono","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#dpairs-count-down-down-mono","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Down-compressing both families at `i` does not decrease the number of disjoint pairs.","title":"Down-compression does not decrease the disjoint-pair count"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/pair-kernel","kind":"node","lean":"StatMech.pairKernel'","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#pair-kernel","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"`fwt' w i S * fwt' w i T` for disjoint `S, T`, and `0` otherwise.","title":"Weight kernel of disjoint pairs off a coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/wdpairs-down-down-mono","kind":"node","lean":"StatMech.wdpairs_downDown_mono","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#wdpairs-down-down-mono","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Down-compressing both families at `i` does not decrease the weighted number of disjoint pairs when `w i true \u2264 w i false`.","title":"Down-compression does not decrease weighted disjoint pairs"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/wdpairs-down-down-not-mono-general","kind":"node","lean":"StatMech.wdpairs_downDown_not_mono_general","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#wdpairs-down-down-not-mono-general","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Without `w i true \u2264 w i false`, `wdpairs_downDown_mono` can fail.","title":"Weighted down-compression monotonicity fails for general weights"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/iter-down-comp","kind":"node","lean":"StatMech.iterDownComp","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#iter-down-comp","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Down-compression at each coordinate of `cs` in turn (the head of `cs` is applied last).","title":"Iterated down-compression along a list of coordinates"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/cmap","kind":"node","lean":"StatMech.cmap","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#cmap","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"The compression map: erase `i` from `s` when `s` contains `i` and `s.erase i \u2209 \ud835\udc9c`.","title":"Down-compression map on a single finset"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/total-size","kind":"node","lean":"StatMech.totalSize","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#total-size","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"The total size `\u2211 S \u2208 \ud835\udc9c, #S` of a family.","title":"Total size of a family of finsets"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/exists-iter-down-comp-is-lower-set","kind":"node","lean":"StatMech.exists_iterDownComp_isLowerSet","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#exists-iter-down-comp-is-lower-set","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Some iterated down-compression of `\ud835\udc9c` is a lower set.","title":"Iterated down-compression reaches a lower set"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/box-family-filter-not-mem-coord","kind":"node","lean":"StatMech.boxFamily_filter_notMem_coord","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#box-family-filter-not-mem-coord","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"The members of `\ud835\udc9c \u25cb \u212c` avoiding `i` are exactly the disjoint unions of members of `\ud835\udc9c` and `\u212c` avoiding `i`.","title":"Members of a disjoint-union family avoiding a coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/is-down-at-coord","kind":"node","lean":"StatMech.IsDownAtCoord","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#is-down-at-coord","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"`IsDownAtCoord i \ud835\udc9c`: the family `\ud835\udc9c` is closed under erasing `i`.","title":"Family closed under erasing a coordinate"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/reimer-step-cex-a","kind":"node","lean":"StatMech.reimerStepCexA","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#reimer-step-cex-a","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"The family `{\u2205, {1}}` on `Fin 2`, the first family of the counterexample `StatMech.reimerStep_box_compression_false`.","title":"First family in the box-compression counterexample"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/reimer-step-cex-b","kind":"node","lean":"StatMech.reimerStepCexB","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#reimer-step-cex-b","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"The family `{\u2205, {0, 1}}` on `Fin 2`, the second family of the counterexample `StatMech.reimerStep_box_compression_false`.","title":"Second family in the box-compression counterexample"},{"id":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/reimer-step-box-compression-false","kind":"node","lean":"StatMech.reimerStep_box_compression_false","parent":20656,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs/index.html#reimer-step-box-compression-false","scope":"Probability/correlation-inequalities/reimer-inequality/down-compression-and-disjoint-pairs","status":"fully_proved","summary":"Down-compressing both families at the same coordinate can strictly decrease the size of `\ud835\udc9c \u25cb \u212c`.","title":"Down-compression can shrink the disjoint-union family"},{"data_url":"scopes/Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets.json","id":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets","kind":"scope","navigation":"module","parent":20591,"result":false,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/index.html","scope":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets","status":"fully_proved","title":"Events determined by disjoint coordinate sets"},{"id":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/determined-by","kind":"node","lean":"StatMech.DeterminedBy","parent":20682,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/index.html#determined-by","scope":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets","status":"fully_proved","summary":"`DeterminedBy A K`: whether a configuration lies in `A` depends only on its values on `K`. This is definitionally `StatMech.DependsOn A K`.","title":"Event determined by a set of coordinates"},{"id":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/determined-by-iff-depends-on","kind":"node","lean":"StatMech.determinedBy_iff_dependsOn","parent":20682,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/index.html#determined-by-iff-depends-on","scope":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets","status":"fully_proved","summary":"`DeterminedBy` and `DependsOn` are the same predicate.","title":"Determined-by and depends-on are the same predicate"},{"id":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/wprob-inter-eq-mul-of-determined","kind":"node","lean":"StatMech.wprob_inter_eq_mul_of_determined","parent":20682,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/index.html#wprob-inter-eq-mul-of-determined","scope":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets","status":"fully_proved","summary":"Events determined by disjoint coordinate sets are independent under a product probability weight: `wprob \u03c6 (A \u2229 B) = wprob \u03c6 A * wprob \u03c6 B`.","title":"Independence of events determined by disjoint coordinate sets"},{"id":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/reimer-bernoulli-eq-of-determined-disjoi","kind":"node","lean":"StatMech.reimer_bernoulli_eq_of_determinedDisjoint","parent":20682,"result":true,"route":"../roadmap/Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets/index.html#reimer-bernoulli-eq-of-determined-disjoi","scope":"Probability/correlation-inequalities/reimer-inequality/events-determined-by-disjoint-coordinate-sets","status":"fully_proved","summary":"Reimer's inequality with equality for events determined by disjoint coordinate sets: under the product Bernoulli(`p`) measure, `P_p(A \u25a1 B) = P_p(A) * P_p(B)`.","title":"Reimer equality for events determined by disjoint coordinates"},{"data_url":"scopes/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem.json","id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem","kind":"scope","parent":20449,"result":false,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/index.html","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem","status":"fully_proved","title":"Stochastic domination and Strassen's theorem"},{"data_url":"scopes/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati.json","id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati","kind":"scope","navigation":"module","parent":20687,"result":false,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/index.html","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati","status":"fully_proved","title":"Stochastic domination of measures on configuration space"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/stochastically-dominated","kind":"node","lean":"StatMech.StochasticallyDominated","parent":20688,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/index.html#stochastically-dominated","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati","status":"fully_proved","summary":"`\u03bc` is stochastically dominated by `\u03bd` if their actual `ENNReal` masses satisfy `\u03bc(A) \u2264 \u03bd(A)` for every measurable increasing event `A \u2286 ConfigSpace E`.","title":"Stochastic domination of measures on configurations"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/refl","kind":"node","lean":"StatMech.StochasticallyDominated.refl","parent":20688,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/index.html#refl","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati","status":"fully_proved","summary":"Stochastic domination is reflexive.","title":"Stochastic domination is reflexive"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/trans","kind":"node","lean":"StatMech.StochasticallyDominated.trans","parent":20688,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/index.html#trans","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati","status":"fully_proved","summary":"Stochastic domination is transitive.","title":"Stochastic domination is transitive"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/real","kind":"node","lean":"StatMech.StochasticallyDominated.real","parent":20688,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati/index.html#real","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/stochastic-domination-of-measures-on-configurati","status":"fully_proved","summary":"For a finite upper measure, stochastic domination also compares real-valued masses.","title":"Stochastic domination compares real-valued masses"},{"data_url":"scopes/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination.json","id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination","kind":"scope","navigation":"module","parent":20687,"result":false,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/index.html","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination","status":"fully_proved","title":"Monotone couplings and stochastic domination"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/order-set","kind":"node","lean":"StatMech.orderSet","parent":20693,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/index.html#order-set","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination","status":"fully_proved","summary":"The set of ordered pairs `(\u03c9, \u03c9')` of configurations with `\u03c9 \u2264 \u03c9'` coordinatewise.","title":"Set of coordinatewise ordered configuration pairs"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/monotone-coupling","kind":"node","lean":"StatMech.MonotoneCoupling","parent":20693,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/index.html#monotone-coupling","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination","status":"fully_proved","summary":"`\u03ba` is a monotone coupling of `\u03bc` and `\u03bd`: a measure on pairs of configurations with first marginal `\u03bc`, second marginal `\u03bd`, and no mass off the ordered pairs `\u03c9 \u2264 \u03c9'`.","title":"Monotone coupling of two measures"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/total-mass-eq","kind":"node","lean":"StatMech.MonotoneCoupling.total_mass_eq","parent":20693,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/index.html#total-mass-eq","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination","status":"fully_proved","summary":"The two marginals of a coupling have the same total mass.","title":"Marginals of a coupling have equal total mass"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/stochastically-dominated","kind":"node","lean":"StatMech.MonotoneCoupling.stochasticallyDominated","parent":20693,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/index.html#stochastically-dominated","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination","status":"fully_proved","summary":"A finite monotone coupling of `\u03bc` and `\u03bd` shows `\u03bc \u227c \u03bd`: the easy direction of Strassen's theorem.","title":"Strassen's theorem, easy direction"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/stochastically-dominated-of-exists-monot","kind":"node","lean":"StatMech.stochasticallyDominated_of_exists_monotoneCoupling","parent":20693,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination/index.html#stochastically-dominated-of-exists-monot","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/monotone-couplings-and-stochastic-domination","status":"fully_proved","summary":"If a finite monotone coupling of `\u03bc` and `\u03bd` exists, then `\u03bc \u227c \u03bd`.","title":"Existence of a monotone coupling implies stochastic domination"},{"data_url":"scopes/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination.json","id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination","kind":"scope","navigation":"module","parent":20687,"result":false,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination/index.html","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination","status":"fully_proved","title":"Finite Strassen stochastic domination"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination/is-upper-set","kind":"node","lean":"MathlibExt.Probability.StrassenCouplingWanted.IsUpperSet","parent":20699,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination/index.html#is-upper-set","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination","status":"fully_proved","summary":"A finite subset $U$ of a preordered set $\\alpha$ is an upper set (is upward closed) if for all $x, y \\in \\alpha$ with $x \\in U$ and $x \\le y$, also $y \\in U$.","title":"Finite upper sets"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination/forall-is-upper-set-sum-le-iff-exists-co","kind":"node","lean":"MathlibExt.Probability.StrassenCouplingWanted.forall_isUpperSet_sum_le_iff_exists_coupling","parent":20699,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination/index.html#forall-is-upper-set-sum-le-iff-exists-co","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination","status":"fully_proved","summary":"Let $\\alpha$ be a finite partially ordered set and let $p, q : \\alpha \\to \\mathbb{R}$ be nonnegative functions with $\\sum_{x \\in \\alpha} p(x) = \\sum_{x \\in \\alpha} q(x)$. Then $\\sum_{x \\in U} p(x) \\le \\sum_{x \\in U} q(x)$ for every upper set $U \\subseteq \\alpha$ if and only if there is a function $\\pi : \\alpha \\times \\alpha \\to \\mathbb{R}$ such that $\\pi(x, y) \\ge 0$ for all $x, y \\in \\alpha$,\u2026","title":"Strassen's theorem on a finite poset, equal-mass form"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination/strassen-coupling-finite-poset","kind":"node","lean":"MathlibExt.Probability.StrassenCouplingWanted.strassen_coupling_finite_poset","parent":20699,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination/index.html#strassen-coupling-finite-poset","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/finite-strassen-stochastic-domination","status":"fully_proved","summary":"Let $\\alpha$ be a finite partially ordered set and let $p, q : \\alpha \\to \\mathbb{R}$ be probability mass functions, that is, nonnegative functions with $\\sum_{x \\in \\alpha} p(x) = 1$ and $\\sum_{x \\in \\alpha} q(x) = 1$. Then $\\sum_{x \\in U} p(x) \\le \\sum_{x \\in U} q(x)$ for every upper set $U \\subseteq \\alpha$ if and only if there is a function $\\pi : \\alpha \\times \\alpha \\to \\mathbb{R}$ such that $\\pi(x, y) \\ge 0$\u2026","title":"Strassen's theorem on a finite poset"},{"data_url":"scopes/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit.json","id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit","kind":"scope","navigation":"module","parent":20687,"result":false,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/index.html","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit","status":"fully_proved","title":"Strassen's theorem for configurations on a finite set"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/is-down","kind":"node","lean":"StatMech.Transport.IsDown","parent":20703,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/index.html#is-down","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit","status":"fully_proved","summary":"`D` is a down-set of `P`: `x \u2264 y` and `y \u2208 D` imply `x \u2208 D`.","title":"Down-set of a partial order"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/order-pairs","kind":"node","lean":"StatMech.Transport.OrderPairs","parent":20703,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/index.html#order-pairs","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit","status":"fully_proved","summary":"The ordered pairs `(x, y)` with `x \u2264 y`; couplings supported on the order are vectors indexed by `OrderPairs P`.","title":"Ordered pairs of a partial order"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/exists-coupling-vec","kind":"node","lean":"StatMech.Transport.exists_coupling_vec","parent":20703,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/index.html#exists-coupling-vec","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit","status":"fully_proved","summary":"Strassen's theorem on a finite poset `P`, for nonnegative vectors: if `\u03bc` and `\u03bd` have equal total mass and `\u03bd(D) \u2264 \u03bc(D)` for every down-set `D`, there is a nonnegative `c` on the ordered pairs `x \u2264 y` with first marginal `\u03bc` and second marginal `\u03bd`. This is the ordered-pair vector form of `MathlibExt.Probability.StrassenCouplingWanted.forall_isUpperSet_sum_le_iff_exists_coupling`.","title":"Strassen's theorem"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/kappa-of-vec","kind":"node","lean":"StatMech.kappaOfVec","parent":20703,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/index.html#kappa-of-vec","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit","status":"fully_proved","summary":"The measure on pairs of configurations that puts mass `c q` on each ordered pair `q`.","title":"Measure on ordered pairs from a mass vector"},{"id":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/exists-monotone-coupling-of-stochastical","kind":"node","lean":"StatMech.exists_monotoneCoupling_of_stochasticallyDominated","parent":20703,"result":true,"route":"../roadmap/Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit/index.html#exists-monotone-coupling-of-stochastical","scope":"Probability/correlation-inequalities/stochastic-domination-and-strassen-s-theorem/strassen-s-theorem-for-configurations-on-a-finit","status":"fully_proved","summary":"Strassen's theorem for configurations on a finite set: if `\u03bc \u227c \u03bd` are probability measures on `ConfigSpace E`, there is a finite monotone coupling of `\u03bc` and `\u03bd` (V. Strassen, *The existence of probability measures with given marginals*, 1965). The converse is `StatMech.MonotoneCoupling.stochasticallyDominated`.","title":"Strassen's theorem"},{"data_url":"scopes/Probability/high-dimensional-probability.json","id":"Probability/high-dimensional-probability","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/high-dimensional-probability/index.html","scope":"Probability/high-dimensional-probability","status":"can_prove","summary":"Probability in high dimension and on geometric objects: log-concave and uniform convex-body laws, isotropic constants, the KLS and thin-shell conjectures and their bounds, random matrices (singularity of Bernoulli matrices, Tracy\u2013Widom limits), and random Euclidean optimization (Beardwood\u2013Halton\u2013Hammersley).","title":"High-dimensional and geometric probability"},{"id":"Probability/high-dimensional-probability/johnstone-largest-eigenvalue-tracy-widom","kind":"node","lean":"MathlibExt.Probability.TracyWidomWanted.johnstone_largest_eigenvalue_tracyWidom","parent":20709,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/index.html#johnstone-largest-eigenvalue-tracy-widom","scope":"Probability/high-dimensional-probability","status":"can_prove","summary":"Let `q` be the Hastings\u2013McLeod solution of Painlev\u00e9 II (`q'' = s q + 2 q\u00b3` with `q(s) \u00b7 2 \u221a\u03c0 s^{1/4} e^{(2/3) s^{3/2}} \u2192 1` as `s \u2192 \u221e`). Let `n k, p k \u2192 \u221e` with `n k / p k \u2192 \u03b3 \u2265 1`, and let `X` have independent standard Gaussian entries (`Measure.pi` of `gaussianReal 0 1` over `Fin (n k) \u00d7 Fin (p k)`). The largest eigenvalue of `X\u1d40 X` is `\u2113\u2081 = sup_{\u2016v\u2016 = 1} \u2016X v\u2016\u00b2`. Then for every `s`, the function\u2026","title":"Johnstone's Tracy\u2013Widom limit"},{"data_url":"scopes/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem.json","id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","kind":"scope","navigation":"module","parent":20709,"result":false,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"can_prove","title":"Beardwood\u2013Halton\u2013Hammersley theorem"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/planar-point","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.PlanarPoint","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#planar-point","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"A planar point is a point of the Euclidean plane $\\mathbb{R}^2$: a pair $(x, y)$ of real numbers, with the Euclidean norm $\\lVert (x, y) \\rVert = \\sqrt{x^2 + y^2}$.","title":"Points of the Euclidean plane"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/cyclic-next","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.cyclicNext","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#cyclic-next","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"For a natural number $n$ and an index $i$ with $0 \\le i < n$, the cyclic successor of $i$ is $(i + 1) \\bmod n$; in particular the successor of $n - 1$ is $0$. For $n = 0$ there are no indices.","title":"Cyclic successor"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/tour-length","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.tourLength","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#tour-length","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"For points $x_0, \\dots, x_{n-1}$ in the Euclidean plane and a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$, the tour length is $\\sum_{i=0}^{n-1} \\lVert x_{\\sigma(i)} - x_{\\sigma(i+1 \\bmod n)} \\rVert$, the length of the closed tour that visits $x_{\\sigma(0)}, \\dots, x_{\\sigma(n-1)}$ in this order and returns to its start.","title":"Length of a closed tour"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/shortest-tour-length","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.shortestTourLength","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#shortest-tour-length","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"For points $x_0, \\dots, x_{n-1}$ in the Euclidean plane, $L_n$ is the minimum, over all permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$, of the length of the closed tour that visits $x_{\\sigma(0)}, \\dots, x_{\\sigma(n-1)}$ in this order and returns to its start.","title":"Shortest tour length"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/normalized-tour-length","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.normalizedTourLength","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#normalized-tour-length","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"Let $\\Omega$ be a set and let $X_0, X_1, X_2, \\dots \\colon \\Omega \\to \\mathbb{R}^2$ be functions. For an outcome $\\omega \\in \\Omega$ and $n \\ge 1$, the normalized tour length is $L_n(\\omega)/\\sqrt{n}$, where $L_n(\\omega)$ is the length of a shortest closed tour through $X_0(\\omega), \\dots, X_{n-1}(\\omega)$. For $n = 0$ it is defined to be $0$.","title":"Normalized tour length"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/is-iidwith-law","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.IsIIDWithLaw","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#is-iidwith-law","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"Let $\\Omega$ be a measurable space with a measure $\\mathbb{P}$, let $X_0, X_1, X_2, \\dots \\colon \\Omega \\to \\mathbb{R}^2$ be functions and let $\\nu$ be a measure on the Borel sets of the plane. The $X_i$ are i.i.d. with law $\\nu$ when $\\mathbb{P}$ and $\\nu$ are probability measures, every $X_i$ is measurable, the $X_i$ are mutually independent under $\\mathbb{P}$, and the distribution of each $X_i$ under\u2026","title":"I.i.d. points with a given law"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/iidrealization","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.IIDRealization","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#iidrealization","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"For a measure $\\nu$ on the plane, an i.i.d. realization with law $\\nu$ consists of a measurable space $\\Omega$ with a measure $\\mathbb{P}$, together with a sequence $X_0, X_1, X_2, \\dots \\colon \\Omega \\to \\mathbb{R}^2$ of points that is i.i.d. with common law $\\nu$ under $\\mathbb{P}$.","title":"I.i.d. realization"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/has-bhhlimit","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.HasBHHLimit","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#has-bhhlimit","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"Let $\\Omega$ be a measurable space with a measure $\\mu$, let $X_0, X_1, X_2, \\dots \\colon \\Omega \\to \\mathbb{R}^2$ be functions into the plane, and let $\\beta$ be a real number. The sequence has BHH limit $\\beta$ when, for $\\mu$-almost every $\\omega \\in \\Omega$, the normalized tour length $L_n(\\omega)/\\sqrt{n}$ tends to $\\beta$ as $n \\to \\infty$, where $L_n(\\omega)$ is the length of a shortest closed tour through\u2026","title":"Almost-sure BHH limit"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/is-bhhconstant-for","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.IsBHHConstantFor","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#is-bhhconstant-for","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"For a measure $\\nu$ on the plane and a real number $\\beta$, $\\beta$ is a BHH constant for $\\nu$ when an i.i.d. realization with law $\\nu$ exists and every i.i.d. realization with law $\\nu$ has almost-sure BHH limit $\\beta$: almost surely, $L_n/\\sqrt{n} \\to \\beta$, where $L_n$ is the length of a shortest closed tour through the first $n$ points.","title":"BHH constant of a law"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/bhh-limit-values","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.bhhLimitValues","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#bhh-limit-values","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"For a measure $\\mu$ on the plane, the BHH limit values of $\\mu$ form the set of real numbers $\\beta$ that are BHH limits for $\\mu$.","title":"BHH limit values of a sampling law"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/mem-bhh-limit-values-iff","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.mem_bhhLimitValues_iff","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#mem-bhh-limit-values-iff","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"For a measure $\\nu$ on the plane and a real number $\\beta$, $\\beta$ belongs to the set of BHH limit values of $\\nu$ if and only if $\\beta$ is a BHH constant for $\\nu$.","title":"Membership in the set of BHH limit values"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/has-unique-bhhconstant-for","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.HasUniqueBHHConstantFor","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#has-unique-bhhconstant-for","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"A measure $\\nu$ on the plane has a unique BHH constant when there is exactly one real number $\\beta$ that is a BHH constant for $\\nu$.","title":"Unique BHH constant of a law"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/bhh-constant-for","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.bhhConstantFor","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#bhh-constant-for","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"For a measure $\\mu$ on the plane, the BHH constant of $\\mu$ is the infimum of the set of real numbers $\\beta$ that are BHH limits for $\\mu$. The infimum of a set of reals that is empty or not bounded below is taken to be $0$.","title":"BHH constant of a sampling law"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/bhh-constant-for-eq-of-unique-characteri","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.bhhConstantFor_eq_of_unique_characterization","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#bhh-constant-for-eq-of-unique-characteri","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on the plane and $\\beta$ a real number. If, for every real $\\gamma$, $\\gamma$ is a BHH limit for $\\mu$ if and only if $\\gamma = \\beta$, then the BHH constant of $\\mu$ (the infimum of its BHH limits) equals $\\beta$.","title":"The BHH constant equals the unique BHH limit"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/unit-square","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.unitSquare","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#unit-square","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"The closed unit square $[0,1]^2$: the set of points of the Euclidean plane both of whose coordinates lie in the interval $[0,1]$.","title":"Unit square"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/unit-square-uniform-law","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.unitSquareUniformLaw","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#unit-square-uniform-law","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"Lebesgue measure on the Euclidean plane restricted to the closed unit square $[0,1]^2$.","title":"Uniform law on the unit square"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/beardwood-halton-hammersley-constant","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.beardwoodHaltonHammersleyConstant","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#beardwood-halton-hammersley-constant","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"The Beardwood\u2013Halton\u2013Hammersley constant is the BHH constant of the uniform law on the closed unit square $[0,1]^2$ (Lebesgue measure restricted to $[0,1]^2$): the infimum of the set of real numbers $\\beta$ such that i.i.d. samples with this law exist and, for every such sample $X_1, X_2, \\dots$, the length of the shortest closed tour through $X_1, \\dots, X_n$ divided by $\\sqrt{n}$ tends to $\\beta$ almost surely.\u2026","title":"Beardwood\u2013Halton\u2013Hammersley constant"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/untitled","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.\u03b2\u2082","parent":20711,"reading_title":{"format":"inline-parts-v1","parts":[["text","The notation "],["math","\\(\\beta_2\\)"]],"source":"The notation $\\beta_2$"},"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#untitled","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"We write $\\beta_2$ for the real-valued Beardwood\u2013Halton\u2013Hammersley constant of the unit square.","title":"The notation $\\beta_2$"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/c","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.C\u2081\u2082","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#c","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"The constant $C_{12}$ is defined to be the Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ for uniform points in the unit square; $C_{12}$ is the name the source uses for it.","title":"Notation C\u2081\u2082 for the BHH constant"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/unit-square-bhhtheorem","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.unitSquareBHHTheorem","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#unit-square-bhhtheorem","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"The proposition that the Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ is a BHH constant for the uniform law on the closed unit square $[0,1]^2$, and that every real number $\\beta$ that is a BHH constant for this law equals $\\beta_2$. Here a real number $\\beta$ is a BHH constant for a law $\\mu$ on the plane when a sequence $X_0, X_1, \\dots$ of independent random points with common law $\\mu$ exists on some\u2026","title":"Statement of the Beardwood\u2013Halton\u2013Hammersley theorem"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/bhh1959-upper-bound","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.bhh1959UpperBound","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#bhh1959-upper-bound","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"This is the proposition that the Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ for uniform points in the unit square satisfies $\\beta_2 \\le 0.92117$, the upper bound reported by Beardwood, Halton and Hammersley (1959).","title":"Statement of the Beardwood\u2013Halton\u2013Hammersley upper bound"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/steinerberger2015-improved-upper-bound","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.steinerberger2015ImprovedUpperBound","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#steinerberger2015-improved-upper-bound","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"The proposition that the Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ satisfies $\\beta_2 \\le 0.92117 - \\frac{9}{16} \\cdot 10^{-6}$.","title":"Statement of Steinerberger's upper bound"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/yu-carlsson2023-strict-upper-bound","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.yuCarlsson2023StrictUpperBound","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#yu-carlsson2023-strict-upper-bound","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"This is the proposition that the real-valued Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ of the unit square satisfies $\\beta_2 < 0.90304$.","title":"Statement of the Yu\u2013Carlsson upper bound"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/bhh1959-lower-bound","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.bhh1959LowerBound","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#bhh1959-lower-bound","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"This is the proposition that the Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ for uniform points in the unit square satisfies $\\beta_2 \\ge 5/8 = 0.625$, the lower bound reported by Beardwood, Halton and Hammersley (1959).","title":"Statement of the Beardwood\u2013Halton\u2013Hammersley lower bound"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/steinerberger2015-corrected-symbolic-low","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.steinerberger2015CorrectedSymbolicLowerBound","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#steinerberger2015-corrected-symbolic-low","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"The proposition that the Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ satisfies $\\frac{5}{8} + \\frac{19}{10368} \\le \\beta_2$.","title":"Statement of the corrected Steinerberger lower bound"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/gaudio-jaillet2020-lower-bound","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.gaudioJaillet2020LowerBound","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#gaudio-jaillet2020-lower-bound","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"fully_proved","summary":"The proposition that the Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ of the unit square is at least $0.6277$, that is, $0.6277 \\le \\beta_2$.","title":"Statement of the Gaudio\u2013Jaillet lower bound"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/unit-square-bhhtheorem-holds","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.unitSquareBHHTheorem_holds","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#unit-square-bhhtheorem-holds","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"can_prove","summary":"Let $\\beta_2$ be the Beardwood\u2013Halton\u2013Hammersley constant and, for points $X_0, X_1, \\dots$ of the plane, let $L_n$ be the length of the shortest closed tour through $X_0, \\dots, X_{n-1}$. Then a sequence of independent random points $X_0, X_1, \\dots$, each uniformly distributed on the closed unit square $[0,1]^2$, exists on some probability space; for every such sequence, almost surely $L_n / \\sqrt{n} \\to \\beta_2$\u2026","title":"Beardwood\u2013Halton\u2013Hammersley theorem"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/bhh1959-upper-bound-holds","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.bhh1959UpperBound_holds","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#bhh1959-upper-bound-holds","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"can_prove","summary":"The Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ for uniform points in the unit square satisfies $\\beta_2 \\le 0.92117$.","title":"Beardwood\u2013Halton\u2013Hammersley upper bound"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/steinerberger2015-improved-upper-bound-h","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.steinerberger2015ImprovedUpperBound_holds","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#steinerberger2015-improved-upper-bound-h","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"can_prove","summary":"The Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ satisfies $\\beta_2 \\le 0.92117 - \\frac{9}{16} \\cdot 10^{-6}$.","title":"Steinerberger's upper bound for the BHH constant"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/yu-carlsson2023-strict-upper-bound-holds","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.yuCarlsson2023StrictUpperBound_holds","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#yu-carlsson2023-strict-upper-bound-holds","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"can_prove","summary":"The real-valued Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ of the unit square satisfies $\\beta_2 < 0.90304$.","title":"Yu\u2013Carlsson upper bound on the BHH constant"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/bhh1959-lower-bound-holds","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.bhh1959LowerBound_holds","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#bhh1959-lower-bound-holds","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"can_prove","summary":"The Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ for uniform points in the unit square satisfies $\\beta_2 \\ge 5/8 = 0.625$.","title":"Beardwood\u2013Halton\u2013Hammersley lower bound"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/steinerberger2015-corrected-symbolic-low-2","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.steinerberger2015CorrectedSymbolicLowerBound_holds","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#steinerberger2015-corrected-symbolic-low-2","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"can_prove","summary":"The Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ satisfies $\\frac{5}{8} + \\frac{19}{10368} \\le \\beta_2$.","title":"Gaudio\u2013Jaillet corrected lower bound for the BHH constant"},{"id":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/gaudio-jaillet2020-lower-bound-holds","kind":"node","lean":"MathlibExt.Probability.BeardwoodHaltonHammersleyConstantWanted.gaudioJaillet2020LowerBound_holds","parent":20711,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem/index.html#gaudio-jaillet2020-lower-bound-holds","scope":"Probability/high-dimensional-probability/beardwood-halton-hammersley-theorem","status":"can_prove","summary":"The Beardwood\u2013Halton\u2013Hammersley constant $\\beta_2$ of the unit square satisfies $0.6277 \\le \\beta_2$.","title":"Gaudio\u2013Jaillet lower bound for the BHH constant"},{"data_url":"scopes/Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices.json","id":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices","kind":"scope","navigation":"module","parent":20709,"result":false,"route":"../roadmap/Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/index.html","scope":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices","status":"can_prove","title":"Singularity of random Bernoulli matrices"},{"id":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/min","kind":"node","lean":"MathlibExt.Probability.RandomBernoulliMatricesWanted.RandomBernoulliMatrices.\u03c3_min","parent":20745,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/index.html#min","scope":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices","status":"fully_proved","summary":"Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$, and let $M$ be a matrix over $\\mathbb{K}$ whose rows and columns are indexed by finite sets $n$ and $m$. Its smallest singular value is the real number $\\sigma_{\\min}(M) = \\inf_{x \\in \\mathbb{K}^m,\\ \\lVert x \\rVert_2 = 1} \\lVert Mx \\rVert_2$, where $\\lVert \\cdot \\rVert_2$ is the Euclidean norm. When $m$ is empty there are no unit vectors, and $\\sigma_{\\min}(M) = 0$.","title":"Smallest singular value of a matrix"},{"id":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/rademacher-measure","kind":"node","lean":"MathlibExt.Probability.RandomBernoulliMatricesWanted.RandomBernoulliMatrices.rademacherMeasure","parent":20745,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/index.html#rademacher-measure","scope":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices","status":"fully_proved","summary":"The Rademacher measure is the probability measure on $\\mathbb{R}$ that assigns mass $1/2$ to $1$ and mass $1/2$ to $-1$.","title":"Rademacher measure on the real line"},{"id":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/theorem-a","kind":"node","lean":"MathlibExt.Probability.RandomBernoulliMatricesWanted.RandomBernoulliMatrices.theorem_A","parent":20745,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/index.html#theorem-a","scope":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices","status":"can_prove","summary":"Let $0 < p \\le 1/2$ and $\\varepsilon > 0$, and write $\\sigma_{\\min}(A)$ for the smallest singular value of a real square matrix $A$. There exist a natural number $N > 0$ and a real number $C > 0$ such that for every $n \\ge N$ the following holds. Let $B$ be a random $n \\times n$ real matrix on a probability space $(\\Omega, P)$ whose $n^2$ entries are mutually independent and each equal to $1$ with probability $p$\u2026","title":"Tikhomirov's Theorem A: smallest singular value of shifted Bernoulli matrices"},{"id":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/corollary-1","kind":"node","lean":"MathlibExt.Probability.RandomBernoulliMatricesWanted.RandomBernoulliMatrices.corollary_1","parent":20745,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/index.html#corollary-1","scope":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices","status":"can_prove","summary":"Let $0 < p \\le 1/2$. For each $n$, let $B_n$ be a random $n \\times n$ real matrix on a probability space $(\\Omega_n, P_n)$ whose $n^2$ entries are mutually independent and each equal to $1$ with probability $p$ and to $0$ with probability $1 - p$. Then there is a function $o : \\mathbb{N} \\to \\mathbb{R}$ with $o(n) \\to 0$ as $n \\to \\infty$ such that $P_n(\\det B_n = 0) = (1 - p + o(n))^n$ for every $n \\ge 1$.","title":"Singularity probability of Bernoulli matrices"},{"id":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/corollary-2","kind":"node","lean":"MathlibExt.Probability.RandomBernoulliMatricesWanted.RandomBernoulliMatrices.corollary_2","parent":20745,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices/index.html#corollary-2","scope":"Probability/high-dimensional-probability/singularity-of-random-bernoulli-matrices","status":"can_prove","summary":"For each $n$, let $M_n$ be a random $n \\times n$ real matrix on a probability space $(\\Omega_n, P_n)$ whose $n^2$ entries are mutually independent and each have the Rademacher law, with mass $1/2$ at each of $1$ and $-1$. Then there is a function $o : \\mathbb{N} \\to \\mathbb{R}$ with $o(n) \\to 0$ as $n \\to \\infty$ such that $P_n(\\det M_n = 0) = (1/2 + o(n))^n$ for every $n \\ge 1$.","title":"Singularity probability of Rademacher matrices"},{"data_url":"scopes/Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes.json","id":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes","kind":"scope","navigation":"module","parent":20709,"result":false,"route":"../roadmap/Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/index.html","scope":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes","status":"can_prove","title":"Suprema of selector and empirical processes"},{"id":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/binomial-set-distribution","kind":"node","lean":"MathlibExt.Analysis.SupremumOfSelectorProcessesWanted.SupremumOfSelectorProcesses.binomialSetDistribution","parent":20751,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/index.html#binomial-set-distribution","scope":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes","status":"fully_proved","summary":"For a type $X$ and $p \\in [0, 1]$, the binomial random subset distribution on $X$ with parameter $p$ is the probability measure on the subsets of $X$ under which each element of $X$ belongs to the random subset independently with probability $p$. A random subset with this law is denoted $X_p$.","title":"Binomial random subset"},{"id":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/is-small","kind":"node","lean":"MathlibExt.Analysis.SupremumOfSelectorProcessesWanted.SupremumOfSelectorProcesses.IsSmall","parent":20751,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-small families of sets"]],"source":"$p$-small families of sets"},"result":true,"route":"../roadmap/Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/index.html#is-small","scope":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes","status":"fully_proved","summary":"Let $X$ be a finite type and $p \\in [0, 1]$. A family $\\mathcal{F}$ of subsets of $X$ is $p$-small when there is a family $\\mathcal{G}$ of subsets of $X$ such that every member of $\\mathcal{F}$ contains some member of $\\mathcal{G}$ and $\\sum_{S \\in \\mathcal{G}} p^{|S|} \\le 1/2$.","title":"$p$-small families of sets"},{"id":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/theorem-1-2","kind":"node","lean":"MathlibExt.Analysis.SupremumOfSelectorProcessesWanted.SupremumOfSelectorProcesses.theorem_1_2","parent":20751,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/index.html#theorem-1-2","scope":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes","status":"can_prove","summary":"There is a real number $L > 0$ with the following property. Let $0 < p < 1$, let $X$ be a finite type, let $\\Lambda$ be a set of functions from $X$ to $[0, \\infty)$, and let $X_p$ be a binomial random subset of $X$ with parameter $p$. Write $E = \\mathbb{E}\\bigl[\\sup_{f \\in \\Lambda} \\sum_{i \\in X_p} f(i)\\bigr]$, computed in $[0, \\infty]$. If $0 < E < \\infty$, then the family of subsets $s \\subseteq X$ with\u2026","title":"Park\u2013Pham theorem on selector processes (Talagrand's selector process conjecture)"},{"id":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/z","kind":"node","lean":"MathlibExt.Analysis.SupremumOfSelectorProcessesWanted.SupremumOfSelectorProcesses.Z","parent":20751,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/index.html#z","scope":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes","status":"fully_proved","summary":"Given random variables $Y_1, \\dots, Y_N$ on $\\Omega$ with values in $\\mathbb{T}$ and a function $f : \\mathbb{T} \\to [0, \\infty)$, $Z_N(f)$ is the random variable $Z_N(f)(\\omega) = \\frac{1}{N} \\sum_{i=1}^{N} f(Y_i(\\omega))$, which is $0$ when $N = 0$.","title":"Empirical averages"},{"id":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/theorem-1-3","kind":"node","lean":"MathlibExt.Analysis.SupremumOfSelectorProcessesWanted.SupremumOfSelectorProcesses.theorem_1_3","parent":20751,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes/index.html#theorem-1-3","scope":"Probability/high-dimensional-probability/suprema-of-selector-and-empirical-processes","status":"can_prove","summary":"There is a real number $L > 0$ with the following property. Let $N > 0$, let $\\mathbb{T}$ be a Polish space with its Borel $\\sigma$-algebra and a probability measure $\\nu$, and let $Y_1, \\dots, Y_N$ be independent random variables on a measure space $(\\Omega, P)$ with values in $\\mathbb{T}$, each with law $\\nu$. For $f : \\mathbb{T} \\to [0, \\infty)$ let $Z_f = \\frac{1}{N} \\sum_{i=1}^{N} f(Y_i)$. Let $\\mathcal{F}$ be\u2026","title":"Park\u2013Pham theorem on positive empirical processes"},{"data_url":"scopes/Probability/high-dimensional-probability/thin-shell-conjecture.json","id":"Probability/high-dimensional-probability/thin-shell-conjecture","kind":"scope","navigation":"module","parent":20709,"result":false,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","title":"Thin-shell conjecture"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/is-log-concave-density","kind":"node","lean":"MathlibExt.Probability.ThinShell.IsLogConcaveDensity","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#is-log-concave-density","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"A function $f : \\mathbb{R}^n \\to \\mathbb{R}$ is a log-concave density when it is measurable, $f(x) \\ge 0$ for every $x$, and $f(x)^{1-t} f(y)^t \\le f((1-t)x + ty)$ for all $x, y \\in \\mathbb{R}^n$ and every real $t$ with $0 < t < 1$.","title":"Log-concave densities"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/is-isotropic-log-concave","kind":"node","lean":"MathlibExt.Probability.ThinShell.IsIsotropicLogConcave","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#is-isotropic-log-concave","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"A measure $\\mu$ on $\\mathbb{R}^n$ is isotropic log-concave when it is a probability measure, it has density $f$ with respect to Lebesgue measure for some log-concave density $f$, and for all coordinates $i, j$ the functions $x \\mapsto x_i$ and $x \\mapsto x_i x_j$ are $\\mu$-integrable with $\\int x_i \\, d\\mu(x) = 0$ and $\\int x_i x_j \\, d\\mu(x) = \\delta_{ij}$, that is, $1$ if $i = j$ and $0$ otherwise.","title":"Isotropic log-concave probability measures"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/normalized-radius-variance","kind":"node","lean":"MathlibExt.Probability.ThinShell.normalizedRadiusVariance","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#normalized-radius-variance","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}^n$, the normalized radius variance of $\\mu$ is $\\frac{1}{n} \\int (\\|x\\|^2 - n)^2 \\, d\\mu(x)$, computed in $[0, \\infty]$, where $\\|x\\|$ is the Euclidean norm (for $n = 0$ the integral is $0$ and so is the value).","title":"Normalized radius variance"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/thin-shell-variance-constant","kind":"node","lean":"MathlibExt.Probability.ThinShell.thinShellVarianceConstant","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#thin-shell-variance-constant","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"Write $|x|$ for the Euclidean norm. The thin-shell variance constant is the supremum, taken in $[0, \\infty]$, of the normalized radius variance $\\frac{1}{n} \\int (|x|^2 - n)^2 \\, d\\mu(x)$ over all dimensions $n \\ge 1$ and all isotropic log-concave probability measures $\\mu$ on $\\mathbb{R}^n$.","title":"Thin-shell variance constant"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/thin-shell-width","kind":"node","lean":"MathlibExt.Probability.ThinShell.thinShellWidth","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#thin-shell-width","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"Write $|x|$ for the Euclidean norm. For $n \\in \\mathbb{N}$, the thin-shell width in dimension $n$ is the square root of the supremum, taken in $[0, \\infty]$, of $\\int (|x| - \\sqrt{n})^2 \\, d\\mu(x)$ over all isotropic log-concave probability measures $\\mu$ on $\\mathbb{R}^n$, with $\\sqrt{\\infty} = \\infty$.","title":"Thin-shell width"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/radius-deviation-sq-le","kind":"node","lean":"MathlibExt.Probability.ThinShell.radius_deviation_sq_le","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#radius-deviation-sq-le","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"For all real numbers $r \\ge 0$ and $d > 0$, $(r - \\sqrt{d})^2 \\le (r^2 - d)^2 / d$.","title":"Elementary radius deviation bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/radius-fluctuation-le-normalized-varianc","kind":"node","lean":"MathlibExt.Probability.ThinShell.radius_fluctuation_le_normalized_variance","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#radius-fluctuation-le-normalized-varianc","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"For every $n > 0$ and every measure $\\mu$ on $\\mathbb{R}^n$, the integral $\\int (\\|x\\| - \\sqrt{n})^2 \\, d\\mu(x)$, taken in $[0, \\infty]$ with $\\|x\\|$ the Euclidean norm, is at most the normalized radius variance of $\\mu$.","title":"Radius fluctuation is bounded by the normalized radius variance"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/normalized-radius-variance-le-constant","kind":"node","lean":"MathlibExt.Probability.ThinShell.normalizedRadiusVariance_le_constant","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#normalized-radius-variance-le-constant","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"The thin-shell variance constant bounds the normalized radius variance of isotropic log-concave measures. For every $n > 0$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$, the normalized radius variance of $\\mu$ is at most the thin-shell variance constant.","title":"The thin-shell variance constant bounds the normalized radius variance of isotropic log-concave measures"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/thin-shell-variance-constant-le-iff","kind":"node","lean":"MathlibExt.Probability.ThinShell.thinShellVarianceConstant_le_iff","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#thin-shell-variance-constant-le-iff","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"For every $C \\in [0, \\infty]$, the thin-shell variance constant is at most $C$ if and only if, for every $n > 0$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$, the normalized radius variance of $\\mu$ is at most $C$.","title":"Characterization of bounds on the thin-shell variance constant"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/radius-fluctuation-le-constant","kind":"node","lean":"MathlibExt.Probability.ThinShell.radius_fluctuation_le_constant","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#radius-fluctuation-le-constant","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"For every $n > 0$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$, the integral $\\int (\\|x\\| - \\sqrt{n})^2 \\, d\\mu(x)$, taken in $[0, \\infty]$ with $\\|x\\|$ the Euclidean norm, is at most the thin-shell variance constant.","title":"The thin-shell variance constant bounds the radius fluctuation"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/thin-shell-variance-constant-nonneg","kind":"node","lean":"MathlibExt.Probability.ThinShell.thinShellVarianceConstant_nonneg","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#thin-shell-variance-constant-nonneg","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"The thin-shell variance constant, a value in $[0, \\infty]$, is at least $0$.","title":"The thin-shell variance constant is nonnegative"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/thin-shell-width-le-sqrt-constant","kind":"node","lean":"MathlibExt.Probability.ThinShell.thinShellWidth_le_sqrt_constant","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#thin-shell-width-le-sqrt-constant","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"For every dimension $n \\ge 1$, the thin-shell width in dimension $n$ is at most the square root of the thin-shell variance constant, computed in $[0, \\infty]$ with $\\sqrt{\\infty} = \\infty$.","title":"Thin-shell width is at most the square root of the variance constant"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/thin-shell-variance-constant-lt-top-of-b","kind":"node","lean":"MathlibExt.Probability.ThinShell.thinShellVarianceConstant_lt_top_of_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#thin-shell-variance-constant-lt-top-of-b","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"fully_proved","summary":"If $C$ is a real number such that for every $n > 0$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$ the normalized radius variance of $\\mu$ is at most $\\max(C, 0)$, then the thin-shell variance constant is finite.","title":"A uniform variance bound makes the thin-shell variance constant finite"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/klartag2007a-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.klartag2007a_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#klartag2007a-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ of dimension $n$ satisfies $\\sigma_n \\le C \\sqrt{n / \\log n}$.","title":"Klartag's first thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/klartag2007b-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.klartag2007b_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#klartag2007b-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"For every $\\varepsilon > 0$ there are a constant $C > 0$ and an integer $N \\ge 3$ such that for every dimension $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ satisfies $\\sigma_n \\le C n^{2/5 + \\varepsilon}$. Here $\\sigma_n$ is the square root of the supremum of $\\mathbb{E}\\,(|X| - \\sqrt{n})^2$ over isotropic log-concave random vectors $X$ in $\\mathbb{R}^n$.","title":"Klartag's power-law thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/fleury2010-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.fleury2010_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#fleury2010-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ of dimension $n$ satisfies $\\sigma_n \\le C n^{3/8}$.","title":"Fleury's thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/guedon-milman2011-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.guedonMilman2011_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#guedon-milman2011-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ of dimension $n$ satisfies $\\sigma_n \\le C n^{1/3}$.","title":"Gu\u00e9don\u2013Milman thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/lee-vempala2017-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.leeVempala2017_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#lee-vempala2017-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There are a constant $C > 0$ and an integer $N \\ge 3$ such that for every dimension $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ satisfies $\\sigma_n \\le C n^{1/4}$. Here $\\sigma_n$ is the square root of the supremum of $\\mathbb{E}\\,(|X| - \\sqrt{n})^2$ over isotropic log-concave random vectors $X$ in $\\mathbb{R}^n$.","title":"Lee\u2013Vempala thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/chen2021-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.chen2021_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#chen2021-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"For every real $\\varepsilon > 0$ there is a natural number $N \\ge 3$ such that for every natural number $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ of dimension $n$ satisfies $\\sigma_n \\le \\exp\\big((\\log n)^{1/2 + \\varepsilon}\\big)$.","title":"Chen's thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/klartag-lehec2022-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.klartagLehec2022_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#klartag-lehec2022-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ of dimension $n$ satisfies $\\sigma_n \\le C (\\log n)^4$.","title":"Klartag\u2013Lehec polylogarithmic thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/jambulapati-lee-vempala2022-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.jambulapatiLeeVempala2022_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#jambulapati-lee-vempala2022-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"Let $\\alpha = \\big(1 + 7\\sqrt{2} + \\sqrt{53 - 4\\sqrt{2}}\\big)/8$. There are a real $C > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ of dimension $n$ satisfies $\\sigma_n \\le C (\\log n)^{\\alpha}$.","title":"Jambulapati\u2013Lee\u2013Vempala thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/klartag2023-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.klartag2023_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#klartag2023-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There are a constant $C > 0$ and an integer $N \\ge 3$ such that for every dimension $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ satisfies $\\sigma_n \\le C \\sqrt{\\log n}$. Here $\\sigma_n$ is the square root of the supremum of $\\mathbb{E}\\,(|X| - \\sqrt{n})^2$ over isotropic log-concave random vectors $X$ in $\\mathbb{R}^n$, and $\\log$ is the natural logarithm.","title":"Klartag's square-root-logarithmic thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/guan2024-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.guan2024_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#guan2024-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ of dimension $n$ satisfies $\\sigma_n \\le C \\log \\log n$.","title":"Guan's thin-shell bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/klartag-lehec2025-variance-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.klartagLehec2025_variance_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#klartag-lehec2025-variance-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There is a real $C > 0$ such that for every natural number $n \\ge 1$ and every isotropic log-concave measure $\\mu$ on $\\mathbb{R}^n$, the normalized radius variance $\\frac{1}{n} \\int (|x|^2 - n)^2 \\, d\\mu(x)$ is at most $C$.","title":"Klartag\u2013Lehec thin-shell variance bound"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/klartag-lehec2025-width-bound","kind":"node","lean":"MathlibExt.Probability.ThinShell.klartagLehec2025_width_bound","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#klartag-lehec2025-width-bound","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"There is a real $C > 0$ such that for every natural number $n \\ge 1$ the thin-shell width $\\sigma_n \\in [0, \\infty]$ of dimension $n$ satisfies $\\sigma_n \\le C$.","title":"Klartag\u2013Lehec bounded thin-shell width"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/thin-shell-variance-constant-lt-top","kind":"node","lean":"MathlibExt.Probability.ThinShell.thinShellVarianceConstant_lt_top","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#thin-shell-variance-constant-lt-top","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"The thin-shell variance constant, the supremum in $[0, \\infty]$ of the normalized radius variances $\\frac{1}{n} \\int (|x|^2 - n)^2 \\, d\\mu(x)$ over all dimensions $n \\ge 1$ and all isotropic log-concave probability measures $\\mu$ on $\\mathbb{R}^n$, is finite.","title":"The thin-shell variance constant is finite"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/std-gaussian-isotropic-variance","kind":"node","lean":"MathlibExt.Probability.ThinShell.stdGaussian_isotropic_variance","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#std-gaussian-isotropic-variance","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"For every dimension $n \\ge 1$, the standard Gaussian measure $\\gamma_n$ on $\\mathbb{R}^n$ is isotropic log-concave (a probability measure with a log-concave Lebesgue density, mean $0$ and identity covariance), and its normalized radius variance $\\frac{1}{n} \\int (|x|^2 - n)^2 \\, d\\gamma_n(x)$ equals $2$.","title":"The standard Gaussian has normalized radius variance 2"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/uniform-cube-isotropic-variance","kind":"node","lean":"MathlibExt.Probability.ThinShell.uniformCube_isotropic_variance","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#uniform-cube-isotropic-variance","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"For every dimension $n \\ge 1$, let $\\mu$ be the uniform probability measure on the cube $[-\\sqrt{3}, \\sqrt{3}]^n \\subset \\mathbb{R}^n$ (Lebesgue measure restricted to the cube and divided by its volume). Then $\\mu$ is isotropic log-concave (a probability measure with a log-concave Lebesgue density, mean $0$ and identity covariance), and its normalized radius variance $\\frac{1}{n} \\int (|x|^2 - n)^2 \\, d\\mu(x)$\u2026","title":"The isotropic cube has normalized radius variance 4/5"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/four-fifths-le-thin-shell-variance-const","kind":"node","lean":"MathlibExt.Probability.ThinShell.fourFifths_le_thinShellVarianceConstant","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#four-fifths-le-thin-shell-variance-const","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"The thin-shell variance constant, an element of $[0, \\infty]$, is at least $4/5$.","title":"Cube lower bound for the thin-shell variance constant"},{"id":"Probability/high-dimensional-probability/thin-shell-conjecture/two-le-thin-shell-variance-constant","kind":"node","lean":"MathlibExt.Probability.ThinShell.two_le_thinShellVarianceConstant","parent":20757,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/thin-shell-conjecture/index.html#two-le-thin-shell-variance-constant","scope":"Probability/high-dimensional-probability/thin-shell-conjecture","status":"can_prove","summary":"The thin-shell variance constant, the supremum in $[0, \\infty]$ of the normalized radius variances $\\frac{1}{n} \\int (|x|^2 - n)^2 \\, d\\mu(x)$ over all dimensions $n \\ge 1$ and all isotropic log-concave probability measures $\\mu$ on $\\mathbb{R}^n$, is at least $2$.","title":"The thin-shell variance constant is at least 2"},{"data_url":"scopes/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws.json","id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","kind":"scope","navigation":"module","parent":20709,"result":false,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","title":"Isotropic constant of log-concave laws"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/covariance-matrix","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.covarianceMatrix","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#covariance-matrix","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}^n$, the covariance matrix of $\\mu$ is the $n \\times n$ real matrix whose $(i, j)$ entry is $\\int (x_i - m_i)(x_j - m_j) \\, d\\mu(x)$, where $m_i = \\int x_i \\, d\\mu(x)$ is the mean of the $i$-th coordinate.","title":"Covariance matrix"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/differential-entropy","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.differentialEntropy","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#differential-entropy","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a function $f \\colon \\mathbb{R}^n \\to \\mathbb{R}$, usually the Lebesgue density of a probability law, the differential entropy of $f$ is $-\\int_{\\mathbb{R}^n} f(x) \\log f(x) \\, dx$, with the natural logarithm and $0 \\log 0 = 0$.","title":"Differential entropy"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/full-dimensional-body","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.FullDimensionalBody","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#full-dimensional-body","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a natural number $n$, a full-dimensional convex body in dimension $n$ is a nonempty compact convex set $K$ in the Euclidean space $\\mathbb{R}^n$ whose interior is nonempty, where $n$ is required to be positive.","title":"Full-dimensional convex body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/log-concave-law","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.LogConcaveLaw","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#log-concave-law","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a positive integer $n$, a log-concave law on $\\mathbb{R}^n$ is a probability measure $\\mu$ on $\\mathbb{R}^n$ together with a Lebesgue density $f$ of $\\mu$ that is log-concave: $f$ is measurable and nonnegative, and $f((1-t)x + ty) \\ge f(x)^{1-t} f(y)^t$ for all $x, y \\in \\mathbb{R}^n$ and $0 < t < 1$. Moreover every coordinate $x_i$ and every product $x_i x_j$ is $\\mu$-integrable, and $f \\log f$ is Lebesgue\u2026","title":"Log-concave law"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/covariance-matrix-pos-semidef","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.covarianceMatrix_posSemidef","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#covariance-matrix-pos-semidef","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For every log-concave law $\\mu$ on $\\mathbb{R}^n$, the covariance matrix of $\\mu$ is positive semidefinite.","title":"Covariance matrices of log-concave laws are positive semidefinite"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/covariance-matrix-eq-one-of-is-isotropic","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.covarianceMatrix_eq_one_of_isIsotropic","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#covariance-matrix-eq-one-of-is-isotropic","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"If $\\mu$ is an isotropic log-concave probability measure on $\\mathbb{R}^n$, then the covariance matrix of $\\mu$ is the $n \\times n$ identity matrix.","title":"Isotropic log-concave laws have identity covariance"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-law","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodyLaw","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-law","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a full-dimensional convex body $K$ in $\\mathbb{R}^n$, the uniform law on $K$ is the probability measure obtained by restricting Lebesgue measure to $K$ and dividing by the volume of $K$; it gives a set $A$ the mass $\\operatorname{vol}(A \\cap K) / \\operatorname{vol}(K)$.","title":"Uniform distribution on a convex body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-isotropic-constant","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodyIsotropicConstant","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-isotropic-constant","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a full-dimensional convex body $K$ in $\\mathbb{R}^n$, let $\\operatorname{Cov}(K)$ be the covariance matrix of the uniform probability law on $K$ and let $\\operatorname{vol}(K)$ be the Lebesgue volume of $K$. The isotropic constant of $K$ is the real number $L_K = \\left(\\det \\operatorname{Cov}(K) / \\operatorname{vol}(K)^2\\right)^{1/(2n)}$.","title":"Isotropic constant of a convex body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/law-isotropic-constant","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.lawIsotropicConstant","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#law-isotropic-constant","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a log-concave law $\\mu$ on $\\mathbb{R}^n$ with Lebesgue density $f$, the isotropic constant of $\\mu$ is $L_\\mu = e^{-h(f)/n} (\\det \\mathrm{Cov}(\\mu))^{1/(2n)}$, where $h(f) = -\\int_{\\mathbb{R}^n} f \\log f$ is the differential entropy of $f$ and $\\mathrm{Cov}(\\mu)$ is the covariance matrix of the coordinates under $\\mu$.","title":"Isotropic constant of a log-concave law"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/dimension-body-supremum","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.dimensionBodySupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#dimension-body-supremum","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, this is $\\sup_K L_K \\in [0, \\infty]$, the supremum of the isotropic constants $L_K$ over all full-dimensional convex bodies $K$ in $\\mathbb{R}^n$. Full-dimensional convex bodies are taken only in dimension $n \\ge 1$, so for $n = 0$ the supremum is over the empty family and is $0$.","title":"Supremum of isotropic constants in a fixed dimension"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/dimension-law-supremum","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.dimensionLawSupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#dimension-law-supremum","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, this is $\\sup_\\mu L_\\mu \\in [0, \\infty]$, the supremum of the isotropic constants $L_\\mu = e^{-\\operatorname{Ent}(f)/n} (\\det \\operatorname{Cov}(\\mu))^{1/(2n)}$ over all log-concave laws $\\mu$ on $\\mathbb{R}^n$, where $f$ is the log-concave Lebesgue density of $\\mu$, $\\operatorname{Ent}(f)$ its differential entropy and $\\operatorname{Cov}(\\mu)$ its covariance matrix. Here a log-concave law\u2026","title":"Supremum of entropic isotropic constants in a fixed dimension"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-supremum","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodySupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-supremum","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"This is the value $\\sup_{n \\ge 1} \\sup_K L_K \\in [0, \\infty]$, where for each dimension $n \\ge 1$ the inner supremum runs over all full-dimensional convex bodies $K$ in $\\mathbb{R}^n$ and $L_K$ is the isotropic constant of $K$.","title":"Supremum of isotropic constants of convex bodies"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/is-centrally-symmetric","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.IsCentrallySymmetric","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#is-centrally-symmetric","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a natural number $n$, a full-dimensional convex body $K \\subseteq \\mathbb{R}^n$ is centrally symmetric when $K = -K$, that is, for every $x \\in \\mathbb{R}^n$, $x \\in K$ if and only if $-x \\in K$.","title":"Centrally symmetric convex body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/centrally-symmetric-body-supremum","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.centrallySymmetricBodySupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#centrally-symmetric-body-supremum","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"This is the value $\\sup_{n \\ge 1} \\sup_K L_K \\in [0, \\infty]$, where for each dimension $n \\ge 1$ the inner supremum runs over the full-dimensional convex bodies $K$ in $\\mathbb{R}^n$ that are centrally symmetric, that is $K = -K$, and $L_K$ is the isotropic constant of $K$.","title":"Supremum of isotropic constants of symmetric convex bodies"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/squared-radius-thin-shell-width","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.squaredRadiusThinShellWidth","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#squared-radius-thin-shell-width","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a natural number $n$, the squared-radius thin-shell width in dimension $n$ is the supremum, in $[0, \\infty]$, of $\\left(\\int (|x|^2 - n)^2 \\, d\\mu(x) / n\\right)^{1/2}$ (with $0/0 = 0$) over all isotropic log-concave probability measures $\\mu$ on $\\mathbb{R}^n$, that is, those with a log-concave Lebesgue density, mean zero and identity covariance.","title":"Squared-radius thin-shell width"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/full-dimensional-law-supremum","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.fullDimensionalLawSupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#full-dimensional-law-supremum","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"This is the supremum, in $[0, \\infty]$, of the isotropic constant $L_\\mu = e^{-h(f)/n} (\\det \\mathrm{Cov}(\\mu))^{1/(2n)}$ over all positive integers $n$ and all log-concave laws $\\mu$ on $\\mathbb{R}^n$ with Lebesgue density $f$, where $h(f) = -\\int_{\\mathbb{R}^n} f \\log f$ is the differential entropy of $f$ and $\\mathrm{Cov}(\\mu)$ is the covariance matrix of the coordinates under $\\mu$.","title":"Supremum of the isotropic constant over full-dimensional log-concave laws"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/is-log-concave-density-indicator-const","kind":"node","lean":"MathlibExt.Probability.ThinShell.isLogConcaveDensity_indicator_const","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#is-log-concave-density-indicator-const","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $K \\subseteq \\mathbb{R}^n$ be a measurable convex set and let $c > 0$. Then the function equal to $c$ on $K$ and to $0$ outside $K$ is a log-concave density.","title":"Constant densities on convex sets are log-concave"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/cond-eq-with-density-indicator","kind":"node","lean":"MathlibExt.Probability.ThinShell.cond_eq_withDensity_indicator","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#cond-eq-with-density-indicator","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$ and let $K \\subseteq \\alpha$ be a measurable set with $\\mu(K) \\neq 0$. Then the conditional measure $\\mu(\\cdot \\mid K)$, that is $\\mu(K)^{-1}$ times the restriction of $\\mu$ to $K$, equals the measure with density $x \\mapsto \\mu(K)^{-1} \\mathbf{1}_K(x)$ with respect to $\\mu$. Here $\\mu(K)^{-1}$ is taken in $[0, \\infty]$, so it is $0$ when $\\mu(K) = \\infty$.","title":"Conditioning on a set is the normalized indicator density"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-density","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodyDensity","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-density","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $n \\ge 1$ and let $K \\subseteq \\mathbb{R}^n$ be a full-dimensional convex body, so that its Lebesgue measure $\\mathrm{Vol}(K)$ is finite and positive. The uniform density of $K$ is the function $f_K \\colon \\mathbb{R}^n \\to \\mathbb{R}$ equal to $1/\\mathrm{Vol}(K)$ on $K$ and to $0$ outside $K$.","title":"Uniform density of a convex body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-log-concave-law","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodyLogConcaveLaw","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-log-concave-law","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $n \\ge 1$ and let $K \\subseteq \\mathbb{R}^n$ be a full-dimensional convex body of Lebesgue measure $\\mathrm{Vol}(K)$. The uniform law of $K$ is the log-concave law on $\\mathbb{R}^n$ whose probability measure is Lebesgue measure restricted to $K$ and divided by $\\mathrm{Vol}(K)$, and whose density is the uniform density of $K$.","title":"Uniform law of a convex body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-density-entropy","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodyDensity_entropy","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-density-entropy","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $n \\ge 1$ and let $K \\subseteq \\mathbb{R}^n$ be a full-dimensional convex body with Lebesgue measure $\\mathrm{Vol}(K)$ and uniform density $f_K$, equal to $1/\\mathrm{Vol}(K)$ on $K$ and to $0$ outside $K$. Then the differential entropy $-\\int_{\\mathbb{R}^n} f_K(x) \\log f_K(x) \\, dx$ of $f_K$ equals $\\log \\mathrm{Vol}(K)$.","title":"Entropy of the uniform density of a convex body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-log-concave-law-isotropic-constant","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodyLogConcaveLaw_isotropicConstant","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-log-concave-law-isotropic-constant","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $n \\ge 1$ and let $K \\subseteq \\mathbb{R}^n$ be a full-dimensional convex body of Lebesgue measure $\\mathrm{Vol}(K)$, with uniform probability measure $\\lambda_K$, uniform density $f_K$, and covariance matrix $C_K$ of the coordinates under $\\lambda_K$. Then the entropy isotropic constant of the uniform law of $K$ equals the isotropic constant of $K$, that is,\u2026","title":"The entropy and volume isotropic constants of a convex body agree"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/dimension-body-supremum-le-law-supremum","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.dimensionBodySupremum_le_lawSupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#dimension-body-supremum-le-law-supremum","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the supremum of the isotropic constants of full-dimensional convex bodies in $\\mathbb{R}^n$ is at most the supremum of the entropy isotropic constants of probability laws on $\\mathbb{R}^n$ with a log-concave Lebesgue density $f$ such that every coordinate $x_i$ and every product $x_i x_j$ is integrable under the law and $f \\log f$ is Lebesgue integrable, both suprema taken in\u2026","title":"Body isotropic constants are bounded by the law supremum"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/intrinsic-log-concave-law","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.IntrinsicLogConcaveLaw","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#intrinsic-log-concave-law","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Fix an ambient dimension $N$. An intrinsic log-concave law on $\\mathbb{R}^N$ consists of a dimension $d$, a log-concave law $\\nu$ on $\\mathbb{R}^d$ (that is, $d > 0$ and $\\nu$ is a probability measure with a log-concave Lebesgue density $f$, integrable coordinates and coordinate products, and integrable $f \\log f$), an affine isometric embedding $\\varphi : \\mathbb{R}^d \\to \\mathbb{R}^N$ of Euclidean spaces, and a\u2026","title":"Log-concave law in intrinsic coordinates"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/intrinsic-law-isotropic-constant","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.intrinsicLawIsotropicConstant","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#intrinsic-law-isotropic-constant","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $\\mu$ be an intrinsic log-concave law on $\\mathbb{R}^N$, given by a log-concave law $\\nu$ on $\\mathbb{R}^d$ with density $f$ and an affine isometric embedding of $\\mathbb{R}^d$ onto the affine span of the support of $\\mu$. The isotropic constant of $\\mu$ is the entropy isotropic constant of $\\nu$,","title":"Isotropic constant of an intrinsic log-concave law"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/dimension-intrinsic-law-supremum","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.dimensionIntrinsicLawSupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#dimension-intrinsic-law-supremum","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a natural number $N$, this is the supremum in $[0, \\infty]$ of the isotropic constants $L_\\mu$ over all intrinsic log-concave laws $\\mu$ on $\\mathbb{R}^N$.","title":"Supremum of intrinsic isotropic constants in a fixed ambient dimension"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/entropy-isotropic-constant","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.entropyIsotropicConstant","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#entropy-isotropic-constant","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"This is the supremum, in $[0, \\infty]$, over all dimensions $N > 0$ of the supremum of the isotropic constants $L_\\mu$ over all intrinsic log-concave laws $\\mu$ on $\\mathbb{R}^N$. A law supported on a proper affine subspace thus contributes the isotropic constant computed in orthonormal coordinates on its affine span.","title":"Entropy-convention isotropic constant"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/entropy-isotropic-constant-le-full-dimen","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.entropyIsotropicConstant_le_fullDimensionalSupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#entropy-isotropic-constant-le-full-dimen","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"The entropy isotropic constant, the supremum of the isotropic constants of intrinsic log-concave laws over all ambient dimensions $N > 0$, is at most the supremum in $[0, \\infty]$, over all dimensions $n > 0$ and all log-concave laws $\\nu$ on $\\mathbb{R}^n$, of the entropy isotropic constant $L_\\nu = e^{-h(f)/n} \\det(\\operatorname{Cov} \\nu)^{1/(2n)}$, where $f$ is the density of $\\nu$ and $h(f) = -\\int f \\log f$.","title":"The entropy isotropic constant is at most the full-dimensional supremum"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-law-affine-span-support","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodyLaw_affineSpan_support","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-law-affine-span-support","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a full-dimensional convex body $K \\subseteq \\mathbb{R}^n$ (a convex body with nonempty interior, $n > 0$), the affine span of the support of the uniform probability law on $K$ is all of $\\mathbb{R}^n$.","title":"The uniform law on a convex body spans the whole space"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/body-intrinsic-log-concave-law","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.bodyIntrinsicLogConcaveLaw","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#body-intrinsic-log-concave-law","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a full-dimensional convex body $K \\subseteq \\mathbb{R}^n$, this is the intrinsic log-concave law on $\\mathbb{R}^n$ of intrinsic dimension $n$ whose chart is the identity map of $\\mathbb{R}^n$, whose law is the uniform probability law on $K$, and whose coordinate law is that uniform law viewed as a log-concave law with density $\\mathbf{1}_K / \\operatorname{vol}(K)$.","title":"The uniform law on a convex body as an intrinsic log-concave law"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/line-isometry","kind":"node","lean":"MathlibExt.Probability.ThinShell.lineIsometry","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#line-isometry","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"The linear isometric isomorphism from $\\mathbb{R}$ onto one-dimensional Euclidean space $\\mathbb{R}^1$ that sends a real number $x$ to the vector whose only coordinate is $x$.","title":"Identification of the line with one-dimensional Euclidean space"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/symmetric-interval","kind":"node","lean":"MathlibExt.Probability.ThinShell.symmetricInterval","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#symmetric-interval","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a real number $a$, the set of points of one-dimensional Euclidean space $\\mathbb{R}^1$ whose only coordinate lies in the closed interval $[-a, a]$. It is empty when $a < 0$.","title":"Symmetric interval in one-dimensional Euclidean space"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/uniform-interval","kind":"node","lean":"MathlibExt.Probability.ThinShell.uniformInterval","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#uniform-interval","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a real number $a$, the measure on one-dimensional Euclidean space $\\mathbb{R}^1$ obtained by conditioning Lebesgue measure on the set $\\{x : -a \\le x_1 \\le a\\}$ of points whose only coordinate $x_1$ lies in $[-a, a]$, that is, Lebesgue measure restricted to this set and divided by its measure. For $a > 0$ this is the uniform probability distribution with density $1/(2a)$ on the interval; for $a \\le 0$ the set\u2026","title":"Uniform distribution on a symmetric interval"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/uniform-interval-sqrt-three-is-isotropic","kind":"node","lean":"MathlibExt.Probability.ThinShell.uniformInterval_sqrt_three_isIsotropicLogConcave","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#uniform-interval-sqrt-three-is-isotropic","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"The uniform distribution on the symmetric interval $[-\\sqrt{3}, \\sqrt{3}]$ of one-dimensional Euclidean space $\\mathbb{R}^1$ is an isotropic log-concave probability measure: it has a log-concave density with respect to Lebesgue measure, mean zero and identity covariance.","title":"The uniform law on (\u2212\u221a3, \u221a3) is isotropic log-concave"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/uniform-interval-variance-le-thin-shell","kind":"node","lean":"MathlibExt.Probability.ThinShell.uniformInterval_variance_le_thinShellConstant","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#uniform-interval-variance-le-thin-shell","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $\\mu$ be the uniform distribution on the symmetric interval $[-\\sqrt{3}, \\sqrt{3}]$ of one-dimensional Euclidean space $\\mathbb{R}^1$. Its normalized radius variance $\\int (\\|x\\|^2 - 1)^2 \\, d\\mu(x)$ is at most the thin-shell variance constant, the supremum of the normalized radius variance over all isotropic log-concave probability measures in all positive dimensions.","title":"Thin-shell variance bound for the uniform law on (\u2212\u221a3, \u221a3)"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/uniform-interval-radius-fluctuation-le-t","kind":"node","lean":"MathlibExt.Probability.ThinShell.uniformInterval_radiusFluctuation_le_thinShellConstant","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#uniform-interval-radius-fluctuation-le-t","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"Let $\\mu$ be the uniform distribution on the symmetric interval $[-\\sqrt{3}, \\sqrt{3}]$ of one-dimensional Euclidean space $\\mathbb{R}^1$. Then $\\int (\\|x\\| - \\sqrt{1})^2 \\, d\\mu(x)$, the mean squared deviation of the Euclidean norm from $\\sqrt{1} = 1$, is at most the thin-shell variance constant.","title":"Radius fluctuation of the uniform law on (\u2212\u221a3, \u221a3)"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/uniform-interval-one-not-is-isotropic-lo","kind":"node","lean":"MathlibExt.Probability.ThinShell.uniformInterval_one_not_isIsotropicLogConcave","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#uniform-interval-one-not-is-isotropic-lo","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"The uniform distribution on the symmetric interval $[-1, 1]$ of one-dimensional Euclidean space $\\mathbb{R}^1$ is not an isotropic log-concave probability measure.","title":"The uniform law on (\u22121, 1) is not isotropic log-concave"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/interval-body","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.intervalBody","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#interval-body","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"For a real number $a > 0$, this is the closed interval $[-a, a]$ in the one-dimensional Euclidean space, regarded as a full-dimensional convex body.","title":"Symmetric interval as a convex body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/unit-interval-body","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.unitIntervalBody","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#unit-interval-body","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"The unit interval body is the closed interval $[-1/2, 1/2]$, of length one and centred at the origin, in the one-dimensional Euclidean space, regarded as a full-dimensional convex body.","title":"Unit interval body"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/inv-sqrt-twelve-le-body-supremum","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.inv_sqrt_twelve_le_bodySupremum","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#inv-sqrt-twelve-le-body-supremum","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"The supremum, over all positive integers $n$ and all full-dimensional convex bodies $K \\subseteq \\mathbb{R}^n$, of the isotropic constant $L_K = \\left(\\det \\mathrm{Cov}(K) / \\mathrm{Vol}(K)^2\\right)^{1/(2n)}$, where $\\mathrm{Cov}(K)$ is the covariance matrix of the uniform distribution on $K$, is at least $1/\\sqrt{12}$.","title":"Cube lower bound for the supremal isotropic constant"},{"id":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/unit-interval-body-mem-iff","kind":"node","lean":"MathlibExt.Probability.IsotropicConstant.unitIntervalBody_mem_iff","parent":20788,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws/index.html#unit-interval-body-mem-iff","scope":"Probability/high-dimensional-probability/isotropic-constant-of-log-concave-laws","status":"fully_proved","summary":"A point $x$ of the one-dimensional Euclidean space lies in the unit interval body if and only if its single coordinate $x_1$ satisfies $-1/2 \\le x_1 \\le 1/2$.","title":"Membership in the unit interval body"},{"data_url":"scopes/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture.json","id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","kind":"scope","navigation":"module","parent":20709,"result":false,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","title":"Kannan\u2013Lov\u00e1sz\u2013Simonovits conjecture"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/outer-minkowski-boundary","kind":"node","lean":"MathlibExt.Probability.KLS.outerMinkowskiBoundary","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#outer-minkowski-boundary","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"Let $\\mu$ be a measure on a pseudometric space $E$ and let $A \\subseteq E$. For $\\varepsilon > 0$ let $A_\\varepsilon = \\{x \\in E : \\inf_{a \\in A} d(x, a) \\le \\varepsilon\\}$ be the closed $\\varepsilon$-neighbourhood of $A$, with $\\inf \\emptyset = \\infty$. The outer Minkowski boundary measure of $A$ with respect to $\\mu$ is","title":"Outer Minkowski boundary measure"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/cheeger-coefficient","kind":"node","lean":"MathlibExt.Probability.KLS.cheegerCoefficient","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#cheeger-coefficient","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For a measure $\\mu$ on a pseudometric space $E$ equipped with a \u03c3-algebra, the Cheeger coefficient of $\\mu$ is the infimum in $[0, \\infty]$ of $\\mu^+(A) / \\min(\\mu(A), 1 - \\mu(A))$ over all measurable sets $A \\subseteq E$ with $0 < \\mu(A) < 1$, where $\\mu^+(A) = \\liminf_{\\varepsilon \\to 0^+} (\\mu(A_\\varepsilon) - \\mu(A)) / \\varepsilon$ is the outer Minkowski boundary measure of $A$ and $A_\\varepsilon$ is the closed\u2026","title":"Cheeger coefficient"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/inverse-cheeger-constant","kind":"node","lean":"MathlibExt.Probability.KLS.inverseCheegerConstant","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#inverse-cheeger-constant","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For a measure $\\mu$ on a pseudometric space $E$ equipped with a \u03c3-algebra, the inverse Cheeger constant of $\\mu$ is $1 / h(\\mu) \\in [0, \\infty]$, where $h(\\mu)$ is the Cheeger coefficient of $\\mu$, with the conventions $1 / 0 = \\infty$ and $1 / \\infty = 0$.","title":"Inverse Cheeger constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/is-halfspace","kind":"node","lean":"MathlibExt.Probability.KLS.IsHalfspace","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#is-halfspace","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"A set $A \\subseteq \\mathbb{R}^n$ is a half-space if there are a nonzero vector $v \\in \\mathbb{R}^n$ and a real number $a$ such that $A = \\{x \\in \\mathbb{R}^n : \\langle v, x \\rangle \\le a\\}$.","title":"Half-spaces"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/halfspace-cheeger-coefficient","kind":"node","lean":"MathlibExt.Probability.KLS.halfspaceCheegerCoefficient","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#halfspace-cheeger-coefficient","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}^n$ with the Euclidean metric, the half-space Cheeger coefficient of $\\mu$ is the infimum in $[0, \\infty]$ of $\\mu^+(H) / \\min(\\mu(H), 1 - \\mu(H))$ over all closed half-spaces $H = \\{x : \\langle v, x \\rangle \\le a\\}$ with $v \\ne 0$ and $a \\in \\mathbb{R}$ such that $0 < \\mu(H) < 1$, where $\\mu^+(H)$ is the outer Minkowski boundary measure of $H$. The infimum of the empty family is\u2026","title":"Half-space Cheeger coefficient"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/inverse-halfspace-cheeger-constant","kind":"node","lean":"MathlibExt.Probability.KLS.inverseHalfspaceCheegerConstant","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#inverse-halfspace-cheeger-constant","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}^n$, the inverse half-space Cheeger constant of $\\mu$ is $h^{-1}$, where $h \\in [0, \\infty]$ is the half-space Cheeger coefficient of $\\mu$ and the inverse is taken in $[0, \\infty]$, so that $0^{-1} = \\infty$ and $\\infty^{-1} = 0$.","title":"Inverse half-space Cheeger constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/dimensional-kls-constant","kind":"node","lean":"MathlibExt.Probability.KLS.dimensionalKlsConstant","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#dimensional-kls-constant","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the dimensional KLS constant in dimension $n$ is the supremum in $[0, \\infty]$ of the inverse Cheeger constants of all isotropic log-concave probability measures on $\\mathbb{R}^n$ with the Euclidean metric.","title":"Dimensional KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/kls-constant","kind":"node","lean":"MathlibExt.Probability.KLS.klsConstant","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#kls-constant","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"The KLS constant is the supremum in $[0, \\infty]$ of the dimensional KLS constants in dimension $n$, taken over all positive integers $n$.","title":"KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/is-smooth-poincare-bound","kind":"node","lean":"MathlibExt.Probability.KLS.IsSmoothPoincareBound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#is-smooth-poincare-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}^n$ and let $C \\in [0, \\infty]$. For a differentiable $g : \\mathbb{R}^n \\to \\mathbb{R}$ write $\\|\\nabla g(x)\\|$ for the Euclidean norm of the gradient of $g$ at $x$ (equivalently, the operator norm of the derivative of $g$ at $x$). Then $C$ is a smooth Poincar\u00e9 bound for $\\mu$ if for every infinitely differentiable function $f : \\mathbb{R}^n \\to \\mathbb{R}$ such that $f$, $f^2$\u2026","title":"Smooth Poincar\u00e9 bounds"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/smooth-poincare-constant","kind":"node","lean":"MathlibExt.Probability.KLS.smoothPoincareConstant","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#smooth-poincare-constant","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"The smooth Poincar\u00e9 constant of a measure $\\mu$ on $\\mathbb{R}^n$ is the infimum in $[0, \\infty]$ of all $C \\in [0, \\infty]$ that are smooth Poincar\u00e9 bounds for $\\mu$; it is $\\infty$ when $\\mu$ has no smooth Poincar\u00e9 bound.","title":"Smooth Poincar\u00e9 constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/outer-minkowski-boundary-empty","kind":"node","lean":"MathlibExt.Probability.KLS.outerMinkowskiBoundary_empty","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#outer-minkowski-boundary-empty","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For every measure $\\mu$ on a pseudometric space $E$, the outer Minkowski boundary measure of the empty set with respect to $\\mu$ is $0$.","title":"The empty set has zero outer Minkowski boundary"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/cheeger-coefficient-zero","kind":"node","lean":"MathlibExt.Probability.KLS.cheegerCoefficient_zero","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#cheeger-coefficient-zero","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"On every pseudometric space $E$ equipped with a \u03c3-algebra, the zero measure has Cheeger coefficient $\\infty$.","title":"Cheeger coefficient of the zero measure"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/inverse-cheeger-constant-zero","kind":"node","lean":"MathlibExt.Probability.KLS.inverseCheegerConstant_zero","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#inverse-cheeger-constant-zero","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"On every pseudometric space $E$ equipped with a \u03c3-algebra, the zero measure has inverse Cheeger constant $0$.","title":"Inverse Cheeger constant of the zero measure"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/cheeger-coefficient-le-halfspace","kind":"node","lean":"MathlibExt.Probability.KLS.cheegerCoefficient_le_halfspace","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#cheeger-coefficient-le-halfspace","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}^n$ with the Euclidean metric, the Cheeger coefficient of $\\mu$ is at most its half-space Cheeger coefficient.","title":"The Cheeger coefficient is at most the half-space Cheeger coefficient"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/inverse-cheeger-constant-le-dimensional","kind":"node","lean":"MathlibExt.Probability.KLS.inverseCheegerConstant_le_dimensional","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#inverse-cheeger-constant-le-dimensional","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$ with the Euclidean metric, the inverse Cheeger constant of $\\mu$ is at most the dimensional KLS constant in dimension $n$.","title":"Inverse Cheeger constant bounded by the dimensional KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/dimensional-kls-constant-le-constant","kind":"node","lean":"MathlibExt.Probability.KLS.dimensionalKlsConstant_le_constant","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#dimensional-kls-constant-le-constant","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For every positive integer $n$, the dimensional KLS constant in dimension $n$ is at most the KLS constant.","title":"Dimensional KLS constants are bounded by the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/inverse-cheeger-constant-le-constant","kind":"node","lean":"MathlibExt.Probability.KLS.inverseCheegerConstant_le_constant","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#inverse-cheeger-constant-le-constant","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For every positive integer $n$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$ with the Euclidean metric, the inverse Cheeger constant of $\\mu$ is at most the KLS constant.","title":"Inverse Cheeger constant bounded by the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/kls-constant-le-iff","kind":"node","lean":"MathlibExt.Probability.KLS.klsConstant_le_iff","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#kls-constant-le-iff","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For every $C \\in [0, \\infty]$, the KLS constant is at most $C$ if and only if, for every positive integer $n$, the dimensional KLS constant in dimension $n$ is at most $C$.","title":"Bounding the KLS constant dimension by dimension"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/kls-constant-le-iff-measures","kind":"node","lean":"MathlibExt.Probability.KLS.klsConstant_le_iff_measures","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#kls-constant-le-iff-measures","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"For every $C \\in [0, \\infty]$, the KLS constant is at most $C$ if and only if, for every positive integer $n$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$, the inverse Cheeger constant of $\\mu$ is at most $C$.","title":"Bounding the KLS constant measure by measure"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/kls-constant-lt-top-iff","kind":"node","lean":"MathlibExt.Probability.KLS.klsConstant_lt_top_iff","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#kls-constant-lt-top-iff","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"The KLS constant is finite if and only if there is a real number $C$ such that, for every positive integer $n$ and every isotropic log-concave probability measure $\\mu$ on $\\mathbb{R}^n$, the inverse Cheeger constant of $\\mu$ is at most $\\max(C, 0)$.","title":"Finiteness of the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/mono","kind":"node","lean":"MathlibExt.Probability.KLS.IsSmoothPoincareBound.mono","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#mono","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}^n$ and let $C \\le D$ in $[0, \\infty]$. If $C$ is a smooth Poincar\u00e9 bound for $\\mu$, then so is $D$.","title":"Smooth Poincar\u00e9 bounds are upward closed"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/smooth-poincare-constant-le-of-bound","kind":"node","lean":"MathlibExt.Probability.KLS.smoothPoincareConstant_le_of_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#smooth-poincare-constant-le-of-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}^n$ and let $C \\in [0, \\infty]$. If $C$ is a smooth Poincar\u00e9 bound for $\\mu$, then the smooth Poincar\u00e9 constant of $\\mu$ is at most $C$.","title":"The smooth Poincar\u00e9 constant is at most every smooth Poincar\u00e9 bound"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/kannan-lovasz-simonovits1995-bound","kind":"node","lean":"MathlibExt.Probability.KLS.kannanLovaszSimonovits1995_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#kannan-lovasz-simonovits1995-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and an integer $N \\ge 3$ such that for every $n \\ge N$ the dimensional KLS constant $\\psi_n \\in [0, \\infty]$ satisfies $\\psi_n \\le C \\sqrt{n}$.","title":"Kannan\u2013Lov\u00e1sz\u2013Simonovits bound on the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/lee-vempala2024-bound","kind":"node","lean":"MathlibExt.Probability.KLS.leeVempala2024_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#lee-vempala2024-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and an integer $N \\ge 3$ such that for every $n \\ge N$ the dimensional KLS constant $\\psi_n \\in [0, \\infty]$ satisfies $\\psi_n \\le C n^{1/4}$.","title":"Lee\u2013Vempala bound on the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/chen2021-bound","kind":"node","lean":"MathlibExt.Probability.KLS.chen2021_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#chen2021-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and an integer $N \\ge 3$ such that for every $n \\ge N$ the dimensional KLS constant $\\psi_n \\in [0, \\infty]$ satisfies $\\psi_n \\le \\exp\\bigl(C \\sqrt{\\log n} \\, \\log \\log n\\bigr)$.","title":"Chen's subpolynomial bound on the KLS constant, weaker form"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/klartag-lehec2022-bound","kind":"node","lean":"MathlibExt.Probability.KLS.klartagLehec2022_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#klartag-lehec2022-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and an integer $N \\ge 3$ such that for every $n \\ge N$ the dimensional KLS constant $\\psi_n \\in [0, \\infty]$ satisfies $\\psi_n \\le C (\\log n)^5$.","title":"Klartag\u2013Lehec polylogarithmic bound on the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/jambulapati-lee-vempala2022-bound","kind":"node","lean":"MathlibExt.Probability.KLS.jambulapatiLeeVempala2022_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#jambulapati-lee-vempala2022-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"Let $\\alpha = 1 + \\bigl(1 + 7\\sqrt{2} + \\sqrt{53 - 4\\sqrt{2}}\\bigr)/8$. There are a real $C > 0$ and an integer $N \\ge 3$ such that for every $n \\ge N$ the dimensional KLS constant $\\psi_n \\in [0, \\infty]$ satisfies $\\psi_n \\le C (\\log n)^{\\alpha}$.","title":"Jambulapati\u2013Lee\u2013Vempala polylogarithmic bound on the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/lehec-reported-bound-weaker-specializati","kind":"node","lean":"MathlibExt.Probability.KLS.lehec_reported_bound_weaker_specialization","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#lehec-reported-bound-weaker-specializati","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"There are a real $C > 0$ and an integer $N \\ge 3$ such that for every $n \\ge N$ the dimensional KLS constant $\\psi_n \\in [0, \\infty]$ satisfies $\\psi_n \\le C (\\log n)^{3.083}$.","title":"Lehec's reported polylogarithmic bound on the KLS constant, weakened"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/klartag2023-bound","kind":"node","lean":"MathlibExt.Probability.KLS.klartag2023_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#klartag2023-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"There is a real $C > 0$ such that for every integer $n \\ge 2$ the dimensional KLS constant $\\psi_n \\in [0, \\infty]$ satisfies $\\psi_n \\le C \\sqrt{\\log n}$.","title":"Klartag's square-root-logarithmic bound on the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/std-gaussian-inverse-cheeger","kind":"node","lean":"MathlibExt.Probability.KLS.stdGaussian_inverseCheeger","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#std-gaussian-inverse-cheeger","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"For every $n \\ge 1$, the standard Gaussian measure on $\\mathbb{R}^n$ has inverse Cheeger constant exactly $\\sqrt{\\pi/2}$.","title":"Inverse Cheeger constant of the standard Gaussian"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/sqrt-pi-div-two-le-kls-constant","kind":"node","lean":"MathlibExt.Probability.KLS.sqrtPiDivTwo_le_klsConstant","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#sqrt-pi-div-two-le-kls-constant","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"The KLS constant is at least $\\sqrt{\\pi/2}$.","title":"A lower bound for the KLS constant"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/catalogue-poincare-lower-bound","kind":"node","lean":"MathlibExt.Probability.KLS.catalogue_poincare_lower_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#catalogue-poincare-lower-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"Let $n \\ge 1$ and let $\\mu$ be an isotropic log-concave probability measure on $\\mathbb{R}^n$, with inverse Cheeger constant $\\psi(\\mu) \\in [0, \\infty]$ and smooth Poincar\u00e9 constant $C_P(\\mu) \\in [0, \\infty]$. Then $\\frac{1}{\\pi} \\psi(\\mu)^2 \\le C_P(\\mu)$.","title":"Lower Poincar\u00e9\u2013Cheeger comparison with coefficient 1/\u03c0"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/klartag2023-poincare-lower-bound","kind":"node","lean":"MathlibExt.Probability.KLS.klartag2023_poincare_lower_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#klartag2023-poincare-lower-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"Let $n \\ge 1$ and let $\\mu$ be an isotropic log-concave probability measure on $\\mathbb{R}^n$, with inverse Cheeger constant $\\psi(\\mu) \\in [0, \\infty]$ and smooth Poincar\u00e9 constant $C_P(\\mu) \\in [0, \\infty]$. Then $\\frac{1}{9} \\psi(\\mu)^2 \\le C_P(\\mu)$.","title":"Lower Poincar\u00e9\u2013Cheeger comparison with coefficient 1/9"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/poincare-cheeger-upper-bound","kind":"node","lean":"MathlibExt.Probability.KLS.poincare_cheeger_upper_bound","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#poincare-cheeger-upper-bound","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"Let $n \\ge 1$, let $\\mu$ be an isotropic log-concave probability measure on $\\mathbb{R}^n$, and let $\\psi_\\mu \\in [0, \\infty]$ be the inverse Cheeger constant of $\\mu$. Then the smooth Poincar\u00e9 constant of $\\mu$ is at most $4 \\psi_\\mu^2$.","title":"Cheeger's inequality for isotropic log-concave measures"},{"id":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/kls-le-thin-shell-variance-log-squared","kind":"node","lean":"MathlibExt.Probability.KLS.kls_le_thinShell_variance_log_squared","parent":20830,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture/index.html#kls-le-thin-shell-variance-log-squared","scope":"Probability/high-dimensional-probability/kannan-lov-sz-simonovits-conjecture","status":"can_prove","summary":"Let $\\Sigma \\in [0, \\infty]$ be the thin-shell variance constant: the supremum, over all dimensions $m \\ge 1$ and all isotropic log-concave probability measures $\\mu$ on $\\mathbb{R}^m$, of $\\frac{1}{m} \\int (\\lVert x \\rVert^2 - m)^2 \\, d\\mu(x)$. There is a real $C > 0$ such that for every integer $n \\ge 3$ the dimensional KLS constant $\\psi_n \\in [0, \\infty]$ satisfies $\\psi_n^2 \\le C (\\log n)^2 \\, \\Sigma$.","title":"Dimensional KLS constant versus thin-shell variance, up to a squared logarithm"},{"data_url":"scopes/Probability/high-dimensional-probability/wigner-s-semicircle-law.json","id":"Probability/high-dimensional-probability/wigner-s-semicircle-law","kind":"scope","navigation":"module","parent":20709,"result":false,"route":"../roadmap/Probability/high-dimensional-probability/wigner-s-semicircle-law/index.html","scope":"Probability/high-dimensional-probability/wigner-s-semicircle-law","status":"can_prove","title":"Wigner's semicircle law"},{"id":"Probability/high-dimensional-probability/wigner-s-semicircle-law/empirical-spectral-measure-herm","kind":"node","lean":"MathlibExt.Analysis.WignerSemicircleWanted.Analysis.WignerSemicircleProblem.empiricalSpectralMeasureHerm","parent":20866,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/wigner-s-semicircle-law/index.html#empirical-spectral-measure-herm","scope":"Probability/high-dimensional-probability/wigner-s-semicircle-law","status":"fully_proved","summary":"Let $W$ be a Hermitian $n \\times n$ complex matrix with real eigenvalues $\\lambda_1, \\dots, \\lambda_n$, listed with multiplicity. The empirical spectral measure of $W$ is the measure $\\frac{1}{n} \\sum_{j=1}^{n} \\delta_{\\lambda_j}$ on $\\mathbb{R}$, where $\\delta_\\lambda$ is the Dirac measure at $\\lambda$; for $n = 0$ it is the zero measure.","title":"Empirical spectral measure of a Hermitian matrix"},{"id":"Probability/high-dimensional-probability/wigner-s-semicircle-law/semicircle-law","kind":"node","lean":"MathlibExt.Analysis.WignerSemicircleWanted.Analysis.WignerSemicircleProblem.semicircleLaw","parent":20866,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/wigner-s-semicircle-law/index.html#semicircle-law","scope":"Probability/high-dimensional-probability/wigner-s-semicircle-law","status":"fully_proved","summary":"The semicircle distribution is the measure $\\sigma$ on $\\mathbb{R}$ given by $\\sigma(A) = \\int_{A \\cap [-2, 2]} \\frac{\\sqrt{4 - x^2}}{2\\pi} \\, dx$ for every measurable set $A \\subseteq \\mathbb{R}$.","title":"Semicircle distribution"},{"id":"Probability/high-dimensional-probability/wigner-s-semicircle-law/wigner-matrix","kind":"node","lean":"MathlibExt.Analysis.WignerSemicircleWanted.Analysis.WignerSemicircleProblem.wignerMatrix","parent":20866,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/wigner-s-semicircle-law/index.html#wigner-matrix","scope":"Probability/high-dimensional-probability/wigner-s-semicircle-law","status":"fully_proved","summary":"Let $X_{ij} : \\Omega \\to \\mathbb{R}$, for $i, j \\in \\mathbb{N}$, be real functions on a set $\\Omega$. For $n \\in \\mathbb{N}$ and $\\omega \\in \\Omega$, the Wigner matrix $W_n(\\omega)$ is the $n \\times n$ complex matrix, with rows and columns indexed by $\\{0, \\dots, n - 1\\}$, whose entry at $(i, j)$ is the real number $X_{\\min(i, j)\\, \\max(i, j)}(\\omega)$. Only the $X_{ij}$ with $i \\le j$ are used, and the matrix is\u2026","title":"Wigner matrix"},{"id":"Probability/high-dimensional-probability/wigner-s-semicircle-law/wigner-matrix-is-hermitian","kind":"node","lean":"MathlibExt.Analysis.WignerSemicircleWanted.Analysis.WignerSemicircleProblem.wignerMatrix_isHermitian","parent":20866,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/wigner-s-semicircle-law/index.html#wigner-matrix-is-hermitian","scope":"Probability/high-dimensional-probability/wigner-s-semicircle-law","status":"fully_proved","summary":"For every family of real functions $X_{ij} : \\Omega \\to \\mathbb{R}$, $i, j \\in \\mathbb{N}$, on a set $\\Omega$, every $n \\in \\mathbb{N}$ and every $\\omega \\in \\Omega$, the Wigner matrix $W_n(\\omega)$ is Hermitian.","title":"Wigner matrices are Hermitian"},{"id":"Probability/high-dimensional-probability/wigner-s-semicircle-law/wigner-semicircle","kind":"node","lean":"MathlibExt.Analysis.WignerSemicircleWanted.Analysis.WignerSemicircleProblem.wigner_semicircle","parent":20866,"result":true,"route":"../roadmap/Probability/high-dimensional-probability/wigner-s-semicircle-law/index.html#wigner-semicircle","scope":"Probability/high-dimensional-probability/wigner-s-semicircle-law","status":"can_prove","summary":"Let $(\\Omega, \\mu)$ be a probability space and let $X_{ij} : \\Omega \\to \\mathbb{R}$, for $i, j \\in \\mathbb{N}$, be measurable functions. Suppose that the family $(X_{ij})_{i \\le j}$ is mutually independent, that all $X_{ij}$ with $i \\le j$ have the same distribution, and that for $i \\le j$ both $X_{ij}$ and $X_{ij}^2$ are integrable with $\\mathbb{E}[X_{ij}] = 0$ and $\\mathbb{E}[X_{ij}^2] = 1$. For\u2026","title":"Wigner's semicircle law"},{"data_url":"scopes/Probability/distributions.json","id":"Probability/distributions","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/distributions/index.html","scope":"Probability/distributions","status":"can_prove","summary":"Named discrete and continuous laws and their moments, densities, mass functions and tails, together with distribution functions, quantiles, medians and joint laws; sampling laws used in tests (t, F) go to Linear models and multivariate analysis.","title":"Probability distributions"},{"data_url":"scopes/Probability/distributions/discrete.json","id":"Probability/distributions/discrete","kind":"scope","parent":20872,"result":false,"route":"../roadmap/Probability/distributions/discrete/index.html","scope":"Probability/distributions/discrete","status":"fully_proved","summary":"Laws on countable sets, with their mass functions, moments and tails: discrete uniform, binomial, Poisson binomial, hypergeometric, multinomial, negative binomial and beta-binomial laws, and expectation under a probability mass function.","title":"Discrete distributions"},{"data_url":"scopes/Probability/distributions/discrete/binomial-distribution.json","id":"Probability/distributions/discrete/binomial-distribution","kind":"scope","navigation":"module","parent":20873,"result":false,"route":"../roadmap/Probability/distributions/discrete/binomial-distribution/index.html","scope":"Probability/distributions/discrete/binomial-distribution","status":"fully_proved","title":"Binomial distribution"},{"id":"Probability/distributions/discrete/binomial-distribution/integral-mul-sub-one-map-cast-binomial","kind":"node","lean":"ProbabilityTheory.integral_mul_sub_one_map_cast_binomial","parent":20874,"result":true,"route":"../roadmap/Probability/distributions/discrete/binomial-distribution/index.html#integral-mul-sub-one-map-cast-binomial","scope":"Probability/distributions/discrete/binomial-distribution","status":"fully_proved","summary":"`E[X (X - 1)] = n (n - 1) p\u00b2`.","title":"Second factorial moment of the binomial law"},{"id":"Probability/distributions/discrete/binomial-distribution/integral-sq-map-cast-binomial","kind":"node","lean":"ProbabilityTheory.integral_sq_map_cast_binomial","parent":20874,"result":true,"route":"../roadmap/Probability/distributions/discrete/binomial-distribution/index.html#integral-sq-map-cast-binomial","scope":"Probability/distributions/discrete/binomial-distribution","status":"fully_proved","summary":"`E[X\u00b2] = n p (1 - p) + (n p)\u00b2`.","title":"Second moment of the binomial law"},{"id":"Probability/distributions/discrete/binomial-distribution/variance-id-map-cast-binomial","kind":"node","lean":"ProbabilityTheory.variance_id_map_cast_binomial","parent":20874,"result":true,"route":"../roadmap/Probability/distributions/discrete/binomial-distribution/index.html#variance-id-map-cast-binomial","scope":"Probability/distributions/discrete/binomial-distribution","status":"fully_proved","summary":"`Var X = n p (1 - p)`.","title":"Variance of the binomial law"},{"id":"Probability/distributions/discrete/binomial-distribution/integral-mul-sub-one-of-has-law-binomial","kind":"node","lean":"ProbabilityTheory.integral_mul_sub_one_of_hasLaw_binomial","parent":20874,"result":true,"route":"../roadmap/Probability/distributions/discrete/binomial-distribution/index.html#integral-mul-sub-one-of-has-law-binomial","scope":"Probability/distributions/discrete/binomial-distribution","status":"fully_proved","summary":"If `X` has law `Bin(\u211d, n, p)` then `E[X (X - 1)] = n (n - 1) p\u00b2`.","title":"Second factorial moment of a binomial random variable"},{"id":"Probability/distributions/discrete/binomial-distribution/integral-sq-of-has-law-binomial","kind":"node","lean":"ProbabilityTheory.integral_sq_of_hasLaw_binomial","parent":20874,"result":true,"route":"../roadmap/Probability/distributions/discrete/binomial-distribution/index.html#integral-sq-of-has-law-binomial","scope":"Probability/distributions/discrete/binomial-distribution","status":"fully_proved","summary":"If `X` has law `Bin(\u211d, n, p)` then `E[X\u00b2] = n p (1 - p) + (n p)\u00b2`.","title":"Second moment of a binomial random variable"},{"id":"Probability/distributions/discrete/binomial-distribution/variance-of-has-law-binomial","kind":"node","lean":"ProbabilityTheory.variance_of_hasLaw_binomial","parent":20874,"result":true,"route":"../roadmap/Probability/distributions/discrete/binomial-distribution/index.html#variance-of-has-law-binomial","scope":"Probability/distributions/discrete/binomial-distribution","status":"fully_proved","summary":"If `X` has law `Bin(\u211d, n, p)` then `Var X = n p (1 - p)`.","title":"Variance of a binomial random variable"},{"data_url":"scopes/Probability/distributions/discrete/discrete-uniform-distribution.json","id":"Probability/distributions/discrete/discrete-uniform-distribution","kind":"scope","navigation":"module","parent":20873,"result":false,"route":"../roadmap/Probability/distributions/discrete/discrete-uniform-distribution/index.html","scope":"Probability/distributions/discrete/discrete-uniform-distribution","status":"fully_proved","title":"Discrete uniform distribution"},{"id":"Probability/distributions/discrete/discrete-uniform-distribution/uniform-finset","kind":"node","lean":"PMF.uniformFinset","parent":20881,"result":true,"route":"../roadmap/Probability/distributions/discrete/discrete-uniform-distribution/index.html#uniform-finset","scope":"Probability/distributions/discrete/discrete-uniform-distribution","status":"fully_proved","summary":"For a nonempty finite subset $s$ of a type $\\alpha$, the uniform distribution on $s$ is the probability mass function on $\\alpha$ that gives mass $1/|s|$ to each element of $s$ and mass $0$ to every other element of $\\alpha$.","title":"Uniform distribution on a finite set"},{"id":"Probability/distributions/discrete/discrete-uniform-distribution/uniform-fintype","kind":"node","lean":"PMF.uniformFintype","parent":20881,"result":true,"route":"../roadmap/Probability/distributions/discrete/discrete-uniform-distribution/index.html#uniform-fintype","scope":"Probability/distributions/discrete/discrete-uniform-distribution","status":"fully_proved","summary":"For a nonempty finite type $\\alpha$, the uniform distribution on $\\alpha$ is the probability mass function on $\\alpha$ that gives mass $1/|\\alpha|$ to every element of $\\alpha$.","title":"Uniform distribution on a finite type"},{"id":"Probability/distributions/discrete/discrete-uniform-distribution/uniform-finset-univ","kind":"node","lean":"PMF.uniformFinset_univ","parent":20881,"result":true,"route":"../roadmap/Probability/distributions/discrete/discrete-uniform-distribution/index.html#uniform-finset-univ","scope":"Probability/distributions/discrete/discrete-uniform-distribution","status":"fully_proved","summary":"For a nonempty finite type $\\alpha$, the uniform distribution on the finite set of all elements of $\\alpha$ equals the uniform distribution on $\\alpha$.","title":"Uniform distribution on the whole finite type"},{"id":"Probability/distributions/discrete/discrete-uniform-distribution/to-outer-measure-uniform-fintype-apply","kind":"node","lean":"PMF.toOuterMeasure_uniformFintype_apply","parent":20881,"result":true,"route":"../roadmap/Probability/distributions/discrete/discrete-uniform-distribution/index.html#to-outer-measure-uniform-fintype-apply","scope":"Probability/distributions/discrete/discrete-uniform-distribution","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite type. For every subset $s \\subseteq \\alpha$, the outer measure associated to the uniform distribution on $\\alpha$ assigns to $s$ the value $|s| / |\\alpha|$, computed in the extended nonnegative reals.","title":"Outer measure of an event under the uniform distribution"},{"id":"Probability/distributions/discrete/discrete-uniform-distribution/to-measure-uniform-fintype-apply","kind":"node","lean":"PMF.toMeasure_uniformFintype_apply","parent":20881,"result":true,"route":"../roadmap/Probability/distributions/discrete/discrete-uniform-distribution/index.html#to-measure-uniform-fintype-apply","scope":"Probability/distributions/discrete/discrete-uniform-distribution","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite type carrying a \u03c3-algebra. For every measurable set $s \\subseteq \\alpha$, the probability measure associated to the uniform distribution on $\\alpha$ gives $s$ the measure $|s| / |\\alpha|$, computed in the extended nonnegative reals.","title":"Probability of an event under the uniform distribution"},{"data_url":"scopes/Probability/distributions/discrete/expectation-under-a-probability-mass-function.json","id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","kind":"scope","navigation":"module","parent":20873,"result":false,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","title":"Expectation under a probability mass function"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect","kind":"node","lean":"PMF.expect","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $\\alpha$ be a type carrying a \u03c3-algebra in which singletons are measurable, let $p$ be a probability mass function on $\\alpha$ and let $f : \\alpha \\to \\mathbb{R}$. The expectation of $f$ under $p$ is the Bochner integral $\\int f \\, d\\mu_p$, where $\\mu_p$ is the probability measure on $\\alpha$ associated to $p$.","title":"Expectation under a probability mass function"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-eq-tsum","kind":"node","lean":"PMF.expect_eq_tsum","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-eq-tsum","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a measurable space $\\alpha$ in which singletons are measurable, and let $f : \\alpha \\to \\mathbb{R}$ be integrable with respect to the probability measure associated to $p$. Then the expectation of $f$ under $p$ equals the unordered sum $\\sum_{a \\in \\alpha} p(a) f(a)$.","title":"Expectation as a series"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-eq-sum","kind":"node","lean":"PMF.expect_eq_sum","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-eq-sum","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a finite measurable space $\\alpha$ in which singletons are measurable, and let $f : \\alpha \\to \\mathbb{R}$. Then the expectation of $f$ under $p$ equals the finite sum $\\sum_{a \\in \\alpha} p(a) f(a)$.","title":"Expectation on a finite type"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-const","kind":"node","lean":"PMF.expect_const","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-const","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $\\alpha$ be a finite set with a $\\sigma$-algebra in which singletons are measurable and let $p$ be a probability mass function on $\\alpha$. For every $c \\in \\mathbb{R}$, the expectation of the constant function $a \\mapsto c$ under $p$ is $c$.","title":"Expectation of a constant on a finite type"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-uniform-of-fintype","kind":"node","lean":"PMF.expect_uniformOfFintype","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-uniform-of-fintype","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite type carrying a \u03c3-algebra in which singletons are measurable. For every function $f : \\alpha \\to \\mathbb{R}$, the expectation of $f$ under the uniform distribution on $\\alpha$ equals the average $\\frac{1}{|\\alpha|} \\sum_{a \\in \\alpha} f(a)$.","title":"Expectation under the uniform distribution"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-add","kind":"node","lean":"PMF.expect_add","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-add","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $\\alpha$ be a finite set with a $\\sigma$-algebra in which singletons are measurable, let $p$ be a probability mass function on $\\alpha$, and write $\\mathbb{E}_p[h]$ for the expectation of $h : \\alpha \\to \\mathbb{R}$ under $p$. For all $f, g : \\alpha \\to \\mathbb{R}$, $\\mathbb{E}_p[f + g] = \\mathbb{E}_p[f] + \\mathbb{E}_p[g]$.","title":"Additivity of expectation on a finite type"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-sum","kind":"node","lean":"PMF.expect_sum","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-sum","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a finite type $\\alpha$ carrying a \u03c3-algebra in which singletons are measurable, and write $\\mathbb{E}_p[g]$ for the expectation of $g : \\alpha \\to \\mathbb{R}$ under $p$. For every finite subset $s$ of an index type $\\iota$ and every family of functions $f_i : \\alpha \\to \\mathbb{R}$, $i \\in \\iota$, we have\u2026","title":"Expectation of a finite sum"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-const-mul","kind":"node","lean":"PMF.expect_const_mul","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-const-mul","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $\\alpha$ be a finite set with a $\\sigma$-algebra in which singletons are measurable, let $p$ be a probability mass function on $\\alpha$, and write $\\mathbb{E}_p[h]$ for the expectation of $h : \\alpha \\to \\mathbb{R}$ under $p$. For every $c \\in \\mathbb{R}$ and $f : \\alpha \\to \\mathbb{R}$, the function $a \\mapsto c\\,f(a)$ has expectation $c\\,\\mathbb{E}_p[f]$.","title":"Homogeneity of expectation on a finite type"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-smul","kind":"node","lean":"PMF.expect_smul","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-smul","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a finite measurable space $\\alpha$ in which singletons are measurable, let $c \\in \\mathbb{R}$ and let $f : \\alpha \\to \\mathbb{R}$. The expectation under $p$ of the scalar multiple $c f$ (the function $a \\mapsto c \\cdot f(a)$) equals $c$ times the expectation of $f$ under $p$.","title":"Expectation of a scalar multiple"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-nonneg","kind":"node","lean":"PMF.expect_nonneg","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-nonneg","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a finite measurable space $\\alpha$ in which singletons are measurable, and let $f : \\alpha \\to \\mathbb{R}$ satisfy $f(a) \\ge 0$ for every $a \\in \\alpha$. Then the expectation of $f$ under $p$ is nonnegative.","title":"Nonnegativity of expectation"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-le-expect","kind":"node","lean":"PMF.expect_le_expect","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-le-expect","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a finite measurable space $\\alpha$ in which singletons are measurable, and let $f, g : \\alpha \\to \\mathbb{R}$ satisfy $f(a) \\le g(a)$ for every $a \\in \\alpha$. Then the expectation of $f$ under $p$ is at most the expectation of $g$ under $p$.","title":"Monotonicity of expectation"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-eq-sum-of-finite-support","kind":"node","lean":"PMF.expect_eq_sum_of_finite_support","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-eq-sum-of-finite-support","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a measurable space $\\alpha$ in which singletons are measurable, and let $s$ be a finite subset of $\\alpha$ such that $p(a) = 0$ for every $a \\notin s$. Then for every function $f : \\alpha \\to \\mathbb{R}$, the expectation of $f$ under $p$ equals $\\sum_{a \\in s} p(a) f(a)$.","title":"Expectation of a finitely supported distribution"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-map","kind":"node","lean":"PMF.expect_map","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-map","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a finite measurable space $\\alpha$ in which singletons are measurable, let $\\beta$ be a measurable space in which singletons are measurable, and let $g : \\alpha \\to \\beta$ and $f : \\beta \\to \\mathbb{R}$. Write $g_* p$ for the pushforward of $p$ along $g$, which gives each $b \\in \\beta$ the mass $\\sum_{a :\\, g(a) = b} p(a)$. Then the expectation of $f$ under $g_* p$ equals\u2026","title":"Change of variables for expectation"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-indicator-decide","kind":"node","lean":"PMF.expect_indicator_decide","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-indicator-decide","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a finite measurable space $\\alpha$ in which singletons are measurable, and let $q$ be a property of elements of $\\alpha$. The expectation under $p$ of the function equal to $1$ where $q$ holds and $0$ elsewhere is $\\sum_{a \\in \\alpha,\\ q(a)} p(a)$, the total mass of the elements satisfying $q$.","title":"Expectation of an indicator"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn","kind":"node","lean":"PMF.expectENN","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"For a probability mass function $p$ on a type $\\alpha$ and a function $f : \\alpha \\to [0, \\infty]$, the extended expectation of $f$ under $p$ is the unordered sum $\\sum_{a \\in \\alpha} p(a) f(a)$ in $[0, \\infty]$, which always exists.","title":"Extended expectation"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn-eq-sum","kind":"node","lean":"PMF.expectENN_eq_sum","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn-eq-sum","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a finite set $\\alpha$. For every $f : \\alpha \\to [0, \\infty]$, the extended expectation of $f$ under $p$ equals the finite sum $\\sum_{a \\in \\alpha} p(a)\\, f(a)$.","title":"Extended expectation on a finite set"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn-const","kind":"node","lean":"PMF.expectENN_const","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn-const","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a set $\\alpha$. For every $c \\in [0, \\infty]$, the extended expectation of the constant function $a \\mapsto c$ under $p$ is $c$.","title":"Extended expectation of a constant"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn-uniform-of-fintype","kind":"node","lean":"PMF.expectENN_uniformOfFintype","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn-uniform-of-fintype","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite type and $f : \\alpha \\to [0, \\infty]$. The extended expectation of $f$ under the uniform distribution on $\\alpha$ equals $|\\alpha|^{-1} \\sum_{a \\in \\alpha} f(a)$, computed in $[0, \\infty]$.","title":"Extended expectation under the uniform distribution"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn-add","kind":"node","lean":"PMF.expectENN_add","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn-add","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a set $\\alpha$, and write $\\overline{\\mathbb{E}}_p[h]$ for the extended expectation of $h : \\alpha \\to [0, \\infty]$ under $p$. For all $f, g : \\alpha \\to [0, \\infty]$, $\\overline{\\mathbb{E}}_p[f + g] = \\overline{\\mathbb{E}}_p[f] + \\overline{\\mathbb{E}}_p[g]$.","title":"Additivity of extended expectation"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn-sum","kind":"node","lean":"PMF.expectENN_sum","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn-sum","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a set $\\alpha$, and write $\\overline{\\mathbb{E}}_p[h]$ for the extended expectation of $h : \\alpha \\to [0, \\infty]$ under $p$. For every index set $\\iota$, finite subset $s \\subseteq \\iota$ and family of functions $f_i : \\alpha \\to [0, \\infty]$ ($i \\in \\iota$), $\\overline{\\mathbb{E}}_p\\bigl[a \\mapsto \\sum_{i \\in s} f_i(a)\\bigr] = \\sum_{i \\in s} \\overline{\\mathbb{E}}_p[f_i]$.","title":"Extended expectation of a finite sum"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn-eq-sum-of-finite-support","kind":"node","lean":"PMF.expectENN_eq_sum_of_finite_support","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn-eq-sum-of-finite-support","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a set $\\alpha$, and let $s \\subseteq \\alpha$ be a finite set with $p(a) = 0$ for every $a \\notin s$. For every $f : \\alpha \\to [0, \\infty]$, the extended expectation of $f$ under $p$ equals $\\sum_{a \\in s} p(a)\\, f(a)$.","title":"Extended expectation under a finitely supported distribution"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn-map","kind":"node","lean":"PMF.expectENN_map","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn-map","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $p$ be a probability mass function on a set $\\alpha$, let $g : \\alpha \\to \\beta$ be a map to a set $\\beta$, and let $g_* p$ be the pushforward of $p$ along $g$, the probability mass function $b \\mapsto \\sum_{a : g(a) = b} p(a)$. For every $f : \\beta \\to [0, \\infty]$, the extended expectation of $f$ under $g_* p$ equals the extended expectation of $f \\circ g$ under $p$.","title":"Change of variables for extended expectation"},{"id":"Probability/distributions/discrete/expectation-under-a-probability-mass-function/expect-enn-nat-cast-to-real","kind":"node","lean":"PMF.expectENN_natCast_toReal","parent":20887,"result":true,"route":"../roadmap/Probability/distributions/discrete/expectation-under-a-probability-mass-function/index.html#expect-enn-nat-cast-to-real","scope":"Probability/distributions/discrete/expectation-under-a-probability-mass-function","status":"fully_proved","summary":"Let $\\alpha$ be a finite set with a $\\sigma$-algebra in which singletons are measurable, let $p$ be a probability mass function on $\\alpha$, and let $g : \\alpha \\to \\mathbb{N}$. Let $x \\in [0, \\infty]$ be the extended expectation of $g$, viewed as a function to $[0, \\infty]$, under $p$, and send $x$ to $\\mathbb{R}$ by the map that is the identity on $[0, \\infty)$ and sends $\\infty$ to $0$. The result is the\u2026","title":"Extended and real expectations agree on natural-valued functions"},{"data_url":"scopes/Probability/distributions/discrete/hypergeometric-distribution.json","id":"Probability/distributions/discrete/hypergeometric-distribution","kind":"scope","navigation":"module","parent":20873,"result":false,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","title":"Hypergeometric distribution"},{"id":"Probability/distributions/discrete/hypergeometric-distribution/hypergeometric-real","kind":"node","lean":"ProbabilityTheory.hypergeometricReal","parent":20911,"result":true,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html#hypergeometric-real","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","summary":"The hypergeometric mass function: the probability `C(K, k) C(N - K, n - k) / C(N, n)` of drawing exactly `k` marked items in `n` draws without replacement from `N` items of which `K` are marked; it is `0` for `k > n`. The parameter domain is `K \u2264 N`, `n \u2264 N`; outside it the value is not a probability (`N - K` truncates, and `C(N, n) = 0` makes it `0` for `n > N`).","title":"Hypergeometric mass function"},{"id":"Probability/distributions/discrete/hypergeometric-distribution/hypergeometric-real-nonneg","kind":"node","lean":"ProbabilityTheory.hypergeometricReal_nonneg","parent":20911,"result":true,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html#hypergeometric-real-nonneg","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","summary":"The hypergeometric mass function is nonnegative.","title":"Nonnegativity of the hypergeometric mass function"},{"id":"Probability/distributions/discrete/hypergeometric-distribution/hypergeometric-real-of-lt","kind":"node","lean":"ProbabilityTheory.hypergeometricReal_of_lt","parent":20911,"result":true,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html#hypergeometric-real-of-lt","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","summary":"The hypergeometric mass function vanishes above the number of draws.","title":"Hypergeometric mass vanishes above the number of draws"},{"id":"Probability/distributions/discrete/hypergeometric-distribution/hypergeometric-real-of-lt-2","kind":"node","lean":"ProbabilityTheory.hypergeometricReal_of_lt'","parent":20911,"result":true,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html#hypergeometric-real-of-lt-2","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","summary":"The hypergeometric mass function vanishes above the number of marked items.","title":"Hypergeometric mass vanishes above the number of marked items"},{"id":"Probability/distributions/discrete/hypergeometric-distribution/sum-hypergeometric-real","kind":"node","lean":"ProbabilityTheory.sum_hypergeometricReal","parent":20911,"result":true,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html#sum-hypergeometric-real","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","summary":"For `K \u2264 N` and `n \u2264 N` the hypergeometric probabilities sum to one; this is Vandermonde's identity `\u2211\u2096 C(K, k) C(N - K, n - k) = C(N, n)`.","title":"Normalization"},{"id":"Probability/distributions/discrete/hypergeometric-distribution/sum-mul-hypergeometric-real","kind":"node","lean":"ProbabilityTheory.sum_mul_hypergeometricReal","parent":20911,"result":true,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html#sum-mul-hypergeometric-real","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","summary":"For `0 < N`, `K \u2264 N` and `n \u2264 N` the hypergeometric law has mean `n K / N`.","title":"Mean"},{"id":"Probability/distributions/discrete/hypergeometric-distribution/hypergeometric","kind":"node","lean":"ProbabilityTheory.hypergeometric","parent":20911,"result":true,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html#hypergeometric","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","summary":"The hypergeometric law on `\u2115` with population size `N`, `K \u2264 N` marked items, and `n \u2264 N` draws without replacement.","title":"Hypergeometric distribution"},{"id":"Probability/distributions/discrete/hypergeometric-distribution/hypergeometric-apply","kind":"node","lean":"ProbabilityTheory.hypergeometric_apply","parent":20911,"result":true,"route":"../roadmap/Probability/distributions/discrete/hypergeometric-distribution/index.html#hypergeometric-apply","scope":"Probability/distributions/discrete/hypergeometric-distribution","status":"fully_proved","summary":"The mass of `k` under `hypergeometric N K n` is `hypergeometricReal N K n k`.","title":"Point masses of the hypergeometric distribution"},{"data_url":"scopes/Probability/distributions/discrete/multinomial-distribution.json","id":"Probability/distributions/discrete/multinomial-distribution","kind":"scope","navigation":"module","parent":20873,"result":false,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","title":"Multinomial distribution"},{"id":"Probability/distributions/discrete/multinomial-distribution/multinomial-real","kind":"node","lean":"ProbabilityTheory.multinomialReal","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#multinomial-real","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"The multinomial mass function: the probability `N! / \u220f\u1d62 x\u1d62! \u00b7 \u220f\u1d62 p\u1d62 ^ x\u1d62` that `N` independent throws, landing in cell `i` with probability `p\u1d62`, produce the cell counts `x`; it is `0` unless `\u2211\u1d62 x\u1d62 = N`. The formula is a polynomial in `p` and is defined for every real vector `p`; it is a probability mass function when `p` is a probability vector (`0 \u2264 p\u1d62`, `\u2211\u1d62 p\u1d62 = 1`), which the normalization theorems assume. The\u2026","title":"Multinomial mass function"},{"id":"Probability/distributions/discrete/multinomial-distribution/multinomial-real-nonneg","kind":"node","lean":"ProbabilityTheory.multinomialReal_nonneg","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#multinomial-real-nonneg","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"The multinomial mass function is nonnegative for a nonnegative vector `p`.","title":"Nonnegativity of the multinomial mass function"},{"id":"Probability/distributions/discrete/multinomial-distribution/multinomial-real-of-sum-ne","kind":"node","lean":"ProbabilityTheory.multinomialReal_of_sum_ne","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#multinomial-real-of-sum-ne","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"The multinomial mass function vanishes off the simplex `\u2211\u1d62 x\u1d62 = N`.","title":"Multinomial mass vanishes off the simplex"},{"id":"Probability/distributions/discrete/multinomial-distribution/multinomial-real-eq-factorial-div","kind":"node","lean":"ProbabilityTheory.multinomialReal_eq_factorial_div","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#multinomial-real-eq-factorial-div","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"On the simplex the multinomial mass is `N! / \u220f\u1d62 x\u1d62! \u00b7 \u220f\u1d62 p\u1d62 ^ x\u1d62`.","title":"Multinomial mass formula on the simplex"},{"id":"Probability/distributions/discrete/multinomial-distribution/multinomial-real-of-not-mem","kind":"node","lean":"ProbabilityTheory.multinomialReal_of_notMem","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#multinomial-real-of-not-mem","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"The multinomial mass function vanishes outside the finite set `piAntidiag univ N` of count vectors with total `N`: the support is contained in the antidiagonal. Equality of the support with the antidiagonal needs all `p i > 0` (e.g. for `p = (1, 0)`, `N = 1` the vector `(0, 1)` is in the antidiagonal but has mass `0`).","title":"Support of the multinomial mass lies in the antidiagonal"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-multinomial-real-eq-pow","kind":"node","lean":"ProbabilityTheory.sum_multinomialReal_eq_pow","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-multinomial-real-eq-pow","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"`\u2211\u2093 P(x) = (\u2211\u1d62 p\u1d62) ^ N` for every real vector `p`.","title":"Multinomial theorem"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-multinomial-real","kind":"node","lean":"ProbabilityTheory.sum_multinomialReal","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-multinomial-real","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"For `\u2211\u1d62 p\u1d62 = 1` the multinomial masses sum to one.","title":"Normalization"},{"id":"Probability/distributions/discrete/multinomial-distribution/multinomial","kind":"node","lean":"ProbabilityTheory.multinomial","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#multinomial","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"The multinomial law of `N` throws with cell probabilities `p` (a probability vector), as a probability mass function on count vectors `\u03b9 \u2192 \u2115`.","title":"Multinomial distribution as a probability mass function"},{"id":"Probability/distributions/discrete/multinomial-distribution/multinomial-apply","kind":"node","lean":"ProbabilityTheory.multinomial_apply","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#multinomial-apply","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"The mass of the multinomial PMF at `x` is `multinomialReal N p x`.","title":"Mass of the multinomial distribution at a count vector"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-mul-multinomial-real-eq","kind":"node","lean":"ProbabilityTheory.sum_mul_multinomialReal_eq","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-mul-multinomial-real-eq","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"`\u2211\u2093 x\u1d62 P(x) = N p\u1d62 (\u2211\u2c7c p\u2c7c) ^ (N - 1)` for every real vector `p`.","title":"Mean, polynomial form"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-mul-mul-multinomial-real-of-ne-eq","kind":"node","lean":"ProbabilityTheory.sum_mul_mul_multinomialReal_of_ne_eq","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-mul-mul-multinomial-real-of-ne-eq","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"For `i \u2260 j`, `\u2211\u2093 x\u1d62 x\u2c7c P(x) = N (N - 1) p\u1d62 p\u2c7c (\u2211\u2097 p\u2097) ^ (N - 2)` for every real vector `p`.","title":"Second factorial moment, polynomial form"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-mul-sub-one-multinomial-real-eq","kind":"node","lean":"ProbabilityTheory.sum_mul_sub_one_multinomialReal_eq","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-mul-sub-one-multinomial-real-eq","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"`\u2211\u2093 x\u1d62 (x\u1d62 - 1) P(x) = N (N - 1) p\u1d62\u00b2 (\u2211\u2097 p\u2097) ^ (N - 2)` for every real vector `p`.","title":"Second factorial moment, polynomial form"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-mul-multinomial-real","kind":"node","lean":"ProbabilityTheory.sum_mul_multinomialReal","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-mul-multinomial-real","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"For a probability vector (`0 \u2264 p\u1d62`, `\u2211\u2c7c p\u2c7c = 1`), `E X\u1d62 = N p\u1d62`. The underlying `sum_mul_multinomialReal_eq` is a polynomial identity holding for every real `p`.","title":"Mean"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-mul-mul-multinomial-real-of-ne","kind":"node","lean":"ProbabilityTheory.sum_mul_mul_multinomialReal_of_ne","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-mul-mul-multinomial-real-of-ne","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"For a probability vector and `i \u2260 j`, `E[X\u1d62 X\u2c7c] = N (N - 1) p\u1d62 p\u2c7c`.","title":"Mixed second moment"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-mul-sub-one-multinomial-real","kind":"node","lean":"ProbabilityTheory.sum_mul_sub_one_multinomialReal","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-mul-sub-one-multinomial-real","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"For a probability vector, `E[X\u1d62 (X\u1d62 - 1)] = N (N - 1) p\u1d62\u00b2`.","title":"Second factorial moment"},{"id":"Probability/distributions/discrete/multinomial-distribution/sum-covariance-multinomial-real","kind":"node","lean":"ProbabilityTheory.sum_covariance_multinomialReal","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#sum-covariance-multinomial-real","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"For a probability vector (`0 \u2264 p\u1d62`, `\u2211\u2097 p\u2097 = 1`), `Cov(X\u1d62, X\u2c7c) = \u2211\u2093 (x\u1d62 - N p\u1d62) (x\u2c7c - N p\u2c7c) P(x) = N (\u03b4\u1d62\u2c7c p\u1d62 - p\u1d62 p\u2c7c)`; in particular `Var X\u1d62 = N p\u1d62 (1 - p\u1d62)` and `Cov(X\u1d62, X\u2c7c) = -N p\u1d62 p\u2c7c` for `i \u2260 j`.","title":"Covariance"},{"id":"Probability/distributions/discrete/multinomial-distribution/cond-measure-real-eq-multinomial-real","kind":"node","lean":"ProbabilityTheory.cond_measureReal_eq_multinomialReal","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#cond-measure-real-eq-multinomial-real","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"Let `X\u1d62 ~ Po(\u03bb\u1d62)`, `i \u2208 \u03b9`, be independent Poisson counts with `\u2211\u1d62 \u03bb\u1d62 > 0`. Given their total `\u2211\u1d62 X\u1d62 = N`, the count vector is multinomial with `N` throws and cell probabilities `\u03bb\u1d62 / \u2211\u2c7c \u03bb\u2c7c`: `P(X = x | \u2211\u1d62 X\u1d62 = N) = multinomialReal N (\u03bb / \u2211 \u03bb) x`.","title":"Poisson conditioning"},{"id":"Probability/distributions/discrete/multinomial-distribution/cond-measure-real-eq-binomial","kind":"node","lean":"ProbabilityTheory.cond_measureReal_eq_binomial","parent":20920,"result":true,"route":"../roadmap/Probability/distributions/discrete/multinomial-distribution/index.html#cond-measure-real-eq-binomial","scope":"Probability/distributions/discrete/multinomial-distribution","status":"fully_proved","summary":"For independent `X ~ Po(a)` and `Y ~ Po(b)` with `a + b > 0`, the law of `X` given `X + Y = N` is binomial `Bin(N, a / (a + b))`.","title":"Poisson conditioning, two cells"},{"data_url":"scopes/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution.json","id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","kind":"scope","navigation":"module","parent":20873,"result":false,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","title":"Negative binomial and beta-binomial distributions"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/neg-binomial-pmf","kind":"node","lean":"ProbabilityTheory.negBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#neg-binomial-pmf","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"Negative binomial mass `P(X = k) = r^{(k)} / k! \u00b7 p ^ r \u00b7 (1 - p) ^ k` with real shape `r` (rising factorial `r^{(k)} = (ascPochhammer \u211d k).eval r`) and success probability `p`, using the real power `p ^ r`. A probability mass function for `r > 0` and `p \u2208 (0, 1]`.","title":"Negative binomial mass function with real shape"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/beta-binomial-pmf","kind":"node","lean":"ProbabilityTheory.betaBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#beta-binomial-pmf","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"Beta-binomial mass `C(n, k) \u03b1^{(k)} \u03b2^{(n - k)} / (\u03b1 + \u03b2)^{(n)}` (zero for `k > n`, since `C(n, k) = 0`). A probability mass function for `\u03b1, \u03b2 > 0`.","title":"Beta-binomial mass function"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/asc-pochhammer-eval-add-eq-sum","kind":"node","lean":"ProbabilityTheory.ascPochhammer_eval_add_eq_sum","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#asc-pochhammer-eval-add-eq-sum","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"`(\u03b1 + \u03b2)^{(n)} = \u2211_{k \u2264 n} C(n, k) \u03b1^{(k)} \u03b2^{(n - k)}`, for all real `\u03b1, \u03b2`.","title":"Chu\u2013Vandermonde identity for rising factorials"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/sum-beta-binomial-pmf","kind":"node","lean":"ProbabilityTheory.sum_betaBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#sum-beta-binomial-pmf","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The beta-binomial masses sum to one for `\u03b1, \u03b2 > 0`.","title":"Beta-binomial masses sum to one"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/has-sum-neg-binomial-pmf","kind":"node","lean":"ProbabilityTheory.hasSum_negBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#has-sum-neg-binomial-pmf","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The negative binomial masses sum to one for `r > 0` and `p \u2208 (0, 1]` (the binomial series `\u2211\u2096 r^{(k)} x\u1d4f / k! = (1 - x)^{-r}` at `x = 1 - p`).","title":"Negative binomial masses sum to one"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/sum-neg-binomial-pmf-mul","kind":"node","lean":"ProbabilityTheory.sum_negBinomialPMF_mul","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#sum-neg-binomial-pmf-mul","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"`\u2211_{k \u2264 n} NB(r\u2081, p)(k) NB(r\u2082, p)(n - k) = NB(r\u2081 + r\u2082, p)(n)` for `p > 0` and all real shapes.","title":"Convolution of negative binomial laws with a common `p`"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/neg-binomial-pmf-mul-div-eq-beta-binomia","kind":"node","lean":"ProbabilityTheory.negBinomialPMF_mul_div_eq_betaBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#neg-binomial-pmf-mul-div-eq-beta-binomia","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `r\u2081, r\u2082 > 0`, `0 < p < 1` and `k \u2264 n`, `NB(r\u2081, p)(k) NB(r\u2082, p)(n - k) / NB(r\u2081 + r\u2082, p)(n) = BB(n; r\u2081, r\u2082)(k)`; the conditional law of one component given the total is beta-binomial and does not depend on `p`.","title":"Negative binomial splitting"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/measure-cond-eq-beta-binomial-pmf","kind":"node","lean":"ProbabilityTheory.measure_cond_eq_betaBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#measure-cond-eq-beta-binomial-pmf","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"If `X` and `Y` are independent with `P(X = j) = NB(r\u2081, p)(j)` and `P(Y = j) = NB(r\u2082, p)(j)` for all `j`, where `r\u2081, r\u2082 > 0` and `0 < p < 1`, then `P(X = k \u2223 X + Y = n) = BB(n; r\u2081, r\u2082)(k)` for `k \u2264 n`.","title":"Negative binomial splitting for random variables"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/sum-mul-beta-binomial-pmf","kind":"node","lean":"ProbabilityTheory.sum_mul_betaBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#sum-mul-beta-binomial-pmf","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0`, `E M = n \u03b1 / (\u03b1 + \u03b2) = n \u03c0`.","title":"Mean"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/sum-mul-sub-one-beta-binomial-pmf","kind":"node","lean":"ProbabilityTheory.sum_mul_sub_one_betaBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#sum-mul-sub-one-beta-binomial-pmf","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0`, `E[M (M - 1)] = n (n - 1) \u03b1 (\u03b1 + 1) / ((\u03b1 + \u03b2) (\u03b1 + \u03b2 + 1))`.","title":"Second factorial moment"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/variance-beta-binomial","kind":"node","lean":"ProbabilityTheory.variance_betaBinomial","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#variance-beta-binomial","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"Variance (Dolzhenko\u2013Smith parametrization). For `\u03b1, \u03b2 > 0`, with `\u03c0 = \u03b1 / (\u03b1 + \u03b2)` and `\u03b3 = 1 / (\u03b1 + \u03b2 + 1)`, `Var M = E M\u00b2 - (E M)\u00b2 = n \u03c0 (1 - \u03c0) (1 + (n - 1) \u03b3)`.","title":"Variance"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/poisson-mixture-real","kind":"node","lean":"ProbabilityTheory.poissonMixtureReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#poisson-mixture-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The mixed Poisson mass function `p(k) = \u222b e^{-\u03bb} \u03bb^k / k! d\u03bd(\u03bb)` for a mixing law `\u03bd` of the rate. It is a probability mass function when `\u03bd` is a probability measure concentrated on `[0, \u221e)`, which the theorems below assume; for such `\u03bd` the integrand lies in `[0, 1]`, so the integral is never a junk value.","title":"Mixed Poisson mass function"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/poisson-mixture-real-zero","kind":"node","lean":"ProbabilityTheory.poissonMixtureReal_zero","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#poisson-mixture-real-zero","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The zero class of a mixed Poisson law is `E e^{-\u039b}`.","title":"Zero class of a mixed Poisson law"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/poisson-mixture-real-nonneg","kind":"node","lean":"ProbabilityTheory.poissonMixtureReal_nonneg","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#poisson-mixture-real-nonneg","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The mixed Poisson mass function is nonnegative when `\u03bd` is concentrated on `[0, \u221e)`.","title":"Nonnegativity of the mixed Poisson mass function"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/poisson-mixture-real-dirac","kind":"node","lean":"ProbabilityTheory.poissonMixtureReal_dirac","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#poisson-mixture-real-dirac","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The Poisson mixture over a point mass `\u03b4_m` is the Poisson law with mean `m`.","title":"Poisson mixture over a point mass is Poisson"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/has-sum-poisson-mixture-real","kind":"node","lean":"ProbabilityTheory.hasSum_poissonMixtureReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#has-sum-poisson-mixture-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For a probability measure `\u03bd` concentrated on `[0, \u221e)`, the mixed Poisson masses sum to one.","title":"Normalization"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/exp-neg-integral-le-poisson-mixture-real","kind":"node","lean":"ProbabilityTheory.exp_neg_integral_le_poissonMixtureReal_zero","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#exp-neg-integral-le-poisson-mixture-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"Zero inflation by mixing (Jensen's inequality for `\u03bb \u21a6 e^{-\u03bb}`). For a probability measure `\u03bd` with integrable rate, `P(N = 0) = E e^{-\u039b} \u2265 e^{-E \u039b}`, the zero class of the Poisson law with the same mean.","title":"Zero inflation by mixing"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/has-sum-mul-poisson-mixture-real","kind":"node","lean":"ProbabilityTheory.hasSum_mul_poissonMixtureReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#has-sum-mul-poisson-mixture-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For a mixing measure `\u03bd` on `[0, \u221e)` with integrable rate, `E N = E \u039b` (finiteness of `\u03bd` is not needed for this identity).","title":"Mean"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/has-sum-mul-sub-one-poisson-mixture-real","kind":"node","lean":"ProbabilityTheory.hasSum_mul_sub_one_poissonMixtureReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#has-sum-mul-sub-one-poisson-mixture-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For a mixing measure `\u03bd` on `[0, \u221e)` with square-integrable rate, `E[N (N - 1)] = E \u039b\u00b2` (finiteness of `\u03bd` is not needed for this identity).","title":"Second factorial moment"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/has-sum-sq-poisson-mixture-real","kind":"node","lean":"ProbabilityTheory.hasSum_sq_poissonMixtureReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#has-sum-sq-poisson-mixture-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For a finite mixing measure `\u03bd` on `[0, \u221e)` with square-integrable rate, `E N\u00b2 = E \u039b\u00b2 + E \u039b`.","title":"Second moment"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/variance-poisson-mixture","kind":"node","lean":"ProbabilityTheory.variance_poissonMixture","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#variance-poisson-mixture","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For a probability measure `\u03bd` on `[0, \u221e)` with square-integrable rate, `Var N = E N\u00b2 - (E N)\u00b2 = E \u039b + Var \u039b`: Poisson noise plus the variance of the rate.","title":"Variance decomposition"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/cv-sq-poisson-mixture","kind":"node","lean":"ProbabilityTheory.cv_sq_poissonMixture","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#cv-sq-poisson-mixture","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"Squared coefficient of variation (Brennecke et al. 2013; Svensson 2020). For a probability measure `\u03bd` on `[0, \u221e)` with square-integrable rate and `E \u039b > 0`, `CV\u00b2(N) = Var N / (E N)\u00b2 = 1 / E \u039b + Var \u039b / (E \u039b)\u00b2 = 1 / E \u039b + CV\u00b2(\u039b)`.","title":"Squared coefficient of variation"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/neg-binomial-real","kind":"node","lean":"ProbabilityTheory.negBinomialReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#neg-binomial-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The negative binomial mass function with size `\u03b8` and mean `\u03bc`: `NB(k; \u03b8, \u03bc) = \u03b8\u207d\u1d4f\u207e / k! \u00b7 (\u03b8 / (\u03b8 + \u03bc)) ^ \u03b8 \u00b7 (\u03bc / (\u03b8 + \u03bc)) ^ k`, with the rising factorial `\u03b8\u207d\u1d4f\u207e = \u0393(\u03b8 + k) / \u0393(\u03b8)` and a real power `(\u03b8 / (\u03b8 + \u03bc)) ^ \u03b8`. The parameter domain is `\u03b8 > 0`, `\u03bc \u2265 0`; at `\u03bc = 0` it is the point mass at `0`. Outside the domain the formula is not a probability mass function.","title":"Negative binomial mass function with size and mean"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/neg-binomial-real-eq-neg-binomial-pmf","kind":"node","lean":"ProbabilityTheory.negBinomialReal_eq_negBinomialPMF","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#neg-binomial-real-eq-neg-binomial-pmf","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b8 > 0` and `\u03bc \u2265 0`, `negBinomialReal \u03b8 \u03bc` is `negBinomialPMF \u03b8 p` at `p = \u03b8 / (\u03b8 + \u03bc)`, since `1 - p = \u03bc / (\u03b8 + \u03bc)`. The two APIs are the same law; reuse the splitting and conditioning results proved for `negBinomialPMF` through this equivalence.","title":"Mean parametrization is the canonical `p`-parametrization"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/neg-binomial-real-nonneg","kind":"node","lean":"ProbabilityTheory.negBinomialReal_nonneg","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#neg-binomial-real-nonneg","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The negative binomial mass function is nonnegative on the parameter domain.","title":"Nonnegativity of the negative binomial mass function"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/neg-binomial-real-zero","kind":"node","lean":"ProbabilityTheory.negBinomialReal_zero","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#neg-binomial-real-zero","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The zero class of the negative binomial law is `(\u03b8 / (\u03b8 + \u03bc)) ^ \u03b8`.","title":"Zero class of the negative binomial law"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/neg-binomial-real-mean-zero","kind":"node","lean":"ProbabilityTheory.negBinomialReal_mean_zero","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#neg-binomial-real-mean-zero","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"With mean `0` the negative binomial law is the point mass at `0`.","title":"Negative binomial law with mean zero is a point mass"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/exp-neg-le-neg-binomial-real-zero","kind":"node","lean":"ProbabilityTheory.exp_neg_le_negBinomialReal_zero","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#exp-neg-le-neg-binomial-real-zero","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b8 > 0` and `\u03bc \u2265 0`, `P(0) = (\u03b8 / (\u03b8 + \u03bc)) ^ \u03b8 \u2265 e^{-\u03bc}`: a negative binomial law puts at least as much mass at zero as the Poisson law with the same mean.","title":"Zero class versus the Poisson law"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/poisson-mixture-real-gamma-measure","kind":"node","lean":"ProbabilityTheory.poissonMixtureReal_gammaMeasure","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#poisson-mixture-real-gamma-measure","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The Poisson mixture over a Gamma rate of shape `a > 0` and rate `b > 0` is negative binomial with size `a` and mean `a / b`.","title":"Greenwood\u2013Yule"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/poisson-mixture-real-gamma-measure-mean","kind":"node","lean":"ProbabilityTheory.poissonMixtureReal_gammaMeasure_mean","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#poisson-mixture-real-gamma-measure-mean","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b8, \u03bc > 0`, the Poisson mixture over `Gamma(\u03b8, \u03b8 / \u03bc)` (mean `\u03bc`, variance `\u03bc\u00b2 / \u03b8`) is `NB(\u03b8, \u03bc)`.","title":"Greenwood\u2013Yule, mean form"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/has-sum-neg-binomial-real","kind":"node","lean":"ProbabilityTheory.hasSum_negBinomialReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#has-sum-neg-binomial-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b8 > 0` and `\u03bc \u2265 0` the negative binomial masses sum to one. For `\u03bc > 0` this is `hasSum_negBinomialPMF` transported along `negBinomialReal_eq_negBinomialPMF`, so the mean parametrization shares the canonical law's normalization instead of re-proving it.","title":"Normalization"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/has-sum-mul-neg-binomial-real","kind":"node","lean":"ProbabilityTheory.hasSum_mul_negBinomialReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#has-sum-mul-neg-binomial-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b8 > 0` and `\u03bc \u2265 0` the negative binomial law has mean `\u03bc`.","title":"Mean"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/has-sum-sq-neg-binomial-real","kind":"node","lean":"ProbabilityTheory.hasSum_sq_negBinomialReal","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#has-sum-sq-neg-binomial-real","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b8 > 0` and `\u03bc \u2265 0`, `E N\u00b2 = \u03bc + \u03bc\u00b2 / \u03b8 + \u03bc\u00b2`.","title":"Second moment"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/variance-neg-binomial","kind":"node","lean":"ProbabilityTheory.variance_negBinomial","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#variance-neg-binomial","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b8 > 0` and `\u03bc \u2265 0` the negative binomial law has variance `\u03bc + \u03bc\u00b2 / \u03b8`.","title":"Variance"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/gamma-pdfreal-posterior","kind":"node","lean":"ProbabilityTheory.gammaPDFReal_posterior","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#gamma-pdfreal-posterior","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"With prior `\u039b ~ Gamma(a, b)` (`a, b > 0`) and one observation `N = k` from `Po(\u039b)`, the posterior density (likelihood times prior over the marginal `NB(k; a, a / b)`) is the `Gamma(a + k, b + 1)` density.","title":"Gamma\u2013Poisson conjugacy, posterior"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/posterior-predictive-gamma-poisson","kind":"node","lean":"ProbabilityTheory.posterior_predictive_gammaPoisson","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#posterior-predictive-gamma-poisson","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"With prior `\u039b ~ Gamma(a, b)` (`a, b > 0`) and counts `N, N'` that are independent `Po(\u039b)` given `\u039b`, `P(N' = j | N = k) = P(N = k, N' = j) / P(N = k) = NB(j; a + k, (a + k) / (b + 1))`.","title":"Gamma\u2013Poisson conjugacy, posterior predictive"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/immigration-death-flow","kind":"node","lean":"ProbabilityTheory.immigrationDeathFlow","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#immigration-death-flow","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The net probability flow into state `n` of the immigration\u2013death chain on `\u2115`, in which molecules are made at rate `\u03ba` (`n \u2192 n + 1`) and each decays at rate `\u03b3` (`n \u2192 n - 1` at rate `\u03b3 n`): `(\u03ba p(n - 1) + \u03b3 (n + 1) p(n + 1)) - (\u03ba + \u03b3 n) p(n)`, with no inflow from below at `n = 0`. A law `p` is stationary (global balance) iff this flow vanishes for every `n`.","title":"Net probability flow of the immigration\u2013death chain"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/immigration-death-flow-poisson","kind":"node","lean":"ProbabilityTheory.immigrationDeathFlow_poisson","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#immigration-death-flow-poisson","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `\u03b3 \u2260 0` the Poisson law with mean `\u03ba / \u03b3` satisfies global balance for the immigration\u2013death chain. The identity holds for all real `\u03ba`; the chain is meaningful for `\u03ba \u2265 0`, `\u03b3 > 0`.","title":"Immigration\u2013death stationary law"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/fano-poisson","kind":"node","lean":"ProbabilityTheory.fano_poisson","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#fano-poisson","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For mean `m > 0`, `Var N / E N = 1`; in particular the stationary law of the immigration\u2013death chain has Fano factor `1`.","title":"Fano factor of the Poisson law"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/burst-death-flow","kind":"node","lean":"ProbabilityTheory.burstDeathFlow","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#burst-death-flow","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"The net probability flow into state `n` of the bursty production\u2013degradation chain: bursts arrive at rate `\u03ba` and add `B` molecules, `B` geometric with mean `b`, `P(B = j) = b ^ j / (1 + b) ^ (j + 1)` for `j \u2265 0`; each molecule decays at rate `\u03b3`. A burst of size `0` changes nothing, so state `n` is left by a burst at rate `\u03ba P(B \u2265 1) = \u03ba b / (1 + b)`. The flow is\u2026","title":"Net probability flow of the bursty production\u2013degradation chain"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/burst-death-flow-neg-binomial","kind":"node","lean":"ProbabilityTheory.burstDeathFlow_negBinomial","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#burst-death-flow-neg-binomial","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"Bursty expression stationary law (Shahrezaei\u2013Swain). For burst rate `\u03ba > 0`, decay rate `\u03b3 > 0` and mean burst size `b \u2265 0`, the negative binomial law with size `a = \u03ba / \u03b3` (bursts per lifetime) and mean `a b` satisfies global balance for the bursty production\u2013degradation chain.","title":"Bursty expression stationary law"},{"id":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/fano-neg-binomial","kind":"node","lean":"ProbabilityTheory.fano_negBinomial","parent":20939,"result":true,"route":"../roadmap/Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution/index.html#fano-neg-binomial","scope":"Probability/distributions/discrete/negative-binomial-and-beta-binomial-distribution","status":"fully_proved","summary":"For `a > 0` and `b > 0`, the negative binomial law with size `a` and mean `a b` has `Var N / E N = 1 + b`.","title":"Fano factor of bursty expression"},{"data_url":"scopes/Probability/distributions/discrete/poisson-binomial-distribution.json","id":"Probability/distributions/discrete/poisson-binomial-distribution","kind":"scope","navigation":"module","parent":20873,"result":false,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","title":"Poisson binomial distribution"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/is-bernoulli-rv","kind":"node","lean":"ProbabilityTheory.IsBernoulliRV","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#is-bernoulli-rv","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, let $X : \\Omega \\to \\mathbb{N}$ and let $p$ be a real number. Then $X$ is a Bernoulli random variable with parameter $p$ under $\\mu$ when $0 \\le p \\le 1$, $X$ is $\\mu$-almost everywhere measurable, and the pushforward of $\\mu$ along $X$ is the Bernoulli distribution $p\\,\\delta_1 + (1 - p)\\,\\delta_0$ on $\\mathbb{N}$, which gives mass $p$ to $1$, mass $1 - p$ to\u2026","title":"Bernoulli random variable"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/nonneg","kind":"node","lean":"ProbabilityTheory.IsBernoulliRV.nonneg","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#nonneg","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"If $X$ is a Bernoulli random variable with parameter $p$ under a measure $\\mu$, then $0 \\le p$.","title":"Bernoulli parameters are nonnegative"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/le-one","kind":"node","lean":"ProbabilityTheory.IsBernoulliRV.le_one","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#le-one","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"If $X$ is a Bernoulli random variable with parameter $p$ under a measure $\\mu$, then $p \\le 1$.","title":"Bernoulli parameters are at most one"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/has-law","kind":"node","lean":"ProbabilityTheory.IsBernoulliRV.hasLaw","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#has-law","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"If $X : \\Omega \\to \\mathbb{N}$ is a Bernoulli random variable with parameter $p$ under a measure $\\mu$, then $X$ is $\\mu$-almost everywhere measurable and its law under $\\mu$, the pushforward of $\\mu$ along $X$, is the Bernoulli distribution $p\\,\\delta_1 + (1 - p)\\,\\delta_0$ on $\\mathbb{N}$.","title":"Law of a Bernoulli random variable"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/measure-zero","kind":"node","lean":"ProbabilityTheory.IsBernoulliRV.measure_zero","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#measure-zero","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"If $X : \\Omega \\to \\mathbb{N}$ is a Bernoulli random variable with parameter $p$ under a measure $\\mu$, then $\\mu(\\{\\omega : X(\\omega) = 0\\}) = 1 - p$, as an extended nonnegative real.","title":"Probability that a Bernoulli variable equals zero"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/measure-one","kind":"node","lean":"ProbabilityTheory.IsBernoulliRV.measure_one","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#measure-one","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"If $X : \\Omega \\to \\mathbb{N}$ is a Bernoulli random variable with parameter $p$ under a measure $\\mu$, then $\\mu(\\{\\omega : X(\\omega) = 1\\}) = p$, as an extended nonnegative real.","title":"Probability that a Bernoulli variable equals one"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/bernoulli-sum-pmf","kind":"node","lean":"ProbabilityTheory.bernoulli_sum_pmf","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#bernoulli-sum-pmf","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $m \\in \\mathbb{N}$, and let $X_1, \\dots, X_m : \\Omega \\to \\mathbb{N}$ be measurable, mutually independent random variables such that each $X_i$ is a Bernoulli random variable with real parameter $p_i$ under $\\mu$. Then for every $j \\in \\mathbb{N}$ the probability that $X_1 + \\dots + X_m = j$ equals","title":"Distribution of a sum of independent Bernoulli variables"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/bernoulli-sum-pmf-iid","kind":"node","lean":"ProbabilityTheory.bernoulli_sum_pmf_iid","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#bernoulli-sum-pmf-iid","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $m \\in \\mathbb{N}$, let $p$ be a real number, and let $X_1, \\dots, X_m : \\Omega \\to \\mathbb{N}$ be measurable, mutually independent random variables, each a Bernoulli random variable with parameter $p$ under $\\mu$. Then for every $j \\in \\mathbb{N}$ the probability that $X_1 + \\dots + X_m = j$ is $\\binom{m}{j} p^j (1 - p)^{m - j}$, where the\u2026","title":"Binomial distribution of a sum of i.i.d. Bernoulli variables"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-pmf","kind":"node","lean":"ProbabilityTheory.poissonBinomialPMF","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-pmf","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Poisson-binomial mass function of the number of successes among independent trials with success probabilities `p : Fin m \u2192 \u211d`: `\u2211_{A \u2286 [m], |A| = j} \u220f_{i \u2208 A} p i \u220f_{i \u2209 A} (1 - p i)`. A probability mass function when every `p i \u2208 [0, 1]`; the algebraic identities below hold for all real `p`.","title":"Poisson-binomial mass function"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-tail","kind":"node","lean":"ProbabilityTheory.poissonBinomialTail","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-tail","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Upper tail `P(S \u2265 k) = \u2211_{k \u2264 j \u2264 m} P(S = j)` of the Poisson-binomial law (the mass function vanishes above `m`).","title":"Upper tail of the Poisson-binomial law"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-pmf-nonneg","kind":"node","lean":"ProbabilityTheory.poissonBinomialPMF_nonneg","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-pmf-nonneg","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"The Poisson-binomial mass function is nonnegative for probabilities in `[0, 1]`.","title":"Nonnegativity of the Poisson-binomial mass function"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-pmf-eq-zero-of-lt","kind":"node","lean":"ProbabilityTheory.poissonBinomialPMF_eq_zero_of_lt","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-pmf-eq-zero-of-lt","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"The Poisson-binomial mass function vanishes above the number of trials.","title":"Poisson-binomial mass vanishes above the number of trials"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-pmf-cons-zero","kind":"node","lean":"ProbabilityTheory.poissonBinomialPMF_cons_zero","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-pmf-cons-zero","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Adding a trial with probability `a`: no successes requires a failure of the new trial.","title":"Poisson-binomial mass at zero after adding a trial"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-pmf-cons-succ","kind":"node","lean":"ProbabilityTheory.poissonBinomialPMF_cons_succ","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-pmf-cons-succ","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Adding a trial with probability `a`: first-step decomposition of the mass function, `P(S' = j + 1) = a P(S = j) + (1 - a) P(S = j + 1)`.","title":"First-step recurrence for the Poisson-binomial mass function"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/sum-poisson-binomial-pmf","kind":"node","lean":"ProbabilityTheory.sum_poissonBinomialPMF","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#sum-poisson-binomial-pmf","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"The Poisson-binomial masses sum to one (an identity for all real `p`; it is the expansion of `\u220f ((1 - p i) + p i) = 1`).","title":"Poisson-binomial masses sum to one"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-tail-eq-add","kind":"node","lean":"ProbabilityTheory.poissonBinomialTail_eq_add","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-tail-eq-add","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Peeling off the lowest term of the tail: `P(S \u2265 k) = P(S = k) + P(S \u2265 k + 1)`.","title":"Peeling off the lowest term of the Poisson-binomial tail"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-tail-cons-succ","kind":"node","lean":"ProbabilityTheory.poissonBinomialTail_cons_succ","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-tail-cons-succ","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Adding a trial with probability `a` raises the upper tail by exactly `a P(S = k)`: `P(S' \u2265 k + 1) = P(S \u2265 k + 1) + a P(S = k)`. So the tail is affine in each trial's probability, with nonnegative slope a point mass of the remaining trials.","title":"Poisson-binomial tail after adding a trial"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-tail-mono","kind":"node","lean":"ProbabilityTheory.poissonBinomialTail_mono","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-tail-mono","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"If `p i \u2264 q i` for all `i`, with all probabilities in `[0, 1]`, then `P_p(S \u2265 k) \u2264 P_q(S \u2265 k)`.","title":"The burden-test p-value is monotone in every background rate"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/poisson-binomial-pmf-const","kind":"node","lean":"ProbabilityTheory.poissonBinomialPMF_const","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#poisson-binomial-pmf-const","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"With equal probabilities the Poisson-binomial law is the binomial law.","title":"Poisson-binomial law with equal probabilities is binomial"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/binomial-tail-mono","kind":"node","lean":"ProbabilityTheory.binomialTail_mono","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#binomial-tail-mono","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"For `0 \u2264 p \u2264 q \u2264 1`, `\u2211_{k \u2264 j \u2264 n} C(n, j) p^j (1 - p)^{n - j} \u2264 \u2211_{k \u2264 j \u2264 n} C(n, j) q^j (1 - q)^{n - j}`.","title":"Binomial tail monotonicity"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/binomial-real-ici-mono","kind":"node","lean":"ProbabilityTheory.binomial_real_Ici_mono","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#binomial-real-ici-mono","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"The upper tail `Bin(n, p)([k, \u221e))` of Mathlib's binomial law is monotone in `p`.","title":"Binomial upper tail is monotone in the success probability"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/measure-le-sum-eq-poisson-binomial-tail","kind":"node","lean":"ProbabilityTheory.measure_le_sum_eq_poissonBinomialTail","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#measure-le-sum-eq-poisson-binomial-tail","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"For independent Bernoulli variables `X i` with parameters `p i`, the one-sided burden p-value `P(\u2211 X i \u2265 k)` is the Poisson-binomial tail.","title":"Tail of a sum of independent Bernoulli variables is Poisson-binomial"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/sum-range-abs-poisson-binomial-pmf-sub-p","kind":"node","lean":"ProbabilityTheory.sum_range_abs_poissonBinomialPMF_sub_poisson_le","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#sum-range-abs-poisson-binomial-pmf-sub-p","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"For probabilities `p i \u2208 [0, 1]` and `\u03bb = \u2211 p i`, every partial sum of `\u2211_j |P(S = j) - e^{-\u03bb} \u03bb^j / j!|` is at most `2 \u2211 (p i)\u00b2`.","title":"Le Cam's inequality, finite truncations"},{"id":"Probability/distributions/discrete/poisson-binomial-distribution/tsum-abs-poisson-binomial-pmf-sub-poisso","kind":"node","lean":"ProbabilityTheory.tsum_abs_poissonBinomialPMF_sub_poisson_le","parent":20982,"result":true,"route":"../roadmap/Probability/distributions/discrete/poisson-binomial-distribution/index.html#tsum-abs-poisson-binomial-pmf-sub-poisso","scope":"Probability/distributions/discrete/poisson-binomial-distribution","status":"fully_proved","summary":"Le Cam's inequality (Le Cam 1960): the series `\u2211_j |P(S = j) - e^{-\u03bb} \u03bb^j / j!|` is summable and its sum is at most `2 \u2211 (p i)\u00b2`, where `\u03bb = \u2211 p i`.","title":"Le Cam's inequality"},{"data_url":"scopes/Probability/distributions/continuous.json","id":"Probability/distributions/continuous","kind":"scope","parent":20872,"result":false,"route":"../roadmap/Probability/distributions/continuous/index.html","scope":"Probability/distributions/continuous","status":"can_prove","summary":"Laws with densities, with their moments and tails: beta, gamma, exponential, chi-square, log-normal, Weibull and Dirichlet laws, the multivariate normal law with Gaussian conditioning, orthant probabilities and L\u00b2 limits, and the Poisson\u2013Gamma duality.","title":"Continuous distributions"},{"data_url":"scopes/Probability/distributions/continuous/beta-distribution.json","id":"Probability/distributions/continuous/beta-distribution","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","title":"Beta distribution"},{"id":"Probability/distributions/continuous/beta-distribution/beta-pdfreal-nonneg","kind":"node","lean":"ProbabilityTheory.betaPDFReal_nonneg","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#beta-pdfreal-nonneg","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"The beta density is nonnegative for positive shape parameters.","title":"Nonnegativity of the beta density"},{"id":"Probability/distributions/continuous/beta-distribution/integral-beta-pdfreal","kind":"node","lean":"ProbabilityTheory.integral_betaPDFReal","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#integral-beta-pdfreal","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"The beta density integrates to `1` (the Bochner-integral form of `lintegral_betaPDF_eq_one`).","title":"Beta density integrates to one"},{"id":"Probability/distributions/continuous/beta-distribution/integral-beta-measure","kind":"node","lean":"ProbabilityTheory.integral_betaMeasure","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#integral-beta-measure","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"An integral against `betaMeasure \u03b1 \u03b2` is the integral of the function times the beta density. No integrability hypothesis is needed: both sides are the junk value `0` together.","title":"Integrals against the beta law"},{"id":"Probability/distributions/continuous/beta-distribution/beta-add-one-left","kind":"node","lean":"ProbabilityTheory.beta_add_one_left","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#beta-add-one-left","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"`B(\u03b1 + 1, \u03b2) = B(\u03b1, \u03b2) \u00b7 \u03b1 / (\u03b1 + \u03b2)`.","title":"Recurrence of the beta function in the first argument"},{"id":"Probability/distributions/continuous/beta-distribution/beta-comm","kind":"node","lean":"ProbabilityTheory.beta_comm","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#beta-comm","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"The beta function is symmetric.","title":"Symmetry of the beta function"},{"id":"Probability/distributions/continuous/beta-distribution/beta-pdfreal-mul-pow","kind":"node","lean":"ProbabilityTheory.betaPDFReal_mul_pow","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#beta-pdfreal-mul-pow","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"Multiplying the `Beta(\u03b1, \u03b2)` density by `x ^ k (1 - x) ^ m` gives `B(\u03b1 + k, \u03b2 + m) / B(\u03b1, \u03b2)` times the `Beta(\u03b1 + k, \u03b2 + m)` density.","title":"Beta-function identity for densities"},{"id":"Probability/distributions/continuous/beta-distribution/integral-pow-mul-one-sub-pow-beta-measur","kind":"node","lean":"ProbabilityTheory.integral_pow_mul_one_sub_pow_betaMeasure","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#integral-pow-mul-one-sub-pow-beta-measur","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0` and natural `k, m`, `E[X ^ k (1 - X) ^ m] = B(\u03b1 + k, \u03b2 + m) / B(\u03b1, \u03b2)`.","title":"Mixed moments of the beta law"},{"id":"Probability/distributions/continuous/beta-distribution/integral-id-beta-measure","kind":"node","lean":"ProbabilityTheory.integral_id_betaMeasure","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#integral-id-beta-measure","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0`, `E[X] = \u03b1 / (\u03b1 + \u03b2)`.","title":"Mean of the beta law"},{"id":"Probability/distributions/continuous/beta-distribution/integral-sq-beta-measure","kind":"node","lean":"ProbabilityTheory.integral_sq_betaMeasure","parent":21008,"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#integral-sq-beta-measure","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0`, `E[X ^ 2] = \u03b1 (\u03b1 + 1) / ((\u03b1 + \u03b2) (\u03b1 + \u03b2 + 1))`.","title":"Second moment of the beta law"},{"id":"Probability/distributions/continuous/beta-distribution/integral-mul-one-sub-beta-measure","kind":"node","lean":"ProbabilityTheory.integral_mul_one_sub_betaMeasure","parent":21008,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean of "],["math","\\(X(1 - X)\\)"],["text"," under a beta law"]],"source":"Mean of $X(1 - X)$ under a beta law"},"result":true,"route":"../roadmap/Probability/distributions/continuous/beta-distribution/index.html#integral-mul-one-sub-beta-measure","scope":"Probability/distributions/continuous/beta-distribution","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0`, `E[X (1 - X)] = \u03b1 \u03b2 / ((\u03b1 + \u03b2) (\u03b1 + \u03b2 + 1))` (half the expected heterozygosity of a `Beta(\u03b1, \u03b2)` allele frequency).","title":"Mean of $X(1 - X)$ under a beta law"},{"data_url":"scopes/Probability/distributions/continuous/chi-square-distribution.json","id":"Probability/distributions/continuous/chi-square-distribution","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","title":"Chi-square distribution"},{"id":"Probability/distributions/continuous/chi-square-distribution/chi-square-measure","kind":"node","lean":"ProbabilityTheory.chiSquareMeasure","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#chi-square-measure","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"For a natural number $p$, the chi-square law with $p$ degrees of freedom is the gamma law on $\\mathbb{R}$ with shape $p/2$ and rate $1/2$ (equivalently, scale $2$): the measure with density $x \\mapsto \\frac{(1/2)^{p/2}}{\\Gamma(p/2)} x^{p/2 - 1} e^{-x/2}$ for $x > 0$ and $0$ for $x < 0$ with respect to Lebesgue measure.","title":"Chi-square distribution"},{"id":"Probability/distributions/continuous/chi-square-distribution/mgf-sq-std-gaussian","kind":"node","lean":"ProbabilityTheory.mgf_sq_stdGaussian","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#mgf-sq-std-gaussian","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"Let $\\gamma$ be the standard Gaussian measure on $\\mathbb{R}$ (mean $0$, variance $1$). For every real $t$, $\\int_{\\mathbb{R}} e^{t x^2} \\, d\\gamma(x) = 1/\\sqrt{1 - 2t}$.","title":"Moment generating function of the square of a standard normal"},{"id":"Probability/distributions/continuous/chi-square-distribution/integrable-exp-mul-sq-std-gaussian","kind":"node","lean":"ProbabilityTheory.integrable_exp_mul_sq_stdGaussian","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#integrable-exp-mul-sq-std-gaussian","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"Let $\\gamma$ be the standard Gaussian measure on $\\mathbb{R}$ (mean $0$, variance $1$). For every real $t < 1/2$, the function $x \\mapsto e^{t x^2}$ is integrable with respect to $\\gamma$.","title":"Exponential moments of the square of a standard normal"},{"id":"Probability/distributions/continuous/chi-square-distribution/iio-subset-integrable-exp-set-map-sq-std","kind":"node","lean":"ProbabilityTheory.Iio_subset_integrableExpSet_map_sq_stdGaussian","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#iio-subset-integrable-exp-set-map-sq-std","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"Let $\\gamma$ be the standard Gaussian measure on $\\mathbb{R}$ (mean $0$, variance $1$) and let $\\nu$ be its image under $x \\mapsto x^2$. For every real $t < 1/2$, the function $y \\mapsto e^{t y}$ is integrable with respect to $\\nu$.","title":"Exponential moments of the law of the square of a standard normal"},{"id":"Probability/distributions/continuous/chi-square-distribution/mgf-id-map-sq-std-gaussian","kind":"node","lean":"ProbabilityTheory.mgf_id_map_sq_stdGaussian","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#mgf-id-map-sq-std-gaussian","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"Let $\\gamma$ be the standard Gaussian measure on $\\mathbb{R}$ (mean $0$, variance $1$) and let $\\nu$ be its image under $x \\mapsto x^2$. For every real $t$, $\\int_{\\mathbb{R}} e^{t y} \\, d\\nu(y) = 1/\\sqrt{1 - 2t}$.","title":"Moment generating function of the law of the square of a standard normal"},{"id":"Probability/distributions/continuous/chi-square-distribution/mgf-gamma-measure-half-half-eq-sqrt","kind":"node","lean":"ProbabilityTheory.mgf_gammaMeasure_half_half_eq_sqrt","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#mgf-gamma-measure-half-half-eq-sqrt","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"Let $\\mu$ be the gamma law on $\\mathbb{R}$ with shape $1/2$ and rate $1/2$. For every real $t < 1/2$, $\\int_{\\mathbb{R}} e^{t x} \\, d\\mu(x) = 1/\\sqrt{1 - 2t}$.","title":"Moment generating function of the gamma law with shape and rate one half"},{"id":"Probability/distributions/continuous/chi-square-distribution/mgf-chi-square-measure-of-lt","kind":"node","lean":"ProbabilityTheory.mgf_chiSquareMeasure_of_lt","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#mgf-chi-square-measure-of-lt","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"Let $p$ be a positive integer and let $\\chi^2_p$ be the chi-square law with $p$ degrees of freedom. For every real $t < 1/2$, $\\int_{\\mathbb{R}} e^{t x} \\, d\\chi^2_p(x) = \\left(\\frac{1/2}{1/2 - t}\\right)^{p/2}$.","title":"Moment generating function of the chi-square distribution"},{"id":"Probability/distributions/continuous/chi-square-distribution/integrable-exp-mul-chi-square-measure","kind":"node","lean":"ProbabilityTheory.integrable_exp_mul_chiSquareMeasure","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#integrable-exp-mul-chi-square-measure","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"Let $p$ be a positive integer and let $\\chi^2_p$ be the chi-square law with $p$ degrees of freedom. For every real $t < 1/2$, the function $x \\mapsto e^{t x}$ is integrable with respect to $\\chi^2_p$.","title":"Exponential moments of the chi-square distribution"},{"id":"Probability/distributions/continuous/chi-square-distribution/gaussian-real-map-sq","kind":"node","lean":"ProbabilityTheory.gaussianReal_map_sq","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#gaussian-real-map-sq","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"The square of a standard normal variable is chi-square with one degree of freedom, as an identity of measures on `\u211d`. (Moved here from the likelihood-ratio development so that both that development and Cochran's theorem reuse one declaration.)","title":"The square of a standard normal variable is chi-square with one degree of freedom"},{"id":"Probability/distributions/continuous/chi-square-distribution/prod-mgf-chi-square-measure-eq-mgf-sum","kind":"node","lean":"ProbabilityTheory.prod_mgf_chiSquareMeasure_eq_mgf_sum","parent":21019,"result":true,"route":"../roadmap/Probability/distributions/continuous/chi-square-distribution/index.html#prod-mgf-chi-square-measure-eq-mgf-sum","scope":"Probability/distributions/continuous/chi-square-distribution","status":"fully_proved","summary":"Chi-square MGFs multiply by adding degrees, below `1/2`. Requires a nonempty index set so the summed degree stays positive.","title":"Chi-square moment generating functions multiply by adding degrees"},{"data_url":"scopes/Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits.json","id":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/index.html","scope":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits","status":"fully_proved","title":"Closure of Gaussian laws under L\u00b2 limits"},{"id":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/of-tendsto-e-lp-norm-real","kind":"node","lean":"ProbabilityTheory.HasGaussianLaw.of_tendsto_eLpNorm_real","parent":21030,"result":true,"route":"../roadmap/Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/index.html#of-tendsto-e-lp-norm-real","scope":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits","status":"fully_proved","summary":"If each `X n` is a real Gaussian random variable and `X n \u2192 Y` in `L\u00b2(P)`, then `Y` is Gaussian.","title":"Closure of real Gaussian laws under `L\u00b2` limits"},{"id":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/of-tendsto-e-lp-norm","kind":"node","lean":"ProbabilityTheory.HasGaussianLaw.of_tendsto_eLpNorm","parent":21030,"result":true,"route":"../roadmap/Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/index.html#of-tendsto-e-lp-norm","scope":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits","status":"fully_proved","summary":"If each `X n` is a Gaussian random vector in a separable Banach space `E` and `X n \u2192 Y` in `L\u00b2(P)`, then `Y` is Gaussian.","title":"Closure of Gaussian laws under `L\u00b2` limits"},{"id":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/of-tendsto-e-lp-norm-2","kind":"node","lean":"ProbabilityTheory.IsGaussianProcess.of_tendsto_eLpNorm","parent":21030,"result":true,"route":"../roadmap/Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/index.html#of-tendsto-e-lp-norm-2","scope":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits","status":"fully_proved","summary":"If each `X n` is a real Gaussian process and `X n t \u2192 Y t` in `L\u00b2(P)` for every `t`, then `Y` is a Gaussian process.","title":"Closure of Gaussian processes under `L\u00b2` limits"},{"id":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/tendsto-integral-mul-of-tendsto-e-lp-nor","kind":"node","lean":"ProbabilityTheory.tendsto_integral_mul_of_tendsto_eLpNorm","parent":21030,"result":true,"route":"../roadmap/Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits/index.html#tendsto-integral-mul-of-tendsto-e-lp-nor","scope":"Probability/distributions/continuous/closure-of-gaussian-laws-under-l-limits","status":"fully_proved","summary":"If `X n \u2192 Y` and `Z n \u2192 W` in `L\u00b2(P)` (all square integrable), then `E[X n Z n] \u2192 E[Y W]`.","title":"Continuity of the second moment pairing in `L\u00b2`"},{"data_url":"scopes/Probability/distributions/continuous/dirichlet-distribution.json","id":"Probability/distributions/continuous/dirichlet-distribution","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","title":"Dirichlet distribution"},{"id":"Probability/distributions/continuous/dirichlet-distribution/lintegral-ioo-rpow-mul-rpow","kind":"node","lean":"ProbabilityTheory.lintegral_Ioo_rpow_mul_rpow","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#lintegral-ioo-rpow-mul-rpow","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"For real numbers $a > 0$ and $b > 0$, the integral over the open interval $(0, 1)$, taken as a Lebesgue integral with values in $[0, \\infty]$, is","title":"The Beta integral"},{"id":"Probability/distributions/continuous/dirichlet-distribution/dirichlet-density","kind":"node","lean":"ProbabilityTheory.dirichletDensity","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#dirichlet-density","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"For parameters $\\alpha = (\\alpha_0, \\dots, \\alpha_n) \\in \\mathbb{R}^{n+1}$, the unnormalized Dirichlet density is the function $D_\\alpha \\colon \\mathbb{R}^n \\to [0, \\infty]$ defined by","title":"Unnormalized Dirichlet density"},{"id":"Probability/distributions/continuous/dirichlet-distribution/multi-beta","kind":"node","lean":"ProbabilityTheory.multiBeta","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#multi-beta","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"For real parameters $\\alpha = (\\alpha_0, \\dots, \\alpha_{n-1})$, the multivariate Beta function is","title":"Multivariate Beta function"},{"id":"Probability/distributions/continuous/dirichlet-distribution/lintegral-dirichlet-density-zero","kind":"node","lean":"ProbabilityTheory.lintegral_dirichletDensity_zero","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#lintegral-dirichlet-density-zero","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $\\alpha = (\\alpha_0) \\in \\mathbb{R}^1$ with $\\alpha_0 > 0$, let $D_\\alpha$ be the unnormalized Dirichlet density with parameters $\\alpha$, a function on $\\mathbb{R}^0$, and let $B(\\alpha)$ be the multivariate Beta function. Then $\\int_{\\mathbb{R}^0} D_\\alpha(x) \\, dx = B(\\alpha)$, the integral being taken with respect to Lebesgue measure.","title":"The Dirichlet integral for a single parameter"},{"id":"Probability/distributions/continuous/dirichlet-distribution/lintegral-fin-succ","kind":"node","lean":"ProbabilityTheory.lintegral_fin_succ","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#lintegral-fin-succ","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $G \\colon \\mathbb{R}^{n+1} \\to [0, \\infty]$ be measurable. Then","title":"Tonelli's theorem for the first coordinate"},{"id":"Probability/distributions/continuous/dirichlet-distribution/lintegral-comp-smul-fin","kind":"node","lean":"ProbabilityTheory.lintegral_comp_smul_fin","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#lintegral-comp-smul-fin","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $F \\colon \\mathbb{R}^n \\to [0, \\infty]$ be measurable and let $c > 0$ be a real number. Then","title":"Scaling the variable of a Lebesgue integral"},{"id":"Probability/distributions/continuous/dirichlet-distribution/dirichlet-density-cons-smul","kind":"node","lean":"ProbabilityTheory.dirichletDensity_cons_smul","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#dirichlet-density-cons-smul","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $\\alpha = (\\alpha_0, \\dots, \\alpha_{n+1}) \\in \\mathbb{R}^{n+2}$, write $D_\\alpha$ for the unnormalized Dirichlet density with parameters $\\alpha$ (a function on $\\mathbb{R}^{n+1}$), and let $\\alpha' = (\\alpha_1, \\dots, \\alpha_{n+1}) \\in \\mathbb{R}^{n+1}$. For every real $t$ with $0 < t < 1$ and every $u \\in \\mathbb{R}^n$,","title":"Splitting off the first coordinate of the Dirichlet density"},{"id":"Probability/distributions/continuous/dirichlet-distribution/lintegral-dirichlet-density","kind":"node","lean":"ProbabilityTheory.lintegral_dirichletDensity","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#lintegral-dirichlet-density","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $\\alpha = (\\alpha_0, \\dots, \\alpha_n) \\in \\mathbb{R}^{n+1}$ with $\\alpha_i > 0$ for every $i$, and let $\\Delta = \\{x \\in \\mathbb{R}^n : x_i > 0 \\text{ for all } i \\text{ and } \\sum_{i=0}^{n-1} x_i < 1\\}$ be the open corner simplex. Then","title":"The Dirichlet integral"},{"id":"Probability/distributions/continuous/dirichlet-distribution/simplex-completion","kind":"node","lean":"ProbabilityTheory.simplexCompletion","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#simplex-completion","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"For $x = (x_0, \\dots, x_{n-1}) \\in \\mathbb{R}^n$, the simplex completion of $x$ is the vector $(x_0, \\dots, x_{n-1}, 1 - \\sum_{i=0}^{n-1} x_i) \\in \\mathbb{R}^{n+1}$, whose coordinates sum to $1$.","title":"Simplex completion"},{"id":"Probability/distributions/continuous/dirichlet-distribution/dirichlet-measure","kind":"node","lean":"ProbabilityTheory.dirichletMeasure","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#dirichlet-measure","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $\\alpha = (\\alpha_0, \\dots, \\alpha_n) \\in \\mathbb{R}^{n+1}$, let $D_\\alpha$ be the unnormalized Dirichlet density on $\\mathbb{R}^n$ with parameters $\\alpha$, and let $B(\\alpha) = \\prod_{i=0}^{n} \\Gamma(\\alpha_i) / \\Gamma(\\sum_{i=0}^{n} \\alpha_i)$ be the multivariate Beta function. The Dirichlet distribution $\\mathrm{Dir}(\\alpha)$ is the measure on $\\mathbb{R}^{n+1}$ obtained by pushing forward the measure on\u2026","title":"Dirichlet distribution"},{"id":"Probability/distributions/continuous/dirichlet-distribution/ae-dirichlet-measure","kind":"node","lean":"ProbabilityTheory.ae_dirichletMeasure","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#ae-dirichlet-measure","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"For every $n$ and every parameter vector $\\alpha \\in \\mathbb{R}^{n+1}$, almost every $\\theta \\in \\mathbb{R}^{n+1}$ with respect to the Dirichlet distribution $\\mathrm{Dir}(\\alpha)$ is a probability vector with positive entries: $\\theta_i > 0$ for every $i$ and $\\sum_{i=0}^{n} \\theta_i = 1$.","title":"The Dirichlet distribution is concentrated on the open simplex"},{"id":"Probability/distributions/continuous/dirichlet-distribution/dirichlet-density-mul-prod-pow","kind":"node","lean":"ProbabilityTheory.dirichletDensity_mul_prod_pow","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#dirichlet-density-mul-prod-pow","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $\\alpha = (\\alpha_0, \\dots, \\alpha_n) \\in \\mathbb{R}^{n+1}$ and let $k = (k_0, \\dots, k_n)$ be natural numbers; write $D_\\beta$ for the unnormalized Dirichlet density with parameters $\\beta$. For every $x \\in \\mathbb{R}^n$, let $\\theta = (x_0, \\dots, x_{n-1}, 1 - \\sum_{i=0}^{n-1} x_i)$ be its completion to a vector summing to $1$, and write $\\theta_i^+ = \\max(\\theta_i, 0)$. Then","title":"Multiplying the Dirichlet density by a multinomial likelihood"},{"id":"Probability/distributions/continuous/dirichlet-distribution/with-density-prod-pow-dirichlet-measure","kind":"node","lean":"ProbabilityTheory.withDensity_prod_pow_dirichletMeasure","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#with-density-prod-pow-dirichlet-measure","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $\\alpha = (\\alpha_0, \\dots, \\alpha_n)$ have positive real entries and let $k = (k_0, \\dots, k_n)$ be natural numbers (counts). The measure with density $\\theta \\mapsto \\prod_{i=0}^{n} \\max(\\theta_i, 0)^{k_i}$ with respect to the Dirichlet distribution $\\mathrm{Dir}(\\alpha)$, that is, $\\mathrm{Dir}(\\alpha)$ weighted by the likelihood of the counts $k$, equals","title":"Dirichlet\u2013multinomial conjugacy"},{"id":"Probability/distributions/continuous/dirichlet-distribution/lintegral-dirichlet-measure-apply","kind":"node","lean":"ProbabilityTheory.lintegral_dirichletMeasure_apply","parent":21035,"result":true,"route":"../roadmap/Probability/distributions/continuous/dirichlet-distribution/index.html#lintegral-dirichlet-measure-apply","scope":"Probability/distributions/continuous/dirichlet-distribution","status":"fully_proved","summary":"Let $\\alpha = (\\alpha_0, \\dots, \\alpha_n) \\in \\mathbb{R}^{n+1}$ with $\\alpha_j > 0$ for every $j$. For every $i$, the $i$-th coordinate of a random vector $\\theta$ with the Dirichlet distribution $\\mathrm{Dir}(\\alpha)$ has mean","title":"Mean of the Dirichlet distribution"},{"data_url":"scopes/Probability/distributions/continuous/exponential-distribution.json","id":"Probability/distributions/continuous/exponential-distribution","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","title":"Exponential distribution"},{"id":"Probability/distributions/continuous/exponential-distribution/gamma-pdfreal-one-eq-indicator","kind":"node","lean":"ProbabilityTheory.gammaPDFReal_one_eq_indicator","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#gamma-pdfreal-one-eq-indicator","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"The rate-`r` exponential density times `f`, as an indicator on `Ici 0`.","title":"Exponential density as an indicator on the half-line"},{"id":"Probability/distributions/continuous/exponential-distribution/integral-exp-measure-eq-set-integral","kind":"node","lean":"ProbabilityTheory.integral_expMeasure_eq_setIntegral","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integral-exp-measure-eq-set-integral","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"Integrals against the exponential law of rate `r > 0` are integrals of `r * exp (-(r * x)) * f x` over `(0, \u221e)`.","title":"Integrals against the exponential law"},{"id":"Probability/distributions/continuous/exponential-distribution/integrable-exp-measure-iff","kind":"node","lean":"ProbabilityTheory.integrable_expMeasure_iff","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integrable-exp-measure-iff","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"A function is integrable against the exponential law of rate `r > 0` iff `r * exp (-(r * x)) * f x` is integrable on `(0, \u221e)`.","title":"Integrability criterion for the exponential law"},{"id":"Probability/distributions/continuous/exponential-distribution/pow-eq-rpow-mul-exp","kind":"node","lean":"ProbabilityTheory.pow_eq_rpow_mul_exp","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#pow-eq-rpow-mul-exp","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"Rewriting `x ^ n` times the exponential density in the gamma-integrand form.","title":"Power times exponential density in gamma-integrand form"},{"id":"Probability/distributions/continuous/exponential-distribution/integrable-pow-exp-measure","kind":"node","lean":"ProbabilityTheory.integrable_pow_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integrable-pow-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"All moments of the exponential law are finite.","title":"Finiteness of all moments of the exponential law"},{"id":"Probability/distributions/continuous/exponential-distribution/integral-pow-exp-measure","kind":"node","lean":"ProbabilityTheory.integral_pow_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integral-pow-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"The `n`-th moment of the exponential law of rate `r > 0` is `n! / r ^ n`.","title":"Moments of the exponential law"},{"id":"Probability/distributions/continuous/exponential-distribution/integrable-id-exp-measure","kind":"node","lean":"ProbabilityTheory.integrable_id_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integrable-id-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"The identity is integrable against the exponential law.","title":"Integrability of the identity under the exponential law"},{"id":"Probability/distributions/continuous/exponential-distribution/integral-id-exp-measure","kind":"node","lean":"ProbabilityTheory.integral_id_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integral-id-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"The mean of the exponential law of rate `r > 0` is `r\u207b\u00b9`.","title":"Mean of the exponential law"},{"id":"Probability/distributions/continuous/exponential-distribution/integral-sq-exp-measure","kind":"node","lean":"ProbabilityTheory.integral_sq_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integral-sq-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"The second moment of the exponential law of rate `r > 0` is `2 / r ^ 2`.","title":"Second moment of the exponential law"},{"id":"Probability/distributions/continuous/exponential-distribution/variance-id-exp-measure","kind":"node","lean":"ProbabilityTheory.variance_id_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#variance-id-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"The variance of the exponential law of rate `r > 0` is `r\u207b\u00b9 ^ 2`.","title":"Variance of the exponential law"},{"id":"Probability/distributions/continuous/exponential-distribution/integrable-of-exp-measure","kind":"node","lean":"ProbabilityTheory.HasLaw.integrable_of_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integrable-of-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"A random variable with exponential law of rate `r > 0` is integrable.","title":"Exponentially distributed random variables are integrable"},{"id":"Probability/distributions/continuous/exponential-distribution/integral-eq-inv-of-exp-measure","kind":"node","lean":"ProbabilityTheory.HasLaw.integral_eq_inv_of_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#integral-eq-inv-of-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"A random variable with exponential law of rate `r > 0` has mean `r\u207b\u00b9`.","title":"Mean of an exponentially distributed random variable"},{"id":"Probability/distributions/continuous/exponential-distribution/variance-eq-of-exp-measure","kind":"node","lean":"ProbabilityTheory.HasLaw.variance_eq_of_expMeasure","parent":21050,"result":true,"route":"../roadmap/Probability/distributions/continuous/exponential-distribution/index.html#variance-eq-of-exp-measure","scope":"Probability/distributions/continuous/exponential-distribution","status":"fully_proved","summary":"A random variable with exponential law of rate `r > 0` has variance `r\u207b\u00b9 ^ 2`.","title":"Variance of an exponentially distributed random variable"},{"data_url":"scopes/Probability/distributions/continuous/gamma-distribution.json","id":"Probability/distributions/continuous/gamma-distribution","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","title":"Gamma distribution"},{"id":"Probability/distributions/continuous/gamma-distribution/integrable-on-rpow-mul-exp-neg-mul-ioi","kind":"node","lean":"ProbabilityTheory.integrableOn_rpow_mul_exp_neg_mul_Ioi","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integrable-on-rpow-mul-exp-neg-mul-ioi","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"For real numbers $a > 0$ and $s > 0$, the function $x \\mapsto x^{a-1} e^{-sx}$ is Lebesgue integrable on the open half-line $(0, \\infty)$.","title":"Integrability of the scaled gamma integrand"},{"id":"Probability/distributions/continuous/gamma-distribution/integrable-exp-mul-gamma-measure","kind":"node","lean":"ProbabilityTheory.integrable_exp_mul_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integrable-exp-mul-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"Let $a > 0$ and $r > 0$. For every real $t < r$, the function $x \\mapsto e^{tx}$ is integrable with respect to the gamma distribution with shape $a$ and rate $r$.","title":"Exponential moments of the gamma distribution"},{"id":"Probability/distributions/continuous/gamma-distribution/mgf-gamma-measure-of-lt","kind":"node","lean":"ProbabilityTheory.mgf_gammaMeasure_of_lt","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#mgf-gamma-measure-of-lt","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"Let $a > 0$ and $r > 0$, and let $X$ have the gamma distribution with shape $a$ and rate $r$. For every real $t < r$, the moment generating function of $X$ at $t$ is $\\mathbb{E}[e^{tX}] = \\left(\\frac{r}{r - t}\\right)^a$.","title":"Moment generating function of the gamma distribution"},{"id":"Probability/distributions/continuous/gamma-distribution/iio-subset-integrable-exp-set-gamma-meas","kind":"node","lean":"ProbabilityTheory.Iio_subset_integrableExpSet_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#iio-subset-integrable-exp-set-gamma-meas","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"Let $a > 0$ and $r > 0$, and let $S$ be the set of real $t$ such that $x \\mapsto e^{tx}$ is integrable with respect to the gamma distribution with shape $a$ and rate $r$. Then the half-line $(-\\infty, r)$ is contained in $S$.","title":"Half-line of exponential integrability for the gamma distribution"},{"id":"Probability/distributions/continuous/gamma-distribution/ioo-subset-interior-integrable-exp-set-g","kind":"node","lean":"ProbabilityTheory.Ioo_subset_interior_integrableExpSet_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#ioo-subset-interior-integrable-exp-set-g","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"Let $a > 0$, $r > 0$ and $0 < \\varepsilon < r$, and let $S$ be the set of real $t$ such that $x \\mapsto e^{tx}$ is integrable with respect to the gamma distribution with shape $a$ and rate $r$. Then the interval $(-\\varepsilon, \\varepsilon)$ is contained in the interior of $S$.","title":"A neighbourhood of zero of exponential integrability for the gamma distribution"},{"id":"Probability/distributions/continuous/gamma-distribution/gamma-add-nat-eq","kind":"node","lean":"ProbabilityTheory.Gamma_add_nat_eq","parent":21064,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gamma function at "],["math","\\(a + k\\)"],["text"," via the rising factorial"]],"source":"Gamma function at $a + k$ via the rising factorial"},"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#gamma-add-nat-eq","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"`\u0393(a + k) = a\u207d\u1d4f\u207e \u0393(a)` for `a > 0`.","title":"Gamma function at $a + k$ via the rising factorial"},{"id":"Probability/distributions/continuous/gamma-distribution/pow-mul-rpow-sub-one","kind":"node","lean":"ProbabilityTheory.pow_mul_rpow_sub_one","parent":21064,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(r^k r^{a-1} = r^{a+k-1}\\)"],["text"," for nonnegative "],["math","\\(r\\)"]],"source":"$r^k r^{a-1} = r^{a+k-1}$ for nonnegative $r$"},"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#pow-mul-rpow-sub-one","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"`r\u1d4f r^(a - 1) = r^(a + k - 1)` for `r \u2265 0` and `a > 0`.","title":"$r^k r^{a-1} = r^{a+k-1}$ for nonnegative $r$"},{"id":"Probability/distributions/continuous/gamma-distribution/ae-nonneg-gamma-measure","kind":"node","lean":"ProbabilityTheory.ae_nonneg_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#ae-nonneg-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The Gamma law is concentrated on `[0, \u221e)`.","title":"Gamma law is concentrated on the nonnegative reals"},{"id":"Probability/distributions/continuous/gamma-distribution/integrable-pow-mul-exp-neg-mul-gamma-mea","kind":"node","lean":"ProbabilityTheory.integrable_pow_mul_exp_neg_mul_gammaMeasure","parent":21064,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrability of "],["math","\\(x^k e^{-cx}\\)"],["text"," under the Gamma law"]],"source":"Integrability of $x^k e^{-cx}$ under the Gamma law"},"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integrable-pow-mul-exp-neg-mul-gamma-mea","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"For `a, b > 0` and `c \u2265 0`, `x \u21a6 x ^ k e^{-c x}` is integrable for `Gamma(a, b)`.","title":"Integrability of $x^k e^{-cx}$ under the Gamma law"},{"id":"Probability/distributions/continuous/gamma-distribution/integral-pow-mul-exp-neg-mul-gamma-measu","kind":"node","lean":"ProbabilityTheory.integral_pow_mul_exp_neg_mul_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integral-pow-mul-exp-neg-mul-gamma-measu","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"For shape `a > 0`, rate `b > 0` and `c \u2265 0`, `\u222b x ^ k e^{-c x} dGamma(a, b) = a\u207d\u1d4f\u207e b\u1d43 / (b + c) ^ (a + k)`.","title":"Gamma integrals"},{"id":"Probability/distributions/continuous/gamma-distribution/integrable-id-gamma-measure","kind":"node","lean":"ProbabilityTheory.integrable_id_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integrable-id-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"`x \u21a6 x` is integrable for `Gamma(a, b)`.","title":"Gamma law has finite first moment"},{"id":"Probability/distributions/continuous/gamma-distribution/integrable-sq-gamma-measure","kind":"node","lean":"ProbabilityTheory.integrable_sq_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integrable-sq-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"`x \u21a6 x\u00b2` is integrable for `Gamma(a, b)`.","title":"Gamma law has finite second moment"},{"id":"Probability/distributions/continuous/gamma-distribution/integrable-exp-set-gamma-measure","kind":"node","lean":"ProbabilityTheory.integrableExpSet_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integrable-exp-set-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The exponential moment `E[exp (t X)]` of a gamma `(a, r)` law is finite exactly for `t < r`.","title":"Exponential moments of the gamma law are finite exactly below the rate"},{"id":"Probability/distributions/continuous/gamma-distribution/eq-gamma-measure-of-mgf-eq","kind":"node","lean":"ProbabilityTheory.eq_gammaMeasure_of_mgf_eq","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#eq-gamma-measure-of-mgf-eq","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"MGF uniqueness for gamma laws: a real probability law whose MGF equals the gamma `(a, r)` MGF as a function on all of `\u211d` (both vanish where the exponential moment is infinite) is that gamma law.","title":"Moment generating function determines the gamma law"},{"id":"Probability/distributions/continuous/gamma-distribution/upper-incomplete-gamma","kind":"node","lean":"Real.upperIncompleteGamma","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#upper-incomplete-gamma","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The upper incomplete gamma function `\u0393(a, x) = \u222b_x^\u221e t ^ (a - 1) e^{-t} dt` (DLMF 8.2.2). The domain used below is `a > 0` and `x \u2265 0`, where the integral converges. Outside it the Bochner integral may be the junk value `0` (for `a \u2264 0`, `x = 0` the integrand is not integrable).","title":"Upper incomplete gamma function"},{"id":"Probability/distributions/continuous/gamma-distribution/upper-incomplete-gamma-zero","kind":"node","lean":"Real.upperIncompleteGamma_zero","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#upper-incomplete-gamma-zero","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"`\u0393(a, 0) = \u0393(a)` for `a > 0`.","title":"Upper incomplete gamma function at zero"},{"id":"Probability/distributions/continuous/gamma-distribution/upper-incomplete-gamma-one","kind":"node","lean":"Real.upperIncompleteGamma_one","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#upper-incomplete-gamma-one","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"`\u0393(1, x) = e^{-x}`.","title":"Upper incomplete gamma function with shape one"},{"id":"Probability/distributions/continuous/gamma-distribution/upper-incomplete-gamma-add-one","kind":"node","lean":"Real.upperIncompleteGamma_add_one","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#upper-incomplete-gamma-add-one","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The recurrence `\u0393(a + 1, x) = x ^ a e^{-x} + a \u0393(a, x)` for `a > 0` and `x \u2265 0` (DLMF 8.8.2), from integration by parts.","title":"Recurrence for the upper incomplete gamma function"},{"id":"Probability/distributions/continuous/gamma-distribution/upper-incomplete-gamma-nat-cast-add-one","kind":"node","lean":"Real.upperIncompleteGamma_natCast_add_one","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#upper-incomplete-gamma-nat-cast-add-one","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The closed form at natural shapes (DLMF 8.4.8): for `x \u2265 0`, `\u0393(n + 1, x) = n! e^{-x} \u2211_{k = 0}^{n} x ^ k / (k!)`.","title":"Upper incomplete gamma function at natural shapes"},{"id":"Probability/distributions/continuous/gamma-distribution/integral-rpow-gamma-measure","kind":"node","lean":"ProbabilityTheory.integral_rpow_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integral-rpow-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"Real moments of the gamma law: for shape `a > 0`, rate `r > 0` and `p > -a`, `E[X ^ p] = \u0393(a + p) / (\u0393(a) r ^ p)`. (The law is carried by `[0, \u221e)`, so the values of `x ^ p` for `x < 0` play no role.)","title":"Real moments of the gamma law"},{"id":"Probability/distributions/continuous/gamma-distribution/integral-id-gamma-measure","kind":"node","lean":"ProbabilityTheory.integral_id_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integral-id-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The mean of the gamma law with shape `a > 0` and rate `r > 0` is `a / r`.","title":"Mean of the gamma law"},{"id":"Probability/distributions/continuous/gamma-distribution/integral-sq-gamma-measure","kind":"node","lean":"ProbabilityTheory.integral_sq_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#integral-sq-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The second moment of the gamma law with shape `a > 0` and rate `r > 0` is `a (a + 1) / r ^ 2`.","title":"Second moment of the gamma law"},{"id":"Probability/distributions/continuous/gamma-distribution/mem-lp-id-gamma-measure","kind":"node","lean":"ProbabilityTheory.memLp_id_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#mem-lp-id-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The gamma law has finite second moment.","title":"Gamma law has finite second moment"},{"id":"Probability/distributions/continuous/gamma-distribution/variance-id-gamma-measure","kind":"node","lean":"ProbabilityTheory.variance_id_gammaMeasure","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#variance-id-gamma-measure","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The variance of the gamma law with shape `a > 0` and rate `r > 0` is `a / r ^ 2`.","title":"Variance of the gamma law"},{"id":"Probability/distributions/continuous/gamma-distribution/gamma-measure-map-const-mul","kind":"node","lean":"ProbabilityTheory.gammaMeasure_map_const_mul","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#gamma-measure-map-const-mul","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"Scaling a gamma variable by `c > 0` keeps its shape and divides its rate by `c`: `c X ~ Gamma(a, r / c)` when `X ~ Gamma(a, r)`.","title":"Scaling a gamma random variable"},{"id":"Probability/distributions/continuous/gamma-distribution/gamma-measure-real-ioi","kind":"node","lean":"ProbabilityTheory.gammaMeasure_real_Ioi","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#gamma-measure-real-ioi","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The upper tail of the gamma law: `P(X > c) = \u0393(a, r c) / \u0393(a)` for `c \u2265 0`.","title":"Upper tail of the gamma law"},{"id":"Probability/distributions/continuous/gamma-distribution/gamma-measure-real-ioi-nat-cast-add-one","kind":"node","lean":"ProbabilityTheory.gammaMeasure_real_Ioi_natCast_add_one","parent":21064,"result":true,"route":"../roadmap/Probability/distributions/continuous/gamma-distribution/index.html#gamma-measure-real-ioi-nat-cast-add-one","scope":"Probability/distributions/continuous/gamma-distribution","status":"fully_proved","summary":"The Erlang tail: for shape `n + 1`, `P(X > c) = e^{-r c} \u2211_{k = 0}^{n} (r c) ^ k / (k!)` when `c \u2265 0`.","title":"Erlang tail formula"},{"data_url":"scopes/Probability/distributions/continuous/gaussian-conditioning.json","id":"Probability/distributions/continuous/gaussian-conditioning","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","title":"Gaussian conditioning"},{"id":"Probability/distributions/continuous/gaussian-conditioning/noise-kernel","kind":"node","lean":"MetaMathlibExt.noiseKernel","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#noise-kernel","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $E$ be an additive commutative group with a measurable structure for which addition $E \\times E \\to E$ is measurable, let $F$ be a measurable space, let $f \\colon F \\to E$ be a measurable map and let $\\nu$ be a measure on $E$. If $\\nu$ is s-finite, the additive-noise kernel from $F$ to $E$ sends $y \\in F$ to the image of the product measure $\\delta_{f(y)} \\otimes \\nu$ (where $\\delta_{f(y)}$ is the Dirac measure\u2026","title":"The additive-noise kernel"},{"id":"Probability/distributions/continuous/gaussian-conditioning/cond-distrib-add-of-indep-fun","kind":"node","lean":"MetaMathlibExt.condDistrib_add_of_indepFun","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#cond-distrib-add-of-indep-fun","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $E$ be a nonempty standard Borel space that is an abelian group with measurable addition, let $F$ be a measurable space, and let $Y \\colon \\Omega \\to F$ and $Z \\colon \\Omega \\to E$ be independent measurable random variables on a probability space. For every measurable $f \\colon F \\to E$ and almost every $y$ with respect to the law of $Y$, the conditional distribution of $f(Y) + Z$ given $Y = y$ is the law of\u2026","title":"Conditioning on a signal observed in independent additive noise"},{"id":"Probability/distributions/continuous/gaussian-conditioning/cond-distrib-of-has-gaussian-law","kind":"node","lean":"MetaMathlibExt.condDistrib_of_hasGaussianLaw","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#cond-distrib-of-has-gaussian-law","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $E$ and $F$ be finite-dimensional real inner product spaces, and let $X$ and $Y$ be measurable random vectors in $E$ and $F$ on a probability space such that $(X, Y)$ is jointly Gaussian. Let $A \\colon F \\to E$ be a linear map whose residual $X - A Y$ is uncorrelated with $Y$ (the normal equations): for all $u \\in E$ and $w \\in F$, $\\operatorname{Cov}(\\langle u, X - A Y \\rangle, \\langle w, Y \\rangle) = 0$. Then\u2026","title":"Gaussian conditioning"},{"id":"Probability/distributions/continuous/gaussian-conditioning/map-to-euclidean-lin-multivariate-gaussi","kind":"node","lean":"MetaMathlibExt.map_toEuclideanLin_multivariateGaussian","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#map-to-euclidean-lin-multivariate-gaussi","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Write $\\mathcal{N}(a, \\Sigma)$ for the multivariate Gaussian distribution with mean $a$ and covariance matrix $\\Sigma$. Let $\\iota$ and $\\kappa$ be finite index sets, let $M$ be a real $\\iota \\times \\kappa$ matrix, let $\\mu \\in \\mathbb{R}^\\kappa$ and let $S$ be a positive semidefinite real $\\kappa \\times \\kappa$ matrix. Then the image of $\\mathcal{N}(\\mu, S)$ under the linear map $z \\mapsto M z$ is\u2026","title":"Linear images of multivariate Gaussians"},{"id":"Probability/distributions/continuous/gaussian-conditioning/map-add-multivariate-gaussian","kind":"node","lean":"MetaMathlibExt.map_add_multivariateGaussian","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#map-add-multivariate-gaussian","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $\\iota$ be a finite index set. For all $c, \\mu \\in \\mathbb{R}^\\iota$ and every real $\\iota \\times \\iota$ matrix $S$, the pushforward of $N(\\mu, S)$ under the translation $x \\mapsto c + x$ is $N(c + \\mu, S)$.","title":"Translating a multivariate Gaussian"},{"id":"Probability/distributions/continuous/gaussian-conditioning/block-row","kind":"node","lean":"MetaMathlibExt.blockRow","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#block-row","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $\\rho$, $\\iota$ and $\\kappa$ be index sets with $\\iota$ and $\\kappa$ finite. For real matrices $P$ with rows indexed by $\\rho$ and columns by $\\iota$, and $Q$ with rows indexed by $\\rho$ and columns by $\\kappa$, the block row $[P\\ Q]$ is the real matrix with rows indexed by $\\rho$ and columns indexed by the disjoint union $\\iota \\sqcup \\kappa$, whose entry in row $r$ is $P_{ri}$ in the column of $i \\in \\iota$\u2026","title":"Horizontal block matrix"},{"id":"Probability/distributions/continuous/gaussian-conditioning/covariance-inner-of-map-eq","kind":"node","lean":"MetaMathlibExt.covariance_inner_of_map_eq","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#covariance-inner-of-map-eq","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $\\nu$ be a finite index set and let $V$ be a measurable random vector in $\\mathbb{R}^\\nu$ on a probability space with law $N(\\mu, S)$, where $S$ is positive semidefinite. Then for all $a, b \\in \\mathbb{R}^\\nu$, $\\operatorname{Cov}(\\langle a, V \\rangle, \\langle b, V \\rangle) = a^\\top S b$.","title":"Covariances of linear functionals of a Gaussian vector"},{"id":"Probability/distributions/continuous/gaussian-conditioning/cond-distrib-multivariate-gaussian","kind":"node","lean":"MetaMathlibExt.condDistrib_multivariateGaussian","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#cond-distrib-multivariate-gaussian","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $\\iota$ and $\\kappa$ be finite index sets and let $V$ be a measurable random vector in $\\mathbb{R}^{\\iota \\sqcup \\kappa}$ on a probability space, with blocks $X \\in \\mathbb{R}^\\iota$ and $Y \\in \\mathbb{R}^\\kappa$. Suppose $V$ has law $N(\\mu, S)$ with $S$ positive semidefinite, write $\\mu_X, \\mu_Y$ for the blocks of $\\mu$ and $S_{11}, S_{12}, S_{21}, S_{22}$ for the blocks of $S$ indexed by $\\iota \\times \\iota$,\u2026","title":"Gaussian conditioning (Schur complement)"},{"id":"Probability/distributions/continuous/gaussian-conditioning/pos-semidef-from-blocks-zero","kind":"node","lean":"MetaMathlibExt.posSemidef_fromBlocks_zero","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#pos-semidef-from-blocks-zero","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $\\iota$ and $\\kappa$ be finite index sets and let $S$ and $R$ be positive semidefinite real matrices indexed by $\\iota$ and by $\\kappa$. Then the block-diagonal matrix $\\begin{pmatrix} S & 0 \\\\ 0 & R \\end{pmatrix}$, indexed by $\\iota \\sqcup \\kappa$, is positive semidefinite.","title":"Block-diagonal matrices with positive semidefinite blocks"},{"id":"Probability/distributions/continuous/gaussian-conditioning/inner-sum-equiv-prod-symm","kind":"node","lean":"MetaMathlibExt.inner_sumEquivProd_symm","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#inner-sum-equiv-prod-symm","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $\\iota$ and $\\kappa$ be finite index sets. For $p_1 \\in \\mathbb{R}^\\iota$, $p_2 \\in \\mathbb{R}^\\kappa$ and $t \\in \\mathbb{R}^{\\iota \\sqcup \\kappa}$ with blocks $t_\\iota \\in \\mathbb{R}^\\iota$ and $t_\\kappa \\in \\mathbb{R}^\\kappa$, the inner product of the concatenation of $p_1$ and $p_2$ with $t$ is $\\langle p_1, t_\\iota \\rangle + \\langle p_2, t_\\kappa \\rangle$.","title":"The inner product splits over blocks"},{"id":"Probability/distributions/continuous/gaussian-conditioning/char-fun-map-sum-equiv-prod-symm-prod","kind":"node","lean":"MetaMathlibExt.charFun_map_sumEquivProd_symm_prod","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#char-fun-map-sum-equiv-prod-symm-prod","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $\\iota$ and $\\kappa$ be finite index sets, let $\\mu$ and $\\nu$ be finite measures on $\\mathbb{R}^\\iota$ and $\\mathbb{R}^\\kappa$, and let $\\pi$ be the pushforward of the product measure $\\mu \\otimes \\nu$ to $\\mathbb{R}^{\\iota \\sqcup \\kappa}$ under concatenation of the two blocks. Write $\\varphi_\\rho$ for the characteristic function of a measure $\\rho$. Then for every $t \\in \\mathbb{R}^{\\iota \\sqcup \\kappa}$ with\u2026","title":"Characteristic function of a product of blocks"},{"id":"Probability/distributions/continuous/gaussian-conditioning/map-sum-equiv-prod-symm-prod-multivariat","kind":"node","lean":"MetaMathlibExt.map_sumEquivProd_symm_prod_multivariateGaussian","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#map-sum-equiv-prod-symm-prod-multivariat","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Write $\\mathcal{N}(a, \\Sigma)$ for the multivariate Gaussian distribution with mean $a$ and covariance matrix $\\Sigma$. Let $\\iota$ and $\\kappa$ be finite index sets, let $m \\in \\mathbb{R}^\\iota$ and $m' \\in \\mathbb{R}^\\kappa$, and let $S$ and $R$ be positive semidefinite real matrices indexed by $\\iota$ and by $\\kappa$. Then the image of the product measure $\\mathcal{N}(m, S) \\otimes \\mathcal{N}(m', R)$ under the\u2026","title":"Independent Gaussian blocks"},{"id":"Probability/distributions/continuous/gaussian-conditioning/map-sum-equiv-prod-symm-of-indep-fun","kind":"node","lean":"MetaMathlibExt.map_sumEquivProd_symm_of_indepFun","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#map-sum-equiv-prod-symm-of-indep-fun","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Write $\\mathcal{N}(a, \\Sigma)$ for the multivariate Gaussian distribution with mean $a$ and covariance matrix $\\Sigma$. Let $\\iota$ and $\\kappa$ be finite index sets, and let $x$ and $v$ be independent measurable random vectors in $\\mathbb{R}^\\iota$ and $\\mathbb{R}^\\kappa$ on a probability space, with laws $\\mathcal{N}(m, S)$ and $\\mathcal{N}(m', R)$ for positive semidefinite real matrices $S$ and $R$. Then the\u2026","title":"Concatenation of independent Gaussian random vectors"},{"id":"Probability/distributions/continuous/gaussian-conditioning/from-blocks-one-zero-mul-mul-transpose","kind":"node","lean":"MetaMathlibExt.fromBlocks_one_zero_mul_mul_transpose","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#from-blocks-one-zero-mul-mul-transpose","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $n$ and $p$ be finite index sets and let $S$, $R$ and $H$ be real matrices of sizes $n \\times n$, $p \\times p$ and $p \\times n$. Let $M$ be the block matrix with block rows $(I_n, 0)$ and $(H, I_p)$, and let $D$ be the block-diagonal matrix with diagonal blocks $S$ and $R$. Then $M D M^\\top$ is the block matrix with block rows $(S, S H^\\top)$ and $(H S, H S H^\\top + R)$.","title":"Block-triangular congruence of a block-diagonal matrix"},{"id":"Probability/distributions/continuous/gaussian-conditioning/pos-semidef-from-blocks-linear","kind":"node","lean":"MetaMathlibExt.posSemidef_fromBlocks_linear","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#pos-semidef-from-blocks-linear","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $n$ and $p$ be finite index sets, let $S$ and $R$ be positive semidefinite real matrices indexed by $n$ and by $p$, and let $H$ be a real $p \\times n$ matrix. Then the block matrix $\\begin{pmatrix} S & S H^\\top \\\\ H S & H S H^\\top + R \\end{pmatrix}$, indexed by $n \\sqcup p$, is positive semidefinite.","title":"The joint covariance of a linear-Gaussian model is positive semidefinite"},{"id":"Probability/distributions/continuous/gaussian-conditioning/pos-semidef-sub-mul-inv-mul","kind":"node","lean":"MetaMathlibExt.posSemidef_sub_mul_inv_mul","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#pos-semidef-sub-mul-inv-mul","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $n$ and $p$ be finite index sets, let $S$ be a positive semidefinite real $n \\times n$ matrix, let $R$ be a positive definite real $p \\times p$ matrix and let $H$ be a real $p \\times n$ matrix. Then $S - S H^\\top (H S H^\\top + R)^{-1} H S$ is positive semidefinite.","title":"The posterior covariance is positive semidefinite"},{"id":"Probability/distributions/continuous/gaussian-conditioning/map-linear-add-of-indep-fun","kind":"node","lean":"MetaMathlibExt.map_linear_add_of_indepFun","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#map-linear-add-of-indep-fun","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Write $\\mathcal{N}(a, \\Sigma)$ for the multivariate Gaussian distribution with mean $a$ and covariance matrix $\\Sigma$. Let $n$ and $p$ be finite index sets, and let $x$ and $w$ be independent measurable random vectors in $\\mathbb{R}^n$ and $\\mathbb{R}^p$ on a probability space, where $x$ has law $\\mathcal{N}(m, S)$ and $w$ has law $\\mathcal{N}(c, Q)$ for positive semidefinite real matrices $S$ and $Q$. Then for\u2026","title":"The marginal of a linear-Gaussian model"},{"id":"Probability/distributions/continuous/gaussian-conditioning/cond-distrib-linear-add-of-indep-fun","kind":"node","lean":"MetaMathlibExt.condDistrib_linear_add_of_indepFun","parent":21092,"result":true,"route":"../roadmap/Probability/distributions/continuous/gaussian-conditioning/index.html#cond-distrib-linear-add-of-indep-fun","scope":"Probability/distributions/continuous/gaussian-conditioning","status":"fully_proved","summary":"Let $n$ and $p$ be finite index sets, and let $x$ and $v$ be independent measurable random vectors in $\\mathbb{R}^n$ and $\\mathbb{R}^p$ on a probability space, with $x \\sim N(m, S)$ and $v \\sim N(0, R)$ for positive semidefinite matrices $S$ and $R$. Let $H$ be a real $p \\times n$ matrix such that $H S H^\\top + R$ is invertible, and let $Y = H x + v$. Then for almost every $y$ with respect to the law of $Y$, the\u2026","title":"Bayes' theorem for a linear-Gaussian observation"},{"data_url":"scopes/Probability/distributions/continuous/multivariate-normal-distribution.json","id":"Probability/distributions/continuous/multivariate-normal-distribution","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"can_prove","title":"Multivariate normal distribution"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/std-gaussian-pdfreal","kind":"node","lean":"MetaMathlibExt.stdGaussianPDFReal","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#std-gaussian-pdfreal","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $\\iota$ be a finite set of size $n$. The standard Gaussian density on the Euclidean space $\\mathbb{R}^\\iota$ is the real-valued function $v \\mapsto (2\\pi)^{-n/2} \\exp(-\\|v\\|^2/2)$.","title":"Standard Gaussian density function"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/std-gaussian-density-measure","kind":"node","lean":"MetaMathlibExt.stdGaussianDensityMeasure","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#std-gaussian-density-measure","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $\\iota$ be a finite set of size $n$. This is the measure on the Euclidean space $\\mathbb{R}^\\iota$ with density $v \\mapsto (2\\pi)^{-n/2} \\exp(-\\|v\\|^2/2)$ with respect to Lebesgue measure.","title":"Standard Gaussian measure via its density"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/std-gaussian-eq-with-density","kind":"node","lean":"MetaMathlibExt.stdGaussian_eq_withDensity","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#std-gaussian-eq-with-density","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $\\iota$ be a finite set of size $n$. The standard Gaussian measure on the Euclidean space $\\mathbb{R}^\\iota$ has density $v \\mapsto (2\\pi)^{-n/2} \\exp(-\\|v\\|^2/2)$ with respect to Lebesgue measure.","title":"The standard Gaussian has a density"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/map-with-density-measurable-equiv","kind":"node","lean":"MetaMathlibExt.map_withDensity_measurableEquiv","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#map-with-density-measurable-equiv","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $e : \\alpha \\to \\beta$ be a measurable equivalence between measurable spaces, $\\nu$ a measure on $\\alpha$ and $g : \\alpha \\to [0, \\infty]$ any function. Then the pushforward under $e$ of the measure with density $g$ with respect to $\\nu$ is the measure with density $g \\circ e^{-1}$ with respect to the pushforward $e_*\\nu$ of $\\nu$.","title":"Pushforward of a measure with density"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/affine-equiv","kind":"node","lean":"MetaMathlibExt.affineEquiv","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#affine-equiv","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $\\iota$ be a finite type, $\\mu \\in \\mathbb{R}^\\iota$ and $A$ a real $\\iota \\times \\iota$ matrix with nonzero determinant. The map $x \\mapsto \\mu + Ax$ is a measurable equivalence of the Euclidean space $\\mathbb{R}^\\iota$ with itself, with inverse $y \\mapsto A^{-1}(y - \\mu)$.","title":"The affine change of variables as a measurable equivalence"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/map-affine-equiv-volume","kind":"node","lean":"MetaMathlibExt.map_affineEquiv_volume","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#map-affine-equiv-volume","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $\\iota$ be a finite type, $\\mu \\in \\mathbb{R}^\\iota$ and $A$ a real $\\iota \\times \\iota$ matrix with nonzero determinant. The pushforward of Lebesgue measure on the Euclidean space $\\mathbb{R}^\\iota$ under the measurable equivalence $x \\mapsto \\mu + Ax$ equals $|\\det A|^{-1}$ times Lebesgue measure.","title":"Lebesgue measure under an invertible affine map"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/multivariate-gaussian-pdfreal","kind":"node","lean":"MetaMathlibExt.multivariateGaussianPDFReal","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#multivariate-gaussian-pdfreal","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $\\iota$ be a finite set of size $n$. For $\\mu, x \\in \\mathbb{R}^\\iota$ and a real $\\iota \\times \\iota$ matrix $S$, this is the real number","title":"Multivariate Gaussian density function"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/norm-sq-sqrt-inv-apply","kind":"node","lean":"MetaMathlibExt.norm_sq_sqrt_inv_apply","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#norm-sq-sqrt-inv-apply","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $\\iota$ be a finite type, $S$ a positive definite real $\\iota \\times \\iota$ matrix and $S^{1/2}$ its positive semidefinite square root. Then for every $u$ in the Euclidean space $\\mathbb{R}^\\iota$, $\\|(S^{1/2})^{-1} u\\|^2 = u^\\top S^{-1} u$.","title":"Quadratic form of the inverse square root"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/multivariate-gaussian-eq-with-density","kind":"node","lean":"MetaMathlibExt.multivariateGaussian_eq_withDensity","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#multivariate-gaussian-eq-with-density","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"fully_proved","summary":"Let $\\iota$ be a finite set of size $n$, let $\\mu \\in \\mathbb{R}^\\iota$ and let $S$ be a positive definite real $\\iota \\times \\iota$ matrix. Then the multivariate Gaussian measure on the Euclidean space $\\mathbb{R}^\\iota$ with mean $\\mu$ and covariance matrix $S$ (the image of the standard Gaussian under $x \\mapsto \\mu + S^{1/2} x$) has density","title":"The multivariate Gaussian density"},{"id":"Probability/distributions/continuous/multivariate-normal-distribution/sheppard-quadrant-probability-arcsine","kind":"node","lean":"MathlibExt.Probability.SheppardArcsineWanted.sheppard_quadrant_probability_arcsine","parent":21111,"result":true,"route":"../roadmap/Probability/distributions/continuous/multivariate-normal-distribution/index.html#sheppard-quadrant-probability-arcsine","scope":"Probability/distributions/continuous/multivariate-normal-distribution","status":"can_prove","summary":"Let $\\rho$ be a real number with $-1 < \\rho < 1$, and let $\\gamma$ be the multivariate Gaussian measure on $\\mathbb{R}^2$ with mean $0$ and covariance matrix whose diagonal entries are $1$ and whose off-diagonal entries are $\\rho$ (the law of a standard bivariate normal vector with correlation $\\rho$). Then the open first quadrant $\\{x \\in \\mathbb{R}^2 : x_1 > 0,\\ x_2 > 0\\}$ has $\\gamma$-probability\u2026","title":"Sheppard's formula (Gaussian arcsine law)"},{"data_url":"scopes/Probability/distributions/continuous/poisson-gamma-duality.json","id":"Probability/distributions/continuous/poisson-gamma-duality","kind":"scope","navigation":"module","parent":21007,"result":false,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","title":"Poisson\u2013Gamma duality"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-cdf","kind":"node","lean":"ProbabilityTheory.poissonCDF","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-cdf","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The Poisson distribution function `F_k(x) = \u2211_{j \u2264 k} e^{-x} x^j / (j!)` as a function of the mean `x` (defined for every real `x`; it is a probability only for `0 \u2264 x`).","title":"Poisson distribution function as a function of the mean"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-measure-real-iic","kind":"node","lean":"ProbabilityTheory.poissonMeasure_real_Iic","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-measure-real-iic","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"`poissonCDF r k` is the distribution function of Mathlib's `poissonMeasure r` at `k`.","title":"Poisson distribution function agrees with the Poisson law"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-cdf-zero-left","kind":"node","lean":"ProbabilityTheory.poissonCDF_zero_left","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-cdf-zero-left","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"At mean `0` the Poisson law is the point mass at `0`: `F_k(0) = 1`.","title":"Poisson distribution function at mean zero"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-cdf-zero-right","kind":"node","lean":"ProbabilityTheory.poissonCDF_zero_right","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-cdf-zero-right","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"`F_0(x) = e^{-x}`.","title":"Poisson distribution function at count zero"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-cdf-pos","kind":"node","lean":"ProbabilityTheory.poissonCDF_pos","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-cdf-pos","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The distribution function is positive at every nonnegative real mean.","title":"Positivity of the Poisson distribution function"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/has-deriv-at-poisson-cdf","kind":"node","lean":"ProbabilityTheory.hasDerivAt_poissonCDF","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#has-deriv-at-poisson-cdf","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The derivative in the mean telescopes: `\u2202_x F_k(x) = -e^{-x} x^k / (k!)`.","title":"Derivative of the Poisson distribution function in the mean"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/continuous-poisson-cdf","kind":"node","lean":"ProbabilityTheory.continuous_poissonCDF","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#continuous-poisson-cdf","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The distribution function is continuous in the mean.","title":"Continuity of the Poisson distribution function in the mean"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/gamma-measure-real-ioi-poisson-cdf","kind":"node","lean":"ProbabilityTheory.gammaMeasure_real_Ioi_poissonCDF","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#gamma-measure-real-ioi-poisson-cdf","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The Gamma law of shape `k + 1` and rate `r` (Mathlib's `gammaMeasure`) has upper tail `P(Gamma(k + 1, r) > x) = F_k(r x)`. This is the Erlang tail `gammaMeasure_real_Ioi_natCast_add_one` in `GammaMoments`, rewritten with the definition of `poissonCDF`.","title":"Poisson\u2013Gamma duality"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-cdf-strict-anti-on","kind":"node","lean":"ProbabilityTheory.poissonCDF_strictAntiOn","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-cdf-strict-anti-on","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The distribution function is strictly decreasing in the mean on `[0, \u221e)`.","title":"Poisson distribution function is strictly decreasing in the mean"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-cdf-lt-poisson-cdf-succ","kind":"node","lean":"ProbabilityTheory.poissonCDF_lt_poissonCDF_succ","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-cdf-lt-poisson-cdf-succ","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The distribution function is strictly increasing in `k` at every positive mean.","title":"Poisson distribution function is strictly increasing in the count"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-cdf-mono","kind":"node","lean":"ProbabilityTheory.poissonCDF_mono","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-cdf-mono","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The distribution function is monotone in `k` at every nonnegative mean.","title":"Poisson distribution function is monotone in the count"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/tendsto-poisson-cdf-at-top","kind":"node","lean":"ProbabilityTheory.tendsto_poissonCDF_atTop","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#tendsto-poisson-cdf-at-top","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"For fixed `k`, `F_k(x) \u2192 0` as the mean `x \u2192 \u221e`.","title":"Poisson distribution function vanishes as the mean grows"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/tendsto-poisson-cdf-nat","kind":"node","lean":"ProbabilityTheory.tendsto_poissonCDF_nat","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#tendsto-poisson-cdf-nat","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"For a fixed nonnegative mean, `F_k(x) \u2192 1` as `k \u2192 \u221e`.","title":"Poisson distribution function tends to one as the count grows"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/garwood-upper","kind":"node","lean":"ProbabilityTheory.garwoodUpper","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#garwood-upper","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"Garwood's exact upper confidence bound for a Poisson mean after observing `k`, at level `\u03b1`: `u_\u03b1(k) = sup {x \u2265 0 | \u03b1 \u2264 F_k(x)}`. For `0 < \u03b1 < 1` it is the root of `F_k(u) = \u03b1` (`poissonCDF_garwoodUpper`).","title":"Garwood's exact upper confidence bound"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/garwood-upper-pos","kind":"node","lean":"ProbabilityTheory.garwoodUpper_pos","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#garwood-upper-pos","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The bound is positive for `0 < \u03b1 < 1`.","title":"Positivity of the Garwood upper bound"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-cdf-garwood-upper","kind":"node","lean":"ProbabilityTheory.poissonCDF_garwoodUpper","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-cdf-garwood-upper","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The bound solves `F_k(u_\u03b1(k)) = \u03b1`.","title":"Garwood upper bound solves the distribution function equation"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/le-garwood-upper-iff","kind":"node","lean":"ProbabilityTheory.le_garwoodUpper_iff","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#le-garwood-upper-iff","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"A nonnegative mean lies below the bound iff it is not rejected: `x \u2264 u_\u03b1(k) \u2194 \u03b1 \u2264 F_k(x)`.","title":"Means below the Garwood upper bound are exactly the non-rejected ones"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/garwood-upper-strict-mono","kind":"node","lean":"ProbabilityTheory.garwoodUpper_strictMono","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#garwood-upper-strict-mono","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"The bound is strictly increasing in the observed count.","title":"Garwood upper bound is strictly increasing in the count"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/garwood-upper-zero","kind":"node","lean":"ProbabilityTheory.garwoodUpper_zero","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#garwood-upper-zero","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"After observing `0` the bound is `u_\u03b1(0) = log (1 / \u03b1)` (`e^{-u} = \u03b1`).","title":"Garwood upper bound after observing zero"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/poisson-measure-real-poisson-cdf-le","kind":"node","lean":"ProbabilityTheory.poissonMeasure_real_poissonCDF_le","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#poisson-measure-real-poisson-cdf-le","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"For `K ~ Poisson(r)` and every `\u03b1 \u2265 0`, `P(F_r(K) \u2264 \u03b1) \u2264 \u03b1`. (For `\u03b1 < 0` the event is empty.)","title":"Super-uniformity of the distribution-function transform"},{"id":"Probability/distributions/continuous/poisson-gamma-duality/garwood-upper-coverage","kind":"node","lean":"ProbabilityTheory.garwoodUpper_coverage","parent":21122,"result":true,"route":"../roadmap/Probability/distributions/continuous/poisson-gamma-duality/index.html#garwood-upper-coverage","scope":"Probability/distributions/continuous/poisson-gamma-duality","status":"fully_proved","summary":"For every mean `r \u2265 0` and `0 < \u03b1 < 1`, `P_r(r \u2264 u_\u03b1(K)) \u2265 1 - \u03b1` when `K ~ Poisson(r)`.","title":"Exact coverage of Garwood's bound"},{"data_url":"scopes/Probability/distributions/distribution-functions.json","id":"Probability/distributions/distribution-functions","kind":"scope","parent":20872,"result":false,"route":"../roadmap/Probability/distributions/distribution-functions/index.html","scope":"Probability/distributions/distribution-functions","status":"fully_proved","summary":"Cumulative distribution functions and their continuity points, quantile functions and inverse transform sampling, medians, and joint laws.","title":"Distribution functions and quantiles"},{"id":"Probability/distributions/distribution-functions/map-prod-mk-apply-prod","kind":"node","lean":"MathlibExt.Probability.map_prodMk_apply_prod","parent":21144,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/index.html#map-prod-mk-apply-prod","scope":"Probability/distributions/distribution-functions","status":"fully_proved","summary":"Let $P$ be a measure on a measurable space $\\Omega$, and let $X : \\Omega \\to \\mathcal{X}$ and $Y : \\Omega \\to \\mathcal{Y}$ be $P$-almost everywhere measurable maps into measurable spaces. For measurable sets $s \\subseteq \\mathcal{X}$ and $t \\subseteq \\mathcal{Y}$, the pushforward of $P$ under $\\omega \\mapsto (X(\\omega), Y(\\omega))$ gives $s \\times t$ the measure\u2026","title":"Joint law of two random variables on a rectangle"},{"data_url":"scopes/Probability/distributions/distribution-functions/cumulative-distribution-function.json","id":"Probability/distributions/distribution-functions/cumulative-distribution-function","kind":"scope","navigation":"module","parent":21144,"result":false,"route":"../roadmap/Probability/distributions/distribution-functions/cumulative-distribution-function/index.html","scope":"Probability/distributions/distribution-functions/cumulative-distribution-function","status":"fully_proved","title":"Cumulative distribution function"},{"id":"Probability/distributions/distribution-functions/cumulative-distribution-function/measure-singleton-eq-cdf-sub-left-lim","kind":"node","lean":"MathlibExt.Probability.CdfContinuityPoints.measure_singleton_eq_cdf_sub_leftLim","parent":21146,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/cumulative-distribution-function/index.html#measure-singleton-eq-cdf-sub-left-lim","scope":"Probability/distributions/distribution-functions/cumulative-distribution-function","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with cumulative distribution function $F(y) = \\mu((-\\infty, y])$, and for $x \\in \\mathbb{R}$ let $F(x^-) = \\lim_{y \\to x^-} F(y)$ be its left limit at $x$. Then $\\mu(\\{x\\})$ equals the jump $F(x) - F(x^-)$, which is a nonnegative real number since $F$ is nondecreasing.","title":"Atoms are the jumps of the cdf"},{"id":"Probability/distributions/distribution-functions/cumulative-distribution-function/continuous-at-cdf-iff-measure-singleton","kind":"node","lean":"MathlibExt.Probability.CdfContinuityPoints.continuousAt_cdf_iff_measure_singleton_eq_zero","parent":21146,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/cumulative-distribution-function/index.html#continuous-at-cdf-iff-measure-singleton","scope":"Probability/distributions/distribution-functions/cumulative-distribution-function","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with cumulative distribution function $F(y) = \\mu((-\\infty, y])$. For every $x \\in \\mathbb{R}$, $F$ is continuous at $x$ if and only if $\\mu(\\{x\\}) = 0$.","title":"The cdf is continuous exactly at the non-atoms"},{"id":"Probability/distributions/distribution-functions/cumulative-distribution-function/cdf-continuous-at-singleton-null","kind":"node","lean":"MathlibExt.Probability.CdfContinuityPoints.cdf_continuousAt_singleton_null","parent":21146,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/cumulative-distribution-function/index.html#cdf-continuous-at-singleton-null","scope":"Probability/distributions/distribution-functions/cumulative-distribution-function","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with cumulative distribution function $F(y) = \\mu((-\\infty, y])$. If $F$ is continuous at a point $x \\in \\mathbb{R}$, then $\\mu(\\{x\\}) = 0$.","title":"No atom at a continuity point of the cdf"},{"id":"Probability/distributions/distribution-functions/cumulative-distribution-function/cdf-continuous-at-frontier-iic-null","kind":"node","lean":"MathlibExt.Probability.CdfContinuityPoints.cdf_continuousAt_frontier_Iic_null","parent":21146,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/cumulative-distribution-function/index.html#cdf-continuous-at-frontier-iic-null","scope":"Probability/distributions/distribution-functions/cumulative-distribution-function","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with cumulative distribution function $F(y) = \\mu((-\\infty, y])$. If $F$ is continuous at a point $x \\in \\mathbb{R}$, then the boundary of the half-line $(-\\infty, x]$, which is $\\{x\\}$, has $\\mu$-measure zero.","title":"Continuity points of the cdf have null boundary"},{"id":"Probability/distributions/distribution-functions/cumulative-distribution-function/measure-ioc-eq-of-real-cdf","kind":"node","lean":"MathlibExt.Probability.CdfContinuityPoints.measure_Ioc_eq_ofReal_cdf","parent":21146,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/cumulative-distribution-function/index.html#measure-ioc-eq-of-real-cdf","scope":"Probability/distributions/distribution-functions/cumulative-distribution-function","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with cumulative distribution function $F(y) = \\mu((-\\infty, y])$. For all $a, b \\in \\mathbb{R}$, the measure $\\mu((a, b])$ equals $\\max(F(b) - F(a), 0)$, as an extended nonnegative real.","title":"Mass of a half-open interval via the cdf"},{"id":"Probability/distributions/distribution-functions/cumulative-distribution-function/dense-cdf-continuity-points","kind":"node","lean":"MathlibExt.Probability.CdfContinuityPoints.dense_cdf_continuityPoints","parent":21146,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/cumulative-distribution-function/index.html#dense-cdf-continuity-points","scope":"Probability/distributions/distribution-functions/cumulative-distribution-function","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with cumulative distribution function $F(y) = \\mu((-\\infty, y])$. The set of points $x \\in \\mathbb{R}$ at which $F$ is continuous is dense in $\\mathbb{R}$.","title":"The continuity points of the cdf are dense"},{"data_url":"scopes/Probability/distributions/distribution-functions/median-of-a-distribution.json","id":"Probability/distributions/distribution-functions/median-of-a-distribution","kind":"scope","navigation":"module","parent":21144,"result":false,"route":"../roadmap/Probability/distributions/distribution-functions/median-of-a-distribution/index.html","scope":"Probability/distributions/distribution-functions/median-of-a-distribution","status":"fully_proved","title":"Median of a distribution"},{"id":"Probability/distributions/distribution-functions/median-of-a-distribution/is-median","kind":"node","lean":"ProbabilityTheory.IsMedian","parent":21153,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/median-of-a-distribution/index.html#is-median","scope":"Probability/distributions/distribution-functions/median-of-a-distribution","status":"fully_proved","summary":"Let $\\nu$ be a measure on $\\mathbb{R}$, typically a probability measure. A real number $m$ is a median of $\\nu$ when $\\nu((-\\infty, m]) \\ge 1/2$ and $\\nu([m, \\infty)) \\ge 1/2$, where a set of infinite measure counts as having measure $0$.","title":"Median of a measure on the real line"},{"id":"Probability/distributions/distribution-functions/median-of-a-distribution/integrable-abs-sub-iff","kind":"node","lean":"ProbabilityTheory.integrable_abs_sub_iff","parent":21153,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/median-of-a-distribution/index.html#integrable-abs-sub-iff","scope":"Probability/distributions/distribution-functions/median-of-a-distribution","status":"fully_proved","summary":"Let $\\nu$ be a finite measure on $\\mathbb{R}$ and let $c \\in \\mathbb{R}$. Then $\\theta \\mapsto |\\theta - c|$ is $\\nu$-integrable if and only if $\\theta \\mapsto \\theta$ is $\\nu$-integrable.","title":"Absolute deviations are integrable exactly when the mean exists"},{"id":"Probability/distributions/distribution-functions/median-of-a-distribution/integral-abs-sub-le-of-is-median","kind":"node","lean":"ProbabilityTheory.integral_abs_sub_le_of_isMedian","parent":21153,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/median-of-a-distribution/index.html#integral-abs-sub-le-of-is-median","scope":"Probability/distributions/distribution-functions/median-of-a-distribution","status":"fully_proved","summary":"Let $\\nu$ be a probability measure on $\\mathbb{R}$ and let $m$ be a median of $\\nu$. Then for every $y \\in \\mathbb{R}$,","title":"A median minimizes the expected absolute loss"},{"data_url":"scopes/Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling.json","id":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling","kind":"scope","navigation":"module","parent":21144,"result":false,"route":"../roadmap/Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/index.html","scope":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling","status":"fully_proved","title":"Quantile function and inverse transform sampling"},{"id":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/cdf-quantile","kind":"node","lean":"ProbabilityTheory.cdfQuantile","parent":21157,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/index.html#cdf-quantile","scope":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}$ with cumulative distribution function $F_\\mu$ (for a probability measure, $F_\\mu(x) = \\mu((-\\infty, x])$). The quantile function of $\\mu$, the generalized inverse of $F_\\mu$, is the function $Q_\\mu : \\mathbb{R} \\to \\mathbb{R}$ given by $Q_\\mu(u) = \\inf \\lbrace x \\in \\mathbb{R} : u \\le F_\\mu(x) \\rbrace$ for $u \\in (0, 1)$ and $Q_\\mu(u) = 0$ for $u \\notin (0, 1)$.","title":"Quantile function of a real measure"},{"id":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/cdf-quantile-mem","kind":"node","lean":"ProbabilityTheory.cdfQuantile_mem","parent":21157,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/index.html#cdf-quantile-mem","scope":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}$ with cumulative distribution function $F_\\mu$ and quantile function $Q_\\mu$. For every $u \\in (0, 1)$, $u \\le F_\\mu(Q_\\mu(u))$.","title":"The cdf at the quantile is at least the level"},{"id":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/cdf-quantile-le-iff","kind":"node","lean":"ProbabilityTheory.cdfQuantile_le_iff","parent":21157,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/index.html#cdf-quantile-le-iff","scope":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}$ with cumulative distribution function $F_\\mu$ and quantile function $Q_\\mu$. For every $u \\in (0, 1)$ and every real $x$, $Q_\\mu(u) \\le x$ if and only if $u \\le F_\\mu(x)$.","title":"The quantile function as a generalized inverse of the cdf"},{"id":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/cdf-quantile-mono-on","kind":"node","lean":"ProbabilityTheory.cdfQuantile_monoOn","parent":21157,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/index.html#cdf-quantile-mono-on","scope":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}$, the quantile function $Q_\\mu$ is monotone on $(0, 1)$: if $0 < u \\le v < 1$, then $Q_\\mu(u) \\le Q_\\mu(v)$.","title":"Monotonicity of the quantile function"},{"id":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/measurable-cdf-quantile","kind":"node","lean":"ProbabilityTheory.measurable_cdfQuantile","parent":21157,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/index.html#measurable-cdf-quantile","scope":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}$, the quantile function $Q_\\mu : \\mathbb{R} \\to \\mathbb{R}$ is measurable.","title":"Measurability of the quantile function"},{"id":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/map-cdf-quantile-restrict-icc","kind":"node","lean":"ProbabilityTheory.map_cdfQuantile_restrict_Icc","parent":21157,"result":true,"route":"../roadmap/Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling/index.html#map-cdf-quantile-restrict-icc","scope":"Probability/distributions/distribution-functions/quantile-function-and-inverse-transform-sampling","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with quantile function $Q_\\mu$. The pushforward under $Q_\\mu$ of Lebesgue measure restricted to $[0, 1]$ is $\\mu$.","title":"Inverse transform sampling"},{"data_url":"scopes/Probability/hypothesis-testing.json","id":"Probability/hypothesis-testing","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/hypothesis-testing/index.html","scope":"Probability/hypothesis-testing","status":"fully_proved","summary":"Tests and their validity and power: Neyman\u2013Pearson, likelihood-ratio, score and Wald tests, monotone likelihood ratio, exact conditional tests (Fisher, Hardy\u2013Weinberg), rank tests and ROC area (Mann\u2013Whitney), permutation p-values, multiple testing (Bonferroni, Holm, closed testing, Benjamini\u2013Hochberg, false discovery rates), and meta-analysis and heterogeneity statistics.","title":"Hypothesis testing"},{"id":"Probability/hypothesis-testing/neyman-pearson-dominated-two-simple","kind":"node","lean":"MathlibExt.Probability.Statistics.NeymanPearson.neyman_pearson_dominated_two_simple","parent":21164,"result":true,"route":"../roadmap/Probability/hypothesis-testing/index.html#neyman-pearson-dominated-two-simple","scope":"Probability/hypothesis-testing","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, and let $p, q : \\Omega \\to \\mathbb{R}$ be measurable, nonnegative and $\\mu$-integrable. Let $k \\ge 0$ be a real number and let $\\varphi^\\star : \\Omega \\to [0, 1]$ be measurable with $\\varphi^\\star(x) = 1$ whenever $q(x) > k\\,p(x)$ and $\\varphi^\\star(x) = 0$ whenever $q(x) < k\\,p(x)$. Then every measurable test $\\varphi : \\Omega \\to [0, 1]$ satisfying\u2026","title":"Neyman\u2013Pearson lemma"},{"data_url":"scopes/Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium.json","id":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/index.html","scope":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium","status":"fully_proved","title":"Exact test for Hardy\u2013Weinberg equilibrium"},{"id":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/levene-prob","kind":"node","lean":"MathlibExt.Probability.Statistics.ExactTests.leveneProb","parent":21166,"result":true,"route":"../roadmap/Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/index.html#levene-prob","scope":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium","status":"fully_proved","summary":"Levene's conditional probability of `h` heterozygotes among `N` diploid individuals given `nA` copies of allele `A` (and `nB = 2N - nA` of `B`) under Hardy\u2013Weinberg equilibrium: `N! nA! nB! 2^h / (nAA! h! nBB! (2N)!)` with `nAA = (nA - h) / 2`, `nBB = (nB - h) / 2`, when `h \u2264 nA`, `h \u2264 nB` and `nA - h` is even, and `0` otherwise. The parameter domain is `nA \u2264 2N`; for `nA > 2N` the value is `0`.","title":"Levene's exact Hardy\u2013Weinberg probability"},{"id":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/het-count","kind":"node","lean":"MathlibExt.Probability.Statistics.ExactTests.hetCount","parent":21166,"result":true,"route":"../roadmap/Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/index.html#het-count","scope":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium","status":"fully_proved","summary":"The number of heterozygous individuals when the `A` gametes occupy the slots `s`: individuals `j` carrying exactly one of the slots `(j, 0)`, `(j, 1)`.","title":"Number of heterozygous individuals"},{"id":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/card-het-count-div-choose","kind":"node","lean":"MathlibExt.Probability.Statistics.ExactTests.card_hetCount_div_choose","parent":21166,"result":true,"route":"../roadmap/Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/index.html#card-het-count-div-choose","scope":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium","status":"fully_proved","summary":"Among the `C(2N, nA)` equally likely placements of the `nA` copies of `A` in the `2N` gamete slots, the fraction with exactly `h` heterozygous individuals is `leveneProb N nA h`.","title":"Levene's formula is the conditional law under random union of gametes"},{"id":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/sum-levene-prob","kind":"node","lean":"MathlibExt.Probability.Statistics.ExactTests.sum_leveneProb","parent":21166,"result":true,"route":"../roadmap/Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium/index.html#sum-levene-prob","scope":"Probability/hypothesis-testing/exact-test-for-hardy-weinberg-equilibrium","status":"fully_proved","summary":"For `nA \u2264 2N` the probabilities `leveneProb N nA h` over the admissible heterozygote counts `h` sum to one.","title":"Normalization of Levene's distribution"},{"data_url":"scopes/Probability/hypothesis-testing/fisher-s-exact-test.json","id":"Probability/hypothesis-testing/fisher-s-exact-test","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/fisher-s-exact-test/index.html","scope":"Probability/hypothesis-testing/fisher-s-exact-test","status":"fully_proved","title":"Fisher's exact test"},{"id":"Probability/hypothesis-testing/fisher-s-exact-test/measure-real-add-eq-binomial","kind":"node","lean":"MathlibExt.Probability.Statistics.ExactTests.measureReal_add_eq_binomial","parent":21171,"result":true,"route":"../roadmap/Probability/hypothesis-testing/fisher-s-exact-test/index.html#measure-real-add-eq-binomial","scope":"Probability/hypothesis-testing/fisher-s-exact-test","status":"fully_proved","summary":"If `X ~ Bin(n\u2081, p)` and `Y ~ Bin(n\u2082, p)` are independent, then `X + Y ~ Bin(n\u2081 + n\u2082, p)`: `P(X + Y = t) = C(n\u2081 + n\u2082, t) p^t (1 - p)^(n\u2081 + n\u2082 - t)`.","title":"Sum of independent binomials"},{"id":"Probability/hypothesis-testing/fisher-s-exact-test/cond-measure-real-eq-hypergeometric-real","kind":"node","lean":"MathlibExt.Probability.Statistics.ExactTests.cond_measureReal_eq_hypergeometricReal","parent":21171,"result":true,"route":"../roadmap/Probability/hypothesis-testing/fisher-s-exact-test/index.html#cond-measure-real-eq-hypergeometric-real","scope":"Probability/hypothesis-testing/fisher-s-exact-test","status":"fully_proved","summary":"Let `X ~ Bin(n\u2081, p)` and `Y ~ Bin(n\u2082, p)` be independent with a common `p \u2208 (0, 1)`. Given the margin `X + Y = t` with `t \u2264 n\u2081 + n\u2082`, the cell `X` is hypergeometric: `P(X = k | X + Y = t) = C(n\u2081, k) C(n\u2082, t - k) / C(n\u2081 + n\u2082, t)`, whatever `p`.","title":"Fisher's exact conditional law"},{"data_url":"scopes/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area.json","id":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/index.html","scope":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","status":"fully_proved","title":"Mann\u2013Whitney statistic and ROC area"},{"id":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/roc-rate","kind":"node","lean":"ProbabilityTheory.rocRate","parent":21174,"result":true,"route":"../roadmap/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/index.html#roc-rate","scope":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","status":"fully_proved","summary":"Empirical rate `#{i | t \u2264 z i} / k` of scores at or above the threshold `t` (the true- or false-positive rate when `z` are the positive or negative scores).","title":"Empirical ROC rate at or above a threshold"},{"id":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/roc-rate-above","kind":"node","lean":"ProbabilityTheory.rocRateAbove","parent":21174,"result":true,"route":"../roadmap/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/index.html#roc-rate-above","scope":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","status":"fully_proved","summary":"Empirical rate `#{i | t < z i} / k` of scores strictly above `t`: the ROC coordinate at the next larger distinct score.","title":"Empirical ROC rate strictly above a threshold"},{"id":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/empirical-auc","kind":"node","lean":"ProbabilityTheory.empiricalAUC","parent":21174,"result":true,"route":"../roadmap/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/index.html#empirical-auc","scope":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","status":"fully_proved","summary":"Trapezoidal area under the empirical ROC curve of positive scores `x` and negative scores `y`: the sum over the distinct observed scores `t` of the trapezoid from the point `(rocRateAbove y t, rocRateAbove x t)` to `(rocRate y t, rocRate x t)`.","title":"Empirical area under the ROC curve"},{"id":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/mann-whitney-u","kind":"node","lean":"ProbabilityTheory.mannWhitneyU","parent":21174,"result":true,"route":"../roadmap/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/index.html#mann-whitney-u","scope":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","status":"fully_proved","summary":"Mann\u2013Whitney count `U = \u2211\u1d62 \u2211\u2c7c (1[y j < x i] + \u00bd 1[y j = x i])` of (positive, negative) pairs ranked correctly, with ties counted one half.","title":"Mann\u2013Whitney U statistic"},{"id":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/roc-rate-above-eq-roc-rate-of-forall-not","kind":"node","lean":"ProbabilityTheory.rocRateAbove_eq_rocRate_of_forall_not_mem_Ioo","parent":21174,"result":true,"route":"../roadmap/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/index.html#roc-rate-above-eq-roc-rate-of-forall-not","scope":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","status":"fully_proved","summary":"If no score lies strictly between `t < t'`, the strict rate at `t` is the rate at `t'`: the left end of the trapezoid at `t` is the ROC point of the next larger distinct score.","title":"Strict ROC rate equals the rate at the next score"},{"id":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/roc-rate-above-eq-zero-of-forall-le","kind":"node","lean":"ProbabilityTheory.rocRateAbove_eq_zero_of_forall_le","parent":21174,"result":true,"route":"../roadmap/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/index.html#roc-rate-above-eq-zero-of-forall-le","scope":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","status":"fully_proved","summary":"Above the largest score the strict rate is zero: the first trapezoid starts at `(0, 0)`.","title":"Strict ROC rate vanishes above the largest score"},{"id":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/empirical-auc-eq-mann-whitney-u","kind":"node","lean":"ProbabilityTheory.empiricalAUC_eq_mannWhitneyU","parent":21174,"result":true,"route":"../roadmap/Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area/index.html#empirical-auc-eq-mann-whitney-u","scope":"Probability/hypothesis-testing/mann-whitney-statistic-and-roc-area","status":"fully_proved","summary":"Empirical AUC = Mann\u2013Whitney with ties (Bamber 1975): the trapezoidal area under the empirical ROC curve equals `U / (m n)`, for nonempty samples.","title":"Empirical AUC = Mann\u2013Whitney with ties"},{"data_url":"scopes/Probability/hypothesis-testing/meta-analysis.json","id":"Probability/hypothesis-testing/meta-analysis","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","title":"Meta-analysis"},{"id":"Probability/hypothesis-testing/meta-analysis/fisher-statistic","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.fisherStatistic","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#fisher-statistic","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"Fisher's combined statistic `-2 \u2211 log p\u1d62` of `k` p-values.","title":"Fisher's combined statistic"},{"id":"Probability/hypothesis-testing/meta-analysis/has-law-neg-two-mul-log-uniform","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.hasLaw_neg_two_mul_log_uniform","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#has-law-neg-two-mul-log-uniform","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"`-2 log U` has the exponential law of rate `1/2`, i.e. chi-square with two degrees of freedom, when `U` is uniform on `[0, 1]`.","title":"Minus twice the log of a uniform variable is chi-square"},{"id":"Probability/hypothesis-testing/meta-analysis/has-law-fisher-statistic","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.hasLaw_fisherStatistic","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#has-law-fisher-statistic","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"Fisher (1932), \u00a721.1: for `k \u2265 1` independent p-values, each uniform on `[0, 1]`, Fisher's statistic `-2 \u2211 log p\u1d62` is chi-square with `2k` degrees of freedom.","title":"Fisher's method for combining p-values"},{"id":"Probability/hypothesis-testing/meta-analysis/stouffer-statistic","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.stoufferStatistic","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#stouffer-statistic","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"Stouffer's weighted z statistic `\u2211 w\u1d62 z\u1d62 / \u221a(\u2211 w\u1d62\u00b2)` of the study z-scores `z\u1d62`.","title":"Stouffer's weighted z statistic"},{"id":"Probability/hypothesis-testing/meta-analysis/has-law-stouffer-statistic","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.hasLaw_stoufferStatistic","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#has-law-stouffer-statistic","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"Stouffer's method (Stouffer et al. 1949; Liptak 1958): for independent standard normal z-scores and a nonzero weight vector, the weighted z statistic is standard normal.","title":"Stouffer's method"},{"id":"Probability/hypothesis-testing/meta-analysis/inverse-variance-weights","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.inverseVarianceWeights","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#inverse-variance-weights","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"The inverse-variance (fixed-effect) weights `v\u1d62\u207b\u00b9 / \u2211\u2c7c v\u2c7c\u207b\u00b9` for study variances `v`.","title":"Inverse-variance weights"},{"id":"Probability/hypothesis-testing/meta-analysis/sum-inverse-variance-weights","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.sum_inverseVarianceWeights","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#sum-inverse-variance-weights","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"The inverse-variance weights sum to one.","title":"Inverse-variance weights sum to one"},{"id":"Probability/hypothesis-testing/meta-analysis/sum-sq-mul-inverse-variance-weights","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.sum_sq_mul_inverseVarianceWeights","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#sum-sq-mul-inverse-variance-weights","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"The variance `\u2211 w\u1d62\u00b2 v\u1d62` of the inverse-variance-weighted average is `(\u2211 v\u1d62\u207b\u00b9)\u207b\u00b9`.","title":"Variance of the inverse-variance-weighted average"},{"id":"Probability/hypothesis-testing/meta-analysis/inv-sum-inv-le-sum-sq-mul","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.inv_sum_inv_le_sum_sq_mul","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#inv-sum-inv-le-sum-sq-mul","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"Optimality of inverse-variance weighting: for positive variances `v` and weights `c` summing to one, `\u2211 c\u1d62\u00b2 v\u1d62 \u2265 (\u2211 v\u1d62\u207b\u00b9)\u207b\u00b9`. The gap is `\u2211 v\u1d62 (c\u1d62 - w\u1d62)\u00b2` for the inverse-variance weights `w`.","title":"Optimality of inverse-variance weights"},{"id":"Probability/hypothesis-testing/meta-analysis/variance-inverse-variance-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.variance_inverseVariance_le","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#variance-inverse-variance-le","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"Cochran (1954): for pairwise independent square-integrable estimates `X i` with positive variances, the inverse-variance-weighted average has variance `(\u2211 Var[X\u1d62]\u207b\u00b9)\u207b\u00b9`, the smallest variance among all weighted averages `\u2211 c\u1d62 X\u1d62` with `\u2211 c\u1d62 = 1`.","title":"Cochran's minimum-variance theorem for weighted averages"},{"id":"Probability/hypothesis-testing/meta-analysis/cochran-q","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.cochranQ","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#cochran-q","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"Cochran's heterogeneity statistic `Q = \u2211\u1d62 v\u1d62\u207b\u00b9 (X\u1d62 - \u2211\u2c7c w\u2c7c X\u2c7c)\u00b2` of the estimates `X` with variances `v`, where `w = inverseVarianceWeights v`.","title":"Cochran's heterogeneity statistic Q"},{"id":"Probability/hypothesis-testing/meta-analysis/cochran-q-eq-norm-sq","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.cochranQ_eq_norm_sq","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#cochran-q-eq-norm-sq","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"Algebraic core of Cochran's theorem for `Q`: if `X\u1d62 = \u03b8 + \u221av\u1d62 z\u1d62` with `v\u1d62 > 0`, then `Q` is the squared distance from `z` to the line spanned by `a = (v\u1d62^{-1/2})\u1d62`, i.e. the squared norm of the projection of `z` onto the orthogonal complement of that line.","title":"Cochran's Q as a squared projection norm"},{"id":"Probability/hypothesis-testing/meta-analysis/has-law-cochran-q","kind":"node","lean":"MathlibExt.Probability.Statistics.MetaAnalysis.hasLaw_cochranQ","parent":21182,"result":true,"route":"../roadmap/Probability/hypothesis-testing/meta-analysis/index.html#has-law-cochran-q","scope":"Probability/hypothesis-testing/meta-analysis","status":"fully_proved","summary":"For `k \u2265 2` independent Gaussian estimates `X\u1d62 ~ N(\u03b8, v\u1d62)` with positive known variances, Cochran's `Q` has the chi-square law with `k - 1` degrees of freedom.","title":"Cochran (1954)"},{"data_url":"scopes/Probability/hypothesis-testing/monotone-likelihood-ratio.json","id":"Probability/hypothesis-testing/monotone-likelihood-ratio","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","title":"Monotone likelihood ratio"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/has-monotone-likelihood-ratio-on","kind":"node","lean":"ProbabilityTheory.HasMonotoneLikelihoodRatioOn","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#has-monotone-likelihood-ratio-on","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"The likelihood family `L : \u2115 \u2192 \u211d \u2192 \u211d` (observation, parameter) has a monotone likelihood ratio on the parameter set `S`: `L` is nonnegative on `S`, and for observations `n \u2264 m` and parameters `\u03b8 \u2264 \u03b8'` in `S`, `L m \u03b8 * L n \u03b8' \u2264 L n \u03b8 * L m \u03b8'` (total positivity of order two). For positive likelihoods this is monotonicity of `\u03b8 \u21a6 L m \u03b8 / L n \u03b8`. The nonnegativity is part of the predicate: without it the downstream\u2026","title":"Monotone likelihood ratio"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/has-antitone-likelihood-ratio-on","kind":"node","lean":"ProbabilityTheory.HasAntitoneLikelihoodRatioOn","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#has-antitone-likelihood-ratio-on","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"The likelihood family has an antitone likelihood ratio on `S`: `L` is nonnegative on `S`, and for `n \u2264 m` and `\u03b8 \u2264 \u03b8'` in `S`, `L n \u03b8 * L m \u03b8' \u2264 L m \u03b8 * L n \u03b8'`. For positive likelihoods this says `\u03b8 \u21a6 L m \u03b8 / L n \u03b8` is nonincreasing (larger counts favor smaller parameters).","title":"Antitone likelihood ratio"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/set-integral-ioi-mul-integral-le-of-has","kind":"node","lean":"ProbabilityTheory.setIntegral_Ioi_mul_integral_le_of_hasMLR","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#set-integral-ioi-mul-integral-le-of-has","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"Posterior stochastic ordering, cross-multiplied form (Milgrom 1981, Prop. 1). If the family `L` has a monotone likelihood ratio on a set `S` carrying the prior `\u03bc` (`\u03bc S\u1d9c = 0`), and every `L n` is `\u03bc`-integrable, then for `n \u2264 m` and every threshold `t`, `(\u222b_{\u03b8 > t} L n d\u03bc) (\u222b L m d\u03bc) \u2264 (\u222b_{\u03b8 > t} L m d\u03bc) (\u222b L n d\u03bc)`.","title":"Posterior stochastic ordering, cross-multiplied form"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/set-integral-ioi-mul-integral-le-of-has-2","kind":"node","lean":"ProbabilityTheory.setIntegral_Ioi_mul_integral_le_of_hasAntitoneLR","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#set-integral-ioi-mul-integral-le-of-has-2","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"Under the hypotheses of `setIntegral_Ioi_mul_integral_le_of_hasMLR` with the reversed ratio condition, for `n \u2264 m`: `(\u222b_{\u03b8 > t} L m d\u03bc) (\u222b L n d\u03bc) \u2264 (\u222b_{\u03b8 > t} L n d\u03bc) (\u222b L m d\u03bc)`.","title":"Posterior stochastic ordering for an antitone likelihood ratio, cross-multiplied form"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/posterior-tail","kind":"node","lean":"ProbabilityTheory.posteriorTail","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#posterior-tail","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"Posterior probability of `{\u03b8 > t}` after observing `n`, by Bayes' formula for the prior `\u03bc` and likelihood `L`: `(\u222b_{\u03b8 > t} L n \u03b8 d\u03bc) / \u222b L n \u03b8 d\u03bc`. Meaningful when the marginal likelihood `\u222b L n d\u03bc` is positive (otherwise the division returns the junk value `0`), and a genuine posterior probability only when `L` is nonnegative (as in the MLR predicates: a signed `L` can give a value outside `[0, 1]`).","title":"Posterior tail probability"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/posterior-tail-mono-of-has-mlr","kind":"node","lean":"ProbabilityTheory.posteriorTail_mono_of_hasMLR","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#posterior-tail-mono-of-has-mlr","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"The posterior is stochastically nondecreasing in the observation for an MLR family and every prior (Milgrom 1981, Prop. 2): if `n \u2264 m` and both marginal likelihoods are positive, then `P(\u03b8 > t \u2223 n) \u2264 P(\u03b8 > t \u2223 m)` for every `t`.","title":"The posterior is stochastically nondecreasing in the observation"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/posterior-tail-anti-of-has-antitone-lr","kind":"node","lean":"ProbabilityTheory.posteriorTail_anti_of_hasAntitoneLR","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#posterior-tail-anti-of-has-antitone-lr","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"The posterior is stochastically nonincreasing in the observation for a family with an antitone likelihood ratio and every prior: if `n \u2264 m` and both marginal likelihoods are positive, then `P(\u03b8 > t \u2223 m) \u2264 P(\u03b8 > t \u2223 n)` for every `t`.","title":"The posterior is stochastically nonincreasing in the observation"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/poisson-lik","kind":"node","lean":"ProbabilityTheory.poissonLik","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#poisson-lik","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"Poisson likelihood with parameter-dependent mean `\u03bb \u03b8`: `L n \u03b8 = e^{-\u03bb \u03b8} \u03bb \u03b8 ^ n / (n !)`.","title":"Poisson likelihood"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/poisson-lik-has-mlr","kind":"node","lean":"ProbabilityTheory.poissonLik_hasMLR","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#poisson-lik-has-mlr","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"A Poisson family whose mean is nonnegative and monotone on `S` has a monotone likelihood ratio on `S`.","title":"Poisson family with monotone mean has monotone likelihood ratio"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/poisson-lik-has-antitone-lr","kind":"node","lean":"ProbabilityTheory.poissonLik_hasAntitoneLR","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#poisson-lik-has-antitone-lr","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"A Poisson family whose mean is nonnegative and antitone on `S` has an antitone likelihood ratio on `S`.","title":"Poisson family with antitone mean has antitone likelihood ratio"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/binomial-lik","kind":"node","lean":"ProbabilityTheory.binomialLik","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#binomial-lik","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"Binomial likelihood with `N` trials and parameter-dependent success probability `p \u03b8`: `L n \u03b8 = C(N, n) p \u03b8 ^ n (1 - p \u03b8) ^ (N - n)` (zero for `n > N`, since `C(N, n) = 0`).","title":"Binomial likelihood"},{"id":"Probability/hypothesis-testing/monotone-likelihood-ratio/binomial-lik-has-mlr","kind":"node","lean":"ProbabilityTheory.binomialLik_hasMLR","parent":21196,"result":true,"route":"../roadmap/Probability/hypothesis-testing/monotone-likelihood-ratio/index.html#binomial-lik-has-mlr","scope":"Probability/hypothesis-testing/monotone-likelihood-ratio","status":"fully_proved","summary":"A binomial family whose success probability lies in `[0, 1]` and is monotone on `S` has a monotone likelihood ratio on `S`.","title":"Binomial family has monotone likelihood ratio"},{"data_url":"scopes/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests.json","id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","title":"Score, Wald and likelihood-ratio tests"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/gaussian-real-map-sq-div","kind":"node","lean":"ProbabilityTheory.gaussianReal_map_sq_div","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#gaussian-real-map-sq-div","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"If `Y ~ N(0, v)` with `v \u2260 0`, then `Y\u00b2 / v ~ \u03c7\u00b2\u2081`.","title":"Squared normalized Gaussian is chi-square with one degree of freedom"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/gaussian-real-map-max-zero-sq","kind":"node","lean":"ProbabilityTheory.gaussianReal_map_max_zero_sq","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#gaussian-real-map-max-zero-sq","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"`max(Z, 0)\u00b2` has the mixture law `\u00bd \u03b4\u2080 + \u00bd \u03c7\u00b2\u2081` (chi-bar-square with weights `\u00bd, \u00bd`).","title":"Law of the positive part squared of a standard normal variable"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-distribution-score-statistic","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInDistribution_scoreStatistic","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-distribution-score-statistic","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"Under the conditions of `tendstoInDistribution_score` and `0 < I(\u03b8\u2080)`, the score statistic `(n^{-1/2} \u2211_{i<n} score p \u03b8\u2080 (X i))\u00b2 / I(\u03b8\u2080)` converges in distribution to `\u03c7\u00b2\u2081`.","title":"Rao's score test, known information"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-distribution-score-statistic-2","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInDistribution_scoreStatistic_of_estimator","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-distribution-score-statistic-2","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"As `tendstoInDistribution_scoreStatistic`, with `I(\u03b8\u2080)` replaced by any estimators `\u00ce n` converging to `I(\u03b8\u2080)` in probability. Where `\u00ce n \u03c9 = 0` the statistic takes Lean's value `x / 0 = 0`; this event has probability tending to `0` because `0 < I(\u03b8\u2080)`.","title":"Rao's score test with an estimated information"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-measure-observed-information","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInMeasure_observedInformation","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-measure-observed-information","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"For i.i.d. observations with law `p \u03b8\u2080 \u03bd`, if `\u2202_\u03b8 score p \u03b8\u2080` is measurable and integrable with `E \u2202_\u03b8 score p \u03b8\u2080 = -I(\u03b8\u2080)` (the information identity), then `-(1/n) \u2211_{i<n} \u2202_\u03b8 score p \u03b8\u2080 (X i) \u2192 I(\u03b8\u2080)` in probability.","title":"Consistency of the observed information"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-distribution-wald-statistic","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInDistribution_waldStatistic","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-distribution-wald-statistic","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"Under the conditions of `tendstoInDistribution_sqrt_mul_sub_of_likelihoodEquation` (without the normalization `\u222b p \u03b8 d\u03bd = 1`, which that proof does not use), the Wald statistic `n (\u03b8\u0302\u2099 - \u03b8\u2080)\u00b2 I(\u03b8\u2080)` of a consistent root `\u03b8\u0302\u2099` of the likelihood equation converges in distribution to `\u03c7\u00b2\u2081`.","title":"Wald's test"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-distribution-wald-statistic-o","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInDistribution_waldStatistic_of_estimator","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-distribution-wald-statistic-o","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"As `tendstoInDistribution_waldStatistic`, with `I(\u03b8\u2080)` replaced by any estimators `\u00ce n \u2192 I(\u03b8\u2080)` in probability (for example the observed information of `tendstoInMeasure_observedInformation`; `I(\u03b8\u0302\u2099)` also qualifies when `\u03b8 \u21a6 I(\u03b8)` is continuous at `\u03b8\u2080`, by the continuous mapping theorem).","title":"Wald's test with an estimated information"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-measure-two-mul-log-likelihoo","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInMeasure_two_mul_logLikelihood_sub_sub","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-measure-two-mul-log-likelihoo","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"Let `\u03b8\u0302\u2099` be `\u221an`-consistent (`\u221an (\u03b8\u0302\u2099 - \u03b8\u2080)` converges in distribution) and solve the likelihood equation with probability tending to one. If on the ball of radius `\u03b4 > 0` about `\u03b8\u2080` the log-likelihood is twice differentiable in `\u03b8` with `|\u2202\u00b3_\u03b8 log p \u03b8 x| \u2264 M x` for an integrable `M`, and `\u2202\u00b2_\u03b8 log p \u03b8\u2080` is integrable with mean `-I(\u03b8\u2080)`, then `2 (\u2113\u2099(\u03b8\u0302\u2099) - \u2113\u2099(\u03b8\u2080)) - n (\u03b8\u0302\u2099 - \u03b8\u2080)\u00b2 I(\u03b8\u2080) \u2192 0` in probability, where\u2026","title":"Quadratic expansion of the log-likelihood ratio"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-distribution-two-mul-log-like","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInDistribution_two_mul_logLikelihood_sub","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-distribution-two-mul-log-like","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"Under the conditions of `tendstoInDistribution_waldStatistic` and joint measurability of `(\u03b8, x) \u21a6 p \u03b8 x`, the log-likelihood-ratio statistic `2 (\u2113\u2099(\u03b8\u0302\u2099) - \u2113\u2099(\u03b8\u2080))`, `\u2113\u2099 \u03b8 = \u2211_{i<n} log p \u03b8 (X i)`, of a consistent root `\u03b8\u0302\u2099` of the likelihood equation converges in distribution to `\u03c7\u00b2\u2081`.","title":"Wilks' theorem (simple null)"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-distribution-two-mul-i-sup-lo","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInDistribution_two_mul_iSup_logLikelihood_sub","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-distribution-two-mul-i-sup-lo","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"If moreover `\u03b8\u0302\u2099 \u03c9` maximizes the log-likelihood `\u2113\u2099 \u03b8 = \u2211_{i<n} log p \u03b8 (X i \u03c9)` over all `\u03b8` for every `n` and `\u03c9`, then the likelihood-ratio statistic `2 (sup_\u03b8 \u2113\u2099 \u03b8 - \u2113\u2099 \u03b8\u2080)` converges in distribution to `\u03c7\u00b2\u2081`.","title":"Wilks' theorem, supremum form"},{"id":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/tendsto-in-distribution-two-mul-log-like-2","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioTests.tendstoInDistribution_two_mul_logLikelihood_max_zero_sub","parent":21209,"result":true,"route":"../roadmap/Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests/index.html#tendsto-in-distribution-two-mul-log-like-2","scope":"Probability/hypothesis-testing/score-wald-and-likelihood-ratio-tests","status":"fully_proved","summary":"Likelihood ratio on the boundary (one-sided parameter space `\u03b8 \u2208 [0, \u221e)`, true value `\u03b8\u2080 = 0`). Let the model be defined and regular on a ball about `0` as in `tendstoInDistribution_two_mul_logLikelihood_sub` (so `0` is interior to the domain of the family, though not to the parameter space), and let `\u03b8\u0302\u2099` be a consistent root of the likelihood equation. The statistic `2 (\u2113\u2099(max \u03b8\u0302\u2099 0) - \u2113\u2099(0))` of the estimator\u2026","title":"Likelihood ratio on the boundary"},{"data_url":"scopes/Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test.json","id":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/index.html","scope":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test","status":"fully_proved","title":"Local power of the likelihood-ratio test"},{"id":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/map-with-density-infinite-pi-prod","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioLocalPower.map_withDensity_infinitePi_prod","parent":21221,"result":true,"route":"../roadmap/Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/index.html#map-with-density-infinite-pi-prod","scope":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test","status":"fully_proved","summary":"If `\u03bc.withDensity r` is a probability measure, then reweighting the product measure `\u03bc^\u2115` by `\u220f_{i<n} r (x\u1d62)` and projecting onto the first `n` coordinates gives the product measure `(\u03bc.withDensity r)^n`.","title":"Reweighting an infinite product measure by a product density"},{"id":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/tendsto-in-measure-log-likelihood-ratio","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioLocalPower.tendstoInMeasure_logLikelihoodRatio_sub","parent":21221,"result":true,"route":"../roadmap/Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/index.html#tendsto-in-measure-log-likelihood-ratio","scope":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test","status":"fully_proved","summary":"Suppose that on the ball of radius `\u03b4 > 0` about `\u03b8\u2080` the log-likelihood `log p \u03b8 x` is differentiable in `\u03b8`, the score `\u2202_\u03b8 log p \u03b8 x` is twice differentiable in `\u03b8`, and `|\u2202\u00b3_\u03b8 log p \u03b8 x| \u2264 M x`. Let `Y 0, Y 1, \u2026` be random variables such that `-(n\u207b\u00b9 \u2211_{i<n} \u2202\u00b2_\u03b8 log p \u03b8\u2080 (Y i)) \u2192 I` and `n\u207b\u00b9 \u2211_{i<n} M (Y i) \u2192 c` in probability (laws of large numbers). Then, for the local parameters `\u03b8\u2099 = \u03b8\u2080 + h / \u221an`,\u2026","title":"Local asymptotic normality of the log-likelihood ratio"},{"id":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/tendsto-in-distribution-two-mul-log-like","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioLocalPower.tendstoInDistribution_two_mul_logLikelihood_sub_of_triangular","parent":21221,"result":true,"route":"../roadmap/Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/index.html#tendsto-in-distribution-two-mul-log-like","scope":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test","status":"fully_proved","summary":"Let `X n i` be random variables (row `n` is the sample of size `n`) and `\u03b8\u0302\u2099` estimators that solve the likelihood equation `\u2211_{i<n} \u2202_\u03b8 log p \u03b8 (X n i) = 0` with probability tending to one and converge to `\u03b8\u2080` in probability. Suppose that on the ball of radius `\u03b4 > 0` about `\u03b8\u2080` the log-likelihood is differentiable in `\u03b8`, the score is twice differentiable, and `|\u2202\u00b3_\u03b8 log p \u03b8 x| \u2264 M x`. Assume the laws of large\u2026","title":"Wilks' expansion for a triangular array"},{"id":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/tendsto-in-distribution-two-mul-log-like-2","kind":"node","lean":"MathlibExt.Probability.Statistics.LikelihoodRatioLocalPower.tendstoInDistribution_two_mul_logLikelihood_sub_local","parent":21221,"result":true,"route":"../roadmap/Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test/index.html#tendsto-in-distribution-two-mul-log-like-2","scope":"Probability/hypothesis-testing/local-power-of-the-likelihood-ratio-test","status":"fully_proved","summary":"Let row `n` of the triangular array `X n 0, X n 1, \u2026` be i.i.d. with law `p \u03b8\u2099 \u03bd`, `\u03b8\u2099 = \u03b8\u2080 + h / \u221an`, under the Cram\u00e9r conditions at `\u03b8\u2080` of `LikelihoodRatioTests.tendstoInDistribution_two_mul_logLikelihood_sub` (all moment and envelope conditions are under the null law `p \u03b8\u2080 \u03bd`). If `\u03b8\u0302\u2099` solves the likelihood equation for row `n` with probability tending to one and `\u03b8\u0302\u2099 \u2192 \u03b8\u2080` in probability, then\u2026","title":"Local power of the likelihood-ratio test"},{"data_url":"scopes/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion.json","id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","title":"Valid p-values and false discovery proportion"},{"id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/is-valid-pvalue","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.IsValidPValue","parent":21226,"result":true,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html#is-valid-pvalue","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","summary":"A real random variable `X` is a *valid* (super-uniform) p-value under `\u03bc` if `\u03bc {X \u2264 t} \u2264 t` for every level `t \u2208 [0, 1]`. Uniform p-values are the equality case.","title":"Valid p-value"},{"id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/is-valid-pvalue-of-has-law-uniform","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.isValidPValue_of_hasLaw_uniform","parent":21226,"result":true,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html#is-valid-pvalue-of-has-law-uniform","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","summary":"A p-value distributed uniformly on `[0, 1]` is valid.","title":"Uniform p-values are valid"},{"id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/num-below","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.numBelow","parent":21226,"result":true,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html#num-below","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","summary":"The number of p-values that are at most `t`.","title":"Number of p-values below a threshold"},{"id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/num-below-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.numBelow_le","parent":21226,"result":true,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html#num-below-le","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","summary":"At most `m` p-values lie below any threshold.","title":"Number of p-values below a threshold is at most the total"},{"id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/num-below-mono","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.numBelow_mono","parent":21226,"result":true,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html#num-below-mono","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","summary":"`numBelow p` is monotone in the threshold.","title":"Number of p-values below a threshold is monotone"},{"id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/false-discovery-proportion","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.falseDiscoveryProportion","parent":21226,"result":true,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html#false-discovery-proportion","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","summary":"The false discovery proportion of a rejection set `R` when `H0` is the set of true nulls: `|R \u2229 H0| / max |R| 1` (zero when nothing is rejected).","title":"False discovery proportion"},{"id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/false-discovery-proportion-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.falseDiscoveryProportion_nonneg","parent":21226,"result":true,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html#false-discovery-proportion-nonneg","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","summary":"The false discovery proportion is nonnegative.","title":"False discovery proportion is nonnegative"},{"id":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/false-discovery-proportion-le-one","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.falseDiscoveryProportion_le_one","parent":21226,"result":true,"route":"../roadmap/Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion/index.html#false-discovery-proportion-le-one","scope":"Probability/hypothesis-testing/valid-p-values-and-false-discovery-proportion","status":"fully_proved","summary":"The false discovery proportion is at most one.","title":"False discovery proportion is at most one"},{"data_url":"scopes/Probability/hypothesis-testing/benjamini-hochberg-procedure.json","id":"Probability/hypothesis-testing/benjamini-hochberg-procedure","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/benjamini-hochberg-procedure/index.html","scope":"Probability/hypothesis-testing/benjamini-hochberg-procedure","status":"fully_proved","title":"Benjamini\u2013Hochberg procedure"},{"id":"Probability/hypothesis-testing/benjamini-hochberg-procedure/bh-index","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.bhIndex","parent":21235,"result":true,"route":"../roadmap/Probability/hypothesis-testing/benjamini-hochberg-procedure/index.html#bh-index","scope":"Probability/hypothesis-testing/benjamini-hochberg-procedure","status":"fully_proved","summary":"The Benjamini\u2013Hochberg step-up index: the largest `k \u2208 {0, \u2026, m}` such that at least `k` p-values are at most `q k / m` (`k = 0` always qualifies).","title":"Benjamini\u2013Hochberg step-up index"},{"id":"Probability/hypothesis-testing/benjamini-hochberg-procedure/bh-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.bhRejections","parent":21235,"result":true,"route":"../roadmap/Probability/hypothesis-testing/benjamini-hochberg-procedure/index.html#bh-rejections","scope":"Probability/hypothesis-testing/benjamini-hochberg-procedure","status":"fully_proved","summary":"The hypotheses rejected by the Benjamini\u2013Hochberg procedure at level `q`: those whose p-value is at most `q * bhIndex q p / m`.","title":"Benjamini\u2013Hochberg rejection set"},{"id":"Probability/hypothesis-testing/benjamini-hochberg-procedure/bh-index-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.bhIndex_le","parent":21235,"result":true,"route":"../roadmap/Probability/hypothesis-testing/benjamini-hochberg-procedure/index.html#bh-index-le","scope":"Probability/hypothesis-testing/benjamini-hochberg-procedure","status":"fully_proved","summary":"The BH step-up index is at most `m`.","title":"Benjamini\u2013Hochberg index is at most the number of hypotheses"},{"id":"Probability/hypothesis-testing/benjamini-hochberg-procedure/bh-index-le-num-below","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.bhIndex_le_numBelow","parent":21235,"result":true,"route":"../roadmap/Probability/hypothesis-testing/benjamini-hochberg-procedure/index.html#bh-index-le-num-below","scope":"Probability/hypothesis-testing/benjamini-hochberg-procedure","status":"fully_proved","summary":"At the BH step-up index `k`, at least `k` p-values are at most `q k / m`.","title":"Benjamini\u2013Hochberg index satisfies the step-up condition"},{"id":"Probability/hypothesis-testing/benjamini-hochberg-procedure/le-bh-index","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.le_bhIndex","parent":21235,"result":true,"route":"../roadmap/Probability/hypothesis-testing/benjamini-hochberg-procedure/index.html#le-bh-index","scope":"Probability/hypothesis-testing/benjamini-hochberg-procedure","status":"fully_proved","summary":"The BH step-up index is the largest `k \u2264 m` with at least `k` p-values at most `q k / m`.","title":"Maximality of the Benjamini\u2013Hochberg index"},{"id":"Probability/hypothesis-testing/benjamini-hochberg-procedure/card-bh-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.card_bhRejections","parent":21235,"result":true,"route":"../roadmap/Probability/hypothesis-testing/benjamini-hochberg-procedure/index.html#card-bh-rejections","scope":"Probability/hypothesis-testing/benjamini-hochberg-procedure","status":"fully_proved","summary":"The BH procedure at level `q \u2265 0` rejects exactly `bhIndex q p` hypotheses.","title":"Number of Benjamini\u2013Hochberg rejections"},{"id":"Probability/hypothesis-testing/benjamini-hochberg-procedure/integral-false-discovery-proportion-bh-r","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.integral_falseDiscoveryProportion_bhRejections_le","parent":21235,"result":true,"route":"../roadmap/Probability/hypothesis-testing/benjamini-hochberg-procedure/index.html#integral-false-discovery-proportion-bh-r","scope":"Probability/hypothesis-testing/benjamini-hochberg-procedure","status":"fully_proved","summary":"If the p-values are independent and every true-null p-value is uniformly distributed on `[0, 1]`, the BH procedure at level `q \u2208 [0, 1]` has false discovery rate at most `q |H0| / m`. Nothing is assumed about the false-null p-values beyond measurability and independence. (For `m = 0` both sides are `0`.)","title":"Benjamini\u2013Hochberg FDR control"},{"data_url":"scopes/Probability/hypothesis-testing/bonferroni-correction.json","id":"Probability/hypothesis-testing/bonferroni-correction","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/bonferroni-correction/index.html","scope":"Probability/hypothesis-testing/bonferroni-correction","status":"fully_proved","title":"Bonferroni correction"},{"id":"Probability/hypothesis-testing/bonferroni-correction/bonferroni-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.bonferroniRejections","parent":21243,"result":true,"route":"../roadmap/Probability/hypothesis-testing/bonferroni-correction/index.html#bonferroni-rejections","scope":"Probability/hypothesis-testing/bonferroni-correction","status":"fully_proved","summary":"The hypotheses rejected by the Bonferroni procedure at level `\u03b1`: those whose p-value is at most `\u03b1 / m`.","title":"Bonferroni rejection set"},{"id":"Probability/hypothesis-testing/bonferroni-correction/measure-exists-mem-le-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.measure_exists_mem_le_le","parent":21243,"result":true,"route":"../roadmap/Probability/hypothesis-testing/bonferroni-correction/index.html#measure-exists-mem-le-le","scope":"Probability/hypothesis-testing/bonferroni-correction","status":"fully_proved","summary":"If every true-null p-value is valid and `c \u2208 [0, 1]`, the probability that some true null has p-value at most `c` is at most `|H0| c`. No independence is assumed.","title":"Union bound for valid p-values"},{"id":"Probability/hypothesis-testing/bonferroni-correction/measure-exists-le-div-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.measure_exists_le_div_le","parent":21243,"result":true,"route":"../roadmap/Probability/hypothesis-testing/bonferroni-correction/index.html#measure-exists-le-div-le","scope":"Probability/hypothesis-testing/bonferroni-correction","status":"fully_proved","summary":"If every true-null p-value is valid, the probability that some true null has p-value at most `\u03b1 / m` is at most `\u03b1 |H0| / m`, for `\u03b1 \u2208 [0, 1]`. No independence is assumed. (For `m = 0` there is no hypothesis and both sides are `0`.)","title":"Bonferroni inequality"},{"id":"Probability/hypothesis-testing/bonferroni-correction/measure-bonferroni-familywise-error-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.measure_bonferroni_familywiseError_le","parent":21243,"result":true,"route":"../roadmap/Probability/hypothesis-testing/bonferroni-correction/index.html#measure-bonferroni-familywise-error-le","scope":"Probability/hypothesis-testing/bonferroni-correction","status":"fully_proved","summary":"If every true-null p-value is valid, the probability that the Bonferroni procedure at level `\u03b1 \u2208 [0, 1]` rejects at least one true null is at most `\u03b1`.","title":"The Bonferroni procedure controls the FWER"},{"data_url":"scopes/Probability/hypothesis-testing/holm-bonferroni-method.json","id":"Probability/hypothesis-testing/holm-bonferroni-method","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","title":"Holm\u2013Bonferroni method"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/holm-index","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.holmIndex","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#holm-index","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"Holm's step-down count at level `\u03b1`: the largest `r \u2208 {0, \u2026, m}` such that for every `k \u2208 {1, \u2026, r}` the `k`-th smallest p-value is at most `\u03b1 / (m - k + 1)`, i.e. at least `k` p-values are at most `\u03b1 / (m - k + 1)`. (`r = 0` always qualifies.)","title":"Holm's step-down count"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/holm-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.holmRejections","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#holm-rejections","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"The hypotheses rejected by Holm's step-down procedure at level `\u03b1`: those whose p-value is at most `\u03b1 / (m - r + 1)` with `r = holmIndex \u03b1 p`; these are the hypotheses with the `r` smallest p-values.","title":"Hypotheses rejected by Holm's step-down procedure"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/holm-index-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.holmIndex_le","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#holm-index-le","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"Holm's step-down count is at most `m`.","title":"Holm's step-down count is at most the number of tests"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/le-num-below-of-le-holm-index","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.le_numBelow_of_le_holmIndex","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#le-num-below-of-le-holm-index","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"Every step up to Holm's count succeeds: for `1 \u2264 k \u2264 holmIndex \u03b1 p`, at least `k` p-values are at most `\u03b1 / (m - k + 1)`.","title":"Every step up to Holm's count succeeds"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/le-holm-index","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.le_holmIndex","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#le-holm-index","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"Holm's count is the largest index up to which every step succeeds.","title":"Holm's count is the largest index where every step succeeds"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/mem-holm-rejections-of-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.mem_holmRejections_of_le","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#mem-holm-rejections-of-le","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"The rejection set is closed under smaller p-values.","title":"Holm's rejection set is closed under smaller p-values"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/card-holm-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.card_holmRejections","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#card-holm-rejections","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"Holm's procedure at level `\u03b1 \u2265 0` rejects exactly `holmIndex \u03b1 p` hypotheses.","title":"Number of hypotheses rejected by Holm's procedure"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/exists-le-div-card-of-holm-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.exists_le_div_card_of_holmRejections","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#exists-le-div-card-of-holm-rejections","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"If Holm rejects a hypothesis in `J`, then some `j \u2208 J` has p-value at most `\u03b1 / |J|`: the first hypothesis of `J` in sorted order sits at a position `k \u2264 m - |J| + 1`. This is the shared Holm rejection bound, used both for Holm's FWER control below and for identifying Holm's procedure with the closure of the Bonferroni local tests.","title":"Holm rejection bound for a set of hypotheses"},{"id":"Probability/hypothesis-testing/holm-bonferroni-method/measure-holm-familywise-error-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.measure_holm_familywiseError_le","parent":21248,"result":true,"route":"../roadmap/Probability/hypothesis-testing/holm-bonferroni-method/index.html#measure-holm-familywise-error-le","scope":"Probability/hypothesis-testing/holm-bonferroni-method","status":"fully_proved","summary":"If every true-null p-value is valid, the probability that Holm's step-down procedure at level `\u03b1 \u2208 [0, 1]` rejects at least one true null is at most `\u03b1`. No independence is assumed.","title":"Holm's procedure controls the FWER"},{"data_url":"scopes/Probability/hypothesis-testing/closed-testing-principle.json","id":"Probability/hypothesis-testing/closed-testing-principle","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/closed-testing-principle/index.html","scope":"Probability/hypothesis-testing/closed-testing-principle","status":"fully_proved","title":"Closed testing principle"},{"id":"Probability/hypothesis-testing/closed-testing-principle/closed-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.closedRejections","parent":21258,"result":true,"route":"../roadmap/Probability/hypothesis-testing/closed-testing-principle/index.html#closed-rejections","scope":"Probability/hypothesis-testing/closed-testing-principle","status":"fully_proved","summary":"The closed testing procedure: given the decisions `\u03c6 J` of local tests of the intersection hypotheses `H_J`, reject `H\u1d62` exactly when `\u03c6 J` holds for every `J` containing `i`.","title":"Closed testing procedure"},{"id":"Probability/hypothesis-testing/closed-testing-principle/mem-closed-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.mem_closedRejections","parent":21258,"result":true,"route":"../roadmap/Probability/hypothesis-testing/closed-testing-principle/index.html#mem-closed-rejections","scope":"Probability/hypothesis-testing/closed-testing-principle","status":"fully_proved","summary":"Membership in the closed rejection set.","title":"Membership in the closed rejection set"},{"id":"Probability/hypothesis-testing/closed-testing-principle/inter-nonempty-closed-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.inter_nonempty_closedRejections","parent":21258,"result":true,"route":"../roadmap/Probability/hypothesis-testing/closed-testing-principle/index.html#inter-nonempty-closed-rejections","scope":"Probability/hypothesis-testing/closed-testing-principle","status":"fully_proved","summary":"If the closed procedure rejects a hypothesis in `H0`, then the local test of the intersection hypothesis `H_{H0}` rejects.","title":"Closed testing rejection implies the intersection test rejects"},{"id":"Probability/hypothesis-testing/closed-testing-principle/measure-closed-familywise-error-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.measure_closed_familywiseError_le","parent":21258,"result":true,"route":"../roadmap/Probability/hypothesis-testing/closed-testing-principle/index.html#measure-closed-familywise-error-le","scope":"Probability/hypothesis-testing/closed-testing-principle","status":"fully_proved","summary":"If `H0` is the set of true nulls and the local test of `H_{H0}` rejects with probability at most `\u03b1`, then the closed procedure rejects some true null with probability at most `\u03b1`. Applied to every possible `H0`, local level-`\u03b1` tests give strong control of the family-wise error rate. No assumption on the dependence between local tests is made.","title":"The closed testing principle"},{"id":"Probability/hypothesis-testing/closed-testing-principle/bonferroni-local-test","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.bonferroniLocalTest","parent":21258,"result":true,"route":"../roadmap/Probability/hypothesis-testing/closed-testing-principle/index.html#bonferroni-local-test","scope":"Probability/hypothesis-testing/closed-testing-principle","status":"fully_proved","summary":"The Bonferroni local test of the intersection hypothesis `H_J` at level `\u03b1`: reject when some `j \u2208 J` has `p j \u2264 \u03b1 / |J|`. (It never rejects `H_\u2205`.)","title":"Bonferroni local test of an intersection hypothesis"},{"id":"Probability/hypothesis-testing/closed-testing-principle/measure-bonferroni-local-test-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.measure_bonferroniLocalTest_le","parent":21258,"result":true,"route":"../roadmap/Probability/hypothesis-testing/closed-testing-principle/index.html#measure-bonferroni-local-test-le","scope":"Probability/hypothesis-testing/closed-testing-principle","status":"fully_proved","summary":"The Bonferroni local test of `H_{H0}` has level `\u03b1 \u2208 [0, 1]` when the p-values in `H0` are valid.","title":"Level of the Bonferroni local test"},{"id":"Probability/hypothesis-testing/closed-testing-principle/holm-rejections-eq-closed-rejections","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.holmRejections_eq_closedRejections","parent":21258,"result":true,"route":"../roadmap/Probability/hypothesis-testing/closed-testing-principle/index.html#holm-rejections-eq-closed-rejections","scope":"Probability/hypothesis-testing/closed-testing-principle","status":"fully_proved","summary":"For `\u03b1 \u2265 0`, the hypotheses rejected by Holm's step-down procedure are exactly those rejected by the closed testing procedure built on the Bonferroni local tests.","title":"Holm's procedure is a closed test"},{"data_url":"scopes/Probability/hypothesis-testing/permutation-tests.json","id":"Probability/hypothesis-testing/permutation-tests","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/permutation-tests/index.html","scope":"Probability/hypothesis-testing/permutation-tests","status":"fully_proved","title":"Permutation tests"},{"id":"Probability/hypothesis-testing/permutation-tests/permutation-pvalue","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.permutationPValue","parent":21266,"result":true,"route":"../roadmap/Probability/hypothesis-testing/permutation-tests/index.html#permutation-pvalue","scope":"Probability/hypothesis-testing/permutation-tests","status":"fully_proved","summary":"The permutation (group-invariance) p-value of the statistic `T` at `\u03c9`: the fraction of group elements `g` with `T (g \u2022 \u03c9) \u2265 T \u03c9`.","title":"Permutation p-value"},{"id":"Probability/hypothesis-testing/permutation-tests/card-filter-rank-count-le","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.card_filter_rankCount_le","parent":21266,"result":true,"route":"../roadmap/Probability/hypothesis-testing/permutation-tests/index.html#card-filter-rank-count-le","scope":"Probability/hypothesis-testing/permutation-tests","status":"fully_proved","summary":"Counting lemma: for any family of reals `t` indexed by `G` and any `k`, at most `k` elements `h` have rank count `#{g | t h \u2264 t g}` at most `k`.","title":"Counting lemma for rank counts"},{"id":"Probability/hypothesis-testing/permutation-tests/is-valid-pvalue-permutation-pvalue","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.isValidPValue_permutationPValue","parent":21266,"result":true,"route":"../roadmap/Probability/hypothesis-testing/permutation-tests/index.html#is-valid-pvalue-permutation-pvalue","scope":"Probability/hypothesis-testing/permutation-tests","status":"fully_proved","summary":"Lehmann and Romano (2005), \u00a715.2; Hemerik and Goeman (2018): if the law `\u03bc` is invariant under the action of a finite group `G`, the permutation p-value of any measurable statistic is valid.","title":"Validity of permutation p-values"},{"id":"Probability/hypothesis-testing/permutation-tests/monte-carlo-permutation-pvalue","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.monteCarloPermutationPValue","parent":21266,"result":true,"route":"../roadmap/Probability/hypothesis-testing/permutation-tests/index.html#monte-carlo-permutation-pvalue","scope":"Probability/hypothesis-testing/permutation-tests","status":"fully_proved","summary":"The Monte Carlo permutation p-value `(1 + #{b | T(g_b \u2022 \u03c9) \u2265 T(\u03c9)}) / (B + 1)` of the statistic `T` at the data `\u03c9` with the `B` drawn group elements `g`.","title":"Monte Carlo permutation p-value"},{"id":"Probability/hypothesis-testing/permutation-tests/is-valid-pvalue-monte-carlo-permutation","kind":"node","lean":"MathlibExt.Probability.Statistics.MultipleTesting.isValidPValue_monteCarloPermutationPValue","parent":21266,"result":true,"route":"../roadmap/Probability/hypothesis-testing/permutation-tests/index.html#is-valid-pvalue-monte-carlo-permutation","scope":"Probability/hypothesis-testing/permutation-tests","status":"fully_proved","summary":"Phipson and Smyth (2010); Hemerik and Goeman (2018): if the null law `\u03bc` is invariant under a finite group `G` and `g\u2081, \u2026, g_B` are drawn independently and uniformly from `G`, independently of the data, then `(1 + #{b | T(g_b \u2022 \u03c9) \u2265 T(\u03c9)}) / (B + 1)` is a valid p-value under the joint law.","title":"Validity of Monte Carlo permutation p-values"},{"data_url":"scopes/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates.json","id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","kind":"scope","navigation":"module","parent":21164,"result":false,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","title":"Two-groups model and local false discovery rates"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/two-groups-model","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#two-groups-model","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The two-groups model of Efron et al. (2001): prior null probability `\u03c0\u2080 \u2208 [0, 1]`, null density `f\u2080` and non-null density `f\u2081` of the z-value (probability densities on `\u211d`).","title":"Two-groups model"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/mixture-density","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.mixtureDensity","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#mixture-density","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The mixture density `f = \u03c0\u2080 f\u2080 + (1 - \u03c0\u2080) f\u2081` of the z-values.","title":"Mixture density of the two-groups model"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/null-law","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.nullLaw","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#null-law","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The law `P\u2080` of a null z-value.","title":"Law of a null z-value"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/alt-law","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.altLaw","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#alt-law","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The law `P\u2081` of a non-null z-value.","title":"Law of a non-null z-value"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/joint-law","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.jointLaw","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#joint-law","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Joint law of (null indicator, z-value) on `Bool \u00d7 \u211d`; `true` means the case is null.","title":"Joint law of null indicator and z-value"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/local-fdr","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.localFdr","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#local-fdr","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Efron's local false discovery rate `fdr(z) = \u03c0\u2080 f\u2080(z) / f(z)` (zero where `f(z) = 0`).","title":"Local false discovery rate"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/tail-fdr","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.tailFdr","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#tail-fdr","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The tail-area false discovery rate of a region `\u0393`: `Fdr(\u0393) = \u03c0\u2080 P\u2080(\u0393) / P(\u0393)`, written with the densities. Efron's `Fdr(z)` is `tailFdr (Set.Iic z)`. It is `0` when `P(\u0393) = 0`.","title":"Tail-area false discovery rate"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/mixture-density-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.mixtureDensity_nonneg","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#mixture-density-nonneg","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The mixture density is nonnegative.","title":"Mixture density is nonnegative"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/mul-f-le-mixture-density","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.mul_f\u2080_le_mixtureDensity","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#mul-f-le-mixture-density","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The null part of the mixture is dominated by the mixture: `\u03c0\u2080 f\u2080 \u2264 f`.","title":"Null part of the mixture is dominated by the mixture"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/measurable-mixture-density","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.measurable_mixtureDensity","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#measurable-mixture-density","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The mixture density is measurable.","title":"Mixture density is measurable"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/integrable-mixture-density","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.integrable_mixtureDensity","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#integrable-mixture-density","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The mixture density is integrable.","title":"Mixture density is integrable"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/measurable-local-fdr","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.measurable_localFdr","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#measurable-local-fdr","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The local false discovery rate is measurable.","title":"Local false discovery rate is measurable"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/local-fdr-mul-mixture-density","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.localFdr_mul_mixtureDensity","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#local-fdr-mul-mixture-density","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The basic identity `fdr(z) f(z) = \u03c0\u2080 f\u2080(z)`, for every `z` (where `f(z) = 0` both sides vanish because `\u03c0\u2080 f\u2080 \u2264 f`).","title":"Local fdr times mixture density equals the null part"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/local-fdr-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.localFdr_nonneg","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#local-fdr-nonneg","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The local false discovery rate is nonnegative.","title":"Local false discovery rate is nonnegative"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/local-fdr-le-one","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.localFdr_le_one","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#local-fdr-le-one","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The local false discovery rate is at most one.","title":"Local false discovery rate is at most one"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/null-law-apply","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.nullLaw_apply","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#null-law-apply","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"`P\u2080(A) = \u222b_A f\u2080`.","title":"Null law as the integral of the null density"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/alt-law-apply","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.altLaw_apply","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#alt-law-apply","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"`P\u2081(A) = \u222b_A f\u2081`.","title":"Non-null law as the integral of the non-null density"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/joint-law-null-prod","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.jointLaw_null_prod","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#joint-law-null-prod","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"`P(H = 0, Z \u2208 A) = \u03c0\u2080 P\u2080(A)`.","title":"Joint probability of a null case in a set"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/joint-law-alt-prod","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.jointLaw_alt_prod","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#joint-law-alt-prod","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"`P(H = 1, Z \u2208 A) = (1 - \u03c0\u2080) P\u2081(A)`.","title":"Joint probability of a non-null case in a set"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/joint-law-univ-prod","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.jointLaw_univ_prod","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#joint-law-univ-prod","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The z-value has the mixture law: `P(Z \u2208 A) = \u222b_A f`.","title":"z-value has the mixture law"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/map-snd-joint-law","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.map_snd_jointLaw","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#map-snd-joint-law","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The law of the z-value under the joint law is the mixture `f \u00b7 Lebesgue`.","title":"Marginal z-value law of the joint law is the mixture"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/joint-law-null-prod-eq-lintegral-local-f","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.jointLaw_null_prod_eq_lintegral_localFdr","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#joint-law-null-prod-eq-lintegral-local-f","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Efron et al. (2001): `P(H = 0, Z \u2208 A) = \u222b_A fdr(z) dP_Z(z)` for every measurable `A`, so `fdr(Z)` is a version of the posterior null probability `P(H = 0 | Z)`.","title":"Null probability as an integral of the local fdr"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/cond-exp-null-eq-local-fdr","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.condExp_null_eq_localFdr","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#cond-exp-null-eq-local-fdr","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Efron et al. (2001): `fdr(Z) = P(H = 0 | Z)` almost surely, as a conditional expectation of the null indicator given the z-value.","title":"Local fdr is the posterior null probability"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/cond-null-eq-tail-fdr","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.cond_null_eq_tailFdr","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#cond-null-eq-tail-fdr","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"`P(H = 0 | Z \u2208 \u0393) = Fdr(\u0393)`: the tail-area false discovery rate is the posterior null probability given `Z \u2208 \u0393` (Storey 2002, Theorem 1; Efron 2010, ch. 2). Both sides are `0` when `P(Z \u2208 \u0393) = 0`.","title":"Tail-area fdr is the posterior null probability given a region"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/tail-fdr-eq-set-integral-local-fdr","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.tailFdr_eq_setIntegral_localFdr","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#tail-fdr-eq-set-integral-local-fdr","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"`E[fdr(Z) | Z \u2208 \u0393] = Fdr(\u0393)`, written with densities: the tail-area false discovery rate is the `f`-weighted average of the local false discovery rate over `\u0393` (Efron 2010, ch. 5).","title":"Tail-area fdr is the average local fdr over the region"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/null-set","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.nullSet","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#null-set","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The true-null set of an outcome vector `x : Fin m \u2192 Bool \u00d7 Bool`, where `(x i).1` records that hypothesis `i` is null and `(x i).2` that it is rejected.","title":"True-null set of a test outcome vector"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/rejected-set","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.rejectedSet","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#rejected-set","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The rejection set of an outcome vector.","title":"Rejection set of a test outcome vector"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/outcome-weight","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.outcomeWeight","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#outcome-weight","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The probability `\u220f\u1d62 q(x\u1d62)` of an outcome vector when the `m` (null, rejected) pairs are i.i.d. with law `q`.","title":"Probability of a test outcome vector"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/positive-fdr","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.positiveFDR","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#positive-fdr","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Storey's positive false discovery rate `pFDR = E[V / R | R > 0]` for `m` tests whose (null, rejected) indicator pairs are i.i.d. with law `q`: `V / R` is the false discovery proportion.","title":"Storey's positive false discovery rate"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/positive-fdr-eq","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.positiveFDR_eq","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#positive-fdr-eq","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Storey (2002), Theorem 1: for `m \u2265 1` i.i.d. tests, `pFDR = P(H = 0 | T \u2208 \u0393)`, where `q (true, true) = P(H = 0, T \u2208 \u0393)` and `q (false, true) = P(H = 1, T \u2208 \u0393)`. The identity is homogeneous in `q`, so only `q \u2265 0` is assumed; when `P(T \u2208 \u0393) = 0` both sides are `0`.","title":"Storey's theorem: pFDR is a posterior null probability"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/pair-law","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.pairLaw","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#pair-law","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The law of one test's (null, rejected) pair in the two-groups model when the test rejects iff its z-value lies in `\u0393`.","title":"Law of a test's null and rejection pair"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/positive-fdr-pair-law-eq-tail-fdr","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.TwoGroupsModel.positiveFDR_pairLaw_eq_tailFdr","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#positive-fdr-pair-law-eq-tail-fdr","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Storey (2002), Theorem 1, in the two-groups model: for `m \u2265 1` i.i.d. tests rejecting on `\u0393`, `pFDR(\u0393) = \u03c0\u2080 P\u2080(\u0393) / P(\u0393) = Fdr(\u0393)`.","title":"Storey's theorem in the two-groups model"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/storey-pi0","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.storeyPi0","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#storey-pi0","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Storey's estimator `\u03c0\u0302\u2080(t) = #{i | p\u1d62 > t} / (m (1 - t))` of the proportion of true nulls, for a tuning parameter `t \u2208 [0, 1)`.","title":"Storey's estimator of the null proportion"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/le-integral-storey-pi0","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.le_integral_storeyPi0","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#le-integral-storey-pi0","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Conservative bias of Storey's (2002) estimator for a fixed set `H0` of true nulls: if their p-values are valid, then `E[\u03c0\u0302\u2080(t)] \u2265 m\u2080 / m` for every `t \u2208 [0, 1)`. No independence is assumed; the non-null p-values are arbitrary measurable functions.","title":"Conservative bias of Storey's null proportion estimator"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/q-value","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.qValue","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#q-value","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"Storey's (2003) q-value for a nested family of rejection regions `\u0393_s = {T \u2264 s}` with `pFDR(\u0393_s) = pfdr s`: `q(t) = inf_{s \u2265 t} pfdr s`, the smallest pFDR over the regions that contain the observed statistic `t`.","title":"Storey's q-value"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/q-value-le","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.qValue_le","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#q-value-le","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The q-value is at most the pFDR of the smallest region containing the statistic.","title":"q-value is at most the pFDR of the smallest region"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/monotone-q-value","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.monotone_qValue","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#monotone-q-value","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"The q-value is monotone in the statistic.","title":"q-value is monotone in the statistic"},{"id":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/q-value-eq-of-monotone","kind":"node","lean":"MathlibExt.Probability.Statistics.TwoGroupsTesting.qValue_eq_of_monotone","parent":21272,"result":true,"route":"../roadmap/Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates/index.html#q-value-eq-of-monotone","scope":"Probability/hypothesis-testing/two-groups-model-and-local-false-discovery-rates","status":"fully_proved","summary":"When the pFDR is nondecreasing along the nested regions, the q-value is the pFDR.","title":"q-value equals the pFDR when the pFDR is monotone"},{"data_url":"scopes/Probability/linear-models-multivariate.json","id":"Probability/linear-models-multivariate","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/index.html","scope":"Probability/linear-models-multivariate","status":"can_prove","summary":"Multivariate normal theory and linear models: Gaussian maximum likelihood, Cochran's theorem on quadratic forms, t and F sampling laws, Gauss\u2013Markov and generalized least squares, principal component analysis, and Gaussian graphical models.","title":"Linear models and multivariate analysis"},{"id":"Probability/linear-models-multivariate/gauss-markov-covariance-minimality","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussMarkov.gaussMarkov_covariance_minimality","parent":21311,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/index.html#gauss-markov-covariance-minimality","scope":"Probability/linear-models-multivariate","status":"fully_proved","summary":"Let $X$ be a real $m \\times n$ matrix such that $X^\\top X$ is invertible, and let $B = (X^\\top X)^{-1} X^\\top$. Then $BX = I_n$, and for every real $n \\times m$ matrix $A$ with $AX = I_n$ the matrix $AA^\\top - BB^\\top$ is positive semidefinite.","title":"Gauss\u2013Markov covariance minimality"},{"data_url":"scopes/Probability/linear-models-multivariate/cochran-s-theorem.json","id":"Probability/linear-models-multivariate/cochran-s-theorem","kind":"scope","navigation":"module","parent":21311,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","title":"Cochran's theorem"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/cochran-rank-criterion","kind":"node","lean":"Matrix.cochran_rank_criterion","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#cochran-rank-criterion","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"Rank criterion (Cochran 1934): square matrices `A k` over a field that sum to the identity have ranks adding up to the size iff they are idempotent and pairwise annihilating.","title":"Rank criterion"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/i-indep-fun-star-projection-std-gaussian","kind":"node","lean":"ProbabilityTheory.iIndepFun_starProjection_stdGaussian","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#i-indep-fun-star-projection-std-gaussian","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"Under the standard Gaussian law, orthogonal projections onto pairwise orthogonal subspaces are mutually independent.","title":"Independence of orthogonal projections of a standard Gaussian"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/i-indep-fun-norm-sq-star-projection-std","kind":"node","lean":"ProbabilityTheory.iIndepFun_norm_sq_starProjection_stdGaussian","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#i-indep-fun-norm-sq-star-projection-std","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"Under the standard Gaussian law, the squared lengths `\u2016P_{K k} Z\u2016\u00b2` of the projections onto pairwise orthogonal subspaces are mutually independent.","title":"Independence of squared lengths of orthogonal Gaussian projections"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/group-mean","kind":"node","lean":"ProbabilityTheory.groupMean","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#group-mean","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"The mean of the observations `x i` with `grp i = g` (`0` for an empty group).","title":"Group mean of observations"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/indep-fun-group-mean-residual","kind":"node","lean":"ProbabilityTheory.indepFun_groupMean_residual","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#indep-fun-group-mean-residual","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"For i.i.d. `N(\u03bc, v)` observations `x\u1d62` labelled by groups `grp i`, the vector of group means is independent of the vector of within-group residuals `x\u1d62 - x\u0304_{grp i}`.","title":"Gaussian one-way layout"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/indep-fun-sum-sq-between-sum-sq-within","kind":"node","lean":"ProbabilityTheory.indepFun_sumSqBetween_sumSqWithin","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#indep-fun-sum-sq-between-sum-sq-within","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"For i.i.d. `N(\u03bc, v)` observations labelled by groups, the between-group sum of squares `\u2211\u1d62 (x\u0304_{grp i} - x\u0304)\u00b2` is independent of the within-group sum of squares `\u2211\u1d62 (x\u1d62 - x\u0304_{grp i})\u00b2`.","title":"One-way analysis of variance"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/has-law-sample-mean","kind":"node","lean":"ProbabilityTheory.hasLaw_sampleMean","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#has-law-sample-mean","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"For an i.i.d. sample `X\u2081, \u2026, X\u2099 ~ N(\u03bc, v)` with `n \u2265 1`, the sample mean has the law `N(\u03bc, v / n)`.","title":"Law of the Gaussian sample mean"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/i-indep-fun-dot-product-mul-vec-pi-gauss","kind":"node","lean":"ProbabilityTheory.iIndepFun_dotProduct_mulVec_pi_gaussianReal","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#i-indep-fun-dot-product-mul-vec-pi-gauss","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"Cochran's theorem, independence part: if `Z` has independent standard normal coordinates and symmetric matrices `A k` sum to the identity with ranks adding up to the dimension, the quadratic forms `Z\u1d40 (A k) Z` are mutually independent.","title":"Cochran's theorem"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/indep-fun-sample-mean-residual","kind":"node","lean":"ProbabilityTheory.indepFun_sampleMean_residual","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#indep-fun-sample-mean-residual","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"For an i.i.d. sample `X\u2081, \u2026, X\u2099 ~ N(\u03bc, v)`, the sample mean is independent of the residual vector `(X\u1d62 - X\u0304)\u1d62`.","title":"Independence of the Gaussian sample mean and residuals"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/indep-fun-sample-mean-sum-sq-dev","kind":"node","lean":"ProbabilityTheory.indepFun_sampleMean_sumSqDev","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#indep-fun-sample-mean-sum-sq-dev","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"For an i.i.d. sample `X\u2081, \u2026, X\u2099 ~ N(\u03bc, v)`, the sample mean `X\u0304` is independent of the sum of squared deviations `\u2211 (X\u1d62 - X\u0304)\u00b2 = (n - 1) S\u00b2`.","title":"Independence of the Gaussian sample mean and sum of squared deviations"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/pi-gaussian-real-map-sum-sq","kind":"node","lean":"ProbabilityTheory.pi_gaussianReal_map_sum_sq","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#pi-gaussian-real-map-sum-sq","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"For independent standard normal coordinates, the sum of the squares of the coordinates in a nonempty finite set `s` has the chi-square law with `|s|` degrees of freedom.","title":"Sum of squares of independent standard normals is chi-square"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/std-gaussian-map-sum-sq-inner","kind":"node","lean":"ProbabilityTheory.stdGaussian_map_sum_sq_inner","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#std-gaussian-map-sum-sq-inner","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"Coordinates of the standard Gaussian law in an orthonormal basis `b`: the sum of the squared coordinates `\u27eab i, x\u27eb` over a nonempty set `s` of indices is chi-square with `|s|` degrees of freedom.","title":"Squared orthonormal coordinates of a standard Gaussian are chi-square"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/std-gaussian-map-norm-sq-star-projection","kind":"node","lean":"ProbabilityTheory.stdGaussian_map_norm_sq_starProjection","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#std-gaussian-map-norm-sq-star-projection","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"Cochran's theorem, projection form: for the standard Gaussian law on a finite-dimensional real inner product space `E` and a subspace `K` of positive dimension, the squared norm of the orthogonal projection onto `K` is chi-square with `dim K` degrees of freedom.","title":"Cochran's theorem"},{"id":"Probability/linear-models-multivariate/cochran-s-theorem/pi-gaussian-real-map-dot-product-mul-vec","kind":"node","lean":"ProbabilityTheory.pi_gaussianReal_map_dotProduct_mulVec","parent":21313,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/cochran-s-theorem/index.html#pi-gaussian-real-map-dot-product-mul-vec","scope":"Probability/linear-models-multivariate/cochran-s-theorem","status":"fully_proved","summary":"Cochran's theorem, matrix form: if `Z` has independent standard normal coordinates and `A` is a symmetric idempotent matrix of positive rank, the quadratic form `Z\u1d40 A Z` is chi-square with `rank A` degrees of freedom.","title":"Cochran's theorem"},{"data_url":"scopes/Probability/linear-models-multivariate/frisch-waugh-lovell-theorem.json","id":"Probability/linear-models-multivariate/frisch-waugh-lovell-theorem","kind":"scope","navigation":"module","parent":21311,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/frisch-waugh-lovell-theorem/index.html","scope":"Probability/linear-models-multivariate/frisch-waugh-lovell-theorem","status":"fully_proved","title":"Frisch\u2013Waugh\u2013Lovell theorem"},{"id":"Probability/linear-models-multivariate/frisch-waugh-lovell-theorem/frisch-waugh-lovell","kind":"node","lean":"Matrix.frischWaughLovell","parent":21328,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/frisch-waugh-lovell-theorem/index.html#frisch-waugh-lovell","scope":"Probability/linear-models-multivariate/frisch-waugh-lovell-theorem","status":"fully_proved","summary":"Frisch\u2013Waugh\u2013Lovell for weighted least squares with a symmetric weight matrix `W`: if `(b, t)` solves the normal equations of the regression of `y` on `[A | z]` and `z\u22a5 = z - A u` is the weighted residual of `z` on `A` (`A\u1d40 W z\u22a5 = 0`), then `t \u27eaz\u22a5, z\u22a5\u27eb_W = \u27eaz\u22a5, y\u27eb_W`, where `\u27eax, y\u27eb_W = x \u2b1d\u1d65 W y`.","title":"Frisch\u2013Waugh\u2013Lovell"},{"id":"Probability/linear-models-multivariate/frisch-waugh-lovell-theorem/frisch-waugh-lovell-residual","kind":"node","lean":"Matrix.frischWaughLovell_residual","parent":21328,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/frisch-waugh-lovell-theorem/index.html#frisch-waugh-lovell-residual","scope":"Probability/linear-models-multivariate/frisch-waugh-lovell-theorem","status":"fully_proved","summary":"Under the hypotheses of `frischWaughLovell`, if also `y\u22a5 = y - A v` is the weighted residual of `y` on `A` (`A\u1d40 W y\u22a5 = 0`), then `t \u27eaz\u22a5, z\u22a5\u27eb_W = \u27eaz\u22a5, y\u22a5\u27eb_W`: the coefficient of `z` in the full regression is the coefficient of the regression of the residualized response on the residualized regressor.","title":"Frisch\u2013Waugh\u2013Lovell, residualized form"},{"id":"Probability/linear-models-multivariate/frisch-waugh-lovell-theorem/frisch-waugh-lovell-eq-div","kind":"node","lean":"Matrix.frischWaughLovell_eq_div","parent":21328,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/frisch-waugh-lovell-theorem/index.html#frisch-waugh-lovell-eq-div","scope":"Probability/linear-models-multivariate/frisch-waugh-lovell-theorem","status":"fully_proved","summary":"Under the hypotheses of `frischWaughLovell_residual`, if `\u27eaz\u22a5, z\u22a5\u27eb_W \u2260 0` then `t = \u27eaz\u22a5, y\u22a5\u27eb_W / \u27eaz\u22a5, z\u22a5\u27eb_W`.","title":"Frisch\u2013Waugh\u2013Lovell, solved form"},{"data_url":"scopes/Probability/linear-models-multivariate/gaussian-graphical-models.json","id":"Probability/linear-models-multivariate/gaussian-graphical-models","kind":"scope","navigation":"module","parent":21311,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","title":"Gaussian graphical models"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/inv-to-blocks-inv","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.inv_toBlocks\u2081\u2081_inv","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#inv-to-blocks-inv","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"For positive definite `S` on `\u03b9 \u2295 \u03ba`, `S\u2081\u2081 - S\u2081\u2082 S\u2082\u2082\u207b\u00b9 S\u2082\u2081 = ((S\u207b\u00b9)\u2081\u2081)\u207b\u00b9`.","title":"The conditional covariance is the inverse of a precision block"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/partial-correlation","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.partialCorrelation","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#partial-correlation","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"The partial correlation of coordinates `0` and `1` given the `\u03ba`-block, computed from the conditional covariance `C = S\u2081\u2081 - S\u2081\u2082 S\u2082\u2082\u207b\u00b9 S\u2082\u2081`: `C\u2080\u2081 / \u221a(C\u2080\u2080 C\u2081\u2081)`.","title":"Partial correlation from the conditional covariance"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/partial-correlation-eq","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.partialCorrelation_eq","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#partial-correlation-eq","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"The partial correlation from the precision matrix: `\u03c1\u2081\u2082\u00b7rest = -\u03a9\u2080\u2081 / \u221a(\u03a9\u2080\u2080 \u03a9\u2081\u2081)`.","title":"Partial correlation from the precision matrix"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/rest","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.rest","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#rest","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"The `\u03ba`-block (the conditioning coordinates) of a vector on `Fin 2 \u2295 \u03ba`.","title":"Conditioning block of a vector"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/measurable-rest","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.measurable_rest","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#measurable-rest","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"The `\u03ba`-block is a measurable function of the vector.","title":"Measurability of the conditioning block"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/cond-indep-fun-iff-inv-apply-eq-zero","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.condIndepFun_iff_inv_apply_eq_zero","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#cond-indep-fun-iff-inv-apply-eq-zero","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"If `V = (X\u2080, X\u2081, Y) ~ N(\u03bc, S)` with `S` positive definite, then `X\u2080` and `X\u2081` are conditionally independent given `Y` iff `(S\u207b\u00b9)\u2080\u2081 = 0`. (The standard Borel sample space is the setting of Mathlib's conditional independence.)","title":"Pairwise Markov property of Gaussian vectors"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/sample-cov","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.sampleCov","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#sample-cov","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"The sample second-moment matrix `m\u207b\u00b9 \u2211\u209b x\u209b x\u209b\u1d40` of `m` observations in `\u211d^\u03b9`.","title":"Sample second-moment matrix"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/sum-dot-product-mul-vec-eq","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.sum_dotProduct_mulVec_eq","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#sum-dot-product-mul-vec-eq","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"The quadratic forms of a sample add up to `m \u00b7 tr (S K)`.","title":"Sum of sample quadratic forms as a trace"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/gauss-log-lik","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.gaussLogLik","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#gauss-log-lik","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"The Gaussian log-likelihood objective `log det K - tr (S K)` of a precision matrix `K`.","title":"Gaussian log-likelihood of a precision matrix"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/is-supported-on","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.IsSupportedOn","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#is-supported-on","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"A matrix vanishes off the edges of `G`: `K i j = 0` for distinct non-adjacent `i, j`.","title":"Matrix supported on the edges of a graph"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/gauss-log-lik-le-of-inv-eq-on-graph","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.gaussLogLik_le_of_inv_eq_on_graph","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#gauss-log-lik-le-of-inv-eq-on-graph","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"Dempster's covariance-selection MLE (sufficiency): a positive definite `K\u0302` supported on `G` whose inverse agrees with `S` on the diagonal and the edges maximizes `log det K - tr (S K)` over positive definite `K` supported on `G`.","title":"Dempster's covariance-selection MLE"},{"id":"Probability/linear-models-multivariate/gaussian-graphical-models/eq-of-gauss-log-lik-eq","kind":"node","lean":"MathlibExt.Probability.Statistics.GaussianGraphicalModel.eq_of_gaussLogLik_eq","parent":21332,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-graphical-models/index.html#eq-of-gauss-log-lik-eq","scope":"Probability/linear-models-multivariate/gaussian-graphical-models","status":"fully_proved","summary":"Under the hypotheses of `gaussLogLik_le_of_inv_eq_on_graph`, a positive definite `K` supported on `G` with the same objective value is `K\u0302`.","title":"Uniqueness of Dempster's MLE"},{"data_url":"scopes/Probability/linear-models-multivariate/generalized-least-squares.json","id":"Probability/linear-models-multivariate/generalized-least-squares","kind":"scope","navigation":"module","parent":21311,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/generalized-least-squares/index.html","scope":"Probability/linear-models-multivariate/generalized-least-squares","status":"fully_proved","title":"Generalized least squares"},{"id":"Probability/linear-models-multivariate/generalized-least-squares/gls-matrix","kind":"node","lean":"MathlibExt.Probability.Statistics.GeneralizedLeastSquares.glsMatrix","parent":21345,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/generalized-least-squares/index.html#gls-matrix","scope":"Probability/linear-models-multivariate/generalized-least-squares","status":"fully_proved","summary":"The generalized least-squares matrix `(X\u1d40 V\u207b\u00b9 X)\u207b\u00b9 X\u1d40 V\u207b\u00b9`; `glsMatrix X V *\u1d65 y` is the GLS estimate of `b` in `y = X b + e` with `Cov e = V`.","title":"Generalized least-squares matrix"},{"id":"Probability/linear-models-multivariate/generalized-least-squares/gls-matrix-mul-self","kind":"node","lean":"MathlibExt.Probability.Statistics.GeneralizedLeastSquares.glsMatrix_mul_self","parent":21345,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/generalized-least-squares/index.html#gls-matrix-mul-self","scope":"Probability/linear-models-multivariate/generalized-least-squares","status":"fully_proved","summary":"The GLS estimator is unbiased: `B X = 1` when `X\u1d40 V\u207b\u00b9 X` is invertible.","title":"Unbiasedness of the generalized least-squares estimator"},{"id":"Probability/linear-models-multivariate/generalized-least-squares/gls-matrix-mul-mul-transpose","kind":"node","lean":"MathlibExt.Probability.Statistics.GeneralizedLeastSquares.glsMatrix_mul_mul_transpose","parent":21345,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/generalized-least-squares/index.html#gls-matrix-mul-mul-transpose","scope":"Probability/linear-models-multivariate/generalized-least-squares","status":"fully_proved","summary":"The covariance of the GLS estimator: `B V B\u1d40 = (X\u1d40 V\u207b\u00b9 X)\u207b\u00b9`.","title":"Covariance of the generalized least-squares estimator"},{"id":"Probability/linear-models-multivariate/generalized-least-squares/aitken","kind":"node","lean":"MathlibExt.Probability.Statistics.GeneralizedLeastSquares.aitken","parent":21345,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/generalized-least-squares/index.html#aitken","scope":"Probability/linear-models-multivariate/generalized-least-squares","status":"fully_proved","summary":"The GLS estimator has minimal covariance among linear unbiased estimators: for every `A` with `A X = 1`, `A V A\u1d40 - B V B\u1d40` is positive semidefinite, where `B = glsMatrix X V`.","title":"Aitken's theorem"},{"data_url":"scopes/Probability/linear-models-multivariate/principal-component-analysis.json","id":"Probability/linear-models-multivariate/principal-component-analysis","kind":"scope","navigation":"module","parent":21311,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","title":"Principal component analysis"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/col-mean","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.colMean","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#col-mean","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The sample mean of column `j` of an `n \u00d7 p` data matrix (rows are observations). For `n = 0` it is the junk value `0`.","title":"Sample mean of a data matrix column"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/center","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.center","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#center","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The column-centered data matrix `X i j - x\u0304\u2c7c`.","title":"Column-centered data matrix"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sample-cov","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sampleCov","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sample-cov","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The unbiased sample covariance matrix `(n - 1)\u207b\u00b9 (X - x\u0304)\u1d40 (X - x\u0304)` of an `n \u00d7 p` data matrix. The divisor needs `2 \u2264 n`; for `n = 1` the factor is the junk value `(0 : \u211d)\u207b\u00b9 = 0` (and for `n = 0` the product is the zero matrix).","title":"Unbiased sample covariance matrix"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/scores","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.scores","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#scores","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The principal-component scores `(X - x\u0304) w` of the observations along a direction `w`.","title":"Principal component scores"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sum-center","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sum_center","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sum-center","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The centered columns sum to zero (for at least one observation).","title":"Centered columns sum to zero"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sample-cov-apply","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sampleCov_apply","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sample-cov-apply","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"Entries of the sample covariance matrix: `S\u2c7c\u2096 = (n - 1)\u207b\u00b9 \u2211\u1d62 (X\u1d62\u2c7c - x\u0304\u2c7c)(X\u1d62\u2096 - x\u0304\u2096)`.","title":"Entries of the sample covariance matrix"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sample-cov-is-hermitian","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sampleCov_isHermitian","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sample-cov-is-hermitian","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The sample covariance matrix is symmetric.","title":"Sample covariance matrix is symmetric"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sample-cov-pos-semidef","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sampleCov_posSemidef","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sample-cov-pos-semidef","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The sample covariance matrix is positive semidefinite (for every `n`).","title":"Sample covariance matrix is positive semidefinite"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/trace-sample-cov","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.trace_sampleCov","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#trace-sample-cov","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The trace of the sample covariance matrix is the total sample variance, the sum of the column sample variances.","title":"Trace of the sample covariance is the total sample variance"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sum-scores","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sum_scores","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sum-scores","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The scores along any direction have sample mean zero.","title":"Principal component scores have sample mean zero"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sample-var-scores","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sampleVar_scores","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sample-var-scores","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The sample variance of the scores along `w` is the quadratic form `w\u1d40 S w`.","title":"Sample variance of scores is a quadratic form"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/ordered-eigenvalues","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.orderedEigenvalues","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#ordered-eigenvalues","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The eigenvalues `\u03bb\u2080 \u2265 \u03bb\u2081 \u2265 \u22ef \u2265 \u03bb_{p-1}` of a real symmetric `p \u00d7 p` matrix, in decreasing order (Mathlib's `LinearMap.IsSymmetric.eigenvalues` of `Matrix.toEuclideanLin S`).","title":"Eigenvalues of a symmetric matrix in decreasing order"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/ordered-eigenbasis","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.orderedEigenbasis","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#ordered-eigenbasis","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"An orthonormal eigenbasis of a real symmetric matrix, ordered to match `orderedEigenvalues`; for a covariance matrix these are the principal axes.","title":"Ordered orthonormal eigenbasis of a symmetric matrix"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/rayleigh","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.rayleigh","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#rayleigh","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The Rayleigh quotient `w\u1d40 S w / w\u1d40 w` of a real matrix (junk value `0` at `w = 0`).","title":"Rayleigh quotient of a matrix"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/ordered-eigenvalues-antitone","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.orderedEigenvalues_antitone","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#ordered-eigenvalues-antitone","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The ordered eigenvalues are decreasing.","title":"Ordered eigenvalues are decreasing"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/mul-vec-ordered-eigenbasis","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.mulVec_orderedEigenbasis","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#mul-vec-ordered-eigenbasis","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The `k`-th basis vector is an eigenvector for the `k`-th eigenvalue.","title":"Ordered eigenbasis vectors are eigenvectors"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/dot-product-mul-vec-eq-sum","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.dotProduct_mulVec_eq_sum","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#dot-product-mul-vec-eq-sum","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The quadratic form expanded in the ordered eigenbasis: `x\u1d40 S x = \u2211 \u03bb\u1d62 \u27eav\u1d62, x\u27eb\u00b2`.","title":"Quadratic form expanded in the ordered eigenbasis"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sum-ordered-eigenvalues","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sum_orderedEigenvalues","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sum-ordered-eigenvalues","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The ordered eigenvalues sum to the trace.","title":"Ordered eigenvalues sum to the trace"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/ordered-eigenvalues-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.orderedEigenvalues_nonneg","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#ordered-eigenvalues-nonneg","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The eigenvalues of a positive semidefinite matrix are nonnegative.","title":"Eigenvalues of a positive semidefinite matrix are nonnegative"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/rayleigh-ordered-eigenbasis","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.rayleigh_orderedEigenbasis","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#rayleigh-ordered-eigenbasis","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The Rayleigh quotient of the `k`-th eigenvector is the `k`-th eigenvalue.","title":"Rayleigh quotient of an eigenvector is its eigenvalue"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/rayleigh-le-of-orthogonal","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.rayleigh_le_of_orthogonal","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#rayleigh-le-of-orthogonal","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"On nonzero directions orthogonal to the first `k` eigenvectors, the Rayleigh quotient is at most the `k`-th eigenvalue; the bound is attained at the `k`-th eigenvector (`rayleigh_orderedEigenbasis`).","title":"Hotelling's characterization of the principal axes"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/le-rayleigh-of-orthogonal","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.le_rayleigh_of_orthogonal","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#le-rayleigh-of-orthogonal","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"Dual of `rayleigh_le_of_orthogonal`: on nonzero directions orthogonal to the eigenvectors after index `k`, the Rayleigh quotient is at least the `k`-th eigenvalue.","title":"Lower bound on the Rayleigh quotient for orthogonal directions"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/exists-le-rayleigh-of-finrank","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.exists_le_rayleigh_of_finrank","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#exists-le-rayleigh-of-finrank","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"Every subspace of dimension `p - k` contains a nonzero vector whose Rayleigh quotient is at least `\u03bb\u2096`.","title":"Courant\u2013Fischer min\u2013max, matrix form"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/exists-rayleigh-le-of-finrank","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.exists_rayleigh_le_of_finrank","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#exists-rayleigh-le-of-finrank","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"Every subspace of dimension `k + 1` contains a nonzero vector whose Rayleigh quotient is at most `\u03bb\u2096`.","title":"Courant\u2013Fischer max\u2013min, matrix form"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/variance-explained","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.varianceExplained","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#variance-explained","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The proportion of total sample variance explained by the first `k` principal components: `(\u03bb\u2080 + \u22ef + \u03bb_{k-1}) / tr S` for the sample covariance matrix `S`. The junk value `0` is returned when the total variance `tr S` is `0` (constant data).","title":"Proportion of variance explained by principal components"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/variance-explained-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.varianceExplained_nonneg","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#variance-explained-nonneg","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The proportion of variance explained is nonnegative.","title":"Proportion of variance explained is nonnegative"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/variance-explained-le-one","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.varianceExplained_le_one","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#variance-explained-le-one","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The proportion of variance explained is at most one.","title":"Proportion of variance explained is at most one"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/variance-explained-mono","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.varianceExplained_mono","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#variance-explained-mono","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"Keeping more components explains more variance.","title":"More components explain more variance"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/variance-explained-of-le","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.varianceExplained_of_le","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#variance-explained-of-le","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"All `p` components explain all of the variance, when the total variance is positive.","title":"All components explain all of the variance"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/right-singular-basis","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.rightSingularBasis","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#right-singular-basis","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"Right singular vectors of `A`: an orthonormal eigenbasis of `A\u1d40 A`, ordered so that the `i`-th vector has eigenvalue `\u03c3\u1d62\u00b2` with `\u03c3\u2080 \u2265 \u03c3\u2081 \u2265 \u22ef` the singular values.","title":"Right singular vectors of a matrix"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/trunc-svd","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.truncSVD","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#trunc-svd","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The rank-`k` truncated singular value decomposition `A V\u2096 V\u2096\u1d40 = \u2211_{i<k} \u03c3\u1d62 u\u1d62 v\u1d62\u1d40`, where `V\u2096` collects the first `k` right singular vectors.","title":"Truncated singular value decomposition"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/rank-trunc-svd-le","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.rank_truncSVD_le","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#rank-trunc-svd-le","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The truncated SVD has rank at most `k`.","title":"Rank of the truncated SVD is at most k"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/frobenius-sub-trunc-svd-sq","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.frobenius_sub_truncSVD_sq","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#frobenius-sub-trunc-svd-sq","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The squared Frobenius error of the truncated SVD is the sum of the discarded squared singular values.","title":"Frobenius error of the truncated SVD"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/sum-sq-singular-values-le-frobenius-sub","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.sum_sq_singularValues_le_frobenius_sub_sq","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#sum-sq-singular-values-le-frobenius-sub","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"Every matrix of rank at most `k` has squared Frobenius distance from `A` at least the sum of the discarded squared singular values.","title":"Eckart\u2013Young (Frobenius norm)"},{"id":"Probability/linear-models-multivariate/principal-component-analysis/is-least-frobenius-sub-sq","kind":"node","lean":"MathlibExt.Probability.Statistics.PCA.isLeast_frobenius_sub_sq","parent":21350,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/principal-component-analysis/index.html#is-least-frobenius-sub-sq","scope":"Probability/linear-models-multivariate/principal-component-analysis","status":"fully_proved","summary":"The truncated SVD is a best rank-`k` approximation of `A` in the Frobenius norm: the least squared distance from `A` to a matrix of rank at most `k` is `\u2211_{k \u2264 i < n} \u03c3\u1d62\u00b2`, attained by `truncSVD A k`.","title":"Eckart\u2013Young, best-approximation form"},{"data_url":"scopes/Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions.json","id":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions","kind":"scope","navigation":"module","parent":21311,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/index.html","scope":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions","status":"can_prove","title":"Student's t and Snedecor's F distributions"},{"id":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/map-div-sqrt-chi-square-eq-student-t","kind":"node","lean":"ProbabilityTheory.map_div_sqrt_chiSquare_eq_studentT","parent":21386,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/index.html#map-div-sqrt-chi-square-eq-student-t","scope":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions","status":"fully_proved","summary":"Student's `t` law (Casella\u2013Berger, Definition 5.3.4): for independent `Z ~ N(0, 1)` and `V ~ \u03c7\u00b2_k` with `k \u2265 1`, `Z / \u221a(V / k)` has density `\u0393((k + 1)/2) / (\u221a(k \u03c0) \u0393(k/2)) \u00b7 (1 + t\u00b2/k)^{-(k+1)/2}`.","title":"Student's `t` law"},{"id":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/tendsto-in-distribution-u-statistic","kind":"node","lean":"ProbabilityTheory.NormalTheoryWanted.tendstoInDistribution_uStatistic","parent":21386,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/index.html#tendsto-in-distribution-u-statistic","scope":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions","status":"can_prove","summary":"Hoeffding's central limit theorem (1948, \u00a77): for an i.i.d. sample from `\u03bc` and a symmetric square-integrable kernel `h` of degree `m`, `\u221an (U_n - \u03b8)` converges in distribution to `N(0, m\u00b2 \u03c3\u2081\u00b2)`, where `\u03b8 = E h` and `\u03c3\u2081\u00b2 = Var h\u2081`.","title":"Hoeffding's central limit theorem"},{"id":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/map-div-chi-square-eq-snedecor-f","kind":"node","parent":21386,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/map-div-chi-square-eq-snedecor-f.html","scope":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions","status":"can_state","summary":"Snedecor's `F` law (Casella\u2013Berger, Definition 5.3.6): for independent `V\u2081 ~ \u03c7\u00b2_{d\u2081}` and `V\u2082 ~ \u03c7\u00b2_{d\u2082}` with `d\u2081, d\u2082 \u2265 1`, `(V\u2081 / d\u2081) / (V\u2082 / d\u2082)` has density `\u0393((d\u2081 + d\u2082)/2) / (\u0393(d\u2081/2) \u0393(d\u2082/2)) \u00b7 (d\u2081/d\u2082)^{d\u2081/2} x^{d\u2081/2 - 1} (1 + d\u2081 x / d\u2082)^{-(d\u2081+d\u2082)/2}` on `x > 0`.","title":"Snedecor's `F` law"},{"id":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/map-t-statistic-pi-gaussian-real","kind":"node","parent":21386,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions/map-t-statistic-pi-gaussian-real.html","scope":"Probability/linear-models-multivariate/student-s-t-and-snedecor-s-f-distributions","status":"can_state","summary":"For an i.i.d. sample `X\u2081, \u2026, X\u2099 ~ N(\u03bc, v)` with `n \u2265 2` and `v > 0`, `\u221an (X\u0304 - \u03bc) / S` with `S\u00b2 = \u2211 (X\u1d62 - X\u0304)\u00b2 / (n - 1)` has the Student `t` law with `n - 1` degrees of freedom.","title":"One-sample `t` statistic"},{"data_url":"scopes/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices.json","id":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","kind":"scope","navigation":"module","parent":21311,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/index.html","scope":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","status":"fully_proved","title":"Weighted means and scatter matrices"},{"id":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/weighted-mean","kind":"node","lean":"MetaMathlibExt.weightedMean","parent":21391,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/index.html#weighted-mean","scope":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","status":"fully_proved","summary":"Let $I$ be a finite set and $J$ an index set, and let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$. Their weighted mean is the point $\\big(\\sum_{i \\in I} w_i\\big)^{-1} \\sum_{i \\in I} w_i x_i$ of $\\mathbb{R}^J$, that is $\\sum_i w_i x_i / \\sum_i w_i$, with the convention $0^{-1} = 0$ when the weights sum to $0$.","title":"Weighted mean"},{"id":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/weighted-scatter","kind":"node","lean":"MetaMathlibExt.weightedScatter","parent":21391,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/index.html#weighted-scatter","scope":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","status":"fully_proved","summary":"Let $I$ be a finite set and $J$ an index set, let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$, and let $\\mu \\in \\mathbb{R}^J$. The weighted scatter matrix of the $x_i$ about $\\mu$ is the real $J \\times J$ matrix","title":"Weighted scatter matrix"},{"id":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/sum-smul-sub-weighted-mean","kind":"node","lean":"MetaMathlibExt.sum_smul_sub_weightedMean","parent":21391,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/index.html#sum-smul-sub-weighted-mean","scope":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","status":"fully_proved","summary":"Let $I$ be a finite set and $J$ any index set, let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$ such that $\\sum_i w_i \\ne 0$, and let $m$ be their weighted mean. Then $\\sum_{i \\in I} w_i (x_i - m) = 0$.","title":"Weighted residuals sum to zero"},{"id":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/sum-mul-quad-sub-eq","kind":"node","lean":"MetaMathlibExt.sum_mul_quad_sub_eq","parent":21391,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/index.html#sum-mul-quad-sub-eq","scope":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","status":"fully_proved","summary":"Let $I$ and $J$ be finite sets, let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$ such that $\\sum_i w_i \\ne 0$, let $m$ be their weighted mean, let $A$ be a symmetric real $J \\times J$ matrix, and let $\\mu \\in \\mathbb{R}^J$. Then","title":"The bias-variance split of a weighted quadratic form"},{"id":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/sum-mul-quad-eq-trace","kind":"node","lean":"MetaMathlibExt.sum_mul_quad_eq_trace","parent":21391,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/index.html#sum-mul-quad-eq-trace","scope":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","status":"fully_proved","summary":"Let $I$ and $J$ be finite sets, let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$ such that $\\sum_i w_i \\ne 0$, let $A$ be a real $J \\times J$ matrix, let $\\mu \\in \\mathbb{R}^J$, and let $C$ be the weighted scatter matrix of the $x_i$ about $\\mu$. Then","title":"Weighted quadratic forms as traces against the scatter matrix"},{"id":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/dot-product-weighted-scatter-mul-vec","kind":"node","lean":"MetaMathlibExt.dotProduct_weightedScatter_mulVec","parent":21391,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/index.html#dot-product-weighted-scatter-mul-vec","scope":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","status":"fully_proved","summary":"Let $I$ and $J$ be finite sets, let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$, let $\\mu \\in \\mathbb{R}^J$, and let $C$ be the weighted scatter matrix of the $x_i$ about $\\mu$. Then for every $u \\in \\mathbb{R}^J$,","title":"Quadratic form of a weighted scatter matrix"},{"id":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/weighted-scatter-pos-def","kind":"node","lean":"MetaMathlibExt.weightedScatter_posDef","parent":21391,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/weighted-means-and-scatter-matrices/index.html#weighted-scatter-pos-def","scope":"Probability/linear-models-multivariate/weighted-means-and-scatter-matrices","status":"fully_proved","summary":"Let $I$ and $J$ be finite sets and let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with positive weights $w_i > 0$ that do not all lie in an affine hyperplane: for every nonzero $v \\in \\mathbb{R}^J$ there are $i, j \\in I$ with $\\langle v, x_i \\rangle \\ne \\langle v, x_j \\rangle$. Then for every $\\mu \\in \\mathbb{R}^J$ the weighted scatter matrix of the $x_i$ about $\\mu$ is positive definite.","title":"Weighted scatter matrices of points in general position are positive definite"},{"data_url":"scopes/Probability/linear-models-multivariate/gaussian-maximum-likelihood.json","id":"Probability/linear-models-multivariate/gaussian-maximum-likelihood","kind":"scope","navigation":"module","parent":21311,"result":false,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-maximum-likelihood/index.html","scope":"Probability/linear-models-multivariate/gaussian-maximum-likelihood","status":"fully_proved","title":"Gaussian maximum likelihood"},{"id":"Probability/linear-models-multivariate/gaussian-maximum-likelihood/log-multivariate-gaussian-pdfreal","kind":"node","lean":"MetaMathlibExt.log_multivariateGaussianPDFReal","parent":21399,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-maximum-likelihood/index.html#log-multivariate-gaussian-pdfreal","scope":"Probability/linear-models-multivariate/gaussian-maximum-likelihood","status":"fully_proved","summary":"Let $J$ be a finite set with $n$ elements, let $\\mu, x \\in \\mathbb{R}^J$, let $S$ be a positive definite real $J \\times J$ matrix, and write $\\mathcal{N}(x \\mid \\mu, S)$ for the multivariate Gaussian density at $x$ with mean $\\mu$ and covariance $S$. Then $\\log \\mathcal{N}(x \\mid \\mu, S) = -\\frac{n}{2} \\log (2\\pi) - \\frac{1}{2} \\log \\det S - \\frac{1}{2} (x - \\mu)^{\\mathsf T} S^{-1} (x - \\mu)$.","title":"Log-density of the multivariate Gaussian"},{"id":"Probability/linear-models-multivariate/gaussian-maximum-likelihood/sum-mul-log-multivariate-gaussian-pdfrea","kind":"node","lean":"MetaMathlibExt.sum_mul_log_multivariateGaussianPDFReal_eq","parent":21399,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-maximum-likelihood/index.html#sum-mul-log-multivariate-gaussian-pdfrea","scope":"Probability/linear-models-multivariate/gaussian-maximum-likelihood","status":"fully_proved","summary":"Let $I$ be a finite set and $J$ a finite set with $n$ elements, let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$ whose sum $W = \\sum_{i \\in I} w_i$ is nonzero, let $\\mu \\in \\mathbb{R}^J$, let $S$ be a positive definite real $J \\times J$ matrix, and let $C(\\mu)$ be the weighted scatter matrix of the $x_i$ about $\\mu$. Write $\\mathcal{N}(x \\mid \\mu, S)$ for the multivariate Gaussian density at\u2026","title":"The weighted Gaussian log-likelihood via the scatter matrix"},{"id":"Probability/linear-models-multivariate/gaussian-maximum-likelihood/sum-mul-log-multivariate-gaussian-pdfrea-2","kind":"node","lean":"MetaMathlibExt.sum_mul_log_multivariateGaussianPDFReal_le_weightedMean","parent":21399,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-maximum-likelihood/index.html#sum-mul-log-multivariate-gaussian-pdfrea-2","scope":"Probability/linear-models-multivariate/gaussian-maximum-likelihood","status":"fully_proved","summary":"Let $I$ and $J$ be finite sets, let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$ such that $\\sum_i w_i > 0$, let $m$ be their weighted mean, and let $S$ be a positive definite real $J \\times J$ matrix. Write $\\mathcal{N}(x \\mid \\mu, S)$ for the multivariate Gaussian density at $x$ with mean $\\mu$ and covariance $S$. Then for every $\\mu \\in \\mathbb{R}^J$,\u2026","title":"The weighted mean is the Gaussian MLE of the mean"},{"id":"Probability/linear-models-multivariate/gaussian-maximum-likelihood/sum-mul-log-multivariate-gaussian-pdfrea-3","kind":"node","lean":"MetaMathlibExt.sum_mul_log_multivariateGaussianPDFReal_le_of_posDef","parent":21399,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-maximum-likelihood/index.html#sum-mul-log-multivariate-gaussian-pdfrea-3","scope":"Probability/linear-models-multivariate/gaussian-maximum-likelihood","status":"fully_proved","summary":"Let $I$ and $J$ be finite sets, let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with real weights $w_i$ such that $\\sum_i w_i > 0$, let $m$ be their weighted mean, and let $C$ be their weighted scatter matrix about $m$ (the weighted sample covariance); assume $C$ is positive definite. Write $\\mathcal{N}(x \\mid \\mu, S)$ for the multivariate Gaussian density at $x$ with mean $\\mu$ and covariance $S$. Then for every\u2026","title":"The weighted Gaussian MLE when the sample covariance is positive definite"},{"id":"Probability/linear-models-multivariate/gaussian-maximum-likelihood/sum-mul-log-multivariate-gaussian-pdfrea-4","kind":"node","lean":"MetaMathlibExt.sum_mul_log_multivariateGaussianPDFReal_le","parent":21399,"result":true,"route":"../roadmap/Probability/linear-models-multivariate/gaussian-maximum-likelihood/index.html#sum-mul-log-multivariate-gaussian-pdfrea-4","scope":"Probability/linear-models-multivariate/gaussian-maximum-likelihood","status":"fully_proved","summary":"Let $I$ be a nonempty finite set and $J$ a finite set, and let $x_i \\in \\mathbb{R}^J$ ($i \\in I$) be points with positive weights $w_i > 0$ that do not all lie in an affine hyperplane: for every nonzero $v \\in \\mathbb{R}^J$ there are $i, j \\in I$ with $\\langle v, x_i \\rangle \\ne \\langle v, x_j \\rangle$. Let $m$ be their weighted mean and $C$ their weighted scatter matrix about $m$ (the weighted sample covariance),\u2026","title":"The weighted Gaussian MLE for points in general position"},{"data_url":"scopes/Probability/statistical-estimation.json","id":"Probability/statistical-estimation","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/statistical-estimation/index.html","scope":"Probability/statistical-estimation","status":"can_prove","summary":"Point estimation theory: bias, risk and mean squared error, likelihood and maximum likelihood (consistency and asymptotic normality), Fisher information and Cram\u00e9r\u2013Rao, sufficiency, factorization (Fisher\u2013Neyman, Halmos\u2013Savage) and Rao\u2013Blackwell, Stein's lemma and James\u2013Stein shrinkage, and contiguity and local asymptotic normality.","title":"Statistical estimation"},{"data_url":"scopes/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators.json","id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","kind":"scope","navigation":"module","parent":21405,"result":false,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","title":"Bias and mean squared error of estimators"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/bias","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.bias","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#bias","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $\\mu$ a probability measure on $S$ for which $W$ is integrable, and $\\theta \\in \\mathbb{R}$ a target. The bias of $W$ under $\\mu$ at $\\theta$ is $\\int_S W \\, d\\mu - \\theta$, the expected value of $W$ under $\\mu$ minus $\\theta$.","title":"Bias of an estimator"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/bias-eq-integral","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.bias_eq_integral","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#bias-eq-integral","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $\\mu$ a probability measure on $S$ for which $W$ is integrable, and $\\theta \\in \\mathbb{R}$ a target. Then the bias of $W$ under $\\mu$ at $\\theta$ equals $\\int_S W \\, d\\mu - \\theta$.","title":"Bias as an integral"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/mean-squared-error","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.meanSquaredError","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#mean-squared-error","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, let $W : S \\to \\mathbb{R}$ be an estimator and $\\theta \\in \\mathbb{R}$ a target value, and suppose the squared error $x \\mapsto (W(x) - \\theta)^2$ is $\\mu$-integrable. The mean squared error of $W$ at $\\theta$ under $\\mu$ is the real number $\\int_S (W(x) - \\theta)^2 \\, d\\mu(x)$.","title":"Mean squared error"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/mean-squared-error-eq-integral","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.meanSquaredError_eq_integral","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#mean-squared-error-eq-integral","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, let $W : S \\to \\mathbb{R}$ and $\\theta \\in \\mathbb{R}$, and suppose the squared error $x \\mapsto (W(x) - \\theta)^2$ is $\\mu$-integrable. Then the mean squared error of $W$ at $\\theta$ under $\\mu$ equals $\\int_S (W(x) - \\theta)^2 \\, d\\mu(x)$.","title":"Mean squared error as an integral"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/mean-squared-error-enn","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.meanSquaredErrorENN","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#mean-squared-error-enn","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, let $W : S \\to \\mathbb{R}$ be an estimator and let $\\theta \\in \\mathbb{R}$ be a target value. The extended mean squared error of $W$ at $\\theta$ under $\\mu$ is the (lower) Lebesgue integral $\\int_S (W(x) - \\theta)^2 \\, d\\mu(x)$ of the nonnegative squared error, computed in $[0, \\infty]$. It is defined for every $W$, with no integrability hypothesis.","title":"Extended mean squared error"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/of-real-mean-squared-error","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.ofReal_meanSquaredError","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#of-real-mean-squared-error","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, let $W : S \\to \\mathbb{R}$ and $\\theta \\in \\mathbb{R}$, and suppose the squared error $x \\mapsto (W(x) - \\theta)^2$ is $\\mu$-integrable. Then the extended mean squared error $\\int_S (W(x) - \\theta)^2 \\, d\\mu(x)$, computed in $[0, \\infty]$, is finite and equals the mean squared error of $W$ at $\\theta$ under $\\mu$.","title":"Finite and extended mean squared error agree"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/mean-squared-error-enn-eq-top-of-not-fin","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.meanSquaredErrorENN_eq_top_of_not_finite","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#mean-squared-error-enn-eq-top-of-not-fin","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, $W : S \\to \\mathbb{R}$ an estimator and $\\theta \\in \\mathbb{R}$. If the squared error $x \\mapsto (W(x) - \\theta)^2$ is almost everywhere strongly measurable with respect to $\\mu$ but not $\\mu$-integrable, then the extended mean squared error $\\int_S (W(x) - \\theta)^2 \\, d\\mu(x) \\in [0, \\infty]$ of $W$ at $\\theta$ equals $\\infty$.","title":"Extended mean squared error of a measurable non-integrable squared error is infinite"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/family-bias","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.familyBias","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#family-bias","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$ such that $W$ is integrable for every $P_\\theta$, and $\\tau : \\Theta \\to \\mathbb{R}$ a target function. The family bias of $W$ is the function on $\\Theta$ whose value at $\\theta$ is the bias of $W$ under $P_\\theta$ at the target $\\tau(\\theta)$, that is,\u2026","title":"Family bias of an estimator"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/family-bias-apply","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.familyBias_apply","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#family-bias-apply","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$ such that $W$ is integrable for every $P_\\theta$, and $\\tau : \\Theta \\to \\mathbb{R}$ a target function. Then for every $\\theta \\in \\Theta$ the family bias of $W$ at $\\theta$ equals $\\int_S W \\, dP_\\theta - \\tau(\\theta)$.","title":"Family bias as an integral"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/family-mean-squared-error","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.familyMeanSquaredError","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#family-mean-squared-error","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$ and $\\tau : \\Theta \\to \\mathbb{R}$ a target function such that $(W - \\tau(\\theta))^2$ is integrable for $P_\\theta$ for every $\\theta \\in \\Theta$. The family mean squared error of $W$ is the function on $\\Theta$ whose value at $\\theta$ is the mean squared error of $W$ under\u2026","title":"Family mean squared error of an estimator"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/family-mean-squared-error-apply","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.familyMeanSquaredError_apply","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#family-mean-squared-error-apply","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$ and $\\tau : \\Theta \\to \\mathbb{R}$ a target function such that $(W - \\tau(\\theta))^2$ is integrable for $P_\\theta$ for every $\\theta \\in \\Theta$. Then for every $\\theta \\in \\Theta$ the family mean squared error of $W$ at $\\theta$ equals\u2026","title":"Family mean squared error as an integral"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/is-unbiased","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.IsUnbiased","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#is-unbiased","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $(P_\\theta)_{\\theta \\in \\Theta}$ be a family of probability measures on a measurable space $S$ and $\\tau : \\Theta \\to \\mathbb{R}$ a target function. An estimator $W : S \\to \\mathbb{R}$ is unbiased for $\\tau$ if for every $\\theta \\in \\Theta$ the function $W$ is $P_\\theta$-integrable and $\\int_S W \\, dP_\\theta = \\tau(\\theta)$.","title":"Unbiased estimator"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/integrable","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.IsUnbiased.integrable","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#integrable","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $(P_\\theta)_{\\theta \\in \\Theta}$ be a family of probability measures on $S$ and $\\tau : \\Theta \\to \\mathbb{R}$ a target function. If an estimator $W : S \\to \\mathbb{R}$ is unbiased for $\\tau$, then $W$ is $P_\\theta$-integrable for every $\\theta \\in \\Theta$.","title":"Unbiased estimators are integrable"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/expect-eq","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.IsUnbiased.expect_eq","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#expect-eq","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$ and $\\tau : \\Theta \\to \\mathbb{R}$ a target function. If $W$ is unbiased for $\\tau$ under the family $(P_\\theta)$, then $\\int_S W \\, dP_\\theta = \\tau(\\theta)$ for every $\\theta \\in \\Theta$.","title":"An unbiased estimator has the target as its expectation"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/is-unbiased-iff-forall-family-bias-eq-ze","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.isUnbiased_iff_forall_familyBias_eq_zero","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#is-unbiased-iff-forall-family-bias-eq-ze","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $(P_\\theta)_{\\theta \\in \\Theta}$ be a family of probability measures on $S$, $\\tau : \\Theta \\to \\mathbb{R}$ a target function, and $W : S \\to \\mathbb{R}$ an estimator that is $P_\\theta$-integrable for every $\\theta \\in \\Theta$. Then $W$ is unbiased for $\\tau$ if and only if its family bias $\\int_S W \\, dP_\\theta - \\tau(\\theta)$ is zero for every $\\theta \\in \\Theta$.","title":"An estimator is unbiased exactly when its bias vanishes"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/integrable-sq-sub-of-mem-lp-two","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.integrable_sq_sub_of_memLp_two","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#integrable-sq-sub-of-mem-lp-two","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $S$ and $W : S \\to \\mathbb{R}$ an estimator in $L^2(\\mu)$. Then for every $\\theta \\in \\mathbb{R}$ the squared error $x \\mapsto (W(x) - \\theta)^2$ is $\\mu$-integrable.","title":"A finite second moment makes the squared error integrable"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/mean-squared-error-eq-variance-add-sq-bi","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.meanSquaredError_eq_variance_add_sq_bias","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#mean-squared-error-eq-variance-add-sq-bi","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, let $\\theta \\in \\mathbb{R}$, and let $W : S \\to \\mathbb{R}$ be square-integrable with respect to $\\mu$, that is, $W \\in L^2(\\mu)$. Then the mean squared error of $W$ at $\\theta$ under $\\mu$ equals $\\operatorname{Var}_\\mu(W) + b^2$, where $b = \\int_S W \\, d\\mu - \\theta$ is the bias of $W$ at $\\theta$ under $\\mu$.","title":"Bias\u2013variance decomposition of the mean squared error"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/mean-squared-error-eq-variance-add-sq-ex","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.meanSquaredError_eq_variance_add_sq_expectation_sub","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#mean-squared-error-eq-variance-add-sq-ex","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, let $\\theta \\in \\mathbb{R}$, and let $W : S \\to \\mathbb{R}$ be square-integrable with respect to $\\mu$, that is, $W \\in L^2(\\mu)$. Then the mean squared error of $W$ at $\\theta$ under $\\mu$ equals $\\operatorname{Var}_\\mu(W) + \\left(\\int_S W \\, d\\mu - \\theta\\right)^2$.","title":"Mean squared error as variance plus squared mean deviation"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/of-real-variance-add-sq-bias","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.ofReal_variance_add_sq_bias","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#of-real-variance-add-sq-bias","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, let $\\theta \\in \\mathbb{R}$, and let $W : S \\to \\mathbb{R}$ be square-integrable with respect to $\\mu$, that is, $W \\in L^2(\\mu)$. Then the extended mean squared error $\\int_S (W(x) - \\theta)^2 \\, d\\mu(x)$, computed in $[0, \\infty]$, is finite and equals $\\operatorname{Var}_\\mu(W) + b^2$, where $b = \\int_S W \\, d\\mu - \\theta$ is the bias of $W$ at\u2026","title":"Bias\u2013variance decomposition of the extended mean squared error"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/family-mean-squared-error-eq-variance-ad","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.familyMeanSquaredError_eq_variance_add_sq_familyBias","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#family-mean-squared-error-eq-variance-ad","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$ and $\\tau : \\Theta \\to \\mathbb{R}$ a target function. Suppose that for every $\\theta \\in \\Theta$, $W$ has a finite second moment under $P_\\theta$, and $W$ and $(W - \\tau(\\theta))^2$ are integrable for $P_\\theta$. Then for every $\\theta \\in \\Theta$ the family mean squared error\u2026","title":"Bias\u2013variance decomposition for a parametric family"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/mean-squared-error-eq-variance-of-expect","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.meanSquaredError_eq_variance_of_expect_eq","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#mean-squared-error-eq-variance-of-expect","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, let $\\theta \\in \\mathbb{R}$, and let $W : S \\to \\mathbb{R}$ be square-integrable with respect to $\\mu$ (that is, $W \\in L^2(\\mu)$) and satisfy $\\int_S W \\, d\\mu = \\theta$. Then the mean squared error of $W$ at $\\theta$ under $\\mu$ equals $\\operatorname{Var}_\\mu(W)$.","title":"Mean squared error of an unbiased estimator"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/family-mean-squared-error-eq-variance","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.IsUnbiased.familyMeanSquaredError_eq_variance","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#family-mean-squared-error-eq-variance","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $W : S \\to \\mathbb{R}$ an estimator, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$ and $\\tau : \\Theta \\to \\mathbb{R}$ a target function. Suppose that $W$ is unbiased for $\\tau$ under the family, and that for every $\\theta \\in \\Theta$, $W$ has a finite second moment under $P_\\theta$ and $(W - \\tau(\\theta))^2$ is integrable for $P_\\theta$. Then for every\u2026","title":"An unbiased estimator has mean squared error equal to its variance"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/frequentist-risk","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.frequentistRisk","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#frequentist-risk","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $\\delta : S \\to A$ a decision rule, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$, and $\\ell : \\Theta \\times A \\to [0, \\infty)$ a loss such that $x \\mapsto \\ell(\\theta, \\delta(x))$ is $P_\\theta$-integrable for every $\\theta \\in \\Theta$. The frequentist risk of $\\delta$ is the function $\\Theta \\to \\mathbb{R}$ given by\u2026","title":"Frequentist risk"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/frequentist-risk-apply","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.frequentistRisk_apply","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#frequentist-risk-apply","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $\\delta : S \\to A$ a decision rule, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$ and $\\ell : \\Theta \\times A \\to [0, \\infty)$ a loss function such that $x \\mapsto \\ell(\\theta, \\delta(x))$ is integrable for $P_\\theta$ for every $\\theta \\in \\Theta$. Then for every $\\theta \\in \\Theta$ the frequentist risk of $\\delta$ at $\\theta$ equals\u2026","title":"Frequentist risk as an integral"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/frequentist-risk-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.frequentistRisk_nonneg","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#frequentist-risk-nonneg","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\delta : S \\to A$ be a decision rule, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$, and $\\ell : \\Theta \\times A \\to [0, \\infty)$ a loss such that $x \\mapsto \\ell(\\theta, \\delta(x))$ is $P_\\theta$-integrable for every $\\theta \\in \\Theta$. Then the frequentist risk $R(\\theta, \\delta)$ of $\\delta$ at $\\theta$ is nonnegative for every $\\theta \\in \\Theta$.","title":"Frequentist risk is nonnegative"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/frequentist-risk-enn","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.frequentistRiskENN","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#frequentist-risk-enn","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $\\delta : S \\to A$ a decision rule, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$, and $\\ell : \\Theta \\times A \\to [0, \\infty)$ a loss. The extended frequentist risk of $\\delta$ at $\\theta \\in \\Theta$ is the Lebesgue integral $\\int_S \\ell(\\theta, \\delta(x)) \\, dP_\\theta(x)$, taken in $[0, \\infty]$.","title":"Extended frequentist risk"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/of-real-frequentist-risk","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.ofReal_frequentistRisk","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#of-real-frequentist-risk","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $\\Theta$ and $A$ be sets, let $(P_\\vartheta)_{\\vartheta \\in \\Theta}$ be a family of probability measures on a measurable space $S$, let $\\delta : S \\to A$ be a decision rule and $\\ell : \\Theta \\times A \\to [0, \\infty)$ a loss, and suppose $x \\mapsto \\ell(\\vartheta, \\delta(x))$ is $P_\\vartheta$-integrable for every $\\vartheta \\in \\Theta$. Then for every $\\theta \\in \\Theta$ the extended frequentist risk\u2026","title":"Finite and extended frequentist risk agree"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/frequentist-risk-enn-eq-top-of-not-integ","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.frequentistRiskENN_eq_top_of_not_integrable","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#frequentist-risk-enn-eq-top-of-not-integ","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $S$ be a measurable space, $\\delta : S \\to A$ a decision rule, $(P_\\theta)_{\\theta \\in \\Theta}$ a family of probability measures on $S$, $\\ell : \\Theta \\times A \\to [0, \\infty)$ a loss, and $\\theta \\in \\Theta$. If the function $x \\mapsto \\ell(\\theta, \\delta(x))$ is almost everywhere strongly measurable with respect to $P_\\theta$ but not $P_\\theta$-integrable, then the extended frequentist risk\u2026","title":"Extended frequentist risk of a measurable non-integrable loss is infinite"},{"id":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/frequentist-risk-squared-eq-family-mean","kind":"node","lean":"MathlibExt.Probability.Statistics.EstimationRisk.frequentistRisk_squared_eq_familyMeanSquaredError","parent":21406,"result":true,"route":"../roadmap/Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators/index.html#frequentist-risk-squared-eq-family-mean","scope":"Probability/statistical-estimation/bias-and-mean-squared-error-of-estimators","status":"fully_proved","summary":"Let $(P_\\theta)_{\\theta \\in \\Theta}$ be a family of probability measures on $S$, $W : S \\to \\mathbb{R}$ an estimator and $\\tau : \\Theta \\to \\mathbb{R}$ a target function such that $x \\mapsto (W(x) - \\tau(\\theta))^2$ is $P_\\theta$-integrable for every $\\theta \\in \\Theta$. Then for every $\\theta$, the frequentist risk at $\\theta$ of $W$, as a decision rule with actions in $\\mathbb{R}$, under the squared-error loss\u2026","title":"Mean squared error is frequentist risk under squared-error loss"},{"data_url":"scopes/Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas.json","id":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas","kind":"scope","navigation":"module","parent":21405,"result":false,"route":"../roadmap/Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/index.html","scope":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas","status":"fully_proved","title":"Contiguity and Le Cam's lemmas"},{"id":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/is-contiguous","kind":"node","lean":"MathlibExt.Probability.Statistics.Contiguity.IsContiguous","parent":21436,"result":true,"route":"../roadmap/Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/index.html#is-contiguous","scope":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas","status":"fully_proved","summary":"The sequence of measures `Q n` is contiguous with respect to `P n` (`Q \u25c1 P`): every sequence of measurable sets whose `P n`-measure tends to `0` has `Q n`-measure tending to `0`.","title":"Contiguity of sequences of measures"},{"id":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/gaussian-real-with-density-exp","kind":"node","lean":"MathlibExt.Probability.Statistics.Contiguity.gaussianReal_withDensity_exp","parent":21436,"result":true,"route":"../roadmap/Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/index.html#gaussian-real-with-density-exp","scope":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas","status":"fully_proved","summary":"The law `N(0, v)` reweighted by the density `exp(h x - h\u00b2 v / 2)` is `N(h v, v)`. For `v = 0` both sides are the Dirac mass at `0`.","title":"Exponential tilting of a centered Gaussian"},{"id":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/is-contiguous-with-density-exp","kind":"node","lean":"MathlibExt.Probability.Statistics.Contiguity.isContiguous_withDensity_exp","parent":21436,"result":true,"route":"../roadmap/Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/index.html#is-contiguous-with-density-exp","scope":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas","status":"fully_proved","summary":"Le Cam's first lemma (LAN case). Let `Q\u2099 = exp(\u039b\u2099) \u00b7 P` be probability measures. If `S\u2099 \u2192 N(0, v)` in distribution under `P` and `\u039b\u2099 - h S\u2099 \u2192 -h\u00b2 v / 2` in `P`-probability, then `Q\u2099` is contiguous with respect to `P`. This is the case of Le Cam's first lemma in which `\u039b\u2099 \u2192 N(-\u03c3\u00b2/2, \u03c3\u00b2)` under `P`, `\u03c3\u00b2 = h\u00b2 v`; the general form (any limit `L` of `dQ\u2099/dP` with `E L = 1`) is not stated here.","title":"Le Cam's first lemma"},{"id":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/tendsto-in-distribution-with-density-exp","kind":"node","lean":"MathlibExt.Probability.Statistics.Contiguity.tendstoInDistribution_withDensity_exp","parent":21436,"result":true,"route":"../roadmap/Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/index.html#tendsto-in-distribution-with-density-exp","scope":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas","status":"fully_proved","summary":"Le Cam's third lemma (LAN case). Let `Q\u2099 = exp(\u039b\u2099) \u00b7 P` be probability measures. If `S\u2099 \u2192 N(0, v)` in distribution under `P` and `\u039b\u2099 - h S\u2099 \u2192 -h\u00b2 v / 2` in `P`-probability, then `S\u2099 \u2192 N(h v, v)` in distribution under `Q\u2099`. This is the case of Le Cam's third lemma for the central sequence itself (covariance `h v` with `\u039b\u2099`); the general form, for an arbitrary statistic jointly asymptotically normal with `\u039b\u2099`, is not\u2026","title":"Le Cam's third lemma"},{"id":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/tendsto-measure-abs-sub-ge","kind":"node","lean":"MathlibExt.Probability.Statistics.Contiguity.IsContiguous.tendsto_measure_abs_sub_ge","parent":21436,"result":true,"route":"../roadmap/Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas/index.html#tendsto-measure-abs-sub-ge","scope":"Probability/statistical-estimation/contiguity-and-le-cam-s-lemmas","status":"fully_proved","summary":"If `Q \u25c1 P` (a fixed `P`) and the measurable functions `T n` converge to `c` in `P`-probability, then `Q n {\u03c9 | \u03b5 \u2264 |T n \u03c9 - c|} \u2192 0` for every `\u03b5 > 0`.","title":"Contiguity transfers convergence in probability"},{"data_url":"scopes/Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound.json","id":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound","kind":"scope","navigation":"module","parent":21405,"result":false,"route":"../roadmap/Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/index.html","scope":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound","status":"fully_proved","title":"Fisher information and the Cram\u00e9r\u2013Rao bound"},{"id":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/fisher-information","kind":"node","lean":"MathlibExt.Probability.Statistics.FisherInformation.fisherInformation","parent":21442,"result":true,"route":"../roadmap/Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/index.html#fisher-information","scope":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound","status":"fully_proved","summary":"The Fisher information at `\u03b8` of the one-parameter family of densities `p \u03b8` with respect to `\u03bd`: `\u222b (\u2202_\u03b8 p \u03b8 x)\u00b2 / p \u03b8 x d\u03bd(x)`. Points with `p \u03b8 x = 0` contribute `0` (for a nonnegative family, `\u2202_\u03b8 p \u03b8 x = 0` there).","title":"Fisher information"},{"id":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/law-of","kind":"node","lean":"MathlibExt.Probability.Statistics.FisherInformation.lawOf","parent":21442,"result":true,"route":"../roadmap/Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/index.html#law-of","scope":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound","status":"fully_proved","summary":"The probability law with density `p \u03b8` with respect to `\u03bd`.","title":"Probability law of a density family member"},{"id":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/sq-deriv-le-fisher-information-mul","kind":"node","lean":"MathlibExt.Probability.Statistics.FisherInformation.sq_deriv_le_fisherInformation_mul","parent":21442,"result":true,"route":"../roadmap/Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/index.html#sq-deriv-le-fisher-information-mul","scope":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound","status":"fully_proved","summary":"Under the hypotheses of `cramerRao_general`, `g'(\u03b8)\u00b2 \u2264 I(\u03b8) \u00b7 Var_\u03b8 T`. Unlike the quotient form, this is informative when `I(\u03b8) = 0`: it then says `g'(\u03b8) = 0`.","title":"Cram\u00e9r\u2013Rao inequality, multiplicative form"},{"id":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/cramer-rao-general","kind":"node","lean":"MathlibExt.Probability.Statistics.FisherInformation.cramerRao_general","parent":21442,"result":true,"route":"../roadmap/Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/index.html#cramer-rao-general","scope":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound","status":"fully_proved","summary":"Cram\u00e9r\u2013Rao inequality, with only the hypotheses the proof needs. Let `p \u03b8'` be nonnegative probability densities with respect to `\u03bd` for `\u03b8'` near `\u03b8`, with square-integrable score at `\u03b8`, and let `T` be an estimator with finite second moment under `p \u03b8` that is unbiased for `g \u03b8'` near `\u03b8`. If `\u222b p \u03b8' d\u03bd` and `\u222b T p \u03b8' d\u03bd` can be differentiated under the integral sign at `\u03b8`, then `g'(\u03b8)\u00b2 / I(\u03b8) \u2264 Var_\u03b8 T`. When\u2026","title":"Cram\u00e9r\u2013Rao inequality"},{"id":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/cramer-rao","kind":"node","lean":"MathlibExt.Probability.Statistics.FisherInformation.cramerRao","parent":21442,"result":true,"route":"../roadmap/Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/index.html#cramer-rao","scope":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound","status":"fully_proved","summary":"Let `p \u03b8'` be probability densities with respect to `\u03bd` for `\u03b8'` near `\u03b8`, differentiable in the parameter at `\u03b8`, with finite positive Fisher information `I(\u03b8)`, and let `T` be an estimator with finite second moment that is unbiased for `g \u03b8'` near `\u03b8`. If `\u222b p \u03b8' d\u03bd` and `\u222b T p \u03b8' d\u03bd` can be differentiated under the integral sign at `\u03b8`, then `Var_\u03b8 T \u2265 g'(\u03b8)\u00b2 / I(\u03b8)`. The hypotheses `hp_meas`, `hp_diff` and\u2026","title":"Cram\u00e9r\u2013Rao inequality"},{"id":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/ae-tendsto-maximum-likelihood","kind":"node","lean":"MathlibExt.Probability.Statistics.FisherInformation.ae_tendsto_maximumLikelihood","parent":21442,"result":true,"route":"../roadmap/Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound/index.html#ae-tendsto-maximum-likelihood","scope":"Probability/statistical-estimation/fisher-information-and-the-cram-r-rao-bound","status":"fully_proved","summary":"Wald's consistency theorem for maximum likelihood (Ferguson's form, positive densities). Let `\u0398` be a compact metric space and `p \u03b8` probability densities with respect to `\u03bd`, positive, measurable in `x` and continuous in `\u03b8`. Assume the model is identifiable (`p \u03b8 = p \u03b8\u2080` `\u03bd`-a.e. only for `\u03b8 = \u03b8\u2080`) and that the log-likelihood ratios have an upper envelope `K` integrable under `p \u03b8\u2080`. Let `X 0, X 1, \u2026` be\u2026","title":"Wald's consistency theorem for maximum likelihood"},{"data_url":"scopes/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood.json","id":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","kind":"scope","navigation":"module","parent":21405,"result":false,"route":"../roadmap/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/index.html","scope":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","status":"fully_proved","title":"Asymptotic normality of maximum likelihood"},{"id":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/mul-tendsto-in-measure-zero","kind":"node","lean":"MeasureTheory.TendstoInDistribution.mul_tendstoInMeasure_zero","parent":21449,"result":true,"route":"../roadmap/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/index.html#mul-tendsto-in-measure-zero","scope":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","status":"fully_proved","summary":"A sequence `A i` converging in distribution, times a sequence `\u03b7 i` converging in probability to `0`, converges in probability to `0` (a consequence of Slutsky's theorem).","title":"Product of a distributionally convergent and a vanishing sequence"},{"id":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/tendsto-in-measure-zero-of-abs-le","kind":"node","lean":"MeasureTheory.tendstoInMeasure_zero_of_abs_le","parent":21449,"result":true,"route":"../roadmap/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/index.html#tendsto-in-measure-zero-of-abs-le","scope":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","status":"fully_proved","summary":"If `|W i| \u2264 V i` on events `G i` whose complements have probability tending to `0`, and `V i` tends to `0` in probability, then `W i` tends to `0` in probability.","title":"Convergence to zero in probability by domination on likely events"},{"id":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/score","kind":"node","lean":"MathlibExt.Probability.Statistics.MLEAsymptoticNormality.score","parent":21449,"result":true,"route":"../roadmap/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/index.html#score","scope":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","status":"fully_proved","summary":"The score `\u2202_\u03b8 log p \u03b8 x` of a one-parameter family of densities `p`.","title":"Score function"},{"id":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/integral-score-sq","kind":"node","lean":"MathlibExt.Probability.Statistics.MLEAsymptoticNormality.integral_score_sq","parent":21449,"result":true,"route":"../roadmap/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/index.html#integral-score-sq","scope":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","status":"fully_proved","summary":"For a positive density family, `E_\u03b8 [score\u00b2] = I(\u03b8)`: the Fisher information of `FisherInformation.fisherInformation` is the second moment of the score under `p \u03b8 \u03bd`.","title":"Fisher information is the second moment of the score"},{"id":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/tendsto-in-distribution-score","kind":"node","lean":"MathlibExt.Probability.Statistics.MLEAsymptoticNormality.tendstoInDistribution_score","parent":21449,"result":true,"route":"../roadmap/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/index.html#tendsto-in-distribution-score","scope":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","status":"fully_proved","summary":"For i.i.d. observations with law `p \u03b8\u2080 \u03bd`, a positive density family differentiable in `\u03b8` at `\u03b8\u2080`, a measurable score with mean zero and `\u222b score\u00b2 d(p \u03b8\u2080 \u03bd) < \u221e`, the normalized score `n^{-1/2} \u2211_{i<n} score p \u03b8\u2080 (X i)` converges in distribution to `N(0, I(\u03b8\u2080))`.","title":"Central limit theorem for the score"},{"id":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/tendsto-in-distribution-sqrt-mul-sub-of","kind":"node","lean":"MathlibExt.Probability.Statistics.MLEAsymptoticNormality.tendstoInDistribution_sqrt_mul_sub_of_estimatingEquation","parent":21449,"result":true,"route":"../roadmap/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/index.html#tendsto-in-distribution-sqrt-mul-sub-of","scope":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","status":"fully_proved","summary":"Asymptotic normality of a consistent root of an estimating equation (one-dimensional Z-estimator under Cram\u00e9r-type conditions). Let `X 0, X 1, \u2026` be i.i.d. with law `\u03bcX` and let `\u03c8 \u03b8 x` satisfy, for every `x` and every `\u03b8` in the ball of radius `\u03b4 > 0` about `\u03b8\u2080`: `\u03c8 \u00b7 x` and `\u2202_\u03b8 \u03c8 \u00b7 x` are differentiable at `\u03b8`, and `|\u2202\u00b2_\u03b8 \u03c8 \u03b8 x| \u2264 M x` with `M` integrable. Assume `E \u03c8 \u03b8\u2080 = 0`, `E \u03c8 \u03b8\u2080\u00b2 < \u221e`, `\u2202_\u03b8 \u03c8 \u03b8\u2080`\u2026","title":"Asymptotic normality of a consistent root of an estimating equation"},{"id":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/tendsto-in-distribution-sqrt-mul-sub-of-2","kind":"node","lean":"MathlibExt.Probability.Statistics.MLEAsymptoticNormality.tendstoInDistribution_sqrt_mul_sub_of_likelihoodEquation","parent":21449,"result":true,"route":"../roadmap/Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood/index.html#tendsto-in-distribution-sqrt-mul-sub-of-2","scope":"Probability/statistical-estimation/asymptotic-normality-of-maximum-likelihood","status":"fully_proved","summary":"Asymptotic normality of consistent roots of the likelihood equation (Cram\u00e9r's conditions). Let `p \u03b8` be positive densities with respect to `\u03bd`, and on the ball of radius `\u03b4 > 0` about `\u03b8\u2080` let each `p \u03b8` be measurable with `\u222b p \u03b8 d\u03bd = 1`, so that the family is a statistical model there (as in `FisherInformation.cramerRao`). The proof uses measurability only at `\u03b8\u2080` and does not use the normalization `_hp_int`: the\u2026","title":"Asymptotic normality of consistent roots of the likelihood equation"},{"data_url":"scopes/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators.json","id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","kind":"scope","navigation":"module","parent":21405,"result":false,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","title":"Likelihood and maximum likelihood estimators"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/normalized-density-family","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#normalized-density-family","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"Let $\\Omega$ be a measurable space with a measure $\\mu$ and let $\\Theta$ be a set of parameters. A normalized density family on $\\Omega$ indexed by $\\Theta$ with dominating measure $\\mu$ consists of a function $f_\\theta : \\Omega \\to [0, \\infty]$ for each $\\theta \\in \\Theta$ such that $f_\\theta$ is $\\mu$-almost everywhere measurable and $\\int_\\Omega f_\\theta \\, d\\mu = 1$.","title":"Normalized density family"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/ext","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily.ext","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#ext","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"Let $F$ and $G$ be normalized density families on a measurable space $\\Omega$ with the same parameter set $\\Theta$ and the same dominating measure $\\mu$. If for every $\\theta \\in \\Theta$ and every $x \\in \\Omega$ the density of $F$ at $\\theta$ and the density of $G$ at $\\theta$ take the same value at $x$, then $F = G$.","title":"Normalized density families are determined by their densities"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/law","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily.law","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#law","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"Let $F$ be a normalized density family on a measurable space $\\Omega$ with parameter set $\\Theta$, dominating measure $\\mu$ and densities $f_\\theta$. The population law at $\\theta \\in \\Theta$ is the measure $P_\\theta$ on $\\Omega$ with density $f_\\theta$ with respect to $\\mu$, so that $P_\\theta(A) = \\int_A f_\\theta \\, d\\mu$ for every measurable set $A \\subseteq \\Omega$.","title":"Population law"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/is-probability-measure-law","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily.isProbabilityMeasure_law","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#is-probability-measure-law","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"Let $F$ be a normalized density family on $\\Omega$ with parameter set $\\Theta$, dominating measure $\\mu$ and densities $f_\\theta$. For every $\\theta \\in \\Theta$, the population law $P_\\theta$, the measure with density $f_\\theta$ with respect to $\\mu$, is a probability measure.","title":"Population laws are probability measures"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/iid-likelihood","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily.iidLikelihood","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#iid-likelihood","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"Let $F$ be a normalized density family on $\\Omega$ with parameter set $\\Theta$ and densities $f_\\theta$ for $\\theta \\in \\Theta$. For a sample size $n$, a parameter $\\theta \\in \\Theta$ and a sample $x = (x_0, \\dots, x_{n-1}) \\in \\Omega^n$, the iid likelihood of $\\theta$ at $x$ is $L_n(\\theta; x) = \\prod_{i < n} f_\\theta(x_i)$, a value in $[0, \\infty]$. For $n = 0$ this is the empty product $1$.","title":"Likelihood of an iid sample"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/iid-likelihood-nil","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily.iidLikelihood_nil","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#iid-likelihood-nil","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"For a normalized density family $F$, every parameter $\\theta$ and the empty sample $x$ (sample size $n = 0$), the iid likelihood of $\\theta$ at $x$ equals $1$, the empty product.","title":"Likelihood of the empty sample"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/is-maximum-likelihood-estimator","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily.IsMaximumLikelihoodEstimator","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#is-maximum-likelihood-estimator","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"Let $F$ be a normalized density family on $\\Omega$ with parameter set $\\Theta$, let $n$ be a sample size, and let $L_n(\\theta; x)$ denote the iid likelihood of $\\theta \\in \\Theta$ at a sample $x \\in \\Omega^n$. A map $\\hat\\theta$ from $\\Omega^n$ to $\\Theta$ is a maximum likelihood estimator for samples of size $n$ if for every sample $x \\in \\Omega^n$ the parameter $\\hat\\theta(x)$ maximizes\u2026","title":"Maximum likelihood estimator"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/is-maximum-likelihood-estimator-iff","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily.isMaximumLikelihoodEstimator_iff","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#is-maximum-likelihood-estimator-iff","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"Let $F$ be a normalized density family on $\\Omega$ with parameter set $\\Theta$, let $n$ be a sample size, let $L_n(\\theta; x)$ denote the iid likelihood of $\\theta \\in \\Theta$ at a sample $x \\in \\Omega^n$, and let $\\hat\\theta$ be a map from $\\Omega^n$ to $\\Theta$. Then $\\hat\\theta$ is a maximum likelihood estimator for samples of size $n$ if and only if $L_n(\\theta; x) \\le L_n(\\hat\\theta(x); x)$ for every sample\u2026","title":"Characterization of maximum likelihood estimators"},{"id":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/induced-likelihood","kind":"node","lean":"ProbabilityTheory.NormalizedDensityFamily.inducedLikelihood","parent":21457,"result":true,"route":"../roadmap/Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators/index.html#induced-likelihood","scope":"Probability/statistical-estimation/likelihood-and-maximum-likelihood-estimators","status":"fully_proved","summary":"Let $F$ be a normalized density family on $\\Omega$ with parameter set $\\Theta$, let $n$ be a sample size, let $\\tau : \\Theta \\to H$ be a map into a set $H$, and let $x \\in \\Omega^n$ be a sample. For $\\eta$ in the image of $\\tau$, the induced likelihood of $\\eta$ at $x$ is the supremum, in $[0, \\infty]$, of the iid likelihood $L_n(\\theta; x)$ over the fiber of $\\tau$ above $\\eta$:","title":"Induced likelihood"},{"data_url":"scopes/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation.json","id":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","kind":"scope","navigation":"module","parent":21405,"result":false,"route":"../roadmap/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/index.html","scope":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","status":"fully_proved","title":"Stein's lemma and James\u2013Stein estimation"},{"id":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/integrable-and-integral-sub-mul-gaussian","kind":"node","lean":"ProbabilityTheory.integrable_and_integral_sub_mul_gaussianReal","parent":21467,"result":true,"route":"../roadmap/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/index.html#integrable-and-integral-sub-mul-gaussian","scope":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","status":"fully_proved","summary":"For `X ~ N(\u03bc, v)` and an everywhere differentiable `g` with integrable derivative, `(X - \u03bc) g(X)` is integrable and `E[(X - \u03bc) g(X)] = v E[g'(X)]`. For `v = 0` both sides vanish.","title":"Stein's lemma"},{"id":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/integral-sub-mul-pi-gaussian-real","kind":"node","lean":"ProbabilityTheory.integral_sub_mul_pi_gaussianReal","parent":21467,"result":true,"route":"../roadmap/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/index.html#integral-sub-mul-pi-gaussian-real","scope":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","status":"fully_proved","summary":"Stein's lemma, coordinate form: for `X ~ N(\u03b8, I_p)` and a function `h` whose slices along the `i`-th axis are differentiable everywhere (for almost every base point) with integrable partial derivative `dh`, `E[(X\u1d62 - \u03b8\u1d62) h(X)] = E[dh(X)]` whenever `(X\u1d62 - \u03b8\u1d62) h(X)` is integrable. Unlike the one-dimensional case, this integrability does not follow from that of `dh`: `h(x) = exp(x\u2081\u00b2)` has `\u2202\u2080 h = 0` but `x\u2080 h(x)` is\u2026","title":"Stein's lemma"},{"id":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/integral-sum-sq-sub-pi-gaussian-real","kind":"node","lean":"ProbabilityTheory.integral_sum_sq_sub_pi_gaussianReal","parent":21467,"result":true,"route":"../roadmap/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/index.html#integral-sum-sq-sub-pi-gaussian-real","scope":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","status":"fully_proved","summary":"The quadratic risk of the usual estimator `X` of the mean of `N(\u03b8, I_p)` is `p`.","title":"Quadratic risk of the usual Gaussian mean estimator"},{"id":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/integral-sum-sq-add-sub-pi-gaussian-real","kind":"node","lean":"ProbabilityTheory.integral_sum_sq_add_sub_pi_gaussianReal","parent":21467,"result":true,"route":"../roadmap/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/index.html#integral-sum-sq-add-sub-pi-gaussian-real","scope":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","status":"fully_proved","summary":"For `X ~ N(\u03b8, I_p)` and an estimator `X + g(X)` whose coordinates `g\u1d62` are square integrable and satisfy the hypotheses of Stein's lemma in the `i`-th coordinate with partial derivative `dg\u1d62`, the quadratic risk is `p + E[\u2211 g\u1d62(X)\u00b2 + 2 \u2211 dg\u1d62(X)]`.","title":"Stein's unbiased risk estimate"},{"id":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/james-stein","kind":"node","lean":"ProbabilityTheory.jamesStein","parent":21467,"result":true,"route":"../roadmap/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/index.html#james-stein","scope":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","status":"fully_proved","summary":"The James\u2013Stein estimator `\u03b4(x) = (1 - (p - 2) / \u2016x\u2016\u00b2) x` of the mean of `N(\u03b8, I_p)`. At `x = 0` the factor is `1` by Lean's `a / 0 = 0` convention, so `\u03b4(0) = 0`.","title":"James\u2013Stein estimator"},{"id":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/integrable-and-integral-james-stein-risk","kind":"node","lean":"ProbabilityTheory.integrable_and_integral_jamesStein_risk","parent":21467,"result":true,"route":"../roadmap/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/index.html#integrable-and-integral-james-stein-risk","scope":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","status":"fully_proved","summary":"James\u2013Stein (1961): for `p \u2265 3` and `X ~ N(\u03b8, I_p)`, `\u2016X\u2016\u207b\u00b2` is integrable and the risk of the James\u2013Stein estimator is `p - (p - 2)\u00b2 E[\u2016X\u2016\u207b\u00b2]`.","title":"James\u2013Stein"},{"id":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/integral-james-stein-risk-lt","kind":"node","lean":"ProbabilityTheory.integral_jamesStein_risk_lt","parent":21467,"result":true,"route":"../roadmap/Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation/index.html#integral-james-stein-risk-lt","scope":"Probability/statistical-estimation/stein-s-lemma-and-james-stein-estimation","status":"fully_proved","summary":"James\u2013Stein dominance (Stein 1956, James\u2013Stein 1961): for `p \u2265 3`, the James\u2013Stein estimator has strictly smaller quadratic risk than `X` at every mean `\u03b8`.","title":"James\u2013Stein dominance"},{"data_url":"scopes/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo.json","id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","kind":"scope","navigation":"module","parent":21405,"result":false,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"can_prove","title":"Sufficient statistics and the factorization theorem"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/is-sufficient","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.IsSufficient","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#is-sufficient","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"fully_proved","summary":"A sub-\u03c3-algebra `m` is *sufficient* for the family of measures `P` on `(\u03a9, m\u2080)` if `m \u2264 m\u2080`, every `P \u03b8` is finite, and for every `m\u2080`-measurable set `s` one `m`-measurable function is a version of the conditional probability `P_\u03b8[1_s | m]` under every `P \u03b8`. Bundling `m \u2264 m\u2080` and finiteness into the predicate means an `IsSufficient` hypothesis itself represents a sufficient sub-\u03c3-algebra for statistical laws: the\u2026","title":"Sufficient sub-\u03c3-algebra"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/exists-common-cond-exp","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.IsSufficient.exists_common_condExp","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#exists-common-cond-exp","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"fully_proved","summary":"If `m` is sufficient for a family of probability measures, every `m\u2080`-measurable real function `W` that is integrable under all the measures has one `m`-measurable version of the conditional expectation `P_\u03b8[W | m]` for every `\u03b8`. (Measurability of `W` itself, not only under each `P \u03b8`, is what lets one approximating sequence serve every parameter.)","title":"Common conditional expectation under a sufficient \u03c3-algebra"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/is-sufficient-of-with-density","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.isSufficient_of_withDensity","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#is-sufficient-of-with-density","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"fully_proved","summary":"Factorization implies sufficiency (Halmos\u2013Savage, the sufficiency half). If every `P \u03b8` has a density `g \u03b8` with respect to one measure `\u03bd` that is \u03c3-finite on `m`, and each `g \u03b8` is `m`-measurable, then `m` is sufficient for `P`.","title":"Factorization implies sufficiency"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/is-sufficient-comap-of-factorization-of","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.isSufficient_comap_of_factorization_of_sigmaFinite_trim","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#is-sufficient-comap-of-factorization-of","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"fully_proved","summary":"Factorization criterion, sufficiency direction, for a statistic, under an extra \u03c3-finiteness hypothesis. If `dP_\u03b8/d\u03bc = g_\u03b8(T) \u00b7 h` with `g_\u03b8` and `h` measurable and `h \u00b7 \u03bc` \u03c3-finite on the \u03c3-algebra generated by `T`, then `T` is sufficient for `P`.","title":"Factorization criterion, sufficiency direction, for a statistic, under an extra \u03c3-finiteness hypothesis"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/is-sufficient-pmf","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.IsSufficientPMF","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#is-sufficient-pmf","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"fully_proved","summary":"A statistic `T : X \u2192 \u03c4` is *sufficient* for the discrete family `P : \u0398 \u2192 PMF X` if the conditional probability `P \u03b8 x / P_\u03b8(T = T x)` of `x` given `T = T x` is the same under any two parameters `\u03b8, \u03b8'` for which `{T = T x}` has positive probability. Here `P_\u03b8(T = t) = (P \u03b8).map T t`.","title":"Sufficient statistic for a discrete family"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/factorization","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.IsSufficientPMF.factorization","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#factorization","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"fully_proved","summary":"If `T` is sufficient for `P`, then `P \u03b8 x = g \u03b8 (T x) * h x` with `g \u03b8 t = P_\u03b8(T = t)` and a parameter-free `h` with values in `[0, 1]`.","title":"Factorization, necessity"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/is-sufficient-pmf-of-factorization","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.isSufficientPMF_of_factorization","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#is-sufficient-pmf-of-factorization","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"fully_proved","summary":"If `P \u03b8 x = g \u03b8 (T x) * h x` for nonnegative real `g`, `h` with `h` free of `\u03b8`, then `T` is sufficient for `P`.","title":"Factorization, sufficiency"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/is-sufficient-pmf-iff-exists-factorizati","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.isSufficientPMF_iff_exists_factorization","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#is-sufficient-pmf-iff-exists-factorizati","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"fully_proved","summary":"The Fisher\u2013Neyman factorization theorem (discrete case). A statistic `T` is sufficient for the family of mass functions `P` if and only if there are nonnegative functions `g` and `h`, with `h` free of the parameter, such that `P \u03b8 x = g \u03b8 (T x) * h x` for every `\u03b8`, `x`.","title":"The Fisher\u2013Neyman factorization theorem"},{"id":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/exists-factorization-of-is-sufficient-co","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.exists_factorization_of_isSufficient_comap","parent":21475,"result":true,"route":"../roadmap/Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo/index.html#exists-factorization-of-is-sufficient-co","scope":"Probability/statistical-estimation/sufficient-statistics-and-the-factorization-theo","status":"can_prove","summary":"If the family of probability measures `P` is dominated by a \u03c3-finite measure `\u03bc` and the statistic `T` is sufficient for `P`, then the densities factor as `dP_\u03b8/d\u03bc = g_\u03b8(T) \u00b7 h` with measurable `g_\u03b8` and `h`, `h` free of `\u03b8`.","title":"Halmos\u2013Savage factorization, necessity"},{"data_url":"scopes/Probability/statistical-estimation/rao-blackwell-theorem.json","id":"Probability/statistical-estimation/rao-blackwell-theorem","kind":"scope","navigation":"module","parent":21405,"result":false,"route":"../roadmap/Probability/statistical-estimation/rao-blackwell-theorem/index.html","scope":"Probability/statistical-estimation/rao-blackwell-theorem","status":"fully_proved","title":"Rao\u2013Blackwell theorem"},{"id":"Probability/statistical-estimation/rao-blackwell-theorem/integrable-comp-cond-exp","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.integrable_comp_condExp","parent":21485,"result":true,"route":"../roadmap/Probability/statistical-estimation/rao-blackwell-theorem/index.html#integrable-comp-cond-exp","scope":"Probability/statistical-estimation/rao-blackwell-theorem","status":"fully_proved","summary":"For a convex loss `L`, `L \u2218 \u03bc[W | m]` is integrable whenever `W` and `L \u2218 W` are.","title":"Integrability of a convex loss of a conditional expectation"},{"id":"Probability/statistical-estimation/rao-blackwell-theorem/integral-comp-cond-exp-le","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.integral_comp_condExp_le","parent":21485,"result":true,"route":"../roadmap/Probability/statistical-estimation/rao-blackwell-theorem/index.html#integral-comp-cond-exp-le","scope":"Probability/statistical-estimation/rao-blackwell-theorem","status":"fully_proved","summary":"For a convex loss `L` and an integrable `W` with integrable loss, `\u222b L(\u03bc[W | m]) d\u03bc \u2264 \u222b L(W) d\u03bc`.","title":"Rao\u2013Blackwell inequality for one law"},{"id":"Probability/statistical-estimation/rao-blackwell-theorem/integral-sq-sub-eq-add","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.integral_sq_sub_eq_add","parent":21485,"result":true,"route":"../roadmap/Probability/statistical-estimation/rao-blackwell-theorem/index.html#integral-sq-sub-eq-add","scope":"Probability/statistical-estimation/rao-blackwell-theorem","status":"fully_proved","summary":"For `W` with a finite second moment and any target `c`, `E (W - c)\u00b2 = E (\u03bc[W | m] - c)\u00b2 + E (W - \u03bc[W | m])\u00b2`. This is the Pythagorean identity `MeasureTheory.integral_sq_sub_eq_integral_sq_sub_condExp_add` with the constant predictor `fun _ => c`, with the two summands commuted; it is not a second Pythagorean proof.","title":"Squared-error decomposition"},{"id":"Probability/statistical-estimation/rao-blackwell-theorem/integral-sq-sub-cond-exp-eq-iff","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.integral_sq_sub_condExp_eq_iff","parent":21485,"result":true,"route":"../roadmap/Probability/statistical-estimation/rao-blackwell-theorem/index.html#integral-sq-sub-cond-exp-eq-iff","scope":"Probability/statistical-estimation/rao-blackwell-theorem","status":"fully_proved","summary":"Conditioning leaves the squared-error risk at `c` unchanged if and only if `W` already equals `\u03bc[W | m]` almost everywhere.","title":"Equality case under squared error"},{"id":"Probability/statistical-estimation/rao-blackwell-theorem/rao-blackwell","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.IsSufficient.raoBlackwell","parent":21485,"result":true,"route":"../roadmap/Probability/statistical-estimation/rao-blackwell-theorem/index.html#rao-blackwell","scope":"Probability/statistical-estimation/rao-blackwell-theorem","status":"fully_proved","summary":"Let `m` be sufficient for the family of probability measures `P`, and let `W` be a measurable estimator that is integrable under every `P \u03b8`. There is one parameter-free `m`-measurable estimator `\u03b7`, a version of `P_\u03b8[W | m]` for every `\u03b8`, with the same mean as `W` under every `\u03b8` and no larger risk under every loss `\u2113 \u03b8` that is convex in the estimate (for which `\u2113 \u03b8 \u2218 W` is integrable).","title":"The Rao\u2013Blackwell theorem"},{"id":"Probability/statistical-estimation/rao-blackwell-theorem/rao-blackwell-comap","kind":"node","lean":"MathlibExt.Probability.Statistics.Sufficiency.raoBlackwell_comap","parent":21485,"result":true,"route":"../roadmap/Probability/statistical-estimation/rao-blackwell-theorem/index.html#rao-blackwell-comap","scope":"Probability/statistical-estimation/rao-blackwell-theorem","status":"fully_proved","summary":"If the statistic `T` is sufficient for `P` and `W` is a measurable estimator with finite second moment under every `P \u03b8`, there is a measurable `\u03c6` such that `\u03c6 \u2218 T` is a version of every `E_\u03b8[W | T]`, has the same mean as `W`, and for every target `\u03c4 \u03b8` satisfies `E_\u03b8 (W - \u03c4 \u03b8)\u00b2 = E_\u03b8 (\u03c6(T) - \u03c4 \u03b8)\u00b2 + E_\u03b8 (W - \u03c6(T))\u00b2`.","title":"The Rao\u2013Blackwell theorem for a sufficient statistic"},{"data_url":"scopes/Probability/bayesian-statistics.json","id":"Probability/bayesian-statistics","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/bayesian-statistics/index.html","scope":"Probability/bayesian-statistics","status":"can_prove","summary":"Bayesian inference and nonparametric priors: the Dirichlet process and stick-breaking, P\u00f3lya and Blackwell\u2013MacQueen urns, Ferguson conjugacy, and posterior computations.","title":"Bayesian statistics"},{"data_url":"scopes/Probability/bayesian-statistics/dirichlet-process.json","id":"Probability/bayesian-statistics/dirichlet-process","kind":"scope","navigation":"module","parent":21492,"result":false,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html","scope":"Probability/bayesian-statistics/dirichlet-process","status":"can_prove","title":"Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/stick-remainder","kind":"node","lean":"ProbabilityTheory.stickRemainder","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#stick-remainder","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The stick left after breaking off the first `k` pieces at proportions `b`: `\u220f_{j<k} (1 - b\u2c7c)`.","title":"Leftover stick in stick breaking"},{"id":"Probability/bayesian-statistics/dirichlet-process/stick-weight","kind":"node","lean":"ProbabilityTheory.stickWeight","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#stick-weight","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The `k`-th piece of a stick broken at proportions `b`: `b\u2096 \u220f_{j<k} (1 - b\u2c7c)`.","title":"Stick-breaking weights"},{"id":"Probability/bayesian-statistics/dirichlet-process/stick-breaking-measure","kind":"node","lean":"ProbabilityTheory.stickBreakingMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#stick-breaking-measure","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The discrete measure `\u2211\u2096 \u03c0\u2096 \u03b4_{\u03b8\u2096}` with the stick-breaking weights `\u03c0\u2096 = stickWeight b k` (negative weights, which do not occur for proportions in `[0, 1]`, are read as `0`).","title":"Stick-breaking measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/sum-range-stick-weight-add-stick-remaind","kind":"node","lean":"ProbabilityTheory.sum_range_stickWeight_add_stickRemainder","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#sum-range-stick-weight-add-stick-remaind","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The first `n` pieces and the leftover stick make up the whole stick. Holds for all real `b`.","title":"Stick-breaking pieces and leftover make up the whole stick"},{"id":"Probability/bayesian-statistics/dirichlet-process/stick-remainder-mem-icc","kind":"node","lean":"ProbabilityTheory.stickRemainder_mem_Icc","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#stick-remainder-mem-icc","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The leftover stick is in `[0, 1]` for proportions in `[0, 1]`.","title":"Leftover stick lies in the unit interval"},{"id":"Probability/bayesian-statistics/dirichlet-process/stick-weight-nonneg","kind":"node","lean":"ProbabilityTheory.stickWeight_nonneg","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#stick-weight-nonneg","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The pieces are nonnegative for proportions in `[0, 1]`.","title":"Nonnegativity of stick-breaking weights"},{"id":"Probability/bayesian-statistics/dirichlet-process/antitone-stick-remainder","kind":"node","lean":"ProbabilityTheory.antitone_stickRemainder","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#antitone-stick-remainder","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The leftover stick shrinks for proportions in `[0, 1]`.","title":"Leftover stick is decreasing"},{"id":"Probability/bayesian-statistics/dirichlet-process/has-sum-stick-weight-iff","kind":"node","lean":"ProbabilityTheory.hasSum_stickWeight_iff","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#has-sum-stick-weight-iff","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"For proportions in `[0, 1]`, the pieces sum to `1` iff the leftover stick tends to `0`.","title":"Stick-breaking weights sum to one iff the leftover stick tends to zero"},{"id":"Probability/bayesian-statistics/dirichlet-process/stick-breaking-measure-apply","kind":"node","lean":"ProbabilityTheory.stickBreakingMeasure_apply","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#stick-breaking-measure-apply","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The mass of a measurable set under the stick-breaking measure.","title":"Mass of a set under the stick-breaking measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/is-probability-measure-stick-breaking-me","kind":"node","lean":"ProbabilityTheory.isProbabilityMeasure_stickBreakingMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#is-probability-measure-stick-breaking-me","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The stick-breaking measure is a probability measure when the pieces sum to `1`.","title":"Stick-breaking measure is a probability measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/stick-breaking-measure-apply-le-one","kind":"node","lean":"ProbabilityTheory.stickBreakingMeasure_apply_le_one","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#stick-breaking-measure-apply-le-one","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The stick-breaking measure has total mass at most `1` for proportions in `[0, 1]`.","title":"Stick-breaking measure has total mass at most one"},{"id":"Probability/bayesian-statistics/dirichlet-process/measurable-stick-breaking-measure","kind":"node","lean":"ProbabilityTheory.measurable_stickBreakingMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#measurable-stick-breaking-measure","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The stick-breaking measure depends measurably (for the Giry \u03c3-algebra on `Measure X`) on the proportions and the atoms.","title":"Measurability of the stick-breaking measure in its parameters"},{"id":"Probability/bayesian-statistics/dirichlet-process/stick-breaking-process","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#stick-breaking-process","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"A stick-breaking process on the probability space `(\u03a9, \u03bc)`: independent proportions `\u03b2\u2096 \u2208 [0, 1]` (almost surely, with arbitrary laws), i.i.d. atoms `atom k` with law `H`, and the proportion sequence independent of the atom sequence. Ishwaran\u2013James (2001) take `\u03b2\u2096 \u223c Beta(a\u2096, b\u2096)`; Sethuraman's Dirichlet process is `IsDirichlet`.","title":"Stick-breaking process"},{"id":"Probability/bayesian-statistics/dirichlet-process/weight","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.weight","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#weight","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The random piece `\u03c0\u2096 = \u03b2\u2096 \u220f_{j<k} (1 - \u03b2\u2c7c)`.","title":"Random stick-breaking weight"},{"id":"Probability/bayesian-statistics/dirichlet-process/remainder","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.remainder","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#remainder","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The random leftover stick `\u220f_{j<k} (1 - \u03b2\u2c7c)`.","title":"Random leftover stick"},{"id":"Probability/bayesian-statistics/dirichlet-process/g","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.G","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#g","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The random measure `G = \u2211\u2096 \u03c0\u2096 \u03b4_{\u03b8\u2096}`.","title":"Random measure of a stick-breaking process"},{"id":"Probability/bayesian-statistics/dirichlet-process/sum-sq-weight","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.sumSqWeight","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#sum-sq-weight","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"`\u2211\u2096 E[\u03c0\u2096\u00b2]`: given `G`, two independent draws from `G` pick the same piece `k` with probability `\u2211\u2096 \u03c0\u2096\u00b2`. For an atomless base measure the atoms are almost surely distinct, so this is also the probability that the two draws coincide (not used here).","title":"Expected sum of squared stick-breaking weights"},{"id":"Probability/bayesian-statistics/dirichlet-process/measurable-weight","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.measurable_weight","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#measurable-weight","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The pieces are measurable.","title":"Measurability of the random stick-breaking weights"},{"id":"Probability/bayesian-statistics/dirichlet-process/ae-forall-mem-icc","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.ae_forall_mem_Icc","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#ae-forall-mem-icc","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Almost surely every proportion is in `[0, 1]`.","title":"Stick-breaking proportions lie in the unit interval almost surely"},{"id":"Probability/bayesian-statistics/dirichlet-process/ae-weight-mem-icc","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.ae_weight_mem_Icc","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#ae-weight-mem-icc","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Almost surely every piece is in `[0, 1]`.","title":"Random stick-breaking weights lie in the unit interval almost surely"},{"id":"Probability/bayesian-statistics/dirichlet-process/measurable-g","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.measurable_G","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#measurable-g","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The random measure depends measurably on `\u03c9`.","title":"Measurability of the random stick-breaking measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/is-probability-measure-h","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.isProbabilityMeasure_H","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#is-probability-measure-h","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The base measure is a probability measure.","title":"Base measure of a stick-breaking process is a probability measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/integral-remainder","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.integral_remainder","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#integral-remainder","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"`E[\u220f_{j<k} (1 - \u03b2\u2c7c)] = \u220f_{j<k} (1 - E[\u03b2\u2c7c])`.","title":"Expected leftover stick is the product of expected complements"},{"id":"Probability/bayesian-statistics/dirichlet-process/integral-weight","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.integral_weight","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#integral-weight","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"`E[\u03c0\u2096] = E[\u03b2\u2096] \u220f_{j<k} (1 - E[\u03b2\u2c7c])`.","title":"Mean of a stick-breaking weight"},{"id":"Probability/bayesian-statistics/dirichlet-process/integral-weight-sq","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.integral_weight_sq","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#integral-weight-sq","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"`E[\u03c0\u2096\u00b2] = E[\u03b2\u2096\u00b2] \u220f_{j<k} E[(1 - \u03b2\u2c7c)\u00b2]`.","title":"Second moment of a stick-breaking weight"},{"id":"Probability/bayesian-statistics/dirichlet-process/ae-has-sum-weight","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.ae_hasSum_weight","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#ae-has-sum-weight","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"If the expected leftover stick `\u220f_{j<k} (1 - E[\u03b2\u2c7c])` tends to `0`, the pieces sum to `1` almost surely.","title":"Random stick-breaking weights sum to one almost surely"},{"id":"Probability/bayesian-statistics/dirichlet-process/ae-is-probability-measure-g-of-has-sum","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.ae_isProbabilityMeasure_G_of_hasSum","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#ae-is-probability-measure-g-of-has-sum","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"When the pieces sum to `1` almost surely, `G` is almost surely a probability measure.","title":"Stick-breaking random measure is almost surely a probability measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/lintegral-g-apply","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.lintegral_G_apply","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#lintegral-g-apply","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"If the pieces sum to `1` almost surely, then `E[G(A)] = H(A)` for every measurable `A`.","title":"Mean of the random measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/lintegral-g-mul-g","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.lintegral_G_mul_G","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#lintegral-g-mul-g","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"If the pieces sum to `1` almost surely, then `E[G(A) G(B)] = (1 - s) H(A) H(B) + s H(A \u2229 B)` with `s = \u2211\u2096 E[\u03c0\u2096\u00b2]` (`sumSqWeight`, at most `1` under this hypothesis): two draws from `G` pick the same piece with probability `s`, and otherwise land on two independent atoms.","title":"Mixed second moment of the random measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/is-dirichlet","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#is-dirichlet","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Sethuraman's stick-breaking representation of the Dirichlet process `DP(\u03b1, H)` (Sethuraman 1994): the proportions are `Beta(1, \u03b1)` with `\u03b1 > 0`. We take this construction as the definition of `DP(\u03b1, H)`; Sethuraman's theorem that it has Ferguson's Dirichlet marginals is recorded in `WantedExt/Probability/DirichletProcessWanted.lean`.","title":"Sethuraman's stick-breaking representation of the Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/polya-urn-predictive","kind":"node","lean":"ProbabilityTheory.polyaUrnPredictive","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#polya-urn-predictive","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The predictive law in the Blackwell\u2013MacQueen urn: after observations `x\u2080, \u2026, x_{n-1}`, the next observation has law `(\u03b1 H + \u2211\u1d62 \u03b4_{x\u1d62}) / (\u03b1 + n)`. It is a probability measure for `\u03b1 > 0` and a probability measure `H` (`isProbabilityMeasure_polyaUrnPredictive`); outside that domain the definition is total but meaningless (e.g. `\u03b1 = n = 0` gives the zero measure).","title":"Blackwell\u2013MacQueen urn predictive law"},{"id":"Probability/bayesian-statistics/dirichlet-process/polya-urn-predictive-apply","kind":"node","lean":"ProbabilityTheory.polyaUrnPredictive_apply","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#polya-urn-predictive-apply","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The predictive probability of a measurable set: `(\u03b1 H(A) + #{i | x\u1d62 \u2208 A}) / (\u03b1 + n)`.","title":"Predictive probability of a set in the Blackwell\u2013MacQueen urn"},{"id":"Probability/bayesian-statistics/dirichlet-process/is-probability-measure-polya-urn-predict","kind":"node","lean":"ProbabilityTheory.isProbabilityMeasure_polyaUrnPredictive","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#is-probability-measure-polya-urn-predict","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The predictive law is a probability measure for `\u03b1 > 0` and a probability measure `H`.","title":"Blackwell\u2013MacQueen predictive law is a probability measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/ae-mem-icc-beta-measure","kind":"node","lean":"ProbabilityTheory.ae_mem_Icc_betaMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#ae-mem-icc-beta-measure","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The Beta law is concentrated on `[0, 1]`.","title":"Beta law is concentrated on the unit interval"},{"id":"Probability/bayesian-statistics/dirichlet-process/integral-one-sub-sq-beta-measure-one","kind":"node","lean":"ProbabilityTheory.integral_one_sub_sq_betaMeasure_one","parent":21493,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second moment of "],["math","\\(1 - X\\)"],["text"," under "],["math","\\(\\mathrm{Beta}(1, \\alpha)\\)"]],"source":"Second moment of $1 - X$ under $\\mathrm{Beta}(1, \\alpha)$"},"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#integral-one-sub-sq-beta-measure-one","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"`E[(1 - x)\u00b2] = \u03b1 / (\u03b1 + 2)` under `Beta(1, \u03b1)`.","title":"Second moment of $1 - X$ under $\\mathrm{Beta}(1, \\alpha)$"},{"id":"Probability/bayesian-statistics/dirichlet-process/integral","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.integral_\u03b2","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#integral","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The proportions of a Dirichlet process have mean `1 / (1 + \u03b1)`.","title":"Mean proportion of a Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/integral-weight-2","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.integral_weight","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#integral-weight-2","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Mean of a Dirichlet-process weight (Sethuraman 1994): `E[\u03c0\u2096] = \u03b1\u1d4f / (1 + \u03b1)^(k+1)` (`\u03b1^{k-1} / (1 + \u03b1)^k` for Sethuraman's `\u03c0_k`, indexed from `1`).","title":"Mean of a Dirichlet-process weight"},{"id":"Probability/bayesian-statistics/dirichlet-process/ae-has-sum-weight-2","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.ae_hasSum_weight","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#ae-has-sum-weight-2","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The Dirichlet-process weights sum to `1` almost surely (Sethuraman 1994): the leftover stick tends to `0`.","title":"The Dirichlet-process weights sum to `1` almost surely"},{"id":"Probability/bayesian-statistics/dirichlet-process/ae-is-probability-measure-g","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.ae_isProbabilityMeasure_G","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#ae-is-probability-measure-g","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"A Dirichlet process is almost surely a probability measure.","title":"Dirichlet process is almost surely a probability measure"},{"id":"Probability/bayesian-statistics/dirichlet-process/lintegral-g-apply-2","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.lintegral_G_apply","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#lintegral-g-apply-2","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Mean of the Dirichlet process (Ferguson 1973): `E[G(A)] = H(A)`.","title":"Mean of the Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/sum-sq-weight-eq","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.sumSqWeight_eq","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#sum-sq-weight-eq","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Two draws from a Dirichlet process pick the same piece with probability `1 / (1 + \u03b1)`: `\u2211\u2096 E[\u03c0\u2096\u00b2] = 1 / (1 + \u03b1)`. This is the probability that the second customer of the Chinese restaurant process with parameter `\u03b1` joins the first one's table.","title":"Probability that two Dirichlet process draws pick the same piece"},{"id":"Probability/bayesian-statistics/dirichlet-process/lintegral-g-mul-g-2","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.lintegral_G_mul_G","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#lintegral-g-mul-g-2","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Mixed second moment of the Dirichlet process (Ferguson 1973): `E[G(A) G(B)] = (\u03b1 H(A) H(B) + H(A \u2229 B)) / (\u03b1 + 1)`.","title":"Mixed second moment of the Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/lintegral-g-mul-g-eq-predictive","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.lintegral_G_mul_G_eq_predictive","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#lintegral-g-mul-g-eq-predictive","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The Blackwell\u2013MacQueen urn for two draws (Blackwell\u2013MacQueen 1973): if `X\u2081, X\u2082` are i.i.d. draws from a Dirichlet process `G`, then `P(X\u2081 \u2208 A, X\u2082 \u2208 B) = E[G(A) G(B)]` equals `\u222b_A P(X\u2082 \u2208 B | X\u2081 = x) dH(x)` with the predictive law `polyaUrnPredictive \u03b1 H ![x] = (\u03b1 H + \u03b4\u2093) / (\u03b1 + 1)`.","title":"The Blackwell\u2013MacQueen urn for two draws"},{"id":"Probability/bayesian-statistics/dirichlet-process/variance-g-apply","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.IsDirichlet.variance_G_apply","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#variance-g-apply","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Variance of the Dirichlet process (Ferguson 1973): `Var(G(A)) = H(A) (1 - H(A)) / (\u03b1 + 1)`.","title":"Variance of the Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/sethuraman-measure","kind":"node","lean":"ProbabilityTheory.sethuramanMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#sethuraman-measure","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The product probability space `(\u2115 \u2192 \u211d) \u00d7 (\u2115 \u2192 X)` carrying i.i.d. `Beta(1, \u03b1)` proportions and, independently, i.i.d. `H` atoms.","title":"Sethuraman's canonical probability space"},{"id":"Probability/bayesian-statistics/dirichlet-process/measure-preserving-fst-sethuraman-measur","kind":"node","lean":"ProbabilityTheory.measurePreserving_fst_sethuramanMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#measure-preserving-fst-sethuraman-measur","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The proportion coordinate of the canonical space has law `\u2297\u2096 Beta(1, \u03b1)`.","title":"Law of the proportion coordinate of Sethuraman's space"},{"id":"Probability/bayesian-statistics/dirichlet-process/measure-preserving-snd-sethuraman-measur","kind":"node","lean":"ProbabilityTheory.measurePreserving_snd_sethuramanMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#measure-preserving-snd-sethuraman-measur","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The atom coordinate of the canonical space has law `\u2297\u2096 H`.","title":"Law of the atom coordinate of Sethuraman's space"},{"id":"Probability/bayesian-statistics/dirichlet-process/has-law-fst-sethuraman-measure","kind":"node","lean":"ProbabilityTheory.hasLaw_fst_sethuramanMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#has-law-fst-sethuraman-measure","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Each proportion of the canonical space has law `Beta(1, \u03b1)`.","title":"Law of each proportion in Sethuraman's space"},{"id":"Probability/bayesian-statistics/dirichlet-process/has-law-snd-sethuraman-measure","kind":"node","lean":"ProbabilityTheory.hasLaw_snd_sethuramanMeasure","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#has-law-snd-sethuraman-measure","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Each atom of the canonical space has law `H`.","title":"Law of each atom in Sethuraman's space"},{"id":"Probability/bayesian-statistics/dirichlet-process/sethuraman","kind":"node","lean":"ProbabilityTheory.sethuraman","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#sethuraman","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Sethuraman's construction on the canonical space: the coordinates of the two factors are the proportions and the atoms.","title":"Sethuraman's construction"},{"id":"Probability/bayesian-statistics/dirichlet-process/is-dirichlet-sethuraman","kind":"node","lean":"ProbabilityTheory.StickBreakingProcess.isDirichlet_sethuraman","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#is-dirichlet-sethuraman","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The canonical model is a Dirichlet process.","title":"Sethuraman's canonical model is a Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/dirichlet-process","kind":"node","lean":"ProbabilityTheory.dirichletProcess","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#dirichlet-process","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"The Dirichlet process `DP(\u03b1, H)` as a law on `Measure X`, defined as the law of Sethuraman's (1994) stick-breaking measure with `Beta(1, \u03b1)` proportions and `H` atoms. Ferguson's (1973) definition by Dirichlet marginals is the wanted statement `map_dirichletProcess_partition`.","title":"Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/lintegral-dirichlet-process-apply","kind":"node","lean":"ProbabilityTheory.lintegral_dirichletProcess_apply","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#lintegral-dirichlet-process-apply","scope":"Probability/bayesian-statistics/dirichlet-process","status":"fully_proved","summary":"Mean of the Dirichlet process, law form: `\u222b \u03bd(A) dDP(\u03b1, H)(\u03bd) = H(A)`.","title":"Mean of the Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/map-dirichlet-process-partition","kind":"node","lean":"ProbabilityTheory.map_dirichletProcess_partition","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#map-dirichlet-process-partition","scope":"Probability/bayesian-statistics/dirichlet-process","status":"can_prove","summary":"Finite-dimensional marginals of the Dirichlet process (Ferguson 1973, \u00a73; Sethuraman 1994, Theorem 3.4): on a measurable partition `A\u2080, \u2026, A_m` of `X` with `H(A\u1d62) > 0`, the vector `(G(A\u2080), \u2026, G(A_m))` has the Dirichlet law `Dir(\u03b1 H(A\u2080), \u2026, \u03b1 H(A_m))`.","title":"Finite-dimensional marginals of the Dirichlet process"},{"id":"Probability/bayesian-statistics/dirichlet-process/lintegral-dirichlet-process-prod","kind":"node","lean":"ProbabilityTheory.lintegral_dirichletProcess_prod","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#lintegral-dirichlet-process-prod","scope":"Probability/bayesian-statistics/dirichlet-process","status":"can_prove","summary":"The moments of the Dirichlet process are the Blackwell\u2013MacQueen urn (Blackwell\u2013MacQueen 1973): `E[\u220f\u1d62 G(A\u1d62)] = P(X\u2080 \u2208 A\u2080, \u2026, X_{n-1} \u2208 A_{n-1})` for the urn's first `n` draws.","title":"The moments of the Dirichlet process are the Blackwell\u2013MacQueen urn"},{"id":"Probability/bayesian-statistics/dirichlet-process/set-lintegral-dirichlet-process-prod","kind":"node","lean":"ProbabilityTheory.setLIntegral_dirichletProcess_prod","parent":21493,"result":true,"route":"../roadmap/Probability/bayesian-statistics/dirichlet-process/index.html#set-lintegral-dirichlet-process-prod","scope":"Probability/bayesian-statistics/dirichlet-process","status":"can_prove","summary":"Ferguson's conjugacy (Ferguson 1973, Theorem 1): given `n` observations `xs` from `G \u223c DP(\u03b1, H)`, the posterior of `G` is `DP(\u03b1 + n, (\u03b1 H + \u2211\u1d62 \u03b4_{xs\u1d62}) / (\u03b1 + n))`. Stated as the disintegration `P(G \u2208 S, X\u2080 \u2208 A\u2080, \u2026) = \u222b_{\u220f A\u1d62} DP(\u03b1 + n, posterior base)(S) dP(xs)` over the urn law of the observations.","title":"Ferguson's conjugacy"},{"data_url":"scopes/Probability/bayesian-statistics/blackwell-macqueen-urn.json","id":"Probability/bayesian-statistics/blackwell-macqueen-urn","kind":"scope","navigation":"module","parent":21492,"result":false,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","title":"Blackwell\u2013MacQueen urn"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/measurable-polya-urn-predictive","kind":"node","lean":"ProbabilityTheory.measurable_polyaUrnPredictive","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#measurable-polya-urn-predictive","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"The predictive law depends measurably on the previous draws.","title":"Measurability of the P\u00f3lya urn predictive law"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/blackwell-mac-queen","kind":"node","lean":"ProbabilityTheory.blackwellMacQueen","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#blackwell-mac-queen","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"The Blackwell\u2013MacQueen urn with parameter `\u03b1` and base measure `H`: the joint law of the first `n` draws, where draw `n` has law `polyaUrnPredictive \u03b1 H xs` given the previous draws `xs` (Blackwell\u2013MacQueen 1973).","title":"Blackwell\u2013MacQueen urn"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/is-probability-measure-blackwell-mac-que","kind":"node","lean":"ProbabilityTheory.isProbabilityMeasure_blackwellMacQueen","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#is-probability-measure-blackwell-mac-que","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"The urn law is a probability measure.","title":"Blackwell\u2013MacQueen urn law is a probability measure"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/blackwell-mac-queen-succ-pi","kind":"node","lean":"ProbabilityTheory.blackwellMacQueen_succ_pi","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#blackwell-mac-queen-succ-pi","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"`P(X\u2080 \u2208 A\u2080, \u2026, X_n \u2208 A_n)` is the integral over `A\u2080 \u00d7 \u22ef \u00d7 A_{n-1}` of the predictive probability `(\u03b1 H(A_n) + #{i < n | x\u1d62 \u2208 A_n}) / (\u03b1 + n)`.","title":"The predictive rule of the urn"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/polya-urn-predictive-comp-perm","kind":"node","lean":"ProbabilityTheory.polyaUrnPredictive_comp_perm","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#polya-urn-predictive-comp-perm","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"The predictive law does not depend on the order of the previous draws.","title":"P\u00f3lya urn predictive law is invariant under reordering"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/map-perm-blackwell-mac-queen","kind":"node","lean":"ProbabilityTheory.map_perm_blackwellMacQueen","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#map-perm-blackwell-mac-queen","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"Exchangeability of the Blackwell\u2013MacQueen urn (Blackwell\u2013MacQueen 1973): relabeling the draws by a permutation does not change their joint law.","title":"Exchangeability of the Blackwell\u2013MacQueen urn"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/tie-partition","kind":"node","lean":"ProbabilityTheory.tiePartition","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#tie-partition","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"The partition of the indices `{0, \u2026, n-1}` into groups of equal entries of `xs`.","title":"Tie partition of a sequence of draws"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/mem-part-tie-partition","kind":"node","lean":"ProbabilityTheory.mem_part_tiePartition","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#mem-part-tie-partition","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"Two indices lie in the same block of `tiePartition xs` iff the entries agree.","title":"Blocks of the tie partition are classes of equal entries"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/tie-partition-eq-iff","kind":"node","lean":"ProbabilityTheory.tiePartition_eq_iff","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#tie-partition-eq-iff","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"`tiePartition xs = \u03c0` iff equal entries are exactly the pairs in a common block of `\u03c0`.","title":"Characterization of the tie partition"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/comap-cast-succ-tie-partition-snoc","kind":"node","lean":"ProbabilityTheory.comap_castSucc_tiePartition_snoc","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#comap-cast-succ-tie-partition-snoc","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"Forgetting the last draw restricts the tie partition to the earlier draws.","title":"Tie partition after forgetting the last draw"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/tie-partition-snoc-eq-seat-none-iff","kind":"node","lean":"ProbabilityTheory.tiePartition_snoc_eq_seat_none_iff","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#tie-partition-snoc-eq-seat-none-iff","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"Where the last draw sits: a new table iff it differs from all earlier draws.","title":"Last draw opens a new block iff it is new"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/tie-partition-snoc-eq-seat-some-iff","kind":"node","lean":"ProbabilityTheory.tiePartition_snoc_eq_seat_some_iff","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#tie-partition-snoc-eq-seat-some-iff","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"Where the last draw sits: at the table `B` iff it equals the earlier draws in `B`.","title":"Last draw joins a block iff it matches that block"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/measurable-set-tie-partition-eq","kind":"node","lean":"ProbabilityTheory.measurableSet_tiePartition_eq","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#measurable-set-tie-partition-eq","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"The event that the draws have a given tie partition is measurable.","title":"Measurability of the tie partition event"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/blackwell-mac-queen-tie-partition","kind":"node","lean":"ProbabilityTheory.blackwellMacQueen_tiePartition","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#blackwell-mac-queen-tie-partition","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"The ties of the Blackwell\u2013MacQueen urn follow the Chinese restaurant process (Pitman 2006, \u00a73.1): for an atomless base measure, the partition of the first `n` draws into groups of equal values has the law `\u03b1^{|\u03c0|} \u220f_{B \u2208 \u03c0} (|B| - 1)! / \u03b1^{(n)}`.","title":"The ties of the Blackwell\u2013MacQueen urn follow the Chinese restaurant process"},{"id":"Probability/bayesian-statistics/blackwell-macqueen-urn/lintegral-dirichlet-process-prod-of-le-t","kind":"node","lean":"ProbabilityTheory.lintegral_dirichletProcess_prod_of_le_two","parent":21550,"result":true,"route":"../roadmap/Probability/bayesian-statistics/blackwell-macqueen-urn/index.html#lintegral-dirichlet-process-prod-of-le-t","scope":"Probability/bayesian-statistics/blackwell-macqueen-urn","status":"fully_proved","summary":"For `n \u2264 2`, `E[\u220f_{i<n} G(A\u1d62)] = P(X\u2080 \u2208 A\u2080, \u2026, X_{n-1} \u2208 A_{n-1})` for the Blackwell\u2013MacQueen urn. The case `n \u2265 3` is the wanted `lintegral_dirichletProcess_prod`.","title":"Moments of the Dirichlet process for at most two draws"},{"data_url":"scopes/Probability/nonparametric-survival.json","id":"Probability/nonparametric-survival","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/nonparametric-survival/index.html","scope":"Probability/nonparametric-survival","status":"fully_proved","summary":"Distribution-free and semiparametric methods: empirical distribution functions (Glivenko\u2013Cantelli), U-statistics, the Kaplan\u2013Meier estimator, Cox partial likelihood and the log-rank test, and design-based causal inference with potential outcomes.","title":"Nonparametric statistics and survival analysis"},{"data_url":"scopes/Probability/nonparametric-survival/cox-proportional-hazards-model.json","id":"Probability/nonparametric-survival/cox-proportional-hazards-model","kind":"scope","navigation":"module","parent":21566,"result":false,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","title":"Cox proportional hazards model"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/risk-set","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.riskSet","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#risk-set","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The risk set at the observed time of subject `i`: subjects still under observation, `{l : T_l \u2265 T\u1d62}`.","title":"Risk set in survival analysis"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/mem-risk-set","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.mem_riskSet","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#mem-risk-set","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"Subject `i` is in its own risk set.","title":"A subject belongs to its own risk set"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/risk-set-nonempty","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.riskSet_nonempty","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#risk-set-nonempty","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The risk set of subject `i` is nonempty.","title":"Risk sets are nonempty"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/partial-likelihood","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.partialLikelihood","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#partial-likelihood","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"Cox's partial likelihood `\u220f_{i : \u03b4\u1d62} exp \u27ea\u03b2, x\u1d62\u27eb / \u2211_{l \u2208 R(T\u1d62)} exp \u27ea\u03b2, x_l\u27eb`.","title":"Cox partial likelihood"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/log-partial-likelihood","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.logPartialLikelihood","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#log-partial-likelihood","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The log partial likelihood `\u2211_{i : \u03b4\u1d62} (\u27ea\u03b2, x\u1d62\u27eb - log \u2211_{l \u2208 R(T\u1d62)} exp \u27ea\u03b2, x_l\u27eb)`.","title":"Cox log partial likelihood"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/log-partial-likelihood-2","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.log_partialLikelihood","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#log-partial-likelihood-2","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The logarithm of the partial likelihood is the log partial likelihood.","title":"Log of the Cox partial likelihood is the log partial likelihood"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/concave-on-log-partial-likelihood","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.concaveOn_logPartialLikelihood","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#concave-on-log-partial-likelihood","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"Concavity of the Cox log partial likelihood on `\u211d\u1d56`.","title":"Concavity of the Cox log partial likelihood"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/risk-mean","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.riskMean","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#risk-mean","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The `exp (b z)`-weighted mean of a scalar covariate `z` over the risk set of subject `i`.","title":"Weighted mean of a covariate over a risk set"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/risk-var","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.riskVar","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#risk-var","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The `exp (b z)`-weighted variance of a scalar covariate over the risk set of subject `i`.","title":"Weighted variance of a covariate over a risk set"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/cox-score","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.coxScore","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#cox-score","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The Cox score for a scalar covariate: `\u2211_{i : \u03b4\u1d62} (z\u1d62 - z\u0304\u1d62(b))`.","title":"Cox score for a scalar covariate"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/cox-information","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.coxInformation","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#cox-information","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The Cox observed information for a scalar covariate: `\u2211_{i : \u03b4\u1d62} Var\u1d62(b)`.","title":"Cox observed information for a scalar covariate"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/log-partial-likelihood-3","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.logPartialLikelihood\u2081","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#log-partial-likelihood-3","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The log partial likelihood of a scalar covariate `z` as a function of `b \u2208 \u211d`.","title":"Log partial likelihood of a scalar covariate"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/has-deriv-at-log-partial-likelihood","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.hasDerivAt_logPartialLikelihood","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#has-deriv-at-log-partial-likelihood","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The derivative of the scalar log partial likelihood is the Cox score.","title":"Derivative of the log partial likelihood is the Cox score"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/has-deriv-at-cox-score","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.hasDerivAt_coxScore","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#has-deriv-at-cox-score","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The derivative of the Cox score is minus the observed information.","title":"Derivative of the Cox score is minus the information"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/num-at-risk","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.numAtRisk","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#num-at-risk","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The number at risk at the time of subject `i`.","title":"Number at risk at a subject's time"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/num-at-risk-in","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.numAtRiskIn","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#num-at-risk-in","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The number of group-1 subjects at risk at the time of subject `i`.","title":"Number of group-1 subjects at risk"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/group-indicator","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.groupIndicator","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#group-indicator","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The group indicator `z\u1d62 = 1` for group 1 (`g i = true`) and `0` otherwise.","title":"Group indicator covariate"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/observed-minus-expected","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.observedMinusExpected","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#observed-minus-expected","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The log-rank numerator `O - E`: observed minus expected group-1 events, `\u2211_{i : \u03b4\u1d62} (z\u1d62 - n\u2081(T\u1d62) / n(T\u1d62))`.","title":"Log-rank numerator: observed minus expected events"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/breslow-information","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.breslowInformation","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#breslow-information","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The Breslow (Cox) information `\u2211_{i : \u03b4\u1d62} n\u2081 n\u2080 / n\u00b2` at the risk sets of the events. This is `-\u2113''(0)` for a group indicator, not the tied log-rank variance: for one tied risk set with `n = 4`, `n\u2081 = 2` and `d = 2` events it gives `1 / 2`, while the hypergeometric log-rank variance is `1 / 3`. It equals the log-rank variance only when the event times are distinct (so each risk set contributes at most one event).","title":"Breslow information"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/cox-score-zero","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.coxScore_zero","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#cox-score-zero","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"`\u2113'(0) = O - E`.","title":"The log-rank statistic is the Cox score at `\u03b2 = 0`"},{"id":"Probability/nonparametric-survival/cox-proportional-hazards-model/cox-information-zero","kind":"node","lean":"MathlibExt.Probability.Statistics.CoxPartialLikelihood.coxInformation_zero","parent":21567,"result":true,"route":"../roadmap/Probability/nonparametric-survival/cox-proportional-hazards-model/index.html#cox-information-zero","scope":"Probability/nonparametric-survival/cox-proportional-hazards-model","status":"fully_proved","summary":"The Cox information at `\u03b2 = 0` is the Breslow information `\u2211 n\u2081 n\u2080 / n\u00b2` (not the tied log-rank variance; see `breslowInformation`).","title":"The Cox information at `\u03b2 = 0` is the Breslow information"},{"data_url":"scopes/Probability/nonparametric-survival/glivenko-cantelli-theorem.json","id":"Probability/nonparametric-survival/glivenko-cantelli-theorem","kind":"scope","navigation":"module","parent":21566,"result":false,"route":"../roadmap/Probability/nonparametric-survival/glivenko-cantelli-theorem/index.html","scope":"Probability/nonparametric-survival/glivenko-cantelli-theorem","status":"fully_proved","title":"Glivenko\u2013Cantelli theorem"},{"id":"Probability/nonparametric-survival/glivenko-cantelli-theorem/empirical-count","kind":"node","lean":"MathlibExt.Probability.Statistics.GlivenkoCantelliWanted.empiricalCount","parent":21589,"result":true,"route":"../roadmap/Probability/nonparametric-survival/glivenko-cantelli-theorem/index.html#empirical-count","scope":"Probability/nonparametric-survival/glivenko-cantelli-theorem","status":"fully_proved","summary":"Let $X_0, X_1, X_2, \\dots$ be real-valued functions on a set $\\Omega$. For a natural number $n$, a point $\\omega \\in \\Omega$ and a real number $x$, the empirical count is the number of indices $i \\in \\{0, 1, \\dots, n\\}$ with $X_i(\\omega) \\le x$. Divided by $n + 1$, it is the value at $x$ of the empirical distribution function of $X_0(\\omega), \\dots, X_n(\\omega)$.","title":"Empirical count"},{"id":"Probability/nonparametric-survival/glivenko-cantelli-theorem/glivenko-cantelli","kind":"node","lean":"MathlibExt.Probability.Statistics.GlivenkoCantelliWanted.glivenko_cantelli","parent":21589,"result":true,"route":"../roadmap/Probability/nonparametric-survival/glivenko-cantelli-theorem/index.html#glivenko-cantelli","scope":"Probability/nonparametric-survival/glivenko-cantelli-theorem","status":"fully_proved","summary":"Let $P$ be a probability measure on a measurable space $\\Omega$, and let $X_0, X_1, X_2, \\dots$ be measurable real-valued functions on $\\Omega$ that are mutually independent under $P$ and each have the same distribution as $X_0$. Write $F(x) = P(X_0 \\le x)$ for the distribution function of $X_0$, and $C_n(\\omega, x)$ for the empirical count, the number of indices $i \\in \\{0, 1, \\dots, n\\}$ with $X_i(\\omega) \\le x$.\u2026","title":"Glivenko\u2013Cantelli theorem"},{"data_url":"scopes/Probability/nonparametric-survival/kaplan-meier-estimator.json","id":"Probability/nonparametric-survival/kaplan-meier-estimator","kind":"scope","navigation":"module","parent":21566,"result":false,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","title":"Kaplan\u2013Meier estimator"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/censored-data","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#censored-data","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"Right-censored survival data grouped at `k` distinct, increasing observation times: `deaths j` subjects die and `censored j` subjects are censored at `time j`.","title":"Grouped right-censored survival data"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/size","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.size","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#size","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The number of subjects in the sample.","title":"Sample size of censored survival data"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/at-risk","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.atRisk","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#at-risk","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The number at risk at `time j`: subjects whose observation time is at least `time j`.","title":"Number at risk at an observation time"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-factor","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmFactor","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-factor","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier factor `1 - d\u2c7c / n\u2c7c` at `time j` (it is `1` if nobody is at risk, where also `d\u2c7c = 0`).","title":"Kaplan\u2013Meier factor"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-survival","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmSurvival","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-survival","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier estimate `\u015c(t) = \u220f_{t\u2c7c \u2264 t} (1 - d\u2c7c / n\u2c7c)` of the survival function.","title":"Kaplan\u2013Meier survival estimate"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-mass","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmMass","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-mass","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier jump at `time j`: `\u015c(t\u2c7c\u208b) \u00b7 d\u2c7c / n\u2c7c`.","title":"Kaplan\u2013Meier jump at an observation time"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-tail","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmTail","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-tail","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier mass left after the last observation time, `\u220f\u2c7c (1 - d\u2c7c / n\u2c7c)`.","title":"Kaplan\u2013Meier mass after the last observation"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-measure","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmMeasure","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-measure","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier measure: the jumps `kmMass j` at `time j` and the leftover mass `kmTail` at a chosen point `\u03c4` (taken after the last observation time).","title":"Kaplan\u2013Meier measure"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/likelihood","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.likelihood","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#likelihood","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The censored-data likelihood of a law `\u03bc` of the survival time: `\u220f\u2c7c \u03bc{t\u2c7c}^{d\u2c7c} \u00b7 \u03bc(t\u2c7c, \u221e)^{c\u2c7c}`.","title":"Likelihood of right-censored data"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/deaths-le-at-risk","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.deaths_le_atRisk","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#deaths-le-at-risk","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The deaths at a time are among those at risk.","title":"Deaths at a time are among those at risk"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-factor-le-one","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmFactor_le_one","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-factor-le-one","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier factor is at most `1`.","title":"Kaplan\u2013Meier factor is at most one"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-factor-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmFactor_nonneg","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-factor-nonneg","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier factor is non-negative.","title":"Kaplan\u2013Meier factor is nonnegative"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-mass-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmMass_nonneg","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-mass-nonneg","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier jumps are non-negative.","title":"Kaplan\u2013Meier jumps are nonnegative"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-tail-nonneg","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmTail_nonneg","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-tail-nonneg","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The leftover Kaplan\u2013Meier mass is non-negative.","title":"Leftover Kaplan\u2013Meier mass is nonnegative"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/sum-km-mass-add-km-tail","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.sum_kmMass_add_kmTail","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#sum-km-mass-add-km-tail","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier jumps and the leftover mass add up to `1`.","title":"Kaplan\u2013Meier jumps and leftover mass sum to one"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-measure-real-apply","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmMeasure_real_apply","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-measure-real-apply","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The mass of a measurable set under the Kaplan\u2013Meier measure.","title":"Mass of a set under the Kaplan\u2013Meier measure"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-measure-is-probability-measure","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmMeasure_isProbabilityMeasure","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-measure-is-probability-measure","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The Kaplan\u2013Meier measure is a probability measure.","title":"Kaplan\u2013Meier measure is a probability measure"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-measure-real-ioi","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmMeasure_real_Ioi","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-measure-real-ioi","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"The survival function of the Kaplan\u2013Meier measure is the Kaplan\u2013Meier estimate on `(-\u221e, \u03c4)`: `P(T > t) = \u015c(t)` for `t < \u03c4`.","title":"Survival function of the Kaplan\u2013Meier measure"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/likelihood-le-km-measure","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.likelihood_le_kmMeasure","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#likelihood-le-km-measure","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"Kaplan\u2013Meier is the nonparametric MLE (Kaplan\u2013Meier 1958): every probability measure on `\u211d` has censored-data likelihood at most that of the Kaplan\u2013Meier measure.","title":"Kaplan\u2013Meier is the nonparametric MLE"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-survival-pos-of-censored-pos","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmSurvival_pos_of_censored_pos","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-survival-pos-of-censored-pos","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"A time at which some subject is censored has positive Kaplan\u2013Meier survival.","title":"Kaplan\u2013Meier survival is positive at a censoring time"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/size-mul-km-survival","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.size_mul_kmSurvival","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#size-mul-km-survival","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"Self-consistency (Efron 1967, (7.4)): `n \u015c(t)` is the number of observations exceeding `t` plus, for each subject censored at or before `t`, the conditional survival `\u015c(t) / \u015c(t\u2c7c)`.","title":"Self-consistency"},{"id":"Probability/nonparametric-survival/kaplan-meier-estimator/km-survival-of-censored-eq-zero","kind":"node","lean":"MathlibExt.Probability.Statistics.KaplanMeier.CensoredData.kmSurvival_of_censored_eq_zero","parent":21592,"result":true,"route":"../roadmap/Probability/nonparametric-survival/kaplan-meier-estimator/index.html#km-survival-of-censored-eq-zero","scope":"Probability/nonparametric-survival/kaplan-meier-estimator","status":"fully_proved","summary":"Without censoring, the Kaplan\u2013Meier estimate is the empirical survival function: the fraction `#{observations > t} / n = 1 - F\u2099(t)`.","title":"Kaplan\u2013Meier estimate without censoring is the empirical survival"},{"data_url":"scopes/Probability/nonparametric-survival/potential-outcomes-and-causal-identification.json","id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","kind":"scope","navigation":"module","parent":21566,"result":false,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","title":"Potential outcomes and causal identification"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/sample-mean","kind":"node","lean":"MathlibExt.Probability.Causality.sampleMean","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#sample-mean","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The mean of `y` over a set of units `S`, `(1/|S|) \u2211_{i \u2208 S} y\u1d62` (the junk value `0` for `S = \u2205`).","title":"Sample mean over a set of units"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/sample-var","kind":"node","lean":"MathlibExt.Probability.Causality.sampleVar","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#sample-var","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The sample variance of `y` over `S` with divisor `|S| - 1`; meaningful for `|S| \u2265 2` (for `|S| \u2264 1` it is the junk value `0`).","title":"Sample variance over a set of units"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/pop-mean","kind":"node","lean":"MathlibExt.Probability.Causality.popMean","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#pop-mean","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The mean of a finite population of `N = Fintype.card \u03b9` units, `(1/N) \u2211\u1d62 y\u1d62`.","title":"Mean of a finite population"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/pop-var","kind":"node","lean":"MathlibExt.Probability.Causality.popVar","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#pop-var","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The variance of a finite population with divisor `N - 1`, `S\u00b2 = \u2211\u1d62 (y\u1d62 - \u0233)\u00b2 / (N - 1)`; meaningful for `N \u2265 2`.","title":"Variance of a finite population"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/ate","kind":"node","lean":"MathlibExt.Probability.Causality.ate","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#ate","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The finite-population average treatment effect `(1/N) \u2211\u1d62 (y\u2081\u1d62 - y\u2080\u1d62)`.","title":"Finite-population average treatment effect"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/diff-in-means","kind":"node","lean":"MathlibExt.Probability.Causality.diffInMeans","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#diff-in-means","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The difference in means for the treated set `S`: the mean of the treated outcomes `y\u2081` over `S` minus the mean of the control outcomes `y\u2080` over the complement.","title":"Difference-in-means estimator"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/neyman-var","kind":"node","lean":"MathlibExt.Probability.Causality.neymanVar","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#neyman-var","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Neyman's variance estimator `s\u2081\u00b2 / n\u2081 + s\u2080\u00b2 / n\u2080` for the treated set `S`.","title":"Neyman's variance estimator"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/rand-exp","kind":"node","lean":"MathlibExt.Probability.Causality.randExp","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#rand-exp","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The expectation of a statistic `f` of the treated set under complete randomization: the treated set is uniform among the `(N choose n\u2081)` sets of `n\u2081` units (`n\u2081 \u2264 N`; for `n\u2081 > N` there is no such set and the value is `0`).","title":"Expectation of a statistic under complete randomization"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/rand-exp-diff-in-means","kind":"node","lean":"MathlibExt.Probability.Causality.randExp_diffInMeans","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#rand-exp-diff-in-means","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Under complete randomization with `n\u2081 \u2265 1` treated and `n\u2080 \u2265 1` control units, the difference in means is unbiased for the finite-population average treatment effect.","title":"Neyman (1923)"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/rand-exp-sq-diff-in-means-sub-ate","kind":"node","lean":"MathlibExt.Probability.Causality.randExp_sq_diffInMeans_sub_ate","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#rand-exp-sq-diff-in-means-sub-ate","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The variance of the difference in means under complete randomization is `S\u2081\u00b2/n\u2081 + S\u2080\u00b2/n\u2080 - S_\u03c4\u00b2/N`, where `S_\u03c4\u00b2` is the population variance of the unit-level effects `\u03c4\u1d62 = y\u2081\u1d62 - y\u2080\u1d62` and `n\u2080 = N - n\u2081`.","title":"Neyman (1923)"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/rand-exp-neyman-var","kind":"node","lean":"MathlibExt.Probability.Causality.randExp_neymanVar","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#rand-exp-neyman-var","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The expectation of Neyman's variance estimator under complete randomization with at least two treated and two control units is `S\u2081\u00b2/n\u2081 + S\u2080\u00b2/n\u2080` (the sample variances are unbiased for the population variances under sampling without replacement).","title":"Expectation of Neyman's variance estimator under complete randomization"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/rand-var-le-rand-exp-neyman-var","kind":"node","lean":"MathlibExt.Probability.Causality.randVar_le_randExp_neymanVar","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#rand-var-le-rand-exp-neyman-var","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Its expectation under complete randomization is at least the variance of the difference in means, with equality exactly when the treatment effect `y\u2081\u1d62 - y\u2080\u1d62` is the same for all units.","title":"Neyman's variance estimator is conservative"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/observed","kind":"node","lean":"MathlibExt.Probability.Causality.observed","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#observed","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The observed outcome: `Y\u2081` on the treated event `T`, `Y\u2080` off it (consistency).","title":"Observed outcome from potential outcomes"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/propensity","kind":"node","lean":"MathlibExt.Probability.Causality.propensity","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#propensity","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The propensity of the event `T` given `W = w`, `P(T \u2229 {W = w}) / P(W = w)`. Meaningful for `P(W = w) > 0`; it is `0` (a junk value) when `P(W = w) = 0`.","title":"Propensity score"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/cond-indep-discrete","kind":"node","lean":"MathlibExt.Probability.Causality.CondIndepDiscrete","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#cond-indep-discrete","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The event `T` is conditionally independent of `V` given `W`, stated through the values of `W`: `P(T, V \u2208 B, W = w) P(W = w) = P(T, W = w) P(V \u2208 B, W = w)` for all measurable `B` and all `w`. For a `W` with countably many values this is the conditional independence of the indicator of `T` and `V` given `W`.","title":"Conditional independence given a discrete variable"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/strongly-ignorable","kind":"node","lean":"MathlibExt.Probability.Causality.StronglyIgnorable","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#strongly-ignorable","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Strong ignorability of the treatment `T` given `W` for the potential outcomes `V` (Rosenbaum & Rubin 1983): `T \u22a5 V | W` and overlap, `0 < P(T | W = w) < 1` for every value `w` of positive probability.","title":"Strong ignorability"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/comp","kind":"node","lean":"MathlibExt.Probability.Causality.StronglyIgnorable.comp","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#comp","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Strong ignorability is inherited by measurable functions of the potential outcomes.","title":"Strong ignorability passes to functions of the potential outcomes"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/cond-indep-discrete-propensity","kind":"node","lean":"MathlibExt.Probability.Causality.condIndepDiscrete_propensity","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#cond-indep-discrete-propensity","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The propensity score `e(X) = P(T | X)` is a balancing score: given `e(X)`, the treatment is conditionally independent of the covariate `X`.","title":"Rosenbaum & Rubin (1983), Theorem 1"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/propensity-propensity","kind":"node","lean":"MathlibExt.Probability.Causality.propensity_propensity","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#propensity-propensity","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"The propensity score is calibrated: among units with score `c` (an event of positive probability), the proportion treated is `c`.","title":"Calibration of the propensity score"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/cond-indep-discrete-comp-iff","kind":"node","lean":"MathlibExt.Probability.Causality.condIndepDiscrete_comp_iff","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#cond-indep-discrete-comp-iff","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"A function `b(X)` of the covariate is a balancing score (`T \u22a5 X | b(X)`) if and only if it is finer than the propensity score: covariate values of positive probability with the same `b` have the same propensity. (Theorem 1 is the case `b = e`.)","title":"Rosenbaum & Rubin (1983), Theorem 2"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/comp-of-balancing","kind":"node","lean":"MathlibExt.Probability.Causality.StronglyIgnorable.comp_of_balancing","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#comp-of-balancing","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"If treatment assignment is strongly ignorable given the covariate `X`, it is strongly ignorable given any balancing score `b(X)`.","title":"Rosenbaum & Rubin (1983), Theorem 3"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/strongly-ignorable-propensity","kind":"node","lean":"MathlibExt.Probability.Causality.stronglyIgnorable_propensity","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#strongly-ignorable-propensity","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Rosenbaum & Rubin (1983), Theorem 3 for the propensity score: if treatment assignment is strongly ignorable given the covariate `X`, it is strongly ignorable given `e(X)`.","title":"Rosenbaum & Rubin (1983), Theorem 3"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/integral-ipw-treated","kind":"node","lean":"MathlibExt.Probability.Causality.integral_ipw_treated","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#integral-ipw-treated","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Inverse probability weighting (Horvitz & Thompson 1952; Rosenbaum 1987): under strong ignorability of `T` given a finitely-valued covariate `X` for the treated potential outcome `Y\u2081`, `E[1_T Y / e(X)] = E[Y\u2081]`, where `Y` is the observed outcome.","title":"Inverse probability weighting"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/integral-ipw-control","kind":"node","lean":"MathlibExt.Probability.Causality.integral_ipw_control","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#integral-ipw-control","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Inverse probability weighting, control arm: under strong ignorability of `T` given `X` for `Y\u2080`, `E[1_{T\u1d9c} Y / (1 - e(X))] = E[Y\u2080]`.","title":"Inverse probability weighting"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/integral-ipw-ate","kind":"node","lean":"MathlibExt.Probability.Causality.integral_ipw_ate","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#integral-ipw-ate","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Under strong ignorability of `T` given a finitely-valued `X` for the pair `(Y\u2080, Y\u2081)`, `E[1_T Y / e(X) - 1_{T\u1d9c} Y / (1 - e(X))] = E[Y\u2081 - Y\u2080]`.","title":"Identification of the average treatment effect by inverse probability weighting"},{"id":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification/integral-ipw-treated-cond-exp","kind":"node","lean":"MathlibExt.Probability.Causality.integral_ipw_treated_condExp","parent":21615,"result":true,"route":"../roadmap/Probability/nonparametric-survival/potential-outcomes-and-causal-identification/index.html#integral-ipw-treated-cond-exp","scope":"Probability/nonparametric-survival/potential-outcomes-and-causal-identification","status":"fully_proved","summary":"Let the covariate information be a sub-\u03c3-algebra `m` (for example the one generated by `X`), and `e = P(T | m)` the propensity score. If `Y\u2081` is integrable, the treatment is mean-ignorable for `Y\u2081`, `E[1_T Y\u2081 | m] = e E[Y\u2081 | m]` (implied by `Y\u2081 \u22a5 T | m`), and overlap holds in the strong form `e \u2265 \u03b5 > 0`, then `E[1_T Y / e] = E[Y\u2081]` for the observed outcome `Y`.","title":"Inverse probability weighting with a general covariate"},{"data_url":"scopes/Probability/nonparametric-survival/u-statistics.json","id":"Probability/nonparametric-survival/u-statistics","kind":"scope","navigation":"module","parent":21566,"result":false,"route":"../roadmap/Probability/nonparametric-survival/u-statistics/index.html","scope":"Probability/nonparametric-survival/u-statistics","status":"fully_proved","title":"U-statistics"},{"id":"Probability/nonparametric-survival/u-statistics/u-statistic","kind":"node","lean":"ProbabilityTheory.uStatistic","parent":21642,"result":true,"route":"../roadmap/Probability/nonparametric-survival/u-statistics/index.html#u-statistic","scope":"Probability/nonparametric-survival/u-statistics","status":"fully_proved","summary":"The U-statistic with kernel `h` of degree `m` on a sample `x` of size `n`: the average of `h (x \u2218 f)` over the `n.descFactorial m` injective maps `f : Fin m \u21aa Fin n`. For `m > n` there are no such maps and the value is `0` (Lean's `a / 0 = 0`).","title":"U-statistic"},{"id":"Probability/nonparametric-survival/u-statistics/hoeffding-proj","kind":"node","lean":"ProbabilityTheory.hoeffdingProj","parent":21642,"result":true,"route":"../roadmap/Probability/nonparametric-survival/u-statistics/index.html#hoeffding-proj","scope":"Probability/nonparametric-survival/u-statistics","status":"fully_proved","summary":"Hoeffding's `c`-th projection of a kernel `h` of degree `m` under `\u03bc`: `h_c(y) = E h(y\u2081, \u2026, y_c, X_{c+1}, \u2026, X_m)` with `X_{c+1}, \u2026, X_m` i.i.d. `\u03bc`.","title":"Hoeffding projection of a kernel"},{"id":"Probability/nonparametric-survival/u-statistics/integral-u-statistic","kind":"node","lean":"ProbabilityTheory.integral_uStatistic","parent":21642,"result":true,"route":"../roadmap/Probability/nonparametric-survival/u-statistics/index.html#integral-u-statistic","scope":"Probability/nonparametric-survival/u-statistics","status":"fully_proved","summary":"For an i.i.d. sample of size `n \u2265 m` and an integrable kernel, the U-statistic has mean `E h(X\u2081, \u2026, X_m)`.","title":"Unbiasedness"},{"id":"Probability/nonparametric-survival/u-statistics/variance-u-statistic","kind":"node","lean":"ProbabilityTheory.variance_uStatistic","parent":21642,"result":true,"route":"../roadmap/Probability/nonparametric-survival/u-statistics/index.html#variance-u-statistic","scope":"Probability/nonparametric-survival/u-statistics","status":"fully_proved","summary":"Hoeffding's variance formula (1948, \u00a75): for an i.i.d. sample of size `n \u2265 m` and a symmetric square-integrable kernel `h` of degree `m`, `Var U = C(n, m)\u207b\u00b9 \u2211_{c=1}^{m} C(m, c) C(n - m, m - c) Var h_c`.","title":"Hoeffding's variance formula"},{"data_url":"scopes/Probability/foundations.json","id":"Probability/foundations","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/foundations/index.html","scope":"Probability/foundations","status":"fully_proved","summary":"Construction of probability measures and processes (Kolmogorov extension, measures from Stieltjes functions), independence, conditional expectation and regular conditional distributions, exchangeability and de Finetti's theorem, and zero\u2013one laws for sequences (Hewitt\u2013Savage); zero\u2013one laws for continuous processes go to Brownian motion and stochastic calculus, tail sigma-algebras of random walks to Random walks.","title":"Foundations of probability"},{"id":"Probability/foundations/de-finetti-binary","kind":"node","lean":"MathlibExt.Probability.DeFinettiWanted.deFinetti_binary","parent":21647,"result":true,"route":"../roadmap/Probability/foundations/index.html#de-finetti-binary","scope":"Probability/foundations","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on the space $\\{0, 1\\}^{\\mathbb{N}}$ of binary sequences $\\omega = (\\omega_i)_{i \\in \\mathbb{N}}$, with $0$ and $1$ standing for false and true. Suppose $\\mu$ is exchangeable: for every permutation $\\pi$ of $\\mathbb{N}$ that moves only finitely many points, the image of $\\mu$ under the map $\\omega \\mapsto \\omega \\circ \\pi$ is $\\mu$. Then there is a probability measure $\\nu$ on\u2026","title":"De Finetti's theorem for binary sequences"},{"id":"Probability/foundations/kolmogorov-extension","kind":"node","lean":"MathlibExt.Probability.KolmogorovExtensionWanted.kolmogorov_extension","parent":21647,"result":true,"route":"../roadmap/Probability/foundations/index.html#kolmogorov-extension","scope":"Probability/foundations","status":"fully_proved","summary":"Let $(\\alpha_i)_{i \\in \\iota}$ be a family of Polish spaces, each with its Borel \u03c3-algebra, indexed by an arbitrary type $\\iota$. For each finite set $J \\subseteq \\iota$ let $P_J$ be a probability measure on $\\prod_{j \\in J} \\alpha_j$, and suppose the family is projective: whenever $J \\subseteq I$ are finite, the image of $P_I$ under the restriction map $\\prod_{i \\in I} \\alpha_i \\to \\prod_{j \\in J} \\alpha_j$ is\u2026","title":"Kolmogorov extension theorem"},{"data_url":"scopes/Probability/foundations/conditional-expectation-as-the-best-predictor.json","id":"Probability/foundations/conditional-expectation-as-the-best-predictor","kind":"scope","navigation":"module","parent":21647,"result":false,"route":"../roadmap/Probability/foundations/conditional-expectation-as-the-best-predictor/index.html","scope":"Probability/foundations/conditional-expectation-as-the-best-predictor","status":"fully_proved","title":"Conditional expectation as the best predictor"},{"id":"Probability/foundations/conditional-expectation-as-the-best-predictor/integral-sub-cond-exp-mul-eq-zero","kind":"node","lean":"MeasureTheory.integral_sub_condExp_mul_eq_zero","parent":21650,"result":true,"route":"../roadmap/Probability/foundations/conditional-expectation-as-the-best-predictor/index.html#integral-sub-cond-exp-mul-eq-zero","scope":"Probability/foundations/conditional-expectation-as-the-best-predictor","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $(\\Omega, \\mathcal{F})$, let $\\mathcal{G} \\subseteq \\mathcal{F}$ be a sub-\u03c3-algebra, and let $Y : \\Omega \\to \\mathbb{R}$ be square-integrable with respect to $\\mu$. Write $\\mathbb{E}[Y \\mid \\mathcal{G}]$ for the conditional expectation of $Y$ with respect to $\\mu$ given $\\mathcal{G}$. Then for every strongly $\\mathcal{G}$-measurable $h : \\Omega \\to \\mathbb{R}$\u2026","title":"Orthogonality of the conditional-expectation residual"},{"id":"Probability/foundations/conditional-expectation-as-the-best-predictor/integral-sq-sub-eq-integral-sq-sub-cond","kind":"node","lean":"MeasureTheory.integral_sq_sub_eq_integral_sq_sub_condExp_add","parent":21650,"result":true,"route":"../roadmap/Probability/foundations/conditional-expectation-as-the-best-predictor/index.html#integral-sq-sub-eq-integral-sq-sub-cond","scope":"Probability/foundations/conditional-expectation-as-the-best-predictor","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $(\\Omega, \\mathcal{F})$, let $\\mathcal{G} \\subseteq \\mathcal{F}$ be a sub-\u03c3-algebra, and let $Y : \\Omega \\to \\mathbb{R}$ be square-integrable with respect to $\\mu$. Write $\\mathbb{E}[Y \\mid \\mathcal{G}]$ for the conditional expectation of $Y$ with respect to $\\mu$ given $\\mathcal{G}$. Then for every strongly $\\mathcal{G}$-measurable $g : \\Omega \\to \\mathbb{R}$\u2026","title":"Pythagorean identity for the conditional mean"},{"id":"Probability/foundations/conditional-expectation-as-the-best-predictor/integral-sq-sub-cond-exp-le","kind":"node","lean":"MeasureTheory.integral_sq_sub_condExp_le","parent":21650,"result":true,"route":"../roadmap/Probability/foundations/conditional-expectation-as-the-best-predictor/index.html#integral-sq-sub-cond-exp-le","scope":"Probability/foundations/conditional-expectation-as-the-best-predictor","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $(\\Omega, \\mathcal{F})$, let $\\mathcal{G} \\subseteq \\mathcal{F}$ be a sub-\u03c3-algebra, and let $Y : \\Omega \\to \\mathbb{R}$ be square-integrable with respect to $\\mu$. Write $\\mathbb{E}[Y \\mid \\mathcal{G}]$ for the conditional expectation of $Y$ with respect to $\\mu$ given $\\mathcal{G}$. Then for every strongly $\\mathcal{G}$-measurable $g : \\Omega \\to \\mathbb{R}$\u2026","title":"The conditional mean minimizes the expected squared error"},{"id":"Probability/foundations/conditional-expectation-as-the-best-predictor/integral-sq-sub-cond-exp-comap-le","kind":"node","lean":"MeasureTheory.integral_sq_sub_condExp_comap_le","parent":21650,"result":true,"route":"../roadmap/Probability/foundations/conditional-expectation-as-the-best-predictor/index.html#integral-sq-sub-cond-exp-comap-le","scope":"Probability/foundations/conditional-expectation-as-the-best-predictor","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $\\Omega$, let $X : \\Omega \\to \\mathcal{X}$ be a measurable map into a measurable space $\\mathcal{X}$, and let $Y : \\Omega \\to \\mathbb{R}$ be square-integrable with respect to $\\mu$. Write $\\mathbb{E}[Y \\mid X]$ for the conditional expectation of $Y$ with respect to $\\mu$ given the \u03c3-algebra generated by $X$. Then for every measurable\u2026","title":"The regression function is the Bayes-optimal regressor"},{"data_url":"scopes/Probability/foundations/hewitt-savage-zero-one-law.json","id":"Probability/foundations/hewitt-savage-zero-one-law","kind":"scope","navigation":"module","parent":21647,"result":false,"route":"../roadmap/Probability/foundations/hewitt-savage-zero-one-law/index.html","scope":"Probability/foundations/hewitt-savage-zero-one-law","status":"fully_proved","title":"Hewitt\u2013Savage zero\u2013one law"},{"id":"Probability/foundations/hewitt-savage-zero-one-law/hewitt-savage-zero-one-general","kind":"node","lean":"MathlibExt.Probability.HewittSavageWanted.hewitt_savage_zero_one_general","parent":21655,"result":true,"route":"../roadmap/Probability/foundations/hewitt-savage-zero-one-law/index.html#hewitt-savage-zero-one-general","scope":"Probability/foundations/hewitt-savage-zero-one-law","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $S$, and let $E \\subseteq S^{\\mathbb{N}}$ be a set that is measurable for the product \u03c3-algebra and invariant under every finitely supported permutation $\\sigma$ of $\\mathbb{N}$, that is, $\\{\\omega : \\omega \\circ \\sigma \\in E\\} = E$ whenever $\\sigma$ moves only finitely many points. Then $E$ has measure $0$ or $1$ under the product measure\u2026","title":"Hewitt\u2013Savage zero\u2013one law"},{"id":"Probability/foundations/hewitt-savage-zero-one-law/hewitt-savage-zero-one","kind":"node","lean":"MathlibExt.Probability.HewittSavageWanted.hewitt_savage_zero_one","parent":21655,"result":true,"route":"../roadmap/Probability/foundations/hewitt-savage-zero-one-law/index.html#hewitt-savage-zero-one","scope":"Probability/foundations/hewitt-savage-zero-one-law","status":"fully_proved","summary":"Let $S$ be a Polish space with its Borel \u03c3-algebra, let $\\mu$ be a probability measure on $S$, and let $E \\subseteq S^{\\mathbb{N}}$ be a set that is measurable for the product \u03c3-algebra and invariant under every finitely supported permutation $\\sigma$ of $\\mathbb{N}$, that is, $\\{\\omega : \\omega \\circ \\sigma \\in E\\} = E$ whenever $\\sigma$ moves only finitely many points. Then $E$ has measure $0$ or $1$ under the\u2026","title":"Hewitt\u2013Savage zero\u2013one law on a Polish space"},{"id":"Probability/foundations/hewitt-savage-zero-one-law/is-exchangeable-event","kind":"node","lean":"MathlibExt.Probability.HewittSavageSequence.IsExchangeableEvent","parent":21655,"result":true,"route":"../roadmap/Probability/foundations/hewitt-savage-zero-one-law/index.html#is-exchangeable-event","scope":"Probability/foundations/hewitt-savage-zero-one-law","status":"fully_proved","summary":"Let $S$ be a measurable space. A set $A \\subseteq S^{\\mathbb{N}}$ is an exchangeable (or permutable) event if it is measurable for the product \u03c3-algebra and $\\{x : x \\circ \\sigma \\in A\\} = A$ for every permutation $\\sigma$ of $\\mathbb{N}$ that moves only finitely many indices.","title":"Exchangeable event"},{"id":"Probability/foundations/hewitt-savage-zero-one-law/is-exchangeable-event-wrt","kind":"node","lean":"MathlibExt.Probability.HewittSavageSequence.IsExchangeableEventWrt","parent":21655,"result":true,"route":"../roadmap/Probability/foundations/hewitt-savage-zero-one-law/index.html#is-exchangeable-event-wrt","scope":"Probability/foundations/hewitt-savage-zero-one-law","status":"fully_proved","summary":"Let $\\Omega$ and $S$ be measurable spaces and let $X_0, X_1, \\ldots : \\Omega \\to S$ be a sequence of maps. An event $E \\subseteq \\Omega$ is exchangeable with respect to $(X_n)$ if there is an exchangeable event $A \\subseteq S^{\\mathbb{N}}$ such that $E = \\{\\omega \\in \\Omega : (X_n(\\omega))_{n \\in \\mathbb{N}} \\in A\\}$.","title":"Event exchangeable with respect to a sequence"},{"id":"Probability/foundations/hewitt-savage-zero-one-law/measurable-set","kind":"node","lean":"MathlibExt.Probability.HewittSavageSequence.IsExchangeableEventWrt.measurableSet","parent":21655,"result":true,"route":"../roadmap/Probability/foundations/hewitt-savage-zero-one-law/index.html#measurable-set","scope":"Probability/foundations/hewitt-savage-zero-one-law","status":"fully_proved","summary":"Let $\\Omega$ and $S$ be measurable spaces and let $X_0, X_1, \\ldots : \\Omega \\to S$ be measurable maps. If $E \\subseteq \\Omega$ is exchangeable with respect to $(X_n)$, then $E$ is measurable.","title":"Exchangeable events of a measurable sequence are measurable"},{"id":"Probability/foundations/hewitt-savage-zero-one-law/hewitt-savage-zero-one-2","kind":"node","lean":"MathlibExt.Probability.HewittSavageSequence.hewittSavage_zeroOne","parent":21655,"result":true,"route":"../roadmap/Probability/foundations/hewitt-savage-zero-one-law/index.html#hewitt-savage-zero-one-2","scope":"Probability/foundations/hewitt-savage-zero-one-law","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $S$ be a measurable space, and let $X_0, X_1, X_2, \\ldots : \\Omega \\to S$ be measurable maps that are independent under $\\mu$ and each have the same distribution under $\\mu$ as $X_0$. If an event $E \\subseteq \\Omega$ is exchangeable with respect to $(X_n)$, then $\\mu(E) = 0$ or $\\mu(E) = 1$.","title":"Hewitt\u2013Savage zero\u2013one law for i.i.d. sequences"},{"data_url":"scopes/Probability/foundations/probability-measures-from-distribution-functions.json","id":"Probability/foundations/probability-measures-from-distribution-functions","kind":"scope","navigation":"module","parent":21647,"result":false,"route":"../roadmap/Probability/foundations/probability-measures-from-distribution-functions/index.html","scope":"Probability/foundations/probability-measures-from-distribution-functions","status":"fully_proved","title":"Probability measures from distribution functions"},{"id":"Probability/foundations/probability-measures-from-distribution-functions/stieltjes-measure-construction","kind":"node","lean":"MathlibExt.Probability.StieltjesMeasureConstruction.stieltjes_measure_construction","parent":21662,"result":true,"route":"../roadmap/Probability/foundations/probability-measures-from-distribution-functions/index.html#stieltjes-measure-construction","scope":"Probability/foundations/probability-measures-from-distribution-functions","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a Stieltjes function, that is, non-decreasing and right-continuous, with $f(x) \\to 0$ as $x \\to -\\infty$ and $f(x) \\to 1$ as $x \\to \\infty$, and let $\\mu_f$ be its Stieltjes measure on $\\mathbb{R}$. Then $\\mu_f$ is a probability measure, $\\mu_f((a, b]) = f(b) - f(a)$ for all real $a \\le b$, and every finite measure $\\nu$ on $\\mathbb{R}$ with $\\nu((a, b]) = f(b) - f(a)$ for all\u2026","title":"Construction of measures on \u211d"},{"id":"Probability/foundations/probability-measures-from-distribution-functions/exists-unique-probability-measure-of-sti","kind":"node","lean":"MathlibExt.Probability.StieltjesMeasureConstruction.exists_unique_probabilityMeasure_of_stieltjes","parent":21662,"result":true,"route":"../roadmap/Probability/foundations/probability-measures-from-distribution-functions/index.html#exists-unique-probability-measure-of-sti","scope":"Probability/foundations/probability-measures-from-distribution-functions","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a Stieltjes function, that is, non-decreasing and right-continuous, with $f(x) \\to 0$ as $x \\to -\\infty$ and $f(x) \\to 1$ as $x \\to \\infty$. Then there is exactly one measure $\\mu$ on $\\mathbb{R}$ that is a probability measure and satisfies $\\mu((a, b]) = f(b) - f(a)$ for all real $a \\le b$.","title":"Construction of measures on \u211d, as unique existence"},{"data_url":"scopes/Probability/foundations/regular-conditional-distributions.json","id":"Probability/foundations/regular-conditional-distributions","kind":"scope","navigation":"module","parent":21647,"result":false,"route":"../roadmap/Probability/foundations/regular-conditional-distributions/index.html","scope":"Probability/foundations/regular-conditional-distributions","status":"fully_proved","title":"Regular conditional distributions"},{"id":"Probability/foundations/regular-conditional-distributions/regular-conditional-distribution","kind":"node","lean":"ProbabilityTheory.RegularConditionalDistribution","parent":21665,"result":true,"route":"../roadmap/Probability/foundations/regular-conditional-distributions/index.html#regular-conditional-distribution","scope":"Probability/foundations/regular-conditional-distributions","status":"fully_proved","summary":"Let $P$ be a measure on a measurable space $(\\Omega, \\mathcal{F})$, let $S$ be a measurable space, $X : \\Omega \\to S$ a map, $\\mathcal{G}$ a \u03c3-algebra on $\\Omega$, and $\\rho$ a family assigning a measure $\\rho_\\omega$ on $S$ to each $\\omega \\in \\Omega$. Then $\\rho$ is a regular conditional distribution of $X$ given $\\mathcal{G}$ under $P$ if $\\mathcal{G} \\subseteq \\mathcal{F}$; $X$ is $\\mathcal{F}$-measurable; for\u2026","title":"Regular conditional distribution (measure-valued family)"},{"id":"Probability/foundations/regular-conditional-distributions/is-regular-conditional-distribution","kind":"node","lean":"ProbabilityTheory.IsRegularConditionalDistribution","parent":21665,"result":true,"route":"../roadmap/Probability/foundations/regular-conditional-distributions/index.html#is-regular-conditional-distribution","scope":"Probability/foundations/regular-conditional-distributions","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\alpha$ and let $X : \\alpha \\to \\beta$ and $Y : \\alpha \\to \\Omega$ be maps. A kernel $\\kappa$ from $\\beta$ to $\\Omega$ is a regular conditional distribution of $Y$ given $X$ under $\\mu$ if $X$ and $Y$ are measurable, $\\kappa$ is a Markov kernel, and for every measurable set $s \\subseteq \\Omega$ the real-valued function $a \\mapsto \\kappa(X(a))(s)$ is $\\mu$-almost everywhere equal to the\u2026","title":"Regular conditional distribution (kernel form)"},{"id":"Probability/foundations/regular-conditional-distributions/cond-distrib-is-regular-conditional-dist","kind":"node","lean":"ProbabilityTheory.condDistrib_isRegularConditionalDistribution","parent":21665,"result":true,"route":"../roadmap/Probability/foundations/regular-conditional-distributions/index.html#cond-distrib-is-regular-conditional-dist","scope":"Probability/foundations/regular-conditional-distributions","status":"fully_proved","summary":"Let $\\Omega$ be a nonempty standard Borel space, $\\mu$ a finite measure on $\\alpha$, and $X : \\alpha \\to \\beta$, $Y : \\alpha \\to \\Omega$ measurable maps. Then the conditional distribution of $Y$ given $X$ under $\\mu$, the kernel from $\\beta$ to $\\Omega$ obtained by disintegrating the joint law of $(X, Y)$ under $\\mu$, is a regular conditional distribution of $Y$ given $X$ under $\\mu$.","title":"The conditional distribution kernel is a regular conditional distribution"},{"id":"Probability/foundations/regular-conditional-distributions/exists-regular-conditional-distribution","kind":"node","lean":"ProbabilityTheory.exists_regularConditionalDistribution","parent":21665,"result":true,"route":"../roadmap/Probability/foundations/regular-conditional-distributions/index.html#exists-regular-conditional-distribution","scope":"Probability/foundations/regular-conditional-distributions","status":"fully_proved","summary":"Let $\\Omega$ be a nonempty standard Borel space, $\\mu$ a finite measure on $\\alpha$, and $X : \\alpha \\to \\beta$, $Y : \\alpha \\to \\Omega$ measurable maps. Then there exists a kernel $\\kappa$ from $\\beta$ to $\\Omega$ that is a regular conditional distribution of $Y$ given $X$ under $\\mu$; in particular $\\kappa$ is a Markov kernel.","title":"Existence of regular conditional distributions (kernel form)"},{"id":"Probability/foundations/regular-conditional-distributions/exists-regular-conditional-distribution-2","kind":"node","lean":"ProbabilityTheory.exists_regularConditionalDistribution_family_of_le","parent":21665,"result":true,"route":"../roadmap/Probability/foundations/regular-conditional-distributions/index.html#exists-regular-conditional-distribution-2","scope":"Probability/foundations/regular-conditional-distributions","status":"fully_proved","summary":"Let $\\Omega$ be a nonempty standard Borel space, $\\mu$ a finite measure on a measurable space $(\\alpha, \\mathcal{F})$, $\\mathcal{G} \\subseteq \\mathcal{F}$ a sub-\u03c3-algebra, and $Y : \\alpha \\to \\Omega$ an $\\mathcal{F}$-measurable map. Then some family of measures on $\\Omega$ indexed by $\\alpha$ is a measure-valued regular conditional distribution of $Y$ given $\\mathcal{G}$ under $\\mu$.","title":"Existence of measure-valued regular conditional distributions given a sub-\u03c3-algebra"},{"id":"Probability/foundations/regular-conditional-distributions/exists-regular-conditional-distribution-3","kind":"node","lean":"ProbabilityTheory.exists_regularConditionalDistribution_family","parent":21665,"result":true,"route":"../roadmap/Probability/foundations/regular-conditional-distributions/index.html#exists-regular-conditional-distribution-3","scope":"Probability/foundations/regular-conditional-distributions","status":"fully_proved","summary":"Let $\\Omega$ be a nonempty standard Borel space, $\\mu$ a finite measure on $\\alpha$, and $X : \\alpha \\to \\beta$, $Y : \\alpha \\to \\Omega$ measurable maps. Then some family of measures on $\\Omega$ indexed by $\\alpha$ is a measure-valued regular conditional distribution of $Y$ given the \u03c3-algebra $\\sigma(X)$ on $\\alpha$ generated by $X$, under $\\mu$.","title":"Existence of measure-valued regular conditional distributions given a random variable"},{"data_url":"scopes/Probability/characteristic-functions.json","id":"Probability/characteristic-functions","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/characteristic-functions/index.html","scope":"Probability/characteristic-functions","status":"fully_proved","summary":"Characteristic functions, probability generating functions and positive-definite functions (Bochner), Taylor bounds for characteristic functions, lattice support, L\u00e9vy's continuity theorem and Cram\u00e9r's decomposition theorem for the normal law.","title":"Characteristic functions and generating functions"},{"id":"Probability/characteristic-functions/cramer-decomposition","kind":"node","lean":"MathlibExt.Probability.CramerDecompositionWanted.cramer_decomposition","parent":21672,"result":true,"route":"../roadmap/Probability/characteristic-functions/index.html#cramer-decomposition","scope":"Probability/characteristic-functions","status":"fully_proved","summary":"Let $X$ and $Y$ be measurable real random variables on a probability space. If $X$ and $Y$ are independent and $X + Y$ has a Gaussian law, then $X$ and $Y$ each have a Gaussian law. Here a Gaussian law may be degenerate, that is, a point mass.","title":"Cram\u00e9r's decomposition theorem"},{"id":"Probability/characteristic-functions/levy-continuity-of-char-fun","kind":"node","lean":"MathlibExt.Probability.LevyContinuityWanted.levy_continuity_of_charFun","parent":21672,"result":true,"route":"../roadmap/Probability/characteristic-functions/index.html#levy-continuity-of-char-fun","scope":"Probability/characteristic-functions","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$, let $(\\mu_n)_{n \\in \\mathbb{N}}$ be a sequence of probability measures on the Euclidean space $\\mathbb{R}^d$, and let $\\varphi : \\mathbb{R}^d \\to \\mathbb{C}$. Suppose that for every $t \\in \\mathbb{R}^d$ the characteristic functions $\\hat\\mu_n$ of the $\\mu_n$ satisfy $\\hat\\mu_n(t) \\to \\varphi(t)$ as $n \\to \\infty$, and that $\\varphi$ is continuous at $0$. Then there is a probability measure\u2026","title":"L\u00e9vy's continuity theorem"},{"data_url":"scopes/Probability/characteristic-functions/bochner-s-theorem.json","id":"Probability/characteristic-functions/bochner-s-theorem","kind":"scope","navigation":"module","parent":21672,"result":false,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","title":"Bochner's theorem"},{"id":"Probability/characteristic-functions/bochner-s-theorem/bochner","kind":"node","lean":"MathlibExt.Analysis.Fourier.BochnerWanted.bochner","parent":21675,"result":true,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html#bochner","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","summary":"Let $f : \\mathbb{R} \\to \\mathbb{C}$. Then $f$ is continuous, satisfies $f(0) = 1$, and is positive definite, meaning that for every $n \\in \\mathbb{N}$, all points $x_1, \\dots, x_n \\in \\mathbb{R}$ and all coefficients $c_1, \\dots, c_n \\in \\mathbb{C}$ the sum $S = \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\overline{c_i}\\, c_j\\, f(x_i - x_j)$ is a nonnegative real number, if and only if there is a Borel probability measure $\\mu$\u2026","title":"Bochner's theorem"},{"id":"Probability/characteristic-functions/bochner-s-theorem/is-positive-definite","kind":"node","lean":"ProbabilityTheory.IsPositiveDefinite","parent":21675,"result":true,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html#is-positive-definite","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","summary":"A function $\\varphi : \\mathbb{R} \\to \\mathbb{C}$ is positive definite when for every $n \\in \\mathbb{N}$, all $t_1, \\dots, t_n \\in \\mathbb{R}$ and all $z_1, \\dots, z_n \\in \\mathbb{C}$, the sum $S = \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\varphi(t_i - t_j) \\, z_i \\, \\overline{z_j}$ is real and nonnegative, that is, $\\operatorname{Im} S = 0$ and $\\operatorname{Re} S \\ge 0$.","title":"Positive definite functions"},{"id":"Probability/characteristic-functions/bochner-s-theorem/is-positive-definite-real","kind":"node","lean":"ProbabilityTheory.IsPositiveDefiniteReal","parent":21675,"result":true,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html#is-positive-definite-real","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","summary":"A function $\\varphi : \\mathbb{R} \\to \\mathbb{R}$ is positive definite when its composition with the inclusion $\\mathbb{R} \\to \\mathbb{C}$ is positive definite, that is, when for every $n \\in \\mathbb{N}$, all $t_1, \\dots, t_n \\in \\mathbb{R}$ and all $z_1, \\dots, z_n \\in \\mathbb{C}$, the sum $\\sum_{i=1}^{n} \\sum_{j=1}^{n} \\varphi(t_i - t_j) \\, z_i \\, \\overline{z_j}$ is a nonnegative real number.","title":"Positive definite real-valued functions"},{"id":"Probability/characteristic-functions/bochner-s-theorem/is-characteristic-fun","kind":"node","lean":"ProbabilityTheory.IsCharacteristicFun","parent":21675,"result":true,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html#is-characteristic-fun","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","summary":"A function $\\varphi : \\mathbb{R} \\to \\mathbb{C}$ is a characteristic function when there is a probability measure $\\mu$ on $\\mathbb{R}$ whose characteristic function equals $\\varphi$, that is, $\\int_{\\mathbb{R}} e^{itx} \\, d\\mu(x) = \\varphi(t)$ for every $t \\in \\mathbb{R}$.","title":"Characteristic functions"},{"id":"Probability/characteristic-functions/bochner-s-theorem/is-characteristic-fun-real","kind":"node","lean":"ProbabilityTheory.IsCharacteristicFunReal","parent":21675,"result":true,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html#is-characteristic-fun-real","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","summary":"A function $\\varphi : \\mathbb{R} \\to \\mathbb{R}$ is a real-valued characteristic function when its composition with the inclusion $\\mathbb{R} \\to \\mathbb{C}$ is a characteristic function, that is, when there is a probability measure $\\mu$ on $\\mathbb{R}$ with $\\int_{\\mathbb{R}} e^{itx} \\, d\\mu(x) = \\varphi(t)$ for every $t \\in \\mathbb{R}$.","title":"Real-valued characteristic functions"},{"id":"Probability/characteristic-functions/bochner-s-theorem/bochner-forward","kind":"node","lean":"ProbabilityTheory.bochner_forward","parent":21675,"result":true,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html#bochner-forward","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","summary":"Let $\\varphi : \\mathbb{R} \\to \\mathbb{C}$ be the characteristic function of a probability measure on $\\mathbb{R}$. Then $\\varphi$ is continuous, positive definite, and satisfies $\\varphi(0) = 1$.","title":"Forward direction of Bochner's theorem"},{"id":"Probability/characteristic-functions/bochner-s-theorem/char-fun-is-positive-definite","kind":"node","lean":"ProbabilityTheory.charFun_isPositiveDefinite","parent":21675,"result":true,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html#char-fun-is-positive-definite","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with characteristic function $\\varphi_\\mu(t) = \\int_{\\mathbb{R}} e^{itx} \\, d\\mu(x)$. Then $\\varphi_\\mu$ is positive definite: for every $n \\in \\mathbb{N}$, all $t_1, \\dots, t_n \\in \\mathbb{R}$ and all $z_1, \\dots, z_n \\in \\mathbb{C}$, the sum $\\sum_{i=1}^{n} \\sum_{j=1}^{n} \\varphi_\\mu(t_i - t_j) \\, z_i \\, \\overline{z_j}$ is a nonnegative real number.","title":"Characteristic functions are positive definite"},{"id":"Probability/characteristic-functions/bochner-s-theorem/bochner-forward-real","kind":"node","lean":"ProbabilityTheory.bochner_forward_real","parent":21675,"result":true,"route":"../roadmap/Probability/characteristic-functions/bochner-s-theorem/index.html#bochner-forward-real","scope":"Probability/characteristic-functions/bochner-s-theorem","status":"fully_proved","summary":"Let $\\varphi : \\mathbb{R} \\to \\mathbb{R}$ be a real-valued characteristic function, that is, a function whose composition with the inclusion $\\mathbb{R} \\to \\mathbb{C}$ is the characteristic function of a probability measure on $\\mathbb{R}$. Then $\\varphi$ is continuous, $\\varphi$ is positive definite in the sense that its composition with the inclusion $\\mathbb{R} \\to \\mathbb{C}$ is positive definite, and\u2026","title":"Real-valued characteristic functions are continuous and positive definite"},{"data_url":"scopes/Probability/characteristic-functions/lattice-distributions.json","id":"Probability/characteristic-functions/lattice-distributions","kind":"scope","navigation":"module","parent":21672,"result":false,"route":"../roadmap/Probability/characteristic-functions/lattice-distributions/index.html","scope":"Probability/characteristic-functions/lattice-distributions","status":"fully_proved","title":"Lattice distributions"},{"id":"Probability/characteristic-functions/lattice-distributions/is-lattice-rv","kind":"node","lean":"MathlibExt.Probability.CharFunLattice.IsLatticeRV","parent":21684,"result":true,"route":"../roadmap/Probability/characteristic-functions/lattice-distributions/index.html#is-lattice-rv","scope":"Probability/characteristic-functions/lattice-distributions","status":"fully_proved","summary":"A measure $\\mu$ on $\\mathbb{R}$ with characteristic function $\\varphi(t) = \\int e^{itx} \\, d\\mu(x)$ is a lattice random variable if $\\varphi(l) = 1$ for some real $l \\ne 0$ and $\\varphi(t) \\ne 1$ for some real $t$.","title":"Lattice random variable"},{"id":"Probability/characteristic-functions/lattice-distributions/char-fun-eq-one-iff-measure-eq-one-of-la","kind":"node","lean":"MathlibExt.Probability.CharFunLattice.charFun_eq_one_iff_measure_eq_one_of_lattice","parent":21684,"result":true,"route":"../roadmap/Probability/characteristic-functions/lattice-distributions/index.html#char-fun-eq-one-iff-measure-eq-one-of-la","scope":"Probability/characteristic-functions/lattice-distributions","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ with characteristic function $\\varphi(t) = \\int e^{itx} \\, d\\mu(x)$, and let $l \\ne 0$ be real. Then $\\varphi(l) = 1$ if and only if $\\mu$ gives mass $1$ to the lattice $\\{(2\\pi / l) k : k \\in \\mathbb{Z}\\}$.","title":"A characteristic function equal to one detects lattice support"},{"id":"Probability/characteristic-functions/lattice-distributions/exists-lattice-support","kind":"node","lean":"MathlibExt.Probability.CharFunLattice.IsLatticeRV.exists_lattice_support","parent":21684,"result":true,"route":"../roadmap/Probability/characteristic-functions/lattice-distributions/index.html#exists-lattice-support","scope":"Probability/characteristic-functions/lattice-distributions","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\mathbb{R}$ that is a lattice random variable. Then there is a real number $h \\ne 0$ such that $\\mu$ gives mass $1$ to the lattice $\\{hk : k \\in \\mathbb{Z}\\}$.","title":"Lattice random variables are supported on a lattice"},{"data_url":"scopes/Probability/characteristic-functions/probability-generating-functions.json","id":"Probability/characteristic-functions/probability-generating-functions","kind":"scope","navigation":"module","parent":21672,"result":false,"route":"../roadmap/Probability/characteristic-functions/probability-generating-functions/index.html","scope":"Probability/characteristic-functions/probability-generating-functions","status":"fully_proved","title":"Probability generating functions"},{"id":"Probability/characteristic-functions/probability-generating-functions/pgf","kind":"node","lean":"MathlibExt.Probability.pgf","parent":21688,"result":true,"route":"../roadmap/Probability/characteristic-functions/probability-generating-functions/index.html#pgf","scope":"Probability/characteristic-functions/probability-generating-functions","status":"fully_proved","summary":"For a measure $\\mu$ on $\\Omega$ and a function $X : \\Omega \\to \\mathbb{N}$, the probability generating function of $X$ under $\\mu$ is the function $G_X : \\mathbb{R} \\to \\mathbb{R}$ given by","title":"Probability generating function"},{"id":"Probability/characteristic-functions/probability-generating-functions/integrable-pow-of-abs-le-one","kind":"node","lean":"MathlibExt.Probability.integrable_pow_of_abs_le_one","parent":21688,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrability of "],["math","\\(s^X\\)"],["text"," for "],["math","\\(|s| \\le 1\\)"]],"source":"Integrability of $s^X$ for $|s| \\le 1$"},"result":true,"route":"../roadmap/Probability/characteristic-functions/probability-generating-functions/index.html#integrable-pow-of-abs-le-one","scope":"Probability/characteristic-functions/probability-generating-functions","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on $\\Omega$, let $X : \\Omega \\to \\mathbb{N}$ be measurable, and let $s$ be a real number with $|s| \\le 1$. Then the function $\\omega \\mapsto s^{X(\\omega)}$ is integrable with respect to $\\mu$.","title":"Integrability of $s^X$ for $|s| \\le 1$"},{"id":"Probability/characteristic-functions/probability-generating-functions/pgf-one","kind":"node","lean":"MathlibExt.Probability.pgf_one","parent":21688,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generating function at "],["math","\\(s = 1\\)"]],"source":"The generating function at $s = 1$"},"result":true,"route":"../roadmap/Probability/characteristic-functions/probability-generating-functions/index.html#pgf-one","scope":"Probability/characteristic-functions/probability-generating-functions","status":"fully_proved","summary":"For every probability measure $\\mu$ on $\\Omega$ and every function $X : \\Omega \\to \\mathbb{N}$, the probability generating function $G_X(s) = \\int_\\Omega s^{X(\\omega)} \\, d\\mu(\\omega)$ of $X$ under $\\mu$ satisfies $G_X(1) = 1$.","title":"The generating function at $s = 1$"},{"id":"Probability/characteristic-functions/probability-generating-functions/pgf-zero","kind":"node","lean":"MathlibExt.Probability.pgf_zero","parent":21688,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generating function at "],["math","\\(s = 0\\)"]],"source":"The generating function at $s = 0$"},"result":true,"route":"../roadmap/Probability/characteristic-functions/probability-generating-functions/index.html#pgf-zero","scope":"Probability/characteristic-functions/probability-generating-functions","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$ and let $X : \\Omega \\to \\mathbb{N}$ be measurable. Then the probability generating function $G_X(s) = \\int_\\Omega s^{X(\\omega)} \\, d\\mu(\\omega)$ of $X$ under $\\mu$ satisfies $G_X(0) = \\mu(\\lbrace \\omega : X(\\omega) = 0 \\rbrace)$, where the measure of the event is read as a real number, an infinite value being read as $0$.","title":"The generating function at $s = 0$"},{"id":"Probability/characteristic-functions/probability-generating-functions/pgf-eq-tsum","kind":"node","lean":"MathlibExt.Probability.pgf_eq_tsum","parent":21688,"result":true,"route":"../roadmap/Probability/characteristic-functions/probability-generating-functions/index.html#pgf-eq-tsum","scope":"Probability/characteristic-functions/probability-generating-functions","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on $\\Omega$, let $X : \\Omega \\to \\mathbb{N}$ be measurable, and let $s$ be a real number with $|s| \\le 1$. Then the probability generating function $G_X(s) = \\int_\\Omega s^{X(\\omega)} \\, d\\mu(\\omega)$ of $X$ under $\\mu$ satisfies","title":"Series form of the probability generating function"},{"id":"Probability/characteristic-functions/probability-generating-functions/pgf-add-of-indep-fun","kind":"node","lean":"MathlibExt.Probability.pgf_add_of_indepFun","parent":21688,"result":true,"route":"../roadmap/Probability/characteristic-functions/probability-generating-functions/index.html#pgf-add-of-indep-fun","scope":"Probability/characteristic-functions/probability-generating-functions","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on $\\Omega$, and for $Z : \\Omega \\to \\mathbb{N}$ write $G_Z(s) = \\int_\\Omega s^{Z(\\omega)} \\, d\\mu(\\omega)$ for the probability generating function of $Z$ under $\\mu$. If $X, Y : \\Omega \\to \\mathbb{N}$ are measurable and independent under $\\mu$, then $G_{X+Y}(s) = G_X(s) \\, G_Y(s)$ for every real $s$ with $|s| \\le 1$.","title":"Generating function of a sum of independent variables"},{"data_url":"scopes/Probability/characteristic-functions/taylor-bounds-for-characteristic-functions.json","id":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions","kind":"scope","navigation":"module","parent":21672,"result":false,"route":"../roadmap/Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/index.html","scope":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions","status":"fully_proved","title":"Taylor bounds for characteristic functions"},{"id":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/norm-exp-of-real-mul-i-sub-one-sub-le","kind":"node","lean":"MathlibExt.Probability.CharFunTaylorBounds.norm_exp_ofReal_mul_I_sub_one_sub_le","parent":21695,"result":true,"route":"../roadmap/Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/index.html#norm-exp-of-real-mul-i-sub-one-sub-le","scope":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions","status":"fully_proved","summary":"For every real number $u$, $|e^{iu} - 1 - iu| \\le u^2$, with no restriction on the size of $u$.","title":"Global second-order bound for the imaginary exponential"},{"id":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/norm-char-fun-sub-one-sub-le","kind":"node","lean":"MathlibExt.Probability.CharFunTaylorBounds.norm_charFun_sub_one_sub_le","parent":21695,"result":true,"route":"../roadmap/Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/index.html#norm-char-fun-sub-one-sub-le","scope":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions","status":"fully_proved","summary":"Let $X$ be a measurable real random variable on a probability space with $X^2$ integrable, and let $\\varphi_X$ be the characteristic function of the law of $X$. Then for every real $t$, $|\\varphi_X(t) - 1 - it\\,\\mathbb{E}[X]| \\le t^2 \\, \\mathbb{E}[X^2]$.","title":"Second-order Taylor bound for the characteristic function"},{"id":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/norm-char-fun-sub-one-le","kind":"node","lean":"MathlibExt.Probability.CharFunTaylorBounds.norm_charFun_sub_one_le","parent":21695,"result":true,"route":"../roadmap/Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/index.html#norm-char-fun-sub-one-le","scope":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions","status":"fully_proved","summary":"Let $X$ be a measurable real random variable on a probability space with $|X|$ integrable, and let $\\varphi_X$ be the characteristic function of the law of $X$. Then for every real $t$, $|\\varphi_X(t) - 1| \\le |t| \\, \\mathbb{E}|X|$.","title":"First-order Taylor bound for the characteristic function"},{"id":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/norm-char-fun-sub-one-le-sq-mul-variance","kind":"node","lean":"MathlibExt.Probability.CharFunTaylorBounds.norm_charFun_sub_one_le_sq_mul_variance","parent":21695,"result":true,"route":"../roadmap/Probability/characteristic-functions/taylor-bounds-for-characteristic-functions/index.html#norm-char-fun-sub-one-le-sq-mul-variance","scope":"Probability/characteristic-functions/taylor-bounds-for-characteristic-functions","status":"fully_proved","summary":"Let $X$ be a measurable real random variable on a probability space with $\\mathbb{E}[X] = 0$ and $X^2$ integrable, and let $\\varphi_X$ be the characteristic function of the law of $X$. Then for every real $t$, $|\\varphi_X(t) - 1| \\le t^2 \\, \\mathbb{E}[X^2]$.","title":"Characteristic-function second-order bound"},{"data_url":"scopes/Probability/infinite-divisibility-regular-variation.json","id":"Probability/infinite-divisibility-regular-variation","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/index.html","scope":"Probability/infinite-divisibility-regular-variation","status":"fully_proved","summary":"Infinitely divisible and stable distributions, convolution powers, the L\u00e9vy\u2013Khintchin representation, and regularly varying and slowly varying functions (Karamata's theorem, Potter bounds, generalized inverses) with the truncated moments and tail integrals of heavy-tailed laws.","title":"Infinitely divisible laws and regular variation"},{"id":"Probability/infinite-divisibility-regular-variation/levy-khintchin","kind":"node","lean":"MetaMathlibExt.levy_Khintchin","parent":21700,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/index.html#levy-khintchin","scope":"Probability/infinite-divisibility-regular-variation","status":"fully_proved","summary":"A probability measure $\\mu$ on $\\mathbb{R}$ is infinitely divisible, meaning that for every integer $n \\ge 1$ there is a probability measure $\\rho$ on $\\mathbb{R}$ whose $n$-fold convolution power $\\rho^{\\ast n}$ equals $\\mu$, if and only if there are real numbers $\\gamma$ and $\\sigma \\ge 0$ and a measure $\\nu$ on $\\mathbb{R}$ with $\\nu(\\{0\\}) = 0$ and $\\int \\min(1, x^2)\\, d\\nu(x) < \\infty$ such that the\u2026","title":"L\u00e9vy\u2013Khintchin representation on the real line"},{"data_url":"scopes/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions.json","id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","kind":"scope","navigation":"module","parent":21700,"result":false,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","title":"Infinitely divisible distributions"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow","kind":"node","lean":"MeasureTheory.Measure.convPow","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a \u03c3-algebra and let $\\nu$ be a measure on $M$. The $n$-th convolution power $\\nu^{*n}$ of $\\nu$ is defined by recursion on $n$: $\\nu^{*0} = \\delta_0$, the Dirac measure at the identity $0$ of $M$, and $\\nu^{*(n+1)} = \\nu^{*n} * \\nu$, where $*$ denotes convolution of measures.","title":"Convolution power of a measure"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow-zero","kind":"node","lean":"MeasureTheory.Measure.convPow_zero","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow-zero","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"For every measure $\\nu$ on an additive commutative monoid $M$ with a \u03c3-algebra, the zeroth convolution power of $\\nu$ is the Dirac measure $\\delta_0$ at the identity element $0$ of $M$.","title":"The zeroth convolution power"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow-succ","kind":"node","lean":"MeasureTheory.Measure.convPow_succ","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow-succ","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a \u03c3-algebra and let $\\nu$ be a measure on $M$. For every natural number $n$, the $(n+1)$-st convolution power of $\\nu$ is the convolution of the $n$-th convolution power of $\\nu$ with $\\nu$.","title":"Recursion for convolution powers"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow-one","kind":"node","lean":"MeasureTheory.Measure.convPow_one","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow-one","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a \u03c3-algebra for which addition $M \\times M \\to M$ is measurable, and let $\\nu$ be an s-finite measure on $M$. Then the first convolution power of $\\nu$ is $\\nu$ itself.","title":"The first convolution power"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/sfinite-conv-pow","kind":"node","lean":"MeasureTheory.Measure.sfinite_convPow","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#sfinite-conv-pow","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a measurable structure. If $\\nu$ is an s-finite measure on $M$, then for every natural number $n$ its $n$-fold convolution power $\\nu^{*n}$ is s-finite.","title":"Convolution powers of an s-finite measure are s-finite"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/is-probability-measure-conv-pow","kind":"node","lean":"MeasureTheory.Measure.isProbabilityMeasure_convPow","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#is-probability-measure-conv-pow","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a measurable structure. If $\\nu$ is a probability measure on $M$, then for every natural number $n$ its $n$-fold convolution power $\\nu^{*n}$ is a probability measure.","title":"Convolution powers of a probability measure are probability measures"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow-succ-eq-iterate","kind":"node","lean":"MeasureTheory.Measure.convPow_succ_eq_iterate","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow-succ-eq-iterate","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a \u03c3-algebra for which addition $M \\times M \\to M$ is measurable, and let $\\nu$ be an s-finite measure on $M$. For every natural number $n$, the $(n+1)$-st convolution power of $\\nu$ equals the measure obtained from $\\nu$ by applying the map $\\mu \\mapsto \\nu * \\mu$ exactly $n$ times, where $*$ denotes convolution of measures.","title":"Convolution powers as iterated left convolution"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow-add","kind":"node","lean":"MeasureTheory.Measure.convPow_add","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow-add","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a \u03c3-algebra for which addition $M \\times M \\to M$ is measurable, and let $\\nu$ be an s-finite measure on $M$. Writing $*$ for convolution of measures and $\\nu^{*n}$ for the $n$-th convolution power of $\\nu$, for all natural numbers $m$ and $n$ we have $\\nu^{*(m+n)} = \\nu^{*m} * \\nu^{*n}$.","title":"Convolution powers add exponents"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-mul-conv","kind":"node","lean":"MeasureTheory.Measure.conv_mul_conv","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-mul-conv","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a \u03c3-algebra for which addition $M \\times M \\to M$ is measurable. For all s-finite measures $a, b, c, d$ on $M$ we have $(a * b) * (c * d) = (a * c) * (b * d)$, where $*$ denotes convolution of measures.","title":"Interchange law for convolution"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow-conv","kind":"node","lean":"MeasureTheory.Measure.convPow_conv","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow-conv","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a \u03c3-algebra for which addition $M \\times M \\to M$ is measurable, and let $\\nu$ and $\\rho$ be s-finite measures on $M$. Writing $*$ for convolution of measures and $\\mu^{*n}$ for the $n$-th convolution power of a measure $\\mu$, for every natural number $n$ we have $(\\nu * \\rho)^{*n} = \\nu^{*n} * \\rho^{*n}$.","title":"Convolution power of a convolution"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/is-infinitely-divisible","kind":"node","lean":"MeasureTheory.Measure.IsInfinitelyDivisible","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#is-infinitely-divisible","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a measurable structure in which addition is jointly measurable. A measure $\\mu$ on $M$ is infinitely divisible if for every natural number $n \\ge 1$ there is a probability measure $\\nu$ on $M$ whose $n$-fold convolution power $\\nu^{*n}$ equals $\\mu$.","title":"Infinitely divisible measures"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/is-probability-measure","kind":"node","lean":"MeasureTheory.Measure.IsInfinitelyDivisible.isProbabilityMeasure","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#is-probability-measure","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a measurable structure in which addition is jointly measurable. Every infinitely divisible measure on $M$ is a probability measure.","title":"Infinitely divisible measures are probability measures"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/is-infinitely-divisible-iff-exists-itera","kind":"node","lean":"MeasureTheory.Measure.isInfinitelyDivisible_iff_exists_iterate","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#is-infinitely-divisible-iff-exists-itera","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a measurable structure in which addition is jointly measurable. A measure $\\mu$ on $M$ is infinitely divisible if and only if for every natural number $n \\ne 0$ there is a probability measure $\\nu$ on $M$ such that $\\mu = \\nu * (\\nu * (\\cdots * (\\nu * \\nu)))$ with $n$ factors $\\nu$, that is, $\\mu$ is obtained from $\\nu$ by convolving on the left with $\\nu$ exactly\u2026","title":"Infinite divisibility through iterated convolution"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv","kind":"node","lean":"MeasureTheory.Measure.IsInfinitelyDivisible.conv","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a measurable structure in which addition is jointly measurable. If $\\mu$ and $\\eta$ are infinitely divisible measures on $M$, then their convolution $\\mu * \\eta$ is infinitely divisible.","title":"Infinite divisibility is closed under convolution"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow-dirac","kind":"node","lean":"MeasureTheory.Measure.convPow_dirac","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow-dirac","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid with a \u03c3-algebra for which addition $M \\times M \\to M$ is measurable. For every point $x \\in M$ and every natural number $n$, the $n$-th convolution power of the Dirac measure $\\delta_x$ is $\\delta_{nx}$, where $nx$ is the sum of $n$ copies of $x$ (so $0x = 0$).","title":"Convolution powers of a Dirac measure"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/is-infinitely-divisible-dirac","kind":"node","lean":"MeasureTheory.Measure.isInfinitelyDivisible_dirac","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#is-infinitely-divisible-dirac","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"For every real number $a$, the Dirac measure $\\delta_a$ on $\\mathbb{R}$ is infinitely divisible: for every $n \\ge 1$, $\\delta_a$ is the $n$-fold convolution power of $\\delta_{a/n}$.","title":"Dirac measures on the real line are infinitely divisible"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/conv-pow-succ-poisson-measure","kind":"node","lean":"MeasureTheory.Measure.convPow_succ_poissonMeasure","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#conv-pow-succ-poisson-measure","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"For every real rate $r \\ge 0$ and every natural number $n$, the $(n+1)$-st convolution power of the Poisson measure on $\\mathbb{N}$ with rate $r$ is the Poisson measure on $\\mathbb{N}$ with rate $(n+1)r$.","title":"Convolution powers of a Poisson measure"},{"id":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/is-infinitely-divisible-poisson-measure","kind":"node","lean":"MeasureTheory.Measure.isInfinitelyDivisible_poissonMeasure","parent":21702,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions/index.html#is-infinitely-divisible-poisson-measure","scope":"Probability/infinite-divisibility-regular-variation/infinitely-divisible-distributions","status":"fully_proved","summary":"For every real $r \\ge 0$, the Poisson distribution with parameter $r$, as a measure on $\\mathbb{N}$, is infinitely divisible: for every $n \\ge 1$ it is the $n$-fold convolution power of the Poisson distribution with parameter $r/n$.","title":"Poisson distributions are infinitely divisible"},{"data_url":"scopes/Probability/infinite-divisibility-regular-variation/regular-variation.json","id":"Probability/infinite-divisibility-regular-variation/regular-variation","kind":"scope","navigation":"module","parent":21700,"result":false,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","title":"Regular variation"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/is-slowly-varying","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsSlowlyVarying","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#is-slowly-varying","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"A function $L : \\mathbb{R} \\to \\mathbb{R}$ is slowly varying (at infinity) if $L(x) > 0$ for all sufficiently large $x$ and, for every real $c > 0$, $L(cx) / L(x) \\to 1$ as $x \\to \\infty$.","title":"Slowly varying function"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/of-tendsto","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsSlowlyVarying.of_tendsto","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#of-tendsto","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Let $L : \\mathbb{R} \\to \\mathbb{R}$ and $a > 0$. If $L(x) > 0$ for all sufficiently large $x$ and $L(x) \\to a$ as $x \\to \\infty$, then $L$ is slowly varying.","title":"Functions with a positive limit are slowly varying"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/is-slowly-varying-one","kind":"node","lean":"MathlibExt.Probability.RegularVariation.isSlowlyVarying_one","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#is-slowly-varying-one","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"The constant function $x \\mapsto 1$ on $\\mathbb{R}$ is slowly varying.","title":"The constant function one is slowly varying"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/is-slowly-varying-const","kind":"node","lean":"MathlibExt.Probability.RegularVariation.isSlowlyVarying_const","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#is-slowly-varying-const","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every real $c > 0$, the constant function $x \\mapsto c$ on $\\mathbb{R}$ is slowly varying.","title":"Positive constants are slowly varying"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/mul","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsSlowlyVarying.mul","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#mul","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"If $L_1, L_2 : \\mathbb{R} \\to \\mathbb{R}$ are slowly varying, then so is their pointwise product $x \\mapsto L_1(x) L_2(x)$.","title":"Products of slowly varying functions"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/const-mul","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsSlowlyVarying.const_mul","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#const-mul","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"If $L : \\mathbb{R} \\to \\mathbb{R}$ is slowly varying and $c > 0$, then $x \\mapsto c L(x)$ is slowly varying.","title":"Positive multiples of slowly varying functions"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/is-slowly-varying-one-add-inv","kind":"node","lean":"MathlibExt.Probability.RegularVariation.isSlowlyVarying_one_add_inv","parent":21721,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function "],["math","\\(1 + (1 + x)^{-1}\\)"],["text"," is slowly varying"]],"source":"The function $1 + (1 + x)^{-1}$ is slowly varying"},"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#is-slowly-varying-one-add-inv","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"The function $x \\mapsto 1 + (1 + x)^{-1}$ on $\\mathbb{R}$ is slowly varying; it tends to the positive limit $1$ at infinity.","title":"The function $1 + (1 + x)^{-1}$ is slowly varying"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/of-eventually-eq","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsSlowlyVarying.of_eventuallyEq","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#of-eventually-eq","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Let $f, g : \\mathbb{R} \\to \\mathbb{R}$. If $f$ is slowly varying and $f(x) = g(x)$ for all sufficiently large $x$, then $g$ is slowly varying.","title":"Slow variation is preserved by eventual equality"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/is-regularly-varying","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsRegularlyVarying","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#is-regularly-varying","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"A function $f : \\mathbb{R} \\to \\mathbb{R}$ is regularly varying with index $\\rho \\in \\mathbb{R}$ if there is a slowly varying function $L : \\mathbb{R} \\to \\mathbb{R}$ such that $f(x) = x^{\\rho} L(x)$ for all sufficiently large $x$.","title":"Regularly varying function"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/is-regularly-varying-rpow","kind":"node","lean":"MathlibExt.Probability.RegularVariation.isRegularlyVarying_rpow","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#is-regularly-varying-rpow","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every real number $\\rho$, the function $x \\mapsto x^\\rho$ on $\\mathbb{R}$ is regularly varying with index $\\rho$.","title":"Powers are regularly varying"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/mul-2","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsRegularlyVarying.mul","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#mul-2","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"If $f : \\mathbb{R} \\to \\mathbb{R}$ is regularly varying with index $\\rho$ and $g : \\mathbb{R} \\to \\mathbb{R}$ is regularly varying with index $\\sigma$, then the pointwise product $fg$ is regularly varying with index $\\rho + \\sigma$.","title":"Products of regularly varying functions"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/const-mul-2","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsRegularlyVarying.const_mul","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#const-mul-2","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"If $f : \\mathbb{R} \\to \\mathbb{R}$ is regularly varying with index $\\rho$ and $c > 0$, then $x \\mapsto c f(x)$ is regularly varying with index $\\rho$.","title":"Positive multiples of regularly varying functions"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/is-regularly-varying-const-mul-rpow","kind":"node","lean":"MathlibExt.Probability.RegularVariation.isRegularlyVarying_const_mul_rpow","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#is-regularly-varying-const-mul-rpow","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For real numbers $c > 0$ and $\\rho$, the function $x \\mapsto c x^\\rho$ on $\\mathbb{R}$ is regularly varying with index $\\rho$.","title":"Positive multiples of powers are regularly varying"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/of-eventually-eq-2","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsRegularlyVarying.of_eventuallyEq","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#of-eventually-eq-2","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Let $f, g : \\mathbb{R} \\to \\mathbb{R}$ and $\\rho \\in \\mathbb{R}$. If $f$ is regularly varying with index $\\rho$ and $f(x) = g(x)$ for all sufficiently large $x$, then $g$ is regularly varying with index $\\rho$.","title":"Regular variation is preserved by eventual equality"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/eventually-pos","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsRegularlyVarying.eventually_pos","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#eventually-pos","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"If $f : \\mathbb{R} \\to \\mathbb{R}$ is regularly varying with some index $\\rho$, then $f(x) > 0$ for all sufficiently large $x$.","title":"Regularly varying functions are eventually positive"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/right-tail","kind":"node","lean":"MathlibExt.Probability.RegularVariation.rightTail","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#right-tail","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}$ and $x \\in \\mathbb{R}$, the right tail of $\\mu$ at $x$ is the real number $\\mu((x, \\infty))$ when $\\mu((x, \\infty)) < \\infty$, and $0$ when $\\mu((x, \\infty)) = \\infty$.","title":"Right tail of a measure"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/left-tail","kind":"node","lean":"MathlibExt.Probability.RegularVariation.leftTail","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#left-tail","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}$ and a real $x$, the left tail of $\\mu$ at $x$ is the real number $\\mu((-\\infty, -x))$, where an infinite value is sent to $0$.","title":"Left tail of a measure"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/combined-tail","kind":"node","lean":"MathlibExt.Probability.RegularVariation.combinedTail","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#combined-tail","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For a measure $\\mu$ on $\\mathbb{R}$ and a real number $x$, the combined tail of $\\mu$ at $x$ is the real number $\\mu((x, \\infty)) + \\mu((-\\infty, -x))$, where each of the two measures is converted from $[0, \\infty]$ to a real number, an infinite value becoming $0$.","title":"Combined tail of a measure"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/right-tail-le-one","kind":"node","lean":"MathlibExt.Probability.RegularVariation.rightTail_le_one","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#right-tail-le-one","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every probability measure $\\mu$ on $\\mathbb{R}$ and every $x \\in \\mathbb{R}$, the right tail of $\\mu$ at $x$ is at most $1$.","title":"The right tail of a probability measure is at most one"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/left-tail-le-one","kind":"node","lean":"MathlibExt.Probability.RegularVariation.leftTail_le_one","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#left-tail-le-one","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every probability measure $\\mu$ on $\\mathbb{R}$ and every real $x$, the left tail of $\\mu$ at $x$ is at most $1$.","title":"The left tail of a probability measure is at most one"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/combined-tail-le-two","kind":"node","lean":"MathlibExt.Probability.RegularVariation.combinedTail_le_two","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#combined-tail-le-two","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every probability measure $\\mu$ on $\\mathbb{R}$ and every real number $x$, the combined tail of $\\mu$ at $x$ satisfies $\\mu((x, \\infty)) + \\mu((-\\infty, -x)) \\le 2$.","title":"The combined tail of a probability measure is at most two"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/right-tail-antitone","kind":"node","lean":"MathlibExt.Probability.RegularVariation.rightTail_antitone","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#right-tail-antitone","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on $\\mathbb{R}$. If $x \\le y$, then the right tail of $\\mu$ at $y$ is at most the right tail of $\\mu$ at $x$.","title":"The right tail of a finite measure is antitone"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/left-tail-antitone","kind":"node","lean":"MathlibExt.Probability.RegularVariation.leftTail_antitone","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#left-tail-antitone","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every finite measure $\\mu$ on $\\mathbb{R}$, the left tail of $\\mu$ is antitone in the threshold: if $a \\le b$, then $\\mu((-\\infty, -b)) \\le \\mu((-\\infty, -a))$.","title":"Monotonicity of the left tail"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/combined-tail-antitone","kind":"node","lean":"MathlibExt.Probability.RegularVariation.combinedTail_antitone","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#combined-tail-antitone","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on $\\mathbb{R}$ and write $T(x) = \\mu((x, \\infty)) + \\mu((-\\infty, -x))$ for its combined tail at a real number $x$. Then $T$ is antitone: $T(b) \\le T(a)$ whenever $a \\le b$.","title":"The combined tail is antitone"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/right-tail-le-combined-tail","kind":"node","lean":"MathlibExt.Probability.RegularVariation.rightTail_le_combinedTail","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#right-tail-le-combined-tail","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}$ and every $x \\in \\mathbb{R}$, the right tail of $\\mu$ at $x$ is at most the combined tail of $\\mu$ at $x$.","title":"The right tail is at most the combined tail"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/left-tail-le-combined-tail","kind":"node","lean":"MathlibExt.Probability.RegularVariation.leftTail_le_combinedTail","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#left-tail-le-combined-tail","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every measure $\\mu$ on $\\mathbb{R}$ and every real $x$, the left tail of $\\mu$ at $x$ is at most the combined tail of $\\mu$ at $x$.","title":"The left tail is at most the combined tail"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/tendsto-right-tail-at-top","kind":"node","lean":"MathlibExt.Probability.RegularVariation.tendsto_rightTail_atTop","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#tendsto-right-tail-at-top","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every finite measure $\\mu$ on $\\mathbb{R}$, the right tail $\\mu((x, \\infty))$, taken as a real number, tends to $0$ as $x \\to \\infty$.","title":"The right tail of a finite measure vanishes at infinity"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/tendsto-left-tail-at-top","kind":"node","lean":"MathlibExt.Probability.RegularVariation.tendsto_leftTail_atTop","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#tendsto-left-tail-at-top","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every finite measure $\\mu$ on $\\mathbb{R}$, the left tail $\\mu((-\\infty, -x))$, taken as a real number, tends to $0$ as $x \\to \\infty$.","title":"The left tail of a finite measure vanishes at infinity"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/tendsto-combined-tail-at-top","kind":"node","lean":"MathlibExt.Probability.RegularVariation.tendsto_combinedTail_atTop","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#tendsto-combined-tail-at-top","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every finite measure $\\mu$ on $\\mathbb{R}$, the combined tail $\\mu((x, \\infty)) + \\mu((-\\infty, -x))$, with both masses taken as real numbers, tends to $0$ as $x \\to \\infty$.","title":"The combined tail of a finite measure vanishes at infinity"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/has-tail-balance","kind":"node","lean":"MathlibExt.Probability.RegularVariation.HasTailBalance","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#has-tail-balance","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on $\\mathbb{R}$ and $p$ a real number. Write $R(x) = \\mu((x, \\infty))$ for the right tail and $T(x) = \\mu((x, \\infty)) + \\mu((-\\infty, -x))$ for the combined tail of $\\mu$. The measure $\\mu$ has tail balance $p$ when $0 \\le p \\le 1$, $T(x) > 0$ for all sufficiently large $x$, and $R(x) / T(x) \\to p$ as $x \\to \\infty$.","title":"Tail balance"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/has-tail-balance-zero-of-eventually-righ","kind":"node","lean":"MathlibExt.Probability.RegularVariation.hasTailBalance_zero_of_eventually_rightTail_eq_zero","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#has-tail-balance-zero-of-eventually-righ","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on $\\mathbb{R}$. If $\\mu((x, \\infty)) = 0$ for all sufficiently large $x$ and the combined tail $\\mu((x, \\infty)) + \\mu((-\\infty, -x))$ is positive for all sufficiently large $x$, then $\\mu$ has tail balance $0$.","title":"Tail balance zero from a vanishing right tail"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/not-has-tail-balance-of-eventually-combi","kind":"node","lean":"MathlibExt.Probability.RegularVariation.not_hasTailBalance_of_eventually_combinedTail_eq_zero","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#not-has-tail-balance-of-eventually-combi","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on $\\mathbb{R}$ and let $p \\in \\mathbb{R}$. If the combined tail of $\\mu$ at $x$ equals $0$ for all sufficiently large $x$, then $\\mu$ does not have tail balance $p$.","title":"Eventually vanishing combined tail rules out tail balance"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/not-has-tail-balance-dirac","kind":"node","lean":"MathlibExt.Probability.RegularVariation.not_hasTailBalance_dirac","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#not-has-tail-balance-dirac","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For every $a \\in \\mathbb{R}$ and every $p \\in \\mathbb{R}$, the Dirac measure $\\delta_a$ at $a$ does not have tail balance $p$.","title":"Dirac masses have no tail balance"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/regularly-varying-generalized-inverse","kind":"node","lean":"MathlibExt.Probability.RegularVariation.regularlyVarying_generalized_inverse","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#regularly-varying-generalized-inverse","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"For a regularly varying function `f` of index `-\u03b1` with `\u03b1 > 0` that is eventually non-increasing, there exists a sequence `a : \u2115 \u2192 \u211d` of positive reals such that `n \u00b7 f (a n) \u2192 1` as `n \u2192 \u221e`.","title":"Generalized inverse of a regularly varying function"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/ratio-tendsto","kind":"node","lean":"MathlibExt.Probability.RegularVariation.IsRegularlyVarying.ratio_tendsto","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#ratio-tendsto","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"A regularly varying function has the expected ratio limit.","title":"Ratio limit of a regularly varying function"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/potter-upper","kind":"node","lean":"MathlibExt.Probability.RegularVariation.potter_upper","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#potter-upper","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Potter's bound for antitone functions with a ratio limit. If `W` is antitone, bounded, eventually positive and `W (c * x) / W x \u2192 c ^ \u03c1` for every `c > 0`, then for every `\u03b4 > 0` there are `C > 0` and `Y` such that `W (x * u) / W x \u2264 C * u ^ (\u03c1 - \u03b4)` for all `x \u2265 Y` and `u \u2208 (0, 1]`. The proof chains the ratio limit at the dyadic scales `1/2` and `2`.","title":"Potter's bound"},{"id":"Probability/infinite-divisibility-regular-variation/regular-variation/karamata-integral","kind":"node","lean":"MathlibExt.Probability.RegularVariation.karamata_integral","parent":21721,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/regular-variation/index.html#karamata-integral","scope":"Probability/infinite-divisibility-regular-variation/regular-variation","status":"fully_proved","summary":"Karamata's integral theorem for antitone functions with a ratio limit. If `W` is antitone, nonnegative, bounded, eventually positive and `W (c * x) / W x \u2192 c ^ \u03c1` for every `c > 0`, then for `\u03b2 \u2265 0` and `\u03b2 + 1 + \u03c1 > 0`, `(\u222b t in (0)..x, t ^ \u03b2 * W t) / (x ^ (\u03b2 + 1) * W x) \u2192 1 / (\u03b2 + 1 + \u03c1)`. The proof substitutes `t = x * u` and applies dominated convergence with Potter's bound as the dominator.","title":"Karamata's integral theorem"},{"data_url":"scopes/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction.json","id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","kind":"scope","navigation":"module","parent":21700,"result":false,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","title":"Stable distributions and domains of attraction"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/conv-pow-gaussian-real","kind":"node","lean":"MathlibExt.Probability.StableDistribution.convPow_gaussianReal","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#conv-pow-gaussian-real","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"The convolution power of a Gaussian is Gaussian with the mean and the variance multiplied by `n`: the law of the sum of `n` i.i.d. `N(\u03bc, v)` variables is `N(n\u03bc, nv)`.","title":"Convolution powers of a Gaussian are Gaussian"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/is-stable","kind":"node","lean":"MathlibExt.Probability.StableDistribution.IsStable","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#is-stable","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"Indexed stability fixes the positive-n dilation to `n^(1/\u03b1)` and allows arbitrary centering. Degenerate Dirac identities may satisfy several indices.","title":"Stable distribution of a given index"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/is-stable-dirac","kind":"node","lean":"MathlibExt.Probability.StableDistribution.isStable_dirac","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#is-stable-dirac","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"A Dirac law satisfies the fixed-dilation identity at every admissible index. These are degenerate rescaling identities, with centering `na - n^(1/\u03b1)a`.","title":"Dirac measures are stable of every index"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/is-stable-gaussian-real","kind":"node","lean":"MathlibExt.Probability.StableDistribution.isStable_gaussianReal","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#is-stable-gaussian-real","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"Every real Gaussian measure is stable of index `2`: the sum of `n` i.i.d. `N(\u03bc, v)` variables is `N(n\u03bc, nv)`, which is the pushforward of `N(\u03bc, v)` under `x \u21a6 \u221an \u00b7 x + (n - \u221an) \u03bc`.","title":"Gaussian measures are stable of index 2"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/not-is-stable-gaussian-real-one","kind":"node","lean":"MathlibExt.Probability.StableDistribution.not_isStable_gaussianReal_one","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#not-is-stable-gaussian-real-one","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"A nondegenerate Gaussian cannot have fixed stability index one. The n=4 convolution variance is `4v`, while every such affine dilation has variance `16v`.","title":"Nondegenerate Gaussian is not stable of index 1"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/is-nondegenerate","kind":"node","lean":"MathlibExt.Probability.StableDistribution.IsNondegenerate","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#is-nondegenerate","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"A measure `\u03bc` on `\u211d` is *nondegenerate* if it is not a point mass `\u03b4_a` for any `a`.","title":"Nondegenerate measure on the real line"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/is-nondegenerate-gaussian-real","kind":"node","lean":"MathlibExt.Probability.StableDistribution.isNondegenerate_gaussianReal","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#is-nondegenerate-gaussian-real","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"A Gaussian with nonzero variance is nondegenerate: it gives mass `0` to every singleton, while `\u03b4_a` gives mass `1` to `{a}`.","title":"Gaussian with nonzero variance is nondegenerate"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/not-is-nondegenerate-dirac","kind":"node","lean":"MathlibExt.Probability.StableDistribution.not_isNondegenerate_dirac","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#not-is-nondegenerate-dirac","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"A Dirac measure is degenerate, by definition.","title":"Dirac measure is degenerate"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/in-domain-of-attraction","kind":"node","lean":"MathlibExt.Probability.StableDistribution.InDomainOfAttraction","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#in-domain-of-attraction","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"The probability measure `\u03bc` is *in the domain of attraction* of `G` if there exist sequences of normalizing constants `a n > 0` and centering constants `b n` such that the law of `(S_n - b n) / a n` (where `S_n` has law `\u03bc^{*n}`) converges weakly to `G`, expressed here as integration against arbitrary bounded continuous functions.","title":"Domain of attraction"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/in-domain-of-attraction-self","kind":"node","lean":"MathlibExt.Probability.StableDistribution.IsStable.inDomainOfAttraction_self","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#in-domain-of-attraction-self","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"A stable law is a normalized weak limit of its own convolution powers. The positive-n inverse affine identity is exact; order zero uses normalizer one.","title":"Stable law lies in its own domain of attraction"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/in-domain-of-attraction-index","kind":"node","lean":"MathlibExt.Probability.StableDistribution.InDomainOfAttractionIndex","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#in-domain-of-attraction-index","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"Indexed attraction requires a nondegenerate stable limit of that index. The general normalized weak-limit relation alone does not assign an index.","title":"Domain of attraction of a given index"},{"id":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/in-domain-of-attraction-dirac","kind":"node","lean":"MathlibExt.Probability.StableDistribution.inDomainOfAttraction_dirac","parent":21759,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction/index.html#in-domain-of-attraction-dirac","scope":"Probability/infinite-divisibility-regular-variation/stable-distributions-and-domains-of-attraction","status":"fully_proved","summary":"The Dirac measure is in its own domain of attraction: with `a n = 1` and `b n = (n - 1) a`, the normalized law of `S_n` is `\u03b4_a` for every `n`.","title":"Dirac measure lies in its own domain of attraction"},{"data_url":"scopes/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails.json","id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","kind":"scope","navigation":"module","parent":21700,"result":false,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","title":"Truncated moments of regularly varying tails"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/interval-integrable-of-antitone-bdd","kind":"node","lean":"MathlibExt.Probability.TailIntegralRV.intervalIntegrable_of_antitone_bdd","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#interval-integrable-of-antitone-bdd","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"Antitone bounded functions are interval-integrable.","title":"Antitone bounded functions are interval integrable"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/layer-cake-first-moment","kind":"node","lean":"MathlibExt.Probability.TailIntegralRV.layerCake_firstMoment","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#layer-cake-first-moment","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"Layer-cake formula for the first moment on `(0, a]`: the integral of `y` over `(0, a]` equals the integral of the right tail minus the endpoint correction.","title":"Layer-cake formula for the first moment"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/tail-integral-tendsto","kind":"node","lean":"MathlibExt.Probability.TailIntegralRV.tailIntegral_tendsto","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#tail-integral-tendsto","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"For a probability measure `\u03bc` on `\u211d` with regularly varying combined tail of index `-\u03b1`, `0 < \u03b1 < 1`, and tail balance `p`, the first-moment tail integral satisfies `(\u222b y in Ioc 0 (c * x), y \u2202\u03bc) / (x * combinedTail \u03bc x) \u2192 p * \u03b1 * c ^ (1 - \u03b1) / (1 - \u03b1)`.","title":"Corrected regularly-varying tail integral"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/tail-ge","kind":"node","lean":"MathlibExt.Probability.TruncatedMoment.tailGe","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#tail-ge","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"The upper tail `(\u03bc {y | s \u2264 |y|}).toReal`.","title":"Closed upper tail of a measure"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/tail-gt","kind":"node","lean":"MathlibExt.Probability.TruncatedMoment.tailGt","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#tail-gt","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"The lower tail `(\u03bc {y | s < |y|}).toReal`.","title":"Strict upper tail of a measure"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/tail-gt-eq-combined-tail","kind":"node","lean":"MathlibExt.Probability.TruncatedMoment.tailGt_eq_combinedTail","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#tail-gt-eq-combined-tail","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"At nonnegative thresholds the strict absolute tail is the canonical combined tail.","title":"Strict absolute tail equals the combined tail"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/tail-ge-eq-combined-tail-add-atoms","kind":"node","lean":"MathlibExt.Probability.TruncatedMoment.tailGe_eq_combinedTail_add_atoms","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#tail-ge-eq-combined-tail-add-atoms","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"The non-strict absolute tail adds both endpoint atoms to the canonical strict tail.","title":"Non-strict absolute tail as combined tail plus endpoint atoms"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/tail-ge-ratio-tendsto","kind":"node","lean":"MathlibExt.Probability.TruncatedMoment.tailGe_ratio_tendsto","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#tail-ge-ratio-tendsto","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"The upper tail is asymptotically the combined tail.","title":"Upper tail is asymptotic to the combined tail"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/tail-gt-ratio-tendsto","kind":"node","lean":"MathlibExt.Probability.TruncatedMoment.tailGt_ratio_tendsto","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#tail-gt-ratio-tendsto","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"The lower tail is asymptotically the combined tail.","title":"Lower tail is asymptotic to the combined tail"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/layer-cake","kind":"node","lean":"MathlibExt.Probability.TruncatedMoment.layerCake","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#layer-cake","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"Layer-cake formula for the truncated second moment, in terms of the upper tail `tailGe`. The term `x ^ 2 * tailGt \u03bc x` removes the contribution from points with `|y| > x`, which lie outside the closed truncation interval.","title":"Layer-cake formula for the truncated second moment"},{"id":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/truncated-moment-tendsto","kind":"node","lean":"MathlibExt.Probability.TruncatedMoment.truncatedMoment_tendsto","parent":21772,"result":true,"route":"../roadmap/Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails/index.html#truncated-moment-tendsto","scope":"Probability/infinite-divisibility-regular-variation/truncated-moments-of-regularly-varying-tails","status":"fully_proved","summary":"The corrected truncated-second-moment asymptotic: for a probability measure on `\u211d` with regularly varying combined tail of index `-\u03b1`, `0 < \u03b1 < 2`, the ratio `(\u222b y in Icc (-x) x, y ^ 2 \u2202\u03bc) / (x ^ 2 * combinedTail \u03bc x)` tends to `\u03b1 / (2 - \u03b1)`.","title":"Truncated second moment asymptotic for regularly varying tails"},{"data_url":"scopes/Probability/convergence-of-measures.json","id":"Probability/convergence-of-measures","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/convergence-of-measures/index.html","scope":"Probability/convergence-of-measures","status":"fully_proved","summary":"Weak convergence and convergence in probability, tightness and Prokhorov's theorem, Helly selection, Skorokhod representation, the continuous mapping theorem, the Cram\u00e9r\u2013Wold device and total variation distance; specific limit laws go to the limit-theorem fields.","title":"Convergence of probability measures"},{"id":"Probability/convergence-of-measures/cramer-wold-device","kind":"node","lean":"MathlibExt.Probability.CramerWoldWanted.cramer_wold_device","parent":21784,"result":true,"route":"../roadmap/Probability/convergence-of-measures/index.html#cramer-wold-device","scope":"Probability/convergence-of-measures","status":"fully_proved","summary":"Let $d$ be a natural number, let $(\\Omega, \\mu)$ and $(\\Omega', \\mu')$ be probability spaces, and let $X_n : \\Omega \\to \\mathbb{R}^d$ for $n \\in \\mathbb{N}$ and $Z : \\Omega' \\to \\mathbb{R}^d$ be maps, where $\\mathbb{R}^d$ carries the Euclidean inner product. If for every $t \\in \\mathbb{R}^d$ the real-valued maps $\\langle t, X_n \\rangle$ converge in distribution to $\\langle t, Z \\rangle$ as $n \\to \\infty$, then\u2026","title":"Cram\u00e9r\u2013Wold device"},{"id":"Probability/convergence-of-measures/prokhorov-sequential","kind":"node","lean":"MathlibExt.Probability.ProkhorovSequential.prokhorov_sequential","parent":21784,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prokhorov's theorem, sequential form on "],["math","\\(\\mathbb{R}\\)"]],"source":"Prokhorov's theorem, sequential form on $\\mathbb{R}$"},"result":true,"route":"../roadmap/Probability/convergence-of-measures/index.html#prokhorov-sequential","scope":"Probability/convergence-of-measures","status":"fully_proved","summary":"A sequence $(\\mu_n)$ of probability measures on $\\mathbb{R}$ is tight if and only if every subsequence of $(\\mu_n)$ has a further subsequence converging weakly to a probability measure on $\\mathbb{R}$.","title":"Prokhorov's theorem, sequential form on $\\mathbb{R}$"},{"id":"Probability/convergence-of-measures/skorokhod-representation","kind":"node","lean":"MathlibExt.Probability.SkorokhodRepresentationWanted.skorokhod_representation","parent":21784,"result":true,"route":"../roadmap/Probability/convergence-of-measures/index.html#skorokhod-representation","scope":"Probability/convergence-of-measures","status":"fully_proved","summary":"Let $X$ be a Polish space with its Borel $\\sigma$-algebra, let $(\\mu_n)_{n \\in \\mathbb{N}}$ be a sequence of probability measures on $X$, and let $\\mu$ be a probability measure on $X$ such that $\\mu_n$ converges weakly to $\\mu$. Write $\\lambda$ for Lebesgue measure on $\\mathbb{R}$ restricted to $[0, 1]$. Then there exist measurable maps $Y_n : \\mathbb{R} \\to X$ for $n \\in \\mathbb{N}$ and $Y : \\mathbb{R} \\to X$ such\u2026","title":"Skorokhod representation theorem"},{"data_url":"scopes/Probability/convergence-of-measures/continuous-mapping-theorem.json","id":"Probability/convergence-of-measures/continuous-mapping-theorem","kind":"scope","navigation":"module","parent":21784,"result":false,"route":"../roadmap/Probability/convergence-of-measures/continuous-mapping-theorem/index.html","scope":"Probability/convergence-of-measures/continuous-mapping-theorem","status":"fully_proved","title":"Continuous mapping theorem"},{"id":"Probability/convergence-of-measures/continuous-mapping-theorem/continuous-mapping","kind":"node","lean":"ProbabilityTheory.continuous_mapping","parent":21788,"result":true,"route":"../roadmap/Probability/convergence-of-measures/continuous-mapping-theorem/index.html#continuous-mapping","scope":"Probability/convergence-of-measures/continuous-mapping-theorem","status":"fully_proved","summary":"Let $\\mu_n$ ($n \\in \\mathbb{N}$) and $\\mu$ be probability measures on $\\mathbb{R}$, let $g : \\mathbb{R} \\to \\mathbb{R}$ be a measurable function whose set of discontinuity points has $\\mu$-measure zero, and write $\\nu \\circ g^{-1}$ for the pushforward of a measure $\\nu$ on $\\mathbb{R}$ under $g$. If $\\int f \\, d\\mu_n \\to \\int f \\, d\\mu$ as $n \\to \\infty$ for every bounded continuous function\u2026","title":"Continuous mapping theorem, integral form"},{"id":"Probability/convergence-of-measures/continuous-mapping-theorem/tendsto-map-of-tendsto-of-ae-continuous","kind":"node","lean":"MeasureTheory.ProbabilityMeasure.tendsto_map_of_tendsto_of_ae_continuous","parent":21788,"result":true,"route":"../roadmap/Probability/convergence-of-measures/continuous-mapping-theorem/index.html#tendsto-map-of-tendsto-of-ae-continuous","scope":"Probability/convergence-of-measures/continuous-mapping-theorem","status":"fully_proved","summary":"Let $\\mu_n$ ($n \\in \\mathbb{N}$) and $\\mu$ be probability measures on $\\mathbb{R}$, and let $g : \\mathbb{R} \\to \\mathbb{R}$ be a measurable function whose set of discontinuity points has $\\mu$-measure zero. If $\\mu_n$ converges weakly to $\\mu$ as $n \\to \\infty$, then the pushforward probability measures $\\mu_n \\circ g^{-1}$ converge weakly to $\\mu \\circ g^{-1}$.","title":"Continuous mapping theorem for weak convergence of probability measures"},{"data_url":"scopes/Probability/convergence-of-measures/convergence-in-probability.json","id":"Probability/convergence-of-measures/convergence-in-probability","kind":"scope","navigation":"module","parent":21784,"result":false,"route":"../roadmap/Probability/convergence-of-measures/convergence-in-probability/index.html","scope":"Probability/convergence-of-measures/convergence-in-probability","status":"fully_proved","title":"Convergence in probability"},{"id":"Probability/convergence-of-measures/convergence-in-probability/tendsto-in-measure-iff-tendsto-measure-a","kind":"node","lean":"MathlibExt.Probability.ConvergenceInProbability.tendstoInMeasure_iff_tendsto_measure_abs_sub","parent":21791,"result":true,"route":"../roadmap/Probability/convergence-of-measures/convergence-in-probability/index.html#tendsto-in-measure-iff-tendsto-measure-a","scope":"Probability/convergence-of-measures/convergence-in-probability","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, and let $X_n : \\Omega \\to \\mathbb{R}$ ($n \\in \\mathbb{N}$) and $Y : \\Omega \\to \\mathbb{R}$ be functions. Then $X_n$ converges to $Y$ in $\\mu$-measure, that is, $\\mu(\\{\\omega : |X_n(\\omega) - Y(\\omega)| \\ge \\varepsilon\\}) \\to 0$ as $n \\to \\infty$ for every $\\varepsilon \\in (0, \\infty]$, if and only if\u2026","title":"Convergence in measure in the textbook \u03b5-formulation"},{"id":"Probability/convergence-of-measures/convergence-in-probability/convergence-in-probability-iff-subsequen","kind":"node","lean":"MathlibExt.Probability.ConvergenceInProbability.convergence_in_probability_iff_subsequential_ae_convergence","parent":21791,"result":true,"route":"../roadmap/Probability/convergence-of-measures/convergence-in-probability/index.html#convergence-in-probability-iff-subsequen","scope":"Probability/convergence-of-measures/convergence-in-probability","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $\\Omega$, let $X_n : \\Omega \\to \\mathbb{R}$ ($n \\in \\mathbb{N}$) be $\\mu$-almost everywhere strongly measurable functions, and let $Y : \\Omega \\to \\mathbb{R}$ be any function. Then $\\mu(\\{\\omega : |X_n(\\omega) - Y(\\omega)| > \\varepsilon\\}) \\to 0$ as $n \\to \\infty$ for every real $\\varepsilon > 0$ if and only if for every strictly increasing sequence $(n_k)$ of\u2026","title":"Convergence in probability via subsequential a.s. convergence"},{"data_url":"scopes/Probability/convergence-of-measures/helly-s-selection-theorem.json","id":"Probability/convergence-of-measures/helly-s-selection-theorem","kind":"scope","navigation":"module","parent":21784,"result":false,"route":"../roadmap/Probability/convergence-of-measures/helly-s-selection-theorem/index.html","scope":"Probability/convergence-of-measures/helly-s-selection-theorem","status":"fully_proved","title":"Helly's selection theorem"},{"id":"Probability/convergence-of-measures/helly-s-selection-theorem/is-distribution-function","kind":"node","lean":"ProbabilityTheory.IsDistributionFunction","parent":21794,"result":true,"route":"../roadmap/Probability/convergence-of-measures/helly-s-selection-theorem/index.html#is-distribution-function","scope":"Probability/convergence-of-measures/helly-s-selection-theorem","status":"fully_proved","summary":"A nondecreasing, right-continuous function $F : \\mathbb{R} \\to \\mathbb{R}$ (a Stieltjes function) is a (cumulative) distribution function if $F(x) \\to 0$ as $x \\to -\\infty$ and $F(x) \\to 1$ as $x \\to +\\infty$.","title":"Distribution function"},{"id":"Probability/convergence-of-measures/helly-s-selection-theorem/nonneg","kind":"node","lean":"ProbabilityTheory.IsDistributionFunction.nonneg","parent":21794,"result":true,"route":"../roadmap/Probability/convergence-of-measures/helly-s-selection-theorem/index.html#nonneg","scope":"Probability/convergence-of-measures/helly-s-selection-theorem","status":"fully_proved","summary":"If $F$ is a distribution function on $\\mathbb{R}$, then $0 \\le F(x)$ for every $x \\in \\mathbb{R}$.","title":"Distribution functions are nonnegative"},{"id":"Probability/convergence-of-measures/helly-s-selection-theorem/le-one","kind":"node","lean":"ProbabilityTheory.IsDistributionFunction.le_one","parent":21794,"result":true,"route":"../roadmap/Probability/convergence-of-measures/helly-s-selection-theorem/index.html#le-one","scope":"Probability/convergence-of-measures/helly-s-selection-theorem","status":"fully_proved","summary":"If $F$ is a distribution function on $\\mathbb{R}$, then $F(x) \\le 1$ for every $x \\in \\mathbb{R}$.","title":"Distribution functions are at most one"},{"id":"Probability/convergence-of-measures/helly-s-selection-theorem/mem-icc","kind":"node","lean":"ProbabilityTheory.IsDistributionFunction.mem_Icc","parent":21794,"result":true,"route":"../roadmap/Probability/convergence-of-measures/helly-s-selection-theorem/index.html#mem-icc","scope":"Probability/convergence-of-measures/helly-s-selection-theorem","status":"fully_proved","summary":"If $F$ is a distribution function on $\\mathbb{R}$, then $F(x) \\in [0, 1]$ for every $x \\in \\mathbb{R}$.","title":"Distribution functions take values in the unit interval"},{"id":"Probability/convergence-of-measures/helly-s-selection-theorem/helly-selection","kind":"node","lean":"ProbabilityTheory.helly_selection","parent":21794,"result":true,"route":"../roadmap/Probability/convergence-of-measures/helly-s-selection-theorem/index.html#helly-selection","scope":"Probability/convergence-of-measures/helly-s-selection-theorem","status":"fully_proved","summary":"Let $(F_n)_{n \\in \\mathbb{N}}$ be a sequence of distribution functions on $\\mathbb{R}$. Then there are a strictly increasing map $\\varphi : \\mathbb{N} \\to \\mathbb{N}$ and a nondecreasing, right-continuous function $G : \\mathbb{R} \\to \\mathbb{R}$ such that $F_{\\varphi(k)}(x) \\to G(x)$ as $k \\to \\infty$ at every point $x \\in \\mathbb{R}$ at which $G$ is continuous.","title":"Helly's selection theorem"},{"data_url":"scopes/Probability/convergence-of-measures/slutsky-s-theorem-and-the-delta-method.json","id":"Probability/convergence-of-measures/slutsky-s-theorem-and-the-delta-method","kind":"scope","navigation":"module","parent":21784,"result":false,"route":"../roadmap/Probability/convergence-of-measures/slutsky-s-theorem-and-the-delta-method/index.html","scope":"Probability/convergence-of-measures/slutsky-s-theorem-and-the-delta-method","status":"fully_proved","title":"Slutsky's theorem and the delta method"},{"id":"Probability/convergence-of-measures/slutsky-s-theorem-and-the-delta-method/tendsto-in-measure-of-tendsto-in-distrib","kind":"node","lean":"MeasureTheory.tendstoInMeasure_of_tendstoInDistribution_const","parent":21800,"result":true,"route":"../roadmap/Probability/convergence-of-measures/slutsky-s-theorem-and-the-delta-method/index.html#tendsto-in-measure-of-tendsto-in-distrib","scope":"Probability/convergence-of-measures/slutsky-s-theorem-and-the-delta-method","status":"fully_proved","summary":"If the random variables `X i` on one probability space converge in distribution to the constant `c`, then they converge to `c` in probability.","title":"Convergence in distribution to a constant implies convergence in probability"},{"data_url":"scopes/Probability/convergence-of-measures/tightness-of-probability-measures.json","id":"Probability/convergence-of-measures/tightness-of-probability-measures","kind":"scope","navigation":"module","parent":21784,"result":false,"route":"../roadmap/Probability/convergence-of-measures/tightness-of-probability-measures/index.html","scope":"Probability/convergence-of-measures/tightness-of-probability-measures","status":"fully_proved","title":"Tightness of probability measures"},{"id":"Probability/convergence-of-measures/tightness-of-probability-measures/is-tight-family","kind":"node","lean":"MathlibExt.Probability.TightnessFamily.IsTightFamily","parent":21802,"result":true,"route":"../roadmap/Probability/convergence-of-measures/tightness-of-probability-measures/index.html#is-tight-family","scope":"Probability/convergence-of-measures/tightness-of-probability-measures","status":"fully_proved","summary":"Let $\\alpha$ be a space carrying a \u03c3-algebra and a topology. A family $(\\mu_i)_{i \\in \\iota}$ of probability measures on $\\alpha$ is tight if its range $\\{\\mu_i : i \\in \\iota\\}$ is a tight set of measures: for every $\\varepsilon \\in (0, \\infty]$ there is a compact set $K \\subseteq \\alpha$ such that $\\mu_i(\\alpha \\setminus K) \\le \\varepsilon$ for every index $i$.","title":"Tight family of probability measures"},{"id":"Probability/convergence-of-measures/tightness-of-probability-measures/is-tight-family-real","kind":"node","lean":"MathlibExt.Probability.TightnessFamily.IsTightFamilyReal","parent":21802,"result":true,"route":"../roadmap/Probability/convergence-of-measures/tightness-of-probability-measures/index.html#is-tight-family-real","scope":"Probability/convergence-of-measures/tightness-of-probability-measures","status":"fully_proved","summary":"A family $(\\mu_i)_{i \\in \\iota}$ of probability measures on $\\mathbb{R}$ is tight in the real-line sense if for every $\\varepsilon \\in (0, \\infty]$ there is $M \\in \\mathbb{R}$ such that $\\mu_i([-M, M]) \\ge 1 - \\varepsilon$ for every index $i$.","title":"Tightness on the real line"},{"id":"Probability/convergence-of-measures/tightness-of-probability-measures/is-tight-family-2","kind":"node","lean":"MathlibExt.Probability.TightnessFamily.IsTightFamilyReal.isTightFamily","parent":21802,"result":true,"route":"../roadmap/Probability/convergence-of-measures/tightness-of-probability-measures/index.html#is-tight-family-2","scope":"Probability/convergence-of-measures/tightness-of-probability-measures","status":"fully_proved","summary":"Let $(\\mu_i)_{i \\in \\iota}$ be a family of probability measures on $\\mathbb{R}$. If the family is tight in the real-line sense, that is, for every $\\varepsilon \\in (0, \\infty]$ there is $M \\in \\mathbb{R}$ with $\\mu_i([-M, M]) \\ge 1 - \\varepsilon$ for every $i$, then the family is tight, that is, for every $\\varepsilon \\in (0, \\infty]$ there is a compact set $K \\subseteq \\mathbb{R}$ with\u2026","title":"Real-line tightness implies tightness"},{"id":"Probability/convergence-of-measures/tightness-of-probability-measures/is-tight-family-real-2","kind":"node","lean":"MathlibExt.Probability.TightnessFamily.IsTightFamily.isTightFamilyReal","parent":21802,"result":true,"route":"../roadmap/Probability/convergence-of-measures/tightness-of-probability-measures/index.html#is-tight-family-real-2","scope":"Probability/convergence-of-measures/tightness-of-probability-measures","status":"fully_proved","summary":"Let $(\\mu_i)_{i \\in \\iota}$ be a family of probability measures on $\\mathbb{R}$. If the family is tight, that is, for every $\\varepsilon \\in (0, \\infty]$ there is a compact set $K \\subseteq \\mathbb{R}$ with $\\mu_i(\\mathbb{R} \\setminus K) \\le \\varepsilon$ for every $i$, then it is tight in the real-line sense: for every $\\varepsilon \\in (0, \\infty]$ there is $M \\in \\mathbb{R}$ with\u2026","title":"Tight families on the real line concentrate on bounded intervals"},{"id":"Probability/convergence-of-measures/tightness-of-probability-measures/is-tight-family-iff-is-tight-measure-set","kind":"node","lean":"MathlibExt.Probability.TightnessFamily.isTightFamily_iff_isTightMeasureSet","parent":21802,"result":true,"route":"../roadmap/Probability/convergence-of-measures/tightness-of-probability-measures/index.html#is-tight-family-iff-is-tight-measure-set","scope":"Probability/convergence-of-measures/tightness-of-probability-measures","status":"fully_proved","summary":"Let $\\alpha$ be a space carrying a \u03c3-algebra and a topology. A family $(\\mu_i)_{i \\in \\iota}$ of probability measures on $\\alpha$ is tight exactly when its range $\\{\\mu_i : i \\in \\iota\\}$ is a tight set of measures.","title":"Tightness of a family is tightness of its range"},{"id":"Probability/convergence-of-measures/tightness-of-probability-measures/is-tight-family-of-singleton","kind":"node","lean":"MathlibExt.Probability.TightnessFamily.isTightFamily_of_singleton","parent":21802,"result":true,"route":"../roadmap/Probability/convergence-of-measures/tightness-of-probability-measures/index.html#is-tight-family-of-singleton","scope":"Probability/convergence-of-measures/tightness-of-probability-measures","status":"fully_proved","summary":"For every index type $\\iota$ and every probability measure $\\mu$ on $\\mathbb{R}$, the constant family $(\\mu)_{i \\in \\iota}$ is tight.","title":"Constant families on the real line are tight"},{"data_url":"scopes/Probability/convergence-of-measures/total-variation-distance.json","id":"Probability/convergence-of-measures/total-variation-distance","kind":"scope","navigation":"module","parent":21784,"result":false,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","title":"Total variation distance"},{"id":"Probability/convergence-of-measures/total-variation-distance/total-variation-dist","kind":"node","lean":"MathlibExt.Probability.TotalVariationDistance.totalVariationDist","parent":21809,"result":true,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html#total-variation-dist","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","summary":"For finite measures $\\mu$ and $\\nu$ on a measurable space $\\Omega$, the total variation distance $d_{TV}(\\mu, \\nu)$ is the real number $\\sup_B |\\mu(B) - \\nu(B)|$, where the supremum runs over all measurable sets $B \\subseteq \\Omega$ and the masses $\\mu(B)$ and $\\nu(B)$ are taken as real numbers.","title":"Total variation distance between finite measures"},{"id":"Probability/convergence-of-measures/total-variation-distance/total-variation-dist-nonneg","kind":"node","lean":"MathlibExt.Probability.TotalVariationDistance.totalVariationDist_nonneg","parent":21809,"result":true,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html#total-variation-dist-nonneg","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","summary":"For finite measures $\\mu$ and $\\nu$ on a measurable space, the total variation distance satisfies $0 \\le d_{TV}(\\mu, \\nu)$.","title":"Total variation distance is nonnegative"},{"id":"Probability/convergence-of-measures/total-variation-distance/total-variation-dist-self","kind":"node","lean":"MathlibExt.Probability.TotalVariationDistance.totalVariationDist_self","parent":21809,"result":true,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html#total-variation-dist-self","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","summary":"For every finite measure $\\mu$ on a measurable space, the total variation distance $d_{TV}(\\mu, \\mu)$ is $0$.","title":"Total variation distance from a measure to itself"},{"id":"Probability/convergence-of-measures/total-variation-distance/total-variation-dist-comm","kind":"node","lean":"MathlibExt.Probability.TotalVariationDistance.totalVariationDist_comm","parent":21809,"result":true,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html#total-variation-dist-comm","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","summary":"For finite measures $\\mu$ and $\\nu$ on a measurable space, the total variation distances satisfy $d_{TV}(\\mu, \\nu) = d_{TV}(\\nu, \\mu)$.","title":"Symmetry of total variation distance"},{"id":"Probability/convergence-of-measures/total-variation-distance/total-variation-dist-triangle","kind":"node","lean":"MathlibExt.Probability.TotalVariationDistance.totalVariationDist_triangle","parent":21809,"result":true,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html#total-variation-dist-triangle","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","summary":"For finite measures $\\mu$, $\\nu$ and $\\rho$ on a measurable space, the total variation distance satisfies $d_{TV}(\\mu, \\rho) \\le d_{TV}(\\mu, \\nu) + d_{TV}(\\nu, \\rho)$.","title":"Triangle inequality for total variation distance"},{"id":"Probability/convergence-of-measures/total-variation-distance/total-variation-dist-le-one","kind":"node","lean":"MathlibExt.Probability.TotalVariationDistance.totalVariationDist_le_one","parent":21809,"result":true,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html#total-variation-dist-le-one","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","summary":"For probability measures $\\mu$ and $\\nu$ on a measurable space, the total variation distance satisfies $d_{TV}(\\mu, \\nu) \\le 1$.","title":"Total variation distance between probability measures is at most one"},{"id":"Probability/convergence-of-measures/total-variation-distance/total-variation-dist-eq-zero-iff","kind":"node","lean":"MathlibExt.Probability.TotalVariationDistance.totalVariationDist_eq_zero_iff","parent":21809,"result":true,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html#total-variation-dist-eq-zero-iff","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","summary":"For finite measures $\\mu$ and $\\nu$ on a measurable space, the total variation distance $d_{TV}(\\mu, \\nu)$ is $0$ if and only if $\\mu = \\nu$.","title":"Total variation distance vanishes only for equal measures"},{"id":"Probability/convergence-of-measures/total-variation-distance/total-variation-dist-dirac","kind":"node","lean":"MathlibExt.Probability.TotalVariationDistance.totalVariationDist_dirac","parent":21809,"result":true,"route":"../roadmap/Probability/convergence-of-measures/total-variation-distance/index.html#total-variation-dist-dirac","scope":"Probability/convergence-of-measures/total-variation-distance","status":"fully_proved","summary":"Let $\\Omega$ be a measurable space in which every singleton is measurable, and let $a, b \\in \\Omega$. The total variation distance between the Dirac measures $\\delta_a$ and $\\delta_b$ is $0$ if $a = b$ and $1$ if $a \\neq b$.","title":"Total variation distance between Dirac measures"},{"data_url":"scopes/Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func.json","id":"Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func","kind":"scope","navigation":"module","parent":21784,"result":false,"route":"../roadmap/Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func/index.html","scope":"Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func","status":"fully_proved","title":"Weak convergence in the space of continuous functions"},{"id":"Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func/continuous-path-measure-ext-of-finset-ev","kind":"node","lean":"MathlibExt.Probability.ContinuousPathConvergence.continuousPath_measure_ext_of_finset_eval","parent":21818,"result":true,"route":"../roadmap/Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func/index.html#continuous-path-measure-ext-of-finset-ev","scope":"Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func","status":"fully_proved","summary":"Probability measures on continuous real paths over `[0, T]` are determined by all of their finite evaluation pushforwards. The empty `Finset` is included, and repeated coordinates are identified by `Finset`.","title":"Continuous-path laws are determined by finite-dimensional marginals"},{"id":"Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func/tendsto-continuous-path-of-tight-of-fins","kind":"node","lean":"MathlibExt.Probability.ContinuousPathConvergence.tendsto_continuousPath_of_tight_of_finset_eval","parent":21818,"result":true,"route":"../roadmap/Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func/index.html#tendsto-continuous-path-of-tight-of-fins","scope":"Probability/convergence-of-measures/weak-convergence-in-the-space-of-continuous-func","status":"fully_proved","summary":"Let `T > 0`. A tight sequence of probability laws on `C(Set.Icc 0 T, \u211d)` converges weakly to `\u03bc` in the uniform topology if every finite evaluation pushforward converges weakly to the corresponding pushforward of `\u03bc`.","title":"Tightness plus finite-dimensional convergence gives weak convergence"},{"data_url":"scopes/Probability/large-deviations.json","id":"Probability/large-deviations","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/large-deviations/index.html","scope":"Probability/large-deviations","status":"fully_proved","summary":"Large deviation principles and rate functions, Cram\u00e9r's theorem and Chernoff bounds for empirical means, log-moment generating functions and the Legendre\u2013Fenchel transform used to build them.","title":"Large deviations"},{"data_url":"scopes/Probability/large-deviations/large-deviation-principle.json","id":"Probability/large-deviations/large-deviation-principle","kind":"scope","navigation":"module","parent":21821,"result":false,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","title":"Large deviation principle"},{"id":"Probability/large-deviations/large-deviation-principle/is-rate-function","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.IsRateFunction","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#is-rate-function","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a topological space. A rate function on $X$ is a lower semicontinuous function $I : X \\to [0, \\infty]$.","title":"Rate function"},{"id":"Probability/large-deviations/large-deviation-principle/is-good-rate-function","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.IsGoodRateFunction","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#is-good-rate-function","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a topological space. A function $I : X \\to [0, \\infty]$ is a good rate function if it is a rate function and, for every real number $a \\ge 0$, the sub-level set $\\{x \\in X : I(x) \\le a\\}$ is compact.","title":"Good rate function"},{"id":"Probability/large-deviations/large-deviation-principle/is-convex-rate-function","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.IsConvexRateFunction","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#is-convex-rate-function","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a real vector space carrying a topology (no compatibility between the topology and the linear structure is assumed). A function $I : X \\to [0, \\infty]$ is a convex rate function if it is a rate function and $I(ax + by) \\le a I(x) + b I(y)$ for all $x, y \\in X$ and all real numbers $a, b \\ge 0$ with $a + b = 1$.","title":"Convex rate function"},{"id":"Probability/large-deviations/large-deviation-principle/satisfies-ldp","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.SatisfiesLDP","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#satisfies-ldp","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a topological space with a $\\sigma$-algebra containing every open set, let $(\\mu_n)_{n \\in \\mathbb{N}}$ be a sequence of measures on $X$, and let $I : X \\to [0, \\infty]$. The sequence $(\\mu_n)$ satisfies the large deviation principle with rate function $I$ and speed $n$ if $I$ is a rate function, every open set $U \\subseteq X$ satisfies","title":"Large deviation principle for a sequence of measures"},{"id":"Probability/large-deviations/large-deviation-principle/is-rate-function-2","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.IsGoodRateFunction.isRateFunction","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#is-rate-function-2","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Every good rate function on a topological space is a rate function.","title":"Good rate functions are rate functions"},{"id":"Probability/large-deviations/large-deviation-principle/is-rate-function-3","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.IsConvexRateFunction.isRateFunction","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#is-rate-function-3","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Every convex rate function on a real vector space with a topology is a rate function.","title":"Convex rate functions are rate functions"},{"id":"Probability/large-deviations/large-deviation-principle/borel-lower-bound","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.SatisfiesLDP.borel_lower_bound","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#borel-lower-bound","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a topological space with a $\\sigma$-algebra in which every open set is measurable, and let $(\\mu_n)_{n \\in \\mathbb{N}}$ be measures on $X$ satisfying the large deviation principle with rate function $I : X \\to [0, \\infty]$. Then for every set $\\Gamma \\subseteq X$, not necessarily measurable (for non-measurable $\\Gamma$, $\\mu_n(\\Gamma)$ is its outer measure), with interior $\\Gamma^\\circ$,","title":"Large deviation lower bound for arbitrary sets"},{"id":"Probability/large-deviations/large-deviation-principle/borel-upper-bound","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.SatisfiesLDP.borel_upper_bound","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#borel-upper-bound","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a topological space with a $\\sigma$-algebra in which every open set is measurable, and let $(\\mu_n)_{n \\in \\mathbb{N}}$ be measures on $X$ satisfying the large deviation principle with rate function $I : X \\to [0, \\infty]$. Then for every set $\\Gamma \\subseteq X$, not necessarily measurable (for non-measurable $\\Gamma$, $\\mu_n(\\Gamma)$ is its outer measure), with closure $\\overline{\\Gamma}$,","title":"Large deviation upper bound for arbitrary sets"},{"id":"Probability/large-deviations/large-deviation-principle/borel-bounds","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.SatisfiesLDP.borel_bounds","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#borel-bounds","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a topological space with a $\\sigma$-algebra in which every open set is measurable, and let $(\\mu_n)_{n \\in \\mathbb{N}}$ be measures on $X$ satisfying the large deviation principle with rate function $I : X \\to [0, \\infty]$. Then for every set $\\Gamma \\subseteq X$, not necessarily measurable (for non-measurable $\\Gamma$, $\\mu_n(\\Gamma)$ is its outer measure), with interior $\\Gamma^\\circ$ and closure\u2026","title":"Large deviation bounds for arbitrary sets"},{"id":"Probability/large-deviations/large-deviation-principle/satisfies-ldp-of-borel-bounds","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.satisfiesLDP_of_borel_bounds","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#satisfies-ldp-of-borel-bounds","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a topological space with a $\\sigma$-algebra containing every open set, let $(\\mu_n)_{n \\in \\mathbb{N}}$ be a sequence of measures on $X$, and let $I$ be a rate function on $X$. Suppose that every measurable set $\\Gamma \\subseteq X$, with interior $\\Gamma^\\circ$ and closure $\\overline{\\Gamma}$, satisfies","title":"Large deviation principle from bounds on measurable sets"},{"id":"Probability/large-deviations/large-deviation-principle/dirac-rate","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.diracRate","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#dirac-rate","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"For a point $x_0$ of a set $X$, the Dirac rate function at $x_0$ is the function $X \\to [0, \\infty]$ that equals $0$ at $x_0$ and $\\infty$ at every other point.","title":"Rate function of a Dirac mass"},{"id":"Probability/large-deviations/large-deviation-principle/dirac-rate-apply-self","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.diracRate_apply_self","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#dirac-rate-apply-self","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"For every point $x_0$ of a set $X$, the Dirac rate function at $x_0$ takes the value $0$ at $x_0$.","title":"The Dirac rate function vanishes at its center"},{"id":"Probability/large-deviations/large-deviation-principle/dirac-rate-apply-of-ne","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.diracRate_apply_of_ne","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#dirac-rate-apply-of-ne","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"For points $x \\ne x_0$ of a set $X$, the Dirac rate function at $x_0$ takes the value $\\infty$ at $x$.","title":"The Dirac rate function is infinite away from its center"},{"id":"Probability/large-deviations/large-deviation-principle/i-inf-dirac-rate-of-mem","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.iInf_diracRate_of_mem","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#i-inf-dirac-rate-of-mem","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"For a point $x_0$ of a set $X$ and a subset $s \\subseteq X$ with $x_0 \\in s$, the infimum over $x \\in s$ of the Dirac rate function at $x_0$, evaluated at $x$, is $0$.","title":"Infimum of the Dirac rate function over a set containing the center"},{"id":"Probability/large-deviations/large-deviation-principle/i-inf-dirac-rate-of-not-mem","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.iInf_diracRate_of_notMem","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#i-inf-dirac-rate-of-not-mem","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"For a point $x_0$ of a set $X$ and a subset $s \\subseteq X$ with $x_0 \\notin s$, the infimum over $x \\in s$ of the Dirac rate function at $x_0$, evaluated at $x$, is $\\infty$.","title":"Infimum of the Dirac rate function over a set avoiding the center"},{"id":"Probability/large-deviations/large-deviation-principle/dirac-ldp-term-eq-zero-of-mem","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.dirac_ldp_term_eq_zero_of_mem","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#dirac-ldp-term-eq-zero-of-mem","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a measurable space, let $\\delta_{x_0}$ be the Dirac measure at a point $x_0 \\in X$, and let $s \\subseteq X$ be a set with $x_0 \\in s$. Then $\\frac{1}{n} \\log \\delta_{x_0}(s) = 0$ in the extended reals for every natural number $n$, the factor $1/n$ being read as $0$ when $n = 0$.","title":"Log-scaled Dirac mass of a set containing the center"},{"id":"Probability/large-deviations/large-deviation-principle/dirac-ldp-term-eq-bot-of-not-mem","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.dirac_ldp_term_eq_bot_of_notMem","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#dirac-ldp-term-eq-bot-of-not-mem","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a measurable space in which every singleton is measurable, let $\\delta_{x_0}$ be the Dirac measure at a point $x_0 \\in X$, and let $s \\subseteq X$ be a set with $x_0 \\notin s$. Then for every natural number $n \\ge 1$, $\\frac{1}{n} \\log \\delta_{x_0}(s) = -\\infty$ in the extended reals, with $\\log 0 = -\\infty$.","title":"Log-scaled Dirac mass of a set avoiding the center"},{"id":"Probability/large-deviations/large-deviation-principle/satisfies-ldp-dirac","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.satisfiesLDP_dirac","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#satisfies-ldp-dirac","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"Let $X$ be a $T_1$ topological space equipped with its Borel $\\sigma$-algebra, and let $x_0 \\in X$. The constant sequence of measures $\\mu_n = \\delta_{x_0}$ satisfies the large deviation principle with the Dirac rate function of $x_0$, which is $0$ at $x_0$ and $\\infty$ elsewhere.","title":"Large deviation principle for a Dirac mass"},{"id":"Probability/large-deviations/large-deviation-principle/is-good-rate-function-abs","kind":"node","lean":"MathlibExt.Probability.LargeDeviationPrinciple.isGoodRateFunction_abs","parent":21822,"result":true,"route":"../roadmap/Probability/large-deviations/large-deviation-principle/index.html#is-good-rate-function-abs","scope":"Probability/large-deviations/large-deviation-principle","status":"fully_proved","summary":"The function $x \\mapsto |x|$, viewed as a map $\\mathbb{R} \\to [0, \\infty]$, is a good rate function on $\\mathbb{R}$.","title":"The absolute value is a good rate function"},{"data_url":"scopes/Probability/large-deviations/legendre-transform.json","id":"Probability/large-deviations/legendre-transform","kind":"scope","navigation":"module","parent":21821,"result":false,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","title":"Legendre transform"},{"id":"Probability/large-deviations/legendre-transform/legendre-transform","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.legendreTransform","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#legendre-transform","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$, give $\\mathbb{R}^d$ its standard inner product, and let $\\Lambda : \\mathbb{R}^d \\to [-\\infty, \\infty]$. The Legendre transform of $\\Lambda$ at $x \\in \\mathbb{R}^d$ is the Fenchel conjugate of $\\Lambda$ at the linear functional $y \\mapsto \\langle x, y \\rangle$, that is, the extended real number $\\Lambda^*(x) = \\sup_{y \\in \\mathbb{R}^d} \\big(\\langle x, y \\rangle - \\Lambda(y)\\big)$.","title":"Legendre transform"},{"id":"Probability/large-deviations/legendre-transform/legendre-transform-eq-i-sup","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.legendreTransform_eq_iSup","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#legendre-transform-eq-i-sup","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $\\Lambda : \\mathbb{R}^d \\to [-\\infty, \\infty]$. For every $x \\in \\mathbb{R}^d$, the Legendre transform $\\Lambda^*$ of $\\Lambda$ satisfies $\\Lambda^*(x) = \\sup_{l \\in \\mathbb{R}^d} \\big(\\langle l, x \\rangle - \\Lambda(l)\\big)$.","title":"Supremum formula for the Legendre transform"},{"id":"Probability/large-deviations/legendre-transform/le-legendre-transform","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.le_legendreTransform","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#le-legendre-transform","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $\\Lambda : \\mathbb{R}^d \\to [-\\infty, \\infty]$, with Legendre transform $\\Lambda^*$. For all $x, l \\in \\mathbb{R}^d$, we have $\\langle l, x \\rangle - \\Lambda(l) \\le \\Lambda^*(x)$ in $[-\\infty, \\infty]$.","title":"Fenchel\u2013Young inequality for the Legendre transform"},{"id":"Probability/large-deviations/legendre-transform/legendre-transform-antitone","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.legendreTransform_antitone","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#legendre-transform-antitone","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $\\Lambda_1, \\Lambda_2 : \\mathbb{R}^d \\to [-\\infty, \\infty]$, with Legendre transforms $\\Lambda_1^*$ and $\\Lambda_2^*$. If $\\Lambda_1(y) \\le \\Lambda_2(y)$ for every $y \\in \\mathbb{R}^d$, then $\\Lambda_2^*(x) \\le \\Lambda_1^*(x)$ for every $x \\in \\mathbb{R}^d$.","title":"The Legendre transform is order-reversing"},{"id":"Probability/large-deviations/legendre-transform/convex-epi-legendre-transform","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.convex_epi_legendreTransform","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#convex-epi-legendre-transform","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $\\Lambda : \\mathbb{R}^d \\to [-\\infty, \\infty]$, with Legendre transform $\\Lambda^*$. Then the set $\\{(x, t) \\in \\mathbb{R}^d \\times \\mathbb{R} : \\Lambda^*(x) \\le t\\}$ is convex.","title":"The epigraph of the Legendre transform is convex"},{"id":"Probability/large-deviations/legendre-transform/convex-on-legendre-transform","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.convexOn_legendreTransform","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#convex-on-legendre-transform","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $\\Lambda : \\mathbb{R}^d \\to [-\\infty, \\infty]$, with Legendre transform $\\Lambda^*$. If $\\Lambda^*(x)$ is a real number for every $x \\in \\mathbb{R}^d$, then $\\Lambda^*$, viewed as a function $\\mathbb{R}^d \\to \\mathbb{R}$, is convex on $\\mathbb{R}^d$.","title":"A finite Legendre transform is convex"},{"id":"Probability/large-deviations/legendre-transform/legendre-transform-legendre-transform-le","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.legendreTransform_legendreTransform_le","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#legendre-transform-legendre-transform-le","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $\\Lambda : \\mathbb{R}^d \\to [-\\infty, \\infty]$ be a function that never takes the value $-\\infty$, with Legendre transform $\\Lambda^*$. Then the Legendre transform $\\Lambda^{**}$ of $\\Lambda^*$ satisfies $\\Lambda^{**}(x) \\le \\Lambda(x)$ for every $x \\in \\mathbb{R}^d$.","title":"The double Legendre transform lies below the function"},{"id":"Probability/large-deviations/legendre-transform/legendre-transform-norm-sq-div-two","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.legendreTransform_norm_sq_div_two","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#legendre-transform-norm-sq-div-two","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $q : \\mathbb{R}^d \\to [-\\infty, \\infty]$ be $q(y) = \\|y\\|^2 / 2$, where $\\|\\cdot\\|$ is the Euclidean norm. Its Legendre transform $q^*$ satisfies $q^*(x) = \\|x\\|^2 / 2$ for every $x \\in \\mathbb{R}^d$; in particular every value of $q^*$ is finite.","title":"The half squared norm is its own Legendre transform"},{"id":"Probability/large-deviations/legendre-transform/legendre-transform-zero","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.legendreTransform_zero","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#legendre-transform-zero","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $\\Lambda : \\mathbb{R}^d \\to [-\\infty, \\infty]$ be the zero function. Its Legendre transform $\\Lambda^*$ satisfies $\\Lambda^*(0) = 0$.","title":"Legendre transform of the zero function at the origin"},{"id":"Probability/large-deviations/legendre-transform/legendre-transform-zero-of-ne","kind":"node","lean":"MathlibExt.Probability.LegendreTransform.legendreTransform_zero_of_ne","parent":21842,"result":true,"route":"../roadmap/Probability/large-deviations/legendre-transform/index.html#legendre-transform-zero-of-ne","scope":"Probability/large-deviations/legendre-transform","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $\\Lambda : \\mathbb{R}^d \\to [-\\infty, \\infty]$ be the zero function. Its Legendre transform $\\Lambda^*$ satisfies $\\Lambda^*(x) = +\\infty$ for every nonzero $x \\in \\mathbb{R}^d$.","title":"Legendre transform of the zero function away from the origin"},{"data_url":"scopes/Probability/large-deviations/cram-r-s-theorem.json","id":"Probability/large-deviations/cram-r-s-theorem","kind":"scope","navigation":"module","parent":21821,"result":false,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","title":"Cram\u00e9r's theorem"},{"id":"Probability/large-deviations/cram-r-s-theorem/empirical-mean-measure-vec","kind":"node","lean":"MathlibExt.Probability.CramerUpperBound.empiricalMeanMeasureVec","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#empirical-mean-measure-vec","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Law of the empirical mean `A\u2099 = (1/n) \u2211_{i<n} X\u1d62` for an `\u211d\u1d48`-valued sample.","title":"Law of the empirical mean"},{"id":"Probability/large-deviations/cram-r-s-theorem/log-mgfvec","kind":"node","lean":"MathlibExt.Probability.CramerUpperBound.logMGFVec","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#log-mgfvec","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Real-valued log-MGF expression `\u039b(t) = log E[exp \u27eat, X\u27eb]`.","title":"Real-valued log-MGF expression"},{"id":"Probability/large-deviations/cram-r-s-theorem/measure-empirical-mean-le-of-threshold","kind":"node","lean":"MathlibExt.Probability.CramerUpperBound.measure_empiricalMean_le_of_threshold","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#measure-empirical-mean-le-of-threshold","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Chernoff bound for the empirical mean of real-valued i.i.d. variables, in event form: if on the event `E` the empirical mean `(1/n) \u2211_{i<n} X\u1d62` satisfies the threshold inequality `a \u2264 t * (empirical mean) - log (mgf of X\u2081 at t)`, then `\u03bc(E) \u2264 exp(-n * a)`.","title":"Chernoff bound for the empirical mean of real-valued i.i.d. variables"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-upper-bound","kind":"node","lean":"MathlibExt.Probability.CramerUpperBound.cramer_upper_bound","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-upper-bound","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"For i.i.d. random vectors in `\u211d\u1d48`, the law `\u03bc\u2099` of the empirical mean satisfies `\u03bc\u2099(F) \u2264 exp(-n * a)` whenever `a` lower-bounds `\u27ea\u03be, \u00b7\u27eb - \u039b(\u03be)` on measurable `F`. Assume a probability measure, measurable coordinates and integrability of `exp \u27ea\u03be, X\u2080\u27eb` at this direction.","title":"Fixed-direction Chernoff bound"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-upper-bound-rate","kind":"node","lean":"MathlibExt.Probability.CramerUpperBound.cramer_upper_bound_rate","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-upper-bound-rate","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"For a fixed `\u03be` and `s = inf_{x \u2208 F} (\u27ea\u03be, x\u27eb - \u039b(\u03be))`, the `n`-th root of `\u03bc\u2099(F)` has limsup at most `exp(-s)`. Assume measurable i.i.d. coordinates under a probability measure, measurable `F`, exponential integrability at `\u03be`, and a bounded-below rate image on `F`, with its real-valued `sInf` used for `s`. Optimizing over admissible `\u03be` gives the supremum of these fixed-direction rates; the Legendre-rate Cram\u00e9r\u2026","title":"Fixed-direction Chernoff bound, rate form"},{"id":"Probability/large-deviations/cram-r-s-theorem/extended-log-mgfvec","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.extendedLogMGFVec","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#extended-log-mgfvec","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Extended log moment: nonintegrable measurable exponential moments give `\u22a4`.","title":"Extended log moment generating function of a random vector"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-rate-function","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.cramerRateFunction","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-rate-function","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"The canonical extended conjugate, converted with top preserved.","title":"Cram\u00e9r rate function"},{"id":"Probability/large-deviations/cram-r-s-theorem/log-mgfvec-zero","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.logMGFVec_zero","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#log-mgfvec-zero","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"The canonical real log moment is zero at the zero direction.","title":"Log moment generating function vanishes at zero"},{"id":"Probability/large-deviations/cram-r-s-theorem/log-mgfvec-const","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.logMGFVec_const","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#log-mgfvec-const","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"The canonical real log moment of a constant vector.","title":"Log moment generating function of a constant vector"},{"id":"Probability/large-deviations/cram-r-s-theorem/extended-log-mgfvec-eq-coe-log-mgfvec","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.extendedLogMGFVec_eq_coe_logMGFVec","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#extended-log-mgfvec-eq-coe-log-mgfvec","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Finite directions compose with the existing fixed-direction Chernoff API.","title":"Extended log moment agrees with the real one at integrable directions"},{"id":"Probability/large-deviations/cram-r-s-theorem/extended-log-mgfvec-eq-top-of-not-integr","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.extendedLogMGFVec_eq_top_of_not_integrable","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#extended-log-mgfvec-eq-top-of-not-integr","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Measurability separates infinite moments from nonmeasurable totalized integrals.","title":"Extended log moment is infinite at non-integrable directions"},{"id":"Probability/large-deviations/cram-r-s-theorem/extended-log-mgfvec-zero","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.extendedLogMGFVec_zero","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#extended-log-mgfvec-zero","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"The zero direction has extended log moment zero under probability.","title":"Extended log moment vanishes at zero"},{"id":"Probability/large-deviations/cram-r-s-theorem/extended-log-mgfvec-const","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.extendedLogMGFVec_const","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#extended-log-mgfvec-const","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Extended log moments of constant vectors retain the exact inner product.","title":"Extended log moment of a constant vector"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-rate-function-eq-i-sup","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.cramerRateFunction_eq_iSup","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-rate-function-eq-i-sup","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Supporting terms are obtained from the canonical conjugate, not a second transform.","title":"Cram\u00e9r rate function as a supremum"},{"id":"Probability/large-deviations/cram-r-s-theorem/le-cramer-rate-function","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.le_cramerRateFunction","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#le-cramer-rate-function","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Extended Fenchel\u2013Young, valid for every measure and every direction.","title":"Extended Fenchel\u2013Young inequality for the Cram\u00e9r rate function"},{"id":"Probability/large-deviations/cram-r-s-theorem/le-cramer-rate-function-of-integrable","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.le_cramerRateFunction_of_integrable","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#le-cramer-rate-function-of-integrable","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"The original real supporting term remains valid at integrable directions.","title":"Fenchel\u2013Young lower bound at integrable directions"},{"id":"Probability/large-deviations/cram-r-s-theorem/legendre-transform-extended-log-mgfvec-n","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.legendreTransform_extendedLogMGFVec_nonneg","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#legendre-transform-extended-log-mgfvec-n","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Probability makes the canonical conjugate nonnegative via the zero direction.","title":"Nonnegativity of the Legendre transform of the log moment"},{"id":"Probability/large-deviations/cram-r-s-theorem/coe-cramer-rate-function","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.coe_cramerRateFunction","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#coe-cramer-rate-function","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Under probability the conversion preserves the entire nonnegative conjugate.","title":"Cram\u00e9r rate function agrees with the extended conjugate"},{"id":"Probability/large-deviations/cram-r-s-theorem/lower-semicontinuous-cramer-rate-functio","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.lowerSemicontinuous_cramerRateFunction","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#lower-semicontinuous-cramer-rate-functio","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Converted canonical conjugates are lower semicontinuous for arbitrary input data.","title":"Lower semicontinuity of the Cram\u00e9r rate function"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-rate-function-convex","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.cramerRateFunction_convex","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-rate-function-convex","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Convexity is transported from the actual canonical conjugate epigraph.","title":"Convexity of the Cram\u00e9r rate function"},{"id":"Probability/large-deviations/cram-r-s-theorem/is-convex-rate-function-cramer-rate-func","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.isConvexRateFunction_cramerRateFunction","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#is-convex-rate-function-cramer-rate-func","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"The existing LDP convex-rate structure is available without global MGF finiteness.","title":"Cram\u00e9r rate function is a convex rate function"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-rate-function-eq-i-sup-of-integra","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.cramerRateFunction_eq_iSup_of_integrable","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-rate-function-eq-i-sup-of-integra","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"With all moments integrable, the canonical real expression represents every direction.","title":"Cram\u00e9r rate function as a supremum when all moments are integrable"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-rate-function-eq-of-real-i-sup","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.cramerRateFunction_eq_ofReal_iSup","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-rate-function-eq-of-real-i-sup","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Physical finite-domain agreement: bounded real supporting values and integrable moments. The finiteness premise belongs to this corollary, not the corrected definitions.","title":"Cram\u00e9r rate function as a real supremum under finite moments"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-rate-function-eq-zero-at-mean","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.cramerRateFunction_eq_zero_at_mean","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-rate-function-eq-zero-at-mean","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"The original integrable-vector and moment premises imply zero rate at the mean.","title":"Cram\u00e9r rate function vanishes at the mean"},{"id":"Probability/large-deviations/cram-r-s-theorem/cramer-rate-function-const","kind":"node","lean":"MathlibExt.Probability.CramerRateFunction.cramerRateFunction_const","parent":21853,"result":true,"route":"../roadmap/Probability/large-deviations/cram-r-s-theorem/index.html#cramer-rate-function-const","scope":"Probability/large-deviations/cram-r-s-theorem","status":"fully_proved","summary":"Constant laws retain zero at their center and top elsewhere, including dimension zero.","title":"Cram\u00e9r rate function of a constant law"},{"data_url":"scopes/Probability/concentration-inequalities.json","id":"Probability/concentration-inequalities","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/concentration-inequalities/index.html","scope":"Probability/concentration-inequalities","status":"can_prove","summary":"Non-asymptotic tail and moment bounds: Hoeffding, McDiarmid, Efron\u2013Stein, Khintchine and Paley\u2013Zygmund inequalities, sampling without replacement, sub-Gaussian convex comparison, and hypercontractivity (Bonami\u2013Beckner, p-biased two-point inequalities) on the discrete cube.","title":"Concentration inequalities"},{"id":"Probability/concentration-inequalities/khintchine-inequality","kind":"node","lean":"MetaMathlibExt.khintchine_inequality","parent":21879,"result":true,"route":"../roadmap/Probability/concentration-inequalities/index.html#khintchine-inequality","scope":"Probability/concentration-inequalities","status":"fully_proved","summary":"For every real $p > 0$ there are constants $A_p > 0$ and $B_p > 0$ such that the following holds for every $N \\in \\mathbb{N}$, all real numbers $a_1, \\dots, a_N$ and every probability space. Let $\\varepsilon_1, \\dots, \\varepsilon_N$ be independent measurable random variables, each almost surely equal to $1$ or $-1$ and each with mean $0$, and let $S = \\sum_{i=1}^N a_i \\varepsilon_i$ and\u2026","title":"Khintchine inequality"},{"id":"Probability/concentration-inequalities/paley-zygmund","kind":"node","lean":"ProbabilityTheory.paley_zygmund","parent":21879,"result":true,"route":"../roadmap/Probability/concentration-inequalities/index.html#paley-zygmund","scope":"Probability/concentration-inequalities","status":"fully_proved","summary":"Let $Z$ be a real random variable on a probability space with probability measure $\\mathbb{P}$, such that $Z \\ge 0$ almost surely and $\\mathbb{E}[Z^2] < \\infty$, and let $\\theta$ be a real number with $0 \\le \\theta \\le 1$. Then","title":"The Paley\u2013Zygmund inequality"},{"data_url":"scopes/Probability/concentration-inequalities/bonami-beckner-hypercontractivity.json","id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity","kind":"scope","parent":21879,"result":false,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/index.html","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity","status":"fully_proved","title":"Bonami\u2013Beckner hypercontractivity"},{"data_url":"scopes/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on.json","id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","kind":"scope","navigation":"module","parent":21882,"result":false,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","title":"Tensorization of two-point hypercontractivity on the uniform cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-uexp","kind":"node","lean":"StatMech.Probability.bnt_uexp","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-uexp","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"The expectation of `f` under the uniform measure on `\u03b9 \u2192 Bool`. This is `\ud835\udd3c \u03c9, f \u03c9` (`Finset.expect`).","title":"Uniform expectation on the Boolean cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-qnorm-pow","kind":"node","lean":"StatMech.Probability.bnt_qnormPow","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-qnorm-pow","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"`E|f|^q` under the uniform measure.","title":"q-th moment under the uniform measure on the cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-qnorm","kind":"node","lean":"StatMech.Probability.bnt_qnorm","parent":21883,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^q\\)"],["text"," norm under the uniform measure on the cube"]],"source":"$L^q$ norm under the uniform measure on the cube"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-qnorm","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"The `L^q` norm `(E|f|^q)^(1/q)` under the uniform measure.","title":"$L^q$ norm under the uniform measure on the cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-noise-kernel","kind":"node","lean":"StatMech.Probability.bnt_noiseKernel","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-noise-kernel","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"The noise kernel: `\u220f i, (1 + \u03c1)/2` or `(1 - \u03c1)/2` according as `x i = y i` or not.","title":"Noise kernel on the Boolean cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-noise-op","kind":"node","lean":"StatMech.Probability.bnt_noiseOp","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-noise-op","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"The noise operator `T_\u03c1 f x = \u2211 y, bnt_noiseKernel \u03c1 x y * f y`: for `\u03c1 \u2208 [-1, 1]`, the expectation of `f y` where each coordinate of `y` equals that of `x` with probability `(1 + \u03c1)/2`.","title":"Noise operator on the Boolean cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-uexp-eq-expect","kind":"node","lean":"StatMech.Probability.bnt_uexp_eq_expect","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-uexp-eq-expect","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"`bnt_uexp` is the uniform average `Finset.expect`.","title":"Uniform expectation on the cube equals the finite average"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-noise-op-eq-bool-noise-op","kind":"node","lean":"StatMech.Probability.bnt_noiseOp_eq_boolNoiseOp","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-noise-op-eq-bool-noise-op","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"On `Fin n \u2192 Bool`, `bnt_noiseOp` is the noise operator `BooleanCube.boolNoiseOp`.","title":"Agreement of the two noise operators on the Boolean cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-vector-two-point","kind":"node","lean":"StatMech.Probability.bnt_VectorTwoPoint","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-vector-two-point","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"The vector-valued two-point inequality at noise `\u03c1`, with `q = 1 + \u03c1\u00b2`: for all `n` and `u v : (Fin n \u2192 Bool) \u2192 \u211d`, the average of `\u2016((1+\u03c1)/2) u + ((1-\u03c1)/2) v\u2016\u2082\u00b2` and `\u2016((1-\u03c1)/2) u + ((1+\u03c1)/2) v\u2016\u2082\u00b2` is at most `((\u2016u\u2016_q^q + \u2016v\u2016_q^q) / 2)^(2/q)`.","title":"Vector-valued two-point hypercontractive inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-tensor-step","kind":"node","lean":"StatMech.Probability.bnt_TensorStep","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-tensor-step","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"The induction step of tensorization: `\u2016T_\u03c1 g\u2016\u2082 \u2264 \u2016g\u2016_(1+\u03c1\u00b2)` in dimension `n` implies the same in dimension `n + 1`.","title":"Induction step of hypercontractivity tensorization"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-tensor-step-of-vec","kind":"node","lean":"StatMech.Probability.bnt_tensorStep_of_vec","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-tensor-step-of-vec","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"The vector two-point inequality gives the tensorization step.","title":"Vector two-point inequality implies the tensorization step"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-hypercontractivity","kind":"node","lean":"StatMech.Probability.bnt_hypercontractivity","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-hypercontractivity","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"Hypercontractivity `\u2016T_\u03c1 f\u2016\u2082 \u2264 \u2016f\u2016_(1+\u03c1\u00b2)` on `Fin n \u2192 Bool`, from the vector two-point inequality.","title":"Hypercontractivity from the vector two-point inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/bnt-hypercontractivity-of-two-point","kind":"node","lean":"StatMech.Probability.bnt_hypercontractivity_of_twoPoint","parent":21883,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on/index.html#bnt-hypercontractivity-of-two-point","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/tensorization-of-two-point-hypercontractivity-on","status":"fully_proved","summary":"Hypercontractivity `\u2016T_\u03c1 f\u2016\u2082 \u2264 \u2016f\u2016_(1+\u03c1\u00b2)` on `Fin n \u2192 Bool`, from the tensorization step.","title":"Hypercontractivity from the tensorization step"},{"data_url":"scopes/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality.json","id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality","kind":"scope","navigation":"module","parent":21882,"result":false,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality/index.html","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality","status":"fully_proved","title":"The two-point Bonami inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality/bts-taylor-lower","kind":"node","lean":"StatMech.Probability.bts_taylor_lower","parent":21896,"reading_title":{"format":"inline-parts-v1","parts":[["text","Taylor lower bound for "],["math","\\((1+t)^q + (1-t)^q\\)"]],"source":"Taylor lower bound for $(1+t)^q + (1-t)^q$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality/index.html#bts-taylor-lower","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality","status":"fully_proved","summary":"For `1 \u2264 q \u2264 2` and `|t| \u2264 1`, `2 + q(q-1) t\u00b2 \u2264 (1 + t)^q + (1 - t)^q`.","title":"Taylor lower bound for $(1+t)^q + (1-t)^q$"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality/bts-two-point-scalar","kind":"node","lean":"StatMech.Probability.bts_twoPointScalar","parent":21896,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality/index.html#bts-two-point-scalar","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-two-point-bonami-inequality","status":"fully_proved","summary":"The two-point Bonami inequality (Bonami 1970; Beckner 1975): for `0 \u2264 \u03c1 \u2264 1` and real `a, b`, with `q = 1 + \u03c1\u00b2`, `a\u00b2 + \u03c1\u00b2 b\u00b2 \u2264 ((|a + b|^q + |a - b|^q) / 2)^(2/q)`, i.e. `\u2016a + \u03c1 b x\u2016\u2082 \u2264 \u2016a + b x\u2016_q` for a uniform random sign `x`.","title":"Two-point Bonami inequality"},{"data_url":"scopes/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one.json","id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one","kind":"scope","navigation":"module","parent":21882,"result":false,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/index.html","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one","status":"fully_proved","title":"The p-biased two-point inequality, reduced to one variable"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/pbt-g","kind":"node","lean":"StatMech.Probability.pbt_g","parent":21899,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/index.html#pbt-g","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one","status":"fully_proved","summary":"`((1-p) + p z)\u00b2 + 4(q-1) p\u00b2(1-p)\u00b2(z - 1)\u00b2`: the left side of the `p`-biased two-point inequality at `u = 1`, `v = z`.","title":"Left side of the one-variable p-biased inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/pbt-one-var-core","kind":"node","lean":"StatMech.Probability.pbt_OneVarCore","parent":21899,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/index.html#pbt-one-var-core","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one","status":"fully_proved","summary":"The one-variable `p`-biased inequality: `pbt_g p q z \u2264 ((1-p) + p z^q)^(2/q)` for `z \u2265 0`.","title":"Statement of the one-variable p-biased inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/pbt-lhs","kind":"node","lean":"StatMech.Probability.pbt_lhs","parent":21899,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/index.html#pbt-lhs","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one","status":"fully_proved","summary":"The left side `((1-p) u + p v)\u00b2 + 4(q-1) p\u00b2(1-p)\u00b2(u - v)\u00b2` of the `p`-biased two-point inequality.","title":"Left side of the p-biased two-point inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/pbt-two-point-value-of-core","kind":"node","lean":"StatMech.Probability.pbt_two_point_value_of_core","parent":21899,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one/index.html#pbt-two-point-value-of-core","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality-reduced-to-one","status":"fully_proved","summary":"The `p`-biased two-point inequality, assuming the one-variable inequality for `p` and `1 - p`.","title":"Reduction of the p-biased two-point inequality to one variable"},{"data_url":"scopes/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin.json","id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin","kind":"scope","navigation":"module","parent":21882,"result":false,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/index.html","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin","status":"fully_proved","title":"The one-variable p-biased inequality from a pointwise bound"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/pih-m","kind":"node","lean":"StatMech.Probability.pih_m","parent":21904,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/index.html#pih-m","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin","status":"fully_proved","summary":"The mean `(1-p) + p z` of the `p`-biased two-valued function with values `1` and `z`.","title":"Mean of a p-biased two-valued function"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/pih-v","kind":"node","lean":"StatMech.Probability.pih_V","parent":21904,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/index.html#pih-v","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin","status":"fully_proved","summary":"The variance term `4(q-1) p\u00b2(1-p)\u00b2(z - 1)\u00b2`.","title":"Variance term in the p-biased inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/pih-key","kind":"node","lean":"StatMech.Probability.pih_KEY","parent":21904,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/index.html#pih-key","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin","status":"fully_proved","summary":"The pointwise bound `(q/2) m^(q-2) V \u2264 (1-p) + p z^q - m^q` for all `z \u2265 0`, with `m = pih_m p z`, `V = pih_V p q z`.","title":"Pointwise bound behind the p-biased inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/pih-core-of-key","kind":"node","lean":"StatMech.Probability.pih_core_of_KEY","parent":21904,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/index.html#pih-core-of-key","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin","status":"fully_proved","summary":"The pointwise bound `pih_KEY p q` implies the one-variable inequality `pbt_OneVarCore p q`.","title":"Pointwise bound implies the one-variable p-biased inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/pih-two-point-value-of-key","kind":"node","lean":"StatMech.Probability.pih_two_point_value_of_KEY","parent":21904,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin/index.html#pih-two-point-value-of-key","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-one-variable-p-biased-inequality-from-a-poin","status":"fully_proved","summary":"The `p`-biased two-point inequality from `pih_KEY p q` and `pih_KEY (1 - p) q`.","title":"p-biased two-point inequality from the pointwise bound"},{"data_url":"scopes/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality.json","id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality","kind":"scope","navigation":"module","parent":21882,"result":false,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/index.html","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality","status":"fully_proved","title":"The p-biased two-point inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/pcx-g","kind":"node","lean":"StatMech.Probability.pcx_g","parent":21910,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/index.html#pcx-g","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality","status":"fully_proved","summary":"`(1-p)(1 - p t)^q + p(1 + (1-p) t)^q - 1 - 2q(q-1) p\u00b2(1-p)\u00b2 t\u00b2`, convex and nonnegative on `[-1/(1-p), 1/p]` for `1 \u2264 q \u2264 2`.","title":"Convex auxiliary function for the p-biased inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/pcx-key-pos","kind":"node","lean":"StatMech.Probability.pcx_KEY_pos","parent":21910,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/index.html#pcx-key-pos","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality","status":"fully_proved","summary":"The pointwise bound `pih_KEY p q` for `0 < p < 1` and `1 \u2264 q \u2264 2`.","title":"Pointwise bound for the p-biased inequality holds"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/pcx-one-var-core","kind":"node","lean":"StatMech.Probability.pcx_OneVarCore","parent":21910,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/index.html#pcx-one-var-core","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality","status":"fully_proved","summary":"The one-variable `p`-biased inequality for `0 < p < 1` and `1 \u2264 q \u2264 2`.","title":"One-variable p-biased inequality"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/pcx-two-point-value","kind":"node","lean":"StatMech.Probability.pcx_two_point_value","parent":21910,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality/index.html#pcx-two-point-value","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/the-p-biased-two-point-inequality","status":"fully_proved","summary":"The `p`-biased two-point inequality: for `0 < p < 1`, `1 \u2264 q \u2264 2` and real `u, v`, `((1-p) u + p v)\u00b2 + 4(q-1) p\u00b2(1-p)\u00b2(u - v)\u00b2 \u2264 ((1-p)|u|^q + p|v|^q)^(2/q)`.","title":"p-biased two-point inequality"},{"data_url":"scopes/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform.json","id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform","kind":"scope","navigation":"module","parent":21882,"result":false,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform/index.html","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform","status":"fully_proved","title":"Bonami\u2013Beckner hypercontractivity on the uniform cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform/vtp-vector-two-point","kind":"node","lean":"StatMech.Probability.vtp_vectorTwoPoint","parent":21915,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform/index.html#vtp-vector-two-point","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform","status":"fully_proved","summary":"The vector two-point inequality holds for `0 \u2264 \u03c1 \u2264 1`.","title":"Vector two-point inequality for nonnegative noise"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform/vtp-hypercontractivity","kind":"node","lean":"StatMech.Probability.vtp_hypercontractivity","parent":21915,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform/index.html#vtp-hypercontractivity","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/bonami-beckner-hypercontractivity-on-the-uniform","status":"fully_proved","summary":"The Bonami\u2013Beckner hypercontractive inequality (Bonami 1970; Beckner 1975): for `0 \u2264 \u03c1 \u2264 1` and `f : (Fin n \u2192 Bool) \u2192 \u211d`, `\u2016T_\u03c1 f\u2016\u2082 \u2264 \u2016f\u2016_(1 + \u03c1\u00b2)` under the uniform measure.","title":"Bonami\u2013Beckner hypercontractive inequality"},{"data_url":"scopes/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize.json","id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","kind":"scope","navigation":"module","parent":21882,"result":false,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","title":"PBiased Tensorize"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-val","kind":"node","lean":"StatMech.Probability.ptn_val","parent":21918,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-val","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The `0/1` value of a bit.","title":"Zero-one value of a bit"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-sigma","kind":"node","lean":"StatMech.Probability.ptn_sigma","parent":21918,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-sigma","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The standard deviation `\u221a(p(1-p))` of a Bernoulli(`p`) bit.","title":"Standard deviation of a Bernoulli bit"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-psi","kind":"node","lean":"StatMech.Probability.ptn_psi","parent":21918,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-psi","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The normalized Bernoulli(`p`) bit `\u03c8(b) = (b - p) / \u221a(p(1-p))`, which has mean `0` and variance `1`.","title":"Normalized Bernoulli bit"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-expect-prod","kind":"node","lean":"StatMech.Probability.ptn_expect_prod","parent":21918,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-expect-prod","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"Expectations of product functions factor under the product measure.","title":"Expectations of product functions factor under product measure"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-pchar","kind":"node","lean":"StatMech.Probability.ptn_pchar","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased character"]],"source":"$p$-biased character"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-pchar","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The `p`-biased character `\u03c6_S(\u03c9) = \u220f_{i \u2208 S} \u03c8(\u03c9 i)`.","title":"$p$-biased character"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-orthonormal","kind":"node","lean":"StatMech.Probability.ptn_orthonormal","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Orthonormality of the "],["math","\\(p\\)"],["text","-biased characters"]],"source":"Orthonormality of the $p$-biased characters"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-orthonormal","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The `p`-biased characters are orthonormal in `L\u00b2(\u03bc_p)`.","title":"Orthonormality of the $p$-biased characters"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-coeff","kind":"node","lean":"StatMech.Probability.ptn_coeff","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased Fourier coefficient"]],"source":"$p$-biased Fourier coefficient"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-coeff","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The `p`-biased Fourier coefficient `f\u0302(S) = \ud835\udd3c_p[f \u03c6_S]`.","title":"$p$-biased Fourier coefficient"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-expansion","kind":"node","lean":"StatMech.Probability.ptn_expansion","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased Fourier inversion"]],"source":"$p$-biased Fourier inversion"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-expansion","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"Fourier inversion under `\u03bc_p`: `f = \u2211_S f\u0302(S) \u03c6_S` for `0 < p < 1`.","title":"$p$-biased Fourier inversion"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-parseval","kind":"node","lean":"StatMech.Probability.ptn_parseval","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parseval's identity under the "],["math","\\(p\\)"],["text","-biased measure"]],"source":"Parseval's identity under the $p$-biased measure"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-parseval","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"Parseval's identity under `\u03bc_p`: `\ud835\udd3c_p[f g] = \u2211_S f\u0302(S) \u011d(S)`.","title":"Parseval's identity under the $p$-biased measure"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-var-eq-fourier-weight","kind":"node","lean":"StatMech.Probability.ptn_var_eq_fourierWeight","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased variance as Fourier weight off the empty set"]],"source":"$p$-biased variance as Fourier weight off the empty set"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-var-eq-fourier-weight","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The variance under `\u03bc_p` is the Fourier weight off the empty set.","title":"$p$-biased variance as Fourier weight off the empty set"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-infl-eq-fourier-weight","kind":"node","lean":"StatMech.Probability.ptn_infl_eq_fourierWeight","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased influence of a coordinate as Fourier weight"]],"source":"$p$-biased influence of a coordinate as Fourier weight"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-infl-eq-fourier-weight","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The influence of `e` on a Boolean function under `\u03bc_p` is `\u2211_{S \u220b e} f\u0302(S)\u00b2 / (p(1-p))`.","title":"$p$-biased influence of a coordinate as Fourier weight"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-total-infl-eq-fourier-weight","kind":"node","lean":"StatMech.Probability.ptn_totalInfl_eq_fourierWeight","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased total influence as Fourier weight"]],"source":"$p$-biased total influence as Fourier weight"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-total-infl-eq-fourier-weight","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The total influence of a Boolean function under `\u03bc_p` is `\u2211_S |S| f\u0302(S)\u00b2 / (p(1-p))`.","title":"$p$-biased total influence as Fourier weight"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-qnorm-pow","kind":"node","lean":"StatMech.Probability.ptn_qnormPow","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(q\\)"],["text","-th moment under the "],["math","\\(p\\)"],["text","-biased measure"]],"source":"$q$-th moment under the $p$-biased measure"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-qnorm-pow","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The `q`-th moment `\ud835\udd3c_p |f|^q`.","title":"$q$-th moment under the $p$-biased measure"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-qnorm","kind":"node","lean":"StatMech.Probability.ptn_qnorm","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(L^q\\)"],["text"," norm under the "],["math","\\(p\\)"],["text","-biased measure"]],"source":"$L^q$ norm under the $p$-biased measure"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-qnorm","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The `L^q(\u03bc_p)` norm `(\ud835\udd3c_p |f|^q)^{1/q}`.","title":"$L^q$ norm under the $p$-biased measure"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-noise-op","kind":"node","lean":"StatMech.Probability.ptn_noiseOp","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased noise operator on the Fourier side"]],"source":"$p$-biased noise operator on the Fourier side"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-noise-op","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The `p`-biased noise operator on the Fourier side: `T_\u03c1 f = \u2211_S \u03c1^{|S|} f\u0302(S) \u03c6_S`.","title":"$p$-biased noise operator on the Fourier side"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-kbit","kind":"node","lean":"StatMech.Probability.ptn_kbit","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-coordinate "],["math","\\(p\\)"],["text","-biased noise kernel"]],"source":"One-coordinate $p$-biased noise kernel"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-kbit","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The one-coordinate noise kernel: keep the bit with probability `\u03c1`, otherwise resample it from Bernoulli(`p`).","title":"One-coordinate $p$-biased noise kernel"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-kernel","kind":"node","lean":"StatMech.Probability.ptn_kernel","parent":21918,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-kernel","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The product noise kernel on `{0, 1}^\u03b9`.","title":"Product noise kernel on the Boolean cube"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-kop","kind":"node","lean":"StatMech.Probability.ptn_kop","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased noise operator via its Markov kernel"]],"source":"$p$-biased noise operator via its Markov kernel"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-kop","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The `p`-biased noise operator `T_\u03c1 f (x) = \u2211_y K_\u03c1(x, y) f y`, defined by its Markov kernel.","title":"$p$-biased noise operator via its Markov kernel"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-kop-eq-noise-op","kind":"node","lean":"StatMech.Probability.ptn_kop_eq_noiseOp","parent":21918,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-kop-eq-noise-op","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The kernel and Fourier definitions of the `p`-biased noise operator agree for `0 < p < 1`.","title":"Kernel and Fourier noise operators agree"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-tensor-step","kind":"node","lean":"StatMech.Probability.ptn_tensorStep","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tensorization step for "],["math","\\(p\\)"],["text","-biased hypercontractivity"]],"source":"Tensorization step for $p$-biased hypercontractivity"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-tensor-step","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"Induction step of the tensorization: hypercontractivity on `{0, 1}^n` implies it on `{0, 1}^{n+1}`.","title":"Tensorization step for $p$-biased hypercontractivity"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-hypercontractivity-fin","kind":"node","lean":"StatMech.Probability.ptn_hypercontractivity_fin","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased hypercontractivity on "],["math","\\(\\{0, 1\\}^n\\)"]],"source":"$p$-biased hypercontractivity on $\\{0, 1\\}^n$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-hypercontractivity-fin","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"`p`-biased hypercontractivity on `{0, 1}^n`: `\u2016T_\u03c1 f\u2016\u2082 \u2264 \u2016f\u2016_q` for `0 < p < 1`, `1 \u2264 q \u2264 2` and `\u03c1\u00b2 = 4p(1-p)(q - 1)`. The noise rate is not optimal for `p \u2260 1/2`.","title":"$p$-biased hypercontractivity on $\\{0, 1\\}^n$"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-low-degree-weight-fin","kind":"node","lean":"StatMech.Probability.ptn_lowDegreeWeight_fin","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourier form of "],["math","\\(p\\)"],["text","-biased hypercontractivity on "],["math","\\(\\{0, 1\\}^n\\)"]],"source":"Fourier form of $p$-biased hypercontractivity on $\\{0, 1\\}^n$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-low-degree-weight-fin","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"Fourier form of `ptn_hypercontractivity_fin` on `{0, 1}^n`.","title":"Fourier form of $p$-biased hypercontractivity on $\\{0, 1\\}^n$"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-low-degree-weight","kind":"node","lean":"StatMech.Probability.ptn_lowDegreeWeight","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourier form of "],["math","\\(p\\)"],["text","-biased hypercontractivity on finite index types"]],"source":"Fourier form of $p$-biased hypercontractivity on finite index types"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-low-degree-weight","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"`p`-biased hypercontractivity in Fourier form on any finite index type: `\u2211_S \u03c1^{2|S|} f\u0302(S)\u00b2 \u2264 \u2016f\u2016_q\u00b2` for `\u03c1\u00b2 = 4p(1-p)(q - 1)`.","title":"Fourier form of $p$-biased hypercontractivity on finite index types"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-per-coord-hc","kind":"node","lean":"StatMech.Probability.ptn_perCoord_hc","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased hypercontractivity for a discrete derivative"]],"source":"$p$-biased hypercontractivity for a discrete derivative"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-per-coord-hc","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"`p`-biased hypercontractivity applied to the discrete derivative `\u2202\u2091 1_A`.","title":"$p$-biased hypercontractivity for a discrete derivative"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-summed-hc","kind":"node","lean":"StatMech.Probability.ptn_summed_hc","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Summed per-coordinate "],["math","\\(p\\)"],["text","-biased hypercontractive bound"]],"source":"Summed per-coordinate $p$-biased hypercontractive bound"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-summed-hc","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"`ptn_perCoord_hc` summed over all coordinates.","title":"Summed per-coordinate $p$-biased hypercontractive bound"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-capped-hc","kind":"node","lean":"StatMech.Probability.ptn_capped_hc","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped per-coordinate estimate under the "],["math","\\(p\\)"],["text","-biased measure"]],"source":"Capped per-coordinate estimate under the $p$-biased measure"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-capped-hc","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The capped per-coordinate estimate under `\u03bc_p`: `(1/(p(1-p))) \u2211_S |S| \u03c1^{2(|S|-1)} f\u0302(S)\u00b2 \u2264 \u03b4^{2/q - 1} \u2211\u2091 Inf\u2091`, with `\u03b4 = max\u2091 Inf\u2091`.","title":"Capped per-coordinate estimate under the $p$-biased measure"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-rho-optimise","kind":"node","lean":"StatMech.Probability.ptn_RhoOptimise","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased KKL inequality conditional on the capped estimate"]],"source":"$p$-biased KKL inequality conditional on the capped estimate"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-rho-optimise","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The KKL inequality with constant `2` for every `p \u2208 (1/2, q)` with `p < 1`, with the capped estimate `ptn_capped_hc` as a hypothesis.","title":"$p$-biased KKL inequality conditional on the capped estimate"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-kkl-hc-of-rho-optimise","kind":"node","lean":"StatMech.Probability.ptn_kklHC_of_rhoOptimise","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional "],["math","\\(p\\)"],["text","-biased KKL statement yields KKL with constant "],["math","\\(2\\)"]],"source":"Conditional $p$-biased KKL statement yields KKL with constant $2$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-kkl-hc-of-rho-optimise","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"`ptn_RhoOptimise q` gives the KKL inequality with constant `2` at every `p \u2208 (1/2, q)`.","title":"Conditional $p$-biased KKL statement yields KKL with constant $2$"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-pbiased-hc-of-rho-optimise","kind":"node","lean":"StatMech.Probability.ptn_PBiasedHC_of_rhoOptimise","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional KKL gives KKL with constant 2 on "],["math","\\((1/2, q)\\)"],["text"," for "],["math","\\(q \\le 1\\)"]],"source":"Conditional KKL gives KKL with constant 2 on $(1/2, q)$ for $q \\le 1$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-pbiased-hc-of-rho-optimise","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"`ptn_RhoOptimise q` implies `kpb_PBiasedHC q`.","title":"Conditional KKL gives KKL with constant 2 on $(1/2, q)$ for $q \\le 1$"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-rho-optimise-c0","kind":"node","lean":"StatMech.Probability.ptn_rhoOptimise_c0","parent":21918,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trivial KKL inequality with constant "],["math","\\(0\\)"]],"source":"Trivial KKL inequality with constant $0$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-rho-optimise-c0","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The KKL inequality with constant `0` holds trivially.","title":"Trivial KKL inequality with constant $0$"},{"id":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/ptn-sharp-threshold-of-rho-optimise","kind":"node","lean":"StatMech.Probability.ptn_sharpThreshold_of_rhoOptimise","parent":21918,"result":true,"route":"../roadmap/Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize/index.html#ptn-sharp-threshold-of-rho-optimise","scope":"Probability/concentration-inequalities/bonami-beckner-hypercontractivity/pbiased-tensorize","status":"fully_proved","summary":"The sharp-threshold bound `kpb_sharpThreshold` under `ptn_RhoOptimise q`.","title":"Sharp-threshold bound from the conditional KKL statement"},{"data_url":"scopes/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails.json","id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","kind":"scope","navigation":"module","parent":21879,"result":false,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"can_prove","title":"Convex comparison under sub-Gaussian tails"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/is-one-sub-gaussian-tail","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.IsOneSubGaussianTail","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#is-one-sub-gaussian-tail","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$. A function $X : \\Omega \\to \\mathbb{R}$ has a 1-sub-Gaussian tail when it is measurable and integrable, has mean $\\int X \\, d\\mu = 0$, and satisfies $\\mu\\{\\omega : |X(\\omega)| > t\\} \\le 2 e^{-t^2/2}$ for every real $t \\ge 0$.","title":"Random variables with a 1-sub-Gaussian tail"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/convex-dominated","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.ConvexDominated","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#convex-dominated","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $\\mu_X$ and $\\mu_Y$ be measures on measurable spaces $\\Omega_X$ and $\\Omega_Y$, and let $X : \\Omega_X \\to \\mathbb{R}$ and $Y : \\Omega_Y \\to \\mathbb{R}$. Then $X$ (under $\\mu_X$) is convexly dominated by $Y$ (under $\\mu_Y$) when every convex function $f : \\mathbb{R} \\to \\mathbb{R}$ for which $f \\circ X$ is $\\mu_X$-integrable and $f \\circ Y$ is $\\mu_Y$-integrable satisfies\u2026","title":"Convex domination"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/is-admissible-constant","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.IsAdmissibleConstant","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#is-admissible-constant","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"An extended nonnegative real $C \\in [0, \\infty]$ is an admissible comparison constant when $0 < C < \\infty$ and, for every probability measure $\\mu$ on a measurable space $\\Omega$ and every function $X : \\Omega \\to \\mathbb{R}$ with a 1-sub-Gaussian tail under $\\mu$, $X$ under $\\mu$ is convexly dominated by the function $g \\mapsto \\sqrt{C}\\, g$ under the standard Gaussian measure $\\mathcal{N}(0, 1)$ on $\\mathbb{R}$.","title":"Admissible convex sub-Gaussian comparison constants"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/comparison-constant","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.comparisonConstant","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#comparison-constant","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"The comparison constant is the infimum, in $[0, \\infty]$, of the set of admissible comparison constants $C$; it equals $\\infty$ when no admissible constant exists.","title":"The convex sub-Gaussian comparison constant"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/gaussian-tail-envelope","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.gaussianTailEnvelope","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#gaussian-tail-envelope","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"For a real number $t$, the capped Gaussian tail envelope at $t$ is $\\min\\{1, 2 e^{-t^2/2}\\}$.","title":"The capped Gaussian tail envelope"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/one-le-of-is-admissible-constant","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.one_le_of_isAdmissibleConstant","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#one-le-of-is-admissible-constant","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Every admissible comparison constant $C$ satisfies $C \\ge 1$.","title":"Admissible comparison constants are at least one"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/one-le-comparison-constant","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.one_le_comparisonConstant","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#one-le-comparison-constant","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"The convex sub-Gaussian comparison constant is at least $1$. This holds also when no admissible comparison constant exists, in which case the constant is $\\infty$.","title":"The comparison constant is at least one"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/elim","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.IsAdmissibleConstant.elim","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#elim","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $C$ be an admissible comparison constant, let $\\mu$ be a probability measure on a measurable space $\\Omega$, and let $X : \\Omega \\to \\mathbb{R}$ have a 1-sub-Gaussian tail. Then $X$ is convexly dominated by $\\sqrt{C}\\,G$, where $G$ is a standard Gaussian random variable (the identity on $\\mathbb{R}$ under the standard normal law $N(0, 1)$).","title":"Admissible constants compare variables on every sample space"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/saturates-gaussian-tail","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparison.SaturatesGaussianTail","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#saturates-gaussian-tail","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$. A function $X : \\Omega \\to \\mathbb{R}$ saturates the Gaussian tail when $\\mu\\{\\omega : |X(\\omega)| > t\\} = \\min\\{1, 2 e^{-t^2/2}\\}$ for every real $t \\ge 0$, that is, its strict tails equal the capped Gaussian tail envelope.","title":"Saturating the Gaussian tail envelope"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/standard-normal-tail","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.standardNormalTail","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#standard-normal-tail","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"For a real number $x$, the standard Gaussian upper tail at $x$ is $Q(x) = \\int_x^\\infty \\varphi(t)\\,dt$, where $\\varphi(t) = e^{-t^2/2}/\\sqrt{2\\pi}$ is the standard Gaussian density.","title":"Standard Gaussian upper tail"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-t0","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.davisPowerT0","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-t0","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"The transition threshold is the real number $t_0 = \\sqrt{2\\log 2}$, where $\\log$ is the natural logarithm.","title":"The Davis\u2013Power transition threshold"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-a","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.davisPowerA","parent":21950,"reading_title":{"format":"inline-parts-v1","parts":[["text","The envelope integral "],["math","\\(A\\)"]],"source":"The envelope integral $A$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-a","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $A = \\int_0^\\infty \\min(1, 2e^{-t^2/2})\\,dt$, the integral over $[0, \\infty)$ of the capped Gaussian tail envelope $t \\mapsto \\min(1, 2e^{-t^2/2})$.","title":"The envelope integral $A$"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-b","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.davisPowerB","parent":21950,"reading_title":{"format":"inline-parts-v1","parts":[["text","Half the envelope integral, "],["math","\\(B\\)"]],"source":"Half the envelope integral, $B$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-b","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $B = A/2$, where $A = \\int_0^\\infty \\min(1, 2e^{-t^2/2})\\,dt$ is the integral over $[0, \\infty)$ of the capped Gaussian tail envelope.","title":"Half the envelope integral, $B$"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-h","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.davisPowerH","parent":21950,"reading_title":{"format":"inline-parts-v1","parts":[["text","The function "],["math","\\(h\\)"],["text"," of Davis\u2013Power"]],"source":"The function $h$ of Davis\u2013Power"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-h","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"For real $x$, let $h(x) = 2x e^{-x^2/2} + 2\\int_x^\\infty e^{-t^2/2}\\,dt$.","title":"The function $h$ of Davis\u2013Power"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-solution","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.DavisPowerSolution","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-solution","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"A parameter quadruple is a quadruple $(a, p_0, z, c_0)$ of real numbers, the unknowns of the Davis\u2013Power equation system. Being a solution of the system is a separate condition.","title":"Parameter quadruples of the Davis\u2013Power system"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/is-valid","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.DavisPowerSolution.IsValid","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#is-valid","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $\\varphi(t) = e^{-t^2/2}/\\sqrt{2\\pi}$ be the standard Gaussian density, $Q(x) = \\int_x^\\infty \\varphi(t)\\,dt$ its upper tail, $t_0 = \\sqrt{2\\log 2}$ with $\\log$ the natural logarithm, $B = \\frac{1}{2}\\int_0^\\infty \\min(1, 2e^{-t^2/2})\\,dt$ and $h(x) = 2x e^{-x^2/2} + 2\\int_x^\\infty e^{-t^2/2}\\,dt$. A quadruple $(a, p_0, z, c_0)$ of real numbers is a valid solution of the Davis\u2013Power system when $a > t_0$,\u2026","title":"Valid solutions of the Davis\u2013Power system"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-c0-values","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.davisPowerC0Values","parent":21950,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of "],["math","\\(c_0\\)"],["text"," solving the Davis\u2013Power system"]],"source":"Values of $c_0$ solving the Davis\u2013Power system"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-c0-values","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $\\varphi(t) = e^{-t^2/2}/\\sqrt{2\\pi}$ be the standard Gaussian density, $B = \\frac12 \\int_0^\\infty \\min(1, 2e^{-t^2/2})\\,dt$ and $h(x) = 2x e^{-x^2/2} + 2\\int_x^\\infty e^{-t^2/2}\\,dt$. This is the set of reals $c_0$ for which there are reals $a$, $p_0$, $z$ with $a > \\sqrt{2 \\log 2}$, $h(a) = B$, $p_0 = 2e^{-a^2/2}$, $0 < p_0 < 1$, $\\int_z^\\infty \\varphi(t)\\,dt = p_0$ and $c_0 = B / \\varphi(z)$.","title":"Values of $c_0$ solving the Davis\u2013Power system"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-c0","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWantedDavis.davisPowerC0","parent":21950,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Davis\u2013Power constant "],["math","\\(c_0\\)"]],"source":"The Davis\u2013Power constant $c_0$"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-c0","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"fully_proved","summary":"Let $\\varphi(t) = e^{-t^2/2}/\\sqrt{2\\pi}$ be the standard Gaussian density, $B = \\frac12 \\int_0^\\infty \\min(1, 2e^{-t^2/2})\\,dt$ and $h(x) = 2x e^{-x^2/2} + 2\\int_x^\\infty e^{-t^2/2}\\,dt$. The constant $c_0$ is the infimum of the set of reals $c$ for which there are reals $a$, $p_0$, $z$ with $a > \\sqrt{2 \\log 2}$, $h(a) = B$, $p_0 = 2e^{-a^2/2}$, $0 < p_0 < 1$, $\\int_z^\\infty \\varphi(t)\\,dt = p_0$ and\u2026","title":"The Davis\u2013Power constant $c_0$"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-system-exists-unique","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWanted.davisPowerSystem_existsUnique","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-system-exists-unique","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"can_prove","summary":"There is exactly one quadruple $(a, p_0, z, c_0)$ of real numbers that is a valid solution of the Davis\u2013Power system.","title":"Existence and uniqueness for the Davis\u2013Power system"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/historical-finiteness-upper-bound","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWanted.historicalFinitenessUpperBound","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#historical-finiteness-upper-bound","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"can_prove","summary":"The comparison constant is finite. Here the comparison constant is the infimum, in $[0, \\infty]$, of the variance factors $C \\in (0, \\infty)$ such that every real random variable $X$ on a probability space that is integrable, has mean zero and satisfies $P(|X| > t) \\le 2e^{-t^2/2}$ for all $t \\ge 0$ satisfies $E f(X) \\le E f(\\sqrt{C}\\,G)$ for every convex $f : \\mathbb{R} \\to \\mathbb{R}$ for which both expectations\u2026","title":"Finiteness of the convex sub-Gaussian comparison constant"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-sharp-upper","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWanted.davisPowerSharpUpper","parent":21950,"reading_title":{"format":"inline-parts-v1","parts":[["text","Davis\u2013Power upper bound: "],["math","\\(c_0^2\\)"],["text"," is an admissible variance factor"]],"source":"Davis\u2013Power upper bound: $c_0^2$ is an admissible variance factor"},"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-sharp-upper","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"can_prove","summary":"Let $c_0$ be the Davis\u2013Power constant and $G$ a standard Gaussian. Then $c_0^2 > 0$, and every real random variable $X$ on a probability space that is measurable and integrable, has mean $0$ and satisfies $\\mathbb{P}(|X| > t) \\le 2e^{-t^2/2}$ for all $t \\ge 0$ is dominated by $|c_0|\\,G$ in the convex order. Here a real random variable $X$ is dominated by $Y$ in the convex order when\u2026","title":"Davis\u2013Power upper bound: $c_0^2$ is an admissible variance factor"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-sharp-lower","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWanted.davisPowerSharpLower","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-sharp-lower","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"can_prove","summary":"Every admissible variance factor $C \\in [0, \\infty]$ satisfies $c_0^2 \\le C$, where $c_0$ is the Davis\u2013Power constant. Here $C$ is admissible when $0 < C < \\infty$ and every real random variable $X$ on a probability space that is measurable and integrable, has mean $0$ and satisfies $\\mathbb{P}(|X| > t) \\le 2e^{-t^2/2}$ for all $t \\ge 0$ is dominated by $\\sqrt{C}\\,G$ in the convex order, with $G$ a standard\u2026","title":"Davis\u2013Power lower bound for admissible variance factors"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/comparison-constant-eq-davis-power-c0-sq","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWanted.comparisonConstant_eq_davisPowerC0_sq","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#comparison-constant-eq-davis-power-c0-sq","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"can_prove","summary":"The convex sub-Gaussian comparison constant, the infimum in $[0, \\infty]$ of all admissible variance factors, equals $c_0^2$, where $c_0$ is the Davis\u2013Power constant.","title":"Davis\u2013Power value of the convex sub-Gaussian comparison constant"},{"id":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/davis-power-extremal-attainment","kind":"node","lean":"MathlibExt.Probability.ConvexSubGaussianComparisonWanted.davisPowerExtremalAttainment","parent":21950,"result":true,"route":"../roadmap/Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails/index.html#davis-power-extremal-attainment","scope":"Probability/concentration-inequalities/convex-comparison-under-sub-gaussian-tails","status":"can_prove","summary":"Let $c_0$ be the Davis\u2013Power constant and $G$ a standard Gaussian, and say that a real random variable $X$ is dominated by $Y$ in the convex order when $\\mathbb{E} f(X) \\le \\mathbb{E} f(Y)$ for every convex $f : \\mathbb{R} \\to \\mathbb{R}$ such that $f(X)$ and $f(Y)$ are integrable. There is a real random variable $X$ on a probability space that is measurable and integrable, has mean $0$ and satisfies\u2026","title":"Davis\u2013Power extremal variable"},{"data_url":"scopes/Probability/concentration-inequalities/efron-stein-inequality.json","id":"Probability/concentration-inequalities/efron-stein-inequality","kind":"scope","navigation":"module","parent":21879,"result":false,"route":"../roadmap/Probability/concentration-inequalities/efron-stein-inequality/index.html","scope":"Probability/concentration-inequalities/efron-stein-inequality","status":"fully_proved","title":"Efron\u2013Stein inequality"},{"id":"Probability/concentration-inequalities/efron-stein-inequality/efron-stein-finite-iid-general","kind":"node","lean":"MathlibExt.Probability.EfronStein.efron_stein_finite_iid_general","parent":21975,"result":true,"route":"../roadmap/Probability/concentration-inequalities/efron-stein-inequality/index.html#efron-stein-finite-iid-general","scope":"Probability/concentration-inequalities/efron-stein-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a finite nonempty set, let $p : \\alpha \\to \\mathbb{R}$ satisfy $p(a) \\ge 0$ for every $a \\in \\alpha$ and $\\sum_{a \\in \\alpha} p(a) = 1$, let $n \\in \\mathbb{N}$ and let $f : \\alpha^n \\to \\mathbb{R}$. For $x = (x_1, \\dots, x_n) \\in \\alpha^n$ write $w(x) = \\prod_{j=1}^{n} p(x_j)$, and for a coordinate $i \\in \\{1, \\dots, n\\}$ and $a \\in \\alpha$ let $x^{(i,a)}$ be $x$ with its $i$-th coordinate replaced\u2026","title":"Efron\u2013Stein inequality for a finite product weight"},{"id":"Probability/concentration-inequalities/efron-stein-inequality/efron-stein-finite-iid","kind":"node","lean":"MathlibExt.Probability.EfronStein.efron_stein_finite_iid","parent":21975,"result":true,"route":"../roadmap/Probability/concentration-inequalities/efron-stein-inequality/index.html#efron-stein-finite-iid","scope":"Probability/concentration-inequalities/efron-stein-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a finite nonempty set, let $p : \\alpha \\to \\mathbb{R}$ satisfy $p(a) \\ge 0$ for every $a \\in \\alpha$ and $\\sum_{a \\in \\alpha} p(a) = 1$, let $n \\in \\mathbb{N}$ and let $f : \\alpha^n \\to \\mathbb{R}$. For $x = (x_1, \\dots, x_n) \\in \\alpha^n$ write $w(x) = \\prod_{j=1}^{n} p(x_j)$, and for a coordinate $i \\in \\{1, \\dots, n\\}$ and $a \\in \\alpha$ let $x^{(i,a)}$ be $x$ with its $i$-th coordinate replaced\u2026","title":"Efron\u2013Stein inequality for finite i.i.d. coordinates"},{"data_url":"scopes/Probability/concentration-inequalities/hoeffding-s-inequality.json","id":"Probability/concentration-inequalities/hoeffding-s-inequality","kind":"scope","navigation":"module","parent":21879,"result":false,"route":"../roadmap/Probability/concentration-inequalities/hoeffding-s-inequality/index.html","scope":"Probability/concentration-inequalities/hoeffding-s-inequality","status":"fully_proved","title":"Hoeffding's inequality"},{"id":"Probability/concentration-inequalities/hoeffding-s-inequality/hoeffding-bernoulli-sum-abs-tail","kind":"node","lean":"ProbabilityTheory.hoeffding_bernoulli_sum_abs_tail","parent":21978,"result":true,"route":"../roadmap/Probability/concentration-inequalities/hoeffding-s-inequality/index.html#hoeffding-bernoulli-sum-abs-tail","scope":"Probability/concentration-inequalities/hoeffding-s-inequality","status":"fully_proved","summary":"Let $X_0, \\dots, X_{n-1}$ be independent real random variables on a probability space, each almost everywhere measurable, almost surely equal to $0$ or $1$, and with expectation $p$, and let $\\varepsilon > 0$. Then the probability that $\\lvert X_0 + \\dots + X_{n-1} - np \\rvert > \\varepsilon n$ is at most $2 e^{-2 \\varepsilon^2 n}$.","title":"Two-sided Hoeffding inequality for Bernoulli sums"},{"id":"Probability/concentration-inequalities/hoeffding-s-inequality/hoeffding-binomial-lower-tail","kind":"node","lean":"ProbabilityTheory.hoeffding_binomial_lower_tail","parent":21978,"result":true,"route":"../roadmap/Probability/concentration-inequalities/hoeffding-s-inequality/index.html#hoeffding-binomial-lower-tail","scope":"Probability/concentration-inequalities/hoeffding-s-inequality","status":"fully_proved","summary":"Let $X_0, \\dots, X_{n-1}$ be independent real random variables on a probability space, each almost everywhere measurable, almost surely equal to $0$ or $1$, and with expectation $p$, and let $\\varepsilon > 0$. Then the probability that $X_0 + \\dots + X_{n-1} \\le n(p - \\varepsilon)$ is at most $e^{-2 \\varepsilon^2 n}$.","title":"Hoeffding lower-tail bound for Bernoulli sums"},{"id":"Probability/concentration-inequalities/hoeffding-s-inequality/card-sample-tail-le","kind":"node","lean":"Finset.card_sample_tail_le","parent":21978,"result":true,"route":"../roadmap/Probability/concentration-inequalities/hoeffding-s-inequality/index.html#card-sample-tail-le","scope":"Probability/concentration-inequalities/hoeffding-s-inequality","status":"fully_proved","summary":"Hoeffding's upper-tail bound for sampling without replacement, stated as a count.","title":"Hoeffding's bound for sampling without replacement"},{"data_url":"scopes/Probability/concentration-inequalities/mcdiarmid-s-inequality.json","id":"Probability/concentration-inequalities/mcdiarmid-s-inequality","kind":"scope","navigation":"module","parent":21879,"result":false,"route":"../roadmap/Probability/concentration-inequalities/mcdiarmid-s-inequality/index.html","scope":"Probability/concentration-inequalities/mcdiarmid-s-inequality","status":"fully_proved","title":"McDiarmid's inequality"},{"id":"Probability/concentration-inequalities/mcdiarmid-s-inequality/mcdiarmid-one-sided-finite-iid-general","kind":"node","lean":"MathlibExt.Probability.McDiarmid.mcdiarmid_one_sided_finite_iid_general","parent":21982,"result":true,"route":"../roadmap/Probability/concentration-inequalities/mcdiarmid-s-inequality/index.html#mcdiarmid-one-sided-finite-iid-general","scope":"Probability/concentration-inequalities/mcdiarmid-s-inequality","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $\\alpha$ be a finite nonempty set with weights $p : \\alpha \\to \\mathbb{R}$ such that $p(a) \\ge 0$ for all $a$ and $\\sum_{a \\in \\alpha} p(a) = 1$, and give each $x \\in \\alpha^n$ the product weight $P(x) = \\prod_{i\\lt n} p(x_i)$. Let $f : \\alpha^n \\to \\mathbb{R}$ and let $c_0, \\dots, c_{n-1} \\ge 0$ be such that $|f(x) - f(y)| \\le c_i$ whenever $x, y \\in \\alpha^n$ agree in every coordinate\u2026","title":"McDiarmid's inequality when the sensitivities are not all zero"},{"id":"Probability/concentration-inequalities/mcdiarmid-s-inequality/mcdiarmid-one-sided-finite-iid-of-sum-sq","kind":"node","lean":"MathlibExt.Probability.McDiarmid.mcdiarmid_one_sided_finite_iid_of_sum_sq_eq_zero","parent":21982,"result":true,"route":"../roadmap/Probability/concentration-inequalities/mcdiarmid-s-inequality/index.html#mcdiarmid-one-sided-finite-iid-of-sum-sq","scope":"Probability/concentration-inequalities/mcdiarmid-s-inequality","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $\\alpha$ be a finite nonempty set with weights $p : \\alpha \\to \\mathbb{R}$ such that $p(a) \\ge 0$ for all $a$ and $\\sum_{a \\in \\alpha} p(a) = 1$, and give each $x \\in \\alpha^n$ the product weight $P(x) = \\prod_{i\\lt n} p(x_i)$. Let $f : \\alpha^n \\to \\mathbb{R}$ and let $c_0, \\dots, c_{n-1} \\ge 0$ be such that $|f(x) - f(y)| \\le c_i$ whenever $x, y \\in \\alpha^n$ agree in every coordinate\u2026","title":"The upper tail vanishes when every sensitivity is zero"},{"id":"Probability/concentration-inequalities/mcdiarmid-s-inequality/mcdiarmid-one-sided-finite-iid","kind":"node","lean":"MathlibExt.Probability.McDiarmid.mcdiarmid_one_sided_finite_iid","parent":21982,"result":true,"route":"../roadmap/Probability/concentration-inequalities/mcdiarmid-s-inequality/index.html#mcdiarmid-one-sided-finite-iid","scope":"Probability/concentration-inequalities/mcdiarmid-s-inequality","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $\\alpha$ be a finite nonempty set with weights $p : \\alpha \\to \\mathbb{R}$ such that $p(a) \\ge 0$ for all $a$ and $\\sum_{a \\in \\alpha} p(a) = 1$, and give each $x \\in \\alpha^n$ the product weight $P(x) = \\prod_{i\\lt n} p(x_i)$. Let $f : \\alpha^n \\to \\mathbb{R}$ and let $c_0, \\dots, c_{n-1} \\ge 0$ be such that $|f(x) - f(y)| \\le c_i$ whenever $x, y \\in \\alpha^n$ agree in every coordinate\u2026","title":"McDiarmid's bounded-differences inequality, finite i.i.d. case"},{"data_url":"scopes/Probability/martingales.json","id":"Probability/martingales","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/martingales/index.html","scope":"Probability/martingales","status":"fully_proved","summary":"Discrete-time martingales, sub- and supermartingales: convergence theorems (almost sure, L^p, backwards, L\u00e9vy's downward theorem), uniform integrability, Doob's maximal inequalities, orthogonality and conditional variance of increments, optional stopping in all its forms, and Wald's equations for stopped sums. Martingale measures and the fundamental theorem of asset pricing.","title":"Martingales and optional stopping"},{"data_url":"scopes/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore.json","id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","kind":"scope","navigation":"module","parent":21986,"result":false,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","title":"Backwards martingales and L\u00e9vy's downward theorem"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/levy-downward-ae","kind":"node","lean":"MeasureTheory.levy_downward_ae","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#levy-downward-ae","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Let $(\\Omega, \\mathcal F, \\mu)$ be a probability space, let $\\mathcal F_0 \\supseteq \\mathcal F_1 \\supseteq \\cdots$ be a decreasing sequence of sub-\u03c3-algebras of $\\mathcal F$, and let $\\mathcal F_\\infty = \\bigcap_n \\mathcal F_n$. For every function $g : \\Omega \\to \\mathbb R$, the conditional expectations $\\mathbb E[g \\mid \\mathcal F_n]$ converge to $\\mathbb E[g \\mid \\mathcal F_\\infty]$ as $n \\to \\infty$,\u2026","title":"L\u00e9vy's downward theorem, almost sure convergence"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/levy-downward-tendsto-e-lp-norm-one","kind":"node","lean":"MeasureTheory.levy_downward_tendsto_eLpNorm_one","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#levy-downward-tendsto-e-lp-norm-one","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Let $(\\Omega, \\mathcal F, \\mu)$ be a probability space, let $\\mathcal F_0 \\supseteq \\mathcal F_1 \\supseteq \\cdots$ be a decreasing sequence of sub-\u03c3-algebras of $\\mathcal F$, and let $\\mathcal F_\\infty = \\bigcap_n \\mathcal F_n$. For every integrable function $g : \\Omega \\to \\mathbb R$, the $L^1(\\mu)$ norm $\\lVert \\mathbb E[g \\mid \\mathcal F_n] - \\mathbb E[g \\mid \\mathcal F_\\infty] \\rVert_1$ tends to $0$ as\u2026","title":"L\u00e9vy's downward theorem in L\u00b9"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/is-backwards-martingale","kind":"node","lean":"MeasureTheory.IsBackwardsMartingale","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#is-backwards-martingale","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $(\\Omega, \\mathcal F)$ and let $(\\mathcal F_n)_{n \\ge 0}$ be a sequence of \u03c3-algebras on $\\Omega$. A sequence of functions $M_n : \\Omega \\to \\mathbb R$ is a backwards martingale with respect to $(\\mathcal F_n)$ and $\\mu$ if $M_0$ is $\\mu$-integrable and, for every $n$, $M_n = \\mathbb E[M_0 \\mid \\mathcal F_n]$ $\\mu$-almost everywhere.","title":"Backwards martingale"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/backwards-martingale-ae-tendsto","kind":"node","lean":"MeasureTheory.backwards_martingale_ae_tendsto","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#backwards-martingale-ae-tendsto","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Let $(\\Omega, \\mathcal F, \\mu)$ be a probability space, let $\\mathcal F_0 \\supseteq \\mathcal F_1 \\supseteq \\cdots$ be a decreasing sequence of sub-\u03c3-algebras of $\\mathcal F$, and let $\\mathcal F_\\infty = \\bigcap_n \\mathcal F_n$. If the real-valued process $(M_n)_{n \\ge 0}$ is a backwards martingale with respect to $(\\mathcal F_n)$ and $\\mu$, then $M_n \\to \\mathbb E[M_0 \\mid \\mathcal F_\\infty]$ as $n \\to \\infty$,\u2026","title":"Backwards martingale convergence theorem, almost sure convergence"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/backwards-martingale-tendsto-e-lp-norm-o","kind":"node","lean":"MeasureTheory.backwards_martingale_tendsto_eLpNorm_one","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#backwards-martingale-tendsto-e-lp-norm-o","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Let $(\\Omega, \\mathcal F, \\mu)$ be a probability space, let $\\mathcal F_0 \\supseteq \\mathcal F_1 \\supseteq \\cdots$ be a decreasing sequence of sub-\u03c3-algebras of $\\mathcal F$, and let $\\mathcal F_\\infty = \\bigcap_n \\mathcal F_n$. If the real-valued process $(M_n)_{n \\ge 0}$ is a backwards martingale with respect to $(\\mathcal F_n)$ and $\\mu$, then the $L^1(\\mu)$ norm\u2026","title":"Backwards martingale convergence theorem in L\u00b9"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/tail-algebra","kind":"node","lean":"MeasureTheory.Filtration.tailAlgebra","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#tail-algebra","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"The tail sigma-algebra of a decreasing filtration, the intersection of its levels.","title":"Tail sigma-algebra of a decreasing filtration"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/tail-algebra-le","kind":"node","lean":"MeasureTheory.Filtration.tailAlgebra_le","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#tail-algebra-le","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"The tail sigma-algebra lies in the ambient sigma-algebra.","title":"Tail sigma-algebra lies in the ambient sigma-algebra"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/tail-algebra-le-seq","kind":"node","lean":"MeasureTheory.Filtration.tailAlgebra_le_seq","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#tail-algebra-le-seq","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"The tail sigma-algebra lies in every level of the decreasing filtration.","title":"Tail sigma-algebra lies in every level of the filtration"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/tail-algebra-const","kind":"node","lean":"MeasureTheory.Filtration.tailAlgebra_const","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#tail-algebra-const","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"A constant decreasing filtration has its constant value as tail algebra.","title":"Tail sigma-algebra of a constant filtration"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/strongly-adapted-cond-exp","kind":"node","lean":"MeasureTheory.Filtration.stronglyAdapted_condExp","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#strongly-adapted-cond-exp","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Conditional-expectation representatives are strongly adapted separately from their canonical backwards-martingale membership. No integrability of g is needed.","title":"Conditional expectations along a filtration are strongly adapted"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/integrable","kind":"node","lean":"MeasureTheory.IsBackwardsMartingale.integrable","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#integrable","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Every term is integrable by its canonical conditional-expectation representation.","title":"Terms of a backwards martingale are integrable"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/uniform-integrable","kind":"node","lean":"MeasureTheory.IsBackwardsMartingale.uniformIntegrable","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#uniform-integrable","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Canonical backwards membership is uniformly integrable in L1 on a finite measure. The ambient bounds suffice; representative adaptedness and antitonicity are unnecessary.","title":"Backwards martingales are uniformly integrable"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/cond-exp-succ-ae-eq","kind":"node","lean":"MeasureTheory.IsBackwardsMartingale.condExp_succ_ae_eq","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#cond-exp-succ-ae-eq","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Conditioning a term on the next, smaller level gives the next term a.e.","title":"Conditioning a backwards martingale on the next level"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/is-backwards-martingale-cond-exp","kind":"node","lean":"MeasureTheory.isBackwardsMartingale_condExp","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#is-backwards-martingale-cond-exp","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Conditional expectations of arbitrary g form a canonical backwards martingale along decreasing sub-sigma-algebras. The nonintegrable-g convention is retained.","title":"Conditional expectations form a backwards martingale"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/tendsto-e-lp-norm-cond-exp-sub-of-ae-ten","kind":"node","lean":"MathlibExt.Probability.CondExpAntitoneLimit.tendsto_eLpNorm_condExp_sub_of_ae_tendsto","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#tendsto-e-lp-norm-cond-exp-sub-of-ae-ten","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $(\\Omega, \\mathcal{F})$, let $(\\mathcal{G}_n)_{n \\in \\mathbb{N}}$ be a sequence of sub-\u03c3-algebras of $\\mathcal{F}$, let $g : \\Omega \\to \\mathbb{R}$ be integrable, and write $\\mu[g \\mid \\mathcal{G}_n]$ for the conditional expectation of $g$ given $\\mathcal{G}_n$ with respect to $\\mu$. If $h : \\Omega \\to \\mathbb{R}$ is integrable and\u2026","title":"Almost-everywhere convergent conditional expectations converge in L\u00b9"},{"id":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/cond-exp-antitone-limit-eq","kind":"node","lean":"MathlibExt.Probability.CondExpAntitoneLimit.condExp_antitone_limit_eq","parent":21987,"result":true,"route":"../roadmap/Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore/index.html#cond-exp-antitone-limit-eq","scope":"Probability/martingales/backwards-martingales-and-l-vy-s-downward-theore","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $(\\Omega, \\mathcal{F})$, let $(\\mathcal{G}_n)_{n \\in \\mathbb{N}}$ be any sequence of sub-\u03c3-algebras of $\\mathcal{F}$ (not assumed monotone), and let $g : \\Omega \\to \\mathbb{R}$ be integrable. Write $\\mu[g \\mid \\mathcal{G}]$ for the conditional expectation of $g$ given a sub-\u03c3-algebra $\\mathcal{G}$ with respect to $\\mu$, and\u2026","title":"Identifying the a.e. limit of conditional expectations"},{"data_url":"scopes/Probability/martingales/convex-functions-of-martingales.json","id":"Probability/martingales/convex-functions-of-martingales","kind":"scope","navigation":"module","parent":21986,"result":false,"route":"../roadmap/Probability/martingales/convex-functions-of-martingales/index.html","scope":"Probability/martingales/convex-functions-of-martingales","status":"fully_proved","title":"Convex functions of martingales"},{"id":"Probability/martingales/convex-functions-of-martingales/submartingale-comp-convex","kind":"node","lean":"MeasureTheory.Martingale.submartingale_comp_convex","parent":22004,"result":true,"route":"../roadmap/Probability/martingales/convex-functions-of-martingales/index.html#submartingale-comp-convex","scope":"Probability/martingales/convex-functions-of-martingales","status":"fully_proved","summary":"Let $\\mu$ be a probability measure, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration, and let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$. Let $\\varphi : \\mathbb{R} \\to \\mathbb{R}$ be convex, and suppose $\\varphi(X_n)$ is integrable for every $n$. Then $(\\varphi(X_n))_{n \\in \\mathbb{N}}$ is a submartingale with respect to $(\\mathcal{F}_n)$ and $\\mu$.","title":"Convex functions of martingales are submartingales"},{"id":"Probability/martingales/convex-functions-of-martingales/comp-convex-of-monotone-on","kind":"node","lean":"MeasureTheory.Submartingale.comp_convex_of_monotoneOn","parent":22004,"result":true,"route":"../roadmap/Probability/martingales/convex-functions-of-martingales/index.html#comp-convex-of-monotone-on","scope":"Probability/martingales/convex-functions-of-martingales","status":"fully_proved","summary":"Let $\\mu$ be a probability measure, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration, and let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued submartingale with respect to $(\\mathcal{F}_n)$ and $\\mu$. Let $\\varphi : \\mathbb{R} \\to \\mathbb{R}$ be convex and nondecreasing, and suppose $\\varphi(X_n)$ is integrable for every $n$. Then $(\\varphi(X_n))_{n \\in \\mathbb{N}}$ is a submartingale with respect to\u2026","title":"Convex nondecreasing functions preserve submartingales"},{"id":"Probability/martingales/convex-functions-of-martingales/supermartingale-comp-concave","kind":"node","lean":"MeasureTheory.Martingale.supermartingale_comp_concave","parent":22004,"result":true,"route":"../roadmap/Probability/martingales/convex-functions-of-martingales/index.html#supermartingale-comp-concave","scope":"Probability/martingales/convex-functions-of-martingales","status":"fully_proved","summary":"Let $\\mu$ be a probability measure, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration, and let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$. Let $\\varphi : \\mathbb{R} \\to \\mathbb{R}$ be concave, and suppose $\\varphi(X_n)$ is integrable for every $n$. Then $(\\varphi(X_n))_{n \\in \\mathbb{N}}$ is a supermartingale with respect to $(\\mathcal{F}_n)$ and $\\mu$.","title":"Concave functions of martingales are supermartingales"},{"data_url":"scopes/Probability/martingales/doob-s-maximal-inequalities.json","id":"Probability/martingales/doob-s-maximal-inequalities","kind":"scope","navigation":"module","parent":21986,"result":false,"route":"../roadmap/Probability/martingales/doob-s-maximal-inequalities/index.html","scope":"Probability/martingales/doob-s-maximal-inequalities","status":"fully_proved","title":"Doob's maximal inequalities"},{"id":"Probability/martingales/doob-s-maximal-inequalities/doob-sq-maximal-ineq","kind":"node","lean":"MeasureTheory.Martingale.doob_sq_maximal_ineq","parent":22008,"result":true,"route":"../roadmap/Probability/martingales/doob-s-maximal-inequalities/index.html#doob-sq-maximal-ineq","scope":"Probability/martingales/doob-s-maximal-inequalities","status":"fully_proved","summary":"Let $\\mu$ be a probability measure and let $(f_k)_{k \\in \\mathbb{N}}$ be a real-valued martingale with respect to a filtration $(\\mathcal{F}_k)$ and $\\mu$ such that every $f_k$ is in $L^2(\\mu)$. Then for every $n \\in \\mathbb{N}$, $\\int (\\max_{k \\le n} |f_k|)^2 \\, d\\mu \\le 4 \\int f_n^2 \\, d\\mu$.","title":"Doob's L\u00b2 maximal inequality, finite horizon"},{"id":"Probability/martingales/doob-s-maximal-inequalities/predictable-quadratic-variation","kind":"node","lean":"MeasureTheory.predictableQuadraticVariation","parent":22008,"result":true,"route":"../roadmap/Probability/martingales/doob-s-maximal-inequalities/index.html#predictable-quadratic-variation","scope":"Probability/martingales/doob-s-maximal-inequalities","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$, let $(\\mathcal{F}_k)_{k \\in \\mathbb{N}}$ be a filtration and let $(f_k)_{k \\in \\mathbb{N}}$ be a real-valued process. The predictable quadratic variation of $f$ with respect to $(\\mathcal{F}_k)$ and $\\mu$ is the process defined pointwise by $A_n(\\omega) = \\sum_{k=0}^{n-1} \\mathbb{E}_\\mu(f_{k+1} - f_k)^2 \\mid \\mathcal{F}_k$, so $A_0 = 0$. For a square-integrable martingale this is\u2026","title":"Predictable quadratic variation"},{"id":"Probability/martingales/doob-s-maximal-inequalities/integral-predictable-quadratic-variation","kind":"node","lean":"MeasureTheory.Martingale.integral_predictableQuadraticVariation","parent":22008,"result":true,"route":"../roadmap/Probability/martingales/doob-s-maximal-inequalities/index.html#integral-predictable-quadratic-variation","scope":"Probability/martingales/doob-s-maximal-inequalities","status":"fully_proved","summary":"Let $\\mu$ be a probability measure and let $(f_k)_{k \\in \\mathbb{N}}$ be a real-valued martingale with respect to a filtration $(\\mathcal{F}_k)$ and $\\mu$ such that every $f_k$ is in $L^2(\\mu)$, and let $A_n$ be its predictable quadratic variation. Then for every $n \\in \\mathbb{N}$, $\\int A_n \\, d\\mu = \\int f_n^2 \\, d\\mu - \\int f_0^2 \\, d\\mu$.","title":"Expectation of the predictable quadratic variation"},{"id":"Probability/martingales/doob-s-maximal-inequalities/doob-sq-maximal-ineq-predictable-quadrat","kind":"node","lean":"MeasureTheory.Martingale.doob_sq_maximal_ineq_predictableQuadraticVariation","parent":22008,"result":true,"route":"../roadmap/Probability/martingales/doob-s-maximal-inequalities/index.html#doob-sq-maximal-ineq-predictable-quadrat","scope":"Probability/martingales/doob-s-maximal-inequalities","status":"fully_proved","summary":"Doob's L\u00b2 maximal inequality in terms of the predictable quadratic variation, finite horizon. Let $\\mu$ be a probability measure and let $(f_k)_{k \\in \\mathbb{N}}$ be a real-valued martingale with respect to a filtration $(\\mathcal{F}_k)$ and $\\mu$ such that every $f_k$ is in $L^2(\\mu)$ and $f_0 = 0$ almost everywhere, and let $A_n$ be its predictable quadratic variation. Then for every $n \\in \\mathbb{N}$,\u2026","title":"Doob's L\u00b2 maximal inequality in terms of the predictable quadratic variation, finite horizon"},{"id":"Probability/martingales/doob-s-maximal-inequalities/doob-sq-maximal-ineq-i-sup","kind":"node","lean":"MeasureTheory.Martingale.doob_sq_maximal_ineq_iSup","parent":22008,"result":true,"route":"../roadmap/Probability/martingales/doob-s-maximal-inequalities/index.html#doob-sq-maximal-ineq-i-sup","scope":"Probability/martingales/doob-s-maximal-inequalities","status":"fully_proved","summary":"Let $\\mu$ be a probability measure, let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to a filtration $(\\mathcal{F}_n)$ and $\\mu$ such that every $f_n$ is in $L^2(\\mu)$ and $f_0 = 0$ almost everywhere, and let $A_n$ be its predictable quadratic variation. Then $\\int \\sup_n |f_n|^2 \\, d\\mu \\le 4 \\int \\sup_n \\max(A_n, 0) \\, d\\mu$, where both integrals are of $[0, \\infty]$-valued functions and\u2026","title":"Doob's L\u00b2 maximal inequality, infinite horizon"},{"data_url":"scopes/Probability/martingales/fundamental-theorem-of-asset-pricing.json","id":"Probability/martingales/fundamental-theorem-of-asset-pricing","kind":"scope","navigation":"module","parent":21986,"result":false,"route":"../roadmap/Probability/martingales/fundamental-theorem-of-asset-pricing/index.html","scope":"Probability/martingales/fundamental-theorem-of-asset-pricing","status":"fully_proved","title":"Fundamental theorem of asset pricing"},{"id":"Probability/martingales/fundamental-theorem-of-asset-pricing/one-period-fundamental-theorem-asset-pri","kind":"node","lean":"MathlibExt.Probability.Finance.FundamentalTheoremAssetPricing.one_period_fundamental_theorem_asset_pricing_general","parent":22014,"result":true,"route":"../roadmap/Probability/martingales/fundamental-theorem-of-asset-pricing/index.html#one-period-fundamental-theorem-asset-pri","scope":"Probability/martingales/fundamental-theorem-of-asset-pricing","status":"fully_proved","summary":"Let $\\Omega$ be a nonempty finite set of states and $A$ a finite set of assets, and let $g(a, \\omega) \\in \\mathbb{R}$ be the gain of asset $a$ in state $\\omega$. Call a portfolio $h \\colon A \\to \\mathbb{R}$ an arbitrage if its gain $\\sum_{a \\in A} h(a)\\, g(a, \\omega)$ is nonnegative in every state $\\omega$ and positive in some state. There is no arbitrage if and only if there is a function\u2026","title":"Fundamental theorem of asset pricing for a finite one-period market"},{"id":"Probability/martingales/fundamental-theorem-of-asset-pricing/one-period-fundamental-theorem-asset-pri-2","kind":"node","lean":"MathlibExt.Probability.Finance.FundamentalTheoremAssetPricing.one_period_fundamental_theorem_asset_pricing","parent":22014,"result":true,"route":"../roadmap/Probability/martingales/fundamental-theorem-of-asset-pricing/index.html#one-period-fundamental-theorem-asset-pri-2","scope":"Probability/martingales/fundamental-theorem-of-asset-pricing","status":"fully_proved","summary":"Let $\\Omega$ be a nonempty finite set of states and $A$ a finite set of assets, and let $g(a, \\omega) \\in \\mathbb{R}$ be the gain of asset $a$ in state $\\omega$. Call a portfolio $h \\colon A \\to \\mathbb{R}$ an arbitrage if its gain $\\sum_{a \\in A} h(a)\\, g(a, \\omega)$ is nonnegative in every state $\\omega$ and positive in some state. There is no arbitrage if and only if there is a function\u2026","title":"One-period fundamental theorem of asset pricing"},{"data_url":"scopes/Probability/martingales/martingale-convergence-theorems.json","id":"Probability/martingales/martingale-convergence-theorems","kind":"scope","navigation":"module","parent":21986,"result":false,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","title":"Martingale convergence theorems"},{"id":"Probability/martingales/martingale-convergence-theorems/tendsto-e-lp-norm-limit-process-of-e-lp","kind":"node","lean":"MeasureTheory.Martingale.tendsto_eLpNorm_limitProcess_of_eLpNorm_bdd","parent":22017,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(L^p\\)"],["text"," martingale convergence theorem"]],"source":"The $L^p$ martingale convergence theorem"},"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#tendsto-e-lp-norm-limit-process-of-e-lp","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(\\Omega, \\mu)$ be a probability space with a filtration $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$, write $\\mathcal{F}_\\infty = \\bigvee_n \\mathcal{F}_n$, let $1 < p < \\infty$, and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$ whose $L^p$ norms are uniformly bounded: there is a constant $R \\ge 0$ with $\\|f_n\\|_{L^p(\\mu)} \\le R$ for every $n$. Let $f_\\infty$ be\u2026","title":"The $L^p$ martingale convergence theorem"},{"id":"Probability/martingales/martingale-convergence-theorems/uniform-integrable-tfae","kind":"node","lean":"MeasureTheory.Submartingale.uniformIntegrable_tfae","parent":22017,"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#uniform-integrable-tfae","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(\\Omega, \\mu)$ be a probability space with a filtration $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$, and let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued submartingale with respect to $(\\mathcal{F}_n)$ and $\\mu$. The following are equivalent: 1. $(X_n)$ is uniformly integrable. 2. There is an integrable $X_\\infty$ such that $X_n \\to X_\\infty$ almost surely and $\\int |X_n - X_\\infty| \\, d\\mu \\to 0$. 3. There is an\u2026","title":"Equivalent conditions for uniform integrability of a submartingale"},{"id":"Probability/martingales/martingale-convergence-theorems/uniform-integrable-tfae-2","kind":"node","lean":"MeasureTheory.Martingale.uniformIntegrable_tfae","parent":22017,"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#uniform-integrable-tfae-2","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(\\Omega, \\mu)$ be a probability space with a filtration $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$, and let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$. The following are equivalent: 1. $(X_n)$ is uniformly integrable. 2. There is an integrable $X_\\infty$ such that $X_n \\to X_\\infty$ almost surely and $\\int |X_n - X_\\infty| \\, d\\mu \\to 0$. 3. There is an\u2026","title":"Equivalent conditions for uniform integrability of a martingale"},{"id":"Probability/martingales/martingale-convergence-theorems/ae-tendsto-of-tendsto-predictable-quad-v","kind":"node","lean":"MeasureTheory.Martingale.ae_tendsto_of_tendsto_predictableQuadVar","parent":22017,"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#ae-tendsto-of-tendsto-predictable-quad-v","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to a filtration $(\\mathcal{F}_n)$ on a probability space, with $E[X_n^2] < \\infty$ for every $n$, and let $(A_n)$ be its predictable quadratic variation. Then for almost every $\\omega$: if $A_n(\\omega)$ converges to a real limit as $n \\to \\infty$, then $X_n(\\omega)$ converges to a real limit as well.","title":"Square-integrable martingale convergence"},{"id":"Probability/martingales/martingale-convergence-theorems/integral-mono","kind":"node","lean":"MeasureTheory.Submartingale.integral_mono","parent":22017,"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#integral-mono","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(X_n)_{n \\ge 0}$ be a real-valued submartingale with respect to a filtration $(\\mathcal F_n)_{n \\ge 0}$ on a probability space $(\\Omega, \\mu)$. Then $\\int X_i \\, d\\mu \\le \\int X_j \\, d\\mu$ for all natural numbers $i \\le j$.","title":"Expectations increase along a submartingale"},{"id":"Probability/martingales/martingale-convergence-theorems/integral-norm-le","kind":"node","lean":"MeasureTheory.Submartingale.integral_norm_le","parent":22017,"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#integral-norm-le","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(X_n)_{n \\ge 0}$ be a real-valued submartingale with respect to a filtration $(\\mathcal F_n)_{n \\ge 0}$ on a probability space $(\\Omega, \\mu)$, and let $C$ be a real number with $\\int X_n^+ \\, d\\mu \\le C$ for every $n$, where $X_n^+$ is the positive part of $X_n$. Then for every $n$, $\\int |X_n| \\, d\\mu \\le 2C - \\int X_0 \\, d\\mu$.","title":"L\u00b9 bound for a submartingale with bounded positive parts"},{"id":"Probability/martingales/martingale-convergence-theorems/exists-ae-tendsto-of-pos-part-bdd","kind":"node","lean":"MeasureTheory.Submartingale.exists_ae_tendsto_of_posPart_bdd","parent":22017,"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#exists-ae-tendsto-of-pos-part-bdd","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(X_n)_{n \\ge 0}$ be a real-valued submartingale with respect to a filtration $(\\mathcal F_n)_{n \\ge 0}$ on a probability space $(\\Omega, \\mu)$, and let $C$ be a real number with $\\int X_n^+ \\, d\\mu \\le C$ for every $n$, where $X_n^+$ is the positive part of $X_n$. Then there is an integrable function $X_\\infty : \\Omega \\to \\mathbb R$ such that $X_n(\\omega) \\to X_\\infty(\\omega)$ as $n \\to \\infty$ for\u2026","title":"Submartingale almost-sure convergence theorem"},{"id":"Probability/martingales/martingale-convergence-theorems/exists-ae-tendsto-of-neg-part-bdd","kind":"node","lean":"MeasureTheory.Supermartingale.exists_ae_tendsto_of_negPart_bdd","parent":22017,"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#exists-ae-tendsto-of-neg-part-bdd","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(X_n)_{n \\ge 0}$ be a real-valued supermartingale with respect to a filtration $(\\mathcal F_n)_{n \\ge 0}$ on a probability space $(\\Omega, \\mu)$, and let $C$ be a real number with $\\int X_n^- \\, d\\mu \\le C$ for every $n$, where $X_n^-$ is the negative part of $X_n$. Then there is an integrable function $X_\\infty : \\Omega \\to \\mathbb R$ such that $X_n(\\omega) \\to X_\\infty(\\omega)$ as $n \\to \\infty$ for\u2026","title":"Supermartingale almost-sure convergence theorem"},{"id":"Probability/martingales/martingale-convergence-theorems/exists-ae-tendsto-of-pos-part-bdd-2","kind":"node","lean":"MeasureTheory.Martingale.exists_ae_tendsto_of_posPart_bdd","parent":22017,"result":true,"route":"../roadmap/Probability/martingales/martingale-convergence-theorems/index.html#exists-ae-tendsto-of-pos-part-bdd-2","scope":"Probability/martingales/martingale-convergence-theorems","status":"fully_proved","summary":"Let $(X_n)_{n \\ge 0}$ be a real-valued martingale with respect to a filtration $(\\mathcal F_n)_{n \\ge 0}$ on a probability space $(\\Omega, \\mu)$, and let $C$ be a real number with $\\int X_n^+ \\, d\\mu \\le C$ for every $n$, where $X_n^+$ is the positive part of $X_n$. Then there is an integrable function $X_\\infty : \\Omega \\to \\mathbb R$ such that $X_n(\\omega) \\to X_\\infty(\\omega)$ as $n \\to \\infty$ for $\\mu$-almost\u2026","title":"Martingale almost-sure convergence theorem"},{"data_url":"scopes/Probability/martingales/optional-stopping-theorem.json","id":"Probability/martingales/optional-stopping-theorem","kind":"scope","navigation":"module","parent":21986,"result":false,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","title":"Optional stopping theorem"},{"id":"Probability/martingales/optional-stopping-theorem/stopped-value-le-cond-exp-stopped-value","kind":"node","lean":"MeasureTheory.Submartingale.stoppedValue_le_condExp_stoppedValue","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#stopped-value-le-cond-exp-stopped-value","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration such that $\\mu$ is \u03c3-finite on every $\\mathcal{F}_n$, and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued submartingale with respect to $(\\mathcal{F}_n)$ and $\\mu$. Let $L$ and $M$ be stopping times with values in $\\mathbb{N} \\cup \\{\\infty\\}$ such that $L(\\omega) \\le M(\\omega) \\le N$ for every $\\omega$,\u2026","title":"Bounded optional stopping, conditional form"},{"id":"Probability/martingales/optional-stopping-theorem/cond-exp-stopped-value-le-stopped-value","kind":"node","lean":"MeasureTheory.Supermartingale.condExp_stoppedValue_le_stoppedValue","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#cond-exp-stopped-value-le-stopped-value","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration such that $\\mu$ is \u03c3-finite on every $\\mathcal{F}_n$, and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued supermartingale with respect to $(\\mathcal{F}_n)$ and $\\mu$. Let $L$ and $M$ be stopping times with values in $\\mathbb{N} \\cup \\{\\infty\\}$ such that $L(\\omega) \\le M(\\omega) \\le N$ for every $\\omega$,\u2026","title":"Bounded optional stopping, conditional form, supermartingale dual"},{"id":"Probability/martingales/optional-stopping-theorem/stopped-value-at-top","kind":"node","lean":"MeasureTheory.stoppedValueAtTop","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#stopped-value-at-top","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"For a real-valued process $(f_n)_{n \\in \\mathbb{N}}$ on $\\Omega$ and a map $\\tau : \\Omega \\to \\mathbb{N} \\cup \\{\\infty\\}$, the value $f_\\tau$ of $f$ at $\\tau$ is the function on $\\Omega$ given by $f_\\tau(\\omega) = f_{\\tau(\\omega)}(\\omega)$ when $\\tau(\\omega) < \\infty$ and $f_\\tau(\\omega) = \\lim_{n \\to \\infty} f_n(\\omega)$ when $\\tau(\\omega) = \\infty$, where the limit is an unspecified real number if\u2026","title":"Value of a process at a possibly infinite time"},{"id":"Probability/martingales/optional-stopping-theorem/uniform-integrable-stopped-process-of-un-2","kind":"node","lean":"MeasureTheory.Martingale.uniformIntegrable_stoppedProcess_of_uniformIntegrable","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#uniform-integrable-stopped-process-of-un-2","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to a filtration $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ that is uniformly integrable. Then for every stopping time $\\tau$ with values in $\\mathbb{N} \\cup \\{\\infty\\}$, the stopped process $(f_{\\tau \\wedge n})_{n \\in \\mathbb{N}}$ is uniformly integrable.","title":"Optional stopping preserves uniform integrability of martingales"},{"id":"Probability/martingales/optional-stopping-theorem/integral-stopped-value-at-top-eq-of-unif","kind":"node","lean":"MeasureTheory.Martingale.integral_stoppedValueAtTop_eq_of_uniformIntegrable","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#integral-stopped-value-at-top-eq-of-unif","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ that is uniformly integrable. Let $f_\\infty$ be the limit process of $f$ (its almost-everywhere limit, by the martingale convergence theorem). Then for every stopping time $\\tau$ with values in $\\mathbb{N} \\cup \\{\\infty\\}$, the\u2026","title":"General optional stopping for uniformly integrable martingales"},{"id":"Probability/martingales/optional-stopping-theorem/stopped-filtration","kind":"node","lean":"MeasureTheory.IsStoppingTime.stoppedFiltration","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#stopped-filtration","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on a measurable space $\\Omega$ and let $\\tau$ be a stopping time for it with values in $\\mathbb{N} \\cup \\{\\infty\\}$. The stopped filtration is the filtration $n \\mapsto \\mathcal{F}_{\\tau \\wedge n}$, where $\\mathcal{F}_{\\tau \\wedge n}$ is the \u03c3-algebra of the stopping time $\\tau \\wedge n$.","title":"Stopped filtration"},{"id":"Probability/martingales/optional-stopping-theorem/stopped-filtration-apply","kind":"node","lean":"MeasureTheory.IsStoppingTime.stoppedFiltration_apply","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#stopped-filtration-apply","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on a measurable space $\\Omega$ and let $\\tau$ be a stopping time for it with values in $\\mathbb{N} \\cup \\{\\infty\\}$. For every $n \\in \\mathbb{N}$, the $n$-th \u03c3-algebra of the stopped filtration is $\\mathcal{F}_{\\tau \\wedge n}$, the \u03c3-algebra of the stopping time $\\tau \\wedge n$.","title":"Levels of the stopped filtration"},{"id":"Probability/martingales/optional-stopping-theorem/measurable-set-eq-top-i-sup-stopped-filt","kind":"node","lean":"MeasureTheory.IsStoppingTime.measurableSet_eq_top_iSup_stoppedFiltration","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#measurable-set-eq-top-i-sup-stopped-filt","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on a measurable space $\\Omega$ and let $\\tau$ be a stopping time for it with values in $\\mathbb{N} \\cup \\{\\infty\\}$. For each $n$, let $\\mathcal{F}_{\\tau \\wedge n}$ be the \u03c3-algebra of the stopping time $\\tau \\wedge n$. Then the set $\\{\\omega : \\tau(\\omega) = \\infty\\}$ is measurable with respect to $\\bigvee_n \\mathcal{F}_{\\tau \\wedge n}$.","title":"The event that the stopping time is infinite is measurable for the stopped filtration"},{"id":"Probability/martingales/optional-stopping-theorem/measurable-set-inter-eq-top-i-sup-stoppe","kind":"node","lean":"MeasureTheory.IsStoppingTime.measurableSet_inter_eq_top_iSup_stoppedFiltration","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#measurable-set-inter-eq-top-i-sup-stoppe","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on a measurable space $\\Omega$ and let $\\tau$ be a stopping time for it with values in $\\mathbb{N} \\cup \\{\\infty\\}$. For each $n$, let $\\mathcal{F}_{\\tau \\wedge n}$ be the \u03c3-algebra of the stopping time $\\tau \\wedge n$. If a set $s \\subseteq \\Omega$ is measurable with respect to $\\bigvee_n \\mathcal{F}_n$, then $s \\cap \\{\\tau = \\infty\\}$ is measurable with\u2026","title":"Infinite-time parts of limit-\u03c3-algebra sets are measurable for the stopped filtration"},{"id":"Probability/martingales/optional-stopping-theorem/i-sup-stopped-filtration","kind":"node","lean":"MeasureTheory.IsStoppingTime.iSup_stoppedFiltration","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#i-sup-stopped-filtration","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on a measurable space $\\Omega$ and let $\\tau$ be a stopping time for it with values in $\\mathbb{N} \\cup \\{\\infty\\}$. For each $n$, let $\\mathcal{F}_{\\tau \\wedge n}$ be the \u03c3-algebra of the stopping time $\\tau \\wedge n$, the $n$-th \u03c3-algebra of the stopped filtration, and let $\\mathcal{F}_\\tau$ be the \u03c3-algebra of $\\tau$. Then\u2026","title":"The stopped filtration generates the stopping-time \u03c3-algebra"},{"id":"Probability/martingales/optional-stopping-theorem/stopped-value-at-top-ae-eq-cond-exp-limi","kind":"node","lean":"MeasureTheory.Martingale.stoppedValueAtTop_ae_eq_condExp_limitProcess","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#stopped-value-at-top-ae-eq-cond-exp-limi","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration, and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$ that is uniformly integrable in $L^1(\\mu)$. Let $f_\\infty$ be its limit process, a function measurable with respect to $\\bigvee_n \\mathcal{F}_n$ to which $f_n$ converges $\\mu$-almost everywhere. Then for\u2026","title":"Stopped value as conditional expectation of the limit"},{"id":"Probability/martingales/optional-stopping-theorem/stopped-value-at-top-stopped-process-of","kind":"node","lean":"MeasureTheory.stoppedValueAtTop_stoppedProcess_of_le","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#stopped-value-at-top-stopped-process-of","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $(f_n)_{n \\in \\mathbb{N}}$ be a sequence of real-valued functions on $\\Omega$ and let $L, M : \\Omega \\to \\mathbb{N} \\cup \\{\\infty\\}$ satisfy $L(\\omega) \\le M(\\omega)$ for every $\\omega$. Then the value at $L$ of the stopped process $n \\mapsto f_{M \\wedge n}$ equals the value $f_L$ of $f$ at $L$ at every point of $\\Omega$. Here the value of a sequence $(g_n)$ at a point $\\omega$ where the time is infinite is\u2026","title":"Stopping at a later time does not change the stopped value"},{"id":"Probability/martingales/optional-stopping-theorem/stopped-value-at-top-ae-eq-cond-exp-stop","kind":"node","lean":"MeasureTheory.Martingale.stoppedValueAtTop_ae_eq_condExp_stoppedValueAtTop","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#stopped-value-at-top-ae-eq-cond-exp-stop","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration, and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$. Let $L \\le M$ be stopping times with values in $\\mathbb{N} \\cup \\{\\infty\\}$, and suppose the stopped process $(f_{M \\wedge n})_{n \\in \\mathbb{N}}$ is uniformly integrable in $L^1(\\mu)$. Then\u2026","title":"Conditional optional stopping with UI of the stopped process"},{"id":"Probability/martingales/optional-stopping-theorem/stopped-value-at-top-le-cond-exp-stopped","kind":"node","lean":"MeasureTheory.Submartingale.stoppedValueAtTop_le_condExp_stoppedValueAtTop","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#stopped-value-at-top-le-cond-exp-stopped","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Conditional optional stopping for uniformly integrable submartingales. For stopping times `L \u2264 M` (possibly infinite), `f_L \u2264\u1d50 \u03bc[f_M | \u2131_L]`.","title":"Conditional optional stopping for uniformly integrable submartingales"},{"id":"Probability/martingales/optional-stopping-theorem/integral-stopped-value-at-top-mono-of-un","kind":"node","lean":"MeasureTheory.Submartingale.integral_stoppedValueAtTop_mono_of_uniformIntegrable","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#integral-stopped-value-at-top-mono-of-un","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Expectation form of conditional optional stopping for uniformly integrable submartingales. For stopping times `L \u2264 M` (possibly infinite), `\u222b f_L \u2264 \u222b f_M`.","title":"Expectation form of conditional optional stopping for uniformly integrable submartingales"},{"id":"Probability/martingales/optional-stopping-theorem/uniform-integrable-stopped-process-of-un","kind":"node","lean":"MeasureTheory.Submartingale.uniformIntegrable_stoppedProcess_of_uniformIntegrable","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#uniform-integrable-stopped-process-of-un","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued submartingale with respect to a filtration $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ that is uniformly integrable. Then for every stopping time $\\tau$ with values in $\\mathbb{N} \\cup \\{\\infty\\}$, the stopped process $(f_{\\tau \\wedge n})_{n \\in \\mathbb{N}}$ is uniformly integrable.","title":"General optional stopping, uniform integrability preservation"},{"id":"Probability/martingales/optional-stopping-theorem/integral-stopped-value-at-top-of-uniform","kind":"node","lean":"MeasureTheory.Submartingale.integral_stoppedValueAtTop_of_uniformIntegrable","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#integral-stopped-value-at-top-of-uniform","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration and let $(f_n)_{n \\in \\mathbb{N}}$ be a real-valued submartingale with respect to $(\\mathcal{F}_n)$ that is uniformly integrable. Let $f_\\infty$ be the limit process of $f$ (its almost-everywhere limit, by the martingale convergence theorem). Then for every stopping time $\\tau$ with values in $\\mathbb{N} \\cup \\{\\infty\\}$,\u2026","title":"General optional stopping, expectation inequality"},{"id":"Probability/martingales/optional-stopping-theorem/expected-stopped-value-anti-mono","kind":"node","lean":"MeasureTheory.Supermartingale.expected_stoppedValue_anti_mono","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#expected-stopped-value-anti-mono","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$ and $(\\mathcal G_n)_{n \\ge 0}$ a filtration such that the restriction of $\\mu$ to each $\\mathcal G_n$ is \u03c3-finite, and let $(f_n)_{n \\ge 0}$ be a real-valued supermartingale with respect to $(\\mathcal G_n)$ and $\\mu$. Let $\\tau$ and $\\pi$ be stopping times for $(\\mathcal G_n)$ with values in $\\mathbb N \\cup \\{\\infty\\}$, and let $N$ be a natural number such that\u2026","title":"Optional stopping for supermartingales at two bounded stopping times"},{"id":"Probability/martingales/optional-stopping-theorem/expected-stopped-value-le","kind":"node","lean":"MeasureTheory.Supermartingale.expected_stoppedValue_le","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#expected-stopped-value-le","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$ and $(\\mathcal G_n)_{n \\ge 0}$ a filtration such that the restriction of $\\mu$ to each $\\mathcal G_n$ is \u03c3-finite, and let $(f_n)_{n \\ge 0}$ be a real-valued supermartingale with respect to $(\\mathcal G_n)$ and $\\mu$. Let $\\tau$ be a stopping time for $(\\mathcal G_n)$ with values in $\\mathbb N \\cup \\{\\infty\\}$, and let $N$ be a natural number such that $\\tau(\\omega) \\le N$ for\u2026","title":"Optional stopping for supermartingales at a bounded stopping time"},{"id":"Probability/martingales/optional-stopping-theorem/expected-stopped-value-eq","kind":"node","lean":"MeasureTheory.Martingale.expected_stoppedValue_eq","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#expected-stopped-value-eq","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$ and $(\\mathcal G_n)_{n \\ge 0}$ a filtration such that the restriction of $\\mu$ to each $\\mathcal G_n$ is \u03c3-finite, and let $(f_n)_{n \\ge 0}$ be a real-valued martingale with respect to $(\\mathcal G_n)$ and $\\mu$. Let $\\tau$ and $\\pi$ be stopping times for $(\\mathcal G_n)$ with values in $\\mathbb N \\cup \\{\\infty\\}$, and let $N$ be a natural number such that\u2026","title":"Optional stopping for martingales at two bounded stopping times"},{"id":"Probability/martingales/optional-stopping-theorem/expected-stopped-value-eq-zero","kind":"node","lean":"MeasureTheory.Martingale.expected_stoppedValue_eq_zero","parent":22027,"result":true,"route":"../roadmap/Probability/martingales/optional-stopping-theorem/index.html#expected-stopped-value-eq-zero","scope":"Probability/martingales/optional-stopping-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$ and $(\\mathcal G_n)_{n \\ge 0}$ a filtration such that the restriction of $\\mu$ to each $\\mathcal G_n$ is \u03c3-finite, and let $(f_n)_{n \\ge 0}$ be a real-valued martingale with respect to $(\\mathcal G_n)$ and $\\mu$. Let $\\tau$ be a stopping time for $(\\mathcal G_n)$ with values in $\\mathbb N \\cup \\{\\infty\\}$, and let $N$ be a natural number such that $\\tau(\\omega) \\le N$ for every\u2026","title":"Optional stopping for martingales at a bounded stopping time"},{"data_url":"scopes/Probability/martingales/square-integrable-martingales.json","id":"Probability/martingales/square-integrable-martingales","kind":"scope","navigation":"module","parent":21986,"result":false,"route":"../roadmap/Probability/martingales/square-integrable-martingales/index.html","scope":"Probability/martingales/square-integrable-martingales","status":"fully_proved","title":"Square-integrable martingales"},{"id":"Probability/martingales/square-integrable-martingales/cond-exp-sub-sq-eq","kind":"node","lean":"MeasureTheory.Martingale.condExp_sub_sq_eq","parent":22049,"result":true,"route":"../roadmap/Probability/martingales/square-integrable-martingales/index.html#cond-exp-sub-sq-eq","scope":"Probability/martingales/square-integrable-martingales","status":"fully_proved","summary":"Let $(\\Omega, \\mu)$ be a probability space with a filtration $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$, and let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$ such that $X_n$ is square-integrable for every $n$. Then for all $m \\le n$, almost surely","title":"Conditional variance theorem for martingales"},{"id":"Probability/martingales/square-integrable-martingales/integral-sub-mul-eq-zero","kind":"node","lean":"MeasureTheory.Martingale.integral_sub_mul_eq_zero","parent":22049,"result":true,"route":"../roadmap/Probability/martingales/square-integrable-martingales/index.html#integral-sub-mul-eq-zero","scope":"Probability/martingales/square-integrable-martingales","status":"fully_proved","summary":"Let $(\\Omega, \\mu)$ be a probability space with a filtration $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$, and let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$ such that $\\mathbb{E}[X_k^2] < \\infty$ for every $k$. Let $m \\le n$ and let $Y$ be an $\\mathcal{F}_m$-measurable real random variable with $\\mathbb{E}[Y^2] < \\infty$. Then $\\mathbb{E}[(X_n - X_m) Y] = 0$. In\u2026","title":"Orthogonality of martingale increments"},{"id":"Probability/martingales/square-integrable-martingales/integral-sub-sq-eq","kind":"node","lean":"MeasureTheory.Martingale.integral_sub_sq_eq","parent":22049,"result":true,"route":"../roadmap/Probability/martingales/square-integrable-martingales/index.html#integral-sub-sq-eq","scope":"Probability/martingales/square-integrable-martingales","status":"fully_proved","summary":"Let $(\\Omega, \\mu)$ be a probability space with a filtration $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$, and let $(X_n)_{n \\in \\mathbb{N}}$ be a real-valued martingale with respect to $(\\mathcal{F}_n)$ and $\\mu$ such that $\\mathbb{E}[X_k^2] < \\infty$ for every $k$. Then for all $m \\le n$,","title":"Pythagorean identity for martingale increments"},{"data_url":"scopes/Probability/martingales/wald-s-equations.json","id":"Probability/martingales/wald-s-equations","kind":"scope","navigation":"module","parent":21986,"result":false,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","title":"Wald's equations"},{"id":"Probability/martingales/wald-s-equations/cross-pair-integral-zero","kind":"node","lean":"MathlibExt.Probability.WaldCrossTerms.cross_pair_integral_zero","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#cross-pair-integral-zero","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"A single cross term `E[X_j \u00b7 X_i \u00b7 1_{T > i}]` (with `j < i`) vanishes when the `X_i` are identically distributed with zero mean and `X_i` is independent of `(X_j, 1_{T > i})`.","title":"Vanishing of a single cross term in Wald's second equation"},{"id":"Probability/martingales/wald-s-equations/wald-cross-terms-zero","kind":"node","lean":"MathlibExt.Probability.WaldCrossTerms.wald_cross_terms_zero","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#wald-cross-terms-zero","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"The full cross-term contribution to `E[(S_T)^2]` vanishes: `E[\u2211_{i < T} \u2211_{j < i} X_j X_i] = 0` for identically distributed, mean-zero variables, with each `X_i` independent of the filtration at time `i` and measurable at time `i + 1`, given absolute summability of the stopped cross-term series.","title":"Vanishing of the cross terms in Wald's second equation"},{"id":"Probability/martingales/wald-s-equations/sum-sq-expansion","kind":"node","lean":"MathlibExt.Probability.WaldCrossTerms.sum_sq_expansion","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#sum-sq-expansion","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"Algebraic identity expanding the square of a finite sum into diagonal and cross-term contributions: `(\u2211_{i<n} f i)^2 = \u2211_{i<n} (f i)^2 + 2 \u2211_{i<n} \u2211_{j<i} f j \u00b7 f i`.","title":"Square of a finite sum as diagonal plus cross terms"},{"id":"Probability/martingales/wald-s-equations/wald-cross-terms-zero-of-bounded","kind":"node","lean":"MathlibExt.Probability.WaldCrossTerms.wald_cross_terms_zero_of_bounded","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#wald-cross-terms-zero-of-bounded","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"The total cross-term contribution vanishes for a stopping time bounded by a constant, with no extra summability or measurability hypotheses: the sums are finite, and every term past the bound is zero.","title":"Cross terms vanish for a bounded stopping time"},{"id":"Probability/martingales/wald-s-equations/wald-integrable-stopped-sum","kind":"node","lean":"MeasureTheory.wald_integrable_stopped_sum","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#wald-integrable-stopped-sum","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"Under Wald's hypotheses, the stopped sum is integrable. This reuses Wald's summable `L\u00b9` series and applies Mathlib's sum measurability and norm bound.","title":"Integrability of the stopped sum in Wald's equation"},{"id":"Probability/martingales/wald-s-equations/wald-equation","kind":"node","lean":"MeasureTheory.wald_equation","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#wald-equation","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"Let `X\u1d62` be identically distributed integrable real random variables and `T` a stopping time with `E T < \u221e`, with each `X\u2096` independent of the filtration at time `k`. Then `E[S_T] = E[X\u2081] \u00b7 E[T]`, where `S_T = \u2211_{i < T} X\u1d62`.","title":"Wald's equation"},{"id":"Probability/martingales/wald-s-equations/integrable-stopped-sum-sq-diag","kind":"node","lean":"MathlibExt.Probability.WaldDiagonalTerms.integrable_stopped_sum_sq_diag","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#integrable-stopped-sum-sq-diag","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"The stopped sum of squares is integrable under the stated diagonal hypotheses.","title":"Integrability of the stopped sum of squares"},{"id":"Probability/martingales/wald-s-equations/wald-diagonal-terms","kind":"node","lean":"MathlibExt.Probability.WaldDiagonalTerms.wald_diagonal_terms","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#wald-diagonal-terms","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"Wald's equation applied to the squared increments gives the diagonal identity.","title":"Diagonal identity in Wald's second equation"},{"id":"Probability/martingales/wald-s-equations/wald-second-equation","kind":"node","lean":"MeasureTheory.wald_second_equation","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#wald-second-equation","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"Let $(\\Omega, \\mathbb{P})$ be a probability space with a filtration $(\\mathcal{F}_k)_{k \\ge 0}$, and let $X_0, X_1, \\dots$ be identically distributed real random variables with $\\mathbb{E}|X_0| < \\infty$, $\\mathbb{E}[X_0] = 0$ and $\\mathbb{E}[X_0^2] < \\infty$. Let $T$ be an $\\mathbb{N}$-valued stopping time for $(\\mathcal{F}_k)$ with $\\mathbb{E}[T] < \\infty$. Suppose each $X_k$ is independent of $\\mathcal{F}_k$ and\u2026","title":"Wald's second equation"},{"id":"Probability/martingales/wald-s-equations/predictable-filtration","kind":"node","lean":"MathlibExt.Probability.StoppedSumMaximal.predictableFiltration","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#predictable-filtration","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"The \"predictable\" filtration associated to a sequence `X : \u2115 \u2192 \u03a9 \u2192 \u211d`: `\u2131_k = \u03c3(X_0, \u2026, X_{k-1})`. Ported verbatim from v1 `WaldSecondEquation.lean`.","title":"Predictable filtration of a sequence"},{"id":"Probability/martingales/wald-s-equations/predictable-filtration-zero","kind":"node","lean":"MathlibExt.Probability.StoppedSumMaximal.predictableFiltration_zero","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#predictable-filtration-zero","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"The predictable filtration contains no observations at time zero.","title":"Predictable filtration at time zero is trivial"},{"id":"Probability/martingales/wald-s-equations/predictable-filtration-succ","kind":"node","lean":"MathlibExt.Probability.StoppedSumMaximal.predictableFiltration_succ","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#predictable-filtration-succ","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"A successor step adds exactly the next observation to the predictable filtration.","title":"Successor step of the predictable filtration"},{"id":"Probability/martingales/wald-s-equations/integrable-stopped-partial-sum-sq","kind":"node","lean":"MathlibExt.Probability.StoppedSumMaximal.integrable_stopped_partialSum_sq","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#integrable-stopped-partial-sum-sq","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"Square-integrability of a stopped partial sum, general-filtration form.","title":"Square-integrability of a stopped partial sum"},{"id":"Probability/martingales/wald-s-equations/integrable-maximal-stopped-partial-sum-s","kind":"node","lean":"MathlibExt.Probability.StoppedSumMaximal.integrable_maximal_stopped_partial_sum_sq","parent":22053,"result":true,"route":"../roadmap/Probability/martingales/wald-s-equations/index.html#integrable-maximal-stopped-partial-sum-s","scope":"Probability/martingales/wald-s-equations","status":"fully_proved","summary":"Square-integrability of the maximal stopped partial sum, in v1's form: `max_{0 \u2264 k \u2264 T} S\u2096\u00b2` is integrable. This is v1's `integrable_maximal_stopped_partial_sum_sq`, whose proof in v1 was left unproved.","title":"Square-integrability of the maximal stopped partial sum"},{"data_url":"scopes/Probability/random-walks.json","id":"Probability/random-walks","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/random-walks/index.html","scope":"Probability/random-walks","status":"can_prove","summary":"Sums of i.i.d. steps as processes: the drift/oscillation dichotomy, limsup and liminf of partial sums, tail measurability of asymptotic events, the strong Markov property of i.i.d. sequences at stopping times, and ballot theorems.","title":"Random walks"},{"id":"Probability/random-walks/bertrand-ballot","kind":"node","lean":"MetaMathlibExt.bertrand_ballot","parent":22068,"result":true,"route":"../roadmap/Probability/random-walks/index.html#bertrand-ballot","scope":"Probability/random-walks","status":"fully_proved","summary":"Let $p$ and $q$ be natural numbers with $1 \\le q < p$. Call a sequence of $p + q$ votes, each for A or for B, favourable if it contains exactly $p$ votes for A and, for every $k$ with $1 \\le k \\le p + q$, strictly more of its first $k$ votes are for A than for B. Then the number of favourable sequences, multiplied by $p + q$, equals $(p - q)\\binom{p+q}{q}$.","title":"Bertrand's ballot theorem"},{"data_url":"scopes/Probability/random-walks/drift-and-oscillation-of-random-walks.json","id":"Probability/random-walks/drift-and-oscillation-of-random-walks","kind":"scope","navigation":"module","parent":22068,"result":false,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","title":"Drift and oscillation of random walks"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/tail-measurable-space-from","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.tailMeasurableSpaceFrom","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#tail-measurable-space-from","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Given \u03c3-algebras $\\mathcal{A}_0, \\mathcal{A}_1, \\dots$ on $\\Omega$ and $n \\in \\mathbb{N}$, this is $\\sigma(\\mathcal{A}_n, \\mathcal{A}_{n+1}, \\dots)$, the smallest \u03c3-algebra on $\\Omega$ containing every $\\mathcal{A}_i$ with $i \\ge n$.","title":"\u03c3-algebra generated by a tail of a sequence of \u03c3-algebras"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/tail-measurable-space","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.tailMeasurableSpace","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#tail-measurable-space","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Given \u03c3-algebras $\\mathcal{A}_0, \\mathcal{A}_1, \\dots$ on $\\Omega$, the tail \u03c3-algebra is","title":"Tail \u03c3-algebra of a sequence of \u03c3-algebras"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/tail-measurable-space-of-fun","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.tailMeasurableSpaceOfFun","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#tail-measurable-space-of-fun","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\beta$ be a measurable space and let $X_n : \\Omega \\to \\beta$ for $n \\in \\mathbb{N}$. The tail \u03c3-algebra of $X$ is","title":"Tail \u03c3-algebra of a sequence of functions"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/random-walk-partial-sum","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.randomWalkPartialSum","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#random-walk-partial-sum","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For a sequence of functions $X_i : \\Omega \\to \\mathbb{R}$ and $n \\in \\mathbb{N}$, the $n$-th partial sum is $S_n(\\omega) = \\sum_{i < n} X_i(\\omega)$, the position at time $n$ of the random walk on $\\mathbb{R}$ with increments $X_i$.","title":"Partial sums of a random walk"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/sigma-alg","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.sigmaAlg","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#sigma-alg","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For a sequence of functions $X_n : \\Omega \\to \\mathbb{R}$, this is the sequence of \u03c3-algebras $\\sigma(X_n)$ on $\\Omega$, where $\\sigma(X_n)$ consists of the preimages under $X_n$ of Borel subsets of $\\mathbb{R}$.","title":"\u03c3-algebras generated by a sequence of functions"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/meas-xi","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.meas_Xi","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#meas-xi","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$, and for $n \\in \\mathbb{N}$ let $\\sigma(X_n, X_{n+1}, \\dots)$ be the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel measurable. If $n \\le i$, then $X_i$ is measurable with respect to $\\sigma(X_n, X_{n+1}, \\dots)$.","title":"Later terms are measurable for the \u03c3-algebra from a given index"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/meas-t-sum","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.meas_tSum","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#meas-t-sum","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$, and for $n \\in \\mathbb{N}$ let $\\sigma(X_n, X_{n+1}, \\dots)$ be the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel measurable. For all $n, m \\in \\mathbb{N}$, the function $\\omega \\mapsto \\sum_{n \\le i < m} X_i(\\omega)$ is measurable with respect to $\\sigma(X_n, X_{n+1}, \\dots)$.","title":"Tail partial sums are measurable for the \u03c3-algebra from their first index"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-drifts-to-top-from","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_driftsToTop_from","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-drifts-to-top-from","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$, let $S_m = \\sum_{i < m} X_i$ be their partial sums, and for $n \\in \\mathbb{N}$ let $\\sigma(X_n, X_{n+1}, \\dots)$ be the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel measurable. For every $n$, the set of $\\omega \\in \\Omega$ with $S_m(\\omega) \\to +\\infty$ as $m \\to \\infty$ is measurable with respect to\u2026","title":"Drift of a random walk to +\u221e is measurable from every index"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-drifts-to-bot-from","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_driftsToBot_from","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-drifts-to-bot-from","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$, let $S_m = \\sum_{i < m} X_i$ be their partial sums, and for $n \\in \\mathbb{N}$ let $\\sigma(X_n, X_{n+1}, \\dots)$ be the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel measurable. For every $n$, the set of $\\omega \\in \\Omega$ with $S_m(\\omega) \\to -\\infty$ as $m \\to \\infty$ is measurable with respect to\u2026","title":"Drift of a random walk to \u2212\u221e is measurable from every index"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-limsup-top-from","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_limsupTop_from","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-limsup-top-from","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$, let $S_m = \\sum_{i < m} X_i$ be their partial sums, and for $n \\in \\mathbb{N}$ let $\\sigma(X_n, X_{n+1}, \\dots)$ be the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel measurable. For every $n$, the set of $\\omega \\in \\Omega$ with $\\limsup_{m \\to \\infty} S_m(\\omega) = +\\infty$, the limit superior taken in the extended\u2026","title":"A limit superior +\u221e of a random walk is measurable from every index"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-liminf-bot-from","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_liminfBot_from","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-liminf-bot-from","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$, let $S_m = \\sum_{i < m} X_i$ be their partial sums, and for $n \\in \\mathbb{N}$ let $\\sigma(X_n, X_{n+1}, \\dots)$ be the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel measurable. For every $n$, the set of $\\omega \\in \\Omega$ with $\\liminf_{m \\to \\infty} S_m(\\omega) = -\\infty$, the limit inferior taken in the extended\u2026","title":"A limit inferior \u2212\u221e of a random walk is measurable from every index"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-converges-finite-from","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_convergesFinite_from","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-converges-finite-from","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$, let $S_m = \\sum_{i < m} X_i$ be their partial sums, and for $n \\in \\mathbb{N}$ let $\\sigma(X_n, X_{n+1}, \\dots)$ be the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel measurable. For every $n$, the set of $\\omega \\in \\Omega$ for which $S_m(\\omega)$ converges to a real limit as $m \\to \\infty$ is measurable with respect to\u2026","title":"Convergence of a random walk is measurable from every index"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-drifts-to-top","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_driftsToTop","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-drifts-to-top","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$ and let $S_m = \\sum_{i < m} X_i$ be their partial sums. The set of $\\omega \\in \\Omega$ with $S_m(\\omega) \\to +\\infty$ as $m \\to \\infty$ belongs to the tail \u03c3-algebra $\\bigcap_{n} \\sigma(X_n, X_{n+1}, \\dots)$ of the sequence, where $\\sigma(X_n, X_{n+1}, \\dots)$ is the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel\u2026","title":"Drift of a random walk to +\u221e is a tail event"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-drifts-to-bot","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_driftsToBot","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-drifts-to-bot","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$ and let $S_m = \\sum_{i < m} X_i$ be their partial sums. The set of $\\omega \\in \\Omega$ with $S_m(\\omega) \\to -\\infty$ as $m \\to \\infty$ belongs to the tail \u03c3-algebra $\\bigcap_{n} \\sigma(X_n, X_{n+1}, \\dots)$ of the sequence, where $\\sigma(X_n, X_{n+1}, \\dots)$ is the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with $i \\ge n$ is Borel\u2026","title":"Drift of a random walk to \u2212\u221e is a tail event"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-oscillates","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_oscillates","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-oscillates","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_n : \\Omega \\to \\mathbb{R}$ for $n \\in \\mathbb{N}$ be a sequence of functions with partial sums $S_m = \\sum_{i < m} X_i$. Then the event that $\\limsup_{m \\to \\infty} S_m = +\\infty$ and $\\liminf_{m \\to \\infty} S_m = -\\infty$, with the limit superior and inferior taken in the extended reals, belongs to the tail \u03c3-algebra of $X$.","title":"Tail measurability of the oscillation event"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-converges-finite","kind":"node","lean":"MathlibExt.Probability.RandomWalkTailMeasurability.measurableSet_convergesFinite","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-converges-finite","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real-valued functions on a set $\\Omega$ and let $S_m = \\sum_{i < m} X_i$ be their partial sums. The set of $\\omega \\in \\Omega$ for which $S_m(\\omega)$ converges to a real limit as $m \\to \\infty$ belongs to the tail \u03c3-algebra $\\bigcap_{n} \\sigma(X_n, X_{n+1}, \\dots)$ of the sequence, where $\\sigma(X_n, X_{n+1}, \\dots)$ is the smallest \u03c3-algebra on $\\Omega$ for which every $X_i$ with\u2026","title":"Convergence of a random walk is a tail event"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/random-walk-limsup","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.randomWalkLimsup","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#random-walk-limsup","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For real-valued functions $X_0, X_1, \\dots$ on a set $\\Omega$ and a point $\\omega \\in \\Omega$, this is $\\limsup_{n \\to \\infty} S_n(\\omega)$, taken in the extended reals $[-\\infty, +\\infty]$, where $S_n(\\omega) = \\sum_{i < n} X_i(\\omega)$ is the $n$-th partial sum.","title":"Limsup of a random walk"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/random-walk-liminf","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.randomWalkLiminf","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#random-walk-liminf","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For real-valued functions $X_0, X_1, \\dots$ on a set $\\Omega$ and a point $\\omega \\in \\Omega$, this is $\\liminf_{n \\to \\infty} S_n(\\omega)$, taken in the extended reals $[-\\infty, +\\infty]$, where $S_n(\\omega) = \\sum_{i < n} X_i(\\omega)$ is the $n$-th partial sum.","title":"Liminf of a random walk"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/shift-seq","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.shiftSeq","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#shift-seq","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"The one-step shift sends a real sequence $\\omega = (\\omega_0, \\omega_1, \\omega_2, \\dots)$ to the sequence $(\\omega_1, \\omega_2, \\dots)$ whose $i$-th term is $\\omega_{i+1}$.","title":"Shift of a real sequence"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/coord-seq","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.coordSeq","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#coord-seq","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For $i \\in \\mathbb{N}$, the $i$-th coordinate is the map $\\mathbb{R}^{\\mathbb{N}} \\to \\mathbb{R}$ sending a real sequence $\\omega = (\\omega_0, \\omega_1, \\dots)$ to its $i$-th term $\\omega_i$.","title":"Coordinate process on sequence space"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/limsup-succ-eq","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.limsup_succ_eq","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#limsup-succ-eq","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For every sequence $(u_n)_{n \\in \\mathbb{N}}$ of extended reals, $\\limsup_{n \\to \\infty} u_{n+1} = \\limsup_{n \\to \\infty} u_n$.","title":"Dropping the first term preserves the limsup"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/liminf-succ-eq","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.liminf_succ_eq","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#liminf-succ-eq","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For every sequence $(u_n)$ in the extended reals $[-\\infty, +\\infty]$, $\\liminf_{n \\to \\infty} u_{n+1} = \\liminf_{n \\to \\infty} u_n$.","title":"Dropping the first term does not change the liminf"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/limsup-add-const","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.limsup_add_const","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#limsup-add-const","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For every real $c$ and every real sequence $(u_n)$, with limits superior taken in the extended reals $[-\\infty, +\\infty]$, $\\limsup_{n \\to \\infty} (c + u_n) = c + \\limsup_{n \\to \\infty} u_n$.","title":"Limsup commutes with adding a constant"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/liminf-add-const","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.liminf_add_const","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#liminf-add-const","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For every real $c$ and every real sequence $(u_n)$, with limits inferior taken in the extended reals $[-\\infty, +\\infty]$, $\\liminf_{n \\to \\infty} (c + u_n) = c + \\liminf_{n \\to \\infty} u_n$.","title":"Liminf commutes with adding a constant"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/limsup-shift","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.limsup_shift","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#limsup-shift","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For a real sequence $\\omega = (\\omega_0, \\omega_1, \\dots)$ write $S_n(\\omega) = \\sum_{i < n} \\omega_i$, and let $\\theta\\omega = (\\omega_1, \\omega_2, \\dots)$ be the one-step shift. Then for every $\\omega$, $\\limsup_{n \\to \\infty} S_n(\\omega) = \\omega_0 + \\limsup_{n \\to \\infty} S_n(\\theta\\omega)$, where both limits superior are taken in the extended reals $[-\\infty, +\\infty]$.","title":"One-step shift identity for the limsup of partial sums"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/liminf-shift","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.liminf_shift","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#liminf-shift","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"For a real sequence $\\omega = (\\omega_0, \\omega_1, \\dots)$ write $S_n(\\omega) = \\sum_{i\\lt n} \\omega_i$, and let $\\omega'$ be its shift, $\\omega'_i = \\omega_{i+1}$. Then, with limits inferior taken in $[-\\infty, +\\infty]$, we have $\\liminf_{n \\to \\infty} S_n(\\omega) = \\omega_0 + \\liminf_{n \\to \\infty} S_n(\\omega')$.","title":"One-step shift identity for the liminf of the partial sums"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/partial-sum-comp-perm-eq","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.partialSum_comp_perm_eq","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#partial-sum-comp-perm-eq","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\mathbb{N}$ with $\\sigma(m) = m$ for every $m \\ge N$. Then for every real sequence $(\\omega_i)_{i \\in \\mathbb{N}}$ and every $n \\ge N$, $\\sum_{i < n} \\omega_{\\sigma(i)} = \\sum_{i < n} \\omega_i$.","title":"Partial sums are unchanged by permuting the first coordinates"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/limsup-comp-perm","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.limsup_comp_perm","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#limsup-comp-perm","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\mathbb{N}$ that moves only finitely many points. For every real sequence $\\omega = (\\omega_0, \\omega_1, \\dots)$, the limit superior in $[-\\infty, +\\infty]$ of the partial sums $\\sum_{i\\lt n} \\omega_{\\sigma(i)}$ equals the limit superior of the partial sums $\\sum_{i\\lt n} \\omega_i$.","title":"Finite permutations do not change the limsup of the partial sums"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/liminf-comp-perm","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.liminf_comp_perm","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#liminf-comp-perm","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\mathbb{N}$ that moves only finitely many points. For every real sequence $\\omega = (\\omega_0, \\omega_1, \\dots)$, the limit inferior in $[-\\infty, +\\infty]$ of the partial sums $\\sum_{i\\lt n} \\omega_{\\sigma(i)}$ equals the limit inferior of the partial sums $\\sum_{i\\lt n} \\omega_i$.","title":"Finite permutations do not change the liminf of the partial sums"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/map-shift-seq-infinite-pi","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.map_shiftSeq_infinitePi","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#map-shift-seq-infinite-pi","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\rho$ be a probability measure on $\\mathbb{R}$, and let $\\rho^{\\otimes \\mathbb{N}}$ be the product measure on the sequence space $\\mathbb{R}^{\\mathbb{N}}$. The image of $\\rho^{\\otimes \\mathbb{N}}$ under the one-step shift $(\\omega_0, \\omega_1, \\dots) \\mapsto (\\omega_1, \\omega_2, \\dots)$ is $\\rho^{\\otimes \\mathbb{N}}$.","title":"Shift invariance of the i.i.d. product measure"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/exists-ae-eq-const-of-forall-le-zero-or","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.exists_ae_eq_const_of_forall_le_zero_or_one","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#exists-ae-eq-const-of-forall-le-zero-or","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $\\Omega$ and let $F : \\Omega \\to [-\\infty, +\\infty]$ be measurable. If for every real $a$ the set $\\{\\omega : F(\\omega) \\le a\\}$ has $\\mu$-measure $0$ or $1$, then there is $c \\in [-\\infty, +\\infty]$ with $F = c$ $\\mu$-almost everywhere.","title":"Zero\u2013one sublevel sets force an almost surely constant variable"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/tendsto-at-bot-of-limsup-eq-bot","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.tendsto_atBot_of_limsup_eq_bot","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#tendsto-at-bot-of-limsup-eq-bot","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $(S_n)$ be a sequence of real numbers. If $\\limsup_{n \\to \\infty} S_n = -\\infty$ in the extended reals, then $S_n \\to -\\infty$ as $n \\to \\infty$.","title":"Limsup minus infinity forces divergence to minus infinity"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/limsup-eq-bot-of-tendsto-at-bot","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.limsup_eq_bot_of_tendsto_atBot","parent":22070,"reading_title":{"format":"inline-parts-v1","parts":[["text","A sequence tending to "],["math","\\(-\\infty\\)"],["text"," has limsup "],["math","\\(-\\infty\\)"]],"source":"A sequence tending to $-\\infty$ has limsup $-\\infty$"},"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#limsup-eq-bot-of-tendsto-at-bot","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $(S_n)_{n \\in \\mathbb{N}}$ be a sequence of real numbers with $S_n \\to -\\infty$ as $n \\to \\infty$. Then the limit superior of $(S_n)$, taken in the extended reals $[-\\infty, +\\infty]$, equals $-\\infty$.","title":"A sequence tending to $-\\infty$ has limsup $-\\infty$"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/tendsto-at-top-of-liminf-eq-top","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.tendsto_atTop_of_liminf_eq_top","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#tendsto-at-top-of-liminf-eq-top","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $(S_n)$ be a sequence of real numbers. If $\\liminf_{n \\to \\infty} S_n = +\\infty$ in the extended reals, then $S_n \\to +\\infty$ as $n \\to \\infty$.","title":"Liminf plus infinity forces divergence to plus infinity"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/liminf-eq-top-of-tendsto-at-top","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.liminf_eq_top_of_tendsto_atTop","parent":22070,"reading_title":{"format":"inline-parts-v1","parts":[["text","A real sequence tending to "],["math","\\(+\\infty\\)"],["text"," has liminf "],["math","\\(+\\infty\\)"]],"source":"A real sequence tending to $+\\infty$ has liminf $+\\infty$"},"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#liminf-eq-top-of-tendsto-at-top","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"If a real sequence $(S_n)$ tends to $+\\infty$, then its limit inferior in the extended reals $[-\\infty, +\\infty]$ is $+\\infty$.","title":"A real sequence tending to $+\\infty$ has liminf $+\\infty$"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-coord-drifts-to-top","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.measurableSet_coord_driftsToTop","parent":22070,"reading_title":{"format":"inline-parts-v1","parts":[["text","The event of drift to "],["math","\\(+\\infty\\)"],["text"," on sequence space is measurable"]],"source":"The event of drift to $+\\infty$ on sequence space is measurable"},"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-coord-drifts-to-top","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Equip $\\mathbb{R}^{\\mathbb{N}}$ with its product \u03c3-algebra and write $S_n(\\omega) = \\sum_{i < n} \\omega_i$. The set of $\\omega \\in \\mathbb{R}^{\\mathbb{N}}$ with $S_n(\\omega) \\to +\\infty$ as $n \\to \\infty$ is measurable.","title":"The event of drift to $+\\infty$ on sequence space is measurable"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-coord-drifts-to-bot","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.measurableSet_coord_driftsToBot","parent":22070,"reading_title":{"format":"inline-parts-v1","parts":[["text","The event of drift to "],["math","\\(-\\infty\\)"],["text"," on sequence space is measurable"]],"source":"The event of drift to $-\\infty$ on sequence space is measurable"},"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-coord-drifts-to-bot","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Equip $\\mathbb{R}^{\\mathbb{N}}$ with its product \u03c3-algebra and write $S_n(\\omega) = \\sum_{i < n} \\omega_i$. The set of $\\omega \\in \\mathbb{R}^{\\mathbb{N}}$ with $S_n(\\omega) \\to -\\infty$ as $n \\to \\infty$ is measurable.","title":"The event of drift to $-\\infty$ on sequence space is measurable"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-coord-oscillates","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.measurableSet_coord_oscillates","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-coord-oscillates","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Equip $\\mathbb{R}^{\\mathbb{N}}$ with its product \u03c3-algebra and write $S_n(\\omega) = \\sum_{i < n} \\omega_i$. The set of $\\omega \\in \\mathbb{R}^{\\mathbb{N}}$ with $\\limsup_{n \\to \\infty} S_n(\\omega) = +\\infty$ and $\\liminf_{n \\to \\infty} S_n(\\omega) = -\\infty$ (limits taken in the extended reals) is measurable.","title":"The oscillation event on sequence space is measurable"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-coord-forall-eq-zero","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.measurableSet_coord_forall_eq_zero","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-coord-forall-eq-zero","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"In the sequence space $\\mathbb{R}^{\\mathbb{N}}$ with its product \u03c3-algebra, the set of sequences $\\omega$ with $\\omega_i = 0$ for every $i$ is measurable.","title":"The zero-sequence event is measurable"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/converges-finite-of-ae-forall-eq-zero","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.convergesFinite_of_ae_forall_eq_zero","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#converges-finite-of-ae-forall-eq-zero","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$ and let $X_0, X_1, \\dots$ be real-valued functions on $\\Omega$, with partial sums $S_n = \\sum_{i\\lt n} X_i$. If for $\\mu$-almost every $\\omega$ we have $X_i(\\omega) = 0$ for all $i$, then for $\\mu$-almost every $\\omega$ the sequence $S_n(\\omega)$ converges to a finite real limit.","title":"Vanishing steps give a convergent random walk"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/measurable-set-coord-converges-finite","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.measurableSet_coord_convergesFinite","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#measurable-set-coord-converges-finite","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Equip $\\mathbb{R}^{\\mathbb{N}}$ with its product \u03c3-algebra and write $S_n(\\omega) = \\sum_{i < n} \\omega_i$. The set of $\\omega \\in \\mathbb{R}^{\\mathbb{N}}$ for which $S_n(\\omega)$ converges to a finite real limit as $n \\to \\infty$ is measurable.","title":"The convergence event on sequence space is measurable"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/zero-walk","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.ZeroWalk","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#zero-walk","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $X_0, X_1, \\ldots$ be real-valued functions on $\\Omega$, and let $S_n = \\sum_{i < n} X_i$ be the partial sums of the random walk. The walk is a zero walk when, for $\\mu$-almost every $\\omega \\in \\Omega$, we have $S_n(\\omega) = 0$ for every $n \\in \\mathbb{N}$.","title":"Zero random walk"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/drifts-to-top","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.DriftsToTop","parent":22070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Random walk drifting to "],["math","\\(+\\infty\\)"]],"source":"Random walk drifting to $+\\infty$"},"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#drifts-to-top","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real random variables on a probability space $(\\Omega, \\mu)$ and $S_n = \\sum_{i < n} X_i$. The random walk $(S_n)$ drifts to $+\\infty$ if for $\\mu$-almost every $\\omega$, $S_n(\\omega) \\to +\\infty$ as $n \\to \\infty$.","title":"Random walk drifting to $+\\infty$"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/drifts-to-bot","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.DriftsToBot","parent":22070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Random walk drifting to "],["math","\\(-\\infty\\)"]],"source":"Random walk drifting to $-\\infty$"},"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#drifts-to-bot","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real random variables on a probability space $(\\Omega, \\mu)$ and $S_n = \\sum_{i < n} X_i$. The random walk $(S_n)$ drifts to $-\\infty$ if for $\\mu$-almost every $\\omega$, $S_n(\\omega) \\to -\\infty$ as $n \\to \\infty$.","title":"Random walk drifting to $-\\infty$"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/oscillates","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.Oscillates","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#oscillates","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real random variables on a probability space $(\\Omega, \\mu)$ and $S_n = \\sum_{i < n} X_i$. The random walk $(S_n)$ oscillates if for $\\mu$-almost every $\\omega$, $\\limsup_n S_n(\\omega) = +\\infty$ and $\\liminf_n S_n(\\omega) = -\\infty$, the limits taken in the extended reals.","title":"Oscillating random walk"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/converges-finite","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.ConvergesFinite","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#converges-finite","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real random variables on a probability space $(\\Omega, \\mu)$ and $S_n = \\sum_{i < n} X_i$. The random walk $(S_n)$ converges to a finite limit if for $\\mu$-almost every $\\omega$ there is $l \\in \\mathbb{R}$ with $S_n(\\omega) \\to l$ as $n \\to \\infty$.","title":"Random walk with a finite limit"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/zero-walk-iff-ae-forall-eq-zero","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.zeroWalk_iff_ae_forall_eq_zero","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#zero-walk-iff-ae-forall-eq-zero","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real random variables on a probability space and $S_n = \\sum_{i < n} X_i$. Then almost surely $S_n = 0$ for every $n$ if and only if almost surely $X_i = 0$ for every $i$.","title":"A random walk is zero exactly when its steps are"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/converges-finite-2","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.ZeroWalk.convergesFinite","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#converges-finite-2","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be real random variables on a probability space and $S_n = \\sum_{i < n} X_i$. If almost surely $S_n = 0$ for every $n$, then almost surely there is $l \\in \\mathbb{R}$ with $S_n \\to l$ as $n \\to \\infty$.","title":"A zero walk converges to a finite limit"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/dichotomy-product-zero","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.dichotomy_product_zero","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#dichotomy-product-zero","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\rho$ be a probability measure on $\\mathbb{R}$, let $\\nu = \\rho^{\\otimes \\mathbb{N}}$ be the product measure on $\\mathbb{R}^{\\mathbb{N}}$, and let $S_n(\\omega) = \\sum_{i < n} \\omega_i$ be the partial sums of the coordinates. Then one of the following holds: $\\nu$-almost surely $S_n = 0$ for every $n$; $\\nu$-almost surely $S_n \\to +\\infty$; $\\nu$-almost surely $S_n \\to -\\infty$; or $\\nu$-almost surely\u2026","title":"Random-walk dichotomy on sequence space"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/dichotomy-product","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.dichotomy_product","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#dichotomy-product","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $\\rho$ be a probability measure on $\\mathbb{R}$, let $\\nu = \\rho^{\\otimes \\mathbb{N}}$ be the product measure on $\\mathbb{R}^{\\mathbb{N}}$, and let $S_n(\\omega) = \\sum_{i < n} \\omega_i$ be the partial sums of the coordinates. Then one of the following holds: $\\nu$-almost surely $S_n$ converges to a finite limit (which may depend on $\\omega$); $\\nu$-almost surely $S_n \\to +\\infty$; $\\nu$-almost surely\u2026","title":"Random-walk dichotomy on sequence space, finite-limit form"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/random-walk-dichotomy-zero","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.random_walk_dichotomy_zero","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#random-walk-dichotomy-zero","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent, identically distributed real random variables on a probability space and $S_n = \\sum_{i < n} X_i$. Then one of the following events has probability one: $S_n = 0$ for every $n$; $S_n \\to +\\infty$; $S_n \\to -\\infty$; or $\\limsup_n S_n = +\\infty$ and $\\liminf_n S_n = -\\infty$.","title":"Random-walk dichotomy"},{"id":"Probability/random-walks/drift-and-oscillation-of-random-walks/random-walk-dichotomy","kind":"node","lean":"MathlibExt.Probability.RandomWalkDichotomy.random_walk_dichotomy","parent":22070,"result":true,"route":"../roadmap/Probability/random-walks/drift-and-oscillation-of-random-walks/index.html#random-walk-dichotomy","scope":"Probability/random-walks/drift-and-oscillation-of-random-walks","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be independent, identically distributed real random variables on a probability space and $S_n = \\sum_{i < n} X_i$. Then one of the following events has probability one: $S_n$ converges to a finite limit (which may depend on the outcome); $S_n \\to +\\infty$; $S_n \\to -\\infty$; or $\\limsup_n S_n = +\\infty$ and $\\liminf_n S_n = -\\infty$.","title":"Random-walk dichotomy, finite-limit form"},{"data_url":"scopes/Probability/random-walks/random-products-in-compact-groups.json","id":"Probability/random-walks/random-products-in-compact-groups","kind":"scope","navigation":"module","parent":22068,"result":false,"route":"../roadmap/Probability/random-walks/random-products-in-compact-groups/index.html","scope":"Probability/random-walks/random-products-in-compact-groups","status":"can_prove","title":"Random products in compact groups"},{"id":"Probability/random-walks/random-products-in-compact-groups/coin-flip","kind":"node","lean":"MathlibExt.Dynamics.ScottishBook116bWanted.coinFlip","parent":22123,"result":true,"route":"../roadmap/Probability/random-walks/random-products-in-compact-groups/index.html#coin-flip","scope":"Probability/random-walks/random-products-in-compact-groups","status":"fully_proved","summary":"The fair coin measure is the measure $\\tfrac12 \\delta_{\\mathrm{false}} + \\tfrac12 \\delta_{\\mathrm{true}}$ on the Booleans $\\{\\mathrm{false}, \\mathrm{true}\\}$: it gives mass $1/2$ to each of the two values.","title":"Fair coin measure"},{"id":"Probability/random-walks/random-products-in-compact-groups/coin-measure","kind":"node","lean":"MathlibExt.Dynamics.ScottishBook116bWanted.coinMeasure","parent":22123,"result":true,"route":"../roadmap/Probability/random-walks/random-products-in-compact-groups/index.html#coin-measure","scope":"Probability/random-walks/random-products-in-compact-groups","status":"fully_proved","summary":"The fair coin product measure is the measure on the space of Boolean sequences $c : \\mathbb{N} \\to \\{\\mathrm{false}, \\mathrm{true}\\}$ given by the infinite product, over $n \\in \\mathbb{N}$, of the fair coin measure, which gives mass $1/2$ to each of $\\mathrm{false}$ and $\\mathrm{true}$.","title":"Fair coin product measure"},{"id":"Probability/random-walks/random-products-in-compact-groups/step","kind":"node","lean":"MathlibExt.Dynamics.ScottishBook116bWanted.step","parent":22123,"result":true,"route":"../roadmap/Probability/random-walks/random-products-in-compact-groups/index.html#step","scope":"Probability/random-walks/random-products-in-compact-groups","status":"fully_proved","summary":"Let $\\varphi$ and $\\psi$ be elements of a type $G$ and let $c : \\mathbb{N} \\to \\{\\mathrm{false}, \\mathrm{true}\\}$ be a sequence. For $n \\in \\mathbb{N}$, the $n$-th chosen generator is $\\varphi$ if $c(n) = \\mathrm{false}$ and $\\psi$ if $c(n) = \\mathrm{true}$.","title":"Generator chosen by a coin sequence"},{"id":"Probability/random-walks/random-products-in-compact-groups/partial-prod","kind":"node","lean":"MathlibExt.Dynamics.ScottishBook116bWanted.partialProd","parent":22123,"result":true,"route":"../roadmap/Probability/random-walks/random-products-in-compact-groups/index.html#partial-prod","scope":"Probability/random-walks/random-products-in-compact-groups","status":"fully_proved","summary":"Let $G$ be a group, let $\\varphi, \\psi \\in G$, and let $c : \\mathbb{N} \\to \\{\\mathrm{false}, \\mathrm{true}\\}$. For $i \\ge 1$ let $f_i$ be the generator chosen at index $i - 1$: $f_i = \\varphi$ if $c(i - 1) = \\mathrm{false}$ and $f_i = \\psi$ if $c(i - 1) = \\mathrm{true}$. The successive products are defined by $P_0 = 1$ and $P_{n+1} = P_n f_{n+1}$, so $P_n = f_1 f_2 \\cdots f_n$ for every $n \\in \\mathbb{N}$, with\u2026","title":"Successive products along a coin sequence"},{"id":"Probability/random-walks/random-products-in-compact-groups/scottish-book-problem116b-claim","kind":"node","lean":"MathlibExt.Dynamics.ScottishBook116bWanted.ScottishBookProblem116bClaim","parent":22123,"result":true,"route":"../roadmap/Probability/random-walks/random-products-in-compact-groups/index.html#scottish-book-problem116b-claim","scope":"Probability/random-walks/random-products-in-compact-groups","status":"fully_proved","summary":"The claim is the following. Let $G$ be any compact Hausdorff topological group with its Borel $\\sigma$-algebra, and let $\\mu$ be a Haar measure on $G$ that is a probability measure. For $\\varphi, \\psi \\in G$ and a Boolean sequence $c : \\mathbb{N} \\to \\{\\mathrm{false}, \\mathrm{true}\\}$, let $f_{n+1} = \\varphi$ if $c(n) = \\mathrm{false}$ and $f_{n+1} = \\psi$ if $c(n) = \\mathrm{true}$, and define the successive\u2026","title":"Scottish Book Problem 116(b), measurable-set form"},{"id":"Probability/random-walks/random-products-in-compact-groups/scottish-book116b-refuted","kind":"node","lean":"MathlibExt.Dynamics.ScottishBook116bWanted.scottishBook116b_refuted","parent":22123,"result":true,"route":"../roadmap/Probability/random-walks/random-products-in-compact-groups/index.html#scottish-book116b-refuted","scope":"Probability/random-walks/random-products-in-compact-groups","status":"can_prove","summary":"The measurable-set form of Scottish Book Problem 116(b) is false. For $\\varphi, \\psi$ in a group $G$ and a Boolean sequence $c : \\mathbb{N} \\to \\{\\mathrm{false}, \\mathrm{true}\\}$, let $f_{n+1} = \\varphi$ if $c(n) = \\mathrm{false}$ and $f_{n+1} = \\psi$ if $c(n) = \\mathrm{true}$, and define the successive products by $P_0 = 1$ and $P_{n+1} = P_n f_{n+1}$. It is not the case that for every compact Hausdorff\u2026","title":"Scottish Book Problem 116(b) fails for arbitrary measurable sets"},{"data_url":"scopes/Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g.json","id":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g","kind":"scope","navigation":"module","parent":22068,"result":false,"route":"../roadmap/Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/index.html","scope":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g","status":"can_prove","title":"Return probabilities of random walks on Cayley graphs"},{"id":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/regular-cayley-graph","kind":"node","lean":"MathlibExt.GraphTheory.AMR11Problem33Wanted.AMR11Q33.RegularCayleyGraph","parent":22130,"result":true,"route":"../roadmap/Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/index.html#regular-cayley-graph","scope":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g","status":"fully_proved","summary":"For a group $G$ and a natural number $k$, the data of a $k$-regular Cayley graph on $G$ is a finite set $S \\subseteq G$ of generators such that $S$ is closed under inverses ($s \\in S$ implies $s^{-1} \\in S$), the identity $1$ does not belong to $S$, every element of $G$ is the product of a finite (possibly empty) list of elements of $S$, and $S$ has exactly $k$ elements.","title":"Regular Cayley graph"},{"id":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/first-return-prob","kind":"node","lean":"MathlibExt.GraphTheory.AMR11Problem33Wanted.AMR11Q33.firstReturnProb","parent":22130,"result":true,"route":"../roadmap/Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/index.html#first-return-prob","scope":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g","status":"fully_proved","summary":"Let $\\Gamma$ be a $k$-regular Cayley graph on a group $G$ with generating set $S$. For $n \\ge 1$, the first-return probability of $\\Gamma$ at time $n$ is the real number obtained by dividing by $k^n$ the number of words $(s_1, \\dots, s_n) \\in S^n$ with $s_1 s_2 \\cdots s_n = 1$ and $s_1 \\cdots s_m \\ne 1$ for every $0 < m < n$; at time $n = 0$ it is $0$.","title":"First-return probability"},{"id":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/return-prob","kind":"node","lean":"MathlibExt.GraphTheory.AMR11Problem33Wanted.AMR11Q33.returnProb","parent":22130,"result":true,"route":"../roadmap/Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/index.html#return-prob","scope":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g","status":"fully_proved","summary":"The return probability of a $k$-regular Cayley graph $\\Gamma$ is the sum $\\sum_{n \\ge 0} p_n$, where $p_n$ is the first-return probability of $\\Gamma$ at time $n$.","title":"Return probability"},{"id":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/return-prob-uniform-gap","kind":"node","lean":"MathlibExt.GraphTheory.AMR11Problem33Wanted.AMR11Q33.returnProbUniformGap","parent":22130,"result":true,"route":"../roadmap/Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/index.html#return-prob-uniform-gap","scope":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g","status":"fully_proved","summary":"The statement that for every natural number $k > 1$ there is a real number $C < 1$ such that for every group $G$ and every $k$-regular Cayley graph $\\Gamma$ on $G$, if the return probability of $\\Gamma$ is less than $1$ (that is, $\\Gamma$ is transient), then it is at most $C$.","title":"Uniform return-probability gap"},{"id":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/return-prob-uniform-gap-holds","kind":"node","lean":"MathlibExt.GraphTheory.AMR11Problem33Wanted.returnProbUniformGap_holds","parent":22130,"result":true,"route":"../roadmap/Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g/index.html#return-prob-uniform-gap-holds","scope":"Probability/random-walks/return-probabilities-of-random-walks-on-cayley-g","status":"can_prove","summary":"For every natural number $k > 1$ there is a real number $C < 1$ such that for every group $G$ and every $k$-regular Cayley graph $\\Gamma$ on $G$ whose return probability is less than $1$, the return probability of $\\Gamma$ is at most $C$.","title":"Uniform return-probability gap for transient Cayley graphs"},{"data_url":"scopes/Probability/random-walks/strong-markov-property-of-i-i-d-sequences.json","id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","kind":"scope","navigation":"module","parent":22068,"result":false,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","title":"Strong Markov property of i.i.d. sequences"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/post-stopping-time-seq","kind":"node","lean":"ProbabilityTheory.postStoppingTimeSeq","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#post-stopping-time-seq","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Given maps $X_0, X_1, X_2, \\dots$ from a set $\\Omega$ to a set $S$ and a map $T : \\Omega \\to \\mathbb{N}$, the post-stopping-time sequence is the sequence $(Y_k)_{k \\in \\mathbb{N}}$ of maps from $\\Omega$ to $S$ given by $Y_k(\\omega) = X_{T(\\omega)+k+1}(\\omega)$. When $T$ is a stopping time, this is the sequence of observations immediately after time $T$.","title":"Post-stopping-time sequence"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/ident-distrib-tail","kind":"node","lean":"ProbabilityTheory.identDistrib_tail","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#ident-distrib-tail","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$ and let $X_0, X_1, X_2, \\dots$ be maps from $\\Omega$ to a measurable space $S$ such that $X_i$ and $X_j$ are identically distributed under $\\mu$ for all $i, j$. Then for all $n, k \\in \\mathbb{N}$ the map $X_{n+k+1}$ has the same distribution under $\\mu$ as $X_0$.","title":"Shifted terms of an identically distributed sequence"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/indep-stopping-time-tail","kind":"node","lean":"ProbabilityTheory.indep_stopping_time_tail","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#indep-stopping-time-tail","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on $\\Omega$, and let $X_0, X_1, X_2, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$ that are mutually independent under $\\mu$. Suppose that for every $n$ the \u03c3-algebra $\\mathcal{F}_n$ is contained in the \u03c3-algebra generated by $X_0, \\dots, X_n$, and let\u2026","title":"A stopping-time event is independent of a later observation"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/measurable-set-stopping-time-eq","kind":"node","lean":"ProbabilityTheory.measurableSet_stoppingTime_eq","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#measurable-set-stopping-time-eq","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on a measurable space $\\Omega$ and let $T : \\Omega \\to \\mathbb{N}$ be a stopping time for $(\\mathcal{F}_n)$. Then for every $n \\in \\mathbb{N}$ the set $\\{\\omega : T(\\omega) = n\\}$ is measurable in the \u03c3-algebra of $\\Omega$.","title":"Level sets of a stopping time are measurable"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/aemeasurable-post-stopping-time-seq","kind":"node","lean":"ProbabilityTheory.aemeasurable_postStoppingTimeSeq","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#aemeasurable-post-stopping-time-seq","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on $\\Omega$, let $X_0, X_1, X_2, \\dots$ be maps from $\\Omega$ to a measurable space $S$ that are each $\\mu$-almost-everywhere measurable, and let $T : \\Omega \\to \\mathbb{N}$ be a stopping time for $(\\mathcal{F}_n)$. Then for every $k \\in \\mathbb{N}$ the map $\\omega \\mapsto X_{T(\\omega)+k+1}(\\omega)$ is\u2026","title":"The post-stopping-time sequence is a.e. measurable"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/strong-markov-iid-ident-distrib","kind":"node","lean":"ProbabilityTheory.strong_markov_iid_identDistrib","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#strong-markov-iid-ident-distrib","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on $\\Omega$, and let $X_0, X_1, X_2, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$ that are mutually independent and identically distributed under $\\mu$. Suppose that for every $n$ the \u03c3-algebra $\\mathcal{F}_n$ is contained in the \u03c3-algebra generated by $X_0, \\dots, X_n$,\u2026","title":"Strong Markov property for i.i.d. sequences: the law after a stopping time"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/indep-stopping-time-tail-finset","kind":"node","lean":"ProbabilityTheory.indep_stopping_time_tail_finset","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#indep-stopping-time-tail-finset","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on $\\Omega$, and let $X_0, X_1, X_2, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$ that are mutually independent under $\\mu$. Suppose that for every $n$ the \u03c3-algebra $\\mathcal{F}_n$ is contained in the \u03c3-algebra generated by $X_0, \\dots, X_n$, and let\u2026","title":"A stopping-time event is independent of finitely many later observations"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/post-stopping-time-seq-preimage-eq","kind":"node","lean":"ProbabilityTheory.postStoppingTimeSeq_preimage_eq","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#post-stopping-time-seq-preimage-eq","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on $\\Omega$, and let $X_0, X_1, X_2, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$ that are mutually independent and identically distributed under $\\mu$. Suppose that for every $n$ the \u03c3-algebra $\\mathcal{F}_n$ is contained in the \u03c3-algebra generated by $X_0, \\dots, X_n$,\u2026","title":"Preimage measures of a post-stopping-time variable"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/strong-markov-iid-i-indep-fun","kind":"node","lean":"ProbabilityTheory.strong_markov_iid_iIndepFun","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#strong-markov-iid-i-indep-fun","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on $\\Omega$, and let $X_0, X_1, X_2, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$ that are mutually independent and identically distributed under $\\mu$. Suppose that for every $n$ the \u03c3-algebra $\\mathcal{F}_n$ is contained in the \u03c3-algebra generated by $X_0, \\dots, X_n$,\u2026","title":"Strong Markov property for i.i.d. sequences: independence after a stopping time"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/strong-markov-iid-post-stopping-time-seq","kind":"node","lean":"ProbabilityTheory.strong_markov_iid_postStoppingTimeSeq","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#strong-markov-iid-post-stopping-time-seq","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $(\\mathcal{F}_n)_{n \\in \\mathbb{N}}$ be a filtration on $\\Omega$, and let $X_0, X_1, X_2, \\dots$ be measurable maps from $\\Omega$ to a measurable space $S$ that are mutually independent and identically distributed under $\\mu$. Suppose that for every $n$ the \u03c3-algebra $\\mathcal{F}_n$ is contained in the \u03c3-algebra generated by $X_0, \\dots, X_n$,\u2026","title":"Strong Markov property for i.i.d. sequences: the post-stopping-time sequence is i.i.d"},{"id":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences/strong-markov-iid-indep-past","kind":"node","lean":"ProbabilityTheory.strong_markov_iid_indep_past","parent":22136,"result":true,"route":"../roadmap/Probability/random-walks/strong-markov-property-of-i-i-d-sequences/index.html#strong-markov-iid-indep-past","scope":"Probability/random-walks/strong-markov-property-of-i-i-d-sequences","status":"fully_proved","summary":"Let $X_0, X_1, \\dots$ be measurable random variables on a probability space $(\\Omega, \\mu)$ with values in a measurable space $S$, mutually independent and all with the same distribution. Let $(\\mathcal F_n)_{n \\ge 0}$ be a filtration with $\\mathcal F_n \\subseteq \\sigma(X_0, \\dots, X_n)$ for every $n$, and let $T : \\Omega \\to \\mathbb N$ be a stopping time for $(\\mathcal F_n)$, finite everywhere. Then the stopped\u2026","title":"Strong Markov property (independence from the past)"},{"data_url":"scopes/Probability/branching-processes.json","id":"Probability/branching-processes","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/branching-processes/index.html","scope":"Probability/branching-processes","status":"fully_proved","summary":"Galton\u2013Watson processes, extinction probabilities, total progeny and the hitting-time theorem, and linear birth\u2013death processes.","title":"Branching processes"},{"data_url":"scopes/Probability/branching-processes/galton-watson-process.json","id":"Probability/branching-processes/galton-watson-process","kind":"scope","navigation":"module","parent":22148,"result":false,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","title":"Galton\u2013Watson process"},{"id":"Probability/branching-processes/galton-watson-process/law","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.law","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#law","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The law of the generation size `Z_n` of the Galton\u2013Watson process with offspring law `\u03bd` started from one individual: `Z_0 = 1` and, given `Z_n = z`, `Z_{n+1}` has law `\u03bd^{\u2217z}` (`Measure.convPow \u03bd z`).","title":"Generation-size law of a Galton\u2013Watson process"},{"id":"Probability/branching-processes/galton-watson-process/mean","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.mean","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#mean","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The mean offspring number `m = E \u03be = \u2211 k \u03bd{k} \u2208 [0, \u221e]`.","title":"Mean offspring number"},{"id":"Probability/branching-processes/galton-watson-process/extinction-prob","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.extinctionProb","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#extinction-prob","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The extinction probability `q = sup_n P(Z_n = 0)`, the limit of the probabilities of extinction by generation `n` (`tendsto_law_real_zero`). On a path space carrying the process the events `{Z_n = 0}` increase with `n` (`0` is absorbing), so `q` is the probability of eventual extinction; this file describes the process through its generation laws and does not construct the path space.","title":"Extinction probability of a Galton\u2013Watson process"},{"id":"Probability/branching-processes/galton-watson-process/iterate-pgf-mem-icc","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.iterate_pgf_mem_Icc","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#iterate-pgf-mem-icc","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The iterates `f^[n](s)` of the offspring generating function stay in `[0, 1]` for `s \u2208 [0, 1]`.","title":"Iterates of the offspring generating function stay in (0, 1)"},{"id":"Probability/branching-processes/galton-watson-process/monotone-iterate-pgf-zero","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.monotone_iterate_pgf_zero","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#monotone-iterate-pgf-zero","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The iterates `f^[n](0)` of the offspring generating function are non-decreasing.","title":"Iterates of the generating function at zero are nondecreasing"},{"id":"Probability/branching-processes/galton-watson-process/is-least-i-sup-iterate-pgf-zero","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.isLeast_iSup_iterate_pgf_zero","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#is-least-i-sup-iterate-pgf-zero","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"`sup_n f^[n](0)` is the least fixed point of `f` in `[0, 1]` (Athreya\u2013Ney 1972, ch. I).","title":"Limit of generating-function iterates is the least fixed point"},{"id":"Probability/branching-processes/galton-watson-process/i-sup-iterate-pgf-zero-eq-one-iff","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.iSup_iterate_pgf_zero_eq_one_iff","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#i-sup-iterate-pgf-zero-eq-one-iff","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The least fixed point is `1` iff the mean offspring number is at most `1`, provided the offspring number is not almost surely `1` (Athreya\u2013Ney 1972, ch. I).","title":"Least fixed point equals one iff mean offspring is at most one"},{"id":"Probability/branching-processes/galton-watson-process/pgf-law","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.pgf_law","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#pgf-law","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The generating function of `Z_n` is the `n`-th iterate of the offspring generating function on `[0, 1]` (Athreya\u2013Ney 1972, ch. I).","title":"Generating function of the n-th generation size"},{"id":"Probability/branching-processes/galton-watson-process/law-real-zero","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.law_real_zero","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#law-real-zero","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"`P(Z_n = 0) = f^[n](0)`.","title":"Probability of extinction by generation n"},{"id":"Probability/branching-processes/galton-watson-process/lintegral-law","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.lintegral_law","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#lintegral-law","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The mean generation size is `E Z_n = m ^ n` (Athreya\u2013Ney 1972, ch. I).","title":"Mean generation size of a Galton\u2013Watson process"},{"id":"Probability/branching-processes/galton-watson-process/monotone-law-real-zero","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.monotone_law_real_zero","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#monotone-law-real-zero","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"`P(Z_n = 0)` is non-decreasing in `n`.","title":"Extinction by generation n is nondecreasing in n"},{"id":"Probability/branching-processes/galton-watson-process/tendsto-law-real-zero","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.tendsto_law_real_zero","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#tendsto-law-real-zero","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"`P(Z_n = 0)` converges to the extinction probability.","title":"Extinction by generation n converges to the extinction probability"},{"id":"Probability/branching-processes/galton-watson-process/is-least-extinction-prob","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.isLeast_extinctionProb","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#is-least-extinction-prob","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"The extinction probability is the least fixed point of the offspring generating function in `[0, 1]` (Athreya\u2013Ney 1972, ch. I).","title":"Extinction probability is the least fixed point"},{"id":"Probability/branching-processes/galton-watson-process/extinction-prob-eq-one-iff","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.extinctionProb_eq_one_iff","parent":22149,"result":true,"route":"../roadmap/Probability/branching-processes/galton-watson-process/index.html#extinction-prob-eq-one-iff","scope":"Probability/branching-processes/galton-watson-process","status":"fully_proved","summary":"Extinction is certain iff the mean offspring number is at most `1`, provided the offspring number is not almost surely `1` (Athreya\u2013Ney 1972, ch. I).","title":"Galton\u2013Watson extinction criterion"},{"data_url":"scopes/Probability/branching-processes/linear-birth-death-process.json","id":"Probability/branching-processes/linear-birth-death-process","kind":"scope","navigation":"module","parent":22148,"result":false,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","title":"Linear birth\u2013death process"},{"id":"Probability/branching-processes/linear-birth-death-process/extinction-prob","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.extinctionProb","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#extinction-prob","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Kendall's extinction probability `\u03b1 t = P(X t = 0 | X 0 = 1)` of the linear birth\u2013death process with per-capita birth rate `b` and death rate `d`: `d (E - 1) / (b E - d)` with `E = exp ((b - d) t)` if `b \u2260 d`, and `b t / (1 + b t)` if `b = d`. Meaningful for `0 \u2264 b`, `0 \u2264 d`, `0 \u2264 t`.","title":"Kendall's extinction probability for the birth\u2013death process"},{"id":"Probability/branching-processes/linear-birth-death-process/geom-ratio","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.geomRatio","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#geom-ratio","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Kendall's ratio `\u03b2 t` of the geometric tail of the law of `X t`: `b (E - 1) / (b E - d)` with `E = exp ((b - d) t)` if `b \u2260 d`, and `b t / (1 + b t)` if `b = d`.","title":"Kendall's geometric tail ratio"},{"id":"Probability/branching-processes/linear-birth-death-process/prob","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.prob","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#prob","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Kendall's law of the linear birth\u2013death process at time `t` started from one individual: `prob b d t 0 = \u03b1 t` and `prob b d t (n + 1) = (1 - \u03b1 t) (1 - \u03b2 t) (\u03b2 t) ^ n`.","title":"Kendall's law of the linear birth\u2013death process"},{"id":"Probability/branching-processes/linear-birth-death-process/prob-zero-time","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.prob_zero_time","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#prob-zero-time","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"At time `0` the population is one individual: `prob b d 0 = \u03b4\u2081`.","title":"Linear birth\u2013death law at time zero"},{"id":"Probability/branching-processes/linear-birth-death-process/prob-nonneg","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.prob_nonneg","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#prob-nonneg","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Kendall's probabilities are nonnegative.","title":"Nonnegativity of Kendall's probabilities"},{"id":"Probability/branching-processes/linear-birth-death-process/has-sum-prob","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.hasSum_prob","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#has-sum-prob","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Kendall's probabilities sum to `1`: no mass escapes to infinity.","title":"Kendall's probabilities sum to one"},{"id":"Probability/branching-processes/linear-birth-death-process/has-sum-mul-prob","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.hasSum_mul_prob","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#has-sum-mul-prob","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"The mean population size at time `t` started from one individual is `E[X t] = exp ((b - d) t)` (Kendall 1948; Bailey 1964, ch. 8).","title":"Mean population size of the linear birth\u2013death process"},{"id":"Probability/branching-processes/linear-birth-death-process/has-deriv-at-prob-zero","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.hasDerivAt_prob_zero","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#has-deriv-at-prob-zero","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Forward Kolmogorov equation for state `0`: `p_0' = d p_1`.","title":"Forward Kolmogorov equation for the empty state"},{"id":"Probability/branching-processes/linear-birth-death-process/has-deriv-at-prob-succ","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.hasDerivAt_prob_succ","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#has-deriv-at-prob-succ","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Forward Kolmogorov equation for state `n + 1`: `p_(n+1)' = b n p_n - (b + d) (n + 1) p_(n+1) + d (n + 2) p_(n+2)`.","title":"Forward Kolmogorov equation for positive states"},{"id":"Probability/branching-processes/linear-birth-death-process/has-deriv-at-extinction-prob","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.hasDerivAt_extinctionProb","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#has-deriv-at-extinction-prob","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Backward Kolmogorov equation for the extinction probability: `\u03b1' = d - (b + d) \u03b1 + b \u03b1\u00b2` (Karlin\u2013Taylor 1975, ch. 4).","title":"Backward Kolmogorov equation for the extinction probability"},{"id":"Probability/branching-processes/linear-birth-death-process/eq-extinction-prob-of-has-deriv-within-a","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.eq_extinctionProb_of_hasDerivWithinAt","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#eq-extinction-prob-of-has-deriv-within-a","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"The extinction probability is the only `[0, 1]`-valued solution `q` on `[0, \u221e)` of the backward equation `q' = d - (b + d) q + b q\u00b2` with `q 0 = 0`.","title":"Uniqueness of the extinction probability solving the backward equation"},{"id":"Probability/branching-processes/linear-birth-death-process/tendsto-extinction-prob-of-lt","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.tendsto_extinctionProb_of_lt","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#tendsto-extinction-prob-of-lt","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Supercritical case: if `d < b`, the extinction probability tends to `d / b`.","title":"Limiting extinction probability in the supercritical case"},{"id":"Probability/branching-processes/linear-birth-death-process/tendsto-extinction-prob-of-le","kind":"node","lean":"MathlibExt.Probability.LinearBirthDeath.tendsto_extinctionProb_of_le","parent":22164,"result":true,"route":"../roadmap/Probability/branching-processes/linear-birth-death-process/index.html#tendsto-extinction-prob-of-le","scope":"Probability/branching-processes/linear-birth-death-process","status":"fully_proved","summary":"Critical and subcritical cases: if `b \u2264 d` and `0 < d`, extinction is certain in the limit.","title":"Certain extinction in the critical and subcritical cases"},{"data_url":"scopes/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem.json","id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","kind":"scope","navigation":"module","parent":22148,"result":false,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","title":"Total progeny and the hitting-time theorem"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/is-first-passage","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.IsFirstPassage","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#is-first-passage","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"The exploration walk started at `k` with steps `x_i - 1` stays positive at every time `j < n`: `j < k + \u2211_{i < j} x_i`. Together with `\u2211 x_i + k = n` this says that the walk first hits `0` at time `n`.","title":"First passage of the exploration walk"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/rotate","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.rotate","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#rotate","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"The cyclic shift `i \u21a6 x_{(i + r) mod n}` of a tuple.","title":"Cyclic shift of a tuple"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/card-filter-is-first-passage-rotate","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.card_filter_isFirstPassage_rotate","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#card-filter-is-first-passage-rotate","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"Cycle lemma (Dvoretzky\u2013Motzkin 1947). If `x : Fin n \u2192 \u2115` satisfies `\u2211 x_i + k = n` with `1 \u2264 k`, then exactly `k` of the `n` cyclic shifts of `x` (counted by shift `r < n`) are first-passage sequences.","title":"Cycle lemma"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/walk-weight","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.walkWeight","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#walk-weight","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"The weight `W_p(k, n) = \u2211 \u220f_i p(x_i)` over the step sequences `x : Fin n \u2192 \u2115` with `\u2211 x_i + k = n` (the walk from `k` is at `0` at time `n`). For `p(j) = P(\u03be = j)` this is `P(\u03be_1 + \u22ef + \u03be_n = n - k)` when `k \u2264 n`, and `0` when `k > n`.","title":"Walk weight of step sequences ending at zero"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/hitting-weight","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.hittingWeight","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#hitting-weight","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"The weight `H_p(k, n) = \u2211 \u220f_i p(x_i)` over the step sequences with `\u2211 x_i + k = n` along which the walk from `k` first hits `0` at time `n`. For `p(j) = P(\u03be = j)` this is the probability `P(T = n)` that the total progeny of `k` ancestors is `n`.","title":"Hitting weight of first-passage step sequences"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/nat-cast-mul-hitting-weight","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.natCast_mul_hittingWeight","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#nat-cast-mul-hitting-weight","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"Hitting-time theorem, weighted form: `n \u00b7 H_p(k, n) = k \u00b7 W_p(k, n)` for every weight `p` with values in a commutative semiring and all `k, n`. For probability weights this is the hitting-time theorem of van der Hofstad\u2013Keane (2008); the proof double counts cyclic shifts with the cycle lemma.","title":"Hitting-time theorem"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/measure-first-passage","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.measure_firstPassage","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#measure-first-passage","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"Under `\u03bd^{\u2297n}`, the event that the walk from `k` first hits `0` at time `n` has measure `H_p(k, n)` with `p(j) = \u03bd{j}`. When `\u03bd` is a probability measure this is the probability `P(T = n)`.","title":"Probability of first passage at time n"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/measure-walk","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.measure_walk","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#measure-walk","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"Under `\u03bd^{\u2297n}`, the event `\u03be_1 + \u22ef + \u03be_n + k = n` has measure `W_p(k, n)` with `p(j) = \u03bd{j}`. When `\u03bd` is a probability measure this is the probability `P(\u03be_1 + \u22ef + \u03be_n = n - k)`.","title":"Probability of the offspring walk event as a walk weight"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/nat-cast-mul-measure-first-passage","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.natCast_mul_measure_firstPassage","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#nat-cast-mul-measure-first-passage","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"For i.i.d. offspring numbers with law `\u03bd` and `k` ancestors, the finite-measure identity `n \u00b7 P(T = n) = k \u00b7 P(\u03be_1 + \u22ef + \u03be_n = n - k)` holds as an identity of event masses (Otter 1949; Dwass 1969); the two sides are probabilities when `\u03bd` is a probability measure.","title":"Otter\u2013Dwass formula"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/walk-weight-poisson","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.walkWeight_poisson","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#walk-weight-poisson","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"For Poisson(`r`) step weights, `W_p(k, n) = e^{-rn} (rn)^{n-k} / (n-k)!` when `k \u2264 n`: the sum of `n` independent Poisson(`r`) variables is Poisson(`rn`).","title":"Walk weights for Poisson offspring"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/nat-cast-mul-hitting-weight-poisson","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.natCast_mul_hittingWeight_poisson","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#nat-cast-mul-hitting-weight-poisson","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"For Poisson(`r`) offspring and `k \u2264 n` ancestors, `n \u00b7 P(T = n) = k e^{-rn} (rn)^{n-k} / (n-k)!` (Tanner 1961; Haight\u2013Breuer 1960).","title":"Borel\u2013Tanner law"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/hitting-weight-poisson-one","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.hittingWeight_poisson_one","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#hitting-weight-poisson-one","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"For Poisson(`r`) offspring and one ancestor, the total progeny satisfies `P(T = n) = e^{-rn} (rn)^{n-1} / n!` for `n \u2265 1` (Borel 1942).","title":"Borel law"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/mean-total-progeny","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.meanTotalProgeny","parent":22178,"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#mean-total-progeny","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"The mean total progeny `\u2211_n E Z_n` of the Galton\u2013Watson process started from one individual. By Tonelli this is `E \u2211_n Z_n` for any process with the generation laws `law \u03bd n`.","title":"Mean total progeny of a Galton\u2013Watson process"},{"id":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/mean-total-progeny-eq","kind":"node","lean":"MathlibExt.Probability.GaltonWatson.meanTotalProgeny_eq","parent":22178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean total progeny equals "],["math","\\(1/(1-m)\\)"]],"source":"Mean total progeny equals $1/(1-m)$"},"result":true,"route":"../roadmap/Probability/branching-processes/total-progeny-and-the-hitting-time-theorem/index.html#mean-total-progeny-eq","scope":"Probability/branching-processes/total-progeny-and-the-hitting-time-theorem","status":"fully_proved","summary":"The mean total progeny is `(1 - m)\u207b\u00b9` in `[0, \u221e]`: it is `1 / (1 - m)` for `m < 1` and `\u221e` for `m \u2265 1` (`m` the mean offspring number).","title":"Mean total progeny equals $1/(1-m)$"},{"data_url":"scopes/Probability/point-processes.json","id":"Probability/point-processes","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/point-processes/index.html","scope":"Probability/point-processes","status":"can_prove","summary":"Poisson processes on the line and Poisson point processes on general spaces (arrival times, counts, restriction theorem), and stationary random functions.","title":"Point processes and stationary processes"},{"data_url":"scopes/Probability/point-processes/poisson-point-processes.json","id":"Probability/point-processes/poisson-point-processes","kind":"scope","navigation":"module","parent":22193,"result":false,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html","scope":"Probability/point-processes/poisson-point-processes","status":"can_prove","title":"Poisson point processes"},{"id":"Probability/point-processes/poisson-point-processes/counting-measure","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.countingMeasure","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#counting-measure","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"The counting measure `\u2211_{i < n} \u03b4_{x_i}` of a finite tuple of points.","title":"Counting measure of a finite tuple of points"},{"id":"Probability/point-processes/poisson-point-processes/poisson-point-process","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.poissonPointProcess","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#poisson-point-process","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"The law of the Poisson point process with finite intensity `\u03bc` on `E`: `\u2211_n e^{-\u03bc(E)} / n! \u2022 (\u03bc^{\u2297n}).map countingMeasure`, a probability measure on `Measure E` (Last\u2013Penrose Prop. 3.5, mixed binomial construction).","title":"Poisson point process with finite intensity"},{"id":"Probability/point-processes/poisson-point-processes/counting-measure-apply","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.countingMeasure_apply","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#counting-measure-apply","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"`(\u2211 \u03b4_{x_i})(s) = #{i | x_i \u2208 s}`, as a sum of indicators.","title":"Counting measure of a set as a sum of indicators"},{"id":"Probability/point-processes/poisson-point-processes/measurable-counting-measure","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.measurable_countingMeasure","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#measurable-counting-measure","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"The counting measure of a tuple depends measurably on the tuple.","title":"Measurability of the counting measure in the points"},{"id":"Probability/point-processes/poisson-point-processes/lintegral-counting-measure","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.lintegral_countingMeasure","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#lintegral-counting-measure","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"`\u222b f d(\u2211 \u03b4_{x_i}) = \u2211 f(x_i)`.","title":"Integral against a counting measure"},{"id":"Probability/point-processes/poisson-point-processes/tsum-coeff-mul-pow","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.tsum_coeff_mul_pow","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#tsum-coeff-mul-pow","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"The Poisson weights sum to one: `\u2211_n e^{-\u03bc(E)} \u03bc(E)^n / n! = 1`.","title":"Poisson weights sum to one"},{"id":"Probability/point-processes/poisson-point-processes/lintegral-pi-eval","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.lintegral_pi_eval","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#lintegral-pi-eval","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"`\u222b f(x_i) d\u03bc^{\u2297n}(x) = \u03bc(E)^{n-1} \u222b f d\u03bc`.","title":"Integral of a coordinate function under a product measure"},{"id":"Probability/point-processes/poisson-point-processes/counting-measure-comp","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.countingMeasure_comp","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#counting-measure-comp","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"The image of `\u2211 \u03b4_{x_i}` under `T` is `\u2211 \u03b4_{T(x_i)}`.","title":"Image of a counting measure under a map"},{"id":"Probability/point-processes/poisson-point-processes/map-apply-eq-poisson-measure","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.map_apply_eq_poissonMeasure","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#map-apply-eq-poisson-measure","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"The number of points `\u03b7(B)` in a measurable set `B` has the Poisson law of mean `\u03bc(B)`.","title":"Point counts of a Poisson point process are Poisson"},{"id":"Probability/point-processes/poisson-point-processes/map-apply-pi","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.map_apply_pi","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#map-apply-pi","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"The counts `\u03b7(A i)` of pairwise disjoint measurable sets are independent, and `\u03b7(A i)` has the Poisson law of mean `\u03bc(A i)`: the mixed binomial process is a Poisson process with intensity `\u03bc` in the sense of Last\u2013Penrose Def. 3.1 (Last\u2013Penrose Prop. 3.5).","title":"Mixed binomial process is a Poisson process"},{"id":"Probability/point-processes/poisson-point-processes/lintegral-lintegral","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.lintegral_lintegral","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#lintegral-lintegral","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"`E \u222b f d\u03b7 = \u222b f d\u03bc` for measurable `f : E \u2192 [0, \u221e]` (Last\u2013Penrose Prop. 2.7).","title":"Campbell's formula"},{"id":"Probability/point-processes/poisson-point-processes/integral-exp-neg","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.integral_exp_neg","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#integral-exp-neg","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"For measurable `f : E \u2192 [0, \u221e)`, `E exp(-\u222b f d\u03b7) = exp(-\u222b (1 - e^{-f}) d\u03bc)` (Last\u2013Penrose Thm. 3.9). The integral `\u222b f d\u03b7` is a finite sum `\u2211 f(x_i)` for every configuration charged by the process.","title":"Laplace functional"},{"id":"Probability/point-processes/poisson-point-processes/lintegral-lintegral-mecke","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.lintegral_lintegral_mecke","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#lintegral-lintegral-mecke","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"For measurable `h : E \u00d7 Measure E \u2192 [0, \u221e]`, `E \u222b h(x, \u03b7) \u03b7(dx) = \u222b E h(x, \u03b7 + \u03b4_x) \u03bc(dx)` (Mecke 1967; Last\u2013Penrose Thm. 4.1).","title":"Mecke equation"},{"id":"Probability/point-processes/poisson-point-processes/map-map","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.map_map","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#map-map","scope":"Probability/point-processes/poisson-point-processes","status":"fully_proved","summary":"The image `\u03b7 \u2218 T\u207b\u00b9` of a Poisson point process under a measurable map `T : E \u2192 F` is the Poisson point process with intensity `\u03bc \u2218 T\u207b\u00b9` (Last\u2013Penrose Thm. 5.1).","title":"Mapping theorem"},{"id":"Probability/point-processes/poisson-point-processes/map-restrict-pi","kind":"node","lean":"MathlibExt.Probability.PoissonPointProcess.map_restrict_pi","parent":22194,"result":true,"route":"../roadmap/Probability/point-processes/poisson-point-processes/index.html#map-restrict-pi","scope":"Probability/point-processes/poisson-point-processes","status":"can_prove","summary":"The restrictions `\u03b7|_{A i}` of a Poisson point process with finite intensity `\u03bc` to pairwise disjoint measurable sets `A i` are independent Poisson point processes with intensities `\u03bc|_{A i}` (Last\u2013Penrose Thm. 5.2).","title":"Restriction theorem"},{"data_url":"scopes/Probability/point-processes/poisson-processes.json","id":"Probability/point-processes/poisson-processes","kind":"scope","navigation":"module","parent":22193,"result":false,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","title":"Poisson processes"},{"id":"Probability/point-processes/poisson-processes/is-poisson-process","kind":"node","lean":"ProbabilityTheory.IsPoissonProcess","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#is-poisson-process","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"`N : \u211d\u22650 \u2192 \u03a9 \u2192 \u2115` is a Poisson process of rate `r` under `P` (Kingman 1993, ch. 4; Karlin\u2013Taylor 1975, ch. 4): `N 0 = 0` and the paths are nondecreasing almost surely, the increments over consecutive intervals are independent, and for `s \u2264 t` the increment `N t - N s` has the Poisson law of mean `r (t - s)`.","title":"Poisson process"},{"id":"Probability/point-processes/poisson-processes/arrival-time","kind":"node","lean":"ProbabilityTheory.PoissonProcess.arrivalTime","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#arrival-time","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"The `n`-th arrival time `S n = T 0 + \u22ef + T (n - 1)` of the interarrival times `T`.","title":"Arrival times from interarrival times"},{"id":"Probability/point-processes/poisson-processes/count","kind":"node","lean":"ProbabilityTheory.PoissonProcess.count","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#count","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"The counting process of the interarrival times `T`: `count T t \u03c9` is the number of arrivals `n \u2265 1` with `S n \u2264 t`. When that set is infinite (a null event for exponential interarrivals of positive rate) `Set.ncard` returns `0`.","title":"Counting process of interarrival times"},{"id":"Probability/point-processes/poisson-processes/has-law-arrival-time","kind":"node","lean":"ProbabilityTheory.PoissonProcess.hasLaw_arrivalTime","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#has-law-arrival-time","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"The `(n + 1)`-th arrival time of i.i.d. rate-`r` exponential interarrivals has the Gamma (Erlang) law with shape `n + 1` and rate `r` (Karlin\u2013Taylor 1975, \u00a74.2).","title":"Arrival times of exponential interarrivals are Erlang"},{"id":"Probability/point-processes/poisson-processes/has-law-count","kind":"node","lean":"ProbabilityTheory.PoissonProcess.hasLaw_count","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#has-law-count","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"The number of arrivals in `[0, t]` of i.i.d. rate-`r` exponential interarrivals has the Poisson law of mean `r t` (Karlin\u2013Taylor 1975, \u00a74.2, Theorem 2.1).","title":"Arrival counts of exponential interarrivals are Poisson"},{"id":"Probability/point-processes/poisson-processes/ae-count-zero","kind":"node","lean":"ProbabilityTheory.PoissonProcess.ae_count_zero","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#ae-count-zero","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"The counting process of i.i.d. rate-`r` exponential interarrivals starts at `0` almost surely (all interarrival times are positive almost surely).","title":"Counting process starts at zero almost surely"},{"id":"Probability/point-processes/poisson-processes/ae-count-monotone","kind":"node","lean":"ProbabilityTheory.PoissonProcess.ae_count_monotone","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#ae-count-monotone","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"The counting process of i.i.d. rate-`r` exponential interarrivals has nondecreasing paths almost surely (the arrival times tend to infinity almost surely, so every arrival set in `[0, t]` is finite).","title":"Counting process has nondecreasing paths almost surely"},{"id":"Probability/point-processes/poisson-processes/add","kind":"node","lean":"ProbabilityTheory.IsPoissonProcess.add","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#add","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"Superposition (Kingman 1993, \u00a72.2): the sum of two independent Poisson processes of rates `r\u2081` and `r\u2082` is a Poisson process of rate `r\u2081 + r\u2082`. Independence is of the two processes as random paths.","title":"Superposition"},{"id":"Probability/point-processes/poisson-processes/thinning","kind":"node","lean":"ProbabilityTheory.PoissonProcess.thinning","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#thinning","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"The thinning of `N` by the marks `B`: the number of indices `i < N t` with `B i`, i.e. the number of points of `N` in `[0, t]` that are kept when the `i`-th point (in time order, from `0`) is kept iff `B i`.","title":"Thinning of a Poisson process"},{"id":"Probability/point-processes/poisson-processes/thinning-2","kind":"node","lean":"ProbabilityTheory.IsPoissonProcess.thinning","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#thinning-2","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"Thinning (Kingman 1993, \u00a75.1; Karlin\u2013Taylor 1975, \u00a74.1): keeping each point of a Poisson process of rate `r` independently with probability `p`, independently of the process, gives a Poisson process of rate `p r`. The marks `B i` are i.i.d. with `P(B i) = p` and independent of `N` as a random path.","title":"Thinning"},{"id":"Probability/point-processes/poisson-processes/has-law-count-increments","kind":"node","lean":"ProbabilityTheory.PoissonProcess.hasLaw_count_increments","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#has-law-count-increments","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"The increments of the counting process of i.i.d. rate-`r` exponential interarrivals over consecutive intervals `(t i, t (i + 1)]` are independent with the Poisson laws of means `r (t (i + 1) - t i)`: the increment vector has the product law (Kingman 1993, ch. 4; Karlin\u2013Taylor 1975, ch. 4).","title":"Independent Poisson increments of the counting process"},{"id":"Probability/point-processes/poisson-processes/is-poisson-process-count","kind":"node","lean":"ProbabilityTheory.PoissonProcess.isPoissonProcess_count","parent":22210,"result":true,"route":"../roadmap/Probability/point-processes/poisson-processes/index.html#is-poisson-process-count","scope":"Probability/point-processes/poisson-processes","status":"fully_proved","summary":"Exponential interarrivals give a Poisson process (Kingman 1993, ch. 4; Karlin\u2013Taylor 1975, ch. 4): the counting process of i.i.d. rate-`r` exponential interarrival times is a Poisson process of rate `r`.","title":"Exponential interarrivals give a Poisson process"},{"data_url":"scopes/Probability/point-processes/stationary-random-functions.json","id":"Probability/point-processes/stationary-random-functions","kind":"scope","navigation":"module","parent":22193,"result":false,"route":"../roadmap/Probability/point-processes/stationary-random-functions/index.html","scope":"Probability/point-processes/stationary-random-functions","status":"can_prove","title":"Stationary random functions"},{"id":"Probability/point-processes/stationary-random-functions/is-stationary","kind":"node","lean":"MathlibExt.Probability.ScottishBook180Wanted.IsStationary","parent":22223,"result":true,"route":"../roadmap/Probability/point-processes/stationary-random-functions/index.html#is-stationary","scope":"Probability/point-processes/stationary-random-functions","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$, and let $v$ assign to each real $t$ a function $v_t : \\Omega \\to \\mathbb{R}$. Then $v$ is stationary if there are a real number $c$ and a function $R : \\mathbb{R} \\to \\mathbb{R}$ such that for all real $t$ and $h$, $\\int v_t \\, d\\mu = 0$, $\\int v_t^2 \\, d\\mu = c$ and $\\int v_t \\, v_{t+h} \\, d\\mu = R(h)$.","title":"Stationary random functions (Slutsky\u2013Khinchin)"},{"id":"Probability/point-processes/stationary-random-functions/scottish-book-problem180-claim","kind":"node","lean":"MathlibExt.Probability.ScottishBook180Wanted.ScottishBookProblem180Claim","parent":22223,"result":true,"route":"../roadmap/Probability/point-processes/stationary-random-functions/index.html#scottish-book-problem180-claim","scope":"Probability/point-processes/stationary-random-functions","status":"fully_proved","summary":"The claim that for every probability space $(\\Omega, \\mu)$ that carries a measurable random variable $U : \\Omega \\to \\mathbb{R}$ uniformly distributed on $[0, 1]$, and every stationary random function $v$ on $\\Omega$ (in the sense of Slutsky and Khinchin) such that each $v_t : \\Omega \\to \\mathbb{R}$ is measurable, there exist a measurable random variable $A : \\Omega \\to \\mathbb{R}$ uniformly distributed on $[0, 1]$\u2026","title":"Scottish Book Problem 180 (Kamp\u00e9 de F\u00e9riet's claim)"},{"id":"Probability/point-processes/stationary-random-functions/scottish-book180-refuted","kind":"node","lean":"MathlibExt.Probability.ScottishBook180Wanted.scottishBook180_refuted","parent":22223,"result":true,"route":"../roadmap/Probability/point-processes/stationary-random-functions/index.html#scottish-book180-refuted","scope":"Probability/point-processes/stationary-random-functions","status":"can_prove","summary":"The claim of Scottish Book Problem 180 is false: there are a probability space $(\\Omega, \\mu)$ carrying a measurable random variable uniformly distributed on $[0, 1]$ and a stationary random function $v$ on $\\Omega$ with every $v_t$ measurable, such that no measurable random variable $A : \\Omega \\to \\mathbb{R}$ uniformly distributed on $[0, 1]$ and no map $\\varphi : \\mathbb{R} \\to \\Omega$ satisfy both\u2026","title":"Scottish Book Problem 180 fails"},{"data_url":"scopes/Probability/random-fields.json","id":"Probability/random-fields","kind":"scope","parent":19384,"result":false,"route":"../roadmap/Probability/random-fields/index.html","scope":"Probability/random-fields","status":"fully_proved","summary":"Spin and bond configuration spaces, Bernoulli product measures and cylinder events, Gibbs-type conditional distributions given the outside of a region, translation invariance and ergodicity, weak and monotone limits of measures on configuration space, and marginals and conditional independence for weights on finite configuration spaces.","title":"Random fields on lattices"},{"data_url":"scopes/Probability/random-fields/boundary-conditions-for-bond-configurations.json","id":"Probability/random-fields/boundary-conditions-for-bond-configurations","kind":"scope","navigation":"module","parent":22227,"result":false,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","title":"Boundary conditions for bond configurations"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/boundary-condition","kind":"node","lean":"StatMech.Lattice.BoundaryCondition","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#boundary-condition","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"Boundary conditions for a bond model on a finite graph.","title":"Boundary conditions for a bond model"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/is-open","kind":"node","lean":"StatMech.Lattice.IsOpen","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#is-open","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"The edge `{x, y}` of `G` is open in the configuration `\u03c9`.","title":"Open edge in a bond configuration"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/open-graph","kind":"node","lean":"StatMech.Lattice.openGraph","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#open-graph","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"The spanning subgraph of `G` consisting of the open edges of `\u03c9`.","title":"Open subgraph of a bond configuration"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/num-clusters-free","kind":"node","lean":"StatMech.Lattice.numClustersFree","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#num-clusters-free","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"The number of open clusters (connected components of the open graph) with free boundary conditions.","title":"Number of open clusters with free boundary conditions"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/wire-rel","kind":"node","lean":"StatMech.Lattice.wireRel","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#wire-rel","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"Both vertices lie on the boundary.","title":"Boundary wiring relation"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/boundary-clique-graph","kind":"node","lean":"StatMech.Lattice.boundaryCliqueGraph","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#boundary-clique-graph","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"The complete graph on the boundary vertices, used to wire the boundary together.","title":"Complete graph on the boundary vertices"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/wired-graph","kind":"node","lean":"StatMech.Lattice.wiredGraph","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#wired-graph","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"The wired open graph: the open edges of `\u03c9` together with an edge between every two distinct boundary vertices.","title":"Wired open graph"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/num-clusters-wired","kind":"node","lean":"StatMech.Lattice.numClustersWired","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#num-clusters-wired","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"The number of open clusters with wired boundary conditions: all boundary vertices lie in a single cluster.","title":"Number of open clusters with wired boundary conditions"},{"id":"Probability/random-fields/boundary-conditions-for-bond-configurations/num-clusters","kind":"node","lean":"StatMech.Lattice.numClusters","parent":22228,"result":true,"route":"../roadmap/Probability/random-fields/boundary-conditions-for-bond-configurations/index.html#num-clusters","scope":"Probability/random-fields/boundary-conditions-for-bond-configurations","status":"fully_proved","summary":"The number of open clusters under a boundary condition. The periodic case uses the free count: the periodic identification is built into the graph `G`.","title":"Number of open clusters under a boundary condition"},{"data_url":"scopes/Probability/random-fields/conditional-independence-on-finite-configuration.json","id":"Probability/random-fields/conditional-independence-on-finite-configuration","kind":"scope","navigation":"module","parent":22227,"result":false,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","title":"Conditional independence on finite configuration spaces"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/marginal","kind":"node","lean":"MetaMathlibExt.FiniteWeight.marginal","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#marginal","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, a configuration being a map $x : V \\to \\alpha$, and let $p$ be a real-valued weight on configurations. The (unnormalized) marginal of $p$ on a set of variables $U \\subseteq V$ at a configuration $x$ is","title":"Marginal of a weight"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/marginal-anti","kind":"node","lean":"MetaMathlibExt.FiniteWeight.marginal_anti","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#marginal-anti","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, let $p$ be a real-valued weight on the configurations $x : V \\to \\alpha$ with $p(x) \\ge 0$ for every $x$, and let $U \\subseteq U' \\subseteq V$. Then for every configuration $x$, the marginal of $p$ on $U'$ at $x$ is at most the marginal of $p$ on $U$ at $x$.","title":"Monotonicity of marginals in the fixed variables"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/cond-indep","kind":"node","lean":"MetaMathlibExt.FiniteWeight.CondIndep","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#cond-indep","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, let $p$ be a real-valued weight on the configurations $x : V \\to \\alpha$, and for $U \\subseteq V$ write $p(x_U)$ for the marginal of $p$ on $U$ at $x$, the total weight of the configurations that agree with $x$ on $U$. For $A, B, S \\subseteq V$, the variables in $A$ and in $B$ are conditionally independent given those in $S$ under $p$ when","title":"Conditional independence for a finite weight"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/cond-eq-mul","kind":"node","lean":"MetaMathlibExt.FiniteWeight.CondIndep.cond_eq_mul","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#cond-eq-mul","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, let $p$ be a real-valued weight on the configurations $x : V \\to \\alpha$, and for $U \\subseteq V$ write $p(x_U)$ for the marginal of $p$ on $U$ at $x$. If, for $A, B, S \\subseteq V$, the variables in $A$ and in $B$ are conditionally independent given those in $S$ under $p$, then every configuration $x$ with $p(x_S) \\ne 0$ satisfies","title":"Conditional independence in ratio form"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/marginal-mul","kind":"node","lean":"MetaMathlibExt.FiniteWeight.marginal_mul","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#marginal-mul","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, let $f$ and $g$ be real-valued weights on the configurations $x : V \\to \\alpha$, and let $P, Q, U \\subseteq V$ with $P \\cup Q = V$ and $P \\cap Q \\subseteq U$, such that $f(x)$ depends only on the values of $x$ on $P$ and $g(x)$ only on its values on $Q$. Write $fg$ for the pointwise product, $P^c$ and $Q^c$ for the complements in $V$, and, for a weight $h$, a set\u2026","title":"Marginals of a product"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/cond-indep-of-depends-on","kind":"node","lean":"MetaMathlibExt.FiniteWeight.condIndep_of_dependsOn","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#cond-indep-of-depends-on","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, let $f$ and $g$ be real-valued weights on the configurations $x : V \\to \\alpha$, and let $P, Q \\subseteq V$ with $P \\cup Q = V$ such that $f(x)$ depends only on the values of $x$ on $P$ and $g(x)$ only on its values on $Q$. If $A \\subseteq V$ is disjoint from $Q$ and $B \\subseteq V$ is disjoint from $P$, then the variables in $A$ and in $B$ are conditionally independent given\u2026","title":"The factorization criterion for conditional independence"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/cond-indep-of-sdiff","kind":"node","lean":"MetaMathlibExt.FiniteWeight.condIndep_of_sdiff","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#cond-indep-of-sdiff","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, let $p$ be a real-valued weight on the configurations $x : V \\to \\alpha$, and let $A, B, S \\subseteq V$. If the variables in $A \\setminus S$ and in $B \\setminus S$ are conditionally independent given those in $S$ under $p$, then the variables in $A$ and in $B$ are conditionally independent given those in $S$ under $p$.","title":"Conditional independence from the variables outside the conditioning set"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/card-filter-agree","kind":"node","lean":"MetaMathlibExt.FiniteWeight.card_filter_agree","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#card-filter-agree","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types and $W \\subseteq V$. For any two configurations $x, z : V \\to \\alpha$, the number of configurations $y : V \\to \\alpha$ with $y(v) = z(v)$ for all $v \\in W$ equals the number of configurations $y$ with $y(v) = x(v)$ for all $v \\in W$.","title":"Agreement classes of configurations have equal size"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/marginal-marginal","kind":"node","lean":"MetaMathlibExt.FiniteWeight.marginal_marginal","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#marginal-marginal","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, let $p$ be a real-valued weight on the configurations $x : V \\to \\alpha$, let $U \\subseteq W \\subseteq V$, and let $p_W$ be the $W$-marginal of $p$, the weight sending $x$ to the total weight of the configurations that agree with $x$ on $W$. Then for every configuration $x$, the $U$-marginal of $p_W$ at $x$ equals $N$ times the $U$-marginal of $p$ at $x$, where $N$ is the\u2026","title":"Marginalizing in two steps"},{"id":"Probability/random-fields/conditional-independence-on-finite-configuration/cond-indep-marginal-iff","kind":"node","lean":"MetaMathlibExt.FiniteWeight.condIndep_marginal_iff","parent":22238,"result":true,"route":"../roadmap/Probability/random-fields/conditional-independence-on-finite-configuration/index.html#cond-indep-marginal-iff","scope":"Probability/random-fields/conditional-independence-on-finite-configuration","status":"fully_proved","summary":"Let $V$ and $\\alpha$ be finite types, let $p$ be a real-valued weight on the configurations $x : V \\to \\alpha$, let $W \\subseteq V$, and let $p_W$ be the $W$-marginal of $p$, the weight sending $x$ to the total weight of the configurations that agree with $x$ on $W$. For all $A, B, S \\subseteq W$, the variables in $A$ and in $B$ are conditionally independent given those in $S$ under $p_W$ if and only if they are\u2026","title":"Conditional independence can be read off a marginal"},{"data_url":"scopes/Probability/random-fields/spin-and-bond-configuration-spaces.json","id":"Probability/random-fields/spin-and-bond-configuration-spaces","kind":"scope","navigation":"module","parent":22227,"result":false,"route":"../roadmap/Probability/random-fields/spin-and-bond-configuration-spaces/index.html","scope":"Probability/random-fields/spin-and-bond-configuration-spaces","status":"fully_proved","title":"Spin and bond configuration spaces"},{"id":"Probability/random-fields/spin-and-bond-configuration-spaces/config-space","kind":"node","lean":"StatMech.ConfigSpace","parent":22249,"result":true,"route":"../roadmap/Probability/random-fields/spin-and-bond-configuration-spaces/index.html#config-space","scope":"Probability/random-fields/spin-and-bond-configuration-spaces","status":"fully_proved","summary":"The configuration space `E \u2192 Bool` of a statistical-mechanics model with sites (or bonds) indexed by `E`, carrying the product topology and the product \u03c3-algebra.","title":"Configuration space of a statistical-mechanics model"},{"id":"Probability/random-fields/spin-and-bond-configuration-spaces/eval","kind":"node","lean":"StatMech.ConfigSpace.eval","parent":22249,"result":true,"route":"../roadmap/Probability/random-fields/spin-and-bond-configuration-spaces/index.html#eval","scope":"Probability/random-fields/spin-and-bond-configuration-spaces","status":"fully_proved","summary":"The coordinate projection `\u03c9 \u21a6 \u03c9 e` of a configuration at the site `e`.","title":"Coordinate projection of a configuration"},{"id":"Probability/random-fields/spin-and-bond-configuration-spaces/measurable-space-eq-i-sup-comap","kind":"node","lean":"StatMech.ConfigSpace.measurableSpace_eq_iSup_comap","parent":22249,"result":true,"route":"../roadmap/Probability/random-fields/spin-and-bond-configuration-spaces/index.html#measurable-space-eq-i-sup-comap","scope":"Probability/random-fields/spin-and-bond-configuration-spaces","status":"fully_proved","summary":"The product \u03c3-algebra on `ConfigSpace E` is generated by the coordinate projections; this holds by definition of `MeasurableSpace.pi`.","title":"Product \u03c3-algebra on configuration space is generated by the coordinates"},{"data_url":"scopes/Probability/random-fields/bernoulli-product-measures-and-cylinder-events.json","id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","kind":"scope","navigation":"module","parent":22227,"result":false,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","title":"Bernoulli product measures and cylinder events"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/bernoulli-measure","kind":"node","lean":"StatMech.bernoulliMeasure","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#bernoulli-measure","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The Bernoulli distribution on `Bool` with success probability `p`: mass `p` on `true` and `1 - p` on `false`. This is Mathlib's `ProbabilityTheory.bernoulliMeasure true false p`, with the parameter given as an `\u211d\u22650` bounded by `1`.","title":"Bernoulli distribution on the Booleans"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/bernoulli-measure-eq","kind":"node","lean":"StatMech.bernoulliMeasure_eq","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#bernoulli-measure-eq","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"`bernoulliMeasure p hp` is Mathlib's `ProbabilityTheory.bernoulliMeasure true false p`.","title":"Bernoulli distribution agrees with the library Bernoulli measure"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/bernoulli-product-measure","kind":"node","lean":"StatMech.bernoulliProductMeasure","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#bernoulli-product-measure","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The Bernoulli product measure on `ConfigSpace E`: the coordinates are independent and each is `true` with probability `p` (`Measure.infinitePi` of `bernoulliMeasure p hp`). See also Mathlib's `ProbabilityTheory.setBernoulli`, the same product measure on `Set E`.","title":"Bernoulli product measure on configuration space"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/bernoulli-product-measure-eq-gbernoulli","kind":"node","lean":"StatMech.bernoulliProductMeasure_eq_gbernoulliLaw","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#bernoulli-product-measure-eq-gbernoulli","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"On bond configurations `Sym2 V \u2192 Bool`, `bernoulliProductMeasure` is the bond percolation law `MathlibExt.Probability.ProductBondPercolation.gbernoulliLaw`.","title":"Bernoulli product measure on bonds is the bond percolation law"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/apply-singleton","kind":"node","lean":"StatMech.bernoulliProductMeasure.apply_singleton","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#apply-singleton","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"For finite `E`, the mass of a configuration is the product of its coordinate masses.","title":"Mass of a configuration under a finite Bernoulli product measure"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/real-singleton","kind":"node","lean":"StatMech.bernoulliProductMeasure.real_singleton","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#real-singleton","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"For finite `E`, the mass of a configuration is the product of its coordinate masses.","title":"Mass of a configuration under a finite Bernoulli product measure, real form"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/real-eq-sum","kind":"node","lean":"StatMech.bernoulliProductMeasure.real_eq_sum","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#real-eq-sum","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"For finite `E`, the probability of an event is the sum of the masses of its configurations.","title":"Probability of an event as a sum of configuration masses"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/real-singleton-nonneg","kind":"node","lean":"StatMech.bernoulliProductMeasure.real_singleton_nonneg","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#real-singleton-nonneg","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"Configuration masses are nonnegative.","title":"Nonnegativity of configuration masses"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/sum-real-singleton","kind":"node","lean":"StatMech.bernoulliProductMeasure.sum_real_singleton","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#sum-real-singleton","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"For finite `E`, the configuration masses sum to `1`.","title":"Configuration masses sum to one"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/real-singleton-log-modular","kind":"node","lean":"StatMech.bernoulliProductMeasure.real_singleton_logModular","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#real-singleton-log-modular","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The configuration weights of a Bernoulli product measure are log-modular: `w(a) w(b) = w(a \u2294 b) w(a \u2293 b)`. In particular they satisfy the FKG lattice condition.","title":"Log-modularity of Bernoulli product weights"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/bernoulli-measure-real-singleton","kind":"node","lean":"StatMech.bernoulliMeasure_real_singleton","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#bernoulli-measure-real-singleton","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The mass of a point under the Bernoulli distribution: `p` at `true`, `1 - p` at `false`.","title":"Point masses of the Bernoulli distribution"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/real-singleton-eq-config-weight","kind":"node","lean":"StatMech.realSingleton_eq_configWeight","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#real-singleton-eq-config-weight","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"On a finite index set, the mass of a configuration is its product weight `configWeight`.","title":"Mass of a configuration on a finite index set is its product weight"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/finite-real-prob-eq-prob","kind":"node","lean":"StatMech.finiteRealProb_eq_prob","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#finite-real-prob-eq-prob","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"On a finite index set, the Bernoulli product measure of an event is its weight `prob p`.","title":"Bernoulli product measure of an event on a finite index set"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/extend-off","kind":"node","lean":"StatMech.extendOff","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#extend-off","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The configuration that agrees with `\u03b7` on the finite set `F` and with `\u03c9` off `F`.","title":"Configuration patched from one configuration on a finite set and another off it"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/eq-cylinder-restrict-image","kind":"node","lean":"StatMech.eq_cylinder_restrict_image","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#eq-cylinder-restrict-image","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"An event whose indicator depends only on the coordinates in `F` is the cylinder over its projection onto `F`.","title":"Events depending on finitely many coordinates are cylinders"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/bernoulli-product-measure-cylinder","kind":"node","lean":"StatMech.bernoulliProductMeasure_cylinder","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#bernoulli-product-measure-cylinder","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The infinite Bernoulli product measure of a cylinder over `F` is the finite Bernoulli product measure of its base.","title":"Bernoulli product measure of a cylinder event"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/real-prob-cylinder","kind":"node","lean":"StatMech.realProb_cylinder","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#real-prob-cylinder","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"Real-valued form of `bernoulliProductMeasure_cylinder`.","title":"Bernoulli product measure of a cylinder event, real-valued form"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/real-prob-cylinder-eq-prob","kind":"node","lean":"StatMech.realProb_cylinder_eq_prob","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#real-prob-cylinder-eq-prob","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The probability of an event depending only on the coordinates in `F` is the finite weight `prob p` of its projection onto `F`.","title":"Probability of a finitely dependent event as a finite weight"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/real-prob-cylinder-eq-finsum","kind":"node","lean":"StatMech.realProb_cylinder_eq_finsum","parent":22253,"reading_title":{"format":"inline-parts-v1","parts":[["text","Probability of a finitely dependent event as a polynomial in "],["math","\\(p\\)"]],"source":"Probability of a finitely dependent event as a polynomial in $p$"},"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#real-prob-cylinder-eq-finsum","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The probability of an event depending only on the coordinates in `F`, as an explicit polynomial in `p`.","title":"Probability of a finitely dependent event as a polynomial in $p$"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/differentiable-real-prob-cylinder","kind":"node","lean":"StatMech.differentiable_realProb_cylinder","parent":22253,"reading_title":{"format":"inline-parts-v1","parts":[["text","Differentiability in "],["math","\\(p\\)"],["text"," of the probability of a cylinder event"]],"source":"Differentiability in $p$ of the probability of a cylinder event"},"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#differentiable-real-prob-cylinder","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The weight of the projection of an event onto finitely many coordinates is differentiable in `p`.","title":"Differentiability in $p$ of the probability of a cylinder event"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/is-increasing-restrict-image","kind":"node","lean":"StatMech.isIncreasing_restrict_image","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#is-increasing-restrict-image","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"The projection of an increasing event onto finitely many coordinates is increasing.","title":"Finite projections of increasing events are increasing"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/has-deriv-at-real-prob-cylinder-pivotal","kind":"node","lean":"StatMech.hasDerivAt_realProb_cylinder_pivotal","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#has-deriv-at-real-prob-cylinder-pivotal","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"Russo's formula for an increasing event `A` projected onto finitely many coordinates `F`: the derivative in `p` of its weight is the sum over `e \u2208 F` of the weight of the event that `e` is pivotal (L. Russo, 1981).","title":"Russo's formula"},{"id":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events/deriv-real-prob-cylinder-pivotal","kind":"node","lean":"StatMech.deriv_realProb_cylinder_pivotal","parent":22253,"result":true,"route":"../roadmap/Probability/random-fields/bernoulli-product-measures-and-cylinder-events/index.html#deriv-real-prob-cylinder-pivotal","scope":"Probability/random-fields/bernoulli-product-measures-and-cylinder-events","status":"fully_proved","summary":"Russo's formula, `deriv` form of `hasDerivAt_realProb_cylinder_pivotal`.","title":"Russo's formula"},{"data_url":"scopes/Probability/random-fields/gibbs-conditional-distributions.json","id":"Probability/random-fields/gibbs-conditional-distributions","kind":"scope","navigation":"module","parent":22227,"result":false,"route":"../roadmap/Probability/random-fields/gibbs-conditional-distributions/index.html","scope":"Probability/random-fields/gibbs-conditional-distributions","status":"fully_proved","title":"Gibbs conditional distributions"},{"id":"Probability/random-fields/gibbs-conditional-distributions/outside-sigma","kind":"node","lean":"StatMech.outsideSigma","parent":22277,"result":true,"route":"../roadmap/Probability/random-fields/gibbs-conditional-distributions/index.html#outside-sigma","scope":"Probability/random-fields/gibbs-conditional-distributions","status":"fully_proved","summary":"The \u03c3-algebra on `ConfigSpace E` generated by the coordinates outside `\u039b`: the information about a configuration that is visible from outside the region `\u039b`.","title":"\u03c3-algebra of the configuration outside a region"},{"id":"Probability/random-fields/gibbs-conditional-distributions/outside-sigma-le","kind":"node","lean":"StatMech.outsideSigma_le","parent":22277,"result":true,"route":"../roadmap/Probability/random-fields/gibbs-conditional-distributions/index.html#outside-sigma-le","scope":"Probability/random-fields/gibbs-conditional-distributions","status":"fully_proved","summary":"The outside \u03c3-algebra is a sub-\u03c3-algebra of the product \u03c3-algebra.","title":"Outside \u03c3-algebra is a sub-\u03c3-algebra of the product \u03c3-algebra"},{"id":"Probability/random-fields/gibbs-conditional-distributions/gibbs-conditional","kind":"node","lean":"StatMech.gibbsConditional","parent":22277,"result":true,"route":"../roadmap/Probability/random-fields/gibbs-conditional-distributions/index.html#gibbs-conditional","scope":"Probability/random-fields/gibbs-conditional-distributions","status":"fully_proved","summary":"The conditional distribution of a finite measure `\u03bc` on `ConfigSpace E` given the configuration outside `\u039b`: a regular version of `\u03bc(\u00b7 | outsideSigma \u039b)`, namely Mathlib's `condExpKernel \u03bc (outsideSigma \u039b)`. For a Gibbs measure this is the Gibbs specification in `\u039b` (DLR condition).","title":"Conditional distribution given the outside of a region"},{"id":"Probability/random-fields/gibbs-conditional-distributions/gibbs-conditional-comp-trim","kind":"node","lean":"StatMech.gibbsConditional_comp_trim","parent":22277,"result":true,"route":"../roadmap/Probability/random-fields/gibbs-conditional-distributions/index.html#gibbs-conditional-comp-trim","scope":"Probability/random-fields/gibbs-conditional-distributions","status":"fully_proved","summary":"Disintegration: averaging the conditional distribution over `\u03bc` gives back `\u03bc`.","title":"Averaging the outside conditional distribution recovers the measure"},{"id":"Probability/random-fields/gibbs-conditional-distributions/gibbs-conditional-comp-prod-trim","kind":"node","lean":"StatMech.gibbsConditional_compProd_trim","parent":22277,"result":true,"route":"../roadmap/Probability/random-fields/gibbs-conditional-distributions/index.html#gibbs-conditional-comp-prod-trim","scope":"Probability/random-fields/gibbs-conditional-distributions","status":"fully_proved","summary":"Disintegration in product form: `\u03bc.trim \u2297\u2098 gibbsConditional \u03bc \u039b` is the law of `(\u03c9, \u03c9)`.","title":"Disintegration in product form for the outside conditional distribution"},{"id":"Probability/random-fields/gibbs-conditional-distributions/gibbs-conditional-ae-eq-cond-exp","kind":"node","lean":"StatMech.gibbsConditional_ae_eq_condExp","parent":22277,"result":true,"route":"../roadmap/Probability/random-fields/gibbs-conditional-distributions/index.html#gibbs-conditional-ae-eq-cond-exp","scope":"Probability/random-fields/gibbs-conditional-distributions","status":"fully_proved","summary":"`gibbsConditional \u03bc \u039b` is a version of the conditional probability `\u03bc\u27e6s | outsideSigma \u039b\u27e7`.","title":"Outside conditional distribution is a version of the conditional probability"},{"data_url":"scopes/Probability/random-fields/translation-invariance-and-ergodicity-of-random.json","id":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","kind":"scope","navigation":"module","parent":22227,"result":false,"route":"../roadmap/Probability/random-fields/translation-invariance-and-ergodicity-of-random/index.html","scope":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","status":"fully_proved","title":"Translation invariance and ergodicity of random fields"},{"id":"Probability/random-fields/translation-invariance-and-ergodicity-of-random/shift","kind":"node","lean":"StatMech.ConfigSpace.shift","parent":22284,"result":true,"route":"../roadmap/Probability/random-fields/translation-invariance-and-ergodicity-of-random/index.html#shift","scope":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","status":"fully_proved","summary":"The shift of a configuration by `g : G`: `(shift g \u03c9) e = \u03c9 (g\u207b\u00b9 \u2022 e)`. This is the action `arrowAction` of Mathlib on functions out of `E`.","title":"Shift of a configuration by a group element"},{"id":"Probability/random-fields/translation-invariance-and-ergodicity-of-random/shift-eq-arrow-action-smul","kind":"node","lean":"StatMech.ConfigSpace.shift_eq_arrowAction_smul","parent":22284,"result":true,"route":"../roadmap/Probability/random-fields/translation-invariance-and-ergodicity-of-random/index.html#shift-eq-arrow-action-smul","scope":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","status":"fully_proved","summary":"`shift` is Mathlib's `arrowAction` of `G` on functions `E \u2192 Bool`.","title":"Configuration shift is the action on functions"},{"id":"Probability/random-fields/translation-invariance-and-ergodicity-of-random/is-translation-invariant","kind":"node","lean":"StatMech.ConfigSpace.IsTranslationInvariant","parent":22284,"result":true,"route":"../roadmap/Probability/random-fields/translation-invariance-and-ergodicity-of-random/index.html#is-translation-invariant","scope":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","status":"fully_proved","summary":"A measure on `ConfigSpace E` is translation invariant if every shift `shift g`, `g : G`, preserves it.","title":"Translation-invariant measure on configuration space"},{"id":"Probability/random-fields/translation-invariance-and-ergodicity-of-random/is-ergodic","kind":"node","lean":"StatMech.ConfigSpace.IsErgodic","parent":22284,"result":true,"route":"../roadmap/Probability/random-fields/translation-invariance-and-ergodicity-of-random/index.html#is-ergodic","scope":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","status":"fully_proved","summary":"A measure on `ConfigSpace E` is ergodic for the shift action of `G` if it is translation invariant and every measurable set with `shift g \u207b\u00b9' s = s` for all `g` has measure `0` or `\u03bc univ`. See also Mathlib's `ErgodicSMul`, which asks the same for a.e.-invariant sets.","title":"Ergodic measure on configuration space"},{"id":"Probability/random-fields/translation-invariance-and-ergodicity-of-random/bernoulli-i-indep-fun","kind":"node","lean":"StatMech.FK.bernoulli_iIndepFun","parent":22284,"result":true,"route":"../roadmap/Probability/random-fields/translation-invariance-and-ergodicity-of-random/index.html#bernoulli-i-indep-fun","scope":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","status":"fully_proved","summary":"Coordinates are independent under the Bernoulli product measure. See also Mathlib's `ProbabilityTheory.iIndepFun_infinitePi`, of which this is a special case.","title":"Independence of coordinates under the Bernoulli product measure"},{"id":"Probability/random-fields/translation-invariance-and-ergodicity-of-random/bernoulli-ergodic","kind":"node","lean":"StatMech.FK.bernoulli_ergodic","parent":22284,"result":true,"route":"../roadmap/Probability/random-fields/translation-invariance-and-ergodicity-of-random/index.html#bernoulli-ergodic","scope":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","status":"fully_proved","summary":"Invariant events of the Bernoulli product measure have probability `0` or `1`.","title":"Bernoulli zero-one law for infinite groups acting finite-to-one"},{"id":"Probability/random-fields/translation-invariance-and-ergodicity-of-random/bernoulli-is-ergodic","kind":"node","lean":"StatMech.FK.bernoulli_isErgodic","parent":22284,"result":true,"route":"../roadmap/Probability/random-fields/translation-invariance-and-ergodicity-of-random/index.html#bernoulli-is-ergodic","scope":"Probability/random-fields/translation-invariance-and-ergodicity-of-random","status":"fully_proved","summary":"The Bernoulli product measure is ergodic under every action of an infinite group with injective orbit maps.","title":"Ergodicity of Bernoulli measures under free infinite-group actions"},{"data_url":"scopes/Probability/random-fields/weak-and-monotone-limits-on-configuration-space.json","id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","kind":"scope","navigation":"module","parent":22227,"result":false,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","title":"Weak and monotone limits on configuration space"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/weak-converges-to","kind":"node","lean":"StatMech.WeakConvergesTo","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#weak-converges-to","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"Weak convergence of a sequence of probability measures on `ConfigSpace E`: convergence in the topology of `ProbabilityMeasure (ConfigSpace E)`, whose open sets are generated by the integrals of bounded continuous functions.","title":"Weak convergence of probability measures on configuration space"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/tendsto-integral","kind":"node","lean":"StatMech.WeakConvergesTo.tendsto_integral","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#tendsto-integral","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"Under weak convergence, the integral of every bounded continuous function converges.","title":"Weak convergence implies convergence of integrals"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/weak-converges-to-of-forall-tendsto-inte","kind":"node","lean":"StatMech.weakConvergesTo_of_forall_tendsto_integral","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#weak-converges-to-of-forall-tendsto-inte","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"If the integral of every bounded continuous function converges, the measures converge weakly.","title":"Convergence of integrals implies weak convergence"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/is-clopen-cylinder-event","kind":"node","lean":"StatMech.isClopen_cylinderEvent","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#is-clopen-cylinder-event","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"A cylinder event of `ConfigSpace E`, determined by finitely many coordinates, is clopen.","title":"Cylinder events are clopen"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/tendsto-real-of-is-clopen","kind":"node","lean":"StatMech.WeakConvergesTo.tendsto_real_of_isClopen","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#tendsto-real-of-is-clopen","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"Under weak convergence on `ConfigSpace E`, the probability of every clopen event converges (portmanteau theorem).","title":"Portmanteau theorem for clopen events"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/monotone-real-of-chain","kind":"node","lean":"StatMech.monotone_real_of_chain","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#monotone-real-of-chain","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"Along a chain `\u03bc\u2080 \u227c \u03bc\u2081 \u227c \u22ef`, the probability of an increasing event is monotone.","title":"Increasing-event probabilities along an increasing chain are monotone"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/antitone-real-of-chain","kind":"node","lean":"StatMech.antitone_real_of_chain","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#antitone-real-of-chain","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"Along a chain `\u22ef \u227c \u03bc\u2081 \u227c \u03bc\u2080`, the probability of an increasing event is antitone.","title":"Increasing-event probabilities along a decreasing chain are antitone"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/le-real-of-monotone-is-increasing","kind":"node","lean":"StatMech.WeakConvergesTo.le_real_of_monotone_isIncreasing","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#le-real-of-monotone-is-increasing","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"The weak limit of an increasing chain gives each increasing clopen event at least the probability any term of the chain gives it.","title":"Weak limit of an increasing chain dominates its terms"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/ge-real-of-antitone-is-increasing","kind":"node","lean":"StatMech.WeakConvergesTo.ge_real_of_antitone_isIncreasing","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#ge-real-of-antitone-is-increasing","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"The weak limit of a decreasing chain gives each increasing clopen event at most the probability any term of the chain gives it.","title":"Weak limit of a decreasing chain is dominated by its terms"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/le-real-of-weak-limit","kind":"node","lean":"StatMech.le_real_of_weakLimit","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#le-real-of-weak-limit","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"A lower bound on the probabilities of a clopen event along a weakly convergent sequence passes to the limit.","title":"Lower bounds on clopen-event probabilities pass to weak limits"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/real-le-of-weak-limit","kind":"node","lean":"StatMech.real_le_of_weakLimit","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#real-le-of-weak-limit","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"An upper bound on the probabilities of a clopen event along a weakly convergent sequence passes to the limit.","title":"Upper bounds on clopen-event probabilities pass to weak limits"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/dominated-weak-limit-of-forall-dominated","kind":"node","lean":"StatMech.dominated_weakLimit_of_forall_dominated","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#dominated-weak-limit-of-forall-dominated","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"If `\u03c1 \u227c \u03bc\u2099` for all `n` and `\u03bc\u2099 \u2192 \u03bd` weakly, then `\u03c1(A) \u2264 \u03bd(A)` for every increasing clopen event `A`.","title":"Stochastic domination from below passes to weak limits"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/weak-limit-dominated-of-forall-dominated","kind":"node","lean":"StatMech.weakLimit_dominated_of_forall_dominated","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#weak-limit-dominated-of-forall-dominated","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"If `\u03bc\u2099 \u227c \u03c1` for all `n` and `\u03bc\u2099 \u2192 \u03bd` weakly, then `\u03bd(A) \u2264 \u03c1(A)` for every increasing clopen event `A`.","title":"Stochastic domination from above passes to weak limits"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/ext-of-forall-is-clopen","kind":"node","lean":"StatMech.ProbabilityMeasure.ext_of_forall_isClopen","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#ext-of-forall-is-clopen","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"Two probability measures on `ConfigSpace E` that agree on clopen sets are equal: the clopen cylinder events form a generating \u03c0-system.","title":"Probability measures agreeing on clopen sets are equal"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/is-clopen-apply-eq-of-subseq-tendsto","kind":"node","lean":"StatMech.isClopen_apply_eq_of_subseq_tendsto","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#is-clopen-apply-eq-of-subseq-tendsto","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"If the probabilities of clopen events converge to `L`, every subsequential weak limit `\u03be` satisfies `\u03be(A) = L(A)` on clopen `A`.","title":"Subsequential weak limits are determined by clopen-event limits"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/tendsto-weakly-of-forall-is-clopen-tends","kind":"node","lean":"StatMech.tendsto_weakly_of_forall_isClopen_tendsto","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#tendsto-weakly-of-forall-is-clopen-tends","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"If the probability of every clopen event converges along a sequence of probability measures on `ConfigSpace E`, the sequence converges weakly: all subsequential limits (which exist by Prokhorov) agree on clopen sets, hence coincide.","title":"Convergence on clopen events implies weak convergence"},{"id":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space/tendsto-nnreal-of-monotone-is-increasing","kind":"node","lean":"StatMech.tendsto_NNReal_of_monotone_isIncreasing","parent":22292,"result":true,"route":"../roadmap/Probability/random-fields/weak-and-monotone-limits-on-configuration-space/index.html#tendsto-nnreal-of-monotone-is-increasing","scope":"Probability/random-fields/weak-and-monotone-limits-on-configuration-space","status":"fully_proved","summary":"Along a chain `\u03bc\u2080 \u227c \u03bc\u2081 \u227c \u22ef`, the probability of an increasing clopen event converges to its supremum.","title":"Convergence of increasing-event probabilities along an increasing chain"},{"area":"Analysis & Probability","data_url":"chapters/Dynamics.json","id":"Dynamics","kind":"scope","parent":0,"result":false,"route":"../roadmap/Dynamics/index.html","scope":"Dynamics","status":"can_prove","summary":"Dynamics studies systems evolving through iteration or time, including recurrence, stability, and long-term behavior.","title":"Dynamical Systems"},{"data_url":"scopes/Dynamics/ergodic-theory.json","id":"Dynamics/ergodic-theory","kind":"scope","parent":22310,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/index.html","scope":"Dynamics/ergodic-theory","status":"can_prove","summary":"Ergodic theory studies measure-preserving transformations and their statistical behaviour: ergodic averages, recurrence, mixing and the structure of measure-preserving actions.","title":"Ergodic theory"},{"id":"Dynamics/ergodic-theory/furstenberg-multiple-recurrence","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.MultipleRecurrenceWanted.furstenberg_multipleRecurrence","parent":22311,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/index.html#furstenberg-multiple-recurrence","scope":"Dynamics/ergodic-theory","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ be a measurable map preserving $\\mu$, and let $A$ be a measurable set with $\\mu(A) > 0$. For $m \\in \\mathbb{N}$ write $T^{-m}(A)$ for the preimage of $A$ under the $m$-th iterate of $T$. Then for every integer $k > 0$ there is an integer $n > 0$ such that","title":"Furstenberg multiple recurrence theorem"},{"id":"Dynamics/ergodic-theory/krylov-bogolyubov","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.KrylovBogolyubovWanted.krylov_bogolyubov","parent":22311,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/index.html#krylov-bogolyubov","scope":"Dynamics/ergodic-theory","status":"fully_proved","summary":"Every continuous map $T : X \\to X$ of a nonempty compact metric space $X$ to itself has an invariant Borel probability measure: there is a probability measure $\\mu$ on the Borel $\\sigma$-algebra of $X$ whose pushforward under $T$ equals $\\mu$.","title":"Krylov\u2013Bogolyubov theorem"},{"data_url":"scopes/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages.json","id":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","kind":"scope","navigation":"module","parent":22311,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/index.html","scope":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","status":"can_prove","title":"Bilinear polynomial ergodic averages"},{"id":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/lacunary","kind":"node","lean":"MathlibExt.Dynamics.PointwiseErgodicTheoremsWanted.PointwiseErgodicTheorems.Lacunary","parent":22314,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/index.html#lacunary","scope":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","status":"fully_proved","summary":"For a real number $\\Lambda \\ge 0$, a sequence $(a_n)_{n \\in \\mathbb{N}}$ of nonnegative real numbers is $\\Lambda$-lacunary if $a_{n+1} / a_n > \\Lambda$ for every $n \\in \\mathbb{N}$, where division by $0$ gives $0$; in particular every term of a $\\Lambda$-lacunary sequence is positive.","title":"Lacunary sequence"},{"id":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/variational-norm","kind":"node","lean":"MathlibExt.Dynamics.PointwiseErgodicTheoremsWanted.PointwiseErgodicTheorems.variationalNorm","parent":22314,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/index.html#variational-norm","scope":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","status":"fully_proved","summary":"For a real number $r \\ge 0$ and a sequence $a = (a_n)_{n \\in \\mathbb{N}}$ of complex numbers, the $r$-variational norm of $a$ is the element of $[0, \\infty]$","title":"Variational norm"},{"id":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/a","kind":"node","lean":"MathlibExt.Dynamics.PointwiseErgodicTheoremsWanted.PointwiseErgodicTheorems.A","parent":22314,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/index.html#a","scope":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","status":"fully_proved","summary":"Let $X$ be a measurable space and let $T : X \\to X$ be a measurable bijection with measurable inverse. For a real number $N \\ge 0$, polynomials $P_1, P_2 \\in \\mathbb{Z}[t]$, functions $f, g : X \\to \\mathbb{C}$ and a point $x \\in X$, the bilinear average is the complex number","title":"Bilinear polynomial ergodic averages"},{"id":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/theorem-1-17-i","kind":"node","lean":"MathlibExt.Dynamics.PointwiseErgodicTheoremsWanted.PointwiseErgodicTheorems.theorem_1_17_i","parent":22314,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/index.html#theorem-1-17-i","scope":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","status":"can_prove","summary":"Let $(X, \\mu)$ be a $\\sigma$-finite measure space and let $T : X \\to X$ be a measurable bijection with measurable inverse that preserves $\\mu$. Let $P \\in \\mathbb{Z}[t]$ have degree at least $2$, and let $p_1, p_2 \\in (1, \\infty)$ and $p \\in [0, \\infty]$ satisfy $p_1^{-1} + p_2^{-1} = p^{-1}$ and $p^{-1} \\le 1$. For a real number $N \\ge 0$, functions $f, g : X \\to \\mathbb{C}$ and $x \\in X$ let","title":"Mean ergodic theorem for bilinear polynomial averages"},{"id":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/theorem-1-17-ii","kind":"node","lean":"MathlibExt.Dynamics.PointwiseErgodicTheoremsWanted.PointwiseErgodicTheorems.theorem_1_17_ii","parent":22314,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/index.html#theorem-1-17-ii","scope":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","status":"can_prove","summary":"Let $(X, \\mu)$ be a $\\sigma$-finite measure space and let $T : X \\to X$ be a measurable bijection with measurable inverse that preserves $\\mu$. Let $P \\in \\mathbb{Z}[t]$ have degree at least $2$, and let $p_1, p_2 \\in (1, \\infty)$ and $p \\in [0, \\infty]$ satisfy $p_1^{-1} + p_2^{-1} = p^{-1}$ and $p^{-1} \\le 1$. For a real number $N \\ge 0$, functions $f, g : X \\to \\mathbb{C}$ and $x \\in X$ let","title":"Pointwise ergodic theorem for bilinear polynomial averages"},{"id":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/theorem-1-17-iii","kind":"node","lean":"MathlibExt.Dynamics.PointwiseErgodicTheoremsWanted.PointwiseErgodicTheorems.theorem_1_17_iii","parent":22314,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/index.html#theorem-1-17-iii","scope":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","status":"can_prove","summary":"Let $(X, \\mu)$ be a $\\sigma$-finite measure space and let $T : X \\to X$ be a measurable bijection with measurable inverse that preserves $\\mu$. Let $P \\in \\mathbb{Z}[t]$ have degree at least $2$, and let $p_1, p_2 \\in (1, \\infty)$ and $p \\in [0, \\infty]$ satisfy $p_1^{-1} + p_2^{-1} = p^{-1}$ and $p^{-1} \\le 1$. For a real number $N \\ge 0$, functions $f, g : X \\to \\mathbb{C}$ and $x \\in X$ let","title":"Maximal inequality for bilinear polynomial averages"},{"id":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/theorem-1-17-iv","kind":"node","lean":"MathlibExt.Dynamics.PointwiseErgodicTheoremsWanted.PointwiseErgodicTheorems.theorem_1_17_iv","parent":22314,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages/index.html#theorem-1-17-iv","scope":"Dynamics/ergodic-theory/bilinear-polynomial-ergodic-averages","status":"can_prove","summary":"Let $(X, \\mu)$ be a $\\sigma$-finite measure space and let $T : X \\to X$ be a measurable bijection with measurable inverse that preserves $\\mu$. Let $P \\in \\mathbb{Z}[t]$ have degree at least $2$, and let $p_1, p_2 \\in (1, \\infty)$ and $p \\in [0, \\infty]$ satisfy $p_1^{-1} + p_2^{-1} = p^{-1}$ and $p^{-1} \\le 1$. For a real number $N \\ge 0$, functions $f, g : X \\to \\mathbb{C}$ and $x \\in X$ let","title":"Long variational inequality for bilinear polynomial averages"},{"data_url":"scopes/Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem.json","id":"Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem","kind":"scope","navigation":"module","parent":22311,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem/index.html","scope":"Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem","status":"can_prove","title":"Bourgain's polynomial ergodic theorem"},{"id":"Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem/is-integer-polynomial-sequence","kind":"node","lean":"MathlibExt.Dynamics.BourgainErgodicWanted.Dynamics.BourgainErgodic.IsIntegerPolynomialSequence","parent":22322,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem/index.html#is-integer-polynomial-sequence","scope":"Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem","status":"fully_proved","summary":"A sequence $a : \\mathbb{N} \\to \\mathbb{N}$ is an integer polynomial sequence if there is a polynomial $P$ with integer coefficients such that $P(n) \\ge 0$ and $a(n) = P(n)$ for every $n \\in \\mathbb{N}$.","title":"Integer polynomial sequences"},{"id":"Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem/polynomial-ergodic-average","kind":"node","lean":"MathlibExt.Dynamics.BourgainErgodicWanted.Dynamics.BourgainErgodic.polynomialErgodicAverage","parent":22322,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem/index.html#polynomial-ergodic-average","scope":"Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem","status":"fully_proved","summary":"Given a sequence $a : \\mathbb{N} \\to \\mathbb{N}$, a map $T : X \\to X$, a function $f : X \\to \\mathbb{R}$, $n \\in \\mathbb{N}$ and $x \\in X$, the $n$-th polynomial ergodic average of $f$ at $x$ is","title":"Polynomial ergodic averages"},{"id":"Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem/bourgain-polynomial-ergodic","kind":"node","lean":"MathlibExt.Dynamics.BourgainErgodicWanted.Dynamics.BourgainErgodic.bourgain_polynomial_ergodic","parent":22322,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem/index.html#bourgain-polynomial-ergodic","scope":"Dynamics/ergodic-theory/bourgain-s-polynomial-ergodic-theorem","status":"can_prove","summary":"Let $(X, \\mu)$ be a probability space, let $T : X \\to X$ be a measure-preserving map, let $a : \\mathbb{N} \\to \\mathbb{N}$ be an integer polynomial sequence, and let $f : X \\to \\mathbb{R}$ be a function in $L^p(\\mu)$ for some $p$ with $1 < p < \\infty$. Then there is a function $F : X \\to \\mathbb{R}$ such that for $\\mu$-almost every $x \\in X$ the polynomial ergodic averages\u2026","title":"Bourgain's polynomial ergodic theorem"},{"data_url":"scopes/Dynamics/ergodic-theory/genericity-of-weak-mixing.json","id":"Dynamics/ergodic-theory/genericity-of-weak-mixing","kind":"scope","navigation":"module","parent":22311,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/genericity-of-weak-mixing/index.html","scope":"Dynamics/ergodic-theory/genericity-of-weak-mixing","status":"can_prove","title":"Genericity of weak mixing"},{"id":"Dynamics/ergodic-theory/genericity-of-weak-mixing/automorphism","kind":"node","lean":"MathlibExt.Dynamics.HalmosGenericWeakMixingWanted.Dynamics.HalmosGenericWeakMixingProblem.Automorphism","parent":22326,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/genericity-of-weak-mixing/index.html#automorphism","scope":"Dynamics/ergodic-theory/genericity-of-weak-mixing","status":"fully_proved","summary":"An automorphism of a measure space $(X, m)$ is a bijection $e \\colon X \\to X$ such that $e$ and its inverse are measurable and $e$ preserves $m$, that is, $m(e^{-1}(A)) = m(A)$ for every measurable set $A \\subseteq X$.","title":"Measure-preserving automorphism"},{"id":"Dynamics/ergodic-theory/genericity-of-weak-mixing/is-weakly-mixing","kind":"node","lean":"MathlibExt.Dynamics.HalmosGenericWeakMixingWanted.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing","parent":22326,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/genericity-of-weak-mixing/index.html#is-weakly-mixing","scope":"Dynamics/ergodic-theory/genericity-of-weak-mixing","status":"fully_proved","summary":"Let $(X, m)$ be a measure space. An automorphism $T$ of $(X, m)$ is weakly mixing if for all measurable sets $A, B \\subseteq X$,","title":"Weakly mixing automorphism"},{"id":"Dynamics/ergodic-theory/genericity-of-weak-mixing/generic-weakly-mixing","kind":"node","lean":"MathlibExt.Dynamics.HalmosGenericWeakMixingWanted.Dynamics.HalmosGenericWeakMixingProblem.generic_weakly_mixing","parent":22326,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/genericity-of-weak-mixing/index.html#generic-weakly-mixing","scope":"Dynamics/ergodic-theory/genericity-of-weak-mixing","status":"can_prove","summary":"Let $X$ be a standard Borel space and $m$ a probability measure on $X$ for which every singleton has measure $0$. Give the set of automorphisms of $(X, m)$ the weak topology, the topology generated by the sets of automorphisms $T$ with $m(T(A) \\mathbin{\\triangle} T_0(A)) < \\varepsilon$, for each automorphism $T_0$, measurable set $A \\subseteq X$ and real number $\\varepsilon > 0$. Then there is a dense $G_\\delta$\u2026","title":"Halmos's theorem: weak mixing is generic"},{"data_url":"scopes/Dynamics/ergodic-theory/lax-approximation-theorem.json","id":"Dynamics/ergodic-theory/lax-approximation-theorem","kind":"scope","navigation":"module","parent":22311,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"can_prove","title":"Lax approximation theorem"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/torus","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.Torus","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#torus","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"fully_proved","summary":"For a natural number $d$, the $d$-dimensional torus is $\\mathbb{T}^d = (\\mathbb{R}/\\mathbb{Z})^d$, the product of $d$ copies of the circle $\\mathbb{R}/\\mathbb{Z}$.","title":"Torus"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/volume-preserving-equiv","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.VolumePreservingEquiv","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#volume-preserving-equiv","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"fully_proved","summary":"Let $d$ be a natural number. A volume-preserving transformation of the torus $\\mathbb{T}^d = (\\mathbb{R}/\\mathbb{Z})^d$ is a bijection $T$ of $\\mathbb{T}^d$ such that $T$ and $T^{-1}$ are measurable and $T$ preserves the volume measure.","title":"Volume-preserving transformations of the torus"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/delta-dist","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.deltaDist","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#delta-dist","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"fully_proved","summary":"Let $d$ be a natural number and let $T$ and $S$ be measurable, invertible, volume-preserving transformations of the torus $\\mathbb{T}^d = (\\mathbb{R}/\\mathbb{Z})^d$. Let $\\rho$ be the distance on $\\mathbb{T}^d$ given by the maximum over the coordinates of the distance on the circle $\\mathbb{R}/\\mathbb{Z}$ (taken to be $0$ when $d = 0$). The distance $\\delta(T, S) \\in [0, \\infty]$ is the essential supremum, with\u2026","title":"Essential-supremum distance between volume-preserving maps"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/toral-dynamical-system","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.ToralDynamicalSystem","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#toral-dynamical-system","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"fully_proved","summary":"Let $d$ be a natural number. A toral dynamical system on the torus $\\mathbb{T}^d = (\\mathbb{R}/\\mathbb{Z})^d$ is a homeomorphism of $\\mathbb{T}^d$ that preserves the volume measure.","title":"Toral dynamical system"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/to-volume-preserving-equiv","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.ToralDynamicalSystem.toVolumePreservingEquiv","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#to-volume-preserving-equiv","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"fully_proved","summary":"Let $d$ be a natural number. A volume-preserving homeomorphism $T$ of the torus $\\mathbb{T}^d = (\\mathbb{R}/\\mathbb{Z})^d$ is regarded as a measurable, invertible, volume-preserving transformation of $\\mathbb{T}^d$: the same map, with its inverse, viewed as a measurable equivalence.","title":"Toral dynamical systems as volume-preserving transformations"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/cube","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.cube","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#cube","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"fully_proved","summary":"Let $n$ and $d$ be natural numbers and let $k = (k_0, \\dots, k_{d-1})$ with each $k_i \\in \\{0, \\dots, n-1\\}$. The grid cube with index $k$ is the set of points $x$ of the torus $\\mathbb{T}^d = (\\mathbb{R}/\\mathbb{Z})^d$ such that, for every $i < d$, the coordinate $x_i$ is the class modulo $1$ of some real number $r$ with $k_i/n \\le r < (k_i + 1)/n$.","title":"Grid cube in the torus"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/cube-shift","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.cubeShift","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#cube-shift","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"fully_proved","summary":"Let $n$ and $d$ be natural numbers, let $\\sigma$ be a permutation of the index set $\\{0, \\dots, n-1\\}^d$, let $k$ be an index and let $i < d$. The shift along axis $i$ determined by $\\sigma$ and $k$ is the point $(\\sigma(k)_i - k_i)/n$ of the circle $\\mathbb{R}/\\mathbb{Z}$, that is, the class modulo $1$ of this real number.","title":"Coordinate shift between grid cubes"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/is-cyclic-cube-exchange","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.IsCyclicCubeExchange","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#is-cyclic-cube-exchange","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"fully_proved","summary":"Let $n$ and $d$ be natural numbers. For an index $k \\in \\{0, \\dots, n-1\\}^d$, the grid cube with index $k$ is the set of points of the torus $\\mathbb{T}^d = (\\mathbb{R}/\\mathbb{Z})^d$ whose $i$-th coordinate is the class of a real number in $[k_i/n, (k_i + 1)/n)$ for every $i < d$. A measurable, invertible, volume-preserving transformation $T$ of $\\mathbb{T}^d$ is a cyclic cube exchange with $n$ subdivisions if\u2026","title":"Cyclic cube exchange"},{"id":"Dynamics/ergodic-theory/lax-approximation-theorem/lax-approximation","kind":"node","lean":"MathlibExt.Dynamics.LaxApproximationWanted.Dynamics.LaxApproximation.lax_approximation","parent":22330,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/lax-approximation-theorem/index.html#lax-approximation","scope":"Dynamics/ergodic-theory/lax-approximation-theorem","status":"can_prove","summary":"Let $d \\ge 1$ be a natural number and let $T$ be a toral dynamical system on the torus $\\mathbb{T}^d = (\\mathbb{R}/\\mathbb{Z})^d$, regarded as a measurable, invertible, volume-preserving transformation of $\\mathbb{T}^d$. Then for every $\\varepsilon \\in (0, \\infty]$ there are a natural number $n$ and a measurable, invertible, volume-preserving transformation $S$ of $\\mathbb{T}^d$ such that $S$ is a cyclic cube\u2026","title":"Lax's approximation theorem"},{"data_url":"scopes/Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair.json","id":"Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair","kind":"scope","navigation":"module","parent":22311,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair/index.html","scope":"Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair","status":"can_prove","title":"Multiplicative recurrence along Pythagorean pairs"},{"id":"Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair/conjecture","kind":"node","lean":"MathlibExt.Dynamics.MultStructureRecurrenceWanted.conjecture","parent":22340,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair/index.html#conjecture","scope":"Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair","status":"fully_proved","summary":"The proposition that for every probability space $(X, \\mu)$ and every family $(T_a)_{a \\ge 1}$, indexed by the positive integers, of measurable bijections of $X$ with measurable inverses that preserve $\\mu$, such that $T_1$ is the identity and $T_{ab} = T_a \\circ T_b$ for all positive integers $a, b$, every measurable set $A \\subseteq X$ with $\\mu(A) > 0$ admits positive integers $m > n$ with","title":"Pythagorean-pair recurrence problem (AMR-047-0034)"},{"id":"Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair/conjecture-holds","kind":"node","lean":"MathlibExt.Dynamics.MultStructureRecurrenceWanted.conjecture_holds","parent":22340,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair/index.html#conjecture-holds","scope":"Dynamics/ergodic-theory/multiplicative-recurrence-along-pythagorean-pair","status":"can_prove","summary":"Let $(X, \\mu)$ be a probability space and let $(T_a)_{a \\ge 1}$, indexed by the positive integers, be measurable bijections of $X$ with measurable inverses that preserve $\\mu$, such that $T_1$ is the identity and $T_{ab} = T_a \\circ T_b$ for all positive integers $a, b$. Then for every measurable set $A \\subseteq X$ with $\\mu(A) > 0$ there exist positive integers $m > n$ such that","title":"Multiplicative recurrence along Pythagorean pairs"},{"data_url":"scopes/Dynamics/ergodic-theory/rokhlin-lemma.json","id":"Dynamics/ergodic-theory/rokhlin-lemma","kind":"scope","navigation":"module","parent":22311,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"can_prove","title":"Rokhlin lemma"},{"id":"Dynamics/ergodic-theory/rokhlin-lemma/is-free-action","kind":"node","lean":"MathlibExt.Dynamics.OrnsteinWeissWanted.Dynamics.IsFreeAction","parent":22343,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(\\mathbb{Z}^d\\)"],["text","-action"]],"source":"Free $\\mathbb{Z}^d$-action"},"result":true,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html#is-free-action","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"fully_proved","summary":"Let $d$ be a natural number, let $\\mu$ be a measure on a measurable space $\\Omega$, and let $T$ assign to each $v \\in \\mathbb{Z}^d$ a map $T_v \\colon \\Omega \\to \\Omega$. Then $T$ is free (in the Ornstein\u2013Weiss sense) with respect to $\\mu$ if, for every nonzero $v \\in \\mathbb{Z}^d$, the set of points $x \\in \\Omega$ with $T_v(x) = x$ has $\\mu$-measure zero.","title":"Free $\\mathbb{Z}^d$-action"},{"id":"Dynamics/ergodic-theory/rokhlin-lemma/box-shape","kind":"node","lean":"MathlibExt.Dynamics.OrnsteinWeissWanted.Dynamics.boxShape","parent":22343,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html#box-shape","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"fully_proved","summary":"For natural numbers $d$ and $N$, the integer box $[0, N)^d$ is the finite set of vectors $v \\in \\mathbb{Z}^d$ with $0 \\le v_i < N$ for every coordinate $i$.","title":"Integer box"},{"id":"Dynamics/ergodic-theory/rokhlin-lemma/ornstein-weiss-rokhlin","kind":"node","lean":"MathlibExt.Dynamics.OrnsteinWeissWanted.Dynamics.ornstein_weiss_rokhlin","parent":22343,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ornstein\u2013Weiss Rokhlin lemma for "],["math","\\(\\mathbb{Z}^d\\)"],["text","-actions"]],"source":"Ornstein\u2013Weiss Rokhlin lemma for $\\mathbb{Z}^d$-actions"},"result":true,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html#ornstein-weiss-rokhlin","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"can_prove","summary":"Let $d \\ge 1$, let $\\Omega$ be a standard Borel space with a probability measure $\\mu$, and let $T$ assign to each $v \\in \\mathbb{Z}^d$ a map $T_v \\colon \\Omega \\to \\Omega$ such that $T_0$ is the identity, $T_{u + v} = T_u \\circ T_v$ for all $u, v \\in \\mathbb{Z}^d$, every $T_v$ is measure preserving for $\\mu$, and the action is free (for every $v \\ne 0$ the set of fixed points of $T_v$ has $\\mu$-measure zero). Then\u2026","title":"Ornstein\u2013Weiss Rokhlin lemma for $\\mathbb{Z}^d$-actions"},{"id":"Dynamics/ergodic-theory/rokhlin-lemma/is-aperiodic","kind":"node","lean":"MathlibExt.Dynamics.RokhlinLemmaWanted.Dynamics.IsAperiodic","parent":22343,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html#is-aperiodic","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\Omega$. A map $T : \\Omega \\to \\Omega$ is aperiodic with respect to $\\mu$ if the set of periodic points of $T$, the points $x$ with $T^k(x) = x$ for some $k \\ge 1$, where $T^k$ is the $k$-fold iterate of $T$, has (outer) $\\mu$-measure zero. Equivalently, for $\\mu$-almost every $x$, no positive iterate of $T$ fixes $x$.","title":"Aperiodic transformation"},{"id":"Dynamics/ergodic-theory/rokhlin-lemma/tower-floor","kind":"node","lean":"MathlibExt.Dynamics.RokhlinLemmaWanted.Dynamics.towerFloor","parent":22343,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html#tower-floor","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"fully_proved","summary":"For a map $T : \\Omega \\to \\Omega$, a set $B \\subseteq \\Omega$ and $k \\in \\mathbb{N}$, the level-$k$ floor with base $B$ is the image $T^k(B)$ of $B$ under the $k$-fold iterate $T^k$ of $T$; the level-$0$ floor is $B$.","title":"Floor of a Rokhlin tower"},{"id":"Dynamics/ergodic-theory/rokhlin-lemma/tower-union","kind":"node","lean":"MathlibExt.Dynamics.RokhlinLemmaWanted.Dynamics.towerUnion","parent":22343,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html#tower-union","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"fully_proved","summary":"For a map $T : \\Omega \\to \\Omega$, a set $B \\subseteq \\Omega$ and $n \\in \\mathbb{N}$, the union of the tower with base $B$ and height $n$ is the set $\\bigcup_{k=0}^{n-1} T^k(B)$ of its floors $B, T(B), \\dots, T^{n-1}(B)$, where $T^k(B)$ is the image of $B$ under the $k$-fold iterate of $T$. It is empty when $n = 0$, and no disjointness of the floors is required.","title":"Union of a Rokhlin tower"},{"id":"Dynamics/ergodic-theory/rokhlin-lemma/is-rokhlin-tower","kind":"node","lean":"MathlibExt.Dynamics.RokhlinLemmaWanted.Dynamics.IsRokhlinTower","parent":22343,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html#is-rokhlin-tower","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"fully_proved","summary":"Let $\\Omega$ be a measurable space, $T : \\Omega \\to \\Omega$ a map and $n \\in \\mathbb{N}$. A set $B \\subseteq \\Omega$ is the base of a Rokhlin tower of height $n$ for $T$ if $B$ is measurable and the floors $B, T(B), \\dots, T^{n-1}(B)$ are pairwise disjoint, that is, $T^i(B) \\cap T^j(B) = \\emptyset$ for all $i \\ne j$ with $i, j < n$; here $T^k(B)$ is the image of $B$ under the $k$-fold iterate of $T$.","title":"Rokhlin tower"},{"id":"Dynamics/ergodic-theory/rokhlin-lemma/rokhlin-lemma","kind":"node","lean":"MathlibExt.Dynamics.RokhlinLemmaWanted.Dynamics.rokhlin_lemma","parent":22343,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/rokhlin-lemma/index.html#rokhlin-lemma","scope":"Dynamics/ergodic-theory/rokhlin-lemma","status":"can_prove","summary":"Let $\\Omega$ be a standard Borel space with a probability measure $\\mu$, and let $T : \\Omega \\to \\Omega$ be a measure-preserving map that is aperiodic with respect to $\\mu$. Then for every integer $n \\ge 1$ and every $\\varepsilon \\in (0, \\infty]$ there is a measurable set $B \\subseteq \\Omega$ such that the floors $B, T(B), \\dots, T^{n-1}(B)$ are pairwise disjoint, where $T^k(B)$ is the image of $B$ under the\u2026","title":"Rokhlin lemma for non-invertible maps"},{"data_url":"scopes/Dynamics/ergodic-theory/birkhoff-ergodic-theorem.json","id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","kind":"scope","navigation":"module","parent":22311,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","title":"Birkhoff ergodic theorem"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/maximal-ergodic","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.maximal_ergodic","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#maximal-ergodic","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a finite measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ preserve $\\mu$, and let $f : \\alpha \\to \\mathbb{R}$ be measurable and $\\mu$-integrable. For a natural number $n$ write $S_n f(x) = \\sum_{k=0}^{n-1} f(T^k x)$. Then the integral of $f$ with respect to $\\mu$ over the set of points $x \\in \\alpha$ such that $S_n f(x) > 0$ for some natural number $n$ is nonnegative.","title":"Hopf maximal ergodic theorem"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/dick-set","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.DickSet","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#dick-set","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a measurable space, $T : \\alpha \\to \\alpha$ a map, $g : \\alpha \\to \\mathbb{R}$ a function and $c$ a real number. The maximal divergence set of $T$, $g$ and $c$ is the set of points $x \\in \\alpha$ for which there is a rational number $q > c$ such that $n q < \\sum_{k=0}^{n-1} g(T^k x)$ for infinitely many natural numbers $n$.","title":"Maximal divergence set"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/dick-measurable","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.dick_measurable","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#dick-measurable","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ preserve $\\mu$, and let $g : \\alpha \\to \\mathbb{R}$ be measurable. Then for every real number $c$, the maximal divergence set of $T$, $g$ and $c$ is a measurable set.","title":"Measurability of the maximal divergence set"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/dick-mem","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.dick_mem","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#dick-mem","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a measurable space, $T : \\alpha \\to \\alpha$ a map, $g : \\alpha \\to \\mathbb{R}$ a function, $c$ a real number and $x \\in \\alpha$. Then $x$ lies in the maximal divergence set of $T$, $g$ and $c$ if and only if there is a rational number $q > c$ such that for every natural number $N$ there is a natural number $n \\ge N$ with $n q < \\sum_{k=0}^{n-1} g(T^k x)$.","title":"Membership in the maximal divergence set"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/dick-inv","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.dick_inv","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#dick-inv","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a measurable space. For every map $T : \\alpha \\to \\alpha$, every function $g : \\alpha \\to \\mathbb{R}$ and every real number $c$, the preimage under $T$ of the maximal divergence set of $T$, $g$ and $c$ equals that set.","title":"Invariance of the maximal divergence set"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/dick-iter","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.dick_iter","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#dick-iter","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a measurable space, $T : \\alpha \\to \\alpha$ a map, $g : \\alpha \\to \\mathbb{R}$ a function and $c$ a real number. For every natural number $k$ and every $x \\in \\alpha$, the point $T^k x$ lies in the maximal divergence set of $T$, $g$ and $c$ if and only if $x$ does.","title":"Invariance of the maximal divergence set under iterates"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/dick-sum-in","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.dick_sum_in","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#dick-sum-in","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a measurable space, $T : \\alpha \\to \\alpha$ a map, $g : \\alpha \\to \\mathbb{R}$ a function, $c$ a real number, and $D \\subseteq \\alpha$ a set such that for every natural number $k$ and every $x \\in \\alpha$, $T^k x \\in D$ if and only if $x \\in D$. Let $h : \\alpha \\to \\mathbb{R}$ satisfy $h(y) = g(y) - c$ for $y \\in D$ and $h(y) = 0$ for $y \\notin D$. Then for every $x \\in D$ and every natural number\u2026","title":"Birkhoff sums of a function cut off to an invariant set, inside the set"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/dick-sum-out","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.dick_sum_out","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#dick-sum-out","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a measurable space, $T : \\alpha \\to \\alpha$ a map, $g : \\alpha \\to \\mathbb{R}$ a function, $c$ a real number, and $D \\subseteq \\alpha$ a set such that for every natural number $k$ and every $x \\in \\alpha$, $T^k x \\in D$ if and only if $x \\in D$. Let $h : \\alpha \\to \\mathbb{R}$ satisfy $h(y) = g(y) - c$ for $y \\in D$ and $h(y) = 0$ for $y \\notin D$. Then for every $x \\notin D$ and every natural\u2026","title":"Birkhoff sums of a function cut off to an invariant set, outside the set"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/birkhoff-pointwise-ergodic","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.BirkhoffWanted.birkhoff_pointwise_ergodic","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#birkhoff-pointwise-ergodic","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ be ergodic with respect to $\\mu$ (in particular $T$ preserves $\\mu$), and let $g : \\alpha \\to \\mathbb{R}$ be $\\mu$-integrable. Then for $\\mu$-almost every $x \\in \\alpha$, the Birkhoff averages $\\frac{1}{n} \\sum_{k=0}^{n-1} g(T^k x)$ converge to $\\int g \\, d\\mu$ as $n \\to \\infty$.","title":"Birkhoff pointwise ergodic theorem (ergodic case)"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/integrable-birkhoff-sum-of-measure-prese","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.integrable_birkhoffSum_of_measurePreserving","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#integrable-birkhoff-sum-of-measure-prese","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ preserve $\\mu$, and let $g : \\alpha \\to \\mathbb{R}$ be $\\mu$-integrable. Then for every $n \\in \\mathbb{N}$ the Birkhoff sum $x \\mapsto \\sum_{k=0}^{n-1} g(T^k x)$ is $\\mu$-integrable.","title":"Integrability of Birkhoff sums"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/integrable-birkhoff-average-of-measure-p","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.integrable_birkhoffAverage_of_measurePreserving","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#integrable-birkhoff-average-of-measure-p","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ preserve $\\mu$, and let $g : \\alpha \\to \\mathbb{R}$ be $\\mu$-integrable. Then for every $n \\in \\mathbb{N}$ the Birkhoff average $x \\mapsto \\frac{1}{n} \\sum_{k=0}^{n-1} g(T^k x)$ is $\\mu$-integrable (for $n = 0$ it is the zero function).","title":"Integrability of Birkhoff averages"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/integral-comp-iterate-of-measure-preserv","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.integral_comp_iterate_of_measurePreserving","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#integral-comp-iterate-of-measure-preserv","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ preserve $\\mu$, and let $u : \\alpha \\to \\mathbb{R}$ be $\\mu$-integrable. Then for every $k \\in \\mathbb{N}$, writing $T^k$ for the $k$-th iterate of $T$, we have $\\int u(T^k x) \\, d\\mu(x) = \\int u \\, d\\mu$.","title":"Invariance of the integral under iterates"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/integral-birkhoff-average-of-measure-pre","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.integral_birkhoffAverage_of_measurePreserving","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#integral-birkhoff-average-of-measure-pre","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ preserve $\\mu$, and let $u : \\alpha \\to \\mathbb{R}$ be $\\mu$-integrable. Then for every $n \\ge 1$, $\\int \\frac{1}{n} \\sum_{k=0}^{n-1} u(T^k x) \\, d\\mu(x) = \\int u \\, d\\mu$.","title":"Birkhoff averages preserve the integral"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/norm-birkhoff-average-le-birkhoff-averag","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.norm_birkhoffAverage_le_birkhoffAverage_norm","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#norm-birkhoff-average-le-birkhoff-averag","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $T : \\alpha \\to \\alpha$ be a map. For $f : \\alpha \\to \\mathbb{R}$ and $n \\ge 1$ write $A_n f(x) = \\frac{1}{n} \\sum_{k=0}^{n-1} f(T^k x)$ for the $n$-th Birkhoff average of $f$, and $A_0 f = 0$. Then for every $u : \\alpha \\to \\mathbb{R}$, every $n \\in \\mathbb{N}$ and every $x \\in \\alpha$, writing $|u|$ for the function $y \\mapsto |u(y)|$, we have $|A_n u(x)| \\le A_n |u|(x)$.","title":"Triangle inequality for Birkhoff averages"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/birkhoff-average-le-birkhoff-average","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.birkhoffAverage_le_birkhoffAverage","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#birkhoff-average-le-birkhoff-average","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $T : \\alpha \\to \\alpha$ be a map. For $u : \\alpha \\to \\mathbb{R}$, $n \\ge 1$ and $x \\in \\alpha$, write $A_n u(x) = \\frac{1}{n} \\sum_{k=0}^{n-1} u(T^k x)$ for the $n$-th Birkhoff average of $u$ at $x$, and $A_0 u(x) = 0$. If $u, v : \\alpha \\to \\mathbb{R}$ satisfy $u(y) \\le v(y)$ for every $y \\in \\alpha$, then $A_n u(x) \\le A_n v(x)$ for every $n \\in \\mathbb{N}$ and every $x \\in \\alpha$.","title":"Monotonicity of Birkhoff averages"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/norm-birkhoff-average-le-of-bounded","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.norm_birkhoffAverage_le_of_bounded","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#norm-birkhoff-average-le-of-bounded","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $T : \\alpha \\to \\alpha$ be a map, let $M \\ge 0$, and let $g : \\alpha \\to \\mathbb{R}$ satisfy $|g(x)| \\le M$ for every $x \\in \\alpha$. Then for every $n \\in \\mathbb{N}$ and every $x \\in \\alpha$, the Birkhoff average $\\frac{1}{n} \\sum_{k=0}^{n-1} g(T^k x)$ (taken to be $0$ for $n = 0$) has absolute value at most $M$.","title":"Birkhoff averages of a bounded function are bounded"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/integral-norm-birkhoff-average-sub-le","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.integral_norm_birkhoffAverage_sub_le","parent":22352,"reading_title":{"format":"inline-parts-v1","parts":[["text","Birkhoff averages are "],["math","\\(L^1\\)"],["text","-contractions"]],"source":"Birkhoff averages are $L^1$-contractions"},"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#integral-norm-birkhoff-average-sub-le","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ preserve $\\mu$, and let $g, h : \\alpha \\to \\mathbb{R}$ be $\\mu$-integrable. For a function $f : \\alpha \\to \\mathbb{R}$ and $n \\ge 1$ write $A_n f(x) = \\frac{1}{n} \\sum_{k=0}^{n-1} f(T^k x)$, and $A_0 f = 0$. Then for every $n \\in \\mathbb{N}$, $\\int |A_n g(x) - A_n h(x)| \\, d\\mu(x) \\le \\int |g(x) - h(x)| \\, d\\mu(x)$.","title":"Birkhoff averages are $L^1$-contractions"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/tendsto-integral-norm-birkhoff-average-o","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.tendsto_integral_norm_birkhoffAverage_of_bounded","parent":22352,"reading_title":{"format":"inline-parts-v1","parts":[["text","Birkhoff's ergodic theorem in "],["math","\\(L^1\\)"],["text"," for bounded functions"]],"source":"Birkhoff's ergodic theorem in $L^1$ for bounded functions"},"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#tendsto-integral-norm-birkhoff-average-o","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ be ergodic with respect to $\\mu$ (in particular $T$ preserves $\\mu$), let $M \\ge 0$, and let $g : \\alpha \\to \\mathbb{R}$ be $\\mu$-integrable with $|g(x)| \\le M$ for every $x \\in \\alpha$. For $n \\ge 1$ let $A_n g(x) = \\frac{1}{n} \\sum_{k=0}^{n-1} g(T^k x)$. Then $\\int \\left| A_n g(x) - \\int g \\, d\\mu \\right| d\\mu(x) \\to 0$\u2026","title":"Birkhoff's ergodic theorem in $L^1$ for bounded functions"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/birkhoff-ergodic-theorem-l1","kind":"node","lean":"MathlibExt.Probability.BirkhoffErgodicL1.birkhoff_ergodic_theorem_L1","parent":22352,"reading_title":{"format":"inline-parts-v1","parts":[["text","Birkhoff's ergodic theorem in "],["math","\\(L^1\\)"],["text"," (ergodic case)"]],"source":"Birkhoff's ergodic theorem in $L^1$ (ergodic case)"},"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#birkhoff-ergodic-theorem-l1","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\alpha$, let $T : \\alpha \\to \\alpha$ be ergodic with respect to $\\mu$ (in particular $T$ preserves $\\mu$), and let $g : \\alpha \\to \\mathbb{R}$ be $\\mu$-integrable. For $n \\ge 1$ let $A_n g(x) = \\frac{1}{n} \\sum_{k=0}^{n-1} g(T^k x)$ be the Birkhoff averages of $g$. Then $\\int \\left| A_n g(x) - \\int g \\, d\\mu \\right| d\\mu(x) \\to 0$ as $n \\to \\infty$; that is,\u2026","title":"Birkhoff's ergodic theorem in $L^1$ (ergodic case)"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/max-partial-sum","kind":"node","lean":"StatMech.GarsiaMax.maxPartialSum","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#max-partial-sum","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The maximal Birkhoff sum `max_{1 \u2264 n \u2264 N} \u2211_{k<n} f (T^[k] x)` over the first `N` times.","title":"Maximal Birkhoff partial sum"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/garsia-maximal-inequality","kind":"node","lean":"StatMech.GarsiaMax.garsia_maximal_inequality","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#garsia-maximal-inequality","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Maximal ergodic theorem (E. Hopf, 1954), in the finite form of A. M. Garsia, *A simple proof of E. Hopf's maximal ergodic theorem*, 1965: for measure-preserving `T` and integrable `f`, the integral of `f` over `{max_{1 \u2264 n \u2264 N} S_n f > 0}` is nonnegative. See also `MathlibExt.Dynamics.Ergodic.BirkhoffWanted.maximal_ergodic`, the limiting form for measurable `f`.","title":"Maximal ergodic theorem"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/is-invariant-set","kind":"node","lean":"StatMech.BirkhoffAE.IsInvariantSet","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#is-invariant-set","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"`E` is invariant under `T`: `x \u2208 E \u2194 T x \u2208 E`, i.e. `T \u207b\u00b9' E = E` (see `isInvariantSet_iff_preimage_eq`).","title":"Invariant set of a map"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/is-invariant-set-iff-preimage-eq","kind":"node","lean":"StatMech.BirkhoffAE.isInvariantSet_iff_preimage_eq","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#is-invariant-set-iff-preimage-eq","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"`IsInvariantSet T E` is the condition `T \u207b\u00b9' E = E` used by Mathlib's `PreErgodic` and `MeasurableSpace.invariants`.","title":"Invariant set as a set equal to its preimage"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/iterate","kind":"node","lean":"StatMech.BirkhoffAE.IsInvariantSet.iterate","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#iterate","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"An invariant set is invariant under every iterate.","title":"Invariant set is invariant under every iterate"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/f-star","kind":"node","lean":"StatMech.BirkhoffAE.fStar","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#f-star","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The upper limit `limsup_n A_n f x` of the Birkhoff averages, in `EReal`.","title":"Upper limit of Birkhoff averages"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/f-star-low","kind":"node","lean":"StatMech.BirkhoffAE.fStarLow","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#f-star-low","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The lower limit `liminf_n A_n f x` of the Birkhoff averages, in `EReal`.","title":"Lower limit of Birkhoff averages"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/f-star-comp","kind":"node","lean":"StatMech.BirkhoffAE.fStar_comp","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#f-star-comp","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The upper limit of the Birkhoff averages is `T`-invariant.","title":"Upper limit of Birkhoff averages is invariant"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/set-integral-nonneg-of-invariant","kind":"node","lean":"StatMech.BirkhoffAE.setIntegral_nonneg_of_invariant","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#set-integral-nonneg-of-invariant","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The maximal inequality on an invariant set: if every point of the invariant set `E` has some positive Birkhoff sum of `g`, then `\u222b_E g d\u03bc \u2265 0`.","title":"Maximal ergodic inequality on an invariant set"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/level-set","kind":"node","lean":"StatMech.BirkhoffAE.levelSet","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#level-set","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The invariant set `{f_* < a} \u2229 {b < f*}` where the Birkhoff averages oscillate across `[a, b]`.","title":"Oscillation set of Birkhoff averages"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/measure-level-set-eq-zero","kind":"node","lean":"StatMech.BirkhoffAE.measure_levelSet_eq_zero","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#measure-level-set-eq-zero","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"For `a < b` the oscillation set `levelSet T f a b` is null.","title":"Oscillation set of Birkhoff averages is null"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/birkhoff-limit","kind":"node","lean":"StatMech.BirkhoffAE.birkhoffLimit","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#birkhoff-limit","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The limit of the Birkhoff averages of `f` (defined as `f*` cast to `\u211d`; it is the limit almost everywhere by `tendsto_birkhoffAverage_ae`).","title":"Birkhoff limit of a function"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/tendsto-birkhoff-average-ae","kind":"node","lean":"StatMech.BirkhoffAE.tendsto_birkhoffAverage_ae","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#tendsto-birkhoff-average-ae","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The Birkhoff averages of an integrable `f` converge a.e. to `birkhoffLimit T f`.","title":"Birkhoff averages converge almost everywhere to the Birkhoff limit"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/e-lp-norm-birkhoff-average-le","kind":"node","lean":"StatMech.BirkhoffAE.eLpNorm_birkhoffAverage_le","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#e-lp-norm-birkhoff-average-le","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Birkhoff averages are `L\u00b9`-contractions. See also `MathlibExt.Probability.BirkhoffErgodicL1.integral_norm_birkhoffAverage_sub_le`, the same bound for differences, stated with integrals of norms.","title":"Birkhoff averages are L1 contractions"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/integrable-birkhoff-limit","kind":"node","lean":"StatMech.BirkhoffAE.integrable_birkhoffLimit","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#integrable-birkhoff-limit","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The Birkhoff limit of an integrable function is integrable.","title":"Integrability of the Birkhoff limit"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/set-integral-birkhoff-limit","kind":"node","lean":"StatMech.BirkhoffAE.setIntegral_birkhoffLimit","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#set-integral-birkhoff-limit","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"On every measurable invariant set `E`, `\u222b_E birkhoffLimit T f d\u03bc = \u222b_E f d\u03bc`.","title":"Birkhoff limit preserves integrals over invariant sets"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/birkhoff-limit-ae-eq-cond-exp","kind":"node","lean":"StatMech.BirkhoffAE.birkhoffLimit_ae_eq_condExp","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#birkhoff-limit-ae-eq-cond-exp","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"The Birkhoff limit is the conditional expectation of `f` given the invariant \u03c3-algebra.","title":"Birkhoff limit is a conditional expectation on invariant sets"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/tendsto-birkhoff-average-cond-exp-ae","kind":"node","lean":"StatMech.BirkhoffAE.tendsto_birkhoffAverage_condExp_ae","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#tendsto-birkhoff-average-cond-exp-ae","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"Birkhoff's pointwise ergodic theorem (G. D. Birkhoff, *Proof of the ergodic theorem*, 1931): for `T` preserving a finite measure `\u03bc` and integrable `f`, the averages `(1/n) \u2211_{k<n} f \u2218 T^k` converge `\u03bc`-a.e. to `\u03bc[f | MeasurableSpace.invariants T]`. See also `MathlibExt.Dynamics.Ergodic.BirkhoffWanted.birkhoff_pointwise_ergodic`, the ergodic case, and `birkhoffLimit_ae_eq_integral_of_ergodic`.","title":"Birkhoff's pointwise ergodic theorem"},{"id":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem/birkhoff-limit-ae-eq-integral-of-ergodic","kind":"node","lean":"StatMech.BirkhoffAE.birkhoffLimit_ae_eq_integral_of_ergodic","parent":22352,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/birkhoff-ergodic-theorem/index.html#birkhoff-limit-ae-eq-integral-of-ergodic","scope":"Dynamics/ergodic-theory/birkhoff-ergodic-theorem","status":"fully_proved","summary":"For ergodic `T` on a probability space the Birkhoff limit is the constant `\u222b f d\u03bc`; this is `MathlibExt.Dynamics.Ergodic.BirkhoffWanted.birkhoff_pointwise_ergodic` combined with `tendsto_birkhoffAverage_ae`.","title":"Birkhoff limit of an ergodic map is the space average"},{"data_url":"scopes/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem.json","id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","kind":"scope","navigation":"module","parent":22311,"result":false,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","title":"Kingman's subadditive ergodic theorem"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/kingman-subadditive","kind":"node","lean":"MathlibExt.Dynamics.Ergodic.KingmanWanted.kingman_subadditive","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#kingman-subadditive","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\alpha$ and let $T : \\alpha \\to \\alpha$ preserve $\\mu$. Let $(f_n)_{n \\in \\mathbb{N}}$ be functions $\\alpha \\to \\mathbb{R}$, each measurable and $\\mu$-integrable, such that for every $n$ the inequality $f_n \\ge 0$ holds $\\mu$-almost everywhere, $f_0 = 0$ holds $\\mu$-almost everywhere, and for all natural numbers $m$ and $n$ the inequality\u2026","title":"Kingman's subadditive ergodic theorem (nonnegative case)"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/integral-comp-measure-preserving","kind":"node","lean":"StatMech.Kingman.integral_comp_measurePreserving","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#integral-comp-measure-preserving","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The integral is invariant under a measure-preserving map: `\u222b g \u2218 T d\u03bc = \u222b g d\u03bc` for a.e.-strongly measurable `g`.","title":"Integral is invariant under a measure-preserving map"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/integral-comp-iterate","kind":"node","lean":"StatMech.Kingman.integral_comp_iterate","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#integral-comp-iterate","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The integral is invariant under iterates of a measure-preserving map: `\u222b g \u2218 T^[m] d\u03bc = \u222b g d\u03bc` for a.e.-strongly measurable `g`. See also `MathlibExt.Probability.BirkhoffErgodicL1.integral_comp_iterate_of_measurePreserving`, which assumes `g` integrable.","title":"Integral is invariant under iterates of a measure-preserving map"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/subadditive-cocycle","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#subadditive-cocycle","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"`X` is a subadditive cocycle over `T`: `T` preserves `\u03bc`, every `X n` is integrable, and `X (m + n) \u2264 X m + X n \u2218 T^[m]` almost everywhere.","title":"Subadditive cocycle over a measure-preserving map"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/exp-seq","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.expSeq","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#exp-seq","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The expectation `\u222b X n d\u03bc` of the `n`-th term.","title":"Expectation sequence of a cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/exp-seq-subadditive","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.expSeq_subadditive","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#exp-seq-subadditive","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The expectations of a subadditive cocycle are subadditive.","title":"Expectations of a subadditive cocycle are subadditive"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/subadditive-exp-seq","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.subadditive_expSeq","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#subadditive-exp-seq","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The expectations of a subadditive cocycle form a `Subadditive` sequence.","title":"Expectations of a subadditive cocycle form a subadditive sequence"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/ae-le-birkhoff-sum","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.ae_le_birkhoffSum","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#ae-le-birkhoff-sum","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Iterating subadditivity: `X n \u2264 \u2211_{k<n} X 1 \u2218 T^[k]` almost everywhere, for all `n \u2265 1`.","title":"Subadditive cocycle bounded by Birkhoff sums of its first term"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/exp-seq-le-nsmul","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.expSeq_le_nsmul","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#exp-seq-le-nsmul","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"`\u222b X n d\u03bc \u2264 n \u222b X 1 d\u03bc` for `n \u2265 1`.","title":"Expectation of a subadditive cocycle grows at most linearly"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/gamma","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.gamma","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#gamma","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The time constant `lim_n (\u222b X n d\u03bc) / n = inf_n (\u222b X n d\u03bc) / n` (Fekete's lemma); it is meaningful when `expSeq n / n` is bounded below.","title":"Time constant of a subadditive cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/tendsto-exp-seq-div","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.tendsto_expSeq_div","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#tendsto-exp-seq-div","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Fekete's lemma for the expectations: `expSeq n / n \u2192 gamma` when bounded below.","title":"Fekete's lemma for the expectations of a subadditive cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/gamma-le-div","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.gamma_le_div","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#gamma-le-div","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"`gamma \u2264 expSeq n / n` for every `n \u2260 0`.","title":"Time constant is a lower bound for normalized expectations"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/gamma-le-exp-seq-one","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.gamma_le_expSeq_one","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#gamma-le-exp-seq-one","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"`gamma \u2264 \u222b X 1 d\u03bc`.","title":"Time constant is at most the first expectation"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/bdd-below-of-nonneg","kind":"node","lean":"StatMech.Kingman.SubadditiveCocycle.bddBelow_of_nonneg","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#bdd-below-of-nonneg","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"For a nonnegative cocycle, `expSeq n / n` is bounded below by `0`.","title":"Normalized expectations of a nonnegative cocycle are bounded below"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/ae-all-subadditive","kind":"node","lean":"StatMech.KingmanAE.ae_all_subadditive","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#ae-all-subadditive","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The cocycle inequality holds for all `m, n` simultaneously, almost everywhere.","title":"Cocycle inequality holds for all indices almost everywhere"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/ae-le-block-birkhoff-sum","kind":"node","lean":"StatMech.KingmanAE.ae_le_block_birkhoffSum","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#ae-le-block-birkhoff-sum","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Block decomposition: `X (q k) \u2264 \u2211_{j<q} X k \u2218 T^{j k}` almost everywhere.","title":"Block decomposition bound for a subadditive cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/tendsto-comp-div-at-top","kind":"node","lean":"StatMech.KingmanAE.tendsto_comp_div_atTop","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#tendsto-comp-div-at-top","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"For integrable `g`, `g (T^[m] x) / m \u2192 0` almost everywhere.","title":"Integrable function along an orbit divided by time tends to zero"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/rem-dom","kind":"node","lean":"StatMech.KingmanAE.remDom","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#rem-dom","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The integrable dominating function `\u2211_{r<k} (X r)\u207a` for the remainder terms `X r`, `r < k`.","title":"Dominating function for remainder terms in Kingman's theorem"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/g-star-le","kind":"node","lean":"StatMech.KingmanAE.gStar_le","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#g-star-le","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"For each `k \u2265 1`, `limsup X n / n \u2264 birkhoffLimit (T^[k]) (X k) / k` almost everywhere.","title":"Upper limit of a cocycle bounded by Birkhoff limits of blocks"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/g-star","kind":"node","lean":"StatMech.KingmanAE.gStar","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#g-star","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The upper limit `limsup_n X n x / n` in `EReal`. The map `T` is a parameter only so that the notation records the cocycle's base map.","title":"Upper limit of a normalized subadditive cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/g-star-low","kind":"node","lean":"StatMech.KingmanAE.gStarLow","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#g-star-low","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The lower limit `liminf_n X n x / n` in `EReal`.","title":"Lower limit of a normalized subadditive cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/g-star-lt-top-ae","kind":"node","lean":"StatMech.KingmanAE.gStar_lt_top_ae","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#g-star-lt-top-ae","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"`limsup X n / n < \u22a4` almost everywhere.","title":"Upper limit of a normalized cocycle is finite almost everywhere"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/kingman-limit","kind":"node","lean":"StatMech.KingmanAE.kingmanLimit","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#kingman-limit","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The limit `lim X n / n` in Kingman's theorem, defined as `limsup X n / n` cast to `\u211d` (so it is `0` where the `limsup` is infinite).","title":"Kingman limit of a subadditive cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/tendsto-div-kingman-limit-ae","kind":"node","lean":"StatMech.KingmanAE.tendsto_div_kingmanLimit_ae","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#tendsto-div-kingman-limit-ae","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"If `g* \u2264 g_*` and `g* > \u22a5` almost everywhere, then `X n / n \u2192 kingmanLimit` almost everywhere.","title":"Almost-everywhere convergence to the Kingman limit"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/kingman-limit-comp-ae","kind":"node","lean":"StatMech.KingmanAE.kingmanLimit_comp_ae","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#kingman-limit-comp-ae","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Under the hypotheses of `tendsto_div_kingmanLimit_ae` and integrability of the limit, the limit is `T`-invariant almost everywhere.","title":"Invariance of the Kingman limit"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/integrable-kingman-limit","kind":"node","lean":"StatMech.KingmanAE.integrable_kingmanLimit","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#integrable-kingman-limit","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"For a nonnegative cocycle, `kingmanLimit` is integrable.","title":"Integrability of the Kingman limit for nonnegative cocycles"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/integral-kingman-limit-le-gamma","kind":"node","lean":"StatMech.KingmanAE.integral_kingmanLimit_le_gamma","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#integral-kingman-limit-le-gamma","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"For a nonnegative cocycle, `\u222b kingmanLimit \u2264 gamma`.","title":"Integral of the Kingman limit is at most the time constant"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/g-low","kind":"node","lean":"StatMech.KingmanFilling.gLow","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#g-low","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The lower limit `liminf_n X n x / n`, cast to `\u211d`.","title":"Real lower limit of a normalized cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/eq-of-le-to-real-eq","kind":"node","lean":"StatMech.KingmanFilling.eq_of_le_toReal_eq","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#eq-of-le-to-real-eq","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Two extended reals `x \u2264 y` with `x \u2260 \u22a5`, `y \u2260 \u22a4` and equal real parts are equal.","title":"Ordered extended reals with equal real parts are equal"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/g-star-low-le-comp","kind":"node","lean":"StatMech.KingmanFilling.gStarLow_le_comp","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#g-star-low-le-comp","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The lower limit is nondecreasing along `T`: `g_* \u2264 g_* \u2218 T` almost everywhere.","title":"Lower limit of a cocycle is nondecreasing along the map"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/bad-set","kind":"node","lean":"StatMech.KingmanFilling.badSet","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#bad-set","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The bad set: points where `X m x > (gLow x + \u03b5) m` for every block length `1 \u2264 m \u2264 N`.","title":"Bad set in the greedy decomposition for Kingman's theorem"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/dep","kind":"node","lean":"StatMech.KingmanFilling.dep","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#dep","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The per-step budget `gLow + \u03b5 + 1_{badSet} X 1` of the greedy decomposition.","title":"Per-step budget of the greedy decomposition"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/bad-cost","kind":"node","lean":"StatMech.KingmanFilling.badCost","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#bad-cost","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The cost `1_{badSet} X 1` of a bad step.","title":"Cost of a bad step in the greedy decomposition"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/greedy","kind":"node","lean":"StatMech.KingmanFilling.greedy","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#greedy","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Greedy decomposition along one orbit: `X n (T^[s] x)` is at most the sum of the budgets `dep` over `n` steps plus one remainder term, for every `n` and starting time `s`.","title":"Greedy decomposition along an orbit"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/greedy-windowed","kind":"node","lean":"StatMech.KingmanFilling.greedy_windowed","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#greedy-windowed","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The greedy bound from time `0`, with the remainder bounded by a sum over the last window.","title":"Windowed greedy bound from time zero"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/gamma-le-integral-g-low","kind":"node","lean":"StatMech.KingmanFilling.gamma_le_integral_gLow","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#gamma-le-integral-g-low","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The time constant is at most `\u222b g_*`.","title":"Time constant is at most the integral of the lower limit"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/g-star-le-g-star-low-ae","kind":"node","lean":"StatMech.KingmanFilling.gStar_le_gStarLow_ae","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#g-star-le-g-star-low-ae","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"For a nonnegative cocycle the upper and lower limits of `X n / n` agree almost everywhere.","title":"Upper and lower limits of a nonnegative cocycle agree"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/tendsto-div-kingman-limit-ae-of-nonneg","kind":"node","lean":"StatMech.KingmanFilling.tendsto_div_kingmanLimit_ae_of_nonneg","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#tendsto-div-kingman-limit-ae-of-nonneg","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Kingman's subadditive ergodic theorem for nonnegative cocycles: `X n / n` converges almost everywhere to `kingmanLimit T X` (J. F. C. Kingman, 1968; proof of J. M. Steele, 1989).","title":"Kingman's subadditive ergodic theorem"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/add-coc","kind":"node","lean":"StatMech.KingmanSigned.addCoc","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#add-coc","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The additive cocycle `n \u21a6 \u2211_{k<n} X 1 \u2218 T^[k]` (the Birkhoff sums of `X 1`).","title":"Additive cocycle of Birkhoff sums of the first term"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/zc","kind":"node","lean":"StatMech.KingmanSigned.Zc","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#zc","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The subadditivity defect `addCoc n - X n` (`0` at `n = 0`); it is nonnegative a.e.","title":"Subadditivity defect of a cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/wc","kind":"node","lean":"StatMech.KingmanSigned.Wc","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#wc","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The negative defect `X n - addCoc n` (`0` at `n = 0`), a nonpositive subadditive cocycle.","title":"Negative subadditivity defect of a cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/wc-subadd-cocycle","kind":"node","lean":"StatMech.KingmanSigned.Wc_subaddCocycle","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#wc-subadd-cocycle","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The negative defect is a subadditive cocycle.","title":"Negative defect is a subadditive cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/sc","kind":"node","lean":"StatMech.KingmanSigned.Sc","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#sc","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The shifted defect `X n + M n - addCoc n`, a subadditive cocycle for every `M`.","title":"Shifted subadditivity defect of a cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/vc","kind":"node","lean":"StatMech.KingmanSigned.Vc","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#vc","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The truncated defect `(Sc M n)\u207a = (M n - Zc n)\u207a`, a nonnegative subadditive cocycle.","title":"Truncated subadditivity defect of a cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/vc-subadd-cocycle","kind":"node","lean":"StatMech.KingmanSigned.Vc_subaddCocycle","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#vc-subadd-cocycle","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"The truncated defect is a nonnegative subadditive cocycle.","title":"Truncated defect is a nonnegative subadditive cocycle"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/ae-liminf-lt-top","kind":"node","lean":"StatMech.KingmanSigned.ae_liminf_lt_top","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#ae-liminf-lt-top","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"If the time constant is finite, `liminf_n Zc n / n < \u221e` almost everywhere (Fatou).","title":"Lower limit of the normalized defect is finite almost everywhere"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/ae-tendsto-x-div","kind":"node","lean":"StatMech.KingmanSigned.ae_tendsto_X_div","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#ae-tendsto-x-div","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"If the time constant is finite, `X n / n` converges almost everywhere.","title":"Kingman's theorem: normalized cocycle converges almost everywhere"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/kingman-ae-general","kind":"node","lean":"StatMech.KingmanSigned.kingman_ae_general","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#kingman-ae-general","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Kingman's subadditive ergodic theorem (J. F. C. Kingman, *The ergodic theory of subadditive stochastic processes*, 1968): if `X` is a subadditive cocycle over a map `T` preserving the finite measure `\u03bc` and `inf_n (\u222b X n d\u03bc) / n > -\u221e`, then `X n / n` converges `\u03bc`-almost everywhere to the finite limit `kingmanLimit T X`. The sign of `X` is arbitrary. Invariance and integrability of the limit, and\u2026","title":"Kingman's subadditive ergodic theorem"},{"id":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/exists-invariant-limit-of-nonneg","kind":"node","lean":"StatMech.KingmanSigned.exists_invariant_limit_of_nonneg","parent":22391,"result":true,"route":"../roadmap/Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem/index.html#exists-invariant-limit-of-nonneg","scope":"Dynamics/ergodic-theory/kingman-s-subadditive-ergodic-theorem","status":"fully_proved","summary":"Kingman's theorem for nonnegative cocycles, with integrable and invariant limit: if `T` preserves the finite measure `\u03bc` and `X` is a nonnegative integrable subadditive cocycle, then `X n / n` converges a.e. to an integrable `T`-invariant function. This is `MathlibExt.Dynamics.Ergodic.KingmanWanted.kingman_subadditive` without its hypotheses that each `X n` is measurable and `X 0 = 0`, and for finite rather than\u2026","title":"Kingman's subadditive ergodic theorem for nonnegative cocycles"},{"data_url":"scopes/Dynamics/topological-symbolic-dynamics.json","id":"Dynamics/topological-symbolic-dynamics","kind":"scope","parent":22310,"result":false,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/index.html","scope":"Dynamics/topological-symbolic-dynamics","status":"can_prove","summary":"Topological dynamics studies continuous maps of topological spaces through orbits, periodic points and recurrence, and symbolic dynamics studies shift spaces and cellular automata on sequences of symbols.","title":"Topological and symbolic dynamics"},{"id":"Dynamics/topological-symbolic-dynamics/binaryrec-realizable-subspace-dim","kind":"node","lean":"MetaMathlibExt.binaryrec_realizable_subspace_dim","parent":22441,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/index.html#binaryrec-realizable-subspace-dim","scope":"Dynamics/topological-symbolic-dynamics","status":"can_prove","summary":"Let $B$ and $C$ be integers with $C \\neq 0$, and suppose the recurrence is nondegenerate: whenever $\\alpha \\neq \\beta$ are complex roots of $x^2 - Bx + C$, we have $(\\alpha/\\beta)^n \\neq 1$ for every integer $n \\geq 1$. Put $\\Delta = B^2 - 4C$. Let $R$ be the set of complex sequences $(u_n)_{n \\geq 0}$ coming from integer sequences $u$ that satisfy $u_{n+2} = B u_{n+1} - C u_n$ for all $n \\geq 0$ and for which\u2026","title":"Realizable span of a nondegenerate binary recurrence"},{"data_url":"scopes/Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem.json","id":"Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem","kind":"scope","navigation":"module","parent":22441,"result":false,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem/index.html","scope":"Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem","status":"fully_proved","title":"Curtis\u2013Hedlund\u2013Lyndon theorem"},{"id":"Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem/curtis-hedlund-lyndon-general","kind":"node","lean":"MetaMathlibExt.curtis_hedlund_lyndon_general","parent":22443,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem/index.html#curtis-hedlund-lyndon-general","scope":"Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem","status":"fully_proved","summary":"Let $A$ be a finite set and $B$ any set, both with the discrete topology, and give $A^{\\mathbb{Z}}$ and $B^{\\mathbb{Z}}$ the product topology. For $k \\in \\mathbb{Z}$ let $\\sigma^k$ be the shift on either space given by $(\\sigma^k x)_n = x_{n+k}$ for all $n \\in \\mathbb{Z}$. A map $F : A^{\\mathbb{Z}} \\to B^{\\mathbb{Z}}$ is a sliding block code, that is, there are a radius $r \\in \\mathbb{N}$ and a local rule\u2026","title":"Curtis\u2013Hedlund\u2013Lyndon theorem with an arbitrary discrete output alphabet"},{"id":"Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem/curtis-hedlund-lyndon","kind":"node","lean":"MetaMathlibExt.curtis_hedlund_lyndon","parent":22443,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem/index.html#curtis-hedlund-lyndon","scope":"Dynamics/topological-symbolic-dynamics/curtis-hedlund-lyndon-theorem","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets with the discrete topology, and give $A^{\\mathbb{Z}}$ and $B^{\\mathbb{Z}}$ the product topology. For $k \\in \\mathbb{Z}$ let $\\sigma^k$ be the shift on either space given by $(\\sigma^k x)_n = x_{n+k}$ for all $n \\in \\mathbb{Z}$. A map $F : A^{\\mathbb{Z}} \\to B^{\\mathbb{Z}}$ is a sliding block code, that is, there are a radius $r \\in \\mathbb{N}$ and a local rule\u2026","title":"Curtis\u2013Hedlund\u2013Lyndon theorem"},{"data_url":"scopes/Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences.json","id":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences","kind":"scope","navigation":"module","parent":22441,"result":false,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/index.html","scope":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences","status":"fully_proved","title":"Eventually periodic sequences"},{"id":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/is-eventually-periodic","kind":"node","lean":"MetaMathlibExt.IsEventuallyPeriodic","parent":22446,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/index.html#is-eventually-periodic","scope":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences","status":"fully_proved","summary":"A sequence $u : \\mathbb{N} \\to \\alpha$ with values in a type $\\alpha$ is eventually periodic if there are natural numbers $p > 0$ (a period) and $n_0$ (a preperiod) such that $u(n + p) = u(n)$ for every natural number $n \\ge n_0$.","title":"Eventually periodic sequences"},{"id":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/of-witness","kind":"node","lean":"MetaMathlibExt.IsEventuallyPeriodic.of_witness","parent":22446,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/index.html#of-witness","scope":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences","status":"fully_proved","summary":"Let $\\alpha$ be a type and $u : \\mathbb{N} \\to \\alpha$ a sequence. If there are natural numbers $p > 0$ and $n_0$ such that $u(n + p) = u(n)$ for every natural number $n \\ge n_0$, then $u$ is eventually periodic.","title":"Explicit period and preperiod give eventual periodicity"},{"id":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/witness-spec","kind":"node","lean":"MetaMathlibExt.IsEventuallyPeriodic.witness_spec","parent":22446,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/index.html#witness-spec","scope":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences","status":"fully_proved","summary":"Let $\\alpha$ be a type. If a sequence $u : \\mathbb{N} \\to \\alpha$ is eventually periodic, then there are natural numbers $p > 0$ and $n_0$ such that $u(n + p) = u(n)$ for every natural number $n \\ge n_0$.","title":"Period and preperiod of an eventually periodic sequence"},{"id":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/of-periodic","kind":"node","lean":"MetaMathlibExt.IsEventuallyPeriodic.of_periodic","parent":22446,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/index.html#of-periodic","scope":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences","status":"fully_proved","summary":"Let $\\alpha$ be a type, let $u : \\mathbb{N} \\to \\alpha$ be a sequence and let $p > 0$ be a natural number with $u(n + p) = u(n)$ for every $n \\in \\mathbb{N}$. Then $u$ is eventually periodic.","title":"Periodic sequences are eventually periodic"},{"id":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/is-ultimately-periodic","kind":"node","lean":"MetaMathlibExt.IsUltimatelyPeriodic","parent":22446,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/index.html#is-ultimately-periodic","scope":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences","status":"fully_proved","summary":"Let $\\alpha$ be a type. A sequence $u : \\mathbb{N} \\to \\alpha$ is called ultimately periodic exactly when it is eventually periodic; the two terms name the same notion.","title":"Ultimately periodic sequences"},{"id":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/of-finite-orbit","kind":"node","lean":"MetaMathlibExt.IsEventuallyPeriodic.of_finite_orbit","parent":22446,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences/index.html#of-finite-orbit","scope":"Dynamics/topological-symbolic-dynamics/eventually-periodic-sequences","status":"fully_proved","summary":"Let $S$ be a finite set, $F : S \\to S$ a map and $r : \\mathbb{N} \\to S$ a sequence with $r(n + 1) = F(r(n))$ for every $n \\in \\mathbb{N}$. Then $r$ is eventually periodic.","title":"Orbits in a finite set are eventually periodic"},{"data_url":"scopes/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata.json","id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","kind":"scope","navigation":"module","parent":22441,"result":false,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","title":"Full shift and cellular automata"},{"id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/full-shift-space","kind":"node","lean":"SymbolicDynamics.FullShift.FullShiftSpace","parent":22453,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html#full-shift-space","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","summary":"For $N \\in \\mathbb{N}$, the configuration space of the two-sided full shift on $N$ symbols is the set $\\{0, \\dots, N-1\\}^{\\mathbb{Z}}$ of all bi-infinite sequences $x = (x_i)_{i \\in \\mathbb{Z}}$ of symbols. It carries the product of the discrete topologies.","title":"Full shift space"},{"id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/unit-shift","kind":"node","lean":"SymbolicDynamics.FullShift.unitShift","parent":22453,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html#unit-shift","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","summary":"The unit shift on the full shift on $N$ symbols is the map $\\sigma$ from $\\{0, \\dots, N-1\\}^{\\mathbb{Z}}$ to itself given by $(\\sigma x)_i = x_{i+1}$ for all $i \\in \\mathbb{Z}$, the left translation by one step.","title":"Unit shift"},{"id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/shift-homeomorph","kind":"node","lean":"SymbolicDynamics.FullShift.shiftHomeomorph","parent":22453,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html#shift-homeomorph","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","summary":"The unit shift $\\sigma$ of the full shift on $N$ symbols, given by $(\\sigma x)_i = x_{i+1}$, is a homeomorphism of $\\{0, \\dots, N-1\\}^{\\mathbb{Z}}$ onto itself, with inverse given by $(\\sigma^{-1} x)_i = x_{i-1}$.","title":"The unit shift as a homeomorphism"},{"id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/automorphism-group","kind":"node","lean":"SymbolicDynamics.FullShift.automorphismGroup","parent":22453,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html#automorphism-group","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","summary":"The automorphism group of the full shift on $N$ symbols is the subgroup of the group of homeomorphisms of $\\{0, \\dots, N-1\\}^{\\mathbb{Z}}$ onto itself consisting of the homeomorphisms $\\varphi$ that commute with the unit shift $\\sigma$, given by $(\\sigma x)_i = x_{i+1}$, that is, $\\varphi \\circ \\sigma = \\sigma \\circ \\varphi$.","title":"Automorphism group of the full shift"},{"id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/is-one-dimensional-ca","kind":"node","lean":"SymbolicDynamics.FullShift.IsOneDimensionalCA","parent":22453,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html#is-one-dimensional-ca","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","summary":"A map $f$ from the full shift $\\{0, \\dots, N-1\\}^{\\mathbb{Z}}$ on $N$ symbols to itself is a one-dimensional cellular automaton if $f$ is continuous and commutes with the unit shift $\\sigma$, given by $(\\sigma x)_i = x_{i+1}$, that is, $f \\circ \\sigma = \\sigma \\circ f$.","title":"One-dimensional cellular automaton"},{"id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/jointly-periodic-points","kind":"node","lean":"SymbolicDynamics.FullShift.jointlyPeriodicPoints","parent":22453,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html#jointly-periodic-points","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","summary":"For a map $f$ from the full shift $\\{0, \\dots, N-1\\}^{\\mathbb{Z}}$ on $N$ symbols to itself, the jointly periodic points of $f$ are the configurations $x$ that are periodic for $f$ and periodic for the unit shift $\\sigma$, given by $(\\sigma x)_i = x_{i+1}$: there are $m \\ge 1$ and $n \\ge 1$ with $f^m(x) = x$ and $\\sigma^n(x) = x$.","title":"Jointly periodic points"},{"id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/joint-periodic-count","kind":"node","lean":"SymbolicDynamics.FullShift.jointPeriodicCount","parent":22453,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html#joint-periodic-count","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$ and a map $f$ from the full shift $\\{0, \\dots, N-1\\}^{\\mathbb{Z}}$ on $N$ symbols to itself, $P(f, k)$ is the number of configurations $x$ with $\\sigma^k(x) = x$, where $\\sigma$ is the unit shift given by $(\\sigma x)_i = x_{i+1}$, that are also periodic for $f$, that is, $f^n(x) = x$ for some $n \\ge 1$. Here $k$ need not be the least period of $x$ under $\\sigma$.","title":"Joint periodic count"},{"id":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/joint-periodic-growth-rate","kind":"node","lean":"SymbolicDynamics.FullShift.jointPeriodicGrowthRate","parent":22453,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata/index.html#joint-periodic-growth-rate","scope":"Dynamics/topological-symbolic-dynamics/full-shift-and-cellular-automata","status":"fully_proved","summary":"For a map $f$ from the full shift $S_N$ on $N$ symbols to itself, the joint periodic growth rate is the real number","title":"Joint periodic growth rate"},{"data_url":"scopes/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts.json","id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","kind":"scope","navigation":"module","parent":22441,"result":false,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"can_prove","title":"Conjugacy of aperiodic parts of shifts"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/x","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.X","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#x","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"fully_proved","summary":"$X$ is the space of all two-sided sequences $x = (x_i)_{i \\in \\mathbb{Z}}$ with every $x_i \\in \\{0, 1\\}$, carrying the product topology.","title":"The full 2-shift"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/yamb","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.YAmb","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#yamb","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"fully_proved","summary":"The space of all two-sided sequences $y = (y_i)_{i \\in \\mathbb{Z}}$ with every $y_i \\in \\{0, 1, 2\\}$, carrying the product topology. It is the ambient space of the proper 3-colouring shift.","title":"The full 3-shift"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/y","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.Y","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#y","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"fully_proved","summary":"$Y$ is the set of sequences $y = (y_i)_{i \\in \\mathbb{Z}}$ in the full 3-shift (so every $y_i \\in \\{0, 1, 2\\}$) such that $y_i \\ne y_{i+1}$ for every $i \\in \\mathbb{Z}$.","title":"The proper 3-colouring shift"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/aperiodic-x","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.AperiodicX","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#aperiodic-x","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"fully_proved","summary":"Let $\\sigma$ be the left shift on the full 2-shift $X$, given by $(\\sigma x)_i = x_{i+1}$. The aperiodic part of $X$ is the set of all $x \\in X$ that are not periodic for $\\sigma$, that is, with $\\sigma^n x \\ne x$ for every $n \\ge 1$.","title":"Aperiodic part of the full 2-shift"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/aperiodic-y","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.AperiodicY","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#aperiodic-y","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"fully_proved","summary":"Let $\\sigma$ be the left shift on the full 3-shift, given by $(\\sigma y)_i = y_{i+1}$. The aperiodic part of the proper 3-colouring shift $Y$ is the set of all $y \\in Y$ that are not periodic for $\\sigma$, that is, with $\\sigma^n y \\ne y$ for every $n \\ge 1$.","title":"Aperiodic part of the 3-colouring shift"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/shift-xres","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.shiftXRes","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#shift-xres","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"fully_proved","summary":"The left shift $\\sigma$, given by $(\\sigma x)_i = x_{i+1}$, maps the aperiodic part of the full 2-shift $X$ into itself; this is the self-map of the aperiodic part of $X$ obtained by restricting $\\sigma$ to it.","title":"Shift on the aperiodic part of the full 2-shift"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/shift-yres","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.shiftYRes","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#shift-yres","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"fully_proved","summary":"The left shift $\\sigma$, given by $(\\sigma y)_i = y_{i+1}$, maps the aperiodic part of the proper 3-colouring shift $Y$ into itself; this is the self-map of the aperiodic part of $Y$ obtained by restricting $\\sigma$ to it.","title":"Shift on the aperiodic part of the 3-colouring shift"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/conjecture","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.conjecture","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#conjecture","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"fully_proved","summary":"Let $X$ be the full 2-shift and $Y$ the proper 3-colouring shift, write $X \\setminus \\mathrm{Per}(X)$ and $Y \\setminus \\mathrm{Per}(Y)$ for their aperiodic parts with the subspace topologies, and let $\\sigma$ be the left shift $(\\sigma z)_i = z_{i+1}$ on each. The statement is that there is a homeomorphism $e \\colon X \\setminus \\mathrm{Per}(X) \\to Y \\setminus \\mathrm{Per}(Y)$ with $e(\\sigma a) = \\sigma(e(a))$ for\u2026","title":"Hochman's aperiodic conjugacy question"},{"id":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/conjecture-refuted","kind":"node","lean":"MathlibExt.Dynamics.AMR43Problem1Wanted.conjecture_refuted","parent":22462,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts/index.html#conjecture-refuted","scope":"Dynamics/topological-symbolic-dynamics/conjugacy-of-aperiodic-parts-of-shifts","status":"can_prove","summary":"Let $X$ be the full 2-shift, $Y$ the proper 3-colouring shift, and $\\sigma$ the left shift on each. There is no homeomorphism $e$ from the aperiodic part of $X$ onto the aperiodic part of $Y$, each with the subspace topology, such that $e(\\sigma a) = \\sigma(e(a))$ for every $a$ in the aperiodic part of $X$; that is, Hochman's aperiodic conjugacy question has a negative answer.","title":"The aperiodic parts of the full 2-shift and the 3-colouring shift are not conjugate"},{"data_url":"scopes/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits.json","id":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","kind":"scope","navigation":"module","parent":22441,"result":false,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/index.html","scope":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","status":"fully_proved","title":"Preperiodic points and grand orbits"},{"id":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/forward-orbit","kind":"node","lean":"Function.forwardOrbit","parent":22472,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/index.html#forward-orbit","scope":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","status":"fully_proved","summary":"For a self-map $f$ of a set $X$ and a point $x \\in X$, the forward orbit of $x$ under $f$ is the set $\\{f^n(x) : n \\in \\mathbb{N}\\}$ of all iterates of $x$, including $f^0(x) = x$.","title":"Forward orbit"},{"id":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/is-preperiodic-pt","kind":"node","lean":"Function.IsPreperiodicPt","parent":22472,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/index.html#is-preperiodic-pt","scope":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","status":"fully_proved","summary":"A point $x$ is preperiodic for a self-map $f$ when its forward orbit $\\{f^n(x) : n \\in \\mathbb{N}\\}$ under $f$ is a finite set.","title":"Preperiodic point"},{"id":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/is-strictly-preperiodic-pt","kind":"node","lean":"Function.IsStrictlyPreperiodicPt","parent":22472,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/index.html#is-strictly-preperiodic-pt","scope":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","status":"fully_proved","summary":"A point $x$ is strictly preperiodic for a self-map $f$ when it is preperiodic for $f$ but not periodic, that is, there is no $n > 0$ with $f^n(x) = x$.","title":"Strictly preperiodic point"},{"id":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/is-preperiodic-pt-of-type","kind":"node","lean":"Function.IsPreperiodicPtOfType","parent":22472,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/index.html#is-preperiodic-pt-of-type","scope":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","status":"fully_proved","summary":"Let $f$ be a self-map of a set $X$ and let $m$ and $n$ be natural numbers. A point $x \\in X$ is preperiodic of type $(m, n)$ when $n > 0$ and $f^{m+n}(x) = f^m(x)$, that is, $f^m(x)$ is periodic with period $n$. Neither $m$ nor $n$ is required to be minimal.","title":"Preperiodic point of a given type"},{"id":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/grand-orbit-rel","kind":"node","lean":"Function.GrandOrbitRel","parent":22472,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/index.html#grand-orbit-rel","scope":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","status":"fully_proved","summary":"For a self-map $f$ of a set $X$, points $x, y \\in X$ are related, that is, lie in the same grand orbit of $f$, when $f^m(x) = f^n(y)$ for some natural numbers $m$ and $n$.","title":"Grand orbit relation"},{"id":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/grand-orbit-setoid","kind":"node","lean":"Function.grandOrbitSetoid","parent":22472,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/index.html#grand-orbit-setoid","scope":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","status":"fully_proved","summary":"For a self-map $f$ of a set $X$, the relation on $X$ in which $x \\sim y$ when $f^m(x) = f^n(y)$ for some natural numbers $m$ and $n$ is an equivalence relation; its equivalence classes are the grand orbits of $f$.","title":"Grand orbit equivalence relation"},{"id":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/grand-orbit","kind":"node","lean":"Function.grandOrbit","parent":22472,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits/index.html#grand-orbit","scope":"Dynamics/topological-symbolic-dynamics/preperiodic-points-and-grand-orbits","status":"fully_proved","summary":"For a self-map $f$ of a set $X$ and a point $x \\in X$, the grand orbit of $x$ is the set of all $y \\in X$ such that $f^m(x) = f^n(y)$ for some natural numbers $m$ and $n$, that is, the equivalence class of $x$ under the grand orbit relation.","title":"Grand orbit"},{"data_url":"scopes/Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence.json","id":"Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence","kind":"scope","navigation":"module","parent":22441,"result":false,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence/index.html","scope":"Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence","status":"can_prove","title":"Topological multiple recurrence"},{"id":"Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence/is-multiply-recurrent","kind":"node","lean":"MathlibExt.Dynamics.FurstenbergTopologicalWanted.Dynamics.IsMultiplyRecurrent","parent":22480,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence/index.html#is-multiply-recurrent","scope":"Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence","status":"fully_proved","summary":"Let $X$ be a topological space and $T \\colon X \\to X$ a map. A point $x \\in X$ is multiply recurrent for $T$ if for every integer $d \\ge 1$ there is a strictly increasing sequence of natural numbers $(n_k)$ such that $T^{j n_k}(x) \\to x$ as $k \\to \\infty$ for every integer $j$ with $1 \\le j \\le d$, where $T^{m}$ denotes the $m$-th iterate of $T$.","title":"Multiply recurrent point"},{"id":"Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence/furstenberg-topological-recurrence","kind":"node","lean":"MathlibExt.Dynamics.FurstenbergTopologicalWanted.Dynamics.furstenberg_topological_recurrence","parent":22480,"result":true,"route":"../roadmap/Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence/index.html#furstenberg-topological-recurrence","scope":"Dynamics/topological-symbolic-dynamics/topological-multiple-recurrence","status":"can_prove","summary":"Every homeomorphism $T$ of a nonempty compact metric space $X$ has a multiply recurrent point: there is a point $x \\in X$ such that for every integer $d \\ge 1$ there is a strictly increasing sequence of natural numbers $(n_k)$ with $T^{j n_k}(x) \\to x$ as $k \\to \\infty$ for every integer $j$ with $1 \\le j \\le d$, where $T^{m}$ denotes the $m$-th iterate of $T$.","title":"Furstenberg\u2013Weiss multiple recurrence for a single homeomorphism"},{"data_url":"scopes/Dynamics/smooth-dynamics.json","id":"Dynamics/smooth-dynamics","kind":"scope","parent":22310,"result":false,"route":"../roadmap/Dynamics/smooth-dynamics/index.html","scope":"Dynamics/smooth-dynamics","status":"can_prove","summary":"Smooth dynamics studies differentiable maps and flows: hyperbolicity, stable manifolds and shadowing, Lyapunov exponents and entropy, Hamiltonian perturbation theory, and stability of flows given by differential equations.","title":"Smooth dynamics"},{"data_url":"scopes/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents.json","id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","kind":"scope","navigation":"module","parent":22483,"result":false,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"can_prove","title":"Entropy, dimension and Lyapunov exponents"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/euc-plane","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.EucPlane","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#euc-plane","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"The Euclidean plane $\\mathbb{R}^2$ with its Euclidean inner product and norm. The theorems of this module model a compact surface by a compact invariant subset of this plane.","title":"Euclidean plane"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/dim-measure","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.dimMeasure","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#dim-measure","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"For a measure $\\mu$ on an extended metric space $M$ equipped with a \u03c3-algebra, the Hausdorff dimension of $\\mu$ is the infimum, in $[0, \\infty]$, of the Hausdorff dimensions of the measurable sets $s \\subseteq M$ with $\\mu(M \\setminus s) = 0$.","title":"Hausdorff dimension of a measure"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/is-measurable-partition","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.IsMeasurablePartition","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#is-measurable-partition","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $M$. A finite family $P$ of subsets of $M$ is a measurable partition of $M$ modulo $\\mu$-null sets when every member of $P$ is measurable, the complement of the union of the members of $P$ has $\\mu$-measure $0$, and any two distinct members $A \\ne B$ of $P$ satisfy $\\mu(A \\cap B) = 0$.","title":"Measurable partition modulo null sets"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/partition-entropy","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.partitionEntropy","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#partition-entropy","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"For a measure $\\mu$ on a measurable space $M$ and a finite collection $P$ of subsets of $M$, the entropy of $P$ is $H_\\mu(P) = \\sum_{A \\in P} -\\mu(A) \\log \\mu(A)$, where $\\log$ is the natural logarithm, with the convention $0 \\log 0 = 0$. Here $\\mu(A)$ is taken as a real number, so a set of infinite measure contributes $0$.","title":"Shannon entropy of a finite family of sets"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/iterated-join","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.iteratedJoin","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#iterated-join","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"For a map $T : M \\to M$, a finite family $P$ of subsets of $M$ and $n \\in \\mathbb{N}$, the iterated join $\\bigvee_{k=0}^{n-1} T^{-k} P$ is the finite family of all sets $\\bigcap_{k=0}^{n-1} T^{-k}(A_k)$ with $A_0, \\dots, A_{n-1} \\in P$, where $T^{-k}(A)$ is the preimage of $A$ under the $k$-th iterate $T^k$. For $n = 0$ it is $\\{M\\}$.","title":"Iterated join of a partition"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/entropy-w","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.entropyW","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#entropy-w","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $M$, let $T : M \\to M$ be a map and let $P$ be a finite family of subsets of $M$. The entropy of $T$ relative to $P$ is","title":"Entropy of a map relative to a partition"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/kolmogorov-sinai-entropy","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.kolmogorovSinaiEntropy","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#kolmogorov-sinai-entropy","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"Let $\\mu$ be a measure on a measurable space $M$ and let $T : M \\to M$ be a map. The Kolmogorov\u2013Sinai entropy $h_\\mu(T)$ is the supremum of the entropies $h_\\mu(T, P)$ of $T$ relative to $P$ over all finite measurable partitions $P$ of $M$ modulo $\\mu$-null sets, taken to be $0$ when these entropies are not bounded above.","title":"Kolmogorov\u2013Sinai entropy"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/lyapunov-upper-at","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.lyapunovUpperAt","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#lyapunov-upper-at","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"For a map $T : \\mathbb{R}^2 \\to \\mathbb{R}^2$ of the Euclidean plane and a point $x \\in \\mathbb{R}^2$, the upper Lyapunov exponent of $T$ at $x$ is","title":"Upper Lyapunov exponent"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/lyapunov-lower-at","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.lyapunovLowerAt","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#lyapunov-lower-at","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"For a map $T : \\mathbb{R}^2 \\to \\mathbb{R}^2$ of the Euclidean plane and a point $x \\in \\mathbb{R}^2$, the lower Lyapunov exponent of $T$ at $x$ is","title":"Lower Lyapunov exponent"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/harmonic-mean-lyapunov","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.harmonicMeanLyapunov","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#harmonic-mean-lyapunov","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"fully_proved","summary":"For real numbers $\\lambda_1$ and $\\lambda_2$, this is the harmonic mean $\\frac{2 \\lambda_1 (-\\lambda_2)}{\\lambda_1 - \\lambda_2}$ of $\\lambda_1$ and $-\\lambda_2$, taken to be $0$ when $\\lambda_1 = \\lambda_2$.","title":"Harmonic mean of Lyapunov exponents"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/entropy-dimension-lyapunov","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.entropy_dimension_lyapunov","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#entropy-dimension-lyapunov","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"can_prove","summary":"Let $T$ be a $C^2$ diffeomorphism of the Euclidean plane $\\mathbb{R}^2$: $T$ has a two-sided inverse $T^{-1}$ and both $T$ and $T^{-1}$ are $C^2$. Let $K \\subseteq \\mathbb{R}^2$ be a compact set with $T(K) = K$, and let $\\mu$ be a Borel probability measure on $\\mathbb{R}^2$ with $\\mu(\\mathbb{R}^2 \\setminus K) = 0$ that is $T$-invariant and ergodic for $T$. Let $\\lambda_1$ and $\\lambda_2$ be the $\\mu$-integrals of\u2026","title":"Young's entropy\u2013dimension\u2013Lyapunov formula (planar case)"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/pesin-formula","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.pesin_formula","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#pesin-formula","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"can_prove","summary":"Let $T$ be a $C^2$ diffeomorphism of the Euclidean plane $\\mathbb{R}^2$, that is, a $C^2$ map with a $C^2$ two-sided inverse. Let $K \\subseteq \\mathbb{R}^2$ be a compact set with $T(K) = K$, and let $\\mu$ be an ergodic $T$-invariant Borel probability measure on $\\mathbb{R}^2$ with $\\mu(\\mathbb{R}^2 \\setminus K) = 0$. Write $h_\\mu(T)$ for the Kolmogorov\u2013Sinai entropy of $T$ with respect to $\\mu$, and $\\lambda_1(x)$\u2026","title":"Pesin entropy formula (planar case)"},{"id":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/margulis-ruelle","kind":"node","lean":"MathlibExt.Dynamics.LaiSangYoungWanted.Dynamics.margulis_ruelle","parent":22484,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents/index.html#margulis-ruelle","scope":"Dynamics/smooth-dynamics/entropy-dimension-and-lyapunov-exponents","status":"can_prove","summary":"Let $T$ be a $C^2$ diffeomorphism of the Euclidean plane $\\mathbb{R}^2$, that is, a $C^2$ map with a $C^2$ two-sided inverse. Let $K \\subseteq \\mathbb{R}^2$ be a compact set with $T(K) = K$, and let $\\mu$ be an ergodic $T$-invariant Borel probability measure on $\\mathbb{R}^2$ with $\\mu(\\mathbb{R}^2 \\setminus K) = 0$. Write $h_\\mu(T)$ for the Kolmogorov\u2013Sinai entropy of $T$ with respect to $\\mu$, and $\\lambda_1(x)$\u2026","title":"Margulis\u2013Ruelle inequality (planar case)"},{"data_url":"scopes/Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem.json","id":"Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem","kind":"scope","navigation":"module","parent":22483,"result":false,"route":"../roadmap/Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem/index.html","scope":"Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem","status":"can_prove","title":"Kolmogorov\u2013Arnold\u2013Moser theorem"},{"id":"Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem/is-diophantine","kind":"node","lean":"MathlibExt.Dynamics.KAMWanted.Dynamics.IsDiophantine","parent":22498,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem/index.html#is-diophantine","scope":"Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem","status":"fully_proved","summary":"A real number $\\alpha$ is Diophantine (with exponent $2$, as used in KAM theory) if there is $C > 0$ such that $C / q^2 \\le |\\alpha - p/q|$ for all integers $p$ and $q$ with $q \\neq 0$.","title":"Diophantine number"},{"id":"Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem/kam-invariant-curve","kind":"node","lean":"MathlibExt.Dynamics.KAMWanted.Dynamics.kam_invariant_curve","parent":22498,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem/index.html#kam-invariant-curve","scope":"Dynamics/smooth-dynamics/kolmogorov-arnold-moser-theorem","status":"can_prove","summary":"Let $\\alpha$ be a Diophantine real number and let $f : \\mathbb{R} \\to \\mathbb{R}$ be real-analytic, $1$-periodic and not constant, with $\\int_0^1 f(x)\\,dx = 0$. Then there is $c_0 > 0$ such that for every real $c$ with $|c| < c_0$ there is a $C^\\infty$, strictly increasing function $q : \\mathbb{R} \\to \\mathbb{R}$ such that $t \\mapsto q(t) - t$ is $1$-periodic and, for every real $t$,","title":"KAM theorem for twist maps (persistence of an invariant curve)"},{"data_url":"scopes/Dynamics/smooth-dynamics/lasalle-s-invariance-principle.json","id":"Dynamics/smooth-dynamics/lasalle-s-invariance-principle","kind":"scope","navigation":"module","parent":22483,"result":false,"route":"../roadmap/Dynamics/smooth-dynamics/lasalle-s-invariance-principle/index.html","scope":"Dynamics/smooth-dynamics/lasalle-s-invariance-principle","status":"fully_proved","title":"LaSalle's invariance principle"},{"id":"Dynamics/smooth-dynamics/lasalle-s-invariance-principle/omega-limit-singleton-subset-of-is-close","kind":"node","lean":"Flow.omegaLimit_singleton_subset_of_isClosed","parent":22501,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/lasalle-s-invariance-principle/index.html#omega-limit-singleton-subset-of-is-close","scope":"Dynamics/smooth-dynamics/lasalle-s-invariance-principle","status":"fully_proved","summary":"The \u03c9-limit set of a point under a semiflow is contained in every closed set that contains the forward orbit.","title":"Omega-limit set lies in every closed set containing the orbit"},{"id":"Dynamics/smooth-dynamics/lasalle-s-invariance-principle/exists-forall-eq-of-mem-omega-limit","kind":"node","lean":"Flow.exists_forall_eq_of_mem_omegaLimit","parent":22501,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/lasalle-s-invariance-principle/index.html#exists-forall-eq-of-mem-omega-limit","scope":"Dynamics/smooth-dynamics/lasalle-s-invariance-principle","status":"fully_proved","summary":"If the forward orbit of `x` lies in a compact set `K` and the continuous function `V` is nonincreasing along it, then there is a constant `c` (the limit of `V(\u03d5 t x)`) such that `V(\u03d5 t y) = c` for every `y \u2208 \u03c9\u207a(x)` and every `t \u2265 0`.","title":"`V` is constant on the \u03c9-limit set"},{"id":"Dynamics/smooth-dynamics/lasalle-s-invariance-principle/lasalle","kind":"node","lean":"Flow.lasalle","parent":22501,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/lasalle-s-invariance-principle/index.html#lasalle","scope":"Dynamics/smooth-dynamics/lasalle-s-invariance-principle","status":"fully_proved","summary":"LaSalle's invariance principle (LaSalle 1960; Khalil 2002, Theorem 4.4). Let the forward orbit of `x` under the semiflow `\u03d5` lie in a compact set `K`, let `V` be continuous and nonincreasing along that orbit, and let `E` contain every `y \u2208 K` whose orbit keeps `V` constant. Then the \u03c9-limit set `\u03c9\u207a(x)` is nonempty and contained in the largest forward-invariant subset `M = {y | \u2200 t, \u03d5 t y \u2208 E}` of `E`, and the orbit\u2026","title":"LaSalle's invariance principle"},{"data_url":"scopes/Dynamics/smooth-dynamics/shadowing-lemma.json","id":"Dynamics/smooth-dynamics/shadowing-lemma","kind":"scope","navigation":"module","parent":22483,"result":false,"route":"../roadmap/Dynamics/smooth-dynamics/shadowing-lemma/index.html","scope":"Dynamics/smooth-dynamics/shadowing-lemma","status":"can_prove","title":"Shadowing lemma"},{"id":"Dynamics/smooth-dynamics/shadowing-lemma/e","kind":"node","lean":"MathlibExt.Dynamics.HyperbolicShadowingWanted.Dynamics.HyperbolicShadowingProblem.E","parent":22505,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/shadowing-lemma/index.html#e","scope":"Dynamics/smooth-dynamics/shadowing-lemma","status":"fully_proved","summary":"For a natural number $d$, $E_d$ is the Euclidean space $\\mathbb{R}^d$, the real $d$-tuples with the Euclidean norm.","title":"Euclidean model space for the shadowing lemma"},{"id":"Dynamics/smooth-dynamics/shadowing-lemma/hyperbolic-structure","kind":"node","lean":"MathlibExt.Dynamics.HyperbolicShadowingWanted.Dynamics.HyperbolicShadowingProblem.HyperbolicStructure","parent":22505,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/shadowing-lemma/index.html#hyperbolic-structure","scope":"Dynamics/smooth-dynamics/shadowing-lemma","status":"fully_proved","summary":"Let $T : \\mathbb{R}^d \\to \\mathbb{R}^d$ be a homeomorphism and let $K \\subseteq \\mathbb{R}^d$. Write $DF_x$ for the derivative of a map $F$ at $x$, $T^n$ for the $n$-fold iterate of $T$ and $T^{-n}$ for the $n$-fold iterate of $T^{-1}$. A hyperbolic structure for $T$ on $K$ consists of the following: $T$ and $T^{-1}$ are $C^1$; $T(K) = K$; to every $x \\in \\mathbb{R}^d$ are assigned linear subspaces $E^s(x)$ and\u2026","title":"Hyperbolic structure"},{"id":"Dynamics/smooth-dynamics/shadowing-lemma/is-hyperbolic","kind":"node","lean":"MathlibExt.Dynamics.HyperbolicShadowingWanted.Dynamics.HyperbolicShadowingProblem.IsHyperbolic","parent":22505,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/shadowing-lemma/index.html#is-hyperbolic","scope":"Dynamics/smooth-dynamics/shadowing-lemma","status":"fully_proved","summary":"Let $T : \\mathbb{R}^d \\to \\mathbb{R}^d$ be a homeomorphism. A set $K \\subseteq \\mathbb{R}^d$ is a hyperbolic invariant set for $T$ if there exists a hyperbolic structure for $T$ on $K$.","title":"Hyperbolic invariant set"},{"id":"Dynamics/smooth-dynamics/shadowing-lemma/is-pseudo-orbit","kind":"node","lean":"MathlibExt.Dynamics.HyperbolicShadowingWanted.Dynamics.HyperbolicShadowingProblem.IsPseudoOrbit","parent":22505,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/shadowing-lemma/index.html#is-pseudo-orbit","scope":"Dynamics/smooth-dynamics/shadowing-lemma","status":"fully_proved","summary":"Let $d$ be a natural number, let $\\mathbb{R}^d$ carry the Euclidean norm, let $T : \\mathbb{R}^d \\to \\mathbb{R}^d$ be a map and let $\\varepsilon$ be a real number. A sequence $(x_n)_{n \\in \\mathbb{N}}$ in $\\mathbb{R}^d$ is an $\\varepsilon$-pseudo orbit of $T$ if $\\|x_{n+1} - T(x_n)\\| < \\varepsilon$ for every $n \\in \\mathbb{N}$.","title":"Pseudo-orbit"},{"id":"Dynamics/smooth-dynamics/shadowing-lemma/has-shadowing","kind":"node","lean":"MathlibExt.Dynamics.HyperbolicShadowingWanted.Dynamics.HyperbolicShadowingProblem.HasShadowing","parent":22505,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/shadowing-lemma/index.html#has-shadowing","scope":"Dynamics/smooth-dynamics/shadowing-lemma","status":"fully_proved","summary":"Let $T : \\mathbb{R}^d \\to \\mathbb{R}^d$ be a map, write $T^n$ for its $n$-fold iterate, and let $K \\subseteq \\mathbb{R}^d$. The set $K$ has the shadowing property for $T$ if there is an open set $U \\subseteq \\mathbb{R}^d$ containing $K$ such that for every $\\delta > 0$ there is $\\varepsilon > 0$ with the following property: for every $\\varepsilon$-pseudo orbit $(x_n)_{n \\in \\mathbb{N}}$ of $T$ with $x_n \\in U$ for\u2026","title":"Shadowing property"},{"id":"Dynamics/smooth-dynamics/shadowing-lemma/hyperbolic-has-shadowing","kind":"node","lean":"MathlibExt.Dynamics.HyperbolicShadowingWanted.Dynamics.HyperbolicShadowingProblem.hyperbolic_has_shadowing","parent":22505,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/shadowing-lemma/index.html#hyperbolic-has-shadowing","scope":"Dynamics/smooth-dynamics/shadowing-lemma","status":"can_prove","summary":"Let $T : \\mathbb{R}^d \\to \\mathbb{R}^d$ be a homeomorphism and let $K \\subseteq \\mathbb{R}^d$ be a compact set that is a hyperbolic invariant set for $T$. Then $K$ has the shadowing property for $T$.","title":"Anosov\u2013Bowen shadowing lemma"},{"data_url":"scopes/Dynamics/smooth-dynamics/stable-manifold-theorem.json","id":"Dynamics/smooth-dynamics/stable-manifold-theorem","kind":"scope","navigation":"module","parent":22483,"result":false,"route":"../roadmap/Dynamics/smooth-dynamics/stable-manifold-theorem/index.html","scope":"Dynamics/smooth-dynamics/stable-manifold-theorem","status":"can_prove","title":"Stable manifold theorem"},{"id":"Dynamics/smooth-dynamics/stable-manifold-theorem/e","kind":"node","lean":"MathlibExt.Dynamics.StableUnstableManifoldsWanted.Dynamics.StableUnstableManifoldsProblem.E","parent":22512,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euclidean space "],["math","\\(\\mathbb{R}^n\\)"],["text"," for local stable and unstable sets"]],"source":"Euclidean space $\\mathbb{R}^n$ for local stable and unstable sets"},"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/stable-manifold-theorem/index.html#e","scope":"Dynamics/smooth-dynamics/stable-manifold-theorem","status":"fully_proved","summary":"For a natural number $n$, this is the space $\\mathbb{R}^n$ of real $n$-tuples with the Euclidean inner product and norm.","title":"Euclidean space $\\mathbb{R}^n$ for local stable and unstable sets"},{"id":"Dynamics/smooth-dynamics/stable-manifold-theorem/complex-eigenvalues","kind":"node","lean":"MathlibExt.Dynamics.StableUnstableManifoldsWanted.Dynamics.StableUnstableManifoldsProblem.complexEigenvalues","parent":22512,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/stable-manifold-theorem/index.html#complex-eigenvalues","scope":"Dynamics/smooth-dynamics/stable-manifold-theorem","status":"fully_proved","summary":"For a continuous linear map $L$ from $\\mathbb{R}^n$ to itself, the complex eigenvalues of $L$ form the multiset of complex roots of the characteristic polynomial of $L$, each counted with its multiplicity as a root.","title":"Complex eigenvalues of a real linear map"},{"id":"Dynamics/smooth-dynamics/stable-manifold-theorem/is-hyperbolic-linear","kind":"node","lean":"MathlibExt.Dynamics.StableUnstableManifoldsWanted.Dynamics.StableUnstableManifoldsProblem.IsHyperbolicLinear","parent":22512,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/stable-manifold-theorem/index.html#is-hyperbolic-linear","scope":"Dynamics/smooth-dynamics/stable-manifold-theorem","status":"fully_proved","summary":"A continuous linear map $L$ from $\\mathbb{R}^n$ to itself is hyperbolic when none of its complex eigenvalues lies on the unit circle, that is, every complex root $\\mu$ of the characteristic polynomial of $L$ satisfies $|\\mu| \\ne 1$.","title":"Hyperbolic linear map"},{"id":"Dynamics/smooth-dynamics/stable-manifold-theorem/stable-unstable-manifolds-exist","kind":"node","lean":"MathlibExt.Dynamics.StableUnstableManifoldsWanted.Dynamics.StableUnstableManifoldsProblem.stable_unstable_manifolds_exist","parent":22512,"result":true,"route":"../roadmap/Dynamics/smooth-dynamics/stable-manifold-theorem/index.html#stable-unstable-manifolds-exist","scope":"Dynamics/smooth-dynamics/stable-manifold-theorem","status":"can_prove","summary":"Let $f : \\mathbb{R}^n \\to \\mathbb{R}^n$ be continuously differentiable on a neighbourhood of a point $x_0$ with $f(x_0) = x_0$, and suppose that the derivative $Df(x_0)$ is invertible and hyperbolic. Then there is an open set $U$ containing $x_0$ such that the local stable set $W^s = \\{x : f^k(x) \\in U \\text{ for all } k \\ge 0 \\text{ and } f^k(x) \\to x_0 \\text{ as } k \\to \\infty\\}$ and the local unstable set $W^u$,\u2026","title":"Local stable and unstable sets at a hyperbolic fixed point"},{"data_url":"scopes/Dynamics/low-dimensional-dynamics.json","id":"Dynamics/low-dimensional-dynamics","kind":"scope","parent":22310,"result":false,"route":"../roadmap/Dynamics/low-dimensional-dynamics/index.html","scope":"Dynamics/low-dimensional-dynamics","status":"can_prove","summary":"Low-dimensional dynamics covers maps of the interval and circle, continued-fraction-type maps, flows in the plane and homeomorphisms of surfaces.","title":"Low-dimensional dynamics"},{"id":"Dynamics/low-dimensional-dynamics/li-yorke-period-three-implies-periods","kind":"node","lean":"MathlibExt.Dynamics.Periodic.LiYorkeWanted.liYorke_period_three_implies_periods","parent":22517,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/index.html#li-yorke-period-three-implies-periods","scope":"Dynamics/low-dimensional-dynamics","status":"fully_proved","summary":"Let $f \\colon [0, 1] \\to [0, 1]$ be a continuous map. If some point of $[0, 1]$ has minimal period $3$ under $f$, then for every positive integer $n$ some point of $[0, 1]$ has minimal period $n$ under $f$.","title":"Li\u2013Yorke theorem: period three implies all periods"},{"data_url":"scopes/Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms.json","id":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms","kind":"scope","navigation":"module","parent":22517,"result":false,"route":"../roadmap/Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/index.html","scope":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms","status":"can_prove","title":"Linearization of circle diffeomorphisms"},{"id":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/is-diophantine","kind":"node","lean":"MathlibExt.Analysis.HermanProblem290Wanted.IsDiophantine","parent":22519,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/index.html#is-diophantine","scope":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms","status":"fully_proved","summary":"A real number $\\alpha$ is Diophantine when there are real numbers $\\gamma > 0$ and $\\tau \\ge 0$ such that $|\\alpha - p/q| \\ge \\gamma / q^{2 + \\tau}$ for every integer $p$ and every integer $q \\ge 1$.","title":"Diophantine numbers"},{"id":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/analytic-circle-diffeo","kind":"node","lean":"MathlibExt.Analysis.HermanProblem290Wanted.AnalyticCircleDiffeo","parent":22519,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/index.html#analytic-circle-diffeo","scope":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms","status":"fully_proved","summary":"A real-analytic orientation-preserving diffeomorphism of the circle is given by a lift $F : \\mathbb{R} \\to \\mathbb{R}$ together with an inverse lift $G : \\mathbb{R} \\to \\mathbb{R}$, both real-analytic on all of $\\mathbb{R}$, such that for every real $x$ we have $F(x + 1) = F(x) + 1$, $G(x + 1) = G(x) + 1$, $G(F(x)) = x$ and $F(G(x)) = x$, and $F$ is strictly increasing.","title":"Real-analytic circle diffeomorphisms"},{"id":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/has-rotation-number","kind":"node","lean":"MathlibExt.Analysis.HermanProblem290Wanted.HasRotationNumber","parent":22519,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/index.html#has-rotation-number","scope":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms","status":"fully_proved","summary":"Let $F : \\mathbb{R} \\to \\mathbb{R}$ be a function, and for a natural number $n$ write $F^n$ for the $n$-th iterate of $F$. Then $F$ has rotation number $\\alpha \\in \\mathbb{R}$ when, for every real $x$, $(F^n(x) - x)/n$ tends to $\\alpha$ as $n$ tends to infinity.","title":"Rotation number of a lift"},{"id":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/is-analytically-conjugate-to-rotation","kind":"node","lean":"MathlibExt.Analysis.HermanProblem290Wanted.IsAnalyticallyConjugateToRotation","parent":22519,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/index.html#is-analytically-conjugate-to-rotation","scope":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms","status":"fully_proved","summary":"A function $F : \\mathbb{R} \\to \\mathbb{R}$ is analytically conjugate to the rotation by $\\alpha \\in \\mathbb{R}$ when there is a real-analytic circle diffeomorphism with lift $H$ such that $H(F(x)) = H(x) + \\alpha$ for every real $x$.","title":"Analytic conjugacy to a rotation"},{"id":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/conjecture","kind":"node","lean":"MathlibExt.Analysis.HermanProblem290Wanted.conjecture","parent":22519,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/index.html#conjecture","scope":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms","status":"fully_proved","summary":"The statement that there is an irrational real number $\\alpha$ that is not Diophantine such that, for every real-analytic circle diffeomorphism whose lift $F$ has rotation number $\\alpha$, $F$ is analytically conjugate to the rotation by $\\alpha$.","title":"Herman's analytic linearization problem"},{"id":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/conjecture-holds","kind":"node","lean":"MathlibExt.Analysis.HermanProblem290Wanted.conjecture_holds","parent":22519,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms/index.html#conjecture-holds","scope":"Dynamics/low-dimensional-dynamics/linearization-of-circle-diffeomorphisms","status":"can_prove","summary":"There is an irrational real number $\\alpha$ that is not Diophantine such that, for every real-analytic circle diffeomorphism whose lift $F$ has rotation number $\\alpha$, $F$ is analytically conjugate to the rotation by $\\alpha$.","title":"Herman's analytic linearization problem has a positive answer"},{"data_url":"scopes/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions.json","id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","kind":"scope","navigation":"module","parent":22517,"result":false,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"can_prove","title":"N-continued fraction expansions"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/n-expansion-digit","kind":"node","lean":"MetaMathlibExt.nExpansionDigit","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#n-expansion-digit","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"fully_proved","summary":"Let $N$ be a natural number and $\\alpha$ a real number, and write $I_\\alpha = [\\alpha, \\alpha + 1]$. The digit of a real number $x$ is $d(x) = \\lfloor N/x - \\alpha \\rfloor$, except that when $N/\\alpha - \\alpha$ is an integer, the digit of $\\alpha$ itself is $d(\\alpha) = N/\\alpha - \\alpha - 1$.","title":"Digit of the N-expansion map"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/n-expansion-map","kind":"node","lean":"MetaMathlibExt.nExpansionMap","parent":22526,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(N\\)"],["text","-expansion map"]],"source":"The $N$-expansion map"},"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#n-expansion-map","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"fully_proved","summary":"For a natural number $N$ and a real number $\\alpha$, the $N$-expansion map is $T_\\alpha(x) = N/x - d_\\alpha(x)$ for real $x$, where $d_\\alpha(x)$ is the $N$-expansion digit: $d_\\alpha(x) = \\lfloor N/x - \\alpha \\rfloor$, except that $d_\\alpha(\\alpha) = \\lfloor N/\\alpha - \\alpha \\rfloor - 1$ when $N/\\alpha - \\alpha$ is an integer. The map is meant to act on the interval $I_\\alpha = [\\alpha, \\alpha + 1]$, but it is\u2026","title":"The $N$-expansion map"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/n-expansion-digit-of-ne","kind":"node","lean":"MetaMathlibExt.nExpansionDigit_of_ne","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#n-expansion-digit-of-ne","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"fully_proved","summary":"Let $N$ be a natural number and let $\\alpha$ and $x$ be real numbers with $x \\ne \\alpha$. Then the digit of $x$ for the $N$-expansion map on $[\\alpha, \\alpha + 1]$ is $\\lfloor N/x - \\alpha \\rfloor$.","title":"The digit away from the left endpoint"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/n-expansion-map-mem-ico-of-ne","kind":"node","lean":"MetaMathlibExt.nExpansionMap_mem_Ico_of_ne","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#n-expansion-map-mem-ico-of-ne","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"fully_proved","summary":"Let $N$ be a natural number and let $\\alpha$ and $x$ be real numbers with $x \\ne \\alpha$. Then the $N$-expansion map $T_\\alpha$ satisfies $\\alpha \\le T_\\alpha(x) < \\alpha + 1$.","title":"The N-expansion map away from the left endpoint"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/n-expansion-map-self-of-int","kind":"node","lean":"MetaMathlibExt.nExpansionMap_self_of_int","parent":22526,"reading_title":{"format":"inline-parts-v1","parts":[["text","The left endpoint goes to the right endpoint when "],["math","\\(N/\\alpha - \\alpha\\)"],["text"," is an integer"]],"source":"The left endpoint goes to the right endpoint when $N/\\alpha - \\alpha$ is an integer"},"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#n-expansion-map-self-of-int","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"fully_proved","summary":"Let $N$ be a natural number and $\\alpha$ a real number such that $N/\\alpha - \\alpha$ is an integer, with the convention $N/0 = 0$. Then the $N$-expansion map $T_\\alpha$ satisfies $T_\\alpha(\\alpha) = \\alpha + 1$.","title":"The left endpoint goes to the right endpoint when $N/\\alpha - \\alpha$ is an integer"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/ideal-period-word","kind":"node","lean":"MetaMathlibExt.idealPeriodWord","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#ideal-period-word","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"fully_proved","summary":"For $n \\ge 1$, the ideal period word $d_n$ is a finite list of blocks of partial quotients, each block being $a = 2, 1, 1$ or $b = 2, 2$. Writing lists of blocks by juxtaposition, with commas between parts, $d_1 = [ab]$, $d_2 = [abb]$, $d_3 = [abb, ab, ab]$ and $d_4 = [abb, ab, ab, abb, abb]$; for $n \\ge 5$, if $d_{n-1} = [j, ab, ab]$ for a list $j$ then $d_n = [j, ab, ab, j, abb]$, and if $d_{n-1} = [j, abb]$ then\u2026","title":"Ideal period words"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/ideal-period-word-five","kind":"node","lean":"MetaMathlibExt.idealPeriodWord_five","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#ideal-period-word-five","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"fully_proved","summary":"Writing $a$ for the block $2, 1, 1$ and $b$ for the block $2, 2$ of partial quotients, the ideal period word $d_5$ is $d_4$ followed by the blocks $abb\\,ab\\,ab\\,abb\\,ab\\,ab$, in this order.","title":"The fifth ideal period word"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/ideal-period-word-flatten-length-small","kind":"node","lean":"MetaMathlibExt.idealPeriodWord_flatten_length_small","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#ideal-period-word-flatten-length-small","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"fully_proved","summary":"The digit strings of the ideal period words $d_1, d_2, d_3, d_4, d_5$ have lengths $5, 7, 17, 31, 65$ respectively.","title":"Lengths of the first ideal period words"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/n-expansion-orbit-gt-iff-odd","kind":"node","lean":"MetaMathlibExt.nExpansion_orbit_gt_iff_odd","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#n-expansion-orbit-gt-iff-odd","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"can_prove","summary":"Let $N$ be a positive integer and let $0 < \\alpha < \\alpha'$ be real numbers. Write $T_\\alpha$ for the $N$-expansion map and $d_\\alpha$ for the $N$-expansion digit with parameter $\\alpha$, and let $m$ be a natural number. If the orbits $\\alpha_j = T_\\alpha^j(\\alpha)$ and $\\alpha'_j = T_{\\alpha'}^j(\\alpha')$ satisfy $d_\\alpha(\\alpha_j) = d_{\\alpha'}(\\alpha'_j)$ for all $j \\in \\{0, \\dots, m\\}$, then\u2026","title":"Orbits with the same digits alternate in order"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/ideal-period-word-point-lt-iff","kind":"node","lean":"MetaMathlibExt.idealPeriodWord_point_lt_iff","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#ideal-period-word-point-lt-iff","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"can_prove","summary":"Let $N$ be a natural number, let $\\alpha$ be a real number with $0 < \\alpha \\le \\sqrt N - 1$, and let $T_\\alpha$ be the $N$-expansion map on $I_\\alpha = [\\alpha, \\alpha + 1]$. For $m \\ge 1$ write $|d_m|$ for the length of the digit string of the ideal period word $d_m$. Let $k, \\ell \\ge 1$ and let $x, y \\in I_\\alpha$ be such that for every $i \\ge 0$ the digit of $T_\\alpha^i(x)$ is entry $i \\bmod |d_k|$ (counting\u2026","title":"The ideal periodic points decrease"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/ideal-period-word-point-le-orbit","kind":"node","lean":"MetaMathlibExt.idealPeriodWord_point_le_orbit","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#ideal-period-word-point-le-orbit","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"can_prove","summary":"Let $N$ be a natural number, let $\\alpha$ be a real number with $0 < \\alpha \\le \\sqrt N - 1$, and let $T_\\alpha$ be the $N$-expansion map on $I_\\alpha = [\\alpha, \\alpha + 1]$. Let $n \\ge 1$, write $|d_n|$ for the length of the digit string of the ideal period word $d_n$, and let $x \\in I_\\alpha$ be such that for every $i \\ge 0$ the digit of $T_\\alpha^i(x)$ is entry $i \\bmod |d_n|$ (counting from $0$) of that digit\u2026","title":"The ideal periodic point is the least point of its orbit"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/ideal-period-word-flatten-length","kind":"node","lean":"MetaMathlibExt.idealPeriodWord_flatten_length","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#ideal-period-word-flatten-length","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"can_prove","summary":"For every $n \\ge 1$, the digit string of the ideal period word $d_n$ (the concatenation of its blocks of partial quotients) has length $|d_n| = 2^{n+1} - (-1)^n$.","title":"Length of the ideal period words"},{"id":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/ideal-period-word-count-blocks","kind":"node","lean":"MetaMathlibExt.idealPeriodWord_count_blocks","parent":22526,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions/index.html#ideal-period-word-count-blocks","scope":"Dynamics/low-dimensional-dynamics/n-continued-fraction-expansions","status":"can_prove","summary":"For every $n \\ge 1$, the ideal period word $d_n$ contains exactly $2^{n-1}$ blocks $2, 2$, and the number $c_n$ of its blocks $2, 1, 1$ satisfies $3 c_n = 2^n - (-1)^n$; that is, $c_n = (2^n + 1)/3$ for odd $n$ and $c_n = (2^n - 1)/3$ for even $n$.","title":"Block counts of the ideal period words"},{"data_url":"scopes/Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps.json","id":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps","kind":"scope","navigation":"module","parent":22517,"result":false,"route":"../roadmap/Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/index.html","scope":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps","status":"can_prove","title":"Natural duals of M\u00f6bius maps"},{"id":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/moebius-fixed-point-eq-iff","kind":"node","lean":"MetaMathlibExt.moebius_fixedPoint_eq_iff","parent":22540,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/index.html#moebius-fixed-point-eq-iff","scope":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps","status":"can_prove","summary":"Let $\\beta, \\gamma, \\eta$ be real numbers with $\\beta \\ne 2$ and $3 - 2\\beta + (2 - \\beta)\\eta \\ne 0$. Then $(-2 + 2\\beta + \\gamma + (-1 + \\beta + \\gamma)\\eta)/(3 - 2\\beta + (2 - \\beta)\\eta) = \\eta$ if and only if $\\eta = -1$ or $\\eta = (-2 + 2\\beta + \\gamma)/(2 - \\beta)$.","title":"Fixed points of a Moebius dual branch"},{"id":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/moebius-dual-endpoint-ne-neg-one","kind":"node","lean":"MetaMathlibExt.moebius_dual_endpoint_ne_neg_one","parent":22540,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/index.html#moebius-dual-endpoint-ne-neg-one","scope":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps","status":"can_prove","summary":"Let $\\alpha, \\beta, \\delta$ be real numbers with $\\alpha > 0$, $\\beta < 1$ and $\\delta \\ge 0$, and define the dual branches $V^\\#_{\\alpha\\delta}(y) = (2\\alpha\\delta - 1 + \\alpha\\delta y)/(2\\alpha + 1 + \\alpha y)$ and $V^\\#_{\\beta\\delta}(y) = (2\\delta - 2\\beta\\delta - 1 + (\\delta - \\beta\\delta - 1) y)/ (3 - 2\\beta + (2 - \\beta) y)$. Then $V^\\#_{\\beta\\delta}(-1) \\ne V^\\#_{\\alpha\\delta}(-1)$.","title":"Moebius dual branches disagree at \u22121"},{"id":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/moebius-dual-central-equations","kind":"node","lean":"MetaMathlibExt.moebius_dual_central_equations","parent":22540,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/index.html#moebius-dual-central-equations","scope":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps","status":"can_prove","summary":"Let $\\alpha, \\beta, \\gamma, \\delta$ and $\\xi$ be real numbers with $\\alpha > 0$, $\\beta < 1$, $\\gamma > -1$ and $\\delta \\ge 0$. Define the dual branches $V^\\#_{\\alpha\\gamma}(y) = (\\gamma - 2\\alpha - \\alpha y)/(2\\alpha + 1 + \\alpha y)$, $V^\\#_{\\alpha\\delta}(y) = (2\\alpha\\delta - 1 + \\alpha\\delta y)/(2\\alpha + 1 + \\alpha y)$,\u2026","title":"Central equations for natural duals of Moebius maps"},{"id":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/moebius-natural-dual-singular","kind":"node","lean":"MetaMathlibExt.moebius_naturalDual_singular","parent":22540,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps/index.html#moebius-natural-dual-singular","scope":"Dynamics/low-dimensional-dynamics/natural-duals-of-m-bius-maps","status":"can_prove","summary":"Let $\\alpha, \\beta, \\gamma, \\delta$ be real numbers with $\\alpha > 0$, $\\beta < 1$, $\\gamma > -1$ and $\\delta \\ge 0$. Define the dual branches $V^\\#_{\\alpha\\gamma}(y) = (\\gamma - 2\\alpha - \\alpha y)/(2\\alpha + 1 + \\alpha y)$, $V^\\#_{\\alpha\\delta}(y) = (2\\alpha\\delta - 1 + \\alpha\\delta y)/(2\\alpha + 1 + \\alpha y)$, $V^\\#_{\\beta\\gamma}(y) = (2\\beta + \\gamma - 2 + (\\beta + \\gamma - 1) y)/ (3 - 2\\beta + (2 - \\beta) y)$\u2026","title":"Singularity of natural duals of Moebius maps (algebraic form)"},{"data_url":"scopes/Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem.json","id":"Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem","kind":"scope","navigation":"module","parent":22517,"result":false,"route":"../roadmap/Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem/index.html","scope":"Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem","status":"can_prove","title":"Poincar\u00e9\u2013Bendixson theorem"},{"id":"Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem/plane","kind":"node","lean":"MathlibExt.Dynamics.PoincareBendixsonWanted.Dynamics.Plane","parent":22545,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem/index.html#plane","scope":"Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem","status":"fully_proved","summary":"The ambient space is the plane $\\mathbb{R}^2$ with its Euclidean inner product and norm.","title":"The Euclidean plane"},{"id":"Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem/poincare-bendixson","kind":"node","lean":"MathlibExt.Dynamics.PoincareBendixsonWanted.Dynamics.poincare_bendixson","parent":22545,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem/index.html#poincare-bendixson","scope":"Dynamics/low-dimensional-dynamics/poincar-bendixson-theorem","status":"can_prove","summary":"Let $F \\colon \\mathbb{R}^2 \\to \\mathbb{R}^2$ be a continuously differentiable vector field on the Euclidean plane, and let $\\gamma \\colon \\mathbb{R} \\to \\mathbb{R}^2$ be a curve such that for every $t \\ge 0$ the derivative of $\\gamma$ at $t$ within $[0, \\infty)$ is $F(\\gamma(t))$. Let $\\omega(\\gamma) = \\bigcap_{s \\in \\mathbb{R}} \\overline{\\gamma([s, \\infty))}$ be the $\\omega$-limit set of $\\gamma$. Then at least\u2026","title":"Poincar\u00e9\u2013Bendixson theorem"},{"data_url":"scopes/Dynamics/low-dimensional-dynamics/simplicity-conjecture.json","id":"Dynamics/low-dimensional-dynamics/simplicity-conjecture","kind":"scope","navigation":"module","parent":22517,"result":false,"route":"../roadmap/Dynamics/low-dimensional-dynamics/simplicity-conjecture/index.html","scope":"Dynamics/low-dimensional-dynamics/simplicity-conjecture","status":"can_prove","title":"Simplicity conjecture"},{"id":"Dynamics/low-dimensional-dynamics/simplicity-conjecture/homeo","kind":"node","lean":"MathlibExt.Dynamics.SimplicityConjectureWanted.SimplicityConjecture.Homeo","parent":22548,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/simplicity-conjecture/index.html#homeo","scope":"Dynamics/low-dimensional-dynamics/simplicity-conjecture","status":"fully_proved","summary":"Let $\\mathbb{D}$ be the closed unit disc in $\\mathbb{R}^2$ and let $\\omega$ be the restriction of Lebesgue measure to $\\mathbb{D}$. $\\mathrm{Homeo}_c(\\mathbb{D}, \\omega)$ is the subgroup of the group of homeomorphisms of $\\mathbb{D}$ consisting of the homeomorphisms $\\phi$ that preserve $\\omega$ and whose support, the closure of $\\{x \\in \\mathbb{D} : \\phi(x) \\neq x\\}$, is contained in the open unit disc.","title":"Area-preserving homeomorphisms of the disc supported in the interior"},{"id":"Dynamics/low-dimensional-dynamics/simplicity-conjecture/theorem-1-2","kind":"node","lean":"MathlibExt.Dynamics.SimplicityConjectureWanted.SimplicityConjecture.theorem_1_2","parent":22548,"result":true,"route":"../roadmap/Dynamics/low-dimensional-dynamics/simplicity-conjecture/index.html#theorem-1-2","scope":"Dynamics/low-dimensional-dynamics/simplicity-conjecture","status":"can_prove","summary":"Let $\\mathbb{D} \\subseteq \\mathbb{R}^2$ be the closed unit disc and let $\\omega$ be the restriction of Lebesgue measure to $\\mathbb{D}$. The group $\\mathrm{Homeo}_c(\\mathbb{D}, \\omega)$ of homeomorphisms of $\\mathbb{D}$ that preserve $\\omega$ and whose support, the closure of the set of non-fixed points, lies in the open unit disc is not simple: it is trivial or it has a normal subgroup other than the trivial\u2026","title":"Simplicity conjecture"},{"data_url":"scopes/Dynamics/complex-dynamics.json","id":"Dynamics/complex-dynamics","kind":"scope","parent":22310,"result":false,"route":"../roadmap/Dynamics/complex-dynamics/index.html","scope":"Dynamics/complex-dynamics","status":"can_prove","summary":"Complex dynamics studies the iteration of holomorphic maps, especially polynomials, together with their Julia sets and parameter spaces such as the Mandelbrot set.","title":"Complex dynamics"},{"data_url":"scopes/Dynamics/complex-dynamics/fatou-julia-dichotomy.json","id":"Dynamics/complex-dynamics/fatou-julia-dichotomy","kind":"scope","navigation":"module","parent":22551,"result":false,"route":"../roadmap/Dynamics/complex-dynamics/fatou-julia-dichotomy/index.html","scope":"Dynamics/complex-dynamics/fatou-julia-dichotomy","status":"can_prove","title":"Fatou\u2013Julia dichotomy"},{"id":"Dynamics/complex-dynamics/fatou-julia-dichotomy/tc","kind":"node","lean":"MathlibExt.Analysis.FatouJuliaWanted.ComplexAnalysis.FatouJuliaProblem.Tc","parent":22552,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quadratic map "],["math","\\(T_c\\)"]],"source":"Quadratic map $T_c$"},"result":true,"route":"../roadmap/Dynamics/complex-dynamics/fatou-julia-dichotomy/index.html#tc","scope":"Dynamics/complex-dynamics/fatou-julia-dichotomy","status":"fully_proved","summary":"For $c \\in \\mathbb{C}$, $T_c \\colon \\mathbb{C} \\to \\mathbb{C}$ is the map $T_c(z) = z^2 + c$.","title":"Quadratic map $T_c$"},{"id":"Dynamics/complex-dynamics/fatou-julia-dichotomy/mandelbrot","kind":"node","lean":"MathlibExt.Analysis.FatouJuliaWanted.ComplexAnalysis.FatouJuliaProblem.Mandelbrot","parent":22552,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/fatou-julia-dichotomy/index.html#mandelbrot","scope":"Dynamics/complex-dynamics/fatou-julia-dichotomy","status":"fully_proved","summary":"Let $T_c(z) = z^2 + c$ for $c \\in \\mathbb{C}$ and let $T_c^n$ be its $n$-fold iterate. The Mandelbrot set $M$ is the set of $c \\in \\mathbb{C}$ for which the orbit of $0$ under $T_c$ is bounded: there is a real $R$ such that $|T_c^n(0)| \\le R$ for every $n \\ge 0$.","title":"Mandelbrot set of the quadratic family"},{"id":"Dynamics/complex-dynamics/fatou-julia-dichotomy/filled-julia","kind":"node","lean":"MathlibExt.Analysis.FatouJuliaWanted.ComplexAnalysis.FatouJuliaProblem.FilledJulia","parent":22552,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/fatou-julia-dichotomy/index.html#filled-julia","scope":"Dynamics/complex-dynamics/fatou-julia-dichotomy","status":"fully_proved","summary":"For $c \\in \\mathbb{C}$, let $T_c(z) = z^2 + c$ and let $T_c^n$ be its $n$-fold iterate. The filled Julia set $K_c$ is the set of $z \\in \\mathbb{C}$ whose orbit under $T_c$ is bounded: there is a real $M$ such that $|T_c^n(z)| \\le M$ for every $n \\ge 0$.","title":"Filled Julia set"},{"id":"Dynamics/complex-dynamics/fatou-julia-dichotomy/julia-cantor-dichotomy","kind":"node","lean":"MathlibExt.Analysis.FatouJuliaWanted.ComplexAnalysis.FatouJuliaProblem.julia_cantor_dichotomy","parent":22552,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/fatou-julia-dichotomy/index.html#julia-cantor-dichotomy","scope":"Dynamics/complex-dynamics/fatou-julia-dichotomy","status":"can_prove","summary":"For $c \\in \\mathbb{C}$, let $K_c$ be the filled Julia set of $T_c(z) = z^2 + c$ and let $M$ be the Mandelbrot set. If $c \\in M$, then $K_c$ is nonempty and connected. If $c \\notin M$, then $K_c$ is homeomorphic to the Cantor space $\\{0, 1\\}^{\\mathbb{N}}$ of binary sequences with the product topology.","title":"Fatou\u2013Julia dichotomy for quadratic polynomials"},{"data_url":"scopes/Dynamics/complex-dynamics/mandelbrot-set.json","id":"Dynamics/complex-dynamics/mandelbrot-set","kind":"scope","navigation":"module","parent":22551,"result":false,"route":"../roadmap/Dynamics/complex-dynamics/mandelbrot-set/index.html","scope":"Dynamics/complex-dynamics/mandelbrot-set","status":"can_prove","title":"Mandelbrot set"},{"id":"Dynamics/complex-dynamics/mandelbrot-set/tc","kind":"node","lean":"MathlibExt.Analysis.MandelbrotWanted.ComplexAnalysis.MandelbrotProblem.Tc","parent":22557,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/mandelbrot-set/index.html#tc","scope":"Dynamics/complex-dynamics/mandelbrot-set","status":"fully_proved","summary":"For $c \\in \\mathbb{C}$, $T_c \\colon \\mathbb{C} \\to \\mathbb{C}$ is the map $T_c(z) = z^2 + c$.","title":"The quadratic family"},{"id":"Dynamics/complex-dynamics/mandelbrot-set/mandelbrot","kind":"node","lean":"MathlibExt.Analysis.MandelbrotWanted.ComplexAnalysis.MandelbrotProblem.Mandelbrot","parent":22557,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/mandelbrot-set/index.html#mandelbrot","scope":"Dynamics/complex-dynamics/mandelbrot-set","status":"fully_proved","summary":"The Mandelbrot set is the set of all $c \\in \\mathbb{C}$ for which the orbit of $0$ under $T_c(z) = z^2 + c$ is bounded: there is a real number $M$ with $|T_c^n(0)| \\le M$ for every $n \\in \\mathbb{N}$, where $T_c^n$ is the $n$-th iterate of $T_c$ and $T_c^0$ is the identity.","title":"The Mandelbrot set"},{"id":"Dynamics/complex-dynamics/mandelbrot-set/mandelbrot-connected","kind":"node","lean":"MathlibExt.Analysis.MandelbrotWanted.ComplexAnalysis.MandelbrotProblem.mandelbrot_connected","parent":22557,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/mandelbrot-set/index.html#mandelbrot-connected","scope":"Dynamics/complex-dynamics/mandelbrot-set","status":"can_prove","summary":"The Mandelbrot set, the set of all $c \\in \\mathbb{C}$ for which the sequence $(T_c^n(0))_{n \\ge 0}$ of iterates of $0$ under $T_c(z) = z^2 + c$ is bounded, is a nonempty connected subset of $\\mathbb{C}$.","title":"The Mandelbrot set is connected (Douady\u2013Hubbard)"},{"id":"Dynamics/complex-dynamics/mandelbrot-set/tc-2","kind":"node","lean":"MathlibExt.Dynamics.MandelbrotBoundaryWanted.Dynamics.MandelbrotBoundary.Tc","parent":22557,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/mandelbrot-set/index.html#tc-2","scope":"Dynamics/complex-dynamics/mandelbrot-set","status":"fully_proved","summary":"For each complex number $c$, the map $T_c : \\mathbb{C} \\to \\mathbb{C}$ is given by $T_c(z) = z^2 + c$.","title":"Quadratic family"},{"id":"Dynamics/complex-dynamics/mandelbrot-set/mandelbrot-2","kind":"node","lean":"MathlibExt.Dynamics.MandelbrotBoundaryWanted.Dynamics.MandelbrotBoundary.Mandelbrot","parent":22557,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/mandelbrot-set/index.html#mandelbrot-2","scope":"Dynamics/complex-dynamics/mandelbrot-set","status":"fully_proved","summary":"The Mandelbrot set is the set $M$ of complex numbers $c$ for which the orbit of $0$ under the quadratic map $T_c(z) = z^2 + c$ is bounded: there is a real number $R$ with $|T_c^n(0)| \\le R$ for every $n \\in \\mathbb{N}$, where $T_c^n$ is the $n$-th iterate of $T_c$ and $T_c^0$ is the identity.","title":"Mandelbrot set"},{"id":"Dynamics/complex-dynamics/mandelbrot-set/mandelbrot-boundary-dimh","kind":"node","lean":"MathlibExt.Dynamics.MandelbrotBoundaryWanted.Dynamics.MandelbrotBoundary.mandelbrot_boundary_dimh","parent":22557,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/mandelbrot-set/index.html#mandelbrot-boundary-dimh","scope":"Dynamics/complex-dynamics/mandelbrot-set","status":"can_prove","summary":"The boundary of the Mandelbrot set $M \\subseteq \\mathbb{C}$ has Hausdorff dimension $2$.","title":"Hausdorff dimension of the Mandelbrot boundary (Shishikura)"},{"data_url":"scopes/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials.json","id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","kind":"scope","navigation":"module","parent":22551,"result":false,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"can_prove","title":"Parameter space of cubic polynomials"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/cubic-family","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.cubicFamily","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#cubic-family","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"fully_proved","summary":"For a parameter $\\lambda \\in \\mathbb{C}$, the cubic polynomial map $f_\\lambda : \\mathbb{C} \\to \\mathbb{C}$ is given by $f_\\lambda(z) = \\lambda z^2 + z^3$.","title":"Cubic family"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/is-critical-point","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.IsCriticalPoint","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#is-critical-point","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"fully_proved","summary":"For $\\lambda \\in \\mathbb{C}$, a point $c \\in \\mathbb{C}$ is a critical point of $f_\\lambda(z) = \\lambda z^2 + z^3$ when $f_\\lambda$ has complex derivative $0$ at $c$, that is, $f_\\lambda'(c) = 0$.","title":"Critical points of the cubic family"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/basin-of-zero","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.basinOfZero","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#basin-of-zero","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"fully_proved","summary":"For $\\lambda \\in \\mathbb{C}$, the basin of zero of $f_\\lambda(z) = \\lambda z^2 + z^3$ is the set of $z \\in \\mathbb{C}$ whose forward orbit tends to $0$, that is, $f_\\lambda^{n}(z) \\to 0$ as $n \\to \\infty$, where $f_\\lambda^{n}$ denotes the $n$-th iterate of $f_\\lambda$.","title":"Basin of attraction of zero"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/is-immediate-basin","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.IsImmediateBasin","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#is-immediate-basin","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"fully_proved","summary":"For $\\lambda \\in \\mathbb{C}$, a set $V \\subseteq \\mathbb{C}$ is the immediate basin of zero of $f_\\lambda(z) = \\lambda z^2 + z^3$ when $0 \\in V$, $V$ is contained in the basin of zero of $f_\\lambda$ (the set of $z$ with $f_\\lambda^{n}(z) \\to 0$), $V$ is connected, and every connected set $W \\subseteq \\mathbb{C}$ with $0 \\in W$ that is contained in the basin of zero of $f_\\lambda$ satisfies $W \\subseteq V$. Thus $V$\u2026","title":"Immediate basin of zero"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/parameter-set","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.parameterSet","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#parameter-set","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"fully_proved","summary":"The parameter set is the set of $\\lambda \\in \\mathbb{C}$ for which some immediate basin of zero $V$ of $f_\\lambda(z) = \\lambda z^2 + z^3$ contains every critical point of $f_\\lambda$.","title":"Parameters with both critical points in the immediate basin"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/is-parameter-component","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.IsParameterComponent","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#is-parameter-component","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"fully_proved","summary":"A set $U \\subseteq \\mathbb{C}$ is a parameter component when it is a connected component of the parameter set, the set of $\\lambda \\in \\mathbb{C}$ for which every critical point of $f_\\lambda(z) = \\lambda z^2 + z^3$ lies in the immediate basin of zero of $f_\\lambda$: $U$ is nonempty, contained in the parameter set and connected, and every connected set $W \\subseteq \\mathbb{C}$ with $U \\subseteq W$ that is contained\u2026","title":"Parameter components"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/is-jordan-curve","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.IsJordanCurve","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#is-jordan-curve","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"fully_proved","summary":"A set $S \\subseteq \\mathbb{C}$ is a Jordan curve when there is a continuous injective map $f$ from the unit circle $\\{z \\in \\mathbb{C} : |z| = 1\\}$ to $\\mathbb{C}$ whose image is exactly $S$.","title":"Jordan curves in the plane"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/conjecture","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.conjecture","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#conjecture","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"fully_proved","summary":"Let $f_\\lambda(z) = \\lambda z^2 + z^3$ for $\\lambda \\in \\mathbb{C}$. The statement is that for every parameter component $U$, that is, every connected component of the set of $\\lambda$ for which every critical point of $f_\\lambda$ lies in the immediate basin of zero of $f_\\lambda$, the boundary $\\partial U$ of $U$ in $\\mathbb{C}$ is a Jordan curve.","title":"McMullen's question on Jordan boundaries of cubic parameter components"},{"id":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/conjecture-holds","kind":"node","lean":"MathlibExt.Dynamics.CubicParameterJordanBoundaryWanted.conjecture_holds","parent":22564,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials/index.html#conjecture-holds","scope":"Dynamics/complex-dynamics/parameter-space-of-cubic-polynomials","status":"can_prove","summary":"For the cubic family $f_\\lambda(z) = \\lambda z^2 + z^3$ with $\\lambda \\in \\mathbb{C}$, the boundary of every parameter component is a Jordan curve. Here a parameter component is a connected component of the set of $\\lambda$ for which every critical point of $f_\\lambda$ lies in the immediate basin of zero of $f_\\lambda$.","title":"Roesch's theorem on the boundaries of cubic parameter components"},{"data_url":"scopes/Dynamics/complex-dynamics/siegel-linearization-theorem.json","id":"Dynamics/complex-dynamics/siegel-linearization-theorem","kind":"scope","navigation":"module","parent":22551,"result":false,"route":"../roadmap/Dynamics/complex-dynamics/siegel-linearization-theorem/index.html","scope":"Dynamics/complex-dynamics/siegel-linearization-theorem","status":"can_prove","title":"Siegel linearization theorem"},{"id":"Dynamics/complex-dynamics/siegel-linearization-theorem/is-diophantine","kind":"node","lean":"MathlibExt.Analysis.PoincareSiegelWanted.ComplexAnalysis.IsDiophantine","parent":22574,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/siegel-linearization-theorem/index.html#is-diophantine","scope":"Dynamics/complex-dynamics/siegel-linearization-theorem","status":"fully_proved","summary":"A real number $\\alpha$ is Diophantine if there are real numbers $C > 0$ and $\\tau$ such that $C / |q|^{\\tau} \\le |\\alpha - p/q|$ for all integers $p$ and $q$ with $q \\neq 0$.","title":"Diophantine real numbers"},{"id":"Dynamics/complex-dynamics/siegel-linearization-theorem/poincare-siegel","kind":"node","lean":"MathlibExt.Analysis.PoincareSiegelWanted.ComplexAnalysis.poincare_siegel","parent":22574,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/siegel-linearization-theorem/index.html#poincare-siegel","scope":"Dynamics/complex-dynamics/siegel-linearization-theorem","status":"can_prove","summary":"Let $\\alpha$ be a Diophantine real number and let $\\lambda = e^{2\\pi i \\alpha}$. If $f : \\mathbb{C} \\to \\mathbb{C}$ is analytic at $0$ with $f(0) = 0$ and $f'(0) = \\lambda$, then there is a function $u : \\mathbb{C} \\to \\mathbb{C}$, analytic at $0$ with $u(0) = 0$ and $u'(0) = 1$, such that $f(u(z)) = u(\\lambda z)$ for all $z$ in some neighbourhood of $0$.","title":"Poincar\u00e9\u2013Siegel linearisation theorem"},{"data_url":"scopes/Dynamics/complex-dynamics/tricorn.json","id":"Dynamics/complex-dynamics/tricorn","kind":"scope","navigation":"module","parent":22551,"result":false,"route":"../roadmap/Dynamics/complex-dynamics/tricorn/index.html","scope":"Dynamics/complex-dynamics/tricorn","status":"can_prove","title":"Tricorn"},{"id":"Dynamics/complex-dynamics/tricorn/tantibar","kind":"node","lean":"MathlibExt.Analysis.MandelbarWanted.ComplexAnalysis.Tantibar","parent":22577,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/tricorn/index.html#tantibar","scope":"Dynamics/complex-dynamics/tricorn","status":"fully_proved","summary":"For $c \\in \\mathbb{C}$, the map $T_c : \\mathbb{C} \\to \\mathbb{C}$ is $T_c(z) = \\bar z^2 + c$, where $\\bar z$ is the complex conjugate of $z$.","title":"The antiholomorphic quadratic map"},{"id":"Dynamics/complex-dynamics/tricorn/mandelbar","kind":"node","lean":"MathlibExt.Analysis.MandelbarWanted.ComplexAnalysis.Mandelbar","parent":22577,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/tricorn/index.html#mandelbar","scope":"Dynamics/complex-dynamics/tricorn","status":"fully_proved","summary":"The Mandelbar set is the set of $c \\in \\mathbb{C}$ for which the orbit of $0$ under the map $T_c(z) = \\bar z^2 + c$ is bounded: there is a real number $M$ with $|T_c^n(0)| \\le M$ for all $n \\ge 0$, where $T_c^n$ is the $n$-th iterate of $T_c$ and $T_c^0$ is the identity.","title":"The Mandelbar set (tricorn)"},{"id":"Dynamics/complex-dynamics/tricorn/mandelbar-not-path-connected","kind":"node","lean":"MathlibExt.Analysis.MandelbarWanted.ComplexAnalysis.mandelbar_not_path_connected","parent":22577,"result":true,"route":"../roadmap/Dynamics/complex-dynamics/tricorn/index.html#mandelbar-not-path-connected","scope":"Dynamics/complex-dynamics/tricorn","status":"can_prove","summary":"The Mandelbar set (tricorn), the set of $c \\in \\mathbb{C}$ for which the orbit of $0$ under $z \\mapsto \\bar z^2 + c$ is bounded, is not path-connected.","title":"The Mandelbar set is not path-connected"},{"area":"Discrete Mathematics & Computer Science","data_url":"chapters/Combinatorics.json","id":"Combinatorics","kind":"scope","parent":0,"result":false,"route":"../roadmap/Combinatorics/index.html","scope":"Combinatorics","status":"can_state","summary":"Combinatorics studies finite and discrete configurations through counting, extremal structure, and constructive arguments.","title":"Combinatorics"},{"data_url":"scopes/Combinatorics/additive-combinatorics.json","id":"Combinatorics/additive-combinatorics","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/index.html","scope":"Combinatorics/additive-combinatorics","status":"can_prove","summary":"The additive structure of sets in integers and abelian groups: sumsets and inverse theorems, sum-product growth, Sidon sets and difference sets, sum-free sets, additive bases, zero-sum problems, and arithmetic progressions in dense sets; colouring (Ramsey-type) versions go to Ramsey theory.","title":"Additive combinatorics"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems","kind":"scope","parent":22582,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems","status":"can_prove","summary":"Size and structure of A+B, hA and restricted or signed sumsets: Cauchy\u2013Davenport and Vosper, Freiman-type and polynomial Freiman\u2013Ruzsa theorems, small doubling, sums of dilates, h-fold sumsets with distinct terms, rectification, sets determined by k-sums, max-min (lunar) decompositions, and sum-product and product-set growth.","title":"Sumsets and inverse theorems"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/cauchy-davenport-chowla","kind":"node","lean":"MathlibExt.Combinatorics.Additive.CauchyDavenportChowlaWanted.cauchy_davenport_chowla","parent":22583,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/index.html#cauchy-davenport-chowla","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems","status":"fully_proved","summary":"Let $q$ be a positive integer and let $A$ and $B$ be nonempty subsets of $\\mathbb{Z}/q\\mathbb{Z}$ such that $0 \\in B$ and every nonzero element of $B$ is a unit of $\\mathbb{Z}/q\\mathbb{Z}$. Then $|A + B| \\ge \\min(q, |A| + |B| - 1)$, where $A + B = \\{a + b : a \\in A, b \\in B\\}$.","title":"Cauchy\u2013Davenport\u2013Chowla theorem"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets","status":"can_prove","title":"Difference sets of full-dimensional lattice sets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets/is-full-dimensional","kind":"node","lean":"MathlibExt.Combinatorics.StanchescuDifferenceSetsAIMCombinatoricsProblem231Wanted.IsFullDimensional","parent":22585,"reading_title":{"format":"inline-parts-v1","parts":[["text","Full-dimensional sets in "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Full-dimensional sets in $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets/index.html#is-full-dimensional","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets","status":"fully_proved","summary":"A finite set $A \\subseteq \\mathbb{Z}^d$ is full-dimensional when it is not contained in an affine hyperplane: for every real vector $a \\in \\mathbb{R}^d$ and every real number $c$, if $\\sum_i a_i x_i = c$ for all $x \\in A$, then $a = 0$.","title":"Full-dimensional sets in $\\mathbb{Z}^d$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets/stanchescu-difference-bound","kind":"node","lean":"MathlibExt.Combinatorics.StanchescuDifferenceSetsAIMCombinatoricsProblem231Wanted.stanchescuDifferenceBound","parent":22585,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets/index.html#stanchescu-difference-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets","status":"fully_proved","summary":"The proposition that for every integer $d \\ge 2$ there is a natural number $N$ such that every full-dimensional finite set $A \\subseteq \\mathbb{Z}^d$ (one not contained in an affine hyperplane) with $|A| \\ge N$ satisfies $|A - A| \\ge \\left(2d - 2 + \\frac{1}{d - 1}\\right)|A| - (2d^2 - 4d + 3)$, where $A - A = \\{a - b : a, b \\in A\\}$ is the difference set of $A$.","title":"Statement of Stanchescu's difference set bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets/stanchescu-difference-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.StanchescuDifferenceSetsAIMCombinatoricsProblem231Wanted.stanchescuDifferenceBound_holds","parent":22585,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets/index.html#stanchescu-difference-bound-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/difference-sets-of-full-dimensional-lattice-sets","status":"can_prove","summary":"For every integer $d \\ge 2$ there is a natural number $N$ such that every finite set $A \\subseteq \\mathbb{Z}^d$ with $|A| \\ge N$ that is not contained in an affine hyperplane (no nonzero real linear functional is constant on $A$) satisfies $|A - A| \\ge \\left(2d - 2 + \\frac{1}{d - 1}\\right)|A| - (2d^2 - 4d + 3)$, where $A - A = \\{a - b : a, b \\in A\\}$ is the difference set of $A$.","title":"Conlon\u2013Lim theorem: Stanchescu's difference set bound holds"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums","status":"can_prove","title":"Distinct consecutive sums"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums/consecutive-sums","kind":"node","lean":"MathlibExt.Combinatorics.IncreasingSequenceConsecutiveSumsErdosProblem356Wanted.consecutiveSums","parent":22589,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums/index.html#consecutive-sums","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums","status":"fully_proved","summary":"Let $k \\in \\mathbb{N}$ and let $a_0, \\dots, a_{k-1}$ be natural numbers. The consecutive sums of $a$ form the finite set of natural numbers $a_u + a_{u+1} + \\dots + a_v$ for indices $0 \\le u \\le v \\le k - 1$.","title":"Consecutive sums"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums/many-distinct-consecutive-sums","kind":"node","lean":"MathlibExt.Combinatorics.IncreasingSequenceConsecutiveSumsErdosProblem356Wanted.manyDistinctConsecutiveSums","parent":22589,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums/index.html#many-distinct-consecutive-sums","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums","status":"fully_proved","summary":"The proposition that there is a real number $c > 0$ such that for all sufficiently large $n \\in \\mathbb{N}$ there exist $k \\in \\mathbb{N}$ and natural numbers $a_0 < a_1 < \\dots < a_{k-1}$ with $1 \\le a_i \\le n$ for every $i$, whose set of consecutive sums $a_u + a_{u+1} + \\dots + a_v$ (for $u \\le v$) has at least $c n^2$ elements.","title":"Erd\u0151s Problem 356"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums/many-distinct-consecutive-sums-holds","kind":"node","lean":"MathlibExt.Combinatorics.IncreasingSequenceConsecutiveSumsErdosProblem356Wanted.manyDistinctConsecutiveSums_holds","parent":22589,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums/index.html#many-distinct-consecutive-sums-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/distinct-consecutive-sums","status":"can_prove","summary":"There is a real number $c > 0$ such that for all sufficiently large $n \\in \\mathbb{N}$ there exist $k \\in \\mathbb{N}$ and natural numbers $a_0 < a_1 < \\dots < a_{k-1}$ with $1 \\le a_i \\le n$ for every $i$, whose set of consecutive sums $a_u + a_{u+1} + \\dots + a_v$ (for $u \\le v$) has at least $c n^2$ elements.","title":"Erd\u0151s Problem 356 has a positive answer"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields","status":"can_prove","title":"Explicit sum-product expanders over prime fields"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields/bourgain-expander-square-plus-product","kind":"node","lean":"MathlibExt.Combinatorics.BourgainExplicitExpanderAIMCombinatoricsProblem238Wanted.bourgainExpanderSquarePlusProduct","parent":22593,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields/index.html#bourgain-expander-square-plus-product","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields","status":"fully_proved","summary":"The statement that there is a real $\\varepsilon > 0$ such that for every real $C \\ge 1$ there are a real $c > 0$ and a natural number $p_0$ such that for every prime $p \\ge p_0$ and all sets $A, B \\subseteq \\mathbb{F}_p$ with $\\sqrt{p}/C \\le |A| \\le C\\sqrt{p}$ and $\\sqrt{p}/C \\le |B| \\le C\\sqrt{p}$, the set $\\{a^2 + ab : a \\in A,\\ b \\in B\\}$ has at least $c\\, p^{1/2 + \\varepsilon}$ elements.","title":"Bourgain's expander property for x\u00b2 + xy"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields/bourgain-expander-square-plus-product-ho","kind":"node","lean":"MathlibExt.Combinatorics.BourgainExplicitExpanderAIMCombinatoricsProblem238Wanted.bourgainExpanderSquarePlusProduct_holds","parent":22593,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields/index.html#bourgain-expander-square-plus-product-ho","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/explicit-sum-product-expanders-over-prime-fields","status":"can_prove","summary":"Bourgain's expander property for $f(x, y) = x^2 + xy$ holds: there is a real $\\varepsilon > 0$ such that for every real $C \\ge 1$ there are a real $c > 0$ and a natural number $p_0$ such that for every prime $p \\ge p_0$ and all sets $A, B \\subseteq \\mathbb{F}_p$ with $\\sqrt{p}/C \\le |A| \\le C\\sqrt{p}$ and $\\sqrt{p}/C \\le |B| \\le C\\sqrt{p}$, the set $f(A, B) = \\{a^2 + ab : a \\in A,\\ b \\in B\\}$ has at least\u2026","title":"x\u00b2 + xy is an explicit expander over prime fields"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets","status":"can_prove","title":"Large subspaces in sumsets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.AlmostSubspaceInSumsetWanted.conjecture","parent":22596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Almost-subspace-in-"],["math","\\(A + A\\)"],["text"," conjecture"]],"source":"Almost-subspace-in-$A + A$ conjecture"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets","status":"fully_proved","summary":"The statement that there are real constants $C > 0$ and $E > 0$ such that for every $n \\in \\mathbb{N}$, every set $A \\subseteq \\mathbb{F}_2^n$ and all real numbers $\\alpha$ and $\\varepsilon$ with $\\alpha > 0$, $0 < \\varepsilon < 1$ and $|A| / 2^n = \\alpha$, there is an additive subgroup $V$ of $\\mathbb{F}_2^n$ whose codimension $\\log_2(2^n / |V|)$ is at most $C \\, (\\log(\\varepsilon^{-1} \\alpha^{-1}) + 1)^E$ and\u2026","title":"Almost-subspace-in-$A + A$ conjecture"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.AlmostSubspaceInSumsetWanted.conjecture_holds","parent":22596,"reading_title":{"format":"inline-parts-v1","parts":[["text","A subspace almost inside "],["math","\\(A + A\\)"],["text"," (Kelley\u2013Meka)"]],"source":"A subspace almost inside $A + A$ (Kelley\u2013Meka)"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/large-subspaces-in-sumsets","status":"can_prove","summary":"There are real constants $C > 0$ and $E > 0$ such that for every $n \\in \\mathbb{N}$, every set $A \\subseteq \\mathbb{F}_2^n$ and all real numbers $\\alpha$ and $\\varepsilon$ with $\\alpha > 0$, $0 < \\varepsilon < 1$ and $|A| / 2^n = \\alpha$, there is an additive subgroup $V$ of $\\mathbb{F}_2^n$ whose codimension $\\log_2(2^n / |V|)$ is at most $C \\, (\\log(\\varepsilon^{-1} \\alpha^{-1}) + 1)^E$ and which satisfies\u2026","title":"A subspace almost inside $A + A$ (Kelley\u2013Meka)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","title":"Max-min indecomposable sequences and lunar primes"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/max-min-mul","kind":"node","lean":"MetaMathlibExt.maxMinMul","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#max-min-mul","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"The max-min product of sequences $g, h \\colon \\mathbb{N} \\to \\mathbb{N}$ is the sequence $g \\otimes h$ with $(g \\otimes h)_k = \\max_{i + j = k} \\min(g_i, h_j)$ for every $k \\in \\mathbb{N}$, the maximum running over pairs of natural numbers $(i, j)$. This is the product of the max-min semiring $(\\mathbb{N}, \\max, \\min)$ on coefficient sequences.","title":"Max-min product"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/max-min-mul-zero","kind":"node","lean":"MetaMathlibExt.maxMinMul_zero","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#max-min-mul-zero","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"For all sequences $g, h \\colon \\mathbb{N} \\to \\mathbb{N}$, the max-min product satisfies $(g \\otimes h)_0 = \\min(g_0, h_0)$.","title":"Constant coefficient of a max-min product"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/max-min-mul-one","kind":"node","lean":"MetaMathlibExt.maxMinMul_one","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#max-min-mul-one","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"For all sequences $g, h \\colon \\mathbb{N} \\to \\mathbb{N}$, the max-min product satisfies $(g \\otimes h)_1 = \\max(\\min(g_0, h_1), \\min(g_1, h_0))$.","title":"Linear coefficient of a max-min product"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/is-max-min-monomial","kind":"node","lean":"MetaMathlibExt.IsMaxMinMonomial","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#is-max-min-monomial","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"A coefficient sequence $g : \\mathbb{N} \\to \\mathbb{N}$ over the max-min semiring is a monomial when exactly one of its entries is nonzero, that is, when it is $c x^i$ for some index $i$ and some coefficient $c \\neq 0$.","title":"Monomials over the max-min semiring"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/not-is-max-min-monomial-zero","kind":"node","lean":"MetaMathlibExt.not_isMaxMinMonomial_zero","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#not-is-max-min-monomial-zero","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"The zero coefficient sequence $\\mathbb{N} \\to \\mathbb{N}$ is not a max-min monomial: a monomial has exactly one nonzero entry, and the zero sequence has none.","title":"The zero sequence is not a monomial"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/is-max-min-indecomposable","kind":"node","lean":"MetaMathlibExt.IsMaxMinIndecomposable","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#is-max-min-indecomposable","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"A coefficient sequence $f : \\mathbb{N} \\to \\mathbb{N}$ is indecomposable (in the arXiv version: irreducible) when for all sequences $g, h : \\mathbb{N} \\to \\mathbb{N}$ with $g \\otimes h = f$, where $\\otimes$ is the max-min product, $g$ or $h$ is a max-min monomial, that is, a sequence with exactly one nonzero term.","title":"Max-min indecomposable sequences"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/not-is-max-min-indecomposable-zero","kind":"node","lean":"MetaMathlibExt.not_isMaxMinIndecomposable_zero","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#not-is-max-min-indecomposable-zero","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"The zero coefficient sequence $\\mathbb{N} \\to \\mathbb{N}$ is not indecomposable over the max-min semiring: it factors as $0 = 0 \\otimes 0$, where $\\otimes$ is the max-min product, and the zero sequence is not a monomial.","title":"The zero sequence is decomposable"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/digit-seq","kind":"node","lean":"MetaMathlibExt.digitSeq","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#digit-seq","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"Let $n$ and $b$ be natural numbers. The coefficient sequence of a digit string $a = (a_0, \\dots, a_{n-1})$ with digits in $\\{0, \\dots, b - 1\\}$ is the sequence $\\mathbb{N} \\to \\mathbb{N}$ whose $k$-th term is $a_k$ for $k < n$ and $0$ for $k \\ge n$.","title":"Coefficient sequence of a digit string"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/digit-seq-of-le","kind":"node","lean":"MetaMathlibExt.digitSeq_of_le","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#digit-seq-of-le","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"Let $n$ and $b$ be natural numbers and let $a = (a_0, \\dots, a_{n-1})$ be a digit string with digits in $\\{0, \\dots, b - 1\\}$. Then for every $k \\ge n$, the $k$-th term of the coefficient sequence of $a$ is $0$.","title":"A digit string has no coefficients beyond its length"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-unit","kind":"node","lean":"MetaMathlibExt.lunarUnit","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-unit","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"For a natural number $b$, the lunar unit of base $b$ is the coefficient sequence $\\mathbb{N} \\to \\mathbb{N}$ whose constant term is $b - 1$ and whose other terms are all $0$, that is, the constant polynomial $b - 1$ over the max-min semiring. The subtraction is truncated, so the lunar unit of base $0$ is the zero sequence.","title":"Lunar unit"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/is-max-min-monomial-lunar-unit","kind":"node","lean":"MetaMathlibExt.isMaxMinMonomial_lunarUnit","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#is-max-min-monomial-lunar-unit","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"For every natural number $b \\ge 2$, the lunar unit of base $b$, the sequence with constant term $b - 1$ and all other terms $0$, is a max-min monomial: it has exactly one nonzero term.","title":"The lunar unit is a monomial"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/is-lunar-prime","kind":"node","lean":"MetaMathlibExt.IsLunarPrime","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#is-lunar-prime","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"Let $b$ be a natural number. The lunar unit of base $b$ is the constant polynomial $b - 1$, the sequence with constant term $b - 1$ and all other terms $0$. A coefficient sequence $f : \\mathbb{N} \\to \\mathbb{N}$ is a lunar prime of base $b$ (Applegate, LeBrun and Sloane) when for all sequences $g, h : \\mathbb{N} \\to \\mathbb{N}$ with all terms less than $b$ and $g \\otimes h = f$, where $\\otimes$ is the max-min\u2026","title":"Lunar primes"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/not-is-lunar-prime-zero","kind":"node","lean":"MetaMathlibExt.not_isLunarPrime_zero","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#not-is-lunar-prime-zero","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"For every integer $b \\ge 2$, the zero sequence is not a lunar prime of base $b$: $0 = 0 \\otimes 0$, where $\\otimes$ is the max-min product, and $0$ is not the lunar unit, the constant polynomial $b - 1$ with coefficient sequence $(b - 1, 0, 0, \\dots)$.","title":"Zero is not a lunar prime"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/is-lunar-prime-candidate","kind":"node","lean":"MetaMathlibExt.IsLunarPrimeCandidate","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#is-lunar-prime-candidate","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"Let $b$ be a natural number. A coefficient sequence $f : \\mathbb{N} \\to \\mathbb{N}$ is a prime candidate in base $b$ when its constant term $f_0$ is nonzero and some term $f_i$ equals $b - 1$, where the subtraction is truncated, so that $b - 1 = 0$ when $b = 0$. For $f$ with all terms less than $b$, the second condition says that the largest coefficient of $f$ is $b - 1$.","title":"Lunar prime candidates"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/is-lunar-prime-candidate-lunar-unit","kind":"node","lean":"MetaMathlibExt.isLunarPrimeCandidate_lunarUnit","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#is-lunar-prime-candidate-lunar-unit","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"For every natural number $b \\ge 2$, the lunar unit of base $b$, the sequence with constant term $b - 1$ and all other terms $0$, is a prime candidate in base $b$: its constant term is nonzero and one of its terms equals $b - 1$.","title":"The lunar unit is a prime candidate"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/shitov-boolean-prime-density","kind":"node","lean":"MetaMathlibExt.shitov_boolean_prime_density","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#shitov-boolean-prime-density","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"For $n \\in \\mathbb{N}$, consider the Boolean polynomials of degree exactly $n$ with nonzero constant term, that is, the base-$2$ lunar numbers given by digit strings $(a_0, \\dots, a_n) \\in \\{0, 1\\}^{n+1}$ with $a_0 \\ne 0$ and $a_n \\ne 0$. The proportion of these which are lunar primes in base $2$ tends to $1$ as $n \\to \\infty$.","title":"Shitov's theorem: almost all Boolean polynomials with nonzero constant term are prime"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/max-min-indecomposable-density","kind":"node","lean":"MetaMathlibExt.maxMin_indecomposable_density","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#max-min-indecomposable-density","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Fix an integer $b \\ge 2$. For each $n$, among the polynomials of degree $n$ with coefficients in $\\mathcal{B}_b = \\{0, \\dots, b - 1\\}$, that is, coefficient strings $(a_0, \\dots, a_n)$ with $a_n \\ne 0$, the proportion of those that are indecomposable over the max-min semiring tends to $1$ as $n \\to \\infty$.","title":"Almost all max-min polynomials are indecomposable"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-prime-count-is-equivalent","kind":"node","lean":"MetaMathlibExt.lunar_prime_count_isEquivalent","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-prime-count-is-equivalent","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Fix an integer $b \\ge 2$. For $n \\ge 1$ let $\\pi_b(n)$ be the number of digit strings $(a_0, \\dots, a_{n-1})$ with $a_k \\in \\{0, \\dots, b - 1\\}$ and leading digit $a_{n-1} \\ne 0$ whose coefficient sequence $(a_0, \\dots, a_{n-1}, 0, 0, \\dots)$ is a lunar prime of base $b$; these are the prime lunar numbers of base $b$ with $n$ digits. Then $\\pi_b(n) \\sim (b - 1)^2 b^{n-2}$ as $n \\to \\infty$, as conjectured by\u2026","title":"Asymptotic count of lunar primes"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/digit-map-lunar-hom","kind":"node","lean":"MetaMathlibExt.digitMap_lunar_hom","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#digit-map-lunar-hom","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Let $d : \\mathbb{N} \\to \\mathbb{N}$ be nondecreasing, and for a coefficient sequence $f : \\mathbb{N} \\to \\mathbb{N}$ write $d(f)$ for the sequence $k \\mapsto d(f_k)$. Write $f \\oplus g$ for the pointwise maximum and $f \\otimes g$ for the max-min product of coefficient sequences $f$ and $g$. Then for all $f$ and $g$, $d(f) \\oplus d(g) = d(f \\oplus g)$ and $d(f) \\otimes d(g) = d(f \\otimes g)$.","title":"Monotone digit maps are lunar homomorphisms"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/boolean-close-factor-pairs-card-le","kind":"node","lean":"MetaMathlibExt.boolean_close_factor_pairs_card_le","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#boolean-close-factor-pairs-card-le","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers with $0 < k \\le n$, and let $d > 0$ be a real number. Then the number of pairs $(A, B)$ of finite sets of natural numbers such that $0 \\in A$, the largest element of $A$ is $k$, the largest element of $B$ is $n - k$, and $|A + B| \\le |A| + |B| + d$, where $A + B = \\{a + b : a \\in A, b \\in B\\}$ is the sumset, is at most $n^{2d+2} 2^k$.","title":"Counting Boolean factor pairs with a small sumset"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/is-lunar-prime-of-indecomposable-candida","kind":"node","lean":"MetaMathlibExt.isLunarPrime_of_indecomposable_candidate","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#is-lunar-prime-of-indecomposable-candida","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Let $n$ and $b \\ge 2$ be natural numbers, let $a = (a_0, \\dots, a_{n-1})$ be a digit string with digits in $\\{0, \\dots, b - 1\\}$, and let $f$ be its coefficient sequence. If $f$ is a prime candidate in base $b$ and $f$ is max-min indecomposable, then $f$ is a lunar prime of base $b$.","title":"Indecomposable prime candidates are lunar primes"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-level-count","kind":"node","lean":"MetaMathlibExt.lunarLevelCount","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-level-count","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"Let $f$ be a lunar number of base $b$ and degree at most $n$, with digits $c_0, \\dots, c_n \\in \\{0, \\dots, b - 1\\}$, and let $i$ be a natural number. The level count $|f_i|$ is the number of indices $k \\le n$ with $c_k \\ge i$, that is, the number of nonzero coefficients of the $i$-level support $f_i = s_i(f)$, the Boolean polynomial with coefficient $1$ at $x^k$ when $c_k \\ge i$ and $0$ otherwise.","title":"Level counts of a lunar number"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-level-count-example","kind":"node","lean":"MetaMathlibExt.lunarLevelCount_example","parent":22599,"reading_title":{"format":"inline-parts-v1","parts":[["text","Level counts of "],["math","\\(1 \\oplus 2x^2\\)"],["text"," in base "],["math","\\(3\\)"]],"source":"Level counts of $1 \\oplus 2x^2$ in base $3$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-level-count-example","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"For the base-$3$ lunar number $f = 1 \\oplus 2x^2$ of degree at most $2$, with digits $1, 0, 2$, exactly two digits are at least $1$ and exactly one digit is at least $2$: $|f_1| = 2$ and $|f_2| = 1$.","title":"Level counts of $1 \\oplus 2x^2$ in base $3$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-decomposable-set","kind":"node","lean":"MetaMathlibExt.lunarDecomposableSet","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-decomposable-set","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"For natural numbers $b$ and $n$, let $\\mathcal{L}_b^n$ be the set of lunar numbers of base $b$ and degree at most $n$, that is, polynomials $c_0 \\oplus c_1 x \\oplus \\dots \\oplus c_n x^n$ with digits $c_k \\in \\{0, \\dots, b - 1\\}$. The set $\\Sigma_b^n$ of decomposable lunar numbers consists of the $h \\in \\mathcal{L}_b^n$ that factor as $h = f \\otimes g$ under the max-min product, with $f, g \\in \\mathcal{L}_b^n$ each\u2026","title":"Decomposable lunar numbers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-level-decomposable-set","kind":"node","lean":"MetaMathlibExt.lunarLevelDecomposableSet","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-level-decomposable-set","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"Let $b$ and $n$ be natural numbers and $a = \\lfloor b/2 \\rfloor$. For a lunar number $f$ of base $b$ and degree at most $n$, given by its digits $f_0, \\dots, f_n \\in \\{0, \\dots, b - 1\\}$, write $|f_a|$ for the number of indices $k \\le n$ with $f_k \\ge a$. The set $\\mathcal{D}_b^n$ consists of the lunar numbers $h$ of base $b$ and degree at most $n$ such that $h = f \\otimes g$ for some lunar numbers $f$ and $g$ of\u2026","title":"Level-decomposable lunar numbers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-hoeffding-extremal-set","kind":"node","lean":"MetaMathlibExt.lunarHoeffdingExtremalSet","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-hoeffding-extremal-set","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"Let $b$ and $n$ be natural numbers and $d$ a real number. Let $\\mathcal{L}_b^n$ be the set of lunar numbers of base $b$ and degree at most $n$ (digit strings $c_0, \\dots, c_n$ with digits in $\\{0, \\dots, b - 1\\}$), with the max-min product $\\otimes$, and for $f \\in \\mathcal{L}_b^n$ let $|f_i|$ be the number of its digits that are at least $i$. The set $\\mathcal{H}_b^n(d)$ of $d$-Hoeffding extremal lunar numbers\u2026","title":"Hoeffding extremal lunar numbers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/mem-lunar-decomposable-set-example","kind":"node","lean":"MetaMathlibExt.mem_lunarDecomposableSet_example","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#mem-lunar-decomposable-set-example","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"In base $2$, the lunar number with digit string $(1, 1, 1)$, that is $1 \\oplus x \\oplus x^2$ with $\\oplus$ the lunar sum (digitwise maximum), lies both in the set $\\Sigma_2^2$ of decomposable lunar numbers of degree at most $2$ and in the set $\\mathcal{D}_2^2$ of level-decomposable lunar numbers of degree at most $2$.","title":"A decomposable lunar number in base 2"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/zero-mem-lunar-hoeffding-extremal-set","kind":"node","lean":"MetaMathlibExt.zero_mem_lunarHoeffdingExtremalSet","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#zero-mem-lunar-hoeffding-extremal-set","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"fully_proved","summary":"In base $2$, the zero lunar number of degree at most $2$ (the digit string $(0, 0, 0)$) lies in the $1$-Hoeffding extremal set $\\mathcal{H}_2^2(1)$: for the level $i = 1$, the number $|0_1|$ of its digits that are at least $1$ is $0$, and $\\left|0 - (2 - 1) \\cdot 2 / 2\\right| = 1 > 1/2$.","title":"An example of a Hoeffding extremal lunar number"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-decomposable-set-ncard-le","kind":"node","lean":"MetaMathlibExt.lunarDecomposableSet_ncard_le","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-decomposable-set-ncard-le","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Fix an integer $b \\ge 2$. There is a constant $C > 0$ such that for every integer $n \\ge 2$ and all reals $d > 0$ and $v > 0$, the set $\\Sigma_b^n$ of decomposable lunar numbers of base $b$ and degree at most $n$ satisfies","title":"Upper bound on the number of decomposable lunar numbers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-hoeffding-extremal-set-ncard-le","kind":"node","lean":"MetaMathlibExt.lunarHoeffdingExtremalSet_ncard_le","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-hoeffding-extremal-set-ncard-le","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Fix an integer $b \\ge 2$. There is a constant $C > 0$ such that for every integer $n \\ge 1$ and every real $d > 0$, the set $\\mathcal{H}_b^n(d)$ of $d$-Hoeffding extremal lunar numbers of base $b$ and degree at most $n$ has at most $C n^2 e^{-d^2/(4(n+2))} b^n$ elements.","title":"Bound on the number of Hoeffding extremal lunar numbers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-level-count-le-of-not-mem-lunar-ho","kind":"node","lean":"MetaMathlibExt.lunarLevelCount_le_of_not_mem_lunarHoeffdingExtremalSet","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-level-count-le-of-not-mem-lunar-ho","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Let $b$ and $n$ be natural numbers, $d$ a real number and $f, g, h$ lunar numbers of base $b$ and degree at most $n$ with $h = f \\otimes g$ under the max-min product. For a natural number $i$ write $|f_i|$, $|g_i|$ and $|h_i|$ for the numbers of digits of $f$, $g$ and $h$ that are at least $i$. If $h$ is not in the set $\\mathcal{H}_b^n(d)$ of $d$-Hoeffding extremal lunar numbers of base $b$ and degree at most $n$,\u2026","title":"Level counts of a product that is not Hoeffding extremal"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-level-decomposable-set-diff-ncard","kind":"node","lean":"MetaMathlibExt.lunarLevelDecomposableSet_diff_ncard_le","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-level-decomposable-set-diff-ncard","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Let $b \\ge 2$ and $n$ be natural numbers and $d > 0$ a real number. The number of elements of $\\mathcal{D}_b^n \\setminus \\mathcal{H}_b^n(d)$ is at most $n^{2d+4} 2^{d/2-n/3} b^{n+1}$, where $\\mathcal{D}_b^n$ is the set of level-decomposable lunar numbers of base $b$ and degree at most $n$ and $\\mathcal{H}_b^n(d)$ is the set of $d$-Hoeffding extremal ones.","title":"Bound on level-decomposable lunar numbers that are not Hoeffding extremal"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-low-degree-factor-ncard-le","kind":"node","lean":"MetaMathlibExt.lunar_low_degree_factor_ncard_le","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-low-degree-factor-ncard-le","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Let $b \\ge 2$ and $n$ be natural numbers and let $d > 0$ and $v > 0$ be reals. Count the lunar numbers $h$ of base $b$ and degree at most $n$ that lie in $\\Sigma_b^n \\setminus \\mathcal{H}_b^n(d)$ (decomposable but not $d$-Hoeffding extremal) and can be written as $h = f \\otimes g$, with $\\otimes$ the max-min product, where $f$ and $g$ are lunar numbers of base $b$ and degree at most $n$, each with at least two\u2026","title":"Decomposable lunar numbers with a low-degree factor"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/lunar-indecomposable-level-diff-ncard-le","kind":"node","lean":"MetaMathlibExt.lunarIndecomposableLevel_diff_ncard_le","parent":22599,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime/index.html#lunar-indecomposable-level-diff-ncard-le","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/max-min-indecomposable-sequences-and-lunar-prime","status":"can_prove","summary":"Fix an integer $b \\ge 2$. For a natural number $n$ let $\\mathcal{I}_b^n = \\Sigma_b^n \\setminus \\mathcal{D}_b^n$, where $\\Sigma_b^n$ is the set of decomposable lunar numbers of base $b$ and degree at most $n$ and $\\mathcal{D}_b^n$ the set of level-decomposable ones, and for a real $d$ let $\\mathcal{H}_b^n(d)$ be the set of $d$-Hoeffding extremal ones. There is a constant $C > 0$ such that for every natural number\u2026","title":"Bound on decomposable lunar numbers neither level decomposable nor extremal"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"can_prove","title":"Polynomial Freiman\u2013Ruzsa theorem"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/f","kind":"node","lean":"MathlibExt.Combinatorics.ConjectureOfMartonWanted.ConjectureOfMarton.F","parent":22634,"reading_title":{"format":"inline-parts-v1","parts":[["text","The vector space "],["math","\\(\\mathbb{F}_2^n\\)"]],"source":"The vector space $\\mathbb{F}_2^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#f","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"For a natural number $n$, $\\mathbb{F}_2^n$ is the space of functions from $\\{0, \\dots, n - 1\\}$ to the field $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$ with two elements, the $n$-dimensional vector space over $\\mathbb{F}_2$.","title":"The vector space $\\mathbb{F}_2^n$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/theorem-1-2","kind":"node","lean":"MathlibExt.Combinatorics.ConjectureOfMartonWanted.ConjectureOfMarton.theorem_1_2","parent":22634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial Freiman\u2013Ruzsa theorem over "],["math","\\(\\mathbb{F}_2\\)"]],"source":"Polynomial Freiman\u2013Ruzsa theorem over $\\mathbb{F}_2$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#theorem-1-2","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"can_prove","summary":"Let $n$ be a natural number, let $A$ be a nonempty subset of $\\mathbb{F}_2^n$, and let $K$ be a real number such that the sumset $A + A = \\{a + b : a, b \\in A\\}$ satisfies $|A + A| \\le K |A|$. Then there is a subgroup $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ such that $A$ is covered by at most $2K^{12}$ cosets of $H$: there is a finite set $T \\subseteq \\mathbb{F}_2^n$ with $|T| \\le 2K^{12}$ and $A \\subseteq T + H$.","title":"Polynomial Freiman\u2013Ruzsa theorem over $\\mathbb{F}_2$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.conjecture","parent":22634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial Freiman\u2013Ruzsa conjecture over "],["math","\\(\\mathbb{F}_2\\)"]],"source":"Polynomial Freiman\u2013Ruzsa conjecture over $\\mathbb{F}_2$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"There are real polynomials $P$ and $Q$, with $P(K) > 0$ and $Q(K) > 0$ for every real $K \\ge 1$, such that for every $n \\in \\mathbb{N}$, every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ and every real $K \\ge 1$ with $|A + A| \\le K|A|$, there is a subset $A' \\subseteq A$ with $|A| / P(K) \\le |A'|$ and $|\\operatorname{span}(A')| \\le Q(K)\\,|A'|$, where $\\operatorname{span}(A')$ is the $\\mathbb{F}_2$-linear span\u2026","title":"Polynomial Freiman\u2013Ruzsa conjecture over $\\mathbb{F}_2$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.conjecture_holds","parent":22634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gowers\u2013Green\u2013Manners\u2013Tao theorem (polynomial Freiman\u2013Ruzsa over "],["math","\\(\\mathbb{F}_2\\)"],["text",")"]],"source":"Gowers\u2013Green\u2013Manners\u2013Tao theorem (polynomial Freiman\u2013Ruzsa over $\\mathbb{F}_2$)"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"can_prove","summary":"The polynomial Freiman\u2013Ruzsa conjecture over $\\mathbb{F}_2$ holds: there are real polynomials $P$ and $Q$, with $P(K) > 0$ and $Q(K) > 0$ for every real $K \\ge 1$, such that for every $n \\in \\mathbb{N}$, every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ and every real $K \\ge 1$ with $|A + A| \\le K|A|$, there is a subset $A' \\subseteq A$ with $|A| / P(K) \\le |A'|$ and $|\\operatorname{span}(A')| \\le Q(K)\\,|A'|$,\u2026","title":"Gowers\u2013Green\u2013Manners\u2013Tao theorem (polynomial Freiman\u2013Ruzsa over $\\mathbb{F}_2$)"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/binary-ambient-space","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.BinaryAmbientSpace","parent":22634,"reading_title":{"format":"inline-parts-v1","parts":[["text","The binary vector space "],["math","\\(\\mathbb{F}_2^n\\)"]],"source":"The binary vector space $\\mathbb{F}_2^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#binary-ambient-space","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"For a natural number $n$, the ambient space $\\mathbb{F}_2^n$ of all functions from $\\{0, 1, \\dots, n - 1\\}$ to the two-element field $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$.","title":"The binary vector space $\\mathbb{F}_2^n$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/is-asymptotic-coset-cover-exponent","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.IsAsymptoticCosetCoverExponent","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#is-asymptotic-coset-cover-exponent","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"A real number $C$ is an admissible asymptotic coset-cover exponent if for every real $\\varepsilon > 0$ there is a real $K_0 \\ge 1$ such that for every real $K \\ge K_0$, every $n \\in \\mathbb{N}$ and every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ with $|A + A| \\le K|A|$, there is an $\\mathbb{F}_2$-linear subspace $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ such that $A$ is covered by at most\u2026","title":"Admissible asymptotic coset-cover exponents"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/asymptotic-coset-cover-exponents","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.asymptoticCosetCoverExponents","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#asymptotic-coset-cover-exponents","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"This is the set of all real numbers $C$ that are admissible asymptotic coset-cover exponents: for every real $\\varepsilon > 0$ there is a real $K_0 \\ge 1$ such that for every real $K \\ge K_0$, every $n \\in \\mathbb{N}$ and every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ with $|A + A| \\le K|A|$, there are an $\\mathbb{F}_2$-linear subspace $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ and a finite set\u2026","title":"The set of admissible coset-cover exponents"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/marton-pfrcover-exponent","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.martonPFRCoverExponent","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#marton-pfrcover-exponent","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"Call a real number $C$ an admissible asymptotic coset-cover exponent if for every $\\varepsilon > 0$ there is $K_0 \\ge 1$ such that for every real $K \\ge K_0$, every $n \\in \\mathbb{N}$ and every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ with $|A + A| \\le K|A|$, there is a subspace $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ such that $A$ is covered by at most $K^{C + \\varepsilon}$ translates of $H$. The Marton\u2026","title":"The Marton coset-cover exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/least-cover-exponent-claim","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.leastCoverExponentClaim","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#least-cover-exponent-claim","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"The proposition that the Marton coset-cover exponent $C^*$, the infimum of the admissible asymptotic coset-cover exponents, is the least admissible exponent: $C^*$ is itself an admissible asymptotic coset-cover exponent, and $C^* \\le C$ for every admissible asymptotic coset-cover exponent $C$.","title":"Statement that the Marton coset-cover exponent is attained"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/marton-pfrfiniteness-theorem","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.martonPFRFinitenessTheorem","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#marton-pfrfiniteness-theorem","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"The proposition that some real number $C$ is an admissible asymptotic coset-cover exponent: for every $\\varepsilon > 0$ there is $K_0 \\ge 1$ such that for every real $K \\ge K_0$, every $n \\in \\mathbb{N}$ and every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ with $|A + A| \\le K|A|$, there is a subspace $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ such that $A$ is covered by at most $K^{C + \\varepsilon}$\u2026","title":"Statement that an admissible coset-cover exponent exists"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/ggmt2025-upper-bound7-sqrt17","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.ggmt2025UpperBound7Sqrt17","parent":22634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the Gowers\u2013Green\u2013Manners\u2013Tao exponent "],["math","\\(7 + \\sqrt{17}\\)"]],"source":"Statement of the Gowers\u2013Green\u2013Manners\u2013Tao exponent $7 + \\sqrt{17}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#ggmt2025-upper-bound7-sqrt17","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"The proposition that $7 + \\sqrt{17}$ is an admissible asymptotic coset-cover exponent.","title":"Statement of the Gowers\u2013Green\u2013Manners\u2013Tao exponent $7 + \\sqrt{17}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/liao2024-upper-bound9","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.liao2024UpperBound9","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#liao2024-upper-bound9","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"The proposition that $9$ is an admissible asymptotic coset-cover exponent: for every $\\varepsilon > 0$ there is $K_0 \\ge 1$ such that for every real $K \\ge K_0$, every $n \\in \\mathbb{N}$ and every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ with $|A + A| \\le K|A|$, there is a subspace $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ such that $A$ is covered by at most $K^{9 + \\varepsilon}$ translates of $H$.","title":"Statement of Liao's exponent-9 coset-cover bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/liao2024-simplified-upper-bound11","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.liao2024SimplifiedUpperBound11","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#liao2024-simplified-upper-bound11","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"The proposition that $11$ is an admissible asymptotic coset-cover exponent: for every $\\varepsilon > 0$ there is $K_0 \\ge 1$ such that for every real $K \\ge K_0$, every $n \\in \\mathbb{N}$ and every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ with $|A + A| \\le K|A|$, there is a subspace $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ such that $A$ is covered by at most $K^{11 + \\varepsilon}$ translates of $H$.","title":"Statement of Liao's simplified exponent-11 coset-cover bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/basis-vectors-lower-bound1","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.basisVectorsLowerBound1","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#basis-vectors-lower-bound1","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"fully_proved","summary":"The proposition that every admissible asymptotic coset-cover exponent $C$ satisfies $C \\ge 1$.","title":"Statement of the basis-vector lower bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/least-cover-exponent-claim-holds","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.leastCoverExponentClaim_holds","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#least-cover-exponent-claim-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"can_prove","summary":"The set of admissible asymptotic coset-cover exponents has a least element, namely its infimum $C^*$ (the Marton cover exponent): $C^*$ is itself admissible, and $C^* \\le C$ for every admissible asymptotic coset-cover exponent $C$.","title":"The least coset-cover exponent is attained"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/ggmt2025-upper-bound7-sqrt17-holds","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.ggmt2025UpperBound7Sqrt17_holds","parent":22634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gowers\u2013Green\u2013Manners\u2013Tao exponent "],["math","\\(7 + \\sqrt{17}\\)"]],"source":"Gowers\u2013Green\u2013Manners\u2013Tao exponent $7 + \\sqrt{17}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#ggmt2025-upper-bound7-sqrt17-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"can_prove","summary":"The number $7 + \\sqrt{17}$ is an admissible asymptotic coset-cover exponent.","title":"Gowers\u2013Green\u2013Manners\u2013Tao exponent $7 + \\sqrt{17}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/liao2024-upper-bound9-holds","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.liao2024UpperBound9_holds","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#liao2024-upper-bound9-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"can_prove","summary":"The number $9$ is an admissible asymptotic coset-cover exponent: for every $\\varepsilon > 0$ there is $K_0 \\ge 1$ such that for every real $K \\ge K_0$, every $n \\in \\mathbb{N}$ and every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ with $|A + A| \\le K|A|$, there is a subspace $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ such that $A$ is covered by at most $K^{9 + \\varepsilon}$ translates of $H$.","title":"Liao's coset-cover bound with exponent 9"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/liao2024-simplified-upper-bound11-holds","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.liao2024SimplifiedUpperBound11_holds","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#liao2024-simplified-upper-bound11-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"can_prove","summary":"The number $11$ is an admissible asymptotic coset-cover exponent: for every $\\varepsilon > 0$ there is $K_0 \\ge 1$ such that for every real $K \\ge K_0$, every $n \\in \\mathbb{N}$ and every nonempty finite set $A \\subseteq \\mathbb{F}_2^n$ with $|A + A| \\le K|A|$, there is a subspace $H$ of $\\mathbb{F}_2^n$ with $|H| \\le |A|$ such that $A$ is covered by at most $K^{11 + \\varepsilon}$ translates of $H$.","title":"Liao's simplified coset-cover bound with exponent 11"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/basis-vectors-lower-bound1-holds","kind":"node","lean":"MathlibExt.Combinatorics.PolynomialFreimanRuzsaBinaryWanted.basisVectorsLowerBound1_holds","parent":22634,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem/index.html#basis-vectors-lower-bound1-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/polynomial-freiman-ruzsa-theorem","status":"can_prove","summary":"Every admissible asymptotic coset-cover exponent $C$ satisfies $C \\ge 1$.","title":"Basis-vector lower bound for coset-cover exponents"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","title":"Possible sizes of h-fold sumsets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/sumset-trivial-elements","kind":"node","lean":"MetaMathlibExt.sumsetTrivialElements","parent":22654,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trivial elements of an "],["math","\\(h\\)"],["text","-fold sumset"]],"source":"Trivial elements of an $h$-fold sumset"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#sumset-trivial-elements","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, let $h, k \\in \\mathbb{N}$ and let $a \\colon \\mathbb{N} \\to G$ list the set $A = \\{a_0, a_1, \\dots, a_{k-1}\\}$; write $n x$ for the $n$-fold sum $x + \\dots + x$. The trivial elements of the $h$-fold sumset $hA$ form the finite set that is empty when $k = 0$ and otherwise consists of $h a_0$ together with the sums $(h - i) a_j + i a_{j+1}$ for all natural numbers $i \\le h$\u2026","title":"Trivial elements of an $h$-fold sumset"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/sumset-trivial-elements-zero","kind":"node","lean":"MetaMathlibExt.sumsetTrivialElements_zero","parent":22654,"reading_title":{"format":"inline-parts-v1","parts":[["text","No trivial elements for "],["math","\\(k = 0\\)"]],"source":"No trivial elements for $k = 0$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#sumset-trivial-elements-zero","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $h$ a natural number and $a \\colon \\mathbb{N} \\to G$. For $k = 0$, when $\\{a_0, \\dots, a_{k-1}\\}$ is the empty set, the set of trivial elements of its $h$-fold sumset is empty.","title":"No trivial elements for $k = 0$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/sumset-trivial-elements-one","kind":"node","lean":"MetaMathlibExt.sumsetTrivialElements_one","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#sumset-trivial-elements-one","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $h$ a natural number and $a \\colon \\mathbb{N} \\to G$. For $k = 1$, when $\\{a_0, \\dots, a_{k-1}\\}$ is the one-element set $\\{a_0\\}$, the set of trivial elements of its $h$-fold sumset is exactly $\\{h a_0\\}$, where $h a_0 = a_0 + \\dots + a_0$ is the $h$-fold sum of $a_0$.","title":"Trivial elements of a one-element set"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/adjacent-sum-mem-sumset-trivial-elements","kind":"node","lean":"MetaMathlibExt.adjacent_sum_mem_sumsetTrivialElements","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#adjacent-sum-mem-sumset-trivial-elements","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $a_0, a_1, \\ldots$ a sequence in $G$, and $h, k, i, j$ natural numbers with $j + 1 < k$ and $i \\le h$. Then $(h - i)a_j + i a_{j+1}$ is a trivial element of the $h$-fold sumset of $a_0, \\ldots, a_{k-1}$.","title":"Adjacent sums are trivial elements"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/mem-sumset-trivial-elements-iff","kind":"node","lean":"MetaMathlibExt.mem_sumsetTrivialElements_iff","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#mem-sumset-trivial-elements-iff","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $a_0, a_1, \\ldots$ a sequence in $G$, and $h, k$ natural numbers. An element $x \\in G$ is a trivial element of the $h$-fold sumset of $a_0, \\ldots, a_{k-1}$ if and only if either $k > 0$ and $x = h a_0$, or $x = (h - i)a_j + i a_{j+1}$ for some natural numbers $j$ and $i$ with $j + 1 < k$ and $i \\le h$.","title":"Characterization of the trivial elements"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/nsmul-sumset-erase-max-nontrivial","kind":"node","lean":"MetaMathlibExt.nsmul_sumset_erase_max_nontrivial","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#nsmul-sumset-erase-max-nontrivial","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","summary":"Let $G$ be a linearly ordered additive commutative group, let $h$ and $k > 1$ be natural numbers, and let $a_0, a_1, \\ldots$ be a sequence in $G$ with $a_0 < a_1 < \\cdots < a_{k-1}$. Put $A = \\{a_0, \\ldots, a_{k-1}\\}$ and $B = \\{a_0, \\ldots, a_{k-2}\\}$, and write $hX$ for the $h$-fold sumset of a set $X$, the set of sums of $h$ elements of $X$. Then $hB \\subseteq hA$, and every element of $hB$ that is not a trivial\u2026","title":"Dropping the largest element keeps nontrivial sums nontrivial"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/nontrivial-sumset-card-of-last-gap-ne-ns","kind":"node","lean":"MetaMathlibExt.nontrivial_sumset_card_of_last_gap_ne_nsmul","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#nontrivial-sumset-card-of-last-gap-ne-ns","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","summary":"Let $G$ be a linearly ordered additive commutative group, let $h$ and $k \\ge 4$ be natural numbers, and let $a_0, a_1, \\ldots$ be a sequence in $G$ with $0 = a_0 < a_1 < \\cdots < a_{k-3}$. Suppose that $a_{k-2} = a_{k-3} + b$ and $a_{k-1} = a_{k-3} + b + c$, where $b, c \\in G$ satisfy $b > 0$, $c > 0$ and $c \\ne db$ for every integer $d \\ge 1$. Then at least $h - 1$ elements of the $h$-fold sumset\u2026","title":"Many nontrivial sums when the last gap is not a multiple of the previous one"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/nontrivial-sumset-card-of-last-gap-eq-ns","kind":"node","lean":"MetaMathlibExt.nontrivial_sumset_card_of_last_gap_eq_nsmul","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#nontrivial-sumset-card-of-last-gap-eq-ns","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","summary":"Let $G$ be a linearly ordered additive commutative group, let $h$ and $k \\ge 4$ be natural numbers, and let $a_0, a_1, \\ldots$ be a sequence in $G$ with $0 = a_0 < a_1 < \\cdots < a_{k-3}$. Suppose that $a_{k-2} = a_{k-3} + b$ and $a_{k-1} = a_{k-3} + b + db$, where $b \\in G$ satisfies $b > 0$ and $d > 1$ is an integer. Then at least $h - 1$ elements of the $h$-fold sumset $h\\{a_0, \\ldots, a_{k-1}\\}$ (the set of\u2026","title":"Many nontrivial sums when the last gap is a proper multiple of the previous one"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/arith-prog-or-nontrivial-sumset-card-of","kind":"node","lean":"MetaMathlibExt.arithProg_or_nontrivial_sumset_card_of_last_gaps_eq","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#arith-prog-or-nontrivial-sumset-card-of","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","summary":"Let $G$ be a linearly ordered additive commutative group, let $h$ and $k \\ge 4$ be natural numbers, and let $a_0, a_1, \\ldots$ be a sequence in $G$ with $0 = a_0 < a_1 < \\cdots < a_{k-1}$ and $a_{k-1} - a_{k-2} = a_{k-2} - a_{k-3}$. Then either $a_i = i a_1$ for every $i < k$, or at least $h - 1$ elements of the $h$-fold sumset $hB$ of $B = \\{a_0, \\ldots, a_{k-2}\\}$ (the set of sums of $h$ elements of $B$) are not\u2026","title":"Equal last two gaps: an arithmetic progression or many nontrivial sums"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/card-nsmul-not-mem-icc-of-is-add-torsion","kind":"node","lean":"MetaMathlibExt.card_nsmul_notMem_Icc_of_isAddTorsionFree","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#card-nsmul-not-mem-icc-of-is-add-torsion","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","summary":"Let $G$ be a torsion-free additive commutative group and let $h$ and $k \\ge 4$ be natural numbers. For every $k$-element set $A \\subseteq G$, the $h$-fold sumset $hA$ (the set of sums of $h$ elements of $A$) does not satisfy $hk - h + 2 \\le |hA| \\le hk - 1$.","title":"A gap in the possible sumset sizes"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/sumset-sizes-inter-icc-of-is-add-torsion","kind":"node","lean":"MetaMathlibExt.sumsetSizes_inter_Icc_of_isAddTorsionFree","parent":22654,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sumset sizes between "],["math","\\(hk - h + 1\\)"],["text"," and "],["math","\\(hk\\)"]],"source":"Sumset sizes between $hk - h + 1$ and $hk$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#sumset-sizes-inter-icc-of-is-add-torsion","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","summary":"Let $G$ be a nontrivial torsion-free additive commutative group and let $h \\ge 1$ and $k \\ge 4$ be natural numbers. For a finite set $A \\subseteq G$ write $hA$ for its $h$-fold sumset, the set of sums of $h$ elements of $A$, and write $R_G(h, k)$ for the set of sizes $|hA|$ of $k$-element sets $A \\subseteq G$. Then $R_G(h, k) \\cap [hk - h + 1, hk] = \\{hk - h + 1, hk\\}$.","title":"Sumset sizes between $hk - h + 1$ and $hk$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/card-nsmul-arith-prog-skip-penultimate","kind":"node","lean":"MetaMathlibExt.card_nsmul_arithProg_skip_penultimate","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#card-nsmul-arith-prog-skip-penultimate","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","summary":"Let $G$ be a linearly ordered additive commutative group, let $a \\in G$ with $a > 0$, and let $k \\ge 3$ and $h \\ge 1$ be natural numbers. The set $A = \\{0, a, 2a, \\ldots, (k - 2)a, ka\\}$, obtained from $\\{0, a, 2a, \\ldots, ka\\}$ by removing $(k - 1)a$, has $|hA| = hk$, where $hA$ is the $h$-fold sumset of $A$, the set of sums of $h$ elements of $A$.","title":"Sumset of an arithmetic progression with its second-to-last term removed"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/card-nsmul-not-mem-icc-add-mul-of-is-add","kind":"node","lean":"MetaMathlibExt.card_nsmul_notMem_Icc_add_mul_of_isAddTorsionFree","parent":22654,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets/index.html#card-nsmul-not-mem-icc-add-mul-of-is-add","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/possible-sizes-of-h-fold-sumsets","status":"can_prove","summary":"Let $G$ be a torsion-free additive commutative group and let $h$, $k$ and $\\ell$ be natural numbers with $\\ell + 3 \\le \\min(h, k)$. For every $k$-element set $A \\subseteq G$, the $h$-fold sumset $hA$ (the set of sums of $h$ elements of $A$) does not satisfy $hk - h + 2 + \\ell h \\le |hA| \\le hk - h + 1 + \\ell h + (h - 2 - \\ell)$.","title":"Rajagopal's family of missing sumset sizes"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling","status":"can_prove","title":"Progressions in sets of small doubling"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/katz-sumset","kind":"node","lean":"MathlibExt.Combinatorics.KatzSmallDoublingProgressionsWanted.katzSumset","parent":22668,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/index.html#katz-sumset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling","status":"fully_proved","summary":"For a finite set $A \\subseteq \\mathbb{Z}$, the sumset $A + A = \\{a + b : a, b \\in A\\}$, a finite set of integers.","title":"Sumset of a finite set of integers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/katz-apset","kind":"node","lean":"MathlibExt.Combinatorics.KatzSmallDoublingProgressionsWanted.katzAPSet","parent":22668,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/index.html#katz-apset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling","status":"fully_proved","summary":"For a natural number $k$ and a finite set $A \\subseteq \\mathbb{Z}$, the set of pairs $(a, d)$ of integers such that $a + i d \\in A$ for every natural number $i < k$. The pair $(a, d)$ stands for the $k$-term arithmetic progression $a, a + d, \\dots, a + (k - 1)d$; the difference $d$ may be zero or negative.","title":"Arithmetic progressions in a finite set of integers, as pairs"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.KatzSmallDoublingProgressionsWanted.conjecture","parent":22668,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling","status":"fully_proved","summary":"For a finite set $A \\subseteq \\mathbb{Z}$ write $A + A = \\{a + b : a, b \\in A\\}$. This is the statement that for every natural number $k \\ge 3$ and every real number $C \\ge 2$ there is a real number $c > 0$ such that every finite set $A \\subseteq \\mathbb{Z}$ with $|A + A| < C|A|$ has at least $c|A|^2$ pairs $(a, d)$ of integers with $a + i d \\in A$ for all natural numbers $i < k$.","title":"Katz's question on progressions in sets of small doubling"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.KatzSmallDoublingProgressionsWanted.conjecture_holds","parent":22668,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/progressions-in-sets-of-small-doubling","status":"can_prove","summary":"For a finite set $A \\subseteq \\mathbb{Z}$ write $A + A = \\{a + b : a, b \\in A\\}$. For every natural number $k \\ge 3$ and every real number $C \\ge 2$ there is a real number $c > 0$ such that every finite set $A \\subseteq \\mathbb{Z}$ with $|A + A| < C|A|$ has at least $c|A|^2$ pairs $(a, d)$ of integers with $a + i d \\in A$ for all natural numbers $i < k$.","title":"Sets of small doubling contain many arithmetic progressions"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","title":"Restricted signed sumsets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/restricted-sumset","kind":"node","lean":"MetaMathlibExt.restrictedSumset","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#restricted-sumset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a natural number $h$ and a finite subset $A$ of an additive commutative monoid $G$, the restricted sumset $h^{\\wedge} A$ is the set of all sums $\\sum_{a \\in B} a$ with $B$ an $h$-element subset of $A$, that is, the sums of $h$ distinct elements of $A$. In particular $0^{\\wedge} A = \\{0\\}$.","title":"Restricted sumset"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/restricted-signed-sumset","kind":"node","lean":"MetaMathlibExt.restrictedSignedSumset","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#restricted-signed-sumset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a natural number $h$ and a finite set $A$ of integers, the restricted signed sumset $h^{\\wedge}_{\\pm} A$ is the set of all integers $\\sum_{a \\in B} a - 2 \\sum_{a \\in N} a$, where $B$ ranges over the $h$-element subsets of $A$ and $N$ ranges over the subsets of $B$. These are the sums of the elements of $B$ in which the elements of $N$ are given a negative sign.","title":"Restricted signed sumset"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/mem-restricted-sumset-iff","kind":"node","lean":"MetaMathlibExt.mem_restrictedSumset_iff","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#mem-restricted-sumset-iff","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a natural number $h$, a finite set $A$ of integers and an integer $x$, $x$ lies in the restricted sumset $h^{\\wedge} A$ if and only if there is a function $c : \\mathbb{Z} \\to \\mathbb{Z}$ with $c(a) \\in \\{0, 1\\}$ for every $a \\in A$, $\\sum_{a \\in A} c(a) = h$ and $x = \\sum_{a \\in A} c(a)\\, a$.","title":"Coefficient description of the restricted sumset"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/mem-restricted-signed-sumset-iff","kind":"node","lean":"MetaMathlibExt.mem_restrictedSignedSumset_iff","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#mem-restricted-signed-sumset-iff","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a natural number $h$, a finite set $A$ of integers and an integer $x$, $x$ lies in the restricted signed sumset $h^{\\wedge}_{\\pm} A$ if and only if there is a function $c : \\mathbb{Z} \\to \\mathbb{Z}$ with $c(a) \\in \\{-1, 0, 1\\}$ for every $a \\in A$, $\\sum_{a \\in A} |c(a)| = h$ and $x = \\sum_{a \\in A} c(a)\\, a$.","title":"Coefficient description of the restricted signed sumset"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/ordered-finset","kind":"node","lean":"MetaMathlibExt.orderedFinset","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#ordered-finset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a sequence $a_0, \\dots, a_{k-1}$ of integers, this is the finite set $\\{a_i : 0 \\le i < k\\}$ of its values.","title":"Set of values of a finite integer sequence"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/initial-segment","kind":"node","lean":"MetaMathlibExt.initialSegment","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#initial-segment","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a sequence $a_0, \\dots, a_{k-1}$ of integers and a natural number $n$, the initial segment of length $n$ is the set $\\{a_i : 0 \\le i < k,\\ i < n\\}$ of the values at indices below $n$; it is the set of all values of the sequence when $n \\ge k$.","title":"Initial segment of a finite integer sequence"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/is-indexed-ap","kind":"node","lean":"MetaMathlibExt.IsIndexedAP","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#is-indexed-ap","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"A sequence $a_0, \\dots, a_{k-1}$ of integers is an arithmetic progression with common difference $d$ if $d > 0$ and $a_{i+1} = a_i + d$ for every $i$ with $i + 1 < k$.","title":"Finite integer sequences in arithmetic progression"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/scale-finset","kind":"node","lean":"MetaMathlibExt.scaleFinset","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#scale-finset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For an integer $d$ and a finite set $B$ of integers, the dilation $d \\cdot B$ is the set $\\{db : b \\in B\\}$.","title":"Dilation of a finite set of integers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/odd-progression","kind":"node","lean":"MetaMathlibExt.oddProgression","parent":22673,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first "],["math","\\(k\\)"],["text"," odd positive integers"]],"source":"The first $k$ odd positive integers"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#odd-progression","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a natural number $k$, this is the set of integers $\\{1, 3, \\dots, 2k - 1\\} = \\{2i + 1 : 0 \\le i < k\\}$; it is empty when $k = 0$.","title":"The first $k$ odd positive integers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/tail-finset","kind":"node","lean":"MetaMathlibExt.tailFinset","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#tail-finset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a sequence $a_0, \\dots, a_{k-1}$ of integers, this is the set $\\{a_i : 1 \\le i < k\\}$. When the sequence is strictly increasing, it is the set $A' = A \\setminus \\{a_0\\}$ obtained from $A = \\{a_0, \\dots, a_{k-1}\\}$ by removing its least element.","title":"Values of a finite integer sequence after its first term"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/zero-interval","kind":"node","lean":"MetaMathlibExt.zeroInterval","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#zero-interval","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For a natural number $k$, the integer interval $[0, k-1]$ is the set $\\{0, 1, \\dots, k-1\\}$ of integers $i$ with $0 \\le i < k$; it is empty when $k = 0$.","title":"Zero-based integer interval"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-odd-f","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_odd_five","parent":22673,"reading_title":{"format":"inline-parts-v1","parts":[["text","The restricted signed 4-fold sumset of "],["math","\\(\\{1, 3, 5, 7, 9\\}\\)"]],"source":"The restricted signed 4-fold sumset of $\\{1, 3, 5, 7, 9\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-odd-f","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"The restricted signed 4-fold sumset $4^{\\wedge}_{\\pm} \\{1, 3, 5, 7, 9\\}$ has exactly $25$ elements.","title":"The restricted signed 4-fold sumset of $\\{1, 3, 5, 7, 9\\}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-count","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_counterexample","parent":22673,"reading_title":{"format":"inline-parts-v1","parts":[["text","The restricted signed 4-fold sumset of "],["math","\\(\\{0, 1, 2, 4, 6\\}\\)"]],"source":"The restricted signed 4-fold sumset of $\\{0, 1, 2, 4, 6\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-count","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"The restricted signed 4-fold sumset $4^{\\wedge}_{\\pm} \\{0, 1, 2, 4, 6\\}$ has exactly $21 = 8 \\cdot 5 - 19$ elements.","title":"The restricted signed 4-fold sumset of $\\{0, 1, 2, 4, 6\\}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/ne-scale-finset-zero-interval-five","kind":"node","lean":"MetaMathlibExt.ne_scaleFinset_zeroInterval_five","parent":22673,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\{0, 1, 2, 4, 6\\}\\)"],["text"," is not a dilate of "],["math","\\(\\{0, 1, 2, 3, 4\\}\\)"]],"source":"$\\{0, 1, 2, 4, 6\\}$ is not a dilate of $\\{0, 1, 2, 3, 4\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#ne-scale-finset-zero-interval-five","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"For every integer $d$, the set $\\{0, 1, 2, 4, 6\\}$ differs from the dilation $d \\cdot \\{0, 1, 2, 3, 4\\} = \\{0, d, 2d, 3d, 4d\\}$.","title":"$\\{0, 1, 2, 4, 6\\}$ is not a dilate of $\\{0, 1, 2, 3, 4\\}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-three-four","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_three_four","parent":22673,"reading_title":{"format":"inline-parts-v1","parts":[["text","The restricted signed 3-fold sumset of "],["math","\\(\\{1, 2, 3, 4\\}\\)"]],"source":"The restricted signed 3-fold sumset of $\\{1, 2, 3, 4\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-three-four","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"fully_proved","summary":"The restricted signed 3-fold sumset $3^{\\wedge}_{\\pm} \\{1, 2, 3, 4\\}$ has exactly $19$ elements.","title":"The restricted signed 3-fold sumset of $\\{1, 2, 3, 4\\}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-general-lo","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_general_lower_bounds","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-general-lo","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h$ and $k$ be integers with $1 \\le h \\le k$, let $A$ be a set of $k$ nonnegative integers, and write $h^{\\wedge}_{\\pm} A$ for the restricted signed $h$-fold sumset of $A$. If $0 \\notin A$, then $2|h^{\\wedge}_{\\pm} A| \\ge 4(hk - h^2) + h(h+1) + 2$; if $0 \\in A$, then $2|h^{\\wedge}_{\\pm} A| \\ge 4(hk - h^2) + h(h-1) + 2$.","title":"Lower bounds for restricted signed sumsets of nonnegative integers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-sumset-ge-and-eq-iff-ap","kind":"node","lean":"MetaMathlibExt.card_restrictedSumset_ge_and_eq_iff_ap","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-sumset-ge-and-eq-iff-ap","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $A$ be a nonempty finite set of $k$ integers and let $h$ be a natural number with $1 \\le h \\le k$. Then the restricted sumset $h^{\\wedge} A$ has at least $hk - h^2 + 1$ elements. Moreover, if $k \\ge 5$ and $2 \\le h \\le k - 2$, then $h^{\\wedge} A$ has exactly $hk - h^2 + 1$ elements if and only if there are integers $a$ and $d > 0$ such that $A$ is the arithmetic progression of length $k$ with first term $a$ and\u2026","title":"Nathanson's bound for restricted sumsets and its inverse"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-ap-h-add-o","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_ap_h_add_one","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-ap-h-add-o","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h \\ge 3$ be a natural number and let $a_1 < a_2 < \\cdots < a_{h+1}$ be positive integers that form an arithmetic progression with common difference $d > 0$, that is, $a_{i+1} = a_i + d$ for $1 \\le i \\le h$. Let $A = \\{a_1, \\ldots, a_{h+1}\\}$ and let $h^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. Then $|h^{\\wedge}_{\\pm} A| \\ge (h+1)^2$ if $d = 2a_1$, and $|h^{\\wedge}_{\\pm} A| \\ge (h+1)^2 + 1$\u2026","title":"Restricted signed sumsets of arithmetic progressions with one more term than summands"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-ap-ge","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_ap_ge","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-ap-ge","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h \\ge 3$ and $k \\ge h + 1$ be natural numbers, and let $a_1 < a_2 < \\cdots < a_k$ be integers with $a_1 > 0$ that form an arithmetic progression with common difference $d > 0$, that is, $a_{i+1} = a_i + d$ for $1 \\le i < k$. Let $A = \\{a_1, \\ldots, a_k\\}$ and let $h^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. Then $|h^{\\wedge}_{\\pm} A| \\ge 2hk - h^2 + 1$ if $d = 2a_1$, and\u2026","title":"Lower bound for restricted signed sumsets of arithmetic progressions"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-ap-zero","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_ap_zero","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-ap-zero","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h$ and $k$ be natural numbers with $4 \\le h \\le k - 1$, and let $0 = a_1 < a_2 < \\cdots < a_k$ be nonnegative integers that form an arithmetic progression with common difference $d > 0$, that is, $a_{i+1} = a_i + d$ for $1 \\le i < k$. Let $A = \\{a_1, \\ldots, a_k\\}$ and let $h^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. Then $|h^{\\wedge}_{\\pm} A| = 2hk - h(h+1) + 1$.","title":"Restricted signed sumsets of arithmetic progressions containing zero"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/eq-scaled-odd-progression-of-extremal","kind":"node","lean":"MetaMathlibExt.eq_scaled_oddProgression_of_extremal","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#eq-scaled-odd-progression-of-extremal","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h$ and $k$ be natural numbers with $4 \\le h \\le k - 1$, and let $A = \\{a_1 < a_2 < \\cdots < a_k\\}$ be a set of $k$ positive integers. Write $A_{h+1} = \\{a_1, \\dots, a_{h+1}\\}$, $A' = \\{a_2, \\dots, a_k\\}$ and $-A' = \\{-x : x \\in A'\\}$. Suppose that the restricted signed sumset $h^{\\wedge}_{\\pm} A$ has exactly $2hk - h^2 + 1$ elements and that at least one of the following holds: (a) there is $d > 0$ with\u2026","title":"Inverse theorem for the extremal restricted signed sumset of positive integers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/eq-scaled-zero-interval-of-extremal","kind":"node","lean":"MetaMathlibExt.eq_scaled_zeroInterval_of_extremal","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#eq-scaled-zero-interval-of-extremal","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h$ and $k$ be natural numbers with $4 \\le h \\le k - 1$, and let $A = \\{a_0 < a_1 < \\cdots < a_{k-1}\\}$ be a set of $k$ nonnegative integers with $a_0 = 0$. Write $A_h = \\{a_0, \\dots, a_h\\}$, $A' = \\{a_1, \\dots, a_{k-1}\\}$ and $-A' = \\{-x : x \\in A'\\}$. Suppose that the restricted signed sumset $h^{\\wedge}_{\\pm} A$ has exactly $2hk - h(h+1) + 1$ elements and that at least one of the following holds: (a) there\u2026","title":"Inverse theorem for the extremal restricted signed sumset of sets containing zero"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-ge-of-fibo","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_ge_of_fibonacci_growth","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-ge-of-fibo","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h \\ge 3$ and let $a_1 < a_2 < \\dots < a_{h+1}$ be positive integers such that $a_i \\ge a_{i-1} + a_{i-2}$ for every $i$ with $4 \\le i \\le h + 1$. Write $h^{\\wedge}_{\\pm} A$ for the restricted signed $h$-fold sumset of $A = \\{a_1, \\dots, a_{h+1}\\}$. Then $|h^{\\wedge}_{\\pm} A| \\ge (h+1)^2 + 1$.","title":"Restricted signed sumsets of sets with Fibonacci-type growth"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-ge-of-gap","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_ge_of_gap_conditions","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-ge-of-gap","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h \\ge 3$ and let $a_1 < a_2 < \\dots < a_{h+1}$ be positive integers such that $a_3 - a_2 < 2a_1$ and $a_2 - a_1 < 2(a_i - a_{i-1})$ for every $i$ with $4 \\le i \\le h + 1$. Write $h^{\\wedge}_{\\pm} A$ for the restricted signed $h$-fold sumset of $A = \\{a_1, \\dots, a_{h+1}\\}$. Then $|h^{\\wedge}_{\\pm} A| \\ge (h+1)^2 + 1$.","title":"Restricted signed sumsets under gap conditions"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-ge","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_ge","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-ge","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $k \\ge 5$, let $A$ be a set of $k$ positive integers, and write $4^{\\wedge}_{\\pm} A$ for the restricted signed 4-fold sumset of $A$. Then $|4^{\\wedge}_{\\pm} A| \\ge 8k - 15$. Moreover, $|4^{\\wedge}_{\\pm} A| = 8k - 15$ if and only if $A = m \\cdot \\{1, 3, \\dots, 2k - 1\\}$, where $m = \\min A$ and $m \\cdot B = \\{mb : b \\in B\\}$.","title":"Lower bound for restricted signed 4-fold sumsets of positive integers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-five","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_five_ge","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-five","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $A$ be a set of five positive integers and let $4^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. For an integer $m$ and a set $B$ of integers write $m \\cdot B = \\{mb : b \\in B\\}$. Then $|4^{\\wedge}_{\\pm} A| \\ge 25$, and $|4^{\\wedge}_{\\pm} A| = 25$ if and only if $A = m \\cdot \\{1, 3, 5, 7, 9\\}$ for $m = \\min A$.","title":"Restricted signed fourfold sumsets of five positive integers"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-nonne","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_nonnegative_ge","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-nonne","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $k \\ge 5$, let $A$ be a set of $k$ nonnegative integers with $0 \\in A$, and write $4^{\\wedge}_{\\pm} A$ for the restricted signed 4-fold sumset of $A$. Then $|4^{\\wedge}_{\\pm} A| \\ge 8k - 19$.","title":"Lower bound for restricted signed 4-fold sumsets of nonnegative sets containing zero"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-ge-of-init","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_ge_of_initialSegment","parent":22673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extending a restricted signed sumset bound from the first "],["math","\\(h+1\\)"],["text"," elements"]],"source":"Extending a restricted signed sumset bound from the first $h+1$ elements"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-ge-of-init","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $h \\ge 3$ and $k \\ge h + 1$, let $a_1 < a_2 < \\dots < a_k$ be positive integers, and let $t \\ge 0$ be an integer. Put $A = \\{a_1, \\dots, a_k\\}$ and $A_{h+1} = \\{a_1, \\dots, a_{h+1}\\}$, and write $h^{\\wedge}_{\\pm} B$ for the restricted signed $h$-fold sumset of a finite set $B$. If $|h^{\\wedge}_{\\pm} A_{h+1}| \\ge (h+1)^2 + t$, then $|h^{\\wedge}_{\\pm} A| \\ge 2hk - h^2 + 1 + t$.","title":"Extending a restricted signed sumset bound from the first $h+1$ elements"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-ge-of","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_ge_of_one_exceptional_gap","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-ge-of","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $a_1 < a_2 < a_3 < a_4 < a_5$ be integers with $a_1 > 0$, let $A = \\{a_1, \\dots, a_5\\}$, and write $4^{\\wedge}_{\\pm} A$ for the restricted signed 4-fold sumset of $A$. If exactly one of the four gaps $a_{i+1} - a_i$ with $i \\in \\{1, 2, 3, 4\\}$ differs from $2a_1$, then $|4^{\\wedge}_{\\pm} A| \\ge 26$.","title":"Restricted signed 4-fold sumsets when exactly one gap differs from twice the minimum"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-ge-of-2","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_ge_of_two_exceptional_gaps","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-ge-of-2","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Restricted signed fourfold sumsets when exactly two gaps differ from twice the least element. Let $a_1 < a_2 < a_3 < a_4 < a_5$ be integers with $a_1 > 0$, let $A = \\{a_1, \\ldots, a_5\\}$ and let $4^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. If exactly two of the four gaps $a_{i+1} - a_i$ ($1 \\le i \\le 4$) differ from $2a_1$, then $|4^{\\wedge}_{\\pm} A| \\ge 26$.","title":"Restricted signed fourfold sumsets when exactly two gaps differ from twice the least element"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-ge-of-3","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_ge_of_one_regular_gap","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-ge-of-3","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $a_1 < a_2 < a_3 < a_4 < a_5$ be integers with $a_1 > 0$, let $A = \\{a_1, \\ldots, a_5\\}$ and let $4^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. If exactly one of the four gaps $a_{i+1} - a_i$ ($1 \\le i \\le 4$) equals $2a_1$, then $|4^{\\wedge}_{\\pm} A| \\ge 26$.","title":"Restricted signed fourfold sumsets when exactly one gap equals twice the least element"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-ge-of-4","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_ge_of_unequal_outer_gaps","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-ge-of-4","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Restricted signed fourfold sumsets when no gap equals twice the least element and the first and third gaps differ. Let $a_1 < a_2 < a_3 < a_4 < a_5$ be integers with $a_1 > 0$, let $A = \\{a_1, \\ldots, a_5\\}$ and let $4^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. If none of the four gaps $a_{i+1} - a_i$ ($1 \\le i \\le 4$) equals $2a_1$ and $a_4 - a_3 \\ne a_2 - a_1$, then $|4^{\\wedge}_{\\pm} A| \\ge 26$.","title":"Restricted signed fourfold sumsets when no gap equals twice the least element and the first and third gaps differ"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-ge-of-5","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_ge_of_equal_outer_gaps_ne","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-ge-of-5","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Restricted signed fourfold sumsets when no gap equals twice the least element, the first and third gaps agree, and the last two gaps do not sum to twice the least element. Let $a_1 < a_2 < a_3 < a_4 < a_5$ be integers with $a_1 > 0$, let $A = \\{a_1, \\ldots, a_5\\}$ and let $4^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. If none of the four gaps $a_{i+1} - a_i$ ($1 \\le i \\le 4$) equals $2a_1$,\u2026","title":"Restricted signed fourfold sumsets when no gap equals twice the least element, the first and third gaps agree, and the last two gaps do not sum to twice the least element"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-ge-of-6","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_ge_of_equal_outer_gaps_eq","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-ge-of-6","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Restricted signed fourfold sumsets when no gap equals twice the least element, the first and third gaps agree, and the last two gaps sum to twice the least element. Let $a_1 < a_2 < a_3 < a_4 < a_5$ be integers with $a_1 > 0$, let $A = \\{a_1, \\ldots, a_5\\}$ and let $4^{\\wedge}_{\\pm} A$ be the restricted signed sumset of $A$. If none of the four gaps $a_{i+1} - a_i$ ($1 \\le i \\le 4$) equals $2a_1$,\u2026","title":"Restricted signed fourfold sumsets when no gap equals twice the least element, the first and third gaps agree, and the last two gaps sum to twice the least element"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/card-restricted-signed-sumset-four-ge-of-7","kind":"node","lean":"MetaMathlibExt.card_restrictedSignedSumset_four_ge_of_no_regular_gap","parent":22673,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets/index.html#card-restricted-signed-sumset-four-ge-of-7","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/restricted-signed-sumsets","status":"can_prove","summary":"Let $a_1 < a_2 < a_3 < a_4 < a_5$ be integers with $a_1 > 0$, let $A = \\{a_1, \\dots, a_5\\}$, and write $4^{\\wedge}_{\\pm} A$ for the restricted signed 4-fold sumset of $A$. If $a_{i+1} - a_i \\ne 2a_1$ for every $i \\in \\{1, 2, 3, 4\\}$, then $|4^{\\wedge}_{\\pm} A| \\ge 26$.","title":"Restricted signed 4-fold sumsets when no gap equals twice the minimum"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums","status":"can_prove","title":"Sets determined by their k-sums"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums/k-sum-multiset","kind":"node","lean":"MathlibExt.Combinatorics.KSumMultisetDeterminesSetErdosProblem494Wanted.kSumMultiset","parent":22709,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiset of "],["math","\\(k\\)"],["text","-sums"]],"source":"Multiset of $k$-sums"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums/index.html#k-sum-multiset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums","status":"fully_proved","summary":"For a finite set $A \\subseteq \\mathbb{C}$ and a natural number $k$, $A_k$ is the multiset of the sums $\\sum_{z \\in S} z$ over the $k$-element subsets $S$ of $A$, that is, of the sums of $k$ distinct elements of $A$, each subset counted once.","title":"Multiset of $k$-sums"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums/k-sum-multiset-determines-large-set","kind":"node","lean":"MathlibExt.Combinatorics.KSumMultisetDeterminesSetErdosProblem494Wanted.kSumMultisetDeterminesLargeSet","parent":22709,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums/index.html#k-sum-multiset-determines-large-set","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums","status":"fully_proved","summary":"For a finite set $A \\subseteq \\mathbb{C}$ and a natural number $k$, let $A_k$ be the multiset of the sums of the $k$-element subsets of $A$. This is the statement that for every natural number $k \\ge 3$ there is a natural number $n_0$ such that any two finite sets $A, B \\subseteq \\mathbb{C}$ with $|A| = |B| \\ge n_0$ and $A_k = B_k$ are equal.","title":"Erd\u0151s Problem 494 for large sets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums/k-sum-multiset-determines-large-set-hold","kind":"node","lean":"MathlibExt.Combinatorics.KSumMultisetDeterminesSetErdosProblem494Wanted.kSumMultisetDeterminesLargeSet_holds","parent":22709,"reading_title":{"format":"inline-parts-v1","parts":[["text","Large sets of complex numbers are determined by their "],["math","\\(k\\)"],["text","-sums"]],"source":"Large sets of complex numbers are determined by their $k$-sums"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums/index.html#k-sum-multiset-determines-large-set-hold","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sets-determined-by-their-k-sums","status":"can_prove","summary":"For a finite set $A \\subseteq \\mathbb{C}$ and a natural number $k$, let $A_k$ be the multiset of the sums $\\sum_{z \\in S} z$ over the $k$-element subsets $S$ of $A$. For every natural number $k \\ge 3$ there is a natural number $n_0$ such that any two finite sets $A, B \\subseteq \\mathbb{C}$ with $|A| = |B| \\ge n_0$ and $A_k = B_k$ are equal.","title":"Large sets of complex numbers are determined by their $k$-sums"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling","status":"can_prove","title":"Small doubling and small tripling"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling/small-doubling-implies-small-tripling","kind":"node","lean":"MathlibExt.Combinatorics.SmallDoublingTriplingAIMCombinatoricsProblem242Wanted.smallDoublingImpliesSmallTripling","parent":22713,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling/index.html#small-doubling-implies-small-tripling","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling","status":"fully_proved","summary":"The proposition that small doubling forces small tripling uniformly in all groups: there is a function $f : \\mathbb{R} \\to \\mathbb{R}$ such that for every group $G$, every finite nonempty subset $A \\subseteq G$ and every real number $K$, if $|A \\cdot A| \\le K |A|$ then $|A \\cdot A \\cdot A| \\le f(K) |A|$. Here $A \\cdot A = \\{ab : a, b \\in A\\}$ and $A \\cdot A \\cdot A = \\{abc : a, b, c \\in A\\}$.","title":"Tao's small-doubling question (AIM Problem 3.5)"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling/small-doubling-implies-small-tripling-re","kind":"node","lean":"MathlibExt.Combinatorics.SmallDoublingTriplingAIMCombinatoricsProblem242Wanted.smallDoublingImpliesSmallTripling_refuted","parent":22713,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling/index.html#small-doubling-implies-small-tripling-re","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/small-doubling-and-small-tripling","status":"can_prove","summary":"There is no function $f : \\mathbb{R} \\to \\mathbb{R}$ such that for every group $G$, every finite nonempty subset $A \\subseteq G$ and every real number $K$ with $|A \\cdot A| \\le K |A|$ one has $|A \\cdot A \\cdot A| \\le f(K) |A|$. Here $A \\cdot A = \\{ab : a, b \\in A\\}$ and $A \\cdot A \\cdot A = \\{abc : a, b, c \\in A\\}$.","title":"Small doubling does not imply small tripling"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling","status":"can_prove","title":"Sum-product estimates for sets of small doubling"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem818Wanted.conjecture","parent":22716,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling","status":"fully_proved","summary":"The statement that for every real $K > 0$ there are reals $C > 0$ and $c > 0$ such that every finite set $A$ of integers with $|A| \\ge 2$ and $|A + A| \\le K|A|$ satisfies $c|A|^2 / (\\log |A|)^C \\le |A \\cdot A|$. Here $A + A = \\{a + b : a, b \\in A\\}$, $A \\cdot A = \\{ab : a, b \\in A\\}$ and $\\log$ is the natural logarithm.","title":"Erd\u0151s Problem 818"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling/conjecture-holds","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem818Wanted.conjecture_holds","parent":22716,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-estimates-for-sets-of-small-doubling","status":"can_prove","summary":"For every real $K > 0$ there are reals $C > 0$ and $c > 0$ such that every finite set $A$ of integers with $|A| \\ge 2$ and $|A + A| \\le K|A|$ satisfies $c|A|^2 / (\\log |A|)^C \\le |A \\cdot A|$. Here $A + A = \\{a + b : a, b \\in A\\}$, $A \\cdot A = \\{ab : a, b \\in A\\}$ and $\\log$ is the natural logarithm.","title":"Erd\u0151s Problem 818 is true"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","title":"Sum-product problem over the reals"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-max-card","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductMaxCard","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-max-card","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A$ of real numbers, this is the natural number $\\max(|A+A|, |AA|)$, where $A+A = \\{a+b : a, b \\in A\\}$ is the sumset and $AA = \\{ab : a, b \\in A\\}$ is the product set of $A$.","title":"Size of the larger of the sumset and the product set"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-ratio","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductRatio","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-ratio","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A$ of real numbers, the sum-product ratio of $A$ is $\\log \\max(|A+A|, |AA|) / \\log |A|$, where $A+A$ and $AA$ are the sumset and the product set of $A$. With the conventions $\\log 0 = 0$ and $x / 0 = 0$, the ratio is $0$ when $|A| \\le 1$.","title":"Sum-product ratio"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/fixed-cardinality-ratios","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.fixedCardinalityRatios","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#fixed-cardinality-ratios","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$. For a natural number $n \\ge 2$, this is the set of the real numbers $\\log \\max(|A + A|, |AA|) / \\log |A|$ for all sets $A \\subset \\mathbb{R}$ with $|A| = n$; for $n \\le 1$ it consists of the single number $0$.","title":"Sum-product ratios at a fixed cardinality"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-level-minimum","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductLevelMinimum","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-level-minimum","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a natural number $n$, the level minimum $m(n)$ is the infimum of the sum-product ratios $\\log \\max(|A+A|, |AA|) / \\log |A|$ over all sets $A$ of $n$ real numbers, where $A+A$ and $AA$ are the sumset and the product set of $A$, and the ratio is taken to be $0$ when $|A| \\le 1$ (conventions $\\log 0 = 0$ and $x / 0 = 0$).","title":"Least sum-product ratio at a fixed cardinality"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-level-minimum-is-least","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductLevelMinimum_isLeast","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-level-minimum-is-least","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For every natural number $n$, let $m(n)$ be the infimum of the sum-product ratios $\\log \\max(|A+A|, |AA|) / \\log |A|$ over all sets $A$ of $n$ real numbers, where $A+A$ and $AA$ are the sumset and the product set of $A$ and the ratio is taken to be $0$ when $|A| \\le 1$ (conventions $\\log 0 = 0$ and $x / 0 = 0$). Then $m(n)$ is the least element of the set of these ratios: it is the ratio of some set of $n$ real\u2026","title":"The level minimum is attained"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-ratio-nonneg","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductRatio_nonneg","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-ratio-nonneg","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For every nonempty finite set $A$ of real numbers, the sum-product ratio $\\log \\max(|A+A|, |AA|) / \\log |A|$ is at least $0$, where $A+A$ and $AA$ are the sumset and the product set of $A$, and the ratio is taken to be $0$ when $|A| = 1$ (conventions $\\log 0 = 0$ and $x / 0 = 0$).","title":"The sum-product ratio is nonnegative"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-ratio-le-two","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductRatio_le_two","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-ratio-le-two","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For every finite set $A$ of real numbers with $|A| \\ge 2$, the sum-product ratio $\\log \\max(|A+A|, |AA|) / \\log |A|$ is at most $2$, where $A+A$ and $AA$ are the sumset and the product set of $A$.","title":"The sum-product ratio is at most two"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-level-minimum-nonneg","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductLevelMinimum_nonneg","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-level-minimum-nonneg","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the level minimum $m(n)$, the infimum of the sum-product ratios $\\log \\max(|A+A|, |AA|) / \\log |A|$ over all sets $A$ of $n$ real numbers, satisfies $m(n) \\ge 0$. Here $A+A$ and $AA$ are the sumset and the product set of $A$, and the ratio is taken to be $0$ when $|A| = 1$ (conventions $\\log 0 = 0$, $x / 0 = 0$).","title":"The level minimum is nonnegative"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-level-minimum-le-two","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductLevelMinimum_le_two","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-level-minimum-le-two","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For every natural number $n \\ge 2$, the level minimum $m(n)$, the infimum of the sum-product ratios $\\log \\max(|A+A|, |AA|) / \\log |A|$ over all sets $A$ of $n$ real numbers, where $A+A$ and $AA$ are the sumset and the product set of $A$, satisfies $m(n) \\le 2$.","title":"The level minimum is at most two"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/eventually-sum-product-level-minimum-mem","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.eventually_sumProductLevelMinimum_mem_Icc","parent":22719,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum-product level minima eventually lie in "],["math","\\((0, 2)\\)"]],"source":"The sum-product level minima eventually lie in $(0, 2)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#eventually-sum-product-level-minimum-mem","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$. For a natural number $n \\ge 2$ let $m_n$ be the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. Then $0 \\le m_n \\le 2$ for every sufficiently large $n$.","title":"The sum-product level minima eventually lie in $(0, 2)$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-level-minimum-is-bounded-und","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductLevelMinimum_isBoundedUnder_le","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-level-minimum-is-bounded-und","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"There is a real number $b$ such that $m(n) \\le b$ for all sufficiently large natural numbers $n$, where the level minimum $m(n)$ is the infimum of the sum-product ratios $\\log \\max(|A+A|, |AA|) / \\log |A|$ over all sets $A$ of $n$ real numbers, and $A+A$ and $AA$ are the sumset and the product set of $A$.","title":"The level minima are eventually bounded above"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-level-minimum-is-bounded-und-2","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductLevelMinimum_isBoundedUnder_ge","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-level-minimum-is-bounded-und-2","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"There is a real number $b$ such that the sum-product level minimum $m(n)$ satisfies $m(n) \\ge b$ for all sufficiently large $n$.","title":"The sum-product level minima are eventually bounded below"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-exponent","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductExponent","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-exponent","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ let $m(n)$ be the sum-product level minimum: the least value of $\\log \\max(|A + A|, |AA|) / \\log |A|$ over sets $A \\subset \\mathbb{R}$ with $n$ elements, where $A + A$ and $AA$ are the sets of sums $a + b$ and products $ab$ of elements $a, b \\in A$. The real sum-product exponent is $\\liminf_{n \\to \\infty} m(n)$.","title":"Sum-product exponent of the reals"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/sum-product-exponent-eq-liminf-add-two","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.sumProductExponent_eq_liminf_add_two","parent":22719,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum-product exponent ignores sets of size "],["math","\\(0\\)"],["text"," and "],["math","\\(1\\)"]],"source":"The sum-product exponent ignores sets of size $0$ and $1$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#sum-product-exponent-eq-liminf-add-two","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"Let $m(n)$ be the sum-product level minimum at $n$. The real sum-product exponent equals $\\liminf_{n \\to \\infty} m(n + 2)$, so the levels $n = 0$ and $n = 1$ do not affect it.","title":"The sum-product exponent ignores sets of size $0$ and $1$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/has-uniform-sum-product-lower-growth","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.HasUniformSumProductLowerGrowth","parent":22719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniform sum-product lower growth at rate "],["math","\\(s\\)"]],"source":"Uniform sum-product lower growth at rate $s$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#has-uniform-sum-product-lower-growth","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"A real number $s$ has uniform sum-product lower growth when for every $\\varepsilon > 0$ there is a constant $K > 0$ such that every nonempty finite set $A \\subset \\mathbb{R}$ satisfies $K \\cdot |A|^{s - \\varepsilon} \\le \\max(|A + A|, |AA|)$, where $A + A$ and $AA$ are the sets of sums $a + b$ and products $ab$ of elements $a, b \\in A$.","title":"Uniform sum-product lower growth at rate $s$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/epsilon-uniform-lower-growth-characteriz","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.epsilonUniformLowerGrowthCharacterization","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#epsilon-uniform-lower-growth-characteriz","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$, and let the real sum-product exponent $C_{84b}$ be the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. This is the proposition that $C_{84b}$ is the largest real number $s$ with the\u2026","title":"Statement of the uniform-growth characterization of the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/erdos-szemeredi-historical-conjecture","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.erdosSzemerediHistoricalConjecture","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#erdos-szemeredi-historical-conjecture","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$, and let the real sum-product exponent $C_{84b}$ be the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. This is the proposition that $C_{84b} = 2$.","title":"Statement of the Erd\u0151s\u2013Szemer\u00e9di conjecture for real sets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/bloom-sawin-schildkraut-zhelezov-refutat","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.bloomSawinSchildkrautZhelezovRefutation","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#bloom-sawin-schildkraut-zhelezov-refutat","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"The statement that the real sum-product exponent $C$ satisfies $C < 2$.","title":"Statement of the Bloom\u2013Sawin\u2013Schildkraut\u2013Zhelezov refutation"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/erdos-szemeredi-historical-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.erdosSzemerediHistoricalUpperBound","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#erdos-szemeredi-historical-upper-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$, and let the real sum-product exponent $C_{84b}$ be the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. This is the proposition that $C_{84b} \\le 2$.","title":"Statement of the upper bound two for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/bloom-sawin-schildkraut-zhelezov-upper-b","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.bloomSawinSchildkrautZhelezovUpperBound","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#bloom-sawin-schildkraut-zhelezov-upper-b","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"The statement that there is a real number $c > 0$ such that the real sum-product exponent $C$ satisfies $C \\le 2 - c$.","title":"Statement of the Bloom\u2013Sawin\u2013Schildkraut\u2013Zhelezov upper bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/bloom-sawin-schildkraut-zhelezov-numeric","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.bloomSawinSchildkrautZhelezovNumericalUpperBound","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#bloom-sawin-schildkraut-zhelezov-numeric","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"The statement that the real sum-product exponent $C$ satisfies $C \\le 199999913/100000000$.","title":"Statement of the Bloom\u2013Sawin\u2013Schildkraut\u2013Zhelezov numerical upper bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/trivial-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.trivialLowerBound","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#trivial-lower-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$. This is the proposition that the real sum-product exponent, the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$, is at least $1$.","title":"Statement of the trivial lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/elekes-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.elekesLowerBound","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#elekes-lower-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$, and let the real sum-product exponent $C_{84b}$ be the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. This is the proposition that $C_{84b} \\ge 5/4$.","title":"Statement of Elekes's lower bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/solymosi-fourteen-elevenths-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.solymosiFourteenEleventhsLowerBound","parent":22719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of Solymosi's "],["math","\\(14/11\\)"],["text"," lower bound"]],"source":"Statement of Solymosi's $14/11$ lower bound"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#solymosi-fourteen-elevenths-lower-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"The proposition that the real sum-product exponent is at least $14/11$.","title":"Statement of Solymosi's $14/11$ lower bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/solymosi-four-thirds-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.solymosiFourThirdsLowerBound","parent":22719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of Solymosi's "],["math","\\(4/3\\)"],["text"," lower bound"]],"source":"Statement of Solymosi's $4/3$ lower bound"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#solymosi-four-thirds-lower-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"The proposition that the real sum-product exponent is at least $4/3$.","title":"Statement of Solymosi's $4/3$ lower bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/rudnev-stevens-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.rudnevStevensLowerBound","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#rudnev-stevens-lower-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"The proposition that the real sum-product exponent is at least $1558/1167 = 4/3 + 2/1167$.","title":"Statement of the Rudnev\u2013Stevens lower bound"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/bloom-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.bloomLowerBound","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#bloom-lower-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"The statement that the real sum-product exponent $C$ satisfies $1270/951 \\le C$.","title":"Statement of Bloom's lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/cushman-current-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.cushmanCurrentLowerBound","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#cushman-current-lower-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"fully_proved","summary":"The statement that the real sum-product exponent $C$ satisfies $4/3 + 10/4407 \\le C$.","title":"Statement of Cushman's lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/epsilon-uniform-lower-growth-characteriz-2","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.epsilonUniformLowerGrowthCharacterization_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#epsilon-uniform-lower-growth-characteriz-2","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$, and let the real sum-product exponent $C_{84b}$ be the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. Then $C_{84b}$ is the largest real number $s$ with the following property: for every\u2026","title":"The real sum-product exponent is the largest uniform growth rate"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/erdos-szemeredi-historical-conjecture-re","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.erdosSzemerediHistoricalConjecture_refuted","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#erdos-szemeredi-historical-conjecture-re","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$, and let the real sum-product exponent $C_{84b}$ be the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. Then $C_{84b} \\ne 2$.","title":"The Erd\u0151s\u2013Szemer\u00e9di conjecture fails for real sets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/bloom-sawin-schildkraut-zhelezov-refutat-2","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.bloomSawinSchildkrautZhelezovRefutation_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#bloom-sawin-schildkraut-zhelezov-refutat-2","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"The real sum-product exponent $C$ satisfies $C < 2$. Here, for a finite set $A \\subset \\mathbb{R}$, $A + A$ and $A \\cdot A$ are the sets of sums $a + b$ and products $ab$ with $a, b \\in A$, and $M(A) = \\max(|A + A|, |A \\cdot A|)$; for $n \\ge 2$, $m(n)$ is the minimum of $\\log M(A) / \\log |A|$ over all $A \\subset \\mathbb{R}$ with $|A| = n$; and $C = \\liminf_{n \\to \\infty} m(n)$.","title":"The real sum-product exponent is less than 2 (Bloom\u2013Sawin\u2013Schildkraut\u2013Zhelezov)"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/erdos-szemeredi-historical-upper-bound-h","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.erdosSzemerediHistoricalUpperBound_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#erdos-szemeredi-historical-upper-bound-h","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$, and let the real sum-product exponent $C_{84b}$ be the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. Then $C_{84b} \\le 2$.","title":"The real sum-product exponent is at most two"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/bloom-sawin-schildkraut-zhelezov-upper-b-2","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.bloomSawinSchildkrautZhelezovUpperBound_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#bloom-sawin-schildkraut-zhelezov-upper-b-2","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"There is a real number $c > 0$ such that the real sum-product exponent $C$ satisfies $C \\le 2 - c$. Here, for a finite set $A \\subset \\mathbb{R}$, $A + A$ and $A \\cdot A$ are the sets of sums $a + b$ and products $ab$ with $a, b \\in A$, and $M(A) = \\max(|A + A|, |A \\cdot A|)$; for $n \\ge 2$, $m(n)$ is the minimum of $\\log M(A) / \\log |A|$ over all $A \\subset \\mathbb{R}$ with $|A| = n$; and\u2026","title":"Bloom\u2013Sawin\u2013Schildkraut\u2013Zhelezov upper bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/bloom-sawin-schildkraut-zhelezov-numeric-2","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.bloomSawinSchildkrautZhelezovNumericalUpperBound_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#bloom-sawin-schildkraut-zhelezov-numeric-2","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"The real sum-product exponent $C$ satisfies $C \\le 199999913/100000000 = 1.99999913$. Here, for a finite set $A \\subset \\mathbb{R}$, $A + A$ and $A \\cdot A$ are the sets of sums $a + b$ and products $ab$ with $a, b \\in A$, and $M(A) = \\max(|A + A|, |A \\cdot A|)$; for $n \\ge 2$, $m(n)$ is the minimum of $\\log M(A) / \\log |A|$ over all $A \\subset \\mathbb{R}$ with $|A| = n$; and $C = \\liminf_{n \\to \\infty} m(n)$.","title":"Bloom\u2013Sawin\u2013Schildkraut\u2013Zhelezov numerical upper bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/trivial-lower-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.trivialLowerBound_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#trivial-lower-bound-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"The real sum-product exponent is at least $1$. Here the exponent is $\\liminf_{n \\to \\infty} m(n)$, where $m(n)$ is the infimum of $\\log \\max(|A+A|, |AA|) / \\log |A|$ over all sets $A$ of $n$ real numbers, and $A+A$ and $AA$ are the sumset and the product set of $A$.","title":"Trivial lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/elekes-lower-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.elekesLowerBound_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#elekes-lower-bound-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"For a finite set $A \\subset \\mathbb{R}$ write $A + A$ and $AA$ for the sets of all sums $a + b$ and all products $ab$ with $a, b \\in A$, and let the real sum-product exponent $C_{84b}$ be the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log n$ over sets $A \\subset \\mathbb{R}$ with $|A| = n$. Then $C_{84b} \\ge 5/4$.","title":"Elekes's lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/solymosi-fourteen-elevenths-lower-bound-2","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.solymosiFourteenEleventhsLowerBound_holds","parent":22719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solymosi's "],["math","\\(14/11\\)"],["text"," lower bound for the real sum-product exponent"]],"source":"Solymosi's $14/11$ lower bound for the real sum-product exponent"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#solymosi-fourteen-elevenths-lower-bound-2","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"Let $C$ be the real sum-product exponent: the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log |A|$ over $n$-element sets $A \\subset \\mathbb{R}$, where $A + A$ and $AA$ are the sets of sums $a + b$ and products $ab$ of elements $a, b \\in A$. Then $C \\ge 14/11$.","title":"Solymosi's $14/11$ lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/solymosi-four-thirds-lower-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.solymosiFourThirdsLowerBound_holds","parent":22719,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solymosi's "],["math","\\(4/3\\)"],["text"," lower bound for the real sum-product exponent"]],"source":"Solymosi's $4/3$ lower bound for the real sum-product exponent"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#solymosi-four-thirds-lower-bound-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"Let $C$ be the real sum-product exponent: the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log |A|$ over $n$-element sets $A \\subset \\mathbb{R}$, where $A + A$ and $AA$ are the sets of sums $a + b$ and products $ab$ of elements $a, b \\in A$. Then $C \\ge 4/3$.","title":"Solymosi's $4/3$ lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/rudnev-stevens-lower-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.rudnevStevensLowerBound_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#rudnev-stevens-lower-bound-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"Let $C$ be the real sum-product exponent: the limit inferior, as $n \\to \\infty$, of the least value of $\\log \\max(|A + A|, |AA|) / \\log |A|$ over $n$-element sets $A \\subset \\mathbb{R}$, where $A + A$ and $AA$ are the sets of sums $a + b$ and products $ab$ of elements $a, b \\in A$. Then $C \\ge 1558/1167$.","title":"Rudnev\u2013Stevens lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/bloom-lower-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.bloomLowerBound_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#bloom-lower-bound-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"The real sum-product exponent $C$ is at least $1270/951$. Here, for a finite set $A \\subset \\mathbb{R}$, $A + A$ and $A \\cdot A$ are the sets of sums $a + b$ and products $ab$ with $a, b \\in A$, and $M(A) = \\max(|A + A|, |A \\cdot A|)$; for $n \\ge 2$, $m(n)$ is the minimum of $\\log M(A) / \\log |A|$ over all $A \\subset \\mathbb{R}$ with $|A| = n$; and $C = \\liminf_{n \\to \\infty} m(n)$.","title":"Bloom's lower bound for the real sum-product exponent"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/cushman-current-lower-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.RealSumProductExponentWanted.cushmanCurrentLowerBound_holds","parent":22719,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals/index.html#cushman-current-lower-bound-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sum-product-problem-over-the-reals","status":"can_prove","summary":"The real sum-product exponent $C$ is at least $4/3 + 10/4407$. Here, for a finite set $A \\subset \\mathbb{R}$, $A + A$ and $A \\cdot A$ are the sets of sums $a + b$ and products $ab$ with $a, b \\in A$, and $M(A) = \\max(|A + A|, |A \\cdot A|)$; for $n \\ge 2$, $m(n)$ is the minimum of $\\log M(A) / \\log |A|$ over all $A \\subset \\mathbb{R}$ with $|A| = n$; and $C = \\liminf_{n \\to \\infty} m(n)$.","title":"Cushman's lower bound for the real sum-product exponent"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs","status":"can_prove","title":"Sums and products along graphs"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs/restricted-sum-product-along-graphs","kind":"node","lean":"MathlibExt.Combinatorics.GraphRestrictedSumProductErdosProblem808Wanted.restrictedSumProductAlongGraphs","parent":22761,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs/index.html#restricted-sum-product-along-graphs","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs","status":"fully_proved","summary":"The statement that for all real numbers $c > 0$ and $\\varepsilon > 0$ and all sufficiently large natural numbers $n$ the following holds. Let $A$ be a set of $n$ positive integers and let $E$ be a finite set of pairs $(a, b)$ with $a, b \\in A$ and $a < b$ (the edges of a graph on $A$) such that $|E| \\ge n^{1+c}$. Write $A +_E A = \\{a + b : (a, b) \\in E\\}$ and $A \\cdot_E A = \\{ab : (a, b) \\in E\\}$. Then\u2026","title":"Erd\u0151s Problem 808 (sums and products along graphs)"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs/restricted-sum-product-along-graphs-refu","kind":"node","lean":"MathlibExt.Combinatorics.GraphRestrictedSumProductErdosProblem808Wanted.restrictedSumProductAlongGraphs_refuted","parent":22761,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs/index.html#restricted-sum-product-along-graphs-refu","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-and-products-along-graphs","status":"can_prove","summary":"There are real numbers $c > 0$ and $\\varepsilon > 0$ such that for infinitely many natural numbers $n$ there are a set $A$ of $n$ positive integers and a finite set $E$ of pairs $(a, b)$ with $a, b \\in A$ and $a < b$ (the edges of a graph on $A$) such that $|E| \\ge n^{1+c}$ and $\\max(|\\{a + b : (a, b) \\in E\\}|, |\\{ab : (a, b) \\in E\\}|) < n^{1+c-\\varepsilon}$.","title":"Erd\u0151s Problem 808 is false"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates","status":"can_prove","title":"Sums of dilates"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates/transcendental-dilate-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.TranscendentalDilateSumsetAIMCombinatoricsProblem264Wanted.transcendentalDilateLowerBound","parent":22764,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates/index.html#transcendental-dilate-lower-bound","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates","status":"fully_proved","summary":"There is a real constant $c > 0$ such that for every real number $\\alpha$ transcendental over $\\mathbb{Q}$ and every finite set $A$ of real numbers,","title":"Lower bound for sums of transcendental dilates"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates/transcendental-dilate-lower-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.TranscendentalDilateSumsetAIMCombinatoricsProblem264Wanted.transcendentalDilateLowerBound_holds","parent":22764,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates/index.html#transcendental-dilate-lower-bound-holds","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sums-of-dilates","status":"can_prove","summary":"There is a real constant $c > 0$ such that for every real number $\\alpha$ transcendental over $\\mathbb{Q}$ and every finite set $A$ of real numbers,","title":"Conlon\u2013Lim theorem on sums of transcendental dilates"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","title":"Sumsets with at least r distinct terms"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/restricted-sumset-two-zero-one-two","kind":"node","lean":"MetaMathlibExt.restrictedSumset_two_zero_one_two","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Restricted twofold sumset of "],["math","\\(\\{0, 1, 2\\}\\)"]],"source":"Restricted twofold sumset of $\\{0, 1, 2\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#restricted-sumset-two-zero-one-two","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"fully_proved","summary":"In the integers, the restricted sumset $2^{\\wedge}\\{0, 1, 2\\}$, the set of sums of two distinct elements of $\\{0, 1, 2\\}$, equals $\\{1, 2, 3\\}$.","title":"Restricted twofold sumset of $\\{0, 1, 2\\}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/at-least-distinct-sumset","kind":"node","lean":"MetaMathlibExt.atLeastDistinctSumset","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums with at least "],["math","\\(r\\)"],["text"," distinct terms"]],"source":"Sums with at least $r$ distinct terms"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#at-least-distinct-sumset","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $A$ a finite subset of $G$ and $h, r$ natural numbers. The sumset $h^{(\\ge r)}A$ is the set of all sums of $h$ elements of $A$, repetitions allowed, in which at least $r$ distinct elements occur, that is, the elements $\\sum_{a \\in B} x_a a$ with $B \\subseteq A$, $|B| \\ge r$, positive integers $x_a$ and $\\sum_{a \\in B} x_a = h$.","title":"Sums with at least $r$ distinct terms"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/at-least-distinct-sumset-three-two-zero","kind":"node","lean":"MetaMathlibExt.atLeastDistinctSumset_three_two_zero_one_two","parent":22767,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#at-least-distinct-sumset-three-two-zero","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"fully_proved","summary":"In the integers, the set of sums of three elements of $\\{0, 1, 2\\}$, repetitions allowed, in which at least two distinct elements occur is $3^{(\\ge 2)}\\{0, 1, 2\\} = \\{1, 2, 3, 4, 5\\}$; the sums $0$ and $6$ need a single repeated term.","title":"Example of a sumset with at least two distinct terms"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/cauchy-davenport-nsmul","kind":"node","lean":"MetaMathlibExt.cauchy_davenport_nsmul","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cauchy\u2013Davenport theorem for "],["math","\\(h\\)"],["text","-fold sumsets"]],"source":"Cauchy\u2013Davenport theorem for $h$-fold sumsets"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#cauchy-davenport-nsmul","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $p$ be a prime, $A$ a nonempty subset of $\\mathbb{Z}/p\\mathbb{Z}$ with $|A| = k$, and $h \\ge 1$ a natural number. Then the $h$-fold sumset $hA = \\{a_1 + \\dots + a_h : a_1, \\dots, a_h \\in A\\}$ satisfies $|hA| \\ge \\min(p,\\ hk - h + 1)$.","title":"Cauchy\u2013Davenport theorem for $h$-fold sumsets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/dias-da-silva-hamidoune","kind":"node","lean":"MetaMathlibExt.dias_da_silva_hamidoune","parent":22767,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#dias-da-silva-hamidoune","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $p$ be a prime, $A$ a nonempty subset of $\\mathbb{Z}/p\\mathbb{Z}$ with $|A| = k$, and $h$ a natural number with $1 \\le h \\le k$. Then the restricted sumset $h^{\\wedge}A$ of sums of $h$ distinct elements of $A$ satisfies $|h^{\\wedge}A| \\ge \\min(p,\\ hk - h^2 + 1)$.","title":"Dias da Silva\u2013Hamidoune theorem"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/vosper-nsmul","kind":"node","lean":"MetaMathlibExt.vosper_nsmul","parent":22767,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#vosper-nsmul","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $p$ be a prime, let $A$ be a subset of the cyclic group $\\mathbb{Z}/p\\mathbb{Z}$ with $k = |A| \\ge 2$ elements, let $h \\ge 2$ be an integer, and let $hA = \\{a_1 + \\dots + a_h : a_1, \\dots, a_h \\in A\\}$ be the $h$-fold sumset of $A$. If $|hA| = hk - h + 1$ and $hk - h + 1 \\le p - 2$, then $A$ is an arithmetic progression of length $k$ in $\\mathbb{Z}/p\\mathbb{Z}$.","title":"Vosper's theorem for iterated sumsets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/nathanson-restricted-sumset-inverse","kind":"node","lean":"MetaMathlibExt.nathanson_restrictedSumset_inverse","parent":22767,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#nathanson-restricted-sumset-inverse","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $A$ be a finite set of integers with $k = |A| \\ge 5$ elements and let $h$ be an integer with $2 \\le h \\le k - 2$. Write $h^{\\wedge} A$ for the restricted sumset of $A$, the set of sums of $h$ distinct elements of $A$. If $|h^{\\wedge} A| = hk - h^2 + 1$, then $A$ is an arithmetic progression of length $k$.","title":"Nathanson's inverse theorem for restricted sumsets"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/freiman-three-k-sub-four","kind":"node","lean":"MetaMathlibExt.freiman_three_k_sub_four","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Freiman's "],["math","\\(3k - 4\\)"],["text"," theorem"]],"source":"Freiman's $3k - 4$ theorem"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#freiman-three-k-sub-four","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $A$ be a finite set of integers with $|A| = k \\ge 3$, $\\min A = 0$ and $\\gcd(A) = 1$, and let $b$ be a natural number. If $|2A| = 2k - 1 + b \\le 3k - 4$, where $2A = \\{a + a' : a, a' \\in A\\}$, then $A$ is contained in an arithmetic progression of length at most $k + b$.","title":"Freiman's $3k - 4$ theorem"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/plagne-min-nsmul-card","kind":"node","lean":"MetaMathlibExt.plagne_min_nsmul_card","parent":22767,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#plagne-min-nsmul-card","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $m$, $k$ and $h$ be natural numbers with $1 \\le k \\le m$ and $h \\ge 1$. For a subset $A$ of $\\mathbb{Z}_m = \\mathbb{Z}/m\\mathbb{Z}$ let $hA = \\{a_1 + \\dots + a_h : a_1, \\dots, a_h \\in A\\}$ be its $h$-fold sumset. Then the minimum of $|hA|$ over all $k$-element subsets $A \\subseteq \\mathbb{Z}_m$ is","title":"Plagne's theorem on the smallest iterated sumsets in cyclic groups"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/exists-card-eq-sum-eq-zmod","kind":"node","lean":"MetaMathlibExt.exists_card_eq_sum_eq_zmod","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bajnok's lemma on subset sums in "],["math","\\(\\mathbb{Z}/d\\mathbb{Z}\\)"]],"source":"Bajnok's lemma on subset sums in $\\mathbb{Z}/d\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#exists-card-eq-sum-eq-zmod","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $d$ and $t$ be natural numbers with $1 \\le t \\le d - 1$. Then every $\\ell \\in \\mathbb{Z}/d\\mathbb{Z}$ is the sum of the elements of some $t$-element subset $L \\subseteq \\mathbb{Z}/d\\mathbb{Z}$.","title":"Bajnok's lemma on subset sums in $\\mathbb{Z}/d\\mathbb{Z}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/at-least-distinct-sumset-eq-nsmul-add-re","kind":"node","lean":"MetaMathlibExt.atLeastDistinctSumset_eq_nsmul_add_restrictedSumset","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Decomposition of sums with at least "],["math","\\(r\\)"],["text"," distinct terms"]],"source":"Decomposition of sums with at least $r$ distinct terms"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#at-least-distinct-sumset-eq-nsmul-add-re","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $G$ be an abelian group, $A$ a nonempty finite subset of $G$, and $r, h$ natural numbers with $1 \\le r \\le |A|$ and $r \\le h$. Write $h^{(\\ge r)}A$ for the set of sums of $h$ elements of $A$, repetitions allowed, in which at least $r$ distinct elements occur, $(h - r)A$ for the set of sums of $h - r$ elements of $A$, repetitions allowed (so $0A = \\{0\\}$), and $r^{\\wedge}A$ for the restricted sumset of sums of\u2026","title":"Decomposition of sums with at least $r$ distinct terms"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/at-least-distinct-sumset-card-ge-zmod","kind":"node","lean":"MetaMathlibExt.atLeastDistinctSumset_card_ge_zmod","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sharp lower bound for sums with at least "],["math","\\(r\\)"],["text"," distinct terms modulo a prime"]],"source":"Sharp lower bound for sums with at least $r$ distinct terms modulo a prime"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#at-least-distinct-sumset-card-ge-zmod","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $p$ be a prime, and for natural numbers $h, r$ and a finite set $A \\subseteq \\mathbb{Z}/p\\mathbb{Z}$ write $h^{(\\ge r)}A$ for the set of sums of $h$ elements of $A$, repetitions allowed, in which at least $r$ distinct elements occur. If $A$ is nonempty with $|A| = k$, $1 \\le r \\le k$ and $r \\le h$, then $|h^{(\\ge r)}A| \\ge \\min\\{p,\\ h(k - 1) - r(r - 1) + 1\\}$. The bound is sharp: for all natural numbers\u2026","title":"Sharp lower bound for sums with at least $r$ distinct terms modulo a prime"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/at-least-distinct-sumset-inverse-zmod","kind":"node","lean":"MetaMathlibExt.atLeastDistinctSumset_inverse_zmod","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inverse theorem for sums with at least "],["math","\\(r\\)"],["text"," distinct terms modulo a prime"]],"source":"Inverse theorem for sums with at least $r$ distinct terms modulo a prime"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#at-least-distinct-sumset-inverse-zmod","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $p$ be a prime, let $A \\subseteq \\mathbb{Z}/p\\mathbb{Z}$ be a set with $|A| = k \\ge 2$, and let $r, h$ be natural numbers with $1 \\le r \\le k$ and $h \\ge r + 2$. Write $h^{(\\ge r)}A$ for the set of sums of $h$ elements of $A$, repetitions allowed, in which at least $r$ distinct elements occur. If $|h^{(\\ge r)}A| = h(k - 1) - r(r - 1) + 1$ and $h(k - 1) - r(r - 1) + 1 \\le p - 2$, then $A$ is an arithmetic\u2026","title":"Inverse theorem for sums with at least $r$ distinct terms modulo a prime"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/at-least-distinct-sumset-card-ge-int","kind":"node","lean":"MetaMathlibExt.atLeastDistinctSumset_card_ge_int","parent":22767,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound and inverse theorem for sums with at least "],["math","\\(r\\)"],["text"," distinct terms in "],["math","\\(\\mathbb{Z}\\)"]],"source":"Lower bound and inverse theorem for sums with at least $r$ distinct terms in $\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#at-least-distinct-sumset-card-ge-int","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $A$ be a nonempty finite set of integers with $|A| = k$, and let $r, h$ be natural numbers with $1 \\le r \\le k$, $r \\le h$ and $h \\ge 2$. Write $h^{(\\ge r)}A$ for the set of sums of $h$ elements of $A$, repetitions allowed, in which at least $r$ distinct elements occur. Then $|h^{(\\ge r)}A| \\ge h(k - 1) - r(r - 1) + 1$. If equality holds and either $k \\ge 2$ and $h \\ge r + 2$, or $k \\ge 5$, $2 \\le r \\le k - 2$\u2026","title":"Lower bound and inverse theorem for sums with at least $r$ distinct terms in $\\mathbb{Z}$"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/lev-nsmul-card-ge","kind":"node","lean":"MetaMathlibExt.lev_nsmul_card_ge","parent":22767,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms/index.html#lev-nsmul-card-ge","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/sumsets-with-at-least-r-distinct-terms","status":"can_prove","summary":"Let $A$ be a finite set of integers with $k = |A| \\ge 3$ elements, smallest element $\\min A = 0$ and $\\gcd(A) = 1$. For $n \\ge 1$ let $nA = \\{a_1 + \\dots + a_n : a_1, \\dots, a_n \\in A\\}$ be the $n$-fold sumset of $A$. Then for every integer $h \\ge 2$,","title":"Lev's lower bound for iterated sumsets"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem.json","id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem","kind":"scope","navigation":"module","parent":22583,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem/index.html","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem","status":"fully_proved","title":"Vosper's theorem"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem/are-finite-arithmetic-progressions-with","kind":"node","lean":"MetaMathlibExt.AreFiniteArithmeticProgressionsWithCommonDifference","parent":22783,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem/index.html#are-finite-arithmetic-progressions-with","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid and let $A$ and $B$ be finite subsets of $G$. We say that $A$ and $B$ are arithmetic progressions with a common difference if there are $d, a, b \\in G$ such that $A = \\{a + nd : n \\in \\mathbb{N},\\ n < |A|\\}$ and $B = \\{b + nd : n \\in \\mathbb{N},\\ n < |B|\\}$, where $nd$ is the sum of $n$ copies of $d$. The two progressions may have different initial terms and different\u2026","title":"Arithmetic progressions with a common difference"},{"id":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem/vosper-eq-iff","kind":"node","lean":"MetaMathlibExt.vosper_eq_iff","parent":22783,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem/index.html#vosper-eq-iff","scope":"Combinatorics/additive-combinatorics/sumsets-inverse-theorems/vosper-s-theorem","status":"fully_proved","summary":"Let $p$ be a prime and let $A$ and $B$ be nonempty subsets of $\\mathbb{Z}/p\\mathbb{Z}$ whose sumset $A + B = \\{a + b : a \\in A, b \\in B\\}$ is not all of $\\mathbb{Z}/p\\mathbb{Z}$. Then $|A + B| = |A| + |B| - 1$ if and only if one of the following holds: (i) $\\min(|A|, |B|) = 1$; (ii) $|A + B| = p - 1$ and there is an element $c \\in \\mathbb{Z}/p\\mathbb{Z}$ such that the complement of $A + B$ is $\\{c\\}$ and $B$ is the\u2026","title":"Vosper's theorem"},{"data_url":"scopes/Combinatorics/additive-combinatorics/more-sums-than-differences.json","id":"Combinatorics/additive-combinatorics/more-sums-than-differences","kind":"scope","parent":22582,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/index.html","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences","status":"can_prove","summary":"Comparison of |A+A| with |A\u2212A|: sum-dominant (MSTD) and difference-dominated sets, minimal MSTD and MPTQ sets, Chu's constructions, unions of progressions, and the single-set sum-difference exponent.","title":"Sets with more sums than differences"},{"data_url":"scopes/Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets.json","id":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets","kind":"scope","navigation":"module","parent":22786,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/index.html","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets","status":"can_prove","title":"Alternating sum- and difference-dominated sets"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/interval-erase-add-sub","kind":"node","lean":"MetaMathlibExt.interval_erase_add_sub","parent":22787,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/index.html#interval-erase-add-sub","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets","status":"can_prove","summary":"Let $m$ and $r$ be integers with $m \\ge 4$ and $2 \\le r \\le m - 3$, and let $B = \\{0, 1, \\dots, m - 1\\} \\setminus \\{r\\}$. Then the sumset $B + B = \\{b + b' : b, b' \\in B\\}$ is $\\{0, 1, \\dots, 2m - 2\\}$ and the difference set $B - B = \\{b - b' : b, b' \\in B\\}$ is the set of integers from $-(m - 1)$ to $m - 1$.","title":"Sums and differences of an interval with one point removed"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/alternating-sequence-of-copies","kind":"node","lean":"MetaMathlibExt.alternating_sequence_of_copies","parent":22787,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/index.html#alternating-sequence-of-copies","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets","status":"can_prove","summary":"For a set $S$ of integers write $S + S = \\{s + s' : s, s' \\in S\\}$ and $S - S = \\{s - s' : s, s' \\in S\\}$. Let $A_1$ be a finite set of integers with $|A_1 - A_1| < |A_1 + A_1|$ and $0 \\in A_1$, and let $n$ be an integer with $0 \\le a < n$ for every $a \\in A_1$. Suppose that (a) $A_1 + A_1$ and $A_1 - A_1$ meet the same number of residue classes modulo $n$, and (b) with\u2026","title":"Alternating sum- and difference-dominated sets from shifted copies"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/nathanson-mstd-construction","kind":"node","lean":"MetaMathlibExt.nathanson_mstd_construction","parent":22787,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/index.html#nathanson-mstd-construction","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets","status":"can_prove","summary":"For a set $S$ of integers write $S + S = \\{s + s' : s, s' \\in S\\}$ and $S - S = \\{s - s' : s, s' \\in S\\}$. Let $m$, $d$, $k$ be integers with $m \\ge 4$, $1 \\le d \\le m - 1$ and $2d \\ne m$, and suppose $k \\ge 3$ if $2d < m$ and $k \\ge 4$ if $2d > m$. Let $B = \\{0, 1, \\dots, m - 1\\} \\setminus \\{d\\}$, $L = \\{jm - d : 1 \\le j \\le k\\}$, $A^* = B \\cup L \\cup \\{(k + 1)m - 2d - b : b \\in B\\}$ and $A = A^* \\cup \\{m\\}$. Then\u2026","title":"Nathanson's sum-dominated set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/alternating-sequence-from-nathanson","kind":"node","lean":"MetaMathlibExt.alternating_sequence_from_nathanson","parent":22787,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets/index.html#alternating-sequence-from-nathanson","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/alternating-sum-and-difference-dominated-sets","status":"can_prove","summary":"For a set $S$ of integers write $S + S = \\{s + s' : s, s' \\in S\\}$, $S - S = \\{s - s' : s, s' \\in S\\}$, $c + S = \\{c + s : s \\in S\\}$ and $c - S = \\{c - s : s \\in S\\}$. Let $m$, $d$, $k$ be integers with $m \\ge 4$, $m \\equiv 0 \\pmod 4$ and $4d = m$ or $4d = 3m$, and suppose $k \\ge 3$ if $2d < m$ and $k \\ge 4$ if $2d > m$. Let $B = \\{0, 1, \\dots, m - 1\\} \\setminus \\{d\\}$, $L = \\{jm - d : 1 \\le j \\le k\\}$ and\u2026","title":"Alternating sum- and difference-dominated sets from Nathanson's construction"},{"data_url":"scopes/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets.json","id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","kind":"scope","navigation":"module","parent":22786,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","title":"Chu's constructions of sum-dominant sets"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/set-of-consecutive-differences","kind":"node","lean":"MetaMathlibExt.setOfConsecutiveDifferences","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#set-of-consecutive-differences","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"fully_proved","summary":"For an integer $a$ and natural numbers $g_1, \\dots, g_r$, the set $(a \\mid g_1, \\dots, g_r)$ is the finite set of integers $\\{a, a + g_1, a + g_1 + g_2, \\dots, a + g_1 + \\dots + g_r\\}$; for $r = 0$ it is $\\{a\\}$.","title":"Sets given by consecutive differences"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/set-of-consecutive-differences-cons","kind":"node","lean":"MetaMathlibExt.setOfConsecutiveDifferences_cons","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#set-of-consecutive-differences-cons","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"fully_proved","summary":"For an integer $a$ and natural numbers $g_1, \\dots, g_r$ ($r \\ge 0$), write $(a \\mid g_1, \\dots, g_r)$ for the set of partial sums $\\{a, a + g_1, \\dots, a + g_1 + \\dots + g_r\\}$. Then for every natural number $g$, $(a \\mid g, g_1, \\dots, g_r) = \\{a\\} \\cup (a + g \\mid g_1, \\dots, g_r)$.","title":"Prepending a gap"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/set-of-consecutive-differences-example","kind":"node","lean":"MetaMathlibExt.setOfConsecutiveDifferences_example","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#set-of-consecutive-differences-example","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"fully_proved","summary":"The set with least element $0$ and consecutive differences $1, 5, 1, 1, 9, 7$ is $(0 \\mid 1, 5, 1, 1, 9, 7) = \\{0, 1, 6, 7, 8, 17, 24\\}$.","title":"An example of difference notation"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-block","kind":"node","lean":"MetaMathlibExt.chuBlock","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-block","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"fully_proved","summary":"For a natural number $k$, the difference block $M^k$ is the list of natural numbers $1, 4, \\ldots, 4, 3$ of length $k + 2$: a $1$, then $k$ copies of $4$, then a $3$.","title":"Chu difference block"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-block-one-two","kind":"node","lean":"MetaMathlibExt.chuBlock_one_two","parent":22792,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Chu difference blocks "],["math","\\(M^1\\)"],["text"," and "],["math","\\(M^2\\)"]],"source":"The Chu difference blocks $M^1$ and $M^2$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-block-one-two","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"fully_proved","summary":"The Chu difference block $M^1$ is the list $1, 4, 3$, and the Chu difference block $M^2$ is the list $1, 4, 4, 3$.","title":"The Chu difference blocks $M^1$ and $M^2$"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-balanced","kind":"node","lean":"MetaMathlibExt.chu_family_balanced","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-balanced","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$ be a positive integer and let $A = (0 \\mid 1, 1, 2, M^k, M^{k+1}, 1, 1) \\subseteq \\mathbb{Z}$ be the set of partial sums, starting at $0$, of the difference sequence formed by $1, 1, 2$, the Chu difference blocks $M^k$ and $M^{k+1}$, and $1, 1$. Then $A$ is balanced: $|A + A| = |A - A|$.","title":"A balanced family of Chu et al"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-difference-dominated","kind":"node","lean":"MetaMathlibExt.chu_family_difference_dominated","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-difference-dominated","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$ be a positive integer and let $A = (0 \\mid 1, 1, 2, M^k, M^{k+1}, M^{k+1}, 1, 1) \\subseteq \\mathbb{Z}$ be the set of partial sums, starting at $0$, of the difference sequence formed by $1, 1, 2$, the Chu difference blocks $M^k$, $M^{k+1}$ and $M^{k+1}$, and $1, 1$. Then $A$ is difference-dominated: $|A + A| < |A - A|$.","title":"A difference-dominated family of Chu et al"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-mstd-increasing","kind":"node","lean":"MetaMathlibExt.chu_family_mstd_increasing","parent":22792,"reading_title":{"format":"inline-parts-v1","parts":[["text","MSTD sets with "],["math","\\(M^k\\)"],["text"," before "],["math","\\(M^{k+1}\\)"],["text"," (Theorem 3)"]],"source":"MSTD sets with $M^k$ before $M^{k+1}$ (Theorem 3)"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-mstd-increasing","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$ be a positive integer, let $M^n$ denote the block $1, 4, \\dots, 4, 3$ with $n$ fours, and let $M_1$ be one of the sequences $(1, 1, 2)$, $(1, 1, 2, 1)$. Let $A = (0 \\mid 1, 1, 2, M^k, M^{k+1}, M_1)$ be the set of partial sums, starting from $0$, of the concatenated sequence $1, 1, 2, M^k, M^{k+1}, M_1$. Then $A$ is sum-dominated: $|A - A| < |A + A|$.","title":"MSTD sets with $M^k$ before $M^{k+1}$ (Theorem 3)"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-mstd-decreasing","kind":"node","lean":"MetaMathlibExt.chu_family_mstd_decreasing","parent":22792,"reading_title":{"format":"inline-parts-v1","parts":[["text","MSTD sets with "],["math","\\(M^{k+1}\\)"],["text"," before "],["math","\\(M^k\\)"],["text"," (Theorem 4)"]],"source":"MSTD sets with $M^{k+1}$ before $M^k$ (Theorem 4)"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-mstd-decreasing","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$ be a positive integer, let $M^n$ denote the block $1, 4, \\dots, 4, 3$ with $n$ fours, and let $M_1$ be one of the sequences $(1, 1)$, $(1, 1, 2)$, $(1, 1, 2, 1)$. Let $A = (0 \\mid 1, 1, 2, M^{k+1}, M^k, M_1)$ be the set of partial sums, starting from $0$, of the concatenated sequence $1, 1, 2, M^{k+1}, M^k, M_1$. Then $A$ is sum-dominated: $|A - A| < |A + A|$.","title":"MSTD sets with $M^{k+1}$ before $M^k$ (Theorem 4)"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"fully_proved","summary":"For natural numbers $k$, $l$, $t$, let $M^n$ be the block $1, 4, \\dots, 4, 3$ with $n$ fours, and let $D$ be the sequence made of $t$ copies of the block $1, 1, 2, M^k, \\dots, M^k$ (with $l$ copies of $M^k$) followed by $1, 1, 2, 1$. The set $A_{k,l,t}$ is $(0 \\mid D)$, the set of partial sums of $D$ starting from $0$.","title":"Kumar\u2013Mohan\u2013Pandey sets"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-one-one-one","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_one_one_one","parent":22792,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Kumar\u2013Mohan\u2013Pandey set "],["math","\\(A_{1,1,1}\\)"]],"source":"The Kumar\u2013Mohan\u2013Pandey set $A_{1,1,1}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-one-one-one","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"fully_proved","summary":"For $k = l = t = 1$, the Kumar\u2013Mohan\u2013Pandey set $A_{1,1,1}$, the set of partial sums starting from $0$ of the sequence $1, 1, 2, 1, 4, 3, 1, 1, 2, 1$, is $\\{0, 1, 2, 4, 5, 9, 12, 13, 14, 16, 17\\}$.","title":"The Kumar\u2013Mohan\u2013Pandey set $A_{1,1,1}$"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-mem-sub","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_mem_sub","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-mem-sub","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be positive integers, let $A = A_{k,l,t}$ be the Kumar\u2013Mohan\u2013Pandey set, and put $T = t(kl + l + 1)$. Then the difference set $A - A$ contains $4i$ and $1 + 4i$ for every integer $i$ with $1 \\le i \\le 1 + T$, and contains $3 + 4i$ for every integer $i$ with $0 \\le i \\le T$.","title":"Differences in the Kumar\u2013Mohan\u2013Pandey sets (Lemma 1)"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-card-sub-six-add-four-mul","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_card_sub_six_add_four_mul","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-card-sub-six-add-four-mul","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be positive integers and let $A \\subseteq \\mathbb{Z}$ be the Chu family set with parameters $k$, $l$, $t$. Then exactly $lt + t$ elements $d$ of $A - A$ satisfy $d \\ge 6$ and $d \\equiv 2 \\pmod 4$; that is, exactly $lt + t$ positive differences of $A$ have the form $6 + 4m$ with $m \\ge 0$ an integer.","title":"Positive differences of the form 6 + 4m in the Chu family set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-mem","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_mem","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-mem","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be positive integers, let $A = A_{k,l,t}$ be the Kumar\u2013Mohan\u2013Pandey set, and put $T = t(kl + l + 1)$. Then $1 + 4i \\in A$ for every integer $i$ with $0 \\le i \\le 1 + T$, and $2 + 4T \\in A$ and $4 + 4T \\in A$.","title":"Elements of the Kumar\u2013Mohan\u2013Pandey sets (Lemma 4)"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-mem-add","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_mem_add","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-mem-add","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be positive integers, let $A \\subseteq \\mathbb{Z}$ be the Chu family set with parameters $k$, $l$, $t$, and let $T = t(kl + l + 1)$. Then $1 + 4i$ and $2 + 4i$ lie in $A + A$ for every integer $i$ with $0 \\le i \\le 2 + 2T$, and $3 + 4i$ lies in $A + A$ for every integer $i$ with $0 \\le i \\le 1 + 2T$.","title":"Elements of the sumset of the Chu family set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-mem-block-starts","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_mem_block_starts","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-mem-block-starts","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be positive integers and let $A \\subseteq \\mathbb{Z}$ be the Chu family set with parameters $k$, $l$, $t$. Then $4 + (4k + 4)i + (4kl + 4l + 4)j \\in A$ for all integers $i$, $j$ with $0 \\le i \\le l$ and $0 \\le j \\le t - 1$.","title":"Block starting points of the Chu family set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-card-sub","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_card_sub","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-card-sub","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be positive integers and let $A \\subseteq \\mathbb{Z}$ be the Chu family set with parameters $k$, $l$, $t$. Then $|A - A| = 11 + 6klt + 8lt + 8t$.","title":"Size of the difference set of the Chu family set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-card-add","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_card_add","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-card-add","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be positive integers and let $A \\subseteq \\mathbb{Z}$ be the Chu family set with parameters $k$, $l$, $t$. Then $|A + A| \\ge 11 + 6klt + 10lt + 8t$.","title":"Lower bound for the sumset of the Chu family set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-add-supset-six-families","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_add_supset_six_families","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-add-supset-six-families","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be positive integers, let $A \\subseteq \\mathbb{Z}$ be the Chu family set with parameters $k$, $l$, $t$, and let $P = 8kl + 8l + 8$. Let $X_1$, $X_2$, $X_4$, $X_5$ be the sets of the integers $Pj + (4k + 4)i + 4$, $Pj + (4k + 4)i + 8$, $Pj + 4kl + 4l + (4k + 4)i + 8$ and $Pj + 4kl + 4l + (4k + 4)i + 12$ respectively, for $1 \\le i \\le l$ and $0 \\le j \\le t - 1$, and let $X_3$, $X_6$ be the sets of\u2026","title":"Six disjoint families in the sumset of the Chu family set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/chu-family-klt-run-sums-r1-eq-r3","kind":"node","lean":"MetaMathlibExt.chuFamilyKLT_run_sums_R1_eq_R3","parent":22792,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets/index.html#chu-family-klt-run-sums-r1-eq-r3","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/chu-s-constructions-of-sum-dominant-sets","status":"can_prove","summary":"Let $k$, $l$, $t$ be natural numbers, let $M^k$ be the block $1, 4, \\dots, 4, 3$ with $k$ fours, and let $B$ be the concatenation of $l$ copies of $M^k$. For $j \\ge 1$ let $R_1(j)$ be the sequence $1, 1$ followed by $j$ copies of $2, B, 1, 1$, and let $R_3(j)$ be the sequence $2$ followed by $j$ copies of $B, 1, 1, 2$. Then the sums of the entries of $R_1(j)$ for $1 \\le j \\le t$ form the same set as the sums of the\u2026","title":"Two families of runs with the same sums (Proposition 1)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set.json","id":"Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set","kind":"scope","navigation":"module","parent":22786,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set/index.html","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set","status":"fully_proved","title":"Minimum size of a sum-dominant set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set/card-add-le-card-sub-of-card-lt-six","kind":"node","lean":"MetaMathlibExt.card_add_le_card_sub_of_card_lt_six","parent":22813,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set/index.html#card-add-le-card-sub-of-card-lt-six","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set","status":"fully_proved","summary":"Let $A$ be a finite set of integers with $|A| < 6$. Then $|A + A| \\le |A - A|$, where $A + A = \\{a + b : a, b \\in A\\}$ is the sumset and $A - A = \\{a - b : a, b \\in A\\}$ is the difference set of $A$.","title":"Chu's theorem for arbitrary finite sets of integers"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set/no-small-sum-dominant","kind":"node","lean":"MetaMathlibExt.no_small_sum_dominant","parent":22813,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set/index.html#no-small-sum-dominant","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/minimum-size-of-a-sum-dominant-set","status":"fully_proved","summary":"Let $A$ be a finite set of nonnegative integers with $|A| < 6$. Then $|A + A| \\le |A - A|$, where $A + A = \\{a + b : a, b \\in A\\}$ is the sumset and $A - A = \\{a - b : a, b \\in A\\}$ is the difference set of $A$; that is, $A$ is not sum-dominant.","title":"Chu's theorem: sets with fewer than six elements are not sum-dominant"},{"data_url":"scopes/Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set.json","id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set","kind":"scope","navigation":"module","parent":22786,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/index.html","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set","status":"can_prove","title":"Sum-difference exponent of a single set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/is-admissible-exponent","kind":"node","lean":"MathlibExt.Combinatorics.SingleSetSumDifferenceExponentWanted.IsAdmissibleExponent","parent":22816,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/index.html#is-admissible-exponent","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set","status":"fully_proved","summary":"A real number $\\theta$ is an admissible exponent when every nonempty finite set $A$ of integers satisfies $|A+A|/|A| \\le (|A-A|/|A|)^{\\theta}$, where $A+A$ and $A-A$ are the sumset and the difference set of $A$.","title":"Admissible exponents for sums and differences"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/single-set-sum-difference-exponent","kind":"node","lean":"MathlibExt.Combinatorics.SingleSetSumDifferenceExponentWanted.singleSetSumDifferenceExponent","parent":22816,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/index.html#single-set-sum-difference-exponent","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set","status":"fully_proved","summary":"The single-set sum-difference exponent $C_{3d}$ is the infimum of the admissible exponents, that is, of the set of real numbers $\\theta$ such that $|A+A|/|A| \\le (|A-A|/|A|)^{\\theta}$ for every nonempty finite set $A$ of integers, where $A+A$ and $A-A$ are the sumset and the difference set of $A$.","title":"The single-set sum-difference exponent"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/single-set-sum-difference-exponent-eq-tw","kind":"node","lean":"MathlibExt.Combinatorics.SingleSetSumDifferenceExponentWanted.singleSetSumDifferenceExponent_eq_two","parent":22816,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/index.html#single-set-sum-difference-exponent-eq-tw","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set","status":"fully_proved","summary":"This is the statement that the single-set sum-difference exponent $C_{3d}$ is equal to $2$.","title":"The proposition that the single-set sum-difference exponent equals 2"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/single-set-sum-difference-exponent-eq-tw-2","kind":"node","lean":"MathlibExt.Combinatorics.SingleSetSumDifferenceExponentWanted.singleSetSumDifferenceExponent_eq_two_holds","parent":22816,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set/index.html#single-set-sum-difference-exponent-eq-tw-2","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sum-difference-exponent-of-a-single-set","status":"can_prove","summary":"The single-set sum-difference exponent $C_{3d}$ is equal to $2$.","title":"The single-set sum-difference exponent equals 2"},{"data_url":"scopes/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant.json","id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","kind":"scope","navigation":"module","parent":22786,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","title":"Unions of two progressions are not sum-dominant"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/real-interval-set","kind":"node","lean":"MetaMathlibExt.realIntervalSet","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#real-interval-set","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"fully_proved","summary":"For a natural number $n$, $I_n = \\{0, 1, \\ldots, n - 1\\}$ is the set of the first $n$ natural numbers, viewed as a finite set of real numbers; $I_0$ is empty.","title":"Initial integer interval"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/card-real-interval-set","kind":"node","lean":"MetaMathlibExt.card_realIntervalSet","parent":22821,"reading_title":{"format":"inline-parts-v1","parts":[["text","Size of the interval "],["math","\\(I_n\\)"]],"source":"Size of the interval $I_n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#card-real-interval-set","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"fully_proved","summary":"For every natural number $n$, the set $I_n = \\{0, 1, \\dots, n - 1\\}$ of real numbers has exactly $n$ elements.","title":"Size of the interval $I_n$"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/real-apset","kind":"node","lean":"MetaMathlibExt.realAPSet","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#real-apset","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"fully_proved","summary":"For a real number $x$, a natural number $m$ and a real number $d$, $J(x, m, d)$ is the finite set of real numbers $\\{x + i d : i = 0, 1, \\ldots, m\\}$, that is, $x, x + d, \\ldots, x + m d$.","title":"Finite arithmetic progression"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/card-real-apset","kind":"node","lean":"MetaMathlibExt.card_realAPSet","parent":22821,"reading_title":{"format":"inline-parts-v1","parts":[["text","Size of the progression "],["math","\\(J(x, m, d)\\)"]],"source":"Size of the progression $J(x, m, d)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#card-real-apset","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"fully_proved","summary":"For a real number $x$, a natural number $m$ and a real number $d \\ne 0$, the progression $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$ has exactly $m + 1$ elements.","title":"Size of the progression $J(x, m, d)$"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/is-mstdset","kind":"node","lean":"MetaMathlibExt.IsMSTDSet","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#is-mstdset","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"fully_proved","summary":"A finite set $A$ of real numbers is an *MSTD set* (more sums than differences) when its difference set has fewer elements than its sumset: $|A - A| < |A + A|$, where $A + A = \\{a + b : a, b \\in A\\}$ and $A - A = \\{a - b : a, b \\in A\\}$.","title":"MSTD set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-singleton","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_singleton","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-singleton","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"fully_proved","summary":"For every real number $a$, the set $\\{a\\}$ is not an MSTD set.","title":"A single point is not an MSTD set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-union-two-ap","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_union_two_ap","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-union-two-ap","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"For real numbers $x, d$ and a natural number $m$, write $J(x, m, d) = \\{x + i d : i = 0, 1, \\ldots, m\\}$. Let $a, b, d_1, d_2$ be real numbers with $d_1 > 0$ and $d_2 > 0$, and let $m_1, m_2$ be natural numbers with $a + m_1 d_1 < b$, so that the largest element of $J(a, m_1, d_1)$ is less than the smallest element of $J(b, m_2, d_2)$. Then $J(a, m_1, d_1) \\cup J(b, m_2, d_2)$ is not an MSTD set.","title":"The union of two separated arithmetic progressions is not an MSTD set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-nat-of-le-two","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_nat_of_le_two_mul_sub_two","parent":22821,"reading_title":{"format":"inline-parts-v1","parts":[["text","Interval plus integer progression starting in "],["math","\\((n + 1, 2n - 2)\\)"],["text"," is not MSTD"]],"source":"Interval plus integer progression starting in $(n + 1, 2n - 2)$ is not MSTD"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-nat-of-le-two","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. For natural numbers $n \\ge 3$, $m \\ge 2$, $d \\ge 2$ and $x$ with $n + 1 \\le x \\le 2n - 2$, the set $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus integer progression starting in $(n + 1, 2n - 2)$ is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-nat-of-two-mu","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_nat_of_two_mul_sub_one_le","parent":22821,"reading_title":{"format":"inline-parts-v1","parts":[["text","Interval plus integer progression starting at "],["math","\\(2n - 1\\)"],["text"," or later is not MSTD"]],"source":"Interval plus integer progression starting at $2n - 1$ or later is not MSTD"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-nat-of-two-mu","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. For natural numbers $n \\ge 3$, $m \\ge 2$, $d \\ge 2$ and $x \\ge 2n - 1$, the set $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus integer progression starting at $2n - 1$ or later is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-nat-of-lt","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_nat_of_lt","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-nat-of-lt","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. For natural numbers $n \\ge 3$, $m \\ge 2$, $d \\ge 2$ and $x > n - 1$, the set $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus integer progression starting after the interval is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-irrational-di","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_irrational_diff","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-irrational-di","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. For natural numbers $n \\ge 3$, $m \\ge 2$ and $x \\ge n + 1$, and a positive irrational real number $d$, the set $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus irrational-step progression with natural start is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-rat-diff","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_rat_diff","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-rat-diff","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$, $m$ and $x$ be natural numbers with $x \\ge n + 1$, and let $d = s + p/q$, where $s, p, q$ are natural numbers with $0 < p < q$ and $\\gcd(p, q) = 1$. Suppose that either $2 \\le m \\le q - 1$, or $m \\ge q$ and $qd > n - 1$. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$. Then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"An interval and a later natural-start progression with few terms or large step are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-rat-diff-of-l","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_rat_diff_of_le","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-rat-diff-of-l","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$, $m$ and $x$ be natural numbers with $x \\ge n + 1$, and let $d = s + p/q$, where $s, p, q$ are natural numbers with $0 < p < q$ and $\\gcd(p, q) = 1$. Suppose that $m \\ge q$ and $qd \\le n - 1$. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$. Then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"An interval and a later progression with natural start and small step are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-irrational-of","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_irrational_of_forall_not_nat","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-irrational-of","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. Let $n \\ge 3$ and $m$ be natural numbers, $d$ a positive irrational real number and $x > n - 1$ a real number such that none of $x + id$ for $i = 0, 1, \\dots, m$ is a natural number. Then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus irrational-step progression avoiding the naturals is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/eq-of-add-mul-irrational-mem-nat","kind":"node","lean":"MetaMathlibExt.eq_of_add_mul_irrational_mem_nat","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#eq-of-add-mul-irrational-mem-nat","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $x$ and $d$ be real numbers with $d$ irrational. If $i$ and $j$ are natural numbers such that $x + id$ and $x + jd$ are both nonnegative integers, then $i = j$.","title":"At most one nonnegative integer point on a progression with irrational difference"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-irrational-of-2","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_irrational_of_exists_nat","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-irrational-of-2","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. Let $n \\ge 3$ and $m$ be natural numbers, $d$ a positive irrational real number and $x > n - 1$ a real number that is not a natural number, and suppose that $x + jd$ is a natural number for some integer $j$ with $1 \\le j \\le m$. Then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus irrational-step progression meeting the naturals is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-real-rat-diff","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_real_rat_diff_of_forall_not_nat","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-real-rat-diff","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$ and $m$ be natural numbers, let $x > n - 1$ be a real number that is not a natural number, and let $d = s + p/q$, where $s, p, q$ are natural numbers with $0 < p < q$ and $\\gcd(p, q) = 1$. Suppose that $x + id$ is not a natural number for any natural number $i \\le m$, and that either $2 \\le m \\le q - 1$, or $m \\ge q$ and $qd > n - 1$. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and\u2026","title":"An interval and a later rational-step progression with no integer term are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/eq-of-add-mul-rat-mem-nat","kind":"node","lean":"MetaMathlibExt.eq_of_add_mul_rat_mem_nat","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#eq-of-add-mul-rat-mem-nat","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $x$ be a real number, let $s, p, q$ be natural numbers with $0 < p < q$ and $\\gcd(p, q) = 1$, and let $d = s + p/q$. If $i, j \\in \\{1, \\dots, q - 1\\}$ and $x + id$ and $x + jd$ are both nonnegative integers, then $i = j$.","title":"Uniqueness of a nonnegative integer point for a rational difference"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-real-rat-diff-2","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_real_rat_diff_of_exists_nat","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-real-rat-diff-2","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$ and $m$ be natural numbers, let $x > n - 1$ be a real number that is not a natural number, and let $d = s + p/q$, where $s, p, q$ are natural numbers with $0 < p < q$ and $\\gcd(p, q) = 1$. Suppose that $x + jd$ is a natural number for some natural number $j$ with $1 \\le j \\le q - 1$, and that either $2 \\le m \\le q - 1$, or $m \\ge q$ and $qd > n - 1$. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and\u2026","title":"An interval and a later rational-step progression with an integer term are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-real-rat-diff-3","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_real_rat_diff_of_le","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-real-rat-diff-3","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$ and $m$ be natural numbers, let $x > n - 1$ be a real number that is not a natural number, and let $d = s + p/q$, where $s, p, q$ are natural numbers with $0 < p < q$ and $\\gcd(p, q) = 1$. Suppose that $m \\ge q$ and $qd \\le n - 1$. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$. Then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"An interval and a later progression with non-integer start and small step are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-nat-of-le-sub","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_nat_of_le_sub_one","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-nat-of-le-sub","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. For natural numbers $n \\ge 3$, $m \\ge 2$, $d \\ge 2$ and $x \\le n - 1$, the set $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus integer progression starting inside the interval is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-overlap-of-de","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_overlap_of_den_ne_two","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-overlap-of-de","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$, $m \\ge 2$ and $d$ be natural numbers with $2 \\le d \\le n - 1$, and let $x$ be a real number with $0 < x < n - 1$ such that either $x$ is irrational or $x = s + p/q$ for natural numbers $s, p, q$ with $0 < p < q$, $q \\ne 2$ and $\\gcd(p, q) = 1$. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$. Then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"An interval and an overlapping progression with small integer step are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-overlap-half","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_overlap_half","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-overlap-half","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. Let $n \\ge 3$, $m \\ge 2$, $s$ and $d$ be natural numbers with $2 \\le d \\le n - 1$, and let $x = s + 1/2$. If $x < n - 1$, then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus integer-step progression with half-integer start is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-overlap-large","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_overlap_large_diff","parent":22821,"reading_title":{"format":"inline-parts-v1","parts":[["text","Interval plus progression with non-integer start and integer step at least "],["math","\\(n\\)"],["text"," is not MSTD"]],"source":"Interval plus progression with non-integer start and integer step at least $n$ is not MSTD"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-overlap-large","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$, $m \\ge 2$ and $d \\ge n$ be natural numbers, and let $x$ be a real number with $0 < x < n - 1$ that is not a natural number. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$. Then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus progression with non-integer start and integer step at least $n$ is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-overlap-of-co","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_overlap_of_coprime_den","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-overlap-of-co","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$ and $m \\ge 2$ be natural numbers, and let $x$ and $d$ be real numbers with $0 < x < n - 1$. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$. Suppose that one of the following holds: (i) $x$ is irrational, and $d$ is rational but not a positive integer; (ii) $x$ is rational but not a natural number, and $d$ is irrational; (iii) there are natural numbers\u2026","title":"An interval and an overlapping progression with incompatible start and step are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/add-mul-mem-nat-unique-of-one-lt-gcd","kind":"node","lean":"MetaMathlibExt.add_mul_mem_nat_unique_of_one_lt_gcd","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#add-mul-mem-nat-unique-of-one-lt-gcd","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $s_1, s_2, p_1, p_2, q_1, q_2$ be natural numbers with $p_1, p_2 > 0$, $\\gcd(p_1, q_1) = 1$, $\\gcd(p_2, q_2) = 1$ and $\\gcd(q_1, q_2) > 1$, and let $x = s_1 + p_1/q_1$ and $d = s_2 + p_2/q_2$. For indices $i, i', j, j' \\in \\{1, \\dots, q_2 - 1\\}$: if $x + id$ and $x + i'd$ are both nonnegative integers then $i = i'$; if $2x + jd$ and $2x + j'd$ are both nonnegative integers then $j = j'$; and if $x + id$ and\u2026","title":"Uniqueness of integer points when the denominators share a factor"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-overlap-of-on","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_overlap_of_one_lt_gcd","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-overlap-of-on","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $n \\ge 3$ and $m \\ge 2$ be natural numbers, and let $s_1, s_2, p_1, p_2, q_1, q_2$ be natural numbers with $p_1 > 0$, $p_2 > 0$, $\\gcd(p_1, q_1) = \\gcd(p_2, q_2) = 1$ and $\\gcd(q_1, q_2) > 1$. Let $x = s_1 + p_1/q_1$ and $d = s_2 + p_2/q_2$, where a fraction with denominator $0$ is read as $0$, and suppose that $x < n - 1 < d$. Write $I_n = \\{0, 1, \\dots, n - 1\\}$ and\u2026","title":"An interval and an overlapping progression whose denominators share a factor are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/add-mul-not-mem-nat-of-half","kind":"node","lean":"MetaMathlibExt.add_mul_not_mem_nat_of_half","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#add-mul-not-mem-nat-of-half","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $s_1, s_2, p_2, q_2$ be natural numbers with $q_2$ odd, and let $x = s_1 + 1/2$ and $d = s_2 + p_2/q_2$. Then $x + id$ is not a nonnegative integer for any natural number $i$.","title":"No integer points for a half-integer start and an odd denominator"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-interval-ap-overlap-half-2","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_intervalAP_overlap_half_odd_den","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-interval-ap-overlap-half-2","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $I_n = \\{0, 1, \\dots, n - 1\\}$ and $J(x, m, d) = \\{x + id : i = 0, 1, \\dots, m\\}$, as finite sets of reals. Let $n \\ge 3$, $m \\ge 2$, $s_1$, $s_2$ and $p_2$ be natural numbers and $q_2$ an odd natural number, and let $x = s_1 + 1/2$ and $d = s_2 + p_2/q_2$. If $x < n - 1 < d$, then $I_n \\cup J(x, m, d)$ is not an MSTD set.","title":"Interval plus progression with half-integer start and odd-denominator step is not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/not-is-mstdset-union-two-ap-overlap","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_union_two_ap_overlap","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#not-is-mstdset-union-two-ap-overlap","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"For real numbers $x, d$ and a natural number $m$, write $J(x, m, d) = \\{x + id : i = 0, 1, \\ldots, m\\}$. Let $x_1, x_2, d_1, d_2$ be real numbers with $d_1 > 0$, let $m_1, m_2$ be natural numbers with $x_2 < x_1 + m_1 d_1$, and let $s_1, s_2, p_1, p_2, q_1, q_2$ be natural numbers with $p_1, p_2, q_1, q_2 > 0$, $\\gcd(p_1, q_1) = \\gcd(p_2, q_2) = 1$, $(x_2 - x_1)/d_1 = s_1 + p_1/q_1$ and $d_2/d_1 = s_2 + p_2/q_2$.\u2026","title":"Two overlapping progressions with rational ratios, outside two open cases, are not MSTD"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/add-mul-not-mem-nat-of-irrational-of-rat","kind":"node","lean":"MetaMathlibExt.add_mul_not_mem_nat_of_irrational_of_rat","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#add-mul-not-mem-nat-of-irrational-of-rat","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $x$ be an irrational real number and $d$ a rational number. Then for all natural numbers $i$ and $j$, neither $x + id$ nor $2x + jd$ is a nonnegative integer.","title":"No nonnegative integer points for an irrational start and a rational difference"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/add-mul-not-mem-nat-of-rat-of-irrational","kind":"node","lean":"MetaMathlibExt.add_mul_not_mem_nat_of_rat_of_irrational","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#add-mul-not-mem-nat-of-rat-of-irrational","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $x$ be a rational number that is not a nonnegative integer, and let $d$ be an irrational real number. Then $x + id$ is not a nonnegative integer for any natural number $i$, and $2x + jd$ is not a nonnegative integer for any natural number $j \\ge 1$.","title":"Nonnegative integer points for a rational start and an irrational difference"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/add-mul-not-mem-nat-of-coprime-den","kind":"node","lean":"MetaMathlibExt.add_mul_not_mem_nat_of_coprime_den","parent":22821,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant/index.html#add-mul-not-mem-nat-of-coprime-den","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/unions-of-two-progressions-are-not-sum-dominant","status":"can_prove","summary":"Let $s_1, s_2, p_1, p_2, q_1, q_2$ be natural numbers with $\\gcd(p_1, q_1) = 1$, $\\gcd(q_1, q_2) = 1$ and $q_1 \\ge 3$, and let $x = s_1 + p_1/q_1$ and $d = s_2 + p_2/q_2$. Then for all natural numbers $i$ and $j$, neither $x + id$ nor $2x + jd$ is a nonnegative integer.","title":"No integer points for coprime denominators, the first at least 3"},{"data_url":"scopes/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients.json","id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","kind":"scope","navigation":"module","parent":22786,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","title":"Sets with more products than quotients"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/is-mptqset","kind":"node","lean":"MetaMathlibExt.IsMPTQSet","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#is-mptqset","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"fully_proved","summary":"A finite set $A$ of real numbers is an MPTQ (more products than quotients) set when $0 \\notin A$ and $|A/A| < |A \\cdot A|$, where $A \\cdot A = \\{ab : a, b \\in A\\}$ is the product set, $A/A = \\{a/b : a, b \\in A\\}$ is the quotient set, and $|X|$ is the number of elements of $X$.","title":"MPTQ set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/not-is-mptqset-singleton","kind":"node","lean":"MetaMathlibExt.not_isMPTQSet_singleton","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#not-is-mptqset-singleton","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"fully_proved","summary":"For every real number $a$, the one-element set $\\{a\\}$ is not an MPTQ set.","title":"A singleton is not an MPTQ set"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/not-is-mptqset-of-zero-mem","kind":"node","lean":"MetaMathlibExt.not_isMPTQSet_of_zero_mem","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#not-is-mptqset-of-zero-mem","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"fully_proved","summary":"If $A$ is a finite set of real numbers with $0 \\in A$, then $A$ is not an MPTQ set.","title":"Sets containing zero are not MPTQ sets"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/not-is-mstdset-int-of-card-eq-seven","kind":"node","lean":"MetaMathlibExt.not_isMSTDSet_int_of_card_eq_seven","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#not-is-mstdset-int-of-card-eq-seven","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"No set of seven integers, viewed as a set of real numbers, is an MSTD set.","title":"Hegarty: no seven-element MSTD set of integers"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/is-mstdset-int-of-card-eq-eight-iff","kind":"node","lean":"MetaMathlibExt.isMSTDSet_int_of_card_eq_eight_iff","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#is-mstdset-int-of-card-eq-eight-iff","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"Let $A$ be a set of eight integers, regarded as a set of real numbers, and let $S = \\{0, 2, 3, 4, 7, 11, 12, 14\\}$. Then $A$ is an MSTD set if and only if $A = \\{as + b : s \\in S\\}$ for some integers $a \\neq 0$ and $b$.","title":"Hegarty's classification of MSTD sets of size eight"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/eight-le-card-of-is-mptqset-of-pos","kind":"node","lean":"MetaMathlibExt.eight_le_card_of_isMPTQSet_of_pos","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#eight-le-card-of-is-mptqset-of-pos","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"Every MPTQ set whose elements are all positive real numbers has at least $8$ elements.","title":"Chu's bound for MPTQ sets of positive reals"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/five-le-card-of-is-mptqset-of-neg","kind":"node","lean":"MetaMathlibExt.five_le_card_of_isMPTQSet_of_neg","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#five-le-card-of-is-mptqset-of-neg","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"Every MPTQ set of real numbers that contains a negative number has at least $5$ elements.","title":"Chu's bound for MPTQ sets with a negative element"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/not-is-mptqset-insert-of-le","kind":"node","lean":"MetaMathlibExt.not_isMPTQSet_insert_of_le","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#not-is-mptqset-insert-of-le","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"Let $A$ be a finite set of real numbers that is not an MPTQ set, let $x$ be a real number, let $n \\le m$ be natural numbers, and write $A' = A \\cup \\{x\\}$. If $|A' \\cdot A'| \\le |A \\cdot A| + n$ and $|A/A| + m \\le |A'/A'|$, where $X \\cdot X$ and $X/X$ denote the product set and quotient set of $X$, then $A'$ is not an MPTQ set.","title":"Adjoining an element that adds no more products than quotients"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/mul-ne-of-mul-eq-of-distinct","kind":"node","lean":"MetaMathlibExt.mul_ne_of_mul_eq_of_distinct","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#mul-ne-of-mul-eq-of-distinct","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"Let $a$, $b$, $c$, $d$ be pairwise distinct positive real numbers with $ab = cd$. Then $ad \\neq bc$ and $ac \\neq bd$.","title":"One equal pairing of products excludes the other two"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/at-most-one-sq-eq-mul-of-distinct","kind":"node","lean":"MetaMathlibExt.at_most_one_sq_eq_mul_of_distinct","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#at-most-one-sq-eq-mul-of-distinct","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"Let $a$, $b$, $c$ be pairwise distinct positive real numbers. Then at most one of the equations $a^2 = bc$, $b^2 = ac$ and $c^2 = ab$ holds; that is, no two of them hold simultaneously.","title":"At most one square is the product of the other two"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/not-is-mptqset-of-card-five-one-neg","kind":"node","lean":"MetaMathlibExt.not_isMPTQSet_of_card_five_one_neg","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#not-is-mptqset-of-card-five-one-neg","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"If $A$ is a set of five real numbers exactly one of which is negative, then $A$ is not an MPTQ set.","title":"Five-element sets with one negative element are not MPTQ sets"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/not-is-mptqset-of-card-five-two-neg","kind":"node","lean":"MetaMathlibExt.not_isMPTQSet_of_card_five_two_neg","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#not-is-mptqset-of-card-five-two-neg","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"If $A$ is a set of five real numbers exactly two of which are negative, then $A$ is not an MPTQ set.","title":"Five-element sets with two negative elements are not MPTQ sets"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/is-mptqset-union-neg-iff","kind":"node","lean":"MetaMathlibExt.isMPTQSet_union_neg_iff","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#is-mptqset-union-neg-iff","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"Let $A$ be a finite set of positive real numbers and $-A = \\{-a : a \\in A\\}$. Then $A \\cup (-A)$ is an MPTQ set if and only if $A$ is an MPTQ set.","title":"Adjoining negatives preserves the MPTQ property"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/not-is-mptqset-of-card-eq-five","kind":"node","lean":"MetaMathlibExt.not_isMPTQSet_of_card_eq_five","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#not-is-mptqset-of-card-eq-five","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"No set of five real numbers is an MPTQ set.","title":"No MPTQ set has five elements"},{"id":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/is-mptqset-union-prime-mul-iff","kind":"node","lean":"MetaMathlibExt.isMPTQSet_union_prime_mul_iff","parent":22855,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients/index.html#is-mptqset-union-prime-mul-iff","scope":"Combinatorics/additive-combinatorics/more-sums-than-differences/sets-with-more-products-than-quotients","status":"can_prove","summary":"Let $A$ be a finite set of integers and $p$ a prime number such that every element of $A$ is coprime to $p$, and let $pA = \\{pa : a \\in A\\}$. Regarding sets of integers as sets of real numbers, $A \\cup pA$ is an MPTQ set if and only if $A$ is an MPTQ set.","title":"Adjoining multiples by a coprime prime preserves the MPTQ property"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets","kind":"scope","parent":22582,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets","status":"can_prove","summary":"Sidon and B_h-sets (bounds, counts, greedy and thick constructions, isolated sums, the Sidon ideal), Golomb rulers, difference bases, cyclic and perfect difference sets including Singer's theorem, and difference sets of sparse sets.","title":"Sidon sets and difference sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/singer-cyclic-difference-set-exists","kind":"node","lean":"MetaMathlibExt.singer_cyclic_difference_set_exists","parent":22871,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/index.html#singer-cyclic-difference-set-exists","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets","status":"fully_proved","summary":"For every prime power $q$, there is a subset $D$ of the cyclic group $\\mathbb{Z}/(q^2 + q + 1)\\mathbb{Z}$ with $q + 1$ elements such that every nonzero element $x$ of this group can be written as $x = a - b$ for exactly one ordered pair $(a, b)$ of elements of $D$.","title":"Singer's difference set theorem"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets","status":"fully_proved","title":"B_h-sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets/sum-ne","kind":"node","lean":"AdditiveCombinatorics.sumNE","parent":22873,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets/index.html#sum-ne","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets","status":"fully_proved","summary":"For a nonempty list $(a_1, \\dots, a_k)$ of elements of a type with an addition, its sum is obtained by adding the entries from left to right starting from the head: it is $a_1$ when $k = 1$, and $(\\cdots((a_1 + a_2) + a_3) + \\cdots) + a_k$ in general. No zero element is needed, and the proof that the list is nonempty does not affect the value.","title":"Sum of a nonempty list"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets/sum-ne-proof-irrel","kind":"node","lean":"AdditiveCombinatorics.sumNE_proof_irrel","parent":22873,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets/index.html#sum-ne-proof-irrel","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets","status":"fully_proved","summary":"For every list $l$ of elements of a type with an addition and any two proofs $e_1$ and $e_2$ that $l$ is nonempty, the sum of $l$ computed with $e_1$ equals the sum of $l$ computed with $e_2$.","title":"The sum of a nonempty list does not depend on the nonemptiness proof"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets/is-bh-set","kind":"node","lean":"AdditiveCombinatorics.IsBhSet","parent":22873,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(B_h\\)"],["text","-set"]],"source":"$B_h$-set"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets/index.html#is-bh-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/b-h-sets","status":"fully_proved","summary":"Let $G$ be an additive commutative semigroup, $A \\subseteq G$ and $h$ a natural number. $A$ is a $B_h$-set if $A$ is nonempty, $h \\ge 1$, and whenever two sequences $(a_1, \\dots, a_h)$ and $(b_1, \\dots, b_h)$ of elements of $A$ satisfy $a_1 + \\dots + a_h = b_1 + \\dots + b_h$, the sequence $(b_1, \\dots, b_h)$ is a rearrangement of $(a_1, \\dots, a_h)$.","title":"$B_h$-set"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","title":"Constructions of thick B_h-sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bose-set","kind":"node","lean":"MetaMathlibExt.boseSet","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bose-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, $K$ a field extension of $F$, $h$ and $b$ natural numbers and $\\theta \\in K$. The generalized Bose set $\\mathrm{Bose}_h(q, b)$ is the set of residues $a \\in \\mathbb{Z}/(q^h - 1)\\mathbb{Z}$ such that $\\theta^a = \\theta^b + v$ for some $v \\in F$, where $\\theta^a$ is computed with the least nonnegative representative of $a$ (for $h = 0$ the modulus is $0$, $a$ ranges over\u2026","title":"Generalized Bose set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/mem-bose-set","kind":"node","lean":"MetaMathlibExt.mem_boseSet","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#mem-bose-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $K$ be a field extension of $F$, let $\\theta \\in K$, and let $h$ and $b$ be natural numbers. A residue $a \\in \\mathbb{Z}/(q^h - 1)\\mathbb{Z}$ lies in the Bose set $\\mathrm{Bose}_h(q, b)$ built from $F$, $K$ and $\\theta$ if and only if $\\theta^a = \\theta^b + v$ for some $v \\in F$, where $\\theta^a$ means $\\theta$ raised to the least nonnegative representative of $a$\u2026","title":"Membership in a Bose set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/singer-set","kind":"node","lean":"MetaMathlibExt.singerSet","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#singer-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $K$ be a field extension of $F$, let $\\theta \\in K$, let $h$ and $b$ be natural numbers, and put $M = (q^{h+1} - 1)/(q - 1)$. The generalized Singer set $\\mathrm{Singer}_h(q, b)$ is the set of residues $a \\in \\mathbb{Z}/M\\mathbb{Z}$ such that $\\theta^a = u\\theta^b + v$ for some $u, v \\in F$, where $\\theta^a$ means $\\theta$ raised to the least nonnegative\u2026","title":"Generalized Singer set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/mem-singer-set","kind":"node","lean":"MetaMathlibExt.mem_singerSet","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#mem-singer-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements, let $K$ be a field extension of $F$, let $\\theta \\in K$, let $h$ and $b$ be natural numbers, and put $M = (q^{h+1} - 1)/(q - 1)$. A residue $a \\in \\mathbb{Z}/M\\mathbb{Z}$ lies in the Singer set $\\mathrm{Singer}_h(q, b)$ built from $F$, $K$ and $\\theta$ if and only if $\\theta^a = u\\theta^b + v$ for some $u, v \\in F$, where $\\theta^a$ means $\\theta$ raised to the least\u2026","title":"Membership in a Singer set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/zmod-affinely-equivalent","kind":"node","lean":"MetaMathlibExt.ZModAffinelyEquivalent","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#zmod-affinely-equivalent","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"For a natural number $M$, sets $A_1, A_2 \\subseteq \\mathbb{Z}/M\\mathbb{Z}$ are affinely equivalent, written $A_1 \\sim A_2$, if there are a unit $d$ of $\\mathbb{Z}/M\\mathbb{Z}$ and an element $s \\in \\mathbb{Z}/M\\mathbb{Z}$ such that $A_2 = \\{d a + s : a \\in A_1\\}$.","title":"Affine equivalence of sets of residues"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/z-mod-affinely-equivalent-refl","kind":"node","lean":"MetaMathlibExt.zModAffinelyEquivalent_refl","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#z-mod-affinely-equivalent-refl","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"For every natural number $M$, every subset $A$ of $\\mathbb{Z}/M\\mathbb{Z}$ is affinely equivalent to itself (take $d = 1$ and $s = 0$).","title":"Affine equivalence is reflexive"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-max-card","kind":"node","lean":"MetaMathlibExt.bhMaxCard","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximum size of a "],["math","\\(B_h\\)"],["text","-set in "],["math","\\(\\{1, \\dots, n\\}\\)"]],"source":"Maximum size of a $B_h$-set in $\\{1, \\dots, n\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-max-card","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"For natural numbers $h$ and $n$, $R_h(n)$ is the largest number of elements of a $B_h$-set of natural numbers contained in $\\{1, \\dots, n\\}$. If no $B_h$-set is contained in $\\{1, \\dots, n\\}$ (for example when $n = 0$), then $R_h(n) = 0$.","title":"Maximum size of a $B_h$-set in $\\{1, \\dots, n\\}$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-min-length","kind":"node","lean":"MetaMathlibExt.bhMinLength","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shortest interval containing a "],["math","\\(B_h\\)"],["text","-set of given size"]],"source":"Shortest interval containing a $B_h$-set of given size"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-min-length","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"For natural numbers $h$ and $k$, $R_h^{-1}(k)$ is the smallest natural number $n$ such that some $B_h$-set of natural numbers with exactly $k$ elements is contained in $\\{1, \\dots, n\\}$; it is $0$ when no such $n$ exists, for example when $k = 0$, since a $B_h$-set is nonempty.","title":"Shortest interval containing a $B_h$-set of given size"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-max-card-le","kind":"node","lean":"MetaMathlibExt.bhMaxCard_le","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(R_h(n)\\)"],["text"," is at most "],["math","\\(n\\)"]],"source":"$R_h(n)$ is at most $n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-max-card-le","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"For all natural numbers $h$ and $n$, let $R_h(n)$ be the largest size of a $B_h$-set contained in $\\{1, \\dots, n\\}$, or $0$ if there is no such set (for example when $h = 0$ or $n = 0$). Then $R_h(n) \\le n$.","title":"$R_h(n)$ is at most $n$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-max-card-one","kind":"node","lean":"MetaMathlibExt.bhMaxCard_one","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(R_h(1) = 1\\)"]],"source":"$R_h(1) = 1$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-max-card-one","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"For every natural number $h \\ge 1$, the largest size $R_h(1)$ of a $B_h$-set contained in $\\{1\\}$ equals $1$.","title":"$R_h(1) = 1$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-min-length-one","kind":"node","lean":"MetaMathlibExt.bhMinLength_one","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","The shortest interval holding a one-element "],["math","\\(B_h\\)"],["text","-set"]],"source":"The shortest interval holding a one-element $B_h$-set"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-min-length-one","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"fully_proved","summary":"For every natural number $h \\ge 1$, the smallest natural number $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with exactly one element is $1$; that is, $R_h^{-1}(1) = 1$.","title":"The shortest interval holding a one-element $B_h$-set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/is-bh-set-bose-set","kind":"node","lean":"MetaMathlibExt.isBhSet_boseSet","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bose's construction of "],["math","\\(B_h\\)"],["text","-sets"]],"source":"Bose's construction of $B_h$-sets"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#is-bh-set-bose-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $h \\ge 2$, let $F$ be a finite field with $q$ elements, let $K$ be a field extension of $F$ of degree $h$, let $\\theta$ be a generator of the multiplicative group $K^\\times$, and let $b$ be a natural number such that $\\theta^b$ has degree $h$ over $F$. Then the generalized Bose set $\\mathrm{Bose}_h(q, b)$, a subset of $\\mathbb{Z}/(q^h - 1)\\mathbb{Z}$, is a $B_h$-set with exactly $q$ elements.","title":"Bose's construction of $B_h$-sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/is-bh-set-singer-set","kind":"node","lean":"MetaMathlibExt.isBhSet_singerSet","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Singer's construction of "],["math","\\(B_h\\)"],["text","-sets"]],"source":"Singer's construction of $B_h$-sets"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#is-bh-set-singer-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $h \\ge 2$, let $F$ be a finite field with $q$ elements, let $K$ be a field extension of $F$ of degree $h + 1$, let $\\theta$ be a generator of the multiplicative group $K^\\times$, and let $b$ be a natural number such that $\\theta^b$ has degree $h + 1$ over $F$. Then the generalized Singer set $\\mathrm{Singer}_h(q, b)$, a subset of $\\mathbb{Z}/M\\mathbb{Z}$ with $M = (q^{h+1} - 1)/(q - 1)$, is a $B_h$-set with\u2026","title":"Singer's construction of $B_h$-sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/z-mod-affinely-equivalent-bose-set","kind":"node","lean":"MetaMathlibExt.zModAffinelyEquivalent_boseSet","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#z-mod-affinely-equivalent-bose-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and characteristic $p$, let $h \\ge 2$, let $K$ be a field extension of $F$ of degree $h$, and let $\\theta \\in K$ have order $|K| - 1$, that is, generate $K^\\times$. Let $b$ and $e$ be natural numbers such that the minimal polynomials of $\\theta^b$ and $\\theta^e$ over $F$ both have degree $h$. Suppose that (i) $b \\equiv e \\pmod{(q^h - 1)/(q - 1)}$, or (ii)\u2026","title":"Affine equivalences between Bose sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/z-mod-affinely-equivalent-singer-set","kind":"node","lean":"MetaMathlibExt.zModAffinelyEquivalent_singerSet","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#z-mod-affinely-equivalent-singer-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $F$ be a finite field with $q$ elements and characteristic $p$, let $h \\ge 2$, let $K$ be a field extension of $F$ of degree $h + 1$, and let $\\theta \\in K$ have order $|K| - 1$, that is, generate $K^\\times$. Put $M = (q^{h+1} - 1)/(q - 1)$. Let $b$ and $e$ be natural numbers such that the minimal polynomials of $\\theta^b$ and $\\theta^e$ over $F$ both have degree $h + 1$. Suppose that (iv)\u2026","title":"Affine equivalences between Singer sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-max-card-bh-min-length-bounds-uncondi","kind":"node","lean":"MetaMathlibExt.bhMaxCard_bhMinLength_bounds_unconditional","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unconditional bounds for "],["math","\\(R_h(n)\\)"],["text"," and "],["math","\\(R_h^{-1}(k)\\)"]],"source":"Unconditional bounds for $R_h(n)$ and $R_h^{-1}(k)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-max-card-bh-min-length-bounds-uncondi","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $h \\ge 2$. Write $R_h(n)$ for the largest size of a $B_h$-set contained in $\\{1, \\dots, n\\}$, with $R_h(0) = 0$, and $R_h^{-1}(k)$ for the least $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with $k$ elements, with $R_h^{-1}(0) = 0$ (a $B_h$-set is nonempty). Then every natural number $n$ satisfies $R_h(n) \\ge n^{1/h} - 2^{44} n^{154/(155h)}$, and every natural number $k$ satisfies\u2026","title":"Unconditional bounds for $R_h(n)$ and $R_h^{-1}(k)$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-max-card-bh-min-length-bounds-large","kind":"node","lean":"MetaMathlibExt.bhMaxCard_bhMinLength_bounds_large","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds for "],["math","\\(R_h(n)\\)"],["text"," and "],["math","\\(R_h^{-1}(k)\\)"],["text"," beyond "],["math","\\(e^{e^{34}}\\)"]],"source":"Bounds for $R_h(n)$ and $R_h^{-1}(k)$ beyond $e^{e^{34}}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-max-card-bh-min-length-bounds-large","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $h \\ge 2$. Write $R_h(n)$ for the largest size of a $B_h$-set contained in $\\{1, \\dots, n\\}$ and $R_h^{-1}(k)$ for the least $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with $k$ elements. Then every natural number $n \\ge e^{e^{34}}$ satisfies $R_h(n) \\ge n^{1/h} - 7 n^{2/(3h)}$, and every natural number $k \\ge e^{e^{34}}$ satisfies $R_h^{-1}(k) \\le k^h + (3k)^{h - 1/3}$.","title":"Bounds for $R_h(n)$ and $R_h^{-1}(k)$ beyond $e^{e^{34}}$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-max-card-bh-min-length-bounds-eventua","kind":"node","lean":"MetaMathlibExt.bhMaxCard_bhMinLength_bounds_eventually","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual bounds for "],["math","\\(R_h(n)\\)"],["text"," and "],["math","\\(R_h^{-1}(k)\\)"]],"source":"Eventual bounds for $R_h(n)$ and $R_h^{-1}(k)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-max-card-bh-min-length-bounds-eventua","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $h \\ge 2$. Write $R_h(n)$ for the largest size of a $B_h$-set contained in $\\{1, \\dots, n\\}$ and $R_h^{-1}(k)$ for the least $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with $k$ elements. Then $R_h(n) \\ge n^{1/h} - n^{21/(40h)}$ for all sufficiently large natural numbers $n$, and $R_h^{-1}(k) \\le k^h + 2^h k^{h - 19/40}$ for all sufficiently large natural numbers $k$.","title":"Eventual bounds for $R_h(n)$ and $R_h^{-1}(k)$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-max-card-bh-min-length-bounds-of-rh","kind":"node","lean":"MetaMathlibExt.bhMaxCard_bhMinLength_bounds_of_RH","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Riemann Hypothesis bounds for "],["math","\\(R_h(n)\\)"],["text"," and "],["math","\\(R_h^{-1}(k)\\)"]],"source":"Riemann Hypothesis bounds for $R_h(n)$ and $R_h^{-1}(k)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-max-card-bh-min-length-bounds-of-rh","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Assume the Riemann Hypothesis and let $h \\ge 2$. Write $R_h(n)$ for the largest size of a $B_h$-set contained in $\\{1, \\dots, n\\}$, with $R_h(0) = 0$ (a $B_h$-set is nonempty), and $R_h^{-1}(k)$ for the least $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with $k$ elements, and let $\\log$ be the natural logarithm, with the convention $\\log 0 = 0$. Then every natural number $k \\ge 1$ satisfies\u2026","title":"Riemann Hypothesis bounds for $R_h(n)$ and $R_h^{-1}(k)$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/exists-prime-between-pow-155","kind":"node","lean":"MetaMathlibExt.exists_prime_between_pow_155","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#exists-prime-between-pow-155","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"For every natural number $n \\ge 1$ there is a prime $p$ with $n^{155} < p < (n + 1)^{155}$.","title":"Cully-Hugill's theorem for 155th powers"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-min-length-lt-and-bh-max-card-ge-expl","kind":"node","lean":"MetaMathlibExt.bhMinLength_lt_and_bhMaxCard_ge_explicit","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unconditional bounds for "],["math","\\(R_h^{-1}(k)\\)"],["text",", "],["math","\\(k \\ge 4\\)"],["text",", and "],["math","\\(R_h(n)\\)"],["text",", "],["math","\\(n \\ge h + 3\\)"]],"source":"Unconditional bounds for $R_h^{-1}(k)$, $k \\ge 4$, and $R_h(n)$, $n \\ge h + 3$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-min-length-lt-and-bh-max-card-ge-expl","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $h \\ge 2$. Write $R_h(n)$ for the largest size of a $B_h$-set contained in $\\{1, \\dots, n\\}$ and $R_h^{-1}(k)$ for the least $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with $k$ elements. Then every natural number $k \\ge 4$ satisfies $R_h^{-1}(k) < k^h + 3^{155h} k^{h - 1/155}$, and every natural number $n \\ge h + 3$ satisfies $R_h(n) \\ge n^{1/h} - 2^{44} n^{154/(155h)}$.","title":"Unconditional bounds for $R_h^{-1}(k)$, $k \\ge 4$, and $R_h(n)$, $n \\ge h + 3$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/exists-prime-between-cubes-of-large","kind":"node","lean":"MetaMathlibExt.exists_prime_between_cubes_of_large","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#exists-prime-between-cubes-of-large","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"For every natural number $n > \\exp(\\exp(32.537))$ there is a prime $p$ with $n^3 < p < (n + 1)^3$.","title":"Cully-Hugill's theorem for cubes"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-min-length-lt-and-bh-max-card-gt-larg","kind":"node","lean":"MetaMathlibExt.bhMinLength_lt_and_bhMaxCard_gt_large","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Strict bounds for thick "],["math","\\(B_h\\)"],["text","-sets beyond "],["math","\\(e^{e^{34}}\\)"]],"source":"Strict bounds for thick $B_h$-sets beyond $e^{e^{34}}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-min-length-lt-and-bh-max-card-gt-larg","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $h \\ge 2$. For natural numbers $n$ and $k$, let $R_h(n)$ be the largest size of a $B_h$-set of natural numbers contained in $\\{1, \\dots, n\\}$, and let $R_h^{-1}(k)$ be the smallest $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with exactly $k$ elements. Then every natural number $k > e^{e^{34}}$ satisfies $R_h^{-1}(k) < k^h + (3k)^{h - 1/3}$, and every natural number $n > e^{e^{34}}$ satisfies\u2026","title":"Strict bounds for thick $B_h$-sets beyond $e^{e^{34}}$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/eventually-exists-prime-short-intervals","kind":"node","lean":"MetaMathlibExt.eventually_exists_prime_short_intervals","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#eventually-exists-prime-short-intervals","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"There is a real number $x_0$ such that for every real $x \\ge x_0$ there is a prime $p$ with $x - x^{21/40} \\le p \\le x$ and a prime $p'$ with $x \\le p' \\le x + x^{21/40}$.","title":"Baker\u2013Harman\u2013Pintz theorem"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-min-length-lt-and-bh-max-card-ge-even","kind":"node","lean":"MetaMathlibExt.bhMinLength_lt_and_bhMaxCard_ge_eventually","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual strict upper bound for "],["math","\\(R_h^{-1}(k)\\)"],["text"," and lower bound for "],["math","\\(R_h(n)\\)"]],"source":"Eventual strict upper bound for $R_h^{-1}(k)$ and lower bound for $R_h(n)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-min-length-lt-and-bh-max-card-ge-even","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Let $h \\ge 2$. Write $R_h(n)$ for the largest size of a $B_h$-set contained in $\\{1, \\dots, n\\}$ and $R_h^{-1}(k)$ for the least $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with $k$ elements. Then $R_h^{-1}(k) < k^h + 2^h k^{h - 19/40}$ for all sufficiently large natural numbers $k$, and $R_h(n) \\ge n^{1/h} - n^{21/(40h)}$ for all sufficiently large natural numbers $n$.","title":"Eventual strict upper bound for $R_h^{-1}(k)$ and lower bound for $R_h(n)$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/exists-prime-between-sq-of-rh","kind":"node","lean":"MetaMathlibExt.exists_prime_between_sq_of_RH","parent":22877,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#exists-prime-between-sq-of-rh","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Assume the Riemann Hypothesis. Then for every natural number $n \\ge 2$ there is a prime $p$ with $n^2 < p < \\bigl(n + (1 + 1/\\log n)^2 \\log n\\bigr)^2$, where $\\log$ is the natural logarithm.","title":"Dudek\u2013Greni\u00e9\u2013Molteni theorem"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/bh-min-length-lt-and-bh-max-card-ge-of-r","kind":"node","lean":"MetaMathlibExt.bhMinLength_lt_and_bhMaxCard_ge_of_RH","parent":22877,"reading_title":{"format":"inline-parts-v1","parts":[["text","Riemann Hypothesis bounds for "],["math","\\(R_h^{-1}(k)\\)"],["text",", "],["math","\\(k \\ge 4\\)"],["text",", and "],["math","\\(R_h(n)\\)"],["text",", "],["math","\\(n \\ge h + 3\\)"]],"source":"Riemann Hypothesis bounds for $R_h^{-1}(k)$, $k \\ge 4$, and $R_h(n)$, $n \\ge h + 3$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets/index.html#bh-min-length-lt-and-bh-max-card-ge-of-r","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/constructions-of-thick-b-h-sets","status":"can_prove","summary":"Assume the Riemann Hypothesis and let $h \\ge 2$. Write $R_h(n)$ for the largest size of a $B_h$-set contained in $\\{1, \\dots, n\\}$ and $R_h^{-1}(k)$ for the least $n$ such that $\\{1, \\dots, n\\}$ contains a $B_h$-set with $k$ elements, and let $\\log$ be the natural logarithm. Then every natural number $k \\ge 4$ satisfies $R_h^{-1}(k) < k^h + \\log(20k)\\, k^{h - 1/2} + 2 k^{h - 1} (\\log(20k))^{2h}$, and every natural\u2026","title":"Riemann Hypothesis bounds for $R_h^{-1}(k)$, $k \\ge 4$, and $R_h(n)$, $n \\ge h + 3$"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","title":"Difference bases"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/sub-rep-count","kind":"node","lean":"MetaMathlibExt.subRepCount","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#sub-rep-count","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"For a finite subset $A$ of an abelian group $G$ and an element $x \\in G$, the difference representation function $r_{A-A}(x)$ is the number of ordered pairs $(a, a') \\in A \\times A$ with $a - a' = x$.","title":"Difference representation function"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/diff-basis-number","kind":"node","lean":"MetaMathlibExt.diffBasisNumber","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(g\\)"],["text","-difference basis number"]],"source":"The $g$-difference basis number"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#diff-basis-number","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"For a finite set $A \\subseteq \\mathbb{Z}$ and $x \\in \\mathbb{Z}$, write $r_{A-A}(x)$ for the number of ordered pairs $(a, a') \\in A \\times A$ with $a - a' = x$. For natural numbers $g$ and $n$, a finite set $A \\subseteq \\mathbb{Z}$ is a $g$-difference basis for $[n] = \\{1, \\dots, n\\}$ if $r_{A-A}(x) \\ge g$ for every $x \\in [n]$, and $\\eta_g(n)$ is the least size of such a set.","title":"The $g$-difference basis number"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/group-diff-basis-number","kind":"node","lean":"MetaMathlibExt.groupDiffBasisNumber","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#group-diff-basis-number","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"For an abelian group $G$ and $g \\in \\mathbb{N}$, a $g$-difference basis of $G$ is a finite set $A \\subseteq G$ such that every $x \\in G$ equals $a - a'$ for at least $g$ ordered pairs $(a, a') \\in A \\times A$. The difference basis number $\\eta_g(G)$ is the least size of a $g$-difference basis of $G$, and $0$ if $G$ has none.","title":"Difference basis number of an abelian group"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/bounded-diff-set-number","kind":"node","lean":"MetaMathlibExt.boundedDiffSetNumber","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#bounded-diff-set-number","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"For a finite set $A \\subseteq \\mathbb{Z}$ and $x \\in \\mathbb{Z}$, write $r_{A-A}(x)$ for the number of ordered pairs $(a, a') \\in A \\times A$ with $a - a' = x$. For natural numbers $g$ and $n$, $\\alpha_g(n)$ is the largest size of a set $A \\subseteq [n] = \\{1, \\dots, n\\}$ with $r_{A-A}(x) \\le g$ for every nonzero integer $x$.","title":"Maximal sets with bounded difference multiplicities"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/sub-rep-count-zero","kind":"node","lean":"MetaMathlibExt.subRepCount_zero","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#sub-rep-count-zero","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"For every finite subset $A$ of an abelian group, $r_{A-A}(0) = |A|$: the number of ordered pairs $(a, a') \\in A \\times A$ with $a - a' = 0$ equals the number of elements of $A$.","title":"Representations of zero as a difference"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/diff-basis-number-one-one","kind":"node","lean":"MetaMathlibExt.diffBasisNumber_one_one","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(\\eta_1(1) = 2\\)"]],"source":"The value $\\eta_1(1) = 2$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#diff-basis-number-one-one","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"Let $\\eta_1(1)$ be the least size of a finite set $A \\subseteq \\mathbb{Z}$ containing elements $a, a'$ with $a - a' = 1$. Then $\\eta_1(1) = 2$: such a set needs two elements, and $\\{0, 1\\}$ is one.","title":"The value $\\eta_1(1) = 2$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/group-diff-basis-number-one-zmod-three","kind":"node","lean":"MetaMathlibExt.groupDiffBasisNumber_one_zmod_three","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Difference basis number of "],["math","\\(\\mathbb{Z}/3\\mathbb{Z}\\)"]],"source":"Difference basis number of $\\mathbb{Z}/3\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#group-diff-basis-number-one-zmod-three","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"The least size $\\eta_1(\\mathbb{Z}/3\\mathbb{Z})$ of a set $A \\subseteq \\mathbb{Z}/3\\mathbb{Z}$ such that every element of $\\mathbb{Z}/3\\mathbb{Z}$ is a difference $a - a'$ of two elements of $A$ equals $2$: the set $\\{0, 1\\}$ has differences $0, 1, 2$, and one element is not enough.","title":"Difference basis number of $\\mathbb{Z}/3\\mathbb{Z}$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/bounded-diff-set-number-le","kind":"node","lean":"MetaMathlibExt.boundedDiffSetNumber_le","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#bounded-diff-set-number-le","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"For natural numbers $g$ and $n$, let $\\alpha_g(n)$ be the largest size of a set $A \\subseteq [n] = \\{1, \\dots, n\\}$ such that every nonzero integer $x$ is the difference $a - a'$ of at most $g$ ordered pairs $(a, a') \\in A \\times A$. Then $\\alpha_g(n) \\le n$, since these sets are subsets of $[n]$.","title":"Trivial upper bound for sets with bounded difference multiplicities"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/bounded-diff-set-number-one","kind":"node","lean":"MetaMathlibExt.boundedDiffSetNumber_one","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(\\alpha_g(1) = 1\\)"]],"source":"The value $\\alpha_g(1) = 1$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#bounded-diff-set-number-one","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"fully_proved","summary":"For natural numbers $g$ and $n$, let $\\alpha_g(n)$ be the largest size of a set $A \\subseteq \\{1, \\dots, n\\}$ such that every nonzero integer $x$ is the difference $a - a'$ of at most $g$ ordered pairs $(a, a') \\in A \\times A$. Then for every natural number $g$ we have $\\alpha_g(1) = 1$: the singleton $\\{1\\}$ has no nonzero differences.","title":"The value $\\alpha_g(1) = 1$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/sqrt-two-mul-le-diff-basis-number-div-sq","kind":"node","lean":"MetaMathlibExt.sqrt_two_mul_le_diffBasisNumber_div_sqrt","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#sqrt-two-mul-le-diff-basis-number-div-sq","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"Let $g, n \\in \\mathbb{N}$ with $n > 1$, and let $\\eta_g(n)$ be the least size of a finite set $A \\subseteq \\mathbb{Z}$ such that every $x \\in \\{1, \\dots, n\\}$ equals $a - a'$ for at least $g$ ordered pairs $(a, a') \\in A \\times A$. Then $\\eta_g(n) / \\sqrt{n} \\ge \\sqrt{2g}$.","title":"Trivial lower bound for difference bases of an interval"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/redei-renyi-diff-basis","kind":"node","lean":"MetaMathlibExt.redeiRenyi_diffBasis","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#redei-renyi-diff-basis","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"Let $n \\ge 1$ and $k = \\lceil \\sqrt{n} \\rceil$. The set of integers $B_n = \\{1, \\dots, k - 1\\} \\cup \\{k, 2k, \\dots, k^2, (k + 1)k\\}$ has $2k$ elements, and every $x \\in \\{1, \\dots, n\\}$ is a difference $b - b'$ of two elements of $B_n$.","title":"R\u00e9dei\u2013R\u00e9nyi difference basis"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/diff-basis-number-div-sqrt-le","kind":"node","lean":"MetaMathlibExt.diffBasisNumber_div_sqrt_le","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for the "],["math","\\(g\\)"],["text","-difference basis number"]],"source":"Upper bound for the $g$-difference basis number"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#diff-basis-number-div-sqrt-le","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For natural numbers $g$ and $n$, let $\\eta_g(n)$ be the least size of a finite set $A \\subseteq \\mathbb{Z}$ such that every $x \\in \\{1, \\dots, n\\}$ is the difference $a - a'$ of at least $g$ ordered pairs $(a, a') \\in A \\times A$. Then for all natural numbers $g$ and $n$,","title":"Upper bound for the $g$-difference basis number"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/diff-basis-number-one-le-and-le-mul","kind":"node","lean":"MetaMathlibExt.diffBasisNumber_one_le_and_le_mul","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Comparing "],["math","\\(g\\)"],["text","-difference bases with difference bases"]],"source":"Comparing $g$-difference bases with difference bases"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#diff-basis-number-one-le-and-le-mul","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For natural numbers $g$ and $n$, let $\\eta_g(n)$ be the least size of a finite set $A \\subseteq \\mathbb{Z}$ such that every $x \\in \\{1, \\dots, n\\}$ is the difference $a - a'$ of at least $g$ ordered pairs $(a, a') \\in A \\times A$. If $n > 1$ and $g \\ge 1$, then $\\eta_1(n) \\le \\eta_g(n) \\le g\\,\\eta_1(n)$.","title":"Comparing $g$-difference bases with difference bases"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/tendsto-diff-basis-number-div-sqrt","kind":"node","lean":"MetaMathlibExt.tendsto_diffBasisNumber_div_sqrt","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#tendsto-diff-basis-number-div-sqrt","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For a finite set $A$ of integers and an integer $x$, write $r_{A-A}(x)$ for the number of ordered pairs $(a, a') \\in A \\times A$ with $a - a' = x$. For every integer $g \\ge 1$, let $\\eta_g(n)$ be the least size of a finite set $A$ of integers with $r_{A-A}(x) \\ge g$ for every integer $x \\in [1, n]$. Then there is a real number $c > 0$ with $\\eta_g(n) / \\sqrt{n} \\to c$ as $n \\to \\infty$.","title":"Kravitz's question: the normalized difference basis numbers converge"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/singer-product-diff-basis","kind":"node","lean":"MetaMathlibExt.singer_product_diffBasis","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#singer-product-diff-basis","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"Let $g, q, v \\in \\mathbb{N}$ and $m = q^2 + q + 1$. Let $a_0, \\dots, a_q$ be integers in $\\{1, \\dots, m\\}$ such that every $r \\in \\{1, \\dots, m - 1\\}$ is congruent modulo $m$ to $a_i - a_j$ for exactly one ordered pair $(i, j)$ with $i \\ne j$ (a Singer difference set modulo $m$). Let $B$ be a finite set of integers such that every $x \\in \\{1, \\dots, v\\}$ equals $b - b'$ for at least $g$ ordered pairs\u2026","title":"Singer product construction of difference bases"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/diff-basis-number-le-succ-mul","kind":"node","lean":"MetaMathlibExt.diffBasisNumber_le_succ_mul","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Scaling bound for the "],["math","\\(g\\)"],["text","-difference basis number"]],"source":"Scaling bound for the $g$-difference basis number"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#diff-basis-number-le-succ-mul","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For natural numbers $g$ and $m$, let $\\eta_g(m)$ be the least size of a finite set $A \\subseteq \\mathbb{Z}$ such that every $x \\in \\{1, \\dots, m\\}$ is the difference $a - a'$ of at least $g$ ordered pairs $(a, a') \\in A \\times A$. If $q$ is a prime and $n$ and $v$ are natural numbers with $n \\le q^2 v$, then $\\eta_g(n) \\le (q + 1)\\,\\eta_g(v)$.","title":"Scaling bound for the $g$-difference basis number"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/sub-rep-count-prod-ge","kind":"node","lean":"MetaMathlibExt.subRepCount_prod_ge","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#sub-rep-count-prod-ge","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"Let $G_1, G_2$ be abelian groups, $g_1, g_2 \\in \\mathbb{N}$, and $A_1 \\subseteq G_1$, $A_2 \\subseteq G_2$ finite sets. If every element of $G_1$ equals $a - a'$ for at least $g_1$ ordered pairs $(a, a') \\in A_1 \\times A_1$, and every element of $G_2$ equals $a - a'$ for at least $g_2$ ordered pairs $(a, a') \\in A_2 \\times A_2$, then every element of $G_1 \\times G_2$ equals $z - z'$ for at least $g_1 g_2$ ordered\u2026","title":"Products of difference bases"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/group-diff-basis-number-zmod-le","kind":"node","lean":"MetaMathlibExt.groupDiffBasisNumber_zmod_le","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#group-diff-basis-number-zmod-le","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For every prime $p$, the least size $\\eta_1(\\mathbb{Z}/p\\mathbb{Z})$ of a set $A \\subseteq \\mathbb{Z}/p\\mathbb{Z}$ such that every residue modulo $p$ is a difference $a - a'$ of two elements of $A$ satisfies $\\eta_1(\\mathbb{Z}/p\\mathbb{Z}) \\le 2 \\lceil \\sqrt{p} \\rceil$.","title":"Difference bases of the integers modulo a prime"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/group-diff-basis-number-prod-self-le","kind":"node","lean":"MetaMathlibExt.groupDiffBasisNumber_prod_self_le","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#group-diff-basis-number-prod-self-le","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For an abelian group $G$ let $\\eta_1(G)$ be the least size of a set $A \\subseteq G$ such that every element of $G$ is a difference $a - a'$ of two elements of $A$. If $K$ is a finite field with $q$ elements whose characteristic is not $2$, then $\\eta_1(K \\times K) \\le q + \\eta_1(K)$, where $K$ and $K \\times K$ are taken as additive groups.","title":"Difference bases of the square of a finite field"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/group-diff-basis-number-le-hundred-sqrt","kind":"node","lean":"MetaMathlibExt.groupDiffBasisNumber_le_hundred_sqrt","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#group-diff-basis-number-le-hundred-sqrt","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For every finite field $K$ with $q$ elements, the least size $\\eta_1(K)$ of a set $A \\subseteq K$ such that every element of $K$ is a difference $a - a'$ of two elements of $A$ satisfies $\\eta_1(K) \\le 100 \\sqrt{q}$.","title":"Square-root bound for difference bases of finite fields"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/tendsto-group-diff-basis-number-galois-f","kind":"node","lean":"MetaMathlibExt.tendsto_groupDiffBasisNumber_galoisField_prod","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Difference bases of "],["math","\\(\\mathbb{F}_{p^t} \\times \\mathbb{F}_{p^t}\\)"]],"source":"Difference bases of $\\mathbb{F}_{p^t} \\times \\mathbb{F}_{p^t}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#tendsto-group-diff-basis-number-galois-f","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"Let $p$ be an odd prime. For a finite abelian group $G$, a set $A \\subseteq G$ and $x \\in G$, write $r_{A-A}(x)$ for the number of ordered pairs $(a, a') \\in A \\times A$ with $a - a' = x$, and let $\\eta_1(G)$ be the least size of a set $A \\subseteq G$ with $r_{A-A}(x) \\ge 1$ for every $x \\in G$. Then $\\eta_1(\\mathbb{F}_{p^t} \\times \\mathbb{F}_{p^t}) / p^t \\to 1$ as $t \\to \\infty$, that is,\u2026","title":"Difference bases of $\\mathbb{F}_{p^t} \\times \\mathbb{F}_{p^t}$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/tendsto-group-diff-basis-number-even-odd","kind":"node","lean":"MetaMathlibExt.tendsto_groupDiffBasisNumber_even_odd","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even and odd limits for difference bases of "],["math","\\(\\mathbb{F}_p^n\\)"]],"source":"Even and odd limits for difference bases of $\\mathbb{F}_p^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#tendsto-group-diff-basis-number-even-odd","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"Let $p$ be an odd prime and $g \\ge 1$ an integer. For a finite abelian group $G$, a set $A \\subseteq G$ and $x \\in G$, write $r_{A-A}(x)$ for the number of ordered pairs $(a, a') \\in A \\times A$ with $a - a' = x$, and let $\\eta_g(G)$ be the least size of a set $A \\subseteq G$ with $r_{A-A}(x) \\ge g$ for every $x \\in G$. Then there are real numbers $L_e > 0$ and $L_o > 0$ such that, as $k \\to \\infty$,\u2026","title":"Even and odd limits for difference bases of $\\mathbb{F}_p^n$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/bose-chowla-sub-mem-subgroup-eq","kind":"node","lean":"MetaMathlibExt.boseChowla_sub_mem_subgroup_eq","parent":22905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonzero differences of the Bose\u2013Chowla set avoid the subgroup of order "],["math","\\(g\\)"]],"source":"Nonzero differences of the Bose\u2013Chowla set avoid the subgroup of order $g$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#bose-chowla-sub-mem-subgroup-eq","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"Let $q$ and $g$ be natural numbers with $g \\mid q - 1$, let $K$ be a finite field with $q^2$ elements, and let $\\theta \\in K$ be a primitive $(q^2 - 1)$-th root of unity, that is, a generator of $K^\\times$. If $a$ and $b$ are natural numbers less than $q^2 - 1$ with $(\\theta^a - \\theta)^q = \\theta^a - \\theta$ and $(\\theta^b - \\theta)^q = \\theta^b - \\theta$, and $(q^2 - 1)/g$ divides $a - b$, then $a = b$.","title":"Nonzero differences of the Bose\u2013Chowla set avoid the subgroup of order $g$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/bounded-diff-set-number-le-one-add-mul-s","kind":"node","lean":"MetaMathlibExt.boundedDiffSetNumber_le_one_add_mul_sqrt","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#bounded-diff-set-number-le-one-add-mul-s","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For natural numbers $g$ and $n$, let $\\alpha_g(n)$ be the largest size of a set $A \\subseteq \\{1, \\dots, n\\}$ such that every nonzero integer $x$ is the difference $a - a'$ of at most $g$ ordered pairs $(a, a') \\in A \\times A$. If $g \\ge 1$, then for every real $\\varepsilon > 0$ we have $\\alpha_g(n) \\le (1 + \\varepsilon)\\sqrt{gn}$ for all sufficiently large $n$; that is, $\\alpha_g(n) \\le (1 + o(1))\\sqrt{gn}$.","title":"Upper bound for sets with bounded difference multiplicities"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/exists-bounded-diff-set-card-div-sqrt-te","kind":"node","lean":"MetaMathlibExt.exists_boundedDiffSet_card_div_sqrt_tendsto","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#exists-bounded-diff-set-card-div-sqrt-te","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For every integer $g \\ge 1$ there are finite sets $A_n \\subseteq \\{1, \\dots, n\\}$ of integers, one for each $n \\in \\mathbb{N}$, such that for every $n$ each nonzero integer $x$ equals $a - a'$ for at most $g$ ordered pairs $(a, a') \\in A_n \\times A_n$, and $|A_n| / \\sqrt{gn} \\to 1$ as $n \\to \\infty$.","title":"Lower-bound construction for sets with bounded difference multiplicities"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/tendsto-bounded-diff-set-number-div-sqrt","kind":"node","lean":"MetaMathlibExt.tendsto_boundedDiffSetNumber_div_sqrt","parent":22905,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases/index.html#tendsto-bounded-diff-set-number-div-sqrt","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-bases","status":"can_prove","summary":"For a finite set $A$ of integers and an integer $x$, write $r_{A-A}(x)$ for the number of ordered pairs $(a, a') \\in A \\times A$ with $a - a' = x$. For every integer $g \\ge 1$, let $\\alpha_g(n)$ be the largest size of a set $A$ of integers in the interval $[1, n]$ with $r_{A-A}(x) \\le g$ for every nonzero integer $x$. Then $\\alpha_g(n) / \\sqrt{gn} \\to 1$ as $n \\to \\infty$, that is,\u2026","title":"Asymptotics of the largest sets with bounded difference counts"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets","status":"can_prove","title":"Difference sets of sparse sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem899Wanted.conjecture","parent":22932,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets","status":"fully_proved","summary":"The proposition asserts the following for every infinite set $A$ of natural numbers. Write $A(N) = |A \\cap \\{1, \\dots, N\\}|$ for its counting function and $D(N)$ for the number of integers $d \\in \\{1, \\dots, N\\}$ of the form $d = a - b$ with $a, b \\in A$. If $A(N)/N \\to 0$ as $N \\to \\infty$, then $\\limsup_{N \\to \\infty} D(N)/A(N) = +\\infty$.","title":"Erd\u0151s Problem 899 on difference sets of sparse sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem899Wanted.conjecture_holds","parent":22932,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/difference-sets-of-sparse-sets","status":"can_prove","summary":"Let $A$ be an infinite set of natural numbers whose counting function $A(N) = |A \\cap \\{1, \\dots, N\\}|$ satisfies $A(N)/N \\to 0$ as $N \\to \\infty$, and let $D(N)$ be the number of integers $d \\in \\{1, \\dots, N\\}$ of the form $d = a - b$ with $a, b \\in A$. Then $\\limsup_{N \\to \\infty} D(N)/A(N) = +\\infty$, where the ratio is computed in the extended reals and taken to be $0$ when $A(N) = 0$.","title":"Ruzsa's theorem on difference sets of sparse sets"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets","status":"can_prove","title":"Greedy B_h-sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/is-next-greedy-bh","kind":"node","lean":"MathlibExt.Combinatorics.GreedyBhFourthWanted.IsNextGreedyBh","parent":22935,"reading_title":{"format":"inline-parts-v1","parts":[["text","Next term of a greedy "],["math","\\(B_h\\)"],["text","-set"]],"source":"Next term of a greedy $B_h$-set"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/index.html#is-next-greedy-bh","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets","status":"fully_proved","summary":"Let $h$ and $a$ be natural numbers and $A$ a set of natural numbers. Then $a$ is the next greedy $B_h$ choice after $A$ if $a$ is greater than every element of $A$, the set $A \\cup \\{a\\}$ is a $B_h$-set, and for every natural number $b < a$ that is greater than every element of $A$, the set $A \\cup \\{b\\}$ is not a $B_h$-set.","title":"Next term of a greedy $B_h$-set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/is-fourth-greedy-bh","kind":"node","lean":"MathlibExt.Combinatorics.GreedyBhFourthWanted.IsFourthGreedyBh","parent":22935,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourth positive term of the greedy "],["math","\\(B_h\\)"],["text","-set"]],"source":"Fourth positive term of the greedy $B_h$-set"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/index.html#is-fourth-greedy-bh","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets","status":"fully_proved","summary":"For natural numbers $h$ and $a$, we say that $a$ is the fourth positive element $a_4(h)$ of the greedy $B_h$-set if $1$ is the next greedy $B_h$ choice after $\\{0\\}$, $h + 1$ is the next greedy $B_h$ choice after $\\{0, 1\\}$, $h^2 + h + 1$ is the next greedy $B_h$ choice after $\\{0, 1, h + 1\\}$, and $a$ is the next greedy $B_h$ choice after $\\{0, 1, h + 1, h^2 + h + 1\\}$.","title":"Fourth positive term of the greedy $B_h$-set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/fourth-value","kind":"node","lean":"MathlibExt.Combinatorics.GreedyBhFourthWanted.fourthValue","parent":22935,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quasipolynomial for the fourth greedy "],["math","\\(B_h\\)"],["text"," term"]],"source":"Quasipolynomial for the fourth greedy $B_h$ term"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/index.html#fourth-value","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets","status":"fully_proved","summary":"For a natural number $h$, the value $v(h)$ is $(h^3 + 3h^2 + 3h + 1)/2$ if $h$ is odd and $(h^3 + 2h^2 + 3h + 2)/2$ if $h$ is even.","title":"Quasipolynomial for the fourth greedy $B_h$ term"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/is-fourth-greedy-bh-fourth-value","kind":"node","lean":"MathlibExt.Combinatorics.GreedyBhFourthWanted.isFourthGreedyBh_fourthValue","parent":22935,"reading_title":{"format":"inline-parts-v1","parts":[["text","Formula for the fourth greedy "],["math","\\(B_h\\)"],["text"," term"]],"source":"Formula for the fourth greedy $B_h$ term"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/index.html#is-fourth-greedy-bh-fourth-value","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets","status":"can_prove","summary":"For every integer $h \\ge 1$, the fourth positive element $a_4(h)$ of the greedy $B_h$-set is $(h^3 + 3h^2 + 3h + 1)/2$ if $h$ is odd and $(h^3 + 2h^2 + 3h + 2)/2$ if $h$ is even.","title":"Formula for the fourth greedy $B_h$ term"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/fourth-value-le-of-is-fourth-greedy-bh","kind":"node","lean":"MathlibExt.Combinatorics.GreedyBhFourthWanted.fourthValue_le_of_isFourthGreedyBh","parent":22935,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound for the fourth greedy "],["math","\\(B_h\\)"],["text"," term"]],"source":"Lower bound for the fourth greedy $B_h$ term"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/index.html#fourth-value-le-of-is-fourth-greedy-bh","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets","status":"can_prove","summary":"Let $h$ and $a$ be natural numbers with $h \\ge 2$. If $a$ is the fourth positive element $a_4(h)$ of the greedy $B_h$-set, then $a \\ge v(h)$, where $v(h) = (h^3 + 3h^2 + 3h + 1)/2$ if $h$ is odd and $v(h) = (h^3 + 2h^2 + 3h + 2)/2$ if $h$ is even.","title":"Lower bound for the fourth greedy $B_h$ term"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/is-fourth-greedy-bh-le-fourth-value","kind":"node","lean":"MathlibExt.Combinatorics.GreedyBhFourthWanted.isFourthGreedyBh_le_fourthValue","parent":22935,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for the fourth greedy "],["math","\\(B_h\\)"],["text"," term"]],"source":"Upper bound for the fourth greedy $B_h$ term"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets/index.html#is-fourth-greedy-bh-le-fourth-value","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/greedy-b-h-sets","status":"can_prove","summary":"Let $h$ and $a$ be natural numbers with $h \\ge 1$. If $a$ is the fourth positive element $a_4(h)$ of the greedy $B_h$-set, then $a \\le v(h)$, where $v(h) = (h^3 + 3h^2 + 3h + 1)/2$ if $h$ is odd and $v(h) = (h^3 + 2h^2 + 3h + 2)/2$ if $h$ is even.","title":"Upper bound for the fourth greedy $B_h$ term"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","status":"can_prove","title":"Sidon sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/is-sidon","kind":"node","lean":"Set.IsSidon","parent":22942,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/index.html#is-sidon","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","status":"fully_proved","summary":"A set in an additive commutative monoid is Sidon when every equality of two pairwise sums is trivial up to swapping the summands. Repeated summands are included.","title":"Sidon set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/is-sidon-2","kind":"node","lean":"MathlibExt.Combinatorics.ErdosSidonIsolatedSumsWanted.IsSidon","parent":22942,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/index.html#is-sidon-2","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","status":"fully_proved","summary":"A finite set $A$ of natural numbers is a Sidon set when every equality $a + b = c + d$ with $a, b, c, d \\in A$ is trivial, that is, $a = c$ and $b = d$, or $a = d$ and $b = c$.","title":"Finite Sidon sets of natural numbers"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/sumset","kind":"node","lean":"MathlibExt.Combinatorics.ErdosSidonIsolatedSumsWanted.Sumset","parent":22942,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/index.html#sumset","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","status":"fully_proved","summary":"For a finite set $A$ of natural numbers, the sumset $A + A$ is the finite set of all sums $x + y$ with $x, y \\in A$.","title":"Sumset of a finite set of natural numbers"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/is-isolated","kind":"node","lean":"MathlibExt.Combinatorics.ErdosSidonIsolatedSumsWanted.IsIsolated","parent":22942,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/index.html#is-isolated","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","status":"fully_proved","summary":"Let $S$ be a finite set of natural numbers. A natural number $a$ is isolated in $S$ when $a \\in S$, $a + 1 \\notin S$, and either $a = 0$ or $a - 1 \\notin S$.","title":"Isolated elements of a set of natural numbers"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/isolated-count","kind":"node","lean":"MathlibExt.Combinatorics.ErdosSidonIsolatedSumsWanted.IsolatedCount","parent":22942,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/index.html#isolated-count","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","status":"fully_proved","summary":"For a finite set $S$ of natural numbers, the isolated count of $S$ is the number of elements $a \\in S$ with $a + 1 \\notin S$ and either $a = 0$ or $a - 1 \\notin S$.","title":"Number of isolated elements"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosSidonIsolatedSumsWanted.conjecture","parent":22942,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","status":"fully_proved","summary":"A finite set $A$ of natural numbers is a Sidon set when $a + b = c + d$ with $a, b, c, d \\in A$ implies $\\{a, b\\} = \\{c, d\\}$, and its sumset is $A + A = \\{x + y : x, y \\in A\\}$. An element $s$ of a finite set $S$ of natural numbers is isolated in $S$ when $s + 1 \\notin S$ and either $s = 0$ or $s - 1 \\notin S$. The proposition asserts that for every natural number $M \\ge 1$ there is a natural number $K$ such that\u2026","title":"Erd\u0151s Problem 152 on isolated sums of Sidon sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosSidonIsolatedSumsWanted.conjecture_holds","parent":22942,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets","status":"can_prove","summary":"For every natural number $M \\ge 1$ there is a natural number $K$ such that every finite Sidon set $A$ of natural numbers with at least $K$ elements has at least $M$ isolated elements in its sumset $A + A = \\{x + y : x, y \\in A\\}$. Here $A$ is a Sidon set when $a + b = c + d$ with $a, b, c, d \\in A$ implies $\\{a, b\\} = \\{c, d\\}$, and an element $s$ of $A + A$ is isolated when $s + 1 \\notin A + A$ and either $s = 0$\u2026","title":"Large Sidon sets have many isolated sums"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers","status":"can_prove","title":"Golomb rulers"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/golomb-ruler-lengths","kind":"node","lean":"MathlibExt.Combinatorics.GolombRulersWanted.golombRulerLengths","parent":22950,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/index.html#golomb-ruler-lengths","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers","status":"fully_proved","summary":"For a natural number $m$, the set of Golomb ruler lengths with $m$ marks consists of the natural numbers $L$ for which there is a finite Sidon set $S$ of natural numbers with $S \\subseteq \\{0, 1, \\dots, L\\}$, exactly $m$ elements, and $L \\in S$. A mark at $0$ is not required.","title":"Golomb ruler lengths"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/optimal-golomb-length","kind":"node","lean":"MathlibExt.Combinatorics.GolombRulersWanted.optimalGolombLength","parent":22950,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/index.html#optimal-golomb-length","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers","status":"fully_proved","summary":"For $m \\ge 1$, the optimal Golomb ruler length $G(m)$ is the infimum in $\\mathbb{N}$ of the set of Golomb ruler lengths with $m$ marks, that is, of the natural numbers $L$ such that some $m$-element Sidon set of natural numbers contained in $\\{0, 1, \\dots, L\\}$ contains $L$; for $m = 0$ we set $G(0) = 0$.","title":"Optimal Golomb ruler length"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/optimal-golomb-length-three-to-eleven","kind":"node","lean":"MathlibExt.Combinatorics.GolombRulersWanted.optimalGolombLength_three_to_eleven","parent":22950,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/index.html#optimal-golomb-length-three-to-eleven","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers","status":"can_prove","summary":"The optimal Golomb ruler lengths with $m = 3, 4, \\dots, 11$ marks are, in order, $3, 6, 11, 17, 25, 34, 44, 55, 72$.","title":"Optimal Golomb rulers with 3 to 11 marks"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/optimal-golomb-length-twenty-seven","kind":"node","lean":"MathlibExt.Combinatorics.GolombRulersWanted.optimalGolombLength_twentySeven","parent":22950,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/index.html#optimal-golomb-length-twenty-seven","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers","status":"can_prove","summary":"The optimal Golomb ruler length with $27$ marks is $553$.","title":"Optimal 27-mark Golomb ruler"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/optimal-golomb-length-twenty-eight","kind":"node","lean":"MathlibExt.Combinatorics.GolombRulersWanted.optimalGolombLength_twentyEight","parent":22950,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers/index.html#optimal-golomb-length-twenty-eight","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/golomb-rulers","status":"can_prove","summary":"The optimal Golomb ruler length with $28$ marks is $585$.","title":"Optimal 28-mark Golomb ruler"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets","status":"can_prove","title":"Number of Sidon sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/sidon-subsets-up-to","kind":"node","lean":"MathlibExt.Combinatorics.SidonSubsetCountErdosProblem861Wanted.sidonSubsetsUpTo","parent":22956,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sidon subsets of "],["math","\\(\\{1, \\dots, N\\}\\)"]],"source":"Sidon subsets of $\\{1, \\dots, N\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/index.html#sidon-subsets-up-to","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets","status":"fully_proved","summary":"For a natural number $N$, this is the finite family of all subsets of $\\{1, \\dots, N\\}$ that are Sidon sets of natural numbers, including the empty set and the singletons.","title":"Sidon subsets of $\\{1, \\dots, N\\}$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/max-sidon-size","kind":"node","lean":"MathlibExt.Combinatorics.SidonSubsetCountErdosProblem861Wanted.maxSidonSize","parent":22956,"reading_title":{"format":"inline-parts-v1","parts":[["text","Largest Sidon subset of "],["math","\\(\\{1, \\dots, N\\}\\)"],["text"," (Erd\u0151s Problem 861)"]],"source":"Largest Sidon subset of $\\{1, \\dots, N\\}$ (Erd\u0151s Problem 861)"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/index.html#max-sidon-size","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets","status":"fully_proved","summary":"For a natural number $N$, $f(N)$ is the largest cardinality of a Sidon subset of $\\{1, \\dots, N\\}$.","title":"Largest Sidon subset of $\\{1, \\dots, N\\}$ (Erd\u0151s Problem 861)"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/sidon-count-over-two-pow-max-diverges","kind":"node","lean":"MathlibExt.Combinatorics.SidonSubsetCountErdosProblem861Wanted.sidonCountOverTwoPowMaxDiverges","parent":22956,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/index.html#sidon-count-over-two-pow-max-diverges","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets","status":"fully_proved","summary":"Let $A(N)$ be the number of Sidon subsets of $\\{1, \\dots, N\\}$, including the empty set and the singletons, and let $f(N)$ be the largest size of a Sidon subset of $\\{1, \\dots, N\\}$. The statement is that $A(N) / 2^{f(N)} \\to \\infty$ as $N \\to \\infty$.","title":"Erd\u0151s Problem 861, first question"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/sidon-count-exponent-asymptotic","kind":"node","lean":"MathlibExt.Combinatorics.SidonSubsetCountErdosProblem861Wanted.sidonCountExponentAsymptotic","parent":22956,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/index.html#sidon-count-exponent-asymptotic","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets","status":"fully_proved","summary":"Let $A(N)$ be the number of Sidon subsets of $\\{1, \\dots, N\\}$, including the empty set and the singletons, and let $f(N)$ be the largest size of a Sidon subset of $\\{1, \\dots, N\\}$. The statement is that $\\log_2 A(N) / f(N) \\to 1$ as $N \\to \\infty$, that is, $A(N) = 2^{(1 + o(1)) f(N)}$.","title":"Erd\u0151s Problem 861, second question"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/sidon-count-over-two-pow-max-diverges-ho","kind":"node","lean":"MathlibExt.Combinatorics.SidonSubsetCountErdosProblem861Wanted.sidonCountOverTwoPowMaxDiverges_holds","parent":22956,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/index.html#sidon-count-over-two-pow-max-diverges-ho","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets","status":"can_prove","summary":"Let $A(N)$ be the number of Sidon subsets of $\\{1, \\dots, N\\}$, including the empty set and the singletons, and let $f(N)$ be the largest size of a Sidon subset of $\\{1, \\dots, N\\}$. Then $A(N) / 2^{f(N)} \\to \\infty$ as $N \\to \\infty$.","title":"Erd\u0151s Problem 861, first question: positive answer"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/sidon-count-exponent-asymptotic-refuted","kind":"node","lean":"MathlibExt.Combinatorics.SidonSubsetCountErdosProblem861Wanted.sidonCountExponentAsymptotic_refuted","parent":22956,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets/index.html#sidon-count-exponent-asymptotic-refuted","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/number-of-sidon-sets","status":"can_prove","summary":"Let $A(N)$ be the number of Sidon subsets of $\\{1, \\dots, N\\}$, including the empty set and the singletons, and let $f(N)$ be the largest size of a Sidon subset of $\\{1, \\dots, N\\}$. Then the ratio $\\log_2 A(N) / f(N)$ does not tend to $1$ as $N \\to \\infty$; that is, $A(N) = 2^{(1 + o(1)) f(N)}$ fails.","title":"Erd\u0151s Problem 861, second question: negative answer"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","title":"Sidon ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/is-weak-sidon-set","kind":"node","lean":"MetaMathlibExt.IsWeakSidonSet","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#is-weak-sidon-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"A set $A \\subseteq \\mathbb{N}$ is a *weak Sidon set* (or *well spread*) if sums of two distinct elements of $A$ are distinct: whenever $a, b, c, d \\in A$ satisfy $a < b$, $c < d$ and $a + b = c + d$, then $a = c$.","title":"Weak Sidon set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/is-weak-sidon-set-of-is-sidon","kind":"node","lean":"MetaMathlibExt.isWeakSidonSet_of_isSidon","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#is-weak-sidon-set-of-is-sidon","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"Every Sidon set $A \\subseteq \\mathbb{N}$ is a weak Sidon set.","title":"Sidon sets are weak Sidon sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/generated-ideal","kind":"node","lean":"MetaMathlibExt.generatedIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#generated-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"Let $P$ be a property of subsets of $\\mathbb{N}$. The ideal generated by $P$ is the family of all sets $A \\subseteq \\mathbb{N}$ that are contained in the union of finitely many sets with property $P$. The empty union is allowed, so the empty set always belongs to it.","title":"Ideal generated by a property"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/mem-generated-ideal","kind":"node","lean":"MetaMathlibExt.mem_generatedIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#mem-generated-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"Let $P$ be a property of subsets of $\\mathbb{N}$. Every set $A \\subseteq \\mathbb{N}$ with property $P$ lies in the ideal generated by $P$, the family of all subsets of finite unions of sets with property $P$.","title":"Sets with a property lie in the ideal it generates"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/generated-ideal-of-subset","kind":"node","lean":"MetaMathlibExt.generatedIdeal_of_subset","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#generated-ideal-of-subset","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"Let $P$ be a property of subsets of $\\mathbb{N}$. If $A$ lies in the ideal generated by $P$ and $B \\subseteq A$, then $B$ also lies in the ideal generated by $P$.","title":"Generated ideals are closed under subsets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/union-mem-generated-ideal","kind":"node","lean":"MetaMathlibExt.union_mem_generatedIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#union-mem-generated-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"Let $P$ be a property of subsets of $\\mathbb{N}$, and consider the ideal generated by $P$, the family of all subsets of finite unions of sets with property $P$. If $A$ and $B$ lie in this ideal, then so does $A \\cup B$.","title":"Generated ideals are closed under unions"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/sidon-ideal","kind":"node","lean":"MetaMathlibExt.sidonIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#sidon-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"The Sidon ideal $\\mathrm{Si}$ is the ideal on $\\mathbb{N}$ generated by the Sidon sets: it consists of all sets $A \\subseteq \\mathbb{N}$ contained in a finite union of Sidon subsets of $\\mathbb{N}$.","title":"Sidon ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/is-gap-lacunary-set","kind":"node","lean":"MetaMathlibExt.IsGapLacunarySet","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#is-gap-lacunary-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"A set $A \\subseteq \\mathbb{N}$ is *gap-lacunary* (called *lacunary* in the source) if it is infinite and, writing $A(n)$ for its $n$-th element in increasing order (counting from $n = 0$), the gaps $A(n+1) - A(n)$ tend to infinity as $n \\to \\infty$.","title":"Gap-lacunary set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/infinite","kind":"node","lean":"MetaMathlibExt.IsGapLacunarySet.infinite","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#infinite","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"Every gap-lacunary set $A \\subseteq \\mathbb{N}$ is infinite.","title":"Gap-lacunary sets are infinite"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/is-almost-thin-set","kind":"node","lean":"MetaMathlibExt.IsAlmostThinSet","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#is-almost-thin-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"A set $A \\subseteq \\mathbb{N}$ is *almost thin* if it is infinite and, writing $A(n)$ for its $n$-th element in increasing order (counting from $n = 0$), there is a real number $\\alpha < 1$ such that $A(n)/A(n+1) \\le \\alpha$ for all sufficiently large $n$.","title":"Almost thin set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/infinite-2","kind":"node","lean":"MetaMathlibExt.IsAlmostThinSet.infinite","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#infinite-2","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"Every almost thin set $A \\subseteq \\mathbb{N}$ is infinite.","title":"Almost thin sets are infinite"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/van-der-waerden-ideal","kind":"node","lean":"MetaMathlibExt.vanDerWaerdenIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#van-der-waerden-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"The van der Waerden ideal $W$ consists of the sets $A \\subseteq \\mathbb{N}$ for which there is an $n > 0$ such that $A$ contains no arithmetic progression $a, a + r, \\dots, a + (n - 1) r$ of $n$ terms with $a \\in \\mathbb{N}$ and $r > 0$.","title":"Van der Waerden ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/empty-mem-van-der-waerden-ideal","kind":"node","lean":"MetaMathlibExt.empty_mem_vanDerWaerdenIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#empty-mem-van-der-waerden-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"The empty subset of $\\mathbb{N}$ belongs to the van der Waerden ideal.","title":"The empty set lies in the van der Waerden ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/positive-differences","kind":"node","lean":"MetaMathlibExt.positiveDifferences","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#positive-differences","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"For a set $Z \\subseteq \\mathbb{N}$, the positive difference set $D(Z)$ is the set of all differences $a - b$ with $a, b \\in Z$ and $a > b$.","title":"Positive difference set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/d-fin-ideal","kind":"node","lean":"MetaMathlibExt.dFinIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#d-fin-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"For a set $Z \\subseteq \\mathbb{N}$, let $D(Z) = \\{a - b : a, b \\in Z,\\ b < a\\}$ be its set of positive differences. The ideal $D_{\\mathrm{fin}}$ consists of the sets $A \\subseteq \\mathbb{N}$ for which there is a natural number $n$ such that $D(Z) \\not\\subseteq A$ for every $n$-element set $Z \\subseteq \\mathbb{N}$.","title":"The ideal of sets avoiding large difference sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/empty-mem-d-fin-ideal","kind":"node","lean":"MetaMathlibExt.empty_mem_dFinIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#empty-mem-d-fin-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"The empty set belongs to the ideal $D_{\\mathrm{fin}}$ of sets $A \\subseteq \\mathbb{N}$ for which some natural number $n$ satisfies $D(Z) \\not\\subseteq A$ for every $n$-element set $Z \\subseteq \\mathbb{N}$, where $D(Z)$ is the set of positive differences $a - b$ with $a, b \\in Z$ and $b < a$.","title":"The empty set lies in the difference-set ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/sidon-block","kind":"node","lean":"MetaMathlibExt.sidonBlock","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#sidon-block","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"For natural numbers $n$, $C$, $M$ and $K$, the block $\\mathrm{Block}(n, C, M, K)$ is the set of natural numbers $M l + C \\cdot 2^k$ with $0 \\le k < n$ and $1 \\le l \\le K$.","title":"Block sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/sidon-block-one-one","kind":"node","lean":"MetaMathlibExt.sidonBlock_one_one","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#sidon-block-one-one","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"fully_proved","summary":"For all natural numbers $C$ and $M$, the block $\\mathrm{Block}(1, C, M, 1) = \\{M l + C \\cdot 2^k : 0 \\le k < 1,\\ 1 \\le l \\le 1\\}$ is the singleton $\\{M + C\\}$.","title":"One-term blocks"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/exists-partition-is-sidon-of-is-weak-sid","kind":"node","lean":"MetaMathlibExt.exists_partition_isSidon_of_isWeakSidonSet","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#exists-partition-is-sidon-of-is-weak-sid","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"Every weak Sidon set $A \\subseteq \\mathbb{N}$ is the union of two disjoint Sidon sets $B$ and $C$.","title":"Weak Sidon sets split into two Sidon sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/generated-ideal-is-weak-sidon-set-eq-sid","kind":"node","lean":"MetaMathlibExt.generatedIdeal_isWeakSidonSet_eq_sidonIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#generated-ideal-is-weak-sidon-set-eq-sid","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"The ideal on $\\mathbb{N}$ generated by the weak Sidon sets equals the Sidon ideal.","title":"Weak Sidon sets generate the Sidon ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/sidon-ideal-is-fsigma","kind":"node","lean":"MetaMathlibExt.sidonIdeal_isFSigma","parent":22963,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Sidon ideal is "],["math","\\(F_\\sigma\\)"]],"source":"The Sidon ideal is $F_\\sigma$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#sidon-ideal-is-fsigma","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"Give the Cantor space $\\{0, 1\\}^{\\mathbb{N}}$ the product topology and identify each set $A \\subseteq \\mathbb{N}$ with its indicator function. Then the set of indicator functions of the members of the Sidon ideal is a countable union of closed subsets of $\\{0, 1\\}^{\\mathbb{N}}$.","title":"The Sidon ideal is $F_\\sigma$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/is-sidon-of-two-mul-nth-le","kind":"node","lean":"MetaMathlibExt.isSidon_of_two_mul_nth_le","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#is-sidon-of-two-mul-nth-le","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"Let $A \\subseteq \\mathbb{N}$ with $0 \\notin A$, and write $A(n)$ for the $n$-th element of $A$ in increasing order, counting from $n = 0$, with $A(n) = 0$ when $A$ has at most $n$ elements. If $2 A(n) \\le A(n+1)$ for every $n \\in \\mathbb{N}$, then $A$ is a Sidon set.","title":"Positive sets whose elements at least double are Sidon"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/sidon-ideal-subset-lacunary-ideal","kind":"node","lean":"MetaMathlibExt.sidonIdeal_subset_lacunaryIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#sidon-ideal-subset-lacunary-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"Call a set $A \\subseteq \\mathbb{N}$ lacunary if it is infinite and the gaps $A(n + 1) - A(n)$ between consecutive elements tend to infinity, where $A(n)$ is the $n$-th element of $A$ in increasing order. Every set in the Sidon ideal is contained in a finite union of lacunary sets, that is, it lies in the ideal $\\mathrm{Lac}$ generated by the lacunary sets.","title":"The Sidon ideal is contained in the lacunary ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/exists-is-sidon-card-gt-cbrt","kind":"node","lean":"MetaMathlibExt.exists_isSidon_card_gt_cbrt","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#exists-is-sidon-card-gt-cbrt","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"There exist a Sidon set $A \\subseteq \\mathbb{N}$ and a real constant $c > 0$ such that $|A \\cap \\{1, \\dots, n\\}| > c\\, n^{1/3}$ for every natural number $n \\ge 1$.","title":"Mian\u2013Chowla lower bound for an infinite Sidon set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/exists-is-sidon-card-gt-cbrt-mul-log","kind":"node","lean":"MetaMathlibExt.exists_isSidon_card_gt_cbrt_mul_log","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#exists-is-sidon-card-gt-cbrt-mul-log","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"There exist a Sidon set $A \\subseteq \\mathbb{N}$ and a real constant $c > 0$ such that $|A \\cap \\{1, \\dots, n\\}| > c\\, (n \\log n)^{1/3}$ for every natural number $n \\ge 1$, where $\\log$ is the natural logarithm.","title":"Ajtai\u2013Koml\u00f3s\u2013Szemer\u00e9di lower bound for an infinite Sidon set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/exists-is-sidon-card-gt-rpow-sqrt-two-su","kind":"node","lean":"MetaMathlibExt.exists_isSidon_card_gt_rpow_sqrt_two_sub_one","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#exists-is-sidon-card-gt-rpow-sqrt-two-su","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"There is a Sidon set $A \\subseteq \\mathbb{N}$ such that for every real $\\varepsilon > 0$, the inequality $|A \\cap \\{1, \\dots, n\\}| > n^{\\sqrt{2} - 1 - \\varepsilon}$ holds for all sufficiently large natural numbers $n$; that is, $|A \\cap \\{1, \\dots, n\\}| > n^{\\sqrt{2} - 1 - o(1)}$.","title":"Ruzsa's lower bound for an infinite Sidon set"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/exists-is-sidon-not-mem-almost-thin-idea","kind":"node","lean":"MetaMathlibExt.exists_isSidon_not_mem_almostThinIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#exists-is-sidon-not-mem-almost-thin-idea","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"There is a Sidon set $A \\subseteq \\mathbb{N}$ that is not contained in any finite union of almost thin sets; that is, $A$ does not belong to the ideal generated by the almost thin sets.","title":"A Sidon set outside the almost thin ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/almost-thin-ideal-subset-sidon-ideal","kind":"node","lean":"MetaMathlibExt.almostThinIdeal_subset_sidonIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#almost-thin-ideal-subset-sidon-ideal","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"Every set $A \\subseteq \\mathbb{N}$ that is contained in a finite union of almost thin sets is contained in a finite union of Sidon sets; that is, the ideal generated by the almost thin sets is contained in the Sidon ideal.","title":"The almost thin ideal is contained in the Sidon ideal"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/not-sidon-block-subset-i-union-is-sidon","kind":"node","lean":"MetaMathlibExt.not_sidonBlock_subset_iUnion_isSidon","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#not-sidon-block-subset-i-union-is-sidon","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"Let $n \\ge 1$ and $C \\ge 1$ be natural numbers, and put $M = C \\cdot 2^n + 1$ and $K = (n - 1)\\binom{n}{2} + 1$. Then the block $\\mathrm{Block}(n, C, M, K) = \\{M l + C \\cdot 2^k : 0 \\le k < n,\\ 1 \\le l \\le K\\}$ is not contained in the union of any $n - 1$ Sidon subsets of $\\mathbb{N}$.","title":"A block not covered by few Sidon sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/exists-mem-van-der-waerden-lacunary-d-fi","kind":"node","lean":"MetaMathlibExt.exists_mem_vanDerWaerden_lacunary_dFin_not_mem_sidonIdeal","parent":22963,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal/index.html#exists-mem-van-der-waerden-lacunary-d-fi","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-ideal","status":"can_prove","summary":"There is a set $A \\subseteq \\mathbb{N}$ that belongs to the van der Waerden ideal, is contained in a finite union of gap-lacunary sets (infinite sets whose gaps between consecutive elements tend to infinity), and belongs to the ideal $D_{\\mathrm{fin}}$ of sets avoiding large difference sets, but is not contained in any finite union of Sidon sets.","title":"A set in the van der Waerden, lacunary and difference-set ideals outside the Sidon ideal"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets","status":"can_prove","title":"Sidon sets and perfect difference sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/is-perfect-difference-set-mod","kind":"node","lean":"MathlibExt.Combinatorics.SidonPerfectDifferenceSetErdosProblem707Wanted.IsPerfectDifferenceSetMod","parent":22994,"reading_title":{"format":"inline-parts-v1","parts":[["text","Perfect difference sets modulo "],["math","\\(v\\)"]],"source":"Perfect difference sets modulo $v$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/index.html#is-perfect-difference-set-mod","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets","status":"fully_proved","summary":"A set $B \\subseteq \\mathbb{N}$ is a perfect difference set modulo a natural number $v$ when every nonzero residue $x$ modulo $v$ satisfies $x \\equiv a - b \\pmod{v}$ for exactly one ordered pair $(a, b)$ of elements of $B$.","title":"Perfect difference sets modulo $v$"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/sidon-extends-to-prime-perfect-differenc","kind":"node","lean":"MathlibExt.Combinatorics.SidonPerfectDifferenceSetErdosProblem707Wanted.sidonExtendsToPrimePerfectDifferenceSet","parent":22994,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/index.html#sidon-extends-to-prime-perfect-differenc","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets","status":"fully_proved","summary":"The proposition that for every finite Sidon set $A \\subseteq \\mathbb{N}$ there are a prime $p$ and a set $B \\subseteq \\mathbb{N}$ with $A \\subseteq B$ such that $B$ is a perfect difference set modulo $p^2 + p + 1$.","title":"Erd\u0151s Problem 707"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/sidon-extends-to-perfect-difference-set","kind":"node","lean":"MathlibExt.Combinatorics.SidonPerfectDifferenceSetErdosProblem707Wanted.sidonExtendsToPerfectDifferenceSet","parent":22994,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/index.html#sidon-extends-to-perfect-difference-set","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets","status":"fully_proved","summary":"The proposition that for every finite Sidon set $A \\subseteq \\mathbb{N}$ there are a natural number $m \\ge 1$ and a set $B \\subseteq \\mathbb{N}$ with $A \\subseteq B$ such that $B$ is a perfect difference set modulo $m^2 + m + 1$.","title":"Erd\u0151s Problem 707, weak form"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/sidon-extends-to-prime-perfect-differenc-2","kind":"node","lean":"MathlibExt.Combinatorics.SidonPerfectDifferenceSetErdosProblem707Wanted.sidonExtendsToPrimePerfectDifferenceSet_refuted","parent":22994,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/index.html#sidon-extends-to-prime-perfect-differenc-2","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets","status":"can_prove","summary":"It is not the case that every finite Sidon set $A \\subseteq \\mathbb{N}$ is contained in a perfect difference set modulo $p^2 + p + 1$ for some prime $p$: there is a finite Sidon set $A \\subseteq \\mathbb{N}$ such that for every prime $p$, no set $B \\subseteq \\mathbb{N}$ with $A \\subseteq B$ is a perfect difference set modulo $p^2 + p + 1$. Here $A$ is Sidon when $w + x = y + z$ with $w, x, y, z \\in A$ forces\u2026","title":"Erd\u0151s Problem 707 is false"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/sidon-extends-to-perfect-difference-set-2","kind":"node","lean":"MathlibExt.Combinatorics.SidonPerfectDifferenceSetErdosProblem707Wanted.sidonExtendsToPerfectDifferenceSet_refuted","parent":22994,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets/index.html#sidon-extends-to-perfect-difference-set-2","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-and-perfect-difference-sets","status":"can_prove","summary":"It is not the case that every finite Sidon set $A \\subseteq \\mathbb{N}$ is contained in a perfect difference set modulo $m^2 + m + 1$ for some natural number $m \\ge 1$: there is a finite Sidon set $A \\subseteq \\mathbb{N}$ such that for every $m \\ge 1$, no set $B \\subseteq \\mathbb{N}$ with $A \\subseteq B$ is a perfect difference set modulo $m^2 + m + 1$. Here $A$ is Sidon when $w + x = y + z$ with $w, x, y, z \\in A$\u2026","title":"The weak form of Erd\u0151s Problem 707 is false"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression","status":"can_prove","title":"Sidon sets meeting every infinite progression"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression/sidon-complement-contains-progression","kind":"node","lean":"MathlibExt.Combinatorics.SidonSetComplementProgressionErdosProblem198Wanted.sidonComplementContainsProgression","parent":23000,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression/index.html#sidon-complement-contains-progression","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression","status":"fully_proved","summary":"The proposition that the complement of every Sidon set $A \\subseteq \\mathbb{N}$ contains an infinite arithmetic progression: for every Sidon set $A \\subseteq \\mathbb{N}$ there are natural numbers $a$ and $d \\ge 1$ such that $a + kd \\notin A$ for every natural number $k$.","title":"Erd\u0151s Problem 198"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression/sidon-complement-contains-progression-re","kind":"node","lean":"MathlibExt.Combinatorics.SidonSetComplementProgressionErdosProblem198Wanted.sidonComplementContainsProgression_refuted","parent":23000,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression/index.html#sidon-complement-contains-progression-re","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/sidon-sets-meeting-every-infinite-progression","status":"can_prove","summary":"It is not the case that the complement of every Sidon set $A \\subseteq \\mathbb{N}$ contains an infinite arithmetic progression: there is a Sidon set $A \\subseteq \\mathbb{N}$ such that for all natural numbers $a$ and $d \\ge 1$ some term $a + kd$, $k \\in \\mathbb{N}$, lies in $A$. Here $A$ is Sidon when $w + x = y + z$ with $w, x, y, z \\in A$ forces $\\{w, x\\} = \\{y, z\\}$.","title":"Erd\u0151s Problem 198 is false"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets.json","id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets","kind":"scope","navigation":"module","parent":22871,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets/index.html","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets","status":"can_prove","title":"Upper bounds for Sidon sets"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets/max-sidon-subset-size","kind":"node","lean":"MathlibExt.Combinatorics.SidonUpperBoundImprovementGreenProblem31Wanted.maxSidonSubsetSize","parent":23003,"reading_title":{"format":"inline-parts-v1","parts":[["text","Largest Sidon subset of "],["math","\\(\\{1, \\dots, N\\}\\)"],["text"," (Green's Problem 31)"]],"source":"Largest Sidon subset of $\\{1, \\dots, N\\}$ (Green's Problem 31)"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets/index.html#max-sidon-subset-size","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets","status":"fully_proved","summary":"For a natural number $N$, $F(N)$ is the largest cardinality of a Sidon subset of $[N] = \\{1, \\dots, N\\}$.","title":"Largest Sidon subset of $\\{1, \\dots, N\\}$ (Green's Problem 31)"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets/sidon-upper-bound-improved","kind":"node","lean":"MathlibExt.Combinatorics.SidonUpperBoundImprovementGreenProblem31Wanted.sidonUpperBoundImproved","parent":23003,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets/index.html#sidon-upper-bound-improved","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets","status":"fully_proved","summary":"Let $F(N)$ be the largest size of a Sidon subset of $\\{1, \\dots, N\\}$. The statement is that there are a real constant $c < 1$ and a natural number $N_0$ such that $F(N) \\le \\sqrt{N} + c N^{1/4}$ for all $N \\ge N_0$.","title":"Green's Problem 31: improving Lindstr\u00f6m's upper bound"},{"id":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets/sidon-upper-bound-improved-holds","kind":"node","lean":"MathlibExt.Combinatorics.SidonUpperBoundImprovementGreenProblem31Wanted.sidonUpperBoundImproved_holds","parent":23003,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets/index.html#sidon-upper-bound-improved-holds","scope":"Combinatorics/additive-combinatorics/sidon-sets-difference-sets/upper-bounds-for-sidon-sets","status":"can_prove","summary":"Let $F(N)$ be the largest size of a Sidon subset of $\\{1, \\dots, N\\}$. There are a real constant $c < 1$ and a natural number $N_0$ such that $F(N) \\le \\sqrt{N} + c N^{1/4}$ for all $N \\ge N_0$.","title":"Lindstr\u00f6m's upper bound for Sidon sets can be improved"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sum-free-sets.json","id":"Combinatorics/additive-combinatorics/sum-free-sets","kind":"scope","parent":22582,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/index.html","scope":"Combinatorics/additive-combinatorics/sum-free-sets","status":"can_prove","summary":"Sets with no solution to x+y=z or similar: sum-free sets in abelian groups and intervals, counts of sum-free and maximal sum-free sets, large sum-free subsets, square-sum-free, divisor-sum-free and non-averaging sets.","title":"Sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/is-product-free","kind":"node","lean":"Set.IsProductFree","parent":23007,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/index.html#is-product-free","scope":"Combinatorics/additive-combinatorics/sum-free-sets","status":"fully_proved","summary":"A set $S$ in a type $M$ with a multiplication is product-free if $xy \\notin S$ for all $x, y \\in S$, where $x$ and $y$ need not be distinct.","title":"Product-free set"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets.json","id":"Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets","kind":"scope","navigation":"module","parent":23007,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets/index.html","scope":"Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets","status":"can_prove","title":"Divisor-sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets/is-divisor-sum-free","kind":"node","lean":"MathlibExt.Combinatorics.DivisorSumFreeSetLowerBoundWanted.IsDivisorSumFree","parent":23009,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets/index.html#is-divisor-sum-free","scope":"Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets","status":"fully_proved","summary":"A finite set $S$ of natural numbers is divisor-sum-free if no element $x \\in S$ divides the sum of the elements of any nonempty subset of $S \\setminus \\{x\\}$.","title":"Divisor-sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.DivisorSumFreeSetLowerBoundWanted.conjecture","parent":23009,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets","status":"fully_proved","summary":"The proposition that for every real $\\varepsilon > 0$ there is a natural number $N_0$ such that for every natural number $N \\ge 1$ with $N \\ge N_0$ there is a set $S \\subseteq \\{1, \\dots, N\\}$ with $|S| > N^{1/2 - \\varepsilon}$ that is divisor-sum-free: no element $x \\in S$ divides the sum of the elements of any nonempty subset of $S \\setminus \\{x\\}$.","title":"Erd\u0151s Problem 131: large divisor-sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.DivisorSumFreeSetLowerBoundWanted.conjecture_refuted","parent":23009,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets/index.html#conjecture-refuted","scope":"Combinatorics/additive-combinatorics/sum-free-sets/divisor-sum-free-sets","status":"can_prove","summary":"It is not true that for every real $\\varepsilon > 0$ there is a natural number $N_0$ such that for every natural number $N \\ge 1$ with $N \\ge N_0$ there is a set $S \\subseteq \\{1, \\dots, N\\}$ with $|S| > N^{1/2 - \\varepsilon}$ that is divisor-sum-free, meaning that no element $x \\in S$ divides the sum of the elements of any nonempty subset of $S \\setminus \\{x\\}$.","title":"Erd\u0151s Problem 131 has a negative answer"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets.json","id":"Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets","kind":"scope","navigation":"module","parent":23007,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets/index.html","scope":"Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets","status":"can_prove","title":"Large sum-free subsets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets/is-sum-free","kind":"node","lean":"MathlibExt.Combinatorics.GreenProblem1LargeSumFreeWanted.IsSumFree","parent":23013,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets/index.html#is-sum-free","scope":"Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets","status":"fully_proved","summary":"A finite set $S$ of natural numbers is sum-free if $x + y \\notin S$ for all $x, y \\in S$, where $x = y$ is allowed.","title":"Sum-free sets of natural numbers"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.GreenProblem1LargeSumFreeWanted.conjecture","parent":23013,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets","status":"fully_proved","summary":"The statement that there is a function $\\omega$ from the natural numbers to the real numbers with $\\omega(n) \\to \\infty$ as $n \\to \\infty$ such that every finite set $A$ of positive integers contains a sum-free subset $B \\subseteq A$ with $|B| \\ge |A|/3 + \\omega(|A|)$.","title":"Green's Problem 1 on large sum-free subsets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.GreenProblem1LargeSumFreeWanted.conjecture_holds","parent":23013,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/sum-free-sets/large-sum-free-subsets","status":"can_prove","summary":"There is a function $\\omega$ from the natural numbers to the real numbers with $\\omega(n) \\to \\infty$ as $n \\to \\infty$ such that every finite set $A$ of positive integers contains a subset $B \\subseteq A$ with $x + y \\notin B$ for all $x, y \\in B$ (a sum-free subset) and $|B| \\ge |A|/3 + \\omega(|A|)$.","title":"Bedert's theorem on large sum-free subsets"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets.json","id":"Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets","kind":"scope","navigation":"module","parent":23007,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets/index.html","scope":"Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets","status":"can_prove","title":"Non-averaging sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets/is-non-averaging","kind":"node","lean":"MathlibExt.Combinatorics.NonAveragingSetSizeErdosProblem186Wanted.IsNonAveraging","parent":23017,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets/index.html#is-non-averaging","scope":"Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets","status":"fully_proved","summary":"A finite set $A$ of natural numbers is non-averaging if no $a \\in A$ is the arithmetic mean of a nonempty set $S \\subseteq A \\setminus \\{a\\}$, that is, $\\sum_{s \\in S} s \\ne |S| \\cdot a$ for every $a \\in A$ and every nonempty $S \\subseteq A \\setminus \\{a\\}$.","title":"Non-averaging sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets/non-averaging-max-size-quarter-exponent","kind":"node","lean":"MathlibExt.Combinatorics.NonAveragingSetSizeErdosProblem186Wanted.nonAveragingMaxSizeQuarterExponent","parent":23017,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets/index.html#non-averaging-max-size-quarter-exponent","scope":"Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets","status":"fully_proved","summary":"For every $\\varepsilon > 0$ and every sufficiently large $N$, the largest non-averaging subset of $\\{1, \\dots, N\\}$ has size between $N^{1/4 - \\varepsilon}$ and $N^{1/4 + \\varepsilon}$: some non-averaging subset of $\\{1, \\dots, N\\}$ has at least $N^{1/4 - \\varepsilon}$ elements, and every non-averaging subset of $\\{1, \\dots, N\\}$ has at most $N^{1/4 + \\varepsilon}$ elements. In short, the largest non-averaging\u2026","title":"Erd\u0151s Problem 186 on non-averaging sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets/non-averaging-max-size-quarter-exponent-2","kind":"node","lean":"MathlibExt.Combinatorics.NonAveragingSetSizeErdosProblem186Wanted.nonAveragingMaxSizeQuarterExponent_holds","parent":23017,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets/index.html#non-averaging-max-size-quarter-exponent-2","scope":"Combinatorics/additive-combinatorics/sum-free-sets/non-averaging-sets","status":"can_prove","summary":"For every $\\varepsilon > 0$ and every sufficiently large $N$, some non-averaging subset of $\\{1, \\dots, N\\}$ has at least $N^{1/4 - \\varepsilon}$ elements, and every non-averaging subset of $\\{1, \\dots, N\\}$ has at most $N^{1/4 + \\varepsilon}$ elements. Thus the largest non-averaging subset of $\\{1, \\dots, N\\}$ has size $N^{1/4 + o(1)}$.","title":"Pham\u2013Zakharov theorem on non-averaging sets"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets.json","id":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets","kind":"scope","navigation":"module","parent":23007,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/index.html","scope":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets","status":"can_prove","title":"Number of sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/maximal-sum-free-count","kind":"node","lean":"MathlibExt.Combinatorics.MaximalSumFreeSetsCountErdosProblem877Wanted.maximalSumFreeCount","parent":23021,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of maximal sum-free subsets of "],["math","\\(\\{1, \\dots, n\\}\\)"]],"source":"Number of maximal sum-free subsets of $\\{1, \\dots, n\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/index.html#maximal-sum-free-count","scope":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets","status":"fully_proved","summary":"For a natural number $n$, $f_m(n)$ is the number of sets $A \\subseteq \\{1, \\dots, n\\}$ that are sum-free, meaning $x + y \\notin A$ for all $x, y \\in A$ (including $x = y$), and maximal with this property: every sum-free set $B$ with $A \\subseteq B \\subseteq \\{1, \\dots, n\\}$ equals $A$.","title":"Number of maximal sum-free subsets of $\\{1, \\dots, n\\}$"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/maximal-sum-free-count-small","kind":"node","lean":"MathlibExt.Combinatorics.MaximalSumFreeSetsCountErdosProblem877Wanted.maximalSumFreeCountSmall","parent":23021,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/index.html#maximal-sum-free-count-small","scope":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets","status":"fully_proved","summary":"Let $f_m(n)$ be the number of maximal sum-free subsets of $\\{1, \\dots, n\\}$. This is the statement, asked by Cameron and Erd\u0151s, that $f_m(n) / 2^{n/2} \\to 0$ as $n \\to \\infty$, that is, $f_m(n) = o(2^{n/2})$.","title":"Erd\u0151s Problem 877 on maximal sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/maximal-sum-free-count-small-holds","kind":"node","lean":"MathlibExt.Combinatorics.MaximalSumFreeSetsCountErdosProblem877Wanted.maximalSumFreeCountSmall_holds","parent":23021,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal sum-free subsets of "],["math","\\(\\{1, \\dots, n\\}\\)"],["text"," number "],["math","\\(o(2^{n/2})\\)"]],"source":"Maximal sum-free subsets of $\\{1, \\dots, n\\}$ number $o(2^{n/2})$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/index.html#maximal-sum-free-count-small-holds","scope":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets","status":"can_prove","summary":"Let $f_m(n)$ be the number of maximal sum-free subsets of $\\{1, \\dots, n\\}$. Then $f_m(n) / 2^{n/2} \\to 0$ as $n \\to \\infty$, that is, $f_m(n) = o(2^{n/2})$.","title":"Maximal sum-free subsets of $\\{1, \\dots, n\\}$ number $o(2^{n/2})$"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/sum-free-count","kind":"node","lean":"MathlibExt.Combinatorics.SumFreeSubsetCountErdosProblem748Wanted.sumFreeCount","parent":23021,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of sum-free subsets of "],["math","\\(\\{1, \\dots, n\\}\\)"]],"source":"Number of sum-free subsets of $\\{1, \\dots, n\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/index.html#sum-free-count","scope":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets","status":"fully_proved","summary":"For a natural number $n$, $f(n)$ is the number of sets $A \\subseteq \\{1, \\dots, n\\}$ that are sum-free, meaning that there are no $a, b, c \\in A$ with $a = b + c$ (the case $b = c$ included). The empty set is counted.","title":"Number of sum-free subsets of $\\{1, \\dots, n\\}$"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/sum-free-count-exponent","kind":"node","lean":"MathlibExt.Combinatorics.SumFreeSubsetCountErdosProblem748Wanted.sumFreeCountExponent","parent":23021,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/index.html#sum-free-count-exponent","scope":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets","status":"fully_proved","summary":"Let $f(n)$ be the number of sum-free subsets of $\\{1, \\dots, n\\}$. This is the statement that $\\log_2 f(n) / n \\to 1/2$ as $n \\to \\infty$, that is, $f(n) = 2^{(1 + o(1)) n / 2}$.","title":"Erd\u0151s Problem 748 on the number of sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/sum-free-count-exponent-holds","kind":"node","lean":"MathlibExt.Combinatorics.SumFreeSubsetCountErdosProblem748Wanted.sumFreeCountExponent_holds","parent":23021,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum-free subsets of "],["math","\\(\\{1, \\dots, n\\}\\)"],["text"," number "],["math","\\(2^{(1 + o(1)) n / 2}\\)"]],"source":"Sum-free subsets of $\\{1, \\dots, n\\}$ number $2^{(1 + o(1)) n / 2}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets/index.html#sum-free-count-exponent-holds","scope":"Combinatorics/additive-combinatorics/sum-free-sets/number-of-sum-free-sets","status":"can_prove","summary":"Let $f(n)$ be the number of sum-free subsets of $\\{1, \\dots, n\\}$, that is, of sets $A \\subseteq \\{1, \\dots, n\\}$ with no $a, b, c \\in A$ satisfying $a = b + c$ (the case $b = c$ included). Then $\\log_2 f(n) / n \\to 1/2$ as $n \\to \\infty$, that is, $f(n) = 2^{(1 + o(1)) n / 2}$.","title":"Sum-free subsets of $\\{1, \\dots, n\\}$ number $2^{(1 + o(1)) n / 2}$"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets.json","id":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets","kind":"scope","navigation":"module","parent":23007,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/index.html","scope":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets","status":"can_prove","title":"Square-sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/is-square-sum-free","kind":"node","lean":"MathlibExt.Combinatorics.SquareSumFreeSetsErdosProblem438Wanted.IsSquareSumFree","parent":23028,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/index.html#is-square-sum-free","scope":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets","status":"fully_proved","summary":"A finite set $A$ of natural numbers is square-sum-free if for all $a, b \\in A$, not necessarily distinct, the sum $a + b$ is not a perfect square.","title":"Square-sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/max-square-sum-free-size","kind":"node","lean":"MathlibExt.Combinatorics.SquareSumFreeSetsErdosProblem438Wanted.maxSquareSumFreeSize","parent":23028,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/index.html#max-square-sum-free-size","scope":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets","status":"fully_proved","summary":"For a natural number $N$, the largest size of a set $A \\subseteq \\{1, \\dots, N\\}$ such that no sum $a + b$ with $a, b \\in A$, not necessarily distinct, is a perfect square.","title":"Maximum size of a square-sum-free set"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/max-square-sum-free-density","kind":"node","lean":"MathlibExt.Combinatorics.SquareSumFreeSetsErdosProblem438Wanted.maxSquareSumFreeDensity","parent":23028,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/index.html#max-square-sum-free-density","scope":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets","status":"fully_proved","summary":"For a natural number $N$, let $M(N)$ be the largest size of a set $A \\subseteq \\{1, \\dots, N\\}$ such that no sum $a + b$ with $a, b \\in A$, not necessarily distinct, is a perfect square. This is the statement that $M(N) / N \\to 11/32$ as $N \\to \\infty$.","title":"Erd\u0151s Problem 438 (statement)"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/max-square-sum-free-density-holds","kind":"node","lean":"MathlibExt.Combinatorics.SquareSumFreeSetsErdosProblem438Wanted.maxSquareSumFreeDensity_holds","parent":23028,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets/index.html#max-square-sum-free-density-holds","scope":"Combinatorics/additive-combinatorics/sum-free-sets/square-sum-free-sets","status":"can_prove","summary":"For a natural number $N$, let $M(N)$ be the largest size of a set $A \\subseteq \\{1, \\dots, N\\}$ such that no sum $a + b$ with $a, b \\in A$, not necessarily distinct, is a perfect square. Then $M(N) / N \\to 11/32$ as $N \\to \\infty$.","title":"Maximal density of square-sum-free sets (Erd\u0151s Problem 438)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups.json","id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","kind":"scope","navigation":"module","parent":23007,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","title":"Sum-free sets in abelian groups"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/is-sum-free-set","kind":"node","lean":"MetaMathlibExt.IsSumFreeSet","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#is-sum-free-set","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a set with an addition. A subset $S \\subseteq G$ is sum-free if $x + y \\notin S$ for all $x, y \\in S$, where $x = y$ is allowed; that is, $S$ contains no Schur triple $x + y = z$ with $x, y, z \\in S$.","title":"Sum-free set"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/is-distinct-sum-free-set","kind":"node","lean":"MetaMathlibExt.IsDistinctSumFreeSet","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#is-distinct-sum-free-set","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a set with an addition. A subset $S \\subseteq G$ is distinct sum-free if there are no pairwise distinct $x, y, z \\in S$ with $x + y = z$, that is, if $S$ contains no distinct Schur triple.","title":"Distinct sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/is-distinct-sum-free-set-2","kind":"node","lean":"MetaMathlibExt.IsSumFreeSet.isDistinctSumFreeSet","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#is-distinct-sum-free-set-2","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"For every set $G$ with an addition, every sum-free subset of $G$ is distinct sum-free.","title":"Sum-free sets are distinct sum-free"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/is-sum-free-set-empty","kind":"node","lean":"MetaMathlibExt.isSumFreeSet_empty","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#is-sum-free-set-empty","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"For every set $G$ with an addition, the empty subset of $G$ is sum-free.","title":"The empty set is sum-free"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/largest-sum-free-card","kind":"node","lean":"MetaMathlibExt.largestSumFreeCard","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#largest-sum-free-card","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a set with an addition and $S \\subseteq G$. Write $\\mu(S)$ for the supremum in $\\mathbb{N}$ of the sizes of the finite sum-free subsets of $S$ (sets $T$ with $x + y \\notin T$ for all $x, y \\in T$), taken to be $0$ if these sizes are unbounded. For finite $S$ this is the largest size of a sum-free subset of $S$, and $\\mu(G)$ is the value for $S = G$.","title":"Size of a largest sum-free subset"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/largest-sum-free-card-le","kind":"node","lean":"MetaMathlibExt.largestSumFreeCard_le","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#largest-sum-free-card-le","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a set with an addition. For every finite subset $S \\subseteq G$, the largest size of a sum-free subset of $S$ is at most $|S|$.","title":"Trivial bound on sum-free subsets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-sum-free-count","kind":"node","lean":"MetaMathlibExt.maxSumFreeCount","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-sum-free-count","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"For a set $G$ with an addition, $f_{\\max}(G)$ is the number of maximal sum-free subsets of $G$, that is, of sets $S \\subseteq G$ with $x + y \\notin S$ for all $x, y \\in S$ (including $x = y$) that are not properly contained in another such set; it is $0$ if there are infinitely many.","title":"Number of maximal sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-distinct-sum-free-count","kind":"node","lean":"MetaMathlibExt.maxDistinctSumFreeCount","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-distinct-sum-free-count","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a set with an addition. The number $f^*_{\\max}(G)$ is the number of subsets of $G$ that are maximal, with respect to inclusion, among the distinct sum-free subsets of $G$, and it is $0$ if there are infinitely many such subsets. Here $S \\subseteq G$ is distinct sum-free if $x + y \\notin S$ for all $x, y \\in S$ such that $x$, $y$ and $x + y$ are pairwise distinct.","title":"Number of maximal distinct sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/distinct-schur-triple-count","kind":"node","lean":"MetaMathlibExt.distinctSchurTripleCount","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#distinct-schur-triple-count","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"For a subset $F$ of a set $G$ with an addition, this is the number of ordered pairs $(x, y)$ with $x, y \\in F$ such that $x$, $y$ and $x + y$ are pairwise distinct and $x + y \\in F$, that is, the number of ordered distinct Schur triples $(x, y, x + y)$ in $F$. If there are infinitely many such pairs, the value is $0$ by convention.","title":"Number of distinct Schur triples"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/is-type-one-group","kind":"node","lean":"MetaMathlibExt.IsTypeOneGroup","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#is-type-one-group","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite set and $p$ a natural number. $G$ is of type $\\mathrm{I}(p)$ if $p$ is the smallest prime with $p \\equiv 2 \\pmod 3$ dividing $|G|$: that is, $p$ is prime, $p \\equiv 2 \\pmod 3$, $p$ divides $|G|$, and $p \\le q$ for every prime $q$ with $q \\equiv 2 \\pmod 3$ that divides $|G|$.","title":"Groups of type I(p)"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/is-type-one-group-iff","kind":"node","lean":"MetaMathlibExt.isTypeOneGroup_iff","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#is-type-one-group-iff","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite set and $p$ a natural number. Then $G$ is of type $\\mathrm{I}(p)$ if and only if $p$ is prime, $p \\equiv 2 \\pmod 3$, $p$ divides $|G|$, and $p \\le q$ for every prime $q$ with $q \\equiv 2 \\pmod 3$ that divides $|G|$.","title":"Characterization of type I(p)"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/largest-sum-free-card-univ-eq","kind":"node","lean":"MetaMathlibExt.largestSumFreeCard_univ_eq","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#largest-sum-free-card-univ-eq","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $G$ be a finite abelian group of order $n$, and let $\\mu(G)$ be the largest size of a sum-free subset of $G$. If $G$ is of type $\\mathrm{I}(p)$ for a natural number $p$, then $\\mu(G) = n\\left(\\frac{1}{3} + \\frac{1}{3p}\\right)$. If no prime $q \\equiv 2 \\pmod 3$ divides $n$ and $3 \\mid n$ (type $\\mathrm{II}$), then $\\mu(G) = n/3$. If no prime $q \\equiv 2 \\pmod 3$ divides $n$ and $3 \\nmid n$ (type $\\mathrm{III}$),\u2026","title":"Diananda\u2013Yap and Green\u2013Ruzsa theorem"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-indep-set-count","kind":"node","lean":"MetaMathlibExt.maxIndepSetCount","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-indep-set-count","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"For a graph $\\Gamma$, $\\mathrm{mis}(\\Gamma)$ is the number of maximal independent sets of $\\Gamma$, that is, of sets of pairwise non-adjacent vertices that are not properly contained in another such set; it is $0$ if there are infinitely many.","title":"Number of maximal independent sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/distinct-link-adj","kind":"node","lean":"MetaMathlibExt.DistinctLinkAdj","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#distinct-link-adj","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a set with an addition and $S \\subseteq G$. Two elements $x, y \\in G$ are adjacent in the distinct link graph of $S$ if $x \\ne y$ and there is some $s \\in S$, different from both $x$ and $y$, with $x + y = s$, $x + s = y$ or $y + s = x$. In an abelian group this says exactly that $\\{x, y, s\\}$ is a distinct Schur triple for some $s \\in S$. In the paper's graph $L^*_S[B]$ with vertex set $B$, these are\u2026","title":"Edges of the distinct link graph"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/distinct-link-loop","kind":"node","lean":"MetaMathlibExt.DistinctLinkLoop","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#distinct-link-loop","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a set with an addition and $S \\subseteq G$. An element $x \\in G$ carries a loop in the distinct link graph of $S$ if there are distinct $s, s' \\in S$, both different from $x$, with $s + s' = x$, $x + s = s'$ or $x + s' = s$. In an abelian group this says exactly that $\\{x, s, s'\\}$ is a distinct Schur triple for some distinct $s, s' \\in S$.","title":"Loops of the distinct link graph"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/is-distinct-link-indep","kind":"node","lean":"MetaMathlibExt.IsDistinctLinkIndep","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#is-distinct-link-indep","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"Let $G$ be a set with an addition and let $S, B, I \\subseteq G$. The set $I$ is an independent set of the distinct link graph $L^*_S[B]$ (the graph on vertex set $B$ given by the edges and loops of the distinct link graph of $S$) if $I \\subseteq B$, no element of $I$ carries a loop, and no two elements of $I$ are adjacent.","title":"Independent sets of the distinct link graph"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/is-distinct-link-indep-empty","kind":"node","lean":"MetaMathlibExt.isDistinctLinkIndep_empty","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#is-distinct-link-indep-empty","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"fully_proved","summary":"For every set $G$ with an addition and all $S, B \\subseteq G$, the empty set is an independent set of the distinct link graph $L^*_S[B]$.","title":"The empty set is independent in the distinct link graph"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/exists-distinct-sum-free-containers","kind":"node","lean":"MetaMathlibExt.exists_distinct_sumFree_containers","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#exists-distinct-sum-free-containers","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"For every real $\\delta > 0$ there is a natural number $n_0$ such that every finite abelian group $G$ of order $n \\ge n_0$ has a family $\\mathcal{F}$ of subsets of $G$ with the following properties: (i) each $F \\in \\mathcal{F}$ contains at most $\\delta n^2$ ordered distinct Schur triples $(x, y, x + y)$; (ii) every distinct sum-free subset of $G$ is contained in some $F \\in \\mathcal{F}$; (iii)\u2026","title":"Container lemma"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/exists-add-monoid-hom-zmod-of-large-sum","kind":"node","lean":"MetaMathlibExt.exists_addMonoidHom_zmod_of_large_sumFree","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#exists-add-monoid-hom-zmod-of-large-sum","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $G$ be a finite abelian group of type $\\mathrm{I}(p)$ and let $k$ be a natural number with $p = 3k + 2$. If $A \\subseteq G$ is sum-free and $|A| > \\left(\\frac{1}{3} + \\frac{1}{3(p + 1)}\\right)|G|$, then there is a group homomorphism $\\psi : G \\to \\mathbb{Z}_p$ with $\\psi(a) \\in \\{k + 1, \\dots, 2k + 1\\}$ for every $a \\in A$, where $\\mathbb{Z}_p$ is identified with $\\{0, 1, \\dots, p - 1\\}$.","title":"Green\u2013Ruzsa structure lemma"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/ncard-add-subgroup-card-eq-ncard-index-e","kind":"node","lean":"MetaMathlibExt.ncard_addSubgroup_card_eq_ncard_index_eq","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#ncard-add-subgroup-card-eq-ncard-index-e","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"In a finite abelian group $G$, for every natural number $k$ the number of subgroups of order $k$ equals the number of subgroups of index $k$.","title":"Counting subgroups by order and by index"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/ncard-index-two-add-subgroup-rank","kind":"node","lean":"MetaMathlibExt.ncard_index_two_addSubgroup_rank","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#ncard-index-two-add-subgroup-rank","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $r_1, r_2$ be natural numbers with $r = r_1 + r_2 \\ge 1$, let $\\alpha_1, \\dots, \\alpha_{r_1} \\ge 2$, and let $G = \\mathbb{Z}_{2^{\\alpha_1}} \\oplus \\dots \\oplus \\mathbb{Z}_{2^{\\alpha_{r_1}}} \\oplus \\mathbb{Z}_2^{r_2}$. Then exactly $2^{r_1} - 1$ subgroups of $G$ of index $2$ have rank $r$, and exactly $2^r - 2^{r_1}$ subgroups of $G$ of index $2$ have rank $r - 1$, where the rank of a group is the minimal number\u2026","title":"Ranks of the index-two subgroups of a finite abelian 2-group"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-indep-set-count-le-of-max-degree-le","kind":"node","lean":"MetaMathlibExt.maxIndepSetCount_le_of_maxDegree_le","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-indep-set-count-le-of-max-degree-le","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $k \\ge 1$ be a real number and let $\\Gamma$ be a finite graph with $n$ vertices, minimum degree $\\delta(\\Gamma)$ and maximum degree $\\Delta(\\Gamma) \\le k\\,\\delta(\\Gamma)$, and put $b = \\sqrt{\\delta(\\Gamma)}$. Then the number $\\mathrm{mis}(\\Gamma)$ of maximal independent sets of $\\Gamma$ satisfies\u2026","title":"Balogh\u2013Liu\u2013Sharifzadeh\u2013Treglown bound"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-indep-set-count-le-of-almost-triangl","kind":"node","lean":"MetaMathlibExt.maxIndepSetCount_le_of_almost_triangleFree","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-indep-set-count-le-of-almost-triangl","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $\\Gamma$ be a finite graph, let $T$ be a set of vertices of $\\Gamma$, let $n \\ge 1$ and $D \\ge 1$ be integers, and let $k$ be a real number. Suppose that the graph $\\Gamma'$ obtained from $\\Gamma$ by deleting $T$ is triangle-free, has $n$ vertices and has at least $n/2 + k$ edges, and that $\\Gamma$ has maximum degree $\\Delta(\\Gamma) \\le D$. Then the number $\\mathrm{mis}(\\Gamma)$ of maximal independent sets of\u2026","title":"Almost triangle-free bound"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-indep-set-count-le-of-edge-count","kind":"node","lean":"MetaMathlibExt.maxIndepSetCount_le_of_edge_count","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-indep-set-count-le-of-edge-count","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $\\Gamma$ be a graph with $n$ vertices, $n + k$ edges for an integer $k$, and maximum degree $\\Delta \\ge 1$. Then the number $\\mathrm{mis}(\\Gamma)$ of maximal independent sets of $\\Gamma$ satisfies $\\mathrm{mis}(\\Gamma) \\le 3^{\\Delta/13} \\cdot 3^{n/3 - k/(13\\Delta)}$.","title":"Liu\u2013Sharifzadeh stability bound"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/maximal-is-distinct-link-indep-of-maxima","kind":"node","lean":"MetaMathlibExt.maximal_isDistinctLinkIndep_of_maximal","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#maximal-is-distinct-link-indep-of-maxima","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $G$ be an abelian group and let $B, S \\subseteq G$ be disjoint distinct sum-free sets. If $I \\subseteq B$ and $S \\cup I$ is a maximal distinct sum-free subset of $G$, then $I$ is a maximal independent set of the distinct link graph $L^*_S[B]$: a subset of $B$ with no looped vertex and no two adjacent vertices that is not properly contained in another such set.","title":"Link-graph extension lemma"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/two-rpow-le-max-distinct-sum-free-count","kind":"node","lean":"MetaMathlibExt.two_rpow_le_maxDistinctSumFreeCount_of_typeOne","parent":23033,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal distinct sum-free sets in groups of type "],["math","\\(I(p)\\)"]],"source":"Maximal distinct sum-free sets in groups of type $I(p)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#two-rpow-le-max-distinct-sum-free-count","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $G$ be a finite abelian group and let $p \\ge 5$ be such that $G$ is of type $I(p)$, that is, $p$ is the smallest prime with $p \\equiv 2 \\pmod 3$ that divides $|G|$. Call $S \\subseteq G$ sum-free if $x + y \\notin S$ for all $x, y \\in S$, and distinct sum-free if $x + y \\notin S$ for all $x, y \\in S$ such that $x$, $y$ and $x + y$ are pairwise distinct. Let $\\mu(G)$ be the size of a largest sum-free subset of\u2026","title":"Maximal distinct sum-free sets in groups of type $I(p)$"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/tendsto-prob-largest-sum-free-card-rando","kind":"node","lean":"MetaMathlibExt.tendsto_prob_largestSumFreeCard_random","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#tendsto-prob-largest-sum-free-card-rando","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"For a finite abelian group $G$ let $\\mu(G)$ be the size of a largest sum-free subset of $G$. Let $(G_n)_{n \\ge 0}$ be finite abelian groups with $|G_i| < |G_j|$ and $\\mu(G_i)/|G_i| = \\mu(G_j)/|G_j|$ whenever $i < j$. Then for every $\\varepsilon > 0$ there is $C > 0$ such that for every real sequence $(p_n)$ with $C |G_n|^{-1/2} \\le p_n \\le 1$ for all $n$, the probability that the largest sum-free subset of the\u2026","title":"Random sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-sum-free-count-elementary-two-three","kind":"node","lean":"MetaMathlibExt.maxSumFreeCount_elementary_two_three","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-sum-free-count-elementary-two-three","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Let $f_{\\max}(G)$ be the number of maximal sum-free subsets of $G$. As $k \\to \\infty$, with $n = 2^k$, $f_{\\max}(\\mathbb{Z}_2^k) = \\bigl(\\binom{n-1}{2} + o(1)\\bigr) 2^{n/4}$, and with $n = 3^k$, $f_{\\max}(\\mathbb{Z}_3^k) = \\bigl((n-3)(n-1)/3 + o(1)\\bigr) 3^{n/9}$; that is, $f_{\\max}(\\mathbb{Z}_2^k)/2^{n/4} - \\binom{n-1}{2} \\to 0$ and $f_{\\max}(\\mathbb{Z}_3^k)/3^{n/9} - (n-3)(n-1)/3 \\to 0$.","title":"Counting maximal sum-free sets in elementary abelian 2- and 3-groups"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-distinct-sum-free-count-even-order","kind":"node","lean":"MetaMathlibExt.maxDistinctSumFreeCount_even_order","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-distinct-sum-free-count-even-order","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Write $f^*_{\\max}(G)$ for the number of maximal distinct sum-free subsets of an abelian group $G$. There is a natural number $C$ such that for every real $\\varepsilon > 0$ there is $n_0$ with the following property. Let $r \\ge 1$, let $\\alpha_1, \\dots, \\alpha_r \\ge 1$, let $K$ be a finite abelian group of odd order, and let $G = \\mathbb{Z}_{2^{\\alpha_1}} \\oplus \\dots \\oplus \\mathbb{Z}_{2^{\\alpha_r}} \\oplus K$ have\u2026","title":"Maximal distinct sum-free sets in groups of even order"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/max-distinct-sum-free-count-le-sharp","kind":"node","lean":"MetaMathlibExt.maxDistinctSumFreeCount_le_sharp","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#max-distinct-sum-free-count-le-sharp","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Write $f^*_{\\max}(G)$ for the number of maximal distinct sum-free subsets of an abelian group $G$. There is a natural number $C$ such that for every real $\\eta > 0$ there is $n_0$ with the following property. Let $r_1, r_2 \\ge 0$ with $r = r_1 + r_2 \\ge 1$, let $\\alpha_1, \\dots, \\alpha_{r_1} \\ge 2$, let $K$ be a finite abelian group of odd order, and let\u2026","title":"Sharp upper bound on maximal distinct sum-free sets"},{"id":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/le-max-distinct-sum-free-count-sharp","kind":"node","lean":"MetaMathlibExt.le_maxDistinctSumFreeCount_sharp","parent":23033,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups/index.html#le-max-distinct-sum-free-count-sharp","scope":"Combinatorics/additive-combinatorics/sum-free-sets/sum-free-sets-in-abelian-groups","status":"can_prove","summary":"Write $f^*_{\\max}(G)$ for the number of maximal distinct sum-free subsets of an abelian group $G$. There is a natural number $C$ such that for every real $\\eta > 0$ there is $n_0$ with the following property. Let $r_1, r_2 \\ge 0$ with $r = r_1 + r_2 \\ge 1$, let $\\alpha_1, \\dots, \\alpha_{r_1} \\ge 2$, let $K$ be a finite abelian group of odd order, and let\u2026","title":"Sharp lower bound on maximal distinct sum-free sets"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases.json","id":"Combinatorics/additive-combinatorics/additive-bases","kind":"scope","parent":22582,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases","status":"can_prove","summary":"Sets whose h-fold sums cover the integers or a group: asymptotic, exact and restricted bases, Schnirelmann density, Rohrbach's theorem, representation counts and convolutions, sums of cubes covering vector spaces, knight-move bases of the plane, and sumsets of infinite sets of positive or zero density.","title":"Additive bases"},{"id":"Combinatorics/additive-combinatorics/additive-bases/exists-add-eq-of-image-sub-eq-self","kind":"node","lean":"Finset.exists_add_eq_of_image_sub_eq_self","parent":23065,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/index.html#exists-add-eq-of-image-sub-eq-self","scope":"Combinatorics/additive-combinatorics/additive-bases","status":"fully_proved","summary":"Let $A$ be a finite set of natural numbers and $a$ a natural number with $\\{a - x : x \\in A\\} = A$, where $a - x$ denotes truncated subtraction ($a - x = 0$ when $x > a$). If every natural number $m \\le a$ is a sum $x + y$ of two elements $x, y \\in A$, then every natural number $m \\le 2a$ is such a sum.","title":"Rohrbach's symmetric basis theorem"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/additive-complements.json","id":"Combinatorics/additive-combinatorics/additive-bases/additive-complements","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/additive-complements/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/additive-complements","status":"can_prove","title":"Additive complements"},{"id":"Combinatorics/additive-combinatorics/additive-bases/additive-complements/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem477Wanted.conjecture","parent":23067,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/additive-complements/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/additive-bases/additive-complements","status":"fully_proved","summary":"The statement that there are a polynomial $f$ with integer coefficients of degree at least $2$ and a set $A$ of integers such that for every integer $n$ there is exactly one pair $(a, b)$ with $a \\in A$, $b = f(k)$ for some integer $k$, and $n = a + b$. Only the pair is required to be unique, not the integer $k$.","title":"Erd\u0151s Problem 477"},{"id":"Combinatorics/additive-combinatorics/additive-bases/additive-complements/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem477Wanted.conjecture_holds","parent":23067,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/additive-complements/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/additive-bases/additive-complements","status":"can_prove","summary":"There are a polynomial $f$ with integer coefficients of degree at least $2$ and a set $A$ of integers such that for every integer $n$ there is exactly one pair $(a, b)$ with $a \\in A$, $b = f(k)$ for some integer $k$, and $n = a + b$.","title":"Erd\u0151s Problem 477 has a positive answer"},{"id":"Combinatorics/additive-combinatorics/additive-bases/additive-complements/conjecture-holds-2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem31Wanted.conjecture_holds","parent":23067,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/additive-complements/index.html#conjecture-holds-2","scope":"Combinatorics/additive-combinatorics/additive-bases/additive-complements","status":"can_prove","summary":"For every infinite set $A$ of natural numbers there is a set $B$ of natural numbers of natural density $0$ such that every sufficiently large natural number lies in $A + B = \\{a + b \\mid a \\in A,\\ b \\in B\\}$.","title":"Lorentz's theorem on additive complements (Erd\u0151s Problem 31)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions.json","id":"Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions","status":"fully_proved","title":"Additive representation functions"},{"id":"Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions/sum-conv","kind":"node","lean":"AdditiveCombinatorics.sumConv","parent":23071,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions/index.html#sum-conv","scope":"Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions","status":"fully_proved","summary":"Let $R$ be a semiring. The sum convolution, or Cauchy product, of two functions $f, g \\colon \\mathbb{N} \\to R$ is the function $\\mathbb{N} \\to R$ whose value at $n \\in \\mathbb{N}$ is","title":"Sum convolution"},{"id":"Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions/ordered-representation-count","kind":"node","lean":"AdditiveCombinatorics.orderedRepresentationCount","parent":23071,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions/index.html#ordered-representation-count","scope":"Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions","status":"fully_proved","summary":"For a set $A$ of natural numbers and a natural number $n$, the ordered representation count of $n$ by $A$ is the number of ordered pairs $(a, b) \\in A \\times A$ with $a + b = n$. Pairs are not identified under swapping: when $a \\ne b$, the pairs $(a, b)$ and $(b, a)$ are counted separately.","title":"Ordered representation count"},{"id":"Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions/ordered-representation-count-pos-iff","kind":"node","lean":"AdditiveCombinatorics.orderedRepresentationCount_pos_iff","parent":23071,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions/index.html#ordered-representation-count-pos-iff","scope":"Combinatorics/additive-combinatorics/additive-bases/additive-representation-functions","status":"fully_proved","summary":"For a set $A$ of natural numbers and a natural number $n$, the number of ordered pairs $(a, b) \\in A \\times A$ with $a + b = n$ is positive if and only if $n = a + b$ for some $a \\in A$ and $b \\in A$.","title":"Positivity of the ordered representation count"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases.json","id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"can_prove","title":"Asymptotic additive bases"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/is-asymptotic-add-basis-of-order","kind":"node","lean":"Set.IsAsymptoticAddBasisOfOrder","parent":23075,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic additive basis of order "],["math","\\(n\\)"]],"source":"Asymptotic additive basis of order $n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#is-asymptotic-add-basis-of-order","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, $A \\subseteq M$ and $n$ a natural number. $A$ is an asymptotic additive basis of order $n$ if all but finitely many elements of $M$ are sums of exactly $n$ (not necessarily distinct) elements of $A$, that is, $M \\setminus nA$ is finite, where $nA = \\{a_1 + \\dots + a_n : a_1, \\dots, a_n \\in A\\}$ and $0A = \\{0\\}$.","title":"Asymptotic additive basis of order $n$"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/is-add-basis-of-order","kind":"node","lean":"Set.IsAddBasisOfOrder","parent":23075,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exact additive basis of order "],["math","\\(n\\)"]],"source":"Exact additive basis of order $n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#is-add-basis-of-order","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, $A \\subseteq M$ and $n$ a natural number. $A$ is an exact additive basis of order $n$ if every element of $M$ is a sum of exactly $n$ (not necessarily distinct) elements of $A$, that is, $M = nA$, where $nA = \\{a_1 + \\dots + a_n : a_1, \\dots, a_n \\in A\\}$ and $0A = \\{0\\}$.","title":"Exact additive basis of order $n$"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/is-asymptotic-add-basis","kind":"node","lean":"Set.IsAsymptoticAddBasis","parent":23075,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#is-asymptotic-add-basis","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"A subset $A$ of an additive commutative monoid $M$ is an asymptotic additive basis if there is a natural number $n$ such that $A$ is an asymptotic additive basis of order $n$, that is, all but finitely many elements of $M$ are sums of exactly $n$ (not necessarily distinct) elements of $A$.","title":"Asymptotic additive basis"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/is-weak-add-basis-of-order","kind":"node","lean":"Set.IsWeakAddBasisOfOrder","parent":23075,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weak additive basis of order "],["math","\\(n\\)"]],"source":"Weak additive basis of order $n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#is-weak-add-basis-of-order","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, $A \\subseteq M$ and $n$ a natural number. $A$ is a weak additive basis of order $n$ if every element of $M$ is a sum of at most $n$ (not necessarily distinct) elements of $A$, that is, lies in $mA$ for some $m \\le n$, where $mA = \\{a_1 + \\dots + a_m : a_1, \\dots, a_m \\in A\\}$ and $0A = \\{0\\}$.","title":"Weak additive basis of order $n$"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/is-weak-add-basis","kind":"node","lean":"Set.IsWeakAddBasis","parent":23075,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#is-weak-add-basis","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"A subset $A$ of an additive commutative monoid $M$ is a weak additive basis if there is a natural number $n$ such that $A$ is a weak additive basis of order $n$, that is, every element of $M$ is a sum of at most $n$ (not necessarily distinct) elements of $A$.","title":"Weak additive basis"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/add-sumset","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem869Wanted.addSumset","parent":23075,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#add-sumset","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"For a set $A$ of natural numbers, $A + A$ is the set of natural numbers $n$ such that $n = a + b$ for some $a, b \\in A$.","title":"Sumset of a set of naturals"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/is-additive-basis-order2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem869Wanted.IsAdditiveBasisOrder2","parent":23075,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#is-additive-basis-order2","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"A set $A$ of natural numbers is an additive basis of order $2$ if there is a natural number $N$ such that every natural number $n \\ge N$ can be written as $n = a + b$ with $a, b \\in A$.","title":"Additive basis of order 2"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/is-minimal-additive-basis-order2","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem869Wanted.IsMinimalAdditiveBasisOrder2","parent":23075,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#is-minimal-additive-basis-order2","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"A set $A$ of natural numbers is a minimal additive basis of order $2$ if it is an additive basis of order $2$ and, for every $a \\in A$, infinitely many natural numbers do not lie in the sumset $(A \\setminus \\{a\\}) + (A \\setminus \\{a\\})$.","title":"Minimal additive basis of order 2"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/statement","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem869Wanted.Statement","parent":23075,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#statement","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"fully_proved","summary":"The statement that for all disjoint sets $A_1$ and $A_2$ of natural numbers that are both additive bases of order $2$, there is a set $B \\subseteq A_1 \\cup A_2$ that is a minimal additive basis of order $2$.","title":"Erd\u0151s Problem 869 (disjoint bases and minimal bases)"},{"id":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/statement-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem869Wanted.Statement_refuted","parent":23075,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases/index.html#statement-refuted","scope":"Combinatorics/additive-combinatorics/additive-bases/asymptotic-additive-bases","status":"can_prove","summary":"There are disjoint sets $A_1$ and $A_2$ of natural numbers, both additive bases of order $2$, such that no subset of $A_1 \\cup A_2$ is a minimal additive basis of order $2$.","title":"Erd\u0151s Problem 869 is false (Larsen)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences.json","id":"Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences","status":"can_prove","title":"Bounded-gap sumsets avoiding lacunary sequences"},{"id":"Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences/has-lacunary-threshold","kind":"node","lean":"MathlibExt.Combinatorics.LacunaryAvoidingBoundedGapSumsetErdosProblem1112Wanted.HasLacunaryThreshold","parent":23086,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences/index.html#has-lacunary-threshold","scope":"Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences","status":"fully_proved","summary":"For natural numbers $d_1$, $d_2$ and $k$, the threshold $r_k(d_1, d_2)$ of Erd\u0151s Problem 1112 exists when there is a natural number $r$ such that for every strictly increasing sequence $b_0 < b_1 < \\cdots$ of positive integers with $b_{i+1} \\ge r b_i$ for all $i$, there is a sequence $a_0, a_1, \\dots$ of positive integers with $a_i + d_1 \\le a_{i+1} \\le a_i + d_2$ for all $i$ such that no $b_i$ lies in the $k$-fold\u2026","title":"Lacunary threshold for bounded-gap sumsets"},{"id":"Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences/lacunary-threshold-gaps-two-three-three","kind":"node","lean":"MathlibExt.Combinatorics.LacunaryAvoidingBoundedGapSumsetErdosProblem1112Wanted.lacunaryThresholdGapsTwoThreeThreeFold","parent":23086,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of Erd\u0151s Problem 1112 for gaps in "],["math","\\((2, 3)\\)"],["text"," and triple sums"]],"source":"Statement of Erd\u0151s Problem 1112 for gaps in $(2, 3)$ and triple sums"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences/index.html#lacunary-threshold-gaps-two-three-three","scope":"Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences","status":"fully_proved","summary":"The threshold $r_3(2, 3)$ exists: there is a natural number $r$ such that for every strictly increasing sequence $b_0 < b_1 < \\cdots$ of positive integers with $b_{i+1} \\ge r b_i$ for all $i$, there is a sequence $a_0, a_1, \\dots$ of positive integers with $a_i + 2 \\le a_{i+1} \\le a_i + 3$ for all $i$ such that no $b_i$ is a sum $x + y + z$ of three, not necessarily distinct, terms $x, y, z$ of the sequence $a$.","title":"Statement of Erd\u0151s Problem 1112 for gaps in $(2, 3)$ and triple sums"},{"id":"Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences/lacunary-threshold-gaps-two-three-three-2","kind":"node","lean":"MathlibExt.Combinatorics.LacunaryAvoidingBoundedGapSumsetErdosProblem1112Wanted.lacunaryThresholdGapsTwoThreeThreeFold_refuted","parent":23086,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences/index.html#lacunary-threshold-gaps-two-three-three-2","scope":"Combinatorics/additive-combinatorics/additive-bases/bounded-gap-sumsets-avoiding-lacunary-sequences","status":"can_prove","summary":"There is no natural number $r$ such that for every strictly increasing sequence $b_0 < b_1 < \\cdots$ of positive integers with $b_{i+1} \\ge r b_i$ for all $i$, there is a sequence $a_0, a_1, \\dots$ of positive integers with $a_i + 2 \\le a_{i+1} \\le a_i + 3$ for all $i$ such that no $b_i$ is a sum of three, not necessarily distinct, terms of the sequence $a$. In other words, the threshold $r_3(2, 3)$ of Erd\u0151s\u2026","title":"Bollob\u00e1s\u2013Hegyv\u00e1ri\u2013Jin theorem on Erd\u0151s Problem 1112"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture.json","id":"Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture","status":"can_prove","title":"Erd\u0151s sumset conjecture"},{"id":"Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture/sumset-in-positive-upper-density","kind":"node","lean":"MathlibExt.Combinatorics.SumsetInPositiveDensityErdosProblem109Wanted.sumsetInPositiveUpperDensity","parent":23090,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture/index.html#sumset-in-positive-upper-density","scope":"Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture","status":"fully_proved","summary":"The upper density of a set $A \\subseteq \\mathbb{N}$ is $\\limsup_{N \\to \\infty} \\lvert A \\cap \\{0, \\dots, N - 1\\} \\rvert / N$. This is the statement that every set $A \\subseteq \\mathbb{N}$ of positive upper density contains $B + C = \\{b + c \\mid b \\in B, c \\in C\\}$ for some infinite sets $B, C \\subseteq \\mathbb{N}$.","title":"Erd\u0151s Problem 109 (the Erd\u0151s sumset conjecture)"},{"id":"Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture/sumset-in-positive-upper-density-holds","kind":"node","lean":"MathlibExt.Combinatorics.SumsetInPositiveDensityErdosProblem109Wanted.sumsetInPositiveUpperDensity_holds","parent":23090,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture/index.html#sumset-in-positive-upper-density-holds","scope":"Combinatorics/additive-combinatorics/additive-bases/erd-s-sumset-conjecture","status":"can_prove","summary":"The upper density of a set $A \\subseteq \\mathbb{N}$ is $\\limsup_{N \\to \\infty} \\lvert A \\cap \\{0, \\dots, N - 1\\} \\rvert / N$. Every set $A \\subseteq \\mathbb{N}$ of positive upper density contains $B + C = \\{b + c \\mid b \\in B, c \\in C\\}$ for some infinite sets $B, C \\subseteq \\mathbb{N}$.","title":"Moreira\u2013Richter\u2013Robertson theorem (the Erd\u0151s sumset conjecture)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice.json","id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"can_prove","title":"Knight and king distances on the integer lattice"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/knight-moves","kind":"node","lean":"MetaMathlibExt.knightMoves","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#knight-moves","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"For integers $a$ and $b$, the $(a,b)$-knight's move set is","title":"Knight moves"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/king-moves","kind":"node","lean":"MetaMathlibExt.kingMoves","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#king-moves","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"The king's move set $K \\subset \\mathbb{Z}^2$ consists of the eight unit horizontal, vertical and diagonal steps:","title":"King moves"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/piece-distance","kind":"node","lean":"MetaMathlibExt.pieceDistance","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#piece-distance","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"Let $A \\subseteq \\mathbb{Z}^2$ be a finite set of moves and, for $h \\ge 1$, let $hA = \\{a_1 + \\dots + a_h : a_1, \\dots, a_h \\in A\\}$ be its $h$-fold sumset. The piece distance of a point $p \\in \\mathbb{Z}^2$ for $A$ is the natural number that is $0$ if $p = (0,0)$ and otherwise the infimum of the set of integers $h \\ge 1$ with $p \\in hA$, where the infimum of the empty set is taken to be $0$.","title":"Piece distance on the integer lattice"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/max-norm-ball","kind":"node","lean":"MetaMathlibExt.maxNormBall","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#max-norm-ball","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"For a natural number $h$, the max-norm ball $B_h \\subset \\mathbb{Z}^2$ is the product of two copies of the integer interval $\\{-h, \\dots, h\\}$, that is, the set of lattice points $(x, y)$ with $\\max(|x|, |y|) \\le h$.","title":"Max-norm ball"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/piece-distance-eq-of-mem-of-forall","kind":"node","lean":"MetaMathlibExt.pieceDistance_eq_of_mem_of_forall","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#piece-distance-eq-of-mem-of-forall","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"Let $A \\subset \\mathbb{Z}^2$ be a finite set, let $p \\in \\mathbb{Z}^2$ be nonzero and let $n \\ge 1$ be a natural number. If $p$ lies in the $n$-fold sumset $nA = \\{a_1 + \\dots + a_n : a_1, \\dots, a_n \\in A\\}$, and every natural number $h \\ge 1$ with $p \\in hA$ satisfies $n \\le h$, then the piece distance of $p$ for $A$ is $n$.","title":"Piece distance from a minimal fold"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/piece-distance-king-moves","kind":"node","lean":"MetaMathlibExt.pieceDistance_kingMoves","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#piece-distance-king-moves","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"For every lattice point $p = (x, y) \\in \\mathbb{Z}^2$, the piece distance of $p$ for the king's move set $K$ is $\\max(|x|, |y|)$.","title":"King distance is the max norm"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/piece-distance-knight-moves-one-two","kind":"node","lean":"MetaMathlibExt.pieceDistance_knightMoves_one_two","parent":23093,"reading_title":{"format":"inline-parts-v1","parts":[["text","The knight reaches "],["math","\\((1,2)\\)"],["text"," in one move"]],"source":"The knight reaches $(1,2)$ in one move"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#piece-distance-knight-moves-one-two","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"Let $N \\subseteq \\mathbb{Z}^2$ be the move set of the usual knight, consisting of the eight vectors $(\\pm 1, \\pm 2)$ and $(\\pm 2, \\pm 1)$. The piece distance of the point $(1,2)$ for $N$ is $1$.","title":"The knight reaches $(1,2)$ in one move"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/piece-distance-knight-moves-one-zero","kind":"node","lean":"MetaMathlibExt.pieceDistance_knightMoves_one_zero","parent":23093,"reading_title":{"format":"inline-parts-v1","parts":[["text","The knight needs three moves to reach "],["math","\\((1,0)\\)"]],"source":"The knight needs three moves to reach $(1,0)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#piece-distance-knight-moves-one-zero","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"Let $N \\subseteq \\mathbb{Z}^2$ be the move set of the usual knight, consisting of the eight vectors $(\\pm 1, \\pm 2)$ and $(\\pm 2, \\pm 1)$. The piece distance of the point $(1,0)$ for $N$ is $3$.","title":"The knight needs three moves to reach $(1,0)$"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/piece-distance-knight-moves-one-one","kind":"node","lean":"MetaMathlibExt.pieceDistance_knightMoves_one_one","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#piece-distance-knight-moves-one-one","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"For the usual knight's move set $N_{1,2} = \\{(\\pm 1, \\pm 2), (\\pm 2, \\pm 1)\\}$, the piece distance of the point $(1, 1)$ is $2$.","title":"The usual knight reaches (1, 1) in two moves"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/piece-distance-knight-moves-zero-zero","kind":"node","lean":"MetaMathlibExt.pieceDistance_knightMoves_zero_zero","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#piece-distance-knight-moves-zero-zero","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"Let $N \\subseteq \\mathbb{Z}^2$ be the move set of the usual knight, consisting of the eight vectors $(\\pm 1, \\pm 2)$ and $(\\pm 2, \\pm 1)$. The piece distance of the origin $(0,0)$ for $N$ is $0$.","title":"The knight's distance to the origin is zero"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/card-max-norm-ball","kind":"node","lean":"MetaMathlibExt.card_maxNormBall","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#card-max-norm-ball","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"fully_proved","summary":"For every natural number $h$, the max-norm ball $B_h = \\{(x, y) \\in \\mathbb{Z}^2 : \\max(|x|, |y|) \\le h\\}$ has exactly $(2h+1)^2$ points.","title":"Number of points in the max-norm ball"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/knight-distance-uniform-on-box","kind":"node","lean":"MetaMathlibExt.knight_distance_uniform_on_box","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#knight-distance-uniform-on-box","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"can_prove","summary":"There is an absolute constant $M \\in \\mathbb{R}$ such that for all natural numbers $a$ and $b$ with $1 \\le a < b$, $\\gcd(a, b) = 1$ and $a + b$ odd, every lattice point $(x, y) \\in \\mathbb{Z}^2$ with $\\max(|x|, |y|) \\le a + b$ has piece distance at most $Mb$ for the $(a,b)$-knight's move set $N_{a,b} = \\{(\\pm a, \\pm b), (\\pm b, \\pm a)\\}$.","title":"Uniform bound for knight distance on a box"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/knight-distance-asymptotics","kind":"node","lean":"MetaMathlibExt.knight_distance_asymptotics","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#knight-distance-asymptotics","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"can_prove","summary":"There is an absolute constant $M \\in \\mathbb{R}$ with the following property. Let $a$ and $b$ be natural numbers with $1 \\le a < b$, $\\gcd(a, b) = 1$ and $a + b$ odd, let $N_{a,b} = \\{(\\pm a, \\pm b), (\\pm b, \\pm a)\\}$ be the $(a,b)$-knight's move set, and for $p \\in \\mathbb{Z}^2$ let $d_{a,b}(p)$ be the piece distance of $p$ for $N_{a,b}$. Then for all integers $x$ and $y$ with $0 \\le y \\le x$: if $by \\le ax$, then\u2026","title":"Knight distance asymptotics"},{"id":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/knight-velocity-formula","kind":"node","lean":"MetaMathlibExt.knight_velocity_formula","parent":23093,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice/index.html#knight-velocity-formula","scope":"Combinatorics/additive-combinatorics/additive-bases/knight-and-king-distances-on-the-integer-lattice","status":"can_prove","summary":"Let $a$ and $b$ be natural numbers with $1 \\le a < b$, $\\gcd(a, b) = 1$ and $a + b$ odd, let $N_{a,b} = \\{(\\pm a, \\pm b), (\\pm b, \\pm a)\\}$ be the $(a,b)$-knight's move set, and for $p \\in \\mathbb{Z}^2$ let $d_{a,b}(p)$ be the piece distance of $p$ for $N_{a,b}$. For a natural number $h$, let $B_h = \\{(x, y) \\in \\mathbb{Z}^2 : \\max(|x|, |y|) \\le h\\}$ be the max-norm ball and let\u2026","title":"Velocity formula for the (a, b)-knight"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases.json","id":"Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases","status":"fully_proved","title":"Restricted additive bases"},{"id":"Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases/minimum-restricted-additive-basis-card","kind":"node","lean":"MetaMathlibExt.minimumRestrictedAdditiveBasisCard","parent":23108,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases/index.html#minimum-restricted-additive-basis-card","scope":"Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases","status":"fully_proved","summary":"Let $s$ be a natural number. A restricted additive basis of $\\{0, 1, \\dots, s\\}$ is a finite set $A$ of natural numbers with $2a \\le s$ for every $a \\in A$ such that every natural number $x \\le s$ can be written as $x = a + b$ with $a, b \\in A$, where $a = b$ is allowed. The minimum restricted additive basis size $k^*(s)$ is the smallest cardinality of a restricted additive basis of $\\{0, 1, \\dots, s\\}$; it is\u2026","title":"Minimum size of a restricted additive basis"},{"id":"Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases/minimum-restricted-additive-basis-card-e","kind":"node","lean":"MetaMathlibExt.minimumRestrictedAdditiveBasisCard_eq_none_iff","parent":23108,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases/index.html#minimum-restricted-additive-basis-card-e","scope":"Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases","status":"fully_proved","summary":"For every natural number $s$, the minimum size $k^*(s)$ of a restricted additive basis of $\\{0, 1, \\dots, s\\}$, that is, of a finite set $A$ of natural numbers with $2a \\le s$ for every $a \\in A$ such that every natural number $x \\le s$ equals $a + b$ for some $a, b \\in A$, is undefined if and only if $s$ is odd.","title":"The minimum restricted additive basis size is undefined exactly in the odd case"},{"id":"Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases/minimum-restricted-additive-basis-card-i","kind":"node","lean":"MetaMathlibExt.minimumRestrictedAdditiveBasisCard_isSome_iff","parent":23108,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases/index.html#minimum-restricted-additive-basis-card-i","scope":"Combinatorics/additive-combinatorics/additive-bases/restricted-additive-bases","status":"fully_proved","summary":"For every natural number $s$, the minimum size $k^*(s)$ of a restricted additive basis of $\\{0, 1, \\dots, s\\}$, that is, of a finite set $A$ of natural numbers with $2a \\le s$ for every $a \\in A$ such that every natural number $x \\le s$ equals $a + b$ for some $a, b \\in A$, is defined if and only if $s$ is even.","title":"The minimum restricted additive basis size is defined exactly in the even case"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density.json","id":"Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density","status":"fully_proved","title":"Schnirelmann density"},{"id":"Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density/schnirelmann-density-add-lower-bound","kind":"node","lean":"schnirelmannDensity_add_lower_bound","parent":23112,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density/index.html#schnirelmann-density-add-lower-bound","scope":"Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density","status":"fully_proved","summary":"For a set $A$ of natural numbers, let $\\sigma(A) = \\inf_{n \\ge 1} |A \\cap \\{1, \\dots, n\\}| / n$ be its Schnirelmann density. If $A$ and $B$ are sets of natural numbers that both contain $0$, then their sumset $A + B = \\{a + b : a \\in A,\\ b \\in B\\}$ satisfies $\\sigma(A) + \\sigma(B) - \\sigma(A)\\sigma(B) \\le \\sigma(A + B)$.","title":"Schnirelmann's inequality"},{"id":"Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density/exists-additive-basis-of-pos-schnirelman","kind":"node","lean":"exists_additive_basis_of_pos_schnirelmannDensity","parent":23112,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density/index.html#exists-additive-basis-of-pos-schnirelman","scope":"Combinatorics/additive-combinatorics/additive-bases/schnirelmann-density","status":"fully_proved","summary":"Let $A$ be a set of natural numbers with $0 \\in A$ whose Schnirelmann density $\\sigma(A) = \\inf_{n \\ge 1} |A \\cap \\{1, \\dots, n\\}| / n$ is positive. Then $A$ is an additive basis of finite order: there is a natural number $h$ such that every natural number $n$ can be written as $n = a_1 + \\dots + a_h$ with $a_1, \\dots, a_h \\in A$, repetitions allowed.","title":"Schnirelmann's basis theorem"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density.json","id":"Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density","status":"can_prove","title":"Shifted sumsets in sets of positive density"},{"id":"Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density/restricted-sumset-shift-in-dense-set","kind":"node","lean":"MathlibExt.Combinatorics.DensityRestrictedSumsetShiftErdosProblem656Wanted.restrictedSumsetShiftInDenseSet","parent":23115,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density/index.html#restricted-sumset-shift-in-dense-set","scope":"Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density","status":"fully_proved","summary":"The statement that for every set $A$ of natural numbers with positive upper density, that is, with $\\limsup_{N \\to \\infty} |A \\cap \\{0, \\dots, N - 1\\}| / N > 0$, there are an infinite set $B \\subseteq A$ and an integer $t$ such that $b_1 + b_2 + t \\in A$ for all distinct $b_1, b_2 \\in B$.","title":"Erd\u0151s Problem 656 (shifted restricted sumsets in dense sets)"},{"id":"Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density/restricted-sumset-shift-in-dense-set-hol","kind":"node","lean":"MathlibExt.Combinatorics.DensityRestrictedSumsetShiftErdosProblem656Wanted.restrictedSumsetShiftInDenseSet_holds","parent":23115,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density/index.html#restricted-sumset-shift-in-dense-set-hol","scope":"Combinatorics/additive-combinatorics/additive-bases/shifted-sumsets-in-sets-of-positive-density","status":"can_prove","summary":"For every set $A$ of natural numbers with positive upper density, that is, with $\\limsup_{N \\to \\infty} |A \\cap \\{0, \\dots, N - 1\\}| / N > 0$, there are an infinite set $B \\subseteq A$ and an integer $t$ such that $b_1 + b_2 + t \\in A$ for all distinct $b_1, b_2 \\in B$.","title":"Erd\u0151s Problem 656 is true"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets.json","id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"can_prove","title":"Splitting sets with dense sumsets"},{"id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/upper-density-sumset-splits","kind":"node","lean":"MathlibExt.Combinatorics.SumsetPartitionErdosProblem741Wanted.upperDensitySumsetSplits","parent":23118,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html#upper-density-sumset-splits","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"fully_proved","summary":"This proposition asserts that for every set $A \\subseteq \\mathbb{N}$ whose sumset $A + A = \\{a + b : a, b \\in A\\}$ has positive upper density there are disjoint sets $A_1, A_2 \\subseteq \\mathbb{N}$ with $A_1 \\cup A_2 = A$ such that both $A_1 + A_1$ and $A_2 + A_2$ have positive upper density.","title":"Erd\u0151s Problem 741, upper-density question"},{"id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/has-bounded-gaps","kind":"node","lean":"MathlibExt.Combinatorics.SumsetPartitionErdosProblem741Wanted.HasBoundedGaps","parent":23118,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html#has-bounded-gaps","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"fully_proved","summary":"A set $S \\subseteq \\mathbb{N}$ has bounded gaps (is syndetic) if there is a natural number $g$ such that for every natural number $n$ some $s \\in S$ satisfies $n \\le s < n + g$.","title":"Sets with bounded gaps"},{"id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/basis-without-bounded-gap-split","kind":"node","lean":"MathlibExt.Combinatorics.SumsetPartitionErdosProblem741Wanted.basisWithoutBoundedGapSplit","parent":23118,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html#basis-without-bounded-gap-split","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"fully_proved","summary":"This proposition asserts that there is a set $A \\subseteq \\mathbb{N}$ that is an asymptotic additive basis of order $2$ (all but finitely many natural numbers are sums of two elements of $A$) such that for all disjoint sets $A_1, A_2 \\subseteq \\mathbb{N}$ with $A_1 \\cup A_2 = A$, the sumsets $A_1 + A_1 = \\{a + b : a, b \\in A_1\\}$ and $A_2 + A_2$ do not both have bounded gaps.","title":"Erd\u0151s Problem 741, bounded-gaps question"},{"id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/upper-density-sumset-splits-holds","kind":"node","lean":"MathlibExt.Combinatorics.SumsetPartitionErdosProblem741Wanted.upperDensitySumsetSplits_holds","parent":23118,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html#upper-density-sumset-splits-holds","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"can_prove","summary":"For every set $A \\subseteq \\mathbb{N}$ whose sumset $A + A = \\{a + b : a, b \\in A\\}$ has positive upper density, there are disjoint sets $A_1, A_2 \\subseteq \\mathbb{N}$ with $A_1 \\cup A_2 = A$ such that both $A_1 + A_1$ and $A_2 + A_2$ have positive upper density.","title":"Splitting a set whose sumset has positive upper density"},{"id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/basis-without-bounded-gap-split-holds","kind":"node","lean":"MathlibExt.Combinatorics.SumsetPartitionErdosProblem741Wanted.basisWithoutBoundedGapSplit_holds","parent":23118,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html#basis-without-bounded-gap-split-holds","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"can_prove","summary":"There is a set $A \\subseteq \\mathbb{N}$ that is an asymptotic additive basis of order $2$ (all but finitely many natural numbers are sums of two elements of $A$) such that for all disjoint sets $A_1, A_2 \\subseteq \\mathbb{N}$ with $A_1 \\cup A_2 = A$, the sumsets $A_1 + A_1 = \\{a + b : a, b \\in A_1\\}$ and $A_2 + A_2$ do not both have bounded gaps, where a set $S \\subseteq \\mathbb{N}$ has bounded gaps if for some\u2026","title":"An asymptotic basis of order 2 admitting no bounded-gap splitting"},{"id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem871Wanted.conjecture","parent":23118,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"fully_proved","summary":"The statement that for every set $A$ of natural numbers such that every sufficiently large $n$ is $a + b$ with $a, b \\in A$ and, for every natural number $t$, every sufficiently large $n$ has at least $t$ representations $n = a + b$ with $a, b \\in A$ and $a \\le b$, the set $A$ is the disjoint union of two sets $B$ and $C$ such that every sufficiently large $n$ is $a + b$ with $a, b \\in B$ and every sufficiently\u2026","title":"Erd\u0151s Problem 871 (splitting rich bases of order 2)"},{"id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/disproved-form","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem871Wanted.disprovedForm","parent":23118,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html#disproved-form","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"fully_proved","summary":"This is the proposition that there is a set $A$ of natural numbers such that every sufficiently large $n$ is $a + b$ with $a, b \\in A$ and, for every natural number $t$, every sufficiently large $n$ has at least $t$ representations $n = a + b$ with $a, b \\in A$ and $a \\le b$, but $A$ is not the disjoint union of two sets $B$ and $C$ such that every sufficiently large $n$ is $a + b$ with $a, b \\in B$ and every\u2026","title":"Negation of the Erd\u0151s Problem 871 statement"},{"id":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem871Wanted.conjecture_refuted","parent":23118,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets/index.html#conjecture-refuted","scope":"Combinatorics/additive-combinatorics/additive-bases/splitting-sets-with-dense-sumsets","status":"can_prove","summary":"There is a set $A$ of natural numbers such that every sufficiently large $n$ is $a + b$ with $a, b \\in A$ and, for every natural number $t$, every sufficiently large $n$ has at least $t$ representations $n = a + b$ with $a, b \\in A$ and $a \\le b$, but $A$ is not the disjoint union of two sets $B$ and $C$ such that every sufficiently large $n$ is $a + b$ with $a, b \\in B$ and every sufficiently large $n$ is $a + b$\u2026","title":"Erd\u0151s Problem 871 is false (Larsen)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces.json","id":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces","status":"can_prove","title":"Sums of cubes in ternary vector spaces"},{"id":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/cube-origin","kind":"node","lean":"MathlibExt.Combinatorics.GreenProblem69Wanted.cubeOrigin","parent":23127,"reading_title":{"format":"inline-parts-v1","parts":[["text","The standard cube in "],["math","\\(\\mathbb{F}_3^n\\)"]],"source":"The standard cube in $\\mathbb{F}_3^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/index.html#cube-origin","scope":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the standard cube is the set $\\{0, 1\\}^n \\subseteq \\mathbb{F}_3^n$ of vectors each of whose coordinates is $0$ or $1$.","title":"The standard cube in $\\mathbb{F}_3^n$"},{"id":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/is-cube","kind":"node","lean":"MathlibExt.Combinatorics.GreenProblem69Wanted.IsCube","parent":23127,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cubes in "],["math","\\(\\mathbb{F}_3^n\\)"]],"source":"Cubes in $\\mathbb{F}_3^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/index.html#is-cube","scope":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces","status":"fully_proved","summary":"A set $A \\subseteq \\mathbb{F}_3^n$ is a cube if there is an $\\mathbb{F}_3$-linear automorphism $e$ of $\\mathbb{F}_3^n$ with $A = e(\\{0, 1\\}^n)$, the image of the standard cube $\\{0, 1\\}^n$ under $e$.","title":"Cubes in $\\mathbb{F}_3^n$"},{"id":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/sumset-eq-univ","kind":"node","lean":"MathlibExt.Combinatorics.GreenProblem69Wanted.SumsetEqUniv","parent":23127,"reading_title":{"format":"inline-parts-v1","parts":[["text","A sumset of "],["math","\\(100\\)"],["text"," sets equal to "],["math","\\(\\mathbb{F}_3^n\\)"]],"source":"A sumset of $100$ sets equal to $\\mathbb{F}_3^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/index.html#sumset-eq-univ","scope":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces","status":"fully_proved","summary":"For sets $A_1, \\dots, A_{100} \\subseteq \\mathbb{F}_3^n$, the property that every $v \\in \\mathbb{F}_3^n$ can be written as $v = a_1 + \\dots + a_{100}$ with $a_i \\in A_i$ for each $i$, that is, $A_1 + \\dots + A_{100} = \\mathbb{F}_3^n$.","title":"A sumset of $100$ sets equal to $\\mathbb{F}_3^n$"},{"id":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.GreenProblem69Wanted.conjecture","parent":23127,"reading_title":{"format":"inline-parts-v1","parts":[["text","Green's problem on sums of "],["math","\\(100\\)"],["text"," cubes in "],["math","\\(\\mathbb{F}_3^n\\)"]],"source":"Green's problem on sums of $100$ cubes in $\\mathbb{F}_3^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces","status":"fully_proved","summary":"The statement that for every $n \\ge 1$ and every family $A_1, \\dots, A_{100}$ of cubes in $\\mathbb{F}_3^n$ (not necessarily distinct), that is, images of $\\{0, 1\\}^n$ under linear automorphisms of $\\mathbb{F}_3^n$, the sumset $A_1 + \\dots + A_{100}$ is all of $\\mathbb{F}_3^n$.","title":"Green's problem on sums of $100$ cubes in $\\mathbb{F}_3^n$"},{"id":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.GreenProblem69Wanted.conjecture_holds","parent":23127,"reading_title":{"format":"inline-parts-v1","parts":[["text","Any "],["math","\\(100\\)"],["text"," cubes sum to all of "],["math","\\(\\mathbb{F}_3^n\\)"]],"source":"Any $100$ cubes sum to all of $\\mathbb{F}_3^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces/index.html#conjecture-holds","scope":"Combinatorics/additive-combinatorics/additive-bases/sums-of-cubes-in-ternary-vector-spaces","status":"can_prove","summary":"For every $n \\ge 1$ and every family $A_1, \\dots, A_{100}$ of cubes in $\\mathbb{F}_3^n$ (not necessarily distinct), that is, images of $\\{0, 1\\}^n$ under linear automorphisms of $\\mathbb{F}_3^n$, the sumset $A_1 + \\dots + A_{100}$ is all of $\\mathbb{F}_3^n$.","title":"Any $100$ cubes sum to all of $\\mathbb{F}_3^n$"},{"data_url":"scopes/Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero.json","id":"Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero","kind":"scope","navigation":"module","parent":23065,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero/index.html","scope":"Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero","status":"can_prove","title":"Sumsets of sets of density zero"},{"id":"Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero/sumset-ratio-limsup-at-least-three","kind":"node","lean":"MathlibExt.Combinatorics.SumsetGrowthZeroDensityErdosProblem245Wanted.sumsetRatioLimsupAtLeastThree","parent":23133,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero/index.html#sumset-ratio-limsup-at-least-three","scope":"Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero","status":"fully_proved","summary":"For $N \\in \\mathbb{N}$ write $[1, N] = \\{1, \\dots, N\\}$, and for $A \\subseteq \\mathbb{N}$ write $A + A = \\{a + b \\mid a, b \\in A\\}$. This is the statement that for every infinite set $A \\subseteq \\mathbb{N}$ with $\\lvert A \\cap [1, N] \\rvert / N \\to 0$ as $N \\to \\infty$ and every real $\\varepsilon > 0$, there are infinitely many $N$ with\u2026","title":"Erd\u0151s Problem 245 on sumsets of sets of density zero"},{"id":"Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero/sumset-ratio-limsup-at-least-three-holds","kind":"node","lean":"MathlibExt.Combinatorics.SumsetGrowthZeroDensityErdosProblem245Wanted.sumsetRatioLimsupAtLeastThree_holds","parent":23133,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero/index.html#sumset-ratio-limsup-at-least-three-holds","scope":"Combinatorics/additive-combinatorics/additive-bases/sumsets-of-sets-of-density-zero","status":"can_prove","summary":"For $N \\in \\mathbb{N}$ write $[1, N] = \\{1, \\dots, N\\}$, and for $A \\subseteq \\mathbb{N}$ write $A + A = \\{a + b \\mid a, b \\in A\\}$. Let $A \\subseteq \\mathbb{N}$ be an infinite set with $\\lvert A \\cap [1, N] \\rvert / N \\to 0$ as $N \\to \\infty$. Then for every real $\\varepsilon > 0$ there are infinitely many $N$ with $\\lvert (A + A) \\cap [1, N] \\rvert > (3 - \\varepsilon) \\lvert A \\cap [1, N] \\rvert$; that is,\u2026","title":"Freiman's theorem on sumsets of sets of density zero"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums","kind":"scope","parent":22582,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums","status":"can_prove","summary":"Davenport and weighted Davenport constants, zero-sum subsequences and cross numbers, Erd\u0151s\u2013Ginzburg\u2013Ziv-type results, perfect subset-sum covers, zero subset sums and balanced subset sums in finite abelian groups, and subset sums with bounded numbers of terms.","title":"Zero-sum problems and subset sums"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups","kind":"scope","navigation":"module","parent":23136,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups","status":"can_prove","title":"Balanced subset sums in finite abelian groups"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/subset-sum-count","kind":"node","lean":"MathlibExt.Combinatorics.RandomSubsetSumsBalancedErdosProblem1179Wanted.subsetSumCount","parent":23137,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/index.html#subset-sum-count","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups","status":"fully_proved","summary":"For a finite subset $A$ of an abelian group $G$ and an element $g \\in G$, $F_A(g)$ is the number of subsets $S \\subseteq A$, the empty set included, whose elements sum to $g$, that is, with $\\sum_{x \\in S} x = g$.","title":"Number of subsets with a given sum"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/is-balanced","kind":"node","lean":"MathlibExt.Combinatorics.RandomSubsetSumsBalancedErdosProblem1179Wanted.IsBalanced","parent":23137,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\varepsilon\\)"],["text","-balanced subsets of a finite abelian group"]],"source":"$\\varepsilon$-balanced subsets of a finite abelian group"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/index.html#is-balanced","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups","status":"fully_proved","summary":"Let $G$ be a finite abelian group with $N = |G|$ elements and let $\\varepsilon$ be a real number. A subset $A \\subseteq G$ is $\\varepsilon$-balanced when, for every $g \\in G$, the number $F_A(g)$ of subsets of $A$ whose elements sum to $g$ satisfies $\\left| F_A(g) - 2^{|A|}/N \\right| \\le \\varepsilon \\cdot 2^{|A|}/N$.","title":"$\\varepsilon$-balanced subsets of a finite abelian group"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/balanced-probability","kind":"node","lean":"MathlibExt.Combinatorics.RandomSubsetSumsBalancedErdosProblem1179Wanted.balancedProbability","parent":23137,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/index.html#balanced-probability","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups","status":"fully_proved","summary":"For a finite abelian group $G$, a real number $\\varepsilon$ and a natural number $k$, this is the number of $\\varepsilon$-balanced $k$-element subsets of $G$ divided by the number of all $k$-element subsets of $G$, that is, the probability that a uniformly random $k$-element subset of $G$ is $\\varepsilon$-balanced. When $k > |G|$ there are no $k$-element subsets and the value is $0$.","title":"Probability that a random subset is balanced"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/balanced-subset-sums-threshold-log-two","kind":"node","lean":"MathlibExt.Combinatorics.RandomSubsetSumsBalancedErdosProblem1179Wanted.balancedSubsetSumsThresholdLogTwo","parent":23137,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/index.html#balanced-subset-sums-threshold-log-two","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups","status":"fully_proved","summary":"The statement of Erd\u0151s Problem 1179 is the proposition that, for every real $\\varepsilon$ with $0 < \\varepsilon < 1$, the following two bounds hold. Upper bound: there is a function $k : \\mathbb{N} \\to \\mathbb{N}$ with $k(N) / \\log_2 N \\to 1$ as $N \\to \\infty$ such that, for every sequence $(G_i)_{i \\in \\mathbb{N}}$ of finite abelian groups with $|G_i| \\to \\infty$, the probability that a uniformly random\u2026","title":"Erd\u0151s Problem 1179 (balanced subset sums of random subsets)"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/balanced-subset-sums-threshold-log-two-h","kind":"node","lean":"MathlibExt.Combinatorics.RandomSubsetSumsBalancedErdosProblem1179Wanted.balancedSubsetSumsThresholdLogTwo_holds","parent":23137,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups/index.html#balanced-subset-sums-threshold-log-two-h","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/balanced-subset-sums-in-finite-abelian-groups","status":"can_prove","summary":"For every real $\\varepsilon$ with $0 < \\varepsilon < 1$, the following two bounds hold. Upper bound: there is a function $k : \\mathbb{N} \\to \\mathbb{N}$ with $k(N) / \\log_2 N \\to 1$ as $N \\to \\infty$ such that, for every sequence $(G_i)_{i \\in \\mathbb{N}}$ of finite abelian groups with $|G_i| \\to \\infty$, the probability that a uniformly random $k(|G_i|)$-element subset of $G_i$ is $\\varepsilon$-balanced tends to\u2026","title":"Erd\u0151s Problem 1179 holds"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture","kind":"scope","navigation":"module","parent":23136,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture","status":"can_prove","title":"Graham's zero-sum conjecture"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture/fixed-zero-sum-length-two-residues","kind":"node","lean":"MathlibExt.Combinatorics.ZeroSumFixedLengthTwoResiduesErdosProblem541Wanted.fixedZeroSumLengthTwoResidues","parent":23143,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture/index.html#fixed-zero-sum-length-two-residues","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture","status":"fully_proved","summary":"This is the statement that for every prime $p$ and every sequence $a_0, \\dots, a_{p-1}$ of residues modulo $p$, not necessarily distinct, if there is a natural number $r$ such that every nonempty set $S \\subseteq \\{0, \\dots, p-1\\}$ with $\\sum_{i \\in S} a_i = 0$ in $\\mathbb{Z}/p\\mathbb{Z}$ has exactly $r$ elements, then at most two distinct residues occur among $a_0, \\dots, a_{p-1}$.","title":"Graham's zero-sum statement (Erd\u0151s Problem 541)"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture/fixed-zero-sum-length-two-residues-holds","kind":"node","lean":"MathlibExt.Combinatorics.ZeroSumFixedLengthTwoResiduesErdosProblem541Wanted.fixedZeroSumLengthTwoResidues_holds","parent":23143,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture/index.html#fixed-zero-sum-length-two-residues-holds","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/graham-s-zero-sum-conjecture","status":"can_prove","summary":"For every prime $p$ and every sequence $a_0, \\dots, a_{p-1}$ of residues modulo $p$, not necessarily distinct, if there is a natural number $r$ such that every nonempty set $S \\subseteq \\{0, \\dots, p-1\\}$ with $\\sum_{i \\in S} a_i = 0$ in $\\mathbb{Z}/p\\mathbb{Z}$ has exactly $r$ elements, then at most two distinct residues occur among $a_0, \\dots, a_{p-1}$.","title":"Zero sums of a single length use at most two residues (Erd\u0151s Problem 541)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","kind":"scope","navigation":"module","parent":23136,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","title":"Perfect subset-sum covers of abelian 2-groups"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/cover-count","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.coverCount","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#cover-count","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"Let $G$ be an abelian group, $s = (s_i)_{i \\in \\iota}$ a family in $G$ indexed by a finite type $\\iota$ (a colored multiset), and $g \\in G$. The cover count of $g$ by $s$ is the number of index sets $I \\subseteq \\iota$ with $\\sum_{i \\in I} s_i = g$: the number of times $s$ covers $g$ by its subset sums (the empty set covers $0$).","title":"Cover count"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/is-perfect-mcover","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.IsPerfectMCover","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Perfect "],["math","\\(m\\)"],["text","-cover"]],"source":"Perfect $m$-cover"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#is-perfect-mcover","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"Let $G$ be an abelian group, $m$ a natural number and $s = (s_i)_{i \\in \\iota}$ a family of elements of $G$ indexed by a finite set $\\iota$ (a colored multiset: repeated values count separately). The family $s$ is a perfect $m$-cover of $G$ when every $g \\in G$ equals $\\sum_{i \\in I} s_i$ for exactly $m$ subsets $I \\subseteq \\iota$ (the empty subset has sum $0$).","title":"Perfect $m$-cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/is-perfect-cover","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.IsPerfectCover","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#is-perfect-cover","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"Let $G$ be an abelian group. A finite set $S \\subseteq G$ is a perfect cover of $G$ when every element of $G$ is the sum of exactly one subset of $S$ (the empty subset has sum $0$); that is, the family of the elements of $S$ is a perfect $1$-cover of $G$.","title":"Perfect cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/count","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.count","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#count","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"For an abelian group $G$, $C(G)$ is the number of perfect covers of $G$: finite subsets $S \\subseteq G$ such that every element of $G$ is the sum of exactly one subset of $S$ (the empty subset has sum $0$).","title":"Number of perfect covers"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/count-eq-card-filter","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.count_eq_card_filter","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#count-eq-card-filter","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"For a finite abelian group $G$, the number $C(G)$ of perfect covers of $G$ equals the number of elements of the finite set of all subsets of $G$ that are perfect covers.","title":"Perfect cover count as a finset cardinality"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/count-small","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.count_small","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#count-small","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"Writing $C(K)$ for the number of perfect covers of a group $K$, we have $C(\\mathbb{Z}_2) = 1$, $C(\\mathbb{Z}_4) = 2$ and $C(\\mathbb{Z}_2 \\oplus \\mathbb{Z}_2) = 3$.","title":"Small values of the perfect cover count"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/is-perfect-cover-zmod-four","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.isPerfectCover_zmod_four","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#is-perfect-cover-zmod-four","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"In the cyclic group $\\mathbb{Z}_4 = \\mathbb{Z}/4\\mathbb{Z}$, the set $\\{1, 2\\}$ is a perfect cover and the set $\\{1, 3\\}$ is not, since $1 + 3 = 0$.","title":"Perfect covers of \u2124\u2084: an example and a non-example"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/odd-combinations","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.oddCombinations","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#odd-combinations","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"For a finite subset $A$ of an abelian group $G$, $\\mathcal{T}(A)$ is the set of all sums $\\sum_{x \\in A} c_x x$ with integer coefficients $c_x$ such that every $c_x$ is either $0$ or odd.","title":"Odd combinations"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/zero-mem-odd-combinations","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.zero_mem_oddCombinations","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#zero-mem-odd-combinations","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"For every finite subset $A$ of an abelian group $G$, $0$ lies in the set $\\mathcal{T}(A)$ of odd combinations of $A$ (take every coefficient equal to $0$).","title":"Zero is an odd combination"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/mem-odd-combinations-of-mem","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.mem_oddCombinations_of_mem","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#mem-odd-combinations-of-mem","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"Let $G$ be an abelian group and $A$ a finite subset of $G$, and let $\\mathcal{T}(A)$ be the set of combinations $\\sum_{y \\in A} c_y y$ in which every integer coefficient $c_y$ is $0$ or odd. Every $x \\in A$ lies in $\\mathcal{T}(A)$: take $c_x = 1$ and $c_y = 0$ for $y \\neq x$.","title":"Elements of a finite set are odd combinations of it"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/odd-combinations-empty","kind":"node","lean":"MetaMathlibExt.PerfectSubsetSumCover.oddCombinations_empty","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#odd-combinations-empty","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"fully_proved","summary":"Let $G$ be an abelian group and, for a finite $A \\subseteq G$, let $\\mathcal{T}(A)$ be the set of combinations $\\sum_{x \\in A} c_x x$ in which every integer coefficient $c_x$ is $0$ or odd. Then $\\mathcal{T}(\\emptyset) = \\{0\\}$.","title":"Odd combinations of the empty set"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/exists-is-perfect-cover-mem","kind":"node","lean":"MetaMathlibExt.exists_isPerfectCover_mem","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#exists-is-perfect-cover-mem","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Every finite abelian group $G$ of order $2^n$ has a perfect cover, and every nonzero $x \\in G$ lies in some perfect cover of $G$.","title":"Existence of perfect covers"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/count-zmod-two-pow","kind":"node","lean":"MetaMathlibExt.count_zmod_two_pow","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#count-zmod-two-pow","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"For every $n \\ge 0$, the cyclic group $\\mathbb{Z}_{2^n}$ has exactly $2^{n(n-1)/2}$ perfect covers.","title":"Number of perfect covers of a cyclic 2-group"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/count-zmod-two-pow-prod-recurrence","kind":"node","lean":"MetaMathlibExt.count_zmod_two_pow_prod_recurrence","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#count-zmod-two-pow-prod-recurrence","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Write $C(K)$ for the number of perfect covers of a group $K$, and $a_{m,n} = C(\\mathbb{Z}_{2^m} \\oplus \\mathbb{Z}_{2^n})$. For all $m \\ge n \\ge 1$, $a_{m,n} = 2^{m+n-1} (a_{m-1,n} + 2 a_{m,n-1}) - 3 \\cdot 2^{2m+2n-4} \\, a_{m-1,n-1}$. Moreover, for every $n \\ge 0$ there is a polynomial $p$ of degree $n$ with rational coefficients such that $a_{m,n} = 2^{(m-1)n} \\, 2^{m(m-1)/2} \\, p(m)$ for all $m \\ge n$.","title":"Perfect covers of a product of two cyclic 2-groups"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/count-elementary-prod-zmod-two-pow","kind":"node","lean":"MetaMathlibExt.count_elementary_prod_zmod_two_pow","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Perfect covers of "],["math","\\((\\mathbb{Z}_2)^m \\oplus \\mathbb{Z}_{2^n}\\)"]],"source":"Perfect covers of $(\\mathbb{Z}_2)^m \\oplus \\mathbb{Z}_{2^n}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#count-elementary-prod-zmod-two-pow","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Write $C(K)$ for the number of perfect covers of a group $K$, and for natural numbers $m$ and $n$ let $b_{m,n} = C((\\mathbb{Z}_2)^m \\oplus \\mathbb{Z}_{2^n})$. Then: for every $m \\ge 0$, $C((\\mathbb{Z}_2)^m) \\cdot m! = \\prod_{i=0}^{m-1} (2^m - 2^i)$; for all $m, n \\ge 1$, writing $B_{m,k} = 2^m \\prod_{i=0}^{k-1} (2^m - 2^i) / (k+1)!$ for $1 \\le k \\le m$, we have\u2026","title":"Perfect covers of $(\\mathbb{Z}_2)^m \\oplus \\mathbb{Z}_{2^n}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/count-zmod-two-pow-prod-small-special","kind":"node","lean":"MetaMathlibExt.count_zmod_two_pow_prod_small_special","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#count-zmod-two-pow-prod-small-special","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Write $C(K)$ for the number of perfect covers of a group $K$. For every $n \\ge 0$, $C(\\mathbb{Z}_{2^n} \\oplus \\mathbb{Z}_2) = 2^{n-1} \\, 2^{n(n-1)/2} \\, (n + 2)$, where $2^{n-1}$ is an integer power of $2$ (equal to $1/2$ at $n = 0$). For every $n \\ge 1$, $C(\\mathbb{Z}_{2^n} \\oplus \\mathbb{Z}_4) = 2^{2n-2} \\, 2^{n(n-1)/2} \\, (n^2 + 11n + 4)$. For every $n \\ge 2$,\u2026","title":"Perfect covers of a cyclic 2-group times a cyclic group of order 2, 4 or 8"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/count-elementary-two-three-prod-zmod","kind":"node","lean":"MetaMathlibExt.count_elementary_two_three_prod_zmod","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#count-elementary-two-three-prod-zmod","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Write $C(K)$ for the number of perfect covers of a group $K$. For every $n \\ge 0$, $C((\\mathbb{Z}_2)^2 \\oplus \\mathbb{Z}_{2^n}) = 2^{2n-2} \\, 2^{n(n-1)/2} \\, (3n^2 + 13n + 12)$ and $C((\\mathbb{Z}_2)^3 \\oplus \\mathbb{Z}_{2^n}) = 2^{3n-3} \\, 2^{n(n-1)/2} \\, (28n^3 + 196n^2 + 392n + 224)$, where $2^{2n-2}$ and $2^{3n-3}$ are integer powers of $2$ (fractional at $n = 0$).","title":"Perfect covers of an elementary abelian group of order 4 or 8 times a cyclic 2-group"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/is-perfect-mcover-cons","kind":"node","lean":"MetaMathlibExt.isPerfectMCover_cons","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#is-perfect-mcover-cons","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be an abelian group and let $s = (s_1, \\dots, s_k)$ be a perfect $m$-cover of $G$. For every $g \\in G$, the family $(g, s_1, \\dots, s_k)$ is a perfect $2m$-cover of $G$.","title":"Adjoining an element doubles the multiplicity of a perfect cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/is-perfect-mcover-fst-snd","kind":"node","lean":"MetaMathlibExt.isPerfectMCover_fst_snd","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#is-perfect-mcover-fst-snd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $H$ and $K$ be finite abelian groups with $|H| = 2^m$ and $|K| = 2^n$, and let $s = (s_i)_{i \\in \\iota}$, indexed by a finite set $\\iota$, be a perfect $l$-cover of $H \\oplus K$. Then the first coordinates of the $s_i$ form a perfect $2^n l$-cover of $H$, and the second coordinates form a perfect $2^m l$-cover of $K$.","title":"Projections of a perfect cover of a direct sum"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/perfect-cover-erase-cover-count","kind":"node","lean":"MetaMathlibExt.perfectCover_erase_coverCount","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#perfect-cover-erase-cover-count","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be a finite abelian group of order $2^n$, let $S$ be a perfect cover of $G$, let $z \\in S$ and let $S_1 = S \\setminus \\{z\\}$. Then (a) every element of $G$ is the sum of at most one subset of $S_1$, and exactly half of the $2^n$ elements of $G$ are the sum of a subset of $S_1$; (b) if $g \\in G$ is the sum of a subset of $S_1$ and $j$ is an integer with $j \\ge 1$, then $g + 2jz$ is the sum of a subset of\u2026","title":"Removing one element from a perfect cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/cover-count-init-add-last","kind":"node","lean":"MetaMathlibExt.coverCount_init_add_last","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Removing the last element of a perfect "],["math","\\(2^j\\)"],["text","-cover"]],"source":"Removing the last element of a perfect $2^j$-cover"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#cover-count-init-add-last","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be an abelian group, let $s = (x_1, \\dots, x_{k+1})$ be a family in $G$ that is a perfect $2^j$-cover of $G$, and let $S_1 = (x_1, \\dots, x_k)$. For every $a \\in G$, if $S_1$ covers $a$ exactly $c$ times, then $S_1$ covers $a + x_{k+1}$ exactly $2^j - c$ times.","title":"Removing the last element of a perfect $2^j$-cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/cover-count-init-add-zsmul-last","kind":"node","lean":"MetaMathlibExt.coverCount_init_add_zsmul_last","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shifting by multiples of the last element of a perfect "],["math","\\(2^j\\)"],["text","-cover"]],"source":"Shifting by multiples of the last element of a perfect $2^j$-cover"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#cover-count-init-add-zsmul-last","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be an abelian group, let $s = (x_1, \\dots, x_{k+1})$ be a family in $G$ that is a perfect $2^j$-cover of $G$, and let $S_1 = (x_1, \\dots, x_k)$. For every $b \\in G$, if $S_1$ covers $b$ exactly $c$ times, then for every integer $t$ it covers $b + 2t\\,x_{k+1}$ exactly $c$ times and $b + (2t + 1)\\,x_{k+1}$ exactly $2^j - c$ times.","title":"Shifting by multiples of the last element of a perfect $2^j$-cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/odd-multiple-iff-perfect-cover-replace","kind":"node","lean":"MetaMathlibExt.oddMultiple_iff_perfectCover_replace","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#odd-multiple-iff-perfect-cover-replace","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be a finite abelian group of order $2^n$ and let $a, b \\in G$ be nonzero. Then $b = (2j+1)a$ for some integer $j$ if and only if, for every perfect cover $S$ of $G$ with $a \\in S$, the set $(S \\setminus \\{a\\}) \\cup \\{b\\}$ is again a perfect cover of $G$. If moreover $a \\ne b$, then $b = (2j+1)a$ for some integer $j$ if and only if no perfect cover of $G$ contains both $a$ and $b$.","title":"Odd multiples and replacement in perfect covers"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/is-perfect-mcover-cons-odd-multiple","kind":"node","lean":"MetaMathlibExt.isPerfectMCover_cons_oddMultiple","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Replacing an entry of a perfect "],["math","\\(2^k\\)"],["text","-cover by an odd multiple"]],"source":"Replacing an entry of a perfect $2^k$-cover by an odd multiple"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#is-perfect-mcover-cons-odd-multiple","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be an abelian group, let $k$ and $N$ be natural numbers, and let $a, b, s_1, \\dots, s_N \\in G$. If the family $(a, s_1, \\dots, s_N)$ is a perfect $2^k$-cover of $G$ and $b = (2j + 1)a$ for some integer $j$, then the family $(b, s_1, \\dots, s_N)$ is also a perfect $2^k$-cover of $G$.","title":"Replacing an entry of a perfect $2^k$-cover by an odd multiple"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/exists-is-perfect-mcover-zmod-iff","kind":"node","lean":"MetaMathlibExt.exists_isPerfectMCover_zmod_iff","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Which families are perfect "],["math","\\(m\\)"],["text","-covers of "],["math","\\(\\mathbb{Z}_{2^n}\\)"]],"source":"Which families are perfect $m$-covers of $\\mathbb{Z}_{2^n}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#exists-is-perfect-mcover-zmod-iff","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $s = (s_i)_{i \\in \\iota}$ be a family in $\\mathbb{Z}_{2^n}$ indexed by a finite type $\\iota$ (a colored multiset). Then $s$ is a perfect $m$-cover of $\\mathbb{Z}_{2^n}$ for some $m$ if and only if for every $j$ with $1 \\le j \\le n$ some $s_i$ has additive order $2^j$.","title":"Which families are perfect $m$-covers of $\\mathbb{Z}_{2^n}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/exists-perfect-cover-subfamily-zmod","kind":"node","lean":"MetaMathlibExt.exists_perfectCover_subfamily_zmod","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Perfect "],["math","\\(m\\)"],["text","-covers of "],["math","\\(\\mathbb{Z}_{2^n}\\)"],["text"," contain perfect covers"]],"source":"Perfect $m$-covers of $\\mathbb{Z}_{2^n}$ contain perfect covers"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#exists-perfect-cover-subfamily-zmod","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $s = (s_i)_{i \\in \\iota}$ be a family in $\\mathbb{Z}_{2^n}$ indexed by a finite type $\\iota$. If $s$ is a perfect $m$-cover of $\\mathbb{Z}_{2^n}$, then there is a finite set $T \\subseteq \\iota$ such that the subfamily $(s_i)_{i \\in T}$ is a perfect $1$-cover of $\\mathbb{Z}_{2^n}$.","title":"Perfect $m$-covers of $\\mathbb{Z}_{2^n}$ contain perfect covers"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/exists-snd-add-order-of-of-is-perfect-mc","kind":"node","lean":"MetaMathlibExt.exists_snd_addOrderOf_of_isPerfectMCover","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Orders of second coordinates in perfect "],["math","\\(m\\)"],["text","-covers"]],"source":"Orders of second coordinates in perfect $m$-covers"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#exists-snd-add-order-of-of-is-perfect-mc","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $H$ be a finite abelian group of order $2^r$ and let $s = (s_i)_{i \\in \\iota}$ be a family in $H \\oplus \\mathbb{Z}_{2^n}$ indexed by a finite type $\\iota$. If $s$ is a perfect $m$-cover of $H \\oplus \\mathbb{Z}_{2^n}$, then for every $j$ with $1 \\le j \\le n$ the second coordinate of some $s_i$ has additive order $2^j$.","title":"Orders of second coordinates in perfect $m$-covers"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/odd-combinations-inter-of-is-perfect-cov","kind":"node","lean":"MetaMathlibExt.oddCombinations_inter_of_isPerfectCover","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#odd-combinations-inter-of-is-perfect-cov","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be an abelian group of order $2^n$ and $S$ a perfect cover of $G$. For a finite $A \\subseteq G$, let $\\mathcal{T}(A)$ be the set of combinations $\\sum_{x \\in A} c_x x$ in which every integer coefficient $c_x$ is $0$ or odd. Then: (a) for disjoint $A, B \\subseteq S$, $\\mathcal{T}(A) \\cap \\mathcal{T}(B) = \\{0\\}$; (b) if $A \\subseteq S$ and $s \\in S \\setminus A$, then $s \\notin \\mathcal{T}(A)$.","title":"Odd combinations of disjoint parts of a perfect cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/zsmul-mem-odd-combinations","kind":"node","lean":"MetaMathlibExt.zsmul_mem_oddCombinations","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#zsmul-mem-odd-combinations","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $A$ be a finite subset of an abelian group $G$. If $g$ lies in the set $\\mathcal{T}(A)$ of odd combinations of $A$, then $(2j+1)g \\in \\mathcal{T}(A)$ for every integer $j$.","title":"Odd combinations are closed under odd multiples"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/cover-count-le-one-of-not-mem-odd-combin","kind":"node","lean":"MetaMathlibExt.coverCount_le_one_of_not_mem_oddCombinations","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#cover-count-le-one-of-not-mem-odd-combin","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be an abelian group, and for a finite set $A \\subseteq G$ write $\\mathcal{T}(A)$ for the set of odd combinations of $A$. Let $x = (x_1, \\dots, x_m)$ be a family in $G$ such that $x_i \\notin \\mathcal{T}(\\{x_1, \\dots, x_{i-1}\\})$ for every $i$ (for $i = 1$ this says $x_1 \\ne 0$, and for $i \\ge 2$ it forces $x_i \\ne 0$). Then $x$ covers every element of $G$ at most once, that is, all subset sums of $x$ are\u2026","title":"Distinct subset sums from the odd-combination condition"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/card-perfect-cover-superset-subgroup","kind":"node","lean":"MetaMathlibExt.card_perfectCover_superset_subgroup","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#card-perfect-cover-superset-subgroup","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be an abelian group of order $2^n$ and $H$ a subgroup of order $2^m$. Then the number of perfect covers of $G$ that contain a perfect cover of $H$ is $2^{m(n-m)} \\cdot C(G/H) \\cdot C(H)$, where $C(K)$ denotes the number of perfect covers of a group $K$.","title":"Perfect covers containing a perfect cover of a subgroup"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/not-exists-unique-add-order-of-two-mem-o","kind":"node","lean":"MetaMathlibExt.not_existsUnique_addOrderOf_two_mem_oddCombinations","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#not-exists-unique-add-order-of-two-mem-o","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $G$ be an abelian group of order $2^n$, let $S$ be a perfect cover of $G$, and let $A \\subseteq S$ contain no element of order $2$. Let $\\mathcal{T}(A)$ be the set of combinations $\\sum_{x \\in A} c_x x$ in which every integer coefficient $c_x$ is $0$ or odd. Then $\\mathcal{T}(A)$ does not contain exactly one element of order $2$: it contains none or at least two.","title":"Elements of order two among odd combinations of part of a perfect cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/exists-add-order-of-eq-two-of-is-perfect","kind":"node","lean":"MetaMathlibExt.exists_addOrderOf_eq_two_of_isPerfectCover","parent":23146,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of order "],["math","\\(2\\)"],["text"," in perfect covers of "],["math","\\(\\mathbb{Z}_{2^m} \\oplus \\mathbb{Z}_{2^n}\\)"]],"source":"Elements of order $2$ in perfect covers of $\\mathbb{Z}_{2^m} \\oplus \\mathbb{Z}_{2^n}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#exists-add-order-of-eq-two-of-is-perfect","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $m, n$ be natural numbers with $m + n > 0$. Every perfect cover of $\\mathbb{Z}_{2^m} \\oplus \\mathbb{Z}_{2^n}$ contains an element of order $2$.","title":"Elements of order $2$ in perfect covers of $\\mathbb{Z}_{2^m} \\oplus \\mathbb{Z}_{2^n}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/exists-is-perfect-cover-card-odd","kind":"node","lean":"MetaMathlibExt.exists_isPerfectCover_card_odd","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#exists-is-perfect-cover-card-odd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $H$ be a finite abelian group of order $2^m$, let $n \\ge 1$, and let $G = H \\oplus \\mathbb{Z}_{2^n}$. For every $k$ with $1 \\le k \\le m + 1$, some perfect cover $S$ of $G$ has exactly $k$ elements $(h, y)$ whose coordinate $y \\in \\mathbb{Z}_{2^n}$ is odd.","title":"Perfect covers with a prescribed number of odd last coordinates"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/card-perfect-cover-one-odd","kind":"node","lean":"MetaMathlibExt.card_perfectCover_one_odd","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#card-perfect-cover-one-odd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $H$ be an abelian group of order $2^m$, let $n \\ge 1$ and let $G = H \\oplus \\mathbb{Z}_{2^n}$. Call an element of $G$ odd when its $\\mathbb{Z}_{2^n}$-coordinate, taken as a representative in $\\{0, 1, \\dots, 2^n - 1\\}$, is odd. Then the number of perfect covers of $G$ with exactly one odd element is $2^m \\cdot 2^{n-1} \\cdot C(H \\oplus \\mathbb{Z}_{2^{n-1}})$, where $C(K)$ denotes the number of perfect covers of a\u2026","title":"Perfect covers with exactly one odd element"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/perfect-cover-odd-even-span","kind":"node","lean":"MetaMathlibExt.perfectCover_odd_even_span","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#perfect-cover-odd-even-span","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $m$, $n$ and $k$ be natural numbers with $k + 1 \\le m + n$, and let $s = (s_1, \\dots, s_{m+n})$, $s_i = (x_i, y_i)$, be a family in $(\\mathbb{Z}_2)^m \\oplus \\mathbb{Z}_{2^n}$ that is a perfect $1$-cover: every element of the group is the sum of the $s_i$ over exactly one set of indices. Suppose that $y_i$, read as an integer in $\\{0, \\dots, 2^n - 1\\}$, is odd exactly when $i \\le k + 1$. Let $U$ be the\u2026","title":"Spans of the odd and even entries of a perfect cover"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/perfect-cover-even-tail-cover-count","kind":"node","lean":"MetaMathlibExt.perfectCover_even_tail_coverCount","parent":23146,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups/index.html#perfect-cover-even-tail-cover-count","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/perfect-subset-sum-covers-of-abelian-2-groups","status":"can_prove","summary":"Let $s = (s_1, \\dots, s_{m+n})$, $s_i = (x_i, y_i)$, be a family in $(\\mathbb{Z}_2)^m \\oplus \\mathbb{Z}_{2^n}$ that is a perfect $1$-cover: every element of the group is the sum of the $s_i$ over exactly one set of indices. Let $k + 1 \\le m + n$ and suppose that $y_i$, read as an integer in $\\{0, \\dots, 2^n - 1\\}$, is odd exactly when $i \\le k + 1$. Let $V$ be the $\\mathbb{Z}_2$-span of $x_{k+2}, \\dots, x_{m+n}$ in\u2026","title":"The even tail of a perfect cover"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums","kind":"scope","navigation":"module","parent":23136,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums","status":"can_prove","title":"Rearrangements with distinct partial sums"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums/exists-distinct-partial-sums-ordering","kind":"node","lean":"MathlibExt.Combinatorics.ValidOrderingWanted.exists_distinct_partial_sums_ordering","parent":23184,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums/index.html#exists-distinct-partial-sums-ordering","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums","status":"can_prove","summary":"Every finite set $A$ of nonzero integers has a valid ordering: its elements can be listed as $a_1, \\dots, a_m$, each exactly once, so that the partial sums $a_1, a_1 + a_2, \\dots, a_1 + \\cdots + a_m$ are pairwise distinct. The empty sum $0$ is not counted among the partial sums.","title":"Valid orderings of finite sets of nonzero integers"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums/exists-freiman-iso-to-int","kind":"node","lean":"MathlibExt.Combinatorics.ValidOrderingWanted.exists_freimanIso_to_int","parent":23184,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums/index.html#exists-freiman-iso-to-int","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/rearrangements-with-distinct-partial-sums","status":"can_prove","summary":"Let $\\ell \\ge 2$ be a natural number, $p$ a prime and $A$ a subset of $\\mathbb{Z}/p\\mathbb{Z}$ with $|A| \\le \\lceil \\log p / \\log \\ell \\rceil$. Then there are a finite set $B$ of integers and a map $f \\colon \\mathbb{Z}/p\\mathbb{Z} \\to \\mathbb{Z}$ that is a Freiman isomorphism of order $\\ell$ from $A$ to $B$: $f$ maps $A$ bijectively onto $B$, and for all $a_1, \\dots, a_\\ell, b_1, \\dots, b_\\ell \\in A$ we have\u2026","title":"Lev's rectification principle"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","kind":"scope","navigation":"module","parent":23136,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","title":"Subset sums with a bounded number of terms"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/generalized-hfold-sumset","kind":"node","lean":"MetaMathlibExt.generalizedHFoldSumset","parent":23187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized "],["math","\\(h\\)"],["text","-fold sumset"]],"source":"Generalized $h$-fold sumset"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#generalized-hfold-sumset","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For elements $a_0, \\dots, a_{k-1}$ of an additive commutative monoid $G$ (not necessarily distinct), natural numbers $\\bar r = (r_0, \\dots, r_{k-1})$ and a natural number $h$, the generalized $h$-fold sumset of $A = (a_0, \\dots, a_{k-1})$ is the finite set $h^{(\\bar r)}A = \\{\\sum_j s_j a_j : s_j \\in \\mathbb{N}, s_j \\le r_j, \\sum_j s_j = h\\}$, where $s_j a_j$ is the sum of $s_j$ copies of $a_j$.","title":"Generalized $h$-fold sumset"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/yang-chen-bound","kind":"node","lean":"MetaMathlibExt.yangChenBound","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#yang-chen-bound","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"Let $k, h \\in \\mathbb{N}$ and $\\bar r = (r_0, \\dots, r_{k-1})$ with all $r_j \\in \\mathbb{N}$; all sums below run over indices $j \\in \\{0, \\dots, k-1\\}$. Let $\\mu$ be the largest $m \\in \\{0, \\dots, k\\}$ with $\\sum_{j < m} r_j \\le h$, let $\\eta$ be the integer such that $\\eta + 1$ is the least $m \\in \\{0, \\dots, k\\}$ with $\\sum_{j \\ge m} r_j \\le h$ (so $\\eta = -1$ when $h \\ge \\sum_j r_j$), and set\u2026","title":"The Yang\u2013Chen bound"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/yang-chen-bound-one-one-one-two","kind":"node","lean":"MetaMathlibExt.yangChenBound_one_one_one_two","parent":23187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Yang\u2013Chen bound at "],["math","\\((1, 1, 1)\\)"],["text"," and "],["math","\\(h = 2\\)"]],"source":"Yang\u2013Chen bound at $(1, 1, 1)$ and $h = 2$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#yang-chen-bound-one-one-one-two","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"The Yang\u2013Chen bound $L(\\bar r, h)$ for $\\bar r = (1, 1, 1)$ and $h = 2$ equals $3$.","title":"Yang\u2013Chen bound at $(1, 1, 1)$ and $h = 2$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/yang-chen-bound-one-one-two-two","kind":"node","lean":"MetaMathlibExt.yangChenBound_one_one_two_two","parent":23187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Yang\u2013Chen bound at "],["math","\\((1, 1, 2)\\)"],["text"," and "],["math","\\(h = 2\\)"]],"source":"Yang\u2013Chen bound at $(1, 1, 2)$ and $h = 2$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#yang-chen-bound-one-one-two-two","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"The Yang\u2013Chen bound $L(\\bar r, h)$ for $\\bar r = (1, 1, 2)$ and $h = 2$ equals $4$.","title":"Yang\u2013Chen bound at $(1, 1, 2)$ and $h = 2$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/generalized-hfold-sumset-zero-one-two","kind":"node","lean":"MetaMathlibExt.generalizedHFoldSumset_zero_one_two","parent":23187,"reading_title":{"format":"inline-parts-v1","parts":[["text","A generalized "],["math","\\(2\\)"],["text","-fold sumset of "],["math","\\(\\{0, 1, 2\\}\\)"]],"source":"A generalized $2$-fold sumset of $\\{0, 1, 2\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#generalized-hfold-sumset-zero-one-two","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For $A = (0, 1, 2)$ in the integers and $\\bar r = (1, 1, 1)$, the generalized $2$-fold sumset $2^{(\\bar r)}A$, the set of sums $s_0 \\cdot 0 + s_1 \\cdot 1 + s_2 \\cdot 2$ with $s_0, s_1, s_2 \\in \\{0, 1\\}$ and $s_0 + s_1 + s_2 = 2$, equals $\\{1, 2, 3\\}$.","title":"A generalized $2$-fold sumset of $\\{0, 1, 2\\}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subset-sums","kind":"node","lean":"MetaMathlibExt.boundedSubsetSums","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subset-sums","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For natural numbers $\\alpha, \\beta$ and a finite subset $A$ of an additive commutative monoid $G$, the set $\\Sigma^{\\beta}_{\\alpha}(A)$ is the union of the restricted sumsets $m^{\\wedge} A$ over $\\alpha \\le m \\le |A| - \\beta$, that is, the set of sums $\\sum_{b \\in B} b$ over subsets $B \\subseteq A$ with $\\alpha \\le |B| \\le |A| - \\beta$. Here $|A| - \\beta$ is read as $0$ when $\\beta > |A|$.","title":"Subset sums with a bounded number of terms"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subset-sums-one-one-neg-one-one","kind":"node","lean":"MetaMathlibExt.boundedSubsetSums_one_one_neg_one_one_two","parent":23187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounded subset sums of "],["math","\\(\\{-1, 1, 2\\}\\)"]],"source":"Bounded subset sums of $\\{-1, 1, 2\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subset-sums-one-one-neg-one-one","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"The set $\\Sigma^{1}_{1}(\\{-1, 1, 2\\})$ of sums of subsets of the integer set $\\{-1, 1, 2\\}$ with at least $1$ and at most $2$ elements is $\\{-1, 0, 1, 2, 3\\}$.","title":"Bounded subset sums of $\\{-1, 1, 2\\}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/repetition-sequence","kind":"node","lean":"MetaMathlibExt.repetitionSequence","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#repetition-sequence","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"Let $G$ be a type, $k \\in \\mathbb{N}$, $A = (a_0, \\dots, a_{k-1})$ a $k$-tuple of elements of $G$ and $\\bar t = (t_0, \\dots, t_{k-1})$ a $k$-tuple of natural numbers. The sequence $(A, \\bar t)$ is the finite multiset of elements of $G$ obtained by taking $t_i$ copies of $a_i$ for each $i < k$; an element $x \\in G$ thus occurs $\\sum_{i : a_i = x} t_i$ times.","title":"Sequence with prescribed multiplicities"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/card-repetition-sequence","kind":"node","lean":"MetaMathlibExt.card_repetitionSequence","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#card-repetition-sequence","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"Let $G$ be a set, $a_0, \\dots, a_{k-1} \\in G$ (not necessarily distinct), $t_0, \\dots, t_{k-1}$ natural numbers, $A = (a_0, \\dots, a_{k-1})$ and $\\bar t = (t_0, \\dots, t_{k-1})$. The multiset $(A, \\bar t)$, made of $t_i$ copies of $a_i$ for each $i$, has $t_0 + \\dots + t_{k-1}$ elements counted with multiplicity.","title":"Length of a sequence with repetitions"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subsequence-sums","kind":"node","lean":"MetaMathlibExt.boundedSubsequenceSums","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subsequence-sums","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For natural numbers $\\alpha, \\beta$ and a finite sequence $\\mathcal{A}$ (a multiset) in an additive commutative monoid $G$, $\\Sigma^\\beta_\\alpha(\\mathcal{A})$ is the finite set of the sums $\\sigma(\\mathcal{B})$ of the terms of the subsequences $\\mathcal{B}$ of $\\mathcal{A}$ (sub-multisets) with $\\alpha \\le |\\mathcal{B}|$ and $|\\mathcal{B}| + \\beta \\le |\\mathcal{A}|$, that is, with at least $\\alpha$ and at most\u2026","title":"Subsequence sums with a bounded number of terms"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subsequence-sums-one-one-example","kind":"node","lean":"MetaMathlibExt.boundedSubsequenceSums_one_one_example","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subsequence-sums-one-one-example","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For the sequence of integers $\\mathcal{A} = (-2, 7, 7)$, in which $-2$ occurs once and $7$ twice, $\\Sigma^1_1(\\mathcal{A}) = \\{-2, 5, 7, 14\\}$: the sums of the subsequences with at least one and at most two terms are $-2$, $7$, $-2 + 7 = 5$ and $7 + 7 = 14$.","title":"An example of bounded subsequence sums"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/mixed-sign-subset-sum-bound","kind":"node","lean":"MetaMathlibExt.mixedSignSubsetSumBound","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#mixed-sign-subset-sum-bound","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For natural numbers $n$, $p$, $\\alpha$, $\\beta$ (the numbers of negative and positive elements of a set $A$, and the bounds on subset sizes), let $\\kappa = n + p - \\beta$, computed in the integers, $T(x) = x(x+1)/2$ and $S = T(p) + T(n) + 1$. The integer $L_{n,p}(\\alpha, \\beta)$, written $L(\\alpha, \\beta, A)$ in the source, is $S - T(\\beta - n) - T(\\beta - p)$ if $1 \\le \\alpha < \\kappa < n \\le p$;\u2026","title":"Mixed-sign subset-sum bound"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/mixed-sign-subset-sum-bound-one-two-one","kind":"node","lean":"MetaMathlibExt.mixedSignSubsetSumBound_one_two_one_one","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#mixed-sign-subset-sum-bound-one-two-one","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For $n = 1$, $p = 2$ and $\\alpha = \\beta = 1$, the mixed-sign subset-sum bound is $L_{1,2}(1, 1) = 5$; this value comes from its second case.","title":"A value of the mixed-sign subset-sum bound"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/mixed-sign-zero-subset-sum-bound","kind":"node","lean":"MetaMathlibExt.mixedSignZeroSubsetSumBound","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#mixed-sign-zero-subset-sum-bound","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For natural numbers $n$, $p$, $\\alpha$, $\\beta$ (the numbers of negative and positive elements of a set $A$ that also contains $0$, and the bounds on subset sizes), let $\\kappa = n + p + 1 - \\beta$, computed in the integers, $T(x) = x(x+1)/2$, $T_0(x) = x(x-1)/2$ and $S = T(p) + T(n) + 1$. The integer $L^0_{n,p}(\\alpha, \\beta)$, written $L_0(\\alpha, \\beta, A)$ in the source, is $S - T_0(\\beta - n) - T_0(\\beta - p)$\u2026","title":"Mixed-sign subset-sum bound with zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/mixed-sign-zero-subset-sum-bound-one-one","kind":"node","lean":"MetaMathlibExt.mixedSignZeroSubsetSumBound_one_one_one_one","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#mixed-sign-zero-subset-sum-bound-one-one","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"fully_proved","summary":"For $n = p = 1$ and $\\alpha = \\beta = 1$, the mixed-sign subset-sum bound with zero is $L^0_{1,1}(1, 1) = 3$; this value comes from its third case.","title":"A value of the mixed-sign subset-sum bound with zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/yang-chen-card-generalized-hfold-sumset","kind":"node","lean":"MetaMathlibExt.yangChen_card_generalizedHFoldSumset_ge","parent":23187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Yang\u2013Chen lower bound for generalized "],["math","\\(h\\)"],["text","-fold sumsets"]],"source":"Yang\u2013Chen lower bound for generalized $h$-fold sumsets"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#yang-chen-card-generalized-hfold-sumset","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $k, h \\in \\mathbb{N}$ and let $\\bar r = (r_0, \\dots, r_{k-1})$ be positive integers with $2 \\le h \\le r_0 + \\dots + r_{k-1}$, and let $L(\\bar r, h)$ be the Yang\u2013Chen bound. For all integers $a_0 < a_1 < \\dots < a_{k-1}$, the generalized $h$-fold sumset $h^{(\\bar r)}A$ of $A = (a_0, \\dots, a_{k-1})$ has at least $L(\\bar r, h)$ elements; and there exist integers $a_0 < a_1 < \\dots < a_{k-1}$ for which\u2026","title":"Yang\u2013Chen lower bound for generalized $h$-fold sumsets"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/yang-chen-card-generalized-hfold-sumset-2","kind":"node","lean":"MetaMathlibExt.yangChen_card_generalizedHFoldSumset_eq_iff_isAP","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#yang-chen-card-generalized-hfold-sumset-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $k \\ge 5$ and $h \\ge 2$ be integers, let $\\bar r = (r_0, \\dots, r_{k-1})$ be positive integers with $h \\le r_0 + \\dots + r_{k-1} - 2$, and let $L(\\bar r, h)$ be the Yang\u2013Chen bound. For all integers $a_0 < a_1 < \\dots < a_{k-1}$, with $A = \\{a_0, \\dots, a_{k-1}\\}$, the generalized $h$-fold sumset $h^{(\\bar r)}A$ has exactly $L(\\bar r, h)$ elements if and only if $A$ is an arithmetic progression of length $k$.","title":"Yang\u2013Chen inverse theorem for at least five terms"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/yang-chen-card-generalized-hfold-sumset-3","kind":"node","lean":"MetaMathlibExt.yangChen_card_generalizedHFoldSumset_eq_three","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#yang-chen-card-generalized-hfold-sumset-3","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $a_0 < a_1 < a_2$ be integers, $A = \\{a_0, a_1, a_2\\}$, $\\bar r = (r_0, r_1, r_2)$ positive integers and $h \\ge 2$ an integer with $h \\le r_0 + r_1 + r_2 - 2$, and let $L(\\bar r, h)$ be the Yang\u2013Chen bound. If $r_1 = 1$, then the generalized $h$-fold sumset $h^{(\\bar r)}A$ has exactly $L(\\bar r, h)$ elements. If $r_1 \\ge 2$, then $h^{(\\bar r)}A$ has exactly $L(\\bar r, h)$ elements if and only if $A$ is an\u2026","title":"Yang\u2013Chen inverse theorem for three terms"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/yang-chen-card-generalized-hfold-sumset-4","kind":"node","lean":"MetaMathlibExt.yangChen_card_generalizedHFoldSumset_eq_four","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#yang-chen-card-generalized-hfold-sumset-4","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $a_0 < a_1 < a_2 < a_3$ be integers, $A = \\{a_0, a_1, a_2, a_3\\}$, $\\bar r = (r_0, r_1, r_2, r_3)$ positive integers and $h \\ge 2$ an integer with $h \\le r_0 + r_1 + r_2 + r_3 - 2$, and let $L(\\bar r, h)$ be the Yang\u2013Chen bound. If $r_1 = r_2 = 1$, then the generalized $h$-fold sumset $h^{(\\bar r)}A$ has exactly $L(\\bar r, h)$ elements if and only if $a_1 - a_0 = a_3 - a_2$. If $r_1 \\ge 2$ or $r_2 \\ge 2$, then\u2026","title":"Yang\u2013Chen inverse theorem for four terms"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subset-sums-eq-generalized-hfold","kind":"node","lean":"MetaMathlibExt.boundedSubsetSums_eq_generalizedHFoldSumset_of_zero_not_mem","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subset-sums-eq-generalized-hfold","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $G$ be an abelian group and let $A = \\{a_1, \\dots, a_k\\}$ be a set of $k$ distinct nonzero elements of $G$. If $\\alpha, \\beta$ are natural numbers with $\\alpha + \\beta \\le k - 1$, then $\\Sigma^\\beta_\\alpha(A)$, the set of sums of subsets $B \\subseteq A$ with $\\alpha \\le |B| \\le k - \\beta$, equals the generalized $(k - \\beta)$-fold sumset $(k - \\beta)^{(\\bar r)} A_0$ of $A_0 = (a_0, a_1, \\dots, a_k)$, where\u2026","title":"Bounded subset sums as a generalized h-fold sumset when zero is not in the set"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subset-sums-eq-generalized-hfold-2","kind":"node","lean":"MetaMathlibExt.boundedSubsetSums_eq_generalizedHFoldSumset_of_zero_mem","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subset-sums-eq-generalized-hfold-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $G$ be an abelian group, let $m$ be a natural number, let $a_1, \\dots, a_m$ be distinct nonzero elements of $G$, put $a_0 = 0$ and $k = m + 1$, and let $A = \\{a_0, a_1, \\dots, a_m\\}$, a set of $k$ elements. If $\\alpha, \\beta$ are natural numbers with $\\alpha \\ge 1$, $\\beta \\le k - 1$ and $\\alpha + \\beta \\le k$, then $\\Sigma^\\beta_\\alpha(A)$, the set of sums of subsets $B \\subseteq A$ with\u2026","title":"Bounded subset sums as a generalized h-fold sumset when zero is in the set"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/generalized-hfold-sumset-comp-perm","kind":"node","lean":"MetaMathlibExt.generalizedHFoldSumset_comp_perm","parent":23187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reordering invariance of the generalized "],["math","\\(h\\)"],["text","-fold sumset"]],"source":"Reordering invariance of the generalized $h$-fold sumset"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#generalized-hfold-sumset-comp-perm","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $G$ be an additive commutative monoid, $a_0, \\dots, a_{k-1} \\in G$, $r_0, \\dots, r_{k-1}$ and $h$ natural numbers, and write $h^{(\\bar r)}A = \\{\\sum_j s_j a_j : 0 \\le s_j \\le r_j, \\sum_j s_j = h\\}$ for the generalized $h$-fold sumset of $A = (a_0, \\dots, a_{k-1})$ with $\\bar r = (r_0, \\dots, r_{k-1})$. For every permutation $\\pi$ of $\\{0, \\dots, k-1\\}$, the generalized $h$-fold sumset of\u2026","title":"Reordering invariance of the generalized $h$-fold sumset"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subsequence-sums-eq-generalized","kind":"node","lean":"MetaMathlibExt.boundedSubsequenceSums_eq_generalizedHFoldSumset_of_zero_not_mem","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subsequence-sums-eq-generalized","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $G$ be an abelian group and let $a_1, \\dots, a_k$ be distinct nonzero elements of $G$. Let $t_1, \\dots, t_k$ be positive integers, let $\\mathcal{A}$ be the sequence in which each $a_i$ occurs $t_i$ times, and let $h = t_1 + \\dots + t_k$ be its length. If $\\alpha, \\beta$ are natural numbers with $\\alpha \\ge 1$ and $\\alpha + \\beta \\le h - 1$, then $\\Sigma^\\beta_\\alpha(\\mathcal{A})$, the set of sums of\u2026","title":"Bounded subsequence sums as a generalized h-fold sumset when zero is not a term"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subsequence-sums-eq-generalized-2","kind":"node","lean":"MetaMathlibExt.boundedSubsequenceSums_eq_generalizedHFoldSumset_of_zero_mem","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subsequence-sums-eq-generalized-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $G$ be an abelian group, let $m$ be a natural number, let $a_1, \\dots, a_m$ be distinct nonzero elements of $G$, and put $a_0 = 0$. Let $t_0, \\dots, t_m$ be positive integers, let $\\mathcal{A}$ be the sequence in which each $a_i$ occurs $t_i$ times, and let $h = t_0 + \\dots + t_m$ be its length. If $\\alpha, \\beta$ are natural numbers with $\\alpha \\ge 1$, $\\beta \\le h - 1$ and $\\alpha + \\beta \\le h$, then\u2026","title":"Bounded subsequence sums as a generalized h-fold sumset when zero is a term"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/card-bounded-subset-sums-ge-of-nonneg","kind":"node","lean":"MetaMathlibExt.card_boundedSubsetSums_ge_of_nonneg","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#card-bounded-subset-sums-ge-of-nonneg","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $k$, $\\alpha$, $\\beta$ be natural numbers with $k \\ge 2$, $\\alpha \\ge 1$ and $\\alpha + \\beta + 1 \\le k$. For a finite set $A$ of integers write $\\Sigma_\\alpha^\\beta(A)$ for the set of sums of subsets $B \\subseteq A$ with $\\alpha \\le |B| \\le |A| - \\beta$. Every set $A$ of $k$ positive integers satisfies $|\\Sigma_\\alpha^\\beta(A)| \\ge \\binom{k+1}{2} - \\binom{\\alpha+1}{2} - \\binom{\\beta+1}{2} + 1$, and every set\u2026","title":"Sharp lower bounds for bounded subset sums of nonnegative sets"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/card-bounded-subset-sums-ge-mixed-sign","kind":"node","lean":"MetaMathlibExt.card_boundedSubsetSums_ge_mixedSign","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#card-bounded-subset-sums-ge-mixed-sign","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $n$, $p$, $\\alpha$, $\\beta$ be natural numbers with $1 \\le n \\le p$, $1 \\le \\alpha$ and $\\alpha + \\beta + 1 \\le n + p$, and let $L_{n,p}(\\alpha, \\beta)$ be the mixed-sign subset-sum bound. For a finite set $A$ of integers write $\\Sigma_\\alpha^\\beta(A)$ for the set of sums of subsets $B \\subseteq A$ with $\\alpha \\le |B| \\le |A| - \\beta$. Every finite set $A$ of nonzero integers with exactly $n$ negative and\u2026","title":"Lower bound for bounded subset sums of a mixed-sign set"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/card-bounded-subset-sums-ge-mixed-sign-z","kind":"node","lean":"MetaMathlibExt.card_boundedSubsetSums_ge_mixedSign_zero","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#card-bounded-subset-sums-ge-mixed-sign-z","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $n, p, \\alpha, \\beta$ be natural numbers with $1 \\le n \\le p$, $1 \\le \\alpha$ and $\\alpha + \\beta \\le n + p$. Every finite set $A$ of integers that contains $0$, exactly $n$ negative integers and exactly $p$ positive integers satisfies $|\\Sigma^{\\beta}_{\\alpha}(A)| \\ge L_0(\\alpha, \\beta, A)$, and some such set $A$ attains equality. Here $\\Sigma^{\\beta}_{\\alpha}(A)$ is the set of sums of subsets $B \\subseteq A$\u2026","title":"Lower bound for subset sums of a mixed-sign set containing zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subset-sums-inverse-of-nonneg","kind":"node","lean":"MetaMathlibExt.boundedSubsetSums_inverse_of_nonneg","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subset-sums-inverse-of-nonneg","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $k \\ge 3$ and let $\\alpha, \\beta$ be natural numbers with $1 \\le \\alpha \\le k - 2$, $\\beta \\le k - 2$ and $\\alpha + \\beta \\le k - 1$. Let $A$ be a set of $k$ integers, and let $\\Sigma^{\\beta}_{\\alpha}(A)$ be the set of sums of subsets $B \\subseteq A$ with $\\alpha \\le |B| \\le k - \\beta$. If all elements of $A$ are positive and\u2026","title":"Inverse theorem for subset sums of nonnegative sets"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subset-sums-inverse-mixed-sign","kind":"node","lean":"MetaMathlibExt.boundedSubsetSums_inverse_mixedSign","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subset-sums-inverse-mixed-sign","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $n, p, \\alpha, \\beta$ be natural numbers with $1 \\le n \\le p$, put $k = n + p$, and assume $1 \\le \\alpha \\le k - 2$, $\\beta \\le k - 2$ and $\\alpha + \\beta \\le k - 1$. Let $A$ be a finite set of integers that does not contain $0$ and has exactly $n$ negative and exactly $p$ positive elements, and write $E$ for the equation $|\\Sigma^{\\beta}_{\\alpha}(A)| = L(\\alpha, \\beta, A)$. If $k = 3$, $\\alpha = 1$ and\u2026","title":"Inverse theorem for subset sums of a mixed-sign set"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subset-sums-inverse-mixed-sign-z","kind":"node","lean":"MetaMathlibExt.boundedSubsetSums_inverse_mixedSign_zero","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subset-sums-inverse-mixed-sign-z","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $n, p, \\alpha, \\beta$ be natural numbers with $1 \\le n \\le p$, put $k = n + p + 1$, and assume $1 \\le \\alpha \\le k - 2$, $\\beta \\le k - 2$ and $\\alpha + \\beta \\le k - 1$. Let $A$ be a finite set of integers that contains $0$, exactly $n$ negative integers and exactly $p$ positive integers. Then $|\\Sigma^{\\beta}_{\\alpha}(A)| = L_0(\\alpha, \\beta, A)$ if and only if $A = d \\cdot \\{-n, -n + 1, \\dots, p\\}$ for some\u2026","title":"Inverse theorem for subset sums of a mixed-sign set containing zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/card-bounded-subsequence-sums-ge-of-pos","kind":"node","lean":"MetaMathlibExt.card_boundedSubsequenceSums_ge_of_pos","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#card-bounded-subsequence-sums-ge-of-pos","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $k \\ge 2$, let $t_1, \\dots, t_k$ be positive integers, let $h = t_1 + \\dots + t_k$, and let $\\alpha, \\beta$ be natural numbers with $1 \\le \\alpha$ and $\\alpha + \\beta \\le h - 1$. Put $\\bar r = (h - \\alpha - \\beta, t_1, \\dots, t_k)$. For integers $0 < b_1 < \\dots < b_k$, let $\\mathcal{A}$ be the sequence in which $b_i$ occurs $t_i$ times. Then $|\\Sigma^{\\beta}_{\\alpha}(\\mathcal{A})| \\ge L(\\bar r, h - \\beta)$ for\u2026","title":"Lower bound for bounded subsequence sums of a positive sequence"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/card-bounded-subsequence-sums-ge-of-nonn","kind":"node","lean":"MetaMathlibExt.card_boundedSubsequenceSums_ge_of_nonneg_zero","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#card-bounded-subsequence-sums-ge-of-nonn","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $m \\ge 1$, let $t_0, \\dots, t_m$ be positive integers, let $h = t_0 + \\dots + t_m$, and let $\\alpha, \\beta$ be natural numbers with $1 \\le \\alpha$, $\\beta \\le h - 1$ and $\\alpha + \\beta \\le h$. Put $\\bar r = (h - \\alpha - \\beta + t_0, t_1, \\dots, t_m)$. For integers $0 < b_1 < \\dots < b_m$, let $\\mathcal{A}$ be the sequence in which $0$ occurs $t_0$ times and $b_i$ occurs $t_i$ times. Then\u2026","title":"Lower bound for bounded subsequence sums of a nonnegative sequence containing zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/card-bounded-subsequence-sums-ge-mixed-s","kind":"node","lean":"MetaMathlibExt.card_boundedSubsequenceSums_ge_mixedSign","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#card-bounded-subsequence-sums-ge-mixed-s","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $1 \\le n \\le p$, let $t_0, \\dots, t_{n-1}$ and $t_{n+1}, \\dots, t_{n+p}$ be positive integers, let $h$ be their sum, and let $\\alpha, \\beta$ be natural numbers with $1 \\le \\alpha$ and $\\alpha + \\beta \\le h - 1$. Put $\\bar r = (t_0, \\dots, t_{n-1}, h - \\alpha - \\beta, t_{n+1}, \\dots, t_{n+p})$. For integers $c_0 < \\dots < c_{n-1} < 0 < d_1 < \\dots < d_p$, let $\\mathcal{A}$ be the sequence in which $c_i$ occurs\u2026","title":"Lower bound for bounded subsequence sums of a mixed-sign sequence"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/card-bounded-subsequence-sums-ge-mixed-s-2","kind":"node","lean":"MetaMathlibExt.card_boundedSubsequenceSums_ge_mixedSign_zero","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#card-bounded-subsequence-sums-ge-mixed-s-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $1 \\le n \\le p$, let $t_0, \\dots, t_{n-1}$, $t_n$ and $t_{n+1}, \\dots, t_{n+p}$ be positive integers, let $h = t_0 + t_1 + \\dots + t_{n+p}$, and let $\\alpha, \\beta$ be natural numbers with $1 \\le \\alpha$ and $\\alpha + \\beta \\le h - 1$. Put $\\bar r = (t_0, \\dots, t_{n-1}, h - \\alpha - \\beta + t_n, t_{n+1}, \\dots, t_{n+p})$. For integers $c_0 < \\dots < c_{n-1} < 0 < d_1 < \\dots < d_p$, let $\\mathcal{A}$ be the\u2026","title":"Lower bound for bounded subsequence sums of a mixed-sign sequence containing zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subsequence-sums-inverse-of-pos","kind":"node","lean":"MetaMathlibExt.boundedSubsequenceSums_inverse_of_pos","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subsequence-sums-inverse-of-pos","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $k \\ge 2$, let $a_1 < \\dots < a_k$ be positive integers and $A = \\{a_1, \\dots, a_k\\}$, let $t_1, \\dots, t_k$ be positive integers, let $\\mathcal{A}$ be the sequence in which each $a_i$ occurs $t_i$ times, and let $h = t_1 + \\dots + t_k$. Let $\\alpha, \\beta$ be natural numbers with $1 \\le \\alpha \\le h - 2$, $\\beta \\le h - 2$ and $\\alpha + \\beta \\le h - 1$, put $\\bar r = (h - \\alpha - \\beta, t_1, \\dots, t_k)$,\u2026","title":"Inverse theorem for bounded subsequence sums of positive sequences"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subsequence-sums-inverse-of-nonn","kind":"node","lean":"MetaMathlibExt.boundedSubsequenceSums_inverse_of_nonneg_zero","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subsequence-sums-inverse-of-nonn","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $m \\ge 2$, let $0 < a_1 < \\dots < a_m$ be integers, put $a_0 = 0$ and $A = \\{a_0, a_1, \\dots, a_m\\}$, let $t_0, t_1, \\dots, t_m$ be positive integers, let $\\mathcal{A}$ be the sequence in which each $a_i$ occurs $t_i$ times, and let $h = t_0 + t_1 + \\dots + t_m$. Let $\\alpha, \\beta$ be natural numbers with $1 \\le \\alpha \\le h - 2$, $\\beta \\le h - 2$ and $\\alpha + \\beta \\le h - 1$, put\u2026","title":"Inverse theorem for bounded subsequence sums of nonnegative sequences containing zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subsequence-sums-inverse-mixed-s","kind":"node","lean":"MetaMathlibExt.boundedSubsequenceSums_inverse_mixedSign","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subsequence-sums-inverse-mixed-s","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $1 \\le n \\le p$, let $c_1 < \\dots < c_n < 0 < d_1 < \\dots < d_p$ be integers and $A = \\{c_1, \\dots, c_n, d_1, \\dots, d_p\\}$, let $s_1, \\dots, s_n, t_1, \\dots, t_p$ be positive integers, let $\\mathcal{A}$ be the sequence in which each $c_i$ occurs $s_i$ times and each $d_j$ occurs $t_j$ times, and let $h = s_1 + \\dots + s_n + t_1 + \\dots + t_p$, with $h \\ge 3$. Let $\\alpha, \\beta$ be natural numbers with\u2026","title":"Inverse theorem for bounded subsequence sums of mixed-sign sequences without zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/bounded-subsequence-sums-inverse-mixed-s-2","kind":"node","lean":"MetaMathlibExt.boundedSubsequenceSums_inverse_mixedSign_zero","parent":23187,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms/index.html#bounded-subsequence-sums-inverse-mixed-s-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/subset-sums-with-a-bounded-number-of-terms","status":"can_prove","summary":"Let $1 \\le n \\le p$, let $c_1 < \\dots < c_n < 0 < d_1 < \\dots < d_p$ be integers and $A = \\{c_1, \\dots, c_n, 0, d_1, \\dots, d_p\\}$, a set of $k = n + p + 1$ elements, let $s_1, \\dots, s_n, z, t_1, \\dots, t_p$ be positive integers, let $\\mathcal{A}$ be the sequence in which each $c_i$ occurs $s_i$ times, $0$ occurs $z$ times and each $d_j$ occurs $t_j$ times, and let $h = s_1 + \\dots + s_n + z + t_1 + \\dots + t_p$.\u2026","title":"Inverse theorem for bounded subsequence sums of mixed-sign sequences containing zero"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant","kind":"scope","navigation":"module","parent":23136,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant","status":"can_prove","title":"Weighted Davenport constant"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","kind":"scope","navigation":"transparent","parent":23226,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","title":"Weighted zero-sum free sequences to Weighted zero-sum subsequences are monotone in the weight set"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/is-weighted-zero-sum-free","kind":"node","lean":"MetaMathlibExt.IsWeightedZeroSumFree","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#is-weighted-zero-sum-free","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid and $A$ a finite set of natural numbers. A finite sequence $s = (s_1, \\dots, s_n)$ over $G$ is $A$-weighted zero-sum free when, for every non-empty set $I \\subseteq \\{1, \\dots, n\\}$ of positions and every choice of weights $a_i \\in A$ for $i \\in I$, the weighted sum $\\sum_{i \\in I} a_i s_i$ is non-zero, where $a_i s_i$ is the sum of $a_i$ copies of $s_i$. Terms are indexed\u2026","title":"Weighted zero-sum free sequences"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/weighted-davenport","kind":"node","lean":"MetaMathlibExt.weightedDavenport","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#weighted-davenport","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid and $A$ a finite set of natural numbers. The $A$-weighted Davenport constant $D_A(G) \\in \\mathbb{N} \\cup \\{\\infty\\}$ is the least length $\\ell \\in \\mathbb{N}$ such that no sequence over $G$ of length $\\ell$ is $A$-weighted zero-sum free, that is, such that every sequence over $G$ of length $\\ell$ has a non-empty $A$-weighted zero-sum subsequence; it is $\\infty$ when no such\u2026","title":"Weighted Davenport constant"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/weighted-davenport-eq-nat-cast-s-inf","kind":"node","lean":"MetaMathlibExt.weightedDavenport_eq_natCast_sInf","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#weighted-davenport-eq-nat-cast-s-inf","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $A$ a finite set of natural numbers and $D_A(G)$ the $A$-weighted Davenport constant of $G$. If there is a length $\\ell \\in \\mathbb{N}$ such that no sequence over $G$ of length $\\ell$ is $A$-weighted zero-sum free, then $D_A(G)$ is finite and equals the least such length.","title":"The weighted Davenport constant as a least length"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/take","kind":"node","lean":"MetaMathlibExt.IsWeightedZeroSumFree.take","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#take","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid and $A$ a finite set of natural numbers. If a sequence $s$ over $G$ is $A$-weighted zero-sum free, then for every $\\ell \\in \\mathbb{N}$ the prefix of $s$ formed by its first $\\ell$ terms (all of $s$ when $\\ell$ is at least its length) is $A$-weighted zero-sum free.","title":"Prefixes of weighted zero-sum free sequences"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/weighted-davenport-le","kind":"node","lean":"MetaMathlibExt.weightedDavenport_le","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#weighted-davenport-le","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $A$ a finite set of natural numbers, $D_A(G)$ the $A$-weighted Davenport constant of $G$ and $\\ell$ a natural number. If no sequence over $G$ of length $\\ell$ is $A$-weighted zero-sum free, that is, every sequence over $G$ of length $\\ell$ has a non-empty $A$-weighted zero-sum subsequence, then $D_A(G) \\le \\ell$.","title":"Upper bound for the weighted Davenport constant"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/lt-weighted-davenport","kind":"node","lean":"MetaMathlibExt.lt_weightedDavenport","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#lt-weighted-davenport","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $A$ a finite set of natural numbers and $D_A(G)$ the $A$-weighted Davenport constant of $G$. If a sequence over $G$ of length $L$ is $A$-weighted zero-sum free, then $L < D_A(G)$.","title":"Lower bound for the weighted Davenport constant"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/weighted-davenport-eq-top-iff","kind":"node","lean":"MetaMathlibExt.weightedDavenport_eq_top_iff","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#weighted-davenport-eq-top-iff","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $G$ be an additive commutative monoid, $A$ a finite set of natural numbers and $D_A(G)$ the $A$-weighted Davenport constant of $G$. Then $D_A(G) = \\infty$ if and only if for every $\\ell \\in \\mathbb{N}$ there is an $A$-weighted zero-sum free sequence over $G$ of length $\\ell$.","title":"When the weighted Davenport constant is infinite"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/weighted-davenport-zmod-one","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_one","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#weighted-davenport-zmod-one","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"For every non-empty finite set $A$ of natural numbers, the $A$-weighted Davenport constant of the trivial group $\\mathbb{Z}/1\\mathbb{Z}$ is $D_A(\\mathbb{Z}/1\\mathbb{Z}) = 1$.","title":"Weighted Davenport constant of the trivial group"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/weighted-davenport-singleton-one-zmod-tw","kind":"node","lean":"MetaMathlibExt.weightedDavenport_singleton_one_zmod_two","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#weighted-davenport-singleton-one-zmod-tw","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"The ordinary Davenport constant of $\\mathbb{Z}/2\\mathbb{Z}$, that is, its $\\{1\\}$-weighted Davenport constant, is $D_{\\{1\\}}(\\mathbb{Z}/2\\mathbb{Z}) = 2$.","title":"Davenport constant of the group of order two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/unit-set","kind":"node","lean":"MetaMathlibExt.unitSet","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#unit-set","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"For a natural number $n$, $U(n)$ is the set of units of the ring $\\mathbb{Z}/n\\mathbb{Z}$ of integers modulo $n$.","title":"Units modulo n"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/unit-square-set","kind":"node","lean":"MetaMathlibExt.unitSquareSet","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#unit-square-set","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"For a natural number $n$, $U(n)^2$ is the set of elements of $\\mathbb{Z}/n\\mathbb{Z}$ of the form $u^2$ with $u$ a unit of $\\mathbb{Z}/n\\mathbb{Z}$.","title":"Squares of units modulo n"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/square-set","kind":"node","lean":"MetaMathlibExt.squareSet","parent":23227,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonzero squares modulo "],["math","\\(n\\)"]],"source":"Nonzero squares modulo $n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#square-set","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"For a natural number $n$, $S(n)$ is the set of elements $x$ of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$ with $x \\neq 0$ and $x = y^2$ for some $y \\in \\mathbb{Z}_n$.","title":"Nonzero squares modulo $n$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/has-weighted-zero-sum","kind":"node","lean":"MetaMathlibExt.HasWeightedZeroSum","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#has-weighted-zero-sum","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$ and let $S = (x_1, \\dots, x_k)$ be a sequence in $\\mathbb{Z}/n\\mathbb{Z}$. Then $S$ is an $A$-weighted zero-sum sequence if there are weights $a_1, \\dots, a_k \\in A$ with $a_1 x_1 + a_2 x_2 + \\dots + a_k x_k = 0$.","title":"Weighted zero-sum sequences"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/has-weighted-zero-sum-subseq","kind":"node","lean":"MetaMathlibExt.HasWeightedZeroSumSubseq","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#has-weighted-zero-sum-subseq","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$ and let $S = (x_1, \\dots, x_k)$ be a sequence in $\\mathbb{Z}/n\\mathbb{Z}$. Then $S$ has an $A$-weighted zero-sum subsequence if there are a nonempty set $J \\subseteq \\{1, \\dots, k\\}$ of indices and weights $a_i \\in A$ for $i \\in J$ with $\\sum_{i \\in J} a_i x_i = 0$.","title":"Weighted zero-sum subsequences"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/has-consecutive-weighted-zero-sum-subseq","kind":"node","lean":"MetaMathlibExt.HasConsecutiveWeightedZeroSumSubseq","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#has-consecutive-weighted-zero-sum-subseq","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$ and let $S = (x_1, \\dots, x_k)$ be a sequence in $\\mathbb{Z}/n\\mathbb{Z}$. Then $S$ has an $A$-weighted zero-sum consecutive subsequence if there are indices $1 \\le l \\le h \\le k$ and weights $a_l, \\dots, a_h \\in A$ with $\\sum_{i=l}^{h} a_i x_i = 0$.","title":"Consecutive weighted zero-sum subsequences"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/zmod-weight-reps","kind":"node","lean":"MetaMathlibExt.zmodWeightReps","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#zmod-weight-reps","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $n$ be a natural number and $A$ a subset of $\\mathbb{Z}_n$. The representatives of $A$ form the finite set of natural numbers $a$ with $0 \\le a < n$ whose residue modulo $n$ lies in $A$.","title":"Representatives of a set of residues"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/mem-zmod-weight-reps","kind":"node","lean":"MetaMathlibExt.mem_zmodWeightReps","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#mem-zmod-weight-reps","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$. A natural number $a$ lies in the set of weight representatives of $A$ if and only if $a < n$ and the residue of $a$ modulo $n$ lies in $A$.","title":"Membership in the weight representatives"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/weighted-consecutive-defining-set","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveDefiningSet","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#weighted-consecutive-defining-set","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $n$ be a natural number and $A$ a subset of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$. This is the set of positive integers $k$ such that every sequence $(x_1, \\dots, x_k)$ of elements of $\\mathbb{Z}_n$ has a nonempty block of consecutive terms $x_i, \\dots, x_j$ with weights $a_i, \\dots, a_j \\in A$ such that $a_i x_i + \\dots + a_j x_j = 0$. Its least element, or $0$ when it is empty, is the $A$-weighted\u2026","title":"Lengths forcing a consecutive weighted zero sum"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/weighted-consecutive-constant","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#weighted-consecutive-constant","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $n$ be a natural number and $A$ a subset of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$. The $A$-weighted consecutive constant $C_A$ of $\\mathbb{Z}_n$ is the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ of elements of $\\mathbb{Z}_n$ has a nonempty block of consecutive terms $x_i, \\dots, x_j$ with weights $a_i, \\dots, a_j \\in A$ such that $a_i x_i + \\dots + a_j x_j = 0$; it is $0$ when no\u2026","title":"Weighted consecutive zero-sum constant"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/mono","kind":"node","lean":"MetaMathlibExt.HasWeightedZeroSumSubseq.mono","parent":23227,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01/index.html#mono","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-01","status":"fully_proved","summary":"Let $A \\subseteq B \\subseteq \\mathbb{Z}/n\\mathbb{Z}$ and let $S$ be a finite sequence in $\\mathbb{Z}/n\\mathbb{Z}$. If $S$ has an $A$-weighted zero-sum subsequence, then $S$ has a $B$-weighted zero-sum subsequence.","title":"Weighted zero-sum subsequences are monotone in the weight set"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","kind":"scope","navigation":"transparent","parent":23226,"reading_title":{"format":"inline-parts-v1","parts":[["text","The one-term zero sequence has a square-weighted zero sum to Square-weighted constants of nonsquares with even "],["math","\\(2\\)"],["text","-adic valuation"]],"source":"The one-term zero sequence has a square-weighted zero sum to Square-weighted constants of nonsquares with even $2$-adic valuation"},"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","title":"The one-term zero sequence has a square-weighted zero sum to Square-weighted constants of nonsquares with even $2$-adic valuation"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/zero-singleton-has-weighted-zero-sum","kind":"node","lean":"MetaMathlibExt.zeroSingleton_hasWeightedZeroSum","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#zero-singleton-has-weighted-zero-sum","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"fully_proved","summary":"Let $n \\ge 2$ and let $S(n)$ be the set of nonzero squares in $\\mathbb{Z}_n$. The sequence over $\\mathbb{Z}_n$ whose only term is $0$ has an $S(n)$-weighted zero sum: there is a weight $a \\in S(n)$ with $a \\cdot 0 = 0$.","title":"The one-term zero sequence has a square-weighted zero sum"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/square-set-prime-power-decomposition","kind":"node","lean":"MetaMathlibExt.squareSet_primePower_decomposition","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#square-set-prime-power-decomposition","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $p$ be a prime, let $r \\ge 1$, and let $l = \\lfloor (r - 1)/2 \\rfloor$. Write $S(p^r)$ for the set of nonzero squares in $\\mathbb{Z}_{p^r}$, $U(p^r)^2$ for the set of squares of units in $\\mathbb{Z}_{p^r}$, and $p^{2k} U(p^r)^2 = \\{p^{2k} x : x \\in U(p^r)^2\\}$. Then","title":"Decomposition of the nonzero squares modulo a prime power"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/ncard-scale-unit-squares-prime-power","kind":"node","lean":"MetaMathlibExt.ncard_scale_unitSquares_primePower","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#ncard-scale-unit-squares-prime-power","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $p$ be a prime and $r, k$ natural numbers with $k = 0$ or $2k < r$, and write $U(m)^2$ for the set of squares of units of $\\mathbb{Z}/m\\mathbb{Z}$. Then the set $p^{2k} U(p^r)^2 = \\{p^{2k} x : x \\in U(p^r)^2\\}$ in $\\mathbb{Z}/p^r\\mathbb{Z}$ has the same number of elements as $U(p^{r-2k})^2$.","title":"Scaled unit squares modulo a prime power"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/ncard-square-set-prime-power","kind":"node","lean":"MetaMathlibExt.ncard_squareSet_primePower","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#ncard-square-set-prime-power","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $p$ be a prime and $r \\ge 1$, and write $S(m)$ for the set of nonzero squares and $U(m)^2$ for the set of squares of units in $\\mathbb{Z}/m\\mathbb{Z}$. If $r$ is even, then $|S(p^r)| = \\sum_{k=0}^{r/2-1} |U(p^{r-2k})^2|$, and if $r$ is odd, then $|S(p^r)| = \\sum_{k=0}^{(r-1)/2} |U(p^{r-2k})^2|$.","title":"Number of nonzero squares modulo a prime power"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/square-weighted-zero-sum-of-prime-power","kind":"node","lean":"MetaMathlibExt.squareWeightedZeroSum_of_primePower_reduction","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#square-weighted-zero-sum-of-prime-power","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $n \\ge 2$, let $p$ be a prime dividing $n$, and let $r$ be the exponent of $p$ in $n$. Write $S(m)$ for the set of nonzero squares in $\\mathbb{Z}_m$. Let $x_1, \\dots, x_k$ be a sequence in $\\mathbb{Z}_n$ whose reduction $\\bar x_1, \\dots, \\bar x_k$ modulo $p^r$ has an $S(p^r)$-weighted zero sum, that is, there are $b_1, \\dots, b_k \\in S(p^r)$ with $\\sum_i b_i \\bar x_i = 0$. Then there are\u2026","title":"Square-weighted zero sums lift from a prime-power component"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/weighted-consecutive-constant-square-set","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_le_primePower","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#weighted-consecutive-constant-square-set","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $n \\ge 2$, let $p$ be a prime dividing $n$ and let $r = v_p(n)$ be the exponent of $p$ in $n$. For a natural number $m$, write $S(m)$ for the set of nonzero squares in $\\mathbb{Z}_m = \\mathbb{Z}/m\\mathbb{Z}$ and $C_{S(m)}$ for the $S(m)$-weighted consecutive constant of $\\mathbb{Z}_m$. Then $C_{S(n)} \\le C_{S(p^r)}$.","title":"Prime-power bound for the square-weighted consecutive constant"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/square-weighted-zero-sum-of-reduction-od","kind":"node","lean":"MetaMathlibExt.squareWeightedZeroSum_of_reduction_oddPower","parent":23248,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-weighted zero sums lift from "],["math","\\(p\\)"],["text"," to odd powers "],["math","\\(p^r\\)"]],"source":"Square-weighted zero sums lift from $p$ to odd powers $p^r$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#square-weighted-zero-sum-of-reduction-od","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $p \\ge 2$ be a natural number and $r$ an odd natural number. Write $S(m)$ for the set of nonzero squares in $\\mathbb{Z}_m$. Let $x_1, \\dots, x_k$ be a sequence in $\\mathbb{Z}_{p^r}$ whose reduction $\\bar x_1, \\dots, \\bar x_k$ modulo $p$ has an $S(p)$-weighted zero sum, that is, there are $b_1, \\dots, b_k \\in S(p)$ with $\\sum_i b_i \\bar x_i = 0$. Then there are $a_1, \\dots, a_k \\in S(p^r)$ with\u2026","title":"Square-weighted zero sums lift from $p$ to odd powers $p^r$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/no-square-weighted-subseq-three-units-tw","kind":"node","lean":"MetaMathlibExt.no_squareWeightedSubseq_three_units_twoPower","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#no-square-weighted-subseq-three-units-tw","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $r$ be a nonzero even natural number, and let $x_1, x_2, x_3$ be units of $\\mathbb{Z}_{2^r}$ that are each congruent to $1$ modulo $4$. Write $S(2^r)$ for the set of nonzero squares in $\\mathbb{Z}_{2^r}$. Then $(x_1, x_2, x_3)$ has no $S(2^r)$-weighted zero-sum subsequence: there are no non-empty set $J$ of indices and weights $a_j \\in S(2^r)$ for $j \\in J$ with $\\sum_{j \\in J} a_j x_j = 0$.","title":"Square-weighted zero-sum free unit triples modulo even powers of two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/no-square-weighted-subseq-pair-and-prime","kind":"node","lean":"MetaMathlibExt.no_squareWeightedSubseq_pair_and_prime_of_reduction","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#no-square-weighted-subseq-pair-and-prime","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $p$ be a prime and $r$ a nonzero even natural number, and write $S(m)$ for the set of nonzero squares in $\\mathbb{Z}/m\\mathbb{Z}$. Let $z_1, z_2$ be units of $\\mathbb{Z}/p^2\\mathbb{Z}$ whose reduction $(z_1 \\bmod p, z_2 \\bmod p)$ is not an $S(p)$-weighted zero-sum sequence. Let $T = (x_1, x_2, y_1)$ be a sequence in $\\mathbb{Z}/p^r\\mathbb{Z}$ whose reduction modulo $p^2$ is $(z_1, z_2, p)$. Then $T$ has no\u2026","title":"Three-term sequences without square-weighted zero-sum subsequences"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/weighted-davenport-square-set-even-squar","kind":"node","lean":"MetaMathlibExt.weightedDavenport_squareSet_evenSquare_lowerBound","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#weighted-davenport-square-set-even-squar","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $n \\ge 2$ be an even perfect square. Write $S(n)$ for the set of nonzero squares in $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$ and $D_{S(n)} \\in \\mathbb{N} \\cup \\{\\infty\\}$ for the weighted Davenport constant of $\\mathbb{Z}_n$ with weight set $S(n)$, a weight acting by multiplication. Then $D_{S(n)} \\ge 4$.","title":"Square-weighted Davenport constant of even squares is at least four"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/square-weighted-zero-sum-of-reduction-sq","kind":"node","lean":"MetaMathlibExt.squareWeightedZeroSum_of_reduction_square","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#square-weighted-zero-sum-of-reduction-sq","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $p \\ge 2$, $r \\ne 0$ and $k$ be natural numbers with $r$ even, and let $t_1, \\dots, t_k$ be elements of $\\mathbb{Z}/p^r\\mathbb{Z}$, with reductions $\\bar t_1, \\dots, \\bar t_k$ modulo $p^2$. Write $U(p^2)^2$ for the set of squares of units of $\\mathbb{Z}/p^2\\mathbb{Z}$ and $S(p^r)$ for the set of nonzero squares of $\\mathbb{Z}/p^r\\mathbb{Z}$. If there are $a_1, \\dots, a_k \\in U(p^2)^2$ with\u2026","title":"Lifting weighted zero sums from p\u00b2 to an even power of p"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/square-set-constants-le-unit-squares-mod","kind":"node","lean":"MetaMathlibExt.squareSet_constants_le_unitSquares_modSquare","parent":23248,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-weighted constants modulo "],["math","\\(p^r\\)"],["text",", for "],["math","\\(p \\ge 2\\)"],["text"," and nonzero even "],["math","\\(r\\)"],["text",", are at most unit-square constants modulo "],["math","\\(p^2\\)"]],"source":"Square-weighted constants modulo $p^r$, for $p \\ge 2$ and nonzero even $r$, are at most unit-square constants modulo $p^2$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#square-set-constants-le-unit-squares-mod","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Square-weighted constants modulo $p^r$, for $p \\ge 2$ and nonzero even $r$, are at most unit-square constants modulo $p^2$. Let $p \\ge 2$ be a natural number and let $r$ be a nonzero even natural number. Write $S(p^r)$ for the set of nonzero squares in $\\mathbb{Z}_{p^r}$ and $U(p^2)^2$ for the set of squares of units in $\\mathbb{Z}_{p^2}$. For a weight set $A \\subseteq \\mathbb{Z}_m$, write $D_A(\\mathbb{Z}_m)$ for\u2026","title":"Square-weighted constants modulo $p^r$, for $p \\ge 2$ and nonzero even $r$, are at most unit-square constants modulo $p^2$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/weighted-consecutive-constant-square-set-2","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_twoPower_le_four","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#weighted-consecutive-constant-square-set-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"For every even natural number $r \\ne 0$, the consecutive $S(2^r)$-weighted zero-sum constant satisfies $C_{S(2^r)} \\le 4$, where $S(2^r)$ is the set of nonzero squares of $\\mathbb{Z}/2^r\\mathbb{Z}$.","title":"Consecutive square-weighted constant of an even power of two is at most four"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/square-set-constants-even-square","kind":"node","lean":"MetaMathlibExt.squareSet_constants_evenSquare","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#square-set-constants-even-square","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $n \\ge 2$ be an even perfect square and let $S(n)$ be the set of nonzero squares in $\\mathbb{Z}_n$. Then the $S(n)$-weighted Davenport constant of $\\mathbb{Z}_n$, with weights the representatives in $\\{0, \\dots, n - 1\\}$ of the elements of $S(n)$, equals $4$. Moreover, some positive length $k$ forces every sequence of length $k$ in $\\mathbb{Z}_n$ to have a non-empty consecutive $S(n)$-weighted zero-sum\u2026","title":"Square-weighted constants of even squares"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/exists-pair-units-avoiding-prime-square","kind":"node","lean":"MetaMathlibExt.exists_pair_units_avoiding_prime_squareWeights","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#exists-pair-units-avoiding-prime-square","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $n \\ge 2$ be odd, and write $S(m)$ for the set of nonzero squares in $\\mathbb{Z}/m\\mathbb{Z}$. Then there are units $u, v$ of $\\mathbb{Z}/n\\mathbb{Z}$ such that for every prime $p$ dividing $n$, the sequence $(u \\bmod p, v \\bmod p)$ in $\\mathbb{Z}/p\\mathbb{Z}$ has no $S(p)$-weighted zero-sum subsequence.","title":"Two units without square-weighted zero sums modulo any prime divisor"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/no-square-weighted-subseq-pair-of-reduct","kind":"node","lean":"MetaMathlibExt.no_squareWeightedSubseq_pair_of_reduction","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#no-square-weighted-subseq-pair-of-reduct","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $p$ be an odd prime, let $r \\ge 1$, and let $x_1, x_2$ be units of $\\mathbb{Z}_{p^r}$. Write $S(m)$ for the set of nonzero squares in $\\mathbb{Z}_m$, and say that a sequence $(y_i)$ in $\\mathbb{Z}_m$ has an $A$-weighted zero-sum subsequence when there are a non-empty set $J$ of indices and weights $a_j \\in A$ for $j \\in J$ with $\\sum_{j \\in J} a_j y_j = 0$. If the reduction $(\\bar x_1, \\bar x_2)$ of\u2026","title":"Square-weighted zero-sum free unit pairs lift to odd prime powers"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/weighted-davenport-square-set-odd-lower","kind":"node","lean":"MetaMathlibExt.weightedDavenport_squareSet_odd_lowerBound","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#weighted-davenport-square-set-odd-lower","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $n \\ge 2$ be odd, let $S(n)$ be the set of nonzero squares in $\\mathbb{Z}_n$, and let $A$ be the set of integers $a$ with $0 \\le a < n$ whose residue modulo $n$ lies in $S(n)$. Then the $A$-weighted Davenport constant $D_{S(n)}$ of $\\mathbb{Z}_n$ is at least $3$.","title":"The square-weighted Davenport constant of an odd modulus is at least three"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/weighted-davenport-square-set-even-two-v","kind":"node","lean":"MetaMathlibExt.weightedDavenport_squareSet_even_twoValuation_lowerBound","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#weighted-davenport-square-set-even-two-v","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $n \\ge 2$ be a natural number whose exponent $v_2(n)$ of $2$ is even and at least $2$. Write $S(n)$ for the set of nonzero squares in $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$ and $D_{S(n)} \\in \\mathbb{N} \\cup \\{\\infty\\}$ for the weighted Davenport constant of $\\mathbb{Z}_n$ with weight set $S(n)$, a weight acting by multiplication. Then $D_{S(n)} \\ge 3$.","title":"Square-weighted Davenport constant for an even 2-adic valuation of at least two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/weighted-consecutive-constant-square-set-3","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_oddPrimePower_le_three","parent":23248,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#weighted-consecutive-constant-square-set-3","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"For every odd prime $p$ and every odd natural number $r$, the consecutive $S(p^r)$-weighted zero-sum constant satisfies $C_{S(p^r)} \\le 3$, where $S(p^r)$ is the set of nonzero squares of $\\mathbb{Z}/p^r\\mathbb{Z}$.","title":"Consecutive square-weighted constant of an odd prime to an odd power is at most three"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/square-set-constants-nonsquare-even-two","kind":"node","lean":"MetaMathlibExt.squareSet_constants_nonsquare_even_twoValuation","parent":23248,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-weighted constants of nonsquares with even "],["math","\\(2\\)"],["text","-adic valuation"]],"source":"Square-weighted constants of nonsquares with even $2$-adic valuation"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02/index.html#square-set-constants-nonsquare-even-two","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-02","status":"can_prove","summary":"Let $n \\ge 2$ be a natural number that is not a perfect square and whose $2$-adic valuation $v_2(n)$ is even (possibly zero), and let $S(n)$ be the set of nonzero squares in $\\mathbb{Z}_n$. Then the $S(n)$-weighted Davenport constant of $\\mathbb{Z}_n$, with weights the representatives in $\\{0, \\dots, n - 1\\}$ of the elements of $S(n)$, equals $3$. Moreover, some positive length $k$ forces every sequence of length\u2026","title":"Square-weighted constants of nonsquares with even $2$-adic valuation"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","kind":"scope","navigation":"transparent","parent":23226,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","title":"Consecutive square-weighted constant of an odd power of two is at most two to A nested ceiling identity for divisors"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-consecutive-constant-square-set","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_odd_twoPower_le_two","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-consecutive-constant-square-set","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"For every odd natural number $r$, the consecutive $S(2^r)$-weighted zero-sum constant satisfies $C_{S(2^r)} \\le 2$, where $S(2^r)$ is the set of nonzero squares of $\\mathbb{Z}/2^r\\mathbb{Z}$.","title":"Consecutive square-weighted constant of an odd power of two is at most two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/square-set-constants-odd-two-valuation","kind":"node","lean":"MetaMathlibExt.squareSet_constants_odd_twoValuation","parent":23269,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-weighted constants of numbers with odd "],["math","\\(2\\)"],["text","-adic valuation"]],"source":"Square-weighted constants of numbers with odd $2$-adic valuation"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#square-set-constants-odd-two-valuation","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $n$ be a positive even natural number whose $2$-adic valuation $v_2(n)$ is odd, and let $S(n)$ be the set of nonzero squares in $\\mathbb{Z}_n$. Then the $S(n)$-weighted Davenport constant of $\\mathbb{Z}_n$, with weights the representatives in $\\{0, \\dots, n - 1\\}$ of the elements of $S(n)$, equals $2$. Moreover, some positive length $k$ forces every sequence of length $k$ in $\\mathbb{Z}_n$ to have a non-empty\u2026","title":"Square-weighted constants of numbers with odd $2$-adic valuation"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/no-square-weighted-subseq-four-term-patt","kind":"node","lean":"MetaMathlibExt.no_squareWeightedSubseq_fourTerm_pattern","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#no-square-weighted-subseq-four-term-patt","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $p$ be a prime and $r$ a nonzero even natural number, and write $S(m)$ for the set of nonzero squares in $\\mathbb{Z}/m\\mathbb{Z}$. Let $w_1, w_2, u$ be units of $\\mathbb{Z}/p^r\\mathbb{Z}$ such that the reduction $(w_1 \\bmod p, w_2 \\bmod p)$ has no $S(p)$-weighted zero-sum subsequence. Then the sequence $(u w_1, u w_2, p w_1, p w_2)$ in $\\mathbb{Z}/p^r\\mathbb{Z}$ has no $S(p^r)$-weighted zero-sum subsequence.","title":"Four-term sequences without square-weighted zero-sum subsequences"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-davenport-square-set-odd-square","kind":"node","lean":"MetaMathlibExt.weightedDavenport_squareSet_oddSquare_lowerBound","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-davenport-square-set-odd-square","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $n \\ge 2$ be an odd perfect square, let $S(n)$ be the set of nonzero squares in $\\mathbb{Z}_n$, and let $A$ be the set of integers $a$ with $0 \\le a < n$ whose residue modulo $n$ lies in $S(n)$. Then the $A$-weighted Davenport constant $D_{S(n)}$ of $\\mathbb{Z}_n$ is at least $5$.","title":"The square-weighted Davenport constant of an odd square is at least five"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/three-unit-square-cosets-cover","kind":"node","lean":"MetaMathlibExt.three_unitSquare_cosets_cover","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#three-unit-square-cosets-cover","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $p$ be an odd prime, let $r \\ge 1$, and let $A = U(p^r)^2$ be the set of squares of units of $\\mathbb{Z}/p^r\\mathbb{Z}$; for $y \\in \\mathbb{Z}/p^r\\mathbb{Z}$ write $Ay = \\{ay : a \\in A\\}$. For all units $y_1, y_2, y_3$ of $\\mathbb{Z}/p^r\\mathbb{Z}$, every element of $\\mathbb{Z}/p^r\\mathbb{Z}$ is a sum $x_1 + x_2 + x_3$ with $x_1 \\in Ay_1$, $x_2 \\in Ay_2 \\cup \\{0\\}$ and $x_3 \\in Ay_3 \\cup \\{0\\}$.","title":"Three unit-square cosets sum to every residue modulo an odd prime power"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-davenport-square-set-odd-prime","kind":"node","lean":"MetaMathlibExt.weightedDavenport_squareSet_oddPrimePower_le_five","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-davenport-square-set-odd-prime","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $p$ be an odd prime, let $r$ be a nonzero even natural number, let $S(p^r)$ be the set of nonzero squares in $\\mathbb{Z}_{p^r}$, and let $A$ be the set of integers $a$ with $0 \\le a < p^r$ whose residue modulo $p^r$ lies in $S(p^r)$. Then the $A$-weighted Davenport constant $D_{S(p^r)}$ of $\\mathbb{Z}_{p^r}$ is at most $5$.","title":"The square-weighted Davenport constant of an even power of an odd prime is at most five"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-davenport-square-set-odd-square-2","kind":"node","lean":"MetaMathlibExt.weightedDavenport_squareSet_oddSquare","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-davenport-square-set-odd-square-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $n \\ge 2$ be an odd perfect square, let $S(n)$ be the set of nonzero squares in $\\mathbb{Z}_n$, and let $A$ be the set of integers $a$ with $0 \\le a < n$ whose residue modulo $n$ lies in $S(n)$. Then the $A$-weighted Davenport constant $D_{S(n)}$ of $\\mathbb{Z}_n$ equals $5$.","title":"The square-weighted Davenport constant of an odd square is five"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-consecutive-constant-unit-squar","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_unitSquares_twentyFive","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-consecutive-constant-unit-squar","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $U(25)^2$ be the set of squares of units of $\\mathbb{Z}_{25} = \\mathbb{Z}/25\\mathbb{Z}$. Some positive length $k$ has the property that every sequence of $k$ elements of $\\mathbb{Z}_{25}$ has a nonempty block of consecutive terms with a $U(25)^2$-weighted sum equal to zero, and the least such length, the $U(25)^2$-weighted consecutive constant $C_{U(25)^2}$ of $\\mathbb{Z}_{25}$, equals $9$.","title":"Unit-square-weighted consecutive constant modulo 25 equals nine"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-consecutive-constant-square-set-2","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_oddSquare_le_nine","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-consecutive-constant-square-set-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"For every odd natural number $n \\ge 2$ that is a perfect square, the consecutive $S(n)$-weighted zero-sum constant satisfies $C_{S(n)} \\le 9$, where $S(n)$ is the set of nonzero squares of $\\mathbb{Z}/n\\mathbb{Z}$.","title":"Consecutive square-weighted constant of an odd square is at most nine"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-consecutive-constant-square-set-3","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_largePrimePower_le_five","parent":23269,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-weighted consecutive constant of "],["math","\\(p^r\\)"],["text"," for prime "],["math","\\(p \\ge 7\\)"],["text"," and even "],["math","\\(r \\ge 4\\)"],["text"," is at most five"]],"source":"Square-weighted consecutive constant of $p^r$ for prime $p \\ge 7$ and even $r \\ge 4$ is at most five"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-consecutive-constant-square-set-3","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Square-weighted consecutive constant of $p^r$ for prime $p \\ge 7$ and even $r \\ge 4$ is at most five. Let $p \\ge 7$ be a prime and $r \\ge 4$ an even natural number. Write $S(p^r)$ for the set of nonzero squares in $\\mathbb{Z}_{p^r} = \\mathbb{Z}/p^r\\mathbb{Z}$ and $C_{S(p^r)}$ for the $S(p^r)$-weighted consecutive constant of $\\mathbb{Z}_{p^r}$. Then $C_{S(p^r)} \\le 5$.","title":"Square-weighted consecutive constant of $p^r$ for prime $p \\ge 7$ and even $r \\ge 4$ is at most five"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-consecutive-constant-square-set-4","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_oddSquare_largePrime","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-consecutive-constant-square-set-4","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $n \\ge 2$ be an odd perfect square and let $p \\ge 7$ be a prime with $p^4 \\mid n$. Write $S(n)$ for the set of nonzero squares in $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$. Then some positive length $k$ has the property that every sequence of $k$ elements of $\\mathbb{Z}_n$ has a nonempty block of consecutive terms with an $S(n)$-weighted sum equal to zero, and the least such length, the $S(n)$-weighted\u2026","title":"Square-weighted consecutive constant five for odd squares with a large prime fourth power"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/unit-square-weighted-zero-sum-mod-five-p","kind":"node","lean":"MetaMathlibExt.unitSquareWeightedZeroSum_modFivePower_of_four_units","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#unit-square-weighted-zero-sum-mod-five-p","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $r \\ge 1$ and let $U(5^r)^2$ be the set of squares of units of $\\mathbb{Z}/5^r\\mathbb{Z}$. If a sequence $s_1, \\dots, s_k$ in $\\mathbb{Z}/5^r\\mathbb{Z}$ has at least four terms, at distinct positions, that are units, then there are $a_1, \\dots, a_k \\in U(5^r)^2$ with $a_1 s_1 + \\dots + a_k s_k = 0$.","title":"Four unit terms force a unit-square-weighted zero sum modulo a power of 5"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-consecutive-constant-square-set-5","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_fivePower_le_seven","parent":23269,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-weighted consecutive constant of "],["math","\\(5^r\\)"],["text"," for even "],["math","\\(r \\ge 4\\)"],["text"," is at most seven"]],"source":"Square-weighted consecutive constant of $5^r$ for even $r \\ge 4$ is at most seven"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-consecutive-constant-square-set-5","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $r \\ge 4$ be an even natural number. Write $S(5^r)$ for the set of nonzero squares in $\\mathbb{Z}_{5^r} = \\mathbb{Z}/5^r\\mathbb{Z}$ and $C_{S(5^r)}$ for the $S(5^r)$-weighted consecutive constant of $\\mathbb{Z}_{5^r}$. Then $C_{S(5^r)} \\le 7$.","title":"Square-weighted consecutive constant of $5^r$ for even $r \\ge 4$ is at most seven"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-consecutive-constant-square-set-6","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_square_divisible_fiveFourth","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-consecutive-constant-square-set-6","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $n \\ge 2$ be a perfect square with $5^4 \\mid n$. Write $S(n)$ for the set of nonzero squares in $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$ and $C_{S(n)}$ for the $S(n)$-weighted consecutive constant of $\\mathbb{Z}_n$. Then $C_{S(n)} \\le 7$.","title":"Square-weighted consecutive constant of squares divisible by 625 is at most seven"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-consecutive-constant-square-set-7","kind":"node","lean":"MetaMathlibExt.weightedConsecutiveConstant_squareSet_square_oddSquarefree","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-consecutive-constant-square-set-7","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $m$ be an odd squarefree natural number and let $n = m^2$ with $n \\ge 2$. Write $S(n)$ for the set of nonzero squares of $\\mathbb{Z}/n\\mathbb{Z}$. Then there is a positive length $k$ such that every sequence $x_1, \\dots, x_k$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a nonempty block of consecutive terms $x_i, \\dots, x_j$ and weights $a_i, \\dots, a_j \\in S(n)$ with $a_i x_i + \\dots + a_j x_j = 0$, and the consecutive\u2026","title":"Consecutive square-weighted constant of the square of an odd squarefree number"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-davenport-zmod-arithmetic-progr","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_arithmeticProgression","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-davenport-zmod-arithmetic-progr","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $n \\ge 2$ and $a$ be natural numbers, and let $A = \\{(a + i) \\bmod n : 0 \\le i < \\lceil n/2 \\rceil\\}$ be the set of remainders modulo $n$ of the $\\lceil n/2 \\rceil$ consecutive integers $a, a + 1, \\dots, a + \\lceil n/2 \\rceil - 1$. If $0 \\notin A$, then the $A$-weighted Davenport constant of $\\mathbb{Z}/n\\mathbb{Z}$ is $D_A(\\mathbb{Z}/n\\mathbb{Z}) = 2$.","title":"Xia's theorem on arithmetic progressions of weights"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-davenport-zmod-not-dvd-prime","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_not_dvd_prime","parent":23269,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Davenport constant of "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"],["text"," for the weights prime to "],["math","\\(p\\)"]],"source":"Weighted Davenport constant of $\\mathbb{Z}/n\\mathbb{Z}$ for the weights prime to $p$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-davenport-zmod-not-dvd-prime","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $n$ be a positive integer, $p$ a prime dividing $n$ and $k = v_p(n)$ the exponent of $p$ in $n$. For the weight set $A = \\{a \\in \\{1, \\dots, n - 1\\} : p \\nmid a\\}$, the $A$-weighted Davenport constant of $\\mathbb{Z}/n\\mathbb{Z}$ is $D_A(\\mathbb{Z}/n\\mathbb{Z}) = k + 1$.","title":"Weighted Davenport constant of $\\mathbb{Z}/n\\mathbb{Z}$ for the weights prime to $p$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-davenport-one-zmod-prod-of-dvd","kind":"node","lean":"MetaMathlibExt.weightedDavenport_one_zmod_prod_of_dvd","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-davenport-one-zmod-prod-of-dvd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"For positive integers $h$ and $k$ with $h \\mid k$, the Davenport constant $D(G) = D_{\\{1\\}}(G)$ of $G = \\mathbb{Z}/h\\mathbb{Z} \\oplus \\mathbb{Z}/k\\mathbb{Z}$ is $D(G) = h + k - 1$.","title":"Olson's theorem for groups of rank at most two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/weighted-davenport-one-pi-zmod-prime-pow","kind":"node","lean":"MetaMathlibExt.weightedDavenport_one_pi_zmod_prime_pow","parent":23269,"reading_title":{"format":"inline-parts-v1","parts":[["text","Olson's theorem for abelian "],["math","\\(p\\)"],["text","-groups"]],"source":"Olson's theorem for abelian $p$-groups"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#weighted-davenport-one-pi-zmod-prime-pow","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"Let $p$ be a prime, $r \\ge 0$ and let $e_1, \\dots, e_r \\ge 1$ be natural numbers. The Davenport constant $D(G) = D_{\\{1\\}}(G)$ of $G = \\mathbb{Z}/p^{e_1}\\mathbb{Z} \\oplus \\dots \\oplus \\mathbb{Z}/p^{e_r}\\mathbb{Z}$ is $D(G) = 1 + \\sum_{i=1}^{r} (p^{e_i} - 1)$. For $r = 0$ the group $G$ is trivial and this reads $D(G) = 1$.","title":"Olson's theorem for abelian $p$-groups"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/ceil-div-ceil-div-sub-one-sub-one-of-dvd","kind":"node","lean":"MetaMathlibExt.ceilDiv_ceilDiv_sub_one_sub_one_of_dvd","parent":23269,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03/index.html#ceil-div-ceil-div-sub-one-sub-one-of-dvd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-03","status":"can_prove","summary":"For all natural numbers $m, n \\ge 2$ with $m \\mid n$, we have $\\lceil n / (\\lceil n/(m-1) \\rceil - 1) \\rceil = m$.","title":"A nested ceiling identity for divisors"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","kind":"scope","navigation":"transparent","parent":23226,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weight sets with weighted Davenport constant "],["math","\\(m\\)"],["text"," on "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"],["text"," to Three-term sums of square-weighted units cover "],["math","\\(\\mathbb{Z}_{p^r}\\)"]],"source":"Weight sets with weighted Davenport constant $m$ on $\\mathbb{Z}/n\\mathbb{Z}$ to Three-term sums of square-weighted units cover $\\mathbb{Z}_{p^r}$"},"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","title":"Weight sets with weighted Davenport constant $m$ on $\\mathbb{Z}/n\\mathbb{Z}$ to Three-term sums of square-weighted units cover $\\mathbb{Z}_{p^r}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-zmod-of-dvd","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_of_dvd","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weight sets with weighted Davenport constant "],["math","\\(m\\)"],["text"," on "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"]],"source":"Weight sets with weighted Davenport constant $m$ on $\\mathbb{Z}/n\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-zmod-of-dvd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $n \\ge 2$ be a natural number and $m \\ge 2$ a divisor of $n$. Every finite set $A$ of natural numbers with $n/m \\in A \\subseteq \\{1, \\dots, \\lceil n/(m-1) \\rceil - 1\\}$ has $A$-weighted Davenport constant $D_A(\\mathbb{Z}/n\\mathbb{Z}) = m$.","title":"Weight sets with weighted Davenport constant $m$ on $\\mathbb{Z}/n\\mathbb{Z}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-zmod-icc","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_Icc","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Davenport constant of "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"],["text"," for the weights "],["math","\\(\\{1, \\dots, r\\}\\)"]],"source":"Weighted Davenport constant of $\\mathbb{Z}/n\\mathbb{Z}$ for the weights $\\{1, \\dots, r\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-zmod-icc","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $n$ and $r$ be natural numbers with $1 < r < n$ and let $A = \\{1, \\dots, r\\}$. Then the $A$-weighted Davenport constant of $\\mathbb{Z}/n\\mathbb{Z}$ is $D_A(\\mathbb{Z}/n\\mathbb{Z}) = \\lceil n/r \\rceil$.","title":"Weighted Davenport constant of $\\mathbb{Z}/n\\mathbb{Z}$ for the weights $\\{1, \\dots, r\\}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-zmod-icc-of-dvd","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_Icc_of_dvd","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial-interval weights with weighted Davenport constant "],["math","\\(m\\)"]],"source":"Initial-interval weights with weighted Davenport constant $m$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-zmod-icc-of-dvd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $m, n \\ge 2$ be natural numbers with $m \\mid n$ and put $r = \\lceil n/(m-1) \\rceil - 1$. Then the $\\{1, \\dots, r\\}$-weighted Davenport constant of $\\mathbb{Z}/n\\mathbb{Z}$ is $D_{\\{1, \\dots, r\\}}(\\mathbb{Z}/n\\mathbb{Z}) = m$.","title":"Initial-interval weights with weighted Davenport constant $m$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/exists-weighted-zero-sum-free-zmod-prod","kind":"node","lean":"MetaMathlibExt.exists_weightedZeroSumFree_zmod_prod","parent":23290,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#exists-weighted-zero-sum-free-zmod-prod","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $m, n \\ge 2$ be natural numbers with $m \\mid n$, let $k \\ge 1$ and put $\\alpha = \\lceil n/(m-1) \\rceil - 1$. Then there is a sequence $s$ over $\\mathbb{Z}/n\\mathbb{Z} \\oplus \\mathbb{Z}/kn\\mathbb{Z}$ of length $m + \\lceil kn/\\alpha \\rceil - 2$ that is $\\{1, \\dots, \\alpha\\}$-weighted zero-sum free: for every non-empty set $I$ of positions and all weights $a_i \\in \\{1, \\dots, \\alpha\\}$, the weighted sum\u2026","title":"Long weighted zero-sum free sequences in groups of rank two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-zmod-prod-of-dvd","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_prod_of_dvd","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weight sets with prescribed weighted Davenport constant on "],["math","\\(\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/kn\\mathbb{Z}\\)"],["text",", divisible case"]],"source":"Weight sets with prescribed weighted Davenport constant on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/kn\\mathbb{Z}$, divisible case"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-zmod-prod-of-dvd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Weight sets with prescribed weighted Davenport constant on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/kn\\mathbb{Z}$, divisible case. Let $n \\ge 2$ and $k \\ge 1$ be natural numbers, let $m \\ge 2$ be a divisor of $n$, let $\\alpha = \\lceil n/(m-1) \\rceil - 1$, and suppose that $\\alpha$ divides $n$. Then for every finite set $A$ of natural numbers with $n/m \\in A \\subseteq \\{1, \\dots, \\alpha\\}$, the $A$-weighted\u2026","title":"Weight sets with prescribed weighted Davenport constant on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/kn\\mathbb{Z}$, divisible case"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-zmod-prod-of-not-dvd","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_prod_of_not_dvd","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weight sets with prescribed weighted Davenport constant on "],["math","\\(\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/kn\\mathbb{Z}\\)"],["text",", non-divisible case"]],"source":"Weight sets with prescribed weighted Davenport constant on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/kn\\mathbb{Z}$, non-divisible case"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-zmod-prod-of-not-dvd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Weight sets with prescribed weighted Davenport constant on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/kn\\mathbb{Z}$, non-divisible case. Let $n \\ge 2$ and $k \\ge 1$ be natural numbers, let $m \\ge 2$ be a divisor of $n$, let $\\alpha = \\lceil n/(m-1) \\rceil - 1$, and suppose that $\\alpha$ does not divide $n$. Let $r$ be the remainder of $n$ modulo $\\alpha$, so that $1 \\le r < \\alpha$, and suppose that\u2026","title":"Weight sets with prescribed weighted Davenport constant on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/kn\\mathbb{Z}$, non-divisible case"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-zmod-prod-self-of-dvd","kind":"node","lean":"MetaMathlibExt.weightedDavenport_zmod_prod_self_of_dvd","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weight sets with weighted Davenport constant "],["math","\\(2m - 1\\)"],["text"," on "],["math","\\(\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}\\)"]],"source":"Weight sets with weighted Davenport constant $2m - 1$ on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-zmod-prod-self-of-dvd","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Weight sets with weighted Davenport constant $2m - 1$ on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$. Let $n \\ge 2$ be a natural number and $m \\ge 2$ a divisor of $n$. Every finite set $A$ of natural numbers with $n/m \\in A \\subseteq \\{1, \\dots, \\lceil n/(m-1) \\rceil - 1\\}$ has $A$-weighted Davenport constant $D_A(\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}) = 2m - 1$.","title":"Weight sets with weighted Davenport constant $2m - 1$ on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/exists-weighted-zero-sum-free-pi-zmod-pr","kind":"node","lean":"MetaMathlibExt.exists_weightedZeroSumFree_pi_zmod_prime_pow","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Long weighted zero-sum free sequences in abelian "],["math","\\(p\\)"],["text","-groups"]],"source":"Long weighted zero-sum free sequences in abelian $p$-groups"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#exists-weighted-zero-sum-free-pi-zmod-pr","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $p$ be a prime, let $r \\ge 1$, let $e_1 \\le e_2 \\le \\dots \\le e_r$ be natural numbers and let $G = \\mathbb{Z}/p^{e_1}\\mathbb{Z} \\oplus \\dots \\oplus \\mathbb{Z}/p^{e_r}\\mathbb{Z}$. Let $c$ be a natural number with $1 \\le c \\le e_1$ and put $\\beta = \\lceil p^{e_1}/(p^c - 1) \\rceil - 1$. Then there is a sequence $s$ over $G$ of length $\\sum_{i=1}^{r} (\\lceil p^{e_i}/\\beta \\rceil - 1)$ that is\u2026","title":"Long weighted zero-sum free sequences in abelian $p$-groups"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-pi-zmod-prime-pow-of","kind":"node","lean":"MetaMathlibExt.weightedDavenport_pi_zmod_prime_pow_of_dvd","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Davenport constant of abelian "],["math","\\(p\\)"],["text","-groups, divisible case"]],"source":"Weighted Davenport constant of abelian $p$-groups, divisible case"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-pi-zmod-prime-pow-of","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $p$ be a prime, let $r \\ge 1$, let $e_1 \\le e_2 \\le \\dots \\le e_r$ be natural numbers and let $G = \\mathbb{Z}/p^{e_1}\\mathbb{Z} \\oplus \\dots \\oplus \\mathbb{Z}/p^{e_r}\\mathbb{Z}$. Let $c$ be a natural number with $1 \\le c \\le e_1$, put $\\beta = \\lceil p^{e_1}/(p^c - 1) \\rceil - 1$ and let $A$ be a finite set of natural numbers with $p^{e_1 - c} \\in A \\subseteq \\{1, \\dots, \\beta\\}$. If $\\beta \\mid p^{e_1}$, then\u2026","title":"Weighted Davenport constant of abelian $p$-groups, divisible case"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-pi-zmod-prime-pow-of-2","kind":"node","lean":"MetaMathlibExt.weightedDavenport_pi_zmod_prime_pow_of_not_dvd","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Davenport constant of abelian "],["math","\\(p\\)"],["text","-groups, non-divisible case"]],"source":"Weighted Davenport constant of abelian $p$-groups, non-divisible case"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-pi-zmod-prime-pow-of-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $p$ be a prime, let $r \\ge 1$, let $e_1 \\le e_2 \\le \\dots \\le e_r$ be natural numbers and let $G = \\mathbb{Z}/p^{e_1}\\mathbb{Z} \\oplus \\dots \\oplus \\mathbb{Z}/p^{e_r}\\mathbb{Z}$. Let $c$ be a natural number with $1 \\le c \\le e_1$, put $\\beta = \\lceil p^{e_1}/(p^c - 1) \\rceil - 1$ and let $A$ be a finite set of natural numbers with $p^{e_1 - c} \\in A \\subseteq \\{1, \\dots, \\beta\\}$. Suppose that\u2026","title":"Weighted Davenport constant of abelian $p$-groups, non-divisible case"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-davenport-pi-zmod-prime-pow-con","kind":"node","lean":"MetaMathlibExt.weightedDavenport_pi_zmod_prime_pow_const","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Davenport constant of "],["math","\\((\\mathbb{Z}/p^e\\mathbb{Z})^r\\)"]],"source":"Weighted Davenport constant of $(\\mathbb{Z}/p^e\\mathbb{Z})^r$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-davenport-pi-zmod-prime-pow-con","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $p$ be a prime and let $a, e, r$ be natural numbers with $1 \\le a \\le e$ and $r \\ge 1$. Let $A$ be a finite set of natural numbers with $p^{e-a} \\in A \\subseteq \\{1, \\dots, \\lceil p^e/(p^a - 1) \\rceil - 1\\}$. Then the $A$-weighted Davenport constant of $(\\mathbb{Z}/p^e\\mathbb{Z})^r$ is $D_A((\\mathbb{Z}/p^e\\mathbb{Z})^r) = 1 + r(p^a - 1)$.","title":"Weighted Davenport constant of $(\\mathbb{Z}/p^e\\mathbb{Z})^r$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/unit-weighted-zero-sum-constants","kind":"node","lean":"MetaMathlibExt.unit_weighted_zero_sum_constants","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit-weighted zero-sum constants of "],["math","\\(\\mathbb{Z}_n\\)"]],"source":"Unit-weighted zero-sum constants of $\\mathbb{Z}_n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#unit-weighted-zero-sum-constants","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $n$ be a positive integer, let $U(n)$ be the set of units of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$, and let $\\Omega(n)$ be the number of prime factors of $n$ counted with multiplicity. Then $D_{U(n)}(n) = \\Omega(n) + 1$ and $C_{U(n)}(n) = 2^{\\Omega(n)}$. Here $D_{U(n)}(n)$ is the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}_n$ has a nonempty set $I$ of indices and\u2026","title":"Unit-weighted zero-sum constants of $\\mathbb{Z}_n$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/quadratic-residue-weighted-zero-sum-cons","kind":"node","lean":"MetaMathlibExt.quadratic_residue_weighted_zero_sum_constants","parent":23290,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#quadratic-residue-weighted-zero-sum-cons","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $p$ be an odd prime and let $Q_p$ be the set of squares of units of $\\mathbb{Z}_p = \\mathbb{Z}/p\\mathbb{Z}$. Then $C_{Q_p}(p) = D_{Q_p}(p) = 3$. Here $D_{Q_p}(p)$ is the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}_p$ has a nonempty set $I$ of indices and weights $a_i \\in Q_p$ for $i \\in I$ with $\\sum_{i \\in I} a_i x_i = 0$, and $C_{Q_p}(p)$ is defined in the same way\u2026","title":"Quadratic-residue-weighted zero-sum constants modulo a prime"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/consecutive-constant-mul-le","kind":"node","lean":"MetaMathlibExt.consecutive_constant_mul_le","parent":23290,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#consecutive-constant-mul-le","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"For a natural number $n$ and $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$, let $C_A(n)$ denote the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a block of consecutive terms $x_s, \\dots, x_t$ with $s \\le t$ and weights $a_s, \\dots, a_t \\in A$ such that $a_s x_s + \\dots + a_t x_t = 0$. Let $m$ and $q$ be natural numbers and let $A \\subseteq \\mathbb{Z}/mq\\mathbb{Z}$,\u2026","title":"Product lower bound for the consecutive weighted zero-sum constant"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/davenport-constant-add-le","kind":"node","lean":"MetaMathlibExt.davenport_constant_add_le","parent":23290,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#davenport-constant-add-le","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"For a natural number $n$ and $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$, let $D_A(n)$ denote the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a nonempty set $J$ of indices and weights $a_i \\in A$ for $i \\in J$ with $\\sum_{i \\in J} a_i x_i = 0$. Let $m$ and $q$ be natural numbers and let $A \\subseteq \\mathbb{Z}/mq\\mathbb{Z}$, $B \\subseteq \\mathbb{Z}/m\\mathbb{Z}$\u2026","title":"Lower bound for the weighted Davenport constant of a product modulus"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/jacobi-kernel-index-two","kind":"node","lean":"MetaMathlibExt.jacobi_kernel_index_two","parent":23290,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#jacobi-kernel-index-two","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $n$ be an odd natural number, let $U(n)$ be the group of units of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$, and let $S(n) = \\{u \\in U(n) : (u/n) = 1\\}$, where $(u/n)$ is the Jacobi symbol $(a/n)$ of the representative $0 \\le a < n$ of $u$. If $n$ is not a perfect square, then $S(n)$ is a subgroup of $U(n)$ of index $2$. If $n$ is a perfect square, then $S(n) = U(n)$.","title":"The index of the Jacobi symbol kernel"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/units-subset-image-jacobi-kernel","kind":"node","lean":"MetaMathlibExt.units_subset_image_jacobi_kernel","parent":23290,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#units-subset-image-jacobi-kernel","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $d$ and $n'$ be coprime natural numbers such that $d n'$ is odd and $d$ is not a perfect square, and put $n = d n'$. Then every unit $y$ of $\\mathbb{Z}/n'\\mathbb{Z}$ is the reduction modulo $n'$ of a unit $x$ of $\\mathbb{Z}/n\\mathbb{Z}$ whose least nonnegative representative $v$ has Jacobi symbol $\\left(\\frac{v}{n}\\right) = 1$.","title":"Units lift to the kernel of the Jacobi symbol across a non-square coprime factor"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/weighted-zero-sum-of-image","kind":"node","lean":"MetaMathlibExt.weighted_zero_sum_of_image","parent":23290,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#weighted-zero-sum-of-image","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $d$ and $n'$ be coprime natural numbers, put $n = d n'$, and let $f : \\mathbb{Z}/n\\mathbb{Z} \\to \\mathbb{Z}/n'\\mathbb{Z}$ be reduction modulo $n'$. Let $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$ and $A' \\subseteq \\mathbb{Z}/n'\\mathbb{Z}$ with $A' \\subseteq f(A)$, and let $x_1, \\dots, x_k \\in \\mathbb{Z}/n\\mathbb{Z}$ each be divisible by $d$ in $\\mathbb{Z}/n\\mathbb{Z}$. If there are $a'_1, \\dots, a'_k \\in A'$ with\u2026","title":"Lifting weighted zero sums along a coprime factor"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/unit-weighted-zero-sum-prime-power","kind":"node","lean":"MetaMathlibExt.unit_weighted_zero_sum_prime_power","parent":23290,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#unit-weighted-zero-sum-prime-power","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $p$ be an odd prime, let $r$ and $k$ be natural numbers, and let $x_1, \\dots, x_k \\in \\mathbb{Z}/p^r\\mathbb{Z}$. Suppose that for at least two indices $i$ the least nonnegative representative of $x_i$ is coprime to $p$. Then there are units $a_1, \\dots, a_k$ of $\\mathbb{Z}/p^r\\mathbb{Z}$ with $a_1 x_1 + \\dots + a_k x_k = 0$.","title":"Unit-weighted zero sums modulo an odd prime power"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/unit-square-three-sumset-eq-univ","kind":"node","lean":"MetaMathlibExt.unit_square_three_sumset_eq_univ","parent":23290,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-term sums of square-weighted units cover "],["math","\\(\\mathbb{Z}_{p^r}\\)"]],"source":"Three-term sums of square-weighted units cover $\\mathbb{Z}_{p^r}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04/index.html#unit-square-three-sumset-eq-univ","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-04","status":"can_prove","summary":"Let $p \\ge 7$ be a prime, $r$ a natural number and $n = p^r$; write $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$ and $A = U(n)^2$ for the set of squares of units of $\\mathbb{Z}_n$. For all units $x_1, x_2, x_3$ of $\\mathbb{Z}_n$, $A x_1 + A x_2 + A x_3 = \\mathbb{Z}_n$: every $z \\in \\mathbb{Z}_n$ equals $a_1 x_1 + a_2 x_2 + a_3 x_3$ for some $a_1, a_2, a_3 \\in A$.","title":"Three-term sums of square-weighted units cover $\\mathbb{Z}_{p^r}$"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","kind":"scope","navigation":"transparent","parent":23226,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-term sums with prescribed square classes cover "],["math","\\(\\mathbb{Z}_{p^r}\\)"],["text"," to Consecutive zero-sum constant of a finite abelian group"]],"source":"Three-term sums with prescribed square classes cover $\\mathbb{Z}_{p^r}$ to Consecutive zero-sum constant of a finite abelian group"},"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","title":"Three-term sums with prescribed square classes cover $\\mathbb{Z}_{p^r}$ to Consecutive zero-sum constant of a finite abelian group"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/unit-square-class-three-sumset-eq-univ","kind":"node","lean":"MetaMathlibExt.unit_square_class_three_sumset_eq_univ","parent":23311,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-term sums with prescribed square classes cover "],["math","\\(\\mathbb{Z}_{p^r}\\)"]],"source":"Three-term sums with prescribed square classes cover $\\mathbb{Z}_{p^r}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#unit-square-class-three-sumset-eq-univ","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $p \\ge 7$ be a prime, $r \\ge 1$ and $n = p^r$; write $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$, $U(n)$ for its group of units, $A_1 = U(n)^2$ for the set of squares of units and $A_2 = U(n) \\setminus U(n)^2$. For all units $x_1, x_2, x_3$ of $\\mathbb{Z}_n$ and every $f : \\{1, 2, 3\\} \\to \\{1, 2\\}$, $A_{f(1)} x_1 + A_{f(2)} x_2 + A_{f(3)} x_3 = \\mathbb{Z}_n$: every $z \\in \\mathbb{Z}_n$ equals\u2026","title":"Three-term sums with prescribed square classes cover $\\mathbb{Z}_{p^r}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/unit-square-weighted-zero-sum-prime-powe","kind":"node","lean":"MetaMathlibExt.unit_square_weighted_zero_sum_prime_power","parent":23311,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three unit terms force a square-weighted zero sum modulo "],["math","\\(p^r\\)"]],"source":"Three unit terms force a square-weighted zero sum modulo $p^r$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#unit-square-weighted-zero-sum-prime-powe","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $p \\ge 7$ be a prime, $r$ a natural number and $n = p^r$, and let $U(n)^2$ be the set of squares of units of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$. If a sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}_n$ has at least three indices $i$ with $x_i$ a unit, then there are $a_1, \\dots, a_k \\in U(n)^2$ with $a_1 x_1 + \\dots + a_k x_k = 0$, that is, the sequence is a $U(n)^2$-weighted zero-sum sequence.","title":"Three unit terms force a square-weighted zero sum modulo $p^r$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-weighted-zero-sum-of-few-units","kind":"node","lean":"MetaMathlibExt.jacobi_weighted_zero_sum_of_few_units","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-weighted-zero-sum-of-few-units","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $n$ be an odd squarefree natural number, $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$, and $S(n)$ the set of units $u$ of $\\mathbb{Z}_n$ whose Jacobi symbol $(a/n)$ is $1$, where $0 \\le a < n$ is the representative of $u$. Let $(x_1, \\dots, x_k)$ be a sequence in $\\mathbb{Z}_n$, and call $x_i$ coprime to a prime $p$ when its representative in $\\{0, \\dots, n-1\\}$ is. Suppose that for every prime divisor $p$ of $n$ at\u2026","title":"Few unit terms force a Jacobi-weighted zero sum"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-weighted-zero-sum-of-three-coprim","kind":"node","lean":"MetaMathlibExt.jacobi_weighted_zero_sum_of_three_coprime","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-weighted-zero-sum-of-three-coprim","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $n$ be a squarefree natural number all of whose prime divisors are at least $7$, let $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$, and let $S(n)$ be the set of units $u$ of $\\mathbb{Z}_n$ whose Jacobi symbol $(a/n)$ is $1$, where $0 \\le a < n$ is the representative of $u$. Let $(x_1, \\dots, x_k)$ be a sequence in $\\mathbb{Z}_n$, and call $x_i$ coprime to a prime $p$ when its representative in $\\{0, \\dots, n-1\\}$ is.\u2026","title":"Three terms coprime to one prime force a Jacobi-weighted zero sum"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-davenport-constant","kind":"node","lean":"MetaMathlibExt.jacobi_davenport_constant","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-davenport-constant","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"For a natural number $n$ and $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$, let $D_A(n)$ denote the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a nonempty set $J$ of indices and weights $a_i \\in A$ for $i \\in J$ with $\\sum_{i \\in J} a_i x_i = 0$. Let $n$ be an odd squarefree natural number and let $S(n)$ be the set of units $x$ of $\\mathbb{Z}/n\\mathbb{Z}$ with\u2026","title":"Davenport constant for the weights of Jacobi symbol one"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-consecutive-constant","kind":"node","lean":"MetaMathlibExt.jacobi_consecutive_constant","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-consecutive-constant","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"For a natural number $n$ and $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$, let $C_A(n)$ denote the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a block of consecutive terms $x_s, \\dots, x_t$ with $s \\le t$ and weights $a_s, \\dots, a_t \\in A$ such that $a_s x_s + \\dots + a_t x_t = 0$. Let $n$ be an odd squarefree natural number and let $S(n)$ be the set of units $x$\u2026","title":"Consecutive constant for the weights of Jacobi symbol one"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-eq-legendre-index-two","kind":"node","lean":"MetaMathlibExt.jacobi_eq_legendre_index_two","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-eq-legendre-index-two","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $n$ be an odd natural number and $p$ a prime divisor of $n$, and suppose that $p$ is not the unique prime divisor of $n$ with odd exponent in $n$; that is, it is not the case that the exponent of $p$ in $n$ is odd and every prime divisor of $n$ with odd exponent in $n$ equals $p$. Then the set of units $u$ of $\\mathbb{Z}/n\\mathbb{Z}$ whose Jacobi symbol $\\left(\\frac{u}{n}\\right)$ equals the Legendre symbol\u2026","title":"Units with equal Jacobi and Legendre symbols form a subgroup of index two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-kernel-subset-image-jacobi-eq-leg","kind":"node","lean":"MetaMathlibExt.jacobi_kernel_subset_image_jacobi_eq_legendre","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-kernel-subset-image-jacobi-eq-leg","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $n$ be an odd natural number and let $p$, $p'$ be prime divisors of $n$ with $p$ coprime to $n' = n/p$. For a residue $x$ modulo $N$ and an odd $m$, write $(x/m)$ for the Jacobi symbol $(a/m)$ of the representative $0 \\le a < N$ of $x$. Let $U(m)$ be the group of units of $\\mathbb{Z}_m = \\mathbb{Z}/m\\mathbb{Z}$, let $S(n') = \\{y \\in U(n') : (y/n') = 1\\}$ and $L(n;p') = \\{x \\in U(n) : (x/n) = (x/p')\\}$, and let\u2026","title":"Lifting the Jacobi kernel into the Jacobi agreement set"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/units-prime-subset-image-jacobi-eq-legen","kind":"node","lean":"MetaMathlibExt.units_prime_subset_image_jacobi_eq_legendre","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#units-prime-subset-image-jacobi-eq-legen","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $n$ be odd and let $p'$ be a prime dividing $n$ such that $p'$ is coprime to $n/p'$. Write $L(n;p')$ for the set of units $x$ of $\\mathbb{Z}/n\\mathbb{Z}$ whose least nonnegative representative $v$ satisfies $\\left(\\frac{v}{n}\\right) = \\left(\\frac{v}{p'}\\right)$, where both are Jacobi symbols. Then every unit $z$ of $\\mathbb{Z}/p'\\mathbb{Z}$ is the reduction modulo $p'$ of some element of $L(n;p')$.","title":"Units modulo a prime divisor lift to units where the Jacobi and Legendre symbols agree"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-kernel-prod-units-subset-crt-imag","kind":"node","lean":"MetaMathlibExt.jacobi_kernel_prod_units_subset_crt_image","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-kernel-prod-units-subset-crt-imag","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $n$ be an odd squarefree natural number, $p'$ a prime divisor of $n$ and $n' = n/p'$. For a residue $x$ modulo $N$ and an odd $m$, write $(x/m)$ for the Jacobi symbol $(a/m)$ of the representative $0 \\le a < N$ of $x$. Let $U(m)$ be the group of units of $\\mathbb{Z}_m = \\mathbb{Z}/m\\mathbb{Z}$, let $S(n') = \\{y \\in U(n') : (y/n') = 1\\}$ and $L(n;p') = \\{x \\in U(n) : (x/n) = (x/p')\\}$, and for $m \\mid n$ let\u2026","title":"Chinese remainder lifts into the Jacobi agreement set"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-eq-legendre-weighted-zero-sum","kind":"node","lean":"MetaMathlibExt.jacobi_eq_legendre_weighted_zero_sum","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-eq-legendre-weighted-zero-sum","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $n$ be a squarefree natural number all of whose prime divisors are at least $7$, let $p'$ be a prime divisor of $n$, let $n' = n/p'$, and let $L(n;p')$ be the set of units $x$ of $\\mathbb{Z}/n\\mathbb{Z}$ whose Jacobi symbol $\\left(\\frac{x}{n}\\right)$ equals the Legendre symbol $\\left(\\frac{x}{p'}\\right)$. Let $(x_1, \\dots, x_k)$ be a sequence in $\\mathbb{Z}/n\\mathbb{Z}$ such that for every prime divisor $p$ of\u2026","title":"Weighted zero-sum criterion for the Jacobi\u2013Legendre weights"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-eq-legendre-davenport-constant","kind":"node","lean":"MetaMathlibExt.jacobi_eq_legendre_davenport_constant","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-eq-legendre-davenport-constant","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"For a natural number $n$ and $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$, let $D_A(n)$ denote the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a nonempty set $J$ of indices and weights $a_i \\in A$ for $i \\in J$ with $\\sum_{i \\in J} a_i x_i = 0$. Let $n$ be a squarefree natural number all of whose prime divisors are at least $7$, with $\\Omega(n) \\ne 2$, where\u2026","title":"Davenport constant of the Jacobi\u2013Legendre weights"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-eq-legendre-davenport-constant-tw","kind":"node","lean":"MetaMathlibExt.jacobi_eq_legendre_davenport_constant_two_primes","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-eq-legendre-davenport-constant-tw","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"For a natural number $n$ and $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$, let $D_A(n)$ denote the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a nonempty set $J$ of indices and weights $a_i \\in A$ for $i \\in J$ with $\\sum_{i \\in J} a_i x_i = 0$. Let $p'$ and $q$ be distinct primes with $p' \\ge 7$ and $q \\ge 7$, let $n = p'q$, and let $L(n;p')$ be the set of units\u2026","title":"Davenport constant of the Jacobi\u2013Legendre weights modulo a product of two primes"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-eq-legendre-consecutive-constant","kind":"node","lean":"MetaMathlibExt.jacobi_eq_legendre_consecutive_constant","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-eq-legendre-consecutive-constant","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"For a natural number $n$ and $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$, let $C_A(n)$ denote the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a block of consecutive terms $x_s, \\dots, x_t$ with $s \\le t$ and weights $a_s, \\dots, a_t \\in A$ such that $a_s x_s + \\dots + a_t x_t = 0$. Let $n$ be a squarefree natural number all of whose prime divisors are at least\u2026","title":"Consecutive constant of the Jacobi\u2013Legendre weights"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/jacobi-eq-legendre-consecutive-constant-2","kind":"node","lean":"MetaMathlibExt.jacobi_eq_legendre_consecutive_constant_two_primes","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#jacobi-eq-legendre-consecutive-constant-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"For a natural number $n$ and $A \\subseteq \\mathbb{Z}/n\\mathbb{Z}$, let $C_A(n)$ denote the least positive integer $k$ such that every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}/n\\mathbb{Z}$ has a block of consecutive terms $x_s, \\dots, x_t$ with $s \\le t$ and weights $a_s, \\dots, a_t \\in A$ such that $a_s x_s + \\dots + a_t x_t = 0$. Let $p'$ and $q$ be distinct primes with $p' \\ge 7$ and $q \\ge 7$, let $n = p'q$,\u2026","title":"Consecutive constant of the Jacobi\u2013Legendre weights modulo a product of two primes"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/davenport-prod-ring-ge","kind":"node","lean":"MetaMathlibExt.davenport_prod_ring_ge","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#davenport-prod-ring-ge","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $R$ and $R'$ be rings, $M$ an $R$-module, $N$ an $R'$-module, $A \\subseteq R$ and $B \\subseteq R'$. Let $D_A(M)$ be the least positive integer $k$, if one exists, such that every sequence $x_1, \\dots, x_k$ in $M$ has a nonempty set of indices $I$ and weights $a_i \\in A$ for $i \\in I$ with $\\sum_{i \\in I} a_i x_i = 0$, and define $D_B(N)$ in the same way. Let $D_{A \\times B}(M \\times N)$ be the least positive\u2026","title":"Weighted Davenport constant of a product with product weights"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/davenport-prod-ge","kind":"node","lean":"MetaMathlibExt.davenport_prod_ge","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#davenport-prod-ge","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $R$ be a ring and $A \\subseteq R$. For an $R$-module $P$, let $D_A(P)$ be the least positive integer $k$, if one exists, such that every sequence $x_1, \\dots, x_k$ in $P$ has a nonempty set of indices $I$ and weights $a_i \\in A$ for $i \\in I$ with $\\sum_{i \\in I} a_i x_i = 0$. If $M$ and $N$ are $R$-modules for which $D_A(M)$, $D_A(N)$ and $D_A(M \\times N)$ all exist, then\u2026","title":"Weighted Davenport constant of a product of modules"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/consecutive-prod-ring-ge","kind":"node","lean":"MetaMathlibExt.consecutive_prod_ring_ge","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#consecutive-prod-ring-ge","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $R$ and $R'$ be rings, $M$ an $R$-module, $N$ an $R'$-module, $A \\subseteq R$ and $B \\subseteq R'$. Let $C_A(M)$ be the least positive integer $k$, if one exists, such that every sequence $x_1, \\dots, x_k$ in $M$ has indices $s \\le t$ and weights $a_s, \\dots, a_t \\in A$ with $a_s x_s + \\dots + a_t x_t = 0$, and define $C_B(N)$ in the same way. Let $C_{A \\times B}(M \\times N)$ be the least positive integer $k$,\u2026","title":"Consecutive weighted zero-sum constant of a product with product weights"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/consecutive-prod-ge","kind":"node","lean":"MetaMathlibExt.consecutive_prod_ge","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#consecutive-prod-ge","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $R$ be a ring and $A \\subseteq R$. For an $R$-module $P$, let $C_A(P)$ be the least positive integer $k$, if one exists, such that every sequence $x_1, \\dots, x_k$ in $P$ has indices $s \\le t$ and weights $a_s, \\dots, a_t \\in A$ with $a_s x_s + \\dots + a_t x_t = 0$. If $M$ and $N$ are $R$-modules for which $C_A(M)$, $C_A(N)$ and $C_A(M \\times N)$ all exist, then $C_A(M \\times N) \\ge C_A(M) \\, C_A(N)$.","title":"Consecutive weighted zero-sum constant of a product of modules"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/consecutive-constant-eq-card","kind":"node","lean":"MetaMathlibExt.consecutive_constant_eq_card","parent":23311,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05/index.html#consecutive-constant-eq-card","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-05","status":"can_prove","summary":"Let $G$ be a finite abelian group and let $C(G)$ be the least positive integer $k$ such that every sequence $x_1, \\dots, x_k$ in $G$ has a block of consecutive terms $x_s, \\dots, x_t$ with $s \\le t$ and $x_s + \\dots + x_t = 0$. Then $C(G) = |G|$.","title":"Consecutive zero-sum constant of a finite abelian group"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","kind":"scope","navigation":"transparent","parent":23226,"reading_title":{"format":"inline-parts-v1","parts":[["text","A block of consecutive coordinates orthogonal to "],["math","\\(a\\)"],["text"," vectors of "],["math","\\(\\mathbb{Z}_2^m\\)"],["text"," to Consecutive weighted zero-sum constant for odd weights"]],"source":"A block of consecutive coordinates orthogonal to $a$ vectors of $\\mathbb{Z}_2^m$ to Consecutive weighted zero-sum constant for odd weights"},"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","title":"A block of consecutive coordinates orthogonal to $a$ vectors of $\\mathbb{Z}_2^m$ to Consecutive weighted zero-sum constant for odd weights"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-consecutive-orthogonal","kind":"node","lean":"MetaMathlibExt.exists_consecutive_orthogonal","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","A block of consecutive coordinates orthogonal to "],["math","\\(a\\)"],["text"," vectors of "],["math","\\(\\mathbb{Z}_2^m\\)"]],"source":"A block of consecutive coordinates orthogonal to $a$ vectors of $\\mathbb{Z}_2^m$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-consecutive-orthogonal","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $\\mathbb{Z}_2 = \\mathbb{Z}/2\\mathbb{Z}$, let $2^a \\le m$ and let $v_1, \\dots, v_a \\in \\mathbb{Z}_2^m$, with coordinates indexed by $\\{0, \\dots, m - 1\\}$. Then there are indices $s \\le t$ in $\\{0, \\dots, m - 1\\}$ such that $\\sum_{j = s}^{t} (v_i)_j = 0$ in $\\mathbb{Z}_2$ for every $i$; that is, the nonzero vector whose ones occupy exactly the positions $s, \\dots, t$ is orthogonal to every $v_i$.","title":"A block of consecutive coordinates orthogonal to $a$ vectors of $\\mathbb{Z}_2^m$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-consecutive-even-inter","kind":"node","lean":"MetaMathlibExt.exists_consecutive_even_inter","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","A block of consecutive integers meeting "],["math","\\(a\\)"],["text"," given sets evenly"]],"source":"A block of consecutive integers meeting $a$ given sets evenly"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-consecutive-even-inter","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $2^a \\le m$ and let $X_1, \\dots, X_a$ be finite sets of natural numbers. Then there are integers $s, t$ with $1 \\le s \\le t \\le m$ such that $|\\{s, s + 1, \\dots, t\\} \\cap X_i|$ is even for every $i$.","title":"A block of consecutive integers meeting $a$ given sets evenly"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/unit-weighted-zero-sum-of-even-non-divis","kind":"node","lean":"MetaMathlibExt.unitWeightedZeroSum_of_even_nonDivisible","parent":23332,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#unit-weighted-zero-sum-of-even-non-divis","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $p$ be a prime, let $r$ and $k$ be natural numbers, put $n = p^r$, and let $x_1, \\dots, x_k \\in \\mathbb{Z}/n\\mathbb{Z}$. Suppose that for each $i \\in [1, r]$ the set $X_i = \\{j : x_j \\not\\equiv 0 \\pmod{p^i}\\}$ has an even number of elements. Then there are units $a_1, \\dots, a_k$ of $\\mathbb{Z}/n\\mathbb{Z}$ with $a_1 x_1 + \\dots + a_k x_k = 0$.","title":"Griffiths' criterion for unit-weighted zero sums modulo a prime power"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/davenport-zmod-two-pow","kind":"node","lean":"MetaMathlibExt.davenport_zmod_two_pow","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Davenport constant of "],["math","\\(\\mathbb{Z}_2^a\\)"]],"source":"Davenport constant of $\\mathbb{Z}_2^a$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#davenport-zmod-two-pow","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $\\mathbb{Z}_2 = \\mathbb{Z}/2\\mathbb{Z}$. For every natural number $a$, the least positive integer $k$ such that every sequence of $k$ elements of the group $\\mathbb{Z}_2^a$ has a nonempty subsequence with sum zero is $a + 1$; that is, the Davenport constant of $\\mathbb{Z}_2^a$ is $a + 1$.","title":"Davenport constant of $\\mathbb{Z}_2^a$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-ne-zero-orthogonal","kind":"node","lean":"MetaMathlibExt.exists_ne_zero_orthogonal","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","A nonzero vector of "],["math","\\(\\mathbb{Z}_2^m\\)"],["text"," orthogonal to "],["math","\\(a\\)"],["text"," vectors when "],["math","\\(a < m\\)"]],"source":"A nonzero vector of $\\mathbb{Z}_2^m$ orthogonal to $a$ vectors when $a < m$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-ne-zero-orthogonal","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $\\mathbb{Z}_2 = \\mathbb{Z}/2\\mathbb{Z}$, let $a + 1 \\le m$ and let $v_1, \\dots, v_a \\in \\mathbb{Z}_2^m$. Then there is a nonzero $w \\in \\mathbb{Z}_2^m$ with $w \\cdot v_i = \\sum_j w_j (v_i)_j = 0$ in $\\mathbb{Z}_2$ for every $i$.","title":"A nonzero vector of $\\mathbb{Z}_2^m$ orthogonal to $a$ vectors when $a < m$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-nonempty-even-inter","kind":"node","lean":"MetaMathlibExt.exists_nonempty_even_inter","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","A nonempty subset of "],["math","\\(\\{1, \\dots, m\\}\\)"],["text"," meeting "],["math","\\(a\\)"],["text"," given sets evenly"]],"source":"A nonempty subset of $\\{1, \\dots, m\\}$ meeting $a$ given sets evenly"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-nonempty-even-inter","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $a + 1 \\le m$ and let $X_1, \\dots, X_a$ be finite sets of natural numbers. Then there is a nonempty set $Y \\subseteq \\{1, \\dots, m\\}$ such that $|Y \\cap X_i|$ is even for every $i$.","title":"A nonempty subset of $\\{1, \\dots, m\\}$ meeting $a$ given sets evenly"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-zero-sum-card-zmod-two-pow","kind":"node","lean":"MetaMathlibExt.exists_zeroSum_card_zmod_two_pow","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero-sum subsequences of length "],["math","\\(m\\)"],["text"," in "],["math","\\(\\mathbb{Z}_2^a\\)"]],"source":"Zero-sum subsequences of length $m$ in $\\mathbb{Z}_2^a$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-zero-sum-card-zmod-two-pow","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $a$ and $m$ be natural numbers with $m$ even and $2^a \\le m$, and write $\\mathbb{Z}_2 = \\mathbb{Z}/2\\mathbb{Z}$. Every sequence $(x_1, \\dots, x_{m+a})$ of length $m + a$ in $\\mathbb{Z}_2^a$ has a zero-sum subsequence of length $m$: there is a set $I$ of exactly $m$ indices with $\\sum_{i \\in I} x_i = 0$.","title":"Zero-sum subsequences of length $m$ in $\\mathbb{Z}_2^a$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-weight-orthogonal","kind":"node","lean":"MetaMathlibExt.exists_weight_orthogonal","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","A weight-"],["math","\\(m\\)"],["text"," vector orthogonal to given vectors of "],["math","\\(\\mathbb{Z}_2^{m+a}\\)"]],"source":"A weight-$m$ vector orthogonal to given vectors of $\\mathbb{Z}_2^{m+a}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-weight-orthogonal","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $a$ and $m$ be natural numbers with $m$ even and $2^a \\le m$, write $\\mathbb{Z}_2 = \\mathbb{Z}/2\\mathbb{Z}$, and let $v_1, \\dots, v_a$ be vectors in $\\mathbb{Z}_2^{m+a}$. Then there is a vector $w \\in \\mathbb{Z}_2^{m+a}$ with exactly $m$ coordinates equal to $1$ such that $w \\cdot v_i = \\sum_j w_j (v_i)_j = 0$ for every $i$.","title":"A weight-$m$ vector orthogonal to given vectors of $\\mathbb{Z}_2^{m+a}$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-card-even-inter","kind":"node","lean":"MetaMathlibExt.exists_card_even_inter","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","An "],["math","\\(m\\)"],["text","-subset of "],["math","\\(\\{1, \\dots, m + a\\}\\)"],["text"," meeting "],["math","\\(a\\)"],["text"," given sets evenly"]],"source":"An $m$-subset of $\\{1, \\dots, m + a\\}$ meeting $a$ given sets evenly"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-card-even-inter","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $m$ be even with $2^a \\le m$, and let $X_1, \\dots, X_a$ be finite sets of natural numbers. Then there is a set $Y \\subseteq \\{1, \\dots, m + a\\}$ with $|Y| = m$ such that $|Y \\cap X_i|$ is even for every $i$.","title":"An $m$-subset of $\\{1, \\dots, m + a\\}$ meeting $a$ given sets evenly"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-unit-weighted-zero-sum-card","kind":"node","lean":"MetaMathlibExt.exists_unitWeightedZeroSum_card","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit-weighted zero sums of even length in "],["math","\\(\\mathbb{Z}_n\\)"]],"source":"Unit-weighted zero sums of even length in $\\mathbb{Z}_n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-unit-weighted-zero-sum-card","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $n \\ge 1$, let $\\Omega(n)$ be the number of prime factors of $n$ counted with multiplicity, and let $m$ be even with $2^{\\Omega(n)} \\le m$. Then every sequence $x_1, \\dots, x_{m + \\Omega(n)}$ in $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$ has a unit-weighted zero-sum subsequence of length $m$: there are a set $I \\subseteq \\{1, \\dots, m + \\Omega(n)\\}$ with $|I| = m$ and units $a_i$ of $\\mathbb{Z}_n$ for $i \\in I$\u2026","title":"Unit-weighted zero sums of even length in $\\mathbb{Z}_n$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/exists-weight-mem-subspace","kind":"node","lean":"MetaMathlibExt.exists_weight_mem_subspace","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every "],["math","\\(m\\)"],["text","-dimensional subspace of "],["math","\\(\\mathbb{Z}_2^{m+a}\\)"],["text"," has a vector of weight "],["math","\\(m\\)"]],"source":"Every $m$-dimensional subspace of $\\mathbb{Z}_2^{m+a}$ has a vector of weight $m$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#exists-weight-mem-subspace","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $\\mathbb{Z}_2 = \\mathbb{Z}/2\\mathbb{Z}$ and let $m$ be even with $2^a \\le m$. Then every subspace $W$ of $\\mathbb{Z}_2^{m+a}$ of dimension $m$ contains a vector with exactly $m$ coordinates equal to $1$.","title":"Every $m$-dimensional subspace of $\\mathbb{Z}_2^{m+a}$ has a vector of weight $m$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/max-finrank-avoiding-weight","kind":"node","lean":"MetaMathlibExt.max_finrank_avoiding_weight","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Largest subspaces of "],["math","\\(\\mathbb{Z}_2^{m+a}\\)"],["text"," with no vector of weight "],["math","\\(m\\)"]],"source":"Largest subspaces of $\\mathbb{Z}_2^{m+a}$ with no vector of weight $m$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#max-finrank-avoiding-weight","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $a$ and $m$ be natural numbers with $m$ even and $2^a \\le m$, and write $\\mathbb{Z}_2 = \\mathbb{Z}/2\\mathbb{Z}$. The largest dimension of a subspace of $\\mathbb{Z}_2^{m+a}$ containing no vector with exactly $m$ coordinates equal to $1$ is $m - 1$.","title":"Largest subspaces of $\\mathbb{Z}_2^{m+a}$ with no vector of weight $m$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/gao-unit-constant-of-even","kind":"node","lean":"MetaMathlibExt.gao_unit_constant_of_even","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit-weighted Erd\u0151s\u2013Ginzburg\u2013Ziv constant of "],["math","\\(\\mathbb{Z}_n\\)"],["text"," for even "],["math","\\(n\\)"]],"source":"Unit-weighted Erd\u0151s\u2013Ginzburg\u2013Ziv constant of $\\mathbb{Z}_n$ for even $n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#gao-unit-constant-of-even","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $n$ be a positive even integer, let $U(n)$ be the set of units of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$, and let $\\Omega(n)$ be the number of prime factors of $n$ counted with multiplicity. Then $E_{U(n)}(n) = n + \\Omega(n)$, where $E_{U(n)}(n)$ is the least positive integer $k$ such that for every sequence $(x_1, \\dots, x_k)$ in $\\mathbb{Z}_n$ there are a set $I$ of exactly $n$ indices and units\u2026","title":"Unit-weighted Erd\u0151s\u2013Ginzburg\u2013Ziv constant of $\\mathbb{Z}_n$ for even $n$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/e-extremal-of-unit-equiv-profile","kind":"node","lean":"MetaMathlibExt.eExtremal_of_unitEquiv_profile","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sequences equivalent to the extremal profile have no unit-weighted zero sum of length "],["math","\\(2^r\\)"]],"source":"Sequences equivalent to the extremal profile have no unit-weighted zero sum of length $2^r$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#e-extremal-of-unit-equiv-profile","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Sequences equivalent to the extremal profile have no unit-weighted zero sum of length $2^r$. Let $r \\ge 1$, $n = 2^r$ and $L = n + r - 1$, and let $U(n)$ be the unit group of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$. Let $y_1, \\dots, y_L \\in \\mathbb{Z}_n$ be a sequence in which, for each $i \\in \\{0, \\dots, r - 2\\}$, exactly one term equals $2^i$, an odd number of terms equal $2^{r-1}$, and every term is $0$ or $2^i$\u2026","title":"Sequences equivalent to the extremal profile have no unit-weighted zero sum of length $2^r$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/unit-equiv-profile-of-e-extremal","kind":"node","lean":"MetaMathlibExt.unitEquiv_profile_of_eExtremal","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Structure of extremal sequences for unit-weighted zero sums of length "],["math","\\(2^r\\)"]],"source":"Structure of extremal sequences for unit-weighted zero sums of length $2^r$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#unit-equiv-profile-of-e-extremal","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $r \\ge 1$, let $n = 2^r$, let $U(n)$ be the set of units of $\\mathbb{Z}_n = \\mathbb{Z}/n\\mathbb{Z}$, and let $x = (x_1, \\dots, x_{n+r-1})$ be a sequence in $\\mathbb{Z}_n$ with no $U(n)$-weighted zero-sum subsequence of length $n$: there is no set $I$ of exactly $n$ indices with units $a_i \\in U(n)$ for $i \\in I$ such that $\\sum_{i \\in I} a_i x_i = 0$. Then $x$ is $U(n)$-equivalent to a sequence\u2026","title":"Structure of extremal sequences for unit-weighted zero sums of length $2^r$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/unit-weighted-zero-sum-of-even-units","kind":"node","lean":"MetaMathlibExt.unitWeightedZeroSum_of_even_units","parent":23332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Luca's lemma on unit-weighted zero sums modulo "],["math","\\(2^r\\)"]],"source":"Luca's lemma on unit-weighted zero sums modulo $2^r$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#unit-weighted-zero-sum-of-even-units","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $r$ and $k$ be natural numbers and let $x_1, \\dots, x_k \\in \\mathbb{Z}/2^r\\mathbb{Z}$. If the number of indices $j$ for which $x_j$ is a unit is even and nonzero, then there are units $a_1, \\dots, a_k$ of $\\mathbb{Z}/2^r\\mathbb{Z}$ with $a_1 x_1 + \\dots + a_k x_k = 0$.","title":"Luca's lemma on unit-weighted zero sums modulo $2^r$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/unit-weighted-zero-sum-of-div-prime","kind":"node","lean":"MetaMathlibExt.unitWeightedZeroSum_of_div_prime","parent":23332,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#unit-weighted-zero-sum-of-div-prime","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $p$ be a prime, let $n'$ and $k$ be natural numbers, put $n = p n'$, and let $z_1, \\dots, z_k \\in \\mathbb{Z}/n\\mathbb{Z}$. Write $\\bar z_i$ for the reduction of $z_i$ modulo $n'$. If there are units $a_1, \\dots, a_k$ of $\\mathbb{Z}/n'\\mathbb{Z}$ with $\\sum_i a_i \\bar z_i = 0$, then there are units $b_1, \\dots, b_k$ of $\\mathbb{Z}/n\\mathbb{Z}$ with $\\sum_i b_i \\, p z_i = 0$.","title":"Unit-weighted zero sums of multiples of a prime"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/c-extremal-iff-halves","kind":"node","lean":"MetaMathlibExt.cExtremal_iff_halves","parent":23332,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#c-extremal-iff-halves","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $s \\ge 0$, $n = 2^{s+1}$ and $n' = 2^s$. Say that a sequence $y_1, \\dots, y_L$ in $\\mathbb{Z}_m$ has no consecutive unit-weighted zero-sum if there are no indices $1 \\le i \\le j \\le L$ and units $a_i, \\dots, a_j$ of $\\mathbb{Z}_m$ with $a_i y_i + \\dots + a_j y_j = 0$. For a sequence $x_1, \\dots, x_{n-1}$ in $\\mathbb{Z}_n$, let $r_l \\in \\{0, \\dots, n - 1\\}$ be the representative of $x_l$ and let\u2026","title":"Halving consecutive unit-weighted zero-sum free sequences modulo a power of two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/consecutive-interval-weights","kind":"node","lean":"MetaMathlibExt.consecutive_interval_weights","parent":23332,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#consecutive-interval-weights","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $n \\ge 2$ and $1 \\le r \\le n - 1$, and let $A \\subseteq \\mathbb{Z}_n$ be the set of residues of $1, 2, \\dots, r$. Let $C_A(n)$ be the least positive integer $k$ such that every sequence $x_1, \\dots, x_k$ in $\\mathbb{Z}_n$ has indices $s \\le t$ and weights $a_s, \\dots, a_t \\in A$ with $a_s x_s + \\dots + a_t x_t = 0$. Then $C_A(n) = \\lceil n / r \\rceil$.","title":"Consecutive weighted zero-sum constant for interval weights"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/consecutive-odd-weights","kind":"node","lean":"MetaMathlibExt.consecutive_odd_weights","parent":23332,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06/index.html#consecutive-odd-weights","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-06","status":"can_prove","summary":"Let $n$ be a positive even integer, let $2^r$ be the largest power of $2$ dividing $n$ and let $m = n / 2^r$. Let $A \\subseteq \\mathbb{Z}_n$ contain the residue of $m$ and consist of odd elements (residues whose representative in $\\{0, \\dots, n - 1\\}$ is odd). Let $C_A(n)$ be the least positive integer $k$ such that every sequence $x_1, \\dots, x_k$ in $\\mathbb{Z}_n$ has indices $s \\le t$ and weights\u2026","title":"Consecutive weighted zero-sum constant for odd weights"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07","kind":"scope","navigation":"transparent","parent":23226,"reading_title":{"format":"inline-parts-v1","parts":[["text","Davenport constant of "],["math","\\(\\mathbb{Z}_p\\)"],["text"," with "],["math","\\(k\\)"],["text","-th power unit weights is at most "],["math","\\(k\\)"],["text"," to Weighted zero-sum free pairs for "],["math","\\(k\\)"],["text","-th power weights modulo a prime"]],"source":"Davenport constant of $\\mathbb{Z}_p$ with $k$-th power unit weights is at most $k$ to Weighted zero-sum free pairs for $k$-th power weights modulo a prime"},"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07","status":"can_prove","title":"Davenport constant of $\\mathbb{Z}_p$ with $k$-th power unit weights is at most $k$ to Weighted zero-sum free pairs for $k$-th power weights modulo a prime"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/davenport-unit-pow-le","kind":"node","lean":"MetaMathlibExt.davenport_unit_pow_le","parent":23353,"reading_title":{"format":"inline-parts-v1","parts":[["text","Davenport constant of "],["math","\\(\\mathbb{Z}_p\\)"],["text"," with "],["math","\\(k\\)"],["text","-th power unit weights is at most "],["math","\\(k\\)"]],"source":"Davenport constant of $\\mathbb{Z}_p$ with $k$-th power unit weights is at most $k$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/index.html#davenport-unit-pow-le","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07","status":"can_prove","summary":"Let $k$ be an odd prime and $p$ a prime with $p \\equiv 1 \\pmod{k}$ and $p \\not\\equiv 1 \\pmod{k^2}$. Let $\\mathbb{Z}_p = \\mathbb{Z}/p\\mathbb{Z}$ and let $U(p)^k$ be the set of $k$-th powers of units of $\\mathbb{Z}_p$. Then every sequence $x_1, \\dots, x_k$ of $k$ elements of $\\mathbb{Z}_p$ has a $U(p)^k$-weighted zero-sum subsequence: there are a nonempty set $I \\subseteq \\{1, \\dots, k\\}$ and weights $a_i \\in U(p)^k$\u2026","title":"Davenport constant of $\\mathbb{Z}_p$ with $k$-th power unit weights is at most $k$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/pair-no-weighted-zero-sum-iff","kind":"node","lean":"MetaMathlibExt.pair_no_weightedZeroSum_iff","parent":23353,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/index.html#pair-no-weighted-zero-sum-iff","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07","status":"can_prove","summary":"Let $F$ be a field, let $A$ be a subgroup of the multiplicative group $F^*$, and let $x, y \\in F$ be nonzero. Then the sequence $(x_1, x_2) = (x, y)$ has no $A$-weighted zero-sum subsequence, that is, there is no nonempty set $I \\subseteq \\{1, 2\\}$ with elements $a_i \\in A$ for $i \\in I$ such that $\\sum_{i \\in I} a_i x_i = 0$, if and only if $x$ and $-y$ lie in different cosets of $A$ in $F^*$.","title":"Weighted zero-sum free pairs in a field"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/davenport-proper-subgroup-ge-three","kind":"node","lean":"MetaMathlibExt.davenport_proper_subgroup_ge_three","parent":23353,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted zero-sum free pairs for a proper subgroup of "],["math","\\(F^\\times\\)"]],"source":"Weighted zero-sum free pairs for a proper subgroup of $F^\\times$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/index.html#davenport-proper-subgroup-ge-three","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07","status":"can_prove","summary":"Let $F$ be a field and $A$ a proper subgroup of the multiplicative group $F^\\times$. Then some sequence $x_1, x_2$ in $F$ has no $A$-weighted zero-sum subsequence: there is no nonempty $I \\subseteq \\{1, 2\\}$ with weights $a_i \\in A$ for $i \\in I$ such that $\\sum_{i \\in I} a_i x_i = 0$.","title":"Weighted zero-sum free pairs for a proper subgroup of $F^\\times$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/davenport-unit-pow-ge-three","kind":"node","lean":"MetaMathlibExt.davenport_unit_pow_ge_three","parent":23353,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted zero-sum free pairs for "],["math","\\(k\\)"],["text","-th power weights modulo a prime"]],"source":"Weighted zero-sum free pairs for $k$-th power weights modulo a prime"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07/index.html#davenport-unit-pow-ge-three","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/weighted-davenport-constant/part-07","status":"can_prove","summary":"Let $k \\ge 2$, let $p$ be a prime with $p \\equiv 1 \\pmod k$, and let $U(p)^k = \\{u^k : u \\in \\mathbb{Z}_p^\\times\\}$. Then some sequence $x_1, x_2$ in $\\mathbb{Z}_p$ has no $U(p)^k$-weighted zero-sum subsequence: there is no nonempty $I \\subseteq \\{1, 2\\}$ with weights $a_i \\in U(p)^k$ for $i \\in I$ such that $\\sum_{i \\in I} a_i x_i = 0$.","title":"Weighted zero-sum free pairs for $k$-th power weights modulo a prime"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups","kind":"scope","navigation":"module","parent":23136,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups","status":"can_prove","title":"Zero subset sums in cyclic groups"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups/sqrt-size-has-zero-subset-sum","kind":"node","lean":"MathlibExt.Combinatorics.ZeroSubsetSumSqrtErdosProblem540Wanted.sqrtSizeHasZeroSubsetSum","parent":23358,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups/index.html#sqrt-size-has-zero-subset-sum","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups","status":"fully_proved","summary":"This is the statement that there is a real number $C > 0$ such that for every integer $N \\ge 1$, every set $A \\subseteq \\mathbb{Z}/N\\mathbb{Z}$ with $|A| \\ge C\\sqrt{N}$ has a nonempty subset $S \\subseteq A$ whose elements sum to $0$ in $\\mathbb{Z}/N\\mathbb{Z}$.","title":"The Erd\u0151s\u2013Heilbronn zero subset sum statement (Erd\u0151s Problem 540)"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups/sqrt-size-has-zero-subset-sum-holds","kind":"node","lean":"MathlibExt.Combinatorics.ZeroSubsetSumSqrtErdosProblem540Wanted.sqrtSizeHasZeroSubsetSum_holds","parent":23358,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero subset sums of large sets in "],["math","\\(\\mathbb{Z}/N\\mathbb{Z}\\)"],["text"," (Erd\u0151s Problem 540)"]],"source":"Zero subset sums of large sets in $\\mathbb{Z}/N\\mathbb{Z}$ (Erd\u0151s Problem 540)"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups/index.html#sqrt-size-has-zero-subset-sum-holds","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-subset-sums-in-cyclic-groups","status":"can_prove","summary":"There is a real number $C > 0$ such that for every integer $N \\ge 1$, every set $A \\subseteq \\mathbb{Z}/N\\mathbb{Z}$ with $|A| \\ge C\\sqrt{N}$ has a nonempty subset $S \\subseteq A$ whose elements sum to $0$ in $\\mathbb{Z}/N\\mathbb{Z}$.","title":"Zero subset sums of large sets in $\\mathbb{Z}/N\\mathbb{Z}$ (Erd\u0151s Problem 540)"},{"data_url":"scopes/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number.json","id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","kind":"scope","navigation":"module","parent":23136,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"can_prove","title":"Zero-sum subsequences of small cross number"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/exists-zero-sum-cross-number-le-one","kind":"node","lean":"MetaMathlibExt.exists_zero_sum_cross_number_le_one","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#exists-zero-sum-cross-number-le-one","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"can_prove","summary":"Let $\\Gamma$ be a finite abelian group of order $|\\Gamma|$. Every sequence $g_1, \\dots, g_{|\\Gamma|}$ of $|\\Gamma|$ elements of $\\Gamma$ has a nonempty zero-sum subsequence with cross number at most one: there is a nonempty set $S \\subseteq \\{1, \\dots, |\\Gamma|\\}$ of indices with $\\sum_{i \\in S} g_i = 0$ and $\\sum_{i \\in S} 1/\\operatorname{ord}(g_i) \\le 1$, where $\\operatorname{ord}(g_i)$ is the order of $g_i$.","title":"Zero-sum subsequences of cross number at most one (Geroldinger; Elledge\u2013Hurlbert)"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/exists-zero-sum-cross-number-le-one-2","kind":"node","lean":"MetaMathlibExt.ZMod.exists_zero_sum_cross_number_le_one","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#exists-zero-sum-cross-number-le-one-2","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"can_prove","summary":"For every positive integer $n$, every sequence $g_1, \\dots, g_n$ of $n$ elements of $\\mathbb{Z}/n\\mathbb{Z}$ has a nonempty zero-sum subsequence with cross number at most one: there is a nonempty set $S \\subseteq \\{1, \\dots, n\\}$ of indices with $\\sum_{i \\in S} g_i = 0$ and $\\sum_{i \\in S} 1/\\operatorname{ord}(g_i) \\le 1$, where $\\operatorname{ord}(g_i)$ is the additive order of $g_i$. The theorem is due to Lemke\u2026","title":"Lemke\u2013Kleitman theorem"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/is-good-sequence","kind":"node","lean":"MetaMathlibExt.IsGoodSequence","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#is-good-sequence","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $\\Gamma$ be a finite abelian group, written additively, and let $g = (g_1, \\dots, g_t)$ be a sequence of $t$ elements of $\\Gamma$. The sequence $g$ is *good* if there is a nonempty set $S \\subseteq \\{1, \\dots, t\\}$ of indices with $\\sum_{i \\in S} g_i = 0$ whose cross number $\\sum_{i \\in S} 1/|g_i|$ is at most $1$, where $|g_i|$ is the order of $g_i$.","title":"Good sequences in a finite abelian group"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/good-sequence-prob","kind":"node","lean":"MetaMathlibExt.goodSequenceProb","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#good-sequence-prob","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $\\Gamma$ be a finite abelian group and $t \\ge 0$ an integer. A sequence $g_1, \\dots, g_t$ in $\\Gamma$ is good if some nonempty set $S \\subseteq \\{1, \\dots, t\\}$ of indices has $\\sum_{i \\in S} g_i = 0$ and $\\sum_{i \\in S} 1/\\operatorname{ord}(g_i) \\le 1$, where $\\operatorname{ord}(g_i)$ is the order of $g_i$. The good-sequence probability of $\\Gamma$ at length $t$ is the number of good sequences of length $t$ in\u2026","title":"Probability that a random sequence is good"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/good-sequence-prob-zero","kind":"node","lean":"MetaMathlibExt.goodSequenceProb_zero","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#good-sequence-prob-zero","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"For every finite abelian group $\\Gamma$, the probability that a uniformly random sequence of length $0$ in $\\Gamma$ is good equals $0$: the empty sequence has no nonempty zero-sum subsequence, so it is not good.","title":"Good-sequence probability at length zero"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/half-good-threshold","kind":"node","lean":"MetaMathlibExt.halfGoodThreshold","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#half-good-threshold","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"For a finite abelian group $\\Gamma$, $P_{1/2}(\\Gamma)$ is the least $t \\ge 0$ such that a uniformly random sequence of $t$ elements of $\\Gamma$ is good (has a nonempty zero-sum subsequence with cross number at most one) with probability at least $1/2$, that is, such that at least half of the $|\\Gamma|^t$ sequences of length $t$ are good. If no such $t$ exists, $P_{1/2}(\\Gamma)$ is defined to be $0$.","title":"Half-good threshold"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/half-good-threshold-eq-one-of-card-le-tw","kind":"node","lean":"MetaMathlibExt.halfGoodThreshold_eq_one_of_card_le_two","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#half-good-threshold-eq-one-of-card-le-tw","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"If $\\Gamma$ is a finite abelian group with at most two elements, then its half-good threshold $P_{1/2}(\\Gamma)$ equals $1$.","title":"Half-good threshold of a group of order at most two"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/is-threshold-function","kind":"node","lean":"MetaMathlibExt.IsThresholdFunction","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#is-threshold-function","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $P : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{R}$, where $P(k, s)$ is read as the probability of an event in the $k$-th space when $s$ random elements are chosen. A function $t : \\mathbb{N} \\to \\mathbb{R}$ is a *threshold function* for $P$ if $t(k) > 0$ for all sufficiently large $k$ and, for every function $f : \\mathbb{N} \\to \\mathbb{N}$, $P(k, f(k)) \\to 1$ as $k \\to \\infty$ whenever $t = o(f)$, and\u2026","title":"Threshold functions"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/const-mul","kind":"node","lean":"MetaMathlibExt.IsThresholdFunction.const_mul","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#const-mul","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $P : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{R}$ and $t : \\mathbb{N} \\to \\mathbb{R}$. If $t$ is a threshold function for $P$ and $c > 0$ is a real number, then $k \\mapsto c \\, t(k)$ is also a threshold function for $P$.","title":"Positive multiples of threshold functions"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/pebbling-step","kind":"node","lean":"MetaMathlibExt.PebblingStep","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#pebbling-step","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex set $V$, and let $w$ assign a natural-number cost $w(uv)$ to each unordered pair of vertices $\\{u, v\\}$. For configurations $C, C' : V \\to \\mathbb{N}$, there is a pebbling step from $C$ to $C'$ if there is an edge $uv$ of $G$ with $0 < w(uv) \\le C(u)$ such that $C'$ is obtained from $C$ by removing $w(uv)$ pebbles from $u$ and adding one pebble to $v$.","title":"Lagarias\u2013Saks pebbling step"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/pebbling-solvable","kind":"node","lean":"MetaMathlibExt.PebblingSolvable","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#pebbling-solvable","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex set $V$ with natural-number edge costs $w$. A configuration $C : V \\to \\mathbb{N}$ of pebbles is *solvable* if for every target vertex $r$ there is a finite, possibly empty, sequence of pebbling steps on $G$ with costs $w$ leading from $C$ to a configuration with at least one pebble on $r$.","title":"Solvable pebbling configurations"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/pebbling-solvable-of-forall-pos","kind":"node","lean":"MetaMathlibExt.pebblingSolvable_of_forall_pos","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#pebbling-solvable-of-forall-pos","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $G$ be a simple graph with natural-number edge costs $w$. Every pebbling configuration $C$ on $G$ with $C(r) \\ge 1$ for every vertex $r$ is solvable with respect to $G$ and $w$.","title":"Configurations covering every vertex are solvable"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/solvable-configuration-prob","kind":"node","lean":"MetaMathlibExt.solvableConfigurationProb","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#solvable-configuration-prob","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ with $n$ vertices, let $w$ assign a natural-number cost to each unordered pair of vertices, and let $t \\ge 0$ be an integer. A configuration of $t$ pebbles is a function $C : V \\to \\mathbb{N}$ with $\\sum_{v \\in V} C(v) = t$. The value is the number of configurations of $t$ pebbles that are solvable with respect to $G$ and $w$, divided by the number of all\u2026","title":"Probability that a random pebbling configuration is solvable"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/solvable-configuration-prob-le-one","kind":"node","lean":"MetaMathlibExt.solvableConfigurationProb_le_one","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#solvable-configuration-prob-le-one","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"For every simple graph $G$ on a finite vertex set with natural-number edge costs $w$ and every $t \\ge 0$, the probability that a uniformly random configuration of $t$ pebbles on $G$ is solvable is at most $1$.","title":"The solvability probability is at most one"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/grid-graph","kind":"node","lean":"MetaMathlibExt.gridGraph","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#grid-graph","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"For natural numbers $d$ and $k$, the $d$-dimensional grid $P_k^d = \\square_{i\\lt d} P_k$ is the simple graph on $\\{0, \\dots, k-1\\}^d$ in which $x$ and $y$ are adjacent when there is a coordinate $i$ with $x_j = y_j$ for all $j \\ne i$ and $|x_i - y_i| = 1$. For $k = 2$ this is the $d$-cube $Q_d$.","title":"Grid graph"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/grid-cost","kind":"node","lean":"MetaMathlibExt.gridCost","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#grid-cost","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $d$ and $k$ be natural numbers and let $w_0, \\dots, w_{d-1}$ be natural numbers. The grid cost with cost $w_i$ in coordinate $i$ assigns to each unordered pair $\\{x, y\\}$ of points of $\\{0, \\dots, k-1\\}^d$ the sum of the $w_i$ over the coordinates $i$ with $x_i \\ne y_i$. On an edge of the grid exactly one coordinate differs, so an edge in coordinate $i$ costs $w_i$.","title":"Grid edge costs"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/grid-cost-eq-of-eq-off","kind":"node","lean":"MetaMathlibExt.gridCost_eq_of_eq_off","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#grid-cost-eq-of-eq-off","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"fully_proved","summary":"Let $d$ and $k$ be natural numbers, let $w_0, \\dots, w_{d-1}$ be natural numbers, let $x, y \\in \\{0, \\dots, k-1\\}^d$, and let $i \\in \\{0, \\dots, d-1\\}$. If $x_i \\ne y_i$ and $x_j = y_j$ for every $j \\ne i$, then the grid cost of the pair $\\{x, y\\}$ with cost $w_j$ in coordinate $j$ is $w_i$.","title":"Grid cost of a pair differing in one coordinate"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/half-good-threshold-pi-zmod-prime-pow-le","kind":"node","lean":"MetaMathlibExt.halfGoodThreshold_pi_zmod_prime_pow_le","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#half-good-threshold-pi-zmod-prime-pow-le","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"can_prove","summary":"Let $d \\ge 0$, let $p_1, \\dots, p_d$ be primes, not necessarily distinct, and for each $k$ let $\\Gamma_k = \\prod_{i=1}^d \\mathbb{Z}/p_i^k\\mathbb{Z}$, and let $P_{1/2}(\\Gamma_k)$ be its half-good threshold. Write $\\lg = \\log_2$. Then there is a real constant $C$ such that for all sufficiently large $k$,","title":"Threshold version of Geroldinger's theorem"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/grid-graph-pebbling-threshold-eq","kind":"node","lean":"MetaMathlibExt.gridGraph_pebbling_threshold_eq","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#grid-graph-pebbling-threshold-eq","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"can_prove","summary":"Let $d \\ge 1$ and let $w_0, \\dots, w_{d-1}$ be natural numbers with $w_i \\ge 2$ for every $i$. For each $k$ give the grid $P_k^d$ on $\\{0, \\dots, k-1\\}^d$ the cost $w_i$ on every edge in coordinate $i$, and let $t : \\mathbb{N} \\to \\mathbb{R}$ be a threshold function, as $k \\to \\infty$, for the probability that a uniformly random configuration of $s$ pebbles on $P_k^d$ is solvable. Write $\\lg = \\log_2$ and","title":"Bushaw\u2013Kettle pebbling threshold for grids"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/cube-pebbling-threshold-bounds","kind":"node","lean":"MetaMathlibExt.cube_pebbling_threshold_bounds","parent":23361,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#cube-pebbling-threshold-bounds","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"can_prove","summary":"For $d \\ge 0$ let $Q_d$ be the $d$-cube, the grid graph on $\\{0, 1\\}^d$, with cost $2$ on every edge, and let $t : \\mathbb{N} \\to \\mathbb{R}$ be a threshold function, as $d \\to \\infty$, for the probability that a uniformly random configuration of $s$ pebbles on $Q_d$ is solvable. Write $n = 2^d$ and $\\lg = \\log_2$. Then for every real $\\varepsilon > 0$, as $d \\to \\infty$, $n^{1-\\varepsilon} = O(t(d))$ and\u2026","title":"Pebbling threshold bounds for the cubes (Czygrinow\u2013Wagner, Alon)"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/zmod-two-pi-good-threshold-is-big-o","kind":"node","lean":"MetaMathlibExt.zmod_two_pi_good_threshold_isBigO","parent":23361,"reading_title":{"format":"inline-parts-v1","parts":[["text","Threshold for good sequences in "],["math","\\(\\mathbb{Z}_2^d\\)"]],"source":"Threshold for good sequences in $\\mathbb{Z}_2^d$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#zmod-two-pi-good-threshold-is-big-o","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"can_prove","summary":"Write $\\lg = \\log_2$. Let $t$ be a threshold function, in $d$, for the probability that a uniformly random sequence of $s$ elements of $\\mathbb{Z}_2^d$ is good, and write $n = 2^d$. Then for every $\\varepsilon > 0$, $t(d) = O\\left(n / (\\lg \\lg n)^{1-\\varepsilon}\\right)$ as $d \\to \\infty$.","title":"Threshold for good sequences in $\\mathbb{Z}_2^d$"},{"id":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/zmod-thirty-five-pow-good-threshold-le","kind":"node","lean":"MetaMathlibExt.zmod_thirtyFive_pow_good_threshold_le","parent":23361,"reading_title":{"format":"inline-parts-v1","parts":[["text","Threshold for good sequences in "],["math","\\(\\mathbb{Z}/35^k\\mathbb{Z}\\)"]],"source":"Threshold for good sequences in $\\mathbb{Z}/35^k\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number/index.html#zmod-thirty-five-pow-good-threshold-le","scope":"Combinatorics/additive-combinatorics/zero-sum-subset-sums/zero-sum-subsequences-of-small-cross-number","status":"can_prove","summary":"Write $\\lg = \\log_2$. Let $t$ be a threshold function, in $k$, for the probability that a uniformly random sequence of $s$ elements of $\\mathbb{Z}/35^k\\mathbb{Z}$ is good. Then there is a real constant $C$ such that for all sufficiently large $k$,","title":"Threshold for good sequences in $\\mathbb{Z}/35^k\\mathbb{Z}$"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets","kind":"scope","parent":22582,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets","status":"can_prove","summary":"Density versions: Szemer\u00e9di's and Roth's theorems and Roth-number bounds, S\u00e1rk\u00f6zy's theorem, cap sets, counts of three-term progressions, progressions in complements and in walks, arithmetic progression API, and polynomial triangle removal in vector spaces.","title":"Arithmetic progressions in dense sets"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/szemeredi","kind":"node","lean":"MathlibExt.Combinatorics.Additive.SzemerediWanted.szemeredi","parent":23384,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/index.html#szemeredi","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets","status":"fully_proved","summary":"For every natural number $k \\ge 1$ and every real number $\\delta > 0$ there is a natural number $N$ such that for every $n \\ge N$, every set $A \\subseteq \\{0, 1, \\dots, n - 1\\}$ with $\\delta n \\le |A|$ contains an arithmetic progression of length $k$: there are natural numbers $a$ and $d > 0$ with $a + i d \\in A$ for all $i < k$.","title":"Szemer\u00e9di's theorem"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","title":"Arithmetic progressions and progression-free sets"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/is-apof-length-with","kind":"node","lean":"Set.IsAPOfLengthWith","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#is-apof-length-with","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, $l \\in \\mathbb{N} \\cup \\{\\infty\\}$ and $a, d \\in M$. A set $s \\subseteq M$ is an arithmetic progression of length $l$ with first term $a$ and difference $d$ if $s = \\{a + n d : n \\in \\mathbb{N},\\ n < l\\}$ and $s$ has exactly $l$ elements, counting the size of an infinite set as $\\infty$; here $n d$ is the sum of $n$ copies of $d$.","title":"Set arithmetic progressions with given first term and difference"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/is-apof-length","kind":"node","lean":"Set.IsAPOfLength","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#is-apof-length","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid and $l \\in \\mathbb{N} \\cup \\{\\infty\\}$. A set $s \\subseteq \\alpha$ is an arithmetic progression of length $l$ if there are $a, d \\in \\alpha$ such that $s = \\{a + nd : n \\in \\mathbb{N},\\ n < l\\}$ and $s$ has exactly $l$ elements (infinitely many when $l = \\infty$). Here $nd$ denotes the sum of $n$ copies of $d$.","title":"Sets that are arithmetic progressions"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/card","kind":"node","lean":"Set.IsAPOfLength.card","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#card","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid and $l \\in \\mathbb{N} \\cup \\{\\infty\\}$. If a set $s \\subseteq M$ is an arithmetic progression of length $l$, then $s$ has exactly $l$ elements, counting the size of an infinite set as $\\infty$.","title":"Cardinality of a set arithmetic progression"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/eq","kind":"node","lean":"Set.IsAPOfLength.eq","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#eq","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid and $l \\in \\mathbb{N} \\cup \\{\\infty\\}$. If a set $s \\subseteq M$ is an arithmetic progression of length $l$, then there are $a, d \\in M$ with $s = \\{a + n d : n \\in \\mathbb{N},\\ n < l\\}$, where $n d$ is the sum of $n$ copies of $d$.","title":"First term and difference of a set arithmetic progression"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/is-apof-length-with-2","kind":"node","lean":"List.IsAPOfLengthWith","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#is-apof-length-with-2","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, $l \\in \\mathbb{N}$ and $a, d \\in M$. A list $s$ of elements of $M$ is an arithmetic progression of length $l$ starting from $a$ with difference $d$, in either direction, if $s = (a, a + d, \\dots, a + (l - 1)d)$ or $s = (a + (l - 1)d, \\dots, a + d, a)$, where $n d$ is the sum of $n$ copies of $d$; when $l = 0$ both lists are empty.","title":"List arithmetic progressions with given start and difference"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/is-apof-length-2","kind":"node","lean":"List.IsAPOfLength","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#is-apof-length-2","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid and $l \\in \\mathbb{N}$. A list $s$ of elements of $M$ is an arithmetic progression of length $l$ if there are $a, d \\in M$ such that $s = (a, a + d, \\dots, a + (l - 1)d)$ or $s = (a + (l - 1)d, \\dots, a + d, a)$, where $n d$ is the sum of $n$ copies of $d$.","title":"Arithmetic progressions of lists"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/has-monotone-ap","kind":"node","lean":"HasMonotoneAP","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#has-monotone-ap","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, $\\beta$ a preordered type, $f : \\beta \\to M$ and $k \\in \\mathbb{N}$. Then $f$ contains a monotone arithmetic progression of length $k$ if there are $x_1 < x_2 < \\dots < x_k$ in $\\beta$ such that the list $(f(x_1), \\dots, f(x_k))$ is an arithmetic progression of length $k$: for some $a, d \\in M$ it equals $(a, a + d, \\dots, a + (k - 1)d)$ or its reverse.","title":"Monotone arithmetic progressions in a function"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/is-apof-length-free","kind":"node","lean":"Set.IsAPOfLengthFree","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#is-apof-length-free","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid and $l \\in \\mathbb{N} \\cup \\{\\infty\\}$. A set $s \\subseteq M$ is free of nontrivial arithmetic progressions of length $l$ if every subset of $s$ that is an arithmetic progression of length $l$ forces $l \\le 1$; equivalently, $l \\le 1$ or no subset of $s$ is an arithmetic progression of length $l$.","title":"Progression-free sets"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/not-is-apof-length-free","kind":"node","lean":"Set.IsAPOfLength.not_isAPOfLengthFree","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#not-is-apof-length-free","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $s$ be a set in an additive commutative monoid and $l \\in \\mathbb{N} \\cup \\{\\infty\\}$ with $l > 1$. If $s$ is an arithmetic progression of length $l$, then $s$ is not free of nontrivial arithmetic progressions of length $l$.","title":"A progression of length greater than one is not progression-free"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/is-apof-length-free-2","kind":"node","lean":"Finset.IsAPOfLengthFree","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#is-apof-length-free-2","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid and $l \\in \\mathbb{N} \\cup \\{\\infty\\}$. A finite set $s \\subseteq M$ is free of nontrivial arithmetic progressions of length $l$ if it is so as a set: $l \\le 1$, or no subset of $s$ is an arithmetic progression of length $l$.","title":"Progression-free finite sets"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/is-apof-length-free-three","kind":"node","lean":"ThreeAPFree.isAPOfLengthFree_three","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#is-apof-length-free-three","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $s$ be a set of natural numbers such that $a = b$ whenever $a, b, c \\in s$ satisfy $a + c = 2b$. Then $s$ is free of nontrivial arithmetic progressions of length $3$: no subset of $s$ is an arithmetic progression of length $3$.","title":"Sets of naturals without three-term progressions"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/max-card","kind":"node","lean":"Set.IsAPOfLengthFree.maxCard","parent":23386,"reading_title":{"format":"inline-parts-v1","parts":[["text","Largest size of a progression-free subset of "],["math","\\(\\{1, \\dots, N\\}\\)"]],"source":"Largest size of a progression-free subset of $\\{1, \\dots, N\\}$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#max-card","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"For natural numbers $k$ and $N$, $r_k(N)$ is the largest cardinality of a subset $S \\subseteq \\{1, \\dots, N\\}$ that is free of nontrivial arithmetic progressions of length $k$, meaning that if $k \\ge 2$ then no subset of $S$ is an arithmetic progression of length $k$ (for $k \\le 1$ every subset qualifies).","title":"Largest size of a progression-free subset of $\\{1, \\dots, N\\}$"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/max-card-le","kind":"node","lean":"Set.IsAPOfLengthFree.maxCard_le","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#max-card-le","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"For all natural numbers $k$ and $N$, the largest size $r_k(N)$ of a subset of $\\{1, \\dots, N\\}$ free of nontrivial arithmetic progressions of length $k$ satisfies $r_k(N) \\le N$.","title":"Trivial upper bound for the largest progression-free subset"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/max-card-spec","kind":"node","lean":"Set.IsAPOfLengthFree.maxCard_spec","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#max-card-spec","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"For all natural numbers $k$ and $N$, let $r_k(N)$ be the largest size of a subset of $\\{1, \\dots, N\\}$ free of nontrivial arithmetic progressions of length $k$. Then some subset $S \\subseteq \\{1, \\dots, N\\}$ free of nontrivial arithmetic progressions of length $k$ has exactly $r_k(N)$ elements.","title":"A largest progression-free subset is attained"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/card-le-max-card","kind":"node","lean":"Set.IsAPOfLengthFree.card_le_maxCard","parent":23386,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set/index.html#card-le-max-card","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/arithmetic-progressions-and-progression-free-set","status":"fully_proved","summary":"Let $k$ and $N$ be natural numbers, and let $S$ be a finite set of natural numbers with $S \\subseteq \\{1, \\dots, N\\}$ that is free of nontrivial arithmetic progressions of length $k$. Then $|S| \\le r_k(N)$, where $r_k(N)$ is the largest size of a subset of $\\{1, \\dots, N\\}$ containing no nontrivial $k$-term arithmetic progression.","title":"Progression-free subsets are bounded by the extremal size"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers","status":"can_prove","title":"Bounds for Roth numbers"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers/roth-number-log-power-decay","kind":"node","lean":"MathlibExt.Combinatorics.RothNumberLogPowerErdosProblem140Wanted.rothNumberLogPowerDecay","parent":23402,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers/index.html#roth-number-log-power-decay","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers","status":"fully_proved","summary":"The statement that for every real $C > 0$ there is a real constant $K$ such that $r_3(N) \\le K N / (\\log N)^C$ for every natural number $N \\ge 2$, where $r_3(N)$ is the largest size of a subset of $\\{0, 1, \\dots, N - 1\\}$ containing no non-trivial three-term arithmetic progression.","title":"Roth numbers below every power of the logarithm (Erd\u0151s Problem 140)"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers/roth-number-log-power-decay-holds","kind":"node","lean":"MathlibExt.Combinatorics.RothNumberLogPowerErdosProblem140Wanted.rothNumberLogPowerDecay_holds","parent":23402,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers/index.html#roth-number-log-power-decay-holds","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/bounds-for-roth-numbers","status":"can_prove","summary":"For every real $C > 0$ there is a real constant $K$ such that $r_3(N) \\le K N / (\\log N)^C$ for every natural number $N \\ge 2$, where $r_3(N)$ is the largest size of a subset of $\\{0, 1, \\dots, N - 1\\}$ containing no non-trivial three-term arithmetic progression.","title":"Kelley\u2013Meka: Roth numbers are below every power of the logarithm"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem","status":"fully_proved","title":"Furstenberg\u2013S\u00e1rk\u00f6zy theorem"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem/initial-segment-count","kind":"node","lean":"MathlibExt.Combinatorics.Additive.SarkozyWanted.initialSegmentCount","parent":23405,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem/index.html#initial-segment-count","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem","status":"fully_proved","summary":"For a set $A$ of natural numbers and a natural number $N$, the initial segment count of $A$ at $N$ is the number of elements of $A$ that are less than $N$, that is, $|A \\cap \\{0, 1, \\dots, N - 1\\}|$.","title":"Initial segment count"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem/furstenberg-sarkozy","kind":"node","lean":"MathlibExt.Combinatorics.Additive.SarkozyWanted.furstenberg_sarkozy","parent":23405,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem/index.html#furstenberg-sarkozy","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/furstenberg-s-rk-zy-theorem","status":"fully_proved","summary":"Let $A$ be a set of natural numbers, and suppose there is a real number $\\delta > 0$ such that for every natural number $M$ there is a natural number $N \\ge M$ with $\\delta N \\le |A \\cap \\{0, 1, \\dots, N - 1\\}|$. Then there are $a, b \\in A$ and a natural number $k > 0$ with $a < b$ and $b = a + k^2$.","title":"Furstenberg\u2013S\u00e1rk\u00f6zy theorem"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","status":"can_prove","title":"Number of arithmetic progressions in dense sets"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/has-ap","kind":"node","lean":"MathlibExt.Combinatorics.ManyThreeTermProgressionsErdosProblem179Wanted.HasAP","parent":23408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Containing a "],["math","\\(k\\)"],["text","-term arithmetic progression"]],"source":"Containing a $k$-term arithmetic progression"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/index.html#has-ap","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","status":"fully_proved","summary":"A finite set $A \\subseteq \\mathbb{N}$ contains a $k$-term arithmetic progression when there are natural numbers $a$ and $d \\ge 1$ with $a + i d \\in A$ for every natural number $i < k$, that is, $a, a + d, \\dots, a + (k - 1) d$ all lie in $A$. For $k = 0$ this holds vacuously.","title":"Containing a $k$-term arithmetic progression"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/ap-count","kind":"node","lean":"MathlibExt.Combinatorics.ManyThreeTermProgressionsErdosProblem179Wanted.apCount","parent":23408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-term arithmetic progressions in a set"]],"source":"Number of $k$-term arithmetic progressions in a set"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/index.html#ap-count","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","status":"fully_proved","summary":"For a finite set $A \\subseteq \\mathbb{N}$ and a natural number $k$, $\\mathrm{AP}_k(A)$ is the number of pairs $(a, d)$ of natural numbers with $a \\in A$ and $1 \\le d \\le \\max A$ such that $a + i d \\in A$ for every natural number $i < k$. For $k \\ge 2$ the conditions $a \\in A$ and $d \\le \\max A$ follow from the others, so $\\mathrm{AP}_k(A)$ counts the $k$-term progressions $a, a + d, \\dots, a + (k - 1) d$ in $A$\u2026","title":"Number of $k$-term arithmetic progressions in a set"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/ap-threshold","kind":"node","lean":"MathlibExt.Combinatorics.ManyThreeTermProgressionsErdosProblem179Wanted.apThreshold","parent":23408,"reading_title":{"format":"inline-parts-v1","parts":[["text","The progression threshold "],["math","\\(F_k(N, \\ell)\\)"]],"source":"The progression threshold $F_k(N, \\ell)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/index.html#ap-threshold","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","status":"fully_proved","summary":"For natural numbers $k$, $\\ell$ and $N$, $F_k(N, \\ell)$ is the least natural number $m$ such that every $N$-element set $A \\subseteq \\mathbb{N}$ with $\\mathrm{AP}_k(A) \\ge m$ contains an $\\ell$-term arithmetic progression, where $\\mathrm{AP}_k(A)$ is the number of pairs $(a, d)$ with $a \\in A$, $1 \\le d \\le \\max A$ and $a + i d \\in A$ for all $i < k$ (for $k \\ge 2$, the number of $k$-term progressions in $A$ with\u2026","title":"The progression threshold $F_k(N, \\ell)$"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/three-apthreshold-for-four-aplittle-o","kind":"node","lean":"MathlibExt.Combinatorics.ManyThreeTermProgressionsErdosProblem179Wanted.threeAPThresholdForFourAPLittleO","parent":23408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s Problem 179: is "],["math","\\(F_3(N, 4) = o(N^2)\\)"],["text","?"]],"source":"Erd\u0151s Problem 179: is $F_3(N, 4) = o(N^2)$?"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/index.html#three-apthreshold-for-four-aplittle-o","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","status":"fully_proved","summary":"Erd\u0151s Problem 179: is $F_3(N, 4) = o(N^2)$? The statement that $F_3(N, 4) = o(N^2)$ as $N \\to \\infty$ over the natural numbers, that is, $F_3(N, 4) / N^2 \\to 0$, where $F_3(N, 4)$ is the least $m$ such that every $N$-element set $A \\subseteq \\mathbb{N}$ containing at least $m$ three-term arithmetic progressions $a, a + d, a + 2d$ with $d \\ge 1$ contains a $4$-term arithmetic progression.","title":"Erd\u0151s Problem 179: is $F_3(N, 4) = o(N^2)$?"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/three-apthreshold-exponent-two","kind":"node","lean":"MathlibExt.Combinatorics.ManyThreeTermProgressionsErdosProblem179Wanted.threeAPThresholdExponentTwo","parent":23408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s Problem 179: does "],["math","\\(\\log F_3(N, \\ell) / \\log N\\)"],["text"," tend to "],["math","\\(2\\)"],["text","?"]],"source":"Erd\u0151s Problem 179: does $\\log F_3(N, \\ell) / \\log N$ tend to $2$?"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/index.html#three-apthreshold-exponent-two","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","status":"fully_proved","summary":"Erd\u0151s Problem 179: does $\\log F_3(N, \\ell) / \\log N$ tend to $2$? The statement that for every natural number $\\ell > 3$, $\\log F_3(N, \\ell) / \\log N \\to 2$ as $N \\to \\infty$ over the natural numbers, where $F_3(N, \\ell)$ is the least $m$ such that every $N$-element set $A \\subseteq \\mathbb{N}$ containing at least $m$ three-term arithmetic progressions $a, a + d, a + 2d$ with $d \\ge 1$ contains an $\\ell$-term\u2026","title":"Erd\u0151s Problem 179: does $\\log F_3(N, \\ell) / \\log N$ tend to $2$?"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/three-apthreshold-for-four-aplittle-o-ho","kind":"node","lean":"MathlibExt.Combinatorics.ManyThreeTermProgressionsErdosProblem179Wanted.threeAPThresholdForFourAPLittleO_holds","parent":23408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s Problem 179: "],["math","\\(F_3(N, 4) = o(N^2)\\)"]],"source":"Erd\u0151s Problem 179: $F_3(N, 4) = o(N^2)$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/index.html#three-apthreshold-for-four-aplittle-o-ho","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","status":"can_prove","summary":"As $N \\to \\infty$ over the natural numbers, $F_3(N, 4) / N^2 \\to 0$, that is, $F_3(N, 4) = o(N^2)$, where $F_3(N, 4)$ is the least natural number $m$ such that every $N$-element set $A \\subseteq \\mathbb{N}$ containing at least $m$ three-term arithmetic progressions $a, a + d, a + 2d$ with $d \\ge 1$ contains a $4$-term arithmetic progression.","title":"Erd\u0151s Problem 179: $F_3(N, 4) = o(N^2)$"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/three-apthreshold-exponent-two-holds","kind":"node","lean":"MathlibExt.Combinatorics.ManyThreeTermProgressionsErdosProblem179Wanted.threeAPThresholdExponentTwo_holds","parent":23408,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s Problem 179: "],["math","\\(\\log F_3(N, \\ell) / \\log N\\)"],["text"," tends to "],["math","\\(2\\)"]],"source":"Erd\u0151s Problem 179: $\\log F_3(N, \\ell) / \\log N$ tends to $2$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets/index.html#three-apthreshold-exponent-two-holds","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/number-of-arithmetic-progressions-in-dense-sets","status":"can_prove","summary":"For every natural number $\\ell > 3$, $\\log F_3(N, \\ell) / \\log N \\to 2$ as $N \\to \\infty$ over the natural numbers, where $F_3(N, \\ell)$ is the least natural number $m$ such that every $N$-element set $A \\subseteq \\mathbb{N}$ containing at least $m$ three-term arithmetic progressions $a, a + d, a + 2d$ with $d \\ge 1$ contains an $\\ell$-term arithmetic progression.","title":"Erd\u0151s Problem 179: $\\log F_3(N, \\ell) / \\log N$ tends to $2$"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces","status":"can_prove","title":"Polynomial triangle removal in vector spaces"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/is-triangle-free","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRemovalVectorSpaceAMR127Problem0013Wanted.IsTriangleFree","parent":23416,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triangle-free subsets of "],["math","\\(\\mathbb{F}_2^n\\)"]],"source":"Triangle-free subsets of $\\mathbb{F}_2^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/index.html#is-triangle-free","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces","status":"fully_proved","summary":"A set $A \\subseteq \\mathbb{F}_2^n$ is triangle-free if there are no $x, y \\in A$ with $x + y \\in A$ (the case $x = y$ is allowed).","title":"Triangle-free subsets of $\\mathbb{F}_2^n$"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/triangle-count","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRemovalVectorSpaceAMR127Problem0013Wanted.triangleCount","parent":23416,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of triangles in a subset of "],["math","\\(\\mathbb{F}_2^n\\)"]],"source":"Number of triangles in a subset of $\\mathbb{F}_2^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/index.html#triangle-count","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces","status":"fully_proved","summary":"For $A \\subseteq \\mathbb{F}_2^n$, the triangle count of $A$ is the number of ordered pairs $(x, y) \\in A \\times A$ with $x + y \\in A$. Divided by $4^n$, it is the probability that $x, y, x + y \\in A$ for independent uniform $x, y \\in \\mathbb{F}_2^n$.","title":"Number of triangles in a subset of $\\mathbb{F}_2^n$"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/polynomial-triangle-removal","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRemovalVectorSpaceAMR127Problem0013Wanted.polynomialTriangleRemoval","parent":23416,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial triangle removal in "],["math","\\(\\mathbb{F}_2^n\\)"]],"source":"Polynomial triangle removal in $\\mathbb{F}_2^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/index.html#polynomial-triangle-removal","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces","status":"fully_proved","summary":"This is the statement that there are real constants $c > 0$ and $C > 0$ such that for every $n$, every $A \\subseteq \\mathbb{F}_2^n$ and every real $\\varepsilon$ with $0 < \\varepsilon \\le 1$: if every triangle-free $B \\subseteq A$ misses at least $\\varepsilon 2^n$ elements of $A$, then the triangle count of $A$ is at least $c \\varepsilon^C 4^n$, that is, $\\Pr_{x,y}[x, y, x + y \\in A] \\ge c \\varepsilon^C$ for\u2026","title":"Polynomial triangle removal in $\\mathbb{F}_2^n$"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/polynomial-triangle-removal-holds","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRemovalVectorSpaceAMR127Problem0013Wanted.polynomialTriangleRemoval_holds","parent":23416,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial triangle removal holds in "],["math","\\(\\mathbb{F}_2^n\\)"]],"source":"Polynomial triangle removal holds in $\\mathbb{F}_2^n$"},"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces/index.html#polynomial-triangle-removal-holds","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/polynomial-triangle-removal-in-vector-spaces","status":"can_prove","summary":"There are real constants $c > 0$ and $C > 0$ such that for every natural number $n$, every finite set $A \\subseteq \\mathbb{F}_2^n$ and every real number $\\varepsilon$ with $0 < \\varepsilon \\le 1$ the following holds: if every triangle-free $B \\subseteq A$ misses at least $\\varepsilon 2^n$ elements of $A$, then the triangle count of $A$ is at least $c \\varepsilon^C 4^n$. Here a set $B \\subseteq \\mathbb{F}_2^n$ is\u2026","title":"Polynomial triangle removal holds in $\\mathbb{F}_2^n$"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro","status":"can_prove","title":"Progression-free sets meeting every infinite progression"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro/three-apfree-complement-contains-progres","kind":"node","lean":"MathlibExt.Combinatorics.ThreeAPFreeComplementProgressionErdosProblem199Wanted.threeAPFreeComplementContainsProgression","parent":23421,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro/index.html#three-apfree-complement-contains-progres","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro","status":"fully_proved","summary":"The statement that for every set $A \\subseteq \\mathbb{R}$ containing no three-term arithmetic progression (no $x, y, z \\in A$ with $x + z = 2y$ and $x \\ne y$) there are reals $a$ and $d \\ne 0$ with $a + nd \\notin A$ for every natural number $n$, that is, the complement $\\mathbb{R} \\setminus A$ contains the infinite arithmetic progression $a, a + d, a + 2d, \\dots$.","title":"Erd\u0151s Problem 199"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro/three-apfree-complement-contains-progres-2","kind":"node","lean":"MathlibExt.Combinatorics.ThreeAPFreeComplementProgressionErdosProblem199Wanted.threeAPFreeComplementContainsProgression_refuted","parent":23421,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro/index.html#three-apfree-complement-contains-progres-2","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/progression-free-sets-meeting-every-infinite-pro","status":"can_prove","summary":"There is a set $A \\subseteq \\mathbb{R}$ containing no three-term arithmetic progression (no $x, y, z \\in A$ with $x + z = 2y$ and $x \\ne y$) such that for all reals $a$ and $d \\ne 0$ some term $a + nd$, with $n$ a natural number, lies in $A$: the complement $\\mathbb{R} \\setminus A$ contains no infinite arithmetic progression $a, a + d, a + 2d, \\dots$ with $d \\ne 0$.","title":"Erd\u0151s Problem 199 is false"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference","status":"can_prove","title":"Progressions with differences from a difference set"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference/progression-with-difference-in-differenc","kind":"node","lean":"MathlibExt.Combinatorics.ProgressionDifferenceSetErdosProblem1185Wanted.progressionWithDifferenceInDifferenceSet","parent":23424,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference/index.html#progression-with-difference-in-differenc","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference","status":"fully_proved","summary":"This is the statement that for every real $\\delta > 0$ and every natural number $k \\ge 3$ there is a natural number $m \\ge 1$ such that for all sufficiently large natural numbers $N$, whenever $A$ and $B$ are subsets of $\\{1, \\dots, N\\}$ with $|A| \\ge \\delta N$ and $|B| \\ge m$, there are natural numbers $a$ and $d > 0$ such that $b_1 = b_2 + d$ for some $b_1, b_2 \\in B$ and $a + i d \\in A$ for every natural number\u2026","title":"Erd\u0151s Problem 1185: progressions with difference in a difference set"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference/progression-with-difference-in-differenc-2","kind":"node","lean":"MathlibExt.Combinatorics.ProgressionDifferenceSetErdosProblem1185Wanted.progressionWithDifferenceInDifferenceSet_refuted","parent":23424,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference/index.html#progression-with-difference-in-differenc-2","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/progressions-with-differences-from-a-difference","status":"can_prove","summary":"The following statement is false: for every real $\\delta > 0$ and every natural number $k \\ge 3$ there is a natural number $m \\ge 1$ such that for all sufficiently large natural numbers $N$, whenever $A$ and $B$ are subsets of $\\{1, \\dots, N\\}$ with $|A| \\ge \\delta N$ and $|B| \\ge m$, there are natural numbers $a$ and $d > 0$ such that $b_1 = b_2 + d$ for some $b_1, b_2 \\in B$ and $a + i d \\in A$ for every natural\u2026","title":"Erd\u0151s Problem 1185 is false"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers","status":"can_prove","title":"Small values of cap set sizes and Roth numbers"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/cap-set-dim-three","kind":"node","lean":"MathlibExt.Combinatorics.Additive.ThreeAPFreeValuesWanted.capSet_dim_three","parent":23427,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/index.html#cap-set-dim-three","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers","status":"can_prove","summary":"Let $\\mathbb{F}_3$ be the field with three elements. The largest subset of $\\mathbb{F}_3^3$ containing no three-term arithmetic progression, that is, no elements $a, b, c$ of the subset with $a + c = 2b$ and $a \\neq b$, has exactly $9$ elements.","title":"Largest cap set in dimension three"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/cap-set-dim-four","kind":"node","lean":"MathlibExt.Combinatorics.Additive.ThreeAPFreeValuesWanted.capSet_dim_four","parent":23427,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/index.html#cap-set-dim-four","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers","status":"can_prove","summary":"Let $\\mathbb{F}_3$ be the field with three elements. The largest subset of $\\mathbb{F}_3^4$ containing no three-term arithmetic progression, that is, no elements $a, b, c$ of the subset with $a + c = 2b$ and $a \\neq b$, has exactly $20$ elements.","title":"Largest cap set in dimension four"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/cap-set-dim-five","kind":"node","lean":"MathlibExt.Combinatorics.Additive.ThreeAPFreeValuesWanted.capSet_dim_five","parent":23427,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/index.html#cap-set-dim-five","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers","status":"can_prove","summary":"Let $\\mathbb{F}_3$ be the field with three elements. The largest subset of $\\mathbb{F}_3^5$ containing no three-term arithmetic progression, that is, no elements $a, b, c$ of the subset with $a + c = 2b$ and $a \\neq b$, has exactly $45$ elements.","title":"Largest cap set in dimension five"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/cap-set-dim-six","kind":"node","lean":"MathlibExt.Combinatorics.Additive.ThreeAPFreeValuesWanted.capSet_dim_six","parent":23427,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/index.html#cap-set-dim-six","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers","status":"can_prove","summary":"Let $\\mathbb{F}_3$ be the field with three elements. The largest subset of $\\mathbb{F}_3^6$ containing no three-term arithmetic progression, that is, no elements $a, b, c$ of the subset with $a + c = 2b$ and $a \\neq b$, has exactly $112$ elements.","title":"Largest cap set in dimension six"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/roth-number-nat-five-to-fourteen","kind":"node","lean":"MathlibExt.Combinatorics.Additive.ThreeAPFreeValuesWanted.rothNumberNat_five_to_fourteen","parent":23427,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers/index.html#roth-number-nat-five-to-fourteen","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/small-values-of-cap-set-sizes-and-roth-numbers","status":"can_prove","summary":"For a natural number $n$, let $r_3(n)$ be the largest size of a subset of $\\{0, 1, \\dots, n - 1\\}$ containing no three-term arithmetic progression, that is, no elements $a, b, c$ of the subset with $a + c = 2b$ and $a \\neq b$. Then the values $r_3(5), r_3(6), \\dots, r_3(14)$ are $4, 4, 4, 4, 5, 5, 6, 6, 7, 8$, in this order.","title":"Roth numbers from five to fourteen"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid","status":"can_prove","title":"Squares in dense subsets of the grid"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid/dense-grid-contains-square","kind":"node","lean":"MathlibExt.Combinatorics.DenseGridSquareErdosProblem658Wanted.denseGridContainsSquare","parent":23433,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid/index.html#dense-grid-contains-square","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid","status":"fully_proved","summary":"The statement that for every real number $\\delta > 0$ there is a natural number $N_0$ such that for every $N \\ge N_0$, every set $A \\subseteq \\{1, \\dots, N\\}^2$ of pairs of natural numbers with $|A| \\ge \\delta N^2$ contains the four vertices $(a, b)$, $(a + d, b)$, $(a, b + d)$ and $(a + d, b + d)$ of an axis-parallel square, for some natural numbers $a$, $b$ and $d \\ge 1$.","title":"Erd\u0151s Problem 658 (axis-parallel squares in dense subsets of the grid)"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid/dense-grid-contains-square-holds","kind":"node","lean":"MathlibExt.Combinatorics.DenseGridSquareErdosProblem658Wanted.denseGridContainsSquare_holds","parent":23433,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid/index.html#dense-grid-contains-square-holds","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/squares-in-dense-subsets-of-the-grid","status":"can_prove","summary":"For every real number $\\delta > 0$ there is a natural number $N_0$ such that for every $N \\ge N_0$, every set $A \\subseteq \\{1, \\dots, N\\}^2$ of pairs of natural numbers with $|A| \\ge \\delta N^2$ contains the four vertices $(a, b)$, $(a + d, b)$, $(a, b + d)$ and $(a + d, b + d)$ of an axis-parallel square, for some natural numbers $a$, $b$ and $d \\ge 1$.","title":"Erd\u0151s Problem 658 is true"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks","status":"can_prove","title":"Three-term progressions in unit-step walks"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/is-unit-step-walk","kind":"node","lean":"MathlibExt.Combinatorics.UnitStepWalkThreeTermProgressionErdosProblem192Wanted.IsUnitStepWalk","parent":23436,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/index.html#is-unit-step-walk","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks","status":"fully_proved","summary":"Let $d$ be a natural number. A sequence $a_0, a_1, a_2, \\dots$ of points of $\\mathbb{R}^d$ is a unit-step walk when for every $i$ the step $a_{i+1} - a_i$ is a standard unit vector $e_j$ of $\\mathbb{R}^d$ for some $j$.","title":"Unit-step walks"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/has-three-ap","kind":"node","lean":"MathlibExt.Combinatorics.UnitStepWalkThreeTermProgressionErdosProblem192Wanted.HasThreeAP","parent":23436,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/index.html#has-three-ap","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks","status":"fully_proved","summary":"Let $d$ be a natural number and let $a_0, a_1, a_2, \\dots$ be a sequence of points of $\\mathbb{R}^d$. The sequence contains a three-term arithmetic progression when there are indices $i$, $j$, $k$ with $a_i \\ne a_j$ and $a_i + a_k = 2 a_j$.","title":"Three-term arithmetic progressions in a sequence of points"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/unit-step-walk-three-apiff","kind":"node","lean":"MathlibExt.Combinatorics.UnitStepWalkThreeTermProgressionErdosProblem192Wanted.unitStepWalkThreeAPIff","parent":23436,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/index.html#unit-step-walk-three-apiff","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks","status":"fully_proved","summary":"The statement that for every natural number $d$, every sequence $a_0, a_1, a_2, \\dots$ of points of $\\mathbb{R}^d$ whose steps $a_{i+1} - a_i$ are all standard unit vectors contains a three-term arithmetic progression (indices $i$, $j$, $k$ with $a_i \\ne a_j$ and $a_i + a_k = 2 a_j$) if and only if $d \\le 3$.","title":"Erd\u0151s Problem 192 on three-term progressions in unit-step walks"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/unit-step-walk-three-apiff-holds","kind":"node","lean":"MathlibExt.Combinatorics.UnitStepWalkThreeTermProgressionErdosProblem192Wanted.unitStepWalkThreeAPIff_holds","parent":23436,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks/index.html#unit-step-walk-three-apiff-holds","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/three-term-progressions-in-unit-step-walks","status":"can_prove","summary":"For every natural number $d$, every sequence $a_0, a_1, a_2, \\dots$ of points of $\\mathbb{R}^d$ whose steps $a_{i+1} - a_i$ are all standard unit vectors contains a three-term arithmetic progression (indices $i$, $j$, $k$ with $a_i \\ne a_j$ and $a_i + a_k = 2 a_j$) if and only if $d \\le 3$.","title":"Solution of Erd\u0151s Problem 192 on three-term progressions in unit-step walks"},{"data_url":"scopes/Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets.json","id":"Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets","kind":"scope","navigation":"module","parent":23384,"result":false,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets/index.html","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets","status":"can_prove","title":"Unions of progression-free sets"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets/has-few3-aps","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem847Wanted.HasFew3APs","parent":23441,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets/index.html#has-few3-aps","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets","status":"fully_proved","summary":"A set $A$ of natural numbers has this property if there is a real number $\\varepsilon > 0$ such that every finite subset $B \\subseteq A$ contains a subset $C \\subseteq B$ with $|C| \\ge \\varepsilon |B|$ that contains no nontrivial three-term arithmetic progression, that is, no $a, b, c \\in C$ with $a + c = 2b$ and $a \\ne b$.","title":"Sets with uniformly large 3AP-free subsets"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets/conjecture","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem847Wanted.conjecture","parent":23441,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets/index.html#conjecture","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets","status":"fully_proved","summary":"Call a set of natural numbers 3AP-free if it contains no $a, b, c$ with $a + c = 2b$ and $a \\ne b$. The proposition asserts that every infinite set $A$ of natural numbers such that, for some real number $\\varepsilon > 0$, every finite subset $B \\subseteq A$ has a 3AP-free subset of size at least $\\varepsilon |B|$, is a finite union of 3AP-free sets: there are a natural number $n$ and 3AP-free sets\u2026","title":"Erd\u0151s Problem 847"},{"id":"Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets/conjecture-refuted","kind":"node","lean":"MathlibExt.NumberTheory.ErdosProblem847Wanted.conjecture_refuted","parent":23441,"result":true,"route":"../roadmap/Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets/index.html#conjecture-refuted","scope":"Combinatorics/additive-combinatorics/progressions-dense-sets/unions-of-progression-free-sets","status":"can_prove","summary":"Call a set of natural numbers 3AP-free if it contains no $a, b, c$ with $a + c = 2b$ and $a \\ne b$. There is an infinite set $A$ of natural numbers such that, for some real number $\\varepsilon > 0$, every finite subset $B \\subseteq A$ has a 3AP-free subset of size at least $\\varepsilon |B|$, but $A$ is not the union of finitely many 3AP-free sets of natural numbers.","title":"Erd\u0151s Problem 847 is false"},{"data_url":"scopes/Combinatorics/ramsey-theory.json","id":"Combinatorics/ramsey-theory","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/index.html","scope":"Combinatorics/ramsey-theory","status":"can_prove","summary":"Unavoidable monochromatic structure under finite colourings of integers, words, sets and semigroups; graph and ordered-graph Ramsey problems go to GraphTheory, ordinal and infinite-cardinal partition relations to SetTheory, density statements without colouring to additive combinatorics.","title":"Ramsey theory"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems","kind":"scope","parent":23445,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems","status":"can_prove","summary":"Monochromatic arithmetic progressions: van der Waerden's theorem and van der Waerden numbers (off-diagonal, W(3,k)), short or large-difference progressions, rainbow progressions, descending waves, accessible sets and diffsequences, and progression-free Ramsey sets.","title":"Van der Waerden-type theorems"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden","kind":"node","lean":"MetaMathlibExt.vanDerWaerden","parent":23446,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/index.html#van-der-waerden","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems","status":"fully_proved","summary":"For all positive integers $r$ and $k$ there is a natural number $N$ such that for every coloring of $\\{1, \\dots, N\\}$ with $r$ colors there are natural numbers $a \\ge 1$ and $d \\ge 1$ with $a + (k - 1)d \\le N$ such that each of the $k$ terms $a, a + d, \\dots, a + (k - 1)d$ has the same color as $a$.","title":"Van der Waerden's theorem"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","kind":"scope","navigation":"module","parent":23446,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","title":"Accessible sets and diffsequences"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-monochromatic","kind":"node","lean":"MetaMathlibExt.IsMonochromatic","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-monochromatic","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $r$ and $k$ be natural numbers and let $c$ color the positive integers with the colors $0, 1, \\dots, r - 1$. A sequence $x_0, \\dots, x_{k-1}$ of positive integers is monochromatic for $c$ when $c(x_i) = c(x_j)$ for all $i, j < k$.","title":"Monochromatic sequences of positive integers"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-dap","kind":"node","lean":"MetaMathlibExt.IsDAP","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arithmetic progressions with gap in "],["math","\\(D\\)"]],"source":"Arithmetic progressions with gap in $D$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-dap","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $D$ be a set of natural numbers and $k$ a natural number. A sequence $x_0, \\dots, x_{k-1}$ of positive integers is a $k$-term arithmetic progression with gap in $D$ if there are a positive integer $a$ and an element $d \\in D$ such that $x_i = a + i d$ for every $i < k$.","title":"Arithmetic progressions with gap in $D$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-dap-iff","kind":"node","lean":"MetaMathlibExt.isDAP_iff","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of arithmetic progressions with gap in "],["math","\\(D\\)"]],"source":"Characterization of arithmetic progressions with gap in $D$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-dap-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $D$ be a set of natural numbers, $k$ a natural number and $x_0, \\dots, x_{k-1}$ positive integers. Then $x_0, \\dots, x_{k-1}$ is a $k$-term arithmetic progression with gap in $D$ if and only if there are a positive integer $a$ and an element $d \\in D$ such that $x_i = a + i d$ for every $i < k$.","title":"Characterization of arithmetic progressions with gap in $D$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-rlarge","kind":"node","lean":"MetaMathlibExt.IsRLarge","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(r\\)"],["text","-large sets"]],"source":"$r$-large sets"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-rlarge","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $r$ be a natural number. A set $D$ of natural numbers is $r$-large when every element of $D$ is positive, $r \\ge 1$, and for every coloring $c$ of the positive integers with the colors $0, 1, \\dots, r - 1$ and every natural number $k \\ge 1$ there are positive integers $x_0, \\dots, x_{k-1}$ that are monochromatic for $c$ and form a $D$-AP: there are a positive integer $a$ and some $d \\in D$ with $x_i = a + i d$\u2026","title":"$r$-large sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-rlarge-iff","kind":"node","lean":"MetaMathlibExt.isRLarge_iff","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(r\\)"],["text","-large sets"]],"source":"Characterization of $r$-large sets"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-rlarge-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $r$ be a natural number and $D$ a set of natural numbers. Then $D$ is $r$-large if and only if every element of $D$ is positive, $r \\ge 1$, and for every coloring $c$ of the positive integers with the colors $0, 1, \\dots, r - 1$ and every natural number $k \\ge 1$ there are positive integers $x_0, \\dots, x_{k-1}$ that are monochromatic for $c$ and form a $D$-AP: there are a positive integer $a$ and some\u2026","title":"Characterization of $r$-large sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-large","kind":"node","lean":"MetaMathlibExt.IsLarge","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-large","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is large when it is $r$-large for every natural number $r \\ge 1$.","title":"Large set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-large-iff","kind":"node","lean":"MetaMathlibExt.isLarge_iff","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-large-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is large if and only if it is $r$-large for every natural number $r \\ge 1$.","title":"Characterization of large sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-ddiff-sequence","kind":"node","lean":"MetaMathlibExt.IsDDiffSequence","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(D\\)"],["text","-diffsequences"]],"source":"$D$-diffsequences"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-ddiff-sequence","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $D$ be a set of natural numbers and $k$ a natural number. A finite sequence $x_0, \\dots, x_{k-1}$ of integers is a $D$-diffsequence when for every $i$ with $i + 1 < k$ there is $d \\in D$ with $x_{i+1} - x_i = d$.","title":"$D$-diffsequences"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-ddiff-sequence-iff","kind":"node","lean":"MetaMathlibExt.isDDiffSequence_iff","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(D\\)"],["text","-diffsequences"]],"source":"Characterization of $D$-diffsequences"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-ddiff-sequence-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $D$ be a set of natural numbers and $x_0, \\dots, x_{k-1}$ integers. Then $x_0, \\dots, x_{k-1}$ is a $D$-diffsequence if and only if for every $i$ with $i + 1 < k$ there is $d \\in D$ such that $x_{i+1} - x_i = d$.","title":"Characterization of $D$-diffsequences"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-raccessible","kind":"node","lean":"MetaMathlibExt.IsRAccessible","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(r\\)"],["text","-accessibility for colorings of the positive integers"]],"source":"$r$-accessibility for colorings of the positive integers"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-raccessible","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $r$ be a natural number. A set $D$ of natural numbers is $r$-accessible when every element of $D$ is positive, $r \\ge 1$, and for every coloring $c$ of the positive integers with the colors $0, 1, \\dots, r - 1$ and every natural number $k \\ge 1$ there are positive integers $x_0, \\dots, x_{k-1}$ that are monochromatic for $c$ and form a $D$-diffsequence: $x_{i+1} - x_i \\in D$, the difference taken in the\u2026","title":"$r$-accessibility for colorings of the positive integers"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-raccessible-iff","kind":"node","lean":"MetaMathlibExt.isRAccessible_iff","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(r\\)"],["text","-accessible sets"]],"source":"Characterization of $r$-accessible sets"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-raccessible-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $r$ be a natural number and $D$ a set of natural numbers. Then $D$ is $r$-accessible if and only if every element of $D$ is positive, $r \\ge 1$, and for every coloring $c$ of the positive integers with the colors $0, 1, \\dots, r - 1$ and every natural number $k \\ge 1$ there are positive integers $x_0, \\dots, x_{k-1}$ that are monochromatic for $c$ and form a $D$-diffsequence: $x_{i+1} - x_i \\in D$, the\u2026","title":"Characterization of $r$-accessible sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/mono","kind":"node","lean":"MetaMathlibExt.IsRAccessible.mono","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#mono","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $D$ be a set of natural numbers and let $r, s$ be natural numbers with $1 \\le r \\le s$. If $D$ is $s$-accessible, then $D$ is $r$-accessible.","title":"Accessibility with more colors implies accessibility with fewer"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-accessible","kind":"node","lean":"MetaMathlibExt.IsAccessible","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-accessible","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is accessible if it is $r$-accessible for every natural number $r \\ge 1$.","title":"Accessible sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-accessible-iff","kind":"node","lean":"MetaMathlibExt.isAccessible_iff","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Accessibility means "],["math","\\(r\\)"],["text","-accessibility for every "],["math","\\(r \\ge 1\\)"]],"source":"Accessibility means $r$-accessibility for every $r \\ge 1$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-accessible-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is accessible if and only if it is $r$-accessible for every natural number $r \\ge 1$.","title":"Accessibility means $r$-accessibility for every $r \\ge 1$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-lacunary","kind":"node","lean":"MetaMathlibExt.IsLacunary","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-lacunary","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A sequence $(m_n)_{n \\ge 0}$ of natural numbers is lacunary when $m_n > 0$ for every $n$ and there is a real number $\\lambda > 1$ such that $m_{n+1} \\ge \\lambda m_n$ for every $n \\ge 0$.","title":"Lacunary sequence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-lacunary-iff","kind":"node","lean":"MetaMathlibExt.isLacunary_iff","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-lacunary-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A sequence $(m_n)_{n \\ge 0}$ of natural numbers is lacunary if and only if $m_n > 0$ for every $n$ and there is a real number $q > 1$ such that $m_{n+1} \\ge q \\, m_n$ for every $n \\ge 0$.","title":"Characterization of lacunary sequences"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-lacunary-set","kind":"node","lean":"MetaMathlibExt.IsLacunarySet","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-lacunary-set","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is lacunary when $D = \\{m_n : n \\ge 0\\}$ for some lacunary sequence $(m_n)_{n \\ge 0}$ of natural numbers.","title":"Lacunary set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-lacunary-set-iff","kind":"node","lean":"MetaMathlibExt.isLacunarySet_iff","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-lacunary-set-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is lacunary if and only if there is a lacunary sequence $(m_n)_{n \\ge 0}$ of natural numbers with $D = \\{m_n : n \\ge 0\\}$.","title":"Characterization of lacunary sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-minimal-system","kind":"node","lean":"MetaMathlibExt.IsMinimalSystem","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-minimal-system","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $X$ be a compact metric space and $T$ a homeomorphism of $X$. The system $(X, T)$ is minimal when there is no closed set $K \\subseteq X$ that is nonempty, different from $X$ and invariant, meaning $T(K) = K$.","title":"Minimal system"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-minimal-system-iff","kind":"node","lean":"MetaMathlibExt.isMinimalSystem_iff","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-minimal-system-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $X$ be a compact metric space and $T \\colon X \\to X$ a homeomorphism. Then $T$ is minimal if and only if there is no closed set $K \\subseteq X$ with $K \\ne \\emptyset$, $K \\ne X$ and $T(K) = K$.","title":"Characterization of minimal systems"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/iterate-preimage","kind":"node","lean":"MetaMathlibExt.iteratePreimage","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#iterate-preimage","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"For a homeomorphism $T$ of a topological space $X$, an integer $s$ and a subset $U \\subseteq X$, the iterated preimage $T^{-s} U$ is defined as the preimage $(T^s)^{-1}(U)$ of $U$ under the $s$-th power $T^s$ of $T$ in the group of homeomorphisms of $X$.","title":"Iterated preimage"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/iterate-preimage-eq","kind":"node","lean":"MetaMathlibExt.iteratePreimage_eq","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#iterate-preimage-eq","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"For a homeomorphism $T$ of a topological space $X$, an integer $s$ and a subset $U \\subseteq X$, the iterated preimage $T^{-s} U$ is the preimage $(T^s)^{-1}(U)$ of $U$ under the $s$-th power $T^s$ of $T$ in the group of homeomorphisms of $X$.","title":"Iterated preimage as a preimage under a power"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-ltopological-recurrence","kind":"node","lean":"MetaMathlibExt.IsLTopologicalRecurrence","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sets of "],["math","\\(\\ell\\)"],["text","-topological recurrence"]],"source":"Sets of $\\ell$-topological recurrence"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-ltopological-recurrence","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $\\ell$ be a natural number. A set $D$ of natural numbers is a set of $\\ell$-topological recurrence when every element of $D$ is positive and, for every compact metric space $X$, every minimal homeomorphism $T \\colon X \\to X$ and every nonempty open set $U \\subseteq X$, there is $d \\in D$ such that $\\bigcap_{j=0}^{\\ell} T^{-jd}(U)$ is nonempty, where $T^{-jd}(U)$ is the preimage of $U$ under the $jd$-th iterate\u2026","title":"Sets of $\\ell$-topological recurrence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-ltopological-recurrence-iff","kind":"node","lean":"MetaMathlibExt.isLTopologicalRecurrence_iff","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(\\ell\\)"],["text","-topological recurrence"]],"source":"Characterization of $\\ell$-topological recurrence"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-ltopological-recurrence-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $\\ell$ be a natural number and $D$ a set of natural numbers. Then $D$ is a set of $\\ell$-topological recurrence if and only if every element of $D$ is positive and, for every compact metric space $X$, every minimal homeomorphism $T \\colon X \\to X$ and every nonempty open set $U \\subseteq X$, there is $d \\in D$ such that $\\bigcap_{j=0}^{\\ell} T^{-jd}(U)$ is nonempty, where $T^{-jd}(U)$ is the preimage of $U$\u2026","title":"Characterization of $\\ell$-topological recurrence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-topological-recurrence","kind":"node","lean":"MetaMathlibExt.IsTopologicalRecurrence","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-topological-recurrence","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is a set of topological recurrence when it is a set of $\\ell$-topological recurrence for $\\ell = 1$.","title":"Sets of topological recurrence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-topological-recurrence-iff","kind":"node","lean":"MetaMathlibExt.isTopologicalRecurrence_iff","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Topological recurrence is "],["math","\\(1\\)"],["text","-topological recurrence"]],"source":"Topological recurrence is $1$-topological recurrence"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-topological-recurrence-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is a set of topological recurrence if and only if it is a set of $\\ell$-topological recurrence for $\\ell = 1$.","title":"Topological recurrence is $1$-topological recurrence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-multiple-topological-recurrence","kind":"node","lean":"MetaMathlibExt.IsMultipleTopologicalRecurrence","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-multiple-topological-recurrence","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is a set of multiple topological recurrence when it is a set of $\\ell$-topological recurrence for every natural number $\\ell \\ge 1$.","title":"Sets of multiple topological recurrence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-multiple-topological-recurrence-iff","kind":"node","lean":"MetaMathlibExt.isMultipleTopologicalRecurrence_iff","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-multiple-topological-recurrence-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is a set of multiple topological recurrence if and only if $D$ is a set of $\\ell$-topological recurrence for every natural number $\\ell \\ge 1$.","title":"Characterization of multiple topological recurrence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-chromatically-intersective","kind":"node","lean":"MetaMathlibExt.IsChromaticallyIntersective","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-chromatically-intersective","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is chromatically intersective if every element of $D$ is positive and, for every natural number $r \\ge 1$ and every coloring of the positive integers with $r$ colors, there are positive integers $a$ and $b$ of the same color whose difference $a - b$, computed in the integers, lies in $D$.","title":"Chromatically intersective sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-chromatically-intersective-iff","kind":"node","lean":"MetaMathlibExt.isChromaticallyIntersective_iff","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-chromatically-intersective-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $D$ of natural numbers is chromatically intersective if and only if every element of $D$ is positive and, for every natural number $r \\ge 1$ and every coloring of the positive integers with $r$ colors, there are positive integers $a$ and $b$ of the same color whose difference $a - b$, computed in the integers, lies in $D$.","title":"Characterization of chromatically intersective sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/positive-fibonacci-set","kind":"node","lean":"MetaMathlibExt.positiveFibonacciSet","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#positive-fibonacci-set","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"The positive Fibonacci set is $\\{F_n : n \\ge 2\\}$, where the Fibonacci numbers are $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$. Equivalently, it is the set $\\{1, 2, 3, 5, 8, \\dots\\}$ of positive Fibonacci numbers, each counted once.","title":"Positive Fibonacci set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/mem-positive-fibonacci-set-iff","kind":"node","lean":"MetaMathlibExt.mem_positiveFibonacciSet_iff","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#mem-positive-fibonacci-set-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Write $F_n$ for the $n$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. A natural number $d$ belongs to the positive Fibonacci set if and only if $d = F_n$ for some natural number $n \\ge 2$.","title":"Membership in the positive Fibonacci set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/even-fibonacci-set","kind":"node","lean":"MetaMathlibExt.evenFibonacciSet","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#even-fibonacci-set","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $F_m$ denote the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. The even Fibonacci set is $\\{F_{3n+3} : n \\ge 0\\}$; its first terms are $2, 8, 34, \\dots$.","title":"Even Fibonacci set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/mem-even-fibonacci-set-iff","kind":"node","lean":"MetaMathlibExt.mem_evenFibonacciSet_iff","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#mem-even-fibonacci-set-iff","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Write $F_m$ for the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. A natural number $d$ belongs to the even Fibonacci set if and only if $d = F_{3n+3}$ for some natural number $n$.","title":"Membership in the even Fibonacci set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/accessibility-degree-growth-bound","kind":"node","lean":"MetaMathlibExt.accessibility_degree_growth_bound","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#accessibility-degree-growth-bound","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $r \\ge 2$, let $d_0 < d_1 < d_2 < \\cdots$ be a strictly increasing sequence of natural numbers with $d_0 > 0$, and let $D = \\{d_n : n \\ge 0\\}$. If there are a real number $\\delta > 0$ and an integer $N \\ge 1$ such that $d_{n+1} \\ge \\left(2 + \\frac{1}{r-1} + \\delta\\right) d_n$ for every $n \\ge N - 1$, then $D$ is not $r$-accessible.","title":"Growth bound for the degree of accessibility"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/not-two-accessible-of-growth","kind":"node","lean":"MetaMathlibExt.not_two_accessible_of_growth","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#not-two-accessible-of-growth","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $d_0 < d_1 < d_2 < \\cdots$ be a strictly increasing sequence of natural numbers with $d_0 > 0$, and let $D = \\{d_n : n \\in \\mathbb{N}\\}$. Suppose there are a real number $\\delta > 0$ and a natural number $N \\ge 1$ such that $d_{n+1} \\ge (3 + \\delta) d_n$ for every $n \\in \\mathbb{N}$ with $n + 1 \\ge N$. Then $D$ is not $2$-accessible.","title":"Fast eventual growth rules out 2-accessibility"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/two-large-contains-multiples","kind":"node","lean":"MetaMathlibExt.two_large_contains_multiples","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#two-large-contains-multiples","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Every $2$-large set $D$ of positive integers contains a multiple of each positive integer $m$.","title":"2-large sets contain multiples of every positive integer"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/r-large-sdiff-finite","kind":"node","lean":"MetaMathlibExt.rLarge_sdiff_finite","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite removal preserves "],["math","\\(r\\)"],["text","-largeness"]],"source":"Finite removal preserves $r$-largeness"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#r-large-sdiff-finite","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $r$ be a natural number. If a set $D \\subseteq \\mathbb{N}$ is $r$-large and $F \\subseteq \\mathbb{N}$ is finite, then $D \\setminus F$ is $r$-large.","title":"Finite removal preserves $r$-largeness"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/large-union-partition","kind":"node","lean":"MetaMathlibExt.large_union_partition","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#large-union-partition","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"If $D_1, D_2 \\subseteq \\mathbb{N}$ and their union $D = D_1 \\cup D_2$ is large, then $D_1$ is large or $D_2$ is large.","title":"Largeness is partition regular"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/not-two-large-of-triple-growth","kind":"node","lean":"MetaMathlibExt.not_two_large_of_triple_growth","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#not-two-large-of-triple-growth","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $d_0 < d_1 < d_2 < \\cdots$ be a strictly increasing sequence of natural numbers with $d_0 > 0$ and $d_{n+1} \\ge 3 d_n$ for every $n \\in \\mathbb{N}$, and let $D = \\{d_n : n \\in \\mathbb{N}\\}$. Then $D$ is not $2$-large.","title":"Sets growing by a factor of three are not 2-large"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/lacunary-union-avoids-integers","kind":"node","lean":"MetaMathlibExt.lacunary_union_avoids_integers","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#lacunary-union-avoids-integers","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $\\ell \\ge 1$ and let $S_0, \\dots, S_{\\ell-1} \\subseteq \\mathbb{N}$ be lacunary sets with union $S = \\bigcup_i S_i$. Then there are real numbers $\\alpha$ and $\\varepsilon > 0$ such that the infimum over $s \\in S$ of the distance $|\\alpha s - \\operatorname{round}(\\alpha s)|$ from $\\alpha s$ to its nearest integer $\\operatorname{round}(\\alpha s)$ equals $\\varepsilon$.","title":"Dilates of a finite union of lacunary sets stay away from the integers"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/two-coloring-avoids-long-dap","kind":"node","lean":"MetaMathlibExt.two_coloring_avoids_long_DAP","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-colorings avoiding long "],["math","\\(D\\)"],["text","-progressions"]],"source":"Two-colorings avoiding long $D$-progressions"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#two-coloring-avoids-long-dap","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $D$ be a set of positive integers, let $\\alpha$ be a real number and let $\\varepsilon > 0$ be such that $\\lVert \\alpha d \\rVert \\ge \\varepsilon$ for every $d \\in D$, where $\\lVert t \\rVert$ denotes the distance from $t$ to the nearest integer. Then there is a $2$-coloring of the positive integers with no monochromatic $D$-arithmetic progression of length $\\lceil 1/(2\\varepsilon) \\rceil + 1$.","title":"Two-colorings avoiding long $D$-progressions"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/finite-union-lacunary-not-two-large","kind":"node","lean":"MetaMathlibExt.finite_union_lacunary_not_two_large","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#finite-union-lacunary-not-two-large","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"For every $n \\ge 1$ and all lacunary sets $D_1, \\dots, D_n$, the union $D_1 \\cup \\dots \\cup D_n$ is not 2-large.","title":"Finite unions of lacunary sets are not 2-large"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/polynomial-range-large","kind":"node","lean":"MetaMathlibExt.polynomial_range_large","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#polynomial-range-large","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $p$ be a polynomial with integer coefficients, positive leading coefficient and $p(0) = 0$. Then the set of positive integers $d$ such that $p(n) = d$ for some positive integer $n$ is large.","title":"Positive values of a polynomial vanishing at zero form a large set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/accessible-of-colored-subset-diffsequenc","kind":"node","lean":"MetaMathlibExt.accessible_of_colored_subset_diffsequences","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#accessible-of-colored-subset-diffsequenc","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $r \\ge 2$ and $c$ be natural numbers, let $D$ be a set of positive integers, and write $D + c = \\{d + c : d \\in D\\}$. Suppose that for every coloring of $D$ with $r - 1$ colors and every $k \\ge 1$ there are $x_0, \\dots, x_{k-1} \\in D$, all of the same color, that form a $(D + c)$-diffsequence, that is, $x_{i+1} - x_i \\in D + c$ for every $i < k - 1$. Then $D + c$ is $r$-accessible.","title":"A coloring criterion for accessibility of a shifted set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/two-accessible-of-internal-diffsequences","kind":"node","lean":"MetaMathlibExt.two_accessible_of_internal_diffsequences","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#two-accessible-of-internal-diffsequences","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $D$ be a set of positive integers. If for every $k \\ge 1$ there are positive integers $x_1, \\dots, x_k$, all in $D$, that form a $D$-diffsequence, then $D$ is $2$-accessible.","title":"Sets containing their own long diffsequences are 2-accessible"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/accessible-integral-quotients","kind":"node","lean":"MetaMathlibExt.accessible_integral_quotients","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#accessible-integral-quotients","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $d \\ge 1$ and $r$ be natural numbers and let $A$ be a set of natural numbers. If $A$ is $rd$-accessible, then the set $\\{n \\ge 1 : nd \\in A\\}$, that is, the positive integers of the form $a/d$ with $a \\in A$, is $r$-accessible.","title":"Integral quotients of an accessible set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/accessible-divisible-subset","kind":"node","lean":"MetaMathlibExt.accessible_divisible_subset","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiples of "],["math","\\(d\\)"],["text"," in an accessible set"]],"source":"Multiples of $d$ in an accessible set"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#accessible-divisible-subset","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $d \\ge 1$ and $r$ be natural numbers and let $A$ be a set of natural numbers. If $A$ is $rd$-accessible, then the set $\\{a \\in A : d \\mid a\\}$ of members of $A$ divisible by $d$ is $r$-accessible.","title":"Multiples of $d$ in an accessible set"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/coloring-avoids-diffsequence-of-fraction","kind":"node","lean":"MetaMathlibExt.coloring_avoids_diffsequence_of_fractional_bounds","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#coloring-avoids-diffsequence-of-fraction","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $r \\ge 1$, let $d_0 < d_1 < \\cdots$ be a strictly increasing sequence of natural numbers with $d_0 > 0$, and let $D = \\{d_n : n \\ge 0\\}$. Suppose there are real numbers $\\alpha$ and $\\varepsilon > 0$ such that for every $n$ the fractional part of $\\alpha d_n$ lies in the closed interval $[\\varepsilon, (r-1)/r]$. Then there is a coloring of the positive integers with $r$ colors that has no monochromatic\u2026","title":"Colorings avoiding long diffsequences under fractional-part bounds"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fibonacci-two-coloring-avoids-six-term-a","kind":"node","lean":"MetaMathlibExt.fibonacci_two_coloring_avoids_six_term_AP","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fibonacci-two-coloring-avoids-six-term-a","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"There is a coloring of the positive integers with two colors such that no 6-term arithmetic progression $a, a + d, \\dots, a + 5d$ of positive integers whose gap $d$ is a positive Fibonacci number is monochromatic.","title":"A 2-coloring with no monochromatic 6-term progression with positive Fibonacci gap"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fibonacci-not-two-large","kind":"node","lean":"MetaMathlibExt.fibonacci_not_two_large","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fibonacci-not-two-large","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $F_n$ denote the $n$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. The positive Fibonacci set $\\{F_n : n \\ge 2\\} = \\{1, 2, 3, 5, 8, \\dots\\}$ is not $2$-large.","title":"The positive Fibonacci set is not 2-large"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/even-fibonacci-two-coloring-avoids-four","kind":"node","lean":"MetaMathlibExt.even_fibonacci_two_coloring_avoids_four_term_diffsequence","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#even-fibonacci-two-coloring-avoids-four","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $F_m$ denote the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$, and let $E = \\{F_{3n+3} : n \\ge 0\\} = \\{2, 8, 34, \\dots\\}$ be the even Fibonacci set. There is a coloring of the positive integers with $2$ colors under which no four positive integers $x_0, x_1, x_2, x_3$ of the same color satisfy $x_{i+1} - x_i \\in E$ for $i = 0, 1, 2$.","title":"A 2-coloring with no monochromatic 4-term even-Fibonacci diffsequence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/even-fibonacci-not-two-accessible","kind":"node","lean":"MetaMathlibExt.even_fibonacci_not_two_accessible","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#even-fibonacci-not-two-accessible","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $F_m$ denote the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. The even Fibonacci set $\\{F_{3n+3} : n \\ge 0\\} = \\{2, 8, 34, \\dots\\}$ is not $2$-accessible.","title":"The even Fibonacci set is not 2-accessible"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/furstenberg-weiss-correspondence","kind":"node","lean":"MetaMathlibExt.furstenberg_weiss_correspondence","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#furstenberg-weiss-correspondence","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $\\mathcal{A}$ be a collection of finite sets of integers that is closed under translation, that is, $A + t \\in \\mathcal{A}$ for all $A \\in \\mathcal{A}$ and $t \\in \\mathbb{Z}$, and let $r \\ge 1$. For a homeomorphism $T$ and an integer $a$, write $T^{-a}(U)$ for the preimage of $U$ under the $a$-th power of $T$. Then the following three conditions are equivalent: (a) for every coloring of $\\mathbb{Z}$ with $r$\u2026","title":"Furstenberg\u2013Weiss correspondence for translation-invariant families"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/recurrence-accessibility-equivalences","kind":"node","lean":"MetaMathlibExt.recurrence_accessibility_equivalences","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#recurrence-accessibility-equivalences","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $D$ be a set of positive integers. Say that $D$ has *diffsequence recurrence* if for every compact metric space $X$, every minimal homeomorphism $T$ of $X$, every nonempty open set $U \\subseteq X$ and every $k \\ge 1$ there are integers $s_1, \\dots, s_k$ forming a $D$-diffsequence such that $U \\cap T^{-s_1}U \\cap \\dots \\cap T^{-s_k}U$ is nonempty, where $T^{-s}U$ is the preimage of $U$ under the integer power\u2026","title":"Topological recurrence, accessibility and chromatic intersectivity"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/topological-recurrence-union-partition","kind":"node","lean":"MetaMathlibExt.topological_recurrence_union_partition","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","One part of a union inherits "],["math","\\(k\\)"],["text","-topological recurrence"]],"source":"One part of a union inherits $k$-topological recurrence"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#topological-recurrence-union-partition","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $k$ be a natural number and let $D_1, D_2 \\subseteq \\mathbb{N}$. If $D_1 \\cup D_2$ is a set of $k$-topological recurrence, then $D_1$ or $D_2$ is a set of $k$-topological recurrence.","title":"One part of a union inherits $k$-topological recurrence"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/accessible-union-partition","kind":"node","lean":"MetaMathlibExt.accessible_union_partition","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#accessible-union-partition","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $D$, $D_1$ and $D_2$ be sets of natural numbers with $D = D_1 \\cup D_2$. If $D$ is accessible, then $D_1$ is accessible or $D_2$ is accessible.","title":"One part of an accessible union is accessible"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-monochromatic-2","kind":"node","lean":"MathlibExt.Combinatorics.PrimeTwoAccessibilityWanted.IsMonochromatic","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-monochromatic-2","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Given a colouring $c : \\mathbb{N} \\to \\{0, \\dots, r - 1\\}$ of the natural numbers with $r$ colours, a finite sequence $x_0, \\dots, x_{k-1}$ of natural numbers is monochromatic when all its terms have the same colour: $c(x_i) = c(x_j)$ for all $i, j < k$.","title":"Monochromatic finite sequences"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-sdiff-sequence","kind":"node","lean":"MathlibExt.Combinatorics.PrimeTwoAccessibilityWanted.IsSDiffSequence","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite "],["math","\\(S\\)"],["text","-diffsequences"]],"source":"Finite $S$-diffsequences"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-sdiff-sequence","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"For a set $S$ of natural numbers, a finite sequence $x_0, \\dots, x_{k-1}$ of natural numbers is an $S$-diffsequence when each term is the previous one plus an element of $S$: for every $i$ with $i + 1 < k$ there is $d \\in S$ with $x_{i+1} = x_i + d$.","title":"Finite $S$-diffsequences"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/is-raccessible-2","kind":"node","lean":"MathlibExt.Combinatorics.PrimeTwoAccessibilityWanted.IsRAccessible","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(r\\)"],["text","-accessible sets of natural numbers"]],"source":"$r$-accessible sets of natural numbers"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#is-raccessible-2","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"A set $S \\subseteq \\mathbb{N}$ is $r$-accessible when for every colouring $c : \\mathbb{N} \\to \\{0, \\dots, r - 1\\}$ with $r$ colours and every length $k \\ne 1$ there is a sequence $x_0, \\dots, x_{k-1}$ of natural numbers that is monochromatic for $c$ and is an $S$-diffsequence: for each $i$ with $i + 1 < k$ there is $d \\in S$ with $x_{i+1} = x_i + d$.","title":"$r$-accessible sets of natural numbers"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.PrimeTwoAccessibilityWanted.conjecture","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#conjecture","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"The statement that the set of prime numbers is $2$-accessible: for every colouring $c : \\mathbb{N} \\to \\{0, 1\\}$ of the natural numbers with two colours and every length $k \\ne 1$ there is a sequence $x_0, \\dots, x_{k-1}$ of natural numbers, all of the same colour, such that each difference $x_{i+1} - x_i$ is a prime number. Open Problem Garden question OPG-1797 asks whether this holds.","title":"2-accessibility of the primes"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.PrimeTwoAccessibilityWanted.conjecture_holds","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#conjecture-holds","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"The set of prime numbers is $2$-accessible: for every colouring $c : \\mathbb{N} \\to \\{0, 1\\}$ of the natural numbers with two colours and every length $k \\ne 1$ there is a sequence $x_0, \\dots, x_{k-1}$ of natural numbers, all of the same colour, such that each difference $x_{i+1} - x_i$ is a prime number.","title":"The primes are 2-accessible"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/half-even-fibonacci-set","kind":"node","lean":"MetaMathlibExt.halfEvenFibonacciSet","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#half-even-fibonacci-set","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $F_m$ be the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. The set of halves of the positive even Fibonacci numbers is $G = \\{F_{3n}/2 : n \\ge 1\\} = \\{1, 4, 17, 72, \\dots\\}$; its elements are written $g_n = F_{3n}/2$ for $n \\ge 1$.","title":"Halves of even Fibonacci numbers"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/four-mem-half-even-fibonacci-set","kind":"node","lean":"MetaMathlibExt.four_mem_halfEvenFibonacciSet","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(4\\)"],["text"," is half of an even Fibonacci number"]],"source":"$4$ is half of an even Fibonacci number"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#four-mem-half-even-fibonacci-set","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $F_m$ be the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. The number $4$ belongs to the set $G = \\{F_{3n}/2 : n \\ge 1\\} = \\{1, 4, 17, 72, \\dots\\}$ of halves of the positive even Fibonacci numbers; indeed $4 = F_6/2$.","title":"$4$ is half of an even Fibonacci number"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/thue-morse-morphism","kind":"node","lean":"MetaMathlibExt.thueMorseMorphism","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Thue\u2013Morse morphism "],["math","\\(\\mu\\)"]],"source":"The Thue\u2013Morse morphism $\\mu$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#thue-morse-morphism","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"The morphism $\\mu$ on binary words sends the letter $0$ to the word $10$ and the letter $1$ to the word $01$.","title":"The Thue\u2013Morse morphism $\\mu$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fibonacci-nu-morphism","kind":"node","lean":"MetaMathlibExt.fibonacciNuMorphism","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","The morphism "],["math","\\(\\nu\\)"]],"source":"The morphism $\\nu$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fibonacci-nu-morphism","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"The morphism $\\nu$ is given on the letters $0$ and $1$ by $\\nu(0) = 1$ and $\\nu(1) = 00$.","title":"The morphism $\\nu$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/wesley-word-s","kind":"node","lean":"MetaMathlibExt.wesleyWordS","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wesley's word "],["math","\\(S\\)"]],"source":"Wesley's word $S$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#wesley-word-s","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $\\mu$ be the morphism of binary words with $0 \\mapsto 10$ and $1 \\mapsto 01$, and let $f_0 = 0$, $f_1 = 01$, $f_2 = 010$, $\\dots$ be the finite Fibonacci words. The word $S$ is the infinite binary word, indexed from $0$, whose letter at index $n$ is the letter at index $n$ of $\\mu(f_{n+2})$.","title":"Wesley's word $S$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/wesley-word-t","kind":"node","lean":"MetaMathlibExt.wesleyWordT","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wesley's word "],["math","\\(T\\)"]],"source":"Wesley's word $T$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#wesley-word-t","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Let $\\nu$ be the morphism of binary words with $0 \\mapsto 1$ and $1 \\mapsto 00$, and let $f_0 = 0$, $f_1 = 01$, $f_2 = 010$, $\\dots$ be the finite Fibonacci words. The word $T$ is the infinite binary word, indexed from $0$, whose letter at index $n$ is the letter at index $n$ of $\\nu(f_{n+2})$.","title":"Wesley's word $T$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/wesley-words-first-ten","kind":"node","lean":"MetaMathlibExt.wesleyWords_first_ten","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first ten letters of Wesley's words "],["math","\\(S\\)"],["text"," and "],["math","\\(T\\)"]],"source":"The first ten letters of Wesley's words $S$ and $T$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#wesley-words-first-ten","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"Indexing letters from $0$, the letters of $S$ at indices $0, 1, \\dots, 9$ are $1, 0, 0, 1, 1, 0, 1, 0, 0, 1$ and the letters of $T$ at indices $0, 1, \\dots, 9$ are $1, 0, 0, 1, 1, 0, 0, 1, 0, 0$; that is, $S$ begins with $1001101001$ and $T$ begins with $1001100100$.","title":"The first ten letters of Wesley's words $S$ and $T$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/wesley-fractional-constant","kind":"node","lean":"MetaMathlibExt.wesleyFractionalConstant","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wesley's constants "],["math","\\(c_{n,\\varepsilon}\\)"]],"source":"Wesley's constants $c_{n,\\varepsilon}$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#wesley-fractional-constant","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"For a natural number $n$ and an integer $\\varepsilon$, the real number $c_{n,\\varepsilon}$ is defined by the residue of $n$ modulo $4$: if $n \\equiv 0 \\pmod 4$, then $c_{n,\\varepsilon} = (4 + (2\\sqrt5 - 5)\\varepsilon)/10$; if $n \\equiv 1 \\pmod 4$, then $c_{n,\\varepsilon} = (11 - 4\\sqrt5)/5$ when $\\varepsilon = -4$ and $c_{n,\\varepsilon} = (4 + (4\\sqrt5 - 5)\\varepsilon)/20$ otherwise; if $n \\equiv 2 \\pmod 4$, then\u2026","title":"Wesley's constants $c_{n,\\varepsilon}$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/wesley-fractional-constant-zero-of-mod-f","kind":"node","lean":"MetaMathlibExt.wesleyFractionalConstant_zero_of_mod_four_eq_zero","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wesley's constant "],["math","\\(c_{n,0}\\)"],["text"," for "],["math","\\(n \\equiv 0 \\pmod 4\\)"]],"source":"Wesley's constant $c_{n,0}$ for $n \\equiv 0 \\pmod 4$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#wesley-fractional-constant-zero-of-mod-f","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"fully_proved","summary":"For every natural number $n$ with $n \\equiv 0 \\pmod 4$, Wesley's constant satisfies $c_{n,0} = 2/5$.","title":"Wesley's constant $c_{n,0}$ for $n \\equiv 0 \\pmod 4$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fibonacci-set-degree-of-accessibility-le","kind":"node","lean":"MetaMathlibExt.fibonacciSet_degreeOfAccessibility_le_three","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Fibonacci numbers have degree of accessibility at most "],["math","\\(3\\)"]],"source":"The Fibonacci numbers have degree of accessibility at most $3$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fibonacci-set-degree-of-accessibility-le","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"The positive Fibonacci set $\\{1, 2, 3, 5, 8, \\dots\\}$ is not $r$-accessible for any natural number $r \\ge 4$. Moreover, some coloring of the positive integers with four colors admits no monochromatic $4$-term diffsequence with differences in the positive Fibonacci set: there are no positive integers $x_0, x_1, x_2, x_3$ of one color such that $x_{i+1} - x_i$ is a positive Fibonacci number for $i = 0, 1, 2$.","title":"The Fibonacci numbers have degree of accessibility at most $3$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fibonacci-gap-progressions-not-two-regul","kind":"node","lean":"MetaMathlibExt.fibonacci_gap_progressions_not_two_regular","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arithmetic progressions with Fibonacci gaps are not "],["math","\\(2\\)"],["text","-regular"]],"source":"Arithmetic progressions with Fibonacci gaps are not $2$-regular"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fibonacci-gap-progressions-not-two-regul","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"There is a coloring of the positive integers with two colors that admits no monochromatic $5$-term arithmetic progression with gap in the positive Fibonacci set $\\{1, 2, 3, 5, 8, \\dots\\}$, that is, no positive integers $x_0, \\dots, x_4$ of one color with $x_i = a + i d$ for all $i < 5$, for a positive integer $a$ and a positive Fibonacci number $d$. Moreover, the positive Fibonacci set is $1$-large, and it is not\u2026","title":"Arithmetic progressions with Fibonacci gaps are not $2$-regular"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fib-div-golden-ratio-sub-fib","kind":"node","lean":"MetaMathlibExt.fib_div_goldenRatio_sub_fib","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fib-div-golden-ratio-sub-fib","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $F_n$ be the $n$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$, and let $\\varphi = (1 + \\sqrt{5})/2$ be the golden ratio. For every natural number $n \\ge 2$,","title":"Fibonacci numbers divided by the golden ratio"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fibonacci-word-beatty-properties","kind":"node","lean":"MetaMathlibExt.fibonacciWord_beatty_properties","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Beatty-sequence descriptions of the words "],["math","\\(F_\\infty\\)"],["text",", "],["math","\\(S\\)"],["text"," and "],["math","\\(T\\)"]],"source":"Beatty-sequence descriptions of the words $F_\\infty$, $S$ and $T$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fibonacci-word-beatty-properties","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $\\varphi = (1 + \\sqrt{5})/2$ be the golden ratio and $F_\\infty = 010010100\\cdots$ the infinite Fibonacci word. Let $S = \\mu(F_\\infty)$ be its image under the morphism $\\mu$ with $\\mu(0) = 10$ and $\\mu(1) = 01$, and $T = \\nu(F_\\infty)$ its image under the morphism $\\nu$ with $\\nu(0) = 1$ and $\\nu(1) = 00$. For a word $W$ and an integer $n \\ge 1$, write $W(n)$ for the $n$-th letter of $W$, counting from $1$.\u2026","title":"Beatty-sequence descriptions of the words $F_\\infty$, $S$ and $T$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/not-accessible-four-of-half-even-fibonac","kind":"node","lean":"MetaMathlibExt.not_accessible_four_of_halfEvenFibonacci","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Halved even Fibonacci numbers and "],["math","\\(4\\)"],["text","-accessibility of the Fibonacci numbers"]],"source":"Halved even Fibonacci numbers and $4$-accessibility of the Fibonacci numbers"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#not-accessible-four-of-half-even-fibonac","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Write $f_n$ for the $n$-th Fibonacci number ($f_0 = 0$, $f_1 = 1$). Let $F = \\{1, 2, 3, 5, 8, \\dots\\}$ be the set of positive Fibonacci numbers and $G = \\{f_{3n}/2 : n \\ge 1\\} = \\{1, 4, 17, 72, \\dots\\}$ the set of halves of the even Fibonacci numbers. If $G$ is not $2$-accessible, then $F$ is not $4$-accessible.","title":"Halved even Fibonacci numbers and $4$-accessibility of the Fibonacci numbers"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fract-golden-ratio-differences","kind":"node","lean":"MetaMathlibExt.fract_goldenRatio_differences","parent":23448,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fract-golden-ratio-differences","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $F_m$ be the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$, let $\\varphi = (1 + \\sqrt{5})/2$ be the golden ratio, and write $\\{y\\} = y - \\lfloor y \\rfloor$ for the fractional part. Let $i \\ge 1$ be a natural number, put $g_i = F_{3i}/2$ (an integer, since $F_{3i}$ is even), and let $x_1, x_2, m_1, m_2$ be integers. Then: (i) if $g_i$ is even, $2(x_2 - x_1) = g_i$, $x_1 = \\lfloor m_1\\varphi \\rfloor$ and\u2026","title":"Differences of fractional parts of multiples of the golden ratio"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/fract-fib-add-div-sqrt-five","kind":"node","lean":"MetaMathlibExt.fract_fib_add_div_sqrt_five","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fractional parts of shifted Fibonacci numbers divided by "],["math","\\(\\sqrt{5}\\)"]],"source":"Fractional parts of shifted Fibonacci numbers divided by $\\sqrt{5}$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#fract-fib-add-div-sqrt-five","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $F_n$ be the $n$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$, let $\\varphi = (1 + \\sqrt{5})/2$ be the golden ratio, so that $2\\varphi - 1 = \\sqrt{5}$, and write $\\{y\\} = y - \\lfloor y \\rfloor$ for the fractional part. Let $n \\ge 13$ be a natural number and $\\varepsilon \\in \\{-4, 0, 4\\}$, let $c_{n,\\varepsilon}$ be Wesley's constant, which depends on $n \\bmod 4$ and $\\varepsilon$, and put\u2026","title":"Fractional parts of shifted Fibonacci numbers divided by $\\sqrt{5}$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/lucas-set-not-four-accessible","kind":"node","lean":"MetaMathlibExt.lucasSet_not_four_accessible","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Lucas numbers are not "],["math","\\(4\\)"],["text","-accessible"]],"source":"The Lucas numbers are not $4$-accessible"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#lucas-set-not-four-accessible","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"Let $L = \\{2, 1, 3, 4, 7, \\dots\\}$ be the set of Lucas numbers $L_0 = 2$, $L_1 = 1$, $L_{n+2} = L_{n+1} + L_n$. There is a coloring of the positive integers with the colors $0, 1, 2, 3$ for which no sequence $x_0, x_1, x_2$ of positive integers is both monochromatic and an $L$-diffsequence; and $L$ is not $r$-accessible for any $r \\ge 4$.","title":"The Lucas numbers are not $4$-accessible"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/even-fibonacci-set-degree-of-accessibili","kind":"node","lean":"MetaMathlibExt.evenFibonacciSet_degreeOfAccessibility_eq_one","parent":23448,"reading_title":{"format":"inline-parts-v1","parts":[["text","The even Fibonacci numbers have degree of accessibility "],["math","\\(1\\)"]],"source":"The even Fibonacci numbers have degree of accessibility $1$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences/index.html#even-fibonacci-set-degree-of-accessibili","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/accessible-sets-and-diffsequences","status":"can_prove","summary":"The even Fibonacci set $\\{2, 8, 34, \\dots\\}$ is $1$-accessible, and it is not $r$-accessible for any natural number $r \\ge 2$.","title":"The even Fibonacci numbers have degree of accessibility $1$"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres","kind":"scope","navigation":"module","parent":23446,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres","status":"can_prove","title":"Colourings with only short monochromatic progressions"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres/two-colouring-short-progressions","kind":"node","lean":"MathlibExt.Combinatorics.ShortMonochromaticProgressionsErdosProblem984Wanted.twoColouringShortProgressions","parent":23529,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres/index.html#two-colouring-short-progressions","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres","status":"fully_proved","summary":"The proposition that there is a colouring of the positive integers with two colours such that for every real $\\varepsilon > 0$ there is a real constant $C_\\varepsilon$ with the following property: whenever $a \\ge 1$, $d \\ge 1$ and $k \\ge 0$ are integers and $a, a + d, \\dots, a + (k - 1)d$ all have the same colour, $k \\le C_\\varepsilon a^\\varepsilon$.","title":"Statement of Erd\u0151s Problem 984 on short monochromatic progressions"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres/two-colouring-short-progressions-holds","kind":"node","lean":"MathlibExt.Combinatorics.ShortMonochromaticProgressionsErdosProblem984Wanted.twoColouringShortProgressions_holds","parent":23529,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres/index.html#two-colouring-short-progressions-holds","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/colourings-with-only-short-monochromatic-progres","status":"can_prove","summary":"There is a colouring of the positive integers with two colours such that for every real $\\varepsilon > 0$ there is a real constant $C_\\varepsilon$ with the following property: whenever $a \\ge 1$, $d \\ge 1$ and $k \\ge 0$ are integers and $a, a + d, \\dots, a + (k - 1)d$ all have the same colour, $k \\le C_\\varepsilon a^\\varepsilon$.","title":"Erd\u0151s Problem 984: a two-colouring with short monochromatic progressions exists"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves","kind":"scope","navigation":"module","parent":23446,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves","status":"can_prove","title":"Monochromatic descending waves"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/is-descending-wave","kind":"node","lean":"MathlibExt.Combinatorics.DescendingWaveColoringErdosProblem781Wanted.IsDescendingWave","parent":23532,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/index.html#is-descending-wave","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves","status":"fully_proved","summary":"A $k$-term descending wave is a sequence $x_0, x_1, \\dots, x_{k-1}$ of natural numbers with $x_0 < x_1 < \\dots < x_{k-1}$ and $x_{j-1} + x_{j+1} \\le 2 x_j$ for every $j$ with $0 < j < k - 1$.","title":"Descending wave"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/forces-monochromatic-wave","kind":"node","lean":"MathlibExt.Combinatorics.DescendingWaveColoringErdosProblem781Wanted.ForcesMonochromaticWave","parent":23532,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/index.html#forces-monochromatic-wave","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the statement that for every colouring of the natural numbers with two colours there is a monochromatic $k$-term descending wave all of whose terms lie in $\\{1, \\dots, n\\}$. Equivalently, every colouring of $\\{1, \\dots, n\\}$ with two colours has a monochromatic $k$-term descending wave, so the statement says $f(k) \\le n$, where $f(k)$ is the least $n$ with this property.","title":"Forcing a monochromatic descending wave"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/wave-number-quadratic","kind":"node","lean":"MathlibExt.Combinatorics.DescendingWaveColoringErdosProblem781Wanted.waveNumberQuadratic","parent":23532,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/index.html#wave-number-quadratic","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves","status":"fully_proved","summary":"For natural numbers $k$ and $n$, say that $n$ forces a $k$-term wave if every colouring of $\\{1, \\dots, n\\}$ with two colours has a monochromatic $k$-term descending wave. The statement that for every natural number $k \\ge 1$ the number $k^2 - k + 1$ forces a $k$-term wave and the number $k^2 - k$ does not; that is, $f(k) = k^2 - k + 1$ for every $k \\ge 1$, where $f(k)$ is the least $n$ that forces a $k$-term wave.","title":"Erd\u0151s Problem 781, the exact-value question"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/wave-number-cubic-order","kind":"node","lean":"MathlibExt.Combinatorics.DescendingWaveColoringErdosProblem781Wanted.waveNumberCubicOrder","parent":23532,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/index.html#wave-number-cubic-order","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves","status":"fully_proved","summary":"For natural numbers $k$ and $n$, say that $n$ forces a $k$-term wave if every colouring of $\\{1, \\dots, n\\}$ with two colours has a monochromatic $k$-term descending wave. The statement that there are real numbers $c > 0$ and $C$ such that for all sufficiently large $k$, every natural number $n$ that forces a $k$-term wave satisfies $n \\ge c k^3$, and the natural number $\\lfloor \\max(C k^3, 0) \\rfloor$ forces a\u2026","title":"Erd\u0151s Problem 781, cubic order of the wave number"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/wave-number-quadratic-refuted","kind":"node","lean":"MathlibExt.Combinatorics.DescendingWaveColoringErdosProblem781Wanted.waveNumberQuadratic_refuted","parent":23532,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/index.html#wave-number-quadratic-refuted","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves","status":"can_prove","summary":"For natural numbers $k$ and $n$, say that $n$ forces a $k$-term wave if every colouring of $\\{1, \\dots, n\\}$ with two colours has a monochromatic $k$-term descending wave. It is not the case that for every natural number $k \\ge 1$ the number $k^2 - k + 1$ forces a $k$-term wave and the number $k^2 - k$ does not. Writing $f(k)$ for the least $n$ that forces a $k$-term wave, this says that $f(k) \\ne k^2 - k + 1$ for\u2026","title":"Erd\u0151s Problem 781 is false"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/wave-number-cubic-order-holds","kind":"node","lean":"MathlibExt.Combinatorics.DescendingWaveColoringErdosProblem781Wanted.waveNumberCubicOrder_holds","parent":23532,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves/index.html#wave-number-cubic-order-holds","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-descending-waves","status":"can_prove","summary":"For natural numbers $k$ and $n$, say that $n$ forces a $k$-term wave if every colouring of $\\{1, \\dots, n\\}$ with two colours has a monochromatic $k$-term descending wave. There are real numbers $c > 0$ and $C$ such that for all sufficiently large $k$, every natural number $n$ that forces a $k$-term wave satisfies $n \\ge c k^3$, and the natural number $\\lfloor \\max(C k^3, 0) \\rfloor$ forces a $k$-term wave. Writing\u2026","title":"Alon\u2013Spencer: the descending-wave number has cubic order"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions","kind":"scope","navigation":"module","parent":23446,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions","status":"can_prove","title":"Monochromatic or rainbow arithmetic progressions"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/has-good-ap","kind":"node","lean":"MathlibExt.Combinatorics.ErdosRainbowAPGrowthWanted.HasGoodAP","parent":23539,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/index.html#has-good-ap","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ be a colouring and let $N$ and $k$ be natural numbers. The colouring $f$ has a good $k$-term arithmetic progression below $N$ if there are natural numbers $a$ and $d > 0$ with $a + d(k - 1) < N$, where $k - 1$ is natural-number subtraction (so it is $0$ when $k = 0$), such that the colours $f(a + di)$ for $i < k$ are either all equal (the progression is monochromatic) or pairwise\u2026","title":"Good arithmetic progressions"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/h","kind":"node","lean":"MathlibExt.Combinatorics.ErdosRainbowAPGrowthWanted.H","parent":23539,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/index.html#h","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions","status":"fully_proved","summary":"For a natural number $k$, $H(k)$ is the least natural number $N$ such that every colouring $f : \\mathbb{N} \\to \\mathbb{N}$ has a good $k$-term arithmetic progression below $N$, that is, there are natural numbers $a$ and $d > 0$ with $a + d(k - 1) < N$, where $k - 1$ is natural-number subtraction, such that the colours $f(a + di)$ for $i < k$ are all equal or pairwise distinct. If no such $N$ exists, then $H(k) = 0$.","title":"Monochromatic-or-rainbow progression number"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosRainbowAPGrowthWanted.conjecture","parent":23539,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/index.html#conjecture","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions","status":"fully_proved","summary":"For a natural number $k$, let $H(k)$ be the least natural number $N$ such that every colouring $f : \\mathbb{N} \\to \\mathbb{N}$ has a good $k$-term arithmetic progression below $N$, that is, there are natural numbers $a$ and $d > 0$ with $a + d(k - 1) < N$, where $k - 1$ is natural-number subtraction, such that the colours $f(a + di)$ for $i < k$ are all equal or pairwise distinct; let $H(k) = 0$ if no such $N$\u2026","title":"Erd\u0151s Problem 190 on monochromatic or rainbow progressions"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosRainbowAPGrowthWanted.conjecture_holds","parent":23539,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions/index.html#conjecture-holds","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-or-rainbow-arithmetic-progressions","status":"can_prove","summary":"For a natural number $k$, let $H(k)$ be the least natural number $N$ such that every colouring $f : \\mathbb{N} \\to \\mathbb{N}$ has a good $k$-term arithmetic progression below $N$, that is, there are natural numbers $a$ and $d > 0$ with $a + d(k - 1) < N$, where $k - 1$ is natural-number subtraction, such that the colours $f(a + di)$ for $i < k$ are all equal or pairwise distinct; let $H(k) = 0$ if no such $N$\u2026","title":"Solution of Erd\u0151s Problem 190 on monochromatic or rainbow progressions"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif","kind":"scope","navigation":"module","parent":23446,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif","status":"can_prove","title":"Monochromatic progressions with large common difference"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif/monochromatic-progression-large-differen","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticProgressionLargeDifferenceErdosProblem645Wanted.monochromaticProgressionLargeDifference","parent":23544,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif/index.html#monochromatic-progression-large-differen","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif","status":"fully_proved","summary":"The statement that for every colouring of the positive integers with two colours there are positive integers $x$ and $d$ with $d > x$ such that $x$, $x + d$ and $x + 2d$ all have the same colour.","title":"Erd\u0151s Problem 645"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif/monochromatic-progression-large-differen-2","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticProgressionLargeDifferenceErdosProblem645Wanted.monochromaticProgressionLargeDifference_holds","parent":23544,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif/index.html#monochromatic-progression-large-differen-2","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/monochromatic-progressions-with-large-common-dif","status":"can_prove","summary":"For every colouring of the positive integers with two colours there are positive integers $x$ and $d$ with $d > x$ such that $x$, $x + d$ and $x + 2d$ all have the same colour.","title":"Erd\u0151s Problem 645 is true"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets","kind":"scope","navigation":"module","parent":23446,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets","status":"can_prove","title":"Progression-free Ramsey sets"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets/progression-free-ramsey-set","kind":"node","lean":"MathlibExt.Combinatorics.ProgressionFreeRamseySetErdosProblem966Wanted.progressionFreeRamseySet","parent":23547,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets/index.html#progression-free-ramsey-set","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets","status":"fully_proved","summary":"This is the statement that for all natural numbers $k \\ge 2$ and $r \\ge 2$ there is a set $A$ of natural numbers that contains no arithmetic progression of $k + 1$ distinct terms and such that for every colouring $c$ of the natural numbers with $r$ colours there are a subset $t \\subseteq A$ that is an arithmetic progression of $k$ distinct terms and a colour $i$ with $c(x) = i$ for every $x \\in t$.","title":"Erd\u0151s Problem 966: progression-free sets that are Ramsey for shorter progressions"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets/progression-free-ramsey-set-holds","kind":"node","lean":"MathlibExt.Combinatorics.ProgressionFreeRamseySetErdosProblem966Wanted.progressionFreeRamseySet_holds","parent":23547,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets/index.html#progression-free-ramsey-set-holds","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/progression-free-ramsey-sets","status":"can_prove","summary":"For all natural numbers $k \\ge 2$ and $r \\ge 2$ there is a set $A$ of natural numbers that contains no arithmetic progression of $k + 1$ distinct terms and such that for every colouring $c$ of the natural numbers with $r$ colours there are a subset $t \\subseteq A$ that is an arithmetic progression of $k$ distinct terms and a colour $i$ with $c(x) = i$ for every $x \\in t$.","title":"Erd\u0151s Problem 966 holds"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","kind":"scope","navigation":"module","parent":23446,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"can_prove","title":"Van der Waerden numbers W(3, k)"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/has-monochromatic-ap","kind":"node","lean":"MathlibExt.Combinatorics.OffDiagonalVanDerWaerdenGrowthAIMCombinatoricsProblem90Wanted.HasMonochromaticAP","parent":23550,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#has-monochromatic-ap","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"fully_proved","summary":"Let $c$ be a colouring of the natural numbers with the two colours true and false, let $b$ be a colour and let $l, N$ be natural numbers. Then $c$ has a monochromatic progression of colour $b$ and length $l$ in $\\{1, \\dots, N\\}$ when there are natural numbers $a \\ge 1$ and $d \\ge 1$ with $a + (l - 1) d \\le N$, where $l - 1$ is read as $0$ when $l = 0$, such that $c(a + j d) = b$ for every $j < l$.","title":"Monochromatic arithmetic progression"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/off-diagonal-van-der-waerden-polynomial","kind":"node","lean":"MathlibExt.Combinatorics.OffDiagonalVanDerWaerdenGrowthAIMCombinatoricsProblem90Wanted.offDiagonalVanDerWaerdenPolynomial","parent":23550,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#off-diagonal-van-der-waerden-polynomial","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"fully_proved","summary":"There are natural numbers $C$ and $A$ such that for every $k \\ge 1$, every colouring of the natural numbers in red and blue has a red $3$-term or a blue $k$-term arithmetic progression $a, a + d, \\dots$ with $a \\ge 1$, $d \\ge 1$ and all terms in $\\{1, \\dots, C k^A\\}$. Equivalently, the van der Waerden number $W(3, k)$, the least $n$ such that every red/blue colouring of $\\{1, \\dots, n\\}$ has a red $3$-term or a\u2026","title":"Polynomial growth of the off-diagonal van der Waerden numbers (claim)"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/off-diagonal-van-der-waerden-polynomial-2","kind":"node","lean":"MathlibExt.Combinatorics.OffDiagonalVanDerWaerdenGrowthAIMCombinatoricsProblem90Wanted.offDiagonalVanDerWaerdenPolynomial_refuted","parent":23550,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#off-diagonal-van-der-waerden-polynomial-2","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"can_prove","summary":"There are no natural numbers $C$ and $A$ such that for every $k \\ge 1$, every colouring of the natural numbers in red and blue has a red $3$-term or a blue $k$-term arithmetic progression $a, a + d, \\dots$ with $a \\ge 1$, $d \\ge 1$ and all terms in $\\{1, \\dots, C k^A\\}$.","title":"The off-diagonal van der Waerden numbers are not polynomially bounded"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/has-mono-ap","kind":"node","lean":"MathlibExt.Combinatorics.VanDerWaerdenThreeKErdosProblem721Wanted.HasMonoAP","parent":23550,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#has-mono-ap","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"fully_proved","summary":"Let $c$ be a red/blue colouring of the natural numbers, given as a map to the Booleans ($\\mathrm{true}$ is red, $\\mathrm{false}$ is blue), let $b$ be a colour and let $\\ell$ and $N$ be natural numbers. Then $c$ has an $\\ell$-term arithmetic progression of colour $b$ inside $\\{1, \\dots, N\\}$ if there are natural numbers $a$ and $d > 0$ such that for every $i < \\ell$ we have $1 \\le a + id \\le N$ and $c(a + id) = b$.","title":"Monochromatic arithmetic progressions in an initial interval"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/forces-three-k","kind":"node","lean":"MathlibExt.Combinatorics.VanDerWaerdenThreeKErdosProblem721Wanted.ForcesThreeK","parent":23550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Intervals forcing the van der Waerden number "],["math","\\(W(3, k)\\)"]],"source":"Intervals forcing the van der Waerden number $W(3, k)$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#forces-three-k","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"fully_proved","summary":"For natural numbers $N$ and $k$, the interval $\\{1, \\dots, N\\}$ forces $W(3, k)$ if every red/blue colouring of the natural numbers has, inside $\\{1, \\dots, N\\}$, a red $3$-term arithmetic progression or a blue $k$-term arithmetic progression (progressions have common difference $d > 0$). Equivalently $W(3, k) \\le N$, where the van der Waerden number $W(3, k)$ is the least $n$ such that every red/blue colouring of\u2026","title":"Intervals forcing the van der Waerden number $W(3, k)$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/van-der-waerden-three-subexponential","kind":"node","lean":"MathlibExt.Combinatorics.VanDerWaerdenThreeKErdosProblem721Wanted.vanDerWaerdenThreeSubexponential","parent":23550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subexponential upper bound for "],["math","\\(W(3, k)\\)"],["text"," (statement)"]],"source":"Subexponential upper bound for $W(3, k)$ (statement)"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#van-der-waerden-three-subexponential","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"fully_proved","summary":"Write $W(3, k)$ for the van der Waerden number: the least $n$ such that every red/blue colouring of $\\{1, \\dots, n\\}$ has a red $3$-term or a blue $k$-term arithmetic progression (progressions have common difference $d > 0$). Say that $N$ forces $W(3, k)$ if every red/blue colouring of $\\{1, \\dots, N\\}$ has a red $3$-term or a blue $k$-term arithmetic progression. This is the proposition that there is a real\u2026","title":"Subexponential upper bound for $W(3, k)$ (statement)"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/van-der-waerden-three-superpolynomial","kind":"node","lean":"MathlibExt.Combinatorics.VanDerWaerdenThreeKErdosProblem721Wanted.vanDerWaerdenThreeSuperpolynomial","parent":23550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Green's superpolynomial lower bound for "],["math","\\(W(3, k)\\)"],["text"," (statement)"]],"source":"Green's superpolynomial lower bound for $W(3, k)$ (statement)"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#van-der-waerden-three-superpolynomial","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"fully_proved","summary":"Write $W(3, k)$ for the van der Waerden number: the least $n$ such that every red/blue colouring of $\\{1, \\dots, n\\}$ has a red $3$-term or a blue $k$-term arithmetic progression (progressions have common difference $d > 0$). Say that $N$ forces $W(3, k)$ if every red/blue colouring of $\\{1, \\dots, N\\}$ has a red $3$-term or a blue $k$-term arithmetic progression. This is the proposition that there is a real\u2026","title":"Green's superpolynomial lower bound for $W(3, k)$ (statement)"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/van-der-waerden-three-subexponential-hol","kind":"node","lean":"MathlibExt.Combinatorics.VanDerWaerdenThreeKErdosProblem721Wanted.vanDerWaerdenThreeSubexponential_holds","parent":23550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Schoen's subexponential upper bound for "],["math","\\(W(3, k)\\)"]],"source":"Schoen's subexponential upper bound for $W(3, k)$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#van-der-waerden-three-subexponential-hol","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"can_prove","summary":"Write $W(3, k)$ for the van der Waerden number: the least $n$ such that every red/blue colouring of $\\{1, \\dots, n\\}$ has a red $3$-term or a blue $k$-term arithmetic progression (progressions have common difference $d > 0$). Say that $N$ forces $W(3, k)$ if every red/blue colouring of $\\{1, \\dots, N\\}$ has a red $3$-term or a blue $k$-term arithmetic progression. There is a real constant $c < 1$ such that for all\u2026","title":"Schoen's subexponential upper bound for $W(3, k)$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/van-der-waerden-three-superpolynomial-ho","kind":"node","lean":"MathlibExt.Combinatorics.VanDerWaerdenThreeKErdosProblem721Wanted.vanDerWaerdenThreeSuperpolynomial_holds","parent":23550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Green's superpolynomial lower bound for "],["math","\\(W(3, k)\\)"]],"source":"Green's superpolynomial lower bound for $W(3, k)$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k/index.html#van-der-waerden-three-superpolynomial-ho","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/van-der-waerden-numbers-w-3-k","status":"can_prove","summary":"Write $W(3, k)$ for the van der Waerden number: the least $n$ such that every red/blue colouring of $\\{1, \\dots, n\\}$ has a red $3$-term or a blue $k$-term arithmetic progression (progressions have common difference $d > 0$). Say that $N$ forces $W(3, k)$ if every red/blue colouring of $\\{1, \\dots, N\\}$ has a red $3$-term or a blue $k$-term arithmetic progression. There is a real constant $c > 0$ such that for all\u2026","title":"Green's superpolynomial lower bound for $W(3, k)$"},{"data_url":"scopes/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric.json","id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","kind":"scope","navigation":"module","parent":23446,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","title":"Zero-sum van der Waerden theorems for symmetric functions"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-mvanishing-family","kind":"node","lean":"MetaMathlibExt.IsMVanishingFamily","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(m\\)"],["text","-vanishing family"]],"source":"$m$-vanishing family"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-mvanishing-family","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"fully_proved","summary":"Let $n$ be a natural number and let $F = (f^{(l)})_{l \\ge 0}$ be a family of functions $f^{(l)} : (\\mathbb{Z}/n\\mathbb{Z})^l \\to M$ into a type $M$ with a zero element. For a natural number $m$, $F$ is $m$-vanishing if for every sequence $a_0, a_1, a_2, \\dots$ of elements of $\\mathbb{Z}/n\\mathbb{Z}$ there are a length $l > 1$ and a start $s \\ge 0$ such that $f^{(l)}(a_{s+jl}, a_{s+jl+1}, \\dots, a_{s+jl+l-1}) = 0$\u2026","title":"$m$-vanishing family"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-vanishing-family","kind":"node","lean":"MetaMathlibExt.IsVanishingFamily","parent":23560,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-vanishing-family","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"fully_proved","summary":"Let $n$ be a natural number. A family $F$ of functions on blocks over $\\mathbb{Z}/n\\mathbb{Z}$ with values in a type with a zero is vanishing if it is $m$-vanishing for every positive integer $m$.","title":"Vanishing family"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/sum-add-mul-prod-family","kind":"node","lean":"MetaMathlibExt.sumAddMulProdFamily","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum-plus-product family "],["math","\\(F_c\\)"]],"source":"The sum-plus-product family $F_c$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#sum-add-mul-prod-family","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"fully_proved","summary":"For $c \\in \\mathbb{Z}/n$, the family $F_c$ over $\\mathbb{Z}/n$ sends a block $(b_1, \\dots, b_l)$ of elements of $\\mathbb{Z}/n$, of any length $l \\ge 0$, to $b_1 + \\dots + b_l + c\\, b_1 b_2 \\cdots b_l$, where an empty sum is $0$ and an empty product is $1$.","title":"The sum-plus-product family $F_c$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/mono","kind":"node","lean":"MetaMathlibExt.IsMVanishingFamily.mono","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity of "],["math","\\(m\\)"],["text","-vanishing"]],"source":"Monotonicity of $m$-vanishing"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#mono","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"fully_proved","summary":"Let $F$ be a family of functions on blocks over $\\mathbb{Z}/n$ with values in a type $M$ with a zero. If $F$ is $m$-vanishing and $m' \\le m$, then $F$ is $m'$-vanishing.","title":"Monotonicity of $m$-vanishing"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-mvanishing-family-zero","kind":"node","lean":"MetaMathlibExt.isMVanishingFamily_zero","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every family is "],["math","\\(0\\)"],["text","-vanishing"]],"source":"Every family is $0$-vanishing"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-mvanishing-family-zero","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"fully_proved","summary":"For every natural number $n$ and every family $F$ of functions $f^{(l)} : (\\mathbb{Z}/n\\mathbb{Z})^l \\to M$, $l \\ge 0$, into a type $M$ with a zero element, $F$ is $0$-vanishing.","title":"Every family is $0$-vanishing"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/sum-add-mul-prod-family-two","kind":"node","lean":"MetaMathlibExt.sumAddMulProdFamily_two","parent":23560,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#sum-add-mul-prod-family-two","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"fully_proved","summary":"For $c \\in \\mathbb{Z}/n$ and $b_0, b_1 \\in \\mathbb{Z}/n$, the sum-plus-product family $F_c$ over $\\mathbb{Z}/n$ satisfies $F_c(b_0, b_1) = b_0 + b_1 + c\\, b_0 b_1$.","title":"The sum-plus-product family on blocks of length two"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-vanishing-family-sum","kind":"node","lean":"MetaMathlibExt.isVanishingFamily_sum","parent":23560,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-vanishing-family-sum","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"For every positive integer $n$, the family over $\\mathbb{Z}/n\\mathbb{Z}$ sending a block $(b_1, \\dots, b_l)$ to $\\sum_i b_i$ is vanishing: for every $m > 0$, every sequence over $\\mathbb{Z}/n\\mathbb{Z}$ contains $m$ consecutive blocks of a common length $l > 1$, each with sum zero.","title":"Zero-sum van der Waerden theorem"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/not-is-mvanishing-family-sum-add-mul-pro","kind":"node","lean":"MetaMathlibExt.not_isMVanishingFamily_sumAddMulProd_of_ne","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(F_c\\)"],["text"," with "],["math","\\(c \\notin \\{0, \\pm 1\\}\\)"],["text"," is not "],["math","\\(m\\)"],["text","-vanishing for "],["math","\\(m > 0\\)"]],"source":"$F_c$ with $c \\notin \\{0, \\pm 1\\}$ is not $m$-vanishing for $m > 0$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#not-is-mvanishing-family-sum-add-mul-pro","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $n > 1$ and let $c \\in \\mathbb{Z}/n$ with $c \\ne 0$, $c \\ne 1$ and $c \\ne -1$. Then for every $m > 0$ the sum-plus-product family $F_c$ over $\\mathbb{Z}/n$, $F_c(b_1, \\dots, b_l) = b_1 + \\dots + b_l + c\\, b_1 \\cdots b_l$, is not $m$-vanishing.","title":"$F_c$ with $c \\notin \\{0, \\pm 1\\}$ is not $m$-vanishing for $m > 0$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/sum-add-mul-prod-one-classification","kind":"node","lean":"MetaMathlibExt.sumAddMulProd_one_classification","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["text","Classification of vanishing for the sum-plus-product family "],["math","\\(F_1\\)"]],"source":"Classification of vanishing for the sum-plus-product family $F_1$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#sum-add-mul-prod-one-classification","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $n > 1$ and let $F_1$ be the family over $\\mathbb{Z}/n$ given by $F_1(b_1, \\dots, b_l) = b_1 + \\dots + b_l + b_1 \\cdots b_l$. If $n \\notin \\{2, 3, 4, 6, 8\\}$, then $F_1$ is not $m$-vanishing for any $m > 0$. If $n \\in \\{2, 3, 4, 8\\}$, then $F_1$ is vanishing. If $n = 6$, then $F_1$ is $1$-vanishing but not $m$-vanishing for any $m > 1$.","title":"Classification of vanishing for the sum-plus-product family $F_1$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-vanishing-family-sum-add-mul-prod-one","kind":"node","lean":"MetaMathlibExt.isVanishingFamily_sumAddMulProd_one_three","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum-plus-product family is vanishing over "],["math","\\(\\mathbb{Z}/3\\mathbb{Z}\\)"]],"source":"The sum-plus-product family is vanishing over $\\mathbb{Z}/3\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-vanishing-family-sum-add-mul-prod-one","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $F_1$ be the family over $\\mathbb{Z}/3\\mathbb{Z}$ sending a block $(b_1, \\dots, b_l)$ to $\\sum_i b_i + \\prod_i b_i$. Then $F_1$ is vanishing.","title":"The sum-plus-product family is vanishing over $\\mathbb{Z}/3\\mathbb{Z}$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-vanishing-family-sum-add-mul-prod-one-2","kind":"node","lean":"MetaMathlibExt.isVanishingFamily_sumAddMulProd_one_four_eight","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum-plus-product family is vanishing over "],["math","\\(\\mathbb{Z}/4\\mathbb{Z}\\)"],["text"," and "],["math","\\(\\mathbb{Z}/8\\mathbb{Z}\\)"]],"source":"The sum-plus-product family is vanishing over $\\mathbb{Z}/4\\mathbb{Z}$ and $\\mathbb{Z}/8\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-vanishing-family-sum-add-mul-prod-one-2","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"The sum-plus-product family is vanishing over $\\mathbb{Z}/4\\mathbb{Z}$ and $\\mathbb{Z}/8\\mathbb{Z}$. Let $n \\in \\{4, 8\\}$ and let $F_1$ be the family over $\\mathbb{Z}/n\\mathbb{Z}$ sending a block $(b_1, \\dots, b_l)$ to $\\sum_i b_i + \\prod_i b_i$. Then $F_1$ is vanishing.","title":"The sum-plus-product family is vanishing over $\\mathbb{Z}/4\\mathbb{Z}$ and $\\mathbb{Z}/8\\mathbb{Z}$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-mvanishing-family-one-sum-add-mul-pro","kind":"node","lean":"MetaMathlibExt.isMVanishingFamily_one_sumAddMulProd_one_six","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum-plus-product family is "],["math","\\(1\\)"],["text","-vanishing over "],["math","\\(\\mathbb{Z}/6\\mathbb{Z}\\)"]],"source":"The sum-plus-product family is $1$-vanishing over $\\mathbb{Z}/6\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-mvanishing-family-one-sum-add-mul-pro","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $F_1$ be the family over $\\mathbb{Z}/6\\mathbb{Z}$ sending a block $(b_1, \\dots, b_l)$ to $\\sum_i b_i + \\prod_i b_i$. Then $F_1$ is $1$-vanishing: every sequence over $\\mathbb{Z}/6\\mathbb{Z}$ contains a block of consecutive terms, of some length $l > 1$, on which $F_1$ vanishes.","title":"The sum-plus-product family is $1$-vanishing over $\\mathbb{Z}/6\\mathbb{Z}$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/not-is-mvanishing-family-sum-add-mul-pro-2","kind":"node","lean":"MetaMathlibExt.not_isMVanishingFamily_sumAddMulProd_six","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(F_{\\pm 1}\\)"],["text"," over "],["math","\\(\\mathbb{Z}/6\\)"],["text"," is not "],["math","\\(m\\)"],["text","-vanishing for "],["math","\\(m > 1\\)"]],"source":"$F_{\\pm 1}$ over $\\mathbb{Z}/6$ is not $m$-vanishing for $m > 1$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#not-is-mvanishing-family-sum-add-mul-pro-2","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"For every $m > 1$, neither of the sum-plus-product families $F_{-1}$ and $F_1$ over $\\mathbb{Z}/6$, where $F_c(b_1, \\dots, b_l) = b_1 + \\dots + b_l + c\\, b_1 \\cdots b_l$, is $m$-vanishing.","title":"$F_{\\pm 1}$ over $\\mathbb{Z}/6$ is not $m$-vanishing for $m > 1$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/not-is-mvanishing-family-sum-add-mul-pro-3","kind":"node","lean":"MetaMathlibExt.not_isMVanishingFamily_sumAddMulProd_one_prime_one_mod_four","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(F_1\\)"],["text"," modulo a prime "],["math","\\(n \\equiv 1 \\pmod 4\\)"],["text"," is not "],["math","\\(m\\)"],["text","-vanishing for "],["math","\\(m > 0\\)"]],"source":"$F_1$ modulo a prime $n \\equiv 1 \\pmod 4$ is not $m$-vanishing for $m > 0$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#not-is-mvanishing-family-sum-add-mul-pro-3","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $n$ be a prime with $n \\equiv 1 \\pmod 4$. Then for every $m > 0$ the sum-plus-product family $F_1$ over $\\mathbb{Z}/n$, $F_1(b_1, \\dots, b_l) = b_1 + \\dots + b_l + b_1 \\cdots b_l$, is not $m$-vanishing.","title":"$F_1$ modulo a prime $n \\equiv 1 \\pmod 4$ is not $m$-vanishing for $m > 0$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/exists-add-mul-eq-zero-mul-pow-eq-one","kind":"node","lean":"MetaMathlibExt.exists_add_mul_eq_zero_mul_pow_eq_one","parent":23560,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#exists-add-mul-eq-zero-mul-pow-eq-one","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Nonzero solutions of a two-equation system modulo primes greater than 3 congruent to 3 mod 4. Let $n > 3$ be a prime with $n \\equiv 3 \\pmod 4$. Then there are nonzero $x, y \\in \\mathbb{Z}/n\\mathbb{Z}$ and an integer $r \\in \\{2, 3\\}$ such that $x + ry = 0$ and $xy^r = 1$ in $\\mathbb{Z}/n\\mathbb{Z}$.","title":"Nonzero solutions of a two-equation system modulo primes greater than 3 congruent to 3 mod 4"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/not-is-mvanishing-family-sum-add-mul-pro-4","kind":"node","lean":"MetaMathlibExt.not_isMVanishingFamily_sumAddMulProd_one_prime_three_mod_four","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(F_1\\)"],["text"," modulo a prime "],["math","\\(n > 3\\)"],["text"," with "],["math","\\(n \\equiv 3 \\pmod 4\\)"],["text"," is not "],["math","\\(m\\)"],["text","-vanishing for "],["math","\\(m > 0\\)"]],"source":"$F_1$ modulo a prime $n > 3$ with $n \\equiv 3 \\pmod 4$ is not $m$-vanishing for $m > 0$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#not-is-mvanishing-family-sum-add-mul-pro-4","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $n > 3$ be a prime with $n \\equiv 3 \\pmod 4$. Then for every $m > 0$ the sum-plus-product family $F_1$ over $\\mathbb{Z}/n$, $F_1(b_1, \\dots, b_l) = b_1 + \\dots + b_l + b_1 \\cdots b_l$, is not $m$-vanishing.","title":"$F_1$ modulo a prime $n > 3$ with $n \\equiv 3 \\pmod 4$ is not $m$-vanishing for $m > 0$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/not-is-mvanishing-family-sum-add-mul-pro-5","kind":"node","lean":"MetaMathlibExt.not_isMVanishingFamily_sumAddMulProd_one_eight_mul","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(F_1\\)"],["text"," modulo "],["math","\\(8u\\)"],["text"," with "],["math","\\(u > 1\\)"],["text"," is not "],["math","\\(m\\)"],["text","-vanishing for "],["math","\\(m > 0\\)"]],"source":"$F_1$ modulo $8u$ with $u > 1$ is not $m$-vanishing for $m > 0$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#not-is-mvanishing-family-sum-add-mul-pro-5","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $u > 1$ be a natural number and $n = 8u$. Then for every $m > 0$ the sum-plus-product family $F_1$ over $\\mathbb{Z}/n$, $F_1(b_1, \\dots, b_l) = b_1 + \\dots + b_l + b_1 \\cdots b_l$, is not $m$-vanishing.","title":"$F_1$ modulo $8u$ with $u > 1$ is not $m$-vanishing for $m > 0$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/not-is-mvanishing-family-sum-add-mul-pro-6","kind":"node","lean":"MetaMathlibExt.not_isMVanishingFamily_sumAddMulProd_one_nine_mul","parent":23560,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(F_1\\)"],["text"," modulo "],["math","\\(9u\\)"],["text"," with "],["math","\\(u > 0\\)"],["text"," is not "],["math","\\(m\\)"],["text","-vanishing for "],["math","\\(m > 0\\)"]],"source":"$F_1$ modulo $9u$ with $u > 0$ is not $m$-vanishing for $m > 0$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#not-is-mvanishing-family-sum-add-mul-pro-6","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $u > 0$ be a natural number and $n = 9u$. Then for every $m > 0$ the sum-plus-product family $F_1$ over $\\mathbb{Z}/n$, $F_1(b_1, \\dots, b_l) = b_1 + \\dots + b_l + b_1 \\cdots b_l$, is not $m$-vanishing.","title":"$F_1$ modulo $9u$ with $u > 0$ is not $m$-vanishing for $m > 0$"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-vanishing-family-sum-add-mul-prod-neg","kind":"node","lean":"MetaMathlibExt.isVanishingFamily_sumAddMulProd_neg_one_prime_pow","parent":23560,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-vanishing-family-sum-add-mul-prod-neg","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $p$ be a prime, $k > 0$ and $n = p^k$, and let $F_{-1}$ be the family over $\\mathbb{Z}/n\\mathbb{Z}$ sending a block $(b_1, \\dots, b_l)$ to $\\sum_i b_i - \\prod_i b_i$. Then $F_{-1}$ is vanishing.","title":"The sum-minus-product family is vanishing modulo prime powers"},{"id":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/is-vanishing-family-transformation-sums","kind":"node","lean":"MetaMathlibExt.isVanishingFamily_transformation_sums","parent":23560,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric/index.html#is-vanishing-family-transformation-sums","scope":"Combinatorics/ramsey-theory/van-der-waerden-theorems/zero-sum-van-der-waerden-theorems-for-symmetric","status":"can_prove","summary":"Let $n$ be a positive integer, $d$ a natural number and $g_1, \\dots, g_d : \\mathbb{Z}/n\\mathbb{Z} \\to \\mathbb{Z}/n\\mathbb{Z}$ maps. Then the family over $\\mathbb{Z}/n\\mathbb{Z}$ sending a block $(b_1, \\dots, b_l)$ to the vector $\\left(\\sum_i g_1(b_i), \\dots, \\sum_i g_d(b_i)\\right) \\in (\\mathbb{Z}/n\\mathbb{Z})^d$ is vanishing, where a block counts as vanishing when all $d$ coordinates of this vector are zero.","title":"Vanishing of families of sums of transformed entries"},{"data_url":"scopes/Combinatorics/ramsey-theory/schur-rado-theorems.json","id":"Combinatorics/ramsey-theory/schur-rado-theorems","kind":"scope","parent":23445,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/index.html","scope":"Combinatorics/ramsey-theory/schur-rado-theorems","status":"can_prove","summary":"Monochromatic solutions of equations: Schur's theorem and Schur and modular Schur numbers, Rado's theorem and disjunctive Rado, monochromatic linear equations, monochromatic sums, sums of squares and products, and rainbow-free colourings of x\u2212y=z^k.","title":"Schur and Rado theorems"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/exists-bounded-rep","kind":"node","lean":"MathlibExt.Combinatorics.DisjunctiveRadoWanted.exists_bounded_rep","parent":23581,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/index.html#exists-bounded-rep","scope":"Combinatorics/ramsey-theory/schur-rado-theorems","status":"can_prove","summary":"Let $a$, $\\lambda$, $n$ and $N$ be natural numbers with $a \\ge 3$, $n \\ge 1$, $\\lambda \\ge a - 1$ and $N \\le \\lambda (a + n)$. Then there are natural numbers $x_0, \\dots, x_{n-1}$ and $y$, each at most $\\lambda$, such that $x_0 + \\dots + x_{n-1} + a y = N$.","title":"Bounded representations with one scaled part"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-ramsey","kind":"node","lean":"MetaMathlibExt.schur_ramsey","parent":23581,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/index.html#schur-ramsey","scope":"Combinatorics/ramsey-theory/schur-rado-theorems","status":"fully_proved","summary":"For every integer $r \\ge 1$ there is a natural number $S$ such that for every coloring of the natural numbers with $r$ colors there are $x, y, z \\in \\{1, \\dots, S\\}$, not necessarily distinct, with the same color and $x + y = z$.","title":"Schur's theorem"},{"data_url":"scopes/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers.json","id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","kind":"scope","navigation":"module","parent":23581,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","title":"Modular Schur numbers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/is-sum-free-mod","kind":"node","lean":"MetaMathlibExt.IsSumFreeMod","parent":23584,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum-free sets modulo "],["math","\\(m\\)"]],"source":"Sum-free sets modulo $m$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#is-sum-free-mod","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"fully_proved","summary":"For natural numbers $m$ and $l$, a set $A$ of integers is $l$-sum-free modulo $m$ if there are no $x_1, \\dots, x_l, y \\in A$ (repetitions allowed) with $x_1 + \\dots + x_l \\equiv y \\pmod{m}$.","title":"Sum-free sets modulo $m$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/has-sum-free-partition-mod","kind":"node","lean":"MetaMathlibExt.HasSumFreePartitionMod","parent":23584,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum-free partitions modulo "],["math","\\(m\\)"]],"source":"Sum-free partitions modulo $m$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#has-sum-free-partition-mod","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"fully_proved","summary":"For natural numbers $m$, $k$, $l$ and $n$, the range $\\{1, \\dots, n\\}$ admits a partition into $k$ classes that are $l$-sum-free modulo $m$ if there is a coloring $c$ of the natural numbers with $k$ colors such that, for each color $j$, the set of integers $a$ with $1 \\le a \\le n$ and $c(a) = j$ is $l$-sum-free modulo $m$. Classes may be empty; for $k = 0$ there is no coloring, so the property fails for every $n$,\u2026","title":"Sum-free partitions modulo $m$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number","kind":"node","lean":"MetaMathlibExt.modularSchurNumber","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"fully_proved","summary":"For natural numbers $m$, $k$ and $l$, the generalized Schur number modulo $m$, written $S_m(k, l)$, is the largest natural number $n$ such that $\\{1, \\dots, n\\}$ admits a partition into $k$ (possibly empty) classes that are each $l$-sum-free modulo $m$; it is $0$ when there is no largest such $n$.","title":"Modular Schur number"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/has-sum-free-partition-mod-zero","kind":"node","lean":"MetaMathlibExt.hasSumFreePartitionMod_zero","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#has-sum-free-partition-mod-zero","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"fully_proved","summary":"For all natural numbers $m$, $l$ and $k \\ge 1$, the empty range $\\{1, \\dots, 0\\}$ admits a partition into $k$ classes that are $l$-sum-free modulo $m$.","title":"The empty range has a sum-free partition"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/not-is-sum-free-mod-of-dvd-mem","kind":"node","lean":"MetaMathlibExt.not_isSumFreeMod_of_dvd_mem","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#not-is-sum-free-mod-of-dvd-mem","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"fully_proved","summary":"Let $m$ and $l$ be natural numbers and let $A$ be a set of integers. If $A$ contains an integer $a$ divisible by $m$, then $A$ is not $l$-sum-free modulo $m$: the sum of $l$ copies of $a$ is congruent to $a$ modulo $m$.","title":"Sets containing a multiple of the modulus are not sum-free"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/is-sum-free-mod-three-two-singleton-one","kind":"node","lean":"MetaMathlibExt.isSumFreeMod_three_two_singleton_one","parent":23584,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\{1\\}\\)"],["text"," is "],["math","\\(2\\)"],["text","-sum-free modulo "],["math","\\(3\\)"]],"source":"$\\{1\\}$ is $2$-sum-free modulo $3$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#is-sum-free-mod-three-two-singleton-one","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"fully_proved","summary":"The singleton $\\{1\\}$ is $2$-sum-free modulo $3$, since $1 + 1 \\not\\equiv 1 \\pmod{3}$.","title":"$\\{1\\}$ is $2$-sum-free modulo $3$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/not-is-sum-free-mod-singleton-one-iff","kind":"node","lean":"MetaMathlibExt.not_isSumFreeMod_singleton_one_iff","parent":23584,"reading_title":{"format":"inline-parts-v1","parts":[["text","When the singleton "],["math","\\(\\{1\\}\\)"],["text"," is sum-free"]],"source":"When the singleton $\\{1\\}$ is sum-free"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#not-is-sum-free-mod-singleton-one-iff","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $m \\ge 1$ and $l$ be natural numbers. The singleton $\\{1\\}$ is not $l$-sum-free modulo $m$ if and only if $l \\equiv 1 \\pmod{m}$.","title":"When the singleton $\\{1\\}$ is sum-free"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-eq-zero-iff","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_eq_zero_iff","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-eq-zero-iff","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $m$, $k$ and $l$ be positive integers. The generalized Schur number $S_m(k, l)$, the largest $n$ such that $\\{1, \\dots, n\\}$ admits a partition into $k$ classes that are $l$-sum-free modulo $m$, equals $0$ if and only if $l \\equiv 1 \\pmod{m}$.","title":"Vanishing of modular Schur numbers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-lt-of-mod-eq","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_lt_of_modEq","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-lt-of-mod-eq","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $m$, $k$, $l$ and $a$ be natural numbers with $k \\ge 1$, $l \\ge 1$ and $1 \\le a \\le m - 1$, and let $d = \\gcd(a, m)$. If $l \\equiv 1 \\pmod{m/d}$, then the generalized Schur number $S_m(k, l)$ modulo $m$ satisfies $S_m(k, l) < a$.","title":"Congruence upper bound for modular Schur numbers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-two-mul","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_two_mul","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-two-mul","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $k$, $l$ and $n$ be natural numbers with $k \\ge 2$, $n \\ge 2$, $l \\ge 1$ and $l \\not\\equiv 1 \\pmod{2n}$, and let $S_{2n}(k, l)$ be the generalized Schur number modulo $2n$. If $l \\equiv n + 1 \\pmod{2n}$, then $S_{2n}(k, l) = 1$; otherwise $S_{2n}(k, l) \\ge 2$.","title":"Modular Schur numbers modulo an even number"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-eq-sub-one","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_eq_sub_one","parent":23584,"reading_title":{"format":"inline-parts-v1","parts":[["text","Modular Schur numbers attaining "],["math","\\(m - 1\\)"]],"source":"Modular Schur numbers attaining $m - 1$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-eq-sub-one","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $l \\ge 2$ and $m \\ge 2$ be natural numbers such that $l \\not\\equiv 1 \\pmod{d}$ for every divisor $d \\ne 1$ of $m$, and let $k \\ge m - 1$. Then the generalized Schur number $S_m(k, l)$, the largest $n$ such that $\\{1, \\dots, n\\}$ admits a partition into $k$ classes that are $l$-sum-free modulo $m$, equals $m - 1$.","title":"Modular Schur numbers attaining $m - 1$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/le-modular-schur-number-prime-pow","kind":"node","lean":"MetaMathlibExt.le_modularSchurNumber_prime_pow","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#le-modular-schur-number-prime-pow","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $i$ and $l$ be positive integers with $l \\not\\equiv 1 \\pmod{p}$, and let $1 \\le k \\le p - 1$. Then the generalized Schur number $S_{p^i}(k, l)$, the largest $n$ such that $\\{1, \\dots, n\\}$ admits a partition into $k$ classes that are $l$-sum-free modulo $p^i$, is at least $k$.","title":"Lower bound for modular Schur numbers modulo a prime power"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-prime-pow-eq","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_prime_pow_eq","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-prime-pow-eq","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $i$ and $l$ be positive integers with $l \\not\\equiv 1 \\pmod p$, and let $k$ be a natural number with $k \\ge p^i - 1$. Then the generalized Schur number modulo $p^i$ satisfies $S_{p^i}(k, l) = p^i - 1$.","title":"Modular Schur numbers modulo a prime power for many classes"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-prime","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_prime","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-prime","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $p$ be a prime and let $l$ be a positive integer with $l \\ge p - 1$ and $l \\not\\equiv 1 \\pmod p$. Then the generalized Schur number modulo $p$ satisfies $S_p(k, l) = k$ for every natural number $k$ with $1 \\le k \\le p - 1$, and $S_p(k, l) = p - 1$ for every natural number $k \\ge p - 1$.","title":"Modular Schur numbers modulo a prime"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/is-sum-free-mod-singleton-prime-pow","kind":"node","lean":"MetaMathlibExt.isSumFreeMod_singleton_prime_pow","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#is-sum-free-mod-singleton-prime-pow","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $i$ and $l$ be positive integers with $l \\not\\equiv 1 \\pmod{p}$, and let $a$ be a natural number not divisible by $p^i$. Then the singleton $\\{a\\}$ is $l$-sum-free modulo $p^i$.","title":"Singletons are sum-free modulo a prime power"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/mod-eq-of-is-sum-free-mod-pair","kind":"node","lean":"MetaMathlibExt.modEq_of_isSumFreeMod_pair","parent":23584,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elements of a sum-free pair are congruent modulo "],["math","\\(p\\)"]],"source":"Elements of a sum-free pair are congruent modulo $p$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#mod-eq-of-is-sum-free-mod-pair","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $p$ be a prime, let $i$ be a positive integer, and let $l$ be a natural number with $l \\ge p^i - 1$ and $l \\not\\equiv 1 \\pmod{p}$. If $a$ and $b$ are natural numbers such that $\\{a, b\\}$ is $l$-sum-free modulo $p^i$, then $a \\equiv b \\pmod{p}$.","title":"Elements of a sum-free pair are congruent modulo $p$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-two","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_two","parent":23584,"reading_title":{"format":"inline-parts-v1","parts":[["text","Modular Schur numbers modulo "],["math","\\(2\\)"]],"source":"Modular Schur numbers modulo $2$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-two","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"For all positive integers $k$ and $l$, the modular Schur number $S_2(k, l)$ equals $0$ when $l$ is odd and equals $1$ when $l$ is even.","title":"Modular Schur numbers modulo $2$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-three","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_three","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-three","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $k$ and $l$ be positive integers and let $S_3(k, l)$ be the generalized Schur number modulo $3$. If $l \\equiv 1 \\pmod 3$, then $S_3(k, l) = 0$. Otherwise $S_3(1, l) = 1$, and $S_3(k, l) = 2$ for every $k \\ge 2$.","title":"Modular Schur numbers modulo 3"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/linear-congruence-solutions","kind":"node","lean":"MetaMathlibExt.linear_congruence_solutions","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#linear-congruence-solutions","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $a$ and $b$ be integers, let $m$ be a positive integer, and let $d = \\gcd(a, m)$. The congruence $a x \\equiv b \\pmod{m}$ has an integer solution if and only if $d \\mid b$. If $d \\mid b$, then exactly $d$ of the numbers $x \\in \\{0, 1, \\dots, m - 1\\}$ are solutions. Moreover, if $x_0$ is a solution, then an integer $x$ is a solution if and only if $x \\equiv x_0 + u \\cdot (m / d) \\pmod{m}$ for some integer $u$\u2026","title":"Solutions of a linear congruence"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/not-is-sum-free-mod-singleton-iff","kind":"node","lean":"MetaMathlibExt.not_isSumFreeMod_singleton_iff","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#not-is-sum-free-mod-singleton-iff","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $a$, $l$ and $m$ be natural numbers with $l \\ge 1$, $m \\ge 1$ and $m \\nmid a$, and let $d = \\gcd(a, m)$. If $d = 1$, then $\\{a\\}$ is not $l$-sum-free modulo $m$ if and only if $l \\equiv 1 \\pmod{m}$. If $d > 1$, then $\\{a\\}$ is not $l$-sum-free modulo $m$ if and only if $l \\equiv 1 \\pmod{m/d}$.","title":"When a singleton set is sum-free"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-one-eq-one-iff","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_one_eq_one_iff","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-one-eq-one-iff","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $l$ and $m$ be natural numbers with $l \\ge 2$, $m \\ge 2$ and $l \\not\\equiv 1 \\pmod m$. Then the generalized Schur number $S_m(1, l)$ modulo $m$ equals $1$ if and only if $l = 2$ or $2l \\ge m + 1$.","title":"When the one-class modular Schur number equals one"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-four","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_four","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-four","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $k$ and $l$ be positive integers and let $S_4(k, l)$ be the generalized Schur number modulo $4$. If $l \\equiv 1 \\pmod 4$, then $S_4(k, l) = 0$. Otherwise, if $k = 1$, or if $k \\ge 2$ and $l \\equiv 3 \\pmod 4$, then $S_4(k, l) = 1$; and if $k \\ge 2$ and $l \\equiv 0$ or $2 \\pmod 4$, then $S_4(k, l) = 3$.","title":"Modular Schur numbers modulo 4"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-six","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_six","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-six","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $k$ and $l$ be positive integers and let $S_6(k, l)$ be the generalized Schur number modulo $6$. If $l \\equiv 1 \\pmod 6$, then $S_6(k, l) = 0$. Otherwise $S_6(k, l) = 1$ if $k = 1$ and $l \\ne 3$, or $k \\ge 2$ and $l \\equiv 4 \\pmod 6$; $S_6(k, l) = 2$ if $k = 1$ and $l = 3$, or $k \\ge 2$ and $l \\equiv 3$ or $5 \\pmod 6$; $S_6(k, l) = 3$ if $k = 2$, $l \\equiv 0$ or $2 \\pmod 6$, and $l \\ge 6$; $S_6(k, l) = 4$ if\u2026","title":"Modular Schur numbers modulo 6"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-five","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_five","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-five","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $k$ and $l$ be positive integers and let $S_5(k, l)$ be the generalized Schur number modulo $5$. If $l \\equiv 1 \\pmod 5$, then $S_5(k, l) = 0$. Otherwise $S_5(1, l) = 1$; if $k \\in \\{2, 3\\}$ and $l \\ne 2$, then $S_5(k, l) = k$; and if $k \\in \\{2, 3\\}$ and $l = 2$, or if $k \\ge 4$, then $S_5(k, l) = 4$.","title":"Modular Schur numbers modulo 5"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/modular-schur-number-seven","kind":"node","lean":"MetaMathlibExt.modularSchurNumber_seven","parent":23584,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers/index.html#modular-schur-number-seven","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/modular-schur-numbers","status":"can_prove","summary":"Let $k$ and $l$ be positive integers and let $S_7(k, l)$ be the generalized Schur number modulo $7$. If $l \\equiv 1 \\pmod 7$, then $S_7(k, l) = 0$. Otherwise $S_7(k, l) = 1$ if $k = 1$ and $l \\ne 3$; $S_7(k, l) = 2$ if $k = 1$ and $l = 3$, or $k = 2$ and $l \\ge 4$; $S_7(k, l) = 3$ if $k = 2$ and $l \\in \\{2, 3\\}$, or $k = 3$ and $l \\ge 5$; $S_7(k, l) = 4$ if $k = 4$ and $l \\ge 5$; $S_7(k, l) = 5$ if $k = 3$ and\u2026","title":"Modular Schur numbers modulo 7"},{"data_url":"scopes/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations.json","id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","kind":"scope","navigation":"module","parent":23581,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"can_prove","title":"Monochromatic solutions of linear equations"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/has-canceling-partition","kind":"node","lean":"MetaMathlibExt.HasCancelingPartition","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#has-canceling-partition","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"fully_proved","summary":"Let $a_1, \\dots, a_k$ be integers, the coefficients of the linear equation $a_1x_1 + \\cdots + a_kx_k = 0$. The equation has a canceling partition when its indices split into pairs of distinct indices $i, j$ with $a_i + a_j = 0$; precisely, when there is a permutation $\\sigma$ of the indices $1, \\dots, k$ such that for every $i$ we have $\\sigma(\\sigma(i)) = i$, $\\sigma(i) \\ne i$ and $a_i + a_{\\sigma(i)} = 0$.","title":"Canceling partition"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/has-canceling-partition-one-two-neg-one","kind":"node","lean":"MetaMathlibExt.hasCancelingPartition_one_two_neg_one_neg_two","parent":23610,"reading_title":{"format":"inline-parts-v1","parts":[["text","A canceling partition for "],["math","\\(x_1 + 2x_2 - x_3 - 2x_4 = 0\\)"]],"source":"A canceling partition for $x_1 + 2x_2 - x_3 - 2x_4 = 0$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#has-canceling-partition-one-two-neg-one","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"fully_proved","summary":"The equation $x_1 + 2x_2 - x_3 - 2x_4 = 0$, with coefficient vector $(1, 2, -1, -2)$, has a canceling partition.","title":"A canceling partition for $x_1 + 2x_2 - x_3 - 2x_4 = 0$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/not-has-canceling-partition-one-one","kind":"node","lean":"MetaMathlibExt.not_hasCancelingPartition_one_one","parent":23610,"reading_title":{"format":"inline-parts-v1","parts":[["text","The equation "],["math","\\(x_1 + x_2 = 0\\)"],["text"," has no canceling partition"]],"source":"The equation $x_1 + x_2 = 0$ has no canceling partition"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#not-has-canceling-partition-one-one","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"fully_proved","summary":"The equation $x_1 + x_2 = 0$, with coefficient vector $(1, 1)$, has no canceling partition.","title":"The equation $x_1 + x_2 = 0$ has no canceling partition"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/is-uncommon-over-finite-group","kind":"node","lean":"MetaMathlibExt.IsUncommonOverFiniteGroup","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#is-uncommon-over-finite-group","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"fully_proved","summary":"Let $G$ be a finite abelian group, let $a_1, \\dots, a_k$ be integers, let $r$ be a natural number, and let $S \\subseteq G^k$ be the set of solutions of $a_1x_1 + \\cdots + a_kx_k = 0$. The equation is $r$-uncommon over $G$ when there are functions $w_1, \\dots, w_r \\colon G \\to \\mathbb{R}$ with $w_c(x) \\ge 0$ for all $c$ and $x$, and $w_1(x) + \\cdots + w_r(x) = 1$ for every $x \\in G$, such that","title":"Uncommon equations over a finite abelian group"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/not-is-uncommon-over-finite-group-one","kind":"node","lean":"MetaMathlibExt.not_isUncommonOverFiniteGroup_one","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#not-is-uncommon-over-finite-group-one","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"fully_proved","summary":"For every finite abelian group $G$, every $k \\ge 0$ and all integers $a_1, \\dots, a_k$, the equation $a_1 x_1 + \\dots + a_k x_k = 0$ is not $1$-uncommon over $G$.","title":"No equation is 1-uncommon over a finite abelian group"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/is-uncommon-over-integers","kind":"node","lean":"MetaMathlibExt.IsUncommonOverIntegers","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#is-uncommon-over-integers","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"fully_proved","summary":"Let $a_1, \\dots, a_k$ be integers and let $r$ be a natural number. For each $n$, write $[n]$ for the set of integers $1, \\dots, n$ and $S_n$ for the set of solutions $(x_1, \\dots, x_k) \\in [n]^k$ of $a_1x_1 + \\cdots + a_kx_k = 0$; for a colouring $\\chi$ of the integers with $r$ colours, let $M_\\chi(n)$ be the number of solutions in $S_n$ whose entries all have the same colour. The equation is $r$-uncommon over the\u2026","title":"Uncommon equations over the integers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/not-is-uncommon-over-integers-one","kind":"node","lean":"MetaMathlibExt.not_isUncommonOverIntegers_one","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#not-is-uncommon-over-integers-one","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"fully_proved","summary":"For every $k \\ge 0$ and all integers $a_1, \\dots, a_k$, the equation $a_1 x_1 + \\dots + a_k x_k = 0$ is not $1$-uncommon over the integers.","title":"No equation is 1-uncommon over the integers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/is-uncommon-over-finite-group-of-odd","kind":"node","lean":"MetaMathlibExt.isUncommonOverFiniteGroup_of_odd","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#is-uncommon-over-finite-group-of-odd","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"can_prove","summary":"Let $G$ be a nontrivial finite abelian group, let $m \\ge 1$, and let $a_1, \\dots, a_{2m+1}$ be integers each coprime to the order $|G|$. Then for every $r \\ge 3$ the equation $a_1x_1 + \\cdots + a_{2m+1}x_{2m+1} = 0$ is $r$-uncommon over $G$.","title":"Uncommonness of odd-length equations over finite abelian groups"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/is-uncommon-over-integers-of-odd","kind":"node","lean":"MetaMathlibExt.isUncommonOverIntegers_of_odd","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#is-uncommon-over-integers-of-odd","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"can_prove","summary":"Let $m \\ge 1$ and let $a_1, \\dots, a_{2m+1}$ be nonzero integers, at least one negative and at least one positive. Then for every $r \\ge 3$ the equation $a_1x_1 + \\cdots + a_{2m+1}x_{2m+1} = 0$ is $r$-uncommon over the integers.","title":"Uncommonness of odd-length equations over the integers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/is-uncommon-over-integers-of-two","kind":"node","lean":"MetaMathlibExt.isUncommonOverIntegers_of_two","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#is-uncommon-over-integers-of-two","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"can_prove","summary":"Let $k \\ge 3$ and let $a_1, \\dots, a_k$ be nonzero integers. If the equation $a_1x_1 + \\cdots + a_kx_k = 0$ is $2$-uncommon over the integers, then it is $r$-uncommon over the integers for every $r \\ge 3$.","title":"Two-colour uncommonness over the integers extends to more colours"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/is-uncommon-over-integers-of-not-has-can","kind":"node","lean":"MetaMathlibExt.isUncommonOverIntegers_of_not_hasCancelingPartition","parent":23610,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#is-uncommon-over-integers-of-not-has-can","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"can_prove","summary":"Let $m \\ge 2$ and let $a_1, \\dots, a_{2m}$ be nonzero integers, at least one negative and at least one positive, such that the equation $a_1x_1 + \\cdots + a_{2m}x_{2m} = 0$ has no canceling partition. Then this equation is $r$-uncommon over the integers for every $r \\ge 2$.","title":"Uncommonness of even-length equations without a canceling partition"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/is-uncommon-over-integers-one-two-neg-on","kind":"node","lean":"MetaMathlibExt.isUncommonOverIntegers_one_two_neg_one_neg_two","parent":23610,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uncommonness of "],["math","\\(x + 2y - z - 2w = 0\\)"],["text"," over the integers"]],"source":"Uncommonness of $x + 2y - z - 2w = 0$ over the integers"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations/index.html#is-uncommon-over-integers-one-two-neg-on","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-solutions-of-linear-equations","status":"can_prove","summary":"For every $r \\ge 2$, the equation $x + 2y - z - 2w = 0$ is $r$-uncommon over the integers.","title":"Uncommonness of $x + 2y - z - 2w = 0$ over the integers"},{"data_url":"scopes/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums.json","id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums","kind":"scope","navigation":"module","parent":23581,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums/index.html","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums","status":"can_prove","title":"Monochromatic sums"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums/monochromatic-sums-linear-count","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticSumsLinearErdosProblem484Wanted.monochromaticSumsLinearCount","parent":23623,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums/index.html#monochromatic-sums-linear-count","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums","status":"fully_proved","summary":"This is the statement that there is a real constant $c > 0$ such that for every integer $k \\ge 1$ there is an $N_0 \\in \\mathbb{N}$ with the following property: for every $N \\ge N_0$ and every colouring $f : \\mathbb{N} \\to \\{0, \\dots, k-1\\}$, at least $cN$ of the integers $n \\in \\{1, \\dots, N\\}$ can be written as $n = a + b$ with $a, b \\in \\{1, \\dots, N\\}$, $a \\ne b$ and $f(a) = f(b)$.","title":"Erd\u0151s Problem 484: linearly many monochromatic sums"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums/monochromatic-sums-linear-count-holds","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticSumsLinearErdosProblem484Wanted.monochromaticSumsLinearCount_holds","parent":23623,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums/index.html#monochromatic-sums-linear-count-holds","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums","status":"can_prove","summary":"There is a real constant $c > 0$ such that for every integer $k \\ge 1$ there is an $N_0 \\in \\mathbb{N}$ such that for every $N \\ge N_0$ and every colouring $f : \\mathbb{N} \\to \\{0, \\dots, k-1\\}$, at least $cN$ of the integers $n \\in \\{1, \\dots, N\\}$ can be written as $n = a + b$ with $a, b \\in \\{1, \\dots, N\\}$, $a \\ne b$ and $f(a) = f(b)$.","title":"Erd\u0151s\u2013S\u00e1rk\u00f6zy\u2013S\u00f3s theorem on monochromatic sums (Erd\u0151s Problem 484)"},{"data_url":"scopes/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers.json","id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers","kind":"scope","navigation":"module","parent":23581,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/index.html","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers","status":"can_prove","title":"Monochromatic sums equal to perfect powers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/monochromatic-square-sum","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticSquareSumErdosProblem439Wanted.monochromaticSquareSum","parent":23626,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/index.html#monochromatic-square-sum","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers","status":"fully_proved","summary":"The statement that for every colouring of the positive integers with finitely many colours there are distinct positive integers $x$ and $y$ of the same colour such that $x + y$ is a perfect square.","title":"Erd\u0151s Problem 439 for squares"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/monochromatic-power-sum","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticSquareSumErdosProblem439Wanted.monochromaticPowerSum","parent":23626,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s Problem 439 for "],["math","\\(k\\)"],["text","-th powers"]],"source":"Erd\u0151s Problem 439 for $k$-th powers"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/index.html#monochromatic-power-sum","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers","status":"fully_proved","summary":"The statement that for every integer $k \\ge 2$ and every colouring of the positive integers with finitely many colours there are distinct positive integers $x$ and $y$ of the same colour such that $x + y = z^k$ for some natural number $z$.","title":"Erd\u0151s Problem 439 for $k$-th powers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/monochromatic-square-sum-holds","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticSquareSumErdosProblem439Wanted.monochromaticSquareSum_holds","parent":23626,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/index.html#monochromatic-square-sum-holds","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers","status":"can_prove","summary":"For every colouring of the positive integers with finitely many colours there are distinct positive integers $x$ and $y$ of the same colour such that $x + y$ is a perfect square.","title":"Erd\u0151s Problem 439 for squares is true"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/monochromatic-power-sum-holds","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticSquareSumErdosProblem439Wanted.monochromaticPowerSum_holds","parent":23626,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s Problem 439 for "],["math","\\(k\\)"],["text","-th powers is true"]],"source":"Erd\u0151s Problem 439 for $k$-th powers is true"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers/index.html#monochromatic-power-sum-holds","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-equal-to-perfect-powers","status":"can_prove","summary":"For every integer $k \\ge 2$ and every colouring of the positive integers with finitely many colours there are distinct positive integers $x$ and $y$ of the same colour such that $x + y = z^k$ for some natural number $z$.","title":"Erd\u0151s Problem 439 for $k$-th powers is true"},{"data_url":"scopes/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products.json","id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products","kind":"scope","navigation":"module","parent":23581,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products/index.html","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products","status":"can_prove","title":"Monochromatic sums of products"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products/monochromatic-sums-of-products","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticSumsOfProductsErdosProblem1198Wanted.monochromaticSumsOfProducts","parent":23631,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products/index.html#monochromatic-sums-of-products","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products","status":"fully_proved","summary":"This is the statement that for every colouring $c : \\mathbb{N} \\to \\{0, 1\\}$ there are a strictly increasing sequence $0 < a_0 < a_1 < \\cdots$ of natural numbers and a colour $\\gamma \\in \\{0, 1\\}$ such that $c\\bigl(\\sum_{j < k} \\prod_{i \\in S_j} a_i\\bigr) = \\gamma$ for every $k \\ge 1$ and all pairwise disjoint nonempty finite sets $S_0, \\dots, S_{k-1} \\subseteq \\mathbb{N}$, except when $k = 1$ and $S_0$ has exactly\u2026","title":"Erd\u0151s Problem 1198: monochromatic sums of products"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products/monochromatic-sums-of-products-refuted","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticSumsOfProductsErdosProblem1198Wanted.monochromaticSumsOfProducts_refuted","parent":23631,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products/index.html#monochromatic-sums-of-products-refuted","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/monochromatic-sums-of-products","status":"can_prove","summary":"There is a colouring $c : \\mathbb{N} \\to \\{0, 1\\}$ such that for every strictly increasing sequence $0 < a_0 < a_1 < \\cdots$ of natural numbers, the numbers $\\sum_{j < k} \\prod_{i \\in S_j} a_i$, taken over all $k \\ge 1$ and all pairwise disjoint nonempty finite sets $S_0, \\dots, S_{k-1} \\subseteq \\mathbb{N}$ other than those with $k = 1$ and $|S_0| = 1$, do not all have the same colour.","title":"Smith's counterexample to Erd\u0151s Problem 1198"},{"data_url":"scopes/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k.json","id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","kind":"scope","navigation":"module","parent":23581,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","title":"Rainbow-free colorings for x \u2212 y = z^k"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/exists-card-color-class-eq-one-of-rainbo","kind":"node","lean":"MetaMathlibExt.exists_card_colorClass_eq_one_of_rainbowFree_powDiff","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#exists-card-color-class-eq-one-of-rainbo","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime and $k$ a natural number, and let $c$ be a coloring of $\\mathbb{Z}/p\\mathbb{Z}$ with three colors that uses every color and is rainbow-free for $x - y = z^k$: no $x, y, z \\in \\mathbb{Z}/p\\mathbb{Z}$ with $x - y = z^k$ have $c(x), c(y), c(z)$ pairwise distinct. Then some color is taken by exactly one element of $\\mathbb{Z}/p\\mathbb{Z}$.","title":"Rainbow-free 3-colorings modulo a prime have a singleton color class"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/eq-zero-of-color-eq-color-zero-of-rainbo","kind":"node","lean":"MetaMathlibExt.eq_zero_of_color_eq_color_zero_of_rainbowFree_powDiff","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#eq-zero-of-color-eq-color-zero-of-rainbo","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime and $k$ a natural number, and let $c$ be a coloring of $\\mathbb{Z}/p\\mathbb{Z}$ with three colors that uses every color and is rainbow-free for $x - y = z^k$: no $x, y, z \\in \\mathbb{Z}/p\\mathbb{Z}$ with $x - y = z^k$ have $c(x), c(y), c(z)$ pairwise distinct. If $a \\in \\mathbb{Z}/p\\mathbb{Z}$ satisfies $c(a) = c(0)$, then $a = 0$.","title":"Zero has a color class of its own"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/rainbow-free-pow-diff-lift-of-zmod","kind":"node","lean":"MetaMathlibExt.rainbowFree_powDiff_lift_of_zmod","parent":23634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rainbow-free colorings of "],["math","\\(\\mathbb{Z}/n\\mathbb{Z}\\)"],["text"," lift to the positive integers"]],"source":"Rainbow-free colorings of $\\mathbb{Z}/n\\mathbb{Z}$ lift to the positive integers"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#rainbow-free-pow-diff-lift-of-zmod","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers and let $c$ be a coloring of $\\mathbb{Z}/n\\mathbb{Z}$ with three colors that is rainbow-free for $x - y = z^k$: there are no $x, y, z \\in \\mathbb{Z}/n\\mathbb{Z}$ with $x - y = z^k$ whose colors $c(x)$, $c(y)$, $c(z)$ are pairwise distinct. Then the coloring $i \\mapsto c(i \\bmod n)$ of the positive integers is rainbow-free for $x - y = z^k$: for all positive integers $x, y, z$ with\u2026","title":"Rainbow-free colorings of $\\mathbb{Z}/n\\mathbb{Z}$ lift to the positive integers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/exists-coprime-same-colors-of-coprime-di","kind":"node","lean":"MetaMathlibExt.exists_coprime_same_colors_of_coprime_dist","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#exists-coprime-same-colors-of-coprime-di","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $n$ be a natural number and let $c$ be a coloring of $\\mathbb{Z}/n\\mathbb{Z}$ with three colors, used to color every positive integer by its residue modulo $n$. If $i_1, i_2$ are positive integers with $\\gcd(|i_1 - i_2|, n) = 1$, then there are coprime positive integers $j_1, j_2$ with $|j_1 - j_2| = |i_1 - i_2|$ such that $j_1$ has the color of $i_1$ and $j_2$ has the color of $i_2$.","title":"Coprime representatives at a coprime distance"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/color-neg-eq-of-rainbow-free-pow-diff","kind":"node","lean":"MetaMathlibExt.color_neg_eq_of_rainbowFree_powDiff","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#color-neg-eq-of-rainbow-free-pow-diff","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime and $k$ a natural number, and let $c$ be a coloring of $\\mathbb{Z}/p\\mathbb{Z}$ with three colors that uses every color and is rainbow-free for $x - y = z^k$: no $x, y, z \\in \\mathbb{Z}/p\\mathbb{Z}$ with $x - y = z^k$ have $c(x), c(y), c(z)$ pairwise distinct. Then $c(-a) = c(a)$ for every $a \\in \\mathbb{Z}/p\\mathbb{Z}$.","title":"Rainbow-free colorings modulo a prime are symmetric under negation"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/pow-map-component-unique-cycle","kind":"node","lean":"MetaMathlibExt.powMap_component_unique_cycle","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#pow-map-component-unique-cycle","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime and $k$ a natural number, and let $G^k_p$ be the digraph on $\\mathbb{Z}/p\\mathbb{Z}$ with edges $a \\to a^k$, so that $m$ steps lead from $a$ to $a^{k^m}$. Call $a$ a cycle vertex if $a^{k^r} = a$ for some $r > 0$, and say that $a$ and $b$ lie in the same component if $a^{k^m} = b^{k^n}$ for some $m, n \\ge 0$. Then every $a$ has a cycle vertex $a^{k^m}$ among its iterates, and if two cycle\u2026","title":"Each component of the power digraph has exactly one cycle"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/card-pow-map-components-eq-two-iff-of-ev","kind":"node","lean":"MetaMathlibExt.card_powMap_components_eq_two_iff_of_even","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#card-pow-map-components-eq-two-iff-of-ev","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime and $k > 0$ an even natural number, and let $G^k_p$ be the directed graph on $\\mathbb{Z}/p\\mathbb{Z}$ with an edge $a \\to a^k$ from every vertex $a$. Its (weak) components are the classes of the equivalence relation \"$a^{k^m} = b^{k^n}$ for some natural numbers $m, n$\". Then $G^k_p$ has exactly two components if and only if every prime factor of $p - 1$ divides $k$.","title":"Power digraphs modulo a prime with two components"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/card-pow-map-components-eq-three-iff-of","kind":"node","lean":"MetaMathlibExt.card_powMap_components_eq_three_iff_of_odd","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#card-pow-map-components-eq-three-iff-of","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime and $k \\ge 3$ an odd natural number, and let $G^k_p$ be the directed graph on $\\mathbb{Z}/p\\mathbb{Z}$ with an edge $a \\to a^k$ from every vertex $a$. Its (weak) components are the classes of the equivalence relation \"$a^{k^m} = b^{k^n}$ for some natural numbers $m, n$\". Then $G^k_p$ has exactly three components if and only if $p - 1 = 2m$ for some natural number $m$ every prime factor of which\u2026","title":"Power digraphs modulo a prime with three components"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/exists-forall-gcd-dvd-eq-add-mul","kind":"node","lean":"MetaMathlibExt.exists_forall_gcd_dvd_eq_add_mul","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#exists-forall-gcd-dvd-eq-add-mul","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $i$ and $j$ be positive integers. There is an $n_0$ such that every integer $m > n_0$ divisible by $\\gcd(i, j)$ can be written as $m = ui + vj$ with nonnegative integers $u$ and $v$.","title":"Large multiples of the gcd are nonnegative combinations"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/exists-rainbow-pow-diff-of-upper-density","kind":"node","lean":"MetaMathlibExt.exists_rainbow_powDiff_of_upperDensity","parent":23634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper density threshold for rainbow solutions of "],["math","\\(x - y = z^k\\)"]],"source":"Upper density threshold for rainbow solutions of $x - y = z^k$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#exists-rainbow-pow-diff-of-upper-density","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $k \\ge 2$ be an integer, let $s = \\lfloor k / 2 \\rfloor$ and $\\delta = (4^s - 1) / (3 \\cdot 4^s)$, and let $c$ be a coloring of the positive integers by three colors. Suppose that for each color $e$ there is an $\\varepsilon > 0$ such that, for arbitrarily large $N$, more than $(\\delta + \\varepsilon) N$ of the integers $x \\in \\{1, \\dots, N\\}$ have $c(x) = e$; that is, every color class has upper density greater\u2026","title":"Upper density threshold for rainbow solutions of $x - y = z^k$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/exists-card-color-class-le-div-min-fac-o","kind":"node","lean":"MetaMathlibExt.exists_card_colorClass_le_div_minFac_of_rainbowFree_powDiff","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#exists-card-color-class-le-div-min-fac-o","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $n$ be an odd natural number, let $r_1$ be its smallest prime factor (with $r_1 = 1$ when $n = 1$), let $k$ be a natural number, and let $c$ be a coloring of $\\mathbb{Z}/n\\mathbb{Z}$ with three colors that uses every color and is rainbow-free for $x - y = z^k$: no $x, y, z \\in \\mathbb{Z}/n\\mathbb{Z}$ with $x - y = z^k$ have $c(x), c(y), c(z)$ pairwise distinct. Then some color class has at most $n / r_1$\u2026","title":"Small color classes in rainbow-free 3-colorings modulo an odd integer"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/rainbow-free-pow-diff-iff","kind":"node","lean":"MetaMathlibExt.rainbowFree_powDiff_iff","parent":23634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rainbow-free colorings of "],["math","\\(\\mathbb{Z}/p\\mathbb{Z}\\)"],["text"," for "],["math","\\(x - y = z^k\\)"]],"source":"Rainbow-free colorings of $\\mathbb{Z}/p\\mathbb{Z}$ for $x - y = z^k$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#rainbow-free-pow-diff-iff","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime, $k$ a natural number, and $c$ a coloring of $\\mathbb{Z}/p\\mathbb{Z}$ by three colors that uses all three. Then $c$ has no solution of $x - y = z^k$ in $\\mathbb{Z}/p\\mathbb{Z}$ with $c(x)$, $c(y)$, $c(z)$ pairwise distinct if and only if the following three conditions hold: $0$ is the only element of its color class; $c(a) = c(b)$ whenever $a^{k^m} = b^{k^n}$ for some $m, n \\ge 0$, that is, every\u2026","title":"Rainbow-free colorings of $\\mathbb{Z}/p\\mathbb{Z}$ for $x - y = z^k$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/rainbow-number-pow-diff-of-even","kind":"node","lean":"MetaMathlibExt.rainbowNumber_powDiff_of_even","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#rainbow-number-pow-diff-of-even","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime and $k$ an even natural number. A coloring of $\\mathbb{Z}/p\\mathbb{Z}$ with $r$ colors is exact if it uses every color, and a solution of $x - y = z^k$ in $\\mathbb{Z}/p\\mathbb{Z}$ is rainbow if $x$, $y$ and $z$ receive pairwise distinct colors. Then the least $r \\ge 3$ such that every exact coloring of $\\mathbb{Z}/p\\mathbb{Z}$ with $r$ colors has a rainbow solution of $x - y = z^k$ is $3$ if\u2026","title":"Rainbow number of the power-difference equation modulo a prime, even exponent"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/rainbow-number-pow-diff-of-odd","kind":"node","lean":"MetaMathlibExt.rainbowNumber_powDiff_of_odd","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#rainbow-number-pow-diff-of-odd","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be an odd prime and $k \\ge 3$ an odd natural number. A coloring of $\\mathbb{Z}/p\\mathbb{Z}$ with $r$ colors is exact if it uses every color, and a solution of $x - y = z^k$ in $\\mathbb{Z}/p\\mathbb{Z}$ is rainbow if $x$, $y$ and $z$ receive pairwise distinct colors. Then the least $r \\ge 3$ such that every exact coloring of $\\mathbb{Z}/p\\mathbb{Z}$ with $r$ colors has a rainbow solution of $x - y = z^k$ is\u2026","title":"Rainbow number of the power-difference equation modulo a prime, odd exponent"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/color-one-dominant-of-rainbow-free-pow-d","kind":"node","lean":"MetaMathlibExt.color_one_dominant_of_rainbowFree_powDiff","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#color-one-dominant-of-rainbow-free-pow-d","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $r$ and $k$ be natural numbers and let $c$ be a coloring of the positive integers with $r$ colors that uses every color on some positive integer and is rainbow-free for $x - y = z^k$: no positive integers $x, y, z$ with $x = y + z^k$ have $c(x), c(y), c(z)$ pairwise distinct. Then for every positive integer $i$ and every natural number $\\ell$, if $c$ takes exactly two colors on $i, i + 1, \\dots, i + \\ell - 1$,\u2026","title":"The color of one dominates bichromatic strings"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/monochromatic-shift-of-rainbow-free-pow","kind":"node","lean":"MetaMathlibExt.monochromatic_shift_of_rainbowFree_powDiff","parent":23634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shifting monochromatic strings by "],["math","\\(k\\)"],["text","-th powers"]],"source":"Shifting monochromatic strings by $k$-th powers"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#monochromatic-shift-of-rainbow-free-pow","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $k$ be a natural number and let $c$ be a coloring of the positive integers by three distinct colors $R$, $B$, $G$ with no rainbow solution of $x - y = z^k$: there are no positive integers $x, y, z$ with $x = y + z^k$ and $c(x)$, $c(y)$, $c(z)$ pairwise distinct. Assume that $R$ is dominant: for all positive integers $i$ and all $\\ell \\ge 0$, if $c$ takes exactly two colors on\u2026","title":"Shifting monochromatic strings by $k$-th powers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/exists-small-gap-of-limsup-count","kind":"node","lean":"MetaMathlibExt.exists_small_gap_of_limsup_count","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#exists-small-gap-of-limsup-count","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $S$ be a set of natural numbers, let $n_0 \\ge 2$ be an integer, and write $S(n)$ for the number of elements of $S \\cap \\{1, \\dots, n\\}$. If $\\limsup_{n \\to \\infty} (S(n) - n / n_0) = \\infty$, that is, for every real $M$ there are arbitrarily large $n$ with $S(n) - n / n_0 > M$, then there is an integer $d$ with $1 \\le d \\le n_0 - 1$ such that $j$ and $j + d$ both lie in $S$ for arbitrarily large $j$.","title":"Zhan's small-gap lemma"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/nondominant-string-length-bounded","kind":"node","lean":"MetaMathlibExt.nondominant_string_length_bounded","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#nondominant-string-length-bounded","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $k$ be a natural number, let $n_0 \\ge 2$ be an integer, and let $c$ be a coloring of the positive integers by three colors with no rainbow solution of $x - y = z^k$: there are no positive integers $x, y, z$ with $x = y + z^k$ and $c(x)$, $c(y)$, $c(z)$ pairwise distinct. For a color $e$ write $N_e(n)$ for the number of $x \\in \\{1, \\dots, n\\}$ with $c(x) = e$, and assume that for every real $M$ there are\u2026","title":"Non-dominant monochromatic strings have bounded length"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/not-both-a-property-of-coprime","kind":"node","lean":"MetaMathlibExt.not_both_aProperty_of_coprime","parent":23634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coprime numbers of a non-dominant color cannot both have the "],["math","\\(A\\)"],["text","-property"]],"source":"Coprime numbers of a non-dominant color cannot both have the $A$-property"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#not-both-a-property-of-coprime","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $k$ be a natural number, let $n_0 \\ge 2$ be an integer, and let $c$ be a coloring of the positive integers by three distinct colors $R$, $B$, $G$ with no rainbow solution of $x - y = z^k$: there are no positive integers $x, y, z$ with $x = y + z^k$ and $c(x)$, $c(y)$, $c(z)$ pairwise distinct. For a color $e$ write $N_e(n)$ for the number of $x \\in \\{1, \\dots, n\\}$ with $c(x) = e$, and assume that for every\u2026","title":"Coprime numbers of a non-dominant color cannot both have the $A$-property"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/exists-rainbow-pow-diff-of-coprime-pairs","kind":"node","lean":"MetaMathlibExt.exists_rainbow_powDiff_of_coprime_pairs","parent":23634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coprime pairs in both non-dominant colors force a rainbow solution of "],["math","\\(x - y = z^k\\)"]],"source":"Coprime pairs in both non-dominant colors force a rainbow solution of $x - y = z^k$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#exists-rainbow-pow-diff-of-coprime-pairs","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $k \\ge 2$ and $n_0 \\ge 2$ be integers, and let $c$ be a coloring of the positive integers by three distinct colors $R$, $B$, $G$. For a color $e$ write $N_e(n)$ for the number of $x \\in \\{1, \\dots, n\\}$ with $c(x) = e$. Assume that for every real $M$ there are arbitrarily large $n$ with $N_e(n) - n / n_0 > M$ for all three colors $e$, that is,\u2026","title":"Coprime pairs in both non-dominant colors force a rainbow solution of $x - y = z^k$"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/color-dominant-pow-strings-of-rainbow-fr","kind":"node","lean":"MetaMathlibExt.color_dominant_powStrings_of_rainbowFree_zmod","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#color-dominant-pow-strings-of-rainbow-fr","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be a prime and $k$ a natural number, and let $c$ be a coloring of $\\mathbb{Z}/p\\mathbb{Z}$ with three colors that uses every color and is rainbow-free for $x - y = z^k$: no $x, y, z \\in \\mathbb{Z}/p\\mathbb{Z}$ with $x - y = z^k$ have $c(x), c(y), c(z)$ pairwise distinct. Let $a \\ne 0$ and $i$ be elements of $\\mathbb{Z}/p\\mathbb{Z}$ and let $\\ell$ be a natural number. If the elements $(i + t)a^k$ for\u2026","title":"Dominant colors on power-step strings modulo a prime"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/component-monochromatic-of-rainbow-free","kind":"node","lean":"MetaMathlibExt.component_monochromatic_of_rainbowFree","parent":23634,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#component-monochromatic-of-rainbow-free","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $n$ and $t$ be natural numbers, let $f$ be a map from $\\mathbb{Z}/n\\mathbb{Z}$ to itself, and let $c$ be a coloring of $\\mathbb{Z}/n\\mathbb{Z}$ with $t$ colors that uses every color and is rainbow-free for $x - y = f(z)$: no $x, y, z \\in \\mathbb{Z}/n\\mathbb{Z}$ with $x - y = f(z)$ have $c(x), c(y), c(z)$ pairwise distinct. For $v \\in \\mathbb{Z}/n\\mathbb{Z}$, call the set of $a$ with $f^m(a) = f^{m'}(v)$ for\u2026","title":"Components avoiding the color of zero are monochromatic"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/is-cycle-vertex-pow-map-iff-order-of-dvd","kind":"node","lean":"MetaMathlibExt.isCycleVertex_powMap_iff_orderOf_dvd","parent":23634,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycle vertices of the "],["math","\\(k\\)"],["text","-th power map modulo "],["math","\\(p\\)"]],"source":"Cycle vertices of the $k$-th power map modulo $p$"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k/index.html#is-cycle-vertex-pow-map-iff-order-of-dvd","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/rainbow-free-colorings-for-x-y-z-k","status":"can_prove","summary":"Let $p$ be an odd prime, $k$ a natural number, and $t$ the largest divisor of $p - 1$ that is coprime to $k$. A nonzero $a \\in \\mathbb{Z}/p\\mathbb{Z}$ lies on a cycle of the map $x \\mapsto x^k$, that is, $a^{k^m} = a$ for some $m \\ge 1$, if and only if the multiplicative order of $a$ divides $t$.","title":"Cycle vertices of the $k$-th power map modulo $p$"},{"data_url":"scopes/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers.json","id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","kind":"scope","navigation":"module","parent":23581,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"can_prove","title":"Schur numbers"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/schur-forces","kind":"node","lean":"MathlibExt.Combinatorics.SchurNumbersWanted.schurForces","parent":23658,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html#schur-forces","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"fully_proved","summary":"For natural numbers $k$ and $N$, we say that $N$ forces a monochromatic Schur triple in $k$ colours when every colouring $c$ of the natural numbers with $k$ colours has $x, y, z \\in \\{1, \\dots, N\\}$ with $x + y = z$ and $c(x) = c(y) = c(z)$.","title":"Schur forcing property"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/schur-number","kind":"node","lean":"MathlibExt.Combinatorics.SchurNumbersWanted.schurNumber","parent":23658,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html#schur-number","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"fully_proved","summary":"For $k \\ge 1$, the Schur number $S(k)$ is $N_0 - 1$, where $N_0$ is the least natural number $N$ that forces a monochromatic Schur triple in $k$ colours: every colouring $c$ of the natural numbers with $k$ colours has $x, y, z \\in \\{1, \\dots, N\\}$ with $x + y = z$ and $c(x) = c(y) = c(z)$. For $k = 0$, $S(0) = 0$.","title":"Schur number"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/schur-number-two","kind":"node","lean":"MathlibExt.Combinatorics.SchurNumbersWanted.schurNumber_two","parent":23658,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html#schur-number-two","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"can_prove","summary":"The Schur number $S(2)$ equals $4$.","title":"The second Schur number"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/schur-number-three","kind":"node","lean":"MathlibExt.Combinatorics.SchurNumbersWanted.schurNumber_three","parent":23658,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html#schur-number-three","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"can_prove","summary":"The Schur number $S(3)$ equals $13$.","title":"The third Schur number"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/schur-number-four","kind":"node","lean":"MathlibExt.Combinatorics.SchurNumbersWanted.schurNumber_four","parent":23658,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html#schur-number-four","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"can_prove","summary":"The Schur number $S(4)$ equals $44$.","title":"The fourth Schur number"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/schur-number-five","kind":"node","lean":"MathlibExt.Combinatorics.SchurNumbersWanted.schurNumber_five","parent":23658,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html#schur-number-five","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"can_prove","summary":"The Schur number $S(5)$ equals $160$.","title":"The fifth Schur number"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/le-schur-number-six","kind":"node","lean":"MathlibExt.Combinatorics.SchurNumbersWanted.le_schurNumber_six","parent":23658,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html#le-schur-number-six","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"can_prove","summary":"The Schur number $S(6)$ is at least $536$.","title":"Lower bound for the sixth Schur number"},{"id":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/schur-number-six-le","kind":"node","lean":"MathlibExt.Combinatorics.SchurNumbersWanted.schurNumber_six_le","parent":23658,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers/index.html#schur-number-six-le","scope":"Combinatorics/ramsey-theory/schur-rado-theorems/schur-numbers","status":"can_prove","summary":"The Schur number $S(6)$ is at most $1836$.","title":"Upper bound for the sixth Schur number"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey.json","id":"Combinatorics/ramsey-theory/structural-ramsey","kind":"scope","parent":23445,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey","status":"can_prove","summary":"Ramsey's theorem for sets and hypergraphs (including size Ramsey numbers and Paris\u2013Harrington), Hales\u2013Jewett and density Hales\u2013Jewett, Hindman's and the pairwise sum theorem via ultrafilters, central, J-, C- and piecewise syndetic sets in semigroups, and grid Ramsey properties.","title":"Ramsey's theorem, Hales\u2013Jewett and Hindman"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/ramsey-two-uniform","kind":"node","lean":"MathlibExt.Combinatorics.Ramsey.ramsey_two_uniform","parent":23667,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/index.html#ramsey-two-uniform","scope":"Combinatorics/ramsey-theory/structural-ramsey","status":"fully_proved","summary":"For every number of colors $r \\ge 1$ and every natural number $k$ there is a natural number $N$ such that for every coloring of the subsets of $\\{0, 1, \\dots, N - 1\\}$ with $r$ colors there are a $k$-element subset $S$ and a color $c$ such that every $2$-element subset of $S$ has color $c$.","title":"Finite Ramsey theorem for pairs"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/paris-harrington","kind":"node","lean":"MathlibExt.Combinatorics.ParisHarringtonWanted.paris_harrington","parent":23667,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/index.html#paris-harrington","scope":"Combinatorics/ramsey-theory/structural-ramsey","status":"fully_proved","summary":"For all integers $c \\ge 1$, $k \\ge 1$ and $m \\ge 1$ there is a natural number $N$ such that for every coloring $f$ of the finite subsets of $\\mathbb{N}$ with $c$ colors there is a nonempty set $H \\subseteq \\{1, \\dots, N\\}$ with $|H| \\ge m$ and $\\min H \\le |H|$ such that all $k$-element subsets of $H$ have the same color: some color $t$ satisfies $f(S) = t$ for every $k$-element subset $S \\subseteq H$.","title":"Strengthened finite Ramsey theorem (Paris\u2013Harrington principle)"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets.json","id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","title":"Arithmetic progressions in CR-sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-add-thick","kind":"node","lean":"MetaMathlibExt.IsAddThick","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-add-thick","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"A subset $A$ of a commutative semigroup $(S, +)$ is thick when for every finite set $E \\subseteq S$ there is an $x \\in S$ with $E + x \\subseteq A$, that is, $e + x \\in A$ for every $e \\in E$.","title":"Thick set in a commutative semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-add-piecewise-syndetic","kind":"node","lean":"MetaMathlibExt.IsAddPiecewiseSyndetic","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-add-piecewise-syndetic","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"A subset $A$ of a commutative semigroup $(S, +)$ is piecewise syndetic when there is a finite set $F \\subseteq S$ such that the union $\\bigcup_{t \\in F} (-t + A)$ is thick, where $-t + A = \\{s \\in S : t + s \\in A\\}$.","title":"Piecewise syndetic set in a commutative semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-jset","kind":"node","lean":"MetaMathlibExt.IsJSet","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-jset","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"Let $S$ be a commutative semigroup and let $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$. A set $A \\subseteq S$ is a $J$-set when for every finite set $F$ of sequences $f \\colon \\mathbb{N} \\to S$ there are $a \\in S$ and a nonempty finite set $H \\subseteq \\mathbb{N}$ such that $a + \\sum_{n \\in H} f(n) \\in A$ for every $f \\in F$.","title":"J-sets in commutative semigroups"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-crset","kind":"node","lean":"MetaMathlibExt.IsCRSet","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-crset","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"A subset $A$ of a commutative semigroup $(S, +)$ is a CR-set (combinatorially rich) when for every natural number $n$, including $n = 0$, there is a natural number $r$ such that for every $r \\times n$ matrix $(m_{ij})$ with entries in $S$ there are a nonempty set $\\alpha \\subseteq \\{1, \\ldots, r\\}$ and an element $s \\in S$ such that $s + \\sum_{i \\in \\alpha} m_{ij} \\in A$ for every column $j \\in \\{1, \\ldots, n\\}$.","title":"CR-set"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-essential-crset","kind":"node","lean":"MetaMathlibExt.IsEssentialCRSet","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-essential-crset","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"Let $(S, +)$ be a commutative semigroup, and add ultrafilters on $S$ by the rule that $E \\in p + q$ if and only if $\\{x \\in S : \\{y \\in S : x + y \\in E\\} \\in q\\} \\in p$. A subset $A$ of $S$ is an essential CR-set when there is an ultrafilter $p$ on $S$ with $p + p = p$, all of whose members are CR-sets, such that $A \\in p$.","title":"Essential CR-set"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-add-thick-univ","kind":"node","lean":"MetaMathlibExt.isAddThick_univ","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-add-thick-univ","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"If $S$ is a nonempty commutative semigroup, then $S$ itself is a thick subset of $S$.","title":"The whole commutative semigroup is thick"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/not-is-add-thick-empty","kind":"node","lean":"MetaMathlibExt.not_isAddThick_empty","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#not-is-add-thick-empty","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"In a nonempty commutative semigroup, the empty set is not thick.","title":"The empty set is not thick"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-add-piecewise-syndetic-univ","kind":"node","lean":"MetaMathlibExt.isAddPiecewiseSyndetic_univ","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-add-piecewise-syndetic-univ","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"If $S$ is a nonempty commutative semigroup, then $S$ itself is a piecewise syndetic subset of $S$.","title":"The whole commutative semigroup is piecewise syndetic"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-jset-univ","kind":"node","lean":"MetaMathlibExt.isJSet_univ","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-jset-univ","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"In every nonempty commutative semigroup $S$, the set $S$ itself is a $J$-set.","title":"The whole semigroup is a J-set"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-crset-univ","kind":"node","lean":"MetaMathlibExt.isCRSet_univ","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-crset-univ","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"If $S$ is a nonempty commutative semigroup, then $S$ itself is a CR-set in $S$.","title":"The whole commutative semigroup is a CR-set"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/not-is-crset-empty","kind":"node","lean":"MetaMathlibExt.not_isCRSet_empty","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#not-is-crset-empty","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"In every commutative semigroup, the empty set is not a CR-set.","title":"The empty set is not a CR-set"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-ramsey-family","kind":"node","lean":"MetaMathlibExt.IsRamseyFamily","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-ramsey-family","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"Let $S$ be a set. A collection $F$ of subsets of $S$ is a family with the Ramsey property when $F$ is nonempty, $\\emptyset \\notin F$, $F$ is closed under supersets (if $A \\in F$ and $A \\subseteq B$, then $B \\in F$), and whenever $A \\cup B \\in F$, then $A \\in F$ or $B \\in F$.","title":"Families with the Ramsey property"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/beta-family","kind":"node","lean":"MetaMathlibExt.betaFamily","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#beta-family","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"For a family $\\mathcal F$ of subsets of a set $S$, $\\beta(\\mathcal F)$ is the set of ultrafilters $p$ on $S$ all of whose members lie in $\\mathcal F$, that is, $A \\in \\mathcal F$ for every $A \\in p$.","title":"The set \u03b2(F) of ultrafilters inside a family"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/mem-beta-family","kind":"node","lean":"MetaMathlibExt.mem_betaFamily","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#mem-beta-family","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"Let $F$ be a collection of subsets of a set $S$ and let $p$ be an ultrafilter on $S$. Then $p \\in \\beta(F)$ if and only if every member of $p$ belongs to $F$.","title":"Membership in \u03b2(F)"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/beta-family-nonempty-sets","kind":"node","lean":"MetaMathlibExt.betaFamily_nonempty_sets","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#beta-family-nonempty-sets","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"fully_proved","summary":"For every set $S$, every ultrafilter on $S$ lies in $\\beta(\\mathcal F)$, where $\\mathcal F$ is the family of all nonempty subsets of $S$; that is, $\\beta(\\mathcal F) = \\beta S$, the set of all ultrafilters on $S$.","title":"\u03b2 of the nonempty sets is all of \u03b2S"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-closed-beta-family","kind":"node","lean":"MetaMathlibExt.isClosed_betaFamily","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-closed-beta-family","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $F$ be a family of subsets of a set $S$ with the Ramsey property. Then the set $\\beta(F)$ of ultrafilters on $S$ all of whose members lie in $F$ is closed in the space $\\beta S$ of all ultrafilters on $S$.","title":"Ultrafilters inside a Ramsey family form a closed set"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/mem-iff-exists-mem-beta-family","kind":"node","lean":"MetaMathlibExt.mem_iff_exists_mem_betaFamily","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#mem-iff-exists-mem-beta-family","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $F$ be a family of subsets of a set $S$ with the Ramsey property, and let $\\beta(F)$ be the set of ultrafilters on $S$ all of whose members lie in $F$. Then a set $A \\subseteq S$ lies in $F$ if and only if $A \\in p$ for some $p \\in \\beta(F)$; that is, $F = \\bigcup \\beta(F)$.","title":"Theorem 3, part 2: a Ramsey family is the union of its ultrafilters"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-ramsey-family-and-beta-family-eq-of-i","kind":"node","lean":"MetaMathlibExt.isRamseyFamily_and_betaFamily_eq_of_isClosed","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-ramsey-family-and-beta-family-eq-of-i","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $S$ be a set, let $\\beta S$ be the space of ultrafilters on $S$, and let $K$ be a nonempty closed subset of $\\beta S$. Let $\\mathcal F_K$ be the collection of the sets $E \\subseteq S$ with $E \\in p$ for some $p \\in K$. Then $\\mathcal F_K$ is a family with the Ramsey property, and $\\beta(\\mathcal F_K) = K$, where $\\beta(\\mathcal F_K)$ is the set of ultrafilters on $S$ all of whose members lie in $\\mathcal F_K$.","title":"Nonempty closed sets of ultrafilters come from Ramsey families"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/exists-idempotent-mem-beta-family","kind":"node","lean":"MetaMathlibExt.exists_idempotent_mem_betaFamily","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#exists-idempotent-mem-beta-family","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $S$ be a semigroup and $\\mathcal F$ a family of subsets of $S$ with the Ramsey property, and multiply ultrafilters on $S$ by $E \\in pq$ iff $\\{x \\in S : \\{y \\in S : xy \\in E\\} \\in q\\} \\in p$. If $pq \\in \\beta(\\mathcal F)$ for all $p, q \\in \\beta(\\mathcal F)$, then there is $p \\in \\beta(\\mathcal F)$ with $pp = p$.","title":"\u03b2(F) contains an idempotent when it is a subsemigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/beta-family-is-left-ideal-iff","kind":"node","lean":"MetaMathlibExt.betaFamily_isLeftIdeal_iff","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#beta-family-is-left-ideal-iff","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $S$ be a semigroup and $\\mathcal F$ a family of subsets of $S$ with the Ramsey property. Multiply ultrafilters on $S$ by $E \\in pq$ iff $\\{x \\in S : \\{y \\in S : xy \\in E\\} \\in q\\} \\in p$. Then $pq \\in \\beta(\\mathcal F)$ for every ultrafilter $p$ on $S$ and every $q \\in \\beta(\\mathcal F)$ (that is, $\\beta(\\mathcal F)$ is a left ideal of $\\beta S$) if and only if $sE = \\{se : e \\in E\\}$ lies in $\\mathcal F$ for\u2026","title":"Left-ideal criterion for \u03b2(F)"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/beta-family-is-subsemigroup-iff","kind":"node","lean":"MetaMathlibExt.betaFamily_isSubsemigroup_iff","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#beta-family-is-subsemigroup-iff","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $S$ be a semigroup and $\\mathcal F$ a family of subsets of $S$ with the Ramsey property. For $x \\in S$ and $E \\subseteq S$ write $x^{-1}E = \\{y \\in S : xy \\in E\\}$, and multiply ultrafilters on $S$ by $E \\in pq$ iff $\\{x \\in S : x^{-1}E \\in q\\} \\in p$. Then $pq \\in \\beta(\\mathcal F)$ for all $p, q \\in \\beta(\\mathcal F)$ if and only if every $E \\subseteq S$ has the following property: if there is\u2026","title":"Subsemigroup criterion for \u03b2(F)"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/exists-idempotent-mem-of-decreasing-sequ","kind":"node","lean":"MetaMathlibExt.exists_idempotent_mem_of_decreasing_sequence","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#exists-idempotent-mem-of-decreasing-sequ","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $S$ be a semigroup and let $F$ be a family of subsets of $S$ with the Ramsey property, and write $\\beta(F)$ for the set of ultrafilters on $S$ all of whose members lie in $F$. For $x \\in S$ and $E \\subseteq S$ write $x^{-1}E = \\{y \\in S : xy \\in E\\}$, and multiply ultrafilters on $S$ by the rule that $E \\in p \\cdot q$ if and only if $\\{x \\in S : x^{-1}E \\in q\\} \\in p$. Suppose that $p \\cdot q \\in \\beta(F)$\u2026","title":"Decreasing shift-closed sequences give essential sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/exists-decreasing-sequence-of-idempotent","kind":"node","lean":"MetaMathlibExt.exists_decreasing_sequence_of_idempotent_mem","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#exists-decreasing-sequence-of-idempotent","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $S$ be a countable semigroup and $\\mathcal F$ a family of subsets of $S$ with the Ramsey property. For $x \\in S$ and $E \\subseteq S$ write $x^{-1}E = \\{y \\in S : xy \\in E\\}$, multiply ultrafilters on $S$ by $E \\in pq$ iff $\\{x \\in S : x^{-1}E \\in q\\} \\in p$, and assume $pq \\in \\beta(\\mathcal F)$ for all $p, q \\in \\beta(\\mathcal F)$. If $A \\subseteq S$ belongs to some $p \\in \\beta(\\mathcal F)$ with $pp = p$ (an\u2026","title":"Essential F-sets contain shift-closed decreasing sequences (countable case)"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/arithmetic-progressions-int","kind":"node","lean":"MetaMathlibExt.IsAddPiecewiseSyndetic.arithmeticProgressions_int","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#arithmetic-progressions-int","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"If $S \\subseteq \\mathbb{Z}$ is piecewise syndetic, then for every natural number $k$ the set of pairs $(a, d) \\in \\mathbb{Z} \\times \\mathbb{Z}$ such that $a + i d \\in S$ for all $i = 0, 1, \\dots, k$ is piecewise syndetic in $\\mathbb{Z} \\times \\mathbb{Z}$.","title":"Furstenberg\u2013Glasner theorem on progressions in piecewise syndetic sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/arithmetic-progressions-pnat","kind":"node","lean":"MetaMathlibExt.IsJSet.arithmeticProgressions_pnat","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#arithmetic-progressions-pnat","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $\\mathbb{N}^+$ denote the positive integers under addition. If $S \\subseteq \\mathbb{N}^+$ is a $J$-set, then for every natural number $k$ the set of pairs $(a, d) \\in \\mathbb{N}^+ \\times \\mathbb{N}^+$ such that $a + i d \\in S$ for all $i = 0, 1, \\dots, k$ is a $J$-set in $\\mathbb{N}^+ \\times \\mathbb{N}^+$.","title":"Debnath\u2013Goswami theorem on progressions in J-sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/is-essential-crset-iff-exists-decreasing","kind":"node","lean":"MetaMathlibExt.isEssentialCRSet_iff_exists_decreasing_sequence","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#is-essential-crset-iff-exists-decreasing","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $S$ be a countable commutative semigroup and let $A \\subseteq S$. Then $A$ is an essential CR-set if and only if there is a decreasing sequence $C_0 \\supseteq C_1 \\supseteq C_2 \\supseteq \\cdots$ of CR-sets contained in $A$ such that for every $n$ and every $x \\in C_n$ there is an $m$ with $C_m \\subseteq -x + C_n$, where $-x + C_n = \\{y \\in S : x + y \\in C_n\\}$.","title":"Essential CR-sets via decreasing sequences"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/arithmetic-progressions","kind":"node","lean":"MetaMathlibExt.IsCRSet.arithmeticProgressions","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#arithmetic-progressions","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"If $A$ is a CR-set in a commutative semigroup $(S, +)$ and $l$ is a natural number, then the set of pairs $(a, b) \\in S \\times S$ such that $a, a + b, a + 2b, \\dots, a + lb$ all lie in $A$ is a CR-set in $S \\times S$. Here $a + ib$ is $a$ followed by $i$ additions of $b$, since $S$ need not have a zero.","title":"Arithmetic progressions in CR-sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/arithmetic-progressions-pnat-2","kind":"node","lean":"MetaMathlibExt.IsEssentialCRSet.arithmeticProgressions_pnat","parent":23670,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets/index.html#arithmetic-progressions-pnat-2","scope":"Combinatorics/ramsey-theory/structural-ramsey/arithmetic-progressions-in-cr-sets","status":"can_prove","summary":"Let $\\mathbb{N}^+$ denote the positive integers under addition. If $A \\subseteq \\mathbb{N}^+$ is an essential CR-set and $l$ is a natural number, then the set of pairs $(a, b) \\in \\mathbb{N}^+ \\times \\mathbb{N}^+$ such that $a + i b \\in A$ for all $i = 0, 1, \\dots, l$ is an essential CR-set in $\\mathbb{N}^+ \\times \\mathbb{N}^+$.","title":"Arithmetic progressions in essential CR-sets of the positive integers"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem.json","id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","title":"Density Hales\u2013Jewett theorem"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/reindex","kind":"node","lean":"Combinatorics.Line.reindex","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#reindex","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"A combinatorial line $l$ in $\\alpha^\\iota$ assigns to each coordinate $i \\in \\iota$ either a fixed value in $\\alpha$ or the free variable, with at least one free coordinate. Given a bijection $e : \\iota \\to \\iota'$, the reindexing of $l$ along $e$ is the combinatorial line in $\\alpha^{\\iota'}$ that assigns to each coordinate $j \\in \\iota'$ what $l$ assigns to the coordinate $e^{-1}(j)$.","title":"Reindexing a combinatorial line"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/reindex-apply","kind":"node","lean":"Combinatorics.Line.reindex_apply","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#reindex-apply","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"Let $l$ be a combinatorial line in $\\alpha^\\iota$, let $e : \\iota \\to \\iota'$ be a bijection, and for $a \\in \\alpha$ write $l(a) \\in \\alpha^\\iota$ for the point of $l$ whose free coordinates equal $a$. Then for every $a \\in \\alpha$ and every $j \\in \\iota'$, the $j$-th coordinate of the point at $a$ of the reindexing of $l$ along $e$ equals the $e^{-1}(j)$-th coordinate of $l(a)$.","title":"Points of a reindexed line"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/comp","kind":"node","lean":"Combinatorics.Subspace.comp","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#comp","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"A combinatorial subspace $V$ of $\\alpha^\\iota$ with parameter set $\\eta$ assigns to each coordinate $i \\in \\iota$ either a fixed value in $\\alpha$ or a parameter in $\\eta$, with every parameter assigned to some coordinate. Let $V$ be such a subspace and let $W$ be a combinatorial subspace of $\\alpha^\\eta$ with parameter set $\\eta'$. The composite $V \\circ W$ is the combinatorial subspace of $\\alpha^\\iota$ with\u2026","title":"Composition of combinatorial subspaces"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/comp-apply","kind":"node","lean":"Combinatorics.Subspace.comp_apply","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#comp-apply","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"Let $V$ be a combinatorial subspace of $\\alpha^\\iota$ with parameter set $\\eta$ and let $W$ be a combinatorial subspace of $\\alpha^\\eta$ with parameter set $\\eta'$; view each subspace as its map from parameter values to points. Then for every $x \\in \\alpha^{\\eta'}$, the composite $V \\circ W$ sends $x$ to $V(W(x))$.","title":"Evaluating a composite subspace"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/comp-line","kind":"node","lean":"Combinatorics.Subspace.compLine","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#comp-line","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"Let $V$ be a combinatorial subspace of $\\alpha^\\iota$ with parameter set $\\eta$ and let $l$ be a combinatorial line in $\\alpha^\\eta$. The image $V_* l$ is the combinatorial line in $\\alpha^\\iota$ that has the fixed value $a$ at each coordinate $i$ to which $V$ assigns the fixed value $a$, and has at each coordinate $i$ to which $V$ assigns a parameter $e \\in \\eta$ what $l$ has at the coordinate $e$ (the free\u2026","title":"Image of a combinatorial line in a combinatorial subspace"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/comp-line-apply","kind":"node","lean":"Combinatorics.Subspace.compLine_apply","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#comp-line-apply","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"Let $V$ be a combinatorial subspace of $\\alpha^\\iota$ with parameter set $\\eta$, viewed as its map $x \\mapsto V(x)$ from parameter values to points, and let $l$ be a combinatorial line in $\\alpha^\\eta$, with $l(a)$ the point of $l$ whose free coordinates equal $a$. Then for every $a \\in \\alpha$, the point at $a$ of the image line $V_* l$ is $V(l(a))$.","title":"Points of the image of a line"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/reindex-comp-line","kind":"node","lean":"Combinatorics.Subspace.reindex_compLine","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#reindex-comp-line","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"Let $V$ be a combinatorial subspace of $\\alpha^\\iota$ with parameter set $\\eta$, let $e : \\iota \\to \\iota'$ be a bijection, and let $l$ be a combinatorial line in $\\alpha^\\eta$. Let $V^e$ be the combinatorial subspace of $\\alpha^{\\iota'}$ with parameter set $\\eta$ that assigns to each coordinate $j \\in \\iota'$ what $V$ assigns to the coordinate $e^{-1}(j)$. Then the image of $l$ under $V^e$ is the reindexing along\u2026","title":"Reindexing commutes with images of lines"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/comp-comp-line","kind":"node","lean":"Combinatorics.Subspace.comp_compLine","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#comp-comp-line","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"Let $V$ be a combinatorial subspace of $\\alpha^\\iota$ with parameter set $\\eta$, let $W$ be a combinatorial subspace of $\\alpha^\\eta$ with parameter set $\\eta'$, and let $l$ be a combinatorial line in $\\alpha^{\\eta'}$. Then the image of $l$ under the composite $V \\circ W$ equals the image under $V$ of the image of $l$ under $W$, that is, $(V \\circ W)_* l = V_*(W_* l)$.","title":"Images of lines under composite subspaces"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/exists-lines-mono-in-high-dimension-fin","kind":"node","lean":"Combinatorics.Subspace.exists_lines_mono_in_high_dimension_fin","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#exists-lines-mono-in-high-dimension-fin","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a finite nonempty alphabet, let $\\kappa$ be a finite nonempty set of colors and let $m$ be a natural number. Then there is a natural number $N$ such that for every coloring $C$ of the combinatorial lines in $\\alpha^N$ with colors in $\\kappa$ there are a combinatorial subspace $V$ of $\\alpha^N$ with parameter set $\\{0, \\dots, m - 1\\}$ and a color $z \\in \\kappa$ such that for every combinatorial line\u2026","title":"Graham\u2013Rothschild theorem for combinatorial lines"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/density-hales-jewett","kind":"node","lean":"MathlibExt.Combinatorics.DensityHalesJewettWanted.density_hales_jewett","parent":23699,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem/index.html#density-hales-jewett","scope":"Combinatorics/ramsey-theory/structural-ramsey/density-hales-jewett-theorem","status":"fully_proved","summary":"Write $[k] = \\{0, 1, \\dots, k - 1\\}$. A combinatorial line in $[k]^N$ is given by a nonempty set of free coordinates and a fixed value in $[k]$ at every other coordinate; its $k$ points are obtained by giving all free coordinates a common value $a \\in [k]$. For every natural number $k \\ge 2$ and every real $\\delta > 0$ there is a natural number $N$ such that every set $A \\subseteq [k]^N$ with $\\delta k^N \\le |A|$\u2026","title":"Density Hales\u2013Jewett theorem"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight.json","id":"Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight","status":"can_prove","title":"Homogeneous sets of large logarithmic weight"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight/homogeneous-set-large-log-weight","kind":"node","lean":"MathlibExt.Combinatorics.HomogeneousSetLogWeightErdosProblem191Wanted.homogeneousSetLargeLogWeight","parent":23710,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight/index.html#homogeneous-set-large-log-weight","scope":"Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight","status":"fully_proved","summary":"The statement that for every real $C > 0$ there is an $N$ such that for every $n \\ge N$ and every graph $G$ on the natural numbers (its edges and non-edges 2-colour the pairs of $\\{2, \\dots, n\\}$) there is a finite set $X \\subseteq \\{2, \\dots, n\\}$ that is a clique or an independent set of $G$ (a monochromatic set) with $\\sum_{x \\in X} 1 / \\log x \\ge C$.","title":"Erd\u0151s Problem 191"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight/homogeneous-set-large-log-weight-holds","kind":"node","lean":"MathlibExt.Combinatorics.HomogeneousSetLogWeightErdosProblem191Wanted.homogeneousSetLargeLogWeight_holds","parent":23710,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight/index.html#homogeneous-set-large-log-weight-holds","scope":"Combinatorics/ramsey-theory/structural-ramsey/homogeneous-sets-of-large-logarithmic-weight","status":"can_prove","summary":"For every real $C > 0$ there is an $N$ such that for every $n \\ge N$ and every graph $G$ on the natural numbers there is a finite set $X \\subseteq \\{2, \\dots, n\\}$ that is a clique or an independent set of $G$ and satisfies $\\sum_{x \\in X} 1 / \\log x \\ge C$.","title":"Monochromatic sets of large logarithmic weight exist"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups.json","id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","title":"Large sets in semigroups"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/is-mul-piecewise-syndetic","kind":"node","lean":"MetaMathlibExt.IsMulPiecewiseSyndetic","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#is-mul-piecewise-syndetic","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"A set $D$ in a semigroup $S$ is piecewise syndetic when there is a finite set $G \\subseteq S$ such that for every finite set $F \\subseteq S$ there is $x \\in S$ with $Fx \\subseteq \\bigcup_{t \\in G} t^{-1} D$, where $Fx = \\{fx : f \\in F\\}$ and $t^{-1} D = \\{s \\in S : ts \\in D\\}$.","title":"Piecewise syndetic subsets of a semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/is-mul-jset","kind":"node","lean":"MetaMathlibExt.IsMulJSet","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#is-mul-jset","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"A set $D$ in a semigroup $S$ is a $J$-set when for every nonempty finite set $F$ of sequences $f : \\mathbb{N} \\to S$, where $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$, there are an integer $m \\ge 1$, elements $a_1, \\dots, a_{m+1} \\in S$ and indices $t_1 < \\dots < t_m$ in $\\mathbb{N}$ such that $a_1 f(t_1) a_2 f(t_2) \\cdots a_m f(t_m) a_{m+1} \\in D$ for every $f \\in F$.","title":"J-sets in a semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/is-central-set","kind":"node","lean":"MetaMathlibExt.IsCentralSet","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#is-central-set","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"Let $S$ be a semigroup and let $\\beta S$ be the set of ultrafilters on $S$, with the product for which $D \\in pq$ exactly when $\\{x \\in S : \\{y \\in S : xy \\in D\\} \\in q\\} \\in p$. A two-sided ideal of $\\beta S$ is a set $I \\subseteq \\beta S$ with $rq \\in I$ and $qr \\in I$ for all $q \\in I$ and $r \\in \\beta S$. A set $D \\subseteq S$ is central when some $p \\in \\beta S$ with $pp = p$ lies in every nonempty two-sided\u2026","title":"Central sets in a semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/is-cset","kind":"node","lean":"MetaMathlibExt.IsCSet","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#is-cset","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"Let $S$ be a semigroup and let $\\beta S$ be the set of ultrafilters on $S$, with the product for which $D \\in pq$ exactly when $\\{x \\in S : \\{y \\in S : xy \\in D\\} \\in q\\} \\in p$. Let $J(S)$ be the set of ultrafilters on $S$ all of whose members are $J$-sets. A set $D \\subseteq S$ is a $C$-set when $D \\in p$ for some $p \\in J(S)$ with $pp = p$.","title":"C-sets in a semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/is-mul-piecewise-syndetic-univ","kind":"node","lean":"MetaMathlibExt.isMulPiecewiseSyndetic_univ","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#is-mul-piecewise-syndetic-univ","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"If $S$ is a nonempty semigroup, then $S$ is a piecewise syndetic subset of $S$.","title":"A nonempty semigroup is piecewise syndetic in itself"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/not-is-mul-piecewise-syndetic-empty","kind":"node","lean":"MetaMathlibExt.not_isMulPiecewiseSyndetic_empty","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#not-is-mul-piecewise-syndetic-empty","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"In every semigroup $S$, the empty subset of $S$ is not piecewise syndetic.","title":"The empty set is not piecewise syndetic in a semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/is-mul-jset-univ","kind":"node","lean":"MetaMathlibExt.isMulJSet_univ","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#is-mul-jset-univ","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"If $S$ is a nonempty semigroup, then $S$ is a $J$-set in $S$.","title":"A nonempty semigroup is a J-set in itself"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/not-is-mul-jset-empty","kind":"node","lean":"MetaMathlibExt.not_isMulJSet_empty","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#not-is-mul-jset-empty","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"If $S$ is a nonempty semigroup, then the empty subset of $S$ is not a $J$-set.","title":"The empty set is not a J-set in a nonempty semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/not-is-central-set-empty","kind":"node","lean":"MetaMathlibExt.not_isCentralSet_empty","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#not-is-central-set-empty","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"In every semigroup $S$, the empty subset of $S$ is not central.","title":"The empty set is not central in a semigroup"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/not-is-cset-empty","kind":"node","lean":"MetaMathlibExt.not_isCSet_empty","parent":23713,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups/index.html#not-is-cset-empty","scope":"Combinatorics/ramsey-theory/structural-ramsey/large-sets-in-semigroups","status":"fully_proved","summary":"In every semigroup $S$, the empty subset of $S$ is not a $C$-set.","title":"The empty set is not a C-set in a semigroup"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem.json","id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","title":"Infinitary Hales\u2013Jewett theorem"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/infinitary-hales-jewett","kind":"node","lean":"MetaMathlibExt.infinitary_halesJewett","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#infinitary-hales-jewett","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A$ be a finite alphabet, and color the finite words over $A$ with finitely many colors. A variable word over $A$ is a finite word over $A \\cup \\{v\\}$, for a variable $v \\notin A$, in which $v$ occurs, and $w(a)$ is the word obtained from $w$ by replacing every occurrence of $v$ by the letter $a$. Then there are a sequence $w_0, w_1, \\dots$ of variable words over $A$ and a color $k$ such that the word\u2026","title":"Infinitary Hales\u2013Jewett theorem"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/infinitary-hales-jewett-increasing-alpha","kind":"node","lean":"MetaMathlibExt.infinitary_halesJewett_increasing_alphabets","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#infinitary-hales-jewett-increasing-alpha","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $X$ be a set, let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite subsets of $X$ with union $A$, and color the finite words over $X$ with finitely many colors. A variable word over $A$ is a finite word over $A \\cup \\{v\\}$, for a variable $v \\notin X$, in which $v$ occurs, and $w(a)$ is the word obtained from $w$ by replacing every occurrence of $v$ by the letter $a$. Then there are a sequence $w_0, w_1, \\dots$\u2026","title":"Karagiannis's infinitary Hales\u2013Jewett theorem for increasing alphabets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/exists-variable-word-subst-mem-of-is-mul","kind":"node","lean":"MetaMathlibExt.exists_variableWord_subst_mem_of_isMulPiecewiseSyndetic","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#exists-variable-word-subst-mem-of-is-mul","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite sets with union $A$, and let $S_0$ be the semigroup of nonempty finite words over $A$ under concatenation. For $n \\in \\mathbb{N}$, an $n$-variable word over $A$ is a nonempty finite word over $A \\cup \\{v_0, \\dots, v_{n-1}\\}$, for distinct variables $v_i \\notin A$, in which every $v_i$ occurs; for $\\vec a = (a_0, \\dots, a_{n-1})$ in $A^n$, the word\u2026","title":"Piecewise syndetic sets contain Hales\u2013Jewett configurations"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/exists-variable-word-subst-mem-of-is-mul-2","kind":"node","lean":"MetaMathlibExt.exists_variableWord_subst_mem_of_isMulJSet","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#exists-variable-word-subst-mem-of-is-mul-2","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite sets whose union $A$ is nonempty, and let $S_0$ be the semigroup of nonempty finite words over $A$ under concatenation. For $n \\in \\mathbb{N}$, an $n$-variable word over $A$ is a nonempty finite word over $A \\cup \\{v_0, \\dots, v_{n-1}\\}$, for distinct variables $v_i \\notin A$, in which every $v_i$ occurs; for $\\vec a = (a_0, \\dots, a_{n-1})$ in $A^n$, the word\u2026","title":"J-sets contain Hales\u2013Jewett configurations"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/exists-variable-words-prod-mem-of-is-cen","kind":"node","lean":"MetaMathlibExt.exists_variableWords_prod_mem_of_isCentralSet","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#exists-variable-words-prod-mem-of-is-cen","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite sets with union $A$, and let $S_0$ be the semigroup of nonempty finite words over $A$ under concatenation. For $n \\in \\mathbb{N}$, an $n$-variable word over $A$ is a nonempty finite word over $A \\cup \\{v_0, \\dots, v_{n-1}\\}$, for distinct variables $v_i \\notin A$, in which every $v_i$ occurs; for $\\vec a = (a_0, \\dots, a_{n-1})$ in $A^n$, the word\u2026","title":"Infinitary Hales\u2013Jewett configurations in central sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/exists-variable-words-prod-mem-of-is-cse","kind":"node","lean":"MetaMathlibExt.exists_variableWords_prod_mem_of_isCSet","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#exists-variable-words-prod-mem-of-is-cse","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite sets with union $A$, and let $S_0$ be the semigroup of nonempty finite words over $A$ under concatenation. For $n \\in \\mathbb{N}$, an $n$-variable word over $A$ is a nonempty finite word over $A \\cup \\{v_0, \\dots, v_{n-1}\\}$, for distinct variables $v_i \\notin A$, in which every $v_i$ occurs; for $\\vec a = (a_0, \\dots, a_{n-1})$ in $A^n$, the word\u2026","title":"Infinitary Hales\u2013Jewett configurations in C-sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/exists-variable-words-subst-mem-sum-coun","kind":"node","lean":"MetaMathlibExt.exists_variableWords_subst_mem_sum_count_mem_finiteSums_of_isMulJSet","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#exists-variable-words-subst-mem-sum-coun","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite sets with union $A$, and let $S_0$ be the semigroup of nonempty finite words over $A$ under concatenation. A variable word over $A$ is a nonempty finite word over $A \\cup \\{v\\}$, for a variable $v \\notin A$, in which $v$ occurs; for $a \\in A$, the word $w(a) \\in S_0$ is obtained from $w$ by replacing each occurrence of $v$ by $a$, and $|w|_v$ is the number of\u2026","title":"Variable-count sums of variable words in J-sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/exists-variable-words-prod-mem-sum-count","kind":"node","lean":"MetaMathlibExt.exists_variableWords_prod_mem_sum_count_mem_finiteSums_of_isCSet","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#exists-variable-words-prod-mem-sum-count","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite sets with union $A$, and let $S_0$ be the semigroup of nonempty finite words over $A$ under concatenation. A variable word over $A$ is a nonempty finite word over $A \\cup \\{v\\}$, for a variable $v \\notin A$, in which $v$ occurs; for $a \\in A$, the word $w(a) \\in S_0$ is obtained from $w$ by replacing each occurrence of $v$ by $a$, and $|w|_v$ is the number of\u2026","title":"Variable-count sums of variable words in C-sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/exists-variable-words-subst-mem-prod-pat","kind":"node","lean":"MetaMathlibExt.exists_variableWords_subst_mem_prod_pattern_mem_finiteProducts_of_isMulJSet","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#exists-variable-words-subst-mem-prod-pat","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite sets with union $A$, and let $S_0$ be the semigroup of nonempty finite words over $A$ under concatenation. Let $1 \\le k < n$. An $n$-variable word over $A$ is a nonempty finite word over $A \\cup \\{v_0, \\dots, v_{n-1}\\}$, for distinct variables $v_i \\notin A$, in which every $v_i$ occurs; for $\\vec a = (a_0, \\dots, a_{n-1})$ in $A^n$, the word $w(\\vec a) \\in S_0$ is\u2026","title":"Variable patterns of variable words in J-sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/exists-variable-words-prod-mem-prod-patt","kind":"node","lean":"MetaMathlibExt.exists_variableWords_prod_mem_prod_pattern_mem_finiteProducts_of_isCSet","parent":23724,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem/index.html#exists-variable-words-prod-mem-prod-patt","scope":"Combinatorics/ramsey-theory/structural-ramsey/infinitary-hales-jewett-theorem","status":"can_prove","summary":"Let $A_0 \\subseteq A_1 \\subseteq \\cdots$ be finite sets with union $A$, and let $S_0$ be the semigroup of nonempty finite words over $A$ under concatenation. Let $1 \\le k < n$. An $n$-variable word over $A$ is a nonempty finite word over $A \\cup \\{v_0, \\dots, v_{n-1}\\}$, for distinct variables $v_i \\notin A$, in which every $v_i$ occurs; for $\\vec a = (a_0, \\dots, a_{n-1})$ in $A^n$, the word $w(\\vec a) \\in S_0$ is\u2026","title":"Variable patterns of variable words in C-sets"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings.json","id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"can_prove","title":"Monochromatic L-shapes in grid colourings"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/grid-lramsey-property","kind":"node","lean":"MetaMathlibExt.GridLRamseyProperty","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#grid-lramsey-property","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"fully_proved","summary":"Let $k$ and $n$ be natural numbers and write $[n] = \\{1, \\dots, n\\}$. An L in the grid $[n] \\times [n]$ is a triple of points $(x, y)$, $(x + d, y)$, $(x + d, y + d)$ with $x$, $y$, $d$ positive integers, $x + d \\le n$ and $y + d \\le n$. The pair $(k, n)$ has the grid Ramsey property for Ls when every coloring of $\\mathbb{N} \\times \\mathbb{N}$ with $k$ colors gives the three points of some L in $[n] \\times [n]$ the\u2026","title":"Grid Ramsey property for Ls"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/mono","kind":"node","lean":"MetaMathlibExt.GridLRamseyProperty.mono","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#mono","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"fully_proved","summary":"Let $k$, $m$, $n$ be natural numbers with $m \\le n$. For a natural number $N$ write $[N] = \\{1, \\dots, N\\}$; given a coloring, a monochromatic L in $[N] \\times [N]$ is a triple of points $(x, y)$, $(x + d, y)$, $(x + d, y + d)$ of the same color with $x$, $y$, $d$ positive integers, $x + d \\le N$ and $y + d \\le N$. If every coloring of $\\mathbb{N} \\times \\mathbb{N}$ with $k$ colors has a monochromatic L in\u2026","title":"Monotonicity of the grid Ramsey property for Ls"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/not-grid-lramsey-property-one-one","kind":"node","lean":"MetaMathlibExt.not_gridLRamseyProperty_one_one","parent":23735,"reading_title":{"format":"inline-parts-v1","parts":[["text","One color does not force an L on the "],["math","\\(1 \\times 1\\)"],["text"," grid"]],"source":"One color does not force an L on the $1 \\times 1$ grid"},"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#not-grid-lramsey-property-one-one","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"fully_proved","summary":"The grid L-Ramsey property fails for $k = 1$ color and grid size $n = 1$: not every coloring of the grid $\\{1\\} \\times \\{1\\}$ with one color contains a monochromatic L, that is, three points $(x, y), (x + d, y), (x + d, y + d)$ of the grid of the same color, with $x, y, d$ positive integers. In fact this grid contains no L at all, since that needs $x + d \\le 1$ with $x, d \\ge 1$.","title":"One color does not force an L on the $1 \\times 1$ grid"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/grid-lramsey-property-one-two","kind":"node","lean":"MetaMathlibExt.gridLRamseyProperty_one_two","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#grid-lramsey-property-one-two","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"fully_proved","summary":"Write $[n] = \\{1, \\dots, n\\}$. An L in the grid $[n] \\times [n]$ is a triple of points $(x, y)$, $(x + d, y)$, $(x + d, y + d)$ with $x$, $y$, $d$ positive integers, $x + d \\le n$ and $y + d \\le n$. Every coloring of $\\mathbb{N} \\times \\mathbb{N}$ with a single color gives the three points of some L in $[2] \\times [2]$ the same color, namely the L $(1, 1)$, $(2, 1)$, $(2, 2)$.","title":"With one color the 2 \u00d7 2 grid contains an L"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/exists-grid-lramsey-property","kind":"node","lean":"MetaMathlibExt.exists_gridLRamseyProperty","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#exists-grid-lramsey-property","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"can_prove","summary":"Write $[n] = \\{1, \\dots, n\\}$. For every positive integer $k$ there is a positive integer $n$ such that every coloring of the grid $[n] \\times [n]$ with $k$ colors contains a monochromatic L: positive integers $x$, $y$, $d$ with $x + d \\le n$ and $y + d \\le n$ such that the points $(x, y)$, $(x + d, y)$ and $(x + d, y + d)$ all have the same color.","title":"Grid Ramsey theorem for Ls"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/grid-lramsey-property-three-2593","kind":"node","lean":"MetaMathlibExt.gridLRamseyProperty_three_2593","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#grid-lramsey-property-three-2593","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"can_prove","summary":"Write $[n] = \\{1, \\dots, n\\}$. Every coloring of the grid $[2593] \\times [2593]$ with $3$ colors contains a monochromatic L: positive integers $x$, $y$, $d$ with $x + d \\le 2593$ and $y + d \\le 2593$ such that the points $(x, y)$, $(x + d, y)$ and $(x + d, y + d)$ have the same color. Equivalently, the grid Ramsey number $R_3(L)$, the least $n$ for which every coloring of $[n] \\times [n]$ with $3$ colors contains a\u2026","title":"Three-coloring the 2593 \u00d7 2593 grid forces a monochromatic L"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/grid-lramsey-property-three-1804","kind":"node","lean":"MetaMathlibExt.gridLRamseyProperty_three_1804","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#grid-lramsey-property-three-1804","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"can_prove","summary":"Write $[n] = \\{1, \\dots, n\\}$. Every coloring of the grid $[1804] \\times [1804]$ with $3$ colors contains a monochromatic L: positive integers $x$, $y$, $d$ with $x + d \\le 1804$ and $y + d \\le 1804$ such that the points $(x, y)$, $(x + d, y)$ and $(x + d, y + d)$ have the same color. Equivalently, the grid Ramsey number $R_3(L)$, the least $n$ for which every coloring of $[n] \\times [n]$ with $3$ colors contains a\u2026","title":"Three-coloring the 1804 \u00d7 1804 grid forces a monochromatic L"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/grid-lramsey-property-three-1573","kind":"node","lean":"MetaMathlibExt.gridLRamseyProperty_three_1573","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#grid-lramsey-property-three-1573","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"can_prove","summary":"Write $[n] = \\{1, \\dots, n\\}$. Every coloring of the grid $[1573] \\times [1573]$ with $3$ colors contains a monochromatic L: positive integers $x$, $y$, $d$ with $x + d \\le 1573$ and $y + d \\le 1573$ such that the points $(x, y)$, $(x + d, y)$ and $(x + d, y + d)$ have the same color. Equivalently, the grid Ramsey number $R_3(L)$, the least $n$ for which every coloring of $[n] \\times [n]$ with $3$ colors contains a\u2026","title":"Three-coloring the 1573 \u00d7 1573 grid forces a monochromatic L"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/grid-lramsey-property-three-772","kind":"node","lean":"MetaMathlibExt.gridLRamseyProperty_three_772","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#grid-lramsey-property-three-772","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"can_prove","summary":"Write $[n] = \\{1, \\dots, n\\}$. Every coloring of the grid $[772] \\times [772]$ with $3$ colors contains a monochromatic L: positive integers $x$, $y$, $d$ with $x + d \\le 772$ and $y + d \\le 772$ such that the points $(x, y)$, $(x + d, y)$ and $(x + d, y + d)$ have the same color. Equivalently, the grid Ramsey number $R_3(L)$, the least $n$ for which every coloring of $[n] \\times [n]$ with $3$ colors contains a\u2026","title":"Three-coloring the 772 \u00d7 772 grid forces a monochromatic L"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/grid-lramsey-property-three-493","kind":"node","lean":"MetaMathlibExt.gridLRamseyProperty_three_493","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#grid-lramsey-property-three-493","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"can_prove","summary":"Write $[n] = \\{1, \\dots, n\\}$. Every coloring of the grid $[493] \\times [493]$ with $3$ colors contains a monochromatic L: positive integers $x$, $y$, $d$ with $x + d \\le 493$ and $y + d \\le 493$ such that the points $(x, y)$, $(x + d, y)$ and $(x + d, y + d)$ have the same color. Equivalently, the grid Ramsey number $R_3(L)$, the least $n$ for which every coloring of $[n] \\times [n]$ with $3$ colors contains a\u2026","title":"Three-coloring the 493 \u00d7 493 grid forces a monochromatic L"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/exists-mono-l-of-forced-blue-diagonal","kind":"node","lean":"MetaMathlibExt.exists_mono_L_of_forced_blue_diagonal","parent":23735,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings/index.html#exists-mono-l-of-forced-blue-diagonal","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-l-shapes-in-grid-colourings","status":"can_prove","summary":"Let $n \\ge 3$, write $[n] = \\{1, \\dots, n\\}$, let $c$ be a coloring of $\\mathbb{N} \\times \\mathbb{N}$ with $3$ colors and let red and blue be two distinct colors. Let $s \\ge 1$ and let $X$ be a finite set of natural numbers such that every $x \\in X$ satisfies $x \\ge 1$ and $x + s \\le n$, the point $(x + s, x)$ is blue, and the points $(x, x)$ and $(x + s, x + s)$ are red (so red at $(x + s, x)$ would complete the\u2026","title":"Forced blue points on a diagonal give a monochromatic L"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals.json","id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals","status":"can_prove","title":"Monochromatic pairwise sums in the reals"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals/aleph-one-monochromatic-pair-sums","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticPairSumsRealsErdosProblem965Wanted.alephOneMonochromaticPairSums","parent":23747,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals/index.html#aleph-one-monochromatic-pair-sums","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals","status":"fully_proved","summary":"The statement that for every colouring of the real numbers with two colours there is a set $A \\subseteq \\mathbb{R}$ of cardinality $\\aleph_1$ such that all sums $a + b$ with $a, b \\in A$ and $a \\ne b$ have the same colour.","title":"Erd\u0151s Problem 965"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals/aleph-one-monochromatic-pair-sums-refute","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticPairSumsRealsErdosProblem965Wanted.alephOneMonochromaticPairSums_refuted","parent":23747,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals/index.html#aleph-one-monochromatic-pair-sums-refute","scope":"Combinatorics/ramsey-theory/structural-ramsey/monochromatic-pairwise-sums-in-the-reals","status":"can_prove","summary":"It is not the case that for every colouring of the real numbers with two colours there is a set $A \\subseteq \\mathbb{R}$ of cardinality $\\aleph_1$ such that all sums $a + b$ with $a, b \\in A$ and $a \\ne b$ have the same colour.","title":"Erd\u0151s Problem 965 is false"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem.json","id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","title":"Pairwise sum theorem"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/exists-infinite-finite-sums-monochromati","kind":"node","lean":"MetaMathlibExt.exists_infinite_finiteSums_monochromatic","parent":23750,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html#exists-infinite-finite-sums-monochromati","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","summary":"For every finite coloring of $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$ there is an infinite set $A \\subseteq \\mathbb{N}$ whose set of finite sums $\\mathrm{FS}(A) = \\{\\sum_{a \\in F} a : F \\subseteq A \\text{ finite and nonempty}\\}$ is monochromatic.","title":"Hindman's finite sums theorem"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/exists-infinite-pairwise-sums-monochroma","kind":"node","lean":"MetaMathlibExt.exists_infinite_pairwiseSums_monochromatic","parent":23750,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html#exists-infinite-pairwise-sums-monochroma","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","summary":"For every finite coloring of $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$ there is an infinite set $A \\subseteq \\mathbb{N}$ such that $A \\cup \\{a + a' : a, a' \\in A, a \\ne a'\\}$ is monochromatic.","title":"Pairwise sum theorem"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/exists-disjoint-infinite-union-add-monoc","kind":"node","lean":"MetaMathlibExt.exists_disjoint_infinite_union_add_monochromatic","parent":23750,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html#exists-disjoint-infinite-union-add-monoc","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","summary":"For every finite coloring of $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$ there are disjoint infinite sets $A, B \\subseteq \\mathbb{N}$ such that $A \\cup B \\cup (A + B)$ is monochromatic, where $A + B = \\{a + b : a \\in A, b \\in B\\}$.","title":"Disjoint infinite sets with monochromatic union and sumset"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/exists-interleaved-sequences-iff-mem-ten","kind":"node","lean":"MetaMathlibExt.exists_interleaved_sequences_iff_mem_tensor","parent":23750,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html#exists-interleaved-sequences-iff-mem-ten","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","summary":"For ultrafilters $U$, $V$ on $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$ and $X \\subseteq \\mathbb{N} \\times \\mathbb{N}$, write $X \\in U \\otimes V$ when $\\{n : \\{m : (n, m) \\in X\\} \\in V\\} \\in U$. For every $X \\subseteq \\mathbb{N} \\times \\mathbb{N}$ the following are equivalent: (1) there are strictly increasing sequences $(a_i)_{i \\in \\mathbb{N}}$, $(b_j)_{j \\in \\mathbb{N}}$ in $\\mathbb{N}$ with disjoint ranges such that\u2026","title":"Interleaved sequences and tensor products of ultrafilters"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/weakly-partition-regular-iff-exists-ultr","kind":"node","lean":"MetaMathlibExt.weaklyPartitionRegular_iff_exists_ultrafilter","parent":23750,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html#weakly-partition-regular-iff-exists-ultr","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","summary":"Let $\\mathcal{F}$ be a family of subsets of a set $S$. Every finite coloring of $S$ has a monochromatic member of $\\mathcal{F}$ (that is, $\\mathcal{F}$ is weakly partition regular) if and only if there is an ultrafilter $U$ on $S$ such that every $A \\in U$ contains some member of $\\mathcal{F}$ ($U$ is a PR-witness of $\\mathcal{F}$).","title":"Partition regular families and ultrafilters"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/finite-sums-witness-iff-mem-closure-idem","kind":"node","lean":"MetaMathlibExt.finiteSums_witness_iff_mem_closure_idempotents","parent":23750,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html#finite-sums-witness-iff-mem-closure-idem","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","summary":"Let $\\beta\\mathbb{N}^+$ be the space of ultrafilters on the positive integers $\\mathbb{N}^+$, with the Stone\u2013\u010cech topology and the sum for which $A \\in U + V$ if and only if $\\{n : \\{m : n + m \\in A\\} \\in V\\} \\in U$. For $X \\subseteq \\mathbb{N}^+$ let $\\mathrm{FS}(X)$ be the set of sums of nonempty finite subsets of $X$. An ultrafilter $U \\in \\beta\\mathbb{N}^+$ is a PR-witness of the Hindman family, that is, every\u2026","title":"PR-witnesses of the finite sums family"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/ramsey-witness-iff-mem-closure-tensor-sq","kind":"node","lean":"MetaMathlibExt.ramseyWitness_iff_mem_closure_tensor_squares","parent":23750,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html#ramsey-witness-iff-mem-closure-tensor-sq","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","summary":"For an ultrafilter $U$ on $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$, the tensor square $U \\otimes U$ is the ultrafilter on $\\mathbb{N} \\times \\mathbb{N}$ whose members are the sets $X$ with $\\{n : \\{m : (n, m) \\in X\\} \\in U\\} \\in U$. An ultrafilter $W$ on $\\mathbb{N} \\times \\mathbb{N}$ is a Ramsey's witness when every $X \\in W$ contains $[H]^2 = \\{(h, h') \\in H \\times H : h < h'\\}$ for some infinite\u2026","title":"Ramsey's witnesses are the closure of tensor squares"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/pairwise-sums-witness-iff-exists-ramsey","kind":"node","lean":"MetaMathlibExt.pairwiseSums_witness_iff_exists_ramseyWitness","parent":23750,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem/index.html#pairwise-sums-witness-iff-exists-ramsey","scope":"Combinatorics/ramsey-theory/structural-ramsey/pairwise-sum-theorem","status":"can_prove","summary":"An ultrafilter $W$ on $\\mathbb{N} \\times \\mathbb{N}$, where $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$, is a Ramsey's witness when every $X \\in W$ contains $\\{(h, h') \\in H \\times H : h < h'\\}$ for some infinite $H \\subseteq \\mathbb{N}$. For an ultrafilter $U$ on $\\mathbb{N}$ the following are equivalent: (1) every $A \\in U$ contains an infinite set $B$ together with all sums $b + b'$ of distinct $b, b' \\in B$ ($U$ is a\u2026","title":"PR-witnesses of the pairwise sum family"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets.json","id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"can_prove","title":"Piecewise syndetic sets"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/lattice-cube","kind":"node","lean":"MetaMathlibExt.latticeCube","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#lattice-cube","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"For $t, m \\in \\mathbb{N}$ and $a \\in \\mathbb{N}^t$, the lattice cube with least corner $a$ and side $m$ is the set of points $x \\in \\mathbb{N}^t$ such that $a_i \\le x_i < a_i + m$ for every coordinate $i$; it is a subcube of size $m^t$.","title":"Lattice cube"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/mem-lattice-cube","kind":"node","lean":"MetaMathlibExt.mem_latticeCube","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#mem-lattice-cube","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"For $t, m \\in \\mathbb{N}$ and $a, x \\in \\mathbb{N}^t$, the point $x$ lies in the lattice cube with least corner $a$ and side $m$ if and only if $a_i \\le x_i < a_i + m$ for every coordinate $i$.","title":"Membership in a lattice cube"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/lattice-cube-one","kind":"node","lean":"MetaMathlibExt.latticeCube_one","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#lattice-cube-one","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"For $t \\in \\mathbb{N}$ and $a \\in \\mathbb{N}^t$, the lattice cube with least corner $a$ and side $1$ is the singleton $\\{a\\}$.","title":"Lattice cube of side one"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/is-dense-in-cubes","kind":"node","lean":"MetaMathlibExt.IsDenseInCubes","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#is-dense-in-cubes","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$ and let $C, M \\subseteq \\mathbb{N}^t$. The set $C$ is $d$-dense in $M$ when every subcube of $\\mathbb{N}^t$ of side $d$ that is contained in $M$ meets $C$.","title":"Density in subcubes"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/is-dense-in-cubes-one-iff","kind":"node","lean":"MetaMathlibExt.isDenseInCubes_one_iff","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#is-dense-in-cubes-one-iff","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"For sets $C, M \\subseteq \\mathbb{N}^t$, the set $C$ is $1$-dense in $M$ if and only if $M \\subseteq C$.","title":"Density one means containment"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/is-piecewise-syndetic","kind":"node","lean":"MetaMathlibExt.IsPiecewiseSyndetic","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#is-piecewise-syndetic","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"Let $t \\in \\mathbb{N}$. A set $A \\subseteq \\mathbb{N}^t$ is piecewise syndetic if there is $d \\in \\mathbb{N}$ such that, writing $[0, d] = \\{0, 1, \\dots, d\\}$, for every $m \\in \\mathbb{N}$ the set $A + [0, d]^t = \\{y + z : y \\in A,\\ z \\in [0, d]^t\\}$ contains a lattice cube of side $m$.","title":"Piecewise syndetic set"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/is-piecewise-syndetic-univ","kind":"node","lean":"MetaMathlibExt.isPiecewiseSyndetic_univ","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#is-piecewise-syndetic-univ","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"For every $t \\in \\mathbb{N}$, the whole space $\\mathbb{N}^t$ is piecewise syndetic.","title":"The whole space is piecewise syndetic"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/not-is-piecewise-syndetic-empty","kind":"node","lean":"MetaMathlibExt.not_isPiecewiseSyndetic_empty","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#not-is-piecewise-syndetic-empty","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"For every $t \\in \\mathbb{N}$, the empty subset of $\\mathbb{N}^t$ is not piecewise syndetic.","title":"The empty set is not piecewise syndetic"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/exists-subcube-is-dense-in-cubes-color-c","kind":"node","lean":"MetaMathlibExt.exists_subcube_isDenseInCubes_colorClass","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#exists-subcube-is-dense-in-cubes-color-c","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"can_prove","summary":"Let $t, r \\in \\mathbb{N}$ and let $1 = m_0 < m_1 < m_2 < \\cdots$ be a strictly increasing sequence of natural numbers. Let $M$ be a subcube of $\\mathbb{N}^t$ of side $m_r$, and let $c$ be a coloring of $\\mathbb{N}^t$ by natural numbers with $1 \\le c(x) \\le r$ for every $x \\in M$. Then for some color $i$ with $1 \\le i \\le r$ there is a subcube $M' \\subseteq M$ of side $m_i$ in which the class of color $i$ is\u2026","title":"A color class is dense in a subcube (Brown, Lemma 1)"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/exists-cube-bound-is-dense-in-cubes","kind":"node","lean":"MetaMathlibExt.exists_cube_bound_isDenseInCubes","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#exists-cube-bound-is-dense-in-cubes","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"can_prove","summary":"Let $r, t \\in \\mathbb{N}$ and let $f : \\mathbb{N} \\to \\mathbb{N}$. There is $A \\in \\mathbb{N}$ such that for every coloring of $\\mathbb{N}^t$ with $r$ colors there are natural numbers $d \\ge 1$ and $m > f(d)$, a color $j$, and a subcube $M$ of side $m$ with $M \\subseteq [1, A]^t$ in which the class of color $j$ is $d$-dense. Here $[1, A]^t$ is the set of points $x$ with $1 \\le x_i \\le A$ for every coordinate $i$.","title":"A color class is dense in a large subcube (Brown, Lemma 2)"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/exists-of-eq-i-union","kind":"node","lean":"MetaMathlibExt.IsPiecewiseSyndetic.exists_of_eq_iUnion","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#exists-of-eq-i-union","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"can_prove","summary":"Let $A \\subseteq \\mathbb{N}^t$ be piecewise syndetic and let $A_1, \\dots, A_r \\subseteq \\mathbb{N}^t$ with $A = A_1 \\cup \\dots \\cup A_r$. Then some $A_i$ is piecewise syndetic.","title":"Piecewise syndeticity is partition regular"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/cube-gap-size","kind":"node","lean":"MetaMathlibExt.cubeGapSize","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#cube-gap-size","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"For a set $S \\subseteq \\mathbb{N}^t$, the gap size $\\operatorname{gs}(S)$ is the least $d \\ge 1$ such that $S$ is $d$-dense in some subcube of $\\mathbb{N}^t$ that contains $S$. If no such $d$ exists (for instance when $S$ is unbounded, so that no subcube contains it), the value is $0$.","title":"Gap size"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/cube-gap-size-le-of-is-dense-in-cubes","kind":"node","lean":"MetaMathlibExt.cubeGapSize_le_of_isDenseInCubes","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#cube-gap-size-le-of-is-dense-in-cubes","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"Let $S \\subseteq \\mathbb{N}^t$, let $d \\ge 1$, and let $M$ be a subcube of $\\mathbb{N}^t$ with $S \\subseteq M$. If $S$ is $d$-dense in $M$, then the gap size of $S$ satisfies $\\operatorname{gs}(S) \\le d$.","title":"Density bounds the gap size"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/cube-gap-size-pos","kind":"node","lean":"MetaMathlibExt.cubeGapSize_pos","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#cube-gap-size-pos","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"Let $t \\ge 1$. Every finite set $S \\subseteq \\mathbb{N}^t$ has gap size $\\operatorname{gs}(S) > 0$.","title":"Finite sets in positive dimension have positive gap size"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/cube-gap-size-singleton","kind":"node","lean":"MetaMathlibExt.cubeGapSize_singleton","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#cube-gap-size-singleton","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"fully_proved","summary":"For every point $x \\in \\mathbb{N}^t$, the one-point set $\\lbrace x \\rbrace$ has gap size $\\operatorname{gs}(\\lbrace x \\rbrace) = 1$.","title":"Gap size of a singleton"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/exists-cube-bound-monochromatic-card-gt","kind":"node","lean":"MetaMathlibExt.exists_cube_bound_monochromatic_card_gt_gapSize","parent":23759,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets/index.html#exists-cube-bound-monochromatic-card-gt","scope":"Combinatorics/ramsey-theory/structural-ramsey/piecewise-syndetic-sets","status":"can_prove","summary":"Let $r, t \\in \\mathbb{N}$ with $t \\ge 1$, and let $f : \\mathbb{N} \\to \\mathbb{N}$ be nondecreasing. There is $B \\in \\mathbb{N}$ such that for every coloring of $\\mathbb{N}^t$ with $r$ colors there is a finite monochromatic set $S \\subseteq [1, B]^t$ with $|S| > f(\\operatorname{gs}(S))$, where $\\operatorname{gs}(S)$ is the gap size of $S$ and $[1, B]^t$ is the set of points $x$ with $1 \\le x_i \\le B$ for every\u2026","title":"Monochromatic sets that are large relative to their gap size"},{"data_url":"scopes/Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs.json","id":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs","kind":"scope","navigation":"module","parent":23667,"result":false,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/index.html","scope":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs","status":"can_prove","title":"Size Ramsey numbers of hypergraphs"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/arrows3","kind":"node","lean":"MathlibExt.Combinatorics.HypergraphSizeRamseyBinomialAIMCombinatoricsProblem52Wanted.Arrows3","parent":23776,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/index.html#arrows3","scope":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs","status":"fully_proved","summary":"Let $H$ be a family of subsets of $\\{0, \\dots, N-1\\}$ and let $n$ be a natural number. Then $H$ arrows $K_n^{(3)}$ when for every colouring of the subsets of $\\{0, \\dots, N-1\\}$ with two colours there exist an $n$-element set $S \\subseteq \\{0, \\dots, N-1\\}$ and a colour such that every 3-element subset of $S$ belongs to $H$ and has that colour.","title":"Hypergraph arrowing the complete 3-uniform hypergraph"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/ramsey3","kind":"node","lean":"MathlibExt.Combinatorics.HypergraphSizeRamseyBinomialAIMCombinatoricsProblem52Wanted.ramsey3","parent":23776,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/index.html#ramsey3","scope":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs","status":"fully_proved","summary":"For a natural number $n$, the Ramsey number $r_3(n, n)$ is the least natural number $N$ such that for every colouring of the subsets of $\\{0, \\dots, N-1\\}$ with two colours there is an $n$-element set $S \\subseteq \\{0, \\dots, N-1\\}$ all of whose 3-element subsets have the same colour.","title":"Diagonal 3-uniform Ramsey number"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/size-ramsey3","kind":"node","lean":"MathlibExt.Combinatorics.HypergraphSizeRamseyBinomialAIMCombinatoricsProblem52Wanted.sizeRamsey3","parent":23776,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/index.html#size-ramsey3","scope":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs","status":"fully_proved","summary":"For a natural number $n$, the size Ramsey number $\\hat r_3(n, n)$ is the least natural number $m$ such that for some $N$ there is a 3-uniform hypergraph $H$ on $\\{0, \\dots, N-1\\}$ with exactly $m$ edges that arrows $K_n^{(3)}$.","title":"3-uniform size Ramsey number"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/size-ramsey-equals-binomial","kind":"node","lean":"MathlibExt.Combinatorics.HypergraphSizeRamseyBinomialAIMCombinatoricsProblem52Wanted.sizeRamseyEqualsBinomial","parent":23776,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/index.html#size-ramsey-equals-binomial","scope":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs","status":"fully_proved","summary":"The statement that for every natural number $n$ the 3-uniform size Ramsey number $\\hat r_3(n, n)$ equals the binomial coefficient $\\binom{r_3(n, n)}{3}$, where $r_3(n, n)$ is the diagonal 3-uniform Ramsey number (AimPL Graph Ramsey Theory, Problem 1.08).","title":"Dudek\u2013R\u00f6dl size Ramsey question"},{"id":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/size-ramsey-equals-binomial-refuted","kind":"node","lean":"MathlibExt.Combinatorics.HypergraphSizeRamseyBinomialAIMCombinatoricsProblem52Wanted.sizeRamseyEqualsBinomial_refuted","parent":23776,"result":true,"route":"../roadmap/Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs/index.html#size-ramsey-equals-binomial-refuted","scope":"Combinatorics/ramsey-theory/structural-ramsey/size-ramsey-numbers-of-hypergraphs","status":"can_prove","summary":"It is not true that for every natural number $n$ the 3-uniform size Ramsey number $\\hat r_3(n, n)$ equals the binomial coefficient $\\binom{r_3(n, n)}{3}$, where $r_3(n, n)$ is the diagonal 3-uniform Ramsey number.","title":"The Dudek\u2013R\u00f6dl size Ramsey question has a negative answer"},{"data_url":"scopes/Combinatorics/combinatorics-on-words.json","id":"Combinatorics/combinatorics-on-words","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/index.html","scope":"Combinatorics/combinatorics-on-words","status":"can_prove","summary":"Finite and infinite words over finite alphabets: repetitions and avoidability, morphic and Sturmian words, factor complexity, automatic and regular sequences, Lyndon and primitive words, and integer sequences defined by self-description or self-generation.","title":"Combinatorics on words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/repetitions-in-words.json","id":"Combinatorics/combinatorics-on-words/repetitions-in-words","kind":"scope","parent":23782,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/index.html","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words","status":"can_prove","summary":"Squares, overlaps, cubes and fractional powers in words: square-free and overlap-free words, abelian and k-abelian repetitions, power-free morphisms, Crochemore's test and Dejean's repetition threshold.","title":"Repetitions in words"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/dejean-repetition-threshold","kind":"node","lean":"MetaMathlibExt.dejean_repetition_threshold","parent":23783,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/index.html#dejean-repetition-threshold","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words","status":"can_prove","summary":"For a natural number $n$ and a real number $r$, call an infinite word $w = w_0 w_1 w_2 \\cdots$ over the alphabet $\\{0, \\dots, n-1\\}$ *$r^+$-free* if for all natural numbers $s$, $\\ell$ and $p$ with $0 < p \\le \\ell$ such that $w_{s+i+p} = w_{s+i}$ for every natural number $i$ with $i + p < \\ell$, we have $\\ell / p \\le r$; that is, no factor of $w$ has exponent strictly greater than $r$. Let $\\mathrm{RT}(n)$, the\u2026","title":"Dejean's theorem on repetition thresholds"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-cube-free","kind":"node","lean":"MetaMathlibExt.abelianCubeFree","parent":23783,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/index.html#abelian-cube-free","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words","status":"fully_proved","summary":"An abelian cube is a word $xyz$ with $x$, $y$, $z$ nonempty such that each letter occurs equally often in $x$, $y$ and $z$. Let $a_n$ be the number of words of length $n$ over $\\{0, 1, 2\\}$ that contain no abelian cube as a factor. Then $2^{n/24} = O(a_n)$ as $n \\to \\infty$: there is a constant $C$ such that $2^{n/24} \\le C a_n$ for all sufficiently large $n$.","title":"Exponential lower bound for ternary abelian-cube-free words"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/kurosaki-phi-pow-power-free","kind":"node","lean":"MetaMathlibExt.kurosaki_phi_pow_power_free","parent":23783,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kurosaki's words are "],["math","\\((7/4)^+\\)"],["text","-power-free"]],"source":"Kurosaki's words are $(7/4)^+$-power-free"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/index.html#kurosaki-phi-pow-power-free","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words","status":"fully_proved","summary":"On the alphabet $\\{1, 2, 3\\}$, let $\\sigma$ be the permutation that swaps $1$ and $2$ and fixes $3$, let $\\rho$ be the permutation that swaps $2$ and $3$ and fixes $1$, and for a finite word $w$ let $\\varphi(w) = \\sigma(w)\\,w\\,\\rho(w)$, where $\\sigma(w)$ and $\\rho(w)$ apply the permutation to each letter of $w$. Call $p \\ge 1$ a period of a finite word $u = u_0 u_1 \\cdots$ when $u_i = u_{i+p}$ for every $i$ with\u2026","title":"Kurosaki's words are $(7/4)^+$-power-free"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions.json","id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","kind":"scope","navigation":"module","parent":23783,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/index.html","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","status":"fully_proved","title":"Abelian and k-abelian repetitions"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/word-factor-count","kind":"node","lean":"MetaMathlibExt.wordFactorCount","parent":23787,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/index.html#word-factor-count","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","status":"fully_proved","summary":"For words $w$ and $t$ over an alphabet with $t$ nonempty, the factor count of $t$ in $w$ is the number of positions of $w$ at which an occurrence of $t$ starts, overlapping occurrences all being counted. The factor count of the empty word is $0$.","title":"Number of occurrences of a factor"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/is-kabelian-equivalent","kind":"node","lean":"MetaMathlibExt.IsKAbelianEquivalent","parent":23787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-abelian equivalence"]],"source":"$k$-abelian equivalence"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/index.html#is-kabelian-equivalent","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","status":"fully_proved","summary":"For a natural number $k$, two words $u$ and $v$ over an alphabet are $k$-abelian equivalent when $k \\ge 1$, both $u$ and $v$ are nonempty, $u$ and $v$ have the same prefix of length $k - 1$ and the same suffix of length $k - 1$ (a word shorter than $k - 1$ counting as its own prefix and suffix), and every word $t$ of length $k$ occurs as a factor of $u$, counting overlapping occurrences, exactly as often as it\u2026","title":"$k$-abelian equivalence"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/is-kabelian-square","kind":"node","lean":"MetaMathlibExt.IsKAbelianSquare","parent":23787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-abelian squares"]],"source":"$k$-abelian squares"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/index.html#is-kabelian-square","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","status":"fully_proved","summary":"For a natural number $k$, a word $w$ is a $k$-abelian square when $w = uv$ for two nonempty words $u$ and $v$ that are $k$-abelian equivalent.","title":"$k$-abelian squares"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/is-kabelian-square-free","kind":"node","lean":"MetaMathlibExt.IsKAbelianSquareFree","parent":23787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-abelian square-free words"]],"source":"$k$-abelian square-free words"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/index.html#is-kabelian-square-free","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","status":"fully_proved","summary":"For a natural number $k$, a finite word $w$ is $k$-abelian square-free when $k \\ge 1$ and no factor of $w$ (a word $x$ with $w = pxs$ for some words $p$ and $s$) is a $k$-abelian square.","title":"$k$-abelian square-free words"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/is-three-abelian-equivalent","kind":"node","lean":"MetaMathlibExt.IsThreeAbelianEquivalent","parent":23787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(3\\)"],["text","-abelian equivalence of ternary words"]],"source":"$3$-abelian equivalence of ternary words"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/index.html#is-three-abelian-equivalent","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","status":"fully_proved","summary":"Two words $u$ and $v$ over the alphabet $\\{0, 1, 2\\}$ are $3$-abelian equivalent when they are $k$-abelian equivalent for $k = 3$: both are nonempty, they have the same prefix and the same suffix of length $2$ (a shorter word counting as its own prefix and suffix), and every word of length $3$ occurs as a factor of $u$, counting overlapping occurrences, exactly as often as it occurs in $v$.","title":"$3$-abelian equivalence of ternary words"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/is-three-abelian-square","kind":"node","lean":"MetaMathlibExt.IsThreeAbelianSquare","parent":23787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(3\\)"],["text","-abelian squares over a ternary alphabet"]],"source":"$3$-abelian squares over a ternary alphabet"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/index.html#is-three-abelian-square","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","status":"fully_proved","summary":"A word $w$ over the alphabet $\\{0, 1, 2\\}$ is a $3$-abelian square when $w = uv$ for two nonempty words $u$ and $v$ that are $3$-abelian equivalent.","title":"$3$-abelian squares over a ternary alphabet"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/is-three-abelian-square-free","kind":"node","lean":"MetaMathlibExt.IsThreeAbelianSquareFree","parent":23787,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(3\\)"],["text","-abelian square-free ternary words"]],"source":"$3$-abelian square-free ternary words"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions/index.html#is-three-abelian-square-free","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/abelian-and-k-abelian-repetitions","status":"fully_proved","summary":"A finite word $w$ over the alphabet $\\{0, 1, 2\\}$ is $3$-abelian square-free when no factor of $w$ (a word $x$ with $w = pxs$ for some words $p$ and $s$) is a $3$-abelian square.","title":"$3$-abelian square-free ternary words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes.json","id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","kind":"scope","navigation":"module","parent":23783,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"can_prove","title":"Avoidance of additive cubes"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/letter","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.Letter","parent":23795,"reading_title":{"format":"inline-parts-v1","parts":[["text","The alphabet "],["math","\\(\\{0, 1, 3, 4\\}\\)"]],"source":"The alphabet $\\{0, 1, 3, 4\\}$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#letter","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"fully_proved","summary":"The alphabet with four distinct letters, written $0$, $1$, $3$ and $4$.","title":"The alphabet $\\{0, 1, 3, 4\\}$"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/substitution","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.substitution","parent":23795,"reading_title":{"format":"inline-parts-v1","parts":[["text","The substitution "],["math","\\(0 \\mapsto 03\\)"],["text",", "],["math","\\(1 \\mapsto 43\\)"],["text",", "],["math","\\(3 \\mapsto 1\\)"],["text",", "],["math","\\(4 \\mapsto 01\\)"]],"source":"The substitution $0 \\mapsto 03$, $1 \\mapsto 43$, $3 \\mapsto 1$, $4 \\mapsto 01$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#substitution","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"fully_proved","summary":"The map sending each letter of the alphabet $\\{0, 1, 3, 4\\}$ to a word: $0$ to $03$, $1$ to $43$, $3$ to $1$ and $4$ to $01$.","title":"The substitution $0 \\mapsto 03$, $1 \\mapsto 43$, $3 \\mapsto 1$, $4 \\mapsto 01$"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/substitution-iterate","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.substitutionIterate","parent":23795,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterates of the substitution on "],["math","\\(0\\)"]],"source":"Iterates of the substitution on $0$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#substitution-iterate","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"fully_proved","summary":"For $k \\ge 0$, the word $\\sigma^k(0)$ obtained from the one-letter word $0$ by applying the substitution $\\sigma$ ($0 \\mapsto 03$, $1 \\mapsto 43$, $3 \\mapsto 1$, $4 \\mapsto 01$) $k$ times, where $\\sigma$ acts on a word by replacing each letter with its image: $\\sigma^0(0) = 0$ and $\\sigma^{k+1}(0) = \\sigma(\\sigma^k(0))$.","title":"Iterates of the substitution on $0$"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/is-generated-fixed-point","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.IsGeneratedFixedPoint","parent":23795,"reading_title":{"format":"inline-parts-v1","parts":[["text","The fixed point generated from "],["math","\\(0\\)"]],"source":"The fixed point generated from $0$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#is-generated-fixed-point","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"fully_proved","summary":"Let $\\sigma$ be the substitution $0 \\mapsto 03$, $1 \\mapsto 43$, $3 \\mapsto 1$, $4 \\mapsto 01$ on the alphabet $\\{0, 1, 3, 4\\}$, extended to words by replacing each letter with its image. An infinite word $w = w_0 w_1 w_2 \\cdots$ over $\\{0, 1, 3, 4\\}$ is the fixed point generated from $0$ if the word $\\sigma^k(0)$ is a prefix of $w$ for every $k \\ge 0$.","title":"The fixed point generated from $0$"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/letter-value","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.letterValue","parent":23795,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#letter-value","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"fully_proved","summary":"Each letter of the alphabet $\\{0, 1, 3, 4\\}$ has as its value the integer it names: the letters $0$, $1$, $3$ and $4$ have values $0$, $1$, $3$ and $4$.","title":"Value of a letter"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/factor-sum","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.factorSum","parent":23795,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#factor-sum","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"fully_proved","summary":"For an infinite word $w = w_0 w_1 w_2 \\cdots$ over the alphabet $\\{0, 1, 3, 4\\}$ and natural numbers $s$ and $\\ell$, the sum of the factor of length $\\ell$ starting at position $s$ is the integer $\\sum_{i=0}^{\\ell-1} w_{s+i}$, where each letter counts as its value.","title":"Sum of a factor"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/avoids-pvhhcubes","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.AvoidsPVHHCubes","parent":23795,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#avoids-pvhhcubes","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"fully_proved","summary":"An infinite word $w = w_0 w_1 w_2 \\cdots$ over the alphabet $\\{0, 1, 3, 4\\}$ avoids PVHH-cubes (additive cubes) if there are no natural numbers $s$ and $\\ell \\ge 1$ such that the three consecutive factors $w_s \\cdots w_{s+\\ell-1}$, $w_{s+\\ell} \\cdots w_{s+2\\ell-1}$ and $w_{s+2\\ell} \\cdots w_{s+3\\ell-1}$ of length $\\ell$ all have the same sum of letter values.","title":"Avoiding additive cubes"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.conjecture","parent":23795,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#conjecture","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"fully_proved","summary":"Let $\\sigma$ be the substitution $0 \\mapsto 03$, $1 \\mapsto 43$, $3 \\mapsto 1$, $4 \\mapsto 01$ on the alphabet $\\{0, 1, 3, 4\\}$, extended to words by replacing each letter with its image. The proposition that there is an infinite word $w$ over $\\{0, 1, 3, 4\\}$ such that $\\sigma^k(0)$ is a prefix of $w$ for every $k \\ge 0$ and $w$ avoids PVHH-cubes (additive cubes): no three consecutive nonempty factors of $w$ of\u2026","title":"Additive-cube-free fixed point conjecture"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.PVHHCubeMorphicWordWanted.conjecture_holds","parent":23795,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes/index.html#conjecture-holds","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/avoidance-of-additive-cubes","status":"can_prove","summary":"Let $\\sigma$ be the substitution $0 \\mapsto 03$, $1 \\mapsto 43$, $3 \\mapsto 1$, $4 \\mapsto 01$ on the alphabet $\\{0, 1, 3, 4\\}$, extended to words by replacing each letter with its image. There is an infinite word $w = w_0 w_1 w_2 \\cdots$ over $\\{0, 1, 3, 4\\}$ such that $\\sigma^k(0)$ is a prefix of $w$ for every $k \\ge 0$, and $w$ contains no additive cube: there are no natural numbers $s$ and $\\ell \\ge 1$ such\u2026","title":"The Cassaigne\u2013Currie\u2013Schaeffer\u2013Shallit word avoids additive cubes"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words.json","id":"Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words","kind":"scope","navigation":"module","parent":23783,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words/index.html","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words","status":"fully_proved","title":"Overlap-free words"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words/is-word-overlap","kind":"node","lean":"MetaMathlibExt.IsWordOverlap","parent":23805,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words/index.html#is-word-overlap","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words","status":"fully_proved","summary":"A finite word $o$ over an alphabet $\\alpha$ is an overlap if $o = xyxyx$ for some finite words $x$ and $y$ over $\\alpha$ with $x$ nonempty; $y$ may be empty.","title":"Overlaps in words"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words/is-overlap-free-binary-word","kind":"node","lean":"MetaMathlibExt.IsOverlapFreeBinaryWord","parent":23805,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words/index.html#is-overlap-free-binary-word","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/overlap-free-words","status":"fully_proved","summary":"A finite binary word is overlap-free when none of its contiguous factors is an overlap.","title":"Overlap-free binary words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words.json","id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","kind":"scope","navigation":"module","parent":23783,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","title":"Square-free words"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/is-word-square","kind":"node","lean":"MetaMathlibExt.IsWordSquare","parent":23808,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#is-word-square","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"A word $w$ over an alphabet $\\alpha$ is a square if $w = xx$ for some nonempty word $x$, where $xx$ denotes the concatenation of $x$ with itself.","title":"Square word"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/is-square-free-word","kind":"node","lean":"MetaMathlibExt.IsSquareFreeWord","parent":23808,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#is-square-free-word","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"A finite word $w$ over an alphabet $\\alpha$ is square-free if no factor of $w$ is a square, where a factor of $w$ is a word $s$ with $w = psq$ for some words $p$ and $q$, and a square is a word $xx$ with $x$ nonempty.","title":"Square-free finite word"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/is-infinite-square-free-word","kind":"node","lean":"MetaMathlibExt.IsInfiniteSquareFreeWord","parent":23808,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#is-infinite-square-free-word","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"An infinite word $u = u_0 u_1 u_2 \\cdots$ over an alphabet $\\alpha$ is square-free if it contains no two adjacent equal nonempty blocks: for all $i \\ge 0$ and all $k \\ge 1$ there is some $j < k$ with $u_{i+j} \\ne u_{i+k+j}$.","title":"Square-free infinite word"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/ternary-alphabet","kind":"node","lean":"MetaMathlibExt.TernaryAlphabet","parent":23808,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#ternary-alphabet","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"The ternary alphabet is the set $\\{0, 1, 2\\}$ of natural numbers less than $3$; its three letters play the role of $a$, $b$ and $c$.","title":"The ternary alphabet for square-free words"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/is-ternary-square-free-length","kind":"node","lean":"MetaMathlibExt.IsTernarySquareFreeLength","parent":23808,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-free ternary words of length "],["math","\\(n\\)"]],"source":"Square-free ternary words of length $n$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#is-ternary-square-free-length","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"For a natural number $n$, a finite word $w$ over the ternary alphabet is a square-free ternary word of length $n$ if $w$ is square-free and $w$ has length exactly $n$.","title":"Square-free ternary words of length $n$"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/is-infinite-ternary-square-free-word","kind":"node","lean":"MetaMathlibExt.IsInfiniteTernarySquareFreeWord","parent":23808,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#is-infinite-ternary-square-free-word","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"An infinite word $u = u_0 u_1 u_2 \\cdots$ over the ternary alphabet is a square-free infinite ternary word if it is square-free as an infinite word.","title":"Square-free infinite ternary word"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/of-is-infix","kind":"node","lean":"MetaMathlibExt.IsSquareFreeWord.of_isInfix","parent":23808,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#of-is-infix","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"If a finite word $v$ is square-free and $u$ is a factor of $v$, that is, $v = puq$ for some words $p$ and $q$, then $u$ is square-free.","title":"Factors of square-free words are square-free"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/is-square-free-word-iff-inits-tails","kind":"node","lean":"MetaMathlibExt.isSquareFreeWord_iff_inits_tails","parent":23808,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#is-square-free-word-iff-inits-tails","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"A finite word $w$ is square-free if and only if, for every prefix $s$ of a suffix of $w$ and every nonempty prefix $x$ of $s$, we have $s \\ne xx$, where $xx$ denotes the concatenation of $x$ with itself.","title":"Square-freeness via prefixes of suffixes"},{"id":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/crochemore-squarefree-test-set","kind":"node","lean":"MetaMathlibExt.crochemore_squarefree_test_set","parent":23808,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words/index.html#crochemore-squarefree-test-set","scope":"Combinatorics/combinatorics-on-words/repetitions-in-words/square-free-words","status":"fully_proved","summary":"Let $f$ map each letter of the three-letter alphabet $\\{a, b, c\\}$ to a finite word over a set $T$, and extend $f$ to words by concatenation, $f(x_1 \\cdots x_n) = f(x_1) \\cdots f(x_n)$. Then $f$ maps every square-free word over $\\{a, b, c\\}$ to a square-free word if and only if it maps every square-free word over $\\{a, b, c\\}$ of length at most $5$ to a square-free word.","title":"Crochemore's square-freeness test"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words","kind":"scope","parent":23782,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words","status":"can_prove","summary":"Infinite words generated by substitutions and their codings: Fibonacci, Tribonacci and Thue\u2013Morse words and generalizations, pure morphic and morphic words, Sturmian and balanced words, factor complexity and Pansiot's classification, palindromic and relative-position properties, first differences of binary words.","title":"Morphic and Sturmian words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/pansiot-factor-complexity-classification","kind":"node","lean":"MetaMathlibExt.pansiot_factor_complexity_classification","parent":23818,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/index.html#pansiot-factor-complexity-classification","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words","status":"can_prove","summary":"Let $A$ be a finite alphabet, let $\\sigma$ be a morphism of the free monoid on $A$ that is non-erasing (no $\\sigma(x)$ is empty), let $a \\in A$, and let $w = w_0 w_1 w_2 \\cdots$ be an infinite word over $A$ that is a pure fixed point of $\\sigma$ from $a$: $\\sigma^n(a)$ is a prefix of $w$ for every $n \\ge 0$, and the length $|\\sigma^n(a)|$ tends to infinity. Assume that every letter of $A$ occurs in $w$ and that $w$\u2026","title":"Pansiot's classification of the factor complexity of pure morphic words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","title":"Factor complexity of infinite words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/factor","kind":"node","lean":"InfiniteWord.factor","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#factor","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"A right-infinite word over an alphabet $\\alpha$ is a sequence $u = u_0 u_1 u_2 \\cdots$ of letters $u_n \\in \\alpha$. For natural numbers $k$ and $i$, the length-$k$ factor of $u$ at position $i$ is the word $u_i u_{i+1} \\cdots u_{i+k-1}$, given as the map $\\{0, \\ldots, k-1\\} \\to \\alpha$, $s \\mapsto u_{i+s}$.","title":"Factor of an infinite word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/factor-set","kind":"node","lean":"InfiniteWord.factorSet","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#factor-set","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"For a right-infinite word $u = u_0 u_1 u_2 \\cdots$ over an alphabet $\\alpha$ and a natural number $k$, the set of length-$k$ factors of $u$ is the set of the maps $\\{0, \\ldots, k-1\\} \\to \\alpha$, $s \\mapsto u_{i+s}$, for $i \\in \\mathbb{N}$.","title":"Set of factors of a given length"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/initial","kind":"node","lean":"InfiniteWord.initial","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#initial","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"For a right-infinite word $u = u_0 u_1 u_2 \\cdots$ over an alphabet $\\alpha$ and a natural number $k$, the length-$k$ initial segment of $u$ is the word $u_0 u_1 \\cdots u_{k-1}$, given as the map $\\{0, \\ldots, k-1\\} \\to \\alpha$, $s \\mapsto u_s$; it is the length-$k$ factor of $u$ at position $0$.","title":"Initial segment of an infinite word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/finite-factor-set","kind":"node","lean":"InfiniteWord.finite_factorSet","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#finite-factor-set","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"If the alphabet $\\alpha$ is finite, then for every right-infinite word $u$ over $\\alpha$ and every natural number $k$, the set of length-$k$ factors of $u$ is finite.","title":"Finitely many factors of each length"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/factor-list","kind":"node","lean":"InfiniteWord.factorList","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#factor-list","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"For a right-infinite word $u = u_0 u_1 u_2 \\cdots$ over an alphabet $\\alpha$ and natural numbers $k$ and $i$, the length-$k$ factor of $u$ at position $i$, as a list, is $[u_i, u_{i+1}, \\ldots, u_{i+k-1}]$ (the empty list when $k = 0$).","title":"Factor as a list"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/is-factor","kind":"node","lean":"InfiniteWord.IsFactor","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#is-factor","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"A finite list $w$ over an alphabet $\\alpha$ is a factor of a right-infinite word $u = u_0 u_1 u_2 \\cdots$ over $\\alpha$ when, writing $\\ell$ for the length of $w$, there is a position $i \\in \\mathbb{N}$ with $w = [u_i, u_{i+1}, \\ldots, u_{i+\\ell-1}]$. In particular, the empty list is a factor of every right-infinite word over $\\alpha$.","title":"Lists occurring in an infinite word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/is-nonempty-factor","kind":"node","lean":"InfiniteWord.IsNonemptyFactor","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#is-nonempty-factor","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"Let $u = u_0 u_1 u_2 \\cdots$ be a right-infinite word over an alphabet $\\alpha$, viewed as a map $\\mathbb{N} \\to \\alpha$. A finite word $w = w_0 w_1 \\cdots w_{n-1}$ over $\\alpha$ is a nonempty factor of $u$ when $n \\ge 1$ and there is a position $i \\in \\mathbb{N}$ such that $w_j = u_{i+j}$ for every $j < n$.","title":"Nonempty factors of an infinite word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/is-factor-iff-exists-factor-list","kind":"node","lean":"InfiniteWord.isFactor_iff_exists_factorList","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#is-factor-iff-exists-factor-list","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"Let $w$ be a finite list of length $\\ell$ over an alphabet $\\alpha$ and $u$ a right-infinite word over $\\alpha$. Then $w$ is a factor of $u$ if and only if there is a position $i \\in \\mathbb{N}$ such that the length-$\\ell$ factor of $u$ at position $i$, as a list, equals $w$.","title":"Occurrence via list-valued factors"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/is-nonempty-factor-iff-exists-factor-lis","kind":"node","lean":"InfiniteWord.isNonemptyFactor_iff_exists_factorList","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#is-nonempty-factor-iff-exists-factor-lis","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"Let $w$ be a finite list of length $\\ell$ over an alphabet $\\alpha$ and $u = u_0 u_1 u_2 \\cdots$ a right-infinite word over $\\alpha$. Then $w$ is a nonempty factor of $u$ if and only if $w$ is nonempty and there is a position $i \\in \\mathbb{N}$ with $[u_i, u_{i+1}, \\ldots, u_{i+\\ell-1}] = w$.","title":"Nonempty factors via list-valued factors"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/is-nonempty-factor-iff-is-factor-and-ne","kind":"node","lean":"InfiniteWord.isNonemptyFactor_iff_isFactor_and_ne_nil","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#is-nonempty-factor-iff-is-factor-and-ne","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"For a finite list $w$ over an alphabet $\\alpha$ and a right-infinite word $u$ over $\\alpha$, $w$ is a nonempty factor of $u$ if and only if $w$ is nonempty and $w$ is a factor of $u$.","title":"Nonempty factors are the factors that are nonempty"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/is-factor-of-fn-iff-mem-factor-set","kind":"node","lean":"InfiniteWord.isFactor_ofFn_iff_mem_factorSet","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#is-factor-of-fn-iff-mem-factor-set","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"Let $k$ be a natural number, $v$ a map $\\{0, \\ldots, k-1\\} \\to \\alpha$, written $j \\mapsto v_j$, and $u$ a right-infinite word over $\\alpha$. Then the list $[v_0, v_1, \\ldots, v_{k-1}]$ is a factor of $u$ if and only if $v$ belongs to the set of length-$k$ factors of $u$. This holds also for $k = 0$.","title":"Factors as lists and as tuples agree"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/word-factor-complexity","kind":"node","lean":"MetaMathlibExt.wordFactorComplexity","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#word-factor-complexity","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"Let $w = w_0 w_1 w_2 \\cdots$ be an infinite word over a finite alphabet $A$. For $n \\ge 0$ and a position $i \\ge 0$, the factor of $w$ of length $n$ at position $i$ is the word $w_i w_{i+1} \\cdots w_{i+n-1}$. The factor complexity $p_w(n)$ is the number of distinct factors of $w$ of length $n$, each word being counted once however often it occurs in $w$.","title":"Factor complexity of an infinite word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/word-factor-complexity-zero","kind":"node","lean":"MetaMathlibExt.wordFactorComplexity_zero","parent":23820,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words/index.html#word-factor-complexity-zero","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/factor-complexity-of-infinite-words","status":"fully_proved","summary":"For every infinite word $w$ over a finite alphabet, the factor complexity of $w$ at length $0$ equals $1$, that is, $p_w(0) = 1$: the empty word is the only factor of $w$ of length $0$.","title":"Factor complexity at length zero"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","title":"Fibonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-subst","kind":"node","lean":"MetaMathlibExt.fibSubst","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-subst","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"The Fibonacci substitution sends each letter of the binary alphabet $\\{0, 1\\}$ to a finite binary word: it maps $0$ to $01$ and $1$ to $0$.","title":"Fibonacci substitution"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-morph","kind":"node","lean":"MetaMathlibExt.fibMorph","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-morph","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"The Fibonacci morphism $\\varphi$ extends the Fibonacci substitution $\\sigma$, which maps $0$ to $01$ and $1$ to $0$, to finite binary words by concatenation: $\\varphi$ maps the empty word to the empty word, and $\\varphi(bw) = \\sigma(b)\\varphi(w)$ for every letter $b$ and every finite binary word $w$.","title":"Fibonacci morphism"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word","kind":"node","lean":"MetaMathlibExt.fibWord","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"The finite Fibonacci words $f_n$, for natural numbers $n$, are the words in the letters $0$ and $1$ defined by $f_0 = 0$ and $f_{n+1} = \\varphi(f_n)$, where $\\varphi$ is the Fibonacci morphism, which replaces each letter $0$ by $01$ and each letter $1$ by $0$.","title":"Finite Fibonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-inf","kind":"node","lean":"MetaMathlibExt.fibInf","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-inf","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f_0 = 0$ and let $f_{k+1}$ be the image of $f_k$ under the Fibonacci morphism ($0 \\mapsto 01$, $1 \\mapsto 0$), so that $f_0, f_1, f_2, \\ldots$ are the finite Fibonacci words $0, 01, 010, 01001, \\ldots$. The one-sided infinite Fibonacci word is the infinite binary word whose letter at index $n$ (indices start at $0$) is the letter of $f_{n+2}$ at index $n$.","title":"Infinite Fibonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-morph-append","kind":"node","lean":"MetaMathlibExt.fibMorph_append","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-morph-append","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $\\varphi$ be the Fibonacci morphism on finite binary words, induced by $0 \\mapsto 01$ and $1 \\mapsto 0$. For all finite binary words $a$ and $b$, $\\varphi(ab) = \\varphi(a)\\varphi(b)$, where juxtaposition denotes concatenation.","title":"The Fibonacci morphism respects concatenation"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-morph-length-ge","kind":"node","lean":"MetaMathlibExt.fibMorph_length_ge","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-morph-length-ge","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $\\varphi$ be the Fibonacci morphism on finite binary words, induced by $0 \\mapsto 01$ and $1 \\mapsto 0$. For every finite binary word $w$, the length of $\\varphi(w)$ is at least the length of $w$, that is, $|w| \\le |\\varphi(w)|$.","title":"The Fibonacci morphism does not shorten words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word-head","kind":"node","lean":"MetaMathlibExt.fibWord_head","parent":23834,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite Fibonacci words begin with "],["math","\\(0\\)"]],"source":"Finite Fibonacci words begin with $0$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word-head","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f_0 = 0$ and let $f_{k+1}$ be the image of $f_k$ under the Fibonacci morphism ($0 \\mapsto 01$, $1 \\mapsto 0$). For every natural number $n$, there is a finite binary word $t$ such that $f_n = 0t$, that is, the first letter of $f_n$ is $0$.","title":"Finite Fibonacci words begin with $0$"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word-length","kind":"node","lean":"MetaMathlibExt.fibWord_length","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word-length","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f_0 = 0$ and let $f_{k+1}$ be the image of $f_k$ under the Fibonacci morphism ($0 \\mapsto 01$, $1 \\mapsto 0$). For every natural number $n$, the word $f_n$ has length at least $n + 1$.","title":"Lower bound on the length of the finite Fibonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word-nested-succ","kind":"node","lean":"MetaMathlibExt.fibWord_nested_succ","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word-nested-succ","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"For every natural number $n$, the $n$-th finite Fibonacci word $f_n$ is a prefix of $f_{n+1}$: there is a finite word $t$ in the letters $0$ and $1$ with $f_{n+1} = f_n t$.","title":"Consecutive finite Fibonacci words are nested"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word-prefix","kind":"node","lean":"MetaMathlibExt.fibWord_prefix","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word-prefix","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"For all natural numbers $n \\le m$, the $n$-th finite Fibonacci word $f_n$ is a prefix of the $m$-th finite Fibonacci word $f_m$: there is a finite word $t$ in the letters $0$ and $1$ with $f_m = f_n t$.","title":"Earlier finite Fibonacci words are prefixes of later ones"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/get-d-append-prefix","kind":"node","lean":"MetaMathlibExt.getD_append_prefix","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#get-d-append-prefix","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $a$ and $t$ be finite words in the letters $0$ and $1$, and let $k$ be a natural number smaller than the length of $a$. Then the letter at position $k$ of the concatenation $a t$ equals the letter at position $k$ of $a$, counting positions from $0$.","title":"Appending a word does not change earlier letters"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word-get-d-eq","kind":"node","lean":"MetaMathlibExt.fibWord_getD_eq","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word-get-d-eq","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f_0 = 0$ and let $f_{k+1}$ be the image of $f_k$ under the Fibonacci morphism ($0 \\mapsto 01$, $1 \\mapsto 0$). For all natural numbers $n \\le m$ and every index $i$ less than the length of $f_n$ (indices start at $0$), the letter of $f_m$ at index $i$ equals the letter of $f_n$ at index $i$.","title":"Letters of the finite Fibonacci words stabilize"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-inf-bound","kind":"node","lean":"MetaMathlibExt.fibInf_bound","parent":23834,"reading_title":{"format":"inline-parts-v1","parts":[["text","The finite Fibonacci word of index "],["math","\\(n + 2\\)"],["text"," has more than "],["math","\\(n\\)"],["text"," letters"]],"source":"The finite Fibonacci word of index $n + 2$ has more than $n$ letters"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-inf-bound","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f_0 = 0$ and let $f_{k+1}$ be the image of $f_k$ under the Fibonacci morphism ($0 \\mapsto 01$, $1 \\mapsto 0$). For every natural number $n$, the length of $f_{n+2}$ is greater than $n$.","title":"The finite Fibonacci word of index $n + 2$ has more than $n$ letters"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-inf-agrees","kind":"node","lean":"MetaMathlibExt.fibInf_agrees","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-inf-agrees","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f_0 = 0$ and let $f_{k+1}$ be the image of $f_k$ under the Fibonacci morphism ($0 \\mapsto 01$, $1 \\mapsto 0$). For all natural numbers $m$ and $n$ such that $n$ is less than the length of $f_m$, the letter of $f_m$ at index $n$ equals the letter at index $n$ of the one-sided infinite Fibonacci word (indices start at $0$). In other words, every $f_m$ is a prefix of the infinite Fibonacci word.","title":"Finite Fibonacci words agree with the infinite Fibonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word-add-two","kind":"node","lean":"MetaMathlibExt.fibWord_add_two","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word-add-two","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f_0, f_1, f_2, \\ldots$ be the finite Fibonacci words. For every natural number $n$, $f_{n+2} = f_{n+1} f_n$, where juxtaposition denotes concatenation.","title":"Recurrence for the finite Fibonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word-factorization","kind":"node","lean":"MetaMathlibExt.fibWord_factorization","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word-factorization","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"For every natural number $N$, prepending the letters $0$ and $1$ to the concatenation $f_0 f_1 \\cdots f_{N-1}$ of the first $N$ finite Fibonacci words gives $f_{N+1}$: that is, $0\\,1\\,f_0 f_1 \\cdots f_{N-1} = f_{N+1}$, where the concatenation is empty when $N = 0$.","title":"Concatenating the finite Fibonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-word-eq-of-fn-fib-inf","kind":"node","lean":"MetaMathlibExt.fibWord_eq_ofFn_fibInf","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-word-eq-of-fn-fib-inf","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"For every natural number $n$, the finite Fibonacci word $f_n$ is the word formed by the first $|f_n|$ letters of the infinite Fibonacci word.","title":"Finite Fibonacci words are prefixes of the infinite Fibonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fib-inf-factorization","kind":"node","lean":"MetaMathlibExt.fibInf_factorization","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fib-inf-factorization","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f$ be the infinite Fibonacci word and let $f_0, f_1, f_2, \\ldots$ be the finite Fibonacci words. For every natural number $N$, the finite word $w_N = 0\\,1\\,f_0 f_1 \\cdots f_{N-1}$ (so $w_0 = 01$) consists of the first $|w_N|$ letters of $f$, and for every natural number $k$ the first $k$ letters of $f$ form a prefix of $w_N$ for some $N$. In this sense $f = 0 \\cdot 1 \\cdot f_0 \\cdot f_1 \\cdot f_2 \\cdots$.","title":"Factorization of the infinite Fibonacci word into finite Fibonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/fibonacci-infinite-word-factorization","kind":"node","lean":"MetaMathlibExt.fibonacci_infinite_word_factorization","parent":23834,"reading_title":{"format":"inline-parts-v1","parts":[["text","The infinite Fibonacci word as "],["math","\\(0\\)"],["text"," followed by the finite Fibonacci words"]],"source":"The infinite Fibonacci word as $0$ followed by the finite Fibonacci words"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#fibonacci-infinite-word-factorization","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $f$ be an infinite word over $\\{0, 1\\}$ and let $f_1, f_2, f_3, \\ldots$ be finite words over $\\{0, 1\\}$ with $f_1 = 1$, $f_2 = 0$ and $f_{n+3} = f_{n+2} f_{n+1}$ for every natural number $n$, such that for every natural number $n$ the word $f_{n+2}$ consists of the first $|f_{n+2}|$ letters of $f$. Then for every natural number $N$ the word $w_N = 0\\,f_1 f_2 \\cdots f_N$ (so $w_0 = 0$) consists of the first\u2026","title":"The infinite Fibonacci word as $0$ followed by the finite Fibonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/kimberling-set-mod-two-fibonacci-word","kind":"node","lean":"MetaMathlibExt.kimberling_set_mod_two_fibonacci_word","parent":23834,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word/index.html#kimberling-set-mod-two-fibonacci-word","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/fibonacci-word","status":"fully_proved","summary":"Let $K$ be the set of natural numbers obtained from $1$ by finitely many applications of the maps $x \\mapsto 2x$ and $x \\mapsto 4x - 1$. Let $S$ be a strictly increasing sequence of natural numbers whose set of values is $K$, and let $w$ be the binary sequence with $w(n) = 1$ if $S(n + 1)$ is odd and $w(n) = 0$ if $S(n + 1)$ is even. Let $\\varphi$ be the morphism of binary words with $\\varphi(0) = 01$ and\u2026","title":"Kimberling's set modulo 2 is the Fibonacci word"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words","status":"fully_proved","title":"First differences of binary words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words/first-difference","kind":"node","lean":"MetaMathlibExt.firstDifference","parent":23855,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words/index.html#first-difference","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words","status":"fully_proved","summary":"For an infinite word $w = w_0 w_1 w_2 \\cdots$ over $\\mathbb{Z}/2\\mathbb{Z}$, its first difference is the infinite word $\\Delta w$ over $\\mathbb{Z}/2\\mathbb{Z}$ with $(\\Delta w)_i = w_{i+1} - w_i$ for every natural number $i$.","title":"First difference of an infinite binary word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words/first-difference-list","kind":"node","lean":"MetaMathlibExt.firstDifferenceList","parent":23855,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words/index.html#first-difference-list","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/first-differences-of-binary-words","status":"fully_proved","summary":"For a finite word $w = w_0 w_1 \\cdots w_m$ over $\\mathbb{Z}/2\\mathbb{Z}$, its first difference is the word $(w_1 - w_0)(w_2 - w_1) \\cdots (w_m - w_{m-1})$ of length $m$, computed in $\\mathbb{Z}/2\\mathbb{Z}$. The first difference of the empty word is the empty word.","title":"First difference of a finite binary word"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","title":"Morphic words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/morph-extend","kind":"node","lean":"MetaMathlibExt.morphExtend","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#morph-extend","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Given a map $f$ sending each letter of an alphabet $A$ to a finite word over $A$, its extension to finite words sends a word $a_1 a_2 \\cdots a_m$ over $A$ to the concatenation $f(a_1) f(a_2) \\cdots f(a_m)$ of the images of its letters in order; the empty word is sent to the empty word.","title":"Extension of a letter map to finite words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/morph-iterate","kind":"node","lean":"MetaMathlibExt.morphIterate","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#morph-iterate","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $f$ send each letter of an alphabet $A$ to a finite word over $A$, extended to finite words by concatenation. The iterates of $f$ on finite words over $A$ are defined by $f^0(w) = w$ and $f^{n+1}(w) = f(f^n(w))$ for every finite word $w$ over $A$ and every natural number $n$.","title":"Iterates of a letter map"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/morph-extend-append","kind":"node","lean":"MetaMathlibExt.morphExtend_append","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#morph-extend-append","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $f$ send each letter of an alphabet $A$ to a finite word over $A$, extended to finite words by concatenation. Then $f(uv) = f(u)f(v)$ for all finite words $u$ and $v$ over $A$.","title":"Extended letter maps preserve concatenation"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/morph-iterate-length-of-uniform","kind":"node","lean":"MetaMathlibExt.morphIterate_length_of_uniform","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#morph-iterate-length-of-uniform","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $k$ be a natural number and let $f$ send each letter of an alphabet $A$ to a word of length exactly $k$ over $A$, extended to finite words by concatenation, with iterates $f^n$ (where $f^0$ is the identity). Then for every natural number $n$ and every finite word $w$ over $A$, the word $f^n(w)$ has length $k^n$ times the length of $w$.","title":"Length of iterates of a uniform letter map"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/is-non-erasing","kind":"node","lean":"MetaMathlibExt.IsNonErasing","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#is-non-erasing","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"A map $f$ sending each letter of an alphabet $A$ to a finite word over $A$ is non-erasing when the word $f(a)$ is nonempty for every letter $a \\in A$.","title":"Non-erasing letter maps"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/is-prefix-preserving","kind":"node","lean":"MetaMathlibExt.IsPrefixPreserving","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#is-prefix-preserving","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $f$ send each letter of an alphabet $A$ to a finite word over $A$, extended to finite words by concatenation, with iterates $f^n$ (where $f^0$ is the identity). Then $f$ is prefix-preserving (prolongable) at a letter $a \\in A$ when $f(a) = a\\alpha$ for some finite word $\\alpha$ over $A$ such that the word $f^n(\\alpha)$ is nonempty for every natural number $n$, including $n = 0$.","title":"Prolongable letter maps"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/is-omega-limit","kind":"node","lean":"MetaMathlibExt.IsOmegaLimit","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#is-omega-limit","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $f$ send each letter of an alphabet $A$ to a finite word over $A$, extended to finite words by concatenation, with iterates $f^n$ (where $f^0$ is the identity), let $a \\in A$, and let $w = w_0 w_1 w_2 \\cdots$ be an infinite word over $A$. Then $w$ is the limit of the iterates of $f$ at $a$ when the lengths of the words $f^n(a)$ (iterates applied to the one-letter word $a$) are unbounded as $n$ ranges over the\u2026","title":"Infinite word as the limit of iterates"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/is-pure-morphic","kind":"node","lean":"MetaMathlibExt.IsPureMorphic","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#is-pure-morphic","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"An infinite word $w$ over a finite alphabet $A$ is pure morphic when there are a map $f$ sending each letter of $A$ to a finite word over $A$ and a letter $a \\in A$ such that $f$ is prolongable at $a$ and $w$ is the limit of the iterates of $f$ at $a$.","title":"Pure morphic words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/is-morphic","kind":"node","lean":"MetaMathlibExt.IsMorphic","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#is-morphic","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $A$ be a finite alphabet, $B$ any alphabet, $u$ an infinite word over $A$, $\\tau : A \\to B$ a map (a coding) and $w$ an infinite word over $B$. Then $w$ is morphic with generator $u$ and coding $\\tau$ if $u$ is a pure morphic word and $w(n) = \\tau(u(n))$ for every $n \\in \\mathbb{N}$.","title":"Morphic words via a coding"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/intro","kind":"node","lean":"MetaMathlibExt.IsMorphic.intro","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#intro","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $A$ be a finite alphabet, $u$ a pure morphic infinite word over $A$, $\\tau : A \\to B$ a map and $w$ an infinite word over $B$. If $w(n) = \\tau(u(n))$ for every $n \\in \\mathbb{N}$, then $w$ is morphic with generator $u$ and coding $\\tau$.","title":"Building a morphic word from its generator"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/pure","kind":"node","lean":"MetaMathlibExt.IsMorphic.pure","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#pure","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $A$ be a finite alphabet, $u$ an infinite word over $A$, $\\tau : A \\to B$ a map and $w$ an infinite word over $B$. If $w$ is morphic with generator $u$ and coding $\\tau$, then $u$ is a pure morphic word.","title":"The generator of a morphic word is pure morphic"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/coding-eq","kind":"node","lean":"MetaMathlibExt.IsMorphic.coding_eq","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#coding-eq","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $A$ be a finite alphabet, $u$ an infinite word over $A$, $\\tau : A \\to B$ a map and $w$ an infinite word over $B$. If $w$ is morphic with generator $u$ and coding $\\tau$, then $w(n) = \\tau(u(n))$ for every $n \\in \\mathbb{N}$.","title":"A morphic word is the coding of its generator"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/of-pure-morphic","kind":"node","lean":"MetaMathlibExt.IsMorphic.ofPureMorphic","parent":23858,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words/index.html#of-pure-morphic","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/morphic-words","status":"fully_proved","summary":"Let $A$ be a finite alphabet and $u$ a pure morphic infinite word over $A$. Then $u$ is morphic with generator $u$ and the identity coding of $A$.","title":"Pure morphic words are morphic"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"can_prove","title":"Palindromic fixed points of morphisms"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/sigma","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.Sigma","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#sigma","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"The alphabet is the three-letter set $\\{0, 1, 2\\}$.","title":"The ternary alphabet of Labb\u00e9's counterexample"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/morphism","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.Morphism","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#morphism","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"A morphism of the ternary alphabet $\\{0, 1, 2\\}$ is given by its letter map: a function $\\varphi$ that sends each letter $a$ to a finite word $\\varphi(a)$ over $\\{0, 1, 2\\}$, possibly the empty word.","title":"Morphisms of the ternary alphabet"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/extend","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.extend","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#extend","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"A morphism $\\varphi$ acts on a word $w = w_1 w_2 \\cdots w_k$ by $\\varphi(w) = \\varphi(w_1) \\varphi(w_2) \\cdots \\varphi(w_k)$, the concatenation of the images of its letters; the empty word is sent to the empty word.","title":"Image of a word under a morphism"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/extend-iter","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.extendIter","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#extend-iter","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"Let $\\varphi$ be a morphism of the ternary alphabet $\\{0, 1, 2\\}$, acting on finite words letter by letter. Its iterates $\\varphi^n$ on finite words are defined by $\\varphi^0(w) = w$ and $\\varphi^{n+1}(w) = \\varphi(\\varphi^n(w))$ for every natural number $n$ and every finite word $w$ over $\\{0, 1, 2\\}$.","title":"Iterated images of words under a morphism"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/is-primitive","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.IsPrimitive","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#is-primitive","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"Let $\\varphi$ be a morphism of the ternary alphabet $\\{0, 1, 2\\}$, sending each letter to a finite word, and let $\\varphi^n$ be the $n$-th iterate of its letter by letter extension to finite words, with $\\varphi^0$ the identity. Then $\\varphi$ is primitive when there is a natural number $n$ such that for all letters $a$ and $b$, the letter $b$ occurs in the word $\\varphi^n(a)$.","title":"Primitive morphisms"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/is-palindrome","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.IsPalindrome","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#is-palindrome","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"A finite word $w$ over the alphabet $\\{0, 1, 2\\}$ is a palindrome when it equals its reversal, that is, reads the same backwards; in particular the empty word is a palindrome.","title":"Palindromes"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/in-class-p","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.InClassP","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#in-class-p","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"A morphism $\\varphi$ is in class P (Hof\u2013Knill\u2013Simon) when there is a palindrome $p$ such that for every letter $a$, $\\varphi(a) = p q_a$ for some palindrome $q_a$. The palindromes $p$ and $q_a$ may be empty.","title":"Class P"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/inf-word","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.InfWord","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#inf-word","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"An infinite word over the alphabet $\\{0, 1, 2\\}$ is a sequence $u = u_0 u_1 u_2 \\cdots$ of letters indexed by the natural numbers, that is, a map $\\mathbb{N} \\to \\{0, 1, 2\\}$.","title":"Infinite words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/occurs-at","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.OccursAt","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#occurs-at","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"A finite word $w = w_0 w_1 \\cdots w_{n-1}$ over the ternary alphabet $\\{0, 1, 2\\}$ occurs in an infinite word $u = u_0 u_1 u_2 \\cdots$ over the same alphabet at position $k$ when $w_i = u_{k+i}$ for every $i < n$.","title":"Occurrence of a word at a position"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/is-factor","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.IsFactor","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#is-factor","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"A finite word $w$ is a factor of an infinite word $u$ when it occurs in $u$ at some position: there is $k \\in \\mathbb{N}$ with $w_i = u_{k+i}$ for every $i < |w|$ (letters of $w$ indexed from $0$). In particular the empty word is a factor of every infinite word.","title":"Factors of an infinite word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/is-fixed-point","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.IsFixedPoint","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#is-fixed-point","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"An infinite word $u$ is a fixed point of a morphism $\\varphi$ when it is the limit of the iterates of a prolongable letter: there is a letter $a$ such that $\\varphi(a)$ begins with $a$, the words $\\varphi^m(a)$ ($m \\ge 0$) are arbitrarily long, and every $\\varphi^m(a)$ is a prefix of $u$.","title":"Fixed point of a morphism"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/is-palindromic","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.IsPalindromic","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#is-palindromic","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"An infinite word $u = u_0 u_1 u_2 \\cdots$ over the ternary alphabet $\\{0, 1, 2\\}$ is palindromic when it has arbitrarily long palindromic factors: for every natural number $N$ there is a finite word $w = w_0 \\cdots w_{n-1}$ with $n \\ge N$ that reads the same backwards and occurs in $u$: for some position $k$, $w_i = u_{k+i}$ for all $i < n$.","title":"Palindromic infinite words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/is-uniformly-recurrent","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.IsUniformlyRecurrent","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#is-uniformly-recurrent","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"Let $u = u_0 u_1 u_2 \\cdots$ be an infinite word over the ternary alphabet $\\{0, 1, 2\\}$, and say that a finite word $w = w_0 \\cdots w_{n-1}$ occurs in $u$ at position $m$ when $w_i = u_{m+i}$ for every $i < n$. Then $u$ is uniformly recurrent when for every finite word $w$ that occurs in $u$ at some position, there is a natural number $B$ such that for every position $k$, the word $w$ occurs in $u$ at position\u2026","title":"Uniformly recurrent infinite words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/morph-conjugate","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.MorphConjugate","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#morph-conjugate","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"Let $\\psi$ and $\\theta$ be morphisms of the ternary alphabet $\\{0, 1, 2\\}$, each sending every letter to a finite word, and write $xy$ for the concatenation of words $x$ and $y$. Then $\\psi$ and $\\theta$ are conjugate when there is a finite word $w$ such that $\\psi(a) w = w \\theta(a)$ for every letter $a$, or there is a finite word $w$ such that $\\theta(a) w = w \\psi(a)$ for every letter $a$.","title":"Conjugate morphisms"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/is-identity-morphism","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.IsIdentityMorphism","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#is-identity-morphism","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"A morphism $\\psi$ is the identity morphism when $\\psi(a)$ is the one-letter word $a$ for every letter $a$.","title":"Identity morphism"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.conjecture","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#conjecture","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"fully_proved","summary":"The proposition that there exist a primitive morphism $\\varphi$ on the alphabet $\\{0, 1, 2\\}$ and an infinite word $u$ such that $u$ is a fixed point of $\\varphi$, $u$ is palindromic and uniformly recurrent, and no morphism $\\psi$ that has $u$ as a fixed point and is not the identity is conjugate to a morphism in class P.","title":"Palindromic uniformly recurrent fixed point outside class P"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.PalindromicRecurrentOutsidePWanted.conjecture_holds","parent":23872,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms/index.html#conjecture-holds","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/palindromic-fixed-points-of-morphisms","status":"can_prove","summary":"The conjecture holds: there exist a primitive morphism $\\varphi$ on the alphabet $\\{0, 1, 2\\}$ (some power $\\varphi^n$ maps every letter to a word containing every letter) and an infinite word $u$ such that $u$ is a fixed point of $\\varphi$, $u$ is palindromic (it has arbitrarily long palindromic factors) and uniformly recurrent (for every factor $w$ of $u$ there is $B$ such that, for every $k$, $w$ occurs in $u$\u2026","title":"Labb\u00e9's palindromic fixed point outside class P"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words","status":"fully_proved","title":"Sturmian words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/factor-ones","kind":"node","lean":"MetaMathlibExt.factorOnes","parent":23890,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/index.html#factor-ones","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words","status":"fully_proved","summary":"For an infinite binary word $x = x_0 x_1 x_2 \\cdots$ over $\\{0, 1\\}$ and natural numbers $i$ and $n$, this is the number of indices $k < n$ with $x_{i+k} = 1$, that is, the number of $1$s in the factor $x_i x_{i+1} \\cdots x_{i+n-1}$ of length $n$ starting at position $i$ (it is $0$ when $n = 0$).","title":"Number of ones in a factor"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/is-balanced","kind":"node","lean":"MetaMathlibExt.IsBalanced","parent":23890,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/index.html#is-balanced","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words","status":"fully_proved","summary":"An infinite binary word $x = x_0 x_1 x_2 \\cdots$ over $\\{0, 1\\}$ is balanced when for all natural numbers $i$, $j$ and $n$, the numbers of $1$s in the factors $x_i x_{i+1} \\cdots x_{i+n-1}$ and $x_j x_{j+1} \\cdots x_{j+n-1}$ of length $n$ differ by at most $1$.","title":"Balanced infinite words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/is-aperiodic","kind":"node","lean":"MetaMathlibExt.IsAperiodic","parent":23890,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/index.html#is-aperiodic","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words","status":"fully_proved","summary":"An infinite binary word $x = x_0 x_1 x_2 \\cdots$ over $\\{0, 1\\}$ is aperiodic when it is not ultimately periodic: there are no natural numbers $p > 0$ and $N$ such that $x_{n+p} = x_n$ for all $n \\ge N$.","title":"Aperiodic infinite words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/is-sturmian","kind":"node","lean":"MetaMathlibExt.IsSturmian","parent":23890,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words/index.html#is-sturmian","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/sturmian-words","status":"fully_proved","summary":"An infinite binary word $x = x_0 x_1 x_2 \\cdots$ over $\\{0, 1\\}$ is Sturmian when it is both aperiodic and balanced.","title":"Sturmian words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","title":"Thue\u2013Morse sequence and its generalizations"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/shift-letter","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.shiftLetter","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#shift-letter","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"For $L \\in \\mathbb{N}$ and letters $a, t$ of the alphabet $\\{0, \\dots, L-1\\}$, the cyclic shift of $a$ by $t$ is the letter $(a + t) \\bmod L$. Writing the letters as $a_0, \\dots, a_{L-1}$, it sends $a_i$ to $a_{(i + t) \\bmod L}$.","title":"Cyclic letter shift"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/shift-word","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.shiftWord","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#shift-word","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $L$ be a natural number and let $t$ be a letter of the alphabet $\\{0, 1, \\dots, L - 1\\}$. The cyclic shift by $t$ of a word $w = a_1 a_2 \\cdots a_n$ over this alphabet is the word $\\sigma_t(a_1) \\sigma_t(a_2) \\cdots \\sigma_t(a_n)$ of the same length, where $\\sigma_t(a) = (a + t) \\bmod L$ is the cyclic shift of a letter.","title":"Cyclic shift of a word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/shift-word-mem","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.shiftWord_mem","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#shift-word-mem","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $L$ be a natural number and let $t$ and $b$ be letters of the alphabet $\\{0, 1, \\dots, L - 1\\}$. For a letter $a$ write $\\sigma_t(a) = (a + t) \\bmod L$ for its cyclic shift by $t$, and for a word $w$ over this alphabet write $\\sigma_t(w)$ for the word obtained by shifting each letter of $w$. Then $b$ occurs in $\\sigma_t(w)$ if and only if $b = \\sigma_t(a)$ for some letter $a$ occurring in $w$.","title":"Letters of a cyclically shifted word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/shift-word-length","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.shiftWord_length","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#shift-word-length","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"For every $L \\in \\mathbb{N}$, every letter $t$ of the alphabet $\\{0, \\dots, L-1\\}$ and every finite word $w$ over that alphabet, the word obtained from $w$ by shifting each letter cyclically by $t$ has the same length as $w$.","title":"Cyclic shifting preserves word length"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/shift-letter-val","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.shiftLetter_val","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#shift-letter-val","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"For every $L \\in \\mathbb{N}$ and all letters $a, t$ of the alphabet $\\{0, \\dots, L-1\\}$, the cyclic shift of $a$ by $t$ is the letter $(a + t) \\bmod L$.","title":"Value of the cyclic letter shift"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/block","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.block","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#block","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $L > 1$ and $k > 1$ be natural numbers, take the letters $0, 1, \\dots, L-1$ (the elements of $\\mathbb{Z}/L\\mathbb{Z}$), and let $\\kappa$ assign a shift $\\kappa(j, n) \\in \\mathbb{Z}/L\\mathbb{Z}$ to each $j \\in \\{1, \\dots, k-1\\}$ and each $n \\ge 0$. For a word $w$ and $t \\in \\mathbb{Z}/L\\mathbb{Z}$, let $f^t(w)$ be the word obtained from $w$ by adding $t$ modulo $L$ to every letter. The blocks $A_n$ of the\u2026","title":"Blocks of the generalized Thue\u2013Morse sequence"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/length-flatten-eq","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.length_flatten_eq","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#length-flatten-eq","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $l$ be a list of lists over a type and let $m \\in \\mathbb{N}$. If every list in $l$ has length $m$, then the concatenation of the lists in $l$ has length $n \\cdot m$, where $n$ is the number of lists in $l$.","title":"Length of a concatenation of equal-length lists"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/block-length","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.block_length","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#block-length","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $L > 1$ and $k > 1$ be natural numbers and let $\\kappa$ assign a shift $\\kappa(j, n) \\in \\mathbb{Z}/L\\mathbb{Z}$ to each $j \\in \\{1, \\dots, k-1\\}$ and each $n \\ge 0$. For every $n \\ge 0$, the block $A_n$ of the $(L, k, \\kappa)$-Thue\u2013Morse sequence has exactly $k^n$ letters.","title":"Length of the generalized Thue\u2013Morse blocks"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/block-prefix-succ","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.block_prefix_succ","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#block-prefix-succ","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $L > 1$ and $k > 1$ be natural numbers and let $\\kappa$ assign a shift $\\kappa(j, n) \\in \\mathbb{Z}/L\\mathbb{Z}$ to each $j \\in \\{1, \\dots, k-1\\}$ and each $n \\ge 0$. For every $n \\ge 0$, the block $A_n$ of the $(L, k, \\kappa)$-Thue\u2013Morse sequence is a prefix of $A_{n+1}$: there is a word $w$ with $A_{n+1} = A_n w$.","title":"Each generalized Thue\u2013Morse block is a prefix of the next"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/block-prefix","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.block_prefix","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#block-prefix","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $L > 1$ and $k > 1$ be natural numbers and let $\\kappa$ assign a shift $\\kappa(j, n) \\in \\mathbb{Z}/L\\mathbb{Z}$ to each $j \\in \\{1, \\dots, k-1\\}$ and each $n \\ge 0$. For all natural numbers $m \\le n$, the block $A_m$ of the $(L, k, \\kappa)$-Thue\u2013Morse sequence is a prefix of $A_n$: there is a word $w$ with $A_n = A_m w$.","title":"The generalized Thue\u2013Morse blocks form a chain of prefixes"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/lt-pow-succ-self","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.lt_pow_succ_self","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#lt-pow-succ-self","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"For every natural number $k > 1$ and every $m \\in \\mathbb{N}$, we have $m < k^{m+1}$.","title":"A crude exponential bound"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/get-append-left","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.get_append_left","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#get-append-left","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $l_1$ and $l_2$ be lists over a type, and let $m$ be a natural number less than the length of $l_1$. Then the entry at position $m$ of the concatenation of $l_1$ followed by $l_2$ equals the entry at position $m$ of $l_1$ (positions start at $0$).","title":"Indexing into a concatenation"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/inf-letter","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.infLetter","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#inf-letter","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $L, k \\ge 2$, let $\\kappa : \\{1, \\dots, k-1\\} \\times \\mathbb{N} \\to \\{0, \\dots, L-1\\}$ be a shift schedule, and let $A_0, A_1, \\dots$ be the blocks of the $(L, k, \\kappa)$-Thue\u2013Morse sequence. The infinite word $A_\\infty$ over the alphabet $\\{0, \\dots, L-1\\}$ has, for every $m \\in \\mathbb{N}$, as its letter at position $m$ (positions start at $0$) the letter at position $m$ of the block $A_{m+1}$, which has\u2026","title":"The infinite generalized Thue\u2013Morse word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/inf-letter-coherence","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.infLetter_coherence","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#inf-letter-coherence","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $L, k \\ge 2$, let $\\kappa : \\{1, \\dots, k-1\\} \\times \\mathbb{N} \\to \\{0, \\dots, L-1\\}$ be a shift schedule, and let $A_0, A_1, \\dots$ be the blocks and $A_\\infty$ the infinite word of the $(L, k, \\kappa)$-Thue\u2013Morse sequence. For all $m, N \\in \\mathbb{N}$ with $m + 1 \\le N$ and $m$ less than the length of $A_N$, the letter at position $m$ of $A_N$ equals the letter at position $m$ of $A_\\infty$ (positions start\u2026","title":"Later blocks agree with the infinite word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/classical-kappa","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.classicalKappa","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#classical-kappa","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"The classical shift schedule is the map $\\kappa$ that assigns to the index $j = 1$ and every $n \\in \\mathbb{N}$ the letter $\\kappa(1, n) = 1$ of the alphabet $\\{0, 1\\}$. It is the schedule of the classical Thue\u2013Morse sequence, the $(L, k, \\kappa)$-Thue\u2013Morse sequence with $L = k = 2$.","title":"Classical shift schedule"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/classical-block","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.classicalBlock","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#classical-block","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"The classical Thue\u2013Morse blocks are the blocks $A_n$ of the $(2, 2, \\kappa)$-Thue\u2013Morse sequence with $\\kappa(1, n) = 1$ for every $n \\ge 0$. Thus $A_0$ is the one-letter word $0$, and $A_{n+1}$ is $A_n$ followed by the word obtained from $A_n$ by exchanging the letters $0$ and $1$.","title":"Classical Thue\u2013Morse blocks"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/classical-length","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.classical_length","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#classical-length","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the classical Thue\u2013Morse block $A_n$ (the block $A_n$ of the $(L, k, \\kappa)$-Thue\u2013Morse sequence with $L = k = 2$ and $\\kappa(1, i) = 1$ for all $i \\in \\mathbb{N}$) has exactly $2^n$ letters.","title":"Length of the classical Thue\u2013Morse blocks"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/classical-a0","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.classical_A0","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#classical-a0","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"The classical Thue\u2013Morse block $A_0$ is the one-letter word $0$.","title":"Classical Thue\u2013Morse block of order 0"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/classical-a1","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.classical_A1","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#classical-a1","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"The classical Thue\u2013Morse block $A_1$ is the word $01$.","title":"Classical Thue\u2013Morse block of order 1"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/classical-a2","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.classical_A2","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#classical-a2","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"The classical Thue\u2013Morse block $A_2$ is the word $0110$.","title":"Classical Thue\u2013Morse block of order 2"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/classical-a3","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.classical_A3","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#classical-a3","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"The classical Thue\u2013Morse block $A_3$ is the word $01101001$.","title":"Classical Thue\u2013Morse block of order 3"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/classical-inf-init","kind":"node","lean":"MetaMathlibExt.GeneralizedThueMorse.classical_inf_init","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#classical-inf-init","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $A_\\infty$ be the infinite word of the $(2, 2, \\kappa)$-Thue\u2013Morse sequence with $\\kappa(1, n) = 1$ for every $n \\ge 0$. The letters of $A_\\infty$ at positions $0, 1, \\dots, 7$ are $0, 1, 1, 0, 1, 0, 0, 1$, so $A_\\infty$ begins with $01101001$.","title":"The classical Thue\u2013Morse word begins with 01101001"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/thue-morse-substitution","kind":"node","lean":"MetaMathlibExt.thueMorseSubstitution","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#thue-morse-substitution","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"The Thue\u2013Morse substitution $\\sigma$ on the alphabet $\\{0, 1\\}$ sends each letter to a word of length two: $\\sigma(0) = 01$ and $\\sigma(1) = 10$.","title":"Thue\u2013Morse substitution"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/thue-morse-block","kind":"node","lean":"MetaMathlibExt.thueMorseBlock","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#thue-morse-block","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"Let $\\sigma$ be the Thue\u2013Morse substitution, $\\sigma(0) = 01$ and $\\sigma(1) = 10$, applied to finite words over $\\{0, 1\\}$ letter by letter. For a natural number $n$, the $n$th Thue\u2013Morse block is the finite word $\\sigma^n(0)$: the $0$th block is the one-letter word $0$, and the $(n+1)$st block is obtained from the $n$th by replacing each letter $a$ with $\\sigma(a)$.","title":"Thue\u2013Morse blocks"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/thue-morse","kind":"node","lean":"MetaMathlibExt.thueMorse","parent":23895,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations/index.html#thue-morse","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/thue-morse-sequence-and-its-generalizations","status":"fully_proved","summary":"The Thue\u2013Morse sequence $t_0 t_1 t_2 \\cdots$ over $\\{0, 1\\}$ is indexed from $0$, and $t_n = 1$ exactly when the binary expansion of $n$ has an odd number of digits equal to $1$; otherwise $t_n = 0$.","title":"Thue\u2013Morse sequence"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","title":"Reduced factor complexity"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/red-word","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.redWord","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#red-word","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"fully_proved","summary":"The reduction $\\operatorname{red}(w)$ of a binary word $w$ is the word obtained by collapsing each maximal run of equal consecutive letters of $w$ to a single letter; for example $\\operatorname{red}(0011101) = 0101$.","title":"Reduction of a binary word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/alternations","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.alternations","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#alternations","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"fully_proved","summary":"The number of alternations of a binary word $w$ is the number of occurrences of the factors $01$ and $10$ in $w$, that is, the number of positions $i$ with $w_i \\ne w_{i+1}$.","title":"Alternations of a binary word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/mu-word","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.muWord","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#mu-word","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"fully_proved","summary":"The morphism $\\mu$ sends each letter to a word of length two, $\\mu(0) = 01$ and $\\mu(1) = 10$, and a binary word $w_1 w_2 \\cdots w_k$ to the concatenation $\\mu(w_1) \\mu(w_2) \\cdots \\mu(w_k)$.","title":"Extension of the Thue\u2013Morse substitution to binary words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/paperfolding-seq","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.paperfoldingSeq","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#paperfolding-seq","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"fully_proved","summary":"The regular paperfolding sequence $f = f_0 f_1 f_2 \\cdots$ over $\\{0, 1\\}$ is indexed from $0$: for each $i$, write $i + 1 = m \\cdot 2^k$ with $m$ odd; then $f_i = 0$ if $m \\equiv 1 \\pmod 4$ and $f_i = 1$ if $m \\equiv 3 \\pmod 4$.","title":"Regular paperfolding sequence"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"fully_proved","summary":"For a finite binary word $w$, let $\\mathrm{red}(w)$ be the word obtained from $w$ by collapsing each run of equal letters to a single letter. For an infinite binary word $x = x_0 x_1 x_2 \\cdots$ and a natural number $n$, the reduced factor complexity $\\rho^{\\mathrm{red}}_x(n)$ is the number of distinct words $\\mathrm{red}(w)$ as $w$ ranges over the length-$n$ factors of $x$.","title":"Reduced factor complexity"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-ab-red","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoAbRed","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-ab-red","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"fully_proved","summary":"For a binary sequence $x = x_0 x_1 x_2 \\cdots$ and $n \\in \\mathbb{N}$, the reduced abelian complexity $\\rho^{\\mathrm{ab,red}}_x(n)$ is the number of distinct pairs $(|\\mathrm{red}(v)|, |\\mathrm{red}(v)|_0)$, where $v$ ranges over the length-$n$ factors of $x$, $\\mathrm{red}(v)$ is the reduction of $v$ (each run of equal letters collapsed to a single letter) and $|w|_0$ is the number of letters $0$ in $w$.\u2026","title":"Reduced abelian complexity"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/min-alt","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.minAlt","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#min-alt","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"fully_proved","summary":"For a natural number $n$, $m_n$ is the least number of alternations (positions where two adjacent letters differ) among the length-$n$ factors of the Thue\u2013Morse sequence.","title":"Minimum number of alternations in Thue\u2013Morse factors"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/max-alt","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.maxAlt","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#max-alt","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"fully_proved","summary":"For a natural number $n$, $M_n$ is the largest number of alternations (positions where two adjacent letters differ) among the length-$n$ factors of the Thue\u2013Morse sequence.","title":"Maximum number of alternations in Thue\u2013Morse factors"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/alternations-mu-word","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.alternations_muWord","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#alternations-mu-word","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $\\mu$ be the morphism on binary words with $\\mu(0) = 01$ and $\\mu(1) = 10$. If $w$ is a nonempty binary word of length $|w|$ with $\\alpha$ alternations (positions where two adjacent letters differ), then $\\mu(w)$ has exactly $2|w| - 1 - \\alpha$ alternations.","title":"Alternations of the Thue\u2013Morse image of a word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/min-alt-max-alt-two-mul","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.minAlt_maxAlt_two_mul","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#min-alt-max-alt-two-mul","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"For a natural number $k$, let $m_k$ and $M_k$ be the least and the largest number of alternations (positions where two adjacent letters differ) among the length-$k$ factors of the Thue\u2013Morse sequence. Then for every $n \\ge 2$, $m_{2n} = 2n - 1 - M_{n+1}$, $M_{2n} = 2n - 1 - m_n$, $m_{2n+1} = 2n - M_{n+1}$ and $M_{2n+1} = 2n - m_{n+1}$.","title":"Doubling recurrences for Thue\u2013Morse alternation extrema"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/min-alt-max-alt-four-mul","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.minAlt_maxAlt_four_mul","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#min-alt-max-alt-four-mul","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"For a natural number $k$, let $m_k$ and $M_k$ be the least and the largest number of alternations (positions where two adjacent letters differ) among the length-$k$ factors of the Thue\u2013Morse sequence. Then for every $n \\ge 1$, $m_{4n} = 2n - 1 + m_{n+1}$, $M_{4n} = 2n + M_{n+1}$, $m_{4n+2} = 2n + m_{n+1}$ and $M_{4n+2} = 2n + 1 + M_{n+1}$.","title":"Quadrupling recurrences for Thue\u2013Morse alternation extrema"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red-thue-morse-odd","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed_thueMorse_odd","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red-thue-morse-odd","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $t$ be the Thue\u2013Morse sequence and $\\rho^{\\mathrm{red}}_t$ its reduced factor complexity. For every odd natural number $n$, $\\rho^{\\mathrm{red}}_t(n) = \\rho^{\\mathrm{red}}_t((n+1)/2)$.","title":"Reduced complexity of the Thue\u2013Morse word at odd lengths"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red-thue-morse-even","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed_thueMorse_even","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red-thue-morse-even","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $t$ be the Thue\u2013Morse sequence and $\\rho^{\\mathrm{red}}_t$ its reduced factor complexity. For every natural number $m \\ge 1$, $\\rho^{\\mathrm{red}}_t(4m) = \\rho^{\\mathrm{red}}_t(m+1) + 2$, and for every natural number $m \\ge 0$, $\\rho^{\\mathrm{red}}_t(4m+2) = \\rho^{\\mathrm{red}}_t(m+1) + 2$.","title":"Reduced complexity of the Thue\u2013Morse word at even lengths"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/red-word-length-alternating-even","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.redWord_length_alternating_even","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#red-word-length-alternating-even","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $n$ be a natural number and let $w = w_0 w_1 \\cdots w_{2n}$ be a binary word of length $2n + 1$ with $w_{2i} \\ne w_{2i+2}$ for every $i < n$, so that its even positions read $abab\\cdots$ with $a \\ne b$ and its odd positions are arbitrary. Then the reduction of $w$, the word obtained by collapsing each maximal run of equal letters of $w$ to a single letter, has length $n + 1$; that is, $w$ has exactly $n + 1$\u2026","title":"Runs in a word whose even-position letters alternate"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red-paperfolding-two-mul","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed_paperfolding_two_mul","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red-paperfolding-two-mul","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{red}}_f(n)$ be the number of distinct reductions of length-$n$ factors of $f$. For every $n \\ge 1$, $\\rho^{\\mathrm{red}}_f(2n) = 4$.","title":"Reduced complexity of the paperfolding word at even lengths"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red-paperfolding-eight-mul-add-one","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed_paperfolding_eight_mul_add_one","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red-paperfolding-eight-mul-add-one","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{red}}_f(n)$ be the number of distinct reductions of length-$n$ factors of $f$. For every $n \\ge 1$, $\\rho^{\\mathrm{red}}_f(8n+1) = 4$.","title":"Reduced complexity of the paperfolding word at lengths 8n + 1"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red-paperfolding-eight-mul-add-three","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed_paperfolding_eight_mul_add_three","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red-paperfolding-eight-mul-add-three","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{red}}_f(n)$ be the number of distinct reductions of length-$n$ factors of $f$. For every natural number $n$, $\\rho^{\\mathrm{red}}_f(8n+3) = 6$.","title":"Reduced complexity of the paperfolding word at lengths 8n + 3"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red-paperfolding-eight-mul-add-five","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed_paperfolding_eight_mul_add_five","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red-paperfolding-eight-mul-add-five","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{red}}_f(n)$ be the number of distinct reductions of length-$n$ factors of $f$. For every natural number $n$, $\\rho^{\\mathrm{red}}_f(8n+5) = 6$.","title":"Reduced complexity of the paperfolding word at lengths 8n + 5"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red-paperfolding-eight-mul-add-seven","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed_paperfolding_eight_mul_add_seven","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red-paperfolding-eight-mul-add-seven","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{red}}_f(n)$ be the number of distinct reductions of length-$n$ factors of $f$. For every natural number $n$, $\\rho^{\\mathrm{red}}_f(8n+7) = 6$.","title":"Reduced complexity of the paperfolding word at lengths 8n + 7"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-red-paperfolding-mod-eight","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoRed_paperfolding_mod_eight","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-red-paperfolding-mod-eight","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{red}}_f(n)$ be the number of distinct reductions of length-$n$ factors of $f$. For every $n \\ge 2$, $\\rho^{\\mathrm{red}}_f(n) = 4$ if $n \\equiv 0, 1, 2, 4$ or $6 \\pmod 8$, and $\\rho^{\\mathrm{red}}_f(n) = 6$ if $n \\equiv 3, 5$ or $7 \\pmod 8$.","title":"Reduced complexity of the paperfolding word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-ab-red-paperfolding-even","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoAbRed_paperfolding_even","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-ab-red-paperfolding-even","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{ab,red}}_f(n)$ be the number of reductions of length-$n$ factors of $f$, counted up to rearrangement of their letters. For every even $n > 0$, $\\rho^{\\mathrm{ab,red}}_f(n) = 3$.","title":"Reduced abelian complexity of the paperfolding word at even lengths"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-ab-red-paperfolding-mod-four-one","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoAbRed_paperfolding_mod_four_one","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-ab-red-paperfolding-mod-four-one","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{ab,red}}_f(n)$ be the number of reductions of length-$n$ factors of $f$, counted up to rearrangement of their letters. For every $n > 1$ with $n \\equiv 1 \\pmod 4$, $\\rho^{\\mathrm{ab,red}}_f(n) = 4$.","title":"Reduced abelian complexity of the paperfolding word at lengths 1 mod 4"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/rho-ab-red-paperfolding-mod-four-three","kind":"node","lean":"MathlibExt.Combinatorics.ReducedComplexitiesWanted.rhoAbRed_paperfolding_mod_four_three","parent":23921,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity/index.html#rho-ab-red-paperfolding-mod-four-three","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/reduced-factor-complexity","status":"can_prove","summary":"Let $f$ be the regular paperfolding sequence and let $\\rho^{\\mathrm{ab,red}}_f(n)$ be the number of reductions of length-$n$ factors of $f$, counted up to rearrangement of their letters. For every natural number $n \\equiv 3 \\pmod 4$, $\\rho^{\\mathrm{ab,red}}_f(n) = 5$.","title":"Reduced abelian complexity of the paperfolding word at lengths 3 mod 4"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","title":"Relative position in binary words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/both-letters-infinite","kind":"node","lean":"MetaMathlibExt.BothLettersInfinite","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#both-letters-infinite","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"An infinite word $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ over the letters $a$ and $b$ belongs to the set $\\mathcal{W}$ when both letters occur infinitely often in it, that is, when there are infinitely many positions $i$ with $\\ell_i = a$ and infinitely many positions $i$ with $\\ell_i = b$.","title":"Binary words in which both letters occur infinitely often"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position","kind":"node","lean":"MetaMathlibExt.letterPosition","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"For an infinite binary word $w = \\ell_0 \\ell_1 \\cdots$, a letter $\\alpha$ and $n \\ge 1$, $p_\\alpha(n)$ is the position of the $n$-th occurrence of $\\alpha$ in $w$, counting positions from $0$; that is, $p_\\alpha(n)$ is the $n$-th smallest index $i$ with $\\ell_i = \\alpha$.","title":"Position function of a letter"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position","kind":"node","lean":"MetaMathlibExt.relativePosition","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over $\\{a, b\\}$, and for a letter $\\alpha$ and $n \\ge 1$ let $p_\\alpha(n)$ be the position (counting from $0$) of the $n$-th occurrence of $\\alpha$ in $w$, or $0$ if $\\alpha$ occurs fewer than $n$ times. The relative position function of $w$ is the integer-valued function $r(n) = p_b(n) - p_a(n)$, $n \\ge 1$.","title":"Relative position function"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-one-of-head","kind":"node","lean":"MetaMathlibExt.letterPosition_one_of_head","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-one-of-head","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"If an infinite binary word $w$ starts with the letter $\\alpha$, then the first occurrence of $\\alpha$ in $w$ is at position $0$, that is, $p_\\alpha(1) = 0$.","title":"Position of the first occurrence of the initial letter"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-one-of-head-ab","kind":"node","lean":"MetaMathlibExt.relativePosition_one_of_head_ab","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relative position of a word starting with "],["math","\\(ab\\)"]],"source":"Relative position of a word starting with $ab$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-one-of-head-ab","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"If an infinite word $w$ over $\\{a, b\\}$ starts with $ab$, then its relative position function satisfies $r(1) = 1$.","title":"Relative position of a word starting with $ab$"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/exists-word-letter-position-eq-iff","kind":"node","lean":"MetaMathlibExt.exists_word_letterPosition_eq_iff","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#exists-word-letter-position-eq-iff","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $p$ be a function from the natural numbers to the natural numbers. There is an infinite word $w$ over the letters $a$ and $b$ that starts with $a$, in which both letters occur infinitely often, and whose position function for $a$ satisfies $p_a(n) = p(n)$ for all $n \\ge 1$, if and only if $p(n) < p(n+1)$ for all $n \\ge 1$, $p(1) = 0$, and $p(n+1) - p(n) > 1$ for infinitely many $n \\ge 1$. Likewise, there is an\u2026","title":"Characterization of the position functions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/eq-iff-relative-position-eq","kind":"node","lean":"MetaMathlibExt.eq_iff_relativePosition_eq","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#eq-iff-relative-position-eq","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ and $w'$ be infinite words over the letters $a$ and $b$ in each of which both letters occur infinitely often, and let $r_w$ and $r_{w'}$ be their relative position functions. Then $w = w'$ if and only if $r_w(n) = r_{w'}(n)$ for every $n \\ge 1$.","title":"The relative position function determines the word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/exists-word-relative-position-eq-of-mono","kind":"node","lean":"MetaMathlibExt.exists_word_relativePosition_eq_of_monotone","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#exists-word-relative-position-eq-of-mono","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $r$ be a function from the natural numbers to the natural numbers with $r(n) \\ge 1$ and $r(n) \\le r(n+1)$ for every $n \\ge 1$. Then there is an infinite word $w$ over the letters $a$ and $b$ that starts with $a$, in which both letters occur infinitely often, and whose relative position function satisfies $r_w(n) = r(n)$ for every $n \\ge 1$.","title":"Non-decreasing functions as relative position functions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/exists-word-relative-position-eq-of-anti","kind":"node","lean":"MetaMathlibExt.exists_word_relativePosition_eq_of_antitone","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#exists-word-relative-position-eq-of-anti","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $r$ be a function from the natural numbers to the integers with $r(1) < 0$ and $r(n+1) \\le r(n)$ for every $n \\ge 1$. Then there is an infinite word $w$ over the letters $a$ and $b$ that starts with $b$, in which both letters occur infinitely often, and whose relative position function satisfies $r_w(n) = r(n)$ for every $n \\ge 1$.","title":"Non-increasing functions as relative position functions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/subword-run-iff-letter-position-diff","kind":"node","lean":"MetaMathlibExt.subword_run_iff_letterPosition_diff","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#subword-run-iff-letter-position-diff","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over the letters $a$ and $b$ in which both letters occur infinitely often and $\\ell_0 = a$. For a letter $\\alpha$ and $n \\ge 1$, let $p_\\alpha(n)$ be the position of the $n$-th occurrence of $\\alpha$ in $w$, counting positions from $0$. Let $k \\ge 1$. Then the following are equivalent: there is $i$ with $\\ell_{i+j} = b$ for all $0 \\le j < k$ (the block $b^k$\u2026","title":"Runs of one letter and gaps between letter positions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-monotonicity-runs","kind":"node","lean":"MetaMathlibExt.relativePosition_monotonicity_runs","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-monotonicity-runs","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite word over $\\{a, b\\}$ that starts with $a$ and in which both letters occur infinitely often, let $r$ be its relative position function, and write $\\Delta r(n) = r(n+1) - r(n)$. A factor of $w$ is a block of consecutive letters. (a) If $r(n) < r(n+1)$ for all $n \\ge 1$, then $bb$ is not a factor of $w$. (b) If $bb$ is not a factor of $w$, then $r(n) \\le r(n+1)$ for all $n \\ge 1$. (c) For every\u2026","title":"Monotonicity of the relative position function and runs of letters"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/abs-relative-position-diff-sub-le","kind":"node","lean":"MetaMathlibExt.abs_relativePosition_diff_sub_le","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#abs-relative-position-diff-sub-le","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over the letters $a$ and $b$ in which both letters occur infinitely often and whose first letter $\\ell_0$ is $a$. Let $p_b$ be its position function for the letter $b$, let $r$ be its relative position function, and write $\\Delta s(n) = s(n+1) - s(n)$. Let $k$ be a natural number such that $w$ contains no $k$ consecutive letters $b$, that is, there is no $i$\u2026","title":"Relative position increments without long runs of b"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/periodic-iff-letter-position-diff-period","kind":"node","lean":"MetaMathlibExt.periodic_iff_letterPosition_diff_periodic","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#periodic-iff-letter-position-diff-period","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = w_0 w_1 w_2 \\cdots$ be an infinite binary word over $\\{a, b\\}$ that starts with $a$ and in which both letters occur infinitely often. Let $p_\\alpha(n)$ be the position (counted from $0$) of the $n$-th occurrence of the letter $\\alpha$ in $w$, and $\\Delta p_\\alpha(n) = p_\\alpha(n+1) - p_\\alpha(n)$. Call a sequence $s$ periodic if there is $q > 0$ with $s(n + q) = s(n)$ for all $n \\ge 1$. The following are\u2026","title":"Periodicity of a word and of the gaps between occurrences of its letters"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-diff-periodic-of-perio","kind":"node","lean":"MetaMathlibExt.relativePosition_diff_periodic_of_periodic","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-diff-periodic-of-perio","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = w_0 w_1 w_2 \\cdots$ be an infinite word over $\\{a, b\\}$ in which both letters occur infinitely often, let $p_a(n)$ and $p_b(n)$ be the positions of the $n$-th $a$ and the $n$-th $b$ in $w$, let $r(n) = p_b(n) - p_a(n)$, and let $\\Delta r(n) = r(n + 1) - r(n)$. If $w$ is purely periodic (there is $q > 0$ with $w_{i + q} = w_i$ for all $i \\ge 0$), then there is $q > 0$ with $\\Delta r(n + q) = \\Delta r(n)$\u2026","title":"Periodicity of the relative position differences"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-le-of-run-le","kind":"node","lean":"MetaMathlibExt.letterPosition_le_of_run_le","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-le-of-run-le","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite binary word over $\\{a, b\\}$ that starts with $a$ and in which both letters occur infinitely often, and let $c \\ge 0$ be an integer such that every block of consecutive equal letters of $w$ has length at most $c$. Write $p_\\alpha(n)$ for the position (counted from $0$) of the $n$-th occurrence of the letter $\\alpha$ in $w$, and $r(n) = p_b(n) - p_a(n)$. Then for every $n \\ge 1$,\u2026","title":"Bounds on position functions for words with bounded runs"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-le-of-run-le-2","kind":"node","lean":"MetaMathlibExt.letterPosition_le_of_run_le'","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-le-of-run-le-2","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite binary word over the letters $a$ and $b$ in which both letters occur infinitely often, and let $c$ be a natural number such that every block of consecutive equal letters of $w$ has length at most $c$. Let $p_a(n)$ and $p_b(n)$ be the positions (counting from $0$) of the $n$-th $a$ and the $n$-th $b$ in $w$, and let $r(n) = p_b(n) - p_a(n)$. Then for every $n \\ge 1$, $p_a(n) \\le (c+1)n - 1$,\u2026","title":"Linear bounds for words with bounded runs"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/delete-first","kind":"node","lean":"MetaMathlibExt.deleteFirst","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#delete-first","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over the letters $a$ and $b$, let $\\alpha$ be a letter, and let $p$ be the position (counted from $0$) of the first occurrence of $\\alpha$ in $w$, with $p = 0$ if $\\alpha$ does not occur in $w$. The word $D_\\alpha(w)$ has letter $\\ell_i$ at each position $i < p$ and letter $\\ell_{i+1}$ at each position $i \\ge p$. So $D_\\alpha(w)$ is $w$ with its first\u2026","title":"Deleting the first occurrence of a letter"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/deletion","kind":"node","lean":"MetaMathlibExt.deletion","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#deletion","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"For an infinite word $w$ over the letters $a$ and $b$, the word $D(w)$ is $D_a(D_b(w))$, where $D_\\alpha$ deletes the first occurrence of the letter $\\alpha$ (and deletes the first letter of a word in which $\\alpha$ does not occur): first apply $D_b$, then $D_a$. When both letters occur in $w$, $D(w)$ is $w$ with its first $b$ and its first $a$ deleted.","title":"The deletion operator"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/delete-first-of-head","kind":"node","lean":"MetaMathlibExt.deleteFirst_of_head","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#delete-first-of-head","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over the letters $a$ and $b$, let $\\alpha$ be a letter, and let $D_\\alpha$ be the operator that deletes the first occurrence of $\\alpha$ from a word (and deletes the first letter of a word in which $\\alpha$ does not occur). If $\\ell_0 = \\alpha$, then $D_\\alpha$ shifts $w$ by one: $D_\\alpha(w) = \\ell_1 \\ell_2 \\ell_3 \\cdots$, that is, the letter of\u2026","title":"Deleting a leading letter"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-count","kind":"node","lean":"MetaMathlibExt.letterCount","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-count","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite binary word, $\\alpha$ a letter and $m$ a natural number. The counting function $\\#_\\alpha(m)$ is the number of indices $i < m$ with $\\ell_i = \\alpha$, that is, the number of letters $\\alpha$ among the first $m$ letters $\\ell_0, \\dots, \\ell_{m-1}$ of $w$.","title":"Letter-counting function"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-count-zero","kind":"node","lean":"MetaMathlibExt.letterCount_zero","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-count-zero","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"For every infinite binary word $w$ and every letter $\\alpha$, $\\#_\\alpha(0) = 0$, where $\\#_\\alpha(m)$ is the number of letters $\\alpha$ among the first $m$ letters of $w$.","title":"The letter-counting function at zero"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-count-succ","kind":"node","lean":"MetaMathlibExt.letterCount_succ","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-count-succ","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite binary word, $\\alpha$ a letter and $m$ a natural number, and let $\\#_\\alpha(m)$ be the number of letters $\\alpha$ among $\\ell_0, \\dots, \\ell_{m-1}$. Then $\\#_\\alpha(m+1)$ equals $\\#_\\alpha(m) + 1$ if $\\ell_m = \\alpha$, and $\\#_\\alpha(m)$ otherwise.","title":"Recursion for the letter-counting function"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-reconstruction","kind":"node","lean":"MetaMathlibExt.letterPosition_reconstruction","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-reconstruction","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite binary word over $\\{a, b\\}$ in which both letters occur infinitely often, let $p_a(n)$ and $p_b(n)$ be the positions (counted from $0$) of the $n$-th $a$ and the $n$-th $b$ in $w$, and let $r(n) = p_b(n) - p_a(n)$. If $r(1) > 0$ then $p_a(1) = 0$ and $p_b(1) = r(1)$; if $r(1) < 0$ then $p_a(1) = -r(1)$ and $p_b(1) = 0$. For every $n \\ge 1$, let $k_{n+1}$ be the least natural number different\u2026","title":"Reconstruction of a binary word from its relative position function"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/reflection-image-and-relative-position","kind":"node","lean":"MetaMathlibExt.reflection_image_and_relativePosition","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#reflection-image-and-relative-position","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $\\mathcal{W}$ be the set of infinite words over $\\{a, b\\}$ in which both letters occur infinitely often, and let $a\\mathcal{W}$ and $b\\mathcal{W}$ be its words that begin with $a$ and with $b$. For a word $w$ let $\\bar w$ be its reflection, obtained by exchanging $a$ and $b$ at every position, and let $r_w(n) = p_b(n) - p_a(n)$ be its relative position function, where $p_a(n)$ and $p_b(n)$ are the positions of\u2026","title":"Reflection of binary words and relative position"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-diff-periodic-iff-delete","kind":"node","lean":"MetaMathlibExt.letterPosition_diff_periodic_iff_deleteFirst_periodic","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periodicity of the gaps between the "],["math","\\(b\\)"],["text","'s"]],"source":"Periodicity of the gaps between the $b$'s"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-diff-periodic-iff-delete","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite binary word over the letters $a$ and $b$ in which both letters occur infinitely often, let $p_b(n)$ be the position (counting from $0$) of the $n$-th $b$ in $w$, and let $\\Delta p_b(n) = p_b(n+1) - p_b(n)$. Let $D_a$ delete the first $a$ of a word, and let $v = D_a^{p_b(1)}(w)$, the word $w$ with its initial run of $a$'s deleted. Then there is $q > 0$ with $\\Delta p_b(n + q) = \\Delta p_b(n)$\u2026","title":"Periodicity of the gaps between the $b$'s"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/is-equilibrious-word-iff-deletion-iterat","kind":"node","lean":"MetaMathlibExt.isEquilibriousWord_iff_deletion_iterate_prepend","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#is-equilibrious-word-iff-deletion-iterat","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $u$ be a finite word in the letters $a$ and $b$ of length $2k$. Then $u$ is equilibrious, that is, $a$ and $b$ occur equally often in $u$, if and only if $D^k(uw) = w$ for every infinite binary word $w$ in which both letters occur infinitely often, where $D$ deletes the first $b$ and the first $a$ of a word.","title":"Prepending an equilibrious word is undone by deletion"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-prepend-equilibrious","kind":"node","lean":"MetaMathlibExt.letterPosition_prepend_equilibrious","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-prepend-equilibrious","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $u$ be a finite word over $\\{a, b\\}$ of length $2k$ that is equilibrious (it contains as many letters $a$ as letters $b$), and let $w$ be an infinite binary word in which both letters occur infinitely often. For a word $v$, write $p_{\\alpha,v}(n)$ for the position (counted from $0$) of the $n$-th occurrence of the letter $\\alpha$ in $v$, and $r_v(n) = p_{b,v}(n) - p_{a,v}(n)$. Then for every $n \\ge 1$ the\u2026","title":"Prepending an equilibrious word shifts the position functions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-clone","kind":"node","lean":"MetaMathlibExt.letterPosition_clone","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-clone","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite binary word over the letters $a$ and $b$ in which both letters occur infinitely often, let $k$ be a natural number, and let $\\varphi_k(w)$ be the word whose letter at position $i$ is the letter of $w$ at position $\\lfloor i/k \\rfloor$ (for $k \\ge 1$, the word obtained from $w$ by repeating each letter $k$ times). For a word $v$ and a letter $\\alpha$, let $p_{\\alpha,v}(n)$ be the position\u2026","title":"Position functions of a cloned word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/tendsto-letter-count-div-iff-tendsto-let","kind":"node","lean":"MetaMathlibExt.tendsto_letterCount_div_iff_tendsto_letterPosition_div","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#tendsto-letter-count-div-iff-tendsto-let","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over the letters $a$ and $b$ in which both letters occur infinitely often, and let $\\alpha$ be a letter. Write $\\#_\\alpha(m)$ for the number of occurrences of $\\alpha$ among $\\ell_0, \\ldots, \\ell_{m-1}$, and for $n \\ge 1$ write $p_\\alpha(n)$ for the position of the $n$-th occurrence of $\\alpha$ in $w$, counting positions from $0$. Then (a) for every real $d$\u2026","title":"Letter frequency and growth of letter positions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/limsup-liminf-letter-count-div","kind":"node","lean":"MetaMathlibExt.limsup_liminf_letterCount_div","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#limsup-liminf-letter-count-div","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite binary word in which both letters occur infinitely often and let $\\alpha$ be a letter. Write $\\#_\\alpha(m)$ for the number of occurrences of $\\alpha$ among the first $m$ letters of $w$, and $p_\\alpha(n)$ for the position (counted from $0$) of the $n$-th occurrence of $\\alpha$. Then $\\limsup_{m \\to \\infty} \\#_\\alpha(m)/m = \\limsup_{n \\to \\infty} n/p_\\alpha(n)$ and\u2026","title":"Upper and lower densities of a letter from its position function"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-frequency-tfae","kind":"node","lean":"MetaMathlibExt.letter_frequency_tfae","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-frequency-tfae","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite binary word over the letters $a$ and $b$ in which both letters occur infinitely often, and let $0 < d < 1$. For a letter $\\alpha$, let $\\#_\\alpha(m)$ be the number of letters $\\alpha$ among the first $m$ letters of $w$, and let $p_\\alpha(n)$ be the position (counting from $0$) of the $n$-th occurrence of $\\alpha$ in $w$. Then the following are equivalent: $\\#_b(m)/m \\to d$ as $m \\to \\infty$;\u2026","title":"Letter frequencies and the growth of the position functions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/tendsto-relative-position-div-of-frequen","kind":"node","lean":"MetaMathlibExt.tendsto_relativePosition_div_of_frequency","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#tendsto-relative-position-div-of-frequen","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over the letters $a$ and $b$. Let $\\#_b(m)$ be the number of letters $b$ among $\\ell_0, \\ldots, \\ell_{m-1}$, and for $n \\ge 1$ let $r(n) = p_b(n) - p_a(n)$, where $p_\\alpha(n)$ is the position of the $n$-th occurrence of the letter $\\alpha$ in $w$, counting positions from $0$. If $0 < d < 1$ and $\\#_b(m)/m \\to d$ as $m \\to \\infty$, then\u2026","title":"Relative position of a word with given letter frequency"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/frequency-of-tendsto-relative-position-d","kind":"node","lean":"MetaMathlibExt.frequency_of_tendsto_relativePosition_div","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","The frequency of "],["math","\\(b\\)"],["text"," from the growth of the relative position"]],"source":"The frequency of $b$ from the growth of the relative position"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#frequency-of-tendsto-relative-position-d","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite binary word in which both letters $a$ and $b$ occur infinitely often, let $r$ be its relative position function, and suppose that $r(n)/n$ converges to $L$ in the extended real line $[-\\infty, \\infty]$. Let $\\#_b(m)$ be the number of letters $b$ among the first $m$ letters of $w$. Then $\\#_b(m)/m$ converges, and its limit, the frequency of $b$ in $w$, is $0$ if $L = \\infty$, $1$ if\u2026","title":"The frequency of $b$ from the growth of the relative position"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-letter-position-mean-v","kind":"node","lean":"MetaMathlibExt.relativePosition_letterPosition_mean_values","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-letter-position-mean-v","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w$ be an infinite word over $\\{a, b\\}$ in which both letters occur infinitely often, with letter position functions $p_a$, $p_b$ and relative position function $r$, and consider the sequences $r(n)/n$, $p_b(n)/n$ and $p_a(n)/n$ in the extended reals $[-\\infty, \\infty]$. If one of the three sequences converges, then so do the other two. If $p_b(n)/n \\to p$ and $p_a(n)/n \\to q$, then $1/p + 1/q = 1$, where\u2026","title":"Mean values of the position functions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/is-substitution-image","kind":"node","lean":"MetaMathlibExt.IsSubstitutionImage","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#is-substitution-image","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"Let $\\varrho$ be a substitution on the letters $a$ and $b$, sending each letter to a finite word and extended to finite words by concatenation, and let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ and $v$ be infinite binary words. Then $v$ is the image $\\varrho(w)$ of $w$ when, for every $m$, the finite word $\\varrho(\\ell_0 \\cdots \\ell_{m-1})$ is a prefix of $v$, and these words are arbitrarily long: for every $N$ there is an\u2026","title":"Image of an infinite word under a substitution"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/is-substitution-image-id","kind":"node","lean":"MetaMathlibExt.isSubstitutionImage_id","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#is-substitution-image-id","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"fully_proved","summary":"Every infinite binary word $w$ is its own image under the identity substitution $a \\mapsto a$, $b \\mapsto b$.","title":"Every word is fixed by the identity substitution"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/frequency-substitution-image","kind":"node","lean":"MetaMathlibExt.frequency_substitution_image","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#frequency-substitution-image","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $\\mu$ be a substitution sending the letters $a$ and $b$ to nonempty finite words, extended to words by concatenation, such that both $a$ and $b$ occur in $\\mu(a)\\mu(b)$. For a word $x$ and a letter $\\alpha$ write $\\#_\\alpha(x, m)$ for the number of letters $\\alpha$ among the first $m$ letters of $x$; for a finite word $u$ write $|u|$ for its length and $|u|_\\alpha$ for the number of letters $\\alpha$ in it. Let\u2026","title":"Letter frequencies of a substitution image"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/golden-eigenvector-iff","kind":"node","lean":"MetaMathlibExt.golden_eigenvector_iff","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#golden-eigenvector-iff","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $\\tau = (1 + \\sqrt 5)/2$ be the golden ratio and $\\tau' = (1 - \\sqrt 5)/2$ its conjugate, and let $M$ be a $2 \\times 2$ integer matrix. Then $(\\tau, 1)^T$ is a right eigenvector of $M$, that is, $M (\\tau, 1)^T = c\\,(\\tau, 1)^T$ for some real number $c$, if and only if","title":"Integer matrices with the golden eigenvector"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/pisa-eigenvector-iff","kind":"node","lean":"MetaMathlibExt.pisa_eigenvector_iff","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#pisa-eigenvector-iff","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $j, m \\ge 1$ be integers, let $\\tau_{j,m} = (j + \\sqrt{j^2 + 4m})/2$, and let $M$ be a $2 \\times 2$ integer matrix. (a) If $j^2 + 4m$ is not a perfect square, then $(\\tau_{j,m}, 1)^T$ is a right eigenvector of $M$ over $\\mathbb{R}$ if and only if $M$ has rows $(t + sj, ms)$ and $(s, t)$ for some integers $s, t$. (b) If $j^2 + 4m = r^2$ with $r$ a natural number, then $(\\tau_{j,m}, 1)^T$ is a right eigenvector\u2026","title":"Integer matrices with a Pisa eigenvector"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/metallic-eigenvector-iff","kind":"node","lean":"MetaMathlibExt.metallic_eigenvector_iff","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#metallic-eigenvector-iff","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $k \\ge 1$ be an integer, let $\\tau_k = (k + \\sqrt{k^2 + 4})/2$ and $\\tau_k' = (k - \\sqrt{k^2 + 4})/2$, and let $M$ be a $2 \\times 2$ integer matrix. Then $(\\tau_k, 1)^T$ is a right eigenvector of $M$ over $\\mathbb{R}$ if and only if $M$ has rows $(km + n, m)$ and $(m, n)$ for some integers $m, n$. For such $M$ the characteristic polynomial of $M$ is $(X - (n + m\\tau_k))(X - (n + m\\tau_k'))$, so the eigenvalues\u2026","title":"Integer matrices with a metallic-mean eigenvector"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/fibonacci-word-letter-position-diff","kind":"node","lean":"MetaMathlibExt.fibonacciWord_letterPosition_diff","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#fibonacci-word-letter-position-diff","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $f = \\ell_0 \\ell_1 \\ell_2 \\cdots = abaab\\cdots$ be the Fibonacci word, the infinite Fibonacci word with each $0$ written as $a$ and each $1$ as $b$. Let $p_a(n)$ and $p_b(n)$ be the positions (counted from $0$) of the $n$-th $a$ and the $n$-th $b$ in $f$, and let $r(n) = p_b(n) - p_a(n)$. Then for every $n \\ge 1$: $p_a(n+1) - p_a(n)$ is $2$ if $\\ell_{n-1} = a$ and $1$ if $\\ell_{n-1} = b$; $p_b(n+1) - p_b(n)$ is\u2026","title":"Position differences in the Fibonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-deletion-fibonacci-wor","kind":"node","lean":"MetaMathlibExt.relativePosition_deletion_fibonacciWord","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-deletion-fibonacci-wor","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $f = abaababaabaab\\cdots$ be the Fibonacci word, the fixed point of $a \\mapsto ab$, $b \\mapsto a$. Let $D(f)$ be the word obtained from $f$ by deleting its first $b$ and then the first $a$ of the result, and for an infinite word $v$ over $\\{a, b\\}$ let $r_v(n) = p_b(n) - p_a(n)$, where $p_a(n)$ and $p_b(n)$ are the positions of the $n$-th $a$ and the $n$-th $b$ in $v$. Then $r_{D(f)}(n) = n + 1$ for every\u2026","title":"Relative position of the deleted Fibonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-deletion-iterate-fibon","kind":"node","lean":"MetaMathlibExt.relativePosition_deletion_iterate_fibonacciWord","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-deletion-iterate-fibon","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $f = abaababaabaab\\cdots$ be the Fibonacci word, the fixed point of $a \\mapsto ab$, $b \\mapsto a$. Let $D$ be the operator on infinite words over $\\{a, b\\}$ that deletes the first $b$ and then the first $a$ of the result, and for such a word $v$ let $r_v(n) = p_b(n) - p_a(n)$, where $p_a(n)$ and $p_b(n)$ are the positions of the $n$-th $a$ and the $n$-th $b$ in $v$. Then for every natural number $k$ there is an\u2026","title":"Relative position of iterated deletions of the Fibonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/fibonacci-switch","kind":"node","lean":"MetaMathlibExt.fibonacci_switch","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#fibonacci-switch","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $f = abaab\\cdots$ be the Fibonacci word, the infinite Fibonacci word with each $0$ written as $a$ and each $1$ as $b$, and let $\\mu$ be the substitution $a \\mapsto aab$, $b \\mapsto ab$. Then $D(f)$ is the image $\\mu(f)$ of $f$ under $\\mu$, where $D$ deletes the first $b$ and the first $a$ of a word.","title":"The Fibonacci switch"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-diff-codings-iff-fibonac","kind":"node","lean":"MetaMathlibExt.letterPosition_diff_codings_iff_fibonacci","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gap codings single out the Fibonacci word among words starting with "],["math","\\(a\\)"]],"source":"Gap codings single out the Fibonacci word among words starting with $a$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-diff-codings-iff-fibonac","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite binary word over the letters $a$ and $b$ in which both letters occur infinitely often and $\\ell_0 = a$. For a letter $\\alpha$ let $p_\\alpha(n)$ be the position (counting from $0$) of the $n$-th occurrence of $\\alpha$ in $w$ and $\\Delta p_\\alpha(n) = p_\\alpha(n+1) - p_\\alpha(n)$. Call $\\Delta p_a$ and $\\Delta p_b$ codings of $w$ if there are maps\u2026","title":"Gap codings single out the Fibonacci word among words starting with $a$"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-diff-codings-iff-fibonac-2","kind":"node","lean":"MetaMathlibExt.letterPosition_diff_codings_iff_fibonacci_reflect","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gap codings single out the reflected Fibonacci word among words starting with "],["math","\\(b\\)"]],"source":"Gap codings single out the reflected Fibonacci word among words starting with $b$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-diff-codings-iff-fibonac-2","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite binary word over the letters $a$ and $b$ in which both letters occur infinitely often and $\\ell_0 = b$. For a letter $\\alpha$ let $p_\\alpha(n)$ be the position (counting from $0$) of the $n$-th occurrence of $\\alpha$ in $w$ and $\\Delta p_\\alpha(n) = p_\\alpha(n+1) - p_\\alpha(n)$. Call $\\Delta p_a$ and $\\Delta p_b$ codings of $w$ if there are maps\u2026","title":"Gap codings single out the reflected Fibonacci word among words starting with $b$"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/letter-position-diff-isomorphic-iff-fibo","kind":"node","lean":"MetaMathlibExt.letterPosition_diff_isomorphic_iff_fibonacci","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#letter-position-diff-isomorphic-iff-fibo","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite binary word over the letters $a$ and $b$ in which both letters occur infinitely often. For a letter $\\alpha$ let $p_\\alpha(n)$ be the position (counting from $0$) of the $n$-th occurrence of $\\alpha$ in $w$ and $\\Delta p_\\alpha(n) = p_\\alpha(n+1) - p_\\alpha(n)$. Call $\\Delta p_a$ and $\\Delta p_b$ codings of $w$ if there are maps $\\sigma_a, \\sigma_b$ from\u2026","title":"The Fibonacci word and its reflection are the words isomorphic to their gap sequences"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/fibonacci-iccanobi-f-conjugate-a","kind":"node","lean":"MetaMathlibExt.fibonacci_iccanobiF_conjugate_a","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#fibonacci-iccanobi-f-conjugate-a","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $\\varrho_F$ be the Fibonacci substitution $a \\mapsto ab$, $b \\mapsto a$ and let $\\varrho_F'$ be the iccanobiF substitution $a \\mapsto ba$, $b \\mapsto a$, acting on finite words over the letters $a$ and $b$; thus $\\varrho_F^m(a)$ is the finite Fibonacci word $f_m$ with each $0$ written as $a$ and each $1$ as $b$. Then for every $n \\ge 1$, $ab\\,\\varrho_F^{2n}(a) = (\\varrho_F')^{2n}(a)\\,ba$ and\u2026","title":"Fibonacci and iccanobiF iterates of a"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/fibonacci-iccanobi-f-conjugate-b","kind":"node","lean":"MetaMathlibExt.fibonacci_iccanobiF_conjugate_b","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conjugacy of the Fibonacci and iccanobiF iterates of "],["math","\\(b\\)"]],"source":"Conjugacy of the Fibonacci and iccanobiF iterates of $b$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#fibonacci-iccanobi-f-conjugate-b","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $\\varrho_F'$ be the iccanobiF substitution $a \\mapsto ba$, $b \\mapsto a$, extended to finite words by concatenation, and for each $k$ let $g_k$ be the finite Fibonacci word $f_k$ with each $0$ written as $a$ and each $1$ as $b$. For every $n \\ge 1$,","title":"Conjugacy of the Fibonacci and iccanobiF iterates of $b$"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/iccanobi-f-fixed-points","kind":"node","lean":"MetaMathlibExt.iccanobiF_fixed_points","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","The iccanobiF iterates approximate "],["math","\\(abf\\)"],["text"," and "],["math","\\(baf\\)"]],"source":"The iccanobiF iterates approximate $abf$ and $baf$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#iccanobi-f-fixed-points","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $f = abaab\\cdots$ be the Fibonacci word, the infinite Fibonacci word with each $0$ written as $a$ and each $1$ as $b$, and let $\\varrho_F'$ be the iccanobiF substitution $a \\mapsto ba$, $b \\mapsto a$, extended to finite words by concatenation. For every $n \\ge 0$, the word $(\\varrho_F')^{2n}(a)$ is a prefix of the infinite word $abf$ and $(\\varrho_F')^{2n}(b)$ is a prefix of $baf$, and for every $N$ there is an\u2026","title":"The iccanobiF iterates approximate $abf$ and $baf$"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-prepend-equilibrious-f","kind":"node","lean":"MetaMathlibExt.relativePosition_prepend_equilibrious_fibonacci","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-prepend-equilibrious-f","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $f = abaab\\cdots$ be the infinite Fibonacci word with $0$ read as $a$ and $1$ as $b$, let $j \\ge 0$, let $u$ be a finite word over $\\{a, b\\}$ of length $2j$ in which $a$ and $b$ occur equally often (an equilibrious word), and let $r$ be the relative position function of the concatenation $uf$. Then $r(n) = n - j$ for all $n > j$. If moreover $u = (ab)^j$, then $r(n) = 1$ for all $1 \\le n \\le j$.","title":"Relative positions of the Fibonacci word with an equilibrious word prepended"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/pisa-letter-position-identity","kind":"node","lean":"MetaMathlibExt.pisa_letterPosition_identity","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#pisa-letter-position-identity","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $k \\ge 1$, $l \\ge 0$ and $m \\ge 1$ be natural numbers, let $\\sigma$ be the substitution $a \\mapsto a^k b a^l$, $b \\mapsto a^m$, and let $w$ be a one-sided fixed point of $\\sigma$ (so $\\sigma(w) = w$) that begins with $a$. For an infinite word $v$ over $\\{a, b\\}$ and $n \\ge 1$, let $p_a(n)$ and $p_b(n)$ be the positions (counted from $0$) of the $n$-th $a$ and the $n$-th $b$ in $v$, and let\u2026","title":"Position identities for the extended Pisa substitution"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/pisa-relative-position-of-letter-positio","kind":"node","lean":"MetaMathlibExt.pisa_relativePosition_of_letterPosition_b","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relative position of the extended Pisa fixed point via the positions of "],["math","\\(b\\)"]],"source":"Relative position of the extended Pisa fixed point via the positions of $b$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#pisa-relative-position-of-letter-positio","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $k \\ge 1$, $l \\ge 0$ and $m \\ge 1$ be natural numbers, let $\\sigma$ be the substitution $a \\mapsto a^k b a^l$, $b \\mapsto a^m$, and let $w$ be a one-sided fixed point of $\\sigma$ (so $\\sigma(w) = w$) that begins with $a$. Let $p_a(n)$ and $p_b(n)$ be the positions (counted from $0$) of the $n$-th $a$ and the $n$-th $b$ in $w$, and let $r(n) = p_b(n) - p_a(n)$. Then for every $n \\ge 1$,","title":"Relative position of the extended Pisa fixed point via the positions of $b$"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/primitive-fixed-point-mean-values","kind":"node","lean":"MetaMathlibExt.primitive_fixed_point_mean_values","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#primitive-fixed-point-mean-values","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $\\varrho$ be a substitution on the letters $a, b$ with substitution matrix $M$, where $M_{11} = |\\varrho(a)|_a$, $M_{12} = |\\varrho(b)|_a$, $M_{21} = |\\varrho(a)|_b$ and $M_{22} = |\\varrho(b)|_b$ ($|v|_x$ is the number of letters $x$ in $v$). Suppose $\\varrho$ is primitive (some power of $M$ has all entries positive), and let $u > 0$ and $\\lambda$ be real numbers with $M (u, 1)^T = \\lambda (u, 1)^T$. Let $w$ be\u2026","title":"Letter frequencies and mean positions of primitive fixed points"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/primitive-relative-position-div-tendsto","kind":"node","lean":"MetaMathlibExt.primitive_relativePosition_div_tendsto_one_iff","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relative position asymptotic to "],["math","\\(n\\)"],["text"," and the golden ratio"]],"source":"Relative position asymptotic to $n$ and the golden ratio"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#primitive-relative-position-div-tendsto","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $\\varrho$ be a substitution on the letters $a, b$ with substitution matrix $M$, where $M_{11} = |\\varrho(a)|_a$, $M_{12} = |\\varrho(b)|_a$, $M_{21} = |\\varrho(a)|_b$ and $M_{22} = |\\varrho(b)|_b$ ($|v|_x$ is the number of letters $x$ in $v$), and suppose $\\varrho$ is primitive (some power of $M$ has all entries positive). Let $w$ be a one-sided fixed point of $\\varrho$ (so $\\varrho(w) = w$) in which both\u2026","title":"Relative position asymptotic to $n$ and the golden ratio"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/pisa-frequency-tfae","kind":"node","lean":"MetaMathlibExt.pisa_frequency_tfae","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#pisa-frequency-tfae","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $j \\ge 1$ and $m \\ge 1$ be natural numbers with $m \\ne j + 1$, and let $\\tau = (j + \\sqrt{j^2 + 4m})/2$. Let $\\varrho$ be a substitution on the letters $a, b$ with substitution matrix $M$, where $M_{11} = |\\varrho(a)|_a$, $M_{12} = |\\varrho(b)|_a$, $M_{21} = |\\varrho(a)|_b$ and $M_{22} = |\\varrho(b)|_b$ ($|v|_x$ is the number of letters $x$ in $v$). Suppose $\\varrho$ is primitive (some power of $M$ has all\u2026","title":"Frequency and position criteria for an eigenvector of a primitive substitution"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/linear-relative-position-examples","kind":"node","lean":"MetaMathlibExt.linear_relativePosition_examples","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#linear-relative-position-examples","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"For integers $k, l, m \\ge 0$ write $\\sigma_{k,l,m}$ for the substitution $a \\mapsto a^k b a^l$, $b \\mapsto a^m$, and for a binary word $v$ let $r_v(n) = p_{b,v}(n) - p_{a,v}(n)$, where $p_{\\alpha,v}(n)$ is the position (counted from $0$) of the $n$-th occurrence of the letter $\\alpha$ in $v$. (a) For every $j \\ge 1$, the word $v = (a^j b^j)^\\omega$ has $r_v(n) = j$ for all $n \\ge 1$. (b) For integers $0 \\le j < k$,\u2026","title":"Words with linear relative position function"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-eventually-eq-linear","kind":"node","lean":"MetaMathlibExt.relativePosition_eventually_eq_linear","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-eventually-eq-linear","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $k \\ge 1$, $q$ and $c$ be integers with $0 \\le c \\le k - 1$, and put $j = qk - c$. Let $w$ be an infinite word over $\\{a, b\\}$ that starts with $a$ and is a fixed point of the Pisa substitution $\\sigma_{k-c,c,1}$, given by $a \\mapsto a^{k-c} b a^c$ and $b \\mapsto a$. For an infinite word $v$ write $r_v$ for its relative position function, and let $D$ be the deletion operator, which deletes the first $b$ and\u2026","title":"Eventually linear relative position functions with positive slope"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/primitive-asymptotically-linear-classifi","kind":"node","lean":"MetaMathlibExt.primitive_asymptotically_linear_classification","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#primitive-asymptotically-linear-classifi","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $\\varrho$ be a substitution on the letters $a, b$ with substitution matrix $M$, where $M_{11} = |\\varrho(a)|_a$, $M_{12} = |\\varrho(b)|_a$, $M_{21} = |\\varrho(a)|_b$ and $M_{22} = |\\varrho(b)|_b$ ($|v|_x$ is the number of letters $x$ in $v$). Suppose $\\varrho$ is primitive (some power of $M$ has all entries positive), and let $w$ be a one-sided fixed point of $\\varrho$ (so $\\varrho(w) = w$) in which both\u2026","title":"Primitive substitutions with asymptotically linear relative position"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-of-dimers","kind":"node","lean":"MetaMathlibExt.relativePosition_of_dimers","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Relative positions of a word made of "],["math","\\(ab\\)"],["text"," and "],["math","\\(ba\\)"],["text"," blocks"]],"source":"Relative positions of a word made of $ab$ and $ba$ blocks"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-of-dimers","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over $\\{a, b\\}$ such that every block $\\ell_{2i} \\ell_{2i+1}$ ($i \\ge 0$) is $ab$ or $ba$, and let $r$ be its relative position function. Then for every $n \\ge 1$, $r(n) = 1$ if $\\ell_{2n-2} = a$ and $r(n) = -1$ if $\\ell_{2n-2} = b$.","title":"Relative positions of a word made of $ab$ and $ba$ blocks"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-self-coding-iff-thue-m","kind":"node","lean":"MetaMathlibExt.relativePosition_self_coding_iff_thueMorse","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-self-coding-iff-thue-m","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over $\\{a, b\\}$ that starts with $a$ and in which both letters occur infinitely often, and let $r$ be its relative position function. Then $r(n) = 1$ when $\\ell_{n-1} = a$ and $r(n) = -1$ when $\\ell_{n-1} = b$, for all $n \\ge 1$, if and only if $w$ is the Thue\u2013Morse word $t = abbabaab\\cdots$ (the Thue\u2013Morse sequence with $0$ read as $a$ and $1$ as $b$).","title":"Characterization of the Thue\u2013Morse word by its relative positions"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-isomorphic-iff-thue-mo","kind":"node","lean":"MetaMathlibExt.relativePosition_isomorphic_iff_thueMorse","parent":23945,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of the Thue\u2013Morse word among words starting with "],["math","\\(abba\\)"]],"source":"Characterization of the Thue\u2013Morse word among words starting with $abba$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-isomorphic-iff-thue-mo","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over $\\{a, b\\}$ that begins with $abba$ and in which both letters occur infinitely often, and let $r$ be its relative position function. Then there is an injective map $\\sigma \\colon \\{a, b\\} \\to \\mathbb{Z}$ with $r(n) = \\sigma(\\ell_{n-1})$ for all $n \\ge 1$ if and only if $w$ is the Thue\u2013Morse word $t = abbabaab\\cdots$ (the Thue\u2013Morse sequence with $0$ read\u2026","title":"Characterization of the Thue\u2013Morse word among words starting with $abba$"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/relative-position-self-coding-iff-clone","kind":"node","lean":"MetaMathlibExt.relativePosition_self_coding_iff_clone_thueMorse","parent":23945,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words/index.html#relative-position-self-coding-iff-clone","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/relative-position-in-binary-words","status":"can_prove","summary":"Let $k > 1$ be an integer, let $w = \\ell_0 \\ell_1 \\ell_2 \\cdots$ be an infinite word over $\\{a, b\\}$ that starts with $a$ and in which both letters occur infinitely often, and let $r$ be its relative position function. Then $r(n) = k$ when $\\ell_{n-1} = a$ and $r(n) = -k$ when $\\ell_{n-1} = b$, for all $n \\ge 1$, if and only if $w = \\varphi_k(t)$, the word whose letter at position $i$ is $t_{\\lfloor i/k \\rfloor}$,\u2026","title":"Characterization of the cloned Thue\u2013Morse words by their relative positions"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word.json","id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","kind":"scope","navigation":"module","parent":23818,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","title":"Tribonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-letter","kind":"node","lean":"InfiniteWord.TribonacciLetter","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-letter","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"The Tribonacci word is written over the three-letter alphabet whose letters are $a$, $b$ and $c$.","title":"The Tribonacci alphabet"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word-approx","kind":"node","lean":"InfiniteWord.tribonacciWordApprox","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word-approx","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"The finite Tribonacci words $W(n)$, for $n \\ge 0$, are the words over the alphabet $\\{a, b, c\\}$ defined by $W(0) = a$, $W(1) = ab$, $W(2) = abac$ and $W(n + 3) = W(n + 2)\\,W(n + 1)\\,W(n)$ for every $n \\ge 0$, where juxtaposition denotes concatenation.","title":"Finite Tribonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word-approx-add-three","kind":"node","lean":"InfiniteWord.tribonacciWordApprox_add_three","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word-approx-add-three","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"For every $n \\ge 0$, the finite Tribonacci word $W(n + 3)$ is the concatenation $W(n + 2)\\,W(n + 1)\\,W(n)$.","title":"Recursion for the finite Tribonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word-approx-length-add-three","kind":"node","lean":"InfiniteWord.tribonacciWordApprox_length_add_three","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word-approx-length-add-three","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"Writing $|w|$ for the length of a word $w$, for every $n \\ge 0$ the finite Tribonacci words satisfy $|W(n + 3)| = |W(n + 2)| + |W(n + 1)| + |W(n)|$.","title":"Length recurrence for finite Tribonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word-approx-length-ge","kind":"node","lean":"InfiniteWord.tribonacciWordApprox_length_ge","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word-approx-length-ge","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"For every $n \\ge 0$, the finite Tribonacci word $W(n)$ has length at least $n + 1$.","title":"Length lower bound for finite Tribonacci words"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word-approx-prefix-succ","kind":"node","lean":"InfiniteWord.tribonacciWordApprox_prefix_succ","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word-approx-prefix-succ","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"For every $n \\ge 0$, the finite Tribonacci word $W(n)$ is a prefix of $W(n + 1)$.","title":"Each finite Tribonacci word is a prefix of the next"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word-approx-prefix","kind":"node","lean":"InfiniteWord.tribonacciWordApprox_prefix","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word-approx-prefix","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"For all natural numbers $m \\le n$, the finite Tribonacci word $W(m)$ is a prefix of $W(n)$.","title":"Finite Tribonacci words form a prefix chain"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word-approx-length-mono","kind":"node","lean":"InfiniteWord.tribonacciWordApprox_length_mono","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word-approx-length-mono","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"For all natural numbers $m \\le n$, the length of the finite Tribonacci word $W(m)$ is at most the length of $W(n)$.","title":"Lengths of finite Tribonacci words are monotone"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word","kind":"node","lean":"InfiniteWord.tribonacciWord","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"The infinite Tribonacci word is the sequence over the alphabet $\\{a, b, c\\}$ whose letter at each position $k \\ge 0$ (positions counted from $0$) is the letter at position $k$ of the finite Tribonacci word $W(k)$; this position exists because $W(k)$ has length at least $k + 1$.","title":"The infinite Tribonacci word"},{"id":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/tribonacci-word-eq-get-elem-of-le","kind":"node","lean":"InfiniteWord.tribonacciWord_eq_getElem_of_le","parent":24009,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word/index.html#tribonacci-word-eq-get-elem-of-le","scope":"Combinatorics/combinatorics-on-words/morphic-sturmian-words/tribonacci-word","status":"fully_proved","summary":"For all natural numbers $k \\le n$ such that $k$ is less than the length of the finite Tribonacci word $W(n)$, the letter of $W(n)$ at position $k$ equals the letter of the infinite Tribonacci word at position $k$, positions being counted from $0$.","title":"Finite Tribonacci words agree with the infinite word"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/automatic-sequences.json","id":"Combinatorics/combinatorics-on-words/automatic-sequences","kind":"scope","parent":23782,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/automatic-sequences","status":"can_prove","summary":"k-automatic and k-regular sequences, Cobham's theorem and kernels, meta-automatic sequences, results proved with the Walnut prover, and numeration systems (Dumont\u2013Thomas, Tribonacci and Zeckendorf representations) whose digit words define such sequences.","title":"Automatic sequences and numeration systems"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences.json","id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","kind":"scope","navigation":"module","parent":24020,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","title":"Automatic sequences"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/base-word-value-lsd","kind":"node","lean":"MetaMathlibExt.baseWordValueLSD","parent":24021,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value of a base-"],["math","\\(k\\)"],["text"," digit word, least significant digit first"]],"source":"Value of a base-$k$ digit word, least significant digit first"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#base-word-value-lsd","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"For a natural number $k$, the value of a finite word $d_0 d_1 \\cdots d_{m-1}$ of digits $d_i \\in \\{0, 1, \\dots, k-1\\}$ is the natural number $\\sum_{i < m} d_i k^i$, so the first digit is the least significant one; the empty word has value $0$.","title":"Value of a base-$k$ digit word, least significant digit first"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/is-kautomatic-sequence","kind":"node","lean":"MetaMathlibExt.IsKAutomaticSequence","parent":24021,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#is-kautomatic-sequence","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $k$ be a natural number and $B$ a finite alphabet. A sequence $w : \\mathbb{N} \\to B$ is $k$-automatic when $k \\ge 2$ and there is a deterministic finite automaton with output, consisting of a nonempty finite set of states $Q$, an initial state $s \\in Q$, a transition map $\\delta : Q \\times \\{0, 1, \\dots, k-1\\} \\to Q$ and an output map $\\tau : Q \\to B$, such that for every $n \\in \\mathbb{N}$ and every finite\u2026","title":"Automatic sequences via finite automata"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/is-coding-of-prolongable-kuniform-fixed","kind":"node","lean":"MetaMathlibExt.IsCodingOfProlongableKUniformFixedPoint","parent":24021,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#is-coding-of-prolongable-kuniform-fixed","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $k$ be a natural number and $B$ a finite alphabet. A sequence $w : \\mathbb{N} \\to B$ is a coding of a prolongable $k$-uniform fixed point when $k \\ge 2$ and there are a nonempty finite alphabet $A$, a map $f$ sending each letter of $A$ to a word of length exactly $k$ over $A$, a letter $a \\in A$ with $f(a) = a\\alpha$ for some nonempty word $\\alpha$ over $A$, a sequence $u : \\mathbb{N} \\to A$ and a map\u2026","title":"Codings of prolongable uniform fixed points"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/cobham-automatic-iff-uniform-morphic","kind":"node","lean":"MetaMathlibExt.cobham_automatic_iff_uniform_morphic","parent":24021,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#cobham-automatic-iff-uniform-morphic","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $k \\ge 2$ and let $B$ be a finite alphabet. A sequence $w : \\mathbb{N} \\to B$ is $k$-automatic if and only if it is a coding of a prolongable $k$-uniform fixed point.","title":"Cobham's characterization of automatic sequences"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/is-coding-of-prolongable-kuniform-fixed-2","kind":"node","lean":"MetaMathlibExt.isCodingOfProlongableKUniformFixedPoint_iff","parent":24021,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#is-coding-of-prolongable-kuniform-fixed-2","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $k \\ge 2$ and let $B$ be a finite alphabet. A sequence $w : \\mathbb{N} \\to B$ is a coding of a prolongable $k$-uniform fixed point if and only if there are a natural number $q$, a map $f$ sending each letter of the alphabet $A = \\{0, 1, \\dots, q-1\\}$ to a word of length exactly $k$ over $A$, a letter $a \\in A$, an infinite word $u$ over $A$ and a map $g : A \\to B$ such that $f$ is prolongable at $a$, $u$ is the\u2026","title":"Uniform fixed-point codings via iterates of a uniform map"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/exists-is-morphic","kind":"node","lean":"MetaMathlibExt.IsCodingOfProlongableKUniformFixedPoint.exists_isMorphic","parent":24021,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#exists-is-morphic","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $k$ be a natural number and $B$ a finite alphabet. If a sequence $w : \\mathbb{N} \\to B$ is a coding of a prolongable $k$-uniform fixed point, then there are a natural number $q$, an infinite word $u$ over the alphabet $A = \\{0, 1, \\dots, q-1\\}$ and a map $g : A \\to B$ such that $w$ is morphic with generator $u$ and coding $g$.","title":"Uniform fixed-point codings are morphic"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/exists-is-morphic-2","kind":"node","lean":"MetaMathlibExt.IsKAutomaticSequence.exists_isMorphic","parent":24021,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#exists-is-morphic-2","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $k$ be a natural number and $B$ a finite alphabet. If a sequence $w : \\mathbb{N} \\to B$ is $k$-automatic, then there are a natural number $q$, an infinite word $u$ over the alphabet $A = \\{0, 1, \\dots, q-1\\}$ and a map $g : A \\to B$ such that $w$ is morphic with generator $u$ and coding $g$.","title":"Automatic sequences are morphic"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/k-kernel","kind":"node","lean":"MetaMathlibExt.kKernel","parent":24021,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-kernel of a sequence"]],"source":"The $k$-kernel of a sequence"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#k-kernel","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"For a natural number $k$ and a sequence $a$ indexed by $\\mathbb{N}$, the $k$-kernel of $a$ is the set of all sequences $n \\mapsto a(k^e n + r)$ with $e, r \\in \\mathbb{N}$ and $r < k^e$.","title":"The $k$-kernel of a sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/is-kautomatic","kind":"node","lean":"MetaMathlibExt.IsKAutomatic","parent":24021,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-automatic sequence (finite-kernel definition)"]],"source":"$k$-automatic sequence (finite-kernel definition)"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#is-kautomatic","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"For a natural number $k$, a sequence $a$ indexed by $\\mathbb{N}$ is $k$-automatic when $k \\ge 2$ and the $k$-kernel of $a$ is finite.","title":"$k$-automatic sequence (finite-kernel definition)"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/chr-qkernel-subset","kind":"node","lean":"MetaMathlibExt.chr_qkernel_subset","parent":24021,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(p^r\\)"],["text","-kernel lies in the "],["math","\\(p\\)"],["text","-kernel"]],"source":"The $p^r$-kernel lies in the $p$-kernel"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#chr-qkernel-subset","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"For all natural numbers $p$ and $r$ and every sequence $a$ indexed by $\\mathbb{N}$, the $p^r$-kernel of $a$ is contained in the $p$-kernel of $a$.","title":"The $p^r$-kernel lies in the $p$-kernel"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/chr-pkernel-finite","kind":"node","lean":"MetaMathlibExt.chr_pkernel_finite","parent":24021,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finiteness of the "],["math","\\(p\\)"],["text","-kernel from the "],["math","\\(p^r\\)"],["text","-kernel"]],"source":"Finiteness of the $p$-kernel from the $p^r$-kernel"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#chr-pkernel-finite","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $p$ and $r$ be positive natural numbers and let $a$ be a sequence indexed by $\\mathbb{N}$. If the $p^r$-kernel of $a$ is finite, then the $p$-kernel of $a$ is finite.","title":"Finiteness of the $p$-kernel from the $p^r$-kernel"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/chr-kernel-lam-mem","kind":"node","lean":"MetaMathlibExt.chr_kernel_lam_mem","parent":24021,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-kernel is closed under decimation"]],"source":"The $q$-kernel is closed under decimation"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#chr-kernel-lam-mem","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $q$ be a natural number and let $a$ be a sequence indexed by $\\mathbb{N}$. If $b$ belongs to the $q$-kernel of $a$ and $j < q$, then the sequence $n \\mapsto b(qn + j)$ also belongs to the $q$-kernel of $a$.","title":"The $q$-kernel is closed under decimation"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/chr-self-mem-kernel","kind":"node","lean":"MetaMathlibExt.chr_self_mem_kernel","parent":24021,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#chr-self-mem-kernel","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"For every natural number $q$, every sequence $a$ indexed by $\\mathbb{N}$ belongs to its own $q$-kernel.","title":"A sequence lies in its own kernel"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/is-kautomatic-iff-is-kautomatic-sequence","kind":"node","lean":"MetaMathlibExt.isKAutomatic_iff_isKAutomaticSequence","parent":24021,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences/index.html#is-kautomatic-iff-is-kautomatic-sequence","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/automatic-sequences","status":"fully_proved","summary":"Let $k$ be a natural number and let $a$ be a sequence indexed by $\\mathbb{N}$ with values in a finite type $B$. Then $k \\ge 2$ and the $k$-kernel of $a$ is finite (that is, $a$ is $k$-automatic in the finite-kernel sense) if and only if $k \\ge 2$ and some deterministic finite automaton with output, reading base-$k$ digits least significant first, outputs $a(n)$ on every word of base-$k$ digits whose value is $n$,\u2026","title":"Eilenberg's theorem: finite kernel iff automatic"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem.json","id":"Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem","kind":"scope","navigation":"module","parent":24020,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem/index.html","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem","status":"fully_proved","title":"Christol's theorem"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem/christol-theorem","kind":"node","lean":"MetaMathlibExt.christol_theorem","parent":24036,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem/index.html#christol-theorem","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem","status":"fully_proved","summary":"Let $p$ be a prime, $r \\ge 1$ a natural number, and $K$ a finite field of characteristic $p$ with $q = p^r$ elements. A formal power series $f = \\sum_n f_n X^n \\in K[[X]]$ is algebraic over the polynomial ring $K[X]$ (it is a root of a nonzero polynomial with coefficients in $K[X]$) if and only if its coefficient sequence $n \\mapsto f_n$ is $p$-automatic, that is, its $p$-kernel is finite.","title":"Christol's theorem"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem/christol-theorem-is-kautomatic-sequence","kind":"node","lean":"MetaMathlibExt.christol_theorem_isKAutomaticSequence","parent":24036,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem/index.html#christol-theorem-is-kautomatic-sequence","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/christol-s-theorem","status":"fully_proved","summary":"Let $p$ be a prime, $r \\ge 1$ a natural number, and $K$ a finite field of characteristic $p$ with $q = p^r$ elements. A formal power series $f = \\sum_n f_n X^n \\in K[[X]]$ is algebraic over the polynomial ring $K[X]$ (it is a root of a nonzero polynomial with coefficients in $K[X]$) if and only if its coefficient sequence $n \\mapsto f_n$ is $p$-automatic in the sense of finite automata with output.","title":"Christol's theorem, automaton form"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration.json","id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","kind":"scope","navigation":"module","parent":24020,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","title":"Dumont\u2013Thomas numeration"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-substitution","kind":"node","lean":"MetaMathlibExt.IsSubstitution","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-substitution","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"A substitution is a map $\\eta$ sending each letter of an alphabet $A$ to a finite word over $A$, extended to finite words by concatenation, such that $\\eta$ is non-erasing (the word $\\eta(a)$ is nonempty for every letter $a \\in A$) and some letter $a \\in A$ is growing, that is, $|\\eta^k(a)| \\to \\infty$ as $k \\to \\infty$, where $\\eta^k$ is the $k$-th iterate of $\\eta$. The alphabet $A$ need not be finite.","title":"Substitutions in the sense of Labb\u00e9 and Lep\u0161ov\u00e1"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-non-erasing","kind":"node","lean":"MetaMathlibExt.IsSubstitution.isNonErasing","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-non-erasing","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Every substitution $\\eta$ in the sense of Labb\u00e9 and Lep\u0161ov\u00e1 is non-erasing: $\\eta(a)$ is a nonempty word for every letter $a$.","title":"Substitutions are non-erasing"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-admissible-sequence","kind":"node","lean":"MetaMathlibExt.IsAdmissibleSequence","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-admissible-sequence","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ send each letter of an alphabet $A$ to a finite word over $A$, and let $k \\ge 0$. A finite sequence $(m_i, a_i)_{0 \\le i \\le k}$ of pairs, each made of a finite word $m_i$ over $A$ and a letter $a_i \\in A$, is admissible with respect to $\\eta$ when the word $m_i a_i$ is a prefix of $\\eta(a_{i+1})$ for every $i < k$.","title":"Admissible sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-letter-admissible-sequence","kind":"node","lean":"MetaMathlibExt.IsLetterAdmissibleSequence","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-letter-admissible-sequence","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ map each letter of an alphabet $A$ to a finite word over $A$, and let $a \\in A$. A finite sequence $(m_i, a_i)_{0 \\le i \\le k}$ of pairs, each made of a finite word $m_i$ over $A$ and a letter $a_i \\in A$, is $a$-admissible when it is admissible with respect to $\\eta$ and moreover the word $m_k a_k$ is a prefix of $\\eta(a)$.","title":"Admissible sequences starting from a letter"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-admissible-sequence-zero","kind":"node","lean":"MetaMathlibExt.isAdmissibleSequence_zero","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-admissible-sequence-zero","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ send each letter of an alphabet $A$ to a finite word over $A$. Every one-term sequence $(m_0, a_0)$, with $m_0$ a finite word over $A$ and $a_0 \\in A$, is admissible with respect to $\\eta$: the admissibility condition is vacuous for a sequence with the single index $0$.","title":"One-term sequences are admissible"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/substitution-automaton","kind":"node","lean":"MetaMathlibExt.substitutionAutomaton","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#substitution-automaton","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ map each letter of an alphabet $A$ to a finite word over $A$. The substitution automaton $\\mathcal{A}_{\\eta,\\cdot}$ of $\\eta$ is the deterministic automaton with output whose states are the letters of $A$ (the alphabet need not be finite): reading a digit $i \\in \\mathbb{N}$ in the state $b$ moves to the letter $\\eta(b)_i$ of $\\eta(b)$ at position $i$, counting from $0$, and is undefined when\u2026","title":"Substitution automaton"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/substitution-automaton-nil","kind":"node","lean":"MetaMathlibExt.substitutionAutomaton_nil","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#substitution-automaton-nil","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"For every letter map $\\eta$ and every letter $b$, the substitution automaton $\\mathcal{A}_{\\eta,\\cdot}$ started at $b$ ends in the state $b$ after reading the empty digit word.","title":"The substitution automaton on the empty word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/substitution-automaton-cons","kind":"node","lean":"MetaMathlibExt.substitutionAutomaton_cons","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#substitution-automaton-cons","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ be a letter map, $b$ a letter, $i$ a digit and $w$ a digit word. If $i < |\\eta(b)|$, then the value of the substitution automaton started at $b$ on the word $iw$ (the digit $i$ followed by $w$) equals its value started at $\\eta(b)_i$ on $w$, where $\\eta(b)_i$ is the letter of $\\eta(b)$ at position $i$, counting from $0$. If $i \\ge |\\eta(b)|$, the run on $iw$ is undefined.","title":"Reading a nonempty digit word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/substitution-automaton-fibonacci-small","kind":"node","lean":"MetaMathlibExt.substitutionAutomaton_fibonacci_small","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#substitution-automaton-fibonacci-small","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"For the Fibonacci substitution $0 \\mapsto 01$, $1 \\mapsto 0$ on the alphabet $\\{0, 1\\}$, the substitution automaton started at the letter $0$ ends in the state $0$ after reading the digit word $10$, and its run on the digit word $11$ is undefined, so $11$ is rejected.","title":"Two runs of the Fibonacci automaton"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-numeration","kind":"node","lean":"MetaMathlibExt.dumontThomas_numeration","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-numeration","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, extended to finite words by concatenation, and let $u_0 u_1 u_2 \\cdots$ be a right-infinite word over $A$ that is a fixed point of $\\eta$ in the sense that $\\eta(u_0 \\cdots u_{n-1})$ is a prefix of $u_0 u_1 u_2 \\cdots$ for every $n$. Suppose that $a = u_0$ is a growing letter, i.e. $|\\eta^k(a)| \\to \\infty$. Then for every $n \\ge 1$ there are a natural number $k$ and\u2026","title":"Dumont\u2013Thomas numeration"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/admissible-sequence-weighted-length-lt","kind":"node","lean":"MetaMathlibExt.admissibleSequence_weightedLength_lt","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#admissible-sequence-weighted-length-lt","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, extended to finite words by concatenation, and let $(m_i, a_i)_{0 \\le i \\le k}$ be a sequence of pairs of a finite word $m_i$ over $A$ and a letter $a_i \\in A$ that is admissible with respect to $\\eta$. Then $\\sum_{j=0}^{k} |\\eta^j(m_j)| < |\\eta^k(m_k a_k)|$, where $\\eta^j$ is the $j$-th iterate of $\\eta$ and $|w|$ is the length of a word $w$.","title":"Weighted length of an admissible sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/letter-admissible-sequence-eq-of-weighte","kind":"node","lean":"MetaMathlibExt.letterAdmissibleSequence_eq_of_weightedLength_eq","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#letter-admissible-sequence-eq-of-weighte","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, let $b \\in A$ and $k \\in \\mathbb{N}$, and let $(m_i, a_i)_{i = 0, \\dots, k}$ and $(m'_i, a'_i)_{i = 0, \\dots, k}$ be two $b$-admissible sequences with $\\sum_{j=0}^{k} |\\eta^j(m_j)| = \\sum_{j=0}^{k} |\\eta^j(m'_j)|$. Then $(m_i, a_i) = (m'_i, a'_i)$ for every $i \\le k$.","title":"Admissible sequences are determined by their weighted length"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/proper-prefix-morph-iterate-decompositio","kind":"node","lean":"MetaMathlibExt.properPrefix_morphIterate_decomposition","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#proper-prefix-morph-iterate-decompositio","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, let $\\ell \\ge 1$ and $a \\in A$, and let $m$ be a proper prefix of $\\eta^\\ell(a)$, that is, a prefix of strictly smaller length. Then there are words $m'$ and $m''$ over $A$ and a letter $a' \\in A$ such that $m'a'$ is a prefix of $\\eta(a)$, $m''$ is a proper prefix of $\\eta^{\\ell-1}(a')$, and $m = \\eta^{\\ell-1}(m')\\,m''$.","title":"Decomposition of a proper prefix of an iterated image"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/substitution-automaton-admissible-sequen","kind":"node","lean":"MetaMathlibExt.substitutionAutomaton_admissibleSequence","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#substitution-automaton-admissible-sequen","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution, let $x$ be a letter, let $k \\ge 1$, and let $(m_i, a_i)_{i = 0, \\dots, k-1}$ be an $x$-admissible sequence. Then for every $i < k$, the substitution automaton started at $x$ ends in the state $a_i$ after reading the digit word $|m_{k-1}|\\,|m_{k-2}| \\cdots |m_i|$, whose digits are the lengths of $m_{k-1}, \\dots, m_i$ in this order.","title":"The automaton reads off the letters of an admissible sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/exists-letter-admissible-sequence-of-acc","kind":"node","lean":"MetaMathlibExt.exists_letterAdmissibleSequence_of_accepted","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#exists-letter-admissible-sequence-of-acc","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, let $x \\in A$, let $k \\ge 1$, and let $v_0, v_1, \\dots, v_{k-1}$ be natural numbers. If the digit word $v_{k-1} v_{k-2} \\cdots v_0$ is accepted by the automaton $\\mathcal{A}_{\\eta,x}$ (that is, its run from $x$ is defined), then there is an $x$-admissible sequence $(m_i, a_i)_{0 \\le i \\le k-1}$ with $|m_i| = v_i$ for every $i < k$.","title":"Accepted digit words come from admissible sequences"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/exists-unique-letter-admissible-sequence","kind":"node","lean":"MetaMathlibExt.existsUnique_letterAdmissibleSequence_of_properPrefix","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#exists-unique-letter-admissible-sequence","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, let $x \\in A$, let $p \\ge 1$, and let $m$ be a proper prefix of $\\eta^p(x)$. Then there is an $x$-admissible sequence $(m_i, a_i)_{0 \\le i \\le p-1}$ with $|m| = \\sum_{j=0}^{p-1} |\\eta^j(m_j)|$; every $x$-admissible sequence $(m'_i, a'_i)_{0 \\le i \\le p-1}$ with $|m| = \\sum_{j=0}^{p-1} |\\eta^j(m'_j)|$ agrees with it at every index $i < p$; and\u2026","title":"Proper prefixes of iterated images have unique admissible expansions"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/morph-iterate-get-elem-eq-substitution-a","kind":"node","lean":"MetaMathlibExt.morphIterate_getElem_eq_substitutionAutomaton_tail","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#morph-iterate-get-elem-eq-substitution-a","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, let $p \\ge 1$ and $x \\in A$, let $(m_i, a_i)_{i = 0, \\dots, p-1}$ be an $x$-admissible sequence, and put $\\ell = \\sum_{j=0}^{p-1} |\\eta^j(m_j)|$. Let $\\mathcal{A}_{\\eta,x}$ be the automaton of $\\eta$ started at $x$. Then $\\eta^p(x)$ has a letter at position $\\ell$ (counting from $0$) exactly when the run of $\\mathcal{A}_{\\eta,x}$ on the digit word\u2026","title":"Letters of an iterated image are read by the automaton"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/tail-injective-and-strict-mono","kind":"node","lean":"MetaMathlibExt.tail_injective_and_strictMono","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#tail-injective-and-strict-mono","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution, let $x$ be a letter, let $p \\ge 1$, and let $(m_j, a_j)_{j = 0, \\dots, p-1}$ and $(m'_j, a'_j)_{j = 0, \\dots, p-1}$ be $x$-admissible sequences. Put $n = \\sum_{j\\lt p} |\\eta^j(m_j)|$ and $n' = \\sum_{j\\lt p} |\\eta^j(m'_j)|$, and let $t$ and $t'$ be the digit words $|m_{p-1}| \\cdots |m_0|$ and $|m'_{p-1}| \\cdots |m'_0|$. Then $n = n'$ if and only if $t = t'$, and $n < n'$ if and only if\u2026","title":"Tails are injective and order-preserving"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-numeration-right-periodic","kind":"node","lean":"MetaMathlibExt.dumontThomas_numeration_rightPeriodic","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-numeration-right-periodic","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, let $a \\in A$ be a growing letter, i.e. $|\\eta^k(a)| \\to \\infty$, and let $u_0 u_1 u_2 \\cdots$ be a right-infinite word over $A$ with $u_0 = a$. Let $p \\ge 1$ be such that for every $n$ the word $\\eta^p(u_0 \\cdots u_{n-1})$ is a prefix of $u_0 u_1 u_2 \\cdots$. Then for every $n \\ge 1$ there are a natural number $k$ and an $a$-admissible sequence\u2026","title":"Dumont\u2013Thomas numeration for right-infinite periodic points"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-numeration-left-periodic","kind":"node","lean":"MetaMathlibExt.dumontThomas_numeration_leftPeriodic","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-numeration-left-periodic","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, let $b \\in A$ be a growing letter, i.e. $|\\eta^k(b)| \\to \\infty$, and let $\\cdots u_{-2} u_{-1}$ be a left-infinite word over $A$ with $u_{-1} = b$. Let $p \\ge 1$ be such that for every $m$ the word $\\eta^p(u_{-m} \\cdots u_{-1})$ is a suffix of $\\cdots u_{-2} u_{-1}$. Then for every integer $N \\ge 2$ there are a natural number $k$ and a $b$-admissible sequence\u2026","title":"Dumont\u2013Thomas numeration for left-infinite periodic points"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-two-sided-periodic-point","kind":"node","lean":"MetaMathlibExt.IsTwoSidedPeriodicPoint","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-two-sided-periodic-point","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ map each letter of an alphabet $A$ to a finite word over $A$, extended to finite words by concatenation, with iterates $\\eta^j$; let $u = (u_i)_{i \\in \\mathbb{Z}}$ be a two-sided word over $A$ and let $p$ be a natural number. Then $u$ is a two-sided periodic point of $\\eta$ with period $p$ when $p \\ge 1$, for every natural number $n$ the word $\\eta^p(u_0 u_1 \\cdots u_{n-1})$ is a prefix of the\u2026","title":"Two-sided periodic points"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/one-le","kind":"node","lean":"MetaMathlibExt.IsTwoSidedPeriodicPoint.one_le","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#one-le","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ send each letter of an alphabet $A$ to a finite word over $A$. If a two-sided word $u : \\mathbb{Z} \\to A$ is a periodic point of $\\eta$ with period $p$, then $p \\ge 1$.","title":"Periods of two-sided periodic points are positive"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-periodic-point-with-growing-seed","kind":"node","lean":"MetaMathlibExt.IsPeriodicPointWithGrowingSeed","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-periodic-point-with-growing-seed","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ map each letter of an alphabet $A$ to a finite word over $A$, extended to finite words by concatenation, with iterates $\\eta^k$; let $u = (u_i)_{i \\in \\mathbb{Z}}$ be a two-sided word over $A$ and let $p$ be a natural number. Then $u$ is a periodic point of $\\eta$ with growing seed and least period $p$ when $\\eta$ is a substitution, $p$ is the least natural number $q$ such that $u$ is a two-sided\u2026","title":"Two-sided periodic points with growing seed"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-two-sided-periodic-point-2","kind":"node","lean":"MetaMathlibExt.IsPeriodicPointWithGrowingSeed.isTwoSidedPeriodicPoint","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-two-sided-periodic-point-2","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ map each letter of an alphabet $A$ to a finite word over $A$, let $u = (u_i)_{i \\in \\mathbb{Z}}$ be a two-sided word over $A$ and let $p$ be a natural number. If $u$ is a two-sided periodic point of $\\eta$ with growing seed and least period $p$, then $u$ is a two-sided periodic point of $\\eta$ with period $p$.","title":"Periodic points with growing seed are two-sided periodic points"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-dumont-thomas-rep","kind":"node","lean":"MetaMathlibExt.IsDumontThomasRep","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-dumont-thomas-rep","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ map each letter of an alphabet $A$ to a finite word over $A$, extended to finite words by concatenation, with iterates $\\eta^j$; let $u = (u_i)_{i \\in \\mathbb{Z}}$ be a two-sided word over $A$ and let $p$ be a natural number. A finite word $w$ of natural-number digits is a Dumont\u2013Thomas complement representation of an integer $n$ with respect to $\\eta$, $u$ and $p$ when one of the following four\u2026","title":"Dumont\u2013Thomas complement representations"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-dumont-thomas-rep-zero","kind":"node","lean":"MetaMathlibExt.isDumontThomasRep_zero","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-dumont-thomas-rep-zero","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"For every map $\\eta$ sending each letter of an alphabet $A$ to a finite word over $A$, every two-sided word $u = (u_i)_{i \\in \\mathbb{Z}}$ over $A$ and every natural number $p$, the one-digit word $0$ is a Dumont\u2013Thomas complement representation of the integer $0$ with respect to $\\eta$, $u$ and $p$.","title":"A Dumont\u2013Thomas representation of zero"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/is-dumont-thomas-rep-neg-one","kind":"node","lean":"MetaMathlibExt.isDumontThomasRep_neg_one","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#is-dumont-thomas-rep-neg-one","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ send each letter of an alphabet $A$ to a finite word over $A$, let $u$ be a two-sided infinite word over $A$, and let $p$ be a natural number. Then the one-digit word $1$ is the Dumont\u2013Thomas complement representation of $-1$ for $\\eta$, $u$ and $p$, that is, $\\operatorname{rep}_u(-1) = 1$; no hypothesis on $\\eta$, $u$ or $p$ is needed.","title":"Dumont\u2013Thomas representation of minus one"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/seed-automaton","kind":"node","lean":"MetaMathlibExt.seedAutomaton","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#seed-automaton","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ be a letter map and let $s = b|a$ be the seed formed by the letters $b$ and $a$. The automaton $\\mathcal{A}_{\\eta,s}$ has a new initial state, which is not final, with a transition labelled $0$ to $a$ and a transition labelled $1$ to $b$; from a letter it runs like the substitution automaton $\\mathcal{A}_{\\eta,\\cdot}$. So its value on a digit word is: undefined on the empty word;\u2026","title":"Seed automaton"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/seed-automaton-zero-cons","kind":"node","lean":"MetaMathlibExt.seedAutomaton_zero_cons","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#seed-automaton-zero-cons","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ be a letter map, let $a$ and $b$ be letters, and let $\\mathcal{A}_{\\eta,s}$ be the seed automaton for the seed $s = b|a$. For every digit word $w$, the value of $\\mathcal{A}_{\\eta,s}$ on the word $0w$ (the digit $0$ followed by $w$) equals the value $\\mathcal{A}_{\\eta,a}(w)$ of the substitution automaton started at $a$ on $w$: either both are undefined, or both are the same state.","title":"Seed automaton after reading the digit 0"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/seed-automaton-one-cons","kind":"node","lean":"MetaMathlibExt.seedAutomaton_one_cons","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#seed-automaton-one-cons","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ be a letter map, let $a$ and $b$ be letters, and let $\\mathcal{A}_{\\eta,s}$ be the seed automaton for the seed $s = b|a$. For every digit word $w$, the value of $\\mathcal{A}_{\\eta,s}$ on the word $1w$ (the digit $1$ followed by $w$) equals the value $\\mathcal{A}_{\\eta,b}(w)$ of the substitution automaton started at $b$ on $w$: either both are undefined, or both are the same state.","title":"Seed automaton after reading the digit 1"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/seed-automaton-nil","kind":"node","lean":"MetaMathlibExt.seedAutomaton_nil","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#seed-automaton-nil","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ send each letter of an alphabet $A$ to a finite word over $A$, and let $a, b \\in A$. The automaton $\\mathcal{A}_{\\eta,s}$ of the seed $s = b|a$ does not accept the empty word.","title":"The seed automaton rejects the empty word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/last-letter-digits","kind":"node","lean":"MetaMathlibExt.lastLetterDigits","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#last-letter-digits","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ send each letter of an alphabet $A$ to a finite word over $A$. For $k \\in \\mathbb{N}$ and a letter $c \\in A$, the word $\\mathrm{last}_{\\eta,k}(c)$ of natural numbers is defined by recursion on $k$: $\\mathrm{last}_{\\eta,0}(c)$ is the empty word; if $\\eta(c)$ is nonempty with last letter $d$, then $\\mathrm{last}_{\\eta,k+1}(c)$ is the digit $|\\eta(c)| - 1$ followed by $\\mathrm{last}_{\\eta,k}(d)$; and if\u2026","title":"Digit word along last letters"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/last-letter-digits-succ-of-get-last","kind":"node","lean":"MetaMathlibExt.lastLetterDigits_succ_of_getLast?","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#last-letter-digits-succ-of-get-last","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ map each letter of an alphabet $A$ to a finite word over $A$. For a natural number $j$ and a letter $e \\in A$, the last-letter digit word of $j$ steps from $e$ is obtained by repeating the following step $j$ times, starting at $e$ and stopping early if the image of the current letter is empty: from the current letter $x$, record the digit $|\\eta(x)| - 1$ and move to the last letter of $\\eta(x)$. Let\u2026","title":"Recurrence for the last-letter digit word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/last-letter-digits-fibonacci","kind":"node","lean":"MetaMathlibExt.lastLetterDigits_fibonacci","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#last-letter-digits-fibonacci","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\varphi$ be the Fibonacci substitution on the alphabet $\\{0, 1\\}$, given by $\\varphi(0) = 01$ and $\\varphi(1) = 0$. Following the last letter twice from the letter $1$, where each step from a letter $c$ records the digit $|\\varphi(c)| - 1$ and moves to the last letter of $\\varphi(c)$, gives the digit word $01$: the word $\\varphi(1) = 0$ gives the digit $|\\varphi(1)| - 1 = 0$ and ends with $0$, and then the\u2026","title":"Last-letter digit word for the Fibonacci substitution"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/complement-radix-lt","kind":"node","lean":"MetaMathlibExt.ComplementRadixLT","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#complement-radix-lt","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"For finite words $w$ and $w'$ over $\\mathbb{N}$, write $|w|$ for the length of $w$. Then $w \\prec w'$ holds exactly in the following three cases: $w$ starts with $1$ and $w'$ starts with $0$; both words start with $0$, and either $|w| < |w'|$, or $|w| = |w'|$ and $w$ is lexicographically smaller than $w'$ (radix order); both words start with $1$, and either $|w| > |w'|$, or $|w| = |w'|$ and $w$ is lexicographically\u2026","title":"The complement radix order"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/complement-radix-lt-examples","kind":"node","lean":"MetaMathlibExt.complementRadixLT_examples","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#complement-radix-lt-examples","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"In the complement radix order $\\prec$ on finite words over $\\mathbb{N}$, $111 \\prec 10$, $10 \\prec 1$, $1 \\prec 0$, $0 \\prec 00$, $00 \\prec 01$ and $01 \\prec 000$. These are the consecutive comparisons in the chain $\\cdots \\prec 111 \\prec 10 \\prec 1 \\prec 0 \\prec 00 \\prec 01 \\prec 000$ given in the paper.","title":"Examples of the complement radix order"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-language","kind":"node","lean":"MetaMathlibExt.dumontThomasLanguage","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-language","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ send each letter of an alphabet $A$ to a finite word over $A$, let $u : \\mathbb{Z} \\to A$ and let $p \\in \\mathbb{N}$. Write $\\mathcal{A}_{\\eta,s}$ for the seed automaton of $\\eta$ with seed $s = u_{-1} | u_0$, and $W_{\\max}$ for the digit word obtained by following the last letter from $u_{-1}$ for at most $p$ steps: at a letter $c$ it records $|\\eta(c)| - 1$ and moves to the last letter of $\\eta(c)$,\u2026","title":"The Dumont\u2013Thomas image language"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/mem-dumont-thomas-language","kind":"node","lean":"MetaMathlibExt.mem_dumontThomasLanguage","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#mem-dumont-thomas-language","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $\\eta$ send each letter of an alphabet $A$ to a finite word over $A$, let $u : \\mathbb{Z} \\to A$ and $p \\in \\mathbb{N}$, and let $\\mathcal{A}_{\\eta,s}$ be the automaton of the seed $s = u_{-1}|u_0$. A word $w$ of natural numbers lies in the image language of $\\eta$, $u$ and $p$ if and only if the length of $w$ is $\\ell p + 1$ for some $\\ell \\in \\mathbb{N}$, $\\mathcal{A}_{\\eta,s}$ accepts $w$, $w$ does not start\u2026","title":"Membership in the Dumont\u2013Thomas image language"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-rep-quotient-remainder-pos","kind":"node","lean":"MetaMathlibExt.dumontThomasRep_quotient_remainder_pos","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-rep-quotient-remainder-pos","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution and let $u$ be a two-sided periodic point of $\\eta$ with least period $p$ whose seed letters $u_{-1}$ and $u_0$ are growing, that is, $|\\eta^j(u_{-1})|$ and $|\\eta^j(u_0)|$ tend to infinity with $j$. Let $N \\ge 1$, let $k$ be a multiple of $p$, and let $(m_i, a_i)_{0 \\le i \\le k-1}$ be a $u_0$-admissible sequence (admissible with respect to $\\eta$, with $m_{k-1} a_{k-1}$ a prefix of\u2026","title":"Quotient and remainder for positive Dumont\u2013Thomas representations"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-rep-quotient-remainder-neg","kind":"node","lean":"MetaMathlibExt.dumontThomasRep_quotient_remainder_neg","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-rep-quotient-remainder-neg","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$ and let $u : \\mathbb{Z} \\to A$ be a two-sided periodic point of $\\eta$ with least period $p$ whose seed letters $u_{-1}$ and $u_0$ are both growing. Let $N \\ge 2$ and $k$ be natural numbers with $p \\mid k$ and $N \\le |\\eta^k(u_{-1})|$, and let $(m_i, a_i)_{0 \\le i \\le k-1}$ be a $u_{-1}$-admissible sequence with\u2026","title":"Quotient and remainder for negative positions"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/seed-automaton-dumont-thomas-rep","kind":"node","lean":"MetaMathlibExt.seedAutomaton_dumontThomasRep","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#seed-automaton-dumont-thomas-rep","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, and let $u : \\mathbb{Z} \\to A$ be a two-sided periodic point of $\\eta$ whose least period is $p$ and whose seed letters $u_{-1}$ and $u_0$ are growing ($|\\eta^k(u_{-1})| \\to \\infty$ and $|\\eta^k(u_0)| \\to \\infty$ as $k \\to \\infty$). Let $\\mathcal{A}_{\\eta,s}$ be the automaton of the seed $s = u_{-1}|u_0$. Then every $n \\in \\mathbb{Z}$ has a Dumont\u2013Thomas complement\u2026","title":"Two-sided periodic points are automatic"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/seed-automaton-neutral-padding","kind":"node","lean":"MetaMathlibExt.seedAutomaton_neutral_padding","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#seed-automaton-neutral-padding","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, and let $u : \\mathbb{Z} \\to A$ be a two-sided periodic point of $\\eta$ whose least period is $p$ and whose seed letters $u_{-1}$ and $u_0$ are growing ($|\\eta^k(u_{-1})| \\to \\infty$ and $|\\eta^k(u_0)| \\to \\infty$ as $k \\to \\infty$). Let $\\mathcal{A}_{\\eta,s}$ be the automaton of the seed $s = u_{-1}|u_0$, let $W_{\\min} = 0^p$, and let\u2026","title":"Neutral words of the seed automaton"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-rep-strict-mono","kind":"node","lean":"MetaMathlibExt.dumontThomasRep_strictMono","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-rep-strict-mono","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution and let $u$ be a two-sided periodic point of $\\eta$ with least period $p$ whose seed letters $u_{-1}$ and $u_0$ are growing, that is, $|\\eta^j(u_{-1})|$ and $|\\eta^j(u_0)|$ tend to infinity with $j$. Let $n < n'$ be integers, and let $w$ and $w'$ be the Dumont\u2013Thomas complement representations $\\operatorname{rep}_u(n)$ and $\\operatorname{rep}_u(n')$ of $n$ and $n'$ for $\\eta$, $u$ and\u2026","title":"The Dumont\u2013Thomas representation is increasing"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/range-dumont-thomas-rep","kind":"node","lean":"MetaMathlibExt.range_dumontThomasRep","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#range-dumont-thomas-rep","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$, and let $u : \\mathbb{Z} \\to A$ be a two-sided periodic point of $\\eta$ whose least period is $p$ and whose seed letters $u_{-1}$ and $u_0$ are growing ($|\\eta^k(u_{-1})| \\to \\infty$ and $|\\eta^k(u_0)| \\to \\infty$ as $k \\to \\infty$). Let $\\mathcal{A}_{\\eta,s}$ be the automaton of the seed $s = u_{-1}|u_0$. A word $w$ of natural numbers is the Dumont\u2013Thomas complement\u2026","title":"The image of the Dumont\u2013Thomas complement representation"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-rep-iff-strict-mono-range","kind":"node","lean":"MetaMathlibExt.dumontThomasRep_iff_strictMono_range","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-rep-iff-strict-mono-range","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\eta$ be a substitution on an alphabet $A$ and let $u : \\mathbb{Z} \\to A$ be a two-sided periodic point of $\\eta$ with least period $p$ whose seed letters $u_{-1}$ and $u_0$ are both growing. Write $\\prec$ for the complement radix order and $L$ for the Dumont\u2013Thomas image language of $\\eta$, $u$ and $p$. Let $f$ be a map from $\\mathbb{Z}$ to finite words over $\\mathbb{N}$. Then $f(n)$ is the Dumont\u2013Thomas\u2026","title":"Characterization of Dumont\u2013Thomas complement representations"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/twos-complement-value","kind":"node","lean":"MetaMathlibExt.twosComplementValue","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#twos-complement-value","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"For a word $w = w_{k-1} \\cdots w_0$ of natural numbers, written with $w_{k-1}$ first, the two's complement value is the integer $\\mathrm{val}_{2c}(w) = \\sum_{i\\lt k} w_i 2^i - w_{k-1} 2^k$; the empty word has value $0$. For a binary word (all digits in $\\{0, 1\\}$) this is the usual two's complement value.","title":"Two's complement value"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/twos-complement-value-examples","kind":"node","lean":"MetaMathlibExt.twosComplementValue_examples","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#twos-complement-value-examples","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"The two's complement values $\\mathrm{val}_{2c}$ of the binary words $1$, $10$, $101$ and $011$ (most significant digit first) are $-1$, $-2$, $-3$ and $3$ respectively.","title":"Two's complement values of small words"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-rep-uniform-two-iff-twos-c","kind":"node","lean":"MetaMathlibExt.dumontThomasRep_uniform_two_iff_twosComplement","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-rep-uniform-two-iff-twos-c","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\psi$ send each letter of an alphabet $A$ to a word of length $2$ over $A$, and let $\\beta$ be a two-sided infinite word over $A$ that is a periodic point of $\\psi$ with period $1$, that is, $\\psi(\\beta) = \\beta$. For a finite word $w = w_{k-1} \\cdots w_1 w_0$ of natural numbers (most significant digit first), let $\\operatorname{val}_{2c}(w) = \\sum_{i=0}^{k-1} w_i 2^i - w_{k-1} 2^k$ be its two's complement\u2026","title":"Two's complement as a Dumont\u2013Thomas numeration system"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/fibonacci-complement-value","kind":"node","lean":"MetaMathlibExt.fibonacciComplementValue","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#fibonacci-complement-value","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"Let $(F_i)_{i \\ge 0}$ be the Fibonacci numbers with $F_0 = 1$, $F_1 = 2$ and $F_{i+2} = F_{i+1} + F_i$. The Fibonacci complement value of a word $w = w_{k-1} \\cdots w_1 w_0$ of natural numbers, written most significant digit first, is the integer $\\operatorname{val}_{\\mathcal{F}c}(w) = \\sum_{i=0}^{k-1} w_i F_i - w_{k-1} F_k$ when $k \\ge 1$, and $0$ when $w$ is empty.","title":"Fibonacci complement value"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/fibonacci-complement-value-examples","kind":"node","lean":"MetaMathlibExt.fibonacciComplementValue_examples","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#fibonacci-complement-value-examples","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"fully_proved","summary":"The Fibonacci complement values of the digit words $1$, $100$, $001$ and $010$ (most significant digit first) are $-1$, $-2$, $1$ and $2$, respectively.","title":"Values of small Fibonacci complement words"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/dumont-thomas-rep-fibonacci-iff-fibonacc","kind":"node","lean":"MetaMathlibExt.dumontThomasRep_fibonacci_iff_fibonacciComplement","parent":24039,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration/index.html#dumont-thomas-rep-fibonacci-iff-fibonacc","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/dumont-thomas-numeration","status":"can_prove","summary":"Let $\\varphi$ be the Fibonacci substitution on $\\{0, 1\\}$, given by $\\varphi(0) = 01$ and $\\varphi(1) = 0$, and let $\\gamma : \\mathbb{Z} \\to \\{0, 1\\}$ be a two-sided periodic point of $\\varphi$ with period $2$ such that $\\gamma_{-1} = 1$ and $\\gamma_0 = 0$. Then for every integer $n$ and every finite word $w$ over $\\mathbb{N}$, $w$ is the Dumont\u2013Thomas complement representation of $n$ for $\\varphi$, $\\gamma$ and\u2026","title":"Fibonacci complement numeration"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences.json","id":"Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences","kind":"scope","navigation":"module","parent":24020,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences","status":"fully_proved","title":"Regular sequences"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences/is-kregular-sequence","kind":"node","lean":"MetaMathlibExt.IsKRegularSequence","parent":24091,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-regular sequence"]],"source":"$k$-regular sequence"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences/index.html#is-kregular-sequence","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences","status":"fully_proved","summary":"Let $k$ be a natural number. An integer sequence $f : \\mathbb{N} \\to \\mathbb{Z}$ is $k$-regular, in the sense of Allouche and Shallit, if $k \\ge 2$ and the $\\mathbb{Z}$-submodule of the module of all integer sequences spanned by the $k$-kernel of $f$ is finitely generated.","title":"$k$-regular sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences/is-kregular-sequence-const","kind":"node","lean":"MetaMathlibExt.isKRegularSequence_const","parent":24091,"reading_title":{"format":"inline-parts-v1","parts":[["text","Constant sequences are "],["math","\\(k\\)"],["text","-regular"]],"source":"Constant sequences are $k$-regular"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences/index.html#is-kregular-sequence-const","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences","status":"fully_proved","summary":"For every natural number $k \\ge 2$ and every integer $c$, the constant sequence $n \\mapsto c$ is $k$-regular.","title":"Constant sequences are $k$-regular"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences/two-le","kind":"node","lean":"MetaMathlibExt.IsKRegularSequence.two_le","parent":24091,"reading_title":{"format":"inline-parts-v1","parts":[["text","The base of a "],["math","\\(k\\)"],["text","-regular sequence is at least 2"]],"source":"The base of a $k$-regular sequence is at least 2"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences/index.html#two-le","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/regular-sequences","status":"fully_proved","summary":"Let $k$ be a natural number. If an integer sequence $f$ is $k$-regular, then $k \\ge 2$.","title":"The base of a $k$-regular sequence is at least 2"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences.json","id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","kind":"scope","navigation":"module","parent":24020,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","title":"Meta-automatic sequences"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/thue-morse-quarto","kind":"node","lean":"MetaMathlibExt.thueMorseQuarto","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#thue-morse-quarto","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"The Thue\u2013Morse quarto sequence $\\mathcal{Q} : \\mathbb{N} \\to \\{0, 1\\}$ (OEIS A298952, with $\\mathcal{Q}(0) = 0$) is given by $\\mathcal{Q}(0) = 0$, $\\mathcal{Q}(1) = 1$ and, for every $m \\ge 2$, writing $m = 4n + r$ with $n \\ge 0$ and $0 \\le r \\le 3$: $\\mathcal{Q}(m) = \\mathcal{Q}(n)$ if $r = 0$, $\\mathcal{Q}(m) = 1 - \\mathcal{Q}(n)$ if $r = 1$, $\\mathcal{Q}(m) = \\mathcal{Q}(2n)$ if $r = 2$, and\u2026","title":"Thue\u2013Morse quarto sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-one","kind":"node","lean":"MetaMathlibExt.metaAutomaticOne","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","The meta-automatic sequence "],["math","\\(\\mathcal{M}_1\\)"]],"source":"The meta-automatic sequence $\\mathcal{M}_1$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-one","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"The sequence $\\mathcal{M}_1 : \\mathbb{N} \\to \\{0, 1\\}$ (OEIS A392736) is defined by $\\mathcal{M}_1(0) = 0$, $\\mathcal{M}_1(1) = 1$, the recurrences $\\mathcal{M}_1(4n) = \\mathcal{M}_1(n)$ and $\\mathcal{M}_1(4n+1) = 1 - \\mathcal{M}_1(n)$ for every $n \\ge 1$, and the recurrences $\\mathcal{M}_1(4n+2) = \\mathcal{M}_1(2n + 1 - \\mathcal{M}_1(n))$ and $\\mathcal{M}_1(4n+3) = 1 - \\mathcal{M}_1(2n + 1 - \\mathcal{M}_1(n))$ for\u2026","title":"The meta-automatic sequence $\\mathcal{M}_1$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","The meta-automatic sequence "],["math","\\(\\mathcal{M}_2\\)"]],"source":"The meta-automatic sequence $\\mathcal{M}_2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"The meta-automatic sequence $\\mathcal{M}_2 : \\mathbb{N} \\to \\{0, 1\\}$ (OEIS A391614) is defined by $\\mathcal{M}_2(0) = 0$, $\\mathcal{M}_2(1) = 1$, $\\mathcal{M}_2(4n) = \\mathcal{M}_2(2n + \\mathcal{M}_2(n))$ and $\\mathcal{M}_2(4n+1) = 1 - \\mathcal{M}_2(4n)$ for $n \\ge 1$, and $\\mathcal{M}_2(4n+2) = \\mathcal{M}_2(2n + 1 - \\mathcal{M}_2(n))$ and $\\mathcal{M}_2(4n+3) = 1 - \\mathcal{M}_2(4n+2)$ for $n \\ge 0$.","title":"The meta-automatic sequence $\\mathcal{M}_2$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/thue-morse-quarto-small","kind":"node","lean":"MetaMathlibExt.thueMorseQuarto_small","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#thue-morse-quarto-small","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"The values $\\mathcal{Q}(0), \\mathcal{Q}(1), \\ldots, \\mathcal{Q}(15)$ of the Thue\u2013Morse quarto sequence $\\mathcal{Q}$ are $0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1$, in this order.","title":"First sixteen terms of the Thue\u2013Morse quarto sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-one-small","kind":"node","lean":"MetaMathlibExt.metaAutomaticOne_small","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","First sixteen terms of "],["math","\\(\\mathcal{M}_1\\)"]],"source":"First sixteen terms of $\\mathcal{M}_1$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-one-small","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"Writing the values of the meta-automatic sequence $\\mathcal{M}_1$ as $0$ and $1$, the terms $\\mathcal{M}_1(0), \\dots, \\mathcal{M}_1(15)$ are $0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1$.","title":"First sixteen terms of $\\mathcal{M}_1$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-small","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_small","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","First sixteen terms of "],["math","\\(\\mathcal{M}_2\\)"]],"source":"First sixteen terms of $\\mathcal{M}_2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-small","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"The first sixteen terms $\\mathcal{M}_2(0), \\mathcal{M}_2(1), \\ldots, \\mathcal{M}_2(15)$ of the meta-automatic sequence $\\mathcal{M}_2$ are $0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1$.","title":"First sixteen terms of $\\mathcal{M}_2$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/thue-morse-quarto-rec","kind":"node","lean":"MetaMathlibExt.thueMorseQuarto_rec","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#thue-morse-quarto-rec","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"The Thue\u2013Morse quarto sequence $\\mathcal{Q}$ satisfies $\\mathcal{Q}(4n) = \\mathcal{Q}(n)$ and $\\mathcal{Q}(4n+1) = 1 - \\mathcal{Q}(n)$ for every $n \\ge 1$, and $\\mathcal{Q}(4n+2) = \\mathcal{Q}(2n)$ and $\\mathcal{Q}(4n+3) = 1 - \\mathcal{Q}(2n)$ for every $n \\ge 0$.","title":"Recurrences of the Thue\u2013Morse quarto sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-one-rec","kind":"node","lean":"MetaMathlibExt.metaAutomaticOne_rec","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrences of the meta-automatic sequence "],["math","\\(\\mathcal{M}_1\\)"]],"source":"Recurrences of the meta-automatic sequence $\\mathcal{M}_1$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-one-rec","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"Reading the values of the meta-automatic sequence $\\mathcal{M}_1$ as $0$ and $1$, for every $n \\ge 1$ we have $\\mathcal{M}_1(4n) = \\mathcal{M}_1(n)$ and $\\mathcal{M}_1(4n+1) = 1 - \\mathcal{M}_1(n)$, and for every $n \\in \\mathbb{N}$ we have $\\mathcal{M}_1(4n+2) = \\mathcal{M}_1(2n + 1 - \\mathcal{M}_1(n))$ and $\\mathcal{M}_1(4n+3) = 1 - \\mathcal{M}_1(2n + 1 - \\mathcal{M}_1(n))$.","title":"Recurrences of the meta-automatic sequence $\\mathcal{M}_1$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-rec","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_rec","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrences of "],["math","\\(\\mathcal{M}_2\\)"]],"source":"Recurrences of $\\mathcal{M}_2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-rec","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"The meta-automatic sequence $\\mathcal{M}_2$, read as taking values in $\\{0, 1\\}$, satisfies $\\mathcal{M}_2(4n) = \\mathcal{M}_2(2n + \\mathcal{M}_2(n))$ and $\\mathcal{M}_2(4n+1) = 1 - \\mathcal{M}_2(4n)$ for every $n \\ge 1$, and $\\mathcal{M}_2(4n+2) = \\mathcal{M}_2(2n + 1 - \\mathcal{M}_2(n))$ and $\\mathcal{M}_2(4n+3) = 1 - \\mathcal{M}_2(4n+2)$ for every $n \\ge 0$.","title":"Recurrences of $\\mathcal{M}_2$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/right-special-factors","kind":"node","lean":"MetaMathlibExt.rightSpecialFactors","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#right-special-factors","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"Let $a$ be an infinite binary word and $n$ a natural number. The right-special factors of $a$ of length $n$ are the binary words $u$ of length $n$ such that both $u0$ and $u1$ are factors of $a$.","title":"Right-special factors"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/right-special-factors-const-false","kind":"node","lean":"MetaMathlibExt.rightSpecialFactors_const_false","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#right-special-factors-const-false","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"fully_proved","summary":"For every natural number $n$, the infinite binary word $000\\cdots$ has no right-special factors of length $n$.","title":"The all-zero word has no right-special factors"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/is-kautomatic-sequence-of-digit-recurren","kind":"node","lean":"MetaMathlibExt.isKAutomaticSequence_of_digit_recurrence","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#is-kautomatic-sequence-of-digit-recurren","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $A$ be a finite set, $U : \\mathbb{N} \\to A$ a sequence and $q \\ge 2$ a natural number. Let $t, a, b, n_0$ be natural numbers, let $P$ be the set of pairs $(i, j')$ of natural numbers with $i \\le t$, $j' < q^i$ and $(i, j') \\ne (0, 0)$, and for each natural number $j$ let $f_j$ map families of elements of $A$ indexed by the disjoint union of $\\{0, 1, \\dots, a + b\\}$ and $P$ to $A$. Suppose that for all\u2026","title":"Allouche\u2013Shallit criterion for automatic sequences"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/balanced-nested-xor","kind":"node","lean":"MetaMathlibExt.balanced_nested_xor","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#balanced-nested-xor","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $a : \\mathbb{N} \\to \\{0, 1\\}$ satisfy $a(2n) + a(2n+1) = 1$ for every $n \\in \\mathbb{N}$, and write $\\oplus$ for addition modulo $2$. Then for every $n \\in \\mathbb{N}$, $a(2n + a(n)) = a(2n) \\oplus a(n)$ and $a(2n + 1 - a(n)) = a(2n+1) \\oplus a(n)$.","title":"Nested indices in balanced binary sequences"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/balanced-dyadic-nested-eventually-period","kind":"node","lean":"MetaMathlibExt.balanced_dyadic_nested_eventually_periodic","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#balanced-dyadic-nested-eventually-period","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $a : \\mathbb{N} \\to \\{0, 1\\}$ satisfy $a(2n) + a(2n+1) = 1$ for every $n \\in \\mathbb{N}$. (a) If $a(2n) = a(n + a(n))$ for every $n$, then $a(0) = 0$, and $a(2n) = 0$ and $a(2n+1) = 1$ for every $n \\ge 2$. (b) If $a(2n) = a(n + 1 - a(n))$ for every $n$, then $a(0) = 1$, and $a(2n) = 1$ and $a(2n+1) = 0$ for every $n \\ge 2$.","title":"Balanced sequences with a nested doubling recurrence are eventually alternating"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/thue-morse-quarto-eq-xor","kind":"node","lean":"MetaMathlibExt.thueMorseQuarto_eq_xor","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#thue-morse-quarto-eq-xor","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $\\mathcal{Q}$ be the Thue\u2013Morse quarto sequence and $t$ the Thue\u2013Morse sequence, and let $d(0) = 0$ and $d(n) = \\lfloor \\log_2 n \\rfloor \\bmod 2$ for $n \\ge 1$. Then for every $n \\ge 0$, $\\mathcal{Q}(n) = t(n) \\oplus d(n)$, where $\\oplus$ is addition modulo $2$.","title":"Thue\u2013Morse quarto sequence from the Thue\u2013Morse sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-one-balanced","kind":"node","lean":"MetaMathlibExt.metaAutomaticOne_balanced","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","The meta-automatic sequence "],["math","\\(\\mathcal{M}_1\\)"],["text"," is balanced"]],"source":"The meta-automatic sequence $\\mathcal{M}_1$ is balanced"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-one-balanced","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Reading the values of the meta-automatic sequence $\\mathcal{M}_1$ as $0$ and $1$, we have $\\mathcal{M}_1(2n) + \\mathcal{M}_1(2n+1) = 1$ for every $n \\in \\mathbb{N}$.","title":"The meta-automatic sequence $\\mathcal{M}_1$ is balanced"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-one-pair-closure","kind":"node","lean":"MetaMathlibExt.metaAutomaticOne_pair_closure","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Joint recurrence for "],["math","\\(\\mathcal{M}_1\\)"],["text"," and its nested companion"]],"source":"Joint recurrence for $\\mathcal{M}_1$ and its nested companion"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-one-pair-closure","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Read the values of the meta-automatic sequence $\\mathcal{M}_1$ as $0$ and $1$, write $\\oplus$ for addition modulo $2$, and let $b(m) = \\mathcal{M}_1(2m + 1 - \\mathcal{M}_1(m))$ for every $m \\in \\mathbb{N}$. Then for every $n \\ge 1$, with $x = \\mathcal{M}_1(n)$ and $y = b(n)$, the pairs $s(k) = (\\mathcal{M}_1(k), b(k))$ satisfy $s(4n) = (x, y)$, $s(4n+1) = (1 \\oplus x, x \\oplus y)$, $s(4n+2) = (y, 1 \\oplus x)$ and\u2026","title":"Joint recurrence for $\\mathcal{M}_1$ and its nested companion"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-one-dfao","kind":"node","lean":"MetaMathlibExt.metaAutomaticOne_dfao","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","An automaton for the meta-automatic sequence "],["math","\\(\\mathcal{M}_1\\)"]],"source":"An automaton for the meta-automatic sequence $\\mathcal{M}_1$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-one-dfao","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"The meta-automatic sequence $\\mathcal{M}_1$ is $4$-automatic. Moreover, let $\\delta$ be the transition map on the states $\\{0, 1, 2, 3\\}$ and the digits $\\{0, 1, 2, 3\\}$ given by $\\delta(0, \\cdot) = 0112$, $\\delta(1, \\cdot) = 1230$, $\\delta(2, \\cdot) = 2301$ and $\\delta(3, \\cdot) = 3023$, where the symbol at position $d$ (counted from $0$) of the word listed for $q$ is $\\delta(q, d)$. For every $n \\in \\mathbb{N}$,\u2026","title":"An automaton for the meta-automatic sequence $\\mathcal{M}_1$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-one-morphism-coding","kind":"node","lean":"MetaMathlibExt.metaAutomaticOne_morphism_coding","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathcal{M}_1\\)"],["text"," as a coding of a uniform morphism fixed point"]],"source":"$\\mathcal{M}_1$ as a coding of a uniform morphism fixed point"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-one-morphism-coding","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $\\sigma$ be the $4$-uniform morphism on $\\{0, 1, 2, 3\\}$ with $\\sigma(0) = 0112$, $\\sigma(1) = 1230$, $\\sigma(2) = 2301$ and $\\sigma(3) = 3023$, and write $\\sigma(c)_r$ for the letter of $\\sigma(c)$ at position $r$, counted from $0$. There is a sequence $w : \\mathbb{N} \\to \\mathbb{N}$ with $w(0) = 0$ and $w(4n + r) = \\sigma(w(n))_r$ for all $n \\in \\mathbb{N}$ and $r < 4$, such that, reading the values of the\u2026","title":"$\\mathcal{M}_1$ as a coding of a uniform morphism fixed point"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-balanced","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_balanced","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathcal{M}_2\\)"],["text"," is balanced"]],"source":"$\\mathcal{M}_2$ is balanced"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-balanced","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"For every natural number $n$, the meta-automatic sequence $\\mathcal{M}_2$, read as taking values in $\\{0, 1\\}$, satisfies $\\mathcal{M}_2(2n) + \\mathcal{M}_2(2n+1) = 1$.","title":"$\\mathcal{M}_2$ is balanced"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-recover","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_recover","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recovering the even and odd terms of "],["math","\\(\\mathcal{M}_2\\)"]],"source":"Recovering the even and odd terms of $\\mathcal{M}_2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-recover","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"For every natural number $n$, let $x = \\mathcal{M}_2(n)$ and $y = \\mathcal{M}_2(2n + \\mathcal{M}_2(n))$, where $\\mathcal{M}_2$ is the meta-automatic sequence with values in $\\{0, 1\\}$. Then $\\mathcal{M}_2(2n) = x \\oplus y$ and $\\mathcal{M}_2(2n+1) = x \\oplus y \\oplus 1$, where $\\oplus$ is addition modulo $2$.","title":"Recovering the even and odd terms of $\\mathcal{M}_2$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-affine","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_affine","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pair recurrence and "],["math","\\(4\\)"],["text","-automaticity of "],["math","\\(\\mathcal{M}_2\\)"]],"source":"Pair recurrence and $4$-automaticity of $\\mathcal{M}_2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-affine","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Read the meta-automatic sequence $\\mathcal{M}_2$ as taking values in $\\{0, 1\\}$, write $\\oplus$ for addition modulo $2$, and let $s(n) = (\\mathcal{M}_2(n), \\mathcal{M}_2(2n + \\mathcal{M}_2(n)))$. For every $n \\ge 0$, writing $s(n) = (x, y)$, we have $s(4n) = (y, x \\oplus y)$, $s(4n+1) = (y \\oplus 1, x \\oplus y)$, $s(4n+2) = (y \\oplus 1, x \\oplus y \\oplus 1)$ and $s(4n+3) = (y, x \\oplus y \\oplus 1)$. Moreover,\u2026","title":"Pair recurrence and $4$-automaticity of $\\mathcal{M}_2$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-dfao","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_dfao","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","A four-state automaton computing "],["math","\\(\\mathcal{M}_2\\)"]],"source":"A four-state automaton computing $\\mathcal{M}_2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-dfao","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $\\delta : \\{0, 1, 2, 3\\} \\times \\{0, 1, 2, 3\\} \\to \\{0, 1, 2, 3\\}$ be the map for which the word $\\delta(q, 0)\\,\\delta(q, 1)\\,\\delta(q, 2)\\,\\delta(q, 3)$ is $0132$ for $q = 0$, $2310$ for $q = 1$, $3201$ for $q = 2$ and $1023$ for $q = 3$. For every natural number $n$, the meta-automatic sequence $\\mathcal{M}_2$, read as taking values in $\\{0, 1\\}$, satisfies $\\mathcal{M}_2(n) = q \\bmod 2$, where $q$ is the\u2026","title":"A four-state automaton computing $\\mathcal{M}_2$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-morphism-coding","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_morphism_coding","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathcal{M}_2\\)"],["text"," as a coding of a uniform morphic word"]],"source":"$\\mathcal{M}_2$ as a coding of a uniform morphic word"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-morphism-coding","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $\\sigma$ be the $4$-uniform morphism on $\\{0, 1, 2, 3\\}$ with $\\sigma(0) = 0132$, $\\sigma(1) = 2310$, $\\sigma(2) = 3201$ and $\\sigma(3) = 1023$. There is a sequence $w$ with values in $\\{0, 1, 2, 3\\}$ such that $w(0) = 0$, $w(4n + r)$ is the letter of $\\sigma(w(n))$ in position $r$ (counting from $0$) for all $n \\ge 0$ and $0 \\le r < 4$, and, for every $n$, the meta-automatic sequence $\\mathcal{M}_2$ satisfies\u2026","title":"$\\mathcal{M}_2$ as a coding of a uniform morphic word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-digit-formula","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_digit_formula","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digit formula for "],["math","\\(\\mathcal{M}_2\\)"]],"source":"Digit formula for $\\mathcal{M}_2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-digit-formula","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"For a natural number $n$ and $k \\ge 0$, let $d_k = \\lfloor n / 4^k \\rfloor \\bmod 4$ be the $k$-th base-$4$ digit of $n$, and write $d_k = 2 j_k + i_k$ with $i_k, j_k \\in \\{0, 1\\}$. Let $c_k = i_k + j_k$ if $k \\equiv 0 \\pmod 3$, $c_k = j_k$ if $k \\equiv 1 \\pmod 3$, and $c_k = i_k$ if $k \\equiv 2 \\pmod 3$. Then the meta-automatic sequence $\\mathcal{M}_2$, read as taking values in $\\{0, 1\\}$, satisfies\u2026","title":"Digit formula for $\\mathcal{M}_2$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-eq-thue-morse-mask","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_eq_thueMorse_mask","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathcal{M}_2\\)"],["text"," is the Thue\u2013Morse sequence along a binary digit mask"]],"source":"$\\mathcal{M}_2$ is the Thue\u2013Morse sequence along a binary digit mask"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-eq-thue-morse-mask","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"For a natural number $m$ with binary digits $b_i = \\lfloor m / 2^i \\rfloor \\bmod 2$, let $q(m)$ be the sum of $b_i 2^i$ over the $i \\le m$ with $i \\not\\equiv 2 \\pmod 3$, that is, the number obtained from $m$ by setting its binary digits in the positions $i \\equiv 2 \\pmod 3$ to zero. Then for every natural number $n$, the meta-automatic sequence $\\mathcal{M}_2$ and the Thue\u2013Morse sequence $t$ satisfy\u2026","title":"$\\mathcal{M}_2$ is the Thue\u2013Morse sequence along a binary digit mask"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/word-factor-complexity-succ-eq-add-right","kind":"node","lean":"MetaMathlibExt.wordFactorComplexity_succ_eq_add_rightSpecial","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#word-factor-complexity-succ-eq-add-right","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $a$ be an infinite word over $\\{0, 1\\}$, let $p_a$ be its factor complexity, and let $\\mathrm{RS}_a(n)$ be the set of right-special factors of $a$ of length $n$. Then for every $n \\ge 1$, $p_a(n+1) = p_a(n) + \\#\\mathrm{RS}_a(n)$.","title":"First difference of the factor complexity of a binary word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/right-special-count-regular","kind":"node","lean":"MetaMathlibExt.rightSpecial_count_regular","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#right-special-count-regular","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $k$ be a natural number and $a$ an infinite binary word that is $k$-automatic, and for each $n$ let $\\mathrm{RS}_a(n)$ be the set of right-special factors of $a$ of length $n$. Then the integer sequence $n \\mapsto \\#\\mathrm{RS}_a(n)$ is $k$-regular. Moreover, for every natural number $C$ with $\\#\\mathrm{RS}_a(n) \\le C$ for all $n$, the sequence $n \\mapsto \\#\\mathrm{RS}_a(n)$, taken with values in\u2026","title":"Right-special factor counts of automatic sequences are regular"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/thue-morse-factor-complement","kind":"node","lean":"MetaMathlibExt.thueMorse_factor_complement","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#thue-morse-factor-complement","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"For every natural number $n$, if a binary word $v$ of length $n$ is a factor of the Thue\u2013Morse sequence, then so is the word obtained from $v$ by exchanging $0$ and $1$ in every position.","title":"Factors of the Thue\u2013Morse word are closed under complement"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/thue-morse-quarto-complexity-eq","kind":"node","lean":"MetaMathlibExt.thueMorseQuarto_complexity_eq","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#thue-morse-quarto-complexity-eq","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $\\mathcal{Q}$ be the Thue\u2013Morse Quarto sequence and $t$ the Thue\u2013Morse sequence, with factor complexities $p_{\\mathcal{Q}}$ and $p_t$. For every natural number $n$, let $J_n$ be the set of length-$n$ factors of $\\mathcal{Q}$ that are not factors of $t$. Then $p_{\\mathcal{Q}}(n) = p_t(n) + |J_n|$.","title":"Factor complexity of the Thue\u2013Morse Quarto word versus Thue\u2013Morse"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/thue-morse-quarto-complexity-two-pow","kind":"node","lean":"MetaMathlibExt.thueMorseQuarto_complexity_two_pow","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#thue-morse-quarto-complexity-two-pow","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $\\mathcal{Q}$ be the Thue\u2013Morse Quarto sequence and $t$ the Thue\u2013Morse sequence, with factor complexities $p_{\\mathcal{Q}}$ and $p_t$, and for a natural number $n$ let $J_n$ be the set of length-$n$ factors of $\\mathcal{Q}$ that are not factors of $t$. Then for every natural number $k \\ge 4$, $p_{\\mathcal{Q}}(2^k) = 13 \\cdot 2^{k-2} - 2$, $p_{\\mathcal{Q}}(2^k) = p_t(2^k) + 2^{k-2}$ and $|J_{2^k}| = 2^{k-2}$.","title":"Factor complexity of the Thue\u2013Morse Quarto word at powers of two"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/thue-morse-quarto-complexity-formula","kind":"node","lean":"MetaMathlibExt.thueMorseQuarto_complexity_formula","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#thue-morse-quarto-complexity-formula","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $\\mathcal{Q}$ be the Thue\u2013Morse Quarto sequence, $p_{\\mathcal{Q}}$ its factor complexity and $\\mathrm{RS}_{\\mathcal{Q}}(n)$ its set of right-special factors of length $n$. Let $n \\ge 16$ and $k = \\lfloor \\log_2 n \\rfloor$, so that $2^k \\le n < 2^{k+1}$. Then $p_{\\mathcal{Q}}(n) = 13 \\cdot 2^{k-2} - 2$ if $n = 2^k$, $p_{\\mathcal{Q}}(n) = 4n - 3 \\cdot 2^{k-2} - 4$ if $2^k < n \\le 2^k + 2^{k-1}$,\u2026","title":"Factor complexity and right-special factors of the Thue\u2013Morse Quarto word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-complexity-is-theta","kind":"node","lean":"MetaMathlibExt.metaAutomatic_complexity_isTheta","parent":24095,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-complexity-is-theta","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $p_{\\mathcal{M}_1}$ and $p_{\\mathcal{M}_2}$ be the factor complexities of the meta-automatic sequences $\\mathcal{M}_1$ and $\\mathcal{M}_2$. Then $p_{\\mathcal{M}_1}(n) = \\Theta(n)$ and $p_{\\mathcal{M}_2}(n) = \\Theta(n)$ as $n \\to \\infty$: for each $i \\in \\{1, 2\\}$ there are real constants $c, C > 0$ with $c n \\le p_{\\mathcal{M}_i}(n) \\le C n$ for all sufficiently large $n$.","title":"The meta-automatic sequences have linear factor complexity"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-complexity-two-pow","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_complexity_two_pow","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Factor complexity of "],["math","\\(\\mathcal{M}_2\\)"],["text"," at powers of two"]],"source":"Factor complexity of $\\mathcal{M}_2$ at powers of two"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-complexity-two-pow","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $p(n)$ be the factor complexity of the meta-automatic sequence $\\mathcal{M}_2$, and let $k \\ge 3$. If $3 \\mid k$, then $p(2^k) = 5 \\cdot 2^{k-1} - 2$; if $3 \\nmid k$, then $p(2^k) = 3 \\cdot 2^k - 2$. Moreover, $p(2^{k+3}) = 8\\, p(2^k) + 14$.","title":"Factor complexity of $\\mathcal{M}_2$ at powers of two"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/meta-automatic-two-complexity-formula","kind":"node","lean":"MetaMathlibExt.metaAutomaticTwo_complexity_formula","parent":24095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Factor complexity of "],["math","\\(\\mathcal{M}_2\\)"]],"source":"Factor complexity of $\\mathcal{M}_2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences/index.html#meta-automatic-two-complexity-formula","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/meta-automatic-sequences","status":"can_prove","summary":"Let $p(n)$ be the factor complexity of the meta-automatic sequence $\\mathcal{M}_2$. Then $p(1) = 2$, and for every $n \\ge 2$, writing $k = \\lfloor \\log_2 n \\rfloor$ (so that $2^k \\le n < 2^{k+1}$), the following hold. If $k \\equiv 0 \\pmod 3$, then $p(n) = 2n + 2^{k-1} - 2$ when $n \\le 2^k + 1$, $p(n) = 4n - (3 \\cdot 2^{k-1} + 4)$ when $2^k + 1 \\le n \\le 7 \\cdot 2^{k-2} + 1$, and $p(n) = 2n + 2^{k+1} - 2$ when\u2026","title":"Factor complexity of $\\mathcal{M}_2$"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word.json","id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","kind":"scope","navigation":"module","parent":24020,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","title":"Irvine's conjecture on Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-morphism","kind":"node","lean":"MetaMathlibExt.kimberlingMorphism","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-morphism","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Kimberling's morphism $h$ sends each letter of the alphabet $\\{0, 1\\}$ to a finite word over $\\{0, 1\\}$, by $h(1) = 110$ and $h(0) = 11$.","title":"Kimberling's morphism"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-block","kind":"node","lean":"MetaMathlibExt.kimberlingBlock","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-block","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $h$ be Kimberling's morphism on the alphabet $\\{0, 1\\}$, given by $h(1) = 110$ and $h(0) = 11$ and extended to finite words by concatenation. For each natural number $n$, the $n$-th Kimberling block is the finite word $K_n = h^n(1)$, the $n$-th iterate of $h$ applied to the one-letter word $1$. Thus $K_0 = 1$, $K_1 = 110$ and $K_2 = 11011011$.","title":"Kimberling's blocks"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-block-one","kind":"node","lean":"MetaMathlibExt.kimberlingBlock_one","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-block-one","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"For Kimberling's morphism $h$ on $\\{0, 1\\}$, given by $h(1) = 110$ and $h(0) = 11$, the block $K_1 = h(1)$ is the word $110$.","title":"The first Kimberling block"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-block-two","kind":"node","lean":"MetaMathlibExt.kimberlingBlock_two","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-block-two","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"For Kimberling's morphism $h$ on $\\{0, 1\\}$, given by $h(1) = 110$ and $h(0) = 11$, the block $K_2 = h^2(1)$ is the word $11011011$.","title":"The second Kimberling block"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-block-prefix-succ","kind":"node","lean":"MetaMathlibExt.kimberlingBlock_prefix_succ","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-block-prefix-succ","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $h$ be Kimberling's morphism on $\\{0, 1\\}$, $h(1) = 110$ and $h(0) = 11$, and let $K_n = h^n(1)$. Then for every $n \\ge 0$, the word $K_n$ is a prefix of $K_{n+1}$.","title":"Each Kimberling block is a prefix of the next"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-block-prefix","kind":"node","lean":"MetaMathlibExt.kimberlingBlock_prefix","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-block-prefix","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $h$ be Kimberling's morphism on $\\{0, 1\\}$, $h(1) = 110$ and $h(0) = 11$, and let $K_n = h^n(1)$. Then $K_m$ is a prefix of $K_n$ for all natural numbers $m \\le n$.","title":"Kimberling's blocks form a prefix chain"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/two-pow-le-length-kimberling-block","kind":"node","lean":"MetaMathlibExt.two_pow_le_length_kimberlingBlock","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#two-pow-le-length-kimberling-block","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $h$ be Kimberling's morphism on the alphabet $\\{0, 1\\}$, given by $h(1) = 110$ and $h(0) = 11$, and let $K_n = h^n(1)$. For every natural number $n$, the word $K_n$ has length at least $2^n$.","title":"Lower bound on the length of the Kimberling blocks"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-word","kind":"node","lean":"MetaMathlibExt.kimberlingWord","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-word","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $K_n$ be the $n$-th Kimberling block, a finite word over $\\{0, 1\\}$. Kimberling's word $k = k_1 k_2 k_3 \\cdots = 1101101111\\cdots$ (OEIS A285431) is the infinite word over $\\{0, 1\\}$ whose letter $k_{i+1}$, for $i \\ge 0$, is the $(i+1)$-th letter of $K_m$, where $m = \\lfloor \\log_2 i \\rfloor + 1$ for $i \\ge 1$ and $m = 1$ for $i = 0$. The block $K_m$ has length at least $2^m > i$, so this letter exists.","title":"Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-word-eq-get-d","kind":"node","lean":"MetaMathlibExt.kimberlingWord_eq_getD","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-word-eq-get-d","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $k = k_1 k_2 k_3 \\cdots$ be Kimberling's word and $K_n$ the $n$-th Kimberling block. For all natural numbers $i$ and $n$ with $i < |K_n|$, the letter $k_{i+1}$ equals the $(i+1)$-th letter of $K_n$.","title":"Every Kimberling block is a prefix of Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-word-initial","kind":"node","lean":"MetaMathlibExt.kimberlingWord_initial","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-word-initial","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"The first four letters of Kimberling's word $k = k_1 k_2 k_3 \\cdots$ are $k_1 k_2 k_3 k_4 = 1101$.","title":"The first letters of Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-word-hundred","kind":"node","lean":"MetaMathlibExt.kimberlingWord_hundred","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-word-hundred","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"The letter $k_{101}$ of Kimberling's word $k = k_1 k_2 k_3 \\cdots$ is $0$ (OEIS A285431).","title":"The 101st letter of Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy-table","kind":"node","lean":"MetaMathlibExt.gutkovskiyTable","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy-table","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"The Gutkovskiy tables $T_0, T_1, \\dots$ are finite lists of natural numbers, indexed from $0$, defined by $T_0 = [0]$ and, for $n \\ge 0$, $T_{n+1}$ is $T_n$ with the entry $(n + 1) - T_n[\\lfloor T_n[n]/2 \\rfloor]$ appended at the end. Here $T_n[j]$ is the entry of $T_n$ at index $j$, or $0$ if $T_n$ has fewer than $j + 1$ entries, and the subtraction is truncated at $0$.","title":"Table of Gutkovskiy's sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy","kind":"node","lean":"MetaMathlibExt.gutkovskiy","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Gutkovskiy's sequence (OEIS A286389) is the sequence of natural numbers $g_0, g_1, \\dots$ with $g_0 = 0$ and $g_n = n - g_{\\lfloor g_{n-1}/2 \\rfloor}$ for $n \\ge 1$ (subtraction of natural numbers, truncated at $0$).","title":"Gutkovskiy's sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/size-gutkovskiy-table","kind":"node","lean":"MetaMathlibExt.size_gutkovskiyTable","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#size-gutkovskiy-table","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"For every natural number $n$, the Gutkovskiy table up to index $n$, the array of the values $g_0, \\dots, g_n$ of Gutkovskiy's sequence, has exactly $n + 1$ entries.","title":"Number of entries of the Gutkovskiy table"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/get-d-gutkovskiy-table","kind":"node","lean":"MetaMathlibExt.getD_gutkovskiyTable","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#get-d-gutkovskiy-table","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"For natural numbers $k \\le n$, the entry at index $k$ (counting from $0$) of the Gutkovskiy table $T_n$ is the term $g_k$ of Gutkovskiy's sequence.","title":"Entries of the Gutkovskiy table"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy-le","kind":"node","lean":"MetaMathlibExt.gutkovskiy_le","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy-le","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"For every natural number $n$, the term $g_n$ of Gutkovskiy's sequence satisfies $g_n \\le n$.","title":"Gutkovskiy's sequence is bounded by the index"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy-succ","kind":"node","lean":"MetaMathlibExt.gutkovskiy_succ","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy-succ","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"For every natural number $n$, Gutkovskiy's sequence satisfies $g_{n+1} = (n + 1) - g_{\\lfloor g_n/2 \\rfloor}$ (subtraction of natural numbers, truncated at $0$).","title":"Recurrence for Gutkovskiy's sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy-initial","kind":"node","lean":"MetaMathlibExt.gutkovskiy_initial","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy-initial","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"The first four terms of Gutkovskiy's sequence are $g_0 = 0$, $g_1 = 1$, $g_2 = 2$ and $g_3 = 2$.","title":"First terms of Gutkovskiy's sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy-hundred","kind":"node","lean":"MetaMathlibExt.gutkovskiy_hundred","parent":24131,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gutkovskiy's sequence at "],["math","\\(100\\)"]],"source":"Gutkovskiy's sequence at $100$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy-hundred","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"The term $g_{100}$ of Gutkovskiy's sequence is $74$, in agreement with OEIS A286389.","title":"Gutkovskiy's sequence at $100$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-one-position","kind":"node","lean":"MetaMathlibExt.kimberlingOnePosition","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-one-position","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $k = k_1 k_2 k_3 \\cdots$ be Kimberling's word over $\\{0, 1\\}$. For $n \\ge 1$, $A_n$ is the position of the $n$-th $1$ in $k$, that is, the $n$-th smallest index $j \\ge 1$ with $k_j = 1$; by convention $A_0 = 0$.","title":"Positions of the ones in Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-zero-position","kind":"node","lean":"MetaMathlibExt.kimberlingZeroPosition","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-zero-position","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $k = k_1 k_2 k_3 \\cdots$ be Kimberling's word over $\\{0, 1\\}$. For $n \\ge 1$, $B_n$ is the position of the $n$-th $0$ in $k$, that is, the $n$-th smallest index $j \\ge 1$ with $k_j = 0$; by convention $B_0 = 0$.","title":"Positions of the zeros in Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-one-position-one","kind":"node","lean":"MetaMathlibExt.kimberlingOnePosition_one","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-one-position-one","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"The position $A_1$ of the first $1$ in Kimberling's word $k = k_1 k_2 k_3 \\cdots$, counted from $1$, is $A_1 = 1$.","title":"Position of the first one in Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-zero-position-one","kind":"node","lean":"MetaMathlibExt.kimberlingZeroPosition_one","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-zero-position-one","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"fully_proved","summary":"Let $B_n$ be the position of the $n$-th $0$ in Kimberling's word $k = k_1 k_2 k_3 \\cdots$. Then $B_1 = 3$: the first $0$ of $k$ is $k_3$.","title":"The first zero of Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-block-eq-append","kind":"node","lean":"MetaMathlibExt.kimberlingBlock_eq_append","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-block-eq-append","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $h$ be Kimberling's morphism on $\\{0, 1\\}$, $h(1) = 110$ and $h(0) = 11$, and let $K_n = h^n(1)$. Then for every $n \\ge 2$, $K_n = K_{n-1} K_{n-1} K_{n-2} K_{n-2}$ (concatenation of words).","title":"Recursive structure of Kimberling's blocks"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy-eq-sum-kimberling-word","kind":"node","lean":"MetaMathlibExt.gutkovskiy_eq_sum_kimberlingWord","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy-eq-sum-kimberling-word","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $g_0, g_1, \\dots$ be Gutkovskiy's sequence and let $k_1 k_2 k_3 \\cdots$ be Kimberling's word, with its letters read as the numbers $0$ and $1$. Then $g_n = k_1 + \\dots + k_n$ for every $n \\ge 0$ (the empty sum when $n = 0$).","title":"Irvine's conjecture for Gutkovskiy's sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy-eq-two-mul-floor","kind":"node","lean":"MetaMathlibExt.gutkovskiy_eq_two_mul_floor","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy-eq-two-mul-floor","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $g_0, g_1, \\dots$ be Gutkovskiy's sequence, let $|K_i|$ be the length of the Kimberling block $K_i$, and let $\\alpha = (\\sqrt{3} - 1)/2$ and $\\beta = \\sqrt{3}/3$. Let $n$ be a natural number and let $a_0, a_1, \\dots, a_{m-1}$ be digits in $\\{0, 1, 2\\}$ such that no two consecutive digits $a_i, a_{i+1}$ are both $2$ and $n = \\sum_{i < m} a_i |K_i|$. If $m \\ge 1$ and $a_0 = 1$, then\u2026","title":"Closed form for Gutkovskiy's sequence"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/kimberling-one-position-kimberling-zero","kind":"node","lean":"MetaMathlibExt.kimberlingOnePosition_kimberlingZeroPosition","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#kimberling-one-position-kimberling-zero","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $k = k_1 k_2 k_3 \\cdots$ be Kimberling's word, and for $n \\ge 1$ let $A_n$ and $B_n$ be the positions (counted from $1$) of the $n$-th $1$ and of the $n$-th $0$ in $k$. Then for every $n \\ge 1$, either $A_{B_n} = A_n + B_n - 1$ or $A_{B_n} = A_n + B_n$.","title":"Positions of ones at the positions of zeros in Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/is-omega-limit-gutkovskiy-mod-two","kind":"node","lean":"MetaMathlibExt.isOmegaLimit_gutkovskiy_mod_two","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#is-omega-limit-gutkovskiy-mod-two","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $g_0, g_1, g_2, \\ldots$ be Gutkovskiy's sequence, and let $u$ be the map on the alphabet $\\mathbb{N}$ with $u(0) = 01$ and $u(b) = 0010$ for every letter $b \\ne 0$. Then the infinite word $(g_n \\bmod 2)_{n \\ge 0}$ is the limit of the iterates of $u$ at the letter $0$. On the letters $0$ and $1$ that occur, $u$ is the morphism $0 \\mapsto 01$, $1 \\mapsto 0010$, and the word is OEIS A284772.","title":"Gutkovskiy's sequence modulo 2 is generated by a morphism"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/gutkovskiy-eq-two-mul-sum-mod-two-add","kind":"node","lean":"MetaMathlibExt.gutkovskiy_eq_two_mul_sum_mod_two_add","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#gutkovskiy-eq-two-mul-sum-mod-two-add","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $g_0, g_1, \\dots$ be Gutkovskiy's sequence, let $g'_n = g_n \\bmod 2$ and let $h_n = \\sum_{0 \\le i < n} g'_i$. Then $g_n = 2h_n + g'_n$ for every $n \\ge 0$.","title":"Gutkovskiy's sequence from its parities"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/right-special-factors-kimberling-word-nc","kind":"node","lean":"MetaMathlibExt.rightSpecialFactors_kimberlingWord_ncard_eq_two_iff","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#right-special-factors-kimberling-word-nc","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $k$ be Kimberling's word and write $K_m$ for the length of the $m$-th Kimberling block. A word $u$ is a right-special factor of $k$ if both $u0$ and $u1$ are factors of $k$. For every $n \\ge 0$, $k$ has exactly two right-special factors of length $n$ if and only if there is an $i \\ge 0$ such that $x_i \\le n < x_i + K_{2i}$, where $x_i = K_1 + K_3 + \\cdots + K_{2i+1}$, or $y_i \\le n < y_i + K_{2i+1}$, where\u2026","title":"Lengths with two right-special factors in Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/limsup-liminf-word-factor-complexity-kim","kind":"node","lean":"MetaMathlibExt.limsup_liminf_wordFactorComplexity_kimberlingWord","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#limsup-liminf-word-factor-complexity-kim","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $\\rho(n)$ be the factor complexity of Kimberling's word $k$, the number of distinct factors of $k$ of length $n$. Then, as $n \\to \\infty$ over the natural numbers, $\\limsup \\rho(n)/n = (30 + \\sqrt{3})/23$ and $\\liminf \\rho(n)/n = (3 + \\sqrt{3})/4$.","title":"Factor complexity of Kimberling's word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/s-sup-exponent-kimberling-word","kind":"node","lean":"MetaMathlibExt.sSup_exponent_kimberlingWord","parent":24131,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word/index.html#s-sup-exponent-kimberling-word","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/irvine-s-conjecture-on-kimberling-s-word","status":"can_prove","summary":"Let $k = k_1 k_2 k_3 \\cdots$ be Kimberling's word. The supremum of the ratios $m/p$, over all integers $m \\ge 1$ and $p \\ge 1$ for which some factor $k_{i+1} k_{i+2} \\cdots k_{i+m}$ of $k$ ($i \\ge 0$) has period $p$, that is, $k_{i+1+j} = k_{i+1+j+p}$ for every $j \\ge 0$ with $j + p < m$, is $(2\\sqrt{3} + 12)/3$.","title":"Critical exponent of Kimberling's word"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra.json","id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","kind":"scope","navigation":"module","parent":24020,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","title":"Tribonacci representations and the Trithoff array"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/is-tribonacci-repr","kind":"node","lean":"MetaMathlibExt.IsTribonacciRepr","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#is-tribonacci-repr","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"Let $T_n$ be the Tribonacci numbers, $T_0 = T_1 = 0$, $T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$. A list $w = (w_0, \\dots, w_{\\ell-1})$ of natural numbers, read least significant digit first, is a Tribonacci representation of a natural number $n$ when every digit $w_i$ is $0$ or $1$, no three consecutive entries of $w$ are all equal to $1$, and $n = \\sum_{i=0}^{\\ell-1} w_i T_{i+3}$. Leading zeros (trailing\u2026","title":"Tribonacci representation"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/is-tribonacci-repr-nil","kind":"node","lean":"MetaMathlibExt.isTribonacciRepr_nil","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#is-tribonacci-repr-nil","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"The empty digit list is a Tribonacci representation of $0$.","title":"The empty word represents zero"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/is-tribonacci-repr-nine","kind":"node","lean":"MetaMathlibExt.isTribonacciRepr_nine","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#is-tribonacci-repr-nine","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"The digit list $(0, 1, 0, 1)$, read least significant digit first (the word $1010$ written most significant digit first), is a Tribonacci representation of $9$.","title":"A Tribonacci representation of nine"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-word-eq-of-is-tribonacci-repr","kind":"node","lean":"MetaMathlibExt.tribonacciWord_eq_of_isTribonacciRepr","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-word-eq-of-is-tribonacci-repr","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $w$ be a Tribonacci representation of a natural number $m$, written most significant digit first, with missing digits read as $0$. The letter of the infinite Tribonacci word at position $m$ (positions counted from $0$) is $a$ if $w$ ends in $0$, $b$ if $w$ ends in $01$, and $c$ if $w$ ends in $11$. In particular the empty representation of $0$ ends in $0$, and the representation $1$ of $1$ ends in $01$.","title":"Duch\u00eane\u2013Rigo: letters of the Tribonacci word from Tribonacci representations"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/abs-tribonacci-succ-sub-mul-lt","kind":"node","lean":"MetaMathlibExt.abs_tribonacci_succ_sub_mul_lt","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#abs-tribonacci-succ-sub-mul-lt","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $T_0 = T_1 = 0$, $T_2 = 1$, $T_{m+3} = T_{m+2} + T_{m+1} + T_m$ be the Tribonacci numbers, let $\\alpha$ be the real root of $x^3 = x^2 + x + 1$, and let $\\beta$ be a non-real complex root of the same equation, with $\\gamma = \\overline{\\beta}$ the other non-real root. Then for every natural number $n$,","title":"Tribonacci approximation error bound"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-succ-sub-mul-sign-changes","kind":"node","lean":"MetaMathlibExt.tribonacci_succ_sub_mul_sign_changes","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-succ-sub-mul-sign-changes","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $T_n$ be the Tribonacci numbers ($T_0 = T_1 = 0$, $T_2 = 1$, $T_{n+3} = T_{n+2} + T_{n+1} + T_n$) and let $\\alpha$ be a real number with $\\alpha^3 = \\alpha^2 + \\alpha + 1$. Then for every $n \\ge 0$ the three numbers $T_{k+1} - \\alpha T_k$ for $k = n, n + 1, n + 2$ are neither all positive nor all negative.","title":"Tribonacci differences do not keep one strict sign for three steps"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tendsto-trithoff-first-column-div","kind":"node","lean":"MetaMathlibExt.tendsto_trithoff_first_column_div","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tendsto-trithoff-first-column-div","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $\\alpha$ be a real number with $\\alpha^3 = \\alpha^2 + \\alpha + 1$, and let $a_0 < a_1 < a_2 < \\cdots$ be the positive integers that have a Tribonacci representation with least significant digit $1$, in increasing order (the first column of the Trithoff array, OEIS A003265). Then $a_r / (r + 1)$ tends to $\\alpha / (\\alpha - 1)$ as $r \\to \\infty$.","title":"Growth of the first column of the Trithoff array"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/exists-fibternary-sum-four-and-tribterna","kind":"node","lean":"MetaMathlibExt.exists_fibternary_sum_four_and_tribternary_sum_three","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#exists-fibternary-sum-four-and-tribterna","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Call a natural number Fibternary if its base-$3$ digits are all $0$ or $1$ with no two consecutive $1$s, and Tribternary if its base-$3$ digits are all $0$ or $1$ with no three consecutive $1$s. Every natural number is a sum of four Fibternary numbers and also a sum of three Tribternary numbers (the summands may be $0$ and need not be distinct).","title":"Sums of Fibternary and Tribternary numbers"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/exists-fibternary-multiple","kind":"node","lean":"MetaMathlibExt.exists_fibternary_multiple","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#exists-fibternary-multiple","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Every positive integer $n$ has a positive multiple $m$ that is Fibternary: the base-$3$ digits of $m$ are all $0$ or $1$, and no two consecutive digits of $m$ are both $1$.","title":"Fibternary multiples"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/exists-tribbinary-and-tribternary-multip","kind":"node","lean":"MetaMathlibExt.exists_tribbinary_and_tribternary_multiple","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#exists-tribbinary-and-tribternary-multip","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Every positive integer $n$ has a positive multiple whose binary digits contain no three consecutive $1$s (a Tribbinary number), and a positive multiple whose base-$3$ digits are all $0$ or $1$ with no three consecutive $1$s (a Tribternary number).","title":"Tribbinary and Tribternary multiples"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-value","kind":"node","lean":"MetaMathlibExt.tribonacciValue","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-value","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"Let $T_0 = T_1 = 0$, $T_2 = 1$ and $T_{k+3} = T_{k+2} + T_{k+1} + T_k$ be the Tribonacci numbers. The Tribonacci value of a finite word $w$ of natural-number digits $w_0, w_1, \\dots, w_{\\ell-1}$, listed from the least significant digit, is $[w]_T = \\sum_{i=0}^{\\ell-1} w_i T_{i+3}$, so the digits weigh $1, 2, 4, 7, 13, \\dots$. Digits may exceed $1$ and three consecutive ones are allowed, so this also evaluates the\u2026","title":"Tribonacci value of a digit word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-value-nil","kind":"node","lean":"MetaMathlibExt.tribonacciValue_nil","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-value-nil","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"The Tribonacci value of the empty digit word is $0$.","title":"The empty digit word has Tribonacci value zero"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-value-one-one-one-zero","kind":"node","lean":"MetaMathlibExt.tribonacciValue_one_one_one_zero","parent":24164,"reading_title":{"format":"inline-parts-v1","parts":[["text","The non-canonical Tribonacci word "],["math","\\(1110\\)"],["text"," has value "],["math","\\(13\\)"]],"source":"The non-canonical Tribonacci word $1110$ has value $13$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-value-one-one-one-zero","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"The digit word $1110$, written most significant digit first, whose digits weigh the Tribonacci numbers $T_6 = 7$, $T_5 = 4$, $T_4 = 2$ and $T_3 = 1$, has Tribonacci value $7 + 4 + 2 = 13$. It is the paper's example of a non-canonical representation: its digits are $0$ and $1$ but it has three consecutive ones.","title":"The non-canonical Tribonacci word $1110$ has value $13$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-value-one-zero-three-zero","kind":"node","lean":"MetaMathlibExt.tribonacciValue_one_zero_three_zero","parent":24164,"reading_title":{"format":"inline-parts-v1","parts":[["text","The improper Tribonacci word "],["math","\\(1030\\)"],["text"," has value "],["math","\\(13\\)"]],"source":"The improper Tribonacci word $1030$ has value $13$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-value-one-zero-three-zero","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"The digit word $1030$, written most significant digit first, whose digits weigh the Tribonacci numbers $T_6 = 7$, $T_5 = 4$, $T_4 = 2$ and $T_3 = 1$, has Tribonacci value $1 \\cdot 7 + 3 \\cdot 2 = 13$. It is the paper's example of an improper representation, with a digit larger than $1$.","title":"The improper Tribonacci word $1030$ has value $13$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-succ-record-indices","kind":"node","lean":"MetaMathlibExt.tribonacciSucc_record_indices","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-succ-record-indices","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $T_n$ be the Tribonacci numbers ($T_0 = T_1 = 0$, $T_2 = 1$, $T_{n+3} = T_{n+2} + T_{n+1} + T_n$) and let $\\alpha$ be a real number with $\\alpha^3 = \\alpha^2 + \\alpha + 1$. For each $n$ fix a Tribonacci representation $R(n) = (r_0, r_1, \\dots, r_{\\ell-1})$ of $n$: digits in $\\{0, 1\\}$ with no three consecutive ones and $n = \\sum_i r_i T_{i+3}$. Let $\\mathrm{out}(n) = \\sum_i r_i T_{i+4}$ be the Tribonacci\u2026","title":"Indices used by records of the Tribonacci successor"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-succ-record-of-partial-sum","kind":"node","lean":"MetaMathlibExt.tribonacciSucc_record_of_partial_sum","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-succ-record-of-partial-sum","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $T_n$ be the Tribonacci numbers ($T_0 = T_1 = 0$, $T_2 = 1$, $T_{n+3} = T_{n+2} + T_{n+1} + T_n$) and let $\\alpha$ be a real number with $\\alpha^3 = \\alpha^2 + \\alpha + 1$. For each $n$ fix a Tribonacci representation $R(n) = (r_0, r_1, \\dots, r_{\\ell-1})$ of $n$: digits in $\\{0, 1\\}$ with no three consecutive ones and $n = \\sum_i r_i T_{i+3}$. Let $\\mathrm{out}(n) = \\sum_i r_i T_{i+4}$ be the Tribonacci\u2026","title":"Partial sums along the positive and negative indices are records"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-succ-record-eventually-contai","kind":"node","lean":"MetaMathlibExt.tribonacciSucc_record_eventually_contains","parent":24164,"reading_title":{"format":"inline-parts-v1","parts":[["text","Late records use "],["math","\\(T_{P(t)}\\)"],["text"," and "],["math","\\(T_{Q(t)}\\)"]],"source":"Late records use $T_{P(t)}$ and $T_{Q(t)}$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-succ-record-eventually-contai","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $T_n$ be the Tribonacci numbers ($T_0 = T_1 = 0$, $T_2 = 1$, $T_{n+3} = T_{n+2} + T_{n+1} + T_n$) and let $\\alpha$ be a real number with $\\alpha^3 = \\alpha^2 + \\alpha + 1$. For each $n$ fix a Tribonacci representation $R(n) = (r_0, r_1, \\dots, r_{\\ell-1})$ of $n$: digits in $\\{0, 1\\}$ with no three consecutive ones and $n = \\sum_i r_i T_{i+3}$. Let $\\mathrm{out}(n) = \\sum_i r_i T_{i+4}$ be the Tribonacci\u2026","title":"Late records use $T_{P(t)}$ and $T_{Q(t)}$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/trithoff-row-number-eq","kind":"node","lean":"MetaMathlibExt.trithoff_row_number_eq","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#trithoff-row-number-eq","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $T_0 = T_1 = 0$, $T_2 = 1$ and $T_{k+3} = T_{k+2} + T_{k+1} + T_k$ be the Tribonacci numbers, and let $U_i = T_i + T_{i+1}$, so that $U_0, U_1, U_2, \\dots = 0, 1, 2, 3, 6, 11, \\dots$. Let $n$ have Tribonacci representation $a1$, where $a$ is a digit word written most significant digit first, and let $[a]_T$ be the Tribonacci value of $a$. Then $n$ is the $(n - [a]_T)$-th smallest positive integer whose\u2026","title":"Row numbers of the Trithoff array"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/exists-trithoff-row-difference-eq","kind":"node","lean":"MetaMathlibExt.exists_trithoff_row_difference_eq","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#exists-trithoff-row-difference-eq","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $T_0 = T_1 = 0$, $T_2 = 1$ and $T_{m+3} = T_{m+2} + T_{m+1} + T_m$ be the Tribonacci numbers, and for a finite word $w = (w_0, \\dots, w_{\\ell-1})$ of natural-number digits, listed from the least significant digit, let $[w]_T = \\sum_{i=0}^{\\ell-1} w_i T_{i+3}$ be its Tribonacci value. Let $a = (a_0, \\dots, a_{\\ell-1})$ be a Tribonacci representation of a natural number $n$ (all digits $0$ or $1$, no three\u2026","title":"Differences along a Trithoff row"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-value-cons-zero-of-noncanonic","kind":"node","lean":"MetaMathlibExt.tribonacciValue_cons_zero_of_noncanonical","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-value-cons-zero-of-noncanonic","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $T_0 = T_1 = 0$, $T_2 = 1$ and $T_{k+3} = T_{k+2} + T_{k+1} + T_k$ be the Tribonacci numbers, and for a digit word $u$, written most significant digit first, let $[u]_T$ be its Tribonacci value, in which the $i$-th digit from the right (counting from $0$) weighs $T_{i+3}$. Let $w$ be a digit word with all digits in $\\{0, 1\\}$ (three consecutive ones are allowed) and $[w]_T = n$, and let $v$ be a Tribonacci\u2026","title":"Appending a zero to a non-canonical Tribonacci representation"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/trithoff-precolumns","kind":"node","lean":"MetaMathlibExt.trithoff_precolumns","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#trithoff-precolumns","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Write $[u]_T$ for the Tribonacci value of a digit word $u$, written most significant digit first. Let $n$ have Tribonacci representation $abc1$, where $a$ is a digit word and $b$, $c$ are digits. Let $\\mathrm{out}(n) = [abc10]_T$ and $\\mathrm{out}^2(n) = [abc100]_T$ be the next two entries of the Trithoff row of $n$, and extend the row to the left by the Tribonacci rule: in the integers, the wall is\u2026","title":"Precolumns of the Trithoff array"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/trithoff-wall-seed-preseed","kind":"node","lean":"MetaMathlibExt.trithoff_wall_seed_preseed","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#trithoff-wall-seed-preseed","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $f_1 < f_2 < \\cdots$ be the positive integers whose Tribonacci representation ends in $1$ (the first column of the Trithoff array). For each row $r \\ge 1$ let $\\mathrm{out}(f_r)$ and $\\mathrm{out}^2(f_r)$ be the Tribonacci values of the Tribonacci representation of $f_r$, written most significant digit first, followed by one and by two zeros, that is, with one and with two new least significant digits $0$.\u2026","title":"Wall, seed and pre-seed of the Trithoff array via the Tribonacci word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/trithoff-first-column-sub","kind":"node","lean":"MetaMathlibExt.trithoff_first_column_sub","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#trithoff-first-column-sub","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $f_1 < f_2 < \\cdots$ be the positive integers whose Tribonacci representation ends in the digit $1$ (OEIS A003265, the first column of the Trithoff array). For every $j \\ge 1$ the difference $f_{j+1} - f_j$ is $2$ or $3$, and $f_{j+1} - f_j = 3$ exactly when $j = p_a(k) + p_b(k)$ for some $k \\ge 1$, where $p_a(k)$ and $p_b(k)$ are the positions, counted from $1$, of the $k$-th occurrences of the letters $a$ and\u2026","title":"Gaps in the first column of the Trithoff array"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/is-extra-trib","kind":"node","lean":"MetaMathlibExt.IsExtraTrib","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#is-extra-trib","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"An extraTrib is a two-sided integer sequence $S \\colon \\mathbb{Z} \\to \\mathbb{Z}$ that follows the Tribonacci rule $S(n+3) = S(n+2) + S(n+1) + S(n)$ for every $n \\in \\mathbb{Z}$ and whose terms are eventually positive: there is an integer $N$ with $S(n) > 0$ for all $n \\ge N$.","title":"ExtraTribs"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/not-is-extra-trib-zero","kind":"node","lean":"MetaMathlibExt.not_isExtraTrib_zero","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#not-is-extra-trib-zero","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"The constant sequence $n \\mapsto 0$ on $\\mathbb{Z}$ is not an extraTrib.","title":"The zero sequence is not an extraTrib"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/extra-trib-difference-inverse","kind":"node","lean":"MetaMathlibExt.extraTrib_difference_inverse","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#extra-trib-difference-inverse","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $S$ be an extraTrib. There is exactly one sequence $Q \\colon \\mathbb{Z} \\to \\mathbb{Q}$ that follows the Tribonacci rule $Q(n+3) = Q(n+2) + Q(n+1) + Q(n)$ for all $n \\in \\mathbb{Z}$ and satisfies $Q(n+1) - Q(n) = S(n)$ for all $n \\in \\mathbb{Z}$. Moreover, there is an extraTrib $Q$ with $Q(n+1) - Q(n) = S(n)$ for all $n \\in \\mathbb{Z}$ if and only if $S$ is not of Type EEOO (parities repeating even, even, odd,\u2026","title":"Difference-inverses of extraTribs"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/extra-trib-not-infinitely-invertible","kind":"node","lean":"MetaMathlibExt.extraTrib_not_infinitely_invertible","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#extra-trib-not-infinitely-invertible","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $S$ be an extraTrib. There is no sequence $S_0, S_1, S_2, \\dots$ of functions $\\mathbb{Z} \\to \\mathbb{Z}$ with $S_0 = S$ such that for every $k \\ge 0$ the function $S_{k+1}$ is an extraTrib and a difference-inverse of $S_k$, that is, $S_{k+1}(n+1) - S_{k+1}(n) = S_k(n)$ for all $n \\in \\mathbb{Z}$.","title":"ExtraTribs cannot be inverted forever"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/extra-trib-row-number-mod-eq-one","kind":"node","lean":"MetaMathlibExt.extraTrib_row_number_modEq_one","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#extra-trib-row-number-mod-eq-one","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $S$ be an extraTrib and let $n$ be a natural number dividing every term $S(k)$, $k \\in \\mathbb{Z}$. Let $f_0 < f_1 < f_2 < \\dots$ be the positive integers whose Tribonacci representation has least significant digit $1$ (the first column of the Trithoff array), let $r$ be a natural number, and let $w$ be a Tribonacci representation of $f_r$. For a digit list $v$ written least significant digit first, let $[v]_T$\u2026","title":"Row numbers of divisible extraTribs"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/extra-trib-left-extension","kind":"node","lean":"MetaMathlibExt.extraTrib_left_extension","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#extra-trib-left-extension","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $S$ be an extraTrib. Then $S(k) < 0$ for some integer $k$, and whenever $S(k) < 0$, for every integer $n$ with $n + 2 < k$ the three terms $S(n)$, $S(n+1)$, $S(n+2)$ are neither all positive nor all negative.","title":"Signs of an extraTrib to the left"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/not-is-extra-trib-reversal","kind":"node","lean":"MetaMathlibExt.not_isExtraTrib_reversal","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#not-is-extra-trib-reversal","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"If $S : \\mathbb{Z} \\to \\mathbb{Z}$ is an extraTrib, then the sequence $n \\mapsto |S(-n)|$ is not an extraTrib.","title":"The reversal of an extraTrib is not an extraTrib"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-leftmost-carry","kind":"node","lean":"MetaMathlibExt.tribonacciLeftmostCarry","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-leftmost-carry","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"Let $w = w_0 w_1 \\cdots w_{\\ell-1}$ be a word of natural-number digits, written most significant digit first with positions numbered from $0$, and let $k$ be the least index with $k + 3 < \\ell$ and $w_{k+1}, w_{k+2}, w_{k+3} > 0$. Leftmost carrying replaces the block $w_k w_{k+1} w_{k+2} w_{k+3}$ by $(w_k + 1)(w_{k+1} - 1)(w_{k+2} - 1)(w_{k+3} - 1)$ and leaves the other digits unchanged; it is undefined when there\u2026","title":"Leftmost carrying"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-leftmost-carry-zero-one-one-o","kind":"node","lean":"MetaMathlibExt.tribonacciLeftmostCarry_zero_one_one_one","parent":24164,"reading_title":{"format":"inline-parts-v1","parts":[["text","Leftmost carrying of "],["math","\\(0111\\)"]],"source":"Leftmost carrying of $0111$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-leftmost-carry-zero-one-one-o","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"Leftmost carrying takes the word $0111$, written most significant digit first, to $1000$.","title":"Leftmost carrying of $0111$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-improper-weight","kind":"node","lean":"MetaMathlibExt.tribonacciImproperWeight","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-improper-weight","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"Let $w$ be a word of natural-number digits, written most significant digit first, and let $v = 0w$ be $w$ padded with a $0$ on the left, with positions numbered from $0$. Let $p$ be the position of the leftmost digit $\\ge 2$ of $v$ (the improper boundary), or the length of $v$ if there is no such digit. The weight of $w$ is the sum of the digits of $v$ that come after the last $0$ of $v$ before position $p$.","title":"Weight of an improper word"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-improper-weight-example","kind":"node","lean":"MetaMathlibExt.tribonacciImproperWeight_example","parent":24164,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weight of the improper word "],["math","\\(010113\\)"]],"source":"Weight of the improper word $010113$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-improper-weight-example","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"The word $010113$, written most significant digit first, has weight $1 + 1 + 3 = 5$.","title":"Weight of the improper word $010113$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-canonization-step","kind":"node","lean":"MetaMathlibExt.tribonacciCanonizationStep","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-canonization-step","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"Let $w$ be a word of natural-number digits, written most significant digit first, and let $v = 0w$ be $w$ padded with a $0$ on the left, with positions numbered from $0$. If leftmost carrying applies to $v$, the step returns its result (Step 1). Otherwise let $p$ be the position of the leftmost digit $\\ge 2$ of $v$ (the improper boundary), or the length of $v$ if there is no such digit. If $v$ has no position\u2026","title":"One step of Tribonacci canonization"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-canonization-run","kind":"node","lean":"MetaMathlibExt.tribonacciCanonizationRun","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-canonization-run","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"Words are written most significant digit first. The run of at most $n$ canonization steps on a word $w$ of natural numbers is defined by recursion on $n$: it returns $w$ if $n = 0$ or if $w$ is canonical (every digit is $0$ or $1$ and there are no three consecutive $1$s); otherwise it applies one canonization step to $w$ and returns the run of at most $n - 1$ steps on the result, and it is undefined when that step\u2026","title":"Canonization run"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-canonization-run-example","kind":"node","lean":"MetaMathlibExt.tribonacciCanonizationRun_example","parent":24164,"reading_title":{"format":"inline-parts-v1","parts":[["text","Canonizing "],["math","\\(2000000\\)"]],"source":"Canonizing $2000000$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-canonization-run-example","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"fully_proved","summary":"Words are written most significant digit first. Running at most two steps of the canonization procedure on the word $2000000$ yields $10001000$.","title":"Canonizing $2000000$"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-leftmost-carry-boundary-weigh","kind":"node","lean":"MetaMathlibExt.tribonacciLeftmostCarry_boundary_weight","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-leftmost-carry-boundary-weigh","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Let $w$ be a word of natural-number digits, written most significant digit first, with some digit $\\ge 2$. Suppose leftmost carrying applied to the padded word $0w$ gives a word $u$ (of the same length as $0w$) that still has a digit $\\ge 2$. Then the position of the leftmost digit $\\ge 2$ in $u$ is at least its position in $0w$, and the weight of $u$ is at most the weight of $w$.","title":"Leftmost carrying does not move the improper boundary left or raise the weight"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-canonization-run-terminates","kind":"node","lean":"MetaMathlibExt.tribonacciCanonizationRun_terminates","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-canonization-run-terminates","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Words are written most significant digit first. Let $w$ be a word of natural numbers with some digit at least $2$, let $m$ be its improper weight, and let $k \\ge 3m$. Then there is an $n$ such that running at most $n$ steps of the canonization procedure on $w 0^k$ ($w$ followed by $k$ zeros) yields a word whose digits are all $0$ or $1$ and which has no three consecutive $1$s.","title":"Termination of the canonization procedure"},{"id":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/tribonacci-canonization-run-append-zeros","kind":"node","lean":"MetaMathlibExt.tribonacciCanonizationRun_append_zeros","parent":24164,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra/index.html#tribonacci-canonization-run-append-zeros","scope":"Combinatorics/combinatorics-on-words/automatic-sequences/tribonacci-representations-and-the-trithoff-arra","status":"can_prove","summary":"Words are written most significant digit first. Let $w$ be a word of natural numbers with some digit at least $2$, and suppose that running at most $n$ steps of the canonization procedure on $w$ yields a word $v$ whose digits are all $0$ or $1$ and which has no three consecutive $1$s. Then for every $k$, running at most $n$ steps of the canonization procedure on $w 0^k$ ($w$ followed by $k$ zeros) yields $v 0^k$.","title":"Canonizing a word followed by zeros"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words.json","id":"Combinatorics/combinatorics-on-words/finite-words","kind":"scope","parent":23782,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words","status":"can_prove","summary":"Properties of finite words: Lyndon and primitive words, equilibrious words, Ulam words, keyword substitutions, universal cycles for subsets, capacity of Catalan words, rectangles in nondecreasing words, and intersections in pairs of words.","title":"Finite words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words.json","id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","title":"Capacity of Catalan words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/sum-choose-mul-fib-odd-eq-choose-add-sum","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_fib_odd_eq_choose_add_sum","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#sum-choose-mul-fib-odd-eq-choose-add-sum","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers, with the convention $F_{-1} = 1$. For all integers $n \\ge 1$ and $2 \\le i \\le n + 1$,","title":"A recursion for binomial sums of odd-indexed Fibonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/fib-two-mul-eq-sum-sum-choose-mul-fib-od","kind":"node","lean":"MetaMathlibExt.fib_two_mul_eq_sum_sum_choose_mul_fib_odd","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#fib-two-mul-eq-sum-sum-choose-mul-fib-od","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers, with $F_0 = 0$, $F_1 = 1$ and $F_{k+2} = F_{k+1} + F_k$, and set $F_{-1} = 1$. For every natural number $n \\ge 1$,","title":"A double-sum identity for even-indexed Fibonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/word-capacity","kind":"node","lean":"MetaMathlibExt.wordCapacity","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#word-capacity","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"The capacity of a word $w = w_1 \\cdots w_n$ of natural numbers, viewed as a bargraph, is","title":"Capacity of a word"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/word-area","kind":"node","lean":"MetaMathlibExt.wordArea","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#word-area","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"The area of a word $w = w_1 \\cdots w_n$ of natural numbers is the sum $w_1 + \\cdots + w_n$ of its letters, the area of its bargraph.","title":"Area of a word"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/word-capacity-figure-one","kind":"node","lean":"MetaMathlibExt.wordCapacity_figure_one","parent":24206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capacity of the word "],["math","\\(122341123212323\\)"]],"source":"Capacity of the word $122341123212323$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#word-capacity-figure-one","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"The word $w = 1\\,2\\,2\\,3\\,4\\,1\\,1\\,2\\,3\\,2\\,1\\,2\\,3\\,2\\,3$ of length $15$ has capacity $10$ (Mansour and Shattuck's Figure 1).","title":"Capacity of the word $122341123212323$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-words","kind":"node","lean":"MetaMathlibExt.catalanWords","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-words","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For a natural number $n$, the set $\\mathcal{A}_n$ of Catalan words of length $n$ consists of the words $w = w_1 w_2 \\cdots w_n$ of positive integers with $w_1 = 1$ and $w_{i+1} \\le w_i + 1$ for $1 \\le i < n$; for $n = 0$ it contains only the empty word.","title":"Catalan words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-words","kind":"node","lean":"MetaMathlibExt.smoothWords","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-words","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For $n \\ge 0$, the set $\\mathcal{R}_n$ of smooth words of length $n$ consists of the words $w = w_1 \\cdots w_n$ of positive integers with $w_1 = 1$ (when $n \\ge 1$) and $|w_{i+1} - w_i| \\le 1$ for $1 \\le i < n$.","title":"Smooth words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/mem-catalan-words-iff","kind":"node","lean":"MetaMathlibExt.mem_catalanWords_iff","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#mem-catalan-words-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For a natural number $n$, let $\\mathcal{A}_n$ be the set of Catalan words of length $n$, defined as the words $w = w_1 \\cdots w_n$ with letters in $\\{1, \\dots, n\\}$ such that $w_1 = 1$ when $n \\ge 1$ and $w_{k+1} \\le w_k + 1$ for $1 \\le k < n$. A word $w = w_1 \\cdots w_n$ of natural numbers belongs to $\\mathcal{A}_n$ if and only if $w_k \\ge 1$ for every $k$, $w_1 = 1$ when $n \\ge 1$, and $w_{k+1} \\le w_k + 1$ for\u2026","title":"Characterization of Catalan words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/mem-smooth-words-iff","kind":"node","lean":"MetaMathlibExt.mem_smoothWords_iff","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#mem-smooth-words-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For a natural number $n$, let $\\mathcal{R}_n$ be the set of smooth words of length $n$, defined as the words $w = w_1 \\cdots w_n$ with letters in $\\{1, \\dots, n\\}$ such that $w_1 = 1$ when $n \\ge 1$, and $w_{k+1} \\le w_k + 1$ and $w_k \\le w_{k+1} + 1$ for $1 \\le k < n$. A word $w = w_1 \\cdots w_n$ of natural numbers belongs to $\\mathcal{R}_n$ if and only if $w_k \\ge 1$ for every $k$, $w_1 = 1$ when $n \\ge 1$, and\u2026","title":"Characterization of smooth words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-words","kind":"node","lean":"MetaMathlibExt.restrictedSmoothWords","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-words","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For $n \\ge 0$, the set $\\mathcal{R}^{\\ast}_n$ of restricted smooth words of length $n$ consists of the smooth words of length $n$ in which no two adjacent letters are both equal to $1$.","title":"Restricted smooth words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/words-ending-in","kind":"node","lean":"MetaMathlibExt.wordsEndingIn","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#words-ending-in","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"Given a finite set $S$ of words $w = w_1 \\cdots w_n$ of length $n$ with letters in $\\mathbb{N}$ and a natural number $k$, this is the set of words in $S$ whose last letter $w_n$ is $k$. When $n = 0$ this set is empty.","title":"Words ending in a given letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/words-bounded-by","kind":"node","lean":"MetaMathlibExt.wordsBoundedBy","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#words-bounded-by","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"Given a finite set $S$ of words of length $n$ with letters in $\\mathbb{N}$ and a natural number $k$, this is the set of words in $S$ all of whose letters are at most $k$. Applied to the words of $S$ whose last letter is $k$, it gives those whose largest letter is $k$.","title":"Words with all letters at most a bound"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/capacity-dist","kind":"node","lean":"MetaMathlibExt.capacityDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#capacity-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For a finite set $S$ of words of length $n$ with letters in the natural numbers, the capacity distribution of $S$ is the polynomial $\\sum_{w \\in S} q^{\\operatorname{cap}(w)}$ in $\\mathbb{Z}[q]$, where $\\operatorname{cap}(w)$ is the capacity of $w$.","title":"Capacity distribution"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/area-dist","kind":"node","lean":"MetaMathlibExt.areaDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#area-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For a finite set $S$ of words of length $n$ with letters in the natural numbers, the area distribution of $S$ is the polynomial $\\sum_{w \\in S} q^{\\operatorname{area}(w)}$ in $\\mathbb{Z}[q]$, where the area of a word is the sum of its letters.","title":"Area distribution"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-capacity-dist","kind":"node","lean":"MetaMathlibExt.catalanCapacityDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-capacity-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For natural numbers $n$ and $i$, $a(n, i; q) \\in \\mathbb{Z}[q]$ is the capacity distribution of the Catalan words of length $n$ whose last letter is $i$: the sum of $q^{\\mathrm{cap}(w)}$ over all such words $w$, where $\\mathrm{cap}(w)$ is the capacity of $w$.","title":"Capacity distribution of Catalan words with a given last letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-max-end-capacity-dist","kind":"node","lean":"MetaMathlibExt.catalanMaxEndCapacityDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-max-end-capacity-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For natural numbers $n$ and $i$, $b(n, i; q) \\in \\mathbb{Z}[q]$ is the sum of $q^{\\mathrm{cap}(w)}$, where $\\mathrm{cap}(w)$ is the capacity of $w$, over the Catalan words $w$ of length $n$ whose last letter is $i$ and all of whose letters are at most $i$.","title":"Capacity distribution of Catalan words ending in their largest letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-capacity-dist","kind":"node","lean":"MetaMathlibExt.smoothCapacityDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-capacity-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For $n, i \\ge 0$, $r(n, i; q)$ is the polynomial $\\sum_w q^{\\operatorname{cap}(w)} \\in \\mathbb{Z}[q]$, where $w$ runs over the smooth words of length $n$ whose last letter is $i$ and $\\operatorname{cap}(w)$ is the capacity of $w$.","title":"Capacity distribution of smooth words by last letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-max-end-capacity-dist","kind":"node","lean":"MetaMathlibExt.smoothMaxEndCapacityDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-max-end-capacity-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For $n, i \\ge 0$, $s(n, i; q)$ is the polynomial $\\sum_w q^{\\operatorname{cap}(w)} \\in \\mathbb{Z}[q]$, where $w$ runs over the smooth words of length $n$ whose last letter is $i$ and all of whose letters are at most $i$, and $\\operatorname{cap}(w)$ is the capacity of $w$.","title":"Capacity distribution of smooth words ending in their largest letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-capacity-dist","kind":"node","lean":"MetaMathlibExt.restrictedSmoothCapacityDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-capacity-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For natural numbers $n$ and $i$, $r^*(n, i; q)$ is the polynomial $\\sum_w q^{\\mathrm{cap}(w)}$ in $\\mathbb{Z}[q]$, where $w$ ranges over the restricted smooth words of length $n$ (smooth words with no two adjacent letters equal to $1$) whose last letter is $i$, and $\\mathrm{cap}(w)$ is the capacity of $w$.","title":"Capacity distribution of restricted smooth words with a given last letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-max-end-capacity-dist","kind":"node","lean":"MetaMathlibExt.restrictedSmoothMaxEndCapacityDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-max-end-capacity-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For natural numbers $n$ and $i$, $s^*(n, i; q)$ is the polynomial $\\sum_w q^{\\mathrm{cap}(w)}$ in $\\mathbb{Z}[q]$, where $w$ ranges over the restricted smooth words of length $n$ (smooth words with no two adjacent letters equal to $1$) whose last letter is $i$ and all of whose letters are at most $i$, and $\\mathrm{cap}(w)$ is the capacity of $w$.","title":"Capacity distribution of restricted smooth words ending in their largest letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-growth-words","kind":"node","lean":"MetaMathlibExt.catalanGrowthWords","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-growth-words","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For natural numbers $m$ and $j$, $\\mathcal{C}_{m,j}$ is the set of words $w = w_1 \\cdots w_m$ of length $m$ with letters in $\\{1, \\dots, j\\}$ such that $w_{k+1} \\le w_k + 1$ for $1 \\le k < m$ and whose last letter $w_m$ is $j$. It is empty when $m = 0$.","title":"Words with unit-bounded rises ending in their maximum"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-growth-area-dist","kind":"node","lean":"MetaMathlibExt.catalanGrowthAreaDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-growth-area-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For natural numbers $m$, $j$ and $\\ell$, let $\\mathcal{C}_{m,j}$ be the set of words of length $m$ with letters in $\\{1, \\dots, j\\}$, each letter at most one more than the letter before it, and last letter $j$. Then $c(m, j, \\ell; q) \\in \\mathbb{Z}[q]$ is the area distribution of the words in $\\mathcal{C}_{m,j}$ whose first letter is $\\ell$: the sum of $q^{\\mathrm{area}(w)}$ over these words $w$, where\u2026","title":"Area distribution of words with unit-bounded rises, by first letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-plateau-area-dist","kind":"node","lean":"MetaMathlibExt.smoothPlateauAreaDist","parent":24206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Area distribution of smooth words from "],["math","\\(j\\)"],["text"," to "],["math","\\(j\\)"]],"source":"Area distribution of smooth words from $j$ to $j$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-plateau-area-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For natural numbers $m$ and $j$, let $\\mathcal{T}_{m,j}$ be the set of words $w = w_1 \\cdots w_m$ with letters in $\\{1, \\dots, j\\}$ such that $w_1 = w_m = j$ and $|w_{i+1} - w_i| \\le 1$ for $1 \\le i < m$; in particular $\\mathcal{T}_{0,j}$ is empty. The area distribution of $\\mathcal{T}_{m,j}$ is the polynomial $t(m, j; q) = \\sum_{w \\in \\mathcal{T}_{m,j}} q^{w_1 + \\cdots + w_m}$ in $\\mathbb{Z}[q]$.","title":"Area distribution of smooth words from $j$ to $j$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/capacity-dist-eval-one","kind":"node","lean":"MetaMathlibExt.capacityDist_eval_one","parent":24206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capacity distribution at "],["math","\\(q = 1\\)"]],"source":"Capacity distribution at $q = 1$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#capacity-dist-eval-one","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For every finite set $S$ of words of length $n$ with letters in the natural numbers, the capacity distribution of $S$ evaluated at $q = 1$ equals the number of words in $S$.","title":"Capacity distribution at $q = 1$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/area-dist-eval-one","kind":"node","lean":"MetaMathlibExt.areaDist_eval_one","parent":24206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Area distribution at "],["math","\\(q = 1\\)"]],"source":"Area distribution at $q = 1$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#area-dist-eval-one","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For every finite set $S$ of words of length $n$ with letters in the natural numbers, the area distribution of $S$ evaluated at $q = 1$ equals the number of words in $S$.","title":"Area distribution at $q = 1$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/card-catalan-words-four-smooth-words-fou","kind":"node","lean":"MetaMathlibExt.card_catalanWords_four_smoothWords_four","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#card-catalan-words-four-smooth-words-fou","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"There are $14$ Catalan words of length $4$, $13$ smooth words of length $4$, and $7$ restricted smooth words of length $4$. Here $14$ is the Catalan number $C_4$, $13 = C_4 - 1$, and $7$ is the central trinomial coefficient $G_3$.","title":"Catalan and smooth words of length four"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-growth-area-dist-succ","kind":"node","lean":"MetaMathlibExt.catalanGrowthAreaDist_succ","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-growth-area-dist-succ","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For natural numbers $m$, $j$ and $\\ell$, let $c(m, j, \\ell; q) \\in \\mathbb{Z}[q]$ be the area distribution of the words of length $m$ with letters in $\\{1, \\dots, j\\}$, each letter at most one more than the letter before it, that begin with $\\ell$ and end with $j$. For all natural numbers $m \\ge 2$ and $j$, $\\ell$ with $1 \\le \\ell \\le j$,","title":"Recurrence for the area distribution by first letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-growth-den","kind":"node","lean":"MetaMathlibExt.catalanGrowthDen","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-growth-den","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"The polynomials $d_j(x; q)$, viewed as formal power series in $x$ over $\\mathbb{Z}[q]$, are defined by $d_0 = 1$, $d_1 = 1 - xq$, $d_2 = 1 - xq - xq^2$ and $d_j = (1 - xq^j)\\, d_{j-1} - x^2 q^{2j-1} d_{j-3}$ for $j \\ge 3$.","title":"Denominator polynomials for the area of words with unit-bounded rises"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/constant-coeff-catalan-growth-den","kind":"node","lean":"MetaMathlibExt.constantCoeff_catalanGrowthDen","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#constant-coeff-catalan-growth-den","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For every $j \\ge 0$, the power series $d_j(x; q)$ in $x$ over $\\mathbb{Z}[q]$ defined by $d_0 = 1$, $d_1 = 1 - xq$, $d_2 = 1 - xq - xq^2$ and $d_j = (1 - xq^j) d_{j-1} - x^2 q^{2j-1} d_{j-3}$ for $j \\ge 3$ has constant term $1$.","title":"The area denominators have constant term one"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-growth-area-dist-gen-fun","kind":"node","lean":"MetaMathlibExt.catalanGrowthAreaDist_genFun","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-growth-area-dist-gen-fun","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For natural numbers $i$ and $j$ with $1 \\le i \\le j$ and for $n \\ge 0$, let $c(n, j, i; q) \\in \\mathbb{Z}[q]$ be the area distribution of the words of length $n$ with letters in $\\{1, \\dots, j\\}$, each letter at most one more than the letter before it, that begin with $i$ and end with $j$: the sum of $q^{\\mathrm{area}(w)}$ over these words $w$, where $\\mathrm{area}(w)$ is the sum of the letters of $w$. Let\u2026","title":"Area generating function of words with unit-bounded rises, by first letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-growth-words-area-dist-gen-fun","kind":"node","lean":"MetaMathlibExt.catalanGrowthWords_areaDist_genFun","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-growth-words-area-dist-gen-fun","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For natural numbers $n$ and $j$, let $\\mathcal{C}_{n,j}$ be the set of words $w = w_1 \\cdots w_n$ with letters in $\\{1, \\dots, j\\}$, each letter at most one more than the letter before it, and last letter $j$, and let $c(n, j; q) \\in \\mathbb{Z}[q]$ be the sum of $q^{w_1 + \\cdots + w_n}$ over $w \\in \\mathcal{C}_{n,j}$. Let the formal power series $d_k(x; q)$ in $x$ over $\\mathbb{Z}[q]$ be defined by $d_0 = 1$,\u2026","title":"Area generating function of words with unit-bounded rises ending in their maximum"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/dilate-invert-series","kind":"node","lean":"MetaMathlibExt.dilateInvertSeries","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#dilate-invert-series","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For a natural number $j$ and a power series $f(x; q) = \\sum_{n \\ge 0} f_n(q) x^n$ in $x$ over $\\mathbb{Z}[q]$, the power series $f(q^j x; 1/q)$ in $x$ over the Laurent polynomial ring $\\mathbb{Z}[q, q^{-1}]$ is $\\sum_{n \\ge 0} q^{jn} f_n(q^{-1}) x^n$.","title":"Dilating and inverting a power series"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/constant-coeff-dilate-invert-series","kind":"node","lean":"MetaMathlibExt.constantCoeff_dilateInvertSeries","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#constant-coeff-dilate-invert-series","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For every natural number $j$ and every power series $f(x; q)$ in $x$ over $\\mathbb{Z}[q]$ with constant term $1$, the power series $f(q^j x; 1/q)$ in $x$ over $\\mathbb{Z}[q, q^{-1}]$ also has constant term $1$.","title":"Dilating and inverting a power series preserves constant term one"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-max-end-capacity-dist-gen-fun","kind":"node","lean":"MetaMathlibExt.catalanMaxEndCapacityDist_genFun","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-max-end-capacity-dist-gen-fun","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For natural numbers $n$ and $i$, let $b(n, i; q)$ be the capacity distribution of the Catalan words of length $n$ that end in $i$ and have all letters at most $i$, viewed in $\\mathbb{Z}[q, q^{-1}]$. Let the formal power series $d_k(x; q)$ in $x$ over $\\mathbb{Z}[q]$ be defined by $d_0 = 1$, $d_1 = 1 - xq$, $d_2 = 1 - xq - xq^2$ and $d_k = (1 - xq^k)\\, d_{k-1} - x^2 q^{2k-1} d_{k-3}$ for $k \\ge 3$. For a natural\u2026","title":"Generating function for the capacity of Catalan words ending in their largest letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-capacity-dist-recurrence","kind":"node","lean":"MetaMathlibExt.catalanCapacityDist_recurrence","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-capacity-dist-recurrence","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Write $a(n, i)$ for the capacity distribution of the Catalan words of length $n$ ending in $i$, $b(n, i)$ for that of those Catalan words of length $n$ ending in $i$ whose largest letter is $i$, and $c(m, j)$ for the area distribution of the set $\\mathcal{C}_{m,j}$ of words of length $m$ over $\\{1, \\dots, j\\}$ ending in $j$ in which each letter is at most one more than the letter before it. For $1 < i < n$, in the\u2026","title":"Recurrence for the capacity distribution on Catalan words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-max-end-capacity-dist-recurrence","kind":"node","lean":"MetaMathlibExt.catalanMaxEndCapacityDist_recurrence","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-max-end-capacity-dist-recurrence","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For natural numbers $n$ and $i$, let $b(n, i; q)$ be the capacity distribution of the Catalan words of length $n$ that end in $i$ and have all letters at most $i$. For natural numbers $m$ and $k$, let $c(m, k; q) \\in \\mathbb{Z}[q]$ be the area distribution of the words of length $m$ with letters in $\\{1, \\dots, k\\}$, each letter at most one more than the letter before it, and last letter $k$, and let $c(m, k; 1/q)$\u2026","title":"Recurrence for the capacity of Catalan words ending in their largest letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-capacity-dist-boundary","kind":"node","lean":"MetaMathlibExt.catalanCapacityDist_boundary","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-capacity-dist-boundary","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Write $a(n, i)$ for the capacity distribution of the Catalan words of length $n$ ending in $i$, $b(n, i)$ for that of those Catalan words of length $n$ ending in $i$ whose largest letter is $i$, and $a(n)$ for the capacity distribution of all Catalan words of length $n$. For every $n \\ge 1$, $a(n, n) = b(n, 1) = b(n, n) = 1$, and if $n \\ge 2$ then $a(n, 1) = a(n - 1)$.","title":"Boundary values of the capacity distributions on Catalan words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-words-ending-in-gen-fun","kind":"node","lean":"MetaMathlibExt.catalanWordsEndingIn_genFun","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-words-ending-in-gen-fun","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $C(x) = \\sum_{n \\ge 0} C_n x^n$ be the generating function of the Catalan numbers $C_n$. For every $i \\ge 1$, let $a_{n,i}$ be the number of Catalan words of length $n$ whose last letter is $i$. Then, as formal power series with integer coefficients, $\\sum_{n \\ge 0} a_{n,i} x^n = x^i C(x)^i$.","title":"Generating function of Catalan words by last letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-words-capacity-nat-degree","kind":"node","lean":"MetaMathlibExt.catalanWords_capacity_natDegree","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-words-capacity-nat-degree","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For every natural number $m$, the degree of the capacity distribution $\\sum_{w \\in \\mathcal{A}_n} q^{\\operatorname{cap}(w)}$ of the set $\\mathcal{A}_n$ of Catalan words of length $n$, that is, the largest capacity of a Catalan word of length $n$, is $3\\binom{m}{2}$ for $n = 3m$, $m(3m-1)/2$ for $n = 3m + 1$ and $m(3m+1)/2$ for $n = 3m + 2$. The case $m = 0$ is included.","title":"Maximum capacity of Catalan words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-words-capacity-leading-coeff","kind":"node","lean":"MetaMathlibExt.catalanWords_capacity_leadingCoeff","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-words-capacity-leading-coeff","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For every $n \\ge 4$, let $\\mathcal{A}_n$ be the set of Catalan words of length $n$. The leading coefficient of the capacity distribution $\\sum_{w \\in \\mathcal{A}_n} q^{\\operatorname{cap}(w)}$ is $2$ if $3$ divides $n$ and $1$ otherwise; equivalently, exactly two Catalan words of length $n$ have the largest capacity when $3 \\mid n$, and exactly one does otherwise.","title":"Number of Catalan words of maximum capacity"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-words-capacity-subleading-coeff","kind":"node","lean":"MetaMathlibExt.catalanWords_capacity_subleading_coeff","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-words-capacity-subleading-coeff","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For every $n \\ge 4$, let $\\mathcal{A}_n$ be the set of Catalan words of length $n$, let $\\alpha$ be the degree of its capacity distribution $\\sum_{w \\in \\mathcal{A}_n} q^{\\operatorname{cap}(w)}$, and let $d_n$ be the coefficient of $q^{\\alpha - 1}$ in it. Then $d_4 = 13$, $d_5 = 7$, $d_6 = 8$, and for $n \\ge 7$, $d_n = 2m$ if $n = 3m$, $d_n = m + 1$ if $n = 3m + 1$ and $d_n = m + 2$ if $n = 3m + 2$.","title":"Subleading coefficient of the capacity distribution of Catalan words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-capacity-dist-eval-zero","kind":"node","lean":"MetaMathlibExt.catalanCapacityDist_eval_zero","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-capacity-dist-eval-zero","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For $2 \\le i \\le n$, let $a(n, i; q)$ be the capacity distribution of the Catalan words of length $n$ ending in $i$, so that $a(n, i; 0)$ is the number of those words with capacity $0$. Then","title":"Zero-capacity Catalan words with a given last letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/catalan-words-zero-capacity-count","kind":"node","lean":"MetaMathlibExt.catalanWords_zero_capacity_count","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#catalan-words-zero-capacity-count","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $F_k$ be the Fibonacci numbers, with $F_0 = 0$ and $F_1 = 1$. For $n \\ge 1$, let $a(n, 1; q)$ be the capacity distribution on Catalan words of length $n$ ending in $1$, and let $a(n; q) = \\sum_{w \\in \\mathcal{A}_n} q^{\\operatorname{cap}(w)}$ be the capacity distribution on the set $\\mathcal{A}_n$ of all Catalan words of length $n$. Then $a(n, 1; 0)$, the number of Catalan words of length $n$ ending in $1$ with\u2026","title":"Catalan words of capacity zero"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-plateau-area-dist-recurrence","kind":"node","lean":"MetaMathlibExt.smoothPlateauAreaDist_recurrence","parent":24206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the area distribution of smooth words from "],["math","\\(j\\)"],["text"," to "],["math","\\(j\\)"]],"source":"Recurrence for the area distribution of smooth words from $j$ to $j$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-plateau-area-dist-recurrence","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $t(m, j; q)$ be the area distribution of the words of length $m$ over $\\{1, \\dots, j\\}$ that begin and end with $j$ and whose consecutive letters differ by at most $1$. For all $m \\ge 2$ and $j \\ge 2$,","title":"Recurrence for the area distribution of smooth words from $j$ to $j$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-plateau-area-dist","kind":"node","lean":"MetaMathlibExt.restrictedSmoothPlateauAreaDist","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-plateau-area-dist","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"For natural numbers $m$ and $j$, let $\\mathcal{T}^*_{m,j}$ be the set of words $w = w_1 \\cdots w_m$ over $\\{1, \\dots, j\\}$ with $w_1 = w_m = j$, $w_{k+1} \\le w_k + 1$ and $w_k \\le w_{k+1} + 1$ for $1 \\le k < m$, and no two adjacent letters both equal to $1$ (so $\\mathcal{T}^*_{0,j}$ is empty). Then $t^*(m, j; q)$ is the polynomial $\\sum_{w \\in \\mathcal{T}^*_{m,j}} q^{w_1 + \\cdots + w_m}$ in $\\mathbb{Z}[q]$.","title":"Area polynomial of restricted smooth words between two top letters"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/capacity-continued-fraction-den","kind":"node","lean":"MetaMathlibExt.capacityContinuedFractionDen","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#capacity-continued-fraction-den","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"Given a power series $B$ in $x$ over $\\mathbb{Z}[q]$, define power series $K_k$ in $x$ over $\\mathbb{Z}[q]$ by $K_0 = 1$, $K_1 = B$ and, for $k \\ge 2$, $K_k = 1 - q^k x - q^{2k-1} x^2 K_{k-1}^{-1}$. Here $K_{k-1}^{-1}$ is the formal inverse of $K_{k-1}$ computed with constant term $1$. It is the true inverse whenever $K_{k-1}$ has constant term $1$, which always holds for $k \\ge 3$, and for $k = 2$ when $B$ has\u2026","title":"Continued-fraction denominators"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/constant-coeff-capacity-continued-fracti","kind":"node","lean":"MetaMathlibExt.constantCoeff_capacityContinuedFractionDen","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#constant-coeff-capacity-continued-fracti","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"fully_proved","summary":"Let $B$ be a power series in $x$ over $\\mathbb{Z}[q]$ with constant term $1$, and define power series $K_k$ in $x$ over $\\mathbb{Z}[q]$ by $K_0 = 1$, $K_1 = B$ and $K_k = 1 - q^k x - q^{2k-1} x^2 / K_{k-1}$ for $k \\ge 2$. Then $K_k$ has constant term $1$ for every $k \\ge 0$.","title":"Continued-fraction denominators have constant term one"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-plateau-area-dist-gen-fun","kind":"node","lean":"MetaMathlibExt.smoothPlateauAreaDist_genFun","parent":24206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continued fraction for smooth words from "],["math","\\(j\\)"],["text"," to "],["math","\\(j\\)"],["text"," by area"]],"source":"Continued fraction for smooth words from $j$ to $j$ by area"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-plateau-area-dist-gen-fun","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $t(m, j; q)$ be the area distribution of the words of length $m$ over $\\{1, \\dots, j\\}$ that begin and end with $j$ and whose consecutive letters differ by at most $1$. Define power series $K_k$ in $x$ over $\\mathbb{Z}[q]$ by $K_1 = 1 - qx$ and $K_k = 1 - q^k x - q^{2k-1} x^2 / K_{k-1}$ for $k \\ge 2$; each $K_k$ has constant term $1$. Then for every $j \\ge 1$, in $\\mathbb{Z}[q][[x]]$,","title":"Continued fraction for smooth words from $j$ to $j$ by area"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-plateau-area-dist-gen","kind":"node","lean":"MetaMathlibExt.restrictedSmoothPlateauAreaDist_genFun","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-plateau-area-dist-gen","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For $j \\ge 1$, let $t^*(m, j; q) = \\sum_w q^{w_1 + \\cdots + w_m}$, the sum over the words $w = w_1 \\cdots w_m$ over $\\{1, \\dots, j\\}$ with $w_1 = w_m = j$ (there are none for $m = 0$), consecutive letters differing by at most $1$ and no two adjacent letters equal to $1$, and let $T^*_j(x; q) = \\sum_{m \\ge 0} t^*(m, j; q)\\, x^m$. Define power series $K_k$ in $x$ over $\\mathbb{Z}[q]$ by $K_1 = 1$ and\u2026","title":"Continued fraction for the area series of restricted smooth words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-capacity-dist-recurrence","kind":"node","lean":"MetaMathlibExt.smoothCapacityDist_recurrence","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-capacity-dist-recurrence","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $r(n, i; q)$ be the capacity distribution of the smooth words of length $n$ ending in $i$, let $s(n, i; q)$ be that of those smooth words of length $n$ ending in $i$ whose largest letter is $i$, and let $t(m, j; q)$ be the area distribution (the area of a word is the sum of its letters) of the words of length $m$ over $\\{1, \\dots, j\\}$ that begin and end with $j$ and whose adjacent letters differ by at most\u2026","title":"Recurrence for the capacity distribution of smooth words by last letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-max-end-capacity-dist-recurrence","kind":"node","lean":"MetaMathlibExt.smoothMaxEndCapacityDist_recurrence","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-max-end-capacity-dist-recurrence","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $s(n, i; q)$ be the capacity distribution of the smooth words of length $n$ that end in $i$ and have largest letter $i$, and let $t(m, j; q)$ be the area distribution (the area of a word is the sum of its letters) of the words of length $m$ over $\\{1, \\dots, j\\}$ that begin and end with $j$ and whose adjacent letters differ by at most $1$. For $1 < i < n$, in the ring $\\mathbb{Z}[q, q^{-1}]$ of Laurent\u2026","title":"Recurrence for the capacity distribution of smooth words ending in their largest letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-capacity-dist-boundary","kind":"node","lean":"MetaMathlibExt.smoothCapacityDist_boundary","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-capacity-dist-boundary","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For letters $i$, let $r(n, i; q)$ be the capacity distribution of the smooth words of length $n$ ending in $i$, and $s(n, i; q)$ that of those smooth words of length $n$ ending in $i$ whose largest letter is $i$. For $n \\ge 1$, $r(n, n; q) = s(n, n; q) = s(n, 1; q) = 1$, and if $n \\ge 2$ then $r(n, 1; q) = r(n - 1, 1; q) + r(n - 1, 2; q)$.","title":"Boundary values of the capacity distributions of smooth words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-max-end-capacity-dist-gen-fun","kind":"node","lean":"MetaMathlibExt.smoothMaxEndCapacityDist_genFun","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-max-end-capacity-dist-gen-fun","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For $i \\ge 1$, let $s(n, i; q)$ be the capacity distribution of the smooth words of length $n$ that end in $i$ and have largest letter $i$, and let $t(m, j; q)$ be the area distribution (the area of a word is the sum of its letters) of the words of length $m$ over $\\{1, \\dots, j\\}$ that begin and end with $j$ and whose adjacent letters differ by at most $1$. In the ring of formal power series in $x$ over the\u2026","title":"Generating function of smooth words ending in their largest letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-max-end-capacity-dist-2","kind":"node","lean":"MetaMathlibExt.restrictedSmoothMaxEndCapacityDist_genFun","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-max-end-capacity-dist-2","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For $i \\ge 1$, let $s^*(n, i; q)$ be the capacity distribution of the restricted smooth words of length $n$ that end in $i$ and all of whose letters are at most $i$. For $j \\ge 1$, let $t^*(m, j; q) = \\sum_w q^{w_1 + \\cdots + w_m}$, the sum over the words $w = w_1 \\cdots w_m$ over $\\{1, \\dots, j\\}$ with $w_1 = w_m = j$ (there are none for $m = 0$), consecutive letters differing by at most $1$ and no two adjacent\u2026","title":"Generating function for restricted smooth words ending in their largest letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-words-capacity-nat-degree","kind":"node","lean":"MetaMathlibExt.smoothWords_capacity_natDegree","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-words-capacity-nat-degree","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"A smooth word is a word of positive integers with first letter $1$ whose consecutive letters differ by at most $1$. Let $r(n; q) = \\sum_w q^{\\mathrm{cap}(w)}$, summed over the smooth words $w$ of length $n$, where $\\mathrm{cap}(w)$ is the capacity of $w$. For all $m \\ge 1$ and $0 \\le p \\le 3$, the degree of $r(4m + p; q)$ in $q$ is $m(2m + p - 1)$. Since $r(n; q)$ is a nonzero polynomial with nonnegative\u2026","title":"Maximum capacity of smooth words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-capacity-dist-eval-zero","kind":"node","lean":"MetaMathlibExt.smoothCapacityDist_eval_zero","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-capacity-dist-eval-zero","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For $2 \\le i \\le n$, the number of smooth words of length $n$ ending in $i$ with capacity $0$, that is, the value at $q = 0$ of their capacity distribution $r(n, i; q)$, equals $\\sum_{j=i-2}^{n-2} \\binom{n-2}{j}$.","title":"Zero-capacity smooth words with a given last letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/smooth-words-zero-capacity-count","kind":"node","lean":"MetaMathlibExt.smoothWords_zero_capacity_count","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#smooth-words-zero-capacity-count","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $r(n, i; q)$ be the capacity distribution of the smooth words of length $n$ ending in $i$ and $r(n; q)$ that of all smooth words of length $n$, so that $r(n, i; 0)$ and $r(n; 0)$ count the smooth words of capacity $0$. Then $r(1, 1; 0) = 1$ and $r(n, 1; 0) = 2^{n-2}$ for every $n \\ge 2$; moreover $r(1; 0) = 1$, $r(2; 0) = 2$ and $r(n; 0) = (n + 2) \\cdot 2^{n-3}$ for every $n \\ge 3$ (Mansour and Shattuck's\u2026","title":"Number of smooth words of capacity zero"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-words-capacity-nat-deg","kind":"node","lean":"MetaMathlibExt.restrictedSmoothWords_capacity_natDegree","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-words-capacity-nat-deg","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $m \\ge 0$ and $0 \\le p \\le 7$ with $n = 8m + p \\ge 1$. The degree of the capacity distribution of the restricted smooth words of length $n$, that is, the largest capacity of a restricted smooth word of length $n$, equals $8m^2 + (2p - 3)m + e_p$, where $e_p = 0$ for $p \\le 3$, $e_4 = 1$ and $e_p = 2^{p-5}$ for $5 \\le p \\le 7$.","title":"Maximum capacity of restricted smooth words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-words-capacity-leading","kind":"node","lean":"MetaMathlibExt.restrictedSmoothWords_capacity_leadingCoeff","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-words-capacity-leading","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"Let $n = 8m + p \\ge 1$ with natural numbers $m$ and $p \\le 7$, and let $c_n$ be the leading coefficient of the capacity distribution $\\sum_{w} q^{\\mathrm{cap}(w)}$ over the restricted smooth words $w$ of length $n$ (smooth words with no two adjacent letters equal to $1$), that is, the number of restricted smooth words of length $n$ with maximum capacity. If $m = 0$, then $c_1, \\dots, c_7$ are $1, 1, 3, 1, 5, 2, 1$.\u2026","title":"Number of restricted smooth words of maximum capacity"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-capacity-dist-eval-zer","kind":"node","lean":"MetaMathlibExt.restrictedSmoothCapacityDist_eval_zero","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-capacity-dist-eval-zer","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For natural numbers $n$ and $i$ with $3 \\le i \\le n$, let $r^*(n, i; q)$ be the capacity distribution of the restricted smooth words of length $n$ ending in $i$. Its value at $q = 0$, which is the number of such words of capacity $0$, is","title":"Zero-capacity restricted smooth words with a given last letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/restricted-smooth-words-zero-capacity-co","kind":"node","lean":"MetaMathlibExt.restrictedSmoothWords_zero_capacity_count","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#restricted-smooth-words-zero-capacity-co","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For $n \\ge 1$ and a letter $i$, let $r^{\\ast}(n, i)$ be the number of restricted smooth words of length $n$ ending in $i$ that have capacity $0$, and let $r^{\\ast}(n)$ be the number of restricted smooth words of length $n$ with capacity $0$; these are the values at $q = 0$ of the corresponding capacity distributions. Then $r^{\\ast}(n, 1) = 1$ for $n \\in \\{1, 3\\}$, $r^{\\ast}(2, 1) = 0$ and $r^{\\ast}(n, 1) = 2^{n-4}$\u2026","title":"Zero-capacity restricted smooth words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/card-restricted-smooth-words","kind":"node","lean":"MetaMathlibExt.card_restrictedSmoothWords","parent":24206,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words/index.html#card-restricted-smooth-words","scope":"Combinatorics/combinatorics-on-words/finite-words/capacity-of-catalan-words","status":"can_prove","summary":"For every $n \\ge 1$, the number of restricted smooth words of length $n$ is the grand Motzkin number $G_{n-1}$, that is, the central trinomial coefficient: the coefficient of $X^{n-1}$ in $(1 + X + X^2)^{n-1}$.","title":"Restricted smooth words are counted by grand Motzkin numbers"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words.json","id":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words","status":"can_prove","title":"Consecutive patterns in words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/avoid-zz1-z","kind":"node","lean":"MathlibExt.NumberTheory.OeisConjecturesWanted.avoidZZ1Z","parent":24269,"reading_title":{"format":"inline-parts-v1","parts":[["text","Words avoiding the consecutive pattern "],["math","\\(z, z+1, z\\)"]],"source":"Words avoiding the consecutive pattern $z, z+1, z$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/index.html#avoid-zz1-z","scope":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $f_k(n)$ is the number of words $w_0 w_1 \\cdots w_{n-1}$ of length $n$ over the alphabet $\\{0, 1, \\dots, k-1\\}$ with no position $i$ such that $i + 2 < n$, $w_{i+1} = w_i + 1$ and $w_{i+2} = w_i$.","title":"Words avoiding the consecutive pattern $z, z+1, z$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/avoid-zz1-z-spec","kind":"node","lean":"MathlibExt.NumberTheory.OeisConjecturesWanted.avoidZZ1Z_spec","parent":24269,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for words avoiding the pattern "],["math","\\(z, z+1, z\\)"]],"source":"Recurrence for words avoiding the pattern $z, z+1, z$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/index.html#avoid-zz1-z-spec","scope":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words","status":"can_prove","summary":"Let $k \\ge 1$ and, for each natural number $n$, let $f_k(n)$ be the number of words of length $n$ over the alphabet $\\{0, 1, \\dots, k-1\\}$ that avoid the consecutive pattern $z, z+1, z$. Then $f_k(0) = 1$, $f_k(1) = k$, $f_k(2) = k^2$, and $f_k(n) = k f_k(n-1) - f_k(n-2) + f_k(n-3)$ for every $n \\ge 3$.","title":"Recurrence for words avoiding the pattern $z, z+1, z$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/size-choose-two-pow","kind":"node","lean":"MathlibExt.NumberTheory.OeisConjecturesWanted.size_choose_two_pow","parent":24269,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bit length of "],["math","\\(\\binom{2^n}{2^{n-1}}\\)"]],"source":"Bit length of $\\binom{2^n}{2^{n-1}}$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/index.html#size-choose-two-pow","scope":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words","status":"can_prove","summary":"For every integer $n \\ge 1$, the binary expansion of the binomial coefficient $\\binom{2^n}{2^{n-1}}$ has exactly $2^n - \\lfloor n/2 \\rfloor$ digits.","title":"Bit length of $\\binom{2^n}{2^{n-1}}$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/floor-sum-cbrt","kind":"node","lean":"MathlibExt.NumberTheory.OeisConjecturesWanted.floor_sum_cbrt","parent":24269,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words/index.html#floor-sum-cbrt","scope":"Combinatorics/combinatorics-on-words/finite-words/consecutive-patterns-in-words","status":"can_prove","summary":"For every integer $m \\ge 1$, the sum $S_m = \\sum_{i=1}^{10^{3m}} i^{1/3}$ of the real cube roots of $1, 2, \\dots, 10^{3m}$ satisfies","title":"Floor of a sum of cube roots"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/equilibrious-words.json","id":"Combinatorics/combinatorics-on-words/finite-words/equilibrious-words","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/equilibrious-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/equilibrious-words","status":"fully_proved","title":"Equilibrious words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/equilibrious-words/is-equilibrious-word","kind":"node","lean":"MetaMathlibExt.IsEquilibriousWord","parent":24274,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/equilibrious-words/index.html#is-equilibrious-word","scope":"Combinatorics/combinatorics-on-words/finite-words/equilibrious-words","status":"fully_proved","summary":"A finite word $u$ over the binary alphabet $\\{0, 1\\}$ is equilibrious when the letters $0$ and $1$ occur the same number of times in $u$.","title":"Equilibrious binary words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/equilibrious-words/is-equilibrious-word-reverse","kind":"node","lean":"MetaMathlibExt.isEquilibriousWord_reverse","parent":24274,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/equilibrious-words/index.html#is-equilibrious-word-reverse","scope":"Combinatorics/combinatorics-on-words/finite-words/equilibrious-words","status":"fully_proved","summary":"A finite word $u$ over the binary alphabet $\\{0, 1\\}$ is equilibrious if and only if its reversal is equilibrious.","title":"Reversal preserves equilibrious words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words.json","id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","title":"Intersections in pairs of words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/non-intersecting-word-pairs-card-eq-coef","kind":"node","lean":"MetaMathlibExt.nonIntersectingWordPairs_card_eq_coeff","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#non-intersecting-word-pairs-card-eq-coef","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"Fix natural numbers $k$, $n \\ge 1$ and $r \\ge 1$, and call a pair of words over an alphabet of $k$ letters non-intersecting when no letter occurs in both. Write $[x^m] F$ for the coefficient of $x^m$ in a formal power series $F$ over $\\mathbb{Q}$ and $S(m, j)$ for the Stirling numbers of the second kind. Let $N$ be the number of non-intersecting pairs $(u, v)$ with $u$ of length $n$ and $v$ of length $r$. Then\u2026","title":"Non-intersecting pairs of words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/non-intersecting-word-pairs-same-length","kind":"node","lean":"MetaMathlibExt.nonIntersectingWordPairs_sameLength_generatingFunction","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#non-intersecting-word-pairs-same-length","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"Let $k$ be a natural number, let $S(n, j)$ denote the Stirling numbers of the second kind, and for $n \\ge 1$ let $h(n)$ be the number of pairs $(u, v)$ of words of length $n$ over an alphabet of $k$ letters such that no letter occurs in both. Then, as formal power series over $\\mathbb{Q}$, $\\sum_{n \\ge 1} h(n) x^n = \\sum_{n \\ge 0} \\sum_{j=1}^{k-1} \\binom{k}{j} \\, j! \\, S(n, j) \\, (k-j)^n x^n$.","title":"Generating function of non-intersecting pairs of words of equal length"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/non-intersecting-word-pairs-same-length-2","kind":"node","lean":"MetaMathlibExt.nonIntersectingWordPairs_sameLength_card","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#non-intersecting-word-pairs-same-length-2","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"Let $S(n, j)$ denote the Stirling numbers of the second kind. For natural numbers $k$ and $n \\ge 1$, the number of pairs $(u, v)$ of words of length $n$ over an alphabet of $k$ letters such that no letter occurs in both $u$ and $v$ is $\\sum_{j=1}^{k-1} \\binom{k}{j} \\, j! \\, S(n, j) \\, (k-j)^n$.","title":"Non-intersecting pairs of words of equal length"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/word-pairs-overlap-sum-eq-coeff","kind":"node","lean":"MetaMathlibExt.wordPairs_overlap_sum_eq_coeff","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#word-pairs-overlap-sum-eq-coeff","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"Fix natural numbers $k$, $n \\ge 1$, $r \\ge 1$ and a rational number $q$. For words $u$ and $v$ over an alphabet of $k$ letters, the overlap of $(u, v)$ is the number of positions of $v$ whose letter occurs in $u$. Write $[x^m] F$ for the coefficient of $x^m$ in a formal power series $F$ over $\\mathbb{Q}$, $S(m, j)$ for the Stirling numbers of the second kind, and $c_j = qj + k - j$. Then the sum of\u2026","title":"Overlap statistic on pairs of words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/word-pairs-same-length-overlap-sum","kind":"node","lean":"MetaMathlibExt.wordPairs_sameLength_overlap_sum","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#word-pairs-same-length-overlap-sum","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"Fix natural numbers $k$ and $n \\ge 1$ and a rational number $q$. For words $u$ and $v$ over an alphabet of $k$ letters, the overlap of $(u, v)$ is the number of positions of $v$ whose letter occurs in $u$. Let $S(n, j)$ denote the Stirling numbers of the second kind. Then the sum of $q^{\\text{overlap}}$ over all pairs $(u, v)$ of words of length $n$ equals\u2026","title":"Overlap statistic on pairs of words of equal length"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/weakly-decreasing-non-intersecting-pairs","kind":"node","lean":"MetaMathlibExt.weaklyDecreasingNonIntersectingPairs_card_eq_coeff","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#weakly-decreasing-non-intersecting-pairs","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"Fix natural numbers $k$, $n \\ge 1$ and $r \\ge 1$, and order an alphabet of $k$ letters. Write $[x^m] F$ for the coefficient of $x^m$ in a formal power series $F$ over $\\mathbb{Q}$. The number of pairs $(u, v)$ of weakly decreasing words over this alphabet, $u$ of length $n$ and $v$ of length $r$, such that no letter occurs in both is\u2026","title":"Non-intersecting pairs of weakly decreasing words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/weakly-decreasing-non-intersecting-pairs-2","kind":"node","lean":"MetaMathlibExt.weaklyDecreasingNonIntersectingPairs_generatingFunction","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#weakly-decreasing-non-intersecting-pairs-2","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"Let $k$ be a natural number and order an alphabet of $k$ letters. In the ring of formal power series in $x$ and $y$ over $\\mathbb{Q}$ let $f(x, y) = \\bigl(x (1-x)^{-1} + (1-y)^{-1}\\bigr)^k - (1-x)^{-k} - (1-y)^{-k} + 1$. Then for all natural numbers $n$ and $r$, the coefficient of $x^n y^r$ in $f$ is $0$ if $n = 0$ or $r = 0$, and otherwise equals the number of pairs $(u, v)$ of weakly decreasing words over this\u2026","title":"Generating function of non-intersecting pairs of weakly decreasing words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/weakly-decreasing-non-intersecting-pairs-3","kind":"node","lean":"MetaMathlibExt.weaklyDecreasingNonIntersectingPairs_sameLength_card","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#weakly-decreasing-non-intersecting-pairs-3","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"For natural numbers $k$ and $n \\ge 1$, the number of pairs $(u, v)$ of weakly decreasing words of length $n$ over an ordered alphabet of $k$ letters such that no letter occurs in both is $\\sum_{j=1}^{k-1} \\binom{k}{j} \\binom{n-1}{j-1} \\binom{n+k-j-1}{n}$.","title":"Non-intersecting pairs of weakly decreasing words of equal length"},{"id":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/weakly-decreasing-pairs-overlap-sum-eq-c","kind":"node","lean":"MetaMathlibExt.weaklyDecreasingPairs_overlap_sum_eq_coeff","parent":24277,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words/index.html#weakly-decreasing-pairs-overlap-sum-eq-c","scope":"Combinatorics/combinatorics-on-words/finite-words/intersections-in-pairs-of-words","status":"can_prove","summary":"Fix natural numbers $k$, $n \\ge 1$, $r \\ge 1$ and a rational number $q$, and order an alphabet of $k$ letters. For words $u$ and $v$ over this alphabet, the overlap of $(u, v)$ is the number of positions of $v$ whose letter occurs in $u$. Write $[x^m] F$ for the coefficient of $x^m$ in a formal power series $F$ over $\\mathbb{Q}$. Let $T$ be the sum of $q^{\\text{overlap}}$ over all pairs $(u, v)$ of weakly\u2026","title":"Overlap statistic on pairs of weakly decreasing words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words.json","id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","title":"Keyword substitutions on binary words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-applicable","kind":"node","lean":"MetaMathlibExt.KeywordApplicable","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-applicable","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a = a_0 a_1 \\cdots a_m$ be a binary keyword of length $m + 1$ and $u = u_0 u_1 \\cdots u_{n-1}$ a binary word of length $n$. The keyword $a$ is applicable to $u$ at the ($0$-indexed) letter $i$ when $i + m < n$ and the subword $u_i u_{i+1} \\cdots u_{i+m}$ of $u$ equals $a$ or the letterwise negation of $a$.","title":"Applicability of a keyword"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-simple-map","kind":"node","lean":"MetaMathlibExt.keywordSimpleMap","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-simple-map","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a$ be a keyword, a binary word of length $m + 1$, and let $i \\in \\mathbb{N}$. The simple map $\\varphi_i$ sends a binary word $u$ of length $n$ to the word obtained by negating the letters of $u$ at positions $i, \\dots, i + m$ (positions counted from $0$) if $a$ is applicable to $u$ at letter $i$, that is, if $i + m < n$ and the subword of $u$ at positions $i, \\dots, i + m$ equals $a$ or its letterwise negation\u2026","title":"Simple map"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-simple-map-of-not-applicable","kind":"node","lean":"MetaMathlibExt.keywordSimpleMap_of_not_applicable","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-simple-map-of-not-applicable","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a$ be a keyword and $u$ a binary word. If $a$ is not applicable to $u$ at letter $i$, then the simple map $\\varphi_i$ of $a$ leaves $u$ unchanged: $\\varphi_i(u) = u$.","title":"A simple map that is not applicable is the identity"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-simple-map-apply-of-applicable","kind":"node","lean":"MetaMathlibExt.keywordSimpleMap_apply_of_applicable","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-simple-map-apply-of-applicable","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a$ be a keyword of length $m + 1$ and $u$ a binary word of length $n$, and suppose that $a$ is applicable to $u$ at letter $i$. Then the simple map $\\varphi_i$ of $a$ negates the letters of $u$ at positions $i, \\dots, i + m$ and leaves every other letter unchanged: for every position $j$, $\\varphi_i(u)_j = \\neg u_j$ if $i \\le j \\le i + m$, and $\\varphi_i(u)_j = u_j$ otherwise.","title":"An applicable simple map negates its window"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-equiv","kind":"node","lean":"MetaMathlibExt.KeywordEquiv","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-equiv","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a$ be a binary keyword of length $m + 1$. Two binary words $u$ and $v$ of length $n$ satisfy $u \\sim_a v$ when some finite composition of simple maps of $a$, possibly the empty composition, sends $u$ to $v$.","title":"Keyword equivalence of binary words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-equiv-refl","kind":"node","lean":"MetaMathlibExt.keywordEquiv_refl","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-equiv-refl","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"For every binary keyword $a$ and every binary word $u$, we have $u \\sim_a u$: the empty composition of simple maps sends $u$ to itself.","title":"Reflexivity of keyword equivalence"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-graph","kind":"node","lean":"MetaMathlibExt.keywordGraph","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-graph","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a$ be a keyword, a binary word of length $m + 1$, and let $n \\ge 0$. The substitution graph $\\mathcal{G}_n^{(a)}$ is the simple graph on the binary words of length $n$ in which distinct words $u$ and $v$ are adjacent when $\\varphi_i(u) = v$ or $\\varphi_i(v) = u$ for some $i \\in \\mathbb{N}$, where $\\varphi_i$ is the simple map of $a$ at letter $i$. Equivalently, $u$ and $v$ are adjacent when one is obtained\u2026","title":"Substitution graph"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-graph-adj","kind":"node","lean":"MetaMathlibExt.keywordGraph_adj","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-graph-adj","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a$ be a binary keyword and let $u$ and $v$ be binary words of length $n$. Then $u$ and $v$ are adjacent in the graph $\\mathcal{G}_n^{(a)}$ if and only if $u \\ne v$ and some simple map $\\varphi_i$ of $a$ sends $u$ to $v$ or sends $v$ to $u$.","title":"Adjacency in the keyword substitution graph"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-list-apply","kind":"node","lean":"MetaMathlibExt.keywordListApply","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-list-apply","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a$ be a keyword and, for $j \\in \\mathbb{N}$, let $\\varphi_j$ be the simple map of $a$ at letter $j$. A list $[j_1, \\dots, j_r]$ of natural numbers acts on a binary word $u$ of length $n$ by $u \\mapsto \\varphi_{j_r}(\\cdots(\\varphi_{j_1}(u))\\cdots)$: the head of the list acts first, and the empty list acts as the identity.","title":"Action of a list of simple maps"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-acts-completely","kind":"node","lean":"MetaMathlibExt.KeywordActsCompletely","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-acts-completely","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"Let $a$ be a binary keyword and $u$ a binary word. A list $[j_1, \\dots, j_r]$ of indices acts completely on $u$ when applying the simple maps $\\varphi_{j_1}, \\varphi_{j_2}, \\dots, \\varphi_{j_r}$ of $a$ in this order changes the word at every step: with $u_0 = u$ and $u_q = \\varphi_{j_q}(u_{q-1})$, we have $u_q \\ne u_{q-1}$ for $1 \\le q \\le r$. In particular the empty list acts completely on every word.","title":"Lists of simple maps acting completely"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-list-apply-nil","kind":"node","lean":"MetaMathlibExt.keywordListApply_nil","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-list-apply-nil","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"For every keyword $a$ and every binary word $u$, applying the empty list of simple maps of $a$ to $u$ returns $u$.","title":"The empty list of simple maps acts trivially"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-acts-completely-singleton","kind":"node","lean":"MetaMathlibExt.keywordActsCompletely_singleton","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-acts-completely-singleton","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"fully_proved","summary":"For a binary keyword $a$, an index $j$ and a binary word $u$, the one-element list $[j]$ acts completely on $u$ if and only if the simple map $\\varphi_j$ of $a$ changes $u$, that is, $\\varphi_j(u) \\ne u$.","title":"Complete action of a one-element list"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/card-quot-keyword-equiv-eq-sum-m-step-fi","kind":"node","lean":"MetaMathlibExt.card_quot_keywordEquiv_eq_sum_mStepFibonacci","parent":24287,"reading_title":{"format":"inline-parts-v1","parts":[["text","Keyword classes are counted by partial sums of "],["math","\\(m\\)"],["text","-step Fibonacci numbers"]],"source":"Keyword classes are counted by partial sums of $m$-step Fibonacci numbers"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#card-quot-keyword-equiv-eq-sum-m-step-fi","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"Let $m, n \\ge 1$ and let $F_0, F_1, F_2, \\dots$ be the $m$-step Fibonacci numbers: $F_0 = 1$, $F_k = 2^{k-1}$ for $1 \\le k \\le m - 1$, and $F_k = F_{k-1} + F_{k-2} + \\dots + F_{k-m}$ for $k \\ge m$. Then for every binary keyword $a$ of length $m + 1$, the number of equivalence classes of the relation $\\sim_a$ on binary words of length $n$ is $F_0 + F_1 + \\dots + F_n$.","title":"Keyword classes are counted by partial sums of $m$-step Fibonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-equiv-append-iff","kind":"node","lean":"MetaMathlibExt.keywordEquiv_append_iff","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-equiv-append-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"Let $a$ be a binary keyword, let $u$ and $v$ be binary words of length $n$, and let $w$ be a binary word of length $n'$. Writing $xy$ for the concatenation of words $x$ and $y$, we have $u \\sim_a v$ if and only if $uw \\sim_a vw$, and $u \\sim_a v$ if and only if $wu \\sim_a wv$.","title":"Keyword equivalence is stable under concatenation"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/even-count-min-of-keyword-acts-completel","kind":"node","lean":"MetaMathlibExt.even_count_min_of_keywordActsCompletely","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#even-count-min-of-keyword-acts-completel","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"Let $a$ be a binary keyword of length $m + 1$, let $n$ and $i$ be natural numbers with $i < n$, and for each index $k$ let $\\varphi_k$ be the simple map of $a$ at letter $k$ on binary words of length $n$. Let $l = [j_1, \\dots, j_r]$ be a list of indices that are all greater than $i$, and let $\\varphi_l = \\varphi_{j_r} \\circ \\dots \\circ \\varphi_{j_1}$ be the map that applies $\\varphi_{j_1}$ first. Suppose that\u2026","title":"Parity of the smallest index in a non-commuting list"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/sum-m-step-fibonacci-iff-recurrence","kind":"node","lean":"MetaMathlibExt.sum_mStepFibonacci_iff_recurrence","parent":24287,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for partial sums of "],["math","\\(m\\)"],["text","-step Fibonacci numbers"]],"source":"Recurrence for partial sums of $m$-step Fibonacci numbers"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#sum-m-step-fibonacci-iff-recurrence","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"Let $m \\ge 1$ and let $F : \\mathbb{N} \\to \\mathbb{N}$ satisfy $F(0) = 1$, $F(k) = 2^{k-1}$ for $1 \\le k \\le m - 1$, and $F(k) = F(k-1) + \\cdots + F(k-m)$ for $k \\ge m$ (the $m$-step Fibonacci numbers). Then a real sequence $(S_n)_{n \\ge 0}$ satisfies $S_n = F(0) + F(1) + \\cdots + F(n)$ for all $n$ if and only if $S_n = 2^n$ for $0 \\le n \\le m$ and $S_n = 2 S_{n-1} - S_{n-m-1}$ for $n \\ge m + 1$.","title":"Recurrence for partial sums of $m$-step Fibonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/card-singleton-classes-eq-iff","kind":"node","lean":"MetaMathlibExt.card_singleton_classes_eq_iff","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#card-singleton-classes-eq-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"Let $a$ and $b$ be binary keywords of length $m + 1$. The relations $\\sim_a$ and $\\sim_b$ have the same number of one-element equivalence classes on binary words of length $n$ for every $n \\ge 0$ if and only if, for every $i$ with $1 \\le i \\le m + 1$, the following holds for $a$ exactly when it holds for $b$: the prefix of length $i$ of the keyword equals its suffix of length $i$ or the letterwise negation of that\u2026","title":"Keywords with equally many singleton classes"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-simple-map-comm-iff","kind":"node","lean":"MetaMathlibExt.keywordSimpleMap_comm_iff","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-simple-map-comm-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"Let $a$ be a keyword of length $m + 1$ and let $n$ be a natural number. Let $i < j$ be natural numbers with $j + m < n$, and put $\\Delta = j - i$. The simple maps $\\varphi_i$ and $\\varphi_j$ of $a$ on binary words of length $n$ satisfy $\\varphi_i \\circ \\varphi_j = \\varphi_j \\circ \\varphi_i$ if and only if $\\Delta \\ge m + 1$, or the prefix of $a$ of length $m + 1 - \\Delta$ differs both from the suffix of $a$ of\u2026","title":"When two simple maps commute"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-graph-neg-rev-seminegation","kind":"node","lean":"MetaMathlibExt.keywordGraph_neg_rev_seminegation","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-graph-neg-rev-seminegation","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"Let $a = a_0 a_1 \\cdots a_m$ be a keyword of length $m + 1$ and let $n \\ge 0$. Write $\\neg a$ for the letterwise negation of $a$, $\\rho(a) = a_m a_{m-1} \\cdots a_0$ for its reversal, and $\\bar a$ for its seminegation, which keeps $a_t$ for even $t$ and negates $a_t$ for odd $t$. Then the substitution graphs satisfy $\\mathcal{G}_n^{(a)} = \\mathcal{G}_n^{(\\neg a)}$, and $\\mathcal{G}_n^{(a)}$ is isomorphic both to\u2026","title":"Keyword symmetries of the substitution graph"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-graph-edist-eq-of-intertwining","kind":"node","lean":"MetaMathlibExt.keywordGraph_edist_eq_of_intertwining","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-graph-edist-eq-of-intertwining","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"Let $a$ and $b$ be keywords of length $m + 1$, and write $\\varphi_i^{(a)}$ and $\\varphi_i^{(b)}$ for the simple maps of $a$ and $b$ at letter $i$ on binary words of length $n$. Let $f$ be a bijection of the set of binary words of length $n$ and $\\sigma$ a permutation of the positions $\\{0, \\dots, n - 1\\}$, and suppose that $f(\\varphi_i^{(a)}(u)) = \\varphi_{\\sigma(i)}^{(b)}(f(u))$ for every word $u$ and every\u2026","title":"Intertwining bijections preserve distances in the substitution graph"},{"id":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/keyword-graph-is-bipartite","kind":"node","lean":"MetaMathlibExt.keywordGraph_isBipartite","parent":24287,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words/index.html#keyword-graph-is-bipartite","scope":"Combinatorics/combinatorics-on-words/finite-words/keyword-substitutions-on-binary-words","status":"can_prove","summary":"For every keyword $a$ of length $m + 1$ and every $n \\ge 0$, the substitution graph $\\mathcal{G}_n^{(a)}$ on binary words of length $n$ is bipartite.","title":"The substitution graph is bipartite"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words.json","id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","title":"Lyndon and primitive words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/is-primitive-word","kind":"node","lean":"MetaMathlibExt.IsPrimitiveWord","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#is-primitive-word","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"A nonempty finite word is primitive if it is not a proper power of another word.","title":"Primitive word"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/witt-word-content","kind":"node","lean":"MetaMathlibExt.wittWordContent","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#witt-word-content","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Let $r$ be a natural number. The content of a word $w$ over the alphabet $\\{0, \\dots, r-1\\}$ is the function that assigns to each letter $i$ the number of occurrences of $i$ in $w$.","title":"Content of a word"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/is-lyndon-word","kind":"node","lean":"MetaMathlibExt.IsLyndonWord","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#is-lyndon-word","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered alphabet, and order the words over $\\alpha$ lexicographically. A word $w$ over $\\alpha$ of length $n$ is a Lyndon word if it is nonempty and, for every $k$ with $0 < k < n$, $w$ is strictly smaller than its rotation by $k$, the word obtained by moving the first $k$ letters of $w$ to the end.","title":"Lyndon word"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/witt-word-content-append","kind":"node","lean":"MetaMathlibExt.wittWordContent_append","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#witt-word-content-append","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"For a natural number $r$, words $u$ and $v$ over the alphabet $\\{0, \\dots, r-1\\}$, and every letter $i$, the number of occurrences of $i$ in the concatenation $uv$ is the number of occurrences of $i$ in $u$ plus the number of occurrences of $i$ in $v$.","title":"Content is additive under concatenation"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/witt-word-content-singleton","kind":"node","lean":"MetaMathlibExt.wittWordContent_singleton","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#witt-word-content-singleton","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Let $r$ be a natural number. For every letter $i$ of the alphabet $\\{0, \\dots, r-1\\}$, the content of the one-letter word $i$ takes the value $1$ at $i$ and the value $0$ at every other letter.","title":"Content of a one-letter word"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/witt-word-content-cons","kind":"node","lean":"MetaMathlibExt.wittWordContent_cons","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#witt-word-content-cons","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Let $r$ be a natural number, let $w$ be a word over the alphabet $\\{0, \\dots, r-1\\}$, let $i$ be a letter, and write $c(u)$ for the content of a word $u$ (the number of occurrences of each letter in $u$). If $iw$ denotes $w$ with $i$ prepended, then $c(iw)(i) = c(w)(i) + 1$ and $c(iw)(j) = c(w)(j)$ for every letter $j \\ne i$.","title":"Content of a word after prepending a letter"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/sum-witt-word-content","kind":"node","lean":"MetaMathlibExt.sum_wittWordContent","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#sum-witt-word-content","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"For a natural number $r$ and a word $w$ over the alphabet $\\{0, \\dots, r-1\\}$, summing over all letters $i$ the number of occurrences of $i$ in $w$ gives the length of $w$.","title":"The content of a word sums to its length"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/words-of-content","kind":"node","lean":"MetaMathlibExt.WordsOfContent","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#words-of-content","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Let $r$ be a natural number. Given a function $d$ from the alphabet $\\{0, \\dots, r-1\\}$ to the natural numbers, the words of content $d$ are the words $w$ over this alphabet whose content equals $d$, that is, in which each letter $i$ occurs exactly $d(i)$ times.","title":"Words with prescribed content"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/lyndon-words-of-content","kind":"node","lean":"MetaMathlibExt.LyndonWordsOfContent","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#lyndon-words-of-content","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"For a natural number $r$ and natural numbers $n_0, \\dots, n_{r-1}$, the set of Lyndon words over the alphabet $\\{0, \\dots, r-1\\}$, ordered as the natural numbers, in which each letter $i$ occurs exactly $n_i$ times.","title":"Lyndon words of given content"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/card-words-of-content-eq-multinomial","kind":"node","lean":"MetaMathlibExt.card_wordsOfContent_eq_multinomial","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#card-words-of-content-eq-multinomial","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Let $r$ be a natural number, let $n_0, \\dots, n_{r-1}$ be natural numbers, and let $n = n_0 + \\dots + n_{r-1}$. The number of words over the alphabet $\\{0, \\dots, r-1\\}$ in which each letter $i$ occurs exactly $n_i$ times is the multinomial coefficient $\\binom{n}{n_0, \\ldots, n_{r-1}} = \\frac{n!}{n_0! \\cdots n_{r-1}!}$.","title":"Words of given content are counted by the multinomial coefficient"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/is-primitive-word-iff-nodup-cyclic-permu","kind":"node","lean":"MetaMathlibExt.isPrimitiveWord_iff_nodup_cyclicPermutations","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#is-primitive-word-iff-nodup-cyclic-permu","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Let $w$ be a word of length $n$, and for $0 \\le k < n$ let $w_k$ be the rotation of $w$ by $k$, obtained by moving the first $k$ letters of $w$ to the end. Then $w$ is primitive if and only if $w$ is nonempty and the $n$ words $w_0, w_1, \\dots, w_{n-1}$ are pairwise distinct.","title":"Primitivity via distinct rotations"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/rotate-ne-of-is-primitive-word","kind":"node","lean":"MetaMathlibExt.rotate_ne_of_isPrimitiveWord","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#rotate-ne-of-is-primitive-word","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"If $w$ is a primitive word of length $n$ and $0 < k < n$, then the rotation of $w$ by $k$, obtained by moving the first $k$ letters of $w$ to the end, is different from $w$.","title":"Nontrivial rotations move a primitive word"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/primitive-words-of-content","kind":"node","lean":"MetaMathlibExt.PrimitiveWordsOfContent","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#primitive-words-of-content","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"For a natural number $r$ and natural numbers $n_0, \\dots, n_{r-1}$, the set of primitive words over the alphabet $\\{0, \\dots, r-1\\}$ in which each letter $i$ occurs exactly $n_i$ times.","title":"Primitive words of given content"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/is-primitive-word-of-is-lyndon-word","kind":"node","lean":"MetaMathlibExt.isPrimitiveWord_of_isLyndonWord","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#is-primitive-word-of-is-lyndon-word","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Every Lyndon word over a linearly ordered alphabet is primitive.","title":"Lyndon words are primitive"},{"id":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/card-lyndon-words-of-content-eq-witt-mul","kind":"node","lean":"MetaMathlibExt.card_lyndonWordsOfContent_eq_wittMultigradedValue","parent":24309,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words/index.html#card-lyndon-words-of-content-eq-witt-mul","scope":"Combinatorics/combinatorics-on-words/finite-words/lyndon-and-primitive-words","status":"fully_proved","summary":"Let $r$ be a natural number and let $n_0, \\dots, n_{r-1}$ be natural numbers with $n = n_0 + \\dots + n_{r-1} > 0$. Order the alphabet $\\{0, \\dots, r-1\\}$ as the natural numbers and order words over it lexicographically. Then the number of Lyndon words over this alphabet in which each letter $i$ occurs exactly $n_i$ times equals, as a rational number, Witt's multigraded value\u2026","title":"Witt's formula for the number of Lyndon words of given content"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words.json","id":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words","status":"can_prove","title":"Rectangles in nondecreasing words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/rectangle-capacity","kind":"node","lean":"MetaMathlibExt.rectangleCapacity","parent":24325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(r \\times s\\)"],["text"," rectangles in a bargraph"]],"source":"Number of $r \\times s$ rectangles in a bargraph"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/index.html#rectangle-capacity","scope":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words","status":"fully_proved","summary":"Let $w = w_0 w_1 \\cdots w_{m-1}$ be a word of natural numbers, drawn as a bargraph whose column $i$ has height $w_i$. The number of occurrences of an $r \\times s$ rectangle (height $r$, width $s$) in $w$ is the number of pairs $(i, y)$ of natural numbers with $i < m$, $i + s \\le m$ and $y \\le w_0 + \\cdots + w_{m-1}$ such that $y + r \\le w_{i+j}$ for every $j < s$.","title":"Number of $r \\times s$ rectangles in a bargraph"},{"id":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/rectangle-capacity-example","kind":"node","lean":"MetaMathlibExt.rectangleCapacity_example","parent":24325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Example: "],["math","\\(3 \\times 2\\)"],["text"," rectangles in the word "],["math","\\(345134\\)"]],"source":"Example: $3 \\times 2$ rectangles in the word $345134$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/index.html#rectangle-capacity-example","scope":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words","status":"fully_proved","summary":"The bargraph of the word $345134$ contains exactly four rectangles of height $3$ and width $2$.","title":"Example: $3 \\times 2$ rectangles in the word $345134$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/nondecreasing-rectangle-total","kind":"node","lean":"MetaMathlibExt.nondecreasingRectangleTotal","parent":24325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total number of "],["math","\\(r \\times s\\)"],["text"," rectangles in nondecreasing words"]],"source":"Total number of $r \\times s$ rectangles in nondecreasing words"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/index.html#nondecreasing-rectangle-total","scope":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words","status":"fully_proved","summary":"For natural numbers $k$, $r$, $s$ and $n$, this is the sum, over all nondecreasing words $w_1 \\le w_2 \\le \\cdots \\le w_n$ of length $n$ over the alphabet $[k] = \\{1, \\dots, k\\}$ (the set $W^{\\nearrow}_{n,k}$), of the number of $r \\times s$ rectangles in the bargraph of $w$.","title":"Total number of $r \\times s$ rectangles in nondecreasing words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/nondecreasing-rectangle-total-two-one-tw","kind":"node","lean":"MetaMathlibExt.nondecreasingRectangleTotal_two_one_two_two","parent":24325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Example: "],["math","\\(1 \\times 2\\)"],["text"," rectangles in nondecreasing words of length "],["math","\\(2\\)"]],"source":"Example: $1 \\times 2$ rectangles in nondecreasing words of length $2$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/index.html#nondecreasing-rectangle-total-two-one-tw","scope":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words","status":"fully_proved","summary":"The nondecreasing words of length $2$ over $[2] = \\{1, 2\\}$ are $11$, $12$ and $22$, which contain $1$, $1$ and $2$ rectangles of size $1 \\times 2$ respectively, so the total number of $1 \\times 2$ rectangles over all nondecreasing words of length $2$ over $[2]$ is $4$.","title":"Example: $1 \\times 2$ rectangles in nondecreasing words of length $2$"},{"id":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/nondecreasing-rectangle-total-one-by-s","kind":"node","lean":"MetaMathlibExt.nondecreasingRectangleTotal_one_by_s","parent":24325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for "],["math","\\(1 \\times s\\)"],["text"," rectangles in nondecreasing words"]],"source":"Generating function for $1 \\times s$ rectangles in nondecreasing words"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/index.html#nondecreasing-rectangle-total-one-by-s","scope":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words","status":"can_prove","summary":"Let $k \\ge 1$ and $s \\ge 1$, and let $f_k(n)$ be the total number of $1 \\times s$ rectangles over all nondecreasing words of length $n$ over $[k] = \\{1, \\dots, k\\}$. Then, as formal power series over $\\mathbb{Z}$,","title":"Generating function for $1 \\times s$ rectangles in nondecreasing words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/nondecreasing-rectangle-total-r-by-s","kind":"node","lean":"MetaMathlibExt.nondecreasingRectangleTotal_r_by_s","parent":24325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for "],["math","\\(r \\times s\\)"],["text"," rectangles in nondecreasing words"]],"source":"Generating function for $r \\times s$ rectangles in nondecreasing words"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words/index.html#nondecreasing-rectangle-total-r-by-s","scope":"Combinatorics/combinatorics-on-words/finite-words/rectangles-in-nondecreasing-words","status":"can_prove","summary":"Let $k \\ge 1$, $r \\ge 1$ and $s \\ge 1$, let $g_k(n)$ be the total number of $r \\times s$ rectangles over all nondecreasing words of length $n$ over $[k] = \\{1, \\dots, k\\}$, and let $m = \\max(k + 1 - r, 0)$. Then, as formal power series over $\\mathbb{Z}$,","title":"Generating function for $r \\times s$ rectangles in nondecreasing words"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings.json","id":"Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings","status":"fully_proved","title":"Reverse-permute equivalence of strings"},{"id":"Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings/kstring","kind":"node","lean":"MetaMathlibExt.ReversePermute.KString","parent":24332,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-strings"]],"source":"$k$-strings"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings/index.html#kstring","scope":"Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings","status":"fully_proved","summary":"For a natural number $k$, a $k$-string is a finite sequence (possibly empty) of elements of $\\{0, 1, \\ldots, k-1\\}$.","title":"$k$-strings"},{"id":"Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings/rp-relation","kind":"node","lean":"MetaMathlibExt.ReversePermute.rpRelation","parent":24332,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings/index.html#rp-relation","scope":"Combinatorics/combinatorics-on-words/finite-words/reverse-permute-equivalence-of-strings","status":"fully_proved","summary":"For a natural number $k$, a $k$-string $s$ is related to a $k$-string $t$ by the reverse-permute (RP) relation when there is a permutation $\\sigma$ of $\\{0, 1, \\ldots, k-1\\}$ such that $t$ is obtained, either from $s$ or from the reversal of $s$, by replacing each entry $x$ with $\\sigma(x)$.","title":"The reverse-permute relation"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/ulam-words.json","id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","title":"Ulam words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-word-aux","kind":"node","lean":"MetaMathlibExt.ulamWordAux","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-word-aux","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"For a natural number $f$ (the fuel) and a word $w$ in the letters $0$ and $1$, the test returns false when $f = 0$. With fuel $f + 1$ it returns true when $w$ has length $1$; otherwise it returns true exactly when there is exactly one $k$ with $1 \\le k < |w|$ such that the prefix $w_1 \\cdots w_k$ and the suffix $w_{k+1} \\cdots w_{|w|}$ are distinct and both pass the test with fuel $f$.","title":"Fuel-bounded Ulam test"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word","kind":"node","lean":"MetaMathlibExt.IsUlamWord","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"Ulam words are finite words over the alphabet $\\{0, 1\\}$, defined by recursion on the length. The empty word is not an Ulam word, and the two words $0$ and $1$ of length $1$ are Ulam words. A word $w$ of length at least $2$ is an Ulam word if and only if there is exactly one way to write $w = uv$ with $u$ and $v$ nonempty, distinct Ulam words.","title":"Ulam words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-small","kind":"node","lean":"MetaMathlibExt.isUlamWord_small","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-small","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"The empty word is not an Ulam word, and both words $0$ and $1$ of length one are Ulam words. The Ulam words of length two are exactly $01$ and $10$, and the Ulam words of length three are exactly $001$, $011$, $100$ and $110$.","title":"Ulam words of length at most three"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-length-four-examples","kind":"node","lean":"MetaMathlibExt.isUlamWord_length_four_examples","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-length-four-examples","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"The words $0001$ and $1011$ are Ulam words, while $0011$ and $0110$ are not.","title":"Ulam words of length four: examples"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-word-count","kind":"node","lean":"MetaMathlibExt.ulamWordCount","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-word-count","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"For a natural number $n$, $\\#U_n$ denotes the number of Ulam words of length $n$, that is, of words $w_1 \\cdots w_n$ of length $n$ in the letters $0$ and $1$ that are Ulam words.","title":"Number of Ulam words of a given length"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-word-count-small","kind":"node","lean":"MetaMathlibExt.ulamWordCount_small","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-word-count-small","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"There are $2$, $2$, $4$ and $8$ Ulam words of length $1$, $2$, $3$ and $4$ respectively: $\\#U_1 = 2$, $\\#U_2 = 2$, $\\#U_3 = 4$ and $\\#U_4 = 8$, where $\\#U_n$ is the number of Ulam words of length $n$.","title":"Numbers of short Ulam words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-residue-count","kind":"node","lean":"MetaMathlibExt.ulamResidueCount","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-residue-count","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"For natural numbers $N$ and $n$ and a residue class $a \\in \\mathbb{Z}/N\\mathbb{Z}$, this is the number of Ulam words $w$ of length $n$ in the letters $0$ and $1$ whose binary value $\\pi(w)$, read with the most significant letter first, lies in the class $a$ modulo $N$. For $N = 0$, $\\mathbb{Z}/0\\mathbb{Z}$ is $\\mathbb{Z}$ and the condition is $\\pi(w) = a$.","title":"Ulam words in a residue class"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-residue-count-three-two","kind":"node","lean":"MetaMathlibExt.ulamResidueCount_three_two","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-residue-count-three-two","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"Read the Ulam words of length $2$ as binary numerals, most significant letter first. None of their values is congruent to $0$ modulo $3$, exactly one is congruent to $1$, and exactly one is congruent to $2$.","title":"Residues of the Ulam words of length 2 modulo 3"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-values-sorted","kind":"node","lean":"MetaMathlibExt.ulamValuesSorted","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-values-sorted","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"For a natural number $n$, this is the list $u_1 < u_2 < \\dots < u_k$ of the distinct binary values $\\pi(w)$, read with the most significant letter first, of the Ulam words $w$ of length $n$, in increasing order.","title":"Sorted binary values of Ulam words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-values-sorted-three","kind":"node","lean":"MetaMathlibExt.ulamValuesSorted_three","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-values-sorted-three","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"The increasing list of the distinct binary values $\\pi(w)$, read with the most significant letter first, of the Ulam words $w$ of length $3$ has exactly the entries $1, 3, 4, 6$.","title":"Binary values of the Ulam words of length 3"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-gap-mean","kind":"node","lean":"MetaMathlibExt.ulamGapMean","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-gap-mean","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, read each Ulam word of length $n$ as a binary numeral, most significant letter first, and list the resulting values in increasing order as $u_1 < u_2 < \\dots < u_k$. The mean gap of these values is the rational number","title":"Mean gap of the Ulam values"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/discrete-sierpinski-stage","kind":"node","lean":"MetaMathlibExt.discreteSierpinskiStage","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#discrete-sierpinski-stage","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"The stages $S_n \\subseteq \\mathbb{N} \\times \\mathbb{N}$ of the discrete Sierpi\u0144ski triangle are defined by $S_0 = \\{(2, 1)\\}$ and $S_{n+1} = S_n \\cup (S_n + (2^n, 0)) \\cup (S_n + (2^n, 2^n))$, where $S + (u, v) = \\{(x + u, y + v) : (x, y) \\in S\\}$ denotes a translate.","title":"Stages of the discrete Sierpi\u0144ski triangle"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/discrete-sierpinski","kind":"node","lean":"MetaMathlibExt.discreteSierpinski","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#discrete-sierpinski","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"The discrete Sierpi\u0144ski triangle is the set $\\mathcal{S} = \\bigcup_{n \\ge 0} S_n \\subseteq \\mathbb{N} \\times \\mathbb{N}$, the union of all the stages $S_n$ of the discrete Sierpi\u0144ski triangle.","title":"The discrete Sierpi\u0144ski triangle"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/discrete-sierpinski-stage-one","kind":"node","lean":"MetaMathlibExt.discreteSierpinskiStage_one","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#discrete-sierpinski-stage-one","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"fully_proved","summary":"The stage $S_1$ of the discrete Sierpi\u0144ski triangle is $\\{(2, 1), (3, 1), (3, 2)\\}$.","title":"The first stage of the discrete Sierpi\u0144ski triangle"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-zeros-four-ones-zeros-iff","kind":"node","lean":"MetaMathlibExt.isUlamWord_zeros_four_ones_zeros_iff","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-zeros-four-ones-zeros-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For natural numbers $a$ and $b$ with $a + b \\ge 1$, the binary word $0^a 1111 0^b$ ($a$ zeros, then four ones, then $b$ zeros) is an Ulam word if and only if $a + b \\equiv 1 \\pmod 4$.","title":"Ulam words with a block of four ones"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-word-count-sq-le-sum","kind":"node","lean":"MetaMathlibExt.ulamWordCount_sq_le_sum","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-word-count-sq-le-sum","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For every natural number $n$, $(\\#U_n)^2 \\le \\sum_{i=n}^{2n} \\#U_i$, where $\\#U_i$ is the number of Ulam words of length $i$.","title":"Square of the Ulam word count is at most a block sum"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-three-ones-zeros-iff","kind":"node","lean":"MetaMathlibExt.isUlamWord_three_ones_zeros_iff","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-three-ones-zeros-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For every $n \\ge 3$, the word $1^3 0^{n-3}$ (three ones followed by $n - 3$ zeros) is an Ulam word if and only if $n \\equiv 0 \\pmod 4$ or $n \\equiv 1 \\pmod 4$.","title":"Ulam words with three leading ones"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/two-mul-add-four-le-ulam-word-count","kind":"node","lean":"MetaMathlibExt.two_mul_add_four_le_ulamWordCount","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#two-mul-add-four-le-ulam-word-count","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For every natural number $n \\ge 6$, the number of Ulam words of length $n$ is at least $2n + 4$.","title":"Linear lower bound on the number of Ulam words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-word-count-limsup-div-pow-eq-top","kind":"node","lean":"MetaMathlibExt.ulamWordCount_limsup_div_pow_eq_top","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-word-count-limsup-div-pow-eq-top","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"Let $\\#U_n$ be the number of Ulam words of length $n$ and $\\alpha_0 = (\\#U_{30} / (2 \\cdot 31))^{1/30}$. For every real $\\alpha$ with $1 < \\alpha < \\alpha_0$, every real $C$ and every natural number $N$, there is some $n \\ge N$ with $C \\alpha^n \\le \\#U_n$; that is, $\\limsup_{n \\to \\infty} \\#U_n / \\alpha^n = \\infty$.","title":"Exponential lower bound for Ulam word counts along a subsequence"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-density-inv-is-theta-ulam-gap-mean","kind":"node","lean":"MetaMathlibExt.ulamDensity_inv_isTheta_ulamGapMean","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-density-inv-is-theta-ulam-gap-mean","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $U_n$ be the set of Ulam words of length $n$, and let $\\mu_g(n)$ be the mean gap of the binary values of the words in $U_n$. Then $2^n / |U_n|$ (the reciprocal of the density $|U_n| / 2^n$) and $\\mu_g(n)$ have the same order of growth as $n \\to \\infty$: there are constants $c, C > 0$ such that $c \\, \\mu_g(n) \\le 2^n / |U_n| \\le C \\, \\mu_g(n)$ for all sufficiently large $n$.","title":"The inverse Ulam density has the order of the mean gap"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/card-ulam-zeros-one-zeros","kind":"node","lean":"MetaMathlibExt.card_ulam_zeros_one_zeros","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#card-ulam-zeros-one-zeros","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For every $n \\ge 1$, the number of $a \\in \\{0, \\dots, n - 1\\}$ for which the word $0^a 1 0^{n-1-a}$ ($a$ zeros, then a one, then $n - 1 - a$ zeros) is an Ulam word equals $2^{s_2(n-1)}$, where $s_2(m)$ is the sum of the binary digits of $m$. This number is the term $G(n - 1)$ of Gould's sequence.","title":"Counting Ulam words with a single one"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-zeros-two-ones-zeros-iff","kind":"node","lean":"MetaMathlibExt.isUlamWord_zeros_two_ones_zeros_iff","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-zeros-two-ones-zeros-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For natural numbers $a$ and $b$, the binary word $0^a 11 0^b$ ($a$ zeros, then two ones, then $b$ zeros) is an Ulam word if and only if $a + b$ is odd.","title":"Ulam words with a block of two ones"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-zeros-one-zero-one-zeros-if","kind":"node","lean":"MetaMathlibExt.isUlamWord_zeros_one_zero_one_zeros_iff","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-zeros-one-zero-one-zeros-if","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For natural numbers $a$ and $b$ with $a + b \\ge 2$, the binary word $0^a 101 0^b$ ($a$ zeros, then the letters $1, 0, 1$, then $b$ zeros) is an Ulam word if and only if $a + b$ is odd.","title":"Ulam words with one 101 block"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-zeros-one-zeros-iff","kind":"node","lean":"MetaMathlibExt.isUlamWord_zeros_one_zeros_iff","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-zeros-one-zeros-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For natural numbers $a$ and $b$, the binary word $0^a 1 0^b$ ($a$ zeros, then a one, then $b$ zeros) is an Ulam word if and only if the binomial coefficient $\\binom{a + b}{a}$ is odd.","title":"Parity criterion for Ulam words with a single one"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-ones-zeros-iff-mem-discrete","kind":"node","lean":"MetaMathlibExt.isUlamWord_ones_zeros_iff_mem_discreteSierpinski","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-ones-zeros-iff-mem-discrete","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For natural numbers $x$ and $y$ with $1 \\le y \\le x$, the word $1^y 0^{x-y}$ ($y$ ones followed by $x - y$ zeros) is an Ulam word if and only if $(x, y) = (1, 1)$ or $(x, y)$ belongs to the discrete Sierpi\u0144ski triangle $\\mathcal{S}$.","title":"Ulam words made of ones followed by zeros"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-zeros-10101-zeros-iff","kind":"node","lean":"MetaMathlibExt.isUlamWord_zeros_10101_zeros_iff","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-zeros-10101-zeros-iff","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For natural numbers $a$ and $b$ with $a + b \\ge 1$, the word $0^a 10101 0^b$ ($a$ zeros, then $10101$, then $b$ zeros) is an Ulam word if and only if $a$ and $b$ are both even and at least one of $a$ and $b$ is zero.","title":"Padding the word 10101 with zeros"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/is-ulam-word-map-not-and-ulam-residue-co","kind":"node","lean":"MetaMathlibExt.isUlamWord_map_not_and_ulamResidueCount_eq","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#is-ulam-word-map-not-and-ulam-residue-co","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For a word $w$ over $\\{0, 1\\}$, let $\\bar w$ be its complement (every letter swapped) and let $\\pi(w)$ be the integer whose binary expansion is $w$, most significant letter first. If $w$ is an Ulam word of length $n$, then $\\bar w$ is an Ulam word and $\\pi(\\bar w) + \\pi(w) = 2^n - 1$. Moreover, for all natural numbers $N$ and $n$ and every $a \\in \\mathbb{Z}/N\\mathbb{Z}$, the number of Ulam words $w$ of length $n$\u2026","title":"Complement symmetry of Ulam words"},{"id":"Combinatorics/combinatorics-on-words/finite-words/ulam-words/ulam-residue-count-mod-two-three","kind":"node","lean":"MetaMathlibExt.ulamResidueCount_mod_two_three","parent":24335,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/ulam-words/index.html#ulam-residue-count-mod-two-three","scope":"Combinatorics/combinatorics-on-words/finite-words/ulam-words","status":"can_prove","summary":"For natural numbers $N$ and $n$ and a residue $a \\in \\mathbb{Z}/N\\mathbb{Z}$, let $c_{a,N}(n)$ be the number of Ulam words of length $n$ whose value as a binary numeral, most significant letter first, is congruent to $a$ modulo $N$. Then for every $n \\ge 1$ we have $c_{0,2}(n) = c_{1,2}(n)$, and for every $a \\in \\mathbb{Z}/3\\mathbb{Z}$ we have $c_{a,3}(n) = c_{-a,3}(n)$ if $n$ is even and\u2026","title":"Residue symmetries of Ulam words modulo 2 and 3"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets.json","id":"Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets","kind":"scope","navigation":"module","parent":24205,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets/index.html","scope":"Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets","status":"can_prove","title":"Universal cycles for k-subsets"},{"id":"Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets/is-universal-cycle","kind":"node","lean":"MathlibExt.Combinatorics.UniversalCycleSubsetsWanted.IsUniversalCycle","parent":24364,"reading_title":{"format":"inline-parts-v1","parts":[["text","Universal cycle for "],["math","\\(k\\)"],["text","-subsets"]],"source":"Universal cycle for $k$-subsets"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets/index.html#is-universal-cycle","scope":"Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets","status":"fully_proved","summary":"For natural numbers $n$ and $k$, let $N = \\binom{n}{k}$. A sequence $c_0, \\dots, c_{N-1}$ of elements of $\\{0, \\dots, n-1\\}$, read cyclically, is a universal cycle for the $k$-subsets of $\\{0, \\dots, n-1\\}$ when, for every starting position $i < N$, the $k$ consecutive entries $c_i, c_{i+1}, \\dots, c_{i+k-1}$ (indices taken modulo $N$) are pairwise distinct, and every $k$-element subset of $\\{0, \\dots, n-1\\}$ is\u2026","title":"Universal cycle for $k$-subsets"},{"id":"Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.UniversalCycleSubsetsWanted.conjecture","parent":24364,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets/index.html#conjecture","scope":"Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets","status":"fully_proved","summary":"The statement that for every natural number $k$ there is a natural number $N$ such that every $n \\ge N$ for which $k$ divides $\\binom{n-1}{k-1}$ admits a universal cycle for the $k$-subsets of $\\{0, \\dots, n-1\\}$. Here $n - 1$ and $k - 1$ are truncated at $0$, so for $k = 0$ the divisibility condition reads $0 \\mid 1$ and never holds.","title":"Chung\u2013Diaconis\u2013Graham universal cycle conjecture"},{"id":"Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.UniversalCycleSubsetsWanted.conjecture_holds","parent":24364,"reading_title":{"format":"inline-parts-v1","parts":[["text","Existence of universal cycles for "],["math","\\(k\\)"],["text","-subsets"]],"source":"Existence of universal cycles for $k$-subsets"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets/index.html#conjecture-holds","scope":"Combinatorics/combinatorics-on-words/finite-words/universal-cycles-for-k-subsets","status":"can_prove","summary":"For every natural number $k$ there is a natural number $N$ such that for every $n \\ge N$ for which $k$ divides $\\binom{n-1}{k-1}$, there is a universal cycle for the $k$-subsets of $\\{0, \\dots, n-1\\}$. Here $n - 1$ and $k - 1$ are truncated at $0$, so for $k = 0$ the divisibility condition reads $0 \\mid 1$ and never holds.","title":"Existence of universal cycles for $k$-subsets"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences","kind":"scope","parent":23782,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences","status":"can_prove","summary":"Integer sequences defined by self-reference, greedy or mex rules: Kolakoski, Gijswijt (curling numbers), Cloitre, Connell, Aronson, Smarandache, van der Laan, throwback and anti-recurrence sequences, self-generating sets, fractal sequences and inventory loops.","title":"Self-describing and self-generating sequences"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","title":"Anti-recurrence and greedy subsum-free sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-mex","kind":"node","lean":"MetaMathlibExt.antiTribonacciMex","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-mex","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For a finite set $s$ of natural numbers, this is the least positive integer not in $s$ (the source's $\\mathrm{mex}$ over the positive integers, used to build the anti-Tribonacci sequence). It always lies among $1, \\dots, |s| + 1$.","title":"Least positive integer missing from a finite set"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-mex-example","kind":"node","lean":"MetaMathlibExt.antiTribonacciMex_example","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mex of "],["math","\\(\\{1, 2, 3, 6\\}\\)"]],"source":"Mex of $\\{1, 2, 3, 6\\}$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-mex-example","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The least positive integer not in $\\{1, 2, 3, 6\\}$ is $4$.","title":"Mex of $\\{1, 2, 3, 6\\}$"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-mex-pos","kind":"node","lean":"MetaMathlibExt.antiTribonacciMex_pos","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-mex-pos","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Write $\\mathrm{mex}(s)$ for the least positive integer not in a finite set $s$ of natural numbers. For every such $s$, $\\mathrm{mex}(s) > 0$.","title":"The mex of a finite set is positive"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-mex-not-mem","kind":"node","lean":"MetaMathlibExt.antiTribonacciMex_notMem","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-mex-not-mem","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Write $\\mathrm{mex}(s)$ for the least positive integer not in a finite set $s$ of natural numbers. For every such $s$, $\\mathrm{mex}(s) \\notin s$.","title":"The mex of a finite set lies outside the set"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-mex-le-of-not-mem","kind":"node","lean":"MetaMathlibExt.antiTribonacciMex_le_of_notMem","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-mex-le-of-not-mem","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Write $\\mathrm{mex}(s)$ for the least positive integer not in a finite set $s$ of natural numbers. If $m$ is a positive integer with $m \\notin s$, then $\\mathrm{mex}(s) \\le m$.","title":"The mex is at most every missing positive integer"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-state","kind":"node","lean":"MetaMathlibExt.antiTribonacciState","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-state","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Write $\\mathrm{mex}(U)$ for the least positive integer not in a finite set $U$ of natural numbers. Set $(A_1, B_1, C_1, D_1) = (1, 2, 3, 6)$ and $U_1 = \\{1, 2, 3, 6\\}$. For $k \\ge 1$ let $A_{k+1} = \\mathrm{mex}(U_k)$, $B_{k+1} = \\mathrm{mex}(U_k \\cup \\{A_{k+1}\\})$, $C_{k+1} = \\mathrm{mex}(U_k \\cup \\{A_{k+1}, B_{k+1}\\})$, $D_{k+1} = A_{k+1} + B_{k+1} + C_{k+1}$ and\u2026","title":"Anti-Tribonacci construction"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-a","kind":"node","lean":"MetaMathlibExt.antiTribonacciA","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-a","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For $n \\ge 1$, $A_n$ is the first entry of the $n$-th quadruple $(A_n, B_n, C_n, D_n)$ of the anti-Tribonacci construction: $A_1 = 1$, and for $n \\ge 2$, $A_n$ is the least positive integer that is not among the values $A_k$, $B_k$, $C_k$, $D_k$ with $1 \\le k < n$. At $n = 0$ the definition gives $A_0 = A_1 = 1$.","title":"First summands of the anti-Tribonacci construction"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-b","kind":"node","lean":"MetaMathlibExt.antiTribonacciB","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-b","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For $n \\ge 1$, $B_n$ is the second entry of the $n$-th quadruple $(A_n, B_n, C_n, D_n)$ of the anti-Tribonacci construction: $B_1 = 2$, and for $n \\ge 2$, $B_n$ is the second smallest positive integer that is not among the values $A_k$, $B_k$, $C_k$, $D_k$ with $1 \\le k < n$. At $n = 0$ the definition gives $B_0 = B_1 = 2$.","title":"Second summands of the anti-Tribonacci construction"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-c","kind":"node","lean":"MetaMathlibExt.antiTribonacciC","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-c","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For $n \\ge 1$, $C_n$ is the third entry of the $n$-th quadruple $(A_n, B_n, C_n, D_n)$ of the anti-Tribonacci construction: $C_1 = 3$, and for $n \\ge 2$, $C_n$ is the third smallest positive integer that is not among the values $A_k$, $B_k$, $C_k$, $D_k$ with $1 \\le k < n$. At $n = 0$ the definition gives $C_0 = C_1 = 3$.","title":"Third summands of the anti-Tribonacci construction"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-d","kind":"node","lean":"MetaMathlibExt.antiTribonacciD","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-d","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For $n \\ge 1$, the anti-Tribonacci number $D_n$ (OEIS A265389) is the fourth entry of the $n$-th quadruple $(A_n, B_n, C_n, D_n)$ of the anti-Tribonacci construction, so $D_n = A_n + B_n + C_n$ and $D_1 = 6$. At $n = 0$ the definition gives $D_0 = D_1 = 6$.","title":"Anti-Tribonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-tribonacci-d-small","kind":"node","lean":"MetaMathlibExt.antiTribonacciD_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-tribonacci-d-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Writing $D_n$ for the $n$-th anti-Tribonacci number, the first four anti-Tribonacci numbers are $D_1 = 6$, $D_2 = 16$, $D_3 = 27$ and $D_4 = 36$.","title":"First anti-Tribonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/greedy-subsumfree-below","kind":"node","lean":"MetaMathlibExt.greedySubsumfreeBelow","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","Members of a greedy "],["math","\\(3\\)"],["text","-subsumfree sequence below a bound"]],"source":"Members of a greedy $3$-subsumfree sequence below a bound"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#greedy-subsumfree-below","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For natural numbers $x$, $y$ and $z$ (the source takes $x < y < z$), let $S_{x,y,z}$ consist of $x$, $y$, $z$ and every natural number $n > z$ that is not a sum $a + b + c$ of members $a < b < c$ of $S_{x,y,z}$ smaller than $n$. This definition gives, for each natural number $m$, the finite set of members of $S_{x,y,z}$ below $m$.","title":"Members of a greedy $3$-subsumfree sequence below a bound"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/greedy-subsumfree-below-one-two-three","kind":"node","lean":"MetaMathlibExt.greedySubsumfreeBelow_one_two_three","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","First members of "],["math","\\(S_{1,2,3}\\)"]],"source":"First members of $S_{1,2,3}$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#greedy-subsumfree-below-one-two-three","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The members of the greedy $3$-subsumfree sequence $S_{1,2,3}$ below $26$ are exactly $1, 2, 3, 4, 5, 13, 14, 15, 25$.","title":"First members of $S_{1,2,3}$"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/coeff-not-dvd-three-shape","kind":"node","lean":"MetaMathlibExt.coeff_not_dvd_three_shape","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hajdu\u2013Tijdeman\u2013Varga theorem on "],["math","\\((x-1)^c (x+1)^d\\)"],["text"," modulo "],["math","\\(3\\)"]],"source":"Hajdu\u2013Tijdeman\u2013Varga theorem on $(x-1)^c (x+1)^d$ modulo $3$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#coeff-not-dvd-three-shape","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $c$ and $d$ be natural numbers. If none of the coefficients of $x^0, x^1, \\dots, x^{c+d}$ in the integer polynomial $(x-1)^c (x+1)^d$ is divisible by $3$, then there are natural numbers $i > j$ such that $c + d + 1$ equals $3^j$, $2 \\cdot 3^j$, $3^i + 3^j$ or $2 \\cdot 3^i + 3^j$.","title":"Hajdu\u2013Tijdeman\u2013Varga theorem on $(x-1)^c (x+1)^d$ modulo $3$"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/kimberling-anti-tribonacci","kind":"node","lean":"MetaMathlibExt.kimberling_antiTribonacci","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#kimberling-anti-tribonacci","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $A_1 = 1$, $B_1 = 2$, $C_1 = 3$ and $D_1 = 6$, and for $n \\ge 1$ let $A_{n+1} < B_{n+1} < C_{n+1}$ be the three smallest positive integers that are not among $A_i, B_i, C_i, D_i$ for $i \\le n$, and $D_{n+1} = A_{n+1} + B_{n+1} + C_{n+1}$ (the $D_n$ are the anti-Tribonacci numbers). For every integer $n \\ge 1$, $0 \\le 10n - 6 - 3A_n \\le 2$, $0 \\le 10n - 2 - 3B_n \\le 3$, $0 \\le 10n + 1 - 3C_n \\le 3$ and\u2026","title":"Kimberling's conjecture on the anti-Tribonacci sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/mem-greedy-subsumfree-iff","kind":"node","lean":"MetaMathlibExt.mem_greedySubsumfree_iff","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#mem-greedy-subsumfree-iff","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"For $x < y < z$, the greedy $3$-subsumfree sequence $S_{x,y,z}$ consists of $x$, $y$, $z$ and every $m > z$ that is not the sum of three different members of $S_{x,y,z}$ smaller than $m$. For every positive integer $n$: $n \\in S_{1,2,3}$ if and only if $n \\in \\{1, 5, 13\\}$ or $n \\bmod 23 \\in \\{2, 3, 4, 14, 15\\}$; $n \\in S_{1,3,4}$ if and only if $n \\in \\{1, 7, 19\\}$ or $n \\bmod 33 \\in \\{3, 4, 5, 6, 20, 21, 22\\}$;\u2026","title":"Membership in three greedy 3-subsumfree sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/is-upper-wythoff-unsum","kind":"node","lean":"MetaMathlibExt.IsUpperWythoffUnsum","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#is-upper-wythoff-unsum","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"A positive integer $p$ is an *upper Wythoff number* when the letter at index $p - 1$ (indices start at $0$) of the infinite Fibonacci word $0100101001001\\ldots$ is $1$; that is, the upper Wythoff numbers are the positions of the letter $1$ in the Fibonacci word counted from one. A positive integer $x$ is an *unsum* (OEIS A260317) if there are no upper Wythoff numbers $p < q$ with $p + q = x$.","title":"Unsums of upper Wythoff numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-small","kind":"node","lean":"MetaMathlibExt.upperWythoff_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The positive integers $p < 30$ such that the letter at index $p - 1$ (indices start at $0$) of the infinite Fibonacci word is $1$, that is, the upper Wythoff numbers below $30$, are exactly $2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28$.","title":"The upper Wythoff numbers below 30"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/is-upper-wythoff-unsum-small","kind":"node","lean":"MetaMathlibExt.isUpperWythoffUnsum_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#is-upper-wythoff-unsum-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The unsums (OEIS A260317) that are at most $21$ are exactly $1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 14, 16, 19, 21$.","title":"The unsums up to 21"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-gap","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumGap","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-gap","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Let $a_1 < a_2 < \\cdots$ be the unsums (OEIS A260317): the positive integers that are not a sum $p + q$ of upper Wythoff numbers $p < q$. For $i \\ge 0$, the $i$-th gap between consecutive unsums (counted from zero) is $a_{i+2} - a_{i+1}$; this is OEIS A260311 shifted by one.","title":"Gaps between unsums"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-gap-small","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumGap_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-gap-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Let $a_1 < a_2 < \\cdots$ be the unsums (OEIS A260317) and let $g_i = a_{i+2} - a_{i+1}$ for $i \\ge 0$. Then the gaps $g_0, g_1, \\dots, g_{12}$ are $1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 3, 2$.","title":"The first thirteen gaps between unsums"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/is-upper-wythoff-unsum-marker","kind":"node","lean":"MetaMathlibExt.IsUpperWythoffUnsumMarker","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#is-upper-wythoff-unsum-marker","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"A natural number $i$ is a *marker* if $i \\ge 10$ and the $i$-th gap between consecutive unsums of upper Wythoff numbers (counted from zero) equals $2$.","title":"Markers among the unsum gaps"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-marker","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumMarker","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-marker","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Call a gap index $i$ a marker if $i \\ge 10$ and the $i$-th gap between consecutive unsums (counted from zero) equals $2$. For $j \\ge 0$, the $j$-th marker is the $j$-th marker in increasing order, counted from zero, so the $0$-th marker is the smallest one.","title":"Markers in the gaps between unsums"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-word","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumWord","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-word","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Let $g_0, g_1, g_2, \\ldots$ be the gaps between consecutive unsums, and call an index $i \\ge 10$ with $g_i = 2$ a marker. Let $m_0 < m_1 < m_2 < \\cdots$ be the markers in increasing order. For each $j$, the $j$-th word between markers is $g_{m_j + 1} g_{m_j + 2} \\cdots g_{m_{j+1} - 1}$, the word of the gaps strictly between the $j$-th and the $(j+1)$-th marker.","title":"Words between markers in the unsum gap sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-palindrome","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumPalindrome","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","The words "],["math","\\(W_i\\)"],["text"," of the unsum gap sequence"]],"source":"The words $W_i$ of the unsum gap sequence"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-palindrome","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For $i \\ge 1$, the word $W_i$ is the word of gaps strictly between the $(2i-2)$-th and $(2i-1)$-th markers of the unsum gap sequence, with markers counted from zero; for $i = 0$ the definition gives the same word as for $i = 1$.","title":"The words $W_i$ of the unsum gap sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-marker-zero-one","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumMarker_zero_one","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-marker-zero-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Call a gap index $i$ a marker if $i \\ge 10$ and the $i$-th gap between consecutive unsums (counted from zero) equals $2$. The smallest marker is $10$ and the second smallest is $12$.","title":"The first two markers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-palindrome-one","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumPalindrome_one","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-palindrome-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Let $g_k$ be the $k$-th gap between consecutive unsums (OEIS A260317), counted from zero, and let $m_0 < m_1 < \\cdots$ be the markers, the indices $k \\ge 10$ with $g_k = 2$, in increasing order. The word $W_1$ of the gaps $g_k$ with $m_0 < k < m_1$, in increasing order of $k$, consists of the single digit $3$.","title":"The first palindrome"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-gap-ne-one","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumGap_ne_one","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-gap-ne-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $a_1 < a_2 < \\cdots$ be the unsums (OEIS A260317), and for $i \\ge 0$ let $g_i = a_{i+2} - a_{i+1}$ be the $i$-th gap between consecutive unsums, counted from zero. Then $g_9 = 1$, and $g_i \\ne 1$ for every $i \\ge 10$.","title":"The last gap of size one between unsums"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-word-sum-add-two-is","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumWord_sum_add_two_isFib","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-word-sum-add-two-is","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"For every $j$, the digit sum of the $j$-th word between markers of the unsum gap sequence, plus $2$, is a Fibonacci number: it equals $F_m$ for some natural number $m$.","title":"Digit sums between markers are two less than Fibonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-word-sum-add-two","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumWord_sum_add_two","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-word-sum-add-two","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $F_n$ be the Fibonacci numbers, with $F_0 = 0$ and $F_1 = 1$. For every $j$ (words counted from zero), the digit sum of the $j$-th word between markers of the unsum gap sequence, plus $2$, equals $F_{\\lfloor j/2 \\rfloor + 5}$.","title":"Digit sums of the words between markers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-palindrome-reverse","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumPalindrome_reverse","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-palindrome-reverse","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $g_k$ be the $k$-th gap between consecutive unsums (OEIS A260317), counted from zero, and let $m_0 < m_1 < \\cdots$ be the markers, the indices $k \\ge 10$ with $g_k = 2$, in increasing order. For $i \\ge 1$ let $W_i$ be the word of the gaps $g_k$ with $m_{2i-2} < k < m_{2i-1}$, in increasing order of $k$. Then for every $i \\ge 1$, the word $W_i$ read backwards is $W_i$ itself.","title":"The gap words are palindromes"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-palindrome-head-get","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumPalindrome_head_getLast","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-palindrome-head-get","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $g_k$ be the $k$-th gap between consecutive unsums (OEIS A260317), counted from zero, and let $m_0 < m_1 < \\cdots$ be the markers, the indices $k \\ge 10$ with $g_k = 2$, in increasing order. For $i \\ge 1$ let $W_i$ be the word of the gaps $g_k$ with $m_{2i-2} < k < m_{2i-1}$, in increasing order of $k$. Then for every $i \\ge 1$, the word $W_i$ is nonempty, its first digit is $3$, and its last digit is $3$.","title":"The palindromes begin and end with the digit 3"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-word-prefix-next","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumWord_prefix_next","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","A word "],["math","\\(3V3\\)"],["text"," between markers is followed by a word starting with "],["math","\\(3V\\)"]],"source":"A word $3V3$ between markers is followed by a word starting with $3V$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-word-prefix-next","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $V$ be a finite word of natural numbers, and suppose that the $j$-th word between markers of the unsum gap sequence is $3V3$. Then $3V$ is a prefix of the $(j+1)$-th word between markers, and if the $(j+1)$-th word has the same digit sum as the $j$-th word, then the two words are equal.","title":"A word $3V3$ between markers is followed by a word starting with $3V$"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-palindrome-is-prefix","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumPalindrome_isPrefix_succ_iff","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-palindrome-is-prefix","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $g_k$ be the $k$-th gap between consecutive unsums (OEIS A260317), counted from zero, and let $m_0 < m_1 < \\cdots$ be the markers, the indices $k \\ge 10$ with $g_k = 2$, in increasing order. For $i \\ge 1$ let $W_i$ be the word of the gaps $g_k$ with $m_{2i-2} < k < m_{2i-1}$, in increasing order of $k$. Then for every $i \\ge 1$, the word $W_i$ is a prefix of $W_{i+1}$ if and only if $i$ is odd.","title":"When one palindrome is a prefix of the next"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-palindrome-succ","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumPalindrome_succ","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recursion for the words "],["math","\\(W_i\\)"],["text"," of the unsum gap sequence"]],"source":"Recursion for the words $W_i$ of the unsum gap sequence"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-palindrome-succ","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"For each $i$ let $V_i$ be the word $W_i$ with its first and last digit removed, write $|W|$ for the length of a word $W$, and denote concatenation of words by juxtaposition. For every $i \\ge 2$: if $i$ is odd, then $W_{i+1} = W_i \\, 5 \\, V_{i-1} \\, 3$; if $i$ is even, then $W_{i+1} = 3 \\, V_i \\, 5 \\, W_{i-1}$; and in both cases $|W_{i+1}| = |W_i| + |W_{i-1}|$. Moreover $|W_1| = 1$, $|W_2| = 2$, and\u2026","title":"Recursion for the words $W_i$ of the unsum gap sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/upper-wythoff-unsum-palindrome-tendsto","kind":"node","lean":"MetaMathlibExt.upperWythoffUnsumPalindrome_tendsto","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","The words "],["math","\\(W_i\\)"],["text"," converge to "],["math","\\(3\\)"],["text"," followed by the coded Fibonacci word"]],"source":"The words $W_i$ converge to $3$ followed by the coded Fibonacci word"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#upper-wythoff-unsum-palindrome-tendsto","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $u_0 u_1 u_2 \\cdots$ be the infinite Fibonacci word over $\\{0, 1\\}$, indexed from $0$, and let $\\mathbf{V}$ be the infinite word obtained from it by replacing each $0$ by $5$ and each $1$ by $3$. Then the words $W_i$ of the unsum gap sequence converge to the infinite word $3\\mathbf{V}$: for every $k \\ge 0$ there is $N$ such that for every $i \\ge N$ the word $W_i$ has more than $k$ digits and its digit at\u2026","title":"The words $W_i$ converge to $3$ followed by the coded Fibonacci word"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/period-doubling-word","kind":"node","lean":"MetaMathlibExt.periodDoublingWord","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#period-doubling-word","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Write the period-doubling substitution $a \\mapsto ab$, $b \\mapsto aa$ over the letters $a = 1$ and $b = 0$, so that it sends $1$ to $10$ and $0$ to $11$. The $n$-th period-doubling iterate is the word obtained by applying this substitution $n$ times to the one-letter word $1$; the iterates begin $1$, $10$, $1011$, $10111010$, and their limit is the period-doubling word (Walnut's PD).","title":"Iterates of the period-doubling substitution"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/period-doubling","kind":"node","lean":"MetaMathlibExt.periodDoubling","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#period-doubling","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For a natural number $k$, the letter at index $k$ (indices start at $0$) of the period-doubling word is the letter at index $k$ of the word obtained by applying the period-doubling substitution $1 \\mapsto 10$, $0 \\mapsto 11$ to the one-letter word $1$ a total of $\\lfloor \\log_2 k \\rfloor + 1$ times, where $\\lfloor \\log_2 0 \\rfloor$ is taken to be $0$. That word has length\u2026","title":"The period-doubling word"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/length-period-doubling-word","kind":"node","lean":"MetaMathlibExt.length_periodDoublingWord","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#length-period-doubling-word","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For every natural number $n$, the word obtained by applying the period-doubling substitution $1 \\mapsto 10$, $0 \\mapsto 11$ to the one-letter word $1$ a total of $n$ times has length $2^n$.","title":"Length of the period-doubling iterates"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/period-doubling-word-get-d","kind":"node","lean":"MetaMathlibExt.periodDoublingWord_getD","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#period-doubling-word-get-d","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Let $w_m$ be the word obtained by applying the period-doubling substitution $1 \\mapsto 10$, $0 \\mapsto 11$ to the one-letter word $1$ a total of $m$ times. For all natural numbers $m$ and $k$ with $k < 2^m$, the letter of $w_m$ at index $k$ (indices start at $0$) equals the letter of the period-doubling word at index $k$.","title":"Letters of the period-doubling iterates"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/period-doubling-small","kind":"node","lean":"MetaMathlibExt.periodDoubling_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#period-doubling-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The letters at indices $0, 1, \\ldots, 7$ of the period-doubling word (the fixed point of $1 \\mapsto 10$, $0 \\mapsto 11$ that starts with $1$) are $1, 0, 1, 1, 1, 0, 1, 0$.","title":"First letters of the period-doubling word"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-fibonacci-state","kind":"node","lean":"MetaMathlibExt.antiFibonacciState","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-fibonacci-state","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For each $n \\ge 0$, entry $n$ of the construction is the triple $(A_{n+1}, B_{n+1}, C_{n+1})$ of natural numbers together with the finite set $U_{n+1}$ of values used so far. Entry $0$ is $(1, 2, 3)$ with $U_1 = \\{1, 2, 3\\}$. For $n \\ge 1$, $A_{n+1}$ is the least positive integer not in $U_n$, $B_{n+1}$ is the least positive integer not in $U_n \\cup \\{A_{n+1}\\}$, $C_{n+1} = A_{n+1} + B_{n+1}$, and\u2026","title":"The anti-Fibonacci construction"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-fibonacci-a","kind":"node","lean":"MetaMathlibExt.antiFibonacciA","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-fibonacci-a","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For $n \\ge 1$, $A_n$ is the first entry of the $n$-th triple $(A_n, B_n, C_n)$ of the anti-Fibonacci construction: $A_1 = 1$, and for $n \\ge 2$, $A_n$ is the least positive integer that is not among the values $A_k$, $B_k$, $C_k$ with $1 \\le k < n$. At $n = 0$ the definition gives $A_0 = A_1 = 1$.","title":"First summands of the anti-Fibonacci construction"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-fibonacci-b","kind":"node","lean":"MetaMathlibExt.antiFibonacciB","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-fibonacci-b","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For $n \\ge 1$, $B_n$ is the second entry of the $n$-th triple $(A_n, B_n, C_n)$ of the anti-Fibonacci construction: $B_1 = 2$, and for $n \\ge 2$, $B_n$ is the second smallest positive integer that is not among the values $A_k$, $B_k$, $C_k$ with $1 \\le k < n$. At $n = 0$ the definition gives $B_0 = B_1 = 2$.","title":"Second summands of the anti-Fibonacci construction"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-fibonacci-c","kind":"node","lean":"MetaMathlibExt.antiFibonacciC","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-fibonacci-c","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For $n \\ge 1$, the anti-Fibonacci number $X_n = C_n$ is the third entry of the $n$-th triple $(A_n, B_n, C_n)$ of the anti-Fibonacci construction, so $C_n = A_n + B_n$ and $C_1 = 3$. At $n = 0$ the definition gives $C_0 = C_1 = 3$.","title":"Anti-Fibonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-fibonacci-small","kind":"node","lean":"MetaMathlibExt.antiFibonacci_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-fibonacci-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"In the anti-Fibonacci construction with triples $(A_n, B_n, C_n)$, the first four anti-Fibonacci numbers are $C_1 = 3$, $C_2 = 9$, $C_3 = 13$ and $C_4 = 18$, and $(A_1, B_1) = (1, 2)$, $(A_2, B_2) = (4, 5)$, $(A_3, B_3) = (6, 7)$ and $(A_4, B_4) = (8, 10)$.","title":"First anti-Fibonacci and non-anti-Fibonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/zaslavsky-anti-fibonacci","kind":"node","lean":"MetaMathlibExt.zaslavsky_antiFibonacci","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#zaslavsky-anti-fibonacci","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $A_1 = 1$, $B_1 = 2$, $C_1 = 3$, and for $n \\ge 1$ let $A_{n+1} < B_{n+1}$ be the two smallest positive integers not among $A_1, B_1, C_1, \\ldots, A_n, B_n, C_n$ and $C_{n+1} = A_{n+1} + B_{n+1}$. Let $\\mathrm{PD}[0], \\mathrm{PD}[1], \\ldots$ be the letters of the period-doubling word, the fixed point of $1 \\mapsto 10$, $0 \\mapsto 11$ starting with $1$. Then for every $k \\ge 0$, $C_{2k+1} = 10k + 3$,\u2026","title":"Zaslavsky's formulas for the anti-Fibonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-block","kind":"node","lean":"MetaMathlibExt.antiRecurrenceBlock","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-block","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"For a natural number $m$ and a finite set $s$ of natural numbers, the mex block of length $m$ from $s$ is the list $(x_1, \\dots, x_m)$ in which each $x_j$ is the least positive integer not in $s \\cup \\{x_1, \\dots, x_{j-1}\\}$.","title":"Block of successive mex values"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-state","kind":"node","lean":"MetaMathlibExt.antiRecurrenceState","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-state","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Let $a = (a_1, \\dots, a_k)$ be a finite list of natural numbers. The state at step $0$ is the pair of the empty list and the empty set. If the state at step $n$ has set $S$, then the state at step $n + 1$ is the pair $(\\beta, S')$, where $\\beta = (x_1, \\dots, x_k)$ is the mex block of length $k$ from $S$ (each $x_j$ is the least positive integer not in $S \\cup \\{x_1, \\dots, x_{j-1}\\}$) and\u2026","title":"State of the anti-recurrence construction"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence","kind":"node","lean":"MetaMathlibExt.antiRecurrence","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Let $a = (a_1, \\dots, a_k)$ be a linear form with natural number coefficients. The non-anti-recurrence numbers $B_1, B_2, \\dots$ are produced in blocks of $k$ by the mex rule, and each block $B_{(n-1)k+1}, \\dots, B_{nk}$ generates the anti-recurrence number $A_n = \\sum_{j=1}^{k} a_j B_{(n-1)k+j}$. Here $B_m$ is the least positive integer that differs from $B_1, \\dots, B_{m-1}$ and from the anti-recurrence numbers\u2026","title":"Anti-recurrence numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-complement","kind":"node","lean":"MetaMathlibExt.antiRecurrenceComplement","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-complement","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"Let $a = (a_1, \\dots, a_k)$ be a linear form with natural number coefficients. Its non-anti-recurrence numbers $B_1, B_2, \\dots$ are produced in blocks of $k$ by the mex rule: for $n \\ge 1$, each of $B_{(n-1)k+1}, \\dots, B_{nk}$ in turn is the least positive integer not among $B_1, \\dots, B_{(n-1)k}$, the anti-recurrence numbers $A_1, \\dots, A_{n-1}$ and the earlier entries of its block, and then\u2026","title":"Non-anti-recurrence numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-anti-fibonacci-small","kind":"node","lean":"MetaMathlibExt.antiRecurrence_antiFibonacci_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-anti-fibonacci-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The anti-recurrence numbers $A_1, A_2, A_3, A_4$ of the linear form $(1, 1)$ are $3, 9, 13, 18$, and the first four anti-Fibonacci numbers $C_1, C_2, C_3, C_4$ of the Walnut anti-Fibonacci construction (OEIS A075326) are also $3, 9, 13, 18$; so the two sequences agree for $1 \\le n \\le 4$.","title":"First anti-Fibonacci anti-recurrence numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-anti-pell-small","kind":"node","lean":"MetaMathlibExt.antiRecurrence_antiPell_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-anti-pell-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The anti-recurrence numbers $A_1, A_2, A_3, A_4$ of the linear form $(1, 2)$, the anti-Pell numbers (OEIS A304502), are $5, 11, 20, 26$, and its first eight non-anti-recurrence numbers $B_1, \\dots, B_8$ are $1, 2, 3, 4, 6, 7, 8, 9$ (so $B^1_n = B_{2n-1}$ starts $1, 3, 6, 8$ and $B^2_n = B_{2n}$ starts $2, 4, 7, 9$).","title":"First anti-Pell numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-anti-jacobsthal-small","kind":"node","lean":"MetaMathlibExt.antiRecurrence_antiJacobsthal_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-anti-jacobsthal-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The anti-recurrence numbers $A_1, A_2, A_3, A_4$ of the linear form $(2, 1)$, the anti-Jacobsthal numbers (OEIS A304499), are $4, 11, 19, 25$.","title":"First anti-Jacobsthal numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-anti-tribonacci-small","kind":"node","lean":"MetaMathlibExt.antiRecurrence_antiTribonacci_small","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-anti-tribonacci-small","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"fully_proved","summary":"The anti-recurrence numbers $A_1, A_2, A_3, A_4$ of the linear form $(1, 1, 1)$ are $6, 16, 27, 36$, and the first four anti-Tribonacci numbers $D_1, D_2, D_3, D_4$ of the Walnut anti-Tribonacci construction (OEIS A265389) are also $6, 16, 27, 36$; so the two sequences agree for $1 \\le n \\le 4$.","title":"First anti-Tribonacci anti-recurrence numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-add-length-mul-sum-le","kind":"node","lean":"MetaMathlibExt.antiRecurrence_add_length_mul_sum_le","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-add-length-mul-sum-le","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $a = (a_1, \\dots, a_k)$ be a linear form with positive integer coefficients and $k > 1$, let $\\tau = a_1 + \\dots + a_k$ be its trace, and let $A_n$ and $B_m$ be the anti-recurrence and non-anti-recurrence numbers of $a$; for $m \\ge 1$, block $m$ consists of $B_{(m-1)k+1}, \\dots, B_{mk}$, and it is an interval when $B_{mk} = B_{(m-1)k+1} + k - 1$. Then for every $n \\ge 1$ we have $A_{n+1} \\ge A_n + k\\tau$.\u2026","title":"Gaps between successive anti-recurrence numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-unique","kind":"node","lean":"MetaMathlibExt.antiRecurrence_unique","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-unique","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $a = (a_1, \\dots, a_k)$ be a linear form with positive integer coefficients and $k > 1$, with anti-recurrence numbers $A_1, A_2, \\dots$ and non-anti-recurrence numbers $B_1, B_2, \\dots$. Let $(x_i)_{i \\ge 0}$ and $(y_i)_{i \\ge 0}$ be sequences of natural numbers. Then $x$ and $y$ are strictly increasing, every $y_i$ is positive, every positive integer is a term of exactly one of $x$ and $y$, and\u2026","title":"Characterization of anti-recurrence sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-bonacci-mem-icc-and-mod-eq","kind":"node","lean":"MetaMathlibExt.antiBonacci_mem_Icc_and_modEq","parent":24369,"reading_title":{"format":"inline-parts-v1","parts":[["text","Intervals and residues of the anti-"],["math","\\(k\\)"],["text","-bonacci numbers"]],"source":"Intervals and residues of the anti-$k$-bonacci numbers"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-bonacci-mem-icc-and-mod-eq","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $k > 1$, let $A_n$ be the anti-recurrence numbers of the linear form $(1, \\dots, 1)$ of length $k$, and put $i = \\lfloor k/2 \\rfloor$ and $\\kappa = k^2 + 1$. Then for every $n \\ge 1$, $A_n$ lies in the interval $[\\kappa(n - 1) + 1, \\kappa n]$, and $A_n \\equiv ik + c \\pmod{\\kappa}$ for some natural number $c$ with $1 \\le c \\le k$ if $k$ is even and $i + 1 \\le c \\le i + k$ if $k$ is odd.","title":"Intervals and residues of the anti-$k$-bonacci numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-one-ge","kind":"node","lean":"MetaMathlibExt.antiRecurrence_one_ge","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-one-ge","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $a = (a_1, \\dots, a_k)$ be a linear form with positive integer coefficients, of trace $\\tau = a_1 + \\dots + a_k$, and let $A_1$ be its first anti-recurrence number. Then $\\tau + (k - 1)k/2 \\le A_1$, and the inequality is strict if and only if $a_j > 1$ for some $j$ with $2 \\le j \\le k$.","title":"Lower bound for the first anti-recurrence number"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-mem-icc-of-bounded","kind":"node","lean":"MetaMathlibExt.antiRecurrence_mem_Icc_of_bounded","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-mem-icc-of-bounded","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $a = (a_1, \\dots, a_k)$ be a linear form with positive integer coefficients and $k > 1$, of trace $\\tau = a_1 + \\dots + a_k$, with anti-recurrence numbers $A_1, A_2, \\dots$ and non-anti-recurrence numbers $B_1, B_2, \\dots$, and let $\\kappa = k\\tau + 1$. Suppose that $a$ is $A_1$-bounded, that is, $A_1 \\le (k - 1)\\tau + 2$. Then for every $n \\ge 1$, $A_n$ lies in the interval\u2026","title":"Interval location of anti-recurrence numbers for bounded forms"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-pell-kimberling-bounds","kind":"node","lean":"MetaMathlibExt.antiPell_kimberling_bounds","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-pell-kimberling-bounds","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $A_n$ be the anti-recurrence numbers of the linear form $a = (1, 2)$ (the anti-Pell numbers), let $B_m$ be its non-anti-recurrence numbers, and put $B^1_n = B_{2n-1}$ and $B^2_n = B_{2n}$. Then for every $n \\ge 1$ we have $0 \\le A_n - 7n + 3 \\le 2$, $0 \\le 3B^1_n - 7n + 6 \\le 3$ and $0 \\le 3B^2_n - 7n + 2 \\le 3$.","title":"Kimberling's bounds for the anti-Pell numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-jacobsthal-kimberling-bounds","kind":"node","lean":"MetaMathlibExt.antiJacobsthal_kimberling_bounds","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-jacobsthal-kimberling-bounds","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $A_n$ be the anti-recurrence numbers of the linear form $a = (2, 1)$ (the anti-Jacobsthal numbers), let $B_m$ be its non-anti-recurrence numbers, and put $B^1_n = B_{2n-1}$ and $B^2_n = B_{2n}$. Then for every $n \\ge 1$ we have $0 \\le A_n - 7n + 4 \\le 3$, $0 \\le 3B^1_n - 7n + 6 \\le 4$ and $0 \\le 3B^2_n - 7n + 2 \\le 3$.","title":"Kimberling's bounds for the anti-Jacobsthal numbers"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/anti-recurrence-sub-is-kautomatic-of-bou","kind":"node","lean":"MetaMathlibExt.antiRecurrence_sub_isKAutomatic_of_bounded","parent":24369,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences/index.html#anti-recurrence-sub-is-kautomatic-of-bou","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/anti-recurrence-and-greedy-subsum-free-sequences","status":"can_prove","summary":"Let $a = (a_1, \\dots, a_k)$ be a linear form with positive integer coefficients and $k > 1$, of trace $\\tau = a_1 + \\dots + a_k$, with anti-recurrence numbers $A_1, A_2, \\dots$, and let $\\kappa = k\\tau + 1$. If $a$ is $A_1$-bounded, that is, $A_1 \\le (k - 1)\\tau + 2$, then the integer sequence $n \\mapsto A_{n+1} - \\kappa(n + 1)$, indexed from $n = 0$, is $\\tau$-automatic: $\\tau \\ge 2$ and its $\\tau$-kernel is finite.","title":"Bounded anti-recurrence sequences are automatic up to a linear term"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform","status":"fully_proved","title":"Aronson transform"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform/aronson-transform-exists-unique-general","kind":"node","lean":"MetaMathlibExt.aronsonTransform_existsUnique_general","parent":24433,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform/index.html#aronson-transform-exists-unique-general","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform","status":"fully_proved","summary":"Let $n_0$ be a natural number and let $\\beta : \\mathbb{N} \\to \\mathbb{N}$ be strictly increasing such that infinitely many integers $m \\ge n_0$ are not values of $\\beta$. Then there is exactly one sequence $\\alpha : \\mathbb{N} \\to \\mathbb{N}$ such that $\\alpha$ is strictly increasing, $\\alpha_i \\ge n_0$ for every $i$, $\\alpha_0$ is the least integer $m \\ge n_0$ such that $m = n_0$ holds exactly when $m$ is a value\u2026","title":"Existence and uniqueness of the Aronson transform, general form"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform/aronson-transform-exists-unique","kind":"node","lean":"MetaMathlibExt.aronsonTransform_existsUnique","parent":24433,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform/index.html#aronson-transform-exists-unique","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/aronson-transform","status":"fully_proved","summary":"Let $n_0$ be a natural number and let $\\beta : \\mathbb{N} \\to \\mathbb{N}$ be strictly increasing with $\\beta_i \\ge n_0$ for every $i$, such that infinitely many integers $m \\ge n_0$ are not values of $\\beta$. Then there is exactly one sequence $\\alpha : \\mathbb{N} \\to \\mathbb{N}$ such that $\\alpha$ is strictly increasing, $\\alpha_i \\ge n_0$ for every $i$, $\\alpha_0$ is the least integer $m \\ge n_0$ such that\u2026","title":"Existence and uniqueness of the Aronson transform"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","status":"fully_proved","title":"Cloitre's self-describing sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/cloitre-complement-first","kind":"node","lean":"MetaMathlibExt.cloitreComplementFirst","parent":24436,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/index.html#cloitre-complement-first","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","status":"fully_proved","summary":"For a finite list of natural numbers, a leading letter $2$ is replaced by the two letters $1, 1$, and two leading letters $1, 1$ are replaced by the single letter $2$; the empty list and every list beginning otherwise (with a letter other than $1$ or $2$, or with a $1$ not followed by another $1$) are left unchanged.","title":"First-letter complement of a word"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/cloitre-mirror","kind":"node","lean":"MetaMathlibExt.cloitreMirror","parent":24436,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/index.html#cloitre-mirror","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","status":"fully_proved","summary":"For a finite list $A$ of natural numbers, the mirror image $A^r$ is obtained by reversing $A$ and then complementing the first letter: a leading $2$ of the reversed list becomes $1, 1$, two leading letters $1, 1$ become $2$, and any other list is left unchanged. For example $(1, 1, 2)^r = (1, 1, 1, 1)$ and $(1, 1, 2, 1, 1, 1, 1)^r = (2, 1, 1, 2, 1, 1)$.","title":"Mirror image with first-letter complement"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/cloitre-approx","kind":"node","lean":"MetaMathlibExt.cloitreApprox","parent":24436,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/index.html#cloitre-approx","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","status":"fully_proved","summary":"The finite words $A_k$ of natural numbers are defined by $A_0 = (1, 1)$ and $A_{k+1} = A_k A_k^r$, the concatenation of $A_k$ with its mirror image $A_k^r$: reverse $A_k$, then replace a leading $2$ by $1, 1$, or two leading letters $1, 1$ by $2$, and otherwise leave it unchanged.","title":"Approximants of Cloitre's sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/cloitre-sequence","kind":"node","lean":"MetaMathlibExt.cloitreSequence","parent":24436,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/index.html#cloitre-sequence","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","status":"fully_proved","summary":"Let $A_0 = (1, 1)$ and $A_{k+1} = A_k A_k^r$ be the approximants of Cloitre's sequence, where $A_k^r$ is the mirror image of $A_k$: reverse $A_k$, then replace a leading $2$ by $1, 1$, or two leading letters $1, 1$ by $2$, and otherwise leave it unchanged. Cloitre's sequence $s : \\mathbb{N} \\to \\mathbb{N}$ is defined by letting $s(n)$ be the letter at position $n$ (counting from $0$) of $A_{n+2}$, or $0$ if\u2026","title":"Cloitre's sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/is-run-boundary","kind":"node","lean":"MetaMathlibExt.IsRunBoundary","parent":24436,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/index.html#is-run-boundary","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","status":"fully_proved","summary":"For sequences $s, b : \\mathbb{N} \\to \\mathbb{N}$, $b$ marks the maximal runs of $s$ when $b(0) = 0$, $b$ is strictly increasing, $s$ is constant on each block $[b(n), b(n+1))$, and for every $n$ each value of $s$ on the block $[b(n), b(n+1))$ differs from each value of $s$ on the next block $[b(n+1), b(n+2))$.","title":"Run boundaries of a sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/cloitre-run-sum","kind":"node","lean":"MetaMathlibExt.cloitreRunSum","parent":24436,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/index.html#cloitre-run-sum","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","status":"fully_proved","summary":"For sequences $s, b : \\mathbb{N} \\to \\mathbb{N}$ and $n \\in \\mathbb{N}$, the $n$-th run sum is $\\sum_{b(n) \\le j < b(n+1)} s(j)$, the sum of $s$ over the block $[b(n), b(n+1))$; it is $0$ when $b(n+1) \\le b(n)$.","title":"Sum of a sequence over a run"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/is-cloitre-self-describing","kind":"node","lean":"MetaMathlibExt.IsCloitreSelfDescribing","parent":24436,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence/index.html#is-cloitre-self-describing","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/cloitre-s-self-describing-sequence","status":"fully_proved","summary":"A sequence $s : \\mathbb{N} \\to \\mathbb{N}$ is self-describing in Cloitre's sense when there is a sequence $b : \\mathbb{N} \\to \\mathbb{N}$ with $b(0) = 0$, $b$ strictly increasing, $s$ constant on each block $[b(n), b(n+1))$ and each value of $s$ on a block different from each value on the next block (so the blocks are the maximal runs of $s$), such that for every $n$, $2 s(n) = \\sum_{b(n) \\le j < b(n+1)} s(j)$.","title":"Cloitre's self-describing property"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","title":"Connell sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/connell-row-length","kind":"node","lean":"MetaMathlibExt.connellRowLength","parent":24444,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html#connell-row-length","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","summary":"For natural numbers $r$ and $j$, the length of row $j$ is $1 + r(j - 1)$, where $j - 1$ is read as $0$ when $j = 0$. For $j \\ge 1$ this is the number of entries in row $j$ of the Connell $(m, r)$ triangle for every $m$; the source numbers rows from one, and the value at $j = 0$ is a convention.","title":"Row lengths of the Connell triangle"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/connell-row-start","kind":"node","lean":"MetaMathlibExt.connellRowStart","parent":24444,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html#connell-row-start","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","summary":"For natural numbers $m$ and $r$, the numbers $s_j$ are defined by $s_0 = s_1 = 1$ and $s_{j+1} = s_j + m(\\ell_j - 1) + 1$ for $j \\ge 1$, where $\\ell_j = 1 + r(j - 1)$ is the length of row $j$. For $j \\ge 1$, $s_j$ is the first entry of row $j$ of the Connell $(m, r)$ triangle: each row starts one above the last entry $s_j + m(\\ell_j - 1)$ of the row before. The source numbers rows from one, and the value $s_0 = 1$\u2026","title":"First entries of the rows of the Connell triangle"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/connell-entry","kind":"node","lean":"MetaMathlibExt.connellEntry","parent":24444,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html#connell-entry","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","summary":"For natural numbers $m$, $r$, $j$ and $k$, the entry with zero-based index $k$ in row $j$ of the Connell $(m, r)$ triangle is $s_j + mk$, where $s_j$ is the first entry of row $j$. The formula is used for every $k$, also at or beyond the row length.","title":"Entries of the Connell triangle"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/connell-row-entries","kind":"node","lean":"MetaMathlibExt.connellRowEntries","parent":24444,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html#connell-row-entries","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","summary":"For natural numbers $m$, $r$ and $j$, row $j$ of the Connell $(m, r)$ triangle is the finite set of the numbers $s_j + mk$ with $0 \\le k < \\ell_j$, where $s_j$ is the first entry of row $j$ and $\\ell_j = 1 + r(j - 1)$ is its length, with $j - 1$ read as $0$ when $j = 0$.","title":"Rows of the Connell triangle"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/connell-parameters","kind":"node","lean":"MetaMathlibExt.ConnellParameters","parent":24444,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html#connell-parameters","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","summary":"A pair of natural numbers $(m, r)$ with $m \\ge 2$ and $r \\ge 1$; these are the admissible parameters of a Connell $(m, r)$ sequence.","title":"Connell parameters"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/connell-next","kind":"node","lean":"MetaMathlibExt.connellNext","parent":24444,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html#connell-next","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","summary":"For a natural number $r$, the position following row $j$, zero-based column $k$, is row $j$, column $k + 1$ if $k + 1 < 1 + r(j - 1)$ (the length of row $j$, with $j - 1$ read as $0$ when $j = 0$), and row $j + 1$, column $0$ otherwise.","title":"Next position in the Connell triangle"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/connell-position","kind":"node","lean":"MetaMathlibExt.connellPosition","parent":24444,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html#connell-position","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","summary":"For a natural number $r$, the position $p_n = (j_n, k_n)$ (row $j_n$, zero-based column $k_n$) of index $n \\ge 0$ is defined by $p_0 = (1, 0)$ and $p_{n+1} = (j_n, k_n + 1)$ if $k_n + 1 < 1 + r(j_n - 1)$ (the length of row $j_n$, with $j_n - 1$ read as $0$ when $j_n = 0$), and $p_{n+1} = (j_n + 1, 0)$ otherwise.","title":"Position of a term in the Connell triangle"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/connell-sequence","kind":"node","lean":"MetaMathlibExt.connellSequence","parent":24444,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence/index.html#connell-sequence","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/connell-sequence","status":"fully_proved","summary":"For natural numbers $m \\ge 2$ and $r \\ge 1$, the Connell $(m, r)$ sequence $(a_n)_{n \\ge 0}$ reads the Connell triangle row by row: row $j \\ge 1$ has $\\ell_j = 1 + r(j - 1)$ entries $s_j, s_j + m, \\dots, s_j + m(\\ell_j - 1)$, where $s_1 = 1$ and $s_{j+1} = s_j + m(\\ell_j - 1) + 1$. Thus $a_n = s_j + mk$, where row $j$, zero-based column $k$ is the position reached from row $1$, column $0$ after $n$ steps, each step\u2026","title":"Connell sequence"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences","status":"fully_proved","title":"Fractal sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences/fractal-sequence","kind":"node","lean":"MetaMathlibExt.FractalSequence","parent":24453,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences/index.html#fractal-sequence","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences","status":"fully_proved","summary":"A regular fractal sequence is a sequence $(x_n)_{n \\ge 0}$ of positive integers together with a position array $(p_i(j))_{i, j \\ge 0}$ of natural numbers such that: for every $i \\ge 0$, the strictly increasing sequence $p_i(0) < p_i(1) < \\cdots$ lists exactly the positions $n$ with $x_n = i + 1$ (so every positive integer occurs infinitely often); for all $i$ and $n$ with $x_n = i + 2$, some $m < n$ has\u2026","title":"Regular fractal sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences/pos-injective","kind":"node","lean":"MetaMathlibExt.FractalSequence.pos_injective","parent":24453,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences/index.html#pos-injective","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences","status":"fully_proved","summary":"In a regular fractal sequence with position array $p$, for every $i \\ge 0$ the map $j \\mapsto p_i(j)$, sending $j$ to the position of the $j$-th occurrence of $i + 1$ (counting from $j = 0$), is injective.","title":"Occurrence positions are distinct"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences/infinite-occurrences","kind":"node","lean":"MetaMathlibExt.FractalSequence.infinite_occurrences","parent":24453,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences/index.html#infinite-occurrences","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/fractal-sequences","status":"fully_proved","summary":"Let $(x_n)_{n \\ge 0}$ be a regular fractal sequence. Then for every positive integer $k$ there are infinitely many $n$ with $x_n = k$.","title":"Every value occurs infinitely often in a regular fractal sequence"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","title":"Gijswijt's sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/curling-number","kind":"node","lean":"MetaMathlibExt.curlingNumber","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#curling-number","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"The curling number $\\mathcal{C}(S)$ of a finite string $S$ of natural numbers is the largest integer $k$ such that $S = X Y^k$ for some strings $X$ and $Y$ with $Y$ nonempty; for the empty string $\\varepsilon$ it is $\\mathcal{C}(\\varepsilon) = 1$.","title":"Curling number"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/curling-number-example-one","kind":"node","lean":"MetaMathlibExt.curlingNumber_example_one","parent":24457,"reading_title":{"format":"inline-parts-v1","parts":[["text","Curling number of "],["math","\\(112112223\\)"]],"source":"Curling number of $112112223$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#curling-number-example-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"The string $112112223$ has curling number $\\mathcal{C}(112112223) = 1$.","title":"Curling number of $112112223$"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/curling-number-example-three","kind":"node","lean":"MetaMathlibExt.curlingNumber_example_three","parent":24457,"reading_title":{"format":"inline-parts-v1","parts":[["text","Curling number of "],["math","\\(222322232223\\)"]],"source":"Curling number of $222322232223$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#curling-number-example-three","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"The string $222322232223$ has curling number $\\mathcal{C}(222322232223) = 3$.","title":"Curling number of $222322232223$"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/curling-number-nil","kind":"node","lean":"MetaMathlibExt.curlingNumber_nil","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#curling-number-nil","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"The empty string $\\varepsilon$ has curling number $\\mathcal{C}(\\varepsilon) = 1$.","title":"Curling number of the empty string"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-prefix","kind":"node","lean":"MetaMathlibExt.gijswijtPrefix","parent":24457,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prefixes of the "],["math","\\(m\\)"],["text","th order Gijswijt sequence"]],"source":"Prefixes of the $m$th order Gijswijt sequence"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-prefix","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"Let $m$ and $n$ be natural numbers. The list of the first $n$ terms of the $m$th order sequence $A^{(m)}$ is defined by recursion on $n$: for $n = 0$ it is the empty list, and the list for $n + 1$ is the list $P$ for $n$ followed by the term $\\max(m, \\mathcal{C}(P))$, where $\\mathcal{C}(P)$ is the curling number of $P$ (the largest $k$ with $P = XY^k$ for strings $X$, $Y$ with $Y$ nonempty, and $1$ for the empty\u2026","title":"Prefixes of the $m$th order Gijswijt sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-term","kind":"node","lean":"MetaMathlibExt.gijswijtTerm","parent":24457,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terms of the "],["math","\\(m\\)"],["text","th order Gijswijt sequence"]],"source":"Terms of the $m$th order Gijswijt sequence"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-term","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For $k \\ge 1$, $A^{(m)}(k)$ is the $k$th term of the $m$th order sequence $A^{(m)}$, that is, the last entry of the list of its first $k$ terms. For $k = 0$ the value is $0$ by convention.","title":"Terms of the $m$th order Gijswijt sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-prefix-one-nine","kind":"node","lean":"MetaMathlibExt.gijswijtPrefix_one_nine","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-prefix-one-nine","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"The first nine terms of Gijswijt's sequence $A^{(1)}$ are $1, 1, 2, 1, 1, 2, 2, 2, 3$.","title":"First terms of Gijswijt's sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-prefix-two-eight","kind":"node","lean":"MetaMathlibExt.gijswijtPrefix_two_eight","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-prefix-two-eight","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"The first eight terms of the second-order sequence $A^{(2)}$ are $2, 2, 2, 3, 2, 2, 2, 3$.","title":"Initial terms of the second-order Gijswijt sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-first-occurrence","kind":"node","lean":"MetaMathlibExt.gijswijtFirstOccurrence","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-first-occurrence","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For natural numbers $m$ and $n$, the first occurrence of $n$ in the $m$th order Gijswijt sequence $A^{(m)}$ is the least position $k \\ge 1$ with $A^{(m)}(k) = n$, and it is $0$ if $n$ does not occur in $A^{(m)}$.","title":"First occurrence of a value in a Gijswijt sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-length","kind":"node","lean":"MetaMathlibExt.gijswijtBlockLength","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-length","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For a natural number $m$, write $A^{(m)}(k)$ for the $k$th term of the $m$th order sequence. The block lengths $\\beta^{(m)}(t)$ are defined by $\\beta^{(m)}(0) = 0$, $\\beta^{(m)}(1) = 1$ and, for $t \\ge 1$, $\\beta^{(m)}(t+1)$ is the least $k > (m+1)\\,\\beta^{(m)}(t)$ with $A^{(m)}(k+1) = m$, or $0$ if there is no such $k$.","title":"Block lengths of the higher-order Gijswijt sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block","kind":"node","lean":"MetaMathlibExt.gijswijtBlock","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For natural numbers $m$ and $t$, the block $B_t^{(m)}$ is the prefix of length $\\beta^{(m)}(t)$ of the $m$th order Gijswijt sequence $A^{(m)}$, where $\\beta^{(m)}(t)$ is the $t$th block length.","title":"Gijswijt blocks"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-glue","kind":"node","lean":"MetaMathlibExt.gijswijtGlue","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-glue","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For natural numbers $m$ and $t$, the glue string $S^{(m)}_t$ is the block $B^{(m)}_{t+1}$ of the $m$th order Gijswijt sequence with its first $(m+1)\\,\\beta^{(m)}(t)$ elements removed, where $\\beta^{(m)}(t)$ is the length of the block $B^{(m)}_t$.","title":"Glue strings of a Gijswijt sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-glue-concat","kind":"node","lean":"MetaMathlibExt.gijswijtGlueConcat","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-glue-concat","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For natural numbers $m$ and $t$, the string $T^{(m)}_t$ is the concatenation $S^{(m)}_1 S^{(m)}_2 \\cdots S^{(m)}_{t-1}$ of the glue strings of the $m$th order Gijswijt sequence; it is empty when $t \\le 1$.","title":"Concatenated glue strings of a Gijswijt sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-index","kind":"node","lean":"MetaMathlibExt.gijswijtBlockIndex","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-index","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For natural numbers $m$ and $n$, let $p$ be the position of the first occurrence of $n$ in the $m$th order Gijswijt sequence $A^{(m)}$, with $p = 0$ if $n$ does not occur. Then $t(m, n)$ is the least $t \\ge 1$ such that the block length $\\beta^{(m)}(t)$ equals $p$, so that when $n$ occurs, this first $n$ is the last element of the block $B_t^{(m)}$; if there is no such $t$, then $t(m, n) = 0$.","title":"Block index of the first occurrence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-word","kind":"node","lean":"MetaMathlibExt.gijswijtBlockWord","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-word","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For natural numbers $m$ and $u$ and exponents $a_1, a_2, \\ldots$, the block word of order $m$ is the concatenation $B_{u-1}^{a_{u-1}} \\cdots B_2^{a_2} B_1^{a_1}$, where $B_i = B_i^{(m)}$ is the $i$th block of the $m$th order Gijswijt sequence and $B_i^{a_i}$ is $a_i$ copies of $B_i$ in a row. Only $a_1, \\ldots, a_{u-1}$ are used, and the word is empty when $u \\le 1$.","title":"Block words"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-length-one","kind":"node","lean":"MetaMathlibExt.gijswijtBlockLength_one","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-length-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For every natural number $m$, the first block length $\\beta^{(m)}(1)$ of the $m$th order Gijswijt sequence equals $1$.","title":"The first block has length one"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-one","kind":"node","lean":"MetaMathlibExt.gijswijtBlock_one","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For every $m \\ge 1$, the first block $B_1^{(m)}$ of the $m$th order Gijswijt sequence is the one-term string $m$.","title":"The first block"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-glue-concat-one","kind":"node","lean":"MetaMathlibExt.gijswijtGlueConcat_one","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-glue-concat-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For every natural number $m$, the concatenated glue string $T^{(m)}_1$ is the empty string.","title":"The first concatenated glue string is empty"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-word-one","kind":"node","lean":"MetaMathlibExt.gijswijtBlockWord_one","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-word-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"fully_proved","summary":"For every natural number $m$ and every sequence of exponents $a_1, a_2, \\ldots$, the block word $B_{u-1}^{a_{u-1}} \\cdots B_1^{a_1}$ of order $m$ with $u = 1$, where $B_i$ is the $i$th block of the $m$th order Gijswijt sequence, is a product of no blocks and hence the empty string.","title":"The empty block word"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-first-occurrence-tower","kind":"node","lean":"MetaMathlibExt.gijswijtFirstOccurrence_tower","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-first-occurrence-tower","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"For every $n \\ge 4$ there is a real number $\\alpha$ with $n - 2 < \\alpha < n - 1$ such that the first occurrence of $n$ in Gijswijt's sequence is at position $2^{2^{3^{\\cdot^{\\cdot^{(n-2)^{\\alpha}}}}}}$, a power tower whose bases are $2, 2, 3, \\ldots, n-2$ from the bottom up and whose top exponent is $\\alpha$; for $n = 4$ it is $2^{2^{\\alpha}}$.","title":"Van de Pol's tower formula for first occurrences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/curling-number-le-of-suffix","kind":"node","lean":"MetaMathlibExt.curlingNumber_le_of_suffix","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#curling-number-le-of-suffix","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"For strings $S$ and $T$ of natural numbers, if $T$ is a suffix of $S$, then $\\mathcal{C}(T) \\le \\mathcal{C}(S)$.","title":"The curling number is monotone under suffixes"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-glue-concat-suffix-block","kind":"node","lean":"MetaMathlibExt.gijswijtGlueConcat_suffix_block","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-glue-concat-suffix-block","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $t \\ge 2$. The element $m$ followed by the concatenated glue string $T^{(m)}_t$ is a suffix of the block $B^{(m)}_t$, and whenever $B^{(m)}_t = X\\,T^{(m)}_t\\,Y$ for strings $X$ and $Y$, the string $Y$ is empty; that is, $T^{(m)}_t$ occurs in $B^{(m)}_t$ only as a suffix.","title":"Concatenated glue strings occur in a block only at its end"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-glue-concat-iff-prefix","kind":"node","lean":"MetaMathlibExt.gijswijtGlueConcat_iff_prefix","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-glue-concat-iff-prefix","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$. A finite string $P$ of natural numbers is a prefix of the $(m+1)$th order Gijswijt sequence $A^{(m+1)}$ with curling number $\\mathcal{C}(P) \\le m$ if and only if $P = T^{(m)}_t$ for some $t \\ge 1$, where $T^{(m)}_t$ is the concatenated glue string.","title":"Concatenated glue strings are the next-order prefixes with small curling number"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-runs","kind":"node","lean":"MetaMathlibExt.gijswijtBlock_runs","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-runs","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $t \\ge 2$, and consider the block $B_t^{(m)}$ of the $m$th order Gijswijt sequence. Every maximal run of consecutive terms equal to $m$ in $B_t^{(m)}$ has length exactly $m + 1$, and the term right after the run exists and equals $m + 1$.","title":"Runs of the smallest term in a block"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-first-occurrence-eq-block-lengt","kind":"node","lean":"MetaMathlibExt.gijswijtFirstOccurrence_eq_blockLength","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-first-occurrence-eq-block-lengt","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $1 \\le m \\le n$, and write $\\beta^{(m)}(t)$ for the length of the block $B_t^{(m)}$ of the $m$th order Gijswijt sequence $A^{(m)}$. Then there is a $t \\ge 1$ such that $A^{(m)}(\\beta^{(m)}(t)) = n$ and $\\beta^{(m)}(t)$ is the position of the first occurrence of $n$ in $A^{(m)}$; that is, the first $n$ in $A^{(m)}$ is the last term of the block $B_t^{(m)}$.","title":"First occurrences end blocks"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-glue-concat-block-index","kind":"node","lean":"MetaMathlibExt.gijswijtGlueConcat_blockIndex","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-glue-concat-block-index","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $n \\ge m + 1$. For $j \\ge 1$ let $t(j, n)$ be the least $t \\ge 1$ such that the block $B^{(j)}_t$ of the $j$th order Gijswijt sequence has length equal to the position of the first occurrence of $n$ in that sequence (this position is taken to be $0$ if $n$ does not occur), and $t(j, n) = 0$ if there is no such $t$. Then the concatenated glue string $T^{(m)}_{t(m,n)}$ equals the block\u2026","title":"Concatenated glue strings at first occurrences are blocks of the next order"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-proper-prefix-curling-iff","kind":"node","lean":"MetaMathlibExt.gijswijtProperPrefix_curling_iff","parent":24457,"reading_title":{"format":"inline-parts-v1","parts":[["text","Proper prefixes of a block with curling number at most "],["math","\\(m + 1\\)"]],"source":"Proper prefixes of a block with curling number at most $m + 1$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-proper-prefix-curling-iff","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $u \\ge 1$, and write $B_t$ for the block $B_t^{(m+1)}$ of the $(m+1)$th order Gijswijt sequence. A string $P$ is a proper prefix of $B_u$ (a prefix strictly shorter than $B_u$) with curling number $\\mathcal{C}(P) \\le m + 1$ if and only if $P = B_{u-1}^{a_{u-1}} \\cdots B_2^{a_2} B_1^{a_1}$ for some natural numbers $a_1, \\dots, a_{u-1} \\le m + 1$, where $B_t^{a}$ denotes $a$ copies of $B_t$\u2026","title":"Proper prefixes of a block with curling number at most $m + 1$"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/curling-number-block-word-of-max","kind":"node","lean":"MetaMathlibExt.curlingNumber_blockWord_of_max","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#curling-number-block-word-of-max","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $u \\ge 1$, and let $a_1, \\dots, a_{u-1} \\in \\{0, \\dots, m+1\\}$ with $a_i = m + 1$ for some $1 \\le i \\le u - 1$. Writing $B_t = B_t^{(m+1)}$ for the blocks of the $(m+1)$th order Gijswijt sequence $A^{(m+1)}$, the string $P = B_{u-1}^{a_{u-1}} \\cdots B_2^{a_2} B_1^{a_1}$ has curling number $\\mathcal{C}(P) = m + 1$.","title":"Curling number of block words with a maximal exponent"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/curling-number-block-word-le","kind":"node","lean":"MetaMathlibExt.curlingNumber_blockWord_le","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#curling-number-block-word-le","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $u \\ge 2$, and let $a_1, \\dots, a_{u-1} \\in \\{0, \\dots, m\\}$ with $a_1 \\ne 0$. Writing $B_t = B_t^{(m+1)}$ for the blocks of the $(m+1)$th order Gijswijt sequence $A^{(m+1)}$, the string $P = B_{u-1}^{a_{u-1}} \\cdots B_2^{a_2} B_1^{a_1}$ has curling number $\\mathcal{C}(P) \\le m$.","title":"Curling number of block words with small exponents"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-glue-concat-block-word","kind":"node","lean":"MetaMathlibExt.gijswijtGlueConcat_blockWord","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-glue-concat-block-word","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $u \\ge 2$, and for exponents $a_1, \\ldots, a_{u-1}$ let $W(a) = (B^{(m+1)}_{u-1})^{a_{u-1}} \\cdots (B^{(m+1)}_1)^{a_1}$ be the word in the blocks of the $(m+1)$th order Gijswijt sequence. (a) If $t \\ge 1$ and the concatenated glue string $T^{(m)}_t$ is a proper prefix of $B^{(m+1)}_u$, then $T^{(m)}_t = W(a)$ for some $a_1, \\ldots, a_{u-1} \\in \\{0, \\ldots, m\\}$. (b) For all\u2026","title":"Concatenated glue strings inside a block of the next order"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-length-estimates","kind":"node","lean":"MetaMathlibExt.gijswijtBlockLength_estimates","parent":24457,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bounds on the block lengths of order "],["math","\\(m+1\\)"]],"source":"Bounds on the block lengths of order $m+1$"},"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-length-estimates","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $u \\ge 2$. Write $\\beta^{(m+1)}(u)$ for the length of the block $B_u^{(m+1)}$ of the $(m+1)$th order Gijswijt sequence, and $\\tau^{(m)}(t)$ for the length of $T_t^{(m)} = S_1^{(m)} \\cdots S_{t-1}^{(m)}$, the concatenation of the first $t - 1$ glue strings of order $m$. Then $\\tau^{(m)}(m (m+1)^{u-2}) \\le \\beta^{(m+1)}(u) \\le \\tau^{(m)}((m+1)^{u-1} + 1)$, and the second inequality is strict when\u2026","title":"Bounds on the block lengths of order $m+1$"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-glue-concat-length-estimates","kind":"node","lean":"MetaMathlibExt.gijswijtGlueConcat_length_estimates","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-glue-concat-length-estimates","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $t \\ge 2$, write $\\tau^{(m)}(t)$ for the length of the concatenated glue string $T^{(m)}_t$ and $\\beta^{(m+1)}$ for the block lengths of the $(m+1)$th order Gijswijt sequence, and let $c$ be the least natural number with $t \\le (m+1)^c$, that is, $c = \\lceil \\log_{m+1} t \\rceil$. Then $\\beta^{(m+1)}(c) \\le \\tau^{(m)}(t) \\le \\beta^{(m+1)}(c + 2)$.","title":"Length estimates for concatenated glue strings"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-index-estimates","kind":"node","lean":"MetaMathlibExt.gijswijtBlockIndex_estimates","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-index-estimates","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $m \\ge 1$ and $n \\ge m + 3$, and for $k \\ge 1$ let $t(k, n)$ be the index $t$ such that the first occurrence of $n$ in the $k$th order Gijswijt sequence $A^{(k)}$ is the last element of the block $B_t^{(k)}$ of $A^{(k)}$. Then $m (m+1)^{t(m+1,n) - 2} \\le t(m, n) \\le (m+1)^{t(m+1,n) - 1} + 1$, and the second inequality is strict when $n = m + 3$.","title":"Recursive estimate for the block index"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/gijswijt-block-length-div-lt","kind":"node","lean":"MetaMathlibExt.gijswijtBlockLength_div_lt","parent":24457,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence/index.html#gijswijt-block-length-div-lt","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/gijswijt-s-sequence","status":"can_prove","summary":"Let $M_7 = 4$ and $M_{k+1} = M_k (1 + 9 (k+1)^{1.59} / 2^k)$ for $k \\ge 7$. Then for all $m \\ge 1$ and $u \\ge 7$, the block lengths $\\beta^{(m)}(u)$ of the $m$th order Gijswijt sequence $A^{(m)}$ satisfy $\\beta^{(m)}(u) / (m+1)^{u-1} < M_u$.","title":"Upper bound on block lengths"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets","status":"can_prove","title":"Inventory sequences on multisets"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets/step","kind":"node","lean":"MathlibExt.Combinatorics.MultisetSequenceLoopWanted.step","parent":24492,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets/index.html#step","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets","status":"fully_proved","summary":"The inventory step sends a pair of finite lists of natural numbers (the two rows of an array) to the pair consisting of the increasing list of the distinct numbers occurring in either list and the list of how many times each of these numbers occurs in the two lists together.","title":"Inventory step"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.MultisetSequenceLoopWanted.conjecture","parent":24492,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets/index.html#conjecture","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets","status":"fully_proved","summary":"Let $a_1 < \\dots < a_n$ and $b_1, \\dots, b_n$ be positive integers, and iterate the inventory step starting from the pair of rows $(a_1, \\dots, a_n)$ and $(b_1, \\dots, b_n)$, where the inventory step replaces a pair of rows by the increasing list of the distinct integers occurring in either row together with the list of the number of times each of them occurs in the two rows combined. Then there are a natural\u2026","title":"Inventory loop conjecture"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.MultisetSequenceLoopWanted.conjecture_holds","parent":24492,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets/index.html#conjecture-holds","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/inventory-sequences-on-multisets","status":"can_prove","summary":"Let $a_1 < \\dots < a_n$ and $b_1, \\dots, b_n$ be positive integers, and iterate the inventory step starting from the pair of rows $(a_1, \\dots, a_n)$ and $(b_1, \\dots, b_n)$, where the inventory step replaces a pair of rows by the increasing list of the distinct integers occurring in either row together with the list of the number of times each of them occurs in the two rows combined. Then there are a natural\u2026","title":"Inventory sequences end in short loops"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence","status":"fully_proved","title":"Kolakoski sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence/kolakoski-block-start","kind":"node","lean":"MetaMathlibExt.kolakoskiBlockStart","parent":24496,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence/index.html#kolakoski-block-start","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence","status":"fully_proved","summary":"For a sequence $K_1, K_2, K_3, \\ldots$ of natural numbers, the starting positions $s_n$ of its blocks are defined by $s_1 = 1$ and $s_{n+1} = s_n + K_n$ for every $n \\ge 1$, together with the convention $s_0 = 0$.","title":"Starting positions of Kolakoski blocks"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence/is-kolakoski","kind":"node","lean":"MetaMathlibExt.IsKolakoski","parent":24496,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence/index.html#is-kolakoski","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/kolakoski-sequence","status":"fully_proved","summary":"A sequence $K_1, K_2, K_3, \\ldots$ of natural numbers is a Kolakoski sequence when $K_1 = 1$, every $K_n$ with $n \\ge 1$ is $1$ or $2$, and $K$ describes its own runs in the following sense. Let $s_1 = 1$ and $s_{n+1} = s_n + K_n$ for $n \\ge 1$, and call block $n$ the set of positions $i$ with $s_n \\le i < s_n + K_n$. For every $n \\ge 1$, every term $K_i$ with $i$ in block $n$ equals $1$ if $n$ is odd and $2$ if\u2026","title":"Kolakoski sequences"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","title":"Self-generating sets"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/strict-growth","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.StrictGrowth","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#strict-growth","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"A finite set $F$ of functions $\\mathbb{N} \\to \\mathbb{N}$ has strict growth when $x < f(x)$ for every $f \\in F$ and every $x \\in \\mathbb{N}$.","title":"Strict growth of a set of generators"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/sg","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.SG","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#sg","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and let $s \\in \\mathbb{N}$. A set $\\Gamma \\subseteq \\mathbb{N}$ is self-generating for $s$ and $F$ when $s \\in \\Gamma$ and $f(x) \\in \\Gamma$ for every $f \\in F$ and every $x \\in \\Gamma$.","title":"Self-generating sets"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/fixed-point","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.FixedPoint","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#fixed-point","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and let $s \\in \\mathbb{N}$. A set $C \\subseteq \\mathbb{N}$ is a fixed-point set for $F$ and $s$ when, for every $y \\in \\mathbb{N}$, $y \\in C$ if and only if $y = s$ or $y = f(x)$ for some $f \\in F$ and some $x \\in C$; equivalently, $C = \\{s\\} \\cup \\bigcup_{f \\in F} f(C)$.","title":"Fixed-point sets for a seed and generators"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/gen","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.Gen","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#gen","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and let $s \\in \\mathbb{N}$. Reachability from $s$ is the least predicate on $\\mathbb{N}$ that holds at $s$ and holds at $f(w)$ whenever it holds at $w$ and $f \\in F$. Thus $y$ is reachable exactly when $y = f_k(\\cdots f_1(s) \\cdots)$ for some $k \\ge 0$ and $f_1, \\ldots, f_k \\in F$; the case $k = 0$ is $s$ itself.","title":"Reachability from a seed by generators"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/generated-set","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.GeneratedSet","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#generated-set","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"For a finite set $F$ of functions $\\mathbb{N} \\to \\mathbb{N}$ and a seed $s \\in \\mathbb{N}$, the set generated from $s$ by $F$ consists of all numbers $f_k(\\cdots f_1(s) \\cdots)$ with $k \\ge 0$ and $f_1, \\dots, f_k \\in F$; the case $k = 0$ gives $s$ itself.","title":"Set generated from a seed"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/fixed-point-is-sg","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.fixedPoint_isSG","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#fixed-point-is-sg","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and $s \\in \\mathbb{N}$. Every fixed-point set $C$ for $F$ and $s$ is self-generating: $s \\in C$, and $f(x) \\in C$ for every $f \\in F$ and every $x \\in C$.","title":"Fixed-point sets are self-generating"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/generated-is-sg","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.generated_isSG","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#generated-is-sg","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and $s \\in \\mathbb{N}$. The set generated from $s$ by $F$ contains $s$ and is closed under every $f \\in F$: if $x$ is in it and $f \\in F$, then $f(x)$ is in it.","title":"The generated set is self-generating"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/generated-le-of-sg","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.generated_le_of_sg","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#generated-le-of-sg","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and let $s \\in \\mathbb{N}$. If a set $G \\subseteq \\mathbb{N}$ contains $s$ and satisfies $f(x) \\in G$ for every $f \\in F$ and every $x \\in G$, then $G$ contains the set generated from $s$ by $F$, that is, every number obtained from $s$ by applying finitely many (possibly no) functions of $F$ in succession.","title":"The generated set lies in every self-generating set"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/generated-fixed-point","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.generated_fixedPoint","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#generated-fixed-point","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and $s \\in \\mathbb{N}$, and let $\\Gamma$ be the set generated from $s$ by $F$. Then for every $y \\in \\mathbb{N}$, $y \\in \\Gamma$ if and only if $y = s$ or $y = f(x)$ for some $f \\in F$ and $x \\in \\Gamma$.","title":"The generated set is a fixed-point set"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/fixed-point-eq-generated","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.fixedPoint_eq_generated","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#fixed-point-eq-generated","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ with $x < f(x)$ for every $f \\in F$ and every $x \\in \\mathbb{N}$, let $s \\in \\mathbb{N}$, and let $C \\subseteq \\mathbb{N}$ be a fixed-point set for $F$ and $s$: $y \\in C$ exactly when $y = s$ or $y = f(x)$ for some $f \\in F$ and $x \\in C$. Then $C$ equals the set generated from $s$ by $F$, the numbers obtained from $s$ by applying finitely many\u2026","title":"Under strict growth, a fixed-point set is the generated set"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/generated-eq-s-inter","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.generated_eq_sInter","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#generated-eq-s-inter","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and $s \\in \\mathbb{N}$. The set generated from $s$ by $F$ equals the intersection of all sets $G \\subseteq \\mathbb{N}$ that contain $s$ and are closed under every $f \\in F$.","title":"The generated set is the intersection of all self-generating sets"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/fixed-point-eq-s-inter","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.fixedPoint_eq_sInter","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#fixed-point-eq-s-inter","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ with $x < f(x)$ for every $f \\in F$ and every $x \\in \\mathbb{N}$, and let $s \\in \\mathbb{N}$. If $C$ is a fixed-point set for $F$ and $s$, then $C$ is the intersection of all sets $G \\subseteq \\mathbb{N}$ that contain $s$ and are closed under every $f \\in F$.","title":"Under strict growth, a fixed-point set is the intersection of all self-generating sets"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/fixed-point-is-least","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.fixedPoint_isLeast","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#fixed-point-is-least","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ with $x < f(x)$ for every $f \\in F$ and every $x \\in \\mathbb{N}$, and let $s \\in \\mathbb{N}$. If $C$ is a fixed-point set for $F$ and $s$, then $C$ contains $s$ and is closed under every $f \\in F$, and $C \\subseteq G$ for every set $G \\subseteq \\mathbb{N}$ that contains $s$ and is closed under every $f \\in F$.","title":"Under strict growth, a fixed-point set is the least self-generating set"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/is-least-eq-generated","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.isLeast_eq_generated","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#is-least-eq-generated","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ and let $s \\in \\mathbb{N}$. Call a set $G \\subseteq \\mathbb{N}$ self-generating if $s \\in G$ and $f(x) \\in G$ for every $f \\in F$ and every $x \\in G$. If $C$ is self-generating and is contained in every self-generating set, then $C$ equals the set generated from $s$ by $F$.","title":"The least self-generating set is the generated set"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/fixed-point-unique","kind":"node","lean":"MetaMathlibExt.SelfGeneratingSet.fixedPoint_unique","parent":24499,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets/index.html#fixed-point-unique","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/self-generating-sets","status":"fully_proved","summary":"Let $F$ be a finite set of functions $\\mathbb{N} \\to \\mathbb{N}$ with $x < f(x)$ for every $f \\in F$ and every $x \\in \\mathbb{N}$, and let $s \\in \\mathbb{N}$. If $C_1$ and $C_2$ are both fixed-point sets for $F$ and $s$, then $C_1 = C_2$.","title":"Under strict growth, fixed-point sets are unique"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","title":"Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-block","kind":"node","lean":"MetaMathlibExt.smarandacheBlock","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-block","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"For a natural number $n$, the $n$-th block is the list of the odd numbers $1, 3, \\ldots, 2n - 1$ in increasing order followed by the even numbers $2n, 2n - 2, \\ldots, 2$ in decreasing order; the $0$-th block is empty.","title":"Blocks of the Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-prefix","kind":"node","lean":"MetaMathlibExt.smarandachePrefix","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-prefix","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"For a natural number $N$, this is the list of natural numbers obtained by concatenating, in order, the Smarandache blocks numbered $1, 2, \\dots, N$, where block $n$ is the list $(1, 3, \\dots, 2n - 1, 2n, 2n - 2, \\dots, 2)$ of the odd numbers below $2n$ in increasing order followed by the even numbers from $2n$ down to $2$. For $N = 0$ it is the empty list.","title":"Prefix of the Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-permutation","kind":"node","lean":"MetaMathlibExt.smarandachePermutation","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-permutation","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"The Smarandache permutation sequence (OEIS A004741) is the sequence $(a_i)_{i \\ge 0}$ of natural numbers, indexed from $0$, obtained by concatenating the blocks $(1, 3, \\ldots, 2n - 1, 2n, 2n - 2, \\ldots, 2)$ for $n = 1, 2, 3, \\ldots$ in order; it begins $1, 2, 1, 3, 4, 2, 1, 3, 5, 6, 4, 2, \\ldots$, so $a_0 = 1$.","title":"Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-block-one","kind":"node","lean":"MetaMathlibExt.smarandacheBlock_one","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-block-one","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"The first block of the Smarandache permutation sequence is $(1, 2)$.","title":"First block of the Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-block-two","kind":"node","lean":"MetaMathlibExt.smarandacheBlock_two","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-block-two","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"The second block of the Smarandache permutation sequence is $(1, 3, 4, 2)$.","title":"Second block of the Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-block-three","kind":"node","lean":"MetaMathlibExt.smarandacheBlock_three","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-block-three","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"The third block of the Smarandache permutation sequence is $(1, 3, 5, 6, 4, 2)$.","title":"Third block of the Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-block-four","kind":"node","lean":"MetaMathlibExt.smarandacheBlock_four","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-block-four","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"The fourth block of the Smarandache permutation sequence is $(1, 3, 5, 7, 8, 6, 4, 2)$.","title":"Fourth block of the Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-prefix-three","kind":"node","lean":"MetaMathlibExt.smarandachePrefix_three","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-prefix-three","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"Concatenating the first three blocks of the Smarandache permutation sequence gives $(1, 2, 1, 3, 4, 2, 1, 3, 5, 6, 4, 2)$.","title":"Concatenation of the first three Smarandache blocks"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-permutation-zero","kind":"node","lean":"MetaMathlibExt.smarandachePermutation_zero","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-permutation-zero","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"Indexing the Smarandache permutation sequence $(a_i)_{i \\ge 0}$ from $0$, its initial term is $a_0 = 1$.","title":"Initial term of the Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-permutation-five","kind":"node","lean":"MetaMathlibExt.smarandachePermutation_five","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-permutation-five","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"Indexing the Smarandache permutation sequence $(a_i)_{i \\ge 0}$ from $0$, its sixth term is $a_5 = 2$.","title":"Sixth term of the Smarandache permutation sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/smarandache-permutation-eleven","kind":"node","lean":"MetaMathlibExt.smarandachePermutation_eleven","parent":24515,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence/index.html#smarandache-permutation-eleven","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/smarandache-permutation-sequence","status":"fully_proved","summary":"Indexing the Smarandache permutation sequence $(a_i)_{i \\ge 0}$ from $0$, its twelfth term is $a_{11} = 2$.","title":"Twelfth term of the Smarandache permutation sequence"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","title":"Throwback sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/kozar-throwback-recurrence","kind":"node","lean":"MetaMathlibExt.kozar_throwback_recurrence","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#kozar-throwback-recurrence","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"fully_proved","summary":"For a sequence $Y = (Y_0, Y_1, Y_2, \\dots)$ of natural numbers, the throwback step produces the sequence $Y'$ with $Y'_i = Y_{i+1}$ for $i < Y_0$, $Y'_{Y_0} = Y_0$, and $Y'_i = Y_i$ for $i > Y_0$: the leading term $Y_0$ moves to index $Y_0$, the terms $Y_1, \\dots, Y_{Y_0}$ shift one place to the left, and all later terms stay in place. Let $X$ be the sequence with $X_i = i + 3$ (that is, $3, 4, 5, \\dots$), and let\u2026","title":"Kozar's conjecture on the throwback sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback","kind":"node","lean":"MetaMathlibExt.throwback","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"fully_proved","summary":"Let $s = (s_0, s_1, s_2, \\dots)$ be a sequence of elements of a set $A$, each element $a$ carrying a weight $w(a) \\in \\mathbb{N}$. The throwback $\\tau_w(s)$ moves the leader $s_0$ to index $w(s_0)$, moves the entries at indices $1, \\dots, w(s_0)$ one place left and leaves later entries in place: $\\tau_w(s)_i = s_{i+1}$ for $i < w(s_0)$, $\\tau_w(s)_{w(s_0)} = s_0$, and $\\tau_w(s)_i = s_i$ for $i > w(s_0)$.","title":"Throwback operation"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-sequence","kind":"node","lean":"MetaMathlibExt.throwbackSequence","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-sequence","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"fully_proved","summary":"For a sequence $X = (x_0, x_1, \\dots)$ of natural numbers, the throwback sequence $T(X) = (t_0, t_1, \\dots)$ records the successive leaders under the throwback operation: $t_n$ is the leader (the entry at index $0$) of $\\tau^n(X)$, where $\\tau$ is the throwback operation in which each entry's weight is its own value.","title":"Throwback sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-examples","kind":"node","lean":"MetaMathlibExt.throwback_examples","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-examples","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"fully_proved","summary":"Let $X = (3, 4, 5, \\dots)$, that is, $x_i = i + 3$. The first five terms of its throwback $\\tau(X)$ are $4, 5, 6, 3, 7$, and the first six terms of its throwback sequence $T(X)$ are $3, 4, 5, 6, 3, 7$ (OEIS A357081).","title":"Throwback of the integers from 3 on"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/partial-placement","kind":"node","lean":"MetaMathlibExt.partialPlacement","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#partial-placement","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"fully_proved","summary":"Let $X = (x_0, x_1, \\dots)$ be a sequence of natural numbers; the source takes $X$ strictly increasing with positive terms. The partial placement sequences $P_n(X)$, for $n \\ge 0$, are indexed by positions $z \\in \\mathbb{Z}$, and each position is either empty or holds a term of $X$. $P_0(X)$ holds $x_0$ at the multiples of $x_0 + 1$ and is empty elsewhere. $P_{n+1}(X)$ agrees with $P_n(X)$ at the filled positions\u2026","title":"Partial placement sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/partial-placement-zero-examples","kind":"node","lean":"MetaMathlibExt.partialPlacement_zero_examples","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#partial-placement-zero-examples","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"fully_proved","summary":"Let $X = (3, 4, 5, \\dots)$, that is, $x_i = i + 3$. The partial placement sequence $P_0(X)$ holds the term $3$ at positions $0$, $4$ and $-4$, and its positions $1$, $2$ and $3$ are empty.","title":"First partial placement sequence of the integers from 3 on"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-final-position-or-frequently-l","kind":"node","lean":"MetaMathlibExt.throwback_final_position_or_frequently_leader","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-final-position-or-frequently-l","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $X = (x_0, x_1, \\dots)$ be a sequence of positive integers, with positions numbered from $0$, the leader being at position $0$. Track the terms of $X$ by their original indices: start from the arrangement $(0, 1, 2, \\dots)$ and apply the throwback repeatedly, where the leader, with original index $i$, moves to position $x_i$. Then for every index $j$, either there are a position $p > 0$ and a number $N$ such\u2026","title":"Each term eventually settles or leads infinitely often"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-sequence-infinite-eq","kind":"node","lean":"MetaMathlibExt.throwbackSequence_infinite_eq","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-sequence-infinite-eq","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $X = (x_0, x_1, \\dots)$ be a sequence of pairwise distinct positive integers, and let $T(X) = (t_0, t_1, \\dots)$ be its throwback sequence. Then every term $x_j$ of $X$ appears infinitely often in $T(X)$: there are infinitely many $k$ with $t_k = x_j$.","title":"Every term recurs in the throwback sequence"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-leader-time-le","kind":"node","lean":"MetaMathlibExt.throwback_leader_time_le","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-leader-time-le","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $X = (x_0, x_1, \\dots)$ be a sequence of pairwise distinct positive integers, with positions numbered from $0$, the leader being at position $0$. Let $\\tau$ be the throwback and $T(X) = (t_0, t_1, \\dots)$ the throwback sequence, so $t_k$ is the leader of $\\tau^k(X)$. Let $n \\ge 1$, and let $N$ be the number of indices $i$ with $x_i < n$. Suppose $a$ and $b$ are natural numbers such that for every $m \\ge 0$, the\u2026","title":"Recursive bound on the time to become leader"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-sequence-limiting-frequency","kind":"node","lean":"MetaMathlibExt.throwbackSequence_limiting_frequency","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-sequence-limiting-frequency","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $X = (x_0, x_1, \\dots)$ be a sequence of pairwise distinct positive integers with throwback sequence $T(X) = (t_0, t_1, \\dots)$, and for a natural number $w$ let $\\sigma(w)$ be the number of indices $i$ with $x_i < w$. Then for every index $j$, writing $x = x_j$, the number of $k < N$ with $t_k = x$, divided by $N$, converges as $N \\to \\infty$ to","title":"Limiting frequencies in throwback sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-sequence-limiting-frequency-ne","kind":"node","lean":"MetaMathlibExt.throwbackSequence_limiting_frequency_ne_zero","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-sequence-limiting-frequency-ne","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $X = (x_0, x_1, \\dots)$ be a sequence of pairwise distinct positive integers, and let $T(X) = (t_0, t_1, \\dots)$ be its throwback sequence. Then for every $j$, the proportion $\\#\\{k < N : t_k = x_j\\} / N$ of terms equal to $x_j$ among the first $N$ terms of $T(X)$ converges, as $N \\to \\infty$, to a nonzero real number.","title":"Every term has a nonzero limiting frequency"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/partial-placement-periodic","kind":"node","lean":"MetaMathlibExt.partialPlacement_periodic","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#partial-placement-periodic","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $X = (x_0, x_1, \\dots)$ be a strictly increasing sequence of positive integers and let $n \\ge 0$. Then the partial placement sequence $P_n(X)$ is periodic: there is an integer $p > 0$ such that for every position $z \\in \\mathbb{Z}$, the positions $z + p$ and $z$ of $P_n(X)$ are either both empty or hold the same term.","title":"Partial placement sequences are periodic"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-sequence-uniformly-recurrent","kind":"node","lean":"MetaMathlibExt.throwbackSequence_uniformly_recurrent","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-sequence-uniformly-recurrent","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $X = (x_0, x_1, \\dots)$ be a strictly increasing sequence of positive integers with throwback sequence $T(X) = (t_0, t_1, \\dots)$. Then every block of consecutive terms of $T(X)$ recurs with bounded gaps: for all natural numbers $i$ and $L$ there is a natural number $G$ such that for every natural number $m$ there is an index $j$ with $m \\le j \\le m + G$ and $t_{j+s} = t_{i+s}$ for all $s < L$.","title":"Throwback sequences of increasing sequences are uniformly recurrent"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/partial-placement-minimal-complete-block","kind":"node","lean":"MetaMathlibExt.partialPlacement_minimal_complete_blocks_iff_configurations","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#partial-placement-minimal-complete-block","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $X = (x_0, x_1, \\dots)$ be a strictly increasing sequence of positive integers, let $\\tau$ be the throwback, let $n \\ge 0$, and write $P_n(X)_z$ for the entry, possibly empty, of the partial placement sequence $P_n(X)$ at position $z \\in \\mathbb{Z}$. The minimal complete block at $z$ is the word $P_n(X)_z, P_n(X)_{z+1}, \\dots, P_n(X)_{z+L-1}$, empty positions included, where $L$ is the least length for which\u2026","title":"Minimal complete blocks of partial placements match configurations"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/throwback-consecutive-perfect-mixing","kind":"node","lean":"MetaMathlibExt.throwback_consecutive_perfect_mixing","parent":24527,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences/index.html#throwback-consecutive-perfect-mixing","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/throwback-sequences","status":"can_prove","summary":"Let $m \\ge 1$ and $n \\ge 0$, let $X = (m, m + 1, m + 2, \\dots)$, and let $\\tau$ be the throwback. Then the positions of the terms $m, m + 1, \\dots, m + n$ in $\\tau^k(X)$ are different for different $k$ with $0 \\le k < (m + 1)^{n + 1}$: for $0 \\le k < l < (m + 1)^{n + 1}$, some $m + i$ with $i \\le n$ occupies different positions in $\\tau^k(X)$ and $\\tau^l(X)$. Moreover, each of $m, m + 1, \\dots, m + n$ occupies the\u2026","title":"Perfect mixing for consecutive integers"},{"data_url":"scopes/Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences.json","id":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences","kind":"scope","navigation":"module","parent":24368,"result":false,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/index.html","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences","status":"fully_proved","title":"Van der Laan sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/is-van-der-laan-window","kind":"node","lean":"MetaMathlibExt.IsVanDerLaanWindow","parent":24543,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/index.html#is-van-der-laan-window","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences","status":"fully_proved","summary":"Four natural numbers $a, b, c, d$ form a Van der Laan window when the largest of them equals the sum of the two smallest, that is, when $\\max(a, b, c, d) = \\min(a + b, a + c, a + d, b + c, b + d, c + d)$.","title":"Van der Laan windows"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/is-van-der-laan","kind":"node","lean":"MetaMathlibExt.IsVanDerLaan","parent":24543,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/index.html#is-van-der-laan","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences","status":"fully_proved","summary":"A sequence $s \\colon \\mathbb{N} \\to \\mathbb{N}$ is a Van der Laan sequence when, for every $n \\in \\mathbb{N}$, the four consecutive terms $s(n), s(n + 1), s(n + 2), s(n + 3)$ form a Van der Laan window, that is, the largest of them equals the sum of the two smallest.","title":"Van der Laan sequences"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/is-van-der-laan-window-ordered-iff","kind":"node","lean":"MetaMathlibExt.isVanDerLaanWindow_ordered_iff","parent":24543,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/index.html#is-van-der-laan-window-ordered-iff","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences","status":"fully_proved","summary":"Let $a \\le b \\le c \\le d$ be natural numbers. Then $a, b, c, d$ form a Van der Laan window if and only if $d = a + b$.","title":"Van der Laan windows in increasing order"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/is-van-der-laan-window-const-iff","kind":"node","lean":"MetaMathlibExt.isVanDerLaanWindow_const_iff","parent":24543,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/index.html#is-van-der-laan-window-const-iff","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences","status":"fully_proved","summary":"For a natural number $k$, the window $k, k, k, k$ is a Van der Laan window if and only if $k = 0$.","title":"Constant Van der Laan windows"},{"id":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/is-van-der-laan-const-iff","kind":"node","lean":"MetaMathlibExt.isVanDerLaan_const_iff","parent":24543,"result":true,"route":"../roadmap/Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences/index.html#is-van-der-laan-const-iff","scope":"Combinatorics/combinatorics-on-words/self-describing-sequences/van-der-laan-sequences","status":"fully_proved","summary":"For a natural number $k$, the constant sequence $s \\colon \\mathbb{N} \\to \\mathbb{N}$ with $s(n) = k$ for all $n$ is a Van der Laan sequence if and only if $k = 0$.","title":"Constant Van der Laan sequences"},{"data_url":"scopes/Combinatorics/permutations.json","id":"Combinatorics/permutations","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/permutations/index.html","scope":"Combinatorics/permutations","status":"can_prove","summary":"Permutations of a finite set as combinatorial objects: pattern containment and avoidance, and statistics and cycle structure; descent-counting numbers go to Eulerian numbers, Stirling cycle numbers to Stirling numbers.","title":"Permutations"},{"data_url":"scopes/Combinatorics/permutations/permutation-patterns.json","id":"Combinatorics/permutations/permutation-patterns","kind":"scope","parent":24549,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/index.html","scope":"Combinatorics/permutations/permutation-patterns","status":"can_prove","summary":"Classical, consecutive, mesh and signed patterns in permutations and in inversion and ascent sequences, pattern-avoidance counts, pattern-replacement classes, stack sorting, and king and separable permutations.","title":"Permutation patterns"},{"id":"Combinatorics/permutations/permutation-patterns/fibonacci-card-parity-alternating-avoidi","kind":"node","lean":"MetaMathlibExt.fibonacci_card_parity_alternating_avoiding_312_231","parent":24550,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci count of parity-preserving permutations avoiding "],["math","\\(312\\)"],["text"," and "],["math","\\(231\\)"]],"source":"Fibonacci count of parity-preserving permutations avoiding $312$ and $231$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/index.html#fibonacci-card-parity-alternating-avoidi","scope":"Combinatorics/permutations/permutation-patterns","status":"fully_proved","summary":"For every integer $n \\ge 1$, the number of permutations $\\pi$ of $\\{0, 1, \\dots, n-1\\}$ such that $\\pi(i) \\equiv i \\pmod 2$ for every $i$ and that avoid the patterns $312$ and $231$ equals the Fibonacci number $F_n$, where $F_1 = F_2 = 1$. Here $\\pi$ contains $312$ if there are $i < j < k$ with $\\pi(j) < \\pi(k) < \\pi(i)$, and contains $231$ if there are $i < j < k$ with $\\pi(k) < \\pi(i) < \\pi(j)$.","title":"Fibonacci count of parity-preserving permutations avoiding $312$ and $231$"},{"id":"Combinatorics/permutations/permutation-patterns/card-nontrivial-1234-3412-classes","kind":"node","lean":"MetaMathlibExt.card_nontrivial_1234_3412_classes","parent":24550,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/index.html#card-nontrivial-1234-3412-classes","scope":"Combinatorics/permutations/permutation-patterns","status":"fully_proved","summary":"Let $n \\ge 7$ and let $S_n$ be the group of permutations of $\\{0, 1, \\dots, n - 1\\}$. A move takes $\\sigma \\in S_n$ with positions $i_1 < i_2 < i_3 < i_4$ such that $\\sigma(i_1) < \\sigma(i_2) < \\sigma(i_3) < \\sigma(i_4)$ to the permutation $\\tau$ with $\\tau(i_1) = \\sigma(i_3)$, $\\tau(i_2) = \\sigma(i_4)$, $\\tau(i_3) = \\sigma(i_1)$, $\\tau(i_4) = \\sigma(i_2)$ and $\\tau(j) = \\sigma(j)$ for every other position $j$;\u2026","title":"Nontrivial classes under {1234, 3412}-pattern replacement"},{"id":"Combinatorics/permutations/permutation-patterns/card-permutations-avoiding-231-with-peak","kind":"node","lean":"MetaMathlibExt.card_permutations_avoiding_231_with_peak_count","parent":24550,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/index.html#card-permutations-avoiding-231-with-peak","scope":"Combinatorics/permutations/permutation-patterns","status":"fully_proved","summary":"Let $n \\ge 1$ and $k$ be natural numbers. A permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ avoids $231$ if there are no positions $i < j < l$ with $\\sigma(l) < \\sigma(i) < \\sigma(j)$, and $\\sigma$ has a peak at a position $i$ with $0 < i < n - 1$ if $\\sigma(i-1) < \\sigma(i)$ and $\\sigma(i+1) < \\sigma(i)$. The number of 231-avoiding permutations of $\\{0, \\dots, n-1\\}$ with exactly $k$ peaks is\u2026","title":"Peaks of 231-avoiding permutations"},{"data_url":"scopes/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting.json","id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","kind":"scope","navigation":"module","parent":24550,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","title":"Cyclic-pattern-avoiding stack sorting"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/same-relative-order","kind":"node","lean":"MetaMathlibExt.SameRelativeOrder","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#same-relative-order","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"Two finite sequences $s = s_1 \\cdots s_\\ell$ and $p = p_1 \\cdots p_m$ of natural numbers have the same relative order if $\\ell = m$ and, for all $1 \\le i, j \\le m$, $s_i < s_j$ holds if and only if $p_i < p_j$.","title":"Sequences with the same relative order"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/contains-pattern-seq","kind":"node","lean":"MetaMathlibExt.ContainsPatternSeq","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#contains-pattern-seq","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"A sequence $w$ of natural numbers contains the pattern $p = p_1 \\cdots p_k$ if some subsequence $u_1 \\cdots u_k$ of $w$ (whose entries need not be adjacent in $w$) has the same relative order as $p$: for all $a, b \\in \\{1, \\dots, k\\}$, $u_a < u_b$ if and only if $p_a < p_b$.","title":"Classical pattern containment in a sequence"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/contains-consecutive-pattern-seq","kind":"node","lean":"MetaMathlibExt.ContainsConsecutivePatternSeq","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#contains-consecutive-pattern-seq","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"A sequence $w$ of natural numbers contains a sequence $p = p_1 \\cdots p_k$ as a consecutive pattern if some $k$ adjacent entries $w_{i+1}, \\dots, w_{i+k}$ of $w$ have the same relative order as $p$: for all $a, b \\in \\{1, \\dots, k\\}$, $w_{i+a} < w_{i+b}$ if and only if $p_a < p_b$.","title":"Consecutive pattern containment"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/restricted-stack-sort-aux","kind":"node","lean":"MetaMathlibExt.restrictedStackSortAux","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#restricted-stack-sort-aux","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"Fix a property of finite sequences of natural numbers, called bad. Given an input sequence $\\pi$ and a stack (listed from top to bottom), read $\\pi$ from left to right. Before pushing the next entry $x$, pop the least number $k$ of entries from the top of the stack such that the remaining stack with $x$ pushed on top, read from top to bottom, is not bad (if there is no such $k$, the whole stack is popped and $x$ is\u2026","title":"Restricted stack-sorting from a given stack"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/west-stack-sort","kind":"node","lean":"MetaMathlibExt.westStackSort","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#west-stack-sort","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For a finite sequence $\\pi$ of natural numbers, $s(\\pi)$ is computed with a stack that starts empty. The entries of $\\pi$ are read from left to right; before the next entry $x$ is pushed, the fewest entries are popped from the top of the stack so that the stack with $x$ on top, read from top to bottom, avoids the pattern $21$, and the popped entries are appended to the output in the order in which they are popped.\u2026","title":"West's stack-sorting map"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/cyclic-stack-sort","kind":"node","lean":"MetaMathlibExt.cyclicStackSort","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#cyclic-stack-sort","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For a finite sequence $\\sigma$ of natural numbers, the map $s_{[\\sigma]}$ sends a finite sequence $\\pi$ of natural numbers to the output of the following procedure. Read $\\pi$ from left to right, starting with an empty stack. Before pushing the next entry $x$, pop the fewest entries from the top of the stack such that the stack with $x$ pushed on top, read from top to bottom, contains no rotation of $\\sigma$ as a\u2026","title":"Cyclic-pattern-avoiding stack-sorting map"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/consecutive-cyclic-stack-sort","kind":"node","lean":"MetaMathlibExt.consecutiveCyclicStackSort","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#consecutive-cyclic-stack-sort","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For a sequence $\\sigma$, the map $s^{[\\sigma]}$ sends a sequence $\\pi$ of natural numbers to the output of the following stack procedure. Read $\\pi$ from left to right, starting with an empty stack. Before pushing the next entry $x$, pop the fewest entries from the top of the stack, sending them to the output in the order they are removed, so that the stack with $x$ on top, read from top to bottom, contains no\u2026","title":"Consecutive-cyclic-pattern-avoiding stack-sorting map"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/cyclic-sorting-machine","kind":"node","lean":"MetaMathlibExt.cyclicSortingMachine","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#cyclic-sorting-machine","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For a sequence $\\sigma$, the cyclic avoidance machine $SC_{[\\sigma]} = s \\circ s_{[\\sigma]}$ sends a sequence $\\pi$ of natural numbers to $s(s_{[\\sigma]}(\\pi))$: it first applies the cyclic-pattern-avoiding stack-sorting map $s_{[\\sigma]}$, whose stack, read from top to bottom, avoids every rotation of $\\sigma$ as a pattern, and then West's stack-sorting map $s$, whose stack, read from top to bottom, avoids the\u2026","title":"Cyclic avoidance machine"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/stack-sort-examples","kind":"node","lean":"MetaMathlibExt.stackSort_examples","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#stack-sort-examples","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"Let $s$ be West's stack-sorting map, $s_{[123]}$ the cyclic-pattern-avoiding stack-sorting map for $123$, $s^{[123]}$ the consecutive-cyclic-pattern-avoiding stack-sorting map for $123$, and $SC_{[123]} = s \\circ s_{[123]}$ the cyclic sorting machine for $123$. Then $s(3124) = 1234$, $s_{[123]}(3124) = 2143$, $s^{[123]}(3124) = 4213$, $SC_{[123]}(3124) = 1234$, $SC_{[123]}(1324) = 3124$ and $SC_{[123]}(231) = 123$.","title":"Stack-sorting examples"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/standardize-seq","kind":"node","lean":"MetaMathlibExt.standardizeSeq","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#standardize-seq","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"The standardization $\\mathrm{st}(s)$ of a finite sequence $s = s_1 s_2 \\cdots s_k$ of natural numbers is obtained by replacing each entry $s_i$ by the number of indices $j$ with $s_j \\le s_i$. When the entries of $s$ are distinct, this number is the rank of $s_i$: the $r$th smallest entry becomes $r$.","title":"Standardization"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/consecutive-run-heads","kind":"node","lean":"MetaMathlibExt.consecutiveRunHeads","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#consecutive-run-heads","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"Given an optional previous entry $y$ and a sequence $s_1 s_2 \\cdots s_\\ell$ of natural numbers, this lists, in order, the entries $s_i$ with $s_i \\ne s_{i-1} + 1$, where $s_0 = y$; if there is no previous entry, $s_1$ is always listed. With no previous entry, these are the minima $a$ of the maximal blocks $a, a + 1, \\dots, a + k$ of adjacent consecutive integers of the sequence.","title":"Heads of runs of consecutive integers"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/reduction-seq","kind":"node","lean":"MetaMathlibExt.reductionSeq","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#reduction-seq","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"The reduction $\\operatorname{red}(s)$ of a finite sequence $s$ of natural numbers is obtained in two steps. First, every maximal block of adjacent entries of the form $a, a + 1, \\ldots, a + k$ is replaced by its first entry $a$. Then the resulting sequence is standardized: each of its entries $x$ is replaced by the number of its entries that are at most $x$, which for distinct entries is the rank of $x$.","title":"Reduction of a sequence"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/standardize-seq-reduction-seq-examples","kind":"node","lean":"MetaMathlibExt.standardizeSeq_reductionSeq_examples","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#standardize-seq-reduction-seq-examples","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"The standardization of $315$ is $\\mathrm{st}(315) = 213$, and the reduction of $16783425$ is $\\mathrm{red}(16783425) = 15324$. Here $\\mathrm{st}$ replaces each entry by its rank, and $\\mathrm{red}$ replaces each maximal block of adjacent consecutive integers $a, a + 1, \\dots, a + k$ by its minimum $a$ and then standardizes.","title":"Standardization and reduction examples"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/superimpose-push","kind":"node","lean":"MetaMathlibExt.superimposePush","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#superimpose-push","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For finite sequences $a$ and $b$ of natural numbers, $\\mathrm{push}(a, b)$ is the sequence obtained by taking the first entry of $b$, then the next two entries of $a$ (only one if just one is left), and repeating; once one of the two sequences is exhausted, the remaining entries of the other follow in order.","title":"Push order of the superimpose operation"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/superimpose","kind":"node","lean":"MetaMathlibExt.superimpose","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#superimpose","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For finite sequences $\\sigma_1$ and $\\sigma_2$ of natural numbers, let $p = \\mathrm{push}(\\mathrm{rev}\\,\\sigma_1, \\mathrm{rev}\\,\\sigma_2)$ be the push order of the superimpose operation on the reversed sequences. Then $\\mathrm{si}(\\sigma_1, \\sigma_2)$ is the reversal of $p$: the stack obtained by pushing the entries of $p$ in order onto an empty stack, read from top to bottom.","title":"Superimpose operation"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/superimpose-examples","kind":"node","lean":"MetaMathlibExt.superimpose_examples","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#superimpose-examples","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For the superimpose operation $\\mathrm{si}$, $\\mathrm{si}((1, 2, 3, 4, 5), (6, 7, 8, 9, 10)) = (6, 7, 1, 8, 2, 3, 9, 4, 5, 10)$ and $\\mathrm{si}((1, 2), (3, 4, 5, 6)) = (3, 4, 5, 1, 2, 6)$.","title":"Superimpose examples"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/xi-permutation","kind":"node","lean":"MetaMathlibExt.xiPermutation","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","The permutation "],["math","\\(\\xi_n\\)"]],"source":"The permutation $\\xi_n$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#xi-permutation","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For a natural number $n$, $\\xi_n$ is the sequence listing the elements of $\\{1, \\dots, n\\}$ whose parity differs from that of $n$ in increasing order, followed by the elements with the same parity as $n$ in increasing order. Thus $\\xi_n = 1, 3, \\dots, n-1, 2, 4, \\dots, n$ for even $n$, and $\\xi_n = 2, 4, \\dots, n-1, 1, 3, \\dots, n$ for odd $n$.","title":"The permutation $\\xi_n$"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/xi-permutation-six-seven","kind":"node","lean":"MetaMathlibExt.xiPermutation_six_seven","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#xi-permutation-six-seven","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, let $\\xi_n$ list the elements of $[n] = \\{1, \\dots, n\\}$ whose parity differs from that of $n$ in increasing order, followed by the elements with the parity of $n$ in increasing order. Then $\\xi_6 = 135246$ and $\\xi_7 = 2461357$.","title":"The parity-split permutations for six and seven"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/is-good-seq","kind":"node","lean":"MetaMathlibExt.IsGoodSeq","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#is-good-seq","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"A finite sequence $s = s_1 s_2 \\cdots s_\\ell$ of natural numbers is good if every entry $s_i$ is the minimum or the maximum of $s_i, s_{i+1}, \\ldots, s_\\ell$.","title":"Good sequences of suffix extrema"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/sorted-permutations","kind":"node","lean":"MetaMathlibExt.sortedPermutations","parent":24554,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#sorted-permutations","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"For a map $f$ from finite sequences of natural numbers to finite sequences of natural numbers and for $n \\in \\mathbb{N}$, $\\mathrm{Sort}_n(f)$ is the finite set of permutations $\\pi$ of $[n] = \\{1, \\dots, n\\}$, written as sequences, such that $f(\\pi)$ is the identity permutation $\\mathrm{id}_n = 12 \\cdots n$.","title":"Permutations sorted by a map"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/is-good-seq-consecutive-cyclic-stack-sor","kind":"node","lean":"MetaMathlibExt.isGoodSeq_consecutiveCyclicStackSort_examples","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small cases of good sequences and of sorting by the consecutive "],["math","\\(321\\)"],["text","-cyclic stack"]],"source":"Small cases of good sequences and of sorting by the consecutive $321$-cyclic stack"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#is-good-seq-consecutive-cyclic-stack-sor","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"fully_proved","summary":"The sequence $51432$ is good (each entry is the minimum or the maximum of itself and all entries after it), and among the six permutations of $\\{1, 2, 3\\}$ exactly $312$ and $321$ are sent to $123$ by the consecutive-cyclic-pattern-avoiding stack-sorting map $s^{[321]}$.","title":"Small cases of good sequences and of sorting by the consecutive $321$-cyclic stack"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/cyclic-sorting-machine-123-iterate-sorts","kind":"node","lean":"MetaMathlibExt.cyclicSortingMachine_123_iterate_sorts","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(n - 2\\)"],["text"," passes of "],["math","\\(SC_{(123)}\\)"],["text"," sort every permutation, and "],["math","\\(n - 3\\)"],["text"," do not"]],"source":"$n - 2$ passes of $SC_{(123)}$ sort every permutation, and $n - 3$ do not"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#cyclic-sorting-machine-123-iterate-sorts","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"Let $n \\ge 3$ and let $SC_{[123]}^k$ be the $k$-fold iterate of the cyclic avoidance machine $SC_{[123]}$. For every permutation $\\pi$ of $[n] = \\{1, \\dots, n\\}$, $SC_{[123]}^{n-2}(\\pi) = 12 \\cdots n$, and the bound is tight: $SC_{[123]}^{n-3}(\\xi_n) \\ne 12 \\cdots n$, where $\\xi_n$ lists the elements of $[n]$ of parity different from $n$ in increasing order, followed by those of the parity of $n$ in increasing order.","title":"$n - 2$ passes of $SC_{(123)}$ sort every permutation, and $n - 3$ do not"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/cyclic-sorting-machine-321-identity-orbi","kind":"node","lean":"MetaMathlibExt.cyclicSortingMachine_321_identity_orbit_period","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual periodicity of the identity under "],["math","\\(SC_{(321)}\\)"]],"source":"Eventual periodicity of the identity under $SC_{(321)}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#cyclic-sorting-machine-321-identity-orbi","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"Let $n \\ge 4$, let $c = \\lceil (2n-1)/3 \\rceil$, and let $SC_{[321]}^k$ be the $k$-fold iterate of the cyclic avoidance machine $SC_{[321]}$. For every natural number $m \\ge \\lceil (n-1-c)/2 \\rceil$, $SC_{[321]}^m(12 \\cdots n) = SC_{[321]}^{m+c}(12 \\cdots n)$.","title":"Eventual periodicity of the identity under $SC_{(321)}$"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/sorted-permutations-consecutive-cyclic-s","kind":"node","lean":"MetaMathlibExt.sortedPermutations_consecutiveCyclicStackSort_123","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","No permutation of length at least four is sorted by the consecutive cyclic "],["math","\\(123\\)"],["text","-stack"]],"source":"No permutation of length at least four is sorted by the consecutive cyclic $123$-stack"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#sorted-permutations-consecutive-cyclic-s","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"Let $s^{[123]}$ be the consecutive-cyclic-pattern-avoiding stack-sorting map for $123$. For every $n \\ge 4$, no permutation of $[n] = \\{1, \\dots, n\\}$ is mapped by $s^{[123]}$ to the identity $12 \\cdots n$; that is, the set $\\mathrm{Sort}_n(s^{[123]})$ of permutations of $[n]$ that $s^{[123]}$ maps to the identity has $0$ elements.","title":"No permutation of length at least four is sorted by the consecutive cyclic $123$-stack"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/card-sorted-permutations-consecutive-cyc","kind":"node","lean":"MetaMathlibExt.card_sortedPermutations_consecutiveCyclicStackSort_321","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutations sorted by the consecutive "],["math","\\(321\\)"],["text","-cyclic stack"]],"source":"Permutations sorted by the consecutive $321$-cyclic stack"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#card-sorted-permutations-consecutive-cyc","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"Let $n \\ge 2$, let $s^{[321]}$ be the stack-sorting map whose stack, read from top to bottom, avoids every rotation of $321$ as a consecutive pattern, and let $\\mathrm{Sort}_n(s^{[321]})$ be the set of permutations of $[n] = \\{1, \\dots, n\\}$ that $s^{[321]}$ maps to the identity $12 \\cdots n$. Then $|\\mathrm{Sort}_n(s^{[321]})| = 2^{n-2}$.","title":"Permutations sorted by the consecutive $321$-cyclic stack"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/reduction-seq-cyclic-sorting-machine-123","kind":"node","lean":"MetaMathlibExt.reductionSeq_cyclicSortingMachine_123_reductionSeq","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reducing before the "],["math","\\(123\\)"],["text"," cyclic sorting machine does not change the reduced output"]],"source":"Reducing before the $123$ cyclic sorting machine does not change the reduced output"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#reduction-seq-cyclic-sorting-machine-123","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"For every permutation $\\pi$ of $[n] = \\{1, \\ldots, n\\}$, applying the cyclic sorting machine $\\mathrm{SC}_{[123]}$ to the reduction $\\operatorname{red}(\\pi)$ and reducing the result gives the reduction of $\\mathrm{SC}_{[123]}(\\pi)$: $\\operatorname{red}(\\mathrm{SC}_{[123]}(\\operatorname{red}(\\pi))) = \\operatorname{red}(\\mathrm{SC}_{[123]}(\\pi))$.","title":"Reducing before the $123$ cyclic sorting machine does not change the reduced output"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/cyclic-sorting-machine-123-iterate-eq-id","kind":"node","lean":"MetaMathlibExt.cyclicSortingMachine_123_iterate_eq_id_iff_reductionSeq","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reduction and iterates of "],["math","\\(SC_{(123)}\\)"]],"source":"Reduction and iterates of $SC_{(123)}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#cyclic-sorting-machine-123-iterate-eq-id","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"Let $\\pi$ be a permutation of $[n] = \\{1, \\dots, n\\}$, written as a sequence, let $\\mathrm{red}(\\pi)$ be its reduction, of length $\\ell$, and let $SC_{[123]}^k$ be the $k$-fold iterate of the cyclic avoidance machine $SC_{[123]}$. For every $k \\ge 0$, $SC_{[123]}^k(\\pi) = 12 \\cdots n$ if and only if $SC_{[123]}^k(\\mathrm{red}(\\pi)) = 12 \\cdots \\ell$.","title":"Reduction and iterates of $SC_{(123)}$"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/length-reduction-seq-cyclic-sorting-mach","kind":"node","lean":"MetaMathlibExt.length_reductionSeq_cyclicSortingMachine_123_lt","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","After one pass of the "],["math","\\(123\\)"],["text"," cyclic sorting machine, the reduction is shorter"]],"source":"After one pass of the $123$ cyclic sorting machine, the reduction is shorter"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#length-reduction-seq-cyclic-sorting-mach","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"For every $n \\ge 2$ and every permutation $\\pi$ of $[n] = \\{1, \\ldots, n\\}$, the reduction $\\operatorname{red}(\\mathrm{SC}_{[123]}(\\pi))$ of the image of $\\pi$ under the cyclic sorting machine $\\mathrm{SC}_{[123]}$ has fewer than $n$ entries.","title":"After one pass of the $123$ cyclic sorting machine, the reduction is shorter"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/cyclic-sorting-machine-321-superimpose","kind":"node","lean":"MetaMathlibExt.cyclicSortingMachine_321_superimpose","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","One pass of "],["math","\\(SC_{(321)}\\)"],["text"," on superimposed increasing runs"]],"source":"One pass of $SC_{(321)}$ on superimposed increasing runs"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#cyclic-sorting-machine-321-superimpose","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"Let $n \\ge 4$ and $1 \\le m < \\lceil (2n-1)/3 \\rceil = \\lfloor (2n+1)/3 \\rfloor$. Then the cyclic avoidance machine $SC_{[321]}$ maps the superimposition $\\mathrm{si}((1, \\dots, m), (m+1, \\dots, n))$ to $\\mathrm{si}((1, \\dots, m+1), (m+2, \\dots, n))$.","title":"One pass of $SC_{(321)}$ on superimposed increasing runs"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/head-tail-of-mem-sorted-permutations-con","kind":"node","lean":"MetaMathlibExt.head_tail_of_mem_sortedPermutations_consecutiveCyclicStackSort_321","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","First entry and tail of permutations sorted by the consecutive "],["math","\\(321\\)"],["text","-cyclic stack"]],"source":"First entry and tail of permutations sorted by the consecutive $321$-cyclic stack"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#head-tail-of-mem-sorted-permutations-con","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"Let $n \\ge 2$ and let $\\mathrm{Sort}_n(s^{[321]})$ be the set of permutations of $[n] = \\{1, \\ldots, n\\}$ that the consecutive-cyclic-pattern-avoiding stack-sorting map $s^{[321]}$ sends to the identity $12 \\cdots n$. Every $\\pi = \\pi_1 \\pi_2 \\cdots \\pi_n$ in $\\mathrm{Sort}_n(s^{[321]})$ has $\\pi_1 = n$, and $\\pi_2 \\cdots \\pi_n$ is good: each of its entries is the minimum or the maximum of itself and all entries\u2026","title":"First entry and tail of permutations sorted by the consecutive $321$-cyclic stack"},{"id":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/mem-sorted-permutations-consecutive-cycl","kind":"node","lean":"MetaMathlibExt.mem_sortedPermutations_consecutiveCyclicStackSort_321_iff","parent":24554,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of the permutations sorted by the consecutive "],["math","\\(321\\)"],["text","-cyclic stack"]],"source":"Characterization of the permutations sorted by the consecutive $321$-cyclic stack"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting/index.html#mem-sorted-permutations-consecutive-cycl","scope":"Combinatorics/permutations/permutation-patterns/cyclic-pattern-avoiding-stack-sorting","status":"can_prove","summary":"Let $n \\ge 2$ and let $\\pi = \\pi_1 \\pi_2 \\cdots \\pi_n$ be a permutation of $[n] = \\{1, \\ldots, n\\}$. The consecutive-cyclic-pattern-avoiding stack-sorting map $s^{[321]}$ sends $\\pi$ to the identity $12 \\cdots n$ (that is, $\\pi \\in \\mathrm{Sort}_n(s^{[321]})$) if and only if $\\pi_1 = n$ and $\\pi_2 \\cdots \\pi_n$ is good: each of its entries is the minimum or the maximum of itself and all entries after it.","title":"Characterization of the permutations sorted by the consecutive $321$-cyclic stack"},{"data_url":"scopes/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations.json","id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","kind":"scope","navigation":"module","parent":24550,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","title":"Mesh patterns in permutations"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/mesh-pattern-generating-function-eq-fact","kind":"node","lean":"MetaMathlibExt.meshPatternGeneratingFunction_eq_factorial_mul_prod_general","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#mesh-pattern-generating-function-eq-fact","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $x \\in R$, let $0 < k \\le n$ be natural numbers, and let $P(m, r) \\in R$ for natural numbers $m, r$. Natural numbers are mapped into $R$. Suppose that $P(k-1, k) = (k-1)!$ and that $P(m, k) = (x + (m - 1))\\,P(m-1, k)$ for every natural number $m \\ge k$. Then","title":"Product formula from the mesh-pattern recurrence"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/mesh-pattern-generating-function-eq-fact-2","kind":"node","lean":"MetaMathlibExt.meshPatternGeneratingFunction_eq_factorial_mul_prod","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#mesh-pattern-generating-function-eq-fact-2","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"Let $0 < k \\le n$ be natural numbers, let $x$ be a real number, and let $P(m, r, y)$ be real numbers indexed by natural numbers $m, r$ and a real number $y$. Suppose that $P(k-1, k, x) = (k-1)!$ and that $P(m, k, x) = (x + m - 1)\\,P(m-1, k, x)$ for every natural number $m \\ge k$. Then","title":"Product formula from the mesh-pattern recurrence, real case"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/match-pred","kind":"node","lean":"MetaMathlibExt.MatchPred","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#match-pred","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $m$ elements, with positions numbered $1, \\dots, m$, let $p$ be the position of the largest value of $\\sigma$, and let $r$ be a natural number. A position $i$ is a mesh-pattern match of $\\sigma$ with parameter $r$ when $\\sigma(t) < \\sigma(i)$ for every position $t < i$ and at least $r - 1$ positions $t$ with $i < t < p$ have $\\sigma(t) > \\sigma(i)$.","title":"Mesh-pattern match"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/zero-pred","kind":"node","lean":"MetaMathlibExt.ZeroPred","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#zero-pred","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $m$ elements, with positions numbered $1, \\dots, m$, and let $r$ be a natural number. The zero condition holds for $\\sigma$ with parameter $r$ when some position $p \\ge r$ holds the largest value of $\\sigma$.","title":"Mesh-pattern zero condition"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/mesh-n","kind":"node","lean":"MetaMathlibExt.meshN","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#mesh-n","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $m$ elements, with positions numbered $1, \\dots, m$, and let $r$ be a natural number. The statistic $N_r(\\sigma)$ is the number of mesh-pattern matches of $\\sigma$ with parameter $r$, plus $1$ if the mesh-pattern zero condition holds, that is, if the largest value of $\\sigma$ sits at a position $p \\ge r$.","title":"Canonical mesh-pattern statistic"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/poly-n","kind":"node","lean":"MetaMathlibExt.PolyN","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#poly-n","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"For natural numbers $m$ and $r$, the polynomial $P_{m,r}(X) \\in \\mathbb{N}[X]$ is the sum of $X^{N_r(\\sigma)}$ over all permutations $\\sigma$ of $m$ elements, where $N_r(\\sigma)$ is the canonical mesh-pattern statistic with parameter $r$.","title":"Mesh-pattern counting polynomial"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/mesh-pattern-count-stirling-recursion","kind":"node","lean":"MetaMathlibExt.meshPatternCount_stirlingRecursion","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#mesh-pattern-count-stirling-recursion","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"Let $1 < k \\le n$ and $j > 0$. For a permutation $\\sigma$ of $m$ elements, with positions numbered $1, \\dots, m$, let $p$ be the position of its largest value and let $N_r(\\sigma)$ be the number of positions $i$ such that $\\sigma(t) < \\sigma(i)$ for all $t < i$ and at least $r - 1$ positions $t$ with $i < t < p$ have $\\sigma(t) > \\sigma(i)$, plus $1$ if $p \\ge r$. Let $P_{m,r}(X) \\in \\mathbb{N}[X]$ be the sum of\u2026","title":"Stirling-type recursion for the mesh-pattern counting polynomial"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/mesh-pattern-count-eq-factorial-mul-r-st","kind":"node","lean":"MetaMathlibExt.meshPatternCount_eq_factorial_mul_rStirlingNumber_general","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#mesh-pattern-count-eq-factorial-mul-r-st","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"Let $2 \\le k \\le n$ and let $j$ be a natural number. For a permutation $\\sigma$ of $\\{1, \\dots, n\\}$ with positions numbered $1, \\dots, n$, let $p$ be the position of its largest value, and let $N_k(\\sigma)$ be the canonical mesh-pattern statistic: the number of positions $i$ such that $\\sigma(t) < \\sigma(i)$ for all $t < i$ and at least $k - 1$ positions $t$ with $i < t < p$ have $\\sigma(t) > \\sigma(i)$, plus $1$\u2026","title":"Distribution of the canonical mesh-pattern statistic"},{"id":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/mesh-pattern-count-eq-factorial-mul-r-st-2","kind":"node","lean":"MetaMathlibExt.meshPatternCount_eq_factorial_mul_rStirlingNumber","parent":24586,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations/index.html#mesh-pattern-count-eq-factorial-mul-r-st-2","scope":"Combinatorics/permutations/permutation-patterns/mesh-patterns-in-permutations","status":"fully_proved","summary":"Let $2 \\le k \\le n$ and let $j$ be a natural number. For a permutation $\\sigma$ of $\\{1, \\dots, n\\}$, let $p$ be the position of its largest entry, and call a position $i$ a match if $\\sigma(t) < \\sigma(i)$ for every position $t < i$ and at least $k - 1$ positions $q$ with $i < q < p$ have $\\sigma(q) > \\sigma(i)$. Let $N(\\sigma)$ be the number of matches, plus $1$ if $p \\ge k$. Then the number of permutations\u2026","title":"Counting permutations by mesh-pattern matches"},{"data_url":"scopes/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences.json","id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","kind":"scope","navigation":"module","parent":24550,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","title":"Pattern avoidance in inversion sequences"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-contains-pattern","kind":"node","lean":"MetaMathlibExt.InvSeqContainsPattern","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-contains-pattern","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"A sequence $e = e_0 e_1 \\cdots e_{n-1}$ of natural numbers contains the length-3 pattern $p = p_0 p_1 p_2$ of natural numbers when there are positions $i_0 < i_1 < i_2$ in $\\{0, \\dots, n - 1\\}$ such that, for all $a, b \\in \\{0, 1, 2\\}$, $e_{i_a} < e_{i_b}$ if and only if $p_a < p_b$; that is, the subsequence $e_{i_0} e_{i_1} e_{i_2}$ is order-isomorphic to $p$.","title":"Containment of a length-3 pattern"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/not-inv-seq-contains-pattern-of-lt","kind":"node","lean":"MetaMathlibExt.not_invSeqContainsPattern_of_lt","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#not-inv-seq-contains-pattern-of-lt","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"If $n < 3$, then no sequence $e_0 e_1 \\cdots e_{n-1}$ of natural numbers contains a length-3 pattern $p$.","title":"Short sequences contain no length-3 pattern"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-count","kind":"node","lean":"MetaMathlibExt.invSeqAvoidCount","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-count","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"An inversion sequence of length $n$ is a sequence $e = e_0 e_1 \\cdots e_{n-1}$ of natural numbers with $e_i \\le i$ for every $i$. It contains a length-3 pattern $p = p_0 p_1 p_2$ (a triple of natural numbers) when some subsequence $e_a e_b e_c$ with $a < b < c$ is order-isomorphic to $p$. For a set $B$ of length-3 patterns, $|I_n(B)|$ is the number of inversion sequences of length $n$ that contain no pattern of $B$.","title":"Number of inversion sequences avoiding a set of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-count-one","kind":"node","lean":"MetaMathlibExt.invSeqAvoidCount_one","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-count-one","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"For every set $B$ of length-3 patterns, the number $|I_1(B)|$ of inversion sequences of length one that avoid every pattern in $B$ equals $1$: the only inversion sequence of length one is $0$, and it avoids every length-3 pattern.","title":"One inversion sequence of length one"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-gf","kind":"node","lean":"MetaMathlibExt.invSeqAvoidGF","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-gf","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"For a set $B$ of length-3 patterns, $F_B(x) = \\sum_{n \\ge 1} |I_n(B)| x^n$ is the formal power series in $\\mathbb{Q}[[x]]$ with constant coefficient $0$ whose coefficient of $x^n$, for $n \\ge 1$, is the number $|I_n(B)|$ of inversion sequences of length $n$ that avoid every pattern in $B$.","title":"Generating function of inversion sequences avoiding a set of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/coeff-inv-seq-avoid-gf","kind":"node","lean":"MetaMathlibExt.coeff_invSeqAvoidGF","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#coeff-inv-seq-avoid-gf","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"For every set $B$ of length-3 patterns and every $n \\ge 0$, the generating function $F_B(x) = \\sum_{m \\ge 1} |I_m(B)| x^m \\in \\mathbb{Q}[[x]]$ has constant coefficient $0$, and its coefficient of $x^{n+1}$ is $|I_{n+1}(B)|$, the number of inversion sequences of length $n + 1$ that avoid every pattern in $B$.","title":"Coefficients of the generating function of pattern-avoiding inversion sequences"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-000-010-100-120-101-110","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_000_010_100_120_101_110","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","I-Wilf equivalence of "],["math","\\(\\{000, 010, 100, 101, 120\\}\\)"],["text"," and "],["math","\\(\\{000, 010, 100, 110, 120\\}\\)"]],"source":"I-Wilf equivalence of $\\{000, 010, 100, 101, 120\\}$ and $\\{000, 010, 100, 110, 120\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-000-010-100-120-101-110","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write length-3 patterns as digit strings ($101$ is the pattern $(1, 0, 1)$) and let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in the set $B$. Then for every $n \\ge 0$, $|I_n(\\{000, 010, 100, 101, 120\\})| = |I_n(\\{000, 010, 100, 110, 120\\})|$.","title":"I-Wilf equivalence of $\\{000, 010, 100, 101, 120\\}$ and $\\{000, 010, 100, 110, 120\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-000-010-120-five-class","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_000_010_120_five_class","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three I-Wilf-equivalent sets of five patterns containing "],["math","\\(000\\)"],["text",", "],["math","\\(010\\)"],["text"," and "],["math","\\(120\\)"]],"source":"Three I-Wilf-equivalent sets of five patterns containing $000$, $010$ and $120$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-000-010-120-five-class","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write length-3 patterns as digit strings ($101$ is the pattern $(1, 0, 1)$) and let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in the set $B$. For $B_1 = \\{000, 010, 101, 120, 201\\}$, $B_2 = \\{000, 010, 101, 120, 210\\}$ and $B_3 = \\{000, 010, 110, 120, 210\\}$, and for every $n \\ge 0$, $|I_n(B_1)| = |I_n(B_2)| = |I_n(B_3)|$.","title":"Three I-Wilf-equivalent sets of five patterns containing $000$, $010$ and $120$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-gf-011-102-120","kind":"node","lean":"MetaMathlibExt.invSeqAvoidGF_011_102_120","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inversion sequences avoiding "],["math","\\(\\{011, 102, 120\\}\\)"]],"source":"Inversion sequences avoiding $\\{011, 102, 120\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-gf-011-102-120","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $B = \\{011, 102, 120\\}$, where a pattern is written as a digit string ($101$ is the triple $(1, 0, 1)$), and let $F_B(x) = \\sum_{n \\ge 1} |I_n(B)| x^n \\in \\mathbb{Q}[[x]]$ be the generating function of the inversion sequences avoiding every pattern in $B$. Then","title":"Inversion sequences avoiding $\\{011, 102, 120\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-000-102-120-five-class","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_000_102_120_five_class","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three I-Wilf-equivalent sets of five patterns containing "],["math","\\(000\\)"],["text",", "],["math","\\(102\\)"],["text"," and "],["math","\\(120\\)"]],"source":"Three I-Wilf-equivalent sets of five patterns containing $000$, $102$ and $120$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-000-102-120-five-class","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write length-3 patterns as digit strings ($101$ is the pattern $(1, 0, 1)$) and let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in the set $B$. For $B_1 = \\{000, 101, 102, 120, 201\\}$, $B_2 = \\{000, 101, 102, 120, 210\\}$ and $B_3 = \\{000, 102, 110, 120, 210\\}$, and for every $n \\ge 0$, $|I_n(B_1)| = |I_n(B_2)| = |I_n(B_3)|$.","title":"Three I-Wilf-equivalent sets of five patterns containing $000$, $102$ and $120$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-000-100-120-201-210","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_000_100_120_201_210","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two I-Wilf equivalences exchanging "],["math","\\(201\\)"],["text"," and "],["math","\\(210\\)"],["text"," in five-pattern sets"]],"source":"Two I-Wilf equivalences exchanging $201$ and $210$ in five-pattern sets"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-000-100-120-201-210","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write length-3 patterns as digit strings ($101$ is the pattern $(1, 0, 1)$) and let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in the set $B$. For a pattern $\\tau$ let $A_\\tau = \\{000, 100, 101, 120, \\tau\\}$ and $B_\\tau = \\{000, 100, 110, 120, \\tau\\}$. Then for every $n \\ge 0$, $|I_n(A_{201})| = |I_n(A_{210})|$ and $|I_n(B_{201})| = |I_n(B_{210})|$.","title":"Two I-Wilf equivalences exchanging $201$ and $210$ in five-pattern sets"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-010-102-120-five-classes","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_010_102_120_five_classes","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","I-Wilf equivalences among five-pattern sets containing "],["math","\\(010\\)"],["text",", "],["math","\\(102\\)"],["text"," and "],["math","\\(120\\)"]],"source":"I-Wilf equivalences among five-pattern sets containing $010$, $102$ and $120$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-010-102-120-five-classes","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write length-3 patterns as digit strings ($101$ is the pattern $(1, 0, 1)$) and let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in the set $B$. For a pattern $\\tau$ let $A_\\tau = \\{010, \\tau, 102, 120, 201\\}$ and $A'_\\tau = \\{010, \\tau, 102, 120, 210\\}$, and let $B = \\{010, 101, 102, 120, 201\\}$. Then for every $n \\ge 0$, $|I_n(A_{100})| = |I_n(A_{110})|$ and\u2026","title":"I-Wilf equivalences among five-pattern sets containing $010$, $102$ and $120$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-010-100-101-120-201-210","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_010_100_101_120_201_210","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","I-Wilf equivalence of "],["math","\\(\\{010, 100, 101, 120, 201\\}\\)"],["text"," and "],["math","\\(\\{010, 100, 101, 120, 210\\}\\)"]],"source":"I-Wilf equivalence of $\\{010, 100, 101, 120, 201\\}$ and $\\{010, 100, 101, 120, 210\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-010-100-101-120-201-210","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write length-3 patterns as digit strings ($101$ is the pattern $(1, 0, 1)$) and let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in the set $B$. Then for every $n \\ge 0$, $|I_n(\\{010, 100, 101, 120, 201\\})| = |I_n(\\{010, 100, 101, 120, 210\\})|$.","title":"I-Wilf equivalence of $\\{010, 100, 101, 120, 201\\}$ and $\\{010, 100, 101, 120, 210\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-000-010-120-six-class","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_000_010_120_six_class","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Five I-Wilf-equivalent sets of six patterns containing "],["math","\\(000\\)"],["text",", "],["math","\\(010\\)"],["text"," and "],["math","\\(120\\)"]],"source":"Five I-Wilf-equivalent sets of six patterns containing $000$, $010$ and $120$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-000-010-120-six-class","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write length-3 patterns as digit strings ($101$ is the pattern $(1, 0, 1)$) and let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in the set $B$. For a pattern $\\tau$ let $A_\\tau = \\{000, 010, 100, 101, 120, \\tau\\}$ and $B_\\tau = \\{000, 010, 100, 110, 120, \\tau\\}$, and let $C = \\{000, 010, 101, 110, 120, 201\\}$. Then for every $n \\ge 0$,\u2026","title":"Five I-Wilf-equivalent sets of six patterns containing $000$, $010$ and $120$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-102-110-six-class","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_102_110_six_class","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-102-110-six-class","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write a length-3 pattern as the word of its values, so $101$ is the pattern $(1, 0, 1)$, and for a set $B$ of patterns let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in $B$. Let $A_\\tau = \\{011, 101, 102, 110, 120, \\tau\\}$ and $C = \\{011, 021, 102, 110, 201, 210\\}$. Then $|I_n(A_{201})| = |I_n(A_{210})| = |I_n(C)|$ for every $n \\ge 0$.","title":"Three I-Wilf-equivalent six-pattern sets containing 102 and 110"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-gf-010-011-102-201-210-sev","kind":"node","lean":"MetaMathlibExt.invSeqAvoidGF_010_011_102_201_210_seven","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-gf-010-011-102-201-210-sev","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $B_1 = \\{010, 011, 100, 101, 102, 201, 210\\}$, $B_2 = \\{010, 011, 100, 102, 110, 201, 210\\}$ and $B_3 = \\{010, 011, 101, 102, 110, 201, 210\\}$, where a pattern is written as a digit string ($101$ is the triple $(1, 0, 1)$). For a set $B$ of patterns let $F_B(x) = \\sum_{n \\ge 1} |I_n(B)| x^n \\in \\mathbb{Q}[[x]]$ be the generating function of the inversion sequences avoiding every pattern in $B$. Then for each\u2026","title":"A common generating function for three sets of seven patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-eight-201-210-and-gf","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_eight_201_210_and_gf","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-eight-201-210-and-gf","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write a length-3 pattern as the word of its values, so $101$ is the pattern $(1, 0, 1)$. For a set $B$ of patterns let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in $B$, and let $F_B(x) = \\sum_{n \\ge 1} |I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B_\\tau = \\{000, 010, 100, 101, 102, 110, 120, \\tau\\}$. Then $|I_n(B_{201})| = |I_n(B_{210})|$ for every $n \\ge 0$, and for every\u2026","title":"Eight-pattern sets differing in 201 and 210: equivalence and generating function"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-gf-eight-radical-three","kind":"node","lean":"MetaMathlibExt.invSeqAvoidGF_eight_radical_three","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-gf-eight-radical-three","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $B_1 = \\{000, 010, 100, 101, 102, 120, 201, 210\\}$, $B_2 = \\{000, 010, 100, 102, 110, 120, 201, 210\\}$ and $B_3 = \\{000, 010, 101, 102, 110, 120, 201, 210\\}$, where a pattern is written as a digit string ($101$ is the triple $(1, 0, 1)$). For a set $B$ of patterns let $F_B(x) = \\sum_{n \\ge 1} |I_n(B)| x^n \\in \\mathbb{Q}[[x]]$ be the generating function of the inversion sequences avoiding every pattern in $B$.\u2026","title":"A common generating function for three sets of eight patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-gf-000-100-101-102-110-120","kind":"node","lean":"MetaMathlibExt.invSeqAvoidGF_000_100_101_102_110_120_201_210","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inversion sequences avoiding "],["math","\\(\\{000, 100, 101, 102, 110, 120, 201, 210\\}\\)"]],"source":"Inversion sequences avoiding $\\{000, 100, 101, 102, 110, 120, 201, 210\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-gf-000-100-101-102-110-120","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $B = \\{000, 100, 101, 102, 110, 120, 201, 210\\}$, where a pattern is written as a digit string ($101$ is the triple $(1, 0, 1)$), and let $F_B(x) = \\sum_{n \\ge 1} |I_n(B)| x^n \\in \\mathbb{Q}[[x]]$ be the generating function of the inversion sequences avoiding every pattern in $B$. For every power series $s \\in \\mathbb{Q}[[x]]$ with constant term $1$ and $s^2 = 1 - 2x - 3x^2$,","title":"Inversion sequences avoiding $\\{000, 100, 101, 102, 110, 120, 201, 210\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-gf-nine-patterns","kind":"node","lean":"MetaMathlibExt.invSeqAvoidGF_nine_patterns","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-gf-nine-patterns","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $B = \\{000, 010, 100, 101, 102, 110, 120, 201, 210\\}$, where a pattern is written as a digit string ($101$ is the triple $(1, 0, 1)$), and let $F_B(x) = \\sum_{n \\ge 1} |I_n(B)| x^n \\in \\mathbb{Q}[[x]]$ be the generating function of the inversion sequences avoiding every pattern in $B$. For every power series $s \\in \\mathbb{Q}[[x]]$ with constant term $1$ and $s^2 = 1 - 2x - 3x^2$,","title":"Inversion sequences avoiding nine patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-class-count","kind":"node","lean":"MetaMathlibExt.invSeq_wilfClassCount","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-class-count","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $P$ be the set of the thirteen length-3 patterns $000, 001, 010, 011, 012, 021, 100, 101, 102, 110, 120, 201, 210$, each written as the word of its values. For $B \\subseteq P$ let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in $B$, and let $w_d$ be the number of distinct sequences $(|I_n(B)|)_{n \\ge 0}$ as $B$ ranges over the $d$-element subsets of $P$ (the number of\u2026","title":"Number of I-Wilf classes of sets of length-3 patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-reducible-pattern-sets","kind":"node","lean":"MetaMathlibExt.invSeq_reducible_pattern_sets","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-reducible-pattern-sets","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"An inversion sequence of length $n$ is a sequence $e_0 e_1 \\cdots e_{n-1}$ of natural numbers with $e_i \\le i$ for all $i$; write a length-3 pattern as a digit string ($101$ is the pattern $(1, 0, 1)$). For each pair $(C, \\tau)$ listed below and every $n$, every inversion sequence of length $n$ that avoids every pattern in $C$ also avoids $\\tau$, so $C \\cup \\{\\tau\\}$ and $C$ are avoided by the same inversion\u2026","title":"Reducible sets of length-3 patterns for inversion sequences"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-avoid-gf-010-101-102-201-210-cat","kind":"node","lean":"MetaMathlibExt.invSeqAvoidGF_010_101_102_201_210_catalan","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-avoid-gf-010-101-102-201-210-cat","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $B = \\{010, 101, 102, 201, 210\\}$ and $B' = \\{011, 101, 110, 201, 210\\}$, where a pattern is written as a digit string ($101$ is the triple $(1, 0, 1)$). For a set $A$ of patterns let $F_A(x) = \\sum_{n \\ge 1} |I_n(A)| x^n \\in \\mathbb{Q}[[x]]$ be the generating function of the inversion sequences avoiding every pattern in $A$, and let $C(x) = \\sum_{n \\ge 0} C_n x^n$ be the generating function of the Catalan\u2026","title":"A Catalan generating function for two sets of five patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-102-120-101-110-parts-b-c","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_102_120_101_110_parts_B_C","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-102-120-101-110-parts-b-c","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write a length-3 pattern as the word of its values, so $101$ is the pattern $(1, 0, 1)$, and for a set $S$ of patterns let $|I_n(S)|$ be the number of inversion sequences of length $n$ that avoid every pattern in $S$. Let $B_\\tau = \\{100, \\tau, 102, 120, 210\\}$ and $C_\\tau = \\{\\tau, 102, 120, 201, 210\\}$. Then for every $n \\ge 0$, $|I_n(B_{101})| = |I_n(B_{110})|$ and $|I_n(C_{101})| = |I_n(C_{110})|$.","title":"I-Wilf-equivalence of 101 and 110 alongside 102, 120, 210"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-010-120-201-210-101-110","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_010_120_201_210_101_110","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-010-120-201-210-101-110","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write a length-3 pattern as the word of its values, so $101$ is the pattern $(1, 0, 1)$, and for a set $B$ of patterns let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in $B$. For every $n \\ge 0$,","title":"I-Wilf-equivalence of 101 and 110 alongside 010, 120, 201, 210"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/inv-seq-wilf-010-100-101-102-120-201-210","kind":"node","lean":"MetaMathlibExt.invSeq_wilf_010_100_101_102_120_201_210","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exchanging "],["math","\\(201\\)"],["text"," and "],["math","\\(210\\)"],["text"," in "],["math","\\(\\{010, 100, 101, 102, 120, 201\\}\\)"]],"source":"Exchanging $201$ and $210$ in $\\{010, 100, 101, 102, 120, 201\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#inv-seq-wilf-010-100-101-102-120-201-210","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write length-3 patterns as digit strings ($101$ is the pattern $(1, 0, 1)$) and let $|I_n(B)|$ be the number of inversion sequences of length $n$ that avoid every pattern in the set $B$. For a pattern $\\tau$ let $A_\\tau = \\{010, 100, 101, 102, 120, \\tau\\}$. Then for every $n \\ge 0$, $|I_n(A_{201})| = |I_n(A_{210})|$.","title":"Exchanging $201$ and $210$ in $\\{010, 100, 101, 102, 120, 201\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-pattern2","kind":"node","lean":"MetaMathlibExt.SignedPattern2","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-pattern2","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"A length-2 signed pattern is a word $\\tau_1\\tau_2$ whose underlying word $|\\tau_1||\\tau_2|$ is $00$, $01$ or $10$ and each of whose letters may be barred, for example $\\bar10$ or $0\\bar1$. It is given by the comparison of $|\\tau_1|$ with $|\\tau_2|$ (equal for $00$, less for $01$, greater for $10$) together with whether $\\tau_1$ is barred and whether $\\tau_2$ is barred.","title":"Length-2 signed pattern"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/card-signed-pattern2","kind":"node","lean":"MetaMathlibExt.card_signedPattern2","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#card-signed-pattern2","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"There are exactly twelve length-2 signed patterns.","title":"Number of length-2 signed patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-contains","kind":"node","lean":"MetaMathlibExt.SignedInvSeqContains","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-contains","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"Let $e = e_1 \\cdots e_n$ be a sequence whose entries are natural numbers $|e_i|$, each barred or not, and let $\\tau_1\\tau_2$ be a length-2 signed pattern. Then $e$ contains $\\tau_1\\tau_2$ if there are positions $i < j$ such that $|e_i|$ is equal to, smaller than or larger than $|e_j|$ exactly as $|\\tau_1|$ is equal to, smaller than or larger than $|\\tau_2|$, and $e_i$ and $e_j$ are barred exactly when $\\tau_1$ and\u2026","title":"Containment of a length-2 signed pattern"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-avoid-count","kind":"node","lean":"MetaMathlibExt.signedInvAvoidCount","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-avoid-count","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ in which each entry $e_i$ is a natural number $|e_i| \\le i - 1$, either barred or not; that is, $e_i \\in \\lbrace 0, \\bar 0, 1, \\bar 1, \\dots, i - 1, \\overline{i - 1} \\rbrace$. For a finite set $B$ of length-2 signed patterns and $n \\ge 0$, $|\\bar I_n(B)|$ is the number of signed inversion sequences of length $n$ that contain no pattern of $B$.","title":"Number of signed inversion sequences avoiding a set of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-avoid-count-zero","kind":"node","lean":"MetaMathlibExt.signedInvAvoidCount_zero","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-avoid-count-zero","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"The empty sequence is the only signed inversion sequence of length $0$, and it contains no pattern, so for every finite set $B$ of length-2 signed patterns the number $|\\bar I_0(B)|$ of signed inversion sequences of length $0$ avoiding every pattern of $B$ is $1$.","title":"Signed inversion sequences of length zero"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-avoid-count-one","kind":"node","lean":"MetaMathlibExt.signedInvAvoidCount_one","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-avoid-count-one","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"The signed inversion sequences of length $1$ are $0$ and $\\bar 0$, and neither contains a length-2 signed pattern, so for every finite set $B$ of length-2 signed patterns the number $|\\bar I_1(B)|$ of signed inversion sequences of length $1$ avoiding every pattern of $B$ is $2$.","title":"Signed inversion sequences of length one"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-avoid-gf","kind":"node","lean":"MetaMathlibExt.signedInvAvoidGF","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-avoid-gf","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"For a finite set $B$ of length-2 signed patterns, $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n$ is the formal power series with rational coefficients whose coefficient of $x^n$ is the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$.","title":"Generating function of signed inversion sequences avoiding a set of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/coeff-signed-inv-avoid-gf","kind":"node","lean":"MetaMathlibExt.coeff_signedInvAvoidGF","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#coeff-signed-inv-avoid-gf","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"For a finite set $B$ of length-2 signed patterns and $m \\ge 0$, let $|\\bar I_m(B)|$ be the number of signed inversion sequences of length $m$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{m \\ge 0} |\\bar I_m(B)| x^m$ be their generating function, a formal power series with rational coefficients. Then for every $n \\ge 0$ the coefficient of $x^n$ in $F_B(x)$ is $|\\bar I_n(B)|$, viewed as a rational number.","title":"Coefficients of the avoidance generating function"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-wilf-class-count","kind":"node","lean":"MetaMathlibExt.signedWilfClassCount","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-wilf-class-count","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"For $k \\ge 0$, $\\mathrm{sw}_k$ is the number of Wilf-equivalence classes among the sets of $k$ length-2 signed patterns, that is, the number of distinct counting functions $n \\mapsto |\\bar{I}_n(B)|$ (for $n \\ge 0$) over all sets $B$ of exactly $k$ length-2 signed patterns, where $|\\bar{I}_n(B)|$ is the number of signed inversion sequences of length $n$ avoiding every pattern of $B$.","title":"Number of signed Wilf classes"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-wilf-class-count-zero","kind":"node","lean":"MetaMathlibExt.signedWilfClassCount_zero","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-wilf-class-count-zero","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"fully_proved","summary":"The number $\\mathrm{sw}_0$ of Wilf-equivalence classes among the sets of $0$ length-2 signed patterns is $1$: the empty set is the only set of no patterns.","title":"The empty pattern set forms one signed Wilf class"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b0b1-b00-01","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b0b1_b00_01","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b0b1-b00-01","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Lemma 1 (signed patterns 0\u03041\u0304, 0\u03040, 01)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-count-b0b0-b00-0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_count_b0b0_b00_0b1","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-count-b0b0-b00-0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write a length-2 signed pattern as $\\tau_1\\tau_2$, where $00$, $01$ and $10$ record whether $|\\tau_1|$ is equal to, smaller than or larger than $|\\tau_2|$ and a bar marks a barred letter, and write $|\\bar I_n(B)|$ for the number of signed inversion sequences of length $n$ avoiding every pattern of a set $B$ of such patterns. Let $B = \\lbrace \\bar 0\\bar 0, \\bar 0 0, 0\\bar 1 \\rbrace$ and\u2026","title":"Wilf-equivalence and count for {0\u03040\u0304, 0\u03040, 01\u0304} (Lemma 2)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-count-b0b1-b0b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_count_b0b1_b0b0_0b1","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-count-b0b1-b0b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Lemma 3 (signed patterns 0\u03041\u0304, 0\u03040\u0304, 01\u0304)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-count-b0b1-b0b0-2","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_count_b0b1_b0b0_b01","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-count-b0b1-b0b0-2","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Lemma 4 (signed patterns 0\u03041\u0304, 0\u03040\u0304, 0\u03041)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b1-01","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b1_01","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar1, 01\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, 01\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b1-01","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B = \\{\\bar1\\bar0, \\bar0\\bar1, 01\\}$ and $C = \\{\\bar10, \\bar0\\bar1, 01\\}$. Then $|\\bar I_n(B)| = |\\bar I_n(C)|$ for every $n \\ge 0$, and","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, 01\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b0b0-b01","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b0b0_b01","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b0b0-b01","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{1}\\bar{0}, \\bar{0}\\bar{0}, \\bar{0}1\\})| =","title":"Mansour\u2013Safadi Lemma 6"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b10-b0b1-b0b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b10_b0b1_b0b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b10-b0b1-b0b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{1}0, \\bar{0}\\bar{1}, \\bar{0}\\bar{0}\\})| =","title":"Mansour\u2013Safadi Lemma 7"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-avoid-count-eq-factorial-succ","kind":"node","lean":"MetaMathlibExt.signedInvAvoidCount_eq_factorial_succ","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-avoid-count-eq-factorial-succ","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write a length-2 signed pattern as $\\tau_1\\tau_2$, where $00$, $01$ and $10$ record whether $|\\tau_1|$ is equal to, smaller than or larger than $|\\tau_2|$ and a bar marks a barred letter, and write $|\\bar I_n(B)|$ for the number of signed inversion sequences of length $n$ avoiding every pattern of a set $B$ of such patterns. For every $n \\ge 0$ and each of the eight sets\u2026","title":"Eight signed-pattern triples with (n + 1)! avoiders (Lemma 8)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b10-b0b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b10_b0b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b10-b0b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then $\\{\\bar1\\bar0, \\bar10, \\bar0\\bar0\\} \\sim \\{\\bar1\\bar0, \\bar10, \\bar00\\}$.","title":"Mansour\u2013Safadi Lemma 9: a Wilf-equivalence of three-pattern sets"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b0b0-1b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b0b0_1b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b0b0-1b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{1}\\bar{0}, \\bar{0}\\bar{0}, 1\\bar{0}\\})| =","title":"Mansour\u2013Safadi Lemma 10"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-wilf-class-count-three-to-twelve","kind":"node","lean":"MetaMathlibExt.signedWilfClassCount_three_to_twelve","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-wilf-class-count-three-to-twelve","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $\\mathrm{sw}_k$ be the number of Wilf-equivalence classes among the sets of $k$ length-2 signed patterns. Then $\\mathrm{sw}_3 = 93$, $\\mathrm{sw}_4 = 195$, $239 \\le \\mathrm{sw}_5 \\le 240$, $\\mathrm{sw}_6 = 190$, $\\mathrm{sw}_7 = 107$, $\\mathrm{sw}_8 = 49$, $\\mathrm{sw}_9 = 22$, $\\mathrm{sw}_{10} = 8$, $\\mathrm{sw}_{11} = 3$ and $\\mathrm{sw}_{12} = 1$.","title":"Signed Wilf classes of sets of three to twelve patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b10-b0b0-b00-00","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b10_b0b0_b00_00","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b10-b0b0-b00-00","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then $\\{\\bar1\\bar0, \\bar10, \\bar0\\bar0, \\bar00, 00\\} \\sim \\{\\bar1\\bar0, \\bar10, \\bar0\\bar1, 0\\bar1, 00\\}$.","title":"Mansour\u2013Safadi Class 141 Wilf-equivalence"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-wilf-class-count-three","kind":"node","lean":"MetaMathlibExt.signedWilfClassCount_three","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-wilf-class-count-three","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"The number $\\mathrm{sw}_3$ of Wilf-equivalence classes among the sets of $3$ length-2 signed patterns is $93$.","title":"Signed Wilf classes of sets of three patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-wilf-class-count-four","kind":"node","lean":"MetaMathlibExt.signedWilfClassCount_four","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-wilf-class-count-four","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"The number $\\mathrm{sw}_4$ of Wilf-equivalence classes among the sets of $4$ length-2 signed patterns is $195$.","title":"Signed Wilf classes of sets of four patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-wilf-class-count-five-to-twelve","kind":"node","lean":"MetaMathlibExt.signedWilfClassCount_five_to_twelve","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-wilf-class-count-five-to-twelve","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Let $\\mathrm{sw}_k$ be the number of Wilf-equivalence classes among the sets of $k$ length-2 signed patterns. Then $239 \\le \\mathrm{sw}_5 \\le 240$, $\\mathrm{sw}_6 = 190$, $\\mathrm{sw}_7 = 107$, $\\mathrm{sw}_8 = 49$, $\\mathrm{sw}_9 = 22$, $\\mathrm{sw}_{10} = 8$, $\\mathrm{sw}_{11} = 3$ and $\\mathrm{sw}_{12} = 1$.","title":"Signed Wilf classes of sets of five to twelve patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b0b0-b00-b01","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b0b0_b00_b01_01","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b0b0-b00-b01","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Theorem 3, class C2 (signed patterns 0\u03040\u0304, 0\u03040, 0\u03041, 01)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b0b1-b0b0-b01","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b0b1_b0b0_b01_0b1","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b0b1-b0b0-b01","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Theorem 3, class C3 (signed patterns 0\u03041\u0304, 0\u03040\u0304, 0\u03041, 01\u0304)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b0b1-b0b0-b00","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b0b1_b0b0_b00_0b1","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b0b1-b0b0-b00","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Theorem 3, class C4 (signed patterns 0\u03041\u0304, 0\u03040\u0304, 0\u03040, 01\u0304)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b0b1-b0b0-0b1","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b0b1_b0b0_0b1_01","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b0b1-b0b0-0b1","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Theorem 3, class C12 (signed patterns 0\u03041\u0304, 0\u03040\u0304, 01\u0304, 01)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b0b1-b0b0-b01-2","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b0b1_b0b0_b01_01","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b0b1-b0b0-b01-2","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Theorem 3, class C15 (signed patterns 0\u03041\u0304, 0\u03040\u0304, 0\u03041, 01)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b0b1-b00-0b1","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b0b1_b00_0b1_01","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b0b1-b00-0b1","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Theorem 3, class C16 (signed patterns 0\u03041\u0304, 0\u03040, 01\u0304, 01)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b0b1-b01-0b1-0b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b0b1_b01_0b1_0b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b0b1-b01-0b1-0b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{0}\\bar{1}, \\bar{0}1, 0\\bar{1}, 0\\bar{0}\\})| =","title":"Mansour\u2013Safadi Theorem 3, class C21"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b1-b01","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b1_b01_00","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 00\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 00\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b1-b01","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B = \\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 00\\}$ and $C = \\{\\bar10, \\bar0\\bar0, 0\\bar1, 01\\}$. Then $|\\bar I_n(B)| = |\\bar I_n(C)|$ for every $n \\ge 0$, and\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 00\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b0-0b1","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b0_0b1_01","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar0, 0\\bar1, 01\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar0, 0\\bar1, 01\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b0-0b1","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B = \\{\\bar1\\bar0, \\bar0\\bar0, 0\\bar1, 01\\}$ and $C = \\{\\bar10, \\bar0\\bar1, \\bar01, 00\\}$. Then $|\\bar I_n(B)| = |\\bar I_n(C)|$ for every $n \\ge 0$, and\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar0, 0\\bar1, 01\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b1-b0b","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b1_b0b0_00","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar1, \\bar0\\bar0, 00\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar0\\bar0, 00\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b1-b0b","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B_1 = \\{\\bar1\\bar0, \\bar0\\bar1, \\bar0\\bar0, 00\\}$, $B_2 = \\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 01\\}$, $B_3 = \\{\\bar10, \\bar0\\bar1, \\bar01, 01\\}$,\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar0\\bar0, 00\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b1-b00","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b1_b00_01","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, 01\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, 01\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b1-b00","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B = \\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, 01\\}$ and $C = \\{\\bar10, \\bar0\\bar1, \\bar00, 01\\}$. Then $|\\bar I_n(B)| = |\\bar I_n(C)|$ for every $n \\ge 0$, and\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, 01\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b1-0b1","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b1_0b1_01","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar1, 01\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar1, 01\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b1-0b1","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B = \\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar1, 01\\}$ and $C = \\{\\bar10, \\bar0\\bar1, 0\\bar1, 01\\}$. Then $|\\bar I_n(B)| = |\\bar I_n(C)|$ for every $n \\ge 0$, and\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar1, 01\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b0-b00","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b0_b00_01","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar0, \\bar00, 01\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar0, \\bar00, 01\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b0-b00","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B = \\{\\bar1\\bar0, \\bar0\\bar0, \\bar00, 01\\}$ and $C = \\{\\bar1\\bar0, \\bar01, 0\\bar1, 01\\}$. Then $|\\bar I_n(B)| = |\\bar I_n(C)|$ for every $n \\ge 0$, and for\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar0, \\bar00, 01\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b0-0b1-2","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b0_0b1_00","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b0-0b1-2","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"A signed inversion sequence of length $n$ is a word $e_1 \\cdots e_n$ with $e_k \\in \\{0, \\bar 0, 1, \\bar 1, \\dots, k - 1, \\overline{k - 1}\\}$. It avoids a length-2 signed pattern $\\tau_1\\tau_2$ (one of $00$, $01$, $10$, each letter possibly barred) when no $i < j$ have $|e_i|$, $|e_j|$ in the same order as $|\\tau_1|$, $|\\tau_2|$ with $e_i$, $e_j$ barred exactly when $\\tau_1$, $\\tau_2$ are. Write $\\bar I_n(B)$ for\u2026","title":"Mansour\u2013Safadi Theorem 3, class C41 (signed patterns 1\u03040\u0304, 0\u03040\u0304, 01\u0304, 00)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b0b0-b01-01","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b0b0_b01_01","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b0b0-b01-01","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{1}\\bar{0}, \\bar{0}\\bar{0}, \\bar{0}1, 01\\})| =","title":"Mansour\u2013Safadi Theorem 3, class C45"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b1-b00-2","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b1_b00_00","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, 00\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, 00\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b1-b00-2","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B_1 = \\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, 00\\}$, $B_2 = \\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar0, 00\\}$ and $B_3 = \\{\\bar10, \\bar0\\bar1, 0\\bar0, 00\\}$. Then\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, 00\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b1-b01-2","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b1_b01_0b1","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 0\\bar1\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 0\\bar1\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b1-b01-2","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B = \\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 0\\bar1\\}$ and $C = \\{\\bar10, \\bar0\\bar1, \\bar01, 0\\bar1\\}$. Then $|\\bar I_n(B)| = |\\bar I_n(C)|$ for every $n \\ge 0$,\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar01, 0\\bar1\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-and-gf-b1b0-b0b1-0b1-2","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_and_gf_b1b0_b0b1_0b1_00","parent":24596,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed inversion sequences avoiding "],["math","\\(\\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar1, 00\\}\\)"]],"source":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar1, 00\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-and-gf-b1b0-b0b1-0b1-2","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"For a set $B$ of length-2 signed patterns (a bar marks a barred letter), let $|\\bar I_n(B)|$ be the number of signed inversion sequences of length $n$ avoiding every pattern of $B$, and let $F_B(x) = \\sum_{n \\ge 0} |\\bar I_n(B)| x^n \\in \\mathbb{Q}[[x]]$. Let $B = \\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar1, 00\\}$ and $C = \\{\\bar10, \\bar0\\bar1, 0\\bar1, 00\\}$. Then $|\\bar I_n(B)| = |\\bar I_n(C)|$ for every $n \\ge 0$, and for\u2026","title":"Signed inversion sequences avoiding $\\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar1, 00\\}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b0b1-b0b0-b01-0b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b0b1_b0b0_b01_0b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b0b1-b0b0-b01-0b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{0}\\bar{1}, \\bar{0}\\bar{0}, \\bar{0}1, 0\\bar{0}\\})| =","title":"Mansour\u2013Safadi Theorem 4"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b0b0-b01-0b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b0b0_b01_0b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b0b0-b01-0b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{1}\\bar{0}, \\bar{0}\\bar{0}, \\bar{0}1, 0\\bar{0}\\})| = $|\\bar{I}_n(\\{\\bar{1}0, \\bar{0}\\bar{1}, \\bar{0}\\bar{0}, 0\\bar{0}\\})| = $|\\bar{I}_n(\\{\\bar{1}\\bar{0}, \\bar{0}\\bar{0},\u2026","title":"Mansour\u2013Safadi Theorem 5"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b10-b00-0b1-00","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b10_b00_0b1_00","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b10-b00-0b1-00","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{1}0, \\bar{0}0, 0\\bar{1}, 00\\})| =","title":"Mansour\u2013Safadi Theorem 6"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b0b1-b0b0-b00","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b0b1_b0b0_b00","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b0b1-b0b0-b00","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then\u2026","title":"Mansour\u2013Safadi Theorem 7: four Wilf-equivalent four-pattern sets"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b0b1-b0b0-1b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b0b1_b0b0_1b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b0b1-b0b0-1b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar0\\bar0, 1\\bar0\\} \\sim \\{\\bar1\\bar0, \\bar0\\bar1, 0\\bar0, 1\\bar0\\}$ and\u2026","title":"Mansour\u2013Safadi Theorem 8: two Wilf-equivalences of four-pattern sets"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b10-b0b0-0b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b10_b0b0_0b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b10-b0b0-0b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then $\\{\\bar1\\bar0, \\bar10, \\bar0\\bar0, 0\\bar0\\} \\sim \\{\\bar1\\bar0, \\bar10, \\bar00, 0\\bar0\\}$.","title":"Mansour\u2013Safadi Theorem 9: a Wilf-equivalence of four-pattern sets"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b0b1-b00-b01-0b","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b0b1_b00_b01_0b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b0b1-b00-b01-0b","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, \\bar01, 0\\bar0\\} \\sim \\{\\bar1\\bar0, \\bar0\\bar1, \\bar0\\bar0, \\bar01, 0\\bar0\\}$,\u2026","title":"Mansour\u2013Safadi Theorem 12: four Wilf-equivalences of five-pattern sets"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b0b1-b00-b01-1b","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b0b1_b00_b01_1b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b0b1-b00-b01-1b","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then $\\{\\bar1\\bar0, \\bar0\\bar1, \\bar00, \\bar01, 1\\bar0\\} \\sim \\{\\bar1\\bar0, \\bar0\\bar1, \\bar0\\bar0, \\bar01, 1\\bar0\\}$.","title":"Mansour\u2013Safadi Theorem 13: a Wilf-equivalence of five-pattern sets"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b10-b0b1-0b1-0b0-1b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b10_b0b1_0b1_0b0_1b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b10-b0b1-0b1-0b0-1b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Let $B_1 = \\{\\bar{1}0, \\bar{0}\\bar{1}, 0\\bar{1}, 0\\bar{0}, 1\\bar{0}\\}$, $B_2 = \\{\\bar{1}\\bar{0}, \\bar{0}\\bar{1}, 0\\bar{1}, 0\\bar{0}, 1\\bar{0}\\}$,\u2026","title":"Mansour\u2013Safadi Theorem 14"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b00-b01-0b0-1b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b00_b01_0b0_1b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b00-b01-0b0-1b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write $\\bar{I}_n(B)$ for the set of signed inversion sequences of length $n$ that avoid every length-2 signed pattern in $B$, where a pattern is one of $00$, $01$, $10$ with a bar over each barred letter. Then for every $n \\ge 0$, $|\\bar{I}_n(\\{\\bar{1}\\bar{0}, \\bar{0}0, \\bar{0}1, 0\\bar{0}, 1\\bar{0}\\})| =","title":"Mansour\u2013Safadi Theorem 15"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b10-b0b1-b00-b0","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b10_b0b1_b00_b01_0b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b10-b0b1-b00-b0","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then $\\{\\bar1\\bar0, \\bar10, \\bar0\\bar1, \\bar00, \\bar01, 0\\bar0\\} \\sim \\{\\bar1\\bar0, \\bar10, \\bar0\\bar1, \\bar0\\bar0, \\bar01, 0\\bar0\\}$.","title":"Mansour\u2013Safadi Theorem 16: a Wilf-equivalence of six-pattern sets"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-avoid-count-eq-two-mul-factor","kind":"node","lean":"MetaMathlibExt.signedInvAvoidCount_eq_two_mul_factorial","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-avoid-count-eq-two-mul-factor","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Write a length-2 signed pattern as $\\tau_1\\tau_2$, where $00$, $01$ and $10$ record whether $|\\tau_1|$ is equal to, smaller than or larger than $|\\tau_2|$ and a bar marks a barred letter, and write $|\\bar I_n(A)|$ for the number of signed inversion sequences of length $n$ avoiding every pattern of a set $A$ of such patterns. For every $n \\ge 1$ and each of the three sets\u2026","title":"Three six-pattern sets with 2 \u00b7 n! avoiders (Theorem 17)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/signed-inv-seq-wilf-b1b0-b10-b0b1-b0b0-b","kind":"node","lean":"MetaMathlibExt.signedInvSeq_wilf_b1b0_b10_b0b1_b0b0_b00_b01_1b0","parent":24596,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences/index.html#signed-inv-seq-wilf-b1b0-b10-b0b1-b0b0-b","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-inversion-sequences","status":"can_prove","summary":"Length-2 signed patterns are written as in the paper, a bar marking a barred letter, and $B \\sim C$ means that for every $n \\ge 0$ the number $|\\bar I_n(B)|$ of signed inversion sequences of length $n$ avoiding every pattern of $B$ equals $|\\bar I_n(C)|$. Then $\\{\\bar1\\bar0, \\bar10, \\bar0\\bar1, \\bar0\\bar0, \\bar00, \\bar01, 1\\bar0\\} \\sim \\{\\bar1\\bar0, \\bar10, \\bar0\\bar0, \\bar00, \\bar01, 0\\bar0, 1\\bar0\\}$.","title":"Mansour\u2013Safadi Theorem 18: a Wilf-equivalence of seven-pattern sets"},{"data_url":"scopes/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences.json","id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","kind":"scope","navigation":"module","parent":24550,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","title":"Pattern avoidance in weak ascent sequences"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/word-weak-ascents","kind":"node","lean":"MetaMathlibExt.wordWeakAscents","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#word-weak-ascents","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"For a word $a_1 a_2 \\cdots a_m$ of natural numbers, $\\mathrm{wasc}(a_1 \\cdots a_m)$ is the number of weak ascents, that is, of positions $j$ with $1 \\le j < m$ and $a_j \\le a_{j+1}$; in particular the empty word and every word of length one have no weak ascents.","title":"Number of weak ascents of a word"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/is-weak-ascent-seq","kind":"node","lean":"MetaMathlibExt.IsWeakAscentSeq","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#is-weak-ascent-seq","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"A word $a_1 a_2 \\cdots a_n$ over $\\mathbb{N}$ is a weak ascent sequence if either $n = 0$, or $a_1 = 0$ and $a_i \\le 1 + \\operatorname{wasc}(a_1 \\cdots a_{i-1})$ for every $2 \\le i \\le n$, where $\\operatorname{wasc}(u)$ is the number of weak ascents of $u$, that is, of positions $j$ with $u_j \\le u_{j+1}$.","title":"Weak ascent sequence"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/contains-pattern","kind":"node","lean":"MetaMathlibExt.ContainsPattern","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#contains-pattern","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"A word $w$ over $\\mathbb{N}$ contains a pattern $p$ (another word over $\\mathbb{N}$) if $w$ has a subsequence $s$, not necessarily contiguous, of the same length as $p$ that is order-isomorphic to $p$: for all positions $i$ and $j$, $s_i < s_j$ if and only if $p_i < p_j$.","title":"Pattern containment in words"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/length-three-patterns","kind":"node","lean":"MetaMathlibExt.lengthThreePatterns","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#length-three-patterns","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"The set of patterns of length three consists of the thirteen words $000$, $001$, $010$, $011$, $012$, $021$, $100$, $101$, $102$, $110$, $120$, $201$ and $210$.","title":"The thirteen patterns of length three"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/ascent-avoiders","kind":"node","lean":"MetaMathlibExt.ascentAvoiders","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#ascent-avoiders","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"For a finite set $P$ of patterns (words over $\\mathbb{N}$) and $n \\in \\mathbb{N}$, $A_n(P)$ is the set of ascent sequences of length $n$ that contain no pattern of $P$.","title":"Ascent sequences avoiding a set of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-avoiders","kind":"node","lean":"MetaMathlibExt.weakAscentAvoiders","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-avoiders","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"For a finite set $P$ of patterns (words over $\\mathbb{N}$) and a natural number $n$, $\\mathrm{WA}_n(P)$ is the set of weak ascent sequences of length $n$ that contain none of the patterns in $P$.","title":"Weak ascent sequences avoiding a set of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/is-weak-ascent-seq-example","kind":"node","lean":"MetaMathlibExt.isWeakAscentSeq_example","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The weak ascent sequence "],["math","\\(00203413\\)"]],"source":"The weak ascent sequence $00203413$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#is-weak-ascent-seq-example","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"The word $00203413$ has five weak ascents and is a weak ascent sequence.","title":"The weak ascent sequence $00203413$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/contains-pattern-example","kind":"node","lean":"MetaMathlibExt.containsPattern_example","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The word "],["math","\\(00203413\\)"],["text"," contains "],["math","\\(011\\)"]],"source":"The word $00203413$ contains $011$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#contains-pattern-example","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"The word $00203413$ contains the pattern $011$.","title":"The word $00203413$ contains $011$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/card-length-three-patterns","kind":"node","lean":"MetaMathlibExt.card_lengthThreePatterns","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#card-length-three-patterns","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"The set of patterns of length three, $\\{000, 001, 010, 011, 012, 021, 100, 101, 102, 110, 120, 201, 210\\}$, has exactly $13$ elements.","title":"Number of patterns of length three"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/ascent-wilf-classes-quadruples","kind":"node","lean":"MetaMathlibExt.ascent_wilf_classes_quadruples","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#ascent-wilf-classes-quadruples","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"For a set $P$ of patterns, let $A_n(P)$ be the set of ascent sequences of length $n$ avoiding every pattern in $P$. As $P$ ranges over the $4$-element subsets of the thirteen patterns of length three, the counting sequences $n \\mapsto |A_n(P)|$ take exactly $74$ distinct values. Equivalently, there are $74$ A-Wilf-equivalence classes of such quadruples, where $P$ and $Q$ are A-Wilf equivalent when\u2026","title":"Number of Wilf classes of ascent sequences avoiding four length-3 patterns"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-wilf-classes-quadruples","kind":"node","lean":"MetaMathlibExt.weakAscent_wilf_classes_quadruples","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-wilf-classes-quadruples","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Write $\\mathrm{WA}_n(P)$ for the set of weak ascent sequences of length $n$ avoiding every pattern in $P$, and call two sets of patterns $P$ and $Q$ WA-Wilf equivalent when $|\\mathrm{WA}_n(P)| = |\\mathrm{WA}_n(Q)|$ for every $n \\ge 0$. As $P$ ranges over the $4$-element subsets of the thirteen patterns of length three, the counting sequences $n \\mapsto |\\mathrm{WA}_n(P)|$ take exactly $228$ or exactly $229$\u2026","title":"WA-Wilf classes of pattern quadruples"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-weakly-increasing-card","kind":"node","lean":"MetaMathlibExt.weakAscent_weaklyIncreasing_card","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-weakly-increasing-card","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Let $C_n$ denote the Catalan numbers and $M_n$ the Motzkin numbers. For every natural number $n$, (i) the number of weakly increasing weak ascent sequences of length $n$ is $C_n$, and (ii) the number of weakly increasing weak ascent sequences of length $n$ that avoid the pattern $000$ is $M_n$.","title":"Weakly increasing weak ascent sequences: Catalan and Motzkin counts"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-wilf-class99","kind":"node","lean":"MetaMathlibExt.weakAscent_wilf_class99","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-wilf-class99","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Write $\\mathrm{WA}_n(P)$ for the set of weak ascent sequences of length $n$ avoiding every pattern in $P$, with patterns written as words (for example $011$). The pattern sets $\\{011, 100, 102, 201\\}$ and $\\{011, 100, 120, 201\\}$ are WA-Wilf equivalent: for every $n \\ge 0$, $|\\mathrm{WA}_n(\\{011, 100, 102, 201\\})| = |\\mathrm{WA}_n(\\{011, 100, 120, 201\\})|$.","title":"Weak ascent Wilf equivalence, class 99"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-wilf-class93","kind":"node","lean":"MetaMathlibExt.weakAscent_wilf_class93","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-wilf-class93","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Write $\\mathrm{WA}_n(P)$ for the set of weak ascent sequences of length $n$ avoiding every pattern in $P$, with patterns written as words (for example $011$). The pattern sets $P = \\{011, 102, 120, 201\\}$ and $\\{011, 102, 120, 210\\}$ are WA-Wilf equivalent, that is, $|\\mathrm{WA}_n(P)| = |\\mathrm{WA}_n(\\{011, 102, 120, 210\\})|$ for every $n \\ge 0$. Moreover, the formal power series\u2026","title":"Weak ascent Wilf equivalence and generating function, class 93"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-wilf-class162","kind":"node","lean":"MetaMathlibExt.weakAscent_wilf_class162","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-wilf-class162","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"For every natural number $n$, the number of weak ascent sequences of length $n$ avoiding every pattern in $\\{010, 100, 101, 120\\}$ equals the number avoiding every pattern in $\\{010, 100, 110, 120\\}$.","title":"Weak-ascent Wilf equivalence in Class 162"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-wilf-class205","kind":"node","lean":"MetaMathlibExt.weakAscent_wilf_class205","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-wilf-class205","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"For every natural number $n$, the number of weak ascent sequences of length $n$ avoiding every pattern in $\\{100, 102, 120, 201\\}$ equals the number avoiding every pattern in $\\{102, 110, 120, 201\\}$.","title":"Weak-ascent Wilf equivalence in Class 205"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-wilf-class206","kind":"node","lean":"MetaMathlibExt.weakAscent_wilf_class206","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-wilf-class206","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"For every natural number $n$, the number of weak ascent sequences of length $n$ avoiding every pattern in $\\{100, 102, 120, 210\\}$ equals the number avoiding every pattern in $\\{101, 102, 120, 201\\}$, the number avoiding every pattern in $\\{101, 102, 120, 210\\}$ and the number avoiding every pattern in $\\{102, 110, 120, 210\\}$.","title":"Weak-ascent Wilf equivalence in Class 206"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-wilf-class219","kind":"node","lean":"MetaMathlibExt.weakAscent_wilf_class219","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-wilf-class219","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"For every natural number $n$, the number of weak ascent sequences of length $n$ avoiding every pattern in $\\{100, 101, 102, 201\\}$ equals the number avoiding every pattern in $\\{100, 110, 120, 201\\}$.","title":"Weak-ascent Wilf equivalence in Class 219"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/sum-choose-mul-two-pow-add-two-pow-eq-fi","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_two_pow_add_two_pow_eq_fib","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#sum-choose-mul-two-pow-add-two-pow-eq-fi","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Let $F_m$ denote the Fibonacci numbers, with $F_0 = 0$ and $F_1 = 1$. For every natural number $n \\ge 4$,","title":"Fibonacci closed form of a binomial sum with powers of two"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/reducible-pattern-sets","kind":"node","lean":"MetaMathlibExt.reducible_pattern_sets","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#reducible-pattern-sets","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"For a set $P$ of patterns, let $A_n(P)$ and $WA_n(P)$ be the sets of ascent sequences and of weak ascent sequences of length $n$ that avoid every pattern in $P$. For each set $C$ of patterns and each pattern $\\tau$ listed below, with $B = C \\cup \\{\\tau\\}$, we have $A_n(B) = A_n(C)$ and $WA_n(B) = WA_n(C)$ for every natural number $n$:","title":"Reducible pattern sets (Theorem 3)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-class86-card-eq-motzkin","kind":"node","lean":"MetaMathlibExt.weakAscent_class86_card_eq_motzkin","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-class86-card-eq-motzkin","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Let $M_n$ denote the Motzkin numbers. For every natural number $n \\ge 3$, the number of weak ascent sequences of length $n$ avoiding every pattern in $\\{000, 021, 101, 120\\}$, the number avoiding every pattern in $\\{000, 021, 101, 102\\}$ and the number avoiding every pattern in $\\{000, 021, 110, 120\\}$ are each equal to $M_n + M_{n-2}$.","title":"Motzkin counts for weak ascent sequences in Class 86"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-wilf-class61","kind":"node","lean":"MetaMathlibExt.weakAscent_wilf_class61","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-wilf-class61","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"For every natural number $n$, the number of weak ascent sequences of length $n$ avoiding every pattern in $\\{010, 011, 102, 201\\}$ equals the number avoiding every pattern in $\\{010, 011, 120, 201\\}$.","title":"Weak-ascent Wilf equivalence in Class 61"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/weak-ascent-class211-generating-function","kind":"node","lean":"MetaMathlibExt.weakAscent_class211_generatingFunction","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#weak-ascent-class211-generating-function","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Let $a_n$ be the number of weak ascent sequences of length $n$ that avoid each of the patterns $101$, $102$, $110$ and $201$, and let $F = \\sum_{n \\ge 0} a_n X^n$ in $\\mathbb{Q}[[X]]$. For every power series $S \\in \\mathbb{Q}[[X]]$ with $S^2 = 1 - 4X$ and constant term $1$,","title":"Generating function for weak ascent sequences avoiding 101, 102, 110 and 201"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/is-big-entry","kind":"node","lean":"MetaMathlibExt.IsBigEntry","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#is-big-entry","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"Let $w = w_1 \\cdots w_n$ be a word over $\\mathbb{N}$ and $2 \\le k \\le n$. The entry $w_k$ is a big entry of $w$ if it equals $1$ plus the number of weak ascents of $w_1 \\cdots w_{k-1}$, that is, of positions $j < k - 1$ with $w_j \\le w_{j+1}$. In a weak ascent sequence this is the largest value allowed at position $k$. The entry $w_k$ is referred to by its zero-based index $i = k - 1$, so the index ranges over\u2026","title":"Big entry"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/is-big-entry-example","kind":"node","lean":"MetaMathlibExt.isBigEntry_example","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Big entries of "],["math","\\(00203413\\)"]],"source":"Big entries of $00203413$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#is-big-entry-example","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"In the word $w = 00203413$, with positions numbered from $1$, the entries $w_3 = 2$ and $w_6 = 4$ are big entries and the entry $w_7 = 1$ is not.","title":"Big entries of $00203413$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/last-descent-top","kind":"node","lean":"MetaMathlibExt.lastDescentTop","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#last-descent-top","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"For a finite sequence $w = w_1 w_2 \\cdots w_m$ of natural numbers, a descent of $w$ is an index $1 \\le i < m$ with $w_i > w_{i+1}$, and $w_i$ is its descent top. The position of the last descent top of $w$ is the largest index $i$ with $w_i > w_{i+1}$ (positions counted from $1$), and it is $0$ if $w$ has no descent.","title":"Position of the last descent top"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/last-descent-top-example","kind":"node","lean":"MetaMathlibExt.lastDescentTop_example","parent":24679,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#last-descent-top-example","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"Each of the sequences $0010$, $0020$, $0021$ and $0120$ has its last descent top at position $3$, and the sequence $012$, which has no descent, has last descent top position $0$.","title":"Examples of the last descent top"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/quadruple-uset","kind":"node","lean":"MetaMathlibExt.quadrupleUSet","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sets "],["math","\\(U_{n,j}\\)"]],"source":"The sets $U_{n,j}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#quadruple-uset","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"Let $W_n = WA_n(011, 100, 201, 210)$ be the set of weak ascent sequences of length $n$ that avoid each of the patterns $011$, $100$, $201$ and $210$. For natural numbers $n$ and $j$, $U_{n,j}$ is the set of sequences $w = w_1 w_2 \\cdots w_n$ in $W_n$ such that the last descent top of $w$ is at position $n - 1$, $w$ has a big entry, the first $n - 2$ entries of $w$ are not all $0$, $w_{n-1} \\ne n - 2$, and\u2026","title":"The sets $U_{n,j}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/quadruple-uset-zero","kind":"node","lean":"MetaMathlibExt.quadrupleUSet_zero","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(U_{0,j}\\)"],["text"," is empty"]],"source":"$U_{0,j}$ is empty"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#quadruple-uset-zero","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"For every natural number $j$, the set $U_{0,j}$ is empty.","title":"$U_{0,j}$ is empty"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/quadruple-vset","kind":"node","lean":"MetaMathlibExt.quadrupleVSet","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sets "],["math","\\(V_{n,j}\\)"]],"source":"The sets $V_{n,j}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#quadruple-vset","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"Let $W_n = WA_n(011, 100, 201, 210)$ be the set of weak ascent sequences of length $n$ that avoid each of the patterns $011$, $100$, $201$ and $210$. For natural numbers $n$ and $j$, $V_{n,j}$ is the set of sequences $w = w_1 w_2 \\cdots w_n$ in $W_n$ such that the last descent top of $w$ is at position $n - 2$, $w_n \\le n - 3$, $w_{n-1} \\ne 0$, and $w_{n-2} = w_{n-1} + j$.","title":"The sets $V_{n,j}$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/quadruple-vset-subset","kind":"node","lean":"MetaMathlibExt.quadrupleVSet_subset","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(V_{n,j}\\)"],["text"," lies in "],["math","\\(W_n\\)"]],"source":"$V_{n,j}$ lies in $W_n$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#quadruple-vset-subset","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"fully_proved","summary":"For all natural numbers $n$ and $j$, the set $V_{n,j}$ is contained in $W_n = WA_n(011, 100, 201, 210)$, the set of weak ascent sequences of length $n$ that avoid each of the patterns $011$, $100$, $201$ and $210$.","title":"$V_{n,j}$ lies in $W_n$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/is-big-entry-succ-or-descent","kind":"node","lean":"MetaMathlibExt.isBigEntry_succ_or_descent","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Successors of big entries in "],["math","\\(W_n\\)"]],"source":"Successors of big entries in $W_n$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#is-big-entry-succ-or-descent","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Let $W_n$ be the set of weak ascent sequences of length $n$ avoiding the patterns $011$, $100$, $201$ and $210$, and let $w = w_1 \\cdots w_n \\in W_n$. If $2 \\le k \\le n - 1$ and $w_k$ is a big entry of $w$, then either $w_{k+1}$ is a big entry, or $w_{k+1} < w_k$ and $w_j$ is a big entry for every $j$ with $k + 2 \\le j \\le n$.","title":"Successors of big entries in $W_n$"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/quadruple-uset-quadruple-vset-recurrence","kind":"node","lean":"MetaMathlibExt.quadrupleUSet_quadrupleVSet_recurrence","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for "],["math","\\(|U_{n,1}|\\)"],["text"," and "],["math","\\(|V_{n,1}|\\)"],["text"," (Lemma 2)"]],"source":"Recurrence for $|U_{n,1}|$ and $|V_{n,1}|$ (Lemma 2)"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#quadruple-uset-quadruple-vset-recurrence","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"Let $f$ be either of the sequences $u_n = |U_{n,1}|$ and $v_n = |V_{n,1}|$. Then $f_j = 0$ for every $j \\le 5$, $f_6 = 1$, and $f_n = 5 f_{n-1} - 7 f_{n-2} + 2 f_{n-3}$ for every $n \\ge 7$.","title":"Recurrence for $|U_{n,1}|$ and $|V_{n,1}|$ (Lemma 2)"},{"id":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/quadruple-uset-quadruple-vset-card-formu","kind":"node","lean":"MetaMathlibExt.quadrupleUSet_quadrupleVSet_card_formulas","parent":24679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum formulas for "],["math","\\(|U_{n,1}|\\)"],["text"," and "],["math","\\(|V_{n,1}|\\)"],["text"," (Proposition 2)"]],"source":"Sum formulas for $|U_{n,1}|$ and $|V_{n,1}|$ (Proposition 2)"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences/index.html#quadruple-uset-quadruple-vset-card-formu","scope":"Combinatorics/permutations/permutation-patterns/pattern-avoidance-in-weak-ascent-sequences","status":"can_prove","summary":"For every natural number $n$ let $f(n, a) = \\min(n - 3 - a, \\lfloor (a - 1)/2 \\rfloor)$ and","title":"Sum formulas for $|U_{n,1}|$ and $|V_{n,1}|$ (Proposition 2)"},{"data_url":"scopes/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu.json","id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","kind":"scope","navigation":"module","parent":24550,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"can_prove","title":"Permutations counted by odd-indexed Fibonacci numbers"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/has-pattern","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.HasPattern","parent":24714,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#has-pattern","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n-1\\}$ and let $\\pi$ be a map from $\\{0, 1, \\dots, k-1\\}$ to itself. Then $\\sigma$ contains the pattern $\\pi$ if there are positions $e_0 < e_1 < \\dots < e_{k-1}$ in $\\{0, 1, \\dots, n-1\\}$ such that for all $i, j < k$, $\\sigma(e_i) < \\sigma(e_j)$ holds exactly when $\\pi(i) < \\pi(j)$.","title":"Classical pattern containment in a permutation"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/pat321","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.pat321","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pattern "],["math","\\(321\\)"]],"source":"The pattern $321$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#pat321","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"fully_proved","summary":"The pattern $321$ written with values $0, 1, 2$: the map from $\\{0, 1, 2\\}$ to itself sending $0 \\mapsto 2$, $1 \\mapsto 1$ and $2 \\mapsto 0$.","title":"The pattern $321$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/pat231","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.pat231","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pattern "],["math","\\(231\\)"]],"source":"The pattern $231$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#pat231","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"fully_proved","summary":"The pattern $231$ written with values $0, 1, 2$: the map from $\\{0, 1, 2\\}$ to itself sending $0 \\mapsto 1$, $1 \\mapsto 2$ and $2 \\mapsto 0$.","title":"The pattern $231$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/pat4123","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.pat4123","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["text","The permutation pattern "],["math","\\(4123\\)"]],"source":"The permutation pattern $4123$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#pat4123","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"fully_proved","summary":"The pattern $4123$ with its values shifted down by one: the map $\\{0, 1, 2, 3\\} \\to \\{0, 1, 2, 3\\}$ sending $0, 1, 2, 3$ to $3, 0, 1, 2$ respectively.","title":"The permutation pattern $4123$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/pat3412","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.pat3412","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["text","The permutation pattern "],["math","\\(3412\\)"]],"source":"The permutation pattern $3412$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#pat3412","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"fully_proved","summary":"The pattern $3412$ with its values shifted down by one: the map $\\{0, 1, 2, 3\\} \\to \\{0, 1, 2, 3\\}$ sending $0, 1, 2, 3$ to $2, 3, 0, 1$ respectively.","title":"The permutation pattern $3412$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/pat3124","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.pat3124","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["text","The pattern "],["math","\\(3124\\)"]],"source":"The pattern $3124$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#pat3124","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"fully_proved","summary":"The pattern $3124$ written with values $0, 1, 2, 3$: the map from $\\{0, 1, 2, 3\\}$ to itself sending $0 \\mapsto 2$, $1 \\mapsto 0$, $2 \\mapsto 1$ and $3 \\mapsto 3$.","title":"The pattern $3124$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/card-avoid-321-4123","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.card_avoid_321_4123","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((321, 4123)\\)"],["text","-avoiding permutations are counted by odd-indexed Fibonacci numbers"]],"source":"$(321, 4123)$-avoiding permutations are counted by odd-indexed Fibonacci numbers"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#card-avoid-321-4123","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"can_prove","summary":"For every $n \\ge 1$, the number of permutations of $\\{0, 1, \\dots, n-1\\}$ that avoid both patterns $321$ and $4123$, that is, contain neither of them, is the Fibonacci number $F_{2n-1}$, where $F_0 = 0$ and $F_1 = 1$.","title":"$(321, 4123)$-avoiding permutations are counted by odd-indexed Fibonacci numbers"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/card-avoid-321-3412","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.card_avoid_321_3412","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((321, 3412)\\)"],["text","-avoiding permutations are counted by odd-indexed Fibonacci numbers"]],"source":"$(321, 3412)$-avoiding permutations are counted by odd-indexed Fibonacci numbers"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#card-avoid-321-3412","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"can_prove","summary":"For every $n \\ge 1$, the number of permutations of $\\{0, 1, \\dots, n-1\\}$ that avoid both patterns $321$ and $3412$, that is, contain neither of them, is the Fibonacci number $F_{2n-1}$, where $F_0 = 0$ and $F_1 = 1$.","title":"$(321, 3412)$-avoiding permutations are counted by odd-indexed Fibonacci numbers"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/card-avoid-231-3124","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.card_avoid_231_3124","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((231, 3124)\\)"],["text","-avoiding permutations are counted by odd-indexed Fibonacci numbers"]],"source":"$(231, 3124)$-avoiding permutations are counted by odd-indexed Fibonacci numbers"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#card-avoid-231-3124","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"can_prove","summary":"For every $n \\ge 1$, the number of permutations of $\\{0, 1, \\dots, n-1\\}$ that avoid both patterns $231$ and $3124$, that is, contain neither of them, is the Fibonacci number $F_{2n-1}$, where $F_0 = 0$ and $F_1 = 1$.","title":"$(231, 3124)$-avoiding permutations are counted by odd-indexed Fibonacci numbers"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/card-avoid-321-4123-pos-zero","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.card_avoid_321_4123_pos_zero","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["text","Position of the smallest entry in "],["math","\\((321, 4123)\\)"],["text","-avoiding permutations"]],"source":"Position of the smallest entry in $(321, 4123)$-avoiding permutations"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#card-avoid-321-4123-pos-zero","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"can_prove","summary":"For a natural number $m$, write $S_m(321, 4123)$ for the set of permutations of $\\{0, 1, \\dots, m-1\\}$ that avoid both patterns $321$ and $4123$, that is, contain neither of them. Let $n \\ge 1$ and let $j$ be a position with $j + 1 < n$. Then the number of permutations $\\sigma \\in S_n(321, 4123)$ with $\\sigma(j) = 0$ equals $(j + 1) \\cdot |S_{n-j-1}(321, 4123)|$.","title":"Position of the smallest entry in $(321, 4123)$-avoiding permutations"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/card-avoid-231-3124-pos-zero","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.card_avoid_231_3124_pos_zero","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["text","Position of the smallest entry in "],["math","\\((231, 3124)\\)"],["text","-avoiding permutations"]],"source":"Position of the smallest entry in $(231, 3124)$-avoiding permutations"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#card-avoid-231-3124-pos-zero","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"can_prove","summary":"For a natural number $m$, write $S_m(231, 3124)$ for the set of permutations of $\\{0, 1, \\dots, m-1\\}$ that avoid both patterns $231$ and $3124$, that is, contain neither of them. Let $n \\ge 1$ and let $j$ be a position with $j + 1 < n$. Then the number of permutations $\\sigma \\in S_n(231, 3124)$ with $\\sigma(j) = 0$ equals $(j + 1) \\cdot |S_{n-j-1}(231, 3124)|$.","title":"Position of the smallest entry in $(231, 3124)$-avoiding permutations"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/card-avoid-321-3412-pos-zero-le","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.OddFibonacciWanted.card_avoid_321_3412_pos_zero_le","parent":24714,"reading_title":{"format":"inline-parts-v1","parts":[["text","Smallest entry in the first two positions of "],["math","\\((321, 3412)\\)"],["text","-avoiding permutations"]],"source":"Smallest entry in the first two positions of $(321, 3412)$-avoiding permutations"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu/index.html#card-avoid-321-3412-pos-zero-le","scope":"Combinatorics/permutations/permutation-patterns/permutations-counted-by-odd-indexed-fibonacci-nu","status":"can_prove","summary":"Let $n \\ge 2$. Among the permutations $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$ that avoid both patterns $321$ and $3412$, that is, contain neither of them, the number with $\\sigma(0) = 0$ and the number with $\\sigma(1) = 0$ are both equal to the Fibonacci number $F_{2n-3}$, where $F_0 = 0$ and $F_1 = 1$.","title":"Smallest entry in the first two positions of $(321, 3412)$-avoiding permutations"},{"data_url":"scopes/Combinatorics/permutations/permutation-patterns/king-permutations.json","id":"Combinatorics/permutations/permutation-patterns/king-permutations","kind":"scope","navigation":"module","parent":24550,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","title":"King permutations"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-permutation-not-separable","kind":"node","lean":"MetaMathlibExt.king_permutation_not_separable","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","No king permutation of length at least "],["math","\\(2\\)"],["text"," is separable"]],"source":"No king permutation of length at least $2$ is separable"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-permutation-not-separable","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"For every $n \\ge 2$, every permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ in which adjacent entries never differ by exactly $1$, that is, $|\\sigma(i + 1) - \\sigma(i)| \\ne 1$ whenever $i + 1 < n$, contains the classical pattern $2413$ or the classical pattern $3142$. Since the separable permutations are those avoiding both $2413$ and $3142$, no such permutation is separable.","title":"No king permutation of length at least $2$ is separable"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-permutation-has-pattern-classical","kind":"node","lean":"MetaMathlibExt.king_permutation_hasPattern_classical","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","King permutations of length at least "],["math","\\(2\\)"],["text"," contain "],["math","\\(12\\)"],["text",", "],["math","\\(21\\)"],["text",", "],["math","\\(132\\)"],["text",", "],["math","\\(213\\)"],["text",", "],["math","\\(231\\)"],["text"," and "],["math","\\(312\\)"]],"source":"King permutations of length at least $2$ contain $12$, $21$, $132$, $213$, $231$ and $312$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-permutation-has-pattern-classical","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"For every $n \\ge 2$, every permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ in which adjacent entries never differ by exactly $1$, that is, $|\\sigma(i + 1) - \\sigma(i)| \\ne 1$ whenever $i + 1 < n$, contains each of the classical patterns $12$, $21$, $132$, $213$, $231$ and $312$.","title":"King permutations of length at least $2$ contain $12$, $21$, $132$, $213$, $231$ and $312$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/is-king-perm","kind":"node","lean":"MetaMathlibExt.IsKingPerm","parent":24727,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#is-king-perm","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"A permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ is a king permutation if adjacent entries never differ by exactly $1$: for every position $i$ with $i + 1 < n$, neither $\\sigma(i + 1) = \\sigma(i) + 1$ nor $\\sigma(i) = \\sigma(i + 1) + 1$. Equivalently, $|\\sigma(i + 1) - \\sigma(i)| > 1$ for all adjacent positions $i$ and $i + 1$.","title":"King permutations"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/perm-ascent-count","kind":"node","lean":"MetaMathlibExt.permAscentCount","parent":24727,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#perm-ascent-count","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ written in one-line notation, $\\operatorname{asc}(\\sigma)$ is the number of positions $i$ with $i + 1 < n$ and $\\sigma(i) < \\sigma(i + 1)$.","title":"Number of ascents of a permutation"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/perm-descent-count","kind":"node","lean":"MetaMathlibExt.permDescentCount","parent":24727,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#perm-descent-count","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ written in one-line notation, $\\operatorname{des}(\\sigma)$ is the number of positions $i$ with $i + 1 < n$ and $\\sigma(i + 1) < \\sigma(i)$.","title":"Number of descents of a permutation"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/perm-stats-one-fin-three","kind":"node","lean":"MetaMathlibExt.permStats_one_fin_three","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statistics of the permutation "],["math","\\(123\\)"]],"source":"Statistics of the permutation $123$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#perm-stats-one-fin-three","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"The identity permutation of $\\{0, 1, 2\\}$, written $012$ in one-line notation and $123$ after shifting every value up by one, has two ascents, no descents, three left-to-right maxima, one right-to-left maximum, one left-to-right minimum and three right-to-left minima.","title":"Statistics of the permutation $123$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/perm-stats-rev-perm-fin-three","kind":"node","lean":"MetaMathlibExt.permStats_revPerm_fin_three","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statistics of the permutation "],["math","\\(321\\)"]],"source":"Statistics of the permutation $321$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#perm-stats-rev-perm-fin-three","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"The reversal permutation $i \\mapsto 2 - i$ of $\\{0, 1, 2\\}$, written $210$ in one-line notation and $321$ after shifting every value up by one, has no ascents, two descents, one left-to-right maximum, three right-to-left maxima, three left-to-right minima and one right-to-left minimum.","title":"Statistics of the permutation $321$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/contains-flat-popbelow","kind":"node","lean":"MetaMathlibExt.ContainsFlatPOPBelow","parent":24727,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#contains-flat-popbelow","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n - 1\\}$ and let $j \\in \\{0, 1, 2, 3\\}$. Then $\\sigma$ contains the flat partially ordered pattern $P_{j+1}$ of size $4$, in which label $j + 1$ lies below the other three labels (which are mutually incomparable), if there are positions $i_0 < i_1 < i_2 < i_3$ such that $\\sigma(i_j) < \\sigma(i_m)$ for every $m \\in \\{0, 1, 2, 3\\}$ with $m \\neq j$.","title":"Containing a flat pattern with one bottom label"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/contains-flat-popabove","kind":"node","lean":"MetaMathlibExt.ContainsFlatPOPAbove","parent":24727,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#contains-flat-popabove","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n - 1\\}$ and let $j \\in \\{0, 1, 2, 3\\}$. Then $\\sigma$ contains the flat partially ordered pattern $Q_{j+1}$ of size $4$, in which label $j + 1$ lies above the other three labels, if there are positions $i_0 < i_1 < i_2 < i_3$ such that $\\sigma(i_m) < \\sigma(i_j)$ for every $m \\in \\{0, 1, 2, 3\\}$ with $m \\neq j$.","title":"Containing a flat pattern with one top label"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/four-le-of-contains-flat-popbelow","kind":"node","lean":"MetaMathlibExt.four_le_of_containsFlatPOPBelow","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Containing "],["math","\\(P_{j+1}\\)"],["text"," forces length at least "],["math","\\(4\\)"]],"source":"Containing $P_{j+1}$ forces length at least $4$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#four-le-of-contains-flat-popbelow","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n - 1\\}$ and let $j \\in \\{0, 1, 2, 3\\}$. If $\\sigma$ contains the flat partially ordered pattern $P_{j+1}$ of size $4$, in which label $j + 1$ lies below the other three labels, then $n \\ge 4$.","title":"Containing $P_{j+1}$ forces length at least $4$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/four-le-of-contains-flat-popabove","kind":"node","lean":"MetaMathlibExt.four_le_of_containsFlatPOPAbove","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Containing "],["math","\\(Q_{j+1}\\)"],["text"," forces length at least "],["math","\\(4\\)"]],"source":"Containing $Q_{j+1}$ forces length at least $4$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#four-le-of-contains-flat-popabove","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n - 1\\}$ and let $j \\in \\{0, 1, 2, 3\\}$. If $\\sigma$ contains the flat partially ordered pattern $Q_{j+1}$ of size $4$, in which label $j + 1$ lies above the other three labels, then $n \\ge 4$.","title":"Containing $Q_{j+1}$ forces length at least $4$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popbelow-gf","kind":"node","lean":"MetaMathlibExt.kingFlatPOPBelowGF","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of king permutations avoiding "],["math","\\(P_{j+1}\\)"]],"source":"Generating function of king permutations avoiding $P_{j+1}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popbelow-gf","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"For $j \\in \\{0, 1, 2, 3\\}$ let $P_{j+1}$ be the flat partially ordered pattern of length $4$ whose element number $j + 1$ lies below each of the other three: a permutation $\\sigma$ contains $P_{j+1}$ if there are positions $i_0 < i_1 < i_2 < i_3$ with $\\sigma(i_j) < \\sigma(i_m)$ for every $m \\ne j$. A king permutation is one in which adjacent entries never differ by exactly $1$; let $K_n(P_{j+1})$ be the set of\u2026","title":"Generating function of king permutations avoiding $P_{j+1}$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popabove-gf","kind":"node","lean":"MetaMathlibExt.kingFlatPOPAboveGF","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of king permutations avoiding "],["math","\\(Q_{j+1}\\)"]],"source":"Generating function of king permutations avoiding $Q_{j+1}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popabove-gf","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"For $j \\in \\{0, 1, 2, 3\\}$ let $Q_{j+1}$ be the flat partially ordered pattern of length $4$ whose element number $j + 1$ lies above each of the other three: a permutation $\\sigma$ contains $Q_{j+1}$ if there are positions $i_0 < i_1 < i_2 < i_3$ with $\\sigma(i_m) < \\sigma(i_j)$ for every $m \\ne j$. A king permutation is one in which adjacent entries never differ by exactly $1$; let $K_n(Q_{j+1})$ be the set of\u2026","title":"Generating function of king permutations avoiding $Q_{j+1}$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/constant-coeff-king-flat-popbelow-gf","kind":"node","lean":"MetaMathlibExt.constantCoeff_kingFlatPOPBelowGF","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Constant term of "],["math","\\(A_{P_{j+1}}\\)"]],"source":"Constant term of $A_{P_{j+1}}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#constant-coeff-king-flat-popbelow-gf","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"For every $j \\in \\{0, 1, 2, 3\\}$, the coefficient of $x^0$ in the generating function $A_{P_{j+1}}(x; p, q, u, v)$ of king permutations avoiding the flat partially ordered pattern $P_{j+1}$ is the constant polynomial $1$.","title":"Constant term of $A_{P_{j+1}}$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/constant-coeff-king-flat-popabove-gf","kind":"node","lean":"MetaMathlibExt.constantCoeff_kingFlatPOPAboveGF","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Constant term of "],["math","\\(\\bar A_{Q_{j+1}}\\)"]],"source":"Constant term of $\\bar A_{Q_{j+1}}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#constant-coeff-king-flat-popabove-gf","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"fully_proved","summary":"For every $j \\in \\{0, 1, 2, 3\\}$, the coefficient of $x^0$ in the generating function $\\bar A_{Q_{j+1}}(x; p, q, s, t)$ of king permutations avoiding the flat partially ordered pattern $Q_{j+1}$ is the constant polynomial $1$.","title":"Constant term of $\\bar A_{Q_{j+1}}$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popbelow-gf-zero-eq","kind":"node","lean":"MetaMathlibExt.kingFlatPOPBelowGF_zero_eq","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rational generating function for king permutations avoiding "],["math","\\(P_1\\)"]],"source":"Rational generating function for king permutations avoiding $P_1$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popbelow-gf-zero-eq","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"Let $P_1$ be the flat partially ordered pattern of length $4$ whose first element lies below each of the other three, and let $A_{P_1}(x; p, q, u, v)$ be the generating function of king permutations avoiding $P_1$, in which $x$ marks length and $p$, $q$, $u$, $v$ mark ascents, descents, left-to-right maxima and right-to-left maxima. Then $B \\cdot A_{P_1} = A$ in $\\mathbb{Z}[p, q, u, v][[x]]$, where\u2026","title":"Rational generating function for king permutations avoiding $P_1$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popbelow-gf-one-eq","kind":"node","lean":"MetaMathlibExt.kingFlatPOPBelowGF_one_eq","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rational generating function for king permutations avoiding "],["math","\\(P_2\\)"]],"source":"Rational generating function for king permutations avoiding $P_2$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popbelow-gf-one-eq","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"Let $P_2$ be the flat partially ordered pattern of length $4$ whose second element lies below each of the other three, and let $A_{P_2}(x; p, q, u, v)$ be the generating function of king permutations avoiding $P_2$, in which $x$ marks length and $p$, $q$, $u$, $v$ mark ascents, descents, left-to-right maxima and right-to-left maxima. Then $B \\cdot A_{P_2} = A$ in $\\mathbb{Z}[p, q, u, v][[x]]$, where\u2026","title":"Rational generating function for king permutations avoiding $P_2$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popbelow-gf-three-eq-rename","kind":"node","lean":"MetaMathlibExt.kingFlatPOPBelowGF_three_eq_rename","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generating functions for "],["math","\\(P_4\\)"],["text"," and "],["math","\\(P_1\\)"],["text"," differ by a change of variables"]],"source":"The generating functions for $P_4$ and $P_1$ differ by a change of variables"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popbelow-gf-three-eq-rename","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"For $k = 1$ and $k = 4$ let $P_k$ be the flat partially ordered pattern of length $4$ whose $k$-th element lies below each of the other three, and let $A_{P_k}(x; p, q, u, v)$ be the generating function of king permutations avoiding $P_k$, in which $x$ marks length and $p$, $q$, $u$, $v$ mark ascents, descents, left-to-right maxima and right-to-left maxima. Then $A_{P_4}(x; p, q, u, v) = A_{P_1}(x; q, p, v, u)$:\u2026","title":"The generating functions for $P_4$ and $P_1$ differ by a change of variables"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popabove-gf-zero-eq-rename","kind":"node","lean":"MetaMathlibExt.kingFlatPOPAboveGF_zero_eq_rename","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generating functions for "],["math","\\(Q_1\\)"],["text"," and "],["math","\\(P_1\\)"],["text"," differ by a change of variables"]],"source":"The generating functions for $Q_1$ and $P_1$ differ by a change of variables"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popabove-gf-zero-eq-rename","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"Let $P_1$ (respectively $Q_1$) be the flat partially ordered pattern of length $4$ whose first element lies below (respectively above) each of the other three. Let $A_{P_1}(x; p, q, u, v)$ be the generating function of king permutations avoiding $P_1$, in which $x$ marks length and $p$, $q$, $u$, $v$ mark ascents, descents, left-to-right maxima and right-to-left maxima, and let $\\bar A_{Q_1}(x; p, q, s, t)$ be the\u2026","title":"The generating functions for $Q_1$ and $P_1$ differ by a change of variables"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popabove-gf-three-eq-rename","kind":"node","lean":"MetaMathlibExt.kingFlatPOPAboveGF_three_eq_rename","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The king generating function avoiding "],["math","\\(Q_4\\)"]],"source":"The king generating function avoiding $Q_4$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popabove-gf-three-eq-rename","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"Let $A_{P_1}(x, p, q, u, v)$ be the generating function of king permutations avoiding the flat pattern $P_1$ (label $1$ below labels $2, 3, 4$), where $x, p, q, u, v$ mark length, ascents, descents, left-to-right maxima and right-to-left maxima, and let $\\bar A_{Q_4}(x, p, q, s, t)$ be the generating function of king permutations avoiding the flat pattern $Q_4$ (label $4$ above labels $1, 2, 3$), where\u2026","title":"The king generating function avoiding $Q_4$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popbelow-gf-two-eq-rename","kind":"node","lean":"MetaMathlibExt.kingFlatPOPBelowGF_two_eq_rename","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The generating functions for "],["math","\\(P_3\\)"],["text"," and "],["math","\\(P_2\\)"],["text"," differ by a change of variables"]],"source":"The generating functions for $P_3$ and $P_2$ differ by a change of variables"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popbelow-gf-two-eq-rename","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"For $k = 2$ and $k = 3$ let $P_k$ be the flat partially ordered pattern of length $4$ whose $k$-th element lies below each of the other three, and let $A_{P_k}(x; p, q, u, v)$ be the generating function of king permutations avoiding $P_k$, in which $x$ marks length and $p$, $q$, $u$, $v$ mark ascents, descents, left-to-right maxima and right-to-left maxima. Then $A_{P_3}(x; p, q, u, v) = A_{P_2}(x; q, p, v, u)$:\u2026","title":"The generating functions for $P_3$ and $P_2$ differ by a change of variables"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popabove-gf-one-eq-rename","kind":"node","lean":"MetaMathlibExt.kingFlatPOPAboveGF_one_eq_rename","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The king generating function avoiding "],["math","\\(Q_2\\)"]],"source":"The king generating function avoiding $Q_2$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popabove-gf-one-eq-rename","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"Let $A_{P_2}(x, p, q, u, v)$ be the generating function of king permutations avoiding the flat pattern $P_2$ (label $2$ below labels $1, 3, 4$), where $x, p, q, u, v$ mark length, ascents, descents, left-to-right maxima and right-to-left maxima, and let $\\bar A_{Q_2}(x, p, q, s, t)$ be the generating function of king permutations avoiding the flat pattern $Q_2$ (label $2$ above labels $1, 3, 4$), where\u2026","title":"The king generating function avoiding $Q_2$"},{"id":"Combinatorics/permutations/permutation-patterns/king-permutations/king-flat-popabove-gf-two-eq-rename","kind":"node","lean":"MetaMathlibExt.kingFlatPOPAboveGF_two_eq_rename","parent":24727,"reading_title":{"format":"inline-parts-v1","parts":[["text","The king generating function avoiding "],["math","\\(Q_3\\)"]],"source":"The king generating function avoiding $Q_3$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/king-permutations/index.html#king-flat-popabove-gf-two-eq-rename","scope":"Combinatorics/permutations/permutation-patterns/king-permutations","status":"can_prove","summary":"Let $A_{P_2}(x, p, q, u, v)$ be the generating function of king permutations avoiding the flat pattern $P_2$ (label $2$ below labels $1, 3, 4$), where $x, p, q, u, v$ mark length, ascents, descents, left-to-right maxima and right-to-left maxima, and let $\\bar A_{Q_3}(x, p, q, s, t)$ be the generating function of king permutations avoiding the flat pattern $Q_3$ (label $3$ above labels $1, 2, 4$), where\u2026","title":"The king generating function avoiding $Q_3$"},{"data_url":"scopes/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr.json","id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","kind":"scope","navigation":"module","parent":24550,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","title":"Permutations avoiding two patterns of length three"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/pattern3","kind":"node","lean":"MetaMathlibExt.pattern3","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#pattern3","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"Given a bijection $v$ from $\\{0, 1, 2\\}$ to itself, this is the permutation of $\\{0, 1, 2\\}$ that sends $i$ to $v(i)$. The values are listed zero-based, so $v = (0, 2, 1)$ gives the pattern $132$ in the usual one-based one-line notation.","title":"Length-three pattern from one-line values"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/two-pattern-avoider-count","kind":"node","lean":"MetaMathlibExt.twoPatternAvoiderCount","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#two-pattern-avoider-count","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"For a natural number $n$ and permutations $\\tau$ and $\\rho$ of $\\{0, 1, 2\\}$, $A_n(\\tau, \\rho)$ is the number of permutations of $\\{0, 1, \\dots, n - 1\\}$ that contain neither $\\tau$ nor $\\rho$ as a classical pattern.","title":"Number of permutations avoiding two patterns of length three"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/two-pattern-avoider-count-three-123-132","kind":"node","lean":"MetaMathlibExt.twoPatternAvoiderCount_three_123_132","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutations of length three avoiding "],["math","\\(123\\)"],["text"," and "],["math","\\(132\\)"]],"source":"Permutations of length three avoiding $123$ and $132$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#two-pattern-avoider-count-three-123-132","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"Exactly $4$ permutations of $\\{0, 1, 2\\}$ avoid both patterns $123$ and $132$, that is, $A_3(123, 132) = 4$: of the six permutations of length $3$, only $123$ and $132$ themselves are excluded.","title":"Permutations of length three avoiding $123$ and $132$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/two-pattern-avoider-count-four-132-321","kind":"node","lean":"MetaMathlibExt.twoPatternAvoiderCount_four_132_321","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutations of length four avoiding "],["math","\\(132\\)"],["text"," and "],["math","\\(321\\)"]],"source":"Permutations of length four avoiding $132$ and $321$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#two-pattern-avoider-count-four-132-321","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"Exactly $7$ permutations of $\\{0, 1, 2, 3\\}$ avoid both patterns $132$ and $321$, that is, $A_4(132, 321) = 7 = 1 + \\binom{4}{2}$.","title":"Permutations of length four avoiding $132$ and $321$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/two-pattern-avoider-count-length-three","kind":"node","lean":"MetaMathlibExt.twoPatternAvoiderCount_length_three","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#two-pattern-avoider-count-length-three","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","summary":"For every $n \\ge 1$, writing $A_n(\\tau, \\rho)$ for the number of permutations of $\\{0, 1, \\dots, n - 1\\}$ avoiding both patterns $\\tau$ and $\\rho$: (a) $A_n(123, 132) = A_n(123, 213) = A_n(321, 231) = A_n(321, 312) = 2^{n-1}$; (b) $A_n(231, 312) = A_n(132, 213) = 2^{n-1}$; (c) $A_n(213, 312) = A_n(132, 231) = 2^{n-1}$; (d) $A_n(213, 231) = A_n(132, 312) = 2^{n-1}$; (e)\u2026","title":"Simion\u2013Schmidt enumeration of permutations avoiding two patterns of length three"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/perm-ascents","kind":"node","lean":"MetaMathlibExt.permAscents","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#perm-ascents","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$, $\\operatorname{asc}(\\sigma)$ is the number of positions $i$ with $i + 1 < n$ and $\\sigma(i) < \\sigma(i + 1)$.","title":"Ascent number of a permutation"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/perm-descents","kind":"node","lean":"MetaMathlibExt.permDescents","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#perm-descents","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$, $\\operatorname{des}(\\sigma)$ is the number of positions $i$ with $i + 1 < n$ and $\\sigma(i + 1) < \\sigma(i)$.","title":"Descent number of a permutation"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/perm-ascents-perm-descents-small","kind":"node","lean":"MetaMathlibExt.permAscents_permDescents_small","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ascents and descents of "],["math","\\(123\\)"],["text"," and "],["math","\\(132\\)"]],"source":"Ascents and descents of $123$ and $132$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#perm-ascents-perm-descents-small","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"Writing permutations of $\\{0, 1, 2\\}$ in one-based one-line notation, the identity $123$ has two ascents and no descents, $\\operatorname{asc}(123) = 2$ and $\\operatorname{des}(123) = 0$, and $132$ has one of each, $\\operatorname{asc}(132) = \\operatorname{des}(132) = 1$.","title":"Ascents and descents of $123$ and $132$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/asc-des-gf","kind":"node","lean":"MetaMathlibExt.ascDesGF","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#asc-des-gf","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"For permutations $\\tau$ and $\\rho$ of $\\{0, 1, 2\\}$, let $S_n(\\tau, \\rho)$ be the set of permutations of $\\{0, 1, \\dots, n-1\\}$ that contain neither $\\tau$ nor $\\rho$ as a classical pattern. The ascent\u2013descent generating function of $(\\tau, \\rho)$ is the formal power series","title":"Ascent\u2013descent generating function of a pair of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/asc-des-gf-123-132","kind":"node","lean":"MetaMathlibExt.ascDesGF_123_132","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ascents and descents over permutations avoiding "],["math","\\(123\\)"],["text"," and "],["math","\\(132\\)"]],"source":"Ascents and descents over permutations avoiding $123$ and $132$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#asc-des-gf-123-132","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","summary":"Let","title":"Ascents and descents over permutations avoiding $123$ and $132$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/asc-des-gf-132-321","kind":"node","lean":"MetaMathlibExt.ascDesGF_132_321","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ascents and descents over permutations avoiding "],["math","\\(132\\)"],["text"," and "],["math","\\(321\\)"]],"source":"Ascents and descents over permutations avoiding $132$ and $321$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#asc-des-gf-132-321","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","summary":"Let","title":"Ascents and descents over permutations avoiding $132$ and $321$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/asc-des-gf-213-231","kind":"node","lean":"MetaMathlibExt.ascDesGF_213_231","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ascents and descents over permutations avoiding "],["math","\\(213\\)"],["text"," and "],["math","\\(231\\)"]],"source":"Ascents and descents over permutations avoiding $213$ and $231$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#asc-des-gf-213-231","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","summary":"Let","title":"Ascents and descents over permutations avoiding $213$ and $231$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/perm-extrema-small","kind":"node","lean":"MetaMathlibExt.permExtrema_small","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Left-to-right and right-to-left extrema of "],["math","\\(132\\)"],["text"," and "],["math","\\(213\\)"]],"source":"Left-to-right and right-to-left extrema of $132$ and $213$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#perm-extrema-small","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"Writing permutations of $\\{0, 1, 2\\}$ in one-based one-line notation, $132$ has $2$ left-to-right maxima, $2$ right-to-left maxima, $1$ left-to-right minimum and $2$ right-to-left minima, and $213$ has $2$ left-to-right maxima, $1$ right-to-left maximum, $2$ left-to-right minima and $2$ right-to-left minima.","title":"Left-to-right and right-to-left extrema of $132$ and $213$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/perm-double-ascents-peaks-small","kind":"node","lean":"MetaMathlibExt.permDoubleAscents_peaks_small","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#perm-double-ascents-peaks-small","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"Writing permutations of $\\{0, 1, 2\\}$ in one-based one-line notation, and $\\operatorname{dasc}$, $\\operatorname{ddes}$, $\\operatorname{pk}$ and $\\operatorname{vl}$ for the numbers of double ascents, double descents, peaks and valleys, $\\operatorname{dasc}(123) = 1$, $\\operatorname{ddes}(321) = 1$, $\\operatorname{dasc}(132) = \\operatorname{ddes}(132) = 0$, $\\operatorname{pk}(132) = 1$, $\\operatorname{vl}(132) = 0$,\u2026","title":"Double ascents, double descents, peaks and valleys in length three"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/classical-stat-gf","kind":"node","lean":"MetaMathlibExt.classicalStatGF","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#classical-stat-gf","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"For permutations $\\tau$ and $\\rho$ of $\\{0, 1, 2\\}$, let $S_n(\\tau, \\rho)$ be the set of permutations of $\\{0, 1, \\dots, n-1\\}$ that contain neither $\\tau$ nor $\\rho$ as a classical pattern. The generating function $F_{(\\tau, \\rho)}$ is the formal power series","title":"Generating function of ten classical statistics over a pair of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/dasc-ddes-gf","kind":"node","lean":"MetaMathlibExt.dascDdesGF","parent":24751,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#dasc-ddes-gf","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"fully_proved","summary":"For permutations $\\tau$ and $\\rho$ of $\\{0, 1, 2\\}$, let $S_n(\\tau, \\rho)$ be the set of permutations of $\\{0, 1, \\dots, n-1\\}$ that contain neither $\\tau$ nor $\\rho$ as a classical pattern. The double ascent\u2013double descent generating function of $(\\tau, \\rho)$ is the formal power series","title":"Double ascent\u2013double descent generating function of a pair of patterns"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/classical-stat-gf-123-132","kind":"node","lean":"MetaMathlibExt.classicalStatGF_123_132","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ten classical statistics over permutations avoiding "],["math","\\(123\\)"],["text"," and "],["math","\\(132\\)"]],"source":"Ten classical statistics over permutations avoiding $123$ and $132$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#classical-stat-gf-123-132","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","summary":"Let $F$ be the joint generating function of the ten classical statistics over the permutations avoiding both patterns $123$ and $132$: a formal power series in $x$ over $\\mathbb{Q}[p, q, u, v, s, t, y, z, \\ell, m]$ in which $x$ marks the length and $p, q, u, v, s, t, y, z, \\ell, m$ mark the numbers of ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, right-to-left minima, double\u2026","title":"Ten classical statistics over permutations avoiding $123$ and $132$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/classical-stat-gf-132-321","kind":"node","lean":"MetaMathlibExt.classicalStatGF_132_321","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ten classical statistics over permutations avoiding "],["math","\\(132\\)"],["text"," and "],["math","\\(321\\)"]],"source":"Ten classical statistics over permutations avoiding $132$ and $321$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#classical-stat-gf-132-321","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","summary":"Let $F$ be the joint generating function of the ten classical statistics over the permutations avoiding both patterns $132$ and $321$: a formal power series in $x$ over $\\mathbb{Q}[p, q, u, v, s, t, y, z, \\ell, m]$ in which $x$ marks the length and $p, q, u, v, s, t, y, z, \\ell, m$ mark the numbers of ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, right-to-left minima, double\u2026","title":"Ten classical statistics over permutations avoiding $132$ and $321$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/classical-stat-gf-213-231","kind":"node","lean":"MetaMathlibExt.classicalStatGF_213_231","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ten classical statistics over permutations avoiding "],["math","\\(213\\)"],["text"," and "],["math","\\(231\\)"]],"source":"Ten classical statistics over permutations avoiding $213$ and $231$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#classical-stat-gf-213-231","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","summary":"Let $F$ be the joint generating function of the ten classical statistics over the permutations avoiding both patterns $213$ and $231$: a formal power series in $x$ over $\\mathbb{Q}[p, q, u, v, s, t, y, z, \\ell, m]$ in which $x$ marks the length and $p, q, u, v, s, t, y, z, \\ell, m$ mark the numbers of ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, right-to-left minima, double\u2026","title":"Ten classical statistics over permutations avoiding $213$ and $231$"},{"id":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/dasc-ddes-gf-231-312","kind":"node","lean":"MetaMathlibExt.dascDdesGF_231_312","parent":24751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Double ascents and double descents over permutations avoiding "],["math","\\(231\\)"],["text"," and "],["math","\\(312\\)"]],"source":"Double ascents and double descents over permutations avoiding $231$ and $312$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr/index.html#dasc-ddes-gf-231-312","scope":"Combinatorics/permutations/permutation-patterns/permutations-avoiding-two-patterns-of-length-thr","status":"can_prove","summary":"Let","title":"Double ascents and double descents over permutations avoiding $231$ and $312$"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics.json","id":"Combinatorics/permutations/permutation-statistics","kind":"scope","parent":24549,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/index.html","scope":"Combinatorics/permutations/permutation-statistics","status":"can_prove","summary":"Descents, peaks, major index, left-to-right maxima, fixed points and derangements, cycle counts and the smallest cycle of a random permutation, products of cyclic permutations, Stirling and flattened Stirling permutations, Josephus permutations and permutation-guessing puzzles.","title":"Permutation statistics and cycle structure"},{"id":"Combinatorics/permutations/permutation-statistics/sum-mul-num-derangements-eq-floor","kind":"node","lean":"MetaMathlibExt.sum_mul_numDerangements_eq_floor","parent":24772,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/index.html#sum-mul-num-derangements-eq-floor","scope":"Combinatorics/permutations/permutation-statistics","status":"can_prove","summary":"For every natural number $p$, $\\sum_{n=0}^{p} n\\, D(n) = \\lfloor (p + 1)!/e \\rfloor$, where $D(n)$ is the number of derangements of $n$ objects.","title":"A sum rule for derangements"},{"id":"Combinatorics/permutations/permutation-statistics/zagier-probability-product-two-cyclic-pe","kind":"node","lean":"MetaMathlibExt.zagier_probability_product_two_cyclic_permutations","parent":24772,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/index.html#zagier-probability-product-two-cyclic-pe","scope":"Combinatorics/permutations/permutation-statistics","status":"fully_proved","summary":"Let $n$ and $k$ be integers with $n \\ge 1$ and $1 \\le k \\le n$. Call a permutation of an $n$-element set cyclic when it has exactly one cycle, fixed points counted as cycles. Write $c(n+1, k)$ for the unsigned Stirling number of the first kind, the number of permutations of $n + 1$ elements with exactly $k$ cycles. If $\\sigma$ and $\\tau$ are chosen independently and uniformly among the cyclic permutations of an\u2026","title":"Zagier's formula for products of two cyclic permutations"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics.json","id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","kind":"scope","navigation":"module","parent":24772,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","title":"Classical permutation statistics"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-left-to-right-max-count","kind":"node","lean":"MetaMathlibExt.permLeftToRightMaxCount","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-left-to-right-max-count","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$, $\\mathrm{lmax}(\\sigma)$ is the number of positions $i$ such that $\\sigma(j) < \\sigma(i)$ for every position $j < i$, that is, the number of entries larger than every entry to their left.","title":"Number of left-to-right maxima"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-right-to-left-max-count","kind":"node","lean":"MetaMathlibExt.permRightToLeftMaxCount","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-right-to-left-max-count","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ written in one-line notation, $\\operatorname{rmax}(\\sigma)$ is the number of positions $i$ such that $\\sigma(j) < \\sigma(i)$ for every position $j > i$, that is, the number of entries larger than every entry to their right.","title":"Number of right-to-left maxima"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-left-to-right-min-count","kind":"node","lean":"MetaMathlibExt.permLeftToRightMinCount","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-left-to-right-min-count","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ written in one-line notation, $\\operatorname{lmin}(\\sigma)$ is the number of positions $i$ such that $\\sigma(i) < \\sigma(j)$ for every position $j < i$, that is, the number of entries smaller than every entry to their left.","title":"Number of left-to-right minima"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-right-to-left-min-count","kind":"node","lean":"MetaMathlibExt.permRightToLeftMinCount","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-right-to-left-min-count","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ written in one-line notation, $\\operatorname{rmin}(\\sigma)$ is the number of positions $i$ such that $\\sigma(i) < \\sigma(j)$ for every position $j > i$, that is, the number of entries smaller than every entry to their right.","title":"Number of right-to-left minima"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/is-double-ascent-at","kind":"node","lean":"MetaMathlibExt.IsDoubleAscentAt","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#is-double-ascent-at","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n-1\\}$. A position $i$ is a double ascent of $\\sigma$ if $i + 2 < n$ and $\\sigma(i) < \\sigma(i+1) < \\sigma(i+2)$.","title":"Double ascent of a permutation"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/is-double-descent-at","kind":"node","lean":"MetaMathlibExt.IsDoubleDescentAt","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#is-double-descent-at","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n-1\\}$. A position $i$ is a double descent of $\\sigma$ if $i + 2 < n$ and $\\sigma(i) > \\sigma(i+1) > \\sigma(i+2)$.","title":"Double descent of a permutation"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/is-peak-at","kind":"node","lean":"MetaMathlibExt.IsPeakAt","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#is-peak-at","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n-1\\}$. A position $i$ is a peak of $\\sigma$ if $i + 2 < n$ and $\\sigma(i) < \\sigma(i+1) > \\sigma(i+2)$. Thus the peak entry is $\\sigma(i+1)$, and the position recorded is $i$, that of the first of the three entries.","title":"Peak of a permutation"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/is-valley-at","kind":"node","lean":"MetaMathlibExt.IsValleyAt","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#is-valley-at","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n-1\\}$. A position $i$ is a valley of $\\sigma$ if $i + 2 < n$ and $\\sigma(i) > \\sigma(i+1) < \\sigma(i+2)$. Thus the valley entry is $\\sigma(i+1)$, and the position recorded is $i$, that of the first of the three entries.","title":"Valley of a permutation"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-double-ascents","kind":"node","lean":"MetaMathlibExt.permDoubleAscents","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-double-ascents","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$, $\\mathrm{dasc}(\\sigma)$ is the number of positions $i$ that are double ascents of $\\sigma$.","title":"Number of double ascents"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-double-descents","kind":"node","lean":"MetaMathlibExt.permDoubleDescents","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-double-descents","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$, $\\mathrm{ddes}(\\sigma)$ is the number of positions $i$ that are double descents of $\\sigma$.","title":"Number of double descents"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-peaks","kind":"node","lean":"MetaMathlibExt.permPeaks","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-peaks","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ written in one-line notation, $\\operatorname{pk}(\\sigma)$ is the number of positions $i$ with $i + 2 < n$ and $\\sigma(i) < \\sigma(i + 1) > \\sigma(i + 2)$, that is, the number of peaks of $\\sigma$.","title":"Number of peaks of a permutation"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-valleys","kind":"node","lean":"MetaMathlibExt.permValleys","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-valleys","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ written in one-line notation, $\\operatorname{vl}(\\sigma)$ is the number of positions $i$ with $i + 2 < n$ and $\\sigma(i) > \\sigma(i + 1) < \\sigma(i + 2)$, that is, the number of valleys of $\\sigma$.","title":"Number of valleys of a permutation"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/card-filter-add-two-lt","kind":"node","lean":"MetaMathlibExt.card_filter_add_two_lt","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#card-filter-add-two-lt","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"For every natural number $n$, exactly $n - 2$ of the positions $i \\in \\{0, 1, \\dots, n-1\\}$ satisfy $i + 2 < n$, where $n - 2$ is truncated subtraction (it is $0$ when $n < 2$).","title":"Number of positions with two successors"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/ite-is-double-ascent-at-add-ite-is-doubl","kind":"node","lean":"MetaMathlibExt.ite_isDoubleAscentAt_add_ite_isDoubleDescentAt_add_ite_isPeakAt_add_ite_isValleyAt","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#ite-is-double-ascent-at-add-ite-is-doubl","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n-1\\}$ and let $i$ be a position. If $i + 2 < n$, then $i$ is exactly one of a double ascent, a double descent, a peak or a valley of $\\sigma$; otherwise $i$ is none of them. That is, the number of these four properties that $i$ has is $1$ when $i + 2 < n$ and $0$ otherwise.","title":"Local shape of a position in a permutation"},{"id":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/perm-double-ascents-add-perm-double-desc","kind":"node","lean":"MetaMathlibExt.permDoubleAscents_add_permDoubleDescents_add_permPeaks_add_permValleys","parent":24775,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/classical-permutation-statistics/index.html#perm-double-ascents-add-perm-double-desc","scope":"Combinatorics/permutations/permutation-statistics/classical-permutation-statistics","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, 1, \\dots, n-1\\}$, and write $\\mathrm{dasc}(\\sigma)$, $\\mathrm{ddes}(\\sigma)$, $\\mathrm{pk}(\\sigma)$ and $\\mathrm{vl}(\\sigma)$ for its numbers of double ascents, double descents, peaks and valleys. Then $\\mathrm{dasc}(\\sigma) + \\mathrm{ddes}(\\sigma) + \\mathrm{pk}(\\sigma) + \\mathrm{vl}(\\sigma) = n - 2$, where $n - 2$ is truncated subtraction (it is $0$ when $n < 2$).","title":"Sum of the four local statistics of a permutation"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles.json","id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","kind":"scope","navigation":"module","parent":24772,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","title":"Fixed points of permutations with k cycles"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/k-perm-fixed-point-count","kind":"node","lean":"MetaMathlibExt.kPermFixedPointCount","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#k-perm-fixed-point-count","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"fully_proved","summary":"For natural numbers $n$, $k$ and $j$, $a(n, k, j)$ is the number of permutations of $\\{0, 1, \\dots, n - 1\\}$ with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points.","title":"Permutations counted by cycles and fixed points"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/k-perm-fixed-point-count-eq-zero-of-lt","kind":"node","lean":"MetaMathlibExt.kPermFixedPointCount_eq_zero_of_lt","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#k-perm-fixed-point-count-eq-zero-of-lt","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"fully_proved","summary":"Write $a(n, k, j)$ for the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points. If $k < j$, then $a(n, k, j) = 0$.","title":"A permutation has at most as many fixed points as cycles"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/k-perm-fixed-point-count-zero-zero-zero","kind":"node","lean":"MetaMathlibExt.kPermFixedPointCount_zero_zero_zero","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#k-perm-fixed-point-count-zero-zero-zero","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"fully_proved","summary":"Write $a(n, k, j)$ for the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points. Then $a(0, 0, 0) = 1$: the empty permutation is the only permutation of the empty set, and it has no cycles and no fixed points.","title":"Counting the empty permutation"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/k-perm-fixed-point-count-eq-choose-mul-d","kind":"node","lean":"MetaMathlibExt.kPermFixedPointCount_eq_choose_mul_derangement","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#k-perm-fixed-point-count-eq-choose-mul-d","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Write $a(n, k, j)$ for the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points, and $d(m, i) = a(m, i, 0)$ for the number of derangements of an $m$-element set with $i$ cycles. For all natural numbers $n$, $k$ and $j$ with $j \\le k$, $a(n, k, j) = \\binom{n}{j} \\, d(n - j, k - j)$, where $n - j$ is read as $0$ when $j > n$.","title":"Fixed points split off a derangement"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/mul-k-perm-fixed-point-count","kind":"node","lean":"MetaMathlibExt.mul_kPermFixedPointCount","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#mul-k-perm-fixed-point-count","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Write $a(n, k, j)$ for the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points. For all natural numbers $n$, $k$ and $j$ with $1 \\le j \\le k$, $j \\, a(n, k, j) = n \\, a(n - 1, k - 1, j - 1)$, where $n - 1$ is read as $0$ when $n = 0$.","title":"Recurrence for permutations counted by cycles and fixed points"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/sum-k-perm-fixed-point-count","kind":"node","lean":"MetaMathlibExt.sum_kPermFixedPointCount","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#sum-k-perm-fixed-point-count","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"For natural numbers $n$, $k$ and $j$, let $a(n, k, j)$ be the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points. Then for all natural numbers $n$ and $k$,","title":"Stirling numbers from fixed-point counts"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/sum-mul-k-perm-fixed-point-count","kind":"node","lean":"MetaMathlibExt.sum_mul_kPermFixedPointCount","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#sum-mul-k-perm-fixed-point-count","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"For natural numbers $n$, $k$ and $j$, let $a(n, k, j)$ be the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points, and let $c(n, k)$ be the unsigned Stirling number of the first kind. Then for every natural number $n$ and every $k \\ge 1$,","title":"Total number of fixed points of permutations with a given number of cycles"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/k-perm-fixed-point-variance","kind":"node","lean":"MetaMathlibExt.kPermFixedPoint_variance","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#k-perm-fixed-point-variance","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Write $a(n, k, j)$ for the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points, $c(n, k)$ for the unsigned Stirling number of the first kind (the number of permutations of an $n$-element set with $k$ cycles), and $\\mu_{n,k} = n \\, c(n - 1, k - 1) / c(n, k)$. If $2 \\le k \\le n$, then\u2026","title":"Variance of the number of fixed points of a permutation with a given number of cycles"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/newton-inequality-of-real-roots","kind":"node","lean":"MetaMathlibExt.newton_inequality_of_real_roots","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#newton-inequality-of-real-roots","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Let $p(x) = \\sum_{j=0}^{n} a_j x^j$ be a real polynomial of degree $n \\ge 2$ that splits over $\\mathbb{R}$, that is, all $n$ of its zeros, counted with multiplicity, are real. Then for every $j$ with $1 \\le j \\le n - 1$,","title":"Newton's inequalities"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/schur-factorial-hadamard-real-roots","kind":"node","lean":"MetaMathlibExt.schur_factorial_hadamard_real_roots","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#schur-factorial-hadamard-real-roots","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Let $f = \\sum_k a_k x^k$ and $g = \\sum_k c_k x^k$ be real polynomials that split over $\\mathbb{R}$, and suppose that the zeros of $g$ are either all $\\le 0$ or all $\\ge 0$. Let $d = \\min(\\deg f, \\deg g)$, where the zero polynomial is given degree $0$. Then the polynomial $\\sum_{k=0}^{d} k!\\, a_k c_k x^k$ also splits over $\\mathbb{R}$.","title":"Schur's composition theorem for real-rooted polynomials"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/derangement-cycle-polynomial-real-roots","kind":"node","lean":"MetaMathlibExt.derangementCyclePolynomial_real_roots","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#derangement-cycle-polynomial-real-roots","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Write $a(m, k, j)$ for the number of permutations of an $m$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points, so that $d(m, k) = a(m, k, 0)$ is the number of derangements of an $m$-element set with $k$ cycles. For every $n \\ge 2$, the real polynomial $D_n(x) = \\sum_{k=1}^{n} d(n + k, k) x^{k-1}$ has only real zeros.","title":"Real-rootedness of the derangement cycle polynomial"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/unimodal-of-newton-inequalities","kind":"node","lean":"MetaMathlibExt.unimodal_of_newton_inequalities","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#unimodal-of-newton-inequalities","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Let $n$ be a natural number and let $(a_j)_{j \\ge 0}$ be a sequence of real numbers with $a_j > 0$ for every $j \\le n$, such that","title":"Positive sequences satisfying Newton's inequalities are unimodal"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/k-perm-fixed-point-count-unimodal","kind":"node","lean":"MetaMathlibExt.kPermFixedPointCount_unimodal","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#k-perm-fixed-point-count-unimodal","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Write $a(n, k, j)$ for the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points. If $k \\le n$, then there is an $m \\le k$ such that $j \\mapsto a(n, k, j)$ is weakly increasing on $\\{0, \\dots, m\\}$ and weakly decreasing on $\\{m, \\dots, k\\}$, and every $i \\le k$ with $a(n, k, i) = a(n, k, m)$ is $m$ or $m + 1$.","title":"Unimodality of fixed-point counts for permutations with a given number of cycles"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/coeff-tendsto-gaussian-cdf-of-neg-real-r","kind":"node","lean":"MetaMathlibExt.coeff_tendsto_gaussian_cdf_of_neg_real_roots","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#coeff-tendsto-gaussian-cdf-of-neg-real-r","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Let $(Q_n)_{n \\ge 0}$ be real polynomials such that every $Q_n$ has only real zeros and all of them are $\\le 0$. Put $\\mu_n = Q_n'(1) / Q_n(1)$ and $\\sigma_n^2 = Q_n'(1) / Q_n(1) + Q_n''(1) / Q_n(1) - \\mu_n^2$, and let $\\sigma_n = \\sqrt{\\sigma_n^2}$. If $\\sigma_n^2 \\to \\infty$ as $n \\to \\infty$, then for every real $x$, the sum of the coefficients of $X^j$ in $Q_n$ over the indices $0 \\le j \\le \\deg Q_n$ with\u2026","title":"Central limit theorem for polynomials with nonpositive real zeros"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/k-perm-fixed-point-polynomial-real-roots","kind":"node","lean":"MetaMathlibExt.kPermFixedPointPolynomial_real_roots","parent":24791,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles/index.html#k-perm-fixed-point-polynomial-real-roots","scope":"Combinatorics/permutations/permutation-statistics/fixed-points-of-permutations-with-k-cycles","status":"can_prove","summary":"Write $a(n, k, j)$ for the number of permutations of an $n$-element set with exactly $k$ cycles (fixed points counted as cycles of length one) and exactly $j$ fixed points. If $1 \\le k \\le n$, then the real polynomial $P(x) = \\sum_{j=0}^{k} a(n, k, j) x^j$ has only real zeros.","title":"Real zeros of the fixed-point polynomial"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations.json","id":"Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations","kind":"scope","navigation":"module","parent":24772,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations/index.html","scope":"Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations","status":"can_prove","title":"Fixed-point identities for permutations"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations/sum-perm-sign-falling-fixed-points-mul-p","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.PermutationFixedPointIdentitiesWanted.sum_perm_sign_falling_fixedPoints_mul_pow","parent":24807,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations/index.html#sum-perm-sign-falling-fixed-points-mul-p","scope":"Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers with $n \\ge 2$, $k \\ge 1$ and $k + 1 \\le n$, and let $x$ be a real number. For a permutation $\\sigma$ of $\\lbrace 0, 1, \\dots, n-1 \\rbrace$, write $\\operatorname{sgn}(\\sigma)$ for its sign and $\\operatorname{fix}(\\sigma)$ for its number of fixed points, and for a natural number $m$ write $(m)_k = m(m-1)\\cdots(m-k+1)$ for the falling factorial. Then, summing over all permutations\u2026","title":"Pain's signed falling-factorial identity for fixed points"},{"id":"Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations/sum-perm-sign-factorial-fixed-points-div","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.PermutationFixedPointIdentitiesWanted.sum_perm_sign_factorial_fixedPoints_div","parent":24807,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations/index.html#sum-perm-sign-factorial-fixed-points-div","scope":"Combinatorics/permutations/permutation-statistics/fixed-point-identities-for-permutations","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers with $n \\ge 2$ and $k \\ge 1$. For a permutation $\\sigma$ of $\\lbrace 0, 1, \\dots, n-1 \\rbrace$, write $\\operatorname{sgn}(\\sigma)$ for its sign and $\\operatorname{fix}(\\sigma)$ for its number of fixed points. Then, summing over all permutations $\\sigma$ of this set,","title":"Pain's signed factorial-ratio identity for fixed points"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents.json","id":"Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents","kind":"scope","navigation":"module","parent":24772,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents/index.html","scope":"Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents","status":"fully_proved","title":"Minimal permutations with given descents"},{"id":"Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents/descent-minimal-iff","kind":"node","lean":"MetaMathlibExt.descent_minimal_iff","parent":24810,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents/index.html#descent-minimal-iff","scope":"Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{1, \\dots, n\\}$ and let $d$ be a natural number. A descent of a permutation $\\tau$ of $\\{1, \\dots, k\\}$ is an index $1 \\le i \\le k - 1$ with $\\tau(i+1) < \\tau(i)$, and $\\tau$ occurs in $\\sigma$ if there are positions $\\varphi(1) < \\dots < \\varphi(k)$ in $\\{1, \\dots, n\\}$ with $\\tau(a) < \\tau(b)$ exactly when $\\sigma(\\varphi(a)) < \\sigma(\\varphi(b))$. Then $\\sigma$ has exactly $d$\u2026","title":"Characterization of minimal permutations with a given number of descents"},{"id":"Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents/minimal-permutations-card","kind":"node","lean":"MetaMathlibExt.minimal_permutations_card","parent":24810,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents/index.html#minimal-permutations-card","scope":"Combinatorics/permutations/permutation-statistics/minimal-permutations-with-given-descents","status":"fully_proved","summary":"For every natural number $d$, the number of permutations $\\sigma$ of $\\{0, 1, \\dots, d + 1\\}$ such that $\\sigma$ has exactly $d$ descents (positions $0 \\le i \\le d$ with $\\sigma(i) > \\sigma(i + 1)$) and, for every position $j$, deleting the entry at position $j$ leaves a sequence of $d + 1$ entries with fewer than $d$ descents, equals $2^{d+2} - (d + 1)(d + 2) - 2$.","title":"Minimal permutations of length two more than their number of descents"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics/permutation-wordle.json","id":"Combinatorics/permutations/permutation-statistics/permutation-wordle","kind":"scope","navigation":"module","parent":24772,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","title":"Permutation Wordle"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/permute-along","kind":"node","lean":"MetaMathlibExt.permuteAlong","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#permute-along","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $I$ be a subset of $\\{0, \\dots, n-1\\}$ with $k$ elements, listed in increasing order as $i_0 < i_1 < \\dots < i_{k-1}$, let $\\sigma$ be a permutation of $\\{0, \\dots, k-1\\}$, and let $g$ be a map from $\\{0, \\dots, n-1\\}$ to itself. Applying $\\sigma$ to $g$ along $I$ gives the map that takes the value $g(i_j)$ at $i_{\\sigma(j)}$ for every $j < k$ and agrees with $g$ outside $I$.","title":"Permuting the entries of a map along a set of positions"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/permute-along-univ","kind":"node","lean":"MetaMathlibExt.permuteAlong_univ","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#permute-along-univ","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of $\\{0, \\dots, n-1\\}$ and let $g$ be a map from $\\{0, \\dots, n-1\\}$ to itself. Applying $\\sigma$ to the entries of $g$ along the set of all positions $\\{0, \\dots, n-1\\}$, listed in increasing order, gives the map $g \\circ \\sigma^{-1}$.","title":"Permuting along all positions"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-next-guess","kind":"node","lean":"MetaMathlibExt.permWordleNextGuess","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-next-guess","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $S$ be a strategy, that is, a permutation $S_m$ of $\\{0, \\dots, m-1\\}$ for every $m$, let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$ (the secret), and let $g$ be a map from $\\{0, \\dots, n-1\\}$ to itself (the current guess). Let $I$ be the set of incorrect positions, the $k$ with $g(k) \\ne \\pi(k)$, and let $m$ be the number of elements of $I$. The next guess agrees with $g$ at every position outside $I$ and\u2026","title":"One round of permutation Wordle"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-next-guess-of-eq","kind":"node","lean":"MetaMathlibExt.permWordleNextGuess_of_eq","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-next-guess-of-eq","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $S$ be a strategy, that is, a permutation $S_m$ of $\\{0, \\dots, m-1\\}$ for every $m$, let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$, and let $g$ be a map from $\\{0, \\dots, n-1\\}$ to itself. If $g(i) = \\pi(i)$ at a position $i$, then the guess that one round of permutation Wordle with strategy $S$ and secret $\\pi$ produces from $g$ also takes the value $g(i)$ at $i$.","title":"Correct positions stay locked"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-guess","kind":"node","lean":"MetaMathlibExt.permWordleGuess","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-guess","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $S$ be a strategy, that is, a permutation $S_m$ of $\\{0, \\dots, m-1\\}$ for every $m$, and let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$. The guesses $\\gamma_1, \\gamma_2, \\dots$ of permutation Wordle with strategy $S$ and secret $\\pi$ are the maps from $\\{0, \\dots, n-1\\}$ to itself defined recursively: $\\gamma_1$ is the identity, and $\\gamma_{r+1}$ is the guess that one round of the game with strategy $S$\u2026","title":"Guesses of permutation Wordle"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-guess-succ","kind":"node","lean":"MetaMathlibExt.permWordleGuess_succ","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-guess-succ","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $S$ be a strategy, that is, a permutation $S_m$ of $\\{0, \\dots, m-1\\}$ for every $m$, let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$, and let $\\gamma_1, \\gamma_2, \\dots$ be the guesses of permutation Wordle with strategy $S$ and secret $\\pi$. For every $r \\ge 1$, the guess $\\gamma_{r+1}$ is the guess that one round of the game with strategy $S$ and secret $\\pi$ produces from $\\gamma_r$.","title":"Recurrence for the guesses of permutation Wordle"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-guess-one-of-forall-ne","kind":"node","lean":"MetaMathlibExt.permWordleGuess_one_of_forall_ne","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-guess-one-of-forall-ne","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $S$ be a strategy for permutation Wordle, that is, a permutation $S_m$ of $\\{0, \\dots, m-1\\}$ for every $m$, and let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$ with no fixed point. Then the second guess $\\gamma_2$ of permutation Wordle with strategy $S$ and secret $\\pi$ is the inverse permutation $S_n^{-1}$.","title":"The second guess against a derangement"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-solved-in","kind":"node","lean":"MetaMathlibExt.PermWordleSolvedIn","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-solved-in","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $S$ be a strategy, that is, a permutation $S_m$ of $\\{0, \\dots, m-1\\}$ for every $m$, let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$, and let $\\gamma_1, \\gamma_2, \\dots$ be the guesses of permutation Wordle with strategy $S$ and secret $\\pi$. For a natural number $r$, the game ends after exactly $r$ guesses when $r \\ge 1$, $\\gamma_r = \\pi$, and $\\gamma_j \\ne \\pi$ for every $j$ with $1 \\le j < r$.","title":"Permutation Wordle games solved in a given number of guesses"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-solved-in-one-iff","kind":"node","lean":"MetaMathlibExt.permWordleSolvedIn_one_iff","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-solved-in-one-iff","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $S$ be a strategy, that is, a permutation $S_m$ of $\\{0, \\dots, m-1\\}$ for every $m$, and let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$. Then permutation Wordle with strategy $S$ and secret $\\pi$ ends after exactly one guess if and only if $\\pi$ is the identity.","title":"Games won with the first guess"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-coeff","kind":"node","lean":"MetaMathlibExt.permWordleCoeff","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-coeff","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"For a strategy $S$ and natural numbers $n$ and $r$, the coefficient $[x^r] f_S(x)$ of the generating function of $S$ at length $n$ is the number of secret permutations of $\\{0, \\dots, n-1\\}$ that $S$ guesses in exactly $r$ guesses.","title":"Coefficients of the guess-count generating function"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-coeff-zero","kind":"node","lean":"MetaMathlibExt.permWordleCoeff_zero","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-coeff-zero","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"For every strategy $S$ and every $n$, write $f_S$ for the generating function whose coefficient $[x^r] f_S$ is the number of permutations of $\\{0, \\dots, n-1\\}$ that $S$ guesses in exactly $r$ guesses. Then the constant coefficient $[x^0] f_S$ vanishes.","title":"No permutation is guessed in zero guesses"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-first-correct-at","kind":"node","lean":"MetaMathlibExt.PermWordleFirstCorrectAt","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-first-correct-at","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"Let $S$ be a strategy, that is, a permutation $S_m$ of $\\{0, \\dots, m-1\\}$ for every $m$, let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$, and let $\\gamma_1, \\gamma_2, \\dots$ be the guesses of permutation Wordle with strategy $S$ and secret $\\pi$. For a natural number $r$, $\\rho_S(\\pi) = r$ means that $r \\ge 1$, that $\\gamma_r(k) = \\pi(k)$ for some position $k$, and that for every $j$ with $1 \\le j < r$ there is\u2026","title":"First guess with a correct position in permutation Wordle"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-first-correct-at-one-iff","kind":"node","lean":"MetaMathlibExt.permWordleFirstCorrectAt_one_iff","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-first-correct-at-one-iff","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, a strategy $S$ and a permutation $\\pi$ of $\\{0, \\dots, n-1\\}$, write $\\rho_S(\\pi) = r$ when the $r$-th guess is the first guess with a correct position for the secret $\\pi$. Then $\\rho_S(\\pi) = 1$, that is, the first guess (the identity) already has a correct position, exactly when $\\pi$ has a fixed point.","title":"Permutations with a correct entry in the first guess"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/cyclic-shift-strategy","kind":"node","lean":"MetaMathlibExt.cyclicShiftStrategy","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#cyclic-shift-strategy","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"A strategy assigns to each $m$ a permutation of $m$ positions, used to rearrange the entries on the $m$ incorrect positions of a guess. The cyclic shift strategy $\\mathrm{CS}$ has component $\\mathrm{CS}[m] = [2, 3, \\dots, m, 1]$ in one-line notation, that is, the rotation $i \\mapsto i + 1 \\pmod m$ of $\\{0, 1, \\dots, m-1\\}$.","title":"Cyclic shift strategy"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/cyclic-shift-strategy-apply","kind":"node","lean":"MetaMathlibExt.cyclicShiftStrategy_apply","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#cyclic-shift-strategy-apply","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"For every $m$, the component of the cyclic shift strategy on $m$ positions is the rotation $i \\mapsto i + 1 \\pmod m$ of $\\{0, 1, \\dots, m-1\\}$.","title":"Components of the cyclic shift strategy"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/cyclic-shift-left-strategy","kind":"node","lean":"MetaMathlibExt.cyclicShiftLeftStrategy","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#cyclic-shift-left-strategy","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the strategy $\\mathrm{CSL}_n$ assigns to each $m$ a permutation of $m$ positions: for $m = n$ the left cyclic shift $[m, 1, 2, \\dots, m-1]$ in one-line notation, that is, $i \\mapsto i - 1 \\pmod m$, and for every $m \\ne n$ the right cyclic shift $[2, 3, \\dots, m, 1]$, that is, $i \\mapsto i + 1 \\pmod m$.","title":"Cyclic shift left strategy"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/cyclic-shift-left-strategy-self","kind":"node","lean":"MetaMathlibExt.cyclicShiftLeftStrategy_self","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#cyclic-shift-left-strategy-self","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"For every $n$, the component on $n$ positions of the left-shift strategy $\\mathrm{CSL}_n$ is the left cyclic shift $j \\mapsto j - 1 \\bmod n$ of $\\{0, \\dots, n-1\\}$, the inverse of the cyclic shift $j \\mapsto j + 1 \\bmod n$.","title":"Top component of the left-shift strategy"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/cyclic-shift-left-strategy-of-ne","kind":"node","lean":"MetaMathlibExt.cyclicShiftLeftStrategy_of_ne","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#cyclic-shift-left-strategy-of-ne","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"For all $n$ and $m$ with $m \\ne n$, the component on $m$ positions of the left-shift strategy $\\mathrm{CSL}_n$ is the cyclic shift $j \\mapsto j + 1 \\bmod m$ of $\\{0, \\dots, m-1\\}$.","title":"Non-top components of the left-shift strategy"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/is-inductive-perm-wordle-strategy","kind":"node","lean":"MetaMathlibExt.IsInductivePermWordleStrategy","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#is-inductive-perm-wordle-strategy","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"fully_proved","summary":"A strategy $S$ (a permutation $S[m]$ of $\\{0, \\dots, m-1\\}$ for each $m$) is inductive of length $n$ when $S[m]$ is the rotation $i \\mapsto i + 1 \\pmod m$ for every $m < n$, and $S[n]$ is an $n$-cycle, that is, a single cycle moving every element of $\\{0, \\dots, n-1\\}$.","title":"Inductive strategy"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/avg-card-correct-second-guess-derangemen","kind":"node","lean":"MetaMathlibExt.avg_card_correct_second_guess_derangements","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#avg-card-correct-second-guess-derangemen","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Let $n \\ge 2$ and let $S$ be a permutation Wordle strategy (a permutation $S[m]$ of $\\{0, \\dots, m-1\\}$ for each $m$) whose component $S[n]$ has no fixed point. For a secret permutation $\\pi$ of $\\{0, \\dots, n-1\\}$ let $\\gamma_2$ be the second guess of $S$. The sum, over all derangements $\\pi$ of $\\{0, \\dots, n-1\\}$, of the number of positions $i$ with $\\gamma_2(i) = \\pi(i)$, divided by the number of derangements\u2026","title":"Average number of correct positions in the second guess"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-coeff-one-and-two","kind":"node","lean":"MetaMathlibExt.permWordleCoeff_one_and_two","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-coeff-one-and-two","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Let $n \\in \\mathbb{N}$ and let $S$ be a strategy whose component $S[m]$ has no fixed point for every $2 \\le m \\le n$, and write $[x^r] f_S$ for the number of permutations of $\\{0, \\dots, n-1\\}$ that $S$ guesses in exactly $r$ guesses. Then $[x^1] f_S = 1$ and $[x^2] f_S = 2^n - n - 1$, the Eulerian number $A(n, 1)$.","title":"One- and two-guess counts for deranged strategies"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-coeff-three-lt-cyclic-shift","kind":"node","lean":"MetaMathlibExt.permWordleCoeff_three_lt_cyclicShift","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-coeff-three-lt-cyclic-shift","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Let $n \\ge 4$ and let $S$ be an inductive strategy of length $n$ whose component $S[n]$ is not the rotation $i \\mapsto i + 1 \\pmod n$. Then strictly fewer permutations of $\\{0, \\dots, n-1\\}$ are guessed in exactly three guesses with $S$ than with the cyclic shift strategy.","title":"Cyclic shift maximizes three-guess solves among inductive strategies"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/card-derangements-solved-in-three-cyclic","kind":"node","lean":"MetaMathlibExt.card_derangements_solvedIn_three_cyclicShift","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#card-derangements-solved-in-three-cyclic","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"For $n \\ge 3$, exactly $2^n - (2n + 1)$ derangements $\\pi$ of $\\{0, \\dots, n-1\\}$ are guessed by the cyclic shift strategy in exactly three guesses.","title":"Derangements solved in three guesses by cyclic shift"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/card-solved-in-three-first-correct-at-on","kind":"node","lean":"MetaMathlibExt.card_solvedIn_three_firstCorrectAt_one","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#card-solved-in-three-first-correct-at-on","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Let $n \\ge 4$ and let $S$ be a permutation Wordle strategy (a permutation $S[m]$ of $\\{0, \\dots, m-1\\}$ for each $m$) with $S[m]$ equal to the cyclic shift $j \\mapsto j + 1 \\bmod m$ for every $m < n$. The number of permutations $\\pi$ of $\\{0, \\dots, n-1\\}$ that $S$ guesses in exactly three guesses and whose first guess (the identity) has a correct position, that is, $\\pi$ has a fixed point, is\u2026","title":"Three-guess solves of permutations with a fixed point"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/card-solved-in-three-first-correct-at-th","kind":"node","lean":"MetaMathlibExt.card_solvedIn_three_firstCorrectAt_three","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#card-solved-in-three-first-correct-at-th","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Let $n \\ge 4$ and let $S$ be a permutation Wordle strategy (a permutation $S[m]$ of $\\{0, \\dots, m-1\\}$ for each $m$) whose component $S[n]$ is an $n$-cycle. Exactly one permutation $\\pi$ of $\\{0, \\dots, n-1\\}$ is guessed by $S$ in exactly three guesses with the third guess being the first guess that has a correct position.","title":"A unique three-guess solve with no early correct position"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/card-solved-in-three-first-correct-at-tw","kind":"node","lean":"MetaMathlibExt.card_solvedIn_three_firstCorrectAt_two_cyclicShift","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#card-solved-in-three-first-correct-at-tw","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"For $n \\ge 4$, the number of permutations $\\pi$ of $\\{0, \\dots, n-1\\}$ that the cyclic shift strategy guesses in exactly three guesses, with the second guess being the first guess that has a correct position, is $2^n - 2n - 2$.","title":"Three-guess solves of cyclic shift first correct at guess two"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/card-solved-in-three-first-correct-at-tw-2","kind":"node","lean":"MetaMathlibExt.card_solvedIn_three_firstCorrectAt_two_lt","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#card-solved-in-three-first-correct-at-tw-2","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Let $n \\ge 4$ and let $S$ be an inductive strategy of length $n$ whose component $S[n]$ is not the cyclic shift $j \\mapsto j + 1 \\bmod n$ of $\\{0, \\dots, n-1\\}$. Then fewer than $2^n - 2n - 2$ permutations $\\pi$ of $\\{0, \\dots, n-1\\}$ are guessed by $S$ in exactly three guesses with the second guess being the first guess that has a correct position.","title":"Upper bound on three-guess solves first correct at guess two"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/perm-wordle-coeff-three-cyclic-shift-lef","kind":"node","lean":"MetaMathlibExt.permWordleCoeff_three_cyclicShiftLeft_lt","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#perm-wordle-coeff-three-cyclic-shift-lef","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Let $n \\ge 4$ and let $S$ be an inductive strategy of length $n$ whose component $S[n]$ is not the left cyclic shift $i \\mapsto i - 1 \\pmod n$. Then strictly fewer permutations of $\\{0, \\dots, n-1\\}$ are guessed in exactly three guesses with the cyclic shift left strategy $\\mathrm{CSL}_n$ than with $S$.","title":"Cyclic shift left minimizes three-guess solves among inductive strategies"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/card-solved-in-three-first-correct-at-tw-3","kind":"node","lean":"MetaMathlibExt.card_solvedIn_three_firstCorrectAt_two_cyclicShiftLeft","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#card-solved-in-three-first-correct-at-tw-3","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Let $n \\ge 4$ and let $\\mathrm{CSL}_n$ be the strategy whose component on $n$ positions is the left cyclic shift $j \\mapsto j - 1 \\bmod n$ and whose component on $m \\ne n$ positions is the cyclic shift $j \\mapsto j + 1 \\bmod m$. The number of permutations $\\pi$ of $\\{0, \\dots, n-1\\}$ that $\\mathrm{CSL}_n$ guesses in exactly three guesses, with the second guess being the first guess that has a correct position, is\u2026","title":"Lucas-number count for the left-shift strategy"},{"id":"Combinatorics/permutations/permutation-statistics/permutation-wordle/card-first-correct-at-two-cyclic-shift-l","kind":"node","lean":"MetaMathlibExt.card_firstCorrectAt_two_cyclicShiftLeft_lt","parent":24813,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/permutation-wordle/index.html#card-first-correct-at-two-cyclic-shift-l","scope":"Combinatorics/permutations/permutation-statistics/permutation-wordle","status":"can_prove","summary":"Among inductive strategies, cyclic shift left minimizes three-guess solves first correct at guess two. Let $n \\ge 4$ and let $S$ be an inductive strategy of length $n$ whose component $S[n]$ is not the left cyclic shift $i \\mapsto i - 1 \\pmod n$ of $\\{0, \\dots, n-1\\}$. Let $\\mathrm{CSL}_n$ be the cyclic shift left strategy, whose component on $n$ positions is the left cyclic shift and whose component on $m \\ne n$\u2026","title":"Among inductive strategies, cyclic shift left minimizes three-guess solves first correct at guess two"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation.json","id":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation","kind":"scope","navigation":"module","parent":24772,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/index.html","scope":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation","status":"can_prove","title":"Smallest cycle of a random permutation"},{"id":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/smallest-cycle-mean-asymp","kind":"node","lean":"MetaMathlibExt.smallest_cycle_mean_asymp","parent":24844,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/index.html#smallest-cycle-mean-asymp","scope":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation","status":"can_prove","summary":"For $n \\ge 1$ let $X_n$ be the length of the smallest cycle of a uniformly random permutation of $\\{1, \\dots, n\\}$. Then $\\mathbb{E}(X_n) / (e^{-\\gamma} \\log n) \\to 1$ as $n \\to \\infty$, where $\\gamma$ is the Euler\u2013Mascheroni constant.","title":"Mean length of the smallest cycle of a random permutation"},{"id":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/smallest-cycle-moment-asymp","kind":"node","lean":"MetaMathlibExt.smallest_cycle_moment_asymp","parent":24844,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/index.html#smallest-cycle-moment-asymp","scope":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation","status":"can_prove","summary":"Let $\\omega$ be a real function with $\\omega(x) = 1/x$ for $1 \\le x \\le 2$ and $x\\,\\omega(x) = 1 + \\int_1^{x-1} \\omega(t)\\,dt$ for $x \\ge 2$, and for $n \\ge 1$ let $X_n$ be the length of the smallest cycle of a uniformly random permutation of $\\{1, \\dots, n\\}$. For every integer $\\ell \\ge 2$ there are real constants $\\varepsilon > 0$ and $C$ such that for every $n \\ge 1$","title":"Higher moments of the smallest cycle length"},{"id":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/smallest-cycle-variance-asymp","kind":"node","lean":"MetaMathlibExt.smallest_cycle_variance_asymp","parent":24844,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/index.html#smallest-cycle-variance-asymp","scope":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation","status":"can_prove","summary":"Let $\\omega$ be a real function with $\\omega(x) = 1/x$ for $1 \\le x \\le 2$ and $x\\,\\omega(x) = 1 + \\int_1^{x-1} \\omega(t)\\,dt$ for $x \\ge 2$, and for $n \\ge 1$ let $X_n$ be the length of the smallest cycle of a uniformly random permutation of $\\{1, \\dots, n\\}$. There are real constants $\\varepsilon > 0$ and $C$ such that for every $n \\ge 1$","title":"Variance of the smallest cycle length"},{"id":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/buchstab-taylor-coeff-recursion","kind":"node","lean":"MetaMathlibExt.buchstab_taylor_coeff_recursion","parent":24844,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/index.html#buchstab-taylor-coeff-recursion","scope":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation","status":"can_prove","summary":"Let $\\omega$ be a real function with $\\omega(x) = 1/x$ for $1 \\le x \\le 2$ and $x\\,\\omega(x) = 1 + \\int_1^{x-1} \\omega(t)\\,dt$ for $x \\ge 2$, and let $c_{n,i}$ ($n, i \\ge 0$) be real numbers such that for every integer $n \\ge 1$ and every real $z$ with $-1 \\le z < 1$ the series $\\sum_{i \\ge 0} c_{n,i} z^i$ converges absolutely to $\\omega(n + (1+z)/2)$. Then $c_{1,i} = \\frac{2}{3} \\left(-\\frac{1}{3}\\right)^i$ for\u2026","title":"Recursion for the Taylor coefficients of the Buchstab function"},{"id":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/generalized-buchstab-taylor-coeff-recurs","kind":"node","lean":"MetaMathlibExt.generalized_buchstab_taylor_coeff_recursion","parent":24844,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation/index.html#generalized-buchstab-taylor-coeff-recurs","scope":"Combinatorics/permutations/permutation-statistics/smallest-cycle-of-a-random-permutation","status":"can_prove","summary":"Let $K > 0$, let $\\Omega$ be a real function with $\\Omega(x) = 1$ for $1 \\le x < 2$ and $\\Omega(x) = 1 + K \\int_2^x \\frac{\\Omega(u-1)}{u-1}\\,du$ for $x \\ge 2$, and let $c_{n,i}$ ($n, i \\ge 0$) be real numbers such that for every integer $n \\ge 1$ and every real $z$ with $-1 \\le z < 1$ the series $\\sum_{i \\ge 0} c_{n,i} z^i$ converges absolutely to $\\Omega(n + (1+z)/2)$. Then $c_{1,0} = 1$, $c_{1,i} = 0$ for every\u2026","title":"Recursion for the Taylor coefficients of the generalized Buchstab function"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics/stirling-permutations.json","id":"Combinatorics/permutations/permutation-statistics/stirling-permutations","kind":"scope","navigation":"module","parent":24772,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/stirling-permutations/index.html","scope":"Combinatorics/permutations/permutation-statistics/stirling-permutations","status":"fully_proved","title":"Stirling permutations"},{"id":"Combinatorics/permutations/permutation-statistics/stirling-permutations/is-stirling-permutation","kind":"node","lean":"MetaMathlibExt.IsStirlingPermutation","parent":24850,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/stirling-permutations/index.html#is-stirling-permutation","scope":"Combinatorics/permutations/permutation-statistics/stirling-permutations","status":"fully_proved","summary":"Let $n$ be a natural number. A word $w = w_1 w_2 \\cdots w_{2n}$ of natural numbers is a Stirling permutation of order $n$ if every letter lies in $\\{1, \\dots, n\\}$, each $i \\in \\{1, \\dots, n\\}$ occurs exactly twice, and every letter strictly between the two occurrences of $i$ is greater than $i$.","title":"Stirling permutations"},{"id":"Combinatorics/permutations/permutation-statistics/stirling-permutations/stirling-perm-set","kind":"node","lean":"MetaMathlibExt.StirlingPermSet","parent":24850,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/stirling-permutations/index.html#stirling-perm-set","scope":"Combinatorics/permutations/permutation-statistics/stirling-permutations","status":"fully_proved","summary":"For a natural number $n$, $Q_n$ is the set of all Stirling permutations of order $n$, viewed as words $w_1 w_2 \\cdots w_{2n}$ of natural numbers.","title":"The set of Stirling permutations"},{"data_url":"scopes/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations.json","id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","kind":"scope","navigation":"module","parent":24772,"result":false,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","title":"Flattened Stirling permutations"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/word-run-start","kind":"node","lean":"MetaMathlibExt.wordRunStart","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#word-run-start","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"Position $i$ of a word $w = w_0 w_1 \\cdots w_{m-1}$ of natural numbers starts a run when $i = 0$ or the letter before it is larger, that is, $w_{i-1} > w_i$. Runs are the maximal contiguous weakly increasing subwords, so these are exactly the positions where a run begins.","title":"Start of a run"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/word-run-count","kind":"node","lean":"MetaMathlibExt.wordRunCount","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#word-run-count","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"A run of a word $w = w_0 w_1 \\cdots w_{m-1}$ of natural numbers is a maximal contiguous weakly increasing subword. The number of runs of $w$ is the number of positions $i < m$ that start a run, that is, with $i = 0$ or $w_{i-1} > w_i$; the empty word has no runs.","title":"Number of runs of a word"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/is-flattened-word","kind":"node","lean":"MetaMathlibExt.IsFlattenedWord","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#is-flattened-word","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"A word $w = w_1 w_2 \\cdots w_m$ of natural numbers is flattened when the leading letters of its runs, read left to right, are weakly increasing. Here a run is a maximal contiguous weakly increasing subword, so position $i$ starts a run when $i = 1$ or $w_{i-1} > w_i$; the condition is that $w_i \\le w_j$ whenever $i < j$ and both positions start runs.","title":"Flattened words"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-stirling-count","kind":"node","lean":"MetaMathlibExt.flattenedStirlingCount","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-stirling-count","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For a natural number $n$, $|\\mathrm{flat}(Q_n)|$ is the number of flattened Stirling permutations of order $n$: words $w_1 w_2 \\cdots w_{2n}$ of natural numbers that are Stirling permutations of order $n$ and whose runs (maximal contiguous weakly increasing subwords) have weakly increasing leading letters, read from left to right.","title":"Number of flattened Stirling permutations"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-stirling-run-count","kind":"node","lean":"MetaMathlibExt.flattenedStirlingRunCount","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-stirling-run-count","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $|\\mathrm{flat}_k(Q_n)|$ is the number of flattened Stirling permutations of order $n$ with exactly $k$ runs, where a run is a maximal contiguous weakly increasing subword.","title":"Counting flattened Stirling permutations by runs"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-perm-run-count","kind":"node","lean":"MetaMathlibExt.flattenedPermRunCount","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-perm-run-count","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $f_{n,k}$ is the number of permutations $\\sigma$ of $\\{0, 1, \\dots, n - 1\\}$ such that the word $\\sigma(0) \\sigma(1) \\cdots \\sigma(n - 1)$ is flattened (the leading letters of its runs, read left to right, are weakly increasing) and has exactly $k$ runs, where a run is a maximal contiguous weakly increasing subword. These are the flattened partitions of length $n$ of Nabawanda,\u2026","title":"Flattened partitions counted by runs"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/is-type-bset-partition","kind":"node","lean":"MetaMathlibExt.IsTypeBSetPartition","parent":24853,"reading_title":{"format":"inline-parts-v1","parts":[["text","Type "],["math","\\(B\\)"],["text"," set partitions"]],"source":"Type $B$ set partitions"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#is-type-bset-partition","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For a natural number $n$, a set partition $P$ of the integer interval $[-n, n] = \\{-n, \\dots, n\\}$ is of type $B$ if for every block $\\beta$ of $P$ the set $-\\beta = \\{-x : x \\in \\beta\\}$ is again a block of $P$, and exactly one block $\\beta$ of $P$ satisfies $-\\beta = \\beta$.","title":"Type $B$ set partitions"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/is-flattened-word-example","kind":"node","lean":"MetaMathlibExt.isFlattenedWord_example","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#is-flattened-word-example","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"The word $w = 1\\,1\\,2\\,2\\,9\\,9\\,3\\,8\\,8\\,3\\,4\\,6\\,6\\,7\\,7\\,4\\,5\\,5$ of length $18$ is flattened (the leading letters of its runs, read left to right, are weakly increasing), and it has exactly $4$ runs, where a run is a maximal contiguous weakly increasing subword.","title":"A flattened word with four runs"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/not-is-flattened-word-example","kind":"node","lean":"MetaMathlibExt.not_isFlattenedWord_example","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#not-is-flattened-word-example","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"The word $123321445566778899$ is not flattened.","title":"A word that is not flattened"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/is-type-bset-partition-indiscrete","kind":"node","lean":"MetaMathlibExt.isTypeBSetPartition_indiscrete","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#is-type-bset-partition-indiscrete","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For every natural number $n$, the partition of the integer interval $[-n, n] = \\{-n, \\dots, n\\}$ into a single block is a type $B$ set partition: its only block is closed under negation, so it is the unique block equal to its own negative (the zero-block).","title":"The one-block partition is of type B"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-perm-run-count-recursion","kind":"node","lean":"MetaMathlibExt.flattenedPermRunCount_recursion","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-perm-run-count-recursion","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"Let $f_{n,k}$ be the number of flattened partitions of length $n$ with exactly $k$ runs. For all $n \\ge 1$ and $k \\ge 2$,","title":"Recursion for flattened partitions by number of runs"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/nonempty-flattened-stirling-equiv-type-b","kind":"node","lean":"MetaMathlibExt.nonempty_flattenedStirling_equiv_typeBSetPartition","parent":24853,"reading_title":{"format":"inline-parts-v1","parts":[["text","Flattened Stirling permutations and type "],["math","\\(B\\)"],["text"," set partitions"]],"source":"Flattened Stirling permutations and type $B$ set partitions"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#nonempty-flattened-stirling-equiv-type-b","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"For every natural number $n \\ge 1$, the set of flattened Stirling permutations of order $n + 1$ is in bijection with the set of type $B$ set partitions of $[-n, n] = \\{-n, \\dots, n\\}$.","title":"Flattened Stirling permutations and type $B$ set partitions"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-stirling-count-succ-eq-dowling","kind":"node","lean":"MetaMathlibExt.flattenedStirlingCount_succ_eq_dowling","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-stirling-count-succ-eq-dowling","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"For every $n \\ge 1$, the number $|\\mathrm{flat}(Q_{n+1})|$ of flattened Stirling permutations of order $n + 1$ equals the Dowling number $D_n = \\sum_{k=0}^{n} W_{2,1,1}(n, k)$, where $W_{2,1,1}(n, k)$ are the $(q, r)$-Whitney numbers of the second kind with $m = 2$, $r = 1$ and $q = 1$.","title":"Flattened Stirling permutations are counted by Dowling numbers"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/is-greatest-flattened-stirling-run-count","kind":"node","lean":"MetaMathlibExt.isGreatest_flattenedStirling_runCount","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#is-greatest-flattened-stirling-run-count","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"For every natural number $n \\ge 1$, some flattened Stirling permutation of order $n$ has exactly $\\lceil 2n/3 \\rceil$ runs, and no flattened Stirling permutation of order $n$ has more.","title":"Maximum number of runs of a flattened Stirling permutation"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-stirling-run-count-one","kind":"node","lean":"MetaMathlibExt.flattenedStirlingRunCount_one","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-stirling-run-count-one","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"For every natural number $n \\ge 1$, exactly one flattened Stirling permutation of order $n$ has exactly one run.","title":"Exactly one flattened Stirling permutation has one run"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-stirling-run-count-two-recursi","kind":"node","lean":"MetaMathlibExt.flattenedStirlingRunCount_two_recursion","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-stirling-run-count-two-recursi","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"Write $|\\mathrm{flat}_2(Q_n)|$ for the number of flattened Stirling permutations of order $n$ with exactly two runs. Then $|\\mathrm{flat}_2(Q_1)| = 0$, and for every natural number $n \\ge 1$, $|\\mathrm{flat}_2(Q_{n+1})| = 2\\,|\\mathrm{flat}_2(Q_n)| + 2n - 1$.","title":"Recursion for flattened Stirling permutations with two runs"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-stirling-run-count-two-closed","kind":"node","lean":"MetaMathlibExt.flattenedStirlingRunCount_two_closed","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-stirling-run-count-two-closed","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"For every natural number $n \\ge 1$, the number $|\\mathrm{flat}_2(Q_{n+1})|$ of flattened Stirling permutations of order $n + 1$ with exactly two runs equals $3(2^n - 1) - 2n$.","title":"Closed form for flattened Stirling permutations with two runs"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/flattened-stirling-count-succ-eq-sum","kind":"node","lean":"MetaMathlibExt.flattenedStirlingCount_succ_eq_sum","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#flattened-stirling-count-succ-eq-sum","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"For every natural number $n$, the number of flattened Stirling permutations of order $n + 1$ is","title":"A binomial\u2013Stirling formula for flattened Stirling permutations"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/dowling-eq-sum","kind":"node","lean":"MetaMathlibExt.dowling_eq_sum","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#dowling-eq-sum","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"Let $W_{2,1,1}(n, k)$ be the $(q, r)$-Whitney numbers of the second kind with $m = 2$, $r = 1$ and $q = 1$, and let $D_n = \\sum_{k=0}^{n} W_{2,1,1}(n, k)$ be the associated Dowling numbers. For every $n \\ge 1$,","title":"A binomial\u2013Stirling formula for the Dowling numbers"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/has-sum-dowling-egf","kind":"node","lean":"MetaMathlibExt.hasSum_dowling_egf","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#has-sum-dowling-egf","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"For $n \\ge 0$ let $D_n$ be the $n$-th Dowling number of the group of order $2$, that is, the $(q, r)$-Dowling number $D_n = \\sum_{k=0}^{n} W_{2,1,1}(n, k)$ built from the $(q, r)$-Whitney numbers of the second kind with $m = 2$, $r = 1$, $q = 1$. Then for every complex number $x$ the series $\\sum_{n \\ge 0} D_n x^n / n!$ is summable with sum $\\exp\\bigl(x + (e^{2x} - 1)/2\\bigr)$.","title":"Exponential generating function of the Dowling numbers"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/type-bshift","kind":"node","lean":"MetaMathlibExt.typeBShift","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#type-bshift","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For a finite set $S$ of integers, $h(S) = \\{|i| + 1 : i \\in S\\}$, a finite set of natural numbers.","title":"The shift map"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/doubled-word","kind":"node","lean":"MetaMathlibExt.doubledWord","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#doubled-word","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For a finite set $S = \\{s_1 < s_2 < \\dots < s_k\\}$ of natural numbers, the doubled word of $S$ is $f(S) = s_1 s_1 s_2 s_2 \\cdots s_k s_k$, which lists the elements of $S$ in increasing order with each one written twice; $f(\\emptyset)$ is the empty word.","title":"Doubled word"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/wrapped-word","kind":"node","lean":"MetaMathlibExt.wrappedWord","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#wrapped-word","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For a finite set $S = \\{s_1 < s_2 < \\dots < s_k\\}$ of natural numbers, the wrapped word $g(S)$ is $s_1 s_2 s_2 \\cdots s_k s_k s_1$: the smallest element, then each larger element written twice in increasing order, then the smallest element again. For $S = \\emptyset$, $g(S)$ is the empty word.","title":"Wrapped word of a finite set"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/type-bshift-example","kind":"node","lean":"MetaMathlibExt.typeBShift_example","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#type-bshift-example","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For the map $h(S) = \\{|i| + 1 : i \\in S\\}$ on finite sets of integers, $h(\\{-2, 0, 1\\}) = \\{1, 2, 3\\}$.","title":"An example of the shift map"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/doubled-word-singleton","kind":"node","lean":"MetaMathlibExt.doubledWord_singleton","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#doubled-word-singleton","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For every natural number $s$, the doubled word of the singleton $\\{s\\}$ is $f(\\{s\\}) = s\\,s$, the word of length two whose letters are both $s$.","title":"The doubled word of a singleton"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/length-doubled-word","kind":"node","lean":"MetaMathlibExt.length_doubledWord","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#length-doubled-word","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For every finite set $S$ of natural numbers with elements $s_1 < \\dots < s_k$, the doubled word $f(S) = s_1 s_1 s_2 s_2 \\cdots s_k s_k$ has length $2|S|$.","title":"Length of the doubled word"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/wrapped-word-empty","kind":"node","lean":"MetaMathlibExt.wrappedWord_empty","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#wrapped-word-empty","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"The wrapped word $g(\\emptyset)$ of the empty set is the empty word.","title":"Wrapped word of the empty set"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/wrapped-word-singleton","kind":"node","lean":"MetaMathlibExt.wrappedWord_singleton","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#wrapped-word-singleton","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For every natural number $s$, the wrapped word of the singleton $\\{s\\}$ is $g(\\{s\\}) = s\\,s$, the word of length two both of whose letters are $s$.","title":"Wrapped word of a singleton"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/length-wrapped-word","kind":"node","lean":"MetaMathlibExt.length_wrappedWord","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#length-wrapped-word","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For every finite set $S$ of natural numbers with elements $s_1 < \\dots < s_k$, the wrapped word $g(S) = s_1 s_2 s_2 \\cdots s_k s_k s_1$ (the empty word when $S = \\emptyset$) has length $2|S|$.","title":"Length of the wrapped word"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/adler-block-labels","kind":"node","lean":"MetaMathlibExt.adlerBlockLabels","parent":24853,"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#adler-block-labels","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"Let $n$ be a natural number and $P$ a set partition of the integer interval $\\{-n, \\dots, n\\}$; for $x$ in this interval write $P(x)$ for the block of $P$ containing $x$. The Adler block labels of $P$ are the integers $p$ with $1 \\le p \\le n$ such that $p \\le q$ for every positive $q \\in P(p)$ and $p < -q$ for every negative $q \\in P(p)$; that is, $p$ is the least positive element of its block and is smaller than\u2026","title":"Adler block labels"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/adler-block-labels-subset","kind":"node","lean":"MetaMathlibExt.adlerBlockLabels_subset","parent":24853,"reading_title":{"format":"inline-parts-v1","parts":[["text","Adler labels lie in "],["math","\\(\\{1, \\dots, n\\}\\)"]],"source":"Adler labels lie in $\\{1, \\dots, n\\}$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#adler-block-labels-subset","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For every natural number $n$ and every set partition $P$ of $\\{-n, \\dots, n\\}$, every Adler block label $p$ of $P$ is an integer with $1 \\le p \\le n$.","title":"Adler labels lie in $\\{1, \\dots, n\\}$"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/type-bflattened-word-list","kind":"node","lean":"MetaMathlibExt.typeBFlattenedWordList","parent":24853,"reading_title":{"format":"inline-parts-v1","parts":[["text","The word of a set partition of "],["math","\\((-n, n)\\)"],["text",", as a list"]],"source":"The word of a set partition of $(-n, n)$, as a list"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#type-bflattened-word-list","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"Let $n$ be a natural number and let $P$ be a set partition of $[-n, n] = \\{-n, \\dots, n\\}$. For a finite set $S$ of integers let $h(S) = \\{|i| + 1 : i \\in S\\}$, and for a finite set $T$ of natural numbers with elements $t_1 < \\dots < t_k$ let $f(T) = t_1 t_1 t_2 t_2 \\cdots t_k t_k$ and $g(T) = t_1 t_2 t_2 \\cdots t_k t_k t_1$ (both empty when $T = \\emptyset$). Let $P_0$ be the set of nonnegative elements of the\u2026","title":"The word of a set partition of $(-n, n)$, as a list"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/type-bflattened-word","kind":"node","lean":"MetaMathlibExt.typeBFlattenedWord","parent":24853,"reading_title":{"format":"inline-parts-v1","parts":[["text","The word of a set partition of "],["math","\\((-n, n)\\)"]],"source":"The word of a set partition of $(-n, n)$"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#type-bflattened-word","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"fully_proved","summary":"For a natural number $n$ and a set partition $P$ of $[-n, n] = \\{-n, \\dots, n\\}$, let $L(P)$ be the list of $P$: the list of natural numbers built from the blocks of $P$ (via the shift map and the doubled and wrapped words). The word $\\varphi(P) = w_1 w_2 \\cdots w_{2(n+1)}$ has $w_i$ equal to the $i$-th entry of $L(P)$ when $L(P)$ has at least $i$ entries, and $w_i = 0$ otherwise.","title":"The word of a set partition of $(-n, n)$"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/type-bflattened-word-mem-flattened-stirl","kind":"node","lean":"MetaMathlibExt.typeBFlattenedWord_mem_flattenedStirling","parent":24853,"reading_title":{"format":"inline-parts-v1","parts":[["text","The word of a type "],["math","\\(B\\)"],["text"," set partition is a flattened Stirling permutation"]],"source":"The word of a type $B$ set partition is a flattened Stirling permutation"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#type-bflattened-word-mem-flattened-stirl","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"For every natural number $n$ and every type $B$ set partition $P$ of $[-n, n] = \\{-n, \\dots, n\\}$, the word $\\varphi(P) = w_1 w_2 \\cdots w_{2(n+1)}$ of $P$ is a Stirling permutation of order $n + 1$ and is flattened.","title":"The word of a type $B$ set partition is a flattened Stirling permutation"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/bij-on-type-bflattened-word","kind":"node","lean":"MetaMathlibExt.bijOn_typeBFlattenedWord","parent":24853,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(\\varphi\\)"],["text"," is a bijection from type "],["math","\\(B\\)"],["text"," set partitions onto flattened Stirling permutations"]],"source":"The map $\\varphi$ is a bijection from type $B$ set partitions onto flattened Stirling permutations"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#bij-on-type-bflattened-word","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"The map $\\varphi$ is a bijection from type $B$ set partitions onto flattened Stirling permutations. For a natural number $n$, let $\\varphi$ be the map of Buck et al. (Proposition 1) that sends a set partition $P$ of $\\{-n, \\dots, n\\}$ to a word of length $2(n + 1)$ over the natural numbers, built from the zero block of $P$ and its Adler-labelled blocks with each element $i$ replaced by $|i| + 1$. Then $\\varphi$\u2026","title":"The map $\\varphi$ is a bijection from type $B$ set partitions onto flattened Stirling permutations"},{"id":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/word-run-count-type-bflattened-word","kind":"node","lean":"MetaMathlibExt.wordRunCount_typeBFlattenedWord","parent":24853,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of runs of the word of a type "],["math","\\(B\\)"],["text"," set partition"]],"source":"Number of runs of the word of a type $B$ set partition"},"result":true,"route":"../roadmap/Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations/index.html#word-run-count-type-bflattened-word","scope":"Combinatorics/permutations/permutation-statistics/flattened-stirling-permutations","status":"can_prove","summary":"Let $\\pi$ be a type $B$ set partition of $[-n, n]$ and let $\\varphi(\\pi)$ be the word of length $2(n + 1)$ that the map $\\varphi$ of Proposition 1 assigns to it. Write $P_0$ for the set of nonnegative elements of the block of $\\pi$ containing $0$. Let $\\beta_1, \\dots, \\beta_m$ be the blocks kept by Adler's labelling, exactly one from each pair $\\{\\beta, -\\beta\\}$ of distinct blocks, namely the block containing the\u2026","title":"Number of runs of the word of a type $B$ set partition"},{"data_url":"scopes/Combinatorics/boolean-functions.json","id":"Combinatorics/boolean-functions","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/index.html","scope":"Combinatorics/boolean-functions","status":"can_prove","summary":"Functions from {0,1}^n to {0,1}: counting monotone, self-dual and unate Boolean functions and Dedekind numbers, enumeration of k-SAT functions, noise stability and juntas, and parity and small-bias counts over finite fields.","title":"Boolean functions"},{"data_url":"scopes/Combinatorics/boolean-functions/dedekind-numbers.json","id":"Combinatorics/boolean-functions/dedekind-numbers","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/dedekind-numbers/index.html","scope":"Combinatorics/boolean-functions/dedekind-numbers","status":"can_prove","title":"Dedekind numbers"},{"id":"Combinatorics/boolean-functions/dedekind-numbers/antichain-count","kind":"node","lean":"MathlibExt.Combinatorics.DedekindAntichainCountErdosProblem497Wanted.antichainCount","parent":24891,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/dedekind-numbers/index.html#antichain-count","scope":"Combinatorics/boolean-functions/dedekind-numbers","status":"fully_proved","summary":"For a natural number $n$, the Dedekind number $M(n)$ is the number of antichains with respect to inclusion among the subsets of the $n$-element set $\\{0, 1, \\dots, n-1\\}$, that is, of families of subsets in which no member is contained in a different member; the empty family and the family $\\{\\emptyset\\}$ are counted.","title":"Dedekind numbers"},{"id":"Combinatorics/boolean-functions/dedekind-numbers/antichain-count-log-asymptotic","kind":"node","lean":"MathlibExt.Combinatorics.DedekindAntichainCountErdosProblem497Wanted.antichainCountLogAsymptotic","parent":24891,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/dedekind-numbers/index.html#antichain-count-log-asymptotic","scope":"Combinatorics/boolean-functions/dedekind-numbers","status":"fully_proved","summary":"Let $M(n)$ be the number of antichains with respect to inclusion among the subsets of an $n$-element set (the Dedekind number). The statement is that $\\log_2 M(n) / \\binom{n}{\\lfloor n/2 \\rfloor} \\to 1$ as $n \\to \\infty$; equivalently, $M(n) = 2^{(1 + o(1)) \\binom{n}{\\lfloor n/2 \\rfloor}}$.","title":"Erd\u0151s Problem 497: the asymptotic statement for Dedekind numbers"},{"id":"Combinatorics/boolean-functions/dedekind-numbers/antichain-count-log-asymptotic-holds","kind":"node","lean":"MathlibExt.Combinatorics.DedekindAntichainCountErdosProblem497Wanted.antichainCountLogAsymptotic_holds","parent":24891,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/dedekind-numbers/index.html#antichain-count-log-asymptotic-holds","scope":"Combinatorics/boolean-functions/dedekind-numbers","status":"can_prove","summary":"Let $M(n)$ be the number of antichains with respect to inclusion among the subsets of an $n$-element set. Then $\\log_2 M(n) / \\binom{n}{\\lfloor n/2 \\rfloor} \\to 1$ as $n \\to \\infty$; that is, $M(n) = 2^{(1 + o(1)) \\binom{n}{\\lfloor n/2 \\rfloor}}$.","title":"Kleitman's theorem on Dedekind numbers"},{"data_url":"scopes/Combinatorics/boolean-functions/enumeration-of-k-sat-functions.json","id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"can_prove","title":"Enumeration of k-SAT functions"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/clause-satisfied","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.ClauseSatisfied","parent":24895,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#clause-satisfied","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$. A clause on $n$ variables is a finite set $C$ of literals $(i, b)$, where $i \\in \\{0, \\dots, n - 1\\}$ is a variable and $b \\in \\{\\mathrm{true}, \\mathrm{false}\\}$. An assignment $x : \\{0, \\dots, n - 1\\} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ satisfies $C$ if $x_i = b$ for some literal $(i, b) \\in C$.","title":"Satisfied clause"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/is-ksatfunction","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.IsKSATFunction","parent":24895,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-SAT function"]],"source":"$k$-SAT function"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#is-ksatfunction","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"Let $k, n \\in \\mathbb{N}$. A Boolean function $f : \\{\\mathrm{true}, \\mathrm{false}\\}^n \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ is a $k$-SAT function on $n$ variables if there is a finite set $S$ of clauses on $n$ variables such that each clause in $S$ has at most $k$ literals, no two of them on the same variable, and for every assignment $x$, $f(x) = \\mathrm{true}$ if and only if $x$ satisfies every clause in $S$.","title":"$k$-SAT function"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/is-positive-unate-in","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.IsPositiveUnateIn","parent":24895,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#is-positive-unate-in","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"Let $f : \\{\\mathrm{true}, \\mathrm{false}\\}^n \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ be a Boolean function and $i$ one of its $n$ variables. Then $f$ is positively unate in $i$ if for all assignments $x$ and $y$ that agree at every variable other than $i$ and have $x_i = \\mathrm{false}$ and $y_i = \\mathrm{true}$, $f(x) = \\mathrm{true}$ implies $f(y) = \\mathrm{true}$. Equivalently, switching $i$ from false to true\u2026","title":"Positively unate variable"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/is-negative-unate-in","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.IsNegativeUnateIn","parent":24895,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#is-negative-unate-in","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"Let $f : \\{\\mathrm{true}, \\mathrm{false}\\}^n \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ be a Boolean function and $i$ one of its $n$ variables. Then $f$ is negatively unate in $i$ if for all assignments $x$ and $y$ that agree at every variable other than $i$ and have $x_i = \\mathrm{false}$ and $y_i = \\mathrm{true}$, $f(y) = \\mathrm{true}$ implies $f(x) = \\mathrm{true}$. Equivalently, switching $i$ from false to true\u2026","title":"Negatively unate variable"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/is-unate","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.IsUnate","parent":24895,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#is-unate","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"A Boolean function $f : \\{\\mathrm{true}, \\mathrm{false}\\}^n \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ is unate if, for every variable $i$, $f$ is positively unate in $i$ or negatively unate in $i$.","title":"Unate Boolean function"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/count-ksatfunctions","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.countKSATFunctions","parent":24895,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-SAT functions"]],"source":"Number of $k$-SAT functions"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#count-ksatfunctions","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the number of Boolean functions $f : \\{\\mathrm{true}, \\mathrm{false}\\}^n \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ that are $k$-SAT functions on $n$ variables.","title":"Number of $k$-SAT functions"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/count-unate-ksatfunctions","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.countUnateKSATFunctions","parent":24895,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of unate "],["math","\\(k\\)"],["text","-SAT functions"]],"source":"Number of unate $k$-SAT functions"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#count-unate-ksatfunctions","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the number of Boolean functions $f : \\{\\mathrm{true}, \\mathrm{false}\\}^n \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ that are $k$-SAT functions on $n$ variables and are unate.","title":"Number of unate $k$-SAT functions"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.conjecture","parent":24895,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bollob\u00e1s\u2013Brightwell\u2013Leader conjecture for "],["math","\\(k\\)"]],"source":"Bollob\u00e1s\u2013Brightwell\u2013Leader conjecture for $k$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#conjecture","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"For a natural number $k$, the proposition that if $k \\ge 2$, then the number of unate $k$-SAT functions on $n$ Boolean variables divided by the number of $k$-SAT functions on $n$ Boolean variables tends to $1$ as $n \\to \\infty$; that is, a $1 - o(1)$ fraction of the $k$-SAT functions on $n$ variables is unate.","title":"Bollob\u00e1s\u2013Brightwell\u2013Leader conjecture for $k$"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/conjecture-all-k","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.conjectureAllK","parent":24895,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bollob\u00e1s\u2013Brightwell\u2013Leader conjecture for all "],["math","\\(k\\)"]],"source":"Bollob\u00e1s\u2013Brightwell\u2013Leader conjecture for all $k$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#conjecture-all-k","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"fully_proved","summary":"The proposition that for every natural number $k \\ge 2$, the number of unate $k$-SAT functions on $n$ Boolean variables divided by the number of $k$-SAT functions on $n$ Boolean variables tends to $1$ as $n \\to \\infty$.","title":"Bollob\u00e1s\u2013Brightwell\u2013Leader conjecture for all $k$"},{"id":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions/conjecture-all-k-holds","kind":"node","lean":"MathlibExt.Combinatorics.BollobasBrightwellLeaderUnateDensityWanted.conjectureAllK_holds","parent":24895,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nearly all "],["math","\\(k\\)"],["text","-SAT functions are unate"]],"source":"Nearly all $k$-SAT functions are unate"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/enumeration-of-k-sat-functions/index.html#conjecture-all-k-holds","scope":"Combinatorics/boolean-functions/enumeration-of-k-sat-functions","status":"can_prove","summary":"For every natural number $k \\ge 2$, the number of unate $k$-SAT functions on $n$ Boolean variables divided by the number of $k$-SAT functions on $n$ Boolean variables tends to $1$ as $n \\to \\infty$; that is, the Bollob\u00e1s\u2013Brightwell\u2013Leader conjecture holds for every $k$.","title":"Nearly all $k$-SAT functions are unate"},{"data_url":"scopes/Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function.json","id":"Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function/index.html","scope":"Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function","status":"can_prove","title":"Junta approximation of monotone Boolean functions"},{"id":"Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function/servedio-tan-verbin-conjecture","kind":"node","lean":"MathlibExt.Combinatorics.MonotoneJuntaApproximationAMR127Problem0022Wanted.servedioTanVerbinConjecture","parent":24906,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function/index.html#servedio-tan-verbin-conjecture","scope":"Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function","status":"fully_proved","summary":"Order $\\{0,1\\}^n$ coordinatewise with $0 < 1$, and write $\\deg f$ for the Boolean degree of a function $f : \\{0,1\\}^n \\to \\{0,1\\}$. This is the statement that for every real $\\varepsilon > 0$ there is a natural number $C$ such that for every $n \\in \\mathbb{N}$ and every monotone $f : \\{0,1\\}^n \\to \\{0,1\\}$ there are a set $J$ of at most $(\\deg f + 2)^C$ coordinates and a function $g : \\{0,1\\}^n \\to \\{0,1\\}$ that\u2026","title":"The Servedio\u2013Tan\u2013Verbin conjecture"},{"id":"Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function/servedio-tan-verbin-conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.MonotoneJuntaApproximationAMR127Problem0022Wanted.servedioTanVerbinConjecture_refuted","parent":24906,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function/index.html#servedio-tan-verbin-conjecture-refuted","scope":"Combinatorics/boolean-functions/junta-approximation-of-monotone-boolean-function","status":"can_prove","summary":"Order $\\{0,1\\}^n$ coordinatewise with $0 < 1$, and write $\\deg f$ for the Boolean degree of a function $f : \\{0,1\\}^n \\to \\{0,1\\}$. It is not the case that for every real $\\varepsilon > 0$ there is a natural number $C$ such that for every $n \\in \\mathbb{N}$ and every monotone $f : \\{0,1\\}^n \\to \\{0,1\\}$ there are a set $J$ of at most $(\\deg f + 2)^C$ coordinates and a function $g : \\{0,1\\}^n \\to \\{0,1\\}$ that\u2026","title":"The Servedio\u2013Tan\u2013Verbin conjecture is false"},{"data_url":"scopes/Combinatorics/boolean-functions/majority-is-least-stable-conjecture.json","id":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/majority-is-least-stable-conjecture/index.html","scope":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture","status":"can_prove","title":"Majority is least stable conjecture"},{"id":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture/is-linear-threshold-function","kind":"node","lean":"MathlibExt.Combinatorics.MajorityIsLeastStableAMR127Problem0009Wanted.IsLinearThresholdFunction","parent":24909,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/majority-is-least-stable-conjecture/index.html#is-linear-threshold-function","scope":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture","status":"fully_proved","summary":"A function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ is a linear threshold function if there are reals $w_0, w_1, \\dots, w_n$ such that for every $x \\in \\{-1, 1\\}^n$ we have $f(x) = \\operatorname{sgn}(w_0 + w_1 x_1 + \\dots + w_n x_n)$, where $x_i \\in \\{-1, 1\\}$ is the $i$-th coordinate of $x$, $\\operatorname{sgn} z = 1$ for $z \\ge 0$ and $\\operatorname{sgn} z = -1$ for $z < 0$.","title":"Linear threshold function"},{"id":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture/majority","kind":"node","lean":"MathlibExt.Combinatorics.MajorityIsLeastStableAMR127Problem0009Wanted.majority","parent":24909,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/majority-is-least-stable-conjecture/index.html#majority","scope":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the majority function $\\mathrm{Maj}_n : \\{-1, 1\\}^n \\to \\mathbb{R}$ is given by $\\mathrm{Maj}_n(x) = 1$ if $x_1 + \\dots + x_n \\ge 0$ and $\\mathrm{Maj}_n(x) = -1$ otherwise, where $x_i \\in \\{-1, 1\\}$ is the $i$-th coordinate of $x$.","title":"Majority function"},{"id":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture/noise-stability","kind":"node","lean":"MathlibExt.Combinatorics.MajorityIsLeastStableAMR127Problem0009Wanted.noiseStability","parent":24909,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/majority-is-least-stable-conjecture/index.html#noise-stability","scope":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture","status":"fully_proved","summary":"For a real number $\\rho$ and a function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$, the noise stability of $f$ at $\\rho$ is $\\mathrm{Stab}_\\rho[f] = \\sum_{S \\subseteq \\{1, \\dots, n\\}} \\rho^{|S|} \\hat f(S)^2$, where $\\hat f(S)$ is the Fourier coefficient of $f$ at $S$.","title":"Noise stability"},{"id":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture/majority-is-least-stable-for","kind":"node","lean":"MathlibExt.Combinatorics.MajorityIsLeastStableAMR127Problem0009Wanted.MajorityIsLeastStableFor","parent":24909,"reading_title":{"format":"inline-parts-v1","parts":[["text","Majority Is Least Stable conjecture in "],["math","\\(n\\)"],["text"," variables"]],"source":"Majority Is Least Stable conjecture in $n$ variables"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/majority-is-least-stable-conjecture/index.html#majority-is-least-stable-for","scope":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, this is the statement that every linear threshold function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ satisfies $\\mathrm{Stab}_\\rho[f] \\ge \\mathrm{Stab}_\\rho[\\mathrm{Maj}_n]$ for every real $\\rho$ with $0 \\le \\rho \\le 1$, where $\\mathrm{Stab}_\\rho$ is the noise stability and $\\mathrm{Maj}_n$ is the majority function.","title":"Majority Is Least Stable conjecture in $n$ variables"},{"id":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture/majority-is-least-stable-for-some-odd-at","kind":"node","lean":"MathlibExt.Combinatorics.MajorityIsLeastStableAMR127Problem0009Wanted.majorityIsLeastStableForSomeOddAtLeastFive","parent":24909,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/majority-is-least-stable-conjecture/index.html#majority-is-least-stable-for-some-odd-at","scope":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture","status":"fully_proved","summary":"There is an odd natural number $n \\ge 5$ for which the Majority Is Least Stable conjecture in $n$ variables holds: every linear threshold function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ satisfies $\\mathrm{Stab}_\\rho[f] \\ge \\mathrm{Stab}_\\rho[\\mathrm{Maj}_n]$ for every real $\\rho$ with $0 \\le \\rho \\le 1$, where $\\mathrm{Stab}_\\rho$ is the noise stability and $\\mathrm{Maj}_n$ is the majority function.","title":"Majority Is Least Stable in some odd dimension at least five"},{"id":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture/majority-is-least-stable-for-some-odd-at-2","kind":"node","lean":"MathlibExt.Combinatorics.MajorityIsLeastStableAMR127Problem0009Wanted.majorityIsLeastStableForSomeOddAtLeastFive_refuted","parent":24909,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/majority-is-least-stable-conjecture/index.html#majority-is-least-stable-for-some-odd-at-2","scope":"Combinatorics/boolean-functions/majority-is-least-stable-conjecture","status":"can_prove","summary":"For every odd natural number $n \\ge 5$, the Majority Is Least Stable conjecture in $n$ variables fails: there are a linear threshold function $f : \\{-1, 1\\}^n \\to \\mathbb{R}$ and a real number $\\rho$ with $0 \\le \\rho \\le 1$ such that $\\mathrm{Stab}_\\rho[f] < \\mathrm{Stab}_\\rho[\\mathrm{Maj}_n]$, where $\\mathrm{Stab}_\\rho$ is the noise stability and $\\mathrm{Maj}_n$ is the majority function.","title":"Majority Is Least Stable fails in every odd dimension at least five"},{"data_url":"scopes/Combinatorics/boolean-functions/osss-inequality-for-decision-trees.json","id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","title":"OSSS inequality for decision trees"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/weight","kind":"node","lean":"StatMech.OSSS.weight","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#weight","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The probability `\u220f\u2091 \u03bd e (\u03c9 e)` of the configuration `\u03c9` under the product weight `\u03bd`.","title":"Probability of a configuration under a product weight"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/expect","kind":"node","lean":"StatMech.OSSS.expect","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#expect","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The expectation `\u2211_\u03c9 weight \u03bd \u03c9 * g \u03c9` of `g` under the product weight `\u03bd`.","title":"Expectation under a product weight"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/is-prob-weight","kind":"node","lean":"StatMech.OSSS.IsProbWeight","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#is-prob-weight","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"`\u03bd` is a product probability weight: each `\u03bd e` is a probability vector on `Bool`.","title":"Product probability weight"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/pt","kind":"node","lean":"StatMech.OSSS.pt","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#pt","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The point mass at `b`, as a weight on `Bool`. Replacing `\u03bd e` by `pt b` conditions on `\u03c9 e = b`.","title":"Point-mass weight on the Booleans"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/cov","kind":"node","lean":"StatMech.OSSS.cov","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#cov","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The covariance `\ud835\udd3c[f g] - \ud835\udd3c[f] \ud835\udd3c[g]` under the product weight `\u03bd`.","title":"Covariance under a product weight"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/var","kind":"node","lean":"StatMech.OSSS.var","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#var","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The variance of `g` under the product weight `\u03bd`.","title":"Variance under a product weight"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/infl","kind":"node","lean":"StatMech.OSSS.infl","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#infl","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The influence of coordinate `e` on `g`: `\ud835\udd3c |g(\u03c9^{e \u21a6 1}) - g(\u03c9^{e \u21a6 0})|`. For the indicator of an event this is the probability that `e` is pivotal (`infl_indicator_eq_pivotal`).","title":"Influence of a coordinate"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/cond-mean","kind":"node","lean":"StatMech.OSSS.condMean","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#cond-mean","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The conditional expectation of `g` given all coordinates except `e`.","title":"Conditional expectation given the other coordinates"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/decision-tree","kind":"node","lean":"StatMech.OSSS.DecisionTree","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#decision-tree","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"A binary decision tree on the variables `E`: a leaf outputs a Boolean value, and a node queries a variable and continues in the first subtree if it is `true` and in the second otherwise.","title":"Binary decision tree"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/eval","kind":"node","lean":"StatMech.OSSS.DecisionTree.eval","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#eval","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The Boolean function computed by the decision tree.","title":"Boolean function computed by a decision tree"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/queried","kind":"node","lean":"StatMech.OSSS.DecisionTree.queried","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#queried","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The set of variables the decision tree queries on input `\u03c9`.","title":"Variables queried by a decision tree on an input"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/eval-r","kind":"node","lean":"StatMech.OSSS.DecisionTree.evalR","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#eval-r","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The function computed by the decision tree, as a `0/1`-valued real function.","title":"Real-valued function computed by a decision tree"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/reveal","kind":"node","lean":"StatMech.OSSS.reveal","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#reveal","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The revealment `\u03b4\u2091(T)`: the probability that the decision tree `T` queries the variable `e`.","title":"Revealment of a variable by a decision tree"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/flip-at","kind":"node","lean":"StatMech.OSSS.flipAt","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#flip-at","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The configuration `\u03c9` with coordinate `e` flipped.","title":"Configuration with one coordinate flipped"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/claim-a","kind":"node","lean":"StatMech.OSSS.claimA","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#claim-a","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"Root step of the OSSS induction: the covariance of a function whose discrete derivative in `e` is bounded by `1` with the martingale increment `g - \ud835\udd3c[g | \u03c9_{\u2260e}]` is at most `Inf\u2091(g)`.","title":"Root step of the OSSS induction"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/claim-b","kind":"node","lean":"StatMech.OSSS.claimB","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#claim-b","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"Subtree step of the OSSS induction: for `G` independent of the root variable `e`, the covariance with a tree rooted at `e` splits into the covariances with the two subtrees under the conditioned weights.","title":"Subtree step of the OSSS induction"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/osss-cov","kind":"node","lean":"StatMech.OSSS.osss_cov","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#osss-cov","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The OSSS covariance inequality, stated for all product weights at once so that it can be proved by induction on the decision tree. Use `osss` for the usual argument order.","title":"OSSS covariance inequality over all product weights"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/osss-var","kind":"node","lean":"StatMech.OSSS.osss_var","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#osss-var","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"OSSS inequality (O'Donnell, Saks, Schramm, Servedio, *Every decision tree has an influential variable*, FOCS 2005): if the decision tree `T` computes `f`, then `Var(f) \u2264 \u2211\u1d62 \u03b4\u1d62(T) Inf\u1d62(f)` under every product probability measure.","title":"OSSS inequality"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/osss","kind":"node","lean":"StatMech.OSSS.osss","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#osss","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"OSSS inequality, covariance form (O'Donnell, Saks, Schramm, Servedio, *Every decision tree has an influential variable*, FOCS 2005; in this form Duminil-Copin, Raoufi, Tassion, Ann. of Math. 2019): if the decision tree `T` computes the Boolean function `f`, then for every real function `g` and every product probability measure, `Cov(f, g) \u2264 \u2211\u1d62 \u03b4\u1d62(T) Inf\u1d62(g)`, where `Inf\u1d62(g) = \ud835\udd3c |g(\u03c9^{i \u21a6 1}) - g(\u03c9^{i \u21a6 0})|`.","title":"OSSS inequality"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/bernoulli-weight","kind":"node","lean":"StatMech.OSSS.bernoulliWeight","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#bernoulli-weight","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The Bernoulli(`p`) product weight: every coordinate is `true` with probability `p`.","title":"Bernoulli product weight"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/osss-bernoulli","kind":"node","lean":"StatMech.OSSS.osss_bernoulli","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#osss-bernoulli","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The OSSS inequality for the Bernoulli(`p`) product measure.","title":"OSSS inequality for the Bernoulli product measure"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/infl-indicator-eq-pivotal","kind":"node","lean":"StatMech.OSSS.infl_indicator_eq_pivotal","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#infl-indicator-eq-pivotal","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The influence of `e` on the indicator of `A` is the probability that `e` is pivotal for `A`.","title":"Influence on an indicator is the pivotal probability"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/expect-bernoulli-weight-eq-integral","kind":"node","lean":"StatMech.OSSS.expect_bernoulliWeight_eq_integral","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#expect-bernoulli-weight-eq-integral","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The Bernoulli(`p`) product weight computes integrals against `StatMech.bernoulliProductMeasure`, the corresponding product probability measure.","title":"Bernoulli product weight computes product-measure integrals"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/full-tree","kind":"node","lean":"StatMech.OSSS.Poincare.fullTree","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#full-tree","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The decision tree that queries the variables of `l` in order and then outputs the value of `\u03c6` at the configuration that agrees with the answers on `l` and is `false` elsewhere. It computes `\u03c6` when `\u03c6` depends only on the variables in `l` (`eval_fullTree`).","title":"Decision tree querying a list of variables"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/insens-outside","kind":"node","lean":"StatMech.OSSS.Poincare.InsensOutside","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#insens-outside","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"`\u03c6` depends only on the coordinates in the list `l`.","title":"Function depending only on listed coordinates"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/full-tree-univ","kind":"node","lean":"StatMech.OSSS.Poincare.fullTreeUniv","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#full-tree-univ","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"The decision tree that queries every variable and then outputs the value of `\u03c6`; it computes `\u03c6` and has revealment `1` at every variable.","title":"Decision tree querying every variable"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/poincare","kind":"node","lean":"StatMech.OSSS.Poincare.poincare","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#poincare","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"Poincar\u00e9 inequality for Boolean functions under a product measure: `Var(1_A) \u2264 \u2211\u2091 Inf\u2091(1_A)`, where `Inf\u2091(1_A)` is the probability that `e` is pivotal. This is the OSSS inequality for the full decision tree; it is weaker than the Efron\u2013Stein inequality `Var(f) \u2264 \u2211\u2091 p\u2091(1 - p\u2091) \ud835\udd3c[(\u2202\u2091 f)\u00b2]` by the factor `p\u2091(1 - p\u2091) \u2264 1/4`.","title":"Poincar\u00e9 inequality for Boolean functions under a product measure"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/poincare-cov","kind":"node","lean":"StatMech.OSSS.Poincare.poincare_cov","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#poincare-cov","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"Covariance form of `poincare`: `Cov(1_A, g) \u2264 \u2211\u2091 Inf\u2091(g)` for every real function `g`.","title":"Covariance form of the Poincar\u00e9 inequality"},{"id":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees/poincare-bernoulli","kind":"node","lean":"StatMech.OSSS.Poincare.poincare_bernoulli","parent":24916,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/osss-inequality-for-decision-trees/index.html#poincare-bernoulli","scope":"Combinatorics/boolean-functions/osss-inequality-for-decision-trees","status":"fully_proved","summary":"`poincare` for the Bernoulli(`p`) product measure.","title":"Poincar\u00e9 inequality for the Bernoulli product measure"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality","kind":"scope","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality","status":"can_prove","title":"Kahn\u2013Kalai\u2013Linial inequality"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/kkl-pbiased","kind":"node","lean":"MathlibExt.Complexity.BooleanFunctions.KKLPBiasedWanted.kkl_pbiased","parent":24946,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/index.html#kkl-pbiased","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality","status":"can_prove","summary":"KKL inequality for `p`-biased measures (Bourgain\u2013Kahn\u2013Kalai\u2013Katznelson\u2013Linial; Talagrand). For every `p \u2208 (0, 1)` there is `c > 0` such that for every finite nonempty `E` and every Boolean function `\u03c6` on `{0, 1}^E`, `c \u00b7 Var_p(\u03c6) \u00b7 log (1 / max\u2091 Inf\u2091^p(\u03c6)) \u2264 \u2211\u2091 Inf\u2091^p(\u03c6)`, where `Inf\u2091^p(\u03c6)` is the probability under the Bernoulli(`p`) product measure that `e` is pivotal for `\u03c6`.","title":"KKL inequality for `p`-biased measures"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","status":"fully_proved","title":"Sub-additivity of entropy for Bernoulli product measures"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/esd-phi","kind":"node","lean":"StatMech.Probability.esd_Phi","parent":24948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Function "],["math","\\(x \\log x\\)"],["text"," with "],["math","\\(0 \\log 0 = 0\\)"]],"source":"Function $x \\log x$ with $0 \\log 0 = 0$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/index.html#esd-phi","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","status":"fully_proved","summary":"The function `x \u21a6 x log x` (with `0 log 0 = 0`). It equals `-Real.negMulLog x` (`esd_Phi_eq_neg_negMulLog`).","title":"Function $x \\log x$ with $0 \\log 0 = 0$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/esd-phi-eq-neg-neg-mul-log","kind":"node","lean":"StatMech.Probability.esd_Phi_eq_neg_negMulLog","parent":24948,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(x \\log x\\)"],["text"," is the negation of Mathlib's "],["math","\\(-x \\log x\\)"]],"source":"$x \\log x$ is the negation of Mathlib's $-x \\log x$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/index.html#esd-phi-eq-neg-neg-mul-log","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","status":"fully_proved","summary":"`esd_Phi` is the negative of Mathlib's `Real.negMulLog`.","title":"$x \\log x$ is the negation of Mathlib's $-x \\log x$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/esd-ent","kind":"node","lean":"StatMech.Probability.esd_ent","parent":24948,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/index.html#esd-ent","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","status":"fully_proved","summary":"The entropy `Ent_\u03bd(g) = \ud835\udd3c_\u03bd[g log g] - \ud835\udd3c_\u03bd[g] log \ud835\udd3c_\u03bd[g]` of `g` under the product weight `\u03bd`.","title":"Entropy of a function under a product weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/esd-ent-coord","kind":"node","lean":"StatMech.Probability.esd_entCoord","parent":24948,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/index.html#esd-ent-coord","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","status":"fully_proved","summary":"The conditional entropy of `g` in coordinate `e` given the other coordinates, under the Bernoulli(`p`) weight: `p \u03a6(g(\u03c9^{e\u21a61})) + (1-p) \u03a6(g(\u03c9^{e\u21a60})) - \u03a6(\ud835\udd3c[g | \u03c9_{\u2260e}])`, with `\u03a6(x) = x log x`.","title":"Conditional entropy in one coordinate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/esd-pos-weight","kind":"node","lean":"StatMech.Probability.esd_PosWeight","parent":24948,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/index.html#esd-pos-weight","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","status":"fully_proved","summary":"Every coordinate weight of `\u03bd` is strictly positive.","title":"Strictly positive coordinate weights"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/esd-dpi","kind":"node","lean":"StatMech.Probability.esd_dpi","parent":24948,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/index.html#esd-dpi","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","status":"fully_proved","summary":"Data-processing inequality for entropy under a two-point mixture: for nonnegative `g\u2080, g\u2081`, `p \u03a6(\ud835\udd3c g\u2081) + (1-p) \u03a6(\ud835\udd3c g\u2080) - \u03a6(p \ud835\udd3c g\u2081 + (1-p) \ud835\udd3c g\u2080) \u2264 \ud835\udd3c[p \u03a6(g\u2081) + (1-p) \u03a6(g\u2080) - \u03a6(p g\u2081 + (1-p) g\u2080)]`.","title":"Data-processing inequality for entropy under a two-point mixture"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/esd-subadditivity","kind":"node","lean":"StatMech.Probability.esd_subadditivity","parent":24948,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product/index.html#esd-subadditivity","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/sub-additivity-of-entropy-for-bernoulli-product","status":"fully_proved","summary":"Sub-additivity (tensorization) of entropy for the Bernoulli(`p`) product measure on `{0, 1}^n` (see S. Boucheron, G. Lugosi, P. Massart, *Concentration Inequalities*, 2013, Theorem 4.10): for nonnegative `g`, `Ent(g) \u2264 \u2211\u2091 \ud835\udd3c[Ent\u2091(g)]`.","title":"Sub-additivity (tensorization) of entropy"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun","status":"fully_proved","title":"The KKL inequality: statement and elementary bounds"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/total-infl","kind":"node","lean":"StatMech.Probability.totalInfl","parent":24956,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/index.html#total-infl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun","status":"fully_proved","summary":"The total influence `\u2211\u2091 Inf\u2091(g)` of `g` under the product weight `\u03bd`.","title":"Total influence under a product weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/max-infl","kind":"node","lean":"StatMech.Probability.maxInfl","parent":24956,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/index.html#max-infl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun","status":"fully_proved","summary":"The maximal influence `max\u2091 Inf\u2091(g)` of `g` under the product weight `\u03bd`.","title":"Maximal influence under a product weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/kklhypercontractive","kind":"node","lean":"StatMech.Probability.KKLHypercontractive","parent":24956,"reading_title":{"format":"inline-parts-v1","parts":[["text","KKL inequality with constant "],["math","\\(c\\)"],["text"," for the "],["math","\\(p\\)"],["text","-biased measure"]],"source":"KKL inequality with constant $c$ for the $p$-biased measure"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/index.html#kklhypercontractive","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun","status":"fully_proved","summary":"The KKL inequality with constant `c` for the Bernoulli(`p`) product measure: for every finite nonempty index type `E` and every Boolean function `\u03c6` on `{0, 1}^E`, `c \u00b7 Var(\u03c6) \u00b7 log (1 / max\u2091 Inf\u2091(\u03c6)) \u2264 \u2211\u2091 Inf\u2091(\u03c6)`, with `Inf\u2091(\u03c6)` the probability that `e` is pivotal.","title":"KKL inequality with constant $c$ for the $p$-biased measure"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/kkl-inequality-total","kind":"node","lean":"StatMech.Probability.kkl_inequality_total","parent":24956,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/index.html#kkl-inequality-total","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun","status":"fully_proved","summary":"Kahn\u2013Kalai\u2013Linial inequality, total-influence form (J. Kahn, G. Kalai, N. Linial, *The influence of variables on Boolean functions*, FOCS 1988): under `KKLHypercontractive p c`, `c \u00b7 Var(\u03c6) \u00b7 log (1 / max\u2091 Inf\u2091(\u03c6)) \u2264 \u2211\u2091 Inf\u2091(\u03c6)`.","title":"Kahn\u2013Kalai\u2013Linial inequality"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/kkl-inequality-max","kind":"node","lean":"StatMech.Probability.kkl_inequality_max","parent":24956,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/index.html#kkl-inequality-max","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun","status":"fully_proved","summary":"Kahn\u2013Kalai\u2013Linial inequality, maximal-influence form (J. Kahn, G. Kalai, N. Linial, *The influence of variables on Boolean functions*, FOCS 1988): under `KKLHypercontractive p c`, `c \u00b7 Var(\u03c6) \u00b7 log (1 / \u03b4) \u2264 |E| \u00b7 \u03b4` with `\u03b4 = max\u2091 Inf\u2091(\u03c6)`, so `\u03b4` is at least of order `Var(\u03c6) \u00b7 log |E| / |E|`.","title":"Kahn\u2013Kalai\u2013Linial inequality"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/kkl-hypercontractive-zero","kind":"node","lean":"StatMech.Probability.kkl_hypercontractive_zero","parent":24956,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun/index.html#kkl-hypercontractive-zero","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-statement-and-elementary-boun","status":"fully_proved","summary":"`KKLHypercontractive p 0` holds trivially, since total influence is nonnegative.","title":"KKL inequality with constant 0 holds trivially"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","title":"Walsh\u2013Fourier analysis on the uniform cube and the KKL reduction"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-sign","kind":"node","lean":"StatMech.Probability.khc_sign","parent":24963,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\pm 1\\)"],["text"," value of a bit"]],"source":"$\\pm 1$ value of a bit"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-sign","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The `\u00b11` value of a bit: `false \u21a6 1`, `true \u21a6 -1`.","title":"$\\pm 1$ value of a bit"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-walsh-char","kind":"node","lean":"StatMech.Probability.khc_walshChar","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-walsh-char","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The Walsh character `\u03c7_S(x) = \u220f_{i \u2208 S} (-1)^{x i}` on `{0, 1}^\u03b9`.","title":"Walsh character on the Boolean cube"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-fourier-coeff","kind":"node","lean":"StatMech.Probability.khc_fourierCoeff","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-fourier-coeff","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The Fourier\u2013Walsh coefficient `f\u0302(S) = \ud835\udd3c[f \u03c7_S]` under the uniform measure.","title":"Fourier\u2013Walsh coefficient under the uniform measure"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-reindex","kind":"node","lean":"StatMech.Probability.khc_reindex","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-reindex","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The function `f` on `{0, 1}^\u03b9` transported to `{0, 1}^\u03ba` along the bijection `e : \u03ba \u2243 \u03b9`.","title":"Reindexing a cube function along an index bijection"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-cube-equiv","kind":"node","lean":"StatMech.Probability.khc_cubeEquiv","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-cube-equiv","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The bijection `{0, 1}^\u03b9 \u2243 {0, 1}^\u03ba` induced by `e : \u03ba \u2243 \u03b9`.","title":"Cube bijection induced by an index bijection"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-walsh-char-eq-cube-chi","kind":"node","lean":"StatMech.Probability.khc_walshChar_eq_cubeChi","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-walsh-char-eq-cube-chi","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"On `Fin n`, the Walsh character is `BooleanCube.cubeChi` up to the sign `(-1)^{|S|}`: `BooleanCube.cubeVal` sends `true` to `1`, while `khc_sign` sends it to `-1`.","title":"Walsh character agrees with the canonical cube character up to sign"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-fourier-coeff-eq-bool-fourier-coeff","kind":"node","lean":"StatMech.Probability.khc_fourierCoeff_eq_boolFourierCoeff","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-fourier-coeff-eq-bool-fourier-coeff","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"On `Fin n`, the Fourier coefficient is `BooleanCube.boolFourierCoeff` up to the sign `(-1)^{|S|}`.","title":"Walsh coefficient is the canonical Fourier coefficient up to sign"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-walsh-char-reindex","kind":"node","lean":"StatMech.Probability.khc_walshChar_reindex","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-walsh-char-reindex","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Reindexing a Walsh character reindexes its finite set of coordinates.","title":"Reindexing a Walsh character reindexes its coordinate set"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-reindex-cube-equiv","kind":"node","lean":"StatMech.Probability.khc_reindex_cubeEquiv","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-reindex-cube-equiv","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Reindexing back at the corresponding cube point recovers the original value.","title":"Reindexing preserves values at corresponding cube points"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-fourier-coeff-reindex","kind":"node","lean":"StatMech.Probability.khc_fourierCoeff_reindex","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-fourier-coeff-reindex","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Fourier coefficients are unchanged when both the function and its index are reindexed.","title":"Fourier coefficients are invariant under reindexing"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-fourier-expansion","kind":"node","lean":"StatMech.Probability.khc_fourier_expansion","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-fourier-expansion","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Fourier inversion: `f = \u2211_S f\u0302(S) \u03c7_S`.","title":"Fourier inversion on the Boolean cube"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-parseval","kind":"node","lean":"StatMech.Probability.khc_parseval","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-parseval","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Parseval's identity: `\ud835\udd3c[f g] = \u2211_S f\u0302(S) \u011d(S)`.","title":"Parseval's identity on the Boolean cube"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-noise-op-fourier","kind":"node","lean":"StatMech.Probability.khc_noiseOp_fourier","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-noise-op-fourier","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The noise operator acts diagonally on the Walsh basis: `T_\u03c1 f = \u2211_S \u03c1^{|S|} f\u0302(S) \u03c7_S`.","title":"Noise operator is diagonal in the Walsh basis"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-hypercontractivity-general","kind":"node","lean":"StatMech.Probability.khc_hypercontractivity_general","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-hypercontractivity-general","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Bonami\u2013Beckner hypercontractive inequality (A. Bonami, *\u00c9tude des coefficients de Fourier des fonctions de L\u1d56(G)*, Ann. Inst. Fourier 1970; W. Beckner, *Inequalities in Fourier analysis*, Ann. of Math. 1975): `\u2016T_\u03c1 g\u2016\u2082 \u2264 \u2016g\u2016_{1+\u03c1\u00b2}` on `{0, 1}^\u03b9` for `0 \u2264 \u03c1 \u2264 1`.","title":"Bonami\u2013Beckner hypercontractive inequality"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-low-degree-weight-le","kind":"node","lean":"StatMech.Probability.khc_lowDegreeWeight_le","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-low-degree-weight-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Hypercontractivity in Fourier form: `\u2211_S \u03c1^{2|S|} f\u0302(S)\u00b2 \u2264 \u2016f\u2016\u00b2_{1+\u03c1\u00b2}`.","title":"Hypercontractivity in Fourier form"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-var-eq-fourier-weight","kind":"node","lean":"StatMech.Probability.khc_var_eq_fourierWeight","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-var-eq-fourier-weight","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Variance under the uniform measure is the Fourier weight off the empty set.","title":"Variance as Fourier weight off the empty set"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-infl-eq-four-bool-influence","kind":"node","lean":"StatMech.Probability.khc_infl_eq_four_boolInfluence","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-infl-eq-four-bool-influence","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"Event influence is four times canonical quarter-square influence after reindexing.","title":"Event influence is four times the canonical influence"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-infl-eq-fourier-weight","kind":"node","lean":"StatMech.Probability.khc_infl_eq_fourierWeight","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-infl-eq-fourier-weight","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The influence of `e` on a Boolean function is `\u2211_{S \u220b e} 4 f\u0302(S)\u00b2`.","title":"Influence of a coordinate as Fourier weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-total-infl-eq-fourier-weight","kind":"node","lean":"StatMech.Probability.khc_totalInfl_eq_fourierWeight","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-total-infl-eq-fourier-weight","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The total influence of a Boolean function is `\u2211_S 4 |S| f\u0302(S)\u00b2`.","title":"Total influence as Fourier weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-log-sobolev-step","kind":"node","lean":"StatMech.Probability.khc_LogSobolevStep","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-log-sobolev-step","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The KKL inequality at `p = 1/2` in Fourier form, with the hypercontractive bound as a hypothesis: for every Boolean `\u03c6`, if `\u2211_S \u03c1^{2|S|} f\u0302(S)\u00b2 \u2264 \u2016f\u2016\u00b2_{1+\u03c1\u00b2}` for all `\u03c1 \u2208 [0, 1]`, then `c \u00b7 \u2211_{S \u2260 \u2205} f\u0302(S)\u00b2 \u00b7 log (1 / max\u2091 Inf\u2091(f)) \u2264 \u2211_S 4 |S| f\u0302(S)\u00b2`, where `f = 1_\u03c6`. The hypothesis always holds (`khc_lowDegreeWeight_le`).","title":"Fourier-form KKL inequality given the hypercontractive bound"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-log-sobolev-step-zero","kind":"node","lean":"StatMech.Probability.khc_logSobolevStep_zero","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-log-sobolev-step-zero","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"`khc_LogSobolevStep 0` holds trivially.","title":"Hypercontractive form of KKL with constant 0 holds trivially"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-kkl","kind":"node","lean":"StatMech.Probability.khc_KKL","parent":24963,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hypercontractive form of KKL implies the KKL inequality at "],["math","\\(p = 1/2\\)"]],"source":"Hypercontractive form of KKL implies the KKL inequality at $p = 1/2$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-kkl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"The Fourier form `khc_LogSobolevStep c` implies the KKL inequality `KKLHypercontractive (1/2) c`.","title":"Hypercontractive form of KKL implies the KKL inequality at $p = 1/2$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/khc-var-half-eq-bool-variance","kind":"node","lean":"StatMech.Probability.khc_var_half_eq_boolVariance","parent":24963,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t/index.html#khc-var-half-eq-bool-variance","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/walsh-fourier-analysis-on-the-uniform-cube-and-t","status":"fully_proved","summary":"On `Fin n`, the variance under the uniform product weight is `BooleanCube.boolVariance`.","title":"Uniform variance agrees with the canonical Boolean variance"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function","status":"fully_proved","title":"Hypercontractivity bounds for indicator functions"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/kls-fluct-weight-le-pow","kind":"node","lean":"StatMech.Probability.kls_fluctWeight_le_pow","parent":24987,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/index.html#kls-fluct-weight-le-pow","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function","status":"fully_proved","summary":"Bonami\u2013Beckner for an indicator of uniform measure `m`: `\u2211_{S \u2260 \u2205} \u03c1^{2|S|} f\u0302(S)\u00b2 \u2264 m^{2/(1+\u03c1\u00b2)} - m\u00b2` for `0 \u2264 \u03c1 \u2264 1`.","title":"Bonami\u2013Beckner bound for indicator functions"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/kls-var-le-w","kind":"node","lean":"StatMech.Probability.kls_var_le_W","parent":24987,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/index.html#kls-var-le-w","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function","status":"fully_proved","summary":"Poincar\u00e9 inequality in Fourier form: `\u2211_{S \u2260 \u2205} f\u0302(S)\u00b2 \u2264 \u2211_S 4 |S| f\u0302(S)\u00b2`.","title":"Poincar\u00e9 inequality in Fourier form"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/kls-log-optimisation","kind":"node","lean":"StatMech.Probability.kls_LogOptimisation","parent":24987,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/index.html#kls-log-optimisation","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function","status":"fully_proved","summary":"The KKL inequality at `p = 1/2` in Fourier form, with the indicator hypercontractive bound `\u2211_{S \u2260 \u2205} \u03c1^{2|S|} f\u0302(S)\u00b2 \u2264 m^{2/(1+\u03c1\u00b2)} - m\u00b2` (for all `\u03c1 \u2208 [0, 1]`) as a hypothesis, where `f = 1_\u03c6` and `m = \ud835\udd3c f`. The hypothesis always holds (`kls_fluctWeight_le_pow`).","title":"Fourier-form KKL inequality given the indicator Bonami\u2013Beckner bound"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/kls-log-optimisation-zero","kind":"node","lean":"StatMech.Probability.kls_logOptimisation_zero","parent":24987,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/index.html#kls-log-optimisation-zero","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function","status":"fully_proved","summary":"`kls_LogOptimisation 0` holds trivially.","title":"Indicator-bound form of KKL with constant 0 holds trivially"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/kls-log-optimisation-implies-step","kind":"node","lean":"StatMech.Probability.kls_logOptimisation_implies_step","parent":24987,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/index.html#kls-log-optimisation-implies-step","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function","status":"fully_proved","summary":"`kls_LogOptimisation c` implies `khc_LogSobolevStep c`.","title":"Indicator-bound form of KKL implies the hypercontractive form"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/kls-log-sobolev-step","kind":"node","lean":"StatMech.Probability.kls_logSobolevStep","parent":24987,"reading_title":{"format":"inline-parts-v1","parts":[["text","Indicator-bound form of KKL implies the KKL inequality at "],["math","\\(p = 1/2\\)"]],"source":"Indicator-bound form of KKL implies the KKL inequality at $p = 1/2$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function/index.html#kls-log-sobolev-step","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-bounds-for-indicator-function","status":"fully_proved","summary":"`kls_LogOptimisation c` implies the KKL inequality `KKLHypercontractive (1/2) c`.","title":"Indicator-bound form of KKL implies the KKL inequality at $p = 1/2$"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","title":"Hypercontractivity applied to discrete derivatives"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-deriv","kind":"node","lean":"StatMech.Probability.bph_deriv","parent":24994,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-deriv","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"The discrete derivative `\u2202\u2091 f (\u03c9) = f(\u03c9^{e \u21a6 1}) - f(\u03c9^{e \u21a6 0})`.","title":"Discrete derivative in a coordinate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-per-coord-hc","kind":"node","lean":"StatMech.Probability.bph_perCoord_hc","parent":24994,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-per-coord-hc","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"Bonami\u2013Beckner applied to `\u2202\u2091 1_A`: `\u2211_{S \u220b e} \u03c1^{2|S \\ {e}|} \u00b7 4 f\u0302(S)\u00b2 \u2264 Inf\u2091(1_A)^{2/(1+\u03c1\u00b2)}`.","title":"Bonami\u2013Beckner applied to a discrete derivative"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-summed-hc","kind":"node","lean":"StatMech.Probability.bph_summed_hc","parent":24994,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-summed-hc","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"`bph_perCoord_hc` summed over all coordinates.","title":"Summed per-coordinate Bonami\u2013Beckner bound"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-cap-term","kind":"node","lean":"StatMech.Probability.bph_cap_term","parent":24994,"reading_title":{"format":"inline-parts-v1","parts":[["text","Termwise cap "],["math","\\(x^\\theta \\le \\delta^{\\theta-1} x\\)"]],"source":"Termwise cap $x^\\theta \\le \\delta^{\\theta-1} x$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-cap-term","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"`x^\u03b8 \u2264 \u03b4^{\u03b8-1} x` for `0 \u2264 x \u2264 \u03b4` and `\u03b8 \u2265 1`.","title":"Termwise cap $x^\\theta \\le \\delta^{\\theta-1} x$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-cap-sum","kind":"node","lean":"StatMech.Probability.bph_cap_sum","parent":24994,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-cap-sum","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"`\u2211\u2091 Inf\u2091^{\u03b8} \u2264 (max\u2091 Inf\u2091)^{\u03b8-1} \u2211\u2091 Inf\u2091` with `\u03b8 = 2/(1+\u03c1\u00b2)`.","title":"Sum of influence powers capped by the maximal influence"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-capped-hc","kind":"node","lean":"StatMech.Probability.bph_capped_hc","parent":24994,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-capped-hc","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"The capped per-coordinate hypercontractive estimate: `\u2211_S 4 |S| \u03c1^{2(|S|-1)} f\u0302(S)\u00b2 \u2264 \u03b4^{2/(1+\u03c1\u00b2) - 1} \u2211_S 4 |S| f\u0302(S)\u00b2`, with `\u03b4 = max\u2091 Inf\u2091(1_A)`.","title":"Capped per-coordinate hypercontractive estimate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-rho-optimise","kind":"node","lean":"StatMech.Probability.bph_RhoOptimise","parent":24994,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourier-form KKL at "],["math","\\(p = 1/2\\)"],["text"," conditional on the capped estimate"]],"source":"Fourier-form KKL at $p = 1/2$ conditional on the capped estimate"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-rho-optimise","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"The KKL inequality at `p = 1/2` in Fourier form, with the capped estimate `bph_capped_hc` (for all `\u03c1 \u2208 [0, 1]`) as a hypothesis. It holds with `c = 2` (`cro2_rhoOptimise`).","title":"Fourier-form KKL at $p = 1/2$ conditional on the capped estimate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-rho-optimise-zero","kind":"node","lean":"StatMech.Probability.bph_rhoOptimise_zero","parent":24994,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-rho-optimise-zero","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"`bph_RhoOptimise 0` holds trivially.","title":"Conditional Fourier-form KKL holds trivially with constant zero"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-log-optimisation-of-rho","kind":"node","lean":"StatMech.Probability.bph_logOptimisation_of_rho","parent":24994,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-log-optimisation-of-rho","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"`bph_RhoOptimise c` implies `kls_LogOptimisation c`.","title":"Capped-estimate form of KKL implies the indicator-bound form"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/bph-kkl","kind":"node","lean":"StatMech.Probability.bph_KKL","parent":24994,"reading_title":{"format":"inline-parts-v1","parts":[["text","KKL inequality at "],["math","\\(p = 1/2\\)"],["text"," from the conditional statement"]],"source":"KKL inequality at $p = 1/2$ from the conditional statement"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ/index.html#bph-kkl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/hypercontractivity-applied-to-discrete-derivativ","status":"fully_proved","summary":"`bph_RhoOptimise c` implies the KKL inequality `KKLHypercontractive (1/2) c`.","title":"KKL inequality at $p = 1/2$ from the conditional statement"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p","status":"fully_proved","title":"Log-sum estimates for damped influences under p-biased measures"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/fsm2-damp-weight","kind":"node","lean":"StatMech.Probability.fsm2_dampWeight","parent":25005,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/index.html#fsm2-damp-weight","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p","status":"fully_proved","summary":"The damped influence `D\u2091(\u03c1) = \u2211_{S \u220b e} \u03c1^{2|S \\ {e}|} (f\u0302(S) / \u221a(p(1-p)))\u00b2` of coordinate `e` on the Boolean function `\u03c6` under `\u03bc_p`.","title":"Damped influence of a coordinate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/fsm2-damp-weight-one","kind":"node","lean":"StatMech.Probability.fsm2_dampWeight_one","parent":25005,"reading_title":{"format":"inline-parts-v1","parts":[["text","Damped influence at "],["math","\\(\\rho = 1\\)"],["text"," is the influence"]],"source":"Damped influence at $\\rho = 1$ is the influence"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/index.html#fsm2-damp-weight-one","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p","status":"fully_proved","summary":"At `\u03c1 = 1` the damped influence is the influence.","title":"Damped influence at $\\rho = 1$ is the influence"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/fsm2-damp-weight-le-hc","kind":"node","lean":"StatMech.Probability.fsm2_dampWeight_le_hc","parent":25005,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/index.html#fsm2-damp-weight-le-hc","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p","status":"fully_proved","summary":"Hypercontractive bound on the damped influence: `D\u2091(\u03c1) \u2264 Inf\u2091^{2/q}` when `\u03c1\u00b2 = 4p(1-p)(q-1)` and `1 \u2264 q \u2264 2`.","title":"Hypercontractive bound on the damped influence"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/fsm2-damp-weight-le-infl","kind":"node","lean":"StatMech.Probability.fsm2_dampWeight_le_infl","parent":25005,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/index.html#fsm2-damp-weight-le-infl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p","status":"fully_proved","summary":"The damped influence is at most the influence.","title":"Damped influence is at most the influence"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/fsm2-log-sum-gain","kind":"node","lean":"StatMech.Probability.fsm2_logSum_gain","parent":25005,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/index.html#fsm2-log-sum-gain","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p","status":"fully_proved","summary":"The log-sum inequality for the influences against the damped influences, over the coordinates of positive influence.","title":"Log-sum inequality for influences against damped influences"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/fsm2-log-gain-capped","kind":"node","lean":"StatMech.Probability.fsm2_logGain_capped","parent":25005,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p/index.html#fsm2-log-gain-capped","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/log-sum-estimates-for-damped-influences-under-p","status":"fully_proved","summary":"The capped log-sum estimate: `(\u2211\u2091 Inf\u2091) \u00b7 (2/q - 1) \u00b7 log (1 / \u03b4) \u2264 \u2211\u2091 Inf\u2091 \u00b7 log (Inf\u2091 / D\u2091(\u03c1))` (over coordinates of positive influence), with `\u03b4 = max\u2091 Inf\u2091 \u2208 (0, 1)`.","title":"Capped log-sum estimate for damped influences"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","status":"fully_proved","title":"The edge-isoperimetric inequality and the logarithmic KKL reduction"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/klo-one-sub-pow-le","kind":"node","lean":"StatMech.Probability.klo_one_sub_pow_le","parent":25012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli's inequality "],["math","\\(1 - x^k \\le k(1 - x)\\)"]],"source":"Bernoulli's inequality $1 - x^k \\le k(1 - x)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/index.html#klo-one-sub-pow-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","status":"fully_proved","summary":"Bernoulli's inequality in the form `1 - x^k \u2264 k (1 - x)` for `0 \u2264 x` (the hypothesis `x \u2264 1` is not needed). See also `one_add_mul_le_pow`.","title":"Bernoulli's inequality $1 - x^k \\le k(1 - x)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/klo-master-le","kind":"node","lean":"StatMech.Probability.klo_master_le","parent":25012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on "],["math","\\(m - m^{2/(1+\\rho^2)}\\)"],["text"," by total influence"]],"source":"Bound on $m - m^{2/(1+\\rho^2)}$ by total influence"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/index.html#klo-master-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","status":"fully_proved","summary":"For an event of uniform measure `m` and total influence `I = \u2211_S 4 |S| f\u0302(S)\u00b2`, `m - m^{2/(1+\u03c1\u00b2)} \u2264 (1 - \u03c1\u00b2) I / 4` for `0 \u2264 \u03c1 \u2264 1`.","title":"Bound on $m - m^{2/(1+\\rho^2)}$ by total influence"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/klo-edge-isoperimetric","kind":"node","lean":"StatMech.Probability.klo_edge_isoperimetric","parent":25012,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/index.html#klo-edge-isoperimetric","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","status":"fully_proved","summary":"Edge-isoperimetric inequality on the discrete cube (S. Hart, *A note on the edges of the n-cube*, Discrete Math. 1976): an event of uniform measure `m > 0` has total influence at least `2 m log (1 / m)`. Here `log` is the natural logarithm, so this is weaker by the factor `log 2` than the sharp bound `2 m log\u2082 (1 / m)`.","title":"Edge-isoperimetric inequality"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/klo-delta-upgrade","kind":"node","lean":"StatMech.Probability.klo_DeltaUpgrade","parent":25012,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/index.html#klo-delta-upgrade","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","status":"fully_proved","summary":"The KKL inequality at `p = 1/2` in Fourier form, with the bound `m - m^{2/(1+\u03c1\u00b2)} \u2264 (1 - \u03c1\u00b2) I / 4` (for all `\u03c1 \u2208 [0, 1]`) as a hypothesis, where `m` is the measure and `I` the total influence of `\u03c6`. The hypothesis always holds (`klo_master_le`).","title":"Fourier-form KKL inequality given the measure\u2013influence bound"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/klo-delta-upgrade-zero","kind":"node","lean":"StatMech.Probability.klo_deltaUpgrade_zero","parent":25012,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/index.html#klo-delta-upgrade-zero","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","status":"fully_proved","summary":"`klo_DeltaUpgrade 0` holds trivially.","title":"Measure\u2013influence form of KKL with constant 0 holds trivially"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/klo-log-optimisation-of-delta-upgrade","kind":"node","lean":"StatMech.Probability.klo_logOptimisation_of_deltaUpgrade","parent":25012,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/index.html#klo-log-optimisation-of-delta-upgrade","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","status":"fully_proved","summary":"`klo_DeltaUpgrade c` implies `kls_LogOptimisation c`.","title":"Measure\u2013influence form of KKL implies the indicator-bound form"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/klo-log-optimisation","kind":"node","lean":"StatMech.Probability.klo_logOptimisation","parent":25012,"reading_title":{"format":"inline-parts-v1","parts":[["text","Measure\u2013influence form of KKL implies the KKL inequality at "],["math","\\(p = 1/2\\)"]],"source":"Measure\u2013influence form of KKL implies the KKL inequality at $p = 1/2$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari/index.html#klo-log-optimisation","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-edge-isoperimetric-inequality-and-the-logari","status":"fully_proved","summary":"`klo_DeltaUpgrade c` implies the KKL inequality `KKLHypercontractive (1/2) c`.","title":"Measure\u2013influence form of KKL implies the KKL inequality at $p = 1/2$"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","status":"fully_proved","title":"The KKL inequality on the uniform cube with constant 2"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/cro2-integral-exp-mul","kind":"node","lean":"StatMech.Probability.cro2_integral_exp_mul","parent":25020,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(e^{cx}\\)"],["text"," over an interval"]],"source":"Integral of $e^{cx}$ over an interval"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/index.html#cro2-integral-exp-mul","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","status":"fully_proved","summary":"`\u222b_a^b exp (c x) dx = (exp (c b) - exp (c a)) / c` for `c \u2260 0`.","title":"Integral of $e^{cx}$ over an interval"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/cro2-integral-exp-le","kind":"node","lean":"StatMech.Probability.cro2_integral_exp_le","parent":25020,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on the integral of "],["math","\\(e^{-tu/2}\\)"],["text"," over "],["math","\\((0, 1)\\)"]],"source":"Bound on the integral of $e^{-tu/2}$ over $(0, 1)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/index.html#cro2-integral-exp-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","status":"fully_proved","summary":"`\u222b\u2080\u00b9 exp (-(t/2) u) du \u2264 2 / t` for `t > 0`.","title":"Bound on the integral of $e^{-tu/2}$ over $(0, 1)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/cro2-integral-one-sub-pow","kind":"node","lean":"StatMech.Probability.cro2_integral_one_sub_pow","parent":25020,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\((1 - u)^n\\)"],["text"," over "],["math","\\((0, 1)\\)"]],"source":"Integral of $(1 - u)^n$ over $(0, 1)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/index.html#cro2-integral-one-sub-pow","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","status":"fully_proved","summary":"`\u222b\u2080\u00b9 (1 - u)^n du = 1 / (n + 1)`.","title":"Integral of $(1 - u)^n$ over $(0, 1)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/cro2-integral-lhs","kind":"node","lean":"StatMech.Probability.cro2_integral_lhs","parent":25020,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/index.html#cro2-integral-lhs","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","status":"fully_proved","summary":"Integrating the damped total-influence weights over the noise parameter: `\u222b\u2080\u00b9 \u2211_S 4 |S| (1-u)^{|S|-1} c_S du = 4 \u2211_{S \u2260 \u2205} c_S`.","title":"Integral of damped total-influence weights over the noise parameter"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/cro2-integral-rhs-le","kind":"node","lean":"StatMech.Probability.cro2_integral_rhs_le","parent":25020,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on the integral of "],["math","\\(\\delta^{u/(2-u)}\\)"],["text"," over "],["math","\\((0, 1)\\)"]],"source":"Bound on the integral of $\\delta^{u/(2-u)}$ over $(0, 1)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/index.html#cro2-integral-rhs-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","status":"fully_proved","summary":"`\u222b\u2080\u00b9 \u03b4^{u/(2-u)} du \u2264 2 / log (1/\u03b4)` for `0 < \u03b4 < 1`.","title":"Bound on the integral of $\\delta^{u/(2-u)}$ over $(0, 1)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/cro2-master-sub","kind":"node","lean":"StatMech.Probability.cro2_master_sub","parent":25020,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped estimate rewritten in the variable "],["math","\\(u = 1 - \\rho^2\\)"]],"source":"Capped estimate rewritten in the variable $u = 1 - \\rho^2$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/index.html#cro2-master-sub","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","status":"fully_proved","summary":"The capped estimate in the variable `u = 1 - \u03c1\u00b2`: `\u2211_S 4 |S| (1-u)^{|S|-1} f\u0302(S)\u00b2 \u2264 \u03b4^{u/(2-u)} \u2211_S 4 |S| f\u0302(S)\u00b2`.","title":"Capped estimate rewritten in the variable $u = 1 - \\rho^2$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/cro2-rho-optimise","kind":"node","lean":"StatMech.Probability.cro2_rhoOptimise","parent":25020,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons/index.html#cro2-rho-optimise","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-on-the-uniform-cube-with-cons","status":"fully_proved","summary":"KKL inequality for the uniform measure, Fourier form with constant `2` (after J. Kahn, G. Kalai, N. Linial, *The influence of variables on Boolean functions*, FOCS 1988; this proof follows D. Falik, A. Samorodnitsky, *Edge-isoperimetric inequalities and influences*, 2007): `2 \u00b7 \u2211_{S \u2260 \u2205} f\u0302(S)\u00b2 \u00b7 log (1 / max\u2091 Inf\u2091) \u2264 \u2211_S 4 |S| f\u0302(S)\u00b2` for every Boolean function.","title":"KKL inequality"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures","status":"fully_proved","title":"A damped KKL inequality under p-biased measures"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/fpe-integral-lhs","kind":"node","lean":"StatMech.Probability.fpe_integral_lhs","parent":25028,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/index.html#fpe-integral-lhs","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures","status":"fully_proved","summary":"Integrating the damped total-influence weights over the noise parameter, with the `p`-biased scaling `A = 4p(1-p)`.","title":"Integrating damped total-influence weights over the noise parameter"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/fpe-master-sub","kind":"node","lean":"StatMech.Probability.fpe_master_sub","parent":25028,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped "],["math","\\(p\\)"],["text","-biased per-coordinate estimate in the variable "],["math","\\(u\\)"]],"source":"Capped $p$-biased per-coordinate estimate in the variable $u$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/index.html#fpe-master-sub","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures","status":"fully_proved","summary":"The capped per-coordinate estimate under `\u03bc_p` in the variable `u`, with `\u03c1\u00b2 = A (1 - u)` and `A = 4p(1-p)`.","title":"Capped $p$-biased per-coordinate estimate in the variable $u$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/fpe-damped-var-log-gain","kind":"node","lean":"StatMech.Probability.fpe_dampedVarLogGain","parent":25028,"reading_title":{"format":"inline-parts-v1","parts":[["text","Damped KKL inequality under the "],["math","\\(p\\)"],["text","-biased measure"]],"source":"Damped KKL inequality under the $p$-biased measure"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/index.html#fpe-damped-var-log-gain","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures","status":"fully_proved","summary":"Damped KKL inequality under the Bernoulli(`p`) measure, `0 < p < 1`: with `A = 4p(1-p)`, `2 \u00b7 (\u2211_{S \u2260 \u2205} (A^{|S|-1} / A) f\u0302(S)\u00b2) \u00b7 log (1 / max\u2091 Inf\u2091) \u2264 \u2211\u2091 Inf\u2091`.","title":"Damped KKL inequality under the $p$-biased measure"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/fpe-q-eq-var-at-half","kind":"node","lean":"StatMech.Probability.fpe_Q_eq_var_at_half","parent":25028,"reading_title":{"format":"inline-parts-v1","parts":[["text","Damped variance equals the variance at "],["math","\\(p = 1/2\\)"]],"source":"Damped variance equals the variance at $p = 1/2$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/index.html#fpe-q-eq-var-at-half","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures","status":"fully_proved","summary":"At `p = 1/2` the damped variance is the variance.","title":"Damped variance equals the variance at $p = 1/2$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/fpe-kkl-gain-at-half","kind":"node","lean":"StatMech.Probability.fpe_kklGain_at_half","parent":25028,"reading_title":{"format":"inline-parts-v1","parts":[["text","KKL inequality with constant "],["math","\\(2\\)"],["text"," at "],["math","\\(p = 1/2\\)"]],"source":"KKL inequality with constant $2$ at $p = 1/2$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures/index.html#fpe-kkl-gain-at-half","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/a-damped-kkl-inequality-under-p-biased-measures","status":"fully_proved","summary":"The KKL inequality with constant `2` at `p = 1/2`, obtained from `fpe_dampedVarLogGain`. The same statement is `kkl_inequality_total fsr_pbiased_endpoint`, proved there by the uniform-cube argument.","title":"KKL inequality with constant $2$ at $p = 1/2$"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas","status":"fully_proved","title":"The KKL inequality at p = 1/2, and the p-biased basis at p = 1/2"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/fsr-ptn-psi-half","kind":"node","lean":"StatMech.Probability.fsr_ptn_psi_half","parent":25034,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normalized bit at "],["math","\\(p = 1/2\\)"],["text"," is "],["math","\\(-(-1)^b\\)"]],"source":"Normalized bit at $p = 1/2$ is $-(-1)^b$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/index.html#fsr-ptn-psi-half","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas","status":"fully_proved","summary":"At `p = 1/2` the normalized bit is `-(-1)^b`.","title":"Normalized bit at $p = 1/2$ is $-(-1)^b$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/fsr-ptn-pchar-half","kind":"node","lean":"StatMech.Probability.fsr_ptn_pchar_half","parent":25034,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p\\)"],["text","-biased character at "],["math","\\(p = 1/2\\)"],["text"," is the Walsh character up to sign"]],"source":"$p$-biased character at $p = 1/2$ is the Walsh character up to sign"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/index.html#fsr-ptn-pchar-half","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas","status":"fully_proved","summary":"At `p = 1/2` the `p`-biased character is `(-1)^{|S|}` times the Walsh character.","title":"$p$-biased character at $p = 1/2$ is the Walsh character up to sign"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/fsr-ptn-coeff-half-sq","kind":"node","lean":"StatMech.Probability.fsr_ptn_coeff_half_sq","parent":25034,"reading_title":{"format":"inline-parts-v1","parts":[["text","Squared "],["math","\\(p\\)"],["text","-biased and Walsh coefficients agree at "],["math","\\(p = 1/2\\)"]],"source":"Squared $p$-biased and Walsh coefficients agree at $p = 1/2$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/index.html#fsr-ptn-coeff-half-sq","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas","status":"fully_proved","summary":"At `p = 1/2` the squared `p`-biased and Walsh Fourier coefficients agree.","title":"Squared $p$-biased and Walsh coefficients agree at $p = 1/2$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/fsr-total-infl-half-eq","kind":"node","lean":"StatMech.Probability.fsr_totalInfl_half_eq","parent":25034,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total influence at "],["math","\\(p = 1/2\\)"],["text"," via "],["math","\\(p\\)"],["text","-biased Fourier coefficients"]],"source":"Total influence at $p = 1/2$ via $p$-biased Fourier coefficients"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/index.html#fsr-total-infl-half-eq","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas","status":"fully_proved","summary":"Total influence at `p = 1/2` in terms of the `p`-biased Fourier coefficients.","title":"Total influence at $p = 1/2$ via $p$-biased Fourier coefficients"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/fsr-pbiased-endpoint","kind":"node","lean":"StatMech.Probability.fsr_pbiased_endpoint","parent":25034,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas/index.html#fsr-pbiased-endpoint","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-at-p-1-2-and-the-p-biased-bas","status":"fully_proved","summary":"Kahn\u2013Kalai\u2013Linial inequality for the uniform measure with constant `2` (J. Kahn, G. Kalai, N. Linial, *The influence of variables on Boolean functions*, FOCS 1988; proof after D. Falik, A. Samorodnitsky, 2007): for every finite nonempty `E` and every Boolean function `\u03c6` on `{0, 1}^E`, `2 \u00b7 Var(\u03c6) \u00b7 log (1 / max\u2091 Inf\u2091(\u03c6)) \u2264 \u2211\u2091 Inf\u2091(\u03c6)`, where `Inf\u2091(\u03c6)` is the probability that `e` is pivotal. This is the `p = 1/2`\u2026","title":"Kahn\u2013Kalai\u2013Linial inequality"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","title":"The KKL inequality for events, and sharp thresholds"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-weight-eq","kind":"node","lean":"StatMech.TwoDim.kklw_weight_eq","parent":25040,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-weight-eq","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"The OSSS Bernoulli weight of a configuration is `StatMech.configWeight`.","title":"OSSS Bernoulli weight equals the configuration weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-prob-eq-expect","kind":"node","lean":"StatMech.TwoDim.kklw_prob_eq_expect","parent":25040,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-prob-eq-expect","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"`StatMech.prob p A` is the OSSS expectation of the indicator of `A`.","title":"Event probability as the OSSS expectation of its indicator"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-influence-eq-infl","kind":"node","lean":"StatMech.TwoDim.kklw_influence_eq_infl","parent":25040,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-influence-eq-infl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"`StatMech.BeffaraDC.influence` is the OSSS influence of the indicator of `A`.","title":"Pivotal-probability influence as the OSSS influence of an indicator"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-var-eq","kind":"node","lean":"StatMech.TwoDim.kklw_var_eq","parent":25040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Variance of an event indicator is "],["math","\\(P(A)(1 - P(A))\\)"]],"source":"Variance of an event indicator is $P(A)(1 - P(A))$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-var-eq","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"The variance of the indicator of `A` is `P(A)(1 - P(A))`.","title":"Variance of an event indicator is $P(A)(1 - P(A))$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-log-max-infl","kind":"node","lean":"StatMech.TwoDim.kklw_logMaxInfl","parent":25040,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-log-max-infl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"`log (1 / max\u2091 Inf\u2091(A))` under the Bernoulli(`p`) product measure.","title":"Logarithm of the reciprocal maximal influence of an event"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-hkkl-of-hypercontractive","kind":"node","lean":"StatMech.TwoDim.kklw_hkkl_of_hypercontractive","parent":25040,"reading_title":{"format":"inline-parts-v1","parts":[["text","KKL inequality with constant 2 at "],["math","\\(p\\)"],["text"," implies its event form"]],"source":"KKL inequality with constant 2 at $p$ implies its event form"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-hkkl-of-hypercontractive","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"The KKL inequality `KKLHypercontractive p 2` for an event: `2 \u00b7 P_p(A)(1 - P_p(A)) \u00b7 log (1 / max\u2091 Inf\u2091(A)) \u2264 \u2211\u2091 Inf\u2091(A)`.","title":"KKL inequality with constant 2 at $p$ implies its event form"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-hkkl-half","kind":"node","lean":"StatMech.TwoDim.kklw_hkkl_half","parent":25040,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-hkkl-half","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"Kahn\u2013Kalai\u2013Linial inequality for events under the uniform measure (J. Kahn, G. Kalai, N. Linial, *The influence of variables on Boolean functions*, FOCS 1988), with constant `2`: `2 \u00b7 P(A)(1 - P(A)) \u00b7 log (1 / max\u2091 Inf\u2091(A)) \u2264 \u2211\u2091 Inf\u2091(A)`, where `Inf\u2091(A)` is the probability that `e` is pivotal for `A`.","title":"Kahn\u2013Kalai\u2013Linial inequality"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-hkkl","kind":"node","lean":"StatMech.TwoDim.kklw_hkkl","parent":25040,"reading_title":{"format":"inline-parts-v1","parts":[["text","KKL on the interior of "],["math","\\((1/2, q)\\)"],["text"," implies its event form there"]],"source":"KKL on the interior of $(1/2, q)$ implies its event form there"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-hkkl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"`kklw_hkkl_of_hypercontractive` at every `p` in the interior of `[1/2, q]`.","title":"KKL on the interior of $(1/2, q)$ implies its event form there"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-sharp-threshold","kind":"node","lean":"StatMech.TwoDim.kklw_sharpThreshold","parent":25040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sharp-threshold bound assuming the KKL inequality on "],["math","\\((1/2, q)\\)"]],"source":"Sharp-threshold bound assuming the KKL inequality on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-sharp-threshold","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"Sharp-threshold bound: for an increasing event with `P_{1/2}(A) = 1/2`, if the KKL inequality holds for all `p \u2208 (1/2, q)` and there `P_p(A)(1 - P_p(A)) \u2265 v\u2080` and `log (1 / max\u2091 Inf\u2091(A)) \u2265 L`, then `P_q(A) \u2265 1/2 + 2 v\u2080 L (q - 1/2)`.","title":"Sharp-threshold bound assuming the KKL inequality on $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/kklw-sharp-threshold-window","kind":"node","lean":"StatMech.TwoDim.kklw_sharpThreshold_window","parent":25040,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sharp-threshold window bound assuming the KKL inequality on "],["math","\\((1/2, q)\\)"]],"source":"Sharp-threshold window bound assuming the KKL inequality on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol/index.html#kklw-sharp-threshold-window","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-for-events-and-sharp-threshol","status":"fully_proved","summary":"Window form of `kklw_sharpThreshold`: under the same hypotheses, if `P_q(A) \u2264 1 - \u03b5` then `q - 1/2 \u2264 (1/2 - \u03b5) / (2 v\u2080 L)`.","title":"Sharp-threshold window bound assuming the KKL inequality on $(1/2, q)$"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","title":"The KKL inequality under p-biased measures: reflection and threshold consequences"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/kpb-flip-all","kind":"node","lean":"StatMech.Probability.kpb_flipAll","parent":25051,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html#kpb-flip-all","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","summary":"The coordinatewise complement `\u03c9 \u21a6 \u00ac\u03c9` of a configuration.","title":"Coordinatewise complement of a configuration"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/kpb-weight-flip","kind":"node","lean":"StatMech.Probability.kpb_weight_flip","parent":25051,"reading_title":{"format":"inline-parts-v1","parts":[["text","Complement swaps the Bernoulli "],["math","\\(p\\)"],["text"," and "],["math","\\(1 - p\\)"],["text"," weights"]],"source":"Complement swaps the Bernoulli $p$ and $1 - p$ weights"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html#kpb-weight-flip","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","summary":"Complementing sends the Bernoulli(`p`) weight to the Bernoulli(`1 - p`) weight.","title":"Complement swaps the Bernoulli $p$ and $1 - p$ weights"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/kpb-kkl-reflect","kind":"node","lean":"StatMech.Probability.kpb_kkl_reflect","parent":25051,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reflection of the KKL inequality from "],["math","\\(p\\)"],["text"," to "],["math","\\(1 - p\\)"]],"source":"Reflection of the KKL inequality from $p$ to $1 - p$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html#kpb-kkl-reflect","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","summary":"The KKL inequality with constant `2` at `p` implies it at `1 - p`.","title":"Reflection of the KKL inequality from $p$ to $1 - p$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/kpb-pbiased-hc","kind":"node","lean":"StatMech.Probability.kpb_PBiasedHC","parent":25051,"reading_title":{"format":"inline-parts-v1","parts":[["text","KKL inequality with constant 2 on "],["math","\\((1/2, q)\\)"]],"source":"KKL inequality with constant 2 on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html#kpb-pbiased-hc","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","summary":"The KKL inequality with constant `2` for every Bernoulli parameter `p \u2208 (1/2, q)`.","title":"KKL inequality with constant 2 on $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/kpb-residue-c0","kind":"node","lean":"StatMech.Probability.kpb_residue_c0","parent":25051,"reading_title":{"format":"inline-parts-v1","parts":[["text","KKL inequality with constant 0 holds trivially on "],["math","\\((1/2, q)\\)"]],"source":"KKL inequality with constant 0 holds trivially on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html#kpb-residue-c0","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","summary":"The KKL inequality with constant `0` holds for every `p`; this is trivial.","title":"KKL inequality with constant 0 holds trivially on $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/kpb-hbkkkl","kind":"node","lean":"StatMech.Probability.kpb_Hbkkkl","parent":25051,"reading_title":{"format":"inline-parts-v1","parts":[["text","KKL hypothesis on "],["math","\\((1/2, q)\\)"],["text"," restated on the interior of "],["math","\\((1/2, q)\\)"]],"source":"KKL hypothesis on $(1/2, q)$ restated on the interior of $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html#kpb-hbkkkl","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","summary":"`kpb_PBiasedHC q` in the form used by `StatMech.TwoDim.kklw_sharpThreshold`.","title":"KKL hypothesis on $(1/2, q)$ restated on the interior of $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/kpb-sharp-threshold","kind":"node","lean":"StatMech.Probability.kpb_sharpThreshold","parent":25051,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sharp-threshold bound under the "],["math","\\(p\\)"],["text","-biased KKL hypothesis"]],"source":"Sharp-threshold bound under the $p$-biased KKL hypothesis"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html#kpb-sharp-threshold","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","summary":"`StatMech.TwoDim.kklw_sharpThreshold` under `kpb_PBiasedHC q`.","title":"Sharp-threshold bound under the $p$-biased KKL hypothesis"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/kpb-sharp-threshold-window","kind":"node","lean":"StatMech.Probability.kpb_sharpThreshold_window","parent":25051,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sharp-threshold window under the "],["math","\\(p\\)"],["text","-biased KKL hypothesis"]],"source":"Sharp-threshold window under the $p$-biased KKL hypothesis"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle/index.html#kpb-sharp-threshold-window","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-kkl-inequality-under-p-biased-measures-refle","status":"fully_proved","summary":"`StatMech.TwoDim.kklw_sharpThreshold_window` under `kpb_PBiasedHC q`.","title":"Sharp-threshold window under the $p$-biased KKL hypothesis"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","title":"Decomposition of the variance by maximal element"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-max-e","kind":"node","lean":"StatMech.Probability.mxd_maxE","parent":25060,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-max-e","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"The maximal element of `S`, or an arbitrary element if `S` is empty.","title":"Maximal element of a finite set"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-fiber","kind":"node","lean":"StatMech.Probability.mxd_fiber","parent":25060,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-fiber","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"The nonempty subsets of `\u03b9` whose maximal element is `e`.","title":"Nonempty subsets with a given maximal element"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-fiber-weight","kind":"node","lean":"StatMech.Probability.mxd_fiberWeight","parent":25060,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-fiber-weight","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"The `p`-biased Fourier weight `W\u2091 = \u2211_{S : max S = e} f\u0302(S)\u00b2` of the fiber of `e`.","title":"Fourier weight of a maximal-element fiber"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-var-decomp","kind":"node","lean":"StatMech.Probability.mxd_var_decomp","parent":25060,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-var-decomp","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"The variance of a Boolean function is the sum of the fiber weights.","title":"Variance decomposition by maximal element"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-damp-fiber","kind":"node","lean":"StatMech.Probability.mxd_dampFiber","parent":25060,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-damp-fiber","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"The noise-damped fiber weight `\u2211_{S : max S = e} \u03c1^{2|S|} f\u0302(S)\u00b2`.","title":"Noise-damped fiber weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-damp-fiber-le-per-coord","kind":"node","lean":"StatMech.Probability.mxd_dampFiber_le_perCoord","parent":25060,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-damp-fiber-le-per-coord","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"Hypercontractive bound on the damped fiber weight.","title":"Hypercontractive bound on the damped fiber weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-log-sum-gain","kind":"node","lean":"StatMech.Probability.mxd_logSum_gain","parent":25060,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-log-sum-gain","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"The variance log-sum estimate for the fiber weights against their damped versions.","title":"Log-sum estimate for the fiber weights"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-martingale-hc-statement","kind":"node","lean":"StatMech.Probability.mxd_martingaleHC_statement","parent":25060,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped influence estimate for all "],["math","\\(p\\)"],["text"," in "],["math","\\((1/2, q)\\)"]],"source":"Capped influence estimate for all $p$ in $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-martingale-hc-statement","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"The capped estimate `\u2211_S 4 |S| (1-u)^{|S|-1} f\u0302(S)\u00b2 \u2264 \u03b4^{u/(2-u)} \u2211\u2091 Inf\u2091` for all `p \u2208 (1/2, q)` with `p < 1` and all `u \u2208 [0, 1]`.","title":"Capped influence estimate for all $p$ in $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-rho-optimise-of-martingale-hc","kind":"node","lean":"StatMech.Probability.mxd_rhoOptimise_of_martingaleHC","parent":25060,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped estimate implies KKL with constant 2 on "],["math","\\((1/2, q)\\)"]],"source":"Capped estimate implies KKL with constant 2 on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-rho-optimise-of-martingale-hc","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"`mxd_martingaleHC_statement q` implies `ptn_RhoOptimise q`, hence the KKL inequality with constant `2` on `(1/2, q)`.","title":"Capped estimate implies KKL with constant 2 on $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/mxd-martingale-hc-c0","kind":"node","lean":"StatMech.Probability.mxd_martingaleHC_c0","parent":25060,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element/index.html#mxd-martingale-hc-c0","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/decomposition-of-the-variance-by-maximal-element","status":"fully_proved","summary":"The capped estimate with all weights multiplied by `0` holds trivially.","title":"Capped estimate with zero weights holds trivially"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","title":"The capped estimate in the hypercontractive range"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-per-coord-weight","kind":"node","lean":"StatMech.Probability.frh_perCoordWeight","parent":25071,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-per-coord-weight","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"The per-coordinate damped weight `D\u2091(u) = \u2211_{S \u220b e} (1-u)^{|S|-1} (f\u0302(S)/\u221a(p(1-p)))\u00b2` under `\u03bc_p`.","title":"Per-coordinate damped weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-lhs-eq","kind":"node","lean":"StatMech.Probability.frh_lhs_eq","parent":25071,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-lhs-eq","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"The damped total weight is `4p(1-p) \u2211\u2091 D\u2091(u)`.","title":"Damped total weight as a sum of per-coordinate weights"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-at-zero","kind":"node","lean":"StatMech.Probability.frh_at_zero","parent":25071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Damped total weight at "],["math","\\(u = 0\\)"],["text"," is at most the total influence"]],"source":"Damped total weight at $u = 0$ is at most the total influence"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-at-zero","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"At `u = 0` the damped total weight is at most the total influence.","title":"Damped total weight at $u = 0$ is at most the total influence"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-capped-martingale","kind":"node","lean":"StatMech.Probability.frh_capped_martingale","parent":25071,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-capped-martingale","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"The capped estimate in the hypercontractive range: if `1 - u = 4p(1-p)(q'-1)` with `1 \u2264 q' \u2264 2`, then `\u2211_S 4 |S| (1-u)^{|S|-1} f\u0302(S)\u00b2 \u2264 4p(1-p) * \u03b4^{2/q'-1} \u2211\u2091 Inf\u2091`.","title":"Capped estimate in the hypercontractive range"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-capped-param","kind":"node","lean":"StatMech.Probability.frh_capped_param","parent":25071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parametrization of the hypercontractive range by "],["math","\\(q' \\in (1, 2)\\)"]],"source":"Parametrization of the hypercontractive range by $q' \\in (1, 2)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-capped-param","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"Every `u \u2208 [1 - 4p(1-p), 1]` has the form `1 - u = 4p(1-p)(q'-1)` with `q' \u2208 [1, 2]`.","title":"Parametrization of the hypercontractive range by $q' \\in (1, 2)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-open-residue","kind":"node","lean":"StatMech.Probability.frh_openResidue","parent":25071,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-open-residue","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"The capped estimate `\u2211_S 4 |S| (1-u)^{|S|-1} f\u0302(S)\u00b2 \u2264 \u03b4^{u/(2-u)} \u2211\u2091 Inf\u2091` for all `p \u2208 (1/2, q)` with `p < 1` and all `u \u2208 (0, 1)`.","title":"Open-interval residual case of the capped estimate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-endpoint-one","kind":"node","lean":"StatMech.Probability.frh_endpoint_one","parent":25071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped estimate at the endpoint "],["math","\\(u = 1\\)"]],"source":"Capped estimate at the endpoint $u = 1$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-endpoint-one","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"At `u = 1` the capped estimate holds.","title":"Capped estimate at the endpoint $u = 1$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-martingale-hc-of-open-residue","kind":"node","lean":"StatMech.Probability.frh_martingaleHC_of_openResidue","parent":25071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Open-interval case implies the full capped estimate on "],["math","\\((1/2, q)\\)"]],"source":"Open-interval case implies the full capped estimate on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-martingale-hc-of-open-residue","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"`frh_openResidue q` implies `mxd_martingaleHC_statement q`, by adding the endpoints.","title":"Open-interval case implies the full capped estimate on $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/frh-low-noise-range-nonempty","kind":"node","lean":"StatMech.Probability.frh_lowNoise_range_nonempty","parent":25071,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hypercontractive range misses an interval near "],["math","\\(u = 0\\)"]],"source":"Hypercontractive range misses an interval near $u = 0$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang/index.html#frh-low-noise-range-nonempty","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang","status":"fully_proved","summary":"For `p > 1/2` the hypercontractive range `u \u2265 1 - 4p(1-p)` excludes a nonempty interval near `u = 0`.","title":"Hypercontractive range misses an interval near $u = 0$"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","title":"The capped estimate from convexity"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-convex-gap-le","kind":"node","lean":"StatMech.Probability.cvg_convexGap_le","parent":25081,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-convex-gap-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"A convex function on `[0, 1]` that is nonpositive at both endpoints is nonpositive on `[0, 1]`. See also `ConvexOn.le_max_of_mem_Icc`.","title":"Nonpositivity of a convex function from its endpoint values"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-convex-on-sum","kind":"node","lean":"StatMech.Probability.cvg_convexOn_sum","parent":25081,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-convex-on-sum","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"A finite sum of convex functions is convex.","title":"Finite sum of convex functions is convex"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-one-sub-pow-convex-on","kind":"node","lean":"StatMech.Probability.cvg_oneSubPow_convexOn","parent":25081,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convexity of "],["math","\\((1 - u)^k\\)"],["text"," on "],["math","\\((0, 1)\\)"]],"source":"Convexity of $(1 - u)^k$ on $(0, 1)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-one-sub-pow-convex-on","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"`u \u21a6 (1 - u)^k` is convex on `[0, 1]`.","title":"Convexity of $(1 - u)^k$ on $(0, 1)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-l-convex-on","kind":"node","lean":"StatMech.Probability.cvg_L_convexOn","parent":25081,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-l-convex-on","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"The damped total weight `u \u21a6 \u2211_S 4 |S| (1-u)^{|S|-1} f\u0302(S)\u00b2` is convex on `[0, 1]`.","title":"Convexity of the damped total weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-q-concave-on","kind":"node","lean":"StatMech.Probability.cvg_Q_concaveOn","parent":25081,"reading_title":{"format":"inline-parts-v1","parts":[["text","Concavity of "],["math","\\(\\exp(m u/(2-u))\\)"],["text"," for "],["math","\\(-1 \\le m \\le 0\\)"]],"source":"Concavity of $\\exp(m u/(2-u))$ for $-1 \\le m \\le 0$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-q-concave-on","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"`u \u21a6 exp (m \u00b7 u/(2-u))` is concave on `[0, 1]` for `-1 \u2264 m \u2264 0`.","title":"Concavity of $\\exp(m u/(2-u))$ for $-1 \\le m \\le 0$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-r-concave-on","kind":"node","lean":"StatMech.Probability.cvg_R_concaveOn","parent":25081,"reading_title":{"format":"inline-parts-v1","parts":[["text","Concavity of "],["math","\\(\\delta^{u/(2-u)} T\\)"],["text"," for "],["math","\\(e^{-1} \\le \\delta \\le 1\\)"]],"source":"Concavity of $\\delta^{u/(2-u)} T$ for $e^{-1} \\le \\delta \\le 1$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-r-concave-on","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"`u \u21a6 \u03b4^{u/(2-u)} T` is concave on `[0, 1]` for `e^{-1} \u2264 \u03b4 \u2264 1` and `T \u2265 0`.","title":"Concavity of $\\delta^{u/(2-u)} T$ for $e^{-1} \\le \\delta \\le 1$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-open-residue-of-concave-r","kind":"node","lean":"StatMech.Probability.cvg_openResidue_of_concaveR","parent":25081,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped estimate on "],["math","\\((0, 1)\\)"],["text"," from concavity of the right side"]],"source":"Capped estimate on $(0, 1)$ from concavity of the right side"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-open-residue-of-concave-r","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"If the right side of the capped estimate is concave in `u`, the estimate holds on `(0, 1)`.","title":"Capped estimate on $(0, 1)$ from concavity of the right side"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-max-infl-le-one","kind":"node","lean":"StatMech.Probability.cvg_maxInfl_le_one","parent":25081,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-max-infl-le-one","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"The maximal influence of a Boolean function is at most `1`.","title":"Maximal influence of a Boolean function is at most one"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-open-residue-large-delta","kind":"node","lean":"StatMech.Probability.cvg_openResidue_largeDelta","parent":25081,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped estimate when the maximal influence is at least "],["math","\\(e^{-1}\\)"]],"source":"Capped estimate when the maximal influence is at least $e^{-1}$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-open-residue-large-delta","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"The capped estimate holds for all `u \u2208 (0, 1)` when the maximal influence is at least `e^{-1}`.","title":"Capped estimate when the maximal influence is at least $e^{-1}$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-small-delta-residue","kind":"node","lean":"StatMech.Probability.cvg_smallDeltaResidue","parent":25081,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-small-delta-residue","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"The capped estimate for all `p \u2208 (1/2, q)` with `p < 1`, all Boolean functions with maximal influence `\u03b4 < e^{-1}`, and all `u \u2208 (0, 1)`.","title":"Small-influence residual case of the capped estimate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-open-residue-of-small-delta","kind":"node","lean":"StatMech.Probability.cvg_openResidue_of_smallDelta","parent":25081,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-open-residue-of-small-delta","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"`cvg_smallDeltaResidue q` implies `frh_openResidue q`.","title":"Small-influence case implies the open-interval case"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/cvg-martingale-hc-of-small-delta","kind":"node","lean":"StatMech.Probability.cvg_martingaleHC_of_smallDelta","parent":25081,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small-influence case implies the full capped estimate on "],["math","\\((1/2, q)\\)"]],"source":"Small-influence case implies the full capped estimate on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity/index.html#cvg-martingale-hc-of-small-delta","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-from-convexity","status":"fully_proved","summary":"`cvg_smallDeltaResidue q` implies `mxd_martingaleHC_statement q`.","title":"Small-influence case implies the full capped estimate on $(1/2, q)$"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","title":"The capped estimate in the hypercontractive range with a gap condition"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-exponent-le","kind":"node","lean":"StatMech.Probability.hcq_exponent_le","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-exponent-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"In the hypercontractive range the exponent `2/q' - 1` is at most the target exponent `u/(2-u)`.","title":"Hypercontractive exponent is at most the target exponent"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-gap-closed-form","kind":"node","lean":"StatMech.Probability.hcq_gap_closedForm","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-gap-closed-form","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"Closed form of the exponent gap `u/(2-u) - (2/q' - 1)`.","title":"Closed form of the exponent gap"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-gap-le","kind":"node","lean":"StatMech.Probability.hcq_gap_le","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-gap-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"Upper bound on the exponent gap.","title":"Upper bound on the exponent gap"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-capped-full","kind":"node","lean":"StatMech.Probability.hcq_capped_full","parent":25094,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped estimate in the hypercontractive range with matching "],["math","\\(q'\\)"]],"source":"Capped estimate in the hypercontractive range with matching $q'$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-capped-full","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"The capped estimate in the hypercontractive range, with the matching `q'`.","title":"Capped estimate in the hypercontractive range with matching $q'$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-dom-of-threshold","kind":"node","lean":"StatMech.Probability.hcq_dom_of_threshold","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-dom-of-threshold","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"If `\u03b4^{gap} \u2265 4p(1-p)`, the bound with exponent `2/q' - 1` and factor `4p(1-p)` is at most `\u03b4^{u/(2-u)}`.","title":"Under the gap condition the hypercontractive bound is dominated"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-star-of-dominates","kind":"node","lean":"StatMech.Probability.hcq_star_of_dominates","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-star-of-dominates","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"The capped estimate with exponent `u/(2-u)` from a domination of the hypercontractive bound.","title":"Target-exponent capped estimate from a domination bound"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-star-capped-threshold","kind":"node","lean":"StatMech.Probability.hcq_star_capped_threshold","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-star-capped-threshold","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"The capped estimate with exponent `u/(2-u)` in the hypercontractive range, under the gap condition `\u03b4^{gap} \u2265 4p(1-p)`.","title":"Target-exponent capped estimate under the gap condition"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-small-delta-p-half","kind":"node","lean":"StatMech.Probability.hcq_smallDelta_p_half","parent":25094,"reading_title":{"format":"inline-parts-v1","parts":[["text","Capped estimate at "],["math","\\(p = 1/2\\)"],["text"," for all "],["math","\\(u \\in (0, 1)\\)"]],"source":"Capped estimate at $p = 1/2$ for all $u \\in (0, 1)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-small-delta-p-half","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"At `p = 1/2` the capped estimate `\u2211_S 4 |S| (1-u)^{|S|-1} f\u0302(S)\u00b2 \u2264 \u03b4^{u/(2-u)} \u2211\u2091 Inf\u2091` holds for all `u \u2208 (0, 1)`.","title":"Capped estimate at $p = 1/2$ for all $u \\in (0, 1)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-low-range-residue","kind":"node","lean":"StatMech.Probability.hcq_lowRangeResidue","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-low-range-residue","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"The capped estimate for all `p \u2208 (1/2, q)` with `p < 1`, `\u03b4 < e^{-1}`, and `u \u2208 (0, 1)` either below the hypercontractive range or failing the gap condition.","title":"Low-range residual case of the capped estimate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-small-delta-of-low-range","kind":"node","lean":"StatMech.Probability.hcq_smallDelta_of_lowRange","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-small-delta-of-low-range","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"`hcq_lowRangeResidue q` implies `cvg_smallDeltaResidue q`.","title":"Low-range case implies the small-influence case"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-martingale-hc-of-low-range","kind":"node","lean":"StatMech.Probability.hcq_martingaleHC_of_lowRange","parent":25094,"reading_title":{"format":"inline-parts-v1","parts":[["text","Low-range case implies the full capped estimate on "],["math","\\((1/2, q)\\)"]],"source":"Low-range case implies the full capped estimate on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-martingale-hc-of-low-range","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"`hcq_lowRangeResidue q` implies `mxd_martingaleHC_statement q`.","title":"Low-range case implies the full capped estimate on $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-dominated-nonempty","kind":"node","lean":"StatMech.Probability.hcq_dominated_nonempty","parent":25094,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gap condition holds at "],["math","\\(u = 1\\)"]],"source":"Gap condition holds at $u = 1$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-dominated-nonempty","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"At `u = 1` (`q' = 1`) the gap condition holds.","title":"Gap condition holds at $u = 1$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-threshold-of-gap-small","kind":"node","lean":"StatMech.Probability.hcq_threshold_of_gap_small","parent":25094,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power bound "],["math","\\(s \\le \\delta^g\\)"],["text"," from a logarithmic exponent bound"]],"source":"Power bound $s \\le \\delta^g$ from a logarithmic exponent bound"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-threshold-of-gap-small","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"`s \u2264 \u03b4^g` when `g \u2264 log (1/s) / log (1/\u03b4)`.","title":"Power bound $s \\le \\delta^g$ from a logarithmic exponent bound"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-residue-proper","kind":"node","lean":"StatMech.Probability.hcq_residue_proper","parent":25094,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gap condition holds at some "],["math","\\(u < 1\\)"],["text"," in the hypercontractive range"]],"source":"Gap condition holds at some $u < 1$ in the hypercontractive range"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-residue-proper","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"For `p > 1/2` and `0 < \u03b4 < 1`, some `u \u2208 (0, 1)` lies in the hypercontractive range and satisfies the gap condition.","title":"Gap condition holds at some $u < 1$ in the hypercontractive range"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-antecedent-not-tautology","kind":"node","lean":"StatMech.Probability.hcq_antecedent_not_tautology","parent":25094,"reading_title":{"format":"inline-parts-v1","parts":[["text","Low-range case condition fails for some "],["math","\\(u\\)"]],"source":"Low-range case condition fails for some $u$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-antecedent-not-tautology","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"For `p > 1/2` and `0 < \u03b4 < 1`, the case condition in `hcq_lowRangeResidue` fails for some `u \u2208 (0, 1)`.","title":"Low-range case condition fails for some $u$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/hcq-low-range-nonvacuous","kind":"node","lean":"StatMech.Probability.hcq_lowRange_nonvacuous","parent":25094,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2/index.html#hcq-low-range-nonvacuous","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/the-capped-estimate-in-the-hypercontractive-rang-2","status":"fully_proved","summary":"The right side `\u03b4^{u/(2-u)} \u2211\u2091 Inf\u2091` of the capped estimate is nonnegative.","title":"Right side of the capped estimate is nonnegative"},{"data_url":"scopes/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p.json","id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","kind":"scope","navigation":"module","parent":24946,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","title":"Degree truncation of the capped estimate under p-biased measures"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-full-weight-eq","kind":"node","lean":"StatMech.Probability.dtr_full_weight_eq","parent":25111,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fourier total weight equals "],["math","\\(4p(1-p)\\)"],["text"," times the total influence"]],"source":"Fourier total weight equals $4p(1-p)$ times the total influence"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-full-weight-eq","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"The Fourier total weight under `\u03bc_p`: `4 \u2211_S |S| f\u0302(S)\u00b2 = 4p(1-p) \u2211\u2091 Inf\u2091`.","title":"Fourier total weight equals $4p(1-p)$ times the total influence"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-high-part-le","kind":"node","lean":"StatMech.Probability.dtr_high_part_le","parent":25111,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-high-part-le","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"The Fourier part of degree greater than `k` contributes at most `4p(1-p)(1-u)^k \u2211\u2091 Inf\u2091` to the damped total weight.","title":"Bound on the high-degree part of the damped total weight"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-trunc-split","kind":"node","lean":"StatMech.Probability.dtr_trunc_split","parent":25111,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-trunc-split","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"The capped estimate follows from a bound on its part of degree at most `k`.","title":"Capped estimate from its low-degree part"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-qstar","kind":"node","lean":"StatMech.Probability.dtr_qstar","parent":25111,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exponent "],["math","\\(q^{\\ast}(p, u)\\)"],["text"," paired with the damping"]],"source":"Exponent $q^{\\ast}(p, u)$ paired with the damping"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-qstar","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"The exponent `q*(p, u) = 1 + (1-u)/(4p(1-p))` paired with the damping `1 - u`.","title":"Exponent $q^{\\ast}(p, u)$ paired with the damping"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-low-degree-residue","kind":"node","lean":"StatMech.Probability.dtr_lowDegreeResidue","parent":25111,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-low-degree-residue","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"The low-degree form of the capped estimate for every `p \u2208 (1/2, q)` with `p < 1`, small maximal influence `\u03b4 < e^{-1}`, and parameters `u` outside the range handled by `CappedExponentGap`.","title":"Low-degree residual case of the capped estimate"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-low-degree-of-residue","kind":"node","lean":"StatMech.Probability.dtr_lowDegree_of_residue","parent":25111,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-low-degree-of-residue","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"`dtr_lowDegreeResidue q` implies `hcq_lowRangeResidue q`.","title":"Low-degree case implies the low-range case"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-small-delta-of-residue","kind":"node","lean":"StatMech.Probability.dtr_smallDelta_of_residue","parent":25111,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-small-delta-of-residue","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"`dtr_lowDegreeResidue q` implies `cvg_smallDeltaResidue q`.","title":"Low-degree case implies the small-influence case"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-martingale-hc-of-residue","kind":"node","lean":"StatMech.Probability.dtr_martingaleHC_of_residue","parent":25111,"reading_title":{"format":"inline-parts-v1","parts":[["text","Low-degree case implies the full capped estimate on "],["math","\\((1/2, q)\\)"]],"source":"Low-degree case implies the full capped estimate on $(1/2, q)$"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-martingale-hc-of-residue","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"`dtr_lowDegreeResidue q` implies `mxd_martingaleHC_statement q`.","title":"Low-degree case implies the full capped estimate on $(1/2, q)$"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-budget-pos","kind":"node","lean":"StatMech.Probability.dtr_budget_pos","parent":25111,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-budget-pos","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"For `\u03b4 > 0` the truncation budget `\u03b4^{u/(2-u)} - 4p(1-p)(1-u)^k` is nonnegative for some `k`.","title":"Some truncation level gives a nonnegative budget"},{"id":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/dtr-residue-c0","kind":"node","lean":"StatMech.Probability.dtr_residue_c0","parent":25111,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p/index.html#dtr-residue-c0","scope":"Combinatorics/boolean-functions/kahn-kalai-linial-inequality/degree-truncation-of-the-capped-estimate-under-p","status":"fully_proved","summary":"For `\u03b4^{u/(2-u)} > 0` some truncation level `k` makes the right side of the low-degree bound in `dtr_lowDegreeResidue` nonnegative.","title":"Some truncation level makes the low-degree right side nonnegative"},{"data_url":"scopes/Combinatorics/boolean-functions/parity-counts-over-finite-fields.json","id":"Combinatorics/boolean-functions/parity-counts-over-finite-fields","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/parity-counts-over-finite-fields/index.html","scope":"Combinatorics/boolean-functions/parity-counts-over-finite-fields","status":"fully_proved","title":"Parity counts over finite fields"},{"id":"Combinatorics/boolean-functions/parity-counts-over-finite-fields/odd-dot-product-power-count","kind":"node","lean":"FiniteField.odd_dotProduct_power_count","parent":25122,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-counts-over-finite-fields/index.html#odd-dot-product-power-count","scope":"Combinatorics/boolean-functions/parity-counts-over-finite-fields","status":"fully_proved","summary":"Let a finite field be written as `s` binary coordinates. For a nonempty set `V` of exponents below `m`, the number of pairs `(a, b)` for which an odd number of the dot products `bits (a ^ E) \u2b1d\u1d65 bits b` equal one satisfies the cleared-denominator lower bound","title":"Lower bound on odd-parity pairs in a finite field"},{"id":"Combinatorics/boolean-functions/parity-counts-over-finite-fields/odd-dot-product-power-count-of-sixteen-m","kind":"node","lean":"FiniteField.odd_dotProduct_power_count_of_sixteen_mul_le","parent":25122,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-counts-over-finite-fields/index.html#odd-dot-product-power-count-of-sixteen-m","scope":"Combinatorics/boolean-functions/parity-counts-over-finite-fields","status":"fully_proved","summary":"Under `16 * m \u2264 2 ^ s`, at least `15 / 32` of the field pairs have odd parity across any nonempty exponent set `V`. Fractions are cleared in the statement.","title":"At least 15/32 of field pairs have odd parity"},{"id":"Combinatorics/boolean-functions/parity-counts-over-finite-fields/dot-product-power-count-le-half","kind":"node","lean":"FiniteField.dotProduct_power_count_le_half","parent":25122,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-counts-over-finite-fields/index.html#dot-product-power-count-le-half","scope":"Combinatorics/boolean-functions/parity-counts-over-finite-fields","status":"fully_proved","summary":"For one exponent `E`, at most half of all field pairs satisfy `dotProduct (bits (a ^ E)) (bits b) = 1`.","title":"At most half of field pairs have dot product one"},{"data_url":"scopes/Combinatorics/boolean-functions/parity-products-modulo-powers-of-two.json","id":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/index.html","scope":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two","status":"fully_proved","title":"Parity products modulo powers of two"},{"id":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/even-indicator","kind":"node","lean":"ParityProduct.evenIndicator","parent":25126,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/index.html#even-indicator","scope":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two","status":"fully_proved","summary":"The integer indicator that a natural number is even.","title":"Even indicator of a natural number"},{"id":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/parity-coeff","kind":"node","lean":"ParityProduct.parityCoeff","parent":25126,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/index.html#parity-coeff","scope":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two","status":"fully_proved","summary":"The coefficients of the even-indicator expansion: one at zero and `(-1) ^ i * 2 ^ (i - 1)` at positive indices.","title":"Coefficients of the even-indicator expansion"},{"id":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/two-mul-parity-coeff","kind":"node","lean":"ParityProduct.two_mul_parityCoeff","parent":25126,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/index.html#two-mul-parity-coeff","scope":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two","status":"fully_proved","summary":"Twice the parity coefficient is the sign coefficient, with a correction at zero.","title":"Twice a parity coefficient is a sign coefficient"},{"id":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/two-mul-even-indicator","kind":"node","lean":"ParityProduct.two_mul_evenIndicator","parent":25126,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/index.html#two-mul-even-indicator","scope":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two","status":"fully_proved","summary":"Twice the even indicator is one plus the parity sign.","title":"Twice the even indicator is one plus the parity sign"},{"id":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/sum-powerset-parity-coeff","kind":"node","lean":"ParityProduct.sum_powerset_parityCoeff","parent":25126,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/index.html#sum-powerset-parity-coeff","scope":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two","status":"fully_proved","summary":"Summing parity coefficients over all subsets gives the even-cardinality indicator.","title":"Subset sum of parity coefficients is the even-cardinality indicator"},{"id":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/prod-even-indicator-mul-prod-neg-one-pow","kind":"node","lean":"ParityProduct.prod_evenIndicator_mul_prod_neg_one_pow_modEq","parent":25126,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/parity-products-modulo-powers-of-two/index.html#prod-even-indicator-mul-prod-neg-one-pow","scope":"Combinatorics/boolean-functions/parity-products-modulo-powers-of-two","status":"fully_proved","summary":"A product of even-cardinality indicators and parity signs is congruent to its finite subset expansion cut off at total degree `M`. Each even-indicator factor costs at most one power of two, so the cutoff must satisfy `\u03bd - 1 + #J\u2081 \u2264 M`, with `1 \u2264 \u03bd`. Indices outside `J\u2081 \u222a J\u2082` select only the empty subset.","title":"Truncated subset expansion of parity products modulo powers of two"},{"data_url":"scopes/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions.json","id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","title":"Self-dual monotone Boolean functions"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/boolean-vector-complement","kind":"node","lean":"MetaMathlibExt.booleanVectorComplement","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#boolean-vector-complement","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"fully_proved","summary":"The complement of a Boolean vector $x \\in \\{0,1\\}^n$ is the vector $\\bar x \\in \\{0,1\\}^n$ obtained by negating every coordinate: $\\bar x_i = \\lnot x_i$ for every $i < n$.","title":"Complement of a Boolean vector"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/boolean-function-dual","kind":"node","lean":"MetaMathlibExt.booleanFunctionDual","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#boolean-function-dual","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"fully_proved","summary":"For a Boolean function $f \\colon \\{0,1\\}^n \\to \\{0,1\\}$, its dual is the Boolean function $f^\\star \\colon \\{0,1\\}^n \\to \\{0,1\\}$ given by $f^\\star(x) = \\lnot f(\\bar x)$, where $\\bar x$ is the coordinatewise complement of $x$.","title":"Dual of a Boolean function"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/is-self-dual-monotone","kind":"node","lean":"MetaMathlibExt.IsSelfDualMonotone","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#is-self-dual-monotone","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"fully_proved","summary":"A Boolean function $f \\colon \\{0,1\\}^n \\to \\{0,1\\}$ is self-dual monotone when it is monotone, that is, $f(x) \\le f(y)$ for all $x, y \\in \\{0,1\\}^n$ with $x_i \\le y_i$ for every $i < n$, and it equals its dual, that is, $f = f^\\star$, where $f^\\star(x) = \\lnot f(\\bar x)$ and $\\bar x$ is the coordinatewise complement of $x$.","title":"Self-dual monotone Boolean functions"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/boolean-vector-complement-complement","kind":"node","lean":"MetaMathlibExt.booleanVectorComplement_complement","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#boolean-vector-complement-complement","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"fully_proved","summary":"For every $n$ and every Boolean vector $x \\in \\{0,1\\}^n$, writing $\\bar x$ for the coordinatewise complement of $x$, we have $\\bar{\\bar x} = x$.","title":"Complementation of Boolean vectors is an involution"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/boolean-function-dual-dual","kind":"node","lean":"MetaMathlibExt.booleanFunctionDual_dual","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#boolean-function-dual-dual","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"fully_proved","summary":"For every $n$ and every Boolean function $f \\colon \\{0,1\\}^n \\to \\{0,1\\}$, writing $f^\\star$ for the dual of $f$, we have $(f^\\star)^\\star = f$.","title":"Duality of Boolean functions is an involution"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/self-dual-monotone-proj","kind":"node","lean":"MetaMathlibExt.selfDualMonotone_proj","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#self-dual-monotone-proj","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"fully_proved","summary":"For every $n$ and every index $i < n$, the projection $x \\mapsto x_i$ on $\\{0,1\\}^n$ is a self-dual monotone Boolean function.","title":"Coordinate projections are self-dual monotone"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/boolean-cube-concat","kind":"node","lean":"MetaMathlibExt.booleanCubeConcat","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#boolean-cube-concat","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers and let $W = (W_0, W_1, \\dots, W_{m-1})$ be a finite list of Boolean functions on $n$ variables. The concatenation $W_0 \\cdot W_1 \\cdots W_{m-1}$ on $n + k$ variables is the Boolean function that sends $(x_0, \\dots, x_{n-1}, y_0, \\dots, y_{k-1})$ to $W_i(x_0, \\dots, x_{n-1})$, where $i = \\sum_{j < k} y_j 2^j$ is the number whose binary digits are the last $k$ variables (true\u2026","title":"Concatenation of a list of Boolean functions"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/boolean-cube-concat-false","kind":"node","lean":"MetaMathlibExt.booleanCubeConcat_false","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#boolean-cube-concat-false","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and a finite list $W = (W_0, \\dots, W_{m-1})$ of Boolean functions on $n$ variables, the concatenation of $W$ on $n + k$ variables takes the value $W_0(\\mathrm{false}, \\dots, \\mathrm{false})$ at the all-false input when $W$ is nonempty, and the value false when $W$ is empty.","title":"Concatenation at the all-false input"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/monotone-boolean-function-order-iso-curr","kind":"node","lean":"MetaMathlibExt.monotoneBooleanFunction_orderIso_curry","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#monotone-boolean-function-order-iso-curr","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"For a natural number $m$, let $D_m$ be the poset of monotone Boolean functions $\\{0,1\\}^m \\to \\{0,1\\}$, where $0 < 1$, inputs are ordered coordinatewise and functions are ordered pointwise. Then for all natural numbers $n$ and $k$, $D_{n+k}$ is order-isomorphic to the poset of monotone maps $\\{0,1\\}^k \\to D_n$, ordered pointwise.","title":"Currying monotone Boolean functions (Lemma 1)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/boolean-function-dual-antitone-sup-inf","kind":"node","lean":"MetaMathlibExt.booleanFunctionDual_antitone_sup_inf","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#boolean-function-dual-antitone-sup-inf","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $f$ and $g$ be monotone Boolean functions on $n$ variables, ordered pointwise, with duals $f^*$ and $g^*$. If $f \\le g$ then $g^* \\le f^*$. Moreover $(f \\lor g)^* = f^* \\land g^*$ and $(f \\land g)^* = f^* \\lor g^*$, where $\\lor$ and $\\land$ are pointwise disjunction and conjunction.","title":"Duality reverses order and exchanges joins and meets"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/boolean-function-perm-monotone-dual-self","kind":"node","lean":"MetaMathlibExt.booleanFunction_perm_monotone_dual_selfDual","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#boolean-function-perm-monotone-dual-self","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $f$ be a Boolean function on $n$ variables, let $\\pi$ be a permutation of the $n$ coordinates, and let $f^\\pi$ be the function $x \\mapsto f(x \\circ \\pi)$. If $f$ is monotone, then $f^\\pi$ is monotone and $(f^*)^\\pi = (f^\\pi)^*$, where $f^*$ is the dual of $f$. If $f$ is self-dual monotone, then so is $f^\\pi$.","title":"Permuting variables preserves monotonicity, duals and self-duality"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/card-self-dual-monotone-succ","kind":"node","lean":"MetaMathlibExt.card_selfDualMonotone_succ","parent":25133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting self-dual monotone functions on "],["math","\\(n + 1\\)"],["text"," variables"]],"source":"Counting self-dual monotone functions on $n + 1$ variables"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#card-self-dual-monotone-succ","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"For every $n \\ge 0$, the number of self-dual monotone Boolean functions on $n + 1$ variables equals the number of monotone Boolean functions $b$ on $n$ variables with $b \\le b^*$, where $b^*$ is the dual of $b$ and functions are ordered pointwise. Moreover, for a permutation $\\pi$ of the $n$ coordinates write $b^\\pi$ for the function $x \\mapsto b(x \\circ \\pi)$, and let $R$ be any finite set of monotone Boolean\u2026","title":"Counting self-dual monotone functions on $n + 1$ variables"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/card-self-dual-monotone-succ-le","kind":"node","lean":"MetaMathlibExt.card_selfDualMonotone_succ_le","parent":25133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper bound for self-dual monotone functions on "],["math","\\(n + 1\\)"],["text"," variables"]],"source":"Upper bound for self-dual monotone functions on $n + 1$ variables"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#card-self-dual-monotone-succ-le","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"For every $n \\ge 0$, the number of self-dual monotone Boolean functions on $n + 1$ variables is at most the number of monotone Boolean functions on $n$ variables.","title":"Upper bound for self-dual monotone functions on $n + 1$ variables"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/monotone-boolean-function-weight-dual","kind":"node","lean":"MetaMathlibExt.monotoneBooleanFunction_weight_dual","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#monotone-boolean-function-weight-dual","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Order $\\{0,1\\}$ by $0 < 1$, inputs in $\\{0,1\\}^n$ coordinatewise and Boolean functions on $\\{0,1\\}^n$ pointwise. For such a function $f$, let $f^*(x) = \\lnot f(\\bar x)$ be its dual, where $\\bar x$ is the coordinatewise complement of $x$, and let $\\ell(f)$ be the number of $x \\in \\{0,1\\}^n$ with $f(x) = 1$. Let $b$ be a monotone Boolean function on $\\{0,1\\}^n$. Then: if $b \\le b^*$, then $2\\ell(b) \\le 2^n$; if\u2026","title":"Weights of monotone Boolean functions and their duals (Lemma 8)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/card-self-dual-monotone-succ-eq-add","kind":"node","lean":"MetaMathlibExt.card_selfDualMonotone_succ_eq_add","parent":25133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recursion for self-dual monotone functions from "],["math","\\(n\\)"],["text"," to "],["math","\\(n + 1\\)"],["text"," variables"]],"source":"Recursion for self-dual monotone functions from $n$ to $n + 1$ variables"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#card-self-dual-monotone-succ-eq-add","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $n \\ge 1$, let $\\lambda_m$ be the number of self-dual monotone Boolean functions on $m$ variables, and let $d_n$ be the number of monotone Boolean functions on $n$ variables. For a Boolean function $b$ on $n$ variables let $b^*$ be its dual and $\\ell(b)$ the number of inputs at which $b$ is true, order functions pointwise, and for a permutation $\\pi$ of the $n$ coordinates write $b^\\pi$ for the function\u2026","title":"Recursion for self-dual monotone functions from $n$ to $n + 1$ variables"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/card-self-dual-monotone-add-two","kind":"node","lean":"MetaMathlibExt.card_selfDualMonotone_add_two","parent":25133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting self-dual monotone functions on "],["math","\\(n + 2\\)"],["text"," variables"]],"source":"Counting self-dual monotone functions on $n + 2$ variables"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#card-self-dual-monotone-add-two","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $n \\ge 0$ and let $\\lambda_{n+2}$ be the number of self-dual monotone Boolean functions on $n + 2$ variables. For a monotone Boolean function $b$ on $n$ variables let $r(b)$ be the number of monotone Boolean functions $z$ on $n$ variables with $b \\lor b^* \\le z$, where $b^*$ is the dual of $b$, $\\lor$ is pointwise disjunction and functions are ordered pointwise. Then $\\lambda_{n+2}$ is the sum of $r(b)$ over\u2026","title":"Counting self-dual monotone functions on $n + 2$ variables"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/le-card-self-dual-monotone-add-two","kind":"node","lean":"MetaMathlibExt.le_card_selfDualMonotone_add_two","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#le-card-self-dual-monotone-add-two","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"For a natural number $n$, let $d_n$ be the number of monotone Boolean functions $\\{0,1\\}^n \\to \\{0,1\\}$ (for the coordinatewise order with $0 < 1$) and let $\\lambda_{n+2}$ be the number of self-dual monotone Boolean functions $\\{0,1\\}^{n+2} \\to \\{0,1\\}$, that is, monotone functions $f$ with $f(\\bar x) = \\lnot f(x)$ for every $x$, where $\\bar x$ is the coordinatewise complement of $x$. Then $d_n \\le \\lambda_{n+2}$.","title":"Dedekind numbers bound self-dual counts from below (Corollary 11)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/self-dual-monotone-add-four-iff-concat","kind":"node","lean":"MetaMathlibExt.selfDualMonotone_add_four_iff_concat","parent":25133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Self-dual monotone functions on "],["math","\\(n+4\\)"],["text"," variables as concatenations (Theorem 12)"]],"source":"Self-dual monotone functions on $n+4$ variables as concatenations (Theorem 12)"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#self-dual-monotone-add-four-iff-concat","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Order $\\{0,1\\}$ by $0 < 1$, inputs coordinatewise and Boolean functions pointwise, with $\\vee$ the pointwise maximum. Let $D_n$ be the set of monotone Boolean functions on $\\{0,1\\}^n$, and let $f^*(x) = \\lnot f(\\bar x)$ be the dual of $f$, where $\\bar x$ is the coordinatewise complement of $x$; a Boolean function $F$ is self-dual monotone if it is monotone and $F^* = F$. The concatenation\u2026","title":"Self-dual monotone functions on $n+4$ variables as concatenations (Theorem 12)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/card-self-dual-monotone-add-four","kind":"node","lean":"MetaMathlibExt.card_selfDualMonotone_add_four","parent":25133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting self-dual monotone functions on "],["math","\\(n + 4\\)"],["text"," variables"]],"source":"Counting self-dual monotone functions on $n + 4$ variables"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#card-self-dual-monotone-add-four","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $D_n$ be the set of monotone Boolean functions on $n$ variables, ordered pointwise, with $\\lor$ the pointwise disjunction and $f^*$ the dual of $f$. For $x, h \\in D_n$ let $\\mathrm{re}(x, h)$ be the number of $z \\in D_n$ with $x \\le z \\le h$. Then the number of self-dual monotone Boolean functions on $n + 4$ variables equals","title":"Counting self-dual monotone functions on $n + 4$ variables"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/self-dual-monotone-add-four-iff-concat-o","kind":"node","lean":"MetaMathlibExt.selfDualMonotone_add_four_iff_concat_of_dual_le","parent":25133,"reading_title":{"format":"inline-parts-v1","parts":[["text","Self-dual monotone functions on "],["math","\\(n+4\\)"],["text"," variables, interval form (Theorem 14)"]],"source":"Self-dual monotone functions on $n+4$ variables, interval form (Theorem 14)"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#self-dual-monotone-add-four-iff-concat-o","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Order $\\{0,1\\}$ by $0 < 1$, inputs coordinatewise and Boolean functions pointwise, with $\\vee$ the pointwise maximum. Let $D_n$ be the set of monotone Boolean functions on $\\{0,1\\}^n$, and let $f^*(x) = \\lnot f(\\bar x)$ be the dual of $f$, where $\\bar x$ is the coordinatewise complement of $x$; a Boolean function $F$ is self-dual monotone if it is monotone and $F^* = F$. The concatenation\u2026","title":"Self-dual monotone functions on $n+4$ variables, interval form (Theorem 14)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/maximal-intersecting-mem-iff-compl-not-m","kind":"node","lean":"MetaMathlibExt.maximal_intersecting_mem_iff_compl_notMem","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#maximal-intersecting-mem-iff-compl-not-m","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $n \\ge 1$ and let $\\mathcal U$ be a maximal intersecting family of subsets of $X = \\{0, \\dots, n-1\\}$: any two members of $\\mathcal U$, not necessarily distinct, have nonempty intersection, and no family of subsets of $X$ strictly containing $\\mathcal U$ has this property. Then for every $A \\subseteq X$, $A \\in \\mathcal U$ if and only if $X \\setminus A \\notin \\mathcal U$.","title":"Maximal intersecting families split complementary pairs (Lemma 3, L2)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/maximal-intersecting-of-is-upper-set-of","kind":"node","lean":"MetaMathlibExt.maximal_intersecting_of_isUpperSet_of_mem_iff_compl_notMem","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#maximal-intersecting-of-is-upper-set-of","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $n$ be a natural number and let $\\mathcal U$ be a family of subsets of $X = \\{0, \\dots, n-1\\}$ such that $\\mathcal U$ is an upper set (if $A \\in \\mathcal U$ and $A \\subseteq B \\subseteq X$ then $B \\in \\mathcal U$) and, for every $A \\subseteq X$, $A \\in \\mathcal U$ if and only if $X \\setminus A \\notin \\mathcal U$. Then $\\mathcal U$ is a maximal intersecting family: any two members of $\\mathcal U$, not\u2026","title":"A criterion for maximal intersecting families (Lemma 4)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/self-dual-monotone-not-fixed-of-even-min","kind":"node","lean":"MetaMathlibExt.selfDualMonotone_not_fixed_of_even_minimalPeriod","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#self-dual-monotone-not-fixed-of-even-min","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $\\pi$ be a permutation of $\\{0, \\dots, n-1\\}$ all of whose cycles have even length, that is, every point has an orbit of even size under $\\pi$. Let $f : \\{0,1\\}^n \\to \\{0,1\\}$ be self-dual monotone, that is, monotone for the coordinatewise order with $0 < 1$ and such that $f(\\bar x) = \\lnot f(x)$ for every $x$, where $\\bar x$ is the coordinatewise complement of $x$. Then $f$ is not fixed by $\\pi$: the function\u2026","title":"Permutations with only even cycles fix no self-dual monotone function (Lemma 15)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/self-dual-monotone-eight-not-fixed","kind":"node","lean":"MetaMathlibExt.selfDualMonotone_eight_not_fixed","parent":25133,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#self-dual-monotone-eight-not-fixed","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $\\pi$ be one of the permutations $(0\\,1\\,2\\,3\\,4\\,5\\,6\\,7)$, $(0\\,1)(2\\,3\\,4\\,5\\,6\\,7)$, $(0\\,1\\,2\\,3)(4\\,5\\,6\\,7)$, $(0\\,1)(2\\,3)(4\\,5\\,6\\,7)$ and $(0\\,1)(2\\,3)(4\\,5)(6\\,7)$ of $\\{0, \\dots, 7\\}$, in cycle notation. Let $f : \\{0,1\\}^8 \\to \\{0,1\\}$ be self-dual monotone, that is, monotone for the coordinatewise order with $0 < 1$ and such that $f(\\bar x) = \\lnot f(x)$ for every $x$, where $\\bar x$ is the\u2026","title":"Permutations of eight variables fixing no self-dual monotone function (Corollary 16)"},{"id":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/exists-self-dual-monotone-fixed-of-odd","kind":"node","lean":"MetaMathlibExt.exists_selfDualMonotone_fixed_of_odd","parent":25133,"reading_title":{"format":"inline-parts-v1","parts":[["text","For odd "],["math","\\(n\\)"],["text",", every permutation fixes a self-dual monotone function"]],"source":"For odd $n$, every permutation fixes a self-dual monotone function"},"result":true,"route":"../roadmap/Combinatorics/boolean-functions/self-dual-monotone-boolean-functions/index.html#exists-self-dual-monotone-fixed-of-odd","scope":"Combinatorics/boolean-functions/self-dual-monotone-boolean-functions","status":"can_prove","summary":"Let $n$ be odd and let $\\pi$ be a permutation of the $n$ coordinates. Then there is a self-dual monotone Boolean function $f$ on $n$ variables with $f(x \\circ \\pi) = f(x)$ for every input $x$.","title":"For odd $n$, every permutation fixes a self-dual monotone function"},{"data_url":"scopes/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions.json","id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","kind":"scope","navigation":"module","parent":24890,"result":false,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","title":"Unate and monotone Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-weight","kind":"node","lean":"MetaMathlibExt.boolFunWeight","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-weight","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"The weight $\\mathrm{wt}(f)$ of an $n$-variable Boolean function $f \\colon \\{0,1\\}^n \\to \\{0,1\\}$ is the number of inputs $x \\in \\{0,1\\}^n$ with $f(x) = 1$.","title":"Weight of a Boolean function"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-weight-const-true","kind":"node","lean":"MetaMathlibExt.boolFunWeight_const_true","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-weight-const-true","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"For every $n$, the constant function $1$ on $n$ variables has weight $2^n$.","title":"Weight of the constant one function"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-increasing-in","kind":"node","lean":"MetaMathlibExt.IsBoolFunIncreasingIn","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-increasing-in","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"An $n$-variable Boolean function $f$ is monotone increasing in the variable $x_i$ if for every input $x$ with $x_i = 0$, changing $x_i$ to $1$ does not decrease the value: $f(x) \\le f(x')$, where $x'$ agrees with $x$ except that $x'_i = 1$.","title":"Boolean functions increasing in a variable"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-decreasing-in","kind":"node","lean":"MetaMathlibExt.IsBoolFunDecreasingIn","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-decreasing-in","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"An $n$-variable Boolean function $f$ is monotone decreasing in the variable $x_i$ if for every input $x$ with $x_i = 0$, changing $x_i$ to $1$ does not increase the value: $f(x') \\le f(x)$, where $x'$ agrees with $x$ except that $x'_i = 1$.","title":"Boolean functions decreasing in a variable"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-unate-bool-fun","kind":"node","lean":"MetaMathlibExt.IsUnateBoolFun","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-unate-bool-fun","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"A Boolean function $f$ on $n$ variables, with Boolean values $0 < 1$, is unate when for each variable $i$ either $f$ is increasing in $x_i$ (changing $x_i$ from $0$ to $1$, with the other variables fixed, never decreases the value of $f$) or $f$ is decreasing in $x_i$ (that change never increases the value of $f$).","title":"Unate Boolean function"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-unate-bool-fun-of-monotone","kind":"node","lean":"MetaMathlibExt.isUnateBoolFun_of_monotone","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-unate-bool-fun-of-monotone","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"Every Boolean function $f$ on $n$ variables that is monotone, for the coordinatewise order on $\\{0,1\\}^n$ with $0 \\le 1$, is unate.","title":"Monotone Boolean functions are unate"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-nondegenerate-on","kind":"node","lean":"MetaMathlibExt.IsBoolFunNondegenerateOn","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-nondegenerate-on","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"An $n$-variable Boolean function $f$ is non-degenerate on the variable $x_i$ when flipping $x_i$ changes the value of $f$ at some input: there is $x \\in \\{0,1\\}^n$ with $f(x) \\ne f(x')$, where $x'$ is $x$ with its $i$-th coordinate negated.","title":"Non-degeneracy in a variable"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-nondegenerate-bool-fun","kind":"node","lean":"MetaMathlibExt.IsNondegenerateBoolFun","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-nondegenerate-bool-fun","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"An $n$-variable Boolean function $f$ is non-degenerate when it is non-degenerate on every variable $x_i$, $1 \\le i \\le n$: for each $i$ there is an input $x \\in \\{0,1\\}^n$ with $f(x) \\ne f(x')$, where $x'$ is $x$ with its $i$-th coordinate negated. For $n = 0$ the condition is vacuous, so both $0$-variable (constant) functions are non-degenerate, matching the paper's convention.","title":"Non-degenerate Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-nondegenerate-on-proj","kind":"node","lean":"MetaMathlibExt.isBoolFunNondegenerateOn_proj","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-nondegenerate-on-proj","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"For every $n$ and every index $i \\in \\{1, \\ldots, n\\}$, the projection $x \\mapsto x_i$ on $\\{0,1\\}^n$ is non-degenerate on the variable $x_i$.","title":"A projection is non-degenerate on its own variable"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-concat","kind":"node","lean":"MetaMathlibExt.boolFunConcat","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-concat","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"The concatenation $g \\,\\|\\, h$ of two $n$-variable Boolean functions $g$ and $h$ is the $(n+1)$-variable function whose truth table is that of $g$ followed by that of $h$: its value at $(x_1, x_2, \\dots, x_{n+1})$ is $g(x_2, \\dots, x_{n+1})$ if $x_1 = 0$ and $h(x_2, \\dots, x_{n+1})$ if $x_1 = 1$. Equivalently, $(g \\,\\|\\, h)(x) = \\bar{x}_1 g(x_2, \\dots, x_{n+1}) \\oplus x_1 h(x_2, \\dots, x_{n+1})$.","title":"Concatenation of two Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-concat-cons","kind":"node","lean":"MetaMathlibExt.boolFunConcat_cons","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-concat-cons","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"For $n$-variable Boolean functions $g$ and $h$, every $b \\in \\{0, 1\\}$ and every $x \\in \\{0, 1\\}^n$, the concatenation $g \\,\\|\\, h$ takes the value $g(x)$ at the input $(b, x_1, \\dots, x_n)$ if $b = 0$ and the value $h(x)$ there if $b = 1$.","title":"Evaluating a concatenation"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-var-perm-equiv","kind":"node","lean":"MetaMathlibExt.IsBoolFunVarPermEquiv","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-var-perm-equiv","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"An $n$-variable Boolean function $f$ is permutation-equivalent to $g$ when $g$ is obtained from $f$ by permuting the variables: there is a permutation $\\sigma$ of $\\{1, \\ldots, n\\}$ such that $g(x_1, \\ldots, x_n) = f(x_{\\sigma(1)}, \\ldots, x_{\\sigma(n)})$ for every $x \\in \\{0,1\\}^n$.","title":"Permutation equivalence of Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-var-perm-equiv-refl","kind":"node","lean":"MetaMathlibExt.isBoolFunVarPermEquiv_refl","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-var-perm-equiv-refl","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"Every $n$-variable Boolean function is permutation-equivalent to itself.","title":"Permutation equivalence is reflexive"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-nnequiv","kind":"node","lean":"MetaMathlibExt.IsBoolFunNNEquiv","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-nnequiv","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"An $n$-variable Boolean function $f$ is NN-equivalent to $g$ when $g$ is obtained from $f$ by negating some of the inputs and possibly the output: there are $a \\in \\{0,1\\}^n$ and $c \\in \\{0,1\\}$ such that $g(x) = f(x \\oplus a) \\oplus c$ for every $x \\in \\{0,1\\}^n$, where $\\oplus$ is exclusive or (coordinatewise on vectors).","title":"NN-equivalence of Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-nnequiv-refl","kind":"node","lean":"MetaMathlibExt.isBoolFunNNEquiv_refl","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-nnequiv-refl","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"Every Boolean function is NN-equivalent to itself.","title":"NN-equivalence is reflexive"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-npnequiv","kind":"node","lean":"MetaMathlibExt.IsBoolFunNPNEquiv","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-npnequiv","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"An $n$-variable Boolean function $f$ is NPN-equivalent to $g$ when $g$ is obtained from $f$ by permuting the variables, negating some of the inputs and possibly negating the output: there are a permutation $\\sigma$ of $\\{1, \\ldots, n\\}$, a vector $a \\in \\{0,1\\}^n$ and a bit $c \\in \\{0,1\\}$ such that $g(x) = f(x_{\\sigma(1)} \\oplus a_1, \\ldots, x_{\\sigma(n)} \\oplus a_n) \\oplus c$ for every $x \\in \\{0,1\\}^n$, where\u2026","title":"NPN-equivalence of Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-npnequiv-refl","kind":"node","lean":"MetaMathlibExt.isBoolFunNPNEquiv_refl","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-npnequiv-refl","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"Every $n$-variable Boolean function is NPN-equivalent to itself.","title":"NPN-equivalence is reflexive"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-class-count","kind":"node","lean":"MetaMathlibExt.boolFunClassCount","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-class-count","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"Given a relation $R$ and a property $P$ of $n$-variable Boolean functions, the number of $R$-inequivalent $n$-variable functions with $P$ is the number of distinct sets $\\{g : P(g) \\text{ and } R(f, g)\\}$ as $f$ ranges over the functions with $P$. When $R$ is an equivalence relation, this is the number of equivalence classes of $R$ restricted to the functions with $P$. The paper's $[P]_n$ is this count when $R$ is\u2026","title":"Number of classes of Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-class-count-false","kind":"node","lean":"MetaMathlibExt.boolFunClassCount_false","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-class-count-false","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"fully_proved","summary":"For every relation $R$ on $n$-variable Boolean functions, the number of $R$-inequivalent $n$-variable Boolean functions with the property that holds for no function is $0$.","title":"The empty property has no classes"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/card-bool-fun-eq-sum-choose-card-nondege","kind":"node","lean":"MetaMathlibExt.card_boolFun_eq_sum_choose_card_nondegenerate","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#card-bool-fun-eq-sum-choose-card-nondege","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"Let $P$ be a property of Boolean functions on any number of variables that is invariant under renaming variables along injections: for all $m, n$, every injective map $e$ from $\\{1, \\dots, m\\}$ to $\\{1, \\dots, n\\}$ and every $m$-variable function $g$, the $n$-variable function $x \\mapsto g(x_{e(1)}, \\dots, x_{e(m)})$ has $P$ if and only if $g$ has $P$. Write $P_n$ for the number of $n$-variable Boolean functions\u2026","title":"Counting Boolean functions by their non-degenerate restrictions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/card-nondegenerate-bool-fun-eq-alternati","kind":"node","lean":"MetaMathlibExt.card_nondegenerate_boolFun_eq_alternating_sum","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#card-nondegenerate-bool-fun-eq-alternati","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"Let $P$ be a property of Boolean functions on any number of variables that is invariant under renaming variables along injections: for all $m, n$, every injective map $e$ from $\\{1, \\dots, m\\}$ to $\\{1, \\dots, n\\}$ and every $m$-variable function $g$, the $n$-variable function $x \\mapsto g(x_{e(1)}, \\dots, x_{e(m)})$ has $P$ if and only if $g$ has $P$. Write $P_i$ for the number of $i$-variable Boolean functions\u2026","title":"Counting non-degenerate Boolean functions by inclusion\u2013exclusion"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/card-unate-bool-fun-relations","kind":"node","lean":"MetaMathlibExt.card_unate_boolFun_relations","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#card-unate-bool-fun-relations","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"For $n \\ge 0$, write $U_n$ and $M_n$ for the numbers of unate and of monotone $n$-variable Boolean functions, $BU_n$ and $BM_n$ for the numbers of those that are moreover balanced, that is, twice their weight equals $2^n$, and write the prefix $\\mathrm{nd}\\text{-}$ for the same counts restricted to non-degenerate functions. Then $\\mathrm{nd}\\text{-}U_n = 2^n \\, \\mathrm{nd}\\text{-}M_n$,\u2026","title":"Unate versus monotone Boolean function counts"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/monotone-bool-fun-concat-iff","kind":"node","lean":"MetaMathlibExt.monotone_boolFunConcat_iff","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#monotone-bool-fun-concat-iff","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"Let $g$ and $h$ be Boolean functions on $n$ variables, and let $g \\parallel h$ be their concatenation, the function on $n + 1$ variables that equals $g$ of the remaining variables when its first variable is $0$ and $h$ of them when it is $1$. Then $g \\parallel h$ is monotone if and only if $g$ and $h$ are both monotone and $g(x) \\le h(x)$ for every $x \\in \\{0,1\\}^n$.","title":"Monotone concatenations of Boolean functions (Bakoev)"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/monotone-boolean-function-dual","kind":"node","lean":"MetaMathlibExt.monotone_booleanFunctionDual","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#monotone-boolean-function-dual","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"If a Boolean function $f$ on $n$ variables is monotone, then so is its dual $f^\\star$, defined by $f^\\star(x) = \\neg f(\\bar{x})$, where $\\bar{x}$ is the coordinatewise complement of $x \\in \\{0,1\\}^n$.","title":"The dual of a monotone Boolean function is monotone"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-unate-bool-fun-bool-fun-concat-iff","kind":"node","lean":"MetaMathlibExt.isUnateBoolFun_boolFunConcat_iff","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-unate-bool-fun-bool-fun-concat-iff","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"Let $g$ and $h$ be $n$-variable Boolean functions and let $g \\| h$ be their concatenation, the $(n+1)$-variable function equal to $g$ of the remaining variables when the first variable is $0$ and to $h$ of them when it is $1$. Then $g \\| h$ is unate if and only if $g$ and $h$ are unate, for each variable $x_i$ either both $g$ and $h$ are monotone increasing in $x_i$ or both are monotone decreasing in $x_i$, and\u2026","title":"Unateness of a concatenation"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-unate-bool-fun-not","kind":"node","lean":"MetaMathlibExt.isUnateBoolFun_not","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-unate-bool-fun-not","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"If an $n$-variable Boolean function $f$ is unate, then so is its negation $x \\mapsto \\lnot f(x)$.","title":"The negation of a unate function is unate"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/card-monotone-bool-fun-weight-symm","kind":"node","lean":"MetaMathlibExt.card_monotone_boolFun_weight_symm","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#card-monotone-bool-fun-weight-symm","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"For all $n \\ge 0$ and all $w$ with $0 \\le w \\le 2^n$, the number of monotone $n$-variable Boolean functions of weight $w$ equals the number of monotone $n$-variable Boolean functions of weight $2^n - w$.","title":"Weight symmetry of monotone Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/card-unate-bool-fun-weight-symm","kind":"node","lean":"MetaMathlibExt.card_unate_boolFun_weight_symm","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#card-unate-bool-fun-weight-symm","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"For all $n \\ge 0$ and all $w$ with $0 \\le w \\le 2^n$, the number of unate $n$-variable Boolean functions of weight $w$ equals the number of unate $n$-variable Boolean functions of weight $2^n - w$.","title":"Weight symmetry of unate Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-class-count-var-perm-eq-sum-non","kind":"node","lean":"MetaMathlibExt.boolFunClassCount_varPerm_eq_sum_nondegenerate","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-class-count-var-perm-eq-sum-non","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"Let $(P_n)_{n \\ge 0}$ be a family of properties, $P_n$ a property of $n$-variable Boolean functions, that is invariant under renaming variables: for all $m$ and $n$, every injection $e$ from $\\{1, \\dots, m\\}$ to $\\{1, \\dots, n\\}$ and every $m$-variable function $g$, the $n$-variable function $x \\mapsto g(x \\circ e)$ has $P_n$ if and only if $g$ has $P_m$. Write $[P]_n$ for the number of classes of $n$-variable\u2026","title":"Permutation classes counted by non-degenerate ones"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-class-count-var-perm-nondegener","kind":"node","lean":"MetaMathlibExt.boolFunClassCount_varPerm_nondegenerate_eq_sub","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-class-count-var-perm-nondegener","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"Let $(P_n)_{n \\ge 0}$ be a family of properties, $P_n$ a property of $n$-variable Boolean functions, that is invariant under renaming variables: for all $m$ and $n$, every injection $e$ from $\\{1, \\dots, m\\}$ to $\\{1, \\dots, n\\}$ and every $m$-variable function $g$, the $n$-variable function $x \\mapsto g(x \\circ e)$ has $P_n$ if and only if $g$ has $P_m$. Write $[P]_n$ for the number of classes of $n$-variable\u2026","title":"Non-degenerate permutation classes as a difference"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-class-count-var-perm-unate-le","kind":"node","lean":"MetaMathlibExt.boolFunClassCount_varPerm_unate_le","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-class-count-var-perm-unate-le","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"Write $[Q]_n$ for the number of classes of $n$-variable Boolean functions with property $Q$ under permutation of the variables. Let $U$, $M$, $B$ and $\\mathrm{nd}$ be the properties of being unate, monotone (nondecreasing for the coordinatewise order with $0 < 1$), balanced (true of a function $f$ when $2\\,\\mathrm{wt}(f) = 2^n$, where $\\mathrm{wt}(f)$ is the weight of $f$) and non-degenerate, and write\u2026","title":"Unate versus monotone permutation classes"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/is-bool-fun-nnequiv-monotone-iff","kind":"node","lean":"MetaMathlibExt.isBoolFunNNEquiv_monotone_iff","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#is-bool-fun-nnequiv-monotone-iff","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"Let $f$ and $g$ be monotone $n$-variable Boolean functions. Then $f$ and $g$ are NN-equivalent ($g$ is obtained from $f$ by negating some inputs and possibly the output) if and only if either $g = f$, or $g(x) = \\lnot f(x \\oplus a)$ for all $x$, where $a \\in \\{0,1\\}^n$ has $a_i = 1$ exactly for the variables $x_i$ on which $f$ is non-degenerate and $\\oplus$ is coordinatewise exclusive or.","title":"NN-equivalence of monotone Boolean functions"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-class-count-unate-eq-monotone","kind":"node","lean":"MetaMathlibExt.boolFunClassCount_unate_eq_monotone","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-class-count-unate-eq-monotone","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"For every $n$, the number of NN-inequivalent unate $n$-variable Boolean functions equals the number of NN-inequivalent monotone ones, and the number of NPN-inequivalent unate $n$-variable Boolean functions equals the number of NPN-inequivalent monotone ones. Here monotone means nondecreasing for the coordinatewise order with $0 < 1$.","title":"Unate and monotone functions have equally many NN- and NPN-classes"},{"id":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/bool-fun-class-count-balanced-unate-eq-m","kind":"node","lean":"MetaMathlibExt.boolFunClassCount_balanced_unate_eq_monotone","parent":25159,"result":true,"route":"../roadmap/Combinatorics/boolean-functions/unate-and-monotone-boolean-functions/index.html#bool-fun-class-count-balanced-unate-eq-m","scope":"Combinatorics/boolean-functions/unate-and-monotone-boolean-functions","status":"can_prove","summary":"For every $n$, call an $n$-variable Boolean function $f$ balanced when $2\\,\\mathrm{wt}(f) = 2^n$, where $\\mathrm{wt}(f)$ is the weight of $f$. The number of NN-inequivalent balanced unate $n$-variable Boolean functions equals the number of NN-inequivalent balanced monotone ones, and the number of NPN-inequivalent balanced unate $n$-variable Boolean functions equals the number of NPN-inequivalent balanced monotone\u2026","title":"Balanced unate and balanced monotone functions have equally many NN- and NPN-classes"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian.json","id":"Combinatorics/stirling-bell-eulerian","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/index.html","scope":"Combinatorics/stirling-bell-eulerian","status":"can_prove","summary":"The number triangles and sequences that count set partitions, permutations by cycles, descents and excedances, and their relatives: Stirling and Lah numbers and their generalizations, Bell and Fubini numbers and Bell polynomials, Eulerian numbers and polynomials, and Bernoulli, Euler and Genocchi numbers studied combinatorially.","title":"Stirling, Bell and Eulerian numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers","kind":"scope","parent":25194,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers","status":"can_prove","summary":"Classical Stirling numbers of the first and second kind and Lah numbers: recurrences, explicit and addition formulas, rising and falling factorial expansions, orthogonality and inverse Stirling matrices, cycle counts, inequalities and identities such as Lee's and Gy's.","title":"Stirling and Lah numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers","kind":"scope","navigation":"module","parent":25195,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers","status":"fully_proved","title":"Lah numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers/unsigned-lah-number","kind":"node","lean":"MetaMathlibExt.unsignedLahNumber","parent":25196,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers/index.html#unsigned-lah-number","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the unsigned Lah number $L(n, k)$ is $\\binom{n-1}{k-1} \\frac{n!}{k!}$ when $1 \\le k \\le n$, $L(0, 0) = 1$, and $L(n, k) = 0$ otherwise.","title":"Unsigned Lah number"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers/signed-lah-number","kind":"node","lean":"MetaMathlibExt.signedLahNumber","parent":25196,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers/index.html#signed-lah-number","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lah-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the signed Lah number is the integer $(-1)^{n-k} L(n, k)$ when $k \\le n$, where $L(n, k)$ is the unsigned Lah number, and $0$ when $k > n$.","title":"Signed Lah number"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","kind":"scope","navigation":"module","parent":25195,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","status":"can_prove","title":"Lee's identities for Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/stirling-first-succ-eq-sum-choose-pow","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingIdentitiesWanted.stirlingFirst_succ_eq_sum_choose_pow","parent":25199,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/index.html#stirling-first-succ-eq-sum-choose-pow","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","status":"can_prove","summary":"Write $c(n, k)$ for the unsigned Stirling number of the first kind, the number of permutations of $n$ elements with exactly $k$ cycles. For natural numbers $m \\le n$, the following identity of integers holds:","title":"Lee's expansion of shifted Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/stirling-second-eq-sum-choose-pow","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingIdentitiesWanted.stirlingSecond_eq_sum_choose_pow","parent":25199,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/index.html#stirling-second-eq-sum-choose-pow","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling number of the second kind, the number of partitions of an $n$-element set into $k$ nonempty blocks. For natural numbers $m \\le n$, the following identity of integers holds:","title":"Lee's alternating expansion of Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/stirling-second-mul-sub-eq-sum-choose-al","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingIdentitiesWanted.stirlingSecond_mul_sub_eq_sum_choose_alternating","parent":25199,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/index.html#stirling-second-mul-sub-eq-sum-choose-al","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling number of the second kind, the number of partitions of an $n$-element set into $k$ nonempty blocks. For natural numbers $k$ and $n$ with $k \\ge 1$ and $n \\ge k + 1$, the following identity of integers holds:","title":"Lee's identity for Stirling numbers of the second kind from smaller rows"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/sum-stirling-second-factorial-alternatin","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingIdentitiesWanted.sum_stirlingSecond_factorial_alternating","parent":25199,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/index.html#sum-stirling-second-factorial-alternatin","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","status":"can_prove","summary":"Write $S(a, j)$ for the Stirling number of the second kind, the number of partitions of an $a$-element set into $j$ nonempty blocks. For every natural number $a \\ge 1$, the following identity of integers holds:","title":"Lee's factorial-weighted alternating row sum of Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/stirling-second-mul-sub-eq-sum-factorial","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingIdentitiesWanted.stirlingSecond_mul_sub_eq_sum_factorial_negChoose","parent":25199,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/index.html#stirling-second-mul-sub-eq-sum-factorial","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling number of the second kind, the number of partitions of an $n$-element set into $k$ nonempty blocks. For natural numbers $k$ and $n$ with $k \\ge 1$ and $n \\ge k + 1$, the following identity of integers holds:","title":"Lee's identity for Stirling numbers of the second kind from higher columns"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/stirling-first-mul-sub-eq-sum-choose-fac","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingIdentitiesWanted.stirlingFirst_mul_sub_eq_sum_choose_factorial","parent":25199,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/index.html#stirling-first-mul-sub-eq-sum-choose-fac","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","status":"can_prove","summary":"Write $c(n, k)$ for the unsigned Stirling number of the first kind, the number of permutations of $n$ elements with exactly $k$ cycles. For natural numbers $k$ and $n$ with $k \\ge 1$ and $n \\ge k + 1$, the following identity of integers holds:","title":"Lee's identity for Stirling numbers of the first kind from smaller rows"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/stirling-first-mul-sub-eq-sum-stirling-f","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingIdentitiesWanted.stirlingFirst_mul_sub_eq_sum_stirlingFirst_choose","parent":25199,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers/index.html#stirling-first-mul-sub-eq-sum-stirling-f","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/lee-s-identities-for-stirling-numbers","status":"can_prove","summary":"Write $c(n, k)$ for the unsigned Stirling number of the first kind, the number of permutations of $n$ elements with exactly $k$ cycles. For natural numbers $k$ and $n$ with $k \\ge 1$ and $n \\ge k + 1$, the following identity of integers holds:","title":"Lee's identity for Stirling numbers of the first kind from higher columns"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","kind":"scope","navigation":"module","parent":25195,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","status":"fully_proved","title":"Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/signed-stirling-first","kind":"node","lean":"MetaMathlibExt.signedStirlingFirst","parent":25207,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/index.html#signed-stirling-first","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the signed Stirling number of the first kind is the integer $(-1)^{n-k} c(n,k)$, where $c(n,k)$ is the unsigned Stirling number of the first kind (the number of permutations of $n$ elements with exactly $k$ cycles) and $n - k$ is truncated subtraction, equal to $0$ when $k > n$.","title":"Signed Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/cycle-quotient-count","kind":"node","lean":"MetaMathlibExt.cycleQuotientCount","parent":25207,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/index.html#cycle-quotient-count","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","status":"fully_proved","summary":"For a permutation $\\sigma$ of a finite type $\\alpha$, the number of cycles of $\\sigma$, counting each fixed point as a cycle of length one, is the number of equivalence classes of $\\alpha$ under the relation that holds between $x$ and $y$ when $\\sigma^i(x) = y$ for some integer $i$.","title":"Number of cycles of a permutation"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/cycle-quotient-count-eq","kind":"node","lean":"MetaMathlibExt.cycleQuotientCount_eq","parent":25207,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/index.html#cycle-quotient-count-eq","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","status":"fully_proved","summary":"For a permutation $\\sigma$ of a finite type, the number of cycles of $\\sigma$, counting each fixed point as a cycle of length one, equals the number of cycles of $\\sigma$ of length at least $2$ (the length of its cycle type) plus the number of fixed points of $\\sigma$.","title":"Counting cycles via the cycle type"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/card-perm-cycle-count-eq-stirling-first","kind":"node","lean":"MetaMathlibExt.card_perm_cycleCount_eq_stirlingFirst","parent":25207,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/index.html#card-perm-cycle-count-eq-stirling-first","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the number of permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$ such that the number of cycles of $\\sigma$ of length at least $2$ plus the number of fixed points of $\\sigma$ equals $k$ is the unsigned Stirling number of the first kind $c(n, k)$. In other words, $c(n, k)$ counts the permutations of an $n$-element set with exactly $k$ cycles, each fixed point counted as a cycle of\u2026","title":"Stirling numbers of the first kind count permutations by cycles"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/sum-stirling-first-mul-x-pow","kind":"node","lean":"Nat.sum_stirlingFirst_mul_X_pow","parent":25207,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/index.html#sum-stirling-first-mul-x-pow","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","status":"fully_proved","summary":"The rising factorial is the generating polynomial of the unsigned Stirling numbers of the first kind: `ascPochhammer S n = \u2211_{k=0}^{n} stirlingFirst n k * X ^ k` (Graham\u2013Knuth\u2013Patashnik, \u00a76.1).","title":"Rising factorial generates Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/sum-stirling-first-mul-pow","kind":"node","lean":"Nat.sum_stirlingFirst_mul_pow","parent":25207,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/index.html#sum-stirling-first-mul-pow","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","status":"fully_proved","summary":"`\u2211_{k=0}^{n} stirlingFirst n k * x ^ k = x (x + 1) \u22ef (x + n - 1)` (Graham\u2013Knuth\u2013Patashnik, \u00a76.1).","title":"Stirling numbers of the first kind and the rising factorial"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/stirling-first-addition","kind":"node","lean":"MetaMathlibExt.stirlingFirst_addition","parent":25207,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind/index.html#stirling-first-addition","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-first-kind","status":"fully_proved","summary":"Write $c(n,k)$ for the unsigned Stirling numbers of the first kind and $u^{\\overline{i}} = u(u+1)\\cdots(u+i-1)$ for the rising factorial. For all natural numbers $u$, $v$ and $k$,","title":"Addition formula for Stirling numbers of the first kind"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","kind":"scope","navigation":"module","parent":25195,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","title":"Alzer\u2013Richards identities for Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-iterated-deriv-pow-one-sub","kind":"node","lean":"MetaMathlibExt.sum_iteratedDeriv_pow_one_sub","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alzer\u2013Richards derivative identity for powers of "],["math","\\(1 - f\\)"]],"source":"Alzer\u2013Richards derivative identity for powers of $1 - f$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-iterated-deriv-pow-one-sub","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $r, j, n \\ge 0$ be integers and let $f \\colon \\mathbb{R} \\to \\mathbb{R}$ be $r$ times continuously differentiable. Writing $g^{(i)}$ for the $i$-th derivative of a function $g$ and taking powers of $f$ pointwise, for every real $x$","title":"Alzer\u2013Richards derivative identity for powers of $1 - f$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-iterated-deriv-zpow-one-add","kind":"node","lean":"MetaMathlibExt.sum_iteratedDeriv_zpow_one_add","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alzer\u2013Richards derivative identity for powers of "],["math","\\(1 + f\\)"]],"source":"Alzer\u2013Richards derivative identity for powers of $1 + f$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-iterated-deriv-zpow-one-add","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $r, j, n \\ge 0$ be integers and let $f \\colon \\mathbb{R} \\to \\mathbb{R}$ be $r$ times continuously differentiable with $f(y) \\ne 0$ for every real $y$. Writing $g^{(i)}$ for the $i$-th derivative of a function $g$ and taking integer powers of $f$ pointwise, for every real $x$","title":"Alzer\u2013Richards derivative identity for powers of $1 + f$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-div-choose-mul-stirling-secon","kind":"node","lean":"MetaMathlibExt.sum_choose_div_choose_mul_stirlingSecond_eq_bernoulli","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating Stirling sums with "],["math","\\(\\binom{n+1}{k+1}\\)"],["text"," give Bernoulli numbers"]],"source":"Alternating Stirling sums with $\\binom{n+1}{k+1}$ give Bernoulli numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-div-choose-mul-stirling-secon","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling numbers of the second kind and $B_n$ for the Bernoulli numbers, with $B_1 = -1/2$. Let $n$ be a natural number. Then for every natural number $r \\le n$,","title":"Alternating Stirling sums with $\\binom{n+1}{k+1}$ give Bernoulli numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-div-choose-mul-stirling-secon-2","kind":"node","lean":"MetaMathlibExt.sum_choose_div_choose_mul_stirlingSecond_eq_zero","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing alternating Stirling sums with "],["math","\\(\\binom{r+m}{k}\\)"]],"source":"Vanishing alternating Stirling sums with $\\binom{r+m}{k}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-div-choose-mul-stirling-secon-2","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling numbers of the second kind. For every natural number $m \\ge 1$ and every natural number $r$,","title":"Vanishing alternating Stirling sums with $\\binom{r+m}{k}$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-add-two-div-choose-mul-stirli","kind":"node","lean":"MetaMathlibExt.sum_choose_add_two_div_choose_mul_stirlingSecond","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating Stirling sums with "],["math","\\(\\binom{n+1}{k+2}\\)"]],"source":"Alternating Stirling sums with $\\binom{n+1}{k+2}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-add-two-div-choose-mul-stirli","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling numbers of the second kind and $B_n$ for the Bernoulli numbers, with $B_1 = -1/2$. Let $n \\ge 1$ be a natural number. Then for every natural number $r$ with $1 \\le r \\le n$,","title":"Alternating Stirling sums with $\\binom{n+1}{k+2}$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-div-choose-mul-signed-stirlin","kind":"node","lean":"MetaMathlibExt.sum_choose_div_choose_mul_signedStirlingFirst","parent":25215,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-div-choose-mul-signed-stirlin","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Write $s(n, k)$ for the signed Stirling numbers of the first kind. For all natural numbers $j$ and $n$ and every natural number $r \\le n$,","title":"Alternating sums of signed Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-euler-number-mul-sum-central-power-s","kind":"node","lean":"MetaMathlibExt.sum_eulerNumber_mul_sum_centralPowerSum","parent":25215,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-euler-number-mul-sum-central-power-s","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $n \\ge 1$ be an integer. For integers $k, i \\ge 0$ let $T(k, i) = 2^{-k} \\sum_{j=0}^{k} \\binom{k}{j} (k-2j)^i$ (with $0^0 = 1$), and let $E_k$ be the Euler numbers: $E_k = 0$ for odd $k$, and for even $k$ they are given by $E_0 = 1$ and $E_{2m} = -\\sum_{s=0}^{m-1} \\binom{2m}{2s} E_{2s}$ for $m \\ge 1$. Then","title":"Alzer\u2013Richards identity for Euler numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-fubini-number-mul-sum-alternating-st","kind":"node","lean":"MetaMathlibExt.sum_fubiniNumber_mul_sum_alternatingStirling","parent":25215,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-fubini-number-mul-sum-alternating-st","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $n \\ge 1$ be an integer, let $a(k)$ be the $k$-th Fubini (ordered Bell) number and $S(r, j)$ the Stirling numbers of the second kind, and for integers $r, k \\ge 0$ let $A(r, k) = \\sum_{j=0}^{k} (-1)^j \\binom{k}{j} j! \\, S(r, j)$. Then","title":"Alzer\u2013Richards identity for Fubini numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/pow-sub-desc-pochhammer-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.pow_sub_descPochhammer_eq_sum_choose","parent":25215,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#pow-sub-desc-pochhammer-eq-sum-choose","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"For a complex number $w$ and a natural number $j$ write $\\binom{w}{j} = w(w-1)\\cdots(w-j+1)/j!$. For every natural number $p \\ge 1$ and every complex number $z$,","title":"Sabar's identity for complex arguments"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/mul-pow-pred-sub-pow-mul-sum-inv-eq-sum","kind":"node","lean":"MetaMathlibExt.mul_pow_pred_sub_pow_mul_sum_inv_eq_sum_choose","parent":25215,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#mul-pow-pred-sub-pow-mul-sum-inv-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"For a complex number $w$ and a natural number $j$ write $\\binom{w}{j} = w(w-1)\\cdots(w-j+1)/j!$. Let $p \\ge 1$ be a natural number and let $z$ be a complex number with $z \\ne k$ for every natural number $k < p$. Then","title":"A reciprocal-sum identity for generalized binomial coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-mul-choose-div-choose-eq-hype","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_choose_div_choose_eq_hypergeometric3F2","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Alzer\u2013Richards binomial sum as a "],["math","\\({}_3F_2\\)"],["text"," series"]],"source":"The Alzer\u2013Richards binomial sum as a ${}_3F_2$ series"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-mul-choose-div-choose-eq-hype","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $m \\ge 1$ be an integer and let $p, x$ be complex numbers with $p \\ne -j$ and $x \\ne -j$ for every integer $j$ with $1 \\le j \\le \\lfloor m/2 \\rfloor$. Write $\\binom{z}{k} = z(z-1)\\cdots(z-k+1)/k!$ for complex $z$ and $(z)_n = z(z+1)\\cdots(z+n-1)$ for the rising factorial. Then","title":"The Alzer\u2013Richards binomial sum as a ${}_3F_2$ series"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-mul-choose-div-choose-eq-hype-2","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_choose_div_choose_eq_hypergeometric2F1","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","A binomial sum as a terminating "],["math","\\({}_2F_1\\)"],["text"," series"]],"source":"A binomial sum as a terminating ${}_2F_1$ series"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-mul-choose-div-choose-eq-hype-2","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Write $x^{\\overline{n}} = x(x+1)\\cdots(x+n-1)$ for the rising factorial of a complex number $x$, and $\\binom{w}{k} = w(w-1)\\cdots(w-k+1)/k!$ for a complex number $w$ and a natural number $k$. Let $m$ and $\\nu$ be natural numbers with $1 \\le \\nu \\le \\lfloor m/2 \\rfloor$, and let $p$ be a complex number with $p \\ne -j$ for every natural number $j$ with $1 \\le j \\le \\lfloor m/2 \\rfloor$. Then","title":"A binomial sum as a terminating ${}_2F_1$ series"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-mul-choose-div-choose-of-even","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_choose_div_choose_of_even","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form of the Alzer\u2013Richards binomial sum for even "],["math","\\(m\\)"]],"source":"Closed form of the Alzer\u2013Richards binomial sum for even $m$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-mul-choose-div-choose-of-even","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $m \\ge 2$ be an even integer and $x$ a complex number, and write $\\binom{z}{k} = z(z-1)\\cdots(z-k+1)/k!$ for complex $z$. Put","title":"Closed form of the Alzer\u2013Richards binomial sum for even $m$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-mul-choose-div-choose-of-odd","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_choose_div_choose_of_odd","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form of the Alzer\u2013Richards binomial sum for odd "],["math","\\(m\\)"]],"source":"Closed form of the Alzer\u2013Richards binomial sum for odd $m$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-mul-choose-div-choose-of-odd","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $m$ be an odd positive integer and $x$ a complex number, and write $\\binom{z}{k} = z(z-1)\\cdots(z-k+1)/k!$ for complex $z$. Put","title":"Closed form of the Alzer\u2013Richards binomial sum for odd $m$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/sum-choose-mul-choose-div-choose-shifted","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_choose_div_choose_shifted","parent":25215,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#sum-choose-mul-choose-div-choose-shifted","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $m \\ge 1$ be an integer and $x$ a complex number, write $\\binom{z}{k} = z(z-1)\\cdots(z-k+1)/k!$ for complex $z$, and assume $\\binom{k-m+1/2-x}{k} \\ne 0$ for $0 \\le k \\le \\lfloor m/2 \\rfloor$. Put","title":"Closed form of the shifted Alzer\u2013Richards binomial sum"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/iterated-deriv-pow-one-sub-exp-div-self","kind":"node","lean":"MetaMathlibExt.iteratedDeriv_pow_one_sub_exp_div_self_zero","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivatives at zero of powers of "],["math","\\((1 - e^x)/x\\)"]],"source":"Derivatives at zero of powers of $(1 - e^x)/x$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#iterated-deriv-pow-one-sub-exp-div-self","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling numbers of the second kind. Let $f : \\mathbb{R} \\to \\mathbb{R}$ satisfy $f(0) = -1$ and $f(x) = (1 - e^x)/x$ for every $x \\ne 0$. Then for all natural numbers $k$ and $r$, the $r$-th derivative of $f^k$ at $0$ equals $(-1)^k S(r+k, k) / \\binom{r+k}{k}$.","title":"Derivatives at zero of powers of $(1 - e^x)/x$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/iterated-deriv-inv-one-sub-exp-div-self","kind":"node","lean":"MetaMathlibExt.iteratedDeriv_inv_one_sub_exp_div_self_zero","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivatives of "],["math","\\(x/(1 - e^x)\\)"],["text"," at zero"]],"source":"Derivatives of $x/(1 - e^x)$ at zero"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#iterated-deriv-inv-one-sub-exp-div-self","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Write $B_k$ for the $k$-th Bernoulli number, with the convention $B_1 = -1/2$. Let $f : \\mathbb{R} \\to \\mathbb{R}$ satisfy $f(0) = -1$ and $f(x) = (1 - e^x)/x$ for every $x \\ne 0$. Then for every natural number $k$, the $k$-th derivative of $1/f$ at $0$ equals $-B_k$.","title":"Derivatives of $x/(1 - e^x)$ at zero"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/iterated-deriv-one-add-one-sub-exp-div-s","kind":"node","lean":"MetaMathlibExt.iteratedDeriv_one_add_one_sub_exp_div_self_pow_zero","parent":25215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivatives at zero of "],["math","\\((1 + (1 - e^x)/x)^{n+1}\\)"]],"source":"Derivatives at zero of $(1 + (1 - e^x)/x)^{n+1}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers/index.html#iterated-deriv-one-add-one-sub-exp-div-s","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/alzer-richards-identities-for-stirling-numbers","status":"can_prove","summary":"Let $f : \\mathbb{R} \\to \\mathbb{R}$ satisfy $f(0) = -1$ and $f(x) = (1 - e^x)/x$ for every $x \\ne 0$, and let $n$ be a natural number. Then for every natural number $r \\le n$, the $r$-th derivative of $(1 + f)^{n+1}$ at $0$ is $0$, and the $(n+1)$-th derivative of $(1 + f)^{n+1}$ at $0$ equals $(-1/2)^{n+1} (n+1)!$.","title":"Derivatives at zero of $(1 + (1 - e^x)/x)^{n+1}$"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","kind":"scope","navigation":"module","parent":25195,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","title":"Stirling matrices"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/sum-signed-stirling-first-mul-stirling-s","kind":"node","lean":"MetaMathlibExt.sum_signedStirlingFirst_mul_stirlingSecond","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#sum-signed-stirling-first-mul-stirling-s","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"Write $c(n,k)$ for the unsigned Stirling numbers of the first kind, $s(n,k) = (-1)^{n-k} c(n,k)$ for the signed ones, and $S(k,m)$ for the Stirling numbers of the second kind. For all natural numbers $n$ and $m$,","title":"Orthogonality relation for Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-matrices-mutual-inverse-and-low","kind":"node","lean":"MetaMathlibExt.stirlingMatrices_mutualInverse_and_lowerTriangular","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-matrices-mutual-inverse-and-low","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"Let $N$ be a natural number, and let $A$ and $B$ be the $N \\times N$ integer matrices with rows and columns indexed by $0, 1, \\dots, N-1$ and entries $A_{ij} = (-1)^{i-j} c(i,j)$ and $B_{ij} = S(i,j)$, where $c(i,j)$ is the unsigned Stirling number of the first kind and $S(i,j)$ the Stirling number of the second kind. Then $AB = BA = I$, and $A_{ij} = B_{ij} = 0$ whenever $i < j$.","title":"Stirling matrices are mutually inverse and lower triangular"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-second-shift-matrix","kind":"node","lean":"MetaMathlibExt.stirlingSecondShiftMatrix","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-second-shift-matrix","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For natural numbers $m$ and $n$, the shifted Stirling matrix $S^{[m]}(n)$ is the $n \\times n$ matrix over $\\mathbb{Q}$ whose $(i, j)$ entry is the Stirling number of the second kind $S(i + m, j)$, for $1 \\le i, j \\le n$.","title":"Shifted Stirling matrix of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-second-shift-matrix-zero","kind":"node","lean":"MetaMathlibExt.stirlingSecondShiftMatrix_zero","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-second-shift-matrix-zero","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For every natural number $n$, the shifted Stirling matrix $S^{[0]}(n)$, the $n \\times n$ rational matrix with entries $S(i + 0, j)$ for $1 \\le i, j \\le n$, equals the Stirling matrix of the second kind $S(n)$, the $n \\times n$ rational matrix with entries $S(i, j)$; here $S(i, j)$ denotes the Stirling numbers of the second kind.","title":"Shift zero gives the Stirling matrix of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-second-shift-matrix-apply-zero","kind":"node","lean":"MetaMathlibExt.stirlingSecondShiftMatrix_apply_zero_right","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-second-shift-matrix-apply-zero","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"Let $m$ and $n$ be natural numbers with $n > 0$, and let $S^{[m]}(n)$ be the $n \\times n$ rational matrix with entries $S(i + m, j)$ for $1 \\le i, j \\le n$, where $S(i, j)$ denotes the Stirling numbers of the second kind. Every entry of the first column of $S^{[m]}(n)$ equals $1$: $S(i + m, 1) = 1$ for $1 \\le i \\le n$.","title":"First column of a shifted Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-first-matrix","kind":"node","lean":"MetaMathlibExt.stirlingFirstMatrix","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-first-matrix","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For a natural number $n$, the Stirling matrix of the first kind $s(n)$ is the $n \\times n$ matrix with rational entries $s(n)_{i,j} = s(i, j)$ for $1 \\le i, j \\le n$, where $s(i, j)$ is the signed Stirling number of the first kind. Its entries above the diagonal are zero, since $s(i, j) = 0$ for $j > i$.","title":"Stirling matrix of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-first-matrix-apply-self","kind":"node","lean":"MetaMathlibExt.stirlingFirstMatrix_apply_self","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-first-matrix-apply-self","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For every natural number $n$, every diagonal entry of the Stirling matrix of the first kind $s(n)$ equals $1$, that is, $s(n)_{i,i} = 1$ for $1 \\le i \\le n$.","title":"Diagonal of the Stirling matrix of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-vmatrix","kind":"node","lean":"MetaMathlibExt.stirlingVMatrix","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-vmatrix","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For a natural number $n$, $V(n)$ is the $n \\times n$ matrix over $\\mathbb{Q}$, with rows and columns indexed by $1, \\dots, n$, whose entries are $V_{i,i} = i$ on the diagonal, $V_{i,i+1} = 1$ just above the diagonal, and $0$ elsewhere.","title":"Upper bidiagonal matrix V(n)"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-vmatrix-apply-self","kind":"node","lean":"MetaMathlibExt.stirlingVMatrix_apply_self","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-vmatrix-apply-self","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For every natural number $n$, let $V(n)$ be the $n \\times n$ rational upper bidiagonal matrix with $V(n)_{i,i} = i$, $V(n)_{i,i+1} = 1$ and all other entries $0$, indices running over $1, \\dots, n$. Then $V(n)_{i,i} = i$ for every $1 \\le i \\le n$.","title":"Diagonal entries of the bidiagonal matrix V(n)"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-vtilde-matrix","kind":"node","lean":"MetaMathlibExt.stirlingVTildeMatrix","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-vtilde-matrix","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For a natural number $n$, $\\tilde V(n)$ is the $n \\times n$ matrix over $\\mathbb{Q}$, with rows and columns indexed by $1, \\dots, n$, whose $(i, j)$ entry is $(-1)^{j-i} (i-1)!/j!$ if $i \\le j$ and $0$ if $i > j$.","title":"Upper triangular matrix \u1e7c(n)"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-vtilde-matrix-apply-self","kind":"node","lean":"MetaMathlibExt.stirlingVTildeMatrix_apply_self","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-vtilde-matrix-apply-self","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For a natural number $n$, let $\\tilde V(n)$ be the $n \\times n$ upper triangular matrix over $\\mathbb{Q}$, with rows and columns indexed by $1, \\dots, n$, whose $(i, j)$ entry is $(-1)^{j-i} (i-1)!/j!$ if $i \\le j$ and $0$ if $i > j$. Then for every $i$ with $1 \\le i \\le n$, the $(i, i)$ entry of $\\tilde V(n)$ is $1/i$.","title":"Diagonal entries of the matrix \u1e7c(n)"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/shifted-vandermonde-matrix","kind":"node","lean":"MetaMathlibExt.shiftedVandermondeMatrix","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#shifted-vandermonde-matrix","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For a natural number $n$, the shifted Vandermonde matrix $E(n)$ is the $n \\times n$ matrix with rational entries $E(n)_{i,j} = (j + 1)^{i-1}$ for $1 \\le i, j \\le n$.","title":"Shifted Vandermonde matrix"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/shifted-vandermonde-matrix-apply-zero-le","kind":"node","lean":"MetaMathlibExt.shiftedVandermondeMatrix_apply_zero_left","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#shifted-vandermonde-matrix-apply-zero-le","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For every natural number $n \\ge 1$, every entry in the first row of the shifted Vandermonde matrix $E(n)$ equals $1$, that is, $E(n)_{1,j} = 1$ for $1 \\le j \\le n$.","title":"First row of the shifted Vandermonde matrix"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-qmatrix","kind":"node","lean":"MetaMathlibExt.stirlingQMatrix","parent":25234,"reading_title":{"format":"inline-parts-v1","parts":[["text","Strictly upper triangular matrix "],["math","\\(Q^{(N)}(n)\\)"]],"source":"Strictly upper triangular matrix $Q^{(N)}(n)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-qmatrix","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For natural numbers $N$ and $n$, $Q^{[N]}(n)$ is the $n \\times n$ matrix with rational entries","title":"Strictly upper triangular matrix $Q^{(N)}(n)$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-qmatrix-apply-of-le","kind":"node","lean":"MetaMathlibExt.stirlingQMatrix_apply_of_le","parent":25234,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing of "],["math","\\(Q^{(N)}(n)\\)"],["text"," on and below the diagonal"]],"source":"Vanishing of $Q^{(N)}(n)$ on and below the diagonal"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-qmatrix-apply-of-le","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"fully_proved","summary":"For natural numbers $N$ and $n$, let $Q^{[N]}(n)$ be the $n \\times n$ rational matrix with $Q^{[N]}(n)_{i,j} = (-1)^{j-i+1} i^{N-1} / ((i-1)!\\,(j-i-1)!)$ for $1 \\le i < j \\le n$ and $0$ otherwise, where $N - 1$ is truncated subtraction. Then $Q^{[N]}(n)_{i,j} = 0$ for all indices $1 \\le j \\le i \\le n$.","title":"Vanishing of $Q^{(N)}(n)$ on and below the diagonal"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-vmatrix-inv","kind":"node","lean":"MetaMathlibExt.stirlingVMatrix_inv","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-vmatrix-inv","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"For every natural number $n$, let $V(n)$ be the $n \\times n$ rational upper bidiagonal matrix with $V(n)_{i,i} = i$, $V(n)_{i,i+1} = 1$ and all other entries $0$, and let $\\tilde V(n)$ be the upper triangular matrix with $\\tilde V(n)_{i,j} = (-1)^{j-i} (i-1)!/j!$ for $j \\ge i$ and $\\tilde V(n)_{i,j} = 0$ for $j < i$, all indices running over $1, \\dots, n$. Then $V(n)^{-1} = \\tilde V(n)$.","title":"Explicit inverse of the bidiagonal matrix V(n)"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-qmatrix-succ","kind":"node","lean":"MetaMathlibExt.stirlingQMatrix_succ","parent":25234,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for "],["math","\\(Q^{(N)}(n)\\)"],["text"," in "],["math","\\(N\\)"]],"source":"Recurrence for $Q^{(N)}(n)$ in $N$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-qmatrix-succ","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"For natural numbers $n$ and $N$ with $N \\ge 1$, let $V(n)$ be the $n \\times n$ upper bidiagonal matrix with $V(n)_{i,i} = i$, $V(n)_{i,i+1} = 1$ and all other entries $0$, let $I_n$ be the $n \\times n$ identity matrix, and for $M \\ge 1$ let $Q^{[M]}(n)$ be the $n \\times n$ matrix with $Q^{[M]}(n)_{i,j} = (-1)^{j-i+1} i^{M-1} / ((i-1)!\\,(j-i-1)!)$ for $j \\ge i + 1$ and $0$ otherwise. Then, as matrices over\u2026","title":"Recurrence for $Q^{(N)}(n)$ in $N$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-qmatrix-eq-one-mul-pow","kind":"node","lean":"MetaMathlibExt.stirlingQMatrix_eq_one_mul_pow","parent":25234,"reading_title":{"format":"inline-parts-v1","parts":[["text","Factorization of "],["math","\\(Q^{(N)}(n)\\)"],["text"," through "],["math","\\(Q^{(1)}(n)\\)"]],"source":"Factorization of $Q^{(N)}(n)$ through $Q^{(1)}(n)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-qmatrix-eq-one-mul-pow","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"For natural numbers $n$ and $N$ with $N \\ge 1$, let $V(n)$ be the $n \\times n$ upper bidiagonal matrix with $V(n)_{i,i} = i$, $V(n)_{i,i+1} = 1$ and all other entries $0$, let $I_n$ be the $n \\times n$ identity matrix, and for $M \\ge 1$ let $Q^{[M]}(n)$ be the $n \\times n$ matrix with $Q^{[M]}(n)_{i,j} = (-1)^{j-i+1} i^{M-1} / ((i-1)!\\,(j-i-1)!)$ for $j \\ge i + 1$ and $0$ otherwise. Then, as matrices over\u2026","title":"Factorization of $Q^{(N)}(n)$ through $Q^{(1)}(n)$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-matrix2-mul-sub-one","kind":"node","lean":"MetaMathlibExt.stirlingMatrix2_mul_sub_one","parent":25234,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stirling matrix of the second kind times "],["math","\\(V(n) - I_n\\)"]],"source":"Stirling matrix of the second kind times $V(n) - I_n$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-matrix2-mul-sub-one","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"Let $n$ be a natural number. Let $S(n)$ be the $n \\times n$ Stirling matrix of the second kind, with entries the Stirling numbers of the second kind $S(i, j)$ for $1 \\le i, j \\le n$; let $V(n)$ be the $n \\times n$ upper bidiagonal matrix with $V(n)_{i,i} = i$, $V(n)_{i,i+1} = 1$ and all other entries $0$; let $I_n$ be the identity matrix, $E(n)$ the shifted Vandermonde matrix with $E(n)_{i,j} = (j + 1)^{i-1}$, and\u2026","title":"Stirling matrix of the second kind times $V(n) - I_n$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-matrix2-mul-sub-one-pow","kind":"node","lean":"MetaMathlibExt.stirlingMatrix2_mul_sub_one_pow","parent":25234,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stirling matrix of the second kind times powers of "],["math","\\(V(n) - I_n\\)"]],"source":"Stirling matrix of the second kind times powers of $V(n) - I_n$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-matrix2-mul-sub-one-pow","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"Let $n$ and $N$ be natural numbers with $N \\ge 1$. Let $S(n)$ be the $n \\times n$ Stirling matrix of the second kind, with entries the Stirling numbers of the second kind $S(i, j)$ for $1 \\le i, j \\le n$; let $V(n)$ be the $n \\times n$ upper bidiagonal matrix with $V(n)_{i,i} = i$, $V(n)_{i,i+1} = 1$ and all other entries $0$; let $I_n$ be the identity matrix, $E(n)$ the shifted Vandermonde matrix with\u2026","title":"Stirling matrix of the second kind times powers of $V(n) - I_n$"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-second-shift-matrix-eq-mul-pow","kind":"node","lean":"MetaMathlibExt.stirlingSecondShiftMatrix_eq_mul_pow","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-second-shift-matrix-eq-mul-pow","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"For natural numbers $m$ and $n$, let $S(n)$ and $S^{[m]}(n)$ be the $n \\times n$ rational matrices with entries $S(i, j)$ and $S(i + m, j)$, where $S(i, j)$ denotes the Stirling numbers of the second kind, and let $V(n)$ be the upper bidiagonal matrix with $V(n)_{i,i} = i$, $V(n)_{i,i+1} = 1$ and all other entries $0$, all indices running over $1, \\dots, n$. Then $S^{[m]}(n) = S(n)\\, V(n)^m$ and\u2026","title":"Factorization and determinant of shifted Stirling matrices"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-second-shift-matrix-succ","kind":"node","lean":"MetaMathlibExt.stirlingSecondShiftMatrix_succ","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-second-shift-matrix-succ","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"For natural numbers $m$ and $n$, let $S^{[m]}(n)$ be the $n \\times n$ rational matrix with entries $S(i + m, j)$, where $S(i, j)$ denotes the Stirling numbers of the second kind, and let $V(n)$ be the upper bidiagonal matrix with $V(n)_{i,i} = i$, $V(n)_{i,i+1} = 1$ and all other entries $0$, all indices running over $1, \\dots, n$. Then $S^{[m+1]}(n) = S^{[m]}(n)\\, V(n)$ and\u2026","title":"Recurrence for shifted Stirling matrices"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-second-shift-matrix-add","kind":"node","lean":"MetaMathlibExt.stirlingSecondShiftMatrix_add","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-second-shift-matrix-add","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"For natural numbers $m$ and $n$, let $S^{[m]}(n)$ be the $n \\times n$ rational matrix with entries $S(i + m, j)$ for $1 \\le i, j \\le n$, where $S(i, j)$ denotes the Stirling numbers of the second kind, and let $s(n)$ be the $n \\times n$ rational matrix with entries the signed Stirling numbers of the first kind $s(i, j)$ for $1 \\le i, j \\le n$. Then for all natural numbers $m_1$, $m_2$ and $n$,\u2026","title":"Composition of shifted Stirling matrices"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-second-shift-matrix-inv","kind":"node","lean":"MetaMathlibExt.stirlingSecondShiftMatrix_inv","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-second-shift-matrix-inv","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"For natural numbers $m$ and $n$, let $S^{[m]}(n)$ be the $n \\times n$ rational matrix with entries $S(i + m, j)$, where $S(i, j)$ denotes the Stirling numbers of the second kind; let $s(n)$ be the matrix with entries the signed Stirling numbers of the first kind $s(i, j)$; and let $\\tilde V(n)$ be the upper triangular matrix with $\\tilde V(n)_{i,j} = (-1)^{j-i} (i-1)!/j!$ for $j \\ge i$ and $\\tilde V(n)_{i,j} = 0$\u2026","title":"Inverse of a shifted Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/sum-signed-stirling-first-mul-stirling-s-2","kind":"node","lean":"MetaMathlibExt.sum_signedStirlingFirst_mul_stirlingSecond_add","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#sum-signed-stirling-first-mul-stirling-s-2","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"Write $s(n, k)$ for the signed Stirling numbers of the first kind and $S(n, k)$ for the Stirling numbers of the second kind. For all natural numbers $i$, $j$, $m$ with $1 \\le j \\le i$, the integer $\\sum_{l=1}^{i} s(i, l)\\, S(l + m, j)$ equals $i^m$ if $i = j$ and $0$ if $i > j$.","title":"Shifted orthogonality of Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/sum-choose-stirling-second-eq","kind":"node","lean":"MetaMathlibExt.sum_choose_stirlingSecond_eq","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#sum-choose-stirling-second-eq","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling numbers of the second kind. For all natural numbers $i, j, N \\ge 1$, the following identity holds in the rational numbers:","title":"Alternating binomial sums of shifted Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/stirling-second-succ-sub-eq-sum","kind":"node","lean":"MetaMathlibExt.stirlingSecond_succ_sub_eq_sum","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#stirling-second-succ-sub-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"For natural numbers $i \\ge 1$ and $j \\ge 1$, the Stirling numbers of the second kind $S(i, j)$ satisfy, in $\\mathbb{Q}$,","title":"Difference of consecutive Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/sum-sum-stirling-second-signed-stirling","kind":"node","lean":"MetaMathlibExt.sum_sum_stirlingSecond_signedStirlingFirst_eq_ite","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#sum-sum-stirling-second-signed-stirling","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling numbers of the second kind and $s(n, k)$ for the signed Stirling numbers of the first kind. For all natural numbers $i, j, n$ with $i \\ge 1$, $n \\ge i$ and $n \\ge j$, in the rational numbers,","title":"Double-sum Kronecker delta identity for Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/sum-stirling-second-succ-eq-ite","kind":"node","lean":"MetaMathlibExt.sum_stirlingSecond_succ_eq_ite","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#sum-stirling-second-succ-eq-ite","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling numbers of the second kind. For all natural numbers $i \\ge 1$ and $j$, the rational number","title":"Stirling numbers of the second kind from the next row"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/sum-sum-stirling-second-signed-stirling-2","kind":"node","lean":"MetaMathlibExt.sum_sum_stirlingSecond_signedStirlingFirst_stirlingSecond","parent":25234,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices/index.html#sum-sum-stirling-second-signed-stirling-2","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-matrices","status":"can_prove","summary":"Write $S(n, k)$ for the Stirling numbers of the second kind and $s(n, k)$ for the signed Stirling numbers of the first kind. For all natural numbers $i, j, n, m_1, m_2$ with $i \\ge 1$, $n \\ge i$ and $n \\ge j$,","title":"Double-sum shift identity for Stirling numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind","kind":"scope","navigation":"module","parent":25195,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind","status":"fully_proved","title":"Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind/stirling-second-addition","kind":"node","lean":"MetaMathlibExt.stirlingSecond_addition","parent":25265,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind/index.html#stirling-second-addition","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind","status":"fully_proved","summary":"Write $S(n,k)$ for the Stirling numbers of the second kind. For all natural numbers $u$, $v$ and $k$ with $u > 0$,","title":"Addition formula for Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind/stirling-second-mul-factorial-eq-sum","kind":"node","lean":"MetaMathlibExt.stirlingSecond_mul_factorial_eq_sum","parent":25265,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind/index.html#stirling-second-mul-factorial-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind","status":"fully_proved","summary":"Write $S(t,k)$ for the Stirling numbers of the second kind. For all natural numbers $t$ and $k$,","title":"Explicit formula for Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind/has-sum-stirling-second-succ-egf","kind":"node","lean":"MetaMathlibExt.hasSum_stirlingSecond_succ_egf","parent":25265,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind/index.html#has-sum-stirling-second-succ-egf","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/stirling-numbers-of-the-second-kind","status":"fully_proved","summary":"Write $S(n,k)$ for the Stirling numbers of the second kind. For every natural number $j$ and every real number $x$, the series","title":"Exponential generating function of a shifted Stirling column"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair","kind":"scope","navigation":"module","parent":25195,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair","status":"fully_proved","title":"Laguerre\u2013Stirling\u2013Bell connection pair"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair/stirling-bell-polynomial-connection-pair","kind":"node","lean":"MetaMathlibExt.stirling_bell_polynomial_connection_pair_general","parent":25269,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair/index.html#stirling-bell-polynomial-connection-pair","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair","status":"fully_proved","summary":"Let $\\alpha$, $\\beta$ and $x$ be real numbers, and let $n$ and $r$ be natural numbers. Write $\\binom{k}{j}$ for the binomial coefficients, $c(n,k)$ for the unsigned Stirling numbers of the first kind, $S(n,k)$ for the Stirling numbers of the second kind, and $y^{\\overline{m}} = y(y+1)\\cdots(y+m-1)$ for the rising factorial. For natural numbers $m$, $s$ and real $y$, let $B_m(y) = \\sum_{l=0}^{m} S(m,l)\\, y^l$ be the\u2026","title":"Laguerre\u2013Stirling\u2013Bell connection pair"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair/stirling-bell-polynomial-connection-pair-2","kind":"node","lean":"MetaMathlibExt.stirling_bell_polynomial_connection_pair","parent":25269,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair/index.html#stirling-bell-polynomial-connection-pair-2","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/laguerre-stirling-bell-connection-pair","status":"fully_proved","summary":"Let $\\alpha$, $\\beta$ and $x$ be real numbers with $\\beta \\neq 0$, and let $n$ and $r$ be natural numbers. Write $\\binom{k}{j}$ for the binomial coefficients, $c(n,k)$ for the unsigned Stirling numbers of the first kind, $S(n,k)$ for the Stirling numbers of the second kind, and $y^{\\overline{m}} = y(y+1)\\cdots(y+m-1)$ for the rising factorial. For natural numbers $m$, $s$ and real $y$, let\u2026","title":"Laguerre\u2013Stirling\u2013Bell connection pair, source form"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers.json","id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers","kind":"scope","navigation":"module","parent":25195,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers","status":"can_prove","title":"Triple products of Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/stirling-triple-ratio-bounds-choose","kind":"node","lean":"MetaMathlibExt.stirling_triple_ratio_bounds_choose","parent":25272,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/index.html#stirling-triple-ratio-bounds-choose","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers","status":"can_prove","summary":"Let $n$, $p$ and $j$ be positive integers with $p \\le n$. Write $S(m, i)$ for the Stirling number of the second kind (the number of partitions of an $m$-element set into $i$ nonempty blocks) and $c(m, i)$ for the unsigned Stirling number of the first kind (the number of permutations of $m$ elements with exactly $i$ cycles). Then, as rational numbers,\u2026","title":"Triple products of Stirling numbers bracket the binomial coefficient"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/mul-stirling-second-sub-gt","kind":"node","lean":"MetaMathlibExt.mul_stirlingSecond_sub_gt","parent":25272,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/index.html#mul-stirling-second-sub-gt","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers with $0 < k \\le n$, and write $S(m, i)$ for the Stirling number of the second kind, the number of partitions of an $m$-element set into $i$ nonempty blocks. Then $(n - 2k)\\,S(n + 1, n + 1 - k) < n\\,S(n, n - k)$, where the left side is computed in the integers, so that $n - 2k$ may be negative.","title":"Gy's inequality for Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/symm-triple-ratio-of-log-concave","kind":"node","lean":"MetaMathlibExt.symm_triple_ratio_of_logConcave","parent":25272,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/index.html#symm-triple-ratio-of-log-concave","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers","status":"can_prove","summary":"Let $n$, $p$ and $j$ be positive integers, at least two of which equal $1$, and let $x_1, x_2, \\dots$ be positive reals with $x_{k-1} x_{k+1} \\le x_k^2$ for all $k \\ge 2$. For $m \\ge 0$ write $e_i(m)$ and $h_i(m)$ for the elementary and the complete homogeneous symmetric polynomials of degree $i$ evaluated at $(x_1, \\dots, x_m)$, and put $N = n + p + j$. Then","title":"Gy's triple-ratio inequalities for log-concave sequences"},{"id":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/symm-triple-ratio-of-log-convex","kind":"node","lean":"MetaMathlibExt.symm_triple_ratio_of_logConvex","parent":25272,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers/index.html#symm-triple-ratio-of-log-convex","scope":"Combinatorics/stirling-bell-eulerian/stirling-lah-numbers/triple-products-of-stirling-numbers","status":"can_prove","summary":"Let $n$, $p$ and $j$ be positive integers, at least two of which equal $1$, and let $x_1, x_2, \\dots$ be positive reals with $x_k^2 \\le x_{k-1} x_{k+1}$ for all $k \\ge 2$. For $m \\ge 0$ write $e_i(m)$ and $h_i(m)$ for the elementary and the complete homogeneous symmetric polynomials of degree $i$ evaluated at $(x_1, \\dots, x_m)$, and put $N = n + p + j$. Then","title":"Gy's triple-ratio inequalities for log-convex sequences"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers","kind":"scope","parent":25194,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers","status":"can_prove","summary":"Deformations and generalizations of Stirling numbers: r-Stirling, associated Stirling, Hsu\u2013Shiue, Jacobi\u2013Stirling, Legendre\u2013Stirling, meromorphic and degenerate Stirling numbers, \u03b1-Stirling and generalized Stirling matrices, and (q,r)-Whitney and Dowling numbers.","title":"Generalized Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/generalized-stirling-first","kind":"node","lean":"MetaMathlibExt.generalizedStirlingFirst","parent":25277,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/index.html#generalized-stirling-first","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers","status":"fully_proved","summary":"For natural numbers $n$, $m$ and a real number $x$, the generalized Stirling number of the first kind is","title":"Generalized Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/generalized-stirling-second","kind":"node","lean":"MetaMathlibExt.generalizedStirlingSecond","parent":25277,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/index.html#generalized-stirling-second","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers","status":"fully_proved","summary":"For natural numbers $n$, $m$ and a real number $x$, the generalized Stirling number of the second kind is","title":"Generalized Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/generalized-stirling-first-eq-sum","kind":"node","lean":"MetaMathlibExt.generalizedStirlingFirst_eq_sum","parent":25277,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/index.html#generalized-stirling-first-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers","status":"fully_proved","summary":"For natural numbers $m \\le n$ and every real number $x$, the generalized Stirling number of the first kind satisfies","title":"Sum formula for generalized Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/generalized-stirling-second-eq-sum","kind":"node","lean":"MetaMathlibExt.generalizedStirlingSecond_eq_sum","parent":25277,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/index.html#generalized-stirling-second-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers","status":"fully_proved","summary":"For natural numbers $m \\le n$ and every real number $x$, the generalized Stirling number of the second kind satisfies","title":"Sum formula for generalized Stirling numbers of the second kind"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","title":"(q, r)-Whitney and Dowling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/q-brack","kind":"node","lean":"MetaMathlibExt.QRWhitney.qBrack","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#q-brack","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For a complex number $q$ and a natural number $n$, the q-integer $[n]_q$ is the finite geometric sum $[n]_q = \\sum_{i=0}^{n-1} q^i$.","title":"Complex q-integers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-first","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneyFirst","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-first","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $m, r, q$ be complex numbers with $q \\ne 0$, and for a natural number $j$ let $[j]_q = \\sum_{i=0}^{j-1} q^i$ be the q-integer. The (q, r)-Whitney numbers of the first kind $w(n, k) = w_{m,r,q}(n, k)$, for natural numbers $n$ and $k$, are defined by $w(0, 0) = 1$, $w(0, k + 1) = 0$ and, for all $n$ and $k$,","title":"(q, r)-Whitney numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-second","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneySecond","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-second","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For complex numbers $m$, $r$, $q$, the $(q, r)$-Whitney numbers of the second kind $W_{m,r,q}(n, k)$, indexed by natural numbers $n$ and $k$, are defined by $W_{m,r,q}(0, 0) = 1$, $W_{m,r,q}(0, k + 1) = 0$, $W_{m,r,q}(n + 1, 0) = r \\, W_{m,r,q}(n, 0)$ and","title":"(q, r)-Whitney numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-first-zero-zero","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneyFirst_zero_zero","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-first-zero-zero","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For all complex numbers $m, r, q$ with $q \\ne 0$, the (q, r)-Whitney number of the first kind satisfies $w_{m,r,q}(0, 0) = 1$.","title":"Initial value of the first-kind (q, r)-Whitney numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-first-zero-succ","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneyFirst_zero_succ","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-first-zero-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For all complex numbers $m, r, q$ with $q \\ne 0$ and every natural number $k$, the (q, r)-Whitney number of the first kind satisfies $w_{m,r,q}(0, k + 1) = 0$.","title":"Row zero of the first-kind (q, r)-Whitney numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-first-succ-zero","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneyFirst_succ_zero","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-first-succ-zero","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $m, r, q$ be complex numbers with $q \\ne 0$, write $[j]_q$ for the q-integer of $j$, and write $w(n, k)$ for the (q, r)-Whitney number of the first kind $w_{m,r,q}(n, k)$. Then for every natural number $n$,","title":"Column-zero recurrence for first-kind (q, r)-Whitney numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-first-succ-succ","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneyFirst_succ_succ","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-first-succ-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $m, r, q$ be complex numbers with $q \\ne 0$, write $[j]_q$ for the q-integer of $j$, and write $w(n, k)$ for the (q, r)-Whitney number of the first kind $w_{m,r,q}(n, k)$. Then for all natural numbers $n$ and $k$,","title":"Triangular recurrence for first-kind (q, r)-Whitney numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-second-zero-zero","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneySecond_zero_zero","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-second-zero-zero","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For complex numbers $m$, $r$, $q$, let $W_{m,r,q}(i, j)$, for natural numbers $i$ and $j$, denote the $(q, r)$-Whitney numbers of the second kind. Then $W_{m,r,q}(0, 0) = 1$.","title":"Initial value of the second-kind (q, r)-Whitney numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-second-zero-succ","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneySecond_zero_succ","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-second-zero-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For complex numbers $m$, $r$, $q$, let $W_{m,r,q}(i, j)$, for natural numbers $i$ and $j$, denote the $(q, r)$-Whitney numbers of the second kind. Then $W_{m,r,q}(0, k + 1) = 0$ for every natural number $k$.","title":"Row zero of the second-kind (q, r)-Whitney numbers vanishes off the diagonal"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-second-succ-zero","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneySecond_succ_zero","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-second-succ-zero","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For complex numbers $m$, $r$, $q$, let $W_{m,r,q}(i, j)$, for natural numbers $i$ and $j$, denote the $(q, r)$-Whitney numbers of the second kind. Then for every natural number $n$, $W_{m,r,q}(n + 1, 0) = r \\, W_{m,r,q}(n, 0)$.","title":"Column-zero recurrence for second-kind (q, r)-Whitney numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-second-succ-succ","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneySecond_succ_succ","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-second-succ-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For all complex numbers $m$, $r$, $q$ and all natural numbers $n$ and $k$, the $(q, r)$-Whitney numbers of the second kind $W_{m,r,q}(n, k)$ satisfy","title":"Triangular recurrence for second-kind (q, r)-Whitney numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-first-eq-zero-of-lt","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneyFirst_eq_zero_of_lt","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-first-eq-zero-of-lt","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For all complex numbers $m, r, q$ with $q \\ne 0$ and all natural numbers $n$ and $k$ with $n < k$, the (q, r)-Whitney number of the first kind satisfies $w_{m,r,q}(n, k) = 0$.","title":"First-kind (q, r)-Whitney numbers vanish above the diagonal"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-second-eq-zero-of-lt","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneySecond_eq_zero_of_lt","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-second-eq-zero-of-lt","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"For all complex numbers $m, r, q$ and all natural numbers $n$ and $k$ with $n < k$, the (q, r)-Whitney number of the second kind satisfies $W_{m,r,q}(n, k) = 0$.","title":"Second-kind (q, r)-Whitney numbers vanish above the diagonal"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-first-succ-eq","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneyFirst_succ_eq","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-first-succ-eq","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $m, r, q$ be complex numbers with $q \\ne 0$, write $[j]_q$ for the q-integer of $j$, and write $w(n, k)$ for the (q, r)-Whitney number of the first kind $w_{m,r,q}(n, k)$. Then for all natural numbers $n$ and $k$ with $k \\ge 1$,","title":"Triangular recurrence for first-kind (q, r)-Whitney numbers, positive column form"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-second-succ-eq","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneySecond_succ_eq","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-second-succ-eq","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $m, r, q$ be complex numbers, write $[j]_q$ for the q-integer of $j$, and write $W(n, k)$ for the (q, r)-Whitney number of the second kind $W_{m,r,q}(n, k)$. Then for all natural numbers $n$ and $k$ with $k \\ge 1$,","title":"Triangular recurrence for second-kind (q, r)-Whitney numbers, positive column form"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/qr-dowling-polynomial","kind":"node","lean":"MetaMathlibExt.qrDowlingPolynomial","parent":25282,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((q, r)\\)"],["text","-Dowling polynomial"]],"source":"$(q, r)$-Dowling polynomial"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#qr-dowling-polynomial","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $R$ be a semiring and let $W(n, k) \\in R$ for all natural numbers $n$ and $k$ (intended to be the $(q, r)$-Whitney numbers of the second kind, but any family is allowed). For a natural number $n$, the $(q, r)$-Dowling polynomial of $W$ at $n$ is the polynomial $\\sum_{k=0}^{n} W(n, k)\\, x^k$ in $R[x]$.","title":"$(q, r)$-Dowling polynomial"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/qr-dowling-number","kind":"node","lean":"MetaMathlibExt.qrDowlingNumber","parent":25282,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((q, r)\\)"],["text","-Dowling number"]],"source":"$(q, r)$-Dowling number"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#qr-dowling-number","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $R$ be a semiring and let $W(n, k) \\in R$ for all natural numbers $n$ and $k$. For a natural number $n$, the $(q, r)$-Dowling number of $W$ at $n$ is the value at $x = 1$ of the $(q, r)$-Dowling polynomial $\\sum_{k=0}^{n} W(n, k)\\, x^k$ of $W$.","title":"$(q, r)$-Dowling number"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/qr-dowling-number-eq-sum","kind":"node","lean":"MetaMathlibExt.qrDowlingNumber_eq_sum","parent":25282,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((q, r)\\)"],["text","-Dowling numbers as sums of coefficients"]],"source":"$(q, r)$-Dowling numbers as sums of coefficients"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#qr-dowling-number-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $R$ be a semiring and let $W(n, k) \\in R$ for all natural numbers $n$ and $k$. For every natural number $n$, the $(q, r)$-Dowling number of $W$ at $n$ equals $\\sum_{k=0}^{n} W(n, k)$.","title":"$(q, r)$-Dowling numbers as sums of coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/whitney-second-eq-sum-q-stirling-second","kind":"node","lean":"MetaMathlibExt.QRWhitney.whitneySecond_eq_sum_qStirlingSecond","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#whitney-second-eq-sum-q-stirling-second","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $m, r, q \\in \\mathbb{C}$ and write $[k + 1]_q = 1 + q + \\dots + q^k$. Let $S$ be a complex-valued function on pairs of natural numbers satisfying the recurrence of the $q$-Stirling numbers of the second kind: for all $a, k \\in \\mathbb{N}$, $S(0, 0) = 1$, $S(0, k + 1) = 0$, $S(a + 1, 0) = 0$ and $S(a + 1, k + 1) = q^k S(a, k) + [k + 1]_q \\, S(a, k + 1)$. Then for all $a, k \\in \\mathbb{N}$ the $(q, r)$-Whitney\u2026","title":"(q, r)-Whitney numbers of the second kind through q-Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/eval-qr-dowling-polynomial-whitney-secon","kind":"node","lean":"MetaMathlibExt.eval_qrDowlingPolynomial_whitneySecond","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#eval-qr-dowling-polynomial-whitney-secon","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $m, r, q, x \\in \\mathbb{C}$ and $n \\in \\mathbb{N}$, and write $[k + 1]_q = 1 + q + \\dots + q^k$. Let $S$ be a complex-valued function on pairs of natural numbers satisfying the recurrence of the $q$-Stirling numbers of the second kind: for all $a, k \\in \\mathbb{N}$, $S(0, 0) = 1$, $S(0, k + 1) = 0$, $S(a + 1, 0) = 0$ and $S(a + 1, k + 1) = q^k S(a, k) + [k + 1]_q \\, S(a, k + 1)$. Let $W_{m,r,q}$ be the\u2026","title":"Expansion of the (q, r)-Dowling polynomial in q-Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/q-r-dowling-polynomial-q-stirling-expans","kind":"node","lean":"MetaMathlibExt.q_r_dowling_polynomial_q_stirling_expansion","parent":25282,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers/index.html#q-r-dowling-polynomial-q-stirling-expans","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/q-r-whitney-and-dowling-numbers","status":"fully_proved","summary":"Let $q, m, r, x \\in \\mathbb{C}$ and $n \\in \\mathbb{N}$, and write $[k + 1]_q = \\sum_{i=0}^{k} q^i$. Let $S$ and $W$ be complex-valued functions on pairs of natural numbers, and let $D(a, y)$ be a complex-valued function of $a \\in \\mathbb{N}$ and $y \\in \\mathbb{C}$, such that for all $a, k \\in \\mathbb{N}$ and $y \\in \\mathbb{C}$: $S(0, 0) = 1$, $S(0, k + 1) = 0$, $S(a + 1, 0) = 0$ and\u2026","title":"(q, r)-Dowling polynomial expansion from the defining recurrences"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind","status":"fully_proved","title":"Associated Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind/two-assoc-stirling-second","kind":"node","lean":"MetaMathlibExt.twoAssocStirlingSecond","parent":25304,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind/index.html#two-assoc-stirling-second","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the 2-associated Stirling number of the second kind is the integer $S_2(n, k) = \\sum_{j=0}^{k} (-1)^j \\binom{n}{j} S(n - j, k - j)$, where $S$ denotes the Stirling numbers of the second kind.","title":"2-associated Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind/two-assoc-stirling-second-recurrence","kind":"node","lean":"MetaMathlibExt.twoAssocStirlingSecond_recurrence","parent":25304,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind/index.html#two-assoc-stirling-second-recurrence","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind","status":"fully_proved","summary":"Write $S_2(n, k)$ for the 2-associated Stirling number of the second kind. For natural numbers $m$ and $k$ with $1 \\le k \\le m$, we have $S_2(m + 2, k) = k \\, S_2(m + 1, k) + (m + 1) \\, S_2(m, k - 1)$.","title":"Recurrence for 2-associated Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind/two-assoc-stirling-second-eq-zero-of-lt","kind":"node","lean":"MetaMathlibExt.twoAssocStirlingSecond_eq_zero_of_lt_two_mul","parent":25304,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind/index.html#two-assoc-stirling-second-eq-zero-of-lt","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/associated-stirling-numbers-of-the-second-kind","status":"fully_proved","summary":"For natural numbers $n$ and $k$ with $n < 2k$, the 2-associated Stirling number of the second kind $S_2(n, k)$ equals $0$.","title":"Vanishing of 2-associated Stirling numbers of the second kind"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers","status":"fully_proved","title":"Graphical Stirling and Lah numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers/truncated-succ","kind":"node","lean":"MetaMathlibExt.TruncatedSucc","parent":25308,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers/index.html#truncated-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers","status":"fully_proved","summary":"Let $\\sigma$ be a permutation of a finite linearly ordered set $V$ and let $i, j \\in V$. Then $j$ is the truncated-cycle successor $\\sigma^{(i)}(i)$ of $i$ when $j$ is the first of the iterates $\\sigma(i), \\sigma^2(i), \\dots$ that is at most $i$: there is an $m > 0$ with $\\sigma^m(i) = j$ and $j \\le i$, and $\\sigma^{m'}(i) > i$ for every $m'$ with $0 < m' < m$.","title":"Truncated-cycle successor"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers/is-kcyclic-partition","kind":"node","lean":"MetaMathlibExt.IsKCyclicPartition","parent":25308,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-cyclic partition"]],"source":"$k$-cyclic partition"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers/index.html#is-kcyclic-partition","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers","status":"fully_proved","summary":"Let $V$ be a finite linearly ordered set, $\\sigma$ a permutation of $V$, $E$ a relation on $V$ (the directed edges of a labeled graph) and $k$ a natural number. Then $\\sigma$ is a $k$-cyclic partition for $E$ when $\\sigma$ has exactly $k$ cycles, fixed points counted as cycles of length one, and for all $i, j \\in V$, if $j$ is the truncated-cycle successor of $i$ under $\\sigma$ (the first of\u2026","title":"$k$-cyclic partition"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers/linear-partition","kind":"node","lean":"MetaMathlibExt.LinearPartition","parent":25308,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers/index.html#linear-partition","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/graphical-stirling-and-lah-numbers","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex type $V$ and let $k$ be a natural number. A $k$-linear partition of $G$ is a set of exactly $k$ blocks, each block a nonempty finite sequence of pairwise distinct vertices, such that every vertex of $G$ lies in exactly one block and any two consecutive vertices of a block are nonadjacent in $G$. The blocks themselves are unordered; only the vertices within a block are ordered.","title":"Linear partition of a graph"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers","status":"fully_proved","title":"Hsu\u2013Shiue generalized Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers/generalized-stirling-matrix-factorizatio","kind":"node","lean":"MetaMathlibExt.generalized_stirling_matrix_factorization","parent":25312,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers/index.html#generalized-stirling-matrix-factorizatio","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers","status":"fully_proved","summary":"Let $S$ assign to real numbers $a, b, c$ and natural numbers $n, k$ a real number $S(a, b, c; n, k)$, such that for all real $a, b, c, x$ and every natural number $n$","title":"Factorization of the generalized Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers/generalized-falling-factorial","kind":"node","lean":"MetaMathlibExt.generalizedFallingFactorial","parent":25312,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers/index.html#generalized-falling-factorial","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers","status":"fully_proved","summary":"Let $K$ be $\\mathbb{R}$ or $\\mathbb{C}$. For $z, \\alpha \\in K$ the generalized falling factorial with increment $\\alpha$ is the sequence defined by $(z \\mid \\alpha)_0 = 1$ and $(z \\mid \\alpha)_{n+1} = (z \\mid \\alpha)_n \\, (z - n\\alpha)$ for $n \\ge 0$, so that $(z \\mid \\alpha)_n = z (z - \\alpha) \\cdots (z - (n-1)\\alpha)$ for $n \\ge 1$.","title":"Generalized falling factorial"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers/is-hsu-shiue-family","kind":"node","lean":"MetaMathlibExt.IsHsuShiueFamily","parent":25312,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers/index.html#is-hsu-shiue-family","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/hsu-shiue-generalized-stirling-numbers","status":"fully_proved","summary":"Let $K$ be $\\mathbb{R}$ or $\\mathbb{C}$ and let $\\alpha, \\beta, r \\in K$. For $z \\in K$ write $(z \\mid \\alpha)_0 = 1$ and $(z \\mid \\alpha)_n = z (z - \\alpha) \\cdots (z - (n-1)\\alpha)$ for $n \\ge 1$ (the generalized falling factorial). An array $(S(n,k))_{n, k \\ge 0}$ in $K$ is a Hsu\u2013Shiue family with parameters $(\\alpha, \\beta, r)$ when $\\alpha$, $\\beta$ and $r$ are not all zero, $S(n, 0) = (r \\mid \\alpha)_n$ for\u2026","title":"Characterization of the Hsu\u2013Shiue generalized Stirling numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","title":"Jacobi\u2013Stirling and Legendre\u2013Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/jacobi-stirling-first-poly","kind":"node","lean":"MetaMathlibExt.jacobiStirlingFirstPoly","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#jacobi-stirling-first-poly","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For a commutative ring $R$, an element $\\zeta \\in R$ and a natural number $n$, the $n$-th first-kind Jacobi\u2013Stirling generating polynomial is $\\prod_{i=0}^{n-1} (X - i(i + \\zeta))$ in $R[X]$.","title":"Generating polynomial of the first-kind Jacobi\u2013Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/jacobi-stirling-first","kind":"node","lean":"MetaMathlibExt.jacobiStirlingFirst","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#jacobi-stirling-first","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For a commutative ring $R$, an element $\\zeta \\in R$ and natural numbers $n$ and $k$, the first-kind Jacobi\u2013Stirling number $J(n,k;\\zeta)$ is the coefficient of $X^k$ in the polynomial $\\prod_{i=0}^{n-1} (X - i(i + \\zeta))$ over $R$.","title":"First-kind Jacobi\u2013Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/jacobi-stirling-first-initial-row","kind":"node","lean":"MetaMathlibExt.jacobiStirlingFirst_initial_row","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#jacobi-stirling-first-initial-row","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For every commutative ring $R$, every $\\zeta \\in R$ and every natural number $k$, the first-kind Jacobi\u2013Stirling number $J(0,k;\\zeta)$ equals $1$ if $k = 0$ and $0$ otherwise.","title":"Initial row of the first-kind Jacobi\u2013Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/jacobi-stirling-first-recurrence","kind":"node","lean":"MetaMathlibExt.jacobiStirlingFirst_recurrence","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#jacobi-stirling-first-recurrence","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For every commutative ring $R$, every $\\zeta \\in R$ and all natural numbers $n$ and $k$, the first-kind Jacobi\u2013Stirling numbers satisfy $J(n+1,k+1;\\zeta) = J(n,k;\\zeta) - n(n + \\zeta)\\,J(n,k+1;\\zeta)$.","title":"Recurrence for the first-kind Jacobi\u2013Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/jacobi-stirling-first-support","kind":"node","lean":"MetaMathlibExt.jacobiStirlingFirst_support","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#jacobi-stirling-first-support","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For every commutative ring $R$, every $\\zeta \\in R$ and all natural numbers $n < k$, the first-kind Jacobi\u2013Stirling number $J(n,k;\\zeta)$ equals $0$.","title":"First-kind Jacobi\u2013Stirling numbers vanish above the diagonal"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/jacobi-stirling-first-diagonal","kind":"node","lean":"MetaMathlibExt.jacobiStirlingFirst_diagonal","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#jacobi-stirling-first-diagonal","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For every commutative ring $R$, every $\\zeta \\in R$ and every natural number $n$, the first-kind Jacobi\u2013Stirling number $J(n,n;\\zeta)$ equals $1$.","title":"Diagonal of the first-kind Jacobi\u2013Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/legendre-stirling-first-poly","kind":"node","lean":"MetaMathlibExt.legendreStirlingFirstPoly","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#legendre-stirling-first-poly","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For a natural number $n$, this is the polynomial $\\prod_{i=0}^{n-1} \\bigl(X - i(i+1)\\bigr)$ with integer coefficients; for $n = 0$ it is the empty product $1$.","title":"Generating polynomial of the Legendre\u2013Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/legendre-stirling-first","kind":"node","lean":"MetaMathlibExt.legendreStirlingFirst","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#legendre-stirling-first","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the Legendre\u2013Stirling number of the first kind $J(n, k; 1)$ is the coefficient of $x^k$ in the integer polynomial $\\prod_{i=0}^{n-1} \\bigl(x - i(i+1)\\bigr)$, which is $1$ when $n = 0$. These are the Jacobi\u2013Stirling numbers of the first kind $J(n, k; \\zeta)$ at $\\zeta = 1$.","title":"Legendre\u2013Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/legendre-stirling-first-zero","kind":"node","lean":"MetaMathlibExt.legendreStirlingFirst_zero","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#legendre-stirling-first-zero","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For every natural number $k$, the Legendre\u2013Stirling number of the first kind $J(0, k; 1)$ equals $1$ if $k = 0$ and $0$ otherwise.","title":"Initial row of the Legendre\u2013Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/legendre-stirling-first-succ","kind":"node","lean":"MetaMathlibExt.legendreStirlingFirst_succ","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#legendre-stirling-first-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the Legendre\u2013Stirling numbers of the first kind satisfy $J(n + 1, k + 1; 1) = J(n, k; 1) - n(n + 1) \\, J(n, k + 1; 1)$.","title":"Recurrence for the Legendre\u2013Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/legendre-stirling-first-eq-zero-of-lt","kind":"node","lean":"MetaMathlibExt.legendreStirlingFirst_eq_zero_of_lt","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#legendre-stirling-first-eq-zero-of-lt","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For all natural numbers $n < k$, the Legendre\u2013Stirling number of the first kind $J(n, k; 1)$ is $0$.","title":"Legendre\u2013Stirling numbers of the first kind vanish above the diagonal"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/legendre-stirling-first-diag","kind":"node","lean":"MetaMathlibExt.legendreStirlingFirst_diag","parent":25316,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers/index.html#legendre-stirling-first-diag","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/jacobi-stirling-and-legendre-stirling-numbers","status":"fully_proved","summary":"For every natural number $n$, the Legendre\u2013Stirling number of the first kind $J(n, n; 1)$ equals $1$.","title":"Diagonal Legendre\u2013Stirling numbers of the first kind"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers","status":"fully_proved","title":"Meromorphic Stirling numbers and Bessel numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/meromorphic-stirling-first","kind":"node","lean":"MetaMathlibExt.meromorphicStirlingFirst","parent":25329,"reading_title":{"format":"inline-parts-v1","parts":[["text","The meromorphic Stirling numbers "],["math","\\(S_{2;-1}(n,k)\\)"]],"source":"The meromorphic Stirling numbers $S_{2;-1}(n,k)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/index.html#meromorphic-stirling-first","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$ with $1 \\le k \\le n$, the meromorphic Stirling number $S_{2;-1}(n,k)$ is the rational number","title":"The meromorphic Stirling numbers $S_{2;-1}(n,k)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/meromorphic-stirling-second","kind":"node","lean":"MetaMathlibExt.meromorphicStirlingSecond","parent":25329,"reading_title":{"format":"inline-parts-v1","parts":[["text","The meromorphic Stirling numbers "],["math","\\(S_{-1;1}(n,k)\\)"]],"source":"The meromorphic Stirling numbers $S_{-1;1}(n,k)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/index.html#meromorphic-stirling-second","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$ with $k \\le n \\le 2k$, the meromorphic Stirling number $S_{-1;1}(n,k)$ is the rational number","title":"The meromorphic Stirling numbers $S_{-1;1}(n,k)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/bessel-number-first","kind":"node","lean":"MetaMathlibExt.besselNumberFirst","parent":25329,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/index.html#bessel-number-first","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$ with $1 \\le k \\le n$, the Bessel number of the first kind is the rational number","title":"Bessel numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/bessel-number-second","kind":"node","lean":"MetaMathlibExt.besselNumberSecond","parent":25329,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/index.html#bessel-number-second","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$ with $k \\le n \\le 2k$, the Bessel number of the second kind is the rational number","title":"Bessel numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/meromorphic-stirling-first-eq-bessel-num","kind":"node","lean":"MetaMathlibExt.meromorphicStirlingFirst_eq_besselNumberFirst","parent":25329,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/index.html#meromorphic-stirling-first-eq-bessel-num","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers","status":"fully_proved","summary":"For all natural numbers $n$ and $k$ with $1 \\le k \\le n$, the meromorphic Stirling number $S_{2;-1}(n,k)$ equals the Bessel number of the first kind $b(n,k)$.","title":"First identification of meromorphic Stirling and Bessel numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/meromorphic-stirling-second-eq-bessel-nu","kind":"node","lean":"MetaMathlibExt.meromorphicStirlingSecond_eq_besselNumberSecond","parent":25329,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers/index.html#meromorphic-stirling-second-eq-bessel-nu","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/meromorphic-stirling-numbers-and-bessel-numbers","status":"fully_proved","summary":"For all natural numbers $n$ and $k$ with $k \\le n \\le 2k$, the meromorphic Stirling number $S_{-1;1}(n,k)$ equals the Bessel number of the second kind $B(n,k)$.","title":"Second identification of meromorphic Stirling and Bessel numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol","status":"fully_proved","title":"Noncentral Whitney numbers and Tanny\u2013Dowling polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol/noncentral-whitney-second","kind":"node","lean":"MetaMathlibExt.noncentralWhitneySecond","parent":25336,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol/index.html#noncentral-whitney-second","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol","status":"fully_proved","summary":"For elements $m$ and $a$ of a commutative ring $R$, the noncentral Whitney numbers of the second kind $W_{m,a}(n, k) \\in R$, for natural numbers $n$ and $k$, are defined by $W_{m,a}(0, 0) = 1$, $W_{m,a}(0, k) = 0$ for $k \\ge 1$, $W_{m,a}(n + 1, 0) = (-a)^{n+1}$, and","title":"Noncentral Whitney numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol/noncentral-tanny-dowling-polynomial","kind":"node","lean":"MetaMathlibExt.noncentralTannyDowlingPolynomial","parent":25336,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol/index.html#noncentral-tanny-dowling-polynomial","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/noncentral-whitney-numbers-and-tanny-dowling-pol","status":"fully_proved","summary":"For elements $m$ and $a$ of a commutative ring $R$ and a natural number $n$, the noncentral Tanny\u2013Dowling polynomial $D_{m,a,n}(x) \\in R[x]$ is","title":"Noncentral Tanny\u2013Dowling polynomials"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","title":"r-Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-first","kind":"node","lean":"MetaMathlibExt.rStirlingFirst","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-first","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For natural numbers $n$, $m$ and $r$, the unsigned $r$-Stirling number of the first kind $[n, m]_r$ is the coefficient of $x^m$ in the polynomial with natural-number coefficients","title":"Unsigned r-Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-first-self","kind":"node","lean":"MetaMathlibExt.rStirlingFirst_self","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-first-self","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For all natural numbers $m$ and $r$, the unsigned $r$-Stirling number of the first kind $[r, m]_r$ equals $1$ if $m = r$ and $0$ otherwise: when the first index equals $r$, the product in the definition is empty, so the polynomial is $x^r$.","title":"Initial row of the r-Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-first-zero-of-pos","kind":"node","lean":"MetaMathlibExt.rStirlingFirst_zero_of_pos","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-first-zero-of-pos","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For all natural numbers $N$ and $r$ with $r \\ge 1$, the unsigned $r$-Stirling number of the first kind $[N, 0]_r$ is $0$: the defining polynomial is divisible by $x$, so its constant coefficient vanishes.","title":"Vanishing of the r-Stirling numbers of the first kind with second index zero"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-first-succ","kind":"node","lean":"MetaMathlibExt.rStirlingFirst_succ","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-first-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For natural numbers $n$, $m$ and $r$ with $1 \\le r \\le n$, the unsigned $r$-Stirling numbers of the first kind $[n, m]_r$ satisfy $[n+1, m]_r = [n, m-1]_r + n \\, [n, m]_r$, where for $m = 0$ the term $[n, m-1]_r$ is read as $[n, 0]_r$.","title":"Recurrence for the r-Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-first-eq-alternating-sum","kind":"node","lean":"MetaMathlibExt.rStirlingFirst_eq_alternating_sum","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-first-eq-alternating-sum","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"Let $n$, $m$ and $r$ be natural numbers with $1 \\le r \\le n$. Write $[n, m]_r$ for the unsigned $r$-Stirling number of the first kind, $[a, b]$ for the unsigned Stirling number of the first kind and $\\lbrace a, b \\rbrace$ for the Stirling number of the second kind. Then, in the integers,","title":"Broder's alternating expansion of the r-Stirling numbers of the first kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-second","kind":"node","lean":"MetaMathlibExt.rStirlingSecond","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-second","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For natural numbers $n$, $m$ and $r$, Broder's r-Stirling number of the second kind ${n \\brace m}_r$ is $0$ unless $r \\le m$ and $r \\le n$, and otherwise","title":"r-Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-second-add-add","kind":"node","lean":"MetaMathlibExt.rStirlingSecond_add_add","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-second-add-add","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For all natural numbers $n$, $m$ and $r$, the r-Stirling number of the second kind ${n+r \\brace m+r}_r$ satisfies","title":"Finite-sum formula for r-Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-second-eq-zero","kind":"node","lean":"MetaMathlibExt.rStirlingSecond_eq_zero","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-second-eq-zero","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For natural numbers $n$, $m$ and $r$ with $m < r$ or $n < r$, the r-Stirling number of the second kind ${n \\brace m}_r$ is $0$.","title":"Vanishing of r-Stirling numbers of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-second-zero-right","kind":"node","lean":"MetaMathlibExt.rStirlingSecond_zero_right","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-second-zero-right","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For all natural numbers $n$ and $m$, the r-Stirling number of the second kind ${n \\brace m}_0$ equals the Stirling number of the second kind ${n \\brace m}$.","title":"r-Stirling numbers of the second kind for r = 0"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-second-add-self","kind":"node","lean":"MetaMathlibExt.rStirlingSecond_add_self","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-second-add-self","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For all natural numbers $n$ and $r$, the r-Stirling number of the second kind ${n+r \\brace r}_r$ equals $r^n$ (with $0^0 = 1$).","title":"r-Stirling numbers of the second kind with exactly r blocks"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/r-stirling-second-add-add-eq-generalized","kind":"node","lean":"MetaMathlibExt.rStirlingSecond_add_add_eq_generalizedStirlingSecond","parent":25339,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers/index.html#r-stirling-second-add-add-eq-generalized","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/r-stirling-numbers","status":"fully_proved","summary":"For all natural numbers $n$, $m$ and $r$, the r-Stirling number of the second kind ${n+r \\brace m+r}_r$, viewed as a real number, equals the generalized Stirling number of the second kind of $n$ and $m$ (after Carlitz and Broder) at the real parameter $x = r$.","title":"r-Stirling numbers as generalized Stirling numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers","status":"fully_proved","title":"Simsek numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/simsek-number","kind":"node","lean":"MetaMathlibExt.simsekNumber","parent":25351,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/index.html#simsek-number","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and a complex number $\\lambda$, the Simsek number $y_1(n, k; \\lambda)$ is the complex number","title":"Simsek numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/simsek-number-eq-sum","kind":"node","lean":"MetaMathlibExt.simsekNumber_eq_sum","parent":25351,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/index.html#simsek-number-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers","status":"fully_proved","summary":"For all natural numbers $n$ and $k$ and every complex number $\\lambda$, the Simsek number satisfies $y_1(n, k; \\lambda) = \\frac{1}{k!} \\sum_{j=0}^{k} \\binom{k}{j} j^n \\lambda^j$, where $0^0 = 1$.","title":"Finite-sum formula for Simsek numbers"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/simsek-number-zero-k","kind":"node","lean":"MetaMathlibExt.simsekNumber_zero_k","parent":25351,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/index.html#simsek-number-zero-k","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers","status":"fully_proved","summary":"For every natural number $n$ and every complex number $\\lambda$, the Simsek number $y_1(n, 0; \\lambda)$ equals $0^n$; that is, it equals $1$ when $n = 0$ and $0$ otherwise.","title":"Simsek numbers with second index zero"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/simsek-number-zero-zero","kind":"node","lean":"MetaMathlibExt.simsekNumber_zero_zero","parent":25351,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers/index.html#simsek-number-zero-zero","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/simsek-numbers","status":"fully_proved","summary":"For every complex number $\\lambda$, the Simsek number $y_1(0, 0; \\lambda)$ equals $1$.","title":"Simsek numbers with both indices zero"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo","status":"fully_proved","title":"Stirling expansions of the Akiyama\u2013Tanigawa algorithm"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo/chen-stirling-second-expansion","kind":"node","lean":"MetaMathlibExt.chen_stirlingSecond_expansion","parent":25356,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo/index.html#chen-stirling-second-expansion","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo","status":"fully_proved","summary":"Let $a(m, s)$, for natural numbers $m$ and $s$, be real numbers satisfying $a(m, s + 1) = m \\, a(m, s) - (m + 1) \\, a(m + 1, s)$ for all $m$ and $s$. Then for every natural number $t$,","title":"Chen's Stirling expansion"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo/poly-initial-expansion","kind":"node","lean":"MetaMathlibExt.poly_initial_expansion","parent":25356,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo/index.html#poly-initial-expansion","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-expansions-of-the-akiyama-tanigawa-algo","status":"fully_proved","summary":"For every natural number $n$, every real number $x$ and all real numbers $a(m, t)$, indexed by natural numbers $m$ and $t$, such that $a(m, t + 1) = m\\,a(m, t) - (m + 1)\\,a(m + 1, t)$ for all $m$ and $t$, writing ${n \\brace m}_x$ for the generalized Stirling numbers of the second kind,","title":"Expansion by generalized Stirling numbers of the second kind"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices.json","id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","kind":"scope","navigation":"module","parent":25277,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","title":"\u03b1-Stirling matrices"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-stirling-entry","kind":"node","lean":"MetaMathlibExt.alphaStirlingEntry","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-stirling-entry","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $\\alpha_0, \\alpha_1, \\alpha_2, \\dots$ be a sequence in $R$ ($\\alpha_0$ is never used). The \u03b1-Stirling entries $s_{i,j} \\in R$, for $i, j \\ge 0$, are defined recursively by $s_{i,0} = \\alpha_1$ for all $i$, $s_{0,j} = 0$ for $j \\ge 1$, and $s_{i+1,j+1} = s_{i,j} + \\alpha_{j+2} s_{i,j+1}$ for all $i, j \\ge 0$.","title":"Entries of the \u03b1-Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/shifted-alpha-stirling-entry","kind":"node","lean":"MetaMathlibExt.shiftedAlphaStirlingEntry","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#shifted-alpha-stirling-entry","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $\\alpha = (\\alpha_j)_{j \\ge 1}$ be a sequence in $R$. The entries $\\mathsf{S}_{i,j} \\in R$ of the shifted $\\alpha$-Stirling matrix of $\\alpha$, for natural numbers $i$ and $j$ counted from $0$, are defined by $\\mathsf{S}_{i,0} = \\alpha_1$ for all $i$, $\\mathsf{S}_{0,j} = \\alpha_1$ for all $j$, and","title":"Entries of the shifted \u03b1-Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-upper-entry","kind":"node","lean":"MetaMathlibExt.alphaUpperEntry","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-upper-entry","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $\\alpha_0, \\alpha_1, \\alpha_2, \\dots$ be a sequence in $R$ ($\\alpha_0$ is never used). The \u03b1-upper entries $u_{i,j} \\in R$, for $i, j \\ge 0$, are defined recursively by $u_{0,j} = 1$ for all $j$, $u_{i,0} = 0$ for $i \\ge 1$, and $u_{i+1,j+1} = u_{i+1,j} + \\alpha_{i+1} u_{i,j} + (\\alpha_{j+2} - \\alpha_{i+1}) u_{i,j+1}$ for all $i, j \\ge 0$.","title":"Entries of the \u03b1-upper matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-stirling-entry-succ-succ","kind":"node","lean":"MetaMathlibExt.alphaStirlingEntry_succ_succ","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-stirling-entry-succ-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha$ be a sequence in $R$, and let $s_{i,j}$ ($i, j \\ge 0$) be the \u03b1-Stirling entries of $\\alpha$. Then for all $i, j \\ge 0$, $s_{i+1,j+1} = s_{i,j} + \\alpha_{j+2} s_{i,j+1}$.","title":"Recurrence for the \u03b1-Stirling entries"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-stirling-entry-zero-right","kind":"node","lean":"MetaMathlibExt.alphaStirlingEntry_zero_right","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-stirling-entry-zero-right","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha$ be a sequence in $R$, and let $s_{i,j}$ ($i, j \\ge 0$) be the \u03b1-Stirling entries of $\\alpha$. Then $s_{i,0} = \\alpha_1$ for every $i \\ge 0$.","title":"First column of the \u03b1-Stirling entries"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/shifted-alpha-stirling-entry-succ-succ","kind":"node","lean":"MetaMathlibExt.shiftedAlphaStirlingEntry_succ_succ","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#shifted-alpha-stirling-entry-succ-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha = (\\alpha_j)_{j \\ge 1}$ be a sequence in $R$, and write $\\mathsf{S}_{i,j}$ for the entries of the shifted $\\alpha$-Stirling matrix of $\\alpha$, indexed from $0$. For all natural numbers $i$ and $j$,","title":"Recurrence for the shifted \u03b1-Stirling entries"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/shifted-alpha-stirling-entry-zero-left","kind":"node","lean":"MetaMathlibExt.shiftedAlphaStirlingEntry_zero_left","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#shifted-alpha-stirling-entry-zero-left","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha = (\\alpha_j)_{j \\ge 1}$ be a sequence in $R$, and write $\\mathsf{S}_{i,j}$ for the entries of the shifted $\\alpha$-Stirling matrix of $\\alpha$, indexed from $0$. For every natural number $j$, $\\mathsf{S}_{0,j} = \\alpha_1$; that is, the first row is constant, equal to $\\alpha_1$.","title":"First row of the shifted \u03b1-Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-upper-entry-succ-succ","kind":"node","lean":"MetaMathlibExt.alphaUpperEntry_succ_succ","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-upper-entry-succ-succ","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha$ be a sequence in $R$, and let $u_{i,j}$ ($i, j \\ge 0$) be the \u03b1-upper entries of $\\alpha$. Then for all $i, j \\ge 0$, $u_{i+1,j+1} = u_{i+1,j} + \\alpha_{i+1} u_{i,j} + (\\alpha_{j+2} - \\alpha_{i+1}) u_{i,j+1}$.","title":"Recurrence for the \u03b1-upper entries"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-upper-entry-one-one-one-two","kind":"node","lean":"MetaMathlibExt.alphaUpperEntry_one_one_one_two","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-upper-entry-one-one-one-two","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha$ be a sequence in $R$, and let $u_{i,j}$ ($i, j \\ge 0$) be the \u03b1-upper entries of $\\alpha$. Then $u_{1,1} = \\alpha_2$ and $u_{1,2} = \\alpha_2 + \\alpha_3$.","title":"Small \u03b1-upper entries"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-stirling-matrix","kind":"node","lean":"MetaMathlibExt.alphaStirlingMatrix","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-stirling-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha$ be a sequence in $R$, and let $s_{i,j}$ ($i, j \\ge 0$) be the \u03b1-Stirling entries of $\\alpha$. For $n \\ge 0$, the matrix $S_\\alpha(n)$ is the $n \\times n$ matrix over $R$, with rows and columns indexed by $0, 1, \\dots, n - 1$, whose $(i, j)$ entry is $s_{i,j}$.","title":"The \u03b1-Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/shifted-alpha-stirling-matrix","kind":"node","lean":"MetaMathlibExt.shiftedAlphaStirlingMatrix","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#shifted-alpha-stirling-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha = (\\alpha_j)_{j \\ge 1}$ be a sequence in $R$, and let $n$ be a natural number. The shifted $\\alpha$-Stirling matrix $\\mathsf{S}_\\alpha(n)$ is the $n \\times n$ matrix over $R$ whose entry in row $i$ and column $j$ (counted from $0$) is the shifted $\\alpha$-Stirling entry $\\mathsf{S}_{i,j}$, given by $\\mathsf{S}_{i,0} = \\mathsf{S}_{0,j} = \\alpha_1$ and\u2026","title":"Shifted \u03b1-Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-upper-matrix","kind":"node","lean":"MetaMathlibExt.alphaUpperMatrix","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-upper-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $\\alpha$ be a sequence in $R$, and let $u_{i,j}$ ($i, j \\ge 0$) be the \u03b1-upper entries of $\\alpha$. For $n \\ge 0$, the matrix $\\mathsf{U}_\\alpha(n)$ is the $n \\times n$ matrix over $R$, with rows and columns indexed by $0, 1, \\dots, n - 1$, whose $(i, j)$ entry is $u_{i,j}$.","title":"The \u03b1-upper matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/alpha-upper-entry-eq-hsymm","kind":"node","lean":"MetaMathlibExt.alphaUpperEntry_eq_hsymm","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#alpha-upper-entry-eq-hsymm","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $R$ be a commutative ring, let $\\alpha$ be a sequence in $R$, and let $u_{i,j}$ ($i, j \\ge 0$) be the \u03b1-upper entries of $\\alpha$. For all $i \\le j$, $u_{i,j} = h_i(\\alpha_{i+1}, \\alpha_{i+2}, \\dots, \\alpha_{j+1})$, where $h_i$ is the complete homogeneous symmetric polynomial of degree $i$ in $j - i + 1$ variables.","title":"\u03b1-upper entries are complete homogeneous symmetric polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/det-alpha-upper-matrix","kind":"node","lean":"MetaMathlibExt.det_alphaUpperMatrix","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#det-alpha-upper-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $R$ be a commutative ring and let $\\alpha$ be a sequence in $R$. For every $n \\ge 0$, the determinant of the $n \\times n$ \u03b1-upper matrix $\\mathsf{U}_\\alpha(n)$ is $\\det \\mathsf{U}_\\alpha(n) = \\prod_{i=1}^{n} \\alpha_i^{i-1}$.","title":"Determinant of the \u03b1-upper matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/shifted-alpha-stirling-matrix-eq-mul","kind":"node","lean":"MetaMathlibExt.shiftedAlphaStirlingMatrix_eq_mul","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#shifted-alpha-stirling-matrix-eq-mul","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $R$ be a commutative ring, let $\\alpha = (\\alpha_j)_{j \\ge 1}$ be a sequence in $R$, and let $n$ be a natural number. Then","title":"LU decomposition of the shifted \u03b1-Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/det-shifted-alpha-stirling-matrix","kind":"node","lean":"MetaMathlibExt.det_shiftedAlphaStirlingMatrix","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Determinant of the shifted "],["math","\\(\\alpha\\)"],["text","-Stirling matrix"]],"source":"Determinant of the shifted $\\alpha$-Stirling matrix"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#det-shifted-alpha-stirling-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $R$ be a commutative ring and let $(\\alpha_j)_{j \\ge 1}$ be a sequence in $R$ with $\\alpha_1 = 1$. For $n \\ge 0$ let $\\mathsf{S}_\\alpha(n)$ be the $n \\times n$ matrix whose $(i, j)$ entry $\\mathsf{S}_{i,j}$, with rows and columns indexed from $1$, is $\\alpha_1$ when $i = 1$ or $j = 1$, and satisfies $\\mathsf{S}_{i,j} = \\mathsf{S}_{i,j-1} + \\alpha_j \\mathsf{S}_{i-1,j}$ for $i, j \\ge 2$. Then\u2026","title":"Determinant of the shifted $\\alpha$-Stirling matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/pascal-diagonal-matrix","kind":"node","lean":"MetaMathlibExt.pascalDiagonalMatrix","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#pascal-diagonal-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For $m, n \\ge 0$, $\\mathsf{L}_{d^{(m)}}(n)$ is the $n \\times n$ real matrix that carries the $m$-th diagonal $d^{(m)}_b = \\binom{m+b}{b}$ ($b \\ge 0$) of Pascal's triangle on its $m$-th subdiagonal: with rows and columns indexed from $0$, its $(b + m, b)$ entry is $\\binom{m+b}{b}$ for every $b$ with $b + m < n$, and all its other entries are $0$.","title":"Pascal diagonal matrices"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/pascal-diagonal-matrix-zero","kind":"node","lean":"MetaMathlibExt.pascalDiagonalMatrix_zero","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#pascal-diagonal-matrix-zero","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For $m, n \\ge 0$ let $\\mathsf{L}_{d^{(m)}}(n)$ be the $n \\times n$ real matrix whose $(b + m, b)$ entry is $\\binom{m+b}{b}$ for each $b$ with $b + m < n$ and whose other entries are $0$, with rows and columns indexed from $0$. Then $\\mathsf{L}_{d^{(0)}}(n) = I_n$, the $n \\times n$ identity matrix, for every $n \\ge 0$.","title":"The zeroth Pascal diagonal matrix is the identity"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/pascal-exp-first-column","kind":"node","lean":"MetaMathlibExt.pascalExpFirstColumn","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#pascal-exp-first-column","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For $k \\ge 0$, the sequence of natural numbers $\\alpha^{(k)} = (\\alpha^{(k)}_i)_{i \\ge 0}$ is defined by $\\alpha^{(k)}_0 = 0$, $\\alpha^{(k)}_1 = 1$ and, for $i \\ge 2$, $\\alpha^{(k)}_i = \\binom{i-2}{k-1} \\alpha^{(k)}_{i-k}$ if $1 \\le k < i$ and $\\alpha^{(k)}_i = 0$ otherwise. Thus $\\alpha^{(k)}_i = 0$ for $2 \\le i \\le k$, and $\\alpha^{(0)}_i = 0$ for all $i \\ne 1$.","title":"First-column sequence for exponentials of Pascal diagonal matrices"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/pascal-exp-first-column-one","kind":"node","lean":"MetaMathlibExt.pascalExpFirstColumn_one","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#pascal-exp-first-column-one","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For every $k \\ge 0$, the first-column sequence $\\alpha^{(k)}$ satisfies $\\alpha^{(k)}_1 = 1$.","title":"First term of the Pascal first-column sequence"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/pascal-exp-first-column-of-le","kind":"node","lean":"MetaMathlibExt.pascalExpFirstColumn_of_le","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#pascal-exp-first-column-of-le","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For all natural numbers $k$ and $i$ with $2 \\le i \\le k$, the first-column sequence $\\alpha^{(k)}$ satisfies $\\alpha^{(k)}_i = 0$.","title":"Vanishing terms of the Pascal first-column sequence"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/pascal-exp-first-column-of-lt","kind":"node","lean":"MetaMathlibExt.pascalExpFirstColumn_of_lt","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#pascal-exp-first-column-of-lt","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For all natural numbers $k$ and $i$ with $1 \\le k < i$, the first-column sequence $\\alpha^{(k)}$ satisfies $\\alpha^{(k)}_i = \\binom{i-2}{k-1} \\alpha^{(k)}_{i-k}$.","title":"Recurrence for the Pascal first-column sequence"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/exp-pascal-diagonal-matrix","kind":"node","lean":"MetaMathlibExt.exp_pascalDiagonalMatrix","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#exp-pascal-diagonal-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 0$. For $m \\ge 0$ let $\\mathsf{L}_{d^{(m)}}(n)$ be the $n \\times n$ real matrix whose $(b + m, b)$ entry is $\\binom{m+b}{b}$ for each $b$ with $b + m < n$ and whose other entries are $0$ (indices from $0$). Let $\\alpha^{(k)} = (\\alpha^{(k)}_i)_{i \\ge 1}$ be given by $\\alpha^{(k)}_1 = 1$, $\\alpha^{(k)}_i = 0$ for $2 \\le i \\le k$ and $\\alpha^{(k)}_i = \\binom{i-2}{k-1} \\alpha^{(k)}_{i-k}$ for\u2026","title":"Exponential of a Pascal diagonal matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/power-series-factorial-matrix","kind":"node","lean":"MetaMathlibExt.powerSeriesFactorialMatrix","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#power-series-factorial-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $K$ be a field, let $f$ be a formal power series over $K$ in the variable $t$, and let $n$ be a natural number. The factorial matrix $M_n(f)$ is the $n \\times n$ matrix over $K$ whose entry in row $a$ and column $b$ (counted from $0$) is $(a! \\cdot 1_K) / (b! \\cdot 1_K)$ times the coefficient of $t^{a-b}$ in $f$ when $b \\le a$, and $0$ when $b > a$. Here the quotient is taken in $K$ and is read as $0$ when\u2026","title":"Factorial matrix of a power series"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/power-series-factorial-matrix-apply-self","kind":"node","lean":"MetaMathlibExt.powerSeriesFactorialMatrix_apply_self","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#power-series-factorial-matrix-apply-self","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero, let $f$ be a formal power series over $K$ and let $n \\ge 0$. Every diagonal entry of the leading $n \\times n$ section of $M(f)$, the lower triangular matrix whose $(a, b)$ entry, with rows and columns indexed from $0$, is $\\frac{a!}{b!} [t^{a-b}] f$ for $b \\le a$ and $0$ for $b > a$, equals the constant coefficient $[t^0] f$ of $f$.","title":"Diagonal of the factorial matrix of a power series"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/power-series-factorial-matrix-mul","kind":"node","lean":"MetaMathlibExt.powerSeriesFactorialMatrix_mul","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#power-series-factorial-matrix-mul","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $K$ be a field of characteristic zero, let $f$ and $g$ be formal power series over $K$ in the variable $t$, and let $n$ be a natural number. For a power series $h$ over $K$, write $M_n(h)$ for the $n \\times n$ factorial matrix of $h$, whose entry in row $a$ and column $b$ (counted from $0$) is $\\frac{a!}{b!}$ times the coefficient of $t^{a-b}$ in $h$ when $b \\le a$, and $0$ otherwise. Then\u2026","title":"Multiplicativity of the factorial matrix of a power series"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/det-hessenberg-toeplitz-eq-recurrence","kind":"node","lean":"MetaMathlibExt.det_hessenbergToeplitz_eq_recurrence","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#det-hessenberg-toeplitz-eq-recurrence","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $R$ be a commutative ring, let $k \\ge 1$, and let $(c_j)_{j \\ge 1}$ and $(\\beta_m)_{m \\ge 1}$ be sequences in $R$ such that $\\beta_1 = \\cdots = \\beta_{k-1} = 0$, $\\beta_k = 1$ and $\\beta_m = \\sum_{j=1}^{k} c_j \\beta_{m-j}$ for every $m > k$. For $n \\ge 0$ let $H_n$ be the $n \\times n$ upper Hessenberg Toeplitz matrix whose $(i, j)$ entry is $c_{j-i+1}$ if $0 \\le j - i \\le k - 1$, is $-1$ if $i = j + 1$, and is\u2026","title":"Banded Hessenberg\u2013Toeplitz determinants and linear recurrences"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/stirling-like-entry","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.stirlingLikeEntry","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entries of the Stirling-like matrix "],["math","\\(S_\\alpha\\)"]],"source":"Entries of the Stirling-like matrix $S_\\alpha$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#stirling-like-entry","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals, the numbers $S_{i,j}$ for natural numbers $i, j$ are defined recursively by $S_{1,j} = \\alpha_{j-1}$ for $j \\ge 1$, $S_{i,1} = \\alpha_0$ for $i \\ge 2$, $S_{i,j} = S_{i-1,j-1} + j \\, S_{i-1,j}$ for $i, j \\ge 2$, and $S_{i,j} = 0$ when $i = 0$ or $j = 0$.","title":"Entries of the Stirling-like matrix $S_\\alpha$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/u-entry","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.uEntry","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entries of the matrix "],["math","\\(U_\\alpha\\)"]],"source":"Entries of the matrix $U_\\alpha$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#u-entry","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rational numbers, the entries $U_{i,j}$ ($i, j \\ge 1$) of the matrix $U_\\alpha$ are defined by recursion on the row index $i$. The first row is $U_{1,j} = \\alpha_{j-1}$. For $i \\ge 2$ the first column is $U_{i,1} = 0$, and for $j \\ge 2$","title":"Entries of the matrix $U_\\alpha$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/psi","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.psi","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum "],["math","\\(\\Psi_\\alpha(k)\\)"]],"source":"The sum $\\Psi_\\alpha(k)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#psi","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals and a natural number $k$, $\\Psi_\\alpha(k) = \\sum_{l=0}^{k-1} \\frac{\\alpha_l}{(k-1-l)!}$; in particular $\\Psi_\\alpha(0) = 0$.","title":"The sum $\\Psi_\\alpha(k)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/omega","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.omega","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum "],["math","\\(\\Omega(\\alpha, j)\\)"]],"source":"The sum $\\Omega(\\alpha, j)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#omega","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals and a natural number $j$, $\\Omega(\\alpha, j) = \\sum_{l=0}^{j-1} (j-1)^{\\underline{l}} \\, \\alpha_l$, where $m^{\\underline{l}} = m(m-1)\\cdots(m-l+1)$ is the falling factorial; in particular $\\Omega(\\alpha, 0) = 0$.","title":"The sum $\\Omega(\\alpha, j)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/lambda","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.lambda","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sum "],["math","\\(\\Lambda(\\alpha, k_1, k_2)\\)"]],"source":"The sum $\\Lambda(\\alpha, k_1, k_2)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#lambda","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals and natural numbers $k_1, k_2$,","title":"The sum $\\Lambda(\\alpha, k_1, k_2)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/r-entry","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.rEntry","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entries of the matrix "],["math","\\(R_\\alpha\\)"]],"source":"Entries of the matrix $R_\\alpha$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#r-entry","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals and natural numbers $i, j$, the entry $R_{i,j}$ is $\\frac{(-1)^{j-i}}{(j-i)!} \\Psi_\\alpha(i)$ if $i \\le j$ and $0$ if $i > j$, where $\\Psi_\\alpha(k) = \\sum_{l=0}^{k-1} \\frac{\\alpha_l}{(k-1-l)!}$. The matrix $R_\\alpha$ uses the indices $i, j \\ge 1$.","title":"Entries of the matrix $R_\\alpha$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/r-tilde-entry","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.rTildeEntry","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entries of the matrix "],["math","\\(\\tilde R_\\alpha\\)"]],"source":"Entries of the matrix $\\tilde R_\\alpha$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#r-tilde-entry","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals and natural numbers $i, j$, the entry $\\tilde R_{i,j}$ is $(j-1)^{\\underline{i-1}} / \\Omega(\\alpha, j)$ if $i \\le j$ and $0$ if $i > j$, where $\\Omega(\\alpha, j) = \\sum_{l=0}^{j-1} (j-1)^{\\underline{l}} \\, \\alpha_l$, $m^{\\underline{l}} = m(m-1)\\cdots(m-l+1)$ is the falling factorial, $0 - 1$ is read as $0$, and division by $0$ gives $0$. The matrix\u2026","title":"Entries of the matrix $\\tilde R_\\alpha$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/stirling-like-matrix","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.stirlingLikeMatrix","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#stirling-like-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha \\colon \\mathbb{N} \\to \\mathbb{Q}$ and $n \\in \\mathbb{N}$, the Stirling-like matrix $S_\\alpha(n)$ is the $n \\times n$ rational matrix, with rows and columns indexed by $1, \\dots, n$, whose $(i, j)$ entry is $S_{i,j}$, where $S_{1,j} = \\alpha_{j-1}$ for $j \\ge 1$, $S_{i,1} = \\alpha_0$ for $i \\ge 2$, and $S_{i,j} = S_{i-1,j-1} + j \\, S_{i-1,j}$ for $i, j \\ge 2$.","title":"Stirling-like matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/u-matrix","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.uMatrix","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(U_\\alpha(n)\\)"]],"source":"The matrix $U_\\alpha(n)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#u-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals and a natural number $n$, $U_\\alpha(n)$ is the $n \\times n$ rational matrix whose $(i, j)$ entry, for $1 \\le i, j \\le n$, is $U_{i,j}$, where $U_{1,j} = \\alpha_{j-1}$ for $j \\ge 1$, $U_{i,1} = 0$ for $i \\ge 2$, and $U_{i,j} = U_{i-1,j-1} + (j - i + 1) \\, U_{i-1,j}$ for $i, j \\ge 2$.","title":"The matrix $U_\\alpha(n)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/r-matrix","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.rMatrix","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(R_\\alpha(n)\\)"]],"source":"The matrix $R_\\alpha(n)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#r-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals and a natural number $n$, $R_\\alpha(n)$ is the $n \\times n$ rational matrix whose $(i, j)$ entry, for $1 \\le i, j \\le n$, is $\\frac{(-1)^{j-i}}{(j-i)!} \\Psi_\\alpha(i)$ if $i \\le j$ and $0$ if $i > j$, where $\\Psi_\\alpha(k) = \\sum_{l=0}^{k-1} \\frac{\\alpha_l}{(k-1-l)!}$.","title":"The matrix $R_\\alpha(n)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/r-tilde-matrix","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.rTildeMatrix","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(\\tilde R_\\alpha(n)\\)"]],"source":"The matrix $\\tilde R_\\alpha(n)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#r-tilde-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals and a natural number $n$, $\\tilde R_\\alpha(n)$ is the $n \\times n$ rational matrix whose $(i, j)$ entry, for $1 \\le i, j \\le n$, is $(j-1)^{\\underline{i-1}} / \\Omega(\\alpha, j)$ if $i \\le j$ and $0$ if $i > j$, where $\\Omega(\\alpha, j) = \\sum_{l=0}^{j-1} (j-1)^{\\underline{l}} \\, \\alpha_l$, $m^{\\underline{l}} = m(m-1)\\cdots(m-l+1)$ is the falling\u2026","title":"The matrix $\\tilde R_\\alpha(n)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/stirling-matrix2","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.stirlingMatrix2","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#stirling-matrix2","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the Stirling matrix of the second kind $S(n)$ is the $n \\times n$ rational matrix, with rows and columns indexed by $1, \\dots, n$, whose $(i, j)$ entry is the Stirling number of the second kind $S(i, j)$, the number of partitions of an $i$-element set into $j$ nonempty blocks.","title":"Stirling matrix of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/vandermonde-matrix","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.vandermondeMatrix","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Vandermonde matrix "],["math","\\(V(n)\\)"]],"source":"The Vandermonde matrix $V(n)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#vandermonde-matrix","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, $V(n)$ is the $n \\times n$ rational matrix whose entry in row $i$ and column $j$ ($1 \\le i, j \\le n$) is $V_{i,j} = j^{i-1}$.","title":"The Vandermonde matrix $V(n)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/u-entry-eq-lambda","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.uEntry_eq_lambda","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Explicit formula for the entries of "],["math","\\(U_\\alpha\\)"]],"source":"Explicit formula for the entries of $U_\\alpha$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#u-entry-eq-lambda","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $\\alpha \\colon \\mathbb{N} \\to \\mathbb{Q}$ and let $U_\\alpha = (U_{i,j})_{i \\ge 1, j \\ge 0}$ be given by $U_{i,0} = 0$, $U_{1,j} = \\alpha_{j-1}$ for $j \\ge 1$, $U_{i,1} = 0$ for $i \\ge 2$, and $U_{i,j} = U_{i-1,j-1} + (j-i+1) \\, U_{i-1,j}$ for $i, j \\ge 2$. For $k_1 \\le k_2$ let $\\Lambda(\\alpha, k_1, k_2) = \\sum_{l=0}^{k_1-1} \\binom{k_1-1}{l} \\frac{(l+k_2-k_1)!}{(k_2-k_1)!} \\, \\alpha_{l+k_2-k_1}$. Then\u2026","title":"Explicit formula for the entries of $U_\\alpha$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/u-entry-eq-zero-of-lt","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.uEntry_eq_zero_of_lt","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The matrix "],["math","\\(U_\\alpha\\)"],["text"," is upper triangular"]],"source":"The matrix $U_\\alpha$ is upper triangular"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#u-entry-eq-zero-of-lt","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $\\alpha \\colon \\mathbb{N} \\to \\mathbb{Q}$ and let $U_\\alpha = (U_{i,j})_{i \\ge 1, j \\ge 0}$ be given by $U_{i,0} = 0$, $U_{1,j} = \\alpha_{j-1}$ for $j \\ge 1$, $U_{i,1} = 0$ for $i \\ge 2$, and $U_{i,j} = U_{i-1,j-1} + (j-i+1) \\, U_{i-1,j}$ for $i, j \\ge 2$. Then $U_{i,j} = 0$ for all natural numbers $j < i$.","title":"The matrix $U_\\alpha$ is upper triangular"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/sum-r-entry-col","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.sum_rEntry_col","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Column sums of the matrix "],["math","\\(R_\\alpha\\)"]],"source":"Column sums of the matrix $R_\\alpha$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#sum-r-entry-col","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $\\alpha \\colon \\mathbb{N} \\to \\mathbb{Q}$, let $\\Psi_\\alpha(k) = \\sum_{l=0}^{k-1} \\alpha_l / (k-1-l)!$, and let $R_\\alpha$ be the matrix with entries $R_{i,j} = \\frac{(-1)^{j-i}}{(j-i)!} \\Psi_\\alpha(i)$ for $1 \\le i \\le j$ and $R_{i,j} = 0$ for $i > j$. Then for every $j \\ge 1$, $\\sum_{i=1}^{j} R_{i,j} = \\alpha_{j-1}$.","title":"Column sums of the matrix $R_\\alpha$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/stirling-like-matrix-decomp","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.stirlingLikeMatrix_decomp","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#stirling-like-matrix-decomp","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ be a sequence of rationals. For natural numbers $i, j \\ge 1$ define $S_{i,j}$ and $U_{i,j}$ by $S_{1,j} = U_{1,j} = \\alpha_{j-1}$, $S_{i,1} = \\alpha_0$ and $U_{i,1} = 0$ for $i \\ge 2$, and, for $i, j \\ge 2$, $S_{i,j} = S_{i-1,j-1} + j \\, S_{i-1,j}$ and $U_{i,j} = U_{i-1,j-1} + (j-i+1) \\, U_{i-1,j}$; let $R_{i,j} = \\frac{(-1)^{j-i}}{(j-i)!} \\Psi_\\alpha(i)$ if $i \\le j$ and\u2026","title":"Two factorizations of the Stirling-like matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/stirling-like-matrix-det","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.stirlingLikeMatrix_det","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#stirling-like-matrix-det","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"For a sequence $\\alpha \\colon \\mathbb{N} \\to \\mathbb{Q}$ and $j \\ge 1$ let $\\Omega(\\alpha, j) = \\sum_{l=0}^{j-1} (j-1)^{\\underline{l}} \\, \\alpha_l$, where $m^{\\underline{l}} = m(m-1)\\cdots(m-l+1)$ is the falling factorial. Then for every $n \\in \\mathbb{N}$ the determinant of the $n \\times n$ Stirling-like matrix $S_\\alpha(n)$ is $\\det S_\\alpha(n) = \\prod_{j=1}^{n} \\Omega(\\alpha, j)$.","title":"Determinant of the Stirling-like matrix"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/sum-stirling-second-lambda","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.sum_stirlingSecond_lambda","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second-kind Stirling transform of the "],["math","\\(\\Lambda\\)"],["text"," coefficients"]],"source":"Second-kind Stirling transform of the $\\Lambda$ coefficients"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#sum-stirling-second-lambda","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $\\alpha \\colon \\mathbb{N} \\to \\mathbb{Q}$, let $S(i, k)$ denote the Stirling numbers of the second kind, let $\\Psi_\\alpha(k) = \\sum_{l=0}^{k-1} \\alpha_l / (k-1-l)!$, and for $k_1 \\le k_2$ let $\\Lambda(\\alpha, k_1, k_2) = \\sum_{l=0}^{k_1-1} \\binom{k_1-1}{l} \\frac{(l+k_2-k_1)!}{(k_2-k_1)!} \\, \\alpha_{l+k_2-k_1}$. Then for all $i, j \\ge 1$,","title":"Second-kind Stirling transform of the $\\Lambda$ coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/r-tilde-matrix-eq-inv","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.rTildeMatrix_eq_inv","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","The inverse of the matrix "],["math","\\(R_\\alpha(n)\\)"]],"source":"The inverse of the matrix $R_\\alpha(n)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#r-tilde-matrix-eq-inv","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"For a sequence $\\alpha = (\\alpha_0, \\alpha_1, \\dots)$ of rationals, let $\\Psi_\\alpha(k) = \\sum_{l=0}^{k-1} \\frac{\\alpha_l}{(k-1-l)!}$ and $\\Omega(\\alpha, j) = \\sum_{l=0}^{j-1} (j-1)^{\\underline{l}} \\, \\alpha_l$, where $m^{\\underline{l}} = m(m-1)\\cdots(m-l+1)$ is the falling factorial. For a natural number $n$, let $R_\\alpha(n)$ and $\\tilde R_\\alpha(n)$ be the upper triangular $n \\times n$ rational matrices whose\u2026","title":"The inverse of the matrix $R_\\alpha(n)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/sum-stirling-second-desc-factorial-lambd","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.sum_stirlingSecond_descFactorial_lambda","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stirling and falling-factorial sum of the "],["math","\\(\\Lambda\\)"],["text"," coefficients"]],"source":"Stirling and falling-factorial sum of the $\\Lambda$ coefficients"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#sum-stirling-second-desc-factorial-lambd","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $\\alpha \\colon \\mathbb{N} \\to \\mathbb{Q}$, let $S(i, k)$ denote the Stirling numbers of the second kind, let $m^{\\underline{l}} = m(m-1)\\cdots(m-l+1)$ be the falling factorial, let $\\Omega(\\alpha, j) = \\sum_{l=0}^{j-1} (j-1)^{\\underline{l}} \\, \\alpha_l$, and for $k_1 \\le k_2$ let $\\Lambda(\\alpha, k_1, k_2) = \\sum_{l=0}^{k_1-1} \\binom{k_1-1}{l} \\frac{(l+k_2-k_1)!}{(k_2-k_1)!} \\, \\alpha_{l+k_2-k_1}$. If\u2026","title":"Stirling and falling-factorial sum of the $\\Lambda$ coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/sum-signed-psi-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.sum_signed_psi_eq","parent":25359,"reading_title":{"format":"inline-parts-v1","parts":[["text","First-kind Stirling formula for "],["math","\\(\\Lambda(\\alpha, i, j)\\)"]],"source":"First-kind Stirling formula for $\\Lambda(\\alpha, i, j)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#sum-signed-psi-eq","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $\\alpha \\colon \\mathbb{N} \\to \\mathbb{Q}$, let $s(i, k)$ denote the signed Stirling numbers of the first kind, let $\\Psi_\\alpha(k) = \\sum_{l=0}^{k-1} \\alpha_l / (k-1-l)!$, and for $k_1 \\le k_2$ let $\\Lambda(\\alpha, k_1, k_2) = \\sum_{l=0}^{k_1-1} \\binom{k_1-1}{l} \\frac{(l+k_2-k_1)!}{(k_2-k_1)!} \\, \\alpha_{l+k_2-k_1}$. Then for all $i, j \\ge 1$ the sum","title":"First-kind Stirling formula for $\\Lambda(\\alpha, i, j)$"},{"id":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/sum-signed-eq-delta","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.StirlingMatrixWanted.sum_signed_eq_delta","parent":25359,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices/index.html#sum-signed-eq-delta","scope":"Combinatorics/stirling-bell-eulerian/generalized-stirling-numbers/stirling-matrices","status":"can_prove","summary":"Let $s(i, k)$ denote the signed Stirling numbers of the first kind and $\\delta_{i,j}$ the Kronecker delta ($1$ if $i = j$ and $0$ otherwise). Then for all integers $i, j \\ge 1$, in $\\mathbb{Q}$,","title":"Kronecker-delta identity for signed Stirling numbers of the first kind"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers.json","id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers","kind":"scope","parent":25194,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers","status":"can_prove","summary":"Bell numbers (Dobinski, Spivey, Touchard and Tur\u00e1n-type identities, log-convexity), complete, partial and generalized Bell polynomials, potential polynomials, Fubini (ordered Bell) numbers and their congruences, and horse numbers.","title":"Bell numbers, Fubini numbers and Bell polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/partial-bell-polynomial","kind":"node","lean":"MetaMathlibExt.partialBellPolynomial","parent":25410,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/index.html#partial-bell-polynomial","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $n, k \\in \\mathbb{N}$ and let $x_1, x_2, \\dots$ be a sequence in $R$. The partial Bell polynomial $B_{n,k}(x)$ is the sum, over all tuples $(m_1, \\dots, m_{n+1})$ of natural numbers with $\\sum_j j\\, m_j = n$ and $\\sum_j m_j = k$, of","title":"Partial Bell polynomial"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/dobinski-formula-has-sum","kind":"node","lean":"MetaMathlibExt.dobinski_formula_hasSum","parent":25410,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/index.html#dobinski-formula-has-sum","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers","status":"fully_proved","summary":"Let $B_n$ denote the $n$-th Bell number, the number of partitions of an $n$-element set. For every natural number $n$, the real series $\\sum_{k=0}^{\\infty} k^n / k!$ is summable with sum $e \\cdot B_n$, so $B_n = \\frac{1}{e} \\sum_{k=0}^{\\infty} \\frac{k^n}{k!}$. At $n = 0$ the term $k = 0$ is $0^0 / 0! = 1$, matching the exponential series.","title":"Dobinski's formula"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/r-horse-number-eq-stirling-first-sum-fub","kind":"node","lean":"MetaMathlibExt.rHorseNumber_eq_stirlingFirst_sum_fubini","parent":25410,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/index.html#r-horse-number-eq-stirling-first-sum-fub","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers","status":"fully_proved","summary":"Let $r \\le n$ be natural numbers. For each $k$, count the surjections $f : \\{0, \\dots, n - 1\\} \\to \\{0, \\dots, k - 1\\}$ whose values on the first $r$ points are strictly increasing, $f(0) < f(1) < \\dots < f(r - 1)$, and let $H$ (the $r$-horse number) be the sum of these counts over $0 \\le k \\le n$. Then, as rational numbers,","title":"Schreyer's Fubini formula for r-horse numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities.json","id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities","kind":"scope","navigation":"module","parent":25410,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities/index.html","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities","status":"fully_proved","title":"Bell number identities"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities/bell-eq-signed-stirling-first-convolutio","kind":"node","lean":"MetaMathlibExt.bell_eq_signed_stirlingFirst_convolution","parent":25414,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities/index.html#bell-eq-signed-stirling-first-convolutio","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities","status":"fully_proved","summary":"Write $B_n$ for the $n$-th Bell number and $s(p, m) = (-1)^{p-m} {p \\brack m}$ for the signed Stirling number of the first kind, where ${p \\brack m}$ is the number of permutations of $p$ elements with exactly $m$ cycles. For all natural numbers $n$ and $p$, with the convention $0^0 = 1$,","title":"Gould\u2013Quaintance double-sum formula for Bell numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities/bell-eq-sum-stirling-second","kind":"node","lean":"MetaMathlibExt.bell_eq_sum_stirlingSecond","parent":25414,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities/index.html#bell-eq-sum-stirling-second","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/bell-number-identities","status":"fully_proved","summary":"For every natural number $n$, the Bell number $B_n$ equals $\\sum_{k=0}^{n} S(n, k)$, where $S(n, k)$ is the Stirling number of the second kind.","title":"Bell numbers are row sums of Stirling numbers of the second kind"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers.json","id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","kind":"scope","navigation":"module","parent":25410,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","status":"fully_proved","title":"Fubini numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/fubini-number","kind":"node","lean":"MetaMathlibExt.fubiniNumber","parent":25417,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/index.html#fubini-number","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","status":"fully_proved","summary":"The Fubini number (ordered Bell number) of a natural number $n$ is $F(n) = \\sum_{k=0}^{n} S(n,k)\\, k!$, where $S(n,k)$ is the Stirling number of the second kind.","title":"Fubini numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/fubini-number-eq-sum","kind":"node","lean":"MetaMathlibExt.fubiniNumber_eq_sum","parent":25417,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/index.html#fubini-number-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","status":"fully_proved","summary":"For every natural number $n$, the Fubini number $F(n)$ equals $\\sum_{k=0}^{n} S(n,k)\\, k!$, where $S(n,k)$ is the Stirling number of the second kind.","title":"Fubini numbers as sums of Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/fubini-number-zero","kind":"node","lean":"MetaMathlibExt.fubiniNumber_zero","parent":25417,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/index.html#fubini-number-zero","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","status":"fully_proved","summary":"The Fubini number $F(0)$ equals $1$: the empty set has exactly one weak ordering.","title":"Fubini number of zero"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/fubini-number-one","kind":"node","lean":"MetaMathlibExt.fubiniNumber_one","parent":25417,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/index.html#fubini-number-one","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","status":"fully_proved","summary":"The Fubini number $F(1)$ equals $1$.","title":"Fubini number of one"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/fubini-number-two","kind":"node","lean":"MetaMathlibExt.fubiniNumber_two","parent":25417,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/index.html#fubini-number-two","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","status":"fully_proved","summary":"The Fubini number $F(2)$ equals $3$.","title":"Fubini number of two"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/fubini-number-three","kind":"node","lean":"MetaMathlibExt.fubiniNumber_three","parent":25417,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/index.html#fubini-number-three","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","status":"fully_proved","summary":"The Fubini number $F(3)$ equals $13$.","title":"Fubini number of three"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/fubini-number-eq-factorial-sub-sum-signe","kind":"node","lean":"MetaMathlibExt.fubiniNumber_eq_factorial_sub_sum_signed_stirling","parent":25417,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers/index.html#fubini-number-eq-factorial-sub-sum-signe","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers","status":"fully_proved","summary":"For every natural number $n$, the Fubini number $F(n)$ satisfies","title":"Alternating recurrence for the Fubini numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer.json","id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer","kind":"scope","navigation":"module","parent":25410,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/index.html","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer","status":"can_prove","title":"Fubini numbers modulo an integer"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/desc-pochhammer-eq-sum-signed-stirling-f","kind":"node","lean":"MetaMathlibExt.descPochhammer_eq_sum_signedStirlingFirst","parent":25425,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/index.html#desc-pochhammer-eq-sum-signed-stirling-f","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer","status":"can_prove","summary":"For every natural number $n$, the identity of polynomials with integer coefficients","title":"Falling factorial in terms of signed Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/pow-add-carmichael-mod-eq-of-lt","kind":"node","lean":"MetaMathlibExt.pow_add_carmichael_modEq_of_lt","parent":25425,"reading_title":{"format":"inline-parts-v1","parts":[["text","A Carmichael congruence for all residues modulo "],["math","\\(K\\)"]],"source":"A Carmichael congruence for all residues modulo $K$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/index.html#pow-add-carmichael-mod-eq-of-lt","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer","status":"can_prove","summary":"Let $K$ be a natural number, let $\\lambda(K)$ be the value of the Carmichael function at $K$, and let $R$ be the largest exponent in the prime factorization of $K$ ($R = 0$ for $K \\le 1$). Then $a^R \\equiv a^{\\lambda(K) + R} \\pmod K$ for every natural number $a < K$.","title":"A Carmichael congruence for all residues modulo $K$"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/fubini-number-add-carmichael-mod-eq","kind":"node","lean":"MetaMathlibExt.fubiniNumber_add_carmichael_modEq","parent":25425,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual periodicity of the Fubini numbers modulo "],["math","\\(K\\)"]],"source":"Eventual periodicity of the Fubini numbers modulo $K$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/index.html#fubini-number-add-carmichael-mod-eq","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer","status":"can_prove","summary":"Let $K$ be a natural number, let $\\lambda(K)$ be the value of the Carmichael function at $K$, and let $R$ be the largest exponent in the prime factorization of $K$ ($R = 0$ for $K \\le 1$). Then the Fubini numbers $F(n)$ satisfy $F(n + \\lambda(K)) \\equiv F(n) \\pmod K$ for every natural number $n \\ge R$.","title":"Eventual periodicity of the Fubini numbers modulo $K$"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/fubini-number-eventually-periodic-iff-ca","kind":"node","lean":"MetaMathlibExt.fubiniNumber_eventuallyPeriodic_iff_carmichael_dvd","parent":25425,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/index.html#fubini-number-eventually-periodic-iff-ca","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer","status":"can_prove","summary":"Let $K$ be an odd natural number and let $\\lambda(K)$ be the value of the Carmichael function at $K$. Then a natural number $p$ is divisible by $\\lambda(K)$ if and only if there is a natural number $N$ such that the Fubini numbers $F(n)$ satisfy $F(n + p) \\equiv F(n) \\pmod K$ for all $n \\ge N$. In particular, the eventual period of the Fubini numbers modulo $K$ is exactly $\\lambda(K)$.","title":"Eventual periods of the Fubini numbers modulo an odd number"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/r-fubini-add-carmichael-mod-eq","kind":"node","lean":"MetaMathlibExt.rFubini_add_carmichael_modEq","parent":25425,"reading_title":{"format":"inline-parts-v1","parts":[["text","Eventual periodicity of the "],["math","\\(r\\)"],["text","-Fubini numbers modulo "],["math","\\(K\\)"]],"source":"Eventual periodicity of the $r$-Fubini numbers modulo $K$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer/index.html#r-fubini-add-carmichael-mod-eq","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/fubini-numbers-modulo-an-integer","status":"can_prove","summary":"For a natural number $r \\ge 1$ and a natural number $m$, let $F_r(m)$ be the number of pairs $(k, f)$ of a natural number $k \\le m$ and a surjection $f : \\{0, \\dots, m - 1\\} \\to \\{0, \\dots, k - 1\\}$ that is injective on the elements less than $r$; equivalently, $F_r(m)$ counts the weak orderings of $m$ elements in which the elements less than $r$ are pairwise untied. Let $K$ be a natural number, let $\\lambda(K)$ be\u2026","title":"Eventual periodicity of the $r$-Fubini numbers modulo $K$"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials.json","id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials","kind":"scope","navigation":"module","parent":25410,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/index.html","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials","status":"fully_proved","title":"Generalized Bell polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/gen-bell-number","kind":"node","lean":"MetaMathlibExt.genBellNumber","parent":25431,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/index.html#gen-bell-number","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials","status":"fully_proved","summary":"For a commutative ring $R$, elements $s, h \\in R$ and a natural number $n$, the generalized Bell number is","title":"Generalized Bell numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/gen-bell-polynomial","kind":"node","lean":"MetaMathlibExt.genBellPolynomial","parent":25431,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/index.html#gen-bell-polynomial","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials","status":"fully_proved","summary":"For a commutative ring $R$, elements $s, h \\in R$, a natural number $n$ and $x \\in R$, the generalized Bell polynomial evaluated at $x$ is","title":"Generalized Bell polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/gen-stirling-succ-succ","kind":"node","lean":"MetaMathlibExt.genStirling_succ_succ","parent":25431,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/index.html#gen-stirling-succ-succ","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials","status":"fully_proved","summary":"Let $R$ be a commutative ring and $s, h \\in R$. The generalized Stirling numbers $\\mathfrak{S}_{s;h}(n, k) \\in R$ are defined by $\\mathfrak{S}_{s;h}(0, 0) = 1$, $\\mathfrak{S}_{s;h}(0, k + 1) = \\mathfrak{S}_{s;h}(n + 1, 0) = 0$ and the recurrence below. For all natural numbers $n$ and $k$,","title":"Recurrence for generalized Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/gen-bell-polynomial-eval-one","kind":"node","lean":"MetaMathlibExt.genBellPolynomial_eval_one","parent":25431,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials/index.html#gen-bell-polynomial-eval-one","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/generalized-bell-polynomials","status":"fully_proved","summary":"For a commutative ring $R$, elements $s, h \\in R$ and a natural number $n$, the generalized Bell polynomial $\\mathfrak{B}_{s;h|n}$ evaluated at $x = 1$ equals the generalized Bell number $\\mathfrak{B}_{s;h}(n)$.","title":"Generalized Bell polynomials at one"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers.json","id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers","kind":"scope","navigation":"module","parent":25410,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers","status":"fully_proved","title":"Log-convexity of Bell numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers/bell-turan-identity","kind":"node","lean":"MetaMathlibExt.bell_turan_identity","parent":25436,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers/index.html#bell-turan-identity","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers","status":"fully_proved","summary":"For every natural number $n \\ge 2$, the Bell numbers $B_m$ satisfy","title":"Tur\u00e1n-type identity for the Bell numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers/engel-bell-log-convexity","kind":"node","lean":"MetaMathlibExt.engel_bell_log_convexity","parent":25436,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers/index.html#engel-bell-log-convexity","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/log-convexity-of-bell-numbers","status":"fully_proved","summary":"For every integer $n \\ge 1$, the Bell numbers satisfy $B_n^2 \\le B_{n-1} B_{n+1}$, where $B_n$ is the number of partitions of an $n$-element set.","title":"Engel's inequality for Bell numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials.json","id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials","kind":"scope","navigation":"module","parent":25410,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials/index.html","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials","status":"fully_proved","title":"Potential polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials/howard-potential-polynomial-r-two-eq-par","kind":"node","lean":"MetaMathlibExt.howard_potentialPolynomial_rTwo_eq_partialBell","parent":25439,"reading_title":{"format":"inline-parts-v1","parts":[["text","Howard's formula for the series "],["math","\\(\\sum_{k \\ge 0} 2x^k/(k+2)!\\)"]],"source":"Howard's formula for the series $\\sum_{k \\ge 0} 2x^k/(k+2)!$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials/index.html#howard-potential-polynomial-r-two-eq-par","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials","status":"fully_proved","summary":"Let $n$ be a natural number, let","title":"Howard's formula for the series $\\sum_{k \\ge 0} 2x^k/(k+2)!$"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials/howard-potential-polynomial-eq-partial-b","kind":"node","lean":"MetaMathlibExt.howard_potentialPolynomial_eq_partialBell","parent":25439,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials/index.html#howard-potential-polynomial-eq-partial-b","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/potential-polynomials","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero, let $r \\ge 1$ and $n$ be natural numbers, and let $a_0, a_1, a_2, \\dots$ be a sequence in $K$ with $a_r \\ne 0$. Let $g$ be the formal power series","title":"Howard's formula for potential polynomials"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula.json","id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula","kind":"scope","navigation":"module","parent":25410,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula/index.html","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula","status":"fully_proved","title":"Spivey's Bell number formula"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula/spivey-bell-number-add","kind":"node","lean":"MetaMathlibExt.spivey_bell_number_add","parent":25442,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula/index.html#spivey-bell-number-add","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula","status":"fully_proved","summary":"For all natural numbers $m$ and $n$,","title":"Spivey's Bell number formula"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula/spivey-bell-recurrence","kind":"node","lean":"MetaMathlibExt.spivey_bell_recurrence","parent":25442,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula/index.html#spivey-bell-recurrence","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula","status":"fully_proved","summary":"For all natural numbers $n$ and $m$,","title":"Spivey's recurrence for Bell numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula/spivey-pq-bell","kind":"node","lean":"MetaMathlibExt.spivey_pq_bell","parent":25442,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula/index.html#spivey-pq-bell","scope":"Combinatorics/stirling-bell-eulerian/bell-fubini-numbers/spivey-s-bell-number-formula","status":"fully_proved","summary":"Let $p$ and $q$ be real numbers with $0 < q < p \\le 1$, and for $k \\in \\mathbb{N}$ put $[k] = (p^k - q^k)/(p - q)$. Define real numbers $S(n, k)$ for $n, k \\in \\mathbb{N}$ by $S(0, 0) = 1$, $S(0, k) = 0$ for $k \\ge 1$, $S(n + 1, 0) = 0$, and, for $k \\ge 1$, $S(n + 1, k) = p^{n + 1 - k} q^{k - 1} S(n, k - 1) + [k] \\, S(n, k)$, where the exponent $n + 1 - k$ is read as $0$ when $k > n + 1$. Put\u2026","title":"The (p, q)-deformed Spivey formula"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/eulerian-numbers.json","id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers","kind":"scope","parent":25194,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers","status":"can_prove","summary":"Counting permutations by descents or excedances: Eulerian numbers and polynomials, refined, multiset and generalized Eulerian polynomials, associated Stirling\u2013Eulerian numbers, Worpitzky's identity, gamma-positivity and Hankel determinants of Eulerian polynomials, Bell permutations by weak excedances.","title":"Eulerian numbers and polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/worpitzky-identity-multiset","kind":"node","lean":"MetaMathlibExt.worpitzky_identity_multiset","parent":25446,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/index.html#worpitzky-identity-multiset","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers","status":"fully_proved","summary":"Let $m$ and $x$ be natural numbers, let $k_1, \\dots, k_m$ be natural numbers with sum $K = k_1 + \\dots + k_m$, and for $p \\ge 1$ let $a_p$ be the number of words $w_1 w_2 \\cdots w_K$ over the alphabet $\\{1, \\dots, m\\}$ in which each letter $i$ occurs exactly $k_i$ times and which have exactly $p - 1$ descents, that is, indices $j$ with $1 \\le j < K$ and $w_j > w_{j+1}$. Then","title":"Worpitzky identity for multiset permutations"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers.json","id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","kind":"scope","navigation":"module","parent":25446,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","title":"Associated Stirling\u2013Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/excedance-count","kind":"node","lean":"MetaMathlibExt.excedanceCount","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#excedance-count","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$, an excedance of $\\sigma$ is a point $i$ with $\\sigma(i) > i$, and the excedance count $e(\\sigma)$ is the number of excedances of $\\sigma$.","title":"Number of excedances"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/is-rminus-associated-cycle-perm","kind":"node","lean":"MetaMathlibExt.IsRMinusAssociatedCyclePerm","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutations with all cycles of length at most "],["math","\\(r\\)"]],"source":"Permutations with all cycles of length at most $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#is-rminus-associated-cycle-perm","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"A permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$ is an $r$-minus-associated cycle permutation when every cycle of $\\sigma$, fixed points counted as cycles of length one, has length at most $r$: every fixed point of $\\sigma$ forces $1 \\le r$, and every cycle of $\\sigma$ of length at least $2$ has length at most $r$. For $n = 0$ the empty permutation qualifies for every $r$.","title":"Permutations with all cycles of length at most $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/is-rassociated-cycle-perm","kind":"node","lean":"MetaMathlibExt.IsRAssociatedCyclePerm","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutations with all cycles longer than "],["math","\\(r\\)"]],"source":"Permutations with all cycles longer than $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#is-rassociated-cycle-perm","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"A permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$ is an $r$-associated cycle permutation when every cycle of $\\sigma$, fixed points counted as cycles of length one, has length greater than $r$: every fixed point of $\\sigma$ forces $r < 1$, and every cycle of $\\sigma$ of length at least $2$ has length greater than $r$. For $n = 0$ the empty permutation qualifies for every $r$.","title":"Permutations with all cycles longer than $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-stirling-eulerian","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedStirlingEulerian","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(r\\)"],["text","-minus-associated Stirling Eulerian numbers"]],"source":"$r$-minus-associated Stirling Eulerian numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-stirling-eulerian","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"For natural numbers $r$, $n$ and $m$, the $r$-minus-associated Stirling Eulerian number is the number of permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$ all of whose cycles, fixed points counted as cycles of length $1$, have length at most $r$, and that have exactly $m$ excedances, that is, exactly $m$ indices $i$ with $\\sigma(i) > i$.","title":"$r$-minus-associated Stirling Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-stirling-eulerian","kind":"node","lean":"MetaMathlibExt.rAssociatedStirlingEulerian","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(r\\)"],["text","-associated Stirling Eulerian numbers"]],"source":"$r$-associated Stirling Eulerian numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-stirling-eulerian","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"For natural numbers $r$, $n$ and $m$, the $r$-associated Stirling Eulerian number $A^{r}_{n,m}$ is the number of permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$ all of whose cycles, fixed points counted as cycles of length $1$, have length greater than $r$, and that have exactly $m$ excedances, that is, exactly $m$ indices $i$ with $\\sigma(i) > i$.","title":"$r$-associated Stirling Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/circular-eulerian","kind":"node","lean":"MetaMathlibExt.circularEulerian","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#circular-eulerian","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the circular Eulerian number $C_{n,m}$ is the number of permutations $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$ such that $n \\ge 1$, all points lie in a single cycle of $\\sigma$ (for any two points $i$ and $j$ there is an integer $k$ with $\\sigma^k(i) = j$), and $\\sigma$ has exactly $m$ excedances, that is, exactly $m$ points $i$ with $\\sigma(i) > i$. In particular $C_{0,m} = 0$ for every\u2026","title":"Circular Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/circular-eulerian-small","kind":"node","lean":"MetaMathlibExt.circularEulerian_small","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#circular-eulerian-small","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"Writing $C_{n,m}$ for the circular Eulerian number, we have $C_{0,0} = 0$, $C_{1,0} = 1$, $C_{3,1} = 1$ and $C_{3,2} = 1$.","title":"Small circular Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/same-cycle-all-iff-is-cycle-support-eq-u","kind":"node","lean":"MetaMathlibExt.sameCycle_all_iff_isCycle_support_eq_univ","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#same-cycle-all-iff-is-cycle-support-eq-u","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"Let $n \\ge 2$ and let $\\sigma$ be a permutation of $\\{0, \\dots, n-1\\}$. Any two points $i$ and $j$ lie in the same cycle of $\\sigma$ (that is, $j = \\sigma^k(i)$ for some integer $k$) if and only if $\\sigma$ is a cycle, meaning that it moves some point and any two points it moves lie in the same cycle, and $\\sigma$ moves every point.","title":"Circular permutations are the full cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/is-rcycle-perm","kind":"node","lean":"MetaMathlibExt.IsRCyclePerm","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutations with all cycles of length "],["math","\\(r\\)"]],"source":"Permutations with all cycles of length $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#is-rcycle-perm","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"A permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$ is an $r$-cycle permutation when every cycle of $\\sigma$, fixed points counted as cycles of length one, has exactly $r$ elements: for every point $i$, exactly $r$ points $j$ satisfy $\\sigma^k(i) = j$ for some integer $k$. For $n = 0$ the condition holds for every $r$.","title":"Permutations with all cycles of length $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/is-rcycle-perm-iff","kind":"node","lean":"MetaMathlibExt.isRCyclePerm_iff","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycle-type characterization of "],["math","\\(r\\)"],["text","-cycle permutations"]],"source":"Cycle-type characterization of $r$-cycle permutations"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#is-rcycle-perm-iff","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"A permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$ has all its cycles, fixed points counted as cycles of length one, of length exactly $r$ if and only if every fixed point of $\\sigma$ forces $r = 1$ and every cycle of $\\sigma$ of length at least $2$ has length exactly $r$.","title":"Cycle-type characterization of $r$-cycle permutations"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-stirling-eulerian","kind":"node","lean":"MetaMathlibExt.rCycleStirlingEulerian","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(r\\)"],["text","-cycle Stirling Eulerian numbers"]],"source":"$r$-cycle Stirling Eulerian numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-stirling-eulerian","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"For natural numbers $r$, $n$ and $m$, the number $A^{(r)}_{n,m}$ is the number of permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$ all of whose cycles, fixed points counted as cycles of length $1$, have length exactly $r$, and that have exactly $m$ excedances, that is, exactly $m$ indices $i$ with $\\sigma(i) > i$.","title":"$r$-cycle Stirling Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-stirling-eulerian-small","kind":"node","lean":"MetaMathlibExt.rCycleStirlingEulerian_small","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small "],["math","\\(r\\)"],["text","-cycle Stirling Eulerian numbers"]],"source":"Small $r$-cycle Stirling Eulerian numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-stirling-eulerian-small","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"Write $A^{(r)}_{n,m}$ for the number of permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$ all of whose cycles, fixed points counted as cycles of length $1$, have length exactly $r$, and that have exactly $m$ excedances (indices $i$ with $\\sigma(i) > i$). Then $A^{(2)}_{0,0} = 1$ (the empty permutation), $A^{(2)}_{2,1} = 1$ (the transposition), $A^{(2)}_{3,1} = 0$, and $A^{(3)}_{3,1} = A^{(3)}_{3,2} = 1$ (the two\u2026","title":"Small $r$-cycle Stirling Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/perm-excedance-cycle-egf","kind":"node","lean":"MetaMathlibExt.permExcedanceCycleEGF","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#perm-excedance-cycle-egf","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, let $x, q \\in R$, and for each $n$ let $P_n$ be a set of permutations of $\\{0, 1, \\dots, n-1\\}$. The excedance and cycle exponential generating function of the family $(P_n)$ is the formal power series in $u$ over $R$","title":"Excedance and cycle generating function of a permutation family"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/coeff-perm-excedance-cycle-egf","kind":"node","lean":"MetaMathlibExt.coeff_permExcedanceCycleEGF","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#coeff-perm-excedance-cycle-egf","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, let $x, q \\in R$, and for each $n$ let $P_n$ be a set of permutations of $\\{0, 1, \\dots, n-1\\}$. For every $n \\ge 0$, the coefficient of $u^n$ in the excedance and cycle exponential generating function of the family $(P_n)$ is $\\frac{1}{n!} \\sum_{\\sigma \\in P_n} x^{e(\\sigma)} q^{c(\\sigma)}$, where $e(\\sigma)$ is the number of excedances of $\\sigma$ and $c(\\sigma)$ its\u2026","title":"Coefficients of the excedance and cycle generating function"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/coeff-zero-perm-excedance-cycle-egf-true","kind":"node","lean":"MetaMathlibExt.coeff_zero_permExcedanceCycleEGF_true","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#coeff-zero-perm-excedance-cycle-egf-true","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra and let $x, q \\in R$. In the excedance and cycle exponential generating function of the family of all permutations, the coefficient of $u^0$ is $1$.","title":"Constant coefficient of the generating function of all permutations"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/circular-eulerian-eq-eulerian","kind":"node","lean":"MetaMathlibExt.circularEulerian_eq_eulerian","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#circular-eulerian-eq-eulerian","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"An excedance of a permutation $\\sigma$ of $n$ objects $1, \\dots, n$ is an index $i$ with $\\sigma(i) > i$. Let $C_{n,m}$ be the number of circular permutations of $n$ objects (permutations with a single cycle through all $n \\ge 1$ points) with exactly $m$ excedances, and let $A_{n,m}$ be the number of permutations of $n$ objects with exactly $m$ excedances (the $0$-associated Stirling Eulerian number, which counts\u2026","title":"Circular Eulerian numbers are shifted Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-stirling-eulerian-suc","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedStirlingEulerian_succ","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-stirling-eulerian-suc","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $r \\ge 2$. An excedance of a permutation $\\sigma$ is an $i$ with $\\sigma(i) > i$. Let $A^{r-}_{n,m}$ be the number of permutations of $\\{1, \\dots, n\\}$ all of whose cycles have length at most $r$ and which have exactly $m$ excedances, and let $C_{\\ell,i}$ be the number of circular permutations of $\\ell$ objects (permutations with a single cycle through all points) with exactly $i$ excedances. Then for all\u2026","title":"Recurrence for permutations with cycle lengths at most r, by excedances"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-stirling-eulerian-recurrenc","kind":"node","lean":"MetaMathlibExt.rAssociatedStirlingEulerian_recurrence","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the "],["math","\\(r\\)"],["text","-associated Stirling Eulerian numbers"]],"source":"Recurrence for the $r$-associated Stirling Eulerian numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-stirling-eulerian-recurrenc","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"For natural numbers $r$, $n$, $m$ let $A^r_{n,m}$ be the number of permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$ with exactly $m$ excedances (points $i$ with $\\sigma(i) > i$) all of whose cycles (fixed points counted as cycles of length one) have length greater than $r$, and for natural numbers $a$, $b$ let $C_{a,b}$ be the circular Eulerian number, the number of permutations of $\\{0, \\dots, a-1\\}$, $a \\ge 1$, that\u2026","title":"Recurrence for the $r$-associated Stirling Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-stirling-eulerian-succ","kind":"node","lean":"MetaMathlibExt.rAssociatedStirlingEulerian_succ","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convolution recurrence for the "],["math","\\(r\\)"],["text","-associated Stirling Eulerian numbers"]],"source":"Convolution recurrence for the $r$-associated Stirling Eulerian numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-stirling-eulerian-succ","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"For natural numbers $r$, $n$, $m$ let $A^r_{n,m}$ be the number of permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$ with exactly $m$ excedances (points $i$ with $\\sigma(i) > i$) all of whose cycles (fixed points counted as cycles of length one) have length greater than $r$, and for natural numbers $a$, $b$ let $C_{a,b}$ be the circular Eulerian number, the number of permutations of $\\{0, \\dots, a-1\\}$, $a \\ge 1$, that\u2026","title":"Convolution recurrence for the $r$-associated Stirling Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/derangement-excedance-cycle-sum-neg-two","kind":"node","lean":"MetaMathlibExt.derangement_excedance_cycle_sum_neg_two","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derangements weighted by excedances and cycle weight "],["math","\\(-2\\)"]],"source":"Derangements weighted by excedances and cycle weight $-2$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#derangement-excedance-cycle-sum-neg-two","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $n \\ge 1$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ its number of cycles. Summing over the derangements $\\sigma$ of $n$ objects (permutations without fixed points), the polynomial $(x - 1)^2 \\sum_{\\sigma} x^{e(\\sigma)} (-2)^{c(\\sigma)}$ equals $-2x\\,\\big(-2^{n-1} x - 2^{n-1} x^{n-1} + (x+1)^n\\big)$ in $\\mathbb{Z}[x]$; that is,\u2026","title":"Derangements weighted by excedances and cycle weight $-2$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/exp-circular-perm-egf-eq-perm-egf","kind":"node","lean":"MetaMathlibExt.exp_circularPermEGF_eq_permEGF","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#exp-circular-perm-egf-eq-perm-egf","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra and $x \\in R$, and work with formal power series in $u$ over $R$. For a permutation $\\sigma$ write $e(\\sigma)$ for its number of excedances (points $i$ with $\\sigma(i) > i$), and let $S_n$ be the set of permutations of $\\{0, \\dots, n-1\\}$. Let $A(x,u) = \\sum_{n \\ge 0} \\sum_{\\sigma \\in S_n} x^{e(\\sigma)} u^n/n!$ be the exponential generating function of excedances over\u2026","title":"Logarithm of the Eulerian generating function"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/circular-rminus-perm-egf-eq","kind":"node","lean":"MetaMathlibExt.circularRMinusPermEGF_eq","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Excedance generating function of circular permutations of at most "],["math","\\(r\\)"],["text"," objects"]],"source":"Excedance generating function of circular permutations of at most $r$ objects"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#circular-rminus-perm-egf-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $r \\ge 1$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$), and for $j \\ge 0$ let $A_j(x) = \\sum_{\\sigma} x^{e(\\sigma)}$, summed over all permutations $\\sigma$ of $j$ objects. Then in $R[[u]]$ $\\sum_{n=1}^{r} \\sum_{\\sigma} x^{e(\\sigma)} u^n/n! = u + x \\sum_{j=1}^{r-1} A_j(x)\\,u^{j+1}/(j+1)!$, where\u2026","title":"Excedance generating function of circular permutations of at most $r$ objects"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/circular-rcycle-perm-egf-eq","kind":"node","lean":"MetaMathlibExt.circularRCyclePermEGF_eq","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Excedance generating function of circular permutations of "],["math","\\(r\\)"],["text"," objects"]],"source":"Excedance generating function of circular permutations of $r$ objects"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#circular-rcycle-perm-egf-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $r \\ge 2$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$), and for $j \\ge 0$ let $A_j(x) = \\sum_{\\sigma} x^{e(\\sigma)}$, summed over all permutations $\\sigma$ of $j$ objects. Then the power series $\\sum_{\\sigma} x^{e(\\sigma)} u^r/r!$ in $R[[u]]$, summed over the circular permutations $\\sigma$ of\u2026","title":"Excedance generating function of circular permutations of $r$ objects"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/exp-circular-rplus-perm-egf-eq","kind":"node","lean":"MetaMathlibExt.exp_circularRPlusPermEGF_eq","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Excedance generating function of circular permutations of more than "],["math","\\(r\\)"],["text"," objects"]],"source":"Excedance generating function of circular permutations of more than $r$ objects"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#exp-circular-rplus-perm-egf-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $r \\ge 2$, and work with formal power series in $u$ over $R$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$). For $j \\ge 0$ let $A_j(x) = \\sum_{\\sigma} x^{e(\\sigma)}$, summed over all permutations $\\sigma$ of $j$ objects, and let $A(x,u) = \\sum_{n \\ge 0} A_n(x)\\, u^n/n!$. Let\u2026","title":"Excedance generating function of circular permutations of more than $r$ objects"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-perm-egf-eq-exp","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedPermEGF_eq_exp","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exponential generating function of permutations with cycle lengths at most "],["math","\\(r\\)"]],"source":"Exponential generating function of permutations with cycle lengths at most $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-perm-egf-eq-exp","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r \\ge 2$ be a natural number. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$), and let $A_j(x) = \\sum_{\\sigma \\in S_j} x^{e(\\sigma)}$. Let $A^{r-}(x,u) = \\sum_{n \\ge 0} \\sum_{\\sigma} x^{e(\\sigma)} u^n/n! \\in R[[u]]$, the inner sum over the permutations $\\sigma$ of\u2026","title":"Exponential generating function of permutations with cycle lengths at most $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/exp-circular-rminus-perm-egf-eq-r-minus","kind":"node","lean":"MetaMathlibExt.exp_circularRMinusPermEGF_eq_rMinusAssociatedPermEGF","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#exp-circular-rminus-perm-egf-eq-r-minus","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $r \\ge 2$ an integer, and work with formal power series in $u$ over $R$. For a permutation $\\sigma$ write $e(\\sigma)$ for its number of excedances (points $i$ with $\\sigma(i) > i$). Let $C^{r-}(x,u) = \\sum_{n=1}^{r} \\sum_{\\sigma} x^{e(\\sigma)} u^n/n!$, the inner sum over the permutations of $n$ points that consist of a single cycle, and let\u2026","title":"Logarithm of the generating function of permutations with short cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/exp-eulerian-egf-primitive","kind":"node","lean":"MetaMathlibExt.exp_eulerianEGF_primitive","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#exp-eulerian-egf-primitive","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra and $x \\in R$, and work with formal power series in $u$ over $R$. For a permutation $\\sigma$ write $e(\\sigma)$ for its number of excedances (points $i$ with $\\sigma(i) > i$), let $S_n$ be the set of permutations of $\\{0, \\dots, n-1\\}$, and let $A(x,u) = \\sum_{n \\ge 0} \\sum_{\\sigma \\in S_n} x^{e(\\sigma)} u^n/n!$. If $P(u)$ is a formal power series in $u$ over $R$ with\u2026","title":"Primitive of the Eulerian generating function"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-perm-egf-cycle-count","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedPermEGF_cycleCount_eq","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of permutations with cycle lengths at most "],["math","\\(r\\)"],["text"," and "],["math","\\(k\\)"],["text"," cycles"]],"source":"Generating function of permutations with cycle lengths at most $r$ and $k$ cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-perm-egf-cycle-count","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r, k$ be natural numbers with $r \\ge 2$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included, and let $A_j(x) = \\sum_{\\sigma \\in S_j} x^{e(\\sigma)}$. Let\u2026","title":"Generating function of permutations with cycle lengths at most $r$ and $k$ cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-perm-egf-q-eq-exp","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedPermEGF_q_eq_exp","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-perm-egf-q-eq-exp","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, let $x, q \\in R$ and let $r \\ge 2$. For a permutation $\\sigma$ write $e(\\sigma)$ for its number of excedances (the $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included, and for $j \\ge 1$ let $A_j(x) = \\sum_{\\sigma} x^{e(\\sigma)}$, summed over the permutations $\\sigma$ of $\\{1, \\dots, j\\}$. Then, as formal power series in $u$ over $R$,","title":"Cycle-counting generating function for cycle lengths at most r"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-perm-egf-q-eq-pow","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedPermEGF_q_eq_pow","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-perm-egf-q-eq-pow","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, let $x, q \\in R$ and let $r \\ge 2$. For a permutation $\\sigma$ write $e(\\sigma)$ for its number of excedances (the $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included, and for $t \\in R$ let","title":"Cycle lengths at most r: the q-analogue is a q-th power"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-stirling-eulerian-succ","kind":"node","lean":"MetaMathlibExt.rCycleStirlingEulerian_succ","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for "],["math","\\(r\\)"],["text","-cycle Stirling Eulerian numbers"]],"source":"Recurrence for $r$-cycle Stirling Eulerian numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-stirling-eulerian-succ","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"An excedance of a permutation $\\sigma$ of $\\{0, \\dots, N-1\\}$ is an index $i$ with $\\sigma(i) > i$. Let $A^{(r)}_{N,M}$ be the number of permutations of $\\{0, \\dots, N-1\\}$ with exactly $M$ excedances all of whose cycles, fixed points included, have exactly $r$ elements, and let $C_{r,k}$ be the number of permutations of $\\{0, \\dots, r-1\\}$ with exactly $k$ excedances that consist of a single cycle through all $r$\u2026","title":"Recurrence for $r$-cycle Stirling Eulerian numbers"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-perm-egf-eq-exp","kind":"node","lean":"MetaMathlibExt.rCyclePermEGF_eq_exp","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Excedance generating function of permutations with all cycles of length "],["math","\\(r\\)"]],"source":"Excedance generating function of permutations with all cycles of length $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-perm-egf-eq-exp","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $r \\ge 2$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $\\operatorname{exc}(\\sigma)$ be the number of points $i$ with $\\sigma(i) > i$, and let $A_{r-1}(x) = \\sum_{\\sigma} x^{\\operatorname{exc}(\\sigma)}$, summed over all permutations of $\\{0, \\dots, r-2\\}$. Let $A^{(r)}(x,u) = \\sum_{n \\ge 0} \\sum_{\\sigma} x^{\\operatorname{exc}(\\sigma)} u^n/n!$,\u2026","title":"Excedance generating function of permutations with all cycles of length $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-perm-egf-cycle-count-eq","kind":"node","lean":"MetaMathlibExt.rCyclePermEGF_cycleCount_eq","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Excedance generating function of permutations with "],["math","\\(k\\)"],["text"," cycles, all of length "],["math","\\(r\\)"]],"source":"Excedance generating function of permutations with $k$ cycles, all of length $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-perm-egf-cycle-count-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r$, $k$ be natural numbers with $r \\ge 2$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $\\operatorname{exc}(\\sigma)$ be the number of points $i$ with $\\sigma(i) > i$, and let $A_{r-1}(x) = \\sum_{\\sigma} x^{\\operatorname{exc}(\\sigma)}$, summed over all permutations of $\\{0, \\dots, r-2\\}$. Let\u2026","title":"Excedance generating function of permutations with $k$ cycles, all of length $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-perm-egf-q-eq-exp","kind":"node","lean":"MetaMathlibExt.rCyclePermEGF_q_eq_exp","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycle-weighted exponential formula for "],["math","\\(r\\)"],["text","-cycle permutations"]],"source":"Cycle-weighted exponential formula for $r$-cycle permutations"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-perm-egf-q-eq-exp","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$, and let $r \\ge 2$ be a natural number. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included, and let $A_{r-1}(x) = \\sum_{\\sigma \\in S_{r-1}} x^{e(\\sigma)}$. Call $\\sigma$ an $r$-cycle permutation when every cycle\u2026","title":"Cycle-weighted exponential formula for $r$-cycle permutations"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/sum-r-cycle-perm-cycle-count-eq","kind":"node","lean":"MetaMathlibExt.sum_rCyclePerm_cycleCount_eq","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#sum-r-cycle-perm-cycle-count-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative ring, let $x, q \\in R$, let $r \\ge 2$ and let $n, k \\ge 0$. For a permutation $\\pi$ write $e(\\pi)$ for its number of excedances (the $i$ with $\\pi(i) > i$) and $c(\\pi)$ for its number of cycles, fixed points included, and let $A_{r-1}(x) = \\sum_{\\sigma} x^{e(\\sigma)}$, summed over the permutations $\\sigma$ of $\\{1, \\dots, r - 1\\}$. Summing over the permutations $\\pi$ of $\\{1, \\dots, n\\}$\u2026","title":"Excedance and cycle polynomial of permutations with k cycles, all of length r"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/card-r-cycle-perm","kind":"node","lean":"MetaMathlibExt.card_rCyclePerm","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of permutations with all cycles of length "],["math","\\(r\\)"]],"source":"Number of permutations with all cycles of length $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#card-r-cycle-perm","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $r \\ge 2$ and $n \\ge 0$. The number of permutations of $n$ objects all of whose cycles, fixed points counted as cycles of length one, have length exactly $r$ equals $n!/(r^{n/r}\\,(n/r)!)$ if $r$ divides $n$, and $0$ otherwise.","title":"Number of permutations with all cycles of length $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-perm-count-egf-eq-exp","kind":"node","lean":"MetaMathlibExt.rCyclePermCountEGF_eq_exp","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exponential generating function for permutations with all cycles of length "],["math","\\(r\\)"]],"source":"Exponential generating function for permutations with all cycles of length $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-perm-count-egf-eq-exp","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $r \\ge 2$ and, for each $n$, let $S_n^r$ be the set of permutations of $\\{0, \\dots, n-1\\}$ all of whose cycles (fixed points counted as cycles of length one) have length exactly $r$. Then, as formal power series over $\\mathbb{Q}$,","title":"Exponential generating function for permutations with all cycles of length $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-perm-egf-eq","kind":"node","lean":"MetaMathlibExt.rAssociatedPermEGF_eq","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Excedance generating function of permutations with all cycles longer than "],["math","\\(r\\)"]],"source":"Excedance generating function of permutations with all cycles longer than $r$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-perm-egf-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $r \\ge 1$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $\\operatorname{exc}(\\sigma)$ be the number of points $i$ with $\\sigma(i) > i$, let $A_n(x) = \\sum_{\\sigma} x^{\\operatorname{exc}(\\sigma)}$, summed over all permutations of $\\{0, \\dots, n-1\\}$, and let $A(x,u) = \\sum_{n \\ge 0} A_n(x) u^n/n!$. Let\u2026","title":"Excedance generating function of permutations with all cycles longer than $r$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/exp-circular-rplus-perm-egf-eq-r-associa","kind":"node","lean":"MetaMathlibExt.exp_circularRPlusPermEGF_eq_rAssociatedPermEGF","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#exp-circular-rplus-perm-egf-eq-r-associa","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $r \\ge 0$ an integer, and work with formal power series in $u$ over $R$. For a permutation $\\sigma$ write $e(\\sigma)$ for its number of excedances (points $i$ with $\\sigma(i) > i$). Let $C^{r+}(x,u) = \\sum_{n \\ge r+1} \\sum_{\\sigma} x^{e(\\sigma)} u^n/n!$, the inner sum over the permutations of $n$ points that consist of a single cycle, and let\u2026","title":"Logarithm of the generating function of permutations with long cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-perm-egf-cycle-count-eq","kind":"node","lean":"MetaMathlibExt.rAssociatedPermEGF_cycleCount_eq","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutations with "],["math","\\(k\\)"],["text"," cycles, all of length greater than "],["math","\\(r\\)"],["text",", by excedances"]],"source":"Permutations with $k$ cycles, all of length greater than $r$, by excedances"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-perm-egf-cycle-count-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $r, k \\in \\mathbb{N}$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ the number of cycles of $\\sigma$, fixed points included. Let $A^r(x,u) = \\sum_{n \\ge 0} \\frac{u^n}{n!} \\sum_{\\sigma} x^{e(\\sigma)} \\in R[[u]]$, the inner sum over the permutations $\\sigma$ of\u2026","title":"Permutations with $k$ cycles, all of length greater than $r$, by excedances"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-perm-egf-q-eq-pow","kind":"node","lean":"MetaMathlibExt.rAssociatedPermEGF_q_eq_pow","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-analogue for cycles longer than "],["math","\\(r\\)"],["text"," is a "],["math","\\(q\\)"],["text","-th power"]],"source":"The $q$-analogue for cycles longer than $r$ is a $q$-th power"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-perm-egf-q-eq-pow","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$ and $r$ a natural number. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $\\operatorname{exc}(\\sigma)$ be the number of points $i$ with $\\sigma(i) > i$ and $\\operatorname{cyc}(\\sigma)$ its number of cycles (fixed points counted as cycles of length one). Let","title":"The $q$-analogue for cycles longer than $r$ is a $q$-th power"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/involution-egf-q-eq-exp","kind":"node","lean":"MetaMathlibExt.involutionEGF_q_eq_exp","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#involution-egf-q-eq-exp","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra and $x, q \\in R$. For a permutation $\\sigma$ write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ for its number of cycles, fixed points included. Summing over the involutions $\\sigma$ of $\\{0, \\dots, n-1\\}$ (permutations all of whose cycles have length at most $2$), the following identity of formal power series in $u$ over $R$ holds:","title":"Excedance generating function of involutions"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/fixed-point-free-involution-egf-q-eq-exp","kind":"node","lean":"MetaMathlibExt.fixedPointFreeInvolutionEGF_q_eq_exp","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#fixed-point-free-involution-egf-q-eq-exp","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra and $x, q \\in R$. For a permutation $\\sigma$ write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ for its number of cycles, fixed points included. Summing over the permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$ all of whose cycles have length exactly $2$, the following identity of formal power series in $u$ over $R$ holds:","title":"Excedance generating function of fixed-point-free involutions"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/perm-egf-cycle-count-eq","kind":"node","lean":"MetaMathlibExt.permEGF_cycleCount_eq","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Excedance generating function of permutations with "],["math","\\(k\\)"],["text"," cycles"]],"source":"Excedance generating function of permutations with $k$ cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#perm-egf-cycle-count-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra and $x \\in R$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ the number of cycles of $\\sigma$, fixed points included. Let $A(x,u) = \\sum_{n \\ge 0} \\frac{u^n}{n!} \\sum_{\\sigma} x^{e(\\sigma)} \\in R[[u]]$, the inner sum over all permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$, and for $k \\ge 0$ let\u2026","title":"Excedance generating function of permutations with $k$ cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/perm-egf-q-eq-pow","kind":"node","lean":"MetaMathlibExt.permEGF_q_eq_pow","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#perm-egf-q-eq-pow","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra and $x, q \\in R$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ the number of cycles of $\\sigma$, fixed points included. Let $A(q,x,u) = \\sum_{n \\ge 0} \\frac{u^n}{n!} \\sum_{\\sigma} x^{e(\\sigma)} q^{c(\\sigma)} \\in R[[u]]$, the inner sum over all permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$, and\u2026","title":"Brenti's excedance and cycle generating function"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/first-points-in-distinct-cycles","kind":"node","lean":"MetaMathlibExt.FirstPointsInDistinctCycles","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#first-points-in-distinct-cycles","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"For natural numbers $s$ and $n$, a permutation $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$ has its first $s$ points in distinct cycles when any two points $i, j < s$ that lie in the same cycle of $\\sigma$ are equal. When $n < s$, only the points below $n$ are constrained.","title":"First points in distinct cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/first-points-in-distinct-cycles-zero","kind":"node","lean":"MetaMathlibExt.firstPointsInDistinctCycles_zero","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first-points condition for "],["math","\\(s = 0\\)"]],"source":"The first-points condition for $s = 0$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#first-points-in-distinct-cycles-zero","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"For every $n \\ge 0$ and every permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$, the first $0$ points of $\\sigma$ lie in pairwise distinct cycles: there are no such points, so the condition is vacuous.","title":"The first-points condition for $s = 0$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/first-points-in-distinct-cycles-one","kind":"node","lean":"MetaMathlibExt.firstPointsInDistinctCycles_one","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#first-points-in-distinct-cycles-one","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"For all integers $s, n \\ge 0$, the first $s$ points of the identity permutation of $\\{0, \\dots, n-1\\}$ lie in pairwise distinct cycles: every point forms a cycle of its own.","title":"The identity permutation separates its first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/first-points-in-distinct-cycles-swap","kind":"node","lean":"MetaMathlibExt.firstPointsInDistinctCycles_swap","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","First points of the transposition "],["math","\\((0\\,1)\\)"]],"source":"First points of the transposition $(0\\,1)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#first-points-in-distinct-cycles-swap","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"fully_proved","summary":"Let $\\tau = (0\\,1)$ be the transposition of $\\{0, 1, 2\\}$. The points $0, \\dots, s-1$ lie in pairwise distinct cycles of $\\tau$ for $s = 1$ (trivially, as there is only one such point) but not for $s = 2$, since $0$ and $1$ lie in the same cycle of $\\tau$.","title":"First points of the transposition $(0\\,1)$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-perm-egf-first-points","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedPermEGF_firstPoints_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative formula for permutations with cycle lengths at most "],["math","\\(r\\)"],["text"," and separated first points"]],"source":"Derivative formula for permutations with cycle lengths at most $r$ and separated first points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-perm-egf-first-points","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Derivative formula for permutations with cycle lengths at most $r$ and separated first points. Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r, s$ be natural numbers with $r \\ge 1$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$), and let $A_j(x) = \\sum_{\\sigma \\in S_j} x^{e(\\sigma)}$. Say that the first $s$\u2026","title":"Derivative formula for permutations with cycle lengths at most $r$ and separated first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-perm-egf-first-points-2","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedPermEGF_firstPoints_cycleCount_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative formula for permutations with cycle lengths at most "],["math","\\(r\\)"],["text",", "],["math","\\(k\\)"],["text"," cycles and separated first points"]],"source":"Derivative formula for permutations with cycle lengths at most $r$, $k$ cycles and separated first points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-perm-egf-first-points-2","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Derivative formula for permutations with cycle lengths at most $r$, $k$ cycles and separated first points. Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r, s, k$ be natural numbers with $r \\ge 1$ and $s \\le k$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points\u2026","title":"Derivative formula for permutations with cycle lengths at most $r$, $k$ cycles and separated first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-minus-associated-perm-egf-first-points-3","kind":"node","lean":"MetaMathlibExt.rMinusAssociatedPermEGF_firstPoints_q_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycle-weighted derivative formula for permutations with cycle lengths at most "],["math","\\(r\\)"],["text"," and separated first points"]],"source":"Cycle-weighted derivative formula for permutations with cycle lengths at most $r$ and separated first points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-minus-associated-perm-egf-first-points-3","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Cycle-weighted derivative formula for permutations with cycle lengths at most $r$ and separated first points. Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$, and let $r, s$ be natural numbers with $r \\ge 1$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included,\u2026","title":"Cycle-weighted derivative formula for permutations with cycle lengths at most $r$ and separated first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-perm-egf-first-points-iterate-de","kind":"node","lean":"MetaMathlibExt.rCyclePermEGF_firstPoints_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative formula for "],["math","\\(r\\)"],["text","-cycle permutations with separated first points"]],"source":"Derivative formula for $r$-cycle permutations with separated first points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-perm-egf-first-points-iterate-de","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r, s$ be natural numbers with $r \\ge 2$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$), and let $A_j(x) = \\sum_{\\sigma \\in S_j} x^{e(\\sigma)}$. Call $\\sigma$ an $r$-cycle permutation when every cycle of $\\sigma$, fixed points included, has exactly $r$ elements, and\u2026","title":"Derivative formula for $r$-cycle permutations with separated first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-perm-egf-first-points-cycle-coun","kind":"node","lean":"MetaMathlibExt.rCyclePermEGF_firstPoints_cycleCount_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative formula for "],["math","\\(r\\)"],["text","-cycle permutations with "],["math","\\(k\\)"],["text"," cycles and separated first points"]],"source":"Derivative formula for $r$-cycle permutations with $k$ cycles and separated first points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-perm-egf-first-points-cycle-coun","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r, s, k$ be natural numbers with $r \\ge 2$ and $s \\le k$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included, and let $A_j(x) = \\sum_{\\sigma \\in S_j} x^{e(\\sigma)}$. Call $\\sigma$ an $r$-cycle permutation\u2026","title":"Derivative formula for $r$-cycle permutations with $k$ cycles and separated first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-cycle-perm-egf-first-points-q-iterate","kind":"node","lean":"MetaMathlibExt.rCyclePermEGF_firstPoints_q_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycle-weighted derivative formula for "],["math","\\(r\\)"],["text","-cycle permutations with separated first points"]],"source":"Cycle-weighted derivative formula for $r$-cycle permutations with separated first points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-cycle-perm-egf-first-points-q-iterate","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$, and let $r, s$ be natural numbers with $r \\ge 2$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ write $e(\\sigma)$ for its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included, and let $A_j(x) = \\sum_{\\sigma \\in S_j} x^{e(\\sigma)}$. Call $\\sigma$ an $r$-cycle permutation when every\u2026","title":"Cycle-weighted derivative formula for $r$-cycle permutations with separated first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-perm-egf-first-points-itera","kind":"node","lean":"MetaMathlibExt.rAssociatedPermEGF_firstPoints_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterated derivative for cycles longer than "],["math","\\(r\\)"],["text"," with "],["math","\\(s\\)"],["text"," points in distinct cycles"]],"source":"Iterated derivative for cycles longer than $r$ with $s$ points in distinct cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-perm-egf-first-points-itera","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r$, $s$ be natural numbers with $r \\ge 1$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $\\operatorname{exc}(\\sigma)$ be the number of points $i$ with $\\sigma(i) > i$, let $A_n(x) = \\sum_{\\sigma} x^{\\operatorname{exc}(\\sigma)}$, summed over all permutations of $\\{0, \\dots, n-1\\}$, and let $A(x,u) = \\sum_{n \\ge 0} A_n(x) u^n/n!$. Let\u2026","title":"Iterated derivative for cycles longer than $r$ with $s$ points in distinct cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-perm-egf-first-points-cycle","kind":"node","lean":"MetaMathlibExt.rAssociatedPermEGF_firstPoints_cycleCount_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterated derivative for "],["math","\\(k\\)"],["text"," cycles longer than "],["math","\\(r\\)"],["text"," with "],["math","\\(s\\)"],["text"," points in distinct cycles"]],"source":"Iterated derivative for $k$ cycles longer than $r$ with $s$ points in distinct cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-perm-egf-first-points-cycle","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and let $r$, $s$, $k$ be natural numbers with $r \\ge 1$ and $s \\le k$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $\\operatorname{exc}(\\sigma)$ be the number of points $i$ with $\\sigma(i) > i$, let $A_n(x) = \\sum_{\\sigma} x^{\\operatorname{exc}(\\sigma)}$, summed over all permutations of $\\{0, \\dots, n-1\\}$, and let\u2026","title":"Iterated derivative for $k$ cycles longer than $r$ with $s$ points in distinct cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/r-associated-perm-egf-first-points-q-ite","kind":"node","lean":"MetaMathlibExt.rAssociatedPermEGF_firstPoints_q_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterated derivative of the "],["math","\\(q\\)"],["text","-analogue for cycles longer than "],["math","\\(r\\)"],["text",", "],["math","\\(s\\)"],["text"," points apart"]],"source":"Iterated derivative of the $q$-analogue for cycles longer than $r$, $s$ points apart"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#r-associated-perm-egf-first-points-q-ite","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$, and let $r$, $s$ be natural numbers with $r \\ge 1$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $\\operatorname{exc}(\\sigma)$ be the number of points $i$ with $\\sigma(i) > i$ and $\\operatorname{cyc}(\\sigma)$ its number of cycles (fixed points counted as cycles of length one), let $A_n(x) = \\sum_{\\sigma} x^{\\operatorname{exc}(\\sigma)}$, summed\u2026","title":"Iterated derivative of the $q$-analogue for cycles longer than $r$, $s$ points apart"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/involution-excedance-cycle-sum","kind":"node","lean":"MetaMathlibExt.involution_excedance_cycle_sum","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#involution-excedance-cycle-sum","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$ and $n \\ge 0$. For a permutation $\\sigma$ write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ for its number of cycles, fixed points included. For each $m$ let $W(m) = \\sum_{\\sigma} x^{e(\\sigma)} q^{c(\\sigma)}$, summed over the involutions $\\sigma$ of $\\{0, \\dots, m-1\\}$ (permutations all of whose cycles have length at most $2$).\u2026","title":"Excedances and cycles of involutions"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/fixed-point-free-involution-excedance-cy","kind":"node","lean":"MetaMathlibExt.fixedPointFreeInvolution_excedance_cycle_sum","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#fixed-point-free-involution-excedance-cy","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$ and $n \\ge 0$. For a permutation $\\sigma$ write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ for its number of cycles, fixed points included. Summing over the permutations $\\sigma$ of $\\{0, \\dots, 2n-1\\}$ all of whose cycles have length exactly $2$,","title":"Excedances and cycles of fixed-point-free involutions"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/perm-egf-first-points-iterate-derivative","kind":"node","lean":"MetaMathlibExt.permEGF_firstPoints_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative formula for permutations whose first "],["math","\\(s\\)"],["text"," points lie in distinct cycles"]],"source":"Derivative formula for permutations whose first $s$ points lie in distinct cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#perm-egf-first-points-iterate-derivative","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $s \\in \\mathbb{N}$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$. Let $A(x,u) = \\sum_{n \\ge 0} \\frac{u^n}{n!} \\sum_{\\sigma} x^{e(\\sigma)} \\in R[[u]]$, the inner sum over all permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$, and let $A_{[s]}(x,u)$ be the same series with the inner sum restricted to\u2026","title":"Derivative formula for permutations whose first $s$ points lie in distinct cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/perm-egf-first-points-cycle-count-iterat","kind":"node","lean":"MetaMathlibExt.permEGF_firstPoints_cycleCount_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative formula for permutations with "],["math","\\(k\\)"],["text"," cycles whose first "],["math","\\(s\\)"],["text"," points lie in distinct cycles"]],"source":"Derivative formula for permutations with $k$ cycles whose first $s$ points lie in distinct cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#perm-egf-first-points-cycle-count-iterat","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Derivative formula for permutations with $k$ cycles whose first $s$ points lie in distinct cycles. Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $s \\le k$ natural numbers. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ the number of cycles of $\\sigma$, fixed points included. Let\u2026","title":"Derivative formula for permutations with $k$ cycles whose first $s$ points lie in distinct cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/perm-egf-first-points-q-iterate-derivati","kind":"node","lean":"MetaMathlibExt.permEGF_firstPoints_q_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(q\\)"],["text","-analogue of the derivative formula for permutations whose first "],["math","\\(s\\)"],["text"," points lie in distinct cycles"]],"source":"$q$-analogue of the derivative formula for permutations whose first $s$ points lie in distinct cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#perm-egf-first-points-q-iterate-derivati","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"$q$-analogue of the derivative formula for permutations whose first $s$ points lie in distinct cycles. Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$ and $s \\in \\mathbb{N}$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ the number of cycles of $\\sigma$, fixed points included. Let\u2026","title":"$q$-analogue of the derivative formula for permutations whose first $s$ points lie in distinct cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/first-points-excedance-cycle-sum-neg","kind":"node","lean":"MetaMathlibExt.firstPoints_excedance_cycle_sum_neg","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed excedance sum with weight "],["math","\\(-s\\)"],["text"," over permutations with separated first points"]],"source":"Signed excedance sum with weight $-s$ over permutations with separated first points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#first-points-excedance-cycle-sum-neg","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative ring, $x \\in R$ and $n, s \\ge 0$ integers. For a permutation $\\sigma$ write $e(\\sigma)$ for its number of excedances (points $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included. Summing over the permutations $\\sigma$ of $\\{0, \\dots, n+s-1\\}$ in which the points $0, \\dots, s-1$ lie in pairwise distinct cycles,\u2026","title":"Signed excedance sum with weight $-s$ over permutations with separated first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/first-points-excedance-cycle-sum-neg-suc","kind":"node","lean":"MetaMathlibExt.firstPoints_excedance_cycle_sum_neg_succ","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed excedance sum with weight "],["math","\\(-(s+1)\\)"],["text"," over permutations with separated first points"]],"source":"Signed excedance sum with weight $-(s+1)$ over permutations with separated first points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#first-points-excedance-cycle-sum-neg-suc","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative ring, $x \\in R$ and $n \\ge 1$, $s \\ge 0$ integers. For a permutation $\\sigma$ write $e(\\sigma)$ for its number of excedances (points $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ for its number of cycles, fixed points included. Summing over the permutations $\\sigma$ of $\\{0, \\dots, n+s-1\\}$ in which the points $0, \\dots, s-1$ lie in pairwise distinct cycles,\u2026","title":"Signed excedance sum with weight $-(s+1)$ over permutations with separated first points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/derangement-egf-first-points-q-iterate-d","kind":"node","lean":"MetaMathlibExt.derangementEGF_firstPoints_q_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derangements whose first "],["math","\\(s\\)"],["text"," points lie in distinct cycles"]],"source":"Derangements whose first $s$ points lie in distinct cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#derangement-egf-first-points-q-iterate-d","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$ and $s \\ge 0$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ its number of cycles, fixed points included. Let $A(x,u) = \\sum_{n \\ge 0} \\sum_{\\sigma} x^{e(\\sigma)} u^n/n!$, summed over all permutations $\\sigma$ of $n$ objects, and let $L \\in R[[u]]$ have constant term $0$ and\u2026","title":"Derangements whose first $s$ points lie in distinct cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/derangement-first-points-excedance-cycle","kind":"node","lean":"MetaMathlibExt.derangement_firstPoints_excedance_cycle_sum_neg","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derangements with the first "],["math","\\(s\\)"],["text"," points in distinct cycles, cycle weight "],["math","\\(-s\\)"]],"source":"Derangements with the first $s$ points in distinct cycles, cycle weight $-s$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#derangement-first-points-excedance-cycle","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $n, s \\ge 0$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ its number of cycles, fixed points included. Summing over the derangements $\\sigma$ of $n + s$ objects (permutations without fixed points) in which the first $s$ points lie in pairwise distinct cycles, the identity\u2026","title":"Derangements with the first $s$ points in distinct cycles, cycle weight $-s$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/derangement-first-points-excedance-cycle-2","kind":"node","lean":"MetaMathlibExt.derangement_firstPoints_excedance_cycle_sum_neg_succ","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derangements with the first "],["math","\\(s\\)"],["text"," points in distinct cycles, cycle weight "],["math","\\(-(s+1)\\)"]],"source":"Derangements with the first $s$ points in distinct cycles, cycle weight $-(s+1)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#derangement-first-points-excedance-cycle-2","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $n, s \\ge 0$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ its number of cycles, fixed points included. Summing over the derangements $\\sigma$ of $n + s$ objects (permutations without fixed points) in which the first $s$ points lie in pairwise distinct cycles, the identity\u2026","title":"Derangements with the first $s$ points in distinct cycles, cycle weight $-(s+1)$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/fixed-points-perm-egf-iterate-derivative","kind":"node","lean":"MetaMathlibExt.fixedPointsPermEGF_iterate_derivative","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#fixed-points-perm-egf-iterate-derivative","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$ and $s$ a natural number. Call a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ an $s$-fixed-points permutation if every fixed point of $\\sigma$ is less than $s$ and any two points less than $s$ in the same cycle of $\\sigma$ are equal. Write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$, let $A(x,u) = \\sum_{n \\ge 0} \\sum_{\\sigma} x^{e(\\sigma)} u^n/n!$\u2026","title":"Excedance generating function of s-fixed-points permutations"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/fixed-points-perm-egf-cycle-count-iterat","kind":"node","lean":"MetaMathlibExt.fixedPointsPermEGF_cycleCount_iterate_derivative","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#fixed-points-perm-egf-cycle-count-iterat","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, and $s \\le k$ natural numbers. Call a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ an $s$-fixed-points permutation if every fixed point of $\\sigma$ is less than $s$ and any two points less than $s$ in the same cycle of $\\sigma$ are equal. Write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ for the number of cycles of $\\sigma$, fixed\u2026","title":"Excedance generating function of s-fixed-points permutations with k cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/fixed-points-perm-egf-q-iterate-derivati","kind":"node","lean":"MetaMathlibExt.fixedPointsPermEGF_q_iterate_derivative","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#fixed-points-perm-egf-q-iterate-derivati","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$ and $s$ a natural number. Call a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ an $s$-fixed-points permutation if every fixed point of $\\sigma$ is less than $s$ and any two points less than $s$ in the same cycle of $\\sigma$ are equal. Write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ for the number of cycles of $\\sigma$, fixed\u2026","title":"Cycle-counting excedance generating function of s-fixed-points permutations"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/fixed-points-perm-excedance-cycle-sum-ne","kind":"node","lean":"MetaMathlibExt.fixedPointsPerm_excedance_cycle_sum_neg","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#fixed-points-perm-excedance-cycle-sum-ne","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative ring, $x \\in R$, and $s \\le n$ natural numbers. Call a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ an $s$-fixed-points permutation if every fixed point of $\\sigma$ is less than $s$ and any two points less than $s$ in the same cycle of $\\sigma$ are equal. Write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ for the number of cycles of $\\sigma$, fixed points included.\u2026","title":"Signed excedance\u2013cycle sum over s-fixed-points permutations, cycle weight -s"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/fixed-points-perm-excedance-cycle-sum-ne-2","kind":"node","lean":"MetaMathlibExt.fixedPointsPerm_excedance_cycle_sum_neg_succ","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#fixed-points-perm-excedance-cycle-sum-ne-2","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $s \\le n$ be natural numbers. Call a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ an $s$-fixed-points permutation if every fixed point of $\\sigma$ is less than $s$ and any two points less than $s$ in the same cycle of $\\sigma$ are equal. Write $e(\\sigma)$ for the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ for the number of cycles of $\\sigma$, fixed points included. Then, summing over the $s$-fixed-points\u2026","title":"Signed excedance\u2013cycle sum over s-fixed-points permutations, cycle weight -s-1"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/prescribed-cycle-perm-egf-iterate-deriva","kind":"node","lean":"MetaMathlibExt.prescribedCyclePermEGF_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative formula for prescribed cycle lengths through the first "],["math","\\(s\\)"],["text"," points"]],"source":"Derivative formula for prescribed cycle lengths through the first $s$ points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#prescribed-cycle-perm-egf-iterate-deriva","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, $s \\in \\mathbb{N}$ and $r : \\mathbb{N} \\to \\mathbb{N}$ with $r(i) \\ge 2$ for all $i < s$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$, and for $m \\ge 0$ let $A_m(x) = \\sum_{\\sigma} x^{e(\\sigma)}$ over all permutations $\\sigma$ of $\\{0, \\dots, m-1\\}$. Let\u2026","title":"Derivative formula for prescribed cycle lengths through the first $s$ points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/prescribed-cycle-perm-egf-cycle-count-it","kind":"node","lean":"MetaMathlibExt.prescribedCyclePermEGF_cycleCount_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative formula for prescribed cycle lengths through the first "],["math","\\(s\\)"],["text"," points, with "],["math","\\(k\\)"],["text"," cycles"]],"source":"Derivative formula for prescribed cycle lengths through the first $s$ points, with $k$ cycles"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#prescribed-cycle-perm-egf-cycle-count-it","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Derivative formula for prescribed cycle lengths through the first $s$ points, with $k$ cycles. Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x \\in R$, $s \\le k$ natural numbers and $r : \\mathbb{N} \\to \\mathbb{N}$ with $r(i) \\ge 2$ for all $i < s$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ the number of cycles of $\\sigma$, fixed\u2026","title":"Derivative formula for prescribed cycle lengths through the first $s$ points, with $k$ cycles"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/prescribed-cycle-perm-egf-q-iterate-deri","kind":"node","lean":"MetaMathlibExt.prescribedCyclePermEGF_q_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(q\\)"],["text","-analogue of the derivative formula for prescribed cycle lengths through the first "],["math","\\(s\\)"],["text"," points"]],"source":"$q$-analogue of the derivative formula for prescribed cycle lengths through the first $s$ points"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#prescribed-cycle-perm-egf-q-iterate-deri","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"$q$-analogue of the derivative formula for prescribed cycle lengths through the first $s$ points. Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$, $s \\in \\mathbb{N}$ and $r : \\mathbb{N} \\to \\mathbb{N}$ with $r(i) \\ge 2$ for all $i < s$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let $e(\\sigma)$ be the number of $i$ with $\\sigma(i) > i$ and $c(\\sigma)$ the number of cycles of $\\sigma$, fixed\u2026","title":"$q$-analogue of the derivative formula for prescribed cycle lengths through the first $s$ points"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/cycle-lengths-in-egf-q-eq-exp","kind":"node","lean":"MetaMathlibExt.cycleLengthsInEGF_q_eq_exp","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#cycle-lengths-in-egf-q-eq-exp","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$, and let $\\mathcal{R}$ be a finite set of integers, each at least $2$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ its number of cycles, fixed points included, and for $j \\ge 0$ let $A_j(x) = \\sum_{\\sigma} x^{e(\\sigma)}$, summed over all permutations $\\sigma$ of $j$ objects. Then\u2026","title":"Permutations with all cycle lengths in a prescribed set"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/prescribed-cycle-lengths-in-egf-q-iterat","kind":"node","lean":"MetaMathlibExt.prescribedCycleLengthsInEGF_q_iterate_derivative","parent":25448,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(q\\)"],["text","-analogue of the derivative formula for prescribed cycle lengths through the first "],["math","\\(s\\)"],["text"," points, other cycle lengths in a set "],["math","\\(\\mathcal{R}\\)"]],"source":"$q$-analogue of the derivative formula for prescribed cycle lengths through the first $s$ points, other cycle lengths in a set $\\mathcal{R}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#prescribed-cycle-lengths-in-egf-q-iterat","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"$q$-analogue of the derivative formula for prescribed cycle lengths through the first $s$ points, other cycle lengths in a set $\\mathcal{R}$. Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$, $s \\in \\mathbb{N}$, $r : \\mathbb{N} \\to \\mathbb{N}$ with $r(i) \\ge 2$ for all $i < s$, and $\\mathcal{R}$ a finite set of natural numbers, each at least $2$. For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ let\u2026","title":"$q$-analogue of the derivative formula for prescribed cycle lengths through the first $s$ points, other cycle lengths in a set $\\mathcal{R}$"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/cycle-lengths-not-in-egf-q-eq","kind":"node","lean":"MetaMathlibExt.cycleLengthsNotInEGF_q_eq","parent":25448,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers/index.html#cycle-lengths-not-in-egf-q-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/associated-stirling-eulerian-numbers","status":"can_prove","summary":"Let $R$ be a commutative $\\mathbb{Q}$-algebra, $x, q \\in R$, and let $\\mathcal{R}$ be a finite set of integers, each at least $2$. For a permutation $\\sigma$ let $e(\\sigma)$ be its number of excedances (indices $i$ with $\\sigma(i) > i$) and $c(\\sigma)$ its number of cycles, fixed points included; for $j \\ge 0$ let $A_j(x) = \\sum_{\\sigma} x^{e(\\sigma)}$, summed over all permutations $\\sigma$ of $j$ objects; and let\u2026","title":"Permutations avoiding a prescribed set of cycle lengths"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances.json","id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances","kind":"scope","navigation":"module","parent":25446,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances/index.html","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances","status":"fully_proved","title":"Bell permutations and weak excedances"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances/bell-perm-second-kind-weak-excedance-eq","kind":"node","lean":"MetaMathlibExt.bell_perm_second_kind_weakExcedance_eq_stirlingSecond","parent":25528,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances/index.html#bell-perm-second-kind-weak-excedance-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances","status":"fully_proved","summary":"Let $n \\ge 1$ and $k$ be natural numbers, and write $[n] = \\{0, \\dots, n-1\\}$. A function $f : [n] \\to [n]$ is a restricted growth function if for every $i$, every $j < f(i)$ equals $f(i_0)$ for some $i_0 < i$. Its Bell permutation is the product of transpositions $(0\\ f(0))(1\\ f(1)) \\cdots (n-1\\ f(n-1))$, composed so that the rightmost factor acts first, where $(i\\ i)$ denotes the identity. A weak excedance of a\u2026","title":"Weak excedances of Bell permutations of the second kind"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances/bell-count-perm-increasing-weak-excedanc","kind":"node","lean":"MetaMathlibExt.bell_count_perm_increasing_weakExcedance_letters","parent":25528,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances/index.html#bell-count-perm-increasing-weak-excedanc","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/bell-permutations-and-weak-excedances","status":"fully_proved","summary":"A position $i$ is a weak excedance of a permutation $\\sigma$ when $i \\le \\sigma(i)$. For every $n$, the number of permutations $\\sigma$ of $\\{0, 1, \\dots, n-1\\}$ such that $\\sigma(i) < \\sigma(j)$ whenever $i < j$ are both weak excedances of $\\sigma$ is the Bell number $B_n$.","title":"Bell numbers count permutations with increasing weak excedance letters"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations.json","id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations","kind":"scope","navigation":"module","parent":25446,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations/index.html","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations","status":"fully_proved","title":"Eulerian numbers of multiset permutations"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations/multiset-descent-number","kind":"node","lean":"MetaMathlibExt.multisetDescentNumber","parent":25531,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations/index.html#multiset-descent-number","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations","status":"fully_proved","summary":"For a word $w = w_1 \\cdots w_K$ of length $K$ over the alphabet $\\{0, 1, \\dots, m-1\\}$, the descent number of $w$ is one more than its number of descents: $1 + \\#\\{j : 1 \\le j < K,\\ w_j > w_{j+1}\\}$.","title":"Descent number of a word"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations/is-multiset-permutation","kind":"node","lean":"MetaMathlibExt.IsMultisetPermutation","parent":25531,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations/index.html#is-multiset-permutation","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations","status":"fully_proved","summary":"Let $k_0, \\dots, k_{m-1}$ be natural numbers. A word $w = w_1 \\cdots w_K$ of length $K$ over the alphabet $\\{0, 1, \\dots, m-1\\}$ is a permutation of the multiset with $k_i$ copies of each letter $i$ when, for every letter $i$, exactly $k_i$ positions $j$ have $w_j = i$.","title":"Multiset permutation"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations/multiset-eulerian-number","kind":"node","lean":"MetaMathlibExt.multisetEulerianNumber","parent":25531,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations/index.html#multiset-eulerian-number","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/eulerian-numbers-of-multiset-permutations","status":"fully_proved","summary":"Let $k_0, \\dots, k_{m-1}$ be natural numbers with sum $K$, and let $p$ be a natural number. The multiset Eulerian number is the number of words of length $K$ over the alphabet $\\{0, 1, \\dots, m-1\\}$ that are permutations of the multiset with $k_i$ copies of each letter $i$ and whose descent number, one more than the number of descents, equals $p$.","title":"Eulerian numbers of multiset permutations"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials.json","id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials","kind":"scope","navigation":"module","parent":25446,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials/index.html","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials","status":"fully_proved","title":"General arithmetical Eulerian polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials/general-arithmetical-eulerian-number","kind":"node","lean":"MetaMathlibExt.generalArithmeticalEulerianNumber","parent":25535,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials/index.html#general-arithmetical-eulerian-number","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials","status":"fully_proved","summary":"Let $R$ be a ring, let $n$ and $k$ be natural numbers, and let $a, d \\in R$. The general arithmetical Eulerian number $A_{n,k}(a,d) \\in R$ is","title":"General arithmetical Eulerian number"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials/general-arithmetical-eulerian-polynomial","kind":"node","lean":"MetaMathlibExt.generalArithmeticalEulerianPolynomial","parent":25535,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials/index.html#general-arithmetical-eulerian-polynomial","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/general-arithmetical-eulerian-polynomials","status":"fully_proved","summary":"Let $R$ be a ring, let $n$ be a natural number, and let $a, d \\in R$. For each natural number $k$, write $A_{n,k}(a,d) \\in R$ for the general arithmetical Eulerian number. The general arithmetical Eulerian polynomial is the polynomial in the indeterminate $t$","title":"General arithmetical Eulerian polynomial"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit.json","id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","kind":"scope","navigation":"module","parent":25446,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"can_prove","title":"Generalized Eulerian polynomials and gamma-positivity"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/double-excedance-count","kind":"node","lean":"MetaMathlibExt.doubleExcedanceCount","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#double-excedance-count","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"fully_proved","summary":"For a permutation $w$ of $\\{0, 1, \\dots, n-1\\}$, a double excedance of $w$ is a point $i$ with $w^{-1}(i) < i < w(i)$, and $\\mathrm{dexc}(w)$ is the number of double excedances of $w$.","title":"Number of double excedances"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/double-excedance-count-one","kind":"node","lean":"MetaMathlibExt.doubleExcedanceCount_one","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#double-excedance-count-one","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"fully_proved","summary":"For every $n$, the identity permutation of $\\{0, 1, \\dots, n-1\\}$ has no double excedances.","title":"The identity has no double excedances"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/generalized-eulerian-series","kind":"node","lean":"MetaMathlibExt.generalizedEulerianSeries","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#generalized-eulerian-series","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"fully_proved","summary":"For rational numbers $q$, $t$, $x$, $y$ with $x \\ne y$, this is the formal power series in $z$ over $\\mathbb{Q}$","title":"Generating function of the generalized Eulerian polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/generalized-eulerian-p","kind":"node","lean":"MetaMathlibExt.generalizedEulerianP","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#generalized-eulerian-p","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"fully_proved","summary":"For rational numbers $q, t, x, y$ and $n \\ge 0$, the generalized Eulerian polynomial $P_n(q, t; x, y)$ is $n!$ times the coefficient of $z^n$ in the formal power series over $\\mathbb{Q}$","title":"Generalized Eulerian polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/generalized-eulerian-p-zero-left","kind":"node","lean":"MetaMathlibExt.generalizedEulerianP_zero_left","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#generalized-eulerian-p-zero-left","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"fully_proved","summary":"For all rational numbers $t, x, y$ and every $n \\ge 0$, the generalized Eulerian polynomial satisfies $P_0(0, t; x, y) = 1$ and $P_n(0, t; x, y) = 0$ for $n \\ge 1$.","title":"Generalized Eulerian polynomials with zero cycle weight"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/cyclic-excedance-egf-eq","kind":"node","lean":"MetaMathlibExt.cyclicExcedanceEGF_eq","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#cyclic-excedance-egf-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"can_prove","summary":"Let $x \\ne 1$ be a rational number. For $n \\ge 1$ let $C_n(x) = \\sum_{m=0}^{n} C_{n,m} x^m$, where the circular Eulerian number $C_{n,m}$ is the number of permutations $w$ of $\\{0, 1, \\dots, n-1\\}$ whose points all lie in a single cycle and which have exactly $m$ excedances, that is, points $i$ with $w(i) > i$. Then, as formal power series in $z$ over $\\mathbb{Q}$,","title":"Exponential generating function of excedances over cyclic permutations"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/generalized-eulerian-p-eq-sum-perm","kind":"node","lean":"MetaMathlibExt.generalizedEulerianP_eq_sum_perm","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#generalized-eulerian-p-eq-sum-perm","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"can_prove","summary":"Let $P_n(q, t; x, y)$ be the generalized Eulerian polynomial and let $S_n$ be the set of permutations of $\\{0, 1, \\dots, n-1\\}$. For $w \\in S_n$ write $\\mathrm{cyc}(w)$ for the number of cycles of $w$ (fixed points counted as cycles), $\\mathrm{fix}(w)$ for the number of fixed points of $w$, and $\\mathrm{exc}(w)$ for the number of excedances of $w$, the points $i$ with $w(i) > i$. Then for all rational numbers\u2026","title":"Permutation expansion of the generalized Eulerian polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/generalized-eulerian-p-gamma-nonneg","kind":"node","lean":"MetaMathlibExt.generalizedEulerianP_gamma_nonneg","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#generalized-eulerian-p-gamma-nonneg","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"can_prove","summary":"The gamma coefficients of the generalized Eulerian polynomials have nonnegative integer coefficients. Fix $n \\ge 0$ and let $P_n(q, t; x, y)$ be the generalized Eulerian polynomial. There are natural numbers $c_{k,i,j}$, for $k, i, j \\ge 0$, such that, writing $\\gamma_{n,k}(q, t) = \\sum_{i=0}^{n} \\sum_{j=0}^{n} c_{k,i,j} q^i t^j$, for all rational numbers $q, t, x, y$ with $x \\ne y$,","title":"The gamma coefficients of the generalized Eulerian polynomials have nonnegative integer coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/generalized-eulerian-p-unimodal","kind":"node","lean":"MetaMathlibExt.generalizedEulerianP_unimodal","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#generalized-eulerian-p-unimodal","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"can_prove","summary":"Let $P_n(q, t; x, y)$ be the generalized Eulerian polynomial. For all natural numbers $q$, $t$ and $n$ there are a polynomial $p$ with rational coefficients $p_0, p_1, p_2, \\dots$ and an index $m$ such that $p(x) = P_n(q, t; x, 1)$ for every rational $x \\ne 1$, $p_i \\le p_{i+1}$ for every $i < m$, and $p_{i+1} \\le p_i$ for every $i \\ge m$. That is, $P_n(q, t; x, 1)$ is a unimodal polynomial in $x$.","title":"Unimodality of the generalized Eulerian polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/generalized-eulerian-p-gamma-eq","kind":"node","lean":"MetaMathlibExt.generalizedEulerianP_gamma_eq","parent":25538,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#generalized-eulerian-p-gamma-eq","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"can_prove","summary":"Fix $n \\ge 0$ and let $P_n(q, t; x, y)$ be the generalized Eulerian polynomial. For a permutation $w$ of $\\{0, 1, \\dots, n-1\\}$ write $\\mathrm{cyc}(w)$ for its number of cycles (fixed points counted as cycles), $\\mathrm{fix}(w)$ for its number of fixed points, $\\mathrm{exc}(w)$ for its number of excedances (points $i$ with $w(i) > i$) and $\\mathrm{dexc}(w)$ for its number of double excedances. For $k \\ge 0$ and\u2026","title":"The gamma coefficients of the generalized Eulerian polynomials count permutations"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/generalized-eulerian-p-neg-one","kind":"node","lean":"MetaMathlibExt.generalizedEulerianP_neg_one","parent":25538,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Eulerian polynomials at "],["math","\\(x = -1\\)"],["text",", "],["math","\\(y = 1\\)"]],"source":"Generalized Eulerian polynomials at $x = -1$, $y = 1$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit/index.html#generalized-eulerian-p-neg-one","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/generalized-eulerian-polynomials-and-gamma-posit","status":"can_prove","summary":"Let $P_m(q, t; x, y)$ be the generalized Eulerian polynomial. For a permutation $w$ write $\\mathrm{cyc}(w)$ for its number of cycles (fixed points counted as cycles), $\\mathrm{exc}(w)$ for its number of excedances (points $i$ with $w(i) > i$) and $\\mathrm{dexc}(w)$ for its number of double excedances, and let $D_{2n}$ be the set of derangements (permutations without fixed points) of $\\{0, 1, \\dots, 2n-1\\}$. Then\u2026","title":"Generalized Eulerian polynomials at $x = -1$, $y = 1$"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials.json","id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials","kind":"scope","navigation":"module","parent":25446,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials/index.html","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials","status":"fully_proved","title":"Hankel determinants of Eulerian polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials/eulerian-polynomial-hankel-determinant","kind":"node","lean":"MetaMathlibExt.eulerianPolynomial_hankelDeterminant","parent":25550,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials/index.html#eulerian-polynomial-hankel-determinant","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials","status":"fully_proved","summary":"For $m \\ge 0$ let $P_m \\in \\mathbb{Z}[X]$ be the Eulerian polynomial $P_m = \\sum_{k=0}^{m} W_{m,k} X^k$, where $W_{m,k} = \\sum_{i=0}^{m-k} (-1)^i \\binom{m+1}{i} (m-k-i)^m$ with $0^0 = 1$ (so $P_0 = 1$ and $P_2 = 1 + X$). Then for every $n \\ge 0$ the Hankel determinant $\\det(P_{i+j})_{0 \\le i, j \\le n}$ equals $X^{\\binom{n+1}{2}} \\prod_{k=1}^{n} (k!)^2$.","title":"Hankel determinants of the Eulerian polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials/hankel-transform-shifted-eulerian-polyno","kind":"node","lean":"MetaMathlibExt.hankel_transform_shifted_eulerian_polynomials","parent":25550,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials/index.html#hankel-transform-shifted-eulerian-polyno","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/hankel-determinants-of-eulerian-polynomials","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and $x \\in \\mathbb{R}$. For natural numbers $k \\le m$ let $W_{m,k} = \\sum_{i=0}^{m-k} (-1)^i \\binom{m+1}{i} (m-k-i)^m$, with $0^0 = 1$, and let $P_m(X) = \\sum_{k=0}^{m} W_{m,k} X^k$ be the corresponding Eulerian polynomial. Then the $(n+1) \\times (n+1)$ Hankel determinant of the shifted sequence $P_1(x), P_2(x), \\dots$ is","title":"Hankel determinants of the shifted Eulerian polynomials"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials.json","id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials","kind":"scope","navigation":"module","parent":25446,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials/index.html","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials","status":"fully_proved","title":"Refined Eulerian polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials/sun-refined-eulerian-marginals","kind":"node","lean":"MetaMathlibExt.sun_refined_eulerian_marginals","parent":25553,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials/index.html#sun-refined-eulerian-marginals","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials","status":"fully_proved","summary":"Let $n \\ge 1$. For a permutation $\\sigma$ of $\\{1, \\dots, n\\}$, let $\\mathrm{odes}(\\sigma)$ (resp. $\\mathrm{edes}(\\sigma)$) be the number of odd (resp. even) positions $i \\in \\{1, \\dots, n-1\\}$ with $\\sigma(i) > \\sigma(i+1)$, and let $A_n(p, q) = \\sum_{\\sigma} p^{\\mathrm{odes}(\\sigma)} q^{\\mathrm{edes}(\\sigma)}$, the sum over all permutations $\\sigma$ of $\\{1, \\dots, n\\}$. Then for all real numbers $p$ and $q$,","title":"Marginals of the refined Eulerian polynomial"},{"id":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials/refined-eulerian-polynomial-palindromic","kind":"node","lean":"MetaMathlibExt.refinedEulerianPolynomial_palindromic","parent":25553,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials/index.html#refined-eulerian-polynomial-palindromic","scope":"Combinatorics/stirling-bell-eulerian/eulerian-numbers/refined-eulerian-polynomials","status":"fully_proved","summary":"Let $n \\ge 1$. For a permutation $\\sigma$ of $\\{1, \\dots, n\\}$, let $\\mathrm{odes}(\\sigma)$ (resp. $\\mathrm{edes}(\\sigma)$) be the number of odd (resp. even) positions $i \\in \\{1, \\dots, n-1\\}$ with $\\sigma(i) > \\sigma(i+1)$, and let $A_n(p, q) = \\sum_{\\sigma} p^{\\mathrm{odes}(\\sigma)} q^{\\mathrm{edes}(\\sigma)}$, the sum over all permutations $\\sigma$ of $\\{1, \\dots, n\\}$, be the refined Eulerian polynomial with\u2026","title":"Palindromicity of the refined Eulerian polynomials"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi.json","id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi","kind":"scope","parent":25194,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/index.html","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi","status":"can_prove","summary":"Combinatorial formulas for Bernoulli, Euler, tangent and Genocchi numbers (Garabedian's formulas, median Euler and Genocchi numbers, Euler\u2013Seidel matrices, Carlitz q-Bernoulli polynomials), their divisibility, and sums of powers via Faulhaber coefficients.","title":"Bernoulli, Euler and Genocchi numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/tangent-number-two-adic-divisibility-mod","kind":"node","lean":"MetaMathlibExt.tangentNumber_two_adic_divisibility_mod_six","parent":25556,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/index.html#tangent-number-two-adic-divisibility-mod","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi","status":"fully_proved","summary":"Let $T \\colon \\mathbb{N} \\to \\mathbb{Z}$ satisfy $T(0) = 0$ and, for every $m \\in \\mathbb{N}$,","title":"Tangent numbers: 2-adic divisibility and a congruence mod 6"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/weighted-affine-power-sum","kind":"node","lean":"MetaMathlibExt.weightedAffinePowerSum","parent":25556,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/index.html#weighted-affine-power-sum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi","status":"fully_proved","summary":"Let $R$ be a ring, $a, b$ integers, $u \\in R$ and $n, p$ natural numbers. The weighted affine power sum is","title":"Weighted affine power sum"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials.json","id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","kind":"scope","navigation":"module","parent":25556,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","title":"Carlitz q-Bernoulli polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/q-bracket","kind":"node","lean":"MetaMathlibExt.qBracket","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-analogue of a real number"]],"source":"The $q$-analogue of a real number"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#q-bracket","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For real numbers $q$ and $x$, the $q$-analogue of $x$ is","title":"The $q$-analogue of a real number"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/is-carlitz-type-with","kind":"node","lean":"MetaMathlibExt.IsCarlitzTypeWith","parent":25559,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#is-carlitz-type-with","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For real $q$, a sequence of functions $T_n : \\mathbb{R} \\to \\mathbb{R}$, $n \\ge 0$, is Carlitz-type with associated real sequence $(a_k)_{k \\ge 0}$ if for every natural number $n$ and every real $x$, $T_n(x) = (q - 1)^{-n} \\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k} a_k q^{kx}$.","title":"Carlitz-type sequences"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/carlitz-eta-poly","kind":"node","lean":"MetaMathlibExt.carlitzEtaPoly","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Carlitz "],["math","\\(q\\)"],["text","-polynomials "],["math","\\(\\eta_n\\)"]],"source":"Carlitz $q$-polynomials $\\eta_n$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#carlitz-eta-poly","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For real $q$, write $[y] = (q^y - 1)/(q - 1)$ and let $a_0 = 1$ and $a_k = k/[k]$ for $k \\ge 1$. For natural $n$ and real $x$, Carlitz's $q$-polynomial is $\\eta_n(x) = (q - 1)^{-n} \\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k} a_k q^{kx}$; that is, $(\\eta_n)$ is the Carlitz-type sequence with associated sequence $(a_k)$. Its values at $0$ are written $\\eta_n = \\eta_n(0)$.","title":"Carlitz $q$-polynomials $\\eta_n$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/carlitz-beta-poly","kind":"node","lean":"MetaMathlibExt.carlitzBetaPoly","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Carlitz "],["math","\\(q\\)"],["text","-Bernoulli polynomials"]],"source":"Carlitz $q$-Bernoulli polynomials"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#carlitz-beta-poly","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For real $q$, write $[y] = (q^y - 1)/(q - 1)$. For natural $n$ and real $x$, Carlitz's $q$-Bernoulli polynomial is $\\beta_n(x) = (q - 1)^{-n} \\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k} \\frac{k + 1}{[k + 1]} q^{kx}$; that is, $(\\beta_n)$ is the Carlitz-type sequence with associated sequence $a_k = (k + 1)/[k + 1]$. The Carlitz $q$-Bernoulli numbers are $\\beta_n = \\beta_n(0)$.","title":"Carlitz $q$-Bernoulli polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/q-bracket-zero","kind":"node","lean":"MetaMathlibExt.qBracket_zero","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-analogue of "],["math","\\(0\\)"],["text"," is "],["math","\\(0\\)"]],"source":"The $q$-analogue of $0$ is $0$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#q-bracket-zero","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For every real number $q$, $[0]_q = 0$, where $[x]_q = (q^x - 1)/(q - 1)$ is the $q$-analogue of a real number $x$, taken to be $0$ when $q = 1$.","title":"The $q$-analogue of $0$ is $0$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/q-bracket-one","kind":"node","lean":"MetaMathlibExt.qBracket_one","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-number of "],["math","\\(1\\)"]],"source":"The $q$-number of $1$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#q-bracket-one","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For real $q \\ne 1$, the $q$-analogue $[x] = (q^x - 1)/(q - 1)$ satisfies $[1] = 1$.","title":"The $q$-number of $1$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/apply-zero","kind":"node","lean":"MetaMathlibExt.IsCarlitzTypeWith.apply_zero","parent":25559,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#apply-zero","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"Let $q$ be a real number. If $(T_n)$ is a Carlitz-type $q$-polynomial sequence with associated real sequence $(a_k)$, then $T_0(x) = a_0$ for every real $x$.","title":"The zeroth term of a Carlitz-type sequence"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/carlitz-eta-poly-zero","kind":"node","lean":"MetaMathlibExt.carlitzEtaPoly_zero","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zeroth Carlitz "],["math","\\(q\\)"],["text","-polynomial"]],"source":"The zeroth Carlitz $q$-polynomial"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#carlitz-eta-poly-zero","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For all real $q$ and $x$, Carlitz's $q$-polynomial $\\eta_0(x)$ equals $1$.","title":"The zeroth Carlitz $q$-polynomial"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/carlitz-eta-poly-one-zero","kind":"node","lean":"MetaMathlibExt.carlitzEtaPoly_one_zero","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Carlitz's initial value "],["math","\\(\\eta_1 = 0\\)"]],"source":"Carlitz's initial value $\\eta_1 = 0$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#carlitz-eta-poly-one-zero","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For real $q \\ne 1$, the value $\\eta_1 = \\eta_1(0)$ of Carlitz's $q$-polynomial $\\eta_1$ at $0$ equals $0$.","title":"Carlitz's initial value $\\eta_1 = 0$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/carlitz-beta-poly-zero","kind":"node","lean":"MetaMathlibExt.carlitzBetaPoly_zero","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zeroth Carlitz "],["math","\\(q\\)"],["text","-Bernoulli polynomial"]],"source":"The zeroth Carlitz $q$-Bernoulli polynomial"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#carlitz-beta-poly-zero","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For real $q \\ne 1$ and every real $x$, the Carlitz $q$-Bernoulli polynomial $\\beta_0(x)$ equals $1$.","title":"The zeroth Carlitz $q$-Bernoulli polynomial"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/is-carlitz-type-iff-sum-choose","kind":"node","lean":"MetaMathlibExt.isCarlitzType_iff_sum_choose","parent":25559,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#is-carlitz-type-iff-sum-choose","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$ be real and write $[x] = (q^x - 1)/(q - 1)$. A sequence of functions $T_n : \\mathbb{R} \\to \\mathbb{R}$ is Carlitz-type with some associated real sequence $(a_k)$ if and only if for every natural $n$ and every real $x$, $T_n(x) = \\sum_{k=0}^{n} \\binom{n}{k} T_k(0) q^{kx} [x]^{n-k}$.","title":"Characterization of Carlitz-type sequences"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/binomial-inversion-iff","kind":"node","lean":"MetaMathlibExt.binomial_inversion_iff","parent":25559,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#binomial-inversion-iff","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"For real sequences $(u_n)$ and $(v_n)$, the identities $u_n = \\sum_{k=0}^{n} \\binom{n}{k} v_k$ hold for all natural numbers $n$ if and only if the identities $v_n = \\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k} u_k$ hold for all natural numbers $n$.","title":"Binomial inversion"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/sum-qpow-mul-q-bracket-pow","kind":"node","lean":"MetaMathlibExt.sum_qpow_mul_qBracket_pow","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of "],["math","\\(q^{rk} (k)^n\\)"],["text"," via Carlitz-type sequences"]],"source":"Sums of $q^{rk} (k)^n$ via Carlitz-type sequences"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#sum-qpow-mul-q-bracket-pow","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $[x] = (q^x - 1)/(q - 1)$, and let $r \\ge 1$ be a natural number. Let $S_m(x)$ ($m \\in \\mathbb{N}$, $x \\in \\mathbb{R}$) be the Carlitz-type sequence with associated sequence $a_k = 1/((q - 1)[k + r])$, that is,","title":"Sums of $q^{rk} (k)^n$ via Carlitz-type sequences"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/sum-qpow-mul-q-bracket-pow-sub","kind":"node","lean":"MetaMathlibExt.sum_qpow_mul_qBracket_pow_sub","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of "],["math","\\(q^{rk} (k)^{n-d}\\)"],["text"," via Carlitz-type sequences"]],"source":"Sums of $q^{rk} (k)^{n-d}$ via Carlitz-type sequences"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#sum-qpow-mul-q-bracket-pow-sub","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $[x] = (q^x - 1)/(q - 1)$, and let $r$, $d$ and $n$ be natural numbers with $r \\ge 1$, $d \\le r$ and $d \\le n$. Write $(k)_d = k(k-1)\\cdots(k-d+1)$ for the falling factorial (with $(k)_0 = 1$). Let $T_m(x)$ ($m \\in \\mathbb{N}$, $x \\in \\mathbb{R}$) be the Carlitz-type sequence with associated sequence $a_k = (q - 1)^{d-1} (k)_d / [k + r - d]$, that is,","title":"Sums of $q^{rk} (k)^{n-d}$ via Carlitz-type sequences"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/carlitz-type-symbolic-recurrence","kind":"node","lean":"MetaMathlibExt.carlitzType_symbolic_recurrence","parent":25559,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#carlitz-type-symbolic-recurrence","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, write $[x] = (q^x - 1)/(q - 1)$, let $r$ be a natural number, let $P$ be a real polynomial and let $(a_k)$ be a real sequence with $[k + r]\\, a_k = P(k)$ for every natural number $k$. If $(T_n)$ is a Carlitz-type sequence with associated sequence $(a_k)$ and $t_n = T_n(0)$, then for every natural number $n$ greater than the degree of $P$ (taken to be $0$ when $P = 0$),\u2026","title":"Symbolic recurrence for Carlitz-type sequences"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/is-carlitz-type-with-closure","kind":"node","lean":"MetaMathlibExt.isCarlitzTypeWith_closure","parent":25559,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#is-carlitz-type-with-closure","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$ be real, and let $(T_n)$ be a Carlitz-type sequence of functions $\\mathbb{R} \\to \\mathbb{R}$ with associated real sequence $(a_k)$. Then: (i) $n q^x T_{n-1}(x)$ is Carlitz-type with associated sequence $(q - 1) k a_{k-1}$; (ii) if $a_0 = 0$, then $q^{-x} T_{n+1}(x)/(n + 1)$ is Carlitz-type with associated sequence $a_{k+1}/((q - 1)(k + 1))$; (iii) $q^{-x} (T_n(x) + (q - 1) T_{n+1}(x))$ is\u2026","title":"Closure properties of Carlitz-type sequences"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/sum-qpow-mul-q-bracket-pow-eq-carlitz-et","kind":"node","lean":"MetaMathlibExt.sum_qpow_mul_qBracket_pow_eq_carlitzEta","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of "],["math","\\(q^k (k)^{n-1}\\)"],["text"," via Carlitz's "],["math","\\(q\\)"],["text","-polynomials "],["math","\\(\\eta_n\\)"]],"source":"Sums of $q^k (k)^{n-1}$ via Carlitz's $q$-polynomials $\\eta_n$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#sum-qpow-mul-q-bracket-pow-eq-carlitz-et","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $[x] = (q^x - 1)/(q - 1)$, and let $\\eta_n(x)$ be Carlitz's $q$-polynomials, with $\\eta_n = \\eta_n(0)$. Then for every natural number $n \\ge 1$ and every natural number $N$,","title":"Sums of $q^k (k)^{n-1}$ via Carlitz's $q$-polynomials $\\eta_n$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/sum-qpow-mul-q-bracket-pow-eq-carlitz-be","kind":"node","lean":"MetaMathlibExt.sum_qpow_mul_qBracket_pow_eq_carlitzBeta","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted "],["math","\\(q\\)"],["text","-power sums via Carlitz's "],["math","\\(q\\)"],["text","-Bernoulli polynomials"]],"source":"Weighted $q$-power sums via Carlitz's $q$-Bernoulli polynomials"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#sum-qpow-mul-q-bracket-pow-eq-carlitz-be","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $[x] = (q^x - 1)/(q - 1)$, and let $\\beta_n(x)$ be Carlitz's $q$-Bernoulli polynomials, with $\\beta_n = \\beta_n(0)$. Then for all natural numbers $n$ and $N$,","title":"Weighted $q$-power sums via Carlitz's $q$-Bernoulli polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/extended-carlitz-beta-poly","kind":"node","lean":"MetaMathlibExt.extendedCarlitzBetaPoly","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extended Carlitz "],["math","\\(q\\)"],["text","-Bernoulli polynomials"]],"source":"Extended Carlitz $q$-Bernoulli polynomials"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#extended-carlitz-beta-poly","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For real $q$, the extended Carlitz $q$-Bernoulli polynomials $\\beta_n^{(r)}(x)$, indexed by natural numbers $r$ and $n$ and evaluated at real $x$, are defined by recursion on $r$: $\\beta_n^{(0)}(x) = \\eta_n(x)$ is Carlitz's $q$-polynomial, and $\\beta_n^{(r+1)}(x) = q^{-x} \\bigl(\\beta_n^{(r)}(x) + (q - 1) \\beta_{n+1}^{(r)}(x)\\bigr)$. The extended Carlitz $q$-Bernoulli numbers are $\\beta_n^{(r)} = \\beta_n^{(r)}(0)$.","title":"Extended Carlitz $q$-Bernoulli polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/extended-carlitz-beta-poly-zero","kind":"node","lean":"MetaMathlibExt.extendedCarlitzBetaPoly_zero","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extended Carlitz "],["math","\\(q\\)"],["text","-Bernoulli polynomials of order zero"]],"source":"Extended Carlitz $q$-Bernoulli polynomials of order zero"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#extended-carlitz-beta-poly-zero","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For every real $q$, natural number $n$ and real $x$, the extended Carlitz $q$-Bernoulli polynomial $\\beta_n^{(0)}(x)$ equals Carlitz's $q$-polynomial $\\eta_n(x)$.","title":"Extended Carlitz $q$-Bernoulli polynomials of order zero"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/extended-carlitz-beta-poly-succ","kind":"node","lean":"MetaMathlibExt.extendedCarlitzBetaPoly_succ","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the extended Carlitz "],["math","\\(q\\)"],["text","-Bernoulli polynomials"]],"source":"Recurrence for the extended Carlitz $q$-Bernoulli polynomials"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#extended-carlitz-beta-poly-succ","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"fully_proved","summary":"For every real $q$, natural numbers $r$ and $n$, and real $x$, the extended Carlitz $q$-Bernoulli polynomials satisfy $\\beta_n^{(r+1)}(x) = q^{-x} \\bigl(\\beta_n^{(r)}(x) + (q - 1) \\beta_{n+1}^{(r)}(x)\\bigr)$.","title":"Recurrence for the extended Carlitz $q$-Bernoulli polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/extended-carlitz-beta-poly-closed-forms","kind":"node","lean":"MetaMathlibExt.extendedCarlitzBetaPoly_closed_forms","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed forms for the extended Carlitz "],["math","\\(q\\)"],["text","-Bernoulli polynomials"]],"source":"Closed forms for the extended Carlitz $q$-Bernoulli polynomials"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#extended-carlitz-beta-poly-closed-forms","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$ be real, write $[x] = (q^x - 1)/(q - 1)$, let $r$ be a natural number, and let $\\beta_n^{(r)}(x)$ be the extended Carlitz $q$-Bernoulli polynomials. Put $a_k = (k + r)/[k + r]$ for natural $k$ with $k + r \\ne 0$, and $a_0 = 1$ if $r = 0$. Then: (i) for every natural $n$ and real $x$, $\\beta_n^{(r)}(x) = (q - 1)^{-n} \\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k} a_k q^{kx}$; (ii) for every\u2026","title":"Closed forms for the extended Carlitz $q$-Bernoulli polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/sum-qpow-mul-q-bracket-pow-eq-extended-c","kind":"node","lean":"MetaMathlibExt.sum_qpow_mul_qBracket_pow_eq_extendedCarlitzBeta","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted "],["math","\\(q\\)"],["text","-power sums via the extended Carlitz "],["math","\\(q\\)"],["text","-Bernoulli polynomials"]],"source":"Weighted $q$-power sums via the extended Carlitz $q$-Bernoulli polynomials"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#sum-qpow-mul-q-bracket-pow-eq-extended-c","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $[x] = (q^x - 1)/(q - 1)$, and let $\\beta_n^{(r)}(x)$ be the extended Carlitz $q$-Bernoulli polynomials, with $\\beta_n^{(r)} = \\beta_n^{(r)}(0)$. Then for all natural numbers $r$, $n$ and $N$,","title":"Weighted $q$-power sums via the extended Carlitz $q$-Bernoulli polynomials"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/extended-carlitz-beta-eq-tsum","kind":"node","lean":"MetaMathlibExt.extendedCarlitzBeta_eq_tsum","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Series for the extended Carlitz "],["math","\\(q\\)"],["text","-Bernoulli numbers"]],"source":"Series for the extended Carlitz $q$-Bernoulli numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#extended-carlitz-beta-eq-tsum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $0 < q < 1$, write $[x] = (q^x - 1)/(q - 1)$, and let $\\beta_n^{(r)} = \\beta_n^{(r)}(0)$ be the extended Carlitz $q$-Bernoulli numbers. For every natural number $n \\ge 1$, $\\beta_n^{(0)} = -n \\sum_{k=0}^{\\infty} q^k [k]^{n-1} + (1 - q)^{-n}$, and for every natural number $r \\ge 1$,","title":"Series for the extended Carlitz $q$-Bernoulli numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/tendsto-tsum-q-bracket-bernoulli","kind":"node","lean":"MetaMathlibExt.tendsto_tsum_qBracket_bernoulli","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli numbers as a limit as "],["math","\\(q \\to 1^-\\)"]],"source":"Bernoulli numbers as a limit as $q \\to 1^-$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#tendsto-tsum-q-bracket-bernoulli","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $r \\ge 1$ and $n \\ge 1$ be integers and, for real $q$ and $x$, write $[x] = (q^x - 1)/(q - 1)$. Then, as $q$ tends to $1$ from below,","title":"Bernoulli numbers as a limit as $q \\to 1^-$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/sum-qpow-mul-q-falling","kind":"node","lean":"MetaMathlibExt.sum_qpow_mul_qFalling","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of "],["math","\\(q^{ir}\\)"],["text"," times "],["math","\\(q\\)"],["text","-falling factorials"]],"source":"Sums of $q^{ir}$ times $q$-falling factorials"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#sum-qpow-mul-q-falling","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $[x] = (q^x - 1)/(q - 1)$, and write $[x]_m = [x][x-1]\\cdots[x-m+1]$ for the $q$-falling factorial (with $[x]_0 = 1$). Then for all natural numbers $r \\ge 1$, $k$ and $N$,","title":"Sums of $q^{ir}$ times $q$-falling factorials"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/tendsto-q-interpolant-div-q-bracket","kind":"node","lean":"MetaMathlibExt.tendsto_qInterpolant_div_qBracket","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Limit of the "],["math","\\(q\\)"],["text","-power-sum interpolant at "],["math","\\(0\\)"],["text",": first closed form"]],"source":"Limit of the $q$-power-sum interpolant at $0$: first closed form"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#tendsto-q-interpolant-div-q-bracket","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $r \\ge 1$ and $k \\ge 0$ be integers, and for real $x$ write $[x] = (q^x - 1)/(q - 1)$, $[x]_m = [x][x-1]\\cdots[x-m+1]$ (with $[x]_0 = 1$) and $[k]! = [1][2]\\cdots[k]$. Let $P$ be a real polynomial such that $P(q^N) = q^{k(k-1)/2} \\sum_{i=0}^{N-1} q^{ir} [i]_k$ for every positive integer $N$. Then $P(q^x)/[x]$ tends, as $x \\to 0$ with $x \\ne 0$, to","title":"Limit of the $q$-power-sum interpolant at $0$: first closed form"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/qpow-div-q-falling-partial-fractions","kind":"node","lean":"MetaMathlibExt.qpow_div_qFalling_partial_fractions","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial fraction decomposition of "],["math","\\(q^{kx}\\)"],["text"," over a "],["math","\\(q\\)"],["text","-falling factorial"]],"source":"Partial fraction decomposition of $q^{kx}$ over a $q$-falling factorial"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#qpow-div-q-falling-partial-fractions","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $[x] = (q^x - 1)/(q - 1)$, write $[m]! = [1][2]\\cdots[m]$ (with $[0]! = 1$) and $\\binom{k}{i}_q = [k]!/([i]!\\,[k-i]!)$. Then for every natural number $k$ and every real number $x \\notin \\{0, 1, \\dots, k\\}$,","title":"Partial fraction decomposition of $q^{kx}$ over a $q$-falling factorial"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/tendsto-q-interpolant-div-q-bracket-2","kind":"node","lean":"MetaMathlibExt.tendsto_qInterpolant_div_qBracket'","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Second closed form of a limit of "],["math","\\(q\\)"],["text","-interpolated falling-factorial sums"]],"source":"Second closed form of a limit of $q$-interpolated falling-factorial sums"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#tendsto-q-interpolant-div-q-bracket-2","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, let $[x] = (q^x - 1)/(q - 1)$, write $[x]_m = [x][x-1]\\cdots[x-m+1]$ and $[m]! = [1][2]\\cdots[m]$ (both equal to $1$ when $m = 0$), and let $r \\ge 1$ and $k$ be natural numbers. Let $P$ be a real polynomial such that","title":"Second closed form of a limit of $q$-interpolated falling-factorial sums"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/extended-carlitz-beta-eq-sum-q-stirling","kind":"node","lean":"MetaMathlibExt.extendedCarlitzBeta_eq_sum_qStirling","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extended Carlitz "],["math","\\(q\\)"],["text","-Bernoulli numbers via "],["math","\\(q\\)"],["text","-Stirling numbers"]],"source":"Extended Carlitz $q$-Bernoulli numbers via $q$-Stirling numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#extended-carlitz-beta-eq-sum-q-stirling","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$ be real, write $[x] = (q^x - 1)/(q - 1)$ for real $x$, $[x]_m = [x][x-1] \\cdots [x-m+1]$ (with $[x]_0 = 1$) and $[m]! = [1][2] \\cdots [m]$ (with $[0]! = 1$), and let $\\beta_n^{(r)}(x)$ be the extended Carlitz $q$-Bernoulli polynomials, with numbers $\\beta_n^{(r)} = \\beta_n^{(r)}(0)$. Let $r \\ge 1$ and $n$ be natural numbers, and let $S(m, k)$ be real numbers such that\u2026","title":"Extended Carlitz $q$-Bernoulli numbers via $q$-Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/carlitz-beta-add-mul-succ-eq-sum-q-stirl","kind":"node","lean":"MetaMathlibExt.carlitzBeta_add_mul_succ_eq_sum_qStirling","parent":25559,"reading_title":{"format":"inline-parts-v1","parts":[["text","Carlitz "],["math","\\(q\\)"],["text","-Bernoulli numbers via "],["math","\\(q\\)"],["text","-Stirling numbers"]],"source":"Carlitz $q$-Bernoulli numbers via $q$-Stirling numbers"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#carlitz-beta-add-mul-succ-eq-sum-q-stirl","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"Let $q > 0$ with $q \\ne 1$, write $[x] = (q^x - 1)/(q - 1)$, $[x]_k = [x][x - 1] \\cdots [x - k + 1]$ and $[k]! = [1][2] \\cdots [k]$ (empty products are $1$), and let $\\beta_n = \\beta_n(0)$ be Carlitz's $q$-Bernoulli numbers. If real numbers $S_q(m,k)$ satisfy $[x]^m = \\sum_{k=0}^{m} q^{k(k-1)/2} S_q(m,k) [x]_k$ for every natural number $m$ and every real $x$ (this is the defining identity of Carlitz's $q$-Stirling\u2026","title":"Carlitz $q$-Bernoulli numbers via $q$-Stirling numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/bernoulli-add-bernoulli-succ-eq-sum-stir","kind":"node","lean":"MetaMathlibExt.bernoulli_add_bernoulli_succ_eq_sum_stirlingSecond","parent":25559,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials/index.html#bernoulli-add-bernoulli-succ-eq-sum-stir","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/carlitz-q-bernoulli-polynomials","status":"can_prove","summary":"For every natural number $n$, $B_n + B_{n+1} = \\sum_{k=0}^{n} (-1)^k \\frac{k!}{k+2} S(n,k)$, where $B_n$ are the Bernoulli numbers with $B_1 = -1/2$ (generating function $t/(e^t - 1)$, the paper's convention) and $S(n,k)$ are the Stirling numbers of the second kind.","title":"Sum of consecutive Bernoulli numbers via Stirling numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices.json","id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices","kind":"scope","navigation":"module","parent":25556,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices/index.html","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices","status":"fully_proved","title":"Euler\u2013Seidel matrices"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices/euler-seidel-matrix","kind":"node","lean":"MetaMathlibExt.eulerSeidelMatrix","parent":25592,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices/index.html#euler-seidel-matrix","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices","status":"fully_proved","summary":"Let $M$ be a type with an addition and let $(a_n)_{n \\ge 0}$ be a sequence in $M$. The Euler\u2013Seidel matrix of $a$ is the array $(a_n^k)_{n, k \\ge 0}$, with entry $a_n^k$ in position $(n, k)$, defined by $a_n^0 = a_n$ and $a_n^k = a_n^{k-1} + a_{n+1}^{k-1}$ for $k \\ge 1$.","title":"Euler\u2013Seidel matrix"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices/euler-seidel-matrix-eq-sum","kind":"node","lean":"MetaMathlibExt.eulerSeidelMatrix_eq_sum","parent":25592,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices/index.html#euler-seidel-matrix-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/euler-seidel-matrices","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid and let $(a_n)_{n \\ge 0}$ be a sequence in $M$. For all $n, k \\ge 0$, the entry $a_n^k$ of the Euler\u2013Seidel matrix of $a$ equals $\\sum_{i=0}^{k} \\binom{k}{i} a_{n+i}$.","title":"Closed form of the Euler\u2013Seidel matrix"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers.json","id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","kind":"scope","navigation":"module","parent":25556,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","title":"Faulhaber coefficients and sums of powers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/power-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.powerSum","parent":25595,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#power-sum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the power sum $S_k(n)$ is the rational number $\\sum_{i=1}^{n} i^k = 1^k + 2^k + \\cdots + n^k$, which is $0$ when $n = 0$.","title":"Power sums"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/array-r","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.arrayR","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","The array "],["math","\\(R(k, m)\\)"]],"source":"The array $R(k, m)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#array-r","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"fully_proved","summary":"For natural numbers $k$ and $m$, $R(k, m)$ is the integer $\\sum_{j=0}^{m} (-1)^j \\binom{2m}{j} (m - j)^{2k}$.","title":"The array $R(k, m)$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/array-u","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.arrayU","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","The array "],["math","\\(U(k, m)\\)"]],"source":"The array $U(k, m)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#array-u","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"fully_proved","summary":"For natural numbers $k$ and $m$, $U(k, m)$ is the rational number $\\frac{2}{(2m)!} \\sum_{j=0}^{m} (-1)^j \\binom{2m}{j} (m - j)^{2k}$.","title":"The array $U(k, m)$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/faulhaber-b","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.faulhaberB","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Faulhaber coefficients "],["math","\\(b_{k,r}\\)"]],"source":"The Faulhaber coefficients $b_{k,r}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#faulhaber-b","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"fully_proved","summary":"For natural numbers $p$ and $q$, let $R(p, q) = \\sum_{j=0}^{q} (-1)^j \\binom{2q}{j} (q - j)^{2p}$, and for natural numbers $q$ and $s$ let $\\mathrm{Ps}_q^{(s)}$ be the Legendre\u2013Stirling number of the first kind, the coefficient of $x^s$ in $\\prod_{i=0}^{q-1} (x - i(i+1))$. For natural numbers $k$ and $r$, $b_{k,r}$ is the rational number $\\sum_{m=r}^{k} \\frac{3 \\cdot 2^r}{(2m+1)!}\\, R(k, m)\\, \\mathrm{Ps}_m^{(r)}$.","title":"The Faulhaber coefficients $b_{k,r}$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/faulhaber-c","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.faulhaberC","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Faulhaber coefficients "],["math","\\(c_{k,r}\\)"]],"source":"The Faulhaber coefficients $c_{k,r}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#faulhaber-c","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"fully_proved","summary":"For natural numbers $p$ and $q$, let $R(p, q) = \\sum_{j=0}^{q} (-1)^j \\binom{2q}{j} (q - j)^{2p}$, and for natural numbers $q$ and $s$ let $\\mathrm{Ps}_q^{(s)}$ be the Legendre\u2013Stirling number of the first kind, the coefficient of $x^s$ in $\\prod_{i=0}^{q-1} (x - i(i+1))$. For natural numbers $k$ and $r$, $c_{k,r}$ is the rational number\u2026","title":"The Faulhaber coefficients $c_{k,r}$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/thm-q","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.thmQ","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomials "],["math","\\(Q_{k,m}\\)"],["text"," of Theorem 10"]],"source":"The polynomials $Q_{k,m}$ of Theorem 10"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#thm-q","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"fully_proved","summary":"For natural numbers $k$, $m$ and $n$, write $R(a, b) = \\sum_{i=0}^{b} (-1)^i \\binom{2b}{i} (b - i)^{2a}$ for the array of OEIS A304330. The rational number $Q_{k,m}(n)$ is $n^{2k-1}$ if $m = 1$, and otherwise","title":"The polynomials $Q_{k,m}$ of Theorem 10"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/pow-mul-power-sum-even-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.pow_mul_powerSum_even_eq_sum","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even power sums in terms of "],["math","\\(\\binom{2n+m+1}{2m+1}\\)"]],"source":"Even power sums in terms of $\\binom{2n+m+1}{2m+1}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#pow-mul-power-sum-even-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For natural numbers $k \\ge 1$ and $n$, write $S_{2k}(n) = 1^{2k} + 2^{2k} + \\dots + n^{2k}$ and $R(k, m) = \\sum_{j=0}^{m} (-1)^j \\binom{2m}{j} (m - j)^{2k}$. Then $2^{2k} S_{2k}(n) = \\sum_{m=1}^{k} R(k, m) \\binom{2n+m+1}{2m+1}$.","title":"Even power sums in terms of $\\binom{2n+m+1}{2m+1}$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/power-sum-even-div-iff-odd","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.powerSum_even_div_iff_odd","parent":25595,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#power-sum-even-div-iff-odd","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"Let $k \\ge 1$ be a natural number, let $(a_m)_{m \\ge 0}$ be a sequence of real numbers, and write $S_j(n) = 1^j + 2^j + \\dots + n^j$. Then $\\frac{S_{2k}(n)}{2n+1} = \\sum_{m=1}^{k} a_m \\binom{n+m}{2m}$ for every natural number $n$ if and only if $S_{2k-1}(n) = \\sum_{m=1}^{k} \\frac{2m+1}{m}\\, a_m \\binom{n+m}{2m}$ for every natural number $n$.","title":"Knuth's correspondence between even and odd power sums"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/power-sum-even-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.powerSum_even_eq_sum","parent":25595,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#power-sum-even-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For natural numbers $k \\ge 1$ and $n$, write $S_{2k}(n) = 1^{2k} + 2^{2k} + \\dots + n^{2k}$ and $R(k, m) = \\sum_{j=0}^{m} (-1)^j \\binom{2m}{j} (m - j)^{2k}$. Then $S_{2k}(n) = \\sum_{m=1}^{k} R(k, m)\\, \\frac{2n+1}{2m+1} \\binom{n+m}{2m}$.","title":"Knuth's formula for even power sums"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/power-sum-odd-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.powerSum_odd_eq_sum","parent":25595,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#power-sum-odd-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For all natural numbers $k \\ge 1$ and $n$, the power sum $S_{2k-1}(n) = 1^{2k-1} + 2^{2k-1} + \\cdots + n^{2k-1}$ satisfies","title":"Knuth's binomial formula for the odd power sums"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/sum-two-mul-choose-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.sum_two_mul_choose_eq","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(2i \\binom{2i+k-1}{2k-1}\\)"]],"source":"Sum of $2i \\binom{2i+k-1}{2k-1}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#sum-two-mul-choose-eq","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For all natural numbers $k \\ge 1$ and $n \\ge 1$,","title":"Sum of $2i \\binom{2i+k-1}{2k-1}$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/inversion-u-u","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.inversion_uU","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inversion formula for the arrays "],["math","\\(u(k, m)\\)"],["text"," and "],["math","\\(U(k, m)\\)"],["text"," (Butzer et al.)"]],"source":"Inversion formula for the arrays $u(k, m)$ and $U(k, m)$ (Butzer et al.)"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#inversion-u-u","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For natural numbers $k$ and $m$, let $u(k, m)$ be the first-kind Jacobi\u2013Stirling number $J(k, m; 0)$, the coefficient of $x^m$ in $\\prod_{i=0}^{k-1} (x - i^2)$, and let $U(k, m) = \\frac{2}{(2m)!} \\sum_{j=0}^{m} (-1)^j \\binom{2m}{j} (m - j)^{2k}$. For all sequences $(a_k)_{k \\ge 0}$ and $(b_k)_{k \\ge 0}$ of real numbers, $a_k = \\sum_{m=1}^{k} u(k, m)\\, b_m$ holds for every $k \\ge 1$ if and only if\u2026","title":"Inversion formula for the arrays $u(k, m)$ and $U(k, m)$ (Butzer et al.)"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/array-r-eq-closed-form","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.arrayR_eq_closedForm","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form for the array "],["math","\\(R(k, m)\\)"]],"source":"Closed form for the array $R(k, m)$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#array-r-eq-closed-form","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"Let $k$ and $m$ be natural numbers with $1 \\le m \\le k$, and let $R(k, m) = \\sum_{j=0}^{m} (-1)^j \\binom{2m}{j} (m - j)^{2k}$. Then","title":"Closed form for the array $R(k, m)$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/sum-mul-choose-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.sum_mul_choose_eq","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of "],["math","\\(i \\binom{i+k-1}{2k-1}\\)"]],"source":"Sum of $i \\binom{i+k-1}{2k-1}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#sum-mul-choose-eq","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For all natural numbers $k \\ge 1$ and $n \\ge 1$,","title":"Sum of $i \\binom{i+k-1}{2k-1}$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/pow-mul-power-sum-odd-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.pow_mul_powerSum_odd_eq_sum","parent":25595,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd power sums in terms of "],["math","\\(\\binom{n+m}{2m-1}\\)"]],"source":"Odd power sums in terms of $\\binom{n+m}{2m-1}$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#pow-mul-power-sum-odd-eq-sum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For natural numbers $k \\ge 1$ and $n$, write $S_{2k-1}(n) = 1^{2k-1} + 2^{2k-1} + \\dots + n^{2k-1}$, let $R(p, q) = \\sum_{j=0}^{q} (-1)^j \\binom{2q}{j} (q - j)^{2p}$, and let $Q_{k,1}(n) = n^{2k-1}$ and $Q_{k,m}(n) = 2 \\sum_{j=m}^{k} \\binom{2k-1}{2j-2} R(j-1, m-1)\\, n^{2k-2j+1}$ for $m \\ge 2$. Then $2^{2k-1} S_{2k-1}(n) = \\sum_{m=1}^{k} Q_{k,m}(n) \\binom{n+m}{2m-1}$.","title":"Odd power sums in terms of $\\binom{n+m}{2m-1}$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/power-sum-even-faulhaber","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.powerSum_even_faulhaber","parent":25595,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#power-sum-even-faulhaber","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For all natural numbers $k \\ge 1$ and $n$, writing $S_j(n) = 1^j + 2^j + \\cdots + n^j$ for the power sums and $b_{k,r}$ for the Faulhaber coefficients of Proposition 12, the rational numbers satisfy $S_{2k}(n) = S_2(n) \\sum_{r=1}^{k} b_{k,r} \\, S_1(n)^{r-1}$.","title":"Faulhaber form of the even power sums"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/power-sum-odd-faulhaber","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.powerSum_odd_faulhaber","parent":25595,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#power-sum-odd-faulhaber","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For all natural numbers $k \\ge 1$ and $n$, writing $S_j(n) = 1^j + 2^j + \\cdots + n^j$ for the power sums and $c_{k,r}$ for the Faulhaber coefficients of Proposition 12, the rational numbers satisfy $S_{2k+1}(n) = S_1(n)^2 \\sum_{r=1}^{k} c_{k,r} \\, S_1(n)^{r-1}$.","title":"Faulhaber form of the odd power sums"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/sum-neg-factorial-sq-div-array-r-eq-zero","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SumsOfPowersA304330Wanted.sum_neg_factorial_sq_div_arrayR_eq_zero","parent":25595,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers/index.html#sum-neg-factorial-sq-div-array-r-eq-zero","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/faulhaber-coefficients-and-sums-of-powers","status":"can_prove","summary":"For every natural number $k \\ge 2$,","title":"Vanishing alternating sum of the A304330 array"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi.json","id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi","kind":"scope","navigation":"module","parent":25556,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/index.html","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi","status":"fully_proved","title":"Garabedian's formulas for Bernoulli and Genocchi numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/garabedian-genocchi-formula-general","kind":"node","lean":"MetaMathlibExt.garabedian_genocchi_formula_general","parent":25614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Garabedian's formula for the Bernoulli numbers, including "],["math","\\(n = 0\\)"]],"source":"Garabedian's formula for the Bernoulli numbers, including $n = 0$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/index.html#garabedian-genocchi-formula-general","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi","status":"fully_proved","summary":"For every natural number $n$,","title":"Garabedian's formula for the Bernoulli numbers, including $n = 0$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/garabedian-genocchi-formula","kind":"node","lean":"MetaMathlibExt.garabedian_genocchi_formula","parent":25614,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/index.html#garabedian-genocchi-formula","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi","status":"fully_proved","summary":"For every natural number $n \\ge 1$,","title":"Garabedian's formula for the Bernoulli numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/genocchi-number-eq-stirling-second-sum-g","kind":"node","lean":"MetaMathlibExt.genocchiNumber_eq_stirlingSecond_sum_general","parent":25614,"reading_title":{"format":"inline-parts-v1","parts":[["text","Garabedian's formula for the Genocchi numbers, including "],["math","\\(n = 0\\)"]],"source":"Garabedian's formula for the Genocchi numbers, including $n = 0$"},"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/index.html#genocchi-number-eq-stirling-second-sum-g","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi","status":"fully_proved","summary":"For every natural number $n$, the Genocchi number $G_n$, defined by $\\sum_{m \\ge 0} G_m x^m / m! = 2x/(e^x + 1)$, satisfies","title":"Garabedian's formula for the Genocchi numbers, including $n = 0$"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/genocchi-number-eq-stirling-second-sum","kind":"node","lean":"MetaMathlibExt.genocchiNumber_eq_stirlingSecond_sum","parent":25614,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi/index.html#genocchi-number-eq-stirling-second-sum","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/garabedian-s-formulas-for-bernoulli-and-genocchi","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the Genocchi number $G_n$, defined by $\\sum_{m \\ge 0} G_m x^m / m! = 2x/(e^x + 1)$, satisfies","title":"Garabedian's formula for the Genocchi numbers"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers.json","id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","kind":"scope","navigation":"module","parent":25556,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","title":"Median Euler numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/median-euler-coefficients","kind":"node","lean":"MetaMathlibExt.medianEulerCoefficients","parent":25619,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html#median-euler-coefficients","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","summary":"Let $\\varepsilon_N$ be $1$ if $N$ is odd and $0$ if $N$ is even. For each natural number $n$, this is the list $[c_0, c_1, \\dots, c_n]$ of integers with $c_0 = 1$ and, for $1 \\le N \\le n$, $c_N = \\varepsilon_N - \\sum_j \\binom{N}{j} c_{N-j}$, where the sum runs over the even $j$ with $2 \\le j \\le N$. The Euler coefficient $E_n$ is its last entry $c_n$. The list is built one entry at a time, so each step reuses the\u2026","title":"The list of the first Euler coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/median-euler-coeff","kind":"node","lean":"MetaMathlibExt.medianEulerCoeff","parent":25619,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html#median-euler-coeff","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","summary":"Let $\\varepsilon_n$ be $1$ if $n$ is odd and $0$ if $n$ is even. The Euler coefficients $E_n$ are the integers defined by $E_0 = 1$ and, for $n \\ge 1$, $E_n = \\varepsilon_n - \\sum_j \\binom{n}{j} E_{n-j}$, where the sum runs over the even $j$ with $2 \\le j \\le n$.","title":"Euler coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/median-euler-initial-row","kind":"node","lean":"MetaMathlibExt.medianEulerInitialRow","parent":25619,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html#median-euler-initial-row","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","summary":"Let $E_m$ denote the Euler coefficients. This is the integer sequence $(a_m)_{m \\ge 0}$ with $a_0 = 0$ and $a_m = E_m$ for $m \\ge 1$. It is the initial row of the Euler\u2013Seidel matrix from which the median Euler numbers are read off.","title":"Initial row for the median Euler numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/median-euler-number","kind":"node","lean":"MetaMathlibExt.medianEulerNumber","parent":25619,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html#median-euler-number","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","summary":"Let $E_m$ denote the Euler coefficients, and let $(a_n^k)$ be the Euler\u2013Seidel matrix of the initial row $a_0 = 0$, $a_m = E_m$ for $m \\ge 1$, so that $a_n^0 = a_n$ and $a_n^{k+1} = a_n^k + a_{n+1}^k$. The median Euler number $R_n$ is the natural number $|a_{n+1}^n|$. Equivalently, $R_n = \\left|\\sum_{i=0}^{n} \\binom{n}{i} E_{n+1+i}\\right|$.","title":"Median Euler numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/median-euler-coeff-zero","kind":"node","lean":"MetaMathlibExt.medianEulerCoeff_zero","parent":25619,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html#median-euler-coeff-zero","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","summary":"The Euler coefficient $E_0$ equals $1$.","title":"The zeroth Euler coefficient"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/median-euler-coeff-succ","kind":"node","lean":"MetaMathlibExt.medianEulerCoeff_succ","parent":25619,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html#median-euler-coeff-succ","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","summary":"Let $E_m$ denote the Euler coefficients, and let $\\varepsilon_m$ be $1$ if $m$ is odd and $0$ if $m$ is even. For every natural number $n$, $E_{n+1} = \\varepsilon_{n+1} - \\sum_j \\binom{n+1}{j} E_{n+1-j}$, where the sum runs over the even $j$ with $2 \\le j \\le n+1$.","title":"Recurrence for the Euler coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/median-euler-coeff-egf","kind":"node","lean":"MetaMathlibExt.medianEulerCoeff_egf","parent":25619,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html#median-euler-coeff-egf","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","summary":"Let $E_n$ denote the Euler coefficients, let $e^x$ be the exponential series, and let $e^{-x}$ be the series obtained from it by replacing $x$ with $-x$. In the ring $\\mathbb{Q}[[x]]$ of formal power series with rational coefficients, $(e^x + e^{-x}) \\sum_{n \\ge 0} E_n x^n / n! = 2 + e^x - e^{-x}$.","title":"Exponential generating function of the Euler coefficients"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/median-euler-number-div-three-and-mod-fi","kind":"node","lean":"MetaMathlibExt.medianEulerNumber_div_three_and_modFive","parent":25619,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers/index.html#median-euler-number-div-three-and-mod-fi","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-euler-numbers","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the median Euler number $R_n$ is divisible by $3$, and $R_n / 3 \\equiv 3^{n-1} \\pmod 5$.","title":"Median Euler numbers modulo three and five"},{"data_url":"scopes/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers.json","id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers","kind":"scope","navigation":"module","parent":25556,"result":false,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers/index.html","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers","status":"fully_proved","title":"Median Genocchi numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers/genocchi-number-via-bernoulli","kind":"node","lean":"MetaMathlibExt.genocchiNumberViaBernoulli","parent":25628,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers/index.html#genocchi-number-via-bernoulli","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers","status":"fully_proved","summary":"For a natural number $n$, the Genocchi number $G_n$ is the rational number $G_n = 2(1 - 2^n) B_n$, where $B_n$ is the $n$-th Bernoulli number with the convention $B_1 = -\\tfrac{1}{2}$.","title":"Genocchi numbers via Bernoulli numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers/median-genocchi-odd","kind":"node","lean":"MetaMathlibExt.medianGenocchiOdd","parent":25628,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers/index.html#median-genocchi-odd","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers","status":"fully_proved","summary":"For a natural number $m$, let $B_m$ be the $m$-th Bernoulli number and $G_m = 2(1 - 2^m) B_m$ the $m$-th Genocchi number. For a natural number $n$, the median Genocchi number $H_{2n+1}$ is the rational number defined by $H_1 = 1$ and, for $n \\ge 1$,","title":"Odd-indexed median Genocchi numbers"},{"id":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers/median-genocchi-odd-divisibility","kind":"node","lean":"MetaMathlibExt.medianGenocchiOdd_divisibility","parent":25628,"result":true,"route":"../roadmap/Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers/index.html#median-genocchi-odd-divisibility","scope":"Combinatorics/stirling-bell-eulerian/bernoulli-euler-genocchi/median-genocchi-numbers","status":"fully_proved","summary":"Let $H_{2m+1}$ denote the odd-indexed median Genocchi values, with the signs given by their defining sum, so that $H_5 = 2$, $H_7 = -8$ and $H_9 = 56$. For every natural number $n \\ge 1$ there is an integer $z$ with $H_{2n+3} = 2^n z$, and $z \\equiv 1 \\pmod 6$ if $n$ is odd, while $z \\equiv 4 \\pmod 6$ if $n$ is even.","title":"Divisibility of the median Genocchi numbers"},{"data_url":"scopes/Combinatorics/integer-partitions.json","id":"Combinatorics/integer-partitions","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/index.html","scope":"Combinatorics/integer-partitions","status":"can_prove","summary":"Partitions of integers as combinatorial objects: identities equating partition counts and their bijective proofs, Young diagrams and partition statistics, and counting partitions; congruences for partition functions and theta- or eta-quotient identities go to NumberTheory.","title":"Integer partitions"},{"data_url":"scopes/Combinatorics/integer-partitions/partition-identities.json","id":"Combinatorics/integer-partitions/partition-identities","kind":"scope","parent":25632,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/index.html","scope":"Combinatorics/integer-partitions/partition-identities","status":"can_prove","summary":"Theorems equating counts of two classes of partitions: Euler, MacMahon, Andrews, Andrews\u2013Lewis, Rogers\u2013Ramanujan, Stanley and Elder theorems, polarized and 2-distant partitions, colored Euler-type identities and Merca's sums of different parts.","title":"Partition identities"},{"id":"Combinatorics/integer-partitions/partition-identities/card-odd-count-restricted-four-eq-card-m","kind":"node","lean":"MetaMathlibExt.card_odd_countRestricted_four_eq_card_mod_four_pair_excluding","parent":25633,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/index.html#card-odd-count-restricted-four-eq-card-m","scope":"Combinatorics/integer-partitions/partition-identities","status":"fully_proved","summary":"For every natural number $n$, the number of partitions of $n$ into odd parts in which each part occurs at most three times equals the number of partitions of $n$ into parts congruent to $1$ or $3$ modulo $4$ such that for no natural number $j \\le n$ are both $4j + 1$ and $4j + 3$ parts.","title":"Andrews\u2013Lewis identity"},{"id":"Combinatorics/integer-partitions/partition-identities/partitions-two-distant-eq-polarized","kind":"node","lean":"MetaMathlibExt.partitions_two_distant_eq_polarized","parent":25633,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/index.html#partitions-two-distant-eq-polarized","scope":"Combinatorics/integer-partitions/partition-identities","status":"fully_proved","summary":"For every natural number $n$, the number of partitions of $n$ into distinct parts any two of which differ by at least $2$ equals the number of polarized partitions of $n$, that is, partitions of $n$ into distinct parts in which every even part is greater than twice the number of odd parts.","title":"Partitions with 2-distant parts and polarized partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/rogers-ramanujan-difference-two-vs-one-f","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.Partition.RogersRamanujanWanted.rogers_ramanujan_difference_two_vs_one_four","parent":25633,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/index.html#rogers-ramanujan-difference-two-vs-one-f","scope":"Combinatorics/integer-partitions/partition-identities","status":"fully_proved","summary":"For every natural number $n$, the number of partitions of $n$ into distinct parts, any two of which differ by at least $2$, equals the number of partitions of $n$ all of whose parts are congruent to $1$ or $4$ modulo $5$.","title":"Rogers\u2013Ramanujan identity (first form)"},{"data_url":"scopes/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part.json","id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","kind":"scope","navigation":"module","parent":25633,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","title":"Alder-type inequalities for fixed perimeter partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/sym-cons-embedding","kind":"node","lean":"MetaMathlibExt.symConsEmbedding","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#sym-cons-embedding","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"For natural numbers $m$ and $k$, the map that sends a multiset $t$ of $k$ natural numbers to the multiset obtained from $t$ by adding one copy of $m$ is injective, so it embeds the multisets of size $k$ into all multisets of natural numbers.","title":"Prepending a part to a multiset"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/fixed-perimeter-partitions","kind":"node","lean":"MetaMathlibExt.fixedPerimeterPartitions","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#fixed-perimeter-partitions","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"The perimeter of a partition is its largest part plus its number of parts minus $1$. For a natural number $n$, the partitions with perimeter $n$ form the finite set of all finite multisets $s$ of positive integers whose largest element plus number of elements (counted with multiplicity) equals $n + 1$.","title":"Partitions with fixed perimeter"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/mem-fixed-perimeter-partitions","kind":"node","lean":"MetaMathlibExt.mem_fixedPerimeterPartitions","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#mem-fixed-perimeter-partitions","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"Let $n$ be a natural number and $s$ a finite multiset of natural numbers. Then $s$ is a partition with perimeter $n$ if and only if every element of $s$ is positive and the largest element of $s$ (taken to be $0$ if $s$ is empty) plus the number of elements of $s$, counted with multiplicity, equals $n + 1$.","title":"Membership in the partitions with fixed perimeter"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/card-fixed-perimeter-partitions-three","kind":"node","lean":"MetaMathlibExt.card_fixedPerimeterPartitions_three","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#card-fixed-perimeter-partitions-three","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"There are exactly four partitions with perimeter $3$, that is, four finite multisets of positive integers whose largest element plus number of elements (counted with multiplicity) equals $4$.","title":"The partitions of perimeter three"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/is-ddistinct-parts","kind":"node","lean":"MetaMathlibExt.IsDDistinctParts","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#is-ddistinct-parts","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"Let $d$ be a natural number. A finite multiset $s$ of natural numbers has $d$-distinct parts when no element occurs in $s$ more than once and any two elements $x < y$ of $s$ satisfy $x + d \\le y$.","title":"Partitions into d-distinct parts"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/is-ddistinct-parts-examples","kind":"node","lean":"MetaMathlibExt.isDDistinctParts_examples","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#is-ddistinct-parts-examples","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"The partition $5 + 3$ has $2$-distinct parts, and the partition $4 + 3$ does not.","title":"Examples of 2-distinct partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-distinct-count","kind":"node","lean":"MetaMathlibExt.perimeterDistinctCount","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed perimeter partitions into "],["math","\\(d\\)"],["text","-distinct parts"]],"source":"Fixed perimeter partitions into $d$-distinct parts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-distinct-count","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"For natural numbers $d$, $a$ and $n$, $h_d^{(a)}(n)$ is the number of partitions of perimeter $n$ into $d$-distinct parts that are all at least $a$. Here the perimeter of a partition is its largest part plus its number of parts minus $1$, and a partition has $d$-distinct parts when no part repeats and any two parts differ by at least $d$.","title":"Fixed perimeter partitions into $d$-distinct parts"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-distinct-arm-leg-count","kind":"node","lean":"MetaMathlibExt.perimeterDistinctArmLegCount","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-distinct-arm-leg-count","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"For natural numbers $d$, $a$, $m$ and $k$, $h_d^{(a)}(m, k)$ is the number of nonempty finite multisets of positive integers with largest element $m$, exactly $k$ elements counted with multiplicity, all elements at least $a$, and $d$-distinct parts. Such a partition has perimeter $m + k - 1$.","title":"Partitions into d-distinct parts by largest part and length"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-residue-count","kind":"node","lean":"MetaMathlibExt.perimeterResidueCount","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-residue-count","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"For natural numbers $d$, $a$ and $n$, $f_d^{(a)}(n)$ is the number of partitions with perimeter $n$ all of whose parts are congruent to $a$ modulo $d + 1$. Here a partition is a nonempty multiset of positive integers, and its perimeter is its largest part plus its number of parts minus one.","title":"Fixed perimeter partitions into parts in one residue class"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-residue-arm-leg-count","kind":"node","lean":"MetaMathlibExt.perimeterResidueArmLegCount","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-residue-arm-leg-count","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"For natural numbers $d$, $a$, $m$ and $k$, $f_d^{(a)}(m, k)$ is the number of nonempty multisets of positive integers with largest part $m$ and exactly $k$ elements (counted with multiplicity), all of which are congruent to $a$ modulo $d + 1$.","title":"Partitions into parts in one residue class, by largest part and length"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-pmresidue-count","kind":"node","lean":"MetaMathlibExt.perimeterPMResidueCount","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed perimeter partitions into parts "],["math","\\(\\equiv \\pm a \\pmod{d+3}\\)"]],"source":"Fixed perimeter partitions into parts $\\equiv \\pm a \\pmod{d+3}$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-pmresidue-count","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"For natural numbers $d$, $a$ and $n$, $\\ell_d^{(a)}(n)$ is the number of partitions of perimeter $n$ into parts congruent to $a$ or $-a$ modulo $d + 3$, where the perimeter of a partition is its largest part plus its number of parts minus $1$.","title":"Fixed perimeter partitions into parts $\\equiv \\pm a \\pmod{d+3}$"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-pmresidue-arm-leg-count","kind":"node","lean":"MetaMathlibExt.perimeterPMResidueArmLegCount","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partitions into parts "],["math","\\(\\equiv \\pm a \\pmod{d+3}\\)"],["text"," by largest part and number of parts"]],"source":"Partitions into parts $\\equiv \\pm a \\pmod{d+3}$ by largest part and number of parts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-pmresidue-arm-leg-count","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"For natural numbers $d$, $a$, $\\alpha$ and $\\lambda$, $\\ell_d^{(a)}(\\alpha, \\lambda)$ is the number of nonempty partitions into parts congruent to $a$ or $-a$ modulo $d + 3$ with largest part $\\alpha$ and exactly $\\lambda$ parts (so $\\ell_d^{(a)}(0, 0) = 0$: the empty partition is not counted).","title":"Partitions into parts $\\equiv \\pm a \\pmod{d+3}$ by largest part and number of parts"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-counts-small-values","kind":"node","lean":"MetaMathlibExt.perimeterCounts_small_values","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-counts-small-values","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"fully_proved","summary":"Take $d = a = 1$. Among the partitions with perimeter $3$, exactly two have $1$-distinct parts that are all at least $1$, exactly two have all parts congruent to $1$ modulo $2$, and exactly two have all parts congruent to $1$ or $-1$ modulo $4$. Moreover, exactly one partition with largest part $2$ and $2$ parts has $1$-distinct parts that are all at least $1$, exactly one partition with largest part $3$ and $1$\u2026","title":"Small values of the fixed-perimeter partition counts"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-distinct-count-eq-perimeter-re","kind":"node","lean":"MetaMathlibExt.perimeterDistinctCount_eq_perimeterResidueCount","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed-perimeter identity for "],["math","\\(d\\)"],["text","-distinct parts"]],"source":"Fixed-perimeter identity for $d$-distinct parts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-distinct-count-eq-perimeter-re","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"Let $d$ and $n$ be positive integers and let $a$ be an integer with $1 \\le a \\le d + 1$. Then the number $h_d^{(a)}(n)$ of partitions of perimeter $n$ into $d$-distinct parts (no part repeats and any two parts differ by at least $d$), all at least $a$, equals the number $f_d^{(a)}(n)$ of partitions of perimeter $n$ into parts congruent to $a$ modulo $d + 1$. Here the perimeter of a partition is its largest part\u2026","title":"Fixed-perimeter identity for $d$-distinct parts"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-distinct-arm-leg-count-eq-peri","kind":"node","lean":"MetaMathlibExt.perimeterDistinctArmLegCount_eq_perimeterResidueArmLegCount","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-distinct-arm-leg-count-eq-peri","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"Let $d$, $a$, $m$ and $k$ be natural numbers with $d \\ge 1$, $1 \\le a \\le d + 1$, $m \\ge 1$ and $k \\ge 1$. Write $h_d^{(a)}(m, k)$ for the number of partitions into $d$-distinct parts, all at least $a$, with largest part $m$ and $k$ parts, and $f_d^{(a)}(m, k)$ for the number of partitions into parts congruent to $a$ modulo $d + 1$ with largest part $m$ and $k$ parts. Then\u2026","title":"Refined equality of distinct-part and residue counts"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-distinct-arm-leg-count-ne-zero","kind":"node","lean":"MetaMathlibExt.perimeterDistinctArmLegCount_ne_zero_iff_and_eq_choose","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-distinct-arm-leg-count-ne-zero","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"Let $d$, $a$, $m$ and $k$ be positive integers with $a \\le d + 1$, and write $h_d^{(a)}(m, k)$ for the number of partitions into $d$-distinct parts, all at least $a$, with largest part $m$ and $k$ parts. Then $h_d^{(a)}(m, k) \\ne 0$ if and only if $m \\ge a + d(k - 1)$, and in that case $h_d^{(a)}(m, k) = \\binom{m - a - (d - 1)(k - 1)}{k - 1}$.","title":"Counting d-distinct partitions by largest part and length"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-residue-arm-leg-count-ne-zero","kind":"node","lean":"MetaMathlibExt.perimeterResidueArmLegCount_ne_zero_iff_and_eq_choose","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonvanishing and binomial count for parts "],["math","\\(\\equiv a \\pmod{d+1}\\)"]],"source":"Nonvanishing and binomial count for parts $\\equiv a \\pmod{d+1}$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-residue-arm-leg-count-ne-zero","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"Let $d$, $a$, $\\alpha$ and $\\lambda$ be positive integers with $a \\le d + 1$, and let $f_d^{(a)}(\\alpha, \\lambda)$ be the number of partitions into parts congruent to $a$ modulo $d + 1$ with largest part $\\alpha$ and exactly $\\lambda$ parts. Then $f_d^{(a)}(\\alpha, \\lambda) \\ne 0$ if and only if $\\alpha \\ge a$ and $\\alpha \\equiv a \\pmod{d + 1}$, and in that case\u2026","title":"Nonvanishing and binomial count for parts $\\equiv a \\pmod{d+1}$"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/floor-half-div-le-div-of-two-mul-lt","kind":"node","lean":"MetaMathlibExt.floor_half_div_le_div_of_two_mul_lt","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#floor-half-div-le-div-of-two-mul-lt","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"Let $d$, $a$ and $n$ be natural numbers with $d \\ge 1$, $a \\ge 1$, $2a < d + 3$ and $a \\le n$. Then $\\lfloor \\lfloor (n - a)/(d + 1) \\rfloor / 2 \\rfloor \\le \\lfloor (n - a)/(d + 3) \\rfloor$ and $\\lfloor (\\lfloor (n - a)/(d + 1) \\rfloor + 1) / 2 \\rfloor \\le \\lfloor (n + a)/(d + 3) \\rfloor$.","title":"A floor inequality for perimeter counts"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-pmresidue-arm-leg-count-ne-zer","kind":"node","lean":"MetaMathlibExt.perimeterPMResidueArmLegCount_ne_zero_iff_and_eq_choose","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonvanishing and binomial count for parts "],["math","\\(\\equiv \\pm a \\pmod{d+3}\\)"]],"source":"Nonvanishing and binomial count for parts $\\equiv \\pm a \\pmod{d+3}$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-pmresidue-arm-leg-count-ne-zer","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"Let $d$, $a$, $\\alpha$ and $\\lambda$ be positive integers with $a < (d + 3)/2$, and let $\\ell_d^{(a)}(\\alpha, \\lambda)$ be the number of partitions into parts congruent to $a$ or $-a$ modulo $d + 3$ with largest part $\\alpha$ and exactly $\\lambda$ parts. Then $\\ell_d^{(a)}(\\alpha, \\lambda) \\ne 0$ if and only if $\\alpha \\ge a$ and $\\alpha \\equiv \\pm a \\pmod{d + 3}$. Moreover, if $\\alpha \\ge a$ and\u2026","title":"Nonvanishing and binomial count for parts $\\equiv \\pm a \\pmod{d+3}$"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-distinct-count-le-perimeter-pm","kind":"node","lean":"MetaMathlibExt.perimeterDistinctCount_le_perimeterPMResidueCount","parent":25637,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-distinct-count-le-perimeter-pm","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"Let $d$, $n$ and $a$ be positive integers with $a < (d + 3)/2$. Then the number $h_d^{(a)}(n)$ of partitions of perimeter $n$ into $d$-distinct parts (no part repeats and any two parts differ by at least $d$), all at least $a$, is at most the number $\\ell_d^{(a)}(n)$ of partitions of perimeter $n$ into parts congruent to $a$ or $-a$ modulo $d + 3$. Here the perimeter of a partition is its largest part plus its\u2026","title":"Reverse Alder-type inequality for fixed perimeter partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-distinct-count-shift-inequalit","kind":"node","lean":"MetaMathlibExt.perimeterDistinctCount_shift_inequalities","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity of fixed perimeter counts with "],["math","\\(d\\)"],["text","-distinct parts"]],"source":"Monotonicity of fixed perimeter counts with $d$-distinct parts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-distinct-count-shift-inequalit","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"For natural numbers $d$, $a$ and $n$, let $h_d^{(a)}(n)$ be the number of partitions of perimeter $n$ (largest part plus number of parts minus $1$) into $d$-distinct parts, all at least $a$, where parts are $d$-distinct when no part repeats and any two parts differ by at least $d$. If $d$, $n$ and $a$ are positive, then $h_d^{(a+1)}(n) \\le h_d^{(a)}(n)$, $h_{d+1}^{(a)}(n) \\le h_d^{(a)}(n)$ and\u2026","title":"Monotonicity of fixed perimeter counts with $d$-distinct parts"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-pmresidue-count-shift-inequali","kind":"node","lean":"MetaMathlibExt.perimeterPMResidueCount_shift_inequalities","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity of fixed perimeter counts with parts "],["math","\\(\\equiv \\pm a \\pmod{d+3}\\)"]],"source":"Monotonicity of fixed perimeter counts with parts $\\equiv \\pm a \\pmod{d+3}$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-pmresidue-count-shift-inequali","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"For natural numbers $d$, $a$ and $n$, let $\\ell_d^{(a)}(n)$ be the number of partitions of perimeter $n$ (largest part plus number of parts minus $1$) into parts congruent to $a$ or $-a$ modulo $d + 3$. If $d$, $n$ and $a$ are positive with $a < (d + 3)/2$, then $\\ell_d^{(a)}(n) \\le \\ell_d^{(a)}(n + 1)$ and $\\ell_{d+1}^{(a)}(n) \\le \\ell_d^{(a)}(n)$.","title":"Monotonicity of fixed perimeter counts with parts $\\equiv \\pm a \\pmod{d+3}$"},{"id":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/perimeter-pmresidue-count-le-perimeter-d","kind":"node","lean":"MetaMathlibExt.perimeterPMResidueCount_le_perimeterDistinctCount_of_odd","parent":25637,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alder-type inequality in the degenerate case "],["math","\\(a = (d + 3)/2\\)"]],"source":"Alder-type inequality in the degenerate case $a = (d + 3)/2$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part/index.html#perimeter-pmresidue-count-le-perimeter-d","scope":"Combinatorics/integer-partitions/partition-identities/alder-type-inequalities-for-fixed-perimeter-part","status":"can_prove","summary":"Let $n$ be a positive integer, let $d$ be an odd natural number and put $a = (d + 3)/2$. Then the number $\\ell_d^{(a)}(n)$ of partitions of perimeter $n$ into parts congruent to $a$ or $-a$ modulo $d + 3$ is at most the number $h_d^{(a)}(n)$ of partitions of perimeter $n$ into $d$-distinct parts (no part repeats and any two parts differ by at least $d$), all at least $a$. Here the perimeter of a partition is its\u2026","title":"Alder-type inequality in the degenerate case $a = (d + 3)/2$"},{"data_url":"scopes/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities.json","id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","kind":"scope","navigation":"module","parent":25633,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","title":"Colored partitions and Euler-type identities"},{"id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/card-colored-partitions-different-colors","kind":"node","lean":"MetaMathlibExt.card_coloredPartitions_differentColors","parent":25661,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html#card-colored-partitions-different-colors","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","summary":"A colored part is a pair $(m, j)$ of integers with $1 \\le j \\le m$, read as the part $m$ in color $j$. For every $n \\ge 0$, the number of nonempty finite sets of colored parts $(m, j)$ with $m \\le n$ in which no two colored parts have the same color $j$ is $(n + 1)! - 1$.","title":"Colored partitions with distinct colors"},{"id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/card-colored-partitions-fully-different","kind":"node","lean":"MetaMathlibExt.card_coloredPartitions_fullyDifferent","parent":25661,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html#card-colored-partitions-fully-different","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","summary":"A colored part is a pair $(m, j)$ of integers with $1 \\le j \\le m$, read as the part $m$ in color $j$. For $n \\ge 0$, let $\\mathrm{pf}_c(n)$ be the number of finite sets of colored parts whose sizes $m$ sum to $n$ and in which no two colored parts have the same size and no two have the same color. Then $\\mathrm{pf}_c(n)$ is the sum, over all partitions of $n$ into distinct parts $p_1 > p_2 > \\cdots > p_k$, of the\u2026","title":"Formula for colored partitions into fully different parts"},{"id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/card-colored-partitions-fully-different-2","kind":"node","lean":"MetaMathlibExt.card_coloredPartitions_fullyDifferent_le","parent":25661,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html#card-colored-partitions-fully-different-2","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","summary":"A colored part is a pair $(m, j)$ of integers with $1 \\le j \\le m$, read as the part $m$ in color $j$. For every $n \\ge 0$, the number of nonempty finite sets of colored parts $(m, j)$ with $m \\le n$ in which no two colored parts have the same size $m$ and no two have the same color $j$ is $B_{n+1} - 1$, where $B_k$ denotes the $k$-th Bell number.","title":"Colored partitions into fully different parts with bounded size"},{"id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/ncard-constant-block-partitions-eq-doubl","kind":"node","lean":"MetaMathlibExt.ncard_constantBlockPartitions_eq_doubleCompositions","parent":25661,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html#ncard-constant-block-partitions-eq-doubl","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","summary":"For $n \\ge 0$, count the pairs $(A, P)$ where $A$ is a multiset of $n$ positive integers whose distinct elements are exactly $1, \\dots, k$ for some $k \\ge 0$, and $P$ is a partition of $A$ into constant blocks, that is, a multiset of nonempty multisets, each with all its elements equal, whose sum is $A$. The number of such pairs equals the number $d_c(n)$ of double compositions of $n$, that is, compositions of $n$\u2026","title":"Constant-block partitions and double compositions"},{"id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/ncard-distinct-constant-block-partitions","kind":"node","lean":"MetaMathlibExt.ncard_distinctConstantBlockPartitions_eq_coloredCompositions","parent":25661,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html#ncard-distinct-constant-block-partitions","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","summary":"For $n \\ge 0$, count the pairs $(A, P)$ where $A$ is a multiset of $n$ positive integers whose distinct elements are exactly $1, \\dots, k$ for some $k \\ge 0$, and $P$ is a partition of $A$ into pairwise distinct constant blocks, that is, a multiset of pairwise distinct nonempty multisets, each with all its elements equal, whose sum is $A$. The number of such pairs equals the number of compositions of $n$ in which\u2026","title":"Distinct constant-block partitions and colored compositions"},{"id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/generalized-euler-partition-theorem","kind":"node","lean":"MetaMathlibExt.generalized_euler_partition_theorem","parent":25661,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html#generalized-euler-partition-theorem","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","summary":"Let $S$ be a set of compositions (finite sequences of positive integers) such that doubling every summand of a member of $S$ gives a member of $S$, and halving every summand of a member of $S$ whose summands are all even gives a member of $S$. Let $n \\ge 0$. The number of partitions of $n$ into colored parts in which each part $k$ is colored by a composition in $S$ with sum $k$, and no colored part (a pair of part\u2026","title":"Generalized Euler partition theorem"},{"id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/euler-partition-theorem-restricted-compo","kind":"node","lean":"MetaMathlibExt.euler_partition_theorem_restrictedCompositions","parent":25661,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html#euler-partition-theorem-restricted-compo","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","summary":"A composition (a finite sequence of positive integers) is $\\ell$-restricted if it has at most $\\ell$ summands. Let $\\ell \\ge 0$ and $n \\ge 0$. The number of partitions of $n$ into colored parts in which each part $k$ is colored by an $\\ell$-restricted composition of $k$, and no colored part (a pair of part and color) appears twice, equals the number of such partitions of $n$, with repetition allowed, in which no\u2026","title":"Euler partition theorem for restricted compositions"},{"id":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/colored-partitions-different-parts-eq-re","kind":"node","lean":"MetaMathlibExt.coloredPartitions_differentParts_eq_restrictedColors","parent":25661,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities/index.html#colored-partitions-different-parts-eq-re","scope":"Combinatorics/integer-partitions/partition-identities/colored-partitions-and-euler-type-identities","status":"can_prove","summary":"A colored part is a pair $(m, j)$ of integers with $1 \\le j \\le m$, read as the part $m$ in color $j$, and a colored partition of $n$ is a finite multiset of colored parts whose sizes $m$ sum to $n$. For every $n \\ge 0$, the number of colored partitions of $n$ in which no colored part is repeated equals the number of colored partitions of $n$ in which no even part has an even color, and it also equals the number of\u2026","title":"Colored partitions with different parts"},{"data_url":"scopes/Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem.json","id":"Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem","kind":"scope","navigation":"module","parent":25633,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem/index.html","scope":"Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem","status":"fully_proved","title":"MacMahon's partition theorem"},{"id":"Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem/andrews-macmahon-partition-identity","kind":"node","lean":"MetaMathlibExt.andrews_macmahon_partition_identity","parent":25670,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem/index.html#andrews-macmahon-partition-identity","scope":"Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem","status":"fully_proved","summary":"For all natural numbers $r$ and $n$, the number of partitions of $n$ in which every part occurring an odd number of times occurs at least $2r + 1$ times equals the number of partitions of $n$ in which every odd part is congruent to $2r + 1$ modulo $4r + 2$.","title":"Andrews' generalization of MacMahon's partition theorem"},{"id":"Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem/macmahon-multiplicity-one-free","kind":"node","lean":"MetaMathlibExt.macmahon_multiplicity_one_free","parent":25670,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem/index.html#macmahon-multiplicity-one-free","scope":"Combinatorics/integer-partitions/partition-identities/macmahon-s-partition-theorem","status":"fully_proved","summary":"For every natural number $n$, the number of partitions of $n$ in which no part occurs exactly once equals the number of partitions of $n$ all of whose parts are even or congruent to $3$ modulo $6$.","title":"MacMahon's partition theorem"},{"data_url":"scopes/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts.json","id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","kind":"scope","navigation":"module","parent":25633,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","title":"Merca's identities for sums of different parts"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of different parts congruent to "],["math","\\(r\\)"],["text"," modulo "],["math","\\(m\\)"]],"source":"Sum of different parts congruent to $r$ modulo $m$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"fully_proved","summary":"For natural numbers $m$, $r$, $n$, $a_{r,m}(n)$ is the sum, over all partitions $\\mu$ of $n$, of the different parts $x$ of $\\mu$ with $x \\bmod m = r$; a part contributes $x$ once regardless of its multiplicity. It is used with $0 \\le r < m$, where $x \\bmod m = r$ means $x \\equiv r \\pmod m$.","title":"Sum of different parts congruent to $r$ modulo $m$"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mult-at-least","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsMultAtLeast","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of different parts appearing at least "],["math","\\(m\\)"],["text"," times"]],"source":"Sum of different parts appearing at least $m$ times"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mult-at-least","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"fully_proved","summary":"For natural numbers $m$ and $n$, $s_m(n)$ is the sum, over all partitions $\\mu$ of $n$, of the different parts of $\\mu$ that appear at least $m$ times in $\\mu$; such a part $j$ contributes $j$ once regardless of its multiplicity.","title":"Sum of different parts appearing at least $m$ times"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-small-values","kind":"node","lean":"MetaMathlibExt.sumDistinctParts_small_values","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-small-values","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"fully_proved","summary":"For natural numbers $m$, $r$ and $n$, let $a_{r,m}(n)$ be the sum, over all partitions of $n$, of the different parts $x$ of the partition with $x \\bmod m = r$, and let $s_m(n)$ be the sum, over all partitions of $n$, of the different parts that occur at least $m$ times in the partition; each such part counts once per partition. Then $a_{r,m}(0) = 0$ and $s_m(0) = 0$ for all $m$ and $r$, and moreover\u2026","title":"Small values of Merca's distinct-part sums"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/dyson-rank","kind":"node","lean":"MetaMathlibExt.dysonRank","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#dyson-rank","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"fully_proved","summary":"The rank of a partition $\\mu$ of $n$ is the integer $r(\\mu) = \\mu_1 - \\ell(\\mu)$, the largest part $\\mu_1$ of $\\mu$ minus its number of parts $\\ell(\\mu)$, where the largest part of the empty partition is taken to be $0$. In particular the empty partition has rank $0$.","title":"Dyson's rank of a partition"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/dyson-rank-small-values","kind":"node","lean":"MetaMathlibExt.dysonRank_small_values","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank of the partitions of "],["math","\\(0\\)"],["text"," and "],["math","\\(1\\)"]],"source":"Rank of the partitions of $0$ and $1$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#dyson-rank-small-values","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"fully_proved","summary":"The empty partition of $0$ and the partition $(1)$ of $1$ both have Dyson rank $0$.","title":"Rank of the partitions of $0$ and $1$"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/partition-crank","kind":"node","lean":"MetaMathlibExt.partitionCrank","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#partition-crank","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"fully_proved","summary":"Let $\\mu$ be a partition of $n$ and let $\\omega(\\mu)$ be the number of parts of $\\mu$ equal to $1$. The crank of $\\mu$ (Andrews and Garvan) is the integer defined as follows: if $\\omega(\\mu) = 0$, it is the largest part of $\\mu$, where the largest part of the empty partition is taken to be $0$; if $\\omega(\\mu) > 0$, it is the number of parts of $\\mu$ larger than $\\omega(\\mu)$, counted with multiplicity, minus\u2026","title":"Crank of a partition"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/partition-crank-small-values","kind":"node","lean":"MetaMathlibExt.partitionCrank_small_values","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Crank of the partitions of "],["math","\\(0\\)"],["text"," and "],["math","\\(1\\)"]],"source":"Crank of the partitions of $0$ and $1$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#partition-crank-small-values","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"fully_proved","summary":"The empty partition of $0$ has crank $0$, and the partition $(1)$ of $1$ has crank $0 - 1 = -1$.","title":"Crank of the partitions of $0$ and $1$"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq-eq-sum-partiti","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq_eq_sum_partition","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq-eq-sum-partiti","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m$, $r$ and $n$ be natural numbers with $r < m$, let $a_{r,m}(n)$ be the sum, over all partitions of $n$, of the different parts congruent to $r$ modulo $m$, each counted once per partition whatever its multiplicity, and let $p(k)$ be the number of partitions of $k$. Then","title":"Merca's identity for different parts in a residue class"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mult-at-least-eq-sum","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsMultAtLeast_eq_sum_partition","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Merca's identity for "],["math","\\(s_m(n)\\)"]],"source":"Merca's identity for $s_m(n)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mult-at-least-eq-sum","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m \\ge 1$ and $n$ be natural numbers, write $s_m(n)$ for the sum of the different parts that appear at least $m$ times in partitions of $n$, and $p(k)$ for the number of partitions of $k$. Then $s_m(n) = \\sum_{j \\ge 0,\\ mj \\le n} j\\, p(n - mj)$.","title":"Merca's identity for $s_m(n)$"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq-square-eq-even","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq_square_eq_evenOdd","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq-square-eq-even","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m$, $r$ and $n$ be natural numbers with $r < m$, and for $k \\ge 0$ let $a_{r,m}(k)$ be the sum, over all partitions of $k$, of the different parts congruent to $r$ modulo $m$, each counted once per partition whatever its multiplicity. Let $p_{e-o}(k) = \\sum_{\\mu} (-1)^{\\ell(\\mu)}$, the sum over the partitions $\\mu$ of $k$ with $\\ell(\\mu)$ the number of parts of $\\mu$; it is the number of partitions of $k$ with\u2026","title":"Different parts in a residue class and even-minus-odd partition counts"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq-twice-square-e","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq_twiceSquare_eq_oddDistinct","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Twice-square sum of "],["math","\\(a_{r,m}\\)"],["text"," and distinct odd parts"]],"source":"Twice-square sum of $a_{r,m}$ and distinct odd parts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq-twice-square-e","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m, r, n$ be natural numbers with $r < m$. Write $a_{r,m}(k)$ for the sum of the different parts congruent to $r$ modulo $m$ in all partitions of $k$, and $q_{\\mathrm{odd}}(k)$ for the number of partitions of $k$ into distinct odd parts. Then","title":"Twice-square sum of $a_{r,m}$ and distinct odd parts"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq-triangular-eq","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq_triangular_eq_distinct_half","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed triangular sum of "],["math","\\(a_{r,m}\\)"],["text"," and distinct partitions"]],"source":"Signed triangular sum of $a_{r,m}$ and distinct partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq-triangular-eq","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m, r, n$ be natural numbers with $r < m$. Write $a_{r,m}(k)$ for the sum of the different parts congruent to $r$ modulo $m$ in all partitions of $k$, $q(k)$ for the number of partitions of $k$ into distinct parts, and $\\tilde q(\\ell)$ for $q(\\ell/2)$ when $\\ell$ is even and for $0$ when $\\ell$ is odd. Then","title":"Signed triangular sum of $a_{r,m}$ and distinct partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mult-at-least-triangu","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsMultAtLeast_triangular_eq_distinct_half","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed triangular sum of "],["math","\\(s_m\\)"],["text"," and distinct partitions"]],"source":"Signed triangular sum of $s_m$ and distinct partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mult-at-least-triangu","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m \\ge 1$ and $n$ be natural numbers. Write $s_m(k)$ for the sum of the different parts that appear at least $m$ times in partitions of $k$, $q(k)$ for the number of partitions of $k$ into distinct parts, and $\\tilde q(\\ell)$ for $q(\\ell/2)$ when $\\ell$ is even and for $0$ when $\\ell$ is odd. Then","title":"Signed triangular sum of $s_m$ and distinct partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq-pentagonal-eq","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq_pentagonal_eq_nonnegRank","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq-pentagonal-eq","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m$, $r$ and $n$ be natural numbers with $r < m$, and for $k \\ge 0$ let $a_{r,m}(k)$ be the sum, over all partitions of $k$, of the different parts congruent to $r$ modulo $m$, each counted once per partition whatever its multiplicity. Let $N(k)$ be the number of partitions of $k$ whose Dyson rank (largest part minus number of parts) is nonnegative. Then","title":"Different parts in a residue class and partitions of nonnegative rank"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq-pentagonal-eq-2","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq_pentagonal_eq_posRank","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq-pentagonal-eq-2","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m$, $r$ and $n$ be natural numbers with $r < m$, and for $k \\ge 0$ let $a_{r,m}(k)$ be the sum, over all partitions of $k$, of the different parts congruent to $r$ modulo $m$, each counted once per partition whatever its multiplicity. Let $R(k)$ be the number of partitions of $k$ whose Dyson rank (largest part minus number of parts) is positive. Then","title":"Different parts in a residue class and partitions of positive rank"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mult-at-least-pentago","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsMultAtLeast_pentagonal_eq_nonnegRank","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pentagonal sum of "],["math","\\(s_m\\)"],["text"," and nonnegative rank"]],"source":"Pentagonal sum of $s_m$ and nonnegative rank"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mult-at-least-pentago","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m \\ge 1$ and $n$ be natural numbers. Write $s_m(k)$ for the sum of the different parts that appear at least $m$ times in partitions of $k$, and $N(k)$ for the number of partitions of $k$ with nonnegative Dyson rank. Then","title":"Pentagonal sum of $s_m$ and nonnegative rank"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mult-at-least-pentago-2","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsMultAtLeast_pentagonal_eq_posRank","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pentagonal sum of "],["math","\\(s_m\\)"],["text"," and positive rank"]],"source":"Pentagonal sum of $s_m$ and positive rank"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mult-at-least-pentago-2","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m \\ge 1$ and $n$ be natural numbers. Write $s_m(k)$ for the sum of the different parts that appear at least $m$ times in partitions of $k$, and $R(k)$ for the number of partitions of $k$ with positive Dyson rank. Then","title":"Pentagonal sum of $s_m$ and positive rank"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq-eq-garden-of-e","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq_eq_gardenOfEden","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq-eq-garden-of-e","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m$, $r$ and $n$ be natural numbers with $r < m$, and for $k \\ge 0$ let $a_{r,m}(k)$ be the sum, over all partitions of $k$, of the different parts congruent to $r$ modulo $m$, each counted once per partition whatever its multiplicity. Let $G(k)$ be the number of Garden of Eden partitions of $k$, that is, partitions of $k$ whose Dyson rank (largest part minus number of parts) is at most $-2$. Then","title":"Different parts in a residue class and Garden of Eden partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mult-at-least-eq-gard","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsMultAtLeast_eq_gardenOfEden","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Garden of Eden identity for "],["math","\\(s_m\\)"]],"source":"Garden of Eden identity for $s_m$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mult-at-least-eq-gard","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m \\ge 1$ and $n$ be natural numbers. Write $s_m(k)$ for the sum of the different parts that appear at least $m$ times in partitions of $k$, and $G(k)$ for the number of Garden of Eden partitions of $k$, those whose Dyson rank is at most $-2$ (so $G(0) = 0$). Then","title":"Garden of Eden identity for $s_m$"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mod-eq-triangular-eq-2","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsModEq_triangular_eq_nonnegCrank","parent":25673,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mod-eq-triangular-eq-2","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m$, $r$ and $n$ be natural numbers with $r < m$, and for $k \\ge 0$ let $a_{r,m}(k)$ be the sum, over all partitions of $k$, of the different parts congruent to $r$ modulo $m$, each counted once per partition whatever its multiplicity. Let $C(k)$ be the number of partitions of $k$ whose crank is nonnegative. Then","title":"Different parts in a residue class and partitions of nonnegative crank"},{"id":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/sum-distinct-parts-mult-at-least-triangu-2","kind":"node","lean":"MetaMathlibExt.sumDistinctPartsMultAtLeast_triangular_eq_nonnegCrank","parent":25673,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed triangular sum of "],["math","\\(s_m\\)"],["text"," and nonnegative crank"]],"source":"Signed triangular sum of $s_m$ and nonnegative crank"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts/index.html#sum-distinct-parts-mult-at-least-triangu-2","scope":"Combinatorics/integer-partitions/partition-identities/merca-s-identities-for-sums-of-different-parts","status":"can_prove","summary":"Let $m \\ge 1$ and $n$ be natural numbers. Write $s_m(k)$ for the sum of the different parts that appear at least $m$ times in partitions of $k$, and $C(k)$ for the number of partitions of $k$ with nonnegative crank. Then","title":"Signed triangular sum of $s_m$ and nonnegative crank"},{"data_url":"scopes/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti.json","id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","kind":"scope","navigation":"module","parent":25633,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","title":"Stanley's and Elder's theorems on parts of partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/sum-count-eq-sum-card-parts-count-ge","kind":"node","lean":"MetaMathlibExt.sum_count_eq_sum_card_parts_count_ge","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#sum-count-eq-sum-card-parts-count-ge","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers with $k \\ge 1$. The total number of times the part $k$ occurs, counted with multiplicity, over all partitions of $n$ equals the sum, over all partitions $\\lambda$ of $n$, of the number of distinct parts of $\\lambda$ that occur at least $k$ times in $\\lambda$.","title":"Elder's theorem"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/sum-count-one-eq-sum-card-to-finset","kind":"node","lean":"MetaMathlibExt.sum_count_one_eq_sum_card_toFinset","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#sum-count-one-eq-sum-card-to-finset","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For every natural number $n$, the total number of parts equal to $1$, counted with multiplicity, over all partitions of $n$ equals the sum, over all partitions $\\lambda$ of $n$, of the number of distinct parts of $\\lambda$.","title":"Stanley's theorem"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-parts-up-to","kind":"node","lean":"MetaMathlibExt.nColorPartsUpTo","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","The list of colored parts of size at most "],["math","\\(m\\)"]],"source":"The list of colored parts of size at most $m$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-parts-up-to","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"For $m \\ge 0$, this is the list of the colored parts $k_j$, that is, the pairs $(k, j)$ of natural numbers with $1 \\le j \\le k \\le m$, each listed once, in the order $1_1, 2_1, 2_2, 3_1, 3_2, 3_3, \\dots$ (by size $k$, then by color $j$).","title":"The list of colored parts of size at most $m$"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/mem-n-color-parts-up-to","kind":"node","lean":"MetaMathlibExt.mem_nColorPartsUpTo","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#mem-n-color-parts-up-to","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"For all natural numbers $m$, $k$ and $j$, the pair $(k, j)$ appears in the list of colored parts of size at most $m$ if and only if $1 \\le j \\le k \\le m$.","title":"Membership in the colored parts of bounded size"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/multisets-with-size-sum","kind":"node","lean":"MetaMathlibExt.multisetsWithSizeSum","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#multisets-with-size-sum","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"For a list $L$ of pairs of natural numbers, where the size of a pair $(k, j)$ is $k$, and a natural number $n$, this is the finite set of multisets of pairs defined by recursion on $L$. For the empty list it is the set containing only the empty multiset when $n = 0$, and the empty set otherwise. For a list with head $c$ of size $k$ and tail $L'$, it is the set of all multisets obtained by adding $i$ copies of $c$\u2026","title":"Multisets of parts with a given size sum"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/mem-multisets-with-size-sum","kind":"node","lean":"MetaMathlibExt.mem_multisetsWithSizeSum","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#mem-multisets-with-size-sum","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"Let $L$ be a list of pairs of natural numbers in which every pair $(k, j)$ has size $k \\ge 1$, and let $n$ be a natural number. A multiset $s$ of pairs belongs to the set of multisets drawn from $L$ with size sum $n$ if and only if every element of $s$ lies in $L$ and the sizes of the elements of $s$, counted with multiplicity, sum to $n$.","title":"Membership in the multisets with a given size sum"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-partitions","kind":"node","lean":"MetaMathlibExt.nColorPartitions","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-partitions","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"For a natural number $m$, the $n$-color partitions of $m$ form the finite set of multisets of colored parts $k_j$, that is pairs $(k, j)$ of natural numbers with $1 \\le j \\le k$ (a part of size $k$ comes in $k$ colors), whose sizes $k$, counted with multiplicity, sum to $m$.","title":"n-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/mem-n-color-partitions","kind":"node","lean":"MetaMathlibExt.mem_nColorPartitions","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Characterization of $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#mem-n-color-partitions","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"For every natural number $m$, a multiset $s$ of pairs of natural numbers is an $n$-color partition of $m$ if and only if every element $(k, j)$ of $s$ satisfies $1 \\le j \\le k$ and the sizes $k$ of the elements of $s$, counted with multiplicity, sum to $m$.","title":"Characterization of $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-partitions-card-small","kind":"node","lean":"MetaMathlibExt.nColorPartitions_card_small","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-partitions-card-small","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"Let $C(m)$ be the number of $n$-color partitions of $m$, and write $k_j$ for the colored part of size $k$ and color $j$. Then $C(0) = C(1) = 1$, $C(2) = 3$ and $C(3) = 6$, and the $n$-color partitions of $2$ are exactly $2_1$, $2_2$ and $1_1 + 1_1$.","title":"Small values of the number of n-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/sigma-odd","kind":"node","lean":"MetaMathlibExt.sigmaOdd","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#sigma-odd","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"For a natural number $k$, $\\sigma_{\\mathrm{odd}}(k)$ is the sum of the odd positive divisors of $k$. By convention $\\sigma_{\\mathrm{odd}}(0) = 0$.","title":"Sum of odd divisors"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/sigma-even","kind":"node","lean":"MetaMathlibExt.sigmaEven","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#sigma-even","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"For a natural number $k$, $\\sigma_{\\mathrm{even}}(k)$ is the sum of the even divisors of $k$ (so $\\sigma_{\\mathrm{even}}(0) = 0$).","title":"Sum of the even divisors"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/sigma-odd-sigma-even-small","kind":"node","lean":"MetaMathlibExt.sigmaOdd_sigmaEven_small","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#sigma-odd-sigma-even-small","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"fully_proved","summary":"The sums of the odd and of the even divisors of $12$ and $9$ are $\\sigma_{\\mathrm{odd}}(12) = 1 + 3 = 4$, $\\sigma_{\\mathrm{even}}(12) = 2 + 4 + 6 + 12 = 24$, $\\sigma_{\\mathrm{odd}}(9) = 1 + 3 + 9 = 13$ and $\\sigma_{\\mathrm{even}}(9) = 0$.","title":"Odd and even divisor sums of 12 and 9"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-total-parts-eq-sum-sigma","kind":"node","lean":"MetaMathlibExt.nColor_totalParts_eq_sum_sigma","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total number of parts of "],["math","\\(n\\)"],["text","-color partitions via the divisor sum"]],"source":"Total number of parts of $n$-color partitions via the divisor sum"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-total-parts-eq-sum-sigma","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $r \\ge 0$ let $C(r)$ be the number of $n$-color partitions of $r$, and let $T_o(r)$ (resp. $T_e(r)$) be the total number of parts, counted with multiplicity, over the $n$-color partitions of $r$ having an odd (resp. even) number of parts. Then for every $m \\ge 1$,","title":"Total number of parts of $n$-color partitions via the divisor sum"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/sigma-odd-eq-sum-n-color-parts-diff-mul","kind":"node","lean":"MetaMathlibExt.sigmaOdd_eq_sum_nColor_partsDiff_mul_distinct","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd-divisor sum via the total number of parts of "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Odd-divisor sum via the total number of parts of $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#sigma-odd-eq-sum-n-color-parts-diff-mul","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $r \\ge 0$ let $T_o(r)$ (resp. $T_e(r)$) be the total number of parts, counted with multiplicity, over the $n$-color partitions of $r$ having an odd (resp. even) number of parts, and let $C(D, r)$ be the number of $n$-color partitions of $r$ into distinct colored parts. Then for every $m \\ge 1$,","title":"Odd-divisor sum via the total number of parts of $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-even-parts-eq-sum-sigma-even","kind":"node","lean":"MetaMathlibExt.nColor_evenParts_eq_sum_sigmaEven","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even divisor sums count the even parts of "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Even divisor sums count the even parts of $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-even-parts-eq-sum-sigma-even","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $m \\ge 1$, let $A_e(m)$ be the total number of parts of even size, counted with multiplicity, over the $n$-color partitions of $m$, and for $r \\ge 0$ let $C(r)$ be the number of $n$-color partitions of $r$. Then","title":"Even divisor sums count the even parts of $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-odd-parts-eq-sum-sigma-odd","kind":"node","lean":"MetaMathlibExt.nColor_oddParts_eq_sum_sigmaOdd","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-odd-parts-eq-sum-sigma-odd","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $i \\ge 0$ let $A_o(i)$ be the total number of odd parts (parts of odd size, counted with multiplicity) over the $n$-color partitions of $i$, and let $C(i)$ be the number of $n$-color partitions of $i$. Then for every $m \\ge 1$,","title":"Number of odd parts in n-color partitions as a divisor sum"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-parts-multiplicity-ge-eq-sum","kind":"node","lean":"MetaMathlibExt.nColor_partsMultiplicityGe_eq_sum","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-parts-multiplicity-ge-eq-sum","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Let $t \\ge 1$. For $i \\ge 0$ let $N^t(i)$ be the number of distinct colored parts occurring at least $t$ times in an $n$-color partition of $i$, summed over all $n$-color partitions of $i$, and let $C(i)$ be the number of $n$-color partitions of $i$. Then for every $m \\ge t$,","title":"Parts of multiplicity at least t in n-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-parts-mod-eq-eq-sum","kind":"node","lean":"MetaMathlibExt.nColor_partsModEq_eq_sum","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-parts-mod-eq-eq-sum","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Let $r$ and $k$ be natural numbers with $0 < k < r$. For $i \\ge 0$ let $C(k \\bmod r, i)$ be the total number of parts of size congruent to $k$ modulo $r$, counted with multiplicity, over the $n$-color partitions of $i$, and let $C(i)$ be the number of $n$-color partitions of $i$. Then for every $m \\ge 1$,","title":"Parts in a residue class over n-color partitions as a divisor sum"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-distinct-total-parts-eq-sum-sigm","kind":"node","lean":"MetaMathlibExt.nColor_distinctTotalParts_eq_sum_sigmaOdd","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd divisor sums count the parts of distinct "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Odd divisor sums count the parts of distinct $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-distinct-total-parts-eq-sum-sigm","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Call an $n$-color partition distinct when no colored part occurs more than once. For $m \\ge 1$, let $C_o(D, m)$ (resp. $C_e(D, m)$) be the total number of parts over the distinct $n$-color partitions of $m$ with an odd (resp. even) number of parts, and for $r \\ge 0$ let $C(D, r)$ be the number of distinct $n$-color partitions of $r$. Then","title":"Odd divisor sums count the parts of distinct $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/sigma-eq-sum-n-color-distinct-parts-diff","kind":"node","lean":"MetaMathlibExt.sigma_eq_sum_nColor_distinctPartsDiff","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Divisor sum via distinct "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Divisor sum via distinct $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#sigma-eq-sum-n-color-distinct-parts-diff","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $r \\ge 0$ let $C(r)$ be the number of $n$-color partitions of $r$, and let $C_o(D, r)$ (resp. $C_e(D, r)$) be the total number of parts over the $n$-color partitions of $r$ into an odd (resp. even) number of distinct colored parts. Then for every $m \\ge 1$,","title":"Divisor sum via distinct $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-odd-partition-parts-eq-sum-sigma","kind":"node","lean":"MetaMathlibExt.nColor_oddPartitionParts_eq_sum_sigmaOdd","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-odd-partition-parts-eq-sum-sigma","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $i \\ge 0$ let $Q(O, i)$ be the total number of parts, counted with multiplicity, over the $n$-color partitions of $i$ into odd parts (every part has odd size), and let $C(O, i)$ be the number of such partitions. Then for every $m \\ge 1$,","title":"Number of parts in n-color partitions into odd parts as a divisor sum"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-odd-partition-parts-by-parity-eq","kind":"node","lean":"MetaMathlibExt.nColor_oddPartitionParts_byParity_eq_sum_sigmaOdd","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd divisor sums count the parts of "],["math","\\(n\\)"],["text","-color partitions into odd parts"]],"source":"Odd divisor sums count the parts of $n$-color partitions into odd parts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-odd-partition-parts-by-parity-eq","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Say that an $n$-color partition has odd parts when every part has odd size. For $m \\ge 1$, let $U_o(m)$ (resp. $U_e(m)$) be the total number of parts over the $n$-color partitions of $m$ with odd parts and an odd (resp. even) number of parts, and for $r \\ge 0$ let $C(O, r)$ be the number of $n$-color partitions of $r$ with odd parts. Then","title":"Odd divisor sums count the parts of $n$-color partitions into odd parts"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-total-parts-generating-function","kind":"node","lean":"MetaMathlibExt.nColor_totalParts_generatingFunction","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions for the total number of parts of "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Generating functions for the total number of parts of $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-total-parts-generating-function","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $m \\ge 0$ let $T_o(m)$ (resp. $T_e(m)$) be the total number of parts, counted with multiplicity, over the $n$-color partitions of $m$ having an odd (resp. even) number of parts. Then, as formal power series in $q$ with rational coefficients,","title":"Generating functions for the total number of parts of $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-even-parts-generating-function","kind":"node","lean":"MetaMathlibExt.nColor_evenParts_generatingFunction","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for the even parts of "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Generating function for the even parts of $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-even-parts-generating-function","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Let $A_e(m)$ be the total number of parts of even size, counted with multiplicity, over the $n$-color partitions of $m$. Then, as formal power series in $q$ over $\\mathbb{Q}$, with infinite sums and products taken coefficientwise,","title":"Generating function for the even parts of $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-odd-parts-generating-function","kind":"node","lean":"MetaMathlibExt.nColor_oddParts_generatingFunction","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-odd-parts-generating-function","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $i \\ge 0$ let $A_o(i)$ be the total number of odd parts (parts of odd size, counted with multiplicity) over the $n$-color partitions of $i$. Then in the ring $\\mathbb{Q}[[q]]$ of formal power series, with infinite sums and products taken coefficientwise,","title":"Generating function for the number of odd parts in n-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-parts-multiplicity-ge-generating","kind":"node","lean":"MetaMathlibExt.nColor_partsMultiplicityGe_generatingFunction","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for colored parts of multiplicity at least "],["math","\\(t\\)"]],"source":"Generating function for colored parts of multiplicity at least $t$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-parts-multiplicity-ge-generating","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Let $t \\ge 1$. For $m \\ge 0$ let $N^t(m)$ be the sum, over all $n$-color partitions of $m$, of the number of distinct colored parts that occur at least $t$ times. Then, as formal power series in $q$ with rational coefficients,","title":"Generating function for colored parts of multiplicity at least $t$"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-parts-mod-eq-generating-function","kind":"node","lean":"MetaMathlibExt.nColor_partsModEq_generatingFunction","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-parts-mod-eq-generating-function","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Let $r$ and $k$ be natural numbers with $0 < k < r$, and for $i \\ge 0$ let $C(k \\bmod r, i)$ be the total number of parts of size congruent to $k$ modulo $r$, counted with multiplicity, over the $n$-color partitions of $i$. Then in the ring $\\mathbb{Q}[[q]]$ of formal power series, with infinite sums and products taken coefficientwise,","title":"Generating function for parts in a residue class over n-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-distinct-total-parts-generating","kind":"node","lean":"MetaMathlibExt.nColor_distinctTotalParts_generatingFunction","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions for the parts of distinct "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Generating functions for the parts of distinct $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-distinct-total-parts-generating","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Call an $n$-color partition distinct when no colored part occurs more than once, and let $C_o(D, m)$ (resp. $C_e(D, m)$) be the total number of parts over the distinct $n$-color partitions of $m$ with an odd (resp. even) number of parts. Then, as formal power series in $q$ over $\\mathbb{Q}$, with infinite sums and products taken coefficientwise, both","title":"Generating functions for the parts of distinct $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/sigma-odd-eq-sum-n-color-distinct-total","kind":"node","lean":"MetaMathlibExt.sigmaOdd_eq_sum_nColor_distinctTotalParts_mul_signed","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd-divisor sum via distinct "],["math","\\(n\\)"],["text","-color partitions and signed partition counts"]],"source":"Odd-divisor sum via distinct $n$-color partitions and signed partition counts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#sigma-odd-eq-sum-n-color-distinct-total","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $r \\ge 0$ let $C_o(D, r)$ (resp. $C_e(D, r)$) be the total number of parts over the $n$-color partitions of $r$ into an odd (resp. even) number of distinct colored parts, and let $C_{e-o}(r) = \\sum_\\lambda (-1)^{\\ell(\\lambda)}$, the sum running over all $n$-color partitions $\\lambda$ of $r$, where $\\ell(\\lambda)$ is the number of parts of $\\lambda$ counted with multiplicity. Then for every $m \\ge 1$,","title":"Odd-divisor sum via distinct $n$-color partitions and signed partition counts"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-odd-partition-parts-generating-f","kind":"node","lean":"MetaMathlibExt.nColor_oddPartitionParts_generatingFunction","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-odd-partition-parts-generating-f","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $i \\ge 0$ let $Q(O, i)$ be the total number of parts, counted with multiplicity, over the $n$-color partitions of $i$ into odd parts (every part has odd size). Then in the ring $\\mathbb{Q}[[q]]$ of formal power series, with infinite sums and products taken coefficientwise,","title":"Generating function for the number of parts in n-color partitions into odd parts"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-distinct-parts-count-of-generati","kind":"node","lean":"MetaMathlibExt.nColor_distinctPartsCountOf_generatingFunction","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for parts of a given size in distinct "],["math","\\(n\\)"],["text","-color partitions"]],"source":"Generating function for parts of a given size in distinct $n$-color partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-distinct-parts-count-of-generati","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"Call an $n$-color partition distinct when no colored part occurs more than once. For a natural number $j$, let $Q(D, j, m)$ be the total number of parts of size $j$ (of any color) over the distinct $n$-color partitions of $m$, and let $C(D, m)$ be the number of distinct $n$-color partitions of $m$. Then, as formal power series in $q$ over $\\mathbb{Q}$,","title":"Generating function for parts of a given size in distinct $n$-color partitions"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/n-color-odd-partition-parts-by-parity-ge","kind":"node","lean":"MetaMathlibExt.nColor_oddPartitionParts_byParity_generatingFunction","parent":25695,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#n-color-odd-partition-parts-by-parity-ge","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $i \\ge 0$ let $U_o(i)$ (resp. $U_e(i)$) be the total number of parts, counted with multiplicity, over the $n$-color partitions of $i$ into odd parts (every part has odd size) that have an odd (resp. even) number of parts. Then in the ring $\\mathbb{Q}[[q]]$ of formal power series, with infinite sums and products taken coefficientwise,","title":"Parts of n-color partitions into odd parts, by parity of the number of parts"},{"id":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/neg-one-pow-mul-sigma-odd-eq-sum-n-color","kind":"node","lean":"MetaMathlibExt.neg_one_pow_mul_sigmaOdd_eq_sum_nColor_oddPartsDiff","parent":25695,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed odd-divisor sum via "],["math","\\(n\\)"],["text","-color partitions into odd parts"]],"source":"Signed odd-divisor sum via $n$-color partitions into odd parts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti/index.html#neg-one-pow-mul-sigma-odd-eq-sum-n-color","scope":"Combinatorics/integer-partitions/partition-identities/stanley-s-and-elder-s-theorems-on-parts-of-parti","status":"can_prove","summary":"For $r \\ge 0$ let $U_o(r)$ (resp. $U_e(r)$) be the total number of parts, counted with multiplicity, over the $n$-color partitions of $r$ in which every part has odd size and the number of parts is odd (resp. even), and let $C(O, D, r)$ be the number of $n$-color partitions of $r$ into distinct colored parts all of odd size. Then for every $m \\ge 1$,","title":"Signed odd-divisor sum via $n$-color partitions into odd parts"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics","kind":"scope","parent":25632,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics","status":"can_prove","summary":"Statistics and structure of a single partition: conjugation, hook lengths, s-cores, perimeter, diagonal sequences, lit cells, rank and crank, max-rook numbers, perfect and double-perfect partitions, and reconstruction of a partition from pairwise products.","title":"Young diagrams and partition statistics"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/crank-nonneg-partition-count","kind":"node","lean":"MetaMathlibExt.crank_nonneg_partition_count","parent":25729,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/index.html#crank-nonneg-partition-count","scope":"Combinatorics/integer-partitions/young-diagrams-statistics","status":"fully_proved","summary":"For a partition $\\lambda$, let $\\omega(\\lambda)$ be the number of parts equal to $1$. The crank of $\\lambda$ is its largest part if $\\omega(\\lambda) = 0$ (the largest part of the empty partition being $0$), and otherwise the number of parts greater than $\\omega(\\lambda)$, counted with multiplicity, minus $\\omega(\\lambda)$. Let $p(k)$ be the number of partitions of $k$. Then for every natural number $N$, the number\u2026","title":"Partitions with non-negative crank"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions","status":"can_prove","title":"Balanceable partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions/bounded","kind":"node","lean":"MathlibExt.Combinatorics.BalanceableBoundedPartitionsWanted.Combinatorics.Bounded","parent":25731,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-bounded partition"]],"source":"$k$-bounded partition"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions/index.html#bounded","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions","status":"fully_proved","summary":"A partition of $n$ is $k$-bounded if each of its parts is at most $k$.","title":"$k$-bounded partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions/balanceable","kind":"node","lean":"MathlibExt.Combinatorics.BalanceableBoundedPartitionsWanted.Combinatorics.Balanceable","parent":25731,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions/index.html#balanceable","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions","status":"fully_proved","summary":"A partition of $n$ is balanceable if its multiset of parts can be split as $s_1 + s_2$ into two multisets $s_1$ and $s_2$ whose elements have equal sums.","title":"Balanceable partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions/minimal-balanceable-of-bounded","kind":"node","lean":"MathlibExt.Combinatorics.BalanceableBoundedPartitionsWanted.Combinatorics.minimal_balanceable_of_bounded","parent":25731,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions/index.html#minimal-balanceable-of-bounded","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/balanceable-partitions","status":"can_prove","summary":"For every integer $k \\ge 1$, the smallest positive integer $n$ such that every $k$-bounded partition of $n$ is balanceable is $2 \\operatorname{lcm}(1, 2, \\ldots, k)$.","title":"Smallest size forcing bounded partitions to be balanceable"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","title":"Compositions, conjugate partitions and dominance order"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/weak-composition","kind":"node","lean":"MetaMathlibExt.WeakComposition","parent":25735,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html#weak-composition","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","summary":"Let n,k\u2208\u2115. A weak composition of n into k parts is a function","title":"Weak composition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/set-composition","kind":"node","lean":"MetaMathlibExt.SetComposition","parent":25735,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html#set-composition","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","summary":"Let S be a finite set and let k\u2208\u2115. An ordered set partition of S into k blocks is an ordered tuple","title":"Ordered set partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/partition-conjugate","kind":"node","lean":"List.partitionConjugate","parent":25735,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html#partition-conjugate","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","summary":"Let $\\lambda = [\\lambda_1, \\dots, \\lambda_L]$ be a finite list of natural numbers, with $L \\ge 0$. Its conjugate $\\lambda'$ is the empty list if $L = 0$, and otherwise the list $[\\lambda'_1, \\dots, \\lambda'_{\\lambda_1}]$ of length $\\lambda_1$ with $\\lambda'_j = \\#\\{i : 1 \\le i \\le L,\\ \\lambda_i \\ge j\\}$ for $1 \\le j \\le \\lambda_1$. The length is set by the first entry $\\lambda_1$, and the entries of $\\lambda'$ are\u2026","title":"Conjugate partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/is-self-conjugate","kind":"node","lean":"MetaMathlibExt.List.IsSelfConjugate","parent":25735,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html#is-self-conjugate","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","summary":"Let $\\lambda = [\\lambda_1, \\dots, \\lambda_L]$ be a finite list of natural numbers, with $L \\ge 0$. Its conjugate $\\lambda'$ is the empty list if $L = 0$, and otherwise the list $[\\lambda'_1, \\dots, \\lambda'_{\\lambda_1}]$ of length $\\lambda_1$ with $\\lambda'_j = \\#\\{i : 1 \\le i \\le L,\\ \\lambda_i \\ge j\\}$ for $1 \\le j \\le \\lambda_1$. The list $\\lambda$ is self-conjugate when $\\lambda' = \\lambda$.","title":"Self-conjugate partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/partitions-in-box","kind":"node","lean":"MetaMathlibExt.Nat.partitionsInBox","parent":25735,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html#partitions-in-box","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","summary":"Let $j, k, n \\ge 0$ be integers. A partition of $n$ is a finite multiset of positive integers whose sum is $n$; the empty partition is the unique partition of $0$. Then $p(j, k, n)$ is the number of partitions of $n$ with at most $k$ parts, counted with multiplicity, each of which is at most $j$. In particular $p(j, k, 0) = 1$ for all $j$ and $k$, while $p(0, k, n) = p(j, 0, n) = 0$ for $n \\ge 1$.","title":"Number of partitions in a box"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/vpartition","kind":"node","lean":"MetaMathlibExt.Nat.VPartition","parent":25735,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html#vpartition","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","summary":"Let $n \\ge 0$ be an integer. A partition $\\lambda$ is a finite multiset of positive integers, and its weight $|\\lambda|$ is the sum of its parts. A V-partition of weight $n$ is a triple $(c, \\lambda, \\mu)$ in which $c \\ge 0$ is an integer, $\\lambda$ and $\\mu$ are partitions all of whose parts are at most $c$, and $c + |\\lambda| + |\\mu| = n$.","title":"V-partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/is-unimodal","kind":"node","lean":"MetaMathlibExt.List.IsUnimodal","parent":25735,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html#is-unimodal","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","summary":"A finite list $d = [d_1, \\dots, d_m]$ of natural numbers is unimodal when $d_i \\ge 1$ for every $i$ and there is an index $j$ with $1 \\le j \\le m$ such that $d_1 \\le d_2 \\le \\dots \\le d_j$ and $d_j \\ge d_{j+1} \\ge \\dots \\ge d_m$. In particular a unimodal list has $m \\ge 1$, so the empty list is not unimodal; for $j = 1$ the first condition is vacuous and for $j = m$ the second is.","title":"Unimodal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/dominates","kind":"node","lean":"Nat.Partition.Dominates","parent":25735,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance/index.html#dominates","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/compositions-conjugate-partitions-and-dominance","status":"fully_proved","summary":"Let $n \\ge 0$ be an integer. A partition of $n$ is a finite multiset of positive integers with sum $n$; write it as the list $\\lambda = [\\lambda_1, \\dots, \\lambda_m]$ of its parts in weakly decreasing order, so that $\\lambda_1 \\ge \\dots \\ge \\lambda_m \\ge 1$ and $\\lambda_1 + \\dots + \\lambda_m = n$, and set $\\lambda_j = 0$ for every integer $j > m$. The empty partition ($m = 0$) is the unique partition of $0$. For\u2026","title":"Dominance order on partitions"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","title":"Diagonal sequences of integer partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-sequence","kind":"node","lean":"MetaMathlibExt.diagonalSequence","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-sequence","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"Let $\\alpha$ be a partition with parts $\\alpha_1 \\ge \\alpha_2 \\ge \\dots \\ge \\alpha_t$, and set $\\alpha_i = 0$ for $i > t$. Its diagonal sequence $\\delta(\\alpha) = (d_k)_{k \\ge 0}$ has $d_0 = 0$ and, for $k \\ge 1$, $d_k$ equal to the number of indices $i$ with $1 \\le i \\le k$ and $\\alpha_i + i - 1 \\ge k$, that is, the number of cells of the Young diagram on its $k$-th diagonal from upper right to lower left.\u2026","title":"Diagonal sequence of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-sequence-seven-seven-four-one-o","kind":"node","lean":"MetaMathlibExt.diagonalSequence_seven_seven_four_one_one_one","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonal sequence of "],["math","\\((7, 7, 4, 1, 1, 1)\\)"]],"source":"Diagonal sequence of $(7, 7, 4, 1, 1, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-sequence-seven-seven-four-one-o","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"The diagonal sequence of the partition $(7, 7, 4, 1, 1, 1)$ takes the values $0, 1, 2, 3, 4, 4, 4, 2, 1, 0$ at the indices $0, 1, \\dots, 9$. This is the paper's running example.","title":"Diagonal sequence of $(7, 7, 4, 1, 1, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-of-list","kind":"node","lean":"MetaMathlibExt.diagonalOfList","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-of-list","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"A finite list $(d_1, \\dots, d_\\ell)$ of natural numbers determines the sequence of natural numbers whose value at index $k$ is $d_k$ for $1 \\le k \\le \\ell$, and $0$ for $k = 0$ and for $k > \\ell$. This is the indexing of diagonal sequences, which take the value $0$ at index $0$.","title":"Sequence of a finite list"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-of-list-one-two-one","kind":"node","lean":"MetaMathlibExt.diagonalOfList_one_two_one","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","The list sequence of "],["math","\\((1, 2, 1)\\)"]],"source":"The list sequence of $(1, 2, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-of-list-one-two-one","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"The sequence given by the list $(1, 2, 1)$ takes the values $0, 1, 2, 1, 0, 0$ at the indices $0, 1, 2, 3, 4, 5$.","title":"The list sequence of $(1, 2, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-normal-form","kind":"node","lean":"MetaMathlibExt.diagonalNormalForm","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-normal-form","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For a natural number $q$ and a sequence $s = (s_k)$ of natural numbers, the normal form is the list $(1, 2, \\dots, q - 1, q, q^{(s_q)}, (q-1)^{(s_{q-1})}, \\dots, 1^{(s_1)})$, where $v^{(m)}$ denotes $m$ copies of $v$: the numbers $1, \\dots, q$ followed by $s_v$ copies of $v$ for $v = q, q - 1, \\dots, 1$.","title":"Normal form of a diagonal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-normal-form-four-example","kind":"node","lean":"MetaMathlibExt.diagonalNormalForm_four_example","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonal normal form for "],["math","\\(q = 4\\)"],["text"," and "],["math","\\(s = (1, 1, 0, 2)\\)"]],"source":"Diagonal normal form for $q = 4$ and $s = (1, 1, 0, 2)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-normal-form-four-example","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For $q = 4$ and $s_1 = 1$, $s_2 = 1$, $s_3 = 0$, $s_4 = 2$, the diagonal normal form $(1, 2, 3, 4, 4^{(s_4)}, 3^{(s_3)}, 2^{(s_2)}, 1^{(s_1)})$, where $v^{(m)}$ denotes $m$ copies of $v$, is $(1, 2, 3, 4, 4, 4, 2, 1)$.","title":"Diagonal normal form for $q = 4$ and $s = (1, 1, 0, 2)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-max-partition","kind":"node","lean":"MetaMathlibExt.diagonalMaxPartition","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-max-partition","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For a natural number $q$ and a sequence $s = (s_k)$ of natural numbers, $\\bar\\alpha = (\\bar\\alpha_1, \\dots, \\bar\\alpha_q)$ is the list of the $q$ numbers $\\bar\\alpha_i = q - i + 1 + \\sum_{k=i}^{q} s_k$, $1 \\le i \\le q$, in this order.","title":"Maximal partition of a diagonal normal form"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-max-partition-four-example","kind":"node","lean":"MetaMathlibExt.diagonalMaxPartition_four_example","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal partition for "],["math","\\(q = 4\\)"],["text"," and "],["math","\\(s = (1, 1, 0, 2)\\)"]],"source":"Maximal partition for $q = 4$ and $s = (1, 1, 0, 2)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-max-partition-four-example","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For $q = 4$ and $s_1 = 1$, $s_2 = 1$, $s_3 = 0$, $s_4 = 2$, the maximal partition $\\bar\\alpha$, with parts $\\bar\\alpha_i = q - i + 1 + \\sum_{k=i}^{q} s_k$ for $1 \\le i \\le q$, is $(8, 6, 4, 3)$.","title":"Maximal partition for $q = 4$ and $s = (1, 1, 0, 2)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-min-partition","kind":"node","lean":"MetaMathlibExt.diagonalMinPartition","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-min-partition","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For a natural number $q$ and a sequence $s = (s_k)$ of natural numbers, $\\underline\\alpha$ is the list $(q^{(s_q + 1)}, (q-1)^{(s_{q-1} + 1)}, \\dots, 1^{(s_1 + 1)})$, where $v^{(m)}$ denotes $m$ copies of $v$: first $s_q + 1$ copies of $q$, then $s_{q-1} + 1$ copies of $q - 1$, and so on down to $s_1 + 1$ copies of $1$.","title":"Minimal partition of a diagonal normal form"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-min-partition-four-example","kind":"node","lean":"MetaMathlibExt.diagonalMinPartition_four_example","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimal partition for "],["math","\\(q = 4\\)"],["text"," and "],["math","\\(s = (1, 1, 0, 2)\\)"]],"source":"Minimal partition for $q = 4$ and $s = (1, 1, 0, 2)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-min-partition-four-example","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For $q = 4$ and $s_1 = 1$, $s_2 = 1$, $s_3 = 0$, $s_4 = 2$, the minimal partition $\\underline\\alpha = (4^{(s_4 + 1)}, 3^{(s_3 + 1)}, 2^{(s_2 + 1)}, 1^{(s_1 + 1)})$, where $v^{(m)}$ denotes $m$ copies of $v$, is $(4, 4, 4, 3, 2, 2, 1, 1)$.","title":"Minimal partition for $q = 4$ and $s = (1, 1, 0, 2)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/conjugate-parts","kind":"node","lean":"MetaMathlibExt.conjugateParts","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#conjugate-parts","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For a finite multiset $\\alpha$ of natural numbers with largest element $m$ (and $m = 0$ if $\\alpha$ is empty), the conjugate $\\alpha^*$ is the multiset of the $m$ numbers $c_0, c_1, \\dots, c_{m-1}$, where $c_j$ is the number of elements of $\\alpha$ greater than $j$, counted with multiplicity. When $\\alpha$ is the multiset of parts of a partition, these are the parts of the conjugate partition.","title":"Conjugate of a multiset of parts"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/conjugate-parts-three-one","kind":"node","lean":"MetaMathlibExt.conjugateParts_three_one","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","The conjugate of "],["math","\\((3, 1)\\)"],["text"," is "],["math","\\((2, 1, 1)\\)"]],"source":"The conjugate of $(3, 1)$ is $(2, 1, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#conjugate-parts-three-one","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"The conjugate of the partition with parts $\\{3, 1\\}$ is the partition with parts $\\{2, 1, 1\\}$.","title":"The conjugate of $(3, 1)$ is $(2, 1, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-class","kind":"node","lean":"MetaMathlibExt.diagonalClass","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-class","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For a partition $p$ of $n$, its diagonal class $\\delta^{-1}(\\delta(p))$ is the finite set of all partitions $\\beta$ of $n$ whose diagonal sequence $\\delta(\\beta)$ equals the diagonal sequence $\\delta(p)$ of $p$.","title":"Diagonal class of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/mem-diagonal-class-self","kind":"node","lean":"MetaMathlibExt.mem_diagonalClass_self","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#mem-diagonal-class-self","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For every partition $p$ of $n$, $p$ belongs to the diagonal class of $p$, the set of partitions of $n$ whose diagonal sequence equals that of $p$.","title":"Every partition lies in its diagonal class"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-class-of","kind":"node","lean":"MetaMathlibExt.diagonalClassOf","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-class-of","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For a natural number $n$ and a sequence $d = (d_k)_{k \\ge 0}$ of natural numbers, $[d]$ is the finite set of partitions $\\beta$ of $n$ whose diagonal sequence $\\delta(\\beta)$ equals $d$.","title":"Partitions with a prescribed diagonal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/mem-diagonal-class-of-diagonal-sequence","kind":"node","lean":"MetaMathlibExt.mem_diagonalClassOf_diagonalSequence","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#mem-diagonal-class-of-diagonal-sequence","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"Every partition $\\beta$ of $n$ belongs to the set of partitions of $n$ whose diagonal sequence equals that of $\\beta$.","title":"A partition lies in the class of its diagonal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/partition-majorizes","kind":"node","lean":"MetaMathlibExt.PartitionMajorizes","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#partition-majorizes","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For finite lists $a = (a_1, \\dots, a_s)$ and $b = (b_1, \\dots, b_t)$ of natural numbers, $a$ majorizes $b$, written $a \\succ b$, when $b_1 + \\dots + b_k \\le a_1 + \\dots + a_k$ for every natural number $k \\le \\min\\{s, t\\}$.","title":"Majorization of part lists"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/partition-majorizes-three-one-two-two","kind":"node","lean":"MetaMathlibExt.partitionMajorizes_three_one_two_two","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","Majorization of "],["math","\\((2, 2)\\)"],["text"," by "],["math","\\((3, 1)\\)"]],"source":"Majorization of $(2, 2)$ by $(3, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#partition-majorizes-three-one-two-two","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"The partition $(3, 1)$ majorizes the partition $(2, 2)$: $3 \\ge 2$ and $3 + 1 \\ge 2 + 2$.","title":"Majorization of $(2, 2)$ by $(3, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/vn-arrangement-count","kind":"node","lean":"MetaMathlibExt.vnArrangementCount","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-vn-arrangements"]],"source":"Number of $k$-vn-arrangements"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#vn-arrangement-count","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"Let $k$ be a natural number and $M$ a finite multiset of integers. A $k$-vn-arrangement of $M$ is a finite sequence $(v_1, \\dots, v_b)$ of integers whose multiset of entries is $M$ and with $v_{i+1} - v_i \\le k$ for all $1 \\le i < b$. This is the number of distinct such sequences; sequences are compared entry by entry, so exchanging two equal entries gives the same arrangement. For $k = 1$ these are the\u2026","title":"Number of $k$-vn-arrangements"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/vn-arrangement-count-singleton","kind":"node","lean":"MetaMathlibExt.vnArrangementCount_singleton","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","A singleton multiset has one "],["math","\\(k\\)"],["text","-vn-arrangement"]],"source":"A singleton multiset has one $k$-vn-arrangement"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#vn-arrangement-count-singleton","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"fully_proved","summary":"For every natural number $k$ and every integer $a$, the one-element multiset $\\{a\\}$ has exactly one $k$-vn-arrangement.","title":"A singleton multiset has one $k$-vn-arrangement"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-sequence-succ-le-and-antitone","kind":"node","lean":"MetaMathlibExt.diagonalSequence_succ_le_and_antitone","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-sequence-succ-le-and-antitone","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $\\alpha$ be a partition of $n$ with diagonal sequence $(d_k)_{k \\ge 0}$, where $d_0 = 0$, and let $k \\ge 0$. Then $d_{k+1} \\le d_k + 1$, with equality if and only if $d_j = j$ for every $j$ with $1 \\le j \\le k + 1$. Moreover, if $d_{k+1} \\le d_k$, then $d_{k+2} \\le d_{k+1}$.","title":"Increments of the diagonal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/exists-unique-diagonal-normal-form","kind":"node","lean":"MetaMathlibExt.exists_unique_diagonalNormalForm","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#exists-unique-diagonal-normal-form","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Write $v^{(m)}$ for $m$ copies of $v$. For every partition $\\alpha$ of $n \\ge 1$ there are an integer $q > 0$ and natural numbers $s_1, \\dots, s_q$ such that the diagonal sequence of $\\alpha$, indexed from $0$, is $(0, 1, 2, \\dots, q, q^{(s_q)}, (q-1)^{(s_{q-1})}, \\dots, 1^{(s_1)}, 0, 0, \\dots)$ and $q(q+1)/2 + \\sum_{k=1}^{q} k s_k = n$. Moreover $q$ and $s_1, \\dots, s_q$ are unique: if $q' > 0$ and\u2026","title":"Existence and uniqueness of the diagonal normal form"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-max-partition-spec","kind":"node","lean":"MetaMathlibExt.diagonalMaxPartition_spec","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-max-partition-spec","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $q > 0$ and let $s_1, \\dots, s_q$ be natural numbers with $q(q+1)/2 + \\sum_{k=1}^{q} k s_k = n$, and let $\\bar\\alpha = (\\bar\\alpha_1, \\dots, \\bar\\alpha_q)$ with $\\bar\\alpha_i = q - i + 1 + \\sum_{k=i}^{q} s_k$. Then the entries of $\\bar\\alpha$ are positive, sum to $n$ and are strictly decreasing; the diagonal sequence of $\\bar\\alpha$ is $(1, 2, \\dots, q, q^{(s_q)}, (q-1)^{(s_{q-1})}, \\dots, 1^{(s_1)})$ followed\u2026","title":"The maximal partition has the prescribed diagonal normal form"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/ncard-diagonal-sequences-eq-card-distinc","kind":"node","lean":"MetaMathlibExt.ncard_diagonalSequences_eq_card_distincts","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#ncard-diagonal-sequences-eq-card-distinc","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"For every natural number $n$, including $n = 0$, the number of distinct diagonal sequences of partitions of $n$ equals the number of partitions of $n$ into distinct parts, and the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts.","title":"Counting diagonal sequences"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/conjugate-parts-diagonal-max-partition","kind":"node","lean":"MetaMathlibExt.conjugateParts_diagonalMaxPartition","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#conjugate-parts-diagonal-max-partition","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $q \\ge 0$ and let $s_1, \\dots, s_q$ be natural numbers. The conjugate of the partition $\\bar\\alpha$ with parts $\\bar\\alpha_i = q - i + 1 + \\sum_{k=i}^{q} s_k$ for $1 \\le i \\le q$ is the partition $\\underline{\\alpha} = (q^{(s_q + 1)}, (q-1)^{(s_{q-1} + 1)}, \\dots, 1^{(s_1 + 1)})$, where $v^{(r)}$ means $r$ copies of $v$; the parts are compared as multisets.","title":"Conjugate of a partition with prescribed gaps between its parts"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-sequence-max-eq-min-and-multise","kind":"node","lean":"MetaMathlibExt.diagonalSequence_max_eq_min_and_multiset","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-sequence-max-eq-min-and-multise","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $q \\ge 0$ and $s_1, \\dots, s_q \\ge 0$ be natural numbers, write $v^{(m)}$ for $m$ copies of $v$, and let $\\bar\\alpha$ and $\\underline\\alpha$ be the partitions with parts $\\bar\\alpha_i = q - i + 1 + s_i + s_{i+1} + \\dots + s_q$ for $1 \\le i \\le q$ and $\\underline\\alpha = (q^{(s_q + 1)}, (q-1)^{(s_{q-1} + 1)}, \\dots, 1^{(s_1 + 1)})$. Then $\\bar\\alpha$ and $\\underline\\alpha$ have the same diagonal sequence; this\u2026","title":"Equal diagonal sequences of the maximal and minimal partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/exists-eq-succ-of-ne-diagonal-max-partit","kind":"node","lean":"MetaMathlibExt.exists_eq_succ_of_ne_diagonalMaxPartition","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#exists-eq-succ-of-ne-diagonal-max-partit","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $q \\ge 0$ and $s_1, \\dots, s_q \\ge 0$ be natural numbers, write $v^{(m)}$ for $m$ copies of $v$, and let $\\bar\\alpha$ and $\\underline\\alpha$ be the partitions with parts $\\bar\\alpha_i = q - i + 1 + s_i + s_{i+1} + \\dots + s_q$ for $1 \\le i \\le q$ and $\\underline\\alpha = (q^{(s_q + 1)}, (q-1)^{(s_{q-1} + 1)}, \\dots, 1^{(s_1 + 1)})$. Let $\\alpha$ be a partition of $n$ with parts $\\alpha_1 \\ge \\dots \\ge \\alpha_t$,\u2026","title":"Repeated parts and large gaps in a diagonal class"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/sup-parts-mem-diagonal-first-part-set","kind":"node","lean":"MetaMathlibExt.sup_parts_mem_diagonalFirstPartSet","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#sup-parts-mem-diagonal-first-part-set","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $\\alpha$ be a partition of $n$, let $q$ and $s_1, s_2, \\dots$ be natural numbers, and suppose the diagonal sequence of $\\alpha$ is $(1, 2, \\dots, q, q^{(s_q)}, (q-1)^{(s_{q-1})}, \\dots, 1^{(s_1)})$ followed by zeros, where $v^{(m)}$ denotes $m$ copies of $v$. Then the largest part of $\\alpha$ (taken to be $0$ if $\\alpha$ has no parts) lies in the set $A_1 = \\{q + \\sum_{i=j}^{q} s_i : 1 \\le j \\le q + 1\\}$, that\u2026","title":"The largest part of a partition with given diagonal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/sum-sq-add-sum-sq-conjugate-eq","kind":"node","lean":"MetaMathlibExt.sum_sq_add_sum_sq_conjugate_eq","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#sum-sq-add-sum-sq-conjugate-eq","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $\\alpha$ be a partition of $n$ with diagonal sequence $(d_k)_{k \\ge 1}$ and conjugate partition $\\alpha^*$, and write $s(\\gamma)$ for the sum of the squares of the parts of a partition $\\gamma$. Then $s(\\alpha) + s(\\alpha^*) = 2 \\sum_{k=1}^{2n} k \\, d_k$. Consequently every partition $\\beta$ of $n$ with the same diagonal sequence as $\\alpha$ satisfies $s(\\beta) + s(\\beta^*) = s(\\alpha) + s(\\alpha^*)$.","title":"Sums of squared parts and the diagonal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/partition-majorizes-iff-conjugate","kind":"node","lean":"MetaMathlibExt.partitionMajorizes_iff_conjugate","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#partition-majorizes-iff-conjugate","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"For partitions $\\alpha$ and $\\beta$ of $n$, with parts listed in decreasing order, $\\alpha$ majorizes $\\beta$ if and only if the conjugate partition $\\beta^*$ majorizes the conjugate partition $\\alpha^*$, again with parts listed in decreasing order.","title":"Conjugation reverses majorization"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-max-partition-majorizes-majoriz","kind":"node","lean":"MetaMathlibExt.diagonalMaxPartition_majorizes_majorizes_min","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-max-partition-majorizes-majoriz","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $p$ be a partition of $n$, let $q$ be a natural number and $s = (s_k)$ a sequence of natural numbers, and suppose that the diagonal sequence of $p$ is $(1, 2, \\dots, q, q^{(s_q)}, (q-1)^{(s_{q-1})}, \\dots, 1^{(s_1)})$ followed by zeros, where $v^{(m)}$ denotes $m$ copies of $v$. Let $\\alpha$ be the parts of $p$ in decreasing order, let $\\bar\\alpha = (\\bar\\alpha_1, \\dots, \\bar\\alpha_q)$ with\u2026","title":"The diagonal class lies between the maximal and minimal partitions in majorization"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/vn-arrangement-count-one-eq-prod","kind":"node","lean":"MetaMathlibExt.vnArrangementCount_one_eq_prod","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#vn-arrangement-count-one-eq-prod","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $t$ be a natural number, let $b_0, b_1, \\dots$ be natural numbers, and let $M$ be the multiset of integers containing $b_i$ copies of $i$ for each $0 \\le i \\le t$, so that $M$ has $m = b_0 + \\dots + b_t$ elements. Then the number of vn-arrangements of $M$, that is, orderings $(v_1, \\dots, v_m)$ of $M$ with $v_{j+1} - v_j \\le 1$ for all $j$, is $\\prod_{i=0}^{t-1} \\binom{b_i + b_{i+1}}{b_i}$.","title":"Counting vn-arrangements of a multiset"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/vn-arrangement-count-consecutive","kind":"node","lean":"MetaMathlibExt.vnArrangementCount_consecutive","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#vn-arrangement-count-consecutive","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"For every integer $a$ and every natural number $t \\ge 1$, the set $\\{a, a + 1, \\dots, a + t - 1\\}$ of $t$ consecutive integers has exactly $2^{t-1}$ vn-arrangements, that is, orderings $(v_1, \\dots, v_t)$ of its elements with $v_{i+1} - v_i \\le 1$ for all $i$.","title":"Vn-arrangements of consecutive integers"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/vn-arrangement-count-eq-multinomial-mul","kind":"node","lean":"MetaMathlibExt.vnArrangementCount_eq_multinomial_mul_prod","parent":25744,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(k\\)"],["text","-vn-arrangements of a multiset"]],"source":"Counting $k$-vn-arrangements of a multiset"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#vn-arrangement-count-eq-multinomial-mul","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $k \\le t$ be natural numbers, let $b_0, b_1, \\dots$ be natural numbers, and let $M$ be the multiset of integers containing $b_i$ copies of $i$ for each $0 \\le i \\le t$, so that $M$ has $m = b_0 + \\dots + b_t$ elements. Then the number of $k$-vn-arrangements of $M$, that is, orderings $(v_1, \\dots, v_m)$ of $M$ with $v_{j+1} - v_j \\le k$ for all $j$, is","title":"Counting $k$-vn-arrangements of a multiset"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/card-diagonal-class-filter-card-parts-eq","kind":"node","lean":"MetaMathlibExt.card_diagonalClass_filter_card_parts_eq_q","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#card-diagonal-class-filter-card-parts-eq","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $\\alpha$ be a partition of $n$ with diagonal sequence $\\delta(\\alpha) = (d_j)_{j \\ge 1}$, set $d_0 = 0$, and let $q, L \\ge 0$ be integers with $d_j = j$ for all $1 \\le j \\le q$, $d_{q+1} \\ne q + 1$, $d_L \\ne 0$ and $d_{L+1} = 0$. Put $b_i = \\max(d_i - d_{i+1}, 0)$. Then the number of partitions of $n$ with exactly $q$ parts whose diagonal sequence equals $\\delta(\\alpha)$ is","title":"Partitions in a diagonal class with as many parts as the initial staircase"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/card-diagonal-class-filter-card-parts-eq-2","kind":"node","lean":"MetaMathlibExt.card_diagonalClass_filter_card_parts_eq_of_lt","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#card-diagonal-class-filter-card-parts-eq-2","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $\\alpha$ be a partition of $n$ with diagonal sequence $\\delta(\\alpha) = (d_j)_{j \\ge 1}$, set $d_0 = 0$, and let $q, L \\ge 0$ be integers with $d_j = j$ for all $1 \\le j \\le q$, $d_{q+1} \\ne q + 1$, $d_L \\ne 0$ and $d_{L+1} = 0$. Put $b_i = \\max(d_i - d_{i+1}, 0)$. Then for every $k > q$ the number of partitions of $n$ with exactly $k$ parts whose diagonal sequence equals $\\delta(\\alpha)$ is","title":"Partitions in a diagonal class with more parts than the initial staircase"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/card-diagonal-class-eq-prod-choose","kind":"node","lean":"MetaMathlibExt.card_diagonalClass_eq_prod_choose","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#card-diagonal-class-eq-prod-choose","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $\\alpha$ be a partition of $n$ with diagonal sequence $\\delta(\\alpha) = (d_j)_{j \\ge 1}$, set $d_0 = 0$, and let $q, L \\ge 0$ be integers with $d_j = j$ for all $1 \\le j \\le q$, $d_{q+1} \\ne q + 1$, $d_L \\ne 0$ and $d_{L+1} = 0$. Put $b_i = \\max(d_i - d_{i+1}, 0)$. Then the number of partitions of $n$ whose diagonal sequence equals $\\delta(\\alpha)$ is","title":"Number of partitions with a given diagonal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/card-diagonal-class-of-two-le-s","kind":"node","lean":"MetaMathlibExt.card_diagonalClass_of_two_le_s","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#card-diagonal-class-of-two-le-s","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $q \\ge 1$, let $s = (s_1, s_2, \\dots)$ be a sequence of natural numbers, and let $\\alpha$ be a partition of $n$ whose diagonal sequence $\\delta(\\alpha) = (d_i)_{i \\ge 1}$ is $(1, 2, \\dots, q, q^{(s_q)}, (q-1)^{(s_{q-1})}, \\dots, 1^{(s_1)})$ followed by zeros, where $v^{(r)}$ means $r$ copies of $v$. Put $L = q + s_1 + \\dots + s_q$. If $s_i \\ge 2$ for all $1 \\le i < q$, then exactly one partition of $n$ with\u2026","title":"Diagonal classes whose multiplicities are at least two"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/exists-card-diagonal-class-eq","kind":"node","lean":"MetaMathlibExt.exists_card_diagonalClass_eq","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#exists-card-diagonal-class-eq","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"For every positive integer $m$ there are a natural number $n$ and a partition $\\alpha$ of $n$ whose class, the set of partitions of $n$ with the same diagonal sequence as $\\alpha$, has exactly $m$ elements.","title":"Every positive integer is the size of a diagonal class"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/card-diagonal-class-of-truncate","kind":"node","lean":"MetaMathlibExt.card_diagonalClassOf_truncate","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#card-diagonal-class-of-truncate","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"For $m \\ge 0$ and a sequence $s = (s_1, s_2, \\dots)$ of natural numbers, let $d(m, s)$ be the sequence $(1, 2, \\dots, m, m^{(s_m)}, (m-1)^{(s_{m-1})}, \\dots, 1^{(s_1)})$ followed by zeros, where $v^{(r)}$ means $r$ copies of $v$, and let $n(m, s) = m(m+1)/2 + \\sum_{j=1}^{m} j s_j$. Let $1 \\le k < q$ with $s_k \\ge 2$ and $s_j = 0$ for all $k < j \\le q$. Then the number of partitions of $n(q, s)$ with diagonal\u2026","title":"Truncating the staircase of a diagonal sequence"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/card-diagonal-class-of-min-two","kind":"node","lean":"MetaMathlibExt.card_diagonalClassOf_min_two","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#card-diagonal-class-of-min-two","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"For $m \\ge 0$ and a sequence $s = (s_1, s_2, \\dots)$ of natural numbers, let $d(m, s)$ be the sequence $(1, 2, \\dots, m, m^{(s_m)}, (m-1)^{(s_{m-1})}, \\dots, 1^{(s_1)})$ followed by zeros, where $v^{(r)}$ means $r$ copies of $v$, and let $n(m, s) = m(m+1)/2 + \\sum_{j=1}^{m} j s_j$. Let $q \\ge 1$ and $\\sigma_j = \\min(s_j, 2)$ for all $j$. Then the number of partitions of $n(q, s)$ with diagonal sequence $d(q, s)$\u2026","title":"Capping the multiplicities of a diagonal sequence at two"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/diagonal-class-classification-card-le-th","kind":"node","lean":"MetaMathlibExt.diagonalClass_classification_card_le_three","parent":25744,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions/index.html#diagonal-class-classification-card-le-th","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/diagonal-sequences-of-integer-partitions","status":"can_prove","summary":"Let $\\alpha$ be a partition of $n \\ge 1$, let $d = \\delta(\\alpha)$ be its diagonal sequence, written without its trailing zeros, and let $[\\alpha]$ be the set of partitions of $n$ with diagonal sequence $d$, each written by its parts in decreasing order. Here $v^{(r)}$ means $r$ copies of $v$, and a run $a, a - 1, \\dots, b$ with $a < b$ is empty. (1) If $|[\\alpha]| = 1$, then there is $q \\ge 1$ with\u2026","title":"Diagonal classes of size at most three"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","title":"Fixed hooks in a column of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/partition-col-len","kind":"node","lean":"MetaMathlibExt.partitionColLen","parent":25787,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#partition-col-len","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"fully_proved","summary":"For a partition $\\lambda$ of $n$ and a natural number $j$, the column length $\\lambda'_j$ is the number of parts of $\\lambda$, counted with multiplicity, that are at least $j$. For $j \\ge 1$ this is the length of the $j$-th column of the Young diagram of $\\lambda$, with columns numbered from $1$.","title":"Column lengths of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/partition-row-len","kind":"node","lean":"MetaMathlibExt.partitionRowLen","parent":25787,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#partition-row-len","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"fully_proved","summary":"For a partition $\\lambda$ of $n$ and a natural number $s$, the row length $\\lambda_s$ is the number of $j \\in \\{1, \\dots, n\\}$ whose column length $\\lambda'_j$ is at least $s$. For $s \\ge 1$ this is the $s$-th largest part of $\\lambda$ (rows are numbered from $1$), and it is $0$ when $\\lambda$ has fewer than $s$ parts.","title":"Row lengths of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/partition-hook-length","kind":"node","lean":"MetaMathlibExt.partitionHookLength","parent":25787,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#partition-hook-length","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"fully_proved","summary":"For a partition $\\lambda$ of $n$ and natural numbers $s$ and $m$, the hook length of the square in row $s$ and column $m$ is the integer $h_{s,m}(\\lambda) = \\lambda_s + \\lambda'_m - s - m + 1$, where $\\lambda_s$ is the $s$-th row length and $\\lambda'_m$ the $m$-th column length of $\\lambda$ (rows and columns are numbered from $1$). The integer is given by this formula for all $s$ and $m$, also when $(s, m)$ is not\u2026","title":"Hook length of a square of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/is-fixed-column-hook","kind":"node","lean":"MetaMathlibExt.IsFixedColumnHook","parent":25787,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#is-fixed-column-hook","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"fully_proved","summary":"Let $\\lambda$ be a partition of $n$, let $m$ and $s$ be natural numbers and let $h$ be an integer. The partition $\\lambda$ has an $h$-fixed hook in column $m$ at row $s$ if the square $(s, m)$ lies in the Young diagram of $\\lambda$, that is, $s \\ge 1$ and $1 \\le m \\le \\lambda_s$, and its hook length is $h_{s,m}(\\lambda) = s + h$. In other words, the $s$-th term of the sequence\u2026","title":"Fixed hooks in a column of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/partition4431","kind":"node","lean":"MetaMathlibExt.partition4431","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partition "],["math","\\((4, 4, 3, 1)\\)"],["text"," of "],["math","\\(12\\)"]],"source":"The partition $(4, 4, 3, 1)$ of $12$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#partition4431","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"fully_proved","summary":"The partition of $12$ whose parts are $4$, $4$, $3$ and $1$; it is the example partition drawn in Figure 1 of the source.","title":"The partition $(4, 4, 3, 1)$ of $12$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/partition4431-row-col","kind":"node","lean":"MetaMathlibExt.partition4431_row_col","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Row and column lengths of the partition "],["math","\\((4, 4, 3, 1)\\)"]],"source":"Row and column lengths of the partition $(4, 4, 3, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#partition4431-row-col","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"fully_proved","summary":"For the partition $\\lambda = (4, 4, 3, 1)$ of $12$, the row lengths are $\\lambda_1 = 4$, $\\lambda_2 = 4$, $\\lambda_3 = 3$, $\\lambda_4 = 1$ and $\\lambda_5 = 0$, and the column lengths are $\\lambda'_1 = 4$, $\\lambda'_2 = 3$, $\\lambda'_3 = 3$, $\\lambda'_4 = 2$ and $\\lambda'_5 = 0$.","title":"Row and column lengths of the partition $(4, 4, 3, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/partition4431-hook-length","kind":"node","lean":"MetaMathlibExt.partition4431_hookLength","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hook lengths of the partition "],["math","\\((4, 4, 3, 1)\\)"]],"source":"Hook lengths of the partition $(4, 4, 3, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#partition4431-hook-length","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"fully_proved","summary":"For the partition $\\lambda = (4, 4, 3, 1)$ of $12$, write $h_{s,m}$ for the hook length of the square in row $s$ and column $m$. Along the first row $h_{1,1} = 7$, $h_{1,2} = 5$, $h_{1,3} = 4$ and $h_{1,4} = 2$; down the first column $h_{2,1} = 6$, $h_{3,1} = 4$ and $h_{4,1} = 1$; and on the diagonal $h_{2,2} = 4$ and $h_{3,3} = 1$.","title":"Hook lengths of the partition $(4, 4, 3, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/partition4431-is-fixed-column-hook","kind":"node","lean":"MetaMathlibExt.partition4431_isFixedColumnHook","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed hooks in the partition "],["math","\\((4, 4, 3, 1)\\)"]],"source":"Fixed hooks in the partition $(4, 4, 3, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#partition4431-is-fixed-column-hook","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"fully_proved","summary":"Let $\\lambda = (4, 4, 3, 1)$, a partition of $12$. Then $\\lambda$ has a $1$-fixed hook in column $1$ at row $3$ (its hook length is $h_{3,1} = 4 = 3 + 1$), $\\lambda$ has no $0$-fixed hook in column $1$ at row $4$ (its hook length is $h_{4,1} = 1 \\ne 4 + 0$), and $\\lambda$ has no $(-3)$-fixed hook in column $3$ at row $4$ (the square $(4, 3)$ lies outside the Young diagram, since $\\lambda_4 = 1 < 3$).","title":"Fixed hooks in the partition $(4, 4, 3, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/card-zero-fixed-column-hook-eq-two-color","kind":"node","lean":"MetaMathlibExt.card_zeroFixedColumnHook_eq_twoColor_count","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero-fixed hooks in column "],["math","\\(m\\)"],["text"," and two-colored partitions"]],"source":"Zero-fixed hooks in column $m$ and two-colored partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#card-zero-fixed-column-hook-eq-two-color","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$ and $m \\ge 1$ be natural numbers. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. The partition $\\lambda$ has an $h$-fixed hook in column $m$ at row $r$ when $r \\ge 1$,\u2026","title":"Zero-fixed hooks in column $m$ and two-colored partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/card-fixed-column-hook-part-eq-column","kind":"node","lean":"MetaMathlibExt.card_fixedColumnHook_part_eq_column","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed hooks in column "],["math","\\(m\\)"],["text"," arising from parts of size "],["math","\\(m\\)"]],"source":"Fixed hooks in column $m$ arising from parts of size $m$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#card-fixed-column-hook-part-eq-column","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$ and $m \\ge 1$ be natural numbers and let $h$ be an integer. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. The partition $\\lambda$ has an $h$-fixed hook in column $m$ at row $r$\u2026","title":"Fixed hooks in column $m$ arising from parts of size $m$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/column-hook-size-gen-fun","kind":"node","lean":"MetaMathlibExt.columnHook_size_genFun","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for column-"],["math","\\(m\\)"],["text"," hooks of size "],["math","\\(k\\)"]],"source":"Generating function for column-$m$ hooks of size $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#column-hook-size-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$, $m \\ge 1$ and $k \\ge 1$ be natural numbers. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. Let $H$ be the number of pairs $(\\lambda, r)$ of a partition $\\lambda$ of $n$ and a\u2026","title":"Generating function for column-$m$ hooks of size $k$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/column-hook-size-gen-fun-limit","kind":"node","lean":"MetaMathlibExt.columnHook_size_genFun_limit","parent":25787,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#column-hook-size-gen-fun-limit","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $m \\ge 2$ be a natural number and work in $\\mathbb{Q}[[q]]$. For $j \\in \\mathbb{N}$ write $(q;q)_j = \\prod_{i=1}^{j} (1 - q^i)$ and $(q^m;q)_j = \\prod_{i=0}^{j-1} (1 - q^{m+i})$. Then for every natural number $n$ there is a $K$ such that for every $k \\ge K$ the coefficient of $q^n$ in","title":"Limit of the fixed-hook generating functions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-first-column-hook-part-gen-fun","kind":"node","lean":"MetaMathlibExt.fixedFirstColumnHook_part_genFun","parent":25787,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-first-column-hook-part-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers with $k \\ge 1$ and let $h < k$ be an integer. The number of partitions $\\lambda$ of $n$ that have an $h$-fixed hook in column $1$ at some row $s$ with $\\lambda_s = k$ is the coefficient of $q^n$ in","title":"Partitions with a fixed first-column hook arising from a part of given size"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-column-hook-part-gen-fun","kind":"node","lean":"MetaMathlibExt.fixedColumnHook_part_genFun","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for fixed column hooks from a part of size "],["math","\\(k\\)"]],"source":"Generating function for fixed column hooks from a part of size $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-column-hook-part-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$, $m \\ge 1$ and $k \\ge m$ be natural numbers and let $h$ be an integer with $h + m \\le k$. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. The partition $\\lambda$ has an $h$-fixed\u2026","title":"Generating function for fixed column hooks from a part of size $k$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-column-hook-part-odd-gen-fun","kind":"node","lean":"MetaMathlibExt.fixedColumnHook_part_odd_genFun","parent":25787,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-column-hook-part-odd-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$, $m$ and $k$ be natural numbers with $1 \\le m \\le k$ and $k$ odd, and let $h$ be an integer with $h \\le k - m$. Count the partitions $\\lambda$ of $n$ into odd parts that have an $h$-fixed hook in column $m$ at some row $s$ with $\\lambda_s = k$. Write $(q^2;q^2)_j = \\prod_{i\\lt j} (1 - q^{2i+2})$ and $(q;q^2)_j = \\prod_{i\\lt j} (1 - q^{2i+1})$, write $\\binom{a}{b}_{q^2}$ for the Gaussian binomial coefficient\u2026","title":"Odd partitions with a fixed hook in a given column arising from an odd part"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-column-hook-part-distinct-gen-fun","kind":"node","lean":"MetaMathlibExt.fixedColumnHook_part_distinct_genFun","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed column hooks from a part of size "],["math","\\(k\\)"],["text"," in distinct partitions"]],"source":"Fixed column hooks from a part of size $k$ in distinct partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-column-hook-part-distinct-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$, $m \\ge 1$ and $k \\ge m$ be natural numbers and let $h$ be an integer with $h + m \\le k$. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. The partition $\\lambda$ has an $h$-fixed\u2026","title":"Fixed column hooks from a part of size $k$ in distinct partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-first-column-hook-hook-gen-fun","kind":"node","lean":"MetaMathlibExt.fixedFirstColumnHook_hook_genFun","parent":25787,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-first-column-hook-hook-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers with $k \\ge 1$ and let $h < k$ be an integer. The number of partitions $\\lambda$ of $n$ that have an $h$-fixed hook in column $1$ at some row $s$ whose hook length is $h_{s,1}(\\lambda) = k$ is the coefficient of $q^n$ in","title":"Partitions with a fixed first-column hook of given size"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-column-hook-hook-gen-fun","kind":"node","lean":"MetaMathlibExt.fixedColumnHook_hook_genFun","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for fixed column hooks of size "],["math","\\(k\\)"]],"source":"Generating function for fixed column hooks of size $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-column-hook-hook-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$, $m \\ge 1$ and $k \\ge 1$ be natural numbers and let $h < k$ be an integer. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. The partition $\\lambda$ has an $h$-fixed hook in column\u2026","title":"Generating function for fixed column hooks of size $k$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-column-hook-hook-odd-gen-fun","kind":"node","lean":"MetaMathlibExt.fixedColumnHook_hook_odd_genFun","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for fixed column hooks of size "],["math","\\(k\\)"],["text"," in odd partitions"]],"source":"Generating function for fixed column hooks of size $k$ in odd partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-column-hook-hook-odd-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$, $m \\ge 1$ and $k \\ge 1$ be natural numbers and let $h < k$ be an integer. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. The partition $\\lambda$ has an $h$-fixed hook in column\u2026","title":"Generating function for fixed column hooks of size $k$ in odd partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-column-hook-hook-distinct-gen-fun","kind":"node","lean":"MetaMathlibExt.fixedColumnHook_hook_distinct_genFun","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for fixed column hooks of size "],["math","\\(k\\)"],["text"," in distinct partitions"]],"source":"Generating function for fixed column hooks of size $k$ in distinct partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-column-hook-hook-distinct-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$, $m \\ge 1$ and $k \\ge 1$ be natural numbers and let $h < k$ be an integer. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. The partition $\\lambda$ has an $h$-fixed hook in column\u2026","title":"Generating function for fixed column hooks of size $k$ in distinct partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/fixed-column-hook-hook-odd-distinct-gen","kind":"node","lean":"MetaMathlibExt.fixedColumnHook_hook_oddDistinct_genFun_even","parent":25787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed column hooks of size "],["math","\\(k\\)"],["text"," in odd distinct partitions, even column"]],"source":"Fixed column hooks of size $k$ in odd distinct partitions, even column"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition/index.html#fixed-column-hook-hook-odd-distinct-gen","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/fixed-hooks-in-a-column-of-a-partition","status":"can_prove","summary":"Let $n$, $m$ and $k \\ge 1$ be natural numbers with $m \\ge 1$ even, and let $h < k$ be an integer. For a partition $\\lambda$, write $\\lambda_r$ for its $r$-th largest part ($0$ if it has fewer than $r$ parts), $\\lambda'_m$ for the number of its parts that are at least $m$, and $h_{r,m}(\\lambda) = \\lambda_r + \\lambda'_m - r - m + 1$ for the hook length of the square $(r, m)$. The partition $\\lambda$ has an $h$-fixed\u2026","title":"Fixed column hooks of size $k$ in odd distinct partitions, even column"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"can_prove","title":"Lit cells in integer partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/decreasing-parts","kind":"node","lean":"MetaMathlibExt.decreasingParts","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#decreasing-parts","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"For a partition $p$ of $n$ with $\\ell$ parts, this is the list $x_1 \\ge x_2 \\ge \\cdots \\ge x_\\ell$ of the parts of $p$, each repeated according to its multiplicity, sorted in weakly decreasing order.","title":"Parts of a partition in decreasing order"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-cells-increasing","kind":"node","lean":"MetaMathlibExt.litCellsIncreasing","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-cells-increasing","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"For a partition $p$ of $n$, the number of lit cells of $p$, drawn as a weakly increasing bargraph, is defined as the largest part of $p$ plus the number of parts of $p$ minus the number of distinct parts of $p$ (the largest part of the empty partition is $0$).","title":"Lit cells of a weakly increasing bargraph"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-nodes-increasing","kind":"node","lean":"MetaMathlibExt.litNodesIncreasing","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-nodes-increasing","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"The number of lit nodes of a partition $p$, drawn as a weakly increasing Ferrers diagram of nodes, is defined as the largest part of $p$ plus the number of parts of $p$ minus one; for the empty partition it is $0$.","title":"Lit nodes of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-cells-decreasing","kind":"node","lean":"MetaMathlibExt.litCellsDecreasing","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-cells-decreasing","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"Let $p$ be a partition of $n$ with $\\ell$ parts $x_1 \\ge x_2 \\ge \\cdots \\ge x_\\ell$. The number of lit cells of $p$, drawn as a weakly decreasing bargraph, is defined as $\\max_{1 \\le i \\le \\ell} (x_i + i - 1)$, and as $0$ when $p$ has no parts. It is the largest part of the image of $p$ under Prodinger's bijection $x_i \\mapsto x_i + i - 1$.","title":"Lit cells of a weakly decreasing bargraph"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/triangular-qbinomial","kind":"node","lean":"MetaMathlibExt.triangularQBinomial","parent":25809,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triangular "],["math","\\(q\\)"],["text","-binomial coefficient"]],"source":"Triangular $q$-binomial coefficient"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#triangular-qbinomial","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"For natural numbers $r$ and $s$, the triangular $q$-binomial coefficient $t[r, s] \\in \\mathbb{Z}[[q]]$ is the power series whose coefficient of $q^n$ is the number of partitions of $n$ with exactly $s$ parts $x_1 \\ge \\dots \\ge x_s$ that fit in the staircase with column heights $r, r - 1, \\dots, r - s + 1$, that is, with $x_i + (i - 1) \\le r$ for $1 \\le i \\le s$.","title":"Triangular $q$-binomial coefficient"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/triangle-partition-series","kind":"node","lean":"MetaMathlibExt.trianglePartitionSeries","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#triangle-partition-series","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"For a natural number $r$, the triangle partition series is the power series $T_q(r) = \\sum_{s=0}^{r} t[r, s] \\in \\mathbb{Z}[[q]]$, where $t[r, s]$ is the triangular $q$-binomial coefficient. It is the generating function of partitions contained in the triangle with column heights $r, r - 1, \\dots, 1$.","title":"Triangle partition series"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-cells-increasing-indiscrete","kind":"node","lean":"MetaMathlibExt.litCellsIncreasing_indiscrete","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-cells-increasing-indiscrete","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the partition $(n)$ of $n$ with a single part has $n$ lit cells as a weakly increasing bargraph.","title":"Lit cells of a one-part increasing bargraph"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-nodes-increasing-indiscrete","kind":"node","lean":"MetaMathlibExt.litNodesIncreasing_indiscrete","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-nodes-increasing-indiscrete","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the one-part partition $(n)$ of $n$ has exactly $n$ lit nodes.","title":"Lit nodes of a one-part partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-cells-decreasing-indiscrete","kind":"node","lean":"MetaMathlibExt.litCellsDecreasing_indiscrete","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-cells-decreasing-indiscrete","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the partition $(n)$ of $n$ with a single part has $n$ lit cells as a weakly decreasing bargraph.","title":"Lit cells of a one-part decreasing bargraph"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/triangular-qbinomial-zero-right","kind":"node","lean":"MetaMathlibExt.triangularQBinomial_zero_right","parent":25809,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triangular "],["math","\\(q\\)"],["text","-binomial coefficients with no parts"]],"source":"Triangular $q$-binomial coefficients with no parts"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#triangular-qbinomial-zero-right","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"For every natural number $r$, the triangular $q$-binomial coefficient satisfies $t[r, 0] = 1$: only the empty partition has no parts.","title":"Triangular $q$-binomial coefficients with no parts"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/triangle-partition-series-zero","kind":"node","lean":"MetaMathlibExt.trianglePartitionSeries_zero","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#triangle-partition-series-zero","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"fully_proved","summary":"The triangle partition series $T_q(r)$ at $r = 0$ equals $1$, that is, $T_q(0) = 1$ in $\\mathbb{Z}[[q]]$.","title":"Triangle partition series at zero"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-cells-increasing-generating-function","kind":"node","lean":"MetaMathlibExt.litCellsIncreasing_generatingFunction","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-cells-increasing-generating-function","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"can_prove","summary":"Let $P(q) = \\sum_{n \\ge 0} p(n) q^n$ be the partition generating function, and let $L(q) \\in \\mathbb{Z}[[q]]$ be the power series whose coefficient of $q^n$ is the total number of lit cells over all partitions of $n$ drawn as weakly increasing bargraphs. Then","title":"Generating function for lit cells of weakly increasing bargraphs"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-nodes-increasing-generating-function","kind":"node","lean":"MetaMathlibExt.litNodesIncreasing_generatingFunction","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-nodes-increasing-generating-function","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"can_prove","summary":"Let $P(q) = \\sum_{n \\ge 0} p(n) q^n$ be the partition generating function, and let $N(q) \\in \\mathbb{Z}[[q]]$ be the power series whose coefficient of $q^n$ is the total number of lit nodes over all partitions of $n$ drawn as weakly increasing Ferrers diagrams of nodes. Then","title":"Generating function for lit nodes of weakly increasing diagrams"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/triangular-qbinomial-recursion","kind":"node","lean":"MetaMathlibExt.triangularQBinomial_recursion","parent":25809,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recursion for triangular "],["math","\\(q\\)"],["text","-binomial coefficients"]],"source":"Recursion for triangular $q$-binomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#triangular-qbinomial-recursion","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"can_prove","summary":"The triangular $q$-binomial coefficients $t[r, s] \\in \\mathbb{Z}[[q]]$ satisfy the recursion $t[r, s] = q^s \\sum_{i=0}^{s} t[r - 1, i]$ for all natural numbers $r \\ge 1$ and $s \\le r$, together with the initial conditions $t[1, 1] = q$, $t[2, 1] = q(1 + q)$, $t[r, 0] = 1$ for every $r$, and $t[r, s] = 0$ whenever $r < s$.","title":"Recursion for triangular $q$-binomial coefficients"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/triangular-qbinomial-eq-gauss-binom-sub","kind":"node","lean":"MetaMathlibExt.triangularQBinomial_eq_gaussBinom_sub","parent":25809,"reading_title":{"format":"inline-parts-v1","parts":[["text","Triangular "],["math","\\(q\\)"],["text","-binomial coefficients via Gaussian binomials"]],"source":"Triangular $q$-binomial coefficients via Gaussian binomials"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#triangular-qbinomial-eq-gauss-binom-sub","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"can_prove","summary":"For natural numbers $r$ and $s$ with $1 \\le r$ and $s \\le r$, the triangular $q$-binomial coefficient $t[r, s]$ satisfies","title":"Triangular $q$-binomial coefficients via Gaussian binomials"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-cells-decreasing-le-generating-funct","kind":"node","lean":"MetaMathlibExt.litCellsDecreasing_le_generatingFunction","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-cells-decreasing-le-generating-funct","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"can_prove","summary":"Let $k$ and $s$ be natural numbers with $s \\le k$. Write $\\binom{M}{m}_q$ for the Gaussian binomial coefficient, and $t[s, i]$ for the triangular $q$-binomial coefficient: the generating function of partitions with exactly $i$ parts $x_1 \\ge \\cdots \\ge x_i$ such that $x_j + j - 1 \\le s$ for $1 \\le j \\le i$. Let $\\mathrm{lit}[k, s] \\in \\mathbb{Z}[[q]]$ be the power series whose coefficient of $q^n$ is the number of\u2026","title":"Lit cells of weakly decreasing partitions with few parts"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-cells-decreasing-generating-function","kind":"node","lean":"MetaMathlibExt.litCellsDecreasing_generatingFunction","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-cells-decreasing-generating-function","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"can_prove","summary":"Let $P(q) = \\sum_{n \\ge 0} p(n) q^n$ be the partition generating function, for $k \\ge 0$ let $T_q(k)$ be the generating function of partitions contained in the triangle with column heights $k, k - 1, \\ldots, 1$ (so $T_q(0) = 1$), and let $L(q) \\in \\mathbb{Z}[[q]]$ be the power series whose coefficient of $q^n$ is the total number of lit cells over all partitions of $n$ drawn as weakly decreasing bargraphs. Then","title":"Generating function for lit cells of weakly decreasing bargraphs"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/lit-cells-decreasing-first-column-genera","kind":"node","lean":"MetaMathlibExt.litCellsDecreasing_firstColumn_generatingFunction","parent":25809,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions/index.html#lit-cells-decreasing-first-column-genera","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/lit-cells-in-integer-partitions","status":"can_prove","summary":"For a partition $p$ of $n$ with $\\ell \\ge 1$ parts $x_1 \\ge x_2 \\ge \\cdots \\ge x_\\ell$, say that only the first column of its weakly decreasing bargraph is lit when the number of lit cells of $p$ equals its largest part $x_1$, that is, when $x_i + i - 1 \\le x_1$ for $1 \\le i \\le \\ell$. Let $F(q) \\in \\mathbb{Z}[[q]]$ be the power series whose coefficient of $q^n$ is the sum of $x_1$ over the partitions of $n$ in\u2026","title":"Lit cells when only the first column is lit"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","title":"Max-rook numbers of partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/board-max-rook-number","kind":"node","lean":"MetaMathlibExt.boardMaxRookNumber","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#board-max-rook-number","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"fully_proved","summary":"For a finite board $B \\subseteq \\mathbb{N} \\times \\mathbb{N}$, whose cell $(i, j)$ lies in row $i$ and column $j$, the max-rook number of $B$ is the largest size of a subset of $B$ that contains at most one cell of each row and at most one cell of each column, that is, the largest number of non-attacking rooks that can be placed on $B$.","title":"Max-rook number of a board"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/partition-ferrers-board","kind":"node","lean":"MetaMathlibExt.partitionFerrersBoard","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#partition-ferrers-board","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"fully_proved","summary":"Let $p$ be a partition of $n$, and number rows and columns from $0$. The Ferrers board of $p$ is the finite set of cells $(i, j) \\in \\mathbb{N} \\times \\mathbb{N}$ such that the $(i+1)$-st largest part of $p$ exceeds $j$, that is, such that more than $i$ parts of $p$, counted with multiplicity, exceed $j$.","title":"Ferrers board of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/partition-max-rook-number","kind":"node","lean":"MetaMathlibExt.partitionMaxRookNumber","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#partition-max-rook-number","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"fully_proved","summary":"The max-rook number $R(\\pi)$ of a partition $\\pi$ of $n$ is the largest number of non-attacking rooks on the Ferrers board of $\\pi$: the largest number of cells of the board, no two in the same row and no two in the same column.","title":"Max-rook number of a partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $R_k(n)$ is the number of partitions of $n$ whose max-rook number, the largest number of non-attacking rooks on the Ferrers board, equals $k$.","title":"Partitions with a given max-rook number"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/partition-max-rook-number-max","kind":"node","lean":"MetaMathlibExt.partitionMaxRookNumberMax","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Largest max-rook number among partitions of "],["math","\\(n\\)"]],"source":"Largest max-rook number among partitions of $n$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#partition-max-rook-number-max","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"fully_proved","summary":"For a natural number $n \\ge 1$, $F(n)$ is the largest max-rook number $R(\\pi)$ over all partitions $\\pi$ of $n$, where $R(\\pi)$ is the largest number of non-attacking rooks on the Ferrers board of $\\pi$. By convention $F(0) = 1$.","title":"Largest max-rook number among partitions of $n$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/partition-ferrers-board-two-one","kind":"node","lean":"MetaMathlibExt.partitionFerrersBoard_two_one","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ferrers board of the partition "],["math","\\((2, 1)\\)"]],"source":"Ferrers board of the partition $(2, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#partition-ferrers-board-two-one","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"fully_proved","summary":"With rows and columns numbered from $0$, the Ferrers board of the partition $(2, 1)$ of $3$ consists of exactly the three cells $(0, 0)$, $(0, 1)$ and $(1, 0)$.","title":"Ferrers board of the partition $(2, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/partition-max-rook-number-five-three-two","kind":"node","lean":"MetaMathlibExt.partitionMaxRookNumber_five_three_two_one","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Max-rook number of the partition "],["math","\\((5, 3, 2, 1)\\)"]],"source":"Max-rook number of the partition $(5, 3, 2, 1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#partition-max-rook-number-five-three-two","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"fully_proved","summary":"The partition $(5, 3, 2, 1)$ of $11$ has max-rook number $4$.","title":"Max-rook number of the partition $(5, 3, 2, 1)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/partition-max-rook-number-max-zero","kind":"node","lean":"MetaMathlibExt.partitionMaxRookNumberMax_zero","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","The convention "],["math","\\(F(0) = 1\\)"]],"source":"The convention $F(0) = 1$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#partition-max-rook-number-max-zero","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"fully_proved","summary":"The function $F$, whose value $F(n)$ for $n \\ge 1$ is the largest max-rook number of a partition of $n$, takes the value $F(0) = 1$ at $n = 0$.","title":"The convention $F(0) = 1$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/has-sum-partition-max-rook-number-max","kind":"node","lean":"MetaMathlibExt.hasSum_partitionMaxRookNumberMax","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#has-sum-partition-max-rook-number-max","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $F(n)$ be the largest max-rook number of a partition of $n$ for $n \\ge 1$, and $F(0) = 1$. For every complex number $q$ with $|q| < 1$, the series $\\sum_{n \\ge 0} F(n) q^n$ is summable with sum","title":"Generating function of the largest max-rook number"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-three-gen-fun","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_three_genFun","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of "],["math","\\(R_3(n)\\)"]],"source":"Generating function of $R_3(n)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-three-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_3(n)$ be the number of partitions of $n$ with max-rook number $3$. In the ring $\\mathbb{Z}[[q]]$ of formal power series with integer coefficients,","title":"Generating function of $R_3(n)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-three-recurrenc","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_three_recurrence","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear recurrence for "],["math","\\(R_3(n)\\)"]],"source":"Linear recurrence for $R_3(n)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-three-recurrenc","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_3(n)$ be the number of partitions of $n$ with max-rook number $3$. For every natural number $n > 14$, $R_3(n) = 2R_3(n-1) - R_3(n-2) + R_3(n-3) - 2R_3(n-4) + R_3(n-5)$.","title":"Linear recurrence for $R_3(n)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-four-gen-fun","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_four_genFun","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-four-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_4(n)$ be the number of partitions of $n$ with max-rook number $4$. In the ring of formal power series in $q$ with integer coefficients,","title":"Generating function of partitions with max-rook number 4"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-five-gen-fun","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_five_genFun","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-five-gen-fun","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_5(n)$ be the number of partitions of $n$ with max-rook number $5$. In the ring of formal power series in $q$ with integer coefficients,","title":"Generating function of partitions with max-rook number 5"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-three-formula","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_three_formula","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-three-formula","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_3(n)$ be the number of partitions of $n$ with max-rook number $3$. For every natural number $m$ and every $a \\in \\{0, 1, 2\\}$ with $3m + a > 9$, $R_3(3m + a)$ equals $6m^2 - 15m + 7$, $6m^2 - 11m + 2$ or $6m^2 - 7m - 1$ according as $a = 0$, $a = 1$ or $a = 2$.","title":"Quasi-polynomial formula for partitions with max-rook number 3"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-three-mod-eq","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_three_modEq","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Periodicity of "],["math","\\(R_3(n)\\)"],["text"," modulo "],["math","\\(p\\)"]],"source":"Periodicity of $R_3(n)$ modulo $p$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-three-mod-eq","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_3(n)$ be the number of partitions of $n$ with max-rook number $3$. For all natural numbers $p \\ge 2$ and $n > 9$, $R_3(n + 3p) \\equiv R_3(n) \\pmod{p}$.","title":"Periodicity of $R_3(n)$ modulo $p$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-three-is-equiva","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_three_isEquivalent","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotic growth of "],["math","\\(R_3(n)\\)"]],"source":"Asymptotic growth of $R_3(n)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-three-is-equiva","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_3(n)$ be the number of partitions of $n$ with max-rook number $3$. Then $R_3(n) \\sim \\frac{2}{3} n^2$ as $n \\to \\infty$.","title":"Asymptotic growth of $R_3(n)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-four-formula","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_four_formula","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-four-formula","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_4(n)$ be the number of partitions of $n$ with max-rook number $4$. For all natural numbers $m$ and $a$ with $0 \\le a \\le 11$ and $12m + a > 16$,","title":"Quasi-polynomial formula for partitions with max-rook number 4"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-four-mod-eq","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_four_modEq","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-four-mod-eq","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_4(n)$ be the number of partitions of $n$ with max-rook number $4$. For all natural numbers $p \\ge 2$ and $n > 16$, $R_4(n + 12p) \\equiv R_4(n) \\pmod p$.","title":"A congruence for partitions with max-rook number 4"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-four-is-equival","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_four_isEquivalent","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-four-is-equival","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_4(n)$ be the number of partitions of $n$ with max-rook number $4$. Then $R_4(n) \\sim n^3/9$ as $n \\to \\infty$.","title":"Asymptotics of partitions with max-rook number 4"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-three-mod-two","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_three_mod_two","parent":25828,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity of "],["math","\\(R_3(n)\\)"]],"source":"Parity of $R_3(n)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-three-mod-two","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_3(n)$ be the number of partitions of $n$ with max-rook number $3$. For every natural number $n > 9$, $R_3(n)$ is even if $n$ is odd and odd if $n$ is even.","title":"Parity of $R_3(n)$"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-four-mod-two","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_four_mod_two","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-four-mod-two","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_4(n)$ be the number of partitions of $n$ with max-rook number $4$. For every natural number $n > 16$, $R_4(n)$ is even if $n \\equiv 4 \\pmod 8$ or $n \\equiv 6 \\pmod 8$, and odd otherwise.","title":"Parity of the number of partitions with max-rook number 4"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/max-rook-partition-count-five-mod-two","kind":"node","lean":"MetaMathlibExt.maxRookPartitionCount_five_mod_two","parent":25828,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions/index.html#max-rook-partition-count-five-mod-two","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/max-rook-numbers-of-partitions","status":"can_prove","summary":"Let $R_5(n)$ be the number of partitions of $n$ with max-rook number $5$. For every natural number $n > 25$, $R_5(n)$ is even if $n \\equiv 5 \\pmod 8$ or $n \\equiv 7 \\pmod 8$, and odd otherwise.","title":"Parity of the number of partitions with max-rook number 5"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"can_prove","title":"Number of values of diversity indices"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/gini-simpson-count","kind":"node","lean":"MathlibExt.Probability.DiversityStatisticsWanted.giniSimpsonCount","parent":25851,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html#gini-simpson-count","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"fully_proved","summary":"For a natural number $N$, $\\gamma(N)$ is the number of distinct values of $a_1^2 + \\dots + a_k^2$ over all finite sequences $(a_1, \\dots, a_k)$ of natural numbers with $a_1 + \\dots + a_k = N$.","title":"Number of values of the Gini\u2013Simpson index"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/shannon-count","kind":"node","lean":"MathlibExt.Probability.DiversityStatisticsWanted.shannonCount","parent":25851,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html#shannon-count","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"fully_proved","summary":"For a natural number $N$, $\\sigma(N)$ is the number of distinct real numbers $-\\sum_{i=1}^{k} (a_i / N) \\log (a_i / N)$ over all finite sequences $(a_1, \\dots, a_k)$ of natural numbers with $a_1 + \\dots + a_k = N$, where a term with $a_i = 0$ counts as $0$ (the convention $0 \\log 0 = 0$).","title":"Number of values of the Shannon index"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/gini-simpson-count-le","kind":"node","lean":"MathlibExt.Probability.DiversityStatisticsWanted.giniSimpsonCount_le","parent":25851,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html#gini-simpson-count-le","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"can_prove","summary":"Let $\\gamma(N)$ be the number of distinct values of $a_1^2 + \\dots + a_k^2$ over all finite sequences $(a_1, \\dots, a_k)$ of natural numbers with $a_1 + \\dots + a_k = N$ (for $N \\ge 1$, the number of possible values of the Gini\u2013Simpson index across samples of size $N$). Then for every natural number $N$,","title":"Upper bound on the number of Gini\u2013Simpson values"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/gini-simpson-count-ge-triangular","kind":"node","lean":"MathlibExt.Probability.DiversityStatisticsWanted.giniSimpsonCount_ge_triangular","parent":25851,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html#gini-simpson-count-ge-triangular","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"can_prove","summary":"Let $\\gamma(N)$ be the number of distinct values of $a_1^2 + \\dots + a_k^2$ over all finite sequences $(a_1, \\dots, a_k)$ of natural numbers with $a_1 + \\dots + a_k = N$ (for $N \\ge 1$, the number of possible values of the Gini\u2013Simpson index across samples of size $N$). Let $m$ and $r$ be natural numbers with $r \\le m$, and let $N = \\binom{m+1}{2} + r$. Then","title":"Triangular-number lower bound on the number of Gini\u2013Simpson values"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/gini-simpson-count-ge","kind":"node","lean":"MathlibExt.Probability.DiversityStatisticsWanted.giniSimpsonCount_ge","parent":25851,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html#gini-simpson-count-ge","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"can_prove","summary":"Let $\\gamma(N)$ be the number of distinct values of $a_1^2 + \\dots + a_k^2$ over all finite sequences $(a_1, \\dots, a_k)$ of natural numbers with $a_1 + \\dots + a_k = N$ (for $N \\ge 1$, the number of possible values of the Gini\u2013Simpson index across samples of size $N$). Then for every natural number $N$,","title":"Quadratic lower bound on the number of Gini\u2013Simpson values"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/gini-simpson-count-asymp","kind":"node","lean":"MathlibExt.Probability.DiversityStatisticsWanted.giniSimpsonCount_asymp","parent":25851,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html#gini-simpson-count-asymp","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"can_prove","summary":"Let $\\gamma(N)$ be the number of distinct values of $a_1^2 + \\dots + a_k^2$ over all finite sequences $(a_1, \\dots, a_k)$ of natural numbers with $a_1 + \\dots + a_k = N$ (for $N \\ge 1$, the number of possible values of the Gini\u2013Simpson index across samples of size $N$). Then $\\gamma(N) / N^2 \\to 1/2$ as $N \\to \\infty$, that is, $\\gamma(N) \\sim \\tfrac{1}{2} N^2$.","title":"Asymptotics of the number of Gini\u2013Simpson values"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/shannon-count-le","kind":"node","lean":"MathlibExt.Probability.DiversityStatisticsWanted.shannonCount_le","parent":25851,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html#shannon-count-le","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"can_prove","summary":"Let $\\sigma(N)$ be the number of distinct real numbers $-\\sum_{i=1}^{k} (a_i / N) \\log (a_i / N)$ over all finite sequences $(a_1, \\dots, a_k)$ of natural numbers with $a_1 + \\dots + a_k = N$, where a term with $a_i = 0$ counts as $0$ (for $N \\ge 1$, the number of possible values of the Shannon index across samples of size $N$). Then for every natural number $N \\ge 1$,","title":"Upper bound on the number of Shannon values"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/shannon-count-superpolynomial","kind":"node","lean":"MathlibExt.Probability.DiversityStatisticsWanted.shannonCount_superpolynomial","parent":25851,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices/index.html#shannon-count-superpolynomial","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/number-of-values-of-diversity-indices","status":"can_prove","summary":"Let $\\sigma(N)$ be the number of distinct real numbers $-\\sum_{i=1}^{k} (a_i / N) \\log (a_i / N)$ over all finite sequences $(a_1, \\dots, a_k)$ of natural numbers with $a_1 + \\dots + a_k = N$, where a term with $a_i = 0$ counts as $0$ (for $N \\ge 1$, the number of possible values of the Shannon index across samples of size $N$). Then for every nonzero polynomial $q$ with real coefficients,\u2026","title":"The number of Shannon values grows faster than every polynomial"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","title":"Perfect and double-perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/exists-partition-unique-subsums-but-one","kind":"node","lean":"MetaMathlibExt.exists_partition_unique_subsums_but_one_iff","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#exists-partition-unique-subsums-but-one","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"The partitions of $m$ contained in a partition $\\lambda$ are the distinct sub-multisets of the parts of $\\lambda$ with sum $m$. Let $v \\ge 1$. There are a natural number $n$ and a partition $\\lambda$ of $n$ that contains exactly $v$ partitions of some $m$ with $v \\le m \\le n - v$ and exactly one partition of every positive integer $m' < n$ other than $m$ if and only if $v = 1$ or $v = 2$.","title":"Partitions with unique subsums except one"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/subpartition-count","kind":"node","lean":"MetaMathlibExt.subpartitionCount","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#subpartition-count","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"For a multiset $\\lambda$ of natural numbers and $m \\in \\mathbb{N}$, the number of partitions of $m$ contained in $\\lambda$ is the number of distinct sub-multisets of $\\lambda$ whose elements sum to $m$.","title":"Number of subpartitions with a given sum"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-perfect-partition","kind":"node","lean":"MetaMathlibExt.IsPerfectPartition","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-perfect-partition","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"Let $n$ be a natural number. A multiset $\\lambda$ of natural numbers is a perfect partition of $n$ when its elements are positive, they sum to $n$, and for every integer $m$ with $1 \\le m \\le n$ exactly one partition of $m$ is contained in $\\lambda$, that is, exactly one sub-multiset of $\\lambda$ has sum $m$ (equal sub-multisets are counted once).","title":"Perfect partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-double-perfect-of-order","kind":"node","lean":"MetaMathlibExt.IsDoublePerfectOfOrder","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-double-perfect-of-order","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"A double-perfect partition of $N$ of order $r$ is a finite multiset $\\lambda$ of positive integers with sum $N$ such that, for every integer $m$ with $1 \\le m \\le N$, the number of distinct sub-multisets of $\\lambda$ with sum $m$ is $2$ if $r + 1 \\le m \\le N - r - 1$ and $1$ otherwise. Their set is denoted $U_r(N)$; order $1$ gives the double-perfect partitions.","title":"Double-perfect partitions of higher order"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-type-adouble-perfect","kind":"node","lean":"MetaMathlibExt.IsTypeADoublePerfect","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-type-adouble-perfect","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"Let $r$ and $N$ be natural numbers. A multiset $\\lambda$ of natural numbers is a Type-A double-perfect partition of $N$ of order $r$ when $\\lambda$ is a double-perfect partition of $N$ of order $r$ and there is a perfect partition $\\beta$ of $N - r - 1$ (read as $0$ when $N \\le r$) such that $\\lambda$ is $\\beta$ with $r + 1$ parts equal to $1$ added, or $\\beta$ with one part equal to $r + 1$ added. The set of these\u2026","title":"Type-A double-perfect partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/perfect-partition-of-factorization","kind":"node","lean":"MetaMathlibExt.perfectPartitionOfFactorization","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#perfect-partition-of-factorization","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"To a finite sequence $(a_1, \\dots, a_k)$ of natural numbers, read as the ordered factorization $a_1 a_2 \\cdots a_k$, attach the multiset of parts $(1^{a_1-1}, a_1^{a_2-1}, (a_1a_2)^{a_3-1}, \\dots, (a_1\\cdots a_{k-1})^{a_k-1})$, where $x^e$ denotes $e$ parts equal to $x$ and an exponent $a_i - 1$ is read as $0$ when $a_i = 0$. In particular the empty sequence gives the empty multiset.","title":"MacMahon's partition of an ordered factorization"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/perfect-partition-of-factorization-two-t","kind":"node","lean":"MetaMathlibExt.perfectPartitionOfFactorization_two_three","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#perfect-partition-of-factorization-two-t","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"MacMahon's bijection sends the ordered factorization $6 = 2 \\cdot 3$ to the partition $(1, 2^2) = (1, 2, 2)$ of $5$.","title":"MacMahon's partition of the factorization 2 \u00b7 3"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/perfect-partition-of-factorization-sum-a","kind":"node","lean":"MetaMathlibExt.perfectPartitionOfFactorization_sum_add_one","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#perfect-partition-of-factorization-sum-a","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"Let $k \\ge 0$ and let $a_1, \\dots, a_k \\ge 1$ be integers. The parts of the partition $(1^{a_1-1}, a_1^{a_2-1}, (a_1a_2)^{a_3-1}, \\dots, (a_1\\cdots a_{k-1})^{a_k-1})$ that MacMahon's bijection attaches to the ordered factorization $a_1 a_2 \\cdots a_k$, where $x^e$ denotes $e$ parts equal to $x$, sum to $a_1 a_2 \\cdots a_k - 1$.","title":"The partition of a factorization sums to the product minus one"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/ordered-factorization-count-aux","kind":"node","lean":"MetaMathlibExt.orderedFactorizationCountAux","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#ordered-factorization-count-aux","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, define $f_k(n)$ by recursion on $k$: $f_0(n) = 0$, $f_{k+1}(1) = 1$, and for $n \\ne 1$, $f_{k+1}(n)$ is the sum of $f_k(d)$ over the proper divisors $d$ of $n$, that is, the divisors $d$ of $n$ with $1 \\le d < n$ (there are none when $n = 0$).","title":"Truncated recursion for ordered factorizations"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/ordered-factorization-count","kind":"node","lean":"MetaMathlibExt.orderedFactorizationCount","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#ordered-factorization-count","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"For $n \\ge 2$, $f(n)$ is the number of ordered factorizations $n = a_1 a_2 \\cdots a_k$ of $n$ into factors $a_i > 1$ (OEIS A074206); moreover $f(0) = 0$ and $f(1) = 1$.","title":"Number of ordered factorizations"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/ordered-factorization-count-aux-eq","kind":"node","lean":"MetaMathlibExt.orderedFactorizationCountAux_eq","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#ordered-factorization-count-aux-eq","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"Let $f_k(n)$ be the truncated recursion with $f_0(n) = 0$, $f_{k+1}(1) = 1$, and $f_{k+1}(n)$ the sum of $f_k(d)$ over the proper divisors $d$ of $n$ when $n \\ne 1$. For all natural numbers $n$ and $k$ with $n \\le k$, $f_k(n) = f(n)$, where $f(n)$ is the number of ordered factorizations of $n$ into factors greater than $1$.","title":"The truncated ordered-factorization recursion stabilizes"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/ordered-factorization-count-eq-sum","kind":"node","lean":"MetaMathlibExt.orderedFactorizationCount_eq_sum","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#ordered-factorization-count-eq-sum","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"For every integer $n \\ge 2$, the number $f(n)$ of ordered factorizations of $n$ into factors greater than $1$ satisfies $f(n) = \\sum_{d \\mid n,\\ d < n} f(d)$, the sum running over the proper divisors $d$ of $n$.","title":"Recursion for the number of ordered factorizations"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/ordered-factorization-count-small","kind":"node","lean":"MetaMathlibExt.orderedFactorizationCount_small","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#ordered-factorization-count-small","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"The number $f(n)$ of ordered factorizations of $n$ into factors greater than $1$ takes the values $f(0) = 0$, $f(1) = 1$, $f(6) = 3$ and $f(12) = 8$.","title":"Small values of the ordered factorization count"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-perfect-partition-seven","kind":"node","lean":"MetaMathlibExt.isPerfectPartition_seven","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-perfect-partition-seven","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"The partition $(1^3, 4)$ of $7$, with three parts equal to $1$ and one part equal to $4$, is a perfect partition of $7$.","title":"A perfect partition of seven"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-type-adouble-perfect-one-seven","kind":"node","lean":"MetaMathlibExt.isTypeADoublePerfect_one_seven","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-type-adouble-perfect-one-seven","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"fully_proved","summary":"The partition $(1^5, 2)$ of $7$, with five parts equal to $1$ and one part equal to $2$, is a Type-A double-perfect partition of $7$ of order $1$.","title":"A Type-A double-perfect partition of seven"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/double-perfect-form","kind":"node","lean":"MetaMathlibExt.doublePerfect_form","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#double-perfect-form","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $N \\ge 4$ and let $\\lambda$ be a double-perfect partition of $N$ of order $1$. Then there are integers $q_1 \\ge 2$ and $q_2 \\ge 1$, with $q_2 = 1$ unless $q_1 = 3$, an integer $k \\ge 2$ and integers $q_3, \\dots, q_k \\ge 1$ such that, with $B = q_1 + 2q_2 - 1$, $\\lambda = (1^{q_1}, 2^{q_2}, B^{q_3}, (B(q_3+1))^{q_4}, \\dots, (B(q_3+1) \\cdots (q_{k-1}+1))^{q_k})$ (for $k = 2$ the only parts are $1$ and $2$). That\u2026","title":"Lee's form of double-perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/double-perfect-ncard","kind":"node","lean":"MetaMathlibExt.doublePerfect_ncard","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#double-perfect-ncard","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $f(m)$ be the number of ordered factorizations of $m$ into factors greater than $1$, with $f(0) = 0$ and $f(1) = 1$. For every $n \\ge 4$, the number $d(n)$ of double-perfect partitions of $n$ of order $1$ is $f(n-1)$ unless $n \\equiv 1 \\pmod 4$, in which case $d(n) = f(n-1) - f((n-1)/4)$.","title":"Lee's count of double-perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-double-perfect-one-iff","kind":"node","lean":"MetaMathlibExt.isDoublePerfect_one_iff","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-double-perfect-one-iff","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $N > 3$. A multiset $\\lambda$ of natural numbers is a double-perfect partition of $N$ of order $1$ if and only if there is a perfect partition $\\beta$ of $N - 2$ such that either $1$ occurs exactly once in $\\beta$ and $\\lambda$ is $\\beta$ with two more parts equal to $1$, or $2$ is not a part of $\\beta$ and $\\lambda$ is $\\beta$ with one more part equal to $2$.","title":"Double-perfect partitions from perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/double-perfect-lee-forms","kind":"node","lean":"MetaMathlibExt.doublePerfect_lee_forms","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#double-perfect-lee-forms","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"For an integer $B$, an integer $k \\ge 2$ and integers $q_3, \\dots, q_k \\ge 1$, write $T_B(q_3, \\dots, q_k)$ for the parts $B^{q_3}, (B(q_3+1))^{q_4}, \\dots, (B(q_3+1) \\cdots (q_{k-1}+1))^{q_k}$ (no parts when $k = 2$). Let $N \\ge 4$ and let $\\lambda$ be a double-perfect partition of $N$ of order $1$. Then either $\\lambda = (1^3, 2^{q_2}, T_{2(q_2+1)}(q_3, \\dots, q_k))$ for some $q_2 > 1$, some $k \\ge 2$ and some\u2026","title":"Lee's two forms of double-perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-type-adouble-perfect-iff","kind":"node","lean":"MetaMathlibExt.isTypeADoublePerfect_iff","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-type-adouble-perfect-iff","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $r \\ge 1$ and $N \\ge 2(r+1)$ be integers. A multiset $\\lambda$ of natural numbers is a Type-A double-perfect partition of $N$ of order $r$ if and only if there is a perfect partition $\\beta$ of $N - r - 1$ such that either $1$ occurs exactly $r$ times in $\\beta$ and $\\lambda$ is $\\beta$ with $r + 1$ parts equal to $1$ added, or $1$ does not occur exactly $r$ times in $\\beta$, $r + 1$ is not a part of $\\beta$,\u2026","title":"Characterization of Type-A double-perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/type-adouble-perfect-forms","kind":"node","lean":"MetaMathlibExt.typeADoublePerfect_forms","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#type-adouble-perfect-forms","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $r \\ge 1$ and $N \\ge 2(r+1)$. For a finite sequence $(a_1, \\dots, a_k)$ write $P(a_1, \\dots, a_k) = (1^{a_1-1}, a_1^{a_2-1}, \\dots, (a_1\\cdots a_{k-1})^{a_k-1})$ for the partition that MacMahon's bijection attaches to the ordered factorization $a_1 \\cdots a_k$, where $x^e$ denotes $e$ parts equal to $x$, and let $\\cup$ join parts with multiplicity. Every Type-A double-perfect partition $\\lambda$ of $N$ of order\u2026","title":"Shapes of Type-A double-perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/type-adouble-perfect-ncard","kind":"node","lean":"MetaMathlibExt.typeADoublePerfect_ncard","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#type-adouble-perfect-ncard","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $f(n)$ be the number of ordered factorizations of $n$ into factors greater than $1$. Let $r \\ge 1$ and $N \\ge 2(r+1)$, that is, $1 \\le r \\le \\lfloor (N-2)/2 \\rfloor$, and put $M = 2(r+1)$. The number $d_r(N)$ of Type-A double-perfect partitions of $N$ of order $r$ is $f(N-r) - f((N-r)/M) - (f(r+1) - 1) f((N-r)/(r+1))$ if $N \\equiv r \\pmod M$, $f(N-r) - (f(r+1) - 1) f((N-r)/(r+1))$ if $N \\equiv -1 \\pmod M$, and\u2026","title":"Number of Type-A double-perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-type-adouble-perfect-add-replicate","kind":"node","lean":"MetaMathlibExt.isTypeADoublePerfect_add_replicate","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-type-adouble-perfect-add-replicate","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $r \\ge 1$ and $n \\ge 2(r+1)$ be integers, and let $\\lambda$ be a Type-A double-perfect partition of $n$ of order $r$. Then for every natural number $m$, the partition obtained from $\\lambda$ by adding $m$ parts equal to $n - r$ is a Type-A double-perfect partition of $n + m(n - r)$ of order $r$.","title":"Adding equal parts to a Type-A double-perfect partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-double-perfect-of-order-add-replicate","kind":"node","lean":"MetaMathlibExt.isDoublePerfectOfOrder_add_replicate","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-double-perfect-of-order-add-replicate","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $r \\ge 1$, $n \\ge 2(r+1)$, and let $\\lambda$ be a double-perfect partition of $n$ of order $r$. For every $m \\ge 0$, adding $m$ parts equal to $n - r$ to $\\lambda$ gives a double-perfect partition of $n + m(n - r)$ of order $r$.","title":"Inserting copies of a part into a double-perfect partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/is-double-perfect-of-order-add-largest","kind":"node","lean":"MetaMathlibExt.isDoublePerfectOfOrder_add_largest","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#is-double-perfect-of-order-add-largest","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $r \\ge 1$, $n \\ge 2(r+1)$, and let $\\lambda$ be a double-perfect partition of $n$ of order $r$ whose largest part $\\ell$ is not $r + 1$. Then $\\lambda$ with one more part $\\ell$ is a double-perfect partition of $n + \\ell$ of order $r$. The condition $\\ell \\ne r + 1$ cannot be dropped: $(1^2, 3^4, 6)$ is a Type-A double-perfect partition of $20$ of order $5$, but $(1^2, 3^4, 6^2)$ is not a double-perfect\u2026","title":"Adding a copy of the largest part to a double-perfect partition"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/halved-perfect-is-double-perfect-not-typ","kind":"node","lean":"MetaMathlibExt.halvedPerfect_isDoublePerfect_not_typeA","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#halved-perfect-is-double-perfect-not-typ","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"For integers $a_1, \\dots, a_k \\ge 2$ write $P(a_1, \\dots, a_k)$ for the perfect partition $(1^{a_1-1}, a_1^{a_2-1}, (a_1a_2)^{a_3-1}, \\dots, (a_1 \\cdots a_{k-1})^{a_k-1})$. Let $N$ and $r \\ge 1$ be natural numbers, let $a_1, \\dots, a_k \\ge 2$ and $1 < j < k$ with $a_1 \\cdots a_j = 2(r+1)$, $a_1 \\cdots a_k = 2(N - r)$, $a_j > 2$ and $a_{j+1} > 2$. Let $\\lambda$ consist of the parts of $P(a_1, \\dots, a_j)$ together\u2026","title":"Halved perfect partitions are double-perfect but not of Type A"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/type-bdouble-perfect-ncard","kind":"node","lean":"MetaMathlibExt.typeBDoublePerfect_ncard","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#type-bdouble-perfect-ncard","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $f(n)$ be the number of ordered factorizations of $n$ into factors greater than $1$. Let $r \\ge 1$ and $N \\ge 3r + 4$, that is, $1 \\le r \\le \\lfloor (N-4)/3 \\rfloor$, and put $M = 2(r+1)$ and $c = f(M) - 1 - f(r+1)$. The number $e_r(N)$ of Type-B double-perfect partitions of $N$ of order $r$, that is, of double-perfect partitions of $N$ of order $r$ that are not of Type A, is $c\\,(f((N-r)/(r+1)) - f((N-r)/M))$\u2026","title":"Number of Type-B double-perfect partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/double-perfect-of-order-ncard","kind":"node","lean":"MetaMathlibExt.doublePerfectOfOrder_ncard","parent":25860,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions/index.html#double-perfect-of-order-ncard","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/perfect-and-double-perfect-partitions","status":"can_prove","summary":"Let $f(m)$ be the number of ordered factorizations of $m$ into factors greater than $1$. Let $r \\ge 1$ and $N \\ge 2(r+1)$ (equivalently $1 \\le r \\le \\lfloor (N-2)/2 \\rfloor$), put $M = 2(r+1)$, and let $u_r(N)$ be the number of double-perfect partitions of $N$ of order $r$. Then $u_r(N)$ equals $f(N-r) + (f(M) - 2f(r+1)) f((N-r)/(r+1)) - (f(M) - f(r+1)) f((N-r)/M)$ if $N \\equiv r \\pmod M$,\u2026","title":"Number of double-perfect partitions of higher order"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products","status":"can_prove","title":"Reconstructing partitions from pairwise products"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products/pairwise-products","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.Partition.MultisetReconstructionPairwiseProductsWanted.pairwiseProducts","parent":25889,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products/index.html#pairwise-products","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products","status":"fully_proved","summary":"If a multiset $s$ of natural numbers consists of $x_1, \\dots, x_k$, listed with multiplicity, then the pairwise products of $s$ form the multiset of the products $x_i x_j$ over all pairs of indices $1 \\le i < j \\le k$; equal elements at different positions give separate products, and the result is empty when $k < 2$.","title":"Pairwise products of a multiset"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products/partition-eq-of-pairwise-products-eq-of","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.Partition.MultisetReconstructionPairwiseProductsWanted.partition_eq_of_pairwiseProducts_eq_of_max_eq","parent":25889,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products/index.html#partition-eq-of-pairwise-products-eq-of","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products","status":"can_prove","summary":"Let $n$ be a positive integer and let $\\lambda$ and $\\mu$ be partitions of $n$ with a common largest part, that is, some $a$ is a part of both $\\lambda$ and $\\mu$ and no part of either exceeds $a$. If the pairwise products of the parts of $\\lambda$ and the pairwise products of the parts of $\\mu$ form the same multiset, then $\\lambda = \\mu$.","title":"Li's lemma: reconstruction from the largest part and pairwise products"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products/partition-pairwise-products-eq-iff","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.Partition.MultisetReconstructionPairwiseProductsWanted.partition_pairwiseProducts_eq_iff","parent":25889,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products/index.html#partition-pairwise-products-eq-iff","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/reconstructing-partitions-from-pairwise-products","status":"can_prove","summary":"Let $n$ be a positive integer and let $\\lambda$ and $\\mu$ be partitions of $n$. The pairwise products of the parts of $\\lambda$ and the pairwise products of the parts of $\\mu$ form the same multiset if and only if $\\lambda = \\mu$.","title":"Li's theorem: partitions are determined by their pairwise products"},{"data_url":"scopes/Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions.json","id":"Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions","kind":"scope","navigation":"module","parent":25729,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions/index.html","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions","status":"can_prove","title":"Simultaneous core partitions"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions/is-core","kind":"node","lean":"MathlibExt.Combinatorics.SimultaneousCoreContainmentOWR1265Problem006Wanted.IsCore","parent":25893,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(s\\)"],["text","-cores of Young diagrams"]],"source":"$s$-cores of Young diagrams"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions/index.html#is-core","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions","status":"fully_proved","summary":"Let $s$ be a natural number and $\\mu$ a Young diagram, with rows and columns numbered from $0$; write $r_i$ for the length of row $i$ of $\\mu$ and $c_j$ for the length of column $j$ of $\\mu$. The hook length of the cell $(i, j)$ of $\\mu$ (row $i$, column $j$) is $(r_i - j) + (c_j - i) - 1$: the cells to its right in row $i$, the cells below it in column $j$, and the cell itself (both differences count the cell, so\u2026","title":"$s$-cores of Young diagrams"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions/simultaneous-core-containment","kind":"node","lean":"MathlibExt.Combinatorics.SimultaneousCoreContainmentOWR1265Problem006Wanted.simultaneousCoreContainment","parent":25893,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions/index.html#simultaneous-core-containment","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions","status":"fully_proved","summary":"The proposition that for all coprime positive integers $s$ and $t$ and every Young diagram $\\kappa$ that is both an $s$-core and a $t$-core and has at least as many cells as every Young diagram that is both an $s$-core and a $t$-core, every Young diagram that is both an $s$-core and a $t$-core is contained in $\\kappa$, that is, each of its cells is a cell of $\\kappa$.","title":"Containment of simultaneous cores (Olsson\u2013Stanton question)"},{"id":"Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions/simultaneous-core-containment-holds","kind":"node","lean":"MathlibExt.Combinatorics.SimultaneousCoreContainmentOWR1265Problem006Wanted.simultaneousCoreContainment_holds","parent":25893,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions/index.html#simultaneous-core-containment-holds","scope":"Combinatorics/integer-partitions/young-diagrams-statistics/simultaneous-core-partitions","status":"can_prove","summary":"Call a Young diagram an $s$-core when none of its cells has hook length divisible by $s$. For all coprime positive integers $s$ and $t$, if $\\kappa$ is a Young diagram that is both an $s$-core and a $t$-core and has at least as many cells as every Young diagram that is both an $s$-core and a $t$-core, then every Young diagram that is both an $s$-core and a $t$-core is contained in $\\kappa$ (each of its cells is a\u2026","title":"Fayers\u2013Vandehey containment theorem for simultaneous cores"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions.json","id":"Combinatorics/integer-partitions/counting-partitions","kind":"scope","parent":25632,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/index.html","scope":"Combinatorics/integer-partitions/counting-partitions","status":"can_prove","summary":"The partition function and its relatives as counts: bounds and asymptotics for p(n), the pentagonal number theorem, partition polynomials, Landau's function, overpartitions and regular partitions counted (not their congruences), restricted Gaussian and bounded partitions.","title":"Counting partitions"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function.json","id":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","status":"fully_proved","title":"Bounds on the partition function"},{"id":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/card-count-ge","kind":"node","lean":"Nat.Partition.card_count_ge","parent":25898,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bijection removing "],["math","\\(k\\)"],["text"," copies of a part from partitions"]],"source":"Bijection removing $k$ copies of a part from partitions"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/index.html#card-count-ge","scope":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","status":"fully_proved","summary":"Removing `k` copies of the positive part `v` gives a bijection from partitions of `E` containing at least `k` copies of `v` to partitions of `E - k * v`.","title":"Bijection removing $k$ copies of a part from partitions"},{"id":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/card-monotone","kind":"node","lean":"Nat.Partition.card_monotone","parent":25898,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/index.html#card-monotone","scope":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","status":"fully_proved","summary":"The number of integer partitions is monotone in the integer being partitioned.","title":"Monotonicity of the partition function"},{"id":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/mul-card-eq-sum","kind":"node","lean":"Nat.Partition.mul_card_eq_sum","parent":25898,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/index.html#mul-card-eq-sum","scope":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","status":"fully_proved","summary":"A recurrence obtained by summing all parts of all partitions of `E`, first by partition and then by part value and multiplicity threshold.","title":"Sum-of-parts recurrence for the partition function"},{"id":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/exists-exp-div-pow-le-card","kind":"node","lean":"Nat.Partition.exists_exp_div_pow_le_card","parent":25898,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/index.html#exists-exp-div-pow-le-card","scope":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","status":"fully_proved","summary":"The number of partitions of `n` is at least `H * exp (pi * sqrt (2 / 3) * sqrt n) / n ^ B` for some `H > 0` and natural `B`, when `n \u2265 1`.","title":"Exponential lower bound for the partition function"},{"id":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/card-le-exp","kind":"node","lean":"Nat.Partition.card_le_exp","parent":25898,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/index.html#card-le-exp","scope":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","status":"fully_proved","summary":"The number of integer partitions of `n` is at most `exp (\u03c0 * sqrt (2 / 3) * sqrt n)`.","title":"Exponential upper bound for the partition function"},{"id":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/succ-le-two-mul-card","kind":"node","lean":"Nat.Partition.succ_le_two_mul_card","parent":25898,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/index.html#succ-le-two-mul-card","scope":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","status":"fully_proved","summary":"The number of partitions of `n` is at least `(n + 1) / 2`.","title":"Linear lower bound for the partition function"},{"id":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/exists-logb-card-bounds","kind":"node","lean":"Nat.Partition.exists_logb_card_bounds","parent":25898,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function/index.html#exists-logb-card-bounds","scope":"Combinatorics/integer-partitions/counting-partitions/bounds-on-the-partition-function","status":"fully_proved","summary":"There are constants `B` and `C` such that the base-two logarithm of the number of integer partitions of `n` differs from its leading square-root term by at most a logarithmic lower-order term below and a constant above.","title":"Logarithmic bounds for the partition function"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/landau-s-function.json","id":"Combinatorics/integer-partitions/counting-partitions/landau-s-function","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/landau-s-function/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/landau-s-function","status":"fully_proved","title":"Landau's function"},{"id":"Combinatorics/integer-partitions/counting-partitions/landau-s-function/parts-lcm","kind":"node","lean":"Nat.Partition.partsLCM","parent":25906,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/landau-s-function/index.html#parts-lcm","scope":"Combinatorics/integer-partitions/counting-partitions/landau-s-function","status":"fully_proved","summary":"For a natural number $n$ and a partition $p$ of $n$, this is the least common multiple of the parts of $p$, taken with the convention that it equals $1$ when $p$ has no parts.","title":"Least common multiple of the parts of a partition"},{"id":"Combinatorics/integer-partitions/counting-partitions/landau-s-function/landau-function","kind":"node","lean":"Nat.landauFunction","parent":25906,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/landau-s-function/index.html#landau-function","scope":"Combinatorics/integer-partitions/counting-partitions/landau-s-function","status":"fully_proved","summary":"For a natural number $n$, Landau's function at $n$ is the maximum, over all partitions $p$ of $n$, of the least common multiple of the parts of $p$.","title":"Landau's function"},{"id":"Combinatorics/integer-partitions/counting-partitions/landau-s-function/parts-lcm-le-landau-function","kind":"node","lean":"Nat.Partition.partsLCM_le_landauFunction","parent":25906,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/landau-s-function/index.html#parts-lcm-le-landau-function","scope":"Combinatorics/integer-partitions/counting-partitions/landau-s-function","status":"fully_proved","summary":"For every natural number $n$ and every partition $p$ of $n$, the least common multiple of the parts of $p$ is at most the Landau function at $n$.","title":"The Landau function bounds the LCM of the parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/landau-s-function/exists-partition-parts-lcm-eq-landau-fun","kind":"node","lean":"Nat.exists_partition_partsLCM_eq_landauFunction","parent":25906,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/landau-s-function/index.html#exists-partition-parts-lcm-eq-landau-fun","scope":"Combinatorics/integer-partitions/counting-partitions/landau-s-function","status":"fully_proved","summary":"For every natural number $n$, some partition of $n$ has least common multiple of its parts equal to the Landau function at $n$.","title":"The Landau function is attained"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds.json","id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"can_prove","title":"Multifold partition diamonds"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/perm-value-at","kind":"node","lean":"MetaMathlibExt.permValueAt","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#perm-value-at","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For a permutation $\\tau$ of $\\{0, 1, \\dots, d - 1\\}$ and a natural number $j$, this is the natural number $\\tau(j)$ when $j < d$, and $0$ when $j \\ge d$. It is used to read off descents.","title":"Value of a permutation at a natural position"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/perm-descent-set","kind":"node","lean":"MetaMathlibExt.permDescentSet","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#perm-descent-set","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For a permutation $\\tau$ of $\\{0, 1, \\dots, d - 1\\}$, the descent set is $\\mathrm{Des}(\\tau) = \\{p \\in \\{1, \\dots, d - 1\\} : \\tau(p - 1) > \\tau(p)\\}$. If $\\tau$ is read as the permutation $p \\mapsto \\tau(p - 1) + 1$ of $\\{1, \\dots, d\\}$, this is the usual descent set: the positions $p$, numbered from $1$, where the value at $p$ exceeds the value at $p + 1$.","title":"Descent set of a permutation"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/perm-des","kind":"node","lean":"MetaMathlibExt.permDes","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#perm-des","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For a permutation $\\tau$ of $\\{0, 1, \\dots, d - 1\\}$, the number of descents $\\mathrm{des}(\\tau) = |\\mathrm{Des}(\\tau)|$ is the number of elements of the descent set $\\mathrm{Des}(\\tau)$.","title":"Number of descents from the descent set"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/perm-maj","kind":"node","lean":"MetaMathlibExt.permMaj","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#perm-maj","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For a permutation $\\tau$ of $\\{0, 1, \\dots, d - 1\\}$, the major index is $\\mathrm{maj}(\\tau) = \\sum_{p \\in \\mathrm{Des}(\\tau)} p$, the sum of the positions, numbered from $1$, in the descent set $\\mathrm{Des}(\\tau)$.","title":"Major index of a permutation"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/euler-mahonian","kind":"node","lean":"MetaMathlibExt.eulerMahonian","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#euler-mahonian","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For a natural number $d$ and elements $x, y$ of a commutative ring, the Euler\u2013Mahonian polynomial evaluated at $(x, y)$ is $E_d(x, y) = \\sum_{\\tau \\in S_d} x^{\\operatorname{des}(\\tau)} y^{\\operatorname{maj}(\\tau)}$, where $\\operatorname{des}(\\tau)$ is the number of descents and $\\operatorname{maj}(\\tau)$ the major index of the permutation $\\tau$.","title":"Euler\u2013Mahonian polynomial"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/euler-mahonian-one","kind":"node","lean":"MetaMathlibExt.eulerMahonian_one","parent":25911,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first Euler\u2013Mahonian polynomial is "],["math","\\(1\\)"]],"source":"The first Euler\u2013Mahonian polynomial is $1$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#euler-mahonian-one","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For every commutative ring $R$ and all $x, y \\in R$, the Euler\u2013Mahonian polynomial $E_1$ satisfies $E_1(x, y) = 1$.","title":"The first Euler\u2013Mahonian polynomial is $1$"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/euler-mahonian-three-two-three","kind":"node","lean":"MetaMathlibExt.eulerMahonian_three_two_three","parent":25911,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler\u2013Mahonian polynomial "],["math","\\(E_3\\)"],["text"," at "],["math","\\((2, 3)\\)"]],"source":"Euler\u2013Mahonian polynomial $E_3$ at $(2, 3)$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#euler-mahonian-three-two-three","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"Over the integers, the Euler\u2013Mahonian polynomial $E_3$ takes the value $E_3(2, 3) = 157$.","title":"Euler\u2013Mahonian polynomial $E_3$ at $(2, 3)$"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/card-perm-three-des-one-maj-one","kind":"node","lean":"MetaMathlibExt.card_perm_three_des_one_maj_one","parent":25911,"reading_title":{"format":"inline-parts-v1","parts":[["text","Permutations of "],["math","\\(S_3\\)"],["text"," with descent set "],["math","\\(\\{1\\}\\)"]],"source":"Permutations of $S_3$ with descent set $\\{1\\}$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#card-perm-three-des-one-maj-one","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"Exactly two permutations $\\tau \\in S_3$ have exactly one descent and major index $1$, that is, $\\operatorname{des}(\\tau) = 1$ and $\\operatorname{maj}(\\tau) = 1$; equivalently, their descent set is $\\{1\\}$.","title":"Permutations of $S_3$ with descent set $\\{1\\}$"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/diamond-link-pos","kind":"node","lean":"MetaMathlibExt.diamondLinkPos","parent":25911,"reading_title":{"format":"inline-parts-v1","parts":[["text","Position of the "],["math","\\(k\\)"],["text","th link of a multifold partition diamond"]],"source":"Position of the $k$th link of a multifold partition diamond"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#diamond-link-pos","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For natural numbers $d_1, \\dots, d_M$ (the numbers of folds of the $M$ diamonds) and a natural number $k$, the position of the $k$th link, counting positions from $0$, is $L_k = \\sum_{m=1}^{\\min(k, M)} (d_m + 1)$. In particular $L_0 = 0$, and $L_k = d_1 + \\dots + d_M + M$ for every $k \\ge M$.","title":"Position of the $k$th link of a multifold partition diamond"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/diamond-size","kind":"node","lean":"MetaMathlibExt.diamondSize","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#diamond-size","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"The multifold partition diamond whose $M$ diamonds have $d_1, \\dots, d_M$ folds has $c = d_1 + \\dots + d_M + M + 1$ parts.","title":"Number of parts of a multifold partition diamond"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/diamond-links","kind":"node","lean":"MetaMathlibExt.diamondLinks","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#diamond-links","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For natural numbers $d_1, \\dots, d_M$, the set of link positions is $\\{L_0, L_1, \\dots, L_M\\}$, where $L_k = \\sum_{m=1}^{k} (d_m + 1)$ is the position of the $k$th link. These positions satisfy $0 = L_0 < L_1 < \\dots < L_M = c - 1$, where $c = d_1 + \\dots + d_M + M + 1$ is the number of parts.","title":"The set of link positions of a multifold partition diamond"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/is-multifold-diamond","kind":"node","lean":"MetaMathlibExt.IsMultifoldDiamond","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#is-multifold-diamond","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"Let $d_1, \\dots, d_M$ be natural numbers, let $L_k = \\sum_{m=1}^{k} (d_m + 1)$ for $0 \\le k \\le M$ be the link positions, let $c = L_M + 1$, and let $i$ and $j$ be natural numbers. A function $b : \\mathbb{N} \\to \\mathbb{N}$ is a multifold partition diamond with fold sum $i$ and link sum $j$ when the following hold: for every $k$ with $0 \\le k < M$ and every $f$ with $1 \\le f \\le d_{k+1}$, we have\u2026","title":"Multifold partition diamonds with given fold and link sums"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/diamond-part","kind":"node","lean":"MetaMathlibExt.diamondPart","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#diamond-part","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"Let $c = d_1 + \\dots + d_M + M + 1$ be the number of parts of the multifold partition diamond whose diamonds have $d_1, \\dots, d_M$ folds, and let $n$ be a natural number. A vector $(a_0, \\dots, a_{c-1})$ with entries in $\\{0, \\dots, n - 1\\}$ is read as the function $\\mathbb{N} \\to \\mathbb{N}$ that sends $p$ to $a_p$ for $p < c$ and to $0$ for $p \\ge c$.","title":"Extending a part vector by zero"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/multifold-diamond-count","kind":"node","lean":"MetaMathlibExt.multifoldDiamondCount","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#multifold-diamond-count","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"Let $d_1, \\dots, d_M$ be natural numbers. Put $L_k = \\sum_{m=1}^{k} (d_m + 1)$ for $0 \\le k \\le M$ and $c = L_M + 1$. A multifold partition diamond of shape $(d_1, \\dots, d_M)$ is a sequence $(b_0, \\dots, b_{c-1})$ of nonnegative integers such that $b_{L_{k-1}} \\le b_{L_{k-1} + f} \\le b_{L_k}$ for all $1 \\le k \\le M$ and $1 \\le f \\le d_k$. The parts $b_{L_0}, \\dots, b_{L_M}$ are its links and the other parts are\u2026","title":"Number of multifold partition diamonds"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/multifold-diamond-count-one-one-one","kind":"node","lean":"MetaMathlibExt.multifoldDiamondCount_one_one_one","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#multifold-diamond-count-one-one-one","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"There is exactly one multifold partition diamond of shape $(1)$ with fold sum $1$ and link sum $1$. Equivalently, there is exactly one triple of nonnegative integers $b_0 \\le b_1 \\le b_2$ with $b_1 = 1$ and $b_0 + b_2 = 1$.","title":"Single-fold diamonds with fold sum 1 and link sum 1"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/multifold-diamond-count-two-one-two","kind":"node","lean":"MetaMathlibExt.multifoldDiamondCount_two_one_two","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#multifold-diamond-count-two-one-two","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"There are exactly two multifold partition diamonds of shape $(2)$ with fold sum $1$ and link sum $2$. Equivalently, there are exactly two quadruples $(b_0, b_1, b_2, b_3)$ of nonnegative integers with $b_0 \\le b_f \\le b_3$ for $f = 1, 2$, $b_1 + b_2 = 1$ and $b_0 + b_3 = 2$.","title":"Two-fold diamonds with fold sum 1 and link sum 2"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/multifold-diamond-series","kind":"node","lean":"MetaMathlibExt.multifoldDiamondSeries","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#multifold-diamond-series","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"fully_proved","summary":"For natural numbers $d_1, \\dots, d_M$, $\\sigma_{d_1, \\dots, d_M}(a, b)$ is the formal power series in $\\mathbb{Z}[[a, b]]$ whose coefficient of $a^i b^j$ is the number of multifold partition diamonds of shape $(d_1, \\dots, d_M)$ with fold sum $i$ and link sum $j$. That is, $\\sigma_{d_1, \\dots, d_M}(a, b) = \\sum a^{\\text{fold sum}} b^{\\text{link sum}}$, summed over all such diamonds. We write $\\sigma_{d,M}$ for the\u2026","title":"The bivariate series of multifold partition diamonds"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/euler-mahonian-succ-recursion","kind":"node","lean":"MetaMathlibExt.eulerMahonian_succ_recursion","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#euler-mahonian-succ-recursion","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"can_prove","summary":"For every commutative ring $R$, every natural number $d \\ge 1$ and all $x, y \\in R$, the Euler\u2013Mahonian polynomials $E_d$ and $E_{d+1}$ satisfy","title":"Recursion for the Euler\u2013Mahonian polynomials"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/partition-diamond-gen-fun","kind":"node","lean":"MetaMathlibExt.partitionDiamond_genFun","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#partition-diamond-gen-fun","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"can_prove","summary":"Let $d$ and $M$ be positive integers. Let $\\sigma_{d,M}(a, b)$ be the generating function of multifold partition diamonds of shape $(d, \\dots, d)$ ($M$ copies), where $a$ marks the fold sum and $b$ marks the link sum, and let $E_d(x, y) = \\sum_{\\tau} x^{\\mathrm{des}(\\tau)} y^{\\mathrm{maj}(\\tau)}$ be the Euler\u2013Mahonian polynomial, summed over the permutations $\\tau$ of $\\{1, \\dots, d\\}$. Then the following identity\u2026","title":"Product formula for equal-fold partition diamonds"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/multifold-partition-diamond-gen-fun","kind":"node","lean":"MetaMathlibExt.multifoldPartitionDiamond_genFun","parent":25911,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#multifold-partition-diamond-gen-fun","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"can_prove","summary":"Let $d_1, \\dots, d_M$ be positive integers with $M \\ge 1$, and let $\\omega_k = \\sum_{m=k+1}^{M} d_m$ for $0 \\le k \\le M$. Let $\\sigma_{d_1, \\dots, d_M}(a, b)$ be the generating function of multifold partition diamonds of shape $(d_1, \\dots, d_M)$, where $a$ marks the fold sum and $b$ marks the link sum, and let $E_d(x, y) = \\sum_{\\tau} x^{\\mathrm{des}(\\tau)} y^{\\mathrm{maj}(\\tau)}$ be the Euler\u2013Mahonian polynomial,\u2026","title":"Product formula for multifold partition diamonds"},{"id":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/schmidt-partition-diamond-gen-fun","kind":"node","lean":"MetaMathlibExt.schmidtPartitionDiamond_genFun","parent":25911,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of Schmidt type "],["math","\\(d\\)"],["text","-fold partition diamonds"]],"source":"Generating function of Schmidt type $d$-fold partition diamonds"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds/index.html#schmidt-partition-diamond-gen-fun","scope":"Combinatorics/integer-partitions/counting-partitions/multifold-partition-diamonds","status":"can_prove","summary":"Let $d \\ge 1$. A Schmidt type $d$-fold partition diamond is a finitely supported sequence $(a_p)_{p \\ge 0}$ of natural numbers such that $a_{k(d+1)} \\ge a_{k(d+1)+f} \\ge a_{(k+1)(d+1)}$ for all $k \\ge 0$ and all $f$ with $1 \\le f \\le d$; its weight is the sum of its links $a_{k(d+1)}$, $k \\ge 0$. Let $s_d(w)$ be the number of such diamonds of weight $w$, and let\u2026","title":"Generating function of Schmidt type $d$-fold partition diamonds"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/overpartitions.json","id":"Combinatorics/integer-partitions/counting-partitions/overpartitions","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","title":"Overpartitions"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/overpartition","kind":"node","lean":"Nat.Overpartition","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#overpartition","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"An overpartition of a natural number $n$ consists of a partition $p$ of $n$ together with a finite set of overlined part sizes, each of which occurs as a part of $p$.","title":"Overpartition"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/is-overlined","kind":"node","lean":"Nat.Overpartition.IsOverlined","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#is-overlined","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"A natural number $a$ is overlined in an overpartition $O$ of $n$ when $a$ belongs to the set of overlined part sizes of $O$.","title":"Overlined part size"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/is-overlined-iff","kind":"node","lean":"Nat.Overpartition.isOverlined_iff","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#is-overlined-iff","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For an overpartition $O$ of $n$ and a natural number $a$, $a$ is overlined in $O$ if and only if $a$ belongs to the set of overlined part sizes of $O$.","title":"Characterization of overlined sizes"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/is-overlined-mem-parts","kind":"node","lean":"Nat.Overpartition.isOverlined_mem_parts","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#is-overlined-mem-parts","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"If $a$ is overlined in an overpartition $O$ of $n$, then $a$ is a part of the underlying partition of $O$.","title":"Overlined sizes are parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/is-overlined-pos","kind":"node","lean":"Nat.Overpartition.isOverlined_pos","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#is-overlined-pos","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"If $a$ is overlined in an overpartition $O$ of $n$, then $a > 0$.","title":"Overlined sizes are positive"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/of-partition","kind":"node","lean":"Nat.Overpartition.ofPartition","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#of-partition","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For a partition $p$ of a natural number $n$, the overpartition of $n$ attached to $p$ has underlying partition $p$ and no overlined part sizes.","title":"The overpartition attached to a partition"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/of-partition-to-partition","kind":"node","lean":"Nat.Overpartition.ofPartition_toPartition","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#of-partition-to-partition","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For every partition $p$ of a natural number $n$, the underlying partition of the overpartition of $n$ attached to $p$ (underlying partition $p$, no part size overlined) is $p$.","title":"The underlying partition of the overpartition attached to a partition"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/of-partition-overlines","kind":"node","lean":"Nat.Overpartition.ofPartition_overlines","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#of-partition-overlines","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For every partition $p$ of a natural number $n$, the overpartition of $n$ attached to $p$ (underlying partition $p$, no part size overlined) has an empty set of overlined part sizes.","title":"The overpartition attached to a partition has no overlined sizes"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/not-is-overlined-of-partition","kind":"node","lean":"Nat.Overpartition.not_isOverlined_ofPartition","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#not-is-overlined-of-partition","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For every partition $p$ of $n$ and every natural number $a$, $a$ is not overlined in the overpartition whose underlying partition is $p$ and whose set of overlined part sizes is empty.","title":"No overlines in a plain partition"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/full","kind":"node","lean":"Nat.Overpartition.full","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#full","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For a partition $p$ of $n$, the fully overlined overpartition of $p$ is the overpartition of $n$ whose underlying partition is $p$ and whose set of overlined part sizes is the set of all distinct parts of $p$.","title":"Fully overlined overpartition"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/full-to-partition","kind":"node","lean":"Nat.Overpartition.full_toPartition","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#full-to-partition","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For every partition $p$ of $n$, the underlying partition of the fully overlined overpartition of $p$ is $p$.","title":"Underlying partition of the fully overlined overpartition"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/full-overlines","kind":"node","lean":"Nat.Overpartition.full_overlines","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#full-overlines","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For every partition $p$ of $n$, the set of overlined part sizes of the fully overlined overpartition of $p$ is the set of distinct parts of $p$.","title":"Overlined sizes of the fully overlined overpartition"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/is-overlined-full","kind":"node","lean":"Nat.Overpartition.isOverlined_full","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#is-overlined-full","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"For every partition $p$ of $n$ and every natural number $a$, $a$ is overlined in the fully overlined overpartition of $p$ if and only if $a$ is a part of $p$.","title":"Overlined sizes in the fully overlined overpartition"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/overlines-eq-empty-of-zero","kind":"node","lean":"Nat.Overpartition.overlines_eq_empty_of_zero","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#overlines-eq-empty-of-zero","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"Every overpartition of $0$ has no overlined part sizes.","title":"Overpartitions of zero have no overlined sizes"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions/eq-of-zero","kind":"node","lean":"Nat.Overpartition.eq_of_zero","parent":25933,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions/index.html#eq-of-zero","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions","status":"fully_proved","summary":"Any two overpartitions of $0$ are equal.","title":"Uniqueness of the overpartition of zero"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions.json","id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","title":"Minimal excludant of overpartitions"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/overpartition-count","kind":"node","lean":"MetaMathlibExt.overpartitionCount","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#overpartition-count","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"For a natural number $n$, $\\overline{p}(n)$ is the number of overpartitions of $n$, that is, of pairs consisting of a partition $p$ of $n$ and a set of overlined part sizes, each of which occurs as a part of $p$. For example, $\\overline{p}(3) = 8$.","title":"Number of overpartitions"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/non-overlined-count","kind":"node","lean":"MetaMathlibExt.nonOverlinedCount","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#non-overlined-count","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"For an overpartition $O$ of $n$ and a natural number $a$, the number of non-overlined parts of size $a$ in $O$ is the multiplicity of $a$ in the underlying partition of $O$, minus $1$ if $a$ is overlined in $O$ and minus $0$ otherwise.","title":"Number of non-overlined copies of a part size"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/least-gap-overpartition","kind":"node","lean":"MetaMathlibExt.leastGapOverpartition","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#least-gap-overpartition","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"For natural numbers $r \\ge 1$ and $n$ and an overpartition $O$ of $n$, $\\overline{\\mathrm{mex}}_r(O)$ is the least positive integer $k$ that occurs fewer than $r$ times among the non-overlined parts of $O$. Such a $k$ exists, since $n + 1$ is not a part of $O$. For $r = 0$, which the paper does not use, the value is $0$ by convention.","title":"Minimal excludant of an overpartition with a multiplicity bound"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/d3","kind":"node","lean":"MetaMathlibExt.D3","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#d3","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"For a natural number $n$, $D_3(n)$ is the number of triples $(A, B, C)$ of subsets of $\\{1, \\dots, n\\}$ such that the sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ add up to $n$. Here $A$, $B$, $C$ are the parts of the three colors of a partition of $n$ whose parts of each color are distinct.","title":"Three-colored distinct-part partitions"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/sigma-mex-overpartition","kind":"node","lean":"MetaMathlibExt.sigmaMexOverpartition","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#sigma-mex-overpartition","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"For natural numbers $r$ and $n$, $\\sigma_r\\overline{\\mathrm{mex}}(n)$ is the sum, over all overpartitions $\\pi$ of $n$, of $\\overline{\\mathrm{mex}}_r(\\pi)$, the least positive integer occurring fewer than $r$ times among the non-overlined parts of $\\pi$; for $r = 0$ the value $\\overline{\\mathrm{mex}}_0(\\pi)$ is $0$ by convention, so $\\sigma_0\\overline{\\mathrm{mex}}(n) = 0$. We write\u2026","title":"Sums of minimal excludants over overpartitions"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/overpartition-count-zero","kind":"node","lean":"MetaMathlibExt.overpartitionCount_zero","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#overpartition-count-zero","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"The number $\\overline{p}(0)$ of overpartitions of $0$ equals $1$: the empty overpartition is the only one.","title":"The number of overpartitions of zero"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/d3-zero","kind":"node","lean":"MetaMathlibExt.D3_zero","parent":25949,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-colored distinct-part partitions of "],["math","\\(0\\)"]],"source":"Three-colored distinct-part partitions of $0$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#d3-zero","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"The number $D_3(0)$ of partitions of $0$ into parts of three colors, with the parts of each color distinct, equals $1$: the empty partition is the only one.","title":"Three-colored distinct-part partitions of $0$"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/d3-one","kind":"node","lean":"MetaMathlibExt.D3_one","parent":25949,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-colored distinct-part partitions of "],["math","\\(1\\)"]],"source":"Three-colored distinct-part partitions of $1$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#d3-one","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"The number $D_3(1)$ of partitions of $1$ into parts of three colors, with the parts of each color distinct, equals $3$: the single part $1$ takes one of the three colors.","title":"Three-colored distinct-part partitions of $1$"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/d3-two","kind":"node","lean":"MetaMathlibExt.D3_two","parent":25949,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-colored distinct-part partitions of "],["math","\\(2\\)"]],"source":"Three-colored distinct-part partitions of $2$"},"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#d3-two","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"fully_proved","summary":"The number $D_3(2)$ of partitions of $2$ into parts of three colors, with the parts of each color distinct, equals $6$: the single part $2$ in one of three colors, or two parts $1$ in two different colors.","title":"Three-colored distinct-part partitions of $2$"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/sigma-mex-overpartition-eq-d3","kind":"node","lean":"MetaMathlibExt.sigmaMexOverpartition_eq_D3","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#sigma-mex-overpartition-eq-d3","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"For an overpartition $\\pi$, let $\\overline{\\mathrm{mex}}(\\pi)$ be the least positive integer that does not occur among the non-overlined parts of $\\pi$, and let $\\sigma\\overline{\\mathrm{mex}}(n)$ be the sum of $\\overline{\\mathrm{mex}}(\\pi)$ over all overpartitions $\\pi$ of $n$. For every positive integer $n$, $\\sigma\\overline{\\mathrm{mex}}(n) = D_3(n)$, the number of partitions of $n$ into distinct parts in three\u2026","title":"The overpartition mex sum counts three-colored distinct-part partitions"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/sigma-mex-overpartition-asymptotic","kind":"node","lean":"MetaMathlibExt.sigmaMexOverpartition_asymptotic","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#sigma-mex-overpartition-asymptotic","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"Let $\\sigma\\overline{\\mathrm{mex}}(n)$ be the sum, over all overpartitions $\\pi$ of $n$, of $\\overline{\\mathrm{mex}}(\\pi)$, the least positive integer that does not occur among the non-overlined parts of $\\pi$. Then $\\sigma\\overline{\\mathrm{mex}}(n) \\sim e^{\\pi\\sqrt{n}} / (8 n^{3/4})$ as $n \\to \\infty$, that is, the ratio of the two sides tends to $1$.","title":"Asymptotic formula for the overpartition mex sum"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/sigma-mex-overpartition-parity","kind":"node","lean":"MetaMathlibExt.sigmaMexOverpartition_parity","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#sigma-mex-overpartition-parity","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"For an overpartition $\\pi$, let $\\overline{\\mathrm{mex}}(\\pi)$ be the least positive integer that does not occur among the non-overlined parts of $\\pi$, and let $\\sigma\\overline{\\mathrm{mex}}(n)$ be the sum of $\\overline{\\mathrm{mex}}(\\pi)$ over all overpartitions $\\pi$ of $n$. For every positive integer $n$, $\\sigma\\overline{\\mathrm{mex}}(n)$ is odd if $n = j(j+1)/2$ is a triangular number for some natural number\u2026","title":"Parity of the overpartition mex sum"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/sigma-mex-overpartition-generating-funct","kind":"node","lean":"MetaMathlibExt.sigmaMexOverpartition_generating_function","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#sigma-mex-overpartition-generating-funct","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"Let $r$ be a positive integer and $q$ a complex number with $|q| < 1$. Write $(a;q)_\\infty = \\prod_{j \\ge 0} (1 - a q^j)$, and let $\\sigma_r\\overline{\\mathrm{mex}}(n)$ be the sum over all overpartitions $\\pi$ of $n$ of $\\overline{\\mathrm{mex}}_r(\\pi)$, the least positive integer occurring fewer than $r$ times among the non-overlined parts of $\\pi$. Then","title":"Generating function of the overpartition r-mex sums"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/sigma-mex-overpartition-r-asymptotic","kind":"node","lean":"MetaMathlibExt.sigmaMexOverpartition_r_asymptotic","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#sigma-mex-overpartition-r-asymptotic","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"Let $r$ be a positive integer and let $\\sigma_r\\overline{\\mathrm{mex}}(n)$ be the sum over all overpartitions $\\pi$ of $n$ of $\\overline{\\mathrm{mex}}_r(\\pi)$, the least positive integer occurring fewer than $r$ times among the non-overlined parts of $\\pi$. Then $\\sigma_r\\overline{\\mathrm{mex}}(n) \\sim e^{\\pi\\sqrt{n}} / (8 \\sqrt{r}\\, n^{3/4})$ as $n \\to \\infty$, that is, the ratio of the two sides tends to $1$.","title":"Asymptotic formula for the overpartition r-mex sums"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/sigma-mex-overpartition-power-density","kind":"node","lean":"MetaMathlibExt.sigmaMexOverpartition_power_density","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#sigma-mex-overpartition-power-density","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"Let $m$, $e$ and $k$ be natural numbers with $k \\ge 1$, let $r = 2^m 3^e$, and let $\\sigma_r\\overline{\\mathrm{mex}}(n)$ be the sum over all overpartitions $\\pi$ of $n$ of $\\overline{\\mathrm{mex}}_r(\\pi)$, the least positive integer occurring fewer than $r$ times among the non-overlined parts of $\\pi$. Then the natural numbers $n$ with $2^k \\mid \\sigma_r\\overline{\\mathrm{mex}}(n)$ have natural density $1$: the\u2026","title":"Density of overpartition r-mex sums divisible by powers of two"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/ingham-tauberian","kind":"node","lean":"MetaMathlibExt.ingham_tauberian","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#ingham-tauberian","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"Let $a : \\mathbb{N} \\to \\mathbb{R}$ be nonnegative and weakly increasing, let $A : \\mathbb{C} \\to \\mathbb{C}$ be a function, and let $\\alpha$, $\\beta$, $C$ be real numbers with $C > 0$. Suppose that for every complex $q$ with $|q| < 1$ the series $\\sum_{n \\ge 0} a(n) q^n$ is summable with sum $A(q)$, and that for every complex $q$ with $|q| > 1$ it is not summable. Suppose also that\u2026","title":"Ingham's Tauberian theorem"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/sigma-mex-overpartition-eq-sum-shifted-c","kind":"node","lean":"MetaMathlibExt.sigmaMexOverpartition_eq_sum_shifted_count","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#sigma-mex-overpartition-eq-sum-shifted-c","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"Let $n$ and $r$ be natural numbers with $r \\ge 1$, let $\\overline{p}(m)$ be the number of overpartitions of $m$, let $T_k = k(k+1)/2$ be the $k$-th triangular number, and let $\\sigma_r\\overline{\\mathrm{mex}}(n)$ be the sum over all overpartitions $\\pi$ of $n$ of $\\overline{\\mathrm{mex}}_r(\\pi)$, the least positive integer occurring fewer than $r$ times among the non-overlined parts of $\\pi$. Then","title":"The r-mex sum of overpartitions as a sum of overpartition counts"},{"id":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/eta-quotient-sigma-mex-overpartition-mod","kind":"node","lean":"MetaMathlibExt.etaQuotient_sigmaMexOverpartition_modEq","parent":25949,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions/index.html#eta-quotient-sigma-mex-overpartition-mod","scope":"Combinatorics/integer-partitions/counting-partitions/minimal-excludant-of-overpartitions","status":"can_prove","summary":"Let $r \\ge 1$ and $k \\ge 1$ be integers, and for $n \\ge 0$ let $\\sigma_r \\overline{\\mathrm{mex}}(n)$ be the sum, over all overpartitions of $n$, of the least positive integer occurring fewer than $r$ times among the non-overlined parts. Then there are integers $c_0, c_1, c_2, \\dots$ such that for every complex $q$ with $|q| < 1$, writing $f_d = (q^d; q^d)_\\infty = \\prod_{m \\ge 1} (1 - q^{dm})$,","title":"Eta-quotient congruence for sums of overpartition minimal excludants"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts.json","id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","title":"Overpartitions into odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-parts-count","kind":"node","lean":"MetaMathlibExt.overpartitionOddPartsCount","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-parts-count","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{po}(n)$ is the number of overpartitions of $n$ all of whose parts are odd.","title":"Overpartitions into odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-parts-count-zero","kind":"node","lean":"MetaMathlibExt.overpartitionOddPartsCount_zero","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-parts-count-zero","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"fully_proved","summary":"The number $\\mathrm{po}(0)$ of overpartitions of $0$ into odd parts equals $1$: the empty overpartition is the only one.","title":"Overpartitions of zero into odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-pentagonal-recurrence","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_pentagonal_recurrence","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-pentagonal-recurrence","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\mathrm{po}(j)$ be the number of overpartitions of $j$ into odd parts. For every natural number $n$, the sum $\\sum_k (-1)^k \\, \\mathrm{po}(n - k(3k+1)/2)$ over the integers $k$ with $k(3k+1)/2 \\le n$ equals $(-1)^{\\lceil m/2 \\rceil}$ if $n = m(3m+1)/2$ for some integer $m$, and $0$ otherwise.","title":"Pentagonal recurrence for overpartitions into odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-triangular-recurrence","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_triangular_recurrence","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-triangular-recurrence","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Write $\\mathrm{po}(m)$ for the number of overpartitions of $m$ into odd parts. For every natural number $n$, the sum $\\sum_{k \\ge 0,\\ k(k+1)/2 \\le n} (-1)^{\\lceil k/2 \\rceil} \\, \\mathrm{po}(n - k(k+1)/2)$ equals $1$ if $n = m(m+1)/2$ for some natural number $m$, and equals $0$ otherwise.","title":"Triangular recurrence for overpartitions into odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-square-recurrence","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_square_recurrence","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-square-recurrence","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Write $\\mathrm{po}(m)$ for the number of overpartitions of $m$ into odd parts. For every natural number $n$, the sum $\\mathrm{po}(n) + 2 \\sum_{k \\ge 1,\\ 2k^2 \\le n} (-1)^k \\, \\mathrm{po}(n - 2k^2)$ equals $1$ if $n = 0$, equals $2$ if $n = m^2$ for some integer $m \\ge 1$, and equals $0$ otherwise.","title":"Square recurrence for overpartitions into odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-eq-sum-partition-penta","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_eq_sum_partition_pentagonal","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-eq-sum-partition-penta","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\mathrm{po}(m)$ be the number of overpartitions of $m$ into odd parts and $p(m)$ the number of partitions of $m$. For every natural number $n$, $\\mathrm{po}(n) = \\sum_k (-1)^{\\lceil k/2 \\rceil} p(n - k(3k+1)/2)$, the sum running over the integers $k$ with $k(3k+1)/2 \\le n$.","title":"Overpartitions into odd parts from partitions and pentagonal numbers"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/partition-pentagonal-sum-even","kind":"node","lean":"MetaMathlibExt.partition_pentagonal_sum_even","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#partition-pentagonal-sum-even","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Write $p(m)$ for the number of partitions of $m$. For every positive integer $n$, the sum $\\sum_{k \\in \\mathbb{Z},\\ k(3k+1)/2 \\le n} p(n - k(3k+1)/2)$ is even.","title":"Parity form of Euler's pentagonal recurrence"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distinct-even-parts-count","kind":"node","lean":"MetaMathlibExt.distinctEvenPartsCount","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distinct-even-parts-count","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{ped}(n)$ is the number of partitions of $n$ in which every even part occurs exactly once; the odd parts are unrestricted.","title":"Partitions whose even parts are distinct"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distinct-odd-parts-count","kind":"node","lean":"MetaMathlibExt.distinctOddPartsCount","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distinct-odd-parts-count","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{pod}(n)$ is the number of partitions of $n$ in which every odd part occurs exactly once; the even parts are unrestricted.","title":"Partitions whose odd parts are distinct"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/parts-not-two-mod-four-count","kind":"node","lean":"MetaMathlibExt.partsNotTwoModFourCount","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#parts-not-two-mod-four-count","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"fully_proved","summary":"For a natural number $n$, $P_2(n)$ is the number of partitions of $n$ none of whose parts is congruent to $2$ modulo $4$.","title":"Partitions with no part congruent to 2 modulo 4"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distinct-parts-bipartition-count","kind":"node","lean":"MetaMathlibExt.distinctPartsBipartitionCount","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distinct-parts-bipartition-count","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"fully_proved","summary":"Let $\\operatorname{pd}(m)$ be the number of partitions of $m$ into distinct parts. For a natural number $n$, $q(n)$ is the number of ordered pairs of partitions into distinct parts whose sizes add up to $n$, that is, $q(n) = \\sum_{k=0}^{n} \\operatorname{pd}(k) \\operatorname{pd}(n-k)$.","title":"Bipartitions into distinct parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distinct-even-parts-distinct-odd-parts-p","kind":"node","lean":"MetaMathlibExt.distinctEvenParts_distinctOddParts_partsNotTwoModFour_bipartition_zero","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distinct-even-parts-distinct-odd-parts-p","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"fully_proved","summary":"The number $\\operatorname{ped}(0)$ of partitions of $0$ in which no even part is repeated, the number $\\operatorname{pod}(0)$ of partitions of $0$ in which no odd part is repeated, the number $P_2(0)$ of partitions of $0$ with no part congruent to $2$ modulo $4$, and the number $q(0)$ of ordered pairs of partitions into distinct parts with sizes adding up to $0$ are all equal to $1$, as only the empty partition (or\u2026","title":"Values at zero of the restricted partition counts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distinct-even-parts-triangular-recurrenc","kind":"node","lean":"MetaMathlibExt.distinctEvenParts_triangular_recurrence","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distinct-even-parts-triangular-recurrenc","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\operatorname{ped}(m)$ be the number of partitions of $m$ in which no even part is repeated. For every $n \\ge 0$, the sum $\\sum_{j \\ge 0} (-1)^{\\lceil j/2 \\rceil} \\operatorname{ped}(n - j(j+1)/2)$ over the $j$ with $j(j+1)/2 \\le n$ equals $1$ if $n = k(k+1)$ for some integer $k \\ge 0$, and $0$ otherwise.","title":"Merca's triangular recurrence for partitions with distinct even parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distinct-even-parts-square-recurrence","kind":"node","lean":"MetaMathlibExt.distinctEvenParts_square_recurrence","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distinct-even-parts-square-recurrence","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\operatorname{ped}(m)$ be the number of partitions of $m$ in which no even part is repeated. For every $n \\ge 0$, the sum $\\sum_j (-1)^j \\operatorname{ped}(n - 2j^2)$ over the integers $j$ with $2j^2 \\le n$ equals $1$ if $n = k(k+1)/2$ for some integer $k \\ge 0$, and $0$ otherwise.","title":"Merca's square recurrence for partitions with distinct even parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-eq-sum-distinct-odd-pa","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_eq_sum_distinctOddParts_triangular","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-eq-sum-distinct-odd-pa","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\mathrm{po}(m)$ be the number of overpartitions of $m$ into odd parts and $\\mathrm{pod}(m)$ the number of partitions of $m$ whose odd parts are distinct. For every natural number $n$, $\\mathrm{po}(n) = \\sum_k \\mathrm{pod}(n - k(k+1)/2)$, the sum running over the natural numbers $k$ with $k(k+1)/2 \\le n$.","title":"Overpartitions into odd parts from partitions with distinct odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distinct-odd-parts-triangular-sum-even","kind":"node","lean":"MetaMathlibExt.distinctOddParts_triangular_sum_even","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distinct-odd-parts-triangular-sum-even","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\operatorname{pod}(m)$ be the number of partitions of $m$ in which no odd part is repeated. For every $n > 0$, the sum $\\sum_{k \\ge 0} \\operatorname{pod}(n - k(k+1)/2)$ over the $k$ with $k(k+1)/2 \\le n$ is even.","title":"Evenness of a triangular sum of partition counts with distinct odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-eq-sum-overpartition-s","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_eq_sum_overpartition_square","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-eq-sum-overpartition-s","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\mathrm{po}(m)$ be the number of overpartitions of $m$ into odd parts and $\\overline{p}(m)$ the number of overpartitions of $m$. For every natural number $n$, $\\mathrm{po}(n) = \\sum_k (-1)^k \\, \\overline{p}(n - 2k^2)$, the sum running over the integers $k$ with $2k^2 \\le n$.","title":"Overpartitions into odd parts from overpartitions and squares"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-odd-eq-two-mul-sum-ove","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_odd_eq_two_mul_sum_overpartition","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-odd-eq-two-mul-sum-ove","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\mathrm{po}(m)$ be the number of overpartitions of $m$ into odd parts and $\\overline{p}(m)$ the number of overpartitions of $m$. For every natural number $n$, $\\mathrm{po}(2n+1) = 2 \\sum_k \\overline{p}(n - 2k(k+1))$, the sum running over the natural numbers $k$ with $2k(k+1) \\le n$.","title":"Overpartitions of odd numbers into odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-even-eq-sum-overpartit","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_even_eq_sum_overpartition","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-even-eq-sum-overpartit","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\mathrm{po}(m)$ be the number of overpartitions of $m$ into odd parts and $\\overline{p}(m)$ the number of overpartitions of $m$. For every natural number $n$, $\\mathrm{po}(2n) = \\overline{p}(n) + 2 \\sum_k \\overline{p}(n - 2k^2)$, the sum running over the integers $k \\ge 1$ with $2k^2 \\le n$.","title":"Overpartitions of even numbers into odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-eq-sum-parts-not-two-m","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_eq_sum_partsNotTwoModFour_triangular","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-eq-sum-parts-not-two-m","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\mathrm{po}(m)$ be the number of overpartitions of $m$ into odd parts and $P_2(m)$ the number of partitions of $m$ with no part congruent to $2$ modulo $4$. For every natural number $n$, $\\mathrm{po}(n) = \\sum_k P_2(n - k(k+1)/2)$, the sum running over the natural numbers $k$ with $k(k+1)/2 \\le n$.","title":"Overpartitions into odd parts from partitions with no part congruent to 2 modulo 4"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/parts-not-two-mod-four-triangular-sum-ev","kind":"node","lean":"MetaMathlibExt.partsNotTwoModFour_triangular_sum_even","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#parts-not-two-mod-four-triangular-sum-ev","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Write $P_2(m)$ for the number of partitions of $m$ with no part congruent to $2$ modulo $4$. For every positive integer $n$, the sum $\\sum_{k \\ge 0,\\ k(k+1)/2 \\le n} P_2(n - k(k+1)/2)$ is even.","title":"Parity of a triangular sum of partitions with no part congruent to 2 modulo 4"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distinct-parts-bipartition-eq-sum-partit","kind":"node","lean":"MetaMathlibExt.distinctPartsBipartition_eq_sum_partition_triangular","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distinct-parts-bipartition-eq-sum-partit","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $q(n)$ be the number of ordered pairs of partitions into distinct parts whose sizes add up to $n$, and let $p(m)$ be the number of partitions of $m$. For every $n \\ge 0$, $q(n) = \\sum_{k \\ge 0} p(n - k(k+1)/2)$, where the sum runs over the $k$ with $k(k+1)/2 \\le n$.","title":"Bipartitions into distinct parts from the partition function"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-pentagonal-sum-eq-odd","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_pentagonal_sum_eq_oddDistincts","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-pentagonal-sum-eq-odd","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\mathrm{po}(m)$ be the number of overpartitions of $m$ into odd parts and $\\mathrm{pdo}(m)$ the number of partitions of $m$ into distinct odd parts. For every natural number $n$, the sum $\\sum_k (-1)^k \\, \\mathrm{po}(n - k(3k+1)/2)$ over the integers $k$ with $k(3k+1)/2 \\le n$ equals the sum $\\sum_k (-1)^k \\, \\mathrm{pdo}(n - k(3k+1))$ over the integers $k$ with $k(3k+1) \\le n$.","title":"Overpartitions into odd parts and partitions into distinct odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/odd-distincts-pentagonal-recurrence","kind":"node","lean":"MetaMathlibExt.oddDistincts_pentagonal_recurrence","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#odd-distincts-pentagonal-recurrence","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\operatorname{pdo}(m)$ be the number of partitions of $m$ into distinct odd parts. For every $n \\ge 0$, the sum $\\sum_k (-1)^k \\operatorname{pdo}(n - k(3k+1))$ over the integers $k$ with $k(3k+1) \\le n$ equals $(-1)^{\\lceil m/2 \\rceil}$ if $n = m(3m+1)/2$ for some integer $m$, and $0$ otherwise.","title":"Pentagonal recurrence for partitions into distinct odd parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/overpartition-odd-triangular-sum-eq-dist","kind":"node","lean":"MetaMathlibExt.overpartitionOdd_triangular_sum_eq_distincts","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#overpartition-odd-triangular-sum-eq-dist","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Write $\\mathrm{po}(m)$ for the number of overpartitions of $m$ into odd parts and $\\mathrm{pd}(m)$ for the number of partitions of $m$ into distinct parts. For every natural number $n$, $\\sum_{k \\ge 0,\\ k(k+1)/2 \\le n} (-1)^{\\lceil k/2 \\rceil} \\, \\mathrm{po}(n - k(k+1)/2) = \\sum_{k \\in \\mathbb{Z},\\ k(3k+1) \\le n} (-1)^k \\, \\mathrm{pd}(n - k(3k+1))$.","title":"Overpartitions into odd parts and partitions into distinct parts"},{"id":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/distincts-pentagonal-recurrence","kind":"node","lean":"MetaMathlibExt.distincts_pentagonal_recurrence","parent":25968,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts/index.html#distincts-pentagonal-recurrence","scope":"Combinatorics/integer-partitions/counting-partitions/overpartitions-into-odd-parts","status":"can_prove","summary":"Let $\\operatorname{pd}(m)$ be the number of partitions of $m$ into distinct parts. For every $n \\ge 0$, the sum $\\sum_k (-1)^k \\operatorname{pd}(n - k(3k+1))$ over the integers $k$ with $k(3k+1) \\le n$ equals $1$ if $n = m(m+1)/2$ for some integer $m \\ge 0$, and $0$ otherwise.","title":"Pentagonal recurrence for partitions into distinct parts"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the.json","id":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the","status":"fully_proved","title":"Partition function and the pentagonal number theorem"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/partition-function","kind":"node","lean":"MetaMathlibExt.partitionFunction","parent":25995,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/index.html#partition-function","scope":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the","status":"fully_proved","summary":"For a natural number $n$, $p(n)$ is the number of partitions of $n$, that is, of ways of writing $n$ as a sum of positive integers, disregarding order.","title":"The partition function"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/partition-function-zero","kind":"node","lean":"MetaMathlibExt.partitionFunction_zero","parent":25995,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/index.html#partition-function-zero","scope":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the","status":"fully_proved","summary":"Let $p(n)$ be the number of partitions of $n$. Then $p(0) = 1$: the only partition of $0$ is the empty one.","title":"The partition number of zero"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/partition-function-one","kind":"node","lean":"MetaMathlibExt.partitionFunction_one","parent":25995,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/index.html#partition-function-one","scope":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the","status":"fully_proved","summary":"Let $p(n)$ be the number of partitions of $n$. Then $p(1) = 1$: the only partition of $1$ is $1$ itself.","title":"The partition number of one"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/mk-partition-function-eq-gen-fun","kind":"node","lean":"MetaMathlibExt.mk_partitionFunction_eq_genFun","parent":25995,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/index.html#mk-partition-function-eq-gen-fun","scope":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the","status":"fully_proved","summary":"Let $p(n)$ be the number of partitions of $n$. Over the integers, $\\sum_{n \\ge 0} p(n)\\, x^n$ equals the weighted partition generating function $\\sum_{n \\ge 0} \\sum_{\\lambda} \\prod_{i} f(i, c_i(\\lambda))\\, x^n$ with all weights $f(i, c) = 1$, where $\\lambda$ runs over the partitions of $n$, $i$ over the distinct parts of $\\lambda$, and $c_i(\\lambda)$ is the number of times $i$ occurs in $\\lambda$.","title":"Partition generating function with unit weights"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/mk-partition-function-mul-pentagonal-ser","kind":"node","lean":"MetaMathlibExt.mk_partitionFunction_mul_pentagonalSeries","parent":25995,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/index.html#mk-partition-function-mul-pentagonal-ser","scope":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the","status":"fully_proved","summary":"Let $p(n)$ be the number of partitions of $n$. Over the integers, $\\left(\\sum_{n \\ge 0} p(n)\\, x^n\\right) \\cdot \\sum_{k \\in \\mathbb{Z}} (-1)^k x^{k(3k-1)/2} = 1$. By the pentagonal number theorem the second factor equals $\\prod_{i \\ge 1} (1 - x^i)$.","title":"Euler's identity for the partition generating function"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/pentagonal-number-theorem","kind":"node","lean":"MetaMathlibExt.pentagonal_number_theorem","parent":25995,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the/index.html#pentagonal-number-theorem","scope":"Combinatorics/integer-partitions/counting-partitions/partition-function-and-the-pentagonal-number-the","status":"fully_proved","summary":"Let $p(n)$ be the number of partitions of $n$, and for an integer $k$ let $g(k) = k(3k - 1)/2$ be the generalized pentagonal number of $k$. For every natural number $n \\ge 1$, $p(n) = \\sum_{k} (-1)^{k+1}\\, p(n - g(k))$, where the sum runs over the nonzero integers $k$ with $g(k) \\le n$.","title":"Euler's pentagonal recurrence for the partition function"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/partition-polynomials.json","id":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-polynomials/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials","status":"fully_proved","title":"Partition polynomials"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials/partition-polynomial","kind":"node","lean":"MetaMathlibExt.partitionPolynomial","parent":26002,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-polynomials/index.html#partition-polynomial","scope":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials","status":"fully_proved","summary":"For a partition $\\lambda$ of a natural number $n$ with parts $\\lambda_1, \\dots, \\lambda_\\ell$, listed with multiplicity, the (unnormalized) partition polynomial of $\\lambda$ is $f_\\lambda(x) = x^{\\lambda_1} + \\dots + x^{\\lambda_\\ell}$, a polynomial with natural number coefficients.","title":"Partition polynomial"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials/coeff-partition-polynomial","kind":"node","lean":"MetaMathlibExt.coeff_partitionPolynomial","parent":26002,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-polynomials/index.html#coeff-partition-polynomial","scope":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials","status":"fully_proved","summary":"For every partition $\\lambda$ of a natural number $n$ and every $j \\in \\mathbb{N}$, the coefficient of $x^j$ in the partition polynomial $f_\\lambda$ equals the number of parts of $\\lambda$ equal to $j$.","title":"Coefficients of the partition polynomial"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials/eval-partition-polynomial-at-one","kind":"node","lean":"MetaMathlibExt.eval_partitionPolynomial_at_one","parent":26002,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-polynomials/index.html#eval-partition-polynomial-at-one","scope":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials","status":"fully_proved","summary":"For every partition $\\lambda$ of a natural number $n$, the value $f_\\lambda(1)$ of the partition polynomial $f_\\lambda$ at $1$ equals the number of parts of $\\lambda$, counted with multiplicity.","title":"The partition polynomial at one"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials/partition-polynomial-iterated-deriv-eq-s","kind":"node","lean":"MetaMathlibExt.partitionPolynomial_iteratedDeriv_eq_sum_stirlingSecond_general","parent":26002,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-polynomials/index.html#partition-polynomial-iterated-deriv-eq-s","scope":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials","status":"fully_proved","summary":"Let $k$ and $d$ be natural numbers, let $m \\colon \\mathbb{N} \\to \\mathbb{N}$, write $m_i = m(i)$, and let $f(y) = \\sum_{i=1}^{k} m_i y^i$ for real $y$. Then for every real $x \\ne 0$,","title":"Stirling recursion for derivatives, for arbitrary multiplicities"},{"id":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials/partition-polynomial-iterated-deriv-eq-s-2","kind":"node","lean":"MetaMathlibExt.partitionPolynomial_iteratedDeriv_eq_sum_stirlingSecond","parent":26002,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/partition-polynomials/index.html#partition-polynomial-iterated-deriv-eq-s-2","scope":"Combinatorics/integer-partitions/counting-partitions/partition-polynomials","status":"fully_proved","summary":"Let $k \\ge 1$, let $m \\colon \\mathbb{N} \\to \\mathbb{N}$, and write $m_i = m(i)$. Assume $m_k \\ne 0$ and $m_i = 0$ for every $i$ outside $\\{1, \\dots, k\\}$, so that $m$ lists the part multiplicities of a partition $\\lambda = \\langle 1^{m_1}, \\dots, k^{m_k} \\rangle$ with largest part $k$. Let $f(y) = \\sum_{i=1}^{k} m_i y^i$ for real $y$. Then for every natural number $d \\le k$ and every real $x \\ne 0$,","title":"Derivatives of the partition polynomial via Stirling numbers"},{"data_url":"scopes/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions.json","id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","kind":"scope","navigation":"module","parent":25897,"result":false,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","title":"Restricted Gaussian partitions"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/gaussian-partition","kind":"node","lean":"MetaMathlibExt.GaussianPartition","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#gaussian-partition","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"A Gaussian partition is represented by its list of subspace dimensions, which is a nonincreasing list of natural numbers.","title":"Gaussian partition"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/block","kind":"node","lean":"MetaMathlibExt.Block","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#block","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"A block is a pair $(d, m)$ of natural numbers, read as a part size $d$ of an integer partition together with its multiplicity $m$. No condition is imposed on a single block.","title":"Block"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/branch-step","kind":"node","lean":"MetaMathlibExt.BranchStep","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#branch-step","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"A branch step is a triple $(a, b, c)$ of natural numbers. It describes one simultaneous split level: for the ambient parameter $q$, $c$ copies of the parent dimension $a + b$ become $c$ copies of $a$ and $c q^a$ copies of $b$.","title":"Branch step"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/basic-count","kind":"node","lean":"MetaMathlibExt.basicCount","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#basic-count","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"Let $q$ be a natural number and $(d_0, m_0), \\dots, (d_{k-1}, m_{k-1})$ a list of blocks, and put $p_i = \\sum_{j < i} m_j d_j$. The basic multiplicity of block $i$ is $N_i = \\sum_{j=0}^{m_i - 1} q^{p_i + j d_i}$ when $i < k$, and $N_i = 0$ when $i \\ge k$.","title":"Basic block multiplicity"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/branch-bound-exp","kind":"node","lean":"MetaMathlibExt.branchBoundExp","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#branch-bound-exp","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"Let $(d_0, m_0), \\dots, (d_{k-1}, m_{k-1})$ be a list of blocks and put $p_i = \\sum_{j < i} m_j d_j$. The first-split exponent of block $i$ is $u_i = p_i + (m_i - 1) d_i$ when $i < k$, where $m_i - 1$ is taken to be $0$ if $m_i = 0$, and $u_i = 0$ when $i \\ge k$.","title":"First-split exponent"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/remove-first-npos","kind":"node","lean":"MetaMathlibExt.removeFirstNPos","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#remove-first-npos","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"Let $x$ and $c$ be natural numbers and let $l$ be a list of natural numbers. For $c = 0$ the result is the pair of the empty list and $l$. For $c \\ge 1$, if $l$ has at least $c$ entries equal to $x$, the result is the pair of the list of entries different from $x$ that come before the $c$-th entry equal to $x$, in their order in $l$, and the part of $l$ after that entry. If $l$ has fewer than $c$ entries equal to\u2026","title":"Removing the first occurrences of an entry"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/apply-split","kind":"node","lean":"MetaMathlibExt.applySplit","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#apply-split","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"For natural numbers $q, a, b, c$ and a list $l$ of natural numbers, the split replacement is defined exactly when $l$ has at least $c$ entries equal to $a + b$. It removes the last $c$ entries of $l$ equal to $a + b$ and, at the position of the first of these removed entries, inserts $c$ copies of $a$ followed by $c q^a$ copies of $b$; the entries of $l$ before that position stay in front, and the entries after it\u2026","title":"Split replacement in a dimension list"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/block-final-dims","kind":"node","lean":"MetaMathlibExt.blockFinalDims","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#block-final-dims","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"Let $q$ be a natural number, $(d_0, m_0), \\dots, (d_{k-1}, m_{k-1})$ a list of blocks and $C$ a list of chains of branch steps. The final dimension list of block $i$ is defined when $C$ has an entry $s_1, \\dots, s_r$ at index $i$: start from $N_i$ copies of $d_i$, where $N_i$ is the basic multiplicity of block $i$ (the empty list when $i \\ge k$), and for $t = 1, \\dots, r$ in order apply the split replacement of\u2026","title":"Final dimensions of a block"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/stage-segments","kind":"node","lean":"MetaMathlibExt.stageSegments","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#stage-segments","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"Let $q$ be a natural number, let $(d_0, m_0), \\dots, (d_{\\ell-1}, m_{\\ell-1})$ be blocks, let $C_0, C_1, \\dots$ be chains of branch steps, let $F_0, F_1, \\dots$ be lists of natural numbers, and let $i$ and $k$ be natural numbers; blocks, chains and lists are indexed from $0$. With $p_h = \\sum_{g < h} m_g d_g$, the basic segment of block $h$ is the list of $\\sum_{j=0}^{m_h - 1} q^{p_h + j d_h}$ copies of $d_h$. A\u2026","title":"Stage segments of a restricted Gaussian partition"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/chain-adj-valid","kind":"node","lean":"MetaMathlibExt.ChainAdjValid","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#chain-adj-valid","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"A chain of branch steps $(a_1, b_1, c_1), \\dots, (a_r, b_r, c_r)$ is adjacently valid when for every $j$ with $1 \\le j < r$ we have $a_{j+1} + b_{j+1} = a_j$, $1 \\le b_{j+1} \\le a_{j+1}$ and $1 \\le c_{j+1} \\le c_j$. Only consecutive steps are related, and nothing is required of the first step alone; in particular every chain with at most one step is adjacently valid.","title":"Adjacently valid chain"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/branch-valid","kind":"node","lean":"MetaMathlibExt.BranchValid","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#branch-valid","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"Let $q$ be a natural number, $(d_0, m_0), \\dots, (d_{k-1}, m_{k-1})$ a list of blocks and $i$ an index, and let $u_i$ and $N_i$ be the first-split exponent and the basic multiplicity of block $i$, with $d_i$ read as $0$ when $i \\ge k$. A chain of branch steps $(a_1, b_1, c_1), \\dots, (a_r, b_r, c_r)$ is a valid branch for block $i$ when either $r = 0$, or $r \\ge 1$, the first step satisfies $1 \\le b_1 \\le a_1$,\u2026","title":"Valid branch"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/is-restricted-gaussian-partition","kind":"node","lean":"MetaMathlibExt.IsRestrictedGaussianPartition","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#is-restricted-gaussian-partition","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"Let $q$ and $n$ be natural numbers and let $G$ be a Gaussian partition. Then $G$ is a restricted Gaussian partition for $q$ and $n$ when $q$ is a prime power, $n > 0$, and there exist blocks $(d_1, m_1), \\dots, (d_\\ell, m_\\ell)$ and, for each block $i$, a chain $C_i$ of branch steps $(a, b, c)$, with the following properties.","title":"Restricted Gaussian partition"},{"id":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/basic-ancestor-is-restricted","kind":"node","lean":"MetaMathlibExt.basic_ancestor_is_restricted","parent":26008,"result":true,"route":"../roadmap/Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions/index.html#basic-ancestor-is-restricted","scope":"Combinatorics/integer-partitions/counting-partitions/restricted-gaussian-partitions","status":"fully_proved","summary":"Let $q$ be a prime power and $n > 0$. Let $(d_0, m_0), \\dots, (d_{k-1}, m_{k-1})$ be blocks with $d_i \\ge 1$ and $m_i \\ge 1$ for all $i$, $d_0 > d_1 > \\dots > d_{k-1}$ and $m_0 d_0 + \\dots + m_{k-1} d_{k-1} = n$. Put $p_i = \\sum_{j < i} m_j d_j$, and let the basic dimension list be the concatenation, for $i = 0, \\dots, k - 1$ in order, of $\\sum_{j=0}^{m_i - 1} q^{p_i + j d_i}$ copies of $d_i$. If this list is\u2026","title":"Basic ancestors are restricted"},{"data_url":"scopes/Combinatorics/binomial-coefficients.json","id":"Combinatorics/binomial-coefficients","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients","status":"can_prove","summary":"Binomial, multinomial and generalized binomial coefficients and closed-form evaluation of finite sums built from them (alternating, reciprocal, harmonic-weighted, hypergeometric), including Ap\u00e9ry-like sequences such as Franel, Domb and Zagier numbers; q-binomials go to symmetric functions and q-analogues, Catalan-type numbers to Catalan numbers, Stirling-weighted sums to Stirling numbers.","title":"Binomial coefficients and identities"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums.json","id":"Combinatorics/binomial-coefficients/binomial-sums","kind":"scope","parent":26022,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums","status":"can_prove","summary":"Evaluations and bounds of sums of products of binomial coefficients: Chu\u2013Vandermonde, Dixon, Knuth's old sum, Staver, Frisch, Klamkin, hockey-stick and Filipponi identities, partial-sum bounds, multisection identities, and Franel, Domb, Zagier and Catalan\u2013Larcombe\u2013French numbers defined by binomial sums.","title":"Binomial sums and hypergeometric identities"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/binomial-double-sum-eq-pow-sub-fib","kind":"node","lean":"MetaMathlibExt.binomial_double_sum_eq_pow_sub_fib","parent":26023,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/index.html#binomial-double-sum-eq-pow-sub-fib","scope":"Combinatorics/binomial-coefficients/binomial-sums","status":"fully_proved","summary":"For every natural number $n$,","title":"A binomial double sum equal to a power of two minus a Fibonacci number"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/alternating-binomial-partial-sum","kind":"node","lean":"MetaMathlibExt.alternating_binomial_partial_sum","parent":26023,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/index.html#alternating-binomial-partial-sum","scope":"Combinatorics/binomial-coefficients/binomial-sums","status":"fully_proved","summary":"For every sequence $a_0, a_1, a_2, \\dots$ of complex numbers and every natural number $n$,","title":"Alternating binomial sums of partial sums"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/catalan-larcombe-french-number","kind":"node","lean":"MetaMathlibExt.catalanLarcombeFrenchNumber","parent":26023,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/index.html#catalan-larcombe-french-number","scope":"Combinatorics/binomial-coefficients/binomial-sums","status":"fully_proved","summary":"For a natural number $n$, the $n$th Catalan\u2013Larcombe\u2013French number (OEIS A053175) is","title":"Catalan\u2013Larcombe\u2013French numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/sum-choose-two-mul-add","kind":"node","lean":"MathlibExt.Combinatorics.HockeyStickWanted.sum_choose_two_mul_add","parent":26023,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/index.html#sum-choose-two-mul-add","scope":"Combinatorics/binomial-coefficients/binomial-sums","status":"can_prove","summary":"For all natural numbers $n$ and $k$,","title":"Generalized hockey stick identity of Lynch and Weselcouch"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/strehl-franel-binomial-identity","kind":"node","lean":"MetaMathlibExt.strehl_franel_binomial_identity","parent":26023,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/index.html#strehl-franel-binomial-identity","scope":"Combinatorics/binomial-coefficients/binomial-sums","status":"fully_proved","summary":"For every natural number $n$,","title":"Strehl's identity for the Franel numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/chu-vandermonde-terminating-specializati","kind":"node","lean":"MetaMathlibExt.chu_vandermonde_terminating_specialization","parent":26023,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/index.html#chu-vandermonde-terminating-specializati","scope":"Combinatorics/binomial-coefficients/binomial-sums","status":"fully_proved","summary":"For every natural number $n \\ge 1$, the following sum of rational numbers vanishes:","title":"Terminating Chu\u2013Vandermonde specialization"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values.json","id":"Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values","status":"fully_proved","title":"Binomial sums involving absolute values"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values/brent-osborn-binomial-absolute-moment","kind":"node","lean":"MetaMathlibExt.brent_osborn_binomial_absolute_moment","parent":26030,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values/index.html#brent-osborn-binomial-absolute-moment","scope":"Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values","status":"fully_proved","summary":"For every natural number $n$,","title":"Brent\u2013Osborn binomial absolute moment identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values/double-binomial-sum-eq-single-binomial-s","kind":"node","lean":"MetaMathlibExt.double_binomial_sum_eq_single_binomial_sum","parent":26030,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values/index.html#double-binomial-sum-eq-single-binomial-s","scope":"Combinatorics/binomial-coefficients/binomial-sums/binomial-sums-involving-absolute-values","status":"fully_proved","summary":"For all natural numbers $\\beta$, $m$ and $n$,","title":"Reduction of a double binomial absolute-value sum"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients.json","id":"Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients","status":"fully_proved","title":"Bounds on partial sums of binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients/pow-mul-sum-choose-le","kind":"node","lean":"Real.pow_mul_sum_choose_le","parent":26033,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients/index.html#pow-mul-sum-choose-le","scope":"Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients","status":"fully_proved","summary":"A weighted partial binomial sum is bounded by the full binomial expansion.","title":"Weighted partial binomial sum bounded by the full expansion"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients/pow-mul-sum-choose-le-exp","kind":"node","lean":"Real.pow_mul_sum_choose_le_exp","parent":26033,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients/index.html#pow-mul-sum-choose-le-exp","scope":"Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients","status":"fully_proved","summary":"Exponential bound on the weighted partial binomial sum, including the zero-weight boundary.","title":"Exponential bound on a weighted partial binomial sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients/sum-choose-le-pow","kind":"node","lean":"Real.sum_choose_le_pow","parent":26033,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of subsets of size at most "],["math","\\(r\\)"],["text"," is at most "],["math","\\((3D/r)^r\\)"]],"source":"Number of subsets of size at most $r$ is at most $(3D/r)^r$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients/index.html#sum-choose-le-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/bounds-on-partial-sums-of-binomial-coefficients","status":"fully_proved","summary":"For `1 \u2264 r \u2264 D`, the number of subsets of size at most `r` is at most `(3 * D / r) ^ r`.","title":"Number of subsets of size at most $r$ is at most $(3D/r)^r$"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity.json","id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","title":"Dixon's identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/dixon-alternating-cubic-binomial-sum","kind":"node","lean":"MetaMathlibExt.dixon_alternating_cubic_binomial_sum","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#dixon-alternating-cubic-binomial-sum","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For every natural number $n$, $\\sum_{k=0}^{2n} (-1)^k \\binom{2n}{k}^3 = (-1)^n \\binom{2n}{n} \\binom{3n}{2n}$.","title":"Dixon's alternating cubic binomial sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/ext-fact","kind":"node","lean":"MetaMathlibExt.extFact","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#ext-fact","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For an integer $n$, the extended factorial of $n$ is the rational number $n!$ when $n \\ge 0$ and $0$ when $n < 0$.","title":"Extended factorial"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/ext-fact-of-neg","kind":"node","lean":"MetaMathlibExt.extFact_of_neg","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#ext-fact-of-neg","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For every integer $n < 0$, the extended factorial of $n$ is $0$.","title":"The extended factorial vanishes at negative integers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/ext-fact-nat-cast","kind":"node","lean":"MetaMathlibExt.extFact_natCast","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#ext-fact-nat-cast","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For every natural number $n$, the extended factorial of $n$ equals the ordinary factorial $n!$.","title":"The extended factorial of a natural number"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/inv-fact","kind":"node","lean":"MetaMathlibExt.invFact","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#inv-fact","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For an integer $n$, the inverse factorial of $n$ is the rational number $1/n!$ when $n \\ge 0$ and $0$ when $n < 0$.","title":"Inverse factorial on the integers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/inv-fact-of-neg","kind":"node","lean":"MetaMathlibExt.invFact_of_neg","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#inv-fact-of-neg","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For every integer $n < 0$, the inverse factorial of $n$ is $0$.","title":"The inverse factorial vanishes at negative integers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/inv-fact-nat-cast","kind":"node","lean":"MetaMathlibExt.invFact_natCast","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#inv-fact-nat-cast","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For every natural number $n$, the inverse factorial of $n$, viewed as an integer, equals $(n!)^{-1}$ in $\\mathbb{Q}$.","title":"The inverse factorial at natural numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/inv-fact-shift","kind":"node","lean":"MetaMathlibExt.invFact_shift","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#inv-fact-shift","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For every integer $n$, the inverse factorial of $n$ equals $n + 1$ times the inverse factorial of $n + 1$, as rational numbers.","title":"Shift identity for the inverse factorial"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/int-binom","kind":"node","lean":"MetaMathlibExt.intBinom","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#int-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For an integer $m$ write $m!$ for the extended factorial and $\\frac{1}{m!}$ for the inverse factorial; both are $0$ when $m < 0$. For integers $n$ and $k$, the integer binomial coefficient is the rational number $\\binom{n}{k} = n! \\cdot \\frac{1}{k!} \\cdot \\frac{1}{(n - k)!}$.","title":"Integer binomial coefficient"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/int-binom-nat-cast","kind":"node","lean":"MetaMathlibExt.intBinom_natCast","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#int-binom-nat-cast","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For natural numbers $k \\le n$, the integer binomial coefficient of $n$ and $k$ equals the ordinary binomial coefficient $\\binom{n}{k}$.","title":"Integer binomial coefficients of natural numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/int-binom-of-neg-right","kind":"node","lean":"MetaMathlibExt.intBinom_of_neg_right","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#int-binom-of-neg-right","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For integers $n$ and $k$ with $k < 0$, the integer binomial coefficient $\\binom{n}{k}$ is $0$.","title":"Integer binomial coefficients vanish at negative lower index"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/int-binom-of-lt","kind":"node","lean":"MetaMathlibExt.intBinom_of_lt","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#int-binom-of-lt","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For integers $n < k$, the integer binomial coefficient $\\binom{n}{k}$ is $0$.","title":"Integer binomial coefficients vanish when the lower index exceeds the upper"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/dixon-summand","kind":"node","lean":"MetaMathlibExt.dixonSummand","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#dixon-summand","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"Write $\\binom{n}{m}$ for the integer binomial coefficient of integers $n$ and $m$. For integers $a_1, a_2, a_3$ and $k$, the Dixon summand is the rational number $(-1)^k \\binom{2a_1}{a_1 + k} \\binom{2a_2}{a_2 + k} \\binom{2a_3}{a_3 + k}$.","title":"Term of Dixon's alternating sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/dixon-bound","kind":"node","lean":"MetaMathlibExt.dixonBound","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#dixon-bound","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For integers $a_1, a_2, a_3$, the Dixon bound is $\\max(|a_1|, |a_2|, |a_3|)$.","title":"Bound of the Dixon summation window"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/upsilon","kind":"node","lean":"MetaMathlibExt.upsilon\u2083","parent":26037,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Dixon sum "],["math","\\(\\upsilon_3\\)"]],"source":"The Dixon sum $\\upsilon_3$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#upsilon","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For integers $a_1, a_2, a_3$, let $B = \\max(|a_1|, |a_2|, |a_3|)$. Then $\\upsilon_3(a_1, a_2, a_3)$ is the rational number","title":"The Dixon sum $\\upsilon_3$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/dixon-summation","kind":"node","lean":"MetaMathlibExt.dixon_summation","parent":26037,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity/index.html#dixon-summation","scope":"Combinatorics/binomial-coefficients/binomial-sums/dixon-s-identity","status":"fully_proved","summary":"For an integer $m$ write $m!$ for the extended factorial and $\\frac{1}{m!}$ for the inverse factorial; both are $0$ when $m < 0$. For all integers $a_1, a_2, a_3$, the Dixon sum $\\Upsilon_3(a_1, a_2, a_3)$ equals","title":"Dixon's summation theorem"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum.json","id":"Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum","status":"fully_proved","title":"Filipponi's binomial sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum/filipponi-binomial-sum-formula-general","kind":"node","lean":"MetaMathlibExt.filipponi_binomial_sum_formula_general","parent":26054,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum/index.html#filipponi-binomial-sum-formula-general","scope":"Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum","status":"fully_proved","summary":"For natural numbers $r$ and $s$ let $f_r(s) = \\sum_{j=0}^{s} \\binom{r-j}{j}$, where $\\binom{r-j}{j}$ is read as $0$ when $j > r$. Then for all natural numbers $n$, $h$ and $k$, $f_{n+2h}(k+h) = \\sum_{j=0}^{h} \\binom{h}{j} f_{n+j}(k+j)$.","title":"Filipponi's binomial-sum formula, without range conditions"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum/filipponi-binomial-sum-formula","kind":"node","lean":"MetaMathlibExt.filipponi_binomial_sum_formula","parent":26054,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum/index.html#filipponi-binomial-sum-formula","scope":"Combinatorics/binomial-coefficients/binomial-sums/filipponi-s-binomial-sum","status":"fully_proved","summary":"For natural numbers $r$ and $s$ let $f_r(s) = \\sum_{j=0}^{s} \\binom{r-j}{j}$, where $\\binom{r-j}{j}$ is read as $0$ when $j > r$. If $n$, $h$ and $k$ are natural numbers with $h \\le n$ and $2k \\le n - h$, then $f_{n+2h}(k+h) = \\sum_{j=0}^{h} \\binom{h}{j} f_{n+j}(k+j)$.","title":"Filipponi's binomial-sum formula"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum.json","id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","title":"Knuth's old sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/gamma-binom","kind":"node","lean":"MetaMathlibExt.gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"fully_proved","summary":"For complex numbers $u$ and $v$, the generalized binomial coefficient is","title":"Generalized binomial coefficient of complex arguments"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/gamma-binom-nat-cast","kind":"node","lean":"MetaMathlibExt.gammaBinom_natCast","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#gamma-binom-nat-cast","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the generalized binomial coefficient of complex arguments at $(n, k)$, namely $\\Gamma(n+1)/(\\Gamma(k+1)\\,\\Gamma(n-k+1))$, equals the ordinary binomial coefficient $\\binom{n}{k}$; in particular it is $0$ when $k > n$.","title":"The generalized binomial coefficient at natural arguments"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/gamma-binom-zero-right","kind":"node","lean":"MetaMathlibExt.gammaBinom_zero_right","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#gamma-binom-zero-right","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"fully_proved","summary":"For every complex number $u$ with $\\operatorname{Re} u > -1$, the generalized binomial coefficient of complex arguments $\\binom{u}{0} = \\Gamma(u+1)/(\\Gamma(1)\\,\\Gamma(u+1))$ equals $1$.","title":"The generalized binomial coefficient with lower argument zero"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/integral-cos-half-cpow","kind":"node","lean":"MetaMathlibExt.integral_cos_half_cpow","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of a complex power of "],["math","\\(\\cos(x/2)\\)"],["text"," over "],["math","\\((0, \\pi)\\)"]],"source":"Integral of a complex power of $\\cos(x/2)$ over $(0, \\pi)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#integral-cos-half-cpow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every complex number $u$ with $\\operatorname{Re} u > -1$,","title":"Integral of a complex power of $\\cos(x/2)$ over $(0, \\pi)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/integral-sin-half-cpow","kind":"node","lean":"MetaMathlibExt.integral_sin_half_cpow","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(\\sin(x/2)\\)"],["text"," to a complex power over "],["math","\\((0, \\pi)\\)"]],"source":"Integral of $\\sin(x/2)$ to a complex power over $(0, \\pi)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#integral-sin-half-cpow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every complex number $u$ with $\\operatorname{Re} u > -1$,","title":"Integral of $\\sin(x/2)$ to a complex power over $(0, \\pi)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/integral-cos-pow-zero-pi","kind":"node","lean":"MetaMathlibExt.integral_cos_pow_zero_pi","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(\\cos^m x\\)"],["text"," over "],["math","\\((0, \\pi)\\)"]],"source":"Integral of $\\cos^m x$ over $(0, \\pi)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#integral-cos-pow-zero-pi","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $m$, $\\int_0^\\pi \\cos^m x \\, dx$ equals $\\pi \\binom{m}{m/2} / 2^m$ if $m$ is even and $0$ if $m$ is odd.","title":"Integral of $\\cos^m x$ over $(0, \\pi)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/integral-cos-half-cpow-mul-sin-half-cpow","kind":"node","lean":"MetaMathlibExt.integral_cos_half_cpow_mul_sin_half_cpow","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(\\cos^u(x/2)\\sin^v(x/2)\\)"],["text"," over "],["math","\\((0, \\pi)\\)"]],"source":"Integral of $\\cos^u(x/2)\\sin^v(x/2)$ over $(0, \\pi)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#integral-cos-half-cpow-mul-sin-half-cpow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For all complex numbers $u$ and $v$ with $\\operatorname{Re} u > -1$ and $\\operatorname{Re} v > -1$,","title":"Integral of $\\cos^u(x/2)\\sin^v(x/2)$ over $(0, \\pi)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/integral-cos-pow-mul-sin-cpow","kind":"node","lean":"MetaMathlibExt.integral_cos_pow_mul_sin_cpow","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integral of "],["math","\\(\\cos^m x \\, \\sin^v x\\)"],["text"," over "],["math","\\((0, \\pi)\\)"]],"source":"Integral of $\\cos^m x \\, \\sin^v x$ over $(0, \\pi)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#integral-cos-pow-mul-sin-cpow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $m$ and every complex number $v$ with $\\operatorname{Re} v > -1$, the integral $\\int_0^\\pi \\cos^m x \\, \\sin(x)^v \\, dx$ equals $2^{-m-v} \\pi \\binom{m}{m/2} \\binom{v}{v/2} \\binom{(m+v)/2}{m/2}^{-1}$ if $m$ is even and $0$ if $m$ is odd. Here $\\sin(x)^v$ is the principal complex power of the nonnegative real $\\sin x$ and $\\binom{a}{b} = \\Gamma(a+1)/(\\Gamma(b+1)\\,\\Gamma(a-b+1))$ is the\u2026","title":"Integral of $\\cos^m x \\, \\sin^v x$ over $(0, \\pi)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/knuth-old-sum","kind":"node","lean":"MetaMathlibExt.knuth_old_sum","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#knuth-old-sum","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$, the sum $\\sum_{k=0}^{n} (-1)^k \\binom{n}{k} \\binom{2k}{k} 2^{-k}$ equals $2^{-n} \\binom{n}{n/2}$ if $n$ is even and $0$ if $n$ is odd.","title":"Knuth's old sum (the Reed\u2013Dawson identity)"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/knuth-old-sum-gamma-binom-shift","kind":"node","lean":"MetaMathlibExt.knuth_old_sum_gammaBinom_shift","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#knuth-old-sum-gamma-binom-shift","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For all natural numbers $m$ and $n$ and every real number $v > -1$,","title":"Two-parameter generalization of Knuth's old sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/knuth-old-sum-gamma-binom","kind":"node","lean":"MetaMathlibExt.knuth_old_sum_gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#knuth-old-sum-gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and every real number $v > -1$,","title":"One-parameter generalization of Knuth's old sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-two-mul-inv-gamma-binom","kind":"node","lean":"MetaMathlibExt.sum_choose_two_mul_inv_gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-two-mul-inv-gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex $u$, $w$, which for natural $u$, $w$ is the ordinary binomial coefficient. For every natural number $n$ and every real $v > -1$,","title":"Even-index binomial sum with inverse generalized binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-two-mul-sub-one-inv-gamma-bin","kind":"node","lean":"MetaMathlibExt.sum_choose_two_mul_sub_one_inv_gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-two-mul-sub-one-inv-gamma-bin","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex $u$, $w$, which for natural $u$, $w$ is the ordinary binomial coefficient. For every natural number $n$ and every real $v > -1$,","title":"Odd-index binomial sum with inverse generalized binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-odd-div-two-mul-add-one","kind":"node","lean":"MetaMathlibExt.sum_choose_odd_div_two_mul_add_one","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-odd-div-two-mul-add-one","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$,","title":"Odd-index binomial sum with odd reciprocal weights"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-odd-two-pow-mul-catalan","kind":"node","lean":"MetaMathlibExt.sum_choose_odd_two_pow_mul_catalan","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-odd-two-pow-mul-catalan","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $C_m = \\binom{2m}{m}/(m+1)$ for the $m$-th Catalan number. For every natural number $n$,","title":"Odd-index binomial sum of Catalan numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-choose-two-pow-div-add-o","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_two_pow_div_add_one","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating binomial sum of "],["math","\\(2^{k+m}/(k+m+1)\\)"]],"source":"Alternating binomial sum of $2^{k+m}/(k+m+1)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-choose-two-pow-div-add-o","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For all natural numbers $m$ and $n$, the rational number $S = \\sum_{k=0}^{n} (-1)^k \\binom{n}{k} \\frac{2^{k+m}}{k+m+1}$ satisfies","title":"Alternating binomial sum of $2^{k+m}/(k+m+1)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-central-binom-mul-centra","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_centralBinom_mul_centralBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-central-binom-mul-centra","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$, the integer $\\sum_{k=0}^{n} (-1)^k \\binom{2k}{k} \\binom{2(n-k)}{n-k}$ equals $2^n \\binom{n}{n/2}$ when $n$ is even and $0$ when $n$ is odd.","title":"Alternating convolution of central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-choose-gamma-binom-compl","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_gammaBinom_complement","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-choose-gamma-binom-compl","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and every real $v > -1$, the sum","title":"Alternating convolution of generalized central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-gamma-binom-convolution","kind":"node","lean":"MetaMathlibExt.sum_choose_gammaBinom_convolution","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-gamma-binom-convolution","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For complex numbers $u$ and $w$ write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\,\\Gamma(u-w+1))$ for the generalized binomial coefficient, which is the ordinary binomial coefficient at natural arguments. For every natural number $n$ and every real number $v > -1$,","title":"Convolution of generalized central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-inv-gamma-binom-of-binomial-poly-ide","kind":"node","lean":"MetaMathlibExt.sum_inv_gammaBinom_of_binomial_poly_identity","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-inv-gamma-binom-of-binomial-poly-ide","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $s, n, m, r$ be natural numbers, $f, g : \\mathbb{N} \\to \\mathbb{C}$ and $p, q : \\mathbb{N} \\to \\mathbb{N}$ such that $\\sum_{k=s}^{n} f(k) (1+t)^{p(k)} = \\sum_{k=m}^{r} g(k) t^{q(k)}$ for every complex $t$. Then for all complex numbers $u, v$ with $\\operatorname{Re} u > -1$ and $\\operatorname{Re} v > -1$,","title":"Reciprocal binomial transform of a polynomial identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-gamma-binom-shift-of-binomial-poly-i","kind":"node","lean":"MetaMathlibExt.sum_gammaBinom_shift_of_binomial_poly_identity","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-gamma-binom-shift-of-binomial-poly-i","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $s, n, m, r$ be natural numbers, $f, g : \\mathbb{N} \\to \\mathbb{C}$ and $p, q : \\mathbb{N} \\to \\mathbb{N}$ such that $\\sum_{k=s}^{n} f(k) (1+t)^{p(k)} = \\sum_{k=m}^{r} g(k) t^{q(k)}$ for every complex $t$, and let $u, v$ be complex numbers with $\\operatorname{Re} v > -1$, $\\operatorname{Re}(2(u-p(j))+v) > -1$ for $s \\le j \\le n$ and $\\operatorname{Re}(2(u-q(j))+v) > -1$ for $m \\le j \\le r$. Then","title":"Shifted central binomial transform of a polynomial identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-gamma-binom-shift-of-binomial-poly-i-2","kind":"node","lean":"MetaMathlibExt.sum_gammaBinom_shift_of_binomial_poly_identity'","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-gamma-binom-shift-of-binomial-poly-i-2","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $s, n, m, r$ be natural numbers, $f, g : \\mathbb{N} \\to \\mathbb{C}$ and $p, q : \\mathbb{N} \\to \\mathbb{N}$ such that $\\sum_{k=s}^{n} f(k) (1+t)^{p(k)} = \\sum_{k=m}^{r} g(k) t^{q(k)}$ for every complex $t$, and let $u, v$ be complex numbers with $\\operatorname{Re} v > -1$, $\\operatorname{Re}(2(u-p(j))+v) > -1$ for $s \\le j \\le n$ and $\\operatorname{Re}(2(u-q(j))+v) > -1$ for $m \\le j \\le r$. Then","title":"Shifted central binomial transform of a polynomial identity, dual form"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-inv-gamma-binom-of-binom","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_inv_gammaBinom_of_binomial_poly_identity","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-inv-gamma-binom-of-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\,\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex numbers $u$ and $w$. Let $s, n, m, r$ be natural numbers, let $f, g \\colon \\mathbb{N} \\to \\mathbb{C}$ and $p, q \\colon \\mathbb{N} \\to \\mathbb{N}$ satisfy","title":"Alternating reciprocal generalized-binomial sums from a polynomial identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-two-param-gamma-binom-of-binomial-po","kind":"node","lean":"MetaMathlibExt.sum_two_param_gammaBinom_of_binomial_poly_identity","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-two-param-gamma-binom-of-binomial-po","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\,\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex numbers $u$ and $w$. Let $s, n, m, r$ be natural numbers, let $f, g \\colon \\mathbb{N} \\to \\mathbb{C}$ and $p, q \\colon \\mathbb{N} \\to \\mathbb{N}$ satisfy","title":"Two-parameter generalized-binomial sums from a polynomial identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-gamma-binom-of-binomial-poly-identit","kind":"node","lean":"MetaMathlibExt.sum_gammaBinom_of_binomial_poly_identity_q_even","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-gamma-binom-of-binomial-poly-identit","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $s, n, m, r$ be natural numbers, $f, g : \\mathbb{N} \\to \\mathbb{C}$ and $p, q : \\mathbb{N} \\to \\mathbb{N}$ such that $\\sum_{k=s}^{n} f(k) (1+t)^{p(k)} = \\sum_{k=m}^{r} g(k) t^{q(k)}$ for every complex $t$. Suppose that $q(2j)$ is even whenever $m \\le 2j \\le r$ and $q(2j+1)$ is odd whenever $m \\le 2j+1 \\le r$. Then for every complex $v$ with $\\operatorname{Re} v > -1$,","title":"Central binomial transform of a polynomial identity, even exponents at even indices"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-gamma-binom-of-binomial-poly-identit-2","kind":"node","lean":"MetaMathlibExt.sum_gammaBinom_of_binomial_poly_identity_q_odd","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-gamma-binom-of-binomial-poly-identit-2","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex $u$, $w$. Let $s, n, m, r$ be natural numbers, let $f, g$ be complex-valued functions and $p, q$ natural-number-valued functions on the natural numbers, and suppose that for every complex $t$,","title":"Central binomial sums from a polynomial identity: right exponents odd at even indices"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-gamma-binom-of-binomial-poly-identit-3","kind":"node","lean":"MetaMathlibExt.sum_gammaBinom_of_binomial_poly_identity_p_even","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-gamma-binom-of-binomial-poly-identit-3","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex $u$, $w$. Let $s, n, m, r$ be natural numbers, let $f, g$ be complex-valued functions and $p, q$ natural-number-valued functions on the natural numbers, and suppose that for every complex $t$,","title":"Central binomial sums from a polynomial identity: left exponents even at even indices"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-gamma-binom-of-binomial-poly-identit-4","kind":"node","lean":"MetaMathlibExt.sum_gammaBinom_of_binomial_poly_identity_p_odd","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-gamma-binom-of-binomial-poly-identit-4","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex $u$, $w$. Let $s, n, m, r$ be natural numbers, let $f, g$ be complex-valued functions and $p, q$ natural-number-valued functions on the natural numbers, and suppose that for every complex $t$,","title":"Central binomial sums from a polynomial identity: left exponents odd at even indices"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-gamma-binom-of-poly-identity","kind":"node","lean":"MetaMathlibExt.sum_gammaBinom_of_poly_identity","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial transform of a polynomial identity with exponents "],["math","\\(k\\)"]],"source":"Central binomial transform of a polynomial identity with exponents $k$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-gamma-binom-of-poly-identity","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $s, n, m, r$ be natural numbers and $f, g : \\mathbb{N} \\to \\mathbb{C}$ such that $\\sum_{k=s}^{n} f(k) (1+t)^{k} = \\sum_{k=m}^{r} g(k) t^{k}$ for every complex $t$. Then for every real $v > -1$,","title":"Central binomial transform of a polynomial identity with exponents $k$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-gamma-binom-of-poly-iden","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_gammaBinom_of_poly_identity","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating companion of a generalized-binomial sum from a polynomial identity in "],["math","\\(1+t\\)"]],"source":"Alternating companion of a generalized-binomial sum from a polynomial identity in $1+t$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-gamma-binom-of-poly-iden","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\,\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex numbers $u$ and $w$. Let $s, n, m, r$ be natural numbers and let $f, g \\colon \\mathbb{N} \\to \\mathbb{C}$ satisfy","title":"Alternating companion of a generalized-binomial sum from a polynomial identity in $1+t$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-inv-gamma-binom-one-sub-pow","kind":"node","lean":"MetaMathlibExt.sum_inv_gammaBinom_one_sub_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-inv-gamma-binom-one-sub-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and all complex numbers $u, v, x$ with $\\operatorname{Re} u > -1$ and $\\operatorname{Re} v > -1$,","title":"Polynomial identity for reciprocal generalized binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/knuth-poly-gamma-binom-one-sub-pow","kind":"node","lean":"MetaMathlibExt.knuth_poly_gammaBinom_one_sub_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#knuth-poly-gamma-binom-one-sub-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $\\binom{a}{b}$ denote the generalized binomial coefficient $\\Gamma(a+1)/(\\Gamma(b+1)\\Gamma(a-b+1))$. For every natural number $n$, every real $v > -1$ and every complex $x$,","title":"Generalized central binomial polynomial identity, alternating form"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-two-pow-div-succ-one-sub-pow","kind":"node","lean":"MetaMathlibExt.sum_choose_two_pow_div_succ_one_sub_pow","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating polynomial identity in "],["math","\\(1 - x\\)"],["text"," with weights "],["math","\\(2^k/(k+1)\\)"]],"source":"Alternating polynomial identity in $1 - x$ with weights $2^k/(k+1)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-two-pow-div-succ-one-sub-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and every complex number $x$,","title":"Alternating polynomial identity in $1 - x$ with weights $2^k/(k+1)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-central-binom-succ-one-sub-po","kind":"node","lean":"MetaMathlibExt.sum_choose_centralBinom_succ_one_sub_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-central-binom-succ-one-sub-po","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and every complex number $x$,","title":"A polynomial identity for weighted central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/two-param-poly-gamma-binom-one-sub-pow","kind":"node","lean":"MetaMathlibExt.two_param_poly_gammaBinom_one_sub_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#two-param-poly-gamma-binom-one-sub-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{a}{b} = \\Gamma(a+1)/(\\Gamma(b+1)\\,\\Gamma(a-b+1))$ for the generalized binomial coefficient of complex numbers $a, b$. For every natural number $n$ and all complex numbers $u, v, x$ with $\\operatorname{Re} u > -1$ and $\\operatorname{Re} v > -1$,","title":"A two-parameter polynomial identity for generalized central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-central-binom-choose-one-sub-pow-sym","kind":"node","lean":"MetaMathlibExt.sum_centralBinom_choose_one_sub_pow_symm","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-central-binom-choose-one-sub-pow-sym","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and every complex $x$,","title":"Reflection symmetry of the central binomial polynomial sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-catalan-succ-choose-one-sub-pow-symm","kind":"node","lean":"MetaMathlibExt.sum_catalan_succ_choose_one_sub_pow_symm","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-catalan-succ-choose-one-sub-pow-symm","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $C_m = \\binom{2m}{m}/(m+1)$ denote the $m$-th Catalan number. For every natural number $n$ and every complex $x$,","title":"Reflection symmetry of the Catalan polynomial sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/knuth-poly-gamma-binom-mul-pow","kind":"node","lean":"MetaMathlibExt.knuth_poly_gammaBinom_mul_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#knuth-poly-gamma-binom-mul-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $\\binom{a}{b}$ denote the generalized binomial coefficient $\\Gamma(a+1)/(\\Gamma(b+1)\\Gamma(a-b+1))$. For every natural number $n$, every real $v > -1$ and every complex $x$,","title":"Generalized central binomial polynomial identity, Bernstein form"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-central-binom-one-sub-pow-mul","kind":"node","lean":"MetaMathlibExt.sum_choose_centralBinom_one_sub_pow_mul_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-central-binom-one-sub-pow-mul","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and every complex $x$,","title":"Central binomial polynomial identity, Bernstein form"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-two-pow-div-succ-one-sub-pow-2","kind":"node","lean":"MetaMathlibExt.sum_choose_two_pow_div_succ_one_sub_pow_mul_pow","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial identity in "],["math","\\(x\\)"],["text"," and "],["math","\\(1 - x\\)"],["text"," with weights "],["math","\\(2^k/(k+1)\\)"]],"source":"Polynomial identity in $x$ and $1 - x$ with weights $2^k/(k+1)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-two-pow-div-succ-one-sub-pow-2","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and every complex number $x$,","title":"Polynomial identity in $x$ and $1 - x$ with weights $2^k/(k+1)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-catalan-succ-one-sub-pow-mul","kind":"node","lean":"MetaMathlibExt.sum_choose_catalan_succ_one_sub_pow_mul_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-catalan-succ-one-sub-pow-mul","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $C_m = \\binom{2m}{m}/(m+1)$ denote the $m$-th Catalan number. For every natural number $n$ and every complex $x$,","title":"Catalan polynomial identity, Bernstein form"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-choose-inv-gamma-binom","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_inv_gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-choose-inv-gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$ and all complex numbers $u, v$ with $\\operatorname{Re} u > -1$ and $\\operatorname{Re} v > -1$,","title":"Alternating sum of reciprocal generalized binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-choose-two-pow-gamma-bin","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_two_pow_gammaBinom","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-parameter extension of the alternating sum of "],["math","\\(\\binom{n}{k} 2^{n-2k} \\binom{2k}{k}\\)"]],"source":"One-parameter extension of the alternating sum of $\\binom{n}{k} 2^{n-2k} \\binom{2k}{k}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-choose-two-pow-gamma-bin","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\,\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex numbers $u$ and $w$. For every natural number $n$ and every real number $v > -1$, the following identity holds in $\\mathbb{C}$:","title":"One-parameter extension of the alternating sum of $\\binom{n}{k} 2^{n-2k} \\binom{2k}{k}$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-two-pow-gamma-binom","kind":"node","lean":"MetaMathlibExt.sum_choose_two_pow_gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-two-pow-gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex $u$, $w$, which for natural $u$, $w$ is the ordinary binomial coefficient. For every natural number $n$ and every real $v > -1$,","title":"Binomial transform of generalized central binomial ratios"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-choose-two-pow-central-b","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_two_pow_centralBinom","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sum of "],["math","\\(\\binom{n}{k} 2^{n-2k} \\binom{2k}{k}\\)"]],"source":"Alternating sum of $\\binom{n}{k} 2^{n-2k} \\binom{2k}{k}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-choose-two-pow-central-b","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$, the following identity holds in $\\mathbb{Q}$:","title":"Alternating sum of $\\binom{n}{k} 2^{n-2k} \\binom{2k}{k}$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-even-div-eq-sum-choose-odd-di","kind":"node","lean":"MetaMathlibExt.sum_choose_even_div_eq_sum_choose_odd_div","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-even-div-eq-sum-choose-odd-di","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n \\ge 1$,","title":"Even-index against odd-index reciprocal binomial sums"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-choose-two-pow-catalan","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_two_pow_catalan","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-choose-two-pow-catalan","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$, in the rational numbers","title":"Alternating binomial sum of Catalan numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-even-central-binom-div-four-p","kind":"node","lean":"MetaMathlibExt.sum_choose_even_centralBinom_div_four_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-even-central-binom-div-four-p","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$,","title":"Even-index binomial sum equal to a scaled central binomial coefficient"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-even-div-two-mul-add-one","kind":"node","lean":"MetaMathlibExt.sum_choose_even_div_two_mul_add_one","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-even-div-two-mul-add-one","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$,","title":"Even-index binomial sum with odd reciprocal weights"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-even-central-binom-div-succ","kind":"node","lean":"MetaMathlibExt.sum_choose_even_centralBinom_div_succ","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-even-central-binom-div-succ","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $C_m = \\binom{2m}{m}/(m+1)$ for the $m$-th Catalan number. For every natural number $n$,","title":"Even-index binomial sum of weighted central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-central-binom-div-two-pow","kind":"node","lean":"MetaMathlibExt.sum_choose_centralBinom_div_two_pow","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-central-binom-div-two-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$,","title":"Even-index expansion of a weighted central binomial sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-even-two-pow-sub-div","kind":"node","lean":"MetaMathlibExt.sum_choose_even_two_pow_sub_div","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Even-index and odd-index binomial sums weighted by powers of "],["math","\\(2\\)"]],"source":"Even-index and odd-index binomial sums weighted by powers of $2$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-even-two-pow-sub-div","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$,","title":"Even-index and odd-index binomial sums weighted by powers of $2$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-two-mul-add-one-div-catalan","kind":"node","lean":"MetaMathlibExt.sum_choose_two_mul_add_one_div_catalan","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-two-mul-add-one-div-catalan","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $C_m = \\binom{2m}{m}/(m+1)$ for the $m$-th Catalan number. For every natural number $n$,","title":"Binomial transform of weighted Catalan numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-sub-choose-gamma-binom","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_sub_choose_gammaBinom","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-parameter extension of the alternating sum of "],["math","\\(\\binom{n}{k} \\binom{2k}{k}\\)"]],"source":"One-parameter extension of the alternating sum of $\\binom{n}{k} \\binom{2k}{k}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-sub-choose-gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\,\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex numbers $u$ and $w$. For every natural number $n$ and every real number $v > -1$, the following identity holds in $\\mathbb{C}$:","title":"One-parameter extension of the alternating sum of $\\binom{n}{k} \\binom{2k}{k}$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-sub-choose-central-binom","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_sub_choose_centralBinom","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sum of "],["math","\\(\\binom{n}{k} \\binom{2k}{k}\\)"]],"source":"Alternating sum of $\\binom{n}{k} \\binom{2k}{k}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-sub-choose-central-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$,","title":"Alternating sum of $\\binom{n}{k} \\binom{2k}{k}$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-sub-choose-four-pow-div","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_sub_choose_four_pow_div_succ","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sum of "],["math","\\(\\binom{n}{k} 2^{2k}/(k+1)\\)"]],"source":"Alternating sum of $\\binom{n}{k} 2^{2k}/(k+1)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-sub-choose-four-pow-div","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$, the following identity holds in $\\mathbb{Q}$:","title":"Alternating sum of $\\binom{n}{k} 2^{2k}/(k+1)$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-neg-one-pow-sub-choose-catalan","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_sub_choose_catalan","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sum of "],["math","\\(\\binom{n}{k} \\frac{2(2k+1)}{k+2} C_k\\)"]],"source":"Alternating sum of $\\binom{n}{k} \\frac{2(2k+1)}{k+2} C_k$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-neg-one-pow-sub-choose-catalan","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $C_k = \\frac{1}{k+1}\\binom{2k}{k}$ denote the $k$-th Catalan number. For every natural number $n$, the following identity holds in $\\mathbb{Q}$:","title":"Alternating sum of $\\binom{n}{k} \\frac{2(2k+1)}{k+2} C_k$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-two-mul-gamma-binom-even","kind":"node","lean":"MetaMathlibExt.sum_choose_two_mul_gammaBinom_even","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-two-mul-gamma-binom-even","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex $u$, $w$, which for natural $u$, $w$ is the ordinary binomial coefficient. For every natural number $n$ and every real $v > -1$,","title":"Even-index identity for generalized central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/sum-choose-two-mul-sub-one-gamma-binom-o","kind":"node","lean":"MetaMathlibExt.sum_choose_two_mul_sub_one_gammaBinom_odd","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#sum-choose-two-mul-sub-one-gamma-binom-o","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{u}{w} = \\Gamma(u+1)/(\\Gamma(w+1)\\Gamma(u-w+1))$ for the generalized binomial coefficient of complex $u$, $w$, which for natural $u$, $w$ is the ordinary binomial coefficient. For every natural number $n$ and every real $v > -1$,","title":"Odd-index identity for generalized central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/waring-sum-gamma-binom","kind":"node","lean":"MetaMathlibExt.waring_sum_gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#waring-sum-gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Write $\\binom{a}{b} = \\Gamma(a+1)/(\\Gamma(b+1)\\,\\Gamma(a-b+1))$ for the generalized binomial coefficient. For every integer $n \\ge 1$ and every real number $v > -1$,","title":"Waring-type sum of generalized central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/simons-sum-gamma-binom","kind":"node","lean":"MetaMathlibExt.simons_sum_gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#simons-sum-gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $\\binom{a}{b}$ denote the generalized binomial coefficient $\\Gamma(a+1)/(\\Gamma(b+1)\\Gamma(a-b+1))$. For every natural number $n$ and every real $v > -1$,","title":"Simons-type identity for generalized central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/simons-two-param-gamma-binom","kind":"node","lean":"MetaMathlibExt.simons_two_param_gammaBinom","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#simons-two-param-gamma-binom","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $\\binom{a}{b}$ denote the generalized binomial coefficient $\\Gamma(a+1)/(\\Gamma(b+1)\\Gamma(a-b+1))$. For every natural number $n$ and all real $u > -1$ and $v > -1$,","title":"Two-parameter Simons-type symmetry"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/knuth-old-sum-weight","kind":"node","lean":"MathlibExt.Combinatorics.KnuthOldSumWanted.knuthOldSumWeight","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#knuth-old-sum-weight","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"fully_proved","summary":"For a natural number $m$, the weight $w(m)$ is the rational number $(4^m + 2^m)/2$, which equals $(2^m + 1) 2^{m-1}$.","title":"Weight in Knuth's old sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/card-even-acount-words","kind":"node","lean":"MathlibExt.Combinatorics.KnuthOldSumWanted.card_evenACount_words","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting words with an even number of "],["math","\\(a\\)"],["text","'s"]],"source":"Counting words with an even number of $a$'s"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#card-even-acount-words","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"For every natural number $n$, the number of words of length $n$ over the four-letter alphabet $\\{a, b, c, d\\}$ in which the letter $a$ occurs an even number of times is $(4^n + 2^n)/2$.","title":"Counting words with an even number of $a$'s"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/knuth-old-sum-eq-mul-sum-div","kind":"node","lean":"MathlibExt.Combinatorics.KnuthOldSumWanted.knuthOldSum_eq_mul_sum_div","parent":26057,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#knuth-old-sum-eq-mul-sum-div","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $n$ and $j$ be natural numbers, let $N = n + 2j + 1$, and let $w(m) = (4^m + 2^m)/2$ be the weight in Knuth's old sum. Then, in the rational numbers,","title":"Tefera\u2013Zeleke identity with binomial quotients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/knuth-old-sum-eq-sum-pow","kind":"node","lean":"MathlibExt.Combinatorics.KnuthOldSumWanted.knuthOldSum_eq_sum_pow","parent":26057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tefera\u2013Zeleke identity with powers of "],["math","\\(2\\)"],["text",", "],["math","\\(3\\)"],["text"," and "],["math","\\(4\\)"]],"source":"Tefera\u2013Zeleke identity with powers of $2$, $3$ and $4$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum/index.html#knuth-old-sum-eq-sum-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/knuth-s-old-sum","status":"can_prove","summary":"Let $n$ and $j$ be natural numbers, let $N = n + 2j + 1$, and let $w(m) = (4^m + 2^m)/2$ be the weight in Knuth's old sum. Then, in the rational numbers,","title":"Tefera\u2013Zeleke identity with powers of $2$, $3$ and $4$"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients.json","id":"Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients","status":"fully_proved","title":"Partition-sum expansion of binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients/choose-mul-eq-partition-sum-unconditiona","kind":"node","lean":"MetaMathlibExt.choose_mul_eq_partition_sum_unconditional","parent":26124,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partition-sum expansion of "],["math","\\(\\binom{nm}{l}\\)"],["text",", unconditional form"]],"source":"Partition-sum expansion of $\\binom{nm}{l}$, unconditional form"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients/index.html#choose-mul-eq-partition-sum-unconditiona","scope":"Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients","status":"fully_proved","summary":"For all natural numbers $n$, $m$ and $l$, let $P$ be the set of tuples $s = (s_1, \\dots, s_m)$ of natural numbers with $\\sum_{j=1}^{m} j\\,s_j = l$, and for $s \\in P$ write $T_j(s) = s_{j+1} + \\dots + s_m$. Then","title":"Partition-sum expansion of $\\binom{nm}{l}$, unconditional form"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients/choose-mul-eq-partition-sum","kind":"node","lean":"MetaMathlibExt.choose_mul_eq_partition_sum","parent":26124,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partition-sum expansion of "],["math","\\(\\binom{nm}{l}\\)"]],"source":"Partition-sum expansion of $\\binom{nm}{l}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients/index.html#choose-mul-eq-partition-sum","scope":"Combinatorics/binomial-coefficients/binomial-sums/partition-sum-expansion-of-binomial-coefficients","status":"fully_proved","summary":"Let $n$ and $m$ be positive natural numbers and let $l$ be a natural number with $1 \\le l \\le nm$. Let $P$ be the set of tuples $s = (s_1, \\dots, s_m)$ of natural numbers with $\\sum_{j=1}^{m} j\\,s_j = l$, and for $s \\in P$ write $T_j(s) = s_{j+1} + \\dots + s_m$. Then","title":"Partition-sum expansion of $\\binom{nm}{l}$"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums.json","id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","title":"Rationally weighted binomial sums"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-pow-mul-coeff-eq-sum-stirling-second","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_coeff_eq_sum_stirlingSecond_derivative","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-pow-mul-coeff-eq-sum-stirling-second","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Write $S(r, i)$ for the Stirling numbers of the second kind. Let $N$ and $r$ be natural numbers, let $a_0, \\dots, a_N$ be real numbers, and let $f$ be the polynomial $f(y) = \\sum_{k=0}^{N} a_k y^k$, with $i$-th derivative $f^{(i)}$. Then for every real number $x$, with the convention $0^0 = 1$,","title":"Scherk's identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-rising-step-mul-coeff-eq-sum-hsu-shi","kind":"node","lean":"MetaMathlibExt.sum_risingStep_mul_coeff_eq_sum_hsuShiue_derivative","parent":26127,"reading_title":{"format":"inline-parts-v1","parts":[["text","Scherk's identity with "],["math","\\(s\\)"],["text","-rising factorials"]],"source":"Scherk's identity with $s$-rising factorials"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-rising-step-mul-coeff-eq-sum-hsu-shi","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $s$ be a real number and let $S(n, k)$ be the generalized Stirling numbers of Hsu and Shiue with parameters $(-s, 1, 0)$: a real array with $S(0, 0) = 1$, $S(n, 0) = 0$ for $n \\ge 1$, $S(n, k) = 0$ for $n < k$, and $S(n+1, k+1) = S(n, k) + (k + 1 + ns) \\, S(n, k+1)$ for all $n, k \\ge 0$. Let $N$ and $r$ be natural numbers, let $a_0, \\dots, a_N$ be real numbers, and let $f(y) = \\sum_{k=0}^{N} a_k y^k$, with\u2026","title":"Scherk's identity with $s$-rising factorials"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/stirling-second-three-eq","kind":"node","lean":"MetaMathlibExt.stirlingSecond_three_eq","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#stirling-second-three-eq","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"fully_proved","summary":"For every integer $r \\ge 1$, the Stirling number of the second kind $S(r, 3)$ satisfies $S(r, 3) = \\frac{3^{r-1} + 1}{2} - 2^{r-1}$.","title":"Stirling numbers of the second kind with three blocks"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-neg-one-pow-choose-two-pow-mul-sq","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_two_pow_mul_sq","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-neg-one-pow-choose-two-pow-mul-sq","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"For every natural number $N$,","title":"Greenstreet's identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-poly-mul-fib-pow-eq-sum-stirling-sec","kind":"node","lean":"MetaMathlibExt.sum_poly_mul_fib_pow_eq_sum_stirlingSecond_derivative","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-poly-mul-fib-pow-eq-sum-stirling-sec","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers and $S(r, i)$ the Stirling numbers of the second kind. Let $t$ and $N$ be natural numbers, let $A_0, \\dots, A_t$ be complex numbers, let $h(y) = \\sum_{r=0}^{t} A_r y^r$, and let $P$ be the polynomial $P(y) = \\sum_{j=0}^{N} F_j y^j$, with $i$-th derivative $P^{(i)}$. Then for every $x \\in \\mathbb{C}$,","title":"Polynomially weighted Fibonacci sums via derivatives"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-pow-mul-choose-mul-fib","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_choose_mul_fib","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-pow-mul-choose-mul-fib","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers, $S(r, i)$ the Stirling numbers of the second kind and $(N)_i = N(N-1) \\cdots (N-i+1)$ the falling factorial, with $(N)_0 = 1$. For all natural numbers $N$ and $r$,","title":"Power-weighted binomial sums of Fibonacci numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-pow-mul-choose-mul-lucas-number","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_choose_mul_lucasNumber","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-pow-mul-choose-mul-lucas-number","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $L_n$ denote the Lucas numbers ($L_0 = 2$, $L_1 = 1$, $L_{n+2} = L_{n+1} + L_n$), $S(r, i)$ the Stirling numbers of the second kind and $(N)_i = N(N-1) \\cdots (N-i+1)$ the falling factorial, with $(N)_0 = 1$. For all natural numbers $N$ and $r$, with the convention $0^0 = 1$,","title":"Power-weighted binomial sums of Lucas numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-pow-mul-choose-mul-second-order","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_choose_mul_secondOrder","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-pow-mul-choose-mul-second-order","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $a$ and $b$ be real numbers and let $(W_n)_{n \\ge 0}$ be a real sequence with $W_{n+2} = a W_{n+1} + b W_n$ for all $n \\ge 0$. Write $S(r, i)$ for the Stirling numbers of the second kind and $(N)_i = N(N-1) \\cdots (N-i+1)$ for the falling factorial, with $(N)_0 = 1$. Then for all natural numbers $N$ and $r$, with the convention $0^0 = 1$,","title":"Power-weighted binomial sums of a second-order linear recurrence"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-pow-mul-choose-mul-fib-small","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_choose_mul_fib_small","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-pow-mul-choose-mul-fib-small","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers. For every natural number $N$,","title":"Weighted binomial Fibonacci sums for the powers one to four"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-poly-mul-choose-mul-fib","kind":"node","lean":"MetaMathlibExt.sum_poly_mul_choose_mul_fib","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-poly-mul-choose-mul-fib","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers, $S(j, i)$ the Stirling numbers of the second kind and $(N)_i = N(N-1)\\cdots(N-i+1)$ the falling factorial. Let $A_0, A_1, A_2, \\dots$ be complex numbers and $N$, $t$ natural numbers, and put $h(y) = \\sum_{j=0}^{t} A_j y^j$. Then","title":"Polynomially weighted binomial Fibonacci sums"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-poly-mul-choose-mul-second-order-pow","kind":"node","lean":"MetaMathlibExt.sum_poly_mul_choose_mul_secondOrder_pow","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-poly-mul-choose-mul-second-order-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $a$, $b$ be real numbers and $(U_n)_{n \\ge 0}$ a real sequence with $U_{n+2} = a U_{n+1} + b U_n$ for all $n$. Let $\\alpha \\ne \\beta$ be complex numbers with $\\alpha + \\beta = a$ and $\\alpha\\beta = -b$ (the roots of $y^2 = ay + b$), and put $A = (U_1 - U_0\\beta)/(\\alpha - \\beta)$ and $B = (U_1 - U_0\\alpha)/(\\alpha - \\beta)$. Let $S(j, i)$ denote the Stirling numbers of the second kind and\u2026","title":"Polynomially weighted binomial sums of powers of a second-order sequence"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-choose-mul-pow-div-asc-factorial","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_pow_div_ascFactorial","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-choose-mul-pow-div-asc-factorial","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Write $m^{(i)} = m(m+1)\\cdots(m+i-1)$ for the rising factorial. For all natural numbers $N$ and $s \\ge 1$ and every real number $x$,","title":"Binomial sums with rising-factorial denominators"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-coeff-mul-pow-div-asc-factorial-eq-a","kind":"node","lean":"MetaMathlibExt.sum_coeff_mul_pow_div_ascFactorial_eq_antideriv","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-coeff-mul-pow-div-asc-factorial-eq-a","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"Let $N$ and $s$ be natural numbers, $a_0, \\dots, a_N$ real numbers and $f(y) = \\sum_{k=0}^{N} a_k y^k$. Let $I_0, I_1, I_2, \\dots$ be functions $\\mathbb{R} \\to \\mathbb{R}$ with $I_0 = f$ such that for every $j \\ge 0$ the function $I_{j+1}$ is differentiable on $\\mathbb{R}$ with derivative $I_j$ (the constants of integration are arbitrary). Write $m^{(i)} = m(m+1)\\cdots(m+i-1)$ for the rising factorial. Then for\u2026","title":"Rising-factorial weighted sums via iterated antiderivatives"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-choose-div-succ","kind":"node","lean":"MetaMathlibExt.sum_choose_div_succ","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-choose-div-succ","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"For every natural number $n$,","title":"Epsteen's identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-neg-one-pow-choose-div-eq-harmonic","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_div_eq_harmonic","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-neg-one-pow-choose-div-eq-harmonic","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"For every natural number $n$, with $H_n = \\sum_{j=1}^{n} 1/j$ the $n$-th harmonic number,","title":"Finkel's identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-choose-mul-pow-div-sq","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_pow_div_sq","parent":26127,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial sums weighted by "],["math","\\(x^k/k^2\\)"]],"source":"Binomial sums weighted by $x^k/k^2$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-choose-mul-pow-div-sq","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"For every natural number $n$ and every real number $x$,","title":"Binomial sums weighted by $x^k/k^2$"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/sum-neg-one-pow-choose-div-sq-eq-sum-har","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_choose_div_sq_eq_sum_harmonic","parent":26127,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums/index.html#sum-neg-one-pow-choose-div-sq-eq-sum-har","scope":"Combinatorics/binomial-coefficients/binomial-sums/rationally-weighted-binomial-sums","status":"can_prove","summary":"For every natural number $n$, with $H_k = \\sum_{j=1}^{k} 1/j$ the $k$-th harmonic number,","title":"Alternating binomial sum with inverse-square weights"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien.json","id":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","status":"can_prove","title":"Recurrences for step sums of binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/linear-recurrence-of-finite-window","kind":"node","lean":"MetaMathlibExt.linearRecurrence_of_finite_window","parent":26145,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/index.html#linear-recurrence-of-finite-window","scope":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","status":"can_prove","summary":"Let $R$ be a commutative ring, let $x, A, a : \\mathbb{N} \\to R$ and let $r, s, N \\in \\mathbb{N}$. Suppose that $x_n = \\sum_{i=1}^{r} A_i x_{n-i}$ for all $n \\ge r$, that $N \\ge r + s - 1$, and that $x_n = \\sum_{i=1}^{s} a_i x_{n-i}$ for all $n$ with $s \\le n \\le N$. Then $x_n = \\sum_{i=1}^{s} a_i x_{n-i}$ for all $n \\ge s$.","title":"Finite window criterion for linear recurrences"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/roots-of-unity-product-mul-eq-comp-pow","kind":"node","lean":"MetaMathlibExt.rootsOfUnity_product_mul_eq_comp_pow","parent":26145,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/index.html#roots-of-unity-product-mul-eq-comp-pow","scope":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","status":"can_prove","summary":"Let $f \\in \\mathbb{Z}[x]$ have degree $r$, let $k > 0$ be a natural number and let $\\omega = e^{2\\pi i/k}$. Then there are $G, T \\in \\mathbb{Z}[x]$ such that $G(x) = f(\\omega x) f(\\omega^2 x) \\cdots f(\\omega^{k-1} x)$ as polynomials over $\\mathbb{C}$, $T$ has degree $r$, and $f(x) G(x) = T(x^k)$.","title":"Roots-of-unity products of integer polynomials"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/riordan-step-sum-coeff","kind":"node","lean":"MetaMathlibExt.riordan_stepSum_coeff","parent":26145,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/index.html#riordan-step-sum-coeff","scope":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","status":"can_prove","summary":"Let $K$ be a field, let $g, h \\in K[[x]]$ with $h(0) = 0$, and write $d(n, m) = [x^n]\\, g(x) h(x)^m$. Then for all natural numbers $k > 1$, $p < k$ and $n$, $\\sum_{j=0}^{n} d(n, kj + p) = [x^n]\\, g(x) h(x)^p (1 - h(x)^k)^{-1}$.","title":"Step sums along a Riordan array"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/binomial-step-sum-recurrence-of-window","kind":"node","lean":"MetaMathlibExt.binomialStepSum_recurrence_of_window","parent":26145,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/index.html#binomial-step-sum-recurrence-of-window","scope":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","status":"can_prove","summary":"Let $\\alpha, k, \\gamma, p, s$ be natural numbers with $\\alpha > 1$, $k > 1$, $\\gamma \\ge 1$ and $p < k$, and let $\\delta_1, \\delta_2, \\ldots$ be integers. For $n \\in \\mathbb{N}$ let $c(n)$ be the sum of $\\binom{\\alpha n}{m}$ over all $m$ with $0 \\le m \\le \\alpha n$ and $m \\equiv \\gamma n + p \\pmod{k}$, that is, $c(n) = \\sum_{j \\in \\mathbb{Z}} \\binom{\\alpha n}{\\gamma n + p + kj}$. If\u2026","title":"Window criterion for step sums of binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/shifted-binomial-step-sum-recurrence-of","kind":"node","lean":"MetaMathlibExt.shiftedBinomialStepSum_recurrence_of_window","parent":26145,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/index.html#shifted-binomial-step-sum-recurrence-of","scope":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","status":"can_prove","summary":"Let $\\alpha, \\beta, k, \\gamma, p, s$ be natural numbers with $\\alpha > 1$, $k > 1$, $\\gamma > 1$ and $p < k$, and let $\\delta_1, \\delta_2, \\ldots$ be integers. For $n \\in \\mathbb{N}$ let $c(n)$ be the sum of $\\binom{\\alpha n + \\beta}{m}$ over all $m$ with $0 \\le m \\le \\alpha n + \\beta$ and $m \\equiv \\gamma n + p \\pmod{k}$, that is, $c(n) = \\sum_{j \\in \\mathbb{Z}} \\binom{\\alpha n + \\beta}{\\gamma n + p + kj}$. If\u2026","title":"Window criterion for shifted step sums of binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/rational-riordan-step-sum-linear-recurre","kind":"node","lean":"MetaMathlibExt.rationalRiordan_stepSum_linearRecurrence","parent":26145,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/index.html#rational-riordan-step-sum-linear-recurre","scope":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","status":"can_prove","summary":"Let $g_1, g_2, h_1, h_2 \\in \\mathbb{Z}[x]$ with $g_2(0) = h_1(0) = h_2(0) = 1$, let $\\nu \\ge 1$ and $k \\ge 1$ be natural numbers, and in $\\mathbb{Q}[[x]]$ let $g = g_1/g_2$ and $h = x^\\nu h_1/h_2$. For $p, n \\in \\mathbb{N}$ let $a_p(n) = \\sum_{j=0}^{n} [x^n]\\, g h^{kj+p}$. Then there are $r \\in \\mathbb{N}$ and integers $\\alpha_1, \\ldots, \\alpha_r$ such that $a_p(n) = \\sum_{i=1}^{r} \\alpha_i a_p(n - i)$ for every\u2026","title":"Step sums of a rational Riordan array satisfy a common recurrence"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/catalan-triangle-step-five-sub-eq-fib","kind":"node","lean":"MetaMathlibExt.catalanTriangle_stepFive_sub_eq_fib","parent":26145,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien/index.html#catalan-triangle-step-five-sub-eq-fib","scope":"Combinatorics/binomial-coefficients/binomial-sums/recurrences-for-step-sums-of-binomial-coefficien","status":"can_prove","summary":"For natural numbers $n$ and $k$ let $d(n, k) = \\frac{k + 1}{n + 1} \\binom{2(n + 1)}{n - k}$ if $k \\le n$, and $d(n, k) = 0$ if $k > n$. Let $F_m$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). Then for every $n \\in \\mathbb{N}$, $\\sum_{j=0}^{n} \\bigl(d(n, 5j + 1) - d(n, 5j + 2)\\bigr) = F_{2n}$ and $\\sum_{j=0}^{n} \\bigl(d(n, 5j) - d(n, 5j + 3)\\bigr) = F_{2n+1}$.","title":"Alternating step-five sums of Catalan's triangle are Fibonacci numbers"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity.json","id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","title":"Staver's identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/staver-sign-mul","kind":"node","lean":"MetaMathlibExt.staver_sign_mul","parent":26153,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html#staver-sign-mul","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","summary":"For natural numbers $k \\le n$, we have $(-1)^{n-k} \\cdot (-1)^k = (-1)^n$ in $\\mathbb{Q}$.","title":"Splitting a power of minus one"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/staver-sign-sq","kind":"node","lean":"MetaMathlibExt.staver_sign_sq","parent":26153,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html#staver-sign-sq","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","summary":"For every natural number $k$, we have $(-1)^k \\cdot (-1)^k = 1$ in $\\mathbb{Q}$.","title":"Squares of powers of minus one"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/staver-sign-odd","kind":"node","lean":"MetaMathlibExt.staver_sign_odd","parent":26153,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html#staver-sign-odd","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","summary":"If $n$ is an odd natural number and $k \\le n$ is a natural number, then $(-1)^{n-k} = -(-1)^k$ in $\\mathbb{Q}$.","title":"Reflecting a power of minus one, odd case"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/staver-sign-even","kind":"node","lean":"MetaMathlibExt.staver_sign_even","parent":26153,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html#staver-sign-even","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","summary":"If $n$ is an even natural number and $k \\le n$ is a natural number, then $(-1)^{n-k} = (-1)^k$ in $\\mathbb{Q}$.","title":"Reflecting a power of minus one, even case"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/staver-reflect-zero","kind":"node","lean":"MetaMathlibExt.staver_reflect_zero","parent":26153,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html#staver-reflect-zero","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","summary":"For every natural number $n$, $\\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k}^{-1} = \\sum_{k=0}^{n} (-1)^{k} \\binom{n}{k}^{-1}$ in $\\mathbb{Q}$.","title":"Reflection of the alternating reciprocal binomial sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/staver-reflect-one","kind":"node","lean":"MetaMathlibExt.staver_reflect_one","parent":26153,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html#staver-reflect-one","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","summary":"For every natural number $n$, $\\sum_{k=0}^{n} (-1)^{n-k} \\, k \\binom{n}{k}^{-1} = \\sum_{k=0}^{n} (-1)^{k} (n-k) \\binom{n}{k}^{-1}$ in $\\mathbb{Q}$.","title":"Reflection of the weighted alternating reciprocal binomial sum"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/staver-identity","kind":"node","lean":"MetaMathlibExt.staver_identity","parent":26153,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html#staver-identity","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","summary":"For a natural number $n$, let $S_0(n) = \\sum_{k=0}^{n} (-1)^{n-k} \\binom{n}{k}^{-1}$ and $S_1(n) = \\sum_{k=0}^{n} (-1)^{n-k} \\, k \\binom{n}{k}^{-1}$, computed in $\\mathbb{Q}$. If $n$ is even, then $S_1(n) = \\frac{n}{2} S_0(n)$; if $n$ is odd, then $S_0(n) = 0$.","title":"Staver's identity for alternating reciprocal binomial sums"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/staver-reciprocal-binomial-sum","kind":"node","lean":"MetaMathlibExt.staver_reciprocal_binomial_sum","parent":26153,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity/index.html#staver-reciprocal-binomial-sum","scope":"Combinatorics/binomial-coefficients/binomial-sums/staver-s-identity","status":"fully_proved","summary":"For every natural number $n$, the following identity between rational numbers holds:","title":"Staver's identity for reciprocal binomial sums"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients.json","id":"Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients","status":"fully_proved","title":"Sums of reciprocals of binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients/frisch-alternating-reciprocal-binomial","kind":"node","lean":"MetaMathlibExt.frisch_alternating_reciprocal_binomial","parent":26162,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients/index.html#frisch-alternating-reciprocal-binomial","scope":"Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients","status":"fully_proved","summary":"For natural numbers $n$, $b$ and $c$ with $0 < c \\le b$, the rational numbers $\\sum_{k=0}^{n} (-1)^k \\binom{n}{k} \\binom{b+k}{c}^{-1}$ and $\\frac{c}{n+c} \\binom{n+b}{b-c}^{-1}$ are equal.","title":"Frisch's alternating reciprocal-binomial identity"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients/klamkin-binomial-reciprocal-identity","kind":"node","lean":"MetaMathlibExt.klamkin_binomial_reciprocal_identity","parent":26162,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients/index.html#klamkin-binomial-reciprocal-identity","scope":"Combinatorics/binomial-coefficients/binomial-sums/sums-of-reciprocals-of-binomial-coefficients","status":"fully_proved","summary":"For natural numbers $n$, $b$ and $x$ with $n + b \\le x$, the following equality of rational numbers holds:","title":"Klamkin's binomial reciprocal identity"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers.json","id":"Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers","status":"fully_proved","title":"Zagier numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers/domb-type-sum","kind":"node","lean":"MetaMathlibExt.dombTypeSum","parent":26165,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers/index.html#domb-type-sum","scope":"Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers","status":"fully_proved","summary":"For natural numbers $n$, $r$, $s$ and $t$, the Domb-type sum is the natural number","title":"Domb-type sums"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers/zagier-number","kind":"node","lean":"MetaMathlibExt.zagierNumber","parent":26165,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers/index.html#zagier-number","scope":"Combinatorics/binomial-coefficients/binomial-sums/zagier-numbers","status":"fully_proved","summary":"For a natural number $n$, the Zagier number $Z(n)$ is the Domb-type sum $D(n, 1, 1, 1)$, that is,","title":"Zagier numbers"},{"data_url":"scopes/Combinatorics/binomial-coefficients/binomial-sums/domb-numbers.json","id":"Combinatorics/binomial-coefficients/binomial-sums/domb-numbers","kind":"scope","navigation":"module","parent":26023,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/domb-numbers/index.html","scope":"Combinatorics/binomial-coefficients/binomial-sums/domb-numbers","status":"fully_proved","title":"Domb numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/domb-numbers/domb-number","kind":"node","lean":"MetaMathlibExt.dombNumber","parent":26168,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/domb-numbers/index.html#domb-number","scope":"Combinatorics/binomial-coefficients/binomial-sums/domb-numbers","status":"fully_proved","summary":"For a natural number $n$, the Domb number $\\mathrm{Domb}(n)$ is the Domb-type sum $D(n,2,1,1)$, that is, $\\mathrm{Domb}(n) = \\sum_{k=0}^{n} \\binom{n}{k}^2 \\binom{2k}{k} \\binom{2(n-k)}{n-k}$ (OEIS A002895).","title":"Domb numbers"},{"id":"Combinatorics/binomial-coefficients/binomial-sums/domb-numbers/domb-number-unfold","kind":"node","lean":"MetaMathlibExt.dombNumber_unfold","parent":26168,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/binomial-sums/domb-numbers/index.html#domb-number-unfold","scope":"Combinatorics/binomial-coefficients/binomial-sums/domb-numbers","status":"fully_proved","summary":"For every natural number $n$, the Domb number $\\mathrm{Domb}(n)$ equals $\\sum_{k=0}^{n} \\binom{n}{k}^2 \\binom{2k}{k} \\binom{2(n-k)}{n-k}$.","title":"Explicit formula for the Domb numbers"},{"data_url":"scopes/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums.json","id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums","kind":"scope","parent":26022,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/index.html","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums","status":"can_prove","summary":"Finite sums and power series involving central binomial coefficients or their reciprocals, often weighted by harmonic or odd harmonic numbers, including arcsin-type series and Koll\u00e1r's incomplete binomial sums.","title":"Central binomial coefficients and harmonic sums"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/aerated-central-binomial-number","kind":"node","lean":"MetaMathlibExt.aeratedCentralBinomialNumber","parent":26171,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/index.html#aerated-central-binomial-number","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums","status":"fully_proved","summary":"For a natural number $n$, the $n$-th aerated central binomial number is $\\binom{n}{n/2}$ if $n$ is even and $0$ if $n$ is odd. The sequence begins $1, 0, 2, 0, 6, 0, 20, \\dots$.","title":"Aerated central binomial numbers"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binom-power-sum-stirling","kind":"node","lean":"MetaMathlibExt.central_binom_power_sum_stirling","parent":26171,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/index.html#central-binom-power-sum-stirling","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums","status":"fully_proved","summary":"For every natural number $m$ and every real $x$ with $|x| < 1/4$,","title":"Central binomial power sums via Stirling numbers"},{"data_url":"scopes/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating.json","id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating","kind":"scope","navigation":"module","parent":26171,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating/index.html","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating","status":"fully_proved","title":"Central binomial and Catalan\u2013harmonic generating functions"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating/catalan-harmonic-generating-function-can","kind":"node","lean":"MetaMathlibExt.catalan_harmonic_generating_function_canonical","parent":26174,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating/index.html#catalan-harmonic-generating-function-can","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating","status":"fully_proved","summary":"Let $C_n$ be the $n$th Catalan number and $H_n = \\sum_{k=1}^{n} 1/k$ the $n$th harmonic number, so that $H_0 = 0$. For every real number $x$ with $|x| < 1/4$,","title":"Generating function of Catalan\u2013harmonic products"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating/catalan-harmonic-generating-function","kind":"node","lean":"MetaMathlibExt.catalan_harmonic_generating_function","parent":26174,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating/index.html#catalan-harmonic-generating-function","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating","status":"fully_proved","summary":"For every real number $x$ with $|x| < 1/4$,","title":"Catalan\u2013harmonic generating function with explicit coefficients"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating/has-sum-central-binomial-harmonic","kind":"node","lean":"MetaMathlibExt.hasSum_centralBinomial_harmonic","parent":26174,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating/index.html#has-sum-central-binomial-harmonic","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-and-catalan-harmonic-generating","status":"fully_proved","summary":"Let $H_n = \\sum_{k=1}^{n} 1/k$ be the $n$th harmonic number. For every real number $x$ with $|x| < 1/4$, the series $\\sum_{n=1}^{\\infty} \\binom{2n}{n} H_n x^n$ converges unconditionally, with sum","title":"Central binomial\u2013harmonic generating function"},{"data_url":"scopes/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers.json","id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","kind":"scope","navigation":"module","parent":26171,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","title":"Central binomial sums with harmonic numbers"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"fully_proved","summary":"For a sequence $a = (a_k)_{k \\ge 0}$ of rational numbers and natural numbers $r$ and $n$, the weighted central binomial sum is the rational number","title":"Weighted central binomial sum"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum-zero","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum_zero","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum-zero","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"fully_proved","summary":"For every sequence $a$ of rational numbers and every natural number $r$, the weighted central binomial sum $S_r^a(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r a_k$ satisfies $S_r^a(0) = 0^r a_0$, with the convention $0^0 = 1$.","title":"The weighted central binomial sum at zero"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum-succ","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum_succ","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum-succ","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"fully_proved","summary":"For every sequence $a$ of rational numbers and all natural numbers $r$ and $n$, the weighted central binomial sum $S_r^a(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r a_k$ (with $0^0 = 1$) satisfies","title":"Adding one term to the weighted central binomial sum"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum-odd-harmonic-recur","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum_oddHarmonic_recursion","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum-odd-harmonic-recur","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"For a natural number $n$, let $O_m = \\sum_{i=1}^{m} 1/(2i-1)$ be the odd harmonic numbers, $c_n = \\binom{2n}{n}/4^n$, and for natural $r$ let $S_r(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r$ and $S_r^O(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r O_k$, with $0^0 = 1$. Then","title":"Recurrence for central binomial sums with odd harmonic numbers"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum-harmonic-recursion","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum_harmonic_recursion","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum-harmonic-recursion","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"For a natural number $n$, let $H_m = \\sum_{i=1}^{m} 1/i$ be the harmonic numbers, $c_n = \\binom{2n}{n}/4^n$, and for natural $r$ let $S_r(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r$ and $S_r^H(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r H_k$, with $0^0 = 1$. Then","title":"Recurrence for central binomial sums with harmonic numbers"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum-harmonic-two-mul-r","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum_harmonic_two_mul_recursion","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum-harmonic-two-mul-r","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"For a natural number $n$, let $H_m = \\sum_{i=1}^{m} 1/i$ be the harmonic numbers, $c_n = \\binom{2n}{n}/4^n$, and for natural $r$ let $S_r(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r$ and $S_r^{2H}(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r H_{2k}$, with $0^0 = 1$. Then","title":"Recurrence for central binomial sums with harmonic numbers of even index"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum-odd-harmonic-one-t","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum_oddHarmonic_one_two_three","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum-odd-harmonic-one-t","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"For every natural number $n$, let $O_m = \\sum_{i=1}^{m} 1/(2i-1)$ be the odd harmonic numbers, $c_n = \\binom{2n}{n}/4^n$, and $S_r^O(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r O_k$. Then","title":"Odd harmonic central binomial sums for powers one to three"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum-harmonic-one-two-t","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum_harmonic_one_two_three","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum-harmonic-one-two-t","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"For every natural number $n$, let $H_m = \\sum_{i=1}^{m} 1/i$ be the harmonic numbers, $c_n = \\binom{2n}{n}/4^n$, and $S_r^H(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r H_k$. Then","title":"Central binomial sums with harmonic numbers for powers one to three"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/iterated-deriv-one-sub-exp-pow","kind":"node","lean":"MetaMathlibExt.iteratedDeriv_one_sub_exp_pow","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#iterated-deriv-one-sub-exp-pow","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"For all natural numbers $k$, $r$ and $v$, write ${a \\brace b}$ for the Stirling numbers of the second kind and ${a \\brace b}_v$ for the r-Stirling numbers of the second kind with parameter $v$, and use the convention $0^0 = 1$. Then the real functions $x \\mapsto (1 - e^x)^k$ and $x \\mapsto (1 - e^x)^k e^{vx}$ satisfy","title":"Stirling numbers from derivatives of powers of one minus the exponential"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-choose-harmonic-diff-r-stirling","kind":"node","lean":"MetaMathlibExt.sum_choose_harmonic_diff_rStirling","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power sums of "],["math","\\(\\binom{t}{k}(H_t - H_{t-k})\\)"],["text"," in r-Stirling form"]],"source":"Power sums of $\\binom{t}{k}(H_t - H_{t-k})$ in r-Stirling form"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-choose-harmonic-diff-r-stirling","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $t$ be a complex number that is not an integer. For complex $x$ write $\\binom{x}{k} = x(x-1)\\cdots(x-k+1)/k!$, and write $H_x = \\psi(x+1) + \\gamma$, where $\\psi$ is the digamma function and $\\gamma$ is the Euler\u2013Mascheroni constant (only differences $H_x - H_y = \\psi(x+1) - \\psi(y+1)$ occur below). Let ${a \\brace b}_c$ denote the r-Stirling numbers of the second kind. For all natural numbers $n$ and $r$ (with\u2026","title":"Power sums of $\\binom{t}{k}(H_t - H_{t-k})$ in r-Stirling form"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-choose-harmonic-diff-pow-le-three","kind":"node","lean":"MetaMathlibExt.sum_choose_harmonic_diff_pow_le_three","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power sums of "],["math","\\(\\binom{t}{k}(H_t - H_{t-k})\\)"],["text"," for exponents up to three"]],"source":"Power sums of $\\binom{t}{k}(H_t - H_{t-k})$ for exponents up to three"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-choose-harmonic-diff-pow-le-three","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $n$ be a natural number and $t$ a complex number that is not an integer. For complex $x$ write $\\binom{x}{k} = x(x-1)\\cdots(x-k+1)/k!$, and write $H_x = \\psi(x+1) + \\gamma$, where $\\psi$ is the digamma function and $\\gamma$ is the Euler\u2013Mascheroni constant (only differences $H_x - H_y = \\psi(x+1) - \\psi(y+1)$ occur below). For natural $r$ let $L_r = \\sum_{k=0}^{n} (-1)^k k^r \\binom{t}{k} (H_t - H_{t-k})$ (with\u2026","title":"Power sums of $\\binom{t}{k}(H_t - H_{t-k})$ for exponents up to three"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/central-binom-pow-sum-odd-harmonic-eq-r","kind":"node","lean":"MetaMathlibExt.centralBinomPowSum_oddHarmonic_eq_rStirling","parent":26178,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#central-binom-pow-sum-odd-harmonic-eq-r","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"For natural numbers $n$ and $r$, let $O_m = \\sum_{i=1}^{m} 1/(2i-1)$ be the odd harmonic numbers, ${a \\brace b}_s$ the r-Stirling numbers of the second kind, and $S_r^O(n) = \\sum_{k=0}^{n} \\binom{2k}{k} 4^{-k} k^r O_k$, with $0^0 = 1$. Then","title":"Odd harmonic central binomial sums via r-Stirling numbers"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-central-binom-one-sub-odd-harmonic-e","kind":"node","lean":"MetaMathlibExt.sum_centralBinom_one_sub_oddHarmonic_eq_rStirling","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Central binomial sums weighted by "],["math","\\(1 - O_{k-1}\\)"],["text",", in r-Stirling form"]],"source":"Central binomial sums weighted by $1 - O_{k-1}$, in r-Stirling form"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-central-binom-one-sub-odd-harmonic-e","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $O_m = \\sum_{j=1}^{m} \\frac{1}{2j-1}$ be the odd harmonic numbers (so $O_0 = 0$) and let ${a \\brace b}_c$ denote the r-Stirling numbers of the second kind. For all natural numbers $n$ and $r$,","title":"Central binomial sums weighted by $1 - O_{k-1}$, in r-Stirling form"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-four-pow-central-binom-odd-harmonic","kind":"node","lean":"MetaMathlibExt.sum_four_pow_centralBinom_oddHarmonic_eq_stirling","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power-weighted convolution of "],["math","\\(4^k\\)"],["text"," with "],["math","\\(\\binom{2m}{m} O_m\\)"]],"source":"Power-weighted convolution of $4^k$ with $\\binom{2m}{m} O_m$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-four-pow-central-binom-odd-harmonic","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $O_m = \\sum_{j=1}^{m} \\frac{1}{2j-1}$ be the odd harmonic numbers (so $O_0 = 0$) and let ${r \\brace k}$ denote the Stirling numbers of the second kind. For all natural numbers $n$ and $r$ (with $0^0 = 1$),","title":"Power-weighted convolution of $4^k$ with $\\binom{2m}{m} O_m$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-four-pow-central-binom-one-sub-odd-h","kind":"node","lean":"MetaMathlibExt.sum_four_pow_centralBinom_one_sub_oddHarmonic_eq_stirling","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power-weighted convolution of "],["math","\\(4^k\\)"],["text"," with "],["math","\\(\\binom{2m}{m} (1 - O_{m-1})/(2m-1)\\)"]],"source":"Power-weighted convolution of $4^k$ with $\\binom{2m}{m} (1 - O_{m-1})/(2m-1)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-four-pow-central-binom-one-sub-odd-h","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $O_m = \\sum_{j=1}^{m} \\frac{1}{2j-1}$ be the odd harmonic numbers (so $O_0 = 0$) and let ${r \\brace k}$ denote the Stirling numbers of the second kind. For all natural numbers $n$ and $r$ (with $0^0 = 1$),","title":"Power-weighted convolution of $4^k$ with $\\binom{2m}{m} (1 - O_{m-1})/(2m-1)$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-alternating-inv-central-binom-odd-ha","kind":"node","lean":"MetaMathlibExt.sum_alternating_inv_centralBinom_oddHarmonic_eq_stirling","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating reciprocal central binomial sum with "],["math","\\(O_{n+1} - O_{k+1}\\)"]],"source":"Alternating reciprocal central binomial sum with $O_{n+1} - O_{k+1}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-alternating-inv-central-binom-odd-ha","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $O_m = \\sum_{j=1}^{m} \\frac{1}{2j-1}$ be the odd harmonic numbers (so $O_0 = 0$) and let ${r \\brace k}$ denote the Stirling numbers of the second kind. For all natural numbers $n$ and $r$ (with $0^0 = 1$),","title":"Alternating reciprocal central binomial sum with $O_{n+1} - O_{k+1}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-alternating-inv-central-binom-odd-ha-2","kind":"node","lean":"MetaMathlibExt.sum_alternating_inv_centralBinom_oddHarmonic_sub_eq_stirling","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating reciprocal central binomial sum with "],["math","\\(O_n - O_k\\)"]],"source":"Alternating reciprocal central binomial sum with $O_n - O_k$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-alternating-inv-central-binom-odd-ha-2","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $O_m = \\sum_{j=1}^{m} \\frac{1}{2j-1}$ be the odd harmonic numbers (so $O_0 = 0$) and let ${r \\brace k}$ denote the Stirling numbers of the second kind. For all natural numbers $n$ and $r$ (with $0^0 = 1$),","title":"Alternating reciprocal central binomial sum with $O_n - O_k$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-choose-sub-harmonic-diff-stirling","kind":"node","lean":"MetaMathlibExt.sum_choose_sub_harmonic_diff_stirling","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power sums of "],["math","\\(\\binom{t}{n-k}(H_t - H_{t-n+k})\\)"],["text"," in Stirling form"]],"source":"Power sums of $\\binom{t}{n-k}(H_t - H_{t-n+k})$ in Stirling form"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-choose-sub-harmonic-diff-stirling","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $t$ be a complex number that is not an integer. For complex $x$ write $\\binom{x}{k} = x(x-1)\\cdots(x-k+1)/k!$, and write $H_x = \\psi(x+1) + \\gamma$, where $\\psi$ is the digamma function and $\\gamma$ is the Euler\u2013Mascheroni constant (only differences $H_x - H_y = \\psi(x+1) - \\psi(y+1)$ occur below). Let ${r \\brace k}$ denote the Stirling numbers of the second kind. For all natural numbers $n$ and $r$ (with\u2026","title":"Power sums of $\\binom{t}{n-k}(H_t - H_{t-n+k})$ in Stirling form"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/sum-choose-sub-harmonic-diff-stirling-du","kind":"node","lean":"MetaMathlibExt.sum_choose_sub_harmonic_diff_stirling_dual","parent":26178,"reading_title":{"format":"inline-parts-v1","parts":[["text","Power sums of "],["math","\\(\\binom{n-t}{n-k}(H_{k-t} - H_{n-t})\\)"],["text"," in Stirling form"]],"source":"Power sums of $\\binom{n-t}{n-k}(H_{k-t} - H_{n-t})$ in Stirling form"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers/index.html#sum-choose-sub-harmonic-diff-stirling-du","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/central-binomial-sums-with-harmonic-numbers","status":"can_prove","summary":"Let $t$ be a complex number that is not an integer. For complex $x$ write $\\binom{x}{k} = x(x-1)\\cdots(x-k+1)/k!$, and write $H_x = \\psi(x+1) + \\gamma$, where $\\psi$ is the digamma function and $\\gamma$ is the Euler\u2013Mascheroni constant (only differences $H_x - H_y = \\psi(x+1) - \\psi(y+1)$ occur below). Let ${r \\brace k}$ denote the Stirling numbers of the second kind. For all natural numbers $n$ and $r$ (with\u2026","title":"Power sums of $\\binom{n-t}{n-k}(H_{k-t} - H_{n-t})$ in Stirling form"},{"data_url":"scopes/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien.json","id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien","kind":"scope","navigation":"module","parent":26171,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien/index.html","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien","status":"fully_proved","title":"Knuth's identity for central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien/knuth-catalan-log-sq-canonical","kind":"node","lean":"MetaMathlibExt.knuth_catalan_log_sq_canonical","parent":26198,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien/index.html#knuth-catalan-log-sq-canonical","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien","status":"fully_proved","summary":"Let $x$ be a real number with $|x| \\le 1/4$. Write $H_k = \\sum_{j=1}^{k} \\frac{1}{j}$ for the $k$-th harmonic number and $C_n$ for the $n$-th Catalan number. Then the series","title":"Knuth's identity in harmonic-number notation"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien/knuth-catalan-log-sq","kind":"node","lean":"MetaMathlibExt.knuth_catalan_log_sq","parent":26198,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien/index.html#knuth-catalan-log-sq","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/knuth-s-identity-for-central-binomial-coefficien","status":"fully_proved","summary":"Let $x$ be a real number with $|x| \\le 1/4$, and let $C_n$ denote the $n$-th Catalan number. Then the series","title":"Knuth's identity for central binomial coefficients and harmonic numbers"},{"data_url":"scopes/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums.json","id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","kind":"scope","navigation":"module","parent":26171,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/index.html","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","status":"can_prove","title":"Koll\u00e1r's incomplete binomial sums"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/inv-pow-two-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.IncompleteBinomialHarmonicSumsWanted.invPowTwoSum","parent":26201,"reading_title":{"format":"inline-parts-v1","parts":[["text","Koll\u00e1r's auxiliary sum "],["math","\\(C_n\\)"]],"source":"Koll\u00e1r's auxiliary sum $C_n$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/index.html#inv-pow-two-sum","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","status":"fully_proved","summary":"For a natural number $n$, $C_n$ is the rational number $C_n = \\sum_{k=1}^{n} \\frac{1}{k \\, 2^k}$.","title":"Koll\u00e1r's auxiliary sum $C_n$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/central-binom-weighted-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.IncompleteBinomialHarmonicSumsWanted.centralBinomWeightedSum","parent":26201,"reading_title":{"format":"inline-parts-v1","parts":[["text","Koll\u00e1r's auxiliary sum "],["math","\\(D_n\\)"]],"source":"Koll\u00e1r's auxiliary sum $D_n$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/index.html#central-binom-weighted-sum","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","status":"fully_proved","summary":"For a natural number $n$, $D_n$ is the rational number $D_n = \\sum_{k=1}^{n} \\frac{\\binom{2k}{k}}{2k \\cdot 2^{2k}}$.","title":"Koll\u00e1r's auxiliary sum $D_n$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/sum-range-choose-mul-eq-sum-range-choose","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.IncompleteBinomialHarmonicSumsWanted.sum_range_choose_mul_eq_sum_range_choose_add","parent":26201,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/index.html#sum-range-choose-mul-eq-sum-range-choose","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","status":"can_prove","summary":"Let $(a_k)_{k \\ge 0}$ be a sequence of real numbers and let $m$ and $N$ be positive integers with $m \\le N$. Then","title":"Koll\u00e1r's recursion for incomplete binomial sums"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/sum-range-choose-two-mul-sub-one-mul-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.IncompleteBinomialHarmonicSumsWanted.sum_range_choose_two_mul_sub_one_mul_eq","parent":26201,"reading_title":{"format":"inline-parts-v1","parts":[["text","Koll\u00e1r's recursion for incomplete binomial sums with top index "],["math","\\(2n-1\\)"]],"source":"Koll\u00e1r's recursion for incomplete binomial sums with top index $2n-1$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/index.html#sum-range-choose-two-mul-sub-one-mul-eq","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","status":"can_prove","summary":"Let $(a_k)_{k \\ge 0}$ be a sequence of real numbers, let $n$ be a positive integer and let $m$ be a natural number with $m \\le n$. Then","title":"Koll\u00e1r's recursion for incomplete binomial sums with top index $2n-1$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/sum-range-choose-two-mul-mul-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.IncompleteBinomialHarmonicSumsWanted.sum_range_choose_two_mul_mul_eq","parent":26201,"reading_title":{"format":"inline-parts-v1","parts":[["text","Koll\u00e1r's recursion for incomplete binomial sums with top index "],["math","\\(2n\\)"]],"source":"Koll\u00e1r's recursion for incomplete binomial sums with top index $2n$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/index.html#sum-range-choose-two-mul-mul-eq","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","status":"can_prove","summary":"Let $(a_k)_{k \\ge 0}$ be a sequence of real numbers and let $m$ and $n$ be positive integers with $m \\le n$. Then","title":"Koll\u00e1r's recursion for incomplete binomial sums with top index $2n$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/sum-range-choose-two-mul-sub-one-mul-har","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.IncompleteBinomialHarmonicSumsWanted.sum_range_choose_two_mul_sub_one_mul_harmonic","parent":26201,"reading_title":{"format":"inline-parts-v1","parts":[["text","Half-range binomial sum of harmonic numbers with top index "],["math","\\(2n-1\\)"]],"source":"Half-range binomial sum of harmonic numbers with top index $2n-1$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/index.html#sum-range-choose-two-mul-sub-one-mul-har","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","status":"can_prove","summary":"Let $H_i$ denote the $i$-th harmonic number (with $H_0 = 0$), and let Koll\u00e1r's auxiliary sums be $C_n = \\sum_{k=1}^{n} \\frac{1}{k \\, 2^k}$ and $D_n = \\sum_{k=1}^{n} \\frac{\\binom{2k}{k}}{2k \\cdot 2^{2k}}$. For every positive integer $n$,","title":"Half-range binomial sum of harmonic numbers with top index $2n-1$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/sum-range-choose-two-mul-mul-harmonic","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.IncompleteBinomialHarmonicSumsWanted.sum_range_choose_two_mul_mul_harmonic","parent":26201,"reading_title":{"format":"inline-parts-v1","parts":[["text","Half-range binomial sum of harmonic numbers with top index "],["math","\\(2n\\)"]],"source":"Half-range binomial sum of harmonic numbers with top index $2n$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums/index.html#sum-range-choose-two-mul-mul-harmonic","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/koll-r-s-incomplete-binomial-sums","status":"can_prove","summary":"Let $H_i$ denote the $i$-th harmonic number (with $H_0 = 0$), and let Koll\u00e1r's auxiliary sums be $C_n = \\sum_{k=1}^{n} \\frac{1}{k \\, 2^k}$ and $D_n = \\sum_{k=1}^{n} \\frac{\\binom{2k}{k}}{2k \\cdot 2^{2k}}$. For every positive integer $n$,","title":"Half-range binomial sum of harmonic numbers with top index $2n$"},{"data_url":"scopes/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien.json","id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","kind":"scope","navigation":"module","parent":26171,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","title":"Series of reciprocal central binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/tintegrand","kind":"node","lean":"MetaMathlibExt.TIntegrand","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#tintegrand","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"Let $S$ be a sequence of real polynomials indexed by the natural numbers, with terms $S(l)$. For a natural number $l$ and a real number $x$, the integrand is","title":"Integrand of the antiderivative condition"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/is-poly-tfamily","kind":"node","lean":"MetaMathlibExt.IsPolyTFamily","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#is-poly-tfamily","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"Let $S$ and $T$ be sequences of real polynomials indexed by the natural numbers, with terms $S(l)$ and $T(l)$. The pair $(S, T)$ satisfies this predicate when $T(0) = 0$; for every $l \\ge 1$ the polynomial $T(l)$ is nonzero and has degree $l - 1$; and for every $l \\ge 1$ and every real $x$ with $0 < x < 4$, the function $y \\mapsto 2\\sqrt{y}\\,T(l)(y)$ is differentiable at $x$ with derivative","title":"Polynomial family defined by an antiderivative"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/tsum-div-central-binom-mul-eq-multisecti","kind":"node","lean":"MetaMathlibExt.tsum_div_centralBinom_mul_eq_multisection","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multisection formula for "],["math","\\(\\sum_k x^k / \\binom{2mk}{mk}\\)"]],"source":"Multisection formula for $\\sum_k x^k / \\binom{2mk}{mk}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#tsum-div-central-binom-mul-eq-multisecti","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"Let $m \\ge 1$ be a natural number and $x$ a complex number with $|x| < 4^m$. Write $H(y) = \\sum_{n \\ge 0} y^n / \\binom{2n}{n}$, $w = e^{2\\pi i/m}$, and let $x^{1/m}$ be the principal $m$-th root of $x$. Then","title":"Multisection formula for $\\sum_k x^k / \\binom{2mk}{mk}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/tsum-div-mul-central-binom-mul-eq-multis","kind":"node","lean":"MetaMathlibExt.tsum_div_mul_centralBinom_mul_eq_multisection","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multisection formula for "],["math","\\(\\sum_k x^k / (k \\binom{2mk}{mk})\\)"]],"source":"Multisection formula for $\\sum_k x^k / (k \\binom{2mk}{mk})$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#tsum-div-mul-central-binom-mul-eq-multis","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"Let $m \\ge 1$ be a natural number and $x$ a complex number with $|x| < 4^m$. Write $L(y) = \\sum_{n \\ge 1} y^n / (n \\binom{2n}{n})$, $w = e^{2\\pi i/m}$, and let $x^{1/m}$ be the principal $m$-th root of $x$. Then","title":"Multisection formula for $\\sum_k x^k / (k \\binom{2mk}{mk})$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/tsum-div-sq-mul-central-binom-mul-eq-mul","kind":"node","lean":"MetaMathlibExt.tsum_div_sq_mul_centralBinom_mul_eq_multisection","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multisection formula for "],["math","\\(\\sum_k x^k / (k^2 \\binom{2mk}{mk})\\)"]],"source":"Multisection formula for $\\sum_k x^k / (k^2 \\binom{2mk}{mk})$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#tsum-div-sq-mul-central-binom-mul-eq-mul","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"Let $m \\ge 1$ be a natural number and $x$ a complex number with $|x| < 4^m$. Write $K(y) = \\sum_{n \\ge 1} y^n / (n^2 \\binom{2n}{n})$, $w = e^{2\\pi i/m}$, and let $x^{1/m}$ be the principal $m$-th root of $x$. Then","title":"Multisection formula for $\\sum_k x^k / (k^2 \\binom{2mk}{mk})$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/central-binom-reciprocal-h","kind":"node","lean":"MetaMathlibExt.centralBinomReciprocalH","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#central-binom-reciprocal-h","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For every real number $x$, let","title":"Closed form of the reciprocal central binomial series"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/central-binom-reciprocal-h-zero","kind":"node","lean":"MetaMathlibExt.centralBinomReciprocalH_zero","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","The closed form "],["math","\\(H\\)"],["text"," equals "],["math","\\(1\\)"],["text"," at "],["math","\\(0\\)"]],"source":"The closed form $H$ equals $1$ at $0$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#central-binom-reciprocal-h-zero","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"The function $H(x) = \\frac{4\\sqrt{x}\\,\\arcsin(\\sqrt{x}/2)}{(4 - x)\\sqrt{4 - x}} + \\frac{4}{4 - x}$ of a real variable satisfies $H(0) = 1$.","title":"The closed form $H$ equals $1$ at $0$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-qcoeff","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivQCoeff","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-qcoeff","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For natural numbers $l \\ge 1$ and $k$, the rational number $q_k = q_{l,k}$ is given by $q_l = -4/l$, by $q_k = \\frac{4k + 6}{k} q_{k+1}$ for $1 \\le k < l$, by $q_0 = -12 q_1$, and by $q_k = 0$ for $k > l$. These are the coefficients of Zurita's polynomial $Q_l$. For $l = 0$ the definition gives $q_{0,k} = 0$ for every $k$.","title":"Coefficients of Zurita's Q-polynomials"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-qcoeff-self","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivQCoeff_self","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-qcoeff-self","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For every natural number $l \\ge 1$, the coefficient $q_l = q_{l,l}$ of $x^l$ in Zurita's polynomial $Q_l$ equals $-4/l$.","title":"Leading coefficient of a Q-polynomial"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-qcoeff-of-lt","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivQCoeff_of_lt","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-qcoeff-of-lt","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"Write $q_k = q_{l,k}$ for the coefficient of $x^k$ in Zurita's polynomial $Q_l$. For all natural numbers $k$ and $l$ with $1 \\le k < l$,","title":"Recursion for the coefficients of a Q-polynomial"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-qcoeff-zero","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivQCoeff_zero","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-qcoeff-zero","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For every natural number $l \\ge 1$, the constant coefficient $q_0$ and the coefficient $q_1$ of $x$ in Zurita's polynomial $Q_l$ satisfy $q_0 = -12 q_1$.","title":"Constant coefficient of a Q-polynomial"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-q","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivQ","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-q","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $l \\ge 1$, $Q_l$ is the polynomial with rational coefficients $Q_l(x) = \\sum_{i=0}^{l} q_i x^i$, where $q_l = -4/l$, $q_i = \\frac{4i + 6}{i} q_{i+1}$ for $1 \\le i < l$, and $q_0 = -12 q_1$. For $l = 0$ it is the constant polynomial $Q_0 = 8$.","title":"Zurita's Q-polynomials"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-c","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivC","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-c","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $l \\ge 1$, the rational constant $C_l$ is $C_l = -q_0/2$, where $q_0$ is the constant coefficient of Zurita's polynomial $Q_l$. For $l = 0$ it is $C_0 = -4$.","title":"Zurita's arcsine antiderivative constants"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-c-one-two","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivC_one_two","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-c-one-two","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"Zurita's constants satisfy $C_1 = -24$ and $C_2 = -120$, as in Table 1 of the paper.","title":"The first two arcsine antiderivative constants"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-scoeff","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivSCoeff","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of Zurita's polynomial "],["math","\\(S_l\\)"]],"source":"Coefficients of Zurita's polynomial $S_l$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-scoeff","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For natural numbers $l$ and $k$, the rational number $s_k$ (which depends on $l$) is $0$ when $k > l$, and when $k \\le l$ it is $s_k = \\frac{8}{1 - 2l} \\prod_{i=k}^{l-1} \\frac{8(i + 1)}{2i - 1}$. Equivalently, $s_l = 8/(1 - 2l)$ and $s_k = \\frac{8(k + 1)}{2k - 1} s_{k+1}$ for $0 \\le k < l$.","title":"Coefficients of Zurita's polynomial $S_l$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-scoeff-self","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivSCoeff_self","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-scoeff-self","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For every natural number $l$, the coefficient $s_l = s_{l,l}$ of $x^l$ in Zurita's polynomial $S_l$ equals $8/(1 - 2l)$.","title":"Leading coefficient of an S-polynomial"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-scoeff-of-lt","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivSCoeff_of_lt","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-scoeff-of-lt","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"Write $s_k = s_{l,k}$ for the coefficient of $x^k$ in Zurita's polynomial $S_l$. For all natural numbers $k$ and $l$ with $k < l$,","title":"Recursion for the coefficients of an S-polynomial"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/arcsin-antideriv-s","kind":"node","lean":"MetaMathlibExt.arcsinAntiderivS","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#arcsin-antideriv-s","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $l$, $S_l$ is the polynomial with rational coefficients $S_l(x) = \\sum_{i=0}^{l} s_i x^i$, where $s_l = 8/(1 - 2l)$ and $s_i = \\frac{8(i + 1)}{2i - 1} s_{i+1}$ for $0 \\le i < l$.","title":"Zurita's S-polynomials"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/has-deriv-at-arcsin-antideriv-p-pow-mul","kind":"node","lean":"MetaMathlibExt.hasDerivAt_arcsinAntiderivP_pow_mul_centralBinomReciprocalH","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Antiderivative of "],["math","\\(x^l H(x)\\)"]],"source":"Antiderivative of $x^l H(x)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#has-deriv-at-arcsin-antideriv-p-pow-mul","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"Let $l \\ge 1$, let $Q_l$ and $C_l$ be Zurita's polynomial and constant, and let $H$ be the closed form of the reciprocal central binomial series. Then there is a polynomial $P$ with rational coefficients such that $P(0) = 0$ and $2(4 - x) P'(x) = 8x^l - Q_l(x)$ as polynomials. Moreover, for every such $P$ and every real number $x$ with $0 < x < 4$, the real function","title":"Antiderivative of $x^l H(x)$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/has-deriv-at-arcsin-antideriv-t-pow-mul","kind":"node","lean":"MetaMathlibExt.hasDerivAt_arcsinAntiderivT_pow_mul_centralBinomReciprocalH","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Antiderivative of "],["math","\\(x^{l - 1/2} H(x)\\)"]],"source":"Antiderivative of $x^{l - 1/2} H(x)$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#has-deriv-at-arcsin-antideriv-t-pow-mul","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"Let $l \\ge 0$, let $S_l$ be Zurita's polynomial, and let $H$ be the closed form of the reciprocal central binomial series. Then there is a polynomial $T$ with rational coefficients such that $(4 - x)(2x T'(x) + T(x)) = 8x^l - S_l(x)$ as polynomials. Moreover, for every such $T$ and every real number $x$ with $0 < x < 4$, the real function","title":"Antiderivative of $x^{l - 1/2} H(x)$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-scoeff","kind":"node","lean":"MetaMathlibExt.zuritaSCoeff","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of Zurita's polynomial "],["math","\\(S_n\\)"]],"source":"Coefficients of Zurita's polynomial $S_n$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-scoeff","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $n$, the rational numbers $s_0, s_1, s_2, \\dots$ are defined by $s_0 = 1/(1/2 - n)$ and $s_k = \\frac{k - 1 - n}{k - n + 1/2}\\, s_{k-1}$ for every $k \\ge 1$. Their first $n + 1$ terms are the coefficients of $S_n(x) = \\sum_{k=0}^{n} s_k x^k$.","title":"Coefficients of Zurita's polynomial $S_n$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-c","kind":"node","lean":"MetaMathlibExt.zuritaC","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's constant "],["math","\\(c_n\\)"]],"source":"Zurita's constant $c_n$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-c","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $n$, let $s_n$ be the coefficient of $x^n$ in Zurita's polynomial $S_n(x) = \\sum_{k=0}^n s_k x^k$. Then $c_n$ is the rational number $s_n/2 - 1$. For $n = 1$ this gives $c_1 = 1$, the paper's convention.","title":"Zurita's constant $c_n$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-tcoeff-rev","kind":"node","lean":"MetaMathlibExt.zuritaTCoeffRev","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reversed coefficients of Zurita's polynomial "],["math","\\(T_{n-2}\\)"]],"source":"Reversed coefficients of Zurita's polynomial $T_{n-2}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-tcoeff-rev","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"Fix a natural number $n$, let $s_0, s_1, \\dots$ be the coefficients of $S_n$ and let $c_n = s_n/2 - 1$. The rational numbers $u_0, u_1, \\dots$ are defined by $u_0 = -s_{n-1}/2 - c_n$ and $u_{i+1} = \\frac{(k + 1 - n)\\, u_i + s_k/2}{k - n}$ with $k = n - 2 - i$, where the index subtractions $n - 1$ and $n - 2 - i$ are truncated at $0$. For $n \\ge 2$ they list the coefficients of $T_{n-2}$ backwards, $t_{n-2-i} = u_i$\u2026","title":"Reversed coefficients of Zurita's polynomial $T_{n-2}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-tcoeff","kind":"node","lean":"MetaMathlibExt.zuritaTCoeff","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of Zurita's polynomial "],["math","\\(T_{n-2}\\)"]],"source":"Coefficients of Zurita's polynomial $T_{n-2}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-tcoeff","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $t_k = u_{n-2-k}$, where $n - 2 - k$ is truncated at $0$ and $u_0, u_1, \\dots$ are the reversed coefficients of $T_{n-2}$: with $s_j$ the coefficients of $S_n$ and $c_n = s_n/2 - 1$, $u_0 = -s_{n-1}/2 - c_n$ and $u_{i+1} = ((m + 1 - n) u_i + s_m/2)/(m - n)$ for $m = n - 2 - i$, the indices $n - 1$ and $n - 2 - i$ being truncated at $0$. For $k \\le n - 2$ these are the coefficients\u2026","title":"Coefficients of Zurita's polynomial $T_{n-2}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-mcoeff","kind":"node","lean":"MetaMathlibExt.zuritaMCoeff","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's coefficients "],["math","\\(m_k\\)"]],"source":"Zurita's coefficients $m_k$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-mcoeff","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $n$, the rational numbers $m_0, m_1, \\dots$ are defined by $m_0 = 2/(2n - 1)$ and $m_k = \\frac{2(k - n)}{2k - 2n + 1}\\, m_{k-1}$ for $k \\ge 1$. Zurita's polynomial $M_{n-1}$ is $M_{n-1}(x) = \\sum_{k=0}^{n-1} m_k x^k$.","title":"Zurita's coefficients $m_k$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-d","kind":"node","lean":"MetaMathlibExt.zuritaD","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's constant "],["math","\\(d_n\\)"]],"source":"Zurita's constant $d_n$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-d","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $n$, let $m_0, m_1, \\dots$ be Zurita's coefficients, given by $m_0 = 2/(2n - 1)$ and $m_k = \\frac{2(k - n)}{2k - 2n + 1}\\, m_{k-1}$ for $k \\ge 1$, so that $M_{n-1}(x) = \\sum_{k=0}^{n-1} m_k x^k$ for $n \\ge 1$. Then $d_n$ is the rational number $m_{n-1}/2$ for $n \\ge 1$, and $d_0 = m_0/2 = -1$. For $n = 1$ this gives $d_1 = 1$, the paper's convention.","title":"Zurita's constant $d_n$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-ncoeff","kind":"node","lean":"MetaMathlibExt.zuritaNCoeff","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of Zurita's polynomial "],["math","\\(N_{n-2}\\)"]],"source":"Coefficients of Zurita's polynomial $N_{n-2}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-ncoeff","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For natural numbers $n$ and $k$, let $m_k$ be Zurita's coefficient, given by $m_0 = 2/(2n - 1)$ and $m_k = \\frac{2(k - n)}{2k - 2n + 1}\\, m_{k-1}$ for $k \\ge 1$. This is the rational number $m_k / (2(k - n + 1))$, taken to be $0$ when $k = n - 1$. For $k \\le n - 2$ it is the coefficient of $x^k$ in $N_{n-2}(x) = \\frac12 \\sum_{i=0}^{n-2} \\frac{m_i x^i}{i - n + 1}$.","title":"Coefficients of Zurita's polynomial $N_{n-2}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-table-values","kind":"node","lean":"MetaMathlibExt.zurita_table_values","parent":26209,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-table-values","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For $n = 2$ the coefficients $s_0, s_1, s_2$ of $S_2$ are $-2/3$, $-8/3$ and $16/3$, and the constants $c_2$ and $c_4$ equal $5/3$ and $93/35$. For $n = 3$ the coefficients $t_0, t_1$ of $T_1$ are $-1/10$ and $-3/5$, the coefficients $m_0, m_1, m_2$ of $M_2$ are $2/5$, $8/15$ and $16/15$, the constant $d_3$ equals $8/15$, and the coefficients of $x^0$ and $x^1$ in $N_1$ are $-3/30$ and $-8/30$.","title":"Values from Zurita's Tables 1-4"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-s","kind":"node","lean":"MetaMathlibExt.zuritaS","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's polynomial "],["math","\\(S_n\\)"]],"source":"Zurita's polynomial $S_n$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-s","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $n$ and a real number $x$, $S_n(x) = \\sum_{k=0}^{n} s_k x^k$, where $s_0 = 1/(1/2 - n)$ and $s_k = \\frac{k - 1 - n}{k - n + 1/2}\\, s_{k-1}$ for $k \\ge 1$ are the coefficients of $S_n$.","title":"Zurita's polynomial $S_n$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-t","kind":"node","lean":"MetaMathlibExt.zuritaT","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's polynomial "],["math","\\(T_{n-2}\\)"]],"source":"Zurita's polynomial $T_{n-2}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-t","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $n$ and a real number $x$, $T_{n-2}(x) = \\sum_{k=0}^{n-2} t_k x^k$, where $t_0, t_1, \\dots$ are the coefficients of Zurita's polynomial $T_{n-2}$; for $n \\le 1$ the sum is empty and $T_{n-2}(x) = 0$. For $n \\ge 2$ these coefficients are given by $t_{n-2} = -s_{n-1}/2 - c_n$ and $t_{k-1} = ((k + 1 - n) t_k + s_k/2)/(k - n)$ for $k = n - 2, \\dots, 1$, where $s_0 = 1/(1/2 - n)$ and\u2026","title":"Zurita's polynomial $T_{n-2}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-m","kind":"node","lean":"MetaMathlibExt.zuritaM","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's polynomial "],["math","\\(M_{n-1}\\)"]],"source":"Zurita's polynomial $M_{n-1}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-m","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $n$, let $m_0, m_1, \\dots$ be Zurita's coefficients, given by $m_0 = 2/(2n - 1)$ and $m_k = \\frac{2(k - n)}{2k - 2n + 1}\\, m_{k-1}$ for $k \\ge 1$. Then $M_{n-1}$ is the real polynomial function $M_{n-1}(x) = \\sum_{k=0}^{n-1} m_k x^k$ (so $M_{-1} = 0$).","title":"Zurita's polynomial $M_{n-1}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/zurita-n","kind":"node","lean":"MetaMathlibExt.zuritaN","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's polynomial "],["math","\\(N_{n-2}\\)"]],"source":"Zurita's polynomial $N_{n-2}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#zurita-n","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"fully_proved","summary":"For a natural number $n$, let $m_0, m_1, \\dots$ be Zurita's coefficients, given by $m_0 = 2/(2n - 1)$ and $m_k = \\frac{2(k - n)}{2k - 2n + 1}\\, m_{k-1}$ for $k \\ge 1$. Then $N_{n-2}$ is the real polynomial function $N_{n-2}(x) = \\frac12 \\sum_{k=0}^{n-2} \\frac{m_k x^k}{k - n + 1}$, which is $0$ for $n \\le 1$.","title":"Zurita's polynomial $N_{n-2}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/has-deriv-at-zurita-antiderivative-one","kind":"node","lean":"MetaMathlibExt.hasDerivAt_zurita_antiderivative_one","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's antiderivative of "],["math","\\(x^{-n-1/2}\\arcsin(\\sqrt{x})/(1-x)^{3/2}\\)"]],"source":"Zurita's antiderivative of $x^{-n-1/2}\\arcsin(\\sqrt{x})/(1-x)^{3/2}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#has-deriv-at-zurita-antiderivative-one","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"Let $n \\ge 1$, let $c_n$ be Zurita's constant and let $S_n$ and $T_{n-2}$ be Zurita's polynomials (with $T_{-1} = 0$). For every real $x$ with $0 < x < 1$, the function","title":"Zurita's antiderivative of $x^{-n-1/2}\\arcsin(\\sqrt{x})/(1-x)^{3/2}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/has-deriv-at-zurita-antiderivative-two","kind":"node","lean":"MetaMathlibExt.hasDerivAt_zurita_antiderivative_two","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's antiderivative of "],["math","\\(x^{-n-1/2}\\arcsin(\\sqrt{x})/\\sqrt{1-x}\\)"]],"source":"Zurita's antiderivative of $x^{-n-1/2}\\arcsin(\\sqrt{x})/\\sqrt{1-x}$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#has-deriv-at-zurita-antiderivative-two","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"Let $n \\ge 1$, let $d_n$ be Zurita's constant and let $M_{n-1}$ and $N_{n-2}$ be Zurita's polynomials (with $N_{-1} = 0$). For every real $x$ with $0 < x < 1$, the function","title":"Zurita's antiderivative of $x^{-n-1/2}\\arcsin(\\sqrt{x})/\\sqrt{1-x}$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/tsum-four-pow-div-sub-mul-central-binom","kind":"node","lean":"MetaMathlibExt.tsum_four_pow_div_sub_mul_centralBinom","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's series "],["math","\\(\\sum_{k > n} 4^k x^k / ((k - n)\\binom{2k}{k})\\)"]],"source":"Zurita's series $\\sum_{k > n} 4^k x^k / ((k - n)\\binom{2k}{k})$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#tsum-four-pow-div-sub-mul-central-binom","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"Let $n \\ge 1$, let $S_n(x) = \\sum_{k=0}^n s_k x^k$ and $T_{n-2}$ be Zurita's polynomials (with $T_{-1} = 0$), and let $x$ be real with $0 < x < 1$. Then","title":"Zurita's series $\\sum_{k > n} 4^k x^k / ((k - n)\\binom{2k}{k})$"},{"id":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/tsum-four-pow-div-sub-sq-mul-central-bin","kind":"node","lean":"MetaMathlibExt.tsum_four_pow_div_sub_sq_mul_centralBinom","parent":26209,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zurita's series "],["math","\\(\\sum_{k > n} 4^k x^k / ((k - n)^2\\binom{2k}{k})\\)"]],"source":"Zurita's series $\\sum_{k > n} 4^k x^k / ((k - n)^2\\binom{2k}{k})$"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien/index.html#tsum-four-pow-div-sub-sq-mul-central-bin","scope":"Combinatorics/binomial-coefficients/central-binomial-harmonic-sums/series-of-reciprocal-central-binomial-coefficien","status":"can_prove","summary":"For $j \\ge 1$ write Zurita's polynomials as $S_j(x) = \\sum_{k=0}^{j} s^{(j)}_k x^k$, $T_{j-2}(x) = \\sum_{k=0}^{j-2} t^{(j)}_k x^k$ and $M_{j-1}(x) = \\sum_{k=0}^{j-1} m^{(j)}_k x^k$. Let $n \\ge 1$, let $x$ be real with $0 < x < 1$, and put","title":"Zurita's series $\\sum_{k > n} 4^k x^k / ((k - n)^2\\binom{2k}{k})$"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients","kind":"scope","parent":26022,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients","status":"can_prove","summary":"Variants of the binomial coefficient and of Pascal's triangle: b-ary and digital (carry-free) binomial coefficients, Ward and totienomial coefficients from generalized factorials, trinomial and quadrinomial coefficients, rascal, hyperbolic Pascal and 2-Pascal triangles and their diagonals; the Pascal matrix viewed as a Riordan array goes to Riordan arrays.","title":"Generalized binomial coefficients and Pascal-like triangles"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients","status":"fully_proved","title":"b-nomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/b-nomial-base","kind":"node","lean":"MetaMathlibExt.bNomialBase","parent":26247,"reading_title":{"format":"inline-parts-v1","parts":[["text","Base polynomial of the "],["math","\\(b\\)"],["text","-nomial coefficients"]],"source":"Base polynomial of the $b$-nomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/index.html#b-nomial-base","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients","status":"fully_proved","summary":"For a natural number $b \\ge 2$, this is the integer polynomial $1 + X + \\dots + X^{b-1} = \\sum_{i=0}^{b-1} X^i \\in \\mathbb{Z}[X]$.","title":"Base polynomial of the $b$-nomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/b-nomial-coefficient","kind":"node","lean":"MetaMathlibExt.bNomialCoefficient","parent":26247,"reading_title":{"format":"inline-parts-v1","parts":[["text","Type-1 "],["math","\\(b\\)"],["text","-nomial coefficients"]],"source":"Type-1 $b$-nomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/index.html#b-nomial-coefficient","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients","status":"fully_proved","summary":"For natural numbers $b \\ge 2$ and $n$ and an integer $k$, the type-1 $b$-nomial coefficient in row $n$ and column $k$ is the coefficient of $X^k$ in $(1 + X + \\dots + X^{b-1})^n \\in \\mathbb{Z}[X]$ when $k \\ge 0$, and $0$ when $k < 0$.","title":"Type-1 $b$-nomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/b-nomial-digit-at-or-zero","kind":"node","lean":"MetaMathlibExt.bNomialDigitAtOrZero","parent":26247,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/index.html#b-nomial-digit-at-or-zero","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients","status":"fully_proved","summary":"For a $b$-ary string $x = (x_0, \\dots, x_{n-1})$ of length $n$, written from left to right, and a natural number $j$, this is the digit $x_j$ if $j < n$, and $0$ if $j \\ge n$.","title":"Digits of a string extended by zero"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/is-bnomial-indispensable-digit","kind":"node","lean":"MetaMathlibExt.IsBNomialIndispensableDigit","parent":26247,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/index.html#is-bnomial-indispensable-digit","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients","status":"fully_proved","summary":"Let $x = (x_0, \\dots, x_{n-1})$ be a $b$-ary string of length $n$, written from left to right, and extend it by zero on the right, setting $x_j = 0$ for $j \\ge n$. A position $p < n$ is indispensable in $x$ when there is a natural number $r$ with $1 \\le r \\le n + 1 - p$ such that $x_{p+t} = x_p$ for every $t$ with $1 \\le t < r$, and $x_{p+r} < x_p$. That is, some nonempty constant run of digits beginning at\u2026","title":"Indispensable digits"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/b-nomial-indispensable-digit-count","kind":"node","lean":"MetaMathlibExt.bNomialIndispensableDigitCount","parent":26247,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/index.html#b-nomial-indispensable-digit-count","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients","status":"fully_proved","summary":"For a $b$-ary string $x$ of finite length, $\\iota(x)$ is the number of positions of $x$ that are indispensable.","title":"Number of indispensable digits"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/b-nomial-number-type3","kind":"node","lean":"MetaMathlibExt.bNomialNumberType3","parent":26247,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients/index.html#b-nomial-number-type3","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/b-nomial-coefficients","status":"fully_proved","summary":"For a natural number $b \\ge 2$, a natural number $n$ and an integer $k$, the type-3 $b$-nomial number is the number of $b$-ary strings $x$ of length $n$ with exactly $k$ indispensable digits, that is, with $\\iota(x) = k$. In particular it is zero for $k < 0$.","title":"Type-3 b-nomial numbers"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","title":"Base-b digitwise binomial and multinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-binomial-coefficient","kind":"node","lean":"MetaMathlibExt.bAryBinomialCoefficient","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(b\\)"],["text","-ary binomial coefficient"]],"source":"The $b$-ary binomial coefficient"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-binomial-coefficient","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$ and let $n$ and $k$ be natural numbers with base-$b$ digits $n_l$ and $k_l$, taken to be $0$ beyond the last digit (with $0$ having no digits). The $b$-ary binomial coefficient of $n$ and $k$ is $\\prod_{l=0}^{m-1} \\binom{n_l}{k_l}$, where $m$ is the larger of the numbers of base-$b$ digits of $n$ and of $k$.","title":"The $b$-ary binomial coefficient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-binomial-coefficient-eq-prod-get-d","kind":"node","lean":"MetaMathlibExt.bAryBinomialCoefficient_eq_prod_getD","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digitwise product formula for the "],["math","\\(b\\)"],["text","-ary binomial coefficient"]],"source":"Digitwise product formula for the $b$-ary binomial coefficient"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-binomial-coefficient-eq-prod-get-d","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$, let $n$ and $k$ be natural numbers with base-$b$ digits $n_i$ and $k_i$ (taken to be $0$ beyond the last digit, and with $0$ having no digits), and let $L$ be at least the number of base-$b$ digits of $n$ and at least the number of base-$b$ digits of $k$. Then the $b$-ary binomial coefficient of $n$ and $k$ equals $\\prod_{i=0}^{L-1} \\binom{n_i}{k_i}$.","title":"Digitwise product formula for the $b$-ary binomial coefficient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-binomial-coefficient-generating-fu","kind":"node","lean":"MetaMathlibExt.bAryBinomialCoefficient_generating_function","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the "],["math","\\(b\\)"],["text","-ary binomial coefficients"]],"source":"Generating function of the $b$-ary binomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-binomial-coefficient-generating-fu","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $x \\in R$, let $b \\ge 2$, and write $\\binom{n}{k}_b$ for the $b$-ary binomial coefficient of natural numbers $n$ and $k$. Let $n$ have base-$b$ digits $n_0, \\dots, n_{m-1}$, where $m$ is the number of base-$b$ digits of $n$ ($m = 0$ for $n = 0$). Then","title":"Generating function of the $b$-ary binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/base-digits-binomial-generating-function","kind":"node","lean":"MetaMathlibExt.base_digits_binomial_generating_function","parent":26254,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#base-digits-binomial-generating-function","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $x \\in R$, let $b \\ge 2$, and let $n$ have base-$b$ digits $n_0, \\dots, n_{m-1}$, where $m$ is the number of base-$b$ digits of $n$ ($m = 0$ for $n = 0$). For each natural number $k$ let $k_l$ be its $l$-th base-$b$ digit, taken to be $0$ beyond the last digit. Then","title":"Generating function of digitwise binomial products"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-binomial-coefficient-eq-prod-div-p","kind":"node","lean":"MetaMathlibExt.bAryBinomialCoefficient_eq_prod_div_pow","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digit-product formula for "],["math","\\(b\\)"],["text","-ary binomial coefficients"]],"source":"Digit-product formula for $b$-ary binomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-binomial-coefficient-eq-prod-div-p","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$ and let $n$, $k$ and $L$ be natural numbers with $n < b^L$ and $k < b^L$. Then the $b$-ary binomial coefficient $\\binom{n}{k}_b$ equals","title":"Digit-product formula for $b$-ary binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-binomial-coefficient-symm","kind":"node","lean":"MetaMathlibExt.bAryBinomialCoefficient_symm","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Symmetry of "],["math","\\(b\\)"],["text","-ary binomial coefficients"]],"source":"Symmetry of $b$-ary binomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-binomial-coefficient-symm","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$ and write $\\binom{n}{k}_b$ for the $b$-ary binomial coefficient. For all natural numbers $k \\le n$, $\\binom{n}{k}_b = \\binom{n}{n-k}_b$.","title":"Symmetry of $b$-ary binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-binomial-coefficient-pascal","kind":"node","lean":"MetaMathlibExt.bAryBinomialCoefficient_pascal","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pascal's recurrence for "],["math","\\(b\\)"],["text","-ary binomial coefficients"]],"source":"Pascal's recurrence for $b$-ary binomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-binomial-coefficient-pascal","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$ and write $\\binom{n}{k}_b$ for the $b$-ary binomial coefficient. For all natural numbers $n$ and $k$ with $0 < k \\le n$ and $\\binom{n}{k}_b \\ne 0$, $\\binom{n}{k}_b = \\binom{n-1}{k-1}_b + \\binom{n-1}{k}_b$.","title":"Pascal's recurrence for $b$-ary binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/bary-binomial-symmetry-and-pascal","kind":"node","lean":"MetaMathlibExt.baryBinomial_symmetry_and_pascal","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Symmetry and Pascal's recurrence for "],["math","\\(b\\)"],["text","-ary binomial coefficients"]],"source":"Symmetry and Pascal's recurrence for $b$-ary binomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#bary-binomial-symmetry-and-pascal","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$. For a natural number $m$ write $m_i = \\lfloor m / b^i \\rfloor \\bmod b$ for its $i$-th base-$b$ digit. For integers $m$ and $j$, let $C_b(m, j) = \\prod_{i=0}^{m+j} \\binom{m_i}{j_i}$ if $m \\ge 0$ and $j \\ge 0$, and $C_b(m, j) = 0$ otherwise. Then for all integers $n$ and $k$ with $0 < n$ and $0 \\le k \\le n$, we have $C_b(n, k) = C_b(n, n - k)$, and if $C_b(n, k) \\ne 0$ then\u2026","title":"Symmetry and Pascal's recurrence for $b$-ary binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-binomial-coefficient-chu-vandermon","kind":"node","lean":"MetaMathlibExt.bAryBinomialCoefficient_chu_vandermonde","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Chu\u2013Vandermonde identity for "],["math","\\(b\\)"],["text","-ary binomial coefficients"]],"source":"Chu\u2013Vandermonde identity for $b$-ary binomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-binomial-coefficient-chu-vandermon","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$ and write $\\binom{n}{k}_b$ for the $b$-ary binomial coefficient. Let $m$ and $n$ be natural numbers that add without carries in base $b$, that is, $\\lfloor m / b^l \\rfloor \\bmod b + \\lfloor n / b^l \\rfloor \\bmod b < b$ for every natural number $l$. Then for every natural number $r$,","title":"Chu\u2013Vandermonde identity for $b$-ary binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/bary-chu-vandermonde","kind":"node","lean":"MetaMathlibExt.bary_chu_vandermonde","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(b\\)"],["text","-ary Chu\u2013Vandermonde identity"]],"source":"$b$-ary Chu\u2013Vandermonde identity"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#bary-chu-vandermonde","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$. For a natural number $x$ write $x_l = \\lfloor x / b^l \\rfloor \\bmod b$ for its $l$-th base-$b$ digit, and for natural numbers $x$ and $y$ let $C(x, y) = \\prod_{l=0}^{x+y} \\binom{x_l}{y_l}$. Let $m$, $n$ and $r$ be natural numbers such that the addition of $m$ and $n$ in base $b$ has no carries, that is, $m_l + n_l < b$ for every $l \\le m + n$. Then","title":"$b$-ary Chu\u2013Vandermonde identity"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-digit","kind":"node","lean":"MetaMathlibExt.bAryDigit","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Base-"],["math","\\(b\\)"],["text"," digit of a natural number"]],"source":"Base-$b$ digit of a natural number"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-digit","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"For natural numbers $b$, $n$ and $l$, the rank-$l$ base-$b$ digit $n_l$ of $n$ is the entry at position $l$ (counting from $0$) of the list of base-$b$ digits of $n$, least significant digit first, and $0$ when $l$ is at least the length of that list (with $0$ having no digits). For $b \\ge 2$ this is the coefficient of $b^l$ in the base-$b$ expansion of $n$.","title":"Base-$b$ digit of a natural number"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-lower-digits","kind":"node","lean":"MetaMathlibExt.bAryLowerDigits","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank-"],["math","\\(l\\)"],["text"," base-"],["math","\\(b\\)"],["text"," digits of a list"]],"source":"Rank-$l$ base-$b$ digits of a list"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-lower-digits","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"For natural numbers $b$ and $l$ and a list $(k_1, \\dots, k_m)$ of natural numbers, this is the list $((k_1)_l, \\dots, (k_m)_l)$, where $(k_i)_l$ is the rank-$l$ base-$b$ digit of $k_i$.","title":"Rank-$l$ base-$b$ digits of a list"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/ordinary-multinomial-factor","kind":"node","lean":"MetaMathlibExt.ordinaryMultinomialFactor","parent":26254,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#ordinary-multinomial-factor","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"For a natural number $t$ and a list $(a_1, \\dots, a_m)$ of natural numbers, the multinomial factor of $t$ over $(a_1, \\dots, a_m)$ is $\\frac{t!}{a_1! \\cdots a_m!}$ if $a_1 + \\dots + a_m = t$, and $0$ otherwise.","title":"Multinomial coefficient with a prescribed top"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-multinomial-factor","kind":"node","lean":"MetaMathlibExt.bAryMultinomialFactor","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digitwise "],["math","\\(b\\)"],["text","-ary multinomial factor"]],"source":"Digitwise $b$-ary multinomial factor"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-multinomial-factor","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"For natural numbers $b$, $n$ and $l$ and a list $(k_1, \\dots, k_m)$ of natural numbers, let $n_l$ and $(k_i)_l$ be the rank-$l$ base-$b$ digits of $n$ and $k_i$. The factor at rank $l$ is $\\frac{n_l!}{(k_1)_l! \\cdots (k_m)_l!}$ if $(k_1)_l + \\dots + (k_m)_l = n_l$, and $0$ otherwise.","title":"Digitwise $b$-ary multinomial factor"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-multinomial-bound","kind":"node","lean":"MetaMathlibExt.bAryMultinomialBound","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","Digit-length bound for the "],["math","\\(b\\)"],["text","-ary multinomial coefficient"]],"source":"Digit-length bound for the $b$-ary multinomial coefficient"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-multinomial-bound","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"For natural numbers $b$ and $n$ and a list $(k_1, \\dots, k_m)$ of natural numbers, the bound is the largest of the numbers of base-$b$ digits of $n, k_1, \\dots, k_m$ (with $0$ having no digits).","title":"Digit-length bound for the $b$-ary multinomial coefficient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/b-ary-multinomial","kind":"node","lean":"MetaMathlibExt.bAryMultinomial","parent":26254,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(b\\)"],["text","-ary multinomial coefficient"]],"source":"The $b$-ary multinomial coefficient"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi/index.html#b-ary-multinomial","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/base-b-digitwise-binomial-and-multinomial-coeffi","status":"fully_proved","summary":"Let $b \\ge 2$, let $n$ be a natural number and let $(k_1, \\dots, k_m)$ be a list of natural numbers, and write $n_l$ and $(k_i)_l$ for the rank-$l$ base-$b$ digits of $n$ and $k_i$ (taken to be $0$ beyond the last digit, with $0$ having no digits). Let $N$ be the largest of the numbers of base-$b$ digits of $n, k_1, \\dots, k_m$. The $b$-ary multinomial coefficient of $n$ over $(k_1, \\dots, k_m)$ is\u2026","title":"The $b$-ary multinomial coefficient"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","title":"Digital binomial theorem"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/base-b-generalized-digital-binomial-theo","kind":"node","lean":"MetaMathlibExt.base_b_generalized_digital_binomial_theorem","parent":26271,"reading_title":{"format":"inline-parts-v1","parts":[["text","Base-"],["math","\\(b\\)"],["text"," generalized digital binomial theorem"]],"source":"Base-$b$ generalized digital binomial theorem"},"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html#base-b-generalized-digital-binomial-theo","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","summary":"Let $b \\ge 2$, let $N$ and $n$ be natural numbers with $n < b^N$, and let $x$ and $y$ be real numbers. For a natural number $m$ write $m_i = \\lfloor m / b^i \\rfloor \\bmod b$ for its $i$-th base-$b$ digit, and write $z^{\\overline{j}} = z(z+1)\\cdots(z+j-1)$ for the rising factorial. Then","title":"Base-$b$ generalized digital binomial theorem"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/bin-digit-sum","kind":"node","lean":"MetaMathlibExt.binDigitSum","parent":26271,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html#bin-digit-sum","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","summary":"For a natural number $n$, the binary digit sum $s(n)$ is the sum of the digits of $n$ in base $2$, that is, the number of ones in the binary expansion of $n$.","title":"Binary digit sum"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/bin-digit-sum-zero","kind":"node","lean":"MetaMathlibExt.binDigitSum_zero","parent":26271,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html#bin-digit-sum-zero","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","summary":"The binary digit sum of $0$ is $0$.","title":"Binary digit sum of zero"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/bin-digit-sum-two-mul","kind":"node","lean":"MetaMathlibExt.binDigitSum_two_mul","parent":26271,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html#bin-digit-sum-two-mul","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","summary":"For every natural number $k$, the binary digit sum of $2k$ equals the binary digit sum of $k$: doubling appends the binary digit $0$.","title":"Binary digit sum of an even number"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/bin-digit-sum-two-mul-add-one","kind":"node","lean":"MetaMathlibExt.binDigitSum_two_mul_add_one","parent":26271,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html#bin-digit-sum-two-mul-add-one","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","summary":"For every natural number $k$, the binary digit sum of $2k + 1$ is one more than the binary digit sum of $k$: the binary expansion of $2k + 1$ is that of $k$ followed by the digit $1$.","title":"Binary digit sum of an odd number"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/bin-digit-sum-one","kind":"node","lean":"MetaMathlibExt.binDigitSum_one","parent":26271,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html#bin-digit-sum-one","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","summary":"The binary digit sum of $1$ is $1$.","title":"Binary digit sum of one"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/digital-binomial-theorem-general","kind":"node","lean":"MetaMathlibExt.digital_binomial_theorem_general","parent":26271,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html#digital-binomial-theorem-general","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","summary":"Let $s(m)$ denote the binary digit sum of a natural number $m$. For every natural number $n$ and all elements $x$, $y$ of a commutative semiring, $(x + y)^{s(n)}$ equals the sum of $x^{s(m)} y^{s(n - m)}$ over all $m$ with $0 \\le m \\le n$ and $s(m) + s(n - m) = s(n)$.","title":"Digital binomial theorem over a commutative semiring"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/digital-binomial-theorem","kind":"node","lean":"MetaMathlibExt.digital_binomial_theorem","parent":26271,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem/index.html#digital-binomial-theorem","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/digital-binomial-theorem","status":"fully_proved","summary":"Let $s(m)$ denote the binary digit sum of a natural number $m$. For every natural number $n$ and all elements $x$, $y$ of a commutative ring, $(x + y)^{s(n)}$ equals the sum of $x^{s(m)} y^{s(n - m)}$ over all $m$ with $0 \\le m \\le n$ and $s(m) + s(n - m) = s(n)$.","title":"Digital binomial theorem"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle","status":"can_prove","title":"Generalized hyperbolic Pascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/ghpt-interior","kind":"node","lean":"MetaMathlibExt.ghptInterior","parent":26280,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/index.html#ghpt-interior","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle","status":"fully_proved","summary":"Fix $q \\in \\mathbb{N}$ and a type $R$ with an addition. A vertex is recorded as a pair $(c, x)$ of a type code $c \\in \\mathbb{N}$ (code $0$ for the wingers at the ends of a row, $1$ for type $A$ and $2$ for type $B$) and a value $x \\in R$. Given a list $v_0, v_1, \\dots, v_m$ of such pairs with $m \\ge 1$ and $v_i = (c_i, x_i)$, the interior step lists, from left to right and for $i = 1, \\dots, m$, the type-$A$\u2026","title":"Interior step of the generalized hyperbolic Pascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/ghpt-row","kind":"node","lean":"MetaMathlibExt.ghptRow","parent":26280,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/index.html#ghpt-row","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle","status":"fully_proved","summary":"Fix $q \\in \\mathbb{N}$, a type $R$ with an addition, and leg sequences $\\alpha, \\beta \\colon \\mathbb{N} \\to R$ (left and right). Row $n$ of the generalized hyperbolic Pascal triangle linked to the mosaic $\\{4, q\\}$ is the list of its vertices from left to right, each recorded as a pair (type code, value) as in the interior step, with code $0$ for the wingers. Row $1$ is $(0, \\alpha_1), (0, \\beta_1)$, and for\u2026","title":"Rows of the generalized hyperbolic Pascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/ghpt-row-figure-one","kind":"node","lean":"MetaMathlibExt.ghptRow_figure_one","parent":26280,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/index.html#ghpt-row-figure-one","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle","status":"fully_proved","summary":"In the generalized hyperbolic Pascal triangle linked to $\\{4, 5\\}$ (so $q = 5$) with legs $\\alpha_n = 3n$ and $\\beta_n = 3^n$ in $\\mathbb{N}$, the values of the entries of row $3$, from left to right, are $9, 12, 6, 15, 27$, and those of row $4$ are $12, 21, 12, 18, 6, 6, 21, 15, 42, 81$.","title":"Rows 3 and 4 of an example generalized hyperbolic Pascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/linear-recurrence-inhomogeneous-identity","kind":"node","lean":"MetaMathlibExt.linearRecurrence_inhomogeneous_identity","parent":26280,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/index.html#linear-recurrence-inhomogeneous-identity","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle","status":"can_prove","summary":"Let $k \\in \\mathbb{N}$ and let $A, f, G, w \\colon \\mathbb{N} \\to \\mathbb{C}$ be sequences such that, for every $m \\ge k$, $f_m = \\sum_{i=1}^{k} A_i f_{m-i}$ and $G_m = \\sum_{i=1}^{k} A_i G_{m-i} + w_{m-k}$. Then for every $n \\ge k$,","title":"Convolution identity for an inhomogeneous linear recurrence"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/linear-recurrence-inhomogeneous-explicit","kind":"node","lean":"MetaMathlibExt.linearRecurrence_inhomogeneous_explicit","parent":26280,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/index.html#linear-recurrence-inhomogeneous-explicit","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle","status":"can_prove","summary":"Let $k \\in \\mathbb{N}$ and let $A, f, G, w \\colon \\mathbb{N} \\to \\mathbb{C}$ be sequences such that, for every $m \\ge k$, $f_m = \\sum_{i=1}^{k} A_i f_{m-i}$ and $G_m = \\sum_{i=1}^{k} A_i G_{m-i} + w_{m-k}$. Suppose moreover that $f_j = 0$ for every $j < k - 1$ and $f_{k-1} = 1$. Then for every $n \\ge k - 1$,","title":"Explicit formula for an inhomogeneous linear recurrence"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/ghpt-row-weighted-alternating-sum","kind":"node","lean":"MetaMathlibExt.ghptRow_weighted_alternating_sum","parent":26280,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle/index.html#ghpt-row-weighted-alternating-sum","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/generalized-hyperbolic-pascal-triangle","status":"can_prove","summary":"Let $q, n \\in \\mathbb{N}$, let $\\alpha, \\beta \\colon \\mathbb{N} \\to \\mathbb{C}$ and let $v, w \\in \\mathbb{C}$. Let $x_0, x_1, \\dots, x_{\\ell - 1}$ be the values of the entries of row $n$ of the generalized hyperbolic Pascal triangle linked to $\\{4, q\\}$ with legs $\\alpha$ and $\\beta$, from left to right, and put $\\hat s_n = \\sum_i x_i$, $\\tilde s_n = \\sum_i (-1)^i x_i$ and\u2026","title":"Weighted alternating row sums of the generalized hyperbolic Pascal triangle"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients","status":"fully_proved","title":"Quadrinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/polynomial-quadrinomial-coeff","kind":"node","lean":"MetaMathlibExt.polynomialQuadrinomialCoeff","parent":26287,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/index.html#polynomial-quadrinomial-coeff","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the quadrinomial coefficient $\\binom{n}{k}_4$ is the coefficient of $x^k$ in the polynomial $(1 + x + x^2 + x^3)^n$ with natural-number coefficients. These numbers form the triangle OEIS A008287.","title":"Quadrinomial coefficient triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/zhang-vsum","kind":"node","lean":"MetaMathlibExt.zhangVSum","parent":26287,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/index.html#zhang-vsum","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients","status":"fully_proved","summary":"For natural numbers $n$, $r_1$, $r_2$, $s$, $t$, $u$, Zhang's binomial sum is","title":"Zhang's six-parameter binomial sum"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/quadrinomial-coeff","kind":"node","lean":"MetaMathlibExt.quadrinomialCoeff","parent":26287,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/index.html#quadrinomial-coeff","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients","status":"fully_proved","summary":"For a natural number $n$, the quadrinomial number is $Q(n) = \\sum_{k=0}^{n} \\binom{n}{k} \\binom{n}{2k}$. This is OEIS sequence A005725.","title":"Quadrinomial sequence A005725"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/quadrinomial-coeff-eq-zhang-vsum","kind":"node","lean":"MetaMathlibExt.quadrinomialCoeff_eq_zhangVSum","parent":26287,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients/index.html#quadrinomial-coeff-eq-zhang-vsum","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/quadrinomial-coefficients","status":"fully_proved","summary":"For every natural number $n$, the quadrinomial number $Q(n) = \\sum_{k=0}^{n} \\binom{n}{k} \\binom{n}{2k}$ equals Zhang's six-parameter binomial sum $V(n, 1, 1, 0, 0, 0)$.","title":"Quadrinomial numbers as a specialization of Zhang's V-sum"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","title":"Rascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/is-generalized-rascal-triangle","kind":"node","lean":"MetaMathlibExt.IsGeneralizedRascalTriangle","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#is-generalized-rascal-triangle","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"An integer array $T(r, k)$, indexed by natural numbers $r$ and $k$, is a generalized rascal triangle when there exist integers $c$, $d$, $d_1$, $d_2$ such that $T(r, k) = c + k\\,d_1 + r\\,d_2 + r k\\,d$ for all natural numbers $r$ and $k$.","title":"Generalized rascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/word-ascents","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.wordAscents","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#word-ascents","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"For a finite word $w = w_1 w_2 \\cdots w_n$ of natural numbers, the number of ascents of $w$ is the number of indices $i$ with $1 \\le i < n$ and $w_i < w_{i+1}$.","title":"Number of ascents of a word"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-z","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalZ","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-z","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"For integers $n$ and $k$, the Rascal number $R_{n,k}$ is $k(n-k) + 1$ if $0 \\le k \\le n$, and $0$ otherwise.","title":"Rascal numbers on the integers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-word","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.RascalWord","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-word","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"For natural numbers $j$, $k$ and $n$, $B^{(j)}_k(n)$ is the set of words $w$ of length $n$ over the alphabet $\\{0, 1\\}$ with exactly $k$ ones and at most $j$ ascents, an ascent being a position $i$ with $w_i < w_{i+1}$, that is, a $0$ immediately followed by a $1$.","title":"Binary words with few ascents"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-gen","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalGen","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-gen","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"For natural numbers $j$, $k$ and $n$, the generalized Rascal number $R^{(j)}_{n,k}$ is the number of binary words of length $n$ with exactly $k$ ones and at most $j$ ascents, that is, the number of elements of $B^{(j)}_k(n)$; an ascent is a position at which a $0$ is immediately followed by a $1$.","title":"Generalized Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-gen-z","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalGenZ","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-gen-z","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"For a natural number $j$ and integers $n$ and $k$, $R^{(j)}_{n,k}$ is the generalized Rascal number of $n$ and $k$ (the number of binary words of length $n$ with exactly $k$ ones and at most $j$ ascents) when $0 \\le k \\le n$, and $0$ otherwise.","title":"Generalized Rascal numbers at integer indices"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/is-ascent-seq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.IsAscentSeq","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#is-ascent-seq","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"A finite sequence $w_1 w_2 \\cdots w_n$ of natural numbers is an ascent sequence when $w_1 = 0$ (if $n \\ge 1$) and $w_i \\le \\operatorname{asc}(w_1 \\cdots w_{i-1}) + 1$ for every $1 \\le i \\le n$, where $\\operatorname{asc}(v)$ counts the positions $j$ with $v_j < v_{j+1}$. In particular the empty sequence is an ascent sequence.","title":"Ascent sequences"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/avoids001210","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.Avoids001210","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#avoids001210","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"A finite sequence $w$ of natural numbers avoids $001$ and $210$ when it has no subsequence $x, x, y$ with $x < y$ and no subsequence $x, y, z$ with $x > y > z$. Subsequences need not consist of consecutive entries.","title":"Avoiding the patterns 001 and 210"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/ascent-seq-avoid","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.AscentSeqAvoid","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#ascent-seq-avoid","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"For natural numbers $n$ and $k$, this is the set of ascent sequences $w$ of length $n$ that avoid the patterns $001$ and $210$ and have exactly $k$ ascents, an ascent of $w$ being a position $i$ with $w_i < w_{i+1}$.","title":"Ascent sequences avoiding 001 and 210"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/apk-card","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.apkCard","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#apk-card","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"fully_proved","summary":"For natural numbers $n$ and $k$, this counts the ascent sequences of length $n$ with exactly $k$ ascents that avoid the patterns $001$ and $210$.","title":"Number of ascent sequences avoiding 001 and 210"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-z-eq-rascal-gen-z-one","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalZ_eq_rascalGenZ_one","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-z-eq-rascal-gen-z-one","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For all integers $n$ and $k$, $R_{n,k} = b_k(n)$, where $R_{n,k}$ is the Rascal number and $b_k(n)$ is the number of binary words of length $n$ with exactly $k$ ones and at most one ascent (taken to be $0$ unless $0 \\le k \\le n$).","title":"Rascal numbers count binary words with at most one ascent"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-z-symm","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalZ_symm","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-z-symm","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For integers $n$ and $k$ with $0 \\le k \\le n$, the Rascal numbers satisfy $R_{n,k} = R_{n,n-k}$.","title":"Symmetry of Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/card-prefix-ones-suffix-zeros","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.card_prefixOnes_suffixZeros","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#card-prefix-ones-suffix-zeros","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"Let $n, k, \\ell, u$ be natural numbers with $\\ell \\le k$ and $k + u \\le n$, and let $B_k(n)$ be the set of binary words $w$ of length $n$ with exactly $k$ ones and at most one ascent (a position $i$ with $w_i < w_{i+1}$). The number of words in $B_k(n)$ whose first $\\ell$ letters are all $1$ and whose last $u$ letters are all $0$ equals the Rascal number $R_{n-u-\\ell, k-\\ell}$, where $R_{a,b} = b(a-b) + 1$ for\u2026","title":"Binary words with a prefix of ones and a suffix of zeros"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-z-fleron","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalZ_fleron","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-z-fleron","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For integers $n$ and $k$ with $n \\ge 2$ and $0 < k < n$, the Rascal numbers $R_{n,k}$ satisfy $R_{n,k} = R_{n-1,k} + R_{n-1,k-1} - R_{n-2,k-1} + 1$.","title":"Fleron's recurrence for Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-z-mul-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalZ_mul_eq","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-z-mul-eq","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For integers $n$, $k$, $\\ell$, $u$ with $\\ell \\ge 0$, $u \\ge 0$ and $\\ell \\le k \\le n - u$, the Rascal numbers satisfy $R_{n,k} R_{n-u-\\ell,k-\\ell} = R_{n-u,k} R_{n-\\ell,k-\\ell} + u\\ell$.","title":"Generalized diamond recurrence for Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-z-add-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalZ_add_eq","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-z-add-eq","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For integers $n$, $m$ and $k$ with $m \\ge 0$ and $0 \\le k \\le n$,","title":"Row-shift identity for the Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/mul-rascal-z-sub-one","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.mul_rascalZ_sub_one","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#mul-rascal-z-sub-one","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For integers $n$ and $k$ with $0 < k < n$,","title":"Diagonal recurrence for Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-rascal-z-row","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_rascalZ_row","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-rascal-z-row","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For every natural number $n$, the Rascal numbers $R_{n,k} = k(n-k) + 1$ ($0 \\le k \\le n$) satisfy $\\sum_{k=0}^{n} R_{n,k} = \\binom{n+1}{3} + n + 1$.","title":"Row sums of the Rascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-rascal-z-column","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_rascalZ_column","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-rascal-z-column","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For all natural numbers $k$ and $r$, the Rascal numbers $R_{n,k} = k(n-k) + 1$ ($0 \\le k \\le n$) satisfy $\\sum_{i=0}^{r} R_{k+i,k} = k \\binom{r+1}{2} + r + 1$.","title":"Column sums of the Rascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/two-mul-sum-choose-mul-rascal-z","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.two_mul_sum_choose_mul_rascalZ","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#two-mul-sum-choose-mul-rascal-z","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For every natural number $n$, the Rascal numbers $R_{n,k} = k(n-k) + 1$ ($0 \\le k \\le n$) satisfy $2 \\sum_{k=0}^{n} \\binom{n}{k} R_{n,k} = 2^n \\binom{n}{2} + 2^{n+1}$.","title":"Binomial-weighted row sums of the Rascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-sum-rascal-z","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_sum_rascalZ","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-sum-rascal-z","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For every natural number $n \\ge 2$, the Rascal numbers $R_{i,j} = j(i-j) + 1$ ($0 \\le j \\le i$) satisfy $\\sum_{i=1}^{n} \\sum_{j=1}^{i-1} R_{i,j} = \\binom{n+2}{4} + \\binom{n}{2}$.","title":"Sum of the interior Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-neg-one-pow-choose-rascal-z","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_negOnePow_choose_rascalZ","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-neg-one-pow-choose-rascal-z","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For natural numbers $n$, $k$ and $r$ with $k \\le n$ and $r \\ge 2$, the Rascal numbers satisfy $\\sum_{j=0}^{r} (-1)^{r-j} \\binom{r}{j} R_{n+j,k} = 0$.","title":"Vanishing higher finite differences of Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-rascal-z-alternating-odd","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_rascalZ_alternating_odd","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-rascal-z-alternating-odd","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For every natural number $m$, the Rascal numbers $R_{n,k} = k(n-k) + 1$ ($0 \\le k \\le n$) satisfy $\\sum_{k=0}^{2m+1} (-1)^k R_{2m+1,k} = 0$.","title":"Alternating row sums of the Rascal triangle, odd rows"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-rascal-z-alternating-even","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_rascalZ_alternating_even","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-rascal-z-alternating-even","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For every natural number $m$, the Rascal numbers satisfy $\\sum_{k=0}^{2m} (-1)^k R_{2m,k} = 1 - m$.","title":"Alternating row sums of Rascal numbers on even rows"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/prod-rascal-z-sub-one-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.prod_rascalZ_sub_one_eq_sum","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#prod-rascal-z-sub-one-eq-sum","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For natural numbers $m$ and $n$ with $1 \\le m \\le n$, writing $[m] = \\{1, \\dots, m\\}$,","title":"Subset-sum expansion of the product of the shifted Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/prod-rascal-z-sub-one-eq-factorial-mul","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.prod_rascalZ_sub_one_eq_factorial_mul","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#prod-rascal-z-sub-one-eq-factorial-mul","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For natural numbers $m$ and $n$ with $1 \\le m \\le n$,","title":"Closed form for the product of the shifted Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/factorial-sq-mul-choose-eq-prod","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.factorial_sq_mul_choose_eq_prod","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#factorial-sq-mul-choose-eq-prod","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For natural numbers $n$ and $m$ with $1 \\le m \\le n$,","title":"Binomial coefficients from products of Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/ascent-seq-avoid-form","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.ascentSeqAvoid_form","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#ascent-seq-avoid-form","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers and let $w$ be an ascent sequence of length $n$ with exactly $k$ ascents that avoids the patterns $001$ and $210$. Then either $w$ consists of $0, 1, \\ldots, k-1$ followed by $n - k$ copies of $k$, or there are natural numbers $x$ and $y$ with $1 \\le y < n - k$ and $x < k$ such that $w$ consists of $0, 1, \\ldots, k-1$, followed by $y$ copies of $k$, followed by $n - k - y$ copies\u2026","title":"Shape of ascent sequences avoiding 001 and 210"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/apk-card-succ-eq-rascal-z","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.apkCard_succ_eq_rascalZ","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#apk-card-succ-eq-rascal-z","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For all natural numbers $n$ and $k$, the number of ascent sequences of length $n + 1$ with exactly $k$ ascents that avoid the patterns $001$ and $210$ equals the Rascal number $R_{n,k}$, which is $k(n-k) + 1$ when $k \\le n$ and $0$ when $k > n$.","title":"Ascent sequences avoiding 001 and 210 are counted by Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-gen-z-fleron","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalGenZ_fleron","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-gen-z-fleron","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"Let $j \\ge 1$ be a natural number and let $n$ and $k$ be integers with $n \\ge 2$ and $0 \\le k \\le n$. Then","title":"Generalized Fleron recurrence"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-gen-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalGen_eq_sum","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-gen-eq-sum","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For all natural numbers $j$, $n$ and $k$ with $k \\le n$, the generalized Rascal number $R^{(j)}_{n,k}$, the number of binary words of length $n$ with exactly $k$ ones and at most $j$ ascents (positions where a $0$ is immediately followed by a $1$), equals $\\sum_{i=0}^{j} \\binom{k}{i} \\binom{n-k}{i}$.","title":"Generalized Rascal numbers as sums of products of binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-rascal-gen-row","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_rascalGen_row","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-rascal-gen-row","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"Write $R^{(j)}_{n,k}$ for the number of binary words of length $n$ with exactly $k$ ones and at most $j$ ascents (positions where a $0$ is immediately followed by a $1$). For all natural numbers $j$ and $n$, $\\sum_{k=0}^{n} R^{(j)}_{n,k} = \\sum_{k=0}^{2j+1} \\binom{n}{k}$, where $\\binom{n}{k} = 0$ for $k > n$.","title":"Row sums of generalized Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-rascal-gen-four-mul-add-three","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_rascalGen_four_mul_add_three","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-rascal-gen-four-mul-add-three","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"Write $R^{(j)}_{n,k}$ for the number of binary words of length $n$ with exactly $k$ ones and at most $j$ ascents (positions where a $0$ is immediately followed by a $1$). For every natural number $j$, $\\sum_{k=0}^{4j+3} R^{(j)}_{4j+3,k} = 2^{4j+2}$.","title":"A power-of-two row sum of generalized Rascal numbers"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/iterate-forward-diff-sum-rascal-gen","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.iterate_forwardDiff_sum_rascalGen","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#iterate-forward-diff-sum-rascal-gen","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"Fix a natural number $j$. For natural numbers $n$ and $k$ let $R^{(j)}_{n,k}$ be the number of binary words of length $n$ with exactly $k$ ones and at most $j$ ascents, and let $S(n) = \\sum_{k=0}^{n} R^{(j)}_{n,k}$. Then $(\\Delta^{2j+1} S)(n) = 1$ for every natural number $n$, where $\\Delta$ is the forward difference $(\\Delta a)(n) = a(n+1) - a(n)$.","title":"Forward differences of generalized Rascal row sums"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-rascal-gen-alternating-odd","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_rascalGen_alternating_odd","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-rascal-gen-alternating-odd","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"Write $R^{(j)}_{n,k}$ for the number of binary words of length $n$ with exactly $k$ ones and at most $j$ ascents (positions where a $0$ is immediately followed by a $1$). For all natural numbers $j$ and $m$, $\\sum_{k=0}^{2m+1} (-1)^k R^{(j)}_{2m+1,k} = 0$.","title":"Alternating row sums of generalized Rascal numbers vanish on odd rows"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/sum-rascal-gen-alternating-even","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.sum_rascalGen_alternating_even","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#sum-rascal-gen-alternating-even","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"Write $R^{(j)}_{n,k}$ for the number of binary words of length $n$ with exactly $k$ ones and at most $j$ ascents (positions where a $0$ is immediately followed by a $1$). For natural numbers $j$ and $m$ with $m \\ge 1$, $\\sum_{k=0}^{2m} (-1)^k R^{(j)}_{2m,k} = (-1)^j \\binom{m-1}{j}$, and for $m = 0$ the sum equals $1$.","title":"Alternating row sums of generalized Rascal numbers on even rows"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/rascal-gen-eq-card-subset","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.RascalNumbersWanted.rascalGen_eq_card_subset","parent":26292,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle/index.html#rascal-gen-eq-card-subset","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/rascal-triangle","status":"can_prove","summary":"For all natural numbers $j$, $n$ and $k$, the generalized Rascal number $R^{(j)}_{n,k}$ equals the number of $k$-element subsets of $\\{0, 1, \\dots, n-1\\}$ that have at most $j$ elements smaller than $n - k$.","title":"Generalized Rascal numbers count subsets with few small elements"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","status":"fully_proved","title":"Totienomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/generalized-factorial","kind":"node","lean":"MetaMathlibExt.generalizedFactorial","parent":26330,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/index.html#generalized-factorial","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","status":"fully_proved","summary":"For a sequence $f \\colon \\mathbb{N} \\to \\mathbb{N}$ and a natural number $n$, the generalized factorial of $f$ at $n$ is the product $f(1) f(2) \\cdots f(n)$ of the values $f(i)$ for $1 \\le i \\le n$; for $n = 0$ it is the empty product $1$.","title":"Generalized factorial"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/is-positive-sequence","kind":"node","lean":"MetaMathlibExt.IsPositiveSequence","parent":26330,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/index.html#is-positive-sequence","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","status":"fully_proved","summary":"A sequence $f \\colon \\mathbb{N} \\to \\mathbb{N}$ is positive when $f(i) > 0$ for every $i > 0$; the value $f(0)$ is unrestricted.","title":"Positive sequence"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/generalized-factorial-pos","kind":"node","lean":"MetaMathlibExt.generalizedFactorial_pos","parent":26330,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/index.html#generalized-factorial-pos","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","status":"fully_proved","summary":"Let $f \\colon \\mathbb{N} \\to \\mathbb{N}$ be a positive sequence, that is $f(i) > 0$ for every $i > 0$. Then for every natural number $n$, the generalized factorial $f(1) f(2) \\cdots f(n)$ is positive.","title":"Generalized factorials of positive sequences are positive"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/generalized-binomial-coefficient","kind":"node","lean":"MetaMathlibExt.generalizedBinomialCoefficient","parent":26330,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/index.html#generalized-binomial-coefficient","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","status":"fully_proved","summary":"Let $f \\colon \\mathbb{N} \\to \\mathbb{N}$ be a sequence and write $k!_f = f(1) f(2) \\cdots f(k)$ for its generalized factorial, so that $0!_f = 1$. For natural numbers $n$ and $m$, the generalized binomial coefficient of $f$ is the rational number","title":"Generalized binomial coefficient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/generalized-binomial-coefficient-mul-den","kind":"node","lean":"MetaMathlibExt.generalizedBinomialCoefficient_mul_denominator","parent":26330,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/index.html#generalized-binomial-coefficient-mul-den","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","status":"fully_proved","summary":"Let $f \\colon \\mathbb{N} \\to \\mathbb{N}$ be a positive sequence, that is $f(i) > 0$ for every $i > 0$, and write $k!_f = f(1) f(2) \\cdots f(k)$ for its generalized factorial. Then for all natural numbers $n$ and $m$, the generalized binomial coefficient $\\binom{n+m}{n}_f$, defined as $(n+m)!_f / (n!_f \\, m!_f)$ with division by $0$ giving $0$, satisfies $\\binom{n+m}{n}_f \\cdot n!_f \\, m!_f = (n+m)!_f$ in\u2026","title":"Defining equation of the generalized binomial coefficient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/totienomial-coefficient","kind":"node","lean":"MetaMathlibExt.totienomialCoefficient","parent":26330,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/index.html#totienomial-coefficient","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","status":"fully_proved","summary":"Let $\\varphi$ be Euler's totient function and write $k!_\\varphi = \\varphi(1) \\varphi(2) \\cdots \\varphi(k)$. For natural numbers $n$ and $m$, the totienomial coefficient is the generalized binomial coefficient of $\\varphi$, that is the rational number","title":"Totienomial coefficient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/totienomial-integral-factorization","kind":"node","lean":"MetaMathlibExt.totienomial_integral_factorization","parent":26330,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients/index.html#totienomial-integral-factorization","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/totienomial-coefficients","status":"fully_proved","summary":"For all natural numbers $n$ and $m$, the totienomial coefficient $\\binom{n+m}{n}_\\varphi$ of Euler's totient function $\\varphi$ is a natural number, namely","title":"Integrality and prime factorization of totienomial coefficients"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","title":"Trinomial coefficients and the 2-Pascal triangle"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-coefficient","kind":"node","lean":"MetaMathlibExt.trinomialCoefficient","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-coefficient","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the trinomial coefficient in row $n$ and column $k$ is the coefficient of $X^k$ in the polynomial $(1 + X + X^2)^n$.","title":"Trinomial coefficient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-transform-sequence","kind":"node","lean":"MetaMathlibExt.trinomialTransformSequence","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-transform-sequence","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"fully_proved","summary":"Let $R$ be a semiring and $a \\colon \\mathbb{N} \\to R$ a sequence. The trinomial transform of $a$ is the sequence $b \\colon \\mathbb{N} \\to R$ given by","title":"Trinomial transform"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-transform-sequence-zero","kind":"node","lean":"MetaMathlibExt.trinomialTransformSequence_zero","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-transform-sequence-zero","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"fully_proved","summary":"For every semiring $R$ and every sequence $a \\colon \\mathbb{N} \\to R$, the zeroth term of the trinomial transform of $a$ equals $a_0$.","title":"Zeroth term of the trinomial transform"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-transform-sequence-zero-apply","kind":"node","lean":"MetaMathlibExt.trinomialTransformSequence_zero_apply","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-transform-sequence-zero-apply","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"fully_proved","summary":"For every semiring $R$ and every $n \\in \\mathbb{N}$, the $n$th term of the trinomial transform of the zero sequence $\\mathbb{N} \\to R$ is $0$.","title":"Trinomial transform of the zero sequence"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/riordan-array-sum-eq-coeff-subst","kind":"node","lean":"MetaMathlibExt.riordan_array_sum_eq_coeff_subst","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#riordan-array-sum-eq-coeff-subst","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $[t^m]F$ for the coefficient of $t^m$ in a power series $F$. Let $R$ be a commutative ring and let $d$, $h$, $f$ be power series over $R$ with $h(0) = 0$. Then for every $n \\in \\mathbb{N}$,","title":"Fundamental theorem of Riordan arrays"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-coefficient-eq-coeff","kind":"node","lean":"MetaMathlibExt.trinomialCoefficient_eq_coeff","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-coefficient-eq-coeff","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$, and $[t^m]\\,F$ for the coefficient of $t^m$ in a formal power series $F$ over $\\mathbb{Q}$. For all natural numbers $n$ and $k$, $\\binom{n}{k}_2 = [t^{2n}]\\,(1 - t + t^2)^n (1 - t)^{-1} \\left(t/(1-t)\\right)^k$.","title":"Trinomial coefficients as coefficients of a rational series"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-arith-prog","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_arith_prog","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-arith-prog","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1 + t + t^2)^n$. Let $n$, $x$, $b$ be natural numbers with $x < b$, let $z \\ne 0$ be a complex number, and let $w = e^{2\\pi i/b}$. Then","title":"Roots-of-unity filter for trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-coefficient-alternating-sums-m","kind":"node","lean":"MetaMathlibExt.trinomialCoefficient_alternating_sums_mod_three_five","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-coefficient-alternating-sums-m","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$, and $L_m$ for the Lucas numbers ($L_0 = 2$, $L_1 = 1$, $L_{m+2} = L_{m+1} + L_m$). Let $n$ and $x$ be natural numbers.","title":"Alternating trinomial sums over residue classes modulo 3 and 5"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-arith-prog-alt","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_arith_prog_alternating","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-arith-prog-alt","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1 + t + t^2)^n$. For natural numbers $n$, $x$, $b$ with $b$ even and $x < b$,","title":"Alternating sums of trinomial coefficients in an arithmetic progression"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-even","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_even","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-even","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$. For every $n \\in \\mathbb{N}$ and all complex numbers $z$ and $s$ with $s^2 = z$","title":"Generating polynomial of trinomial coefficients with even index"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-odd","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_odd","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-odd","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$. For every $n \\in \\mathbb{N}$ and all complex numbers $z$ and $s$ with $s^2 = z$ and $s \\ne 0$","title":"Generating polynomial of trinomial coefficients with odd index"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-mul-choose-alt","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_mul_choose_alternating","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-mul-choose-alt","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$. For all $n, m \\in \\mathbb{N}$","title":"Alternating binomial transform of a row of trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-mul-choose-eq","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_mul_choose_eq","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-mul-choose-eq","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$. For all $n, m \\in \\mathbb{N}$","title":"Two binomial transforms of a row of trinomial coefficients agree"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-sub-mul-choose","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_sub_mul_choose","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-sub-mul-choose","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$. For natural numbers $n \\le m$, $\\sum_{k=0}^{n} (-1)^k \\binom{n}{k} \\binom{n-k}{m}_2 = \\binom{n}{m-n}$.","title":"Alternating binomial sum down a trinomial column"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-two-mul-sub","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_two_mul_sub","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-two-mul-sub","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$. For every natural number $n$, $\\sum_{k=0}^{n} (-1)^k \\binom{n}{k} \\binom{2n-2k}{n}_2 = 2^n$.","title":"Alternating binomial sum of trinomial coefficients over even rows"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-convolution","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_convolution","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-convolution","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1 + t + t^2)^n$, and put $\\binom{n}{k}_2 = 0$ for negative integers $k$. Then for all natural numbers $n$, $m$ and every integer $l$,","title":"Vandermonde convolution for trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-sq-alternating","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_sq_alternating","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-sq-alternating","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$. For every natural number $n$, $\\sum_{k=0}^{2n} (-1)^k \\binom{n}{k}_2^2 = \\binom{n}{n}_2$.","title":"Alternating sum of squared trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-even-sq","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_even_sq","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-even-sq","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$. For every $n \\in \\mathbb{N}$ let $c_n = \\binom{n}{n/2}_2$ if $n$ is even and $c_n = 0$ if $n$ is odd. Then","title":"Sums of squares and products of trinomial coefficients with even index"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-coefficient-absorption","kind":"node","lean":"MetaMathlibExt.trinomialCoefficient_absorption","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-coefficient-absorption","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$. For natural numbers $n \\ge 1$ and $k$, $\\frac{k}{n} \\binom{n}{k}_2 = \\binom{n-1}{k-1}_2 + 2 \\binom{n-1}{k-2}_2$, where a term whose lower index is negative is read as $0$.","title":"Absorption identity for trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-diagonal-weigh","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_diagonal_weighted","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-diagonal-weigh","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$, $[t^m]\\,g$ for the coefficient of $t^m$ in a power series $g$, and $f_k = [t^k]\\,f$. For every integer $n > 0$ and every formal power series $f \\in \\mathbb{Q}[[t]]$","title":"Weighted diagonal sums of trinomial coefficients by coefficient extraction"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-diagonal-alter","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_diagonal_alternating","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-diagonal-alter","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$. For every integer $n > 0$ let $S_n = \\sum_{k=0}^{\\lfloor 2n/3 \\rfloor} \\binom{n-k}{k}_2 \\frac{n}{n-k} (-1)^k$, a rational number. Then $S_n = 1$ if $n \\equiv 1$ or $n \\equiv 3 \\pmod 4$, $S_n = -1$ if $n \\equiv 2 \\pmod 4$, and $S_n = 3$ if $n \\equiv 0 \\pmod 4$.","title":"Alternating weighted diagonal sum of trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-upper-half","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_upper_half","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-upper-half","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$. For every natural number $n$ and every complex number $w$,","title":"Generating sum of the upper half of a trinomial row"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-coefficient-eq-coeff-sqrt","kind":"node","lean":"MetaMathlibExt.trinomialCoefficient_eq_coeff_sqrt","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-coefficient-eq-coeff-sqrt","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$, and $[t^m]\\,F$ for the coefficient of $t^m$ in a formal power series $F$ over $\\mathbb{Q}$. Let $S$ be a power series over $\\mathbb{Q}$ with $S^2 = 1 - 2t - 3t^2$ and $S(0) = 1$. For natural numbers $n, a, b, k$ with $2a \\le b$, $2^{(b-a)k} \\binom{n+ak}{n+bk}_2 = [t^{n+bk}]\\, S^{-1} (1 - t - S)^{(b-a)k}$.","title":"Trinomial coefficients along lines from a square-root series"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/implicit-riordan-sum-inverse","kind":"node","lean":"MetaMathlibExt.implicit_riordan_sum_inverse","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#implicit-riordan-sum-inverse","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $[t^m]F$ for the coefficient of $t^m$ in a power series $F$ and $F'$ for its formal derivative. Let $K$ be a field and let $f$, $\\psi$, $g$, $b$ be power series over $K$ with $f(0) \\ne 0$, $\\psi(0) \\ne 0$ and $g(0) \\ne 0$. Put $h(t) = t\\,g(t)$ and, for $m \\in \\mathbb{N}$, $a_m = [t^m]\\,f(t)\\,\\psi(t)^m\\,b(h(t))$. Then for every $n \\in \\mathbb{N}$,","title":"Inversion of an implicit Riordan array sum"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-choose-mul-coeff-eq-coeff-subst","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_coeff_eq_coeff_subst","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-choose-mul-coeff-eq-coeff-subst","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $[t^j]F$ for the coefficient of $t^j$ in a power series $F$ over $\\mathbb{C}$, $\\binom{u}{j}$ for the generalized binomial coefficient, and $(1 + t)^u = \\sum_{j \\ge 0} \\binom{u}{j} t^j$ for the binomial series of a complex number $u$. Let $f$ be a power series over $\\mathbb{C}$. For natural numbers $n$, $x$, $a$, $b$ with $a < b$,","title":"Sprugnoli's binomial sum theorem"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/trinomial-coefficient-eq-coeff-alpha-mul","kind":"node","lean":"MetaMathlibExt.trinomialCoefficient_eq_coeff_alpha_mul_beta_pow","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#trinomial-coefficient-eq-coeff-alpha-mul","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$, and $[t^m]\\,F$ for the coefficient of $t^m$ in a formal power series $F$ over $\\mathbb{Q}$. Let $S$ and $Q$ be power series over $\\mathbb{Q}$ with $S^2 = 4 - 3t^2$, $S(0) = 2$, $2Q^2 = 2 - t^2 - tS$ and $Q(0) = 1$, and put","title":"Trinomial coefficients from an algebraic Riordan array"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-mul-coeff-eq-c","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_mul_coeff_eq_coeff_subst","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-mul-coeff-eq-c","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $[t^j]\\,g$ for the coefficient of $t^j$ in a power series $g$, and $f_k = [t^k]\\,f$. For $n \\in \\mathbb{N}$ let $\\binom{n}{j}_2 = [t^j]\\,(1+t+t^2)^n$ be the trinomial coefficient. For complex $u$ and $z$ let $(1 + zt + z^2t^2)^u$ be the binomial series $\\sum_{i \\ge 0} \\binom{u}{i} (zt + z^2t^2)^i$, and for complex $u$ let $\\binom{u}{j}_2 = [t^j]\\,(1+t+t^2)^u$. Let $n, x, a, b, m, c$ be natural numbers with\u2026","title":"Sums of trinomial coefficients along lines by coefficient extraction"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-choose-gould-and-extended-trinomial","kind":"node","lean":"MetaMathlibExt.sum_choose_gould_and_extended_trinomial","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-choose-gould-and-extended-trinomial","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"For complex $u$ and $j \\in \\mathbb{N}$ write $\\binom{u}{j}$ for the generalized binomial coefficient and $\\binom{u}{j}_2$ for the coefficient of $t^j$ in $(1 + t + t^2)^u = \\sum_{i \\ge 0} \\binom{u}{i}(t + t^2)^i$ (Comtet's extension). Then for all complex $x$, $y$ and every $n \\in \\mathbb{N}$,","title":"Gould-type identities for binomial and extended trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-extended-trinomial-alternating-eq","kind":"node","lean":"MetaMathlibExt.sum_extended_trinomial_alternating_eq","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-extended-trinomial-alternating-eq","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"For complex $u$ and $j \\in \\mathbb{N}$ write $\\binom{u}{j}_2$ for the coefficient of $t^j$ in $(1 + t + t^2)^u = \\sum_{i \\ge 0} \\binom{u}{i}(t + t^2)^i$ (Comtet's extension), where $\\binom{u}{i}$ is the generalized binomial coefficient. Then for every complex $x$ and every $n \\in \\mathbb{N}$,","title":"An alternating convolution of extended trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/two-pow-mul-tribonacci-eq","kind":"node","lean":"MetaMathlibExt.two_pow_mul_tribonacci_eq","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#two-pow-mul-tribonacci-eq","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Let $T_n$ be the Tribonacci numbers, $T_0 = T_1 = 0$, $T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$. Let $(y_n)_{n \\ge 0}$ be a sequence of integers with $y_0 = y_1 = 0$, $y_2 = 1$ and $y_{n+3} = 2y_{n+2} - 2y_{n+1} + 2y_n$ for all $n \\ge 0$ (the sequence A077943). Then for every $n \\ge 0$, $2^{n+1} T_{n+2} = y_{3(n+1)}$.","title":"Tribonacci numbers from every third term of A077943"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-diagonal-binet","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_diagonal_binet","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-diagonal-binet","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$. Let $z, w \\in \\mathbb{C}$ and let $r_1, r_2, r_3 \\in \\mathbb{C}$ be pairwise distinct with $r_1 + r_2 + r_3 = 2w$, $r_1 r_2 + r_1 r_3 + r_2 r_3 = 2w^2$ and $r_1 r_2 r_3 = z + w^3$, that is, let $r_1, r_2, r_3$ be the three distinct zeros of $t^3 - 2wt^2 + 2w^2 t - (z + w^3)$. Then for every $n \\in \\mathbb{N}$","title":"Binet-type formula for diagonal sums of trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/two-pow-mul-sum-trinomial-coefficient-di","kind":"node","lean":"MetaMathlibExt.two_pow_mul_sum_trinomialCoefficient_diagonal_weighted","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#two-pow-mul-sum-trinomial-coefficient-di","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $x^k$ in $(1+x+x^2)^n$. Let $(\\lambda_m)$ be the integer sequence with $\\lambda_0 = 3$, $\\lambda_1 = 2$, $\\lambda_2 = 0$ and $\\lambda_{m+3} = 2\\lambda_{m+2} - 2\\lambda_{m+1} + 2\\lambda_m$ (OEIS A075115). For every natural number $n \\ge 1$, $2^n \\sum_{k=0}^{\\lfloor 2n/3 \\rfloor} \\binom{n-k}{k}_2 \\frac{n}{n-k} = \\lambda_{3n}$.","title":"Weighted diagonal sums of trinomial coefficients and the sequence A075115"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-diagonal-weigh-2","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_diagonal_weighted_power_sum","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-diagonal-weigh-2","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1+t+t^2)^n$. Let $z, w \\in \\mathbb{C}$ and let $r_1, r_2, r_3 \\in \\mathbb{C}$ satisfy $r_1 + r_2 + r_3 = 2w$, $r_1 r_2 + r_1 r_3 + r_2 r_3 = 2w^2$ and $r_1 r_2 r_3 = z + w^3$, that is, let $r_1, r_2, r_3$ be the zeros, with multiplicity, of $t^3 - 2wt^2 + 2w^2 t - (z + w^3)$. Then for every integer $n > 0$","title":"Power-sum formula for weighted diagonal sums of trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/sum-trinomial-coefficient-add-sub-eq-tri","kind":"node","lean":"MetaMathlibExt.sum_trinomialCoefficient_add_sub_eq_tribonacci","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#sum-trinomial-coefficient-add-sub-eq-tri","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $\\binom{n}{k}_2$ for the trinomial coefficient, the coefficient of $t^k$ in $(1 + t + t^2)^n$, and $T_n$ for the Tribonacci numbers, $T_0 = T_1 = 0$, $T_2 = 1$ and $T_{n+3} = T_{n+2} + T_{n+1} + T_n$. Then for every $n \\in \\mathbb{N}$,","title":"Tribonacci formula for a diagonal sum of trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/coeff-two-mul-eq-inverse-trinomial-trans","kind":"node","lean":"MetaMathlibExt.coeff_two_mul_eq_inverse_trinomial_transform","parent":26338,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle/index.html#coeff-two-mul-eq-inverse-trinomial-trans","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/trinomial-coefficients-and-the-2-pascal-triangle","status":"can_prove","summary":"Write $[t^m]F$ for the coefficient of $t^m$ in a power series $F$. Let $S$, $Q$, $R$ be power series over $\\mathbb{Q}$ with $S^2 = 4 - 3t^2$, $S(0) = 2$, $2Q^2 = 2 - t^2 - tS$, $Q(0) = 1$, $R^2 = 1 - t + t^2$ and $R(0) = 1$, and put","title":"Recovering even coefficients from the 2-Pascal transform"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients","status":"fully_proved","title":"Central trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients/central-trinomial-coeff-three-term-recur","kind":"node","lean":"MetaMathlibExt.centralTrinomialCoeff_three_term_recurrence","parent":26374,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients/index.html#central-trinomial-coeff-three-term-recur","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients","status":"fully_proved","summary":"For a natural number $m$, let $T_m$ be the central trinomial coefficient, the coefficient of $x^m$ in $(1 + x + x^2)^m$. For every natural number $n$ the following holds, where for $n = 0$ the index $n - 1$ is read as $0$ (the last term then vanishes, since $3n = 0$):","title":"Three-term recurrence for central trinomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients/central-trinomial-coeff-eq-legendre-spec","kind":"node","lean":"MetaMathlibExt.centralTrinomialCoeff_eq_legendre_specialValue","parent":26374,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients/index.html#central-trinomial-coeff-eq-legendre-spec","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/central-trinomial-coefficients","status":"fully_proved","summary":"For a natural number $n$, let $c_n$ be the central trinomial coefficient, the coefficient of $X^n$ in $(1 + X + X^2)^n$, and let $P_n$ be the $n$th Legendre polynomial. Then, in the complex numbers,","title":"Central trinomial coefficients as Legendre values"},{"data_url":"scopes/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients.json","id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients","kind":"scope","navigation":"module","parent":26246,"result":false,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/index.html","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients","status":"fully_proved","title":"Ward's generalized binomial coefficients"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/ward-units-carry","kind":"node","lean":"MetaMathlibExt.wardUnitsCarry","parent":26377,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/index.html#ward-units-carry","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients","status":"fully_proved","summary":"For natural numbers $b$, $n$ and $m$, the units-place carry $\\varepsilon_0^{b,n,m}$ of $n + m$ in base $b$ is $1$ if $b > 1$ and $(n \\bmod b) + (m \\bmod b) \\ge b$, and $0$ otherwise; in particular it is $0$ for the invalid bases $b \\le 1$.","title":"Units-place carry"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/ward-generalized-binomial-coefficient","kind":"node","lean":"MetaMathlibExt.wardGeneralizedBinomialCoefficient","parent":26377,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/index.html#ward-generalized-binomial-coefficient","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a commutative monoid and $s = (s_b)_{b \\in \\mathbb{N}}$ a sequence in $R$. For natural numbers $n$ and $m$, the Ward generalized binomial coefficient is","title":"Ward generalized binomial coefficient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/ward-units-carry-eq-div","kind":"node","lean":"MetaMathlibExt.wardUnitsCarry_eq_div","parent":26377,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/index.html#ward-units-carry-eq-div","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients","status":"fully_proved","summary":"For all natural numbers $b$, $n$ and $m$, the units-place carry of $n + m$ in base $b$ (taken to be $0$ when $b \\le 1$) equals $\\lfloor ((n \\bmod b) + (m \\bmod b)) / b \\rfloor$, where $n \\bmod 0 = n$ and division by $0$ gives $0$.","title":"Units-place carry as a quotient"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/ward-generalized-binomial-coefficient-eq","kind":"node","lean":"MetaMathlibExt.wardGeneralizedBinomialCoefficient_eq_prod_Icc","parent":26377,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/index.html#ward-generalized-binomial-coefficient-eq","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a commutative monoid and $s = (s_b)_{b \\in \\mathbb{N}}$ a sequence in $R$. For all natural numbers $n$ and $m$, the Ward generalized binomial coefficient $\\binom{n+m}{n}_s$ equals","title":"Ward's generalized binomial coefficient as a finite product"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/ward-sbinomial-carry-product-general","kind":"node","lean":"MetaMathlibExt.ward_sbinomial_carry_product_general","parent":26377,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/index.html#ward-sbinomial-carry-product-general","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients","status":"fully_proved","summary":"Let $s, f : \\mathbb{N} \\to \\mathbb{Z}$ satisfy $s(b) \\ne 0$ for every $b \\ge 1$ and $f(k) = \\prod_{d \\mid k} s(d)$ for every $k \\ge 1$, the product running over the positive divisors of $k$. Write $f!(t) = \\prod_{i=1}^{t} f(i)$. Let $n$ and $m$ be natural numbers, and let $\\binom{n+m}{n}_s$ be the Ward generalized binomial coefficient of $s$ at $n$ and $m$. Then the identity","title":"Ward's identity with hypotheses at positive indices"},{"id":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/ward-sbinomial-carry-product","kind":"node","lean":"MetaMathlibExt.ward_sbinomial_carry_product","parent":26377,"result":true,"route":"../roadmap/Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients/index.html#ward-sbinomial-carry-product","scope":"Combinatorics/binomial-coefficients/generalized-binomial-coefficients/ward-s-generalized-binomial-coefficients","status":"fully_proved","summary":"Let $s, f : \\mathbb{N} \\to \\mathbb{Z}$ satisfy $s(b) \\ne 0$ for every natural number $b$ and $f(k) = \\prod_{d \\mid k} s(d)$ for every natural number $k$, the product running over the positive divisors of $k$ (empty for $k = 0$). Write $f!(t) = \\prod_{i=1}^{t} f(i)$. Then for all natural numbers $n$ and $m$, the identity","title":"Ward's identity for generalized binomial coefficients"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths.json","id":"Combinatorics/catalan-numbers-lattice-paths","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths","status":"can_prove","summary":"The Catalan family of counting sequences and the lattice paths they count: Catalan, Fuss\u2013Catalan, Narayana, Motzkin, Schr\u00f6der, Fine and Riordan numbers, Dyck and Motzkin paths and their statistics, Delannoy numbers and walks with prescribed steps, including arithmetic properties (parity, integrality, perfect powers) of these numbers.","title":"Catalan numbers and lattice paths"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers","kind":"scope","parent":26384,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers","status":"can_prove","summary":"Catalan numbers and generalizations defined by formulas or recurrences: Fuss\u2013Catalan, super Catalan, (p,q)-Patalan, Whitt and Sabar arrays, generalized Catalan\u2013Schr\u00f6der, Fine and Riordan numbers, Narayana numbers and polynomials, Touchard's identity, Raney's cycle lemma, and divisibility or growth of these numbers.","title":"Catalan numbers and their generalizations"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/gamma-closed-form","kind":"node","lean":"MetaMathlibExt.gamma_closed_form","parent":26385,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/index.html#gamma-closed-form","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers","status":"fully_proved","summary":"Let $\\gamma_{n,k}$, for natural numbers $n$ and $k$, be real numbers such that $\\gamma_{n,0} = 1$ for every $n$, $\\gamma_{2n-1,n} = 0$ for every $n \\ge 1$, and","title":"Closed form of the Eloe\u2013Kublik gamma array"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/palindromic-narayana-number","kind":"node","lean":"MetaMathlibExt.palindromicNarayanaNumber","parent":26385,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/index.html#palindromic-narayana-number","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the palindromic Narayana number is the rational number $\\tilde{N}(n, k) = \\binom{n}{k} \\binom{n+1}{k} / (k+1)$.","title":"Palindromic Narayana numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/touchard-catalan-identity","kind":"node","lean":"MetaMathlibExt.touchard_catalan_identity","parent":26385,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/index.html#touchard-catalan-identity","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers","status":"fully_proved","summary":"For every natural number $n$, writing $C_k = \\binom{2k}{k}/(k+1)$ for the $k$-th Catalan number,","title":"Touchard's Catalan identity"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/von-szily-super-catalan-identity","kind":"node","lean":"MetaMathlibExt.von_szily_super_catalan_identity","parent":26385,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/index.html#von-szily-super-catalan-identity","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers","status":"fully_proved","summary":"For all natural numbers $m$ and $n$,","title":"Von Szily's identity for super Catalan numbers"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers","status":"fully_proved","title":"Arithmetic properties of Catalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers/catalan-odd-iff","kind":"node","lean":"MetaMathlibExt.catalan_odd_iff","parent":26390,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers/index.html#catalan-odd-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers","status":"fully_proved","summary":"For every natural number $n$, the $n$th Catalan number $C_n$ is odd if and only if $n = 2^a - 1$ for some natural number $a$.","title":"Parity of Catalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers/catalan-not-perfect-power","kind":"node","lean":"MetaMathlibExt.catalan_not_perfect_power","parent":26390,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan numbers other than "],["math","\\(1\\)"],["text"," are not perfect powers"]],"source":"Catalan numbers other than $1$ are not perfect powers"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers/index.html#catalan-not-perfect-power","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/arithmetic-properties-of-catalan-numbers","status":"fully_proved","summary":"For every natural number $n \\ge 2$, the $n$-th Catalan number $C_n$ is not a perfect power: there are no natural numbers $m$ and $k$ with $k \\ge 2$ and $C_n = m^k$.","title":"Catalan numbers other than $1$ are not perfect powers"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers","status":"fully_proved","title":"Catalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers/catalan-le-four-pow","kind":"node","lean":"catalan_le_four_pow","parent":26393,"reading_title":{"format":"inline-parts-v1","parts":[["text","Catalan numbers are at most "],["math","\\(4^n\\)"]],"source":"Catalan numbers are at most $4^n$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers/index.html#catalan-le-four-pow","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers","status":"fully_proved","summary":"The `n`th Catalan number is at most `4 ^ n`.","title":"Catalan numbers are at most $4^n$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers/catalan-generating-function-sqrt","kind":"node","lean":"MetaMathlibExt.CatalanGeneratingFunction.catalan_generating_function_sqrt","parent":26393,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers/index.html#catalan-generating-function-sqrt","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/catalan-numbers","status":"fully_proved","summary":"Let $C_n$ be the $n$th Catalan number. For every real number $x$ with $|x| < 1/4$ and $x \\neq 0$,","title":"Closed form of the Catalan generating function"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers","status":"fully_proved","title":"Fine numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers/fine-number","kind":"node","lean":"MetaMathlibExt.fineNumber","parent":26396,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers/index.html#fine-number","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers","status":"fully_proved","summary":"For a natural number $n$, the Fine number $F(n)$, indexed from $0$ so that $F(0) = 1$, is the rational number","title":"Fine numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers/fine-number-eq-sum","kind":"node","lean":"MetaMathlibExt.fineNumber_eq_sum","parent":26396,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers/index.html#fine-number-eq-sum","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/fine-numbers","status":"fully_proved","summary":"For every natural number $n$, the Fine number $F(n)$, indexed from $0$ so that $F(0) = 1$, satisfies","title":"Explicit formula for the Fine numbers"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers","status":"fully_proved","title":"Generalized Catalan\u2013Schr\u00f6der numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers/generalized-catalan-schroeder-recurrence","kind":"node","lean":"MetaMathlibExt.generalizedCatalanSchroeder_recurrence_functionalEquation","parent":26399,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers/index.html#generalized-catalan-schroeder-recurrence","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $s, t, p \\in R$, and let $(a_n)_{n \\ge 0}$ be a sequence in $R$ with $a_0 = 1$, $a_1 = p$ and","title":"Functional equation of the generalized Catalan\u2013Schr\u00f6der recurrence"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers/generalized-catalan-schroeder-generating","kind":"node","lean":"MetaMathlibExt.generalizedCatalanSchroeder_generatingFunction","parent":26399,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers/index.html#generalized-catalan-schroeder-generating","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/generalized-catalan-schr-der-numbers","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $s, t, p \\in R$, and let $(a_n)_{n \\ge 0}$ be a sequence in $R$ with $a_0 = 1$, $a_1 = p$ and","title":"Generating function of the generalized Catalan\u2013Schr\u00f6der numbers"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials","status":"fully_proved","title":"Narayana numbers and polynomials"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/narayana","kind":"node","lean":"MetaMathlibExt.narayana","parent":26402,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/index.html#narayana","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials","status":"fully_proved","summary":"For positive integers $n$ and $k$, the Narayana number $N(n, k)$ is the natural number $\\lfloor \\binom{n}{k}\\binom{n}{k-1} / n \\rfloor$.","title":"Narayana number"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/narayana-eq","kind":"node","lean":"MetaMathlibExt.narayana_eq","parent":26402,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/index.html#narayana-eq","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials","status":"fully_proved","summary":"For positive integers $n$ and $k$, the Narayana number $N(n, k)$ satisfies $N(n, k) = \\lfloor \\binom{n}{k}\\binom{n}{k-1} / n \\rfloor$.","title":"Defining formula for the Narayana number"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/narayana-eq-second","kind":"node","lean":"MetaMathlibExt.narayana_eq_second","parent":26402,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/index.html#narayana-eq-second","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials","status":"fully_proved","summary":"For positive integers $n$ and $k$, the Narayana number $N(n, k)$ satisfies $N(n, k) = \\lfloor \\binom{n-1}{k-1}\\binom{n}{k-1} / k \\rfloor$.","title":"Second formula for the Narayana number"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/narayana-polynomial-coeff","kind":"node","lean":"MetaMathlibExt.narayanaPolynomialCoeff","parent":26402,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/index.html#narayana-polynomial-coeff","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials","status":"fully_proved","summary":"The Narayana polynomial coefficients are the natural numbers $a(n, k)$, for natural numbers $n$ and $k$, given by $a(0, 0) = 1$, $a(0, k) = 0$ for $k \\ge 1$, $a(n, 0) = 0$ for $n \\ge 1$, and $a(n, k) = N(n, k)$ for $n, k \\ge 1$, where $N(n, k)$ is the Narayana number. So row zero is $1$ and every positive row has zero constant term.","title":"Narayana polynomial coefficients"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/narayana-polynomial-coeff-succ-succ","kind":"node","lean":"MetaMathlibExt.narayanaPolynomialCoeff_succ_succ","parent":26402,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/index.html#narayana-polynomial-coeff-succ-succ","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the Narayana polynomial coefficient in row $n + 1$ and degree $k + 1$ equals the Narayana number $N(n + 1, k + 1)$.","title":"Narayana polynomial coefficients at positive indices"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/narayana-polynomial","kind":"node","lean":"MetaMathlibExt.narayanaPolynomial","parent":26402,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials/index.html#narayana-polynomial","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/narayana-numbers-and-polynomials","status":"fully_proved","summary":"For a semiring $R$ and a natural number $n$, the $n$-th Narayana polynomial is","title":"Narayana polynomials"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers","status":"fully_proved","title":"Patalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/patalan-generalized-binomial","kind":"node","lean":"MetaMathlibExt.patalanGeneralizedBinomial","parent":26409,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/index.html#patalan-generalized-binomial","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers","status":"fully_proved","summary":"For a rational number $r$ and a natural number $k$, the generalized binomial coefficient is the rational number $\\binom{r}{k} = r(r-1)\\cdots(r-k+1)/k!$, which is $1$ for $k = 0$.","title":"Binomial coefficient with rational upper argument"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/pqpatalan-params","kind":"node","lean":"MetaMathlibExt.PQPatalanParams","parent":26409,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/index.html#pqpatalan-params","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers","status":"fully_proved","summary":"An admissible parameter pair for the $(p,q)$-Patalan numbers is a pair of natural numbers $p$ and $q$ with $p > 1$ and $0 < q < p$.","title":"Admissible parameters for the (p,q)-Patalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/pq-patalan-number","kind":"node","lean":"MetaMathlibExt.pqPatalanNumber","parent":26409,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/index.html#pq-patalan-number","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $q$ with $p > 1$ and $0 < q < p$, the $(p,q)$-Patalan number with index $n \\in \\mathbb{N}$ is the rational number $b(n) = -p^{2n+1} \\binom{n - q/p}{n+1}$, where $\\binom{r}{k} = r(r-1)\\cdots(r-k+1)/k!$ is the binomial coefficient with rational upper argument $r$.","title":"The (p,q)-Patalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/pq-super-patalan-number","kind":"node","lean":"MetaMathlibExt.pqSuperPatalanNumber","parent":26409,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/index.html#pq-super-patalan-number","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers","status":"fully_proved","summary":"For natural numbers $p$ and $q$ with $p > 1$ and $0 < q < p$, and natural numbers $i$ and $j$, the $(p,q)$-super Patalan number is the rational number $Q(i,j) = (-1)^j p^{2(i+j)} \\binom{i - q/p}{i+j}$, where $\\binom{r}{k} = r(r-1)\\cdots(r-k+1)/k!$ is the binomial coefficient with rational upper argument $r$.","title":"The (p,q)-super Patalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/super-patalan-number","kind":"node","lean":"MetaMathlibExt.superPatalanNumber","parent":26409,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers/index.html#super-patalan-number","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/patalan-numbers","status":"fully_proved","summary":"For a natural number $p > 1$ and natural numbers $i$ and $j$, the super Patalan number of order $p$ is the $(p,q)$-super Patalan number with $q = 1$, that is, the rational number $(-1)^j p^{2(i+j)} \\binom{i - 1/p}{i+j}$, where $\\binom{r}{k} = r(r-1)\\cdots(r-k+1)/k!$ is the binomial coefficient with rational upper argument $r$.","title":"Super Patalan numbers of order p"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma","status":"fully_proved","title":"Raney's cycle lemma"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma/raney-cycle-lemma-sum-neg-one-general","kind":"node","lean":"MetaMathlibExt.raney_cycle_lemma_sum_neg_one_general","parent":26415,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma/index.html#raney-cycle-lemma-sum-neg-one-general","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be a list of integers whose entries sum to $-1$. Then there is exactly one index $r$ with $0 \\le r < n$ such that every proper prefix of the rotation $(a_r, \\dots, a_{n-1}, a_0, \\dots, a_{r-1})$ has nonnegative sum: for every $t$ with $0 \\le t < n$, the sum of the first $t$ entries of this rotation is at least $0$.","title":"Raney's cycle lemma for arbitrary integer lists"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma/raney-cycle-lemma-sum-neg-one","kind":"node","lean":"MetaMathlibExt.raney_cycle_lemma_sum_neg_one","parent":26415,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma/index.html#raney-cycle-lemma-sum-neg-one","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/raney-s-cycle-lemma","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be a list of integers, each at least $-1$, whose entries sum to $-1$. Then there is exactly one index $r$ with $0 \\le r < n$ such that every proper prefix of the rotation $(a_r, \\dots, a_{n-1}, a_0, \\dots, a_{r-1})$ has nonnegative sum: for every $t$ with $0 \\le t < n$, the sum of the first $t$ entries of this rotation is at least $0$.","title":"Raney's cycle lemma"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers","status":"fully_proved","title":"Riordan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers/riordan-number","kind":"node","lean":"MetaMathlibExt.riordanNumber","parent":26418,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers/index.html#riordan-number","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers","status":"fully_proved","summary":"For a natural number $n$, the $n$-th Riordan number is the integer","title":"Riordan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers/riordan-number-eq-sum","kind":"node","lean":"MetaMathlibExt.riordanNumber_eq_sum","parent":26418,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers/index.html#riordan-number-eq-sum","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/riordan-numbers","status":"fully_proved","summary":"For every natural number $n$, the $n$-th Riordan number $r_n$ equals","title":"Riordan numbers as an alternating Catalan sum"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers","status":"fully_proved","title":"Whitt's generalized Catalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/whitt-generalized-catalan","kind":"node","lean":"MetaMathlibExt.whittGeneralizedCatalan","parent":26421,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/index.html#whitt-generalized-catalan","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers","status":"fully_proved","summary":"For a rational number $\\alpha$ and a natural number $n$, Whitt's generalized Catalan number $C(\\alpha, n)$ is the rational number","title":"Whitt's generalized Catalan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/whitt-generalized-catalan-succ","kind":"node","lean":"MetaMathlibExt.whittGeneralizedCatalan_succ","parent":26421,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/index.html#whitt-generalized-catalan-succ","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers","status":"fully_proved","summary":"For every rational number $\\alpha$ and every natural number $n$, Whitt's generalized Catalan number $C(\\alpha, n + 1)$ is","title":"Whitt's generalized Catalan numbers in the source's indexing"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/whitt-generalized-catalan-zero","kind":"node","lean":"MetaMathlibExt.whittGeneralizedCatalan_zero","parent":26421,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/index.html#whitt-generalized-catalan-zero","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers","status":"fully_proved","summary":"For every rational number $\\alpha$, Whitt's generalized Catalan number $C(\\alpha, 0)$ equals $0$, being an empty sum.","title":"Whitt's generalized Catalan number at zero"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/whitt-generalized-catalan-one","kind":"node","lean":"MetaMathlibExt.whittGeneralizedCatalan_one","parent":26421,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/index.html#whitt-generalized-catalan-one","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers","status":"fully_proved","summary":"For every rational number $\\alpha$, Whitt's generalized Catalan number $C(\\alpha, 1)$ equals $1$.","title":"Whitt's generalized Catalan number at one"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/whitt-generalized-catalan-zero-left","kind":"node","lean":"MetaMathlibExt.whittGeneralizedCatalan_zero_left","parent":26421,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/index.html#whitt-generalized-catalan-zero-left","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers","status":"fully_proved","summary":"For every natural number $n \\ge 1$, Whitt's generalized Catalan number $C(0, n)$ equals $1$: only the term $k = 0$ of the defining sum survives.","title":"Whitt's generalized Catalan numbers with parameter zero"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/whitt-generalized-catalan-two-small","kind":"node","lean":"MetaMathlibExt.whittGeneralizedCatalan_two_small","parent":26421,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers/index.html#whitt-generalized-catalan-two-small","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/whitt-s-generalized-catalan-numbers","status":"fully_proved","summary":"Whitt's generalized Catalan numbers with parameter $\\alpha = 2$ satisfy $C(2, 1) = 1$, $C(2, 2) = 3$, $C(2, 3) = 13$ and $C(2, 4) = 67$.","title":"First values of Whitt's generalized Catalan numbers with parameter two"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle.json","id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","kind":"scope","navigation":"module","parent":26385,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","title":"Sabar's triangle"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sabar-number","kind":"node","lean":"MetaMathlibExt.sabarNumber","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sabar-number","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"fully_proved","summary":"For natural numbers $n$ and $i$, the entry $s(n, i)$ of Sabar's triangle is the integer","title":"Sabar's triangle"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sabar-number-rows-five-ten","kind":"node","lean":"MetaMathlibExt.sabarNumber_rows_five_ten","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sabar-number-rows-five-ten","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"fully_proved","summary":"The entries $s(5, i)$ of Sabar's triangle for $i = 0, 1, \\dots, 4$ are $1, 2, 7, 6, 5$, and the entries $s(10, i)$ for $i = 0, 1, \\dots, 9$ are $0, 6, 24, 71, 129, 167, 147, 92, 36, 10$.","title":"Rows 5 and 10 of Sabar's triangle"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sabar-number-recurrence-small","kind":"node","lean":"MetaMathlibExt.sabarNumber_recurrence_small","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sabar-number-recurrence-small","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"fully_proved","summary":"For all natural numbers $n$ and $i$ with $i < n < 8$, the entries of Sabar's triangle $s$ satisfy $s(n + 2, i + 1) = s(n, i + 1) + \\binom{n+1}{i}$.","title":"A recurrence for Sabar's triangle in small cases"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/harmonic-eq-sum-inv-choose-add-sum-sabar","kind":"node","lean":"MetaMathlibExt.harmonic_eq_sum_inv_choose_add_sum_sabarNumber","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#harmonic-eq-sum-inv-choose-add-sum-sabar","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For every natural number $n \\ge 2$, the harmonic number $H_n = \\sum_{k=1}^{n} \\frac{1}{k}$ and the entries $s(n-1, i)$ of Sabar's triangle satisfy","title":"Harmonic numbers from Sabar's triangle"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sum-inv-choose-succ-eq-one-add-sum-sabar","kind":"node","lean":"MetaMathlibExt.sum_inv_choose_succ_eq_one_add_sum_sabarNumber","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sum-inv-choose-succ-eq-one-add-sum-sabar","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For every natural number $n \\ge 2$, the entries $s(n-1, i)$ of Sabar's triangle satisfy","title":"Reciprocal binomial sums from Sabar's triangle"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sum-sabar-number-div-succ-eq-harmonic","kind":"node","lean":"MetaMathlibExt.sum_sabarNumber_div_succ_eq_harmonic","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sum-sabar-number-div-succ-eq-harmonic","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For every natural number $n \\ge 2$, writing $s(m, i)$ for Sabar's triangle and $H_n = \\sum_{j=1}^{n} 1/j$ for the $n$-th harmonic number, in the rational numbers","title":"The harmonic numbers as an alternating sum of Sabar numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/fine-number-eq-sabar-number-sub","kind":"node","lean":"MetaMathlibExt.fineNumber_eq_sabarNumber_sub","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#fine-number-eq-sabar-number-sub","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For every natural number $n \\ge 2$, the Fine number $F(n)$ is the difference of two entries of row $2n$ of Sabar's triangle $s$: $F(n) = s(2n, n+1) - s(2n, n-1)$.","title":"Fine numbers from Sabar's triangle"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sum-neg-one-pow-mul-choose-eq-self","kind":"node","lean":"MetaMathlibExt.sum_neg_one_pow_mul_choose_eq_self","parent":26428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sabar's alternating binomial sum equal to "],["math","\\(n\\)"]],"source":"Sabar's alternating binomial sum equal to $n$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sum-neg-one-pow-mul-choose-eq-self","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For every natural number $n \\ge 2$, in the rational numbers,","title":"Sabar's alternating binomial sum equal to $n$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/fine-number-eq-half-neg-one-pow-add-sum","kind":"node","lean":"MetaMathlibExt.fineNumber_eq_half_neg_one_pow_add_sum","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#fine-number-eq-half-neg-one-pow-add-sum","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For every natural number $n \\ge 2$, the Fine number $F(n)$ satisfies","title":"Fine numbers as an alternating binomial sum"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sabar-number-central-eq-fine-number","kind":"node","lean":"MetaMathlibExt.sabarNumber_central_eq_fineNumber","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sabar-number-central-eq-fine-number","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"Let $F(m)$ denote the Fine numbers and $C_m$ the Catalan numbers. For every natural number $n \\ge 1$, the entries of Sabar's triangle $s$ satisfy","title":"Central entries of Sabar's triangle and Fine numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sum-choose-mul-two-pow-add-succ","kind":"node","lean":"MetaMathlibExt.sum_choose_mul_two_pow_add_succ","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sum-choose-mul-two-pow-add-succ","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For a natural number $m$, let $V_m(2) = \\sum_{k=0}^{m} \\binom{m+k}{k} 2^k$. For every natural number $n \\ge 1$, $V_n(2) + V_{n+1}(2) = 3 \\cdot 2^n \\binom{2n+2}{n+1}$.","title":"Consecutive binomial sums weighted by powers of 2"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/whitt-generalized-catalan-two-add-succ","kind":"node","lean":"MetaMathlibExt.whittGeneralizedCatalan_two_add_succ","parent":26428,"reading_title":{"format":"inline-parts-v1","parts":[["text","Consecutive Whitt generalized Catalan numbers at "],["math","\\(\\alpha = 2\\)"]],"source":"Consecutive Whitt generalized Catalan numbers at $\\alpha = 2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#whitt-generalized-catalan-two-add-succ","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For every natural number $n \\ge 1$, Whitt's generalized Catalan numbers at $\\alpha = 2$ satisfy","title":"Consecutive Whitt generalized Catalan numbers at $\\alpha = 2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/sabar-number-central-eq-whitt-generalize","kind":"node","lean":"MetaMathlibExt.sabarNumber_central_eq_whittGeneralizedCatalan","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#sabar-number-central-eq-whitt-generalize","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"Let $C(2, m)$ denote Whitt's generalized Catalan numbers at $\\alpha = 2$. For every natural number $n \\ge 1$, the central entry $s(2n, n)$ of Sabar's triangle satisfies","title":"Central entries of Sabar's triangle and Whitt's numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/two-pow-mul-fine-number-eq-whitt-general","kind":"node","lean":"MetaMathlibExt.two_pow_mul_fineNumber_eq_whittGeneralizedCatalan","parent":26428,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle/index.html#two-pow-mul-fine-number-eq-whitt-general","scope":"Combinatorics/catalan-numbers-lattice-paths/catalan-numbers/sabar-s-triangle","status":"can_prove","summary":"For every natural number $n \\ge 1$, the Fine number $F(n)$, indexed so that $F(0) = 1$ and $F(1) = 0$, and Whitt's generalized Catalan number $C(2, n+1)$ at $\\alpha = 2$ satisfy","title":"Fine numbers from Whitt's generalized Catalan numbers"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers","kind":"scope","parent":26384,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers","status":"can_prove","summary":"Motzkin numbers and paths: colored, weighted and (u,l,d) Motzkin paths, humps and peaks, the Motzkin triangle, links with central trinomial coefficients, alternating Motzkin sums, Aigner's inequality, and Motzkin-related sequences such as A005773; weighted Motzkin paths used to compute Hankel determinants go to Hankel determinants and continued fractions.","title":"Motzkin numbers and Motzkin paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/binomial-t-motzkin-generating-function","kind":"node","lean":"MetaMathlibExt.binomial_tMotzkin_generatingFunction","parent":26443,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/index.html#binomial-t-motzkin-generating-function","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers","status":"fully_proved","summary":"Let $t \\ge 1$. A binomial $[t]$-Motzkin step is one of $\\binom{t}{k}$ differently colored steps that change the height by $1 - k$, for some $k \\in \\{0, 1, \\dots, t\\}$. For $n \\ge 0$ let $M_n$ be the number of sequences of $n$ such steps for which, starting from height $0$, the height after the first $m$ steps is nonnegative for every $0 \\le m \\le n$ and the height after all $n$ steps is $0$. Then, as formal power\u2026","title":"Generating function of binomial Motzkin paths"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","status":"fully_proved","title":"Colored Motzkin paths of rank r"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/uldmotzkin-step","kind":"node","lean":"MetaMathlibExt.ULDMotzkin.ULDMotzkinStep","parent":26445,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/index.html#uldmotzkin-step","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","status":"fully_proved","summary":"Let $r$ and $l$ be natural numbers and let $u, d \\colon \\{0, \\dots, r - 1\\} \\to \\mathbb{N}$. The admissible steps of rank $r$ are the up-steps $U_j$ with $0 \\le j < r$ in one of $u(j)$ colors, the level steps in one of $l$ colors, and the down-steps $D_j$ with $0 \\le j < r$ in one of $d(j)$ colors: the disjoint union of the sets $\\{j\\} \\times \\{0, \\dots, u(j) - 1\\}$ for $j < r$, the set $\\{0, \\dots, l - 1\\}$, and\u2026","title":"Colored Motzkin steps of a given rank"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/step-height","kind":"node","lean":"MetaMathlibExt.ULDMotzkin.stepHeight","parent":26445,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/index.html#step-height","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","status":"fully_proved","summary":"The height change of an admissible step of rank $r$ is the integer $j + 1$ for an up-step $U_j$ with $0 \\le j < r$, $0$ for a level step, and $-(j + 1)$ for a down-step $D_j$ with $0 \\le j < r$, whatever its color.","title":"Height change of a colored Motzkin step"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/path-height","kind":"node","lean":"MetaMathlibExt.ULDMotzkin.pathHeight","parent":26445,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/index.html#path-height","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","status":"fully_proved","summary":"For a sequence $p = (p_0, \\dots, p_{n-1})$ of $n$ admissible steps of rank $r$ and a natural number $k$, the height of $p$ after $k$ steps is the integer $\\sum_{i < \\min(k, n)} h(p_i)$, where $h$ is the height change of a step. For $k \\ge n$ it is the height of the whole sequence.","title":"Height of a colored step sequence after k steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/is-motzkin-path","kind":"node","lean":"MetaMathlibExt.ULDMotzkin.IsMotzkinPath","parent":26445,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/index.html#is-motzkin-path","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","status":"fully_proved","summary":"Let $r \\ge 1$. A sequence $p = (p_0, \\dots, p_{n-1})$ of $n$ admissible steps of rank $r$ is a Motzkin path when its height after $k$ steps is nonnegative for every $k$ with $0 \\le k \\le n$ and its height after $n$ steps is $0$.","title":"Colored Motzkin path condition"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/uldmotzkin-path","kind":"node","lean":"MetaMathlibExt.ULDMotzkin.ULDMotzkinPath","parent":26445,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/index.html#uldmotzkin-path","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","status":"fully_proved","summary":"For $r \\ge 1$ and a natural number $n$, the colored Motzkin paths of rank $r$ and length $n$ are the sequences $(p_0, \\dots, p_{n-1})$ of $n$ admissible steps of rank $r$ that satisfy the Motzkin path condition: the height after $k$ steps is nonnegative for every $k \\le n$, and the height after $n$ steps is $0$.","title":"Colored Motzkin paths of rank r and length n"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/uld-motzkin-number","kind":"node","lean":"MetaMathlibExt.ULDMotzkin.uldMotzkinNumber","parent":26445,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/index.html#uld-motzkin-number","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","status":"fully_proved","summary":"Let $r \\ge 1$ and $l$ be natural numbers and let $u, d \\colon \\{0, \\dots, r - 1\\} \\to \\mathbb{N}$. For a natural number $n$, the $(u,l,d)$-Motzkin number of rank $r$ and length $n$ is the number of colored Motzkin paths of rank $r$ and length $n$ whose up-steps $U_j$ come in $u(j)$ colors, whose level steps come in $l$ colors, and whose down-steps $D_j$ come in $d(j)$ colors.","title":"(u,l,d)-Motzkin numbers of rank r"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/uld-motzkin-number-eq-card","kind":"node","lean":"MetaMathlibExt.ULDMotzkin.uldMotzkinNumber_eq_card","parent":26445,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r/index.html#uld-motzkin-number-eq-card","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/colored-motzkin-paths-of-rank-r","status":"fully_proved","summary":"Let $r \\ge 1$ and $l$ be natural numbers and let $u, d \\colon \\{0, \\dots, r - 1\\} \\to \\mathbb{N}$. For every natural number $n$, the $(u,l,d)$-Motzkin number of rank $r$ and length $n$ equals the number of sequences of $n$ admissible steps of rank $r$ with these color counts that satisfy the Motzkin path condition.","title":"The (u,l,d)-Motzkin number counts Motzkin step sequences"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths","status":"fully_proved","title":"Cubic lattice walks and colored Motzkin paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths/cube-lattice-paths-card-eq-uld-motzkin-n","kind":"node","lean":"MetaMathlibExt.cube_lattice_paths_card_eq_uldMotzkinNumber","parent":26453,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths/index.html#cube-lattice-paths-card-eq-uld-motzkin-n","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers, and let $e_0, \\dots, e_n$ be the standard basis vectors of $\\mathbb{Z}^{n+1}$. The number of walks of $k$ steps in the cubic lattice $\\mathbb{Z}^{n+1}$ that start at the origin, each step being $+e_i$ or $-e_i$ for some $i \\in \\{0, \\dots, n\\}$, such that, counting only the steps in the last coordinate $i = n$, every initial segment has at least as many $+e_n$ steps as $-e_n$\u2026","title":"Cubic lattice walks counted by a colored Motzkin number"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths/cube-lattice-paths-card-eq-colored-motzk","kind":"node","lean":"MetaMathlibExt.cube_lattice_paths_card_eq_colored_motzkin_card","parent":26453,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths/index.html#cube-lattice-paths-card-eq-colored-motzk","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/cubic-lattice-walks-and-colored-motzkin-paths","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers, and let $e_0, \\dots, e_n$ be the standard basis vectors of $\\mathbb{Z}^{n+1}$. Count the walks of $k$ steps in the cubic lattice $\\mathbb{Z}^{n+1}$ that start at the origin, in which each step is $+e_i$ or $-e_i$ for some $i \\in \\{0, \\dots, n\\}$ and, counting only the steps in the last coordinate $i = n$, every initial segment has at least as many $+e_n$ steps as $-e_n$ steps and\u2026","title":"Cubic lattice walks and colored Motzkin paths"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","title":"Generalized Motzkin numbers modulo primes"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/trinomial-laurent","kind":"node","lean":"MetaMathlibExt.trinomialLaurent","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#trinomial-laurent","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"For integers $a$ and $b$, this is the Laurent polynomial $a x^{-1} + b + a x$ with integer coefficients, used to define the generalized central trinomial coefficients and the generalized Motzkin numbers.","title":"The symmetric trinomial Laurent polynomial"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/linear-laurent","kind":"node","lean":"MetaMathlibExt.linearLaurent","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#linear-laurent","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"For integers $\\alpha, \\beta, \\gamma$, this is the Laurent polynomial $\\alpha x^{-1} + \\beta + \\gamma x$ with integer coefficients.","title":"Laurent polynomials of degree at most one"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/central-trinomial","kind":"node","lean":"MetaMathlibExt.centralTrinomial","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#central-trinomial","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"For integers $a, b$ and a natural number $n$, the generalized central trinomial coefficient $T_n^{(a/b)}$ is the constant term of the Laurent polynomial $(a x^{-1} + b + a x)^n$.","title":"Generalized central trinomial coefficients"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"For integers $a, b$ and a natural number $n$, the generalized Motzkin number $M^{(a,b)}_n$ is the constant term of the Laurent polynomial $(a x^{-1} + b + a x)^n (1 - x^2)$ with integer coefficients.","title":"Generalized Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/a005717","kind":"node","lean":"MetaMathlibExt.a005717","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#a005717","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"For a natural number $n$, $a_n$ is the constant term of the Laurent polynomial $x (x^{-1} + 1 + x)^n$ with integer coefficients.","title":"OEIS A005717"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/a005773","kind":"node","lean":"MetaMathlibExt.a005773","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#a005773","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"For a natural number $n$, $a_n$ is the constant term of the Laurent polynomial $(1 + x)(x^{-1} + 1 + x)^n$ with integer coefficients.","title":"OEIS A005773"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/trinomial-laurent-one-one-powers","kind":"node","lean":"MetaMathlibExt.trinomialLaurent_one_one_powers","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first five powers of "],["math","\\(x^{-1} + 1 + x\\)"]],"source":"The first five powers of $x^{-1} + 1 + x$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#trinomial-laurent-one-one-powers","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"In the ring of Laurent polynomials over $\\mathbb{Z}$, $(x^{-1} + 1 + x)^0 = 1$, $(x^{-1} + 1 + x)^1 = x^{-1} + 1 + x$, $(x^{-1} + 1 + x)^2 = x^{-2} + 2x^{-1} + 3 + 2x + x^2$, $(x^{-1} + 1 + x)^3 = x^{-3} + 3x^{-2} + 6x^{-1} + 7 + 6x + 3x^2 + x^3$ and $(x^{-1} + 1 + x)^4 = x^{-4} + 4x^{-3} + 10x^{-2} + 16x^{-1} + 19 + 16x + 10x^2 + 4x^3 + x^4$.","title":"The first five powers of $x^{-1} + 1 + x$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/central-trinomial-one-one-values","kind":"node","lean":"MetaMathlibExt.centralTrinomial_one_one_values","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#central-trinomial-one-one-values","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"The central trinomial coefficients $T_n$, the constant terms of $(x^{-1} + 1 + x)^n$, are $1, 1, 3, 7, 19$ for $n = 0, 1, 2, 3, 4$.","title":"First five central trinomial coefficients"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-one-one-values","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_one_one_values","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","First five generalized Motzkin numbers at "],["math","\\(a = b = 1\\)"]],"source":"First five generalized Motzkin numbers at $a = b = 1$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-one-one-values","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"fully_proved","summary":"The generalized Motzkin numbers $M_n$ for $a = b = 1$, the constant terms of $(x^{-1} + 1 + x)^n (1 - x^2)$, are $1, 1, 2, 4, 9$ for $n = 0, 1, 2, 3, 4$.","title":"First five generalized Motzkin numbers at $a = b = 1$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-central-trinomial-re","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_centralTrinomial_relation","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-central-trinomial-re","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"For integers $a$, $b$ let $T_r$ and $M_r$ be the constant terms of $(a x^{-1} + b + a x)^r$ and of $(a x^{-1} + b + a x)^r (1 - x^2)$, the generalized central trinomial coefficients and generalized Motzkin numbers. Then for every natural number $n$,","title":"Generalized Motzkin numbers from three central trinomial coefficients"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/central-trinomial-recurrence","kind":"node","lean":"MetaMathlibExt.centralTrinomial_recurrence","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#central-trinomial-recurrence","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $a, b$ be integers and let $T_n$ be the constant term of $(a x^{-1} + b + a x)^n$. For every natural number $n$,","title":"Recurrence for generalized central trinomial coefficients"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-recurrence","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_recurrence","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-recurrence","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"For integers $a$, $b$ let $M_r$ be the constant term of $(a x^{-1} + b + a x)^r (1 - x^2)$, the generalized Motzkin number. Then for every natural number $n$,","title":"Three-term recurrence for generalized Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/central-trinomial-mod-prime-symmetry","kind":"node","lean":"MetaMathlibExt.centralTrinomial_mod_prime_symmetry","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#central-trinomial-mod-prime-symmetry","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $a, b$ be integers and let $T_n$ be the constant term of $(a x^{-1} + b + a x)^n$. For every prime $p > 2$ and every natural number $k \\le (p - 1)/2$,","title":"Mirror symmetry of central trinomial coefficients modulo a prime"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-mod-prime-symmetry","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_mod_prime_symmetry","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reflection symmetry of generalized Motzkin numbers modulo "],["math","\\(p\\)"]],"source":"Reflection symmetry of generalized Motzkin numbers modulo $p$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-mod-prime-symmetry","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"For integers $a$, $b$ let $M_r$ be the constant term of $(a x^{-1} + b + a x)^r (1 - x^2)$, the generalized Motzkin number. Let $p > 3$ be a prime and $k$ a natural number with $k \\le (p - 3)/2$. Then","title":"Reflection symmetry of generalized Motzkin numbers modulo $p$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/central-trinomial-prime-minus-one-sq","kind":"node","lean":"MetaMathlibExt.centralTrinomial_prime_minus_one_sq","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#central-trinomial-prime-minus-one-sq","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $a, b$ be integers, let $T_n$ be the constant term of $(a x^{-1} + b + a x)^n$, and let $p$ be a prime with $b^2 - 4a^2 \\not\\equiv 0 \\pmod p$. Then $T_{p-1}^2 \\equiv 1 \\pmod p$.","title":"Squared central trinomial coefficient at one less than a prime"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/prime-dvd-motzkin-prime-minus-two-iff","kind":"node","lean":"MetaMathlibExt.prime_dvd_motzkin_prime_minus_two_iff","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primes dividing the Motzkin number "],["math","\\(M_{p-2}\\)"]],"source":"Primes dividing the Motzkin number $M_{p-2}$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#prime-dvd-motzkin-prime-minus-two-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"A prime $p$ divides the Motzkin number $M_{p-2} = M^{(1,1)}_{p-2}$ if and only if $p \\equiv 1 \\pmod 3$.","title":"Primes dividing the Motzkin number $M_{p-2}$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/constant-term-linear-laurent-pow-digits","kind":"node","lean":"MetaMathlibExt.constantTerm_linearLaurent_pow_digits","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#constant-term-linear-laurent-pow-digits","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $p$ be a prime, let $\\alpha$, $\\beta$, $\\gamma$, $a$, $b$ be integers, and let $n$ be a natural number whose base-$p$ digits are $d_0, \\dots, d_s$ (there are none when $n = 0$, and both products below are then $1$). Write $\\operatorname{ct}(F)$ for the constant term of a Laurent polynomial $F$ and put $P = \\alpha x^{-1} + \\beta + \\gamma x$. Then","title":"Lucas property for constant terms of powers of linear Laurent polynomials"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/constant-term-linear-combination-digit-d","kind":"node","lean":"MetaMathlibExt.constantTermLinearCombination_digit_decomposition","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#constant-term-linear-combination-digit-d","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $\\alpha, \\beta, \\gamma$ be integers with $\\alpha \\ne 0$ or $\\gamma \\ne 0$, let $a_r$ be the constant term of $(\\alpha x^{-1} + \\beta + \\gamma x)^r$, let $p$ be a prime, let $h < p$ be a natural number, let $c_0, c_1, \\ldots$ be integers, and let $b_r = \\sum_{i=0}^{h} c_i a_{r+i}$. Let $q, k$ be natural numbers, let $m, n_0 < p$ be natural numbers with $m \\ne p - 1$, put $n = ((qp + m) p^k + p^k - 1) p + n_0$\u2026","title":"Digit decomposition of shifted sums of constant terms modulo a prime"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/constant-term-linear-combination-small-l","kind":"node","lean":"MetaMathlibExt.constantTermLinearCombination_small_last_digit","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#constant-term-linear-combination-small-l","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $\\alpha$, $\\beta$, $\\gamma$ be integers with $\\alpha \\ne 0$ or $\\gamma \\ne 0$, and let $a_r$ be the constant term of $(\\alpha x^{-1} + \\beta + \\gamma x)^r$. Let $c_0, c_1, c_2, \\dots$ be integers, let $h$ be a natural number, and put $b_r = \\sum_{i=0}^{h} c_i a_{r+i}$. Let $p$ be a prime with $h < p$, and let $q$, $m$, $k$, $n_0$ be natural numbers with $m < p$, $m \\ne p - 1$, $n_0 < p$ and $n_0 < p - h$. If\u2026","title":"Digit formula for shifted sums of constant terms, small last digit"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/symmetric-constant-term-zero-density-one","kind":"node","lean":"MetaMathlibExt.symmetricConstantTerm_zero_density_one","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#symmetric-constant-term-zero-density-one","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $\\alpha$ and $\\beta$ be integers with $\\alpha \\neq 0$, and write $a_r$ for the generalized central trinomial coefficient with parameters $\\alpha, \\beta$, the constant term of $(\\alpha x^{-1} + \\beta + \\alpha x)^r$. Let $p$ be a prime, let $h < p$ be a natural number, let $(c_i)_{i \\ge 0}$ be a sequence of integers, and set $b_r = \\sum_{i=0}^{h} c_i a_{r+i}$ for $r \\in \\mathbb{N}$. If $a_r \\equiv 0 \\pmod p$ for\u2026","title":"Density one for zeros of shifted combinations of central trinomial coefficients"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/symmetric-constant-term-zero-density-for","kind":"node","lean":"MetaMathlibExt.symmetricConstantTerm_zero_density_formula","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#symmetric-constant-term-zero-density-for","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $\\alpha$ and $\\beta$ be integers with $\\alpha \\neq 0$, and write $a_r$ for the generalized central trinomial coefficient with parameters $\\alpha, \\beta$, the constant term of $(\\alpha x^{-1} + \\beta + \\alpha x)^r$. Let $p$ be a prime, let $h < p$ be a natural number, let $(c_i)_{i \\ge 0}$ be a sequence of integers, and set $b_r = \\sum_{i=0}^{h} c_i a_{r+i}$ for $r \\in \\mathbb{N}$. Suppose that\u2026","title":"Zero density of shifted combinations of central trinomial coefficients"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-digit-below-prime-mi","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_digit_below_prime_minus_two","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Motzkin numbers modulo "],["math","\\(p\\)"],["text"," with last digit below "],["math","\\(p - 2\\)"]],"source":"Generalized Motzkin numbers modulo $p$ with last digit below $p - 2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-digit-below-prime-mi","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"For integers $a$, $b$ let $T_r$ and $M_r$ be the constant terms of $(a x^{-1} + b + a x)^r$ and of $(a x^{-1} + b + a x)^r (1 - x^2)$, the generalized central trinomial coefficients and generalized Motzkin numbers. Let $p > 2$ be a prime with $b^2 - 4a^2 \\not\\equiv 0 \\pmod p$, and let $q$, $n_0$ be natural numbers with $n_0 < p - 2$. If $n = qp + n_0$, then","title":"Generalized Motzkin numbers modulo $p$ with last digit below $p - 2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-digit-prime-minus-tw","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_digit_prime_minus_two","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Motzkin numbers modulo "],["math","\\(p\\)"],["text"," with last digit "],["math","\\(p - 2\\)"]],"source":"Generalized Motzkin numbers modulo $p$ with last digit $p - 2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-digit-prime-minus-tw","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"For integers $a$, $b$ let $T_r$ and $M_r$ be the constant terms of $(a x^{-1} + b + a x)^r$ and of $(a x^{-1} + b + a x)^r (1 - x^2)$, the generalized central trinomial coefficients and generalized Motzkin numbers. Let $p > 2$ be a prime with $b^2 - 4a^2 \\not\\equiv 0 \\pmod p$, and let $q$, $m$, $k$ be natural numbers with $m < p - 1$. If $n = ((qp + m)p^k + p^k - 1)p + p - 2$, that is, $n$ is obtained from $q$ by\u2026","title":"Generalized Motzkin numbers modulo $p$ with last digit $p - 2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-digit-prime-minus-on","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_digit_prime_minus_one","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized Motzkin numbers modulo "],["math","\\(p\\)"],["text"," with last digit "],["math","\\(p - 1\\)"]],"source":"Generalized Motzkin numbers modulo $p$ with last digit $p - 1$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-digit-prime-minus-on","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"For integers $a$, $b$ let $T_r$ and $M_r$ be the constant terms of $(a x^{-1} + b + a x)^r$ and of $(a x^{-1} + b + a x)^r (1 - x^2)$, the generalized central trinomial coefficients and generalized Motzkin numbers. Let $p > 2$ be a prime with $D = b^2 - 4a^2 \\not\\equiv 0 \\pmod p$, let $q$, $m$, $k$ be natural numbers with $m < p - 1$, and put $\\ell = p - 2 - m$. If $n = ((qp + m)p^k + p^k - 1)p + p - 1$, that is,\u2026","title":"Generalized Motzkin numbers modulo $p$ with last digit $p - 1$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-zero-digit-below-pri","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_zero_digit_below_prime_minus_two","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-zero-digit-below-pri","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $a, b$ be integers and let $p > 2$ be a prime with $b^2 - 4a^2 \\not\\equiv 0 \\pmod p$, and suppose that the generalized central trinomial coefficients satisfy $T^{(a,b)}_r \\not\\equiv 0 \\pmod p$ for every $r < p$. Let $q, n_0$ be natural numbers with $n_0 < p - 2$ and let $n = qp + n_0$. Then $p$ divides the generalized Motzkin number $M^{(a,b)}_n$ if and only if $p$ divides $M^{(a,b)}_{n_0}$.","title":"Zeros of generalized Motzkin numbers with a small last digit"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-zero-digit-prime-min","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_zero_digit_prime_minus_two","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zeros of generalized Motzkin numbers with last digit "],["math","\\(p - 2\\)"]],"source":"Zeros of generalized Motzkin numbers with last digit $p - 2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-zero-digit-prime-min","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $a, b$ be integers and let $p > 2$ be a prime with $a \\not\\equiv 0 \\pmod p$ and $b^2 - 4a^2 \\not\\equiv 0 \\pmod p$. Write $T_r = T^{(a,b)}_r$ for the generalized central trinomial coefficients and suppose that $T_r \\not\\equiv 0 \\pmod p$ for every $r < p$. Let $q, m, k$ be natural numbers with $m < p - 1$ and let","title":"Zeros of generalized Motzkin numbers with last digit $p - 2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-zero-digit-prime-min-2","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_zero_digit_prime_minus_one","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zeros of generalized Motzkin numbers with last digit "],["math","\\(p - 1\\)"]],"source":"Zeros of generalized Motzkin numbers with last digit $p - 1$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-zero-digit-prime-min-2","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $a, b$ be integers and let $p > 2$ be a prime with $a \\not\\equiv 0 \\pmod p$ and $b^2 - 4a^2 \\not\\equiv 0 \\pmod p$. Write $T_r = T^{(a,b)}_r$ for the generalized central trinomial coefficients and suppose that $T_r \\not\\equiv 0 \\pmod p$ for every $r < p$. Let $q, m, k$ be natural numbers with $m < p - 1$, let","title":"Zeros of generalized Motzkin numbers with last digit $p - 1$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-zero-density-one","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_zero_density_one","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-zero-density-one","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $a, b$ be integers and let $p$ be a prime. If $p$ divides the generalized central trinomial coefficient $T^{(a,b)}_r$ for some $r < p$, then the set of $n \\in \\mathbb{N}$ such that $p$ divides the generalized Motzkin number $M^{(a,b)}_n$ has natural density $1$.","title":"Generalized Motzkin numbers vanish modulo a prime with density one"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-zero-density-formula","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_zero_density_formula","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-zero-density-formula","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $a, b$ be integers and let $p > 2$ be a prime with $a \\not\\equiv 0 \\pmod p$. Write $T_r = T^{(a,b)}_r$ for the generalized central trinomial coefficients and $M_n = M^{(a,b)}_n$ for the generalized Motzkin numbers, and suppose that $T_r \\not\\equiv 0 \\pmod p$ for every $r < p$. Let $A$ be the number of $r < p - 2$ with $M_r \\equiv 0 \\pmod p$, let $B$ be the number of $m < p - 1$ with\u2026","title":"Zero density formula for generalized Motzkin numbers modulo a prime"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/generalized-motzkin-nonzero-density","kind":"node","lean":"MetaMathlibExt.generalizedMotzkin_nonzero_density","parent":26456,"reading_title":{"format":"inline-parts-v1","parts":[["text","Density of each nonzero residue among generalized Motzkin numbers modulo "],["math","\\(p\\)"]],"source":"Density of each nonzero residue among generalized Motzkin numbers modulo $p$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#generalized-motzkin-nonzero-density","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"For integers $a$, $b$ let $T_r$ and $M_r$ be the constant terms of $(a x^{-1} + b + a x)^r$ and of $(a x^{-1} + b + a x)^r (1 - x^2)$, the generalized central trinomial coefficients and generalized Motzkin numbers. Let $p > 2$ be a prime with $a \\not\\equiv 0 \\pmod p$ such that $T_r \\not\\equiv 0 \\pmod p$ for every $r < p$, and suppose that the residues of $T_0, \\dots, T_{p-1}$ generate the unit group\u2026","title":"Density of each nonzero residue among generalized Motzkin numbers modulo $p$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/motzkin-zero-density","kind":"node","lean":"MetaMathlibExt.motzkin_zero_density","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#motzkin-zero-density","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $p$ be a prime, and write $M_n = M^{(1,1)}_n$ for the Motzkin numbers and $T_r = T^{(1,1)}_r$ for the central trinomial coefficients. If $T_r \\equiv 0 \\pmod p$ for some $r < p$, then the set of $n \\in \\mathbb{N}$ with $M_n \\equiv 0 \\pmod p$ has natural density $1$. If instead $T_r \\not\\equiv 0 \\pmod p$ for every $r < p$ and $p > 2$, let $A$ be the number of $r < p - 2$ with $M_r \\equiv 0 \\pmod p$, let $B$ be\u2026","title":"Zero density of Motzkin numbers modulo a prime"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/a005717-zero-density","kind":"node","lean":"MetaMathlibExt.a005717_zero_density","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#a005717-zero-density","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $p$ be a prime, let $T_m$ be the central trinomial coefficient, the constant term of $(x^{-1} + 1 + x)^m$, and let $a_n$ be the constant term of $x (x^{-1} + 1 + x)^n$ (OEIS A005717). If $T_r \\equiv 0 \\pmod p$ for some $r < p$, then the set of $n \\in \\mathbb{N}$ with $a_n \\equiv 0 \\pmod p$ has natural density $1$. If instead $T_r \\not\\equiv 0 \\pmod p$ for every $r < p$ and $p > 2$, let $N_1$ be the number of\u2026","title":"Zero density of A005717 modulo a prime"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/riordan-zero-density","kind":"node","lean":"MetaMathlibExt.riordan_zero_density","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#riordan-zero-density","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $p$ be a prime, and write $r_n$ for the Riordan numbers and $T_m = T^{(1,1)}_m$ for the central trinomial coefficients. If $T_m \\equiv 0 \\pmod p$ for some $m < p$, then the set of $n \\in \\mathbb{N}$ with $r_n \\equiv 0 \\pmod p$ has natural density $1$. If instead $T_m \\not\\equiv 0 \\pmod p$ for every $m < p$ and $p > 2$, let $B$ be the number of $m < p - 1$ with $3 T_m \\equiv T_{p-1} T_{m+1} \\pmod p$ and let $C$\u2026","title":"Zero density of Riordan numbers modulo a prime"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/a005773-zero-density","kind":"node","lean":"MetaMathlibExt.a005773_zero_density","parent":26456,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes/index.html#a005773-zero-density","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/generalized-motzkin-numbers-modulo-primes","status":"can_prove","summary":"Let $p$ be a prime, let $T_m$ be the central trinomial coefficient, the constant term of $(x^{-1} + 1 + x)^m$, and let $a_n$ be the constant term of $(1 + x)(x^{-1} + 1 + x)^n$ (OEIS A005773). If $T_r \\equiv 0 \\pmod p$ for some $r < p$, then the set of $n \\in \\mathbb{N}$ with $a_n \\equiv 0 \\pmod p$ has natural density $1$. If instead $T_r \\not\\equiv 0 \\pmod p$ for every $r < p$ and $p > 2$, let $N_1$ be the number\u2026","title":"Zero density of A005773 modulo a prime"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers","status":"can_prove","title":"Log-behavior of Motzkin prefix numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/a005773-recurrence-ratio-bounds","kind":"node","lean":"MetaMathlibExt.a005773Recurrence_ratio_bounds","parent":26491,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/index.html#a005773-recurrence-ratio-bounds","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers","status":"can_prove","summary":"Let $(A_n)_{n \\ge 0}$ be a sequence of real numbers with $A_0 = A_1 = 1$ and $(n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}$ for every $n \\ge 1$. Write $x_n = A_{n+1}/A_n$ and $\\varphi_n = 3(2n+1)/(2(n+1))$. Then for every $n \\ge 0$ we have $x_n \\le \\varphi_n$ and","title":"Ratio bounds for the A005773 recurrence"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/a005773-recurrence-diff-log-balanced","kind":"node","lean":"MetaMathlibExt.a005773Recurrence_diff_logBalanced","parent":26491,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/index.html#a005773-recurrence-diff-log-balanced","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers","status":"can_prove","summary":"Let $(A_n)_{n \\ge 0}$ be a sequence of real numbers with $A_0 = A_1 = 1$ and $(n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}$ for every $n \\ge 1$, and write $\\Delta A_n = A_{n+1} - A_n$ and $\\Delta^2 A_n = A_{n+2} - 2A_{n+1} + A_n$. Then for every $n \\ge 4$, each of the sequences $z = \\Delta A$ and $z = \\Delta^2 A$ satisfies $z_n^2 \\le z_{n-1} z_{n+1}$ and","title":"The first and second differences of A005773 are log-balanced"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/a005773-recurrence-third-diff-log-convex","kind":"node","lean":"MetaMathlibExt.a005773Recurrence_thirdDiff_logConvex_logBalanced","parent":26491,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/index.html#a005773-recurrence-third-diff-log-convex","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers","status":"can_prove","summary":"Let $(A_n)_{n \\ge 0}$ be a sequence of real numbers with $A_0 = A_1 = 1$ and $(n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}$ for every $n \\ge 1$, and write $z_n = \\Delta^3 A_n = A_{n+3} - 3A_{n+2} + 3A_{n+1} - A_n$. Then $z_n^2 \\le z_{n-1} z_{n+1}$ for every $n \\ge 3$, and","title":"The third differences of A005773 are log-convex and log-balanced"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/a005773-recurrence-weighted-log-concave","kind":"node","lean":"MetaMathlibExt.a005773Recurrence_weighted_logConcave","parent":26491,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/index.html#a005773-recurrence-weighted-log-concave","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers","status":"can_prove","summary":"Let $(A_n)_{n \\ge 0}$ be a sequence of real numbers with $A_0 = A_1 = 1$ and $(n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}$ for every $n \\ge 1$, and write $\\Delta A_n = A_{n+1} - A_n$ and $\\Delta^2 A_n = A_{n+2} - 2A_{n+1} + A_n$. Then for every $n \\ge 2$, each of the four sequences $z_m = m A_m$, $z_m = (m+1) A_{m+1} - m A_m$, $z_m = m\\,\\Delta A_m$ and $z_m = m\\,\\Delta^2 A_m$ satisfies $z_{n-1} z_{n+1} \\le z_n^2$.\u2026","title":"Log-concave weighted sequences built from A005773"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/a005773-recurrence-pow-log-convex","kind":"node","lean":"MetaMathlibExt.a005773Recurrence_pow_logConvex","parent":26491,"reading_title":{"format":"inline-parts-v1","parts":[["text","Log-convexity of "],["math","\\((n A_n)^n\\)"],["text"," for A005773"]],"source":"Log-convexity of $(n A_n)^n$ for A005773"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers/index.html#a005773-recurrence-pow-log-convex","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/log-behavior-of-motzkin-prefix-numbers","status":"can_prove","summary":"Let $(A_n)_{n \\ge 0}$ be a sequence of real numbers with $A_0 = A_1 = 1$ and $(n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}$ for every $n \\ge 1$, and let $z_m = (m A_m)^m$. Then $z_n^2 \\le z_{n-1} z_{n+1}$ for every $n \\ge 2$; that is, the sequence $(n^n A_n^n)_{n \\ge 1}$ is log-convex.","title":"Log-convexity of $(n A_n)^n$ for A005773"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","title":"Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/trinomial-coefficient-self-eq-sum","kind":"node","lean":"MetaMathlibExt.trinomialCoefficient_self_eq_sum","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#trinomial-coefficient-self-eq-sum","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"For every natural number $n$, the coefficient of $X^n$ in $(1 + X + X^2)^n$ equals $\\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\binom{n}{2k} \\binom{2k}{k}$.","title":"Central trinomial coefficients as a binomial sum"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/motzkin-central-trinomial-recurrence","kind":"node","lean":"MetaMathlibExt.motzkin_central_trinomial_recurrence","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#motzkin-central-trinomial-recurrence","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ let $T_n$ be the central trinomial coefficient, the coefficient of $X^n$ in $(1 + X + X^2)^n$. Let $M : \\mathbb{N} \\to \\mathbb{N}$ be any function with $M_n = \\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\binom{n}{2k} C_k$ for every $n$, where $C_k$ is the $k$-th Catalan number (these sums are the Motzkin numbers). Then for every $n \\ge 1$, $T_{n+1} = T_n + 2n \\, M_{n-1}$.","title":"Recurrence linking central trinomial coefficients and Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/motzkin-number","kind":"node","lean":"MetaMathlibExt.motzkinNumber","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#motzkin-number","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"Let $C_k$ denote the $k$-th Catalan number (OEIS A000108). For a natural number $n$, the $n$-th Motzkin number (OEIS A001006) is","title":"Motzkin numbers as a Catalan-weighted binomial sum"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/motzkin-number-zero","kind":"node","lean":"MetaMathlibExt.motzkinNumber_zero","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#motzkin-number-zero","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"The $0$-th Motzkin number is $M(0) = 1$.","title":"Value of the Motzkin number at 0"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/motzkin-number-one","kind":"node","lean":"MetaMathlibExt.motzkinNumber_one","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#motzkin-number-one","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"The first Motzkin number is $M(1) = 1$.","title":"Value of the Motzkin number at 1"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/motzkin-number-two","kind":"node","lean":"MetaMathlibExt.motzkinNumber_two","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#motzkin-number-two","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"The second Motzkin number is $M(2) = 2$.","title":"Value of the Motzkin number at 2"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/motzkin-number-eq-sum-range-half","kind":"node","lean":"MetaMathlibExt.motzkinNumber_eq_sum_range_half","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#motzkin-number-eq-sum-range-half","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"Let $C_k$ denote the $k$-th Catalan number. For every natural number $n$, the $n$-th Motzkin number $M(n)$ satisfies","title":"Truncated Catalan-weighted sum for Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/trinomial-coefficient-succ-eq-add-motzki","kind":"node","lean":"MetaMathlibExt.trinomialCoefficient_succ_eq_add_motzkinNumber","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#trinomial-coefficient-succ-eq-add-motzki","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"Write $T(n)$ for the central trinomial coefficient, the coefficient of $X^n$ in $(1 + X + X^2)^n$, and $M(n)$ for the $n$-th Motzkin number. For every natural number $n \\ge 1$,","title":"Central trinomial recurrence via Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/motzkin-number-succ","kind":"node","lean":"MetaMathlibExt.motzkinNumber_succ","parent":26497,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers/index.html#motzkin-number-succ","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-numbers","status":"fully_proved","summary":"`M(n + 1) = M(n) + \u2211_{k < n} M(k) M(n - 1 - k)`. (Combinatorially, a Motzkin path of length `n + 1` starts with a flat step, or with an up step whose matching down step splits the rest into two shorter Motzkin paths.)","title":"The Motzkin recurrence"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers","status":"fully_proved","title":"Aigner's ratio inequality for Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers/aigner-motzkin-ratio-inequality","kind":"node","lean":"MetaMathlibExt.aigner_motzkin_ratio_inequality","parent":26507,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers/index.html#aigner-motzkin-ratio-inequality","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers","status":"fully_proved","summary":"For $r \\ge 0$ let $M_r = \\sum_{k=0}^{\\lfloor r/2 \\rfloor} \\frac{1}{k+1}\\binom{2k}{k}\\binom{r}{2k}$, where each factor $\\frac{1}{k+1}\\binom{2k}{k}$ is the $k$-th Catalan number, an integer. Then for every $n \\ge 1$, in the rational numbers,","title":"Aigner's ratio inequality for Motzkin numbers, explicit sum form"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers/motzkin-number-ratio-inequality","kind":"node","lean":"MetaMathlibExt.motzkinNumber_ratio_inequality","parent":26507,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers/index.html#motzkin-number-ratio-inequality","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/aigner-s-ratio-inequality-for-motzkin-numbers","status":"fully_proved","summary":"Let $M_n$ denote the $n$-th Motzkin number. For every $n \\ge 1$,","title":"Aigner's ratio inequality for Motzkin numbers"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"can_prove","title":"Alternating sums of Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/riordan-number-even-log-convex","kind":"node","lean":"MetaMathlibExt.riordanNumber_even_logConvex","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#riordan-number-even-log-convex","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"can_prove","summary":"Let $r_n$ denote the $n$-th Riordan number. The even-indexed Riordan numbers $r_0, r_2, r_4, \\dots$ form a log-convex sequence: for every natural number $n$,","title":"Log-convexity of the even-indexed Riordan numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/motzkin-alternating-sum","kind":"node","lean":"MetaMathlibExt.motzkinAlternatingSum","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#motzkin-alternating-sum","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"fully_proved","summary":"Let $M_k$ denote the $k$-th Motzkin number. For a natural number $n$, the $n$-th alternating Motzkin sum is the integer","title":"Alternating sums of Motzkin numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/motzkin-alternating-sum-zero","kind":"node","lean":"MetaMathlibExt.motzkinAlternatingSum_zero","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#motzkin-alternating-sum-zero","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"fully_proved","summary":"Let $M_k$ denote the $k$-th Motzkin number and let $T_n = \\sum_{k=0}^{n} (-1)^{n-k} M_k$. Then $T_0 = 1$.","title":"The alternating Motzkin sum at 0"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/motzkin-alternating-sum-succ","kind":"node","lean":"MetaMathlibExt.motzkinAlternatingSum_succ","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#motzkin-alternating-sum-succ","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"fully_proved","summary":"Let $M_k$ denote the $k$-th Motzkin number and let $T_n = \\sum_{k=0}^{n} (-1)^{n-k} M_k$. For every natural number $n$, $T_{n+1} = M_{n+1} - T_n$.","title":"Recurrence for alternating Motzkin sums"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/motzkin-alternating-sum-one-two","kind":"node","lean":"MetaMathlibExt.motzkinAlternatingSum_one_two","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#motzkin-alternating-sum-one-two","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"fully_proved","summary":"Let $M_k$ denote the $k$-th Motzkin number and let $T_n = \\sum_{k=0}^{n} (-1)^{n-k} M_k$. Then $T_1 = 0$ and $T_2 = 2$.","title":"Alternating Motzkin sums at 1 and 2"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/fibonacci-alternating-sum","kind":"node","lean":"MetaMathlibExt.fibonacciAlternatingSum","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#fibonacci-alternating-sum","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"fully_proved","summary":"Let $F_k$ denote the $k$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For a natural number $n$, the $n$-th alternating Fibonacci sum is the integer","title":"Alternating sums of Fibonacci numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/fibonacci-alternating-sum-zero","kind":"node","lean":"MetaMathlibExt.fibonacciAlternatingSum_zero","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#fibonacci-alternating-sum-zero","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"fully_proved","summary":"Let $F_k$ denote the $k$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$, and let $W_n = \\sum_{k=0}^{n} (-1)^{n-k} F_k$. Then $W_0 = 0$.","title":"The alternating Fibonacci sum at 0"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/fibonacci-alternating-sum-succ","kind":"node","lean":"MetaMathlibExt.fibonacciAlternatingSum_succ","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#fibonacci-alternating-sum-succ","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"fully_proved","summary":"Let $F_k$ denote the $k$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$, and let $W_n = \\sum_{k=0}^{n} (-1)^{n-k} F_k$. For every natural number $n$, $W_{n+1} = F_{n+1} - W_n$.","title":"Recurrence for alternating Fibonacci sums"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/motzkin-alternating-sum-log-balanced-tem","kind":"node","lean":"MetaMathlibExt.motzkinAlternatingSum_logBalanced_tempered","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#motzkin-alternating-sum-log-balanced-tem","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"can_prove","summary":"Let $M_k$ denote the $k$-th Motzkin number and let $T_n = \\sum_{k=0}^{n} (-1)^{n-k} M_k$. For every natural number $n \\ge 9$,","title":"The alternating Motzkin sums are eventually log-balanced and tempered"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/motzkin-alternating-sum-mul-succ-log-con","kind":"node","lean":"MetaMathlibExt.motzkinAlternatingSum_mul_succ_logConvex_logBalanced","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#motzkin-alternating-sum-mul-succ-log-con","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"can_prove","summary":"Products of consecutive alternating Motzkin sums are eventually log-convex and log-balanced. Let $M_k$ denote the $k$-th Motzkin number, let $T_n = \\sum_{k=0}^{n} (-1)^{n-k} M_k$ and let $P_n = T_n T_{n+1}$. For every natural number $n \\ge 7$, $P_n^2 \\le P_{n-1} P_{n+1}$; and for every natural number $n \\ge 10$, $P_n^2 \\le P_{n-1} P_{n+1}$ and $n P_{n-1} P_{n+1} \\le (n+1) P_n^2$.","title":"Products of consecutive alternating Motzkin sums are eventually log-convex and log-balanced"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/motzkin-number-add-motzkin-alternating-s","kind":"node","lean":"MetaMathlibExt.motzkinNumber_add_motzkinAlternatingSum_logBalanced","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#motzkin-number-add-motzkin-alternating-s","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"can_prove","summary":"Let $M_n$ denote the $n$-th Motzkin number, let $T_n = \\sum_{k=0}^{n} (-1)^{n-k} M_k$ be the alternating sum of Motzkin numbers, and put $z_n = M_n + T_n$. Then the sequence $(z_n)_{n \\ge 6}$ is log-balanced: for every natural number $n \\ge 7$,","title":"Log-balancedness of Motzkin numbers plus their alternating sums"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/fibonacci-alternating-sum-tempered","kind":"node","lean":"MetaMathlibExt.fibonacciAlternatingSum_tempered","parent":26510,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers/index.html#fibonacci-alternating-sum-tempered","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/alternating-sums-of-motzkin-numbers","status":"can_prove","summary":"Let $F_k$ denote the $k$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$, and let $W_n = \\sum_{k=0}^{n} (-1)^{n-k} F_k$. For every natural number $n \\ge 12$,","title":"The alternating Fibonacci sums are eventually tempered"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"can_prove","title":"Humps and peaks in Motzkin paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-hump-peak-height-distribution","kind":"node","lean":"MetaMathlibExt.motzkin_hump_peak_height_distribution","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-hump-peak-height-distribution","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $n \\ge 2$ and $k \\ge 1$. A Motzkin path of length $n$ is a sequence of $n$ steps, each up ($+1$), flat ($0$) or down ($-1$), whose partial sums are all nonnegative and whose total is $0$. A hump at height $k$ of such a path is a pair of positions $i < j$ such that step $i$ is up, step $j$ is down, every step strictly between them is flat, and the sum of the steps up to and including step $i$ equals $k$; a peak\u2026","title":"Humps and peaks at a given height in Motzkin paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-step","kind":"node","lean":"MetaMathlibExt.motzkinStep","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-step","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"The three Motzkin steps are encoded as $0 = D$ (down), $1 = F$ (flat) and $2 = U$ (up). The height change of the step encoded by $s \\in \\{0, 1, 2\\}$ is the integer $s - 1$; thus $D$ changes the height by $-1$, $F$ by $0$ and $U$ by $+1$.","title":"Height change of a Motzkin step"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-motzkin-prefix","kind":"node","lean":"MetaMathlibExt.IsMotzkinPrefix","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-motzkin-prefix","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $p = p_0 p_1 \\cdots p_{n-1}$ be a word of length $n$ in the steps $D$, $F$, $U$ (encoded as $0$, $1$, $2$), whose vertical displacements are $-1$, $0$ and $+1$, and let $h \\in \\mathbb{Z}$. The word $p$ is a Motzkin prefix ending at height $h$ when, for every natural number $m \\le n$, the sum of the displacements of $p_0, \\dots, p_{m-1}$ is nonnegative, and the sum of the displacements of all $n$ steps is $h$.","title":"Motzkin prefix"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-motzkin-hump","kind":"node","lean":"MetaMathlibExt.IsMotzkinHump","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-motzkin-hump","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $p = p_0 p_1 \\cdots p_{n-1}$ be a word of length $n$ in the steps $D$, $F$, $U$ (encoded as $0$, $1$, $2$), whose vertical displacements are $-1$, $0$ and $+1$. For $k \\in \\mathbb{N}$ and positions $i, j \\in \\{0, \\dots, n-1\\}$, the pair $(i, j)$ is a hump of height $k$ in $p$ when $i < j$, $p_i = U$, $p_j = D$, every $p_r$ with $i < r < j$ is $F$, and the height after step $i$, that is, the sum of the\u2026","title":"Motzkin hump"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-motzkin-prefix-iff","kind":"node","lean":"MetaMathlibExt.isMotzkinPrefix_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-motzkin-prefix-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $p = p_0 p_1 \\cdots p_{n-1}$ be a word of length $n$ in the steps $D$, $F$, $U$, whose vertical displacements are $-1$, $0$ and $+1$, and let $h \\in \\mathbb{Z}$. Then $p$ is a Motzkin prefix ending at height $h$ if and only if, for every natural number $m \\le n$, the sum of the displacements of $p_0, \\dots, p_{m-1}$ is nonnegative, and the sum of the displacements of all $n$ steps is $h$.","title":"Unfolding the Motzkin prefix condition"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-motzkin-prefix-finite","kind":"node","lean":"MetaMathlibExt.IsMotzkinPrefixFinite","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-motzkin-prefix-finite","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $p = p_0 p_1 \\cdots p_{n-1}$ be a word of length $n$ in the steps $D$, $F$, $U$ (encoded as $0$, $1$, $2$), whose vertical displacements are $-1$, $0$ and $+1$, and let $h \\in \\mathbb{Z}$. The word $p$ satisfies the bounded Motzkin prefix condition for $h$ when, for every $m \\in \\{0, 1, \\dots, n\\}$, the sum of the displacements of $p_0, \\dots, p_{m-1}$ is nonnegative, and the sum of the displacements of all $n$\u2026","title":"Bounded Motzkin prefix condition"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-motzkin-prefix-finite-iff","kind":"node","lean":"MetaMathlibExt.isMotzkinPrefixFinite_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-motzkin-prefix-finite-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $p$ be a word of length $n$ in the steps $D$, $F$, $U$ and let $h \\in \\mathbb{Z}$. Then $p$ satisfies the bounded Motzkin prefix condition for $h$, in which the prefix length $m$ ranges over $\\{0, 1, \\dots, n\\}$, if and only if $p$ is a Motzkin prefix ending at height $h$, in which $m$ ranges over all natural numbers with $m \\le n$.","title":"Bounded form of the Motzkin prefix condition"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-motzkin-hump-iff","kind":"node","lean":"MetaMathlibExt.isMotzkinHump_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-motzkin-hump-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Write a path of length $n$ as a sequence of steps $p_0, \\dots, p_{n-1}$, and let $s(p_r) \\in \\{-1, 0, 1\\}$ be the vertical displacement of the step $p_r$ (down, flat or up). For a natural number $k$ and positions $i, j < n$, the pair $(i, j)$ is a hump of height $k$ if and only if $i < j$, $s(p_i) = 1$, $s(p_j) = -1$, $s(p_r) = 0$ for every $r$ with $i < r < j$, and $\\sum_{r \\le i} s(p_r) = k$.","title":"Characterization of Motzkin humps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-motzkin-peak","kind":"node","lean":"MetaMathlibExt.IsMotzkinPeak","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-motzkin-peak","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $p$ be a word of length $n$ in the steps $D$, $F$, $U$, let $k \\in \\mathbb{N}$, and let $i, j \\in \\{0, \\dots, n-1\\}$. The pair $(i, j)$ is a peak of height $k$ in $p$ when it is a hump of height $k$ in $p$ and $j = i + 1$, that is, when the down step $p_j$ immediately follows the up step $p_i$.","title":"Motzkin peak"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-motzkin-peak-iff","kind":"node","lean":"MetaMathlibExt.isMotzkinPeak_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-motzkin-peak-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $p$ be a word of length $n$ in the steps $D$, $F$, $U$, let $k \\in \\mathbb{N}$, and let $i, j \\in \\{0, \\dots, n-1\\}$. Then $(i, j)$ is a peak of height $k$ in $p$ if and only if $(i, j)$ is a hump of height $k$ in $p$ and $j = i + 1$.","title":"Peaks are humps with adjacent endpoints"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/last-nonflat-step-up","kind":"node","lean":"MetaMathlibExt.LastNonflatStepUp","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#last-nonflat-step-up","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"A word $p = p_0 p_1 \\cdots p_{n-1}$ of length $n$ in the steps $U$ (up), $F$ (flat) and $D$ (down) has last non-flat step up when some position $i$ has $p_i = U$ and every later step $p_j$ with $j > i$ is $F$. In particular, such a word contains an up step.","title":"Paths whose last non-flat step is up"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/last-nonflat-step-up-iff","kind":"node","lean":"MetaMathlibExt.lastNonflatStepUp_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#last-nonflat-step-up-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $p = p_0 p_1 \\cdots p_{n-1}$ be a word of length $n$ in the steps $D$, $F$, $U$. Then the last non-flat step of $p$ is $U$ if and only if there is a position $i$ with $p_i = U$ such that $p_j = F$ for every position $j > i$.","title":"Characterization of words whose last non-flat step is up"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/first-nonup-step-down","kind":"node","lean":"MetaMathlibExt.FirstNonupStepDown","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#first-nonup-step-down","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Write a path of length $n$ as a sequence of steps $p_0, \\dots, p_{n-1}$, each $D$ (down), $F$ (flat) or $U$ (up), encoded as $0$, $1$, $2$. The path has first non-up step down when there is a position $i$ with $p_i = D$ and $p_j = U$ for every $j < i$. In particular, a path all of whose steps are up steps, including the empty path, does not qualify.","title":"Paths whose first non-up step is down"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/first-nonup-step-down-iff","kind":"node","lean":"MetaMathlibExt.firstNonupStepDown_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#first-nonup-step-down-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Write a path of length $n$ as a sequence of steps $p_0, \\dots, p_{n-1}$, each $D$ (down), $F$ (flat) or $U$ (up), encoded as $0$, $1$, $2$. The path has first non-up step $D$ if and only if there is a position $i$ with $p_i = D$ and $p_j = U$ for every $j < i$.","title":"Characterization of paths whose first non-up step is down"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/ends-with-up","kind":"node","lean":"MetaMathlibExt.EndsWithUp","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#ends-with-up","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Write a path of length $n$ as a sequence of steps $p_0, \\dots, p_{n-1}$, each $D$ (down), $F$ (flat) or $U$ (up), encoded as $0$, $1$, $2$. The path ends with an up step when it has a last position, that is, some $i < n$ with $i + 1 = n$, and $p_i = U$.","title":"Paths ending with an up step"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/ends-with-up-iff","kind":"node","lean":"MetaMathlibExt.endsWithUp_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#ends-with-up-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Write a path of length $n$ as a sequence of steps $p_0, \\dots, p_{n-1}$, each $D$ (down), $F$ (flat) or $U$ (up), encoded as $0$, $1$, $2$. The path ends with an up step if and only if there is a position $i < n$ with $i + 1 = n$ and $p_i = U$.","title":"Characterization of paths ending with an up step"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-hump-count","kind":"node","lean":"MetaMathlibExt.motzkinHumpCount","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-hump-count","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"A Motzkin path of length $n$ is a word of $n$ steps $U = (1,1)$, $F = (1,0)$ and $D = (1,-1)$ that ends at height $0$ and never goes below height $0$. A hump of height $k$ in such a path is a pair consisting of an up step and a later down step with only flat steps between them, where the up step ends at height $k$. For natural numbers $n$ and $k$, $H_{n,k}$ is the number of pairs $(P, h)$ with $P$ a Motzkin path of\u2026","title":"Number of humps of a given height in Motzkin paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-series","kind":"node","lean":"MetaMathlibExt.motzkinSeries","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-series","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"The Motzkin series is the formal power series $M(x) = \\sum_{n \\ge 0} m_n x^n$ in $\\mathbb{Z}[[x]]$, where $m_n$ is the $n$-th Motzkin number.","title":"Motzkin generating function"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-standard-young-tableau","kind":"node","lean":"MetaMathlibExt.IsStandardYoungTableau","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-standard-young-tableau","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"A standard Young tableau of order $n$ is encoded by a map $T$ that assigns to each entry $i \\in \\{0, \\dots, n-1\\}$ a cell $T(i) = (r_i, c_i)$ with row and column indices in $\\{0, \\dots, n-1\\}$. The map $T$ is a standard Young tableau when it is injective and, for every entry $i$ and every cell $(r, c)$ with $r \\le r_i$ and $c \\le c_i$, there is an entry $j \\le i$ with $T(j) = (r, c)$.","title":"Standard Young tableau"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-standard-young-tableau-iff","kind":"node","lean":"MetaMathlibExt.isStandardYoungTableau_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-standard-young-tableau-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $T$ assign to each entry $i \\in \\{0, \\dots, n-1\\}$ a cell $T(i) = (r_i, c_i)$ with row and column indices in $\\{0, \\dots, n-1\\}$. Then $T$ is a standard Young tableau of order $n$ if and only if $T$ is injective and, for every entry $i$ and every cell $(r, c)$ with $r \\le r_i$ and $c \\le c_i$, there is an entry $j \\le i$ with $T(j) = (r, c)$.","title":"Unfolding the standard Young tableau condition"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-in-two-one-hook","kind":"node","lean":"MetaMathlibExt.IsInTwoOneHook","parent":26523,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tableaux in the "],["math","\\((2,1)\\)"],["text","-hook"]],"source":"Tableaux in the $(2,1)$-hook"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-in-two-one-hook","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $T$ assign to each entry $i < n$ a cell $(r_i, c_i)$ with zero-based row $r_i < n$ and column $c_i < n$. Then $T$ lies in the $(2,1)$-hook when no entry occupies a cell whose row is at least $2$ and whose column is at least $1$, that is, when no $i$ has both $r_i \\ge 2$ and $c_i \\ge 1$.","title":"Tableaux in the $(2,1)$-hook"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/is-in-two-one-hook-iff","kind":"node","lean":"MetaMathlibExt.isInTwoOneHook_iff","parent":26523,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of tableaux in the "],["math","\\((2,1)\\)"],["text","-hook"]],"source":"Characterization of tableaux in the $(2,1)$-hook"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#is-in-two-one-hook-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $T$ assign to each entry $i < n$ a cell $(r_i, c_i)$ with zero-based row $r_i < n$ and column $c_i < n$. Then $T$ lies in the $(2,1)$-hook if and only if for no entry $i$ do both $r_i \\ge 2$ and $c_i \\ge 1$ hold.","title":"Characterization of tableaux in the $(2,1)$-hook"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/tableau-row-length","kind":"node","lean":"MetaMathlibExt.tableauRowLength","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#tableau-row-length","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"For a natural number $n$, let $T$ assign to each entry $i \\in \\{0, \\dots, n-1\\}$ a cell $T(i)$, given by a zero-based row index and a zero-based column index, both in $\\{0, \\dots, n-1\\}$. For a natural number $r$, the length of row $r$ is the number of entries $i$ whose cell $T(i)$ lies in row $r$.","title":"Row length of a tableau"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/has-first-two-row-difference","kind":"node","lean":"MetaMathlibExt.HasFirstTwoRowDifference","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#has-first-two-row-difference","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $T$ assign to each entry $i < n$ a cell $(r_i, c_i)$ with zero-based row $r_i < n$ and column $c_i < n$, and let $\\lambda_1$ and $\\lambda_2$ be the numbers of entries $i$ with $r_i = 0$ and with $r_i = 1$ respectively. For a natural number $k$, $T$ has first-two-row difference $k$ when $\\lambda_1 = \\lambda_2 + k$, that is, when $\\lambda_1 - \\lambda_2 = k$.","title":"First-two-row difference of a tableau"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/has-first-two-row-difference-iff","kind":"node","lean":"MetaMathlibExt.hasFirstTwoRowDifference_iff","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#has-first-two-row-difference-iff","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $T$ assign to each entry $i < n$ a cell $(r_i, c_i)$ with zero-based row $r_i < n$ and column $c_i < n$, and let $\\lambda_1$ and $\\lambda_2$ be the numbers of entries $i$ with $r_i = 0$ and with $r_i = 1$ respectively. For a natural number $k$, $T$ has first-two-row difference $k$ if and only if $\\lambda_1 = \\lambda_2 + k$.","title":"Characterization of the first-two-row difference"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/syt-hook-count","kind":"node","lean":"MetaMathlibExt.sytHookCount","parent":26523,"reading_title":{"format":"inline-parts-v1","parts":[["text","Standard Young tableaux in the "],["math","\\((2,1)\\)"],["text","-hook with given row difference"]],"source":"Standard Young tableaux in the $(2,1)$-hook with given row difference"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#syt-hook-count","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $\\mathrm{SYT}_k(2,1;n)$ is the number of standard Young tableaux of order $n$ whose shape $\\lambda$ lies in the $(2,1)$-hook, that is $\\lambda_3 \\le 1$, and whose first two row lengths satisfy $\\lambda_1 = \\lambda_2 + k$ (a missing row has length $0$).","title":"Standard Young tableaux in the $(2,1)$-hook with given row difference"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-hump-family","kind":"node","lean":"MetaMathlibExt.MotzkinHumpFamily","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-hump-family","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"For a natural number $n$, this is the set of triples $(k, P, h)$ where $k$ is a natural number with $1 \\le k \\le \\lfloor n/2 \\rfloor$, $P$ is a Motzkin path of length $n$ (a word of $n$ steps $U = (1,1)$, $F = (1,0)$ and $D = (1,-1)$ that ends at height $0$ and never goes below height $0$), and $h$ is a hump of height $k$ in $P$: an up step and a later down step with only flat steps between them, where the up step\u2026","title":"Motzkin paths with a marked hump, over all heights"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-last-up-even-family","kind":"node","lean":"MetaMathlibExt.MotzkinLastUpEvenFamily","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-last-up-even-family","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Use the steps $U = (1,1)$, $F = (1,0)$ and $D = (1,-1)$, and call a word in these steps a Motzkin prefix if none of its initial segments ends below height $0$. For a natural number $n$, this is the set of pairs $(k, p)$ where $k$ is a natural number with $1 \\le k \\le \\lfloor n/2 \\rfloor$ and $p$ is a Motzkin prefix of length $n$ ending at height $2k$ that contains a $U$ after which every step is $F$.","title":"Motzkin prefixes of even height whose last non-flat step is up"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-first-down-family","kind":"node","lean":"MetaMathlibExt.MotzkinFirstDownFamily","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-first-down-family","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Use the steps $U = (1,1)$, $F = (1,0)$ and $D = (1,-1)$, and call a word in these steps a Motzkin prefix if none of its initial segments ends below height $0$. For a natural number $n$, this is the set of pairs $(k, p)$ where $k$ is a natural number with $k \\le n - 2$ and $p$ is a Motzkin prefix of length $n$ ending at height $k$ whose first step other than $U$ exists and is $D$.","title":"Motzkin prefixes whose first non-up step is down"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/syt-hook-count-three-one","kind":"node","lean":"MetaMathlibExt.sytHookCount_three_one","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#syt-hook-count-three-one","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"The number $\\mathrm{SYT}_1(2,1;3)$ of standard Young tableaux of order $3$ in the $(2,1)$-hook whose first row is longer than its second row by exactly $1$ equals $2$.","title":"Hook tableaux of order 3 with row difference 1"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/syt-hook-count-three-zero","kind":"node","lean":"MetaMathlibExt.sytHookCount_three_zero","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#syt-hook-count-three-zero","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"The number $\\mathrm{SYT}_0(2,1;3)$ of standard Young tableaux of order $3$ in the $(2,1)$-hook whose first two rows have the same length equals $1$.","title":"Hook tableaux of order 3 with row difference 0"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/syt-hook-count-two-one","kind":"node","lean":"MetaMathlibExt.sytHookCount_two_one","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#syt-hook-count-two-one","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"The number $\\mathrm{SYT}_1(2,1;2)$ of standard Young tableaux of order $2$ in the $(2,1)$-hook whose first row is longer than its second row by exactly $1$ equals $0$.","title":"Hook tableaux of order 2 with row difference 1"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-hump-count-four-one","kind":"node","lean":"MetaMathlibExt.motzkinHumpCount_four_one","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-hump-count-four-one","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"fully_proved","summary":"Let $H_{n,k}$ be the number of pairs $(P, h)$ where $P$ is a Motzkin path of length $n$ and $h$ is a hump of height $k$ in $P$, that is, an up step and a later down step with only flat steps between them, where the up step ends at height $k$. Then $H_{4,1} = 8$.","title":"Humps of height one in Motzkin paths of length four"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/syt-hook-count-closed-form","kind":"node","lean":"MetaMathlibExt.sytHookCount_closedForm","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#syt-hook-count-closed-form","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"can_prove","summary":"For natural numbers $n$ and $k$ with $k + 2 \\le n$, let $\\mathrm{SYT}_k(2,1;n)$ be the number of standard Young tableaux of order $n$ in the $(2,1)$-hook whose first row is longer than its second row by exactly $k$. Then, as an identity of rational numbers,","title":"Closed formula for hook tableau counts"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-hump-family-equiv-last-up-even-f","kind":"node","lean":"MetaMathlibExt.motzkinHumpFamily_equiv_lastUpEvenFamily","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-hump-family-equiv-last-up-even-f","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"can_prove","summary":"Use the steps $U = (1,1)$, $F = (1,0)$ and $D = (1,-1)$. A Motzkin prefix of length $n$ ending at height $h$ is a word of $n$ steps that never goes below height $0$ and ends at height $h$; a Motzkin path is a Motzkin prefix ending at height $0$. A hump of height $k$ in a path is a pair of an up step and a later down step with only flat steps between them, where the up step ends at height $k$; the hump is a peak\u2026","title":"Humps correspond to last-up prefixes of twice the height"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-hump-count-generating-function","kind":"node","lean":"MetaMathlibExt.motzkinHumpCount_generatingFunction","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-hump-count-generating-function","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"can_prove","summary":"Let $H_{n,k}$ be the number of pairs $(P, h)$ where $P$ is a Motzkin path of length $n$ and $h$ is a hump of height $k$ in $P$ (an up step and a later down step with only flat steps between them, the up step ending at height $k$), and let $M(x) = \\sum_{n \\ge 0} m_n x^n$, where $m_n$ is the $n$-th Motzkin number. For every integer $k \\ge 1$, in the ring $\\mathbb{Z}[[x]]$ of formal power series,","title":"Generating function for humps of a fixed height"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/motzkin-first-down-family-equiv-last-up","kind":"node","lean":"MetaMathlibExt.motzkinFirstDownFamily_equiv_lastUpEvenFamily","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#motzkin-first-down-family-equiv-last-up","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"can_prove","summary":"Use the steps $U = (1,1)$, $F = (1,0)$ and $D = (1,-1)$, and call a word in these steps a Motzkin prefix if none of its initial segments ends below height $0$. For every natural number $n$ there is a bijection between the set of pairs $(k, p)$, where $k$ is a natural number with $k + 2 \\le n$ and $p$ is a Motzkin prefix of length $n$ ending at height $k$ whose first step other than $U$ exists and is $D$, and the\u2026","title":"A bijection between first-down and even-height last-up Motzkin prefixes"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/adjusted-syt-hook-count-generating-funct","kind":"node","lean":"MetaMathlibExt.adjustedSytHookCount_generatingFunction","parent":26523,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function for the adjusted "],["math","\\((2,1)\\)"],["text","-hook tableau counts"]],"source":"Generating function for the adjusted $(2,1)$-hook tableau counts"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#adjusted-syt-hook-count-generating-funct","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"can_prove","summary":"Let $k$ be a natural number, and let $\\mathrm{SYT}_k(2,1;n)$ be the number of standard Young tableaux with $n$ cells in the $(2,1)$-hook whose first two rows differ in length by $k$. Set $T_{n,k} = \\mathrm{SYT}_k(2,1;n) + (-1)^{n+k+1}$ for $n \\ge k$ and $T_{n,k} = 0$ for $n < k$, and let $M(x) = \\sum_{n \\ge 0} m_n x^n$ be the generating function of the Motzkin numbers $m_n$. Then in the ring $\\mathbb{Z}[[x]]$ of\u2026","title":"Generating function for the adjusted $(2,1)$-hook tableau counts"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/syt-hook-count-generating-function","kind":"node","lean":"MetaMathlibExt.sytHookCount_generatingFunction","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#syt-hook-count-generating-function","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"can_prove","summary":"For a natural number $k$, let $\\mathrm{SYT}_k(2,1;n)$ be the number of standard Young tableaux of order $n$ in the $(2,1)$-hook whose first row is longer than its second row by exactly $k$, and let $M(x) = \\sum_{n \\ge 0} m_n x^n$ be the generating function of the Motzkin numbers $m_n$. Then, in the ring of formal power series with integer coefficients,","title":"Generating function for hook tableau counts"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/syt-hook-count-add-eq-hump-count-sub","kind":"node","lean":"MetaMathlibExt.sytHookCount_add_eq_humpCount_sub","parent":26523,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths/index.html#syt-hook-count-add-eq-hump-count-sub","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/humps-and-peaks-in-motzkin-paths","status":"can_prove","summary":"For natural numbers $m$ and $j$, let $\\mathrm{SYT}_j(2,1;m)$ be the number of standard Young tableaux of order $m$ in the $(2,1)$-hook whose first row is longer than its second row by exactly $j$, and let $H_{m,j}$ be the number of Motzkin paths of length $m$ with one distinguished hump of height $j$ (a hump is an up step followed by flat steps and then a down step; its height is the height reached by the up step).\u2026","title":"Hook tableau counts as differences of hump counts"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle.json","id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","kind":"scope","navigation":"module","parent":26443,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","title":"Motzkin triangle"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/motzkin-discriminant","kind":"node","lean":"MetaMathlibExt.motzkinDiscriminant","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#motzkin-discriminant","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"The Motzkin discriminant is the formal power series $1 - 2X - 3X^2 \\in \\mathbb{Q}[[X]]$, the discriminant in the generating functions for the Motzkin triangle.","title":"Motzkin discriminant"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/is-motzkin-square-root","kind":"node","lean":"MetaMathlibExt.IsMotzkinSquareRoot","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#is-motzkin-square-root","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"A formal power series $s \\in \\mathbb{Q}[[X]]$ is a Motzkin square root when $s^2 = 1 - 2X - 3X^2$ and the constant coefficient of $s$ is $1$.","title":"Square roots of the Motzkin discriminant"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/motzkin-square-root","kind":"node","lean":"MetaMathlibExt.motzkinSquareRoot","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#motzkin-square-root","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"The canonical square root is the power series $s = \\sum_{n \\ge 0} \\binom{1/2}{n} (-2X - 3X^2)^n \\in \\mathbb{Q}[[X]]$, obtained by substituting $-2X - 3X^2$ for $Y$ in the binomial series $(1 + Y)^{1/2} = \\sum_{n \\ge 0} \\binom{1/2}{n} Y^n$.","title":"Canonical square root of the Motzkin discriminant"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/motzkin-h","kind":"node","lean":"MetaMathlibExt.motzkinH","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#motzkin-h","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"Let $s = \\sum_{n \\ge 0} \\binom{1/2}{n} (-2X - 3X^2)^n \\in \\mathbb{Q}[[X]]$ be the canonical square root of $1 - 2X - 3X^2$. The power series $h \\in \\mathbb{Q}[[X]]$ has $n$-th coefficient $\\frac{1}{2} [X^{n+1}] (1 - X - s)$ for every $n \\ge 0$. Since $1 - X - s$ has constant coefficient $0$, this is the quotient $h = (1 - X - s)/(2X)$.","title":"Second component of the Motzkin Riordan pair"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/motzkin-g","kind":"node","lean":"MetaMathlibExt.motzkinG","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#motzkin-g","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"Let $s = \\sum_{n \\ge 0} \\binom{1/2}{n} (-2X - 3X^2)^n \\in \\mathbb{Q}[[X]]$ be the canonical square root of $1 - 2X - 3X^2$, which has constant coefficient $1$. The power series $g \\in \\mathbb{Q}[[X]]$ is the multiplicative inverse $g = s^{-1}$ of $s$, that is, $g = 1/\\sqrt{1 - 2X - 3X^2}$.","title":"First component of the Motzkin Riordan pair"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/is-motzkin-riordan-pair","kind":"node","lean":"MetaMathlibExt.IsMotzkinRiordanPair","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#is-motzkin-riordan-pair","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"Formal power series $g, h \\in \\mathbb{Q}[[X]]$ form a Motzkin Riordan pair when there is a power series $s \\in \\mathbb{Q}[[X]]$ with $s^2 = 1 - 2X - 3X^2$ and constant coefficient $1$ such that $g s = 1$ and $2 X h = 1 - X - s$.","title":"Motzkin Riordan pairs"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/motzkin-riordan-pair","kind":"node","lean":"MetaMathlibExt.MotzkinRiordanPair","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#motzkin-riordan-pair","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"A bundled Motzkin Riordan pair consists of formal power series $g, h \\in \\mathbb{Q}[[X]]$ together with a proof that they form a Motzkin Riordan pair: there is $s \\in \\mathbb{Q}[[X]]$ with $s^2 = 1 - 2X - 3X^2$ and constant coefficient $1$ such that $g s = 1$ and $2 X h = 1 - X - s$.","title":"Bundled Motzkin Riordan pairs"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/entry","kind":"node","lean":"MetaMathlibExt.MotzkinRiordanPair.entry","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#entry","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"For a Motzkin Riordan pair $(g, h)$ and natural numbers $n$ and $k$, the $(n, k)$ entry is the rational number $[X^n]\\, g h^k$, the coefficient of $X^n$ in the power series $g h^k \\in \\mathbb{Q}[[X]]$.","title":"Entries of a Motzkin Riordan array"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/motzkin-riordan-pair-2","kind":"node","lean":"MetaMathlibExt.motzkinRiordanPair","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#motzkin-riordan-pair-2","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"Let $s \\in \\mathbb{Q}[[X]]$ be the canonical square root of $1 - 2X - 3X^2$, given by the binomial series, which has constant coefficient $1$. The canonical Motzkin Riordan pair is $(g, h)$ with $g = s^{-1}$ and $h = (1 - X - s)/(2X)$, the Motzkin Riordan pair conditions being witnessed by $s$.","title":"The canonical Motzkin Riordan pair"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/motzkin-triangle","kind":"node","lean":"MetaMathlibExt.motzkinTriangle","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#motzkin-triangle","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"Let $s$ be the square root of $1 - 2X - 3X^2$ in the ring of formal power series over $\\mathbb{Q}$ with constant coefficient $1$, and let $g = 1/s$ and $h = (1 - X - s)/(2X)$. For natural numbers $n$ and $k$, the $(n, k)$ entry of the Motzkin triangle (OEIS A094531) is the rational number given by the coefficient of $X^n$ in $g h^k$.","title":"Motzkin triangle"},{"id":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/motzkin-triangle-initial-rows","kind":"node","lean":"MetaMathlibExt.motzkinTriangle_initial_rows","parent":26564,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle/index.html#motzkin-triangle-initial-rows","scope":"Combinatorics/catalan-numbers-lattice-paths/motzkin-numbers/motzkin-triangle","status":"fully_proved","summary":"Writing $T(n, k)$ for the $(n, k)$ entry of the canonical Motzkin triangle, the rows $n = 0, 1, 2, 3$ (with $0 \\le k \\le n$) are $T(0, 0) = 1$; $T(1, 0) = T(1, 1) = 1$; $T(2, 0) = 3$, $T(2, 1) = 2$, $T(2, 2) = 1$; and $T(3, 0) = 7$, $T(3, 1) = 6$, $T(3, 2) = 3$, $T(3, 3) = 1$, matching OEIS A094531.","title":"Initial rows of the Motzkin triangle"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks","kind":"scope","parent":26384,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks","status":"can_prove","summary":"Statistics on Dyck paths (peaks, valleys, marked steps, subwords) and their variants (skew, rational, k-box Dyck paths), lattice walks with unit or diagonal steps, Delannoy numbers, Sulanke arrays, colored Dyck paths, and nonintersecting paths via the Lindstr\u00f6m\u2013Gessel\u2013Viennot lemma.","title":"Dyck paths and lattice walks"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/lindstrom-gessel-viennot","kind":"node","lean":"MetaMathlibExt.lindstrom_gessel_viennot","parent":26576,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/index.html#lindstrom-gessel-viennot","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $n$ be a natural number, and let $E$ be a relation on a type $V$, read as a directed graph on $V$ with an edge from $u$ to $v$ whenever $E(u, v)$. Suppose there is a function $r : V \\to \\mathbb{N}$ with $r(u) < r(v)$ whenever $E(u, v)$, so the graph is acyclic. A path from $u$ to $w$ is a nonempty list of vertices that starts at $u$, ends at $w$, and in which there is an edge from\u2026","title":"Lindstr\u00f6m\u2013Gessel\u2013Viennot lemma"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/minimal-lattice-walk-count","kind":"node","lean":"MetaMathlibExt.minimal_lattice_walk_count","parent":26576,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/index.html#minimal-lattice-walk-count","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks","status":"fully_proved","summary":"For all natural numbers $a$ and $b$, the number of sequences of $\\max(a, b)$ steps, each taken from $\\{(1,0), (0,1), (1,1)\\}$, whose sum is $(a, b)$ equals $\\binom{\\max(a, b)}{\\min(a, b)}$.","title":"Minimal lattice walks with unit and diagonal steps"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","kind":"scope","navigation":"module","parent":26576,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","title":"Generalized Delannoy numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/delannoy-eq-sum-choose-mul-choose-two-po","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.delannoy_eq_sum_choose_mul_choose_two_pow","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#delannoy-eq-sum-choose-mul-choose-two-po","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For all natural numbers $m$ and $n$, the Delannoy number $D(m, n)$, that is, the Delannoy triangle entry $T(m + n, m; 1)$, satisfies $D(m, n) = \\sum_{j=0}^{\\min(m, n)} \\binom{m}{j} \\binom{n}{j} 2^j$.","title":"Delannoy numbers as a sum of powers of two"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/choose-add-choose-succ-mod-eq-two","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.choose_add_choose_succ_modEq_two","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#choose-add-choose-succ-mod-eq-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For all natural numbers $n$ and $k$ with $k \\ge 1$, $\\binom{n}{k} + \\binom{n+1}{k} \\equiv \\binom{n}{k-1} \\pmod 2$.","title":"Parity of a sum of adjacent binomial coefficients"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-upper","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyUpper","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-upper","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"fully_proved","summary":"For natural numbers $k$, $m$ and $n$, the generalized Delannoy number $D^k(m, n)$ is the natural number","title":"Generalized Delannoy number with upper index"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-lower","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyLower","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-lower","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"fully_proved","summary":"For natural numbers $k$, $m$ and $n$, the generalized Delannoy number $D_k(m, n)$ is","title":"Lower generalized Delannoy number"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-upper-zero-two-two","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyUpper_zero_two_two","parent":26579,"reading_title":{"format":"inline-parts-v1","parts":[["text","An upper generalized Delannoy number equal to "],["math","\\(13\\)"]],"source":"An upper generalized Delannoy number equal to $13$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-upper-zero-two-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"fully_proved","summary":"For natural numbers $k$, $m$ and $n$, write $D^k(m, n) = \\sum_{j=0}^{n} \\binom{m + n + k - j}{n + k} \\binom{n}{j}$ for the upper generalized Delannoy numbers. Then $D^0(2, 2) = 13$.","title":"An upper generalized Delannoy number equal to $13$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-upper-one-three-two","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyUpper_one_three_two","parent":26579,"reading_title":{"format":"inline-parts-v1","parts":[["text","An upper generalized Delannoy number equal to "],["math","\\(44\\)"]],"source":"An upper generalized Delannoy number equal to $44$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-upper-one-three-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"fully_proved","summary":"For natural numbers $k$, $m$ and $n$, write $D^k(m, n) = \\sum_{j=0}^{n} \\binom{m + n + k - j}{n + k} \\binom{n}{j}$ for the upper generalized Delannoy numbers. Then $D^1(3, 2) = 44$.","title":"An upper generalized Delannoy number equal to $44$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-lower-one-two-two","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyLower_one_two_two","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-lower-one-two-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"fully_proved","summary":"The lower generalized Delannoy number $D_1(2, 2)$ equals $8$.","title":"A value of the lower generalized Delannoy number"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-lower-zero","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyLower_zero","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-lower-zero","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"fully_proved","summary":"For natural numbers $k$, $m$ and $n$, let $D_k(m, n) = \\sum_{j=0}^{n} \\binom{m + n - k - j}{n - k} \\binom{n}{j}$ (with subtraction truncated at $0$) and $D^k(m, n) = \\sum_{j=0}^{n} \\binom{m + n + k - j}{n + k} \\binom{n}{j}$ be the lower and upper generalized Delannoy numbers. Then $D_0(m, n) = D^0(m, n)$ for all natural numbers $m$ and $n$.","title":"The lower and upper generalized Delannoy numbers agree at index zero"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-upper-add-succ-right","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyUpper_add_succ_right","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-upper-add-succ-right","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For natural numbers $k$, $m$ and $n$, write $D^k(m, n) = \\sum_{j=0}^{n} \\binom{m + n + k - j}{n + k} \\binom{n}{j}$ for the upper generalized Delannoy numbers. For all natural numbers $k$, $m$ and $n$, $D^k(m, n) + D^k(m, n + 1) = 2 D^{k+1}(m, n)$.","title":"Horizontal sum identity for the upper generalized Delannoy numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-upper-add-succ-left","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyUpper_add_succ_left","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-upper-add-succ-left","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For natural numbers $k$, $m$ and $n$, write $D^k(m, n) = \\sum_{j=0}^{n} \\binom{m + n + k - j}{n + k} \\binom{n}{j}$ for the upper generalized Delannoy numbers. For all natural numbers $k$, $m$ and $n$ with $k \\ge 1$, $D^k(m, n) + D^k(m + 1, n) = D^{k-1}(m + 1, n + 1)$.","title":"Vertical sum identity for the upper generalized Delannoy numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-upper-succ-right-sub","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyUpper_succ_right_sub","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-upper-succ-right-sub","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For natural numbers $k$, $m$ and $n$, write $D^k(m, n) = \\sum_{j=0}^{n} \\binom{m + n + k - j}{n + k} \\binom{n}{j}$ for the upper generalized Delannoy numbers. For all natural numbers $k$, $m$ and $n$ with $m \\ge 1$, $D^k(m, n + 1) - D^k(m, n) = 2 D^{k+1}(m - 1, n)$.","title":"Horizontal difference identity for the upper generalized Delannoy numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-upper-succ-left-sub","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyUpper_succ_left_sub","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-upper-succ-left-sub","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For natural numbers $k$, $m$ and $n$, write $D^k(m, n) = \\sum_{j=0}^{n} \\binom{m + n + k - j}{n + k} \\binom{n}{j}$ for the upper generalized Delannoy numbers. For all natural numbers $k$, $m$ and $n$ with $k \\ge 1$, $D^k(m + 1, n) - D^k(m, n) = D^{k-1}(m + 1, n)$.","title":"Vertical difference identity for the upper generalized Delannoy numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-lower-relations","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyLower_relations","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-lower-relations","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"The four relationship identities for `D_k` (`k \u2265 1`, `m, n \u2265 k`): `D_k(m, n) + D_k(m, n + 1) = 2 D_(k-1)(m, n)`, `D_k(m, n) + D_k(m + 1, n) = D_(k+1)(m + 1, n + 1)`, `D_k(m, n + 1) - D_k(m, n) = 2 D_(k-1)(m - 1, n)`, `D_k(m + 1, n) - D_k(m, n) = D_(k+1)(m + 1, n)`.","title":"Relationship identities for generalized Delannoy numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-lower-eq-sum-two-pow","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyLower_eq_sum_two_pow","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-lower-eq-sum-two-pow","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For natural numbers $k$, $m$, $n$ with $m \\ge k$ and $n \\ge k$, the lower generalized Delannoy number $D_k(m, n)$ satisfies $D_k(m, n) = \\sum_{j=0}^{\\min(m-k, n-k)} \\binom{m-k}{j} \\binom{n}{j+k} 2^{j+k}$.","title":"Lower generalized Delannoy numbers as a sum of powers of two"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-upper-eq-sum-two-pow","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoyUpper_eq_sum_two_pow","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-upper-eq-sum-two-pow","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For natural numbers $k$, $m$ and $n$, write $D^k(m, n) = \\sum_{j=0}^{n} \\binom{m + n + k - j}{n + k} \\binom{n}{j}$ for the upper generalized Delannoy numbers. For all natural numbers $k$, $m$ and $n$,","title":"Powers-of-two formula for the upper generalized Delannoy numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/gen-delannoy-eq-sum-two-pow-unified","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.genDelannoy_eq_sum_two_pow_unified","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#gen-delannoy-eq-sum-two-pow-unified","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"For natural numbers $k$, $m$, $n$, write $D_k(m, n)$ and $D^k(m, n)$ for the lower and upper generalized Delannoy numbers. If $m \\ge k$ and $n \\ge k$, then $D_k(m, n) = \\sum_{j=k}^{\\min(m, n)} \\binom{m-k}{j-k} \\binom{n}{j} 2^j$; and, for all $k$, $m$, $n$, $D^k(m, n) = \\sum_{j=0}^{\\min(m, n)} \\binom{m+k}{j+k} \\binom{n}{j} 2^j$.","title":"Unified power-of-two sums for generalized Delannoy numbers"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/delannoy-array-rows-parity","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.delannoyArray_rows_parity","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#delannoy-array-rows-parity","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"Let $R(m, n)$, for $m, n \\in \\mathbb{N}$, be integers with $R(m+1, n+1) = R(m, n+1) + R(m+1, n) + R(m, n)$ for all $m, n$, and let $m \\in \\mathbb{N}$. If $R(m, 0) \\equiv R(m+1, 0) \\pmod 2$, then $R(m, i) \\equiv R(m+1, i) \\pmod 2$ for all $i$; if $R(m, 0) \\not\\equiv R(m+1, 0) \\pmod 2$, then $R(m, i) \\not\\equiv R(m+1, i) \\pmod 2$ for all $i$.","title":"Parity of adjacent rows of a Delannoy array"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/delannoy-array-horizontal-sum-rows-mod-e","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.delannoyArray_horizontalSum_rows_modEq","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#delannoy-array-horizontal-sum-rows-mod-e","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"Let $R(m, n)$, for $m, n \\in \\mathbb{N}$, be integers with $R(m+1, n+1) = R(m, n+1) + R(m+1, n) + R(m, n)$ for all $m, n$. For an array $S$ let $(HS)(m, n) = S(m, n) + S(m, n+1)$, and write $R^i = H^i R$ for the $i$-fold horizontal sum. Then for every $i \\ge 1$ and all $m, m', n \\in \\mathbb{N}$, $R^i(m, n) \\equiv R^i(m', n) \\pmod 2$.","title":"Rows of iterated horizontal sums of a Delannoy array agree mod 2"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/delannoy-array-binomial-seeds-parity","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.delannoyArray_binomial_seeds_parity","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#delannoy-array-binomial-seeds-parity","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"Let $R(m, n)$, for $m, n \\in \\mathbb{N}$, be integers with $R(m+1, n+1) = R(m, n+1) + R(m+1, n) + R(m, n)$ for all $m, n$. For an array $S$ let $(HS)(m, n) = S(m, n) + S(m, n+1)$, and write $R^i = H^i R$ for the $i$-fold horizontal sum, so $R^0 = R$. Let $k \\in \\mathbb{N}$ and suppose $R(0, n) = \\binom{k+n}{k}$ for all $n$; the values $R(m, 0)$ for $m \\ge 1$ are arbitrary. Then every entry of $R^{k+1}$ is even, and\u2026","title":"Parity of iterated horizontal sums of a Delannoy array with binomial seeds"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/delannoy-array-binomial-seeds-horizontal","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.delannoyArray_binomial_seeds_horizontalSum_modEq","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#delannoy-array-binomial-seeds-horizontal","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"Let $R(m, n)$, for $m, n \\in \\mathbb{N}$, be integers with $R(m+1, n+1) = R(m, n+1) + R(m+1, n) + R(m, n)$ for all $m, n$. Let $k \\ge 1$ and suppose $R(0, n) = \\binom{k+n}{k}$ for all $n$. Then for all $m$ and $n$, $R(m, n) + R(m, n+1) \\equiv \\binom{k+n}{k-1} \\pmod 2$.","title":"First horizontal sums of a Delannoy array with binomial seeds"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/delannoy-array-binomial-seeds-two-pow-dv","kind":"node","lean":"MetaMathlibExt.DelannoyConstructions.delannoyArray_binomial_seeds_two_pow_dvd","parent":26579,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers/index.html#delannoy-array-binomial-seeds-two-pow-dv","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/generalized-delannoy-numbers","status":"can_prove","summary":"Let $R(m, n)$, for $m, n \\in \\mathbb{N}$, be integers with $R(m+1, n+1) = R(m, n+1) + R(m+1, n) + R(m, n)$ for all $m, n$. For an array $S$ let $(HS)(m, n) = S(m, n) + S(m, n+1)$, and write $R^i = H^i R$ for the $i$-fold horizontal sum. Let $k, i \\in \\mathbb{N}$ with $i \\ge 1$, and suppose $R(0, n) = \\binom{k+n}{k}$ for all $n$. Then every entry of $R^{k+i}$ is divisible by $2^i$, and some entry of $R^{k+i}$ is not\u2026","title":"Powers of two dividing iterated horizontal sums of a Delannoy array"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","kind":"scope","navigation":"module","parent":26576,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","status":"fully_proved","title":"Gessel words"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/gessel-alphabet","kind":"node","lean":"MetaMathlibExt.GesselAlphabet","parent":26601,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/index.html#gessel-alphabet","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","status":"fully_proved","summary":"For a natural number $n$, the Gessel alphabet over $S = [n] = \\{1, \\dots, n\\}$ is the disjoint union of an unbarred copy and a barred copy of $[n]$: it has an unbarred letter $i$ and a barred letter $\\bar{i}$ for each $i \\in [n]$.","title":"Gessel alphabet"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/gessel-unbarred-count","kind":"node","lean":"MetaMathlibExt.gesselUnbarredCount","parent":26601,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/index.html#gessel-unbarred-count","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","status":"fully_proved","summary":"Let $n$ be a natural number, let $w$ be a word (a finite sequence of letters) over the Gessel alphabet over $[n] = \\{1, \\dots, n\\}$ and let $m$ be a natural number. The unbarred count of $m$ in $w$ is the number of positions of $w$ that hold the unbarred letter $m$; it is $0$ when $m \\notin [n]$.","title":"Unbarred occurrences of a label"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/gessel-barred-count","kind":"node","lean":"MetaMathlibExt.gesselBarredCount","parent":26601,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/index.html#gessel-barred-count","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","status":"fully_proved","summary":"Let $n$ be a natural number, let $w$ be a word (a finite sequence of letters) over the Gessel alphabet over $[n] = \\{1, \\dots, n\\}$ and let $m$ be a natural number. The barred count of $m$ in $w$ is the number of positions of $w$ that hold the barred letter $\\bar{m}$; it is $0$ when $m \\notin [n]$.","title":"Barred occurrences of a label"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/gessel-balance","kind":"node","lean":"MetaMathlibExt.gesselBalance","parent":26601,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/index.html#gessel-balance","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","status":"fully_proved","summary":"Let $n$ be a natural number, let $w$ be a word over the Gessel alphabet over $[n] = \\{1, \\dots, n\\}$ and let $m$ be a natural number. The signed balance of $m$ in $w$ is the integer obtained by subtracting the number of occurrences of the barred letter $\\bar{m}$ in $w$ from the number of occurrences of the unbarred letter $m$ in $w$; it is $0$ when $m \\notin [n]$.","title":"Signed balance of a label"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/is-gessel-word","kind":"node","lean":"MetaMathlibExt.IsGesselWord","parent":26601,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/index.html#is-gessel-word","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","status":"fully_proved","summary":"Let $n$ be a natural number. For a word $u$ over the Gessel alphabet over $S = [n] = \\{1, \\dots, n\\}$ and a natural number $m$, write $\\beta_u(m)$ for the signed balance of $m$ in $u$: the number of occurrences of the unbarred letter $m$ in $u$ minus the number of occurrences of the barred letter $\\bar{m}$ in $u$. A word $w$ over this alphabet is a Gessel word when, for every prefix $w'$ of $w$ (including the empty\u2026","title":"Gessel word"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/is-complete-gessel-word","kind":"node","lean":"MetaMathlibExt.IsCompleteGesselWord","parent":26601,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/index.html#is-complete-gessel-word","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","status":"fully_proved","summary":"A word $w$ over the Gessel alphabet of $S = [n] = \\{1, \\dots, n\\}$ is a complete Gessel word when it is a Gessel word and, for every $m \\in [n]$, the unbarred letter $m$ and the barred letter $\\bar m$ occur the same number of times in $w$.","title":"Complete Gessel words"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/is-complete-gessel-word-iff-balance-eq-z","kind":"node","lean":"MetaMathlibExt.isCompleteGesselWord_iff_balance_eq_zero","parent":26601,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words/index.html#is-complete-gessel-word-iff-balance-eq-z","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/gessel-words","status":"fully_proved","summary":"A word $w$ over the Gessel alphabet of $[n] = \\{1, \\dots, n\\}$ is a complete Gessel word if and only if it is a Gessel word and, for every $m \\in [n]$, the signed balance of $m$ in $w$ (the number of occurrences of the unbarred letter $m$ minus the number of occurrences of the barred letter $\\bar m$) is zero.","title":"Complete Gessel words are the Gessel words of zero balance"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","kind":"scope","navigation":"module","parent":26576,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","title":"k-box paths and skew Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/skew-dyck-step","kind":"node","lean":"MetaMathlibExt.SkewDyckStep","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#skew-dyck-step","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"A skew Dyck path is built from three kinds of step: the up step $U = (1,1)$, the down step $D = (1,-1)$ and the left step $L = (-1,-1)$.","title":"Skew Dyck steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/is-skew-dyck-word","kind":"node","lean":"MetaMathlibExt.IsSkewDyckWord","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#is-skew-dyck-word","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"A word $w$ in the steps $U$, $D$, $L$ is a skew Dyck word when neither $UL$ nor $LU$ occurs in $w$ as two consecutive letters, the number of $U$s in $w$ equals the number of $D$s and $L$s together, and no prefix of $w$ has more $D$s and $L$s together than $U$s.","title":"Skew Dyck words"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/skew-dyck-words","kind":"node","lean":"MetaMathlibExt.skewDyckWords","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Skew Dyck paths of semilength "],["math","\\(m\\)"]],"source":"Skew Dyck paths of semilength $m$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#skew-dyck-words","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"For $m \\in \\mathbb{N}$, the skew Dyck paths of semilength $m$ are the words of length $2m$ over $\\{U, D, L\\}$ that avoid the factors $UL$ and $LU$, have as many $U$s as $D$s and $L$s together, and have no prefix with more $D$s and $L$s together than $U$s.","title":"Skew Dyck paths of semilength $m$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/card-skew-dyck-words-two","kind":"node","lean":"MetaMathlibExt.card_skewDyckWords_two","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three skew Dyck paths of semilength "],["math","\\(2\\)"]],"source":"Three skew Dyck paths of semilength $2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#card-skew-dyck-words-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"The number of skew Dyck paths of semilength $2$ is $3$.","title":"Three skew Dyck paths of semilength $2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/skew-factor-count","kind":"node","lean":"MetaMathlibExt.skewFactorCount","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#skew-factor-count","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"For words $f$ and $w$ over the skew Dyck steps $U$, $D$, $L$, the factor count of $f$ in $w$ is the number of suffixes of $w$, including $w$ itself and the empty suffix, that begin with $f$. Equivalently, it is the number of positions of $w$ at which an occurrence of $f$ as a contiguous block starts, overlapping occurrences all being counted; in particular the empty word has factor count $|w| + 1$ in $w$.","title":"Occurrences of a factor in a skew Dyck word"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/box-factor","kind":"node","lean":"MetaMathlibExt.boxFactor","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-box factor"]],"source":"The $k$-box factor"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#box-factor","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"For $k \\ge 0$, the $k$-box factor is the word $U D^k L$ in the skew Dyck steps: one step $U$, then $k$ steps $D$, then one step $L$.","title":"The $k$-box factor"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-paths","kind":"node","lean":"MetaMathlibExt.kBoxPaths","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-box paths"]],"source":"$k$-box paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-paths","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the set $B_n^{(k)}$ of $k$-box paths of size $n$ consists of the skew Dyck paths of semilength $(k+2)n - 1$ that contain exactly $n$ occurrences of the factor $U D^k L$. The subtraction is truncated at $0$, so $B_0^{(k)}$ consists of the empty path alone.","title":"$k$-box paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-paths-one-one","kind":"node","lean":"MetaMathlibExt.kBoxPaths_one_one","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(1\\)"],["text","-box path of size "],["math","\\(1\\)"]],"source":"The $1$-box path of size $1$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-paths-one-one","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"The only $1$-box path of size $1$ is $UUDL$.","title":"The $1$-box path of size $1$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/box-word","kind":"node","lean":"MetaMathlibExt.boxWord","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#box-word","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"Let $k \\ge 0$ and let $a = (a_1, \\dots, a_n)$ be a list of natural numbers with $n \\ge 1$. The box word of $k$ and $a$ is the word in the skew Dyck steps","title":"Box words"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/box-word-one-three-two","kind":"node","lean":"MetaMathlibExt.boxWord_one_three_two","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","The box word for "],["math","\\(k = 1\\)"],["text"," and "],["math","\\(a = (3, 2)\\)"]],"source":"The box word for $k = 1$ and $a = (3, 2)$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#box-word-one-three-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"For $k = 1$ and $a = (3, 2)$, the box word $U^{a_1} D^k L D \\, U^{a_2} D^k L$ equals $UUUDLDUUDL$.","title":"The box word for $k = 1$ and $a = (3, 2)$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/is-kdyck-word","kind":"node","lean":"MetaMathlibExt.IsKDyckWord","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-Dyck words"]],"source":"$k$-Dyck words"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#is-kdyck-word","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"A word $w$ over the Booleans is a $k$-Dyck word when the number of letters $\\mathrm{true}$ in $w$ is $k$ times the number of letters $\\mathrm{false}$, and every prefix of $w$ contains at least $k$ times as many letters $\\mathrm{true}$ as letters $\\mathrm{false}$.","title":"$k$-Dyck words"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-dyck-words","kind":"node","lean":"MetaMathlibExt.kDyckWords","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-Dyck paths of size "],["math","\\(n\\)"]],"source":"$k$-Dyck paths of size $n$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-dyck-words","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"Write $U = (1,1)$ and $D^{(k)} = (k,-k)$. The $k$-Dyck paths of size $n$ are the words of length $(k+1)n$ over $\\{U, D^{(k)}\\}$ with exactly $n$ steps $D^{(k)}$ that have $k$ times as many $U$s as $D^{(k)}$s and in which every prefix has at least $k$ times as many $U$s as $D^{(k)}$s.","title":"$k$-Dyck paths of size $n$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/card-k-dyck-words-two-two","kind":"node","lean":"MetaMathlibExt.card_kDyckWords_two_two","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three "],["math","\\(2\\)"],["text","-Dyck paths of size "],["math","\\(2\\)"]],"source":"Three $2$-Dyck paths of size $2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#card-k-dyck-words-two-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"The number of $2$-Dyck paths of size $2$ is $3$.","title":"Three $2$-Dyck paths of size $2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/augment-kdyck-word","kind":"node","lean":"MetaMathlibExt.augmentKDyckWord","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Augmentation of a "],["math","\\(k\\)"],["text","-Dyck word"]],"source":"Augmentation of a $k$-Dyck word"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#augment-kdyck-word","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"Let $k \\ge 0$ and let $w$ be a word in the letters $U = (1,1)$ and $D^{(k)} = (k,-k)$, for example a $k$-Dyck path. The augmented path of $w$ is the word in the skew Dyck steps $U$, $D$, $L$ obtained from $w$ by keeping each letter $U$ and replacing each letter $D^{(k)}$ by $U D^{k-1} L D$, where $D^{k-1}$ means $k - 1$ copies of $D$ (none when $k \\le 1$).","title":"Augmentation of a $k$-Dyck word"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/augmented-kdyck-paths","kind":"node","lean":"MetaMathlibExt.augmentedKDyckPaths","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Augmented "],["math","\\(k\\)"],["text","-Dyck paths"]],"source":"Augmented $k$-Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#augmented-kdyck-paths","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"For natural numbers $k$ and $n$, write $U = (1,1)$ and $D^{(k)} = (k,-k)$. A $k$-Dyck path of size $n$ is a word of length $(k+1)n$ in the letters $U$ and $D^{(k)}$ that has exactly $n$ letters $D^{(k)}$ and $k$ times as many letters $U$ as letters $D^{(k)}$, and in which every prefix has at least $k$ times as many letters $U$ as letters $D^{(k)}$. The augmented path of such a word is the word in the skew Dyck\u2026","title":"Augmented $k$-Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/skew-returns","kind":"node","lean":"MetaMathlibExt.skewReturns","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#skew-returns","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"Read a word $w$ over the skew Dyck steps $U = (1,1)$, $D = (1,-1)$ and $L = (-1,-1)$ as a lattice path starting at the origin. The number of returns of $w$ is the number of steps of $w$ that end on the $x$-axis, that is, the number of nonempty prefixes of $w$ that contain as many $U$ steps as $D$ and $L$ steps together.","title":"Returns of a skew Dyck word"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/skew-long-ascents","kind":"node","lean":"MetaMathlibExt.skewLongAscents","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#skew-long-ascents","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"A long ascent of a word $w$ over the skew Dyck steps $U$, $D$, $L$ is a maximal run of consecutive $U$ steps of length at least $2$. The long ascent count of $w$ is the number of long ascents of $w$.","title":"Long ascents of a skew Dyck word"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/skew-returns-skew-long-ascents-example","kind":"node","lean":"MetaMathlibExt.skewReturns_skewLongAscents_example","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#skew-returns-skew-long-ascents-example","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"The word $U D U^5 D L^2 D U D L$ over the skew Dyck steps has exactly two returns and exactly one long ascent.","title":"Returns and long ascents of an example skew Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/augmented-kdyck-gf","kind":"node","lean":"MetaMathlibExt.augmentedKDyckGF","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of augmented "],["math","\\(k\\)"],["text","-Dyck paths"]],"source":"Generating function of augmented $k$-Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#augmented-kdyck-gf","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"Let $k \\ge 0$ and let $\\hat D_n^{(k)}$ be the set of augmented $k$-Dyck paths of size $n$. Then $G_k(x)$ is the formal power series with integer coefficients","title":"Generating function of augmented $k$-Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-return-gf","kind":"node","lean":"MetaMathlibExt.kBoxReturnGF","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of "],["math","\\(k\\)"],["text","-box paths by returns"]],"source":"Generating function of $k$-box paths by returns"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-return-gf","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, $F_k(t, x)$ is the power series in $\\mathbb{Z}[t][[x]]$ with constant term $0$ whose coefficient of $x^n$ for $n \\ge 1$ is the sum of $t^{\\operatorname{ret}(\\mu)}$ over the set $B_n^{(k)}$ of $k$-box paths $\\mu$ of size $n$, where $\\operatorname{ret}(\\mu)$ is the number of returns of $\\mu$; that is, $F_k(t, x) = \\sum_{n \\ge 1} \\sum_{\\mu \\in B_n^{(k)}} t^{\\operatorname{ret}(\\mu)} x^n$.","title":"Generating function of $k$-box paths by returns"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-long-ascent-gf","kind":"node","lean":"MetaMathlibExt.kBoxLongAscentGF","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Long-ascent generating function of "],["math","\\(k\\)"],["text","-box paths"]],"source":"Long-ascent generating function of $k$-box paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-long-ascent-gf","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"Let $B_n^{(k)}$ be the set of $k$-box paths of size $n$, and let $\\mathrm{la}(\\mu)$ be the number of long ascents of a path $\\mu$. The long-ascent generating function of $k$-box paths is the power series","title":"Long-ascent generating function of $k$-box paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/augmented-kdyck-long-ascent-gf","kind":"node","lean":"MetaMathlibExt.augmentedKDyckLongAscentGF","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Long-ascent generating function of augmented "],["math","\\(k\\)"],["text","-Dyck paths"]],"source":"Long-ascent generating function of augmented $k$-Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#augmented-kdyck-long-ascent-gf","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"fully_proved","summary":"Let $k \\ge 0$, let $\\hat D_n^{(k)}$ be the set of augmented $k$-Dyck paths of size $n$, and for a path $\\mu$ let $\\operatorname{asc}(\\mu)$ be its number of long ascents. Then $\\acute{G}_k(t, x)$ is the formal power series","title":"Long-ascent generating function of augmented $k$-Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/min-semilength-box-factor","kind":"node","lean":"MetaMathlibExt.min_semilength_boxFactor","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimal semilength of skew Dyck paths with "],["math","\\(n\\)"],["text"," box factors"]],"source":"Minimal semilength of skew Dyck paths with $n$ box factors"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#min-semilength-box-factor","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 1$. Some skew Dyck path with exactly $n$ occurrences of the factor $U D^k L$ has semilength (number of $U$ steps) $(k+2)n - 1$, and every skew Dyck path with exactly $n$ occurrences of $U D^k L$ has semilength at least $(k+2)n - 1$.","title":"Minimal semilength of skew Dyck paths with $n$ box factors"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-paths-structure","kind":"node","lean":"MetaMathlibExt.kBoxPaths_structure","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Structure of "],["math","\\(k\\)"],["text","-box paths"]],"source":"Structure of $k$-box paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-paths-structure","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 1$. For a list $a = (a_1, \\dots, a_n)$ of natural numbers write $w_k(a) = U^{a_1} D^k L D \\cdots U^{a_{n-1}} D^k L D\\, U^{a_n} D^k L$. Every $k$-box path of size $n$ equals $w_k(a)$ for some list $a$ of $n$ positive integers. Conversely, if $a$ is a list of $n$ positive integers with $a_1 + \\cdots + a_n = (k+2)n - 1$ and no prefix of $w_k(a)$ has more $D$s and $L$s together than $U$s, then\u2026","title":"Structure of $k$-box paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-paths-decomposition","kind":"node","lean":"MetaMathlibExt.kBoxPaths_decomposition","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Decomposition of "],["math","\\(k\\)"],["text","-box paths"]],"source":"Decomposition of $k$-box paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-paths-decomposition","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$. A word $w$ in the steps $U$, $D$, $L$ is a $k$-box path of some size $n \\ge 1$ if and only if $w = \\mu_1 U \\mu_2 U \\cdots \\mu_{k+1} U D^k L$ for some augmented $(k+1)$-Dyck paths $\\mu_1, \\ldots, \\mu_{k+1}$, each of some size.","title":"Decomposition of $k$-box paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/card-k-box-paths","kind":"node","lean":"MetaMathlibExt.card_kBoxPaths","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-box paths of size "],["math","\\(n\\)"]],"source":"Number of $k$-box paths of size $n$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#card-k-box-paths","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 1$, and let $f_{k,n}$ be the number of $k$-box paths of size $n$. Then","title":"Number of $k$-box paths of size $n$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/card-tailed-k-box-paths","kind":"node","lean":"MetaMathlibExt.card_tailed_kBoxPaths","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting tailed "],["math","\\(k\\)"],["text","-box paths"]],"source":"Counting tailed $k$-box paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#card-tailed-k-box-paths","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 1$, and let $T$ be the number of $k$-box paths of size $n$ that end with $U^{k+1} D^k L$. Then $((k+2)(n-1)+1) \\cdot T = \\binom{(k+2)(n-1)+1}{n-1}$, and if $n \\ge 2$ then also $(n-1) \\cdot T = \\binom{(k+2)(n-1)}{n-2}$.","title":"Counting tailed $k$-box paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-return-gf-eq","kind":"node","lean":"MetaMathlibExt.kBoxReturnGF_eq","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-return-gf-eq","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$, let $F_k(t, x) \\in \\mathbb{Z}[t][[x]]$ be the generating function of $k$-box paths of size at least $1$ by number of returns (exponent of $t$) and size (exponent of $x$), and let $G_{k+1}(x)$ be the generating function of augmented $(k+1)$-Dyck paths by size, viewed in $\\mathbb{Z}[t][[x]]$. Then","title":"Functional equation for the returns generating function"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/card-k-box-paths-returns","kind":"node","lean":"MetaMathlibExt.card_kBoxPaths_returns","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-box paths with "],["math","\\(j\\)"],["text"," returns"]],"source":"Number of $k$-box paths with $j$ returns"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#card-k-box-paths-returns","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$ and $1 \\le j \\le n$, and let $r_{k,n,j}$ be the number of $k$-box paths of size $n$ with exactly $j$ returns. Then","title":"Number of $k$-box paths with $j$ returns"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-paths-returns-mean-variance","kind":"node","lean":"MetaMathlibExt.kBoxPaths_returns_mean_variance","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean and variance of the number of returns of "],["math","\\(k\\)"],["text","-box paths"]],"source":"Mean and variance of the number of returns of $k$-box paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-paths-returns-mean-variance","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 1$, and let $R(\\mu)$ be the number of returns (steps ending on the $x$-axis) of a path $\\mu$. The average of $R$ over the $k$-box paths of size $n$ is","title":"Mean and variance of the number of returns of $k$-box paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-long-ascent-gf-functional-equation","kind":"node","lean":"MetaMathlibExt.kBoxLongAscentGF_functional_equations","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Long-ascent functional equations for "],["math","\\(k\\)"],["text","-box paths"]],"source":"Long-ascent functional equations for $k$-box paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-long-ascent-gf-functional-equation","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $\\acute F_k(t, x)$ be the long-ascent generating function of $k$-box paths and $\\acute G_k(t, x)$ that of augmented $k$-Dyck paths, including the empty path, both in $\\mathbb{Z}[t][[x]]$. Then for every $k \\ge 1$","title":"Long-ascent functional equations for $k$-box paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/card-k-box-paths-long-ascents","kind":"node","lean":"MetaMathlibExt.card_kBoxPaths_longAscents","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-box paths with "],["math","\\(j\\)"],["text"," long ascents"]],"source":"Number of $k$-box paths with $j$ long ascents"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#card-k-box-paths-long-ascents","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$, $n \\ge 1$ and $j \\ge 1$, and let $a_{k,n,j}$ be the number of $k$-box paths of size $n$ with exactly $j$ long ascents. Then","title":"Number of $k$-box paths with $j$ long ascents"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/second-polygonal-eq-card-k-box-paths-two","kind":"node","lean":"MetaMathlibExt.second_polygonal_eq_card_kBoxPaths_two_longAscents","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#second-polygonal-eq-card-k-box-paths-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $K \\ge 4$ and $n \\ge 1$. Then $n\\bigl((K-2)n + (K-4)\\bigr)$ is twice the number of $(K-3)$-box paths of size $n + 1$ with exactly two long ascents (maximal runs of at least two $U$ steps). Equivalently, the $n$th second $K$-gonal number $n\\bigl((K-2)n + (K-4)\\bigr)/2$, the $K$-gonal formula $m\\bigl((K-2)m - (K-4)\\bigr)/2$ at $m = -n$, equals this number of paths.","title":"Second polygonal numbers count box paths with two long ascents"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/sum-k-box-paths-long-ascents","kind":"node","lean":"MetaMathlibExt.sum_kBoxPaths_longAscents","parent":26609,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#sum-k-box-paths-long-ascents","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 1$, let $B_n^{(k)}$ be the set of $k$-box paths of size $n$, and for a path $w$ let $\\mathrm{la}(w)$ be its number of long ascents. Then","title":"Long ascents of box paths: sum and sum of squares"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/k-box-paths-long-ascents-mean-variance","kind":"node","lean":"MetaMathlibExt.kBoxPaths_longAscents_mean_variance","parent":26609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean and variance of long ascents in "],["math","\\(k\\)"],["text","-box paths"]],"source":"Mean and variance of long ascents in $k$-box paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths/index.html#k-box-paths-long-ascents-mean-variance","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/k-box-paths-and-skew-dyck-paths","status":"can_prove","summary":"Let $k \\ge 1$ and $n \\ge 1$. Over the $k$-box paths of size $n$, taken with equal weight, the mean number of long ascents is","title":"Mean and variance of long ascents in $k$-box paths"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","kind":"scope","navigation":"module","parent":26576,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","title":"Marked Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/marked-dyck-step","kind":"node","lean":"MetaMathlibExt.MarkedDyckStep","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#marked-dyck-step","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"A step of a marked Dyck path is one of three kinds: a horizontal step $E = (1, 0)$, an unmarked vertical step $N = (0, 1)$, or a marked vertical step $N^* = (0, 1)$.","title":"Steps of a marked Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/marked-dyck-count-e","kind":"node","lean":"MetaMathlibExt.markedDyckCountE","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#marked-dyck-count-e","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $P$ of marked Dyck steps, $E(P)$ is the number of horizontal steps $E$ in $P$.","title":"Number of horizontal steps in a marked Dyck step sequence"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/marked-dyck-count-n","kind":"node","lean":"MetaMathlibExt.markedDyckCountN","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#marked-dyck-count-n","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $P$ of marked Dyck steps, $N(P)$ is the number of unmarked vertical steps $N$ in $P$.","title":"Number of unmarked vertical steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/marked-dyck-count-nstar","kind":"node","lean":"MetaMathlibExt.markedDyckCountNstar","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#marked-dyck-count-nstar","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $P$ of marked Dyck steps, $N^*(P)$ is the number of marked vertical steps $N^*$ in $P$.","title":"Number of marked vertical steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/marked-dyck-path","kind":"node","lean":"MetaMathlibExt.MarkedDyckPath","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#marked-dyck-path","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"Let $E(Q)$, $N(Q)$ and $N^*(Q)$ denote the numbers of horizontal steps $E$, unmarked vertical steps $N$ and marked vertical steps $N^*$ in a finite sequence $Q$ of marked Dyck steps. A marked Dyck path is a finite sequence $P$ of marked Dyck steps such that $E(P) = N(P) + N^*(P)$ and every prefix $Q$ of $P$ satisfies $N(Q) + N^*(Q) \\le E(Q)$: a lattice path from $(0, 0)$ that ends on the diagonal and never goes\u2026","title":"Marked Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/is-marked-step","kind":"node","lean":"MetaMathlibExt.IsMarkedStep","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#is-marked-step","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"Let $P$ be a marked Dyck path with $\\ell$ steps. For a position $i$ with $0 \\le i < \\ell$, the step of $P$ at position $i$ is marked when it is the marked vertical step $N^*$ (rather than an unmarked vertical step $N$ or a horizontal step $E$).","title":"Marked step of a marked Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/path-size","kind":"node","lean":"MetaMathlibExt.MarkedDyckPath.pathSize","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#path-size","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"The size of a marked Dyck path $P$ is $N^*(P) + E(P)$, where $N^*(P)$ is the number of marked vertical steps of $P$ and $E(P)$ the number of its horizontal steps.","title":"Size of a marked Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/size-eq","kind":"node","lean":"MetaMathlibExt.MarkedDyckPath.size_eq","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#size-eq","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For every marked Dyck path $P$, the size of $P$ equals $N(P) + 2N^*(P)$, where $N(P)$ and $N^*(P)$ are the numbers of unmarked and marked vertical steps of $P$.","title":"Size of a marked Dyck path in vertical steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-step","kind":"node","lean":"MetaMathlibExt.MultiMarkedStep","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-step","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"A step of a multi-marked Dyck path is one of the horizontal step $E = (1, 0)$, the plain vertical step $N = (0, 1)$, or, for an integer $t \\ge 2$, the marked vertical step $N^*_t = (0, 1)$ of index $t$. There are no marked steps of index less than $2$.","title":"Steps of a multi-marked Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-count-e","kind":"node","lean":"MetaMathlibExt.multiMarkedCountE","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-count-e","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $P$ of multi-marked steps (horizontal steps $E$, plain vertical steps $N$ and marked vertical steps $N^*_t$ with index $t \\ge 2$), $E(P)$ is the number of horizontal steps $E$ in $P$.","title":"Number of horizontal steps in a multi-marked step sequence"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-count-plain-n","kind":"node","lean":"MetaMathlibExt.multiMarkedCountPlainN","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-count-plain-n","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $P$ of multi-marked steps (horizontal steps $E$, plain vertical steps $N$ and marked vertical steps $N^*_t$ with index $t \\ge 2$), $N(P)$ is the number of plain (unmarked) vertical steps $N$ in $P$.","title":"Number of plain vertical steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-count-vert","kind":"node","lean":"MetaMathlibExt.multiMarkedCountVert","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-count-vert","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $P$ of multi-marked steps (horizontal steps $E$, plain vertical steps $N$ and marked vertical steps $N^*_t$ with index $t \\ge 2$), this is the number of vertical steps in $P$, plain steps $N$ and marked steps $N^*_t$ counted together.","title":"Number of vertical steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-count-total","kind":"node","lean":"MetaMathlibExt.multiMarkedCountTotal","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-count-total","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $P$ of multi-marked steps (horizontal steps $E$, plain vertical steps $N$ and marked vertical steps $N^*_t$ with index $t \\ge 2$), this is the total number of steps in $P$, that is, the length of $P$.","title":"Number of steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-index-sum","kind":"node","lean":"MetaMathlibExt.multiMarkedIndexSum","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-index-sum","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $s$ of steps of a multi-marked Dyck path, the index sum of $s$ is the sum of the indices $t$ over all occurrences of marked steps $N^*_t$ in $s$, each occurrence counted separately; the steps $E$ and $N$ contribute nothing.","title":"Index sum of a multi-marked step sequence"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-shifted-index-sum","kind":"node","lean":"MetaMathlibExt.multiMarkedShiftedIndexSum","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-shifted-index-sum","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $s$ of steps of a multi-marked Dyck path, the shifted index sum of $s$ is the sum of $t - 1$ over all occurrences of marked steps $N^*_t$ in $s$, each occurrence counted separately; the steps $E$ and $N$ contribute nothing.","title":"Shifted index sum of a multi-marked step sequence"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-count-marked-t","kind":"node","lean":"MetaMathlibExt.multiMarkedCountMarkedT","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-count-marked-t","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a finite sequence $P$ of multi-marked steps (horizontal steps $E$, plain vertical steps $N$ and marked vertical steps $N^*_s$ with index $s \\ge 2$) and a natural number $t$, $N^*_t(P)$ is the number of marked steps in $P$ whose index is exactly $t$.","title":"Number of marked steps of a given index"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-size","kind":"node","lean":"MetaMathlibExt.multiMarkedSize","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-size","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"The size of a finite sequence $s$ of steps of a multi-marked Dyck path is $E(s) + \\sum (t - 1)$, where $E(s)$ is the number of horizontal steps $E$ in $s$ and the sum runs over all occurrences of marked steps $N^*_t$ in $s$.","title":"Size of a multi-marked step sequence"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-dyck-path","kind":"node","lean":"MetaMathlibExt.MultiMarkedDyckPath","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-dyck-path","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"A multi-marked Dyck path is a finite sequence $P$ of steps, each a horizontal step $E = (1,0)$, a plain vertical step $N = (0,1)$ or a marked vertical step $N^*_t = (0,1)$ carrying an index $t \\ge 2$, such that every prefix of $P$, including the empty prefix and $P$ itself, has at most as many vertical steps (plain and marked) as horizontal steps, and $P$ has exactly as many vertical steps as horizontal steps.","title":"Multi-marked Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-count-eq","kind":"node","lean":"MetaMathlibExt.multiMarkedCountEq","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-count-eq","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For every multi-marked Dyck path $P$, the number $E(P)$ of horizontal steps of $P$ equals the number of vertical steps of $P$, plain steps $N$ and marked steps $N^*_t$ counted together.","title":"Equal horizontal and vertical step counts"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-vert-shifted-eq","kind":"node","lean":"MetaMathlibExt.multiMarkedVertShifted_eq","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-vert-shifted-eq","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For every finite sequence $s$ of steps of a multi-marked Dyck path, the number of vertical steps of $s$ plus the shifted index sum of $s$ equals the number of plain vertical steps $N$ of $s$ plus the index sum of $s$.","title":"Vertical count plus shifted index sum"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-size-eq","kind":"node","lean":"MetaMathlibExt.multiMarkedSizeEq","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-size-eq","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For every multi-marked Dyck path $P$, the size $E(P) + \\sum (t - 1)$ of $P$ equals $N(P) + \\sum t$, where $E(P)$ is the number of horizontal steps $E$ of $P$, $N(P)$ is the number of plain vertical steps $N$ of $P$, and both sums run over all occurrences of marked steps $N^*_t$ in $P$.","title":"Two expressions for the size of a multi-marked Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-is-vert","kind":"node","lean":"MetaMathlibExt.multiMarkedIsVert","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-is-vert","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"A step of a multi-marked Dyck path is vertical when it is the plain step $N$ or a marked step $N^*_t$; the horizontal step $E$ is not vertical. This test delimits the final run of vertical steps of a step sequence.","title":"Vertical steps of a multi-marked Dyck path"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-trailing-vert","kind":"node","lean":"MetaMathlibExt.multiMarkedTrailingVert","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-trailing-vert","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"The final run of vertical steps of a finite sequence $s$ of steps of a multi-marked Dyck path is the longest suffix of $s$ consisting of vertical steps ($N$ or $N^*_t$): the part of $s$ after its last horizontal step $E$, or all of $s$ if $s$ has no step $E$.","title":"Final run of vertical steps"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/has-marked-tail","kind":"node","lean":"MetaMathlibExt.MultiMarkedDyckPath.HasMarkedTail","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#has-marked-tail","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"A multi-marked Dyck path $P$ has a marked tail when its final run of vertical steps (the steps after its last horizontal step $E$, or all of $P$ if $P$ has no step $E$) contains a marked vertical step $N^*_t$ for some index $t \\ge 2$.","title":"Marked tail"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/has-unmarked-tail","kind":"node","lean":"MetaMathlibExt.MultiMarkedDyckPath.HasUnmarkedTail","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#has-unmarked-tail","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"A multi-marked Dyck path $P$ has an unmarked tail when it does not have a marked tail, that is, when its final run of vertical steps (the steps after its last horizontal step $E$, or all of $P$ if $P$ has no step $E$) contains no marked vertical step.","title":"Unmarked tail"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-dyck-path-of-size","kind":"node","lean":"MetaMathlibExt.MultiMarkedDyckPathOfSize","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-dyck-path-of-size","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a natural number $n$, $\\tilde P_n$ is the set of multi-marked Dyck paths $P$ of size $n$. Here the size of $P$ is $E(P) + \\sum (t - 1)$, where $E(P)$ is the number of horizontal steps of $P$ and the sum runs over the marked steps $N^*_t$ of $P$, each occurrence counted separately.","title":"Multi-marked Dyck paths of a given size"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/multi-marked-unmarked-tail","kind":"node","lean":"MetaMathlibExt.MultiMarkedUnmarkedTail","parent":26645,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths/index.html#multi-marked-unmarked-tail","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/marked-dyck-paths","status":"fully_proved","summary":"For a natural number $n$, $\\tilde R_n$ is the set of multi-marked Dyck paths of size $n$ with an unmarked tail, that is, whose final run of vertical steps contains no marked step.","title":"Multi-marked Dyck paths with an unmarked tail"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths","kind":"scope","navigation":"module","parent":26576,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths","status":"fully_proved","title":"Peak descents of Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths/dyck-peak-descents","kind":"node","lean":"MetaMathlibExt.dyckPeakDescents","parent":26673,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths/index.html#dyck-peak-descents","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths","status":"fully_proved","summary":"Let $l$ be a finite sequence of up steps $U$ and down steps $D$. A peak of $l$ is a position where a step $U$ is immediately followed by a step $D$. The peak descents of $l$ form the list that has, for each peak from left to right, the length $j$ of the maximal run of consecutive steps $D$ that starts at the $D$ of that peak, so that the peak and its descent form the primitive block $U D^j$.","title":"Peak descents"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths/dyck-peak-descents-sample","kind":"node","lean":"MetaMathlibExt.dyckPeakDescents_sample","parent":26673,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths/index.html#dyck-peak-descents-sample","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/peak-descents-of-dyck-paths","status":"fully_proved","summary":"The Dyck path $UUUUDDDUDDUUDUUDDD$, which factors as $U\\,U\\,U\\,(UD^3)(UD^2)\\,U\\,(UD)\\,U\\,(UD^3)$, has four peaks $UD$, and the lengths of the maximal runs of $D$ steps starting at the $D$ of each peak are, from left to right, $3, 2, 1, 3$.","title":"Peak descents of a sample Dyck path"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","kind":"scope","navigation":"module","parent":26576,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"can_prove","title":"Rational Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/is-dyck-height-at-most-two","kind":"node","lean":"MetaMathlibExt.IsDyckHeightAtMostTwo","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dyck words of height at most "],["math","\\(2\\)"]],"source":"Dyck words of height at most $2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#is-dyck-height-at-most-two","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"A Dyck word has height at most $2$ when no prefix of it has more than two more up steps $U$ than down steps $D$.","title":"Dyck words of height at most $2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/dyck-height-two-block-word","kind":"node","lean":"MetaMathlibExt.dyckHeightTwoBlockWord","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["text","Block word of a Dyck path of height at most "],["math","\\(2\\)"]],"source":"Block word of a Dyck path of height at most $2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#dyck-height-two-block-word","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"For a Dyck word $P = P_0 P_1 \\cdots P_{2n-1}$ of semilength $n$, with steps indexed from $0$, the block word of $P$ is the Boolean word of length $n - 1$ (empty when $n = 0$) whose entry $i$, for $0 \\le i < n - 1$, is true exactly when $P_{2i+1} = U$.","title":"Block word of a Dyck path of height at most $2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/bool-runs","kind":"node","lean":"MetaMathlibExt.boolRuns","parent":26676,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#bool-runs","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"The run decomposition of a Boolean word is the list, from left to right, of its maximal runs of equal consecutive letters, each recorded as the pair (letter, length of the run).","title":"Maximal runs of a Boolean word"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/bool-runs-example","kind":"node","lean":"MetaMathlibExt.boolRuns_example","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal runs of the word "],["math","\\(1001100\\)"]],"source":"Maximal runs of the word $1001100$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#bool-runs-example","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, the maximal runs of the Boolean word $1001100$, listed from left to right as pairs (letter, length), are $(1, 1), (0, 2), (1, 2), (0, 2)$.","title":"Maximal runs of the word $1001100$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/run-following-valleys","kind":"node","lean":"MetaMathlibExt.runFollowingValleys","parent":26676,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#run-following-valleys","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"Let $R = ((b_0, m_0), \\dots, (b_{k-1}, m_{k-1}))$ be a list of pairs of a Boolean letter and a natural number, read as the maximal runs of a block word in which the letter $\\mathrm{false}$ stands for a $0$-valley $DU$. For $i \\in \\mathbb{N}$, the number of $0$-valleys immediately after run $i$ is $m_{i+1}$ if $i + 1 < k$ and $b_{i+1} = \\mathrm{false}$, and $0$ otherwise.","title":"Valleys following a run"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/is-tilde-rational-runs","kind":"node","lean":"MetaMathlibExt.IsTildeRationalRuns","parent":26676,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#is-tilde-rational-runs","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"Let $r, s \\in \\mathbb{N}$ and let $R$ be a list of runs, each a pair (letter, length) with a Boolean letter. For the run at position $i$, let $v_i$ be the length of the run at position $i + 1$ if that run exists and has letter false, and $v_i = 0$ otherwise. Then $R$ satisfies the rational run condition for $q = r/s$ when every run of letter true, say of length $p$ at position $i$, has $p \\le r$ and $p s \\le r v_i$.","title":"The rational Dyck path run condition"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/is-qbonacci-runs","kind":"node","lean":"MetaMathlibExt.IsQBonacciRuns","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(Q\\)"],["text","-bonacci run condition"]],"source":"The $Q$-bonacci run condition"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#is-qbonacci-runs","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"Let $r, s \\in \\mathbb{N}$ and let $R$ be a list of runs, each a pair (letter, length) with a Boolean letter. For the run at position $i$, let $v_i$ be the length of the run at position $i + 1$ if that run exists and has letter false, and $v_i = 0$ otherwise. Then $R$ satisfies the $Q$-bonacci run condition for $q = r/s$ when every run of letter true, say of length $p$ at position $i$, has $p \\le r$ and also\u2026","title":"The $Q$-bonacci run condition"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/is-qbonacci-runs-examples","kind":"node","lean":"MetaMathlibExt.isQBonacciRuns_examples","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["text","Examples of the "],["math","\\(Q\\)"],["text","-bonacci run condition for "],["math","\\(q = 4/5\\)"]],"source":"Examples of the $Q$-bonacci run condition for $q = 4/5$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#is-qbonacci-runs-examples","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, for $r = 4$ and $s = 5$ the maximal runs of the Boolean word $1001100$ do not satisfy the $Q$-bonacci run condition, while the maximal runs of $100111100$ and of $100111$ do.","title":"Examples of the $Q$-bonacci run condition for $q = 4/5$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/tilde-rational-dyck-paths","kind":"node","lean":"MetaMathlibExt.tildeRationalDyckPaths","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["text","The path sets "],["math","\\(\\tilde R_n^{r/s}\\)"]],"source":"The path sets $\\tilde R_n^{r/s}$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#tilde-rational-dyck-paths","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"For natural numbers $r$, $s$ and $n$, $\\tilde R_n^{r/s}$ is the finite set of Dyck paths of semilength $n$ and height at most $2$ that satisfy the following run condition. For $n \\ge 1$ such a path factors as $U X_1 \\cdots X_{n-1} D$ with each $X_i$ either a $1$-peak $UD$ or a $0$-valley $DU$, and its block word $X_1 \\cdots X_{n-1}$ (empty for $n = 0$) splits into maximal runs of equal blocks. Every run of $p$\u2026","title":"The path sets $\\tilde R_n^{r/s}$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/q-bonacci-dyck-paths","kind":"node","lean":"MetaMathlibExt.qBonacciDyckPaths","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(Q\\)"],["text","-bonacci paths "],["math","\\(Q_n^{r/s}\\)"]],"source":"The $Q$-bonacci paths $Q_n^{r/s}$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#q-bonacci-dyck-paths","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"For natural numbers $r$, $s$ and $n$, $Q_n^{r/s}$ is the finite set of Dyck paths of semilength $n$ and height at most $2$ that satisfy the following run condition. For $n \\ge 1$ such a path factors as $U X_1 \\cdots X_{n-1} D$ with each $X_i$ either a $1$-peak $UD$ or a $0$-valley $DU$, and its block word $X_1 \\cdots X_{n-1}$ (empty for $n = 0$) splits into maximal runs of equal blocks. Every run of $p$ consecutive\u2026","title":"The $Q$-bonacci paths $Q_n^{r/s}$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/card-q-bonacci-dyck-paths-three-two-four","kind":"node","lean":"MetaMathlibExt.card_qBonacciDyckPaths_three_two_four","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["text","Path counts for "],["math","\\(q = 3/2\\)"]],"source":"Path counts for $q = 3/2$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#card-q-bonacci-dyck-paths-three-two-four","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"fully_proved","summary":"Let $r = 3$ and $s = 2$. The set $Q_4^{3/2}$ of $Q$-bonacci paths of semilength $4$, that is, the Dyck paths of semilength $4$ and height at most $2$ whose block word's maximal runs satisfy the $Q$-bonacci run condition, has $7$ elements, while the set $\\tilde R_5^{3/2}$ of Dyck paths of semilength $5$ and height at most $2$ whose block word's maximal runs satisfy the rational run condition has $6$ elements.","title":"Path counts for $q = 3/2$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/nonempty-equiv-q-bonacci-tilde-rational","kind":"node","lean":"MetaMathlibExt.nonempty_equiv_qBonacci_tildeRational","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(Q\\)"],["text","-bonacci and rational-run paths are in bijection when "],["math","\\(s = t r + 1\\)"]],"source":"$Q$-bonacci and rational-run paths are in bijection when $s = t r + 1$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#nonempty-equiv-q-bonacci-tilde-rational","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"can_prove","summary":"Let $r \\ge 1$ and $t$ be natural numbers and $s = t r + 1$. Then for every $n \\ge 0$ the set $Q_n^{r/s}$ of $Q$-bonacci paths of semilength $n$ is in bijection with the set $\\tilde R_{n+t+1}^{r/s}$ of Dyck paths of semilength $n + t + 1$ and height at most $2$ whose block word's maximal runs satisfy the rational run condition for $q = r/s$.","title":"$Q$-bonacci and rational-run paths are in bijection when $s = t r + 1$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/forall-nonempty-equiv-q-bonacci-tilde-ra","kind":"node","lean":"MetaMathlibExt.forall_nonempty_equiv_qBonacci_tildeRational_iff","parent":26676,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(Q\\)"],["text","-bonacci and rational-run paths correspond exactly when "],["math","\\(s \\equiv 1 \\pmod r\\)"]],"source":"$Q$-bonacci and rational-run paths correspond exactly when $s \\equiv 1 \\pmod r$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths/index.html#forall-nonempty-equiv-q-bonacci-tilde-ra","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/rational-dyck-paths","status":"can_prove","summary":"Let $r, s \\ge 1$ be coprime natural numbers and let $t = \\lfloor (s - 1)/r \\rfloor$, so that $t r + 1 \\le s \\le (t + 1) r$. For $m \\in \\mathbb{N}$ write $Q_m^{r/s}$ for the set of $Q$-bonacci paths of semilength $m$ and $\\tilde R_m^{r/s}$ for the set of Dyck paths of semilength $m$ and height at most $2$ whose block word's maximal runs satisfy the rational run condition for $q = r/s$. Then $\\tilde R_{n+t+1}^{r/s}$\u2026","title":"$Q$-bonacci and rational-run paths correspond exactly when $s \\equiv 1 \\pmod r$"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","kind":"scope","navigation":"module","parent":26576,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","title":"Subwords in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/dyck-valley-heights","kind":"node","lean":"MetaMathlibExt.dyckValleyHeights","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#dyck-valley-heights","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"For a finite sequence $l$ of up steps $U$ and down steps $D$, the valley heights of $l$ form the list with one entry for each factor $DU$ of $l$, from left to right: the number of $U$ steps minus the number of $D$ steps in $l$ up to and including the $D$ of that factor, taken to be $0$ when negative.","title":"Valley heights"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/dyck-valley-heights-uududdud","kind":"node","lean":"MetaMathlibExt.dyckValleyHeights_UUDUDDUD","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#dyck-valley-heights-uududdud","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"The Dyck path $UUDUDDUD$ has exactly two valleys, of heights $1$ and $0$ from left to right.","title":"Valley heights of UUDUDDUD"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/dyck-valley-heights-uduududd","kind":"node","lean":"MetaMathlibExt.dyckValleyHeights_UDUUDUDD","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#dyck-valley-heights-uduududd","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"The Dyck path $UDUUDUDD$ has exactly two valleys, of heights $0$ and $1$ from left to right.","title":"Valley heights of UDUUDUDD"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/is-nondecreasing-dyck-path","kind":"node","lean":"MetaMathlibExt.IsNondecreasingDyckPath","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#is-nondecreasing-dyck-path","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"A Dyck path is non-decreasing when its valley heights (the heights reached after the $D$ of each factor $DU$), read from left to right, form a non-decreasing sequence.","title":"Non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-words","kind":"node","lean":"MetaMathlibExt.nondecreasingDyckWords","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Non-decreasing Dyck paths of semilength "],["math","\\(n\\)"]],"source":"Non-decreasing Dyck paths of semilength $n$"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-words","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"For a natural number $n$, $D_n$ is the finite set of Dyck paths of semilength $n$ that are non-decreasing: the heights of their valleys (the height reached after the $D$ of each factor $DU$), read from left to right, form a non-decreasing sequence.","title":"Non-decreasing Dyck paths of semilength $n$"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/card-nondecreasing-dyck-words-three","kind":"node","lean":"MetaMathlibExt.card_nondecreasingDyckWords_three","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#card-nondecreasing-dyck-words-three","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"There are exactly $5$ non-decreasing Dyck paths of semilength $3$.","title":"Five non-decreasing Dyck paths of semilength 3"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/card-nondecreasing-dyck-words-four","kind":"node","lean":"MetaMathlibExt.card_nondecreasingDyckWords_four","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#card-nondecreasing-dyck-words-four","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"There are exactly $13$ non-decreasing Dyck paths of semilength $4$.","title":"Thirteen non-decreasing Dyck paths of semilength 4"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/dyck-subword-count","kind":"node","lean":"MetaMathlibExt.dyckSubwordCount","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#dyck-subword-count","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"For finite words $w$ and $l$ in the up step $U$ and the down step $D$, the number of occurrences of $w$ in $l$ is the number of suffixes of $l$, including $l$ itself and the empty suffix, that begin with $w$. Thus every occurrence of $w$ as a contiguous subword of $l$ is counted, overlapping occurrences separately.","title":"Number of occurrences of a subword"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/dyck-subword-count-uu-uuuddd","kind":"node","lean":"MetaMathlibExt.dyckSubwordCount_UU_UUUDDD","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#dyck-subword-count-uu-uuuddd","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"The contiguous subword $UU$ occurs exactly $2$ times in $UUUDDD$; the two occurrences overlap.","title":"Occurrences of UU in UUUDDD"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/dyck-subword-count-udu-ududud","kind":"node","lean":"MetaMathlibExt.dyckSubwordCount_UDU_UDUDUD","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#dyck-subword-count-udu-ududud","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"The contiguous subword $UDU$ occurs exactly $2$ times in $UDUDUD$; the two occurrences overlap.","title":"Occurrences of UDU in UDUDUD"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-total","kind":"node","lean":"MetaMathlibExt.nondecreasingDyckSubwordTotal","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-total","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"For a word $w$ in the steps $U$ and $D$ and for $n \\ge 0$, $d_w(n)$ is the total number of occurrences of $w$ as a contiguous subword (overlapping occurrences counted separately), summed over all non-decreasing Dyck paths of semilength $n$.","title":"Total occurrences of a subword in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-total-ud-thre","kind":"node","lean":"MetaMathlibExt.nondecreasingDyckSubwordTotal_UD_three","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-total-ud-thre","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"Let $d_w(n)$ be the total number of occurrences of the subword $w$ over all non-decreasing Dyck paths of semilength $n$. Then $d_{UD}(3) = 10$; that is, the non-decreasing Dyck paths of semilength $3$ have $10$ peaks $UD$ in total.","title":"Peaks in the non-decreasing Dyck paths of semilength three"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-bivariate-gf","kind":"node","lean":"MetaMathlibExt.nondecreasingDyckSubwordBivariateGF","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-bivariate-gf","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"For a word $w$ in the steps $U$ and $D$, let $D_n$ be the set of non-decreasing Dyck paths of semilength $n$, and let $\\#w(P)$ be the number of occurrences of $w$ as a contiguous subword of a path $P$, overlapping occurrences counted separately. The bivariate generating function $F_w(x, y)$ is the power series in $x$ with coefficients in $\\mathbb{Z}[y]$ given by","title":"Bivariate generating function of a subword in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-total-gf","kind":"node","lean":"MetaMathlibExt.nondecreasingDyckSubwordTotalGF","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-total-gf","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"For a word $w$ in the steps $U$ and $D$, let $d_w(n)$ be the total number of occurrences of $w$ as a contiguous subword (overlapping occurrences counted separately) over all non-decreasing Dyck paths of semilength $n$. The univariate generating function of $w$ is the integer power series","title":"Generating function of the total subword count in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/dyck-last-peak-height","kind":"node","lean":"MetaMathlibExt.dyckLastPeakHeight","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#dyck-last-peak-height","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"For a finite sequence $l$ of up steps $U$ and down steps $D$, the last peak height of $l$ is undefined when $l$ has no factor $UD$. Otherwise it is the height of the peak of the rightmost factor $UD$: the number of $U$ steps minus the number of $D$ steps in $l$ up to and including the $U$ of that factor, taken to be $0$ when negative.","title":"Height of the last peak"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/dyck-last-peak-height-uduudd","kind":"node","lean":"MetaMathlibExt.dyckLastPeakHeight_UDUUDD","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#dyck-last-peak-height-uduudd","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"The last peak of the Dyck path $UDUUDD$ has height $2$.","title":"Last peak height of UDUUDD"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-last-peak-count","kind":"node","lean":"MetaMathlibExt.nondecreasingDyckLastPeakCount","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-last-peak-count","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $l_{n,k}$ is the number of non-decreasing Dyck paths of semilength $n$ whose last peak (the last factor $UD$) has height $k$.","title":"Non-decreasing Dyck paths by height of the last peak"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-last-peak-count-four","kind":"node","lean":"MetaMathlibExt.nondecreasingDyckLastPeakCount_four_one","parent":26690,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-last-peak-count-four","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"fully_proved","summary":"Exactly four non-decreasing Dyck paths of semilength $4$ have their last peak at height $1$; that is, $l_{4,1} = 4$, where $l_{n,k}$ is the number of non-decreasing Dyck paths of semilength $n$ whose last peak has height $k$.","title":"Non-decreasing Dyck paths of semilength four with last peak at height one"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-ud","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_UD","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions and closed form for the subword "],["math","\\(UD\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Generating functions and closed form for the subword $UD$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-ud","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"For $n \\ge 0$ let $\\mathcal{D}_n$ be the set of non-decreasing Dyck paths of semilength $n$, let $\\#_{UD}(P)$ be the number of occurrences of the subword $UD$ in a path $P$, and let $d_{UD}(n) = \\sum_{P \\in \\mathcal{D}_n} \\#_{UD}(P)$. Let $F_{UD}(x, y) = \\sum_{n \\ge 1} \\sum_{P \\in \\mathcal{D}_n} y^{\\#_{UD}(P)} x^n$, a power series in $x$ with coefficients in $\\mathbb{Z}[y]$, and\u2026","title":"Generating functions and closed form for the subword $UD$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-du","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_DU","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subword "],["math","\\(DU\\)"],["text"," in non-decreasing Dyck paths"]],"source":"The subword $DU$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-du","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $F_{DU}(x, y)$ and $D_{DU}(x)$ be the bivariate and univariate generating functions of the subword $DU$ in non-decreasing Dyck paths, let $d_{DU}(n)$ be the total number of occurrences of $DU$ over all non-decreasing Dyck paths of semilength $n$, and let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers. Then","title":"The subword $DU$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-uu-dd","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_UU_DD","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions and closed form for the subwords "],["math","\\(UU\\)"],["text"," and "],["math","\\(DD\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Generating functions and closed form for the subwords $UU$ and $DD$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-uu-dd","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Generating functions and closed form for the subwords $UU$ and $DD$ in non-decreasing Dyck paths. Let $w$ be $UU$ or $DD$. For $n \\ge 0$ let $\\mathcal{D}_n$ be the set of non-decreasing Dyck paths of semilength $n$, let $\\#_w(P)$ be the number of occurrences of the subword $w$ in a path $P$, and let $d_w(n) = \\sum_{P \\in \\mathcal{D}_n} \\#_w(P)$. Let\u2026","title":"Generating functions and closed form for the subwords $UU$ and $DD$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-uuu","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_UUU","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subword "],["math","\\(UUU\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Subword $UUU$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-uuu","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $F_{UUU}(x, y) \\in \\mathbb{Z}[y][[x]]$ be the bivariate generating function with constant term $0$ in which, for $n \\ge 1$, the coefficient of $x^n$ is the sum of $y^{k(P)}$ over the non-decreasing Dyck paths $P$ of semilength $n$, where $k(P)$ is the number of occurrences of the subword $UUU$ in $P$ (overlapping occurrences counted separately). Let $d_{UUU}(n)$ be the total number of occurrences of $UUU$ over\u2026","title":"Subword $UUU$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-udu","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_UDU","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions and closed form for the subword "],["math","\\(UDU\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Generating functions and closed form for the subword $UDU$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-udu","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"For $n \\ge 0$ let $\\mathcal{D}_n$ be the set of non-decreasing Dyck paths of semilength $n$, let $\\#_{UDU}(P)$ be the number of occurrences of the subword $UDU$ in a path $P$, and let $d_{UDU}(n) = \\sum_{P \\in \\mathcal{D}_n} \\#_{UDU}(P)$. Let $F_{UDU}(x, y) = \\sum_{n \\ge 1} \\sum_{P \\in \\mathcal{D}_n} y^{\\#_{UDU}(P)} x^n$, a power series in $x$ with coefficients in $\\mathbb{Z}[y]$, and\u2026","title":"Generating functions and closed form for the subword $UDU$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-udd","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_UDD","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions and closed form for the subword "],["math","\\(UDD\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Generating functions and closed form for the subword $UDD$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-udd","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"For $n \\ge 0$ let $\\mathcal{D}_n$ be the set of non-decreasing Dyck paths of semilength $n$, let $\\#_{UDD}(P)$ be the number of occurrences of the subword $UDD$ in a path $P$, and let $d_{UDD}(n) = \\sum_{P \\in \\mathcal{D}_n} \\#_{UDD}(P)$. Let $F_{UDD}(x, y) = \\sum_{n \\ge 1} \\sum_{P \\in \\mathcal{D}_n} y^{\\#_{UDD}(P)} x^n$, a power series in $x$ with coefficients in $\\mathbb{Z}[y]$, and\u2026","title":"Generating functions and closed form for the subword $UDD$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-duu","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_DUU","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subword "],["math","\\(DUU\\)"],["text"," in non-decreasing Dyck paths"]],"source":"The subword $DUU$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-duu","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $F_{DUU}(x, y)$ and $D_{DUU}(x)$ be the bivariate and univariate generating functions of the subword $DUU$ in non-decreasing Dyck paths, let $d_{DUU}(n)$ be the total number of occurrences of $DUU$ over all non-decreasing Dyck paths of semilength $n$, and let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers. Then","title":"The subword $DUU$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-dud","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_DUD","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subword "],["math","\\(DUD\\)"],["text"," in non-decreasing Dyck paths"]],"source":"The subword $DUD$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-dud","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $F_{DUD}(x, y)$ and $D_{DUD}(x)$ be the bivariate and univariate generating functions of the subword $DUD$ in non-decreasing Dyck paths, let $d_{DUD}(n)$ be the total number of occurrences of $DUD$ over all non-decreasing Dyck paths of semilength $n$, and let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers. Then","title":"The subword $DUD$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-ddu","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_DDU","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subword "],["math","\\(DDU\\)"],["text"," in non-decreasing Dyck paths"]],"source":"The subword $DDU$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-ddu","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $F_{DDU}(x, y)$ and $D_{DDU}(x)$ be the bivariate and univariate generating functions of the subword $DDU$ in non-decreasing Dyck paths, let $d_{DDU}(n)$ be the total number of occurrences of $DDU$ over all non-decreasing Dyck paths of semilength $n$, and let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers. Then","title":"The subword $DDU$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-ddd","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_DDD","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","The subword "],["math","\\(DDD\\)"],["text"," in non-decreasing Dyck paths"]],"source":"The subword $DDD$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-ddd","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $F_{DDD}(x, y)$ and $D_{DDD}(x)$ be the bivariate and univariate generating functions of the subword $DDD$ in non-decreasing Dyck paths, let $d_{DDD}(n)$ be the total number of occurrences of $DDD$ over all non-decreasing Dyck paths of semilength $n$, and let $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers. Then","title":"The subword $DDD$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-uud","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_UUD","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions and closed form for the subword "],["math","\\(UUD\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Generating functions and closed form for the subword $UUD$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-uud","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"For $n \\ge 0$ let $\\mathcal{D}_n$ be the set of non-decreasing Dyck paths of semilength $n$, let $\\#_{UUD}(P)$ be the number of occurrences of the subword $UUD$ in a path $P$, and let $d_{UUD}(n) = \\sum_{P \\in \\mathcal{D}_n} \\#_{UUD}(P)$. Let $F_{UUD}(x, y) = \\sum_{n \\ge 1} \\sum_{P \\in \\mathcal{D}_n} y^{\\#_{UUD}(P)} x^n$, a power series in $x$ with coefficients in $\\mathbb{Z}[y]$, and\u2026","title":"Generating functions and closed form for the subword $UUD$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-uuud","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_UUUD","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subword "],["math","\\(UUUD\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Subword $UUUD$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-uuud","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $F_{UUUD}(x, y) \\in \\mathbb{Z}[y][[x]]$ be the bivariate generating function with constant term $0$ in which, for $n \\ge 1$, the coefficient of $x^n$ is the sum of $y^{k(P)}$ over the non-decreasing Dyck paths $P$ of semilength $n$, where $k(P)$ is the number of occurrences of the subword $UUUD$ in $P$. Let $d_{UUUD}(n)$ be the total number of occurrences of $UUUD$ over all non-decreasing Dyck paths of\u2026","title":"Subword $UUUD$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-u-pow","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_U_pow","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the subword "],["math","\\(U^m\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Recurrence for the subword $U^m$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-u-pow","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $m \\ge 1$. Let $d_w(n)$ be the total number of occurrences of the subword $w$ over all non-decreasing Dyck paths of semilength $n$, and let $F_k$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). Then for every $n > m$,","title":"Recurrence for the subword $U^m$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-u-pow-d","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_U_pow_D","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the subword "],["math","\\(U^mD\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Recurrence for the subword $U^mD$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-u-pow-d","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $m \\ge 1$. Let $d_w(n)$ be the total number of occurrences of the subword $w$ over all non-decreasing Dyck paths of semilength $n$, and let $F_k$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). Then for every $n \\ge m + 2$,","title":"Recurrence for the subword $U^mD$ in non-decreasing Dyck paths"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/nondecreasing-dyck-subword-u-d-pow","kind":"node","lean":"MetaMathlibExt.nondecreasingDyck_subword_U_D_pow","parent":26690,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the subword "],["math","\\(UD^m\\)"],["text"," in non-decreasing Dyck paths"]],"source":"Recurrence for the subword $UD^m$ in non-decreasing Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths/index.html#nondecreasing-dyck-subword-u-d-pow","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/subwords-in-non-decreasing-dyck-paths","status":"can_prove","summary":"Let $m \\ge 1$. Let $d_w(n)$ be the total number of occurrences of the subword $w$ over all non-decreasing Dyck paths of semilength $n$, let $l_{n,k}$ be the number of non-decreasing Dyck paths of semilength $n$ whose last peak has height $k$, and let $F_k$ be the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). Then for every $n$ with $n > m$ and $n \\ge 3$,","title":"Recurrence for the subword $UD^m$ in non-decreasing Dyck paths"},{"data_url":"scopes/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays.json","id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays","kind":"scope","navigation":"module","parent":26576,"result":false,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays/index.html","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays","status":"fully_proved","title":"Sulanke arrays"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays/sulanke-b","kind":"node","lean":"MetaMathlibExt.sulankeB","parent":26724,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays/index.html#sulanke-b","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays","status":"fully_proved","summary":"The Sulanke array $B$ assigns a natural number $B(k, i)$ to every pair of natural numbers $k$ and $i$. It is defined by recursion on $k$: $B(0, 0) = 1$, $B(0, i) = 0$ for $i \\ge 1$, $B(k+1, 0) = B(k, 0)$, and for all $k$ and $i$","title":"Sulanke array B"},{"id":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays/sulanke-a","kind":"node","lean":"MetaMathlibExt.sulankeA","parent":26724,"result":true,"route":"../roadmap/Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays/index.html#sulanke-a","scope":"Combinatorics/catalan-numbers-lattice-paths/dyck-paths-lattice-walks/sulanke-arrays","status":"fully_proved","summary":"The Sulanke array $A$ assigns a natural number $A(k, i)$ to every pair of natural numbers $k$ and $i$. It is defined from the Sulanke array $B$ by $A(0, i) = 0$ for every $i$, $A(k+1, 0) = B(k, 0)$ for every $k$, and $A(k+1, i+1) = B(k, i+1) + 2B(k, i)$ for all $k$ and $i$.","title":"Sulanke array A"},{"data_url":"scopes/Combinatorics/integer-compositions.json","id":"Combinatorics/integer-compositions","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/index.html","scope":"Combinatorics/integer-compositions","status":"can_prove","summary":"Ordered sums of positive integers: compositions with restricted, palindromic, colored or n-color parts, Arndt compositions, perfect compositions, shade, symmetric peaks and valleys, and convolution identities for weighted compositions.","title":"Integer compositions"},{"id":"Combinatorics/integer-compositions/munagi-restricted-compositions","kind":"node","lean":"MetaMathlibExt.munagi_restricted_compositions","parent":26727,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/index.html#munagi-restricted-compositions","scope":"Combinatorics/integer-compositions","status":"fully_proved","summary":"For positive integers $n$ and $k$, the number of compositions of $n$ all of whose parts are congruent to $1$ modulo $k$ equals the number of compositions of $n + k - 1$ all of whose parts are at least $k$.","title":"Munagi's theorem on restricted compositions"},{"id":"Combinatorics/integer-compositions/palindromic-compositions-without-two","kind":"node","lean":"MetaMathlibExt.palindromic_compositions_without_two","parent":26727,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/index.html#palindromic-compositions-without-two","scope":"Combinatorics/integer-compositions","status":"fully_proved","summary":"For an integer $m \\ge 0$, let $C(m)$ be the number of compositions of $m$ (finite sequences of positive integers with sum $m$) with no part equal to $2$, so that $C(0) = 1$ counts the empty composition, and set $C(m) = 0$ for $m < 0$. For a natural number $n$, let $P(n)$ be the number of such compositions of $n$ that are palindromes. Then for every natural number $k$, $P(2k) = C(k+1)$ and\u2026","title":"Palindromic compositions without 2's"},{"id":"Combinatorics/integer-compositions/pow-div-one-sub-pow-eq-sum","kind":"node","lean":"MetaMathlibExt.pow_div_one_sub_pow_eq_sum","parent":26727,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/index.html#pow-div-one-sub-pow-eq-sum","scope":"Combinatorics/integer-compositions","status":"can_prove","summary":"Let $\\alpha < \\beta$ be natural numbers and let $\\zeta = e^{2 \\pi i / \\beta}$. For every complex number $u$ with $1 - u^\\beta \\ne 0$,","title":"Roots-of-unity partial fraction identity"},{"data_url":"scopes/Combinatorics/integer-compositions/arndt-compositions.json","id":"Combinatorics/integer-compositions/arndt-compositions","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","title":"Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For a set $S$ of natural numbers and natural numbers $n$ and $k$, $a_S(n, k)$ is the number of $k$-tuples $(c_1, \\dots, c_k)$ of positive integers such that $c_1 + \\dots + c_k = n$, every $c_i$ lies in $S$, and $c_{2i-1} > c_{2i}$ for every $i$ with $2i \\le k$. These tuples are the Arndt compositions of $n$ with $k$ parts in $S$.","title":"Number of Arndt compositions with parts in a set"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count-six-two","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount_six_two","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count-six-two","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"There are exactly $2$ Arndt compositions of $6$ with two parts.","title":"Arndt compositions of 6 with two parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count-six-three","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount_six_three","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count-six-three","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"There are exactly $4$ Arndt compositions of $6$ with three parts.","title":"Arndt compositions of 6 with three parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count-six-four","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount_six_four","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count-six-four","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"There is exactly one Arndt composition of $6$ with four parts.","title":"Arndt compositions of 6 with four parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-gf","kind":"node","lean":"MetaMathlibExt.arndtGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"An Arndt composition is a finite sequence $(\\sigma_1, \\dots, \\sigma_k)$ of positive integers such that $\\sigma_{2i-1} > \\sigma_{2i}$ for every $i$ with $2i \\le k$. For a set $S$ of natural numbers, let $a_S(n, k)$ be the number of Arndt compositions of $n$ with $k$ parts, all in $S$. The bivariate generating function of these compositions, by weight ($x$) and number of parts ($y$), is the power series\u2026","title":"Generating function of Arndt compositions with parts in a set"},{"id":"Combinatorics/integer-compositions/arndt-compositions/coeff-arndt-gf","kind":"node","lean":"MetaMathlibExt.coeff_arndtGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#coeff-arndt-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For a set $S$ of natural numbers and every $n \\ge 0$, the coefficient of $x^n$ in the generating function $A_S(x, y)$ of Arndt compositions with parts in $S$ is the polynomial $\\sum_{k=0}^{n} a_S(n, k)\\, y^k$ in $\\mathbb{Z}[y]$, where $a_S(n, k)$ is the number of Arndt compositions of $n$ with $k$ parts, all in $S$.","title":"Coefficients of the generating function of Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-gf-mul-one-sub-pairs","kind":"node","lean":"MetaMathlibExt.arndtGF_mul_one_sub_pairs","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-gf-mul-one-sub-pairs","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $S$ be a set of positive integers and let $A_S(x, y)$ be the generating function of Arndt compositions with parts in $S$: the formal power series in $x$ over $\\mathbb{Z}[y]$ whose coefficient of $x^n y^k$ is the number of Arndt compositions of $n$ with $k$ parts, all in $S$. Put $P_S(x) = \\sum_{i \\in S} x^i$ and $Q_S(x) = \\sum_{i, j \\in S,\\ j < i} x^{i+j}$. Then","title":"Pair decomposition of the Arndt generating function"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-gf-univ-mul","kind":"node","lean":"MetaMathlibExt.arndtGF_univ_mul","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-gf-univ-mul","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $A(x, y)$ be the formal power series in $x$ over $\\mathbb{Z}[y]$ whose coefficient of $x^n y^k$ is the number of Arndt compositions of $n$ with $k$ parts. Then","title":"Generating function of all Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count-eq-sum-prod","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount_eq_sum_prod","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count-eq-sum-prod","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For all natural numbers $n$ and $\\ell$, the number of Arndt compositions of $n$ with $2\\ell$ parts is","title":"Arndt compositions with a given number of parts as a sum of products"},{"id":"Combinatorics/integer-compositions/arndt-compositions/sum-arndt-composition-count-le-three","kind":"node","lean":"MetaMathlibExt.sum_arndtCompositionCount_le_three","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#sum-arndt-composition-count-le-three","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For every $n \\ge 0$, the number of Arndt compositions of $n$ (compositions $(\\sigma_1, \\dots, \\sigma_k)$ with $\\sigma_{2i-1} > \\sigma_{2i}$ whenever $2i \\le k$) with all parts in $\\{1, 2, 3\\}$, counting the empty composition when $n = 0$, equals the shifted Padovan number $s_{n+2}$, where $s_0 = 1$, $s_1 = 0$, $s_2 = 1$ and $s_{m+3} = s_{m+1} + s_m$. Equivalently, it is the Padovan number $P(n)$ given by\u2026","title":"Arndt compositions with parts at most 3 and the Padovan numbers"},{"id":"Combinatorics/integer-compositions/arndt-compositions/sum-arndt-composition-count-le-four","kind":"node","lean":"MetaMathlibExt.sum_arndtCompositionCount_le_four","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#sum-arndt-composition-count-le-four","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For every $n \\ge 0$, the number of Arndt compositions of $n$ (compositions $(\\sigma_1, \\dots, \\sigma_k)$ with $\\sigma_{2i-1} > \\sigma_{2i}$ whenever $2i \\le k$) with all parts in $\\{1, 2, 3, 4\\}$, counting the empty composition when $n = 0$, equals $N_{n+1}$, where $(N_m)$ is Narayana's cows sequence: $N_0 = 0$, $N_1 = N_2 = 1$ and $N_m = N_{m-1} + N_{m-3}$ for $m \\ge 3$.","title":"Arndt compositions with parts at most 4 and Narayana's cows sequence"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-gf-le-mul","kind":"node","lean":"MetaMathlibExt.arndtGF_le_mul","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-gf-le-mul","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For a natural number $\\ell$, let $A(x, y; \\ell)$ be the formal power series in $x$ over $\\mathbb{Z}[y]$ whose coefficient of $x^n y^k$ is the number of Arndt compositions of $n$ with $k$ parts, all at most $\\ell$, and let","title":"Generating function of Arndt compositions with bounded largest part"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-gf-ge-mul","kind":"node","lean":"MetaMathlibExt.arndtGF_ge_mul","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-gf-ge-mul","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For a natural number $\\ell \\ge 1$, let $B(x, y; \\ell)$ be the formal power series in $x$ over $\\mathbb{Z}[y]$ whose coefficient of $x^n y^k$ is the number of Arndt compositions of $n$ with $k$ parts, all at least $\\ell$. Then","title":"Generating function of Arndt compositions with bounded smallest part"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-gf-even-odd-mul","kind":"node","lean":"MetaMathlibExt.arndtGF_even_odd_mul","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-gf-even-odd-mul","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $A_E(x, y)$ and $A_O(x, y)$ be the formal power series in $x$ over $\\mathbb{Z}[y]$ whose coefficient of $x^n y^k$ is the number of Arndt compositions of $n$ with $k$ parts, all even, respectively all odd. Then","title":"Generating functions of Arndt compositions with even parts and with odd parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/down-up-composition-count","kind":"node","lean":"MetaMathlibExt.downUpCompositionCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#down-up-composition-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"A down/up (descending wiggly) composition of $n$ with $k$ parts is a sequence $(\\sigma_1, \\dots, \\sigma_k)$ of positive integers with sum $n$ such that $\\sigma_1 > \\sigma_2 < \\sigma_3 > \\sigma_4 < \\cdots$, that is, $\\sigma_i > \\sigma_{i+1}$ for odd $i$ and $\\sigma_i < \\sigma_{i+1}$ for even $i$. For $n, k, j \\ge 0$, $w(n, k, j)$ is the number of down/up compositions of $n$ with $k$ parts whose last part $\\sigma_k$\u2026","title":"Down/up compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/down-up-composition-count-five-two","kind":"node","lean":"MetaMathlibExt.downUpCompositionCount_five_two","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#down-up-composition-count-five-two","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"Among the down/up compositions of $5$ with two parts (pairs $(\\sigma_1, \\sigma_2)$ of positive integers with $\\sigma_1 + \\sigma_2 = 5$ and $\\sigma_1 > \\sigma_2$), exactly one has last part $1$, exactly one has last part $2$, and none has last part $3$.","title":"Down/up compositions of 5 with two parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/down-up-composition-count-six-three","kind":"node","lean":"MetaMathlibExt.downUpCompositionCount_six_three","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#down-up-composition-count-six-three","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"Among the down/up compositions of $6$ with three parts (triples $(\\sigma_1, \\sigma_2, \\sigma_3)$ of positive integers with sum $6$ and $\\sigma_1 > \\sigma_2 < \\sigma_3$), exactly one has last part $3$, exactly one has last part $2$, and none has last part $1$.","title":"Down/up compositions of 6 with three parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/down-up-even-gf","kind":"node","lean":"MetaMathlibExt.downUpEvenGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#down-up-even-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"$F(x, y, z)$ is the formal power series in $\\mathbb{Q}[[x, y, z]]$ whose coefficient of $x^n y^k z^j$ is the number of down/up compositions of $n$ with $k$ parts and last part $j$ when $k$ is even, and $0$ when $k$ is odd. The empty composition, with last part $0$, contributes the constant term $1$.","title":"Generating function of down/up compositions with an even number of parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/down-up-odd-gf","kind":"node","lean":"MetaMathlibExt.downUpOddGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#down-up-odd-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"$G(x, y, z)$ is the formal power series in $\\mathbb{Q}[[x, y, z]]$ whose coefficient of $x^n y^k z^j$ is the number of down/up compositions of $n$ with $k$ parts and last part $j$ when $k$ is odd, and $0$ when $k$ is even.","title":"Generating function of down/up compositions with an odd number of parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-pal-alpha","kind":"node","lean":"MetaMathlibExt.arndtPalAlpha","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","The auxiliary series "],["math","\\(\\alpha\\)"],["text"," for down/up compositions"]],"source":"The auxiliary series $\\alpha$ for down/up compositions"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-pal-alpha","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"In $\\mathbb{Q}[[x, y, z]]$ write $(a; x)_m = \\prod_{j=0}^{m-1} (1 - a x^j)$. For a power series $w$ in $\\mathbb{Q}[[x, y, z]]$, the series $\\alpha(x, y, w)$ is the coefficientwise convergent sum $\\alpha(x, y, w) = \\sum_{n \\ge 0} \\frac{(-1)^n x^{n^2+n} y^{2n} w^n}{(xw; x)_{2n}}$.","title":"The auxiliary series $\\alpha$ for down/up compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-pal-beta","kind":"node","lean":"MetaMathlibExt.arndtPalBeta","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","The auxiliary series "],["math","\\(\\beta\\)"],["text"," for down/up compositions"]],"source":"The auxiliary series $\\beta$ for down/up compositions"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-pal-beta","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"In $\\mathbb{Q}[[x, y, z]]$ write $(a; x)_m = \\prod_{j=0}^{m-1} (1 - a x^j)$. For a power series $w$ in $\\mathbb{Q}[[x, y, z]]$, the series $\\beta(x, y, w)$ is the coefficientwise convergent sum $\\beta(x, y, w) = \\sum_{n \\ge 1} \\frac{(-1)^{n+1} x^{n^2+n} y^{2n} w^n}{(1 - x)(xw; x)_{2n-1}}$.","title":"The auxiliary series $\\beta$ for down/up compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/down-up-even-gf-eq","kind":"node","lean":"MetaMathlibExt.downUpEvenGF_eq","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#down-up-even-gf-eq","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $F(x, y, z) \\in \\mathbb{Q}[[x, y, z]]$ be the generating function of down/up compositions with an even number of parts (including the empty composition, with last part $0$) by weight ($x$), number of parts ($y$) and last part ($z$). For a power series $w$, write $(xw; x)_m = \\prod_{i=1}^{m} (1 - x^i w)$ and let","title":"Formula for the generating function of even-length down/up compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/down-up-odd-gf-eq","kind":"node","lean":"MetaMathlibExt.downUpOddGF_eq","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#down-up-odd-gf-eq","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $F(x, y, z)$ and $G(x, y, z)$ in $\\mathbb{Q}[[x, y, z]]$ be the generating functions of down/up compositions with an even, respectively odd, number of parts by weight ($x$), number of parts ($y$) and last part ($z$). Then","title":"Odd-length down/up compositions from even-length ones"},{"id":"Combinatorics/integer-compositions/arndt-compositions/palindromic-arndt-count","kind":"node","lean":"MetaMathlibExt.palindromicArndtCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#palindromic-arndt-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"An Arndt composition of $n$ with $k$ parts is a sequence $(\\sigma_1, \\dots, \\sigma_k)$ of positive integers with sum $n$ such that $\\sigma_{2i-1} > \\sigma_{2i}$ whenever $2i \\le k$; it is palindromic when $\\sigma_i = \\sigma_{k+1-i}$ for all $i$. For $n, k, m \\ge 0$, $p(n, k, m)$ is the number of palindromic Arndt compositions of $n$ with $k \\ge 1$ parts whose middle summand $\\sigma_{\\lfloor k/2 \\rfloor + 1}$ equals\u2026","title":"Palindromic Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/palindromic-arndt-count-five","kind":"node","lean":"MetaMathlibExt.palindromicArndtCount_five","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#palindromic-arndt-count-five","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"There is exactly one palindromic Arndt composition of $5$ with one part and middle summand $5$, exactly one with three parts and middle summand $1$, and none with three parts and middle summand $2$.","title":"Palindromic Arndt compositions of 5"},{"id":"Combinatorics/integer-compositions/arndt-compositions/palindromic-arndt-gf","kind":"node","lean":"MetaMathlibExt.palindromicArndtGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#palindromic-arndt-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"A palindromic Arndt composition of $n$ with $k \\ge 1$ parts is a tuple $(\\sigma_1, \\dots, \\sigma_k)$ of positive integers with sum $n$ such that $\\sigma_{2i-1} > \\sigma_{2i}$ whenever $2i \\le k$, and $\\sigma_i = \\sigma_{k+1-i}$ for all $i$; its middle summand is $\\sigma_{\\lfloor k/2 \\rfloor + 1}$. The generating function $PA(x, y, z)$ is the formal power series in $x, y, z$ with rational coefficients whose\u2026","title":"Palindromic Arndt generating function"},{"id":"Combinatorics/integer-compositions/arndt-compositions/palindromic-arndt-gf-eq","kind":"node","lean":"MetaMathlibExt.palindromicArndtGF_eq","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#palindromic-arndt-gf-eq","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $PA(x, y, z)$ be the generating function of nonempty palindromic Arndt compositions by weight ($x$), number of parts ($y$) and middle summand ($z$), and let $F(x, y, z)$ be the generating function of down/up compositions $\\sigma_1 > \\sigma_2 < \\sigma_3 > \\cdots$ with an even number of parts by weight ($x$), number of parts ($y$) and last part ($z$). Then, as formal power series over $\\mathbb{Q}$,","title":"Generating function of palindromic Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/colored-arndt-count","kind":"node","lean":"MetaMathlibExt.coloredArndtCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#colored-arndt-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"An Arndt composition of $n$ with $k$ parts is a sequence $(\\sigma_1, \\dots, \\sigma_k)$ of positive integers with sum $n$ such that $\\sigma_{2i-1} > \\sigma_{2i}$ for every $i$ with $2i \\le k$. In a colored Arndt composition, each part of size $m$ carries one of $m$ colors. The number $c(n, k)$ of colored Arndt compositions of $n$ with $k$ parts is therefore the sum of $\\sigma_1 \\sigma_2 \\cdots \\sigma_k$ over all\u2026","title":"Colored Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/colored-arndt-count-four","kind":"node","lean":"MetaMathlibExt.coloredArndtCount_four","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#colored-arndt-count-four","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"Write $m_a$ for a part of size $m$ with color $a \\in \\{1, \\dots, m\\}$. The colored Arndt compositions of $4$ are $(4_1)$, $(4_2)$, $(4_3)$, $(4_4)$, $(3_a, 1_1)$ for $a \\le 3$, and $(2_a, 1_1, 1_1)$ for $a \\le 2$. Hence the numbers of colored Arndt compositions of $4$ with $1$, $2$ and $3$ parts are $c(4, 1) = 4$, $c(4, 2) = 3$ and $c(4, 3) = 2$.","title":"Colored Arndt compositions of 4"},{"id":"Combinatorics/integer-compositions/arndt-compositions/colored-arndt-gf","kind":"node","lean":"MetaMathlibExt.coloredArndtGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#colored-arndt-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"The bivariate generating function of colored Arndt compositions, by weight ($x$) and number of parts ($y$), is the power series $C(x, y) = \\sum_{n \\ge 0} \\left(\\sum_{k=0}^{n} c(n, k)\\, y^k\\right) x^n$ in $\\mathbb{Z}[y][[x]]$, where $c(n, k)$ is the number of colored Arndt compositions of $n$ with $k$ parts.","title":"Generating function of colored Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/colored-arndt-gf-mul","kind":"node","lean":"MetaMathlibExt.coloredArndtGF_mul","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#colored-arndt-gf-mul","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $C(x, y) = \\sum_{n, k} c(n, k)\\, x^n y^k$ in $\\mathbb{Z}[y][[x]]$, where $c(n, k)$ is the number of colored Arndt compositions of $n$ with $k$ parts. Then $C(x, y) \\cdot (1 - x - 3x^2 - x^3(2y^2 - 3) - x^4(y^2 - 3) - x^5(3 + y^2) - x^6 + x^7) = (1 - x)^2 (1 + x)^3 (1 + x^2 + x(y - 2))$.","title":"Closed form of the generating function of colored Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/colored-arndt-total-mul","kind":"node","lean":"MetaMathlibExt.coloredArndt_total_mul","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#colored-arndt-total-mul","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For $n \\ge 0$ let $c(n, k)$ be the number of colored Arndt compositions of $n$ with $k$ parts, and let $c(n) = \\sum_{k=0}^{n} c(n, k)$ be the total number of colored Arndt compositions of $n$ (so $c(0) = 1$, counting the empty composition). Then, in the ring $\\mathbb{Z}[[x]]$ of formal power series,","title":"Generating function for the total number of colored Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/colored-arndt-parts-mul","kind":"node","lean":"MetaMathlibExt.coloredArndt_parts_mul","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of colored Arndt compositions with "],["math","\\(k\\)"],["text"," parts"]],"source":"Generating function of colored Arndt compositions with $k$ parts"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#colored-arndt-parts-mul","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For every $k \\ge 0$, let $C_k(x) = \\sum_{n \\ge 0} c(n, k)\\, x^n$ in $\\mathbb{Z}[[x]]$, where $c(n, k)$ is the number of colored Arndt compositions of $n$ with $k$ parts. Then $C_k(x)\\, (1 - x)^{2k} (1 + x)^{3\\lfloor k/2 \\rfloor} = x^{k + \\lfloor k/2 \\rfloor} (x^2 + x + 2)^{\\lfloor k/2 \\rfloor}$.","title":"Generating function of colored Arndt compositions with $k$ parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/colored-arndt-count-is-equivalent","kind":"node","lean":"MetaMathlibExt.coloredArndtCount_isEquivalent","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Asymptotics of colored Arndt compositions with "],["math","\\(k\\)"],["text"," parts"]],"source":"Asymptotics of colored Arndt compositions with $k$ parts"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#colored-arndt-count-is-equivalent","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For every fixed $k \\ge 1$, the number $c(n, k)$ of colored Arndt compositions of $n$ with $k$ parts satisfies $c(n, k) \\sim \\frac{n^{2k-1}}{2^{\\lfloor k/2 \\rfloor} (2k - 1)!}$ as $n \\to \\infty$.","title":"Asymptotics of colored Arndt compositions with $k$ parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/bivariate-count-gf","kind":"node","lean":"MetaMathlibExt.bivariateCountGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#bivariate-count-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For a function $a$ assigning a natural number $a(n, m)$ to every pair of natural numbers $n, m$, its bivariate generating function is the power series $\\sum_{n \\ge 0} \\bigl(\\sum_{m = 0}^{n} a(n, m)\\, y^m\\bigr) x^n$ in $\\mathbb{Z}[y][[x]]$, where $x$ marks the first index (the weight) and $y$ the second. Only the terms with $m \\le n$ enter; for counts with $a(n, m) = 0$ whenever $m > n$ it equals\u2026","title":"Bivariate generating function of a two-index count"},{"id":"Combinatorics/integer-compositions/arndt-compositions/coeff-bivariate-count-gf","kind":"node","lean":"MetaMathlibExt.coeff_bivariateCountGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#coeff-bivariate-count-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For every function $a$ assigning a natural number $a(n, m)$ to each pair of natural numbers and every $n \\ge 0$, the coefficient of $x^n$ in the bivariate generating function of $a$ is the polynomial $\\sum_{m = 0}^{n} a(n, m)\\, y^m \\in \\mathbb{Z}[y]$.","title":"Coefficients of a bivariate generating function"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-gf-eq-bivariate-count-gf","kind":"node","lean":"MetaMathlibExt.arndtGF_eq_bivariateCountGF","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-gf-eq-bivariate-count-gf","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For a set $S$ of natural numbers, let $a_S(n, m)$ be the number of Arndt compositions of $n$ with $m$ parts, all parts in $S$. Then the Arndt generating function $A_S(x, y)$ equals the bivariate generating function $\\sum_{n \\ge 0} \\sum_{m=0}^{n} a_S(n, m) x^n y^m$ in $\\mathbb{Z}[y][[x]]$.","title":"The Arndt generating function as a bivariate count series"},{"id":"Combinatorics/integer-compositions/arndt-compositions/anti-palindromic-composition-count","kind":"node","lean":"MetaMathlibExt.antiPalindromicCompositionCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#anti-palindromic-composition-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the number of anti-palindromic compositions of $n$ with $m$ parts: sequences $(c_1, \\dots, c_m)$ of positive integers with $c_1 + \\dots + c_m = n$ and $c_i \\ne c_{m+1-i}$ for every $i$ with $2i \\ne m + 1$. That is, each part differs from the part in the mirror position, except the middle part when $m$ is odd.","title":"Anti-palindromic compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/reduced-anti-palindromic-composition-cou","kind":"node","lean":"MetaMathlibExt.reducedAntiPalindromicCompositionCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#reduced-anti-palindromic-composition-cou","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For natural numbers $n$ and $m$, this is the number of compositions $(\\sigma_1, \\ldots, \\sigma_m)$ of $n$ into $m$ positive parts such that $\\sigma_i > \\sigma_{m+1-i}$ for every $i$ with $2i < m + 1$.","title":"Reduced anti-palindromic compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/anti-palindromic-composition-count-six-t","kind":"node","lean":"MetaMathlibExt.antiPalindromicCompositionCount_six_three","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#anti-palindromic-composition-count-six-t","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"There are exactly eight anti-palindromic compositions of $6$ with three parts.","title":"Anti-palindromic compositions of 6 with three parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/reduced-anti-palindromic-composition-cou-2","kind":"node","lean":"MetaMathlibExt.reducedAntiPalindromicCompositionCount_six_three","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#reduced-anti-palindromic-composition-cou-2","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"There are exactly four reduced anti-palindromic compositions of $6$ with three parts.","title":"Reduced anti-palindromic compositions of 6 with three parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/anti-palindromic-composition-count-six-f","kind":"node","lean":"MetaMathlibExt.antiPalindromicCompositionCount_six_four","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#anti-palindromic-composition-count-six-f","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"There are exactly four anti-palindromic compositions of $6$ with four parts, and exactly one reduced anti-palindromic composition of $6$ with four parts.","title":"Anti-palindromic compositions of 6 with four parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-last-part-count","kind":"node","lean":"MetaMathlibExt.arndtLastPartCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-last-part-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"An Arndt composition of $n$ is a composition $(\\sigma_1, \\ldots, \\sigma_\\ell)$ of $n$ into positive parts such that $\\sigma_{2i-1} > \\sigma_{2i}$ for every $i$ with $2i \\le \\ell$. For natural numbers $n$ and $j$, $b(n, j)$ is the number of Arndt compositions of $n$, with any number of parts, whose last part is $j$. The empty composition of $0$ counts as having last part $0$, so $b(0, 0) = 1$ and $b(0, j) = 0$ for\u2026","title":"Arndt compositions by last part"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-last-part-count-six","kind":"node","lean":"MetaMathlibExt.arndtLastPartCount_six","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-last-part-count-six","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"Let $b(n, j)$ be the number of Arndt compositions of $n$ with last part $j$. Then $b(6, 1) = 4$, $b(6, 2) = 2$, $b(6, 3) = 1$ and $b(6, 6) = 1$.","title":"Arndt compositions of 6 by last part"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-composition-count","kind":"node","lean":"MetaMathlibExt.kArndtCompositionCount","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-Arndt compositions"]],"source":"Number of $k$-Arndt compositions"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-composition-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For an integer $k$ and $n, m \\ge 0$, $a_k(n, m)$ is the number of $k$-Arndt compositions of $n$ with $m$ parts: sequences $(c_0, \\ldots, c_{m-1})$ of positive integers with $c_0 + \\cdots + c_{m-1} = n$ such that $c_{2i} > c_{2i+1} + k$ for every $i \\ge 0$ with $2i + 1 < m$. For $k = 0$ these are the Arndt compositions.","title":"Number of $k$-Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-composition-count-examples","kind":"node","lean":"MetaMathlibExt.kArndtCompositionCount_examples","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sample values of the "],["math","\\(k\\)"],["text","-Arndt composition counts"]],"source":"Sample values of the $k$-Arndt composition counts"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-composition-count-examples","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"Writing $a_k(n, m)$ for the number of $k$-Arndt compositions of $n$ with $m$ parts, $a_3(7, 2) = 1$, $a_3(7, 3) = 1$ and $a_{-3}(4, 2) = 3$.","title":"Sample values of the $k$-Arndt composition counts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/block-arndt-composition-count","kind":"node","lean":"MetaMathlibExt.blockArndtCompositionCount","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of "],["math","\\(k\\)"],["text","-block Arndt compositions"]],"source":"Number of $k$-block Arndt compositions"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#block-arndt-composition-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For $k \\ge 1$ and $n, m \\ge 0$, $a^{(k)}(n, m)$ is the number of $k$-block Arndt compositions of $n$ with $m$ parts: sequences $(c_0, \\ldots, c_{m-1})$ of positive integers with $c_0 + \\cdots + c_{m-1} = n$ that strictly decrease inside each block of $k$ consecutive positions $ki, ki + 1, \\ldots, ki + k - 1$ (the last block may be shorter), that is, $c_i > c_{i+1}$ for every $i$ with $i + 1 < m$ and\u2026","title":"Number of $k$-block Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/block-arndt-composition-count-three-ten","kind":"node","lean":"MetaMathlibExt.blockArndtCompositionCount_three_ten_four","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#block-arndt-composition-count-three-ten","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"There are exactly $7$ three-block Arndt compositions of $10$ with $4$ parts, that is, $a^{(3)}(10, 4) = 7$.","title":"Four-part 3-block Arndt compositions of 10"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count-eq-alternating-s","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount_eq_alternating_sum_and_recurrence","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count-eq-alternating-s","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $a(n, m)$ be the number of Arndt compositions of $n$ with $m$ parts, and write $K = n - m - \\lfloor m/2 \\rfloor$. For all $n, m \\ge 0$: if $m + \\lfloor m/2 \\rfloor \\le n$, then","title":"Alternating sum and recurrence for Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count-eq-positive-sum","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount_eq_positive_sum","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count-eq-positive-sum","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $a(n, m)$ be the number of Arndt compositions of $n$ with $m$ parts, and write $h = \\lfloor m/2 \\rfloor$. For all $n \\ge 0$ and $m \\ge 1$: if $m + h \\le n$, then","title":"Positive sum for Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count-holonomic-recurr","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount_holonomic_recurrence","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count-holonomic-recurr","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $a(n, m)$ be the number of Arndt compositions of $n$ with $m$ parts. For all $n, m \\ge 0$,","title":"Holonomic recurrence for Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-count-is-equivalent","kind":"node","lean":"MetaMathlibExt.arndtCompositionCount_isEquivalent","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-count-is-equivalent","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $a(n, m)$ be the number of Arndt compositions of $n$ with $m$ parts. For every fixed integer $m \\ge 1$,","title":"Asymptotics of Arndt compositions with a fixed number of parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-total-eq-fib","kind":"node","lean":"MetaMathlibExt.arndtComposition_total_eq_fib","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-total-eq-fib","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $a(n, m)$ be the number of Arndt compositions of $n$ with $m$ parts, let $a(n) = \\sum_{m=0}^{n} a(n, m)$ be the number of all Arndt compositions of $n$, and let $F_n$ be the $n$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$). Then, in $\\mathbb{Z}[[x]]$,","title":"Arndt compositions are counted by Fibonacci numbers"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-composition-expected-parts-is-equi","kind":"node","lean":"MetaMathlibExt.arndtComposition_expected_parts_isEquivalent","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-composition-expected-parts-is-equi","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $a(n, m)$ be the number of Arndt compositions of $n$ with $m$ parts, and $F_n$ the $n$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$). Then","title":"Expected number of parts of an Arndt composition"},{"id":"Combinatorics/integer-compositions/arndt-compositions/bivariate-count-gf-anti-palindromic","kind":"node","lean":"MetaMathlibExt.bivariateCountGF_antiPalindromic","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#bivariate-count-gf-anti-palindromic","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $ap(n, m)$ be the number of anti-palindromic compositions of $n$ with $m$ parts, and let $Ap(x, y) = \\sum_{n \\ge 0} \\sum_{m=0}^{n} ap(n, m)\\, x^n y^m \\in \\mathbb{Z}[y][[x]]$. Then $Ap(x, y)\\,(1 - x - x^2 + x^3 - 2x^3y^2) = 1 - x - x^2 + x^3 + xy - x^3y$, that is, $Ap(x, y) = (1 - x - x^2 + x^3 + xy - x^3y)/(1 - x - x^2 + x^3 - 2x^3y^2)$.","title":"Generating function of anti-palindromic compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/bivariate-count-gf-reduced-anti-palindro","kind":"node","lean":"MetaMathlibExt.bivariateCountGF_reducedAntiPalindromic","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#bivariate-count-gf-reduced-anti-palindro","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $b(n, m)$ be the number of reduced anti-palindromic compositions of $n$ with $m$ parts, and let $B_p(x, y) = \\sum_{n \\ge 0} \\sum_{m = 0}^{n} b(n, m)\\, x^n y^m$ in $\\mathbb{Z}[y][[x]]$, where $x$ marks the weight and $y$ the number of parts. Then","title":"Generating function of reduced anti-palindromic compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/reduced-anti-palindromic-composition-cou-3","kind":"node","lean":"MetaMathlibExt.reducedAntiPalindromicCompositionCount_eq_arndtCompositionCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#reduced-anti-palindromic-composition-cou-3","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For all natural numbers $n$ and $m$, the number of reduced anti-palindromic compositions of $n$ with $m$ parts equals the number of Arndt compositions of $n$ with $m$ parts, all positive integers being allowed as parts.","title":"Arndt compositions and reduced anti-palindromic compositions are equinumerous"},{"id":"Combinatorics/integer-compositions/arndt-compositions/bivariate-count-gf-arndt-last-part","kind":"node","lean":"MetaMathlibExt.bivariateCountGF_arndtLastPart","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#bivariate-count-gf-arndt-last-part","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $b(n, m)$ be the number of Arndt compositions of $n$ with last part $m$, where the empty composition of $0$ counts as having last part $0$ (so $b(0, 0) = 1$), and let $B(x, y) = \\sum_{n \\ge 0} \\sum_{m=0}^{n} b(n, m)\\, x^n y^m \\in \\mathbb{Z}[y][[x]]$. Then $B(x, y)\\,(1 - x - x^2)(1 - xy)(1 - x^2y) = 1 - x - x^2 - x^2y + 2x^3y + 2x^4y - x^5y - x^4y^2$.","title":"Generating function of Arndt compositions by last part"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-last-part-count-eq","kind":"node","lean":"MetaMathlibExt.arndtLastPartCount_eq","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-last-part-count-eq","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $F_r$ be the $r$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$) and $b(n, m)$ the number of Arndt compositions of $n$ with last part $m$. Define $t_1, t_2 : \\mathbb{N} \\to \\mathbb{N}$ by $t_1(0) = 1$, $t_1(1) = 0$, $t_1(r) = F_{r-2}$ and $t_2(0) = 0$, $t_2(1) = 1$, $t_2(r) = F_{r-1}$ for $r \\ge 2$. Then for all integers $n \\ge 1$ and $m \\ge 1$: $b(n, m) = 0$ if $n < m$, $b(n, m) = t_1(n - m)$ if $m \\le n < 2m$, and\u2026","title":"Closed formula for Arndt compositions by last part"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-last-part-count-is-equivalent","kind":"node","lean":"MetaMathlibExt.arndtLastPartCount_isEquivalent","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-last-part-count-is-equivalent","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $b(n, m)$ be the number of Arndt compositions of $n$ with last part $m$, and let $\\varphi$ be the golden ratio. For every fixed integer $m \\ge 1$, $b(n, m) \\sim \\frac{\\varphi^{n-m-2}}{\\sqrt{5}} \\left(1 + \\frac{1}{\\varphi^{m-1}}\\right)$ as $n \\to \\infty$.","title":"Asymptotics of Arndt compositions with a fixed last part"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-last-part-at-most-at-least","kind":"node","lean":"MetaMathlibExt.arndtLastPart_atMost_atLeast","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arndt compositions with last part at most or at least "],["math","\\(k\\)"]],"source":"Arndt compositions with last part at most or at least $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-last-part-at-most-at-least","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $F_n$ be the $n$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$) and $b(n, j)$ the number of Arndt compositions of $n$ with last part $j$. For all integers $k \\ge 1$ and $n \\ge 2k + 2$, the number of Arndt compositions of $n$ with last part at most $k$ is $\\sum_{j=1}^{k} b(n, j) = F_n - F_{n-k-1} - F_{n-2k-2}$, and the number with last part at least $k$ is $\\sum_{j=k}^{n} b(n, j) = F_{n-k} + F_{n-2k}$.","title":"Arndt compositions with last part at most or at least $k$"},{"id":"Combinatorics/integer-compositions/arndt-compositions/bivariate-count-gf-k-arndt-nonneg","kind":"node","lean":"MetaMathlibExt.bivariateCountGF_kArndt_nonneg","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of "],["math","\\(k\\)"],["text","-Arndt compositions for "],["math","\\(k \\ge 0\\)"]],"source":"Generating function of $k$-Arndt compositions for $k \\ge 0$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#bivariate-count-gf-k-arndt-nonneg","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $k \\ge 0$ be an integer, let $a_k(n, m)$ be the number of $k$-Arndt compositions of $n$ with $m$ parts, and let $A_k(x, y) = \\sum_{n \\ge 0} \\sum_{m=0}^{n} a_k(n, m)\\, x^n y^m \\in \\mathbb{Z}[y][[x]]$. Then $A_k(x, y)\\,(1 - x - x^2 + x^3 - x^{3+k}y^2) = (1 - x^2)(1 - x(1 - y))$, that is, $A_k(x, y) = (1 - x^2)(1 - x(1 - y))/(1 - x - x^2 + x^3 - x^{3+k}y^2)$.","title":"Generating function of $k$-Arndt compositions for $k \\ge 0$"},{"id":"Combinatorics/integer-compositions/arndt-compositions/bivariate-count-gf-k-arndt-neg","kind":"node","lean":"MetaMathlibExt.bivariateCountGF_kArndt_neg","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of "],["math","\\(k\\)"],["text","-Arndt compositions for negative "],["math","\\(k\\)"]],"source":"Generating function of $k$-Arndt compositions for negative $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#bivariate-count-gf-k-arndt-neg","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $k' \\ge 1$ and $k = -k'$, let $a_k(n, m)$ be the number of $k$-Arndt compositions of $n$ with $m$ parts, and let $A_k(x, y) = \\sum_{n \\ge 0} \\sum_{m=0}^{n} a_k(n, m)\\, x^n y^m \\in \\mathbb{Z}[y][[x]]$. Then $A_k(x, y)\\,(1 - x - x^2(1 + y^2) + x^3(1 - y^2) + y^2x^{2+k'}) = (1 - x^2)(1 - x(1 - y))$, that is, $A_k(x, y) = (1 - x^2)(1 - x(1 - y))/(1 - x - x^2(1 + y^2) + x^3(1 - y^2) + y^2x^{2-k})$.","title":"Generating function of $k$-Arndt compositions for negative $k$"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-composition-total-gf-nonneg","kind":"node","lean":"MetaMathlibExt.kArndtComposition_total_gf_nonneg","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(k\\)"],["text","-Arndt compositions for nonnegative "],["math","\\(k\\)"]],"source":"Counting $k$-Arndt compositions for nonnegative $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-composition-total-gf-nonneg","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $k \\ge 0$ be an integer and let $a_k(n) = \\sum_{m = 0}^{n} a_k(n, m)$ be the number of $k$-Arndt compositions of $n$. Then, in $\\mathbb{Z}[[x]]$,","title":"Counting $k$-Arndt compositions for nonnegative $k$"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-composition-total-gf-neg","kind":"node","lean":"MetaMathlibExt.kArndtComposition_total_gf_neg","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\(k\\)"],["text","-Arndt compositions for negative "],["math","\\(k\\)"]],"source":"Counting $k$-Arndt compositions for negative $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-composition-total-gf-neg","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $k = -k'$ with $k' \\ge 1$ an integer, and let $a_k(n) = \\sum_{m = 0}^{n} a_k(n, m)$ be the number of $k$-Arndt compositions of $n$. Then, in $\\mathbb{Z}[[x]]$,","title":"Counting $k$-Arndt compositions for negative $k$"},{"id":"Combinatorics/integer-compositions/arndt-compositions/bivariate-count-gf-block-arndt","kind":"node","lean":"MetaMathlibExt.bivariateCountGF_blockArndt","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#bivariate-count-gf-block-arndt","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $k \\ge 1$, let $a^{(k)}(n, m)$ be the number of $k$-block Arndt compositions of $n$ with $m$ parts, and let $A^{(k)}(x, y) = \\sum_{n \\ge 0} \\sum_{m=0}^{n} a^{(k)}(n, m)\\, x^n y^m \\in \\mathbb{Z}[y][[x]]$. Then $A^{(k)}(x, y) \\bigl(\\prod_{\\ell=1}^{k} (1 - x^\\ell) - x^{\\binom{k+1}{2}} y^k\\bigr) = \\sum_{j=0}^{k-1} x^{\\binom{j+1}{2}} y^j \\prod_{\\ell=j+1}^{k} (1 - x^\\ell)$. Equivalently, with\u2026","title":"Generating function of block Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-total-count","kind":"node","lean":"MetaMathlibExt.kArndtTotalCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-total-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For an integer $k$ and a natural number $n$, $a(n, k)$ is the number of compositions $(c_1, \\dots, c_m)$ of $n$, with any number $m \\ge 0$ of parts, such that $c_{2i-1} > c_{2i} + k$ for every $i$ with $2i \\le m$; a final unpaired part is unconstrained. The empty composition gives $a(0, k) = 1$, and $k = 0$ gives Arndt compositions.","title":"Generalized Arndt compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-total-count-five","kind":"node","lean":"MetaMathlibExt.kArndtTotalCount_five","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-total-count-five","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"The numbers $a(5, k)$ of generalized Arndt compositions of $5$ (compositions $(c_1, \\dots, c_m)$ of $5$ with $c_{2i-1} > c_{2i} + k$ whenever $2i \\le m$) are $a(5, 0) = 5$, $a(5, -1) = 10$, $a(5, -2) = 14$ and $a(5, 1) = 3$.","title":"Generalized Arndt counts for compositions of five"},{"id":"Combinatorics/integer-compositions/arndt-compositions/odd-or-small-even-composition-count","kind":"node","lean":"MetaMathlibExt.oddOrSmallEvenCompositionCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#odd-or-small-even-composition-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $c^{2k}_o(n)$ is the number of compositions of $n$, with any number of parts, in which every part is odd or at most $2k$. For $k = 0$ these are De Morgan's compositions of $n$ into odd parts.","title":"Compositions into odd or small even parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/odd-or-small-even-composition-count-five","kind":"node","lean":"MetaMathlibExt.oddOrSmallEvenCompositionCount_five","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#odd-or-small-even-composition-count-five","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"Let $c^{2k}_o(n)$ be the number of compositions of $n$ in which every part is odd or at most $2k$. Then $c^0_o(5) = 5$ and $c^2_o(5) = 14$.","title":"Compositions of five into odd or small even parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/even-or-small-odd-composition-count","kind":"node","lean":"MetaMathlibExt.evenOrSmallOddCompositionCount","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#even-or-small-odd-composition-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $c^{2k+1}_e(n)$ is the number of compositions of $n$ (sequences of positive integers of any length, including the empty one, that sum to $n$) in which every part is even or at most $2k + 1$. In particular $c^{2k+1}_e(0) = 1$.","title":"Compositions into even or small odd parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/even-or-small-odd-composition-count-five","kind":"node","lean":"MetaMathlibExt.evenOrSmallOddCompositionCount_five","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositions of "],["math","\\(5\\)"],["text"," into even or small odd parts"]],"source":"Compositions of $5$ into even or small odd parts"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#even-or-small-odd-composition-count-five","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"$c^1_e(5) = 10$ and $c^3_e(5) = 15$, where $c^{2k+1}_e(n)$ is the number of compositions of $n$ in which every part is even or at most $2k + 1$.","title":"Compositions of $5$ into even or small odd parts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-u","kind":"node","lean":"MetaMathlibExt.arndtU","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(U\\)"],["text"," on lists"]],"source":"The map $U$ on lists"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-u","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"The map $U$ sends a list $(c_1, c_2, \\dots)$ of natural numbers to the concatenation of the images of its consecutive pairs $(c_1, c_2)$, $(c_3, c_4)$, and so on, where $1^j$ denotes $j$ parts equal to $1$: a pair $(a, b)$ with $b < a$ goes to $(1^{a - b - 1}, 2b + 1)$; a pair $(a, b)$ with $b \\ge a$ goes to $(1^{a + b - 2k}, 2k)$, where $k = \\lfloor (b - a)/2 \\rfloor + 1$ is the integer $k \\ge 1$ with\u2026","title":"The map $U$ on lists"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-u-examples","kind":"node","lean":"MetaMathlibExt.arndtU_examples","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of the map "],["math","\\(U\\)"]],"source":"Values of the map $U$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-u-examples","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"The map $U$ sends $(2, 3)$ to $(1, 1, 1, 2)$, $(1, 4)$ to $(1, 4)$, $(3, 2)$ to $(5)$, $(1, 1, 1, 2)$ to $(2, 1, 2)$ and $(1, 3, 1)$ to $(4, 1)$.","title":"Values of the map $U$"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-v","kind":"node","lean":"MetaMathlibExt.arndtV","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(V\\)"],["text"," on lists"]],"source":"The map $V$ on lists"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-v","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"The map $V$ sends a list $(c_1, c_2, \\dots)$ of natural numbers to the concatenation of the images of its consecutive pairs $(c_1, c_2)$, $(c_3, c_4)$, and so on, where $1^j$ denotes $j$ parts equal to $1$: a pair $(a, b)$ with $b \\le a$ goes to $(1^{a - b}, 2b)$; a pair $(a, b)$ with $b > a$ goes to $(1^{a + b - 2k - 1}, 2k + 1)$, where $k = \\lfloor (b - a + 1)/2 \\rfloor$ is the integer $k \\ge 1$ with\u2026","title":"The map $V$ on lists"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-v-examples","kind":"node","lean":"MetaMathlibExt.arndtV_examples","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Values of the map "],["math","\\(V\\)"]],"source":"Values of the map $V$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-v-examples","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"fully_proved","summary":"The map $V$ sends $(2, 3)$ to $(1, 1, 3)$, $(1, 4)$ to $(5)$, $(4, 1)$ to $(1, 1, 1, 2)$, $(1, 3, 1)$ to $(1, 3, 1)$ and $(2, 2, 1)$ to $(4, 1)$.","title":"Values of the map $V$"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-total-count-zero-eq-odd-composit","kind":"node","lean":"MetaMathlibExt.kArndtTotalCount_zero_eq_odd_compositions","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-total-count-zero-eq-odd-composit","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For every natural number $n$, the number $a(n, 0)$ of Arndt compositions of $n$ (compositions $(c_1, \\dots, c_m)$ of $n$ with $c_{2i-1} > c_{2i}$ whenever $2i \\le m$) equals the number $c^0_o(n)$ of compositions of $n$ into odd parts. This includes $n = 0$, where both counts are $1$.","title":"Arndt compositions and compositions into odd parts are equinumerous"},{"id":"Combinatorics/integer-compositions/arndt-compositions/exists-perm-composition-blocks-eq-arndt","kind":"node","lean":"MetaMathlibExt.exists_perm_composition_blocks_eq_arndtU","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#exists-perm-composition-blocks-eq-arndt","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For every natural number $n$ there is a permutation $\\sigma$ of the compositions of $n$ such that, for every composition $c$ of $n$, the list of parts of $\\sigma(c)$ is $U(c)$, the result of applying the pairwise map $U$ to the list of parts of $c$. In particular, $U$ sends compositions of $n$ to compositions of $n$ and permutes them.","title":"The pairwise map U permutes compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/exists-perm-composition-blocks-eq-arndt-2","kind":"node","lean":"MetaMathlibExt.exists_perm_composition_blocks_eq_arndtV","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(V\\)"],["text"," permutes compositions"]],"source":"The map $V$ permutes compositions"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#exists-perm-composition-blocks-eq-arndt-2","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For every natural number $n$, the map $V$ sends compositions of $n$ to compositions of $n$, and it is a permutation of the set of compositions of $n$.","title":"The map $V$ permutes compositions"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-u-fixed-iff-and-count","kind":"node","lean":"MetaMathlibExt.arndtU_fixed_iff_and_count","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositions fixed by "],["math","\\(U\\)"],["text"," and their Padovan count"]],"source":"Compositions fixed by $U$ and their Padovan count"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-u-fixed-iff-and-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"A composition $(c_1, \\dots, c_m)$ of $n$ is fixed by the map $U$ if and only if $c_i = 1$ for every odd $i$ and $c_i$ is even for every even $i$. Let $c_U(n)$ be the number of compositions of $n$ fixed by $U$. Then $c_U(n) = c_U(n - 2) + c_U(n - 3)$ for every $n \\ge 4$, $c_U(1) = 1$, $c_U(2) = 0$, $c_U(3) = 1$, and $c_U(n) = P_{n-1}$ for every $n \\ge 1$, where $P$ is the shifted Padovan sequence, given by\u2026","title":"Compositions fixed by $U$ and their Padovan count"},{"id":"Combinatorics/integer-compositions/arndt-compositions/arndt-v-fixed-iff-and-count","kind":"node","lean":"MetaMathlibExt.arndtV_fixed_iff_and_count","parent":26731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositions fixed by "],["math","\\(V\\)"],["text"," and their Fibonacci count"]],"source":"Compositions fixed by $V$ and their Fibonacci count"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#arndt-v-fixed-iff-and-count","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"A composition $(c_1, \\dots, c_m)$ of $n$ is fixed by the map $V$ if and only if $c_i = 1$ for every odd $i$ and $c_i$ is odd and at least $3$ for every even $i$. Let $c_V(n)$ be the number of compositions of $n$ fixed by $V$. Then $c_V(n) = c_V(n - 2) + c_V(n - 4)$ for every $n \\ge 5$, $c_V(1) = c_V(4) = 1$, $c_V(2) = c_V(3) = 0$, and $c_V(2j) = c_V(2j + 1) = F_{j-1}$ for every $j \\ge 1$, where $F$ is the Fibonacci\u2026","title":"Compositions fixed by $V$ and their Fibonacci count"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-total-count-recurrence","kind":"node","lean":"MetaMathlibExt.kArndtTotalCount_recurrence","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-total-count-recurrence","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"Let $a(n, k)$ be the number of generalized Arndt compositions of $n$ with shift $k$, set $a(m, k) = 0$ for $m < 0$, and note that $a(0, k) = 1$ counts the empty composition. For every natural number $j$ and every $n \\ge 3$,","title":"Recurrences for generalized Arndt counts"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-total-count-neg-even","kind":"node","lean":"MetaMathlibExt.kArndtTotalCount_neg_even","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-total-count-neg-even","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For every natural number $j \\ge 1$ and every natural number $n$, the number $a(n, -2j)$ of compositions $(c_1, \\dots, c_m)$ of $n$ with $c_{2i-1} > c_{2i} - 2j$ whenever $2i \\le m$ equals the number $c^{2j}_o(n)$ of compositions of $n$ in which every part is odd or at most $2j$. This includes $n = 0$.","title":"Generalized Arndt compositions with negative even shift"},{"id":"Combinatorics/integer-compositions/arndt-compositions/k-arndt-total-count-neg-odd","kind":"node","lean":"MetaMathlibExt.kArndtTotalCount_neg_odd","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#k-arndt-total-count-neg-odd","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For all natural numbers $j$ and $n$, the number $a(n, -2j - 1)$ of compositions $(c_1, \\dots, c_m)$ of $n$ with $c_{2i-1} > c_{2i} - 2j - 1$ whenever $2i \\le m$ equals the number $c^{2j+1}_e(n)$ of compositions of $n$ in which every part is even or at most $2j + 1$. This includes $n = 0$.","title":"Generalized Arndt compositions with negative odd shift"},{"id":"Combinatorics/integer-compositions/arndt-compositions/even-or-small-odd-odd-or-small-even-recu","kind":"node","lean":"MetaMathlibExt.evenOrSmallOdd_oddOrSmallEven_recurrence","parent":26731,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/arndt-compositions/index.html#even-or-small-odd-odd-or-small-even-recu","scope":"Combinatorics/integer-compositions/arndt-compositions","status":"can_prove","summary":"For natural numbers $k$ and $n$, let $c^{2k+1}_e(n)$ be the number of compositions of $n$ in which every part is even or at most $2k + 1$, and $c^{2k}_o(n)$ the number in which every part is odd or at most $2k$. If $k \\ge 1$ and $n \\ge 3$, then","title":"Recurrences for even-or-small-odd and odd-or-small-even compositions"},{"data_url":"scopes/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss.json","id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"can_prove","title":"Colored compositions, Dyck paths and polygon dissections"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-composition-count","kind":"node","lean":"MetaMathlibExt.coloredCompositionCount","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-composition-count","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"Let $\\gamma = (\\gamma_1, \\gamma_2, \\dots)$ be a sequence of natural numbers and let $n, k$ be natural numbers. The number $c_{n,k}(\\gamma)$ of $\\gamma$-colored compositions of $n$ with $k$ parts counts the compositions of $n$ with exactly $k$ parts in which each part equal to $j$ carries one of $\\gamma_j$ colors: it is the sum, over all compositions $(a_1, \\dots, a_k)$ of $n$ into $k$ positive parts, of\u2026","title":"Number of colored compositions"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-composition-count-one-three-two","kind":"node","lean":"MetaMathlibExt.coloredCompositionCount_one_three_two","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-composition-count-one-three-two","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"With one color for every part ($\\gamma_j = 1$ for all $j$), the number of colored compositions of $3$ with $2$ parts is $c_{3,2}(\\gamma) = 2$: the compositions $(1,2)$ and $(2,1)$.","title":"Compositions of 3 into two parts"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-dyck-path-count","kind":"node","lean":"MetaMathlibExt.coloredDyckPathCount","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-dyck-path-count","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"Let $\\gamma = (\\gamma_1, \\gamma_2, \\dots)$ be a sequence of natural numbers and let $n, k$ be natural numbers. In a Dyck path, each peak $UD$ together with the maximal run of $D$ steps starting at its $D$ forms a primitive block $UD^j$. The number $d_{n,k}(\\gamma)$ counts the Dyck paths of semilength $n$ with exactly $k$ peaks in which each primitive block $UD^j$ carries one of $\\gamma_j$ colors: it is the sum,\u2026","title":"Number of colored Dyck paths"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-dyck-path-count-one-three-two","kind":"node","lean":"MetaMathlibExt.coloredDyckPathCount_one_three_two","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-dyck-path-count-one-three-two","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"With one color for every block ($\\gamma_j = 1$ for all $j$), the number of colored Dyck paths of semilength $3$ with $2$ peaks is $d_{3,2}(\\gamma) = 3$, a Narayana number.","title":"Dyck paths of semilength 3 with two peaks"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/polygon-diagonals","kind":"node","lean":"MetaMathlibExt.polygonDiagonals","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#polygon-diagonals","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, label the vertices of the convex $(n+2)$-gon $0, 1, \\dots, n+1$ in cyclic order. Its diagonals are the pairs $(a, b)$ of natural numbers with $a + 2 \\le b \\le n + 1$ and $(a, b) \\ne (0, n+1)$; the excluded pair is the side joining the vertices $0$ and $n+1$.","title":"Diagonals of a convex polygon"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/is-noncrossing-diagonal-set","kind":"node","lean":"MetaMathlibExt.IsNoncrossingDiagonalSet","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#is-noncrossing-diagonal-set","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"A finite set $D$ of pairs of natural numbers is noncrossing when there are no $(a, b) \\in D$ and $(c, d) \\in D$ with $a < c < b < d$. For diagonals $(a, b)$ with $a < b$ of a convex polygon whose vertices are labelled by natural numbers in cyclic order, this says that no two diagonals of $D$ cross in the interior of the polygon.","title":"Noncrossing sets of diagonals"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/dissection-face-size","kind":"node","lean":"MetaMathlibExt.dissectionFaceSize","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#dissection-face-size","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"Let $D$ be a finite set of pairs of natural numbers and let $e = (a, b)$ be a pair of natural numbers. The face size of $e$ in $D$ is $2$ plus the number of natural numbers $v$ with $a < v < b$ for which there is no pair $(c, d) \\in D$ other than $e$ with $a \\le c < v < d \\le b$. When $D$ is a noncrossing set of diagonals of the convex $(n+2)$-gon with vertices $0, \\dots, n+1$ in cyclic order and $e$ is a diagonal\u2026","title":"Size of the polygon under a chord"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-polygon-dissection-count","kind":"node","lean":"MetaMathlibExt.coloredPolygonDissectionCount","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-polygon-dissection-count","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"Let $\\gamma : \\mathbb{N} \\to \\mathbb{N}$ and $n, k \\in \\mathbb{N}$. For $n \\ge 1$, label the vertices of the convex $(n+2)$-gon $0, \\dots, n+1$ in cyclic order, with root side $(0, n+1)$. Then $p_{n,k}(\\gamma)$ is the sum, over all sets $D$ of $k - 1$ pairwise noncrossing diagonals of this polygon, of the product, over the root side and the diagonals $e$ of $D$, of $\\gamma(j_e)$, where $j_e + 2$ is the number of\u2026","title":"Colored polygon dissection count"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-polygon-dissection-count-one-thr","kind":"node","lean":"MetaMathlibExt.coloredPolygonDissectionCount_one_three_two","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-polygon-dissection-count-one-thr","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"With one color for every polygon, that is $\\gamma(j) = 1$ for all $j$, the colored polygon dissection count satisfies $p_{3,2}(\\gamma) = 5$: a convex pentagon has five diagonals, and each one cuts it into two polygons.","title":"Dissections of a pentagon into two polygons"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-polygon-dissection-count-hexagon","kind":"node","lean":"MetaMathlibExt.coloredPolygonDissectionCount_hexagon_two","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-polygon-dissection-count-hexagon","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"fully_proved","summary":"Let $\\gamma_1 = 2$, $\\gamma_2 = 3$, $\\gamma_3 = 5$ and $\\gamma_j = 0$ for every other $j$. The number of partitions of a rooted convex hexagon by one diagonal into $2$ polygons in which each $(j+2)$-gon carries one of $\\gamma_j$ colors is $p_{4,2}(\\gamma) = 87$: three diagonals cut the hexagon into two quadrilaterals and six cut it into a triangle and a pentagon, giving $3 \\cdot 3^2 + 6 \\cdot 2 \\cdot 5 = 87$.","title":"Colored dissections of a hexagon by one diagonal"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-composition-count-eq-partial-bel","kind":"node","lean":"MetaMathlibExt.coloredCompositionCount_eq_partialBell","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-composition-count-eq-partial-bel","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"can_prove","summary":"Let $\\gamma = (\\gamma_1, \\gamma_2, \\dots)$ be a sequence of natural numbers and let $c_{n,k}(\\gamma)$ be the number of compositions of $n$ with $k$ parts in which each part $j$ carries one of $\\gamma_j$ colors. For every natural number $n \\ge 1$ and every natural number $k$, as rational numbers,","title":"Hoggatt\u2013Lind formula for colored compositions"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-dyck-path-count-eq-partial-bell","kind":"node","lean":"MetaMathlibExt.coloredDyckPathCount_eq_partialBell","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-dyck-path-count-eq-partial-bell","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"can_prove","summary":"Let $\\gamma = (\\gamma_1, \\gamma_2, \\dots)$ be a sequence of natural numbers and let $d_{n,k}(\\gamma)$ be the number of Dyck paths of semilength $n$ with $k$ peaks in which each block $UD^j$, formed by a peak and the maximal run of $D$ steps starting at its $D$, carries one of $\\gamma_j$ colors. For every natural number $n \\ge 1$ and every natural number $k$, as rational numbers,","title":"Mansour\u2013Sun formula for colored Dyck paths"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/colored-polygon-dissection-count-eq-part","kind":"node","lean":"MetaMathlibExt.coloredPolygonDissectionCount_eq_partialBell","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#colored-polygon-dissection-count-eq-part","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"can_prove","summary":"Let $\\gamma = (\\gamma_1, \\gamma_2, \\dots)$ be a sequence of natural numbers and let $p_{n,k}(\\gamma)$ be the number of partitions of a rooted convex $(n+2)$-gon by $k - 1$ noncrossing diagonals into $k$ polygons in which each $(j+2)$-gon carries one of $\\gamma_j$ colors. For all natural numbers $n \\ge 1$ and $k \\ge 1$, as rational numbers,","title":"Birmajer\u2013Gil\u2013Weiner formula for colored polygon partitions"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/choose-mul-colored-composition-count-eq","kind":"node","lean":"MetaMathlibExt.choose_mul_coloredCompositionCount_eq","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#choose-mul-colored-composition-count-eq","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"can_prove","summary":"Let $\\gamma = (\\gamma_1, \\gamma_2, \\dots)$ be a sequence of natural numbers. Let $c_{n,k}(\\gamma)$ be the number of compositions of $n$ with $k$ parts in which each part $j$ carries one of $\\gamma_j$ colors, and let $d_{n,k}(\\gamma)$ be the number of Dyck paths of semilength $n$ with $k$ peaks in which each block $UD^j$, formed by a peak and the maximal run of $D$ steps starting at its $D$, carries one of\u2026","title":"Colored compositions and colored Dyck paths"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/choose-mul-colored-dyck-path-count-eq","kind":"node","lean":"MetaMathlibExt.choose_mul_coloredDyckPathCount_eq","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#choose-mul-colored-dyck-path-count-eq","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"can_prove","summary":"Let $\\gamma = (\\gamma_1, \\gamma_2, \\dots)$ be a sequence of natural numbers. Let $d_{n,k}(\\gamma)$ be the number of Dyck paths of semilength $n$ with $k$ peaks in which each block $UD^j$, formed by a peak and the maximal run of $D$ steps starting at its $D$, carries one of $\\gamma_j$ colors, and let $p_{n,k}(\\gamma)$ be the number of partitions of a rooted convex $(n+2)$-gon by $k - 1$ noncrossing diagonals into\u2026","title":"Colored Dyck paths and colored polygon partitions"},{"id":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/succ-mul-colored-polygon-dissection-coun","kind":"node","lean":"MetaMathlibExt.succ_mul_coloredPolygonDissectionCount_eq","parent":26817,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss/index.html#succ-mul-colored-polygon-dissection-coun","scope":"Combinatorics/integer-compositions/colored-compositions-dyck-paths-and-polygon-diss","status":"can_prove","summary":"For every coloring sequence $\\gamma : \\mathbb{N} \\to \\mathbb{N}$ and all natural numbers $n$ and $k$ with $1 \\le k \\le n$, the colored polygon dissection count $p_{n,k}(\\gamma)$ and the colored composition count $c_{n,k}(\\gamma)$ satisfy","title":"Polygon partitions and colored compositions"},{"data_url":"scopes/Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4.json","id":"Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4/index.html","scope":"Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4","status":"fully_proved","title":"Compositions avoiding parts 2, 3 and 4"},{"id":"Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4/octagonal-pyramidal-number","kind":"node","lean":"MetaMathlibExt.octagonalPyramidalNumber","parent":26834,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4/index.html#octagonal-pyramidal-number","scope":"Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4","status":"fully_proved","summary":"For a natural number $k$, the $k$-th octagonal pyramidal number is $k(k+1)(2k-1)/2$.","title":"Octagonal pyramidal numbers"},{"id":"Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4/octagonal-pyramidal-compositions","kind":"node","lean":"MetaMathlibExt.octagonalPyramidalCompositions","parent":26834,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4/index.html#octagonal-pyramidal-compositions","scope":"Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4","status":"fully_proved","summary":"For a natural number $k$, the restricted compositions of $k + 14$ are the ordered compositions of $k + 14$ into $k$ positive parts with no part equal to $2$, $3$ or $4$, that is, the tuples $(c_1, \\dots, c_k)$ of positive integers with $c_1 + \\dots + c_k = k + 14$ and $c_i \\notin \\{2, 3, 4\\}$ for every $i$.","title":"Compositions avoiding parts 2, 3 and 4"},{"id":"Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4/octagonal-pyramidal-number-eq-compositio","kind":"node","lean":"MetaMathlibExt.octagonalPyramidalNumber_eq_composition_card","parent":26834,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4/index.html#octagonal-pyramidal-number-eq-compositio","scope":"Combinatorics/integer-compositions/compositions-avoiding-parts-2-3-and-4","status":"fully_proved","summary":"For every positive integer $k$, the number of ordered compositions of $k + 14$ into $k$ positive parts with no part equal to $2$, $3$ or $4$ is the $k$-th octagonal pyramidal number $k(k+1)(2k-1)/2$.","title":"Octagonal pyramidal numbers count restricted compositions"},{"data_url":"scopes/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts.json","id":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/index.html","scope":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","status":"fully_proved","title":"Compositions with an odd number of parts"},{"id":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/composition-half-index-set","kind":"node","lean":"MetaMathlibExt.compositionHalfIndexSet","parent":26838,"reading_title":{"format":"inline-parts-v1","parts":[["text","The index set "],["math","\\(S(m, n, q)\\)"]],"source":"The index set $S(m, n, q)$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/index.html#composition-half-index-set","scope":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","status":"fully_proved","summary":"For natural numbers $m$, $n$ and $q$, the set $S(m, n, q)$ consists of the natural numbers $k$ such that $2k \\equiv n - q \\pmod{m}$, where the difference $n - q$ is taken in the integers, and $2k < n$.","title":"The index set $S(m, n, q)$"},{"id":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/mem-composition-half-index-set","kind":"node","lean":"MetaMathlibExt.mem_compositionHalfIndexSet","parent":26838,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in "],["math","\\(S(m, n, q)\\)"]],"source":"Membership in $S(m, n, q)$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/index.html#mem-composition-half-index-set","scope":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","status":"fully_proved","summary":"For natural numbers $m$, $n$, $q$ and $k$, the number $k$ lies in the index set $S(m, n, q)$ if and only if $2k \\equiv n - q \\pmod{m}$, with the difference taken in the integers, and $2k < n$.","title":"Membership in $S(m, n, q)$"},{"id":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/composition-half-index-set-zero","kind":"node","lean":"MetaMathlibExt.compositionHalfIndexSet_zero","parent":26838,"reading_title":{"format":"inline-parts-v1","parts":[["text","The index set "],["math","\\(S(m, 0, q)\\)"],["text"," is empty"]],"source":"The index set $S(m, 0, q)$ is empty"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/index.html#composition-half-index-set-zero","scope":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","status":"fully_proved","summary":"For all natural numbers $m$ and $q$, the index set $S(m, 0, q)$ is empty.","title":"The index set $S(m, 0, q)$ is empty"},{"id":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/composition-index-set","kind":"node","lean":"MetaMathlibExt.compositionIndexSet","parent":26838,"reading_title":{"format":"inline-parts-v1","parts":[["text","The index set "],["math","\\(T(m, n, q)\\)"]],"source":"The index set $T(m, n, q)$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/index.html#composition-index-set","scope":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","status":"fully_proved","summary":"For natural numbers $m$, $n$ and $q$, the set $T(m, n, q)$ consists of the natural numbers $k$ such that $k \\equiv n - q \\pmod{m}$, where the difference $n - q$ is taken in the integers, and $k < n$.","title":"The index set $T(m, n, q)$"},{"id":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/mem-composition-index-set","kind":"node","lean":"MetaMathlibExt.mem_compositionIndexSet","parent":26838,"reading_title":{"format":"inline-parts-v1","parts":[["text","Membership in "],["math","\\(T(m, n, q)\\)"]],"source":"Membership in $T(m, n, q)$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/index.html#mem-composition-index-set","scope":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","status":"fully_proved","summary":"For natural numbers $m$, $n$, $q$ and $k$, the number $k$ lies in the index set $T(m, n, q)$ if and only if $k \\equiv n - q \\pmod{m}$, with the difference taken in the integers, and $k < n$.","title":"Membership in $T(m, n, q)$"},{"id":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/composition-index-set-zero","kind":"node","lean":"MetaMathlibExt.compositionIndexSet_zero","parent":26838,"reading_title":{"format":"inline-parts-v1","parts":[["text","The index set "],["math","\\(T(m, 0, q)\\)"],["text"," is empty"]],"source":"The index set $T(m, 0, q)$ is empty"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/index.html#composition-index-set-zero","scope":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","status":"fully_proved","summary":"For all natural numbers $m$ and $q$, the index set $T(m, 0, q)$ is empty.","title":"The index set $T(m, 0, q)$ is empty"},{"id":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/composition-half-index-set-double","kind":"node","lean":"MetaMathlibExt.compositionHalfIndexSet_double","parent":26838,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts/index.html#composition-half-index-set-double","scope":"Combinatorics/integer-compositions/compositions-with-an-odd-number-of-parts","status":"fully_proved","summary":"For natural numbers $m$, $n$ and $q$, let $S(m, n, q)$ be the set of natural numbers $k$ with $2k \\equiv n - q \\pmod{m}$ and $2k < n$, and let $T(m, n, q)$ be the set of natural numbers $k$ with $k \\equiv n - q \\pmod{m}$ and $k < n$, where the difference $n - q$ is taken in the integers. Then for all natural numbers $m$, $n$ and $q$ the sets satisfy $S(2m, 2n, 2q) = T(m, n, q)$.","title":"Doubling identity for the composition index sets"},{"data_url":"scopes/Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3.json","id":"Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3/index.html","scope":"Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3","status":"fully_proved","title":"Compositions with parts 1, 2 and 3"},{"id":"Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3/compositions-parts-one-two-three-eq-trib","kind":"node","lean":"MetaMathlibExt.compositions_parts_one_two_three_eq_tribonacci","parent":26846,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositions with parts "],["math","\\(1\\)"],["text",", "],["math","\\(2\\)"],["text",", "],["math","\\(3\\)"],["text"," are counted by Tribonacci numbers"]],"source":"Compositions with parts $1$, $2$, $3$ are counted by Tribonacci numbers"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3/index.html#compositions-parts-one-two-three-eq-trib","scope":"Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3","status":"fully_proved","summary":"Let $T$ be a sequence of natural numbers with $T_1 = T_2 = 1$, $T_3 = 2$ and $T_m = T_{m-1} + T_{m-2} + T_{m-3}$ for every $m > 3$. Then for every positive integer $n$, the number of compositions of $n$ (ordered sequences of positive integers with sum $n$) whose parts are all $1$, $2$ or $3$ is $T_{n+1}$.","title":"Compositions with parts $1$, $2$, $3$ are counted by Tribonacci numbers"},{"id":"Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3/compositions-parts-one-two-three-eq-cano","kind":"node","lean":"MetaMathlibExt.compositions_parts_one_two_three_eq_canonical_tribonacci","parent":26846,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositions with parts "],["math","\\(1\\)"],["text",", "],["math","\\(2\\)"],["text",", "],["math","\\(3\\)"],["text"," and the Tribonacci sequence"]],"source":"Compositions with parts $1$, $2$, $3$ and the Tribonacci sequence"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3/index.html#compositions-parts-one-two-three-eq-cano","scope":"Combinatorics/integer-compositions/compositions-with-parts-1-2-and-3","status":"fully_proved","summary":"For every positive integer $n$, the number of compositions of $n$ (ordered sequences of positive integers with sum $n$) whose parts are all $1$, $2$ or $3$ equals $t_{n+2}$, where $t$ is the Tribonacci sequence given by $t_0 = t_1 = 0$, $t_2 = 1$ and $t_{k+3} = t_{k+2} + t_{k+1} + t_k$ (OEIS A000073).","title":"Compositions with parts $1$, $2$, $3$ and the Tribonacci sequence"},{"data_url":"scopes/Combinatorics/integer-compositions/even-and-odd-compositions.json","id":"Combinatorics/integer-compositions/even-and-odd-compositions","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","title":"Even and odd compositions"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"fully_proved","summary":"For a natural number $N$ and a property $P$ of compositions of $N$, the signed count of the compositions of $N$ satisfying $P$ is the number of them with an odd number of parts minus the number with an even number of parts. Equivalently, it is the sum of $(-1)^{\\ell(c) + 1}$ over all compositions $c$ of $N$ satisfying $P$, where $\\ell(c)$ is the number of parts of $c$.","title":"Signed count of compositions"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-zero-true","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_zero_true","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-zero-true","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"fully_proved","summary":"The only composition of $0$ is the empty one, which has no parts and so is even. Hence the signed count of all compositions of $0$, the number of odd compositions minus the number of even ones, is $-1$.","title":"Signed count of the compositions of zero"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/is-huang-small-part-placement","kind":"node","lean":"MetaMathlibExt.IsHuangSmallPartPlacement","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#is-huang-small-part-placement","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"fully_proved","summary":"Let $k$ be a natural number. A finite sequence $(a_0, \\dots, a_{\\ell - 1})$ of natural numbers satisfies the placement condition for $k$ when every entry $a_i < k$ has a preceding entry and a following entry, the preceding entry satisfies $a_{i-1} \\ge k$, and the following entry $a_{i+1}$ is either the last entry (that is, $i + 2 = \\ell$) or satisfies $a_{i+1} > k$.","title":"Huang's placement condition for small parts"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/is-huang-small-part-placement-of-forall","kind":"node","lean":"MetaMathlibExt.isHuangSmallPartPlacement_of_forall_le","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#is-huang-small-part-placement-of-forall","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"fully_proved","summary":"If every entry of a finite sequence of natural numbers is at least $k$, then the sequence satisfies Huang's placement condition for $k$, vacuously.","title":"Sequences without small parts satisfy Huang's placement condition"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/sum-distincts-neg-one-pow-card","kind":"node","lean":"MetaMathlibExt.sum_distincts_neg_one_pow_card","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#sum-distincts-neg-one-pow-card","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"For a natural number $n$, let $E(n)$ and $O(n)$ be the numbers of partitions of $n$ into distinct parts with an even and an odd number of parts, respectively. If $2n = j(3j + 1)$ or $2n = j(3j - 1)$ for some integer $j \\ge 0$, then $E(n) - O(n) = (-1)^j$; otherwise $E(n) - O(n) = 0$.","title":"Legendre's identity for partitions into distinct parts"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-parts-gt-one","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_parts_gt_one","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-parts-gt-one","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"For every $n \\ge 0$, among the compositions of $n + 1$ whose parts are all greater than $1$, the number with an odd number of parts minus the number with an even number of parts is $(-1)^j$ if $n = 3j + 1$ or $n = 3j + 2$ for some $j \\ge 0$, and $0$ otherwise.","title":"Signed count of compositions with parts greater than one"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-parts-ge","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_parts_ge","parent":26849,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed count of compositions with parts at least "],["math","\\(k\\)"]],"source":"Signed count of compositions with parts at least $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-parts-ge","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $k, n \\ge 1$. Among the compositions of $n + k - 1$ whose parts are all at least $k$, the number with an odd number of parts minus the number with an even number of parts equals $\\sum_{j=0}^{\\lfloor (n-1)/k \\rfloor} (-1)^j \\binom{n-1-j(k-1)}{j}$.","title":"Signed count of compositions with parts at least $k$"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/sum-distincts-mod-four-mul-neg-one-pow-c","kind":"node","lean":"MetaMathlibExt.sum_distincts_mod_four_mul_neg_one_pow_card","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#sum-distincts-mod-four-mul-neg-one-pow-c","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $r \\ge 1$ and $n$ be natural numbers, and let $E(n)$ and $O(n)$ be the numbers of partitions of $n$ into distinct parts, each congruent to $0$, $2r + 1$ or $2r - 1$ modulo $4r$, with an even and an odd number of parts, respectively. If $n = j(2rj + 1)$ or $n = j(2rj - 1)$ for some integer $j \\ge 0$, then $E(n) - O(n) = (-1)^j$; otherwise $E(n) - O(n) = 0$.","title":"Nyirenda's identity for distinct parts modulo a multiple of four"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/sum-distincts-mod-two-mul-add-one-neg-on","kind":"node","lean":"MetaMathlibExt.sum_distincts_mod_two_mul_add_one_neg_one_pow_card","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#sum-distincts-mod-two-mul-add-one-neg-on","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $r \\ge 1$ and $n$ be natural numbers, and let $E(n)$ and $O(n)$ be the numbers of partitions of $n$ into distinct parts, each congruent to $0$, $r$ or $r + 1$ modulo $2r + 1$, with an even and an odd number of parts, respectively. If $2n = j((2r + 1)j + 1)$ or $2n = j((2r + 1)j - 1)$ for some integer $j \\ge 0$, then $E(n) - O(n) = (-1)^j$; otherwise $E(n) - O(n) = 0$.","title":"Nyirenda's identity for distinct parts modulo an odd number"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-parts-ge-mod","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_parts_ge_modEq","parent":26849,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed count of compositions with parts at least "],["math","\\(k\\)"],["text"," in a residue class"]],"source":"Signed count of compositions with parts at least $k$ in a residue class"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-parts-ge-mod","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $k, n \\ge 1$ and $r > s \\ge 0$. Among the compositions of $n + k - 1$ whose parts are all at least $k$ and congruent to $k + s$ modulo $r$, the number with an odd number of parts minus the number with an even number of parts is the sum of $(-1)^j \\binom{i+j}{i}$ over all $i, j \\ge 0$ with $ri + j(k+s) + s + 1 = n$ (an empty sum when $n \\le s$).","title":"Signed count of compositions with parts at least $k$ in a residue class"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-parts-ge-mod-2","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_parts_ge_modEq_sub","parent":26849,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed count of residue-class compositions with parts at least "],["math","\\(r - s\\)"]],"source":"Signed count of residue-class compositions with parts at least $r - s$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-parts-ge-mod-2","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $r > s \\ge 0$, let $k = r - s$ and let $n \\ge 1$. Among the compositions of $n + k - 1$ whose parts are all at least $k$ and congruent to $k + s$ modulo $r$, the number with an odd number of parts minus the number with an even number of parts is $1$ if $n = s + 1$ and $0$ otherwise.","title":"Signed count of residue-class compositions with parts at least $r - s$"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-parts-ge-mod-3","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_parts_ge_modEq_two_mul_sub","parent":26849,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed count of residue-class compositions with parts at least "],["math","\\(2r - s\\)"]],"source":"Signed count of residue-class compositions with parts at least $2r - s$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-parts-ge-mod-3","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $r > s \\ge 0$, let $k = 2r - s$ and let $n \\ge 1$. Among the compositions of $n + k - 1$ whose parts are all at least $k$ and congruent to $k + s$ modulo $r$, let $b^{r,s}_{k,n}$ be the number with an odd number of parts minus the number with an even number of parts. Then $b^{r,s}_{k,n} = (-1)^j$ if $n = 3rj + s + 1$ or $n = 3rj + r + s + 1$ for some $j \\ge 0$, and $b^{r,s}_{k,n} = 0$ otherwise.","title":"Signed count of residue-class compositions with parts at least $2r - s$"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/card-partition-franklin","kind":"node","lean":"MetaMathlibExt.card_partition_franklin","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#card-partition-franklin","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $k \\ge 1$ and let $m$ and $n$ be natural numbers. The number of partitions of $n$ in which exactly $m$ distinct part sizes occur at least $k$ times equals the number of partitions of $n$ in which exactly $m$ distinct part sizes are divisible by $k$.","title":"Franklin's partition identity"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/card-composition-huang-franklin","kind":"node","lean":"MetaMathlibExt.card_composition_huang_franklin","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#card-composition-huang-franklin","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $k \\ge 1$, $m \\ge 0$ and $n \\ge 1$. The number of compositions of $n$ with exactly $m$ parts not congruent to $1$ modulo $k$, all of which are greater than $k$, equals the number of compositions of $n + k - 1$ with exactly $m$ parts less than $k$ in which every part less than $k$ is preceded by a part at least $k$ and is followed by a part that is either the last part or greater than $k$.","title":"Huang's Franklin-type identity for compositions"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-huang-eq-sum","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_huang_eq_sum","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-huang-eq-sum","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $k, n \\ge 1$ and $m \\ge 0$. Among the compositions of $n + k - 1$ with exactly $m$ parts less than $k$ in which every part less than $k$ is preceded by a part at least $k$ and followed by a part that is either the last part or greater than $k$, let $b^{(m)}_{k,n}$ be the number with an odd number of parts minus the number with an even number of parts. Then $b^{(m)}_{k,n}$ is the sum of\u2026","title":"Signed count of Huang's compositions, binomial form"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-huang-eq-sum-2","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_huang_eq_sum_monomial","parent":26849,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-huang-eq-sum-2","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $k \\ge 2$, $m \\ge 0$ and $n \\ge 1$. Among the compositions of $n + k - 1$ with exactly $m$ parts less than $k$ in which every part less than $k$ is preceded by a part at least $k$ and followed by a part that is either the last part or greater than $k$, let $b^{(m)}_{k,n}$ be the number with an odd number of parts minus the number with an even number of parts. Then $b^{(m)}_{k,n}$ is the sum of\u2026","title":"Signed count of Huang's compositions, monomial form"},{"id":"Combinatorics/integer-compositions/even-and-odd-compositions/composition-odd-minus-even-parts-lt-eq-s","kind":"node","lean":"MetaMathlibExt.compositionOddMinusEven_parts_lt_eq_sum","parent":26849,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed count of compositions with exactly "],["math","\\(m\\)"],["text"," parts less than "],["math","\\(k\\)"]],"source":"Signed count of compositions with exactly $m$ parts less than $k$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/even-and-odd-compositions/index.html#composition-odd-minus-even-parts-lt-eq-s","scope":"Combinatorics/integer-compositions/even-and-odd-compositions","status":"can_prove","summary":"Let $k, n \\ge 1$ and $m \\ge 0$. Among the compositions of $n + k - 1$ with exactly $m$ parts less than $k$, the number with an odd number of parts minus the number with an even number of parts is the sum of $(-1)^{i+\\ell+1} \\binom{i+j-1}{j} \\binom{i}{m} \\binom{m}{\\ell}$ over all $i, j, \\ell \\ge 0$ with $i + j + (k-1)(\\ell + i - m - 1) = n$, where $\\binom{-1}{0} = 1$.","title":"Signed count of compositions with exactly $m$ parts less than $k$"},{"data_url":"scopes/Combinatorics/integer-compositions/perfect-compositions.json","id":"Combinatorics/integer-compositions/perfect-compositions","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/perfect-compositions/index.html","scope":"Combinatorics/integer-compositions/perfect-compositions","status":"fully_proved","title":"Perfect compositions"},{"id":"Combinatorics/integer-compositions/perfect-compositions/perfect-composition-iff-symbolic-parts-p","kind":"node","lean":"MetaMathlibExt.perfect_composition_iff_symbolic_parts_perm_perfect_partition_general","parent":26867,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/perfect-compositions/index.html#perfect-composition-iff-symbolic-parts-p","scope":"Combinatorics/integer-compositions/perfect-compositions","status":"fully_proved","summary":"Call a list $L$ of positive integers perfect for $n$ if for every integer $m$ with $0 < m < n$ there is exactly one list of integers that is a sublist of $L$ (its entries taken in order, not necessarily contiguous) and has sum $m$. A symbolic part $(a, u)$ with $a, u > 0$ stands for a run of $u$ copies of $a$, and the expansion of a list of symbolic parts is the concatenation of their runs. Let $n$ be a natural\u2026","title":"Perfect compositions are permuted perfect partitions"},{"id":"Combinatorics/integer-compositions/perfect-compositions/perfect-composition-iff-symbolic-parts-p-2","kind":"node","lean":"MetaMathlibExt.perfect_composition_iff_symbolic_parts_perm_perfect_partition","parent":26867,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/perfect-compositions/index.html#perfect-composition-iff-symbolic-parts-p-2","scope":"Combinatorics/integer-compositions/perfect-compositions","status":"fully_proved","summary":"Call a list $L$ of positive integers perfect for $n$ if for every integer $m$ with $0 < m < n$ there is exactly one list of integers that is a sublist of $L$ (its entries taken in order, not necessarily contiguous) and has sum $m$. A symbolic part $(a, u)$ with $a, u > 0$ stands for a run of $u$ copies of $a$, and the expansion of a list of symbolic parts is the concatenation of their runs. Let $n > 0$ and let $C$\u2026","title":"Munagi's theorem on perfect compositions"},{"data_url":"scopes/Combinatorics/integer-compositions/positional-colored-compositions.json","id":"Combinatorics/integer-compositions/positional-colored-compositions","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","title":"Positional colored compositions"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/positional-colored-composition-count","kind":"node","lean":"MetaMathlibExt.positionalColoredCompositionCount","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#positional-colored-composition-count","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"fully_proved","summary":"For natural numbers $m$, $k$ and $\\ell$, $c_{m,k}(\\ell)$ is the number of $(m, k)$-colored compositions of $\\ell$: compositions of $\\ell$ in which every part whose position, counted from $1$, is congruent to $k$ modulo $m$ gets one of $p$ colors, where $p$ is the part, and the other parts are uncolored. Equivalently, $c_{m,k}(\\ell)$ is the sum, over all compositions $(p_1, \\dots, p_r)$ of $\\ell$, of the product of\u2026","title":"Positional colored compositions"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/positional-colored-composition-count-zer","kind":"node","lean":"MetaMathlibExt.positionalColoredCompositionCount_zero","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#positional-colored-composition-count-zer","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"fully_proved","summary":"For all natural numbers $m$ and $k$, $c_{m,k}(0) = 1$: the only composition of $0$ is the empty one.","title":"Colored compositions of zero"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/color-two-restricted-compositions-eq-eve","kind":"node","lean":"MetaMathlibExt.colorTwoRestrictedCompositions_eq_even","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#color-two-restricted-compositions-eq-eve","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","summary":"For every $k \\ge 0$, the number of $n$-color compositions of $k$ that do not use the color $2$ equals the number $e(k) = c_{2,0}(k)$ of even colored compositions of $k$, in which only the parts at even positions are colored. In such an $n$-color composition a part $p \\ge 2$ has the $p - 1$ colors other than $2$ and a part $1$ has its single color, so their number is the sum, over all compositions\u2026","title":"Avoiding color 2 counts even colored compositions"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/odd-eq-choose-two-color-compositions","kind":"node","lean":"MetaMathlibExt.odd_eq_choose_two_colorCompositions","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#odd-eq-choose-two-color-compositions","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","summary":"For every $k \\ge 1$, the number $o(k) = c_{2,1}(k)$ of odd colored compositions of $k$, in which only the parts at odd positions are colored, equals the sum, over all compositions $(p_1, \\dots, p_r)$ of $k + 1$, of $\\prod_i \\binom{p_i}{2}$; that is, the number of compositions of $k + 1$ in which a part $p$ has $\\binom{p}{2}$ colors.","title":"Odd colored compositions and binomial-color compositions"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/odd-eq-ternary-strings-avoiding","kind":"node","lean":"MetaMathlibExt.odd_eq_ternaryStrings_avoiding","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#odd-eq-ternary-strings-avoiding","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","summary":"For every $k \\ge 1$, the number $o(k) = c_{2,1}(k)$ of odd colored compositions of $k$, in which only the parts at odd positions are colored, equals the number of strings of length $k - 1$ over the alphabet $\\{0, 1, 2\\}$ that contain neither $01$ nor $12$ as two consecutive letters.","title":"Odd colored compositions and ternary strings"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/even-eq-ternary-strings-avoiding","kind":"node","lean":"MetaMathlibExt.even_eq_ternaryStrings_avoiding","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#even-eq-ternary-strings-avoiding","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","summary":"For every $k \\ge 0$, the number $e(k) = c_{2,0}(k)$ of even colored compositions of $k$, in which only the parts at even positions are colored, equals the number of strings of length $k$ over the alphabet $\\{0, 1, 2\\}$ that contain neither $01$ nor $12$ as two consecutive letters, do not begin with $2$ and do not end with $0$.","title":"Even colored compositions and ternary strings"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/even-succ-eq-sum-run-length-prod","kind":"node","lean":"MetaMathlibExt.even_succ_eq_sum_runLength_prod","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#even-succ-eq-sum-run-length-prod","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","summary":"For every $k \\ge 0$, the number $e(k + 1) = c_{2,0}(k + 1)$ of even colored compositions of $k + 1$, in which only the parts at even positions are colored, equals the sum, over all binary strings of length $k$, of the product of the lengths of the maximal runs of $1$s in the string; a string without a $1$ contributes the empty product $1$.","title":"Even colored compositions and runs in binary strings"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/odd-eq-sum-run-length-prod","kind":"node","lean":"MetaMathlibExt.odd_eq_sum_runLength_prod","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#odd-eq-sum-run-length-prod","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","summary":"For every $k \\ge 1$, the number $o(k) = c_{2,1}(k)$ of odd colored compositions of $k$, in which only the parts at odd positions are colored, equals the sum, over the binary strings of length $k$ whose first letter is $1$, of the product of the lengths of the maximal runs of $1$s in the string.","title":"Odd colored compositions and runs in binary strings"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/positional-colored-composition-count-suc","kind":"node","lean":"MetaMathlibExt.positionalColoredCompositionCount_succ","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#positional-colored-composition-count-suc","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","summary":"For natural numbers $m$, $k$ and $\\ell$ with $m \\ge 1$, $m \\nmid k$ and $\\ell \\ge 1$, $c_{m,k+1}(\\ell + 1) = c_{m,k+1}(\\ell) + c_{m,k}(\\ell)$.","title":"A recurrence for positional colored compositions"},{"id":"Combinatorics/integer-compositions/positional-colored-compositions/even-eq-card-separable-321-avoiding","kind":"node","lean":"MetaMathlibExt.even_eq_card_separable_321_avoiding","parent":26870,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/positional-colored-compositions/index.html#even-eq-card-separable-321-avoiding","scope":"Combinatorics/integer-compositions/positional-colored-compositions","status":"can_prove","summary":"For every $k \\ge 0$, the number $e(k) = c_{2,0}(k)$ of even colored compositions of $k$, in which only the parts at even positions are colored, equals the number of permutations of $\\{1, \\dots, k\\}$ that avoid the patterns $321$, $2413$ and $3142$, that is, the number of $321$-avoiding separable permutations of $\\{1, \\dots, k\\}$.","title":"Even colored compositions and 321-avoiding separable permutations"},{"data_url":"scopes/Combinatorics/integer-compositions/restricted-n-color-compositions.json","id":"Combinatorics/integer-compositions/restricted-n-color-compositions","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","title":"Restricted n-color compositions"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/color-comp-allowed","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ColorCompAllowed","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#color-comp-allowed","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For natural numbers $b$ and $n$, $\\mathrm{CCo}_{b,b+1}(n)$ is the set of $n$-color compositions of $n$ that use only the colors $b$ and $b+1$: finite sequences of pairs $(k_1, c_1), \\dots, (k_m, c_m)$ of natural numbers (part $k_i$ with color $c_i$) such that $1 \\le c_i \\le k_i$ and $c_i \\in \\{b, b+1\\}$ for every $i$, and $k_1 + \\dots + k_m = n$.","title":"n-color compositions with two allowed consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/color-comp-prohibited","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ColorCompProhibited","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#color-comp-prohibited","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For natural numbers $b$ and $n$, $\\mathrm{CCo}_{\\widehat{b,b+1}}(n)$ is the set of $n$-color compositions of $n$ in which no part has color $b$ or $b+1$: finite sequences of pairs $(k_1, c_1), \\dots, (k_m, c_m)$ of natural numbers (part $k_i$ with color $c_i$) such that $1 \\le c_i \\le k_i$, $c_i \\ne b$ and $c_i \\ne b+1$ for every $i$, and $k_1 + \\dots + k_m = n$.","title":"n-color compositions with two prohibited consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/color-comp-prohibit2","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ColorCompProhibit2","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#color-comp-prohibit2","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{CCo}_{\\hat 2}(n)$ is the set of $n$-color compositions of $n$ in which no part has color $2$: finite sequences of pairs $(k_1, c_1), \\dots, (k_m, c_m)$ of natural numbers (part $k_i$ with color $c_i$) such that $1 \\le c_i \\le k_i$ and $c_i \\ne 2$ for every $i$, and $k_1 + \\dots + k_m = n$.","title":"n-color compositions without color 2"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-allowed","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoAllowed","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-allowed","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"An $n$-color composition of a natural number $n$ is a finite sequence of parts, each a positive integer $k$ carrying a color $c$ with $1 \\le c \\le k$, whose sizes sum to $n$. For natural numbers $b$ and $n$, $\\operatorname{cco}_{b,b+1}(n)$ is the number of $n$-color compositions of $n$ in which every part has color $b$ or $b+1$.","title":"Count of n-color compositions with two consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-prohibited","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoProhibited","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-prohibited","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For natural numbers $b$ and $n$, $\\mathrm{cco}_{\\widehat{b,b+1}}(n)$ is the number of elements of $\\mathrm{CCo}_{\\widehat{b,b+1}}(n)$, that is, the number of $n$-color compositions of $n$ in which no part has color $b$ or $b+1$.","title":"Number of n-color compositions with two prohibited consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-prohibit2","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoProhibit2","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-prohibit2","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"An $n$-color composition of a natural number $n$ is a finite sequence of parts, each a positive integer $k$ carrying a color $c$ with $1 \\le c \\le k$, whose sizes sum to $n$. For a natural number $n$, $\\operatorname{cco}_{\\hat 2}(n)$ is the number of $n$-color compositions of $n$ in which no part has color $2$.","title":"Count of n-color compositions avoiding color 2"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-allowed-z","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoAllowedZ","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-allowed-z","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For a natural number $b$, this is the function from the integers to the integers that sends $n \\ge 0$ to $\\operatorname{cco}_{b,b+1}(n)$, the number of $n$-color compositions of $n$ in which every part has color $b$ or $b+1$, and sends every $n < 0$ to $0$.","title":"Integer extension of the count of n-color compositions with two consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-prohibited-z","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoProhibitedZ","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-prohibited-z","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For a natural number $b$, this is the function from the integers to the integers that sends $n \\ge 0$ to $\\operatorname{cco}_{\\widehat{b,b+1}}(n)$, the number of $n$-color compositions of $n$ in which no part has color $b$ or $b+1$, and sends every $n < 0$ to $0$.","title":"Integer extension of the count of n-color compositions avoiding two consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/split-zeros-aux","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.splitZerosAux","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#split-zeros-aux","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For strings $w$ and $a$ over $\\{0, 1, 2\\}$, write $w = s_0 \\, 0 \\, s_1 \\, 0 \\cdots 0 \\, s_m$ where $m$ is the number of $0$s in $w$ and each $s_i$ is a (possibly empty) string without $0$. The result is the list of strings $a s_0, s_1, \\dots, s_m$, in this order, where $a s_0$ is the concatenation of $a$ and $s_0$.","title":"Splitting a ternary string at its zeros, with a prefix"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/split-zeros","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.splitZeros","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#split-zeros","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For a string $w$ over $\\{0, 1, 2\\}$, the list of the segments of $w$ obtained by cutting at every $0$: the part before the first $0$, the parts between consecutive $0$s and the part after the last $0$, in order and including empty parts. A string with $m$ zeros gives $m + 1$ segments; the empty string gives one empty segment.","title":"Splitting a ternary string at its zeros"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/ternary-admissible","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.TernaryAdmissible","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#ternary-admissible","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For natural numbers $b$ and $n$, the strings $w$ over $\\{0, 1, 2\\}$ of length $n - 1$ (truncated subtraction, so length $0$ when $n = 0$) such that every segment of $w$ obtained by cutting at its zeros (before the first zero, between consecutive zeros and after the last zero, empty segments included) either consists only of $1$s and has length at least $b - 1$, or consists only of $2$s and has length at least $b$.\u2026","title":"Admissible ternary strings"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/two-type-comp","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.TwoTypeComp","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#two-type-comp","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"For a natural number $n$, $C^*_{1,2}(n)$ is the set of compositions of $n$ whose parts are $1_1$, $1_2$ (two types of part of size $1$) and $2$, such that no two adjacent parts are parts $1$ of the same type.","title":"Compositions with two types of part 1 and no equal adjacent 1s"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib-nat","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.tribNat","parent":26881,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tribonacci numbers with initial values "],["math","\\(1, 0, 1\\)"]],"source":"Tribonacci numbers with initial values $1, 0, 1$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib-nat","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"The sequence $t : \\mathbb{N} \\to \\mathbb{Z}$ with $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for $n \\ge 3$.","title":"Tribonacci numbers with initial values $1, 0, 1$"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.trib","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"fully_proved","summary":"Let $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for natural numbers $n \\ge 3$. This extends $t$ to an integer-valued function on all integers by setting $t(n) = 0$ for $n < 0$.","title":"Hopkins\u2013Wang tribonacci numbers on the integers"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-allowed-z-recurrence","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoAllowedZ_recurrence","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-allowed-z-recurrence","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Fix a natural number $b \\ge 1$. For an integer $m$, let $c(m)$ be the number of $n$-color compositions of $m$ in which every part has color $b$ or $b+1$ when $m \\ge 0$, and $c(m) = 0$ when $m < 0$. Then for every integer $n \\ge 2$,","title":"Recurrence for n-color compositions with two consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-allowed-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoAllowed_eq_sum","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-allowed-eq-sum","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Let $b \\ge 1$ and $n \\ge 1$ be natural numbers, and let $\\operatorname{cco}_{b,b+1}(n)$ be the number of $n$-color compositions of $n$ in which every part has color $b$ or $b+1$. Then","title":"Double-sum formula for n-color compositions with two consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/nonempty-equiv-color-comp-allowed-ternar","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.nonempty_equiv_colorCompAllowed_ternary","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#nonempty-equiv-color-comp-allowed-ternar","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"For all natural numbers $b \\ge 1$ and $n \\ge 1$, there is a bijection between $\\mathrm{CCo}_{b,b+1}(n)$, the $n$-color compositions of $n$ using only the colors $b$ and $b+1$, and the set of strings of length $n-1$ over the alphabet $\\{0, 1, 2\\}$ such that, when the string is cut at each of its $0$s, every resulting segment (including the possibly empty segments before the first $0$, after the last $0$, and between\u2026","title":"Bijection between n-color compositions with colors b, b+1 and ternary strings"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-allowed-one-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoAllowed_one_eq_sum","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-allowed-one-eq-sum","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"For every natural number $n \\ge 1$, the number $\\operatorname{cco}_{1,2}(n)$ of $n$-color compositions of $n$ in which every part has color $1$ or $2$ satisfies","title":"Binomial sum for n-color compositions with colors 1 and 2"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/nonempty-equiv-color-comp-allowed-two-co","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.nonempty_equiv_colorCompAllowed_two_composition_mod_three","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#nonempty-equiv-color-comp-allowed-two-co","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"For every natural number $n \\ge 1$, there is a bijection between $\\mathrm{CCo}_{2,3}(n+1)$, the $n$-color compositions of $n+1$ using only the colors $2$ and $3$, and the compositions of $n$ all of whose parts are congruent to $1$ or $2$ modulo $3$.","title":"n-color compositions with colors 2, 3 and compositions with parts not divisible by 3"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/nonempty-equiv-color-comp-allowed-two-tw","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.nonempty_equiv_colorCompAllowed_two_twoTypeComp","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#nonempty-equiv-color-comp-allowed-two-tw","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"For every natural number $n$, there is a bijection between $\\mathrm{CCo}_{2,3}(n+2)$, the $n$-color compositions of $n+2$ using only the colors $2$ and $3$, and the compositions of $n$ whose parts are $2$ or one of two types of part $1$, with no two adjacent parts $1$ of the same type.","title":"n-color compositions with colors 2, 3 and compositions with two kinds of 1"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib-eq-add-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.trib_eq_add_sum","parent":26881,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tribonacci numbers from "],["math","\\(t(n-2)\\)"],["text"," and a partial sum"]],"source":"Tribonacci numbers from $t(n-2)$ and a partial sum"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib-eq-add-sum","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Let $t$ be the tribonacci numbers with $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for $n \\ge 3$, extended by $t(n) = 0$ for $n < 0$. For every integer $n \\ge 3$,","title":"Tribonacci numbers from $t(n-2)$ and a partial sum"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib-eq-add-one-add-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.trib_eq_add_one_add_sum","parent":26881,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tribonacci numbers from "],["math","\\(t(n-1)\\)"],["text"," and alternate earlier terms"]],"source":"Tribonacci numbers from $t(n-1)$ and alternate earlier terms"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib-eq-add-one-add-sum","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Let $t$ be the tribonacci numbers with $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for $n \\ge 3$, extended by $t(n) = 0$ for $n < 0$. For every integer $n \\ge 3$,","title":"Tribonacci numbers from $t(n-1)$ and alternate earlier terms"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib-three-mul","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.trib_three_mul","parent":26881,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tribonacci partial-sum identity for "],["math","\\(t(3k)\\)"]],"source":"Tribonacci partial-sum identity for $t(3k)$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib-three-mul","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Let $t$ be the tribonacci numbers with $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for $n \\ge 3$, extended by $t(n) = 0$ for $n < 0$. For every natural number $k$,","title":"Tribonacci partial-sum identity for $t(3k)$"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib-three-mul-add-one","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.trib_three_mul_add_one","parent":26881,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tribonacci partial-sum identity for "],["math","\\(t(3k+1)\\)"]],"source":"Tribonacci partial-sum identity for $t(3k+1)$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib-three-mul-add-one","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Let $t$ be the tribonacci numbers with $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for $n \\ge 3$, extended by $t(n) = 0$ for $n < 0$. For every natural number $k$,","title":"Tribonacci partial-sum identity for $t(3k+1)$"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib-three-mul-add-two","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.trib_three_mul_add_two","parent":26881,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tribonacci partial-sum identity for "],["math","\\(t(3k+2)\\)"]],"source":"Tribonacci partial-sum identity for $t(3k+2)$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib-three-mul-add-two","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Let $t$ be the tribonacci numbers with $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for $n \\ge 3$, extended by $t(n) = 0$ for $n < 0$. For every natural number $k$,","title":"Tribonacci partial-sum identity for $t(3k+2)$"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib-add-two-sub-three-mul","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.trib_add_two_sub_three_mul","parent":26881,"reading_title":{"format":"inline-parts-v1","parts":[["text","Identity for "],["math","\\(t(n) + t(n-2)\\)"],["text"," when "],["math","\\(n \\ge 3\\)"],["text"," and "],["math","\\(n \\equiv 0, 1 \\pmod 3\\)"]],"source":"Identity for $t(n) + t(n-2)$ when $n \\ge 3$ and $n \\equiv 0, 1 \\pmod 3$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib-add-two-sub-three-mul","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Let $t$ be the tribonacci numbers with $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for $n \\ge 3$, extended by $t(n) = 0$ for $n < 0$. For every natural number $k \\ge 1$, both $n = 3k$ and $n = 3k + 1$ satisfy","title":"Identity for $t(n) + t(n-2)$ when $n \\ge 3$ and $n \\equiv 0, 1 \\pmod 3$"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/trib-add-two-sub-three-mul-add-two","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.trib_add_two_sub_three_mul_add_two","parent":26881,"reading_title":{"format":"inline-parts-v1","parts":[["text","Identity for "],["math","\\(t(n) + t(n-2)\\)"],["text"," when "],["math","\\(n \\ge 3\\)"],["text"," and "],["math","\\(n \\equiv 2 \\pmod 3\\)"]],"source":"Identity for $t(n) + t(n-2)$ when $n \\ge 3$ and $n \\equiv 2 \\pmod 3$"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#trib-add-two-sub-three-mul-add-two","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Let $t$ be the tribonacci numbers with $t(0) = 1$, $t(1) = 0$, $t(2) = 1$ and $t(n) = t(n-1) + t(n-2) + t(n-3)$ for $n \\ge 3$, extended by $t(n) = 0$ for $n < 0$. For every natural number $k \\ge 1$, with $n = 3k + 2$,","title":"Identity for $t(n) + t(n-2)$ when $n \\ge 3$ and $n \\equiv 2 \\pmod 3$"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-allowed-three-add","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoAllowed_three_add","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-allowed-three-add","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"For a natural number $m$, let $\\operatorname{cco}_{3,4}(m)$ be the number of $n$-color compositions of $m$ in which every part has color $3$ or $4$, and let $F_k$ be the Fibonacci numbers with $F_0 = 0$ and $F_1 = 1$. Then for every natural number $n \\ge 3$, $\\operatorname{cco}_{3,4}(n) + \\operatorname{cco}_{3,4}(n-2) = F_{n-1}$.","title":"Fibonacci identity for n-color compositions with colors 3 and 4"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-prohibited-z-recurrence","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoProhibitedZ_recurrence","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-prohibited-z-recurrence","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"Fix a natural number $b \\ge 1$. For an integer $m$, let $c(m)$ be the number of $n$-color compositions of $m$ in which no part has color $b$ or $b+1$ when $m \\ge 0$, and $c(m) = 0$ when $m < 0$. Then for every integer $n \\ge 3$,","title":"Recurrence for n-color compositions avoiding two consecutive colors"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/nonempty-equiv-color-comp-prohibited-one","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.nonempty_equiv_colorCompProhibited_one_composition_mod_four","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#nonempty-equiv-color-comp-prohibited-one","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"For every natural number $n \\ge 1$, there is a bijection between $\\mathrm{CCo}_{\\widehat{1,2}}(n+2)$, the $n$-color compositions of $n+2$ in which no part has color $1$ or $2$, and the compositions of $n$ all of whose parts are congruent to $1$ or $2$ modulo $4$.","title":"n-color compositions without colors 1, 2 and compositions with parts 1, 2 mod 4"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/nonempty-equiv-color-comp-prohibited-one-2","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.nonempty_equiv_colorCompProhibited_one_palindromic","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#nonempty-equiv-color-comp-prohibited-one-2","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"For every natural number $n \\ge 1$, there is a bijection between $\\mathrm{CCo}_{\\widehat{1,2}}(n)$, the $n$-color compositions of $n$ in which no part has color $1$ or $2$, and the palindromic compositions of $\\max(2n-5, 0)$ that have no part equal to $2$ and an odd number of parts.","title":"n-color compositions without colors 1, 2 and palindromic compositions"},{"id":"Combinatorics/integer-compositions/restricted-n-color-compositions/cco-prohibited-one-three-mul","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.NColorCompositionsWanted.ccoProhibited_one_three_mul","parent":26881,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-n-color-compositions/index.html#cco-prohibited-one-three-mul","scope":"Combinatorics/integer-compositions/restricted-n-color-compositions","status":"can_prove","summary":"For a natural number $m$, let $\\operatorname{cco}_{\\widehat{1,2}}(m)$ be the number of $n$-color compositions of $m$ in which no part has color $1$ or $2$, and let $\\operatorname{cco}_{\\hat 2}(m)$ be the number in which no part has color $2$. Then for every natural number $k \\ge 1$, $\\operatorname{cco}_{\\widehat{1,2}}(3k) = \\operatorname{cco}_{\\hat 2}(2k-1)$.","title":"n-color compositions avoiding colors 1 and 2 versus avoiding color 2"},{"data_url":"scopes/Combinatorics/integer-compositions/restricted-weighted-compositions.json","id":"Combinatorics/integer-compositions/restricted-weighted-compositions","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/restricted-weighted-compositions/index.html","scope":"Combinatorics/integer-compositions/restricted-weighted-compositions","status":"fully_proved","title":"Restricted weighted compositions"},{"id":"Combinatorics/integer-compositions/restricted-weighted-compositions/s-restricted-weighted-composition-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.sRestrictedWeightedCompositionCount","parent":26914,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-weighted-compositions/index.html#s-restricted-weighted-composition-count","scope":"Combinatorics/integer-compositions/restricted-weighted-compositions","status":"fully_proved","summary":"Let $R$ be a commutative ring, $S$ a finite set of natural numbers and $f \\colon \\mathbb{N} \\to R$ a weight function. For natural numbers $n$ and $k$, $d_{S,f}(n, k)$ is the sum, over all $k$-tuples $(\\pi_1, \\dots, \\pi_k)$ of natural numbers with every $\\pi_i \\in S$ and $\\pi_1 + \\dots + \\pi_k = n$, of the weight $f(\\pi_1) \\cdots f(\\pi_k)$.","title":"Restricted weighted composition count"},{"id":"Combinatorics/integer-compositions/restricted-weighted-compositions/vandermonde-type-convolution-for-srestri","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.vandermondeTypeConvolutionForSRestrictedWeightedCompositions","parent":26914,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/restricted-weighted-compositions/index.html#vandermonde-type-convolution-for-srestri","scope":"Combinatorics/integer-compositions/restricted-weighted-compositions","status":"fully_proved","summary":"Let $R$ be a commutative ring, $S$ a finite set of natural numbers and $f \\colon \\mathbb{N} \\to R$ a weight function, and write $d_{S,f}(n, k)$ for the weighted count of $S$-restricted $f$-weighted compositions of $n$ with $k$ parts. Then for all natural numbers $n$, $k$ and $j$ with $j \\le k$,","title":"Vandermonde convolution for restricted weighted compositions"},{"data_url":"scopes/Combinatorics/integer-compositions/shade-of-integer-compositions.json","id":"Combinatorics/integer-compositions/shade-of-integer-compositions","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/shade-of-integer-compositions/index.html","scope":"Combinatorics/integer-compositions/shade-of-integer-compositions","status":"can_prove","title":"Shade of integer compositions"},{"id":"Combinatorics/integer-compositions/shade-of-integer-compositions/composition-shade","kind":"node","lean":"MetaMathlibExt.compositionShade","parent":26917,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/shade-of-integer-compositions/index.html#composition-shade","scope":"Combinatorics/integer-compositions/shade-of-integer-compositions","status":"fully_proved","summary":"Let $x = (x_0, x_1, \\ldots, x_{k-1})$ be a finite list of natural numbers (the parts of a composition), drawn as a bargraph whose column $c$ has height $x_c$, with $x_c = 0$ for $c \\ge k$. A cell $(c, h)$ with column $c \\ge 0$ and height $h \\ge 1$ is shaded when $h > x_c$ and $x_j \\ge h + (c - j)$ for some column $j < c$. The shade of $x$ is the number of shaded cells.","title":"Shade of a composition"},{"id":"Combinatorics/integer-compositions/shade-of-integer-compositions/total-composition-shade","kind":"node","lean":"MetaMathlibExt.totalCompositionShade","parent":26917,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/shade-of-integer-compositions/index.html#total-composition-shade","scope":"Combinatorics/integer-compositions/shade-of-integer-compositions","status":"fully_proved","summary":"For a natural number $n$, the total shade $c(n)$ is the sum of the shades of all compositions of $n$.","title":"Total shade of compositions"},{"id":"Combinatorics/integer-compositions/shade-of-integer-compositions/composition-shade-four","kind":"node","lean":"MetaMathlibExt.compositionShade_four","parent":26917,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/shade-of-integer-compositions/index.html#composition-shade-four","scope":"Combinatorics/integer-compositions/shade-of-integer-compositions","status":"fully_proved","summary":"The one-part composition $4$ has shade $6$: its bargraph has $6$ shaded cells.","title":"Shade of the composition 4"},{"id":"Combinatorics/integer-compositions/shade-of-integer-compositions/composition-shade-one-three","kind":"node","lean":"MetaMathlibExt.compositionShade_one_three","parent":26917,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/shade-of-integer-compositions/index.html#composition-shade-one-three","scope":"Combinatorics/integer-compositions/shade-of-integer-compositions","status":"fully_proved","summary":"The composition $1 + 3$ has shade $3$: its bargraph has $3$ shaded cells.","title":"Shade of the composition 1 + 3"},{"id":"Combinatorics/integer-compositions/shade-of-integer-compositions/composition-shade-le-total-composition-s","kind":"node","lean":"MetaMathlibExt.compositionShade_le_totalCompositionShade","parent":26917,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/shade-of-integer-compositions/index.html#composition-shade-le-total-composition-s","scope":"Combinatorics/integer-compositions/shade-of-integer-compositions","status":"fully_proved","summary":"For every natural number $n$ and every composition $x$ of $n$, the shade of $x$ is at most the total shade $c(n)$ of the compositions of $n$.","title":"The total shade bounds the shade of each composition"},{"id":"Combinatorics/integer-compositions/shade-of-integer-compositions/total-composition-shade-zero","kind":"node","lean":"MetaMathlibExt.totalCompositionShade_zero","parent":26917,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/shade-of-integer-compositions/index.html#total-composition-shade-zero","scope":"Combinatorics/integer-compositions/shade-of-integer-compositions","status":"fully_proved","summary":"The total shade $c(0)$ of the compositions of $0$ is zero: $c(0) = 0$.","title":"Total shade of the compositions of zero"},{"id":"Combinatorics/integer-compositions/shade-of-integer-compositions/total-composition-shade-generating-funct","kind":"node","lean":"MetaMathlibExt.totalCompositionShade_generatingFunction","parent":26917,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/shade-of-integer-compositions/index.html#total-composition-shade-generating-funct","scope":"Combinatorics/integer-compositions/shade-of-integer-compositions","status":"can_prove","summary":"Let $c(n)$ be the total shade of the compositions of $n$, and let $q$ be a real number with $0 < q < 1/4$. For natural numbers $i_2, i_3, s_1, s_3$ put","title":"Generating function for the total shade of compositions"},{"data_url":"scopes/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions.json","id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","kind":"scope","navigation":"module","parent":26727,"result":false,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","title":"Symmetric peaks and valleys in compositions"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-peak-count","kind":"node","lean":"MetaMathlibExt.symmetricPeakCount","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-peak-count","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"fully_proved","summary":"For a finite word $w = w_1 w_2 \\cdots w_m$ of natural numbers, a symmetric peak is a factor $w_i w_{i+1} w_{i+2}$ of three consecutive letters with $w_i = w_{i+2} < w_{i+1}$. The number of symmetric peaks of $w$ is the number of indices $i$ with $1 \\le i \\le m - 2$ at which $w$ has a symmetric peak (overlapping peaks all count); it is $0$ when $m < 3$.","title":"Number of symmetric peaks"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-peak-height-sum","kind":"node","lean":"MetaMathlibExt.symmetricPeakHeightSum","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-peak-height-sum","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"fully_proved","summary":"For a finite word $w = w_1 w_2 \\cdots w_m$ of natural numbers, a symmetric peak is a factor $w_i w_{i+1} w_{i+2}$ of three consecutive letters with $w_i = w_{i+2} < w_{i+1}$, and its height is $w_{i+1} - w_i$. The symmetric peak height sum of $w$ is the sum of the heights of the symmetric peaks of $w$ over all indices $i$ with $1 \\le i \\le m - 2$ at which one occurs; it is $0$ when $m < 3$.","title":"Sum of the heights of symmetric peaks"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-valley-count","kind":"node","lean":"MetaMathlibExt.symmetricValleyCount","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-valley-count","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"fully_proved","summary":"For a word $w = w_1 w_2 \\cdots w_\\ell$ of natural numbers, a symmetric valley is a position $i$ with $1 < i < \\ell$ such that $w_{i-1} = w_{i+1}$ and $w_i < w_{i-1}$, that is, a factor $a\\,b\\,a$ with $b < a$. The symmetric valley count of $w$ is the number of such positions; in particular it is $0$ when $\\ell < 3$.","title":"Number of symmetric valleys of a word"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-valley-depth-sum","kind":"node","lean":"MetaMathlibExt.symmetricValleyDepthSum","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-valley-depth-sum","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"fully_proved","summary":"For a word $w = w_1 w_2 \\cdots w_\\ell$ of natural numbers, a symmetric valley is a position $i$ with $1 < i < \\ell$ such that $w_{i-1} = w_{i+1}$ and $w_i < w_{i-1}$; its depth is $w_{i-1} - w_i$. The symmetric valley depth sum of $w$ is the sum of the depths of all its symmetric valleys; in particular it is $0$ when $\\ell < 3$.","title":"Total depth of the symmetric valleys of a word"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-peak-height-sum-examples","kind":"node","lean":"MetaMathlibExt.symmetricPeakHeightSum_examples","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-peak-height-sum-examples","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"fully_proved","summary":"The symmetric peak height sums of the words $1\\,3\\,1$, $1\\,2\\,1\\,1$, $1\\,1\\,2\\,1$ and $1\\,1\\,1\\,1\\,1$ are $2$, $1$, $1$ and $0$ respectively, and the word $1\\,3\\,1$ has exactly one symmetric peak.","title":"Symmetric peak examples among the compositions of 5"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-valley-depth-sum-examples","kind":"node","lean":"MetaMathlibExt.symmetricValleyDepthSum_examples","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-valley-depth-sum-examples","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"fully_proved","summary":"The word $2\\,1\\,2\\,1\\,2$ has two symmetric valleys and total symmetric valley depth $2$, the word $3\\,1\\,3\\,1$ has total symmetric valley depth $2$, and the word $3\\,2\\,3$ has total symmetric valley depth $1$.","title":"Symmetric valleys in three compositions of eight"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-peak-height-sum-compositions-g","kind":"node","lean":"MetaMathlibExt.symmetricPeakHeightSum_compositions_genFun","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-peak-height-sum-compositions-g","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For $n \\ge 0$ let $\\mathrm{hsp}(n)$ be the sum, over all compositions $c$ of $n$, of the symmetric peak height sum of the sequence of parts of $c$. Then in $\\mathbb{Z}[[x]]$","title":"Generating function of the total symmetric peak height over compositions"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-peak-height-sum-compositions-b","kind":"node","lean":"MetaMathlibExt.symmetricPeakHeightSum_compositions_bivariate_genFun","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-peak-height-sum-compositions-b","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For $n, k \\ge 0$ let $\\mathrm{hsp}(n, k)$ be the sum, over all compositions $c$ of $n$ with exactly $k$ parts, of the symmetric peak height sum of the sequence of parts of $c$. Then in the ring $\\mathbb{Z}[y][[x]]$ of formal power series in $x$ over $\\mathbb{Z}[y]$,","title":"Bivariate generating function of the total symmetric peak height over compositions"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-valley-depth-sum-compositions","kind":"node","lean":"MetaMathlibExt.symmetricValleyDepthSum_compositions_genFun","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-valley-depth-sum-compositions","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For $n \\ge 0$, let $\\operatorname{dsv}(n)$ be the sum, over all compositions $c$ of $n$, of the total depth of the symmetric valleys of the sequence of parts of $c$. Then in the ring $\\mathbb{Z}[[x]]$ of formal power series,","title":"Generating function for the total depth of symmetric valleys over compositions"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-peak-height-sum-compositions-c","kind":"node","lean":"MetaMathlibExt.symmetricPeakHeightSum_compositions_closed_form","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-peak-height-sum-compositions-c","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For $n \\ge 0$ let $\\mathrm{hsp}(n)$ be the sum, over all compositions $c$ of $n$, of the symmetric peak height sum of the sequence of parts of $c$, and let $\\omega_\\pm = 1 \\pm i\\sqrt{3}$. Then for every $n \\ge 0$, as complex numbers,","title":"Closed form for the total symmetric peak height over compositions"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-valley-depth-sum-compositions-2","kind":"node","lean":"MetaMathlibExt.symmetricValleyDepthSum_compositions_closed_form","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-valley-depth-sum-compositions-2","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For $n \\ge 0$, let $\\operatorname{dsv}(n)$ be the sum, over all compositions $c$ of $n$, of the total depth of the symmetric valleys of the sequence of parts of $c$, and write $\\omega_\\pm = 1 \\pm i\\sqrt{3}$. Then, as complex numbers,","title":"Closed form for the total depth of symmetric valleys over compositions"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-peak-height-sum-compositions-l","kind":"node","lean":"MetaMathlibExt.symmetricPeakHeightSum_compositions_length","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-peak-height-sum-compositions-l","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For $n, k \\ge 0$ let $\\mathrm{hsp}(n, k)$ be the sum, over all compositions $c$ of $n$ with exactly $k$ parts, of the symmetric peak height sum of the sequence of parts of $c$. If $k > 3$, then for every $n \\ge 0$","title":"Total symmetric peak height over compositions with a fixed number of parts"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-valley-depth-sum-compositions-3","kind":"node","lean":"MetaMathlibExt.symmetricValleyDepthSum_compositions_length","parent":26925,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total depth of symmetric valleys over compositions with "],["math","\\(k > 3\\)"],["text"," parts"]],"source":"Total depth of symmetric valleys over compositions with $k > 3$ parts"},"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-valley-depth-sum-compositions-3","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For natural numbers $n$ and $k$ with $k > 3$, the sum, over all compositions $c$ of $n$ with exactly $k$ parts, of the total depth of the symmetric valleys of the sequence of parts of $c$ equals","title":"Total depth of symmetric valleys over compositions with $k > 3$ parts"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-valley-depth-sum-compositions-4","kind":"node","lean":"MetaMathlibExt.symmetricValleyDepthSum_compositions_length_three","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-valley-depth-sum-compositions-4","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For every natural number $n$, the sum, over all compositions $c$ of $n$ with exactly three parts, of the total depth of the symmetric valleys of the sequence of parts of $c$ equals $\\sum_{m=1}^{\\lfloor (n-2)/3 \\rfloor} d_m$, where $d_m = (n-3m)/2$ if $n - m$ is even and $d_m = 0$ otherwise.","title":"Total depth of symmetric valleys over compositions with three parts"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/integral-symmetric-peak-valley-geometric","kind":"node","lean":"MetaMathlibExt.integral_symmetricPeakValley_geometric","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#integral-symmetric-peak-valley-geometric","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"Let $n \\ge 2$, let $0 < p < 1$ and $q = 1 - p$, and let $X_1, \\dots, X_n$ be independent random variables, each geometric on $\\{0, 1, 2, \\dots\\}$ with $\\Pr(X_i = k) = q^k p$, and consider the word $X_1 X_2 \\cdots X_n$. The expected number of its symmetric peaks is $p^2 q (n - 2)/(1 - q^3)$, the expected number of its symmetric valleys is $p^2 \\left(\\frac{1}{1 - q^2} - \\frac{1}{1 - q^3}\\right)(n - 2)$, the expected\u2026","title":"Expected symmetric peaks and valleys in geometric words"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-peak-compositions-trivariate-g","kind":"node","lean":"MetaMathlibExt.symmetricPeak_compositions_trivariate_genFun","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-peak-compositions-trivariate-g","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For a composition $c$ write $\\ell(c)$ for its number of parts, $\\mathrm{sp}(c)$ for the number of symmetric peaks and $\\mathrm{hsp}(c)$ for the symmetric peak height sum of its sequence of parts. In the ring $\\mathbb{Z}[y, q, h][[x]]$ of formal power series in $x$ whose coefficients are polynomials in $y, q, h$ with integer coefficients,","title":"Trivariate generating function for symmetric peaks in compositions"},{"id":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/symmetric-valley-compositions-trivariate","kind":"node","lean":"MetaMathlibExt.symmetricValley_compositions_trivariate_genFun","parent":26925,"result":true,"route":"../roadmap/Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions/index.html#symmetric-valley-compositions-trivariate","scope":"Combinatorics/integer-compositions/symmetric-peaks-and-valleys-in-compositions","status":"can_prove","summary":"For a composition $c$ write $\\ell(c)$ for its number of parts, $\\mathrm{sv}(c)$ for the number of symmetric valleys and $\\mathrm{dsv}(c)$ for the symmetric valley depth sum of its sequence of parts. In the ring $\\mathbb{Z}[y, p, d][[x]]$ of formal power series in $x$ whose coefficients are polynomials in $y, p, d$ with integer coefficients,","title":"Trivariate generating function for symmetric valleys in compositions"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms.json","id":"Combinatorics/riordan-arrays-transforms","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/index.html","scope":"Combinatorics/riordan-arrays-transforms","status":"can_prove","summary":"Infinite lower-triangular arrays and operators on sequences studied through generating functions: the Riordan group, number triangles and their diagonals, binomial, INVERT and Catalan transforms, Hankel and Toeplitz determinants and continued fractions; a specific named family (Motzkin triangle, Stirling matrices) stays with that family.","title":"Riordan arrays and sequence transforms"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays","kind":"scope","parent":26943,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays","status":"can_prove","summary":"Riordan arrays and their A- and Z-sequences, inverses, triple factorizations, pseudo-involutions and total positivity, the Pascal and Pell matrices as Riordan arrays, Barry's triangles, triangular recurrence arrays extended to negative rows, and diagonal or row sums of triangles.","title":"Riordan arrays and number triangles"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/r-fibonacci-number","kind":"node","lean":"MetaMathlibExt.rFibonacciNumber","parent":26944,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/index.html#r-fibonacci-number","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays","status":"fully_proved","summary":"For a real number $r$ and a natural number $n$, the $r$-parameter generalized Fibonacci number is the real number","title":"Barry\u2013Hennessy generalized Fibonacci numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/shifted-riordan-array-constant-term-and","kind":"node","lean":"MetaMathlibExt.shiftedRiordanArray_constantTerm_and_sumCoefficients","parent":26944,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/index.html#shifted-riordan-array-constant-term-and","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $(A_{i,k})_{i,k \\in \\mathbb{N}}$ be an array with entries in $R$, and let $r \\in R$ and $n \\in \\mathbb{N}$. For $s \\in R$ let $A^{(s)}$ be the $s$-shifted array with entries $A^{(s)}_{i,j} = \\sum_{k=j}^{i} \\binom{k}{j} A_{i,k} s^{k-j}$ (an empty sum when $j > i$), and let $p(X) = \\sum_{k=0}^{n} A^{(r)}_{n,k} X^k \\in R[X]$ be the $n$-th row polynomial of $A^{(r)}$. Then\u2026","title":"Constant term and row sum of a shifted array"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sums","kind":"node","lean":"MetaMathlibExt.diagonalSums","parent":26944,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/index.html#diagonal-sums","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays","status":"fully_proved","summary":"For a doubly indexed array $(a_{i,j})_{i, j \\ge 0}$ with entries in an additive commutative monoid $M$, the $n$-th diagonal sum is $d_n = \\sum_{i + j = n} a_{i,j} = \\sum_{k=0}^{n} a_{k,n-k}$.","title":"Diagonal sums of an array"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triple-factorization-of-dot-diagram","kind":"node","lean":"MetaMathlibExt.triple_factorization_of_dot_diagram","parent":26944,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/index.html#triple-factorization-of-dot-diagram","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays","status":"fully_proved","summary":"Let $b$ be an integer, let $\\varepsilon$, $\\delta$ and $\\lambda$ be real numbers, and let $g$ and $f$ be formal power series over $\\mathbb{R}$ with $g(0) = 1$ and $f(0) = 0$. Write $r_{n,k}$ for the coefficient of $x^n$ in $g f^k$, and suppose that for all natural numbers $n$ and $k$","title":"Peart\u2013Woodson triple factorization of Riordan arrays"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/twin-powers","kind":"node","lean":"MetaMathlibExt.twin_powers","parent":26944,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/index.html#twin-powers","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays","status":"fully_proved","summary":"Let $l$ be a natural number, let $n = 2l + 1$, and let $g$ be a formal power series over $\\mathbb{R}$ with $g(0) \\ne 0$. Put $f(x) = x\\, g(x^n)$ and $h(x) = x\\, g(x)^n$. Call a power series $p$ pseudo-involutory if $p(0) = 0$, the coefficient of $x$ in $p$ is nonzero, and the compositional inverse $\\bar p$ of $p$ satisfies $\\bar p(x) = -p(-x)$; call $B$ a B-function of $p$ if $p(x) - x = x\\, p(x)\\, B(x\\, p(x))$. Let","title":"Twin Powers Theorem for pseudo-involutions"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays","status":"fully_proved","title":"A- and Z-sequences of Riordan arrays"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays/riordan-iff-az-sequences","kind":"node","lean":"MetaMathlibExt.riordan_iff_AZ_sequences","parent":26950,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays/index.html#riordan-iff-az-sequences","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays","status":"fully_proved","summary":"Let $(d_{n,k})_{n,k \\ge 0}$ be an integer matrix that is lower triangular, that is, $d_{n,k} = 0$ whenever $n < k$. Then the following are equivalent: (i) there are power series $g, f \\in \\mathbb{Z}[[x]]$ with $g(0) = 1$, $f(0) = 0$ and $[x^1]\\, f \\ne 0$ such that $d_{n,k} = [x^n]\\, g f^k$ for all $n, k \\ge 0$; (ii) $d_{0,0} = 1$ and there are rational sequences $(A_j)_{j \\ge 0}$ and $(Z_j)_{j \\ge 0}$ with\u2026","title":"A- and Z-sequence characterization of integer Riordan arrays"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays/riordan-array-inverse-via-a-and-z-sequen","kind":"node","lean":"MetaMathlibExt.riordan_array_inverse_via_a_and_z_sequences_general","parent":26950,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays/index.html#riordan-array-inverse-via-a-and-z-sequen","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays","status":"fully_proved","summary":"Let $K$ be a field, let $g, f, A, Z, \\bar f$ be formal power series over $K$, and let $d_{00} \\in K$ with $d_{00} \\ne 0$. Suppose that $g(0) = d_{00}$, $f(0) = 0$, $A(0) \\ne 0$, $\\bar f(0) = 0$, $f(\\bar f(x)) = x$, $f(x) = x\\, A(f(x))$ and $g(x) = d_{00} \\bigl(1 - x\\, Z(f(x))\\bigr)^{-1}$. Then","title":"Inverse of a Riordan array from its A- and Z-sequences (one-sided form)"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays/riordan-array-inverse-via-a-and-z-sequen-2","kind":"node","lean":"MetaMathlibExt.riordan_array_inverse_via_a_and_z_sequences","parent":26950,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays/index.html#riordan-array-inverse-via-a-and-z-sequen-2","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/a-and-z-sequences-of-riordan-arrays","status":"fully_proved","summary":"Let $K$ be a field, let $g, f, A, Z, \\bar f$ be formal power series over $K$, and let $d_{00} \\in K$ with $d_{00} \\ne 0$. Suppose that $g(0) = d_{00}$, $f(0) = 0$, $A(0) \\ne 0$ and $\\bar f(0) = 0$; that $f(\\bar f(x)) = x$ and $\\bar f(f(x)) = x$, so that $\\bar f$ is the compositional inverse of $f$; that $f(x) = x\\, A(f(x))$, so that $A$ is the generating function of the A-sequence; and that\u2026","title":"Inverse of a Riordan array from its A- and Z-sequences"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle","status":"can_prove","title":"Barry's Stirling-like triangle"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/barry-triangle-aux","kind":"node","lean":"MetaMathlibExt.barryTriangleAux","parent":26954,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/index.html#barry-triangle-aux","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle","status":"fully_proved","summary":"The function $s(\\beta, n, k)$ of natural numbers $\\beta, n, k$, with natural-number values, is defined by $s(\\beta, 0, 0) = 1$, by $s(\\beta, n, k) = 0$ if exactly one of $n, k$ is $0$, and by","title":"Barry's triangle recurrence"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/barry-triangle-aux-succ-succ","kind":"node","lean":"MetaMathlibExt.barryTriangleAux_succ_succ","parent":26954,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/index.html#barry-triangle-aux-succ-succ","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle","status":"fully_proved","summary":"For all natural numbers $\\beta, n, k$, Barry's triangle recurrence $s$ satisfies $s(\\beta, n + 1, k + 1) = s(\\beta + 1, n, k) + \\beta (k + 1)\\, s(\\beta, n, k + 1)$.","title":"Defining recurrence of Barry's triangle"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/barry-triangle-aux-small-values","kind":"node","lean":"MetaMathlibExt.barryTriangleAux_small_values","parent":26954,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/index.html#barry-triangle-aux-small-values","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle","status":"fully_proved","summary":"Barry's triangle recurrence $s$ takes the values $s(0, 2, 2) = 1$, $s(0, 4, 3) = 4$ and $s(0, 6, 4) = 67$.","title":"Small values of Barry's triangle"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/barry-triangle-aux-eq-matrix-pow","kind":"node","lean":"MetaMathlibExt.barryTriangleAux_eq_matrix_pow","parent":26954,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/index.html#barry-triangle-aux-eq-matrix-pow","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle","status":"can_prove","summary":"Let $s$ be Barry's triangle recurrence and let $2 \\le a \\le n$. Let $M$ be the $(a - 1) \\times (a - 1)$ lower bidiagonal matrix of natural numbers with diagonal entries $M_{i,i} = i(a - i)$ for $i = 1, \\dots, a - 1$, entries $M_{i+1,i} = 1$ just below the diagonal, and all other entries $0$. Then $s(0, n, a)$ is the bottom-left entry $(M^{n-2})_{a-1,1}$ of the power $M^{n-2}$.","title":"Matrix power formula for Barry's triangle"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/barry-triangle-aux-eq-sum-antidiagonal-t","kind":"node","lean":"MetaMathlibExt.barryTriangleAux_eq_sum_antidiagonalTuple","parent":26954,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle/index.html#barry-triangle-aux-eq-sum-antidiagonal-t","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/barry-s-stirling-like-triangle","status":"can_prove","summary":"Let $s$ be Barry's triangle recurrence and let $2 \\le a \\le n$. Then","title":"Matrix-free formula for Barry's triangle"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices","status":"fully_proved","title":"Bell matrices"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/bell-matrix","kind":"node","lean":"MetaMathlibExt.BellMatrix","parent":26960,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/index.html#bell-matrix","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices","status":"fully_proved","summary":"Let $R$ be a commutative ring. A Bell matrix over $R$ is given by a formal power series $g \\in R[[X]]$ whose constant coefficient is a unit of $R$. The corresponding Riordan-array pair is $(g, Xg)$.","title":"Bell matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/h","kind":"node","lean":"MetaMathlibExt.BellMatrix.h","parent":26960,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/index.html#h","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices","status":"fully_proved","summary":"For a Bell matrix over a commutative ring $R$ given by the power series $g$, the second component of its Riordan-array pair is the power series $h = Xg \\in R[[X]]$.","title":"Second component of a Bell matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/entry","kind":"node","lean":"MetaMathlibExt.BellMatrix.entry","parent":26960,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/index.html#entry","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices","status":"fully_proved","summary":"For a Bell matrix over a commutative ring $R$ given by the power series $g$, and natural numbers $n$ and $k$, the entry in row $n$ and column $k$ is the coefficient of $X^n$ in $g \\cdot (Xg)^k$.","title":"Entries of a Bell matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/bell-riordan-central-coefficient","kind":"node","lean":"MetaMathlibExt.bell_riordan_central_coefficient","parent":26960,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices/index.html#bell-riordan-central-coefficient","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/bell-matrices","status":"fully_proved","summary":"Let $g \\in \\mathbb{Z}[[X]]$ be the power series of a Bell matrix over $\\mathbb{Z}$, with constant coefficient $g(0) = 1$, and let $u \\in \\mathbb{Z}[[X]]$ be a power series with constant coefficient $0$ such that substituting $u$ for $X$ in $X \\, g(X)^{-1}$ gives $X$. Then for every natural number $n$, the entry of the Bell matrix in row $2n$ and column $n$ equals $(n+1)$ times the coefficient of $X^{n+1}$ in $u$.","title":"Central coefficients of Bell matrices"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles","status":"fully_proved","title":"Binomial interpolated triangles"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles/is-binary-binomial-interpolated-triangle","kind":"node","lean":"MetaMathlibExt.IsBinaryBinomialInterpolatedTriangle","parent":26965,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles/index.html#is-binary-binomial-interpolated-triangle","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles","status":"fully_proved","summary":"For real numbers $a_0, a_1, \\alpha, \\beta$, a real array $(a_{n,k})_{n,k \\ge 0}$ is the binary binomial interpolated triangle $BT(a_0, a_1, \\alpha, \\beta; \\alpha, \\beta)$ when $\\alpha\\beta \\ne 0$, $|a_0| + |a_1| \\ne 0$, $a_{0,0} = a_0$, $a_{1,0} = a_1$, $a_{n,0} = \\alpha\\, a_{n-1,0} + \\beta\\, a_{n-2,0}$ for every $n \\ge 2$, and $a_{n,k} = \\alpha\\, a_{n,k-1} + \\beta\\, a_{n-1,k-1}$ for all $1 \\le k \\le n$. Entries\u2026","title":"Binary binomial interpolated triangle"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles/rising-diagonal","kind":"node","lean":"MetaMathlibExt.risingDiagonal","parent":26965,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles/index.html#rising-diagonal","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles","status":"fully_proved","summary":"For a real array $a = (a_{n,k})_{n,k \\ge 0}$ and $n \\ge 0$, the rising diagonal of $a$ at level $n$ is the finite sequence $k \\mapsto a_{n-k,k}$ indexed by $0 \\le k \\le \\lfloor n/2 \\rfloor$.","title":"Rising diagonal"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles/rising-diagonal-const","kind":"node","lean":"MetaMathlibExt.risingDiagonal_const","parent":26965,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles/index.html#rising-diagonal-const","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/binomial-interpolated-triangles","status":"fully_proved","summary":"Let $a_0, a_1, \\alpha, \\beta$ be real numbers and let $a = (a_{n,k})_{n,k \\ge 0}$ be a real array that is the binary binomial interpolated triangle $BT(a_0, a_1, \\alpha, \\beta; \\alpha, \\beta)$. Then for every $n \\ge 0$ the rising diagonal of $a$ at level $n$ is constant: $a_{n-i,i} = a_{n-j,j}$ for all $0 \\le i, j \\le \\lfloor n/2 \\rfloor$.","title":"Rising diagonals of a binary binomial interpolated triangle are constant"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","title":"Degenerate harmonic numbers and Riordan arrays"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-term","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicTerm","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-term","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For real numbers $\\lambda$ and $\\alpha$ and a natural number $n$, the degenerate harmonic term is $0$ when $n = 0$ and $(-1)^{n-1} \\binom{\\lambda}{n} / (\\lambda \\alpha^n)$ when $n \\ge 1$, where $\\binom{\\lambda}{n} = \\lambda(\\lambda - 1)\\cdots(\\lambda - n + 1)/n!$.","title":"Degenerate harmonic terms"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-term-zero","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicTerm_zero","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-term-zero","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For all real $\\lambda$ and $\\alpha$, the degenerate harmonic term of index $0$, $H^0_{0,\\lambda}(\\alpha)$, is $0$.","title":"Degenerate harmonic term of index zero"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-term-one","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicTerm_one","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-term-one","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For every real $\\lambda \\ne 0$ and every real $\\alpha$, the degenerate harmonic term of index $1$, $H^0_{1,\\lambda}(\\alpha) = \\binom{\\lambda}{1} / (\\lambda \\alpha)$, equals $1/\\alpha$.","title":"Degenerate harmonic term of index one"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/generalized-degenerate-harmonic","kind":"node","lean":"MetaMathlibExt.generalizedDegenerateHarmonic","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#generalized-degenerate-harmonic","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For real numbers $\\lambda$ and $\\alpha$ and a natural number $n$, Da\u011fl\u0131's generalized degenerate harmonic number is","title":"Generalized degenerate harmonic numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/generalized-degenerate-harmonic-zero","kind":"node","lean":"MetaMathlibExt.generalizedDegenerateHarmonic_zero","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#generalized-degenerate-harmonic-zero","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For all real numbers $\\lambda$ and $\\alpha$, the generalized degenerate harmonic number $H_{0,\\lambda}(\\alpha)$ is $0$.","title":"The zeroth generalized degenerate harmonic number"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/generalized-degenerate-harmonic-succ","kind":"node","lean":"MetaMathlibExt.generalizedDegenerateHarmonic_succ","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#generalized-degenerate-harmonic-succ","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For all real numbers $\\lambda$ and $\\alpha$ and every natural number $n$, the generalized degenerate harmonic numbers satisfy","title":"Recurrence for generalized degenerate harmonic numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/generalized-degenerate-hyperharmonic","kind":"node","lean":"MetaMathlibExt.generalizedDegenerateHyperharmonic","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#generalized-degenerate-hyperharmonic","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For real numbers $\\lambda$ and $\\alpha$, the generalized degenerate hyperharmonic numbers $H^r_{n,\\lambda}(\\alpha)$, indexed by natural numbers $r$ (the order) and $n$, are defined by recursion on $r$. At order $0$ they are the degenerate harmonic terms: $H^0_{0,\\lambda}(\\alpha) = 0$ and $H^0_{n,\\lambda}(\\alpha) = (-1)^{n-1} \\binom{\\lambda}{n} / (\\lambda \\alpha^n)$ for $n \\ge 1$, where $\\binom{\\lambda}{n}$ is the\u2026","title":"Generalized degenerate hyperharmonic numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/generalized-degenerate-hyperharmonic-suc","kind":"node","lean":"MetaMathlibExt.generalizedDegenerateHyperharmonic_succ_zero","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#generalized-degenerate-hyperharmonic-suc","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For all real numbers $\\lambda$ and $\\alpha$ and every natural number $r$, the generalized degenerate hyperharmonic number of order $r + 1$ at index $0$ is zero: $H^{r+1}_{0,\\lambda}(\\alpha) = 0$.","title":"Generalized degenerate hyperharmonic numbers of positive order vanish at zero"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/generalized-degenerate-hyperharmonic-one","kind":"node","lean":"MetaMathlibExt.generalizedDegenerateHyperharmonic_one","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#generalized-degenerate-hyperharmonic-one","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For all real numbers $\\lambda$ and $\\alpha$ and every natural number $n$, the generalized degenerate hyperharmonic number of order $1$ equals the generalized degenerate harmonic number: $H^1_{n,\\lambda}(\\alpha) = H_{n,\\lambda}(\\alpha)$.","title":"Generalized degenerate hyperharmonic numbers of order one"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For real $\\lambda$ and $\\alpha$, let $\\ell_0 = 0$ and $\\ell_m = (-1)^{m-1} \\binom{\\lambda}{m} / (\\lambda \\alpha^m)$ for $m \\ge 1$, where $\\binom{\\lambda}{m}$ is the generalized binomial coefficient and a quotient by $0$ is read as $0$, and let $L(t) = \\sum_{m \\ge 0} \\ell_m t^m$ as a formal power series. For $\\lambda, \\alpha > 0$ this is the expansion of $-\\log_\\lambda(1 - t/\\alpha)$, where\u2026","title":"Generalized degenerate harmonic numbers of a given rank"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank-zero-left","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank_zero_left","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank-zero-left","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For all real $\\lambda$ and $\\alpha$ and every natural number $r$, the generalized degenerate harmonic number of rank $r$ satisfies $H_\\lambda(0, r, \\alpha) = 0$.","title":"Degenerate harmonic numbers of rank vanish at index zero"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank-zero-right","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank_zero_right","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank-zero-right","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For all real $\\lambda$ and $\\alpha$ and every natural number $n$, the generalized degenerate harmonic number of rank $0$ equals the generalized degenerate harmonic number: $H_\\lambda(n, 0, \\alpha) = H_{n,\\lambda}(\\alpha)$.","title":"Degenerate harmonic numbers of rank zero"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-exp-series","kind":"node","lean":"MetaMathlibExt.degenerateExpSeries","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-exp-series","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For a real number $\\lambda$, the degenerate exponential series is the formal power series $e_\\lambda(t) = \\sum_{n \\ge 0} (1)_{n,\\lambda} \\, t^n / n!$ over $\\mathbb{R}$, where $(1)_{0,\\lambda} = 1$ and, for $n \\ge 1$, $(1)_{n,\\lambda} = 1 \\cdot (1 - \\lambda) \\cdots (1 - (n - 1)\\lambda)$ is the generalized falling factorial of $1$ with increment $\\lambda$.","title":"Degenerate exponential series"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/constant-coeff-degenerate-exp-series","kind":"node","lean":"MetaMathlibExt.constantCoeff_degenerateExpSeries","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#constant-coeff-degenerate-exp-series","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For every real number $\\lambda$, the constant term of the degenerate exponential series $e_\\lambda(t)$ is $1$.","title":"Constant term of the degenerate exponential series"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-bernoulli","kind":"node","lean":"MetaMathlibExt.degenerateBernoulli","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-bernoulli","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For a real number $\\lambda$, let $(1)_{0,\\lambda} = 1$ and $(1)_{m,\\lambda} = 1 \\cdot (1 - \\lambda) \\cdots (1 - (m - 1)\\lambda)$ for $m \\ge 1$ (the generalized falling factorial of $1$ with increment $\\lambda$). For $n \\in \\mathbb{N}$, the degenerate Bernoulli number $\\beta_{n,\\lambda}$ is $n!$ times the coefficient of $t^n$ in the inverse of the formal power series\u2026","title":"Degenerate Bernoulli numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-bernoulli-zero","kind":"node","lean":"MetaMathlibExt.degenerateBernoulli_zero","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-bernoulli-zero","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For every real number $\\lambda$, the degenerate Bernoulli number $\\beta_{0,\\lambda}$ equals $1$.","title":"The zeroth degenerate Bernoulli number"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-euler","kind":"node","lean":"MetaMathlibExt.degenerateEuler","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-euler","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For a real number $\\lambda$ and $n \\in \\mathbb{N}$, the degenerate Euler number $\\varepsilon_{n,\\lambda}$ is $n!$ times the coefficient of $t^n$ in the formal power series $2 / (e_\\lambda(t) + 1)$ over $\\mathbb{R}$, where $e_\\lambda(t)$ is the degenerate exponential series; that is, $2 / (e_\\lambda(t) + 1) = \\sum_{n \\ge 0} \\varepsilon_{n,\\lambda} \\, t^n / n!$.","title":"Degenerate Euler numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-euler-zero","kind":"node","lean":"MetaMathlibExt.degenerateEuler_zero","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-euler-zero","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For every real number $\\lambda$, the degenerate Euler number $\\varepsilon_{0,\\lambda}$ equals $1$.","title":"The zeroth degenerate Euler number"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-stirling-first","kind":"node","lean":"MetaMathlibExt.degenerateStirlingFirst","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-stirling-first","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For a real number $\\lambda$, let $\\log_\\lambda(1 + t)$ be the formal power series $\\sum_{m \\ge 1} \\binom{\\lambda}{m} t^m / \\lambda$, where $\\binom{\\lambda}{m} = \\lambda(\\lambda - 1)\\cdots(\\lambda - m + 1)/m!$. For natural numbers $n$ and $k$, the degenerate Stirling number of the first kind $S_{1,\\lambda}(n, k)$ is $n!/k!$ times the coefficient of $t^n$ in $(\\log_\\lambda(1 + t))^k$. Equivalently,\u2026","title":"Degenerate Stirling numbers of the first kind"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-stirling-first-zero-right","kind":"node","lean":"MetaMathlibExt.degenerateStirlingFirst_zero_right","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-stirling-first-zero-right","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For every real number $\\lambda$ and every natural number $n$, the degenerate Stirling number of the first kind $S_{1,\\lambda}(n, 0)$ equals $1$ if $n = 0$ and $0$ otherwise.","title":"Degenerate Stirling numbers of the first kind in column zero"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-stirling-second","kind":"node","lean":"MetaMathlibExt.degenerateStirlingSecond","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-stirling-second","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For a real number $\\lambda$, let $e_\\lambda(t) = \\sum_{n \\ge 0} (1)_{n,\\lambda} t^n / n!$ be the degenerate exponential series, where $(1)_{0,\\lambda} = 1$ and $(1)_{n,\\lambda} = 1 (1 - \\lambda) \\cdots (1 - (n - 1)\\lambda)$. For natural numbers $n$ and $k$, the degenerate Stirling number of the second kind $S_{2,\\lambda}(n, k)$ is $n!/k!$ times the coefficient of $t^n$ in $(e_\\lambda(t) - 1)^k$. Equivalently,\u2026","title":"Degenerate Stirling numbers of the second kind"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-stirling-second-zero-right","kind":"node","lean":"MetaMathlibExt.degenerateStirlingSecond_zero_right","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-stirling-second-zero-right","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For every real number $\\lambda$ and every natural number $n$, the degenerate Stirling number of the second kind $S_{2,\\lambda}(n, 0)$ equals $1$ if $n = 0$ and $0$ otherwise.","title":"Degenerate Stirling numbers of the second kind in column zero"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-daehee","kind":"node","lean":"MetaMathlibExt.degenerateDaehee","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-daehee","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For a real number $\\lambda$ and $n \\in \\mathbb{N}$, the degenerate Daehee number is $D_{n,\\lambda} = n! \\binom{\\lambda}{n+1} / \\lambda$, where $\\binom{\\lambda}{k} = \\lambda (\\lambda - 1) \\cdots (\\lambda - k + 1) / k!$ is the generalized binomial coefficient (for $\\lambda = 0$ the value is $0$, by the convention $x / 0 = 0$).","title":"Degenerate Daehee numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-daehee-zero","kind":"node","lean":"MetaMathlibExt.degenerateDaehee_zero","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-daehee-zero","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"fully_proved","summary":"For every real number $\\lambda \\ne 0$, the degenerate Daehee number $D_{0,\\lambda}$ equals $1$.","title":"The zeroth degenerate Daehee number"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-hyperharmonic-alternating-and","kind":"node","lean":"MetaMathlibExt.degenerateHyperharmonic_alternating_and_binomial_sums","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-hyperharmonic-alternating-and","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ and $\\alpha > 0$ be real numbers and let $n$ be a natural number. Write $H_{m,\\lambda}(\\alpha)$ for the generalized degenerate harmonic numbers, $H^s_{m,\\lambda}(\\alpha)$ for the generalized degenerate hyperharmonic numbers of order $s$, and $D_{n,\\lambda}$ for the degenerate Daehee numbers. Then","title":"Alternating and binomial sums of degenerate hyperharmonic numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-alternating-binomial","kind":"node","lean":"MetaMathlibExt.degenerateHarmonic_alternating_binomial_sum","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-alternating-binomial","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ be real and let $H_{k,\\lambda} = H_{k,\\lambda}(1)$ be the degenerate harmonic numbers, that is, the generalized degenerate harmonic numbers at $\\alpha = 1$. Then for every integer $n \\ge 1$, $\\sum_{k=0}^{n} (-1)^{k+1} \\binom{n}{k} H_{k,\\lambda} = \\binom{n + \\lambda - 1}{n} / \\lambda$, where $\\binom{n + \\lambda - 1}{n}$ is the generalized binomial coefficient.","title":"Alternating binomial sum of degenerate harmonic numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank-exp-weighted-su","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank_exp_weighted_sum","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank-exp-weighted-su","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ and $\\alpha > 0$ be real and let $n$ be a natural number. Write $(1)_{k,\\lambda} = 1 \\cdot (1 - \\lambda) \\cdots (1 - (k-1)\\lambda)$ for the generalized falling factorial (so $(1)_{0,\\lambda} = 1$), $H_\\lambda(m, r, \\alpha)$ for the generalized degenerate harmonic numbers of rank $r$, and $H_{m,\\lambda}(\\alpha)$ for the generalized degenerate harmonic numbers. Then","title":"Alternating degenerate exponential sum of degenerate harmonic numbers of rank"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank-succ-partial-su","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank_succ_partial_sum","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank-succ-partial-su","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ and $\\alpha > 0$ be real and let $n$ and $r$ be natural numbers. Write $H_\\lambda(k, r, \\alpha)$ for the generalized degenerate harmonic numbers of rank $r$ and $H_{m,\\lambda}(\\alpha)$ for the generalized degenerate harmonic numbers. Then","title":"Partial sums of degenerate harmonic numbers of rank as a convolution"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank-shifted-exp-wei","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank_shifted_exp_weighted_sum","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank-shifted-exp-wei","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ and $\\alpha > 0$ be real and let $n$ be a natural number. Write $(1)_{k,\\lambda} = 1 \\cdot (1 - \\lambda) \\cdots (1 - (k-1)\\lambda)$ for the generalized falling factorial and $H_\\lambda(m, r, \\alpha)$ for the generalized degenerate harmonic numbers of rank $r$. Then","title":"Shifted alternating degenerate exponential sum of degenerate harmonic numbers of rank"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-hyperharmonic-eq-binomial-sum","kind":"node","lean":"MetaMathlibExt.degenerateHyperharmonic_eq_binomial_sum","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-hyperharmonic-eq-binomial-sum","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ and $\\alpha > 0$ be real numbers and let $r \\le n$ be natural numbers. Write $\\binom{\\lambda}{j} = \\lambda(\\lambda - 1)\\cdots(\\lambda - j + 1)/j!$ and $H^s_{m,\\lambda}(\\alpha)$ for the generalized degenerate hyperharmonic numbers of order $s$. Then","title":"A binomial sum equal to a degenerate hyperharmonic number"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank-exp-sum-eq-stir","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank_exp_sum_eq_stirlingFirst","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank-exp-sum-eq-stir","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ and $\\alpha > 0$ be real and let $n \\ge 1$ be a natural number. Write $(1)_{k,\\lambda} = 1 \\cdot (1 - \\lambda) \\cdots (1 - (k-1)\\lambda)$ for the generalized falling factorial, $H_\\lambda(m, r, \\alpha)$ for the generalized degenerate harmonic numbers of rank $r$, and $S_{1,\\lambda}(k, i)$ for the degenerate Stirling numbers of the first kind. Then","title":"Degenerate harmonic numbers of rank and degenerate Stirling numbers of the first kind"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank-bernoulli-euler","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank_bernoulli_euler_sums","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank-bernoulli-euler","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ and $\\alpha > 0$ be real and let $n \\ge 1$ be an integer. Write $\\beta_{k,\\lambda}$ and $\\varepsilon_{k,\\lambda}$ for the degenerate Bernoulli and Euler numbers, $H_{m,\\lambda}(\\alpha)$ for the generalized degenerate harmonic numbers, and $H_\\lambda(m, r, \\alpha)$ for the generalized degenerate harmonic numbers of rank $r$. Then\u2026","title":"Degenerate Bernoulli and Euler sums of higher-rank degenerate harmonic numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/degenerate-harmonic-rank-stirling-second","kind":"node","lean":"MetaMathlibExt.degenerateHarmonicRank_stirlingSecond_sum","parent":26969,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays/index.html#degenerate-harmonic-rank-stirling-second","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/degenerate-harmonic-numbers-and-riordan-arrays","status":"can_prove","summary":"Let $\\lambda > 0$ and $\\alpha > 0$ be real and let $m$ and $n$ be natural numbers with $1 \\le m \\le n$. Write $S_{2,\\lambda}(k, m)$ for the degenerate Stirling numbers of the second kind and $H_\\lambda(n, r, \\alpha)$ for the generalized degenerate harmonic numbers of rank $r$. Then","title":"Degenerate harmonic numbers of rank and degenerate Stirling numbers of the second kind"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes","status":"fully_proved","title":"Diagonal sequences of square zig-zag shapes"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes/right-down-diagonal-coefficient","kind":"node","lean":"MetaMathlibExt.rightDownDiagonalCoefficient","parent":27003,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes/index.html#right-down-diagonal-coefficient","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes","status":"fully_proved","summary":"For natural numbers $k$ and $i$, the coefficient $c_k(i)$ of the right-down diagonal recurrence is the integer $(-1)^i \\binom{k+1-i}{i+1}$ when $i \\le k+1$, and $0$ when $i > k+1$.","title":"Right-down diagonal coefficient"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes/is-right-down-diagonal-family","kind":"node","lean":"MetaMathlibExt.IsRightDownDiagonalFamily","parent":27003,"reading_title":{"format":"inline-parts-v1","parts":[["text","Right-down diagonal families of order "],["math","\\(k\\)"]],"source":"Right-down diagonal families of order $k$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes/index.html#is-right-down-diagonal-family","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/diagonal-sequences-of-square-zig-zag-shapes","status":"fully_proved","summary":"For a natural number $k$, a family $(A_j)_{0 \\le j \\le k+1}$ of integer sequences $A_j : \\mathbb{N} \\to \\mathbb{Z}$ is a family of right-down diagonal sequences of order $k$ when $k \\ge 1$ and, for every $j$ and every natural number $n \\ge \\lfloor k/2 \\rfloor + 1$,","title":"Right-down diagonal families of order $k$"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type","status":"fully_proved","title":"Exponential Riordan arrays of Pascal type"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type/exponential-pascal-type-series","kind":"node","lean":"MetaMathlibExt.exponentialPascalTypeSeries","parent":27006,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type/index.html#exponential-pascal-type-series","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type","status":"fully_proved","summary":"Let $R$ be a commutative ring that is a $\\mathbb{Q}$-algebra and let $r \\in \\mathbb{Q}$. This is the formal power series $t + \\frac{r}{2} t^2$ over $R$, the second component of the generating pair $[e^t, t + r t^2/2]$ of the exponential Pascal-type matrix.","title":"Second series of the exponential Pascal-type pair"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type/exponential-pascal-type-pair","kind":"node","lean":"MetaMathlibExt.exponentialPascalTypePair","parent":27006,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type/index.html#exponential-pascal-type-pair","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type","status":"fully_proved","summary":"Let $R$ be a commutative ring that is a $\\mathbb{Q}$-algebra and let $r \\in \\mathbb{Q}$. This is the pair of formal power series $\\left(e^t, t + \\frac{r}{2} t^2\\right)$ over $R$, written $[e^t, t + r t^2/2]$ in exponential Riordan array notation.","title":"Generating pair of the exponential Pascal-type matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type/exponential-pascal-type-matrix","kind":"node","lean":"MetaMathlibExt.exponentialPascalTypeMatrix","parent":27006,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type/index.html#exponential-pascal-type-matrix","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/exponential-riordan-arrays-of-pascal-type","status":"fully_proved","summary":"Let $R$ be a commutative ring that is a $\\mathbb{Q}$-algebra and let $r \\in \\mathbb{Q}$. The exponential Pascal-type matrix with parameter $r$ is the exponential Riordan array $[e^t, t + r t^2/2]$ over $R$: the matrix with rows and columns indexed by the natural numbers whose $(n, k)$ entry is","title":"Exponential Pascal-type matrix"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials","status":"fully_proved","title":"Morgan\u2013Voyce polynomials"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials/morgan-voyce-coeff","kind":"node","lean":"MetaMathlibExt.morganVoyceCoeff","parent":27010,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials/index.html#morgan-voyce-coeff","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the Morgan\u2013Voyce coefficient $d(n, k)$ is the binomial coefficient $\\binom{n + k}{n - k}$ if $k \\le n$, and $0$ if $k > n$.","title":"Morgan\u2013Voyce coefficients"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials/morgan-voyce-polynomial","kind":"node","lean":"MetaMathlibExt.morganVoycePolynomial","parent":27010,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials/index.html#morgan-voyce-polynomial","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/morgan-voyce-polynomials","status":"fully_proved","summary":"For a semiring $R$ and a natural number $n$, the Morgan\u2013Voyce polynomial of index $n$ over $R$ is","title":"Morgan\u2013Voyce polynomials"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles","status":"fully_proved","title":"Ordinary Riordan Pascal-like triangles"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles/ordinary-riordan-pascal-like-matrix","kind":"node","lean":"MetaMathlibExt.ordinaryRiordanPascalLikeMatrix","parent":27013,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles/index.html#ordinary-riordan-pascal-like-matrix","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles","status":"fully_proved","summary":"For natural numbers $n, k$ and an integer $r$, the entry $T(n, k; r)$ is the integer","title":"Ordinary Riordan Pascal-like triangle"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles/ordinary-riordan-pascal-like-matrix-def","kind":"node","lean":"MetaMathlibExt.ordinaryRiordanPascalLikeMatrix_def","parent":27013,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles/index.html#ordinary-riordan-pascal-like-matrix-def","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/ordinary-riordan-pascal-like-triangles","status":"fully_proved","summary":"For natural numbers $n, k$ and an integer $r$, the ordinary Riordan Pascal-like matrix entry $T(n, k; r)$ equals $\\sum_{j=0}^{k} \\binom{k}{j} \\binom{\\max(n - j, 0)}{k} r^j$; this holds by definition.","title":"Entry formula for the ordinary Riordan Pascal-like triangle"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","status":"fully_proved","title":"Pascal matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/pascal-entry","kind":"node","lean":"MetaMathlibExt.pascalEntry","parent":27016,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/index.html#pascal-entry","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","status":"fully_proved","summary":"For natural numbers $i$ and $j$, the entry of the infinite Pascal matrix in row $i$ and column $j$ is the binomial coefficient $\\binom{i}{j}$.","title":"Entries of the infinite Pascal matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/pascal-alpha","kind":"node","lean":"MetaMathlibExt.pascalAlpha","parent":27016,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/index.html#pascal-alpha","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","status":"fully_proved","summary":"For a natural number $j$ and a real number $x$, the column generating function of the infinite Pascal matrix is $\\alpha_j(x) = \\sum_{i \\ge 0} \\binom{i}{j} x^i$, where $\\binom{i}{j}$ is the entry in row $i$ and column $j$; the sum is taken to be $0$ when the series is not summable.","title":"Column generating functions of the Pascal matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/pascal-alpha-eq","kind":"node","lean":"MetaMathlibExt.pascalAlpha_eq","parent":27016,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/index.html#pascal-alpha-eq","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","status":"fully_proved","summary":"For every natural number $j$ and every real number $x$ with $|x| < 1$, the column generating function $\\alpha_j$ of the infinite Pascal matrix satisfies $\\alpha_j(x) = x^j / (1 - x)^{j+1}$.","title":"Closed form of the Pascal column generating functions"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/tsum-pascal-alpha-eq","kind":"node","lean":"MetaMathlibExt.tsum_pascalAlpha_eq","parent":27016,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/index.html#tsum-pascal-alpha-eq","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","status":"fully_proved","summary":"For every real number $x$ with $|x| < 1/2$, the sum $\\sum_{j \\ge 0} \\alpha_j(x)$ of the column generating functions of the infinite Pascal matrix equals $\\sum_{j \\ge 0} \\sum_{i \\ge 0} \\binom{i}{j} x^i$, where $\\binom{i}{j}$ is the entry in row $i$ and column $j$, and it also equals $\\sum_{j \\ge 0} x^j / (1 - x)^{j+1}$.","title":"Summing the Pascal column generating functions"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/tsum-pascal-entry-comm","kind":"node","lean":"MetaMathlibExt.tsum_pascalEntry_comm","parent":27016,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/index.html#tsum-pascal-entry-comm","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","status":"fully_proved","summary":"For every real number $x$ with $|x| < 1/2$, writing $\\binom{i}{j}$ for the entry of the infinite Pascal matrix in row $i$ and column $j$, we have $\\sum_{j \\ge 0} \\sum_{i \\ge 0} \\binom{i}{j} x^i = \\sum_{i \\ge 0} \\sum_{j \\ge 0} \\binom{i}{j} x^i$, and this sum equals $\\frac{1}{1 - x} \\sum_{j \\ge 0} \\left(\\frac{x}{1 - x}\\right)^j$.","title":"Reversing the order of summation in the Pascal matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/tsum-pascal-entry-row","kind":"node","lean":"MetaMathlibExt.tsum_pascalEntry_row","parent":27016,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/index.html#tsum-pascal-entry-row","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","status":"fully_proved","summary":"For every real number $x$ with $|x| < 1/2$, writing $\\binom{i}{j}$ for the entry of the infinite Pascal matrix in row $i$ and column $j$, we have $\\sum_{i \\ge 0} x^i \\sum_{j=0}^{i} \\binom{i}{j} = \\frac{1}{1 - 2x}$.","title":"Summing the Pascal matrix row by row"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/tsum-pascal-alpha-eq-geometric","kind":"node","lean":"MetaMathlibExt.tsum_pascalAlpha_eq_geometric","parent":27016,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix/index.html#tsum-pascal-alpha-eq-geometric","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pascal-matrix","status":"fully_proved","summary":"For every real number $x$ with $|x| < 1/2$, the sum $\\sum_{j \\ge 0} \\alpha_j(x)$ of the column generating functions of the infinite Pascal matrix equals $\\sum_{i \\ge 0} (2x)^i$, and $\\sum_{i \\ge 0} (2x)^i = \\frac{1}{1 - 2x}$.","title":"Total generating function of the Pascal matrix"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array","status":"fully_proved","title":"Pell matrix as a Riordan array"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array/pell-riordan-general","kind":"node","lean":"MetaMathlibExt.pell_riordan_general","parent":27024,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array/index.html#pell-riordan-general","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array","status":"fully_proved","summary":"Let $k$ be a natural number and let $p(n, m)$, for natural numbers $n$ and $m$, be natural numbers such that for every $n$ the power series $\\sum_{m \\ge 0} p(n, m)\\, x^m$ over $\\mathbb{Q}$ satisfies $\\left(\\sum_{m \\ge 0} p(n, m)\\, x^m\\right)(1 - x)^{n+1} = (1 + x + \\dots + x^k)^n$. Then for all natural numbers $n$ and $m$, the coefficient of $x^n$ in $\\frac{1}{1-x} \\left(\\frac{x(1 + x + \\dots + x^k)}{1-x}\\right)^m$\u2026","title":"Generalized Pell matrix as a Riordan array, for every k"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array/pell-riordan","kind":"node","lean":"MetaMathlibExt.pell_riordan","parent":27024,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array/index.html#pell-riordan","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/pell-matrix-as-a-riordan-array","status":"fully_proved","summary":"Let $k \\ge 2$ be a natural number and let $p(n, m)$, for natural numbers $n$ and $m$, be natural numbers such that for every $n$ the power series $\\sum_{m \\ge 0} p(n, m)\\, x^m$ over $\\mathbb{Q}$ satisfies $\\left(\\sum_{m \\ge 0} p(n, m)\\, x^m\\right)(1 - x)^{n+1} = (1 + x + \\dots + x^k)^n$. Then for all natural numbers $n$ and $m$, the coefficient of $x^n$ in\u2026","title":"Generalized Pell matrix as a Riordan array"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"can_prove","title":"Total positivity of Riordan arrays"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/bell-riordan-has-nonnegative-minors-of-a","kind":"node","lean":"MetaMathlibExt.bellRiordan_hasNonnegativeMinors_of_aSequence","parent":27027,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html#bell-riordan-has-nonnegative-minors-of-a","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"can_prove","summary":"Let $g$ be a formal power series over $\\mathbb{R}$ with $g(0) \\neq 0$, write $[x^n]F$ for the coefficient of $x^n$ in a power series $F$, and let $D = (d_{n,k})_{n,k \\ge 0}$ be the Bell-type Riordan array $(g(x), x g(x))$, with entries $d_{n,k} = [x^n]\\, g(x) (x g(x))^k$. Let $(a_j)_{j \\ge 0}$ be a real sequence, the $A$-sequence of $D$, such that for all $n, k \\ge 0$","title":"Bell-type Riordan arrays with a P\u00f3lya frequency A-sequence are totally positive"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/is-riordan-array-az","kind":"node","lean":"MetaMathlibExt.IsRiordanArrayAZ","parent":27027,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html#is-riordan-array-az","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"fully_proved","summary":"For real sequences $(a_n)_{n \\ge 0}$ and $(z_n)_{n \\ge 0}$, an infinite real matrix $R = (r_{n,k})_{n,k \\ge 0}$ is the Riordan array with $A$-sequence $(a_n)$ and $Z$-sequence $(z_n)$ when $r_{0,0} = 1$, $r_{0,k} = 0$ for $k \\ge 1$, and for all $n, k \\ge 0$","title":"Riordan array with given A- and Z-sequences"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/one-zero-and-one-one","kind":"node","lean":"MetaMathlibExt.IsRiordanArrayAZ.one_zero_and_one_one","parent":27027,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html#one-zero-and-one-one","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"fully_proved","summary":"If $R = (r_{n,k})_{n,k \\ge 0}$ is the Riordan array with $A$-sequence $(a_n)$ and $Z$-sequence $(z_n)$, for real sequences $(a_n)$ and $(z_n)$, then $r_{1,0} = z_0$ and $r_{1,1} = a_0$.","title":"Entries in row 1 of a Riordan array"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/riordan-coeff-matrix","kind":"node","lean":"MetaMathlibExt.riordanCoeffMatrix","parent":27027,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html#riordan-coeff-matrix","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"fully_proved","summary":"For real sequences $(z_n)_{n \\ge 0}$ and $(a_n)_{n \\ge 0}$ with generating functions $Z(x)$ and $A(x)$, the coefficient matrix $J(Z(x), A(x))$ is the infinite real matrix, with rows and columns indexed by $n, k \\ge 0$, whose entry in row $n$ and column $k$ is $z_n$ if $k = 0$, is $a_{n+1-k}$ if $1 \\le k \\le n + 1$, and is $0$ if $k > n + 1$. Thus row $n$ reads $z_n, a_n, a_{n-1}, \\ldots, a_0, 0, 0, \\ldots$","title":"Coefficient matrix of a Riordan array"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/riordan-coeff-matrix-zero-one","kind":"node","lean":"MetaMathlibExt.riordanCoeffMatrix_zero_one","parent":27027,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html#riordan-coeff-matrix-zero-one","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"fully_proved","summary":"For real sequences $(z_n)$ and $(a_n)$ with generating functions $Z(x)$ and $A(x)$, and for every $n \\ge 0$, the coefficient matrix $J(Z(x), A(x))$ has entry $z_n$ in row $n$, column $0$, and entry $a_n$ in row $n$, column $1$.","title":"First two columns of the coefficient matrix of a Riordan array"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/riordan-array-has-nonnegative-minors-of","kind":"node","lean":"MetaMathlibExt.riordanArray_hasNonnegativeMinors_of_coeffMatrix","parent":27027,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html#riordan-array-has-nonnegative-minors-of","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"can_prove","summary":"Let $(a_n)$ and $(z_n)$ be real sequences with generating functions $A(x)$ and $Z(x)$, and let $R$ be the Riordan array with $A$-sequence $(a_n)$ and $Z$-sequence $(z_n)$. If the coefficient matrix $J(Z(x), A(x))$ is totally positive, that is, every minor of it is nonnegative, then every minor of $R$ is nonnegative.","title":"Total positivity of a Riordan array from its coefficient matrix"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/riordan-array-has-nonnegative-minors-of-2","kind":"node","lean":"MetaMathlibExt.riordanArray_hasNonnegativeMinors_of_params","parent":27027,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html#riordan-array-has-nonnegative-minors-of-2","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"can_prove","summary":"Let $P, Q, A, Z$ be formal power series over $\\mathbb{R}$ with $P(0) \\ne 0$, write $[x^n]F$ for the coefficient of $x^n$ in a power series $F$, and let $a, b, r, s, t \\ge 0$ be real numbers. Suppose that","title":"A total positivity criterion for Riordan arrays"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/bell-riordan-array-has-nonnegative-minor","kind":"node","lean":"MetaMathlibExt.bellRiordanArray_hasNonnegativeMinors_of_pf","parent":27027,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays/index.html#bell-riordan-array-has-nonnegative-minor","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/total-positivity-of-riordan-arrays","status":"can_prove","summary":"Let $P$ and $A$ be formal power series over $\\mathbb{R}$ with $P(0) \\ne 0$, write $[x^n]F$ for the coefficient of $x^n$ in a power series $F$, and let $r, s, t \\ge 0$ be real numbers with $s^2 \\ge 4t$. Suppose that $P$ is a P\u00f3lya frequency series, that is, every minor of the lower Toeplitz matrix of its coefficients (whose entry in row $n$ and column $k$ is $[x^{n-k}]P$ for $k \\le n$ and $0$ otherwise) is\u2026","title":"Total positivity of Bell-type Riordan arrays from a P\u00f3lya frequency series"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices.json","id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","kind":"scope","navigation":"module","parent":26944,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"can_prove","title":"Triangular recurrence arrays at negative indices"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/triangular-recurrence","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#triangular-recurrence","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"A triangular recurrence is a tuple $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ of six integers, the parameters of the linear recurrence","title":"Triangular recurrence"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/array","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.array","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#array","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"Given integer parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$, the array $a_{n,k} \\in \\mathbb{Q}$, $n, k \\ge 0$, is defined by $a_{0,k} = \\delta_{k,0}$ and, for $n \\ge 1$, by the recurrence (1) $a_{n,k} = (\\alpha n + \\beta k + \\gamma)\\, a_{n-1,k} + (\\alpha' n + \\beta' k + \\gamma')\\, a_{n-1,k-1}$, where $a_{n-1,-1} = 0$.","title":"Triangular array of a linear recurrence"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/array-binomial-four-two","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.array_binomial_four_two","parent":27036,"reading_title":{"format":"inline-parts-v1","parts":[["text","The binomial array at "],["math","\\((4, 2)\\)"]],"source":"The binomial array at $(4, 2)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#array-binomial-four-two","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For the parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma') = (0, 0, 1; 0, 0, 1)$ of the binomial coefficients (Table 1), the triangular array $a$ satisfies $a_{4,2} = 6$.","title":"The binomial array at $(4, 2)$"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/weight","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.weight","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#weight","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For a triangular recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ and integers $n, k$, the first weight is the rational number $\\alpha(n + 1) + \\beta k + \\gamma$.","title":"First weight of a triangular recurrence"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/weight-2","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.weight'","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#weight-2","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For a triangular recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ and integers $n, k$, the second weight is the rational number $\\alpha'(n + 1) + \\beta' k + \\gamma'$.","title":"Second weight of a triangular recurrence"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-pos-aux","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negPosAux","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-pos-aux","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For integer parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ and $p, k \\ge 0$, put $n = -(p + 1)$. The rational numbers $c_{p,k}$ are defined by recursion on $p$ and then on $k$ through $c_{p,k} = (r - (\\alpha'(n+1) + \\beta' k + \\gamma')\\, s) / (\\alpha(n+1) + \\beta k + \\gamma)$, where $r = c_{p-1,k}$ if $p \\ge 1$, $r = 1$ if $p = k = 0$, and $r = 0$ if $p = 0 < k$; and $s = c_{p,k-1}$ if $k \\ge 1$,\u2026","title":"Negative-positive numbers at negative rows (auxiliary)"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-pos","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negPos","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-pos","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"Let $a_{n,k}$ ($n, k \\ge 0$) be the triangular array of a triangular recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$. Its negative-positive numbers $a^{NP}_{n,k} \\in \\mathbb{Q}$, for $n, k \\in \\mathbb{Z}$, are defined by $a^{NP}_{n,k} = a_{n,k}$ for $n, k \\ge 0$, $a^{NP}_{n,k} = 0$ for $k < 0$, and, for $n < 0 \\le k$, recursively by","title":"Negative-positive numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-pos-neg-one-zero","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negPos_neg_one_zero","parent":27036,"reading_title":{"format":"inline-parts-v1","parts":[["text","The negative-positive number at "],["math","\\((-1, 0)\\)"]],"source":"The negative-positive number at $(-1, 0)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-pos-neg-one-zero","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For every triangular recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$, its negative-positive number at row $-1$ and column $0$ is $a^{NP}_{-1,0} = 1/\\gamma$, computed in $\\mathbb{Q}$ with $1/0 = 0$.","title":"The negative-positive number at $(-1, 0)$"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-pos-rec","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negPos_rec","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-pos-rec","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For every triangular recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ and all integers $n \\le -2$ and $k \\ge 1$, the negative-positive numbers satisfy","title":"Recurrence for the negative-positive numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-neg-aux","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negNegAux","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-neg-aux","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For integer parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ and $p, q \\ge 0$, put $n = -(p + 1)$ and $k = n - q$. The rational numbers $b_{p,q}$ are defined by recursion on $p$ and then on $q$ through $b_{p,q} = (c - (\\alpha(n+1) + \\beta(k+1) + \\gamma)\\, e) / (\\alpha'(n+1) + \\beta'(k+1) + \\gamma')$, where $c = b_{p-1,q}$ if $p \\ge 1$, $c = 1$ if $p = q = 0$, and $c = 0$ if $p = 0 < q$; and\u2026","title":"Negative-negative numbers on and below the diagonal (auxiliary)"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-neg","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negNeg","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-neg","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ with triangular array $a_{n,k}$ ($n, k \\ge 0$), the negative-negative numbers $a^{NN}_{n,k} \\in \\mathbb{Q}$, $n, k \\in \\mathbb{Z}$, are defined as follows (Definition 2): $a^{NN}_{0,k} = \\delta_{k,0}$ for all $k$; $a^{NN}_{n,k} = a_{n,k}$ for $n > 0$ and $k \\ge 0$; $a^{NN}_{n,k} = 0$ for $n > 0 > k$ and for $n < 0$, $k > n$; and, for $k \\le n < 0$,\u2026","title":"Negative-negative numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-neg-neg-one","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negNeg_neg_one","parent":27036,"reading_title":{"format":"inline-parts-v1","parts":[["text","Negative-negative numbers at "],["math","\\((-1, -1)\\)"],["text"," and "],["math","\\((-1, 0)\\)"]],"source":"Negative-negative numbers at $(-1, -1)$ and $(-1, 0)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-neg-neg-one","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For all parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$, the negative-negative numbers $a^{NN}$ satisfy $a^{NN}_{-1,-1} = 1/\\gamma'$ (read as $0$ when $\\gamma' = 0$) and $a^{NN}_{-1,0} = 0$.","title":"Negative-negative numbers at $(-1, -1)$ and $(-1, 0)$"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-neg-rec","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negNeg_rec","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-neg-rec","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"For all parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ and all integers $k < n \\le -2$, the negative-negative numbers $a^{NN}$ satisfy $a^{NN}_{n,k} = (a^{NN}_{n+1,k+1} - (\\alpha(n+1) + \\beta(k+1) + \\gamma)\\, a^{NN}_{n,k+1}) / (\\alpha'(n+1) + \\beta'(k+1) + \\gamma')$, where division by $0$ gives $0$.","title":"Recurrence for the negative-negative numbers"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/reflect","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.reflect","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#reflect","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"The reflection of a triangular recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ is the triangular recurrence with parameters $(\\alpha' + \\beta', -\\beta', \\gamma'; \\alpha + \\beta, -\\beta, \\gamma)$.","title":"Reflected triangular recurrence"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/reflect-reflect","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.reflect_reflect","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#reflect-reflect","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"fully_proved","summary":"Reflecting a triangular recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$ twice gives back the recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$.","title":"Reflection of triangular recurrences is an involution"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/linear-diophantine-solutions","kind":"node","lean":"MetaMathlibExt.linear_diophantine_solutions","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#linear-diophantine-solutions","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"can_prove","summary":"Let $a, b, c, u, v \\in \\mathbb{Z}$ with $a \\ne 0$ or $b \\ne 0$, let $d = \\gcd(a, b)$, and suppose $a u + b v = d$ (B\u00e9zout). If $d \\mid c$, then the integers $x, y$ with $a x + b y + c = 0$ are exactly those of the form $x = -(c/d) u + m (b/d)$, $y = -(c/d) v - m (a/d)$ with $m \\in \\mathbb{Z}$. If $d \\nmid c$, then $a x + b y + c = 0$ has no solution in integers $x, y$.","title":"Linear Diophantine equations"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-neg-duality","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negNeg_duality","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-neg-duality","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"can_prove","summary":"For integers $\\alpha, \\beta, \\gamma$, let $a$ and $b$ be the arrays with parameters $(\\alpha, \\beta, \\gamma; 0, 0, 1)$ and $(\\beta, \\alpha, -\\gamma; 0, 0, 1)$. Then their negative-negative numbers satisfy $a^{NN}_{-n,-k} = b^{NN}_{k-1,n-1}$ for all $n, k \\in \\mathbb{Z}$.","title":"Knuth-type duality"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/reflect-neg-pos-eq-neg-neg","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.reflect_negPos_eq_negNeg","parent":27036,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#reflect-neg-pos-eq-neg-neg","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"can_prove","summary":"Let $a$ be the array of a triangular recurrence with parameters $(\\alpha, \\beta, \\gamma; \\alpha', \\beta', \\gamma')$, with weights $w(n, k) = \\alpha(n + 1) + \\beta k + \\gamma$ and $w'(n, k) = \\alpha'(n + 1) + \\beta' k + \\gamma'$, and let $\\tilde a$ be the array of the reflected recurrence, with parameters $(\\alpha' + \\beta', -\\beta', \\gamma'; \\alpha + \\beta, -\\beta, \\gamma)$ and weights $\\tilde w$, $\\tilde w'$. Then\u2026","title":"NP/NN duality"},{"id":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/neg-pos-egf-beta-zero","kind":"node","lean":"MetaMathlibExt.TriangularRecurrence.negPos_egf_beta_zero","parent":27036,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating function of the negative-positive numbers for "],["math","\\(\\beta = \\beta' = 0\\)"]],"source":"Generating function of the negative-positive numbers for $\\beta = \\beta' = 0$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices/index.html#neg-pos-egf-beta-zero","scope":"Combinatorics/riordan-arrays-transforms/riordan-arrays/triangular-recurrence-arrays-at-negative-indices","status":"can_prove","summary":"Let $\\alpha, \\gamma, \\alpha', \\gamma'$ be integers and let $a^{NP}$ be the negative-positive numbers for the parameters $(\\alpha, 0, \\gamma; \\alpha', 0, \\gamma')$. For every $n \\ge 0$, in the ring $\\mathbb{Q}[[u]]$ of formal power series,","title":"Generating function of the negative-positive numbers for $\\beta = \\beta' = 0$"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions.json","id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions","kind":"scope","parent":26943,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/index.html","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions","status":"can_prove","summary":"Hankel transforms of sequences and their invariance under binomial transforms, Heilermann's formula linking Jacobi continued fractions to weighted Motzkin paths, Catalan-sum Hankel determinants, and Toeplitz\u2013Hessenberg determinants.","title":"Hankel determinants and continued fractions"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/layman-binomial-transform-hankel-invaria","kind":"node","lean":"MetaMathlibExt.layman_binomial_transform_hankel_invariant","parent":27056,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/index.html#layman-binomial-transform-hankel-invaria","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions","status":"fully_proved","summary":"For every sequence of integers $(a_n)_{n \\ge 0}$, the Hankel transform of the binomial transform of $a$ equals the Hankel transform of $a$. Here the binomial transform of $a$ is the sequence $b_n = \\sum_{i=0}^{n} \\binom{n}{i} a_i$, and the Hankel transform of a sequence $c$ is the sequence whose $n$-th term, for $n \\ge 0$, is the determinant of the $(n+1) \\times (n+1)$ matrix $(c_{i+j})_{0 \\le i, j \\le n}$.","title":"Layman's theorem: the binomial transform preserves the Hankel transform"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums.json","id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums","kind":"scope","navigation":"module","parent":27056,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums/index.html","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums","status":"fully_proved","title":"Hankel determinants of Catalan sums"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums/layman-catalan-sum-hankel-det-unrestrict","kind":"node","lean":"MetaMathlibExt.layman_catalan_sum_hankel_det_unrestricted","parent":27058,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums/index.html#layman-catalan-sum-hankel-det-unrestrict","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums","status":"fully_proved","summary":"Let $C_m$ denote the $m$-th Catalan number and $F_m$ the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For every natural number $n$, the determinant of the $n \\times n$ integer matrix whose $(i, j)$ entry is $C_{i+j+1} + C_{i+j+2}$ for $0 \\le i, j \\le n-1$ equals $F_{2n+2}$. For $n = 0$ the matrix is empty and both sides equal $1$.","title":"Layman's Catalan-sum Hankel determinant, including the empty matrix"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums/layman-catalan-sum-hankel-det","kind":"node","lean":"MetaMathlibExt.layman_catalan_sum_hankel_det","parent":27058,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums/index.html#layman-catalan-sum-hankel-det","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-determinants-of-catalan-sums","status":"fully_proved","summary":"Let $C_m$ denote the $m$-th Catalan number and $F_m$ the $m$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For every integer $n \\ge 1$, the determinant of the $n \\times n$ integer matrix whose $(i, j)$ entry is $C_{i+j+1} + C_{i+j+2}$ for $0 \\le i, j \\le n-1$ equals $F_{2n+2}$.","title":"Layman's Catalan-sum Hankel determinant"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions.json","id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions","kind":"scope","navigation":"module","parent":27056,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions/index.html","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions","status":"fully_proved","title":"Hankel transforms and Jacobi continued fractions"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions/hankel-transform","kind":"node","lean":"MetaMathlibExt.hankelTransform","parent":27061,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions/index.html#hankel-transform","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $(a_k)_{k \\ge 0}$ be a sequence in $R$. The Hankel transform of $a$ is the sequence $(h_n)_{n \\ge 0}$ in $R$ whose term $h_n$ is the determinant of the $(n + 1) \\times (n + 1)$ Hankel matrix $(a_{i+j})_{0 \\le i, j \\le n}$.","title":"Hankel transform"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions/has-jacobi-fraction","kind":"node","lean":"MetaMathlibExt.HasJacobiFraction","parent":27061,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions/index.html#has-jacobi-fraction","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/hankel-transforms-and-jacobi-continued-fractions","status":"fully_proved","summary":"Let $K$ be a ring, not necessarily commutative, let $(a_n)_{n \\ge 0}$, $(\\alpha_n)_{n \\ge 0}$ and $(\\beta_n)_{n \\ge 0}$ be sequences in $K$, let $\\mu_0 \\in K$, and let $(F_i)_{i \\ge 0}$ be a sequence of formal power series over $K$ in the variable $x$. These data form a Jacobi continued fraction expansion of $a$ when $\\sum_{n \\ge 0} a_n x^n = \\mu_0 F_0$ and, for every $i \\ge 0$,\u2026","title":"Jacobi continued fraction expansion"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula.json","id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula","kind":"scope","navigation":"module","parent":27056,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/index.html","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula","status":"fully_proved","title":"Heilermann's formula"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/hankel-transform-eq-prod-of-has-jacobi-f","kind":"node","lean":"MetaMathlibExt.hankelTransform_eq_prod_of_hasJacobiFraction","parent":27064,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/index.html#hankel-transform-eq-prod-of-has-jacobi-f","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula","status":"fully_proved","summary":"Let $K$ be a commutative ring, let $a$, $\\alpha$, $\\beta$ be sequences in $K$ and let $\\mu_0 \\in K$. Suppose that $a$ has Jacobi fraction data $(\\alpha, \\beta, \\mu_0)$: there are power series $F_0, F_1, \\dots$ over $K$ with $\\sum_{n \\ge 0} a_n x^n = \\mu_0 F_0$ and $(1 - \\alpha_i x - \\beta_{i+1} x^2 F_{i+1}) F_i = 1$ for every $i \\ge 0$. Then for every $n \\ge 0$ the Hankel transform of $a$ at $n$, the determinant of\u2026","title":"Heilermann's formula"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/hankel-transform-eq-of-has-jacobi-fracti","kind":"node","lean":"MetaMathlibExt.hankelTransform_eq_of_hasJacobiFraction_same_beta","parent":27064,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/index.html#hankel-transform-eq-of-has-jacobi-fracti","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula","status":"fully_proved","summary":"Let $K$ be a commutative ring. Say that a sequence $a$ in $K$ has Jacobi fraction data $(\\alpha, \\beta, \\mu_0)$, where $\\alpha$, $\\beta$ are sequences in $K$ and $\\mu_0 \\in K$, when there are power series $F_0, F_1, \\dots$ over $K$ with $\\sum_{n \\ge 0} a_n x^n = \\mu_0 F_0$ and $(1 - \\alpha_i x - \\beta_{i+1} x^2 F_{i+1}) F_i = 1$ for every $i \\ge 0$. If $a$ has Jacobi fraction data $(\\alpha, \\beta, \\mu_0)$ and $a'$\u2026","title":"Hankel transforms of Jacobi fractions do not depend on the linear coefficients"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/exists-weighted-motzkin-coefficients-of","kind":"node","lean":"MetaMathlibExt.exists_weightedMotzkin_coefficients_of_hasJacobiFraction","parent":27064,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/index.html#exists-weighted-motzkin-coefficients-of","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula","status":"fully_proved","summary":"Let $K$ be a ring, let $a$, $\\alpha$, $\\beta$ be sequences in $K$ and let $\\mu_0 \\in K$. Suppose that $a$ has Jacobi fraction data $(\\alpha, \\beta, \\mu_0)$: there are power series $F_0, F_1, \\dots$ over $K$ with $\\sum_{n \\ge 0} a_n x^n = \\mu_0 F_0$ and $(1 - \\alpha_i x - \\beta_{i+1} x^2 F_{i+1}) F_i = 1$ for every $i \\ge 0$. Then there is an array $(M_{n,j})_{n, j \\ge 0}$ in $K$ with $M_{0,0} = 1$ and, for all\u2026","title":"Weighted Motzkin recurrence for the coefficients of a Jacobi fraction"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/hankel-transform-eq-prod-of-weighted-mot","kind":"node","lean":"MetaMathlibExt.hankelTransform_eq_prod_of_weightedMotzkin","parent":27064,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula/index.html#hankel-transform-eq-prod-of-weighted-mot","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/heilermann-s-formula","status":"fully_proved","summary":"Let $K$ be a commutative ring, let $\\alpha$, $\\beta$ be sequences in $K$, let $\\mu_0 \\in K$, and let $(M_{n,j})_{n, j \\ge 0}$ be an array in $K$ with $M_{0,0} = \\mu_0$ and, for all $n, j \\ge 0$, $M_{0,j+1} = 0$, $M_{n+1,0} = \\alpha_0 M_{n,0} + \\beta_1 M_{n,1}$ and $M_{n+1,j+1} = M_{n,j} + \\alpha_{j+1} M_{n,j+1} + \\beta_{j+2} M_{n,j+2}$. Then for every $n \\ge 0$ the Hankel transform of the column\u2026","title":"Heilermann's formula for weighted Motzkin arrays"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices.json","id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices","kind":"scope","navigation":"module","parent":27056,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices/index.html","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices","status":"fully_proved","title":"Stieltjes production matrices"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices/stieltjes-matrix","kind":"node","lean":"MetaMathlibExt.stieltjesMatrix","parent":27069,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices/index.html#stieltjes-matrix","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices","status":"fully_proved","summary":"Let $R$ be an additive monoid with one and let $a_1, b_1, a, b \\in R$. The Stieltjes matrix $S$ is the infinite matrix with rows and columns indexed by $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$ whose entries are $S_{i, i+1} = 1$ for every $i$, $S_{0,0} = a_1$, $S_{1,0} = b_1$, $S_{i,i} = a$ and $S_{i+1,i} = b$ for every $i \\ge 1$, and $S_{i,j} = 0$ for all other $i, j$. So row $0$ is $a_1, 1, 0, \\dots$, row $1$ is\u2026","title":"Tridiagonal Stieltjes production matrix"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices/stieltjes-matrix-of","kind":"node","lean":"MetaMathlibExt.stieltjesMatrixOf","parent":27069,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices/index.html#stieltjes-matrix-of","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/stieltjes-production-matrices","status":"fully_proved","summary":"Let $R$ be an additive monoid with one and let $a, b, \\lambda, \\mu \\in R$. The specialized Stieltjes matrix is the tridiagonal Stieltjes production matrix with $a_1 = a + \\lambda$, $b_1 = b + \\mu$ and the same $a$ and $b$: the infinite matrix $S$ with rows and columns indexed by $\\mathbb{N} = \\{0, 1, 2, \\dots\\}$ whose entries are $S_{i, i+1} = 1$ for every $i$, $S_{0,0} = a + \\lambda$, $S_{1,0} = b + \\mu$,\u2026","title":"Specialized Stieltjes matrix"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan.json","id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","kind":"scope","navigation":"module","parent":27056,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"can_prove","title":"Toeplitz\u2013Hessenberg determinants of Fuss\u2013Catalan numbers"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/toeplitz-hessenberg-fuss-catalan-example","kind":"node","lean":"MetaMathlibExt.toeplitzHessenberg_fussCatalan_examples","parent":27072,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html#toeplitz-hessenberg-fuss-catalan-example","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"fully_proved","summary":"For the generalized Fuss\u2013Catalan numbers $A_n(p, r) = \\frac{r}{pn + r} \\binom{pn + r}{n}$, one has $A_3(2, 1) = 5$, $A_2(2, 3) = 9$, $A_3(3, 1) = 12$ and $A_2(3, 2) = 7$.","title":"Small generalized Fuss\u2013Catalan numbers"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/toeplitz-hessenberg-det-gen-fun","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_genFun","parent":27072,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html#toeplitz-hessenberg-det-gen-fun","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"can_prove","summary":"Let $R$ be a commutative ring and $a_0, a_1, a_2, \\dots$ a sequence in $R$. Write $\\det(a_0; a_1, \\dots, a_n)$ for the determinant of the $n \\times n$ Toeplitz\u2013Hessenberg matrix whose entry in row $i$ and column $j$ is $a_{i+1-j}$ for $j \\le i + 1$ and $0$ above the superdiagonal. In the ring of formal power series over $R$ let $f(x) = \\sum_{n \\ge 1} \\det(a_0; a_1, \\dots, a_n)\\, x^n$ and\u2026","title":"Generating function of Toeplitz\u2013Hessenberg determinants"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/toeplitz-hessenberg-det-fuss-catalan","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_fussCatalan","parent":27072,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html#toeplitz-hessenberg-det-fuss-catalan","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"can_prove","summary":"Write $\\det(a_0; a_1, \\dots, a_n)$ for the determinant of the $n \\times n$ Toeplitz\u2013Hessenberg matrix whose entry in row $i$ and column $j$ is $a_{i+1-j}$ for $j \\le i + 1$ and $0$ above the superdiagonal. For natural numbers $p, r, n$ let $A_n(p, r) = \\frac{r}{pn + r} \\binom{pn + r}{n}$ be the generalized Fuss\u2013Catalan number, taken to be $0$ when $pn + r = 0$. Then for all natural numbers $p, r$ and every $n \\ge 1$,","title":"Toeplitz\u2013Hessenberg determinants of Fuss\u2013Catalan numbers"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/toeplitz-hessenberg-det-fuss-catalan-spe","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_fussCatalan_special","parent":27072,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html#toeplitz-hessenberg-det-fuss-catalan-spe","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"can_prove","summary":"Write $\\det(a_0; a_1, \\dots, a_n)$ for the determinant of the $n \\times n$ Toeplitz\u2013Hessenberg matrix whose entry in row $i$ and column $j$ is $a_{i+1-j}$ for $j \\le i + 1$ and $0$ above the superdiagonal. For natural numbers $p, r \\ge 1$ let $A_n(p, r) = \\frac{r}{pn + r} \\binom{pn + r}{n}$ be the generalized Fuss\u2013Catalan number, put $D_n(p, r) = (-1)^{n-1} \\det(1; A_1(p, r), \\dots, A_n(p, r))$, and let $C_n$ be\u2026","title":"Special Toeplitz\u2013Hessenberg determinants of Fuss\u2013Catalan numbers"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/toeplitz-hessenberg-det-catalan","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_catalan","parent":27072,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html#toeplitz-hessenberg-det-catalan","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"can_prove","summary":"Write $\\det(a_0; a_1, \\dots, a_n)$ for the determinant of the $n \\times n$ Toeplitz\u2013Hessenberg matrix whose entry in row $i$ and column $j$ is $a_{i+1-j}$ for $j \\le i + 1$ and $0$ above the superdiagonal (so $a_0$ fills the superdiagonal and $a_1, \\dots, a_n$ the first column); for $n = 0$ it is $1$. Let $C_n$ be the $n$-th Catalan number, $C(x) = \\sum_{n \\ge 0} C_n x^n \\in \\mathbb{Q}[[x]]$, and\u2026","title":"Toeplitz\u2013Hessenberg determinants with Catalan entries"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/toeplitz-hessenberg-det-ternary","kind":"node","lean":"MetaMathlibExt.toeplitzHessenbergDet_ternary","parent":27072,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html#toeplitz-hessenberg-det-ternary","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"can_prove","summary":"Write $\\det(a_0; a_1, \\dots, a_n)$ for the determinant of the $n \\times n$ Toeplitz\u2013Hessenberg matrix whose entry in row $i$ and column $j$ is $a_{i+1-j}$ for $j \\le i + 1$ and $0$ above the superdiagonal, and let $H_n = \\frac{1}{3n + 1} \\binom{3n + 1}{n}$ and $J_n = \\frac{2}{3n + 2} \\binom{3n + 2}{n}$. Then, in $\\mathbb{Q}$, $\\det(1; H_2, H_3, \\dots, H_{n+1}) = \\frac{2 (-1)^{n-1}}{n} \\binom{3n - 3}{n - 1}$ for\u2026","title":"Toeplitz\u2013Hessenberg determinants of ternary Fuss\u2013Catalan numbers"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/trudi-partition-sum-fuss-catalan","kind":"node","lean":"MetaMathlibExt.trudiPartitionSum_fussCatalan","parent":27072,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html#trudi-partition-sum-fuss-catalan","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"can_prove","summary":"For $s \\in \\mathbb{Q}$ and rational numbers $a_1, \\dots, a_n$ let $T_s(a_1, \\dots, a_n)$ be the sum of $s^{|\\alpha|} \\binom{|\\alpha|}{\\alpha_1, \\dots, \\alpha_n} a_1^{\\alpha_1} \\cdots a_n^{\\alpha_n}$ over all $(\\alpha_1, \\dots, \\alpha_n) \\in \\mathbb{N}^n$ with $\\alpha_1 + 2 \\alpha_2 + \\dots + n \\alpha_n = n$, where $|\\alpha| = \\alpha_1 + \\dots + \\alpha_n$. Let $A_n(p, r) = \\frac{r}{pn + r} \\binom{pn + r}{n}$ be the\u2026","title":"Trudi multi-sums of Fuss\u2013Catalan numbers"},{"id":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/nonempty-equiv-even-noncrossing-even-des","kind":"node","lean":"MetaMathlibExt.nonempty_equiv_evenNoncrossing_evenDescentDyck","parent":27072,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan/index.html#nonempty-equiv-even-noncrossing-even-des","scope":"Combinatorics/riordan-arrays-transforms/hankel-continued-fractions/toeplitz-hessenberg-determinants-of-fuss-catalan","status":"can_prove","summary":"For every natural number $n$, there is a bijection between the noncrossing partitions of $\\{0, \\dots, 2n - 1\\}$ all of whose blocks have even size and the Dyck paths of semilength $2n$ all of whose descents have even length. Here the descents of a Dyck path are the lengths of its maximal runs of down steps, one for each peak $UD$: the run of down steps starting at the $D$ of that peak.","title":"Even noncrossing partitions and Dyck paths with even descents"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/sequence-transforms.json","id":"Combinatorics/riordan-arrays-transforms/sequence-transforms","kind":"scope","parent":26943,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/index.html","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms","status":"can_prove","summary":"Operators on integer sequences and coefficient extraction: binomial, multiple and falling k-binomial transforms, INVERT and Catalan transforms, hypersequences, convolutions, C-finite sequences, and recovering terms from generating-function values.","title":"Sequence transforms and generating functions"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/catalan-transform-pair","kind":"node","lean":"MetaMathlibExt.catalan_transform_pair","parent":27081,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/index.html#catalan-transform-pair","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms","status":"fully_proved","summary":"Let $a, b \\colon \\mathbb{N} \\to \\mathbb{Q}$ be sequences such that, for every $n \\ge 0$,","title":"Barry's Catalan transform and its inverse"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/is-cfinite-sequence","kind":"node","lean":"MetaMathlibExt.IsCFiniteSequence","parent":27081,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/index.html#is-cfinite-sequence","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms","status":"fully_proved","summary":"Let $\\sigma$ be a type indexing variables, and write $\\mathbb{Z}[\\sigma]$ for the multivariate polynomials with integer coefficients in the variables $\\sigma$. A sequence $a_0, a_1, a_2, \\dots$ in $\\mathbb{Z}[\\sigma]$ is C-finite when there exist natural numbers $k$ and $N$ and polynomials $p_1, \\dots, p_k \\in \\mathbb{Z}[\\sigma]$ such that for every $n \\ge N$","title":"C-finite polynomial sequences"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hadamard-fibonacci-product","kind":"node","lean":"MetaMathlibExt.hadamard_fibonacci_product","parent":27081,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/index.html#hadamard-fibonacci-product","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms","status":"fully_proved","summary":"Let $R$ be a commutative ring, let $a, b \\in R$, and let $n \\ge 2$ be a natural number. For each $k$, let $f_k$ be the coefficient of $x^k$ in $1/(1 - ax - x^2)$ and $g_k$ the coefficient of $x^k$ in $1/(1 - bx - x^n)$, the inverses taken in $R[[x]]$. Then the Hadamard product $\\sum_{k \\ge 0} f_k g_k x^k$ equals","title":"Hadamard product of two Fibonacci-type series"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/generating-function-term-extraction","kind":"node","lean":"MetaMathlibExt.generating_function_term_extraction","parent":27081,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/index.html#generating-function-term-extraction","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms","status":"fully_proved","summary":"Let $t_0, t_1, t_2, \\dots$ be natural numbers and let $c$, $m$, $N$ be natural numbers with $c \\ge 2$ and $2 \\le m \\le N$. Suppose that the real series $\\sum_{k \\ge 0} t_k \\, (c^{-N})^k$ is summable and that $t_j < c^{j-2}$ for every $j \\ge m$. Then","title":"Recovering a sequence term from its generating function"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms.json","id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","kind":"scope","navigation":"module","parent":27081,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","title":"Binomial transforms"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/exponential-generating-function-binomial","kind":"node","lean":"MetaMathlibExt.exponentialGeneratingFunction_binomialTransform","parent":27086,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html#exponential-generating-function-binomial","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","summary":"Let $(a_n)_{n \\ge 0}$ and $(b_n)_{n \\ge 0}$ be real sequences with $b_n = \\sum_{k=0}^{n} \\binom{n}{k} a_k$ for every $n$, and let $A : \\mathbb{R} \\to \\mathbb{R}$ be any function. For every real number $t$, if the series $\\sum_{n \\ge 0} a_n t^n / n!$ converges absolutely with sum $A(t)$, then the series $\\sum_{n \\ge 0} b_n t^n / n!$ converges absolutely with sum $e^t A(t)$.","title":"Exponential generating function of the binomial transform"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/falling-kbinomial-transform","kind":"node","lean":"MetaMathlibExt.fallingKBinomialTransform","parent":27086,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html#falling-kbinomial-transform","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","summary":"For an integer $k$ and an integer sequence $(a_i)_{i \\ge 0}$, the falling $k$-binomial transform of $a$ is the integer sequence $(b_n)_{n \\ge 0}$ given by","title":"Falling k-binomial transform"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/binomial-transform","kind":"node","lean":"MetaMathlibExt.binomialTransform","parent":27086,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html#binomial-transform","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","summary":"The binomial transform of an integer sequence $(a_i)_{i \\ge 0}$ is the falling k-binomial transform of $a$ with $k = 1$, the sequence whose $n$-th term is $\\sum_{i=0}^{n} \\binom{n}{i} 1^{n-i} a_i$.","title":"Binomial transform"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/binomial-transform-eq-sum","kind":"node","lean":"MetaMathlibExt.binomialTransform_eq_sum","parent":27086,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html#binomial-transform-eq-sum","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","summary":"For every integer sequence $(a_i)_{i \\ge 0}$ and every $n \\in \\mathbb{N}$, the $n$-th term of the binomial transform of $a$ equals $\\sum_{i=0}^{n} \\binom{n}{i} a_i$.","title":"Binomial transform as a binomial sum"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/multiple-binomial-transform","kind":"node","lean":"MetaMathlibExt.multipleBinomialTransform","parent":27086,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html#multiple-binomial-transform","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","summary":"The binomial transform sends an integer sequence $(a_i)_{i \\ge 0}$ to the sequence $(b_n)_{n \\ge 0}$ given by $b_n = \\sum_{i=0}^{n} \\binom{n}{i} a_i$. For a natural number $r$, the $r$-fold binomial transform of $a$ is obtained by applying the binomial transform $r$ times in succession; the $0$-fold transform of $a$ is $a$ itself.","title":"Multiple binomial transform"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/falling-equiv","kind":"node","lean":"MetaMathlibExt.falling_equiv","parent":27086,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html#falling-equiv","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","summary":"For every integer sequence $a$ and every integer $k$: if $k \\ge 0$, applying the binomial transform $k$ times to $a$ gives the falling $k$-binomial transform of $a$; if $k < 0$, applying the inverse binomial transform (the falling $(-1)$-binomial transform) $-k$ times to $a$ gives the falling $k$-binomial transform of $a$. In both cases the equality is one of integer sequences.","title":"The k-fold binomial transform is the falling k-binomial transform"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/mk-sum-choose-mul-eq-inv-one-sub-mul-sub","kind":"node","lean":"MetaMathlibExt.mk_sum_choose_mul_eq_inv_one_sub_mul_subst","parent":27086,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html#mk-sum-choose-mul-eq-inv-one-sub-mul-sub","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","summary":"Let $K$ be a field and let $a : \\mathbb{N} \\to K$ be a sequence with ordinary generating function $A(X) = \\sum_n a_n X^n \\in K[[X]]$. Then the ordinary generating function of the binomial transform $n \\mapsto \\sum_{k=0}^{n} \\binom{n}{k} a_k$ of $a$ equals $(1 - X)^{-1} \\cdot A\\big(X (1 - X)^{-1}\\big)$ in $K[[X]]$, where $A\\big(X (1 - X)^{-1}\\big)$ is the power series obtained by substituting $X (1 - X)^{-1}$ for\u2026","title":"Generating function of the binomial transform over a field"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/ordinary-generating-function-binomial-tr","kind":"node","lean":"MetaMathlibExt.ordinary_generating_function_binomial_transform","parent":27086,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms/index.html#ordinary-generating-function-binomial-tr","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/binomial-transforms","status":"fully_proved","summary":"Let $a : \\mathbb{N} \\to \\mathbb{R}$ be a real sequence with ordinary generating function $A(X) = \\sum_n a_n X^n \\in \\mathbb{R}[[X]]$. Then the ordinary generating function of the binomial transform $n \\mapsto \\sum_{k=0}^{n} \\binom{n}{k} a_k$ of $a$ equals $(1 - X)^{-1} \\cdot A\\big(X (1 - X)^{-1}\\big)$ in $\\mathbb{R}[[X]]$, where $A\\big(X (1 - X)^{-1}\\big)$ is the power series obtained by substituting\u2026","title":"Ordinary generating function of the binomial transform"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences.json","id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","kind":"scope","navigation":"module","parent":27081,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","title":"Hypersequences"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hypersequence","kind":"node","lean":"MetaMathlibExt.hypersequence","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hypersequence","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"Let $(a_n)_{n \\ge 0}$ be a sequence in an additive commutative monoid. Its hypersequence of generation $r$, written $a^{(r)}_n$, is defined by recursion on $r$: $a^{(0)}_n = a_n$ and $a^{(r+1)}_n = \\sum_{k=0}^{n} a^{(r)}_k$ for all natural numbers $r$ and $n$. Thus each generation is the sequence of partial sums of the previous one.","title":"Hypersequence"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hypersequence-succ","kind":"node","lean":"MetaMathlibExt.hypersequence_succ","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hypersequence-succ","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"Let $(a_n)$ be a sequence in an additive commutative monoid and let $a^{(r)}_n$ denote its hypersequence of generation $r$. For all natural numbers $r$ and $n$, $a^{(r+1)}_n = \\sum_{k=0}^{n} a^{(r)}_k$.","title":"Each hypersequence generation sums the previous one"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hypersequence-one-zero","kind":"node","lean":"MetaMathlibExt.hypersequence_one_zero","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hypersequence-one-zero","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"Let $(a_n)$ be a sequence in an additive commutative monoid and let $a^{(1)}_n$ denote its hypersequence of generation $1$. Then $a^{(1)}_0 = a_0$.","title":"First hypersequence at index zero"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyper-fibonacci-eq-hypersequence","kind":"node","lean":"MetaMathlibExt.hyperFibonacci_eq_hypersequence","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyper-fibonacci-eq-hypersequence","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"Let $F_k$ denote the $k$th Fibonacci number. For all natural numbers $r$ and $n$, the hyperfibonacci number $F^{(r)}_n$ equals the $r$th hypersequence of the Fibonacci sequence at $n$, that is, $F^{(r)}_n = a^{(r)}_n$ for $a_k = F_k$, where $a^{(0)}_n = a_n$ and $a^{(r+1)}_n = \\sum_{k=0}^{n} a^{(r)}_k$.","title":"Hyperfibonacci numbers are the hypersequences of the Fibonacci numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/alternating-binomial-power-sum","kind":"node","lean":"MetaMathlibExt.alternatingBinomialPowerSum","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#alternating-binomial-power-sum","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"For natural numbers $\\ell, m$ and complex numbers $x, y$,","title":"Alternating binomial power sum"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/alternating-binomial-power-sum-one-one-o","kind":"node","lean":"MetaMathlibExt.alternatingBinomialPowerSum_one_one_one_one","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#alternating-binomial-power-sum-one-one-o","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"The alternating binomial power sum $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (kx + y)^\\ell$ satisfies $p_{1,1}(1, 1) = 1 - 2 = -1$.","title":"A value of the alternating binomial power sum"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hypersequence-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.hypersequence_eq_sum_choose","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hypersequence-eq-sum-choose","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $(a_n)$ be a sequence of complex numbers and let $a^{(r)}_n$ denote its hypersequence of generation $r$. For every $r \\ge 1$ and every natural number $n$,","title":"Hypersequences as binomial sums"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hypersequence-succ-succ","kind":"node","lean":"MetaMathlibExt.hypersequence_succ_succ","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hypersequence-succ-succ","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $(a_n)$ be a sequence of complex numbers and let $a^{(r)}_n$ denote its hypersequence of generation $r$. For all natural numbers $r$ and $n$, $a^{(r+1)}_{n+1} = a^{(r+1)}_n + a^{(r)}_{n+1}$.","title":"Recurrence between consecutive hypersequence generations"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyper-fibonacci-eq-fib-sub-sum","kind":"node","lean":"MetaMathlibExt.hyperFibonacci_eq_fib_sub_sum","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyper-fibonacci-eq-fib-sub-sum","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $F_k$ denote the $k$th Fibonacci number and $F^{(r)}_n$ the hyperfibonacci numbers, given by $F^{(0)}_n = F_n$ and $F^{(r+1)}_n = \\sum_{k=0}^{n} F^{(r)}_k$. For all natural numbers $r$ and $n$,","title":"Hyperfibonacci numbers in terms of Fibonacci numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/fib-add-two-mul-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.fib_add_two_mul_eq_sum_choose","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#fib-add-two-mul-eq-sum-choose","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $F_k$ denote the $k$th Fibonacci number. For every natural number $r \\ge 1$ and every natural number $n$,","title":"Fibonacci numbers as binomially weighted sums"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyper-fibonacci-recurrence","kind":"node","lean":"MetaMathlibExt.hyperFibonacci_recurrence","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyper-fibonacci-recurrence","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $F^{(r)}_n$ denote the hyperfibonacci numbers. For all natural numbers $r$ and $n$, $F^{(r)}_0 = 0$, $F^{(r)}_1 = 1$, and $F^{(r+1)}_{n+2} = F^{(r+1)}_{n+1} + F^{(r+1)}_n + \\binom{n+r+1}{r}$.","title":"Recurrence for the hyperfibonacci numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyper-lucas-eq-lucas-sub-sum","kind":"node","lean":"MetaMathlibExt.hyperLucas_eq_lucas_sub_sum","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyper-lucas-eq-lucas-sub-sum","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $L_n$ denote the Lucas numbers and let $L^{(r)}_n$ denote the hyperlucas numbers, the hypersequence of generation $r$ of $(L_n)$. For all natural numbers $r$ and $n$, in the integers,","title":"Hyperlucas numbers in terms of Lucas numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/lucas-add-two-mul-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.lucas_add_two_mul_eq_sum_choose","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#lucas-add-two-mul-eq-sum-choose","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $L_n$ denote the Lucas numbers. For every $r \\ge 1$ and every natural number $n$,","title":"Lucas numbers as binomial sums"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyper-lucas-recurrence","kind":"node","lean":"MetaMathlibExt.hyperLucas_recurrence","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyper-lucas-recurrence","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $L^{(r)}_n$ denote the hyperlucas numbers, the hypersequence of generation $r$ of the Lucas numbers. For all natural numbers $r$ and $n$, $L^{(r)}_0 = 2$, $L^{(r)}_1 = 2r + 1$, and, in the rational numbers,","title":"Recurrence for the hyperlucas numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/sum-mul-eq-hypersequence-one-mul-add-sum","kind":"node","lean":"MetaMathlibExt.sum_mul_eq_hypersequence_one_mul_add_sum","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#sum-mul-eq-hypersequence-one-mul-add-sum","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $a$ and $b$ be sequences of complex numbers and let $a^{(1)}_k = \\sum_{i=0}^{k} a_i$ be the first hypersequence (the partial sums) of $a$. Then for every natural number $n$,","title":"Abel summation by parts"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/sum-pow-mul-hypersequence-recurrence","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_hypersequence_recurrence","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#sum-pow-mul-hypersequence-recurrence","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $a$ be a sequence of complex numbers, let $a^{(r)}_k$ denote its hypersequence of generation $r$, and for natural numbers $\\ell$, $r$ and $n$ let $t^{(r)}_\\ell(n) = \\sum_{k=0}^{n} k^\\ell a^{(r)}_k$, with the convention $0^0 = 1$. Then for all natural numbers $\\ell$, $r$ and $n$,","title":"Recurrence for power-weighted sums of hypersequences"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/sum-pow-mul-hypersequence-eq","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_hypersequence_eq","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#sum-pow-mul-hypersequence-eq","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $a$ be a sequence of complex numbers, let $a^{(r)}_n$ denote its hypersequence of generation $r$, let $p_{\\ell,m}(x, y) = \\sum_{i=0}^{m} (-1)^i \\binom{m}{i} (ix + y)^\\ell$ be the alternating binomial power sum, and let $S(j, m)$ denote the Stirling numbers of the second kind. Then for all natural numbers $\\ell$, $n$ and $r$, with the convention $0^0 = 1$,","title":"Power-weighted sums of hypersequences"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/sum-pow-mul-eq","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_eq","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#sum-pow-mul-eq","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $a$ be a sequence of complex numbers, let $a^{(r)}_n$ denote its hypersequence of generation $r$, let $p_{\\ell,m}(x, y) = \\sum_{i=0}^{m} (-1)^i \\binom{m}{i} (ix + y)^\\ell$ be the alternating binomial power sum, and let $S(j, m)$ denote the Stirling numbers of the second kind. Then for all natural numbers $\\ell$ and $n$, with the convention $0^0 = 1$,","title":"Power-weighted sums of a sequence via its hypersequences"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/alternating-binomial-power-sum-eq-sum-st","kind":"node","lean":"MetaMathlibExt.alternatingBinomialPowerSum_eq_sum_stirlingSecond","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#alternating-binomial-power-sum-eq-sum-st","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (kx + y)^\\ell$ and let $S(j, m)$ denote the Stirling numbers of the second kind. For all complex numbers $x, y$ and natural numbers $\\ell, m$: (i) (Boyadzhiev) $p_{\\ell,m}(x, y) = \\sum_{j=m}^{\\ell} \\binom{\\ell}{j} (-1)^m m!\\, S(j, m)\\, x^j y^{\\ell-j}$, where the sum is empty (hence $0$) when $\\ell < m$; (ii) (Katsuura) if $\\ell < m$ then\u2026","title":"Boyadzhiev's and Katsuura's formulas for alternating binomial power sums"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/alternating-binomial-power-sum-special-v","kind":"node","lean":"MetaMathlibExt.alternatingBinomialPowerSum_special_values","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#alternating-binomial-power-sum-special-v","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (kx + y)^\\ell$, with $0^0 = 1$, and let $S(j, m)$ denote the Stirling numbers of the second kind. For all complex numbers $x, y$ and natural numbers $\\ell, m$: (i) $p_{\\ell,m}(x, x) = (-1)^m m!\\, S(\\ell + 1, m + 1)\\, x^\\ell$, and in particular $p_{\\ell,m}(1, 1) = (-1)^m m!\\, S(\\ell + 1, m + 1)$; (ii) $p_{\\ell,m}(0, y) = 0^m y^\\ell$; (iii)\u2026","title":"Special values of alternating binomial power sums"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/alternating-binomial-power-sum-small-cas","kind":"node","lean":"MetaMathlibExt.alternatingBinomialPowerSum_small_cases","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#alternating-binomial-power-sum-small-cas","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (kx + y)^\\ell$, with $0^0 = 1$. For all complex numbers $x, y$ and natural numbers $\\ell, m$: (i) $p_{\\ell,0}(x, y) = y^\\ell$; (ii) $p_{\\ell,1}(x, y) = y^\\ell - (x + y)^\\ell$; (iii) $p_{\\ell,2}(x, y) = y^\\ell - 2(x + y)^\\ell + (2x + y)^\\ell$; (iv) if $\\ell \\ge 1$, then\u2026","title":"Small cases of alternating binomial power sums"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/alternating-binomial-power-sum-recurrenc","kind":"node","lean":"MetaMathlibExt.alternatingBinomialPowerSum_recurrences","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#alternating-binomial-power-sum-recurrenc","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (kx + y)^\\ell$. For all complex numbers $x, y$ and natural numbers $\\ell, m$ (sums over an empty range are $0$, and $m - 1$ means $0$ when $m = 0$): (i) $p_{\\ell,m+1}(x, y) = p_{\\ell,m}(x, y) - p_{\\ell,m}(x, x + y)$; (ii) $p_{\\ell,m}(x, x + y) = \\sum_{j=m}^{\\ell} \\binom{\\ell}{j} p_{j,m}(x, y)\\, x^{\\ell-j}$; (iii)\u2026","title":"Recurrences for alternating binomial power sums"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/carlitz-sum-deriv-and-recurrence","kind":"node","lean":"MetaMathlibExt.carlitzSum_deriv_and_recurrence","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#carlitz-sum-deriv-and-recurrence","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (kx + y)^\\ell$ and, for a complex number $n$, let $c_{\\ell,m}(n) = p_{\\ell,m}(1, n + 1)$. For natural numbers $\\ell, m \\ge 1$ and every complex number $n$, the complex derivative of $c_{\\ell,m}$ at $n$ is $\\ell\\, c_{\\ell-1,m}(n)$, and $c_{\\ell,m}(n) = (m + n + 1)\\, c_{\\ell-1,m}(n) - m\\, c_{\\ell-1,m-1}(n)$.","title":"Derivative and recurrence of the shifted alternating binomial power sums"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/sum-pow-eq-sum-stirling-second","kind":"node","lean":"MetaMathlibExt.sum_pow_eq_sum_stirlingSecond","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sums of "],["math","\\(m\\)"],["text","th powers via Stirling numbers of the second kind"]],"source":"Sums of $m$th powers via Stirling numbers of the second kind"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#sum-pow-eq-sum-stirling-second","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"For all natural numbers $n$ and $m$,","title":"Sums of $m$th powers via Stirling numbers of the second kind"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/ring-choose-sub-eq-sum","kind":"node","lean":"MetaMathlibExt.ringChoose_sub_eq_sum","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#ring-choose-sub-eq-sum","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"For all natural numbers $m$, $n$ and $k$,","title":"Alternating binomial expansion of a shifted binomial coefficient"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/succ-pow-eq-sum-stirling-second","kind":"node","lean":"MetaMathlibExt.succ_pow_eq_sum_stirlingSecond","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Powers of "],["math","\\(k + 1\\)"],["text"," via Stirling numbers of the second kind"]],"source":"Powers of $k + 1$ via Stirling numbers of the second kind"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#succ-pow-eq-sum-stirling-second","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"For all natural numbers $k$ and $\\ell$,","title":"Powers of $k + 1$ via Stirling numbers of the second kind"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/sum-pow-mul-fib-eq","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_fib_eq","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#sum-pow-mul-fib-eq","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $F_k$ denote the Fibonacci numbers and let $p_{\\ell,m}(x, y) = \\sum_{i=0}^{m} (-1)^i \\binom{m}{i} (ix + y)^\\ell$ be the alternating binomial power sum. Then for all natural numbers $\\ell$ and $n$, with the convention $0^0 = 1$,","title":"Power-weighted sums of Fibonacci numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/sum-pow-mul-lucas-eq","kind":"node","lean":"MetaMathlibExt.sum_pow_mul_lucas_eq","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#sum-pow-mul-lucas-eq","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $L_k$ denote the Lucas numbers, so $L_0 = 2$, $L_1 = 1$ and $L_{k+2} = L_{k+1} + L_k$, and let $p_{\\ell,m}(x, y) = \\sum_{i=0}^{m} (-1)^i \\binom{m}{i} (ix + y)^\\ell$ be the alternating binomial power sum. Then for all natural numbers $\\ell$ and $n$, with the convention $0^0 = 1$,","title":"Power-weighted sums of Lucas numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-esymm-recurrence-closed-form-ro","kind":"node","lean":"MetaMathlibExt.hyperseq_esymm_recurrence_closed_form_row_sums","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence, closed form and row sums of "],["math","\\(e_{j,k}(n)\\)"]],"source":"Recurrence, closed form and row sums of $e_{j,k}(n)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-esymm-recurrence-closed-form-ro","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $n \\in \\mathbb{N}$ and let $e_{j,k}(n)$ be the elementary symmetric polynomial of degree $k$ in $n + 1, \\dots, n + j$. Write $[j, \\ell]$ for the unsigned Stirling numbers of the first kind and $x^{\\overline{j}} = x (x + 1) \\cdots (x + j - 1)$ for the rising factorial. Then $e_{0,k}(n) = \\delta_{0,k}$ for all $k$; $e_{j+1,k+1}(n) = e_{j,k+1}(n) + (n + j + 1) \\, e_{j,k}(n)$ for all $j, k \\in \\mathbb{N}$;\u2026","title":"Recurrence, closed form and row sums of $e_{j,k}(n)$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-esymm-zero-recurrence-stirling","kind":"node","lean":"MetaMathlibExt.hyperseq_esymm_zero_recurrence_stirling","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-esymm-zero-recurrence-stirling","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"For $j, k \\in \\mathbb{N}$ let $e_{j,k}$ be the elementary symmetric polynomial of degree $k$ evaluated at $1, 2, \\dots, j$ (the source's $e_{j,k}(0)$), and let $c(j, i)$ be the unsigned Stirling number of the first kind. Then $e_{0,k} = 1$ if $k = 0$ and $e_{0,k} = 0$ otherwise; for all $j, k$, $e_{j+1,k+1} = e_{j,k+1} + (j + 1)\\, e_{j,k}$; and for all $k \\le j$,","title":"Elementary symmetric functions of the first integers via Stirling numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-b","kind":"node","lean":"MetaMathlibExt.hyperseqB","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parisse's matrix "],["math","\\(B(n)\\)"]],"source":"Parisse's matrix $B(n)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-b","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"For an integer $n$ and $j, k \\in \\mathbb{N}$, let $e_{j,i}(n)$ be the elementary symmetric polynomial of degree $i$ in $n + 1, \\dots, n + j$. The entry $b_{j,k}(n)$ of $B(n)$ is $(-1)^k e_{j,j-k}(n)$ for $k \\le j$ and $0$ for $k > j$; it is an integer.","title":"Parisse's matrix $B(n)$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-b-one","kind":"node","lean":"MetaMathlibExt.hyperseqB_one","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Row one of Parisse's matrix "],["math","\\(B(n)\\)"]],"source":"Row one of Parisse's matrix $B(n)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-b-one","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"For every integer $n$, the entries $b_{j,k}(n)$ of $B(n)$ satisfy $b_{1,0}(n) = n + 1$ and $b_{1,1}(n) = -1$.","title":"Row one of Parisse's matrix $B(n)$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-b-row-three","kind":"node","lean":"MetaMathlibExt.hyperseqB_row_three","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Row three of Parisse's matrix "],["math","\\(B(1)\\)"]],"source":"Row three of Parisse's matrix $B(1)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-b-row-three","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"The entries $b_{3,0}(1), b_{3,1}(1), b_{3,2}(1), b_{3,3}(1)$ of $B(1)$ are $24, -26, 9, -1$.","title":"Row three of Parisse's matrix $B(1)$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/mitrinovic-r","kind":"node","lean":"MetaMathlibExt.mitrinovicR","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#mitrinovic-r","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"For complex numbers $a$ and $b$, Mitrinovi\u0107's numbers $R^k_j(a, b)$, $j, k \\in \\mathbb{N}$ (lower index $j$, upper index $k$), are defined by $R^k_0(a, b) = 1$ if $k = 0$ and $R^k_0(a, b) = 0$ otherwise, and, for all $j, k$, by $R^0_{j+1}(a, b) = -(a + b j)\\, R^0_j(a, b)$ and $R^{k+1}_{j+1}(a, b) = R^k_j(a, b) - (a + b j)\\, R^{k+1}_j(a, b)$.","title":"Mitrinovi\u0107's numbers"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/mitrinovic-r-one","kind":"node","lean":"MetaMathlibExt.mitrinovicR_one","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#mitrinovic-r-one","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"fully_proved","summary":"For all complex numbers $a, b$, Mitrinovi\u0107's numbers satisfy $R^0_1(a, b) = -a$ and $R^1_1(a, b) = 1$.","title":"Mitrinovi\u0107's numbers with lower index one"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/factorial-mul-hypersequence-eq-sum-esymm","kind":"node","lean":"MetaMathlibExt.factorial_mul_hypersequence_eq_sum_esymm","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#factorial-mul-hypersequence-eq-sum-esymm","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $a$ be a sequence of complex numbers and let $j, n \\in \\mathbb{N}$. Write $a^{(r)}_n$ for the hypersequence of generation $r$ of $a$, write $e_{j,k}(n)$ for the elementary symmetric polynomial of degree $k$ in $n + 1, \\dots, n + j$, and put $t_i(n) = \\sum_{k=0}^{n} k^i a_k$, with $0^0 = 1$. Then $j! \\, a^{(j+1)}_n = (-1)^j \\sum_{k=0}^{j} (-1)^k e_{j,k}(n) \\, t_{j-k}(n)$ and\u2026","title":"Hypersequences through elementary symmetric polynomials"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-c-inv-eq-hyperseq-b-div-factori","kind":"node","lean":"MetaMathlibExt.hyperseqC_inv_eq_hyperseqB_div_factorial","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","The inverse of Parisse's matrix "],["math","\\(C(n)\\)"]],"source":"The inverse of Parisse's matrix $C(n)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-c-inv-eq-hyperseq-b-div-factori","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $n \\in \\mathbb{N}$. Let $e_{j,i}(n)$ be the elementary symmetric polynomial of degree $i$ in $n + 1, \\dots, n + j$, let $b_{j,k}(n) = (-1)^k e_{j,j-k}(n)$ for $k \\le j$ and $b_{j,k}(n) = 0$ for $k > j$, let $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (k x + y)^\\ell$ for complex $x, y$, and let $c_{\\ell,m}(n) = p_{\\ell,m}(1, n + 1) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (k + n + 1)^\\ell$. Then for all\u2026","title":"The inverse of Parisse's matrix $C(n)$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-b-inv-eq-hyperseq-c-div-factori","kind":"node","lean":"MetaMathlibExt.hyperseqB_inv_eq_hyperseqC_div_factorial","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","The inverse of Parisse's matrix "],["math","\\(B(n)\\)"]],"source":"The inverse of Parisse's matrix $B(n)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-b-inv-eq-hyperseq-c-div-factori","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $n \\in \\mathbb{N}$. Let $e_{j,i}(n)$ be the elementary symmetric polynomial of degree $i$ in $n + 1, \\dots, n + j$, let $b_{j,k}(n) = (-1)^k e_{j,j-k}(n)$ for $k \\le j$ and $b_{j,k}(n) = 0$ for $k > j$, let $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (k x + y)^\\ell$ for complex $x, y$, and let $c_{\\ell,m}(n) = p_{\\ell,m}(1, n + 1) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (k + n + 1)^\\ell$. Then for all\u2026","title":"The inverse of Parisse's matrix $B(n)$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/sum-hyperseq-b-mul-hyperseq-c","kind":"node","lean":"MetaMathlibExt.sum_hyperseqB_mul_hyperseqC","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#sum-hyperseq-b-mul-hyperseq-c","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"For $n, j, m \\in \\mathbb{N}$ let $b_{j,i}(n) = (-1)^i e_{j-i}(n + 1, \\dots, n + j)$ for $i \\le j$ be the entries of Parisse's matrix $B(n)$, where $e_d$ is the elementary symmetric polynomial of degree $d$, and let $c_{i,m}(n) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (k + n + 1)^i$ be the entries of $C(n)$. Then $\\sum_{i=0}^{j} b_{j,i}(n)\\, c_{i,m}(n)$ equals $j!$ if $j = m$ and $0$ otherwise.","title":"The product of Parisse's matrices B(n) and C(n)"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-b-recurrence-closed-form-row-su","kind":"node","lean":"MetaMathlibExt.hyperseqB_recurrence_closed_form_row_sums","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence, closed form and row sums of Parisse's "],["math","\\(b_{j,k}(n)\\)"]],"source":"Recurrence, closed form and row sums of Parisse's $b_{j,k}(n)$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-b-recurrence-closed-form-row-su","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $n \\in \\mathbb{N}$ and let $b_{j,k}(n)$ be the entries of Parisse's matrix $B(n)$. Write $[j, \\ell]$ for the unsigned Stirling numbers of the first kind and $x^{\\overline{j}} = x (x + 1) \\cdots (x + j - 1)$ for the rising factorial. Then $b_{0,k}(n) = \\delta_{0,k}$ for all $k$; for all $j, k \\in \\mathbb{N}$, $b_{j+1,0}(n) = (n + j + 1) \\, b_{j,0}(n)$ and\u2026","title":"Recurrence, closed form and row sums of Parisse's $b_{j,k}(n)$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-b-zero-recurrence-stirling","kind":"node","lean":"MetaMathlibExt.hyperseqB_zero_recurrence_stirling","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parisse's "],["math","\\(b_{j,k}(0)\\)"],["text"," and Stirling numbers of the first kind"]],"source":"Parisse's $b_{j,k}(0)$ and Stirling numbers of the first kind"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-b-zero-recurrence-stirling","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $b_{j,k}(0)$ be the entries of Parisse's matrix $B(0)$ and write $[j, \\ell]$ for the unsigned Stirling numbers of the first kind. Then $b_{0,k}(0) = \\delta_{0,k}$ for all $k$; for all $j, k \\in \\mathbb{N}$, $b_{j+1,0}(0) = (j + 1) \\, b_{j,0}(0)$ and $b_{j+1,k+1}(0) = (j + 1) \\, b_{j,k+1}(0) - b_{j,k}(0)$; and for all $j, k \\in \\mathbb{N}$,\u2026","title":"Parisse's $b_{j,k}(0)$ and Stirling numbers of the first kind"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/hyperseq-entries-eq-sum-at-neg-one","kind":"node","lean":"MetaMathlibExt.hyperseq_entries_eq_sum_at_neg_one","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parisse's coefficients at "],["math","\\(n\\)"],["text"," from their values at "],["math","\\(-1\\)"]],"source":"Parisse's coefficients at $n$ from their values at $-1$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#hyperseq-entries-eq-sum-at-neg-one","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"Let $n \\in \\mathbb{Z}$. For $j, k \\in \\mathbb{N}$ let $e_{j,k}(n)$ be the elementary symmetric polynomial of degree $k$ in $n + 1, \\dots, n + j$, let $b_{j,k}(n) = (-1)^k e_{j,j-k}(n)$ for $k \\le j$ and $b_{j,k}(n) = 0$ for $k > j$, let $p_{\\ell,m}(x, y) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (k x + y)^\\ell$ for complex $x, y$, with $0^0 = 1$, and let\u2026","title":"Parisse's coefficients at $n$ from their values at $-1$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/mitrinovic-r-zero-eq","kind":"node","lean":"MetaMathlibExt.mitrinovicR_zero_eq","parent":27095,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mitrinovi\u0107's numbers at "],["math","\\(a = 0\\)"]],"source":"Mitrinovi\u0107's numbers at $a = 0$"},"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#mitrinovic-r-zero-eq","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"For every complex number $b$, Mitrinovi\u0107's numbers $R^k_j(0, b)$ satisfy the following. First, for every $k \\in \\mathbb{N}$, $R^k_0(0, b) = 1$ if $k = 0$ and $R^k_0(0, b) = 0$ otherwise. Second, for all $j, k \\in \\mathbb{N}$, $R^{k+1}_{j+1}(0, b) = R^k_j(0, b) + (b - b(j + 1))\\, R^{k+1}_j(0, b)$. Third, for all $j, k \\in \\mathbb{N}$, $R^k_j(0, b) = (-b)^{j-k} c(j, k)$, where $c(j, k)$ is the unsigned Stirling\u2026","title":"Mitrinovi\u0107's numbers at $a = 0$"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/mitrinovic-r-eq-sum-mitrinovic-r-zero","kind":"node","lean":"MetaMathlibExt.mitrinovicR_eq_sum_mitrinovicR_zero","parent":27095,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences/index.html#mitrinovic-r-eq-sum-mitrinovic-r-zero","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/hypersequences","status":"can_prove","summary":"For all complex numbers $a, b$ and all $j, k \\in \\mathbb{N}$, Mitrinovi\u0107's numbers satisfy","title":"Binomial expansion of Mitrinovi\u0107's numbers in the first parameter"},{"data_url":"scopes/Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform.json","id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform","kind":"scope","navigation":"module","parent":27081,"result":false,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/index.html","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform","status":"fully_proved","title":"INVERT transform"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/bell-convolution-linear-recurrence-of-ch","kind":"node","lean":"MetaMathlibExt.bell_convolution_linear_recurrence_of_charZero","parent":27140,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/index.html#bell-convolution-linear-recurrence-of-ch","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero and let $(c_m)_{m \\ge 0}$ and $(y_n)_{n \\ge 0}$ be sequences in $K$ such that for every $n \\ge 0$","title":"Linear recurrence for INVERT convolution powers in characteristic zero"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/bell-convolution-linear-recurrence","kind":"node","lean":"MetaMathlibExt.bell_convolution_linear_recurrence","parent":27140,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/index.html#bell-convolution-linear-recurrence","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform","status":"fully_proved","summary":"Let $(c_m)_{m \\ge 0}$ and $(y_n)_{n \\ge 0}$ be real sequences such that for every $n \\ge 0$","title":"Linear recurrence for convolution powers of an INVERT sequence"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/bell-convolution-linear-recurrence-parti","kind":"node","lean":"MetaMathlibExt.bell_convolution_linear_recurrence_partialBellPolynomial","parent":27140,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/index.html#bell-convolution-linear-recurrence-parti","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform","status":"fully_proved","summary":"Let $K$ be a field of characteristic zero, let $(c_m)_{m \\ge 0}$ and $(y_n)_{n \\ge 0}$ be sequences in $K$, and write $B_{n,k}$ for the partial Bell polynomial. Suppose that for every $n \\ge 0$","title":"Linear recurrence for INVERT convolution powers via partial Bell polynomials"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/invert-transform","kind":"node","lean":"MetaMathlibExt.invertTransform","parent":27140,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/index.html#invert-transform","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform","status":"fully_proved","summary":"The INVERT transform of an integer sequence $a = (a_n)_{n \\ge 0}$ is the integer sequence $b = (b_n)_{n \\ge 0}$ in which $b_n$ is the coefficient of $x^{n+1}$ in the inverse of $1 - \\sum_{m \\ge 0} a_m x^{m+1}$ in $\\mathbb{Z}[[x]]$ (the inverse exists because the constant coefficient is $1$). In the one-based indexing of the source, $a_n$ and $b_n$ are the terms of index $n + 1$.","title":"The INVERT transform of an integer sequence"},{"id":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/invert-transform-spec","kind":"node","lean":"MetaMathlibExt.invertTransform_spec","parent":27140,"result":true,"route":"../roadmap/Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform/index.html#invert-transform-spec","scope":"Combinatorics/riordan-arrays-transforms/sequence-transforms/invert-transform","status":"fully_proved","summary":"Let $a = (a_n)_{n \\ge 0}$ be an integer sequence and $b = (b_n)_{n \\ge 0}$ its INVERT transform, and put $A(x) = \\sum_{n \\ge 0} a_n x^{n+1}$ and $B(x) = \\sum_{n \\ge 0} b_n x^{n+1}$. Then $(1 - A(x))(1 + B(x)) = 1$ in $\\mathbb{Z}[[x]]$, that is, $1 + B(x)$ is the inverse of $1 - A(x)$.","title":"Generating function of the INVERT transform"},{"data_url":"scopes/Combinatorics/parking-functions.json","id":"Combinatorics/parking-functions","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/parking-functions/index.html","scope":"Combinatorics/parking-functions","status":"can_prove","summary":"Parking functions and their variants (partial, weakly increasing, k-Naples, metered) and statistics on them: lucky cars, descents, displacement, and lucky statistics of Stirling permutations read as parking preferences.","title":"Parking functions"},{"id":"Combinatorics/parking-functions/is-partial-parking-function","kind":"node","lean":"MetaMathlibExt.IsPartialParkingFunction","parent":27146,"result":true,"route":"../roadmap/Combinatorics/parking-functions/index.html#is-partial-parking-function","scope":"Combinatorics/parking-functions","status":"fully_proved","summary":"Let $s, t \\in \\mathbb{N}$ and number the $t$ spots $0, \\dots, t-1$. A map $f : \\{0, \\dots, s-1\\} \\to \\{0, \\dots, t-1\\}$, giving the preferred spot of each of $s$ cars, is a partial parking function if for every spot $k \\in \\{0, \\dots, t-1\\}$, at most $t - k$ cars $i$ satisfy $f(i) \\ge k$.","title":"Partial parking function"},{"id":"Combinatorics/parking-functions/is-parking-function","kind":"node","lean":"MetaMathlibExt.IsParkingFunction","parent":27146,"result":true,"route":"../roadmap/Combinatorics/parking-functions/index.html#is-parking-function","scope":"Combinatorics/parking-functions","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and number the $n$ spots $0, \\dots, n-1$. A map $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$, giving the preferred spot of each of $n$ cars, is a parking function if it is a partial parking function with $n$ cars and $n$ spots, that is, if for every spot $k$, at most $n - k$ cars $i$ satisfy $f(i) \\ge k$.","title":"Parking function"},{"id":"Combinatorics/parking-functions/is-parking-function-iff-card-filter-le","kind":"node","lean":"MetaMathlibExt.isParkingFunction_iff_card_filter_le","parent":27146,"result":true,"route":"../roadmap/Combinatorics/parking-functions/index.html#is-parking-function-iff-card-filter-le","scope":"Combinatorics/parking-functions","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, number the $n$ spots $0, \\dots, n-1$, and let $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ give the preferred spot of each of $n$ cars. Then $f$ is a parking function if and only if, for every spot $k \\in \\{0, \\dots, n-1\\}$, at least $k + 1$ cars $i$ satisfy $f(i) \\le k$.","title":"Lower-tail characterization of parking functions"},{"id":"Combinatorics/parking-functions/pollak-partial-parking-functions-card","kind":"node","lean":"MetaMathlibExt.PollakPartialParking.pollak_partial_parking_functions_card","parent":27146,"result":true,"route":"../roadmap/Combinatorics/parking-functions/index.html#pollak-partial-parking-functions-card","scope":"Combinatorics/parking-functions","status":"fully_proved","summary":"Let $s$ and $t$ be natural numbers with $0 < s \\le t$. A partial parking function with $s$ cars and $t$ spots is a map $f : \\{0, \\dots, s - 1\\} \\to \\{0, \\dots, t - 1\\}$ such that for every spot $k \\in \\{0, \\dots, t - 1\\}$ at most $t - k$ cars $i$ have $f(i) \\ge k$. The number of partial parking functions with $s$ cars and $t$ spots is $(t + 1 - s)(t + 1)^{s - 1}$.","title":"Number of partial parking functions"},{"data_url":"scopes/Combinatorics/parking-functions/k-naples-parking-functions.json","id":"Combinatorics/parking-functions/k-naples-parking-functions","kind":"scope","navigation":"module","parent":27146,"result":false,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","title":"k-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-spot","kind":"node","lean":"MetaMathlibExt.naplesSpot","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-Naples rule"]],"source":"The $k$-Naples rule"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-spot","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"On the spots $1, \\dots, n$, when the spots in a list $O$ of natural numbers are occupied, a car preferring $a$ parks in $a$ if $a$ is free; otherwise in the first free spot among those of $a - 1, \\dots, a - k$ that are at least $1$, checked in decreasing order; otherwise in the first free spot among $a + 1, \\dots, n$, checked in increasing order. The car cannot park if none of these is free, or if $a$ is not one of\u2026","title":"The $k$-Naples rule"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-park","kind":"node","lean":"MetaMathlibExt.naplesPark","parent":27151,"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-park","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Starting from a list of occupied spots, cars with preferences $a_1, \\dots, a_m \\in \\mathbb{N}$ arrive in order, and each takes the spot given by the $k$-Naples rule on the spots $1, \\dots, n$ for the spots occupied when it arrives. If every car parks, the result is the final list of occupied spots: the spots of the cars, most recent first, followed by the initially occupied spots. If some car cannot park, there is\u2026","title":"Naples parking"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/is-naples-parking-function","kind":"node","lean":"MetaMathlibExt.IsNaplesParkingFunction","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-Naples parking function"]],"source":"$k$-Naples parking function"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#is-naples-parking-function","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"A preference $\\alpha = (a_1, \\dots, a_n)$ of length $n$ is a $k$-Naples parking function if $1 \\le a_i \\le n$ for every $i$ and all $n$ cars park on the spots $1, \\dots, n$ when they arrive in order and follow the $k$-Naples rule: car $i$ takes spot $a_i$ if it is free; otherwise the first free spot among $a_i - 1, \\dots, a_i - k$, checked in decreasing order and only among spots at least $1$; otherwise the first\u2026","title":"$k$-Naples parking function"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-park-example","kind":"node","lean":"MetaMathlibExt.naplesPark_example","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(2\\)"],["text","-Naples parking example"]],"source":"A $2$-Naples parking example"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-park-example","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Under the $2$-Naples rule on the spots $1, \\dots, 7$, the cars with preferences $(6, 6, 6, 5, 5, 2, 1)$, arriving in order into an empty lot, all park, in the spots $6, 5, 4, 3, 7, 2, 1$ respectively.","title":"A $2$-Naples parking example"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/is-naples-parking-function-example","kind":"node","lean":"MetaMathlibExt.isNaplesParkingFunction_example","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(2\\)"],["text","-Naples parking function that is not "],["math","\\(1\\)"],["text","-Naples"]],"source":"A $2$-Naples parking function that is not $1$-Naples"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#is-naples-parking-function-example","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"The preference $(6, 6, 6, 5, 5, 2, 1)$ of length $7$ is a $2$-Naples parking function but not a $1$-Naples parking function.","title":"A $2$-Naples parking function that is not $1$-Naples"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/ascending-naples-count","kind":"node","lean":"MetaMathlibExt.ascendingNaplesCount","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of ascending "],["math","\\(k\\)"],["text","-Naples parking functions"]],"source":"Number of ascending $k$-Naples parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#ascending-naples-count","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $I_{n,k}$ is the number of $k$-Naples parking functions $(a_1, \\dots, a_n)$ of length $n$ that are weakly increasing, that is, $a_1 \\le a_2 \\le \\dots \\le a_n$.","title":"Number of ascending $k$-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/ascending-naples-one-count","kind":"node","lean":"MetaMathlibExt.ascendingNaplesOneCount","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of ascending "],["math","\\(k\\)"],["text","-Naples parking functions starting with "],["math","\\(1\\)"]],"source":"Number of ascending $k$-Naples parking functions starting with $1$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#ascending-naples-one-count","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"For natural numbers $n$ and $k$, $U_{n,k}$ is the number of weakly increasing $k$-Naples parking functions $(a_1, \\dots, a_n)$ of length $n$ with $a_1 = 1$; in particular $U_{0,k} = 0$.","title":"Number of ascending $k$-Naples parking functions starting with $1$"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/descending-naples-count","kind":"node","lean":"MetaMathlibExt.descendingNaplesCount","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of descending "],["math","\\(k\\)"],["text","-Naples parking functions"]],"source":"Number of descending $k$-Naples parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#descending-naples-count","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"For natural numbers $n$ and $k$, this is the number of $k$-Naples parking functions $\\alpha = (a_1, \\dots, a_n)$ of length $n$ (so $a_i \\in \\{1, \\dots, n\\}$ for all $i$) that are descending, that is, weakly decreasing: $a_1 \\ge a_2 \\ge \\dots \\ge a_n$.","title":"Number of descending $k$-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/descending-strict-naples-count","kind":"node","lean":"MetaMathlibExt.descendingStrictNaplesCount","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of descending strictly "],["math","\\(k\\)"],["text","-Naples parking functions"]],"source":"Number of descending strictly $k$-Naples parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#descending-strict-naples-count","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"For natural numbers $n$ and $k$, this is the number of preferences $\\alpha = (a_1, \\dots, a_n) \\in \\{1, \\dots, n\\}^n$ that are descending ($a_1 \\ge a_2 \\ge \\dots \\ge a_n$), are $k$-Naples parking functions, and are not $(k - 1)$-Naples parking functions. It is used for $k \\ge 1$; for $k = 0$, where $k - 1$ is read as $0$, the count is $0$.","title":"Number of descending strictly $k$-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-count-two-one","kind":"node","lean":"MetaMathlibExt.naplesCount_two_one","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Naples counts for "],["math","\\(n = 2\\)"],["text"," and "],["math","\\(k = 1\\)"]],"source":"Naples counts for $n = 2$ and $k = 1$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-count-two-one","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Among the preferences of length $2$ with entries in $\\{1, 2\\}$, exactly $3$ are weakly increasing $1$-Naples parking functions ($I_{2,1} = 3$), and exactly $2$ of these start with $1$ ($U_{2,1} = 2$); exactly $3$ are weakly decreasing $1$-Naples parking functions, and exactly $1$ is weakly decreasing, $1$-Naples and not $0$-Naples.","title":"Naples counts for $n = 2$ and $k = 1$"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/lattice-height","kind":"node","lean":"MetaMathlibExt.latticeHeight","parent":27151,"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#lattice-height","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"For a lattice path $p$ of $m$ steps, each an Up step $(1, 1)$ or a Down step $(1, -1)$, and a natural number $j$, the height of $p$ after $j$ steps is the number of Up steps minus the number of Down steps among the first $\\min(j, m)$ steps of $p$.","title":"Height of a lattice path"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/dyck-step-height","kind":"node","lean":"MetaMathlibExt.dyckStepHeight","parent":27151,"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#dyck-step-height","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"For a finite sequence $l$ of Up steps $U$ and Down steps $D$ and a natural number $j$, the height of $l$ after $j$ steps is the integer obtained as the number of $U$ minus the number of $D$ among the first $j$ steps of $l$ (among all of $l$ when $j$ exceeds its length).","title":"Height of a sequence of Dyck steps"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/is-kdyck-path","kind":"node","lean":"MetaMathlibExt.IsKDyckPath","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-Dyck path"]],"source":"$k$-Dyck path"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#is-kdyck-path","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers. A lattice path of $2n$ steps, each an Up step $(1, 1)$ or a Down step $(1, -1)$, is a $k$-Dyck path of semilength $n$ if its height after all $2n$ steps is $0$, its height after its first $j$ steps is at least $-k$ for every $j \\le 2n$, and, when $n \\ge 1$, its last step is a Down step.","title":"$k$-Dyck path"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/is-kdyck-path-example","kind":"node","lean":"MetaMathlibExt.isKDyckPath_example","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(1\\)"],["text","-Dyck path that is not a "],["math","\\(0\\)"],["text","-Dyck path"]],"source":"A $1$-Dyck path that is not a $0$-Dyck path"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#is-kdyck-path-example","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Write $U$ for an Up step $(1, 1)$ and $D$ for a Down step $(1, -1)$. The path $DUUD$ is a $1$-Dyck path of semilength $2$ but not a $0$-Dyck path, and the path $UUD$ has height $1$ after its three steps.","title":"A $1$-Dyck path that is not a $0$-Dyck path"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/succ","kind":"node","lean":"MetaMathlibExt.IsNaplesParkingFunction.succ","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every "],["math","\\(k\\)"],["text","-Naples parking function is "],["math","\\((k + 1)\\)"],["text","-Naples"]],"source":"Every $k$-Naples parking function is $(k + 1)$-Naples"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#succ","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For all natural numbers $k$ and $n$, every $k$-Naples parking function $\\alpha$ of length $n$ is also a $(k + 1)$-Naples parking function.","title":"Every $k$-Naples parking function is $(k + 1)$-Naples"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-spot-int","kind":"node","lean":"MetaMathlibExt.naplesSpotInt","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(k\\)"],["text","-Naples rule on "],["math","\\(\\mathbb{Z}\\)"]],"source":"The $k$-Naples rule on $\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-spot-int","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"When the spots in a finite list $O$ of integers are occupied, a car preferring $a \\in \\mathbb{Z}$ parks in $a$ if $a$ is free; otherwise in the first free spot among $a - 1, \\dots, a - k$, checked in decreasing order with no lower bound; otherwise in the first free spot among $a + 1, \\dots, a + |O| + 1$, checked in increasing order (one of them is free, so this is the first free spot after $a$).","title":"The $k$-Naples rule on $\\mathbb{Z}$"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-park-int","kind":"node","lean":"MetaMathlibExt.naplesParkInt","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Naples parking on "],["math","\\(\\mathbb{Z}\\)"]],"source":"Naples parking on $\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-park-int","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Starting from a list of occupied spots in $\\mathbb{Z}$, cars with integer preferences $a_1, \\dots, a_m$ arrive in order, and each takes the spot given by the $k$-Naples rule on $\\mathbb{Z}$ for the spots occupied when it arrives. The result is the final list of occupied spots: the spots of the cars, most recent first, followed by the initially occupied spots.","title":"Naples parking on $\\mathbb{Z}$"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-park-int-example","kind":"node","lean":"MetaMathlibExt.naplesParkInt_example","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(2\\)"],["text","-Naples parking example on "],["math","\\(\\mathbb{Z}\\)"]],"source":"A $2$-Naples parking example on $\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-park-int-example","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Under the $2$-Naples rule on $\\mathbb{Z}$, the cars with preferences $(4, 2, 2, 2)$, arriving in order with no spot occupied, park in the spots $4, 2, 1, 0$ respectively.","title":"A $2$-Naples parking example on $\\mathbb{Z}$"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-park-int-example-three","kind":"node","lean":"MetaMathlibExt.naplesParkInt_example_three","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(3\\)"],["text","-Naples parking example on "],["math","\\(\\mathbb{Z}\\)"]],"source":"A $3$-Naples parking example on $\\mathbb{Z}$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-park-int-example-three","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Under the $3$-Naples rule on $\\mathbb{Z}$, the cars with preferences $(1, 2, 2, 5, 7, 7, 7, 3)$, arriving in order with no spot occupied, park in the spots $1, 2, 0, 5, 7, 6, 4, 3$ respectively.","title":"A $3$-Naples parking example on $\\mathbb{Z}$"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-park-int-succ-mem-icc","kind":"node","lean":"MetaMathlibExt.naplesParkInt_succ_mem_Icc","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","One more backward step keeps the cars in "],["math","\\(\\{0, \\dots, n\\}\\)"]],"source":"One more backward step keeps the cars in $\\{0, \\dots, n\\}$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-park-int-succ-mem-icc","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"Let $(a_1, \\dots, a_n)$ be integers with $1 \\le a_j \\le n$ for every $j$. If, under the $k$-Naples rule on $\\mathbb{Z}$, the cars with these preferences, arriving in order with no spot occupied, occupy exactly the spots $1, \\dots, n$, then under the $(k + 1)$-Naples rule on $\\mathbb{Z}$ every car parks in a spot $s$ with $0 \\le s \\le n$.","title":"One more backward step keeps the cars in $\\{0, \\dots, n\\}$"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/descending-naples-count-eq-card-k-dyck-p","kind":"node","lean":"MetaMathlibExt.descendingNaplesCount_eq_card_kDyckPath","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Descending "],["math","\\(k\\)"],["text","-Naples parking functions and "],["math","\\(k\\)"],["text","-Dyck paths"]],"source":"Descending $k$-Naples parking functions and $k$-Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#descending-naples-count-eq-card-k-dyck-p","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $k \\ge 1$, the number of weakly decreasing $k$-Naples parking functions of length $n$ equals the number of $k$-Dyck paths of length $n$: sequences of $2n$ Up and Down steps that end at height $0$, never go below height $-k$, and end with a Down step.","title":"Descending $k$-Naples parking functions and $k$-Dyck paths"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/descending-naples-count-eq-card-dyck-pat","kind":"node","lean":"MetaMathlibExt.descendingNaplesCount_eq_card_dyckPath","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Descending "],["math","\\(k\\)"],["text","-Naples parking functions and Dyck paths"]],"source":"Descending $k$-Naples parking functions and Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#descending-naples-count-eq-card-dyck-pat","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $k \\ge 1$, the number of weakly decreasing $k$-Naples parking functions of length $n$ equals the number of Dyck paths of semilength $n + k$ (that is, with $2(n + k)$ steps) whose first $k$ steps are Up and whose last $k + 1$ steps are Down.","title":"Descending $k$-Naples parking functions and Dyck paths"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/descending-strict-naples-count-eq-card-d","kind":"node","lean":"MetaMathlibExt.descendingStrictNaplesCount_eq_card_dyckPath","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Descending strictly "],["math","\\(k\\)"],["text","-Naples parking functions and Dyck paths with an early return"]],"source":"Descending strictly $k$-Naples parking functions and Dyck paths with an early return"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#descending-strict-naples-count-eq-card-d","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For $n \\ge 1$ and $k \\ge 1$, the number of descending strictly $k$-Naples parking functions of length $n$ (descending preferences that are $k$-Naples but not $(k - 1)$-Naples) equals the number of Dyck paths of semilength $n + k$, that is, with $2(n + k)$ Up and Down steps, whose first $k$ steps are Up, whose last $k + 1$ steps are Down, and whose height after $j$ steps is $0$ for some $j$ with $0 < j < 2(n + k)$.","title":"Descending strictly $k$-Naples parking functions and Dyck paths with an early return"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/ascending-naples-count-recurrence","kind":"node","lean":"MetaMathlibExt.ascendingNaplesCount_recurrence","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for ascending "],["math","\\(k\\)"],["text","-Naples parking functions"]],"source":"Recurrence for ascending $k$-Naples parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#ascending-naples-count-recurrence","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"Let $I_{n,k}$ be the number of weakly increasing $k$-Naples parking functions of length $n$, let $U_{n,k}$ be the number of those whose first entry is $1$, and let $C_k$ be the $k$-th Catalan number. For all natural numbers $n$ and $k$ with $1 \\le k \\le n - 1$,","title":"Recurrence for ascending $k$-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/ascending-naples-one-count-one-eq-fine-n","kind":"node","lean":"MetaMathlibExt.ascendingNaplesOneCount_one_eq_fineNumber","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ascending "],["math","\\(1\\)"],["text","-Naples parking functions starting with "],["math","\\(1\\)"],["text"," and Fine numbers"]],"source":"Ascending $1$-Naples parking functions starting with $1$ and Fine numbers"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#ascending-naples-one-count-one-eq-fine-n","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For every natural number $n$, the number $U_{n,1}$ of weakly increasing $1$-Naples parking functions of length $n$ whose first entry is $1$ equals the Fine number $F(n + 1)$, where the Fine numbers $1, 0, 1, 2, 6, 18, \\dots$ are indexed so that $F(0) = 1$ and $F(1) = 0$ (OEIS A000957).","title":"Ascending $1$-Naples parking functions starting with $1$ and Fine numbers"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/ascending-naples-count-one-eq-sum","kind":"node","lean":"MetaMathlibExt.ascendingNaplesCount_one_eq_sum","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting ascending "],["math","\\(1\\)"],["text","-Naples parking functions"]],"source":"Counting ascending $1$-Naples parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#ascending-naples-count-one-eq-sum","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For every natural number $n$, let $I_{n,1}$ be the number of weakly increasing $1$-Naples parking functions of length $n$. Then","title":"Counting ascending $1$-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/ascending-naples-generating-function","kind":"node","lean":"MetaMathlibExt.ascendingNaples_generatingFunction","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generating functions for ascending "],["math","\\(k\\)"],["text","-Naples parking functions"]],"source":"Generating functions for ascending $k$-Naples parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#ascending-naples-generating-function","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"Let $I_{n,k}$ be the number of weakly increasing $k$-Naples parking functions of length $n$, let $U_{n,k}$ be the number of those whose first entry is $1$, and let $C_k$ be the $k$-th Catalan number. Write $I_k(x) = \\sum_{n \\ge 0} I_{n,k} x^n$ and $U_k(x) = \\sum_{n \\ge 0} U_{n,k} x^n$, formal power series with natural number coefficients. For every natural number $k \\ge 1$,","title":"Generating functions for ascending $k$-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/descending-naples-count-eq-choose","kind":"node","lean":"MetaMathlibExt.descendingNaplesCount_eq_choose","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting descending "],["math","\\(k\\)"],["text","-Naples parking functions"]],"source":"Counting descending $k$-Naples parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#descending-naples-count-eq-choose","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $k \\ge 1$, the number of weakly decreasing $k$-Naples parking functions of length $n$ is $\\binom{2n-1}{n} - \\binom{2n-1}{n+k+1}$, and the number of weakly decreasing strictly $k$-Naples parking functions of length $n$ (those that are $k$-Naples but not $(k-1)$-Naples) is $\\frac{k+1}{n} \\binom{2n}{n+k+1}$.","title":"Counting descending $k$-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/preference-path","kind":"node","lean":"MetaMathlibExt.preferencePath","parent":27151,"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#preference-path","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Let $\\alpha = (a_1, \\dots, a_n)$ be a sequence of natural numbers. Its lattice path is the sequence of $2n$ steps, each an Up step $(1, 1)$ or a Down step $(1, -1)$, obtained by taking, for each spot $s = 1, \\dots, n$ in turn, one Up step for every index $i$ with $a_i = s$ followed by one Down step, and then appending Down steps until there are $2n$ steps. When every $a_i$ lies in $\\{1, \\dots, n\\}$ this gives\u2026","title":"Lattice path of a parking preference"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/preference-path-example","kind":"node","lean":"MetaMathlibExt.preferencePath_example","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","The lattice path of "],["math","\\((1, 3, 3, 5, 6, 6)\\)"]],"source":"The lattice path of $(1, 3, 3, 5, 6, 6)$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#preference-path-example","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"The lattice path of the parking preference $(1, 3, 3, 5, 6, 6)$ is $\\mathrm{UDDUUDDUDUUD}$, where $\\mathrm{U}$ is an Up step and $\\mathrm{D}$ a Down step.","title":"The lattice path of $(1, 3, 3, 5, 6, 6)$"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/naples-park-drop-set-is-some","kind":"node","lean":"MetaMathlibExt.naplesPark_drop_set_isSome","parent":27151,"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#naples-park-drop-set-is-some","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"Let $(a_1, \\dots, a_n)$ be a parking preference with $1 \\le a_j \\le n$ for every $j$, with which all $n$ cars, arriving in order into an empty lot, park under the $k$-Naples rule on the spots $1, \\dots, n$, car $j$ in spot $d_j$. Let $1 \\le l \\le i \\le n$, and let $q$ be a spot with $1 \\le q < d_l$ and $q \\ne d_j$ for every $j \\le i$ with $j \\ne l$. If exactly the spots $d_1, \\dots, d_i$, with $d_l$ replaced by\u2026","title":"Remaining cars still park after moving a car to an earlier spot"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/is-naples-parking-function-iff-of-monoto","kind":"node","lean":"MetaMathlibExt.isNaplesParkingFunction_iff_of_monotone","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lattice-path criterion for ascending "],["math","\\(k\\)"],["text","-Naples parking functions"]],"source":"Lattice-path criterion for ascending $k$-Naples parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#is-naples-parking-function-iff-of-monoto","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"Let $\\alpha = (a_1, \\dots, a_n)$ be a weakly increasing preference with $1 \\le a_i \\le n$ for every $i$, and let $h_j$ be the height after the first $j$ steps of the lattice path of $\\alpha$, which has $2n$ steps. Then $\\alpha$ is a $k$-Naples parking function if and only if for every $j$ with $1 \\le j \\le 2n$ and $h_j < 0$ there is a $j'$ with $j < j' \\le j + 2k$, $j' \\le 2n$ and $h_{j'} > 0$.","title":"Lattice-path criterion for ascending $k$-Naples parking functions"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/forall-perm-is-naples-parking-function-i","kind":"node","lean":"MetaMathlibExt.forall_perm_isNaplesParkingFunction_iff","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","When every rearrangement of a preference is "],["math","\\(k\\)"],["text","-Naples"]],"source":"When every rearrangement of a preference is $k$-Naples"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#forall-perm-is-naples-parking-function-i","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"Let $\\alpha = (a_1, \\dots, a_n) \\in \\{1, \\dots, n\\}^n$ and let $P$ be its lattice path: for each spot $s = 1, \\dots, n$ in turn, one Up step for every car preferring $s$, then one Down step (this is also the lattice path of the ascending rearrangement of $\\alpha$). The height of $P$ after $m$ steps is the number of Up steps minus the number of Down steps among its first $m$ steps. Then every rearrangement of\u2026","title":"When every rearrangement of a preference is $k$-Naples"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/ascending-naples-count-eq-card-dyck-path","kind":"node","lean":"MetaMathlibExt.ascendingNaplesCount_eq_card_dyckPath","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ascending "],["math","\\(k\\)"],["text","-Naples parking functions and Dyck paths"]],"source":"Ascending $k$-Naples parking functions and Dyck paths"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#ascending-naples-count-eq-card-dyck-path","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $k$, the number of weakly increasing $k$-Naples parking functions of length $n$ equals the number of Dyck paths of semilength $n + k$ (that is, with $2(n + k)$ steps) whose first $k$ steps are Up, whose last $k + 1$ steps are Down, and which satisfy the following condition. Writing $h_j$ for the height after the first $j$ steps, whenever the $m$-th step is a Down step that is\u2026","title":"Ascending $k$-Naples parking functions and Dyck paths"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/left-spine-length","kind":"node","lean":"MetaMathlibExt.leftSpineLength","parent":27151,"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#left-spine-length","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"The number of nodes on the left spine of a binary tree: the root, its left child, that child's left child, and so on (`0` for the empty tree).","title":"Left spine length of a binary tree"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/left-spine-length-examples","kind":"node","lean":"MetaMathlibExt.leftSpineLength_examples","parent":27151,"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#left-spine-length-examples","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"The empty binary tree has left spine length $0$, and the binary tree with three nodes, a root whose left child and right child have no children, has left spine length $2$.","title":"Left spine lengths of small binary trees"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/descending-strict-naples-count-eq-card-b","kind":"node","lean":"MetaMathlibExt.descendingStrictNaplesCount_eq_card_binaryTree","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Descending strictly "],["math","\\(k\\)"],["text","-Naples parking functions and binary trees"]],"source":"Descending strictly $k$-Naples parking functions and binary trees"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#descending-strict-naples-count-eq-card-b","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For every $n$ and every $k \\ge 1$, the number of descending strictly $k$-Naples parking functions of length $n$ (descending preferences that are $k$-Naples but not $(k - 1)$-Naples) equals the number of binary trees $t$ with $n + k$ nodes such that the left spine of $t$ (the root, its left child, that child's left child, and so on) has at least $k$ nodes and the left spine of the right subtree of $t$ has at least\u2026","title":"Descending strictly $k$-Naples parking functions and binary trees"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/inscribed-polygon-edges","kind":"node","lean":"MetaMathlibExt.inscribedPolygonEdges","parent":27151,"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#inscribed-polygon-edges","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Let $R$ be a set of vertices of the convex $s$-gon with vertices $0, 1, \\dots, s - 1$. The edges of the polygon with vertex set $R$ are the pairs $(a, b)$ of elements of $R$ with $a < b$ such that either no element of $R$ lies strictly between $a$ and $b$, or $a$ is the least and $b$ the greatest element of $R$.","title":"Edges of an inscribed polygon"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/is-inner-polygon-dissection","kind":"node","lean":"MetaMathlibExt.IsInnerPolygonDissection","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(r\\)"],["text","-in-"],["math","\\(s\\)"],["text"," dissection"]],"source":"$r$-in-$s$ dissection"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#is-inner-polygon-dissection","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"Let $D$ be a set of pairs $(a, b)$ of vertices of the convex $s$-gon with vertices $0, 1, \\dots, s - 1$, and let $R$ be a set of its vertices. Then $D$ is an $r$-in-$s$ dissection with inner polygon $R$ when all of the following hold: the number of elements of $D$ plus $r$ equals $s$, and $R$ has $r$ elements; every $(a, b) \\in D$ is a diagonal, that is, $a + 2 \\le b$ and $(a, b) \\ne (0, s - 1)$; no $(a, b)$ and\u2026","title":"$r$-in-$s$ dissection"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/is-inner-polygon-dissection-examples","kind":"node","lean":"MetaMathlibExt.isInnerPolygonDissection_examples","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Examples of "],["math","\\(r\\)"],["text","-in-"],["math","\\(s\\)"],["text"," dissections"]],"source":"Examples of $r$-in-$s$ dissections"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#is-inner-polygon-dissection-examples","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"fully_proved","summary":"For the convex polygons with vertices $0, 1, \\dots, s - 1$: the square ($s = 4$) with no diagonals is a $4$-in-$4$ dissection whose quadrilateral has all four vertices; the pentagon ($s = 5$) with the single diagonal $(0, 2)$ is a $4$-in-$5$ dissection with quadrilateral on the vertices $0, 2, 3, 4$, but not with quadrilateral on the vertices $0, 1, 2, 3$.","title":"Examples of $r$-in-$s$ dissections"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/mul-descending-strict-naples-count-eq-ca","kind":"node","lean":"MetaMathlibExt.mul_descendingStrictNaplesCount_eq_card_dissection","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Descending strictly "],["math","\\(k\\)"],["text","-Naples parking functions and polygon dissections"]],"source":"Descending strictly $k$-Naples parking functions and polygon dissections"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#mul-descending-strict-naples-count-eq-ca","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"Let $n$ and $k \\ge 1$ be natural numbers and let $s = n + k + 1$. Then $s$ times the number of weakly decreasing preferences of length $n$ that are $k$-Naples but not $(k - 1)$-Naples parking functions equals the number of triples $(D, R, e)$ such that the set of diagonals $D$ and the vertex set $R$ form a $(2k + 2)$-in-$s$ dissection of the convex $s$-gon with vertices $0, \\dots, s - 1$, and $e$ is an edge of the\u2026","title":"Descending strictly $k$-Naples parking functions and polygon dissections"},{"id":"Combinatorics/parking-functions/k-naples-parking-functions/descending-strict-naples-count-eq-card-n","kind":"node","lean":"MetaMathlibExt.descendingStrictNaplesCount_eq_card_noncrossingPartition","parent":27151,"reading_title":{"format":"inline-parts-v1","parts":[["text","Descending strictly "],["math","\\(k\\)"],["text","-Naples parking functions and noncrossing partitions"]],"source":"Descending strictly $k$-Naples parking functions and noncrossing partitions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/k-naples-parking-functions/index.html#descending-strict-naples-count-eq-card-n","scope":"Combinatorics/parking-functions/k-naples-parking-functions","status":"can_prove","summary":"For every $n$ and every $k \\ge 1$, the number of descending strictly $k$-Naples parking functions of length $n$ (descending preferences that are $k$-Naples but not $(k - 1)$-Naples) equals the number of noncrossing partitions of $\\{0, 1, \\dots, n + k\\}$ in which the block containing $0$ has exactly $2k + 2$ elements.","title":"Descending strictly $k$-Naples parking functions and noncrossing partitions"},{"data_url":"scopes/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions.json","id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","kind":"scope","navigation":"module","parent":27146,"result":false,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","title":"Descents, peaks and valleys of parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/total-descents-over-parking-functions","kind":"node","lean":"MetaMathlibExt.total_descents_over_parking_functions","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#total-descents-over-parking-functions","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"fully_proved","summary":"Let $n$ be a natural number. A descent of a function $f \\colon \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ is an index $i$ with $i + 1 < n$ and $f(i) > f(i+1)$. Summed over all parking functions $f \\colon \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ of length $n$, the number of descents is $\\binom{n}{2} (n+1)^{n-2}$, which is $0$ for $n < 2$.","title":"Total number of descents over parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/parking-function-descent-gen-fun","kind":"node","lean":"MetaMathlibExt.parkingFunction_descent_genFun","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#parking-function-descent-gen-fun","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Write a sequence of length $n$ as $(a_1, \\dots, a_n)$; its descents are the $j \\in \\{1, \\dots, n-1\\}$ with $a_j > a_{j+1}$, and $\\operatorname{des}$ is their number. A parking function $f$ of length $n$ is read as the sequence $a_j = f(j-1) + 1$. Let $PF_n$ be the set of parking functions of length $n$, $PF^{\\uparrow}_n$ the set of nondecreasing ones, and $d(n, j)$ the number of $\\alpha \\in PF_n$ with\u2026","title":"Descent generating function of parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/parking-function-descent-set-card-eq-asc","kind":"node","lean":"MetaMathlibExt.parkingFunction_descentSet_card_eq_ascentSet_reflect","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#parking-function-descent-set-card-eq-asc","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Write a parking function $f$ of length $n$ as the sequence $(a_1, \\dots, a_n)$ with $a_j = f(j-1) + 1$. Its descent set is the set of $j \\in \\{1, \\dots, n-1\\}$ with $a_j > a_{j+1}$, and its ascent set the set of such $j$ with $a_j < a_{j+1}$. For every $n$ and every $I \\subseteq \\{1, \\dots, n-1\\}$, let $J = \\{n - i : i \\in I\\}$. Then the number of parking functions of length $n$ with descent set $I$ equals the\u2026","title":"Descent sets versus reflected ascent sets of parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/parking-function-descent-set-card-eq-asc-2","kind":"node","lean":"MetaMathlibExt.parkingFunction_descentSet_card_eq_ascentSet","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#parking-function-descent-set-card-eq-asc-2","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Write a parking function $f$ of length $n$ as the sequence $(a_1, \\dots, a_n)$ with $a_j = f(j-1) + 1$. Its descent set is the set of $j \\in \\{1, \\dots, n-1\\}$ with $a_j > a_{j+1}$, and its ascent set the set of such $j$ with $a_j < a_{j+1}$. For every $n$ and every $I \\subseteq \\{1, \\dots, n-1\\}$, the number of parking functions of length $n$ with descent set $I$ equals the number of parking functions of length\u2026","title":"Descent sets and ascent sets of parking functions are equidistributed"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/parking-function-descent-set-card-reflec","kind":"node","lean":"MetaMathlibExt.parkingFunction_descentSet_card_reflect","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#parking-function-descent-set-card-reflec","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Write a parking function $f$ of length $n$ as the sequence $(a_1, \\dots, a_n)$ with $a_j = f(j-1) + 1$. Its descent set is the set of $j \\in \\{1, \\dots, n-1\\}$ with $a_j > a_{j+1}$. For every $n$ and every $I \\subseteq \\{1, \\dots, n-1\\}$, let $J = \\{n - i : i \\in I\\}$. Then the number of parking functions of length $n$ with descent set $I$ equals the number with descent set $J$.","title":"Reflection symmetry of descent sets of parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/sum-ndpf-multiset-descent-eq","kind":"node","lean":"MetaMathlibExt.sum_ndpf_multisetDescent_eq","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#sum-ndpf-multiset-descent-eq","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"The descent set of a word $w = (w_1, \\dots, w_n)$ is $\\{i : 1 \\le i \\le n - 1,\\ w_i > w_{i+1}\\}$. Let $n \\in \\mathbb{N}$ and $I \\subseteq \\{1, \\dots, n - 1\\}$. Then the number of parking functions of length $n$ with descent set $I$ equals the sum, over the nondecreasing parking functions $\\beta$ of length $n$, of the number of words of length $n$ with descent set $I$ whose multiset of letters is the multiset of\u2026","title":"Counting parking functions with a given descent set by their content"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/parking-function-descent-set-recursion","kind":"node","lean":"MetaMathlibExt.parkingFunction_descentSet_recursion","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#parking-function-descent-set-recursion","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"The descent set of a word $w = (w_1, \\dots, w_\\ell)$ is $\\{i : 1 \\le i \\le \\ell - 1,\\ w_i > w_{i+1}\\}$, and $d(J; n)$ denotes the number of parking functions of length $n$ with descent set $J$. Let $n \\in \\mathbb{N}$, let $I \\subseteq \\{1, \\dots, n - 1\\}$ be nonempty with largest element $m$, and let $I^- = I \\setminus \\{m\\}$. Then","title":"Recursion for parking functions with a given descent set"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/parking-function-descent-set-initial-car","kind":"node","lean":"MetaMathlibExt.parkingFunction_descentSet_initial_card","parent":27194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parking functions with descent set "],["math","\\(\\{1, \\dots, k\\}\\)"]],"source":"Parking functions with descent set $\\{1, \\dots, k\\}$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#parking-function-descent-set-initial-car","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Let $n \\ge 1$ and $0 \\le k \\le n - 1$. The number of parking functions $(a_1, \\dots, a_n)$ of length $n$ whose descent set $\\{i : 1 \\le i \\le n - 1,\\ a_i > a_{i+1}\\}$ is exactly $\\{1, \\dots, k\\}$ (empty when $k = 0$) equals $\\frac{1}{n}\\binom{n}{k}\\binom{2n-k}{n-k-1}$.","title":"Parking functions with descent set $\\{1, \\dots, k\\}$"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/card-peakless-tieless-parking-function","kind":"node","lean":"MetaMathlibExt.card_peakless_tieless_parkingFunction","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#card-peakless-tieless-parking-function","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Write a parking function $f$ of length $n$ as the sequence $(a_1, \\dots, a_n)$ with $a_j = f(j-1) + 1$. It is peakless-tieless if there is no $j$ with $a_{j-1} < a_j > a_{j+1}$ (a peak) and no $j$ with $a_j = a_{j+1}$ (a tie). For every $n \\ge 1$, the number of peakless-tieless parking functions of length $n$ is the Catalan number $C_n$.","title":"Peakless-tieless parking functions are counted by the Catalan numbers"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/card-peakless-tieless-parking-function-l","kind":"node","lean":"MetaMathlibExt.card_peakless_tieless_parkingFunction_last","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#card-peakless-tieless-parking-function-l","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Write a parking function $f$ of length $n$ as the sequence $(a_1, \\dots, a_n)$ with $a_j = f(j-1) + 1 \\in \\{1, \\dots, n\\}$. It is peakless-tieless if there is no $j$ with $a_{j-1} < a_j > a_{j+1}$ (a peak) and no $j$ with $a_j = a_{j+1}$ (a tie). For natural numbers $i$ and $n$ with $1 \\le i \\le n$, the number of peakless-tieless parking functions of length $n$ with last entry $a_n = i$ is","title":"Peakless-tieless parking functions by last entry"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/card-valleyless-tieless-parking-function","kind":"node","lean":"MetaMathlibExt.card_valleyless_tieless_parkingFunction","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#card-valleyless-tieless-parking-function","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Write a parking function $f$ of length $n$ as the sequence $(a_1, \\dots, a_n)$ with $a_j = f(j-1) + 1$. It is valleyless-tieless if there is no $j$ with $a_{j-1} > a_j < a_{j+1}$ (a valley) and no $j$ with $a_j = a_{j+1}$ (a tie). For every $n \\ge 1$, the number of valleyless-tieless parking functions of length $n$ is the Fine number $F(n+1)$, where the Fine numbers are indexed from $0$ with $F(0) = 1$; for\u2026","title":"Valleyless-tieless parking functions are counted by the Fine numbers"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/valleyless-tieless-prepend-one-bij-on","kind":"node","lean":"MetaMathlibExt.valleyless_tieless_prepend_one_bijOn","parent":27194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prepending "],["math","\\(1\\)"],["text"," to valleyless-tieless parking functions"]],"source":"Prepending $1$ to valleyless-tieless parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#valleyless-tieless-prepend-one-bij-on","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Identify a parking function $f$ of length $m$ with the word $(f(0) + 1, \\dots, f(m-1) + 1)$. A word $(a_1, \\dots, a_m)$ has a tie if $a_k = a_{k+1}$ for some $k$, and a valley if $a_{k-1} > a_k < a_{k+1}$ for some $k$; write $\\mathrm{VTPF}_m$ for the set of parking functions of length $m$ with no ties and no valleys. For $n \\ge 1$, the map $(a_1, \\dots, a_n) \\mapsto (1, a_1 + 1, \\dots, a_n + 1)$ is a bijection from\u2026","title":"Prepending $1$ to valleyless-tieless parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/peakless-tieless-ends-card","kind":"node","lean":"MetaMathlibExt.peakless_tieless_ends_card","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#peakless-tieless-ends-card","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"A parking function $(a_1, \\dots, a_m)$ is peakless-tieless if $a_i \\ne a_{i+1}$ for all $1 \\le i \\le m - 1$ and there is no $i$ with $a_{i-1} < a_i > a_{i+1}$. Let $G_m$, $E_m$ and $L_m$ be the sets of peakless-tieless parking functions of length $m$ with $a_1 > a_m$, $a_1 = a_m$ and $a_1 < a_m$, respectively. Then $|E_{n+1}| = |L_n|$ for every $n \\ge 1$, and $|G_{n+1}| = |L_{n+1}|$ for every $n \\ge 0$.","title":"First and last entries of peakless-tieless parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/card-peakless-tieless-first-lt-last","kind":"node","lean":"MetaMathlibExt.card_peakless_tieless_first_lt_last","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#card-peakless-tieless-first-lt-last","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$, the number of parking functions $(a_1, \\dots, a_{n+1})$ of length $n + 1$ with no ties (no $i$ with $a_i = a_{i+1}$), no peaks (no $i$ with $a_{i-1} < a_i$ and $a_i > a_{i+1}$) and $a_1 < a_{n+1}$ is the Fine number $F(n + 1)$, indexed from $0$ so that $F(0) = 1$.","title":"Peakless-tieless parking functions with first entry below the last"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/card-peakless-parking-function-max-ties","kind":"node","lean":"MetaMathlibExt.card_peakless_parkingFunction_max_ties","parent":27194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peakless parking functions with "],["math","\\(n - 1\\)"],["text"," ties"]],"source":"Peakless parking functions with $n - 1$ ties"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#card-peakless-parking-function-max-ties","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"For every $n \\ge 1$, exactly one parking function $(a_1, \\dots, a_n)$ of length $n$ has no peaks (no $i$ with $a_{i-1} < a_i$ and $a_i > a_{i+1}$) and exactly $n - 1$ ties (indices $i \\in \\{1, \\dots, n - 1\\}$ with $a_i = a_{i+1}$).","title":"Peakless parking functions with $n - 1$ ties"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/exists-parking-function-stat-enc","kind":"node","lean":"MetaMathlibExt.exists_parkingFunction_statEnc","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#exists-parking-function-stat-enc","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Write a parking function $f$ of length $n$ as the sequence $(a_1, \\dots, a_n)$ with $a_j = f(j-1) + 1$. Its statistic encoding is the word $w_1 \\cdots w_{n-1}$ over the letters $A$, $D$, $T$ with $w_j = A$ if $a_j < a_{j+1}$ (an ascent), $w_j = D$ if $a_j > a_{j+1}$ (a descent) and $w_j = T$ if $a_j = a_{j+1}$ (a tie). For every natural number $n$, every word of length $n - 1$ over $A$, $D$, $T$ (the empty word\u2026","title":"Every statistic encoding comes from a parking function"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/unique-stat-enc-adjacent-and-downward-cl","kind":"node","lean":"MetaMathlibExt.unique_statEnc_adjacent_and_downward_closed","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#unique-stat-enc-adjacent-and-downward-cl","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"The statistic encoding of $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ is the word over $\\{A, D, T\\}$ whose letter at each position $0 \\le i \\le n - 2$ is $A$, $D$ or $T$ according as $f(i) < f(i+1)$, $f(i) > f(i+1)$ or $f(i) = f(i+1)$. Let $f$ be a parking function of length $n$ such that every parking function of length $n$ with the same statistic encoding equals $f$. Then $|f(i+1) - f(i)| \\le 1$ for all\u2026","title":"Uniquely encoded parking functions have steps of at most one and no gaps"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/unique-stat-enc-sorted-form","kind":"node","lean":"MetaMathlibExt.unique_statEnc_sorted_form","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#unique-stat-enc-sorted-form","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"The statistic encoding of $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ is the word over $\\{A, D, T\\}$ whose letter at each position $0 \\le i \\le n - 2$ is $A$, $D$ or $T$ according as $f(i) < f(i+1)$, $f(i) > f(i+1)$ or $f(i) = f(i+1)$. Let $f$ be a parking function of length $n$ such that every parking function of length $n$ with the same statistic encoding equals $f$, and let\u2026","title":"The sorted rearrangement of a uniquely encoded parking function"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/unique-stat-enc-valley-plateau-eq-one","kind":"node","lean":"MetaMathlibExt.unique_statEnc_valley_plateau_eq_one","parent":27194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Valley plateaus of a uniquely encoded parking function equal "],["math","\\(1\\)"]],"source":"Valley plateaus of a uniquely encoded parking function equal $1$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#unique-stat-enc-valley-plateau-eq-one","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"For a map $g : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$, its statistic encoding is the word of length $n - 1$ whose $k$-th letter, for $1 \\le k \\le n - 1$, is $<$, $=$ or $>$ according as $g(k-1) < g(k)$, $g(k-1) = g(k)$ or $g(k-1) > g(k)$. Let $f$ be a parking function of length $n$ such that every parking function of length $n$ with the same statistic encoding as $f$ equals $f$, and write $\\alpha_k = f(k-1) + 1$\u2026","title":"Valley plateaus of a uniquely encoded parking function equal $1$"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/unique-parking-function-stat-enc-iff","kind":"node","lean":"MetaMathlibExt.unique_parkingFunction_statEnc_iff","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#unique-parking-function-stat-enc-iff","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"The statistic encoding of a parking function $(a_1, \\dots, a_n)$ is the word $w_1 \\cdots w_{n-1}$ over $\\{A, D, T\\}$ with $w_i = A$ if $a_i < a_{i+1}$, $w_i = D$ if $a_i > a_{i+1}$ and $w_i = T$ if $a_i = a_{i+1}$. For every $n \\in \\mathbb{N}$, a word over $\\{A, D, T\\}$ with one letter for each $1 \\le i \\le n - 1$ is the statistic encoding of exactly one parking function of length $n$ if and only if it consists of\u2026","title":"Statistic encodings with a unique parking function"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/peakless-tieless-of-nondecreasing","kind":"node","lean":"MetaMathlibExt.peaklessTielessOfNondecreasing","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#peakless-tieless-of-nondecreasing","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$. The ties of $f$ are the positions $i \\in \\{1, \\dots, n-1\\}$ with $f(i-1) = f(i)$, and its values are the elements of the image of $f$. Form the list of the ties in decreasing order followed by the values in increasing order. Then $\\varphi_n(f) : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ is the map whose value at $k$ is the entry of this list\u2026","title":"Tie-and-value map"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/peakless-tieless-of-nondecreasing-exampl","kind":"node","lean":"MetaMathlibExt.peaklessTielessOfNondecreasing_example","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#peakless-tieless-of-nondecreasing-exampl","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"fully_proved","summary":"For $n = 6$, the tie-and-value map $\\varphi$ sends $(0, 0, 1, 2, 3, 3)$ to $(5, 1, 0, 1, 2, 3)$; in the paper's one-based values this reads $\\varphi(1, 1, 2, 3, 4, 4) = (6, 2, 1, 2, 3, 4)$.","title":"Example of the tie-and-value map"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/peakless-tieless-of-nondecreasing-bij-on","kind":"node","lean":"MetaMathlibExt.peaklessTielessOfNondecreasing_bijOn","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#peakless-tieless-of-nondecreasing-bij-on","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$, the tie-and-value map $\\varphi$ is a bijection from the nondecreasing parking functions of length $n$ onto the peakless-tieless parking functions of length $n$, that is, the parking functions $(a_1, \\dots, a_n)$ with $a_i \\ne a_{i+1}$ for all $1 \\le i \\le n - 1$ and no $i$ with $a_{i-1} < a_i > a_{i+1}$.","title":"Nondecreasing and peakless-tieless parking functions are in bijection"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/ascent-to-descent-rearrangement-aux","kind":"node","lean":"MetaMathlibExt.ascentToDescentRearrangementAux","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#ascent-to-descent-rearrangement-aux","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"fully_proved","summary":"For a fuel $N \\in \\mathbb{N}$ and a word $x = x_1 \\cdots x_m$ over $\\mathbb{N}$, the word $\\nu_N(x)$ is defined by recursion on $N$: $\\nu_0(x) = x$, and $\\nu_{N+1}(x) = x$ when $m \\le 1$. For $m \\ge 2$, let $A$ be the set of positions of the least letter of $x$ and $B$ the set of positions of its greatest letter. If $b \\le a$ for all $a \\in A$ and $b \\in B$ (Case 1), put $i = \\max A$ and $j = \\min B$; then\u2026","title":"Fuel-bounded ascent-to-descent rearrangement"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/ascent-to-descent-rearrangement","kind":"node","lean":"MetaMathlibExt.ascentToDescentRearrangement","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#ascent-to-descent-rearrangement","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"fully_proved","summary":"For a word $x = x_1 \\cdots x_m$ over $\\mathbb{N}$, the ascent-to-descent rearrangement of $x$ is $\\nu(x) = \\nu_m(x)$, the fuel-bounded ascent-to-descent rearrangement $\\nu_N(x)$ with fuel $N = m$, the length of $x$.","title":"Ascent-to-descent rearrangement"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/ascent-to-descent-rearrangement-of-lengt","kind":"node","lean":"MetaMathlibExt.ascentToDescentRearrangement_of_length_le_one","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#ascent-to-descent-rearrangement-of-lengt","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"fully_proved","summary":"Every word $x$ over $\\mathbb{N}$ of length at most $1$ satisfies $\\nu(x) = x$, where $\\nu$ is the ascent-to-descent rearrangement.","title":"Words of length at most one are fixed by the ascent-to-descent rearrangement"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/ascent-to-descent-rearrangement-example","kind":"node","lean":"MetaMathlibExt.ascentToDescentRearrangement_example","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#ascent-to-descent-rearrangement-example","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"fully_proved","summary":"The ascent-to-descent rearrangement $\\nu$ satisfies $\\nu(1\\,4\\,1\\,3\\,4\\,3\\,1\\,2\\,1) = 4\\,1\\,4\\,3\\,1\\,1\\,2\\,1\\,3$, $\\nu(1\\,4) = 4\\,1$ and $\\nu(3\\,1\\,2\\,1) = 1\\,2\\,1\\,3$.","title":"Example of the ascent-to-descent rearrangement"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/asc-list-eq-des-list-ascent-to-descent-r","kind":"node","lean":"MetaMathlibExt.ascList_eq_desList_ascentToDescentRearrangement","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#asc-list-eq-des-list-ascent-to-descent-r","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"For a word $x = x_1 \\cdots x_m$ over $\\mathbb{N}$, let the ascent set $\\mathrm{Asc}(x)$ be the set of $i \\in \\{1, \\dots, m - 1\\}$ with $x_i < x_{i+1}$, and the descent set $\\mathrm{Des}(x)$ the set of $i \\in \\{1, \\dots, m - 1\\}$ with $x_i > x_{i+1}$. Then every word $x$ over $\\mathbb{N}$ satisfies $\\mathrm{Asc}(x) = \\mathrm{Des}(\\nu(x))$, where $\\nu$ is the ascent-to-descent rearrangement.","title":"Ascents of a word are the descents of its rearrangement"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/ascent-to-descent-rearrangement-injectiv","kind":"node","lean":"MetaMathlibExt.ascentToDescentRearrangement_injective","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#ascent-to-descent-rearrangement-injectiv","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"The ascent-to-descent rearrangement $\\nu$ is injective on finite words over $\\mathbb{N}$: if $\\nu(x) = \\nu(x')$, then $x = x'$.","title":"The ascent-to-descent rearrangement is injective"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/ascent-to-descent-rearrangement-bij-on","kind":"node","lean":"MetaMathlibExt.ascentToDescentRearrangement_bijOn","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#ascent-to-descent-rearrangement-bij-on","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Let $y$ be a finite word over $\\mathbb{N}$ and $I$ a finite set of natural numbers. For a word $x = x_1 \\cdots x_m$, let the ascent set $\\mathrm{Asc}(x)$ be the set of $i \\in \\{1, \\dots, m - 1\\}$ with $x_i < x_{i+1}$, and the descent set $\\mathrm{Des}(x)$ the set of those $i$ with $x_i > x_{i+1}$. Then the ascent-to-descent rearrangement $\\nu$ maps the set of rearrangements $x$ of $y$ with $\\mathrm{Asc}(x) = I$\u2026","title":"The ascent-to-descent rearrangement matches ascent and descent sets"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/valleyless-tieless-flick-bij-on","kind":"node","lean":"MetaMathlibExt.valleyless_tieless_flick_bijOn","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#valleyless-tieless-flick-bij-on","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"Identify a parking function $f$ of length $m$ with the word $(f(0) + 1, \\dots, f(m-1) + 1)$. A word $(a_1, \\dots, a_m)$ has a tie if $a_k = a_{k+1}$ for some $k$, a peak if $a_{k-1} < a_k > a_{k+1}$ for some $k$, and a valley if $a_{k-1} > a_k < a_{k+1}$ for some $k$. For $1 \\le i \\le m$, let $\\mathrm{flick}_i(x_1, \\dots, x_m) = (x_i, x_{i-1}, \\dots, x_1, x_m, x_{m-1}, \\dots, x_{i+1})$, the block\u2026","title":"Flicking valleyless-tieless parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/exists-dyck-word-standard-young-tableau","kind":"node","lean":"MetaMathlibExt.exists_dyckWord_standardYoungTableau_bijOn","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#exists-dyck-word-standard-young-tableau","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"For every $n$ there is a map $f$ from Dyck words to semistandard Young tableaux of the two-row shape $(n, n)$ with the following two properties. First, for every Dyck word $w$ of semilength $n$ and every $i \\ge 1$, if the $i$th letter of $w$ is an up step $U$ then $i$ is an entry of the first row of $f(w)$, and if it is a down step $D$ then $i$ is an entry of the second row. Second, $f$ restricts to a bijection\u2026","title":"Gudmundsson's bijection between Dyck words and standard Young tableaux"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/exists-dyck-word-nondecreasing-parking-f","kind":"node","lean":"MetaMathlibExt.exists_dyckWord_nondecreasing_parkingFunction_bijOn","parent":27194,"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#exists-dyck-word-nondecreasing-parking-f","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"For every $n$ there is a map $g$ from Dyck words to maps $\\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ such that, for every Dyck word $w$ of semilength $n$ and every $i < n$, $g(w)(i)$ is the number of down steps $D$ of $w$ before its $(i+1)$st up step $U$, and $g$ restricts to a bijection from the Dyck words of semilength $n$ onto the nondecreasing parking functions of length $n$.","title":"Dyck words and nondecreasing parking functions"},{"id":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/card-peakless-parking-function-ties-eq-s","kind":"node","lean":"MetaMathlibExt.card_peakless_parkingFunction_ties_eq_sub_two","parent":27194,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peakless parking functions with "],["math","\\(n - 2\\)"],["text"," ties"]],"source":"Peakless parking functions with $n - 2$ ties"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions/index.html#card-peakless-parking-function-ties-eq-s","scope":"Combinatorics/parking-functions/descents-peaks-and-valleys-of-parking-functions","status":"can_prove","summary":"For every $n \\ge 2$, the number of parking functions $(a_1, \\dots, a_n)$ of length $n$ that have no peaks (no $i$ with $a_{i-1} < a_i$ and $a_i > a_{i+1}$) and exactly $n - 2$ ties (indices $i \\in \\{1, \\dots, n - 1\\}$ with $a_i = a_{i+1}$) is $(n - 1)n$.","title":"Peakless parking functions with $n - 2$ ties"},{"data_url":"scopes/Combinatorics/parking-functions/lucky-cars-in-parking-functions.json","id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","kind":"scope","navigation":"module","parent":27146,"result":false,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","title":"Lucky cars in parking functions"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/expected-lucky-spots-weakly-increasing-p","kind":"node","lean":"MetaMathlibExt.expected_lucky_spots_weakly_increasing_parking_functions","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#expected-lucky-spots-weakly-increasing-p","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Let $n \\ge 1$. Number cars and spots from $0$, and encode a weakly increasing parking function of length $n$ as a nondecreasing map $a : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ with $a(i) \\le i$ for every $i$. The average, over all such maps $a$, of the number of indices $i$ with $a(i) = i$ equals $\\frac{3n}{n+2}$. Here car $i$ parks in spot $i$ and is lucky exactly when $a(i) = i$, so this number counts both the\u2026","title":"Expected number of lucky cars in weakly increasing parking functions"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/next-free-spot","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.nextFreeSpot","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#next-free-spot","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Number the $n$ spots $0, \\dots, n-1$. Given a set $S$ of occupied spots and a spot $a$, the next free spot is the least spot $s \\ge a$ with $s \\notin S$; when every spot $s \\ge a$ is occupied, it is $a$ itself.","title":"Next free spot"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/park-spots-from","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.parkSpotsFrom","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#park-spots-from","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Number the $n$ spots $0, \\dots, n-1$. Given a list $a_1, \\dots, a_m$ of preferred spots and a set $S$ of occupied spots, cars $1, \\dots, m$ arrive in order and car $k$ takes the least spot at or after $a_k$ that is neither in $S$ nor taken by cars $1, \\dots, k-1$ (or $a_k$ itself if there is none). The result is the list of spots taken, in arrival order.","title":"Greedy parking from an occupied set"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/park-spots","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.parkSpots","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#park-spots","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Let $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ give the preferred spot of each of $n$ cars. Starting from an empty lot with spots $0, \\dots, n-1$, cars $0, 1, \\dots, n-1$ arrive in order, and each takes the least free spot at or after its preference (or its preference itself if there is none). The outcome sends each car $i$ to the spot it takes.","title":"Greedy parking outcome"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/is-lucky-car","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.IsLuckyCar","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#is-lucky-car","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Let $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ give the preferred spot of each of $n$ cars. Cars $0, 1, \\dots, n-1$ arrive in order, and each takes the least spot at or after its preference not taken by an earlier car, or its preferred spot itself when there is no such spot. Car $i$ is lucky if the spot it takes is its preferred spot $f(i)$.","title":"Lucky car under greedy parking"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/lucky-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.luckyCount","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#lucky-count","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Let $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ give the preferred spot of each of $n$ cars, and let cars $0, 1, \\dots, n-1$ arrive in order, each taking the least spot at or after its preference not taken by an earlier car, or its preferred spot itself when there is no such spot. The number of lucky cars of $f$ is the number of cars $i$ whose spot is $f(i)$.","title":"Number of lucky cars"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/q","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.q","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#q","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Number the cars and spots $0, \\dots, n-1$. For $i, j \\in \\{0, \\dots, n-1\\}$, $q_n(i, j)$ is the number of parking functions $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ with $f(i) = j$ in which car $i$ is lucky, that is, parks in spot $f(i)$.","title":"Lucky-car counts"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/q-inc","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.qInc","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#q-inc","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Number the cars and spots $0, \\dots, n-1$. For $i, j \\in \\{0, \\dots, n-1\\}$, $q^{\\mathrm{inc}}_n(i, j)$ is the number of weakly increasing parking functions $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ with $f(i) = j$ in which car $i$ is lucky, that is, parks in spot $f(i)$.","title":"Increasing lucky-car counts"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/q-dec","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.qDec","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#q-dec","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"Number the cars and spots $0, \\dots, n-1$. For $i, j \\in \\{0, \\dots, n-1\\}$, $q^{\\mathrm{dec}}_n(i, j)$ is the number of weakly decreasing parking functions $f : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, n-1\\}$ with $f(i) = j$ in which car $i$ is lucky, that is, parks in spot $f(i)$.","title":"Decreasing lucky-car counts"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/lucky-gen-poly","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.luckyGenPoly","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#lucky-gen-poly","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, $f_n \\in \\mathbb{Q}[x]$ is the polynomial","title":"Lucky-car polynomial"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/card-parking-lucky-on","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.card_parking_lucky_on","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#card-parking-lucky-on","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Number the cars and spots $0, \\dots, n-1$, and let $L$ and $U$ be disjoint sets of cars. The number of parking functions of length $n$ in which every car in $L$ is lucky and no car in $U$ is lucky equals","title":"Parking functions with prescribed lucky and unlucky cars"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/lucky-gen-poly-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.luckyGenPoly_eq_sum","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#lucky-gen-poly-eq-sum","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ and $0 \\le k \\le n$, let $c_k$ be the number of parking functions of length $n$ with exactly $k$ lucky cars. Then in $\\mathbb{Q}[x]$,","title":"Generating polynomial of lucky cars"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/expected-desc-factorial-lucky-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.expected_descFactorial_luckyCount","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#expected-desc-factorial-lucky-count","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Let $n \\ge 1$ and $\\ell \\in \\mathbb{N}$, let $X(f)$ be the number of lucky cars of a parking function $f$ of length $n$, and let $P_n(x) = \\frac{1}{n+1} \\prod_{i=0}^{n-1} \\bigl((n+1-i)x + i\\bigr) \\in \\mathbb{Q}[x]$. Then","title":"Factorial moments of the number of lucky cars"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/q-boundary-values","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.q_boundary_values","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#q-boundary-values","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Let $n \\ge 1$, number the cars and spots $0, \\dots, n-1$, and let $q_n(i, j)$ be the number of parking functions of length $n$ in which car $i$ prefers spot $j$ and is lucky. Then for all $i, j \\in \\{0, \\dots, n-1\\}$, with powers taken in $\\mathbb{Q}$ (the integer exponents may be negative),","title":"Boundary values of lucky-car counts (Lemma 9)"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/sum-q-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.sum_q_eq","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#sum-q-eq","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Number cars and spots $0, \\dots, n-1$, and let $q_n(i, j)$ be the number of parking functions of length $n$ in which car $i$ prefers spot $j$ and parks there. Then for every $n$ and every car $i < n$,","title":"Number of parking functions in which a given car is lucky"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/sum-q-second-spot","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.sum_q_second_spot","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#sum-q-second-spot","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Number cars and spots $0, \\dots, n-1$, and let $q_n(i, j)$ be the number of parking functions of length $n$ in which car $i$ prefers spot $j$ and parks there. Then for every $n \\ge 2$, in $\\mathbb{Q}$,","title":"Number of parking functions in which the second spot is lucky"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/q-inc-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.qInc_eq","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#q-inc-eq","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Number the cars and spots $0, \\dots, n-1$, let $q^{\\mathrm{inc}}_n(i, j)$ be the number of weakly increasing parking functions of length $n$ in which car $i$ prefers spot $j$ and is lucky, and let $C_m$ be the $m$-th Catalan number. For all $i, j \\in \\{0, \\dots, n-1\\}$, $q^{\\mathrm{inc}}_n(i, j) = C_i \\, C_{n-i}$ if $i = j$, and $q^{\\mathrm{inc}}_n(i, j) = 0$ otherwise.","title":"Increasing lucky-car counts (Lemma 18)"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/ballot-path","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.BallotPath","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#ballot-path","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"fully_proved","summary":"For $k, \\ell \\in \\mathbb{N}$, a ballot path to $(k, \\ell)$ is a word of length $k + \\ell$ in up steps and right steps with exactly $\\ell$ up steps such that every prefix has at least as many up steps as right steps; equivalently, a lattice path from $(0,0)$ to $(k, \\ell)$ with unit up and right steps that stays on or above the line $y = x$.","title":"Ballot paths"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/card-ballot-path","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.card_ballotPath","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#card-ballot-path","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"For natural numbers $k \\le \\ell$, the number of lattice paths from $(0,0)$ to $(k, \\ell)$ with unit up and right steps that stay on or above the line $y = x$ is $\\frac{\\ell - k + 1}{\\ell + 1} \\binom{k + \\ell}{k}$.","title":"Number of ballot paths"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/q-dec-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.qDec_eq","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#q-dec-eq","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Number the cars and spots $0, \\dots, n-1$, and let $q^{\\mathrm{dec}}_n(i, j)$ be the number of weakly decreasing parking functions of length $n$ in which car $i$ prefers spot $j$ and is lucky. For all $i, j \\in \\{0, \\dots, n-1\\}$, if $i + j < n$ then","title":"Decreasing lucky-car counts (Lemma 22)"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/card-dec-lucky-spot-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.card_dec_lucky_spot_eq_sum","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#card-dec-lucky-spot-eq-sum","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Number the cars and spots $0, \\dots, n-1$. For cars $i$ and spots $j$, let $q^d_n(i, j)$ be the number of weakly decreasing parking functions $f$ of length $n$ (that is, $f(0) \\ge f(1) \\ge \\dots \\ge f(n-1)$) with $f(i) = j$ in which car $i$ is lucky. Then for every spot $j$,","title":"Lucky spots of weakly decreasing parking functions as a sum"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/card-dec-lucky-spot-eq-sum-catalan","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.card_dec_lucky_spot_eq_sum_catalan","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#card-dec-lucky-spot-eq-sum-catalan","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"Number the cars and spots $0, \\dots, n-1$. For cars $i$ and spots $j$, let $q^d_n(i, j)$ be the number of weakly decreasing parking functions $f$ of length $n$ (that is, $f(0) \\ge f(1) \\ge \\dots \\ge f(n-1)$) with $f(i) = j$ in which car $i$ is lucky. Then for every spot $j$,","title":"Lucky spots of weakly decreasing parking functions as a Catalan convolution"},{"id":"Combinatorics/parking-functions/lucky-cars-in-parking-functions/expected-lucky-dec","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.LuckyCarsWanted.expected_lucky_dec","parent":27229,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-cars-in-parking-functions/index.html#expected-lucky-dec","scope":"Combinatorics/parking-functions/lucky-cars-in-parking-functions","status":"can_prove","summary":"For $n \\ge 1$, the sum of the numbers of lucky cars over all weakly decreasing parking functions $f$ of length $n$ (those with $f(0) \\ge f(1) \\ge \\dots \\ge f(n-1)$), divided by the Catalan number $C_n$, equals $\\frac{n+1}{2}$.","title":"Expected number of lucky cars in weakly decreasing parking functions"},{"data_url":"scopes/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe.json","id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","kind":"scope","navigation":"module","parent":27146,"result":false,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","title":"Lucky and displacement statistics of Stirling permutations"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-pref","kind":"node","lean":"MetaMathlibExt.stirlingPref","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-pref","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"fully_proved","summary":"Let $w = w_1 \\cdots w_{2n}$ be a word of natural numbers, read as the preferences of $2n$ cars for spots numbered $0, \\dots, 2n - 1$. Car $i \\in \\{1, \\dots, 2n\\}$ prefers the spot $(w_i - 1) \\bmod 2n$, where $w_i - 1$ is truncated at $0$. When every $w_i$ lies in $\\{1, \\dots, 2n\\}$, for example when $w$ is a Stirling permutation of order $n$, this is the spot $w_i - 1$, which is spot $w_i$ in the one-based numbering.","title":"Zero-based preferences of a word"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-park-spot","kind":"node","lean":"MetaMathlibExt.stirlingParkSpot","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-park-spot","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"fully_proved","summary":"Let $w = w_1 \\cdots w_{2n}$ be a word of natural numbers. Car $i$ prefers spot $q_i = ((w_i - 1) \\bmod 2n) + 1$, where $w_i - 1$ is truncated at $0$; when every $w_i$ lies in $\\{1, \\dots, 2n\\}$, this is $q_i = w_i$. Cars $1, \\dots, 2n$ arrive in order, and car $i$ parks in the least spot $p(i) \\in \\{q_i, \\dots, 2n\\}$ not taken by an earlier car, or in $p(i) = q_i$ if there is no such spot. The parking spot of car\u2026","title":"Parking spot of a car"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-lucky-set","kind":"node","lean":"MetaMathlibExt.stirlingLuckySet","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-lucky-set","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"fully_proved","summary":"Let $w = w_1 \\cdots w_{2n}$ be a word of natural numbers. Car $i$ prefers spot $q_i = ((w_i - 1) \\bmod 2n) + 1$, where $w_i - 1$ is truncated at $0$; when every $w_i$ lies in $\\{1, \\dots, 2n\\}$, this is $q_i = w_i$. Cars $1, \\dots, 2n$ arrive in order, and car $i$ parks in the least spot $p(i) \\in \\{q_i, \\dots, 2n\\}$ not taken by an earlier car, or in $p(i) = q_i$ if there is no such spot. The lucky set\u2026","title":"The lucky set of a word"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-displacement","kind":"node","lean":"MetaMathlibExt.stirlingDisplacement","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-displacement","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"fully_proved","summary":"Let $w = w_1 \\cdots w_{2n}$ be a word of natural numbers. Car $i$ prefers spot $q_i = ((w_i - 1) \\bmod 2n) + 1$, where $w_i - 1$ is truncated at $0$; when every $w_i$ lies in $\\{1, \\dots, 2n\\}$, this is $q_i = w_i$. Cars $1, \\dots, 2n$ arrive in order, and car $i$ parks in the least spot $p(i) \\in \\{q_i, \\dots, 2n\\}$ not taken by an earlier car, or in $p(i) = q_i$ if there is no such spot. The displacement of car\u2026","title":"Displacement of a car"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-total-displacement","kind":"node","lean":"MetaMathlibExt.stirlingTotalDisplacement","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-total-displacement","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"fully_proved","summary":"Let $w = w_1 \\cdots w_{2n}$ be a word of natural numbers, and for $i \\in \\{1, \\dots, 2n\\}$ let $d(i)$ be the displacement of car $i$ in $w$. The total displacement of $w$ is $d(w) = \\sum_{i=1}^{2n} d(i)$.","title":"Total displacement of a word"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/is-admissible-lucky-set","kind":"node","lean":"MetaMathlibExt.IsAdmissibleLuckySet","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#is-admissible-lucky-set","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"fully_proved","summary":"Let $n$ be a natural number. A finite set $S$ of natural numbers is an $n$-admissible lucky set if $S = \\mathrm{Lucky}(w)$ for some Stirling permutation $w$ of order $n$, where $\\mathrm{Lucky}(w)$ is the set of lucky cars of $w$.","title":"Admissible lucky sets"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-lucky-set-small","kind":"node","lean":"MetaMathlibExt.stirlingLuckySet_small","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-lucky-set-small","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"fully_proved","summary":"Read as parking functions of length $4$, the Stirling permutation $1122$ has car $1$ as its only lucky car, and $1221$ has exactly cars $1$ and $2$ lucky: $\\mathrm{Lucky}(1122) = \\{1\\}$ and $\\mathrm{Lucky}(1221) = \\{1, 2\\}$.","title":"Lucky sets of 1122 and 1221"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-displacement-small","kind":"node","lean":"MetaMathlibExt.stirlingDisplacement_small","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-displacement-small","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"fully_proved","summary":"The displacement composition of a word $w = w_1 \\cdots w_{2n}$ is $\\mathrm{dis}(w) = (d(1), \\dots, d(2n))$, where $d(i)$ is the displacement of car $i$. For the Stirling permutations $1122$ of order $2$ and $11244233$ of order $4$, $\\mathrm{dis}(1122) = (0, 1, 1, 2)$ and $\\mathrm{dis}(11244233) = (0, 1, 1, 0, 1, 4, 4, 5)$ (the paper's Example 5.5).","title":"Displacement compositions of 1122 and 11244233"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-lucky-set-basic","kind":"node","lean":"MetaMathlibExt.stirlingLuckySet_basic","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-lucky-set-basic","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $n \\ge 1$ and let $w$ be a Stirling permutation of order $n$, read as a parking function of length $2n$. Then (a) car $1$ is lucky; (b) the first car $i$ with $w_i = 1$ is lucky; (c) the number of lucky cars of $w$ is at least $1$ and at most $n$.","title":"Basic properties of lucky cars"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-lucky-set-eq-singleton-iff","kind":"node","lean":"MetaMathlibExt.stirlingLuckySet_eq_singleton_iff","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-lucky-set-eq-singleton-iff","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $n \\ge 1$ and let $w$ be a Stirling permutation of order $n$, read as a parking function of length $2n$. Then car $1$ is the only lucky car of $w$ if and only if for every $x \\in \\{2, \\dots, n\\}$ and every position $i \\le x$, we have $w_i < x$.","title":"Characterization of extremely unlucky Stirling permutations"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/card-extremely-unlucky-stirling","kind":"node","lean":"MetaMathlibExt.card_extremely_unlucky_stirling","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#card-extremely-unlucky-stirling","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"For every integer $n \\ge 1$, the number of Stirling permutations $w$ of order $n$ that are extremely unlucky, that is, whose only lucky car is car $1$ ($\\mathrm{Lucky}(w) = \\{1\\}$), is $(n - 1)!$.","title":"Counting extremely unlucky Stirling permutations"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-lucky-set-card-eq-iff","kind":"node","lean":"MetaMathlibExt.stirlingLuckySet_card_eq_iff","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-lucky-set-card-eq-iff","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w$ be a Stirling permutation of order $n$, read as a parking function of length $2n$. Then $w$ has exactly $n$ lucky cars if and only if the second occurrences of the letters of $w$ appear in decreasing order: for all positions $a < b$ such that $w_a$ also occurs before position $a$ and $w_b$ also occurs before position $b$, we have $w_b < w_a$.","title":"Characterization of extremely lucky Stirling permutations"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/extremely-lucky-stirling-equiv-dyck-word","kind":"node","lean":"MetaMathlibExt.extremely_lucky_stirling_equiv_dyckWord","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#extremely-lucky-stirling-equiv-dyck-word","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"For every natural number $n$, there is a bijection between the Stirling permutations $w$ of order $n$ that are extremely lucky, that is, whose set $\\mathrm{Lucky}(w)$ of lucky cars has exactly $n$ elements, and the Dyck words of semilength $n$ (valid parenthesizations of $n$ pairs of parentheses).","title":"Extremely lucky Stirling permutations and Dyck words"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/card-extremely-lucky-stirling","kind":"node","lean":"MetaMathlibExt.card_extremely_lucky_stirling","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#card-extremely-lucky-stirling","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"For every positive integer $n$, the number of Stirling permutations $w$ of order $n$ that are extremely lucky, that is, whose set $\\mathrm{Lucky}(w)$ of lucky cars has exactly $n$ elements, is the Catalan number $C_n$.","title":"Extremely lucky Stirling permutations are counted by the Catalan numbers"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/succ","kind":"node","lean":"MetaMathlibExt.IsAdmissibleLuckySet.succ","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#succ","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $n \\ge 1$. If $S$ is an $n$-admissible lucky set, then both $S$ and $S \\cup \\{2n + 1\\}$ are $(n + 1)$-admissible lucky sets.","title":"Extending admissible lucky sets"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/one-mem-and-two-mul-not-mem","kind":"node","lean":"MetaMathlibExt.IsAdmissibleLuckySet.one_mem_and_two_mul_not_mem","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#one-mem-and-two-mul-not-mem","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $n \\ge 1$. If $S$ is an $n$-admissible lucky set, then $1 \\in S$ and $2n \\notin S$.","title":"Necessary conditions for admissible lucky sets"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-park-spot-before-second-lucky","kind":"node","lean":"MetaMathlibExt.stirlingParkSpot_before_second_lucky","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-park-spot-before-second-lucky","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$, read as a parking function of length $2n$, and let $x$ be a lucky car of $w$ such that exactly one lucky car of $w$ is smaller than $x$, so that $x$ is the second-smallest element of the lucky set $\\mathrm{Lucky}(w)$. Then every car $c$ with $2 \\le c \\le x - 1$ parks in spot $w_1 + c - 1$.","title":"Parking before the second lucky car"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-odd-of-lt-all-before","kind":"node","lean":"MetaMathlibExt.stirling_odd_of_lt_all_before","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-odd-of-lt-all-before","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$ and let $j \\in \\{1, \\dots, 2n\\}$. If $w_j < w_i$ for every $i < j$, then $j$ is odd.","title":"Left-to-right minima of a Stirling permutation sit at odd positions"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-second-lucky-odd","kind":"node","lean":"MetaMathlibExt.stirling_second_lucky_odd","parent":27253,"reading_title":{"format":"inline-parts-v1","parts":[["text","A second lucky car beyond "],["math","\\(n\\)"],["text"," is odd"]],"source":"A second lucky car beyond $n$ is odd"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-second-lucky-odd","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$, read as a parking function of length $2n$, and let $x$ be a lucky car of $w$ such that exactly one lucky car of $w$ is smaller than $x$, so that $x$ is the second-smallest element of the lucky set $\\mathrm{Lucky}(w)$. If $x > n$, then $x$ is odd.","title":"A second lucky car beyond $n$ is odd"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-second-lucky-pref","kind":"node","lean":"MetaMathlibExt.stirling_second_lucky_pref","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-second-lucky-pref","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$, read as a parking function of length $2n$, and let $x$ be a lucky car of $w$ such that exactly one lucky car of $w$ is smaller than $x$, so that $x$ is the second-smallest element of the lucky set $\\mathrm{Lucky}(w)$. If $w_1 = 1$, then $x \\le w_x \\le n$.","title":"Preference of the second lucky car"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-first-pref-ge-lucky","kind":"node","lean":"MetaMathlibExt.stirling_first_pref_ge_lucky","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-first-pref-ge-lucky","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w$ be a Stirling permutation of order $n$, read as a parking function of length $2n$, with at least $3$ lucky cars, and let $x$ be the second-smallest lucky car of $w$. If $x \\ge n$, then the number of lucky cars of $w$ is at most $w_1$, and every lucky car parks in one of the spots $1, \\dots, w_1$.","title":"The first preference bounds the lucky cars"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-penultimate-lucky-iff","kind":"node","lean":"MetaMathlibExt.stirling_penultimate_lucky_iff","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-penultimate-lucky-iff","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $n \\ge 1$ and let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$, read as a parking function of length $2n$. Then car $2n - 1$ is lucky, that is, belongs to the lucky set $\\mathrm{Lucky}(w)$, if and only if $w_{2n-1} = 1$.","title":"When the penultimate car is lucky"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/is-admissible-lucky-set-pair-iff","kind":"node","lean":"MetaMathlibExt.isAdmissibleLuckySet_pair_iff","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#is-admissible-lucky-set-pair-iff","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $n$, $h$ and $i$ be natural numbers with $h < i \\le 2n$. The set $\\{h, i\\}$ is an $n$-admissible lucky set if and only if $h = 1$ and either $i > n$ and $i$ is odd, or $i \\le n$.","title":"Admissible lucky sets with two elements"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/ncard-admissible-lucky-set-card-two","kind":"node","lean":"MetaMathlibExt.ncard_admissibleLuckySet_card_two","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#ncard-admissible-lucky-set-card-two","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"For every integer $n \\ge 1$, the number of $n$-admissible lucky sets with exactly two elements is $n - 1 + \\lceil (n - 1)/2 \\rceil$.","title":"Counting admissible lucky sets with two elements"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-second-lucky-eq-n","kind":"node","lean":"MetaMathlibExt.stirling_second_lucky_eq_n","parent":27253,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stirling permutations whose second lucky car is "],["math","\\(n\\)"]],"source":"Stirling permutations whose second lucky car is $n$"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-second-lucky-eq-n","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$, read as a parking function of length $2n$, such that car $n$ is lucky and exactly one lucky car of $w$ is smaller than $n$, so that $n$ is the second-smallest element of the lucky set $\\mathrm{Lucky}(w)$. (a) If $n$ is even, then $w_1 = 1$, $w_n = n$ and $\\mathrm{Lucky}(w)$ has exactly $2$ elements. (b) If $n$ is odd and $w_1 = 1$, then $w_n = n$\u2026","title":"Stirling permutations whose second lucky car is $n$"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/is-admissible-lucky-set-one-pred-two-mul","kind":"node","lean":"MetaMathlibExt.isAdmissibleLuckySet_one_pred_two_mul_sub_two_iff","parent":27253,"reading_title":{"format":"inline-parts-v1","parts":[["text","When "],["math","\\(\\{1, n - 1, 2n - 2\\}\\)"],["text"," is an admissible lucky set"]],"source":"When $\\{1, n - 1, 2n - 2\\}$ is an admissible lucky set"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#is-admissible-lucky-set-one-pred-two-mul","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"For every integer $n \\ge 4$, the set $\\{1, n - 1, 2n - 2\\}$ is an $n$-admissible lucky set if and only if $n$ is even.","title":"When $\\{1, n - 1, 2n - 2\\}$ is an admissible lucky set"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-total-displacement-comp-perm","kind":"node","lean":"MetaMathlibExt.stirlingTotalDisplacement_comp_perm","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-total-displacement-comp-perm","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$ and let $\\sigma$ be a permutation of $\\{1, \\dots, 2n\\}$. Read words of length $2n$ as parking functions, and let $d(v)$ denote the total displacement of a word $v$, the sum of the displacements of its cars. Then the rearranged word $w \\circ \\sigma = w_{\\sigma(1)} \\cdots w_{\\sigma(2n)}$ satisfies $d(w \\circ \\sigma) = d(w)$.","title":"Total displacement of a Stirling permutation is invariant under rearrangement"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-total-displacement-eq","kind":"node","lean":"MetaMathlibExt.stirlingTotalDisplacement_eq","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-total-displacement-eq","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$, read as a parking function of length $2n$. Then the total displacement of $w$, the sum $d(1) + \\dots + d(2n)$ of the displacements of its cars, is $n^2$.","title":"Total displacement of a Stirling permutation"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/stirling-card-zero-displacement","kind":"node","lean":"MetaMathlibExt.stirling_card_zero_displacement","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#stirling-card-zero-displacement","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $n \\ge 1$ and let $w$ be a Stirling permutation of order $n$, read as a parking function of length $2n$. The number of cars $i \\in \\{1, \\dots, 2n\\}$ with displacement $d(i) = 0$ is at least $1$ and at most $n$.","title":"Number of cars with zero displacement"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/extremely-lucky-unlucky-car","kind":"node","lean":"MetaMathlibExt.extremely_lucky_unlucky_car","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#extremely-lucky-unlucky-car","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w = w_1 \\cdots w_{2n}$ be a Stirling permutation of order $n$ that is extremely lucky, that is, its set $\\mathrm{Lucky}(w)$ of lucky cars has exactly $n$ elements, and let $1 \\le i \\le n$. If car $j$ is unlucky and exactly $i - 1$ of the cars $1, \\dots, j - 1$ are unlucky, then car $j$ prefers spot $w_j = n - i + 1$ and has displacement $d(j) = 2i - 1$.","title":"Unlucky cars of an extremely lucky Stirling permutation"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/extremely-lucky-eq-of-displacement-eq","kind":"node","lean":"MetaMathlibExt.extremely_lucky_eq_of_displacement_eq","parent":27253,"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#extremely-lucky-eq-of-displacement-eq","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"Let $w$ and $w'$ be Stirling permutations of order $n$, and suppose that $w$ is extremely lucky, that is, its set $\\mathrm{Lucky}(w)$ of lucky cars has exactly $n$ elements. If every car has the same displacement in $w'$ as in $w$ (that is, $\\mathrm{dis}(w') = \\mathrm{dis}(w)$), then $w' = w$.","title":"Extremely lucky Stirling permutations are determined by their displacements"},{"id":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/card-stirling-displacement-nonzero-parts","kind":"node","lean":"MetaMathlibExt.card_stirling_displacement_nonzero_parts","parent":27253,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stirling permutations with "],["math","\\(n\\)"],["text"," nonzero displacements"]],"source":"Stirling permutations with $n$ nonzero displacements"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe/index.html#card-stirling-displacement-nonzero-parts","scope":"Combinatorics/parking-functions/lucky-and-displacement-statistics-of-stirling-pe","status":"can_prove","summary":"For every natural number $n$, the number of Stirling permutations $w$ of order $n$ in which exactly $n$ cars have nonzero displacement (equivalently, whose displacement composition $\\mathrm{dis}(w)$ has exactly $n$ nonzero parts) is the Catalan number $C_n$.","title":"Stirling permutations with $n$ nonzero displacements"},{"data_url":"scopes/Combinatorics/parking-functions/metered-parking-functions.json","id":"Combinatorics/parking-functions/metered-parking-functions","kind":"scope","navigation":"module","parent":27146,"result":false,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","title":"Metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-outcome","kind":"node","lean":"MetaMathlibExt.meteredOutcome","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-outcome","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"Let $t, n \\in \\mathbb{N}$, number the $n$ spots $0, \\dots, n-1$, and let $a_0, a_1, \\dots, a_{m-1}$ be a list of preferences in $\\mathbb{N}$. Cars arrive in order, and car $i$ parks in the least spot $s$ with $a_i \\le s < n$ that is not taken by one of the previous $t$ cars (cars $\\max(0, i - t), \\dots, i - 1$) that parked; if there is no such spot, car $i$ fails. The outcome is the list whose $i$-th entry is the\u2026","title":"Outcome of the metered parking scheme"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-outcome-one-two-ones","kind":"node","lean":"MetaMathlibExt.meteredOutcome_one_two_ones","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-outcome-one-two-ones","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"In the $1$-metered parking scheme on two spots $\\{0, 1\\}$, if three cars arrive all preferring spot $0$, then they park in spots $0$, $1$ and $0$, in that order.","title":"A 1-metered example in which every car parks"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-outcome-one-two-fail","kind":"node","lean":"MetaMathlibExt.meteredOutcome_one_two_fail","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-outcome-one-two-fail","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"In the $1$-metered parking scheme on two spots $\\{0, 1\\}$, if three cars arrive with preferences $0$, $0$, $1$, then the first two cars park in spots $0$ and $1$ and the third car fails to park.","title":"A 1-metered example in which the third car fails"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-outcome-example-six","kind":"node","lean":"MetaMathlibExt.meteredOutcome_example_six","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-outcome-example-six","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"On the six spots $\\{0, \\dots, 5\\}$, for seven cars with preferences $(2, 2, 2, 2, 3, 4, 5)$, the $2$-metered parking scheme parks the cars in spots $(2, 3, 4, 2, 3, 4, 5)$, while in the $1$-metered scheme the first six cars park in spots $(2, 3, 2, 3, 4, 5)$ and the seventh car fails to park.","title":"Metered outcomes of a seven-car example on six spots"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-metered-parking-function","kind":"node","lean":"MetaMathlibExt.IsMeteredParkingFunction","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-metered-parking-function","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"Let $m, n, t \\in \\mathbb{N}$, number the $n$ spots $0, \\dots, n-1$, and let $f : \\{0, \\dots, m-1\\} \\to \\{0, \\dots, n-1\\}$ give the preferred spot of each of $m$ cars. In the $t$-metered scheme the cars arrive in the order $0, 1, \\dots, m-1$, and car $i$ parks in the least spot $s$ with $f(i) \\le s \\le n - 1$ that is not occupied by any of the (at most $t$) cars that arrived just before car $i$; it fails to park if\u2026","title":"Metered parking function"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count","kind":"node","lean":"MetaMathlibExt.meteredParkingCount","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"For natural numbers $t$, $m$ and $n$, $\\mathrm{mpf}_{m,n}(t)$ is the number of $t$-metered $(m, n)$-parking functions, that is, of maps $f$ from the $m$ cars $\\{0, \\dots, m-1\\}$ to the $n$ spots $\\{0, \\dots, n-1\\}$ under which every car parks in the $t$-metered parking scheme.","title":"Number of metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-one-two-two","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_one_two_two","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-one-two-two","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"The number of $1$-metered $(2, 2)$-parking functions is $\\mathrm{mpf}_{2,2}(1) = 3$.","title":"1-metered parking functions with two cars and two spots"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-lucky-count","kind":"node","lean":"MetaMathlibExt.meteredLuckyCount","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-lucky-count","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"For natural numbers $t$, $m$, $n$ and a preference function $f : \\{0, \\dots, m - 1\\} \\to \\{0, \\dots, n - 1\\}$ (car $i$ prefers spot $f(i)$), the lucky count of $f$ under the $t$-metered parking scheme on $n$ spots is the number of cars $i$ that park in their preferred spot $f(i)$.","title":"Lucky cars in the metered scheme"},{"id":"Combinatorics/parking-functions/metered-parking-functions/lace-step","kind":"node","lean":"MetaMathlibExt.laceStep","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#lace-step","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"Given a list of blocks (each a list of natural numbers) and a natural number $a$: if the last block is $(p)$ with $a = p$, or the last block has at least two entries and ends with $e$ where $a = e + 1$, then $a$ is appended to the last block; otherwise, including when there is no block, a new block $(a)$ is added at the end.","title":"One step of the lace decomposition"},{"id":"Combinatorics/parking-functions/metered-parking-functions/lace-decomposition","kind":"node","lean":"MetaMathlibExt.laceDecomposition","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#lace-decomposition","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"A lace is a sequence of natural numbers of the form $(p)$ or $(p, p, p+1, \\dots, p+\\ell-2)$ for some natural number $p$ and some length $\\ell \\ge 2$. The lace decomposition of a finite sequence $(a_1, \\dots, a_k)$ of natural numbers is its partition into blocks of consecutive entries built greedily from left to right: starting with no blocks, each entry $a_i$ in turn is appended to the last block if the result is\u2026","title":"Lace decomposition"},{"id":"Combinatorics/parking-functions/metered-parking-functions/lace-decomposition-examples","kind":"node","lean":"MetaMathlibExt.laceDecomposition_examples","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#lace-decomposition-examples","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"The lace decomposition of $(3, 3, 3, 3, 4, 6)$ is $((3, 3), (3, 3, 4), (6))$, and the lace decomposition of $(4, 1, 2, 2, 3, 4, 5, 5, 1, 2)$ is $((4), (1), (2, 2, 3, 4, 5), (5), (1), (2))$.","title":"Examples of lace decompositions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-window-count-ge","kind":"node","lean":"MetaMathlibExt.metered_window_count_ge","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-window-count-ge","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Number the cars $1, \\dots, m$ and the spots $1, \\dots, n$, and let $a_k$ be the preferred spot of car $k$. If $(a_1, \\dots, a_m)$ is a $t$-metered $(m, n)$-parking function, then for every $i$ with $1 \\le i \\le n$ and every $j$ with $1 \\le j \\le m - t$ (truncated subtraction, so there is no such $j$ when $m \\le t$),","title":"Window condition for metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-eq-zero","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_eq_zero","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-eq-zero","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $m, n, t$ be natural numbers with $m > n$ and $t \\ge n$. Then there are no $t$-metered $(m, n)$-parking functions, that is, $\\mathrm{mpf}_{m,n}(t) = 0$.","title":"Vanishing of metered parking counts"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-metered-parking-function-iff-of-le","kind":"node","lean":"MetaMathlibExt.isMeteredParkingFunction_iff_of_le","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-metered-parking-function-iff-of-le","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $m \\le n$ and $t \\ge m - 1$. A map $f : \\{0, \\dots, m-1\\} \\to \\{0, \\dots, n-1\\}$ is a $t$-metered parking function if and only if it is a partial parking function with $m$ cars and $n$ spots.","title":"Metered parking functions with a long meter are classical"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-metered-parking-function-2","kind":"node","lean":"MetaMathlibExt.IsPartialParkingFunction.isMeteredParkingFunction","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-metered-parking-function-2","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $m \\le n$. Every partial parking function $f$ with $m$ cars and $n$ spots is a $t$-metered parking function for every $t \\in \\mathbb{N}$.","title":"Classical parking functions are metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-outcome-displacement-le","kind":"node","lean":"MetaMathlibExt.meteredOutcome_displacement_le","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-outcome-displacement-le","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $t$, $m$, $n$ be natural numbers, let $f : \\{0, \\dots, m - 1\\} \\to \\{0, \\dots, n - 1\\}$ be a preference function and let $i < m$ be a car. Under the $t$-metered parking scheme on $n$ spots, if car $i$ parks in spot $s$ then $s \\le f(i) + t$, that is, its displacement is at most $t$; and if car $i$ fails to park then $f(i) + t \\ge n$.","title":"Displacement bound in the metered scheme"},{"id":"Combinatorics/parking-functions/metered-parking-functions/pow-sub-le-metered-parking-count","kind":"node","lean":"MetaMathlibExt.pow_sub_le_meteredParkingCount","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#pow-sub-le-metered-parking-count","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"For all natural numbers $t$, $m$ and $n$, $\\mathrm{mpf}_{m,n}(t) \\ge (n - t)^m$, where $n - t$ is truncated subtraction (equal to $0$ when $t \\ge n$).","title":"Lower bound for the number of metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-two","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_two","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-two","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"For every natural number $t \\ge 1$ and every natural number $n$, the number of $t$-metered $(2, n)$-parking functions is $\\mathrm{mpf}_{2,n}(t) = \\max(n^2 - 1, 0)$, that is, $n^2 - 1$ for $n \\ge 1$ and $0$ for $n = 0$.","title":"Metered parking functions with two cars"},{"id":"Combinatorics/parking-functions/metered-parking-functions/one-metered-lucky-iff-lace-start","kind":"node","lean":"MetaMathlibExt.one_metered_lucky_iff_lace_start","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#one-metered-lucky-iff-lace-start","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $0 < m \\le n$ and let $\\alpha = (a_1, \\dots, a_m)$, with each $a_i \\in \\{1, \\dots, n\\}$, be any sequence of preferred spots of $m$ cars on $n$ spots. Under the $1$-metered parking scheme, car $i$ is lucky (it parks in its preferred spot $a_i$) if and only if position $i$ is the first position of one of the laces of the lace decomposition $\\mathrm{lace}(\\alpha)$.","title":"Lucky cars of 1-metered parking are the lace starts"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-metered-parking-function-one-iff-lace","kind":"node","lean":"MetaMathlibExt.isMeteredParkingFunction_one_iff_lace","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-metered-parking-function-one-iff-lace","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $0 < m \\le n$ and $f : \\{0, \\dots, m-1\\} \\to \\{0, \\dots, n-1\\}$. Then $f$ is a $1$-metered parking function if and only if, in the lace decomposition of $(f(0), \\dots, f(m-1))$, the last spot $n - 1$ occurs in no lace other than as its first entry.","title":"One-metered parking functions and laces"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-metered-parking-function-one-two-iff","kind":"node","lean":"MetaMathlibExt.isMeteredParkingFunction_one_two_iff_lace","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-metered-parking-function-one-two-iff","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $m \\ge 1$ and $f : \\{0, \\dots, m-1\\} \\to \\{0, 1\\}$. Then $f$ is a $1$-metered parking function with $m$ cars and $2$ spots if and only if every lace of the lace decomposition of $(f(0), \\dots, f(m-1))$ that contains the value $1$ has length $1$.","title":"One-metered parking functions with two spots"},{"id":"Combinatorics/parking-functions/metered-parking-functions/one-metered-last-spot-card-eq","kind":"node","lean":"MetaMathlibExt.one_metered_last_spot_card_eq","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#one-metered-last-spot-card-eq","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $1 \\le m \\le j < k \\le n$ and number the spots $1, \\dots, n$. The number of $1$-metered $(m, n)$-parking functions whose last car parks in spot $j$ equals the number of those whose last car parks in spot $k$.","title":"Equal last-spot counts for 1-metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/one-metered-last-spot-n-card","kind":"node","lean":"MetaMathlibExt.one_metered_last_spot_n_card","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#one-metered-last-spot-n-card","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $1 \\le m \\le n$ and number the spots $1, \\dots, n$. The number of $1$-metered $(m, n)$-parking functions whose last car parks in spot $n$ equals $\\mathrm{mpf}_{m-1,n}(1)$, the number of $1$-metered $(m - 1, n)$-parking functions.","title":"1-metered parking functions whose last car parks in the last spot"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-one-recursion","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_one_recursion","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-one-recursion","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Write $\\mathrm{mpf}_{m,n}(1)$ for the number of $1$-metered $(m, n)$-parking functions. For every natural number $n$, $\\mathrm{mpf}_{1,n}(1) = n$ and $\\mathrm{mpf}_{0,n}(1) = 1$, and for every natural number $m$ with $1 \\le m \\le n$,","title":"Lucas-type recursion for 1-metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-one-diag","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_one_diag","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-one-diag","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"For every natural number $n > 2$, the number of $1$-metered $(n, n)$-parking functions is","title":"1-metered parking functions with as many cars as spots"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-one-binet","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_one_binet","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-one-binet","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $m, n$ be natural numbers with $m \\le n + 1$ and $n > 2$. Then the number $\\mathrm{mpf}_{m,n}(1)$ of $1$-metered $(m, n)$-parking functions satisfies, as real numbers,","title":"Binet formula for 1-metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-one-three","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_one_three","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-one-three","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"For every natural number $n \\ge 2$, the number of $1$-metered $(3, n)$-parking functions is $\\mathrm{mpf}_{3,n}(1) = n^3 - 2n$.","title":"1-metered parking functions with three cars"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-all-lucky-card","kind":"node","lean":"MetaMathlibExt.metered_all_lucky_card","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-all-lucky-card","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $m \\ge 2$, $n \\ge 2$ and $t \\le m - 1$. The number of $t$-metered $(m, n)$-parking functions in which every car parks in its preferred spot is $n(n-1)\\cdots(n-t+1) \\cdot (n-t)^{m-t}$, where the falling factorial $n(n-1)\\cdots(n-t+1)$ has $t$ factors (it equals $1$ for $t = 0$ and $0$ for $t > n$).","title":"Metered parking functions in which every car is lucky"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-sub-two","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_sub_two","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-sub-two","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $m, n$ be natural numbers with $2 < m \\le n + 1$. Then the number $\\mathrm{mpf}_{m,n}(m-2)$ of $(m-2)$-metered $(m, n)$-parking functions is","title":"Closed form for (m-2)-metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/parking-function-first-entry-card","kind":"node","lean":"MetaMathlibExt.parkingFunction_first_entry_card","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#parking-function-first-entry-card","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $1 \\le m \\le n$ and $1 \\le j \\le n$, and number the spots $1, \\dots, n$. Write","title":"Parking functions with a given first entry"},{"id":"Combinatorics/parking-functions/metered-parking-functions/metered-parking-count-sub-two-pred","kind":"node","lean":"MetaMathlibExt.meteredParkingCount_sub_two_pred","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#metered-parking-count-sub-two-pred","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $m \\ge 2$ be a natural number. The number $\\mathrm{mpf}_{m,m-1}(m-2)$ of $(m-2)$-metered $(m, m-1)$-parking functions equals the sum, over all parking functions $f$ of length $m - 1$, of $f(0) + 1$ (the preferred spot of the first car, with spots numbered from $1$), and it also equals","title":"Metered parking functions with one spot fewer than cars"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-parking-shuffle","kind":"node","lean":"MetaMathlibExt.IsParkingShuffle","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-parking-shuffle","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"Let $m'$, $n$, $k$ be natural numbers and let $\\pi : \\{0, \\dots, m' - 1\\} \\to \\{0, \\dots, n - 1\\}$, with one-based letters $w_i = \\pi(i) + 1$. Then $\\pi$ is a parking-function shuffle with parameters $(m', n, k)$ when all of the following hold: no letter $w_i$ equals $k$; exactly $\\max(m' + k - n, 0)$ letters satisfy $w_i < k$; exactly $\\max(n - k, 0)$ letters satisfy $w_i > k$; for every $s$ with\u2026","title":"Parking-function shuffles"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-parking-shuffle-example","kind":"node","lean":"MetaMathlibExt.isParkingShuffle_example","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-parking-shuffle-example","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"fully_proved","summary":"The word $(2, 6, 1, 0, 6, 1)$ with entries in $\\{0, \\dots, 7\\}$, which is $(3, 7, 2, 1, 7, 2)$ in one-based values, is a $(7, 8)$-parking-function shuffle with parameter $k = 6$, that is, it lies in $\\mathrm{Sh}_7(5, 2)$. In one-based values it is a shuffle of the parking function $(3, 2, 1, 2)$ with $4$ cars and $5$ spots and of $(7, 7) = (1, 1) + (6, 6)$.","title":"An example of a parking-function shuffle"},{"id":"Combinatorics/parking-functions/metered-parking-functions/first-entry-set-eq-iff-is-parking-shuffl","kind":"node","lean":"MetaMathlibExt.firstEntrySet_eq_iff_isParkingShuffle","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#first-entry-set-eq-iff-is-parking-shuffl","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $1 \\le m \\le n$ and $1 \\le k \\le n$, and let $w = (w_1, \\dots, w_{m-1})$ be a word with entries in $\\{0, \\dots, n-1\\}$. Then the set of spots $j \\in \\{0, \\dots, n-1\\}$ for which $(j, w_1, \\dots, w_{m-1})$ is a partial parking function with $m$ cars and $n$ spots equals $\\{0, \\dots, k-1\\}$ if and only if $w$ lies in the set $\\mathrm{Sh}_m(k - 1, n - k)$ of $(m, n)$-parking-function shuffles with parameter $k$.","title":"Parking-function shuffles and the first entry"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-parking-shuffle-card","kind":"node","lean":"MetaMathlibExt.isParkingShuffle_card","parent":27286,"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-parking-shuffle-card","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $1 \\le m \\le n$ and $1 \\le k \\le n$, and write $\\mathrm{Sh}_m(k - 1, n - k)$ for the set of $(m, n)$-parking-function shuffles with parameter $k$, words of length $m - 1$ with entries in $\\{0, \\dots, n-1\\}$. Then $|\\mathrm{Sh}_m(k - 1, n - k)|$ equals $\\binom{m-1}{n-k}(n-m+1)\\,k^{m-n+k-2}(n-k+1)^{n-k-1}$ if $k > n - m + 1$, equals $m^{m-2}$ if $k = n - m + 1$, and equals $0$ if $k < n - m + 1$. The exponent\u2026","title":"Number of parking-function shuffles"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-metered-parking-function-sub-two-iff","kind":"node","lean":"MetaMathlibExt.isMeteredParkingFunction_sub_two_iff","parent":27286,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\((m-2)\\)"],["text","-metered parking functions"]],"source":"Characterization of $(m-2)$-metered parking functions"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-metered-parking-function-sub-two-iff","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $2 < m \\le n + 1$, $f : \\{0, \\dots, m-1\\} \\to \\{0, \\dots, n-1\\}$, $1 \\le k \\le n$ and $j \\in \\mathbb{N}$. Write $\\mathrm{Sh}_p(q - 1, N - q)$ for the set of $(p, N)$-parking-function shuffles with parameter $q$, words of length $p - 1$ with entries in $\\{0, \\dots, N-1\\}$. Suppose that the middle word $w = (f(1), \\dots, f(m-2))$ lies in $\\mathrm{Sh}_{m-1}(k - 1, n - k)$, and that either $m = n + 1$ and $j = 0$,\u2026","title":"Characterization of $(m-2)$-metered parking functions"},{"id":"Combinatorics/parking-functions/metered-parking-functions/is-metered-parking-function-pred-iff","kind":"node","lean":"MetaMathlibExt.isMeteredParkingFunction_pred_iff","parent":27286,"reading_title":{"format":"inline-parts-v1","parts":[["text","Metered parking functions with meter "],["math","\\(n - 1\\)"],["text"," and more cars than spots"]],"source":"Metered parking functions with meter $n - 1$ and more cars than spots"},"result":true,"route":"../roadmap/Combinatorics/parking-functions/metered-parking-functions/index.html#is-metered-parking-function-pred-iff","scope":"Combinatorics/parking-functions/metered-parking-functions","status":"can_prove","summary":"Let $k > 0$ and $f : \\{0, \\dots, n+k-1\\} \\to \\{0, \\dots, n-1\\}$, and let $\\pi_0, \\dots, \\pi_{n-1}$ be the $(n-1)$-metered outcome of the first $n$ preferences $(f(0), \\dots, f(n-1))$ on $n$ spots, where $\\pi_c$ is the spot in which car $c$ parks, if it parks. Then $f$ is an $(n-1)$-metered parking function if and only if $(f(0), \\dots, f(n-1))$ is a parking function and, for every $i \\in \\{0, \\dots, k-1\\}$, car\u2026","title":"Metered parking functions with meter $n - 1$ and more cars than spots"},{"data_url":"scopes/Combinatorics/set-partitions.json","id":"Combinatorics/set-partitions","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/set-partitions/index.html","scope":"Combinatorics/set-partitions","status":"can_prove","summary":"Partitions of a finite set into blocks: finite partition API, ordered and cyclic set partitions, restricted-growth words, peaks and rises, noncrossing partitions, merging-free and marriageable-singles partitions, genus of a partition, and linear chord diagrams and hypermonopoles.","title":"Set partitions"},{"id":"Combinatorics/set-partitions/is-basic-set-partition","kind":"node","lean":"MetaMathlibExt.IsBasicSetPartition","parent":27324,"result":true,"route":"../roadmap/Combinatorics/set-partitions/index.html#is-basic-set-partition","scope":"Combinatorics/set-partitions","status":"fully_proved","summary":"A partition $P$ of $\\{1, \\dots, n\\}$ is basic when its blocks can be listed without repetition as $B_1, \\dots, B_k$ with $|B_1| \\ge |B_2| \\ge \\dots \\ge |B_k|$ so that $B_1 \\cup \\dots \\cup B_i = \\{1, \\dots, |B_1| + \\dots + |B_i|\\}$ for every $i$ with $1 \\le i \\le k$.","title":"Basic set partition"},{"id":"Combinatorics/set-partitions/callan-sequence","kind":"node","lean":"MetaMathlibExt.CallanSequence","parent":27324,"result":true,"route":"../roadmap/Combinatorics/set-partitions/index.html#callan-sequence","scope":"Combinatorics/set-partitions","status":"fully_proved","summary":"Let $k$ and $n$ be natural numbers. Form the blue base set $\\{1, \\dots, k\\} \\cup \\{\\star_B\\}$ and the red base set $\\{1, \\dots, n\\} \\cup \\{\\star_R\\}$, where the blue star $\\star_B$ and the red star $\\star_R$ are new elements. An (unbarred) Callan sequence of size $k \\times n$ consists of a natural number $m \\le n$ together with a sequence of $m + 1$ pairs $(B_0, R_0), \\dots, (B_m, R_m)$ such that the blue blocks\u2026","title":"Callan sequences"},{"id":"Combinatorics/set-partitions/set-partition-two-rise-ordinary-generati","kind":"node","lean":"MetaMathlibExt.setPartition_twoRise_ordinaryGeneratingFunction","parent":27324,"result":true,"route":"../roadmap/Combinatorics/set-partitions/index.html#set-partition-two-rise-ordinary-generati","scope":"Combinatorics/set-partitions","status":"fully_proved","summary":"Fix $k \\ge 1$. For $n, r \\ge 0$ let $a_{n,r}$ be the number of words $\\pi = \\pi_1 \\cdots \\pi_n$ on the alphabet $\\{0, \\dots, k-1\\}$ that use every letter, satisfy the restricted growth condition that each $\\pi_i$ is at most the number of distinct letters among $\\pi_1, \\dots, \\pi_{i-1}$, and have exactly $r$ rises (the source's $2$-rises), that is, positions $i < n$ with $\\pi_i < \\pi_{i+1}$. These words encode the\u2026","title":"Generating function for set partitions by rises"},{"id":"Combinatorics/set-partitions/is-merging-free-partition","kind":"node","lean":"MetaMathlibExt.IsMergingFreePartition","parent":27324,"result":true,"route":"../roadmap/Combinatorics/set-partitions/index.html#is-merging-free-partition","scope":"Combinatorics/set-partitions","status":"fully_proved","summary":"Let $n$ be a natural number. A finite sequence $(B_1, \\dots, B_m)$ of subsets of $\\{0, 1, \\dots, n-1\\}$ is a merging-free partition when the sets $B_i$ are nonempty, pairwise disjoint and cover $\\{0, 1, \\dots, n-1\\}$; their minima increase strictly, $\\min B_1 < \\min B_2 < \\dots < \\min B_m$; and $\\max B_i > \\min B_{i+1}$ for every $i < m$.","title":"Merging-free partitions"},{"id":"Combinatorics/set-partitions/is-ordered-set-partition","kind":"node","lean":"MetaMathlibExt.IsOrderedSetPartition","parent":27324,"result":true,"route":"../roadmap/Combinatorics/set-partitions/index.html#is-ordered-set-partition","scope":"Combinatorics/set-partitions","status":"fully_proved","summary":"For a finite set $V$, a list $(B_1, \\dots, B_k)$ of finite sets is an ordered set partition (set composition) of $V$ when every block $B_i$ is nonempty and contained in $V$, the blocks are pairwise disjoint, and $B_1 \\cup \\dots \\cup B_k = V$.","title":"Ordered set partitions of a finite set"},{"id":"Combinatorics/set-partitions/restricted-growth-word-monomial-generati","kind":"node","lean":"MetaMathlibExt.restricted_growth_word_monomial_generating_function","parent":27324,"result":true,"route":"../roadmap/Combinatorics/set-partitions/index.html#restricted-growth-word-monomial-generati","scope":"Combinatorics/set-partitions","status":"fully_proved","summary":"Fix $k \\ge 0$. A word $u_1 u_2 \\cdots u_n$ in the letters $1, \\dots, k$ is a restricted-growth word if for every $i$, either $u_i = 1$ or $u_i \\le u_j + 1$ for some $j < i$ (so a nonempty such word begins with $1$). For $d = (d_1, \\dots, d_k) \\in \\mathbb{N}^k$, let $r(d)$ be the number of restricted-growth words in which each letter $a$ occurs exactly $d_a + 1$ times. Then, in the ring of formal power series in\u2026","title":"Restricted-growth word generating function"},{"data_url":"scopes/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions.json","id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","kind":"scope","navigation":"module","parent":27324,"result":false,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","title":"Block sizes and pullbacks of set partitions"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/card-prod","kind":"node","lean":"Finpartition.cardProd","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#card-prod","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"For a partition $P$ of a finite set $s$, a commutative monoid $R$ and a function $f : \\mathbb{N} \\to R$, the block-size product of $f$ over $P$ is $\\prod_{B \\in P} f(|B|)$, the product over the blocks $B$ of $P$ of $f$ applied to the cardinality of $B$.","title":"Product over block sizes"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/card-le-of-mem-parts","kind":"node","lean":"Finpartition.card_le_of_mem_parts","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#card-le-of-mem-parts","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"If $P$ is a partition of a finite set $s$ and $B$ is a block of $P$, then $|B| \\le |s|$.","title":"Blocks are no larger than the ground set"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/one-le-card-of-mem-parts","kind":"node","lean":"Finpartition.one_le_card_of_mem_parts","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#one-le-card-of-mem-parts","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Every block $B$ of a partition of a finite set $s$ satisfies $|B| \\ge 1$.","title":"Blocks are nonempty"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/card-lt-of-ne-top","kind":"node","lean":"Finpartition.card_lt_of_ne_top","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#card-lt-of-ne-top","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Let $P$ be a partition of a finite set $s$ that is not the coarsest partition $\\top$ of $s$ (the one whose only block is $s$ when $s$ is nonempty). Then every block $B$ of $P$ satisfies $|B| < |s|$.","title":"Blocks of a partition other than the coarsest are smaller than the ground set"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/eq-top-iff-card-parts-eq-one","kind":"node","lean":"Finpartition.eq_top_iff_card_parts_eq_one","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#eq-top-iff-card-parts-eq-one","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"If $P$ is a partition of a nonempty finite set $s$, then $P$ is the coarsest partition $\\top$ of $s$, whose only block is $s$, if and only if $P$ has exactly one block.","title":"Recognizing the one-block partition"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/card-parts-pos-iff","kind":"node","lean":"Finpartition.card_parts_pos_iff","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#card-parts-pos-iff","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"A partition $P$ of a finite set $s$ has at least one block, that is, its number of blocks is positive, if and only if $s$ is nonempty.","title":"A partition has a block exactly when its ground set is nonempty"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/eq-bot-iff-card-parts-eq-card","kind":"node","lean":"Finpartition.eq_bot_iff_card_parts_eq_card","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#eq-bot-iff-card-parts-eq-card","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"A partition $P$ of a finite set $s$ is the finest partition $\\bot$ of $s$, whose blocks are the singletons $\\{a\\}$ for $a \\in s$, if and only if the number of blocks of $P$ equals $|s|$.","title":"Recognizing the partition into singletons by its number of blocks"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/card-prod-map-monoid-hom","kind":"node","lean":"Finpartition.cardProd_mapMonoidHom","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#card-prod-map-monoid-hom","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Let $P$ be a partition of a finite set $s$, let $R$ and $S$ be commutative monoids, let $f : \\mathbb{N} \\to R$, and let $g : R \\to S$ be a monoid homomorphism. Then $g\\bigl(\\prod_{B \\in P} f(|B|)\\bigr) = \\prod_{B \\in P} g(f(|B|))$, where both products run over the blocks $B$ of $P$.","title":"Monoid homomorphisms preserve block-size products"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/comap","kind":"node","lean":"Finpartition.comap","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#comap","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"The pull-back of a partition `P` of the finite type `\u03b2` along a map `f : \u03b1 \u2192 \u03b2`: the blocks are the nonempty preimages `f \u207b\u00b9' B` of the blocks `B` of `P`.","title":"Pull-back of a finite partition along a map"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/mem-comap-parts","kind":"node","lean":"Finpartition.mem_comap_parts","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#mem-comap-parts","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"The blocks of `P.comap f` are the nonempty preimages of the blocks of `P`.","title":"Blocks of a pulled-back partition"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/comap-refl","kind":"node","lean":"Finpartition.comap_refl","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#comap-refl","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Pulling back along the identity does nothing.","title":"Pulling back a partition along the identity"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/comap-comap","kind":"node","lean":"Finpartition.comap_comap","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#comap-comap","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Pulling back along a composite is pulling back twice.","title":"Pulling back a partition along a composite"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/parts-comap-equiv","kind":"node","lean":"Finpartition.parts_comap_equiv","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#parts-comap-equiv","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Along an equivalence `e`, the blocks of `P.comap e` are the images of the blocks of `P` under `e.symm`.","title":"Blocks of a partition pulled back along an equivalence"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/card-parts-comap-equiv","kind":"node","lean":"Finpartition.card_parts_comap_equiv","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#card-parts-comap-equiv","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Pulling back along an equivalence preserves the number of blocks.","title":"Pull-back along an equivalence preserves the number of blocks"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/card-prod-comap-equiv","kind":"node","lean":"Finpartition.cardProd_comap_equiv","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#card-prod-comap-equiv","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Pulling back along an equivalence preserves every product over the block sizes.","title":"Pull-back along an equivalence preserves block-size products"},{"id":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/comap-symm-comap","kind":"node","lean":"Finpartition.comap_symm_comap","parent":27331,"result":true,"route":"../roadmap/Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions/index.html#comap-symm-comap","scope":"Combinatorics/set-partitions/block-sizes-and-pullbacks-of-set-partitions","status":"fully_proved","summary":"Pulling back along `e` and then along `e.symm` gives back the partition.","title":"Pulling back along an equivalence and its inverse"},{"data_url":"scopes/Combinatorics/set-partitions/genus-of-a-set-partition.json","id":"Combinatorics/set-partitions/genus-of-a-set-partition","kind":"scope","navigation":"module","parent":27324,"result":false,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"can_prove","title":"Genus of a set partition"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/is-increasing-cycle-perm","kind":"node","lean":"MetaMathlibExt.IsIncreasingCyclePerm","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#is-increasing-cycle-perm","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"fully_proved","summary":"A permutation $\\tau$ of $\\{0, \\dots, n-1\\}$ is an increasing-cycle permutation when, for all $x$ and $y$ in $\\{0, \\dots, n-1\\}$ such that $\\tau(x) \\le x$ and $y$ lies in the same cycle of $\\tau$ as $x$, we have $\\tau(x) \\le y$. That is, whenever $\\tau(x) \\le x$, the value $\\tau(x)$ is the least element of the cycle of $x$; equivalently, each cycle of $\\tau$ runs through its elements in increasing order and returns\u2026","title":"Increasing-cycle permutation"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/set-partition-genus","kind":"node","lean":"MetaMathlibExt.setPartitionGenus","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#set-partition-genus","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"fully_proved","summary":"Let $\\sigma$ be the cyclic permutation $i \\mapsto i + 1 \\pmod n$ of $\\{0, \\dots, n-1\\}$. For a permutation $\\tau$ of $\\{0, \\dots, n-1\\}$, thought of as encoding the set partition whose blocks are its cycles, let $|\\alpha|$ be the number of cycles of $\\tau$ and $c$ the number of cycles of $\\sigma \\circ \\tau^{-1}$, fixed points counted as cycles in both. The genus of $\\tau$ is the integer part of\u2026","title":"Genus of a set partition"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/partition-genus-count","kind":"node","lean":"MetaMathlibExt.partitionGenusCount","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#partition-genus-count","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"fully_proved","summary":"For natural numbers $n$ and $g$ and a multiset $\\alpha$ of natural numbers, $C_{n,[\\alpha]}^{(g)}$ is the number of increasing-cycle permutations $\\tau$ of $\\{0, \\dots, n-1\\}$ whose multiset of cycle lengths, fixed points counted as cycles of length $1$, is $\\alpha$ and whose genus is $g$.","title":"Number of set partitions of given type and genus"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/is-increasing-cycle-perm-one","kind":"node","lean":"MetaMathlibExt.isIncreasingCyclePerm_one","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#is-increasing-cycle-perm-one","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"fully_proved","summary":"For every natural number $n$, the identity permutation of $\\{0, \\dots, n-1\\}$ is an increasing-cycle permutation.","title":"The identity is an increasing-cycle permutation"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/set-partition-genus-one","kind":"node","lean":"MetaMathlibExt.setPartitionGenus_one","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#set-partition-genus-one","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"fully_proved","summary":"For every natural number $n$, the genus of the identity permutation of $\\{0, \\dots, n-1\\}$ is $0$; the identity encodes the partition of $\\{0, \\dots, n-1\\}$ into singletons.","title":"The partition into singletons has genus zero"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/is-increasing-cycle-perm-fin-rotate","kind":"node","lean":"MetaMathlibExt.isIncreasingCyclePerm_finRotate","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#is-increasing-cycle-perm-fin-rotate","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"fully_proved","summary":"For every natural number $n$, the cyclic permutation $i \\mapsto i + 1 \\pmod n$ of $\\{0, \\dots, n-1\\}$ is an increasing-cycle permutation.","title":"The cyclic rotation is an increasing-cycle permutation"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/set-partition-genus-fin-rotate","kind":"node","lean":"MetaMathlibExt.setPartitionGenus_finRotate","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#set-partition-genus-fin-rotate","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"fully_proved","summary":"For every natural number $n$, the genus of the cyclic permutation $i \\mapsto i + 1 \\pmod n$ of $\\{0, \\dots, n-1\\}$ is $0$.","title":"The one-block partition has genus zero"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/pairing-genus-generating-function","kind":"node","lean":"MetaMathlibExt.pairing_genus_generatingFunction","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#pairing-genus-generating-function","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"can_prove","summary":"Let $g \\ge 1$, and for $k \\ge 0$ let $C_{2k,[2^k]}^{(g)}$ be the number of genus-$g$ set partitions of $\\{0, \\dots, 2k-1\\}$ into $k$ blocks of size $2$. There is a polynomial $Q^{(g)}$ with rational coefficients, of degree $g - 1$, with $Q^{(g)}(0) = \\frac{(4g)!}{2^{2g}\\,(2g+1)!}$, such that, as formal power series in $u$ over $\\mathbb{Q}$,","title":"Genus generating function of pairings"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/two-block-one-face-and-genus","kind":"node","lean":"MetaMathlibExt.twoBlock_oneFace_and_genus","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#two-block-one-face-and-genus","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"can_prove","summary":"Let $\\sigma$ be the cyclic permutation $i \\mapsto i + 1 \\pmod n$ of $\\{0, \\dots, n-1\\}$, and let $\\tau$ be an increasing-cycle permutation of $\\{0, \\dots, n-1\\}$ with exactly two cycles, fixed points counted as cycles (so $\\tau$ encodes a set partition into two blocks). Then $\\sigma \\circ \\tau^{-1}$ has exactly one cycle of length at least $2$. Moreover, for every $x_0$ in $\\{0, \\dots, n-1\\}$, if $B$ is the cycle\u2026","title":"Genus of two-block partitions"},{"id":"Combinatorics/set-partitions/genus-of-a-set-partition/partition-genus-count-two-blocks","kind":"node","lean":"MetaMathlibExt.partitionGenusCount_twoBlocks","parent":27348,"result":true,"route":"../roadmap/Combinatorics/set-partitions/genus-of-a-set-partition/index.html#partition-genus-count-two-blocks","scope":"Combinatorics/set-partitions/genus-of-a-set-partition","status":"can_prove","summary":"Let $n$, $p$ and $g$ be natural numbers with $2 \\le p < n$. The number $C_{n,[p,\\,n-p]}^{(g)}$ of genus-$g$ set partitions of $\\{0, \\dots, n-1\\}$ into two blocks of sizes $p$ and $n - p$ equals","title":"Genus count of two-block partitions"},{"data_url":"scopes/Combinatorics/set-partitions/linear-chord-diagrams.json","id":"Combinatorics/set-partitions/linear-chord-diagrams","kind":"scope","navigation":"module","parent":27324,"result":false,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"can_prove","title":"Linear chord diagrams"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/mean-short-chords-in-linear-k-chord-diag","kind":"node","lean":"MetaMathlibExt.mean_short_chords_in_linear_k_chord_diagrams","parent":27359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mean number of short chords in linear "],["math","\\(k\\)"],["text","-chord diagrams"]],"source":"Mean number of short chords in linear $k$-chord diagrams"},"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#mean-short-chords-in-linear-k-chord-diag","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"fully_proved","summary":"Let $k \\ge 1$ and $n \\ge 0$ be integers. A linear $k$-chord diagram of length $kn$ is a set of $n$ subsets of $\\{0, 1, \\dots, kn - 1\\}$, each with $k$ elements, whose union is $\\{0, 1, \\dots, kn - 1\\}$; it is therefore a partition of this set into $n$ blocks of size $k$, its chords. A chord is short when it consists of $k$ consecutive integers $\\{i, i + 1, \\dots, i + k - 1\\}$. The number of short chords, averaged\u2026","title":"Mean number of short chords in linear $k$-chord diagrams"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/asymptotic-normality-of-short-chords-in","kind":"node","lean":"MetaMathlibExt.noncrossingLinearKChordDiagram_shortChords_asymptoticallyNormal","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#asymptotic-normality-of-short-chords-in","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"can_prove","summary":"The short-chord count in a uniformly chosen non-crossing linear `k`-chord diagram is asymptotically normal with the stated mean and variance.","title":"Asymptotic normality of short chords in non-crossing chord diagrams"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/is-linear-chord-diagram","kind":"node","lean":"MetaMathlibExt.IsLinearChordDiagram","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#is-linear-chord-diagram","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"fully_proved","summary":"A linear chord diagram on the $2n$ vertices $0, \\dots, 2n-1$, laid out on a line, is encoded by a map $\\sigma$ from $\\{0, \\dots, 2n-1\\}$ to itself that is a fixed-point-free involution: $\\sigma(i) \\ne i$ and $\\sigma(\\sigma(i)) = i$ for every vertex $i$. Vertex $i$ is joined by a chord to $\\sigma(i)$.","title":"Linear chord diagrams"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/is-short-chord-vertex","kind":"node","lean":"MetaMathlibExt.IsShortChordVertex","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#is-short-chord-vertex","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"fully_proved","summary":"Let $\\sigma$ be a map from $\\{0, \\dots, 2n-1\\}$ to itself (for a linear chord diagram, vertex $i$ is joined by a chord to $\\sigma(i)$). A vertex $i$ is an endpoint of a short chord of $\\sigma$ when $\\sigma(i) = i - 1$ or $\\sigma(i) = i + 1$, that is, when $i$ is joined to an adjacent vertex.","title":"Endpoints of short chords"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/is-bubble","kind":"node","lean":"MetaMathlibExt.IsBubble","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#is-bubble","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"fully_proved","summary":"Let $\\sigma$ be a map from $\\{0, \\dots, 2n-1\\}$ to itself (for a linear chord diagram, vertex $i$ is joined by a chord to $\\sigma(i)$). For natural numbers $a$ and $p$, the vertices $a, a+1, \\dots, a+p-1$ form a bubble of size $p$ in $\\sigma$ when $p \\ge 1$, $a + p \\le 2n$, none of these vertices is an endpoint of a short chord of $\\sigma$, the vertex $a - 1$ is an endpoint of a short chord unless $a = 0$, and the\u2026","title":"Bubbles"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/bubble-count","kind":"node","lean":"MetaMathlibExt.bubbleCount","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#bubble-count","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"fully_proved","summary":"For natural numbers $n$ and $p$, $B_{n,p}$ is the total number of bubbles of size $p$ in all linear chord diagrams on $2n$ vertices: the sum, over all linear chord diagrams $\\sigma$ on the vertices $0, \\dots, 2n-1$, of the number of vertices $a$ such that the vertices $a, a+1, \\dots, a+p-1$ form a bubble of size $p$ in $\\sigma$. In particular $B_{n,0} = 0$.","title":"Bubble counts"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/bubble-count-one-one","kind":"node","lean":"MetaMathlibExt.bubbleCount_one_one","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#bubble-count-one-one","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"fully_proved","summary":"$B_{1,1} = 0$: the linear chord diagrams on $2$ vertices have no bubble of size $1$. The only such diagram is a single short chord.","title":"No bubbles of size one on two vertices"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/bubble-count-two-one","kind":"node","lean":"MetaMathlibExt.bubbleCount_two_one","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#bubble-count-two-one","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"fully_proved","summary":"$B_{2,1} = 2$: the linear chord diagrams on $4$ vertices have $2$ bubbles of size $1$ in total.","title":"Bubbles of size one on four vertices"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/bubble-count-two-four","kind":"node","lean":"MetaMathlibExt.bubbleCount_two_four","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#bubble-count-two-four","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"fully_proved","summary":"$B_{2,4} = 1$: the linear chord diagrams on $4$ vertices have exactly one bubble of size $4$ in total. It consists of all four vertices of the crossing diagram (chords $\\{0, 2\\}$ and $\\{1, 3\\}$), which has no short chord.","title":"Bubbles of size four on four vertices"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/sum-bubble-count","kind":"node","lean":"MetaMathlibExt.sum_bubbleCount","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#sum-bubble-count","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"can_prove","summary":"For every natural number $n \\ge 2$, let $B_{n,p}$ be the total number of bubbles of size $p$ over all linear chord diagrams on $2n$ vertices. Then","title":"Total number of bubbles in linear chord diagrams"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/sum-mul-bubble-count","kind":"node","lean":"MetaMathlibExt.sum_mul_bubbleCount","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#sum-mul-bubble-count","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"can_prove","summary":"For every natural number $n \\ge 2$, let $B_{n,p}$ be the total number of bubbles of size $p$ over all linear chord diagrams on $2n$ vertices. Then","title":"First moment of bubble sizes in linear chord diagrams"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/mean-bubble-size","kind":"node","lean":"MetaMathlibExt.mean_bubble_size","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#mean-bubble-size","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"can_prove","summary":"Let $B_{n,p}$ be the total number of bubbles of size $p$ in all linear chord diagrams on $2n$ vertices. For every integer $n \\ge 2$, the mean bubble size satisfies","title":"Mean bubble size"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/bubble-count-formal-contour-integral","kind":"node","lean":"MetaMathlibExt.bubbleCount_formal_contour_integral","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#bubble-count-formal-contour-integral","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"can_prove","summary":"Let $G(s, w, y)$ be the power series in $\\mathbb{Q}[[s, w, y]]$ given by","title":"Formal contour-integral formula for bubble counts"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/bubble-count-formal-laplace-closed-form","kind":"node","lean":"MetaMathlibExt.bubbleCount_formal_laplace_closed_form","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#bubble-count-formal-laplace-closed-form","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"can_prove","summary":"In $\\mathbb{Q}[[t, w, y]]$ let $D = 1 - w(1 - wy)$ and let $I(t, w, y)$ be the power series","title":"Closed form of the bubble generating function"},{"id":"Combinatorics/set-partitions/linear-chord-diagrams/tendsto-bubble-count-density","kind":"node","lean":"MetaMathlibExt.tendsto_bubbleCount_density","parent":27359,"result":true,"route":"../roadmap/Combinatorics/set-partitions/linear-chord-diagrams/index.html#tendsto-bubble-count-density","scope":"Combinatorics/set-partitions/linear-chord-diagrams","status":"can_prove","summary":"Let $B_{n,p}$ be the total number of bubbles of size $p$ over all linear chord diagrams on $2n$ vertices. Let $x$ be a real number with $0 < x \\le 2$, and let $(p_n)_{n \\ge 0}$ be natural numbers with $p_0 = 0$, $p_n \\le 2n - 2$ for every $n \\ge 1$, and $p_n / (n - 1) \\to x$ as $n \\to \\infty$. Then","title":"Limiting density of bubble sizes in linear chord diagrams"},{"data_url":"scopes/Combinatorics/set-partitions/noncrossing-partitions.json","id":"Combinatorics/set-partitions/noncrossing-partitions","kind":"scope","parent":27324,"result":false,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/index.html","scope":"Combinatorics/set-partitions/noncrossing-partitions","status":"fully_proved","title":"Noncrossing partitions"},{"data_url":"scopes/Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order.json","id":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order","kind":"scope","navigation":"module","parent":27375,"result":false,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/index.html","scope":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order","status":"fully_proved","title":"Noncrossing partitions and the refinement order"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/set-partition","kind":"node","lean":"NoncrossingPartition.SetPartition","parent":27376,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/index.html#set-partition","scope":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order","status":"fully_proved","summary":"A set partition of $\\{0, \\dots, n-1\\}$ is a finite collection of nonempty, pairwise disjoint subsets of $\\{0, \\dots, n-1\\}$, called blocks, whose union is $\\{0, \\dots, n-1\\}$.","title":"Set partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/has-crossing","kind":"node","lean":"NoncrossingPartition.HasCrossing","parent":27376,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/index.html#has-crossing","scope":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order","status":"fully_proved","summary":"A set partition $P$ of $\\{0, \\dots, n-1\\}$ has a crossing if there are $a < b < c < d$ in $\\{0, \\dots, n-1\\}$ such that $a$ and $c$ lie in the same block of $P$, $b$ and $d$ lie in the same block of $P$, and these two blocks are different.","title":"Crossing in a set partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/is-noncrossing","kind":"node","lean":"NoncrossingPartition.IsNoncrossing","parent":27376,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/index.html#is-noncrossing","scope":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order","status":"fully_proved","summary":"A set partition of $\\{0, \\dots, n-1\\}$ is noncrossing if it has no crossing.","title":"Noncrossing set partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/noncrossing-partition","kind":"node","lean":"NoncrossingPartition","parent":27376,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/index.html#noncrossing-partition","scope":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order","status":"fully_proved","summary":"A noncrossing partition of $\\{0, \\dots, n-1\\}$ is a set partition of $\\{0, \\dots, n-1\\}$ that is noncrossing.","title":"Noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/block","kind":"node","lean":"NoncrossingPartition.block","parent":27376,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/index.html#block","scope":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order","status":"fully_proved","summary":"For a noncrossing partition $P$ of $\\{0, \\dots, n-1\\}$ and a point $a \\in \\{0, \\dots, n-1\\}$, the block of $P$ at $a$ is the block of the underlying set partition that contains $a$.","title":"Block of a noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/is-noncrossing-of-le-three","kind":"node","lean":"NoncrossingPartition.isNoncrossing_of_le_three","parent":27376,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order/index.html#is-noncrossing-of-le-three","scope":"Combinatorics/set-partitions/noncrossing-partitions/noncrossing-partitions-and-the-refinement-order","status":"fully_proved","summary":"For every $n \\le 3$, every set partition of $\\{0, \\dots, n-1\\}$ is noncrossing.","title":"Partitions of at most three points are noncrossing"},{"data_url":"scopes/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind.json","id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","kind":"scope","navigation":"module","parent":27375,"result":false,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","title":"Extremal noncrossing partitions: discrete and indiscrete"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/is-noncrossing-bot","kind":"node","lean":"NoncrossingPartition.isNoncrossing_bot","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#is-noncrossing-bot","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For every natural number $n$, the discrete partition of $\\{0, \\dots, n-1\\}$, whose blocks are the singletons, is noncrossing.","title":"The discrete partition is noncrossing"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/is-noncrossing-top","kind":"node","lean":"NoncrossingPartition.isNoncrossing_top","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#is-noncrossing-top","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For every natural number $n$, the indiscrete partition of $\\{0, \\dots, n-1\\}$, the coarsest partition of this set, is noncrossing.","title":"The indiscrete partition is noncrossing"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/discrete","kind":"node","lean":"NoncrossingPartition.discrete","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#discrete","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For a natural number $n$, the discrete noncrossing partition of $\\{0, \\dots, n-1\\}$ is the finest partition of this set, whose blocks are the singletons $\\{a\\}$ for $a \\in \\{0, \\dots, n-1\\}$; it is noncrossing.","title":"Discrete noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/indiscrete","kind":"node","lean":"NoncrossingPartition.indiscrete","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#indiscrete","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For a natural number $n$, the indiscrete noncrossing partition of $\\{0, \\dots, n-1\\}$ is the coarsest partition of this set: its only block is the whole set when $n > 0$, and it has no blocks when $n = 0$. It is noncrossing.","title":"The indiscrete noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/block-discrete","kind":"node","lean":"NoncrossingPartition.block_discrete","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#block-discrete","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For every natural number $n$ and every $a \\in \\{0, \\dots, n-1\\}$, the block containing $a$ in the discrete noncrossing partition of $\\{0, \\dots, n-1\\}$ is the singleton $\\{a\\}$.","title":"The block of a point in the discrete noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/block-indiscrete","kind":"node","lean":"NoncrossingPartition.block_indiscrete","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#block-indiscrete","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For every natural number $n$ and every $a \\in \\{0, \\dots, n-1\\}$, the block containing $a$ in the indiscrete noncrossing partition of $\\{0, \\dots, n-1\\}$ is the whole set $\\{0, \\dots, n-1\\}$.","title":"The block of a point in the indiscrete noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/discrete-parts","kind":"node","lean":"NoncrossingPartition.discrete_parts","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#discrete-parts","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For every natural number $n$, the set of blocks of the discrete noncrossing partition of $\\{0, \\dots, n-1\\}$ is exactly the set of singletons $\\{a\\}$ with $a \\in \\{0, \\dots, n-1\\}$.","title":"The discrete noncrossing partition consists of the singletons"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/indiscrete-parts-zero","kind":"node","lean":"NoncrossingPartition.indiscrete_parts_zero","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#indiscrete-parts-zero","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For $n = 0$, the set of blocks of the indiscrete noncrossing partition of $\\{0, \\dots, n-1\\}$, the empty set, is empty.","title":"The indiscrete partition of the empty set has no blocks"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/discrete-parts-zero","kind":"node","lean":"NoncrossingPartition.discrete_parts_zero","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#discrete-parts-zero","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For $n = 0$, the discrete noncrossing partition of the empty set $\\{0, \\dots, n-1\\}$ has no blocks: its set of blocks is empty.","title":"The discrete partition of the empty set has no blocks"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/indiscrete-parts-pos","kind":"node","lean":"NoncrossingPartition.indiscrete_parts_pos","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#indiscrete-parts-pos","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For a natural number $n$ with $0 < n$, the set of blocks of the indiscrete noncrossing partition of $\\{0, \\dots, n-1\\}$ is $\\{\\{0, \\dots, n-1\\}\\}$: its only block is the whole set.","title":"The indiscrete partition of a nonempty set has a single block"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/eq-discrete-iff-val-eq-bot","kind":"node","lean":"NoncrossingPartition.eq_discrete_iff_val_eq_bot","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#eq-discrete-iff-val-eq-bot","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"Let $n$ be a natural number. A noncrossing partition $P$ of $\\{0, \\dots, n-1\\}$ equals the discrete noncrossing partition if and only if $P$, viewed as a set partition, is the finest partition of $\\{0, \\dots, n-1\\}$.","title":"Characterization of the discrete noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/eq-indiscrete-iff-val-eq-top","kind":"node","lean":"NoncrossingPartition.eq_indiscrete_iff_val_eq_top","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#eq-indiscrete-iff-val-eq-top","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"Let $n$ be a natural number. A noncrossing partition $P$ of $\\{0, \\dots, n-1\\}$ equals the indiscrete noncrossing partition if and only if $P$, viewed as a set partition, is the coarsest partition of $\\{0, \\dots, n-1\\}$.","title":"Characterization of the indiscrete noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/eq-indiscrete-of-forall-block-eq-univ","kind":"node","lean":"NoncrossingPartition.eq_indiscrete_of_forall_block_eq_univ","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#eq-indiscrete-of-forall-block-eq-univ","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"Let $n$ be a natural number and let $P$ be a noncrossing partition of $\\{0, \\dots, n-1\\}$. If for every $a \\in \\{0, \\dots, n-1\\}$ the block of $P$ containing $a$ is the whole set $\\{0, \\dots, n-1\\}$, then $P$ is the indiscrete noncrossing partition.","title":"A partition whose blocks are all the whole set is indiscrete"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/eq-indiscrete-zero","kind":"node","lean":"NoncrossingPartition.eq_indiscrete_zero","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#eq-indiscrete-zero","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For $n = 0$, every noncrossing partition of $\\{0, \\dots, n-1\\}$, the empty set, equals the indiscrete noncrossing partition of this set.","title":"The empty set has a unique noncrossing partition"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/discrete-zero-eq-indiscrete-zero","kind":"node","lean":"NoncrossingPartition.discrete_zero_eq_indiscrete_zero","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#discrete-zero-eq-indiscrete-zero","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For $n = 0$, the discrete and the indiscrete noncrossing partitions of the empty set $\\{0, \\dots, n-1\\}$ are equal.","title":"Discrete and indiscrete partitions agree on the empty set"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/eq-discrete-zero","kind":"node","lean":"NoncrossingPartition.eq_discrete_zero","parent":27383,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind/index.html#eq-discrete-zero","scope":"Combinatorics/set-partitions/noncrossing-partitions/extremal-noncrossing-partitions-discrete-and-ind","status":"fully_proved","summary":"For $n = 0$, every noncrossing partition of the empty set $\\{0, \\dots, n-1\\}$ equals the discrete noncrossing partition.","title":"The only noncrossing partition of the empty set is the discrete one"},{"data_url":"scopes/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti.json","id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","kind":"scope","navigation":"module","parent":27375,"result":false,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","title":"Small-arity classification for noncrossing partitions"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/eq-indiscrete-one","kind":"node","lean":"NoncrossingPartition.eq_indiscrete_one","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#eq-indiscrete-one","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"Every noncrossing partition of the one-point set $\\{0\\}$ equals the indiscrete noncrossing partition of $\\{0\\}$.","title":"Noncrossing partitions of one point are indiscrete"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/discrete-two-ne-indiscrete-two","kind":"node","lean":"NoncrossingPartition.discrete_two_ne_indiscrete_two","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#discrete-two-ne-indiscrete-two","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"The discrete noncrossing partition of $\\{0, 1\\}$ is not equal to the indiscrete noncrossing partition of $\\{0, 1\\}$.","title":"The discrete and indiscrete partitions of a two-element set differ"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/eq-discrete-or-indiscrete-two","kind":"node","lean":"NoncrossingPartition.eq_discrete_or_indiscrete_two","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#eq-discrete-or-indiscrete-two","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"Every noncrossing partition of $\\{0, 1\\}$ is either the discrete or the indiscrete noncrossing partition of $\\{0, 1\\}$.","title":"Classification of noncrossing partitions of a two-element set"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/card-zero","kind":"node","lean":"NoncrossingPartition.card_zero","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#card-zero","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"There is exactly one noncrossing partition of the empty set, that is, of $\\{0, \\dots, n-1\\}$ for $n = 0$.","title":"Noncrossing partitions of the empty set"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/card-one","kind":"node","lean":"NoncrossingPartition.card_one","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#card-one","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"There is exactly one noncrossing partition of $\\{0\\}$.","title":"Noncrossing partitions of a one-element set"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/card-two","kind":"node","lean":"NoncrossingPartition.card_two","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#card-two","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"There are exactly two noncrossing partitions of $\\{0, 1\\}$.","title":"Noncrossing partitions of a two-element set"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/proper-two-equiv","kind":"node","lean":"NoncrossingPartition.properTwoEquiv","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#proper-two-equiv","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"Call a noncrossing partition of $\\{0, 1\\}$ proper if it is not the indiscrete one. This is the bijection from a one-element set onto the set of proper noncrossing partitions of $\\{0, 1\\}$, sending its element to the discrete noncrossing partition; in particular the discrete partition is the only proper one.","title":"The proper noncrossing partition of two points"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/two-block-partition","kind":"node","lean":"NoncrossingPartition.twoBlockPartition","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#two-block-partition","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"For $a \\in \\{0, 1, 2\\}$, this is the noncrossing partition of $\\{0, 1, 2\\}$ whose blocks are $\\{a\\}$ and $\\{0, 1, 2\\} \\setminus \\{a\\}$.","title":"Two-block partition of three points"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/three-card-one-two-three","kind":"node","lean":"NoncrossingPartition.three_card_one_two_three","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#three-card-one-two-three","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"Every noncrossing partition of $\\{0, 1, 2\\}$ has one, two, or three blocks.","title":"Noncrossing partitions of three points have one, two or three blocks"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/eq-indiscrete-of-card-one-three","kind":"node","lean":"NoncrossingPartition.eq_indiscrete_of_card_one_three","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#eq-indiscrete-of-card-one-three","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"If a noncrossing partition $P$ of $\\{0, 1, 2\\}$ has exactly one block, then $P$ is the indiscrete noncrossing partition of $\\{0, 1, 2\\}$.","title":"A noncrossing partition of three points with one block is indiscrete"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/eq-discrete-of-card-three-three","kind":"node","lean":"NoncrossingPartition.eq_discrete_of_card_three_three","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#eq-discrete-of-card-three-three","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"If a noncrossing partition $P$ of $\\{0, 1, 2\\}$ has exactly three blocks, then $P$ is the discrete noncrossing partition of $\\{0, 1, 2\\}$.","title":"A noncrossing partition of three points with three blocks is discrete"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/eq-discrete-or-indiscrete-or-two-block-t","kind":"node","lean":"NoncrossingPartition.eq_discrete_or_indiscrete_or_twoBlock_three","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#eq-discrete-or-indiscrete-or-two-block-t","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"Every noncrossing partition $P$ of $\\{0, 1, 2\\}$ is the discrete noncrossing partition of $\\{0, 1, 2\\}$, the indiscrete noncrossing partition of $\\{0, 1, 2\\}$, or, for some $a \\in \\{0, 1, 2\\}$, the two-block noncrossing partition whose blocks are $\\{a\\}$ and $\\{0, 1, 2\\} \\setminus \\{a\\}$.","title":"Classification of noncrossing partitions of a three-element set"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/discrete-three-ne-indiscrete-three","kind":"node","lean":"NoncrossingPartition.discrete_three_ne_indiscrete_three","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#discrete-three-ne-indiscrete-three","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"The discrete noncrossing partition of $\\{0, 1, 2\\}$ is not equal to the indiscrete noncrossing partition of $\\{0, 1, 2\\}$.","title":"The discrete and indiscrete partitions of a three-element set differ"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/two-block-partition-ne-indiscrete-three","kind":"node","lean":"NoncrossingPartition.twoBlockPartition_ne_indiscrete_three","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#two-block-partition-ne-indiscrete-three","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"For every $a \\in \\{0, 1, 2\\}$, the noncrossing partition of $\\{0, 1, 2\\}$ with blocks $\\{a\\}$ and $\\{0, 1, 2\\} \\setminus \\{a\\}$ is different from the indiscrete noncrossing partition of $\\{0, 1, 2\\}$.","title":"The two-block partitions of three points are not indiscrete"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/two-block-equiv","kind":"node","lean":"NoncrossingPartition.twoBlockEquiv","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#two-block-equiv","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"The map sending $a \\in \\{0, 1, 2\\}$ to the partition of $\\{0, 1, 2\\}$ with blocks $\\{a\\}$ and $\\{0, 1, 2\\} \\setminus \\{a\\}$ is a bijection from $\\{0, 1, 2\\}$ onto the set of noncrossing partitions of $\\{0, 1, 2\\}$ with exactly two blocks.","title":"Enumerating the two-block noncrossing partitions of three points"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/proper-three-equiv","kind":"node","lean":"NoncrossingPartition.properThreeEquiv","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#proper-three-equiv","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"Call a noncrossing partition of $\\{0, 1, 2\\}$ proper if it is not the indiscrete one. Adjoin an extra element $\\ast$ to $\\{0, 1, 2\\}$; this is the bijection from $\\{\\ast, 0, 1, 2\\}$ onto the set of proper noncrossing partitions of $\\{0, 1, 2\\}$ that sends $\\ast$ to the discrete noncrossing partition and each $a \\in \\{0, 1, 2\\}$ to the noncrossing partition with blocks $\\{a\\}$ and $\\{0, 1, 2\\} \\setminus \\{a\\}$.","title":"Enumerating the proper noncrossing partitions of three points"},{"id":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/card-three","kind":"node","lean":"NoncrossingPartition.card_three","parent":27400,"result":true,"route":"../roadmap/Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti/index.html#card-three","scope":"Combinatorics/set-partitions/noncrossing-partitions/small-arity-classification-for-noncrossing-parti","status":"fully_proved","summary":"There are exactly five noncrossing partitions of $\\{0, 1, 2\\}$.","title":"Noncrossing partitions of a three-element set"},{"data_url":"scopes/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions.json","id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","kind":"scope","navigation":"module","parent":27324,"result":false,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","title":"Marriageable and lonely singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/merged-same-block","kind":"node","lean":"MetaMathlibExt.mergedSameBlock","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#merged-same-block","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"Let $P$ be a set partition of $\\{0, \\dots, n-1\\}$ and let $i, j, x, y \\in \\{0, \\dots, n-1\\}$. Then $x$ and $y$ lie in the same block after merging the blocks of $i$ and $j$ if either $x$ and $y$ lie in the same block of $P$, or each of $x$ and $y$ lies in the block of $i$ or in the block of $j$. For singleton blocks $\\{i\\}$ and $\\{j\\}$ this joins them into the block $\\{i, j\\}$.","title":"Same block after merging two blocks"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/merged-has-crossing","kind":"node","lean":"MetaMathlibExt.mergedHasCrossing","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#merged-has-crossing","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"Let $P$ be a set partition of $\\{0, \\dots, n-1\\}$ and let $i, j \\in \\{0, \\dots, n-1\\}$. The partition obtained from $P$ by merging the blocks of $i$ and $j$ has a crossing if there are $a < b < c < d$ in $\\{0, \\dots, n-1\\}$ such that $a$ and $c$ lie in one of its blocks and $b$ and $d$ lie in a different one of its blocks.","title":"Crossing after merging two blocks"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/is-marriageable-singles","kind":"node","lean":"MetaMathlibExt.IsMarriageableSingles","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#is-marriageable-singles","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"A set partition $P$ of $\\{0, \\dots, n-1\\}$ is a marriageable singles partition if it has two distinct singleton blocks $\\{i\\}$ and $\\{j\\}$ such that the partition obtained from $P$ by joining them into the block $\\{i, j\\}$ has no crossing.","title":"Marriageable singles partition"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/marriageable-singles-count","kind":"node","lean":"MetaMathlibExt.marriageableSinglesCount","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#marriageable-singles-count","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"For $n \\ge 0$, $M_n$ is the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ that are marriageable singles partitions.","title":"Number of marriageable singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/lonely-singles-count","kind":"node","lean":"MetaMathlibExt.lonelySinglesCount","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#lonely-singles-count","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"For $n \\ge 0$, $L_n$ is the number of lonely singles partitions of $\\{0, \\dots, n-1\\}$, that is, of noncrossing partitions of $\\{0, \\dots, n-1\\}$ that are not marriageable singles partitions.","title":"Number of lonely singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/marriageable-singles-count-of-le-one","kind":"node","lean":"MetaMathlibExt.marriageableSinglesCount_of_le_one","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#marriageable-singles-count-of-le-one","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"For every $n \\le 1$, the number of marriageable singles partitions of $\\{0, \\dots, n-1\\}$ is $M_n = 0$: a marriageable singles partition needs two distinct points.","title":"Small values of the marriageable singles count"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/lonely-singles-count-zero-one","kind":"node","lean":"MetaMathlibExt.lonelySinglesCount_zero_one","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#lonely-singles-count-zero-one","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"The numbers of lonely singles partitions of $\\{0, \\dots, n-1\\}$ for $n = 0$ and $n = 1$ are $L_0 = 1$ and $L_1 = 1$.","title":"Small values of the lonely singles count"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/noncrossing-block-singleton-count","kind":"node","lean":"MetaMathlibExt.noncrossingBlockSingletonCount","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#noncrossing-block-singleton-count","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"For natural numbers $n$, $m$ and $k$, $NC_{n,m,k}$ is the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ that have exactly $m$ blocks, exactly $k$ of which are singletons.","title":"Noncrossing partitions by number of blocks and of singleton blocks"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/noncrossing-block-singleton-count-zero","kind":"node","lean":"MetaMathlibExt.noncrossingBlockSingletonCount_zero","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#noncrossing-block-singleton-count-zero","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"Exactly one noncrossing partition of the empty set has $0$ blocks, $0$ of which are singletons, namely the empty partition; that is, $NC_{0,0,0} = 1$, where $NC_{n,m,k}$ is the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ with $m$ blocks of which $k$ are singletons.","title":"Base value of the noncrossing block and singleton count"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/zero-extended-choose","kind":"node","lean":"MetaMathlibExt.zeroExtendedChoose","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#zero-extended-choose","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"For integers $a$ and $b$, the zero-extended binomial coefficient $\\binom{a}{b}$ is the usual binomial coefficient of $a$ and $b$ when $a \\ge 0$ and $b \\ge 0$, and $0$ otherwise.","title":"Zero-extended binomial coefficient"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/zero-extended-choose-nat-cast","kind":"node","lean":"MetaMathlibExt.zeroExtendedChoose_natCast","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#zero-extended-choose-nat-cast","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"Extend the binomial coefficient $\\binom{a}{b}$ to all integers $a$, $b$ by setting it to $0$ when $a < 0$ or $b < 0$. For all natural numbers $a$ and $b$, this extended binomial coefficient at $a$ and $b$ viewed as integers equals the usual binomial coefficient $\\binom{a}{b}$.","title":"The zero-extended binomial coefficient on natural numbers"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/zero-extended-choose-neg-one-zero","kind":"node","lean":"MetaMathlibExt.zeroExtendedChoose_neg_one_zero","parent":27418,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zero-extended binomial coefficient vanishes at "],["math","\\((-1, 0)\\)"]],"source":"The zero-extended binomial coefficient vanishes at $(-1, 0)$"},"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#zero-extended-choose-neg-one-zero","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"Extend the binomial coefficient $\\binom{a}{b}$ to all integers $a$, $b$ by setting it to $0$ when $a < 0$ or $b < 0$. Under this convention $\\binom{-1}{0} = 0$.","title":"The zero-extended binomial coefficient vanishes at $(-1, 0)$"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/lanes-meet","kind":"node","lean":"MetaMathlibExt.lanesMeet","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#lanes-meet","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"The standard road intersection of size $n$ has entries $E_0, \\dots, E_{n-1}$ and exits $X_0, \\dots, X_{n-1}$ on a circle, with $E_i$ at position $2i$ and $X_j$ at position $2j + 1$; the lane $E_iX_j$ is the chord from $E_i$ to $X_j$. Two lanes $E_iX_j$ and $E_{i'}X_{j'}$ have a common point if $i = i'$, or $j = j'$, or exactly one of the positions $2i'$ and $2j' + 1$ lies strictly between $2i$ and $2j + 1$ (that\u2026","title":"Lanes with a common point"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/is-maximal-set-of-lanes","kind":"node","lean":"MetaMathlibExt.IsMaximalSetOfLanes","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#is-maximal-set-of-lanes","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"In the standard road intersection of size $n$, the lane $E_iX_j$ joins the entry $E_i$ to the exit $X_j$, for $i, j \\in \\{0, \\dots, n-1\\}$. Given a set $A$ of allowed lanes, a set $S$ of lanes is a maximal set of lanes (MSL) among $A$ if $S \\subseteq A$, no two distinct lanes of $S$ have a common point, and every lane of $A$ not in $S$ has a common point with some lane of $S$. The standard road intersection allows\u2026","title":"Maximal set of lanes"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/is-absolute-maximal-set-of-lanes","kind":"node","lean":"MetaMathlibExt.IsAbsoluteMaximalSetOfLanes","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#is-absolute-maximal-set-of-lanes","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"In the standard road intersection of size $n$, the lane $E_iX_j$ joins the entry $E_i$ to the exit $X_j$, for $i, j \\in \\{0, \\dots, n-1\\}$. A set $S$ of lanes is absolute if it is a maximal set of lanes among all lanes and there are no $i \\ne j$ such that $S$ contains both U-turns $E_iX_i$ and $E_jX_j$ and replacing these two lanes in $S$ by $E_iX_j$ and $E_jX_i$ gives again a maximal set of lanes among all lanes.","title":"Absolute maximal set of lanes"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/maximal-sets-of-lanes-two","kind":"node","lean":"MetaMathlibExt.maximalSetsOfLanes_two","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#maximal-sets-of-lanes-two","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"fully_proved","summary":"Write $E_iX_j$ for the lane from the $i$-th entry to the $j$-th exit. In the standard road intersection of size $2$, in which every lane is allowed, the maximal sets of lanes are exactly $\\{E_1X_1, E_2X_2\\}$ and $\\{E_1X_2, E_2X_1\\}$.","title":"Maximal sets of lanes for an intersection of size two"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/card-noncrossing-partition-eq-lonely-add","kind":"node","lean":"MetaMathlibExt.card_noncrossingPartition_eq_lonely_add_marriageable","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#card-noncrossing-partition-eq-lonely-add","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"For every $n \\ge 0$, the number $C_n$ of noncrossing partitions of $\\{0, \\dots, n-1\\}$ equals $L_n + M_n$, where $M_n$ is the number of those that are marriageable singles partitions and $L_n$ is the number of those that are not.","title":"Noncrossing partitions split into lonely and marriageable singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/maximal-sets-of-lanes-equiv-noncrossing","kind":"node","lean":"MetaMathlibExt.maximalSetsOfLanes_equiv_noncrossingPartition","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#maximal-sets-of-lanes-equiv-noncrossing","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"For every $n \\ge 0$, the maximal sets of lanes of the standard road intersection of size $n$, in which every lane is allowed, are in bijection with the noncrossing partitions of $\\{0, \\dots, n-1\\}$, and their number is the Catalan number $C_n$.","title":"Maximal sets of lanes and noncrossing partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/card-nonabsolute-maximal-sets-of-lanes","kind":"node","lean":"MetaMathlibExt.card_nonabsolute_maximalSetsOfLanes","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#card-nonabsolute-maximal-sets-of-lanes","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"For every $n \\ge 0$, the number of maximal sets of lanes of the standard road intersection of size $n$ that are not absolute equals $M_n$, the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ that are marriageable singles partitions.","title":"Nonabsolute maximal sets of lanes are counted by marriageable singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/card-absolute-maximal-sets-of-lanes","kind":"node","lean":"MetaMathlibExt.card_absolute_maximalSetsOfLanes","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#card-absolute-maximal-sets-of-lanes","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"For every $n \\ge 0$, the number of absolute maximal sets of lanes of the standard road intersection of size $n$ equals $L_n$, the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ that are not marriageable singles partitions.","title":"Absolute maximal sets of lanes are counted by lonely singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/card-restricted-maximal-sets-of-lanes","kind":"node","lean":"MetaMathlibExt.card_restricted_maximalSetsOfLanes","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#card-restricted-maximal-sets-of-lanes","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"For every $n \\ge 0$, consider the restricted standard road intersection of size $n$, in which the lane $E_iX_j$ from the entry $E_i$ to the exit $X_j$, for $i, j \\in \\{0, \\dots, n-1\\}$, is allowed exactly when $i \\ne j$ (the U-turns $E_iX_i$ are forbidden). The number of maximal sets of lanes among these allowed lanes equals $L_n$, the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ that are not\u2026","title":"Maximal sets of lanes without U-turns are counted by lonely singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/lonely-marriageable-strict-mono","kind":"node","lean":"MetaMathlibExt.lonely_marriageable_strictMono","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#lonely-marriageable-strict-mono","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"$L_n < L_{n+1}$ for every $n \\ge 2$, and $M_n < M_{n+1}$ for every $n \\ge 3$, where $M_n$ is the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ that are marriageable singles partitions and $L_n$ is the number of those that are not.","title":"Strict growth of the lonely and marriageable singles counts"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/tendsto-marriageable-lonely-at-top","kind":"node","lean":"MetaMathlibExt.tendsto_marriageable_lonely_atTop","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#tendsto-marriageable-lonely-at-top","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"Let $M_n$ and $L_n$ be the numbers of noncrossing partitions of $\\{0, \\dots, n-1\\}$ that are marriageable singles partitions and lonely singles partitions, respectively. Then $M_n \\to \\infty$ and $L_n \\to \\infty$ as $n \\to \\infty$.","title":"The marriageable and lonely singles counts tend to infinity"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/card-add-three-mul-marriageable-le","kind":"node","lean":"MetaMathlibExt.card_add_three_mul_marriageable_le","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#card-add-three-mul-marriageable-le","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"For every $n \\ge 0$, $C_n + 3M_n \\le M_{n+2}$, where $C_n$ is the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ and $M_n$ is the number of those that are marriageable singles partitions.","title":"Recursive lower bound for marriageable singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/noncrossing-block-singleton-count-formul","kind":"node","lean":"MetaMathlibExt.noncrossingBlockSingletonCount_formulas","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#noncrossing-block-singleton-count-formul","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"Write $NC_{n,m,k}$ for the number of noncrossing partitions of $\\{0, \\dots, n-1\\}$ with $m$ blocks of which $k$ are singletons, and extend the binomial coefficient $\\binom{a}{b}$ to all integers $a$, $b$ by setting it to $0$ when $a < 0$ or $b < 0$. For all natural numbers $n$ and $m$: if $(n, m) \\ne (0, 0)$, then the following holds in $\\mathbb{Q}$, where $1/0$ is taken to be $0$ (the denominator vanishes only for\u2026","title":"Counting noncrossing partitions with no or one singleton block"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/lonely-singles-count-ge-sum","kind":"node","lean":"MetaMathlibExt.lonelySinglesCount_ge_sum","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#lonely-singles-count-ge-sum","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"For every $n \\ge 2$, the number $L_n$ of lonely singles partitions of $\\{0, \\dots, n-1\\}$ satisfies","title":"Lower bound for the number of lonely singles partitions"},{"id":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/marriageable-singles-count-ge-sum","kind":"node","lean":"MetaMathlibExt.marriageableSinglesCount_ge_sum","parent":27418,"result":true,"route":"../roadmap/Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions/index.html#marriageable-singles-count-ge-sum","scope":"Combinatorics/set-partitions/marriageable-and-lonely-singles-partitions","status":"can_prove","summary":"Extend the binomial coefficient $\\binom{a}{b}$ to all integers $a$, $b$ by setting it to $0$ when $a < 0$ or $b < 0$, and for every natural number $N$ let","title":"Lower bound for the number of marriageable singles partitions"},{"data_url":"scopes/Combinatorics/set-partitions/partition-number-arrays.json","id":"Combinatorics/set-partitions/partition-number-arrays","kind":"scope","navigation":"module","parent":27324,"result":false,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","title":"Partition number arrays"},{"id":"Combinatorics/set-partitions/partition-number-arrays/extend-a","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.extendA","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#extend-a","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"A vector $(a_0, \\dots, a_{n-1})$ of natural numbers extends to the sequence $\\mathbb{N} \\to \\mathbb{N}$ that sends $j$ to $a_j$ for $j < n$ and to $0$ for $j \\ge n$.","title":"Extension of a vector by zero"},{"id":"Combinatorics/set-partitions/partition-number-arrays/weight-sum-aux","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.weightSumAux","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#weight-sum-aux","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and a function $f \\colon \\mathbb{N} \\to \\mathbb{N}$, this is the sum $\\sum_{i=0}^{n-1} (i + 1)\\,f(i)$.","title":"Index-weighted partial sum"},{"id":"Combinatorics/set-partitions/partition-number-arrays/count-sum-aux","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.countSumAux","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#count-sum-aux","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"For a natural number $n$ and a sequence $f \\colon \\mathbb{N} \\to \\mathbb{N}$, this is the natural number $\\sum_{i=0}^{n-1} f(i)$.","title":"Sum of the first n values of a sequence"},{"id":"Combinatorics/set-partitions/partition-number-arrays/mult-weight-sum","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.multWeightSum","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#mult-weight-sum","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and nonnegative integers $a_1, \\dots, a_n$, the weighted size is $\\sum_{j=1}^{n} j\\,a_j$, the sum of the parts of the partition with $a_j$ parts equal to $j$.","title":"Weighted size of a multiplicity vector"},{"id":"Combinatorics/set-partitions/partition-number-arrays/count-sum","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.countSum","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#count-sum","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"For a vector $(a_1, \\dots, a_n)$ of natural numbers, this is the natural number $\\sum_{j=1}^{n} a_j$.","title":"Number of parts of an exponent vector"},{"id":"Combinatorics/set-partitions/partition-number-arrays/partition-multiplicity","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.PartitionMultiplicity","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#partition-multiplicity","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"A partition multiplicity vector of $n \\in \\mathbb{N}$ is a tuple $(a_1, \\dots, a_n)$ of nonnegative integers with $\\sum_{j=1}^{n} j\\,a_j = n$. It encodes the partition of $n$ that has $a_j$ parts equal to $j$.","title":"Partition multiplicity vectors"},{"id":"Combinatorics/set-partitions/partition-number-arrays/part-count","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.partCount","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#part-count","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"Let $(a_1, \\dots, a_n)$ be a partition multiplicity vector of $n$, that is, nonnegative integers with $\\sum_{j=1}^{n} j\\,a_j = n$, where $a_j$ counts the parts equal to $j$. Its part count is $\\sum_{j=1}^{n} a_j$.","title":"Number of parts of a partition"},{"id":"Combinatorics/set-partitions/partition-number-arrays/gen-stirling-second","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.genStirlingSecond","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#gen-stirling-second","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"For a positive integer $k$, the numbers $S_2(k; n, m)$, for natural numbers $n$ and $m$, are defined by $S_2(k; 0, 0) = 1$, by $S_2(k; 0, m) = 0$ and $S_2(k; m, 0) = 0$ for $m \\ge 1$, and by","title":"Generalized Stirling numbers of the second kind with parameter k"},{"id":"Combinatorics/set-partitions/partition-number-arrays/denom-prod-aux","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.denomProdAux","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#denom-prod-aux","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"For a natural number $n$ and a sequence $g \\colon \\mathbb{N} \\to \\mathbb{N}$, this is the natural number $\\prod_{i=0}^{n-1} g(i)!\\,\\bigl((i+1)!\\bigr)^{g(i)}$.","title":"Factorial product of a sequence"},{"id":"Combinatorics/set-partitions/partition-number-arrays/pow-prod-aux","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.powProdAux","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#pow-prod-aux","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and functions $g, f \\colon \\mathbb{N} \\to \\mathbb{N}$, this is the product $\\prod_{i=0}^{n-1} f(i)^{g(i)}$, with the convention $0^0 = 1$.","title":"Finite power product"},{"id":"Combinatorics/set-partitions/partition-number-arrays/m3-denom-prod","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.m3DenomProd","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#m3-denom-prod","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"Let $p$ be a partition of $n$ with multiplicity vector $(a_1, \\dots, a_n)$, so that exactly $a_j$ parts of $p$ equal $j$ and $\\sum_j j a_j = n$. The $M_3$ denominator of $p$ is the natural number $\\prod_{j=1}^{n} a_j!\\,(j!)^{a_j}$.","title":"Denominator of the M3 partition number"},{"id":"Combinatorics/set-partitions/partition-number-arrays/m3","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.M3","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#m3","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"Let $p$ be a partition of $n$ with multiplicity vector $(a_1, \\dots, a_n)$, so that exactly $a_j$ parts of $p$ equal $j$ and $\\sum_j j a_j = n$. The number $M_3(p)$ is the quotient of $n!$ by $\\prod_{j=1}^{n} a_j!\\,(j!)^{a_j}$ in natural-number division.","title":"The M3 partition number"},{"id":"Combinatorics/set-partitions/partition-number-arrays/array-pow-prod","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.arrayPowProd","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#array-pow-prod","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"Let $p$ be a partition of $n$ with multiplicity vector $(a_1, \\dots, a_n)$, so that exactly $a_j$ parts of $p$ equal $j$ and $\\sum_j j a_j = n$. For natural numbers $f_1, \\dots, f_n$, the product of powers of $f$ over $p$ is the natural number $\\prod_{j=1}^{n} f_j^{a_j}$.","title":"Product of powers over a partition"},{"id":"Combinatorics/set-partitions/partition-number-arrays/hat-m32","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.hatM32","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#hat-m32","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"Let $k$ be a positive integer and let $p$ be a partition of $n$ with multiplicity vector $(a_1, \\dots, a_n)$, so that exactly $a_j$ parts of $p$ equal $j$ and $\\sum_j j a_j = n$. Write $S_2(k; j, 1)$ for the generalized Stirling numbers of the second kind with parameter $k$. The entry $\\widehat{M}_{32}(k; p)$ is the natural number $\\prod_{j=1}^{n} S_2(k; j, 1)^{a_j}$.","title":"The hat M32 partition number array"},{"id":"Combinatorics/set-partitions/partition-number-arrays/m32","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.M32","parent":27446,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partition number array "],["math","\\(M_{32}(k)\\)"]],"source":"The partition number array $M_{32}(k)$"},"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#m32","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"Let $k$ be a positive integer and let $a = (a_1, \\dots, a_n)$ be a partition multiplicity vector of $n$, that is, nonnegative integers with $\\sum_{j=1}^{n} j\\,a_j = n$, where $a_j$ counts the parts equal to $j$. Then $M_{32}(k; a) = M_3(a) \\prod_{j=1}^{n} S_2(k; j, 1)^{a_j}$, where $M_3(a)$ is $n!$ divided by $\\prod_{j=1}^{n} a_j!\\,(j!)^{a_j}$ (the division is exact), and\u2026","title":"The partition number array $M_{32}(k)$"},{"id":"Combinatorics/set-partitions/partition-number-arrays/stirling-first-abs-first-col","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.stirlingFirstAbsFirstCol","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#stirling-first-abs-first-col","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"For positive integers $k$ and $j$, the absolute value $|S_1(k; j, 1)|$ is defined as the quotient $(j + k - 2)!/(k - 1)!$.","title":"Absolute first column of the generalized Stirling numbers of the first kind"},{"id":"Combinatorics/set-partitions/partition-number-arrays/m31","kind":"node","lean":"MetaMathlibExt.PartitionNumberArray.M31","parent":27446,"result":true,"route":"../roadmap/Combinatorics/set-partitions/partition-number-arrays/index.html#m31","scope":"Combinatorics/set-partitions/partition-number-arrays","status":"fully_proved","summary":"Let $k$ be a positive integer and let $p$ be a partition of $n$ with multiplicity vector $(a_1, \\dots, a_n)$, so that exactly $a_j$ parts of $p$ equal $j$ and $\\sum_j j a_j = n$. Write $M_3(p)$ for the M3 partition number of $p$, and for $j \\ge 1$ write $c_j = (j + k - 2)!/(k - 1)!$, the absolute value of the generalized Stirling number of the first kind $S_1(k; j, 1)$. The entry $M_{31}(k; p)$ is the natural\u2026","title":"The M31 partition number array"},{"data_url":"scopes/Combinatorics/set-partitions/peaks-in-set-partitions.json","id":"Combinatorics/set-partitions/peaks-in-set-partitions","kind":"scope","navigation":"module","parent":27324,"result":false,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"can_prove","title":"Peaks in set partitions"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/is-surjective-restricted-growth-string","kind":"node","lean":"MetaMathlibExt.IsSurjectiveRestrictedGrowthString","parent":27464,"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#is-surjective-restricted-growth-string","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"fully_proved","summary":"A word $w = w_0 w_1 \\cdots w_{n-1}$ of length $n$ with letters in $\\{0, 1, \\dots, k-1\\}$ is a surjective restricted-growth string if every letter $0, \\dots, k-1$ occurs in $w$ and, for every position $i$ and every letter $m < w_i$, some position $j < i$ has $w_j = m$.","title":"Surjective restricted-growth strings"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/peak-generating-function-set-partitions","kind":"node","lean":"MetaMathlibExt.peak_generating_function_set_partitions","parent":27464,"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#peak-generating-function-set-partitions","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"fully_proved","summary":"Let $k \\ge 1$. Work in the fraction field of the ring $\\mathbb{Z}[q][[x]]$ of power series in $x$ with coefficients in $\\mathbb{Z}[q]$. For a word $w = w_0 \\cdots w_{n-1}$, let $\\mathrm{pk}(w)$ be the number of positions $a$ with $a + 2 < n$ and $w_a < w_{a+1} > w_{a+2}$, and let $W_{n,k}$ be the set of surjective restricted-growth strings of length $n$ with letters in $\\{0, \\dots, k-1\\}$. Let\u2026","title":"Peak generating function for set partitions"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/total-peaks-canonical","kind":"node","lean":"MetaMathlibExt.total_peaks_canonical","parent":27464,"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#total-peaks-canonical","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"fully_proved","summary":"For natural numbers $n$ and $k$, call a word $\\pi = \\pi_1 \\cdots \\pi_n$ on the alphabet $\\{0, \\dots, k-1\\}$ a restricted growth word if it uses every letter and, for every position $i$, every letter smaller than $\\pi_i$ occurs among $\\pi_1, \\dots, \\pi_{i-1}$; these words encode the partitions of $[n]$ into $k$ blocks. A peak of $\\pi$ is a position $i$ with $1 < i < n$ and $\\pi_{i-1} < \\pi_i > \\pi_{i+1}$. Then the\u2026","title":"Total number of peaks over set partitions"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/non-symmetric-peak-count","kind":"node","lean":"MetaMathlibExt.nonSymmetricPeakCount","parent":27464,"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#non-symmetric-peak-count","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"fully_proved","summary":"For a word $w = w_1 w_2 \\cdots w_\\ell$ of natural numbers, a non-symmetric peak is a position $i$ with $1 < i < \\ell$ such that $w_{i-1} < w_i$, $w_{i+1} < w_i$ and $w_{i-1} \\ne w_{i+1}$. The non-symmetric peak count of $w$ is the number of such positions; in particular it is $0$ when $\\ell < 3$.","title":"Non-symmetric peaks of a word"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/symmetric-peak-count-non-symmetric-peak","kind":"node","lean":"MetaMathlibExt.symmetricPeakCount_nonSymmetricPeakCount_example","parent":27464,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peaks of the word "],["math","\\(1322141251\\)"]],"source":"Peaks of the word $1322141251$"},"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#symmetric-peak-count-non-symmetric-peak","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"fully_proved","summary":"The word $1322141251$ (the list $1, 3, 2, 2, 1, 4, 1, 2, 5, 1$) has exactly one symmetric peak, the factor $141$, and exactly two non-symmetric peaks, the factors $132$ and $251$.","title":"Peaks of the word $1322141251$"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/symmetric-peaks-derivative-generating-fu","kind":"node","lean":"MetaMathlibExt.symmetricPeaks_derivative_generatingFunction","parent":27464,"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#symmetric-peaks-derivative-generating-fu","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"can_prove","summary":"Fix $k \\in \\mathbb{N}$ and, for each $n$, let $P_{n,k}$ be the set of surjective restricted-growth strings of length $n$ with letters in $\\{0, \\dots, k-1\\}$; these encode the partitions of $\\{1, \\dots, n\\}$ into exactly $k$ blocks. For $w \\in P_{n,k}$ let $\\operatorname{sp}(w)$ be the number of symmetric peaks of $w$. In the ring $\\mathbb{Z}[[x]]$ of formal power series let $F_k(x) = \\sum_{n \\ge 0} |P_{n,k}| x^n$.\u2026","title":"Generating function for the total number of symmetric peaks in set partitions"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/non-symmetric-peaks-derivative-generatin","kind":"node","lean":"MetaMathlibExt.nonSymmetricPeaks_derivative_generatingFunction","parent":27464,"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#non-symmetric-peaks-derivative-generatin","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"can_prove","summary":"Fix $k \\in \\mathbb{N}$ and, for each $n$, let $P_{n,k}$ be the set of surjective restricted-growth strings of length $n$ with letters in $\\{0, \\dots, k-1\\}$; these encode the partitions of $\\{1, \\dots, n\\}$ into exactly $k$ blocks. For $w \\in P_{n,k}$ let $\\operatorname{nsp}(w)$ be the number of non-symmetric peaks of $w$. In the ring $\\mathbb{Z}[[x]]$ of formal power series let\u2026","title":"Generating function for the total number of non-symmetric peaks in set partitions"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/sum-symmetric-peak-count-set-partitions","kind":"node","lean":"MetaMathlibExt.sum_symmetricPeakCount_setPartitions","parent":27464,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total number of symmetric peaks over set partitions with "],["math","\\(k\\)"],["text"," blocks"]],"source":"Total number of symmetric peaks over set partitions with $k$ blocks"},"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#sum-symmetric-peak-count-set-partitions","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"can_prove","summary":"For natural numbers $n$ and $k$, let $P_{n,k}$ be the set of surjective restricted-growth strings of length $n$ with letters in $\\{0, \\dots, k-1\\}$ (these encode the partitions of $\\{1, \\dots, n\\}$ into exactly $k$ blocks). For $w \\in P_{n,k}$ let $\\operatorname{sp}(w)$ be the number of symmetric peaks of $w$, and denote by $S(a, b)$ the Stirling numbers of the second kind. Then","title":"Total number of symmetric peaks over set partitions with $k$ blocks"},{"id":"Combinatorics/set-partitions/peaks-in-set-partitions/sum-non-symmetric-peak-count-set-partiti","kind":"node","lean":"MetaMathlibExt.sum_nonSymmetricPeakCount_setPartitions","parent":27464,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total number of non-symmetric peaks over set partitions with "],["math","\\(k\\)"],["text"," blocks"]],"source":"Total number of non-symmetric peaks over set partitions with $k$ blocks"},"result":true,"route":"../roadmap/Combinatorics/set-partitions/peaks-in-set-partitions/index.html#sum-non-symmetric-peak-count-set-partiti","scope":"Combinatorics/set-partitions/peaks-in-set-partitions","status":"can_prove","summary":"For natural numbers $n$ and $k$, let $P_{n,k}$ be the set of surjective restricted-growth strings of length $n$ with letters in $\\{0, \\dots, k-1\\}$ (these encode the partitions of $\\{1, \\dots, n\\}$ into exactly $k$ blocks). For $w \\in P_{n,k}$ let $\\operatorname{nsp}(w)$ be the number of non-symmetric peaks of $w$, and denote by $S(a, b)$ the Stirling numbers of the second kind. Then","title":"Total number of non-symmetric peaks over set partitions with $k$ blocks"},{"data_url":"scopes/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles.json","id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","kind":"scope","navigation":"module","parent":27324,"result":false,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","title":"Reduced unicellular hypermonopoles"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/cyclic-product-number","kind":"node","lean":"MetaMathlibExt.cyclicProductNumber","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#cyclic-product-number","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"fully_proved","summary":"For natural numbers $n$ and $k$, write $c(m, j)$ for the unsigned Stirling numbers of the first kind. The cyclic product number $H(n, k)$ is the quotient $c(n + 1, k) / \\binom{n + 1}{2}$, rounded down to a natural number, when $n + k$ is even (equivalently, $n - k$ is even), and $0$ when $n + k$ is odd. Division by zero gives $0$, so $H(0, k) = 0$ for every $k$.","title":"Cyclic product numbers"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/cyclic-product-number-of-odd","kind":"node","lean":"MetaMathlibExt.cyclicProductNumber_of_odd","parent":27474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing of the cyclic product numbers for odd "],["math","\\(n - k\\)"]],"source":"Vanishing of the cyclic product numbers for odd $n - k$"},"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#cyclic-product-number-of-odd","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"fully_proved","summary":"For natural numbers $n$ and $k$ such that $n + k$ is odd (equivalently, $n - k$ is odd), the cyclic product number $H(n, k)$ is $0$.","title":"Vanishing of the cyclic product numbers for odd $n - k$"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/cyclic-product-number-zero-right","kind":"node","lean":"MetaMathlibExt.cyclicProductNumber_zero_right","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#cyclic-product-number-zero-right","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"fully_proved","summary":"For every natural number $n$, the cyclic product number $H(n, 0)$ is $0$, since the unsigned Stirling number of the first kind $c(n + 1, 0)$ is $0$.","title":"Cyclic product numbers with second argument zero"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/cyclic-product-number-small-values","kind":"node","lean":"MetaMathlibExt.cyclicProductNumber_small_values","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#cyclic-product-number-small-values","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"fully_proved","summary":"The cyclic product numbers satisfy $H(3, 1) = 1$, $H(4, 2) = 5$, $H(5, 1) = 8$ and $H(7, 1) = 180$.","title":"Small cyclic product numbers"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/genus-bounds-of-reduced-unicellular-hype","kind":"node","lean":"MetaMathlibExt.genus_bounds_of_reduced_unicellular_hypermonopole","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#genus-bounds-of-reduced-unicellular-hype","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","summary":"Let $\\sigma$ and $\\alpha$ be permutations of $\\{0, 1, \\dots, n - 1\\}$, and write $z(\\pi)$ for the number of cycles of a permutation $\\pi$, fixed points included. Suppose that $z(\\sigma) = 1$, that $z(\\alpha^{-1} \\circ \\sigma) = 1$, that $\\alpha$ has no fixed point, and that the natural number $g$ satisfies $n + 2 = z(\\sigma) + z(\\alpha) + z(\\alpha^{-1} \\circ \\sigma) + 2g$. Then $2g + 1 \\le n \\le 4g$.","title":"Genus bounds for reduced unicellular hypermonopoles"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/sum-cyclic-product-number-mul-x-pow","kind":"node","lean":"MetaMathlibExt.sum_cyclicProductNumber_mul_X_pow","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#sum-cyclic-product-number-mul-x-pow","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","summary":"For every natural number $n$, let $H(n, k)$ be the cyclic product numbers, viewed as rational numbers. Then in the polynomial ring $\\mathbb{Q}[x]$","title":"Generating polynomial of the cyclic product numbers"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/card-unicellular-hypermonopoles-fin-rota","kind":"node","lean":"MetaMathlibExt.card_unicellular_hypermonopoles_finRotate","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#card-unicellular-hypermonopoles-fin-rota","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers, let $\\sigma$ be the cyclic permutation $i \\mapsto i + 1$ (modulo $n$) of $\\{0, 1, \\dots, n - 1\\}$, and write $z(\\pi)$ for the number of cycles of a permutation $\\pi$, fixed points included. Then the number of permutations $\\alpha$ of $\\{0, 1, \\dots, n - 1\\}$ with $z(\\alpha^{-1} \\circ \\sigma) = 1$ and $z(\\alpha) = k$ is the cyclic product number $H(n, k)$.","title":"Unicellular hypermonopoles with a cyclic vertex permutation"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/reduced-iff-forall-face-ne-fin-rotate","kind":"node","lean":"MetaMathlibExt.reduced_iff_forall_face_ne_finRotate","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#reduced-iff-forall-face-ne-fin-rotate","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","summary":"Let $\\sigma$ be the cyclic permutation $i \\mapsto i + 1$ (modulo $n$) of $\\{0, 1, \\dots, n - 1\\}$. A permutation $\\alpha$ of $\\{0, 1, \\dots, n - 1\\}$ has no fixed point if and only if $\\pi = \\alpha^{-1} \\circ \\sigma$ satisfies $\\pi(i) \\ne i + 1$ (modulo $n$) for every $i$.","title":"Criterion for a hypermap with a cyclic vertex permutation to have no bud"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/cycle-count-bounds-of-reduced","kind":"node","lean":"MetaMathlibExt.cycleCount_bounds_of_reduced","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#cycle-count-bounds-of-reduced","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","summary":"Let $n \\ge 1$ and let $\\alpha$ be a permutation of $\\{0, 1, \\dots, n - 1\\}$ with no fixed point. Then the number $z(\\alpha)$ of cycles of $\\alpha$ satisfies $1 \\le z(\\alpha)$ and $2 z(\\alpha) \\le n$.","title":"Cycle count of a fixed-point-free permutation"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/card-reduced-unicellular-hypermonopoles","kind":"node","lean":"MetaMathlibExt.card_reduced_unicellular_hypermonopoles_finRotate","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#card-reduced-unicellular-hypermonopoles","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","summary":"Let $n \\ge 2$ and $k \\ge 1$ be natural numbers with $2k \\le n$, let $\\sigma$ be the cyclic permutation $i \\mapsto i + 1$ (modulo $n$) of $\\{0, 1, \\dots, n - 1\\}$, and write $z(\\pi)$ for the number of cycles of a permutation $\\pi$, fixed points included. Let $r(n, k)$ be the number of permutations $\\alpha$ of $\\{0, 1, \\dots, n - 1\\}$ with no fixed point such that $z(\\alpha^{-1} \\circ \\sigma) = 1$ and\u2026","title":"Counting reduced unicellular hypermonopoles with a cyclic vertex permutation"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/reduced-unicellular-hypermonopoles-fin-r","kind":"node","lean":"MetaMathlibExt.reduced_unicellular_hypermonopoles_finRotate_vanishing","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#reduced-unicellular-hypermonopoles-fin-r","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","summary":"Let $\\sigma_n$ be the cyclic permutation $i \\mapsto i + 1$ (modulo $n$) of $\\{0, 1, \\dots, n - 1\\}$, write $z(\\pi)$ for the number of cycles of a permutation $\\pi$, fixed points included, and let $r(n, k)$ be the number of permutations $\\alpha$ of $\\{0, 1, \\dots, n - 1\\}$ with no fixed point such that $z(\\alpha^{-1} \\circ \\sigma_n) = 1$ and $z(\\alpha) = k$. Then $r(n, k) = 0$ for all natural numbers $n$ and $k$\u2026","title":"Vanishing of the reduced unicellular hypermonopole counts"},{"id":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/reduced-unicellular-hypermonopoles-of-ge","kind":"node","lean":"MetaMathlibExt.reduced_unicellular_hypermonopoles_of_genus_count","parent":27474,"result":true,"route":"../roadmap/Combinatorics/set-partitions/reduced-unicellular-hypermonopoles/index.html#reduced-unicellular-hypermonopoles-of-ge","scope":"Combinatorics/set-partitions/reduced-unicellular-hypermonopoles","status":"can_prove","summary":"Fix $g \\in \\mathbb{N}$. For each $n \\in \\mathbb{N}$ let $\\zeta_n$ be the cyclic permutation of $\\{0, \\ldots, n - 1\\}$ sending $i$ to $i + 1$ modulo $n$, and let $z(\\pi)$ be the number of cycles of a permutation $\\pi$, fixed points included. Let $U_g$ be the set of pairs $(n, \\alpha)$, with $n \\in \\mathbb{N}$ and $\\alpha$ a permutation of $\\{0, \\ldots, n - 1\\}$, such that $z(\\zeta_n) = 1$,\u2026","title":"Counting reduced unicellular hypermonopoles by genus"},{"data_url":"scopes/Combinatorics/tilings-board-placements.json","id":"Combinatorics/tilings-board-placements","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/index.html","scope":"Combinatorics/tilings-board-placements","status":"can_prove","summary":"Covering and placement problems on lattices and boards: domino and lozenge tilings and perfect matchings of grid regions (Aztec diamonds, M\u00f6bius grids, cylinders and tori), polyominoes and polycubes, tilings of groups by translates and the periodic tiling conjecture, nonattacking chess pieces, rook and file placements on Ferrers boards, and self-avoiding walks on lattices.","title":"Tilings, dimers and board placements"},{"id":"Combinatorics/tilings-board-placements/quarter-period-factorizations","kind":"node","lean":"MetaMathlibExt.quarter_period_factorizations","parent":27487,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/index.html#quarter-period-factorizations","scope":"Combinatorics/tilings-board-placements","status":"fully_proved","summary":"Let $Q \\colon \\mathbb{N} \\to \\mathbb{N}$ satisfy $Q(0) = Q(1) = 1$, $Q(2) = 4$ and $Q(n) = Q(n-1) + 3Q(n-2) + Q(n-3)$ for every $n \\ge 3$. Then for every $m \\ge 1$,","title":"Quarter-period factorizations of the triangular tiling numbers"},{"id":"Combinatorics/tilings-board-placements/domino-tilings-l-grid-via-fibonacci","kind":"node","lean":"MetaMathlibExt.domino_tilings_L_grid_via_fibonacci","parent":27487,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/index.html#domino-tilings-l-grid-via-fibonacci","scope":"Combinatorics/tilings-board-placements","status":"can_prove","summary":"Let $f_n$ be the Fibonacci numbers, with $f_0 = 0$ and $f_1 = 1$. For natural numbers $n, m$ put $L_{n,m} = f_{n+1} f_m + f_n f_{m+1}$ and $a_n = f_{n+1} f_{n-1} + f_n^2$, where $n - 1$ is truncated subtraction. Then $L_{n,m} = f_{n+1} f_m + f_n f_{m+1}$ for all $n, m \\ge 1$, $L_{n,n-1} = a_n$ for all $n \\ge 2$, and $L_{n,n-1}^2 = L_{n,n} L_{n-1,n-1} + 1$ for all $n \\ge 2$.","title":"Tauraso's Cassini-type identity"},{"data_url":"scopes/Combinatorics/tilings-board-placements/aztec-dragon-tilings.json","id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"can_prove","title":"Aztec dragon tilings"},{"id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings/aztec-cell","kind":"node","lean":"MathlibExt.Combinatorics.AztecDragonPowerOfTwoAMR122Problem0015Wanted.AztecCell","parent":27490,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html#aztec-cell","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"fully_proved","summary":"For $m \\in \\mathbb{N}$, the cells of the Aztec diamond region of order $m$ are the pairs of integers $(i, j)$ with $|2i + 1| + |2j + 1| \\le 2m$. The pair $(i, j)$ stands for the unit square with lower-left corner $(i, j)$ and center $(i + \\frac12, j + \\frac12)$, so the condition says that this center satisfies $|i + \\frac12| + |j + \\frac12| \\le m$.","title":"Cells of the Aztec diamond"},{"id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings/ciucu-matrix","kind":"node","lean":"MathlibExt.Combinatorics.AztecDragonPowerOfTwoAMR122Problem0015Wanted.ciucuMatrix","parent":27490,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html#ciucu-matrix","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"fully_proved","summary":"Ciucu's periodic weight matrix of Section 7 (the display before (7.1)) is the $4 \\times 4$ matrix with natural number entries, rows and columns indexed by $0, 1, 2, 3$,","title":"Ciucu's weight matrix for the Aztec dragon"},{"id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings/edge-row","kind":"node","lean":"MathlibExt.Combinatorics.AztecDragonPowerOfTwoAMR122Problem0015Wanted.edgeRow","parent":27490,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html#edge-row","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"fully_proved","summary":"For $m \\in \\mathbb{N}$ and pairs of integers $c = (c_1, c_2)$ and $d = (d_1, d_2)$, put $X = c_1 + d_1 + 1$ and $Y = c_2 + d_2 + 1$, the doubled coordinates of the center of the edge joining $c$ and $d$. The row index of this edge is $\\lfloor (X + Y + 2m - 1) / 2 \\rfloor$, so the rows of the $2m \\times 2m$ array of edge centers of the Aztec diamond graph $AD_m$ are the diagonals $X + Y = \\mathrm{const}$, numbered\u2026","title":"Row index of an edge of the Aztec diamond"},{"id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings/edge-col","kind":"node","lean":"MathlibExt.Combinatorics.AztecDragonPowerOfTwoAMR122Problem0015Wanted.edgeCol","parent":27490,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html#edge-col","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"fully_proved","summary":"For $m \\in \\mathbb{N}$ and pairs of integers $c = (c_1, c_2)$ and $d = (d_1, d_2)$, put $X = c_1 + d_1 + 1$ and $Y = c_2 + d_2 + 1$, the doubled coordinates of the center of the edge joining $c$ and $d$. The column index of this edge is $\\lfloor (X - Y + 2m - 1) / 2 \\rfloor$, so the columns of the array of edge centers of the Aztec diamond graph $AD_m$ are the diagonals $X - Y = \\mathrm{const}$, numbered\u2026","title":"Column index of an edge of the Aztec diamond"},{"id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings/ciucu-weight","kind":"node","lean":"MathlibExt.Combinatorics.AztecDragonPowerOfTwoAMR122Problem0015Wanted.ciucuWeight","parent":27490,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html#ciucu-weight","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"fully_proved","summary":"For $m \\in \\mathbb{N}$ and pairs of integers $c$ and $d$, let $r$ and $s$ be the row and column indices of the pair $(c, d)$ in the array of edge centers of the Aztec diamond graph $AD_m$. The weight $\\mathrm{wt}_A(c, d)$ is the entry of Ciucu's matrix $A$ in row $r \\bmod 4$ and column $s \\bmod 4$; that is, $A$ is placed in a corner of the array and repeated with period $4$ in both directions (Ciucu's Definition\u2026","title":"Ciucu's periodic weight"},{"id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings/dragon-graph","kind":"node","lean":"MathlibExt.Combinatorics.AztecDragonPowerOfTwoAMR122Problem0015Wanted.dragonGraph","parent":27490,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html#dragon-graph","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the dragon graph of order $n$ is the graph on the cells of the Aztec diamond of order $2n$ in which two cells $c$ and $d$ are adjacent when they share a side (their coordinates differ by $1$ in exactly one place) and the edge joining them has weight $\\mathrm{wt}_A(c, d) = 1$ under Ciucu's periodic weight. It is the spanning subgraph of the Aztec diamond graph $AD_{2n}$ obtained by deleting\u2026","title":"The Aztec dragon graph"},{"id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings/aztec-dragon-count","kind":"node","lean":"MathlibExt.Combinatorics.AztecDragonPowerOfTwoAMR122Problem0015Wanted.aztecDragonCount","parent":27490,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html#aztec-dragon-count","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"fully_proved","summary":"The statement that for every natural number $n$ the dragon graph of order $n$ (the spanning subgraph of the Aztec diamond graph $AD_{2n}$ that keeps exactly the edges of weight $1$ under Ciucu's periodic weight $\\mathrm{wt}_A$) has exactly $2^{n(n+1)}$ perfect matchings. This is Propp's Problem 15 in the form of Ciucu's (7.1) and Theorem 7.1 at $a = 1$.","title":"Propp's Aztec dragon count"},{"id":"Combinatorics/tilings-board-placements/aztec-dragon-tilings/aztec-dragon-count-holds","kind":"node","lean":"MathlibExt.Combinatorics.AztecDragonPowerOfTwoAMR122Problem0015Wanted.aztecDragonCount_holds","parent":27490,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/aztec-dragon-tilings/index.html#aztec-dragon-count-holds","scope":"Combinatorics/tilings-board-placements/aztec-dragon-tilings","status":"can_prove","summary":"For every natural number $n$, the dragon graph of order $n$ (the spanning subgraph of the Aztec diamond graph $AD_{2n}$ that keeps exactly the edges of weight $1$ under Ciucu's periodic weight $\\mathrm{wt}_A$) has exactly $2^{n(n+1)}$ perfect matchings.","title":"Perfect matchings of the Aztec dragon graph (Ciucu, Wieland)"},{"data_url":"scopes/Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products.json","id":"Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products/index.html","scope":"Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products","status":"fully_proved","title":"Domino tilings and Fibonacci\u2013Pell products"},{"id":"Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products/domino-tiling-sequence-factorization","kind":"node","lean":"MetaMathlibExt.domino_tiling_sequence_factorization","parent":27499,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products/index.html#domino-tiling-sequence-factorization","scope":"Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products","status":"fully_proved","summary":"Let $d$ and $g$ be sequences of natural numbers with $d_1 = 1$, $d_2 = 8$, $d_3 = 95$, $d_4 = 1183$ and $d_{n+4} + 32 d_{n+2} + d_n = 15 d_{n+3} + 15 d_{n+1}$ for all $n \\ge 1$, and with $g_1 = 1$, $g_2 = 4$ and $g_{n+2} + g_n = 5 g_{n+1}$ for all $n \\ge 1$. Then $d_n = g_n F_{2n-1}$ for every $n \\ge 1$, where $F_m$ is the $m$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$).","title":"Factorization of the domino tiling sequence"},{"id":"Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products/box-prod-tiling-card-eq-fib-mul-pell-num","kind":"node","lean":"MetaMathlibExt.boxProd_tiling_card_eq_fib_mul_pellNumber","parent":27499,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products/index.html#box-prod-tiling-card-eq-fib-mul-pell-num","scope":"Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products","status":"fully_proved","summary":"Let $W_4$ be the complete graph $K_4$ with one edge removed and $P_{n-1}$ the path graph on $n - 1$ vertices. For every natural number $n \\ge 2$, the number of perfect matchings (domino tilings) of the Cartesian product $W_4 \\square P_{n-1}$ equals $f_n p_n$, where $f_n$ is the $n$-th Fibonacci number ($f_0 = 0$, $f_1 = 1$) and $p_n$ is the $n$-th Pell number ($p_0 = 0$, $p_1 = 1$, $p_{k+2} = 2 p_{k+1} + p_k$).","title":"Sellers' domino tiling theorem"},{"id":"Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products/box-prod-tiling-card-eq-fib-mul-pell","kind":"node","lean":"MetaMathlibExt.boxProd_tiling_card_eq_fib_mul_pell","parent":27499,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products/index.html#box-prod-tiling-card-eq-fib-mul-pell","scope":"Combinatorics/tilings-board-placements/domino-tilings-and-fibonacci-pell-products","status":"fully_proved","summary":"Let $W_4$ be the complete graph $K_4$ with one edge removed and $P_{n-1}$ the path graph on $n - 1$ vertices. For every natural number $n \\ge 2$, the number of perfect matchings (domino tilings) of the Cartesian product $W_4 \\square P_{n-1}$ equals $f_n p_n$, where $f_n$ is the $n$-th Fibonacci number ($f_0 = 0$, $f_1 = 1$) and $p_n$ is defined by $p_0 = 0$, $p_1 = 1$ and $p_{k+2} = 2 p_{k+1} + p_k$.","title":"Sellers' domino tiling theorem, recursive Pell form"},{"data_url":"scopes/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles.json","id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","title":"Fault-free tilings of 3 \u00d7 n rectangles"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/tiling-cell","kind":"node","lean":"MetaMathlibExt.TilingCell","parent":27503,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cells of the "],["math","\\(3 \\times n\\)"],["text"," rectangle"]],"source":"Cells of the $3 \\times n$ rectangle"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#tiling-cell","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"A cell of the $3 \\times n$ rectangle is a pair $(r, c)$ of a row $r \\in \\{0, 1, 2\\}$ and a column $c \\in \\{0, \\dots, n-1\\}$.","title":"Cells of the $3 \\times n$ rectangle"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/adjacent","kind":"node","lean":"MetaMathlibExt.TilingCell.Adjacent","parent":27503,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#adjacent","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"Two cells $x$ and $y$ of the $3 \\times n$ rectangle are adjacent when they share an edge: they lie in the same row and their columns differ by $1$, or they lie in the same column and their rows differ by $1$.","title":"Adjacent cells"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/is-square-or-domino","kind":"node","lean":"MetaMathlibExt.IsSquareOrDomino","parent":27503,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#is-square-or-domino","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"A set $d$ of cells of the $3 \\times n$ rectangle is a square-or-domino tile when it consists of exactly one cell, or of exactly two cells that are adjacent.","title":"Square and domino tiles"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/is-square-or-domino-iff","kind":"node","lean":"MetaMathlibExt.isSquareOrDomino_iff","parent":27503,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#is-square-or-domino-iff","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"A set $d$ of cells of the $3 \\times n$ rectangle is a square-or-domino tile if and only if either $d = \\{x\\}$ for some cell $x$, or $d = \\{x, y\\}$ for some distinct adjacent cells $x$ and $y$.","title":"Characterization of square and domino tiles"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/is-square-domino-tiling","kind":"node","lean":"MetaMathlibExt.IsSquareDominoTiling","parent":27503,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#is-square-domino-tiling","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"A square-and-domino tiling of the $3 \\times n$ rectangle is a finite family $T$ of sets of cells such that every member of $T$ is a square-or-domino tile (a single cell or two adjacent cells), any two distinct members of $T$ are disjoint, and every cell of the rectangle lies in some member of $T$.","title":"Square-and-domino tilings"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/is-vertical-fault-free","kind":"node","lean":"MetaMathlibExt.IsVerticalFaultFree","parent":27503,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#is-vertical-fault-free","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"A family $T$ of sets of cells of the $3 \\times n$ rectangle, with columns numbered $0, \\dots, n-1$, is vertical-fault-free when for every natural number $k$ with $k + 1 < n$ some member of $T$ contains two cells of the same row lying in columns $k$ and $k + 1$. For a square-and-domino tiling this says that every internal vertical boundary is crossed by a horizontal domino.","title":"Vertical-fault-free families"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/vertical-fault-free-tiling-count","kind":"node","lean":"MetaMathlibExt.verticalFaultFreeTilingCount","parent":27503,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of vertical-fault-free tilings of the "],["math","\\(3 \\times n\\)"],["text"," rectangle"]],"source":"Number of vertical-fault-free tilings of the $3 \\times n$ rectangle"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#vertical-fault-free-tiling-count","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"For every natural number $n$, this is the number of square-and-domino tilings of the $3 \\times n$ rectangle that are vertical-fault-free.","title":"Number of vertical-fault-free tilings of the $3 \\times n$ rectangle"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/tiling-count","kind":"node","lean":"MetaMathlibExt.tilingCount","parent":27503,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence sequence for vertical-fault-free "],["math","\\(3 \\times n\\)"],["text"," tilings"]],"source":"Recurrence sequence for vertical-fault-free $3 \\times n$ tilings"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#tiling-count","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"The integer sequence $(V_n)_{n \\ge 0}$ is defined by $V_0 = 0$, $V_1 = 3$, $V_2 = 13$, $V_3 = 26$, $V_4 = 66$, $V_5 = 154$, $V_6 = 380$ and $V_n = V_{n-1} + 4 V_{n-2} - V_{n-3} - V_{n-4}$ for $n \\ge 7$.","title":"Recurrence sequence for vertical-fault-free $3 \\times n$ tilings"},{"id":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/fault-free-3xn-tiling-recursion","kind":"node","lean":"MetaMathlibExt.fault_free_3xn_tiling_recursion","parent":27503,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for vertical-fault-free tilings of the "],["math","\\(3 \\times n\\)"],["text"," rectangle"]],"source":"Recurrence for vertical-fault-free tilings of the $3 \\times n$ rectangle"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles/index.html#fault-free-3xn-tiling-recursion","scope":"Combinatorics/tilings-board-placements/fault-free-tilings-of-3-n-rectangles","status":"fully_proved","summary":"For $n \\ge 1$, let $V_n$ be the number of vertical-fault-free tilings of the $3 \\times n$ rectangle by unit squares and dominoes. Then $V_1 = 3$, $V_2 = 13$, $V_3 = 26$, $V_4 = 66$, $V_5 = 154$, $V_6 = 380$, and $V_n = V_{n-1} + 4 V_{n-2} - V_{n-3} - V_{n-4}$ for every $n \\ge 7$.","title":"Recurrence for vertical-fault-free tilings of the $3 \\times n$ rectangle"},{"data_url":"scopes/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers.json","id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","title":"Fence tilings and bi-periodic Fibonacci numbers"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/bi-periodic-fibonacci-sq","kind":"node","lean":"MetaMathlibExt.biPeriodicFibonacciSq","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#bi-periodic-fibonacci-sq","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For elements $a$ and $b$ of a field $K$, the bi-periodic Fibonacci numbers squared $q^2_n \\in K$ ($n \\ge 0$) are defined by $q^2_0 = 0$, $q^2_1 = 1$, $q^2_2 = a^2$ and, for $n \\ge 3$,","title":"Bi-periodic Fibonacci numbers squared"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/bi-periodic-fibonacci-sq-zero","kind":"node","lean":"MetaMathlibExt.biPeriodicFibonacciSq_zero","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#bi-periodic-fibonacci-sq-zero","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For elements $a$ and $b$ of a field $K$, the bi-periodic Fibonacci number squared $q^2_0$ equals $0$.","title":"Zeroth bi-periodic Fibonacci number squared"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/bi-periodic-fibonacci-sq-one","kind":"node","lean":"MetaMathlibExt.biPeriodicFibonacciSq_one","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#bi-periodic-fibonacci-sq-one","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For elements $a$ and $b$ of a field $K$, the bi-periodic Fibonacci number squared $q^2_1$ equals $1$.","title":"First bi-periodic Fibonacci number squared"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/bi-periodic-fibonacci-sq-two","kind":"node","lean":"MetaMathlibExt.biPeriodicFibonacciSq_two","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#bi-periodic-fibonacci-sq-two","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For elements $a$ and $b$ of a field $K$, the bi-periodic Fibonacci number squared $q^2_2$ equals $a^2$.","title":"Second bi-periodic Fibonacci number squared"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/bi-periodic-fibonacci-sq-add-three","kind":"node","lean":"MetaMathlibExt.biPeriodicFibonacciSq_add_three","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#bi-periodic-fibonacci-sq-add-three","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For elements $a$ and $b$ of a field $K$ and every $n \\ge 0$, the bi-periodic Fibonacci numbers squared $q^2_m$ satisfy","title":"Recurrence for the bi-periodic Fibonacci numbers squared"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fence-tilings","kind":"node","lean":"MetaMathlibExt.fenceTilings","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fence-tilings","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"Number the half-cells of an $n$-board $0, \\dots, 2n - 1$. The tilings of the $n$-board by half-squares and $(1/2, 1/2)$-fences are encoded as the sets $F$ of fence positions $k < 2n - 2$ (the fence at $k$ has posts on half-cells $k$ and $k + 2$) whose posts are pairwise disjoint, that is, $k + 2 \\notin F$ for every $k \\in F$.","title":"Fence tilings of a board"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fence-tiling-weight","kind":"node","lean":"MetaMathlibExt.fenceTilingWeight","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fence-tiling-weight","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"Let $a$, $b$ be elements of a commutative semiring, let $n \\ge 0$, and let $F$ be a finite set of natural numbers (fence positions; the fence at $k$ has posts on half-cells $k$ and $k + 2$). The weight of $F$ on the $n$-board is the product, over the half-cells $h \\in \\{0, \\dots, 2n - 1\\}$ that neither lie in $F$ nor equal $k + 2$ for some $k \\in F$, of $a$ if $\\lfloor h/2 \\rfloor$ is even (a half-square in the odd\u2026","title":"Weight of a fence tiling"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fence-tiling-sum","kind":"node","lean":"MetaMathlibExt.fenceTilingSum","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fence-tiling-sum","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"Let $a$, $b$ be elements of a commutative semiring and $n \\ge 0$. The weighted number $S_n$ of tilings of an $n$-board by half-squares and $(1/2, 1/2)$-fences is the sum, over all fence tilings $F$ of the $n$-board, of the weight of $F$, in which a half-square weighs $a$ in an odd cell and $b$ in an even cell and fences weigh $1$.","title":"Weighted count of fence tilings"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fence-tiling-sum-zero","kind":"node","lean":"MetaMathlibExt.fenceTilingSum_zero","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fence-tiling-sum-zero","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For elements $a$ and $b$ of a commutative semiring $R$, the weighted number $S_0$ of tilings of the $0$-board by half-squares and $(1/2, 1/2)$-fences equals $1$: the only tiling is the empty one.","title":"Weighted fence tilings of the empty board"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fence-tiling-sum-one","kind":"node","lean":"MetaMathlibExt.fenceTilingSum_one","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fence-tiling-sum-one","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For elements $a$ and $b$ of a commutative semiring $R$, the weighted number $S_1$ of tilings of a $1$-board by half-squares and $(1/2, 1/2)$-fences, a half-square weighing $a$ in an odd cell and $b$ in an even cell, equals $a^2$: the only tiling has two half-squares in cell $1$.","title":"Weighted fence tilings of the one-cell board"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/no-bifence-tiling-sum","kind":"node","lean":"MetaMathlibExt.noBifenceTilingSum","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#no-bifence-tiling-sum","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"Let $a$, $b$ be elements of a commutative semiring and $n \\ge 0$. Two fences at positions $k$ and $k + 1$ form a bifence. The weighted number $C_n$ of tilings with no bifence is the sum of the weights, with half-square weights $a$ and $b$, of the fence tilings $F$ of the $n$-board with no bifence, that is, with $k + 1 \\notin F$ for every $k \\in F$.","title":"Weighted count of bifence-free fence tilings"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/no-bifence-tiling-sum-zero","kind":"node","lean":"MetaMathlibExt.noBifenceTilingSum_zero","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#no-bifence-tiling-sum-zero","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For all elements $a$, $b$ of a commutative semiring, the weighted number $C_0$ of tilings of the $0$-board with no bifence, with half-square weights $a$ and $b$, equals $1$.","title":"Tilings of the empty board with no bifence"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/no-free-bifence-tiling-sum","kind":"node","lean":"MetaMathlibExt.noFreeBifenceTilingSum","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#no-free-bifence-tiling-sum","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For elements $a$, $b$ of a commutative semiring and $n \\ge 0$, number the cells of an $n$-board $1, \\dots, n$ and its $2n$ half-cells $0, \\dots, 2n - 1$, half-cell $h$ lying in cell $\\lfloor h/2 \\rfloor + 1$. A $(1/2, 1/2)$-fence at position $k$ has its posts on half-cells $k$ and $k + 2$. Then $R_n$ is the sum of the weights of the tilings of the board by half-squares and such fences that contain no free bifence,\u2026","title":"Weighted tilings with no free bifence"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/no-free-bifence-tiling-sum-zero","kind":"node","lean":"MetaMathlibExt.noFreeBifenceTilingSum_zero","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#no-free-bifence-tiling-sum-zero","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For all elements $a$, $b$ of a commutative semiring, the weighted number $R_0$ of tilings of the $0$-board by half-squares and $(1/2, 1/2)$-fences with no free bifence, with half-squares weighted by $a$ and $b$, equals $1$.","title":"Tilings of the empty board with no free bifence"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/no-free-bifence-tiling-sum-one","kind":"node","lean":"MetaMathlibExt.noFreeBifenceTilingSum_one","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#no-free-bifence-tiling-sum-one","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"fully_proved","summary":"For all elements $a$, $b$ of a commutative semiring, the weighted number $R_1$ of tilings of the $1$-board with no free bifence, with half-square weights $a$ and $b$, equals $a^2$.","title":"Tilings of the one-cell board with no free bifence"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fence-tiling-sum-even","kind":"node","lean":"MetaMathlibExt.fenceTilingSum_even","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fence-tiling-sum-even","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $a$ and $b$ be elements of a commutative semiring $R$, and let $S_m$ be the weighted number of tilings of an $m$-board, with cells numbered $1, \\dots, m$, by half-squares and $(1/2, 1/2)$-fences, a half-square weighing $a$ in an odd cell and $b$ in an even cell. Then for every even $n \\ge 2$,","title":"Recurrence for weighted fence tilings of even boards"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fence-tiling-sum-odd","kind":"node","lean":"MetaMathlibExt.fenceTilingSum_odd","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fence-tiling-sum-odd","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $a$ and $b$ be elements of a commutative semiring $R$, and let $S_m$ be the weighted number of tilings of an $m$-board, with cells numbered $1, \\dots, m$, by half-squares and $(1/2, 1/2)$-fences, a half-square weighing $a$ in an odd cell and $b$ in an even cell. Then for every odd $n \\ge 3$,","title":"Recurrence for weighted fence tilings of odd boards"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fence-tiling-sum-eq-bi-periodic-fibonacc","kind":"node","lean":"MetaMathlibExt.fenceTilingSum_eq_biPeriodicFibonacciSq","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fence-tiling-sum-eq-bi-periodic-fibonacc","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $a$ and $b$ be nonzero elements of a field $K$. For every $n \\ge 0$, let $S_n$ be the weighted number of tilings of an $n$-board, with cells numbered $1, \\dots, n$, by half-squares and $(1/2, 1/2)$-fences, a half-square weighing $a$ in an odd cell and $b$ in an even cell. Then $S_n$ equals the bi-periodic Fibonacci number squared $q^2_{n+1}$.","title":"Fence tilings and the bi-periodic Fibonacci numbers squared"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fib-succ-sq-eq-sum-choose-mul-choose","kind":"node","lean":"MetaMathlibExt.fib_succ_sq_eq_sum_choose_mul_choose","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fib-succ-sq-eq-sum-choose-mul-choose","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $F_k$ denote the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). For every natural number $n$,","title":"Squared Fibonacci numbers as a double binomial sum"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/two-mul-sum-fib-sq","kind":"node","lean":"MetaMathlibExt.two_mul_sum_fib_sq","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#two-mul-sum-fib-sq","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $F_j$ denote the $j$-th Fibonacci number, with $F_0 = 0$ and $F_1 = 1$. For every integer $n \\ge 3$,","title":"Sum of squared Fibonacci numbers from three consecutive squares"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fib-succ-sq-sub-parity-eq-sum","kind":"node","lean":"MetaMathlibExt.fib_succ_sq_sub_parity_eq_sum","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fib-succ-sq-sub-parity-eq-sum","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $F_k$ denote the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$) and let $\\xi(m) = m \\bmod 2$. For every natural number $n \\ge 1$,","title":"Squared Fibonacci numbers as weighted sums of squares"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/fib-add-sq-eq","kind":"node","lean":"MetaMathlibExt.fib_add_sq_eq","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#fib-add-sq-eq","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $F_k$ denote the Fibonacci numbers ($F_0 = 0$, $F_1 = 1$). For all natural numbers $m \\ge 1$ and $n \\ge 2$,","title":"Addition formula for squared Fibonacci numbers"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/no-free-bifence-tiling-sum-rec","kind":"node","lean":"MetaMathlibExt.noFreeBifenceTilingSum_rec","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#no-free-bifence-tiling-sum-rec","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $a$ and $b$ be nonzero elements of a field, and let $R_n$ be the weighted number of tilings of an $n$-board by half-squares and $(1/2, 1/2)$-fences with no free bifence, where a half-square has weight $a$ in an odd cell and $b$ in an even cell (cells numbered $1, \\dots, n$). For every $n \\ge 2$,","title":"Recurrence for weighted tilings with no free bifence"},{"id":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/no-bifence-tiling-sum-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.noBifenceTilingSum_eq_sum_choose","parent":27513,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers/index.html#no-bifence-tiling-sum-eq-sum-choose","scope":"Combinatorics/tilings-board-placements/fence-tilings-and-bi-periodic-fibonacci-numbers","status":"can_prove","summary":"Let $a$, $b$ be nonzero elements of a field and let $\\xi(n) = n \\bmod 2$. For every natural number $n$, the weighted number $C_n$ of tilings of the $n$-board by half-squares and $(1/2, 1/2)$-fences with no bifence, with half-square weights $a$ and $b$, is","title":"Closed formula for bifence-free fence tilings"},{"data_url":"scopes/Combinatorics/tilings-board-placements/fixed-polycubes.json","id":"Combinatorics/tilings-board-placements/fixed-polycubes","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","title":"Fixed polycubes"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/lattice-cell","kind":"node","lean":"MetaMathlibExt.latticeCell","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#lattice-cell","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"For a natural number $d$, a lattice cell in dimension $d$ is a point of $\\mathbb{Z}^d$, that is, a function assigning an integer to each of the $d$ coordinate axes.","title":"Lattice cells"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/face-adjacent","kind":"node","lean":"MetaMathlibExt.faceAdjacent","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#face-adjacent","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"Two lattice cells $x, y \\in \\mathbb{Z}^d$ are face adjacent when there is an axis $k$ such that $x_j = y_j$ for every axis $j \\ne k$ and $|x_k - y_k| = 1$.","title":"Face adjacency of lattice cells"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/is-polycube-cells","kind":"node","lean":"MetaMathlibExt.isPolycubeCells","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#is-polycube-cells","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"A finite set $S$ of lattice cells in $\\mathbb{Z}^d$ is a polycube when it is nonempty and any two cells $x, y \\in S$ are joined by a finite sequence of cells of $S$, starting at $x$ and ending at $y$, in which consecutive cells are face adjacent.","title":"Polycube"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/is-canonical-cells","kind":"node","lean":"MetaMathlibExt.isCanonicalCells","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#is-canonical-cells","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"A finite set $S$ of lattice cells in $\\mathbb{Z}^d$ is in canonical position when, for every axis $k$, every cell $x \\in S$ has $x_k \\ge 0$ and some cell $x \\in S$ has $x_k = 0$; that is, the minimum of each coordinate over $S$ is $0$.","title":"Canonical position of a cell set"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/proper-dimension","kind":"node","lean":"MetaMathlibExt.properDimension","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#proper-dimension","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"The proper dimension of a finite set $S$ of lattice cells in $\\mathbb{Z}^d$ is the number of axes $k$ on which $S$ varies, that is, for which there are cells $x, y \\in S$ with $x_k \\ne y_k$.","title":"Proper dimension"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/fixed-polycube","kind":"node","lean":"MetaMathlibExt.fixedPolycube","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#fixed-polycube","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"For natural numbers $d$ and $n$, the fixed $n$-cell polycubes in dimension $d$ are represented by the finite sets $S$ of lattice cells in $\\mathbb{Z}^d$ that are polycubes (nonempty and connected through face adjacency), have exactly $n$ cells, and are in canonical position (the minimum of each coordinate over $S$ is $0$).","title":"Fixed polycubes"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/fixed-polycube-count","kind":"node","lean":"MetaMathlibExt.fixedPolycubeCount","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#fixed-polycube-count","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"For natural numbers $d$ and $n$, $A_d(n)$ is the number of fixed $n$-cell polycubes in dimension $d$, that is, the number of $n$-cell polycubes in $\\mathbb{Z}^d$ that are in canonical position.","title":"Number of fixed polycubes"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/proper-fixed-polycube","kind":"node","lean":"MetaMathlibExt.properFixedPolycube","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#proper-fixed-polycube","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"For natural numbers $n$ and $i$, the fixed $n$-cell polycubes proper in dimension $i$ are the fixed $n$-cell polycubes in dimension $i$ whose proper dimension is $i$, that is, which vary along each of the $i$ axes.","title":"Proper fixed polycubes"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/proper-fixed-polycube-count","kind":"node","lean":"MetaMathlibExt.properFixedPolycubeCount","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#proper-fixed-polycube-count","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"For natural numbers $n$ and $i$, $\\mathrm{DX}(n,i)$ is the number of fixed $n$-cell polycubes proper in dimension $i$, that is, of fixed $n$-cell polycubes in dimension $i$ whose proper dimension is $i$.","title":"Number of proper fixed polycubes"},{"id":"Combinatorics/tilings-board-placements/fixed-polycubes/lunnon-fixed-polycube-dimension-decompos","kind":"node","lean":"MetaMathlibExt.lunnon_fixed_polycube_dimension_decomposition","parent":27538,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/fixed-polycubes/index.html#lunnon-fixed-polycube-dimension-decompos","scope":"Combinatorics/tilings-board-placements/fixed-polycubes","status":"fully_proved","summary":"Let $d$ and $n$ be natural numbers with $n > 1$, let $A_d(n)$ be the number of fixed $n$-cell polycubes in dimension $d$, and for each $i$ let $\\mathrm{DX}(n,i)$ be the number of fixed $n$-cell polycubes proper in dimension $i$. Then","title":"Lunnon's dimension decomposition for fixed polycubes"},{"data_url":"scopes/Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons.json","id":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/index.html","scope":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons","status":"can_prove","title":"Lozenge tilings of hexagons"},{"id":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/lattice-triangle","kind":"node","lean":"MathlibExt.Combinatorics.CentralLozengeOneThirdAMR122Problem0001Wanted.LatticeTriangle","parent":27549,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/index.html#lattice-triangle","scope":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons","status":"fully_proved","summary":"Let $e_1 = (1, 0)$ and $e_2 = (1/2, \\sqrt{3}/2)$, and write $p(i, j) = i e_1 + j e_2$ for integers $i, j$. A unit triangle is a triple $(i, j, d)$ of integers $i, j$ and a boolean $d$: for $d$ false it is the upward triangle $(i, j)$ with vertices $p(i, j)$, $p(i + 1, j)$, $p(i, j + 1)$, and for $d$ true it is the downward triangle $(i, j)$ with vertices $p(i + 1, j)$, $p(i, j + 1)$, $p(i + 1, j + 1)$.","title":"Unit triangles of the triangular lattice"},{"id":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/in-hexagon","kind":"node","lean":"MathlibExt.Combinatorics.CentralLozengeOneThirdAMR122Problem0001Wanted.InHexagon","parent":27549,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/index.html#in-hexagon","scope":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons","status":"fully_proved","summary":"Let $e_1 = (1, 0)$, $e_2 = (1/2, \\sqrt{3}/2)$ and $p(i, j) = i e_1 + j e_2$, and call the triangles with vertices $p(i, j), p(i + 1, j), p(i, j + 1)$ and $p(i + 1, j), p(i, j + 1), p(i + 1, j + 1)$ the upward and the downward unit triangle $(i, j)$. For integers $a, b, c$, the upward unit triangle $(i, j)$ lies in the $a, b, c$ hexagon when $0 \\le j \\le b + c - 1$, $-c \\le i \\le a - 1$ and\u2026","title":"Unit triangles of a semiregular hexagon"},{"id":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/shares-edge","kind":"node","lean":"MathlibExt.Combinatorics.CentralLozengeOneThirdAMR122Problem0001Wanted.SharesEdge","parent":27549,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/index.html#shares-edge","scope":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons","status":"fully_proved","summary":"The relation between unit triangles $s$ and $t$ that holds exactly when $s$ is an upward triangle $(i, j)$ and $t$ is one of the downward triangles $(i, j)$, $(i - 1, j)$ and $(i, j - 1)$. These are the three downward triangles that share an edge with the upward triangle $(i, j)$; the relation never holds with $s$ downward or $t$ upward.","title":"Unit triangles sharing an edge"},{"id":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/hexagon-graph","kind":"node","lean":"MathlibExt.Combinatorics.CentralLozengeOneThirdAMR122Problem0001Wanted.hexagonGraph","parent":27549,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/index.html#hexagon-graph","scope":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons","status":"fully_proved","summary":"For integers $a, b, c$, the dual graph of the $a, b, c$ hexagon is the simple graph whose vertices are the unit triangles lying in the $a, b, c$ hexagon, two of them being adjacent when one is an upward unit triangle $(i, j)$ and the other is one of the downward unit triangles $(i, j)$, $(i - 1, j)$ and $(i, j - 1)$, that is, when they share an edge.","title":"Dual honeycomb graph of a hexagon"},{"id":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/central-lozenge-one-third","kind":"node","lean":"MathlibExt.Combinatorics.CentralLozengeOneThirdAMR122Problem0001Wanted.centralLozengeOneThird","parent":27549,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/index.html#central-lozenge-one-third","scope":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons","status":"fully_proved","summary":"Let $e_1 = (1, 0)$ and $e_2 = (1/2, \\sqrt{3}/2)$, write $p(i, j) = i e_1 + j e_2$, and call the triangles with vertices $p(i, j), p(i + 1, j), p(i, j + 1)$ and $p(i + 1, j), p(i, j + 1), p(i + 1, j + 1)$ the upward and the downward unit triangle $(i, j)$. The statement that for every integer $n \\ge 1$, with $a = c = 2n - 1$ and $b = 2n$, in the semiregular hexagon with side lengths $a, b, c, a, b, c$ and vertices\u2026","title":"Propp's central lozenge problem"},{"id":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/central-lozenge-one-third-holds","kind":"node","lean":"MathlibExt.Combinatorics.CentralLozengeOneThirdAMR122Problem0001Wanted.centralLozengeOneThird_holds","parent":27549,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons/index.html#central-lozenge-one-third-holds","scope":"Combinatorics/tilings-board-placements/lozenge-tilings-of-hexagons","status":"can_prove","summary":"Let $e_1 = (1, 0)$ and $e_2 = (1/2, \\sqrt{3}/2)$, write $p(i, j) = i e_1 + j e_2$, and call the triangles with vertices $p(i, j), p(i + 1, j), p(i, j + 1)$ and $p(i + 1, j), p(i, j + 1), p(i + 1, j + 1)$ the upward and the downward unit triangle $(i, j)$. For every integer $n \\ge 1$, let $a = c = 2n - 1$ and $b = 2n$, and consider the semiregular hexagon with side lengths $a, b, c, a, b, c$ and vertices $0$,\u2026","title":"The central lozenge lies in one third of the tilings"},{"data_url":"scopes/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas.json","id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","title":"Nonattacking bishops and anassas"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/nonattacking-placements","kind":"node","lean":"MetaMathlibExt.nonattackingPlacements","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#nonattacking-placements","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"Let $B$ be a board, that is, a finite set of squares $(i, j) \\in \\mathbb{Z}^2$, let $M$ be a list of basic moves $v \\in \\mathbb{Z}^2$, and let $k$ be a natural number. The nonattacking placements of $k$ pieces with basic moves $M$ on $B$ are the $k$-element subsets $S \\subseteq B$ such that for all distinct squares $p, q \\in S$ and every move $v \\in M$, the difference $q - p$ is not parallel to $v$, that is,\u2026","title":"Nonattacking placements"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/card-nonattacking-placements-zero","kind":"node","lean":"MetaMathlibExt.card_nonattackingPlacements_zero","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#card-nonattacking-placements-zero","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For every board $B$, a finite set of squares of $\\mathbb{Z}^2$, and every list $M$ of basic moves in $\\mathbb{Z}^2$, there is exactly one nonattacking placement of $0$ pieces with basic moves $M$ on $B$.","title":"Exactly one nonattacking placement of no pieces"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/square-board","kind":"node","lean":"MetaMathlibExt.squareBoard","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#square-board","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For a natural number $m$, the square board $S_m$ is the set of squares $(i, j) \\in \\mathbb{Z}^2$ with $1 \\le i \\le m$ and $1 \\le j \\le m$.","title":"The square board"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/white-squares","kind":"node","lean":"MetaMathlibExt.whiteSquares","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#white-squares","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For a natural number $m$, the white squares of the square board $S_m = [1, m]^2$ are the integer squares $(i, j) \\in S_m$ for which $i + j$ is even.","title":"White squares of the square board"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/black-squares","kind":"node","lean":"MetaMathlibExt.blackSquares","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#black-squares","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For a natural number $m$, the black squares of the $m \\times m$ board $S_m = [1, m]^2$ are the squares $(i, j) \\in S_m$ with $i + j$ odd.","title":"Black squares of the square board"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishop-moves","kind":"node","lean":"MetaMathlibExt.bishopMoves","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishop-moves","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"The basic moves of the bishop are the two vectors $(1, 1)$ and $(-1, 1)$ in $\\mathbb{Z}^2$.","title":"Basic moves of the bishop"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/anassa-moves","kind":"node","lean":"MetaMathlibExt.anassaMoves","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#anassa-moves","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"The basic moves of the anassa are the two vectors $(0, 1)$ and $(1, 1)$ in $\\mathbb{Z}^2$.","title":"Basic moves of the anassa"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-square","kind":"node","lean":"MetaMathlibExt.bishopsSquare","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-square","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For natural numbers $m$ and $k$, $B_S(m, k)$ is the number of ways to place $k$ nonattacking bishops on the $m \\times m$ board $S_m = [1, m]^2$, that is, the number of $k$-element subsets of $S_m$ no two of whose squares lie on a common line of slope $1$ or $-1$.","title":"Nonattacking bishops on a square board"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-white","kind":"node","lean":"MetaMathlibExt.bishopsWhite","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-white","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For natural numbers $m$ and $k$, $R_W(m, k)$ is the number of $k$-element sets of white squares of the $m \\times m$ board $S_m = [1, m]^2$ (the squares $(i, j)$ with $i + j$ even) no two of which lie on a common line of slope $1$ or $-1$, that is, the number of ways to place $k$ nonattacking bishops on the white squares of $S_m$.","title":"Nonattacking bishops on the white squares"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-black","kind":"node","lean":"MetaMathlibExt.bishopsBlack","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-black","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For natural numbers $m$ and $k$, $R_K(m, k)$ is the number of $k$-element sets of black squares of the $m \\times m$ board $S_m = [1, m]^2$ (the squares $(i, j)$ with $i + j$ odd) no two of which lie on a common line of slope $1$ or $-1$, that is, the number of ways to place $k$ nonattacking bishops on the black squares of $S_m$.","title":"Nonattacking bishops on the black squares"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/anassas-below","kind":"node","lean":"MetaMathlibExt.anassasBelow","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#anassas-below","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For natural numbers $m$, $k$ and $p$, $A_S(m, k, p)$ is the number of nonattacking placements of $k$ anassas on the square board $S_m = \\{1, \\dots, m\\}^2$ (that is, of $k$-element sets of squares of $S_m$ no two of which lie on a common line in the direction $(0, 1)$ or $(1, 1)$) in which exactly $p$ of the squares $(i, j)$ satisfy $j < i$, that is, lie strictly below the main diagonal.","title":"Nonattacking anassas by number below the diagonal"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/ext-stirling-second","kind":"node","lean":"MetaMathlibExt.extStirlingSecond","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#ext-stirling-second","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"Knuth's extended Stirling number of the second kind at integers $n$ and $k$ is the natural number equal to the Stirling number of the second kind $S(n, k)$ if $n \\ge 0$ and $k \\ge 0$, equal to the unsigned Stirling number of the first kind $c(-k, -n)$ if $n < 0$ and $k < 0$, and equal to $0$ otherwise. Here $c(j, i)$ is the number of permutations of $j$ elements with exactly $i$ cycles.","title":"Extended Stirling numbers of the second kind"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/ext-stirling-second-nat-cast","kind":"node","lean":"MetaMathlibExt.extStirlingSecond_natCast","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#ext-stirling-second-nat-cast","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the extended Stirling number of the second kind at $(n, k)$ equals the Stirling number of the second kind $S(n, k)$.","title":"Extended Stirling numbers at natural arguments"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/ext-stirling-second-neg-neg","kind":"node","lean":"MetaMathlibExt.extStirlingSecond_neg_neg","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#ext-stirling-second-neg-neg","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For all positive integers $a$ and $b$, the extended Stirling number of the second kind at $(-a, -b)$ equals the unsigned Stirling number of the first kind $c(b, a)$, the number of permutations of $b$ elements with exactly $a$ cycles.","title":"Extended Stirling numbers at negative arguments"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/santos-beta","kind":"node","lean":"MetaMathlibExt.santosBeta","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#santos-beta","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For natural numbers $i$, $p$, $q$ and a complex number $z$, Santos's coefficient $\\beta_i(p, q, z)$ is the complex number","title":"Santos's coefficients"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/santos-beta-zero","kind":"node","lean":"MetaMathlibExt.santosBeta_zero","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#santos-beta-zero","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For every complex number $z$, Santos's coefficient satisfies $\\beta_0(0, 0, z) = 1$.","title":"Santos's coefficient with all indices zero"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishop-color-quasi-coeff","kind":"node","lean":"MetaMathlibExt.bishopColorQuasiCoeff","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishop-color-quasi-coeff","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For natural numbers $k$, $d$ and a complex number $z$, the constituent coefficient $c_{k,d}(z)$ is","title":"Constituent coefficients for bishops on one color"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishop-color-quasi-coeff-zero-zero","kind":"node","lean":"MetaMathlibExt.bishopColorQuasiCoeff_zero_zero","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishop-color-quasi-coeff-zero-zero","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"fully_proved","summary":"For every complex number $z$, the constituent coefficient $c_{0,0}(z)$ of the bishop quasi-polynomials (the case $k = d = 0$) equals $1$.","title":"Constituent coefficient for no bishops"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-square-eq-arshon","kind":"node","lean":"MetaMathlibExt.bishopsSquare_eq_arshon","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-square-eq-arshon","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"For all natural numbers $m$ and $k$, the number $B_S(m, k)$ of placements of $k$ nonattacking bishops on the $m \\times m$ board satisfies","title":"Arshon's formula for nonattacking bishops (Kot\u011b\u0161ovec's form)"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-white-recurrence","kind":"node","lean":"MetaMathlibExt.bishopsWhite_recurrence","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-white-recurrence","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"Write $R_W(m, k)$ for the number of placements of $k$ nonattacking bishops on the white squares (those $(i, j)$ with $i + j$ even) of the $m \\times m$ board $[1, m]^2$, and $\\pi(n) \\in \\lbrace 0, 1 \\rbrace$ for the remainder of $n$ modulo $2$. For all natural numbers $m$ and $k$,","title":"Recurrence for bishops on white squares"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-black-recurrence","kind":"node","lean":"MetaMathlibExt.bishopsBlack_recurrence","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-black-recurrence","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"Let $R_K(m, k)$ be the number of nonattacking placements of $k$ bishops on the black squares $(i, j)$, with $i + j$ odd, of the board $\\{1, \\dots, m\\}^2$, and let $\\pi(m) \\in \\{0, 1\\}$ be the remainder of $m$ modulo $2$. For $m, k \\ge 1$,","title":"Recurrence for bishops on the black squares"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-white-eq-sum","kind":"node","lean":"MetaMathlibExt.bishopsWhite_eq_sum","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-white-eq-sum","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"For all natural numbers $m$ and $k$, the number $R_W(m, k)$ of placements of $k$ nonattacking bishops on the white squares (those $(i, j)$ with $i + j$ even) of the $m \\times m$ board $[1, m]^2$ satisfies","title":"Extended Stirling formula for bishops on white squares"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-black-eq-sum","kind":"node","lean":"MetaMathlibExt.bishopsBlack_eq_sum","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-black-eq-sum","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"Let $R_K(m, k)$ be the number of nonattacking placements of $k$ bishops on the black squares $(i, j)$, with $i + j$ odd, of the board $\\{1, \\dots, m\\}^2$, and let ${n \\brace j}$ be Knuth's extended Stirling numbers of the second kind. For all natural numbers $m$ and $k$,","title":"Sum formula for bishops on the black squares"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-square-eq-sum","kind":"node","lean":"MetaMathlibExt.bishopsSquare_eq_sum","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-square-eq-sum","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"For all natural numbers $m$ and $k$, the number $B_S(m, k)$ of placements of $k$ nonattacking bishops on the $m \\times m$ board satisfies","title":"Extended Stirling formula for bishops on a square board"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/stirling-second-sub-eq-sum-associated","kind":"node","lean":"MetaMathlibExt.stirlingSecond_sub_eq_sum_associated","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#stirling-second-sub-eq-sum-associated","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"For natural numbers $k \\le m$,","title":"Ward's identity for Stirling numbers of the second kind"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/choose-mul-choose-eq-sum-santos-beta","kind":"node","lean":"MetaMathlibExt.choose_mul_choose_eq_sum_santosBeta","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#choose-mul-choose-eq-sum-santos-beta","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"For natural numbers $p$, $q$ and complex numbers $x$, $z$,","title":"Santos's expansion of a product of binomial coefficients"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-white-eq-quasi","kind":"node","lean":"MetaMathlibExt.bishopsWhite_eq_quasi","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-white-eq-quasi","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"Write $R_W(m, k)$ for the number of placements of $k$ nonattacking bishops on the white squares (those $(i, j)$ with $i + j$ even) of the $m \\times m$ board $[1, m]^2$, $\\pi(m) \\in \\lbrace 0, 1 \\rbrace$ for the remainder of $m$ modulo $2$, and $c_{k,d}(z) \\in \\mathbb{C}$ for the constituent coefficient of the bishop colour quasi-polynomials (defined for natural $k, d$ and complex $z$). For all natural numbers $m$\u2026","title":"Quasi-polynomial formula for bishops on white squares"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-black-eq-quasi","kind":"node","lean":"MetaMathlibExt.bishopsBlack_eq_quasi","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-black-eq-quasi","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"Let $R_K(m, k)$ be the number of nonattacking placements of $k$ bishops on the black squares $(i, j)$, with $i + j$ odd, of the board $\\{1, \\dots, m\\}^2$, and let $c_{k,d}(z)$ be the constituent coefficients of the bishop quasi-polynomials. For all natural numbers $m$ and $k$,","title":"Quasi-polynomial formula for bishops on the black squares"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/bishops-square-eq-quasi","kind":"node","lean":"MetaMathlibExt.bishopsSquare_eq_quasi","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#bishops-square-eq-quasi","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"Write $B_S(m, k)$ for the number of placements of $k$ nonattacking bishops on the $m \\times m$ board $[1, m]^2$, $\\pi(m) \\in \\lbrace 0, 1 \\rbrace$ for the remainder of $m$ modulo $2$, and $c_{k,d}(z) \\in \\mathbb{C}$ for the constituent coefficient of the bishop colour quasi-polynomials (defined for natural $k, d$ and complex $z$). For all natural numbers $m$ and $k$, as complex numbers,","title":"Quasi-polynomial formula for bishops on a square board"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/anassas-below-recurrence","kind":"node","lean":"MetaMathlibExt.anassasBelow_recurrence","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#anassas-below-recurrence","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"Let $A(m, k, p)$ be the number of nonattacking placements of $k$ anassas on the square board $\\{1, \\dots, m\\}^2$ with exactly $p$ of them strictly below the main diagonal, let ${n \\brace j}$ be Knuth's extended Stirling numbers of the second kind, and let $\\delta$ be the Kronecker delta. For $m \\ge 1$, $k \\ge 2$ and $p \\ge 1$,","title":"Recurrence for anassas below the diagonal"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/anassas-below-eq-sum","kind":"node","lean":"MetaMathlibExt.anassasBelow_eq_sum","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#anassas-below-eq-sum","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"Let $A(m, k, p)$ be the number of nonattacking placements of $k$ anassas on the square board $\\{1, \\dots, m\\}^2$ with exactly $p$ of them strictly below the main diagonal, and let ${n \\brace j}$ be Knuth's extended Stirling numbers of the second kind. For natural numbers $m$, $k$, $p$ with $p \\le k$,","title":"Sum formula for anassas below the diagonal"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/sum-anassas-below-eq","kind":"node","lean":"MetaMathlibExt.sum_anassasBelow_eq","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#sum-anassas-below-eq","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"For natural numbers $m$ and $k$ and each $p$, let $A_S(m, k, p)$ be the number of nonattacking placements of $k$ anassas on the square board $[1, m]^2$ with exactly $p$ of them strictly below the main diagonal, that is, the number of $k$-element sets of squares $(i, j) \\in [1, m]^2$ in which no two squares have the same $i$ or the same $i - j$ and exactly $p$ squares satisfy $j < i$. Then, in $\\mathbb{Q}$,","title":"Total number of nonattacking anassas on the square board"},{"id":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/sum-anassas-below-eq-poly","kind":"node","lean":"MetaMathlibExt.sum_anassasBelow_eq_poly","parent":27556,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas/index.html#sum-anassas-below-eq-poly","scope":"Combinatorics/tilings-board-placements/nonattacking-bishops-and-anassas","status":"can_prove","summary":"For natural numbers $m$ and $k$, let $A_S(m, k, p)$ be the number of nonattacking placements of $k$ anassas on the square board $[1, m]^2$ with exactly $p$ of them strictly below the main diagonal, that is, the number of $k$-element sets of squares $(i, j) \\in [1, m]^2$ in which no two squares have the same $i$ or the same $i - j$ and exactly $p$ squares satisfy $j < i$. Then, in $\\mathbb{Q}$,\u2026","title":"Polynomial formula for the number of nonattacking anassas"},{"data_url":"scopes/Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid.json","id":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/index.html","scope":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid","status":"can_prove","title":"Perfect matchings of the M\u00f6bius grid"},{"id":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/mobius-grid-vertex","kind":"node","lean":"MetaMathlibExt.MobiusGridVertex","parent":27590,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/index.html#mobius-grid-vertex","scope":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid","status":"fully_proved","summary":"For natural numbers $M$ and $N$, the vertices of the $2M \\times N$ M\u00f6bius grid are the pairs $(a, b)$ of natural numbers with $a < 2M$ and $b < N$.","title":"Vertices of the M\u00f6bius grid"},{"id":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/mobius-grid-edge","kind":"node","lean":"MetaMathlibExt.MobiusGridEdge","parent":27590,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/index.html#mobius-grid-edge","scope":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid","status":"fully_proved","summary":"For natural numbers $M$ and $N$, the labeled edges of the $2M \\times N$ M\u00f6bius grid are of three kinds: a horizontal edge $h(i, j)$ for each row $i < 2M$ and column $j < N - 1$, a vertical edge $v(i, j)$ for each row $i < 2M - 1$ and column $j < N$, and a seam edge $\\sigma(i)$ for each row $i < 2M$.","title":"Edges of the M\u00f6bius grid"},{"id":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/mobius-grid-incident","kind":"node","lean":"MetaMathlibExt.MobiusGridIncident","parent":27590,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/index.html#mobius-grid-incident","scope":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid","status":"fully_proved","summary":"For natural numbers $M$ and $N$, write a vertex of the $2M \\times N$ M\u00f6bius grid as $(a, b)$ with row $a < 2M$ and column $b < N$. The vertex $(a, b)$ is incident to the horizontal edge $h(i, j)$ (with $i < 2M$, $j < N - 1$) exactly when $a = i$ and $b \\in \\{j, j + 1\\}$; to the vertical edge $v(i, j)$ (with $i < 2M - 1$, $j < N$) exactly when $b = j$ and $a \\in \\{i, i + 1\\}$; and to the seam edge $\\sigma(i)$ (with\u2026","title":"Incidence in the M\u00f6bius grid"},{"id":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/is-mobius-grid-perfect-matching","kind":"node","lean":"MetaMathlibExt.IsMobiusGridPerfectMatching","parent":27590,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/index.html#is-mobius-grid-perfect-matching","scope":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid","status":"fully_proved","summary":"For natural numbers $M$ and $N$, a finite set $S$ of labeled edges of the $2M \\times N$ M\u00f6bius grid is a perfect matching when every vertex of the grid is incident to exactly one edge of $S$.","title":"Perfect matchings of the M\u00f6bius grid"},{"id":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/mobius-grid-perfect-matching-count","kind":"node","lean":"MetaMathlibExt.mobiusGridPerfectMatchingCount","parent":27590,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/index.html#mobius-grid-perfect-matching-count","scope":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid","status":"fully_proved","summary":"For natural numbers $M$ and $N$, this is the number of perfect matchings of the $2M \\times N$ M\u00f6bius grid, counted as sets of labeled edges.","title":"Number of perfect matchings of the M\u00f6bius grid"},{"id":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/lu-wu-mobius-grid-perfect-matching-count","kind":"node","lean":"MetaMathlibExt.lu_wu_mobius_grid_perfect_matching_count","parent":27590,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid/index.html#lu-wu-mobius-grid-perfect-matching-count","scope":"Combinatorics/tilings-board-placements/perfect-matchings-of-the-m-bius-grid","status":"can_prove","summary":"Let $M \\ge 1$ and let $N \\ge 1$ be odd. Then the number of perfect matchings of the $2M \\times N$ M\u00f6bius grid equals","title":"Lu\u2013Wu formula for perfect matchings of the M\u00f6bius grid"},{"data_url":"scopes/Combinatorics/tilings-board-placements/periodic-tiling-conjecture.json","id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"can_prove","title":"Periodic tiling conjecture"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/is-tiling","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.IsTiling","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#is-tiling","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"fully_proved","summary":"Let $G$ be an abelian group and let $F, A \\subseteq G$. Then $F$ tiles $G$ with tiling set $A$ if the map $A \\times F \\to G$, $(a, f) \\mapsto a + f$, is a bijection; equivalently, every element of $G$ lies in $a + F$ for exactly one $a \\in A$.","title":"Tiling of an abelian group by translates"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/is-periodic-tiling-set","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.IsPeriodicTilingSet","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#is-periodic-tiling-set","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"fully_proved","summary":"A subset $A$ of an abelian group $G$ is periodic if there exist a subgroup $H$ of $G$ of finite index and a finite set $S \\subseteq G$ such that $A = S + H$, that is, $A$ is the union of finitely many cosets of $H$.","title":"Periodic tiling set"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/is-aperiodic-tiling-equation","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.IsAperiodicTilingEquation","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#is-aperiodic-tiling-equation","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"fully_proved","summary":"Let $G$ be an abelian group. A set $F \\subseteq G$ forms an aperiodic tiling equation if $F$ tiles $G$ with some tiling set $A \\subseteq G$, and no tiling set $A \\subseteq G$ with which $F$ tiles $G$ is periodic.","title":"Aperiodic tiling equation"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/theorem-1-4","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.theorem_1_4","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#theorem-1-4","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"can_prove","summary":"There exist a finite abelian group $G_0$ and a finite non-empty set $F \\subseteq \\mathbb{Z}^2 \\times G_0$ that forms an aperiodic tiling equation in the group $\\mathbb{Z}^2 \\times G_0$.","title":"Greenfeld\u2013Tao counterexample to the periodic tiling conjecture"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/corollary-1-6","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.corollary_1_6","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#corollary-1-6","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"can_prove","summary":"There is a natural number $d_0$ such that for every natural number $d \\ge d_0$ there exists a finite non-empty set $F \\subseteq \\mathbb{Z}^d$ that forms an aperiodic tiling equation in $\\mathbb{Z}^d$.","title":"Aperiodic tiling equations in high-dimensional integer lattices"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/is-continuous-tiling","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.IsContinuousTiling","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#is-continuous-tiling","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, a set $\\sigma \\subseteq \\mathbb{R}^d$ continuously tiles $\\mathbb{R}^d$ with tiling set $\\Lambda \\subseteq \\mathbb{R}^d$ if for almost every $x \\in \\mathbb{R}^d$ (with respect to Lebesgue measure) there is exactly one $v \\in \\Lambda$ with $x \\in v + \\sigma$.","title":"Continuous tiling of Euclidean space"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/is-periodic-continuous-tiling-set","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.IsPeriodicContinuousTilingSet","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#is-periodic-continuous-tiling-set","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, a set $\\Lambda \\subseteq \\mathbb{R}^d$ is periodic if there exist a lattice $L \\subseteq \\mathbb{R}^d$, that is, a discrete additive subgroup that spans $\\mathbb{R}^d$ over $\\mathbb{R}$, and a finite set $S \\subseteq \\mathbb{R}^d$ such that $\\Lambda = S + L$, so that $\\Lambda$ is the union of finitely many cosets of $L$.","title":"Periodic continuous tiling set"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/is-aperiodic-continuous-tiling-equation","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.IsAperiodicContinuousTilingEquation","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#is-aperiodic-continuous-tiling-equation","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, a set $\\sigma \\subseteq \\mathbb{R}^d$ forms an aperiodic continuous tiling equation if $\\sigma$ continuously tiles $\\mathbb{R}^d$ with some tiling set $\\Lambda \\subseteq \\mathbb{R}^d$, and no tiling set $\\Lambda$ with which $\\sigma$ continuously tiles $\\mathbb{R}^d$ is periodic.","title":"Aperiodic continuous tiling equation"},{"id":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture/corollary-1-7","kind":"node","lean":"MathlibExt.Combinatorics.PeriodicTilingConjectureWanted.PeriodicTilingConjecture.corollary_1_7","parent":27597,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/periodic-tiling-conjecture/index.html#corollary-1-7","scope":"Combinatorics/tilings-board-placements/periodic-tiling-conjecture","status":"can_prove","summary":"There is a natural number $d_0$ such that for every natural number $d \\ge d_0$ there exists a bounded measurable set $\\sigma \\subseteq \\mathbb{R}^d$ of positive Lebesgue measure that forms an aperiodic continuous tiling equation.","title":"Aperiodic continuous tiling equations in high-dimensional Euclidean space"},{"data_url":"scopes/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice.json","id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"can_prove","title":"Self-avoiding walks on the honeycomb lattice"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/honeycomb-vertex","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.HoneycombVertex","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#honeycomb-vertex","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"A vertex of the honeycomb lattice is a triple $(x, y, s)$ with $x, y \\in \\mathbb{Z}$ and $s$ a boolean recording which of the two classes of the bipartition the vertex belongs to.","title":"Vertices of the honeycomb lattice"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/step","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.step","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#step","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"For a vertex $v = (x, y, s)$ and a direction label $d \\in \\{0, 1, 2\\}$, the step from $v$ in direction $d$ is a vertex on the other side of the bipartition. If $s$ is true, it is $(x, y, \\mathrm{false})$, $(x + 1, y, \\mathrm{false})$ or $(x, y + 1, \\mathrm{false})$ for $d = 0, 1, 2$ respectively. If $s$ is false, it is $(x, y, \\mathrm{true})$, $(x - 1, y, \\mathrm{true})$ or $(x, y - 1, \\mathrm{true})$ for\u2026","title":"Steps in the honeycomb lattice"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/step-step","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.step_step","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#step-step","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"For every vertex $v$ of the honeycomb lattice and every direction label $d \\in \\{0, 1, 2\\}$, taking the step in direction $d$ from $v$ and then the step in direction $d$ again returns to $v$.","title":"Steps are involutions"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/step-injective","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.step_injective","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#step-injective","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"For every vertex $v$ of the honeycomb lattice, the steps from $v$ in the directions $0$, $1$ and $2$ are three distinct vertices.","title":"Distinct labels give distinct neighbors"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/origin","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.origin","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#origin","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"The origin is the vertex $(0, 0, \\mathrm{false})$.","title":"Origin of the honeycomb lattice"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/endpoint","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.endpoint","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#endpoint","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"For a finite word $d_1 d_2 \\cdots d_k$ of direction labels in $\\{0, 1, 2\\}$, its endpoint is the vertex reached from the origin by taking the step in direction $d_1$, then the step in direction $d_2$, and so on through $d_k$. The endpoint of the empty word is the origin.","title":"Endpoint of a direction word"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/vertex-at","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.vertexAt","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#vertex-at","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"For a word $(d_1, \\ldots, d_n)$ of direction labels in $\\{0, 1, 2\\}$ and an integer $0 \\le i \\le n$, the $i$-th vertex of the word is the endpoint of its prefix $(d_1, \\ldots, d_i)$, the vertex reached from the origin after the first $i$ steps.","title":"Vertices along a direction word"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/visited-vertices","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.visitedVertices","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#visited-vertices","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"For a word $(d_1, \\ldots, d_n)$ of direction labels in $\\{0, 1, 2\\}$, the set of visited vertices is the finite set of its $i$-th vertices for $0 \\le i \\le n$; it contains the origin (for $i = 0$) and the endpoint of the word.","title":"Vertices visited by a direction word"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/is-self-avoiding","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.IsSelfAvoiding","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#is-self-avoiding","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"A word $(d_1, \\ldots, d_n)$ of direction labels in $\\{0, 1, 2\\}$ is self-avoiding when its set of visited vertices has exactly $n + 1$ elements, that is, when its $n + 1$ vertices, the $i$-th vertices for $0 \\le i \\le n$, are pairwise distinct.","title":"Self-avoiding direction words"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/walk-count","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.walkCount","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#walk-count","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"For a natural number $n$, this is the number of self-avoiding walks with $n$ steps on the honeycomb lattice starting at the origin.","title":"Number of self-avoiding walks on the honeycomb lattice"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/walk-count-three","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.walkCount_three","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#walk-count-three","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"There are exactly $12$ self-avoiding walks with $3$ steps on the honeycomb lattice starting at a fixed vertex.","title":"Three-step self-avoiding walks on the honeycomb lattice"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/walk-count-six","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.walkCount_six","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#walk-count-six","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"fully_proved","summary":"There are exactly $90$ self-avoiding walks with $6$ steps on the honeycomb lattice starting at a fixed vertex.","title":"Six-step self-avoiding walks on the honeycomb lattice"},{"id":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/honeycomb-connective-constant","kind":"node","lean":"MathlibExt.Combinatorics.HoneycombConnectiveConstantWanted.Combinatorics.HoneycombConnectiveConstant.honeycomb_connective_constant","parent":27607,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice/index.html#honeycomb-connective-constant","scope":"Combinatorics/tilings-board-placements/self-avoiding-walks-on-the-honeycomb-lattice","status":"can_prove","summary":"Let $c_n$ be the number of $n$-step self-avoiding walks on the honeycomb lattice starting at the origin. Then $c_n^{1/n} \\to \\sqrt{2 + \\sqrt{2}}$ as $n \\to \\infty$.","title":"The connective constant of the honeycomb lattice"},{"data_url":"scopes/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet.json","id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","title":"Tilings of grids, cylinders and tori up to symmetry"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cell-f","kind":"node","lean":"MetaMathlibExt.cellF","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cell-f","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the reflection $f$ acts on the cells $(x, y) \\in \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$ of the $n \\times m$ grid by $(x, y) \\mapsto (-1 - x, y)$, that is, $(x, y) \\mapsto (n - 1 - x, y)$ with the first coordinate taken modulo $n$.","title":"Cell map of the reflection f"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cell-r2","kind":"node","lean":"MetaMathlibExt.cellR2","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cell-r2","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the half-turn $r^2$ acts on the cells $(x, y) \\in \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$ of the $n \\times m$ grid by $(x, y) \\mapsto (-1 - x, -1 - y)$, that is, $(x, y) \\mapsto (n - 1 - x, m - 1 - y)$ with the coordinates taken modulo $n$ and $m$.","title":"Cell map of the half-turn r\u00b2"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cell-r2-f","kind":"node","lean":"MetaMathlibExt.cellR2F","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cell-r2-f","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the reflection $r^2 f$ acts on the cells $(x, y) \\in \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$ of the $n \\times m$ grid by $(x, y) \\mapsto (x, -1 - y)$, that is, $(x, y) \\mapsto (x, m - 1 - y)$ with the second coordinate taken modulo $m$.","title":"Cell map of the reflection r\u00b2f"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cell-r","kind":"node","lean":"MetaMathlibExt.cellR","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cell-r","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For a natural number $n$, the quarter-turn $r$ acts on the cells $(x, y) \\in \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$ of the $n \\times n$ grid by $(x, y) \\mapsto (-1 - y, x)$, that is, $(x, y) \\mapsto (n - 1 - y, x)$ modulo $n$.","title":"Cell map of the quarter-turn r"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cell-r3","kind":"node","lean":"MetaMathlibExt.cellR3","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cell-r3","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For a natural number $n$, the three-quarter turn $r^3$ acts on the cells $(x, y) \\in \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$ of the $n \\times n$ grid by $(x, y) \\mapsto (y, -1 - x)$, that is, $(x, y) \\mapsto (y, n - 1 - x)$ modulo $n$.","title":"Cell map of the three-quarter turn r\u00b3"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cell-rf","kind":"node","lean":"MetaMathlibExt.cellRF","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cell-rf","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For a natural number $n$, the diagonal reflection $rf$ acts on the cells $(x, y) \\in \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$ of the $n \\times n$ grid by $(x, y) \\mapsto (y, x)$.","title":"Cell map of the diagonal reflection rf"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cell-r3-f","kind":"node","lean":"MetaMathlibExt.cellR3F","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cell-r3-f","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For a natural number $n$, the antidiagonal reflection $r^3 f$ acts on the cells $(x, y) \\in \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$ of the $n \\times n$ grid by $(x, y) \\mapsto (-1 - y, -1 - x)$, that is, $(x, y) \\mapsto (n - 1 - y, n - 1 - x)$ modulo $n$.","title":"Cell map of the antidiagonal reflection r\u00b3f"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/fixed-tiling-count","kind":"node","lean":"MetaMathlibExt.fixedTilingCount","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#fixed-tiling-count","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $\\alpha$ (the cells) and $T$ (the tile designs) be finite sets, let $\\varphi : \\alpha \\to \\alpha$ be a map and let $\\sigma$ be a permutation of $T$. The number of tilings fixed by $(\\varphi, \\sigma)$ is the number of maps $F : \\alpha \\to T$ such that $F(\\varphi(c)) = \\sigma(F(c))$ for every $c \\in \\alpha$.","title":"Number of fixed tilings"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/tile-fixed-count","kind":"node","lean":"MetaMathlibExt.tileFixedCount","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#tile-fixed-count","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For a finite set $T$ of tile designs, a permutation $\\sigma$ of $T$ and a natural number $k$, this is the number of designs $d \\in T$ with $\\sigma^k(d) = d$.","title":"Tile designs fixed by a power of a permutation"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/perm-orbit","kind":"node","lean":"MetaMathlibExt.permOrbit","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#perm-orbit","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $\\alpha$ be a finite set, $\\varphi$ a permutation of $\\alpha$ and $c \\in \\alpha$. The orbit of $c$ under $\\varphi$ is the finite set $\\{\\varphi^k(c) : k \\in \\mathbb{N}\\}$ of images of $c$ under the nonnegative powers of $\\varphi$. Since $\\alpha$ is finite, this is the orbit of $c$ under the cyclic group generated by $\\varphi$.","title":"Orbit of a cell under a permutation"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/perm-orbits","kind":"node","lean":"MetaMathlibExt.permOrbits","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#perm-orbits","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $\\alpha$ be a finite set and $\\varphi$ a permutation of $\\alpha$. For $c \\in \\alpha$ let $O(c) = \\{\\varphi^k(c) : k \\in \\mathbb{N}\\}$ be the orbit of $c$ under $\\varphi$. The set of orbits of $\\varphi$ is the set $\\{O(c) : c \\in \\alpha\\}$ of subsets of $\\alpha$, in which each orbit occurs once.","title":"Orbits of a permutation"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-shift-sum","kind":"node","lean":"MetaMathlibExt.cylinderShiftSum","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-shift-sum","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$, let $g$ be a map from the cells $\\mathbb{Z}/n \\times \\mathbb{Z}/m$ to themselves, and let $\\sigma$ be a permutation of $T$. The cylinder shift sum of $g$ and $\\sigma$ is the sum, over all shifts $a \\in \\mathbb{Z}/n$, of the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(g(x + a, y)) = \\sigma(F(x, y))$ for every cell $(x, y)$.","title":"Cylinder shift sum"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/torus-shift-sum","kind":"node","lean":"MetaMathlibExt.torusShiftSum","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#torus-shift-sum","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $n, m \\ge 1$, let $T$ be a finite set of tile designs, let $g$ be a map from $\\mathbb{Z}/n \\times \\mathbb{Z}/m$ (the cells of the $n \\times m$ torus) to itself and let $\\sigma$ be a permutation of $T$. For a shift $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$, count the tilings $F \\colon \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(g(x + a, y + b)) = \\sigma(F(x, y))$ for every cell $(x, y)$. The torus shift\u2026","title":"Torus shift sum"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cell-f-involutive","kind":"node","lean":"MetaMathlibExt.cellF_involutive","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cell-f-involutive","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For all natural numbers $n$ and $m$ and every cell $c = (x, y) \\in \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$, applying the cell map $(x, y) \\mapsto (-1 - x, y)$ of the reflection $f$ twice returns $c$.","title":"The reflection f of grid cells is an involution"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/fixed-tiling-count-fin-two-identity","kind":"node","lean":"MetaMathlibExt.fixedTilingCount_fin_two_identity","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#fixed-tiling-count-fin-two-identity","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let the set of cells and the set of tile designs both be the two-element set $\\{0, 1\\}$. The number of maps $F : \\{0, 1\\} \\to \\{0, 1\\}$ with $F(\\varphi(c)) = \\sigma(F(c))$ for every cell $c$, where $\\varphi$ is the identity map on cells and $\\sigma$ is the identity permutation of designs, equals $4$; that is, all $2^2$ tilings are fixed.","title":"Fixed tilings of two cells under the identity"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/fixed-tiling-count-cell-f-two-one-swap","kind":"node","lean":"MetaMathlibExt.fixedTilingCount_cellF_two_one_swap","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed tilings of the "],["math","\\(2 \\times 1\\)"],["text"," grid under reflection and a design swap"]],"source":"Fixed tilings of the $2 \\times 1$ grid under reflection and a design swap"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#fixed-tiling-count-cell-f-two-one-swap","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"On the grid $\\mathbb{Z}/2 \\times \\mathbb{Z}/1$ with two tile designs, exactly two tilings $F$ satisfy $F(f(c)) = \\sigma(F(c))$ for every cell $c$, where $f : (x, y) \\mapsto (-1 - x, y)$ is the reflection exchanging the two cells and $\\sigma$ swaps the two designs.","title":"Fixed tilings of the $2 \\times 1$ grid under reflection and a design swap"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/fixed-tiling-count-eq-orbit-product","kind":"node","lean":"MetaMathlibExt.fixedTilingCount_eq_orbit_product","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#fixed-tiling-count-eq-orbit-product","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $\\alpha$ be a finite set of cells, $T$ a finite set of tile designs, $\\varphi$ a permutation of $\\alpha$ and $\\sigma$ a permutation of $T$. The number of tilings $F : \\alpha \\to T$ with $F(\\varphi(c)) = \\sigma(F(c))$ for every cell $c$ equals $\\prod_{\\theta} t(|\\theta|)$, where $\\theta$ runs over the orbits of $\\varphi$ on $\\alpha$, fixed cells included as orbits of size one, and $t(k)$ is the number of designs\u2026","title":"Fixed tilings as a product over orbits"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-grid-identity","kind":"node","lean":"MetaMathlibExt.rectangularGrid_identity","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-grid-identity","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers, let $T$ be a finite set of tiles and let $\\sigma$ be a permutation of $T$ with $\\sigma^1 = 1$, that is, the identity. Then the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(c) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{nm}$.","title":"Rectangular-grid tilings fixed by the identity"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-grid-r2-even","kind":"node","lean":"MetaMathlibExt.rectangularGrid_r2_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangular-grid tilings fixed by the half-turn, "],["math","\\(nm\\)"],["text"," even"]],"source":"Rectangular-grid tilings fixed by the half-turn, $nm$ even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-grid-r2-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers with $nm$ even, let $T$ be a finite set of tiles and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2$ be the half-turn of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$ given by $r^2(x, y) = (-1 - x, -1 - y)$. Then the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(r^2(c)) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{nm/2}$.","title":"Rectangular-grid tilings fixed by the half-turn, $nm$ even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-grid-r2-odd","kind":"node","lean":"MetaMathlibExt.rectangularGrid_r2_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangular-grid tilings fixed by the half-turn, "],["math","\\(nm\\)"],["text"," odd"]],"source":"Rectangular-grid tilings fixed by the half-turn, $nm$ odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-grid-r2-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers with $nm$ odd, let $T$ be a finite set of tiles, let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$, and let $t$ be the number of tiles $d \\in T$ with $\\sigma(d) = d$. Let $r^2$ be the half-turn of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$ given by $r^2(x, y) = (-1 - x, -1 - y)$. Then the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with\u2026","title":"Rectangular-grid tilings fixed by the half-turn, $nm$ odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-grid-f-even","kind":"node","lean":"MetaMathlibExt.rectangularGrid_f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangular-grid tilings fixed by the reflection "],["math","\\(f\\)"],["text",", "],["math","\\(n\\)"],["text"," even"]],"source":"Rectangular-grid tilings fixed by the reflection $f$, $n$ even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-grid-f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers with $n$ even, let $T$ be a finite set of tiles and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $f$ be the reflection of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$ given by $f(x, y) = (-1 - x, y)$. Then the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(f(c)) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{nm/2}$.","title":"Rectangular-grid tilings fixed by the reflection $f$, $n$ even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-grid-f-odd","kind":"node","lean":"MetaMathlibExt.rectangularGrid_f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangular-grid tilings fixed by the reflection "],["math","\\(f\\)"],["text",", "],["math","\\(n\\)"],["text"," odd"]],"source":"Rectangular-grid tilings fixed by the reflection $f$, $n$ odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-grid-f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers with $n$ odd, let $T$ be a finite set of tiles, let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$, and let $t$ be the number of tiles $d \\in T$ with $\\sigma(d) = d$. Let $f$ be the reflection of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$ given by $f(x, y) = (-1 - x, y)$. Then the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(f(c)) = \\sigma(F(c))$ for\u2026","title":"Rectangular-grid tilings fixed by the reflection $f$, $n$ odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-grid-r2f-even","kind":"node","lean":"MetaMathlibExt.rectangularGrid_r2f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangular-grid tilings fixed by the reflection "],["math","\\(r^2 f\\)"],["text",", "],["math","\\(m\\)"],["text"," even"]],"source":"Rectangular-grid tilings fixed by the reflection $r^2 f$, $m$ even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-grid-r2f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers with $m$ even, let $T$ be a finite set of tiles and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $g$ be the reflection $r^2 f$ of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$, given by $g(x, y) = (x, -1 - y)$. Then the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(g(c)) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{nm/2}$.","title":"Rectangular-grid tilings fixed by the reflection $r^2 f$, $m$ even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-grid-r2f-odd","kind":"node","lean":"MetaMathlibExt.rectangularGrid_r2f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangular-grid tilings fixed by the reflection "],["math","\\(r^2 f\\)"],["text",", "],["math","\\(m\\)"],["text"," odd"]],"source":"Rectangular-grid tilings fixed by the reflection $r^2 f$, $m$ odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-grid-r2f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers with $m$ odd, let $T$ be a finite set of tiles, let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$, and let $t$ be the number of tiles $d \\in T$ with $\\sigma(d) = d$. Let $g$ be the reflection $r^2 f$ of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$, given by $g(x, y) = (x, -1 - y)$. Then the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(g(c)) = \\sigma(F(c))$\u2026","title":"Rectangular-grid tilings fixed by the reflection $r^2 f$, $m$ odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-identity","kind":"node","lean":"MetaMathlibExt.squareGrid_identity","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-identity","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive integer, let $T$ be a finite set and let $\\sigma$ be a permutation of $T$ with $\\sigma^1 = 1$. Then the number of maps $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(c) = \\sigma(F(c))$ for every cell $c$ is $|T|^{n^2}$.","title":"Square-grid tilings fixed by the identity"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r2-even","kind":"node","lean":"MetaMathlibExt.squareGrid_r2_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the half-turn for even "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the half-turn for even $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r2-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, let $T$ be a finite set and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2$ be the half-turn of the grid $(\\mathbb{Z}/n\\mathbb{Z})^2$ given by $r^2(x, y) = (-1 - x, -1 - y)$. Then the number of maps $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(r^2(c)) = \\sigma(F(c))$ for every cell $c$ is $|T|^{n^2/2}$.","title":"Square-grid tilings fixed by the half-turn for even $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r2-odd","kind":"node","lean":"MetaMathlibExt.squareGrid_r2_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the half-turn for odd "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the half-turn for odd $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r2-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, let $T$ be a finite set, let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$, and write $t_\\sigma$ for the number of elements $t \\in T$ with $\\sigma(t) = t$. Let $r^2$ be the half-turn of the grid $(\\mathbb{Z}/n\\mathbb{Z})^2$ given by $r^2(x, y) = (-1 - x, -1 - y)$. Then the number of maps $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(r^2(c)) = \\sigma(F(c))$ for every cell\u2026","title":"Square-grid tilings fixed by the half-turn for odd $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-f-even","kind":"node","lean":"MetaMathlibExt.squareGrid_f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the reflection "],["math","\\(f\\)"],["text"," for even "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the reflection $f$ for even $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, let $T$ be a finite set and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $f$ be the reflection of the grid $(\\mathbb{Z}/n\\mathbb{Z})^2$ given by $f(x, y) = (-1 - x, y)$. Then the number of maps $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(f(c)) = \\sigma(F(c))$ for every cell $c$ is $|T|^{n^2/2}$.","title":"Square-grid tilings fixed by the reflection $f$ for even $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-f-odd","kind":"node","lean":"MetaMathlibExt.squareGrid_f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the reflection "],["math","\\(f\\)"],["text"," for odd "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the reflection $f$ for odd $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, let $T$ be a finite set, let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$, and write $t_\\sigma$ for the number of elements $t \\in T$ with $\\sigma(t) = t$. Let $f$ be the reflection of the grid $(\\mathbb{Z}/n\\mathbb{Z})^2$ given by $f(x, y) = (-1 - x, y)$. Then the number of maps $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(f(c)) = \\sigma(F(c))$ for every cell $c$ is\u2026","title":"Square-grid tilings fixed by the reflection $f$ for odd $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r2f-even","kind":"node","lean":"MetaMathlibExt.squareGrid_r2f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the reflection "],["math","\\(r^2 f\\)"],["text"," for even "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the reflection $r^2 f$ for even $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r2f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, let $T$ be a finite set and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2 f$ be the reflection of the grid $(\\mathbb{Z}/n\\mathbb{Z})^2$ given by $(r^2 f)(x, y) = (x, -1 - y)$. Then the number of maps $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F((r^2 f)(c)) = \\sigma(F(c))$ for every cell $c$ is $|T|^{n^2/2}$.","title":"Square-grid tilings fixed by the reflection $r^2 f$ for even $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r2f-odd","kind":"node","lean":"MetaMathlibExt.squareGrid_r2f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the reflection "],["math","\\(r^2f\\)"],["text",", odd "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the reflection $r^2f$, odd $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r2f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2f$ be the cell map $(x, y) \\mapsto (x, -1-y)$ of the $n \\times n$ grid $(\\mathbb{Z}/n\\mathbb{Z})^2$. Then the number of tilings $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(r^2f(c)) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{(n^2-n)/2} \\, t^n$, where $t$ is the\u2026","title":"Square-grid tilings fixed by the reflection $r^2f$, odd $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r-even","kind":"node","lean":"MetaMathlibExt.squareGrid_r_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the quarter-turn for even "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the quarter-turn for even $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, let $T$ be a finite set and let $\\sigma$ be a permutation of $T$ with $\\sigma^4 = 1$. Let $r$ be the quarter-turn of the grid $(\\mathbb{Z}/n\\mathbb{Z})^2$ given by $r(x, y) = (-1 - y, x)$. Then the number of maps $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(r(c)) = \\sigma(F(c))$ for every cell $c$ is $|T|^{n^2/4}$.","title":"Square-grid tilings fixed by the quarter-turn for even $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r-odd","kind":"node","lean":"MetaMathlibExt.squareGrid_r_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the quarter-turn for odd "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the quarter-turn for odd $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, let $T$ be a finite set, let $\\sigma$ be a permutation of $T$ with $\\sigma^4 = 1$, and write $t_\\sigma$ for the number of elements $t \\in T$ with $\\sigma(t) = t$. Let $r$ be the quarter-turn of the grid $(\\mathbb{Z}/n\\mathbb{Z})^2$ given by $r(x, y) = (-1 - y, x)$. Then the number of maps $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(r(c)) = \\sigma(F(c))$ for every cell $c$ is\u2026","title":"Square-grid tilings fixed by the quarter-turn for odd $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r3-even","kind":"node","lean":"MetaMathlibExt.squareGrid_r3_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the rotation "],["math","\\(r^3\\)"],["text",", even "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the rotation $r^3$, even $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r3-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be a permutation of $T$ with $\\sigma^4 = 1$. Let $r^3$ be the cell map $(x, y) \\mapsto (y, -1-x)$ of the $n \\times n$ grid $(\\mathbb{Z}/n\\mathbb{Z})^2$. Then the number of tilings $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(r^3(c)) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{n^2/4}$.","title":"Square-grid tilings fixed by the rotation $r^3$, even $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r3-odd","kind":"node","lean":"MetaMathlibExt.squareGrid_r3_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the rotation "],["math","\\(r^3\\)"],["text",", odd "],["math","\\(n\\)"]],"source":"Square-grid tilings fixed by the rotation $r^3$, odd $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r3-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be a permutation of $T$ with $\\sigma^4 = 1$. Let $r^3$ be the cell map $(x, y) \\mapsto (y, -1-x)$ of the $n \\times n$ grid $(\\mathbb{Z}/n\\mathbb{Z})^2$. Then the number of tilings $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(r^3(c)) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{(n^2-1)/4} \\, t$, where $t$ is the number\u2026","title":"Square-grid tilings fixed by the rotation $r^3$, odd $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-rf","kind":"node","lean":"MetaMathlibExt.squareGrid_rf","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the diagonal reflection "],["math","\\(rf\\)"]],"source":"Square-grid tilings fixed by the diagonal reflection $rf$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-rf","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $rf$ be the cell map $(x, y) \\mapsto (y, x)$ of the $n \\times n$ grid $(\\mathbb{Z}/n\\mathbb{Z})^2$. Then the number of tilings $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(rf(c)) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{(n^2-n)/2} \\, t^n$, where $t$ is the number of\u2026","title":"Square-grid tilings fixed by the diagonal reflection $rf$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-grid-r3f","kind":"node","lean":"MetaMathlibExt.squareGrid_r3f","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-grid tilings fixed by the diagonal reflection "],["math","\\(r^3f\\)"]],"source":"Square-grid tilings fixed by the diagonal reflection $r^3f$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-grid-r3f","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^3f$ be the cell map $(x, y) \\mapsto (-1-y, -1-x)$ of the $n \\times n$ grid $(\\mathbb{Z}/n\\mathbb{Z})^2$. Then the number of tilings $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(r^3f(c)) = \\sigma(F(c))$ for every cell $c$ equals $|T|^{(n^2-n)/2} \\, t^n$, where $t$ is the\u2026","title":"Square-grid tilings fixed by the diagonal reflection $r^3f$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/reflection-shift-solutions-odd","kind":"node","lean":"MetaMathlibExt.reflection_shift_solutions_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solution of "],["math","\\(x = -1-x-a\\)"],["text"," modulo odd "],["math","\\(n\\)"]],"source":"Solution of $x = -1-x-a$ modulo odd $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#reflection-shift-solutions-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive odd integer and $a \\in \\mathbb{Z}/n$. Then the set of $x \\in \\mathbb{Z}/n$ with $x = -1-x-a$ is the singleton $\\{k(-1-a)\\}$, where $k$ is the residue mod $n$ of the integer $(n+1)/2$.","title":"Solution of $x = -1-x-a$ modulo odd $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/reflection-shift-solutions-even-odd","kind":"node","lean":"MetaMathlibExt.reflection_shift_solutions_even_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Solutions of "],["math","\\(x = -1-x-a\\)"],["text"," modulo even "],["math","\\(n\\)"],["text"," for odd "],["math","\\(a\\)"]],"source":"Solutions of $x = -1-x-a$ modulo even $n$ for odd $a$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#reflection-shift-solutions-even-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive even integer and $a \\in \\mathbb{Z}/n$ whose representative $\\bar a \\in \\{0, \\dots, n-1\\}$ is odd. Then the set of $x \\in \\mathbb{Z}/n$ with $x = -1-x-a$ consists of the residues mod $n$ of the integers $(-1-\\bar a)/2$ and $(n-1-\\bar a)/2$.","title":"Solutions of $x = -1-x-a$ modulo even $n$ for odd $a$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/reflection-shift-solutions-even-even","kind":"node","lean":"MetaMathlibExt.reflection_shift_solutions_even_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","No solutions of "],["math","\\(x = -1-x-a\\)"],["text"," modulo "],["math","\\(n\\)"],["text"," when "],["math","\\(n\\)"],["text"," and "],["math","\\(a\\)"],["text"," are even"]],"source":"No solutions of $x = -1-x-a$ modulo $n$ when $n$ and $a$ are even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#reflection-shift-solutions-even-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive even integer and $a \\in \\mathbb{Z}/n$ whose representative in $\\{0, \\dots, n-1\\}$ is even. Then no $x \\in \\mathbb{Z}/n$ satisfies $x = -1-x-a$.","title":"No solutions of $x = -1-x-a$ modulo $n$ when $n$ and $a$ are even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-identity","kind":"node","lean":"MetaMathlibExt.cylinder_identity","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-identity","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$, and let $\\sigma$ be a permutation of $T$ with $\\sigma^1 = 1$, that is, the identity. Then the cylinder shift sum of the identity map of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$ and $\\sigma$ equals $\\sum_{d \\mid n} \\varphi(d)\\, |T|^{nm/d}$, where $d$ runs over the positive divisors of $n$ and $\\varphi$ is Euler's totient function.","title":"Cylinder fixed-tiling count for the identity"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-r2-m-even","kind":"node","lean":"MetaMathlibExt.cylinder_r2_m_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cylinder fixed-tiling count for the half-turn when "],["math","\\(m\\)"],["text"," is even"]],"source":"Cylinder fixed-tiling count for the half-turn when $m$ is even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-r2-m-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$ with $m$ even, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2$ be the cell map $(x, y) \\mapsto (-1 - x, -1 - y)$ of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$. Then the cylinder shift sum of $r^2$ and $\\sigma$ equals $n \\cdot |T|^{nm/2}$.","title":"Cylinder fixed-tiling count for the half-turn when $m$ is even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-r2-m-odd-n-even","kind":"node","lean":"MetaMathlibExt.cylinder_r2_m_odd_n_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cylinder fixed-tiling count for the half-turn when "],["math","\\(m\\)"],["text"," is odd and "],["math","\\(n\\)"],["text"," is even"]],"source":"Cylinder fixed-tiling count for the half-turn when $m$ is odd and $n$ is even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-r2-m-odd-n-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$ with $m$ odd and $n$ even, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2$ be the cell map $(x, y) \\mapsto (-1 - x, -1 - y)$ of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$, and let $t$ be the number of designs fixed by $\\sigma$. Then twice the cylinder shift sum of $r^2$ and $\\sigma$ equals $n \\left(|T|^{nm/2} + |T|^{(nm - 2)/2}\\, t^2\\right)$.","title":"Cylinder fixed-tiling count for the half-turn when $m$ is odd and $n$ is even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-r2-odd-odd","kind":"node","lean":"MetaMathlibExt.cylinder_r2_odd_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cylinder fixed-tiling count for the half-turn when "],["math","\\(m\\)"],["text"," and "],["math","\\(n\\)"],["text"," are odd"]],"source":"Cylinder fixed-tiling count for the half-turn when $m$ and $n$ are odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-r2-odd-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$ both be odd, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2$ be the cell map $(x, y) \\mapsto (-1 - x, -1 - y)$ of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$, and let $t$ be the number of designs fixed by $\\sigma$. Then the cylinder shift sum of $r^2$ and $\\sigma$ equals $n \\cdot |T|^{(nm - 1)/2} \\cdot t$.","title":"Cylinder fixed-tiling count for the half-turn when $m$ and $n$ are odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-f-even","kind":"node","lean":"MetaMathlibExt.cylinder_f_even","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$, and let $n, m \\ge 1$ with $n$ even. For $a \\in \\mathbb{Z}/n\\mathbb{Z}$ count the tilings $F : \\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z} \\to T$ with $F(-1 - x - a, y) = \\sigma(F(x, y))$ for every cell $(x, y)$, and let $N$ be the sum of these counts over all $a \\in \\mathbb{Z}/n\\mathbb{Z}$. If $t_f$ is the\u2026","title":"Fixed cylinder tilings of the reflection f, n even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-f-odd","kind":"node","lean":"MetaMathlibExt.cylinder_f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cylinder fixed-tiling count for the reflection "],["math","\\(f\\)"],["text"," when "],["math","\\(n\\)"],["text"," is odd"]],"source":"Cylinder fixed-tiling count for the reflection $f$ when $n$ is odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$ with $n$ odd, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $f$ be the cell map $(x, y) \\mapsto (-1 - x, y)$ of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$, and let $t$ be the number of designs fixed by $\\sigma$. Then the cylinder shift sum of $f$ and $\\sigma$ equals $n \\cdot |T|^{(nm - m)/2} \\cdot t^m$.","title":"Cylinder fixed-tiling count for the reflection $f$ when $n$ is odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-r2f-even","kind":"node","lean":"MetaMathlibExt.cylinder_r2f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cylinder fixed-tiling count for the reflection "],["math","\\(r^2 f\\)"],["text"," when "],["math","\\(m\\)"],["text"," is even"]],"source":"Cylinder fixed-tiling count for the reflection $r^2 f$ when $m$ is even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-r2f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$ with $m$ even, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2 f$ be the cell map $(x, y) \\mapsto (x, -1 - y)$ of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$. Then the cylinder shift sum of $r^2 f$ and $\\sigma$ equals $\\sum_{d \\mid n} \\varphi(d)\\, |T|^{nm / \\operatorname{lcm}(d, 2)}$, where $d$ runs over the positive divisors of $n$ and\u2026","title":"Cylinder fixed-tiling count for the reflection $r^2 f$ when $m$ is even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-r2f-odd","kind":"node","lean":"MetaMathlibExt.cylinder_r2f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cylinder fixed-tiling count for the reflection "],["math","\\(r^2 f\\)"],["text"," when "],["math","\\(m\\)"],["text"," is odd"]],"source":"Cylinder fixed-tiling count for the reflection $r^2 f$ when $m$ is odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-r2f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$ with $m$ odd, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. Let $r^2 f$ be the cell map $(x, y) \\mapsto (x, -1 - y)$ of $\\mathbb{Z}/n \\times \\mathbb{Z}/m$, and for $k \\ge 0$ let $t_k$ be the number of designs fixed by $\\sigma^k$. Then the cylinder shift sum of $r^2 f$ and $\\sigma$ equals\u2026","title":"Cylinder fixed-tiling count for the reflection $r^2 f$ when $m$ is odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-identity","kind":"node","lean":"MetaMathlibExt.rectangularTorus_identity","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-identity","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers, let $T$ be a finite set of tiles and let $\\sigma$ be a permutation of $T$ with $\\sigma^1 = 1$, that is, the identity. For $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$ let $N(a, b)$ be the number of tilings $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(x + a, y + b) = \\sigma(F(x, y))$ for every cell $(x, y)$. Then, with $\\varphi$ Euler's totient function,","title":"Torus fixed-tiling sum for the identity"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-r2-even-even","kind":"node","lean":"MetaMathlibExt.rectangularTorus_r2_even_even","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-r2-even-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$, and $n, m$ positive even integers. For a shift $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$, count the maps $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(-1-x-a, -1-y-b) = \\sigma(F(x, y))$ for all cells $(x, y)$, and let $\\mathrm{fxpt}$ be the sum of these counts over all $nm$ shifts (the torus fixed-point count of\u2026","title":"Half-turn fixed points on the rectangular torus, both sides even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-r2-odd-odd","kind":"node","lean":"MetaMathlibExt.rectangularTorus_r2_odd_odd","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-r2-odd-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$, and $n, m$ positive odd integers. For a shift $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$, count the maps $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(-1-x-a, -1-y-b) = \\sigma(F(x, y))$ for all cells $(x, y)$, and let $\\mathrm{fxpt}$ be the sum of these counts over all $nm$ shifts (the torus fixed-point count of\u2026","title":"Half-turn fixed points on the rectangular torus, both sides odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-r2-even-odd","kind":"node","lean":"MetaMathlibExt.rectangularTorus_r2_even_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Half-turn fixed points on the rectangular torus, "],["math","\\(n\\)"],["text"," even and "],["math","\\(m\\)"],["text"," odd"]],"source":"Half-turn fixed points on the rectangular torus, $n$ even and $m$ odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-r2-even-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$, $n$ a positive even integer and $m$ a positive odd integer. For a shift $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$, count the maps $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(-1-x-a, -1-y-b) = \\sigma(F(x, y))$ for all cells $(x, y)$, and let $\\mathrm{fxpt}$ be the sum of these counts over all $nm$ shifts (the\u2026","title":"Half-turn fixed points on the rectangular torus, $n$ even and $m$ odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-r2-odd-even","kind":"node","lean":"MetaMathlibExt.rectangularTorus_r2_odd_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Half-turn fixed points on the rectangular torus, "],["math","\\(n\\)"],["text"," odd and "],["math","\\(m\\)"],["text"," even"]],"source":"Half-turn fixed points on the rectangular torus, $n$ odd and $m$ even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-r2-odd-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$, $n$ a positive odd integer and $m$ a positive even integer. For a shift $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$, count the maps $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(-1-x-a, -1-y-b) = \\sigma(F(x, y))$ for all cells $(x, y)$, and let $\\mathrm{fxpt}$ be the sum of these counts over all $nm$ shifts (the\u2026","title":"Half-turn fixed points on the rectangular torus, $n$ odd and $m$ even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-f-even","kind":"node","lean":"MetaMathlibExt.rectangularTorus_f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Torus fixed-tiling sum for the reflection "],["math","\\(f\\)"],["text",", "],["math","\\(n\\)"],["text"," even"]],"source":"Torus fixed-tiling sum for the reflection $f$, $n$ even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers with $n$ even, let $T$ be a finite set of tiles, let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$, and for $k \\ge 0$ let $t_k$ be the number of tiles $d \\in T$ with $\\sigma^k(d) = d$. Let $f(x, y) = (-1 - x, y)$ on $\\mathbb{Z}/n \\times \\mathbb{Z}/m$, and for $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$ let $N(a, b)$ be the number of tilings\u2026","title":"Torus fixed-tiling sum for the reflection $f$, $n$ even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-f-odd","kind":"node","lean":"MetaMathlibExt.rectangularTorus_f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Torus fixed-tiling sum for the reflection "],["math","\\(f\\)"],["text",", "],["math","\\(n\\)"],["text"," odd"]],"source":"Torus fixed-tiling sum for the reflection $f$, $n$ odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ and $m$ be positive integers with $n$ odd, let $T$ be a finite set of tiles, let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$, and for $k \\ge 0$ let $t_k$ be the number of tiles $d \\in T$ with $\\sigma^k(d) = d$. Let $f(x, y) = (-1 - x, y)$ on $\\mathbb{Z}/n \\times \\mathbb{Z}/m$, and for $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$ let $N(a, b)$ be the number of tilings\u2026","title":"Torus fixed-tiling sum for the reflection $f$, $n$ odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-r2f-even","kind":"node","lean":"MetaMathlibExt.rectangularTorus_r2f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed points of "],["math","\\(r^2 f\\)"],["text"," on the rectangular torus, "],["math","\\(m\\)"],["text"," even"]],"source":"Fixed points of $r^2 f$ on the rectangular torus, $m$ even"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-r2f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$, $n$ a positive integer and $m$ a positive even integer. For a shift $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$, count the maps $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(x+a, -1-y-b) = \\sigma(F(x, y))$ for all cells $(x, y)$, and let $\\mathrm{fxpt}$ be the sum of these counts over all $nm$ shifts (the torus\u2026","title":"Fixed points of $r^2 f$ on the rectangular torus, $m$ even"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rectangular-torus-r2f-odd","kind":"node","lean":"MetaMathlibExt.rectangularTorus_r2f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed points of "],["math","\\(r^2 f\\)"],["text"," on the rectangular torus, "],["math","\\(m\\)"],["text"," odd"]],"source":"Fixed points of $r^2 f$ on the rectangular torus, $m$ odd"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rectangular-torus-r2f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$, $n$ a positive integer and $m$ a positive odd integer. For a shift $(a, b) \\in \\mathbb{Z}/n \\times \\mathbb{Z}/m$, count the maps $F : \\mathbb{Z}/n \\times \\mathbb{Z}/m \\to T$ with $F(x+a, -1-y-b) = \\sigma(F(x, y))$ for all cells $(x, y)$, and let $\\mathrm{fxpt}$ be the sum of these counts over all $nm$ shifts (the torus\u2026","title":"Fixed points of $r^2 f$ on the rectangular torus, $m$ odd"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-identity","kind":"node","lean":"MetaMathlibExt.squareTorus_identity","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-identity","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be the identity permutation of $T$. Let $S$ be the sum, over all shifts $v \\in (\\mathbb{Z}/n\\mathbb{Z})^2$, of the number of tilings $F : (\\mathbb{Z}/n\\mathbb{Z})^2 \\to T$ with $F(c + v) = \\sigma(F(c))$ for every cell $c$. Then $S = \\sum_{d_1 \\mid n} \\sum_{d_2 \\mid n} \\varphi(d_1) \\, \\varphi(d_2) \\, $n$ and $\\varphi$ is Euler's\u2026","title":"Square-torus fixed points of the identity"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r2-odd","kind":"node","lean":"MetaMathlibExt.squareTorus_r2_odd","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r2-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$, and write $t_1$ for the number of designs $d \\in T$ with $\\sigma(d) = d$. Let $S$ be the torus shift sum of the half-turn cell map $r^2(x, y) = (-1-x, -1-y)$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts $(a, b) \\in (\\mathbb{Z}/n\\mathbb{Z})^2$ of the number of tilings\u2026","title":"Fixed tilings of the odd square torus under a half-turn"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r2-even","kind":"node","lean":"MetaMathlibExt.squareTorus_r2_even","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r2-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$, and write $t_1$ for the number of designs $d \\in T$ with $\\sigma(d) = d$. Let $S$ be the torus shift sum of the half-turn cell map $r^2(x, y) = (-1-x, -1-y)$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts $(a, b) \\in (\\mathbb{Z}/n\\mathbb{Z})^2$ of the number of tilings\u2026","title":"Fixed tilings of the even square torus under a half-turn"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-f-even","kind":"node","lean":"MetaMathlibExt.squareTorus_f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-torus fixed points of the reflection "],["math","\\(f\\)"],["text",", even "],["math","\\(n\\)"]],"source":"Square-torus fixed points of the reflection $f$, even $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. For $k \\ge 1$ let $t_k$ be the number of designs in $T$ fixed by $\\sigma^k$. Let $f$ be the cell map $(x, y) \\mapsto (-1-x, y)$ of $(\\mathbb{Z}/n\\mathbb{Z})^2$, and let $S$ be the sum, over all shifts $v \\in (\\mathbb{Z}/n\\mathbb{Z})^2$, of the number of tilings\u2026","title":"Square-torus fixed points of the reflection $f$, even $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-f-odd","kind":"node","lean":"MetaMathlibExt.squareTorus_f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-torus fixed points of the reflection "],["math","\\(f\\)"],["text",", odd "],["math","\\(n\\)"]],"source":"Square-torus fixed points of the reflection $f$, odd $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be a permutation of $T$ with $\\sigma^2 = 1$. For $k \\ge 1$ let $t_k$ be the number of designs in $T$ fixed by $\\sigma^k$. Let $f$ be the cell map $(x, y) \\mapsto (-1-x, y)$ of $(\\mathbb{Z}/n\\mathbb{Z})^2$, and let $S$ be the sum, over all shifts $v \\in (\\mathbb{Z}/n\\mathbb{Z})^2$, of the number of tilings\u2026","title":"Square-torus fixed points of the reflection $f$, odd $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r2f-even","kind":"node","lean":"MetaMathlibExt.squareTorus_r2f_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed tilings of the even square torus under the reflection "],["math","\\(r^2 f\\)"]],"source":"Fixed tilings of the even square torus under the reflection $r^2 f$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r2f-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$; for $k \\in \\mathbb{N}$ write $t_k$ for the number of designs $d \\in T$ with $\\sigma^k(d) = d$, and let $\\varphi$ be Euler's totient function. Let $S$ be the torus shift sum of the cell map $g(x, y) = (x, -1-y)$ of $r^2 f$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts\u2026","title":"Fixed tilings of the even square torus under the reflection $r^2 f$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r2f-odd","kind":"node","lean":"MetaMathlibExt.squareTorus_r2f_odd","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fixed tilings of the odd square torus under the reflection "],["math","\\(r^2 f\\)"]],"source":"Fixed tilings of the odd square torus under the reflection $r^2 f$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r2f-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$; for $k \\in \\mathbb{N}$ write $t_k$ for the number of designs $d \\in T$ with $\\sigma^k(d) = d$, and let $\\varphi$ be Euler's totient function. Let $S$ be the torus shift sum of the cell map $g(x, y) = (x, -1-y)$ of $r^2 f$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts\u2026","title":"Fixed tilings of the odd square torus under the reflection $r^2 f$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r-odd","kind":"node","lean":"MetaMathlibExt.squareTorus_r_odd","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^4 = 1$, and write $t_1$ for the number of designs $d \\in T$ with $\\sigma(d) = d$. Let $S$ be the torus shift sum of the quarter-turn cell map $r(x, y) = (-1-y, x)$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts $(a, b) \\in (\\mathbb{Z}/n\\mathbb{Z})^2$ of the number of tilings\u2026","title":"Fixed tilings of the odd square torus under a quarter-turn"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r-even","kind":"node","lean":"MetaMathlibExt.squareTorus_r_even","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square-torus fixed points of the rotation "],["math","\\(r\\)"],["text",", even "],["math","\\(n\\)"]],"source":"Square-torus fixed points of the rotation $r$, even $n$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, let $T$ be a finite set of tile designs, and let $\\sigma$ be a permutation of $T$ with $\\sigma^4 = 1$. For $k \\ge 1$ let $t_k$ be the number of designs in $T$ fixed by $\\sigma^k$. Let $r$ be the cell map $(x, y) \\mapsto (-1-y, x)$ of $(\\mathbb{Z}/n\\mathbb{Z})^2$, and let $S$ be the sum, over all shifts $v \\in (\\mathbb{Z}/n\\mathbb{Z})^2$, of the number of tilings\u2026","title":"Square-torus fixed points of the rotation $r$, even $n$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r3-odd","kind":"node","lean":"MetaMathlibExt.squareTorus_r3_odd","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r3-odd","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an odd positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^4 = 1$, and write $t_1$ for the number of designs $d \\in T$ with $\\sigma(d) = d$. Let $S$ be the torus shift sum of the three-quarter-turn cell map $r^3(x, y) = (y, -1-x)$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts $(a, b) \\in (\\mathbb{Z}/n\\mathbb{Z})^2$ of the number of tilings\u2026","title":"Fixed tilings of the odd square torus under a three-quarter turn"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r3-even","kind":"node","lean":"MetaMathlibExt.squareTorus_r3_even","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r3-even","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be an even positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^4 = 1$, and for $k = 1, 2$ write $t_k$ for the number of designs $d \\in T$ with $\\sigma^k(d) = d$. Let $S$ be the torus shift sum of the three-quarter-turn cell map $r^3(x, y) = (y, -1-x)$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts $(a, b) \\in (\\mathbb{Z}/n\\mathbb{Z})^2$ of the\u2026","title":"Fixed tilings of the even square torus under a three-quarter turn"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-rf","kind":"node","lean":"MetaMathlibExt.squareTorus_rf","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-rf","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$; write $t_1$ for the number of designs $d \\in T$ with $\\sigma(d) = d$, and let $\\varphi$ be Euler's totient function. Let $S$ be the torus shift sum of the cell map $g(x, y) = (y, x)$ of $rf$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts $(a, b) \\in (\\mathbb{Z}/n\\mathbb{Z})^2$ of\u2026","title":"Fixed tilings of the square torus under the diagonal reflection"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-r3f","kind":"node","lean":"MetaMathlibExt.squareTorus_r3f","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-r3f","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive integer, $T$ a finite set of tile designs and $\\sigma$ a permutation of $T$ with $\\sigma^2 = 1$; write $t_1$ for the number of designs $d \\in T$ with $\\sigma(d) = d$, and let $\\varphi$ be Euler's totient function. Let $S$ be the torus shift sum of the cell map $g(x, y) = (-1-y, -1-x)$ of $r^3 f$ on $(\\mathbb{Z}/n\\mathbb{Z})^2$, that is, the sum over all shifts\u2026","title":"Fixed tilings of the square torus under the anti-diagonal reflection"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rect-cell-action","kind":"node","lean":"MetaMathlibExt.rectCellAction","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rect-cell-action","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $n, m \\in \\mathbb{N}$ and view the cells of the $n \\times m$ grid as $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$. Write the elements of the dihedral group of order $8$ as rotations $r_i$ and reflections $sr_i$ for $i \\in \\mathbb{Z}/4\\mathbb{Z}$. Each element acts on cells as follows: $r_0$ by the identity, $r_i$ with $i \\ne 0$ by the half-turn $(x, y) \\mapsto (-1 - x, -1 - y)$, $sr_0$ by the\u2026","title":"Action of the rectangle group on grid cells"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rect-cell-action-r-zero","kind":"node","lean":"MetaMathlibExt.rectCellAction_r_zero","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rect-cell-action-r-zero","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $n, m \\in \\mathbb{N}$, and let $r_i$ ($i \\in \\mathbb{Z}/4\\mathbb{Z}$) denote the rotations of the dihedral group of order $8$. Under the rectangle-group action on the cells $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$ of the $n \\times m$ grid, the identity element $r_0$ acts as the identity map.","title":"Rectangle-group cell action of the identity"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rect-cell-action-r-two","kind":"node","lean":"MetaMathlibExt.rectCellAction_r_two","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rect-cell-action-r-two","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $n, m \\in \\mathbb{N}$, and let $r_i$ ($i \\in \\mathbb{Z}/4\\mathbb{Z}$) denote the rotations of the dihedral group of order $8$. Under the rectangle-group action on the cells $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$ of the $n \\times m$ grid, $r_2$ acts as the half-turn $(x, y) \\mapsto (-1 - x, -1 - y)$.","title":"Rectangle-group cell action of the half-turn"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rect-cell-action-sr-zero","kind":"node","lean":"MetaMathlibExt.rectCellAction_sr_zero","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rect-cell-action-sr-zero","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $n, m \\in \\mathbb{N}$, and let $sr_i$ ($i \\in \\mathbb{Z}/4\\mathbb{Z}$) denote the reflections of the dihedral group of order $8$. Under the rectangle-group action on the cells $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$ of the $n \\times m$ grid, $sr_0$ (the paper's $f$) acts as the reflection $(x, y) \\mapsto (-1 - x, y)$.","title":"Rectangle-group cell action of the reflection f"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rect-cell-action-sr-two","kind":"node","lean":"MetaMathlibExt.rectCellAction_sr_two","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rect-cell-action-sr-two","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $n, m \\in \\mathbb{N}$, and let $sr_i$ ($i \\in \\mathbb{Z}/4\\mathbb{Z}$) denote the reflections of the dihedral group of order $8$. Under the rectangle-group action on the cells $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$ of the $n \\times m$ grid, $sr_2$ (the paper's $r^2 f$) acts as the reflection $(x, y) \\mapsto (x, -1 - y)$.","title":"Rectangle-group cell action of the reflection r\u00b2f"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-cell-action","kind":"node","lean":"MetaMathlibExt.squareCellAction","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-cell-action","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For a natural number $n$, let $r$ and $f$ be the maps of $(\\mathbb{Z}/n\\mathbb{Z})^2$ given by $r(x, y) = (-1 - y, x)$ (the quarter-turn) and $f(x, y) = (-1 - x, y)$ (the reflection). The dihedral group of order $8$ has the rotations $r_i$ and the reflections $sr_i$ for $i \\in \\mathbb{Z}/4\\mathbb{Z}$. The action sends $r_i$ to the $i$-fold iterate $r^i$ and $sr_i$ to $f \\circ r^i$, where $i$ is read as an integer\u2026","title":"The dihedral action on the cells of the square grid"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-cell-action-r-one","kind":"node","lean":"MetaMathlibExt.squareCellAction_r_one","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","The rotation "],["math","\\(r\\)"],["text"," acts on the square grid as the quarter-turn"]],"source":"The rotation $r$ acts on the square grid as the quarter-turn"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-cell-action-r-one","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"For every natural number $n$, the square cell action of the dihedral group of order $8$ on $\\mathbb{Z}/n \\times \\mathbb{Z}/n$ sends the rotation $r$ to the quarter-turn $(x, y) \\mapsto (-1-y, x)$.","title":"The rotation $r$ acts on the square grid as the quarter-turn"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-cell-action-sr-zero","kind":"node","lean":"MetaMathlibExt.squareCellAction_sr_zero","parent":27621,"reading_title":{"format":"inline-parts-v1","parts":[["text","The reflection "],["math","\\(sr_0\\)"],["text"," acts on the square grid as "],["math","\\(f\\)"]],"source":"The reflection $sr_0$ acts on the square grid as $f$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-cell-action-sr-zero","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"fully_proved","summary":"Let $r_i$ and $sr_i$ ($i \\in \\mathbb{Z}/4\\mathbb{Z}$) denote the rotations and the reflections of the dihedral group of order $8$. For every natural number $n$, the dihedral action on the cells $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/n\\mathbb{Z}$ of the $n \\times n$ grid sends the reflection $sr_0$ to the reflection $f$, $(x, y) \\mapsto (-1 - x, y)$.","title":"The reflection $sr_0$ acts on the square grid as $f$"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/cylinder-distinct-tilings-burnside","kind":"node","lean":"MetaMathlibExt.cylinder_distinctTilings_burnside","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#cylinder-distinct-tilings-burnside","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs, let $n, m \\ge 1$, and let $R$ be a subgroup of the dihedral group $\\langle r, f \\rangle$ of order $8$ that is contained in the rectangle group $\\lbrace 1, r^2, f, r^2 f \\rbrace$ and acts on $T$. For $g \\in R$ let $\\phi_g$ be its cell map on $\\mathbb{Z}/n\\mathbb{Z} \\times \\mathbb{Z}/m\\mathbb{Z}$: $\\phi_1(x, y) = (x, y)$, $\\phi_{r^2}(x, y) = (-1 - x, -1 - y)$,\u2026","title":"Burnside count of cylinder tilings under rectangle symmetries"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/rect-torus-distinct-tilings-burnside","kind":"node","lean":"MetaMathlibExt.rectTorus_distinctTilings_burnside","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#rect-torus-distinct-tilings-burnside","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $T$ be a finite set of tile designs and let $n, m \\ge 1$. Let $R$ be a subgroup of the dihedral group of order $8$, whose elements are the rotations $r_i$ and reflections $sr_i$ for $i \\in \\mathbb{Z}/4\\mathbb{Z}$, such that every element of $R$ is one of $r_0$, $r_2$, $sr_0$, $sr_2$, and let $R$ act on $T$. Tilings of the $n \\times m$ torus are maps\u2026","title":"Burnside count of torus tilings under the rectangle group"},{"id":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/square-torus-distinct-tilings-burnside","kind":"node","lean":"MetaMathlibExt.squareTorus_distinctTilings_burnside","parent":27621,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet/index.html#square-torus-distinct-tilings-burnside","scope":"Combinatorics/tilings-board-placements/tilings-of-grids-cylinders-and-tori-up-to-symmet","status":"can_prove","summary":"Let $n$ be a positive integer and let $T$ be a finite set of tile designs. Let $\\rho(x, y) = (-1-y, x)$ and $\\phi(x, y) = (-1-x, y)$ on the cells $(\\mathbb{Z}/n\\mathbb{Z})^2$ of the $n \\times n$ torus, and let the dihedral group $D_8 = \\langle r, f \\rangle$ of the square act on cells by $r^i \\cdot c = \\rho^i(c)$ and $fr^i \\cdot c = \\phi(\\rho^i(c))$ for $i \\in \\mathbb{Z}/4\\mathbb{Z}$. Let $R$ be a subgroup of $D_8$\u2026","title":"Burnside count of tilings of the square torus"},{"data_url":"scopes/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions.json","id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"can_prove","title":"Tilings of the symmetric group by transpositions"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/is-tiling","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.IsTiling","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#is-tiling","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"fully_proved","summary":"For subsets $X$ and $Y$ of a group $G$, the pair $(X, Y)$ is a tiling (or factorisation) of $G$ if every $g \\in G$ can be written as $g = xy$ with $x \\in X$ and $y \\in Y$ for exactly one pair $(x, y)$.","title":"Tiling of a group by two subsets"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/transpositions-with-one","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.transpositionsWithOne","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#transpositions-with-one","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"fully_proved","summary":"$T_n$ is the set of permutations of $\\{0, \\dots, n-1\\}$ consisting of the identity and all transpositions $(i\\ j)$ with $i \\ne j$.","title":"The identity and the transpositions"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/partition-shape","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.PartitionShape","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#partition-shape","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"fully_proved","summary":"A partition of $n$ is encoded as a list $(\\lambda_0, \\lambda_1, \\dots, \\lambda_{k-1})$ of positive integers with $\\lambda_0 \\ge \\lambda_1 \\ge \\dots \\ge \\lambda_{k-1}$ and $\\lambda_0 + \\dots + \\lambda_{k-1} = n$; the $\\lambda_i$ are its row lengths.","title":"Integer partition as a list of row lengths"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/content-sum-aux","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.PartitionShape.contentSumAux","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#content-sum-aux","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"fully_proved","summary":"For a natural number $i$ and a list of natural numbers $a_0, \\dots, a_{k-1}$, this is the integer $\\sum_{j=0}^{k-1} a_j\\,(a_j - 2(i + j) - 1)$; the empty list gives $0$.","title":"Shifted row-length sum"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/content-sum","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.PartitionShape.contentSum","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#content-sum","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"fully_proved","summary":"For a partition $\\lambda$ of $n$ with row lengths $\\lambda_0 \\ge \\lambda_1 \\ge \\dots \\ge \\lambda_{k-1}$, this is the integer $\\sum_{i=0}^{k-1} \\lambda_i\\,(\\lambda_i - 2i - 1)$.","title":"Twice the content sum of a partition"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/ordered-set-partition","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.OrderedSetPartition","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#ordered-set-partition","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"fully_proved","summary":"Let $\\lambda$ be a partition of $n$ with row lengths $\\lambda_0 \\ge \\dots \\ge \\lambda_{k-1}$. An ordered set partition of $\\{0, \\dots, n-1\\}$ of shape $\\lambda$ is a sequence $(B_0, \\dots, B_{k-1})$ of pairwise disjoint subsets of $\\{0, \\dots, n-1\\}$ whose union is $\\{0, \\dots, n-1\\}$ and with $|B_i| = \\lambda_i$ for every $i$.","title":"Ordered set partition of shape \u03bb"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/sends-partition","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.SendsPartition","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#sends-partition","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"fully_proved","summary":"Let $\\lambda$ be a partition of $n$ and let $P = (P_0, \\dots, P_{k-1})$ and $Q = (Q_0, \\dots, Q_{k-1})$ be ordered set partitions of $\\{0, \\dots, n-1\\}$ of shape $\\lambda$. A permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ sends $P$ to $Q$ if $\\sigma(P_i) = Q_i$ for every $i$.","title":"Permutation sending one ordered set partition to another"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/is-partition-transitive","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.IsPartitionTransitive","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#is-partition-transitive","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"fully_proved","summary":"Let $\\lambda$ be a partition of $n$ with row lengths $\\lambda_0 \\ge \\dots \\ge \\lambda_{k-1}$. A set $Y$ of permutations of $\\{0, \\dots, n-1\\}$ is $\\lambda$-transitive, in the sense of Martin and Sagan, if there is a positive integer $r$ such that for any two ordered set partitions $P = (P_0, \\dots, P_{k-1})$ and $Q = (Q_0, \\dots, Q_{k-1})$ of $\\{0, \\dots, n-1\\}$ of shape $\\lambda$, exactly $r$ permutations\u2026","title":"\u03bb-transitive sets of permutations"},{"id":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/fang-xia-partition-transitive-of-tiling","kind":"node","lean":"MathlibExt.Combinatorics.FangXiaTilingWanted.Combinatorics.FangXiaTilingProblem.fang_xia_partition_transitive_of_tiling","parent":27703,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions/index.html#fang-xia-partition-transitive-of-tiling","scope":"Combinatorics/tilings-board-placements/tilings-of-the-symmetric-group-by-transpositions","status":"can_prove","summary":"Let $S_n$ be the symmetric group on $\\{0, \\dots, n-1\\}$, let $T_n \\subseteq S_n$ consist of the identity and all transpositions, and let $Y \\subseteq S_n$ be such that $(T_n, Y)$ is a tiling of $S_n$: every $g \\in S_n$ can be written as $g = xy$ with $x \\in T_n$ and $y \\in Y$ for exactly one pair $(x, y)$. Let $\\lambda$ be a partition of $n$ with row lengths $\\lambda_0 \\ge \\dots \\ge \\lambda_{k-1}$ such that\u2026","title":"Fang\u2013Xia: tilings by transpositions and \u03bb-transitivity"},{"data_url":"scopes/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards.json","id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","kind":"scope","navigation":"module","parent":27487,"result":false,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","title":"Weighted file placements on Ferrers boards"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/ferrers-board-cells","kind":"node","lean":"MetaMathlibExt.ferrersBoardCells","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#ferrers-board-cells","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"For a list $b = (b_1, \\ldots, b_n)$ of natural numbers, the cells of the Ferrers board with column heights $b$ are the pairs $(i, r)$ of natural numbers with $i < n$ and $r < b_{i+1}$; columns and rows are numbered from $0$, so column $i$ has height $b_{i+1}$.","title":"Cells of a Ferrers board"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/ferrers-board-cells-card-1124","kind":"node","lean":"MetaMathlibExt.ferrersBoardCells_card_1124","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","The board "],["math","\\((1, 1, 2, 4)\\)"],["text"," has eight cells"]],"source":"The board $(1, 1, 2, 4)$ has eight cells"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#ferrers-board-cells-card-1124","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"The Ferrers board with column heights $(1, 1, 2, 4)$, the board $B_1$ of Figure 2, has exactly $8$ cells.","title":"The board $(1, 1, 2, 4)$ has eight cells"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/file-placements","kind":"node","lean":"MetaMathlibExt.filePlacements","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#file-placements","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"A file placement on the Ferrers board with column heights $b$ is a set of cells of the board no two of which lie in the same column.","title":"File placements"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/level-rook-placements","kind":"node","lean":"MetaMathlibExt.levelRookPlacements","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#level-rook-placements","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"Let $m$ be a natural number and let $B$ be the board with column heights $b_1, \\dots, b_n$, rows numbered from $0$, where row $r$ lies in level $\\lfloor r/m \\rfloor$ (every row lies in level $0$ when $m = 0$). An $m$-level rook placement on $B$ is a file placement on $B$, that is a set of cells of $B$ with no two in the same column, that has no two cells in the same level.","title":"Level rook placements"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/level-rook-number","kind":"node","lean":"MetaMathlibExt.levelRookNumber","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#level-rook-number","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"Let $m$ and $k$ be natural numbers and let $B$ be the board with column heights $b_1, \\dots, b_n$, with rows numbered from $0$, where row $r$ lies in level $\\lfloor r/m \\rfloor$ (every row lies in level $0$ when $m = 0$). The $k$-th $m$-level rook number $r_{k,m}(B)$ is the number of $m$-level rook placements on $B$ with exactly $k$ cells, that is, of sets of $k$ cells of $B$ with no two in the same column and no\u2026","title":"Level rook number"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/m-falling-factorial","kind":"node","lean":"MetaMathlibExt.mFallingFactorial","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(m\\)"],["text","-falling factorial"]],"source":"The $m$-falling factorial"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#m-falling-factorial","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"For an integer $x$ and natural numbers $k$ and $m$, the $m$-falling factorial is the integer $x\\downarrow_{k,m} = \\prod_{i=0}^{k-1} (x - mi) = x(x-m)\\cdots(x-(k-1)m)$, which is $1$ for $k = 0$.","title":"The $m$-falling factorial"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/m-falling-factorial-one-three-two","kind":"node","lean":"MetaMathlibExt.mFallingFactorial_one_three_two","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(1\\downarrow_{3,2} = 3\\)"]],"source":"The value $1\\downarrow_{3,2} = 3$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#m-falling-factorial-one-three-two","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"The $2$-falling factorial of $1$ with three factors is $1\\downarrow_{3,2} = 1 \\cdot (1 - 2) \\cdot (1 - 4) = 3$.","title":"The value $1\\downarrow_{3,2} = 3$"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/file-placement-weight","kind":"node","lean":"MetaMathlibExt.filePlacementWeight","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(m\\)"],["text","-weight of a file placement"]],"source":"The $m$-weight of a file placement"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#file-placement-weight","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"For a natural number $m$ and a finite set $F$ of cells $(i, j)$ (column $i$, row $j$), let $f_j$ be the number of cells of $F$ in row $j$. The $m$-weight of $F$ is $\\mathrm{wt}_m(F) = \\prod_j 1\\downarrow_{f_j, m}$, the product over the rows $j$ that contain a cell of $F$, where $x\\downarrow_{k, m} = \\prod_{i < k} (x - m i)$ is the $m$-falling factorial.","title":"The $m$-weight of a file placement"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/file-placement-weight-figure-four","kind":"node","lean":"MetaMathlibExt.filePlacementWeight_figure_four","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(3\\)"],["text","-weight of the placement of Figure 4"]],"source":"The $3$-weight of the placement of Figure 4"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#file-placement-weight-figure-four","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"The file placement $F$ consisting of the cells $(0, 1)$, $(1, 1)$, $(2, 3)$, $(3, 3)$ and $(4, 3)$, written as (column, row) and numbered from $0$, has two cells in the second row and three in the fourth, and its $3$-weight is $\\mathrm{wt}_3(F) = -20$.","title":"The $3$-weight of the placement of Figure 4"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/weighted-file-number","kind":"node","lean":"MetaMathlibExt.weightedFileNumber","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#weighted-file-number","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"For a board $B$ given by a list of column heights and for $m, k \\in \\mathbb{N}$, the $k$-th $m$-weighted file placement number of $B$ is the integer $f_{k,m}(B) = \\sum_F \\mathrm{wt}_m(F)$, where $F$ runs over the file placements on $B$ with exactly $k$ cells and $\\mathrm{wt}_m(F)$ is the $m$-weight of $F$.","title":"Weighted file placement numbers"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/level-rook-number-weighted-file-number-t","kind":"node","lean":"MetaMathlibExt.levelRookNumber_weightedFileNumber_two_two","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two rooks on the "],["math","\\(2 \\times 2\\)"],["text"," board with "],["math","\\(m = 2\\)"]],"source":"Two rooks on the $2 \\times 2$ board with $m = 2$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#level-rook-number-weighted-file-number-t","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"For the board $B = (2, 2)$, the number $r_{2,2}(B)$ of $2$-level rook placements of two rooks on $B$ is $0$, and the weighted file placement number $f_{2,2}(B)$, the sum of the $2$-weights of the file placements of two rooks on $B$, is also $0$.","title":"Two rooks on the $2 \\times 2$ board with $m = 2$"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/is-singleton-board","kind":"node","lean":"MetaMathlibExt.IsSingletonBoard","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#is-singleton-board","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"For natural numbers $h$ and $m$, write $\\lfloor h \\rfloor_m$ for the largest multiple of $m$ that is at most $h$. A list $b = (b_1, \\ldots, b_n)$ of natural numbers is a singleton board for $m$ when it is weakly increasing (the column heights of a Ferrers board) and, for every $1 \\le i < n$, $b_i - \\lfloor b_i \\rfloor_m \\ne 0$ implies $\\lfloor b_i \\rfloor_m < \\lfloor b_{i+1} \\rfloor_m$.","title":"Singleton boards"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/is-singleton-board-1223","kind":"node","lean":"MetaMathlibExt.isSingletonBoard_1223","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","A singleton board for "],["math","\\(2\\)"],["text"," but not for "],["math","\\(3\\)"]],"source":"A singleton board for $2$ but not for $3$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#is-singleton-board-1223","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"The Ferrers board with column heights $(1, 2, 2, 3)$ is a singleton board for $m = 2$ and is not a singleton board for $m = 3$.","title":"A singleton board for $2$ but not for $3$"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/rook-number-factorization","kind":"node","lean":"MetaMathlibExt.rookNumber_factorization","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#rook-number-factorization","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $B = (b_1, \\dots, b_n)$ be a Ferrers board, that is $b_1 \\le \\dots \\le b_n$, and let $r_k(B) = r_{k,1}(B)$ be the number of placements of $k$ non-attacking rooks on $B$ (no two in the same row or column). Then for every integer $x$,","title":"Goldman\u2013Joichi\u2013White factorization theorem"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/level-rook-number-factorization-of-singl","kind":"node","lean":"MetaMathlibExt.levelRookNumber_factorization_of_singleton","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#level-rook-number-factorization-of-singl","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $m \\ge 1$ and let $B = (b_1, \\dots, b_n)$ be a singleton board for $m$. Then for every integer $x$,","title":"Briggs\u2013Remmel factorization on singleton boards"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/level-rook-number-factorization-zones","kind":"node","lean":"MetaMathlibExt.levelRookNumber_factorization_zones","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zone factorization of the "],["math","\\(m\\)"],["text","-level rook polynomial"]],"source":"Zone factorization of the $m$-level rook polynomial"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#level-rook-number-factorization-zones","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $m \\ge 1$ and let $B = (b_1, \\dots, b_n)$ be a Ferrers board, that is $b_1 \\le \\dots \\le b_n$. Write $\\lfloor b \\rfloor_m = m \\lfloor b/m \\rfloor$ for the largest multiple of $m$ that is at most $b$, and $\\rho_i = b_i - \\lfloor b_i \\rfloor_m$. The zone of column $i$ is the set $z$ of columns $k$ with $\\lfloor b_k \\rfloor_m = \\lfloor b_i \\rfloor_m$ (a maximal run of consecutive columns, since the heights\u2026","title":"Zone factorization of the $m$-level rook polynomial"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/level-rook-number-factorization-levels","kind":"node","lean":"MetaMathlibExt.levelRookNumber_factorization_levels","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","Level-number factorization of the "],["math","\\(m\\)"],["text","-level rook polynomial"]],"source":"Level-number factorization of the $m$-level rook polynomial"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#level-rook-number-factorization-levels","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $m \\ge 1$ and let $B = (b_1, \\dots, b_n)$ be a Ferrers board, that is $b_1 \\le \\dots \\le b_n$, with $b_i \\le mn$ for every $i$. Number the rows from $0$ at the bottom, so that column $i$ has the cells in rows $0, \\dots, b_i - 1$, and for $1 \\le j \\le n$ let $l_j$ be the number of cells of $B$ whose row $r$ satisfies $\\lfloor r/m \\rfloor = n - j$ (the $j$-th level from the top). Then for every integer $x$,","title":"Level-number factorization of the $m$-level rook polynomial"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/weighted-file-number-factorization","kind":"node","lean":"MetaMathlibExt.weightedFileNumber_factorization","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#weighted-file-number-factorization","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $m \\ge 1$ and let $B$ be a Ferrers board with $n$ columns of weakly increasing heights $b_0 \\le b_1 \\le \\dots \\le b_{n-1}$ (columns indexed from $0$). Then for every integer $x$,","title":"Factorization theorem for weighted file placements"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/weighted-file-number-eq-level-rook-numbe","kind":"node","lean":"MetaMathlibExt.weightedFileNumber_eq_levelRookNumber_of_singleton","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#weighted-file-number-eq-level-rook-numbe","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $m \\ge 1$ and let $B$ be a singleton board for $m$. Then for every $k \\in \\mathbb{N}$ the $k$-th $m$-weighted file placement number of $B$ equals its $k$-th $m$-level rook number: $f_{k,m}(B) = r_{k,m}(B)$.","title":"Weighted file numbers equal level rook numbers on singleton boards"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/is-chosen-level-and-leftmost","kind":"node","lean":"MetaMathlibExt.IsChosenLevelAndLeftmost","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#is-chosen-level-and-leftmost","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"Let $m$ be a natural number and $F_0$ a finite set of cells $(i, r)$ (column $i$, row $r$), and call the rows $r$ with $\\lfloor r/m \\rfloor = l$ level $l$ (for $m = 0$, every row lies in level $0$). A level $l$ and a cell $c_0$ are chosen for $F_0$ when level $l$ contains at least two cells of $F_0$; every level $l'$ containing at least two cells of $F_0$ contains strictly more cells of $F_0$ than level $l$, or\u2026","title":"The chosen level and its leftmost cell"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/file-placement-block","kind":"node","lean":"MetaMathlibExt.filePlacementBlock","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","The block "],["math","\\(P(F_0)\\)"]],"source":"The block $P(F_0)$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#file-placement-block","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"fully_proved","summary":"Let $B$ be the Ferrers board with column heights $b$, let $m$ and $l$ be natural numbers, and call the rows $r$ with $\\lfloor r/m \\rfloor = l$ level $l$ (for $m = 0$, every row lies in level $0$). For a set $F_0$ of cells and a cell $c_0$, the block $P(F_0)$ is the set of file placements $F$ on $B$ such that $F$ and $F_0$ have the same cells outside level $l$, the cells of $F$ in level $l$ occupy exactly the same\u2026","title":"The block $P(F_0)$"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/file-placement-block-card","kind":"node","lean":"MetaMathlibExt.filePlacementBlock_card","parent":27713,"reading_title":{"format":"inline-parts-v1","parts":[["text","Size of the block "],["math","\\(P(F_0)\\)"]],"source":"Size of the block $P(F_0)$"},"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#file-placement-block-card","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $m \\ge 2$, let $B$ be a singleton board for $m$, and call the rows $r$ with $\\lfloor r/m \\rfloor = l$ level $l$. Let $F_0$ be a file placement on $B$ that is not an $m$-level rook placement, let $l$ and $c_0$ be its chosen level and the leftmost cell of $F_0$ in that level, and suppose level $l$ contains $n$ cells of $F_0$. Then the block $P(F_0)$ determined by $l$ and $c_0$ has exactly $m^{n-1}$ members.","title":"Size of the block $P(F_0)$"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/sum-file-placement-weight-block-of-two","kind":"node","lean":"MetaMathlibExt.sum_filePlacementWeight_block_of_two","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#sum-file-placement-weight-block-of-two","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $m \\ge 2$, let $B$ be a singleton board for $m$, with rows numbered from $0$ and row $r$ in level $\\lfloor r/m \\rfloor$, and let $F_0$ be a file placement on $B$. Let $l$ be the chosen level of $F_0$ (among the levels holding at least two cells of $F_0$, one holding the fewest, the lowest on ties) and $c_0$ the leftmost cell of $F_0$ in level $l$, and suppose that level $l$ holds exactly two cells of $F_0$.\u2026","title":"Weights cancel on a block with two rooks in the chosen level"},{"id":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/sum-file-placement-weight-block","kind":"node","lean":"MetaMathlibExt.sum_filePlacementWeight_block","parent":27713,"result":true,"route":"../roadmap/Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards/index.html#sum-file-placement-weight-block","scope":"Combinatorics/tilings-board-placements/weighted-file-placements-on-ferrers-boards","status":"can_prove","summary":"Let $m \\ge 2$ and let $B$ be a singleton board for $m$. Let $F_0$ be a file placement on $B$ that is not an $m$-level rook placement. Let $l$ be a level and $c_0$ a cell such that $l$ holds at least two cells of $F_0$ and holds the fewest cells of $F_0$ among all such levels (the lowest such level on ties), and $c_0$ is the leftmost cell of $F_0$ in level $l$. Then the $m$-weights of the file placements in the\u2026","title":"Cancellation of the weights over a block"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues.json","id":"Combinatorics/symmetric-functions-q-analogues","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/index.html","scope":"Combinatorics/symmetric-functions-q-analogues","status":"can_prove","summary":"Algebraic combinatorics of symmetric functions and Young tableaux, Gaussian binomial coefficients and other q-analogues, alternating sign matrices and plane partitions (q-TSPP, Robbins numbers), Coxeter\u2013Catalan positivity, chromatic symmetric functions, minuscule polynomials, and positivity or real-rootedness of permanents and gamma vectors.","title":"Symmetric functions, tableaux and q-analogues"},{"id":"Combinatorics/symmetric-functions-q-analogues/q-tspp","kind":"node","lean":"MathlibExt.Combinatorics.QTSPPWanted.QTSPP.q_tspp","parent":27738,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(q\\)"],["text","-TSPP theorem"]],"source":"The $q$-TSPP theorem"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/index.html#q-tspp","scope":"Combinatorics/symmetric-functions-q-analogues","status":"can_prove","summary":"Fix $n$ and let $B_n = \\{0, \\dots, n - 1\\}^3$. A totally symmetric plane partition in $B_n$ is a subset $\\pi \\subseteq B_n$ that is a lower set of $\\mathbb{N}^3$ for the componentwise order and is invariant under every permutation of the three coordinates; the empty set is one. Let $T_n$ be the set of triples $(i, j, k) \\in B_n$ with $i \\le j \\le k$, and let $o(\\pi)$ be the number of elements of $\\pi$ in $T_n$ (the\u2026","title":"The $q$-TSPP theorem"},{"id":"Combinatorics/symmetric-functions-q-analogues/zeilberger-gamma-coefficient","kind":"node","lean":"MetaMathlibExt.zeilberger_gamma_coefficient","parent":27738,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/index.html#zeilberger-gamma-coefficient","scope":"Combinatorics/symmetric-functions-q-analogues","status":"fully_proved","summary":"Let $n$ be a natural number, let $\\gamma_0, \\gamma_1, \\gamma_2, \\dots$ be rational numbers, and let $h \\in \\mathbb{Q}[x]$ be the polynomial $h(x) = \\sum_{j=0}^{\\lfloor n/2 \\rfloor} \\gamma_j \\, x^j (1 + x)^{n - 2j}$. Let $c(x) = \\sum_{m \\ge 0} C_m x^m \\in \\mathbb{Q}[[x]]$ be the generating function of the Catalan numbers $C_m$. Then for every natural number $k \\le \\lfloor n/2 \\rfloor$, the number $\\gamma_k$ is the\u2026","title":"Zeilberger's lemma for gamma vectors"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers.json","id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","kind":"scope","navigation":"module","parent":27738,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"can_prove","title":"Alternating sign matrices and Robbins numbers"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/robbins-number","kind":"node","lean":"MetaMathlibExt.robbinsNumber","parent":27741,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#robbins-number","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the Robbins number $A_n$ is the natural number obtained by dividing $\\prod_{k=0}^{n-1} (3k+1)!$ by $\\prod_{k=0}^{n-1} (n+k)!$, where the division is division of natural numbers rounded down.","title":"Robbins numbers"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/signed-robbins-number","kind":"node","lean":"MetaMathlibExt.signedRobbinsNumber","parent":27741,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#signed-robbins-number","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the signed Robbins number is the integer $(-1)^{\\binom{n+1}{2}} A_{n+1}$, where $A_{n+1}$ is the Robbins number.","title":"Signed Robbins numbers"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/row-partial-sum","kind":"node","lean":"MathlibExt.Combinatorics.AlternatingSignMatrixWanted.Combinatorics.AlternatingSignMatrix.rowPartialSum","parent":27741,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#row-partial-sum","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"fully_proved","summary":"For an $n \\times n$ integer matrix $A$ and indices $i, k \\in \\{0, \\dots, n-1\\}$, the row partial sum of $A$ at $(i, k)$ is the sum of the entries of row $i$ in columns $0$ through $k$, that is, $\\sum_{j \\le k} A_{ij}$.","title":"Row partial sums of a matrix"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/col-partial-sum","kind":"node","lean":"MathlibExt.Combinatorics.AlternatingSignMatrixWanted.Combinatorics.AlternatingSignMatrix.colPartialSum","parent":27741,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#col-partial-sum","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"fully_proved","summary":"For an $n \\times n$ integer matrix $A$ and indices $j, k \\in \\{0, \\dots, n-1\\}$, the column partial sum of $A$ at $(j, k)$ is the sum of the entries of column $j$ in rows $0$ through $k$, that is, $\\sum_{i \\le k} A_{ij}$.","title":"Column partial sums of a matrix"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/is-zero-or-one","kind":"node","lean":"MathlibExt.Combinatorics.AlternatingSignMatrixWanted.Combinatorics.AlternatingSignMatrix.IsZeroOrOne","parent":27741,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#is-zero-or-one","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"fully_proved","summary":"An integer $a$ is zero or one when $a = 0$ or $a = 1$. These are the values allowed for the partial sums of an alternating sign matrix.","title":"Integers equal to zero or one"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/is-alternating-sign-matrix","kind":"node","lean":"MathlibExt.Combinatorics.AlternatingSignMatrixWanted.Combinatorics.AlternatingSignMatrix.IsAlternatingSignMatrix","parent":27741,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#is-alternating-sign-matrix","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"fully_proved","summary":"An $n \\times n$ integer matrix $A$ is an alternating sign matrix when every entry of $A$ is $-1$, $0$ or $1$; for all indices $i, k$ the row partial sum $\\sum_{j \\le k} A_{ij}$ is $0$ or $1$; for all indices $j, k$ the column partial sum $\\sum_{i \\le k} A_{ij}$ is $0$ or $1$; every row of $A$ sums to $1$; and every column of $A$ sums to $1$.","title":"Alternating sign matrices"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/asmatrix","kind":"node","lean":"MathlibExt.Combinatorics.AlternatingSignMatrixWanted.Combinatorics.AlternatingSignMatrix.ASMatrix","parent":27741,"reading_title":{"format":"inline-parts-v1","parts":[["text","Alternating sign matrices of size "],["math","\\(n\\)"]],"source":"Alternating sign matrices of size $n$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#asmatrix","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"fully_proved","summary":"For a natural number $n$, this is the collection of all $n \\times n$ integer matrices that are alternating sign matrices.","title":"Alternating sign matrices of size $n$"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/robbins-product","kind":"node","lean":"MathlibExt.Combinatorics.AlternatingSignMatrixWanted.Combinatorics.AlternatingSignMatrix.robbinsProduct","parent":27741,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#robbins-product","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"fully_proved","summary":"For a natural number $n$, the Robbins product is the rational number","title":"The Robbins product"},{"id":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/alternating-sign-matrix-count","kind":"node","lean":"MathlibExt.Combinatorics.AlternatingSignMatrixWanted.Combinatorics.AlternatingSignMatrix.alternating_sign_matrix_count","parent":27741,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers/index.html#alternating-sign-matrix-count","scope":"Combinatorics/symmetric-functions-q-analogues/alternating-sign-matrices-and-robbins-numbers","status":"can_prove","summary":"For every natural number $n$, the number of $n \\times n$ alternating sign matrices equals the Robbins product $\\prod_{k=0}^{n-1} \\frac{(3k+1)!}{(n+k)!}$.","title":"Alternating sign matrix theorem"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients.json","id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","kind":"scope","navigation":"module","parent":27738,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","title":"Gaussian binomial coefficients"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/gauss-binom","kind":"node","lean":"MathlibExt.gaussBinom","parent":27751,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#gauss-binom","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a semiring and $Q \\in R$. For natural numbers $M$ and $k$, the Gaussian binomial coefficient $\\binom{M}{k}_Q \\in R$ is defined recursively by $\\binom{M}{0}_Q = 1$, $\\binom{0}{k+1}_Q = 0$ and $\\binom{M+1}{k+1}_Q = \\binom{M}{k}_Q + Q^{k+1} \\binom{M}{k+1}_Q$.","title":"Gaussian binomial coefficient"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/gauss-binom-zero-right","kind":"node","lean":"MathlibExt.gaussBinom_zero_right","parent":27751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gaussian binomial coefficients with "],["math","\\(k = 0\\)"]],"source":"Gaussian binomial coefficients with $k = 0$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#gauss-binom-zero-right","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a semiring and $Q \\in R$. For every natural number $M$, the Gaussian binomial coefficient $\\binom{M}{0}_Q$ equals $1$.","title":"Gaussian binomial coefficients with $k = 0$"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/gauss-binom-zero-succ","kind":"node","lean":"MathlibExt.gaussBinom_zero_succ","parent":27751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gaussian binomial coefficients with "],["math","\\(M = 0\\)"]],"source":"Gaussian binomial coefficients with $M = 0$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#gauss-binom-zero-succ","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a semiring and $Q \\in R$. For every natural number $k$, the Gaussian binomial coefficient $\\binom{0}{k+1}_Q$ equals $0$.","title":"Gaussian binomial coefficients with $M = 0$"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/gauss-binom-succ-succ","kind":"node","lean":"MathlibExt.gaussBinom_succ_succ","parent":27751,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(Q\\)"],["text","-Pascal recursion for Gaussian binomial coefficients"]],"source":"$Q$-Pascal recursion for Gaussian binomial coefficients"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#gauss-binom-succ-succ","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a semiring and $Q \\in R$. For all natural numbers $M$ and $k$, the Gaussian binomial coefficients satisfy $\\binom{M+1}{k+1}_Q = \\binom{M}{k}_Q + Q^{k+1} \\binom{M}{k+1}_Q$.","title":"$Q$-Pascal recursion for Gaussian binomial coefficients"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/gauss-binom-eq-zero-of-lt","kind":"node","lean":"MathlibExt.gaussBinom_eq_zero_of_lt","parent":27751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gaussian binomial coefficients vanish when "],["math","\\(k > M\\)"]],"source":"Gaussian binomial coefficients vanish when $k > M$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#gauss-binom-eq-zero-of-lt","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a semiring and $Q \\in R$. For natural numbers $M$ and $k$ with $M < k$, the Gaussian binomial coefficient $\\binom{M}{k}_Q$ is $0$.","title":"Gaussian binomial coefficients vanish when $k > M$"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/gauss-binom-self","kind":"node","lean":"MathlibExt.gaussBinom_self","parent":27751,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#gauss-binom-self","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a semiring and $Q \\in R$. For every natural number $M$, the Gaussian binomial coefficient $\\binom{M}{M}_Q$ equals $1$.","title":"Diagonal Gaussian binomial coefficients"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/q-poch-fin","kind":"node","lean":"MathlibExt.qPochFin","parent":27751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite "],["math","\\(Q\\)"],["text","-Pochhammer symbol"]],"source":"Finite $Q$-Pochhammer symbol"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#q-poch-fin","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a commutative ring and $Q \\in R$. For a natural number $m$, the finite $Q$-Pochhammer symbol $P_Q(m) \\in R$ is the product $\\prod_{i < m} (1 - Q^{i+1})$ of $1 - Q^{i+1}$ over the natural numbers $i < m$.","title":"Finite $Q$-Pochhammer symbol"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/q-poch-fin-zero","kind":"node","lean":"MathlibExt.qPochFin_zero","parent":27751,"reading_title":{"format":"inline-parts-v1","parts":[["text","The finite "],["math","\\(Q\\)"],["text","-Pochhammer symbol at "],["math","\\(0\\)"]],"source":"The finite $Q$-Pochhammer symbol at $0$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#q-poch-fin-zero","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a commutative ring and $Q \\in R$. The finite $Q$-Pochhammer symbol $P_Q(m) = \\prod_{i < m} (1 - Q^{i+1})$, the product running over the natural numbers $i < m$, satisfies $P_Q(0) = 1$.","title":"The finite $Q$-Pochhammer symbol at $0$"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/q-poch-fin-succ","kind":"node","lean":"MathlibExt.qPochFin_succ","parent":27751,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recursion for the finite "],["math","\\(Q\\)"],["text","-Pochhammer symbol"]],"source":"Recursion for the finite $Q$-Pochhammer symbol"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#q-poch-fin-succ","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a commutative ring and $Q \\in R$, and write $P_Q(m) = \\prod_{i < m} (1 - Q^{i+1})$ for the finite $Q$-Pochhammer symbol, the product running over the natural numbers $i < m$. For every natural number $m$, $P_Q(m+1) = P_Q(m) \\, (1 - Q^{m+1})$.","title":"Recursion for the finite $Q$-Pochhammer symbol"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/gauss-binom-mul-q-poch-fin-mul-q-poch-fi","kind":"node","lean":"MathlibExt.gaussBinom_mul_qPochFin_mul_qPochFin","parent":27751,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#gauss-binom-mul-q-poch-fin-mul-q-poch-fi","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $R$ be a commutative ring and $Q \\in R$, and write $P_Q(m) = \\prod_{i < m} (1 - Q^{i+1})$ for the finite $Q$-Pochhammer symbol, the product running over the natural numbers $i < m$. For natural numbers $k \\le M$, the Gaussian binomial coefficient satisfies $\\binom{M}{k}_Q \\, P_Q(k) \\, P_Q(M - k) = P_Q(M)$.","title":"Product formula for Gaussian binomial coefficients"},{"id":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/gaussian-pair-sorted-expansion","kind":"node","lean":"MetaMathlibExt.gaussian_pair_sorted_expansion","parent":27751,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients/index.html#gaussian-pair-sorted-expansion","scope":"Combinatorics/symmetric-functions-q-analogues/gaussian-binomial-coefficients","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers and let $W$ be the set of words $v = v_0 v_1 \\cdots v_{n-1}$ over the alphabet $\\{0, 1\\}$ with exactly $k$ letters equal to $1$. For a word $v$, let $\\operatorname{inv}(v)$ be the number of pairs $i < j$ with $v_i > v_j$; call $v$ pair-sorted when $v_{2i} \\le v_{2i+1}$ for every $i$ with $2i + 1 < n$; and let $\\operatorname{asc}_{\\mathrm{odd}}(v)$ be the number of $i$ with\u2026","title":"Ehrenborg\u2013Readdy expansion of the Gaussian coefficient"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a.json","id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","kind":"scope","navigation":"module","parent":27738,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","title":"Minuscule polynomials of type A"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"fully_proved","summary":"For a natural number $n$, the type-A minuscule polynomial $N_n(x)$ is the polynomial with rational coefficients","title":"Type-A minuscule polynomial"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-one","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_one","parent":27763,"reading_title":{"format":"inline-parts-v1","parts":[["text","The minuscule polynomial "],["math","\\(N_1\\)"],["text"," vanishes"]],"source":"The minuscule polynomial $N_1$ vanishes"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-one","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"fully_proved","summary":"The type-A minuscule polynomial $N_1(x)$ is the zero polynomial.","title":"The minuscule polynomial $N_1$ vanishes"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-two","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_two","parent":27763,"reading_title":{"format":"inline-parts-v1","parts":[["text","The minuscule polynomial "],["math","\\(N_2\\)"]],"source":"The minuscule polynomial $N_2$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-two","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"fully_proved","summary":"The type-A minuscule polynomial $N_2(x)$ equals $2x$.","title":"The minuscule polynomial $N_2$"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-three","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_three","parent":27763,"reading_title":{"format":"inline-parts-v1","parts":[["text","The minuscule polynomial "],["math","\\(N_3\\)"]],"source":"The minuscule polynomial $N_3$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-three","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"fully_proved","summary":"The type-A minuscule polynomial $N_3(x)$ equals $8x + 8x^2$.","title":"The minuscule polynomial $N_3$"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-is-real-rooted","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_isRealRooted","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-is-real-rooted","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"For every natural number $n$, the type-A minuscule polynomial $N_n$, viewed as a complex polynomial, is real-rooted: every element of its multiset of roots is real.","title":"Real-rootedness of the minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-interlaces-succ","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_interlaces_succ","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-interlaces-succ","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $n \\ge 2$. There are nonzero real numbers $c, d$ and real numbers $r_0 \\le \\dots \\le r_{n-1}$ and $s_0 \\le \\dots \\le s_{n-2}$ such that the type-A minuscule polynomials factor over $\\mathbb{R}$ as $N_{n+1}(x) = c \\prod_{i=0}^{n-1} (x - r_i)$ and $N_n(x) = d \\prod_{j=0}^{n-2} (x - s_j)$, and $r_j \\le s_j \\le r_{j+1}$ for every $j$ with $0 \\le j \\le n - 2$.","title":"Interlacing of consecutive minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-coeff-log-concave","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_coeff_logConcave","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-coeff-log-concave","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $N_{n,k}$ be the coefficient of $x^k$ in the type-A minuscule polynomial $N_n(x)$. For all natural numbers $n$ and $k$, $N_{n,k-1} N_{n,k+1} \\le N_{n,k}^2$, where for $k = 0$ the index $k - 1$ is read as $0$.","title":"Log-concavity of the coefficients of the minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-gamma-positive","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_gamma_positive","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-gamma-positive","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"For every natural number $n$ there are non-negative integers $\\gamma_{n,0}, \\dots, \\gamma_{n,\\lfloor n/2 \\rfloor}$ such that the type-A minuscule polynomial $N_n$ satisfies $N_n(x) = \\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\gamma_{n,k}\\, x^k (1 + x)^{n-2k}$.","title":"Gamma-positivity of the minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-comp-sq","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_comp_sq","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-comp-sq","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"For every natural number $n$, the type-A minuscule polynomial $N_n$ satisfies the identity of polynomials with rational coefficients $16x\\, N_n(x^2) = 2(n+1)x \\bigl((1+x)^{2n} + (1-x)^{2n}\\bigr) - \\bigl((1+x)^{2n+2} - (1-x)^{2n+2}\\bigr)$.","title":"Binomial closed form of the minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-egf","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_egf","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-egf","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"For all real numbers $x$ and $z$ with $x \\ne 0$, the series $\\sum_{n \\ge 0} N_n(x^2)\\, z^n / n!$ built from the type-A minuscule polynomials $N_n$ has sum $\\frac{2x(1+x)^2 z - 1 - x^2}{16x} e^{(1+x)^2 z} + \\frac{2x(1-x)^2 z + 1 + x^2}{16x} e^{(1-x)^2 z}$.","title":"Exponential generating function of the minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-eval-one","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_eval_one","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-eval-one","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"For every integer $n \\ge 1$, the type-A minuscule polynomial $N_n$ satisfies $N_n(1) = (n - 1)\\, 2^{2n-3}$.","title":"Bourn\u2013Erickson formula for the minuscule polynomials at one"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/bender-asymptotic-normality","kind":"node","lean":"MetaMathlibExt.bender_asymptotic_normality","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#bender-asymptotic-normality","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"For each natural number $n$ let $r_{n,0}, \\dots, r_{n,n-1}$ be non-negative real numbers, and let $a(n,k)$ be the coefficient of $x^k$ in the real polynomial $A_n(x) = \\prod_{i\\lt n} (x + r_{n,i})$. Put $\\mu_n = \\sum_{i\\lt n} 1/(1 + r_{n,i})$ and $\\sigma_n^2 = \\sum_{i\\lt n} r_{n,i}/(1 + r_{n,i})^2$, and let $\\sigma_n \\ge 0$ be the square root of $\\sigma_n^2$. Then for every $n$,\u2026","title":"Bender's central limit theorem for polynomials with non-positive real zeros"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-asymptotically-no","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_asymptotically_normal","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-asymptotically-no","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $N_{n,k}$ be the coefficient of $x^k$ in the type-A minuscule polynomial $N_n(x)$, and let $\\sigma_n = \\sqrt{(n^2 - n - 2)/(8(n - 1))}$. Then for every real $x$, the sum of the coefficients $N_{n,k}$ over $0 \\le k < n$ with $k \\le n/2 + x \\sigma_n$, divided by $N_n(1)$, tends to $\\Phi(x)$ as $n \\to \\infty$, where $\\Phi$ is the distribution function of the standard normal distribution.","title":"Asymptotic normality of the coefficients of the minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-x-log-concave","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_x_logConcave","parent":27763,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(x\\)"],["text","-log-concavity of the minuscule polynomials"]],"source":"$x$-log-concavity of the minuscule polynomials"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-x-log-concave","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $N_n$ denote the type-A minuscule polynomial. For all natural numbers $n$ and $k$, the coefficient of $x^k$ in $N_n(x)^2 - N_{n+1}(x) N_{n-1}(x)$ is non-negative, where $N_{n-1}$ is read as $N_0$ when $n = 0$.","title":"$x$-log-concavity of the minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/minuscule-polynomial-a-coeff-matrix-has","kind":"node","lean":"MetaMathlibExt.minusculePolynomialA_coeffMatrix_hasNonnegativeMinors","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#minuscule-polynomial-a-coeff-matrix-has","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $N_{n,k}$ be the coefficient of $x^k$ in the type-A minuscule polynomial $N_n(x)$. The infinite real matrix $[N_{n,k}]_{n,k \\ge 1}$, whose entry in row $i$ and column $j$ ($i, j \\ge 0$) is $N_{i+1,j+1}$, has nonnegative minors: every square submatrix obtained by choosing finitely many rows and finitely many columns, each in increasing order, has nonnegative determinant.","title":"Total positivity of the coefficient matrix of the minuscule polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/multiplier-n-sequence-iff","kind":"node","lean":"MetaMathlibExt.multiplier_n_sequence_iff","parent":27763,"reading_title":{"format":"inline-parts-v1","parts":[["text","Craven\u2013Csordas characterization of multiplier "],["math","\\(n\\)"],["text","-sequences"]],"source":"Craven\u2013Csordas characterization of multiplier $n$-sequences"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#multiplier-n-sequence-iff","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $n$ be a natural number and $t_0, t_1, \\dots$ real numbers. Say a real polynomial splits over $\\mathbb{R}$ if it is a constant times a product of linear factors $x - r$ with $r$ real (the zero polynomial splits). Then the following are equivalent: for every real polynomial $\\sum_k a_k x^k$ of degree at most $n$ that splits over $\\mathbb{R}$, the polynomial $\\sum_{k=0}^n a_k t_k x^k$ splits over $\\mathbb{R}$;\u2026","title":"Craven\u2013Csordas characterization of multiplier $n$-sequences"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/multiplier-sequence-iff","kind":"node","lean":"MetaMathlibExt.multiplier_sequence_iff","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#multiplier-sequence-iff","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $t_0, t_1, \\dots$ be non-negative real numbers. Say a real polynomial splits over $\\mathbb{R}$ if it is a constant times a product of linear factors $x - r$ with $r$ real (the zero polynomial splits). Then the following are equivalent: for every real polynomial $\\sum_k a_k x^k$ that splits over $\\mathbb{R}$, the polynomial $\\sum_k a_k t_k x^k$ splits over $\\mathbb{R}$; and for every $n \\ge 1$, the polynomial\u2026","title":"P\u00f3lya\u2013Schur characterization of multiplier sequences"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/polya-frequency-iff-has-sum-edrei-produc","kind":"node","lean":"MetaMathlibExt.polyaFrequency_iff_hasSum_edrei_product","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#polya-frequency-iff-has-sum-edrei-produc","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $a_0, a_1, \\dots$ be non-negative real numbers with $a_0 \\ne 0$. All minors of the infinite lower-triangular Toeplitz matrix $(a_{n-k})_{n,k \\ge 0}$, with zeros above the diagonal, are non-negative (that is, the sequence is a P\u00f3lya frequency sequence) if and only if there are a real $\\gamma \\ge 0$, non-negative reals $\\alpha_0, \\alpha_1, \\dots$ and $\\beta_0, \\beta_1, \\dots$ with\u2026","title":"Aissen\u2013Schoenberg\u2013Whitney\u2013Edrei theorem"},{"id":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/multiplier-sequence-iff-has-sum-exp-mul","kind":"node","lean":"MetaMathlibExt.multiplier_sequence_iff_hasSum_exp_mul_tprod","parent":27763,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a/index.html#multiplier-sequence-iff-has-sum-exp-mul","scope":"Combinatorics/symmetric-functions-q-analogues/minuscule-polynomials-of-type-a","status":"can_prove","summary":"Let $t_0, t_1, \\dots$ be non-negative real numbers with $t_0 \\ne 0$. Say a real polynomial splits over $\\mathbb{R}$ if it is a constant times a product of linear factors $x - r$ with $r$ real (the zero polynomial splits). Then the following are equivalent: for every real polynomial $\\sum_k a_k x^k$ that splits over $\\mathbb{R}$, the polynomial $\\sum_k a_k t_k x^k$ splits over $\\mathbb{R}$; and there are a real\u2026","title":"P\u00f3lya\u2013Schur transcendental characterization of multiplier sequences"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture.json","id":"Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture","kind":"scope","navigation":"module","parent":27738,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture/index.html","scope":"Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture","status":"can_prove","title":"Monotone column permanent conjecture"},{"id":"Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture/permanent-add-xj","kind":"node","lean":"MathlibExt.Combinatorics.MonotoneColumnPermanentAMR156Problem0025Wanted.permanentAddXJ","parent":27783,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomial "],["math","\\(\\operatorname{per}(A + xJ)\\)"]],"source":"The polynomial $\\operatorname{per}(A + xJ)$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture/index.html#permanent-add-xj","scope":"Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture","status":"fully_proved","summary":"For a real $n \\times n$ matrix $A$, this is the polynomial $\\operatorname{per}(A + xJ) \\in \\mathbb{R}[x]$, where $J$ is the $n \\times n$ all-ones matrix: the permanent of the $n \\times n$ matrix of polynomials whose $(i, j)$ entry is $A_{ij} + x$.","title":"The polynomial $\\operatorname{per}(A + xJ)$"},{"id":"Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture/monotone-column-permanent-conjecture","kind":"node","lean":"MathlibExt.Combinatorics.MonotoneColumnPermanentAMR156Problem0025Wanted.monotoneColumnPermanentConjecture","parent":27783,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture/index.html#monotone-column-permanent-conjecture","scope":"Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture","status":"fully_proved","summary":"This is the statement that for every $n \\in \\mathbb{N}$ and every real $n \\times n$ matrix $A$ whose columns are weakly increasing ($A_{ij} \\le A_{i'j}$ whenever $i \\le i'$), every complex root of the polynomial $\\operatorname{per}(A + xJ) \\in \\mathbb{R}[x]$ is real, where $J$ is the $n \\times n$ all-ones matrix.","title":"The Haglund\u2013Ono\u2013Wagner monotone column permanent conjecture"},{"id":"Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture/monotone-column-permanent-conjecture-hol","kind":"node","lean":"MathlibExt.Combinatorics.MonotoneColumnPermanentAMR156Problem0025Wanted.monotoneColumnPermanentConjecture_holds","parent":27783,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture/index.html#monotone-column-permanent-conjecture-hol","scope":"Combinatorics/symmetric-functions-q-analogues/monotone-column-permanent-conjecture","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$ and every real $n \\times n$ matrix $A$ whose columns are weakly increasing ($A_{ij} \\le A_{i'j}$ whenever $i \\le i'$), every complex root of the polynomial $\\operatorname{per}(A + xJ) \\in \\mathbb{R}[x]$ is real, where $J$ is the $n \\times n$ all-ones matrix.","title":"Br\u00e4nd\u00e9n\u2013Haglund\u2013Visontai\u2013Wagner theorem (monotone column permanent conjecture)"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa.json","id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","kind":"scope","navigation":"module","parent":27738,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"can_prove","title":"Near-hook tableaux and first-quadrant lattice paths"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/near-hook-theta","kind":"node","lean":"MetaMathlibExt.nearHookTheta","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The near-hook diagram "],["math","\\(\\theta^{(n)}\\)"]],"source":"The near-hook diagram $\\theta^{(n)}$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#near-hook-theta","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, $\\theta^{(n)}$ is the Young diagram of the near-hook partition $(n + 2, 2, 1^n)$ of $2n + 4$: its rows have lengths $n + 2$ and $2$, followed by $n$ rows of length $1$.","title":"The near-hook diagram $\\theta^{(n)}$"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/near-hook-theta-row-lens","kind":"node","lean":"MetaMathlibExt.nearHookTheta_rowLens","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","Row lengths of "],["math","\\(\\theta^{(n)}\\)"]],"source":"Row lengths of $\\theta^{(n)}$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#near-hook-theta-row-lens","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the row lengths of the Young diagram $\\theta^{(n)}$, from the top row down, are $n + 2$, $2$, and then $1$ repeated $n$ times.","title":"Row lengths of $\\theta^{(n)}$"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/first-quadrant-paths","kind":"node","lean":"MetaMathlibExt.firstQuadrantPaths","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","First-quadrant paths "],["math","\\(\\mathcal{P}_n\\)"]],"source":"First-quadrant paths $\\mathcal{P}_n$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#first-quadrant-paths","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, $\\mathcal{P}_n$ is the set of words $(s_1, \\dots, s_{2n+2})$ of $2n + 2$ steps, each one of $N = (0, 1)$, $S = (0, -1)$, $E = (1, 0)$ and $W = (-1, 0)$ in $\\mathbb{Z}^2$, such that every partial sum $s_1 + \\dots + s_k$ with $0 \\le k \\le 2n + 2$ has both coordinates nonnegative and the total $s_1 + \\dots + s_{2n+2}$ equals $(n, n)$.","title":"First-quadrant paths $\\mathcal{P}_n$"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/north-south-mem-first-quadrant-paths-zer","kind":"node","lean":"MetaMathlibExt.north_south_mem_firstQuadrantPaths_zero","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The path "],["math","\\(NS\\)"],["text"," lies in "],["math","\\(\\mathcal{P}_0\\)"]],"source":"The path $NS$ lies in $\\mathcal{P}_0$"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#north-south-mem-first-quadrant-paths-zer","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"fully_proved","summary":"The two-step word consisting of a north step $(0, 1)$ followed by a south step $(0, -1)$ belongs to the set $\\mathcal{P}_0$ of first-quadrant paths.","title":"The path $NS$ lies in $\\mathcal{P}_0$"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/near-hook-tableau-to-path-psi","kind":"node","lean":"MetaMathlibExt.nearHookTableauToPathPsi","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(\\psi\\)"],["text"," from near-hook tableaux to lattice paths"]],"source":"The map $\\psi$ from near-hook tableaux to lattice paths"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#near-hook-tableau-to-path-psi","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and let $T$ be a semistandard Young tableau of shape $\\theta^{(n)}$, with rows and columns numbered from $0$; call row $0$ of $\\theta^{(n)}$ the arm and column $0$ the leg. Then $\\psi(T)$ is the word $(s_1, \\dots, s_{2n+2})$ of steps in $\\mathbb{Z}^2$ in which $s_i$ is $E = (1, 0)$ if $i + 2$ is an entry of $T$ in the arm; otherwise $s_i$ is $N = (0, 1)$ if $i + 2$ is an entry of $T$ in the\u2026","title":"The map $\\psi$ from near-hook tableaux to lattice paths"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/length-near-hook-tableau-to-path-psi","kind":"node","lean":"MetaMathlibExt.length_nearHookTableauToPathPsi","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The path "],["math","\\(\\psi(T)\\)"],["text"," has "],["math","\\(2n + 2\\)"],["text"," steps"]],"source":"The path $\\psi(T)$ has $2n + 2$ steps"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#length-near-hook-tableau-to-path-psi","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every semistandard Young tableau $T$ of shape $\\theta^{(n)}$, the word $\\psi(T)$ has length $2n + 2$.","title":"The path $\\psi(T)$ has $2n + 2$ steps"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/near-hook-tableau-to-path-xi","kind":"node","lean":"MetaMathlibExt.nearHookTableauToPathXi","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(\\xi\\)"],["text"," from near-hook tableaux to lattice paths"]],"source":"The map $\\xi$ from near-hook tableaux to lattice paths"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#near-hook-tableau-to-path-xi","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and let $T$ be a semistandard Young tableau of shape $\\theta^{(n)}$, with rows and columns numbered from $0$; call row $0$ of $\\theta^{(n)}$ the arm and the cell in row $1$ and column $1$ the heart, and let $H$ be the entry of $T$ in the heart. Then $\\xi(T)$ is the word $(s_1, \\dots, s_{2n+2})$ of steps in $\\mathbb{Z}^2$ given as follows, where $E = (1, 0)$, $N = (0, 1)$, $S = (0, -1)$ and\u2026","title":"The map $\\xi$ from near-hook tableaux to lattice paths"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/length-near-hook-tableau-to-path-xi","kind":"node","lean":"MetaMathlibExt.length_nearHookTableauToPathXi","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The path "],["math","\\(\\xi(T)\\)"],["text"," has "],["math","\\(2n + 2\\)"],["text"," steps"]],"source":"The path $\\xi(T)$ has $2n + 2$ steps"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#length-near-hook-tableau-to-path-xi","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every semistandard Young tableau $T$ of shape $\\theta^{(n)}$, the word $\\xi(T)$ has length $2n + 2$.","title":"The path $\\xi(T)$ has $2n + 2$ steps"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/bij-on-near-hook-tableau-to-path-psi","kind":"node","lean":"MetaMathlibExt.bijOn_nearHookTableauToPathPsi","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(\\psi\\)"],["text"," is a bijection from near-hook tableaux to first-quadrant paths"]],"source":"The map $\\psi$ is a bijection from near-hook tableaux to first-quadrant paths"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#bij-on-near-hook-tableau-to-path-psi","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$, the map $\\psi$ restricts to a bijection from the standard Young tableaux of shape $\\theta^{(n)}$ onto the set $\\mathcal{P}_n$ of first-quadrant paths. Here a standard Young tableau of shape $\\theta^{(n)}$ is a semistandard Young tableau of that shape whose entries map the $2n + 4$ cells of $\\theta^{(n)}$ bijectively onto $\\{1, \\dots, 2n + 4\\}$.","title":"The map $\\psi$ is a bijection from near-hook tableaux to first-quadrant paths"},{"id":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/bij-on-near-hook-tableau-to-path-xi","kind":"node","lean":"MetaMathlibExt.bijOn_nearHookTableauToPathXi","parent":27787,"reading_title":{"format":"inline-parts-v1","parts":[["text","The map "],["math","\\(\\xi\\)"],["text"," is a bijection from near-hook tableaux to first-quadrant paths"]],"source":"The map $\\xi$ is a bijection from near-hook tableaux to first-quadrant paths"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa/index.html#bij-on-near-hook-tableau-to-path-xi","scope":"Combinatorics/symmetric-functions-q-analogues/near-hook-tableaux-and-first-quadrant-lattice-pa","status":"can_prove","summary":"For every $n \\in \\mathbb{N}$, the map $\\xi$ restricts to a bijection from the standard Young tableaux of shape $\\theta^{(n)}$ onto the set $\\mathcal{P}_n$ of first-quadrant paths. Here a standard Young tableau of shape $\\theta^{(n)}$ is a semistandard Young tableau of that shape whose entries map the $2n + 4$ cells of $\\theta^{(n)}$ bijectively onto $\\{1, \\dots, 2n + 4\\}$.","title":"The map $\\xi$ is a bijection from near-hook tableaux to first-quadrant paths"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups.json","id":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","kind":"scope","navigation":"module","parent":27738,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/index.html","scope":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","status":"can_prove","title":"q-Fuss\u2013Catalan numbers of Coxeter groups"},{"id":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/q-int","kind":"node","lean":"MathlibExt.Combinatorics.CoxeterFussCatalanPositivityAIMProbabilityProblem148Wanted.qInt","parent":27798,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/index.html#q-int","scope":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","status":"fully_proved","summary":"For a natural number $a$, the $q$-integer is the integer polynomial $[a]_q = 1 + q + \\dots + q^{a-1} = \\sum_{i=0}^{a-1} q^i$; in particular $[0]_q = 0$.","title":"q-integers"},{"id":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/coxeter-type","kind":"node","lean":"MathlibExt.Combinatorics.CoxeterFussCatalanPositivityAIMProbabilityProblem148Wanted.CoxeterType","parent":27798,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/index.html#coxeter-type","scope":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","status":"fully_proved","summary":"The labels $A_n$, $B_n$, $D_n$ and $I_2(m)$ for natural numbers $n$ and $m$, together with $E_6$, $E_7$, $E_8$, $F_4$, $H_3$ and $H_4$. Every natural number parameter is allowed here; the classification of irreducible finite Coxeter groups uses $A_n$ ($n \\ge 1$), $B_n$ ($n \\ge 2$), $D_n$ ($n \\ge 4$), the six exceptional labels and the dihedral types $I_2(m)$ ($m \\ge 3$).","title":"Types of irreducible finite Coxeter groups"},{"id":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/is-valid","kind":"node","lean":"MathlibExt.Combinatorics.CoxeterFussCatalanPositivityAIMProbabilityProblem148Wanted.CoxeterType.IsValid","parent":27798,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/index.html#is-valid","scope":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","status":"fully_proved","summary":"A Coxeter type is valid when it lies in the parameter range of the classification of irreducible finite Coxeter groups: $A_n$ is valid when $n \\ge 1$, $B_n$ when $n \\ge 2$, $D_n$ when $n \\ge 4$ and $I_2(m)$ when $m \\ge 3$, while $E_6$, $E_7$, $E_8$, $F_4$, $H_3$ and $H_4$ are always valid.","title":"Valid Coxeter types"},{"id":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/degrees","kind":"node","lean":"MathlibExt.Combinatorics.CoxeterFussCatalanPositivityAIMProbabilityProblem148Wanted.CoxeterType.degrees","parent":27798,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/index.html#degrees","scope":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","status":"fully_proved","summary":"The list of degrees $d_1, \\dots, d_r$ of the basic invariants of a Coxeter type (so $d_i = e_i + 1$ for the exponents $e_i$) is: $2, 3, \\dots, n + 1$ for $A_n$; $2, 4, \\dots, 2n$ for $B_n$; $n, 2, 4, \\dots, 2n - 2$ for $D_n$; $2, 5, 6, 8, 9, 12$ for $E_6$; $2, 6, 8, 10, 12, 14, 18$ for $E_7$; $2, 8, 12, 14, 18, 20, 24, 30$ for $E_8$; $2, 6, 8, 12$ for $F_4$; $2, 6, 10$ for $H_3$; $2, 12, 20, 30$ for $H_4$; and\u2026","title":"Degrees of the irreducible finite Coxeter groups"},{"id":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/coxeter-number","kind":"node","lean":"MathlibExt.Combinatorics.CoxeterFussCatalanPositivityAIMProbabilityProblem148Wanted.CoxeterType.coxeterNumber","parent":27798,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/index.html#coxeter-number","scope":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","status":"fully_proved","summary":"The Coxeter number $h$ of a Coxeter type is the largest of its degrees, and $0$ if its list of degrees is empty.","title":"Coxeter number"},{"id":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/coxeter-fuss-catalan-positive","kind":"node","lean":"MathlibExt.Combinatorics.CoxeterFussCatalanPositivityAIMProbabilityProblem148Wanted.coxeterFussCatalanPositive","parent":27798,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/index.html#coxeter-fuss-catalan-positive","scope":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","status":"fully_proved","summary":"The claim that for every type $T$ among $A_n$ ($n \\ge 1$), $B_n$ ($n \\ge 2$), $D_n$ ($n \\ge 4$), $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, $H_4$ and $I_2(m)$ ($m \\ge 3$) of irreducible finite Coxeter groups, with degrees $d_1, \\dots, d_r$ and Coxeter number $h$ (the largest degree), and every integer $k \\ge 1$, there is a polynomial $P(q)$ with nonnegative integer coefficients such that\u2026","title":"AIM Problem 2.1: positivity of q-Fuss-Catalan numbers"},{"id":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/coxeter-fuss-catalan-positive-holds","kind":"node","lean":"MathlibExt.Combinatorics.CoxeterFussCatalanPositivityAIMProbabilityProblem148Wanted.coxeterFussCatalanPositive_holds","parent":27798,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups/index.html#coxeter-fuss-catalan-positive-holds","scope":"Combinatorics/symmetric-functions-q-analogues/q-fuss-catalan-numbers-of-coxeter-groups","status":"can_prove","summary":"Let $T$ be one of the types $A_n$ ($n \\ge 1$), $B_n$ ($n \\ge 2$), $D_n$ ($n \\ge 4$), $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, $H_4$ or $I_2(m)$ ($m \\ge 3$) of irreducible finite Coxeter groups, with degrees $d_1, \\dots, d_r$ and Coxeter number $h$ (the largest degree), and let $k \\ge 1$ be an integer. With the $q$-integers $[a]_q = 1 + q + \\dots + q^{a-1}$, there is a polynomial $P(q)$ with nonnegative integer\u2026","title":"Positivity of q-Fuss-Catalan numbers of Coxeter groups"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture.json","id":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture","kind":"scope","navigation":"module","parent":27738,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/index.html","scope":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture","status":"can_prove","title":"Stanley\u2013Stembridge conjecture"},{"id":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/is-three-plus-one-free","kind":"node","lean":"MathlibExt.Combinatorics.StanleyStembridgeEPositivityAMR156Problem0022Wanted.IsThreePlusOneFree","parent":27806,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((3+1)\\)"],["text","-free posets"]],"source":"$(3+1)$-free posets"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/index.html#is-three-plus-one-free","scope":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture","status":"fully_proved","summary":"A partially ordered set $P$ is $(3+1)$-free when there are no elements $a, b, c, d \\in P$ with $a < b < c$ such that $d$ is incomparable to each of $a$, $b$ and $c$ (neither $d \\le x$ nor $x \\le d$ for $x = a, b, c$).","title":"$(3+1)$-free posets"},{"id":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/incomparability-graph","kind":"node","lean":"MathlibExt.Combinatorics.StanleyStembridgeEPositivityAMR156Problem0022Wanted.incomparabilityGraph","parent":27806,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/index.html#incomparability-graph","scope":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture","status":"fully_proved","summary":"The incomparability graph $\\operatorname{inc}(P)$ of a partially ordered set $P$ is the simple graph on $P$ in which two distinct elements $x$ and $y$ are adjacent when they are incomparable, that is, neither $x \\le y$ nor $y \\le x$.","title":"Incomparability graph of a poset"},{"id":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/chromatic-symmetric-poly","kind":"node","lean":"MathlibExt.Combinatorics.StanleyStembridgeEPositivityAMR156Problem0022Wanted.chromaticSymmetricPoly","parent":27806,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/index.html#chromatic-symmetric-poly","scope":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set $V$ with $n = |V|$ elements, $X_G$ is the polynomial with integer coefficients in the variables $x_0, \\dots, x_{n-1}$ given by $X_G = \\sum_{\\kappa} \\prod_{v \\in V} x_{\\kappa(v)}$, where $\\kappa$ ranges over the proper colourings $\\kappa : V \\to \\{0, \\dots, n - 1\\}$ of $G$, that is, the maps with $\\kappa(u) \\ne \\kappa(v)$ whenever $u$ and $v$ are adjacent.","title":"Chromatic symmetric function in as many variables as vertices"},{"id":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/is-epositive","kind":"node","lean":"MathlibExt.Combinatorics.StanleyStembridgeEPositivityAMR156Problem0022Wanted.IsEPositive","parent":27806,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(e\\)"],["text","-positive polynomials"]],"source":"$e$-positive polynomials"},"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/index.html#is-epositive","scope":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture","status":"fully_proved","summary":"A polynomial $f$ with integer coefficients in the $n$ variables $x_0, \\dots, x_{n-1}$ is $e$-positive when $f = \\sum_{\\lambda} c_\\lambda e_\\lambda$ for some natural numbers $c_\\lambda$, where $\\lambda = (\\lambda_1, \\dots, \\lambda_k)$ ranges over the partitions of $n$ and $e_\\lambda = e_{\\lambda_1} \\cdots e_{\\lambda_k}$ is the product of the elementary symmetric polynomials $e_{\\lambda_i}$ in $x_0, \\dots, x_{n-1}$.","title":"$e$-positive polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/stanley-stembridge","kind":"node","lean":"MathlibExt.Combinatorics.StanleyStembridgeEPositivityAMR156Problem0022Wanted.stanleyStembridge","parent":27806,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/index.html#stanley-stembridge","scope":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture","status":"fully_proved","summary":"The proposition that for every finite $(3+1)$-free partially ordered set $P$, the chromatic symmetric function in $|P|$ variables of the incomparability graph of $P$ is $e$-positive.","title":"Statement of the Stanley\u2013Stembridge conjecture"},{"id":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/stanley-stembridge-holds","kind":"node","lean":"MathlibExt.Combinatorics.StanleyStembridgeEPositivityAMR156Problem0022Wanted.stanleyStembridge_holds","parent":27806,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture/index.html#stanley-stembridge-holds","scope":"Combinatorics/symmetric-functions-q-analogues/stanley-stembridge-conjecture","status":"can_prove","summary":"For every finite $(3+1)$-free partially ordered set $P$, the chromatic symmetric function in $|P|$ variables of the incomparability graph of $P$ is $e$-positive.","title":"Hikita's theorem: the Stanley\u2013Stembridge conjecture holds"},{"data_url":"scopes/Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux.json","id":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux","kind":"scope","navigation":"module","parent":27738,"result":false,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/index.html","scope":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux","status":"fully_proved","title":"Symmetric functions and skew Young tableaux"},{"id":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/omega-alg-hom","kind":"node","lean":"MetaMathlibExt.MvPolynomial.omegaAlgHom","parent":27813,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/index.html#omega-alg-hom","scope":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux","status":"fully_proved","summary":"Let $R$ be a commutative ring and let $V$ be a finite set with $n$ elements. A polynomial in $R[x_v : v \\in V]$ is symmetric if it is unchanged by every permutation of the variables. For $k \\ge 0$, the elementary symmetric polynomial $e_k$ is the sum of $\\prod_{v \\in T} x_v$ over all $k$-element subsets $T \\subseteq V$, and the complete homogeneous symmetric polynomial $h_k$ is the sum of $\\prod_{v \\in M} x_v$ over\u2026","title":"The omega homomorphism on symmetric polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/exponential-specialization","kind":"node","lean":"MvPowerSeries.exponentialSpecialization","parent":27813,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/index.html#exponential-specialization","scope":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux","status":"fully_proved","summary":"Let $\\mathbb{Q}[[x_1, x_2, \\dots]]$ be the ring of formal power series in the variables $x_1, x_2, \\dots$ with rational coefficients: its elements are the formal sums $f = \\sum_\\alpha c_\\alpha x^\\alpha$, where $\\alpha = (\\alpha_1, \\alpha_2, \\dots)$ runs over the sequences of nonnegative integers with only finitely many nonzero terms, $c_\\alpha \\in \\mathbb{Q}$ and $x^\\alpha = x_1^{\\alpha_1} x_2^{\\alpha_2} \\cdots$.\u2026","title":"Exponential specialization"},{"id":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/symmetric-scalar-product","kind":"node","lean":"MetaMathlibExt.symmetricScalarProduct","parent":27813,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/index.html#symmetric-scalar-product","scope":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux","status":"fully_proved","summary":"Let $n \\ge 0$, let $R$ be a commutative ring, and call a polynomial in $R[x_1, \\dots, x_n]$ symmetric if it is unchanged by every permutation of the variables. Let $a_\\delta = \\prod_{1 \\le i < j \\le n} (x_i - x_j)$, so that $a_\\delta = 1$ when $n \\le 1$. For a sequence $\\lambda = (\\lambda_1, \\dots, \\lambda_n)$ of integers with $\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge \\lambda_n \\ge 0$, write $c_\\lambda(P)$ for the\u2026","title":"Scalar product of symmetric polynomials"},{"id":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/skew-shape","kind":"node","lean":"MetaMathlibExt.SkewShape","parent":27813,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/index.html#skew-shape","scope":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux","status":"fully_proved","summary":"A partition is a finite weakly decreasing sequence $\\lambda = (\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge \\lambda_m)$ of positive integers, possibly empty ($m = 0$); set $\\lambda_k = 0$ for $k > m$. Its diagram is the set $D(\\lambda) = \\{(i, j) \\in \\mathbb{Z}_{\\ge 1} \\times \\mathbb{Z}_{\\ge 1} : j \\le \\lambda_i\\}$ of cells, where the cell $(i, j)$ lies in row $i$ and column $j$, and a partition is identified with its\u2026","title":"Skew shape"},{"id":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/skew-semistandard-young-tableau","kind":"node","lean":"MetaMathlibExt.SkewSemistandardYoungTableau","parent":27813,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/index.html#skew-semistandard-young-tableau","scope":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux","status":"fully_proved","summary":"Let $\\lambda/\\mu$ be a skew shape, and write $\\lambda_k$ and $\\mu_k$ for the $k$-th parts of $\\lambda$ and $\\mu$, with $\\lambda_k = 0$ ($\\mu_k = 0$) when $\\lambda$ ($\\mu$) has fewer than $k$ parts. The cells of $\\lambda/\\mu$ are the pairs $(i, j) \\in \\mathbb{Z}_{\\ge 1} \\times \\mathbb{Z}_{\\ge 1}$ with $\\mu_i < j \\le \\lambda_i$, that is, the cells of $D(\\lambda)$ that are not cells of $D(\\mu)$. A semistandard Young\u2026","title":"Semistandard Young tableau of skew shape"},{"id":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/skew-standard-young-tableau","kind":"node","lean":"MetaMathlibExt.SkewStandardYoungTableau","parent":27813,"result":true,"route":"../roadmap/Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux/index.html#skew-standard-young-tableau","scope":"Combinatorics/symmetric-functions-q-analogues/symmetric-functions-and-skew-young-tableaux","status":"fully_proved","summary":"Let $\\lambda/\\mu$ be a skew shape, let $n$ be the number of cells of $\\lambda/\\mu$ (the cells of $D(\\lambda)$ that are not cells of $D(\\mu)$), and write $[n] = \\{1, \\dots, n\\}$, so that $[0] = \\emptyset$. A standard Young tableau of shape $\\lambda/\\mu$ is a semistandard Young tableau $T$ of shape $\\lambda/\\mu$ such that the map $(i, j) \\mapsto T_{i,j}$ sends the cells of $\\lambda/\\mu$ bijectively onto $[n]$. If\u2026","title":"Standard Young tableau of skew shape"},{"data_url":"scopes/Combinatorics/extremal-set-theory.json","id":"Combinatorics/extremal-set-theory","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/index.html","scope":"Combinatorics/extremal-set-theory","status":"can_state","summary":"Largest or smallest families of finite sets under intersection, union or containment constraints, and hypergraphs (uniform set systems) studied by Tur\u00e1n, colouring, matching and covering parameters; problems about graphs go to GraphTheory.","title":"Extremal set theory and hypergraphs"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families.json","id":"Combinatorics/extremal-set-theory/intersecting-families","kind":"scope","parent":27820,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families","status":"can_state","summary":"Erd\u0151s\u2013Ko\u2013Rado and its variants, Chv\u00e1tal's conjecture, forbidden and singleton intersections, comparable pairs, union-free and union-closed families, shadows (Kruskal\u2013Katona, Macaulay), Harper's isoperimetric inequality, and sunflower bounds.","title":"Intersecting families, Sperner theory and sunflowers"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/macaulay-multiset-shadow-minimal","kind":"node","lean":"MetaMathlibExt.macaulay_multiset_shadow_minimal","parent":27821,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/index.html#macaulay-multiset-shadow-minimal","scope":"Combinatorics/extremal-set-theory/intersecting-families","status":"fully_proved","summary":"Let $k \\ge 0$ and let $A$ be a finite set of monomials of degree $k + 1$ in the variables $x_0, x_1, x_2, \\dots$, given by their exponent vectors: finitely supported $a \\colon \\mathbb{N} \\to \\mathbb{N}$ with $\\sum_j a_j = k + 1$. Suppose that $|A| = \\sum_{i=0}^{k} \\binom{d_i}{i + 1}$ for natural numbers $d_0 < d_1 < \\dots < d_k$. Let the shadow $\\partial A$ be the set of exponent vectors obtained from some\u2026","title":"Macaulay's theorem for monomials"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture.json","id":"Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture","status":"can_prove","title":"Chv\u00e1tal's conjecture"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture/is-hereditary","kind":"node","lean":"MathlibExt.Combinatorics.ChvatalHereditaryIntersectingWanted.IsHereditary","parent":27823,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture/index.html#is-hereditary","scope":"Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture","status":"fully_proved","summary":"A finite family $F$ of finite subsets of $\\alpha$ is hereditary (an independence system) when it contains every subset of each of its members: if $A \\in F$ and $B \\subseteq A$, then $B \\in F$.","title":"Hereditary families"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ChvatalHereditaryIntersectingWanted.conjecture","parent":27823,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture/index.html#conjecture","scope":"Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture","status":"fully_proved","summary":"The statement that for every finite set $X$ and every hereditary family $F$ of subsets of $X$ that contains a nonempty set, there is a point $x \\in X$ such that every intersecting subfamily $G \\subseteq F$ has at most as many members as the star $\\{A \\in F : x \\in A\\}$ of $x$. Here $F$ is hereditary when it contains every subset of each of its members, and $G$ is intersecting when any two distinct members of $G$\u2026","title":"Chv\u00e1tal's conjecture"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ChvatalHereditaryIntersectingWanted.conjecture_holds","parent":27823,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture/index.html#conjecture-holds","scope":"Combinatorics/extremal-set-theory/intersecting-families/chv-tal-s-conjecture","status":"can_prove","summary":"For every finite set $X$ and every hereditary family $F$ of subsets of $X$ (one containing every subset of each of its members) that contains a nonempty set, there is a point $x \\in X$ such that every intersecting subfamily $G \\subseteq F$ (one in which any two distinct members meet) has at most as many members as the star $\\{A \\in F : x \\in A\\}$.","title":"Chv\u00e1tal's conjecture holds"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families.json","id":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","status":"can_prove","title":"Comparable pairs in set families"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/comparable-pairs","kind":"node","lean":"MathlibExt.Combinatorics.ComparablePairsSetFamilyErdosProblem777Wanted.comparablePairs","parent":27827,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/index.html#comparable-pairs","scope":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and a family $F$ of subsets of $\\{0, \\dots, n - 1\\}$, the number of comparable pairs of $F$ is the number of ordered pairs $(A, B)$ of members of $F$ with $A \\subsetneq B$. It is the number of edges of the comparability graph $G_F$ on $F$, in which two distinct members are adjacent when one contains the other.","title":"Number of comparable pairs"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/few-comparable-pairs-below-two-half-powe","kind":"node","lean":"MathlibExt.Combinatorics.ComparablePairsSetFamilyErdosProblem777Wanted.fewComparablePairsBelowTwoHalfPower","parent":27827,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/index.html#few-comparable-pairs-below-two-half-powe","scope":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","status":"fully_proved","summary":"For every real $\\varepsilon > 0$ there is $N$ such that for every $n \\ge N$, every family $F$ of subsets of $\\{0, \\dots, n - 1\\}$ with $|F| \\le (2 - \\varepsilon) 2^{n/2}$ has fewer than $2^n$ comparable pairs, that is, ordered pairs $(A, B)$ of members of $F$ with $A \\subsetneq B$.","title":"Erd\u0151s Problem 777, first question"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/dense-comparability-family-small","kind":"node","lean":"MathlibExt.Combinatorics.ComparablePairsSetFamilyErdosProblem777Wanted.denseComparabilityFamilySmall","parent":27827,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/index.html#dense-comparability-family-small","scope":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","status":"fully_proved","summary":"For every real $c > 0$ there is a real constant $C$ such that for every $n$ and every family $F$ of subsets of $\\{0, \\dots, n - 1\\}$ with at least $c |F|^2$ comparable pairs (ordered pairs $(A, B)$ of members of $F$ with $A \\subsetneq B$), we have $|F| \\le C \\cdot 2^{n/2}$.","title":"Erd\u0151s Problem 777, second question"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/nearly-complete-comparability-family-sma","kind":"node","lean":"MathlibExt.Combinatorics.ComparablePairsSetFamilyErdosProblem777Wanted.nearlyCompleteComparabilityFamilySmall","parent":27827,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/index.html#nearly-complete-comparability-family-sma","scope":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","status":"fully_proved","summary":"For every real $\\varepsilon > 0$ there is a real $\\delta > 0$ such that for every $n$, every family $F$ of subsets of $\\{0, \\dots, n - 1\\}$ with more than $|F|^{2 - \\delta}$ comparable pairs (ordered pairs $(A, B)$ of members of $F$ with $A \\subsetneq B$) satisfies $|F| < (2 + \\varepsilon)^{n/2}$.","title":"Erd\u0151s Problem 777, third question"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/few-comparable-pairs-below-two-half-powe-2","kind":"node","lean":"MathlibExt.Combinatorics.ComparablePairsSetFamilyErdosProblem777Wanted.fewComparablePairsBelowTwoHalfPower_holds","parent":27827,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/index.html#few-comparable-pairs-below-two-half-powe-2","scope":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","status":"can_prove","summary":"For every real $\\varepsilon > 0$ there is $N$ such that for every $n \\ge N$, every family $F$ of subsets of $\\{0, \\dots, n - 1\\}$ with $|F| \\le (2 - \\varepsilon) 2^{n/2}$ has fewer than $2^n$ comparable pairs, that is, ordered pairs $(A, B)$ of members of $F$ with $A \\subsetneq B$.","title":"Affirmative answer to Erd\u0151s Problem 777, first question"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/dense-comparability-family-small-refuted","kind":"node","lean":"MathlibExt.Combinatorics.ComparablePairsSetFamilyErdosProblem777Wanted.denseComparabilityFamilySmall_refuted","parent":27827,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/index.html#dense-comparability-family-small-refuted","scope":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","status":"can_prove","summary":"It is not the case that for every real $c > 0$ there is a real constant $C$ such that for every $n$ and every family $F$ of subsets of $\\{0, \\dots, n - 1\\}$ with at least $c |F|^2$ comparable pairs (ordered pairs $(A, B)$ of members of $F$ with $A \\subsetneq B$), we have $|F| \\le C \\cdot 2^{n/2}$. Equivalently, there is a real $c > 0$ such that for every real $C$ some $n$ and some family $F$ of subsets of\u2026","title":"Negative answer to Erd\u0151s Problem 777, second question"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/nearly-complete-comparability-family-sma-2","kind":"node","lean":"MathlibExt.Combinatorics.ComparablePairsSetFamilyErdosProblem777Wanted.nearlyCompleteComparabilityFamilySmall_holds","parent":27827,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families/index.html#nearly-complete-comparability-family-sma-2","scope":"Combinatorics/extremal-set-theory/intersecting-families/comparable-pairs-in-set-families","status":"can_prove","summary":"For every real $\\varepsilon > 0$ there is a real $\\delta > 0$ such that for every $n$, every family $F$ of subsets of $\\{0, \\dots, n - 1\\}$ with more than $|F|^{2 - \\delta}$ comparable pairs (ordered pairs $(A, B)$ of members of $F$ with $A \\subsetneq B$) satisfies $|F| < (2 + \\varepsilon)^{n/2}$.","title":"Affirmative answer to Erd\u0151s Problem 777, third question"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families.json","id":"Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families","status":"can_prove","title":"Erd\u0151s\u2013Ko\u2013Rado for d-cluster-free families"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families/is-kd-family","kind":"node","lean":"MathlibExt.Combinatorics.ClusterFreeErdosKoRadoOWR1386Problem011Wanted.IsKdFamily","parent":27835,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(K(d)\\)"],["text","-families"]],"source":"$K(d)$-families"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families/index.html#is-kd-family","scope":"Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families","status":"fully_proved","summary":"Let $n$, $k$ and $d$ be natural numbers and let $[n] = \\{0, 1, \\dots, n - 1\\}$. A family $G$ of subsets of $[n]$ is a $K(d)$-family of $k$-subsets when every member of $G$ has exactly $k$ elements and any $d$ distinct members $A_1, \\dots, A_d$ of $G$ with $|A_1 \\cup \\dots \\cup A_d| \\le 2k$ have a common element.","title":"$K(d)$-families"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families/kd-family-erdos-ko-rado","kind":"node","lean":"MathlibExt.Combinatorics.ClusterFreeErdosKoRadoOWR1386Problem011Wanted.kdFamilyErdosKoRado","parent":27835,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mubayi's Erd\u0151s\u2013Ko\u2013Rado conjecture for "],["math","\\(K(d)\\)"],["text","-families"]],"source":"Mubayi's Erd\u0151s\u2013Ko\u2013Rado conjecture for $K(d)$-families"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families/index.html#kd-family-erdos-ko-rado","scope":"Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families","status":"fully_proved","summary":"For all natural numbers $n$, $k$ and $d$ with $3 \\le d \\le k$ and $n \\ge dk/(d - 1)$, every $K(d)$-family $G$ of $k$-subsets of $[n] = \\{0, 1, \\dots, n - 1\\}$ satisfies $|G| \\le \\binom{n-1}{k-1}$, and if $|G| = \\binom{n-1}{k-1}$ then some element of $[n]$ lies in every member of $G$.","title":"Mubayi's Erd\u0151s\u2013Ko\u2013Rado conjecture for $K(d)$-families"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families/kd-family-erdos-ko-rado-holds","kind":"node","lean":"MathlibExt.Combinatorics.ClusterFreeErdosKoRadoOWR1386Problem011Wanted.kdFamilyErdosKoRado_holds","parent":27835,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mubayi's Erd\u0151s\u2013Ko\u2013Rado conjecture for "],["math","\\(K(d)\\)"],["text","-families holds"]],"source":"Mubayi's Erd\u0151s\u2013Ko\u2013Rado conjecture for $K(d)$-families holds"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families/index.html#kd-family-erdos-ko-rado-holds","scope":"Combinatorics/extremal-set-theory/intersecting-families/erd-s-ko-rado-for-d-cluster-free-families","status":"can_prove","summary":"For all natural numbers $n$, $k$ and $d$ with $3 \\le d \\le k$ and $n \\ge dk/(d - 1)$, every $K(d)$-family $G$ of $k$-subsets of $[n] = \\{0, 1, \\dots, n - 1\\}$ satisfies $|G| \\le \\binom{n-1}{k-1}$, and if $|G| = \\binom{n-1}{k-1}$ then some element of $[n]$ lies in every member of $G$.","title":"Mubayi's Erd\u0151s\u2013Ko\u2013Rado conjecture for $K(d)$-families holds"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection.json","id":"Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection","status":"can_prove","title":"Families with no singleton intersection"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection/no-singleton-intersection-bound","kind":"node","lean":"MathlibExt.Combinatorics.NoSingletonIntersectionErdosProblem702Wanted.noSingletonIntersectionBound","parent":27839,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection/index.html#no-singleton-intersection-bound","scope":"Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection","status":"fully_proved","summary":"For every $k \\ge 4$ and every sufficiently large $n$ (depending on $k$), every family of more than $\\binom{n-2}{k-2}$ subsets of $\\{1, \\dots, n\\}$, each with exactly $k$ elements, contains two sets $A$ and $B$ with $|A \\cap B| = 1$.","title":"Erd\u0151s Problem 702 on singleton intersections"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection/no-singleton-intersection-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.NoSingletonIntersectionErdosProblem702Wanted.noSingletonIntersectionBound_holds","parent":27839,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection/index.html#no-singleton-intersection-bound-holds","scope":"Combinatorics/extremal-set-theory/intersecting-families/families-with-no-singleton-intersection","status":"can_prove","summary":"For every $k \\ge 4$ and every sufficiently large $n$ (depending on $k$), every family of more than $\\binom{n-2}{k-2}$ subsets of $\\{1, \\dots, n\\}$, each with exactly $k$ elements, contains two sets $A$ and $B$ with $|A \\cap B| = 1$.","title":"Frankl's theorem on families with no singleton intersection"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems.json","id":"Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems","status":"can_prove","title":"Forbidden intersection theorems"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems/forbidden-intersection-exponential-gap","kind":"node","lean":"MathlibExt.Combinatorics.ForbiddenIntersectionErdosProblem703Wanted.forbiddenIntersectionExponentialGap","parent":27842,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems/index.html#forbidden-intersection-exponential-gap","scope":"Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems","status":"fully_proved","summary":"The proposition that for every $\\varepsilon > 0$ there is $\\delta > 0$ such that for all natural numbers $n$ and $r$ with $\\varepsilon n < r < (1/2 - \\varepsilon) n$, every family $\\mathcal{F}$ of subsets of $\\{1, \\dots, n\\}$ with $|A \\cap B| \\ne r$ for all $A, B \\in \\mathcal{F}$ (including $A = B$) has $|\\mathcal{F}| < (2 - \\delta)^n$. Equivalently, $T(n, r) < (2 - \\delta)^n$, where $T(n, r)$ is the largest size\u2026","title":"Erd\u0151s Problem 703: exponential gap for a forbidden intersection size"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems/forbidden-intersection-exponential-gap-h","kind":"node","lean":"MathlibExt.Combinatorics.ForbiddenIntersectionErdosProblem703Wanted.forbiddenIntersectionExponentialGap_holds","parent":27842,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems/index.html#forbidden-intersection-exponential-gap-h","scope":"Combinatorics/extremal-set-theory/intersecting-families/forbidden-intersection-theorems","status":"can_prove","summary":"For every $\\varepsilon > 0$ there is $\\delta > 0$ such that for all natural numbers $n$ and $r$ with $\\varepsilon n < r < (1/2 - \\varepsilon) n$, every family $\\mathcal{F}$ of subsets of $\\{1, \\dots, n\\}$ with $|A \\cap B| \\ne r$ for all $A, B \\in \\mathcal{F}$ (including $A = B$) has $|\\mathcal{F}| < (2 - \\delta)^n$. This answers the second question of Erd\u0151s Problem 703 in the affirmative.","title":"Frankl\u2013R\u00f6dl forbidden intersection theorem"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality.json","id":"Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality","status":"fully_proved","title":"Harper's isoperimetric inequality"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality/expansion","kind":"node","lean":"Hamming.expansion","parent":27845,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality/index.html#expansion","scope":"Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality","status":"fully_proved","summary":"Let $\\iota$ and $F$ be finite, and view functions $\\iota \\to F$ as words with the Hamming distance. For a finite set $A$ of words and $t \\in \\mathbb{N}$, the $t$-expansion of $A$ is the set of words at Hamming distance at most $t$ from some element of $A$, that is, the union over $a \\in A$ of the closed Hamming balls of radius $t$ about $a$.","title":"Hamming expansion"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality/is-ball","kind":"node","lean":"Hamming.IsBall","parent":27845,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality/index.html#is-ball","scope":"Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality","status":"fully_proved","summary":"Let $\\iota$ and $F$ be finite, and view functions $\\iota \\to F$ as words with the Hamming distance. A set $B$ of words is a Hamming ball if there are a word $c$ and a radius $e \\in \\mathbb{N}$ such that $B$ is the closed Hamming ball of radius $e$ about $c$.","title":"Hamming ball predicate"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality/harper-isoperimetric-inequality","kind":"node","lean":"MathlibExt.Combinatorics.HarperIsoperimetricWanted.harper_isoperimetric_inequality","parent":27845,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality/index.html#harper-isoperimetric-inequality","scope":"Combinatorics/extremal-set-theory/intersecting-families/harper-s-isoperimetric-inequality","status":"fully_proved","summary":"Let $n$ be a natural number. For points $x, y \\in \\{0,1\\}^n$, write $d_H(x, y)$ for the Hamming distance between them, the number of coordinates in which they differ. For a set $S \\subseteq \\{0,1\\}^n$ and a natural number $t$, write $S_t$ for the set of points $x \\in \\{0,1\\}^n$ with $d_H(x, s) \\le t$ for some $s \\in S$. Let $A$ and $B$ be subsets of $\\{0,1\\}^n$ such that $B$ is a closed Hamming ball, that is,\u2026","title":"Harper's isoperimetric inequality on the Hamming cube"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets.json","id":"Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets","status":"can_prove","title":"Intersecting families of half-size sets"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets/two-intersecting-half-sets-bound","kind":"node","lean":"MathlibExt.Combinatorics.TwoIntersectingHalfSetsErdosProblem83Wanted.twoIntersectingHalfSetsBound","parent":27849,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets/index.html#two-intersecting-half-sets-bound","scope":"Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets","status":"fully_proved","summary":"The assertion that, for every natural number $n$, every family $F$ of $2n$-element subsets of $\\{0, \\dots, 4n-1\\}$ such that $|A \\cap B| \\ge 2$ for all $A, B \\in F$ (including $A = B$) satisfies $2|F| + \\binom{2n}{n}^2 \\le \\binom{4n}{2n}$, that is, $|F| \\le \\frac{1}{2}\\left(\\binom{4n}{2n} - \\binom{2n}{n}^2\\right)$.","title":"Erd\u0151s Problem 83"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets/two-intersecting-half-sets-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.TwoIntersectingHalfSetsErdosProblem83Wanted.twoIntersectingHalfSetsBound_holds","parent":27849,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets/index.html#two-intersecting-half-sets-bound-holds","scope":"Combinatorics/extremal-set-theory/intersecting-families/intersecting-families-of-half-size-sets","status":"can_prove","summary":"For every natural number $n$, every family $F$ of $2n$-element subsets of $\\{0, \\dots, 4n-1\\}$ such that $|A \\cap B| \\ge 2$ for all $A, B \\in F$ (including $A = B$) satisfies $2|F| + \\binom{2n}{n}^2 \\le \\binom{4n}{2n}$, that is, $|F| \\le \\frac{1}{2}\\left(\\binom{4n}{2n} - \\binom{2n}{n}^2\\right)$.","title":"Bound for 2-intersecting families of half-size sets (Erd\u0151s Problem 83)"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture.json","id":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture","status":"can_prove","title":"Multicolored sunflower conjecture"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/ternary-vec","kind":"node","lean":"MathlibExt.Combinatorics.MulticoloredSunflowerTernaryOWR12177Problem010Wanted.TernaryVec","parent":27852,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/index.html#ternary-vec","scope":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture","status":"fully_proved","summary":"For a natural number $n$, the vectors of length $n$ over $\\mathbb{Z}/3$, that is, the functions from $\\{0, 1, \\dots, n - 1\\}$ to $\\mathbb{Z}/3$.","title":"Ternary vectors"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/has-multicolored-sunflower","kind":"node","lean":"MathlibExt.Combinatorics.MulticoloredSunflowerTernaryOWR12177Problem010Wanted.HasMulticoloredSunflower","parent":27852,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/index.html#has-multicolored-sunflower","scope":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture","status":"fully_proved","summary":"Let $n$ be a natural number and let $S$ be a finite set of triples $(x, y, z)$ of vectors in $(\\mathbb{Z}/3)^n$. Then $S$ contains a multicolored sunflower if there are triples $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3) \\in S$, not all three equal, with $x_1 + y_2 + z_3 = 0$.","title":"Multicolored sunflower"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/multicolored-sunflower-conjecture","kind":"node","lean":"MathlibExt.Combinatorics.MulticoloredSunflowerTernaryOWR12177Problem010Wanted.multicoloredSunflowerConjecture","parent":27852,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/index.html#multicolored-sunflower-conjecture","scope":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture","status":"fully_proved","summary":"There are a real number $\\varepsilon > 0$ and a natural number $n_0$ such that for every $n > n_0$, every finite set $S$ of triples $(x, y, z)$ of vectors in $(\\mathbb{Z}/3)^n$ with $x + y + z = 0$ for every $(x, y, z) \\in S$ and with $|S| \\ge 3^{(1 - \\varepsilon) n}$ contains a multicolored sunflower: triples $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3) \\in S$, not all three equal, with $x_1 + y_2 + z_3 = 0$.","title":"Multicolored sunflower conjecture"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/multicolored-sunflower-conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.MulticoloredSunflowerTernaryOWR12177Problem010Wanted.multicoloredSunflowerConjecture_holds","parent":27852,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture/index.html#multicolored-sunflower-conjecture-holds","scope":"Combinatorics/extremal-set-theory/intersecting-families/multicolored-sunflower-conjecture","status":"can_prove","summary":"There are a real number $\\varepsilon > 0$ and a natural number $n_0$ such that for every $n > n_0$, every finite set $S$ of triples $(x, y, z)$ of vectors in $(\\mathbb{Z}/3)^n$ with $x + y + z = 0$ for every $(x, y, z) \\in S$ and with $|S| \\ge 3^{(1 - \\varepsilon) n}$ contains a multicolored sunflower: triples $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3) \\in S$, not all three equal, with $x_1 + y_2 + z_3 = 0$.","title":"The multicolored sunflower conjecture holds"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma.json","id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"can_state","title":"Sunflower lemma"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/contains-ksunflower","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem857WeakSunflowerWanted.ContainsKSunflower","parent":27857,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#contains-ksunflower","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers. A family $F$ of subsets of $\\{0, \\dots, n-1\\}$ contains a $k$-sunflower if some subfamily $T \\subseteq F$ with exactly $k$ members satisfies $A \\cap B = C \\cap D$ for all $A, B, C, D \\in T$ with $A \\ne B$ and $C \\ne D$, that is, all pairwise intersections of distinct members of $T$ are equal.","title":"Families containing a sunflower"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/is-ksunflower-free","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem857WeakSunflowerWanted.IsKSunflowerFree","parent":27857,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#is-ksunflower-free","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers. A family $F$ of subsets of $\\{0, \\dots, n-1\\}$ is $k$-sunflower-free if it does not contain a $k$-sunflower.","title":"Sunflower-free families"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem857WeakSunflowerWanted.conjecture","parent":27857,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s\u2013Szemer\u00e9di sunflower conjecture for "],["math","\\(k \\ge 4\\)"]],"source":"Erd\u0151s\u2013Szemer\u00e9di sunflower conjecture for $k \\ge 4$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#conjecture","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"fully_proved","summary":"The proposition asserts that for every natural number $k \\ge 4$ there is a real number $c$ with $0 < c < 2$ such that, for all sufficiently large $n$, every $k$-sunflower-free family of subsets of $\\{0, \\dots, n-1\\}$ has at most $c^n$ members.","title":"Erd\u0151s\u2013Szemer\u00e9di sunflower conjecture for $k \\ge 4$"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem857WeakSunflowerWanted.conjecture_holds","parent":27857,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Erd\u0151s\u2013Szemer\u00e9di sunflower bound for "],["math","\\(k \\ge 4\\)"]],"source":"The Erd\u0151s\u2013Szemer\u00e9di sunflower bound for $k \\ge 4$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#conjecture-holds","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"can_prove","summary":"For every natural number $k \\ge 4$ there is a real number $c$ with $0 < c < 2$ such that, for all sufficiently large $n$, every family of subsets of $\\{0, \\dots, n-1\\}$ that contains no $k$-sunflower has at most $c^n$ members. Here a $k$-sunflower in a family is a choice of $k$ distinct members whose pairwise intersections are all equal.","title":"The Erd\u0151s\u2013Szemer\u00e9di sunflower bound for $k \\ge 4$"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/is-sunflower","kind":"node","lean":"MathlibExt.Combinatorics.ImprovedBoundsSunflowerLemmaWanted.Set.IsSunflower","parent":27857,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#is-sunflower","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"fully_proved","summary":"Let $r \\in \\mathbb{N}$ and let $S$ be a family of subsets of a type $X$. Then $S$ is an $r$-sunflower if $S$ has exactly $r$ members (its cardinality is taken to be $0$ when $S$ is infinite) and any two distinct members $A \\ne B$ of $S$ satisfy $A \\cap B = \\bigcap S$, the intersection of all members of $S$.","title":"Sunflower"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/is-system","kind":"node","lean":"MathlibExt.Combinatorics.ImprovedBoundsSunflowerLemmaWanted.Set.IsSystem","parent":27857,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#is-system","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"fully_proved","summary":"Let $w \\in \\mathbb{N}$ and let $\\mathcal{F}$ be a family of subsets of a type $X$. Then $\\mathcal{F}$ is a $w$-set system if every member $A \\in \\mathcal{F}$ has at most $w$ elements, where the size of an infinite set is taken to be $0$.","title":"Bounded-size set systems"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.ImprovedBoundsSunflowerLemmaWanted.ImprovedBoundsSunflowerLemma.lowerBound","parent":27857,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#lower-bound","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"fully_proved","summary":"For natural numbers $r$ and $w$ and a real number $C$, the lower bound is the real number","title":"Sunflower lower bound"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/c","kind":"node","lean":"MathlibExt.Combinatorics.ImprovedBoundsSunflowerLemmaWanted.ImprovedBoundsSunflowerLemma.C","parent":27857,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#c","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"can_state","summary":"An unspecified real number $C$, the absolute constant in the Alweiss\u2013Lovett\u2013Wu\u2013Zhang sunflower bound (Theorem 1.4); it does not depend on $r$ or $w$.","title":"Sunflower constant"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/theorem-1-4","kind":"node","lean":"MathlibExt.Combinatorics.ImprovedBoundsSunflowerLemmaWanted.ImprovedBoundsSunflowerLemma.theorem_1_4","parent":27857,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma/index.html#theorem-1-4","scope":"Combinatorics/extremal-set-theory/intersecting-families/sunflower-lemma","status":"stated","summary":"Let $C$ be the sunflower constant, a real number not depending on $r$ or $w$. Let $r \\ge 3$ and $w \\ge 2$ be natural numbers, let $X$ be a finite type, and let $\\mathcal{F}$ be a $w$-set system of subsets of $X$, that is, every member of $\\mathcal{F}$ has at most $w$ elements. If $\\mathcal{F}$ has at least","title":"Alweiss\u2013Lovett\u2013Wu\u2013Zhang sunflower bound"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families.json","id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","title":"Union-closed families"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/bin-rep","kind":"node","lean":"MetaMathlibExt.binRep","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#bin-rep","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"For a natural number $n$ and a subset $A$ of $\\{0, 1, \\dots, n-1\\}$, the binary code of $A$ is the natural number $b(A) = \\sum_{i \\in A} 2^i$.","title":"Binary code of a subset"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/set-lt","kind":"node","lean":"MetaMathlibExt.setLT","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#set-lt","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"For subsets $A$ and $B$ of $\\{0, \\dots, n-1\\}$, $A$ precedes $B$ in the set order, written $A < B$, if $|A| > |B|$, or $|A| = |B|$ and $b(A) < b(B)$, where $b(A) = \\sum_{i \\in A} 2^i$ is the binary code of $A$.","title":"Set order"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/set-key","kind":"node","lean":"MetaMathlibExt.setKey","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#set-key","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"For a subset $A$ of $\\{0, \\dots, n-1\\}$, the sort key of $A$ is the natural number $(n - |A|) \\cdot 2^n + b(A)$, where $b(A) = \\sum_{i \\in A} 2^i$ is the binary code of $A$.","title":"Sort key of a subset"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/fam-string","kind":"node","lean":"MetaMathlibExt.famString","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#fam-string","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"For a family $U$ of subsets of $\\{0, \\dots, n-1\\}$, the string $s(U)$ is the list of the sort keys of the nonempty members of $U$, in increasing order.","title":"String of a family"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/perm-image","kind":"node","lean":"MetaMathlibExt.permImage","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#perm-image","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ and a family $U$ of subsets of $\\{0, \\dots, n-1\\}$, the permuted image $\\sigma(U)$ is the family $\\{\\sigma(A) : A \\in U\\}$, where $\\sigma(A) = \\{\\sigma(a) : a \\in A\\}$.","title":"Permuted image of a family"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/is-union-closed","kind":"node","lean":"MetaMathlibExt.IsUnionClosed","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#is-union-closed","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"A family $U$ of subsets of $\\{0, \\dots, n-1\\}$ is union-closed if $A \\cup B \\in U$ for all $A, B \\in U$.","title":"Union-closed family"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/is-canonical","kind":"node","lean":"MetaMathlibExt.IsCanonical","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#is-canonical","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"A family $U$ of subsets of $\\{0, \\dots, n-1\\}$ is canonical, the canonical representative of its isomorphism class, if for every permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ the string $s(U)$ is lexicographically at most the string $s(\\sigma(U))$ of the permuted family $\\sigma(U)$.","title":"Canonical family"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/set-lt-iff-set-key-lt","kind":"node","lean":"MetaMathlibExt.setLT_iff_setKey_lt","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#set-lt-iff-set-key-lt","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"For all subsets $A$ and $B$ of $\\{0, \\dots, n-1\\}$, $A < B$ in the set order if and only if the sort key of $A$ is less than the sort key of $B$.","title":"The sort key encodes the set order"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/orderly-canonical-prefix-of-sorted","kind":"node","lean":"MetaMathlibExt.orderly_canonical_prefix_of_sorted","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#orderly-canonical-prefix-of-sorted","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"Let $A_1, \\dots, A_k$ be nonempty subsets of $\\{0, \\dots, n-1\\}$ with $A_i < A_j$ in the set order whenever $i < j$, and suppose the family $U = \\{\\emptyset, A_1, \\dots, A_k\\}$ is canonical. Then for every $m \\le k$, including $m = 0$, the family $\\{\\emptyset, A_1, \\dots, A_m\\}$ is canonical.","title":"Prefixes of a sorted canonical family are canonical"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/orderly-canonical-prefix","kind":"node","lean":"MetaMathlibExt.orderly_canonical_prefix","parent":27867,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-closed-families/index.html#orderly-canonical-prefix","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-closed-families","status":"fully_proved","summary":"Let $n$, $k$ and $m$ be natural numbers with $1 \\le m \\le k$, and let $A_1, \\dots, A_k$ be nonempty subsets of $\\Omega_n = \\{0, \\dots, n-1\\}$ with $A_i < A_j$ in the set order whenever $i < j$. Suppose the family $U = \\{\\emptyset, A_1, \\dots, A_k\\}$ is union-closed, every element of $\\Omega_n$ lies in some member of $U$, $U \\ne \\{\\Omega_n, \\emptyset\\}$, and $U$ is canonical. Then the prefix family\u2026","title":"Orderly generation of union-closed families"},{"data_url":"scopes/Combinatorics/extremal-set-theory/intersecting-families/union-free-families.json","id":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","kind":"scope","navigation":"module","parent":27821,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-free-families/index.html","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","status":"can_prove","title":"Union-free families"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families/subfamily-union-free","kind":"node","lean":"Finset.SubfamilyUnionFree","parent":27878,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-free-families/index.html#subfamily-union-free","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","status":"fully_proved","summary":"A family $F$ of finite sets is union-free over its subfamilies if no member $A \\in F$ is the union of a nonempty subfamily of $F$ that does not contain $A$.","title":"Families in which no member is a union of other members"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families/is-union-free","kind":"node","lean":"MathlibExt.Combinatorics.UnionFreeFamiliesErdosProblem447Wanted.IsUnionFree","parent":27878,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-free-families/index.html#is-union-free","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","status":"fully_proved","summary":"A family $F$ of subsets of $\\{0, \\dots, n-1\\}$ is union-free if there are no pairwise distinct $A, B, C \\in F$ with $A \\cup B = C$.","title":"Union-free family"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families/max-union-free-size","kind":"node","lean":"MathlibExt.Combinatorics.UnionFreeFamiliesErdosProblem447Wanted.maxUnionFreeSize","parent":27878,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-free-families/index.html#max-union-free-size","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","status":"fully_proved","summary":"For a natural number $n$, the largest size $|F|$ of a union-free family $F$ of subsets of $\\{0, \\dots, n-1\\}$.","title":"Maximum size of a union-free family"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families/max-union-free-size-little-o","kind":"node","lean":"MathlibExt.Combinatorics.UnionFreeFamiliesErdosProblem447Wanted.maxUnionFreeSizeLittleO","parent":27878,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-free-families/index.html#max-union-free-size-little-o","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","status":"fully_proved","summary":"The assertion that the maximum size of a union-free family of subsets of $\\{0, \\dots, n-1\\}$ is $o(2^n)$ as $n \\to \\infty$.","title":"Erd\u0151s Problem 447, first question"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families/max-union-free-size-central-binomial","kind":"node","lean":"MathlibExt.Combinatorics.UnionFreeFamiliesErdosProblem447Wanted.maxUnionFreeSizeCentralBinomial","parent":27878,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-free-families/index.html#max-union-free-size-central-binomial","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","status":"fully_proved","summary":"The assertion that, for every real $\\varepsilon > 0$ and all sufficiently large $n$, the maximum size of a union-free family of subsets of $\\{0, \\dots, n-1\\}$ is at most $(1 + \\varepsilon)\\binom{n}{\\lfloor n/2 \\rfloor}$; that is, it is at most $(1 + o(1))\\binom{n}{\\lfloor n/2 \\rfloor}$.","title":"Erd\u0151s Problem 447, second question"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families/max-union-free-size-little-o-holds","kind":"node","lean":"MathlibExt.Combinatorics.UnionFreeFamiliesErdosProblem447Wanted.maxUnionFreeSizeLittleO_holds","parent":27878,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-free-families/index.html#max-union-free-size-little-o-holds","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","status":"can_prove","summary":"The maximum size of a union-free family of subsets of $\\{0, \\dots, n-1\\}$ is $o(2^n)$ as $n \\to \\infty$.","title":"Union-free families have vanishing density (Erd\u0151s Problem 447)"},{"id":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families/max-union-free-size-central-binomial-hol","kind":"node","lean":"MathlibExt.Combinatorics.UnionFreeFamiliesErdosProblem447Wanted.maxUnionFreeSizeCentralBinomial_holds","parent":27878,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/intersecting-families/union-free-families/index.html#max-union-free-size-central-binomial-hol","scope":"Combinatorics/extremal-set-theory/intersecting-families/union-free-families","status":"can_prove","summary":"For every real $\\varepsilon > 0$ and all sufficiently large $n$, the maximum size of a union-free family of subsets of $\\{0, \\dots, n-1\\}$ is at most $(1 + \\varepsilon)\\binom{n}{\\lfloor n/2 \\rfloor}$; that is, it is at most $(1 + o(1))\\binom{n}{\\lfloor n/2 \\rfloor}$.","title":"Kleitman's bound for union-free families (Erd\u0151s Problem 447)"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-turan.json","id":"Combinatorics/extremal-set-theory/hypergraph-turan","kind":"scope","parent":27820,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-turan","status":"can_prove","summary":"Tur\u00e1n numbers and densities of uniform hypergraphs: Tur\u00e1n systems, de Caen's question, generalized triangles, the (6,3) problem, locally sparse triple systems, and dense 3-uniform hypergraphs.","title":"Hypergraph Tur\u00e1n problems"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs.json","id":"Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs","kind":"scope","navigation":"module","parent":27886,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs","status":"can_prove","title":"Dense 3-uniform hypergraphs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem794Wanted.conjecture","parent":27887,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs/index.html#conjecture","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs","status":"fully_proved","summary":"The proposition asserts that for every natural number $n$ and every family $H$ of $3$-element subsets of $\\{0, 1, \\dots, 3n - 1\\}$ with at least $n^3 + 1$ members, there is a $4$-element set $S$ such that at least $3$ members of $H$ are subsets of $S$, or there is a $5$-element set $S$ such that at least $7$ members of $H$ are subsets of $S$.","title":"Erd\u0151s Problem 794 on dense 3-uniform hypergraphs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem794Wanted.conjecture_refuted","parent":27887,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs/index.html#conjecture-refuted","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/dense-3-uniform-hypergraphs","status":"can_prove","summary":"It is not true that for every natural number $n$ and every family $H$ of $3$-element subsets of $\\{0, 1, \\dots, 3n - 1\\}$ with at least $n^3 + 1$ members, there is a $4$-element set $S$ such that at least $3$ members of $H$ are subsets of $S$, or there is a $5$-element set $S$ such that at least $7$ members of $H$ are subsets of $S$.","title":"Refutation of Erd\u0151s Problem 794"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems.json","id":"Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems","kind":"scope","navigation":"module","parent":27886,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems","status":"can_prove","title":"Locally sparse triple systems"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems/ex-single","kind":"node","lean":"MathlibExt.Combinatorics.LocallySparseTripleSystemErdosProblem1076Wanted.exSingle","parent":27890,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extremal number for "],["math","\\(k\\)"],["text"," vertices spanning "],["math","\\(k - 2\\)"],["text"," triples"]],"source":"Extremal number for $k$ vertices spanning $k - 2$ triples"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems/index.html#ex-single","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems","status":"fully_proved","summary":"For $k, n \\in \\mathbb{N}$ let $\\mathcal{F}_k$ be the family of all $3$-uniform hypergraphs with $k$ vertices and $k - 2$ edges. Then $\\mathrm{ex}_3(n, \\mathcal{F}_k)$ is the largest number of triples in a $3$-uniform hypergraph $E$ on the vertex set $\\{0, \\dots, n-1\\}$ that contains no member of $\\mathcal{F}_k$, meaning that no set of exactly $k$ vertices contains $k - 2$ or more triples of $E$ (a member may have\u2026","title":"Extremal number for $k$ vertices spanning $k - 2$ triples"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems/single-family-asymptotic","kind":"node","lean":"MathlibExt.Combinatorics.LocallySparseTripleSystemErdosProblem1076Wanted.singleFamilyAsymptotic","parent":27890,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems/index.html#single-family-asymptotic","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems","status":"fully_proved","summary":"For $k, n \\in \\mathbb{N}$ let $\\mathrm{ex}_3(n, \\mathcal{F}_k)$ be the largest number of triples in a $3$-uniform hypergraph on $n$ vertices in which no $k$ vertices span $k - 2$ or more triples, that is, which contains no member of the family $\\mathcal{F}_k$ of $3$-graphs with $k$ vertices and $k - 2$ edges. This is the statement that for every $k \\ge 5$, $\\mathrm{ex}_3(n, \\mathcal{F}_k) / n^2 \\to 1/6$ as\u2026","title":"Erd\u0151s Problem 1076 as worded"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems/single-family-asymptotic-refuted","kind":"node","lean":"MathlibExt.Combinatorics.LocallySparseTripleSystemErdosProblem1076Wanted.singleFamilyAsymptotic_refuted","parent":27890,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems/index.html#single-family-asymptotic-refuted","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/locally-sparse-triple-systems","status":"can_prove","summary":"For $k, n \\in \\mathbb{N}$ let $\\mathcal{F}_k$ be the family of $3$-graphs with $k$ vertices and $k - 2$ edges, and let $\\mathrm{ex}_3(n, \\mathcal{F}_k)$ be the largest number of triples in a $3$-uniform hypergraph on $n$ vertices that contains no member of $\\mathcal{F}_k$, that is, in which no $k$ vertices span $k - 2$ or more triples. It is not true that for every $k \\ge 5$,\u2026","title":"Erd\u0151s Problem 1076 fails as worded"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem.json","id":"Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem","kind":"scope","navigation":"module","parent":27886,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem","status":"can_prove","title":"Ruzsa\u2013Szemer\u00e9di (6,3) theorem"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem/six-three-extremal-number","kind":"node","lean":"MathlibExt.Combinatorics.SixPointsThreeTriplesErdosProblem716Wanted.sixThreeExtremalNumber","parent":27894,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem/index.html#six-three-extremal-number","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, $\\mathrm{ex}_3(n)$ is the largest number of $3$-element subsets of $\\{1, \\dots, n\\}$ such that every set of at most $6$ of these points contains at most $2$ of the chosen subsets.","title":"The (6, 3) extremal number"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem/six-three-extremal-number-little-o","kind":"node","lean":"MathlibExt.Combinatorics.SixPointsThreeTriplesErdosProblem716Wanted.sixThreeExtremalNumberLittleO","parent":27894,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem/index.html#six-three-extremal-number-little-o","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ let $\\mathrm{ex}_3(n)$ be the largest number of $3$-element subsets of $\\{1, \\dots, n\\}$ such that every set of at most $6$ of these points contains at most $2$ of the chosen subsets. The conjecture (Erd\u0151s Problem 716) is the proposition that $\\mathrm{ex}_3(n) / n^2 \\to 0$ as $n \\to \\infty$, that is, $\\mathrm{ex}_3(n) = o(n^2)$.","title":"The Brown\u2013Erd\u0151s\u2013S\u00f3s (6, 3) conjecture"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem/six-three-extremal-number-little-o-holds","kind":"node","lean":"MathlibExt.Combinatorics.SixPointsThreeTriplesErdosProblem716Wanted.sixThreeExtremalNumberLittleO_holds","parent":27894,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem/index.html#six-three-extremal-number-little-o-holds","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/ruzsa-szemer-di-6-3-theorem","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $\\mathrm{ex}_3(n)$ be the largest number of $3$-element subsets of $\\{1, \\dots, n\\}$ such that every set of at most $6$ of these points contains at most $2$ of the chosen subsets. Then $\\mathrm{ex}_3(n) / n^2 \\to 0$ as $n \\to \\infty$, that is, $\\mathrm{ex}_3(n) = o(n^2)$.","title":"The Ruzsa\u2013Szemer\u00e9di (6, 3) theorem"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs.json","id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs","kind":"scope","navigation":"module","parent":27886,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs","status":"can_prove","title":"Tur\u00e1n densities of complete hypergraphs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/complete-hypergraph-edges","kind":"node","lean":"MathlibExt.Combinatorics.DeCaenTuranDefectGrowthWanted.completeHypergraphEdges","parent":27898,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/index.html#complete-hypergraph-edges","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs","status":"fully_proved","summary":"For natural numbers $k$ and $m$, the edge family of the complete $k$-uniform hypergraph on the vertex set $\\{0, 1, \\dots, m-1\\}$ is the family of all $k$-element subsets of $\\{0, 1, \\dots, m-1\\}$.","title":"Edges of the complete uniform hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/turan-number","kind":"node","lean":"MathlibExt.Combinatorics.DeCaenTuranDefectGrowthWanted.turanNumber","parent":27898,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tur\u00e1n number of the complete "],["math","\\(k\\)"],["text","-graph on "],["math","\\(k+1\\)"],["text"," vertices"]],"source":"Tur\u00e1n number of the complete $k$-graph on $k+1$ vertices"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/index.html#turan-number","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs","status":"fully_proved","summary":"For natural numbers $k$ and $n$, let $K_{k+1}^{(k)}$ be the complete $k$-uniform hypergraph on the vertex set $\\{0, 1, \\dots, k\\}$, whose edges are all $k$-element subsets of that set. The Tur\u00e1n number $\\mathrm{ex}(n, K_{k+1}^{(k)})$ is the largest size of a family $E$ of $k$-element subsets of $\\{0, 1, \\dots, n-1\\}$ that contains no copy of $K_{k+1}^{(k)}$, that is, for which there is no injective map\u2026","title":"Tur\u00e1n number of the complete $k$-graph on $k+1$ vertices"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/turan-density-ratio","kind":"node","lean":"MathlibExt.Combinatorics.DeCaenTuranDefectGrowthWanted.turanDensityRatio","parent":27898,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/index.html#turan-density-ratio","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs","status":"fully_proved","summary":"For natural numbers $k$ and $n$, the Tur\u00e1n density ratio is the real number $\\mathrm{ex}(n, K_{k+1}^{(k)}) / \\binom{n}{k}$, where $\\mathrm{ex}(n, K_{k+1}^{(k)})$ is the Tur\u00e1n number of the complete $k$-uniform hypergraph $K_{k+1}^{(k)}$ on $k+1$ vertices, the largest number of $k$-element subsets of an $n$-element set containing no copy of $K_{k+1}^{(k)}$. The ratio is defined to be $0$ when $\\binom{n}{k} = 0$,\u2026","title":"Tur\u00e1n density ratio"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/turan-density-complete","kind":"node","lean":"MathlibExt.Combinatorics.DeCaenTuranDefectGrowthWanted.turanDensityComplete","parent":27898,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tur\u00e1n density of the complete "],["math","\\(k\\)"],["text","-graph on "],["math","\\(k+1\\)"],["text"," vertices"]],"source":"Tur\u00e1n density of the complete $k$-graph on $k+1$ vertices"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/index.html#turan-density-complete","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs","status":"fully_proved","summary":"For a natural number $k$, the Tur\u00e1n density $\\pi(K_{k+1}^{(k)})$ of the complete $k$-uniform hypergraph $K_{k+1}^{(k)}$ on $k+1$ vertices is the limit superior, as $n \\to \\infty$, of the Tur\u00e1n density ratios $\\mathrm{ex}(n, K_{k+1}^{(k)}) / \\binom{n}{k}$, where $\\mathrm{ex}(n, K_{k+1}^{(k)})$ is the Tur\u00e1n number of $K_{k+1}^{(k)}$ and the ratio is taken to be $0$ when $\\binom{n}{k} = 0$.","title":"Tur\u00e1n density of the complete $k$-graph on $k+1$ vertices"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.DeCaenTuranDefectGrowthWanted.conjecture","parent":27898,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/index.html#conjecture","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs","status":"fully_proved","summary":"For a natural number $k$, let $\\pi(K_{k+1}^{(k)})$ be the Tur\u00e1n density of the complete $k$-uniform hypergraph on $k+1$ vertices. De Caen's conjecture asserts that $k \\, (1 - \\pi(K_{k+1}^{(k)})) \\to \\infty$ as $k \\to \\infty$.","title":"De Caen's conjecture on Tur\u00e1n densities of complete hypergraphs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.DeCaenTuranDefectGrowthWanted.conjecture_refuted","parent":27898,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs/index.html#conjecture-refuted","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-densities-of-complete-hypergraphs","status":"can_prove","summary":"It is not the case that $k \\, (1 - \\pi(K_{k+1}^{(k)}))$ tends to $\\infty$ as $k \\to \\infty$, where $\\pi(K_{k+1}^{(k)})$ is the Tur\u00e1n density of the complete $k$-uniform hypergraph $K_{k+1}^{(k)}$ on $k+1$ vertices.","title":"De Caen's conjecture is false"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles.json","id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","kind":"scope","navigation":"module","parent":27886,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"can_prove","title":"Tur\u00e1n numbers of generalized triangles"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/contains-generalized-triangle","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.ContainsGeneralizedTriangle","parent":27905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Generalized triangle "],["math","\\(\\mathcal{T}_r\\)"]],"source":"Generalized triangle $\\mathcal{T}_r$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#contains-generalized-triangle","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"fully_proved","summary":"Let $r, n \\in \\mathbb{N}$ and let $G$ be a family of subsets of $\\{0, \\dots, n-1\\}$, typically an $r$-uniform hypergraph. Then $G$ contains a generalized triangle $\\mathcal{T}_r$ (the book $B_{r,2}$) when there are edges $D_1, D_2, D_3 \\in G$, not necessarily distinct, with $|D_1 \\cap D_2| + 1 = r$, $D_1 \\triangle D_2 \\subseteq D_3$ and $D_3 \\cap D_1 \\cap D_2 = \\emptyset$.","title":"Generalized triangle $\\mathcal{T}_r$"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/generalized-triangle-turan-number","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.generalizedTriangleTuranNumber","parent":27905,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#generalized-triangle-turan-number","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"fully_proved","summary":"For $r, n \\in \\mathbb{N}$, the Tur\u00e1n number $\\mathrm{ex}(n, \\mathcal{T}_r)$ is the largest number of edges of an $r$-uniform hypergraph on the vertex set $\\{0, \\dots, n-1\\}$ that contains no generalized triangle $\\mathcal{T}_r$.","title":"Tur\u00e1n number of the generalized triangle"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/is-steiner-system","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.IsSteinerSystem","parent":27905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Steiner system "],["math","\\(S(t, r, m)\\)"]],"source":"Steiner system $S(t, r, m)$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#is-steiner-system","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"fully_proved","summary":"Let $t, r, m \\in \\mathbb{N}$ and let $D$ be a family of subsets of $\\{0, \\dots, m-1\\}$. Then $D$ is a Steiner system $S(t, r, m)$ when every block of $D$ has exactly $r$ elements and every $t$-element subset of $\\{0, \\dots, m-1\\}$ is contained in exactly one block of $D$.","title":"Steiner system $S(t, r, m)$"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/blow-up","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.blowUp","parent":27905,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#blow-up","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"fully_proved","summary":"Let $D$ be a family of subsets of $\\{0, \\dots, m-1\\}$, let $r \\in \\mathbb{N}$ and let $\\varphi : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, m-1\\}$. The $r$-uniform blow-up of $D$ along $\\varphi$ is the family of $r$-element subsets $e \\subseteq \\{0, \\dots, n-1\\}$ such that $\\varphi$ is injective on $e$ and the image $\\varphi(e)$ is a block of $D$.","title":"Blow-up of a hypergraph along a map"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/max-blow-up-size","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.maxBlowUpSize","parent":27905,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#max-blow-up-size","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"fully_proved","summary":"Let $D$ be a family of subsets of $\\{0, \\dots, m-1\\}$ and let $r, n \\in \\mathbb{N}$. The largest blow-up size of $D$ is the maximum, over all maps $\\varphi : \\{0, \\dots, n-1\\} \\to \\{0, \\dots, m-1\\}$, of the number of edges of the $r$-uniform blow-up of $D$ along $\\varphi$; it is $0$ when there is no such map.","title":"Largest blow-up of a hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/generalized-triangle-five-turan-number","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.generalizedTriangleFiveTuranNumber","parent":27905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tur\u00e1n number of "],["math","\\(\\mathcal{T}_5\\)"],["text"," via blow-ups of "],["math","\\(S(4, 5, 11)\\)"]],"source":"Tur\u00e1n number of $\\mathcal{T}_5$ via blow-ups of $S(4, 5, 11)$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#generalized-triangle-five-turan-number","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"fully_proved","summary":"There exist a Steiner system $D$ of type $S(4, 5, 11)$ on $\\{0, \\dots, 10\\}$ and an $n_0 \\in \\mathbb{N}$ such that for every $n \\ge n_0$, the Tur\u00e1n number $\\mathrm{ex}(n, \\mathcal{T}_5)$ of the generalized triangle $\\mathcal{T}_5$ (the book $B_{5,2}$) equals the largest number of edges of a $5$-uniform blow-up of $D$ along a map $\\{0, \\dots, n-1\\} \\to \\{0, \\dots, 10\\}$.","title":"Tur\u00e1n number of $\\mathcal{T}_5$ via blow-ups of $S(4, 5, 11)$"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/generalized-triangle-six-turan-number","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.generalizedTriangleSixTuranNumber","parent":27905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tur\u00e1n number of "],["math","\\(\\mathcal{T}_6\\)"],["text"," via blow-ups of "],["math","\\(S(5, 6, 12)\\)"]],"source":"Tur\u00e1n number of $\\mathcal{T}_6$ via blow-ups of $S(5, 6, 12)$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#generalized-triangle-six-turan-number","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"fully_proved","summary":"There exist a Steiner system $D$ of type $S(5, 6, 12)$ on $\\{0, \\dots, 11\\}$ and an $n_0 \\in \\mathbb{N}$ such that for every $n \\ge n_0$, the Tur\u00e1n number $\\mathrm{ex}(n, \\mathcal{T}_6)$ of the generalized triangle $\\mathcal{T}_6$ (the book $B_{6,2}$) equals the largest number of edges of a $6$-uniform blow-up of $D$ along a map $\\{0, \\dots, n-1\\} \\to \\{0, \\dots, 11\\}$.","title":"Tur\u00e1n number of $\\mathcal{T}_6$ via blow-ups of $S(5, 6, 12)$"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/generalized-triangle-five-turan-number-h","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.generalizedTriangleFiveTuranNumber_holds","parent":27905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norin\u2013Yepremyan theorem for "],["math","\\(\\mathcal{T}_5\\)"]],"source":"Norin\u2013Yepremyan theorem for $\\mathcal{T}_5$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#generalized-triangle-five-turan-number-h","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"can_prove","summary":"There exist a Steiner system $D$ of type $S(4, 5, 11)$ on $\\{0, \\dots, 10\\}$ and an $n_0 \\in \\mathbb{N}$ such that for every $n \\ge n_0$, the Tur\u00e1n number $\\mathrm{ex}(n, \\mathcal{T}_5)$ of the generalized triangle $\\mathcal{T}_5$ (the book $B_{5,2}$) equals the largest number of edges of a $5$-uniform blow-up of $D$ along a map $\\{0, \\dots, n-1\\} \\to \\{0, \\dots, 10\\}$.","title":"Norin\u2013Yepremyan theorem for $\\mathcal{T}_5$"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/generalized-triangle-six-turan-number-ho","kind":"node","lean":"MathlibExt.Combinatorics.GeneralizedTriangleTuranNumberAIMCombinatoricsProblem151Wanted.generalizedTriangleSixTuranNumber_holds","parent":27905,"reading_title":{"format":"inline-parts-v1","parts":[["text","Norin\u2013Yepremyan theorem for "],["math","\\(\\mathcal{T}_6\\)"]],"source":"Norin\u2013Yepremyan theorem for $\\mathcal{T}_6$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles/index.html#generalized-triangle-six-turan-number-ho","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-numbers-of-generalized-triangles","status":"can_prove","summary":"There exist a Steiner system $D$ of type $S(5, 6, 12)$ on $\\{0, \\dots, 11\\}$ and an $n_0 \\in \\mathbb{N}$ such that for every $n \\ge n_0$, the Tur\u00e1n number $\\mathrm{ex}(n, \\mathcal{T}_6)$ of the generalized triangle $\\mathcal{T}_6$ (the book $B_{6,2}$) equals the largest number of edges of a $6$-uniform blow-up of $D$ along a map $\\{0, \\dots, n-1\\} \\to \\{0, \\dots, 11\\}$.","title":"Norin\u2013Yepremyan theorem for $\\mathcal{T}_6$"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems.json","id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","kind":"scope","navigation":"module","parent":27886,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","title":"Tur\u00e1n systems"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/is-turan-system","kind":"node","lean":"TuranSystem.IsTuranSystem","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#is-turan-system","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers. A family $H$ of subsets of $\\{0, 1, \\dots, n-1\\}$ is a Tur\u00e1n $(n, s, r)$-system when $0 < r < s \\le n$, $H$ is $r$-uniform (every member of $H$ has exactly $r$ elements), and every $s$-element subset of $\\{0, 1, \\dots, n-1\\}$ contains some member of $H$.","title":"Tur\u00e1n system"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/pos-of-is-turan","kind":"node","lean":"TuranSystem.pos_of_isTuran","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#pos-of-is-turan","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers and let $H$ be a Tur\u00e1n system with parameters $n$, $s$, $r$, whose definition includes $0 < r < s \\le n$. Then $0 < r$.","title":"Tur\u00e1n systems have positive uniformity"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/lt-of-is-turan","kind":"node","lean":"TuranSystem.lt_of_isTuran","parent":27915,"reading_title":{"format":"inline-parts-v1","parts":[["text","A Tur\u00e1n system has "],["math","\\(r < s\\)"]],"source":"A Tur\u00e1n system has $r < s$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#lt-of-is-turan","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers and let $H$ be a Tur\u00e1n $(n, s, r)$-system. Then $r < s$.","title":"A Tur\u00e1n system has $r < s$"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/le-of-is-turan","kind":"node","lean":"TuranSystem.le_of_isTuran","parent":27915,"reading_title":{"format":"inline-parts-v1","parts":[["text","A Tur\u00e1n system has "],["math","\\(s \\le n\\)"]],"source":"A Tur\u00e1n system has $s \\le n$"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#le-of-is-turan","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers and let $H$ be a Tur\u00e1n $(n, s, r)$-system. Then $s \\le n$.","title":"A Tur\u00e1n system has $s \\le n$"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/uniform-of-is-turan","kind":"node","lean":"TuranSystem.uniform_of_isTuran","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#uniform-of-is-turan","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers. If $H$ is a Tur\u00e1n system with parameters $n$, $s$, $r$, then $H$ is $r$-uniform: every member of $H$ has exactly $r$ elements.","title":"Tur\u00e1n systems are uniform"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/cover-of-is-turan","kind":"node","lean":"TuranSystem.cover_of_isTuran","parent":27915,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tur\u00e1n systems cover every "],["math","\\(s\\)"],["text","-set"]],"source":"Tur\u00e1n systems cover every $s$-set"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#cover-of-is-turan","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers and let $H$ be a Tur\u00e1n $(n, s, r)$-system. Then every $s$-element subset $T$ of $\\{0, 1, \\dots, n-1\\}$ contains some member $e$ of $H$.","title":"Tur\u00e1n systems cover every $s$-set"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/complete-graph","kind":"node","lean":"TuranSystem.completeGraph","parent":27915,"reading_title":{"format":"inline-parts-v1","parts":[["text","The complete "],["math","\\(r\\)"],["text","-graph"]],"source":"The complete $r$-graph"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#complete-graph","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"For natural numbers $n$ and $r$, the complete $r$-graph on $n$ vertices is the family of all $r$-element subsets of $\\{0, 1, \\dots, n-1\\}$.","title":"The complete $r$-graph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/complete-graph-uniform","kind":"node","lean":"TuranSystem.completeGraph_uniform","parent":27915,"reading_title":{"format":"inline-parts-v1","parts":[["text","The complete "],["math","\\(r\\)"],["text","-graph is "],["math","\\(r\\)"],["text","-uniform"]],"source":"The complete $r$-graph is $r$-uniform"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#complete-graph-uniform","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"For all natural numbers $n$ and $r$, every member of the complete $r$-graph on $\\{0, 1, \\dots, n-1\\}$ has exactly $r$ elements.","title":"The complete $r$-graph is $r$-uniform"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/complete-graph-is-turan","kind":"node","lean":"TuranSystem.completeGraph_isTuran","parent":27915,"reading_title":{"format":"inline-parts-v1","parts":[["text","The complete "],["math","\\(r\\)"],["text","-graph is a Tur\u00e1n system"]],"source":"The complete $r$-graph is a Tur\u00e1n system"},"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#complete-graph-is-turan","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers with $0 < r < s \\le n$. Then the complete $r$-graph on $\\{0, 1, \\dots, n-1\\}$ (the family of all its $r$-element subsets) is a Tur\u00e1n $(n, s, r)$-system.","title":"The complete $r$-graph is a Tur\u00e1n system"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/realized-counts","kind":"node","lean":"TuranSystem.realizedCounts","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#realized-counts","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"For natural numbers $n$, $s$, $r$, the set of realized sizes is the set of natural numbers $k$ such that some Tur\u00e1n system with parameters $n$, $s$, $r$ on the vertex set $\\{0, \\dots, n-1\\}$ has exactly $k$ members.","title":"Realized sizes of Tur\u00e1n systems"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/mem-realized-counts","kind":"node","lean":"TuranSystem.mem_realizedCounts","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#mem-realized-counts","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"For natural numbers $n$, $s$, $r$, $k$, the number $k$ lies in the set of realized sizes for $(n, s, r)$ if and only if some Tur\u00e1n $(n, s, r)$-system $H$ has exactly $k$ members.","title":"Membership in the realized sizes"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/exists-system","kind":"node","lean":"TuranSystem.exists_system","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#exists-system","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"For all natural numbers $n$, $s$, $r$ with $0 < r < s \\le n$, there is a Tur\u00e1n $(n, s, r)$-system.","title":"Tur\u00e1n systems exist"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/realized-counts-nonempty","kind":"node","lean":"TuranSystem.realizedCounts_nonempty","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#realized-counts-nonempty","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"For natural numbers $n$, $s$, $r$ with $0 < r < s \\le n$, the set of realized sizes for $n$, $s$, $r$ is nonempty: some natural number $k$ is the number of members of a Tur\u00e1n system with parameters $n$, $s$, $r$.","title":"Some size of a Tur\u00e1n system is realized"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/turan-number","kind":"node","lean":"TuranSystem.turanNumber","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#turan-number","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"For natural numbers $n$, $s$, $r$, the Tur\u00e1n number $T(n, s, r)$ is the least element of the set of realized sizes for $n$, $s$, $r$, that is, the least number of members of a Tur\u00e1n system with parameters $n$, $s$, $r$, and $T(n, s, r) = 0$ if no such system exists.","title":"Tur\u00e1n number"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/turan-number-mem","kind":"node","lean":"TuranSystem.turanNumber_mem","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#turan-number-mem","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers with $0 < r < s \\le n$ and let $T(n, s, r)$ denote the Tur\u00e1n number. Then $T(n, s, r)$ belongs to the set of realized sizes for $n$, $s$, $r$.","title":"The Tur\u00e1n number is a realized size"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/turan-number-attains","kind":"node","lean":"TuranSystem.turanNumber_attains","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#turan-number-attains","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers with $0 < r < s \\le n$ and let $T(n, s, r)$ denote the Tur\u00e1n number. Then some Tur\u00e1n system with parameters $n$, $s$, $r$ has exactly $T(n, s, r)$ members.","title":"The Tur\u00e1n number is attained"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/turan-number-le","kind":"node","lean":"TuranSystem.turanNumber_le","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#turan-number-le","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$ be natural numbers and let $T(n, s, r)$ denote the Tur\u00e1n number. If $H$ is a Tur\u00e1n system with parameters $n$, $s$, $r$, then $T(n, s, r) \\le |H|$.","title":"Tur\u00e1n systems are at least as large as the Tur\u00e1n number"},{"id":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/le-turan-number","kind":"node","lean":"TuranSystem.le_turanNumber","parent":27915,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems/index.html#le-turan-number","scope":"Combinatorics/extremal-set-theory/hypergraph-turan/tur-n-systems","status":"fully_proved","summary":"Let $n$, $s$, $r$, $b$ be natural numbers with $0 < r < s \\le n$. If $b \\le k$ for every $k$ in the set of realized sizes for $(n, s, r)$ (the sizes of Tur\u00e1n $(n, s, r)$-systems), then $b$ is at most the Tur\u00e1n number $T(n, s, r)$.","title":"Lower bounds on the Tur\u00e1n number"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching","kind":"scope","parent":27820,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching","status":"can_prove","summary":"Property B and hypergraph colouring, Erd\u0151s\u2013Faber\u2013Lov\u00e1sz, Kneser hypergraphs, independence in Steiner and linear triple systems, set mappings, strongly maximal matchings, minimal covers, transversals, Baranyai's theorem and common elements of small set families.","title":"Hypergraph colouring, matchings and covers"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/baranyai-theorem","kind":"node","lean":"MetaMathlibExt.baranyai_theorem","parent":27934,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/index.html#baranyai-theorem","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching","status":"fully_proved","summary":"Let $n$ and $h$ be natural numbers with $h \\mid n$, and let $V = \\{0, \\dots, n-1\\}$. The $h$-element subsets of $V$, the edges of the complete $h$-uniform hypergraph on $V$, can be partitioned into perfect matchings: there is a finite family $\\mathcal{P}$ of sets of subsets of $V$ such that the union of the members of $\\mathcal{P}$ is exactly the set of $h$-element subsets of $V$, distinct members of $\\mathcal{P}$\u2026","title":"Baranyai's theorem"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/has-property-b","kind":"node","lean":"Finset.HasPropertyB","parent":27934,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/index.html#has-property-b","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching","status":"fully_proved","summary":"A family `F` of finite sets has property B (equivalently, it is `2`-colourable, i.e. its chromatic number is at most `2`) if there is a `2`-colouring of the ground set under which no member of `F` is monochromatic.","title":"Property B for families of finite sets"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/finite-sets-common-element","kind":"node","lean":"MathlibExt.Combinatorics.ScottishBook9Wanted.finiteSetsCommonElement","parent":27934,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/index.html#finite-sets-common-element","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching","status":"can_prove","summary":"Let $n$ be a natural number, let $\\alpha$ be a nonempty set, and let $(E_i)_{i \\in I}$ be a family of finite subsets of $\\alpha$, indexed by an arbitrary set $I$, with $|E_i| \\le n$ for every $i \\in I$. Suppose that for all indices $i_0, i_1, \\dots, i_n \\in I$, not necessarily distinct, the sets $E_{i_0}, E_{i_1}, \\dots, E_{i_n}$ have an element in common. Then some element of $\\alpha$ lies in every $E_i$.","title":"Scottish Book Problem 9 (a common element of small finite sets)"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture","status":"can_prove","title":"Erd\u0151s\u2013Faber\u2013Lov\u00e1sz conjecture"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/hypergraph","kind":"node","lean":"MathlibExt.Combinatorics.ErdosFaberLovaszConjectureWanted.ErdosFaberLovaszConjecture.Hypergraph","parent":27938,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/index.html#hypergraph","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture","status":"fully_proved","summary":"A hypergraph on a type $V$ of vertices is a set of subsets of $V$; its elements are the edges of the hypergraph. Edges may be empty, and no finiteness condition is imposed.","title":"Hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/coloring","kind":"node","lean":"MathlibExt.Combinatorics.ErdosFaberLovaszConjectureWanted.ErdosFaberLovaszConjecture.Hypergraph.Coloring","parent":27938,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/index.html#coloring","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture","status":"fully_proved","summary":"Let $\\mathcal{H}$ be a hypergraph on a vertex type $V$ and let $\\alpha$ be a type. An $\\alpha$-coloring of $\\mathcal{H}$ is a map $c$ from the set of edges of $\\mathcal{H}$ to $\\alpha$ such that $c(e_1) \\ne c(e_2)$ for any two distinct edges $e_1$ and $e_2$ of $\\mathcal{H}$ that have a vertex in common.","title":"Edge coloring of a hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/is-linear","kind":"node","lean":"MathlibExt.Combinatorics.ErdosFaberLovaszConjectureWanted.ErdosFaberLovaszConjecture.Hypergraph.IsLinear","parent":27938,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/index.html#is-linear","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture","status":"fully_proved","summary":"A hypergraph $\\mathcal{H}$ is linear if any two distinct edges of $\\mathcal{H}$ have at most one vertex in common.","title":"Linear hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/chromatic-index","kind":"node","lean":"MathlibExt.Combinatorics.ErdosFaberLovaszConjectureWanted.ErdosFaberLovaszConjecture.Hypergraph.chromaticIndex","parent":27938,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/index.html#chromatic-index","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture","status":"fully_proved","summary":"The chromatic index of a hypergraph $\\mathcal{H}$ is the least natural number $n$ such that $\\mathcal{H}$ has a coloring with colors in $\\{0, 1, \\dots, n-1\\}$, and $0$ if there is no such $n$.","title":"Chromatic index of a hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/theorem-1-1","kind":"node","lean":"MathlibExt.Combinatorics.ErdosFaberLovaszConjectureWanted.ErdosFaberLovaszConjecture.theorem_1_1","parent":27938,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture/index.html#theorem-1-1","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/erd-s-faber-lov-sz-conjecture","status":"can_prove","summary":"For every sufficiently large natural number $n$, every linear hypergraph on the vertex set $\\{0, 1, \\dots, n-1\\}$ has chromatic index at most $n$.","title":"Erd\u0151s\u2013Faber\u2013Lov\u00e1sz conjecture for sufficiently many vertices"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs","status":"can_prove","title":"Hitting sets for partial designs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosBlockDesignHittingSetWanted.conjecture","parent":27944,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs/index.html#conjecture","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs","status":"fully_proved","summary":"The proposition that for every real $c$ with $0 < c < 1$ there is a natural number $C$ such that for every natural number $n$ and every family $\\mathcal{B}$ of subsets of $\\{0, \\dots, n - 1\\}$ in which each member has more than $c\\sqrt{n}$ elements and any two distinct members share at most one element, some set $B \\subseteq \\{0, \\dots, n - 1\\}$ meets every member of $\\mathcal{B}$ in at least $1$ and at most $C$\u2026","title":"Bounded hitting sets for large partial designs (Erd\u0151s Problem 664)"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.ErdosBlockDesignHittingSetWanted.conjecture_refuted","parent":27944,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs/index.html#conjecture-refuted","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hitting-sets-for-partial-designs","status":"can_prove","summary":"It is not true that for every real $c$ with $0 < c < 1$ there is a natural number $C$ such that for every natural number $n$ and every family $\\mathcal{B}$ of subsets of $\\{0, \\dots, n - 1\\}$ in which each member has more than $c\\sqrt{n}$ elements and any two distinct members share at most one element, some set $B \\subseteq \\{0, \\dots, n - 1\\}$ meets every member of $\\mathcal{B}$ in at least $1$ and at most $C$\u2026","title":"Large partial designs need not have uniformly bounded hitting sets"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number","status":"can_prove","title":"Hypergraph chromatic number"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number/hypergraph-colorable","kind":"node","lean":"MathlibExt.Combinatorics.HypergraphColoringShared.HypergraphColorable","parent":27947,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number/index.html#hypergraph-colorable","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number","status":"fully_proved","summary":"A hypergraph $H$ on the vertex set $\\{0, \\dots, n-1\\}$, given as a finite set of edges (sets of vertices), is $k$-colorable when some coloring of the vertices with $k$ colors leaves no edge monochromatic: every edge of $H$ contains two vertices of different colors.","title":"Colorable hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number/chromatic-edge-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.HypergraphChromaticEdgeCountErdosProblem832Wanted.chromaticEdgeLowerBound","parent":27947,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number/index.html#chromatic-edge-lower-bound","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number","status":"fully_proved","summary":"The statement that for every integer $r \\ge 3$ there is a $K$ such that for every $k \\ge K$, every $n$ and every $r$-uniform hypergraph $H$ on the vertex set $\\{0, \\dots, n-1\\}$ with chromatic number $k$ (it is $k$-colorable but not $(k-1)$-colorable), $H$ has at least $\\binom{(r-1)(k-1)+1}{r}$ edges, and if it has exactly that many edges then it is the complete $r$-uniform hypergraph on some set of $(r-1)(k-1)+1$\u2026","title":"Erd\u0151s Problem 832"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number/chromatic-edge-lower-bound-refuted","kind":"node","lean":"MathlibExt.Combinatorics.HypergraphChromaticEdgeCountErdosProblem832Wanted.chromaticEdgeLowerBound_refuted","parent":27947,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number/index.html#chromatic-edge-lower-bound-refuted","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/hypergraph-chromatic-number","status":"can_prove","summary":"There is an integer $r \\ge 3$ such that for infinitely many $k$ there are an $n$ and an $r$-uniform hypergraph $H$ on the vertex set $\\{0, \\dots, n-1\\}$ with chromatic number $k$ (it is $k$-colorable but not $(k-1)$-colorable) that either has fewer than $\\binom{(r-1)(k-1)+1}{r}$ edges, or has exactly that many edges but is not the complete $r$-uniform hypergraph on any set of $(r-1)(k-1)+1$ of its vertices.","title":"Alon's refutation of Erd\u0151s Problem 832"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs","status":"can_prove","title":"Degrees in 3-chromatic hypergraphs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs/three-chromatic-high-degree-vertex","kind":"node","lean":"MathlibExt.Combinatorics.ThreeChromaticHypergraphDegreeErdosProblem833Wanted.threeChromaticHighDegreeVertex","parent":27951,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs/index.html#three-chromatic-high-degree-vertex","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs","status":"fully_proved","summary":"The statement that there is a real $c > 0$ such that for every natural number $r \\ge 2$, every $n$ and every hypergraph $H$ on the vertex set $\\{0, \\dots, n-1\\}$ (a finite set of edges, each a set of exactly $r$ vertices) that is $3$-colorable but not $2$-colorable, some vertex lies in at least $(1 + c)^r$ edges of $H$.","title":"Erd\u0151s Problem 833"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs/three-chromatic-high-degree-vertex-holds","kind":"node","lean":"MathlibExt.Combinatorics.ThreeChromaticHypergraphDegreeErdosProblem833Wanted.threeChromaticHighDegreeVertex_holds","parent":27951,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs/index.html#three-chromatic-high-degree-vertex-holds","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/degrees-in-3-chromatic-hypergraphs","status":"can_prove","summary":"There is a real $c > 0$ such that for every natural number $r \\ge 2$, every $n$ and every hypergraph $H$ on the vertex set $\\{0, \\dots, n-1\\}$ (a finite set of edges, each a set of exactly $r$ vertices) that is $3$-colorable but not $2$-colorable, some vertex lies in at least $(1 + c)^r$ edges of $H$.","title":"Erd\u0151s Problem 833 holds"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems","status":"can_prove","title":"Independence in linear triple systems"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/is-linear-triple-system","kind":"node","lean":"MathlibExt.Combinatorics.LinearTripleSystemIndependenceErdosProblem1024Wanted.IsLinearTripleSystem","parent":27954,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/index.html#is-linear-triple-system","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems","status":"fully_proved","summary":"A finite family $H$ of subsets of $\\{0, \\dots, n-1\\}$ is a linear triple system (a $3$-uniform linear hypergraph, or partial Steiner triple system) when every member of $H$ has exactly three elements and any two distinct members of $H$ share at most one element.","title":"Linear triple systems"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/linear-triple-independence","kind":"node","lean":"MathlibExt.Combinatorics.LinearTripleSystemIndependenceErdosProblem1024Wanted.linearTripleIndependence","parent":27954,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/index.html#linear-triple-independence","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, $f(n)$ is the largest $m \\in \\mathbb{N}$ such that every linear triple system $H$ on the vertex set $\\{0, \\dots, n-1\\}$ has an independent set of size at least $m$: a set $I$ of at least $m$ vertices such that no triple of $H$ is a subset of $I$. The largest such $m$ exists because $m = 0$ qualifies and no such $m$ exceeds $n$.","title":"Independence number of linear triple systems"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/linear-triple-independence-asymp","kind":"node","lean":"MathlibExt.Combinatorics.LinearTripleSystemIndependenceErdosProblem1024Wanted.linearTripleIndependenceAsymp","parent":27954,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/index.html#linear-triple-independence-asymp","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ let $f(n)$ be the independence number of linear triple systems on $n$ vertices: the largest $m$ such that every linear triple system on $n$ vertices has an independent set (a set of vertices containing no triple of the system) of at least $m$ vertices. This is the statement that $f(n) = \\Theta(\\sqrt{n \\log n})$ as $n \\to \\infty$, that is, $f(n) = O(\\sqrt{n \\log n})$ and\u2026","title":"Erd\u0151s Problem 1024: the Phelps\u2013R\u00f6dl asymptotic"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/linear-triple-independence-asymp-holds","kind":"node","lean":"MathlibExt.Combinatorics.LinearTripleSystemIndependenceErdosProblem1024Wanted.linearTripleIndependenceAsymp_holds","parent":27954,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems/index.html#linear-triple-independence-asymp-holds","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independence-in-linear-triple-systems","status":"can_prove","summary":"For $n \\in \\mathbb{N}$ let $f(n)$ be the largest $m$ such that every linear triple system on $\\{0, \\dots, n-1\\}$ (a family of $3$-element sets, any two distinct ones sharing at most one element) has an independent set of at least $m$ vertices, that is, a set of at least $m$ vertices containing no triple of the system. Then $f(n) = \\Theta(\\sqrt{n \\log n})$ as $n \\to \\infty$: there are constants $c, C > 0$ such that\u2026","title":"Phelps\u2013R\u00f6dl theorem on independence in linear triple systems"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","title":"Independent sets in hypergraphs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/is-independent","kind":"node","lean":"SetFamily.IsIndependent","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#is-independent","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"Let $H$ be a finite hypergraph, that is a finite family of finite sets of vertices, called its edges. A finite set $I$ of vertices is independent in $H$ when no edge $e$ of $H$ satisfies $e \\subseteq I$.","title":"Independent set in a hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/is-independent-iff","kind":"node","lean":"SetFamily.isIndependent_iff","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#is-independent-iff","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"A finite set $I$ of vertices is independent in a finite hypergraph $H$ if and only if no edge $e$ of $H$ satisfies $e \\subseteq I$.","title":"Independence unfolded"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/is-independent-empty-hypergraph","kind":"node","lean":"SetFamily.isIndependent_empty_hypergraph","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#is-independent-empty-hypergraph","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"Every finite set of vertices is independent in the empty hypergraph, the hypergraph with no edges.","title":"Every set is independent in the empty hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/is-independent-of-subset","kind":"node","lean":"SetFamily.isIndependent_of_subset","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#is-independent-of-subset","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"If a finite set $I$ of vertices is independent in a finite hypergraph $H$ and $J \\subseteq I$, then $J$ is independent in $H$.","title":"Subsets of independent sets are independent"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/is-independent-empty-iff","kind":"node","lean":"SetFamily.isIndependent_empty_iff","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#is-independent-empty-iff","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"For a finite hypergraph $H$, the empty set is independent in $H$ if and only if every edge of $H$ is nonempty.","title":"When the empty set is independent"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/is-steiner-triple-system","kind":"node","lean":"SetFamily.IsSteinerTripleSystem","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#is-steiner-triple-system","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"A finite hypergraph $H$ on a vertex type $V$, that is a finite family of finite subsets of $V$, is a Steiner triple system when it is $3$-uniform (every edge of $H$ has exactly $3$ elements) and any two distinct vertices $u \\ne v$ of $V$ lie together in exactly one edge of $H$.","title":"Steiner triple system"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/uniform-of-is-steiner-triple-system","kind":"node","lean":"SetFamily.uniform_of_isSteinerTripleSystem","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#uniform-of-is-steiner-triple-system","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"If $H$ is a Steiner triple system, then $H$ is $3$-uniform: every edge of $H$ has exactly $3$ elements.","title":"Steiner triple systems are 3-uniform"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/pair-cover-of-is-steiner-triple-system","kind":"node","lean":"SetFamily.pairCover_of_isSteinerTripleSystem","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#pair-cover-of-is-steiner-triple-system","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"If $H$ is a Steiner triple system on a vertex type $V$, then for all vertices $u \\ne v$ of $V$ there is exactly one edge $e$ of $H$ with $u \\in e$ and $v \\in e$.","title":"Pairs in a Steiner triple system"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/exists-unique-mem-singleton","kind":"node","lean":"SetFamily.existsUnique_mem_singleton","parent":27959,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs/index.html#exists-unique-mem-singleton","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-in-hypergraphs","status":"fully_proved","summary":"Let $t$ be a finite set of vertices and let $P$ be a property of finite sets of vertices. If $t$ satisfies $P$, then there is exactly one finite set $e$ of vertices with $e \\in \\{t\\}$ and $P(e)$.","title":"Unique element of a singleton family"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs","status":"can_prove","title":"Independent sets of set mappings on pairs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/is-independent-for","kind":"node","lean":"MathlibExt.Combinatorics.PairSetMappingIndependenceErdosProblem1025Wanted.IsIndependentFor","parent":27969,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/index.html#is-independent-for","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs","status":"fully_proved","summary":"Let $n$ be a natural number and let $f$ assign to each unordered pair $\\{x, y\\}$ of elements of $\\{0, \\dots, n-1\\}$ (possibly with $x = y$) an element of $\\{0, \\dots, n-1\\}$. A finite set $X \\subseteq \\{0, \\dots, n-1\\}$ is independent for $f$ if $f(\\{x, y\\}) \\notin X$ for all distinct $x, y \\in X$.","title":"Independent set of a set mapping on pairs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/set-mapping-independence","kind":"node","lean":"MathlibExt.Combinatorics.PairSetMappingIndependenceErdosProblem1025Wanted.setMappingIndependence","parent":27969,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/index.html#set-mapping-independence","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs","status":"fully_proved","summary":"For a natural number $n$, $g(n)$ is the largest $k$ such that every map $f$ from the unordered pairs $\\{x, y\\}$ of elements of $\\{0, \\dots, n-1\\}$ (possibly with $x = y$) to $\\{0, \\dots, n-1\\}$ with $f(\\{x, y\\}) \\notin \\{x, y\\}$ for all distinct $x, y$ has an independent set of at least $k$ elements, that is, a set $X \\subseteq \\{0, \\dots, n-1\\}$ with $|X| \\ge k$ and $f(\\{x, y\\}) \\notin X$ for all distinct\u2026","title":"Independence number of set mappings on pairs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/set-mapping-independence-asymp","kind":"node","lean":"MathlibExt.Combinatorics.PairSetMappingIndependenceErdosProblem1025Wanted.setMappingIndependenceAsymp","parent":27969,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/index.html#set-mapping-independence-asymp","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs","status":"fully_proved","summary":"For a natural number $n$, let $g(n)$ be the largest $k$ such that every map $f$ from the unordered pairs of elements of $\\{0, \\dots, n-1\\}$ to $\\{0, \\dots, n-1\\}$ with $f(\\{x, y\\}) \\notin \\{x, y\\}$ for all distinct $x, y$ admits a set $X$ of at least $k$ elements with $f(\\{x, y\\}) \\notin X$ for all distinct $x, y \\in X$ (and $g(n) = 0$ if no largest such $k$ exists). This is the proposition that\u2026","title":"Statement of Erd\u0151s Problem 1025"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/set-mapping-independence-asymp-holds","kind":"node","lean":"MathlibExt.Combinatorics.PairSetMappingIndependenceErdosProblem1025Wanted.setMappingIndependenceAsymp_holds","parent":27969,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs/index.html#set-mapping-independence-asymp-holds","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/independent-sets-of-set-mappings-on-pairs","status":"can_prove","summary":"For a natural number $n$, let $g(n)$ be the largest $k$ such that every map $f$ from the unordered pairs of elements of $\\{0, \\dots, n-1\\}$ to $\\{0, \\dots, n-1\\}$ with $f(\\{x, y\\}) \\notin \\{x, y\\}$ for all distinct $x, y$ admits a set $X$ of at least $k$ elements with $f(\\{x, y\\}) \\notin X$ for all distinct $x, y \\in X$ (and $g(n) = 0$ if no largest such $k$ exists). Then there are constants $c, C > 0$ such that\u2026","title":"Erd\u0151s Problem 1025 (Spencer; Conlon, Fox and Sudakov)"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number","status":"can_prove","title":"Intersecting families with large cover number"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number/is-erdos-lovasz-family","kind":"node","lean":"MathlibExt.Combinatorics.ErdosLovaszCoverFamilyErdosProblem21Wanted.IsErdosLovaszFamily","parent":27974,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number/index.html#is-erdos-lovasz-family","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number","status":"fully_proved","summary":"For a natural number $n$, a finite family $\\mathcal{F}$ of finite sets of natural numbers is an Erd\u0151s\u2013Lov\u00e1sz family of order $n$ when every member of $\\mathcal{F}$ has exactly $n$ elements, any two members of $\\mathcal{F}$ (not necessarily distinct) have a common element, and every set of at most $n - 1$ natural numbers is disjoint from some member of $\\mathcal{F}$, where $n - 1$ is read as $0$ when $n = 0$.","title":"Erd\u0151s\u2013Lov\u00e1sz family"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number/intersecting-family-full-cover-linear-si","kind":"node","lean":"MathlibExt.Combinatorics.ErdosLovaszCoverFamilyErdosProblem21Wanted.intersectingFamilyFullCoverLinearSize","parent":27974,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number/index.html#intersecting-family-full-cover-linear-si","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number","status":"fully_proved","summary":"The statement that there is a natural number $C$ such that for every $n \\ge 1$ there is a finite family $\\mathcal{F}$ of finite sets of natural numbers in which every member has exactly $n$ elements, any two members have a common element, every set of at most $n - 1$ natural numbers is disjoint from some member, and $|\\mathcal{F}| \\le C n$. This expresses $f(n) \\ll n$, where $f(n)$ is the least size of such a family.","title":"Erd\u0151s Problem 21 (Erd\u0151s\u2013Lov\u00e1sz)"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number/intersecting-family-full-cover-linear-si-2","kind":"node","lean":"MathlibExt.Combinatorics.ErdosLovaszCoverFamilyErdosProblem21Wanted.intersectingFamilyFullCoverLinearSize_holds","parent":27974,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number/index.html#intersecting-family-full-cover-linear-si-2","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/intersecting-families-with-large-cover-number","status":"can_prove","summary":"There is a natural number $C$ such that for every $n \\ge 1$ there is a finite family $\\mathcal{F}$ of finite sets of natural numbers with the following properties: every member of $\\mathcal{F}$ has exactly $n$ elements, any two members of $\\mathcal{F}$ have a common element, every set of at most $n - 1$ natural numbers is disjoint from some member of $\\mathcal{F}$, and $|\\mathcal{F}| \\le C n$.","title":"Kahn's theorem on the Erd\u0151s\u2013Lov\u00e1sz problem"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs","status":"can_prove","title":"Kneser hypergraphs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs/kneser-hypergraph-coloring","kind":"node","lean":"MathlibExt.Combinatorics.KneserHypergraphColoringErdosProblem780Wanted.kneserHypergraphColoring","parent":27978,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs/index.html#kneser-hypergraph-coloring","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs","status":"fully_proved","summary":"The statement that for all natural numbers $n, k, r, t$ with $r \\ge 1$, $t \\ge 1$ and $n \\ge kr + (t - 1)(k - 1)$, every coloring of the $r$-element subsets of $\\{0, \\dots, n - 1\\}$ with $t$ colors has a color class containing $k$ pairwise disjoint $r$-element subsets.","title":"Erd\u0151s's Kneser hypergraph problem"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs/kneser-hypergraph-coloring-holds","kind":"node","lean":"MathlibExt.Combinatorics.KneserHypergraphColoringErdosProblem780Wanted.kneserHypergraphColoring_holds","parent":27978,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs/index.html#kneser-hypergraph-coloring-holds","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/kneser-hypergraphs","status":"can_prove","summary":"For all natural numbers $n, k, r, t$ with $r \\ge 1$, $t \\ge 1$ and $n \\ge kr + (t - 1)(k - 1)$, every coloring of the $r$-element subsets of $\\{0, \\dots, n - 1\\}$ with $t$ colors has a color class containing $k$ pairwise disjoint $r$-element subsets.","title":"Alon\u2013Frankl\u2013Lov\u00e1sz theorem on Kneser hypergraphs"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam","status":"can_prove","title":"Sets meeting but not containing members of a family"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam/transversal-non-cover-sets","kind":"node","lean":"MathlibExt.Combinatorics.TransversalNonCoverSetsErdosProblem1027Wanted.transversalNonCoverSets","parent":27981,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam/index.html#transversal-non-cover-sets","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam","status":"fully_proved","summary":"For a finite family $F$ of finite sets of natural numbers with union $X$, this is the finite set of all $B \\subseteq X$ that meet every member of $F$ and contain none of them.","title":"Sets meeting but not containing the members of a family"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam/many-transversal-non-cover-sets","kind":"node","lean":"MathlibExt.Combinatorics.TransversalNonCoverSetsErdosProblem1027Wanted.manyTransversalNonCoverSets","parent":27981,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam/index.html#many-transversal-non-cover-sets","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam","status":"fully_proved","summary":"For every real $c > 0$ there is a real $\\delta > 0$ such that, for all sufficiently large $n$, every finite family $F$ of finite sets of natural numbers, each of size $n$, with $|F| \\le c \\cdot 2^n$ and with union $X$, has at least $\\delta \\cdot 2^{|X|}$ subsets $B \\subseteq X$ that meet every member of $F$ and contain none of them.","title":"Erd\u0151s Problem 1027 (sets meeting but not containing members of a family)"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam/many-transversal-non-cover-sets-holds","kind":"node","lean":"MathlibExt.Combinatorics.TransversalNonCoverSetsErdosProblem1027Wanted.manyTransversalNonCoverSets_holds","parent":27981,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam/index.html#many-transversal-non-cover-sets-holds","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/sets-meeting-but-not-containing-members-of-a-fam","status":"can_prove","summary":"For every real $c > 0$ there is a real $\\delta > 0$ such that, for all sufficiently large $n$, every finite family $F$ of finite sets of natural numbers, each of size $n$, with $|F| \\le c \\cdot 2^n$ and with union $X$, has at least $\\delta \\cdot 2^{|X|}$ subsets $B \\subseteq X$ that meet every member of $F$ and contain none of them.","title":"Erd\u0151s Problem 1027 holds"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"can_prove","title":"Strong matchings and covers in hypergraphs"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/is-matching","kind":"node","lean":"MathlibExt.Combinatorics.StronglyMaximalMatchingAMR29Problem0084Wanted.IsMatching","parent":27985,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html#is-matching","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"fully_proved","summary":"For a hypergraph on a vertex set $V$ whose edges form a set $E$ of finite subsets of $V$, a set $M$ is a matching when $M \\subseteq E$ and any two distinct members of $M$ are disjoint.","title":"Matching in a hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/is-strongly-maximal","kind":"node","lean":"MathlibExt.Combinatorics.StronglyMaximalMatchingAMR29Problem0084Wanted.IsStronglyMaximal","parent":27985,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html#is-strongly-maximal","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"fully_proved","summary":"For a hypergraph with edge set $E$, a set $M$ is a strongly maximal matching (Aharoni) when $M$ is a matching of $E$ and every matching $M'$ of $E$ satisfies $|M' \\setminus M| \\le |M \\setminus M'|$, comparing the cardinalities as cardinal numbers.","title":"Strongly maximal matching"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/bounded-edges-strongly-maximal-matching","kind":"node","lean":"MathlibExt.Combinatorics.StronglyMaximalMatchingAMR29Problem0084Wanted.boundedEdgesStronglyMaximalMatching","parent":27985,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html#bounded-edges-strongly-maximal-matching","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"fully_proved","summary":"The statement that for every vertex set $V$ and every set $E$ of finite subsets of $V$ such that some natural number $k$ bounds the number of elements of every edge in $E$, the hypergraph with edge set $E$ has a strongly maximal matching.","title":"Strongly maximal matchings in hypergraphs with bounded edges"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/bounded-edges-strongly-maximal-matching-2","kind":"node","lean":"MathlibExt.Combinatorics.StronglyMaximalMatchingAMR29Problem0084Wanted.boundedEdgesStronglyMaximalMatching_refuted","parent":27985,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html#bounded-edges-strongly-maximal-matching-2","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"can_prove","summary":"There are a vertex set $V$, a set $E$ of finite subsets of $V$ and a natural number $k$ such that every edge in $E$ has at most $k$ elements, but the hypergraph with edge set $E$ has no strongly maximal matching.","title":"Bounded edge size does not guarantee a strongly maximal matching"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/is-vertex-cover","kind":"node","lean":"MathlibExt.Combinatorics.StronglyMinimalCoverOpenGardenProblem690Wanted.IsVertexCover","parent":27985,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html#is-vertex-cover","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"fully_proved","summary":"Let $E$ be a set of finite subsets of a type $V$, the edges of a hypergraph. A set $X \\subseteq V$ is a (vertex) cover of $E$ if it meets every edge: every $e \\in E$ contains some $v \\in X$.","title":"Vertex cover of a hypergraph"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/is-strongly-minimal-cover","kind":"node","lean":"MathlibExt.Combinatorics.StronglyMinimalCoverOpenGardenProblem690Wanted.IsStronglyMinimalCover","parent":27985,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html#is-strongly-minimal-cover","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"fully_proved","summary":"Let $E$ be a set of finite subsets of a type $V$, the edges of a hypergraph. A set $X \\subseteq V$ is a strongly minimal cover of $E$ (Aharoni) if $X$ is a cover of $E$, that is, it meets every edge in $E$, and $|X \\setminus X'| \\le |X' \\setminus X|$, as cardinals, for every cover $X'$ of $E$.","title":"Strongly minimal cover"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/bounded-edges-strongly-minimal-cover","kind":"node","lean":"MathlibExt.Combinatorics.StronglyMinimalCoverOpenGardenProblem690Wanted.boundedEdgesStronglyMinimalCover","parent":27985,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html#bounded-edges-strongly-minimal-cover","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"fully_proved","summary":"The statement that for every type $V$, every set $E$ of finite subsets of $V$ (the edges of a hypergraph, possibly infinitely many) and every natural number $k$ such that each edge in $E$ is nonempty and has at most $k$ elements, the hypergraph with edge set $E$ has a strongly minimal cover.","title":"Aharoni's strongly minimal cover conjecture"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/bounded-edges-strongly-minimal-cover-ref","kind":"node","lean":"MathlibExt.Combinatorics.StronglyMinimalCoverOpenGardenProblem690Wanted.boundedEdgesStronglyMinimalCover_refuted","parent":27985,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs/index.html#bounded-edges-strongly-minimal-cover-ref","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/strong-matchings-and-covers-in-hypergraphs","status":"can_prove","summary":"There are a type $V$, a set $E$ of finite subsets of $V$ (the edges of a hypergraph) and a natural number $k$ such that every edge in $E$ is nonempty and has at most $k$ elements, but no strongly minimal cover of $E$ exists. Here a cover of $E$ is a set $X \\subseteq V$ that meets every edge in $E$, and a cover $X$ is strongly minimal if $|X \\setminus X'| \\le |X' \\setminus X|$, as cardinals, for every cover $X'$ of\u2026","title":"Aharoni's strongly minimal cover conjecture is false"},{"data_url":"scopes/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems.json","id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems","kind":"scope","navigation":"module","parent":27934,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems/index.html","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems","status":"can_prove","title":"Transversal-critical triple systems"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems/transversal-critical-min-degree-seven","kind":"node","lean":"MathlibExt.Combinatorics.TransversalCriticalTripleSystemDegreeErdosProblem834Wanted.transversalCriticalMinDegreeSeven","parent":27994,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems/index.html#transversal-critical-min-degree-seven","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems","status":"fully_proved","summary":"There exist $n \\in \\mathbb{N}$ and a $3$-uniform hypergraph $H$ on the vertex set $\\{0, \\dots, n-1\\}$ such that some set of at most $3$ vertices meets every edge of $H$, no set of at most $2$ vertices meets every edge of $H$, for every edge $e$ of $H$ some set of at most $2$ vertices meets every edge of $H$ other than $e$, and every vertex lies in at least $7$ edges of $H$.","title":"Erd\u0151s Problem 834 for transversal-critical triple systems"},{"id":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems/transversal-critical-min-degree-seven-re","kind":"node","lean":"MathlibExt.Combinatorics.TransversalCriticalTripleSystemDegreeErdosProblem834Wanted.transversalCriticalMinDegreeSeven_refuted","parent":27994,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems/index.html#transversal-critical-min-degree-seven-re","scope":"Combinatorics/extremal-set-theory/hypergraph-colouring-matching/transversal-critical-triple-systems","status":"can_prove","summary":"There are no $n \\in \\mathbb{N}$ and $3$-uniform hypergraph $H$ on the vertex set $\\{0, \\dots, n-1\\}$ such that some set of at most $3$ vertices meets every edge of $H$, no set of at most $2$ vertices meets every edge of $H$, for every edge $e$ of $H$ some set of at most $2$ vertices meets every edge of $H$ other than $e$, and every vertex lies in at least $7$ edges of $H$.","title":"Erd\u0151s Problem 834 fails for transversal-critical triple systems"},{"data_url":"scopes/Combinatorics/extremal-set-theory/uniform-hypergraphs.json","id":"Combinatorics/extremal-set-theory/uniform-hypergraphs","kind":"scope","navigation":"module","parent":27820,"result":false,"route":"../roadmap/Combinatorics/extremal-set-theory/uniform-hypergraphs/index.html","scope":"Combinatorics/extremal-set-theory/uniform-hypergraphs","status":"fully_proved","title":"Uniform hypergraphs"},{"id":"Combinatorics/extremal-set-theory/uniform-hypergraphs/is-uniform","kind":"node","lean":"SetFamily.IsUniform","parent":27997,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/uniform-hypergraphs/index.html#is-uniform","scope":"Combinatorics/extremal-set-theory/uniform-hypergraphs","status":"fully_proved","summary":"Let $H$ be a finite family of finite subsets of a type $V$, viewed as a hypergraph with edge set $H$, and let $r \\in \\mathbb{N}$. Then $H$ is $r$-uniform when every edge $e \\in H$ has exactly $r$ elements.","title":"Uniform hypergraphs"},{"id":"Combinatorics/extremal-set-theory/uniform-hypergraphs/contains-subgraph","kind":"node","lean":"SetFamily.ContainsSubgraph","parent":27997,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/uniform-hypergraphs/index.html#contains-subgraph","scope":"Combinatorics/extremal-set-theory/uniform-hypergraphs","status":"fully_proved","summary":"Let $H$ be a finite family of finite subsets of a type $V$, viewed as a hypergraph with edge set $H$, and let $m, k \\in \\mathbb{N}$. Then $H$ contains a subgraph with $m$ vertices and at least $k$ edges when there is a set $S$ of exactly $m$ elements of $V$ such that at least $k$ edges of $H$ are subsets of $S$.","title":"Subhypergraph containment"},{"id":"Combinatorics/extremal-set-theory/uniform-hypergraphs/is-uniform-iff","kind":"node","lean":"SetFamily.isUniform_iff","parent":27997,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/uniform-hypergraphs/index.html#is-uniform-iff","scope":"Combinatorics/extremal-set-theory/uniform-hypergraphs","status":"fully_proved","summary":"Let $H$ be a finite family of finite subsets of a type $V$ and let $r \\in \\mathbb{N}$. Then $H$ is $r$-uniform if and only if every member $e \\in H$ has exactly $r$ elements.","title":"Characterization of uniform hypergraphs"},{"id":"Combinatorics/extremal-set-theory/uniform-hypergraphs/contains-subgraph-iff","kind":"node","lean":"SetFamily.containsSubgraph_iff","parent":27997,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/uniform-hypergraphs/index.html#contains-subgraph-iff","scope":"Combinatorics/extremal-set-theory/uniform-hypergraphs","status":"fully_proved","summary":"Let $H$ be a finite family of finite subsets of a type $V$ and let $m, k \\in \\mathbb{N}$. Then $H$ contains a subgraph with $m$ vertices and at least $k$ edges if and only if there is a set $S$ of exactly $m$ elements of $V$ such that at least $k$ members of $H$ are subsets of $S$.","title":"Characterization of subhypergraph containment"},{"id":"Combinatorics/extremal-set-theory/uniform-hypergraphs/is-intersecting","kind":"node","lean":"SetFamily.IsIntersecting","parent":27997,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/uniform-hypergraphs/index.html#is-intersecting","scope":"Combinatorics/extremal-set-theory/uniform-hypergraphs","status":"fully_proved","summary":"Let $F$ be a finite family of finite subsets of a type $\\alpha$. Then $F$ is intersecting when every two distinct members meet: for all $A, B \\in F$ with $A \\neq B$, the intersection $A \\cap B$ is nonempty.","title":"Intersecting families"},{"id":"Combinatorics/extremal-set-theory/uniform-hypergraphs/is-intersecting-iff","kind":"node","lean":"SetFamily.isIntersecting_iff","parent":27997,"result":true,"route":"../roadmap/Combinatorics/extremal-set-theory/uniform-hypergraphs/index.html#is-intersecting-iff","scope":"Combinatorics/extremal-set-theory/uniform-hypergraphs","status":"fully_proved","summary":"Let $F$ be a finite family of finite subsets of a type $\\alpha$. Then $F$ is intersecting if and only if for all $A, B \\in F$ with $A \\neq B$, the intersection $A \\cap B$ is nonempty.","title":"Characterization of intersecting families"},{"data_url":"scopes/Combinatorics/designs-latin-squares.json","id":"Combinatorics/designs-latin-squares","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/index.html","scope":"Combinatorics/designs-latin-squares","status":"can_prove","summary":"Block designs, Steiner systems, pairwise balanced designs and linear spaces, Latin squares and transversals, Hadamard and partial Hadamard matrices, magic squares, 0-1 matrices with prescribed margins (Gale\u2013Ryser), binary matrices up to rotation, and sign tables from determinant-moment expansions; cyclic difference sets go to Sidon sets and difference sets.","title":"Designs, Latin squares and combinatorial matrices"},{"id":"Combinatorics/designs-latin-squares/binary-matrix-rotation-orbit-count","kind":"node","lean":"MetaMathlibExt.binaryMatrixRotationOrbitCount","parent":28004,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/index.html#binary-matrix-rotation-orbit-count","scope":"Combinatorics/designs-latin-squares","status":"fully_proved","summary":"Let $m, n \\ge 1$. For an $m \\times n$ matrix $x$ with entries $x_{r,s} \\in \\{0, 1\\}$, rows indexed by $0 \\le r < m$ and columns by $0 \\le s < n$, put $N(x) = \\sum_{r=0}^{m-1} \\sum_{s=0}^{n-1} x_{r,s} \\, 2^{rn+s}$, and for $0 \\le i < m$, $0 \\le j < n$ let $x^{(i,j)}$ be the rotated matrix with entries $x^{(i,j)}_{r,s} = x_{(r+i) \\bmod m, \\, (s+j) \\bmod n}$. The number of such matrices $x$ with\u2026","title":"Counting binary matrices up to cyclic rotations"},{"id":"Combinatorics/designs-latin-squares/mann-product-criterion-for-orthogonal-la","kind":"node","lean":"MetaMathlibExt.mann_product_criterion_for_orthogonal_latin_squares","parent":28004,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/index.html#mann-product-criterion-for-orthogonal-la","scope":"Combinatorics/designs-latin-squares","status":"fully_proved","summary":"Let $n$ be a natural number, write $[n] = \\{0, \\dots, n-1\\}$, and let $A$ and $B$ be Latin squares of order $n$ with rows, columns and entries indexed by $[n]$: every row and every column of each square is a bijection of $[n]$. For a row index $i$ write $A_i$ and $B_i$ for the permutations $j \\mapsto A_{ij}$ and $j \\mapsto B_{ij}$ of $[n]$. Then the map $(i, j) \\mapsto (A_{ij}, B_{ij})$ is a bijection of\u2026","title":"Mann's product criterion for orthogonal Latin squares"},{"data_url":"scopes/Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced.json","id":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/index.html","scope":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced","status":"can_prove","title":"Block-compatible sequences of pairwise balanced designs"},{"id":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/is-pbd","kind":"node","lean":"MathlibExt.Combinatorics.BlockCompatibleSequenceCountWanted.IsPBD","parent":28007,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/index.html#is-pbd","scope":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced","status":"fully_proved","summary":"For a natural number $n$, a finite list of blocks $B_1, \\ldots, B_m \\subseteq \\{0, \\ldots, n - 1\\}$ is a pairwise balanced design on these $n$ points when the blocks are pairwise distinct, every block has at least two points, and every two distinct points lie in exactly one common block.","title":"Pairwise balanced design"},{"id":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/block-compatible","kind":"node","lean":"MathlibExt.Combinatorics.BlockCompatibleSequenceCountWanted.BlockCompatible","parent":28007,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/index.html#block-compatible","scope":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced","status":"fully_proved","summary":"For a natural number $n$, a finite sequence $(x_1, \\ldots, x_m)$ of natural numbers is block-compatible for $n$ when $x_1 \\ge x_2 \\ge \\cdots \\ge x_m$, every $x_k$ satisfies $2 \\le x_k \\le n$, and there is a pairwise balanced design on the point set $\\{0, \\ldots, n - 1\\}$ whose blocks, listed in order as $B_1, \\ldots, B_m$, have sizes $|B_k| = x_k$ for every $k$.","title":"Block-compatible sequences"},{"id":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.BlockCompatibleSequenceCountWanted.conjecture","parent":28007,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/index.html#conjecture","scope":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced","status":"fully_proved","summary":"The statement that there are a real number $c > 0$ and a natural number $N$ such that for every $n \\ge N$ there is a finite set of at least $\\exp(c \\sqrt{n} \\log n)$ distinct finite sequences of natural numbers, each of which is block-compatible for $n$. Here $\\log$ is the natural logarithm.","title":"Many block-compatible sequences"},{"id":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.BlockCompatibleSequenceCountWanted.conjecture_holds","parent":28007,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced/index.html#conjecture-holds","scope":"Combinatorics/designs-latin-squares/block-compatible-sequences-of-pairwise-balanced","status":"can_prove","summary":"There are a real number $c > 0$ and a natural number $N$ such that for every $n \\ge N$ there is a finite set of at least $\\exp(c \\sqrt{n} \\log n)$ distinct finite sequences of natural numbers, each of which is block-compatible for $n$. Here $\\log$ is the natural logarithm.","title":"Many block-compatible sequences (Erd\u0151s Problem 732)"},{"data_url":"scopes/Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments.json","id":"Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments/index.html","scope":"Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments","status":"fully_proved","title":"Even tables in random determinant moments"},{"id":"Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments/column-weight","kind":"node","lean":"MetaMathlibExt.columnWeight","parent":28012,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments/index.html#column-weight","scope":"Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments","status":"fully_proved","summary":"Let $n$ be a natural number, $M$ a commutative monoid and $f \\colon \\mathbb{N} \\to M$ a function. A column is a finite list of entries from $\\{0, 1, \\dots, n - 1\\}$; for each $j$ in this range let $c_j$ be the number of times $j$ occurs in the column. The weight of the column is the product $\\prod_{j=0}^{n-1} f(c_j)$ in $M$.","title":"Column weight"},{"id":"Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments/even-table","kind":"node","lean":"MetaMathlibExt.EvenTable","parent":28012,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments/index.html#even-table","scope":"Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments","status":"fully_proved","summary":"For natural numbers $k$ and $n$, an even $k \\times n$ table is a $k \\times n$ array of entries from $\\{0, 1, \\dots, n-1\\}$ such that $k$ is even, every row is a permutation of $\\{0, 1, \\dots, n-1\\}$, and in every column each value that occurs there occurs an even number of times.","title":"Even table"},{"id":"Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments/even-table-weight","kind":"node","lean":"MetaMathlibExt.evenTableWeight","parent":28012,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments/index.html#even-table-weight","scope":"Combinatorics/designs-latin-squares/even-tables-in-random-determinant-moments","status":"fully_proved","summary":"Let $M$ be a commutative monoid and $f : \\mathbb{N} \\to M$. The weight of an even $k \\times n$ table $t$ with respect to $f$ is the product of the column weights of its $n$ columns with respect to $f$:","title":"Weight of an even table"},{"data_url":"scopes/Combinatorics/designs-latin-squares/existence-of-steiner-systems.json","id":"Combinatorics/designs-latin-squares/existence-of-steiner-systems","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/existence-of-steiner-systems/index.html","scope":"Combinatorics/designs-latin-squares/existence-of-steiner-systems","status":"can_prove","title":"Existence of Steiner systems"},{"id":"Combinatorics/designs-latin-squares/existence-of-steiner-systems/steiner-systems-exist-for-large-n","kind":"node","lean":"MathlibExt.Combinatorics.SteinerSystemExistenceErdosProblem722Wanted.steinerSystemsExistForLargeN","parent":28016,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/existence-of-steiner-systems/index.html#steiner-systems-exist-for-large-n","scope":"Combinatorics/designs-latin-squares/existence-of-steiner-systems","status":"fully_proved","summary":"The statement that for all natural numbers $r < k$ there is an $N$ such that for every $n \\ge N$ with $\\binom{k-i}{r-i} \\mid \\binom{n-i}{r-i}$ for all $0 \\le i < r$, there is a Steiner system $S(r, k, n)$: a family $F$ of $k$-element subsets of $\\{0, \\dots, n-1\\}$ such that every $r$-element subset of $\\{0, \\dots, n-1\\}$ is contained in exactly one member of $F$ (Erd\u0151s Problem 722).","title":"Existence conjecture for Steiner systems"},{"id":"Combinatorics/designs-latin-squares/existence-of-steiner-systems/steiner-systems-exist-for-large-n-holds","kind":"node","lean":"MathlibExt.Combinatorics.SteinerSystemExistenceErdosProblem722Wanted.steinerSystemsExistForLargeN_holds","parent":28016,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/existence-of-steiner-systems/index.html#steiner-systems-exist-for-large-n-holds","scope":"Combinatorics/designs-latin-squares/existence-of-steiner-systems","status":"can_prove","summary":"For all natural numbers $r < k$ there is an $N$ such that for every $n \\ge N$ with $\\binom{k-i}{r-i} \\mid \\binom{n-i}{r-i}$ for all $0 \\le i < r$, there is a Steiner system $S(r, k, n)$: a family $F$ of $k$-element subsets of $\\{0, \\dots, n-1\\}$ such that every $r$-element subset of $\\{0, \\dots, n-1\\}$ is contained in exactly one member of $F$.","title":"Keevash's existence theorem for Steiner systems"},{"data_url":"scopes/Combinatorics/designs-latin-squares/gale-ryser-theorem.json","id":"Combinatorics/designs-latin-squares/gale-ryser-theorem","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/gale-ryser-theorem/index.html","scope":"Combinatorics/designs-latin-squares/gale-ryser-theorem","status":"fully_proved","title":"Gale\u2013Ryser theorem"},{"id":"Combinatorics/designs-latin-squares/gale-ryser-theorem/prefix-sum","kind":"node","lean":"MathlibExt.Combinatorics.GaleRyserWanted.prefixSum","parent":28019,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/gale-ryser-theorem/index.html#prefix-sum","scope":"Combinatorics/designs-latin-squares/gale-ryser-theorem","status":"fully_proved","summary":"For a sequence $r = (r_0, \\dots, r_{m-1})$ of natural numbers and a natural number $k$, the prefix sum of $r$ of length $k$ is $\\sum_{i < \\min(k, m)} r_i$, the sum of the first $k$ entries of $r$, or of all entries when $k \\ge m$.","title":"Prefix sum"},{"id":"Combinatorics/designs-latin-squares/gale-ryser-theorem/prefix-sum-eq-sum-fin","kind":"node","lean":"MathlibExt.Combinatorics.GaleRyserWanted.prefixSum_eq_sum_fin","parent":28019,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/gale-ryser-theorem/index.html#prefix-sum-eq-sum-fin","scope":"Combinatorics/designs-latin-squares/gale-ryser-theorem","status":"fully_proved","summary":"For a sequence $r = (r_0, \\dots, r_{m-1})$ of natural numbers and a natural number $k \\le m$, the prefix sum of $r$ of length $k$ equals $\\sum_{i=0}^{k-1} r_i$.","title":"Prefix sums within the sequence length"},{"id":"Combinatorics/designs-latin-squares/gale-ryser-theorem/gale-ryser","kind":"node","lean":"MathlibExt.Combinatorics.GaleRyserWanted.gale_ryser","parent":28019,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/gale-ryser-theorem/index.html#gale-ryser","scope":"Combinatorics/designs-latin-squares/gale-ryser-theorem","status":"fully_proved","summary":"Let $r = (r_0, \\dots, r_{m-1})$ and $c = (c_0, \\dots, c_{n-1})$ be nonincreasing sequences of natural numbers. There is an $m \\times n$ matrix with entries in $\\{0, 1\\}$ whose $i$-th row sums to $r_i$ for every $i$ and whose $j$-th column sums to $c_j$ for every $j$ if and only if $\\sum_{i < m} r_i = \\sum_{j < n} c_j$ and, for every $k \\le m$,","title":"Gale\u2013Ryser theorem"},{"data_url":"scopes/Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems.json","id":"Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems/index.html","scope":"Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems","status":"can_prove","title":"High-girth Steiner triple systems"},{"id":"Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems/high-girth-steiner-triple-systems-exist","kind":"node","lean":"MathlibExt.Combinatorics.HighGirthSteinerTripleSystemErdosProblem207Wanted.highGirthSteinerTripleSystemsExist","parent":28023,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems/index.html#high-girth-steiner-triple-systems-exist","scope":"Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems","status":"fully_proved","summary":"The statement that for every integer $g \\ge 2$ there is an $N$ such that for every $n \\ge N$ with $n \\equiv 1$ or $n \\equiv 3 \\pmod 6$ there is a Steiner triple system on the $n$ points $\\{0, \\dots, n-1\\}$ (three-point blocks, every pair of distinct points in exactly one block) in which every set of $j$ distinct blocks, with $2 \\le j \\le g$, covers at least $j + 3$ points.","title":"Erd\u0151s Problem 207"},{"id":"Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems/high-girth-steiner-triple-systems-exist-2","kind":"node","lean":"MathlibExt.Combinatorics.HighGirthSteinerTripleSystemErdosProblem207Wanted.highGirthSteinerTripleSystemsExist_holds","parent":28023,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems/index.html#high-girth-steiner-triple-systems-exist-2","scope":"Combinatorics/designs-latin-squares/high-girth-steiner-triple-systems","status":"can_prove","summary":"For every integer $g \\ge 2$ there is an $N$ such that for every $n \\ge N$ with $n \\equiv 1$ or $n \\equiv 3 \\pmod 6$ there is a Steiner triple system on the $n$ points $\\{0, \\dots, n-1\\}$ in which every set of $j$ distinct blocks, with $2 \\le j \\le g$, covers at least $j + 3$ points.","title":"High-girth Steiner triple systems exist"},{"data_url":"scopes/Combinatorics/designs-latin-squares/line-counts-of-linear-spaces.json","id":"Combinatorics/designs-latin-squares/line-counts-of-linear-spaces","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/line-counts-of-linear-spaces/index.html","scope":"Combinatorics/designs-latin-squares/line-counts-of-linear-spaces","status":"can_prove","title":"Line counts of linear spaces"},{"id":"Combinatorics/designs-latin-squares/line-counts-of-linear-spaces/is-two-design","kind":"node","lean":"MathlibExt.Combinatorics.LinearSpaceLineCountGapErdosProblem903Wanted.IsTwoDesign","parent":28026,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/line-counts-of-linear-spaces/index.html#is-two-design","scope":"Combinatorics/designs-latin-squares/line-counts-of-linear-spaces","status":"fully_proved","summary":"A family $A_1, \\dots, A_t$ of subsets of $\\{1, \\dots, v\\}$ is a $2$-design (also called a pairwise balanced design or a linear space) when every block $A_i$ has at least two points and every pair of distinct points of $\\{1, \\dots, v\\}$ lies in exactly one block.","title":"Linear spaces (2-designs)"},{"id":"Combinatorics/designs-latin-squares/line-counts-of-linear-spaces/line-count-gap-above-projective-plane","kind":"node","lean":"MathlibExt.Combinatorics.LinearSpaceLineCountGapErdosProblem903Wanted.lineCountGapAboveProjectivePlane","parent":28026,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/line-counts-of-linear-spaces/index.html#line-count-gap-above-projective-plane","scope":"Combinatorics/designs-latin-squares/line-counts-of-linear-spaces","status":"fully_proved","summary":"This is the statement that for every prime power $p$, with $n = p^2 + p + 1$, every $2$-design $A_1, \\dots, A_t$ on $\\{1, \\dots, n\\}$ with more than $n$ blocks has at least $n + p$ blocks. Here a $2$-design is a family of subsets in which every block has at least two points and every pair of distinct points lies in exactly one block.","title":"Erd\u0151s Problem 903: line counts just above a projective plane"},{"id":"Combinatorics/designs-latin-squares/line-counts-of-linear-spaces/line-count-gap-above-projective-plane-ho","kind":"node","lean":"MathlibExt.Combinatorics.LinearSpaceLineCountGapErdosProblem903Wanted.lineCountGapAboveProjectivePlane_holds","parent":28026,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/line-counts-of-linear-spaces/index.html#line-count-gap-above-projective-plane-ho","scope":"Combinatorics/designs-latin-squares/line-counts-of-linear-spaces","status":"can_prove","summary":"Let $p$ be a prime power and $n = p^2 + p + 1$. Let $A_1, \\dots, A_t$ be a $2$-design on $\\{1, \\dots, n\\}$: every block $A_i$ has at least two points and every pair of distinct points lies in exactly one block. If $t > n$, then $t \\ge n + p$.","title":"Erd\u0151s\u2013Fowler\u2013S\u00f3s\u2013Wilson gap above a projective plane (Erd\u0151s Problem 903)"},{"data_url":"scopes/Combinatorics/designs-latin-squares/magic-squares.json","id":"Combinatorics/designs-latin-squares/magic-squares","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/magic-squares/index.html","scope":"Combinatorics/designs-latin-squares/magic-squares","status":"can_prove","title":"Magic squares"},{"id":"Combinatorics/designs-latin-squares/magic-squares/ramanujan-part1-ch1-entry1-magicsquare3","kind":"node","lean":"MathlibExt.NumberTheory.Ramanujan.Part1Ch1.Entry1Magicsquare3.ramanujan_part1_ch1_entry1_magicsquare3","parent":28030,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/magic-squares/index.html#ramanujan-part1-ch1-entry1-magicsquare3","scope":"Combinatorics/designs-latin-squares/magic-squares","status":"fully_proved","summary":"Let $n \\ge 1$, let $a, b : \\{0, \\dots, n-1\\} \\to \\mathbb{N}$, and let $L$ and $R$ be $n \\times n$ arrays with entries $L_{ij}, R_{ij} \\in \\{0, \\dots, n-1\\}$ (rows $i$ and columns $j$ indexed by $\\{0, \\dots, n-1\\}$) such that, in each of $L$ and $R$, every row, every column, the main diagonal (entries $(i, i)$) and the antidiagonal (entries $(i, n-1-i)$) contains each value exactly once. Suppose also that\u2026","title":"Equal line sums from two orthogonal diagonal Latin squares"},{"id":"Combinatorics/designs-latin-squares/magic-squares/is-magic-square","kind":"node","lean":"MetaMathlibExt.IsMagicSquare","parent":28030,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/magic-squares/index.html#is-magic-square","scope":"Combinatorics/designs-latin-squares/magic-squares","status":"fully_proved","summary":"An $n \\times n$ array $M$ of natural numbers, with rows and columns indexed by $0, 1, \\dots, n - 1$, is a *magic square* when there is a natural number $s$ such that every row of $M$ and every column of $M$ sums to $s$, and also $\\sum_i M_{i,i} = s$ (the main diagonal) and $\\sum_i M_{i,n-1-i} = s$ (the anti-diagonal).","title":"Magic squares"},{"id":"Combinatorics/designs-latin-squares/magic-squares/is-reduced-magic-square","kind":"node","lean":"MetaMathlibExt.IsReducedMagicSquare","parent":28030,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/magic-squares/index.html#is-reduced-magic-square","scope":"Combinatorics/designs-latin-squares/magic-squares","status":"fully_proved","summary":"An $n \\times n$ array $M$ of natural numbers is a *reduced magic square of order $n$* when it is a magic square and its entries are $0, 1, \\dots, n^2 - 1$, each exactly once; equivalently, every entry of $M$ is less than $n^2$ and distinct cells of $M$ hold distinct entries.","title":"Reduced magic squares"},{"id":"Combinatorics/designs-latin-squares/magic-squares/is-reduced-magic-square-lo-shu","kind":"node","lean":"MetaMathlibExt.isReducedMagicSquare_loShu","parent":28030,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/magic-squares/index.html#is-reduced-magic-square-lo-shu","scope":"Combinatorics/designs-latin-squares/magic-squares","status":"fully_proved","summary":"The $3 \\times 3$ square with rows $(7, 0, 5)$, $(2, 4, 6)$ and $(3, 8, 1)$, a form of the Lo Shu square with every entry reduced by $1$, is a reduced magic square of order $3$; its magic sum is $12$.","title":"The reduced Lo Shu square"},{"id":"Combinatorics/designs-latin-squares/magic-squares/not-is-reduced-magic-square-zero-one-two","kind":"node","lean":"MetaMathlibExt.not_isReducedMagicSquare_zero_one_two_three","parent":28030,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/magic-squares/index.html#not-is-reduced-magic-square-zero-one-two","scope":"Combinatorics/designs-latin-squares/magic-squares","status":"fully_proved","summary":"The $2 \\times 2$ square with rows $(0, 1)$ and $(2, 3)$ is not a reduced magic square.","title":"A 2\u00d72 square that is not a reduced magic square"},{"id":"Combinatorics/designs-latin-squares/magic-squares/is-reduced-magic-square-extended-strache","kind":"node","lean":"MetaMathlibExt.isReducedMagicSquare_extendedStrachey","parent":28030,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/magic-squares/index.html#is-reduced-magic-square-extended-strache","scope":"Combinatorics/designs-latin-squares/magic-squares","status":"can_prove","summary":"Let $m$ be a natural number, let $q = (2m+1)^2$, and let $A_0$, $A_1$, $B_1$, $B_2$ be $(2m+1) \\times (2m+1)$ arrays of natural numbers. Suppose that every entry of $A_0$ is $0$ or $3$, every row of $A_0$ has exactly $m+1$ zeros, and the main diagonal and the anti-diagonal of $A_0$ each have exactly $m+1$ threes; that every entry of $A_1$ is $1$ or $2$, every row of $A_1$ has exactly $m+2$ ones, and the main\u2026","title":"Extended Strachey construction"},{"id":"Combinatorics/designs-latin-squares/magic-squares/is-reduced-magic-square-ref-symmetric-st","kind":"node","lean":"MetaMathlibExt.isReducedMagicSquare_refSymmetric_strachey","parent":28030,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/magic-squares/index.html#is-reduced-magic-square-ref-symmetric-st","scope":"Combinatorics/designs-latin-squares/magic-squares","status":"can_prove","summary":"Let $m$ be a natural number, let $q = (2m+1)^2$, and let $A_0$, $A_1$ and $M$ be $(2m+1) \\times (2m+1)$ arrays of natural numbers, with rows and columns indexed by $0, 1, \\dots, 2m$; write $\\bar\\imath = 2m - i$ and $\\bar\\jmath = 2m - j$. Suppose that every entry of $A_0$ is $0$ or $3$, every row of $A_0$ has exactly $m+1$ zeros, and the main diagonal and the anti-diagonal of $A_0$ each have exactly $m+1$ threes;\u2026","title":"Ref-symmetric squares from the Strachey construction"},{"data_url":"scopes/Combinatorics/designs-latin-squares/partial-hadamard-matrices.json","id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"can_prove","title":"Partial Hadamard matrices"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/sign-val","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamard.signVal","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#sign-val","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"fully_proved","summary":"The sign of a bit $b \\in \\{0, 1\\}$ is the integer $\\sigma(b)$ given by $\\sigma(0) = -1$ and $\\sigma(1) = 1$.","title":"Sign of a bit"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/is-partial-hadamard","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamard.IsPartialHadamard","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#is-partial-hadamard","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"fully_proved","summary":"Read a bit $b \\in \\{0, 1\\}$ as the sign $\\sigma(b)$, where $\\sigma(0) = -1$ and $\\sigma(1) = 1$. An $n \\times t$ matrix $M$ with entries in $\\{0, 1\\}$ is a partial Hadamard matrix when its rows are pairwise orthogonal as sign vectors: for all distinct rows $i \\ne i'$, $\\sum_{j < t} \\sigma(M_{ij}) \\sigma(M_{i'j}) = 0$.","title":"Partial Hadamard matrices"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/partial-hadamard-count","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamard.partialHadamardCount","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#partial-hadamard-count","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"fully_proved","summary":"For natural numbers $n$ and $t$, $N_{n,t}$ is the number of $n \\times t$ partial Hadamard matrices, that is, of $n \\times t$ matrices with entries in $\\{0, 1\\}$ whose rows are pairwise orthogonal when each entry $0$ is read as the sign $-1$ and each entry $1$ as the sign $1$.","title":"Number of partial Hadamard matrices"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/partial-hadamard-count-zero-width","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamard.partialHadamardCount_zero_width","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#partial-hadamard-count-zero-width","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"fully_proved","summary":"For every natural number $n$, the number $N_{n,0}$ of $n \\times 0$ partial Hadamard matrices is $1$: the orthogonality conditions are empty sums, so the empty matrix is the only one.","title":"Width-zero partial Hadamard count"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/leading-term","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamard.leadingTerm","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#leading-term","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"fully_proved","summary":"For natural numbers $n$ and $t$, the de Launey\u2013Levin leading term is the real number $A_{n,4t} = 2^{4nt + (n-1)^2} (8\\pi t)^{-n(n-1)/4}$, where the power of $8\\pi t$ is the real power function and $n - 1$ in the exponent of $2$ is truncated subtraction of natural numbers, so that $(n-1)^2 = 0$ when $n = 0$.","title":"De Launey\u2013Levin leading term"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/is-leading-admissible","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamard.IsLeadingAdmissible","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#is-leading-admissible","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"fully_proved","summary":"For natural numbers $n$ and $m$, let $N_{n,m}$ be the number of $n \\times m$ partial Hadamard matrices, and for natural numbers $n$ and $t$ let $A_{n,4t} = 2^{4nt + (n-1)^2} (8\\pi t)^{-n(n-1)/4}$ be the de Launey\u2013Levin leading term. A real number $\\alpha$ is admissible for the leading term when, for all sequences $(n_k)$ and $(t_k)$ of natural numbers with $n_k \\ge 2$ for every $k$ and\u2026","title":"Admissible exponents for the partial Hadamard leading term"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/leading-term-exponent","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamard.leadingTermExponent","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#leading-term-exponent","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"fully_proved","summary":"For natural numbers $n$ and $m$, let $N_{n,m}$ be the number of $n \\times m$ partial Hadamard matrices, and for natural numbers $n$ and $t$ let $A_{n,4t} = 2^{4nt + (n-1)^2} (8\\pi t)^{-n(n-1)/4}$ be the de Launey\u2013Levin leading term. Call a real number $\\alpha$ admissible when $N_{n_k, 4t_k} / A_{n_k, 4t_k} \\to 1$ along all sequences $(n_k)$ and $(t_k)$ of natural numbers with $n_k \\ge 2$ for every $k$ and\u2026","title":"Leading-term counting exponent for partial Hadamard matrices"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/davis-cubic-admissible-wanted","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamardWanted.davisCubicAdmissible_wanted","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#davis-cubic-admissible-wanted","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"can_prove","summary":"The exponent $3$ is admissible for the partial Hadamard leading term. That is, let $N_{n,m}$ be the number of $n \\times m$ partial Hadamard matrices and let $A_{n,4t} = 2^{4nt + (n-1)^2} (8\\pi t)^{-n(n-1)/4}$ be the de Launey\u2013Levin leading term. Then for all sequences $(n_k)$ and $(t_k)$ of natural numbers with $n_k \\ge 2$ for every $k$ and $t_k / n_k^3 \\to \\infty$ as $k \\to \\infty$, the ratio\u2026","title":"Partial Hadamard counts follow the leading term above the cubic scale"},{"id":"Combinatorics/designs-latin-squares/partial-hadamard-matrices/davis-cubic-sharp-wanted","kind":"node","lean":"MathlibExt.Combinatorics.PartialHadamardWanted.davisCubicSharp_wanted","parent":28038,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/partial-hadamard-matrices/index.html#davis-cubic-sharp-wanted","scope":"Combinatorics/designs-latin-squares/partial-hadamard-matrices","status":"can_prove","summary":"Let $N_{n,m}$ be the number of $n \\times m$ partial Hadamard matrices and let $A_{n,4t} = 2^{4nt + (n-1)^2} (8\\pi t)^{-n(n-1)/4}$ be the de Launey\u2013Levin leading term. Then for every real number $\\alpha < 3$ there are sequences $(n_k)$ and $(t_k)$ of natural numbers with $n_k \\ge 2$ for every $k$ and $t_k / n_k^{\\alpha} \\to \\infty$ as $k \\to \\infty$ along which the ratio $N_{n_k, 4t_k} / A_{n_k, 4t_k}$ does not tend\u2026","title":"No exponent below 3 is admissible for the partial Hadamard leading term"},{"data_url":"scopes/Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals.json","id":"Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals","kind":"scope","navigation":"module","parent":28004,"result":false,"route":"../roadmap/Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals/index.html","scope":"Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals","status":"can_prove","title":"Snevily's conjecture on Latin transversals"},{"id":"Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.SnevilyTransversalWanted.conjecture","parent":28048,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals/index.html#conjecture","scope":"Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals","status":"fully_proved","summary":"The statement that for every finite abelian group $G$ of odd order, written additively, and all subsets $A, B \\subseteq G$ with $|A| = |B|$, there is a bijection $f \\colon A \\to B$ such that the sums $a + f(a)$, for $a \\in A$, are pairwise distinct.","title":"Snevily's conjecture"},{"id":"Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.SnevilyTransversalWanted.conjecture_holds","parent":28048,"result":true,"route":"../roadmap/Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals/index.html#conjecture-holds","scope":"Combinatorics/designs-latin-squares/snevily-s-conjecture-on-latin-transversals","status":"can_prove","summary":"Let $G$ be a finite abelian group of odd order, written additively, and let $A, B \\subseteq G$ be subsets with $|A| = |B|$. Then there is a bijection $f \\colon A \\to B$ such that the sums $a + f(a)$, for $a \\in A$, are pairwise distinct.","title":"Snevily's conjecture is true"},{"data_url":"scopes/Combinatorics/matroid-theory.json","id":"Combinatorics/matroid-theory","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/index.html","scope":"Combinatorics/matroid-theory","status":"can_prove","summary":"Matroids and their invariants: algebraic matroids, Edmonds' intersection and partition theorems, Rado's theorem, characteristic polynomials and log-concavity, the Mason conjectures, Lorentzian polynomials, and lines in weighted matroids.","title":"Matroid theory"},{"data_url":"scopes/Combinatorics/matroid-theory/algebraic-matroids.json","id":"Combinatorics/matroid-theory/algebraic-matroids","kind":"scope","navigation":"module","parent":28051,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/algebraic-matroids/index.html","scope":"Combinatorics/matroid-theory/algebraic-matroids","status":"fully_proved","title":"Algebraic matroids"},{"id":"Combinatorics/matroid-theory/algebraic-matroids/algebraic-representation","kind":"node","lean":"Matroid.AlgebraicRepresentation","parent":28052,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/algebraic-matroids/index.html#algebraic-representation","scope":"Combinatorics/matroid-theory/algebraic-matroids","status":"fully_proved","summary":"Let $K$ be a field and $M$ a matroid on a type $E$. An algebraic representation of $M$ over $K$ consists of a field extension $L$ of $K$ and a map $\\varphi \\colon E \\to L$ such that $L$ is generated over $K$ by the image under $\\varphi$ of the ground set of $M$, and a set $I$ is independent in $M$ exactly when $I$ is contained in the ground set of $M$, $\\varphi$ is injective on $I$, and $\\varphi(I)$ is\u2026","title":"Algebraic representation of a matroid"},{"id":"Combinatorics/matroid-theory/algebraic-matroids/is-algebraic-over","kind":"node","lean":"Matroid.IsAlgebraicOver","parent":28052,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/algebraic-matroids/index.html#is-algebraic-over","scope":"Combinatorics/matroid-theory/algebraic-matroids","status":"fully_proved","summary":"Let $K$ be a field. A matroid $M$ on a type $E$ is algebraic over $K$ if it has an algebraic representation over $K$ whose extension field lies in the larger of the universes of $K$ and $E$.","title":"Algebraic matroid over a field"},{"id":"Combinatorics/matroid-theory/algebraic-matroids/algebraic-representation-some-field","kind":"node","lean":"Matroid.AlgebraicRepresentationSomeField","parent":28052,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/algebraic-matroids/index.html#algebraic-representation-some-field","scope":"Combinatorics/matroid-theory/algebraic-matroids","status":"fully_proved","summary":"Let $M$ be a matroid on a type $E$. An algebraic representation of $M$ over some field consists of a field $K$, a field extension $L$ of $K$ and a map $\\varphi \\colon E \\to L$ such that $L$ is generated over $K$ by the image under $\\varphi$ of the ground set of $M$, and a set $I$ is independent in $M$ exactly when $I$ is contained in the ground set of $M$, $\\varphi$ is injective on $I$, and $\\varphi(I)$ is\u2026","title":"Algebraic representation of a matroid over some field"},{"id":"Combinatorics/matroid-theory/algebraic-matroids/is-algebraic","kind":"node","lean":"Matroid.IsAlgebraic","parent":28052,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/algebraic-matroids/index.html#is-algebraic","scope":"Combinatorics/matroid-theory/algebraic-matroids","status":"fully_proved","summary":"A matroid $M$ on a type $E$ is algebraic if it has an algebraic representation over some field whose base field and extension field lie in the same universe as $E$.","title":"Algebraic matroid"},{"id":"Combinatorics/matroid-theory/algebraic-matroids/to-is-algebraic-over","kind":"node","lean":"Matroid.AlgebraicRepresentation.toIsAlgebraicOver","parent":28052,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/algebraic-matroids/index.html#to-is-algebraic-over","scope":"Combinatorics/matroid-theory/algebraic-matroids","status":"fully_proved","summary":"Let $K$ be a field and $M$ a matroid on a type $E$. If $M$ has an algebraic representation over $K$ whose extension field lies in the larger of the universes of $K$ and $E$, then $M$ is algebraic over $K$.","title":"Algebraic representations witness algebraicity over a field"},{"id":"Combinatorics/matroid-theory/algebraic-matroids/to-is-algebraic","kind":"node","lean":"Matroid.AlgebraicRepresentationSomeField.toIsAlgebraic","parent":28052,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/algebraic-matroids/index.html#to-is-algebraic","scope":"Combinatorics/matroid-theory/algebraic-matroids","status":"fully_proved","summary":"If a matroid $M$ on a type $E$ has an algebraic representation over some field whose base field and extension field lie in the same universe as $E$, then $M$ is algebraic.","title":"Representations over some field witness algebraicity"},{"data_url":"scopes/Combinatorics/matroid-theory/heron-rota-welsh-conjecture.json","id":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture","kind":"scope","navigation":"module","parent":28051,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/heron-rota-welsh-conjecture/index.html","scope":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture","status":"can_prove","title":"Heron\u2013Rota\u2013Welsh conjecture"},{"id":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture/fin-rank","kind":"node","lean":"MathlibExt.Combinatorics.MatroidCharacteristicPolynomialLogConcaveAMR156Problem0026Wanted.finRank","parent":28059,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/heron-rota-welsh-conjecture/index.html#fin-rank","scope":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture","status":"fully_proved","summary":"For a matroid $M$ and a set $X$, this is the rank $r(X)$ of $X$ in $M$, the size of a largest independent set of $M$ contained in $X$, taken as a natural number; it is $0$ when this rank is infinite.","title":"Rank of a set in a matroid, as a natural number"},{"id":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture/char-poly-coeff","kind":"node","lean":"MathlibExt.Combinatorics.MatroidCharacteristicPolynomialLogConcaveAMR156Problem0026Wanted.charPolyCoeff","parent":28059,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/heron-rota-welsh-conjecture/index.html#char-poly-coeff","scope":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture","status":"fully_proved","summary":"Let $M$ be a finite matroid on the ground set $E$ with rank function $r$, and let $\\chi_M(\\lambda) = \\sum_{I \\subseteq E} (-1)^{|I|} \\lambda^{r(E) - r(I)}$ be its characteristic polynomial. For a natural number $j$, this is the integer coefficient of $\\lambda^j$ in $\\chi_M$, that is, the sum of $(-1)^{|I|}$ over the subsets $I \\subseteq E$ with $r(E) - r(I) = j$.","title":"Coefficients of the characteristic polynomial of a matroid"},{"id":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture/char-poly-abs-coeff","kind":"node","lean":"MathlibExt.Combinatorics.MatroidCharacteristicPolynomialLogConcaveAMR156Problem0026Wanted.charPolyAbsCoeff","parent":28059,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/heron-rota-welsh-conjecture/index.html#char-poly-abs-coeff","scope":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture","status":"fully_proved","summary":"Let $M$ be a finite matroid on the ground set $E$ with rank function $r$ and characteristic polynomial $\\chi_M(\\lambda) = \\sum_{I \\subseteq E} (-1)^{|I|} \\lambda^{r(E) - r(I)}$. For a natural number $k$, $w_k(M)$ is the absolute value of the coefficient of $\\lambda^{r(E) - k}$ in $\\chi_M$ when $k \\le r(E)$, and $w_k(M) = 0$ when $k > r(E)$.","title":"Absolute coefficients of the characteristic polynomial of a matroid"},{"id":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture/characteristic-polynomial-log-concave","kind":"node","lean":"MathlibExt.Combinatorics.MatroidCharacteristicPolynomialLogConcaveAMR156Problem0026Wanted.characteristicPolynomialLogConcave","parent":28059,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/heron-rota-welsh-conjecture/index.html#characteristic-polynomial-log-concave","scope":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture","status":"fully_proved","summary":"For every finite matroid $M$ on a ground set $E$ with rank function $r$, let $w_k(M)$ be the absolute value of the coefficient of $\\lambda^{r(E) - k}$ in the characteristic polynomial $\\chi_M(\\lambda) = \\sum_{I \\subseteq E} (-1)^{|I|} \\lambda^{r(E) - r(I)}$ when $k \\le r(E)$, and $w_k(M) = 0$ when $k > r(E)$. Then $w_{k-1}(M) \\, w_{k+1}(M) \\le w_k(M)^2$ for every natural number $k \\ge 1$.","title":"Log-concavity of the characteristic polynomial of a matroid, statement"},{"id":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture/characteristic-polynomial-log-concave-ho","kind":"node","lean":"MathlibExt.Combinatorics.MatroidCharacteristicPolynomialLogConcaveAMR156Problem0026Wanted.characteristicPolynomialLogConcave_holds","parent":28059,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/heron-rota-welsh-conjecture/index.html#characteristic-polynomial-log-concave-ho","scope":"Combinatorics/matroid-theory/heron-rota-welsh-conjecture","status":"can_prove","summary":"For every finite matroid $M$ on a ground set $E$ with rank function $r$, let $w_k(M)$ be the absolute value of the coefficient of $\\lambda^{r(E) - k}$ in the characteristic polynomial $\\chi_M(\\lambda) = \\sum_{I \\subseteq E} (-1)^{|I|} \\lambda^{r(E) - r(I)}$ when $k \\le r(E)$, and $w_k(M) = 0$ when $k > r(E)$. Then $w_{k-1}(M) \\, w_{k+1}(M) \\le w_k(M)^2$ for every natural number $k \\ge 1$.","title":"Adiprasito\u2013Huh\u2013Katz theorem (Heron\u2013Rota\u2013Welsh conjecture)"},{"data_url":"scopes/Combinatorics/matroid-theory/lorentzian-polynomials.json","id":"Combinatorics/matroid-theory/lorentzian-polynomials","kind":"scope","navigation":"module","parent":28051,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"can_prove","title":"Lorentzian polynomials"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/h","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.H","parent":28065,"reading_title":{"format":"inline-parts-v1","parts":[["text","Homogeneous polynomials of degree "],["math","\\(d\\)"]],"source":"Homogeneous polynomials of degree $d$"},"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#h","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"For natural numbers $n$ and $d$, $H^d_n$ denotes the space of homogeneous polynomials of degree $d$ in $\\mathbb{R}[w_1, \\dots, w_n]$, including the zero polynomial.","title":"Homogeneous polynomials of degree $d$"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/p","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.P","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#p","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"$P^d_n$ is the set of homogeneous polynomials $f$ of degree $d$ in $\\mathbb{R}[w_1, \\dots, w_n]$ such that the coefficient in $f$ of every monomial of degree $d$ is strictly positive.","title":"Homogeneous polynomials with positive coefficients"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/hessian","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.H.hessian","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#hessian","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"For a homogeneous polynomial $f$ of degree $2$ in $\\mathbb{R}[w_1, \\dots, w_n]$, its Hessian is the real $n \\times n$ matrix $\\mathcal{H}_f$ of second partial derivatives of $f$: the diagonal entry $(i, i)$ is twice the coefficient of $w_i^2$ in $f$, and for $i \\ne j$ the entry $(i, j)$ is the coefficient of $w_i w_j$ in $f$.","title":"Hessian of a quadratic form"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/untitled","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.\u013f","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#untitled","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"For natural numbers $n$ and $d$, the set $\\mathring{L}^d_n$ of strictly Lorentzian polynomials of degree $d$ in $\\mathbb{R}[w_1, \\dots, w_n]$ is defined by recursion on $d$. Let $P^d_n$ be the set of homogeneous polynomials of degree $d$ in which every monomial of degree $d$ has a strictly positive coefficient, and let $\\partial_i$ be the partial derivative with respect to $w_i$. Then $\\mathring{L}^0_n = P^0_n$ and\u2026","title":"Strictly Lorentzian polynomials"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/is-mconvex","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.IsMConvex","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#is-mconvex","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"A subset $J \\subseteq \\mathbb{N}^n$ is M-convex if for all $\\alpha, \\beta \\in J$ and every index $i$ with $\\alpha_i > \\beta_i$ there is an index $j$ with $\\alpha_j < \\beta_j$ and $\\alpha - e_i + e_j \\in J$, where $e_1, \\dots, e_n$ are the standard unit vectors of $\\mathbb{N}^n$.","title":"M-convex sets"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/m","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.M","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#m","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"$M^d_n$ is the set of homogeneous polynomials $f$ of degree $d$ in $\\mathbb{R}[w_1, \\dots, w_n]$ whose coefficients are all nonnegative and whose support, the set of exponent vectors $\\alpha \\in \\mathbb{N}^n$ of the monomials with nonzero coefficient in $f$, is M-convex.","title":"Homogeneous polynomials with M-convex support"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/is-stable","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.IsStable","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#is-stable","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"A polynomial $f \\in \\mathbb{R}[w_1, \\dots, w_n]$ is stable if either $f$ is the zero polynomial or $f(z_1, \\dots, z_n) \\ne 0$ for all $z_1, \\dots, z_n$ in the open upper half plane $\\mathbb{H} = \\{z \\in \\mathbb{C} : \\operatorname{Im} z > 0\\}$.","title":"Stable polynomials"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/s","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.S","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#s","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"$S^d_n$ is the set of homogeneous polynomials $f$ of degree $d$ in $\\mathbb{R}[w_1, \\dots, w_n]$ whose coefficients are all nonnegative and that are stable: either $f$ is the zero polynomial or $f$ has no zero in $\\mathbb{H}^n$, where $\\mathbb{H}$ is the open upper half plane in $\\mathbb{C}$.","title":"Stable homogeneous polynomials with nonnegative coefficients"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/l","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.L","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#l","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"fully_proved","summary":"For natural numbers $n$ and $d$, the set $L^d_n$ of Lorentzian polynomials is a set of homogeneous polynomials of degree $d$ in $\\mathbb{R}[w_1, \\dots, w_n]$, defined by recursion on $d$. For $d \\le 2$, $L^d_n = S^d_n$, the set of homogeneous stable polynomials of degree $d$ with nonnegative coefficients. For $d \\ge 3$, $L^d_n$ is the set of $f \\in M^d_n$ (the homogeneous polynomials of degree $d$ with nonnegative\u2026","title":"The set of Lorentzian polynomials"},{"id":"Combinatorics/matroid-theory/lorentzian-polynomials/theorem-2-25","kind":"node","lean":"MathlibExt.Combinatorics.LorentzianPolynomialsWanted.LorentzianPolynomials.theorem_2_25","parent":28065,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/lorentzian-polynomials/index.html#theorem-2-25","scope":"Combinatorics/matroid-theory/lorentzian-polynomials","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $d$, the closure of the set $\\mathring{L}^d_n$ of strictly Lorentzian polynomials of degree $d$ in $n$ variables, taken in the space $H^d_n$ of homogeneous polynomials of degree $d$ in $\\mathbb{R}[w_1, \\dots, w_n]$ with the Euclidean topology on coefficients, is the set $L^d_n$ of Lorentzian polynomials of degree $d$ in $n$ variables.","title":"Lorentzian polynomials as the closure of strictly Lorentzian polynomials"},{"data_url":"scopes/Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets.json","id":"Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets","kind":"scope","navigation":"module","parent":28051,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets/index.html","scope":"Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets","status":"can_prove","title":"Mason's conjectures on independent sets"},{"id":"Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets/independent-set-count","kind":"node","lean":"MathlibExt.Combinatorics.StrongMasonWanted.Combinatorics.StrongMason.independentSetCount","parent":28076,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets/index.html#independent-set-count","scope":"Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets","status":"fully_proved","summary":"For a finite matroid $M$ and a natural number $k$, this is the number of independent sets of $M$ with exactly $k$ elements.","title":"Number of independent sets of a given size"},{"id":"Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets/strong-mason-conjecture","kind":"node","lean":"MathlibExt.Combinatorics.StrongMasonWanted.Combinatorics.StrongMason.strong_mason_conjecture","parent":28076,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets/index.html#strong-mason-conjecture","scope":"Combinatorics/matroid-theory/mason-s-conjectures-on-independent-sets","status":"can_prove","summary":"Let $M$ be a finite matroid whose ground set has $n$ elements, and for each $j$ let $I_j$ be the number of independent sets of $M$ with $j$ elements. Then for every natural number $k$ with $0 < k < n$,","title":"The strong Mason conjecture"},{"data_url":"scopes/Combinatorics/matroid-theory/matroid-intersection-theorem.json","id":"Combinatorics/matroid-theory/matroid-intersection-theorem","kind":"scope","navigation":"module","parent":28051,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/matroid-intersection-theorem/index.html","scope":"Combinatorics/matroid-theory/matroid-intersection-theorem","status":"fully_proved","title":"Matroid intersection theorem"},{"id":"Combinatorics/matroid-theory/matroid-intersection-theorem/exists-common-indep-of-cut-le","kind":"node","lean":"MathlibExt.Combinatorics.Matroid.MatroidIntersectionWanted.exists_common_indep_of_cut_le","parent":28079,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/matroid-intersection-theorem/index.html#exists-common-indep-of-cut-le","scope":"Combinatorics/matroid-theory/matroid-intersection-theorem","status":"fully_proved","summary":"Let $E$ be a finite subset of a type $\\alpha$, let $k$ be a natural number, and let $r_1, r_2$ be functions from the finite subsets of $\\alpha$ to $\\mathbb{N}$ such that, for $i = 1, 2$ and all finite $X, Y \\subseteq \\alpha$, we have $r_i(X) \\le |X|$, $r_i(X) \\le r_i(Y)$ whenever $X \\subseteq Y$, and $r_i(X \\cap Y) + r_i(X \\cup Y) \\le r_i(X) + r_i(Y)$. If every $X \\subseteq E$ satisfies\u2026","title":"Matroid intersection for abstract rank functions"},{"id":"Combinatorics/matroid-theory/matroid-intersection-theorem/edmonds-matroid-intersection","kind":"node","lean":"MathlibExt.Combinatorics.Matroid.MatroidIntersectionWanted.edmonds_matroid_intersection'","parent":28079,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/matroid-intersection-theorem/index.html#edmonds-matroid-intersection","scope":"Combinatorics/matroid-theory/matroid-intersection-theorem","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be matroids on a finite type $\\alpha$, with arbitrary ground sets, and let $k$ be a natural number. There is a set $I \\subseteq \\alpha$ independent in both $M_1$ and $M_2$ with $|I| \\ge k$ if and only if every $X \\subseteq \\alpha$ satisfies $k \\le r_1(X) + r_2(\\alpha \\setminus X)$, where $r_i(Y)$ is the largest size of a subset of $Y$ independent in $M_i$.","title":"Edmonds' matroid intersection theorem"},{"id":"Combinatorics/matroid-theory/matroid-intersection-theorem/edmonds-matroid-intersection-2","kind":"node","lean":"MathlibExt.Combinatorics.Matroid.MatroidIntersectionWanted.edmonds_matroid_intersection","parent":28079,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/matroid-intersection-theorem/index.html#edmonds-matroid-intersection-2","scope":"Combinatorics/matroid-theory/matroid-intersection-theorem","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be matroids on a finite type $\\alpha$, each with ground set all of $\\alpha$, and let $k$ be a natural number. There is a set $I \\subseteq \\alpha$ independent in both $M_1$ and $M_2$ with $|I| \\ge k$ if and only if every $X \\subseteq \\alpha$ satisfies $k \\le r_1(X) + r_2(\\alpha \\setminus X)$, where $r_i$ is the rank function of $M_i$.","title":"Edmonds' matroid intersection theorem for matroids on the whole type"},{"data_url":"scopes/Combinatorics/matroid-theory/matroid-partition-theorem.json","id":"Combinatorics/matroid-theory/matroid-partition-theorem","kind":"scope","navigation":"module","parent":28051,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/matroid-partition-theorem/index.html","scope":"Combinatorics/matroid-theory/matroid-partition-theorem","status":"fully_proved","title":"Matroid partition theorem"},{"id":"Combinatorics/matroid-theory/matroid-partition-theorem/edmonds-matroid-partition","kind":"node","lean":"MathlibExt.Combinatorics.Matroid.MatroidPartitionWanted.edmonds_matroid_partition'","parent":28083,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/matroid-partition-theorem/index.html#edmonds-matroid-partition","scope":"Combinatorics/matroid-theory/matroid-partition-theorem","status":"fully_proved","summary":"Let $E$ be a finite set, let $M$ be a matroid with ground set $E(M) \\subseteq E$, let $S \\subseteq E$ and let $k$ be a natural number. Then $S = J_1 \\cup \\dots \\cup J_k$ for some sets $J_1, \\dots, J_k$ that are independent in $M$ (not necessarily disjoint) if and only if $|X| \\le k \\, r_M(X)$ for every $X \\subseteq S$, where $r_M(X)$ denotes the rank in $M$ of $X \\cap E(M)$.","title":"Edmonds' matroid partition theorem, arbitrary ground set"},{"id":"Combinatorics/matroid-theory/matroid-partition-theorem/edmonds-matroid-partition-2","kind":"node","lean":"MathlibExt.Combinatorics.Matroid.MatroidPartitionWanted.edmonds_matroid_partition","parent":28083,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/matroid-partition-theorem/index.html#edmonds-matroid-partition-2","scope":"Combinatorics/matroid-theory/matroid-partition-theorem","status":"fully_proved","summary":"Let $E$ be a finite set, let $M$ be a matroid whose ground set is all of $E$, let $S \\subseteq E$ and let $k$ be a natural number. Then $S = J_1 \\cup \\dots \\cup J_k$ for some sets $J_1, \\dots, J_k$ that are independent in $M$ (not necessarily disjoint) if and only if $|X| \\le k \\, r_M(X)$ for every $X \\subseteq S$, where $r_M$ is the rank function of $M$.","title":"Edmonds' matroid partition theorem"},{"data_url":"scopes/Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids.json","id":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids","kind":"scope","navigation":"module","parent":28051,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/index.html","scope":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids","status":"can_prove","title":"Non-negative lines in weighted matroids"},{"id":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/is-simple-matroid","kind":"node","lean":"MathlibExt.Combinatorics.WeightedMatroidNonnegativeLinesConnectedWanted.IsSimpleMatroid","parent":28086,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/index.html#is-simple-matroid","scope":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids","status":"fully_proved","summary":"A matroid $M$ on elements of $\\{0, \\dots, n-1\\}$ is simple when the set $\\{e, f\\}$ is independent in $M$ for all $e, f \\in \\{0, \\dots, n-1\\}$, so that every set of at most two of these elements is independent (no loops, no parallel pairs).","title":"Simple matroid"},{"id":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/is-line","kind":"node","lean":"MathlibExt.Combinatorics.WeightedMatroidNonnegativeLinesConnectedWanted.IsLine","parent":28086,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/index.html#is-line","scope":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids","status":"fully_proved","summary":"A set $L$ of points is a line of a matroid $M$ on elements of $\\{0, \\dots, n-1\\}$ when $L$ is a flat of $M$ of rank $2$.","title":"Line of a matroid"},{"id":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/weight","kind":"node","lean":"MathlibExt.Combinatorics.WeightedMatroidNonnegativeLinesConnectedWanted.weight","parent":28086,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/index.html#weight","scope":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids","status":"fully_proved","summary":"Given real weights $a_0, \\dots, a_{n-1}$ on the points $0, \\dots, n-1$, the weight of a set $S$ of points is $\\sum_{i \\in S} a_i$.","title":"Weight of a set of points"},{"id":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/nonneg-graph","kind":"node","lean":"MathlibExt.Combinatorics.WeightedMatroidNonnegativeLinesConnectedWanted.nonnegGraph","parent":28086,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/index.html#nonneg-graph","scope":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids","status":"fully_proved","summary":"Let $M$ be a matroid on elements of $\\{0, \\dots, n-1\\}$ and let $a_0, \\dots, a_{n-1}$ be real weights, the weight of a set of points being the sum of the weights of its points. The graph of non-negative lines has as vertices the points $i$ with $a_i \\ge 0$, and two distinct vertices are adjacent when some line of $M$ (a flat of rank $2$) contains both and has non-negative weight.","title":"Graph of non-negative lines"},{"id":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.WeightedMatroidNonnegativeLinesConnectedWanted.conjecture","parent":28086,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/index.html#conjecture","scope":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids","status":"fully_proved","summary":"This is the statement that for every $n$, every simple matroid $M$ of rank $3$ on the ground set $\\{0, \\dots, n-1\\}$ (every set of at most two points is independent) and all real weights $a_0, \\dots, a_{n-1}$ with $a_0 + \\dots + a_{n-1} \\ge 0$, the graph on the points $i$ with $a_i \\ge 0$, in which two distinct points are adjacent when some line of $M$ (a flat of rank $2$) contains both and has non-negative weight,\u2026","title":"Connectivity question for weighted rank-3 matroids"},{"id":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.WeightedMatroidNonnegativeLinesConnectedWanted.conjecture_holds","parent":28086,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids/index.html#conjecture-holds","scope":"Combinatorics/matroid-theory/non-negative-lines-in-weighted-matroids","status":"can_prove","summary":"Let $n$ be a natural number, let $M$ be a simple matroid of rank $3$ on the ground set $\\{0, \\dots, n-1\\}$ (every set of at most two points is independent) and let $a_0, \\dots, a_{n-1}$ be real weights with $a_0 + \\dots + a_{n-1} \\ge 0$. Then the graph on the points $i$ with $a_i \\ge 0$, in which two distinct points are adjacent when some line of $M$ (a flat of rank $2$) contains both and has non-negative weight,\u2026","title":"Non-negative points of a weighted rank-3 matroid are connected by non-negative lines"},{"data_url":"scopes/Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals.json","id":"Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals","kind":"scope","navigation":"module","parent":28051,"result":false,"route":"../roadmap/Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals/index.html","scope":"Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals","status":"fully_proved","title":"Rado's theorem on independent transversals"},{"id":"Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals/rado-matroidal-marriage","kind":"node","lean":"MathlibExt.Combinatorics.Matroid.RadoWanted.rado_matroidal_marriage'","parent":28093,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals/index.html#rado-matroidal-marriage","scope":"Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals","status":"fully_proved","summary":"Let $I$ and $E$ be finite sets, let $M$ be a matroid with ground set $E(M) \\subseteq E$, and let $(A_i)_{i \\in I}$ be a family of subsets of $E$. Then there is an injective map $f : I \\to E$ such that $f(i) \\in A_i$ for every $i \\in I$ and the image $f(I)$ is independent in $M$, if and only if $|J| \\le r_M\\bigl(\\bigcup_{i \\in J} A_i\\bigr)$ for every $J \\subseteq I$, where $r_M(X)$ denotes the rank in $M$ of\u2026","title":"Rado's theorem on independent transversals, arbitrary ground set"},{"id":"Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals/rado-matroidal-marriage-2","kind":"node","lean":"MathlibExt.Combinatorics.Matroid.RadoWanted.rado_matroidal_marriage","parent":28093,"result":true,"route":"../roadmap/Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals/index.html#rado-matroidal-marriage-2","scope":"Combinatorics/matroid-theory/rado-s-theorem-on-independent-transversals","status":"fully_proved","summary":"Let $I$ and $E$ be finite sets, let $M$ be a matroid whose ground set is all of $E$, and let $(A_i)_{i \\in I}$ be a family of subsets of $E$. Then there is an injective map $f : I \\to E$ such that $f(i) \\in A_i$ for every $i \\in I$ and the image $f(I)$ is independent in $M$, if and only if $|J| \\le r_M\\bigl(\\bigcup_{i \\in J} A_i\\bigr)$ for every $J \\subseteq I$, where $r_M$ is the rank function of $M$.","title":"Rado's theorem on independent transversals"},{"data_url":"scopes/Combinatorics/partially-ordered-sets.json","id":"Combinatorics/partially-ordered-sets","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/index.html","scope":"Combinatorics/partially-ordered-sets","status":"can_prove","summary":"Finite posets as combinatorial objects: Dilworth's and Mirsky's theorems, Erd\u0151s\u2013Szekeres monotone subsequences, linear extensions, M\u00f6bius algebras and flag vectors, chain polynomials and Neggers' conjecture, Sperner properties of the weak order, rowmotion and toggle homomesy, poset matrices and rooted trees; lattice theory goes to Order.","title":"Partially ordered sets"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/chain-polynomials-of-posets.json","id":"Combinatorics/partially-ordered-sets/chain-polynomials-of-posets","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/chain-polynomials-of-posets/index.html","scope":"Combinatorics/partially-ordered-sets/chain-polynomials-of-posets","status":"can_prove","title":"Chain polynomials of posets"},{"id":"Combinatorics/partially-ordered-sets/chain-polynomials-of-posets/chain-polynomial","kind":"node","lean":"MathlibExt.Combinatorics.DistributiveLatticeChainPolynomialAMR156Problem0021Wanted.chainPolynomial","parent":28097,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/chain-polynomials-of-posets/index.html#chain-polynomial","scope":"Combinatorics/partially-ordered-sets/chain-polynomials-of-posets","status":"fully_proved","summary":"For a finite partially ordered set $P$ and each natural number $k$, let $c_k(P)$ be the number of $k$-element chains of $P$, that is, of $k$-element subsets of $P$ any two of whose elements are comparable; in particular $c_0(P) = 1$, counting the empty chain. The chain polynomial of $P$ is the polynomial $C(P, t) = \\sum_{k \\ge 0} c_k(P)\\, t^k$ with integer coefficients.","title":"Chain polynomial"},{"id":"Combinatorics/partially-ordered-sets/chain-polynomials-of-posets/distributive-lattice-chain-polynomial-re","kind":"node","lean":"MathlibExt.Combinatorics.DistributiveLatticeChainPolynomialAMR156Problem0021Wanted.distributiveLatticeChainPolynomialRealRooted","parent":28097,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/chain-polynomials-of-posets/index.html#distributive-lattice-chain-polynomial-re","scope":"Combinatorics/partially-ordered-sets/chain-polynomials-of-posets","status":"fully_proved","summary":"The proposition that for every finite distributive lattice $L$, every complex root of the chain polynomial $C(L, t)$ is real, where $C(L, t) = \\sum_{k \\ge 0} c_k(L)\\, t^k$ and $c_k(L)$ is the number of $k$-element chains of $L$.","title":"Stanley's Problem 20' on chain polynomials of distributive lattices"},{"id":"Combinatorics/partially-ordered-sets/chain-polynomials-of-posets/distributive-lattice-chain-polynomial-re-2","kind":"node","lean":"MathlibExt.Combinatorics.DistributiveLatticeChainPolynomialAMR156Problem0021Wanted.distributiveLatticeChainPolynomialRealRooted_refuted","parent":28097,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/chain-polynomials-of-posets/index.html#distributive-lattice-chain-polynomial-re-2","scope":"Combinatorics/partially-ordered-sets/chain-polynomials-of-posets","status":"can_prove","summary":"It is not true that for every finite distributive lattice $L$ every complex root of the chain polynomial $C(L, t)$ is real, where $C(L, t) = \\sum_{k \\ge 0} c_k(L)\\, t^k$ and $c_k(L)$ is the number of $k$-element chains of $L$.","title":"Stanley's Problem 20' has a negative answer"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/dilworth-s-theorem.json","id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","title":"Dilworth's theorem"},{"id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem/is-chain-cover-fin","kind":"node","lean":"MathlibExt.Combinatorics.Order.DilworthWanted.IsChainCoverFin","parent":28101,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html#is-chain-cover-fin","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a partially ordered set. A finite set $C$ of finite subsets of $\\alpha$ is a chain cover of $\\alpha$ when every member of $C$ is a chain and every element of $\\alpha$ lies in some member of $C$.","title":"Chain cover"},{"id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem/max-antichain-size","kind":"node","lean":"MathlibExt.Combinatorics.Order.DilworthWanted.maxAntichainSize","parent":28101,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html#max-antichain-size","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","summary":"For a finite partially ordered set $\\alpha$, the maximum antichain size of $\\alpha$ (its width) is the largest number of elements of an antichain in $\\alpha$.","title":"Maximum antichain size"},{"id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem/min-chain-cover-size","kind":"node","lean":"MathlibExt.Combinatorics.Order.DilworthWanted.minChainCoverSize","parent":28101,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html#min-chain-cover-size","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","summary":"For a finite partially ordered set $\\alpha$, the minimum chain cover size of $\\alpha$ is the smallest number of chains in a chain cover of $\\alpha$, that is, in a finite set of chains of $\\alpha$ whose union is $\\alpha$.","title":"Minimum chain cover size"},{"id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem/card-le-max-antichain-size","kind":"node","lean":"MathlibExt.Combinatorics.Order.DilworthWanted.card_le_maxAntichainSize","parent":28101,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html#card-le-max-antichain-size","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a finite partially ordered set. Every antichain $A$ in $\\alpha$ has at most as many elements as the maximum antichain size of $\\alpha$.","title":"Antichains are no larger than the maximum antichain size"},{"id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem/min-chain-cover-size-le-card","kind":"node","lean":"MathlibExt.Combinatorics.Order.DilworthWanted.minChainCoverSize_le_card","parent":28101,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html#min-chain-cover-size-le-card","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a finite partially ordered set. If $C$ is a chain cover of $\\alpha$, that is, a finite set of chains of $\\alpha$ whose union is $\\alpha$, then the minimum chain cover size of $\\alpha$ is at most the number of chains in $C$.","title":"Chain covers bound the minimum chain cover size"},{"id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem/dilworth-decomposition","kind":"node","lean":"MathlibExt.Combinatorics.Order.DilworthWanted.dilworth_decomposition","parent":28101,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html#dilworth-decomposition","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a finite partially ordered set. The maximum antichain size of $\\alpha$ equals its minimum chain cover size: the largest number of elements of an antichain in $\\alpha$ equals the smallest number of chains whose union is $\\alpha$.","title":"Dilworth's theorem"},{"id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem/exists-is-antichain-card-eq-max-antichai","kind":"node","lean":"MathlibExt.Combinatorics.Order.DilworthWanted.exists_isAntichain_card_eq_maxAntichainSize","parent":28101,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html#exists-is-antichain-card-eq-max-antichai","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a finite partially ordered set. Some antichain in $\\alpha$ has exactly as many elements as the maximum antichain size of $\\alpha$.","title":"The maximum antichain size is attained"},{"id":"Combinatorics/partially-ordered-sets/dilworth-s-theorem/exists-is-chain-cover-fin-card-eq-min-ch","kind":"node","lean":"MathlibExt.Combinatorics.Order.DilworthWanted.exists_isChainCoverFin_card_eq_minChainCoverSize","parent":28101,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/dilworth-s-theorem/index.html#exists-is-chain-cover-fin-card-eq-min-ch","scope":"Combinatorics/partially-ordered-sets/dilworth-s-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a finite partially ordered set. Some chain cover $C$ of $\\alpha$, a finite set of chains of $\\alpha$ whose union is $\\alpha$, consists of exactly as many chains as the minimum chain cover size of $\\alpha$.","title":"The minimum chain cover size is attained"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem.json","id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","title":"Erd\u0151s\u2013Szekeres theorem"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/sub-one-lt-of-pos","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.sub_one_lt_of_pos","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#sub-one-lt-of-pos","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For every natural number $k > 0$, $k - 1 < k$.","title":"A positive natural number exceeds its predecessor"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/is-incr-subseq","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.IsIncrSubseq","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#is-incr-subseq","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$ and $k \\in \\mathbb{N}$, a map $\\varphi : \\{0, \\dots, k-1\\} \\to \\{0, \\dots, N-1\\}$ is an increasing subsequence of $f$ of length $k$ when $\\varphi(i) < \\varphi(j)$ and $f(\\varphi(i)) < f(\\varphi(j))$ for all $i < j$.","title":"Increasing subsequence"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/is-decr-subseq","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.IsDecrSubseq","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#is-decr-subseq","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$ and $k \\in \\mathbb{N}$, a map $\\varphi : \\{0, \\dots, k-1\\} \\to \\{0, \\dots, N-1\\}$ is a decreasing subsequence of $f$ of length $k$ when $\\varphi(i) < \\varphi(j)$ and $f(\\varphi(j)) < f(\\varphi(i))$ for all $i < j$.","title":"Decreasing subsequence"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/is-incr-end","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.IsIncrEnd","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#is-incr-end","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$, a position $p$ and $k \\in \\mathbb{N}$, this is the statement that $k \\ge 1$ and $f$ has an increasing subsequence $\\varphi$ of length $k$ (indices and values strictly increasing) ending at $p$, that is, with $\\varphi(k-1) = p$.","title":"Increasing subsequence ending at a position"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/is-decr-end","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.IsDecrEnd","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#is-decr-end","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$, a position $p$ and $k \\in \\mathbb{N}$, this is the statement that $k \\ge 1$ and $f$ has a decreasing subsequence $\\varphi$ of length $k$ (indices strictly increasing and values strictly decreasing) ending at $p$, that is, with $\\varphi(k-1) = p$.","title":"Decreasing subsequence ending at a position"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/fin1-false","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.fin1_false","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#fin1-false","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"There are no $i, j \\in \\{0\\}$ with $i < j$.","title":"No strict pair in a one-element set"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/singleton-incr","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.singleton_incr","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#singleton-incr","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For every real sequence $f$ indexed by $\\{0, \\dots, N-1\\}$ and every position $p$, the subsequence of length $1$ consisting of the term at position $p$ is a strictly increasing subsequence of $f$.","title":"One-term subsequences are increasing"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/singleton-decr","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.singleton_decr","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#singleton-decr","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For every real sequence $f$ indexed by $\\{0, \\dots, N-1\\}$ and every position $p$, the subsequence of length $1$ consisting of the term at position $p$ is a strictly decreasing subsequence of $f$.","title":"One-term subsequences are decreasing"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/incr-len","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.incrLen","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#incr-len","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$ and a position $p$, this is the largest $k \\le N$ such that $f$ has an increasing subsequence of length $k$ ending at $p$, where indices and values both increase strictly.","title":"Length of the longest increasing subsequence ending at a position"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/decr-len","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.decrLen","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#decr-len","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For a real sequence $f$ indexed by $\\{0, \\dots, N-1\\}$ and an index $p$, this is the largest $k$ such that $f$ has a strictly decreasing subsequence of length $k$ ending at $p$, that is, indices $i_1 < \\dots < i_k = p$ with $f(i_1) > \\dots > f(i_k)$. The maximum exists: $k = 1$ qualifies, and every such $k$ is at most $N$.","title":"Length of the longest decreasing subsequence ending at a point"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/subseq-len-le","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.subseqLen_le","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#subseq-len-le","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"If $\\varphi \\colon \\{0, \\dots, k-1\\} \\to \\{0, \\dots, N-1\\}$ is strictly increasing, that is $\\varphi(i) < \\varphi(j)$ whenever $i < j$, then $k \\le N$.","title":"Subsequences are no longer than the sequence"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/incr-len-is-end","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.incrLen_isEnd","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#incr-len-is-end","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For every $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$ and every position $p$, $f$ has an increasing subsequence ending at $p$ whose length equals the length of the longest increasing subsequence of $f$ ending at $p$.","title":"The longest increasing subsequence ending at a position is attained"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/decr-len-is-end","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.decrLen_isEnd","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#decr-len-is-end","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For every real sequence $f$ indexed by $\\{0, \\dots, N-1\\}$ and every index $p$, $f$ has a strictly decreasing subsequence ending at $p$ whose length equals the length of the longest strictly decreasing subsequence of $f$ ending at $p$.","title":"The longest decreasing subsequence ending at a point is attained"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/le-incr-len","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.le_incrLen","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#le-incr-len","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f$ be a real sequence indexed by $\\{0, \\dots, N-1\\}$ and let $p$ be a position. If $f$ has a strictly increasing subsequence of length $k$ ending exactly at $p$, then $k$ is at most the length of the longest strictly increasing subsequence of $f$ ending at $p$.","title":"Bound by the longest increasing subsequence ending at a position"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/le-decr-len","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.le_decrLen","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#le-decr-len","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f$ be a real sequence indexed by $\\{0, \\dots, N-1\\}$ and let $p$ be a position. If $f$ has a strictly decreasing subsequence of length $k$ ending exactly at $p$, then $k$ is at most the length of the longest strictly decreasing subsequence of $f$ ending at $p$.","title":"Bound by the longest decreasing subsequence ending at a position"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/incr-len-ge-one","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.incrLen_ge_one","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#incr-len-ge-one","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For every $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$ and every position $p$, the length of the longest increasing subsequence of $f$ ending at $p$ is at least $1$.","title":"Longest increasing subsequences ending at a position have length at least one"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/decr-len-ge-one","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.decrLen_ge_one","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#decr-len-ge-one","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For every real sequence $f$ indexed by $\\{0, \\dots, N-1\\}$ and every index $p$, the length of the longest strictly decreasing subsequence of $f$ ending at $p$ is at least $1$.","title":"The longest decreasing subsequence ending at a point has length at least one"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/extend-map","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.extendMap","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#extend-map","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Given $\\varphi : \\{0, \\dots, k-1\\} \\to \\{0, \\dots, N-1\\}$ and $p \\in \\{0, \\dots, N-1\\}$, the appended map on $\\{0, \\dots, k\\}$ sends each $i < k$ to $\\varphi(i)$ and sends $k$ to $p$.","title":"Appending an entry to a map of indices"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/idx-le-last","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.idx_le_last","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#idx-le-last","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $k \\ge 1$ and let $\\varphi : \\{0, \\dots, k-1\\} \\to \\{0, \\dots, N-1\\}$ be strictly increasing with $\\varphi(k-1) = q$. Then $\\varphi(i) \\le q$ for every $i < k$.","title":"Entries of a strictly increasing index map lie at or below its last entry"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/val-le-last-incr","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.val_le_last_incr","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#val-le-last-incr","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f$ be a real sequence indexed by $\\{0, \\dots, N-1\\}$, let $k \\ge 1$, and let $\\varphi$ map $\\{0, \\dots, k-1\\}$ to $\\{0, \\dots, N-1\\}$ so that $f(\\varphi(i)) < f(\\varphi(j))$ whenever $i < j < k$. If $\\varphi(k-1) = q$, then $f(\\varphi(i)) \\le f(q)$ for every $i < k$.","title":"The last value is the largest along a strictly increasing selection"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/last-le-val-decr","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.last_le_val_decr","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#last-le-val-decr","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f$ be a real sequence indexed by $\\{0, \\dots, N-1\\}$, let $k \\ge 1$, and let $\\varphi$ map $\\{0, \\dots, k-1\\}$ to $\\{0, \\dots, N-1\\}$ so that $f(\\varphi(j)) < f(\\varphi(i))$ whenever $i < j < k$. If $\\varphi(k-1) = q$, then $f(q) \\le f(\\varphi(i))$ for every $i < k$.","title":"The last value is the smallest along a strictly decreasing selection"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/extend-idx","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.extend_idx","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#extend-idx","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $k \\ge 1$ and let $\\varphi : \\{0, \\dots, k-1\\} \\to \\{0, \\dots, N-1\\}$ be strictly increasing with last value $\\varphi(k-1) = q$. If $p$ is an index with $q < p$, then the map on $\\{0, \\dots, k\\}$ obtained by appending $p$ to $\\varphi$, which sends each $i < k$ to $\\varphi(i)$ and sends $k$ to $p$, is strictly increasing.","title":"Appending a later index keeps indices increasing"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/extend-last","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.extend_last","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#extend-last","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"For every map $\\varphi : \\{0, \\dots, k-1\\} \\to \\{0, \\dots, N-1\\}$ and every $p \\in \\{0, \\dots, N-1\\}$, the map obtained by appending $p$ to $\\varphi$ takes the value $p$ at its last index $k$.","title":"The last entry of an appended map"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/incr-extend","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.incrExtend","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#incr-extend","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$, let $k \\ge 1$, and let $\\varphi$ be an increasing subsequence of $f$ of length $k$ (indices and values strictly increasing) with last index $\\varphi(k-1) = q$. If $p \\in \\{0, \\dots, N-1\\}$ satisfies $q < p$ and $f(q) < f(p)$, then $f$ has an increasing subsequence of length $k + 1$ ending at $p$.","title":"Extending an increasing subsequence"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/decr-extend","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.decrExtend","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#decr-extend","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f$ be a real sequence indexed by $\\{0, \\dots, N-1\\}$, let $k \\ge 1$, and let $\\varphi$ be a decreasing subsequence of $f$ of length $k$, that is, indices $\\varphi(0) < \\dots < \\varphi(k-1)$ with $f(\\varphi(0)) > \\dots > f(\\varphi(k-1))$, whose last index is $\\varphi(k-1) = q$. If $p$ is an index with $q < p$ and $f(p) < f(q)$, then $f$ has a decreasing subsequence of length $k + 1$ that ends at $p$.","title":"Extending a decreasing subsequence"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/incr-mono","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.incrMono","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#incr-mono","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f : \\{0, \\dots, N-1\\} \\to \\mathbb{R}$ and let $q < p$ be positions with $f(q) < f(p)$. Then the length of the longest increasing subsequence of $f$ ending at $q$ is strictly less than the length of the longest increasing subsequence of $f$ ending at $p$.","title":"Longest increasing subsequence lengths increase strictly along increasing pairs"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/decr-mono","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.decrMono","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#decr-mono","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f$ be a real sequence indexed by $\\{0, \\dots, N-1\\}$, and let $q < p$ be indices with $f(p) < f(q)$. Then the length of the longest strictly decreasing subsequence of $f$ ending at $q$ is strictly less than the length of the longest strictly decreasing subsequence of $f$ ending at $p$.","title":"Longest decreasing subsequence lengths grow along a descent"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/trunc-incr","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.truncIncr","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#trunc-incr","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f$ be a real sequence indexed by $\\{0, \\dots, N-1\\}$. If $f$ has a strictly increasing subsequence of length $k$, then for every $r \\le k$ it has a strictly increasing subsequence of length $r$.","title":"Increasing subsequences of every smaller length"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/trunc-decr","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.truncDecr","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#trunc-decr","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $f$ be a real sequence indexed by $\\{0, \\dots, N-1\\}$. If $f$ has a strictly decreasing subsequence of length $k$, then for every $s \\le k$ it has a strictly decreasing subsequence of length $s$.","title":"Decreasing subsequences of every smaller length"},{"id":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/erdos-szekeres","kind":"node","lean":"MathlibExt.Combinatorics.Order.ErdosSzekeres.erdos_szekeres","parent":28110,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem/index.html#erdos-szekeres","scope":"Combinatorics/partially-ordered-sets/erd-s-szekeres-theorem","status":"fully_proved","summary":"Let $r, s \\ge 1$ be natural numbers, let $N = (r - 1)(s - 1) + 1$, and let $f$ be an injective real sequence indexed by $\\{0, \\dots, N-1\\}$. Then $f$ has a strictly increasing subsequence of length $r$, that is, indices $i_1 < \\dots < i_r$ with $f(i_1) < \\dots < f(i_r)$, or a strictly decreasing subsequence of length $s$, that is, indices $j_1 < \\dots < j_s$ with $f(j_1) > \\dots > f(j_s)$.","title":"Erd\u0151s\u2013Szekeres theorem"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/linear-extensions-of-posets.json","id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","title":"Linear extensions of posets"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/linear-extensions","kind":"node","lean":"FinPoset.linearExtensions","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#linear-extensions","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"Let $\\alpha$ be a finite poset with $n$ elements. Its linear extensions are represented as rank functions: the set of linear extensions of $\\alpha$ is the finite set of all maps $\\sigma : \\alpha \\to \\{0, 1, \\dots, n-1\\}$ that are bijective and monotone, meaning that $x \\le y$ implies $\\sigma(x) \\le \\sigma(y)$.","title":"Linear extensions of a finite poset"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/mem-linear-extensions","kind":"node","lean":"FinPoset.mem_linearExtensions","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#mem-linear-extensions","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"Let $\\alpha$ be a finite poset with $n$ elements. A map $\\sigma : \\alpha \\to \\{0, 1, \\dots, n-1\\}$ belongs to the set of linear extensions of $\\alpha$ if and only if $\\sigma$ is bijective and monotone.","title":"Membership in the linear extensions"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/lin-ext-order-iso","kind":"node","lean":"FinPoset.linExtOrderIso","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#lin-ext-order-iso","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"Let $\\alpha$ be a finite poset with $n$ elements, and fix a linear order on $\\alpha$ extending its partial order (one exists by Szpilrajn's extension theorem; it is chosen once and for all). This is the order isomorphism from $\\alpha$ with that linear order onto $\\{0, 1, \\dots, n-1\\}$ with its usual order.","title":"Ranking a Szpilrajn linear extension"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/some-linear-extension","kind":"node","lean":"FinPoset.someLinearExtension","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#some-linear-extension","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"Let $\\alpha$ be a finite poset with $n$ elements, and fix a linear order on $\\alpha$ extending its partial order (one exists by Szpilrajn's extension theorem; it is chosen once and for all). The chosen linear extension of $\\alpha$ is the rank function $\\alpha \\to \\{0, 1, \\dots, n-1\\}$ sending each element to its position in that linear order, counted from $0$.","title":"A chosen linear extension"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/bijective-some-linear-extension","kind":"node","lean":"FinPoset.bijective_someLinearExtension","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#bijective-some-linear-extension","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"For every finite poset $\\alpha$ with $n$ elements, the chosen linear extension of $\\alpha$, viewed as a rank function $\\alpha \\to \\{0, 1, \\dots, n-1\\}$, is a bijection.","title":"The chosen linear extension is bijective"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/monotone-some-linear-extension","kind":"node","lean":"FinPoset.monotone_someLinearExtension","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#monotone-some-linear-extension","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"For every finite poset $\\alpha$ with $n$ elements, the chosen linear extension of $\\alpha$, viewed as a rank function $\\sigma : \\alpha \\to \\{0, 1, \\dots, n-1\\}$, is monotone: $x \\le y$ implies $\\sigma(x) \\le \\sigma(y)$.","title":"The chosen linear extension is monotone"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/nonempty-linear-extensions","kind":"node","lean":"FinPoset.nonempty_linearExtensions","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#nonempty-linear-extensions","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"Every finite poset has a linear extension.","title":"Existence of linear extensions"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/card-linear-extensions-pos","kind":"node","lean":"FinPoset.card_linearExtensions_pos","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#card-linear-extensions-pos","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"For every finite poset $\\alpha$, the number of linear extensions of $\\alpha$ is positive.","title":"The number of linear extensions is positive"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/maximal-elements","kind":"node","lean":"FinPoset.maximalElements","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#maximal-elements","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"For a finite subset $A$ of a partially ordered set $P$, the maximal elements of $A$ form the subset of those $x \\in A$ such that every $y \\in A$ with $x \\le y$ equals $x$.","title":"Maximal elements of a finite set"},{"id":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets/mem-maximal-elements","kind":"node","lean":"FinPoset.mem_maximalElements","parent":28141,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/linear-extensions-of-posets/index.html#mem-maximal-elements","scope":"Combinatorics/partially-ordered-sets/linear-extensions-of-posets","status":"fully_proved","summary":"Let $A$ be a finite subset of a partially ordered set $P$ and let $x \\in P$. Then $x$ belongs to the set of maximal elements of $A$ if and only if $x \\in A$ and every $y \\in A$ with $x \\le y$ equals $x$.","title":"Membership in the maximal elements"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/mirsky-s-theorem.json","id":"Combinatorics/partially-ordered-sets/mirsky-s-theorem","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/mirsky-s-theorem/index.html","scope":"Combinatorics/partially-ordered-sets/mirsky-s-theorem","status":"fully_proved","title":"Mirsky's theorem"},{"id":"Combinatorics/partially-ordered-sets/mirsky-s-theorem/is-antichain-cover-fin","kind":"node","lean":"MathlibExt.Combinatorics.Order.FinitePosets.IsAntichainCoverFin","parent":28152,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/mirsky-s-theorem/index.html#is-antichain-cover-fin","scope":"Combinatorics/partially-ordered-sets/mirsky-s-theorem","status":"fully_proved","summary":"Let $\\alpha$ be a partially ordered set. A finite family $C$ of finite subsets of $\\alpha$ is a finite antichain cover of $\\alpha$ when every member of $C$ is an antichain and every element of $\\alpha$ belongs to some member of $C$.","title":"Finite antichain covers"},{"id":"Combinatorics/partially-ordered-sets/mirsky-s-theorem/max-chain-size","kind":"node","lean":"MathlibExt.Combinatorics.Order.FinitePosets.maxChainSize","parent":28152,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/mirsky-s-theorem/index.html#max-chain-size","scope":"Combinatorics/partially-ordered-sets/mirsky-s-theorem","status":"fully_proved","summary":"For a finite partially ordered set $\\alpha$, the maximum chain size of $\\alpha$ is the largest cardinality of a chain in $\\alpha$, that is, of a set of pairwise comparable elements.","title":"Maximum chain size"},{"id":"Combinatorics/partially-ordered-sets/mirsky-s-theorem/min-antichain-cover-size","kind":"node","lean":"MathlibExt.Combinatorics.Order.FinitePosets.minAntichainCoverSize","parent":28152,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/mirsky-s-theorem/index.html#min-antichain-cover-size","scope":"Combinatorics/partially-ordered-sets/mirsky-s-theorem","status":"fully_proved","summary":"For a finite partially ordered set $\\alpha$, the minimum antichain cover size of $\\alpha$ is the smallest number of antichains in $\\alpha$ whose union is all of $\\alpha$.","title":"Minimum antichain cover size"},{"id":"Combinatorics/partially-ordered-sets/mirsky-s-theorem/mirsky-dual-general","kind":"node","lean":"MathlibExt.Combinatorics.Order.FinitePosets.mirsky_dual_general","parent":28152,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/mirsky-s-theorem/index.html#mirsky-dual-general","scope":"Combinatorics/partially-ordered-sets/mirsky-s-theorem","status":"fully_proved","summary":"For every finite partially ordered set $\\alpha$, the maximum chain size of $\\alpha$ equals its minimum antichain cover size: the largest cardinality of a chain in $\\alpha$ equals the smallest number of antichains whose union is $\\alpha$.","title":"Dual of Dilworth's theorem"},{"id":"Combinatorics/partially-ordered-sets/mirsky-s-theorem/mirsky-dual","kind":"node","lean":"MathlibExt.Combinatorics.Order.FinitePosets.mirsky_dual","parent":28152,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/mirsky-s-theorem/index.html#mirsky-dual","scope":"Combinatorics/partially-ordered-sets/mirsky-s-theorem","status":"fully_proved","summary":"For every finite partially ordered set $\\alpha$, the maximum chain size of $\\alpha$ equals its minimum antichain cover size: the largest cardinality of a chain in $\\alpha$ equals the smallest number of antichains whose union is $\\alpha$.","title":"Mirsky's theorem"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors.json","id":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/index.html","scope":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors","status":"fully_proved","title":"M\u00f6bius algebras and flag vectors"},{"id":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/moebius-algebra-mul","kind":"node","lean":"MetaMathlibExt.moebiusAlgebraMul","parent":28158,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/index.html#moebius-algebra-mul","scope":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors","status":"fully_proved","summary":"Let $L$ be a lattice with meet $\\wedge$ and let $R$ be a commutative semiring. The M\u00f6bius algebra $A(L, R)$ is the set of functions $f \\colon L \\to R$ with $f(s) \\neq 0$ for only finitely many $s \\in L$. The product of $f, g \\in A(L, R)$ is the function $f \\cdot g \\in A(L, R)$ given by","title":"M\u00f6bius algebra product"},{"id":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/flag-fvector","kind":"node","lean":"MetaMathlibExt.flagFVector","parent":28158,"reading_title":{"format":"inline-parts-v1","parts":[["text","Flag "],["math","\\(f\\)"],["text","-vector"]],"source":"Flag $f$-vector"},"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/index.html#flag-fvector","scope":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors","status":"fully_proved","summary":"Let $P$ be a finite partially ordered set with a grade function $\\rho \\colon P \\to \\mathbb{Z}_{\\ge 0}$, that is, $\\rho(s) < \\rho(t)$ whenever $s < t$, $\\rho(t) = \\rho(s) + 1$ whenever $t$ covers $s$ (that is, $s < t$ and no $u \\in P$ satisfies $s < u < t$), and $\\rho(t) = 0$ whenever $t$ is minimal. For a finite set $S \\subseteq \\mathbb{Z}_{\\ge 0}$, let $P_S = \\{t \\in P : \\rho(t) \\in S\\}$. The flag $f$-vector of\u2026","title":"Flag $f$-vector"},{"id":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/flag-hvector","kind":"node","lean":"MetaMathlibExt.flagHVector","parent":28158,"reading_title":{"format":"inline-parts-v1","parts":[["text","Flag "],["math","\\(h\\)"],["text","-vector"]],"source":"Flag $h$-vector"},"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/index.html#flag-hvector","scope":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors","status":"fully_proved","summary":"Let $P$ be a finite partially ordered set with a grade function $\\rho \\colon P \\to \\mathbb{Z}_{\\ge 0}$, that is, $\\rho(s) < \\rho(t)$ whenever $s < t$, $\\rho(t) = \\rho(s) + 1$ whenever $t$ covers $s$, and $\\rho(t) = 0$ whenever $t$ is minimal. Let $\\alpha_P$ be the flag $f$-vector of $P$ with respect to $\\rho$: for a finite set $T \\subseteq \\mathbb{Z}_{\\ge 0}$, $\\alpha_P(T)$ is the number of maximal chains in the\u2026","title":"Flag $h$-vector"},{"id":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/is-hyperplane-arrangement","kind":"node","lean":"MetaMathlibExt.IsHyperplaneArrangement","parent":28158,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors/index.html#is-hyperplane-arrangement","scope":"Combinatorics/partially-ordered-sets/m-bius-algebras-and-flag-vectors","status":"fully_proved","summary":"Let $K$ be a field, let $V$ be a finite-dimensional vector space over $K$, and let $E$ be an affine subspace of $V$, possibly empty. The dimension of a nonempty affine subspace $v + W$, with $v \\in V$ and $W$ a linear subspace of $V$, is $\\dim W$. An affine hyperplane of $E$ is a nonempty affine subspace $H$ of $V$ with $H \\subseteq E$ and $\\dim H + 1 = \\dim E$. A set $\\mathcal{A}$ of affine subspaces of $V$ is a\u2026","title":"Hyperplane arrangement"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/neggers-stanley-conjecture.json","id":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/index.html","scope":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture","status":"can_prove","title":"Neggers\u2013Stanley conjecture"},{"id":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/descent-count","kind":"node","lean":"MathlibExt.Combinatorics.NeggersStanleyPosetConjectureAMR156Problem0020Wanted.descentCount","parent":28163,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of descents of a permutation of "],["math","\\(\\{0, \\dots, n-1\\}\\)"]],"source":"Number of descents of a permutation of $\\{0, \\dots, n-1\\}$"},"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/index.html#descent-count","scope":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture","status":"fully_proved","summary":"For a permutation $w$ of $\\{0, 1, \\dots, n-1\\}$, read as the word $w(0)\\,w(1) \\cdots w(n-1)$, the number of descents $d(w)$ is the number of indices $i$ with $i + 1 < n$ and $w(i) > w(i+1)$.","title":"Number of descents of a permutation of $\\{0, \\dots, n-1\\}$"},{"id":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/w-polynomial","kind":"node","lean":"MathlibExt.Combinatorics.NeggersStanleyPosetConjectureAMR156Problem0020Wanted.wPolynomial","parent":28163,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(W\\)"],["text","-polynomial of a labeled poset"]],"source":"The $W$-polynomial of a labeled poset"},"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/index.html#w-polynomial","scope":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture","status":"fully_proved","summary":"Let $R$ be a relation on the labels $\\{0, 1, \\dots, n-1\\}$, with $R(i, j)$ read as $i \\lt _P j$ in the poset $P$ it generates. A linear extension of $P$ is a permutation $w$ of the labels such that $a < b$ whenever $R(w(a), w(b))$, that is, a word $w(0)\\,w(1) \\cdots w(n-1)$ listing every label once with $w(a)$ before $w(b)$ whenever $R(w(a), w(b))$. The $W$-polynomial is the integer polynomial\u2026","title":"The $W$-polynomial of a labeled poset"},{"id":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/neggers-conjecture","kind":"node","lean":"MathlibExt.Combinatorics.NeggersStanleyPosetConjectureAMR156Problem0020Wanted.neggersConjecture","parent":28163,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/index.html#neggers-conjecture","scope":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture","status":"fully_proved","summary":"For every $n$ and every relation $R$ on the labels $\\{0, 1, \\dots, n-1\\}$ that is naturally labeled, meaning that $R(i, j)$ implies $i < j$, every complex zero of the $W$-polynomial $W(P, x)$ of the poset $P$ generated by $R$ is real.","title":"Neggers' conjecture"},{"id":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/neggers-conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.NeggersStanleyPosetConjectureAMR156Problem0020Wanted.neggersConjecture_refuted","parent":28163,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/neggers-stanley-conjecture/index.html#neggers-conjecture-refuted","scope":"Combinatorics/partially-ordered-sets/neggers-stanley-conjecture","status":"can_prove","summary":"There exist $n$ and a naturally labeled relation $R$ on the labels $\\{0, 1, \\dots, n-1\\}$, one in which $R(i, j)$ implies $i < j$, such that the $W$-polynomial $W(P, x)$ of the poset $P$ generated by $R$ has a complex zero that is not real.","title":"Neggers' conjecture is false"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees.json","id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"can_prove","title":"Poset matrices and quasi-full rooted trees"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/is-poset-matrix","kind":"node","lean":"MetaMathlibExt.IsPosetMatrix","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#is-poset-matrix","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"fully_proved","summary":"An $n \\times n$ Boolean matrix $M$, with rows and columns indexed by $\\{0, \\dots, n-1\\}$, is an upper-triangular $(0, 1)$-poset matrix when its entries strictly below the main diagonal are false ($M_{ij}$ is false whenever $j < i$), its diagonal entries $M_{ii}$ are true, and its true entries are transitive: if $M_{ij}$ and $M_{jk}$ are true, then $M_{ik}$ is true.","title":"Poset matrix"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/rel","kind":"node","lean":"MetaMathlibExt.IsPosetMatrix.Rel","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#rel","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"fully_proved","summary":"Let $M$ be an $n \\times n$ Boolean matrix with rows and columns indexed by $\\{0, \\dots, n-1\\}$. The relation represented by $M$ relates index $i$ to index $j$ exactly when the entry $M_{ij}$ is true.","title":"Relation represented by a Boolean matrix"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/upper","kind":"node","lean":"MetaMathlibExt.IsPosetMatrix.upper","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#upper","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"fully_proved","summary":"If $M$ is an $n \\times n$ poset matrix, then $M_{ij}$ is false for all indices $i, j < n$ with $j < i$.","title":"Poset matrices are upper triangular"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/diagonal","kind":"node","lean":"MetaMathlibExt.IsPosetMatrix.diagonal","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#diagonal","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"fully_proved","summary":"If $M$ is an $n \\times n$ poset matrix, then $M_{ii}$ is true for every index $i < n$.","title":"Diagonal entries of a poset matrix"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/transitive","kind":"node","lean":"MetaMathlibExt.IsPosetMatrix.transitive","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#transitive","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"fully_proved","summary":"If $M$ is an $n \\times n$ poset matrix, then for all indices $i, j, k < n$, if $M_{ij}$ and $M_{jk}$ are true, then $M_{ik}$ is true.","title":"Transitivity of the true entries of a poset matrix"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/rel-refl","kind":"node","lean":"MetaMathlibExt.IsPosetMatrix.rel_refl","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#rel-refl","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"fully_proved","summary":"If $M$ is an $n \\times n$ poset matrix, then the relation represented by $M$ relates every index $i < n$ to itself.","title":"The relation of a poset matrix is reflexive"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/rel-trans","kind":"node","lean":"MetaMathlibExt.IsPosetMatrix.rel_trans","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#rel-trans","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"fully_proved","summary":"If $M$ is an $n \\times n$ poset matrix and $i, j, k < n$ are indices such that the relation represented by $M$ relates $i$ to $j$ and $j$ to $k$, then it relates $i$ to $k$.","title":"The relation of a poset matrix is transitive"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/is-poset-matrix-block-iff-direct-sum-ord","kind":"node","lean":"MetaMathlibExt.isPosetMatrix_block_iff_directSum_ordinalSum","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#is-poset-matrix-block-iff-direct-sum-ord","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"can_prove","summary":"Let $m, k \\ge 1$ and let $M$ be a poset matrix of order $m + k$, a Boolean matrix with rows and columns indexed by $0, \\dots, m + k - 1$. Then $M_{ij}$ is false for all $i < m \\le j$ if and only if there are poset matrices $A$ of order $m$ and $B$ of order $k$ such that $M$ is their direct sum: $M_{ij} = A_{ij}$ for $i, j < m$, $M_{m+i, m+j} = B_{ij}$ for $i, j < k$, and every other entry of $M$ is false. Likewise\u2026","title":"Block decomposition of a poset matrix"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/direct-sum-ordinal-sum-represents","kind":"node","lean":"MetaMathlibExt.directSum_ordinalSum_represents","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#direct-sum-ordinal-sum-represents","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"can_prove","summary":"A poset matrix $M$ of order $n$, a Boolean matrix with rows and columns indexed by $0, \\dots, n - 1$, represents a partial order $P$ when there is a bijection $e$ from $\\{0, \\dots, n - 1\\}$ to $P$ such that $e(i) \\le e(j)$ if and only if $M_{ij}$ is true. Let $A$ and $B$ be poset matrices of orders $m$ and $k$ representing partial orders $P$ and $Q$. Let $A \\oplus B$ and $A \\boxplus B$ be the matrices $N$ of order\u2026","title":"Direct and ordinal sums of poset matrices represent sums of posets"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/connected-or-disconnected-of-block-prope","kind":"node","lean":"MetaMathlibExt.connected_or_disconnected_of_block_property","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#connected-or-disconnected-of-block-prope","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"can_prove","summary":"Let $M$ be a poset matrix of order $n$, a Boolean matrix with rows and columns indexed by $0, \\dots, n - 1$, that represents a partial order $P$: there is a bijection $e$ from $\\{0, \\dots, n - 1\\}$ to $P$ such that $e(i) \\le e(j)$ if and only if $M_{ij}$ is true. Let $1 \\le r < n$. If $M_{ij}$ is true for all $i < r \\le j$ (the block of $1$s property of length $r$), then $P$ is connected: every subset $S$ of $P$\u2026","title":"Block properties decide connectedness of the represented poset"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/quasi-full-tree-arity","kind":"node","lean":"MetaMathlibExt.quasiFullTree_arity","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#quasi-full-tree-arity","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"can_prove","summary":"Let $P$ be a finite partial order with $n \\ge 3$ elements that is a rooted tree: $P$ has a least element and, for every $x \\in P$, the elements below $x$ form a chain. For a natural number $t$, call $P$ quasi-full $t$-ary when every $x \\in P$ is covered by exactly $\\min(|U_x| - 1, t)$ elements of $P$, where $U_x = \\{y \\in P : x \\le y\\}$. Then (i) if $P$ is quasi-full $(n - 1)$-ary, it is quasi-full $t$-ary for\u2026","title":"Arity of quasi-full rooted trees"},{"id":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/not-relabel-equiv-of-block-lengths-ne","kind":"node","lean":"MetaMathlibExt.not_relabel_equiv_of_blockLengths_ne","parent":28168,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees/index.html#not-relabel-equiv-of-block-lengths-ne","scope":"Combinatorics/partially-ordered-sets/poset-matrices-and-quasi-full-rooted-trees","status":"can_prove","summary":"Let $n, t \\ge 2$ and let $M$ and $M'$ be poset matrices of order $n$, Boolean matrices with rows and columns indexed by $0, \\dots, n - 1$. Let $0 = l_0 < l_1 < \\dots < l_t = n$ be such that the gaps $l_{k+1} - l_k$ are nondecreasing in $k$. Suppose that for every $k \\le t$, $M_{ij}$ is false for all $i < l_k \\le j$, and that for every $k < t$ there is $s \\ge 1$ with $l_k + s < l_{k+1}$ such that $M_{ij}$ is true\u2026","title":"Direct sum criterion for relabeling inequivalence"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets.json","id":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/index.html","scope":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","status":"can_prove","title":"Rowmotion on root posets"},{"id":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/is-homomesic","kind":"node","lean":"MathlibExt.Combinatorics.RootPosetRowmotionHomomesyWanted.IsHomomesic","parent":28181,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/index.html#is-homomesic","scope":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","status":"fully_proved","summary":"Let $X$ be a type, $\\sigma \\colon X \\to X$ a map, $f \\colon X \\to \\mathbb{Q}$ a function and $c \\in \\mathbb{Q}$. Then $f$ is homomesic under $\\sigma$ with average $c$ when $X$ is finite, $\\sigma$ is bijective, and for every $x \\in X$ and every $k > 0$ with $\\sigma^k(x) = x$ the average $\\frac{1}{k} \\sum_{i=0}^{k-1} f(\\sigma^i(x))$ equals $c$.","title":"Homomesy"},{"id":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/poset-antichain","kind":"node","lean":"MathlibExt.Combinatorics.RootPosetRowmotionHomomesyWanted.PosetAntichain","parent":28181,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/index.html#poset-antichain","scope":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","status":"fully_proved","summary":"For a partially ordered set $P$, the finite antichains of $P$ are the finite subsets $A \\subseteq P$ in which no element is below a different element.","title":"Finite antichains of a poset"},{"id":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/is-antichain-cardinality","kind":"node","lean":"MathlibExt.Combinatorics.RootPosetRowmotionHomomesyWanted.IsAntichainCardinality","parent":28181,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/index.html#is-antichain-cardinality","scope":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","status":"fully_proved","summary":"For a partially ordered set $P$, a function $f$ from the finite antichains of $P$ to $\\mathbb{Q}$ is the antichain cardinality statistic when $f(A) = |A|$ for every finite antichain $A$ of $P$.","title":"Antichain cardinality statistic"},{"id":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/is-antichain-rowmotion","kind":"node","lean":"MathlibExt.Combinatorics.RootPosetRowmotionHomomesyWanted.IsAntichainRowmotion","parent":28181,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/index.html#is-antichain-rowmotion","scope":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","status":"fully_proved","summary":"For a partially ordered set $P$, a map $\\rho$ from the finite antichains of $P$ to themselves is antichain rowmotion when $\\rho$ is bijective and, for every finite antichain $A$, $\\rho(A)$ is the set of minimal elements of the complement of the down-set of $A$: an element $x \\in P$ lies in $\\rho(A)$ exactly when $x \\not\\le a$ for all $a \\in A$ and every $y \\le x$ with $y \\not\\le a$ for all $a \\in A$ equals $x$.","title":"Rowmotion on antichains"},{"id":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/is-type-aroot-poset","kind":"node","lean":"MathlibExt.Combinatorics.RootPosetRowmotionHomomesyWanted.IsTypeARootPoset","parent":28181,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/index.html#is-type-aroot-poset","scope":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","status":"fully_proved","summary":"A partially ordered set $P$ is a type-A root poset of rank $n$, for a natural number $n$, when there is a bijection $a \\mapsto (i_a, j_a)$ from $P$ onto the set of pairs $(i, j)$ of integers with $0 \\le i \\le j < n$ such that $a \\le b$ holds exactly when the interval $[i_a, j_a]$ is contained in $[i_b, j_b]$, that is, $i_b \\le i_a$ and $j_a \\le j_b$.","title":"Type-A root poset"},{"id":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.RootPosetRowmotionHomomesyWanted.conjecture","parent":28181,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/index.html#conjecture","scope":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","status":"fully_proved","summary":"The proposition that for every natural number $n$, every finite partially ordered set $P$ that is a type-A root poset of rank $n$ (there is a bijection $a \\mapsto (i_a, j_a)$ from $P$ onto the set of pairs $(i, j)$ of integers with $0 \\le i \\le j < n$ such that $a \\le b$ holds exactly when $i_b \\le i_a$ and $j_a \\le j_b$), and every bijection $\\rho$ of the set of finite antichains of $P$ that sends each antichain\u2026","title":"Type-A rowmotion homomesy conjecture"},{"id":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.RootPosetRowmotionHomomesyWanted.conjecture_holds","parent":28181,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/rowmotion-on-root-posets/index.html#conjecture-holds","scope":"Combinatorics/partially-ordered-sets/rowmotion-on-root-posets","status":"can_prove","summary":"Let $n$ be a natural number and let $P$ be a finite partially ordered set that is a type-A root poset of rank $n$: there is a bijection $a \\mapsto (i_a, j_a)$ from $P$ onto the set of pairs $(i, j)$ of integers with $0 \\le i \\le j < n$ such that $a \\le b$ holds exactly when $[i_a, j_a] \\subseteq [i_b, j_b]$, that is, $i_b \\le i_a$ and $j_a \\le j_b$. Let $\\rho$ be a bijection of the set of finite antichains of $P$\u2026","title":"Antichain size is homomesic under type-A rowmotion"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order.json","id":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/index.html","scope":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order","status":"can_prove","title":"Sperner property of the weak order"},{"id":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/inversions","kind":"node","lean":"MathlibExt.Combinatorics.WeakOrderSpernerAMR30Problem0039Wanted.inversions","parent":28189,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/index.html#inversions","scope":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order","status":"fully_proved","summary":"For a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$, the inversion set of $\\sigma$ is the set of pairs of positions $(i, j)$ with $i < j$ and $\\sigma(j) < \\sigma(i)$.","title":"Inversion set of a permutation"},{"id":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/weak-le","kind":"node","lean":"MathlibExt.Combinatorics.WeakOrderSpernerAMR30Problem0039Wanted.weakLE","parent":28189,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/index.html#weak-le","scope":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order","status":"fully_proved","summary":"For permutations $\\sigma$ and $\\tau$ of $\\{0, \\dots, n-1\\}$, $\\sigma \\le \\tau$ in the weak order when the inversion set of $\\sigma$ (its pairs of positions $(i, j)$ with $i < j$ and $\\sigma(j) < \\sigma(i)$) is contained in the inversion set of $\\tau$.","title":"Weak order on the symmetric group"},{"id":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/weak-order-sperner","kind":"node","lean":"MathlibExt.Combinatorics.WeakOrderSpernerAMR30Problem0039Wanted.weakOrderSperner","parent":28189,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/index.html#weak-order-sperner","scope":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order","status":"fully_proved","summary":"This is the statement that for every $n$ and every antichain $A$ of the weak order on the permutations of $\\{0, \\dots, n-1\\}$ (where $\\sigma \\le \\tau$ when the inversion set of $\\sigma$ is contained in that of $\\tau$), there is a $k$ such that $A$ has at most as many elements as there are permutations with exactly $k$ inversions. Equivalently, no antichain is larger than the largest rank level, the rank of a\u2026","title":"Sperner property of the weak order"},{"id":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/weak-order-sperner-holds","kind":"node","lean":"MathlibExt.Combinatorics.WeakOrderSpernerAMR30Problem0039Wanted.weakOrderSperner_holds","parent":28189,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order/index.html#weak-order-sperner-holds","scope":"Combinatorics/partially-ordered-sets/sperner-property-of-the-weak-order","status":"can_prove","summary":"For every $n$ and every antichain $A$ of the weak order on the symmetric group $S_n$ of permutations of $\\{0, \\dots, n-1\\}$ (where $\\sigma \\le \\tau$ when the inversion set of $\\sigma$ is contained in that of $\\tau$), there is a $k$ such that $A$ has at most as many elements as there are permutations with exactly $k$ inversions.","title":"The weak order on the symmetric group is Sperner"},{"data_url":"scopes/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions.json","id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","kind":"scope","navigation":"module","parent":28096,"result":false,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"can_prove","title":"Toggle homomesy for noncrossing partitions"},{"id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/arc-set","kind":"node","lean":"MathlibExt.Combinatorics.NoncrossingToggleBlockHomomesyAIMOtherProblem36Wanted.ArcSet","parent":28194,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html#arc-set","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"fully_proved","summary":"For a natural number $n$, an arc set on $n$ points is a finite set of ordered pairs $(i, j)$, called arcs, with $i, j \\in \\{0, 1, \\dots, n - 1\\}$.","title":"Arc sets"},{"id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/is-noncrossing-arc-set","kind":"node","lean":"MathlibExt.Combinatorics.NoncrossingToggleBlockHomomesyAIMOtherProblem36Wanted.IsNoncrossingArcSet","parent":28194,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html#is-noncrossing-arc-set","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"fully_proved","summary":"An arc set $P$ on $n$ points is noncrossing when every arc $(i, j) \\in P$ has $i < j$ and any two distinct arcs $(i, j)$ and $(k, l)$ of $P$ satisfy $i \\ne k$, $j \\ne l$, and not $i < k < j < l$. So distinct arcs do not cross and share neither a left nor a right endpoint.","title":"Noncrossing arc sets"},{"id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/toggle","kind":"node","lean":"MathlibExt.Combinatorics.NoncrossingToggleBlockHomomesyAIMOtherProblem36Wanted.toggle","parent":28194,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html#toggle","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"fully_proved","summary":"For an arc $a = (i, j)$ and an arc set $P$ on $n$ points, the toggle $\\tau_{i,j}$ sends $P$ to $P \\setminus \\{a\\}$ if $a \\in P$; otherwise to $P \\cup \\{a\\}$ if $P \\cup \\{a\\}$ is a noncrossing arc set; and otherwise it leaves $P$ unchanged.","title":"Toggle at an arc"},{"id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/toggle-order","kind":"node","lean":"MathlibExt.Combinatorics.NoncrossingToggleBlockHomomesyAIMOtherProblem36Wanted.toggleOrder","parent":28194,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html#toggle-order","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"fully_proved","summary":"For a natural number $n$, the toggle order on $n$ points is the list of all arcs $(i, i + d)$ with $d \\ge 1$, $i \\ge 0$ and $i + d \\le n - 1$, ordered by length $d = 1, \\dots, n - 1$ and, for each length, by increasing left endpoint $i$. In $1$-indexed form it is $(1,2), (2,3), \\dots, (n-1,n), (1,3), \\dots, (n-2,n), \\dots, (1,n)$.","title":"Toggle order"},{"id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/toggle-composite","kind":"node","lean":"MathlibExt.Combinatorics.NoncrossingToggleBlockHomomesyAIMOtherProblem36Wanted.toggleComposite","parent":28194,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html#toggle-composite","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"fully_proved","summary":"The toggle composite $\\sigma$ on $n$ points sends an arc set $P$ on $n$ points to the arc set obtained from $P$ by applying the toggles $\\tau_{i,j}$ at the arcs $(i, j)$ of the toggle order on $n$ points one after another, in the order of that list.","title":"Toggle composite"},{"id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/block-count","kind":"node","lean":"MathlibExt.Combinatorics.NoncrossingToggleBlockHomomesyAIMOtherProblem36Wanted.blockCount","parent":28194,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html#block-count","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"fully_proved","summary":"The block count of an arc set $P$ on $n$ points is $n - |P|$, where the subtraction is truncated at $0$.","title":"Block count"},{"id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/noncrossing-toggle-block-homomesy","kind":"node","lean":"MathlibExt.Combinatorics.NoncrossingToggleBlockHomomesyAIMOtherProblem36Wanted.noncrossingToggleBlockHomomesy","parent":28194,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html#noncrossing-toggle-block-homomesy","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"fully_proved","summary":"Let $\\sigma_n$ be the toggle composite on $n$ points and $\\sigma_n^k$ its $k$-fold iterate. The claim states that for every $n \\ge 1$, every noncrossing arc set $P$ on $n$ points and every $L \\ge 1$ with $\\sigma_n^L(P) = P$, twice the sum of the block counts of $P, \\sigma_n(P), \\dots, \\sigma_n^{L-1}(P)$ equals $L(n + 1)$. That is, the number of blocks is homomesic under $\\sigma_n$ with average $(n + 1)/2$.","title":"Block-count homomesy claim for toggling noncrossing partitions"},{"id":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/noncrossing-toggle-block-homomesy-holds","kind":"node","lean":"MathlibExt.Combinatorics.NoncrossingToggleBlockHomomesyAIMOtherProblem36Wanted.noncrossingToggleBlockHomomesy_holds","parent":28194,"result":true,"route":"../roadmap/Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions/index.html#noncrossing-toggle-block-homomesy-holds","scope":"Combinatorics/partially-ordered-sets/toggle-homomesy-for-noncrossing-partitions","status":"can_prove","summary":"The block-count homomesy claim holds: with $\\sigma_n$ the toggle composite on $n$ points and $\\sigma_n^k$ its $k$-fold iterate, for every $n \\ge 1$, every noncrossing arc set $P$ on $n$ points and every $L \\ge 1$ with $\\sigma_n^L(P) = P$, twice the sum of the block counts of $P, \\sigma_n(P), \\dots, \\sigma_n^{L-1}(P)$ equals $L(n + 1)$.","title":"Block-count homomesy for toggling noncrossing partitions"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes.json","id":"Combinatorics/polytopes-simplicial-complexes","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/index.html","scope":"Combinatorics/polytopes-simplicial-complexes","status":"can_prove","summary":"Face numbers and h-vectors of simplicial spheres and polytopes (Dehn\u2013Sommerville, upper bound theorem), Cohen\u2013Macaulay, partitionable and interval-decomposable complexes, simplicial homology, Ehrhart polynomials, h*-vectors and the integer decomposition property, lattice parallelograms, and hyperplane arrangements.","title":"Polytopes, Ehrhart theory and simplicial complexes"},{"id":"Combinatorics/polytopes-simplicial-complexes/ordinary-ehrhart-count","kind":"node","lean":"MetaMathlibExt.ordinaryEhrhartCount","parent":28203,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/index.html#ordinary-ehrhart-count","scope":"Combinatorics/polytopes-simplicial-complexes","status":"fully_proved","summary":"For a natural number $d$, a set $P \\subseteq \\mathbb{Q}^d$ and a natural number $m$, the ordinary Ehrhart count is the number $|mP \\cap \\mathbb{Z}^d|$ of integer points in the dilate $mP = \\{m x : x \\in P\\}$, where $\\mathbb{Z}^d$ is viewed inside $\\mathbb{Q}^d$; it is $0$ when $mP \\cap \\mathbb{Z}^d$ is infinite.","title":"Ordinary Ehrhart count"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms.json","id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","title":"Clean lattice parallelograms"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/lattice-points","kind":"node","lean":"MetaMathlibExt.latticePoints","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#lattice-points","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"The lattice points of $\\mathbb{R}^2$ are the points $(a, b)$ with $a, b \\in \\mathbb{Z}$; they form the image of $\\mathbb{Z}^2$ in $\\mathbb{R}^2$.","title":"The integer lattice in the plane"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/lattice-parallelogram","kind":"node","lean":"MetaMathlibExt.latticeParallelogram","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#lattice-parallelogram","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"For $u, v \\in \\mathbb{R}^2$, the parallelogram spanned by $u$ and $v$ is the set $\\{s u + t v : 0 \\le s \\le 1,\\ 0 \\le t \\le 1\\}$.","title":"Parallelogram spanned by two vectors"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/std-clean-parallelogram","kind":"node","lean":"MetaMathlibExt.stdCleanParallelogram","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parallelogram "],["math","\\(P_{a,n}\\)"]],"source":"The parallelogram $P_{a,n}$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#std-clean-parallelogram","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"For natural numbers $a$ and $n$, the set $P_{a,n} \\subseteq \\mathbb{R}^2$ is the closed parallelogram spanned by $(1, 0)$ and $(a, n)$, that is, the set of points $s (1, 0) + t (a, n)$ with $0 \\le s \\le 1$ and $0 \\le t \\le 1$. For $n \\ge 1$ it is the parallelogram with vertices $(0, 0)$, $(1, 0)$, $(a, n)$ and $(a + 1, n)$.","title":"The parallelogram $P_{a,n}$"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/unimodular-map","kind":"node","lean":"MetaMathlibExt.unimodularMap","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#unimodular-map","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"For a $2 \\times 2$ integer matrix $M$ with entries $m_{ij}$ and an integer vector $u = (u_1, u_2) \\in \\mathbb{Z}^2$, the map $T_{M,u} : \\mathbb{R}^2 \\to \\mathbb{R}^2$ is $T_{M,u}(x, y) = (m_{11} x + m_{12} y + u_1, m_{21} x + m_{22} y + u_2)$, that is, $p \\mapsto M p + u$. It is an affine unimodular map when $\\det M = \\pm 1$; the definition itself puts no condition on $\\det M$.","title":"Integer affine maps of the plane"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/unimodularly-equivalent","kind":"node","lean":"MetaMathlibExt.UnimodularlyEquivalent","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#unimodularly-equivalent","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"Two sets $P, Q \\subseteq \\mathbb{R}^2$ are unimodularly equivalent when some affine unimodular map sends $P$ onto $Q$: there are a $2 \\times 2$ integer matrix $M$ with $\\det M = 1$ or $\\det M = -1$ and an integer vector $u \\in \\mathbb{Z}^2$ such that $\\{M x + u : x \\in P\\} = Q$.","title":"Unimodularly equivalent sets"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/unimodularly-equivalent-refl","kind":"node","lean":"MetaMathlibExt.unimodularlyEquivalent_refl","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#unimodularly-equivalent-refl","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"Every subset $P$ of $\\mathbb{R}^2$ is unimodularly equivalent to itself (take the identity matrix and the zero translation).","title":"Unimodular equivalence is reflexive"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/is-clean-parallelogram","kind":"node","lean":"MetaMathlibExt.IsCleanParallelogram","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#is-clean-parallelogram","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"For points $u, v \\in \\mathbb{R}^2$, the parallelogram spanned by $u$ and $v$ is clean when $u$ and $v$ are lattice points and the lattice points on the boundary of the parallelogram are exactly $0$, $u$, $v$ and $u + v$.","title":"Clean lattice parallelogram"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/std-column","kind":"node","lean":"MetaMathlibExt.stdColumn","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","Interior lattice points of "],["math","\\(P_{a,n}\\)"],["text"," on a vertical line"]],"source":"Interior lattice points of $P_{a,n}$ on a vertical line"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#std-column","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"For natural numbers $a$ and $n$, let $P_{a,n}$ be the parallelogram spanned by $(1, 0)$ and $(a, n)$, with vertices $(0, 0)$, $(1, 0)$, $(a, n)$, $(a + 1, n)$. For an integer $k$, the set $S_k$ is the set of integers $y$ such that $(k, y)$ lies in the interior of $P_{a,n}$, that is, the $y$-coordinates of the lattice points in the interior of $P_{a,n}$ on the line $x = k$.","title":"Interior lattice points of $P_{a,n}$ on a vertical line"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/antidiagonal-triangle-bij-on","kind":"node","lean":"MetaMathlibExt.antidiagonal_triangle_bijOn","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#antidiagonal-triangle-bij-on","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $u$ and $v$ be linearly independent vectors in $\\mathbb{R}^2$ and let $P$ be the parallelogram they span. The map $p \\mapsto u + v - p$ is a bijection from the set of points of the triangle $\\operatorname{conv}\\{0, u, v\\}$ that lie in the interior of $P$ onto the set of points of the triangle $\\operatorname{conv}\\{u, v, u + v\\}$ that lie in the interior of $P$.","title":"Point reflection between the two triangles of a parallelogram"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/std-clean-parallelogram-unimodularly-equ","kind":"node","lean":"MetaMathlibExt.stdCleanParallelogram_unimodularlyEquivalent","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unimodular equivalences between the parallelograms "],["math","\\(P_{b,n}\\)"]],"source":"Unimodular equivalences between the parallelograms $P_{b,n}$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#std-clean-parallelogram-unimodularly-equ","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"For natural numbers $b$ and $n$ let $P_{b,n}$ be the parallelogram spanned by $(1, 0)$ and $(b, n)$, with vertices $(0, 0)$, $(1, 0)$, $(b, n)$, $(b + 1, n)$. Let $n$, $a$ and $a'$ be natural numbers with $n > 1$, $1 \\le a < n$, $\\gcd(a, n) = 1$, $1 \\le a' < n$ and $a a' \\equiv 1 \\pmod{n}$. Then $P_{a,n}$ is unimodularly equivalent to each of $P_{n-a,n}$, $P_{a',n}$ and $P_{n-a',n}$.","title":"Unimodular equivalences between the parallelograms $P_{b,n}$"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/clean-parallelogram-unimodularly-equival","kind":"node","lean":"MetaMathlibExt.cleanParallelogram_unimodularlyEquivalent_std","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#clean-parallelogram-unimodularly-equival","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $u$ and $v$ be lattice points spanning a clean lattice parallelogram $P$ whose area is an integer $n \\ge 2$. Then there is an integer $a$ with $1 \\le a < n$ and $\\gcd(a, n) = 1$ such that $P$ is unimodularly equivalent to the standard clean parallelogram $P_{a,n}$.","title":"Clean parallelograms are unimodularly equivalent to standard ones"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/interior-std-clean-parallelogram-lattice","kind":"node","lean":"MetaMathlibExt.interior_stdCleanParallelogram_latticePoints","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#interior-std-clean-parallelogram-lattice","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $1 \\le a < n$ with $\\gcd(a, n) = 1$, let $P_{a,n} = \\{s (1, 0) + t (a, n) : 0 \\le s \\le 1,\\ 0 \\le t \\le 1\\}$, and write $\\langle x \\rangle = x - \\lfloor x \\rfloor$ for the fractional part of a real number $x$. The lattice points in the interior of $P_{a,n}$ are exactly the points $\\langle k(n - a)/n \\rangle \\, (1, 0) + (k/n) \\, (a, n)$ for $k = 1, \\dots, n - 1$.","title":"Interior lattice points of a standard clean parallelogram"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/std-clean-parallelogram-symmetry","kind":"node","lean":"MetaMathlibExt.stdCleanParallelogram_symmetry","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","The point reflection symmetry of "],["math","\\(P_{a,n}\\)"]],"source":"The point reflection symmetry of $P_{a,n}$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#std-clean-parallelogram-symmetry","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $a$ and $n$ be natural numbers with $1 \\le a < n$ and $\\gcd(a, n) = 1$, let $P_{a,n} \\subseteq \\mathbb{R}^2$ be the parallelogram with vertices $(0, 0)$, $(1, 0)$, $(a, n)$, $(a + 1, n)$, and let $S : \\mathbb{R}^2 \\to \\mathbb{R}^2$ be $S(x, y) = (a + 1 - x, n - y)$. Then $S(P_{a,n}) = P_{a,n}$, and $S$ is an isometry of $\\mathbb{R}^2$ for the maximum distance $d((x, y), (x', y')) = \\max(|x - x'|, |y - y'|)$.","title":"The point reflection symmetry of $P_{a,n}$"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/std-column-least-greatest","kind":"node","lean":"MetaMathlibExt.stdColumn_least_greatest","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lowest and highest interior lattice points of "],["math","\\(P_{a,n}\\)"],["text"," on each vertical line"]],"source":"Lowest and highest interior lattice points of $P_{a,n}$ on each vertical line"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#std-column-least-greatest","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $a$ and $n$ be natural numbers with $1 \\le a < n$ and $\\gcd(a, n) = 1$, let $P_{a,n}$ be the parallelogram with vertices $(0, 0)$, $(1, 0)$, $(a, n)$, $(a + 1, n)$, and for an integer $k$ let $S_k$ be the set of integers $y$ such that $(k, y)$ lies in the interior of $P_{a,n}$. If $a = 1$, the least element of $S_1$ is $1$ and the greatest is $n - 1$. If $a > 1$, then: for every integer $k$ with\u2026","title":"Lowest and highest interior lattice points of $P_{a,n}$ on each vertical line"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/std-column-low-add-high","kind":"node","lean":"MetaMathlibExt.stdColumn_low_add_high","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lowest and highest interior lattice points of "],["math","\\(P_{a,n}\\)"],["text"," on paired vertical lines"]],"source":"Lowest and highest interior lattice points of $P_{a,n}$ on paired vertical lines"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#std-column-low-add-high","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $a$ and $n$ be natural numbers with $1 \\le a < n$ and $\\gcd(a, n) = 1$, let $P_{a,n}$ be the parallelogram with vertices $(0, 0)$, $(1, 0)$, $(a, n)$, $(a + 1, n)$, and for an integer $k$ let $S_k$ be the set of integers $y$ such that $(k, y)$ lies in the interior of $P_{a,n}$. Let $k, l \\ge 1$ be natural numbers with $k + l = a + 1$, and suppose that $y_L$ is the least element of $S_k$ and $y_H$ is the\u2026","title":"Lowest and highest interior lattice points of $P_{a,n}$ on paired vertical lines"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/euclid-steps","kind":"node","lean":"MetaMathlibExt.euclidSteps","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#euclid-steps","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"For natural numbers $u$ and $v$, the number of division steps of Euclid's algorithm on $(u, v)$ is $0$ if $v = 0$, and otherwise one more than the number of division steps on $(v, u \\bmod v)$.","title":"Number of steps of Euclid's algorithm"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/euclid-steps-small-values","kind":"node","lean":"MetaMathlibExt.euclidSteps_small_values","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#euclid-steps-small-values","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"Euclid's algorithm takes exactly one division step on $(2, 1)$, two division steps on $(3, 2)$ and five division steps on $(13, 8)$.","title":"Euclid's algorithm on small inputs"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/lame-euclid-steps-minimal","kind":"node","lean":"MetaMathlibExt.lame_euclidSteps_minimal","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#lame-euclid-steps-minimal","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $F_0 = 0$, $F_1 = 1$, $F_{k+2} = F_{k+1} + F_k$ be the Fibonacci numbers, and let $N \\ge 1$. Let $u > v > 0$ be natural numbers on which Euclid's algorithm takes exactly $N$ division steps, and suppose that $u \\le u'$ whenever $u' > v' > 0$ are natural numbers on which it takes exactly $N$ division steps. Then $u = F_{N+2}$ and $v = F_{N+1}$.","title":"Lam\u00e9's theorem (1844)"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/convergent-vec","kind":"node","lean":"MetaMathlibExt.convergentVec","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#convergent-vec","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"Let $x$ be a rational number with regular continued fraction $[q_0; q_1, q_2, \\dots]$, and let $A_j$ and $B_j$ be the numerators and denominators of its convergents, given by $A_{-2} = 0$, $A_{-1} = 1$, $B_{-2} = 1$, $B_{-1} = 0$ and $A_j = q_j A_{j-1} + A_{j-2}$, $B_j = q_j B_{j-1} + B_{j-2}$ for $j \\ge 0$. Stark's convergent vectors of $x$ are $v_j = (B_j, A_j)$ for $j \\ge -2$, so $v_{-2} = (1, 0)$,\u2026","title":"Convergent vectors of a continued fraction"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/partial-quotient","kind":"node","lean":"MetaMathlibExt.partialQuotient","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#partial-quotient","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"Let $x$ be a rational number. The continued fraction algorithm sets $x_0 = x$ and $q_0 = \\lfloor x_0 \\rfloor$ and, as long as $x_j$ is not an integer, $x_{j+1} = 1/(x_j - q_j)$ and $q_{j+1} = \\lfloor x_{j+1} \\rfloor$; it stops at the first index $m$ for which $x_m$ is an integer, so that $x = [q_0; q_1, \\dots, q_m]$. For a natural number $i$, the $i$-th partial quotient of $x$ is $q_i$ when $i \\le m$ and $0$ when\u2026","title":"Partial quotients of a rational number"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/convergent-vec-initial","kind":"node","lean":"MetaMathlibExt.convergentVec_initial","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#convergent-vec-initial","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"For every rational number $x$, the convergent vectors of the regular continued fraction of $x$ begin with $v_{-2} = (1, 0)$, $v_{-1} = (0, 1)$ and $v_0 = (1, \\lfloor x \\rfloor)$, and its zeroth partial quotient is $q_0 = \\lfloor x \\rfloor$.","title":"Initial convergent vectors"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/convergent-vec-recurrence","kind":"node","lean":"MetaMathlibExt.convergentVec_recurrence","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#convergent-vec-recurrence","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"fully_proved","summary":"Let $x$ be a rational number with regular continued fraction $[q_0; q_1, q_2, \\dots]$ and convergent vectors $v_{-2}, v_{-1}, v_0, \\dots$, and let $i \\ge 0$. If the expansion of $x$ has a term $b$ in position $i + 2$ (it does not terminate before $q_{i+2}$), then $q_{i+2} = b$ and $v_{i+2} = v_i + b \\, v_{i+1}$.","title":"Recurrence for convergent vectors"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/partial-quotient-last-ge-two","kind":"node","lean":"MetaMathlibExt.partialQuotient_last_ge_two","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","The last partial quotient of "],["math","\\(n/a\\)"],["text"," is at least "],["math","\\(2\\)"]],"source":"The last partial quotient of $n/a$ is at least $2$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#partial-quotient-last-ge-two","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $a$ and $n$ be natural numbers with $1 \\le a < n$, and let $n/a = [q_0; q_1, \\dots, q_m]$ be the regular continued fraction of $n/a$, where $q_0 = \\lfloor n/a \\rfloor$ and $m$ is the number of partial quotients after $q_0$ produced by the continued fraction algorithm (so $m = 0$ exactly when $a$ divides $n$). Then $q_m \\ge 2$.","title":"The last partial quotient of $n/a$ is at least $2$"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/convergent-vec-parallelogram-lattice-poi","kind":"node","lean":"MetaMathlibExt.convergentVec_parallelogram_latticePoints","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#convergent-vec-parallelogram-lattice-poi","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $1 \\le a < n$ be integers, let $n / a = [q_0; q_1, \\dots, q_m]$ be the regular continued fraction of $n / a$, where $m$ is the least index at which it terminates, and let $v_i$ be its convergent vectors. For every integer $i$ with $-2 \\le i < m$, the lattice points of the parallelogram spanned by $v_i$ and $v_{i+1}$ are exactly $0$, $v_i$, $v_{i+1}$ and $v_i + v_{i+1}$, and every point of $\\mathbb{Z}^2$ equals\u2026","title":"Lattice points of the parallelogram of two consecutive convergent vectors"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/convergent-vec-dist-strict-anti","kind":"node","lean":"MetaMathlibExt.convergentVec_dist_strictAnti","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","Distances of the convergent vectors to the line "],["math","\\(y = (n / a) x\\)"]],"source":"Distances of the convergent vectors to the line $y = (n / a) x$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#convergent-vec-dist-strict-anti","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $1 \\le a < n$ be coprime integers, let $n / a = [q_0; q_1, \\dots, q_m]$ be the regular continued fraction of $n / a$, where $m$ is the least index at which it terminates, and let $v_j = (B_j, A_j)$ be its convergent vectors. For $p = (x, y) \\in \\mathbb{R}^2$ let $d(p) = |n x - a y| / \\sqrt{n^2 + a^2}$ be the distance from $p$ to the line $y = (n / a) x$. Then: if $m \\ge 1$, then\u2026","title":"Distances of the convergent vectors to the line $y = (n / a) x$"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/convergent-vec-mem-quarters","kind":"node","lean":"MetaMathlibExt.convergentVec_mem_quarters","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergent vectors in the quarters of "],["math","\\(P_{a,n}\\)"]],"source":"Convergent vectors in the quarters of $P_{a,n}$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#convergent-vec-mem-quarters","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $1 \\le a < n$ be coprime integers, let $n / a = [q_0; q_1, \\dots, q_m]$ be the regular continued fraction of $n / a$, where $m$ is the least index at which it terminates, and let $v_i$ be its convergent vectors. The lines $y = (n / a)(x - 1/2)$ and $y = n / 2$ cut the parallelogram $P_{a,n}$ with vertices $(0, 0)$, $(1, 0)$, $(a, n)$, $(a + 1, n)$ into four closed parallelograms: $P_1$ (lower left), spanned by\u2026","title":"Convergent vectors in the quarters of $P_{a,n}$"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/convergent-vec-triangle-lattice-points","kind":"node","lean":"MetaMathlibExt.convergentVec_triangle_latticePoints","parent":28205,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lattice points of the triangle with vertices "],["math","\\(0\\)"],["text",", "],["math","\\(v_{i-2}\\)"],["text",", "],["math","\\(v_i\\)"]],"source":"Lattice points of the triangle with vertices $0$, $v_{i-2}$, $v_i$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#convergent-vec-triangle-lattice-points","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $1 \\le a < n$ be integers, let $n / a = [q_0; q_1, \\dots, q_m]$ be the regular continued fraction of $n / a$, where $m$ is the least index at which it terminates, and let $v_j$ be its convergent vectors. For every integer $i$ with $0 \\le i \\le m$, let $T_i$ be the closed triangle with vertices $0$, $v_{i-2}$ and $v_i$. Then the segment $[v_{i-2}, v_i]$ contains exactly $q_i + 1$ lattice points, the interior of\u2026","title":"Lattice points of the triangle with vertices $0$, $v_{i-2}$, $v_i$"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/convergent-vec-det-three","kind":"node","lean":"MetaMathlibExt.convergentVec_det_three","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#convergent-vec-det-three","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $1 \\le a < n$ be integers, let $n / a = [q_0; q_1, \\dots, q_m]$ be the regular continued fraction of $n / a$, where $m$ is the least index at which it terminates, and let $v_j = (B_j, A_j)$ be its convergent vectors. Then for every natural number $i$ with $i + 2 \\le m$, the determinant of the $3 \\times 3$ matrix with rows $(1, B_{i-2}, A_{i-2})$, $(1, B_i, A_i)$ and $(1, B_{i+2}, A_{i+2})$ equals\u2026","title":"A determinant of three convergent vectors"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/extreme-points-interior-hull-std-clean-p","kind":"node","lean":"MetaMathlibExt.extremePoints_interiorHull_stdCleanParallelogram","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#extreme-points-interior-hull-std-clean-p","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $1 \\le a < n$ with $\\gcd(a, n) = 1$, let $P_{a,n} = \\{s (1, 0) + t (a, n) : 0 \\le s \\le 1,\\ 0 \\le t \\le 1\\}$, and let $n / a = [q_0; q_1, \\dots, q_m]$ be the regular continued fraction of $n / a$, with $m \\ge 2$. Write $A_i / B_i$ for its $i$-th convergent, and put $v_{-1} = (0, 1)$, $v_i = (B_i, A_i)$ for $i \\ge 0$, $u_i = v_i + (1, 0)$ (so $u_{-1} = (1, 1)$) and $w = (a + 1, n)$. Then the vertices of the\u2026","title":"Vertices of the interior hull of a standard clean parallelogram"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/interior-hull-std-clean-parallelogram-ve","kind":"node","lean":"MetaMathlibExt.interiorHull_stdCleanParallelogram_vertices_area","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#interior-hull-std-clean-parallelogram-ve","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $1 \\le a < n$ with $\\gcd(a, n) = 1$, let $P_{a,n} = \\{s (1, 0) + t (a, n) : 0 \\le s \\le 1,\\ 0 \\le t \\le 1\\}$, let $n / a = [q_0; q_1, \\dots, q_m]$ be the regular continued fraction of $n / a$, and let $H$ be the convex hull of the lattice points in the interior of $P_{a,n}$. If $H$ has positive area, then $H$ has $2(m + 1)$ vertices when $q_0 > 1$ and $2m$ vertices otherwise (that is, when $q_0 = 1$), and\u2026","title":"Vertices and area of the interior hull of a standard clean parallelogram"},{"id":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/interior-hull-std-clean-parallelogram-cl","kind":"node","lean":"MetaMathlibExt.interiorHull_stdCleanParallelogram_clean_iff_fib","parent":28205,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms/index.html#interior-hull-std-clean-parallelogram-cl","scope":"Combinatorics/polytopes-simplicial-complexes/clean-lattice-parallelograms","status":"can_prove","summary":"Let $F_0 = 0$, $F_1 = 1$, $F_{k+2} = F_{k+1} + F_k$ be the Fibonacci numbers. Let $1 \\le a < n$ with $\\gcd(a, n) = 1$, let $P_{a,n} = \\{s (1, 0) + t (a, n) : 0 \\le s \\le 1,\\ 0 \\le t \\le 1\\}$, and let $H$ be the convex hull of the lattice points in the interior of $P_{a,n}$. Then the lattice points on the boundary of $H$ are exactly the vertices of $H$ if and only if there is a natural number $k$ with $n = F_k$ and\u2026","title":"Clean interior hulls and Fibonacci numbers"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres.json","id":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres","status":"can_prove","title":"Face numbers of simplicial spheres"},{"id":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/finite-simplicial-sphere","kind":"node","lean":"MathlibExt.Combinatorics.SimplicialSphereShared.FiniteSimplicialSphere","parent":28237,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/index.html#finite-simplicial-sphere","scope":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres","status":"fully_proved","summary":"For a natural number $d$, a finite simplicial $(d-1)$-sphere is a geometric simplicial complex $K$ in the Euclidean space $\\mathbb{R}^d$ with finitely many faces whose underlying space is homeomorphic to the unit sphere $S^{d-1} = \\{x \\in \\mathbb{R}^d : \\|x\\| = 1\\}$.","title":"Finite simplicial sphere"},{"id":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/face-count","kind":"node","lean":"MathlibExt.Combinatorics.SimplicialSphereShared.faceCount","parent":28237,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/index.html#face-count","scope":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres","status":"fully_proved","summary":"For a finite simplicial $(d-1)$-sphere $X$ and a natural number $k$, $f_k(X)$ is the number of $k$-dimensional faces of $X$, that is, of faces with exactly $k+1$ vertices.","title":"Face numbers of a simplicial sphere"},{"id":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/extended-face-count","kind":"node","lean":"MathlibExt.Combinatorics.SimplicialSphereShared.extendedFaceCount","parent":28237,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/index.html#extended-face-count","scope":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres","status":"fully_proved","summary":"For a finite simplicial $(d-1)$-sphere $X$ and a natural number $i$, the $i$-th extended face number of $X$ is $1$ if $i = 0$, and is the number $f_{i-1}(X)$ of $(i-1)$-dimensional faces of $X$ if $i \\ge 1$. Thus it equals $f_{i-1}(X)$ under the convention $f_{-1}(X) = 1$.","title":"Extended face numbers"},{"id":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/h-vector","kind":"node","lean":"MathlibExt.Combinatorics.SimplicialSphereShared.hVector","parent":28237,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/index.html#h-vector","scope":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres","status":"fully_proved","summary":"For a finite simplicial $(d-1)$-sphere $X$ and a natural number $j$,","title":"The h-vector of a simplicial sphere"},{"id":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/dehn-sommerville","kind":"node","lean":"MathlibExt.Combinatorics.DehnSommervilleWanted.Combinatorics.DehnSommerville.dehn_sommerville","parent":28237,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres/index.html#dehn-sommerville","scope":"Combinatorics/polytopes-simplicial-complexes/face-numbers-of-simplicial-spheres","status":"can_prove","summary":"Let $X$ be a finite simplicial $(d-1)$-sphere, that is, a finite geometric simplicial complex in $\\mathbb{R}^d$ whose underlying space is homeomorphic to the unit sphere of $\\mathbb{R}^d$. Then for every natural number $j \\le d$, the h-vector of $X$ satisfies $h_j = h_{d-j}$. Here $h_j = \\sum_{i=0}^{j} (-1)^{j-i} \\binom{d-i}{j-i} f_{i-1}$, where $f_{-1} = 1$ and $f_k$ is the number of $k$-dimensional faces of $X$.","title":"Dehn\u2013Sommerville equations"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations.json","id":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations","status":"can_prove","title":"Colour groups of sphere triangulations"},{"id":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/triangles","kind":"node","lean":"MathlibExt.Combinatorics.SphereTriangulationColourGroupsKourovka17Problem35Wanted.triangles","parent":28243,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/index.html#triangles","scope":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations","status":"fully_proved","summary":"For a finite simplicial 2-sphere $X$ in $\\mathbb{R}^3$, the triangles of $X$ are its faces with exactly three vertices, that is, its 2-dimensional faces.","title":"Triangles of a sphere triangulation"},{"id":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/is-proper-triangle-colouring","kind":"node","lean":"MathlibExt.Combinatorics.SphereTriangulationColourGroupsKourovka17Problem35Wanted.IsProperTriangleColouring","parent":28243,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/index.html#is-proper-triangle-colouring","scope":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations","status":"fully_proved","summary":"Let $X$ be a finite simplicial 2-sphere in $\\mathbb{R}^3$, and let a colouring assign white or black to every finite set of points of $\\mathbb{R}^3$. The colouring is proper on $X$ if any two distinct triangles of $X$ (faces with exactly three vertices) that have exactly two vertices in common receive different colours.","title":"Proper colourings of the triangles of a sphere triangulation"},{"id":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/colour-group","kind":"node","lean":"MathlibExt.Combinatorics.SphereTriangulationColourGroupsKourovka17Problem35Wanted.colourGroup","parent":28243,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/index.html#colour-group","scope":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations","status":"fully_proved","summary":"Let $X$ be a finite simplicial 2-sphere in $\\mathbb{R}^3$, let a colouring assign white or black to every finite set of points of $\\mathbb{R}^3$, and let $c$ be one of the two colours. The colour group of $X$ for $c$ is the abelian group generated by the vertices of $X$ with the relations $u + v + w = 0$ for each triangle $\\{u, v, w\\}$ of $X$ (a face with exactly three vertices) of colour $c$; that is, the free\u2026","title":"Colour group of a 2-coloured sphere triangulation"},{"id":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/sphere-triangulation-colour-groups-iso","kind":"node","lean":"MathlibExt.Combinatorics.SphereTriangulationColourGroupsKourovka17Problem35Wanted.sphereTriangulationColourGroupsIso","parent":28243,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/index.html#sphere-triangulation-colour-groups-iso","scope":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations","status":"fully_proved","summary":"The statement that for every finite simplicial 2-sphere $X$ in $\\mathbb{R}^3$ and every colouring of the finite sets of points of $\\mathbb{R}^3$ white or black such that any two distinct triangles of $X$ (faces with exactly three vertices) with exactly two vertices in common receive different colours, the groups $G_W$ and $G_B$ are isomorphic. Here $G_W$ is the abelian group generated by the vertices of $X$ with\u2026","title":"Kourovka Problem 17.35 (statement)"},{"id":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/sphere-triangulation-colour-groups-iso-h","kind":"node","lean":"MathlibExt.Combinatorics.SphereTriangulationColourGroupsKourovka17Problem35Wanted.sphereTriangulationColourGroupsIso_holds","parent":28243,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations/index.html#sphere-triangulation-colour-groups-iso-h","scope":"Combinatorics/polytopes-simplicial-complexes/colour-groups-of-sphere-triangulations","status":"can_prove","summary":"Let $X$ be a finite simplicial 2-sphere in $\\mathbb{R}^3$, and let a colouring assign white or black to every finite set of points of $\\mathbb{R}^3$ so that any two distinct triangles of $X$ (faces with exactly three vertices) with exactly two vertices in common receive different colours. Let $G_W$ be the abelian group generated by the vertices of $X$ with the relations $u + v + w = 0$ for each white triangle\u2026","title":"Blackburn\u2013McCourt theorem (Kourovka Problem 17.35)"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes.json","id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","title":"Rational homology of simplicial complexes"},{"id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/is-simplicial-complex","kind":"node","lean":"MathlibExt.Combinatorics.RationalSimplicialHomologyShared.IsSimplicialComplex","parent":28249,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html#is-simplicial-complex","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","summary":"A finite family $K$ of finite subsets of $\\mathbb{N}$ is a simplicial complex when it is closed under taking subsets: for every $\\sigma \\in K$ and every $\\tau \\subseteq \\sigma$, we have $\\tau \\in K$.","title":"Finite abstract simplicial complex"},{"id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/faces-of-card","kind":"node","lean":"MathlibExt.Combinatorics.RationalSimplicialHomologyShared.facesOfCard","parent":28249,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html#faces-of-card","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","summary":"For a finite family $K$ of finite subsets of $\\mathbb{N}$ and $k \\in \\mathbb{N}$, the faces of $K$ with $k$ vertices are the members $\\sigma \\in K$ with exactly $k$ elements (the faces of dimension $k - 1$).","title":"Faces with a given number of vertices"},{"id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/boundary-matrix","kind":"node","lean":"MathlibExt.Combinatorics.RationalSimplicialHomologyShared.boundaryMatrix","parent":28249,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html#boundary-matrix","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","summary":"For a finite family $K$ of finite subsets of $\\mathbb{N}$ and $k \\in \\mathbb{N}$, the boundary matrix $\\partial_k$ of $K$ is the matrix over $\\mathbb{Q}$ whose rows are indexed by the faces of $K$ with $k$ vertices and whose columns are indexed by the faces of $K$ with $k + 1$ vertices; its entry at $(\\tau, \\sigma)$ is $(-1)^i$ when $\\tau$ is obtained from $\\sigma$ by removing its $i$-th smallest vertex, counting\u2026","title":"Simplicial boundary matrix"},{"id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/reduced-betti","kind":"node","lean":"MathlibExt.Combinatorics.RationalSimplicialHomologyShared.reducedBetti","parent":28249,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html#reduced-betti","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","summary":"For a finite family $K$ of finite subsets of $\\mathbb{N}$ and $k \\in \\mathbb{N}$, let $f_k$ be the number of faces of $K$ with $k$ vertices and let $\\partial_k$ be the boundary matrix of $K$ from the faces with $k + 1$ vertices to the faces with $k$ vertices. The reduced Betti number of $K$ indexed by $k$ is the integer $f_k - \\operatorname{rank} \\partial_{k-1} - \\operatorname{rank} \\partial_k$, with ranks over\u2026","title":"Reduced rational Betti numbers"},{"id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/is-rationally-acyclic","kind":"node","lean":"MathlibExt.Combinatorics.RationalSimplicialHomologyShared.IsRationallyAcyclic","parent":28249,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html#is-rationally-acyclic","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","summary":"A finite family $K$ of finite subsets of $\\mathbb{N}$ is rationally acyclic when its reduced Betti number indexed by $k$ is $0$ for every $k \\in \\mathbb{N}$.","title":"Rationally acyclic complexes"},{"id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/link","kind":"node","lean":"MathlibExt.Combinatorics.RationalSimplicialHomologyShared.link","parent":28249,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html#link","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","summary":"For a finite family $K$ of finite subsets of $\\mathbb{N}$ and a finite set $F \\subseteq \\mathbb{N}$, the link of $F$ in $K$ is $\\operatorname{link}_K(F) = \\{G \\in K \\mid G \\cap F = \\emptyset,\\ G \\cup F \\in K\\}$.","title":"Link"},{"id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/top-card","kind":"node","lean":"MathlibExt.Combinatorics.RationalSimplicialHomologyShared.topCard","parent":28249,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html#top-card","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","summary":"For a finite family $K$ of finite subsets of $\\mathbb{N}$, the largest face size of $K$ is the maximum number of elements of a member of $K$, and $0$ when $K$ is empty.","title":"Largest face size"},{"id":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/is-facet","kind":"node","lean":"MathlibExt.Combinatorics.RationalSimplicialHomologyShared.IsFacet","parent":28249,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes/index.html#is-facet","scope":"Combinatorics/polytopes-simplicial-complexes/rational-homology-of-simplicial-complexes","status":"fully_proved","summary":"For a finite family $K$ of finite subsets of $\\mathbb{N}$, a finite set $F \\subseteq \\mathbb{N}$ is a facet of $K$ when $F \\in K$ and $F$ is maximal under inclusion in $K$: every $G \\in K$ with $F \\subseteq G$ equals $F$.","title":"Facet"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe.json","id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"can_prove","title":"Partitionable and interval-decomposable complexes"},{"id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/has-rank-interval-decomposition","kind":"node","lean":"MathlibExt.Combinatorics.AcyclicLinksIntervalDecompositionAMR156Problem0004Wanted.HasRankIntervalDecomposition","parent":28258,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rank-"],["math","\\(k\\)"],["text"," interval decomposition with a subcomplex of bottoms"]],"source":"Rank-$k$ interval decomposition with a subcomplex of bottoms"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html#has-rank-interval-decomposition","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"fully_proved","summary":"For a natural number $k$, a finite family $K$ of finite subsets of $\\mathbb{N}$ has a rank-$k$ interval decomposition with a subcomplex of bottoms if there are a finite family $B$ of finite subsets of $\\mathbb{N}$, closed under taking subsets, and an assignment of a finite set $\\mathrm{top}(F) \\subseteq \\mathbb{N}$ to each finite set $F \\subseteq \\mathbb{N}$, such that every $F \\in B$ satisfies\u2026","title":"Rank-$k$ interval decomposition with a subcomplex of bottoms"},{"id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/acyclic-links-interval-decomposition","kind":"node","lean":"MathlibExt.Combinatorics.AcyclicLinksIntervalDecompositionAMR156Problem0004Wanted.acyclicLinksIntervalDecomposition","parent":28258,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html#acyclic-links-interval-decomposition","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"fully_proved","summary":"The statement that for every integer $k \\ge 1$ and every finite simplicial complex $K$ on the vertex set $\\mathbb{N}$ (a finite family of finite subsets of $\\mathbb{N}$ closed under taking subsets): if the link $\\mathrm{lk}_K(F) = \\{G \\in K : G \\cap F = \\emptyset,\\ G \\cup F \\in K\\}$ of every face $F \\in K$ with fewer than $k$ vertices (including $\\mathrm{lk}_K(\\emptyset) = K$) is acyclic over $\\mathbb{Q}$ (all of\u2026","title":"Stanley's interval-decomposition conjecture for complexes with acyclic links"},{"id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/acyclic-links-interval-decomposition-ref","kind":"node","lean":"MathlibExt.Combinatorics.AcyclicLinksIntervalDecompositionAMR156Problem0004Wanted.acyclicLinksIntervalDecomposition_refuted","parent":28258,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html#acyclic-links-interval-decomposition-ref","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"can_prove","summary":"It is not true that for every integer $k \\ge 1$ and every finite simplicial complex $K$ on the vertex set $\\mathbb{N}$ (a finite family of finite subsets of $\\mathbb{N}$ closed under taking subsets) such that the link $\\mathrm{lk}_K(F) = \\{G \\in K : G \\cap F = \\emptyset,\\ G \\cup F \\in K\\}$ of every face $F \\in K$ with fewer than $k$ vertices is acyclic over $\\mathbb{Q}$ (all of its reduced rational homology\u2026","title":"Stanley's interval-decomposition conjecture for complexes with acyclic links is false"},{"id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/is-cohen-macaulay-over-rat","kind":"node","lean":"MathlibExt.Combinatorics.CohenMacaulayPartitionabilityAMR156Problem0005Wanted.IsCohenMacaulayOverRat","parent":28258,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cohen\u2013Macaulay complexes over "],["math","\\(\\mathbb{Q}\\)"]],"source":"Cohen\u2013Macaulay complexes over $\\mathbb{Q}$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html#is-cohen-macaulay-over-rat","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"fully_proved","summary":"A finite family $K$ of finite subsets of $\\mathbb{N}$ is Cohen\u2013Macaulay over $\\mathbb{Q}$ when for every member $F$ of $K$ (the empty face included, if it belongs to $K$) the link $L = \\operatorname{lk}_K F = \\{G \\in K : G \\cap F = \\emptyset,\\ G \\cup F \\in K\\}$ satisfies $\\tilde\\beta_i(L) = 0$ for every integer $i$ with $-1 \\le i < \\dim L$. Here $\\dim L$ is one less than the largest number of elements of a member\u2026","title":"Cohen\u2013Macaulay complexes over $\\mathbb{Q}$"},{"id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/is-pure","kind":"node","lean":"MathlibExt.Combinatorics.CohenMacaulayPartitionabilityAMR156Problem0005Wanted.IsPure","parent":28258,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html#is-pure","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"fully_proved","summary":"A finite family $K$ of finite subsets of $\\mathbb{N}$ is pure when all its facets (maximal members) have the same number of elements.","title":"Pure complexes"},{"id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/is-partitionable","kind":"node","lean":"MathlibExt.Combinatorics.CohenMacaulayPartitionabilityAMR156Problem0005Wanted.IsPartitionable","parent":28258,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html#is-partitionable","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"fully_proved","summary":"A finite family $K$ of finite subsets of $\\mathbb{N}$ is partitionable when there is a map $r$ assigning to each facet $F$ of $K$ a set $r(F) \\subseteq F$ such that for every member $\\sigma$ of $K$ there is exactly one facet $F$ of $K$ with $r(F) \\subseteq \\sigma \\subseteq F$.","title":"Partitionable complexes"},{"id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/cohen-macaulay-partitionable","kind":"node","lean":"MathlibExt.Combinatorics.CohenMacaulayPartitionabilityAMR156Problem0005Wanted.cohenMacaulayPartitionable","parent":28258,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html#cohen-macaulay-partitionable","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"fully_proved","summary":"Every finite simplicial complex on the vertex set $\\mathbb{N}$ that is pure and Cohen\u2013Macaulay over $\\mathbb{Q}$ is partitionable.","title":"Stanley\u2013Garsia partitionability conjecture"},{"id":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/cohen-macaulay-partitionable-refuted","kind":"node","lean":"MathlibExt.Combinatorics.CohenMacaulayPartitionabilityAMR156Problem0005Wanted.cohenMacaulayPartitionable_refuted","parent":28258,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe/index.html#cohen-macaulay-partitionable-refuted","scope":"Combinatorics/polytopes-simplicial-complexes/partitionable-and-interval-decomposable-complexe","status":"can_prove","summary":"There is a finite simplicial complex on the vertex set $\\mathbb{N}$ that is pure and Cohen\u2013Macaulay over $\\mathbb{Q}$ but not partitionable.","title":"The Stanley\u2013Garsia partitionability conjecture is false"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres.json","id":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres","status":"can_prove","title":"Upper bound theorem for simplicial spheres"},{"id":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/e","kind":"node","lean":"MathlibExt.Combinatorics.UpperBoundSimplicialSpheresWanted.Combinatorics.UpperBoundSimplicialSpheresProblem.E","parent":28267,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euclidean space "],["math","\\(\\mathbb{R}^d\\)"]],"source":"Euclidean space $\\mathbb{R}^d$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/index.html#e","scope":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres","status":"fully_proved","summary":"For a natural number $d$, the real Euclidean space $\\mathbb{R}^d$ of dimension $d$ with its standard inner product; it is the ambient space of a $(d-1)$-dimensional sphere.","title":"Euclidean space $\\mathbb{R}^d$"},{"id":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/cyclic-h","kind":"node","lean":"MathlibExt.Combinatorics.UpperBoundSimplicialSpheresWanted.Combinatorics.UpperBoundSimplicialSpheresProblem.cyclicH","parent":28267,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/index.html#cyclic-h","scope":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres","status":"fully_proved","summary":"For natural numbers $n$, $d$ and $j$, let $m = \\min(j, d - j)$; the value is $\\binom{n - d - 1 + m}{m}$, where subtraction is truncated at $0$. For $n \\ge d + 1$ and $j \\le d$ this is the entry $h_j$ of the h-vector of the cyclic polytope $C(n, d)$, namely $h_j = \\binom{n-d-1+j}{j}$ for $j \\le d/2$ and $h_j = h_{d-j}$ otherwise. For $j > d$ the value is $1$.","title":"h-vector of the cyclic polytope"},{"id":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/cyclic-polytope-face-count","kind":"node","lean":"MathlibExt.Combinatorics.UpperBoundSimplicialSpheresWanted.Combinatorics.UpperBoundSimplicialSpheresProblem.cyclicPolytopeFaceCount","parent":28267,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/index.html#cyclic-polytope-face-count","scope":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres","status":"fully_proved","summary":"For natural numbers $n$, $d$ and $k$, the value is","title":"Face numbers of the cyclic polytope"},{"id":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/upper-bound-theorem-simplicial-spheres","kind":"node","lean":"MathlibExt.Combinatorics.UpperBoundSimplicialSpheresWanted.Combinatorics.UpperBoundSimplicialSpheresProblem.upper_bound_theorem_simplicial_spheres","parent":28267,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres/index.html#upper-bound-theorem-simplicial-spheres","scope":"Combinatorics/polytopes-simplicial-complexes/upper-bound-theorem-for-simplicial-spheres","status":"can_prove","summary":"Let $d$, $n$ and $k$ be natural numbers with $k < d$, and let $X$ be a finite simplicial $(d-1)$-sphere, that is, a finite geometric simplicial complex in $\\mathbb{R}^d$ whose underlying space is homeomorphic to the unit sphere of $\\mathbb{R}^d$. If $X$ has $n$ vertices, then the number of $k$-dimensional faces of $X$ (faces with $k + 1$ vertices) is at most","title":"Upper bound theorem for simplicial spheres"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices.json","id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","title":"Weighted projective space simplices"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/weighted-simplex","kind":"node","lean":"MetaMathlibExt.weightedSimplex","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#weighted-simplex","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a tuple $q = (q_1, \\dots, q_n)$ of natural numbers, the weighted projective space simplex $\\Delta_{(1,q)}$ is the convex hull in $\\mathbb{R}^n$ of the standard basis vectors $e_1, \\dots, e_n$ and the vector $-(q_1 e_1 + \\dots + q_n e_n)$, whose $i$th coordinate is $-q_i$.","title":"Weighted projective space simplex"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/polytope-dilate","kind":"node","lean":"MetaMathlibExt.polytopeDilate","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#polytope-dilate","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a natural number $t$ and a set $P \\subseteq \\mathbb{R}^\\iota$ of real vectors indexed by a type $\\iota$, the dilate of $P$ by $t$ is the set $tP = \\{t y : y \\in P\\} \\subseteq \\mathbb{R}^\\iota$.","title":"Dilate of a set"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/is-lattice-point","kind":"node","lean":"MetaMathlibExt.IsLatticePoint","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#is-lattice-point","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"Let $P \\subseteq \\mathbb{R}^\\iota$ for an index set $\\iota$. An integer vector $x \\in \\mathbb{Z}^\\iota$ is a lattice point of $P$ when $x$, viewed as a real vector coordinate by coordinate, lies in $P$; thus the lattice points of $P$ are the points of $P \\cap \\mathbb{Z}^\\iota$.","title":"Lattice points of a set"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/has-idp","kind":"node","lean":"MetaMathlibExt.HasIDP","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#has-idp","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"A set $P \\subseteq \\mathbb{R}^\\iota$, for a finite index set $\\iota$, has the integer decomposition property (IDP) if for every integer $m \\ge 1$, every lattice point $w \\in \\mathbb{Z}^\\iota$ of the dilate $mP = \\{m y : y \\in P\\}$ is a sum $w = x_1 + \\dots + x_m$ of $m$ lattice points $x_1, \\dots, x_m \\in \\mathbb{Z}^\\iota$ of $P$.","title":"Integer decomposition property"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/is-reflexive","kind":"node","lean":"MetaMathlibExt.IsReflexive","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#is-reflexive","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"Let $P \\subseteq \\mathbb{R}^n$, with coordinates indexed by a finite set of size $n$, and write $tP = \\{t y : y \\in P\\}$. Then $P$ is reflexive when there is a lattice point $c$ in the interior of $P$ such that, for every integer $t \\ge 1$, the lattice points in the interior of $tP$ are exactly the points $c + y$ with $y$ a lattice point of $(t-1)P$.","title":"Reflexive polytopes"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/ehrhart","kind":"node","lean":"MetaMathlibExt.ehrhart","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#ehrhart","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a finite index set $\\iota$, a set $P \\subseteq \\mathbb{R}^\\iota$ and a natural number $t$, the Ehrhart function $L_P(t)$ is the number of integer vectors $x \\in \\mathbb{Z}^\\iota$ lying in the dilate $tP = \\{t y : y \\in P\\}$, with the convention that $L_P(t) = 0$ when there are infinitely many such vectors.","title":"Ehrhart function"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/ehrhart-eq-lattice-enumerator","kind":"node","lean":"MetaMathlibExt.ehrhart_eq_latticeEnumerator","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#ehrhart-eq-lattice-enumerator","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For every subset $P \\subseteq \\mathbb{R}^d$ and every natural number $t$, the Ehrhart function of $P$ at $t$, the number of integer vectors $x \\in \\mathbb{Z}^d$ lying in the dilate $tP$, equals the lattice-point enumerator of $P$ at the real dilate $t$, the number of points of $tP \\cap \\mathbb{Z}^d$. Both counts are taken to be $0$ when the set counted is infinite.","title":"The Ehrhart function agrees with the lattice-point enumerator"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/is-hstar-polynomial","kind":"node","lean":"MetaMathlibExt.IsHStarPolynomial","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h^*\\)"],["text","-polynomial"]],"source":"The $h^*$-polynomial"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#is-hstar-polynomial","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"The $h^*$-polynomial. Let $P \\subseteq \\mathbb{R}^n$, with coordinates indexed by a finite set of size $n$, and let $L_P(t)$ be the number of lattice points of the dilate $tP = \\{t y : y \\in P\\}$ (taken to be $0$ if there are infinitely many). A polynomial $h \\in \\mathbb{Z}[z]$ is an $h^*$-polynomial of $P$ when","title":"The $h^*$-polynomial"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/is-unimodal","kind":"node","lean":"MetaMathlibExt.IsUnimodal","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#is-unimodal","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"A polynomial $p = \\sum_k a_k z^k \\in \\mathbb{Z}[z]$ of degree $n$ (with $n = 0$ for $p = 0$) is unimodal when there is an index $t \\le n$ such that $a_0 \\le a_1 \\le \\dots \\le a_t$ and $a_t \\ge a_{t+1} \\ge \\dots \\ge a_n$.","title":"Unimodal polynomials"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/is-hstar-unimodal","kind":"node","lean":"MetaMathlibExt.IsHStarUnimodal","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(h^*\\)"],["text","-unimodal polytopes"]],"source":"$h^*$-unimodal polytopes"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#is-hstar-unimodal","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"$h^*$-unimodal polytopes. Let $P \\subseteq \\mathbb{R}^n$, with coordinates indexed by a finite set of size $n$, and let $L_P(t)$ be the number of lattice points of the dilate $tP = \\{t y : y \\in P\\}$ (taken to be $0$ if there are infinitely many). Then $P$ is $h^*$-unimodal when there is a unimodal polynomial $h \\in \\mathbb{Z}[z]$ with","title":"$h^*$-unimodal polytopes"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/expand-support","kind":"node","lean":"MetaMathlibExt.expandSupport","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#expand-support","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For support values $r = (r_1, \\dots, r_d)$ and multiplicities $x = (x_1, \\dots, x_d)$ in $\\mathbb{N}$, the expansion $(r_1^{x_1}, \\dots, r_d^{x_d})$ is the list consisting of $x_1$ copies of $r_1$, followed by $x_2$ copies of $r_2$, and so on, ending with $x_d$ copies of $r_d$.","title":"Expansion of support values with multiplicities"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/valid-support","kind":"node","lean":"MetaMathlibExt.ValidSupport","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#valid-support","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$, a pair of tuples $r = (r_1, \\dots, r_d)$ and $x = (x_1, \\dots, x_d)$ of natural numbers is valid support data when $d \\ge 1$, the support entries $r_1, \\dots, r_d$ are positive and pairwise distinct, and the multiplicities $x_1, \\dots, x_d$ are positive.","title":"Valid support data"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/weighted-simplex-list","kind":"node","lean":"MetaMathlibExt.weightedSimplexList","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#weighted-simplex-list","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a list $q = (q_1, \\dots, q_n)$ of natural numbers, with coordinates indexed by the positions of the list, this is the convex hull in $\\mathbb{R}^n$ of the standard basis vectors $e_1, \\dots, e_n$ and the vector $-(q_1 e_1 + \\dots + q_n e_n)$, whose $i$th coordinate is $-q_i$.","title":"Weighted projective space simplex of a list of weights"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/weight-lcm","kind":"node","lean":"MetaMathlibExt.weightLcm","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#weight-lcm","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a finite list $q = (q_1, \\dots, q_n)$ of natural numbers, $\\operatorname{lcm}(q)$ is $\\operatorname{lcm}(\\cdots\\operatorname{lcm}(\\operatorname{lcm}(1, q_1), q_2)\\cdots, q_n)$, the usual least common multiple of $q_1, \\dots, q_n$; in particular the empty list has least common multiple $1$.","title":"Least common multiple of a list of weights"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/setup-ell","kind":"node","lean":"MetaMathlibExt.setupEll","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The integer "],["math","\\(\\ell\\)"],["text"," of support data"]],"source":"The integer $\\ell$ of support data"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#setup-ell","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For tuples $r = (r_1, \\dots, r_d)$ and $x = (x_1, \\dots, x_d)$ of natural numbers, let $L = \\operatorname{lcm}(r_1, \\dots, r_d)$, with $L = 1$ when $d = 0$. Then $\\ell(r, x)$ is the natural-number quotient $\\lfloor (1 + \\sum_{i=1}^{d} x_i r_i) / L \\rfloor$, taken to be $0$ when $L = 0$.","title":"The integer $\\ell$ of support data"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/setup-s","kind":"node","lean":"MetaMathlibExt.setupS","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#setup-s","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a tuple $r = (r_1, \\dots, r_d)$ of natural numbers and an index $i$, the complementary factor $s_i$ is the natural-number quotient $\\lfloor \\operatorname{lcm}(r_1, \\dots, r_d) / r_i \\rfloor$, taken to be $0$ when $r_i = 0$.","title":"Complementary factors of support data"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/weight-statistic","kind":"node","lean":"MetaMathlibExt.weightStatistic","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#weight-statistic","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a list $q = (q_1, \\dots, q_n)$ of natural numbers and $b \\in \\mathbb{N}$, the weight statistic is $w(q, b) = b - \\sum_{i=1}^{n} \\left\\lfloor \\frac{q_i b}{1 + q_1 + \\cdots + q_n} \\right\\rfloor$.","title":"The weight statistic"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/h-star-formula","kind":"node","lean":"MetaMathlibExt.hStarFormula","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h^*\\)"],["text"," formula of a weight vector"]],"source":"The $h^*$ formula of a weight vector"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#h-star-formula","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"The $h^*$ formula of a weight vector. For a list $q = (q_1, \\dots, q_n)$ of natural numbers, put $|q| = q_1 + \\dots + q_n$ and define the weight statistic $w(q,b) = b - \\sum_{i=1}^n \\lfloor q_i b / (1 + |q|) \\rfloor$. The $h^*$ formula of $q$ is the integer polynomial $\\sum_{b=0}^{|q|} z^{w(q,b)}$.","title":"The $h^*$ formula of a weight vector"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/setup-u","kind":"node","lean":"MetaMathlibExt.setupU","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The exponent "],["math","\\(u(\\alpha)\\)"],["text"," of support data"]],"source":"The exponent $u(\\alpha)$ of support data"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#setup-u","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For tuples $r = (r_1, \\dots, r_d)$ and $x = (x_1, \\dots, x_d)$ of natural numbers and a natural number $\\alpha$, let $\\ell = \\ell(r, x)$ be the integer of the support data and $s_1, \\dots, s_d$ its complementary factors. Then $u(\\alpha) = \\alpha \\ell - \\sum_{i=1}^{d} x_i \\lfloor \\alpha / s_i \\rfloor$, computed in the natural numbers: the subtraction is truncated at $0$, and $\\lfloor \\alpha / s_i \\rfloor$ is taken\u2026","title":"The exponent $u(\\alpha)$ of support data"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/setup-g","kind":"node","lean":"MetaMathlibExt.setupG","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The polynomial "],["math","\\(g_r^x\\)"]],"source":"The polynomial $g_r^x$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#setup-g","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For tuples $r = (r_1, \\dots, r_d)$ and $x = (x_1, \\dots, x_d)$ of natural numbers, let $L = \\operatorname{lcm}(r_1, \\dots, r_d)$, with $L = 1$ when $d = 0$, and let $u(\\alpha)$ be the exponent of the support data $(r, x)$ at $\\alpha$. Then $g_r^x$ is the polynomial with integer coefficients $g_r^x(z) = \\sum_{\\alpha = 0}^{L - 1} z^{u(\\alpha)}$, which is $0$ when $L = 0$.","title":"The polynomial $g_r^x$"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/satisfies-idpnecessary","kind":"node","lean":"MetaMathlibExt.SatisfiesIDPNecessary","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#satisfies-idpnecessary","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"A vector of natural-number weights $q = (q_1, \\dots, q_n)$ satisfies the IDP-necessary condition when $1 + \\sum_{i=1}^n (q_i \\bmod q_j) = q_j$ for every $j$, where $q_i \\bmod q_j$ is the remainder of $q_i$ on division by $q_j$ (with the convention $q_i \\bmod 0 = q_i$).","title":"IDP-necessary condition"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/satisfies-support-idpnecessary","kind":"node","lean":"MetaMathlibExt.SatisfiesSupportIDPNecessary","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#satisfies-support-idpnecessary","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"Let $r = (r_1, \\dots, r_d)$ and $x = (x_1, \\dots, x_d)$ be tuples of natural numbers. The pair $(r, x)$ satisfies the support form of the necessary condition when, for every index $j$, $1 + \\sum_{i=1}^{d} x_i (r_i \\bmod r_j) = r_j$, where $r_i \\bmod r_j$ is the remainder of $r_i$ on division by $r_j$, with the convention $r_i \\bmod 0 = r_i$.","title":"Support form of the IDP necessary condition"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/reflexive-iff-weight-dvd","kind":"node","lean":"MetaMathlibExt.reflexive_iff_weight_dvd","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#reflexive-iff-weight-dvd","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $q = (q_1, \\dots, q_n)$ be natural-number weights with $q_i \\ge 1$ for every $i$. The weighted projective simplex with weights $q$ is reflexive if and only if $q_i$ divides $1 + \\sum_{j=1}^n q_j$ for every $i$.","title":"Reflexivity criterion for weighted projective simplices"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/reflexive-support-iff-lcm-dvd","kind":"node","lean":"MetaMathlibExt.reflexive_support_iff_lcm_dvd","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#reflexive-support-iff-lcm-dvd","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $d \\ge 1$, let $r_1, \\dots, r_d$ be pairwise distinct natural numbers with $r_i \\ge 1$, and let $x_1, \\dots, x_d$ be natural numbers with $x_i \\ge 1$. Let $q = (r_1^{x_1}, \\dots, r_d^{x_d})$ be the weight vector in which each $r_i$ is repeated $x_i$ times, in this order. The weighted projective simplex with weights $q$ is reflexive if and only if $\\operatorname{lcm}(r_1, \\dots, r_d)$ divides\u2026","title":"Reflexivity criterion in support form"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/support-gcd-and-complementary-lcm","kind":"node","lean":"MetaMathlibExt.support_gcd_and_complementary_lcm","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#support-gcd-and-complementary-lcm","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $r = (r_1, \\dots, r_d)$ and $x = (x_1, \\dots, x_d)$ be valid support data: $d \\ge 1$, the $r_i$ are pairwise distinct positive integers, and the $x_i$ are positive integers. Let $q = (r_1^{x_1}, \\dots, r_d^{x_d})$ be the weight vector in which each $r_i$ is repeated $x_i$ times, and suppose that the weighted projective space simplex $\\Delta_{(1, q)}$ is reflexive. Then $\\gcd(r_1, \\dots, r_d) = 1$ and\u2026","title":"Support gcd and complementary lcm of reflexive support data"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/affine-free-sum","kind":"node","lean":"MetaMathlibExt.affineFreeSum","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#affine-free-sum","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For sets $P \\subseteq \\mathbb{R}^n$ and $Q \\subseteq \\mathbb{R}^m$ and a point $v \\in \\mathbb{R}^m$, the affine free sum $P *_v Q \\subseteq \\mathbb{R}^n \\times \\mathbb{R}^m$ is the convex hull of $(P \\times \\{0\\}) \\cup (\\{0\\} \\times (Q - v))$, where $Q - v = \\{y - v : y \\in Q\\}$.","title":"Affine free sum"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/negative-weight-vertex","kind":"node","lean":"MetaMathlibExt.negativeWeightVertex","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#negative-weight-vertex","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For natural-number weights $q = (q_1, \\dots, q_n)$, the negative weight vertex is the point $v_0 = -\\sum_{i=1}^n q_i e_i$ of $\\mathbb{R}^n$, whose $i$th coordinate is $-q_i$; here $e_1, \\dots, e_n$ is the standard basis of $\\mathbb{R}^n$.","title":"Negative weight vertex"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/weighted-free-sum-reflexive","kind":"node","lean":"MetaMathlibExt.weighted_free_sum_reflexive","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#weighted-free-sum-reflexive","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $p = (p_1, \\dots, p_m)$ and $w = (w_1, \\dots, w_k)$ be lists of positive integers such that the weighted projective space simplices $\\Delta_{(1,p)}$ and $\\Delta_{(1,w)}$ are reflexive, and let $q = (p_1, \\dots, p_m, (1 + \\sum_i p_i) w_1, \\dots, (1 + \\sum_i p_i) w_k)$. Then $\\Delta_{(1,q)}$ is reflexive.","title":"Reflexivity of the weighted free sum"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/affine-free-sum-idp","kind":"node","lean":"MetaMathlibExt.affineFreeSum_idp","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#affine-free-sum-idp","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Affine free sums of IDP weighted simplices at the negative weight vertex of a reflexive summand are IDP. Let $q = (q_1, \\dots, q_n)$ and $p = (p_1, \\dots, p_m)$ be tuples of natural numbers, let $\\Delta_{(1,q)} \\subseteq \\mathbb{R}^n$ and $\\Delta_{(1,p)} \\subseteq \\mathbb{R}^m$ be the weighted projective space simplices of $q$ and $p$, and let $v_0 = -\\sum_{i=1}^m p_i e_i$ be the negative weight vertex of\u2026","title":"Affine free sums of IDP weighted simplices at the negative weight vertex of a reflexive summand are IDP"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/affine-free-sum-reflexive-idp-h-star-uni","kind":"node","lean":"MetaMathlibExt.affineFreeSum_reflexive_idp_hStarUnimodal","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","Affine free sums of weighted simplices at the negative weight vertex preserve reflexivity, IDP and "],["math","\\(h^\\ast\\)"],["text","-unimodality"]],"source":"Affine free sums of weighted simplices at the negative weight vertex preserve reflexivity, IDP and $h^\\ast$-unimodality"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#affine-free-sum-reflexive-idp-h-star-uni","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Affine free sums of weighted simplices at the negative weight vertex preserve reflexivity, IDP and $h^\\ast$-unimodality. Let $q = (q_1, \\dots, q_n)$ and $p = (p_1, \\dots, p_m)$ be tuples of natural numbers, let $\\Delta_{(1,q)} \\subseteq \\mathbb{R}^n$ and $\\Delta_{(1,p)} \\subseteq \\mathbb{R}^m$ be the weighted projective space simplices of $q$ and $p$, and let $v_0 = -\\sum_{i=1}^m p_i e_i$ be the negative weight\u2026","title":"Affine free sums of weighted simplices at the negative weight vertex preserve reflexivity, IDP and $h^\\ast$-unimodality"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/finite-idp-reflexive-multiplicities","kind":"node","lean":"MetaMathlibExt.finite_idp_reflexive_multiplicities","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#finite-idp-reflexive-multiplicities","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $r_1 < r_2 < \\dots < r_d$ be positive integers such that $r_j \\nmid r_d$ for some $j < d$. Then there are only finitely many vectors $x = (x_1, \\dots, x_d)$ of positive integers for which the weighted projective simplex $\\Delta_{(1,q)}$ is reflexive and has the integer decomposition property, where $q = (r_1^{x_1}, \\dots, r_d^{x_d})$ consists of $x_1$ copies of $r_1$, then $x_2$ copies of $r_2$, and so on,\u2026","title":"Finitely many reflexive IDP multiplicity vectors"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/idp-reflexive-floor-characterization","kind":"node","lean":"MetaMathlibExt.idp_reflexive_floor_characterization","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#idp-reflexive-floor-characterization","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $q = (q_1, \\dots, q_n)$ be positive integers such that the weighted simplex $\\Delta_{(1,q)} = \\operatorname{conv}(e_1, \\dots, e_n, -\\sum_i q_i e_i) \\subseteq \\mathbb{R}^n$ is reflexive, and for each $j$ write $S_j = \\sum_{i \\ne j} q_i$. Then $\\Delta_{(1,q)}$ has the integer decomposition property if and only if the following holds for every $j$: for every integer $b$ with $1 \\le b < q_j$ and","title":"Floor criterion for IDP of reflexive weighted simplices"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/idp-reflexive-implies-necessary","kind":"node","lean":"MetaMathlibExt.idp_reflexive_implies_necessary","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#idp-reflexive-implies-necessary","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $q = (q_1, \\dots, q_n)$ be natural numbers. If the weighted simplex $\\Delta_{(1,q)} = \\operatorname{conv}(e_1, \\dots, e_n, -\\sum_i q_i e_i) \\subseteq \\mathbb{R}^n$ is reflexive and has the integer decomposition property, then $1 + \\sum_{i=1}^n (q_i \\bmod q_j) = q_j$ for every $j$.","title":"A necessary condition for IDP reflexive weighted simplices"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/idp-reflexive-implies-support-necessary","kind":"node","lean":"MetaMathlibExt.idp_reflexive_implies_support_necessary","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#idp-reflexive-implies-support-necessary","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $r_1, \\dots, r_d$ be pairwise distinct positive integers with $d \\ge 1$, let $x_1, \\dots, x_d$ be positive integers, let $n = x_1 + \\dots + x_d$, and let $q = (q_1, \\dots, q_n)$ be the weight vector $(r_1^{x_1}, \\dots, r_d^{x_d})$ in which each $r_i$ is repeated $x_i$ times. If the weighted projective space simplex $\\Delta_{(1,q)}$, the convex hull of the standard basis vectors $e_1, \\dots, e_n$ of\u2026","title":"Necessary condition for IDP reflexive weighted simplices, support form"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/weighted-simplex-h-star-formula","kind":"node","lean":"MetaMathlibExt.weightedSimplex_hStar_formula","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h^*\\)"],["text","-polynomial of a weighted projective space simplex"]],"source":"The $h^*$-polynomial of a weighted projective space simplex"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#weighted-simplex-h-star-formula","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"The $h^*$-polynomial of a weighted projective space simplex. Let $q = (q_1, \\dots, q_n)$ be a list of positive integers, let $\\Delta_{(1,q)} \\subset \\mathbb{R}^n$ be the weighted projective space simplex, and let $L(t)$ be the number of lattice points in its $t$-th dilate $t \\Delta_{(1,q)}$. Then, as formal power series over $\\mathbb{Z}$,","title":"The $h^*$-polynomial of a weighted projective space simplex"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/h-star-geometric-factorization","kind":"node","lean":"MetaMathlibExt.hStar_geometric_factorization","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","Geometric factorization of the "],["math","\\(h^*\\)"],["text"," formula"]],"source":"Geometric factorization of the $h^*$ formula"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#h-star-geometric-factorization","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Geometric factorization of the $h^*$ formula. Let $d \\ge 1$, let $r_1, \\dots, r_d$ be pairwise distinct positive integers, let $x_1, \\dots, x_d$ be positive integers, and let $q = (r_1^{x_1}, \\dots, r_d^{x_d})$ be the list in which each $r_i$ is repeated $x_i$ times. Suppose that the weighted simplex $\\Delta_{(1,q)}$, the convex hull of the standard basis vectors and $-q$, is reflexive. Put\u2026","title":"Geometric factorization of the $h^*$ formula"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/idp-reflexive-multiplicity-bounds","kind":"node","lean":"MetaMathlibExt.idp_reflexive_multiplicity_bounds","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#idp-reflexive-multiplicity-bounds","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $r_1 < r_2 < \\dots < r_d$ be positive integers, let $x_1, \\dots, x_d$ be positive integers, and let $q = (r_1^{x_1}, \\dots, r_d^{x_d})$ be the weight vector in which each $r_i$ is repeated $x_i$ times. Suppose the weighted projective space simplex $\\Delta_{(1,q)}$ is reflexive and has the integer decomposition property. Then $x_i \\le \\lfloor r_{i+1} / r_i \\rfloor$ for every $i < d$, and for every $j < d$ with\u2026","title":"Multiplicity bounds for IDP reflexive weighted simplices"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/one-supported-idp-reflexive-support-eq-o","kind":"node","lean":"MetaMathlibExt.one_supported_idp_reflexive_support_eq_one","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#one-supported-idp-reflexive-support-eq-o","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $r \\ge 1$ and $x \\ge 1$ be natural numbers, and let $q = (r, \\dots, r)$ be the weight vector consisting of $x$ copies of $r$. If the weighted projective simplex with weights $q$ is reflexive and has the integer decomposition property, then $r = 1$.","title":"One-supported IDP reflexive weighted simplices have all weights one"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/two-supported-idp-reflexive-iff-necessar","kind":"node","lean":"MetaMathlibExt.two_supported_idp_reflexive_iff_necessary","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#two-supported-idp-reflexive-iff-necessar","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $r = (r_1, r_2)$ and $x = (x_1, x_2)$ be pairs of positive integers with $r_1 < r_2$, and let $q = (r_1^{x_1}, r_2^{x_2})$ be the weight vector consisting of $r_1$ repeated $x_1$ times followed by $r_2$ repeated $x_2$ times. The weighted projective space simplex $\\Delta_{(1,q)}$ is reflexive and has the integer decomposition property if and only if $1 + \\sum_{i=1}^{2} x_i (r_i \\bmod r_j) = r_j$ for $j = 1, 2$.","title":"IDP reflexive two-supported weighted simplices"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/two-supported-classification","kind":"node","lean":"MetaMathlibExt.two_supported_classification","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#two-supported-classification","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $r = (r_1, r_2)$ and $x = (x_1, x_2)$ be pairs of positive integers with $r_1 < r_2$. Then $1 + \\sum_{i=1}^{2} x_i (r_i \\bmod r_j) = r_j$ holds for $j = 1, 2$ (the support form of the IDP necessary condition) if and only if either $r = (1, 1 + x_1)$ or $r = (1 + x_2, 1 + (1 + x_2)x_1)$.","title":"Classification of two-supported vectors satisfying the necessary condition"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/three-support-family","kind":"node","lean":"MetaMathlibExt.threeSupportFamily","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#three-support-family","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a case index $c \\in \\mathbb{N}$, natural numbers $k, s$ and a triple $x = (x_1, x_2, x_3)$ of natural numbers, the support triple $F_c(k, s; x) \\in \\mathbb{N}^3$ is defined case by case, and $k, s$ enter only when $c \\ge 7$. For $c = 0$ it is $(1, 1 + x_1, (1 + x_1)(1 + x_2))$. For $c = 1$ it is $(1 + x_2, 1 + x_1(1 + x_2), (1 + x_1(1 + x_2))(1 + x_2))$. For $c = 2$ it is\u2026","title":"Support triples of the three-supported classification"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/three-supported-classification","kind":"node","lean":"MetaMathlibExt.three_supported_classification","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#three-supported-classification","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"For $c, k, s \\in \\mathbb{N}$ and $x \\in \\mathbb{N}^3$, write $F_c(k, s; x)$ for the support triple of case $c$ with parameters $k, s$ in the three-supported classification. Let $r = (r_1, r_2, r_3)$ and $x = (x_1, x_2, x_3)$ be triples of positive integers with $r_1 < r_2 < r_3$, and suppose that $1 + \\sum_{i=1}^{3} x_i (r_i \\bmod r_j) = r_j$ for every $j$ (the support form of the IDP necessary condition). Then\u2026","title":"Classification of three-supported vectors satisfying the necessary condition"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/first-seven-three-supported-families-are","kind":"node","lean":"MetaMathlibExt.first_seven_three_supported_families_are_idp","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#first-seven-three-supported-families-are","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $x = (x_1, x_2, x_3)$ be positive integers, and let $r = (r_1, r_2, r_3)$ be the support triple of any one of the first seven cases, (i)\u2013(vii), of the three-support family with parameters $x$. Let $q = (r_1^{x_1}, r_2^{x_2}, r_3^{x_3})$ be the list in which each $r_i$ is repeated $x_i$ times. Then the weighted simplex $\\Delta_{(1,q)}$, the convex hull of the standard basis vectors and $-q$, is reflexive and has\u2026","title":"Cases (i)\u2013(vii) of the three-support family give IDP reflexive simplices"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/eighth-three-supported-family-is-not-idp","kind":"node","lean":"MetaMathlibExt.eighth_three_supported_family_is_not_idp","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#eighth-three-supported-family-is-not-idp","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $x_1, x_2, x_3$ and $k, s$ be positive integers with $x_3 = skx_2 + s - k + 1$, and put $r_1 = 1 + kx_2$, $r_2 = (skx_2 + s + k)(1 + x_1(1 + kx_2))$ and $r_3 = (1 + x_1(1 + kx_2))(1 + x_2(skx_2 + s + k))$. Suppose that $1 + \\sum_{i=1}^{3} x_i (r_i \\bmod r_j) = r_j$ for $j = 1, 2, 3$. Let $q$ be the weight vector $(r_1^{x_1}, r_2^{x_2}, r_3^{x_3})$, consisting of $x_1$ copies of $r_1$, then $x_2$ copies of\u2026","title":"Case (viii) of the three-supported classification is reflexive but not IDP"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/reflexive-stabilization-number","kind":"node","lean":"MetaMathlibExt.reflexiveStabilizationNumber","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#reflexive-stabilization-number","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a list of natural-number weights $q = (q_1, \\dots, q_n)$, the reflexive stabilization number $\\mathrm{rsn}(q)$ is the least natural number $k$ such that the weighted projective simplex with weights $(1^k, q_1, \\dots, q_n)$, that is, $k$ entries equal to $1$ followed by $q$, is reflexive. If no such $k$ exists, then $\\mathrm{rsn}(q) = 0$.","title":"Reflexive stabilization number"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/reflexive-stabilization","kind":"node","lean":"MetaMathlibExt.reflexiveStabilization","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#reflexive-stabilization","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a list of natural-number weights $q = (q_1, \\dots, q_n)$, the reflexive stabilization of $q$ is the list $\\mathrm{rs}(q) = (1^k, q_1, \\dots, q_n)$ obtained by prepending $k = \\mathrm{rsn}(q)$ entries equal to $1$, where $\\mathrm{rsn}(q)$ is the reflexive stabilization number of $q$.","title":"Reflexive stabilization"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/reflexive-stabilization-at","kind":"node","lean":"MetaMathlibExt.reflexiveStabilizationAt","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#reflexive-stabilization-at","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a list of natural-number weights $q = (q_1, \\dots, q_n)$ and a natural number $m$, the $m$th reflexive stabilization of $q$ is the list $\\mathrm{rs}(q, m) = (1^k, q_1, \\dots, q_n)$ obtained by prepending $k$ entries equal to $1$, where $k = \\mathrm{rsn}(q) + (m - 1) \\operatorname{lcm}(q)$. Here $\\mathrm{rsn}(q)$ is the reflexive stabilization number of $q$, $\\operatorname{lcm}(q)$ is the least common multiple\u2026","title":"Higher reflexive stabilizations"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/list-ell","kind":"node","lean":"MetaMathlibExt.listEll","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The number "],["math","\\(\\ell(q)\\)"],["text"," of a weight list"]],"source":"The number $\\ell(q)$ of a weight list"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#list-ell","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a list $q = (q_1, \\dots, q_n)$ of natural numbers, $\\ell(q) = \\lfloor (1 + q_1 + \\dots + q_n) / \\operatorname{lcm}(q_1, \\dots, q_n) \\rfloor$, where the least common multiple of the empty list is $1$, and $\\ell(q) = 0$ when some $q_i = 0$ (so that the least common multiple is $0$).","title":"The number $\\ell(q)$ of a weight list"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/higher-reflexive-stabilizations-not-idp","kind":"node","lean":"MetaMathlibExt.higher_reflexive_stabilizations_not_idp","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#higher-reflexive-stabilizations-not-idp","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $q = (q_1, \\dots, q_n)$ be a nonempty list of integers with $q_i \\ge 2$ for all $i$, and let $m \\ge 2$. Let $\\operatorname{rsn}(q)$ be the least $k$ for which the weighted simplex of $(1^k, q)$, the list $q$ preceded by $k$ ones, is reflexive, and let $\\operatorname{rs}(q,m) = (1^{\\operatorname{rsn}(q) + (m-1)\\operatorname{lcm}(q)}, q)$ be the $m$th reflexive stabilization. Then the weighted simplex\u2026","title":"Higher reflexive stabilizations are not IDP"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/stabilization-eventually-not-unimodal-co","kind":"node","lean":"MetaMathlibExt.stabilization_eventually_not_unimodal_coefficients_one_or_two","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","Large reflexive stabilizations of weights at least "],["math","\\(2\\)"],["text"," with lcm at least "],["math","\\(3\\)"],["text"," are not "],["math","\\(h^\\ast\\)"],["text","-unimodal"]],"source":"Large reflexive stabilizations of weights at least $2$ with lcm at least $3$ are not $h^\\ast$-unimodal"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#stabilization-eventually-not-unimodal-co","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Large reflexive stabilizations of weights at least $2$ with lcm at least $3$ are not $h^\\ast$-unimodal. Let $q = (q_1, \\dots, q_n)$ be a list of natural numbers with $q_i \\ge 2$ for every $i$ and $\\operatorname{lcm}(q) \\ge 3$. For $m \\in \\mathbb{N}$ let $\\operatorname{rs}(q, m)$ be the $m$th reflexive stabilization of $q$, let $\\Delta_{(1,\\operatorname{rs}(q,m))}$ be its weighted projective space simplex, and let\u2026","title":"Large reflexive stabilizations of weights at least $2$ with lcm at least $3$ are not $h^\\ast$-unimodal"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/stabilization-weight-statistic-eq","kind":"node","lean":"MetaMathlibExt.stabilization_weightStatistic_eq","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#stabilization-weight-statistic-eq","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $q = (q_1, \\dots, q_n)$ be a vector of natural numbers whose reflexive stabilization number is $1$, so that its first reflexive stabilization is $\\operatorname{rs}(q) = (1, q_1, \\dots, q_n)$. Suppose that $\\ell(\\operatorname{rs}(q)) = 1$, where for a vector $p$ we write $\\ell(p) = \\lfloor (1 + \\sum_i p_i) / \\operatorname{lcm}(p) \\rfloor$. Then for every natural number $b \\le \\sum_i q_i$, the weight statistics\u2026","title":"The first reflexive stabilization preserves the weight statistic"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/stabilization-h-star-add-monomial","kind":"node","lean":"MetaMathlibExt.stabilization_hStar_add_monomial","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first reflexive stabilization adds one monomial to the "],["math","\\(h^*\\)"],["text","-formula"]],"source":"The first reflexive stabilization adds one monomial to the $h^*$-formula"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#stabilization-h-star-add-monomial","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"The first reflexive stabilization adds one monomial to the $h^*$-formula. Let $q = (q_1, \\dots, q_n)$ be a list of natural numbers whose reflexive stabilization number is $1$, so that its first reflexive stabilization is $\\operatorname{rs}(q) = (1, q_1, \\dots, q_n)$. For a list $p = (p_1, \\dots, p_m)$ of natural numbers put $|p| = p_1 + \\dots + p_m$, let $\\ell(p) = \\lfloor (1 + |p|) / \\operatorname{lcm}(p) \\rfloor$\u2026","title":"The first reflexive stabilization adds one monomial to the $h^*$-formula"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/is-cone-lattice-point","kind":"node","lean":"MetaMathlibExt.IsConeLatticePoint","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#is-cone-lattice-point","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a set $P \\subseteq \\mathbb{R}^n$, a pair $(t, w)$ with $t \\in \\mathbb{N}$ and $w \\in \\mathbb{Z}^n$ is a lattice point of the cone over $P$ when $w$ is a lattice point of the dilate $tP = \\{t y : y \\in P\\}$.","title":"Lattice points of the cone over a polytope"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/is-hilbert-basis-element","kind":"node","lean":"MetaMathlibExt.IsHilbertBasisElement","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#is-hilbert-basis-element","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"Let $P \\subseteq \\mathbb{R}^n$. A pair $a = (t, w)$ with $t \\in \\mathbb{N}$ and $w \\in \\mathbb{Z}^n$ is a Hilbert basis element of the cone over $P$ when $a$ is a lattice point of that cone (that is, $w$ is a lattice point of $tP = \\{t y : y \\in P\\}$), $a \\ne 0$, and $a$ is irreducible: whenever $a = b + c$ with $b$ and $c$ lattice points of the cone over $P$, we have $b = 0$ or $c = 0$.","title":"Hilbert basis elements of the cone over a polytope"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/hilbert-basis","kind":"node","lean":"MetaMathlibExt.hilbertBasis","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#hilbert-basis","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For $P \\subseteq \\mathbb{R}^n$, call a pair $(t, w) \\in \\mathbb{N} \\times \\mathbb{Z}^n$ a cone lattice point of $P$ when $w$ is a lattice point of the dilate $tP = \\{t y : y \\in P\\}$. The Hilbert basis of $P$ is the set of nonzero cone lattice points $a$ of $P$ that are irreducible: whenever $a = b + c$ with $b$ and $c$ cone lattice points of $P$, one of $b$ and $c$ is zero.","title":"Hilbert basis of the cone over a polytope"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idpweights","kind":"node","lean":"MetaMathlibExt.almostIDPWeights","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idpweights","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a natural number $n \\ge 1$, the weight vector $q(n) \\in \\mathbb{N}^{2n+2}$ has its first $2n-1$ coordinates equal to $1$ and its last three coordinates equal to $3n$, $10n$ and $15n$, in that order; that is, $q(n) = (1^{2n-1}, 3n, 10n, 15n)$, where $1^{2n-1}$ denotes $2n-1$ entries equal to $1$.","title":"The almost-IDP weight vector"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idpweights-eq-reflexive-stabiliza","kind":"node","lean":"MetaMathlibExt.almostIDPWeights_eq_reflexiveStabilization","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((1^{2n-1}, 3n, 10n, 15n)\\)"],["text"," is a reflexive stabilization"]],"source":"$(1^{2n-1}, 3n, 10n, 15n)$ is a reflexive stabilization"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idpweights-eq-reflexive-stabiliza","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"For every $n \\ge 1$, the weight vector $(1^{2n-1}, 3n, 10n, 15n)$, consisting of $2n - 1$ ones followed by $3n$, $10n$, $15n$, is the first reflexive stabilization of $(3n, 10n, 15n)$.","title":"$(1^{2n-1}, 3n, 10n, 15n)$ is a reflexive stabilization"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/homogenized-weighted-vertices","kind":"node","lean":"MetaMathlibExt.homogenizedWeightedVertices","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#homogenized-weighted-vertices","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For weights $q = (q_1, \\dots, q_n)$, the homogenized vertex set is the subset of $\\mathbb{N} \\times \\mathbb{Z}^n$ consisting of the point $(1, -q) = (1, (-q_1, \\dots, -q_n))$ and the points $(1, e_i)$ for $i = 1, \\dots, n$, where $e_i$ is the $i$th standard basis vector.","title":"Homogenized vertices of a weighted simplex"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/exceptional-hilbert-column","kind":"node","lean":"MetaMathlibExt.exceptionalHilbertColumn","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#exceptional-hilbert-column","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"Let $n \\ge 1$. Write $t_0, \\dots, t_8$ for the integer triples $(0,0,0)$, $(0,0,-1)$, $(0,-1,-1)$, $(0,-1,-2)$, $(0,-2,-3)$, $(0,-3,-5)$, $(-1,-4,-6)$, $(-2,-7,-10)$ and $(-3,-10,-15)$. For $j = 0, \\dots, 8$ the $j$th exceptional column is the pair $(h_j, v_j)$, where the height $h_j$ is $1$ for $j \\le 6$ and $2$ for $j = 7, 8$, and $v_j \\in \\mathbb{Z}^{2n+2}$ has its first $2n-1$ coordinates equal to $0$ and its\u2026","title":"Exceptional Hilbert basis columns"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idp-hilbert-basis","kind":"node","lean":"MetaMathlibExt.almostIDP_hilbertBasis","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hilbert basis of "],["math","\\(\\Delta_{(1,q(n))}\\)"]],"source":"Hilbert basis of $\\Delta_{(1,q(n))}$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idp-hilbert-basis","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $n \\ge 1$ and let $q(n) = (1^{2n-1}, 3n, 10n, 15n)$, that is, $2n - 1$ ones followed by $3n$, $10n$, $15n$. Write lattice points of the cone over the weighted projective simplex $\\Delta_{(1,q(n))} \\subseteq \\mathbb{R}^{2n+2}$ as pairs $(k, w)$ with height $k \\in \\mathbb{N}$ and $w \\in \\mathbb{Z}^{2n+2}$. The Hilbert basis of this cone is exactly the set of homogenized vertices, namely $(1, e_i)$ for each\u2026","title":"Hilbert basis of $\\Delta_{(1,q(n))}$"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idp-exactly-two-height-two","kind":"node","lean":"MetaMathlibExt.almostIDP_exactly_two_height_two","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idp-exactly-two-height-two","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $n \\ge 1$ and let $q(n) = (1^{2n-1}, 3n, 10n, 15n)$, that is, $2n - 1$ ones followed by $3n$, $10n$, $15n$. Write the elements of the Hilbert basis of the cone over the weighted projective simplex $\\Delta_{(1,q(n))}$ as pairs $(k, w)$ with height $k \\in \\mathbb{N}$ and $w \\in \\mathbb{Z}^{2n+2}$. Exactly two elements of this Hilbert basis have height $k > 1$, and every element of height $k > 1$ has height $k = 2$.","title":"Exactly two Hilbert basis elements of height two"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/even-geometric-factor","kind":"node","lean":"MetaMathlibExt.evenGeometricFactor","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#even-geometric-factor","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a natural number $n$, the even geometric factor is the polynomial $\\sum_{k=0}^{n-1} z^{2k} = 1 + z^2 + \\dots + z^{2n-2}$ with integer coefficients; for $n = 0$ it is the zero polynomial.","title":"Even geometric factor"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idpquartic","kind":"node","lean":"MetaMathlibExt.almostIDPQuartic","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The quartic "],["math","\\(1 + 7z + 14z^2 + 7z^3 + z^4\\)"]],"source":"The quartic $1 + 7z + 14z^2 + 7z^3 + z^4$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idpquartic","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"The polynomial $1 + 7z + 14z^2 + 7z^3 + z^4$ with integer coefficients, the quartic factor from Theorem 13.","title":"The quartic $1 + 7z + 14z^2 + 7z^3 + z^4$"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idpcoefficient-vector","kind":"node","lean":"MetaMathlibExt.almostIDPCoefficientVector","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficient vector "],["math","\\((1, 7, 15, 14, \\dots, 15, 7, 1)\\)"]],"source":"Coefficient vector $(1, 7, 15, 14, \\dots, 15, 7, 1)$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idpcoefficient-vector","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For a natural number $n$, the list of integers $(1, 7, 15, 14)$, followed by $n - 2$ copies of the pair $(16, 14)$, followed by $(15, 7, 1)$. The middle block is empty when $n \\le 2$.","title":"Coefficient vector $(1, 7, 15, 14, \\dots, 15, 7, 1)$"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idp-h-star-factorization","kind":"node","lean":"MetaMathlibExt.almostIDP_hStar_factorization","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h^*\\)"],["text","-polynomial of "],["math","\\(\\Delta_{(1,q(n))}\\)"],["text"," factors"]],"source":"The $h^*$-polynomial of $\\Delta_{(1,q(n))}$ factors"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idp-h-star-factorization","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"The $h^*$-polynomial of $\\Delta_{(1,q(n))}$ factors. Let $n \\ge 2$, let $q(n) = (1^{2n-1}, 3n, 10n, 15n)$, that is, $2n - 1$ ones followed by $3n$, $10n$, $15n$, and let $\\Delta_{(1,q(n))} \\subseteq \\mathbb{R}^{2n+2}$ be the weighted projective simplex. For $t \\ge 0$, let $L(t) = \\#\\big(t\\,\\Delta_{(1,q(n))} \\cap \\mathbb{Z}^{2n+2}\\big)$ be the number of lattice points in the dilate $t\\,\\Delta_{(1,q(n))}$. Then\u2026","title":"The $h^*$-polynomial of $\\Delta_{(1,q(n))}$ factors"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idp-h-star-coefficients","kind":"node","lean":"MetaMathlibExt.almostIDP_hStar_coefficients","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficients of "],["math","\\((1 + z^2 + \\dots + z^{2n-2})(1 + 7z + 14z^2 + 7z^3 + z^4)\\)"]],"source":"Coefficients of $(1 + z^2 + \\dots + z^{2n-2})(1 + 7z + 14z^2 + 7z^3 + z^4)$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idp-h-star-coefficients","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"can_prove","summary":"Let $n \\ge 2$. The coefficients of $z^0, z^1, \\dots, z^{2n+2}$ in the integer polynomial $(1 + z^2 + \\dots + z^{2n-2})(1 + 7z + 14z^2 + 7z^3 + z^4)$ are, in order, $1, 7, 15, 14$, then $n - 2$ repetitions of $16, 14$, then $15, 7, 1$.","title":"Coefficients of $(1 + z^2 + \\dots + z^{2n-2})(1 + 7z + 14z^2 + 7z^3 + z^4)$"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/expand-support-example","kind":"node","lean":"MetaMathlibExt.expandSupport_example","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#expand-support-example","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"Expanding the support values $r = (1, 3)$ with multiplicities $x = (2, 1)$, that is, taking $2$ copies of $1$ followed by $1$ copy of $3$, gives the list $(1, 1, 3)$; in the notation $(r_1^{x_1}, r_2^{x_2})$ for this expansion, $(1^2, 3^1) = (1, 1, 3)$.","title":"Example of support expansion"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/weight-lcm-example","kind":"node","lean":"MetaMathlibExt.weightLcm_example","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","Least common multiple of the weights "],["math","\\((2, 2, 3)\\)"]],"source":"Least common multiple of the weights $(2, 2, 3)$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#weight-lcm-example","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"The least common multiple of the list $(2, 2, 3)$ is $\\operatorname{lcm}(2, 2, 3) = 6$.","title":"Least common multiple of the weights $(2, 2, 3)$"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/h-star-formula-two-two-three","kind":"node","lean":"MetaMathlibExt.hStarFormula_two_two_three","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(h^*\\)"],["text"," formula for the weights "],["math","\\((2,2,3)\\)"]],"source":"The $h^*$ formula for the weights $(2,2,3)$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#h-star-formula-two-two-three","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"The $h^*$ formula for the weights $(2,2,3)$. For $q = (2, 2, 3)$ the $h^*$ formula $\\sum_{b=0}^{7} z^{w(q,b)}$, with weight statistic $w(q,b) = b - \\lfloor 2b/8 \\rfloor - \\lfloor 2b/8 \\rfloor - \\lfloor 3b/8 \\rfloor$, equals the integer polynomial $1 + 2z + 4z^2 + z^3$.","title":"The $h^*$ formula for the weights $(2,2,3)$"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/setup-ell-example","kind":"node","lean":"MetaMathlibExt.setupEll_example","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The integer "],["math","\\(\\ell\\)"],["text"," for the weights "],["math","\\((1^7, 3^4, 5^5)\\)"]],"source":"The integer $\\ell$ for the weights $(1^7, 3^4, 5^5)$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#setup-ell-example","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For the support values $r = (1, 3, 5)$ with multiplicities $x = (7, 4, 5)$, that is, for the weight vector $q = (1^7, 3^4, 5^5)$, the integer $\\ell(r, x) = \\lfloor (1 + \\sum_i x_i r_i) / \\operatorname{lcm}(r_1, r_2, r_3) \\rfloor$ equals $3$.","title":"The integer $\\ell$ for the weights $(1^7, 3^4, 5^5)$"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/polytope-dilate-one","kind":"node","lean":"MetaMathlibExt.polytopeDilate_one","parent":28272,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#polytope-dilate-one","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"For every index type $\\iota$ and every set $P \\subseteq \\mathbb{R}^\\iota$, the dilate $1P = \\{1 \\cdot y : y \\in P\\}$ equals $P$.","title":"Dilation by one"},{"id":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/almost-idpweights-one","kind":"node","lean":"MetaMathlibExt.almostIDPWeights_one","parent":28272,"reading_title":{"format":"inline-parts-v1","parts":[["text","The almost-IDP weight vector for "],["math","\\(n = 1\\)"]],"source":"The almost-IDP weight vector for $n = 1$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices/index.html#almost-idpweights-one","scope":"Combinatorics/polytopes-simplicial-complexes/weighted-projective-space-simplices","status":"fully_proved","summary":"The weight vector $q(n) = (1^{2n-1}, 3n, 10n, 15n)$ of Theorem 13, where $1^{2n-1}$ denotes $2n-1$ entries equal to $1$, taken at $n = 1$ and listed in coordinate order, is $(1, 3, 10, 15)$.","title":"The almost-IDP weight vector for $n = 1$"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical.json","id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","title":"Ehrhart polynomials with roots on the canonical line"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/canonical-line","kind":"node","lean":"MetaMathlibExt.canonicalLine","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#canonical-line","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"The canonical line $\\mathrm{CL}$ is the vertical line of complex numbers with real part $-1/2$, that is, $\\mathrm{CL} = \\{z \\in \\mathbb{C} : \\operatorname{Re} z = -1/2\\}$.","title":"Canonical line"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/h-star-coeff","kind":"node","lean":"MetaMathlibExt.hStarCoeff","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#h-star-coeff","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"Let $f$ be a complex polynomial of degree $d$, with $d = 0$ for the zero polynomial. For a natural number $i$, the $i$-th $h^*$-coefficient of $f$ is","title":"The h-star coefficients of a polynomial"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/is-snn","kind":"node","lean":"MetaMathlibExt.IsSNN","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#is-snn","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For a complex polynomial $f$ of degree $d$ and a natural number $i$, let $h^*_i(f)$ be the coefficient of $t^i$ in $(1 - t)^{d+1} \\sum_{k \\ge 0} f(k)\\, t^k$. Then $f$ is SNN when $f \\ne 0$ and its $h^*$-coefficients $h^*_0(f), \\dots, h^*_d(f)$ are real and nonnegative.","title":"SNN polynomials"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/cl-quadratic-product","kind":"node","lean":"MetaMathlibExt.clQuadraticProduct","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#cl-quadratic-product","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For a finite list $c_1, \\dots, c_m$ of real numbers, this is the monic complex polynomial $\\prod_{k=1}^{m} (X^2 + X + c_k)$; the empty list gives the constant polynomial $1$.","title":"Product of quadratic canonical-line factors"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/is-clwith-parameters","kind":"node","lean":"MetaMathlibExt.IsCLWithParameters","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#is-clwith-parameters","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"Let $f$ be a complex polynomial of degree $d$, let $a$ be a real number, and let $c_1, \\dots, c_m$ be a list of real numbers. Then $f$ is a CL-polynomial with scale $a$ and parameters $c_1, \\dots, c_m$ when $a \\ne 0$, every $c_i \\ge 1/4$, $m = \\lfloor d/2 \\rfloor$, and $f = a \\prod_{i=1}^{m} (X^2 + X + c_i)$ if $d$ is even and $f = a (2X + 1) \\prod_{i=1}^{m} (X^2 + X + c_i)$ if $d$ is odd.","title":"CL-polynomials with given parameters"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/is-cl","kind":"node","lean":"MetaMathlibExt.IsCL","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#is-cl","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"A complex polynomial $f$ of degree $d$ is a CL-polynomial when there are a real number $a \\ne 0$ and real numbers $c_1, \\dots, c_m \\ge 1/4$ with $m = \\lfloor d/2 \\rfloor$ such that $f = a \\prod_{i=1}^{m} (X^2 + X + c_i)$ if $d$ is even and $f = a (2X + 1) \\prod_{i=1}^{m} (X^2 + X + c_i)$ if $d$ is odd.","title":"CL-polynomials"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/b-basis","kind":"node","lean":"MetaMathlibExt.bBasis","parent":28345,"reading_title":{"format":"inline-parts-v1","parts":[["text","The basis polynomials "],["math","\\(b^d_j\\)"]],"source":"The basis polynomials $b^d_j$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#b-basis","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For natural numbers $d$ and $j$, $b^d_j$ is the complex polynomial $b^d_j = \\prod_{i=0}^{d-1} (X + d - j - i)$.","title":"The basis polynomials $b^d_j$"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/p-basis","kind":"node","lean":"MetaMathlibExt.pBasis","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#p-basis","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For natural numbers $d$ and $j$, let $b^d_j$ be the complex polynomial $\\prod_{i=0}^{d-1} (X + d - j - i)$. The polynomial $p^d_j$ is $b^d_j$ if $2j = d$, and $b^d_j + b^d_{d-j}$ otherwise, where $d - j$ is truncated at $0$ (so $p^d_j = b^d_j + b^d_0$ when $j > d$).","title":"Symmetric basis polynomials"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/p-basis-one-zero","kind":"node","lean":"MetaMathlibExt.pBasis_one_zero","parent":28345,"reading_title":{"format":"inline-parts-v1","parts":[["text","The basis polynomial "],["math","\\(p^1_0\\)"]],"source":"The basis polynomial $p^1_0$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#p-basis-one-zero","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"The symmetric basis polynomial $p^1_0$ equals the complex polynomial $2X + 1$.","title":"The basis polynomial $p^1_0$"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/p-basis-two-zero","kind":"node","lean":"MetaMathlibExt.pBasis_two_zero","parent":28345,"reading_title":{"format":"inline-parts-v1","parts":[["text","The basis polynomial "],["math","\\(p^2_0\\)"]],"source":"The basis polynomial $p^2_0$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#p-basis-two-zero","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"The symmetric basis polynomial $p^2_0$ equals the complex polynomial $2(X^2 + X + 1)$.","title":"The basis polynomial $p^2_0$"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/list-interlaces","kind":"node","lean":"MetaMathlibExt.ListInterlaces","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#list-interlaces","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"Let $r$ be a binary relation on a type $\\alpha$ and let $d \\ge 0$. A list $(a_1, \\dots, a_d)$ interlaces a list $(b_1, \\dots, b_{d+1})$ with respect to $r$ when $b_1 \\mathrel{r} a_1 \\mathrel{r} b_2 \\mathrel{r} \\cdots \\mathrel{r} a_d \\mathrel{r} b_{d+1}$, that is, $b_k \\mathrel{r} a_k$ and $a_k \\mathrel{r} b_{k+1}$ for $1 \\le k \\le d$. Two lists interlace only when the second has exactly one more entry than the\u2026","title":"Interlacing lists"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/list-alternates","kind":"node","lean":"MetaMathlibExt.ListAlternates","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#list-alternates","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"Let $r$ be a binary relation on a type $\\alpha$ and let $n \\ge 0$. A list $(a_1, \\dots, a_n)$ alternates with a list $(b_1, \\dots, b_n)$ with respect to $r$ when $a_1 \\mathrel{r} b_1 \\mathrel{r} a_2 \\mathrel{r} \\cdots \\mathrel{r} a_n \\mathrel{r} b_n$, that is, $a_k \\mathrel{r} b_k$ for $1 \\le k \\le n$ and $b_k \\mathrel{r} a_{k+1}$ for $1 \\le k < n$. Two empty lists alternate, and lists of different lengths never\u2026","title":"Alternating lists of equal length"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/real-interlaces","kind":"node","lean":"MetaMathlibExt.RealInterlaces","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#real-interlaces","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"Let $f$ and $g$ be real polynomials. Then $f$ interlaces $g$ when $f$ and $g$ are nonzero with $\\deg g = \\deg f + 1$, the real roots of $f$, listed with multiplicity, are exactly $a_1 \\le \\dots \\le a_d$ where $d = \\deg f$, the real roots of $g$, listed with multiplicity, are exactly $b_1 \\le \\dots \\le b_{d+1}$ (so all roots of $f$ and $g$ are real), and the roots alternate as\u2026","title":"Real interlacing"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/real-alternates","kind":"node","lean":"MetaMathlibExt.RealAlternates","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#real-alternates","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"Let $f$ and $g$ be real polynomials. Their roots alternate when $f$ and $g$ are nonzero of the same degree $n$, the real roots of $f$, listed with multiplicity, are exactly $a_1 \\le \\dots \\le a_n$, those of $g$ are exactly $b_1 \\le \\dots \\le b_n$ (so all roots of $f$ and $g$ are real), and either $a_1 \\le b_1 \\le a_2 \\le b_2 \\le \\dots \\le a_n \\le b_n$ or $b_1 \\le a_1 \\le b_2 \\le a_2 \\le \\dots \\le b_n \\le a_n$.","title":"Alternating roots of equal-degree real polynomials"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/clinterlaces","kind":"node","lean":"MetaMathlibExt.CLInterlaces","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#clinterlaces","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"A complex polynomial $f$ CL-interlaces a complex polynomial $g$ when $f$ and $g$ are nonzero, $\\deg g = \\deg f + 1$, all roots of $f$ and of $g$ lie on the canonical line, and, writing $d = \\deg f$ and listing the roots with multiplicity as $a_1, \\dots, a_d$ for $f$ and $b_1, \\dots, b_{d+1}$ for $g$, each list sorted by increasing imaginary part, they alternate:\u2026","title":"Interlacing on the canonical line"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/has-only-real-roots","kind":"node","lean":"MetaMathlibExt.HasOnlyRealRoots","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#has-only-real-roots","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"A real polynomial $f$ has only real roots when $f \\ne 0$ and the number of real roots of $f$, counted with multiplicity, equals the degree of $f$.","title":"Real-rooted polynomials"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/lattice-point-to-real","kind":"node","lean":"MetaMathlibExt.latticePointToReal","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#lattice-point-to-real","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"A lattice point $x = (x_1, \\dots, x_d) \\in \\mathbb{Z}^d$ is regarded as the point of $\\mathbb{R}^d$ with the same coordinates $x_1, \\dots, x_d$.","title":"Lattice points as real points"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/lattice-polytope","kind":"node","lean":"MetaMathlibExt.latticePolytope","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#lattice-polytope","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"The lattice polytope spanned by a finite set $V \\subseteq \\mathbb{Z}^d$ is the convex hull in $\\mathbb{R}^d$ of the points of $V$.","title":"Lattice polytopes"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/is-full-dimensional","kind":"node","lean":"MetaMathlibExt.IsFullDimensional","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#is-full-dimensional","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"A subset $P$ of $\\mathbb{R}^d$ is full-dimensional when it is nonempty and the direction of its affine span is all of $\\mathbb{R}^d$ (equivalently, the affine span of $P$ is $\\mathbb{R}^d$).","title":"Full-dimensional sets"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/ehrhart-counts","kind":"node","lean":"MetaMathlibExt.EhrhartCounts","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#ehrhart-counts","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"Let $V \\subseteq \\mathbb{Z}^d$ be a finite set and let $P \\subseteq \\mathbb{R}^d$ be the lattice polytope spanned by $V$, its convex hull. A complex polynomial $E$ counts the lattice points of $P$ when, for every natural number $k$, $E(k)$ equals the ordinary Ehrhart count $|kP \\cap \\mathbb{Z}^d|$, the number of points of $\\mathbb{Z}^d$ of the form $kx$ with $x$ a rational point of $P$.","title":"Polynomial counting the lattice points of dilates"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/binomial-polynomial","kind":"node","lean":"MetaMathlibExt.binomialPolynomial","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#binomial-polynomial","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For an integer $n$ and a natural number $d$, the binomial polynomial $\\binom{X+n}{d}$ is the real polynomial $\\frac{1}{d!} \\prod_{i=0}^{d-1} (X + n - i)$.","title":"Binomial polynomial"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/complex-binomial-polynomial","kind":"node","lean":"MetaMathlibExt.complexBinomialPolynomial","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#complex-binomial-polynomial","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For an integer $n$ and a natural number $d$, the complex binomial polynomial $\\binom{X+n}{d} \\in \\mathbb{C}[X]$ is","title":"Complex binomial polynomial"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/model-polynomial","kind":"node","lean":"MetaMathlibExt.modelPolynomial","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#model-polynomial","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For a natural number $d$, the model polynomial is the complex polynomial $M_d = \\binom{X+d}{d} + \\binom{X}{d}$, where for an integer $n$ the binomial polynomial $\\binom{X+n}{d}$ is $\\frac{1}{d!}\\prod_{i=0}^{d-1} (X + n - i)$.","title":"Model polynomial"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/has-diamond","kind":"node","lean":"MetaMathlibExt.HasDiamond","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#has-diamond","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For a complex polynomial $f$ of degree $d$ and a natural number $i$, let $h^*_i(f)$ be the coefficient of $t^i$ in $(1 - t)^{d+1} \\sum_{k \\ge 0} f(k)\\, t^k$. Then $f$ has property (\u2662) when $f \\ne 0$ and its $h^*$-coefficients $h^*_0(f), \\dots, h^*_d(f)$ are real, nonnegative and palindromic, that is, $h^*_i(f) = h^*_{d-i}(f)$ for $0 \\le i \\le d$.","title":"Property (\u2662)"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/canonical-line-root-set","kind":"node","lean":"MetaMathlibExt.canonicalLineRootSet","parent":28345,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\Omega_d\\)"],["text"," of attainable canonical-line roots"]],"source":"The set $\\Omega_d$ of attainable canonical-line roots"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#canonical-line-root-set","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For a natural number $d$, $\\Omega_d$ is the set of complex numbers $z$ on the canonical line such that $z$ is a root of some CL-polynomial of degree $d$ satisfying $(\\diamondsuit)$.","title":"The set $\\Omega_d$ of attainable canonical-line roots"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/functional-equation-iff-h-star-palindrom","kind":"node","lean":"MetaMathlibExt.functionalEquation_iff_hStar_palindromic","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#functional-equation-iff-h-star-palindrom","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $f$ be a nonzero complex polynomial of degree $d$ with $h^*$-coefficients $h^*_0(f), \\dots, h^*_d(f)$. Then $f(X-1) = (-1)^d f(-X)$ as polynomials if and only if $h^*_i(f) = h^*_{d-i}(f)$ for every $i \\le d$.","title":"Functional equation versus palindromic h-star coefficients"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/cl-value-on-canonical-line","kind":"node","lean":"MetaMathlibExt.cl_value_on_canonicalLine","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#cl-value-on-canonical-line","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"If $f$ is a CL-polynomial of degree $d$, then for every $z$ on the canonical line there is a real number $r$ with $f(z) = r\\, i^d$; that is, $f$ maps the canonical line into $\\mathbb{R}\\, i^d$.","title":"Values of CL-polynomials on the canonical line"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/p-basis-zero-interlaces-succ","kind":"node","lean":"MetaMathlibExt.pBasis_zero_interlaces_succ","parent":28345,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p^d_0\\)"],["text"," CL-interlaces "],["math","\\(p^{d+1}_0\\)"]],"source":"$p^d_0$ CL-interlaces $p^{d+1}_0$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#p-basis-zero-interlaces-succ","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"For every natural number $d \\ge 1$, the symmetric basis polynomial $p^d_0$ CL-interlaces $p^{d+1}_0$.","title":"$p^d_0$ CL-interlaces $p^{d+1}_0$"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/p-basis-zero-interlaces-perturbation","kind":"node","lean":"MetaMathlibExt.pBasis_zero_interlaces_perturbation","parent":28345,"reading_title":{"format":"inline-parts-v1","parts":[["text","CL-interlacing of "],["math","\\(p^d_0\\)"],["text"," with a perturbation of "],["math","\\(p^{d+1}_0\\)"]],"source":"CL-interlacing of $p^d_0$ with a perturbation of $p^{d+1}_0$"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#p-basis-zero-interlaces-perturbation","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"For every natural number $d \\ge 1$ and every real number $w > 0$, the symmetric basis polynomial $p^d_0$ CL-interlaces the polynomial $p^{d+1}_0 + w \\, (2X + 1) \\, p^d_0$.","title":"CL-interlacing of $p^d_0$ with a perturbation of $p^{d+1}_0$"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/two-mul-x-add-one-preserves-snn","kind":"node","lean":"MetaMathlibExt.two_mul_X_add_one_preserves_snn","parent":28345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplying by "],["math","\\(2X+1\\)"],["text"," preserves SNN"]],"source":"Multiplying by $2X+1$ preserves SNN"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#two-mul-x-add-one-preserves-snn","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"For every complex polynomial $f$, if $f$ is an SNN polynomial, then so is $(2X + 1) f$.","title":"Multiplying by $2X+1$ preserves SNN"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/quadratic-mul-binomial-polynomial","kind":"node","lean":"MetaMathlibExt.quadratic_mul_binomialPolynomial","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#quadratic-mul-binomial-polynomial","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"For an integer $n$ and a natural number $k$, write $\\binom{X+n}{k} = \\frac{1}{k!} \\prod_{i=0}^{k-1} (X + n - i)$ for the real binomial polynomial. Let $d$ and $j$ be natural numbers with $0 < d$ and $j \\le d$, and let $c$ be a real number. Then","title":"Multiplying a binomial polynomial by a quadratic"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/leibniz-rule-interlacing","kind":"node","lean":"MetaMathlibExt.leibniz_rule_interlacing","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#leibniz-rule-interlacing","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $f, f_1, g, g_1$ be real polynomials with positive leading coefficients, each having only real roots, such that $f$ and $g$ have no common real root, $f_1$ interlaces $f$, and $g_1$ interlaces $g$. Then $f_1 g_1$ interlaces $f g_1 + f_1 g$, the polynomial $f g_1 + f_1 g$ interlaces $f g$, the roots of $f g_1 + f_1 g$ alternate with those of $f g_1$ and with those of $f_1 g$, and $f g_1 + f_1 g$ has only real\u2026","title":"Fisk's Leibniz rule for interlacing"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/coefficientwise-limit-interlaces","kind":"node","lean":"MetaMathlibExt.coefficientwise_limit_interlaces","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#coefficientwise-limit-interlaces","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $(f_n)$ and $(g_n)$ be sequences of real polynomials with only real roots, and let $f$ and $g$ be nonzero real polynomials such that, for every $k$, the coefficient of $X^k$ in $f_n$ tends to that in $f$ and the coefficient of $X^k$ in $g_n$ tends to that in $g$ as $n \\to \\infty$. Suppose that for all sufficiently large $n$ the degree of $f_n$ equals the degree of $f$ and the degree of $g_n$ equals the degree\u2026","title":"Coefficientwise limits preserve real interlacing"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/sorted-cl-parameters-are-snn","kind":"node","lean":"MetaMathlibExt.sorted_cl_parameters_are_snn","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#sorted-cl-parameters-are-snn","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $f$ be a complex polynomial of degree $d$, let $a > 0$ be real, and let $c_0 \\le c_1 \\le \\dots \\le c_{m-1}$ be real numbers, each at least $1/4$, with $m = \\lfloor d/2 \\rfloor$, such that $f = a \\prod_{k=0}^{m-1} (X^2 + X + c_k)$ if $d$ is even and $f = a (2X + 1) \\prod_{k=0}^{m-1} (X^2 + X + c_k)$ if $d$ is odd. Suppose that $c_k \\le 2k + 2$ for every $k$ when $d$ is odd, and $c_k \\le 2k + 1$ for every $k$\u2026","title":"Small sorted CL parameters give an SNN polynomial"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/ehrhart-root-classical-bounds","kind":"node","lean":"MetaMathlibExt.ehrhart_root_classical_bounds","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#ehrhart-root-classical-bounds","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^d$ be a full-dimensional lattice polytope, the convex hull of a finite set of points of $\\mathbb{Z}^d$, and let $E$ be a complex polynomial with $E(k) = |kP \\cap \\mathbb{Z}^d|$ for every natural number $k$. Then every complex root $\\alpha$ of $E$ satisfies $|\\alpha| \\le 1 + (d+1)!$, and every real root $\\alpha$ of $E$ satisfies $-d \\le \\alpha < \\lfloor d/2 \\rfloor$.","title":"Classical bounds on Ehrhart roots"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/ehrhart-roots-in-braun-disc","kind":"node","lean":"MetaMathlibExt.ehrhart_roots_in_braun_disc","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#ehrhart-roots-in-braun-disc","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^d$ be a full-dimensional lattice polytope, the convex hull of a finite set of points of $\\mathbb{Z}^d$, and let $E$ be a complex polynomial with $E(k) = |kP \\cap \\mathbb{Z}^d|$ for every natural number $k$. Then every root $\\alpha$ of $E$ lies in the closed disc centered at $-1/2$ with radius $d(d - 1/2)$, that is, $|\\alpha + 1/2| \\le d(d - 1/2)$.","title":"Braun's disc for Ehrhart roots"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/maximal-model-root-asymptotic","kind":"node","lean":"MetaMathlibExt.maximal_model_root_asymptotic","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#maximal-model-root-asymptotic","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"For $d \\ge 0$ let $M_d = \\binom{X+d}{d} + \\binom{X}{d}$ be the model polynomial. Let $(\\beta_d)_{d \\ge 0}$ be a sequence of complex numbers such that for every $d \\ge 1$, $\\beta_d$ is a root of $M_d$ and $|\\gamma| \\le |\\beta_d|$ for every root $\\gamma$ of $M_d$. Then $|\\beta_d + \\tfrac12| - d^2/\\pi = O(1)$ as $d \\to \\infty$.","title":"Asymptotics of the largest root of the model polynomial"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/hibi-lower-bound","kind":"node","lean":"MetaMathlibExt.hibi_lower_bound","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#hibi-lower-bound","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $P \\subseteq \\mathbb{R}^d$ be the lattice polytope spanned by a finite set of points of $\\mathbb{Z}^d$, and suppose that $P$ is full-dimensional. Let $E$ be a complex polynomial with $E(k) = |kP \\cap \\mathbb{Z}^d|$ for every natural number $k$, and write $h^*_i$ for the $h^*$-coefficients of $E$. If $h^*_d \\ne 0$, then $\\operatorname{Re} h^*_1 \\le \\operatorname{Re} h^*_k$ for every $k$ with $1 \\le k < d$.","title":"Hibi's lower bound"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/canonical-line-root-set-is-connected","kind":"node","lean":"MetaMathlibExt.canonicalLineRootSet_isConnected","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#canonical-line-root-set-is-connected","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"For every positive integer $d$, the set $\\Omega_d$ of canonical-line roots of degree-$d$ CL-polynomials satisfying $(\\diamondsuit)$ is a nonempty connected subset of $\\mathbb{C}$.","title":"The set of attainable canonical-line roots is connected"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/cl-parameter-bounded-by-p-basis-root","kind":"node","lean":"MetaMathlibExt.cl_parameter_bounded_by_pBasis_root","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#cl-parameter-bounded-by-p-basis-root","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $f$ be a complex polynomial of degree $d$ that is a CL-polynomial with real scale $a$ and list of real parameters $c_1, \\dots, c_m$, and suppose that $f$ is SNN. Then for each parameter $c_k$ there is a real number $\\beta$ such that $-\\tfrac12 + \\beta i$ is a root of the symmetric basis polynomial $p^d_0$ and $c_k \\le \\tfrac14 + \\beta^2$.","title":"Parameter bound for SNN CL-polynomials"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/clspan","kind":"node","lean":"MetaMathlibExt.clspan","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#clspan","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"The CL-span of a complex polynomial $f$ is the convex hull, in $\\mathbb{C}$ viewed as a real vector space, of the set of roots of $f$ lying on the canonical line, the line of complex numbers with real part $-1/2$. Here the zero polynomial is regarded as having no roots, so its CL-span is empty.","title":"CL-span"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/clspan-zero","kind":"node","lean":"MetaMathlibExt.clspan_zero","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#clspan-zero","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"The zero polynomial has empty CL-span (its root multiset is empty).","title":"CL-span of the zero polynomial"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/standard-reflexive-simplex-vertices","kind":"node","lean":"MetaMathlibExt.standardReflexiveSimplexVertices","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#standard-reflexive-simplex-vertices","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For a natural number $d$, this is the finite set of integer points $\\{e_1, \\dots, e_d, (-1, \\dots, -1)\\} \\subseteq \\mathbb{Z}^d$, where $e_1, \\dots, e_d$ are the standard basis vectors of $\\mathbb{Z}^d$, so that $(-1, \\dots, -1) = -(e_1 + \\dots + e_d)$. These points are the vertices of the standard reflexive simplex $\\Delta^d_{\\mathrm{sr}}$.","title":"Vertices of the standard reflexive simplex"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/neg-one-mem-standard-reflexive-simplex-v","kind":"node","lean":"MetaMathlibExt.neg_one_mem_standardReflexiveSimplexVertices","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#neg-one-mem-standard-reflexive-simplex-v","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"fully_proved","summary":"For every natural number $d$, let $e_1, \\dots, e_d$ be the standard basis vectors of $\\mathbb{Z}^d$. Then the point $-(e_1 + \\dots + e_d) = (-1, \\dots, -1)$ belongs to the vertex set $\\{e_1, \\dots, e_d, -(e_1 + \\dots + e_d)\\}$ of the standard reflexive simplex $\\Delta^d_{sr}$.","title":"The all-minus-one vector is a vertex of the standard reflexive simplex"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/p-basis-canonical-line-properties","kind":"node","lean":"MetaMathlibExt.pBasis_canonicalLine_properties","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#p-basis-canonical-line-properties","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $d \\ge 1$ and $j$ be natural numbers with $2j \\le d$, let $p^d_j$ be the symmetric basis polynomial, and let $i$ be the imaginary unit. Then: (a) the real part of $p^d_j(it - \\tfrac12) / i^d$ tends to $+\\infty$ as the real number $t$ tends to $+\\infty$; (b) if $2j \\ne d$, then for every real $t$ we have $p^d_j(it - \\tfrac12) = 0$ if and only if there is a real number $r$ with\u2026","title":"The symmetric basis polynomials on the canonical line"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/clspan-subset-clspan-p-basis-zero","kind":"node","lean":"MetaMathlibExt.clspan_subset_clspan_pBasis_zero","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#clspan-subset-clspan-p-basis-zero","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $f$ be a complex polynomial of degree $d$ that is both a CL-polynomial and an SNN polynomial. Then the CL-span of $f$ is contained in the CL-span of the symmetric basis polynomial $p^d_0$.","title":"CL-span bound for SNN CL-polynomials"},{"id":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/clspan-ehrhart-subset-standard-reflexive","kind":"node","lean":"MetaMathlibExt.clspan_ehrhart_subset_standardReflexiveSimplex","parent":28345,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical/index.html#clspan-ehrhart-subset-standard-reflexive","scope":"Combinatorics/polytopes-simplicial-complexes/ehrhart-polynomials-with-roots-on-the-canonical","status":"can_prove","summary":"Let $d \\le 9$, let $P \\subseteq \\mathbb{R}^d$ be the convex hull of a finite set of integer points, and suppose that $P$ is reflexive. Let $\\Delta^d_{sr}$ be the standard reflexive simplex, with vertices $e_1, \\dots, e_d$ and $-(e_1 + \\cdots + e_d)$. Let $E$ and $E'$ be complex polynomials such that, for every natural number $k$, $E(k)$ is the number of integer points of $kP$ and $E'(k)$ is the number of integer\u2026","title":"Ehrhart CL-spans of reflexive polytopes in dimension at most 9"},{"data_url":"scopes/Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors.json","id":"Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors","kind":"scope","navigation":"module","parent":28203,"result":false,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors/index.html","scope":"Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors","status":"can_prove","title":"Unimodality of h*-vectors"},{"id":"Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors/lattice-polytope","kind":"node","lean":"MathlibExt.Combinatorics.HibiSymmetricHStarUnimodalOWR741Problem003Wanted.latticePolytope","parent":28393,"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors/index.html#lattice-polytope","scope":"Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors","status":"fully_proved","summary":"The lattice polytope in `\u211d^d` spanned by the finite set `S \u2286 \u2124^d` of integer points.","title":"Lattice polytope spanned by integer points"},{"id":"Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors/hibi-symmetric-unimodal","kind":"node","lean":"MathlibExt.Combinatorics.HibiSymmetricHStarUnimodalOWR741Problem003Wanted.hibiSymmetricUnimodal","parent":28393,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hibi's conjecture on symmetric "],["math","\\(h^*\\)"],["text","-vectors"]],"source":"Hibi's conjecture on symmetric $h^*$-vectors"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors/index.html#hibi-symmetric-unimodal","scope":"Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors","status":"fully_proved","summary":"Hibi's conjecture on symmetric $h^*$-vectors. For every natural number $d$, every finite set $S \\subseteq \\mathbb{Z}^d$ whose convex hull $P \\subseteq \\mathbb{R}^d$ has nonempty interior, and every polynomial $h = \\sum_i h^*_i z^i \\in \\mathbb{Z}[z]$ that is an $h^*$-polynomial of $P$, that is,","title":"Hibi's conjecture on symmetric $h^*$-vectors"},{"id":"Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors/hibi-symmetric-unimodal-refuted","kind":"node","lean":"MathlibExt.Combinatorics.HibiSymmetricHStarUnimodalOWR741Problem003Wanted.hibiSymmetricUnimodal_refuted","parent":28393,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hibi's conjecture on symmetric "],["math","\\(h^*\\)"],["text","-vectors is false"]],"source":"Hibi's conjecture on symmetric $h^*$-vectors is false"},"result":true,"route":"../roadmap/Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors/index.html#hibi-symmetric-unimodal-refuted","scope":"Combinatorics/polytopes-simplicial-complexes/unimodality-of-h-vectors","status":"can_prove","summary":"Hibi's conjecture on symmetric $h^*$-vectors is false. There are a natural number $d$, a finite set $S \\subseteq \\mathbb{Z}^d$ whose convex hull $P \\subseteq \\mathbb{R}^d$ has nonempty interior, and a polynomial $h = \\sum_i h^*_i z^i \\in \\mathbb{Z}[z]$ that is an $h^*$-polynomial of $P$, that is,","title":"Hibi's conjecture on symmetric $h^*$-vectors is false"},{"data_url":"scopes/Combinatorics/discrepancy-theory.json","id":"Combinatorics/discrepancy-theory","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/discrepancy-theory/index.html","scope":"Combinatorics/discrepancy-theory","status":"can_prove","summary":"How evenly a set system, matrix or vector family can be two-coloured: matrix and vector discrepancy, the Koml\u00f3s and Beck\u2013Fiala problems, hereditary discrepancy, vector balancing, balanced weighted colourings and balancing infinite families.","title":"Discrepancy theory"},{"data_url":"scopes/Combinatorics/discrepancy-theory/balanced-weighted-colourings.json","id":"Combinatorics/discrepancy-theory/balanced-weighted-colourings","kind":"scope","navigation":"module","parent":28397,"result":false,"route":"../roadmap/Combinatorics/discrepancy-theory/balanced-weighted-colourings/index.html","scope":"Combinatorics/discrepancy-theory/balanced-weighted-colourings","status":"fully_proved","title":"Balanced weighted colourings"},{"id":"Combinatorics/discrepancy-theory/balanced-weighted-colourings/exists-balanced-coloring","kind":"node","lean":"Matrix.exists_balanced_coloring","parent":28398,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/balanced-weighted-colourings/index.html#exists-balanced-coloring","scope":"Combinatorics/discrepancy-theory/balanced-weighted-colourings","status":"fully_proved","summary":"A finite symmetric natural-valued matrix with zero diagonal and bounded row sums has a coloring with balanced fibers and bounded weight in every ordered color block.","title":"Balanced coloring of a weighted symmetric matrix"},{"id":"Combinatorics/discrepancy-theory/balanced-weighted-colourings/exists-balanced-coloring-off-diagonal","kind":"node","lean":"Matrix.exists_balanced_coloring_offDiagonal","parent":28398,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/balanced-weighted-colourings/index.html#exists-balanced-coloring-off-diagonal","scope":"Combinatorics/discrepancy-theory/balanced-weighted-colourings","status":"fully_proved","summary":"A finite symmetric natural-valued matrix with bounded off-diagonal row sums has a balanced coloring whose blocks between distinct colors have bounded off-diagonal weight. Diagonal entries are unrestricted and ignored.","title":"Balanced coloring with bounded off-diagonal block weight"},{"id":"Combinatorics/discrepancy-theory/balanced-weighted-colourings/exists-balanced-coloring-fin","kind":"node","lean":"Matrix.exists_balanced_coloring_fin","parent":28398,"reading_title":{"format":"inline-parts-v1","parts":[["text","Balanced off-diagonal coloring for matrices indexed by "],["math","\\(\\{0,\\dots,n-1\\}\\)"]],"source":"Balanced off-diagonal coloring for matrices indexed by $\\{0,\\dots,n-1\\}$"},"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/balanced-weighted-colourings/index.html#exists-balanced-coloring-fin","scope":"Combinatorics/discrepancy-theory/balanced-weighted-colourings","status":"fully_proved","summary":"The `Fin`-indexed form of `Matrix.exists_balanced_coloring_offDiagonal`.","title":"Balanced off-diagonal coloring for matrices indexed by $\\{0,\\dots,n-1\\}$"},{"data_url":"scopes/Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences.json","id":"Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences","kind":"scope","navigation":"module","parent":28397,"result":false,"route":"../roadmap/Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences/index.html","scope":"Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences","status":"can_prove","title":"Balancing infinite families of sequences"},{"id":"Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences/bounded-discrepancy-infinite-family","kind":"node","lean":"MathlibExt.Combinatorics.BalancedSignsInfiniteFamilyErdosProblem178Wanted.boundedDiscrepancyInfiniteFamily","parent":28402,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences/index.html#bounded-discrepancy-infinite-family","scope":"Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences","status":"fully_proved","summary":"The statement that for every family $a_0, a_1, a_2, \\ldots$ of strictly increasing sequences of natural numbers there is a function $f : \\mathbb{N} \\to \\mathbb{Z}$ taking only the values $1$ and $-1$ such that for every $d \\in \\mathbb{N}$ there is a natural number $C$ with $\\left|\\sum_{j < m} f(a_i(j))\\right| \\le C$ for all $i < d$ and all $m \\in \\mathbb{N}$.","title":"Erd\u0151s Problem 178 (balancing infinite families of sequences)"},{"id":"Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences/bounded-discrepancy-infinite-family-hold","kind":"node","lean":"MathlibExt.Combinatorics.BalancedSignsInfiniteFamilyErdosProblem178Wanted.boundedDiscrepancyInfiniteFamily_holds","parent":28402,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences/index.html#bounded-discrepancy-infinite-family-hold","scope":"Combinatorics/discrepancy-theory/balancing-infinite-families-of-sequences","status":"can_prove","summary":"For every family $a_0, a_1, a_2, \\ldots$ of strictly increasing sequences of natural numbers there is a function $f : \\mathbb{N} \\to \\mathbb{Z}$ taking only the values $1$ and $-1$ such that for every $d \\in \\mathbb{N}$ there is a natural number $C$ with $\\left|\\sum_{j < m} f(a_i(j))\\right| \\le C$ for all $i < d$ and all $m \\in \\mathbb{N}$.","title":"Beck's theorem on balancing infinite families of sequences (Erd\u0151s Problem 178)"},{"data_url":"scopes/Combinatorics/discrepancy-theory/beck-fiala-problem.json","id":"Combinatorics/discrepancy-theory/beck-fiala-problem","kind":"scope","navigation":"module","parent":28397,"result":false,"route":"../roadmap/Combinatorics/discrepancy-theory/beck-fiala-problem/index.html","scope":"Combinatorics/discrepancy-theory/beck-fiala-problem","status":"can_prove","title":"Beck\u2013Fiala problem"},{"id":"Combinatorics/discrepancy-theory/beck-fiala-problem/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.LinearBeckFialaWanted.conjecture","parent":28405,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/beck-fiala-problem/index.html#conjecture","scope":"Combinatorics/discrepancy-theory/beck-fiala-problem","status":"fully_proved","summary":"This is the statement that there is a real constant $C > 0$ such that the following holds for all natural numbers $n$, $m$ and $t \\ge 1$. Let $S_1, \\dots, S_n$ be subsets of $\\{1, \\dots, m\\}$ such that each $S_i$ has at most $t$ elements, each element of $\\{1, \\dots, m\\}$ lies in at most $t$ of the sets $S_i$, and $S_i$ and $S_{i'}$ have at most one common element whenever $i \\ne i'$. Then there are signs\u2026","title":"Linear Beck\u2013Fiala conjecture"},{"id":"Combinatorics/discrepancy-theory/beck-fiala-problem/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.LinearBeckFialaWanted.conjecture_holds","parent":28405,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linear Beck\u2013Fiala systems have discrepancy "],["math","\\(O(\\sqrt{t})\\)"]],"source":"Linear Beck\u2013Fiala systems have discrepancy $O(\\sqrt{t})$"},"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/beck-fiala-problem/index.html#conjecture-holds","scope":"Combinatorics/discrepancy-theory/beck-fiala-problem","status":"can_prove","summary":"There is a real constant $C > 0$ such that the following holds for all natural numbers $n$, $m$ and $t \\ge 1$. Let $S_1, \\dots, S_n$ be subsets of $\\{1, \\dots, m\\}$ such that each $S_i$ has at most $t$ elements, each element of $\\{1, \\dots, m\\}$ lies in at most $t$ of the sets $S_i$, and $S_i$ and $S_{i'}$ have at most one common element whenever $i \\ne i'$. Then there are signs\u2026","title":"Linear Beck\u2013Fiala systems have discrepancy $O(\\sqrt{t})$"},{"data_url":"scopes/Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy.json","id":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy","kind":"scope","navigation":"module","parent":28397,"result":false,"route":"../roadmap/Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/index.html","scope":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy","status":"can_prove","title":"Colouring lists for hereditary discrepancy"},{"id":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/discrepancy-on","kind":"node","lean":"MathlibExt.Combinatorics.PolysizeColoringListWanted.discrepancyOn","parent":28408,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/index.html#discrepancy-on","scope":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy","status":"fully_proved","summary":"For a real $n \\times m$ matrix $A$ and a set $S$ of columns of $A$, the discrepancy of $A$ on $S$ is the minimum, over all sign vectors $\\varepsilon \\in \\{-1, 1\\}^m$, of $\\max_i \\left|\\sum_{j \\in S} A_{ij} \\varepsilon_j\\right|$, the largest absolute value of a row sum of $A$ restricted to $S$ and weighted by $\\varepsilon$ (taken to be $0$ when $A$ has no rows).","title":"Discrepancy of a matrix on a set of columns"},{"id":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/hereditary-disc","kind":"node","lean":"MathlibExt.Combinatorics.PolysizeColoringListWanted.hereditaryDisc","parent":28408,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/index.html#hereditary-disc","scope":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy","status":"fully_proved","summary":"The hereditary discrepancy $\\operatorname{herdisc}(A)$ of a real $n \\times m$ matrix $A$ is the maximum, over all sets $S$ of columns of $A$, of the discrepancy of $A$ on $S$.","title":"Hereditary discrepancy of a real matrix"},{"id":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.PolysizeColoringListWanted.conjecture","parent":28408,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/index.html#conjecture","scope":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy","status":"fully_proved","summary":"The statement that there exist a real number $C > 0$ and a natural number $k$ such that for all positive integers $n$ and $m$ and every real $n \\times m$ matrix $A$ there is a set $L$ of at most $m^k$ sign vectors in $\\{-1, 1\\}^m$ with the following property: for every set $S$ of columns of $A$ some $\\varepsilon \\in L$ satisfies $\\left|\\sum_{j \\in S} A_{ij} \\varepsilon_j\\right| \\le C \\operatorname{herdisc}(A)$ for\u2026","title":"Polynomial-size colouring list conjecture"},{"id":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.PolysizeColoringListWanted.conjecture_refuted","parent":28408,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy/index.html#conjecture-refuted","scope":"Combinatorics/discrepancy-theory/colouring-lists-for-hereditary-discrepancy","status":"can_prove","summary":"There do not exist a real number $C > 0$ and a natural number $k$ such that for all positive integers $n$ and $m$ and every real $n \\times m$ matrix $A$ there is a set $L$ of at most $m^k$ sign vectors in $\\{-1, 1\\}^m$ with the following property: for every set $S$ of columns of $A$ some $\\varepsilon \\in L$ satisfies $\\left|\\sum_{j \\in S} A_{ij} \\varepsilon_j\\right| \\le C \\operatorname{herdisc}(A)$ for every row\u2026","title":"The polynomial-size colouring list conjecture is false"},{"data_url":"scopes/Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound.json","id":"Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound","kind":"scope","navigation":"module","parent":28397,"result":false,"route":"../roadmap/Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound/index.html","scope":"Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound","status":"can_prove","title":"Vector balancing and Banaszczyk's volume bound"},{"id":"Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound/volume-bound-is-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.VectorBalancingVolumeBoundAIMCombinatoricsProblem42Wanted.volumeBoundIsUpperBound","parent":28413,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound/index.html#volume-bound-is-upper-bound","scope":"Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound","status":"fully_proved","summary":"This is the proposition that there are real constants $C$ and $c$ such that for every natural number $d \\ge 2$, all norms $X$ and $Y$ on $\\mathbb{R}^d$, every natural number $n$ and all vectors $v_1, \\dots, v_n \\in \\mathbb{R}^d$ with $X(v_i) \\le 1$ for every $i$, there are signs $\\varepsilon_1, \\dots, \\varepsilon_n \\in \\{1, -1\\}$ such that $Y(\\varepsilon_1 v_1 + \\dots + \\varepsilon_n v_n)$ is at most\u2026","title":"Banaszczyk's volume bound as an upper bound for vector balancing (statement)"},{"id":"Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound/volume-bound-is-upper-bound-refuted","kind":"node","lean":"MathlibExt.Combinatorics.VectorBalancingVolumeBoundAIMCombinatoricsProblem42Wanted.volumeBoundIsUpperBound_refuted","parent":28413,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound/index.html#volume-bound-is-upper-bound-refuted","scope":"Combinatorics/discrepancy-theory/vector-balancing-and-banaszczyk-s-volume-bound","status":"can_prove","summary":"There are no real constants $C$ and $c$ such that for every natural number $d \\ge 2$, all norms $X$ and $Y$ on $\\mathbb{R}^d$, every natural number $n$ and all vectors $v_1, \\dots, v_n \\in \\mathbb{R}^d$ with $X(v_i) \\le 1$ for every $i$, there are signs $\\varepsilon_1, \\dots, \\varepsilon_n \\in \\{1, -1\\}$ such that $Y(\\varepsilon_1 v_1 + \\dots + \\varepsilon_n v_n)$ is at most\u2026","title":"Banaszczyk's volume bound is not an upper bound for vector balancing"},{"data_url":"scopes/Combinatorics/discrepancy-theory/vector-discrepancy.json","id":"Combinatorics/discrepancy-theory/vector-discrepancy","kind":"scope","navigation":"module","parent":28397,"result":false,"route":"../roadmap/Combinatorics/discrepancy-theory/vector-discrepancy/index.html","scope":"Combinatorics/discrepancy-theory/vector-discrepancy","status":"fully_proved","title":"Vector discrepancy"},{"id":"Combinatorics/discrepancy-theory/vector-discrepancy/vector-discrepancy","kind":"node","lean":"MatrixDiscrepancy.vectorDiscrepancy","parent":28416,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/vector-discrepancy/index.html#vector-discrepancy","scope":"Combinatorics/discrepancy-theory/vector-discrepancy","status":"fully_proved","summary":"For a real $m \\times n$ matrix $A$, the vector discrepancy of $A$ is the infimum, over all families $u_1, \\dots, u_n$ of unit vectors in the Euclidean space $\\mathbb{R}^n$, of $\\max_{1 \\le i \\le m} \\lVert \\sum_{j=1}^{n} A_{ij} u_j \\rVert$ (taken to be $0$ when $m = 0$), where $\\lVert \\cdot \\rVert$ is the Euclidean norm; it is an element of $[0, \\infty]$.","title":"Vector discrepancy"},{"id":"Combinatorics/discrepancy-theory/vector-discrepancy/vector-discrepancy-attained","kind":"node","lean":"MatrixDiscrepancy.vectorDiscrepancy_attained","parent":28416,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/vector-discrepancy/index.html#vector-discrepancy-attained","scope":"Combinatorics/discrepancy-theory/vector-discrepancy","status":"fully_proved","summary":"Let $A$ be a real $m \\times n$ matrix with $n \\ge 1$. Then there are unit vectors $u_1, \\dots, u_n$ in the Euclidean space $\\mathbb{R}^n$ such that the vector discrepancy of $A$ equals $\\max_{1 \\le i \\le m} \\lVert \\sum_{j=1}^{n} A_{ij} u_j \\rVert$ (taken to be $0$ when $m = 0$), where $\\lVert \\cdot \\rVert$ is the Euclidean norm.","title":"The vector discrepancy is attained"},{"data_url":"scopes/Combinatorics/discrepancy-theory/koml-s-conjecture.json","id":"Combinatorics/discrepancy-theory/koml-s-conjecture","kind":"scope","navigation":"module","parent":28397,"result":false,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","title":"Koml\u00f3s conjecture"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/sign-value","kind":"node","lean":"MatrixDiscrepancy.signValue","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#sign-value","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"For $s \\in \\{0, 1\\}$, the associated real sign is $-1$ if $s = 0$ and $1$ if $s = 1$.","title":"Signs indexed by two elements"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/discrepancy","kind":"node","lean":"MatrixDiscrepancy.discrepancy","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#discrepancy","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"For a real $m \\times n$ matrix $A$, the discrepancy of $A$ is the infimum, over all sign vectors $x \\in \\{-1, 1\\}^n$, of the maximum norm $\\|Ax\\|_\\infty = \\max_i |(Ax)_i|$ (which is $0$ when $m = 0$), taken in $[0, \\infty]$.","title":"Discrepancy of a real matrix"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/column-admissible","kind":"node","lean":"MatrixDiscrepancy.ColumnAdmissible","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#column-admissible","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"A real $m \\times n$ matrix $A$ is column admissible when every column of $A$ lies in the closed Euclidean unit ball, that is, $\\sum_{i=1}^{m} A_{ij}^2 \\le 1$ for every column index $j$.","title":"Matrices with columns in the unit ball"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/dimension-constant","kind":"node","lean":"MatrixDiscrepancy.dimensionConstant","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#dimension-constant","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"For a natural number $n$, this is the supremum of the discrepancy of $A$ over all real $n \\times n$ matrices $A$ whose columns all have Euclidean norm at most $1$, taken in $[0, \\infty]$. The definition also applies to $n = 0$.","title":"Koml\u00f3s constant in a fixed dimension"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/komlos-constant","kind":"node","lean":"MatrixDiscrepancy.komlosConstant","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#komlos-constant","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The Koml\u00f3s constant is the supremum, over all integers $n \\ge 1$, of the Koml\u00f3s constant in dimension $n$, namely the supremum of the discrepancy of $A$ over all real $n \\times n$ matrices $A$ whose columns all have Euclidean norm at most $1$. It is taken in $[0, \\infty]$.","title":"Koml\u00f3s constant"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/discrepancy-attained","kind":"node","lean":"MatrixDiscrepancy.discrepancy_attained","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#discrepancy-attained","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"For every real $m \\times n$ matrix $A$, including the cases $m = 0$ and $n = 0$, there is a sign vector $x \\in \\{-1, 1\\}^n$ such that the discrepancy of $A$ equals $\\|Ax\\|_\\infty$.","title":"The discrepancy is attained"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/uniform-rectangular-bound","kind":"node","lean":"MatrixDiscrepancy.UniformRectangularBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#uniform-rectangular-bound","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The statement that there is a real constant $C > 0$ such that for all natural numbers $n$ and $d$ and all vectors $v_1, \\dots, v_n \\in \\mathbb{R}^d$ of Euclidean norm at most $1$, writing $v_{ij}$ for the $j$-th coordinate of $v_i$, there are signs $\\varepsilon_1, \\dots, \\varepsilon_n \\in \\{-1, 1\\}$ with $\\left|\\sum_{i=1}^{n} \\varepsilon_i v_{ij}\\right| \\le C$ for every coordinate index $j$.","title":"Uniform discrepancy bound for vectors in the unit ball"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/constant-formulation-equivalence","kind":"node","lean":"KomlosDiscrepancyWanted.constantFormulationEquivalence","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#constant-formulation-equivalence","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The statement that the uniform discrepancy bound for vectors in the unit ball holds if and only if the Koml\u00f3s constant is finite.","title":"Equivalence of the square and rectangular Koml\u00f3s formulations (statement)"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/beck-spencer-gluskin-log-upper-bound","kind":"node","lean":"KomlosDiscrepancyWanted.beckSpencerGluskinLogUpperBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#beck-spencer-gluskin-log-upper-bound","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The statement that there are a real number $c > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $c \\log n$, where $\\log$ is the natural logarithm.","title":"Beck\u2013Spencer\u2013Gluskin logarithmic upper bound (statement)"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/banaszczyk1998-sqrt-log-upper-bound","kind":"node","lean":"KomlosDiscrepancyWanted.banaszczyk1998SqrtLogUpperBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#banaszczyk1998-sqrt-log-upper-bound","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The proposition that there are a real constant $c > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $c \\sqrt{\\log n}$, where $\\log$ is the natural logarithm.","title":"Statement of Banaszczyk's Koml\u00f3s bound"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/banaszczyk-sublog-upper-bound","kind":"node","lean":"KomlosDiscrepancyWanted.banaszczykSublogUpperBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#banaszczyk-sublog-upper-bound","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The proposition that for every real $\\varepsilon > 0$ there is a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $\\varepsilon \\log n$, where $\\log$ is the natural logarithm. In other words, the Koml\u00f3s constant in dimension $n$ is $o(\\log n)$.","title":"Statement of the sublogarithmic Koml\u00f3s bound"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/bdg2019-numerical-sqrt-log-upper-bound","kind":"node","lean":"KomlosDiscrepancyWanted.bdg2019NumericalSqrtLogUpperBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#bdg2019-numerical-sqrt-log-upper-bound","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The proposition that there are a real constant $c > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $c \\sqrt{\\log n}$, where $\\log$ is the natural logarithm.","title":"Statement of the Bansal\u2013Dadush\u2013Garg Koml\u00f3s bound"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/bansal-jiang2025-quarter-log-upper-bound","kind":"node","lean":"KomlosDiscrepancyWanted.bansalJiang2025QuarterLogUpperBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#bansal-jiang2025-quarter-log-upper-bound","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The proposition that there are a real constant $c > 0$, a natural number $p$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $c (\\log n)^{1/4} (\\log \\log n)^p$, where $\\log$ is the natural logarithm.","title":"Statement of the Bansal\u2013Jiang Koml\u00f3s bound"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/bansal-jiang2025-little-sqrt-log-upper-b","kind":"node","lean":"KomlosDiscrepancyWanted.bansalJiang2025LittleSqrtLogUpperBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#bansal-jiang2025-little-sqrt-log-upper-b","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The proposition that for every real $\\varepsilon > 0$ there is a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $\\varepsilon \\sqrt{\\log n}$, where $\\log$ is the natural logarithm. In other words, the Koml\u00f3s constant in dimension $n$ is $o(\\sqrt{\\log n})$.","title":"Statement of the little-o square-root-log Koml\u00f3s bound"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/kunisky2023-lower-bound","kind":"node","lean":"KomlosDiscrepancyWanted.kunisky2023LowerBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#kunisky2023-lower-bound","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The statement that the Koml\u00f3s constant is at least $1 + \\sqrt{2}$.","title":"Kunisky's lower bound for the Koml\u00f3s constant (statement)"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/kunisky-infinitely-many-dimensions-lower","kind":"node","lean":"KomlosDiscrepancyWanted.kuniskyInfinitelyManyDimensionsLowerBound","parent":28419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many dimensions with Koml\u00f3s constant at least "],["math","\\(1 + \\sqrt{2}\\)"],["text"," (statement)"]],"source":"Infinitely many dimensions with Koml\u00f3s constant at least $1 + \\sqrt{2}$ (statement)"},"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#kunisky-infinitely-many-dimensions-lower","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The statement that there are infinitely many natural numbers $n \\ge 1$ such that the Koml\u00f3s constant in dimension $n$ is at least $1 + \\sqrt{2}$.","title":"Infinitely many dimensions with Koml\u00f3s constant at least $1 + \\sqrt{2}$ (statement)"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/nikolov2013-vector-coloring-bound","kind":"node","lean":"KomlosDiscrepancyWanted.nikolov2013VectorColoringBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#nikolov2013-vector-coloring-bound","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"The statement that for all natural numbers $m$ and $n \\ge 1$ and every real $m \\times n$ matrix $A$ whose columns all lie in the closed Euclidean unit ball, there are unit vectors $u_1, \\dots, u_n$ in the Euclidean space $\\mathbb{R}^n$ such that $\\lVert \\sum_{j=1}^{n} A_{ij} u_j \\rVert \\le 1$ for every row index $i$, where $\\lVert \\cdot \\rVert$ is the Euclidean norm.","title":"Koml\u00f3s conjecture for vector colorings (statement)"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/nikolov2013-vector-discrepancy-upper-bou","kind":"node","lean":"KomlosDiscrepancyWanted.nikolov2013VectorDiscrepancyUpperBound","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#nikolov2013-vector-discrepancy-upper-bou","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"fully_proved","summary":"This is the proposition that for all natural numbers $m$ and $n$ with $n \\ge 1$ and every real $m \\times n$ matrix $A$ whose columns all lie in the closed Euclidean unit ball, that is, $\\sum_{i} A_{ij}^2 \\le 1$ for every column $j$, the vector discrepancy of $A$ is at most $1$.","title":"Statement of Nikolov's bound on vector discrepancy"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/constant-formulation-equivalence-holds","kind":"node","lean":"KomlosDiscrepancyWanted.constantFormulationEquivalence_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#constant-formulation-equivalence-holds","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"The following two conditions are equivalent. (a) There is a real constant $C > 0$ such that for all natural numbers $n$ and $d$ and all vectors $v_1, \\dots, v_n \\in \\mathbb{R}^d$ of Euclidean norm at most $1$, writing $v_{ij}$ for the $j$-th coordinate of $v_i$, there are signs $\\varepsilon_1, \\dots, \\varepsilon_n \\in \\{-1, 1\\}$ with $\\left|\\sum_{i=1}^{n} \\varepsilon_i v_{ij}\\right| \\le C$ for every coordinate\u2026","title":"The square and rectangular Koml\u00f3s formulations are equivalent"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/beck-spencer-gluskin-log-upper-bound-hol","kind":"node","lean":"KomlosDiscrepancyWanted.beckSpencerGluskinLogUpperBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#beck-spencer-gluskin-log-upper-bound-hol","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"There are a real number $c > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ (the supremum of the discrepancy of real $n \\times n$ matrices whose columns all have Euclidean norm at most $1$, taken in $[0, \\infty]$) is at most $c \\log n$, where $\\log$ is the natural logarithm.","title":"Beck\u2013Spencer\u2013Gluskin logarithmic upper bound for the Koml\u00f3s constant"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/banaszczyk1998-sqrt-log-upper-bound-hold","kind":"node","lean":"KomlosDiscrepancyWanted.banaszczyk1998SqrtLogUpperBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#banaszczyk1998-sqrt-log-upper-bound-hold","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"There are a real constant $c > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $c \\sqrt{\\log n}$, where $\\log$ is the natural logarithm.","title":"Banaszczyk's bound for the Koml\u00f3s problem"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/banaszczyk-sublog-upper-bound-holds","kind":"node","lean":"KomlosDiscrepancyWanted.banaszczykSublogUpperBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#banaszczyk-sublog-upper-bound-holds","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"For every real $\\varepsilon > 0$ there is a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $\\varepsilon \\log n$, where $\\log$ is the natural logarithm. In other words, the Koml\u00f3s constant in dimension $n$ is $o(\\log n)$.","title":"Sublogarithmic bound for the Koml\u00f3s problem"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/bdg2019-numerical-sqrt-log-upper-bound-h","kind":"node","lean":"KomlosDiscrepancyWanted.bdg2019NumericalSqrtLogUpperBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#bdg2019-numerical-sqrt-log-upper-bound-h","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"There are a real constant $c > 0$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $c \\sqrt{\\log n}$, where $\\log$ is the natural logarithm.","title":"Bansal\u2013Dadush\u2013Garg bound for the Koml\u00f3s problem, numerical form"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/bansal-jiang2025-quarter-log-upper-bound-2","kind":"node","lean":"KomlosDiscrepancyWanted.bansalJiang2025QuarterLogUpperBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#bansal-jiang2025-quarter-log-upper-bound-2","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"There are a real constant $c > 0$, a natural number $p$ and a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $c (\\log n)^{1/4} (\\log \\log n)^p$, where $\\log$ is the natural logarithm.","title":"Bansal\u2013Jiang bound for the Koml\u00f3s problem"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/bansal-jiang2025-little-sqrt-log-upper-b-2","kind":"node","lean":"KomlosDiscrepancyWanted.bansalJiang2025LittleSqrtLogUpperBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#bansal-jiang2025-little-sqrt-log-upper-b-2","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"For every real $\\varepsilon > 0$ there is a natural number $N \\ge 3$ such that for every natural number $n \\ge N$, the Koml\u00f3s constant in dimension $n$ is at most $\\varepsilon \\sqrt{\\log n}$, where $\\log$ is the natural logarithm. In other words, the Koml\u00f3s constant in dimension $n$ is $o(\\sqrt{\\log n})$.","title":"Bansal\u2013Jiang improvement on Banaszczyk's Koml\u00f3s bound"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/kunisky2023-lower-bound-holds","kind":"node","lean":"KomlosDiscrepancyWanted.kunisky2023LowerBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#kunisky2023-lower-bound-holds","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"The Koml\u00f3s constant (the supremum over all $n \\ge 1$ of the discrepancy of real $n \\times n$ matrices whose columns all have Euclidean norm at most $1$, taken in $[0, \\infty]$) is at least $1 + \\sqrt{2}$.","title":"Kunisky's lower bound for the Koml\u00f3s constant"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/kunisky-infinitely-many-dimensions-lower-2","kind":"node","lean":"KomlosDiscrepancyWanted.kuniskyInfinitelyManyDimensionsLowerBound_holds","parent":28419,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinitely many dimensions have Koml\u00f3s constant at least "],["math","\\(1 + \\sqrt{2}\\)"]],"source":"Infinitely many dimensions have Koml\u00f3s constant at least $1 + \\sqrt{2}$"},"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#kunisky-infinitely-many-dimensions-lower-2","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"There are infinitely many natural numbers $n \\ge 1$ such that the Koml\u00f3s constant in dimension $n$ (the supremum of the discrepancy of real $n \\times n$ matrices whose columns all have Euclidean norm at most $1$, taken in $[0, \\infty]$) is at least $1 + \\sqrt{2}$.","title":"Infinitely many dimensions have Koml\u00f3s constant at least $1 + \\sqrt{2}$"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/nikolov2013-vector-coloring-bound-holds","kind":"node","lean":"KomlosDiscrepancyWanted.nikolov2013VectorColoringBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#nikolov2013-vector-coloring-bound-holds","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"For all natural numbers $m$ and $n \\ge 1$ and every real $m \\times n$ matrix $A$ whose columns all have Euclidean norm at most $1$, there are unit vectors $u_1, \\dots, u_n$ in the Euclidean space $\\mathbb{R}^n$ such that $\\lVert \\sum_{j=1}^{n} A_{ij} u_j \\rVert \\le 1$ for every row index $i$, where $\\lVert \\cdot \\rVert$ is the Euclidean norm.","title":"Nikolov's theorem: the Koml\u00f3s conjecture holds for vector colorings"},{"id":"Combinatorics/discrepancy-theory/koml-s-conjecture/nikolov2013-vector-discrepancy-upper-bou-2","kind":"node","lean":"KomlosDiscrepancyWanted.nikolov2013VectorDiscrepancyUpperBound_holds","parent":28419,"result":true,"route":"../roadmap/Combinatorics/discrepancy-theory/koml-s-conjecture/index.html#nikolov2013-vector-discrepancy-upper-bou-2","scope":"Combinatorics/discrepancy-theory/koml-s-conjecture","status":"can_prove","summary":"For all natural numbers $m$ and $n$ with $n \\ge 1$ and every real $m \\times n$ matrix $A$ whose columns all lie in the closed Euclidean unit ball, that is, $\\sum_{i} A_{ij}^2 \\le 1$ for every column $j$, the vector discrepancy of $A$ is at most $1$.","title":"Nikolov's bound on vector discrepancy"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics.json","id":"Combinatorics/probabilistic-combinatorics","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/index.html","scope":"Combinatorics/probabilistic-combinatorics","status":"can_state","summary":"Probabilistic methods and random discrete structures outside random graphs: the Lov\u00e1sz local lemma, Littlewood\u2013Offord and Tomaszewski anti-concentration for signed sums, concentration of subset sums, the isolation lemma, thresholds and expectation thresholds, random k-SAT thresholds, perfect matchings in random hypergraphs, the Chv\u00e1tal\u2013Sankoff constant for longest common subsequences, and moment-based counting\u2026","title":"Probabilistic combinatorics"},{"id":"Combinatorics/probabilistic-combinatorics/isolation-lemma","kind":"node","lean":"MetaMathlibExt.isolation_lemma","parent":28449,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/index.html#isolation-lemma","scope":"Combinatorics/probabilistic-combinatorics","status":"fully_proved","summary":"Let $n \\ge 1$ and let $\\mathcal{F}$ be a nonempty family of subsets of $\\{0, 1, \\dots, n - 1\\}$. Assign each element $i$ a weight $w(i) \\in \\{1, 2, \\dots, 2n\\}$ and give a set $S$ the weight $w(S) = \\sum_{i \\in S} w(i)$. Then, among all $(2n)^n$ weight functions $w$, the fraction for which exactly one member of $\\mathcal{F}$ has minimum weight is at least $1/2$.","title":"Isolation lemma"},{"id":"Combinatorics/probabilistic-combinatorics/lovasz-local-lemma-symmetric-finite","kind":"node","lean":"MathlibExt.Probability.Combinatorics.LovaszLocalLemma.lovasz_local_lemma_symmetric_finite","parent":28449,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/index.html#lovasz-local-lemma-symmetric-finite","scope":"Combinatorics/probabilistic-combinatorics","status":"fully_proved","summary":"Let $\\Omega$ be a finite set with a probability mass function $P$, that is, $P(\\omega) \\ge 0$ for every $\\omega \\in \\Omega$ and $\\sum_{\\omega \\in \\Omega} P(\\omega) = 1$, and write $\\Pr[E] = \\sum_{\\omega \\in E} P(\\omega)$ for $E \\subseteq \\Omega$. Let $(A_i)_{i \\in I}$ be a family of events indexed by a finite set $I$, let $G$ be a simple graph on $I$ whose maximum degree is at most a natural number $d$, and let\u2026","title":"Symmetric Lov\u00e1sz Local Lemma"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant.json","id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"can_prove","title":"Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/binary-string","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.BinaryString","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#binary-string","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For a natural number $n$, a binary string of length $n$ is a function from $\\{0, 1, \\dots, n - 1\\}$ to $\\{\\mathrm{false}, \\mathrm{true}\\}$, that is, a sequence of $n$ bits listed in order.","title":"Binary strings"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/binary-string-pair","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.BinaryStringPair","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#binary-string-pair","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For a natural number $n$, a pair of binary strings of length $n$ is an ordered pair $(x, y)$ of binary strings $x$ and $y$, each of length $n$. These pairs form the finite sample space over which expected longest common subsequence lengths are taken.","title":"Pairs of binary strings"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/is-common-subsequence","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.IsCommonSubsequence","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#is-common-subsequence","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For binary strings $x$ and $y$ of length $n$, a finite sequence of bits $z$ is a common subsequence of $x$ and $y$ if $z$ is obtained both from $x$ and from $y$ by deleting some (possibly no) entries and keeping the remaining entries in order.","title":"Common subsequence"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/common-subsequence-lengths","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.commonSubsequenceLengths","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#common-subsequence-lengths","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For binary strings $x$ and $y$ of length $n$, the set of natural numbers $k$ such that some binary list of length $k$ is a common subsequence of $x$ and $y$.","title":"Lengths of common subsequences"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lcs-length","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.lcsLength","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lcs-length","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For binary strings $x$ and $y$ of length $n$, $\\mathrm{LCS}(x, y)$ is the length returned by the dynamic-programming algorithm for the longest common subsequence, applied to $x$ and $y$ read as lists of bits.","title":"Length of a longest common subsequence"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lcs-length-is-greatest","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.lcsLength_isGreatest","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lcs-length-is-greatest","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For binary strings $x$ and $y$ of length $n$, let $\\mathrm{LCS}(x, y)$ be the length returned by the dynamic-programming algorithm for the longest common subsequence on $x$ and $y$. Then some common subsequence of $x$ and $y$ has length $\\mathrm{LCS}(x, y)$, and every common subsequence of $x$ and $y$ has length at most $\\mathrm{LCS}(x, y)$.","title":"The LCS length is the maximum common-subsequence length"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/exists-longest-common-subsequence","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.exists_longestCommonSubsequence","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#exists-longest-common-subsequence","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For binary strings $x$ and $y$ of length $n$, there is a binary list $z$ that is a common subsequence of $x$ and $y$, whose length equals the LCS length of $x$ and $y$, and such that every common subsequence $w$ of $x$ and $y$ has length at most the length of $z$.","title":"Existence of a longest common subsequence"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lcs-length-le","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.lcsLength_le","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lcs-length-le","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For binary strings $x$ and $y$ of length $n$, the length $\\mathrm{LCS}(x, y)$ of a longest common subsequence satisfies $\\mathrm{LCS}(x, y) \\le n$.","title":"The LCS length is at most the string length"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lcs-length-eq-find-greatest","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.lcsLength_eq_findGreatest","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lcs-length-eq-find-greatest","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For binary strings $x$ and $y$ of length $n$, the length $\\mathrm{LCS}(x, y)$ of a longest common subsequence equals the largest $k \\le n$ such that $x$ and $y$ have a common subsequence of length $k$ (such $k$ exist, since the empty sequence is a common subsequence).","title":"The LCS length as a bounded maximum"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/uniform-binary-string-pairs","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.uniformBinaryStringPairs","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#uniform-binary-string-pairs","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For a natural number $n$, this is the finite set of all ordered pairs $(x, y)$ of binary strings $x$ and $y$ of length $n$.","title":"The finite set of pairs of binary strings"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/expected-lcslength","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.expectedLCSLength","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#expected-lcslength","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the expected LCS length $E_n$ is the average of $\\mathrm{LCS}(x, y)$, the length of a longest common subsequence of $x$ and $y$, over all ordered pairs $(x, y)$ of binary strings of length $n$: it is the sum of $\\mathrm{LCS}(x, y)$ over these pairs divided by their number $4^n$. Equivalently, $E_n$ is the expectation of $\\mathrm{LCS}(x, y)$ for two independent uniformly random binary strings\u2026","title":"Expected length of a longest common subsequence"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/expected-lcslength-nonneg","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.expectedLCSLength_nonneg","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#expected-lcslength-nonneg","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For every $n$, the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$ is at least $0$.","title":"Expected LCS length is nonnegative"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/expected-lcslength-le","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.expectedLCSLength_le","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#expected-lcslength-le","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For every $n$, the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$ is at most $n$.","title":"Expected LCS length is at most the string length"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/normalized-expected-lcs","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.normalizedExpectedLCS","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#normalized-expected-lcs","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For $n \\ge 1$, the normalized expected LCS length at $n$ is $E_n / n$, where $E_n$ is the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$. At $n = 0$ it is defined to be $0$.","title":"Normalized expected LCS length"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/normalized-expected-lcs-mem-icc","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.normalizedExpectedLCS_mem_Icc","parent":28452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Normalized expected LCS lengths lie in "],["math","\\((0,1)\\)"]],"source":"Normalized expected LCS lengths lie in $(0,1)$"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#normalized-expected-lcs-mem-icc","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For every natural number $n$, the normalized expected LCS length at $n$ (the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$, divided by $n$, and $0$ for $n = 0$) lies in the closed interval $[0,1]$.","title":"Normalized expected LCS lengths lie in $(0,1)$"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/binary-chvatal-sankoff-constant","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.binaryChvatalSankoffConstant","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#binary-chvatal-sankoff-constant","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"Let $L_n$ be the length of a longest common subsequence of two independent uniformly random binary strings of length $n$. The binary Chv\u00e1tal\u2013Sankoff constant is the real number $\\liminf_{n \\to \\infty} \\mathbb{E}[L_n]/n$.","title":"Binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/c31a","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.C31a","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#c31a","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"Let $L_n$ be the length of a longest common subsequence of two independent uniformly random binary strings of length $n$. The notation $C_{31a}$ stands for the binary Chv\u00e1tal\u2013Sankoff constant $\\liminf_{n \\to \\infty} \\mathbb{E}[L_n]/n$.","title":"Catalogue notation for the binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/chvatal-sankoff-limit-exists","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.chvatalSankoffLimitExists","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#chvatal-sankoff-limit-exists","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"Let $L_n$ be the length of a longest common subsequence of two independent uniformly random binary strings of length $n$. This proposition asserts that the sequence $\\mathbb{E}[L_n]/n$ converges to a real number as $n \\to \\infty$. Chv\u00e1tal and Sankoff proved that it holds.","title":"Statement of the Chv\u00e1tal\u2013Sankoff limit theorem"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/binary-chvatal-sankoff-constant-mem-icc","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.binaryChvatalSankoffConstant_mem_Icc","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#binary-chvatal-sankoff-constant-mem-icc","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"Let $L_n$ be the length of a longest common subsequence of two independent uniformly random binary strings of length $n$, and let $C_{31a} = \\liminf_{n \\to \\infty} \\mathbb{E}[L_n]/n$ be the binary Chv\u00e1tal\u2013Sankoff constant. Then $0 \\le C_{31a} \\le 1$.","title":"Trivial bounds for the binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/trivial-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.trivialUpperBound","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#trivial-upper-bound","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"Let $C_{31a}$ be the binary Chv\u00e1tal\u2013Sankoff constant: the limit inferior as $n \\to \\infty$ of $E_n / n$, where $E_n$ is the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$. This is the proposition $C_{31a} \\le 1$.","title":"Trivial upper bound for the binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/dancik-paterson-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.dancikPatersonUpperBound","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#dancik-paterson-upper-bound","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The proposition that the binary Chv\u00e1tal\u2013Sankoff constant $\\gamma$ satisfies $\\gamma \\le 0.837623$. Here $\\gamma$ is the limit inferior, as $n \\to \\infty$, of $\\mathbb{E}[L_n]/n$, where $L_n$ is the length of a longest common subsequence of two independent uniformly random binary strings of length $n$.","title":"Statement of the Dan\u010d\u00edk\u2013Paterson upper bound"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lueker-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.luekerUpperBound","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lueker-upper-bound","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The proposition that the binary Chv\u00e1tal\u2013Sankoff constant $C_{31a}$ is at most $0.826280$.","title":"Statement of Lueker's upper bound"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/trivial-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.trivialLowerBound","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#trivial-lower-bound","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The proposition that the binary Chv\u00e1tal\u2013Sankoff constant $C_{31a}$ satisfies $0 \\le C_{31a}$.","title":"Statement of the trivial lower bound"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/chvatal-sankoff-strict-positive","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.chvatalSankoffStrictPositive","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#chvatal-sankoff-strict-positive","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The proposition that the binary Chv\u00e1tal\u2013Sankoff constant $\\gamma$ is strictly positive, $0 < \\gamma$. Here $\\gamma$ is the limit inferior, as $n \\to \\infty$, of $\\mathbb{E}[L_n]/n$, where $L_n$ is the length of a longest common subsequence of two independent uniformly random binary strings of length $n$.","title":"Statement of the Chv\u00e1tal\u2013Sankoff positivity bound"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/dancik-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.dancikLowerBound","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#dancik-lower-bound","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The proposition that the binary Chv\u00e1tal\u2013Sankoff constant $\\gamma$ satisfies $0.773911 \\le \\gamma$. Here $\\gamma$ is the limit inferior, as $n \\to \\infty$, of $\\mathbb{E}[L_n]/n$, where $L_n$ is the length of a longest common subsequence of two independent uniformly random binary strings of length $n$.","title":"Statement of Dan\u010d\u00edk's lower bound"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lueker-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.luekerLowerBound","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lueker-lower-bound","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The proposition that the binary Chv\u00e1tal\u2013Sankoff constant $C_{31a}$ is at least $0.788071$.","title":"Statement of Lueker's lower bound"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/heineman-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.heinemanLowerBound","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#heineman-lower-bound","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For each $n$, let $E_n$ be the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$, and let $C$, the binary Chv\u00e1tal\u2013Sankoff constant, be the limit inferior of $E_n / n$ as $n \\to \\infty$. This is the proposition $0.792665992 \\le C$.","title":"Statement of the Heineman et al. lower bound"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/source-reported-unverified2026-lower-bou","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.sourceReportedUnverified2026LowerBound","parent":28452,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of the reported 2026 lower bound "],["math","\\(0.79970\\)"]],"source":"Statement of the reported 2026 lower bound $0.79970$"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#source-reported-unverified2026-lower-bou","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The proposition that the binary Chv\u00e1tal\u2013Sankoff constant $C_{31a}$ is at least $0.79970$.","title":"Statement of the reported 2026 lower bound $0.79970$"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/append-binary-string-pair-equiv","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.appendBinaryStringPairEquiv","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#append-binary-string-pair-equiv","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For natural numbers $m$ and $n$, sending a pair $(x_1, y_1)$ of binary strings of length $m$ together with a pair $(x_2, y_2)$ of binary strings of length $n$ to the pair $(x_1 x_2, y_1 y_2)$ of concatenations is a bijection from the product of the set of pairs of length-$m$ binary strings and the set of pairs of length-$n$ binary strings onto the set of pairs of binary strings of length $m + n$. Its inverse splits\u2026","title":"Concatenation of binary string pairs"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lcs-length-append-le","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.lcsLength_append_le","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lcs-length-append-le","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"Write $\\mathrm{LCS}(x, y)$ for the length of a longest common subsequence of binary strings $x$ and $y$ of equal length. Let $x_1, y_1$ be binary strings of length $m$ and $x_2, y_2$ binary strings of length $n$, and write $x_1x_2$ and $y_1y_2$ for the concatenated strings of length $m + n$. Then $\\mathrm{LCS}(x_1, y_1) + \\mathrm{LCS}(x_2, y_2) \\le \\mathrm{LCS}(x_1x_2, y_1y_2)$.","title":"Superadditivity of the LCS length under concatenation"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/tendsto-binary-chvatal-sankoff-constant","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.tendsto_binaryChvatalSankoffConstant","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#tendsto-binary-chvatal-sankoff-constant","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"If the normalized expected LCS lengths converge to some real number as $n \\to \\infty$, then they converge to the binary Chv\u00e1tal\u2013Sankoff constant $C_{31a}$, which is defined as their limit inferior.","title":"Any limit of the normalized expected LCS lengths is the binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/chvatal-sankoff-limit-exists-of-superadd","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.chvatalSankoffLimitExists_of_superadditive","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#chvatal-sankoff-limit-exists-of-superadd","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"For each natural number $n$, let $E_n$ be the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$. If $E_m + E_n \\le E_{m+n}$ for all natural numbers $m$ and $n$, then the sequence $(E_n / n)_{n \\ge 1}$ converges to a real number as $n \\to \\infty$.","title":"Superadditivity implies the binary LCS limit exists"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/trivial-lower-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.trivialLowerBound_holds","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#trivial-lower-bound-holds","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The binary Chv\u00e1tal\u2013Sankoff constant $C_{31a}$ satisfies $0 \\le C_{31a}$.","title":"The binary Chv\u00e1tal\u2013Sankoff constant is nonnegative"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/trivial-upper-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.trivialUpperBound_holds","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#trivial-upper-bound-holds","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"The binary Chv\u00e1tal\u2013Sankoff constant $C_{31a}$ satisfies $C_{31a} \\le 1$.","title":"The binary Chv\u00e1tal\u2013Sankoff constant is at most one"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/expected-lcslength-superadditive","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.expectedLCSLength_superadditive","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#expected-lcslength-superadditive","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"Let $E_n$ be the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$. Then for all natural numbers $m$ and $n$, $E_m + E_n \\le E_{m+n}$.","title":"Superadditivity of the expected LCS length"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/chvatal-sankoff-limit-exists-holds","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.chvatalSankoffLimitExists_holds","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#chvatal-sankoff-limit-exists-holds","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"fully_proved","summary":"Let $L_n$ be the length of a longest common subsequence of two independent uniformly random binary strings of length $n$. Then the sequence $\\mathbb{E}[L_n]/n$ converges to a real number as $n \\to \\infty$.","title":"Chv\u00e1tal\u2013Sankoff theorem: the binary LCS limit exists"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/dancik-paterson-upper-bound-wanted","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.dancikPatersonUpperBound_wanted","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#dancik-paterson-upper-bound-wanted","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"can_prove","summary":"The binary Chv\u00e1tal\u2013Sankoff constant $\\gamma$ satisfies $\\gamma \\le 0.837623$, where $\\gamma$ is the limit inferior, as $n \\to \\infty$, of $\\mathbb{E}[L_n]/n$ and $L_n$ is the length of a longest common subsequence of two independent uniformly random binary strings of length $n$.","title":"Dan\u010d\u00edk\u2013Paterson upper bound for the binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lueker-upper-bound-wanted","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.luekerUpperBound_wanted","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lueker-upper-bound-wanted","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"can_prove","summary":"The binary Chv\u00e1tal\u2013Sankoff constant $C_{31a}$ is at most $0.826280$.","title":"Lueker's upper bound for the binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/chvatal-sankoff-strict-positive-wanted","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.chvatalSankoffStrictPositive_wanted","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#chvatal-sankoff-strict-positive-wanted","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"can_prove","summary":"Let $L_n$ be the length of a longest common subsequence of two independent uniformly random binary strings of length $n$. The binary Chv\u00e1tal\u2013Sankoff constant $C_{31a} = \\liminf_{n \\to \\infty} \\mathbb{E}[L_n]/n$ satisfies $C_{31a} > 0$.","title":"The binary Chv\u00e1tal\u2013Sankoff constant is positive"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/dancik-lower-bound-wanted","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.dancikLowerBound_wanted","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#dancik-lower-bound-wanted","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"can_prove","summary":"The binary Chv\u00e1tal\u2013Sankoff constant $\\gamma$ satisfies $0.773911 \\le \\gamma$, where $\\gamma$ is the limit inferior, as $n \\to \\infty$, of $\\mathbb{E}[L_n]/n$ and $L_n$ is the length of a longest common subsequence of two independent uniformly random binary strings of length $n$.","title":"Dan\u010d\u00edk's lower bound for the binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/lueker-lower-bound-wanted","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.luekerLowerBound_wanted","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#lueker-lower-bound-wanted","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"can_prove","summary":"For each $n$, let $E_n$ be the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$, and let $C$, the binary Chv\u00e1tal\u2013Sankoff constant, be the limit inferior of $E_n / n$ as $n \\to \\infty$. Then $0.788071 \\le C$.","title":"Lueker's lower bound for the binary Chv\u00e1tal\u2013Sankoff constant"},{"id":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/heineman-lower-bound-wanted","kind":"node","lean":"MathlibExt.Combinatorics.BinaryChvatalSankoffWanted.heinemanLowerBound_wanted","parent":28452,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant/index.html#heineman-lower-bound-wanted","scope":"Combinatorics/probabilistic-combinatorics/chv-tal-sankoff-constant","status":"can_prove","summary":"For each $n$, let $E_n$ be the expected length of a longest common subsequence of two independent uniformly random binary strings of length $n$, and let $C$, the binary Chv\u00e1tal\u2013Sankoff constant, be the limit inferior of $E_n / n$ as $n \\to \\infty$. Then $0.792665992 \\le C$.","title":"Heineman et al. lower bound for the binary Chv\u00e1tal\u2013Sankoff constant"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums.json","id":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/index.html","scope":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums","status":"can_prove","title":"Concentration of subset sums"},{"id":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/subset-sum-count-bound","kind":"node","lean":"MathlibExt.Combinatorics.SubsetSumConcentrationErdosProblem362Wanted.subsetSumCountBound","parent":28495,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/index.html#subset-sum-count-bound","scope":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums","status":"fully_proved","summary":"There is a real constant $C$ such that for every nonempty finite set $A \\subseteq \\mathbb{N}$ with $N$ elements and every $t \\in \\mathbb{N}$, the number of subsets $S \\subseteq A$ with $\\sum_{n \\in S} n = t$ is at most $C \\cdot 2^N / N^{3/2}$.","title":"Erd\u0151s Problem 362: subsets with a given sum"},{"id":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/fixed-size-subset-sum-count-bound","kind":"node","lean":"MathlibExt.Combinatorics.SubsetSumConcentrationErdosProblem362Wanted.fixedSizeSubsetSumCountBound","parent":28495,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/index.html#fixed-size-subset-sum-count-bound","scope":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums","status":"fully_proved","summary":"There is a real constant $C$ such that for every nonempty finite set $A \\subseteq \\mathbb{N}$ with $N$ elements and all $l, t \\in \\mathbb{N}$, the number of $l$-element subsets $S \\subseteq A$ with $\\sum_{n \\in S} n = t$ is at most $C \\cdot 2^N / N^2$.","title":"Erd\u0151s Problem 362: subsets of a given size with a given sum"},{"id":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/subset-sum-count-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.SubsetSumConcentrationErdosProblem362Wanted.subsetSumCountBound_holds","parent":28495,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/index.html#subset-sum-count-bound-holds","scope":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums","status":"can_prove","summary":"There is a real constant $C$ such that for every nonempty finite set $A \\subseteq \\mathbb{N}$ with $N$ elements and every $t \\in \\mathbb{N}$, the number of subsets $S \\subseteq A$ with $\\sum_{n \\in S} n = t$ is at most $C \\cdot 2^N / N^{3/2}$.","title":"S\u00e1rk\u00f6zy\u2013Szemer\u00e9di bound on subsets with a given sum"},{"id":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/fixed-size-subset-sum-count-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.SubsetSumConcentrationErdosProblem362Wanted.fixedSizeSubsetSumCountBound_holds","parent":28495,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums/index.html#fixed-size-subset-sum-count-bound-holds","scope":"Combinatorics/probabilistic-combinatorics/concentration-of-subset-sums","status":"can_prove","summary":"There is a real constant $C$ such that for every nonempty finite set $A \\subseteq \\mathbb{N}$ with $N$ elements and all $l, t \\in \\mathbb{N}$, the number of $l$-element subsets $S \\subseteq A$ with $\\sum_{n \\in S} n = t$ is at most $C \\cdot 2^N / N^2$.","title":"Hal\u00e1sz's bound on subsets of a given size with a given sum"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/littlewood-offord-problem.json","id":"Combinatorics/probabilistic-combinatorics/littlewood-offord-problem","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/littlewood-offord-problem/index.html","scope":"Combinatorics/probabilistic-combinatorics/littlewood-offord-problem","status":"can_prove","title":"Littlewood\u2013Offord problem"},{"id":"Combinatorics/probabilistic-combinatorics/littlewood-offord-problem/littlewood-offord-sharp-bound","kind":"node","lean":"MathlibExt.Combinatorics.ScottishBook179Wanted.littlewoodOffordSharpBound","parent":28500,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/littlewood-offord-problem/index.html#littlewood-offord-sharp-bound","scope":"Combinatorics/probabilistic-combinatorics/littlewood-offord-problem","status":"can_prove","summary":"There is a real constant $A \\ge 0$ such that for every integer $n \\ge 1$ and all real numbers $a_0, a_1, \\dots, a_n$ with $|a_i| \\ge 1$ for every $i$, the number of sign vectors $(\\varepsilon_1, \\dots, \\varepsilon_n) \\in \\{-1, 1\\}^n$ with $|a_0 + \\varepsilon_1 a_1 + \\dots + \\varepsilon_n a_n| < 1$ is at most $A \\cdot 2^n / \\sqrt{n}$.","title":"Sharp Littlewood\u2013Offord bound"},{"id":"Combinatorics/probabilistic-combinatorics/littlewood-offord-problem/signed-sums-in-unit-disc-bound","kind":"node","lean":"MathlibExt.Combinatorics.StrongLittlewoodOffordErdosProblem498Wanted.signedSumsInUnitDiscBound","parent":28500,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/littlewood-offord-problem/index.html#signed-sums-in-unit-disc-bound","scope":"Combinatorics/probabilistic-combinatorics/littlewood-offord-problem","status":"fully_proved","summary":"The statement that for every natural number $n$, all complex numbers $z_1, \\dots, z_n$ with $|z_i| \\ge 1$ for every $i$, and every open disc $D$ of radius $1$ in $\\mathbb{C}$, at most $\\binom{n}{\\lfloor n/2 \\rfloor}$ of the $2^n$ sign vectors $(\\varepsilon_1, \\dots, \\varepsilon_n) \\in \\{1, -1\\}^n$ satisfy $\\sum_i \\varepsilon_i z_i \\in D$ (Erd\u0151s Problem 498).","title":"Strong Littlewood\u2013Offord bound in the plane"},{"id":"Combinatorics/probabilistic-combinatorics/littlewood-offord-problem/signed-sums-in-unit-disc-bound-holds","kind":"node","lean":"MathlibExt.Combinatorics.StrongLittlewoodOffordErdosProblem498Wanted.signedSumsInUnitDiscBound_holds","parent":28500,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/littlewood-offord-problem/index.html#signed-sums-in-unit-disc-bound-holds","scope":"Combinatorics/probabilistic-combinatorics/littlewood-offord-problem","status":"can_prove","summary":"For every natural number $n$, all complex numbers $z_1, \\dots, z_n$ with $|z_i| \\ge 1$ for every $i$, and every open disc $D$ of radius $1$ in $\\mathbb{C}$, at most $\\binom{n}{\\lfloor n/2 \\rfloor}$ of the $2^n$ sign vectors $(\\varepsilon_1, \\dots, \\varepsilon_n) \\in \\{1, -1\\}^n$ satisfy $\\sum_i \\varepsilon_i z_i \\in D$.","title":"Kleitman's theorem on signed sums in the plane"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting.json","id":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/index.html","scope":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","status":"fully_proved","title":"Moment bounds for counting"},{"id":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/sum-weights-shared-endpoint-le","kind":"node","lean":"Matrix.sum_weights_shared_endpoint_le","parent":28504,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/index.html#sum-weights-shared-endpoint-le","scope":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","status":"fully_proved","summary":"Pairs sharing an endpoint with a fixed pair have total weight at most `4 * P`. Loops are permitted. Symmetry bounds the two column fibers by the same row bound.","title":"Bound on the weight of pairs sharing an endpoint with a fixed pair"},{"id":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/sum-mul-mul-kernel-le","kind":"node","lean":"Matrix.sum_mul_mul_kernel_le","parent":28504,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/index.html#sum-mul-mul-kernel-le","scope":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","status":"fully_proved","summary":"A kernel vanishing on pairs with no common endpoint has weighted quadratic sum at most `4 * P * totalWeight * m`. Neither nonnegativity nor symmetry of the kernel is required; diagonal weights are permitted, and no separate nonnegativity assumption on `P` is needed.","title":"Weighted quadratic sum bound for a shared-endpoint kernel"},{"id":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/sum-ordered-nat-int-kernel-le","kind":"node","lean":"Matrix.sum_ordered_nat_int_kernel_le","parent":28504,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/index.html#sum-ordered-nat-int-kernel-le","scope":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","status":"fully_proved","summary":"The `Variance.StHence` bound with the kernel hypothesis restricted to distinct pairs: natural weights on `Fin g`, integer kernel, and row sums over erased diagonals. Diagonal weights and kernel entries outside the distinct-pair domain are arbitrary. The four inequalities in `h0` are precisely the negation of the source `Meets` disjunction. The proof zeroes diagonal weights and masks the kernel outside the\u2026","title":"Shared-endpoint kernel bound with hypotheses on distinct pairs"},{"id":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/card-filter-mul-sq-le-sum-sq","kind":"node","lean":"Finset.card_filter_mul_sq_le_sum_sq","parent":28504,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/index.html#card-filter-mul-sq-le-sum-sq","scope":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","status":"fully_proved","summary":"The number of points where a function is at least a nonnegative threshold, times the threshold squared, is at most the sum of squares.","title":"Chebyshev-type count bound from the sum of squares"},{"id":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/card-filter-mul-sq-le-of-moments","kind":"node","lean":"Finset.card_filter_mul_sq_le_of_moments","parent":28504,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/index.html#card-filter-mul-sq-le-of-moments","scope":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","status":"fully_proved","summary":"A denominator-cleared finite count bound from first and second moments.","title":"Count bound from first and second moments"},{"id":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/card-filter-mul-sq-le-sum-sq-univ","kind":"node","lean":"Finset.card_filter_mul_sq_le_sum_sq_univ","parent":28504,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/index.html#card-filter-mul-sq-le-sum-sq-univ","scope":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","status":"fully_proved","summary":"`card_filter_mul_sq_le_sum_sq` specialized to the universal finset.","title":"Sum-of-squares count bound over the whole type"},{"id":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/card-filter-mul-sq-le-of-moments-univ","kind":"node","lean":"Finset.card_filter_mul_sq_le_of_moments_univ","parent":28504,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting/index.html#card-filter-mul-sq-le-of-moments-univ","scope":"Combinatorics/probabilistic-combinatorics/moment-bounds-for-counting","status":"fully_proved","summary":"`card_filter_mul_sq_le_of_moments` specialized to the universal finset.","title":"Moment count bound over the whole type"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold.json","id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"can_prove","title":"Random 3-SAT threshold"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/clause","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.Clause","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#clause","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"A 3-clause on $n$ variables is a pair $(S, p)$ of a three-element set $S$ of variables among the $n$ and a polarity $p$ assigning a Boolean value to every variable, with $p$ equal to false at every variable outside $S$. This normalization gives each clause in the usual sense (a three-element variable set with one of $2^3$ polarity patterns) exactly one representative.","title":"3-clause"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/formula","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.Formula","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#formula","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"A formula with $m$ clauses on $n$ variables is an ordered $m$-tuple of 3-clauses on $n$ variables, that is, a function from $\\{0, \\dots, m - 1\\}$ to the set of 3-clauses on $n$ variables.","title":"Formulas of the random 3-SAT model"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/clause-sat","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.clauseSat","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#clause-sat","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"An assignment $a$ of Boolean values to $n$ variables satisfies a 3-clause $c$ on these variables when $a$ gives at least one of the three variables of $c$ the polarity that $c$ prescribes for it.","title":"Satisfaction of a 3-clause"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/satisfiable","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.Satisfiable","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#satisfiable","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"A formula $F$ with $m$ clauses on $n$ variables is satisfiable when there is an assignment $a$ of Boolean values to the $n$ variables that satisfies every clause of $F$, that is, $a$ satisfies the 3-clause $F(j)$ for every $j \\in \\{0, \\dots, m - 1\\}$.","title":"Satisfiable 3-SAT formulas"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/sat-prob","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.satProb","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#sat-prob","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the satisfiability probability $p(n, m)$ is the number of satisfiable formulas with $m$ clauses on $n$ variables divided by the number of all such formulas, that is, the probability that a uniformly random formula with $m$ clauses on $n$ variables is satisfiable. When there are no such formulas, which happens exactly when $n < 3$ and $m > 0$, the quotient is taken to be $0$; this\u2026","title":"Satisfiability probability of random 3-SAT"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/clause-nonempty","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.clause_nonempty","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#clause-nonempty","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"If $n \\ge 3$, then there is a 3-clause on $n$ variables: take the first three variables, with polarity true on them and the normalized polarity false elsewhere.","title":"3-clauses exist on at least three variables"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/sat-prob-zero","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.satProb_zero","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#sat-prob-zero","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"For every number $n$ of variables, the probability that a uniformly random formula with $0$ clauses on $n$ variables is satisfiable is $1$: the only such formula is the empty one, and every assignment satisfies it.","title":"Satisfiability probability with no clauses"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/is-upper-cutoff","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.IsUpperCutoff","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#is-upper-cutoff","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"A real number $\\rho$ is an upper threshold cutoff for random 3-SAT when, for every real $r > \\rho$, the probability that a uniformly random formula with $\\lfloor r n \\rfloor$ clauses on $n$ variables is satisfiable tends to $0$ as $n \\to \\infty$ (when $r n < 0$ the formula has no clauses); equivalently, such a formula is unsatisfiable with probability tending to $1$.","title":"Upper threshold cutoff"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/is-lower-cutoff","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.IsLowerCutoff","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#is-lower-cutoff","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"A real number $\\rho$ is a lower threshold cutoff for random 3-SAT when, for every real $r < \\rho$, the probability that a uniformly random formula with $\\lfloor r n \\rfloor$ clauses on $n$ variables is satisfiable tends to $1$ as $n \\to \\infty$ (when $r n < 0$ the formula has no clauses).","title":"Lower threshold cutoff"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/upper-threshold","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.upperThreshold","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#upper-threshold","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"The upper threshold is the infimum, in the real numbers, of the set of upper threshold cutoffs: the reals $\\rho$ such that, for every real $r > \\rho$, the probability that a uniformly random formula with $\\lfloor rn \\rfloor$ clauses on $n$ variables (no clauses when $rn < 0$) is satisfiable tends to $0$ as $n \\to \\infty$. The infimum is taken to be $0$ when this set is empty or unbounded below.","title":"Upper threshold of random 3-SAT"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/lower-threshold","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.lowerThreshold","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#lower-threshold","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"The lower threshold is the supremum, in the real numbers, of the set of lower threshold cutoffs: the reals $\\rho$ such that, for every real $r < \\rho$, the probability that a uniformly random formula with $\\lfloor rn \\rfloor$ clauses on $n$ variables (no clauses when $rn < 0$) is satisfiable tends to $1$ as $n \\to \\infty$. The supremum is taken to be $0$ when this set is empty or unbounded above.","title":"Lower threshold of random 3-SAT"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/has-constant-threshold","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.HasConstantThreshold","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#has-constant-threshold","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"There is a real number $c$ that is both a lower and an upper threshold cutoff: for every real $r < c$, the probability that a uniformly random formula with $\\lfloor r n \\rfloor$ clauses on $n$ variables is satisfiable tends to $1$ as $n \\to \\infty$, and for every real $r > c$ this probability tends to $0$ (when $r n < 0$ the formula has no clauses).","title":"Random 3-SAT threshold conjecture"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/clause-equiv","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.clauseEquiv","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#clause-equiv","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"For every $n$, the 3-clauses on $n$ variables correspond bijectively to the pairs $(S, q)$ of a three-element set $S$ of variables among the $n$ and a Boolean polarity $q$ on the variables of $S$: a clause is sent to its variable set together with its polarity restricted to that set.","title":"Clauses as variable sets with polarity patterns"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/to-cnfclause","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.toCNFClause","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#to-cnfclause","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"The CNF clause of a clause $c$ on $n$ variables is the list of literals $(i, b_i)$, one for each of the three variables $i$ of $c$ taken as a natural-number index, where $b_i$ is the polarity that $c$ prescribes for $i$; the literals are listed in some fixed order.","title":"CNF clause of a random 3-SAT clause"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/extend-assign","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.extendAssign","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#extend-assign","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"An assignment $a$ of Boolean values to the variables $0, \\dots, n - 1$ extends to all natural-number variables by giving variable $i$ the value $a(i)$ when $i < n$ and the value false otherwise.","title":"Extension of an assignment by false"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/clause-sat-restrict-iff","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.clauseSat_restrict_iff","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#clause-sat-restrict-iff","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"Let $A$ assign a Boolean value to every natural-number variable, and let $c$ be a 3-clause on the $n$ variables $0, \\dots, n - 1$. Then the restriction of $A$ to these $n$ variables satisfies $c$ if and only if $A$ satisfies the CNF clause of $c$: the list with one literal $(i, p_i)$ for each of the three variables $i$ of $c$, where $p_i$ is the polarity that $c$ prescribes at $i$. A CNF clause is satisfied when\u2026","title":"Satisfying a clause through a restricted assignment"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/to-cnf","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.toCNF","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#to-cnf","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"The CNF formula of a formula $F$ with $m$ clauses $F_0, \\dots, F_{m-1}$ on $n$ variables is the list of the CNF clauses of $F_0, \\dots, F_{m-1}$, in this order.","title":"CNF formula of a random 3-SAT formula"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/satisfiable-iff","kind":"node","lean":"CSLibExt.Combinatorics.RandomThreeSat.satisfiable_iff","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#satisfiable-iff","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"fully_proved","summary":"A formula $F$ with $m$ clauses on $n$ variables is satisfiable if and only if its CNF formula, which lists each clause of $F$ as its three literals (variable index, polarity), is satisfiable as a CNF formula: some assignment of truth values to all natural-number variable indices makes at least one literal of every clause true.","title":"Random 3-SAT satisfiability agrees with CNF satisfiability"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/diaz-kirousis-mitsche-perez-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.RandomThreeSatWanted.diazKirousisMitschePerezUpperBound_wanted","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#diaz-kirousis-mitsche-perez-upper-bound","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"can_prove","summary":"The number $4.49$ is an upper threshold cutoff for random 3-SAT: for every real $r > 4.49$, the probability that a uniformly random formula with $\\lfloor r n \\rfloor$ clauses on $n$ variables is satisfiable tends to $0$ as $n \\to \\infty$.","title":"D\u00edaz\u2013Kirousis\u2013Mitsche\u2013P\u00e9rez-Gim\u00e9nez upper bound for random 3-SAT"},{"id":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/hajiaghayi-sorkin-lower-bound-wanted","kind":"node","lean":"MathlibExt.Combinatorics.RandomThreeSatWanted.hajiaghayiSorkinLowerBound_wanted","parent":28512,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/random-3-sat-threshold/index.html#hajiaghayi-sorkin-lower-bound-wanted","scope":"Combinatorics/probabilistic-combinatorics/random-3-sat-threshold","status":"can_prove","summary":"The number $3.52$ is a lower threshold cutoff for random 3-SAT: for every real $r < 3.52$, the probability that a uniformly random formula with $\\lfloor r n \\rfloor$ clauses on $n$ variables (no clauses when $r < 0$) is satisfiable tends to $1$ as $n \\to \\infty$.","title":"Hajiaghayi\u2013Sorkin lower bound for random 3-SAT"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem.json","id":"Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem/index.html","scope":"Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem","status":"can_prove","title":"Reverse Littlewood\u2013Offord problem"},{"id":"Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem/small-signed-sum-count","kind":"node","lean":"MathlibExt.Combinatorics.ReverseLittlewoodOffordErdosProblem395Wanted.smallSignedSumCount","parent":28533,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem/index.html#small-signed-sum-count","scope":"Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem","status":"fully_proved","summary":"For complex numbers $z_1, \\dots, z_n$, this is the number of sign vectors $(\\varepsilon_1, \\dots, \\varepsilon_n) \\in \\{-1, 1\\}^n$ with $|\\varepsilon_1 z_1 + \\dots + \\varepsilon_n z_n| \\le \\sqrt{2}$.","title":"Number of short signed sums"},{"id":"Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem/reverse-littlewood-offord-sqrt-two","kind":"node","lean":"MathlibExt.Combinatorics.ReverseLittlewoodOffordErdosProblem395Wanted.reverseLittlewoodOffordSqrtTwo","parent":28533,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem/index.html#reverse-littlewood-offord-sqrt-two","scope":"Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem","status":"fully_proved","summary":"The statement that there is a real constant $c > 0$ such that for every natural number $n \\ge 1$ and all complex numbers $z_1, \\dots, z_n$ with $|z_i| = 1$ for every $i$, the number $N$ of sign vectors $(\\varepsilon_1, \\dots, \\varepsilon_n) \\in \\{-1, 1\\}^n$ with $|\\varepsilon_1 z_1 + \\dots + \\varepsilon_n z_n| \\le \\sqrt{2}$ satisfies $c/n \\le N/2^n$. That is, for independent uniformly random signs this inequality\u2026","title":"Reverse Littlewood\u2013Offord statement of Erd\u0151s Problem 395"},{"id":"Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem/reverse-littlewood-offord-sqrt-two-holds","kind":"node","lean":"MathlibExt.Combinatorics.ReverseLittlewoodOffordErdosProblem395Wanted.reverseLittlewoodOffordSqrtTwo_holds","parent":28533,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem/index.html#reverse-littlewood-offord-sqrt-two-holds","scope":"Combinatorics/probabilistic-combinatorics/reverse-littlewood-offord-problem","status":"can_prove","summary":"There is a real constant $c > 0$ such that for every natural number $n \\ge 1$ and all complex numbers $z_1, \\dots, z_n$ with $|z_i| = 1$ for every $i$, at least a $c/n$ fraction of the $2^n$ sign vectors $(\\varepsilon_1, \\dots, \\varepsilon_n) \\in \\{-1, 1\\}^n$ satisfy $|\\varepsilon_1 z_1 + \\dots + \\varepsilon_n z_n| \\le \\sqrt{2}$. That is, for independent uniformly random signs this inequality holds with probability\u2026","title":"Erd\u0151s Problem 395 holds: the reverse Littlewood\u2013Offord bound"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/shamir-s-problem.json","id":"Combinatorics/probabilistic-combinatorics/shamir-s-problem","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/shamir-s-problem/index.html","scope":"Combinatorics/probabilistic-combinatorics/shamir-s-problem","status":"can_prove","title":"Shamir's problem"},{"id":"Combinatorics/probabilistic-combinatorics/shamir-s-problem/matching-probability","kind":"node","lean":"MathlibExt.Combinatorics.ShamirPerfectMatchingThresholdErdosProblem747Wanted.matchingProbability","parent":28537,"reading_title":{"format":"inline-parts-v1","parts":[["text","Probability of a perfect matching in a random "],["math","\\(3\\)"],["text","-uniform hypergraph"]],"source":"Probability of a perfect matching in a random $3$-uniform hypergraph"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/shamir-s-problem/index.html#matching-probability","scope":"Combinatorics/probabilistic-combinatorics/shamir-s-problem","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, let $T$ be the set of $3$-element subsets of $\\{0, 1, \\dots, 3n - 1\\}$. This is the proportion of the $\\ell$-element subsets $E \\subseteq T$ that contain $n$ pairwise disjoint triples, that is, the probability that the uniformly random $3$-uniform hypergraph on $3n$ vertices with $\\ell$ edges has a perfect matching. When $\\ell > \\binom{3n}{3}$ there is no such hypergraph and the\u2026","title":"Probability of a perfect matching in a random $3$-uniform hypergraph"},{"id":"Combinatorics/probabilistic-combinatorics/shamir-s-problem/matching-above-threshold","kind":"node","lean":"MathlibExt.Combinatorics.ShamirPerfectMatchingThresholdErdosProblem747Wanted.matchingAboveThreshold","parent":28537,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/shamir-s-problem/index.html#matching-above-threshold","scope":"Combinatorics/probabilistic-combinatorics/shamir-s-problem","status":"fully_proved","summary":"The proposition that for every real $\\varepsilon > 0$, the probability that the uniformly random $3$-uniform hypergraph on $3n$ vertices with $\\lceil (1 + \\varepsilon) n \\log n \\rceil$ edges has a perfect matching ($n$ pairwise disjoint edges) tends to $1$ as $n \\to \\infty$, where $\\log$ is the natural logarithm.","title":"Statement of the upper half of Shamir's problem"},{"id":"Combinatorics/probabilistic-combinatorics/shamir-s-problem/matching-threshold","kind":"node","lean":"MathlibExt.Combinatorics.ShamirPerfectMatchingThresholdErdosProblem747Wanted.matchingThreshold","parent":28537,"reading_title":{"format":"inline-parts-v1","parts":[["text","Threshold for a perfect matching in a random "],["math","\\(3\\)"],["text","-uniform hypergraph"]],"source":"Threshold for a perfect matching in a random $3$-uniform hypergraph"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/shamir-s-problem/index.html#matching-threshold","scope":"Combinatorics/probabilistic-combinatorics/shamir-s-problem","status":"fully_proved","summary":"For a natural number $n$, the threshold $\\ell_c(n)$ is the least natural number $\\ell$ such that the uniformly random $3$-uniform hypergraph on $3n$ vertices with $\\ell$ edges has a perfect matching ($n$ pairwise disjoint edges) with probability at least $1/2$.","title":"Threshold for a perfect matching in a random $3$-uniform hypergraph"},{"id":"Combinatorics/probabilistic-combinatorics/shamir-s-problem/matching-threshold-asymptotic","kind":"node","lean":"MathlibExt.Combinatorics.ShamirPerfectMatchingThresholdErdosProblem747Wanted.matchingThresholdAsymptotic","parent":28537,"reading_title":{"format":"inline-parts-v1","parts":[["text","Statement of Shamir's problem: the matching threshold is asymptotic to "],["math","\\(n \\log n\\)"]],"source":"Statement of Shamir's problem: the matching threshold is asymptotic to $n \\log n$"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/shamir-s-problem/index.html#matching-threshold-asymptotic","scope":"Combinatorics/probabilistic-combinatorics/shamir-s-problem","status":"fully_proved","summary":"Let $\\ell_c(n)$ be the least number $\\ell$ such that the uniformly random $3$-uniform hypergraph on $3n$ vertices with $\\ell$ edges has a perfect matching with probability at least $1/2$. The proposition that $\\ell_c(n) / (n \\log n) \\to 1$ as $n \\to \\infty$, where $\\log$ is the natural logarithm.","title":"Statement of Shamir's problem: the matching threshold is asymptotic to $n \\log n$"},{"id":"Combinatorics/probabilistic-combinatorics/shamir-s-problem/matching-above-threshold-holds","kind":"node","lean":"MathlibExt.Combinatorics.ShamirPerfectMatchingThresholdErdosProblem747Wanted.matchingAboveThreshold_holds","parent":28537,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/shamir-s-problem/index.html#matching-above-threshold-holds","scope":"Combinatorics/probabilistic-combinatorics/shamir-s-problem","status":"can_prove","summary":"For every real $\\varepsilon > 0$, the probability that the uniformly random $3$-uniform hypergraph on $3n$ vertices with $\\lceil (1 + \\varepsilon) n \\log n \\rceil$ edges has a perfect matching ($n$ pairwise disjoint edges) tends to $1$ as $n \\to \\infty$, where $\\log$ is the natural logarithm.","title":"Kahn's theorem on Shamir's problem, upper half"},{"id":"Combinatorics/probabilistic-combinatorics/shamir-s-problem/matching-threshold-asymptotic-holds","kind":"node","lean":"MathlibExt.Combinatorics.ShamirPerfectMatchingThresholdErdosProblem747Wanted.matchingThresholdAsymptotic_holds","parent":28537,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kahn's theorem: the threshold in Shamir's problem is asymptotic to "],["math","\\(n \\log n\\)"]],"source":"Kahn's theorem: the threshold in Shamir's problem is asymptotic to $n \\log n$"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/shamir-s-problem/index.html#matching-threshold-asymptotic-holds","scope":"Combinatorics/probabilistic-combinatorics/shamir-s-problem","status":"can_prove","summary":"Let $\\ell_c(n)$ be the least number $\\ell$ such that the uniformly random $3$-uniform hypergraph on $3n$ vertices with $\\ell$ edges has a perfect matching with probability at least $1/2$. Then $\\ell_c(n) / (n \\log n) \\to 1$ as $n \\to \\infty$, where $\\log$ is the natural logarithm.","title":"Kahn's theorem: the threshold in Shamir's problem is asymptotic to $n \\log n$"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds.json","id":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/index.html","scope":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds","status":"can_state","title":"Thresholds and expectation thresholds"},{"id":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/is-weakly-small","kind":"node","lean":"MathlibExt.Combinatorics.FractionalExpectationThresholdsWanted.FractionalExpectationThresholds.IsWeaklySmall","parent":28544,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weakly "],["math","\\(p\\)"],["text","-small families"]],"source":"Weakly $p$-small families"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/index.html#is-weakly-small","scope":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds","status":"fully_proved","summary":"Let $X$ be a finite set, $p \\in [0, 1]$, and $\\mathcal F$ a family of subsets of $X$. Then $\\mathcal F$ is weakly $p$-small if there is a function $g$ from subsets of $X$ to the nonnegative reals such that $\\sum_{S \\subseteq T} g(S) \\ge 1$ for every $T \\in \\mathcal F$ and $\\sum_{S \\subseteq X} g(S)\\, p^{|S|} \\le 1/2$.","title":"Weakly $p$-small families"},{"id":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/p-c","kind":"node","lean":"MathlibExt.Combinatorics.FractionalExpectationThresholdsWanted.FractionalExpectationThresholds.p_c","parent":28544,"reading_title":{"format":"inline-parts-v1","parts":[["text","The threshold "],["math","\\(p_c(\\mathcal F)\\)"]],"source":"The threshold $p_c(\\mathcal F)$"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/index.html#p-c","scope":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds","status":"fully_proved","summary":"For a set $X$ and $p \\in [0, 1]$, let $\\mu_p$ be the $p$-biased product measure on subsets of $X$, under which each element of $X$ lies in the random subset independently with probability $p$. For a family $\\mathcal F$ of subsets of $X$, $p_c(\\mathcal F)$ is some $p \\in [0, 1]$ with $\\mu_p(\\mathcal F) = 1/2$ if such a $p$ exists, and $0$ otherwise.","title":"The threshold $p_c(\\mathcal F)$"},{"id":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/q-f","kind":"node","lean":"MathlibExt.Combinatorics.FractionalExpectationThresholdsWanted.FractionalExpectationThresholds.q_f","parent":28544,"reading_title":{"format":"inline-parts-v1","parts":[["text","The fractional expectation-threshold "],["math","\\(q_f(\\mathcal F)\\)"]],"source":"The fractional expectation-threshold $q_f(\\mathcal F)$"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/index.html#q-f","scope":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds","status":"fully_proved","summary":"For a family $\\mathcal F$ of subsets of a finite set $X$, $q_f(\\mathcal F)$ is the supremum, in $[0, 1]$, of the set of all $p \\in [0, 1]$ such that $\\mathcal F$ is weakly $p$-small; it is $0$ if there is no such $p$.","title":"The fractional expectation-threshold $q_f(\\mathcal F)$"},{"id":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/l","kind":"node","lean":"MathlibExt.Combinatorics.FractionalExpectationThresholdsWanted.FractionalExpectationThresholds.l","parent":28544,"reading_title":{"format":"inline-parts-v1","parts":[["text","Size of a largest minimal element, "],["math","\\(\\ell(\\mathcal F)\\)"]],"source":"Size of a largest minimal element, $\\ell(\\mathcal F)$"},"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/index.html#l","scope":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds","status":"fully_proved","summary":"For a family $\\mathcal F$ of subsets of a set $X$, $\\ell(\\mathcal F)$ is the largest cardinality of a minimal element of $\\mathcal F$ (minimal with respect to inclusion), a natural number.","title":"Size of a largest minimal element, $\\ell(\\mathcal F)$"},{"id":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/k","kind":"node","lean":"MathlibExt.Combinatorics.FractionalExpectationThresholdsWanted.FractionalExpectationThresholds.K","parent":28544,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/index.html#k","scope":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds","status":"can_state","summary":"A real number $K$, not yet specified, that is to serve as the universal constant in the bound of the threshold of an increasing family by $K$ times its fractional expectation-threshold times the logarithm of the size of its largest minimal element.","title":"The constant of the Frankston\u2013Kahn\u2013Narayanan\u2013Park theorem"},{"id":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/theorem-1-1","kind":"node","lean":"MathlibExt.Combinatorics.FractionalExpectationThresholdsWanted.FractionalExpectationThresholds.theorem_1_1","parent":28544,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds/index.html#theorem-1-1","scope":"Combinatorics/probabilistic-combinatorics/thresholds-and-expectation-thresholds","status":"stated","summary":"Let $K$ be the universal real constant still to be supplied. For every finite set $X$ and every increasing family $\\mathcal F$ of subsets of $X$ (closed under taking supersets) with $\\ell(\\mathcal F) \\ge 2$, where $\\ell(\\mathcal F)$ is the size of a largest minimal element of $\\mathcal F$,","title":"Thresholds versus fractional expectation-thresholds"},{"data_url":"scopes/Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture.json","id":"Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture","kind":"scope","navigation":"module","parent":28449,"result":false,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture/index.html","scope":"Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture","status":"can_prove","title":"Tomaszewski's conjecture"},{"id":"Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture/tomaszewski-conjecture","kind":"node","lean":"MathlibExt.Combinatorics.TomaszewskiRandomSignedSumAMR127Problem0002Wanted.tomaszewskiConjecture","parent":28551,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture/index.html#tomaszewski-conjecture","scope":"Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture","status":"fully_proved","summary":"The statement that for every natural number $n$ and every vector $a = (a_1, \\dots, a_n) \\in \\mathbb{R}^n$ with $a_1^2 + \\dots + a_n^2 = 1$, at least half of the $2^n$ sign vectors $x \\in \\{-1, 1\\}^n$ satisfy $|a_1 x_1 + \\dots + a_n x_n| \\le 1$.","title":"Tomaszewski's conjecture"},{"id":"Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture/tomaszewski-conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.TomaszewskiRandomSignedSumAMR127Problem0002Wanted.tomaszewskiConjecture_holds","parent":28551,"result":true,"route":"../roadmap/Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture/index.html#tomaszewski-conjecture-holds","scope":"Combinatorics/probabilistic-combinatorics/tomaszewski-s-conjecture","status":"can_prove","summary":"For every natural number $n$ and every vector $a = (a_1, \\dots, a_n) \\in \\mathbb{R}^n$ with $a_1^2 + \\dots + a_n^2 = 1$, at least half of the $2^n$ sign vectors $x \\in \\{-1, 1\\}^n$ satisfy $|a_1 x_1 + \\dots + a_n x_n| \\le 1$; equivalently, for a uniformly random sign vector $x$ this inequality holds with probability at least $1/2$.","title":"Tomaszewski's conjecture holds"},{"data_url":"scopes/Combinatorics/elementary-enumeration.json","id":"Combinatorics/elementary-enumeration","kind":"scope","parent":22581,"result":false,"route":"../roadmap/Combinatorics/elementary-enumeration/index.html","scope":"Combinatorics/elementary-enumeration","status":"can_prove","summary":"Counting basic objects under simple constraints: subsets, multisets and functions with prescribed values, Schreier-type and weighted Schreier sets, combinations avoiding given differences, necklaces and counting up to rotation, fibre and double counting, introductory catalogues of special counting sequences, and small extremal puzzles such as the bridge and torch problem.","title":"Elementary enumeration"},{"data_url":"scopes/Combinatorics/elementary-enumeration/bridge-and-torch-problem.json","id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","kind":"scope","navigation":"module","parent":28554,"result":false,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","title":"Bridge and torch problem"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-weight","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeWeight","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-weight","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"The rational weights $y_i$ are $y_1 = y_2 = y_3 = 0$, $y_4 = 2$, $y_5 = 4$, $y_6 = 5$, and $y_i = (4i - 8)/3$ for $i \\ge 7$. The value $y_0$ is unused and set to $0$.","title":"Weights of the capacity-3 lower bound"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-weight-small","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeWeight_small","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-weight-small","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"The weights $y_1, \\dots, y_7$ of the capacity-3 lower bound are $0, 0, 0, 2, 4, 5, 20/3$.","title":"First seven weights of the capacity-3 lower bound"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-weight-of-seven-le","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeWeight_of_seven_le","parent":28555,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weights of the capacity-3 lower bound from "],["math","\\(7\\)"],["text"," on"]],"source":"Weights of the capacity-3 lower bound from $7$ on"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-weight-of-seven-le","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For every natural number $i \\ge 7$, the weight $y_i$ of the capacity-3 lower bound equals $(4i - 8)/3$.","title":"Weights of the capacity-3 lower bound from $7$ on"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-schedule-from","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeScheduleFrom","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-schedule-from","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Let $c$ be a natural number and let $P$ and $L$ be finite sets of natural numbers, the people and the left bank. Whether a finite sequence $A_1, \\dots, A_k$ of finite sets of natural numbers (the crossings) is valid from the left bank $L$ with the torch on a given side is defined by recursion on $k$. The empty sequence is valid exactly when $L = \\emptyset$ and the torch is on the right. With the torch on the left,\u2026","title":"Valid crossing sequences from a state of the bridge and torch problem"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-bridge-schedule","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.IsBridgeSchedule","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-bridge-schedule","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Let $c$ and $n$ be natural numbers. A finite sequence $\\sigma$ of finite sets of natural numbers (the crossings) is a schedule for the people $1, \\dots, n$ on a bridge of capacity $c$ with one torch when the following holds. Start with all of $1, \\dots, n$ and the torch on the left bank. The crossings at even positions (counting from $0$) are forward moves: each is a nonempty set of at most $c$ people then on the\u2026","title":"Bridge schedules"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/schedule-cost","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.scheduleCost","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#schedule-cost","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For a finite sequence $\\sigma = (A_1, \\dots, A_k)$ of finite sets of natural numbers (the crossings), the total time of $\\sigma$ is $\\sum_{i=1}^{k} \\max A_i$, where $\\max \\emptyset = 0$. Since person $p$ has crossing time $p$, each crossing costs the largest crossing time among the people in it.","title":"Total time of a schedule"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For a bridge of capacity $c$ and $n$ people $1, \\dots, n$, where person $i$ has crossing time $i$, the optimal crossing time is the least total time of a complete schedule with one torch; the total time of a schedule is the sum, over its crossings, of the largest crossing time among the people in that crossing. It is written $T(n)$ for $c = 2$ and $T_3(n)$ for $c = 3$. If no complete schedule exists (for example\u2026","title":"Optimal crossing time of the bridge and torch problem"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/forward-crossings","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.forwardCrossings","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#forward-crossings","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"The forward crossings of a list of crossings are its entries at the even positions $0, 2, 4, \\dots$ (counting from $0$), in their original order. For a schedule these are its forward moves, from left to right.","title":"Forward crossings"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/return-crossings","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.returnCrossings","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#return-crossings","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For a list $\\sigma = (A_0, A_1, \\dots, A_{\\ell - 1})$ whose entries may come from any type, the returns of $\\sigma$ are its entries at odd positions, $A_1, A_3, A_5, \\dots$ in order, with positions counted from $0$. For the list of crossings of a bridge schedule these are the crossings from right to left.","title":"Return crossings"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/right-bank","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.rightBank","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#right-bank","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Let $n$ be a natural number and let $\\tau = (A_1, \\dots, A_k)$ be a finite sequence of finite sets of natural numbers. The right bank after $\\tau$ is the set of those $p \\in \\{1, \\dots, n\\}$ for which the number of indices $i$ with $p \\in A_i$ is odd.","title":"Right bank after a sequence of crossings"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time-le-schedule-cost","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime_le_scheduleCost","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time-le-schedule-cost","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For all natural numbers $c$ and $n$ and every complete schedule $\\sigma$ for the people $1, \\dots, n$ on a bridge of capacity $c$ with one torch, the optimal crossing time of these $n$ people on this bridge is at most the total time of $\\sigma$.","title":"Every schedule bounds the optimal time"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-bridge-schedule-two-four","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.isBridgeSchedule_two_four","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-bridge-schedule-two-four","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For the people $1, 2, 3, 4$ on a bridge of capacity $2$, the crossings $\\{1,2\\}$ forward, $\\{1\\}$ back, $\\{1,3\\}$ forward, $\\{1\\}$ back, $\\{1,4\\}$ forward, in this order, form a schedule, and its total time is $11$.","title":"A four-person schedule of total time 11"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-bridge-schedule-capacity-three","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.isBridgeSchedule_capacity_three","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-bridge-schedule-capacity-three","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For the people $1, 2, 3$, the single forward crossing $\\{1, 2, 3\\}$ is not a schedule on a bridge of capacity $2$, but it is a schedule on a bridge of capacity $3$.","title":"One crossing of three people under capacities two and three"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/forward-crossings-return-crossings-right","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.forwardCrossings_returnCrossings_rightBank_example","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#forward-crossings-return-crossings-right","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For the schedule whose crossings are, in order, $\\{1,2\\}$, $\\{1\\}$, $\\{1,3\\}$, $\\{1\\}$, $\\{1,4\\}$, the forward crossings are $\\{1,2\\}, \\{1,3\\}, \\{1,4\\}$ and the returns are $\\{1\\}, \\{1\\}$. For the people $1, 2, 3, 4$, the right bank after the crossings $\\{1,2\\}$, $\\{1\\}$ (the first two of the schedule) is $\\{2\\}$.","title":"Forward moves, returns and right bank of a four-person schedule"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/parity-interpolation","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.parity_interpolation","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#parity-interpolation","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $f, g, a : \\mathbb{N} \\to \\mathbb{R}$ be sequences with $a_{2m} = f(m)$ and $a_{2m+1} = g(m)$ for every $m \\in \\mathbb{N}$. Then for every $n \\in \\mathbb{N}$,","title":"Parity interpolation"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-weight-local-inequality","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeWeight_local_inequality","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-weight-local-inequality","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $y_i$ be the weights of the capacity-3 lower bound. For every nonempty set $T$ of positive integers with $|T| \\le 3$,","title":"Local inequality for the capacity-3 weights"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-weight-return-inequality","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeWeight_return_inequality","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-weight-return-inequality","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $y_i$ be the weights of the capacity-3 lower bound. For every nonempty set $R$ of positive integers with $|R| \\le 3$,","title":"Return inequality for the capacity-3 weights"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/closed-form-u-integral-and-near-v","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.closedForm_U_integral_and_near_V","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#closed-form-u-integral-and-near-v","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"For every integer $n \\ge 7$, let $U(n) = n^2/6 + 2n - 181/36 + (-1)^n/4 - (2/9)\\cos(2n\\pi/3)$ and $V(n) = n^2/6 + 2n - 11/2$. Then $U(n)$ is an integer and $0 \\le U(n) - V(n) < 1$.","title":"The capacity-3 closed form is an integer close to the lower bound"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time-two-closed-form","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime_two_closed_form","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time-two-closed-form","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $T(n)$ be the optimal crossing time of $n$ people with crossing times $1, \\dots, n$ on a bridge of capacity $2$ with one torch. For every $n \\ge 2$,","title":"Closed form for capacity two"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time-two-least-trips-of-fastest","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime_two_least_trips_of_fastest","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time-two-least-trips-of-fastest","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"For bridge capacity $2$ and $n \\ge 1$ people $1, \\dots, n$ (person $i$ has crossing time $i$), the least number $a(n)$ of crossings that include person $1$, taken over all optimal schedules (complete schedules whose total time equals the optimal crossing time $T(n)$), is $a(n) = 2\\lceil n/2 \\rceil - 1$. Crossings in both directions are counted.","title":"Fewest crossings of the fastest person in an optimal schedule"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/schedule-cost-two-lower-bound","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.scheduleCost_two_lower_bound","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#schedule-cost-two-lower-bound","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $n \\ge 3$ and let $T(n)$ be the optimal crossing time of the $n$ people $1, \\dots, n$ (person $i$ crossing in time $i$) on a bridge of capacity $2$. Let $\\sigma$ be a bridge schedule of capacity $2$ for these people in which person $1$ takes part in exactly $a$ forward moves, exactly $m$ returns carry two people and exactly $s$ forward moves carry one person. Then the total cost of $\\sigma$ is at least\u2026","title":"Lower bound for capacity-two schedules"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time-three-initial-values","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime_three_initial_values","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time-three-initial-values","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $T_3(n)$ be the optimal crossing time of $n$ people with crossing times $1, \\dots, n$ on a bridge of capacity $3$ with one torch. Then $T_3(1) = 1$, $T_3(2) = 2$, $T_3(3) = 3$, $T_3(4) = 7$, $T_3(5) = 9$ and $T_3(6) = 14$.","title":"Initial values for capacity three"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/exists-optimal-schedule-three-fastest-re","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.exists_optimal_schedule_three_fastest_returns","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#exists-optimal-schedule-three-fastest-re","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $n \\ge 1$ and consider the people $1, \\dots, n$, person $i$ having crossing time $i$, on a bridge of capacity $3$ with one torch. Then some schedule has total time equal to the optimal crossing time $T_3(n)$ and the following property: every return (a crossing at an odd position, counting from $0$) carries a single person $p$, and $p \\le q$ for every person $q$ on the right bank just before that return.","title":"Optimal capacity-3 schedules with fastest single returns"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time-three-le-sub-six","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime_three_le_sub_six","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time-three-le-sub-six","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $T_3(n)$ be the optimal crossing time of $n$ people with crossing times $1, \\dots, n$ on a bridge of capacity $3$ with one torch. For every $k \\ge 9$, $T_3(k) \\le T_3(k - 6) + 2k + 6$.","title":"Six-step upper bound for capacity three"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time-three-le-closed-form","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime_three_le_closed_form","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time-three-le-closed-form","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $T_3(n)$ be the optimal crossing time of $n$ people with crossing times $1, \\dots, n$ on a bridge of capacity $3$ with one torch. For every $n \\ge 7$,","title":"Upper bound for capacity three"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/schedule-cost-three-lower-bound","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.scheduleCost_three_lower_bound","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#schedule-cost-three-lower-bound","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $n \\ge 7$. Every bridge schedule of capacity $3$ for the $n$ people $1, \\dots, n$ (person $i$ crossing in time $i$) has total cost at least $V(n) = n^2/6 + 2n - 11/2$.","title":"Lower bound for capacity-three schedules"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time-three-closed-form","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime_three_closed_form","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time-three-closed-form","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $T_3(n)$ be the optimal crossing time of $n$ people with crossing times $1, \\dots, n$ on a bridge of capacity $3$ with one torch. For every $n \\ge 7$,","title":"Closed form for capacity three"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bridge-time-three-recurrence","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bridgeTime_three_recurrence","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bridge-time-three-recurrence","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $T_3(n)$ be the optimal crossing time of $n$ people with crossing times $1, \\dots, n$ on a bridge of capacity $3$ with one torch. For every $n \\ge 13$, $T_3(n) = T_3(n - 6) + 2n + 6$.","title":"Six-step recurrence for capacity three"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-admissible-bag","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.IsAdmissibleBag","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-admissible-bag","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Let $n$ be a natural number. A finite multiset $F$ of finite sets of natural numbers is an admissible bag for $n$ people (bridge capacity $3$) when (i) every $T \\in F$ is a nonempty subset of $\\{1, \\dots, n\\}$ with $|T| \\le 3$, (ii) every $i \\in \\{1, \\dots, n\\}$ lies in some $T \\in F$, and (iii) $\\sum_{T \\in F} (|T| - 1) = n - 1$, where the sum counts each member of $F$ with its multiplicity and $n - 1$ is read as\u2026","title":"Admissible bags"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bag-weight","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bagWeight","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bag-weight","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For a natural number $n$ and a bag $F$, that is, a finite multiset of finite sets of natural numbers, the weight of $F$ for $n$ people is the real number $w(F) = \\sum_{T \\in F} \\left(\\max T + \\sum_{i \\in T} i\\right) - \\frac{n(n+1)}{2}$, where each set counts with its multiplicity in $F$ and $\\max \\emptyset = 0$.","title":"Weight of a bag"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-admissible-bag-three-and-bag-weight","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.isAdmissibleBag_three_and_bagWeight","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-admissible-bag-three-and-bag-weight","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"The multiset whose only member is the set $\\{1, 2, 3\\}$ is an admissible bag for $3$ people, and its weight for $3$ people is $3$.","title":"A one-trip admissible bag for three people"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-admissible-bag-forward-crossings-of-s","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.isAdmissibleBag_forwardCrossings_of_single_returns","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-admissible-bag-forward-crossings-of-s","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Let $n$ be a natural number and let $\\sigma$ be a schedule for the people $1, \\dots, n$ on a bridge of capacity $3$ in which every return carries exactly one person. Let $F(\\sigma)$ be the multiset of the forward crossings of $\\sigma$. Then $F(\\sigma)$ is an admissible bag for $n$ people, and the total time of $\\sigma$ equals the weight $w(F(\\sigma))$ of this bag for $n$ people.","title":"Schedules with single-person returns give admissible bags"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/exists-is-admissible-bag-bag-weight-le-b","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.exists_isAdmissibleBag_bagWeight_le_bridgeTime_three","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#exists-is-admissible-bag-bag-weight-le-b","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"For every natural number $n$, some admissible bag $F$ for $n$ people has weight $w(F)$ at most the capacity-3 optimal crossing time $T_3(n)$.","title":"Some admissible bag weighs at most the capacity-3 optimal time"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/bag-weight-ge-of-is-admissible-bag","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.bagWeight_ge_of_isAdmissibleBag","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#bag-weight-ge-of-is-admissible-bag","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"For every natural number $n \\ge 7$ and every admissible bag $F$ for $n$ people, the weight of $F$ is at least $\\frac{n^2}{6} + 2n - \\frac{11}{2}$.","title":"Lower bound for the weight of an admissible bag"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/star-move","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.StarMove","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#star-move","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Consider the star graph $\\mathcal{S}(k)$ with center $o$ and leaves $u_0, \\dots, u_{k-1}$, and $n$ persons $0, \\dots, n-1$, where person $p$ has crossing time $p + 1$. A movement consists of a set $P$ of persons, a leaf $u_j$ and a direction: the persons in $P$ carry one torch along the edge $\\{o, u_j\\}$, outward from $o$ to $u_j$ or inward from $u_j$ to $o$. No condition is imposed on $P$ here (it may even be\u2026","title":"Movement in the star model"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/src","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.StarMove.src","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#src","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Let $\\mathcal{S}(k)$ be the star graph with center $o$ and $k$ leaves. A movement in the star model on $\\mathcal{S}(k)$ along the edge $\\{o, u\\}$ between the center $o$ and a leaf $u$ starts at $o$ if it goes outward (from the center to the leaf) and at $u$ otherwise.","title":"Start vertex of a star movement"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/dst","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.StarMove.dst","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#dst","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"In the star model on the star graph with center $o$ and leaves $u_0, \\dots, u_{k-1}$, a torch-movement along the edge $\\{o, u_\\ell\\}$ ends at the leaf $u_\\ell$ if it goes outward from $o$, and at the center $o$ otherwise.","title":"End vertex of a star movement"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-star-step","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.IsStarStep","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-star-step","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Let $\\mathcal{S}(k)$ be the star graph with a center and $k$ leaves. Consider a configuration of the star model on $\\mathcal{S}(k)$ with $n$ persons, given by the vertex $\\mathrm{pos}(p)$ of each person $p$ and the number $\\mathrm{tor}(v)$ of torches at each vertex $v$. A parallel step, a list of star movements, is legal from this configuration when every movement carries one or two persons, all standing at the\u2026","title":"Legal parallel steps in the star model"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/star-step-pos","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.starStepPos","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#star-step-pos","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Given the vertex $\\mathrm{pos}(p)$ of each person $p$ and a step $S$, a finite list of movements, the position of $p$ after $S$ is the end vertex of the first movement of $S$ whose persons include $p$ (its leaf if the movement is outward, the center if it is inward), and $\\mathrm{pos}(p)$ if no movement of $S$ includes $p$.","title":"Positions after a step in the star model"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/star-step-torch","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.starStepTorch","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#star-step-torch","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Given the number $\\mathrm{tor}(v)$ of torches at each vertex $v$ of the star graph and a step $S$, a finite list of movements, the number of torches at $v$ after $S$ is $\\max(\\mathrm{tor}(v) - a, 0) + b$, where $a$ is the number of movements of $S$ that start at $v$ and $b$ is the number of movements of $S$ that end at $v$.","title":"Torch counts after a step in the star model"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/star-step-duration","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.starStepDuration","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#star-step-duration","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"A step is a finite list of movements carried out in parallel. Its duration is the largest crossing time $p + 1$ of a person $p$ moving in one of its movements, and $0$ if no person moves.","title":"Duration of a step in the star model"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/star-run-from","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.starRunFrom","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#star-run-from","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"A configuration of the star model assigns to each person $p$ a vertex $\\mathrm{pos}(p)$ of the star graph $\\mathcal{S}(k)$ and to each vertex $v$ a number $\\mathrm{tor}(v)$ of torches. A finite sequence $\\sigma = (S_1, \\dots, S_\\ell)$ of steps, each a finite list of movements, is a legal run from the configuration $(\\mathrm{pos}, \\mathrm{tor})$ when each step $S_i$ is legal from the configuration reached after\u2026","title":"Legal runs in the star model"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-star-schedule","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.IsStarSchedule","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-star-schedule","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Consider the star model on the star graph $\\mathcal{S}(k)$, which has a center and $k$ leaves, with $n$ persons and $t$ torches. A finite list $\\sigma$ of parallel steps (each a list of star movements) is a star schedule when the following holds. Start from the configuration with every person and all $t$ torches at the center and no torch at any leaf, and apply the steps of $\\sigma$ in order, where a step moves the\u2026","title":"Star schedules"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/star-time","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.starTime","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#star-time","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"For $n$ persons, $k$ edges and $t$ torches, $T(n, k, t)$ is the smallest total duration of a star schedule, where the total duration of a schedule is the sum of the durations of its steps. If no star schedule exists (for example when $n \\ge 1$ and $k = 0$ or $t = 0$), then $T(n, k, t) = 0$ by convention.","title":"Optimal time in the star model"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/star-time-le","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.starTime_le","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#star-time-le","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"Let $T(n, k, t)$ be the optimal time of the star model with $n$ persons, $k$ edges and $t$ torches. For every star schedule $\\sigma$ with $n$ persons, $k$ edges and $t$ torches, $T(n, k, t)$ is at most the total duration of $\\sigma$, the sum of the durations of its steps.","title":"Every star schedule bounds the optimal time"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/is-star-schedule-two-one-one","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.isStarSchedule_two_one_one","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#is-star-schedule-two-one-one","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"In the star model with $2$ persons (crossing times $1$ and $2$), a star graph with $1$ edge and $1$ torch, the sequence of one step consisting of a single movement that carries both persons and the torch from the center to the leaf is a star schedule, and its total duration is $2$.","title":"Two persons crossing a one-edge star"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/not-is-star-step-return-without-torch","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.not_isStarStep_return_without_torch","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#not-is-star-step-return-without-torch","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"fully_proved","summary":"In the star model on the star graph with a center and a single leaf, with one person, suppose the person stands at the leaf, the only torch is at the center and no torch is at the leaf. Then the parallel step consisting of the single movement that carries the person from the leaf back to the center is not legal.","title":"Example: no return from a leaf without a torch"},{"id":"Combinatorics/elementary-enumeration/bridge-and-torch-problem/star-time-bounds","kind":"node","lean":"MetaMathlibExt.BridgeTorchGeneralizations.starTime_bounds","parent":28555,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/bridge-and-torch-problem/index.html#star-time-bounds","scope":"Combinatorics/elementary-enumeration/bridge-and-torch-problem","status":"can_prove","summary":"Consider the star model with $n \\ge 2$ persons of crossing times $1, \\dots, n$, $k \\ge 1$ edges and $t \\ge 1$ torches, let $T(n, k, t)$ be its optimal time, and let $m = \\min(k, t)$ and $s = \\lceil n/(2m) \\rceil$. Let $T(n)$ be the optimal crossing time of $n$ persons with crossing times $1, \\dots, n$ on a bridge of capacity $2$ with one torch. Then","title":"Bridge and torch on star graphs"},{"data_url":"scopes/Combinatorics/elementary-enumeration/combinations-without-specified-separations.json","id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","kind":"scope","navigation":"module","parent":28554,"result":false,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","title":"Combinations without specified separations"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combinations","kind":"node","lean":"MetaMathlibExt.restrictedCombinations","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combinations","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For a finite set $Q$ of natural numbers and $n, k \\ge 0$, the restricted combinations are the $k$-element subsets $s$ of $\\{1, \\dots, n\\}$ such that $b - a \\notin Q$ for all $a, b \\in s$ with $a < b$. Since only distinct elements are compared, whether $0 \\in Q$ makes no difference.","title":"Restricted combinations"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-count","kind":"node","lean":"MetaMathlibExt.restrictedCombinationCount","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-count","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For a finite set $Q$ of natural numbers and natural numbers $n$ and $k$, $S^Q_{n,k}$ is the number of $k$-subsets of $\\{1, \\dots, n\\}$ in which no two distinct elements differ by an element of $Q$.","title":"Number of restricted combinations"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-total","kind":"node","lean":"MetaMathlibExt.restrictedCombinationTotal","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-total","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For a finite set $Q$ of natural numbers and $n \\ge 0$, $S^Q_n$ is the number of subsets of $\\{1, \\dots, n\\}$ in which no two distinct elements differ by an element of $Q$. That is, $S^Q_n = \\sum_{k=0}^{n} S^Q_{n,k}$, where $S^Q_{n,k}$ is the number of such subsets with $k$ elements.","title":"Number of subsets avoiding a set of differences"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-count-empty","kind":"node","lean":"MetaMathlibExt.restrictedCombinationCount_empty","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-count-empty","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, the number $S^{\\emptyset}_{n,k}$ of $k$-subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by an element of the empty set is the binomial coefficient $\\binom{n}{k}$.","title":"Restricted combinations with no forbidden differences"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-total-singleton-o","kind":"node","lean":"MetaMathlibExt.restrictedCombinationTotal_singleton_one_four","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subsets of "],["math","\\(\\{1, 2, 3, 4\\}\\)"],["text"," with no two consecutive elements"]],"source":"Subsets of $\\{1, 2, 3, 4\\}$ with no two consecutive elements"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-total-singleton-o","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"There are exactly $8$ subsets of $\\{1, 2, 3, 4\\}$ in which no two elements differ by $1$.","title":"Subsets of $\\{1, 2, 3, 4\\}$ with no two consecutive elements"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-count-one-q-eq-ca","kind":"node","lean":"MetaMathlibExt.restrictedCombinationCount_one_q_eq_card_windowedAdjacentSwapPerms","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-count-one-q-eq-ca","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"For all natural numbers $q$, $n$ and $k$, the number of $k$-subsets of $\\{1, \\dots, n\\}$ in which no two distinct elements differ by $1$ or by $q$ equals the number of permutations $\\pi$ of $\\{0, 1, \\dots, n\\}$ such that $|\\pi(i) - i| \\le 1$ for every $i$; for every $i_0 \\ge 0$ with $i_0 + q \\le n + 1$, all $i, j$ with $i_0 \\le i, j < i_0 + q$ satisfy $\\pi(i) \\le \\pi(j) + q$ (every window of $q$ consecutive values\u2026","title":"Forbidden differences 1 and q and windowed adjacent-swap permutations"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-total-one-four-ge","kind":"node","lean":"MetaMathlibExt.restrictedCombinationTotal_one_four_genFun","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-total-one-four-ge","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $S^{\\{1,4\\}}_n$ be the number of subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by $1$ or by $4$. Then, in the ring $\\mathbb{Z}[[x]]$ of formal power series,","title":"Generating function for subsets avoiding differences 1 and 4"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-total-one-five-ge","kind":"node","lean":"MetaMathlibExt.restrictedCombinationTotal_one_five_genFun","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-total-one-five-ge","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $S^{\\{1,5\\}}_n$ be the number of subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by $1$ or by $5$. Then, in the ring $\\mathbb{Z}[[x]]$ of formal power series,","title":"Generating function for subsets avoiding differences 1 and 5"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-total-one-six-gen","kind":"node","lean":"MetaMathlibExt.restrictedCombinationTotal_one_six_genFun","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-total-one-six-gen","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $S^{\\{1,6\\}}_n$ be the number of subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by $1$ or by $6$. Then, in the ring $\\mathbb{Z}[[x]]$ of formal power series,","title":"Generating function for subsets avoiding differences 1 and 6"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-count-multiples-e","kind":"node","lean":"MetaMathlibExt.restrictedCombinationCount_multiples_eq_card_strongPerms","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-count-multiples-e","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"For natural numbers $m \\ge 1$, $j \\ge 1$, $n$ and $k$, the number of $k$-subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by an element of $\\{m, 2m, \\dots, jm\\}$ equals the number of permutations $\\pi$ of $\\{0, 1, \\dots, n + jm - 1\\}$ such that $\\pi(i) - i \\in \\{-m, 0, jm\\}$ for every $i$ and $\\pi$ has exactly $k$ excedances (points $i$ with $\\pi(i) > i$).","title":"Forbidden differences m, 2m, \u2026, jm and strongly restricted permutations"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/one-tbonacci-polynomial","kind":"node","lean":"MetaMathlibExt.oneTBonacciPolynomial","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\((1,t)\\)"],["text","-bonacci polynomials"]],"source":"The $(1,t)$-bonacci polynomials"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#one-tbonacci-polynomial","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"Fix a natural number $s$ and put $t = s + 1$. The $(1,t)$-bonacci polynomials $f^{(t)}_n(x)$, $n \\ge 0$, are the polynomials with natural-number coefficients defined by $f^{(t)}_0(x) = 1$ and, for every $n \\ge 0$, $f^{(t)}_{n+1}(x) = f^{(t)}_n(x) + x \\, f^{(t)}_{n-s}(x)$ if $s \\le n$ and $f^{(t)}_{n+1}(x) = f^{(t)}_n(x)$ if $n < s$.","title":"The $(1,t)$-bonacci polynomials"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/one-tbonacci-polynomial-zero","kind":"node","lean":"MetaMathlibExt.oneTBonacciPolynomial_zero","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","Initial value of the "],["math","\\((1,t)\\)"],["text","-bonacci polynomials"]],"source":"Initial value of the $(1,t)$-bonacci polynomials"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#one-tbonacci-polynomial-zero","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For every natural number $s$, the $(1,s+1)$-bonacci polynomial of index $0$ is $f^{(s+1)}_0(x) = 1$.","title":"Initial value of the $(1,t)$-bonacci polynomials"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/one-tbonacci-polynomial-succ-of-le","kind":"node","lean":"MetaMathlibExt.oneTBonacciPolynomial_succ_of_le","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","Recurrence for the "],["math","\\((1,t)\\)"],["text","-bonacci polynomials"]],"source":"Recurrence for the $(1,t)$-bonacci polynomials"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#one-tbonacci-polynomial-succ-of-le","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For all natural numbers $s$ and $n$ with $s \\le n$, the $(1,s+1)$-bonacci polynomials $f^{(s+1)}_m(x)$, $m \\ge 0$, satisfy $f^{(s+1)}_{n+1}(x) = f^{(s+1)}_n(x) + x \\, f^{(s+1)}_{n-s}(x)$.","title":"Recurrence for the $(1,t)$-bonacci polynomials"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/one-tbonacci-polynomial-succ-of-lt","kind":"node","lean":"MetaMathlibExt.oneTBonacciPolynomial_succ_of_lt","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\((1,t)\\)"],["text","-bonacci polynomials below the recurrence range"]],"source":"The $(1,t)$-bonacci polynomials below the recurrence range"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#one-tbonacci-polynomial-succ-of-lt","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For all natural numbers $s$ and $n$ with $n < s$, the $(1,s+1)$-bonacci polynomials $f^{(s+1)}_m(x)$, $m \\ge 0$, satisfy $f^{(s+1)}_{n+1}(x) = f^{(s+1)}_n(x)$.","title":"The $(1,t)$-bonacci polynomials below the recurrence range"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/card-strong-perms-multiples-eq-fibonacci","kind":"node","lean":"MetaMathlibExt.card_strongPerms_multiples_eq_fibonacciP","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting strongly restricted permutations with displacements "],["math","\\(-m\\)"],["text",", "],["math","\\(0\\)"],["text",", "],["math","\\(jm\\)"]],"source":"Counting strongly restricted permutations with displacements $-m$, $0$, $jm$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#card-strong-perms-multiples-eq-fibonacci","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $m, j, i, r$ be natural numbers with $j \\ge 1$ and $r < m$, and let $F_j$ be the Fibonacci $j$-numbers: $F_j(0) = 0$, $F_j(n) = 1$ for $1 \\le n \\le j$, and $F_j(n) = F_j(n-1) + F_j(n-j-1)$ for $n > j$. Then the number of permutations $\\pi$ of $\\{0, 1, \\dots, im + r - 1\\}$ such that $\\pi(x) - x \\in \\{-m, 0, jm\\}$ for every $x$ equals $F_j(i+1)^{m-r} \\, F_j(i+2)^r$.","title":"Counting strongly restricted permutations with displacements $-m$, $0$, $jm$"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/card-strong-perms-multiples-excedance-eq","kind":"node","lean":"MetaMathlibExt.card_strongPerms_multiples_excedance_eq_coeff","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","Excedances of strongly restricted permutations with displacements "],["math","\\(-m\\)"],["text",", "],["math","\\(0\\)"],["text",", "],["math","\\(jm\\)"]],"source":"Excedances of strongly restricted permutations with displacements $-m$, $0$, $jm$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#card-strong-perms-multiples-excedance-eq","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $m, j, i, r, k$ be natural numbers with $j \\ge 1$ and $r < m$, and let $f_n(x)$ be the $(1, j+1)$-bonacci polynomials with natural-number coefficients: $f_0(x) = 1$, and $f_{n+1}(x) = f_n(x) + x f_{n-j}(x)$ if $n \\ge j$, $f_{n+1}(x) = f_n(x)$ if $n < j$. Then the number of permutations $\\pi$ of $\\{0, 1, \\dots, im + r - 1\\}$ with $\\pi(a) - a \\in \\{-m, 0, jm\\}$ for every $a$ and with exactly $k$ excedances\u2026","title":"Excedances of strongly restricted permutations with displacements $-m$, $0$, $jm$"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-total-icc-erase-e","kind":"node","lean":"MetaMathlibExt.restrictedCombinationTotal_Icc_erase_eq_card_compositions","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-total-icc-erase-e","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $p$, $q$ and $n$ be natural numbers with $p < q < 2p$, let $Q = \\{1, \\dots, q\\} \\setminus \\{p\\}$, and let $S^Q_n$ be the number of subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by an element of $Q$. Then $S^Q_n$ is the number of compositions of $n + q$ in which every part is either $1$ or of the form $q + 1 + jp$ for some integer $j \\ge 0$.","title":"Combinations avoiding an interval of differences with one value removed"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/is-well-based","kind":"node","lean":"MetaMathlibExt.IsWellBased","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#is-well-based","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"A finite set $Q$ of natural numbers is well based (Kitaev; Valyuzhenich) if for every $x \\in Q$ and every natural number $a$ with $0 < a < x$, at least one of $a$ and $x - a$ lies in $Q$.","title":"Well-based sets"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/is-well-based-icc","kind":"node","lean":"MetaMathlibExt.isWellBased_Icc","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\{1, \\dots, q\\}\\)"],["text"," is well based"]],"source":"The set $\\{1, \\dots, q\\}$ is well based"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#is-well-based-icc","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For every natural number $q$, the set $\\{1, 2, \\dots, q\\}$ is well based.","title":"The set $\\{1, \\dots, q\\}$ is well based"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/not-is-well-based-one-three-four","kind":"node","lean":"MetaMathlibExt.not_isWellBased_one_three_four","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\{1, 3, 4\\}\\)"],["text"," is not well based"]],"source":"The set $\\{1, 3, 4\\}$ is not well based"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#not-is-well-based-one-three-four","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"The set $\\{1, 3, 4\\}$ is not well based: in $4 = 2 + 2$ neither part lies in the set.","title":"The set $\\{1, 3, 4\\}$ is not well based"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/subword-shift-set","kind":"node","lean":"MetaMathlibExt.subwordShiftSet","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#subword-shift-set","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For natural numbers $\\omega$ and $l$, the shift set of $\\omega$ with length $l$ is the set of $j \\in \\{1, \\dots, l - 1\\}$ with $\\lfloor \\omega / 2^j \\rfloor \\ne \\omega \\bmod 2^{l-j}$. When $\\omega < 2^l$ is read as a binary word of length $l$, this says that the leftmost $l - j$ bits of $\\omega$ differ from its rightmost $l - j$ bits.","title":"Shift set of a binary subword"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/subword-shift-set-1010","kind":"node","lean":"MetaMathlibExt.subwordShiftSet_1010","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#subword-shift-set-1010","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"The shift set of the binary subword $1010$ of length $4$ (the number $10$) is $\\{1, 3\\}$.","title":"Shift set of the subword 1010"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/is-well-based-subword-shift-set","kind":"node","lean":"MetaMathlibExt.isWellBased_subwordShiftSet","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#is-well-based-subword-shift-set","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $l$ be a natural number and $\\omega$ a binary word of length $l$, read as a natural number $\\omega < 2^l$. Then the set $Q$ of those $j \\in \\{1, \\dots, l-1\\}$ for which the leftmost $l - j$ bits of $\\omega$ differ from its rightmost $l - j$ bits, that is $\\lfloor \\omega / 2^j \\rfloor \\ne \\omega \\bmod 2^{l-j}$, is well based.","title":"Subword shift sets are well based"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-bivariate-gen-fun","kind":"node","lean":"MetaMathlibExt.restrictedCombination_bivariate_genFun_of_isWellBased","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-bivariate-gen-fun","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $Q$ be a finite set of positive integers that is well based, and write $S^Q_{n,k}$ for the number of $k$-subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by an element of $Q$. In the ring $\\mathbb{Z}[y][[x]]$ of formal power series in $x$ with coefficients in $\\mathbb{Z}[y]$, let $g(x, y) = \\sum_{n \\ge 0} \\sum_{k=0}^{n} S^Q_{n,k} \\, y^k x^n$ and $c = 1 + \\sum_{q \\in Q} x^q y$. Then\u2026","title":"Bivariate generating function for well-based forbidden differences"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-overlap-tilings","kind":"node","lean":"MetaMathlibExt.restrictedOverlapTilings","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-overlap-tilings","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"Let $Q$ be a finite set of natural numbers and let $q$ be its largest element ($q = 0$ if $Q$ is empty). The $Q$-comb is a piece of length $q + 1$ whose cell $0$ and cells $j \\in Q$ are teeth and whose other cells are gaps. A restricted-overlap tiling of an $n$-board (cells $0, \\dots, n - 1$) with squares and $k$ $Q$-combs is recorded by the set $P$ of the start cells of its combs, with squares filling the cells\u2026","title":"Restricted-overlap tilings"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-overlap-tiling-count","kind":"node","lean":"MetaMathlibExt.restrictedOverlapTilingCount","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-overlap-tiling-count","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For a finite set $Q$ of natural numbers and natural numbers $n$ and $k$, $B^Q_{n,k}$ is the number of restricted-overlap tilings of an $n$-board with squares and $k$ $Q$-combs.","title":"Number of restricted-overlap tilings"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-overlap-tiling-count-singleto","kind":"node","lean":"MetaMathlibExt.restrictedOverlapTilingCount_singleton_one_four_two","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-overlap-tiling-count-singleto","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For $Q = \\{1\\}$ the $Q$-comb is a domino, and the number $B^{\\{1\\}}_{4,2}$ of restricted-overlap tilings of a $4$-board with squares and two $Q$-combs is $1$.","title":"Two dominoes on a 4-board"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-overlap-tiling-count-singleto-2","kind":"node","lean":"MetaMathlibExt.restrictedOverlapTilingCount_singleton_two_four_two","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two interlocking "],["math","\\(\\{2\\}\\)"],["text","-combs on a 4-board"]],"source":"Two interlocking $\\{2\\}$-combs on a 4-board"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-overlap-tiling-count-singleto-2","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For $Q = \\{2\\}$, there is exactly one restricted-overlap tiling of a $4$-board with squares and two $Q$-combs.","title":"Two interlocking $\\{2\\}$-combs on a 4-board"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-count-eq-restrict","kind":"node","lean":"MetaMathlibExt.restrictedCombinationCount_eq_restrictedOverlapTilingCount","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-count-eq-restrict","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $Q$ be a finite set of natural numbers and let $q$ be its largest element ($q = 0$ if $Q$ is empty). Write $S^Q_{n,k}$ for the number of $k$-subsets of $\\{1, \\dots, n\\}$ in which no two distinct elements differ by an element of $Q$, $S^Q_n = \\sum_{k=0}^{n} S^Q_{n,k}$, and $B^Q_{m,k}$ for the number of restricted-overlap tilings of an $m$-board with squares and $k$ $Q$-combs. Then for all natural numbers $n$ and\u2026","title":"Restricted combinations and restricted-overlap tilings"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-gen-fun-eq-tiling","kind":"node","lean":"MetaMathlibExt.restrictedCombination_genFun_eq_tiling_genFun","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-gen-fun-eq-tiling","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $Q$ be a finite set of natural numbers and let $q$ be its largest element ($q = 0$ if $Q$ is empty). Write $S^Q_{n,k}$ for the number of $k$-subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by an element of $Q$, and $B^Q_{n,k}$ for the number of restricted-overlap tilings of an $n$-board with squares and $k$ $Q$-combs. In the ring $\\mathbb{Z}[y][[x]]$ of power series in $x$ with coefficients in\u2026","title":"Generating functions of restricted combinations and restricted-overlap tilings"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/fence-tiling-strong-perm-bijection","kind":"node","lean":"MetaMathlibExt.fenceTiling_strongPerm_bijection","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#fence-tiling-strong-perm-bijection","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $D$ be a finite set of integers and $n$ a natural number. Number the cells of an $n$-board $0, 1, \\dots, n-1$ and give cell $i$ a left slot $2i$ and a right slot $2i + 1$. A fence $(s, g)$, with $s$ and $g$ natural numbers, has teeth in slots $s$ and $s + 1 + 2g$. A tiling is a finite set of fences such that every fence $(s, g)$ in it satisfies $s + 1 + 2g < 2n$ and either $s$ is even and $g \\in D$, or $s$ is\u2026","title":"Fence tilings and strongly restricted permutations"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/metatiles-have-at-most-two-combs","kind":"node","lean":"MetaMathlibExt.MetatilesHaveAtMostTwoCombs","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#metatiles-have-at-most-two-combs","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"Let $Q$ be a finite set of natural numbers and $q$ its largest element ($q = 0$ for $Q = \\emptyset$). The set $Q$ has metatiles with at most two combs if for all natural numbers $N$ and $k$ and every restricted-overlap tiling of an $N$-board with squares and $k$ $Q$-combs, with set $P$ of comb start cells, any three combs $a < b < c$ in $P$ are separated by a boundary: there is a natural number $j$ with\u2026","title":"Metatiles with at most two combs"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/metatiles-have-at-most-two-combs-empty","kind":"node","lean":"MetaMathlibExt.metatilesHaveAtMostTwoCombs_empty","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","For "],["math","\\(Q = \\emptyset\\)"],["text",", metatiles have at most two combs"]],"source":"For $Q = \\emptyset$, metatiles have at most two combs"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#metatiles-have-at-most-two-combs-empty","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For $Q = \\emptyset$, every metatile of every restricted-overlap tiling with squares and $Q$-combs contains at most two combs.","title":"For $Q = \\emptyset$, metatiles have at most two combs"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/not-metatiles-have-at-most-two-combs-thr","kind":"node","lean":"MetaMathlibExt.not_metatilesHaveAtMostTwoCombs_three","parent":28604,"reading_title":{"format":"inline-parts-v1","parts":[["text","For "],["math","\\(Q = \\{3\\}\\)"],["text",", a metatile can contain three combs"]],"source":"For $Q = \\{3\\}$, a metatile can contain three combs"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#not-metatiles-have-at-most-two-combs-thr","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"fully_proved","summary":"For $Q = \\{3\\}$ it is not true that every metatile of every restricted-overlap tiling with squares and $Q$-combs contains at most two combs.","title":"For $Q = \\{3\\}$, a metatile can contain three combs"},{"id":"Combinatorics/elementary-enumeration/combinations-without-specified-separations/restricted-combination-total-eq-card-com","kind":"node","lean":"MetaMathlibExt.restrictedCombinationTotal_eq_card_compositions_iff_metatiles","parent":28604,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/combinations-without-specified-separations/index.html#restricted-combination-total-eq-card-com","scope":"Combinatorics/elementary-enumeration/combinations-without-specified-separations","status":"can_prove","summary":"Let $Q$ be a finite set of natural numbers whose largest element $q$ is positive, and let $S^Q_n$ be the number of subsets of $\\{1, \\dots, n\\}$ in which no two elements differ by an element of $Q$. Then the following are equivalent: (i) there is a finite set $A$ of natural numbers such that, for every $n \\ge 0$, $S^Q_n$ is the number of compositions of $n + q$ with all parts in $A$; (ii) every metatile of every\u2026","title":"Restricted combinations counted by compositions"},{"data_url":"scopes/Combinatorics/elementary-enumeration/fibre-and-double-counting.json","id":"Combinatorics/elementary-enumeration/fibre-and-double-counting","kind":"scope","navigation":"module","parent":28554,"result":false,"route":"../roadmap/Combinatorics/elementary-enumeration/fibre-and-double-counting/index.html","scope":"Combinatorics/elementary-enumeration/fibre-and-double-counting","status":"fully_proved","title":"Fibre and double counting"},{"id":"Combinatorics/elementary-enumeration/fibre-and-double-counting/exists-unique-of-sum-comp","kind":"node","lean":"Finset.existsUnique_of_sum_comp","parent":28640,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/fibre-and-double-counting/index.html#exists-unique-of-sum-comp","scope":"Combinatorics/elementary-enumeration/fibre-and-double-counting","status":"fully_proved","summary":"If every integer-valued sum over `T` pulls back to the corresponding sum over `s`, then a property with a unique witness in `T` pulls back to a property with a unique witness in `s`.","title":"Unique witnesses transported by finite-sum equality"},{"id":"Combinatorics/elementary-enumeration/fibre-and-double-counting/exists-perm-of-fiber","kind":"node","lean":"Fintype.exists_perm_of_fiber","parent":28640,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/fibre-and-double-counting/index.html#exists-perm-of-fiber","scope":"Combinatorics/elementary-enumeration/fibre-and-double-counting","status":"fully_proved","summary":"Two labelings of a finite type with equally sized fibers differ by a permutation.","title":"Labelings with equal fiber sizes differ by a permutation"},{"id":"Combinatorics/elementary-enumeration/fibre-and-double-counting/card-filter-le-of-fibers","kind":"node","lean":"Finset.card_filter_le_of_fibers","parent":28640,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/fibre-and-double-counting/index.html#card-filter-le-of-fibers","scope":"Combinatorics/elementary-enumeration/fibre-and-double-counting","status":"fully_proved","summary":"A density bound transfers through a map whose fibers over the target carrier all have the same cardinality.","title":"Density bounds transfer through equal-fiber maps"},{"data_url":"scopes/Combinatorics/elementary-enumeration/functions-with-prescribed-values.json","id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","kind":"scope","navigation":"module","parent":28554,"result":false,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","title":"Functions with prescribed values"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/pi-finset-on","kind":"node","lean":"Fintype.piFinsetOn","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#pi-finset-on","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"Finite tuples constrained on `P` and equal to a prescribed default elsewhere.","title":"Finite tuples with prescribed default values"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/mem-pi-finset-on","kind":"node","lean":"Fintype.mem_piFinsetOn","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#mem-pi-finset-on","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"Membership gives the allowed values on the active indices and the default off them.","title":"Membership in default-padded finite tuples"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/card-pi-finset-on","kind":"node","lean":"Fintype.card_piFinsetOn","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#card-pi-finset-on","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"Inactive coordinates contribute one choice, so only active coordinates enter the product.","title":"Number of default-padded finite tuples"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/sum-pi-finset-on-prod","kind":"node","lean":"Fintype.sum_piFinsetOn_prod","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#sum-pi-finset-on-prod","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"Distribute a product of finite sums over the default-padded tuples of choices.","title":"Product of sums over default-padded tuples"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/prod-sum-eq-sum-pi-finset-on-mul","kind":"node","lean":"Fintype.prod_sum_eq_sum_piFinsetOn_mul","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#prod-sum-eq-sum-pi-finset-on-mul","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"Products of factorized functions factor through the tuples of their labels.","title":"Products of factorized functions over label tuples"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/exists-unique-pi-finset-on","kind":"node","lean":"Fintype.existsUnique_piFinsetOn","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#exists-unique-pi-finset-on","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"Prescribed active coordinates have exactly one completion with the chosen default.","title":"Unique default completion of active coordinates"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/card-filter-pi-finset-eq-on-of-mem","kind":"node","lean":"Fintype.card_filter_piFinset_eqOn_of_mem","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#card-filter-pi-finset-eq-on-of-mem","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"The number of dependent functions in `piFinset s` with prescribed values on `S` is the product of the numbers of choices outside `S`.","title":"Number of dependent functions with prescribed values"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/card-filter-univ-eq-on","kind":"node","lean":"Fintype.card_filter_univ_eqOn","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#card-filter-univ-eq-on","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"The number of functions agreeing with `f` on `S` is the number of unrestricted choices on the remaining coordinates.","title":"Number of functions agreeing with a given one on a set"},{"id":"Combinatorics/elementary-enumeration/functions-with-prescribed-values/card-filter-univ-eq-on-mul-pow-card","kind":"node","lean":"Fintype.card_filter_univ_eqOn_mul_pow_card","parent":28644,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/functions-with-prescribed-values/index.html#card-filter-univ-eq-on-mul-pow-card","scope":"Combinatorics/elementary-enumeration/functions-with-prescribed-values","status":"fully_proved","summary":"Division-free count of finite functions with prescribed values on a set of coordinates.","title":"Division-free count of functions with prescribed values"},{"data_url":"scopes/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product.json","id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","kind":"scope","navigation":"module","parent":28554,"result":false,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","title":"Necklaces over a group with identity product"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/identity-product-necklace-count","kind":"node","lean":"MetaMathlibExt.identity_product_necklace_count","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#identity-product-necklace-count","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"fully_proved","summary":"Let $G$ be a finite group and $n \\ge 1$. For an $n$-tuple $x = (x_0, \\dots, x_{n-1}) \\in G^n$ let its necklace be the set of its $n$ cyclic rotations $(x_m, x_{m+1}, \\dots, x_{m+n-1})$, $0 \\le m < n$, with indices taken mod $n$. Then the number of necklaces of tuples $x$ with $x_0 x_1 \\cdots x_{n-1} = 1$ is","title":"Counting necklaces with identity product"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/tuple-rotation-orbit","kind":"node","lean":"MetaMathlibExt.tupleRotationOrbit","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#tuple-rotation-orbit","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"fully_proved","summary":"Let $x = (x_0, \\dots, x_{n-1})$ be a tuple of length $n$ with entries in a type $\\alpha$, and let $\\sigma$ be the cyclic rotation of $\\{0, \\dots, n-1\\}$ that sends $i$ to $i + 1$ for $i < n - 1$ and sends $n - 1$ to $0$. The rotation orbit of $x$ is the set of tuples $x \\circ \\sigma^m$ for $m \\in \\mathbb{N}$, that is, of the cyclic shifts $(x_m, x_{m+1}, \\dots, x_{m+n-1})$ of $x$, with indices taken modulo $n$.","title":"Rotation orbit of a tuple"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/self-mem-tuple-rotation-orbit","kind":"node","lean":"MetaMathlibExt.self_mem_tupleRotationOrbit","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#self-mem-tuple-rotation-orbit","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"fully_proved","summary":"Every tuple $x$ of length $n$ with entries in a type $\\alpha$ belongs to the rotation orbit of $x$.","title":"A tuple lies in its own rotation orbit"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/comp-fin-rotate-mem-tuple-rotation-orbit","kind":"node","lean":"MetaMathlibExt.comp_finRotate_mem_tupleRotationOrbit","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#comp-fin-rotate-mem-tuple-rotation-orbit","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"fully_proved","summary":"For every tuple $x = (x_0, \\dots, x_{n-1})$ with entries in a type $\\alpha$, the tuple whose $i$-th entry is $x_{i+1}$, indices taken modulo $n$ (that is, $(x_1, \\dots, x_{n-1}, x_0)$ when $n \\ge 1$), lies in the rotation orbit of $x$.","title":"One-step rotation stays in the rotation orbit"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/identity-product-necklace-count-pi-cycli","kind":"node","lean":"MetaMathlibExt.identityProductNecklace_count_pi_cyclic","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#identity-product-necklace-count-pi-cycli","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $k_1, \\dots, k_m$ be positive integers and let $G = C_{k_1} \\times \\dots \\times C_{k_m}$, where $C_k$ is the cyclic group $\\mathbb{Z} / k\\mathbb{Z}$ written multiplicatively. For $n \\ge 1$, the number of rotation orbits of $n$-tuples $(x_1, \\dots, x_n) \\in G^n$ with $x_1 x_2 \\cdots x_n = 1$ (identity-product $n$-necklaces) is","title":"Identity-product necklaces over a product of cyclic groups"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/sum-div-mul-eq-sum-totient-mul","kind":"node","lean":"MetaMathlibExt.sum_div_mul_eq_sum_totient_mul","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#sum-div-mul-eq-sum-totient-mul","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $f$ and $g$ be functions from the natural numbers to the rational numbers such that $g(k) = \\sum_{d \\mid k} f(d)$ for every positive integer $k$, where sums over $d \\mid k$ run over the positive divisors $d$ of $k$. Then for every positive integer $k$,","title":"Totient-weighted divisor sum identity"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/int-dvd-mul-iff-div-gcd-dvd","kind":"node","lean":"MetaMathlibExt.int_dvd_mul_iff_div_gcd_dvd","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#int-dvd-mul-iff-div-gcd-dvd","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $a, b, c$ be integers with $a$ and $b$ not both zero. Then $a$ divides $bc$ if and only if $a / \\gcd(a, b)$ divides $c$.","title":"Divisibility of a product through the gcd"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/card-pow-eq-one-zmod","kind":"node","lean":"MetaMathlibExt.card_pow_eq_one_zmod","parent":28654,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting solutions of "],["math","\\(x^b = 1\\)"],["text"," in a cyclic group"]],"source":"Counting solutions of $x^b = 1$ in a cyclic group"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#card-pow-eq-one-zmod","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"For natural numbers $a \\ge 1$ and $b$, the cyclic group of order $a$ has exactly $\\gcd(a, b)$ elements $x$ with $x^b = 1$.","title":"Counting solutions of $x^b = 1$ in a cyclic group"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/card-identity-product-period","kind":"node","lean":"MetaMathlibExt.card_identityProduct_period","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#card-identity-product-period","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $G$ be a finite group, $n \\ge 1$ and $k$ a divisor of $n$. The number of tuples $x = (x_0, \\dots, x_{n-1}) \\in G^n$ with $x_0 x_1 \\cdots x_{n-1} = 1$ and $x_{i+k} = x_i$ for all $i$ (indices taken mod $n$) is $r_{n/k}(G) \\, |G|^{k-1}$, where $r_m(G)$ is the number of $g \\in G$ with $g^m = 1$.","title":"Counting identity-product tuples with a given period"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/card-identity-product-period-eq-sum-smal","kind":"node","lean":"MetaMathlibExt.card_identityProduct_period_eq_sum_smallestPeriod","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#card-identity-product-period-eq-sum-smal","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $G$ be a finite group, $n$ a natural number and $k \\ge 1$. Say that $x = (x_0, \\dots, x_{n-1}) \\in G^n$ has period $l$ if $x_{i+l} = x_i$ for all $i$, with indices taken mod $n$. Let $A_k$ be the set of tuples $x$ with $x_0 x_1 \\cdots x_{n-1} = 1$ and period $k$, and for $d \\ge 1$ let $P_d$ be the set of tuples $x$ with $x_0 x_1 \\cdots x_{n-1} = 1$ that have period $d$ but no period $l$ with $0 < l < d$. Then\u2026","title":"Splitting periodic identity-product tuples by smallest period"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/aperiodic-identity-product-necklace-coun","kind":"node","lean":"MetaMathlibExt.aperiodic_identityProductNecklace_count","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#aperiodic-identity-product-necklace-coun","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $G$ be a finite group and $n \\ge 1$. For an $n$-tuple $x = (x_0, \\dots, x_{n-1}) \\in G^n$ let its rotation orbit be the set of its cyclic rotations $(x_m, x_{m+1}, \\dots, x_{m+n-1})$, $m \\ge 0$, with indices taken mod $n$. Then the number of rotation orbits with exactly $n$ elements of tuples $x$ with $x_0 x_1 \\cdots x_{n-1} = 1$ is","title":"Counting aperiodic necklaces with identity product"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/identity-product-necklace-count-smallest","kind":"node","lean":"MetaMathlibExt.identityProductNecklace_count_smallestPeriod","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#identity-product-necklace-count-smallest","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $G$ be a finite group, let $n \\ge 1$ and let $k$ be a divisor of $n$. The number of rotation orbits of size $k$ of $n$-tuples $(x_1, \\dots, x_n) \\in G^n$ with $x_1 x_2 \\cdots x_n = 1$ (identity-product $n$-necklaces of smallest period $k$) is","title":"Identity-product necklaces with a given smallest period"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/homogeneous-necklace-count","kind":"node","lean":"MetaMathlibExt.homogeneousNecklace_count","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#homogeneous-necklace-count","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $G$ be a finite group and let $n \\ge 1$. Let $C_n \\times G$ act on $G^n$, the cyclic group $C_n$ rotating tuples and $h \\in G$ multiplying every entry on the left, so that the orbit of $x = (x_0, \\dots, x_{n-1})$ is the set of tuples $(h x_m, h x_{m+1}, \\dots, h x_{m+n-1})$, indices taken modulo $n$, for $m \\in \\mathbb{N}$ and $h \\in G$. The number of these orbits is","title":"Counting tuples up to rotation and left multiplication"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/product-in-necklace-count","kind":"node","lean":"MetaMathlibExt.productInNecklace_count","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#product-in-necklace-count","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $G$ be a finite group, let $K \\subseteq G$ be closed under conjugation ($g k g^{-1} \\in K$ for all $g \\in G$ and $k \\in K$), and let $n \\ge 1$. The number of rotation orbits of $n$-tuples $(x_1, \\dots, x_n) \\in G^n$ with $x_1 x_2 \\cdots x_n \\in K$ ($K$-product $n$-necklaces) is","title":"Counting necklaces with product in a conjugation-closed set"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/card-product-in-period","kind":"node","lean":"MetaMathlibExt.card_productIn_period","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#card-product-in-period","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $G$ be a finite group, let $K \\subseteq G$ satisfy $g c g^{-1} \\in K$ for all $g \\in G$ and $c \\in K$, let $n \\ge 1$ and let $k$ be a divisor of $n$. The number of tuples $x = (x_0, \\dots, x_{n-1}) \\in G^n$ with $x_0 x_1 \\cdots x_{n-1} \\in K$ and $x_{i+k} = x_i$ for all $i$ (indices taken mod $n$) is $\\#\\{a \\in G : a^{n/k} \\in K\\} \\cdot |G|^{k-1}$.","title":"Counting periodic tuples with product in a conjugation-closed set"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/aperiodic-product-in-necklace-count","kind":"node","lean":"MetaMathlibExt.aperiodic_productInNecklace_count","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#aperiodic-product-in-necklace-count","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $G$ be a finite group, let $K \\subseteq G$ satisfy $g c g^{-1} \\in K$ for all $g \\in G$ and $c \\in K$, and let $n \\ge 1$. For an $n$-tuple $x = (x_0, \\dots, x_{n-1}) \\in G^n$ let its rotation orbit be the set of its cyclic rotations $(x_m, x_{m+1}, \\dots, x_{m+n-1})$, $m \\ge 0$, with indices taken mod $n$. Then the number of rotation orbits with exactly $n$ elements of tuples $x$ with\u2026","title":"Counting aperiodic necklaces with product in a conjugation-closed set"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/exists-normal-basis-finite-field","kind":"node","lean":"MetaMathlibExt.exists_normalBasis_finiteField","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#exists-normal-basis-finite-field","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $K$ be a finite field that is an algebra over a finite field $F$. Then there is an element $\\beta \\in K$ whose images $\\sigma(\\beta)$, as $\\sigma$ ranges over the $F$-algebra automorphisms of $K$, are linearly independent over $F$ (as a family indexed by $\\sigma$).","title":"Normal basis theorem for finite fields"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/normal-basis-map-bijective-frobenius","kind":"node","lean":"MetaMathlibExt.normalBasisMap_bijective_frobenius","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#normal-basis-map-bijective-frobenius","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $K$ be a finite field that is an algebra over a finite field $F$, let $q = |F|$ and $n = [K : F]$, and let $\\beta \\in K$ be such that $\\beta, \\beta^q, \\dots, \\beta^{q^{n-1}}$ are linearly independent over $F$. Define $\\Phi : F^n \\to K$ by $\\Phi(a_0, \\dots, a_{n-1}) = \\sum_{i=0}^{n-1} a_i \\beta^{q^i}$. Then $\\Phi$ is a bijection, and for every $a \\in F^n$,","title":"Normal basis coordinates turn Frobenius into rotation"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/normal-basis-map-rotation-iff-conj","kind":"node","lean":"MetaMathlibExt.normalBasisMap_rotation_iff_conj","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#normal-basis-map-rotation-iff-conj","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $K$ be a finite field that is an algebra over a finite field $F$, let $q = |F|$ and $n = [K : F]$, and let $\\beta \\in K$ be such that $\\beta, \\beta^q, \\dots, \\beta^{q^{n-1}}$ are linearly independent over $F$. Define $\\Phi : F^n \\to K$ by $\\Phi(a_0, \\dots, a_{n-1}) = \\sum_{i=0}^{n-1} a_i \\beta^{q^i}$. Then for all $a, b \\in F^n$, the tuple $b$ lies in the rotation orbit of $a$ if and only if $\\Phi(a)$ and\u2026","title":"Rotation orbits correspond to conjugate elements"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/normal-basis-map-sum-eq-zero-iff-trace","kind":"node","lean":"MetaMathlibExt.normalBasisMap_sum_eq_zero_iff_trace","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#normal-basis-map-sum-eq-zero-iff-trace","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $K$ be a finite field that is an algebra over a finite field $F$, let $q = |F|$ and $n = [K : F]$, and let $\\beta \\in K$ be such that $\\beta, \\beta^q, \\dots, \\beta^{q^{n-1}}$ are linearly independent over $F$. Define $\\Phi : F^n \\to K$ by $\\Phi(a_0, \\dots, a_{n-1}) = \\sum_{i=0}^{n-1} a_i \\beta^{q^i}$. Then for every $a = (a_0, \\dots, a_{n-1}) \\in F^n$, we have $a_0 + a_1 + \\dots + a_{n-1} = 0$ if and only if\u2026","title":"Zero-sum tuples correspond to trace-zero elements"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/card-trace-zero-conj-classes","kind":"node","lean":"MetaMathlibExt.card_traceZero_conjClasses","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#card-trace-zero-conj-classes","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $K$ be a finite field extension of a finite field $F$ and $n = [K : F]$. For $z \\in K$ let its Galois orbit be the set of all $\\sigma(z)$ with $\\sigma$ an automorphism of $K$ fixing $F$, and for $a = (a_0, \\dots, a_{n-1}) \\in F^n$ let its rotation orbit be the set of its cyclic rotations $(a_m, a_{m+1}, \\dots, a_{m+n-1})$, $m \\ge 0$, with indices taken mod $n$. The number of Galois orbits of elements $z \\in K$\u2026","title":"Trace-zero Galois orbits and zero-sum necklaces"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/card-trace-zero-conj-classes-size-eq","kind":"node","lean":"MetaMathlibExt.card_traceZero_conjClasses_size_eq","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#card-trace-zero-conj-classes-size-eq","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $K$ be a finite field extension of a finite field $F$ and $n = [K : F]$. For $z \\in K$ let its Galois orbit be the set of all $\\sigma(z)$ with $\\sigma$ an automorphism of $K$ fixing $F$, and for $a = (a_0, \\dots, a_{n-1}) \\in F^n$ let its rotation orbit be the set of its cyclic rotations $(a_m, a_{m+1}, \\dots, a_{m+n-1})$, $m \\ge 0$, with indices taken mod $n$. The number of Galois orbits with exactly $n$\u2026","title":"Trace-zero Galois orbits of full size and aperiodic zero-sum necklaces"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/card-trace-zero-size-eq","kind":"node","lean":"MetaMathlibExt.card_traceZero_size_eq","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#card-trace-zero-size-eq","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $K$ be a finite field extension of a finite field $F$ and $n = [K : F]$. The number of $z \\in K$ with $\\operatorname{Tr}_{K/F}(z) = 0$ whose Galois orbit, the set of all $\\sigma(z)$ with $\\sigma$ an automorphism of $K$ fixing $F$, has exactly $n$ elements equals the number of tuples $a = (a_0, \\dots, a_{n-1}) \\in F^n$ with $a_0 + a_1 + \\dots + a_{n-1} = 0$ for which there is no $l$ with $0 < l < n$ such that\u2026","title":"Trace-zero elements with Galois orbits of full size"},{"id":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/card-monic-irreducible-next-coeff-zero","kind":"node","lean":"MetaMathlibExt.card_monic_irreducible_nextCoeff_zero","parent":28654,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product/index.html#card-monic-irreducible-next-coeff-zero","scope":"Combinatorics/elementary-enumeration/necklaces-over-a-group-with-identity-product","status":"can_prove","summary":"Let $F$ be a finite field and $n \\ge 1$. For $a = (a_0, \\dots, a_{n-1}) \\in F^n$ let its rotation orbit be the set of its cyclic rotations $(a_m, a_{m+1}, \\dots, a_{m+n-1})$, $m \\ge 0$, with indices taken mod $n$. The number of monic irreducible polynomials of degree $n$ over $F$ whose coefficient of $X^{n-1}$ is $0$ equals the number of rotation orbits with exactly $n$ elements of tuples $a \\in F^n$ with\u2026","title":"Monic irreducible polynomials with vanishing second coefficient"},{"data_url":"scopes/Combinatorics/elementary-enumeration/schreier-sets.json","id":"Combinatorics/elementary-enumeration/schreier-sets","kind":"scope","navigation":"module","parent":28554,"result":false,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","title":"Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-count","kind":"node","lean":"MetaMathlibExt.schreierCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $p$, $q$ and $n$, $|S^{p/q}_n|$ is the number of $(p, q)$-Schreier sets with maximum $n$, that is, of subsets $F \\subseteq \\{1, \\dots, n\\}$ with $n \\in F$ and $p|F| \\le qx$ for every $x \\in F$ (equivalently, $p|F| \\le q \\min F$).","title":"Number of (p, q)-Schreier sets with given maximum"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/maximal-schreier-count","kind":"node","lean":"MetaMathlibExt.maximalSchreierCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of maximal "],["math","\\((p, q)\\)"],["text","-Schreier sets"]],"source":"Number of maximal $(p, q)$-Schreier sets"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#maximal-schreier-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $p$, $q$ and $n$, $|M^{p/q}_n|$ is the number of maximal $(p, q)$-Schreier sets with maximum $n$, that is, of sets $F \\subseteq \\{1, \\dots, n\\}$ with $n \\in F$ and $q \\min F = p |F|$.","title":"Number of maximal $(p, q)$-Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-padovan","kind":"node","lean":"MetaMathlibExt.schreierPadovan","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-padovan","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $p$ and $q$, the sequence $a_{p,q,n}$, $n \\ge 0$, is defined as follows. If $q \\ge 1$ and $n \\ge p + q$, then $a_{p,q,n} = a_{p,q,n-q} + a_{p,q,n-p-q}$. Otherwise, if $p \\ge q$, then $a_{p,q,n} = 0$ for $n < p - q$ and $a_{p,q,n} = 1$ for $n \\ge p - q$; and if $p < q$, then $a_{p,q,n} = 1$ for $n < 2p$ and $a_{p,q,n} = 2$ for $n \\ge 2p$.","title":"Padovan-like sequence for Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/maximal-schreier-padovan","kind":"node","lean":"MetaMathlibExt.maximalSchreierPadovan","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#maximal-schreier-padovan","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $p$ and $q$, the sequence $a^{(m)}_{p,q,n}$, $n \\ge 0$, is defined as follows. If $q \\ge 1$ and $n \\ge p + q$, then $a^{(m)}_{p,q,n} = a^{(m)}_{p,q,n-q} + a^{(m)}_{p,q,n-p-q}$. Otherwise, if $q \\le p$, then $a^{(m)}_{p,q,n}$ is $1$ for $n = p - q$ and $0$ for every other $n$, and if $p < q$, then $a^{(m)}_{p,q,n}$ is $1$ for $n = 2p$ and $0$ for every other $n$.","title":"Padovan-like sequence of maximal Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-padovan-rec","kind":"node","lean":"MetaMathlibExt.schreierPadovan_rec","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-padovan-rec","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $p$, $q$ and $n$ with $q \\ge 1$ and $n \\ge p + q$, the Padovan-like sequence satisfies $a_{p,q,n} = a_{p,q,n-q} + a_{p,q,n-p-q}$.","title":"Recurrence for the Padovan-like sequence"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/maximal-schreier-padovan-rec","kind":"node","lean":"MetaMathlibExt.maximalSchreierPadovan_rec","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#maximal-schreier-padovan-rec","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $p$, $q$ and $n$ with $q \\ge 1$ and $n \\ge p + q$, the Padovan-like sequence $a^{(m)}_{p,q,n}$ of maximal Schreier sets satisfies $a^{(m)}_{p,q,n} = a^{(m)}_{p,q,n-q} + a^{(m)}_{p,q,n-p-q}$.","title":"Recurrence for the Padovan-like sequence of maximal Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-padovan-small","kind":"node","lean":"MetaMathlibExt.schreierPadovan_small","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-padovan-small","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For $n = 0, 1, \\dots, 7$, the Padovan-like sequence $a_{1,2,n}$ takes the values $1, 1, 2, 2, 3, 4, 5, 7$, and $a_{2,1,n}$ takes the values $0, 1, 1, 1, 2, 3, 4, 6$.","title":"Small values of the Padovan-like sequences"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/maximal-schreier-padovan-small","kind":"node","lean":"MetaMathlibExt.maximalSchreierPadovan_small","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#maximal-schreier-padovan-small","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For $n = 0, 1, \\dots, 7$, the Padovan-like sequence $a^{(m)}_{1,2,n}$ of maximal Schreier sets takes the values $0, 0, 1, 0, 1, 1, 1, 2$, and $a^{(m)}_{2,1,n}$ takes the values $0, 1, 0, 0, 1, 1, 1, 2$.","title":"First values of the Padovan-like sequences of maximal Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-count-one-two-small","kind":"node","lean":"MetaMathlibExt.schreierCount_one_two_small","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-count-one-two-small","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For $n = 1, 2, 3, 4$, the number $|S^{1/2}_n|$ of $(1, 2)$-Schreier sets with maximum $n$ takes the values $1, 2, 3, 5$, and the number $|M^{1/2}_n|$ of maximal $(1, 2)$-Schreier sets with maximum $n$ takes the values $0, 1, 1, 1$.","title":"Small values of the (1, 2)-Schreier counts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/satisfies-poly","kind":"node","lean":"MetaMathlibExt.SatisfiesPoly","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#satisfies-poly","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"Let $P = c_0 + c_1 x + \\dots + c_r x^r$ be a real polynomial, where $r$ is the degree of $P$ (and $r = 0$ for $P = 0$). A real sequence $(a_n)_{n \\ge 0}$ satisfies $P$ if $c_0 a_n + c_1 a_{n-1} + \\dots + c_r a_{n-r} = 0$ for every $n \\ge r$.","title":"Sequences satisfying a polynomial"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/satisfies-poly-zero","kind":"node","lean":"MetaMathlibExt.satisfiesPoly_zero","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#satisfies-poly-zero","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"Every real sequence $(a_n)_{n \\ge 0}$ satisfies the zero polynomial.","title":"Every sequence satisfies the zero polynomial"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-count-eq-schreier-padovan","kind":"node","lean":"MetaMathlibExt.schreierCount_eq_schreierPadovan","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-count-eq-schreier-padovan","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Write $a_{p,q,k}$ and $a^{(m)}_{p,q,k}$ for the Padovan-like sequences for Schreier sets and for maximal Schreier sets. For positive integers $p$, $q$ and $n$, the number $|S^{p/q}_n|$ of $(p, q)$-Schreier sets with maximum $n$ equals $a_{p,q,(n-1)q}$, and the number $|M^{p/q}_n|$ of maximal $(p, q)$-Schreier sets with maximum $n$ equals $a^{(m)}_{p,q,(n-1)q}$.","title":"Schreier counts as Padovan-like sequences"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-count-formula-and-recurrence","kind":"node","lean":"MetaMathlibExt.schreierCount_formula_and_recurrence","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-count-formula-and-recurrence","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $p$ and $q$ be positive integers, and write $|S^{p/q}_n|$ for the number of $(p, q)$-Schreier sets with maximum $n$. For every natural number $n$,","title":"Closed formula and recurrence for Schreier counts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/maximal-schreier-count-formula-and-recur","kind":"node","lean":"MetaMathlibExt.maximalSchreierCount_formula_and_recurrence","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form and recurrence for maximal "],["math","\\((p, q)\\)"],["text","-Schreier sets"]],"source":"Closed form and recurrence for maximal $(p, q)$-Schreier sets"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#maximal-schreier-count-formula-and-recur","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $p$ and $q$ be positive integers, put $p' = p / \\gcd(p, q)$ and $q' = q / \\gcd(p, q)$, and for each $n$ let $|M^{p/q}_n|$ be the number of maximal $(p, q)$-Schreier sets with maximum $n$, that is, of sets $F \\subseteq \\{1, \\dots, n\\}$ with $n \\in F$ and $q \\min F = p |F|$. For each $n$, let $K_n$ be the set of integers $k$ with $1 \\le k \\le \\lfloor (n+1)/(p'+q') \\rfloor$ and $q'k \\ge 2$, and let $\\delta_n$ be\u2026","title":"Closed form and recurrence for maximal $(p, q)$-Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/of-dvd","kind":"node","lean":"MetaMathlibExt.SatisfiesPoly.of_dvd","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#of-dvd","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $P$ and $Q$ be real polynomials with $P \\mid Q$. If a real sequence $(a_n)_{n \\ge 0}$ satisfies $P$, then it satisfies $Q$.","title":"A sequence satisfying a polynomial satisfies its multiples"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-count-eq-maximal-schreier-count","kind":"node","lean":"MetaMathlibExt.schreierCount_eq_maximalSchreierCount_add","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Counting "],["math","\\((p, 1)\\)"],["text","-Schreier sets by maximal "],["math","\\((p, 1)\\)"],["text","-Schreier sets"]],"source":"Counting $(p, 1)$-Schreier sets by maximal $(p, 1)$-Schreier sets"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-count-eq-maximal-schreier-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Fix a positive integer $p$. For each $m$, let $|S^{p/1}_m|$ be the number of sets $F \\subseteq \\{1, \\dots, m\\}$ with $m \\in F$ and $\\min F \\ge p |F|$, and let $|M^{p/1}_m|$ be the number of those with $\\min F = p |F|$. Then $|S^{p/1}_n| = |M^{p/1}_{n+p+1}|$ for every natural number $n$.","title":"Counting $(p, 1)$-Schreier sets by maximal $(p, 1)$-Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/maximal-schreier-count-eq-two-mul-sub","kind":"node","lean":"MetaMathlibExt.maximalSchreierCount_eq_two_mul_sub","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maximal "],["math","\\((1, q)\\)"],["text","-Schreier sets counted by "],["math","\\((1, q)\\)"],["text","-Schreier sets"]],"source":"Maximal $(1, q)$-Schreier sets counted by $(1, q)$-Schreier sets"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#maximal-schreier-count-eq-two-mul-sub","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Fix a positive integer $q$. For each $m$, let $|S^{1/q}_m|$ be the number of sets $F \\subseteq \\{1, \\dots, m\\}$ with $m \\in F$ and $q \\min F \\ge |F|$, and let $|M^{1/q}_m|$ be the number of those with $q \\min F = |F|$. Then $|M^{1/q}_n| = 2|S^{1/q}_n| - |S^{1/q}_{n+1}|$ for every positive integer $n$.","title":"Maximal $(1, q)$-Schreier sets counted by $(1, q)$-Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/is-schreier","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.IsSchreier","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#is-schreier","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"A finite set $A$ of natural numbers is *Schreier* if $|A| \\le a$ for every $a \\in A$; equivalently, $A$ is empty or $\\min A \\ge |A|$.","title":"Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/is-strong-schreier","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.IsStrongSchreier","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#is-strong-schreier","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For a natural number $\\ell$, a finite set $A$ of natural numbers is *$\\ell$-strong Schreier* if $\\ell |A| + 1 \\le a + \\ell$ for every $a \\in A$; equivalently, $A$ is empty or $\\min A \\ge \\ell |A| - \\ell + 1$.","title":"Strong Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/is-sparse","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.IsSparse","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#is-sparse","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"A finite set of natural numbers is *sparse* if the differences between consecutive elements in increasing order are non-decreasing.","title":"Sparse sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/is-strongly-sparse","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.IsStronglySparse","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#is-strongly-sparse","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"A finite set of natural numbers is *strongly sparse* if the differences between consecutive elements in increasing order are strictly increasing.","title":"Strongly sparse sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-exact-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionExactCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partitions into exactly "],["math","\\(k\\)"],["text"," parts"]],"source":"Partitions into exactly $k$ parts"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-exact-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $u$ and $k$, $p(u, k)$ is the number of partitions of $u$ into exactly $k$ parts.","title":"Partitions into exactly $k$ parts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partition number "],["math","\\(p(u)\\)"]],"source":"The partition number $p(u)$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For a natural number $u$, $p(u)$ is the number of partitions of $u$, that is, of multisets of positive integers summing to $u$; in particular $p(0) = 1$.","title":"The partition number $p(u)$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-sparse-min-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.schreierSparseMinCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-sparse-min-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $m$, $|F_{n,m}|$ is the number of sets $F \\subseteq \\{1, \\dots, n\\}$ that are Schreier and sparse, contain $n$, and have minimum $m$.","title":"Number of Schreier sparse sets with given minimum and maximum"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-sparse-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.schreierSparseCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of Schreier sparse sets with maximum "],["math","\\(n\\)"]],"source":"Number of Schreier sparse sets with maximum $n$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-sparse-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For a natural number $n$, $|F_n|$ is the number of sets $F \\subseteq \\{1, \\dots, n\\}$ with $n \\in F$ that are Schreier and sparse.","title":"Number of Schreier sparse sets with maximum $n$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/sparse-strong-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.sparseStrongCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of sparse strong Schreier sets with maximum "],["math","\\(n\\)"]],"source":"Number of sparse strong Schreier sets with maximum $n$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#sparse-strong-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, $|A_{n,\\ell}|$ is the number of sets $A \\subseteq \\{1, \\dots, n\\}$ with $n \\in A$ that are sparse and $\\ell$-strong Schreier.","title":"Number of sparse strong Schreier sets with maximum $n$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/strongly-sparse-strong-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.stronglySparseStrongCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#strongly-sparse-strong-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, $|A^s_{n,\\ell}|$ is the number of sets $A \\subseteq \\{1, \\dots, n\\}$ with $n \\in A$ that are strongly sparse, meaning that the differences between consecutive elements of $A$ in increasing order are strictly increasing, and $\\ell$-strong Schreier, meaning that every element of $A$ is at least $\\ell|A| - \\ell + 1$.","title":"Number of strongly sparse strong Schreier sets with a given maximum"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/strong-schreier-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.strongSchreierCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#strong-schreier-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, $|B_{n,\\ell}|$ is the number of sets $B \\subseteq \\{1, \\dots, n\\}$ with $n \\in B$ that are $\\ell$-strong Schreier, that is, every element of $B$ is at least $\\ell|B| - \\ell + 1$.","title":"Number of strong Schreier sets with a given maximum"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/strong-schreier-card-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.strongSchreierCardCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of strong Schreier sets of given size with maximum "],["math","\\(n\\)"]],"source":"Number of strong Schreier sets of given size with maximum $n$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#strong-schreier-card-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$, $\\ell$ and $m$, $|B_{n,\\ell,m}|$ is the number of $\\ell$-strong Schreier sets $B \\subseteq \\{1, \\dots, n\\}$ with $n \\in B$ and exactly $m$ elements.","title":"Number of strong Schreier sets of given size with maximum $n$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-min-part-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionMinPartCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-min-part-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, $|E_{n,\\ell}|$ is the number of partitions of $n$ all of whose parts are at least $\\ell$.","title":"Partitions with all parts at least a given size"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-distinct-min-part-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionDistinctMinPartCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partitions into distinct parts at least "],["math","\\(\\ell\\)"]],"source":"Partitions into distinct parts at least $\\ell$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-distinct-min-part-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, let $E^d_{n,\\ell}$ be the set of partitions of $n$ into pairwise distinct parts, each at least $\\ell$. This is its number of elements $|E^d_{n,\\ell}|$.","title":"Partitions into distinct parts at least $\\ell$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-min-part-exact-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionMinPartExactCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-min-part-exact-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$, $\\ell$ and $k$, $|E_{n,\\ell,k}|$ is the number of partitions of $n$ into exactly $k$ parts, all of which are at least $\\ell$.","title":"Partitions with a given number of parts, all at least a given size"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-min-part-gap-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionMinPartGapCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-min-part-gap-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$, $\\ell$, $k$ and $q$, $|E_{n,\\ell,k,q}|$ is the number of partitions of $n$ into exactly $k$ parts, all at least $\\ell$, whose largest part exceeds the second largest part (counted with multiplicity) by exactly $q$; such a partition has at least two parts, so the count is $0$ for $k < 2$.","title":"Partitions with a given gap between the two largest parts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-no-small-part-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionNoSmallPartCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-no-small-part-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, $|G_{n,\\ell}|$ is the number of partitions of $n$ none of whose parts lies in $\\{2, \\dots, \\ell\\}$.","title":"Partitions avoiding parts from 2 up to a bound"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-no-small-no-even-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionNoSmallNoEvenCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-no-small-no-even-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, $|H_{n,\\ell}|$ is the number of partitions of $n$ none of whose parts lies in $\\{2, \\dots, \\ell\\}$ and none of whose parts is an even number greater than $2\\ell$.","title":"Partitions avoiding parts from 2 up to a bound and large even parts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/composition-min-part-exact-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.compositionMinPartExactCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositions with exactly "],["math","\\(s\\)"],["text"," parts, each at least "],["math","\\(v\\)"]],"source":"Compositions with exactly $s$ parts, each at least $v$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#composition-min-part-exact-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $u$, $v$ and $s$, $c(u, v, s)$ is the number of compositions of $u$ (ordered sequences of positive integers summing to $u$) into exactly $s$ parts, each at least $v$.","title":"Compositions with exactly $s$ parts, each at least $v$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/composition-min-part-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.compositionMinPartCount","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Compositions with parts at least "],["math","\\(v\\)"]],"source":"Compositions with parts at least $v$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#composition-min-part-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $u$ and $v$, $c(u, v)$ is the number of compositions of $u$ (ordered sequences of positive integers summing to $u$) all of whose parts are at least $v$.","title":"Compositions with parts at least $v$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/is-refined-schreier","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.IsRefinedSchreier","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The refined Schreier family "],["math","\\(F_{n,\\ell}\\)"]],"source":"The refined Schreier family $F_{n,\\ell}$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#is-refined-schreier","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, a finite set $A$ of natural numbers belongs to $F_{n,\\ell}$ if $n + 1$ is the largest element of $A$, $A$ has at least three elements, $A$ is sparse and $\\ell$-strong Schreier, and the two largest elements $a < b$ of $A$ below $n + 1$ satisfy $n + 1 + a = 2b$.","title":"The refined Schreier family $F_{n,\\ell}$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/refined-schreier-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.refinedSchreierCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#refined-schreier-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, $|F_{n,\\ell}|$ is the number of finite sets $A$ of natural numbers such that $n + 1 \\in A$, every element of $A$ is at most $n + 1$, $A$ has at least three elements, $A$ is sparse and $\\ell$-strong Schreier, and the two largest elements $a < b$ of $A$ below $n + 1$ satisfy $n + 1 + a = 2b$. Every such set is contained in $\\{0, \\dots, n + 1\\}$.","title":"Refined Schreier count"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/refined-schreier-card-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.refinedSchreierCardCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#refined-schreier-card-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, let $F_{n,\\ell}$ be the family of finite sets $A$ of natural numbers such that $A$ has largest element $n + 1$ and at least three elements, $A$ is sparse (the differences of consecutive elements, in increasing order, are non-decreasing) and $\\ell$-strong Schreier ($\\ell|A| + 1 \\le a + \\ell$ for every $a \\in A$), and the second and third largest elements $b > a$ of $A$ satisfy\u2026","title":"Refined Schreier sets with a given number of elements"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/refined-schreier-card-min-count","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.refinedSchreierCardMinCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#refined-schreier-card-min-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $n$ and $\\ell$, let $F_{n,\\ell}$ be the family of finite sets $A$ of natural numbers such that $A$ has largest element $n + 1$ and at least three elements, $A$ is sparse (the differences of consecutive elements, in increasing order, are non-decreasing) and $\\ell$-strong Schreier ($\\ell|A| + 1 \\le a + \\ell$ for every $a \\in A$), and the second and third largest elements $b > a$ of $A$ satisfy\u2026","title":"Refined Schreier sets with a given size and minimum"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-sparse-min-count-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.schreierSparseMinCount_eq","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-sparse-min-count-eq","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For natural numbers $m$ and $n$ with $m + 1 \\le n$, the number of Schreier sparse sets $F \\subseteq \\{1, \\dots, n\\}$ with $n \\in F$ and $\\min F = m + 1$ equals $p(n - 1, m)$, the number of partitions of $n - 1$ into exactly $m$ parts.","title":"Schreier sparse sets with given minimum and maximum"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-sparse-count-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.schreierSparseCount_eq","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-sparse-count-eq","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For every integer $n \\ge 1$, the number of Schreier sparse sets $F \\subseteq \\{1, \\dots, n\\}$ with $n \\in F$ equals $p(n - 1)$, the number of partitions of $n - 1$.","title":"Counting Schreier sparse sets by partition numbers"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/sparse-strong-count-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.sparseStrongCount_eq_sum","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#sparse-strong-count-eq-sum","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For every integer $n \\ge 1$ and every natural number $\\ell$, the number of sparse $\\ell$-strong Schreier sets $A \\subseteq \\{1, \\dots, n\\}$ with $n \\in A$ equals $\\sum_{i=0}^{n-1} |E_{i,\\ell+1}|$, where $|E_{i,\\ell+1}|$ is the number of partitions of $i$ into parts that are all at least $\\ell + 1$.","title":"Sparse strong Schreier sets as a sum of partition counts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/strongly-sparse-strong-count-eq-sum","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.stronglySparseStrongCount_eq_sum","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#strongly-sparse-strong-count-eq-sum","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $\\ell \\ge 0$, the number $|A^s_{n,\\ell}|$ of strongly sparse $\\ell$-strong Schreier sets $A \\subseteq \\{1, \\dots, n\\}$ with $n \\in A$ equals $\\sum_{i=0}^{n-1} |E^d_{i,\\ell+1}|$, where $|E^d_{i,\\ell+1}|$ is the number of partitions of $i$ into distinct parts that are all at least $\\ell + 1$ (so $|E^d_{0,\\ell+1}| = 1$). Here $A$ is strongly sparse when the differences between\u2026","title":"Strongly sparse strong Schreier sets and partitions into distinct parts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/sparse-strong-count-eq-no-small","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.sparseStrongCount_eq_noSmall","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#sparse-strong-count-eq-no-small","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For all integers $n \\ge 1$ and $\\ell \\ge 1$, the number of sparse $\\ell$-strong Schreier sets $A \\subseteq \\{1, \\dots, n\\}$ with $n \\in A$ equals the number of partitions of $n - 1$ with no part in $\\{2, \\dots, \\ell\\}$.","title":"Sparse strong Schreier sets and partitions avoiding small parts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/strongly-sparse-strong-count-eq-no-small","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.stronglySparseStrongCount_eq_noSmallNoEven","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#strongly-sparse-strong-count-eq-no-small","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $\\ell \\ge 1$, the number $|A^s_{n,\\ell}|$ of strongly sparse $\\ell$-strong Schreier sets $A \\subseteq \\{1, \\dots, n\\}$ with $n \\in A$ equals $|H_{n-1,\\ell}|$, the number of partitions of $n - 1$ with no part in $\\{2, \\dots, \\ell\\}$ and no even part greater than $2\\ell$. Here $A$ is strongly sparse when the differences between consecutive elements of $A$, listed in increasing\u2026","title":"Strongly sparse strong Schreier sets and partitions with restricted parts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/strong-schreier-card-count-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.strongSchreierCardCount_eq","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#strong-schreier-card-count-eq","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For all integers $n \\ge 1$ and $m \\ge 1$ and every natural number $\\ell$, the number of $\\ell$-strong Schreier sets $B \\subseteq \\{1, \\dots, n\\}$ with $n \\in B$ and exactly $m$ elements equals the number of compositions of $n + \\ell$ into exactly $m$ parts, each at least $\\ell + 1$.","title":"Strong Schreier sets of given size and compositions"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/strong-schreier-count-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.strongSchreierCount_eq","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#strong-schreier-count-eq","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $\\ell \\ge 0$, the number $|B_{n,\\ell}|$ of $\\ell$-strong Schreier sets $B \\subseteq \\{1, \\dots, n\\}$ with $n \\in B$ equals $c(n + \\ell, \\ell + 1)$, the number of compositions of $n + \\ell$ into parts that are all at least $\\ell + 1$. Here a set $B$ is $\\ell$-strong Schreier when every element of $B$ is at least $\\ell|B| - \\ell + 1$.","title":"Strong Schreier sets with a given maximum are counted by compositions"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/refined-schreier-card-min-count-eq","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.refinedSchreierCardMinCount_eq","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#refined-schreier-card-min-count-eq","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $n$, $\\ell$, $k$ and $q$ be natural numbers with $n \\ge \\ell + 1$, $k \\ge 2$ and $q \\ge 1$. Let $F_{n,\\ell}$ be the family of finite sets $A$ of natural numbers such that $A$ has largest element $n + 1$ and at least three elements, $A$ is sparse (the differences of consecutive elements, in increasing order, are non-decreasing) and $\\ell$-strong Schreier ($\\ell|A| + 1 \\le a + \\ell$ for every $a \\in A$), and the\u2026","title":"Refined Schreier sets with a given minimum and partitions with a given top gap"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-min-part-exact-count-eq-refine","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionMinPartExactCount_eq_refined","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-min-part-exact-count-eq-refine","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $n$, $\\ell$ and $k$ be natural numbers with $n \\ge \\ell + 1$ and $k \\ge 2$. Let $F_{n,\\ell}$ be the family of finite sets $A$ of natural numbers such that $A$ has largest element $n + 1$ and at least three elements, $A$ is sparse (the differences of consecutive elements, in increasing order, are non-decreasing) and $\\ell$-strong Schreier ($\\ell|A| + 1 \\le a + \\ell$ for every $a \\in A$), and the second and third\u2026","title":"Partitions with a fixed number of large parts and refined Schreier sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/partition-min-part-count-eq-refined-add","kind":"node","lean":"MathlibExt.Combinatorics.Enumerative.SchreierTypeSetsWanted.partitionMinPartCount_eq_refined_add_one","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#partition-min-part-count-eq-refined-add","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $n$ and $\\ell$ be natural numbers with $n \\ge \\ell + 1$. Let $F_{n,\\ell}$ be the family of finite sets $A$ of natural numbers such that $A$ has largest element $n + 1$ and at least three elements, $A$ is sparse (the differences of consecutive elements, in increasing order, are non-decreasing) and $\\ell$-strong Schreier ($\\ell|A| + 1 \\le a + \\ell$ for every $a \\in A$), and the second and third largest elements\u2026","title":"Partitions into large parts and the refined Schreier family"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-multiset-pool","kind":"node","lean":"MetaMathlibExt.schreierMultisetPool","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-multiset-pool","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $s$ and $n$, this is the multiset $\\{1^s, \\dots, (n - 1)^s, n\\}$, in which each of $1, \\dots, n - 1$ has multiplicity $s$ and $n$ appears once.","title":"Ground multiset for Schreier-type multisets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-multiset-count","kind":"node","lean":"MetaMathlibExt.schreierMultisetCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-multiset-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $s$, $p$ and $n$, $|A^{(s)}_{p,n}|$ is the number of distinct sub-multisets $F$ of $\\{1^s, \\dots, (n - 1)^s, n\\}$ (each of $1, \\dots, n - 1$ with multiplicity $s$, and $n$ once) such that $n \\in F$ and every element of $F$ is at least $p|F|$, where $|F|$ counts multiplicity.","title":"Number of Schreier-type multisets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-multiset-count-small","kind":"node","lean":"MetaMathlibExt.schreierMultisetCount_small","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-multiset-count-small","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"Let $|A^{(s)}_{p,n}|$ be the number of distinct sub-multisets $F$ of $\\{1^s, \\dots, (n - 1)^s, n\\}$ with $n \\in F$ and every element of $F$ at least $p|F|$, where $|F|$ counts multiplicity. Then $|A^{(2)}_{1,4}| = 4$ and $|A^{(2)}_{2,5}| = 2$.","title":"Small values of the Schreier multiset count"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/colored-schreier-multiset-count","kind":"node","lean":"MetaMathlibExt.coloredSchreierMultisetCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#colored-schreier-multiset-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $s$, $p$ and $n$, give each of $1, \\dots, n - 1$ the $s$ colors $0, \\dots, s - 1$, writing $k_c$ for $k$ with color $c$, and let $n$ appear once, uncolored. Then $|B^{(s)}_{p,n}|$ is the number of sets $F$ of colored integers drawn from $\\{1_0, \\dots, 1_{s-1}, \\dots, (n-1)_0, \\dots, (n-1)_{s-1}, n\\}$ that contain $n$ and satisfy $\\min F \\ge p|F|$, where the minimum is taken over the underlying\u2026","title":"Counting colored Schreier-type multisets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/colored-schreier-multiset-count-small","kind":"node","lean":"MetaMathlibExt.coloredSchreierMultisetCount_small","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#colored-schreier-multiset-count-small","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"The numbers of colored Schreier-type multisets satisfy $|B^{(2)}_{1,4}| = 6$ and $|B^{(2)}_{2,5}| = 3$ (Table 2 of the paper).","title":"Small values of the colored Schreier multiset count"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/first-non-multiples","kind":"node","lean":"MetaMathlibExt.firstNonMultiples","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#first-non-multiples","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $u$ and $n$, $G_{u,n}$ is the set of positive integers $k \\le un$ with $u \\nmid k$ such that at most $n$ of the integers in $[1, k]$ are not divisible by $u$. For $u \\ge 1$ the bound $k \\le un$ is automatic, and $G_{u,n}$ is the set of the first $n$ positive integers not divisible by $u$.","title":"First non-multiples of an integer"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/first-non-multiples-three-four","kind":"node","lean":"MetaMathlibExt.firstNonMultiples_three_four","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#first-non-multiples-three-four","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"The set $G_{3,4}$ of the first four positive integers not divisible by $3$ is $\\{1, 2, 4, 5\\}$.","title":"The first four positive integers not divisible by 3"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-avoiding-multiples-count","kind":"node","lean":"MetaMathlibExt.schreierAvoidingMultiplesCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-avoiding-multiples-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $u$ and $n$, let $G_{u,n}$ be the set of the first $n$ positive integers not divisible by $u$ (taken to be empty when $u = 0$), and call a finite set $F$ of natural numbers Schreier when every element of $F$ is at least $|F|$. Then $|D_{u,n}|$ is the number of Schreier sets $F \\subseteq G_{u,n}$ with $\\lfloor (un - 1)/(u - 1) \\rfloor \\in F$.","title":"Schreier sets avoiding multiples"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-avoiding-multiples-count-small","kind":"node","lean":"MetaMathlibExt.schreierAvoidingMultiplesCount_small","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-avoiding-multiples-count-small","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"The numbers of Schreier-type sets avoiding multiples satisfy $|D_{2,6}| = 12$ and $|D_{3,6}| = 9$ (Table 3 of the paper).","title":"Small values of the count of Schreier sets avoiding multiples"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-b","kind":"node","lean":"MetaMathlibExt.schreierParentB","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-b","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $s$ and $p$, the sequence $b^{(s)}_{p,n}$, $n \\ge 0$, is defined by $b^{(s)}_{p,n} = 0$ for $n \\le (p - 1)s$, $b^{(s)}_{p,n} = 1$ for $(p - 1)s < n \\le ps + 1$, and $b^{(s)}_{p,n} = b^{(s)}_{p,n-1} + b^{(s)}_{p,n-1-ps}$ for $n \\ge ps + 2$, where $p - 1$ is read as $0$ when $p = 0$.","title":"Parent sequence b for Schreier-type multisets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-b-of-le","kind":"node","lean":"MetaMathlibExt.schreierParentB_of_le","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-b-of-le","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $s$ and $p$, let $b^{(s)}_{p,n}$ be the parent sequence: $0$ for $n \\le (p - 1)s$, $1$ for $(p - 1)s < n \\le ps + 1$, and $b^{(s)}_{p,n} = b^{(s)}_{p,n-1} + b^{(s)}_{p,n-1-ps}$ for $n \\ge ps + 2$, where $p - 1$ is read as $0$ when $p = 0$. Then $b^{(s)}_{p,n} = 0$ for every natural number $n$ with $n \\le (p - 1)s$.","title":"The parent sequence b vanishes for small indices"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-b-of-mem","kind":"node","lean":"MetaMathlibExt.schreierParentB_of_mem","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-b-of-mem","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $s$ and $p$, let $b^{(s)}_{p,n}$ be the parent sequence: $0$ for $n \\le (p - 1)s$, $1$ for $(p - 1)s < n \\le ps + 1$, and $b^{(s)}_{p,n} = b^{(s)}_{p,n-1} + b^{(s)}_{p,n-1-ps}$ for $n \\ge ps + 2$, where $p - 1$ is read as $0$ when $p = 0$. Then $b^{(s)}_{p,n} = 1$ for every natural number $n$ with $(p - 1)s < n \\le ps + 1$.","title":"The parent sequence b equals one on its initial window"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-b-rec","kind":"node","lean":"MetaMathlibExt.schreierParentB_rec","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-b-rec","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $s$ and $p$, let $b^{(s)}_{p,n}$ be the parent sequence: $0$ for $n \\le (p - 1)s$, $1$ for $(p - 1)s < n \\le ps + 1$, and $b^{(s)}_{p,n} = b^{(s)}_{p,n-1} + b^{(s)}_{p,n-1-ps}$ for $n \\ge ps + 2$, where $p - 1$ is read as $0$ when $p = 0$. Then $b^{(s)}_{p,n} = b^{(s)}_{p,n-1} + b^{(s)}_{p,n-1-ps}$ for every natural number $n \\ge ps + 2$.","title":"Recurrence for the parent sequence b"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-d","kind":"node","lean":"MetaMathlibExt.schreierParentD","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parent sequence "],["math","\\(d_{u,n}\\)"]],"source":"The parent sequence $d_{u,n}$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-d","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $u$ and $n$, define $d_{u,n}$ by recursion on $n$: $d_{u,n} = 1$ if $n < 2u$ or $u = 0$, and otherwise $d_{u,n} = d_{u,n-u} + d_{u,n-2u+1}$.","title":"The parent sequence $d_{u,n}$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-d-of-lt","kind":"node","lean":"MetaMathlibExt.schreierParentD_of_lt","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-d-of-lt","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For all natural numbers $u$ and $n$ with $n < 2u$, the parent sequence $d_{u,n}$ satisfies $d_{u,n} = 1$.","title":"Initial values of the parent sequence"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-d-rec","kind":"node","lean":"MetaMathlibExt.schreierParentD_rec","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-d-rec","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For all natural numbers $u \\ge 1$ and $n \\ge 2u$, the parent sequence $d_{u,n}$ satisfies $d_{u,n} = d_{u,n-u} + d_{u,n-2u+1}$.","title":"Recurrence for the parent sequence"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-multiset-count-formula-and-recu","kind":"node","lean":"MetaMathlibExt.schreierMultisetCount_formula_and_recurrence","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-multiset-count-formula-and-recu","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $s, p \\ge 1$, and for $n \\ge 1$ let $|A^{(s)}_{p,n}|$ be the number of distinct sub-multisets $F$ of $\\{1^s, \\dots, (n - 1)^s, n\\}$ with $n \\in F$ and $\\min F \\ge p|F|$, where $|F|$ counts multiplicity. For $1 \\le n \\le sp + 1$,","title":"Formula and recurrence for Schreier-type multisets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/colored-schreier-multiset-count-formula","kind":"node","lean":"MetaMathlibExt.coloredSchreierMultisetCount_formula_and_recurrence","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#colored-schreier-multiset-count-formula","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $s \\ge 1$ and $p \\ge 1$, and write $|B^{(s)}_{p,n}|$ for the number of colored Schreier-type multisets. For $1 \\le n \\le sp + 1$,","title":"Closed form and recurrence for colored Schreier multiset counts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-b-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.schreierParentB_eq_sum_choose","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-b-eq-sum-choose","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $b^{(s)}_{p,n}$ be the parent sequence: $0$ for $n \\le (p - 1)s$, $1$ for $(p - 1)s < n \\le ps + 1$, and $b^{(s)}_{p,n} = b^{(s)}_{p,n-1} + b^{(s)}_{p,n-1-ps}$ for $n \\ge ps + 2$. For positive integers $s$, $p$ and $n$,","title":"Binomial-sum formula for the parent sequence b"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/colored-schreier-multiset-count-eq-schre","kind":"node","lean":"MetaMathlibExt.coloredSchreierMultisetCount_eq_schreierParentB","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#colored-schreier-multiset-count-eq-schre","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For all positive integers $s$, $p$ and $n$, the number $|B^{(s)}_{p,n}|$ of colored Schreier-type multisets equals the term $b^{(s)}_{p,ns-s+1}$ of the parent sequence $b^{(s)}_{p,m}$.","title":"Colored Schreier multiset counts are a subsequence of the parent sequence"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/image-floor-eq-non-multiples","kind":"node","lean":"MetaMathlibExt.image_floor_eq_nonMultiples","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#image-floor-eq-non-multiples","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For every integer $u \\ge 2$, the set $\\{\\lfloor (un - 1)/(u - 1) \\rfloor : n \\ge 1\\}$, with $n$ ranging over the positive integers, is the set of positive integers not divisible by $u$.","title":"The non-multiples of an integer as values of a floor function"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-parent-d-eq-sum-choose","kind":"node","lean":"MetaMathlibExt.schreierParentD_eq_sum_choose","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-parent-d-eq-sum-choose","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For all natural numbers $u \\ge 2$ and $n \\ge 1$, the parent sequence $d_{u,n}$ satisfies","title":"Binomial sum formula for the parent sequence"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-avoiding-multiples-count-eq-sch","kind":"node","lean":"MetaMathlibExt.schreierAvoidingMultiplesCount_eq_schreierParentD","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-avoiding-multiples-count-eq-sch","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For every integer $u \\ge 2$ and every integer $n \\ge 1$, the number $|D_{u,n}|$ of Schreier-type sets avoiding multiples of $u$ equals the term $d_{u,un-(u-1)}$ of the parent sequence $d_{u,m}$.","title":"Schreier sets avoiding multiples are counted by a subsequence of the parent sequence"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/schreier-avoiding-multiples-count-fib-an","kind":"node","lean":"MetaMathlibExt.schreierAvoidingMultiplesCount_fib_and_recurrence","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#schreier-avoiding-multiples-count-fib-an","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $u \\ge 2$, write $|D_{u,n}|$ for the number of Schreier-type sets avoiding multiples of $u$, and write $F_n$ for the Fibonacci numbers, with $F_0 = 0$ and $F_1 = 1$. For $1 \\le n \\le 2u - 1$, $|D_{u,n}| = F_n$, and for $n \\ge 2u$, as an identity of integers,","title":"Fibonacci values and a recurrence for Schreier sets avoiding multiples"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"For natural numbers $k$ and $n$, let $\\omega_k(E)$ denote the number of elements of a finite set $E$ that are different from $k$. The weighted Schreier count $a_{k,n}$ is the number of subsets $E \\subseteq \\{1, \\dots, n\\}$ that are empty or satisfy $\\omega_k(E) < \\min E$.","title":"Weighted Schreier-type sets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-one-one","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_one_one","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Schreier count "],["math","\\(a_{1,1}\\)"]],"source":"Weighted Schreier count $a_{1,1}$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-one-one","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"The weighted Schreier count $a_{1,1}$ equals $2$.","title":"Weighted Schreier count $a_{1,1}$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-two-one","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_two_one","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Schreier count "],["math","\\(a_{2,1}\\)"]],"source":"Weighted Schreier count $a_{2,1}$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-two-one","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"The weighted Schreier count $a_{2,1}$ equals $1$.","title":"Weighted Schreier count $a_{2,1}$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-three-three","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_three_three","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Schreier count "],["math","\\(a_{3,3}\\)"]],"source":"Weighted Schreier count $a_{3,3}$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-three-three","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"The weighted Schreier count $a_{3,3}$ equals $4$.","title":"Weighted Schreier count $a_{3,3}$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-one-four","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_one_four","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Schreier count "],["math","\\(a_{1,4}\\)"]],"source":"Weighted Schreier count $a_{1,4}$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-one-four","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"The weighted Schreier count $a_{1,4}$ equals $6$.","title":"Weighted Schreier count $a_{1,4}$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/seeded-partial-sums","kind":"node","lean":"MetaMathlibExt.seededPartialSums","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#seeded-partial-sums","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, and let $s = (s_n)_{n \\ge 0}$ (the seed sequence) and $a = (a_n)_{n \\ge 0}$ be sequences in $M$. The seeded partial sums $t^s_{k,n}$, for $k, n \\ge 0$, are defined by recursion on $k$: $t^s_{0,n} = a_n$ and $t^s_{k+1,n} = s_k + \\sum_{i\\lt n} t^s_{k,i}$.","title":"Seeded partial sums"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/iterated-partial-sum","kind":"node","lean":"MetaMathlibExt.iteratedPartialSum","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#iterated-partial-sum","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid and let $a = (a_n)_{n \\ge 0}$ be a sequence in $M$. For $k, m \\ge 0$, the $k$-fold iterated partial sum $P^{(k)}(a)(m)$ is the seeded partial sum $t^s_{k,m}$ of $a$ for the zero seed sequence $s = 0$. Thus $P^{(0)}(a)(m) = a_m$ and $P^{(k+1)}(a)(m) = \\sum_{i\\lt m} P^{(k)}(a)(i)$.","title":"Iterated partial sums"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/seeded-partial-sums-succ-zero","kind":"node","lean":"MetaMathlibExt.seededPartialSums_succ_zero","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#seeded-partial-sums-succ-zero","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, let $s = (s_n)_{n \\ge 0}$ and $a = (a_n)_{n \\ge 0}$ be sequences in $M$, and write $t^s_{k,n}$ for the seeded partial sums of $a$ with seed sequence $s$. Then $t^s_{k+1,0} = s_k$ for every $k \\ge 0$.","title":"Seeded partial sums start at the seed"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/iterated-partial-sum-one","kind":"node","lean":"MetaMathlibExt.iteratedPartialSum_one","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#iterated-partial-sum-one","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid and let $(a_n)_{n \\ge 0}$ be a sequence in $M$. For every $m \\ge 0$, the $1$-fold iterated partial sum of $(a_n)$ at $m$ is the ordinary partial sum: $P^{(1)}((a_n)_{n \\ge 0})(m) = \\sum_{i\\lt m} a_i$.","title":"The first iterated partial sum"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/iterated-partial-sum-succ-succ","kind":"node","lean":"MetaMathlibExt.iteratedPartialSum_succ_succ","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#iterated-partial-sum-succ-succ","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"fully_proved","summary":"Let $M$ be an additive commutative monoid, let $a = (a_n)_{n \\ge 0}$ be a sequence in $M$, and write $P^{(k)}(a)(m)$ for its $k$-fold iterated partial sum at $m$. Then for all $k, m \\ge 0$, $P^{(k+1)}(a)(m+1) = P^{(k+1)}(a)(m) + P^{(k)}(a)(m)$.","title":"Recurrence for iterated partial sums"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-diag","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_diag","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-diag","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For every natural number $n \\ge 1$, the weighted Schreier count satisfies $a_{n,n} = 2F_n$, where $F_n$ is the $n$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$).","title":"Diagonal weighted Schreier counts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-one","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_one","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Schreier counts for "],["math","\\(k = 1\\)"]],"source":"Weighted Schreier counts for $k = 1$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-one","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For every natural number $\\ell$, the weighted Schreier count satisfies $a_{1,1+\\ell} = F_{\\ell+2} + 1$, where $F_j$ is the $j$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$).","title":"Weighted Schreier counts for $k = 1$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-add","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_add","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-add","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $a_{k,n}$ be the weighted Schreier count and let $F_n$ denote the Fibonacci numbers, with $F_0 = 0$ and $F_1 = 1$. For all $k \\ge 2$ and $\\ell \\ge 0$, $a_{k,k+\\ell} = 2\\sum_{i=0}^{k-2} \\binom{\\ell}{i} F_{k-i} + 2\\binom{\\ell}{k-1} + \\sum_{j=1}^{\\ell} \\binom{j}{\\ell-j+k}$.","title":"Weighted Schreier counts on and above the diagonal"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-of-pos-of-lt","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_of_pos_of_lt","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Schreier counts for "],["math","\\(0 < m < k\\)"]],"source":"Weighted Schreier counts for $0 < m < k$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-of-pos-of-lt","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For natural numbers $k \\ge 2$ and $m$ with $0 < m < k$, the weighted Schreier count satisfies $a_{k,m} = F_{m+1}$, where $F_j$ is the $j$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$).","title":"Weighted Schreier counts for $0 < m < k$"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-add-of-le","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_add_of_le","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-add-of-le","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $a_{k,n}$ be the weighted Schreier count and let $F_n$ denote the Fibonacci numbers, with $F_0 = 0$ and $F_1 = 1$. For all $k, \\ell \\ge 0$ with $k \\ge \\ell + 2$, we have $a_{k,k+\\ell} = 2F_{k+\\ell}$.","title":"Weighted Schreier counts equal twice a Fibonacci number for small offsets"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-recurrence","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_recurrence","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-recurrence","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For natural numbers $k$ and $n$ with $k \\ge 1$, $n > k$ and $n > 2$, the weighted Schreier counts satisfy $a_{k,n} = a_{k,n-1} + a_{k-1,n-2}$. At $k = 1$ the term $a_{0,n-2}$ is the same count with $\\omega_0(E)$, the number of elements of $E$ different from $0$, equal to the number $|E|$ of all elements of $E$, since no element of a set $E \\subseteq \\{1, \\dots, n-2\\}$ equals $0$.","title":"Recurrence for weighted Schreier counts"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/seeded-partial-sums-sub-zero","kind":"node","lean":"MetaMathlibExt.seededPartialSums_sub_zero","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#seeded-partial-sums-sub-zero","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $R$ be a commutative ring, let $a = (a_n)_{n \\ge 0}$ and $b = (b_n)_{n \\ge 0}$ be sequences in $R$, and write $t^b_{k,n}$ for the seeded partial sums of $a$ with seed sequence $b$ and $t^0_{k,n}$ for those with the zero seed sequence. Then for all $k, n \\ge 0$, $t^b_{k,n} - t^0_{k,n} = \\sum_{i\\lt k} \\binom{n}{i} b_{k-1-i}$, where the binomial coefficients are read as elements of $R$.","title":"Effect of the seed on seeded partial sums"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/iterated-partial-sum-add-one-sub","kind":"node","lean":"MetaMathlibExt.iteratedPartialSum_add_one_sub","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#iterated-partial-sum-add-one-sub","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $R$ be a commutative ring, let $(a_n)_{n \\ge 0}$ be a sequence in $R$, and let $k, m \\ge 0$. Writing $P^{(k)}(x)(m)$ for the $k$-fold iterated partial sum of a sequence $x$ at $m$, we have $P^{(k)}((a_n + 1)_{n \\ge 0})(m) - P^{(k)}((a_n)_{n \\ge 0})(m) = \\binom{m}{k}$, where $\\binom{m}{k}$ is read as an element of $R$.","title":"Adding one to a sequence shifts its iterated partial sums by a binomial coefficient"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/iterated-partial-sum-fib-add-two","kind":"node","lean":"MetaMathlibExt.iteratedPartialSum_fib_add_two","parent":28679,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#iterated-partial-sum-fib-add-two","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"Let $F_n$ denote the Fibonacci numbers, with $F_0 = 0$ and $F_1 = 1$, and write $P^{(k)}(x)(m)$ for the $k$-fold iterated partial sum of a sequence $x$ of natural numbers at $m$. Then for all $k, \\ell \\ge 0$, $P^{(k)}((F_{n+2})_{n \\ge 0})(\\ell) = P^{(k)}((F_n)_{n \\ge 0})(\\ell) + P^{(k)}((F_n)_{n \\ge 0})(\\ell + 1)$.","title":"Iterated partial sums of shifted Fibonacci numbers"},{"id":"Combinatorics/elementary-enumeration/schreier-sets/weighted-schreier-count-of-lt","kind":"node","lean":"MetaMathlibExt.weightedSchreierCount_of_lt","parent":28679,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted Schreier counts for "],["math","\\(n < k\\)"]],"source":"Weighted Schreier counts for $n < k$"},"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/schreier-sets/index.html#weighted-schreier-count-of-lt","scope":"Combinatorics/elementary-enumeration/schreier-sets","status":"can_prove","summary":"For all natural numbers $n < k$, the weighted Schreier count satisfies $a_{k,n} = F_{n+1}$, where $F_j$ is the $j$-th Fibonacci number ($F_0 = 0$, $F_1 = 1$).","title":"Weighted Schreier counts for $n < k$"},{"data_url":"scopes/Combinatorics/elementary-enumeration/special-counting-sequences.json","id":"Combinatorics/elementary-enumeration/special-counting-sequences","kind":"scope","navigation":"module","parent":28554,"result":false,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","title":"Special counting sequences"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/lattice-path","kind":"node","lean":"MetaMathlibExt.LatticePath","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#lattice-path","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$, let $D \\subseteq \\mathbb{Z}^d$ and let $p, q \\in \\mathbb{Z}^d$. A lattice path from $p$ to $q$ with step set $D$ is a finite sequence $(v_0, v_1, \\dots, v_m)$ of points of $\\mathbb{Z}^d$, for some $m \\in \\mathbb{N}$, such that $v_0 = p$, $v_m = q$, and $v_i - v_{i-1} \\in D$ for $1 \\le i \\le m$.","title":"Lattice paths with prescribed steps"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/necklace","kind":"node","lean":"MetaMathlibExt.Necklace","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#necklace","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"Let $A$ be a set and let $n \\in \\mathbb{N}$. For $n \\ge 1$, a word of length $n$ over $A$ is a tuple $(a_0, a_1, \\dots, a_{n-1})$ of elements of $A$, and two words $(a_0, \\dots, a_{n-1})$ and $(b_0, \\dots, b_{n-1})$ are rotationally equivalent if there is an integer $r$ with $0 \\le r < n$ such that $b_i = a_{(i+r) \\bmod n}$ for $0 \\le i < n$. A necklace of length $n$ over $A$ is an equivalence class of words of\u2026","title":"Fixed-length necklaces"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/lucas","kind":"node","lean":"MetaMathlibExt.Nat.lucas","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#lucas","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"The Lucas numbers $(L_n)_{n \\ge 0}$ are the natural numbers defined by $L_0 = 2$, $L_1 = 1$, and $L_{n+2} = L_{n+1} + L_n$ for every $n \\in \\mathbb{N}$.","title":"Lucas number sequence"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/motzkin","kind":"node","lean":"MetaMathlibExt.Nat.motzkin","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#motzkin","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"The Motzkin numbers $(M_n)_{n \\ge 0}$ are the natural numbers defined by $M_0 = 1$ and, for every $n \\in \\mathbb{N}$, $M_{n+1} = M_n + \\sum_{k=0}^{n-1} M_k M_{n-1-k}$, where the sum is empty when $n = 0$. Thus $M_0, M_1, M_2, M_3 = 1, 1, 2, 4$.","title":"Motzkin number sequence"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/delannoy","kind":"node","lean":"MetaMathlibExt.Nat.delannoy","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#delannoy","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"The Delannoy numbers $D(m,n)$, for $m, n \\in \\mathbb{N}$, are defined by $D(0,n) = 1$ and $D(m,0) = 1$ for all $m, n \\in \\mathbb{N}$, and $D(m+1,n+1) = D(m,n+1) + D(m+1,n) + D(m,n)$ for all $m, n \\in \\mathbb{N}$.","title":"Delannoy numbers"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/southeast-path","kind":"node","lean":"MetaMathlibExt.SoutheastPath","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#southeast-path","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"A point of the quarter plane is a pair $(a, b) \\in \\mathbb{N}^2$, with $a$ the horizontal coordinate and $b$ the height. Let $p$ and $q$ be points of the quarter plane. A southeast path from $p$ to $q$ is a list $v = [v_0, v_1, \\dots, v_m]$ of points of the quarter plane, for some $m \\in \\mathbb{N}$, with $v_0 = p$, $v_m = q$, and, for each $1 \\le t \\le m$, either $v_t = v_{t-1} + (1, 0)$ (an east step) or\u2026","title":"Southeast paths"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/weight","kind":"node","lean":"MetaMathlibExt.SoutheastPath.weight","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#weight","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"Let $R$ be a commutative monoid, written multiplicatively, let $x = (x_j)_{j \\in \\mathbb{N}}$ be a family of elements of $R$, let $p, q \\in \\mathbb{N}^2$, and let $v = [v_0, \\dots, v_m]$ be a southeast path from $p$ to $q$. For $1 \\le t \\le m$, the step from $v_{t-1}$ to $v_t$ has weight $w_t = x_j$ if it is an east step at height $j$, that is, $v_{t-1} = (a, j)$ and $v_t = (a+1, j)$, and weight $w_t = 1$ if it is\u2026","title":"Weight of a southeast path"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/is-nonintersecting-path-family","kind":"node","lean":"MetaMathlibExt.IsNonintersectingPathFamily","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#is-nonintersecting-path-family","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and let $p_1, \\dots, p_n$ and $q_1, \\dots, q_n$ be points of $\\mathbb{N}^2$. A family $L = (L_1, \\dots, L_n)$, in which $L_i$ is a southeast path from $p_i$ to $q_i$ for each $i$, is nonintersecting when for all $i \\ne j$ no point occurs both in $L_i$ and in $L_j$.","title":"Nonintersecting families of southeast paths"},{"id":"Combinatorics/elementary-enumeration/special-counting-sequences/nonintersecting-path-weight-sum","kind":"node","lean":"MetaMathlibExt.nonintersectingPathWeightSum","parent":28776,"result":true,"route":"../roadmap/Combinatorics/elementary-enumeration/special-counting-sequences/index.html#nonintersecting-path-weight-sum","scope":"Combinatorics/elementary-enumeration/special-counting-sequences","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $x = (x_j)_{j \\in \\mathbb{N}}$ be a family of elements of $R$, let $n \\in \\mathbb{N}$, and let $p = (p_1, \\dots, p_n)$ and $q = (q_1, \\dots, q_n)$ be tuples of points of $\\mathbb{N}^2$. The nonintersecting path weight sum of type $(p, q)$ is $W(p, q) = \\sum_L \\prod_{i=1}^{n} \\operatorname{wt}(L_i)$, the sum taken over all nonintersecting families $L = (L_1, \\dots, L_n)$ in\u2026","title":"Weighted sum over nonintersecting path families"},{"area":"Discrete Mathematics & Computer Science","data_url":"chapters/GraphTheory.json","id":"GraphTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/GraphTheory/index.html","scope":"GraphTheory","status":"can_prove","summary":"Graph theory studies vertices and edges through connectivity, coloring, minors, flows, and extremal structure.","title":"Graph Theory"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors.json","id":"GraphTheory/planar-graphs-and-minors","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/index.html","scope":"GraphTheory/planar-graphs-and-minors","status":"can_prove","summary":"Graphs drawn in the plane and on the square lattice, crossing numbers, Euler's formula, discrete Jordan curve and turning-number theorems, and the theory of graph minors.","title":"Planar graphs and minors"},{"id":"GraphTheory/planar-graphs-and-minors/legendre-regular-crossing","kind":"node","lean":"MetaMathlibExt.legendre_regular_crossing","parent":28787,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/index.html#legendre-regular-crossing","scope":"GraphTheory/planar-graphs-and-minors","status":"fully_proved","summary":"Let $n$ be a natural number. In the regular drawing of $K_{n,n}$ in $\\mathbb{Q}^2$, the left vertices are the points $(0, i)$ and the right vertices are the points $(1, j)$ for $0 \\le i, j < n$, and each edge is the straight segment from $(0, i)$ to $(1, j)$. For $i < i'$ and $j' < j$, the edges from $(0, i)$ to $(1, j)$ and from $(0, i')$ to $(1, j')$ cross at the point $(x, i + x(j - i))$, where\u2026","title":"Crossing points of the regular drawing of the complete bipartite graph"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes.json","id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","title":"Discrete Gauss\u2013Bonnet formula for polyominoes"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/cell","kind":"node","lean":"StatMech.CellEulerChar.Cell","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#cell","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"A unit cell `[a, a + 1] \u00d7 [b, b + 1]` of the plane, labelled by its lower left corner.","title":"Unit cell of the plane"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/vtx","kind":"node","lean":"StatMech.CellEulerChar.Vtx","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#vtx","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"A lattice point of `\u2124\u00b2`, used as a vertex of the cell complex.","title":"Lattice point as a vertex of the cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/cell-verts","kind":"node","lean":"StatMech.CellEulerChar.cellVerts","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#cell-verts","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The four corners of a unit cell.","title":"Corners of a unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/cell-edges","kind":"node","lean":"StatMech.CellEulerChar.cellEdges","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#cell-edges","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The four sides of a unit cell, as unordered pairs of corners.","title":"Sides of a unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/reg-verts","kind":"node","lean":"StatMech.CellEulerChar.regVerts","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#reg-verts","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The vertices of the union of the cells in `K`.","title":"Vertices of a union of unit cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/reg-edges","kind":"node","lean":"StatMech.CellEulerChar.regEdges","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#reg-edges","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The edges of the union of the cells in `K`.","title":"Edges of a union of unit cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/reg-chi","kind":"node","lean":"StatMech.CellEulerChar.regChi","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#reg-chi","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The Euler characteristic `V - E + F` of the union of the cells in `K`, with one face per cell.","title":"Euler characteristic of a union of unit cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/new-verts","kind":"node","lean":"StatMech.CellEulerChar.newVerts","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#new-verts","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of corners of `c` that are not already vertices of the region `K`.","title":"Number of new corners added by a cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/new-edges","kind":"node","lean":"StatMech.CellEulerChar.newEdges","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#new-edges","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of sides of `c` that are not already edges of the region `K`.","title":"Number of new sides added by a cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/euc-buildable","kind":"node","lean":"StatMech.CellEulerChar.EucBuildable","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#euc-buildable","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Regions built from a single cell by repeatedly attaching a new cell along a boundary arc: the new cell adds one more edge than vertex, so the Euler characteristic is unchanged.","title":"Cell regions built by attaching cells along boundary arcs"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/surround-cells","kind":"node","lean":"StatMech.AngleDefect.surroundCells","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#surround-cells","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The four cells having `v` as a corner.","title":"Four cells around a lattice vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/cell-deg","kind":"node","lean":"StatMech.AngleDefect.cellDeg","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#cell-deg","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of cells of `K` having `v` as a corner.","title":"Cell degree of a lattice vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/edge-deg","kind":"node","lean":"StatMech.AngleDefect.edgeDeg","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#edge-deg","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of edges of the region `K` incident to `v`.","title":"Edge degree of a lattice vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/defect","kind":"node","lean":"StatMech.AngleDefect.defect","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#defect","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The angle defect at `v`, in quarter turns: `4 - 2 \u00b7 edgeDeg K v + cellDeg K v`.","title":"Angle defect at a lattice vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/incident-edge-r","kind":"node","lean":"StatMech.AngleDefect.incidentEdgeR","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#incident-edge-r","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The lattice edge from `v` to its right neighbour.","title":"Lattice edge to the right neighbour"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/incident-edge-l","kind":"node","lean":"StatMech.AngleDefect.incidentEdgeL","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#incident-edge-l","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The lattice edge from `v` to its left neighbour.","title":"Lattice edge to the left neighbour"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/incident-edge-u","kind":"node","lean":"StatMech.AngleDefect.incidentEdgeU","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#incident-edge-u","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The lattice edge from `v` to its upper neighbour.","title":"Lattice edge to the upper neighbour"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/incident-edge-d","kind":"node","lean":"StatMech.AngleDefect.incidentEdgeD","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#incident-edge-d","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The lattice edge from `v` to its lower neighbour.","title":"Lattice edge to the lower neighbour"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/wad-n1","kind":"node","lean":"StatMech.AngleDefect.wad_n1","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#wad-n1","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of vertices of the region with defect `1` (convex corners).","title":"Number of convex corners of a lattice region"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/wad-n3","kind":"node","lean":"StatMech.AngleDefect.wad_n3","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#wad-n3","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of vertices of the region with defect `-1` (reflex corners).","title":"Number of reflex corners of a lattice region"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/wad-n2d","kind":"node","lean":"StatMech.AngleDefect.wad_n2d","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#wad-n2d","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of vertices of the region with defect `-2` (pinch points, where two cells meet only at a corner).","title":"Number of pinch points of a lattice region"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corners-of","kind":"node","lean":"StatMech.Onsager.CellCount.cornersOf","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corners-of","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The four corners of the unit cell with lower-left corner `c`.","title":"Four corners of a unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count","kind":"node","lean":"StatMech.Onsager.CellCount.cornerCount","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of cells of `S` having the lattice point `v` as a corner.","title":"Corner count of a lattice point"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/cell-verts-2","kind":"node","lean":"StatMech.Onsager.CellCount.cellVerts","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#cell-verts-2","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The vertices of the cell complex `S`: all corners of its cells.","title":"Vertices of a cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corners-of-card","kind":"node","lean":"StatMech.Onsager.CellCount.cornersOf_card","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corners-of-card","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"A cell has four distinct corners.","title":"A unit cell has four distinct corners"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/mem-corners-of-iff","kind":"node","lean":"StatMech.Onsager.CellCount.mem_cornersOf_iff","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#mem-corners-of-iff","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"`v` is a corner of `c` iff `c` is one of the four cells around `v`.","title":"Characterization of the corners of a unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count-le-four","kind":"node","lean":"StatMech.Onsager.CellCount.cornerCount_le_four","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count-le-four","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"At most four cells meet at a lattice point.","title":"At most four cells meet at a lattice point"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/sum-corner-count","kind":"node","lean":"StatMech.Onsager.CellCount.sum_cornerCount","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#sum-corner-count","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Double counting of cell\u2013corner incidences: `\u2211 v, cornerCount S v = 4 * S.card`.","title":"Double counting of cell\u2013corner incidences"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-diff","kind":"node","lean":"StatMech.Onsager.CellEuler.cornerDiff","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-diff","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The corner defect of `S`: the number of vertices with exactly one incident cell of `S` (convex corners) minus the number with exactly three (reflex corners).","title":"Corner defect of a set of unit cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count-insert","kind":"node","lean":"StatMech.Onsager.CellEuler.cornerCount_insert","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count-insert","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Adding a new cell `c` raises the corner count by one exactly at the corners of `c`.","title":"Corner count after adding a cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/cell-verts-insert","kind":"node","lean":"StatMech.Onsager.CellEuler.cellVerts_insert","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#cell-verts-insert","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The vertices of `insert c S`.","title":"Vertices of a cell complex after adding a cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-weight","kind":"node","lean":"StatMech.Onsager.CellEuler.cornerWeight","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-weight","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The weight of a vertex with `k` incident cells in the corner defect: `1` if `k = 1`, `-1` if `k = 3`, and `0` otherwise.","title":"Corner weight of a vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count-pos-iff-mem","kind":"node","lean":"StatMech.Onsager.CellEuler.cornerCount_pos_iff_mem","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count-pos-iff-mem","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"A lattice point has an incident cell iff it is a vertex of `S`.","title":"A lattice point has an incident cell iff it is a vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-diff-eq-sum","kind":"node","lean":"StatMech.Onsager.CellEuler.cornerDiff_eq_sum","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-diff-eq-sum","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The corner defect as a sum of vertex weights.","title":"Corner defect as a sum of vertex weights"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-diff-insert","kind":"node","lean":"StatMech.Onsager.CellEuler.cornerDiff_insert","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-diff-insert","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Change of the corner defect when a new cell is added.","title":"Change of the corner defect when a cell is added"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-diff-singleton","kind":"node","lean":"StatMech.Onsager.CellEuler.cornerDiff_singleton","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-diff-singleton","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"A single cell has corner defect `4`.","title":"A single cell has corner defect 4"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count-eq-inter","kind":"node","lean":"StatMech.Onsager.CellEuler.cornerCount_eq_inter","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count-eq-inter","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The corner count at `v` counts the cells of `S` among the four cells around `v`.","title":"Corner count as a count of the four surrounding cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/edges-of","kind":"node","lean":"StatMech.Onsager.CellEuler.edgesOf","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#edges-of","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The four edges of cell `c`, each encoded by twice its midpoint.","title":"Four edges of a unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/cell-edges-2","kind":"node","lean":"StatMech.Onsager.CellEuler.cellEdges","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#cell-edges-2","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The edges of the cell complex `S`, encoded by doubled midpoints.","title":"Edges of a cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/euler-char","kind":"node","lean":"StatMech.Onsager.CellEuler.eulerChar","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#euler-char","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The Euler characteristic `V - E + F` of the closed cell complex formed by the cells of `S`.","title":"Euler characteristic of a cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/euler-char-singleton","kind":"node","lean":"StatMech.Onsager.CellEuler.eulerChar_singleton","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#euler-char-singleton","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"A single cell has Euler characteristic `1`.","title":"A single cell has Euler characteristic 1"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/euler-char-insert","kind":"node","lean":"StatMech.Onsager.CellEuler.eulerChar_insert","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#euler-char-insert","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Change of the Euler characteristic when a new cell is added: new vertices minus new edges plus one.","title":"Change of the Euler characteristic when a cell is added"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/card-inter-eq-sum","kind":"node","lean":"StatMech.Onsager.CellConnect.card_inter_eq_sum","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#card-inter-eq-sum","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The size of `S \u2229 T` as the sum over `T` of the indicator of `S`. See also `Finset.sum_boole`.","title":"Cardinality of an intersection as a sum of indicators"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count-bl","kind":"node","lean":"StatMech.Onsager.CellConnect.cornerCount_BL","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count-bl","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The corner count at the bottom-left corner of a cell `(x, y) \u2209 S`.","title":"Corner count at the bottom-left corner of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count-br","kind":"node","lean":"StatMech.Onsager.CellConnect.cornerCount_BR","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count-br","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The corner count at the bottom-right corner of a cell `(x, y) \u2209 S`.","title":"Corner count at the bottom-right corner of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count-tl","kind":"node","lean":"StatMech.Onsager.CellConnect.cornerCount_TL","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count-tl","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The corner count at the top-left corner of a cell `(x, y) \u2209 S`.","title":"Corner count at the top-left corner of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-count-tr","kind":"node","lean":"StatMech.Onsager.CellConnect.cornerCount_TR","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-count-tr","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The corner count at the top-right corner of a cell `(x, y) \u2209 S`.","title":"Corner count at the top-right corner of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/edge-code","kind":"node","lean":"StatMech.Onsager.CellEulerBound.edgeCode","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#edge-code","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The lattice edge `{a, b}` encoded by the sum `a + b`, twice its midpoint; injective on unit edges.","title":"Midpoint encoding of a lattice edge"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/image-cell-edges-edge-code","kind":"node","lean":"StatMech.Onsager.CellEulerBound.image_cellEdges_edgeCode","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#image-cell-edges-edge-code","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The codes of the edges of a cell are `StatMech.Onsager.CellEuler.edgesOf`.","title":"Edge codes of a unit cell are its doubled edge midpoints"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/card-cell-edges-eq-reg-edges","kind":"node","lean":"StatMech.Onsager.CellEulerBound.card_cellEdges_eq_regEdges","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#card-cell-edges-eq-reg-edges","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The two edge counts of a cell complex agree.","title":"Agreement of the two edge counts of a cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/cell-verts-eq-reg-verts","kind":"node","lean":"StatMech.Onsager.CellEulerBound.cellVerts_eq_regVerts","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#cell-verts-eq-reg-verts","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The two vertex sets of a cell complex agree.","title":"Agreement of the two vertex sets of a cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/euler-char-eq-reg-chi","kind":"node","lean":"StatMech.Onsager.CellEulerBound.eulerChar_eq_regChi","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#euler-char-eq-reg-chi","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The Euler characteristic `eulerChar` agrees with `StatMech.CellEulerChar.regChi`.","title":"Agreement of the two Euler characteristics of a cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/top-edge","kind":"node","lean":"StatMech.Onsager.CellEulerBound.topEdge","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#top-edge","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The top edge of the cell `c`.","title":"Top edge of a unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/top-edges","kind":"node","lean":"StatMech.Onsager.CellEulerBound.topEdges","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#top-edges","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The top edges of the cells of `S`.","title":"Top edges of a set of cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/card-top-edges","kind":"node","lean":"StatMech.Onsager.CellEulerBound.card_topEdges","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#card-top-edges","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Distinct cells have distinct top edges.","title":"Distinct cells have distinct top edges"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/open-top-skeleton","kind":"node","lean":"StatMech.Onsager.CellEulerBound.openTopSkeleton","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#open-top-skeleton","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The 1-skeleton of the cell complex `S` with the top edge of every cell removed, as a simple graph on its vertices.","title":"Open-top skeleton of a cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/open-top-edge-equiv","kind":"node","lean":"StatMech.Onsager.CellEulerBound.openTopEdgeEquiv","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#open-top-edge-equiv","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The edges of `openTopSkeleton S` are the edges of `S` other than top edges.","title":"Edges of the open-top skeleton"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/reg-verts-card-le","kind":"node","lean":"StatMech.Onsager.CellEulerBound.regVerts_card_le","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#reg-verts-card-le","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"If the open-top skeleton is connected, it has at least as many edges as vertices minus one.","title":"Edge count bound for a connected open-top skeleton"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/reg-chi-le-one","kind":"node","lean":"StatMech.Onsager.CellEulerBound.regChi_le_one","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#reg-chi-le-one","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"If the open-top skeleton of `S` is connected, then `regChi S \u2264 1`.","title":"Regular Euler characteristic bound via the open-top skeleton"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/euler-char-le-one","kind":"node","lean":"StatMech.Onsager.CellEulerBound.eulerChar_le_one","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#euler-char-le-one","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"If the open-top skeleton of `S` is connected, then `eulerChar S \u2264 1`. See also `StatMech.Onsager.CellEulerFaithful.eulerChar_le_one_of_skeleton_connected`, which assumes connectivity of the full skeleton.","title":"Euler characteristic bound via the open-top skeleton"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-diff-cex","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.cornerDiff_cex","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-diff-cex","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Two diagonally touching cells have corner defect `6`.","title":"Corner defect of two diagonally touching cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/euler-char-cex","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.eulerChar_cex","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#euler-char-cex","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Two diagonally touching cells have Euler characteristic `1`.","title":"Euler characteristic of two diagonally touching cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/gauss-bonnet-stated-false","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.gaussBonnet_stated_false","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#gauss-bonnet-stated-false","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The formula `cornerDiff S = 4 * eulerChar S` fails in general (two diagonally touching cells); see `StatMech.Onsager.CellGaussBonnetGlobal.cornerDiff_eq` for the correct formula with a pinch term.","title":"Failure of the cell Gauss\u2013Bonnet formula without a pinch term"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/b-i","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.bI","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#b-i","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The indicator `1` or `0` of a Boolean.","title":"Indicator of a Boolean"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/icw","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.icw","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#icw","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The vertex weight `cornerWeight` extended to integers.","title":"Integer-valued corner weight"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/ijump","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.ijump","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#ijump","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The change of the vertex weight from `m` to `m + 1` incident cells.","title":"Integer jump of the corner weight"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/is-zero","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.isZero","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#is-zero","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The indicator of `m = 0`.","title":"Indicator of zero"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/lhs-local","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.lhsLocal","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#lhs-local","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The change of the corner defect when a cell is added, given which of its eight neighbours are present (`e, w, n, s, ne, nw, se, sw`).","title":"Local change of the corner defect"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/delta-v","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.deltaV","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#delta-v","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of new vertices when a cell is added, given its eight neighbours.","title":"Number of new vertices when a cell is added"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/delta-e","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.deltaE","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#delta-e","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of new edges when a cell is added, given its four side neighbours.","title":"Number of new edges when a cell is added"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/rhs-stated","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.rhsStated","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#rhs-stated","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Four times the change of the Euler characteristic when a cell is added.","title":"Four times the local change of the Euler characteristic"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/local-identity-stated-false","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.localIdentity_stated_false","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#local-identity-stated-false","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The change of the corner defect is not always four times the change of the Euler characteristic.","title":"Failure of the local identity without a pinch term"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-delta","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.pinchDelta","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-delta","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The change of the number of pinch points at the four corners of a new cell, given its eight neighbours.","title":"Local change of the number of pinch points"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/rhs-corrected","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.rhsCorrected","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#rhs-corrected","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Four times the change of the Euler characteristic plus twice the change of the number of pinch points.","title":"Corrected right side of the local Gauss\u2013Bonnet identity"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/local-identity-corrected","kind":"node","lean":"StatMech.Onsager.CellGaussBonnet.localIdentity_corrected","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#local-identity-corrected","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Local cell Gauss\u2013Bonnet identity: when a cell is added, the change of the corner defect is four times the change of the Euler characteristic plus twice the change of the number of pinch points.","title":"Local cell Gauss\u2013Bonnet identity"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-weight-eq-icw","kind":"node","lean":"StatMech.Onsager.CellConnect2.cornerWeight_eq_icw","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-weight-eq-icw","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"`cornerWeight` agrees with its integer version `icw`.","title":"Corner weight agrees with its integer version"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-weight-jump","kind":"node","lean":"StatMech.Onsager.CellConnect2.cornerWeight_jump","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-weight-jump","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The change of a vertex weight when one more cell becomes incident.","title":"Change of a vertex weight when one more cell is incident"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/b-i-decide","kind":"node","lean":"StatMech.Onsager.CellConnect2.bI_decide","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#b-i-decide","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The indicator `bI` of a decided proposition.","title":"Boolean indicator of a decided proposition"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-diff-jump-sum","kind":"node","lean":"StatMech.Onsager.CellConnect2.cornerDiff_jump_sum","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-diff-jump-sum","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The change of the corner defect when the cell `(x, y)` is added, in terms of its eight neighbours.","title":"Change of the corner defect in terms of eight neighbours"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/not-mem-cell-verts-iff-corner-count-zero","kind":"node","lean":"StatMech.Onsager.CellConnect2.notMem_cellVerts_iff_cornerCount_zero","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#not-mem-cell-verts-iff-corner-count-zero","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"A lattice point is not a vertex of `S` iff no cell of `S` is incident to it.","title":"Non-vertices are the points with no incident cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/card-sdiff-eq-sum","kind":"node","lean":"StatMech.Onsager.CellConnect2.card_sdiff_eq_sum","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#card-sdiff-eq-sum","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The size of `s \\ t` as a sum of indicators.","title":"Cardinality of a set difference as a sum of indicators"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/sdiff-cell-verts-card-eq-delta-v","kind":"node","lean":"StatMech.Onsager.CellConnect2.sdiff_cellVerts_card_eq_deltaV","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#sdiff-cell-verts-card-eq-delta-v","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of new vertices when the cell `(x, y)` is added.","title":"New vertices of an added cell match the local count"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/mem-cell-edges-bottom","kind":"node","lean":"StatMech.Onsager.CellConnect2.mem_cellEdges_bottom","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#mem-cell-edges-bottom","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The horizontal edge with doubled midpoint `(2x + 1, 2y)` belongs to `S` iff one of the two cells it bounds does.","title":"Membership of a cell's bottom edge in the cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/mem-cell-edges-top","kind":"node","lean":"StatMech.Onsager.CellConnect2.mem_cellEdges_top","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#mem-cell-edges-top","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The horizontal edge with doubled midpoint `(2x + 1, 2y + 2)` belongs to `S` iff one of the two cells it bounds does.","title":"Membership of a cell's top edge in the cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/mem-cell-edges-left","kind":"node","lean":"StatMech.Onsager.CellConnect2.mem_cellEdges_left","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#mem-cell-edges-left","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The vertical edge with doubled midpoint `(2x, 2y + 1)` belongs to `S` iff one of the two cells it bounds does.","title":"Membership of a cell's left edge in the cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/mem-cell-edges-right","kind":"node","lean":"StatMech.Onsager.CellConnect2.mem_cellEdges_right","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#mem-cell-edges-right","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The vertical edge with doubled midpoint `(2x + 2, 2y + 1)` belongs to `S` iff one of the two cells it bounds does.","title":"Membership of a cell's right edge in the cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/sdiff-cell-edges-card-eq-delta-e","kind":"node","lean":"StatMech.Onsager.CellConnect2.sdiff_cellEdges_card_eq_deltaE","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#sdiff-cell-edges-card-eq-delta-e","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of new edges when the cell `(x, y)` is added.","title":"New edges of an added cell match the local count"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-at","kind":"node","lean":"StatMech.Onsager.CellPinch.pinchAt","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-at","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The lattice point `v` is a pinch point of `S`: of the four cells around `v`, exactly two diagonally opposite ones belong to `S`.","title":"Pinch point of a set of unit cells"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-w","kind":"node","lean":"StatMech.Onsager.CellPinch.pinchW","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-w","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The indicator of `v` being a pinch point of `S`.","title":"Pinch point indicator"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-count","kind":"node","lean":"StatMech.Onsager.CellPinch.pinchCount","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-count","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The number of pinch points of `S`.","title":"Number of pinch points"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-at-mem-cell-verts","kind":"node","lean":"StatMech.Onsager.CellPinch.pinchAt_mem_cellVerts","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-at-mem-cell-verts","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"A pinch point is a vertex of `S`.","title":"A pinch point is a vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-count-insert","kind":"node","lean":"StatMech.Onsager.CellPinch.pinchCount_insert","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-count-insert","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Adding a cell changes the pinch count only at the corners of that cell.","title":"Pinch count changes only at the corners of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-diff-bl","kind":"node","lean":"StatMech.Onsager.CellGaussBonnetGlobal.pinch_diff_BL","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-diff-bl","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Change of the pinch indicator at the bottom-left corner of a new cell.","title":"Change of the pinch indicator at the bottom-left corner of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-diff-br","kind":"node","lean":"StatMech.Onsager.CellGaussBonnetGlobal.pinch_diff_BR","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-diff-br","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Change of the pinch indicator at the bottom-right corner of a new cell.","title":"Change of the pinch indicator at the bottom-right corner of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-diff-tl","kind":"node","lean":"StatMech.Onsager.CellGaussBonnetGlobal.pinch_diff_TL","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-diff-tl","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Change of the pinch indicator at the top-left corner of a new cell.","title":"Change of the pinch indicator at the top-left corner of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-diff-tr","kind":"node","lean":"StatMech.Onsager.CellGaussBonnetGlobal.pinch_diff_TR","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-diff-tr","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Change of the pinch indicator at the top-right corner of a new cell.","title":"Change of the pinch indicator at the top-right corner of a new cell"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/pinch-jump-sum","kind":"node","lean":"StatMech.Onsager.CellGaussBonnetGlobal.pinch_jump_sum","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#pinch-jump-sum","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"The change of the pinch count when a cell is added, in terms of its eight neighbours.","title":"Change of the pinch count in terms of eight neighbours"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/corner-diff-eq","kind":"node","lean":"StatMech.Onsager.CellGaussBonnetGlobal.cornerDiff_eq","parent":28789,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes/index.html#corner-diff-eq","scope":"GraphTheory/planar-graphs-and-minors/discrete-gauss-bonnet-formula-for-polyominoes","status":"fully_proved","summary":"Cell Gauss\u2013Bonnet formula: for every finite set of unit cells, convex corners minus reflex corners equals four times the Euler characteristic plus twice the number of pinch points.","title":"Cell Gauss\u2013Bonnet formula"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank.json","id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","title":"Euler's formula and cycle rank"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/card-subtype-ne-add-one","kind":"node","lean":"StatMech.Lattice.card_subtype_ne_add_one","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#card-subtype-ne-add-one","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"Removing one element from a finite type lowers its cardinality by one. See also Mathlib's `Fintype.card_subtype_compl`.","title":"Removing one element lowers cardinality by one"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/card-edge-set-sup-edge","kind":"node","lean":"StatMech.Lattice.card_edgeSet_sup_edge","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#card-edge-set-sup-edge","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"Adding a new edge increases the number of edges by one. This is the `Set.ncard` form of Mathlib's `SimpleGraph.card_edgeFinset_sup_edge`.","title":"Adding a new edge increases the edge count by one"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/card-components-bot","kind":"node","lean":"StatMech.Lattice.card_components_bot","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#card-components-bot","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The edgeless graph has one component per vertex.","title":"Edgeless graph has one component per vertex"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/nullity","kind":"node","lean":"StatMech.Lattice.nullity","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#nullity","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The cycle rank (nullity, first Betti number) `|E| + c - |V|` of a finite graph, where `c` is the number of connected components.","title":"Cycle rank of a finite graph"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/face-count","kind":"node","lean":"StatMech.Lattice.faceCount","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#face-count","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The face count `nullity G + 1`. For a finite subgraph of `\u2124\u00b2` it is the number of faces, including the unbounded one (`StatMech.Lattice.jed_faithfulDiscreteJordan` counts the bounded ones); as a definition it is purely combinatorial.","title":"Face count of a finite graph from its cycle rank"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/euler-relation","kind":"node","lean":"StatMech.Lattice.euler_relation","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#euler-relation","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The combinatorial Euler relation `|V| + F = |E| + c + 1` for a finite graph, where `F = faceCount G` is defined from the cycle rank.","title":"Combinatorial Euler relation for finite graphs"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/connected-card-edges-ge","kind":"node","lean":"StatMech.Lattice.connected_card_edges_ge","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Connected finite graph has at least "],["math","\\(|V| - 1\\)"],["text"," edges"]],"source":"Connected finite graph has at least $|V| - 1$ edges"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#connected-card-edges-ge","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"A connected finite graph has at least `|V| - 1` edges. This is the `Fintype` form of Mathlib's `SimpleGraph.Connected.card_vert_le_card_edgeSet_add_one`.","title":"Connected finite graph has at least $|V| - 1$ edges"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/connected-card-edges-eq-iff-is-acyclic","kind":"node","lean":"StatMech.Lattice.connected_card_edges_eq_iff_isAcyclic","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Connected graph with "],["math","\\(|V| - 1\\)"],["text"," edges is acyclic, and conversely"]],"source":"Connected graph with $|V| - 1$ edges is acyclic, and conversely"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#connected-card-edges-eq-iff-is-acyclic","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"A connected finite graph has exactly `|V| - 1` edges iff it is acyclic. See also Mathlib's `SimpleGraph.isTree_iff_connected_and_card` and `SimpleGraph.IsTree.card_edgeFinset`.","title":"Connected graph with $|V| - 1$ edges is acyclic, and conversely"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/cyclomatic-number","kind":"node","lean":"StatMech.Lattice.cyclomaticNumber","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cyclomatic number "],["math","\\(|E| + 1 - |V|\\)"],["text"," of a finite graph"]],"source":"Cyclomatic number $|E| + 1 - |V|$ of a finite graph"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#cyclomatic-number","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The cyclomatic number `|E| + 1 - |V|` of a finite graph. For a connected graph it is the dimension of the cycle space. See `StatMech.Lattice.nullity` for the version for arbitrary finite graphs.","title":"Cyclomatic number $|E| + 1 - |V|$ of a finite graph"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/ecc-dual-cut-model","kind":"node","lean":"StatMech.Lattice.ecc_DualCutModel","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#ecc-dual-cut-model","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"After deleting the edges `B'`, the graph `A` is connected and `{a, b}` is one of its bridges.","title":"Graph stays connected with a given bridge after deleting edges"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/box-verts","kind":"node","lean":"StatMech.Lattice.boxVerts","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vertices of the grid "],["math","\\((-n, n)^2\\)"]],"source":"Vertices of the grid $(-n, n)^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#box-verts","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The vertices of the grid `[-n, n]\u00b2`.","title":"Vertices of the grid $(-n, n)^2$"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/h-edges","kind":"node","lean":"StatMech.Lattice.hEdges","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Horizontal edges of the grid "],["math","\\((-n, n)^2\\)"]],"source":"Horizontal edges of the grid $(-n, n)^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#h-edges","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The horizontal edges of the grid `[-n, n]\u00b2`.","title":"Horizontal edges of the grid $(-n, n)^2$"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/v-edges","kind":"node","lean":"StatMech.Lattice.vEdges","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vertical edges of the grid "],["math","\\((-n, n)^2\\)"]],"source":"Vertical edges of the grid $(-n, n)^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#v-edges","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The vertical edges of the grid `[-n, n]\u00b2`.","title":"Vertical edges of the grid $(-n, n)^2$"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/box-edges","kind":"node","lean":"StatMech.Lattice.boxEdges","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edges of the grid "],["math","\\((-n, n)^2\\)"]],"source":"Edges of the grid $(-n, n)^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#box-edges","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The edges of the grid `[-n, n]\u00b2`.","title":"Edges of the grid $(-n, n)^2$"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/box-cells","kind":"node","lean":"StatMech.Lattice.boxCells","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit squares of the grid "],["math","\\((-n, n)^2\\)"]],"source":"Unit squares of the grid $(-n, n)^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#box-cells","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The unit squares of the grid `[-n, n]\u00b2`, labelled by their lower left corners.","title":"Unit squares of the grid $(-n, n)^2$"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/box-face-count","kind":"node","lean":"StatMech.Lattice.boxFaceCount","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of faces of the grid "],["math","\\((-n, n)^2\\)"]],"source":"Number of faces of the grid $(-n, n)^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#box-face-count","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The number of faces of the grid `[-n, n]\u00b2`: its unit squares and the unbounded face.","title":"Number of faces of the grid $(-n, n)^2$"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/box-euler","kind":"node","lean":"StatMech.Lattice.box_euler","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Euler's formula for the grid graph "],["math","\\((-n, n)^2\\)"]],"source":"Euler's formula for the grid graph $(-n, n)^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#box-euler","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"Euler's formula `V - E + F = 2` for the planar grid graph `[-n, n]\u00b2`, with faces counted geometrically (unit squares plus the unbounded face).","title":"Euler's formula for the grid graph $(-n, n)^2$"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/box-induce","kind":"node","lean":"StatMech.Lattice.boxInduce","parent":28888,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square lattice induced on the box "],["math","\\((-n, n)^2\\)"]],"source":"Square lattice induced on the box $(-n, n)^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#box-induce","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The subgraph of the square lattice induced on the box `[-n, n]\u00b2`.","title":"Square lattice induced on the box $(-n, n)^2$"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/euler-general","kind":"node","lean":"StatMech.Lattice.euler_general","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#euler-general","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"Euler's formula for a finite graph with `c` connected components: `|V| - |E| + F = 1 + c`, where `F = faceCount G`.","title":"Euler's formula"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/euler-general-disconnected-components","kind":"node","lean":"StatMech.Lattice.euler_general_disconnected_components","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#euler-general-disconnected-components","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"Euler's formula solved for the number of components: `c = |V| - |E| + F - 1`.","title":"Euler's formula solved for the number of components"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/tnse-is-non-bridge-edge","kind":"node","lean":"StatMech.Lattice.tnse_IsNonBridgeEdge","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#tnse-is-non-bridge-edge","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"`e` is an edge of `A` whose endpoints remain connected after `e` is deleted.","title":"Non-bridge edge of a graph"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/tnse-delete-list","kind":"node","lean":"StatMech.Lattice.tnse_deleteList","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#tnse-delete-list","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"`A` with the edges of the list `L` deleted.","title":"Deleting a list of edges from a graph"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/tnse-all-non-bridge","kind":"node","lean":"StatMech.Lattice.tnse_AllNonBridge","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#tnse-all-non-bridge","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"Deleting the edges of `L` from the last to the first, each edge is a non-bridge at the moment it is deleted.","title":"Successive deletion of non-bridge edges"},{"id":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/tnse-has-tree-free-detour","kind":"node","lean":"StatMech.Lattice.tnse_HasTreeFreeDetour","parent":28888,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank/index.html#tnse-has-tree-free-detour","scope":"GraphTheory/planar-graphs-and-minors/euler-s-formula-and-cycle-rank","status":"fully_proved","summary":"The edge `e = {a, b}` has a three-step detour `a, a', b', b` in `A` none of whose edges is `e` or lies in `L`.","title":"Three-step detour avoiding an edge"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/graph-minors.json","id":"GraphTheory/planar-graphs-and-minors/graph-minors","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"can_prove","title":"Graph minors"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/fin-graph","kind":"node","lean":"MetaMathlibExt.FinGraph","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#fin-graph","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"A finite graph consists of a finite vertex set $V$ together with a simple graph on $V$, that is, a symmetric and irreflexive adjacency relation on $V$.","title":"Finite simple graph"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/clique-fin-graph","kind":"node","lean":"MetaMathlibExt.cliqueFinGraph","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#clique-fin-graph","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"For a natural number $n$, $K_n$ is the finite graph whose vertices are the natural numbers less than $n$, in which any two distinct vertices are adjacent.","title":"Complete graph as a finite graph"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/clique-fin-graph-adj","kind":"node","lean":"MetaMathlibExt.cliqueFinGraph_adj","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#clique-fin-graph-adj","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"For every natural number $n$ and all vertices $x$ and $y$ of the complete graph $K_n$, whose vertices are the natural numbers less than $n$, $x$ and $y$ are adjacent if and only if $x \\ne y$.","title":"Adjacency in the complete graph"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/minor-step","kind":"node","lean":"MetaMathlibExt.MinorStep","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#minor-step","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"For finite graphs $H$ and $G$, $H$ arises from $G$ by one minor step when one of the following holds. Isomorphism: $H$ is isomorphic to $G$. Vertex deletion: for some vertex $a$ of $G$ there is an injective map $f$ from the vertices of $H$ to the vertices of $G$ whose image consists of all vertices other than $a$, such that vertices $u, v$ of $H$ are adjacent exactly when $f(u), f(v)$ are adjacent in $G$. Edge\u2026","title":"One-step graph minor"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/is-minor","kind":"node","lean":"MetaMathlibExt.IsMinor","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#is-minor","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"For finite graphs $H$ and $G$, $H$ is a minor of $G$ when there is a finite sequence $H = G_0, G_1, \\dots, G_k = G$ of finite graphs, with $k \\ge 0$, such that for each $i < k$, $G_i$ arises from $G_{i+1}$ by one minor step: an isomorphism, the deletion of one vertex, the deletion of one edge, or the contraction of one edge.","title":"Graph minor relation"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/refl","kind":"node","lean":"MetaMathlibExt.IsMinor.refl","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#refl","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"Every finite graph $G$ is a minor of itself.","title":"Reflexivity of the graph minor relation"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/trans","kind":"node","lean":"MetaMathlibExt.IsMinor.trans","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#trans","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"For finite graphs $H$, $K$ and $G$, if $H$ is a minor of $K$ and $K$ is a minor of $G$, then $H$ is a minor of $G$.","title":"Transitivity of the graph minor relation"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/of-step","kind":"node","lean":"MetaMathlibExt.IsMinor.of_step","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#of-step","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"For finite graphs $H$ and $G$, if $H$ arises from $G$ by one minor step, then $H$ is a minor of $G$.","title":"A single minor step gives a minor"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/of-iso","kind":"node","lean":"MetaMathlibExt.IsMinor.of_iso","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#of-iso","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"fully_proved","summary":"For finite graphs $H$ and $G$, if $H$ is isomorphic to $G$, then $H$ is a minor of $G$.","title":"An isomorphic copy is a minor"},{"id":"GraphTheory/planar-graphs-and-minors/graph-minors/robertson-seymour","kind":"node","lean":"MetaMathlibExt.robertson_seymour","parent":28913,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/graph-minors/index.html#robertson-seymour","scope":"GraphTheory/planar-graphs-and-minors/graph-minors","status":"can_prove","summary":"Let $F$ be a family of finite simple graphs that is closed under taking minors: whenever $G \\in F$ and $H$ is a minor of $G$, also $H \\in F$. Then any two finite graphs that are minors of each other are isomorphic, and there is a finite set $O$ of finite graphs such that a finite graph $G$ belongs to $F$ if and only if no graph in $O$ is a minor of $G$.","title":"Robertson\u2013Seymour theorem"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice.json","id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","title":"Self-duality of the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/dual-lattice","kind":"node","lean":"StatMech.Lattice.dualLattice","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#dual-lattice","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The dual square lattice. Its vertices are the face centres of `\u2124\u00b2`, relabelled as integer points, so as a graph it is `hypercubicLattice 2`.","title":"Dual square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/rot90-fun","kind":"node","lean":"StatMech.Lattice.rot90Fun","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#rot90-fun","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The quarter-turn rotation `(a, b) \u21a6 (-b, a)` of `\u2124\u00b2`.","title":"Quarter-turn rotation of the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/rot90-inv","kind":"node","lean":"StatMech.Lattice.rot90Inv","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#rot90-inv","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The inverse quarter-turn rotation `(a, b) \u21a6 (b, -a)` of `\u2124\u00b2`.","title":"Inverse quarter-turn rotation"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/rot90-equiv","kind":"node","lean":"StatMech.Lattice.rot90Equiv","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#rot90-equiv","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The quarter-turn rotation as a permutation of `\u2124\u00b2`.","title":"Quarter-turn rotation as a permutation"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/rot90-iso","kind":"node","lean":"StatMech.Lattice.rot90Iso","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#rot90-iso","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The quarter-turn rotation as a graph automorphism of the square lattice.","title":"Quarter-turn rotation as a lattice automorphism"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/self-dual","kind":"node","lean":"StatMech.Lattice.selfDual","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#self-dual","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The identity isomorphism between the square lattice and its dual.","title":"Identity self-duality of the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/self-dual-cross","kind":"node","lean":"StatMech.Lattice.selfDualCross","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#self-dual-cross","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The isomorphism between the square lattice and its dual given by the quarter-turn rotation.","title":"Rotational self-duality of the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/crossing-edge-equiv","kind":"node","lean":"StatMech.Lattice.crossingEdgeEquiv","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#crossing-edge-equiv","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The bijection between primal and dual edges induced by `rot90Iso`.","title":"Bijection between primal and dual edges"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/sym2-congr","kind":"node","lean":"StatMech.Lattice.sym2Congr","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#sym2-congr","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"An equivalence `\u03b1 \u2243 \u03b2` induces an equivalence of unordered pairs `Sym2 \u03b1 \u2243 Sym2 \u03b2`.","title":"Unordered pairs under an equivalence"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/cross-edge","kind":"node","lean":"StatMech.Lattice.crossEdge","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#cross-edge","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The bijection on unordered pairs of sites induced by the quarter-turn rotation `(a, b) \u21a6 (-b, a)`. It maps lattice edges to lattice edges, but not to the dual edge crossing them: `{(0,0), (1,0)}` goes to `{(0,0), (0,1)}`. For the geometric crossing use `sharedPrimalEdge`.","title":"Quarter-turn rotation of unordered site pairs"},{"id":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/dual-config","kind":"node","lean":"StatMech.Lattice.dualConfig","parent":28924,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice/index.html#dual-config","scope":"GraphTheory/planar-graphs-and-minors/self-duality-of-the-square-lattice","status":"fully_proved","summary":"The configuration in which `e` is open exactly when the inverse quarter turn `crossEdge.symm e` of `e` is closed in `\u03c9`. This is the dual configuration only up to the rotation: applying it twice gives the half-turn of `\u03c9` (`dualConfig_dualConfig`), not `\u03c9`.","title":"Rotated dual configuration"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings.json","id":"GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings/index.html","scope":"GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings","status":"fully_proved","title":"Straight-line planar drawings"},{"id":"GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings/segment","kind":"node","lean":"SimpleGraph.Segment","parent":28936,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings/index.html#segment","scope":"GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings","status":"fully_proved","summary":"For points $a$ and $b$ of $\\mathbb{R}^2$, the closed segment from $a$ to $b$ is the set of points $a + t(b - a)$ with $t \\in \\mathbb{R}$ and $0 \\le t \\le 1$.","title":"Closed segment in the plane"},{"id":"GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings/is-planar","kind":"node","lean":"SimpleGraph.IsPlanar","parent":28936,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings/index.html#is-planar","scope":"GraphTheory/planar-graphs-and-minors/straight-line-planar-drawings","status":"fully_proved","summary":"A simple graph $G$ with vertex set $V$ is straight-line planar if either $V$ is empty or there is an injective map $f$ from $V$ to $\\mathbb{R}^2$ such that, for any two different edges $vw$ and $xy$ of $G$, the closed segment from $f(v)$ to $f(w)$ and the closed segment from $f(x)$ to $f(y)$ meet only in points of $\\{f(v), f(w)\\} \\cap \\{f(x), f(y)\\}$.","title":"Straight-line planar graph"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/two-regular-minors.json","id":"GraphTheory/planar-graphs-and-minors/two-regular-minors","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/two-regular-minors/index.html","scope":"GraphTheory/planar-graphs-and-minors/two-regular-minors","status":"can_prove","title":"Two-regular minors"},{"id":"GraphTheory/planar-graphs-and-minors/two-regular-minors/has-minor","kind":"node","lean":"MathlibExt.GraphTheory.TwoRegularMinorForcingWanted.HasMinor","parent":28939,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/two-regular-minors/index.html#has-minor","scope":"GraphTheory/planar-graphs-and-minors/two-regular-minors","status":"fully_proved","summary":"For finite simple graphs $G$ and $H$, $G$ has $H$ as a minor if there is a family of vertex sets $B_w$ of $G$, one for each vertex $w$ of $H$, such that each $B_w$ is nonempty and induces a connected subgraph of $G$, the sets $B_u$ and $B_v$ are disjoint whenever $u \\ne v$, and for every edge $uv$ of $H$ some vertex of $B_u$ is adjacent in $G$ to some vertex of $B_v$.","title":"Minor via branch sets"},{"id":"GraphTheory/planar-graphs-and-minors/two-regular-minors/conjecture","kind":"node","lean":"MathlibExt.GraphTheory.TwoRegularMinorForcingWanted.conjecture","parent":28939,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/two-regular-minors/index.html#conjecture","scope":"GraphTheory/planar-graphs-and-minors/two-regular-minors","status":"fully_proved","summary":"For every natural number $t$, every finite simple graph whose average degree is at least $4t/3 - 2$ (the average degree of the graph with no vertices being taken as $0$) contains every 2-regular simple graph on $t$ vertices as a minor.","title":"Average-degree conjecture for 2-regular minors"},{"id":"GraphTheory/planar-graphs-and-minors/two-regular-minors/conjecture-holds","kind":"node","lean":"MathlibExt.GraphTheory.TwoRegularMinorForcingWanted.conjecture_holds","parent":28939,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/two-regular-minors/index.html#conjecture-holds","scope":"GraphTheory/planar-graphs-and-minors/two-regular-minors","status":"can_prove","summary":"For every natural number $t$, every finite simple graph $G$ whose average degree (the sum of the vertex degrees divided by the number of vertices, taken to be $0$ when $G$ has no vertices) is at least $4t/3 - 2$ contains every 2-regular simple graph $H$ on $t$ vertices as a minor: there are nonempty, pairwise disjoint sets of vertices of $G$, one for each vertex of $H$, each inducing a connected subgraph of $G$,\u2026","title":"Cs\u00f3ka\u2013Lo\u2013Norin\u2013Wu\u2013Yepremyan theorem on 2-regular minors"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs.json","id":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/index.html","scope":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs","status":"can_prove","title":"Universal point sets for planar graphs"},{"id":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/is-straight-line-planar-embedding","kind":"node","lean":"MathlibExt.GraphTheory.UniversalPointSetsSubquadraticWanted.UPSS.IsStraightLinePlanarEmbedding","parent":28943,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/index.html#is-straight-line-planar-embedding","scope":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs","status":"fully_proved","summary":"A map $p$ from the vertex set of a simple graph $G$ to $\\mathbb{R}^2$ is a crossing-free straight-line embedding of $G$ if $p$ is injective, the closed segments $[p(u), p(v)]$ and $[p(x), p(y)]$ of any two different edges $uv$ and $xy$ meet only in points $p(z)$ with $z$ an endpoint of both edges, and no vertex $w$ other than $u$ and $v$ has $p(w)$ on the segment $[p(u), p(v)]$ of an edge $uv$.","title":"Straight-line planar embedding"},{"id":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/is-universal-point-set","kind":"node","lean":"MathlibExt.GraphTheory.UniversalPointSetsSubquadraticWanted.UPSS.IsUniversalPointSet","parent":28943,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/index.html#is-universal-point-set","scope":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs","status":"fully_proved","summary":"A finite set $P \\subseteq \\mathbb{R}^2$ is universal for $n$-vertex planar graphs if every straight-line planar simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ has a crossing-free straight-line embedding $p$ with $p(v) \\in P$ for every vertex $v$.","title":"Universal point set for planar graphs"},{"id":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/exists-subquadratic-universal-point-set","kind":"node","lean":"MathlibExt.GraphTheory.UniversalPointSetsSubquadraticWanted.UPSS.ExistsSubquadraticUniversalPointSet","parent":28943,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/index.html#exists-subquadratic-universal-point-set","scope":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs","status":"fully_proved","summary":"The statement that there is a function $f : \\mathbb{N} \\to \\mathbb{N}$ with $f(n)/n^2 \\to 0$ as $n \\to \\infty$ such that for every $n$ some finite set of at most $f(n)$ points of $\\mathbb{R}^2$ is a universal point set for $n$-vertex planar graphs.","title":"Subquadratic universal point sets"},{"id":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/exists-subquadratic-universal-point-set-2","kind":"node","lean":"MathlibExt.GraphTheory.UniversalPointSetsSubquadraticWanted.ExistsSubquadraticUniversalPointSet_holds","parent":28943,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs/index.html#exists-subquadratic-universal-point-set-2","scope":"GraphTheory/planar-graphs-and-minors/universal-point-sets-for-planar-graphs","status":"can_prove","summary":"There is a function $f : \\mathbb{N} \\to \\mathbb{N}$ with $f(n)/n^2 \\to 0$ as $n \\to \\infty$ such that for every $n$ some finite set $P$ of at most $f(n)$ points of $\\mathbb{R}^2$ is a universal point set for $n$-vertex planar graphs: every straight-line planar simple graph on the vertex set $\\{0, \\dots, n-1\\}$ has a crossing-free straight-line embedding with all its vertices in $P$.","title":"Planar graphs have subquadratic universal point sets"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice.json","id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","title":"Boundary contours in the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dart","kind":"node","lean":"StatMech.Lattice.Dart","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dart","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"An oriented edge of the square lattice: a pair of nearest-neighbour sites. This is the same data as Mathlib's `SimpleGraph.Dart` of `hypercubicLattice 2`; see `StatMech.Lattice.Dart.equivSimpleGraphDart`.","title":"Dart of the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/rev","kind":"node","lean":"StatMech.Lattice.Dart.rev","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#rev","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The dart with tail and head exchanged.","title":"Reversal of a lattice dart"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dir","kind":"node","lean":"StatMech.Lattice.Dart.dir","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dir","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The direction `head - tail` of a dart, one of the four unit vectors of `\u2124\u00b2`.","title":"Direction of a lattice dart"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/equiv-simple-graph-dart","kind":"node","lean":"StatMech.Lattice.Dart.equivSimpleGraphDart","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#equiv-simple-graph-dart","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Lattice darts are exactly Mathlib's darts of the square lattice.","title":"Lattice darts agree with graph darts of the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/equiv-simple-graph-dart-rev","kind":"node","lean":"StatMech.Lattice.Dart.equivSimpleGraphDart_rev","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#equiv-simple-graph-dart-rev","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Reversing a lattice dart corresponds to `SimpleGraph.Dart.symm`.","title":"Dart correspondence respects reversal"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/unit-wt","kind":"node","lean":"StatMech.Lattice.unitWt","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell^1\\)"],["text"," norm of a vector in "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"$\\ell^1$ norm of a vector in $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#unit-wt","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The `\u2113\u00b9` norm `|z 0| + |z 1|` of a vector of `\u2124\u00b2`.","title":"$\\ell^1$ norm of a vector in $\\mathbb{Z}^2$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/mk-dart","kind":"node","lean":"StatMech.Lattice.mkDart","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#mk-dart","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The dart from `p` to `p + g`, for a unit vector `g`.","title":"Dart from a site in a unit direction"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/is-boundary-dart","kind":"node","lean":"StatMech.Lattice.IsBoundaryDart","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#is-boundary-dart","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"`e` is a boundary dart of `K`: it points from a site of `K` to a site outside `K`.","title":"Boundary dart of a set of sites"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dart-next","kind":"node","lean":"StatMech.Lattice.dartNext","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dart-next","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The successor of the dart `e` along the boundary of `K`. Write `r` for `e.dir` rotated clockwise by a quarter turn. If `e.head + r \u2208 K` (a reflex corner of `K`), the successor shares the head of `e`; if only `e.tail + r \u2208 K`, it is the translate of `e` by `r`; otherwise (a convex corner of `K`) it starts at `e.tail` in direction `r`.","title":"Successor of a boundary dart"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dart-next-sub","kind":"node","lean":"StatMech.Lattice.dartNextSub","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dart-next-sub","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The successor map restricted to boundary darts of `K`.","title":"Successor map on boundary darts"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dart-succ-graph","kind":"node","lean":"StatMech.Lattice.dartSuccGraph","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dart-succ-graph","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The graph on boundary darts of `K` in which each dart is adjacent to its successor.","title":"Successor graph on boundary darts"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/orbit-walk-aux","kind":"node","lean":"StatMech.Lattice.orbitWalkAux","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#orbit-walk-aux","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The walk `a, next a, \u2026, next^[n] a` in `dartSuccGraph K`.","title":"Walk along the first iterates of a boundary dart"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dart-orbit-period","kind":"node","lean":"StatMech.Lattice.dartOrbitPeriod","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dart-orbit-period","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The minimal period of the boundary dart `a` under the successor map.","title":"Minimal period of a boundary dart"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dart-orbit-walk","kind":"node","lean":"StatMech.Lattice.dartOrbitWalk","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dart-orbit-walk","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The closed walk in `dartSuccGraph K` that runs once around the orbit of `a`.","title":"Closed walk around a boundary dart orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/iterate-inj-on-icc-minimal-period","kind":"node","lean":"StatMech.Lattice.iterate_inj_on_Icc_minimalPeriod","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#iterate-inj-on-icc-minimal-period","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The iterates `f^[j] x` for `1 \u2264 j \u2264 minimalPeriod f x` are distinct. See also Mathlib's `Function.iterate_injOn_Iio_minimalPeriod`, the same statement for `0 \u2264 j < minimalPeriod f x`.","title":"Iterates up to the minimal period are distinct"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/is-inner-corner","kind":"node","lean":"StatMech.Lattice.IsInnerCorner","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#is-inner-corner","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The site `e.head + r`, with `r` the clockwise rotation of `e.dir`, lies in `K`; then the successor of `e` has the same head.","title":"Inner corner of a boundary dart"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/same-orbit","kind":"node","lean":"StatMech.Lattice.SameOrbit","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#same-orbit","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The dart `f` is obtained from `e` by iterating the boundary successor map of `K`.","title":"Two darts in the same boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/orbit-heads-saturate","kind":"node","lean":"StatMech.Lattice.OrbitHeadsSaturate","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#orbit-heads-saturate","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Boundary darts whose heads are connected in `K\u1d9c` lie in the same orbit; a reformulation of `StatMech.Lattice.InterfaceConnected`.","title":"Orbit saturation by heads connected in the complement"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/king-adj","kind":"node","lean":"StatMech.Lattice.KingAdj","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#king-adj","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"King adjacency on `\u2124\u00b2`: distinct sites that differ by at most one in each coordinate.","title":"King adjacency on the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/boundary-king-graph","kind":"node","lean":"StatMech.Lattice.boundaryKingGraph","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#boundary-king-graph","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The graph of king moves on the complement of `K`.","title":"King-move graph on the complement of a set"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/orbit-reached","kind":"node","lean":"StatMech.Lattice.OrbitReached","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#orbit-reached","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Some boundary dart in the orbit of `e` has head `v`.","title":"Site reached by a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/orbit-king-transport","kind":"node","lean":"StatMech.Lattice.OrbitKingTransport","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#orbit-king-transport","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"If the orbit of `e` reaches a site outside `K`, it reaches every king neighbour of that site outside `K`.","title":"Boundary orbits spread along king moves"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/orbit-vertex-saturate","kind":"node","lean":"StatMech.Lattice.OrbitVertexSaturate","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#orbit-vertex-saturate","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Every boundary dart whose head is reached by the orbit of `e` lies in that orbit.","title":"Orbit saturation at reached sites"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/boundary-king-connected","kind":"node","lean":"StatMech.Lattice.BoundaryKingConnected","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#boundary-king-connected","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Sites connected by a lattice path in `K\u1d9c` are connected by king moves in `K\u1d9c`.","title":"Lattice connectivity in the complement implies king connectivity"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/interface-connected","kind":"node","lean":"StatMech.Lattice.InterfaceConnected","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#interface-connected","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Boundary darts of `K` whose heads lie in the same connected component of `K\u1d9c` lie in the same boundary orbit.","title":"Boundary darts with heads in one component share an orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/face-corner00","kind":"node","lean":"StatMech.Lattice.faceCorner00","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower left corner of the unit face at "],["math","\\((a, b)\\)"]],"source":"Lower left corner of the unit face at $(a, b)$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#face-corner00","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The lower left corner `(a, b)` of the unit face `[a, a + 1] \u00d7 [b, b + 1]`.","title":"Lower left corner of the unit face at $(a, b)$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/face-corner10","kind":"node","lean":"StatMech.Lattice.faceCorner10","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower right corner of the unit face at "],["math","\\((a, b)\\)"]],"source":"Lower right corner of the unit face at $(a, b)$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#face-corner10","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The lower right corner `(a + 1, b)` of the unit face `[a, a + 1] \u00d7 [b, b + 1]`.","title":"Lower right corner of the unit face at $(a, b)$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/face-corner11","kind":"node","lean":"StatMech.Lattice.faceCorner11","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper right corner of the unit face at "],["math","\\((a, b)\\)"]],"source":"Upper right corner of the unit face at $(a, b)$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#face-corner11","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The upper right corner `(a + 1, b + 1)` of the unit face `[a, a + 1] \u00d7 [b, b + 1]`.","title":"Upper right corner of the unit face at $(a, b)$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/face-corner01","kind":"node","lean":"StatMech.Lattice.faceCorner01","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper left corner of the unit face at "],["math","\\((a, b)\\)"]],"source":"Upper left corner of the unit face at $(a, b)$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#face-corner01","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The upper left corner `(a, b + 1)` of the unit face `[a, a + 1] \u00d7 [b, b + 1]`.","title":"Upper left corner of the unit face at $(a, b)$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/bd-ind","kind":"node","lean":"StatMech.Lattice.bdInd","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#bd-ind","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"`1` if exactly one of `x` and `y` lies in `S`, and `0` otherwise.","title":"Boundary indicator of an edge"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/face-boundary-degree","kind":"node","lean":"StatMech.Lattice.faceBoundaryDegree","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#face-boundary-degree","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The number of the four edges of the unit face at `(a, b)` that have exactly one endpoint in `S`.","title":"Boundary degree of a unit face"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/mpl-orbit-face-loop","kind":"node","lean":"StatMech.Lattice.mpl_orbitFaceLoop","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#mpl-orbit-face-loop","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The closed walk of faces to the left of the darts in one period of the boundary orbit of `a`.","title":"Face loop left of one period of a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/mpl-orbit-loop","kind":"node","lean":"StatMech.Lattice.mpl_orbitLoop","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#mpl-orbit-loop","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The closed face walk of one period of the orbit of `a`, as a walk in the square lattice.","title":"Minimal-period orbit loop as a lattice walk"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/npd-p00","kind":"node","lean":"StatMech.Lattice.npd_P00","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower left corner of a face "],["math","\\(f\\)"]],"source":"Lower left corner of a face $f$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#npd-p00","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The lower left corner of the face `f`. This is `StatMech.Lattice.faceCorner00 (f 0) (f 1)`.","title":"Lower left corner of a face $f$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/npd-p10","kind":"node","lean":"StatMech.Lattice.npd_P10","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower right corner of a face "],["math","\\(f\\)"]],"source":"Lower right corner of a face $f$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#npd-p10","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The lower right corner of the face `f`.","title":"Lower right corner of a face $f$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/npd-p11","kind":"node","lean":"StatMech.Lattice.npd_P11","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper right corner of a face "],["math","\\(f\\)"]],"source":"Upper right corner of a face $f$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#npd-p11","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The upper right corner of the face `f`.","title":"Upper right corner of a face $f$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/npd-p01","kind":"node","lean":"StatMech.Lattice.npd_P01","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Upper left corner of a face "],["math","\\(f\\)"]],"source":"Upper left corner of a face $f$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#npd-p01","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The upper left corner of the face `f`.","title":"Upper left corner of a face $f$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/npd-diag-touch","kind":"node","lean":"StatMech.Lattice.npd_DiagTouch","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#npd-diag-touch","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"`K` contains exactly two opposite corners of the face `f`.","title":"Diagonal touch at a face"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/npd-no-diag-touch-along-orbit","kind":"node","lean":"StatMech.Lattice.npd_NoDiagTouchAlongOrbit","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#npd-no-diag-touch-along-orbit","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"No face visited in one period of the orbit of `a` is a diagonal touch of `K`.","title":"No diagonal touch along a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/npd-no-diag-touch","kind":"node","lean":"StatMech.Lattice.npd_NoDiagTouch","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#npd-no-diag-touch","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"`K` has no diagonal touch at any face.","title":"Set with no diagonal touches"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/npm-king-saturated","kind":"node","lean":"StatMech.Lattice.npm_KingSaturated","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#npm-king-saturated","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Whenever `K` contains two opposite corners of a face, it contains one of the other two corners.","title":"King-saturated set"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/olb-orbit-loop","kind":"node","lean":"StatMech.Lattice.olb_orbitLoop","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#olb-orbit-loop","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The face loop of the boundary orbit of `e`, as a closed walk in the square lattice.","title":"Boundary orbit face loop as a lattice walk"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/ndt-product","kind":"node","lean":"StatMech.Lattice.ndt_product","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#ndt-product","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The product set `{x | x 0 \u2208 A \u2227 x 1 \u2208 B}`.","title":"Product set of two coordinate sets in the plane"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/ndt-sublevel","kind":"node","lean":"StatMech.Lattice.ndt_sublevel","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#ndt-sublevel","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The sublevel set `{x | \u03c6 (x 0) + \u03c8 (x 1) \u2264 c}` of a separable function.","title":"Sublevel set of a separable function"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/ndt-star-hull-mem","kind":"node","lean":"StatMech.Lattice.ndt_StarHullMem","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#ndt-star-hull-mem","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Membership in the star hull of `K`: sites of `K`, closed under filling the lower right corner of a face whose lower left and upper right corners are members, and the lower left corner of a face whose lower right and upper left corners are members.","title":"Membership in the star hull"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/ndt-star-hull","kind":"node","lean":"StatMech.Lattice.ndt_StarHull","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#ndt-star-hull","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The star hull of `K`: the smallest superset of `K` closed under the diagonal fill rules.","title":"Star hull of a set of lattice sites"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/ndt-fill-closed","kind":"node","lean":"StatMech.Lattice.ndt_FillClosed","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#ndt-fill-closed","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"`S` is closed under the two diagonal fill rules of `StatMech.Lattice.ndt_StarHullMem`.","title":"Closure under the diagonal fill rules"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/rot90-fun-iterate-four-eq","kind":"node","lean":"StatMech.Lattice.rot90Fun_iterate_four_eq","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#rot90-fun-iterate-four-eq","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Four counterclockwise quarter turns are the identity, i.e. every site is a periodic point of `rot90Fun` of period `4`.","title":"Four quarter turns of the square lattice are the identity"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/rot-pow","kind":"node","lean":"StatMech.Lattice.rotPow","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rotation of "],["math","\\(\\mathbb{Z}^2\\)"],["text"," by a number of quarter turns"]],"source":"Rotation of $\\mathbb{Z}^2$ by a number of quarter turns"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#rot-pow","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Rotation of `\u2124\u00b2` by `s` counterclockwise quarter turns, `s : ZMod 4`.","title":"Rotation of $\\mathbb{Z}^2$ by a number of quarter turns"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/rot-pow-add","kind":"node","lean":"StatMech.Lattice.rotPow_add","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quarter-turn rotations form an action of "],["math","\\(\\mathbb{Z}/4\\)"]],"source":"Quarter-turn rotations form an action of $\\mathbb{Z}/4$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#rot-pow-add","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"`rotPow` is an action of the additive group `ZMod 4`.","title":"Quarter-turn rotations form an action of $\\mathbb{Z}/4$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/rot-pow-dir-eq-iff","kind":"node","lean":"StatMech.Lattice.rotPow_dir_eq_iff","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#rot-pow-dir-eq-iff","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"A nontrivial rotation moves every lattice direction: `rotPow s` fixes the direction of a dart only for `s = 0`.","title":"Nontrivial quarter-turn rotations fix no lattice direction"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/turn","kind":"node","lean":"StatMech.Lattice.turn","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#turn","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The quarter turn, as an element of `ZMod 4`, from the boundary dart `e` of `K` to its successor `dartNext K e`: `1` for a counterclockwise (left) turn, `0` for going straight, `3` for a clockwise (right) turn.","title":"Quarter turn between consecutive boundary darts modulo 4"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/turn-z","kind":"node","lean":"StatMech.Lattice.turnZ","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#turn-z","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The signed quarter turn from `e` to `dartNext K e`: `1` counterclockwise, `0` straight, `-1` clockwise. Its reduction mod `4` is `turn K e`.","title":"Signed quarter turn between consecutive boundary darts"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dart-next-dir-eq-rot-pow-turn","kind":"node","lean":"StatMech.Lattice.dartNext_dir_eq_rotPow_turn","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dart-next-dir-eq-rot-pow-turn","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The direction of `dartNext K e` is the direction of `e` rotated by `turn K e` quarter turns.","title":"Next boundary dart direction is rotated by the turn"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/cum-turn","kind":"node","lean":"StatMech.Lattice.cumTurn","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#cum-turn","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The cumulative turn, in `ZMod 4`, over the first `k` steps of the `dartNext K`-orbit of `e`.","title":"Cumulative turn along a boundary-tracing orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dir-iterate-eq-rot-pow-cum-turn","kind":"node","lean":"StatMech.Lattice.dir_iterate_eq_rotPow_cumTurn","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dir-iterate-eq-rot-pow-cum-turn","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"After `k` boundary-tracing steps the direction of a dart is its initial direction rotated by the cumulative turn `cumTurn K e k`.","title":"Dart direction after k steps is rotated by the cumulative turn"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/total-turn-z","kind":"node","lean":"StatMech.Lattice.totalTurnZ","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#total-turn-z","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The total signed turning (in quarter turns) over the first `p` steps of the `dartNext K`-orbit of `e`.","title":"Total signed turning along a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/four-dvd-total-turn-z-of-iterate-eq","kind":"node","lean":"StatMech.Lattice.four_dvd_totalTurnZ_of_iterate_eq","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#four-dvd-total-turn-z-of-iterate-eq","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"If the orbit of `e` closes after `p` steps, the total signed turning over these steps is divisible by `4`.","title":"Total turning of a closed orbit segment is divisible by four"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/four-dvd-orbit-total-turn-z","kind":"node","lean":"StatMech.Lattice.four_dvd_orbit_totalTurnZ","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#four-dvd-orbit-total-turn-z","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The total signed turning around a boundary orbit, taken over its minimal period, is divisible by `4`.","title":"Total turning over the minimal period is divisible by four"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/abs-orbit-total-turn-z-le","kind":"node","lean":"StatMech.Lattice.abs_orbit_totalTurnZ_le","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#abs-orbit-total-turn-z-le","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The total signed turning around a boundary orbit is at most the orbit length in absolute value.","title":"Total turning of a boundary orbit is bounded by its length"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/turning-is-full-revolution","kind":"node","lean":"StatMech.Lattice.TurningIsFullRevolution","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#turning-is-full-revolution","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary orbit through `a` turns by exactly one full revolution, i.e. its total signed turning is `4` or `-4` quarter turns.","title":"Boundary orbit making exactly one full revolution"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/orbit-period-ge-four-of-full-revolution","kind":"node","lean":"StatMech.Lattice.orbitPeriod_ge_four_of_fullRevolution","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#orbit-period-ge-four-of-full-revolution","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"A boundary orbit that makes a full revolution has length at least `4`.","title":"Boundary orbit with a full revolution has length at least four"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/right-corner-count","kind":"node","lean":"StatMech.Lattice.rightCornerCount","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#right-corner-count","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The number of steps `i < p` in the boundary orbit of `e` with turn `turnZ = 1` (reflex corners of `K`).","title":"Number of reflex corners in a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/left-corner-count","kind":"node","lean":"StatMech.Lattice.leftCornerCount","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#left-corner-count","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The number of steps `i < p` in the boundary orbit of `e` with turn `turnZ = -1` (convex corners of `K`).","title":"Number of convex corners in a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/unit-cell","kind":"node","lean":"StatMech.Lattice.unitCell","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#unit-cell","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The one-site set `{(0, 0)}`.","title":"Unit cell as a one-site set"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/uc-dart0","kind":"node","lean":"StatMech.Lattice.ucDart0","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#uc-dart0","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the unit cell pointing right.","title":"Rightward boundary dart of the unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/uc-dart1","kind":"node","lean":"StatMech.Lattice.ucDart1","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#uc-dart1","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the unit cell pointing down.","title":"Downward boundary dart of the unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/uc-dart2","kind":"node","lean":"StatMech.Lattice.ucDart2","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#uc-dart2","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the unit cell pointing left.","title":"Leftward boundary dart of the unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/uc-dart3","kind":"node","lean":"StatMech.Lattice.ucDart3","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#uc-dart3","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the unit cell pointing up.","title":"Upward boundary dart of the unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/uc-base","kind":"node","lean":"StatMech.Lattice.ucBase","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#uc-base","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The rightward boundary dart of the unit cell, as an element of the subtype of boundary darts.","title":"Base boundary dart of the unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/straight-step-count","kind":"node","lean":"StatMech.Lattice.straightStepCount","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#straight-step-count","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The number of steps `i < p` in the boundary orbit of `e` with turn `0`.","title":"Number of straight steps in a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/simple-polygon-corner-balance","kind":"node","lean":"StatMech.Lattice.SimplePolygonCornerBalance","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#simple-polygon-corner-balance","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The numbers of reflex and convex corners along the orbit of `a` differ by `4` or `-4`, the corner count of a simple lattice polygon.","title":"Corner balance of a simple lattice polygon"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/domino","kind":"node","lean":"StatMech.Lattice.domino","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#domino","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The two-site set `{(0, 0), (1, 0)}`.","title":"Domino as a two-site set"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dm-d0","kind":"node","lean":"StatMech.Lattice.dmD0","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dm-d0","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the domino from `(0, 0)` downwards.","title":"Downward domino dart at the left site"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dm-d1","kind":"node","lean":"StatMech.Lattice.dmD1","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dm-d1","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the domino from `(0, 0)` to the left.","title":"Leftward domino dart at the left site"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dm-d2","kind":"node","lean":"StatMech.Lattice.dmD2","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dm-d2","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the domino from `(0, 0)` upwards.","title":"Upward domino dart at the left site"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dm-d3","kind":"node","lean":"StatMech.Lattice.dmD3","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dm-d3","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the domino from `(1, 0)` upwards.","title":"Upward domino dart at the right site"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dm-d4","kind":"node","lean":"StatMech.Lattice.dmD4","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dm-d4","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the domino from `(1, 0)` to the right.","title":"Rightward domino dart at the right site"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dm-d5","kind":"node","lean":"StatMech.Lattice.dmD5","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dm-d5","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart of the domino from `(1, 0)` downwards.","title":"Downward domino dart at the right site"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/dm-base","kind":"node","lean":"StatMech.Lattice.dmBase","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#dm-base","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The boundary dart `dmD0` of the domino, as an element of the subtype of boundary darts.","title":"Base boundary dart of the domino"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/orbit-face-no-pinch","kind":"node","lean":"StatMech.Lattice.OrbitFaceNoPinch","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#orbit-face-no-pinch","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Darts in one period of the orbit of `a` that have the same face also have the same direction.","title":"No-pinch condition for a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/unit-cell-face-loop","kind":"node","lean":"StatMech.Lattice.unitCellFaceLoop","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#unit-cell-face-loop","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The closed walk of faces around the unit cell, of length four.","title":"Face loop around the unit cell"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/domino-face-loop","kind":"node","lean":"StatMech.Lattice.dominoFaceLoop","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#domino-face-loop","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The closed walk of faces around the domino, of length six.","title":"Face loop around the domino"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/usc-four-dirs","kind":"node","lean":"StatMech.Lattice.usc_fourDirs","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Four unit vectors of "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Four unit vectors of $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#usc-four-dirs","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The four unit vectors of `\u2124\u00b2`.","title":"Four unit vectors of $\\mathbb{Z}^2$"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/usc-visit-set","kind":"node","lean":"StatMech.Lattice.usc_visitSet","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#usc-visit-set","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The times `i` in one period of the orbit of `a` at which the dart has face `f`.","title":"Visit times of a boundary orbit to a face"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/usc-face-multiplicity-one","kind":"node","lean":"StatMech.Lattice.usc_FaceMultiplicityOne","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#usc-face-multiplicity-one","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"Every face visited by the orbit of `a` is visited exactly once per period.","title":"Each face visited once per period by a boundary orbit"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/usc-local-turn","kind":"node","lean":"StatMech.Lattice.usc_localTurn","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#usc-local-turn","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The sum of the turns `turnZ` at the visits of the orbit of `a` to the face `f`.","title":"Local turn of a boundary orbit at a face"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/usc-orbit-faces","kind":"node","lean":"StatMech.Lattice.usc_orbitFaces","parent":28948,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#usc-orbit-faces","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"The set of faces visited in one period of the orbit of `a`.","title":"Faces visited by a boundary orbit in one period"},{"id":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/usc-pc-lt-one-of-no-pinch","kind":"node","lean":"StatMech.Lattice.usc_pc_lt_one_of_noPinch","parent":28948,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(\\mathbb{Z}^2) < 1\\)"],["text"," under a no-pinch hypothesis"]],"source":"$p_c(\\mathbb{Z}^2) < 1$ under a no-pinch hypothesis"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice/index.html#usc-pc-lt-one-of-no-pinch","scope":"GraphTheory/planar-graphs-and-minors/boundary-contours-in-the-square-lattice","status":"fully_proved","summary":"`p_c(\u2124\u00b2) < 1`, under the hypothesis that for every configuration with a finite origin cluster the boundary orbit of the exit dart has period at least three, has no pinch, and contains the left exit dart.","title":"$p_c(\\mathbb{Z}^2) < 1$ under a no-pinch hypothesis"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic.json","id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","title":"Whitney's cut\u2013cycle duality on the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jce-nbr","kind":"node","lean":"StatMech.Lattice.jce_nbr","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jce-nbr","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The four nearest neighbours of `v`: right, left, up, down.","title":"Four nearest neighbours of a site"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jce-degree","kind":"node","lean":"StatMech.Lattice.jce_degree","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jce-degree","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The number of edges of `E` at the site `v`.","title":"Degree of a site in an edge set"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jce-closed-contour","kind":"node","lean":"StatMech.Lattice.jce_ClosedContour","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jce-closed-contour","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"Every site has even degree in the edge set `E`.","title":"Closed contour as an even-degree edge set"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jce-horiz-below","kind":"node","lean":"StatMech.Lattice.jce_horizBelow","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jce-horiz-below","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The horizontal edges in the column `[a, a + 1]` at heights at most `b`.","title":"Horizontal edges below a face in its column"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jce-phi","kind":"node","lean":"StatMech.Lattice.jce_phi","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jce-phi","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The parity coloring of faces: `f` is colored when an odd number of horizontal edges of `E` lie directly below it in its column.","title":"Parity coloring of faces by an edge set"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jce-horiz-step-residue","kind":"node","lean":"StatMech.Lattice.jce_HorizStepResidue","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jce-horiz-step-residue","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"For every closed contour `E`, the parity coloring changes between horizontally adjacent faces exactly when the vertical edge between them is in `E`.","title":"Parity coloring changes across vertical contour edges"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jce-hc","kind":"node","lean":"StatMech.Lattice.jce_Hc","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jce-hc","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The number of horizontal edges of `E` below height `b` in the column `[c, c + 1]`.","title":"Number of horizontal contour edges below a height"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jce-edge-flip","kind":"node","lean":"StatMech.Lattice.jce_edgeFlip","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jce-edge-flip","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The flip relation of `E`: the primal edge between the faces `a` and `b` is in `E`.","title":"Edge flip relation of an edge set"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whb-face-region","kind":"node","lean":"StatMech.Lattice.whb_faceRegion","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whb-face-region","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The face-region graph of `G`: two adjacent faces of `\u2124\u00b2` are joined unless the primal edge between them (`sharedPrimalEdge`) is an edge of `G`. This is the faithful planar dual complement of `G`.","title":"Face-region graph of a square-lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whb-correct-whitney","kind":"node","lean":"StatMech.Lattice.whb_CorrectWhitney","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whb-correct-whitney","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"Whitney's cut\u2013cycle duality for subgraphs of `\u2124\u00b2` (H. Whitney, *Planar graphs*, 1933): for every `H` and lattice edge `pq` not in `H`, with `fg` the dual edge across `pq`, the endpoints `p` and `q` are connected in `H` iff `f` and `g` are disconnected in the face-region graph of `H` with `fg` deleted.","title":"Whitney's cut\u2013cycle duality"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whc-dual-cross","kind":"node","lean":"StatMech.Lattice.whc_dualCross","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whc-dual-cross","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The number of steps of the face walk `w` across an edge of `H`.","title":"Number of edge crossings of a face walk"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whc-is-cross","kind":"node","lean":"StatMech.Lattice.whc_isCross","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whc-is-cross","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The primal edge between the faces `a` and `b` is an edge of `H`.","title":"Primal edge between two faces lies in a subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whc-compat-coloring","kind":"node","lean":"StatMech.Lattice.whc_CompatColoring","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whc-compat-coloring","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"`\u03c6` changes value exactly across the dual edges of `H`.","title":"Face coloring compatible with a subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whc-cut-coloring","kind":"node","lean":"StatMech.Lattice.whc_CutColoring","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whc-cut-coloring","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"`\u03c6` changes value exactly across the dual edges of `H` and the extra dual edge `f0g0`.","title":"Cut coloring of a subgraph with an extra dual edge"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whc-htest","kind":"node","lean":"StatMech.Lattice.whc_Htest","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whc-htest","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The unit square with its bottom edge removed.","title":"Unit square with its bottom edge removed"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whc-phi-witness","kind":"node","lean":"StatMech.Lattice.whc_phiWitness","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whc-phi-witness","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The face coloring that is true only at the face `(0, 0)`.","title":"Face coloring supported at a single face"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whc-faithful-region-count","kind":"node","lean":"StatMech.Lattice.whc_faithfulRegionCount","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whc-faithful-region-count","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The number of finite components of the face-region graph of `P`: the bounded faces of `P`.","title":"Number of bounded faces of a lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/whc-faithful-discrete-jordan","kind":"node","lean":"StatMech.Lattice.whc_FaithfulDiscreteJordan","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#whc-faithful-discrete-jordan","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The face count of a finite subgraph of `\u2124\u00b2`: the bounded components of its face-region graph are equinumerous with `Fin (nullity P.G)`, so the number of bounded faces is the cycle rank. Only the number of faces is asserted, not which cycle bounds which face.","title":"Faithful discrete Jordan statement for the face count"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jwc-vertex-loop","kind":"node","lean":"StatMech.Lattice.jwc_vertexLoop","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jwc-vertex-loop","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The closed walk through the four faces having the site `(c, d)` as a corner.","title":"Closed face walk around a lattice vertex"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jwc-cycle-path","kind":"node","lean":"StatMech.Lattice.jwc_cyclePath","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jwc-cycle-path","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The spanning subgraph formed by a path from `y` to `x` in `K`.","title":"Spanning subgraph formed by a path"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jwc-keep-residue","kind":"node","lean":"StatMech.Lattice.jwc_KeepResidue","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jwc-keep-residue","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"Whitney's duality in the bridge direction for `P`: for every subgraph `K` of `P` and edge `xy` of `P` not in `K` with dual edge `f0g0`, if `x` and `y` are disconnected in `K`, then `f0` and `g0` are connected in the face-region graph of `K` with `f0g0` deleted.","title":"Whitney duality in the bridge direction"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jeb-induce-lf","kind":"node","lean":"StatMech.Lattice.jeb_induceLF","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jeb-induce-lf","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"An induced subgraph of a locally finite graph is locally finite.","title":"Induced subgraphs of locally finite graphs are locally finite"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jeb-two-odd-reachable","kind":"node","lean":"StatMech.Lattice.jeb_two_odd_reachable","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jeb-two-odd-reachable","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"In a locally finite graph with finite support, two vertices of odd degree whose other vertices all have even degree lie in the same component (the handshake lemma applied to a component).","title":"Two odd-degree vertices lie in one component"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jeb-even-degree-subgraph","kind":"node","lean":"StatMech.Lattice.jeb_EvenDegreeSubgraph","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jeb-even-degree-subgraph","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"Every site has even degree in the image of `P`.","title":"Even-degree image of a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jeb-faithful-count-residue","kind":"node","lean":"StatMech.Lattice.jeb_FaithfulCountResidue","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jeb-faithful-count-residue","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"Every `PlanarZ2Subgraph` with all degrees even satisfies `whc_FaithfulDiscreteJordan`.","title":"Faithful discrete Jordan property for even-degree subgraphs"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/wcl-symm-flip","kind":"node","lean":"StatMech.Lattice.wcl_SymmFlip","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#wcl-symm-flip","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The flip relation is symmetric on lattice edges.","title":"Symmetry of a flip relation on lattice edges"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/wcl-flip-count","kind":"node","lean":"StatMech.Lattice.wcl_flipCount","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#wcl-flip-count","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The number of steps of the walk `w` that are flips (in either direction).","title":"Number of flips along a lattice walk"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/wcl-closed-even","kind":"node","lean":"StatMech.Lattice.wcl_ClosedEven","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#wcl-closed-even","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"Every closed lattice walk has an even number of flips.","title":"Even-flip condition on closed lattice walks"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/wcl-chosen-walk","kind":"node","lean":"StatMech.Lattice.wcl_chosenWalk","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#wcl-chosen-walk","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"A fixed lattice walk from `o` to `f`, chosen using connectedness of `\u2124\u00b2`.","title":"Chosen lattice walk between two faces"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/wcl-winding-coloring","kind":"node","lean":"StatMech.Lattice.wcl_windingColoring","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#wcl-winding-coloring","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The face coloring given by the parity of the number of flips along the chosen walk from `o`.","title":"Face coloring by parity of flips"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/wcl-cut-flip","kind":"node","lean":"StatMech.Lattice.wcl_cutFlip","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#wcl-cut-flip","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The cut flip relation: steps across a dual edge of `H` or across the extra dual edge `f0g0`.","title":"Cut flip relation of a lattice graph and a dual edge"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/wcl-square-loop","kind":"node","lean":"StatMech.Lattice.wcl_squareLoop","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#wcl-square-loop","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The closed walk around the unit square with corners `(0, 0)` and `(1, -1)`.","title":"Closed walk around a unit square"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/wcl-correct-coloring-residue","kind":"node","lean":"StatMech.Lattice.wcl_CorrectColoringResidue","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#wcl-correct-coloring-residue","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"For every graph `H` of `\u2124\u00b2` and dual edge `f0g0` such that the cut flip relation has even loops, there is a cut coloring distinguishing `f0` from `g0`, and `f0`, `g0` are disconnected in the face-region graph of `H` with `f0g0` deleted.","title":"Cut coloring separating the ends of a dual edge"},{"id":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/jwn-cut-count","kind":"node","lean":"StatMech.Lattice.jwn_cutCount","parent":29038,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic/index.html#jwn-cut-count","scope":"GraphTheory/planar-graphs-and-minors/whitney-s-cut-cycle-duality-on-the-square-lattic","status":"fully_proved","summary":"The number of cut flips of `H` and `f0g0` along `w`; equal to `wcl_flipCount (wcl_cutFlip H f0 g0) w`.","title":"Cut-flip count along a walk"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat.json","id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","title":"Discrete Jordan curve theorem for the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/ffc-off-support-lattice","kind":"node","lean":"StatMech.Lattice.ffc_offSupportLattice","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#ffc-off-support-lattice","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The square lattice with every vertex of `B` removed (edges with an endpoint in `B` deleted).","title":"Square lattice with a set of vertices removed"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/ffc-lift-walk","kind":"node","lean":"StatMech.Lattice.ffc_lift_walk","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#ffc-lift-walk","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"A lattice walk avoiding `B`, viewed as a walk in `ffc_offSupportLattice B`.","title":"Lattice walk avoiding a set, lifted to the reduced lattice"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/ffc-flood-fill","kind":"node","lean":"StatMech.Lattice.ffc_floodFill","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#ffc-flood-fill","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The flood fill of `seed` avoiding `B`: the sites reachable from `seed` in the lattice with `B` removed.","title":"Flood fill avoiding a set of sites"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/off-support","kind":"node","lean":"StatMech.Lattice.offSupport","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#off-support","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The square lattice with the vertices of the closed walk `Vc` removed. It is the same graph as `StatMech.Lattice.ffc_offSupportLattice {z | z \u2208 Vc.support}`.","title":"Square lattice with a closed walk's vertices removed"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/off-support-component","kind":"node","lean":"StatMech.Lattice.offSupportComponent","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#off-support-component","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The sites reachable from `seed` in the lattice with the vertices of `Vc` removed.","title":"Component off a closed walk containing a seed"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/exists-cycle-through-edge-of-even","kind":"node","lean":"StatMech.Lattice.EvenDegree.exists_cycle_through_edge_of_even","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#exists-cycle-through-edge-of-even","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"In a finite graph in which every vertex has even degree, every edge lies on a cycle. See also Mathlib's `SimpleGraph.adj_and_reachable_delete_edges_iff_exists_cycle`, which this combines with `StatMech.Lattice.reachable_deleteEdges_of_even`.","title":"Every edge of a finite even-degree graph lies on a cycle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/shared-primal-edge","kind":"node","lean":"StatMech.Lattice.sharedPrimalEdge","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#shared-primal-edge","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The primal edge of `\u2124\u00b2` separating the adjacent unit faces with lower left corners `f` and `g`.","title":"Primal edge shared by adjacent faces"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/bd-edge","kind":"node","lean":"StatMech.Lattice.bdEdge","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#bd-edge","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The edge `e` has exactly one endpoint in `S`.","title":"Boundary edge of a set"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/face-boundary-graph","kind":"node","lean":"StatMech.Lattice.faceBoundaryGraph","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#face-boundary-graph","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The face-boundary graph of `S`: two adjacent faces are joined when the primal edge between them has exactly one endpoint in `S`. Its edges are the dual edges crossing the boundary of `S`.","title":"Face-boundary graph of a set"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/corner-faces","kind":"node","lean":"StatMech.Lattice.cornerFaces","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#corner-faces","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The four unit faces having the site `c` as a corner.","title":"Four unit faces at a lattice site"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/cross-count","kind":"node","lean":"StatMech.Lattice.crossCount","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#cross-count","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The number of edges of the walk `w` with exactly one endpoint in `S`.","title":"Number of boundary crossings of a walk"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/lattice-minus-barrier","kind":"node","lean":"StatMech.Lattice.latticeMinusBarrier","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#lattice-minus-barrier","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The square lattice with every edge having exactly one endpoint in `S` removed.","title":"Square lattice with the boundary edges of a set removed"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jfd-ext-to-deleted-hom","kind":"node","lean":"StatMech.Lattice.jfd_extToDeletedHom","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jfd-ext-to-deleted-hom","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"For `H` and `f0g0` inside `[-R, R]\u00b2`, the lattice on the exterior of `[-(R + 1), R + 1]\u00b2` maps into the face-region graph of `H` with `f0g0` deleted.","title":"Exterior of a box maps into the face-region graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcs-cycle-face-count-eq-two","kind":"node","lean":"StatMech.Lattice.jcs_cycle_faceCount_eq_two","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcs-cycle-face-count-eq-two","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"A cycle, as a graph on its own vertices, has face count two: the combinatorial form of the Jordan curve theorem for cycles (C. Jordan, *Cours d'analyse*, 1887).","title":"Jordan curve theorem for cycles: face count two"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcs-jordan-region","kind":"node","lean":"StatMech.Lattice.jcs_JordanRegion","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcs-jordan-region","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"`S` and its complement are nonempty, and any two sites on the same side of `S` are connected in the lattice with the boundary edges of `S` removed.","title":"Jordan region of the cut lattice"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jes-bounded-region-count","kind":"node","lean":"StatMech.Lattice.jes_boundedRegionCount","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jes-bounded-region-count","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The number of bounded regions of a finite graph, defined as its cycle rank `nullity G`.","title":"Bounded region count as cycle rank"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-quad","kind":"node","lean":"StatMech.Lattice.jec_quad","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-quad","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The quadrant of sites `p` with `p 0 \u2264 z 0 - 1` and `p 1 \u2264 z 1 - 1`.","title":"Quadrant below and left of a site"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-ray-edge","kind":"node","lean":"StatMech.Lattice.jec_rayEdge","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-ray-edge","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The vertical lattice edges in a column left of `z` joining rows `z 1 - 1` and `z 1`: the edges crossed by a horizontal ray going left from the face corner of `z`.","title":"Ray edges left of a site"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-wall-edge","kind":"node","lean":"StatMech.Lattice.jec_wallEdge","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-wall-edge","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The horizontal lattice edges in a row below `z` joining columns `z 0 - 1` and `z 0`: the edges crossed by a vertical ray going down from the face corner of `z`.","title":"Wall edges below a site"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-ray-count","kind":"node","lean":"StatMech.Lattice.jec_rayCount","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-ray-count","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The number of ray edges of `z` traversed by the walk `w`.","title":"Ray-crossing count of a walk"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-wall-count","kind":"node","lean":"StatMech.Lattice.jec_wallCount","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-wall-count","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The number of wall edges of `z` traversed by the walk `w`.","title":"Wall-crossing count of a walk"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-left-region","kind":"node","lean":"StatMech.Lattice.jec_leftRegion","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-left-region","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The inside of a closed walk `Vc` by the even\u2013odd rule: sites whose ray count with respect to `Vc` is odd.","title":"Inside of a closed walk by the even-odd rule"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-vseg-up","kind":"node","lean":"StatMech.Lattice.jec_vsegUp","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-vseg-up","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The vertical lattice path from `(col, r)` up to `(col, r + k)`.","title":"Vertical lattice path upward"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-hseg-right","kind":"node","lean":"StatMech.Lattice.jec_hsegRight","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-hseg-right","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The horizontal lattice path from `(t, row)` right to `(t + k, row)`.","title":"Horizontal lattice path rightward"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-ret-path","kind":"node","lean":"StatMech.Lattice.jec_retPath","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-ret-path","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"A lattice path from `(b0, d)` back to `(a0, c)` running outside the rectangle `[\u03b1, \u03b2] \u00d7 [c, d]`: up, left to column `\u03b1 - 1`, down to row `c - 1`, right, and up.","title":"Return path around a rectangle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-below-set","kind":"node","lean":"StatMech.Lattice.jec_belowSet","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-below-set","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The half-plane of sites in rows below `y`.","title":"Half-plane below a row"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jec-crossing-arc-sides","kind":"node","lean":"StatMech.Lattice.jec_crossingArcSides","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jec-crossing-arc-sides","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"For every configuration, every overlap rectangle, and every `\u03b1 < \u03b2`, `c \u2264 d`, the arc-side property `StatMech.Universality.CrossingArcSides` holds: an open vertical crossing of the overlap separates the left side of `[\u03b1, \u03b2] \u00d7 [c, d]` from its right side.","title":"Open vertical crossings separate the sides of a rectangle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/ccs-orbit-is-loop","kind":"node","lean":"StatMech.Lattice.ccs_OrbitIsLoop","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#ccs-orbit-is-loop","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Some closed lattice walk has odd ray count exactly at the sites of `K`, so `K` is the inside of that walk.","title":"Set of sites is the inside of a closed lattice walk"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/egf-loop-flood-fill","kind":"node","lean":"StatMech.Lattice.egf_LoopFloodFill","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#egf-loop-flood-fill","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The inside and the outside of `Vc` are nonempty and each is connected in the lattice with the boundary edges of the inside removed. This is `StatMech.Lattice.jcs_JordanRegion (jec_leftRegion Vc)`.","title":"Inside and outside of a closed walk are connected"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/fcb-off-compl-graph","kind":"node","lean":"StatMech.Lattice.fcb_offComplGraph","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#fcb-off-compl-graph","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The subgraph of the square lattice induced on the sites not visited by `Vc`.","title":"Lattice off a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/fcb-off-compl-jordan-region","kind":"node","lean":"StatMech.Lattice.fcb_OffComplJordanRegion","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#fcb-off-compl-jordan-region","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Both sides of `Vc` contain sites off `Vc`, and any two sites off `Vc` on the same side are joined by a lattice path avoiding `Vc`.","title":"Jordan region property off a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jvp-is-cycles-veblen-step","kind":"node","lean":"StatMech.Lattice.jvp_isCycles_veblenStep","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jvp-is-cycles-veblen-step","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"One step of Veblen's theorem (O. Veblen, 1912) for graphs whose vertices all have degree zero or two: a nonempty such graph has a subgraph of the same kind with fewer edges and face count one less (obtained by deleting a cycle).","title":"Veblen's theorem step for unions of cycles"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkr-line-rel","kind":"node","lean":"StatMech.Lattice.jkr_lineRel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkr-line-rel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Both sites lie on the horizontal axis and the pair is not `{(0, 0), (1, 0)}`.","title":"Adjacency relation of the cut horizontal axis"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkr-line-h","kind":"node","lean":"StatMech.Lattice.jkr_lineH","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkr-line-h","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The horizontal axis of `\u2124\u00b2` with the edge from `(0, 0)` to `(1, 0)` removed.","title":"Horizontal axis with one edge removed"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkr-finite-keep-reconnect-residue","kind":"node","lean":"StatMech.Lattice.jkr_FiniteKeepReconnectResidue","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkr-finite-keep-reconnect-residue","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"For every graph `H` of `\u2124\u00b2` with finitely many edges and every lattice edge `pq` outside `H` with dual edge `f0g0`: if `p` and `q` are disconnected in `H`, then `f0` and `g0` are connected in the face-region graph of `H` with `f0g0` deleted.","title":"Keep-reconnect residue for finite graphs"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/region-graph","kind":"node","lean":"StatMech.Lattice.regionGraph","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#region-graph","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The square lattice with the edge `crossEdge e` removed for every edge `e` of `G`. Since `crossEdge` is a global rotation, this is not the faithful dual complement; see `StatMech.Lattice.whb_faceRegion` for the version using `sharedPrimalEdge`.","title":"Rotated region graph of a lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/planar-z2-subgraph","kind":"node","lean":"StatMech.Lattice.PlanarZ2Subgraph","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#planar-z2-subgraph","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"A finite graph together with an injective map into `\u2124\u00b2` that sends edges to lattice edges.","title":"Planar lattice subgraph of the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/image-graph","kind":"node","lean":"StatMech.Lattice.imageGraph","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#image-graph","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The image of a `PlanarZ2Subgraph` as a subgraph of the square lattice.","title":"Image of a planar lattice subgraph in the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/geometric-region-count","kind":"node","lean":"StatMech.Lattice.geometricRegionCount","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#geometric-region-count","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The number of finite connected components of the region graph of `P`.","title":"Geometric count of bounded regions"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/exterior-to-region-hom","kind":"node","lean":"StatMech.Lattice.exteriorToRegionHom","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#exterior-to-region-hom","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Outside a box containing the image of `P`, the lattice embeds in the region graph.","title":"Box exterior embeds in the region graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-is-even","kind":"node","lean":"StatMech.Lattice.jcx_IsEven","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-is-even","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Every vertex of `K` has even degree.","title":"Even-degree condition on a subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-is-even-peelable","kind":"node","lean":"StatMech.Lattice.jcx_IsEvenPeelable","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-is-even-peelable","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"`K` reduces to the empty graph by repeatedly deleting a cycle through a base vertex `v` such that, after the deletion, the other vertices of the cycle are isolated.","title":"Even-peelable graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-cross-emb","kind":"node","lean":"StatMech.Lattice.jcx_cross_emb","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-cross-emb","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The vertices of two unit squares `[0, 1]\u00b2` and `[1, 2]\u00b2` sharing the corner `(1, 1)`.","title":"Vertices of two unit squares sharing a corner"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-cross-rel","kind":"node","lean":"StatMech.Lattice.jcx_cross_rel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-cross-rel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Adjacency of the two squares sharing a corner, as a Boolean relation.","title":"Adjacency relation of two corner-sharing squares"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-cross-g","kind":"node","lean":"StatMech.Lattice.jcx_cross_G","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-cross-g","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Two 4-cycles sharing the vertex `2`, on `Fin 7`.","title":"Two 4-cycles sharing a vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-cross-p","kind":"node","lean":"StatMech.Lattice.jcx_cross_P","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-cross-p","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Two unit squares sharing a corner as a `PlanarZ2Subgraph`.","title":"Two corner-sharing unit squares as a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-cross-cyc-b","kind":"node","lean":"StatMech.Lattice.jcx_cross_cycB","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-cross-cyc-b","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The 4-cycle of the upper square, based at the shared vertex.","title":"Upper 4-cycle based at the shared vertex"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-cross-a-rel","kind":"node","lean":"StatMech.Lattice.jcx_cross_A_rel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-cross-a-rel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Adjacency of the lower 4-cycle, as a Boolean relation.","title":"Adjacency relation of the lower 4-cycle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-cross-a","kind":"node","lean":"StatMech.Lattice.jcx_cross_A","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-cross-a","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The lower 4-cycle, which remains after deleting the upper one.","title":"Lower 4-cycle left after deleting the upper one"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jei-push-graph","kind":"node","lean":"StatMech.Lattice.jei_pushGraph","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jei-push-graph","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The image in `\u2124\u00b2` of a subgraph `K` of a `PlanarZ2Subgraph` `P`.","title":"Pushforward of a subgraph into the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-even-count-residue","kind":"node","lean":"StatMech.Lattice.jcx_EvenCountResidue","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-even-count-residue","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Every subgraph `K` of `P` whose image has all degrees even has exactly `faceCount K` face-region components.","title":"Even-degree subgraphs have as many regions as their face count"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcx-even-peel","kind":"node","lean":"StatMech.Lattice.jcx_EvenPeel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcx-even-peel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Every nonempty subgraph `K` of `P` whose image has all degrees even contains a subgraph of the same kind with fewer edges, face count one less, and one fewer face-region component.","title":"Peeling a cycle from an even-degree subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-lattice-minus-edges","kind":"node","lean":"StatMech.Lattice.jbc_latticeMinusEdges","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-lattice-minus-edges","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The square lattice with the edges of `H` deleted.","title":"Square lattice minus the edges of a graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-rot-iso","kind":"node","lean":"StatMech.Lattice.jbc_rotIso","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-rot-iso","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The quarter-turn rotation is an isomorphism from the lattice minus the edges of `H` onto the region graph of `H`.","title":"Quarter-turn isomorphism onto the region graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-bridge-direction","kind":"node","lean":"StatMech.Lattice.jbc_BridgeDirection","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-bridge-direction","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"If `x` and `y` are not connected in `K`, they are connected in the lattice minus the image of `K \u2294 edge x y`.","title":"Disconnected sites are joined in the lattice minus a bridged graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-csq","kind":"node","lean":"StatMech.Lattice.jbc_csq","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-csq","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The four corners `(0, 0), (1, 0), (1, 1), (0, 1)` of the unit square, in cyclic order.","title":"Corners of the unit square in cyclic order"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-cyc4-rel","kind":"node","lean":"StatMech.Lattice.jbc_cyc4Rel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-cyc4-rel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Adjacency of the 4-cycle `0 \u2013 1 \u2013 2 \u2013 3 \u2013 0`, as a Boolean relation.","title":"Adjacency relation of the 4-cycle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-cyc4","kind":"node","lean":"StatMech.Lattice.jbc_cyc4","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-cyc4","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The 4-cycle on `Fin 4`.","title":"The 4-cycle graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-v0","kind":"node","lean":"StatMech.Lattice.jbc_v0","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-v0","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The vertex `0` of the 4-cycle.","title":"Vertex 0 of the 4-cycle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-v1","kind":"node","lean":"StatMech.Lattice.jbc_v1","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-v1","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The vertex `1` of the 4-cycle.","title":"Vertex 1 of the 4-cycle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jbc-pce","kind":"node","lean":"StatMech.Lattice.jbc_Pce","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jbc-pce","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The unit square: the 4-cycle embedded on the corners of `[0, 1]\u00b2`.","title":"Unit square as a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jgc-is-cactus","kind":"node","lean":"StatMech.Lattice.jgc_IsCactus","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jgc-is-cactus","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"`K` reduces to the empty graph by repeatedly deleting a leaf edge or, when `K` is a union of cycles, one of its cycles.","title":"Cactus graph, defined by peeling leaves and cycles"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jgc-pend-emb","kind":"node","lean":"StatMech.Lattice.jgc_pend_emb","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jgc-pend-emb","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The corners of the unit square together with the site `(-1, 0)`.","title":"Unit square corners with a pendant site"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jgc-pend-rel","kind":"node","lean":"StatMech.Lattice.jgc_pend_rel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jgc-pend-rel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Adjacency of the 4-cycle on `0, 1, 2, 3` with the pendant edge `{0, 4}`, as a Boolean relation.","title":"Adjacency relation of a 4-cycle with a pendant edge"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jgc-pend-g","kind":"node","lean":"StatMech.Lattice.jgc_pend_G","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jgc-pend-g","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The 4-cycle with a pendant edge, on `Fin 5`.","title":"4-cycle with a pendant edge"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jgc-pend-p","kind":"node","lean":"StatMech.Lattice.jgc_pend_P","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jgc-pend-p","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The unit square with a pendant edge as a `PlanarZ2Subgraph`.","title":"Unit square with a pendant edge as a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jgc-pend-core","kind":"node","lean":"StatMech.Lattice.jgc_pend_core","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jgc-pend-core","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The 4-cycle on `0, 1, 2, 3` inside `Fin 5`.","title":"4-cycle core of the pendant example"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jic-tc","kind":"node","lean":"StatMech.Lattice.jic_tc","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jic-tc","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The number of connected components of the face-region graph of the image of `K`.","title":"Number of face regions of a subgraph's image"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jic-two-sq-emb","kind":"node","lean":"StatMech.Lattice.jic_twoSq_emb","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jic-two-sq-emb","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The corners of the unit squares at `(0, 0)` and `(10, 0)`.","title":"Corners of two distant unit squares"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jic-two-sq-rel","kind":"node","lean":"StatMech.Lattice.jic_twoSq_rel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jic-two-sq-rel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Adjacency of two disjoint 4-cycles on `Fin 8`, as a Boolean relation.","title":"Adjacency relation of two disjoint 4-cycles"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jic-two-sq-g","kind":"node","lean":"StatMech.Lattice.jic_twoSq_G","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jic-two-sq-g","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Two disjoint 4-cycles on `Fin 8`.","title":"Two disjoint 4-cycles"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jic-two-sq-p","kind":"node","lean":"StatMech.Lattice.jic_twoSq_P","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jic-two-sq-p","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Two disjoint unit squares as a `PlanarZ2Subgraph`.","title":"Two disjoint unit squares as a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/outer-box-radius","kind":"node","lean":"StatMech.Lattice.outerBoxRadius","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#outer-box-radius","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"A radius `R` such that the image of `P` lies in `box 2 R`.","title":"Radius of a box containing a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/outer-region","kind":"node","lean":"StatMech.Lattice.outerRegion","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#outer-region","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The outer region: the component of the region graph containing the far site `beacon 2 R`.","title":"Outer region of the rotated region graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/bounded-region-equiv-ne-outer","kind":"node","lean":"StatMech.Lattice.boundedRegionEquivNeOuter","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#bounded-region-equiv-ne-outer","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The finite components of the region graph are exactly the components other than the outer region.","title":"Bounded regions are the components other than the outer one"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jfc-whb-sync","kind":"node","lean":"StatMech.Lattice.jfc_WhbSync","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jfc-whb-sync","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Whitney's cut\u2013cycle duality for `P`: for every subgraph `K` of `P` and edge `xy` of `P` not in `K`, with `f0g0` the dual edge across it, `x` and `y` are connected in `K` iff `f0` and `g0` are disconnected in the face-region graph of `K` with `f0g0` deleted.","title":"Whitney cut-cycle duality for a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jfc-whb-outer-region","kind":"node","lean":"StatMech.Lattice.jfc_whb_outerRegion","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jfc-whb-outer-region","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The outer region: the component of the face-region graph of `P` containing a face far from `P`.","title":"Outer face region of a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jfc-push-graph-locally-finite","kind":"node","lean":"StatMech.Lattice.jfc_pushGraph_locallyFinite","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jfc-push-graph-locally-finite","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The image of a subgraph of a `PlanarZ2Subgraph` is locally finite.","title":"Pushed-forward subgraphs are locally finite"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jfc-even-degree-threading","kind":"node","lean":"StatMech.Lattice.jfc_EvenDegreeThreading","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jfc-even-degree-threading","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Every `PlanarZ2Subgraph` whose image has all degrees even satisfies `jfc_WhbSync`.","title":"Whitney duality for even-degree planar lattice subgraphs"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jfc-sq-interior","kind":"node","lean":"StatMech.Lattice.jfc_sq_interior","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jfc-sq-interior","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The component of the face `(0, 0)` in the face-region graph of the unit square.","title":"Interior region of the unit square"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jvp-veblen-step","kind":"node","lean":"StatMech.Lattice.jvp_VeblenStep","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jvp-veblen-step","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Every nonempty subgraph `K` of `P` with even-degree image contains an even-degree subgraph with fewer edges and face count one less.","title":"Veblen step property for even-degree subgraphs"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jvp-region-step","kind":"node","lean":"StatMech.Lattice.jvp_RegionStep","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jvp-region-step","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Whenever `K' \u2264 K` are even-degree subgraphs of `P` with fewer edges and face count one less, the face-region graph of `K` has one more component than that of `K'`.","title":"Region step property for even-degree subgraphs"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkd-gr9emb","kind":"node","lean":"StatMech.Lattice.jkd_gr9emb","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkd-gr9emb","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The nine sites of `[0, 2]\u00b2`, indexed row by row.","title":"Nine sites of a 3 by 3 grid"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkd-gr9rel","kind":"node","lean":"StatMech.Lattice.jkd_gr9rel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkd-gr9rel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Adjacency of the `3 \u00d7 3` grid on `Fin 9`, as a Boolean relation.","title":"Adjacency relation of the 3 by 3 grid"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkd-gr9-g","kind":"node","lean":"StatMech.Lattice.jkd_gr9G","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkd-gr9-g","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The `3 \u00d7 3` grid graph on `Fin 9`.","title":"The 3 by 3 grid graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkd-ring-p","kind":"node","lean":"StatMech.Lattice.jkd_ringP","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkd-ring-p","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The `3 \u00d7 3` grid as a `PlanarZ2Subgraph`.","title":"3 by 3 grid as a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkd-r0","kind":"node","lean":"StatMech.Lattice.jkd_r0","parent":29073,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid vertex "],["math","\\((0, 0)\\)"]],"source":"Grid vertex $(0, 0)$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkd-r0","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The vertex `(0, 0)` of the grid.","title":"Grid vertex $(0, 0)$"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jkd-r1","kind":"node","lean":"StatMech.Lattice.jkd_r1","parent":29073,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grid vertex "],["math","\\((1, 0)\\)"]],"source":"Grid vertex $(1, 0)$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jkd-r1","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The vertex `(1, 0)` of the grid.","title":"Grid vertex $(1, 0)$"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jsf-card-components-sup-edge-merge","kind":"node","lean":"StatMech.Lattice.jsf_card_components_sup_edge_merge","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jsf-card-components-sup-edge-merge","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Adding an edge between two components merges them. This assumes only finitely many components; see also `StatMech.Lattice.card_components_sup_edge_of_not_reachable` for finite vertex types.","title":"Adding an edge between two components merges them"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jed-cut-set","kind":"node","lean":"StatMech.Lattice.jed_cutSet","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jed-cut-set","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The primal edges `sharedPrimalEdge a b` between adjacent faces `a`, `b` on which `T` takes different values.","title":"Cut set of a face coloring"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jed-primal-graph","kind":"node","lean":"StatMech.Lattice.jed_primalGraph","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jed-primal-graph","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The graph on `\u2124\u00b2` whose edges are the lattice edges in `E`.","title":"Lattice graph spanned by an edge set"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jed-faithful-discrete-jordan","kind":"node","lean":"StatMech.Lattice.jed_faithfulDiscreteJordan","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jed-faithful-discrete-jordan","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"For a `PlanarZ2Subgraph` `P`, the finite components of the face-region graph `whb_faceRegion` of its image (faces of `\u2124\u00b2`, two adjacent faces joined unless the primal edge between them lies in `P`) are equinumerous with `Fin (nullity P.G)`, so the number of bounded faces is `|E| - |V| + c`. This is a count: it does not identify each bounded face with the interior of a particular cycle of `P`. References: H.\u2026","title":"Bounded faces of a finite lattice graph are counted by its cycle rank"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jed-box-edges","kind":"node","lean":"StatMech.Lattice.jed_boxEdges","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jed-box-edges","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The twelve lattice edges of the boundary of the rectangle `[-1, 2] \u00d7 [-2, 1]`.","title":"Twelve boundary edges of a 3 by 3 rectangle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jed-box-h","kind":"node","lean":"StatMech.Lattice.jed_boxH","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jed-box-h","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The boundary of the rectangle `[-1, 2] \u00d7 [-2, 1]` as a graph.","title":"Boundary of a 3 by 3 rectangle as a graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-bounded-region","kind":"node","lean":"StatMech.Lattice.jcb_BoundedRegion","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-bounded-region","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The bounded (finite) connected components of the face-region graph of `P`.","title":"Bounded regions of a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-bounded-region-equiv","kind":"node","lean":"StatMech.Lattice.jcb_boundedRegionEquiv","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-bounded-region-equiv","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The bijection between bounded regions of `P` and `Fin (nullity P.G)` given by the discrete Jordan\u2013Euler theorem.","title":"Bounded regions indexed by the cycle rank"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-unique-bounded-region-of-nullity-one","kind":"node","lean":"StatMech.Lattice.jcb_uniqueBoundedRegion_of_nullity_one","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-unique-bounded-region-of-nullity-one","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"A `PlanarZ2Subgraph` of cycle rank one has exactly one bounded region.","title":"Cycle rank one gives a unique bounded region"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-unique-bounded-region-of-connected-c","kind":"node","lean":"StatMech.Lattice.jcb_uniqueBoundedRegion_of_connected_cycle","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-unique-bounded-region-of-connected-c","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Jordan curve theorem for lattice cycles: a connected `PlanarZ2Subgraph` with as many edges as vertices has exactly one bounded region. Reference: C. Jordan, *Cours d'analyse* (1887); for the discrete form, C. Thomassen, *The Jordan\u2013Sch\u00f6nflies theorem and the classification of surfaces* (1992).","title":"Jordan curve theorem"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-lemb","kind":"node","lean":"StatMech.Lattice.jcb_Lemb","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-lemb","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The eight lattice points around the L-shaped tromino, in cyclic order.","title":"Lattice points around the L-shaped tromino"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-lcyc-rel","kind":"node","lean":"StatMech.Lattice.jcb_LcycRel","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-lcyc-rel","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Adjacency of the 8-cycle on `Fin 8`, as a Boolean relation.","title":"Adjacency relation of the 8-cycle"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-lcyc","kind":"node","lean":"StatMech.Lattice.jcb_Lcyc","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-lcyc","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The 8-cycle on `Fin 8`.","title":"The 8-cycle graph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-lpce","kind":"node","lean":"StatMech.Lattice.jcb_LPce","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-lpce","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The 8-cycle around the L-shaped tromino as a `PlanarZ2Subgraph`.","title":"8-cycle around the L-tromino as a planar lattice subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcb-diag-k","kind":"node","lean":"StatMech.Lattice.jcb_diagK","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcb-diag-k","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The two-site set `{(0, 0), (1, 1)}`, which touches diagonally.","title":"Two diagonally touching sites"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/rsf-facial-orbit","kind":"node","lean":"StatMech.Lattice.rsf_facialOrbit","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#rsf-facial-orbit","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The orbit of a boundary dart of `C`, as a connected component of the successor graph on boundary darts.","title":"Facial orbit of a boundary dart"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/rsf-component-darts","kind":"node","lean":"StatMech.Lattice.rsf_componentDarts","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#rsf-component-darts","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The set of boundary darts of a connected component of the sites off the support of `Vc`.","title":"Boundary darts of a component off a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/rsf-facial-orbits-cover-components","kind":"node","lean":"StatMech.Lattice.rsf_FacialOrbitsCoverComponents","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#rsf-facial-orbits-cover-components","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"The boundary dart sets of the components of the complement of `Vc` are exactly the members of `\ud835\udcd5`.","title":"Facial orbits cover the components of the complement"},{"id":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/jcu-has-separating-side","kind":"node","lean":"StatMech.Lattice.jcu_HasSeparatingSide","parent":29073,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat/index.html#jcu-has-separating-side","scope":"GraphTheory/planar-graphs-and-minors/discrete-jordan-curve-theorem-for-the-square-lat","status":"fully_proved","summary":"Some set `S` contains `inside`, misses `outside`, and every lattice edge between sites of `D` that leaves `S` has an endpoint in `B`.","title":"Separating side between inside and outside sites"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons","kind":"scope","navigation":"module","parent":28787,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons","status":"fully_proved","title":"Hopf's Umlaufsatz for lattice polygons"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","title":"Unit lattice step in a given direction to Distinct north-then-west corners sit at distinct vertices"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/step-of","kind":"node","lean":"StatMech.Onsager.BaseCase.stepOf","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#step-of","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The unit step of `\u2124\u00b2` in direction `w`: `0 \u21a6 (1, 0)` (east), `1 \u21a6 (0, 1)` (north), `2 \u21a6 (-1, 0)` (west), `3 \u21a6 (0, -1)` (south).","title":"Unit lattice step in a given direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/pos","kind":"node","lean":"StatMech.Onsager.BaseCase.pos","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#pos","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The `k`-th vertex of the lattice walk with step directions `d`: the sum of the steps `stepOf (d i)` for `i < k`, so that `pos d 0 = 0`.","title":"Vertices of a lattice walk from its step directions"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/cc","kind":"node","lean":"StatMech.Onsager.BaseCase.cc","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#cc","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The number of counterclockwise (left) quarter turns of the cyclic direction sequence `d`, i.e. of indices `i` with `d (i + 1) - d i = 1`.","title":"Number of left turns of a direction sequence"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/sum-turns-zero","kind":"node","lean":"StatMech.Onsager.BaseCase.sum_turns_zero","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#sum-turns-zero","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The turns of a cyclic direction sequence sum to zero in `Fin 4`.","title":"Turns of a cyclic direction sequence sum to zero"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/four-dvd-cc","kind":"node","lean":"StatMech.Onsager.BaseCase.four_dvd_cc","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#four-dvd-cc","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"If every turn is straight or left, the number of left turns is divisible by `4`.","title":"Without right turns, the number of left turns is divisible by four"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/cc-pos","kind":"node","lean":"StatMech.Onsager.BaseCase.cc_pos","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#cc-pos","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"A closed walk with only straight steps and left turns turns at least once.","title":"A closed walk without right turns turns at least once"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/four-le-cc","kind":"node","lean":"StatMech.Onsager.BaseCase.four_le_cc","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#four-le-cc","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"A closed walk with only straight steps and left turns has at least four left turns.","title":"A closed walk without right turns has at least four left turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/base-cc-eq-four","kind":"node","lean":"StatMech.Onsager.BaseCase.base_cc_eq_four","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#base-cc-eq-four","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"A closed walk with only straight steps and left turns has exactly four left turns, provided simplicity rules out eight or more.","title":"Exactly four left turns when simplicity rules out eight"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/ons-dart-rev","kind":"node","lean":"StatMech.Onsager.ons_dartRev","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#ons-dart-rev","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The reverse of a dart: it starts at the head of `d` and points in the opposite direction.","title":"Reverse of a dart"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/ons-turn-z","kind":"node","lean":"StatMech.Onsager.ons_turnZ","parent":29182,"reading_title":{"format":"inline-parts-v1","parts":[["text","Turn between two directions as an element of "],["math","\\(\\mathbb{Z}/4\\)"]],"source":"Turn between two directions as an element of $\\mathbb{Z}/4$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#ons-turn-z","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The turn from direction `\u03bc` to `\u03bd` as an element of `ZMod 4`.","title":"Turn between two directions as an element of $\\mathbb{Z}/4$"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/ons-turn-pow","kind":"node","lean":"StatMech.Onsager.ons_turnPow","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#ons-turn-pow","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The integer turn exponent from `\u03bc` to `\u03bd`: `1` for a left turn, `-1` for a right turn, `0` for going straight (and, by convention, for a U-turn).","title":"Integer turn exponent between two directions"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/pos-zero","kind":"node","lean":"StatMech.Onsager.NoDoubleWind.pos_zero","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#pos-zero","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The walk starts at the origin.","title":"Closed lattice walk starts at the origin"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/pos-succ","kind":"node","lean":"StatMech.Onsager.NoDoubleWind.pos_succ","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#pos-succ","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"Vertex recursion of a closed walk: the next vertex is the current one plus the current step (indices taken cyclically).","title":"Forward vertex recursion of a closed lattice walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/pos-pred","kind":"node","lean":"StatMech.Onsager.NoDoubleWind.pos_pred","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#pos-pred","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"Vertex recursion of a closed walk read backwards from vertex `M`.","title":"Backward vertex recursion of a closed lattice walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/step-of-coords","kind":"node","lean":"StatMech.Onsager.NoDoubleWind.stepOf_coords","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#step-of-coords","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"The four lattice steps, case by case.","title":"Coordinates of the four lattice steps"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/lexmax-corner","kind":"node","lean":"StatMech.Onsager.NoDoubleWind.lexmax_corner","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#lexmax-corner","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"A closed walk with only straight steps and left turns has a north step immediately followed by a west step (at its lexicographically largest vertex).","title":"North step followed by west step at the top corner of a walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/ymax-nw-corner","kind":"node","lean":"StatMech.Onsager.UmlaufsatzStrip.ymax_NW_corner","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#ymax-nw-corner","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"A closed walk without right turns has a north-then-west corner at the rightmost of its highest vertices.","title":"North-then-west corner at the highest vertices"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/ymin-se-corner","kind":"node","lean":"StatMech.Onsager.UmlaufsatzStrip.ymin_SE_corner","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#ymin-se-corner","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"A closed walk without right turns has a south-then-east corner at one of its lowest vertices.","title":"South-then-east corner at the lowest vertices"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/nw-corner-neighbours","kind":"node","lean":"StatMech.Onsager.UmlaufsatzStrip.nw_corner_neighbours","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#nw-corner-neighbours","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"At a north-then-west corner `M`, the previous vertex is just below and the next vertex just to the left.","title":"Previous and next vertices of a north-then-west corner"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/nw-corners-distinct-pos","kind":"node","lean":"StatMech.Onsager.UmlaufsatzStrip.nw_corners_distinct_pos","parent":29182,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01/index.html#nw-corners-distinct-pos","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-01","status":"fully_proved","summary":"In a simple walk, distinct north-then-west corners sit at distinct vertices.","title":"Distinct north-then-west corners sit at distinct vertices"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","title":"Contiguous top row of a walk to A closed walk crosses each horizontal line an even number of times"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/top-row-contiguous","kind":"node","lean":"StatMech.Onsager.UmlaufsatzStrip.TopRowContiguous","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#top-row-contiguous","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The vertices of the walk at its maximal height cover every lattice point between any two of them, i.e. the top row of the walk is a contiguous horizontal segment.","title":"Contiguous top row of a walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/xco-succ","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.xco_succ","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#xco-succ","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The `x`-coordinate recursion of a closed walk. See also `StatMech.Onsager.WalkCrossing.fst_step`.","title":"Recursion for the x-coordinate of a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/yco-succ","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.yco_succ","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#yco-succ","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The `y`-coordinate recursion of a closed walk. See also `StatMech.Onsager.WalkCrossing.snd_step`.","title":"Recursion for the y-coordinate of a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/maxx-no-e-out","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.maxx_no_E_out","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#maxx-no-e-out","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"No east step leaves a vertex of maximal `x`-coordinate.","title":"No east step leaves a rightmost vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/maxx-no-w-in","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.maxx_no_W_in","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#maxx-no-w-in","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"No west step arrives at a vertex of maximal `x`-coordinate.","title":"No west step arrives at a rightmost vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/miny-no-s-out","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.miny_no_S_out","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#miny-no-s-out","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"No south step leaves a vertex of minimal `y`-coordinate.","title":"No south step leaves a lowest vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/miny-no-n-in","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.miny_no_N_in","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#miny-no-n-in","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"No north step arrives at a vertex of minimal `y`-coordinate.","title":"No north step arrives at a lowest vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/tcount","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.tcount","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#tcount","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The number of left turns of `d` from direction `c` to direction `c + 1`.","title":"Number of left turns out of a given direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/scount","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.scount","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#scount","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The number of straight continuations of `d` in direction `c`.","title":"Number of straight continuations in a given direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/tcount-shift","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.tcount_shift","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#tcount-shift","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"In a direction sequence without right turns, every direction is left by the same number of left turns.","title":"Every direction is left by equally many left turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/cc-eq-four-mul-tcount","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.cc_eq_four_mul_tcount","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#cc-eq-four-mul-tcount","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"In a direction sequence without right turns, the number of left turns is four times the number of north-to-west turns.","title":"Left turns are four times the north-to-west turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/exists-two-top-corners","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.exists_two_top_corners","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#exists-two-top-corners","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"Eight or more left turns, without right turns, give two distinct north-then-west corners.","title":"Eight left turns give two north-then-west corners"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/cc-lt-eight-of-unique-top-corner","kind":"node","lean":"StatMech.Onsager.NoDoubleWind2.cc_lt_eight_of_uniqueTopCorner","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#cc-lt-eight-of-unique-top-corner","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"If the north-then-west corner is unique, a direction sequence without right turns has fewer than eight left turns.","title":"A unique top corner forces fewer than eight left turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/is-vert-edge","kind":"node","lean":"StatMech.Onsager.WalkCrossing.isVertEdge","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#is-vert-edge","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The direction `w` is vertical (north or south).","title":"Vertical lattice direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/is-horiz-edge","kind":"node","lean":"StatMech.Onsager.WalkCrossing.isHorizEdge","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#is-horiz-edge","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The direction `w` is horizontal (east or west).","title":"Horizontal lattice direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/snd-step","kind":"node","lean":"StatMech.Onsager.WalkCrossing.snd_step","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#snd-step","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The `y`-coordinate recursion of a closed walk. See also `StatMech.Onsager.NoDoubleWind2.yco_succ`.","title":"Recursion for the y-coordinate of a walk step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/fst-step","kind":"node","lean":"StatMech.Onsager.WalkCrossing.fst_step","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#fst-step","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The `x`-coordinate recursion of a closed walk. See also `StatMech.Onsager.NoDoubleWind2.xco_succ`.","title":"Recursion for the x-coordinate of a walk step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/full-line-cross","kind":"node","lean":"StatMech.Onsager.WalkCrossing.fullLineCross","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#full-line-cross","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"The number of vertical steps of the walk crossing the horizontal line `y = m + 1/2`, i.e. vertical steps between heights `m` and `m + 1`.","title":"Number of crossings of a horizontal line"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/up-down-card-eq","kind":"node","lean":"StatMech.Onsager.WalkCrossing.upDown_card_eq","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#up-down-card-eq","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"A closed walk crosses the line `y = m + 1/2` upwards as often as downwards.","title":"A closed walk crosses a horizontal line equally often up and down"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/full-line-cross-even","kind":"node","lean":"StatMech.Onsager.WalkCrossing.fullLineCross_even","parent":29203,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02/index.html#full-line-cross-even","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-02","status":"fully_proved","summary":"A closed walk crosses each horizontal line `y = m + 1/2` an even number of times.","title":"A closed walk crosses each horizontal line an even number of times"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","title":"A lattice point lies on a walk step to Four left turns, given that two corners force a self-intersection"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/on-edge","kind":"node","lean":"StatMech.Onsager.WalkCrossing.onEdge","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#on-edge","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"The lattice point `p` lies on the closed segment traversed by step `k`.","title":"A lattice point lies on a walk step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/edges-cross-not-injective","kind":"node","lean":"StatMech.Onsager.WalkCrossing.edges_cross_not_injective","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#edges-cross-not-injective","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"If a horizontal and a vertical step that share no index endpoint meet at a point, the walk visits some vertex twice.","title":"Crossing steps force a repeated vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/ray-cross","kind":"node","lean":"StatMech.Onsager.WalkCrossing.rayCross","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#ray-cross","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"The number of vertical steps crossing the line `y = m + 1/2` strictly to the right of `x = qx`.","title":"Number of crossings of a horizontal ray"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/up-cross","kind":"node","lean":"StatMech.Onsager.JordanParity.upCross","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#up-cross","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"The number of horizontal steps of the walk on the column `[a, a + 1]` strictly above the cell `(a, b)`: the crossings of the upward vertical ray from the center of that cell.","title":"Upward ray crossings above a cell"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/ray-parity","kind":"node","lean":"StatMech.Onsager.JordanParity.rayParity","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#ray-parity","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"The parity of the number of crossings of the upward ray from the center of cell `(a, b)` with the walk; cells with parity `1` are inside the walk.","title":"Ray-crossing parity of a lattice cell"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/up-cross-vstep","kind":"node","lean":"StatMech.Onsager.JordanParity.upCross_vstep","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#up-cross-vstep","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"The upward crossing count is the same for cells `(a, b)` and `(a, b + 1)` when no horizontal step lies between them.","title":"Upward crossing count is unchanged across a vertical move"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/ray-parity-vstep","kind":"node","lean":"StatMech.Onsager.JordanParity.rayParity_vstep","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#ray-parity-vstep","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"The ray parity is the same for cells `(a, b)` and `(a, b + 1)` when no horizontal step lies between them.","title":"Ray parity is unchanged across a vertical move"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/up-cross-top-outside","kind":"node","lean":"StatMech.Onsager.JordanParity.upCross_topOutside","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#up-cross-top-outside","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"Cells at or above the top height of the walk have no upward crossings.","title":"No upward crossings for cells above a walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/ray-parity-top-outside","kind":"node","lean":"StatMech.Onsager.JordanParity.rayParity_topOutside","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#ray-parity-top-outside","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"Cells at or above the top height of the walk are outside.","title":"Cells above a walk are outside"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/up-cross-hstep","kind":"node","lean":"StatMech.Onsager.JordanParity.upCross_hstep","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#up-cross-hstep","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"When no vertical step separates the cells `(a, b)` and `(a + 1, b)`, their upward crossing counts add up to an even number.","title":"Upward crossing counts of horizontal neighbours sum to an even number"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/ray-parity-hstep","kind":"node","lean":"StatMech.Onsager.JordanParity.rayParity_hstep","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#ray-parity-hstep","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"The ray parity is the same for cells `(a, b)` and `(a + 1, b)` when no vertical step lies between them.","title":"Ray parity is unchanged across a horizontal move"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/inside-climb-x","kind":"node","lean":"StatMech.Onsager.InsideClimb.inside_climb_x","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#inside-climb-x","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"If cell `(a, b)` is inside and no vertical step separates it from `(a + 1, b)`, then `(a + 1, b)` is inside.","title":"Moving horizontally inside a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/inside-climb-y","kind":"node","lean":"StatMech.Onsager.InsideClimb.inside_climb_y","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#inside-climb-y","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"If cell `(a, b)` is inside and no horizontal step separates it from `(a, b + 1)`, then `(a, b + 1)` is inside.","title":"Moving vertically inside a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/nw-corner-coords","kind":"node","lean":"StatMech.Onsager.InsideClimb.nw_corner_coords","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#nw-corner-coords","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"The vertices before and after a north-then-west corner.","title":"Coordinates of the vertices around a north-then-west corner"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/nw-corner-not-climbable-x","kind":"node","lean":"StatMech.Onsager.InsideClimb.nw_corner_not_climbable_x","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#nw-corner-not-climbable-x","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"At a north-then-west corner, the north step separates the cell below-left of the corner from its right neighbour.","title":"The north step at a corner blocks horizontal movement"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/nw-corner-not-climbable-y","kind":"node","lean":"StatMech.Onsager.InsideClimb.nw_corner_not_climbable_y","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#nw-corner-not-climbable-y","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"At a north-then-west corner, the west step separates the cell below-left of the corner from its top neighbour.","title":"The west step at a corner blocks vertical movement"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/lexmax-unique","kind":"node","lean":"StatMech.Onsager.InsideClimb.lexmax_unique","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#lexmax-unique","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"In a simple walk the lexicographically largest vertex is attained at a unique index.","title":"Uniqueness of the lexicographically largest vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/exists-lexmax-nw-corner","kind":"node","lean":"StatMech.Onsager.InsideClimb.exists_lexmax_nw_corner","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#exists-lexmax-nw-corner","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"A closed walk without right turns has a north-then-west corner.","title":"Existence of a north-then-west corner"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/two-nw-corners-not-injective","kind":"node","lean":"StatMech.Onsager.InsideClimb.two_nw_corners_not_injective","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#two-nw-corners-not-injective","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"Any two distinct north-then-west corners of `d` force `pos d` to be non-injective.","title":"Two north-then-west corners force a self-intersection"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/cc-eq-four-of-self-avoidance-wall","kind":"node","lean":"StatMech.Onsager.InsideClimb.cc_eq_four_of_selfAvoidanceWall","parent":29224,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03/index.html#cc-eq-four-of-self-avoidance-wall","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-03","status":"fully_proved","summary":"A simple closed walk without right turns has four left turns, assuming that two distinct north-then-west corners force a self-intersection.","title":"Four left turns, given that two corners force a self-intersection"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","kind":"scope","navigation":"transparent","parent":29181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit cell to the left of a lattice step to "],["math","\\(k + 2 \\ne k\\)"],["text"," in "],["math","\\(\\mathrm{Fin}\\, n\\)"],["text"," for "],["math","\\(n \\ge 3\\)"]],"source":"Unit cell to the left of a lattice step to $k + 2 \\ne k$ in $\\mathrm{Fin}\\, n$ for $n \\ge 3$"},"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","title":"Unit cell to the left of a lattice step to $k + 2 \\ne k$ in $\\mathrm{Fin}\\, n$ for $n \\ge 3$"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/left-cell","kind":"node","lean":"StatMech.Onsager.Orientation.leftCell","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#left-cell","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"The unit cell to the left of the step in direction `w` starting at `p` (cells are named by their lower-left corners).","title":"Unit cell to the left of a lattice step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/left-cell-eq-of-left-turn","kind":"node","lean":"StatMech.Onsager.Orientation.leftCell_eq_of_leftTurn","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#left-cell-eq-of-left-turn","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"At a left turn, the steps before and after the turn have the same left cell.","title":"Steps around a left turn share their left cell"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/no-v-edge-of-not-vertex","kind":"node","lean":"StatMech.Onsager.Orientation.no_vEdge_of_not_vertex","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#no-v-edge-of-not-vertex","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"A vertical step through the point `(c, h)` has `(c, h)` as a vertex; if `(c, h)` is not a walk vertex, no vertical step straddles height `h` on the line `x = c`.","title":"No vertical step through a non-vertex point"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/no-h-edge-of-not-vertex","kind":"node","lean":"StatMech.Onsager.Orientation.no_hEdge_of_not_vertex","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#no-h-edge-of-not-vertex","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"If `(c, h)` is not a walk vertex, no horizontal step starts at `x = c` on height `h`.","title":"No horizontal step starts at a non-vertex point"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/v-edge-lower-index","kind":"node","lean":"StatMech.Onsager.Orientation.vEdge_lower_index","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#v-edge-lower-index","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"In a simple walk, a vertical step on the segment above the vertex `pos d m` is step `m` or step `m - 1`.","title":"Index of a vertical step above a walk vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/h-edge-left-index","kind":"node","lean":"StatMech.Onsager.Orientation.hEdge_left_index","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#h-edge-left-index","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"In a simple walk, a horizontal step on the segment to the right of the vertex `pos d m` is step `m` or step `m - 1`.","title":"Index of a horizontal step right of a walk vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/v-edge-upper-index","kind":"node","lean":"StatMech.Onsager.Orientation.vEdge_upper_index","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#v-edge-upper-index","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"In a simple walk, a vertical step on the segment below the vertex `pos d m` is step `m` or step `m - 1`.","title":"Index of a vertical step below a walk vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/h-edge-right-index","kind":"node","lean":"StatMech.Onsager.Orientation.hEdge_right_index","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#h-edge-right-index","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"In a simple walk, a horizontal step on the segment to the left of the vertex `pos d m` is step `m` or step `m - 1`.","title":"Index of a horizontal step left of a walk vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/left-cell-parity-eq-of-straight","kind":"node","lean":"StatMech.Onsager.Orientation.leftCell_parity_eq_of_straight","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#left-cell-parity-eq-of-straight","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"Going straight, consecutive left cells have the same ray parity.","title":"Left cells keep their ray parity on a straight step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/left-parity","kind":"node","lean":"StatMech.Onsager.Orientation.leftParity","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#left-parity","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"The ray parity of the cell to the left of step `k`.","title":"Ray parity of the cell left of a step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/left-parity-step","kind":"node","lean":"StatMech.Onsager.Orientation.leftParity_step","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#left-parity-step","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"For a simple closed walk without right turns, consecutive steps have the same left parity.","title":"Consecutive steps have the same left parity"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/left-parity-const","kind":"node","lean":"StatMech.Onsager.Orientation.leftParity_const","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#left-parity-const","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"For a simple closed walk without right turns, the left parity is the same for every step.","title":"Left parity is constant along a walk without right turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/straddle-set","kind":"node","lean":"StatMech.Onsager.RayFlipH.straddleSet","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#straddle-set","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"The vertical steps of the walk on the segment `{a + 1} \u00d7 [b, b + 1]` separating the cells `(a, b)` and `(a + 1, b)`.","title":"Vertical steps separating two horizontal neighbour cells"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/up-cross-hstep-gen","kind":"node","lean":"StatMech.Onsager.RayFlipH.upCross_hstep_gen","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#up-cross-hstep-gen","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"The upward crossing counts of cells `(a, b)` and `(a + 1, b)` plus the number of steps between them is even.","title":"Parity of upward crossings across a vertical segment"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/ray-parity-hstep-gen","kind":"node","lean":"StatMech.Onsager.RayFlipH.rayParity_hstep_gen","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#ray-parity-hstep-gen","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"The ray parities of cells `(a, b)` and `(a + 1, b)` add up to the number of separating vertical steps mod `2`.","title":"Ray parity across a vertical segment counts separating steps"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/ray-parity-hflip","kind":"node","lean":"StatMech.Onsager.RayFlipH.rayParity_hflip","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#ray-parity-hflip","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"An odd number of vertical steps between cells `(a, b)` and `(a + 1, b)` flips the ray parity.","title":"An odd number of vertical steps flips the ray parity"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/mid-set","kind":"node","lean":"StatMech.Onsager.RayFlipV.midSet","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#mid-set","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"The horizontal steps of the walk on the segment `[a, a + 1] \u00d7 {b}`.","title":"Horizontal steps on a unit segment"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/up-cross-vsplit","kind":"node","lean":"StatMech.Onsager.RayFlipV.upCross_vsplit","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#up-cross-vsplit","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"The upward crossings from cell `(a, b)` are those from `(a, b + 1)` plus the horizontal steps between the two cells.","title":"Upward crossings of vertically adjacent cells differ by separating steps"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/ray-parity-vflip","kind":"node","lean":"StatMech.Onsager.RayFlipV.rayParity_vflip","parent":29245,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#ray-parity-vflip","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"An odd number of horizontal steps between cells `(a, b)` and `(a, b + 1)` flips the ray parity.","title":"An odd number of horizontal steps flips the ray parity"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/not-add-two-self","kind":"node","lean":"StatMech.Onsager.EdgeUnique.not_add_two_self","parent":29245,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k + 2 \\ne k\\)"],["text"," in "],["math","\\(\\mathrm{Fin}\\, n\\)"],["text"," for "],["math","\\(n \\ge 3\\)"]],"source":"$k + 2 \\ne k$ in $\\mathrm{Fin}\\, n$ for $n \\ge 3$"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04/index.html#not-add-two-self","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-04","status":"fully_proved","summary":"In `Fin n` with `n \u2265 3`, `k + 2 \u2260 k`.","title":"$k + 2 \\ne k$ in $\\mathrm{Fin}\\, n$ for $n \\ge 3$"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","title":"Steps on a horizontal unit segment join its endpoints to A closed walk crosses each vertical line an even number of times"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/mid-set-endpoints","kind":"node","lean":"StatMech.Onsager.EdgeUnique.midSet_endpoints","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#mid-set-endpoints","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A step on the horizontal segment `[a, a + 1] \u00d7 {b}` goes from one endpoint to the other.","title":"Steps on a horizontal unit segment join its endpoints"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/straddle-set-endpoints","kind":"node","lean":"StatMech.Onsager.EdgeUnique.straddleSet_endpoints","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#straddle-set-endpoints","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A step on the vertical segment `{a + 1} \u00d7 [b, b + 1]` goes from one endpoint to the other.","title":"Steps on a vertical unit segment join its endpoints"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/mid-set-card-le-one","kind":"node","lean":"StatMech.Onsager.EdgeUnique.midSet_card_le_one","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#mid-set-card-le-one","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A simple closed walk traverses each horizontal unit segment at most once.","title":"A simple closed walk traverses each horizontal segment at most once"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/straddle-set-card-le-one","kind":"node","lean":"StatMech.Onsager.EdgeUnique.straddleSet_card_le_one","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#straddle-set-card-le-one","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A simple closed walk traverses each vertical unit segment at most once.","title":"A simple closed walk traverses each vertical segment at most once"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/lexmax-corner-cell-inside","kind":"node","lean":"StatMech.Onsager.InsideConnected.lexmax_corner_cell_inside","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#lexmax-corner-cell-inside","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A simple closed walk without right turns has a north-then-west corner `M` whose below-left cell `((pos d M).1 - 1, (pos d M).2 - 1)` has ray parity `1`.","title":"Some north-then-west corner has its below-left cell inside"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/top-out-ne-n","kind":"node","lean":"StatMech.Onsager.TopRow.top_out_ne_N","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#top-out-ne-n","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"No north step leaves a vertex of maximal height.","title":"No north step leaves a highest vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/top-in-ne-s","kind":"node","lean":"StatMech.Onsager.TopRow.top_in_ne_S","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#top-in-ne-s","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"No south step arrives at a vertex of maximal height.","title":"No south step arrives at a highest vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/forward-west-run","kind":"node","lean":"StatMech.Onsager.TopRow.forward_west_run","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#forward-west-run","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"After `j` consecutive west steps from vertex `M`, the walk is `j` units to the left of `M` at the same height.","title":"Position after a run of west steps"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/single-top-run","kind":"node","lean":"StatMech.Onsager.TopRow.SingleTopRun","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#single-top-run","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"Every vertex of the walk at the height of `M` is reached from `M` by a run of consecutive west steps.","title":"Single west run covering the top row"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/top-row-contiguous-of-single-top-run","kind":"node","lean":"StatMech.Onsager.TopRow.topRowContiguous_of_singleTopRun","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#top-row-contiguous-of-single-top-run","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"If the top row is covered by a single west run from a highest vertex, it is contiguous.","title":"A single top run makes the top row contiguous"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/top-row-contiguous-of-residue","kind":"node","lean":"StatMech.Onsager.TopRow.topRowContiguous_of_residue","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#top-row-contiguous-of-residue","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"If every highest north-then-west corner starts a single top run, the top row is contiguous.","title":"Top row is contiguous when every top corner starts a single run"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/top-arrival-west","kind":"node","lean":"StatMech.Onsager.SingleTopRunProof.top_arrival_west","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#top-arrival-west","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"In a closed walk without right turns, a north step arriving at a highest vertex is followed by a west step.","title":"A north step into a highest vertex is followed by a west step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/top-west-next","kind":"node","lean":"StatMech.Onsager.SingleTopRunProof.top_west_next","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#top-west-next","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A west step on the top row is followed by a west or a south step, and stays on the top row.","title":"A west step on the top row is followed by west or south"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/exists-step-ne-west","kind":"node","lean":"StatMech.Onsager.SingleTopRunProof.exists_step_ne_west","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#exists-step-ne-west","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A closed walk does not consist of west steps only.","title":"A closed walk is not all west steps"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/exists-west-run-end","kind":"node","lean":"StatMech.Onsager.SingleTopRunProof.exists_west_run_end","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#exists-west-run-end","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"From a highest vertex where the walk turns west, it makes `J \u2265 1` west steps and then a south step.","title":"A west run from a highest vertex ends in a south step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/single-top-run-of-run-covers-top","kind":"node","lean":"StatMech.Onsager.SingleTopRunProof.singleTopRun_of_runCoversTop","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#single-top-run-of-run-covers-top","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A west run from `M` whose vertices cover the top row is a single top run.","title":"A west run covering the top row is a single top run"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/fst-step-bound","kind":"node","lean":"StatMech.Onsager.WalkCrossingV.fst_step_bound","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#fst-step-bound","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A step changes the `x`-coordinate by `0` or `\u00b11`.","title":"A step changes the x-coordinate by at most one"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/vert-col-cross","kind":"node","lean":"StatMech.Onsager.WalkCrossingV.vertColCross","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#vert-col-cross","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"The number of horizontal steps of the walk crossing the vertical line `x = m + 1/2`.","title":"Number of crossings of a vertical line"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/left-right-card-eq","kind":"node","lean":"StatMech.Onsager.WalkCrossingV.leftRight_card_eq","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#left-right-card-eq","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A walk crosses the line `x = m + 1/2` to the right as often as to the left.","title":"A closed walk crosses a vertical line equally often left and right"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/vert-col-cross-even","kind":"node","lean":"StatMech.Onsager.WalkCrossingV.vertColCross_even","parent":29266,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05/index.html#vert-col-cross-even","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-05","status":"fully_proved","summary":"A closed walk crosses each vertical line `x = m + 1/2` an even number of times.","title":"A closed walk crosses each vertical line an even number of times"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","title":"Interior cells by the even-odd rule to Horizontal neighbours of different parity are separated by a vertical step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/interior-set","kind":"node","lean":"StatMech.Onsager.InteriorCells.interiorSet","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#interior-set","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"The cells inside the walk by the even-odd rule: those with ray parity `1`.","title":"Interior cells by the even-odd rule"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/y-inf","kind":"node","lean":"StatMech.Onsager.InteriorCells.yInf","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#y-inf","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"The minimal height of a vertex of the walk.","title":"Minimal height of a walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/y-sup","kind":"node","lean":"StatMech.Onsager.InteriorCells.ySup","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#y-sup","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"The maximal height of a vertex of the walk.","title":"Maximal height of a walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/exists-edge-of-ray-parity","kind":"node","lean":"StatMech.Onsager.InteriorCells.exists_edge_of_rayParity","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#exists-edge-of-ray-parity","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"An inside cell lies directly below some horizontal step of the walk.","title":"An inside cell lies below a horizontal step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/ray-parity-zero-of-lt-y-inf","kind":"node","lean":"StatMech.Onsager.InteriorCells.rayParity_zero_of_lt_yInf","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#ray-parity-zero-of-lt-y-inf","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"Cells below the walk are outside.","title":"Cells below a walk are outside"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/interior-set-finite","kind":"node","lean":"StatMech.Onsager.InteriorCells.interiorSet_finite","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#interior-set-finite","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"A closed walk has finitely many inside cells.","title":"Finiteness of the interior of a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/interior-cells","kind":"node","lean":"StatMech.Onsager.InteriorCells.interiorCells","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#interior-cells","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"The finite set of cells inside a closed walk (ray parity `1`). See also `StatMech.Onsager.WalkCellRegion.interiorCells`, the bounded face of the walk; the two agree for simple walks without right turns (`StatMech.Onsager.UmlaufsatzFinal.interiorCells_bridge`).","title":"Finite set of interior cells of a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-count-interior-eq","kind":"node","lean":"StatMech.Onsager.WalkCellBridge.cornerCount_interior_eq","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-count-interior-eq","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"The corner count of the interior at `(vx, vy)` counts the inside cells among the four cells around that point.","title":"Corner count of the interior counts inside cells around a point"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/ray-parity-const-of-not-vertex","kind":"node","lean":"StatMech.Onsager.WalkCellBridge.rayParity_const_of_not_vertex","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#ray-parity-const-of-not-vertex","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"At a lattice point that is not a vertex of the walk, the four cells around it have the same ray parity.","title":"Ray parity is constant around a non-vertex point"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-weight-left-turn-e","kind":"node","lean":"StatMech.Onsager.LeftCornerE.cornerWeight_leftTurn_E","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-weight-left-turn-e","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"At a left turn made of a south step followed by an east step, the corner weight of the interior is `1` if the cell to the left of the walk is inside and `-1` otherwise.","title":"Corner weight at a left turn into the east direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-weight-left-turn-n","kind":"node","lean":"StatMech.Onsager.LeftCornerN.cornerWeight_leftTurn_N","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-weight-left-turn-n","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"At a left turn made of an east step followed by a north step, the corner weight of the interior is `1` if the cell to the left of the walk is inside and `-1` otherwise.","title":"Corner weight at a left turn into the north direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-weight-left-turn-s","kind":"node","lean":"StatMech.Onsager.LeftCornerS.cornerWeight_leftTurn_S","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-weight-left-turn-s","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"At a left turn made of a west step followed by a south step, the corner weight of the interior is `1` if the cell to the left of the walk is inside and `-1` otherwise.","title":"Corner weight at a left turn into the south direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-weight-left-turn-w","kind":"node","lean":"StatMech.Onsager.LeftCornerW.cornerWeight_leftTurn_W","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-weight-left-turn-w","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"At a left turn made of a north step followed by a west step, the corner weight of the interior is `1` if the cell to the left of the walk is inside and `-1` otherwise.","title":"Corner weight at a left turn into the west direction"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/not-is-vert-of-is-horiz","kind":"node","lean":"StatMech.Onsager.StraightCorner.not_isVert_of_isHoriz","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#not-is-vert-of-is-horiz","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"A horizontal direction is not vertical.","title":"A horizontal direction is not vertical"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/not-is-horiz-of-is-vert","kind":"node","lean":"StatMech.Onsager.StraightCorner.not_isHoriz_of_isVert","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#not-is-horiz-of-is-vert","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"A vertical direction is not horizontal.","title":"A vertical direction is not horizontal"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-weight-eq-zero-of-two-dvd","kind":"node","lean":"StatMech.Onsager.StraightCorner.cornerWeight_eq_zero_of_two_dvd","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-weight-eq-zero-of-two-dvd","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"An even corner count has weight `0`.","title":"Even corner counts have zero corner weight"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-weight-straight-horiz","kind":"node","lean":"StatMech.Onsager.StraightCorner.cornerWeight_straight_horiz","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-weight-straight-horiz","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"A vertex where the walk continues horizontally has weight `0`.","title":"Zero corner weight where the walk continues horizontally"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-weight-straight-vert","kind":"node","lean":"StatMech.Onsager.StraightCorner.cornerWeight_straight_vert","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-weight-straight-vert","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"A vertex where the walk continues vertically has weight `0`.","title":"Zero corner weight where the walk continues vertically"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/corner-weight-straight","kind":"node","lean":"StatMech.Onsager.StraightCorner.cornerWeight_straight","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#corner-weight-straight","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"A vertex where a simple closed walk goes straight has corner weight `0` for the interior.","title":"Straight steps of a simple closed walk carry no corner weight"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/straddle-nonempty-of-ne","kind":"node","lean":"StatMech.Onsager.PinchFree.straddle_nonempty_of_ne","parent":29287,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06/index.html#straddle-nonempty-of-ne","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-06","status":"fully_proved","summary":"If cells `(a, b)` and `(a + 1, b)` have different ray parities, a vertical step separates them.","title":"Horizontal neighbours of different parity are separated by a vertical step"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","title":"Vertical neighbours of different parity are separated by a horizontal step to Face-graph adjacency as absence of a walk step between cells"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/mid-nonempty-of-ne","kind":"node","lean":"StatMech.Onsager.PinchFree.mid_nonempty_of_ne","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#mid-nonempty-of-ne","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"If cells `(a, b)` and `(a, b + 1)` have different ray parities, a horizontal step separates them.","title":"Vertical neighbours of different parity are separated by a horizontal step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/pinch-free","kind":"node","lean":"StatMech.Onsager.PinchFree.pinch_free","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#pinch-free","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The interior of a simple closed walk has no pinch points.","title":"Interior of a simple closed walk has no pinch points"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/pinch-count-eq-zero","kind":"node","lean":"StatMech.Onsager.PinchFree.pinchCount_eq_zero","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#pinch-count-eq-zero","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The interior of a simple closed walk has pinch count `0`. See also `StatMech.Onsager.WalkCellLocal.pinchCount_interiorCells_eq_zero`.","title":"Pinch count of the ray-parity interior is zero"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/corner-weight-zero-of-not-mem-image","kind":"node","lean":"StatMech.Onsager.WalkCornerSum.cornerWeight_zero_of_notMem_image","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#corner-weight-zero-of-not-mem-image","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"A lattice point off the walk has corner weight `0` for the interior.","title":"Zero corner weight of the interior off the walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/corner-diff-interior-eq-walk-sum","kind":"node","lean":"StatMech.Onsager.WalkCornerSum.cornerDiff_interior_eq_walk_sum","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#corner-diff-interior-eq-walk-sum","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"For a simple closed walk, the corner defect of the interior is the sum of the corner weights at the walk vertices.","title":"Corner defect of the interior as a sum over walk vertices"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/corner-weight-master","kind":"node","lean":"StatMech.Onsager.MasterCorner.cornerWeight_master","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#corner-weight-master","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The corner weight of the interior at the vertex after step `i` is `\u00b11` (sign given by the orientation) at a left turn and `0` at a straight step.","title":"Corner weights at the turns of a walk without right turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/corner-diff-eq-weight-mul-cc","kind":"node","lean":"StatMech.Onsager.MasterCorner.cornerDiff_eq_weight_mul_cc","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#corner-diff-eq-weight-mul-cc","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The corner defect of the interior equals `\u00b1cc d`, the sign given by the orientation of the walk.","title":"Corner defect of the interior is the number of left turns up to sign"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/cc-eq-four","kind":"node","lean":"StatMech.Onsager.MasterCorner.cc_eq_four","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#cc-eq-four","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"A simple closed walk without right turns whose interior is pinch-free with Euler characteristic `1` has four left turns.","title":"Four left turns from a pinch-free interior of Euler characteristic 1"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/pos-site","kind":"node","lean":"StatMech.Onsager.JedBridge.posSite","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#pos-site","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The `i`-th vertex of the walk as a site of `\u2124\u00b2`.","title":"Vertex of a walk as a lattice site"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/pos-site-injective","kind":"node","lean":"StatMech.Onsager.JedBridge.posSite_injective","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#pos-site-injective","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The vertices of a simple walk are distinct sites.","title":"Vertices of a simple walk are distinct sites"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/step-of-nat-abs-sum","kind":"node","lean":"StatMech.Onsager.JedBridge.stepOf_natAbs_sum","parent":29308,"reading_title":{"format":"inline-parts-v1","parts":[["text","Each lattice step has "],["math","\\(\\ell^1\\)"],["text","-length one"]],"source":"Each lattice step has $\\ell^1$-length one"},"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#step-of-nat-abs-sum","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"Each step has `\u2113\u00b9`-length `1`.","title":"Each lattice step has $\\ell^1$-length one"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/walk-is-sub","kind":"node","lean":"StatMech.Onsager.JedBridge.walk_isSub","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#walk-is-sub","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"Consecutive vertices of the walk are adjacent in the square lattice.","title":"Consecutive walk vertices are adjacent in the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/walk-subgraph","kind":"node","lean":"StatMech.Onsager.JedBridge.walkSubgraph","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#walk-subgraph","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"A simple closed walk as a planar subgraph of `\u2124\u00b2`: the cycle graph `cycleGraph (m + 3)` embedded by its vertex positions.","title":"Simple closed walk as an embedded cycle graph"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/cycle-graph-card-components-one","kind":"node","lean":"StatMech.Onsager.JedBridge.cycleGraph_card_components_one","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#cycle-graph-card-components-one","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"A cycle graph has one connected component. A specialization of `SimpleGraph.Preconnected.subsingleton_connectedComponent`.","title":"A cycle graph has one connected component"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/cycle-graph-card-edges","kind":"node","lean":"StatMech.Onsager.JedBridge.cycleGraph_card_edges","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#cycle-graph-card-edges","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The cycle graph on `m + 3` vertices has `m + 3` edges.","title":"Number of edges of a cycle graph"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/cycle-nullity-one","kind":"node","lean":"StatMech.Onsager.JedBridge.cycle_nullity_one","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#cycle-nullity-one","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The cycle graph has nullity (cycle rank) `1`.","title":"Cycle graph has cycle rank one"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/walk-subgraph-nullity-one","kind":"node","lean":"StatMech.Onsager.JedBridge.walkSubgraph_nullity_one","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#walk-subgraph-nullity-one","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The planar subgraph of a simple closed walk has nullity `1`.","title":"Subgraph of a simple closed lattice walk has cycle rank one"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/walk-one-bounded-region","kind":"node","lean":"StatMech.Onsager.JedBridge.walk_one_bounded_region","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#walk-one-bounded-region","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"Lattice Jordan curve theorem for a simple closed walk: its complement in the face graph of `\u2124\u00b2` has exactly one bounded connected component.","title":"Lattice Jordan curve theorem for simple closed walks"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/bridge-parity-membership","kind":"node","lean":"StatMech.Onsager.JedBridge.BridgeParityMembership","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#bridge-parity-membership","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"The ray-parity interior of the walk coincides with the union of bounded face components.","title":"Ray-parity interior equals the bounded face components"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/bridge-adjacency-climb","kind":"node","lean":"StatMech.Onsager.JedBridge.BridgeAdjacencyClimb","parent":29308,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07/index.html#bridge-adjacency-climb","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-07","status":"fully_proved","summary":"Adjacency in the face graph of the walk is the absence of a walk step on the common side of two cells.","title":"Face-graph adjacency as absence of a walk step between cells"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","title":"Unit cell as a face of the square lattice to Membership in the bounded face"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/cell-face","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.cellFace","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#cell-face","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The unit cell with lower-left corner `p`, viewed as a face of the square lattice, i.e. as the site `![p.1, p.2]` of the dual lattice.","title":"Unit cell as a face of the square lattice"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/walk-p","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.walkP","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#walk-p","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"A simple closed walk as a planar subgraph of `\u2124\u00b2`.","title":"Simple closed walk as a planar subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/walk-region-graph","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.walkRegionGraph","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#walk-region-graph","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The face graph of the walk: adjacent faces of the square lattice whose common side is not an edge of the walk.","title":"Face graph of a closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/walk-edge-mem-image-graph","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.walkEdge_mem_imageGraph","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#walk-edge-mem-image-graph","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"Each step of the walk is an edge of its image graph.","title":"Each walk step is an edge of the image graph"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/image-graph-adj-pos-iff","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.imageGraph_adj_pos_iff","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#image-graph-adj-pos-iff","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"In the image graph of a simple closed walk, the neighbours of a vertex are its predecessor and successor on the walk.","title":"Neighbours in the image graph are walk predecessor and successor"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/ray-parity-eq-of-face-adj-right","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.rayParity_eq_of_faceAdj_right","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#ray-parity-eq-of-face-adj-right","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"Horizontally adjacent cells joined in the face graph have the same ray parity.","title":"Horizontally face-adjacent cells have the same ray parity"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/ray-parity-eq-of-face-adj-top","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.rayParity_eq_of_faceAdj_top","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#ray-parity-eq-of-face-adj-top","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"Vertically adjacent cells joined in the face graph have the same ray parity.","title":"Vertically face-adjacent cells have the same ray parity"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/ray-parity-eq-of-face-adj","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.rayParity_eq_of_faceAdj","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#ray-parity-eq-of-face-adj","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"Cells adjacent in the face graph have the same ray parity.","title":"Face-adjacent cells have the same ray parity"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/ray-parity-eq-of-reachable","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.rayParity_eq_of_reachable","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#ray-parity-eq-of-reachable","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"Ray parity is constant on connected components of the face graph.","title":"Ray parity is constant on face-graph components"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/lex-interior-index","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.lexInteriorIndex","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#lex-interior-index","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"A vertex `M` at which the walk turns from north to west and whose below-left cell is inside (chosen from `StatMech.Onsager.InsideConnected.lexmax_corner_cell_inside`).","title":"Top corner with an inside cell below it"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/lex-interior-index-spec","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.lexInteriorIndex_spec","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#lex-interior-index-spec","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The defining property of `lexInteriorIndex`.","title":"Defining property of the inside top corner"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/lex-interior-face","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.lexInteriorFace","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#lex-interior-face","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The face of the cell below-left of the corner `lexInteriorIndex`.","title":"Face below-left of the inside top corner"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/lex-interior-face-parity","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.lexInteriorFace_parity","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#lex-interior-face-parity","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The face `lexInteriorFace` is inside the walk.","title":"The face below the inside top corner is inside"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/inner-component","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.innerComponent","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#inner-component","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The connected component of the face graph containing `lexInteriorFace`.","title":"Inner component of the face graph"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/outer-beacon-parity","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.outerBeacon_parity","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#outer-beacon-parity","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"A face far outside the bounding box of the walk has ray parity `0`.","title":"A face far outside the walk has ray parity zero"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/inner-component-finite","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.innerComponent_finite","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#inner-component-finite","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The component `innerComponent` is bounded.","title":"The inner face component is finite"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/component-cell-set","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.componentCellSet","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#component-cell-set","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The cells whose faces lie in `innerComponent`.","title":"Cells of the inner face component"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/component-cell-set-finite","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.componentCellSet_finite","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#component-cell-set-finite","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"There are finitely many such cells.","title":"The inner component has finitely many cells"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/interior-cells","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.interiorCells","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#interior-cells","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"The cells of the bounded face of a simple closed walk without right turns. See also `StatMech.Onsager.InteriorCells.interiorCells`, the ray-parity interior; they agree by `mem_interiorCells_iff_rayParity`.","title":"Cells of the bounded face of a simple closed walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/mem-interior-cells","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.mem_interiorCells","parent":29329,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08/index.html#mem-interior-cells","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-08","status":"fully_proved","summary":"Membership in the bounded face.","title":"Membership in the bounded face"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","title":"Lattice Jordan curve theorem, even-odd form to Zero corner weight of the bounded face off the walk"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/mem-interior-cells-iff-ray-parity","kind":"node","lean":"StatMech.Onsager.WalkCellRegion.mem_interiorCells_iff_rayParity","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#mem-interior-cells-iff-ray-parity","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"Lattice Jordan curve theorem, even-odd form: a cell lies in the bounded face of a simple closed walk without right turns iff its ray parity is `1`.","title":"Lattice Jordan curve theorem, even-odd form"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/odd-component","kind":"node","lean":"StatMech.Onsager.GeneralInterior.oddComponent","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#odd-component","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"The connected component of the walk's face graph containing an interior cell.","title":"Face-graph component of an interior cell"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/shared-primal-edge-mem-of-membership-dif","kind":"node","lean":"StatMech.Onsager.WalkCellLocal.sharedPrimalEdge_mem_of_membership_diff","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#shared-primal-edge-mem-of-membership-dif","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"A side shared by an interior and an exterior cell is an edge of the walk.","title":"A side between interior and exterior cells is a walk edge"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/pinch-four-edges-impossible","kind":"node","lean":"StatMech.Onsager.WalkCellLocal.pinch_four_edges_impossible","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#pinch-four-edges-impossible","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"Three walk edges cannot meet at one vertex of a simple closed walk.","title":"Three walk edges cannot meet at a vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/not-pinch-at-interior-cells","kind":"node","lean":"StatMech.Onsager.WalkCellLocal.not_pinchAt_interiorCells","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#not-pinch-at-interior-cells","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"The bounded face of the walk has no pinch points.","title":"Bounded face of a simple closed walk has no pinch points"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/pinch-count-interior-cells-eq-zero","kind":"node","lean":"StatMech.Onsager.WalkCellLocal.pinchCount_interiorCells_eq_zero","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#pinch-count-interior-cells-eq-zero","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"The bounded face of the walk has pinch count `0`. See also `StatMech.Onsager.PinchFree.pinchCount_eq_zero`.","title":"Pinch count of the bounded face is zero"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/left-cell","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.leftCell","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#left-cell","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"The cell to the left of step `k` of the walk. See also `StatMech.Onsager.Orientation.leftCell`; they agree by `StatMech.Onsager.UmlaufsatzFinal.leftCell_bridge`.","title":"Cell to the left of a walk step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/right-cell","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.rightCell","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#right-cell","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"The cell to the right of step `k` of the walk.","title":"Cell to the right of a walk step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/image-graph-not-adj-of-neighbours","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.imageGraph_not_adj_of_neighbours","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#image-graph-not-adj-of-neighbours","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"In the image graph of a simple closed walk, a vertex is adjacent only to its walk neighbours.","title":"Vertices of a walk's image graph are adjacent only to walk neighbours"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/left-cell-face-adj-of-straight","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.leftCell_faceAdj_of_straight","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#left-cell-face-adj-of-straight","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"Going straight, consecutive left cells are adjacent in the face graph of the walk.","title":"Consecutive left cells on a straight step are face-adjacent"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/left-cell-parity-succ","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.leftCell_parity_succ","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#left-cell-parity-succ","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"Consecutive left cells of a simple closed walk without right turns have the same ray parity.","title":"Consecutive left cells have the same ray parity"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/left-cell-parity-eq-of-reachable","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.leftCell_parity_eq_of_reachable","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#left-cell-parity-eq-of-reachable","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"All left cells of a simple closed walk without right turns have the same ray parity.","title":"All left cells have the same ray parity"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/left-cell-parity-one","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.leftCell_parity_one","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#left-cell-parity-one","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"Every left cell of a simple closed walk without right turns is inside.","title":"Every left cell has ray parity one"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/left-cell-mem-interior-cells","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.leftCell_mem_interiorCells","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#left-cell-mem-interior-cells","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"Every left cell of a simple closed walk without right turns lies in the bounded face.","title":"Every left cell lies in the bounded face"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/walk-region-flanking-not-reachable","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.walkRegion_flanking_not_reachable","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#walk-region-flanking-not-reachable","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"The two cells on either side of a walk step are in different face components.","title":"Cells on either side of a step lie in different faces"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/left-right-flanking","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.left_right_flanking","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#left-right-flanking","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"The left and right cells of a step are adjacent faces sharing that step.","title":"Left and right cells of a step share that step"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/right-cell-not-mem-interior-cells","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.rightCell_not_mem_interiorCells","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#right-cell-not-mem-interior-cells","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"No right cell of a simple closed walk without right turns lies in the bounded face.","title":"No right cell lies in the bounded face"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/corner-count-eq-indicators","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.cornerCount_eq_indicators","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#corner-count-eq-indicators","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"The corner count at `v` as a sum of four indicators. See also `StatMech.Onsager.CellEuler.cornerCount_eq_inter`.","title":"Corner count as a sum of four indicators"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/corner-count-at-walk-vertex","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.cornerCount_at_walkVertex","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#corner-count-at-walk-vertex","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"At the vertex after step `k`, the bounded face has one cell after a left turn and two after a straight step.","title":"Corner count of the bounded face at a walk vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/corner-weight-zero-of-not-walk-vertex","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.cornerWeight_zero_of_not_walkVertex","parent":29350,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09/index.html#corner-weight-zero-of-not-walk-vertex","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-09","status":"fully_proved","summary":"Points off the walk have corner weight `0` for the bounded face.","title":"Zero corner weight of the bounded face off the walk"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","title":"Corner weight of the bounded face at a walk vertex to Lower-left corner of a cell as a skeleton vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/corner-weight-at-walk-vertex","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.cornerWeight_at_walkVertex","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#corner-weight-at-walk-vertex","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The corner weight of the bounded face at the vertex after step `k` is `1` after a left turn and `0` after a straight step.","title":"Corner weight of the bounded face at a walk vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/corner-diff-interior-cells-eq-cc","kind":"node","lean":"StatMech.Onsager.WalkCellBoundary.cornerDiff_interiorCells_eq_cc","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#corner-diff-interior-cells-eq-cc","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The corner defect of the bounded face of a simple closed walk without right turns equals its number of left turns.","title":"Corner defect of the bounded face equals the number of left turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/cell-skeleton","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.cellSkeleton","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#cell-skeleton","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The 1-skeleton of the cell complex `S`: vertices are the corners of cells, edges the sides of cells.","title":"1-skeleton of a cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/cell-skeleton-p","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.cellSkeletonP","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#cell-skeleton-p","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The 1-skeleton of `S` as a planar subgraph of `\u2124\u00b2`.","title":"1-skeleton of a cell complex as a planar subgraph"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/reg-edge-mem-image-graph","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.regEdge_mem_imageGraph","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#reg-edge-mem-image-graph","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"An edge of the cell complex is an edge of the embedded skeleton.","title":"Cell complex edges lie in the embedded skeleton"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/shared-primal-edge-mem-reg-edges","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.sharedPrimalEdge_mem_regEdges","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#shared-primal-edge-mem-reg-edges","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The side shared by a cell of `S` and an adjacent face is an edge of `S`.","title":"Shared side of a cell and an adjacent face is an edge"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/occupied-face-isolated","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.occupiedFace_isolated","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#occupied-face-isolated","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The face of a cell of `S` is enclosed by the skeleton on all four sides.","title":"The face of a cell is enclosed on all four sides"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/occupied-component","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.occupiedComponent","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#occupied-component","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The face component of the skeleton containing the cell `c`.","title":"Face component containing a cell"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/occupied-face-reachable-eq","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.occupiedFace_reachable_eq","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#occupied-face-reachable-eq","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The face of a cell of `S` is alone in its face component.","title":"The face of a cell is alone in its component"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/occupied-component-finite","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.occupiedComponent_finite","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#occupied-component-finite","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The face component of a cell of `S` is bounded.","title":"The face component of a cell is bounded"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/occupied-bounded-face","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.occupiedBoundedFace","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#occupied-bounded-face","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The bounded face of the skeleton formed by a cell of `S`.","title":"Bounded face formed by a cell"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/occupied-bounded-face-injective","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.occupiedBoundedFace_injective","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#occupied-bounded-face-injective","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"Distinct cells form distinct bounded faces.","title":"Distinct cells form distinct bounded faces"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/card-le-cell-skeleton-nullity","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.card_le_cellSkeleton_nullity","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#card-le-cell-skeleton-nullity","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The number of cells is at most the nullity of the skeleton.","title":"Number of cells is at most the nullity of the skeleton"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/cell-skeleton-edge-equiv","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.cellSkeletonEdgeEquiv","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#cell-skeleton-edge-equiv","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The edges of `cellSkeleton S` are the edges of the cell complex `S`.","title":"Edges of the skeleton are the edges of the cell complex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/card-cell-skeleton-vertices","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.card_cellSkeleton_vertices","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#card-cell-skeleton-vertices","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The skeleton has `(regVerts S).card` vertices.","title":"Vertex count of the skeleton"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/euler-char-le-one-of-skeleton-connected","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.eulerChar_le_one_of_skeleton_connected","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#euler-char-le-one-of-skeleton-connected","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"If the 1-skeleton of `S` is connected, then `eulerChar S \u2264 1`. See also `StatMech.Onsager.CellEulerBound.eulerChar_le_one`.","title":"Euler characteristic at most 1 for a connected skeleton"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/skeleton-vertex-of-corner","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.skeletonVertexOfCorner","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#skeleton-vertex-of-corner","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"A corner of a cell of `S`, as a vertex of the skeleton.","title":"Corner of a cell as a skeleton vertex"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/corner-reachable-lower-left","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.corner_reachable_lowerLeft","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#corner-reachable-lower-left","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"Every corner of a cell is connected in the skeleton to its lower-left corner.","title":"Cell corners are connected to the lower-left corner"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/cell-representatives-reachable-of-adj","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.cell_representatives_reachable_of_adj","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#cell-representatives-reachable-of-adj","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The lower-left corners of two adjacent cells of `S` are connected in the skeleton.","title":"Adjacent cells have connected lower-left corners"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/skeleton-representative","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.skeletonRepresentative","parent":29371,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10/index.html#skeleton-representative","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-10","status":"fully_proved","summary":"The lower-left corner of a cell of `S`, as a vertex of the skeleton.","title":"Lower-left corner of a cell as a skeleton vertex"},{"data_url":"scopes/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11.json","id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","kind":"scope","navigation":"transparent","parent":29181,"result":false,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","title":"Interior cells have connected lower-left corners to A simple closed walk without right turns has under eight left turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/interior-representatives-reachable","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.interior_representatives_reachable","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#interior-representatives-reachable","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"Any two interior cells of a simple closed walk without right turns have skeleton-connected lower-left corners.","title":"Interior cells have connected lower-left corners"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/interior-cell-skeleton-connected","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.interior_cellSkeleton_connected","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#interior-cell-skeleton-connected","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"The skeleton of the interior of a simple closed walk without right turns is connected.","title":"Connectedness of the skeleton of the interior"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/euler-char-interior-cells-le-one","kind":"node","lean":"StatMech.Onsager.CellEulerFaithful.eulerChar_interiorCells_le_one","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#euler-char-interior-cells-le-one","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"The interior of a simple closed walk without right turns has Euler characteristic at most `1`.","title":"Interior of a simple closed walk has Euler characteristic at most 1"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/turn-sum-eq-four-or-neg-four","kind":"node","lean":"StatMech.Onsager.GeneralUmlaufsatz.turnSum_eq_four_or_neg_four","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#turn-sum-eq-four-or-neg-four","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"Hopf's Umlaufsatz for lattice polygons. A simple closed walk in `\u2124\u00b2` with at least three steps and no U-turns has total turning `\u00b14` quarter turns, that is rotation index `\u00b11` (H. Hopf, *\u00dcber die Drehung der Tangenten und Sehnen ebener Kurven*, 1935).","title":"Hopf's Umlaufsatz"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/left-cell-bridge","kind":"node","lean":"StatMech.Onsager.UmlaufsatzFinal.leftCell_bridge","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#left-cell-bridge","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"The two descriptions of the cell to the left of a step agree.","title":"Agreement of the two definitions of the left cell"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/left-parity-zero","kind":"node","lean":"StatMech.Onsager.UmlaufsatzFinal.leftParity_zero","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#left-parity-zero","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"The cell to the left of the first step is inside: the walk is positively oriented.","title":"A walk without right turns is positively oriented"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/interior-cells-bridge","kind":"node","lean":"StatMech.Onsager.UmlaufsatzFinal.interiorCells_bridge","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#interior-cells-bridge","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"The ray-parity interior of a simple closed walk without right turns equals its bounded face.","title":"Ray-parity interior equals the bounded face"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/cc-eq-four","kind":"node","lean":"StatMech.Onsager.UmlaufsatzFinal.cc_eq_four'","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#cc-eq-four","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"The discrete Umlaufsatz for walks of length `m + 3` without right turns.","title":"Discrete Umlaufsatz for walks of length at least three, shifted form"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/cc-eq-four-2","kind":"node","lean":"StatMech.Onsager.UmlaufsatzFinal.cc_eq_four","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#cc-eq-four-2","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"The discrete Umlaufsatz for walks of length at least `3` without right turns.","title":"Discrete Umlaufsatz for walks of length at least three"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/three-le-n","kind":"node","lean":"StatMech.Onsager.UmlaufsatzFinal.three_le_n","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#three-le-n","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"A closed walk without right turns or reversals has length at least `3`.","title":"A closed walk without right turns has length at least three"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/cc-eq-four-of-no-right-turn","kind":"node","lean":"StatMech.Onsager.UmlaufsatzFinal.cc_eq_four_of_no_right_turn","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#cc-eq-four-of-no-right-turn","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"Discrete Umlaufsatz (theorem of turning tangents) for lattice walks without right turns: a simple closed walk on `\u2124\u00b2` whose every turn is straight or a left quarter turn has exactly four left turns, i.e. its direction makes one full counterclockwise revolution. Right turns and reversals are excluded by hypothesis, so the walks covered are exactly counterclockwise boundaries of lattice rectangles (no reflex corner);\u2026","title":"Discrete Umlaufsatz for lattice walks without right turns"},{"id":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/hnodbl-discharged","kind":"node","lean":"StatMech.Onsager.UmlaufsatzFinal.hnodbl_discharged","parent":29392,"result":true,"route":"../roadmap/GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11/index.html#hnodbl-discharged","scope":"GraphTheory/planar-graphs-and-minors/hopf-s-umlaufsatz-for-lattice-polygons/part-11","status":"fully_proved","summary":"The hypothesis of `StatMech.Onsager.BaseCase.base_cc_eq_four` holds: a simple closed walk without right turns has fewer than eight left turns.","title":"A simple closed walk without right turns has under eight left turns"},{"data_url":"scopes/GraphTheory/graph-coloring.json","id":"GraphTheory/graph-coloring","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/index.html","scope":"GraphTheory/graph-coloring","status":"can_prove","summary":"Vertex, edge, list and fractional colorings of graphs and the chromatic number, from Brooks' and Vizing's theorems to perfect graphs and the four color theorem.","title":"Graph coloring"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01.json","id":"GraphTheory/graph-coloring/part-01","kind":"scope","navigation":"transparent","parent":29405,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/index.html","scope":"GraphTheory/graph-coloring/part-01","status":"can_prove","title":"Part 1 of 2"},{"id":"GraphTheory/graph-coloring/part-01/brooks","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.BrooksWanted.brooks","parent":29406,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/index.html#brooks","scope":"GraphTheory/graph-coloring/part-01","status":"fully_proved","summary":"Let $G$ be a finite connected simple graph and let $\\Delta$ be its maximum degree: every vertex of $G$ has degree at most $\\Delta$ and some vertex has degree exactly $\\Delta$. If $G$ is not complete and is not isomorphic to a cycle graph on an odd number of vertices, then $G$ has a proper vertex coloring with $\\Delta$ colors.","title":"Brooks' theorem"},{"id":"GraphTheory/graph-coloring/part-01/erdos-high-girth-chromatic","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.ErdosGirthChromaticWanted.erdos_high_girth_chromatic","parent":29406,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/index.html#erdos-high-girth-chromatic","scope":"GraphTheory/graph-coloring/part-01","status":"fully_proved","summary":"For all natural numbers $k$ and $g$ with $g \\ge 3$, there is a finite simple graph $G$ that has a cycle, whose girth (the length of a shortest cycle) is at least $g$, and whose chromatic number is at least $k$.","title":"Erd\u0151s's theorem on graphs of high girth and high chromatic number"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/centered-colorings.json","id":"GraphTheory/graph-coloring/part-01/centered-colorings","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/centered-colorings/index.html","scope":"GraphTheory/graph-coloring/part-01/centered-colorings","status":"fully_proved","title":"Centered colorings"},{"id":"GraphTheory/graph-coloring/part-01/centered-colorings/is-phi-center","kind":"node","lean":"SimpleGraph.IsPhiCenter","parent":29409,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/centered-colorings/index.html#is-phi-center","scope":"GraphTheory/graph-coloring/part-01/centered-colorings","status":"fully_proved","summary":"Let $\\varphi$ be a proper vertex coloring of a simple graph $G$, let $s$ be a set of vertices of $G$ and let $v$ be a vertex of $G$. Then $v$ is a $\\varphi$-center of $s$ if $v \\in s$ and no other vertex of $s$ has the color $\\varphi(v)$.","title":"Center of a vertex set under a coloring"},{"id":"GraphTheory/graph-coloring/part-01/centered-colorings/is-centered","kind":"node","lean":"SimpleGraph.IsCentered","parent":29409,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/centered-colorings/index.html#is-centered","scope":"GraphTheory/graph-coloring/part-01/centered-colorings","status":"fully_proved","summary":"A proper vertex coloring $\\varphi$ of a simple graph $G$ is centered if, for every set $s$ of vertices of $G$ whose induced subgraph is connected, some vertex is a $\\varphi$-center of $s$.","title":"Centered coloring"},{"id":"GraphTheory/graph-coloring/part-01/centered-colorings/is-phi-center-iff-exists-unique","kind":"node","lean":"SimpleGraph.isPhiCenter_iff_existsUnique","parent":29409,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/centered-colorings/index.html#is-phi-center-iff-exists-unique","scope":"GraphTheory/graph-coloring/part-01/centered-colorings","status":"fully_proved","summary":"Let $\\varphi$ be a proper vertex coloring of a simple graph $G$, let $s$ be a set of vertices of $G$ and let $v$ be a vertex of $G$. Then $v$ is a $\\varphi$-center of $s$ if and only if $v \\in s$ and there is exactly one vertex $u \\in s$ with $\\varphi(u) = \\varphi(v)$.","title":"Centers via unique existence"},{"id":"GraphTheory/graph-coloring/part-01/centered-colorings/is-phi-center-singleton","kind":"node","lean":"SimpleGraph.isPhiCenter_singleton","parent":29409,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/centered-colorings/index.html#is-phi-center-singleton","scope":"GraphTheory/graph-coloring/part-01/centered-colorings","status":"fully_proved","summary":"For every proper vertex coloring $\\varphi$ of a simple graph $G$ and every vertex $v$ of $G$, $v$ is a $\\varphi$-center of $\\{v\\}$.","title":"A vertex is a center of its singleton"},{"id":"GraphTheory/graph-coloring/part-01/centered-colorings/is-centered-of-injective","kind":"node","lean":"SimpleGraph.isCentered_of_injective","parent":29409,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/centered-colorings/index.html#is-centered-of-injective","scope":"GraphTheory/graph-coloring/part-01/centered-colorings","status":"fully_proved","summary":"Every injective proper vertex coloring of a simple graph is centered: any vertex of a set whose induced subgraph is connected is a center of that set.","title":"Injective colorings are centered"},{"id":"GraphTheory/graph-coloring/part-01/centered-colorings/self-coloring-is-centered","kind":"node","lean":"SimpleGraph.selfColoring_isCentered","parent":29409,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/centered-colorings/index.html#self-coloring-is-centered","scope":"GraphTheory/graph-coloring/part-01/centered-colorings","status":"fully_proved","summary":"For every simple graph $G$ with vertex set $V$, the proper vertex coloring of $G$ by $V$ that gives each vertex itself as its color is centered.","title":"The identity coloring is centered"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs.json","id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs/index.html","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs","status":"can_prove","title":"Chromatic number of (0,2)-graphs"},{"id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs/is-zero-two","kind":"node","lean":"MathlibExt.Combinatorics.ThreeChromaticZeroTwoGraphWanted.IsZeroTwo","parent":29416,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((0,2)\\)"],["text","-graphs"]],"source":"$(0,2)$-graphs"},"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs/index.html#is-zero-two","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs","status":"fully_proved","summary":"For a natural number $n$, a simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ is a $(0,2)$-graph when every two distinct vertices of $G$ have either exactly zero or exactly two common neighbors.","title":"$(0,2)$-graphs"},{"id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ThreeChromaticZeroTwoGraphWanted.conjecture","parent":29416,"reading_title":{"format":"inline-parts-v1","parts":[["text","The three-chromatic "],["math","\\((0,2)\\)"],["text","-graph conjecture"]],"source":"The three-chromatic $(0,2)$-graph conjecture"},"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs","status":"fully_proved","summary":"It asserts that there are a natural number $n$ and a simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ in which every two distinct vertices have either zero or two common neighbors, such that the chromatic number of $G$ is $3$.","title":"The three-chromatic $(0,2)$-graph conjecture"},{"id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.ThreeChromaticZeroTwoGraphWanted.conjecture_refuted","parent":29416,"reading_title":{"format":"inline-parts-v1","parts":[["text","No finite "],["math","\\((0,2)\\)"],["text","-graph has chromatic number three"]],"source":"No finite $(0,2)$-graph has chromatic number three"},"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs/index.html#conjecture-refuted","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-0-2-graphs","status":"can_prove","summary":"For every natural number $n$ and every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ in which every two distinct vertices have either zero or two common neighbors, the chromatic number of $G$ is not $3$. In particular, the three-chromatic $(0,2)$-graph conjecture is false.","title":"No finite $(0,2)$-graph has chromatic number three"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs.json","id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/index.html","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs","status":"can_prove","title":"Chromatic number of wheel-free graphs"},{"id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/has-induced-wheel","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.DaviesWheelFreeChromaticWanted.hasInducedWheel","parent":29420,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/index.html#has-induced-wheel","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs","status":"fully_proved","summary":"A simple graph $G$ has an induced wheel if there are a natural number $\\ell \\ge 4$, distinct vertices $r_0, \\dots, r_{\\ell-1}$ of $G$ (the rim) and a vertex $h$ of $G$ (the hub) with $h \\ne r_i$ for every $i$, such that for all $i, j < \\ell$ the vertices $r_i$ and $r_j$ are adjacent in $G$ exactly when $i - j \\equiv \\pm 1 \\pmod{\\ell}$, and $h$ is adjacent to $r_i$, $r_j$ and $r_k$ for some pairwise distinct\u2026","title":"Induced wheels"},{"id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/wheel-free","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.DaviesWheelFreeChromaticWanted.wheelFree","parent":29420,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/index.html#wheel-free","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs","status":"fully_proved","summary":"A simple graph $G$ is wheel-free if it has no induced wheel, that is, no induced cycle of length at least four together with a vertex outside the cycle that is adjacent to at least three vertices of the cycle.","title":"Wheel-free graphs"},{"id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/large-chromatic-triangle-free-wheel-free","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.DaviesWheelFreeChromaticWanted.largeChromaticTriangleFreeWheelFree","parent":29420,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/index.html#large-chromatic-triangle-free-wheel-free","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs","status":"fully_proved","summary":"This is the proposition that for every positive integer $k$ there are a natural number $n$ and a simple graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ such that $G$ contains no triangle, $G$ is wheel-free (it has no induced wheel) and the chromatic number of $G$ is at least $k$.","title":"Triangle-free wheel-free graphs of large chromatic number (proposition)"},{"id":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/large-chromatic-triangle-free-wheel-free-2","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.DaviesWheelFreeChromaticWanted.largeChromaticTriangleFreeWheelFree_holds","parent":29420,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs/index.html#large-chromatic-triangle-free-wheel-free-2","scope":"GraphTheory/graph-coloring/part-01/chromatic-number-of-wheel-free-graphs","status":"can_prove","summary":"For every positive integer $k$ there are a natural number $n$ and a simple graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ such that $G$ contains no triangle, $G$ is wheel-free (it has no induced wheel) and the chromatic number of $G$ is at least $k$.","title":"Davies's theorem on triangle-free wheel-free graphs"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/color-critical-graphs.json","id":"GraphTheory/graph-coloring/part-01/color-critical-graphs","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","title":"Color-critical graphs"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/is-critical-edges","kind":"node","lean":"SimpleGraph.IsCriticalEdges","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#is-critical-edges","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"Let $G$ be a graph and $S$ a set of unordered pairs of its vertices. Then $S$ is critical for $G$ if deleting the edges of $G$ that lie in $S$ strictly lowers the chromatic number, as an element of $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Critical edge sets"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/is-critical-edge","kind":"node","lean":"SimpleGraph.IsCriticalEdge","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#is-critical-edge","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"For a graph $G$ and an unordered pair $e$ of vertices, $e$ is a critical edge of $G$ if removing $e$ from the edge set of $G$ strictly lowers the chromatic number of $G$, taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Critical edge"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/is-critical-verts","kind":"node","lean":"SimpleGraph.Subgraph.IsCriticalVerts","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#is-critical-verts","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"Let $G'$ be a subgraph of a graph $G$. A set $S$ of vertices is critical for $G'$ if deleting the vertices in $S$ from $G'$ strictly lowers the chromatic number of $G'$, viewed as a graph on its own vertex set, with chromatic numbers taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Critical vertex sets of a subgraph"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/is-critical-vertex","kind":"node","lean":"SimpleGraph.Subgraph.IsCriticalVertex","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#is-critical-vertex","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"Let $G'$ be a subgraph of a graph $G$. A vertex $v$ is critical for $G'$ if $\\{v\\}$ is a critical vertex set of $G'$, that is, deleting $v$ from $G'$ strictly lowers the chromatic number of $G'$, viewed as a graph on its own vertex set, with chromatic numbers taken in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"Critical vertices of a subgraph"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/is-critical","kind":"node","lean":"SimpleGraph.IsCritical","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#is-critical","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"Let $k$ be a natural number. A graph $G$ is $k$-critical (or vertex-critical) if its chromatic number equals $k$ and every vertex $v$ of $G$ is critical, that is, deleting $v$ from $G$ strictly lowers the chromatic number.","title":"Vertex-critical graphs"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/cocolorable","kind":"node","lean":"SimpleGraph.Cocolorable","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#cocolorable","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"A graph $G$ is cocolorable with $n$ colors if there is a map from its vertices to the colors $\\{0, \\dots, n-1\\}$ such that each color class induces either an independent set or a complete graph in $G$.","title":"Cocolorable graphs"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/cocolorable-2","kind":"node","lean":"SimpleGraph.Colorable.cocolorable","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#cocolorable-2","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"For every natural number $n$, if a graph $G$ has a proper coloring with $n$ colors, then $G$ is cocolorable with $n$ colors.","title":"Colorable graphs are cocolorable"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/cocolorable-of-fintype","kind":"node","lean":"SimpleGraph.cocolorable_of_fintype","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#cocolorable-of-fintype","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"Every graph $G$ on a finite vertex set $V$ is cocolorable with $|V|$ colors.","title":"Finite graphs are cocolorable"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/cochromatic-number","kind":"node","lean":"SimpleGraph.cochromaticNumber","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#cochromatic-number","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"The cochromatic number of a graph $G$, an element of $\\mathbb{N} \\cup \\{\\infty\\}$, is the least natural number $n$ such that $G$ is cocolorable with $n$ colors, that is, the least number of colors in a coloring of the vertices of $G$ in which each color class induces either an independent set or a complete graph. It is $\\infty$ exactly when $G$ is not cocolorable with any finite number of colors.","title":"Cochromatic number"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/cochromatic-number-eq-s-inf","kind":"node","lean":"SimpleGraph.Cocolorable.cochromaticNumber_eq_sInf","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#cochromatic-number-eq-s-inf","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"If a graph $G$ is cocolorable with $n$ colors for some natural number $n$, then the cochromatic number of $G$ equals the least natural number $m$ such that $G$ is cocolorable with $m$ colors.","title":"Cochromatic number as a minimum"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/cochromatic-number-to-nat-eq-s-inf","kind":"node","lean":"SimpleGraph.cochromaticNumber_toNat_eq_sInf","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#cochromatic-number-to-nat-eq-s-inf","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"For every graph $G$ on a finite vertex set, the cochromatic number of $G$, converted from $\\mathbb{N} \\cup \\{\\infty\\}$ to $\\mathbb{N}$ by sending $\\infty$ to $0$, equals the least natural number $m$ such that $G$ is cocolorable with $m$ colors.","title":"Cochromatic number of a finite graph"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/chromatic-cardinal-of-a-graph","kind":"node","lean":"SimpleGraph.chromaticCardinal","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#chromatic-cardinal-of-a-graph","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"The chromatic cardinal is the minimal number of colors need to color it. In contrast to `chromaticNumber`, which assigns `\u22a4 : \u2115\u221e` to all non-finitely colorable graphs, this definition returns a `Cardinal` and can therefore distinguish between different infinite chromatic numbers.","title":"Chromatic cardinal of a graph"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/indep-num-k","kind":"node","lean":"SimpleGraph.indepNumK","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#indep-num-k","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"For a graph $G$ on a finite vertex set and a natural number $k$, this is the maximum number of vertices in a union of $k$ (possibly empty, possibly overlapping) independent sets of $G$.","title":"Largest union of independent sets"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/cdscolorable","kind":"node","lean":"SimpleGraph.CDSColorable","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#cdscolorable","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"A graph $G$ on a finite vertex set is CDS-colorable if it has a proper coloring $C$ with colors in $\\mathbb{N}$ such that for every $k \\in \\mathbb{N}$, the number of vertices whose color under $C$ is less than $k$ equals the maximum size of a union of $k$ independent sets of $G$.","title":"CDS-colorable graphs"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/is-rainbow","kind":"node","lean":"SimpleGraph.IsRainbow","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#is-rainbow","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"Let $f$ be a graph homomorphism from $H$ to $G$ and let $c$ be a labeling of the edges of $G$ by elements of a type $K$. Then $f$ is rainbow for $c$ if distinct edges of $H$ are mapped to edges of $G$ with distinct labels, that is, the labeling $e \\mapsto c(f(e))$ of the edges of $H$ is injective.","title":"Rainbow homomorphisms"},{"id":"GraphTheory/graph-coloring/part-01/color-critical-graphs/anti-ramsey-num","kind":"node","lean":"SimpleGraph.antiRamseyNum","parent":29425,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/color-critical-graphs/index.html#anti-ramsey-num","scope":"GraphTheory/graph-coloring/part-01/color-critical-graphs","status":"fully_proved","summary":"For a graph $H$ on a finite vertex type and a natural number $n$, the anti-Ramsey number $\\mathrm{AR}(n, H)$ is the largest $k$ for which there is a coloring of the edges of the complete graph $K_n$ on $\\{0, \\dots, n-1\\}$ with colors $\\{0, \\dots, k-1\\}$, every color used, that contains no rainbow copy of $H$. Here a copy of $H$ is an injective homomorphism from $H$ to $K_n$, and it is rainbow when distinct edges of\u2026","title":"Anti-Ramsey number"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe.json","id":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/index.html","scope":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe","status":"can_prove","title":"Cycle lengths in graphs of large chromatic number"},{"id":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem751Wanted.conjecture","parent":29442,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe","status":"fully_proved","summary":"The statement that both of the following hold: (i) for every natural number $k$ there is a simple graph $G$ with chromatic number $4$ in which any two distinct cycle lengths of $G$ differ by at least $k$; (ii) for all natural numbers $k$ and $g$ there is a simple graph $G$ with chromatic number $4$ and girth at least $g$ in which any two distinct cycle lengths of $G$ differ by at least $k$.","title":"Erd\u0151s Problem 751"},{"id":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem751Wanted.conjecture_refuted","parent":29442,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/index.html#conjecture-refuted","scope":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe","status":"can_prove","summary":"It is not the case that both of the following hold: (i) for every natural number $k$ there is a simple graph $G$ with chromatic number $4$ in which any two distinct cycle lengths of $G$ differ by at least $k$; (ii) for all natural numbers $k$ and $g$ there is a simple graph $G$ with chromatic number $4$ and girth at least $g$ in which any two distinct cycle lengths of $G$ differ by at least $k$.","title":"Erd\u0151s Problem 751 is false"},{"id":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/conjecture-2","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem57Wanted.conjecture","parent":29442,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/index.html#conjecture-2","scope":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe","status":"fully_proved","summary":"Let $G$ be a graph on an arbitrary vertex set that has no proper colouring with finitely many colours. Then the family $(1/\\ell)$, indexed by the odd cycle lengths $\\ell$ of $G$ (the odd numbers $\\ell$ such that $G$ contains a cycle of length $\\ell$), is not summable; that is, $\\sum_\\ell 1/\\ell = \\infty$.","title":"Odd cycle lengths and infinite chromatic number (Erd\u0151s Problem 57)"},{"id":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/conjecture-holds","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem57Wanted.conjecture_holds","parent":29442,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/index.html#conjecture-holds","scope":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe","status":"can_prove","summary":"Let $G$ be a graph on an arbitrary vertex set that has no proper colouring with finitely many colours. Then the family $(1/\\ell)$, indexed by the odd cycle lengths $\\ell$ of $G$ (the odd numbers $\\ell$ such that $G$ contains a cycle of length $\\ell$), is not summable; that is, $\\sum_\\ell 1/\\ell = \\infty$.","title":"Resolution of the Erd\u0151s\u2013Hajnal odd cycle problem (Liu\u2013Montgomery)"},{"id":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/conjecture-3","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem63Wanted.conjecture","parent":29442,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/index.html#conjecture-3","scope":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe","status":"fully_proved","summary":"The proposition that every simple graph $G$ with infinite chromatic number has cycles of length $2^n$ for infinitely many $n$: for every natural number $N$ there is some $n \\ge N$ such that $G$ contains a cycle of length $2^n$.","title":"Statement of Erd\u0151s Problem 63"},{"id":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/conjecture-holds-2","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem63Wanted.conjecture_holds","parent":29442,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe/index.html#conjecture-holds-2","scope":"GraphTheory/graph-coloring/part-01/cycle-lengths-in-graphs-of-large-chromatic-numbe","status":"can_prove","summary":"Every simple graph $G$ with infinite chromatic number has cycles of length $2^n$ for infinitely many $n$: for every natural number $N$ there is some $n \\ge N$ such that $G$ contains a cycle of length $2^n$.","title":"Erd\u0151s Problem 63 (Liu\u2013Montgomery)"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring.json","id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","title":"Degeneracy and greedy coloring"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/is-degenerate-on","kind":"node","lean":"SimpleGraph.IsDegenerateOn","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#is-degenerate-on","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex type $V$, let $U \\subseteq V$ and let $k$ be a natural number. Then $G$ is $k$-degenerate on $U$ if every nonempty subset $W \\subseteq U$ contains a vertex $v$ with at most $k$ neighbors in $W$.","title":"Degeneracy on a vertex set"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/is-degenerate","kind":"node","lean":"SimpleGraph.IsDegenerate","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#is-degenerate","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"A simple graph $G$ on a finite vertex type $V$ is $k$-degenerate, for a natural number $k$, if it is $k$-degenerate on the full vertex set $V$: every nonempty set $W$ of vertices contains a vertex with at most $k$ neighbors in $W$.","title":"Degenerate graph"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/mono","kind":"node","lean":"SimpleGraph.IsDegenerateOn.mono","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#mono","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex type $V$, let $U \\subseteq V$, and let $k \\le l$ be natural numbers. If $G$ is $k$-degenerate on $U$, then $G$ is $l$-degenerate on $U$.","title":"Degeneracy on a vertex set is monotone in the degree bound"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/of-subset","kind":"node","lean":"SimpleGraph.IsDegenerateOn.of_subset","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#of-subset","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex type $V$, let $T \\subseteq U \\subseteq V$ and let $k$ be a natural number. If $G$ is $k$-degenerate on $U$, then $G$ is $k$-degenerate on $T$.","title":"Degeneracy on a vertex set passes to subsets"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/is-degenerate-on-2","kind":"node","lean":"SimpleGraph.IsDegenerate.isDegenerateOn","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#is-degenerate-on-2","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex type $V$ and let $k$ be a natural number. If $G$ is $k$-degenerate, then $G$ is $k$-degenerate on every subset $U \\subseteq V$.","title":"A degenerate graph is degenerate on every vertex set"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/exists-small-vertex","kind":"node","lean":"SimpleGraph.IsDegenerateOn.exists_small_vertex","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#exists-small-vertex","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $s \\subseteq V$ be nonempty. If $G$ is $k$-degenerate on $s$, then some vertex $v \\in s$ has at most $k$ neighbors in $s$.","title":"A degenerate vertex set has a vertex of small degree"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/is-degenerate-bot","kind":"node","lean":"SimpleGraph.isDegenerate_bot","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#is-degenerate-bot","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"For every natural number $k$, the edgeless graph on a finite vertex set is $k$-degenerate.","title":"The edgeless graph is degenerate"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/mono-2","kind":"node","lean":"SimpleGraph.IsDegenerate.mono","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#mono-2","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set. If $G$ is $k$-degenerate and $k \\le l$, then $G$ is $l$-degenerate.","title":"Degeneracy is monotone in the bound"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/of-forall-degree-le","kind":"node","lean":"SimpleGraph.IsDegenerate.of_forall_degree_le","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#of-forall-degree-le","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set. If every vertex of $G$ has degree at most $k$, then $G$ is $k$-degenerate.","title":"Bounded maximum degree implies degeneracy"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/is-degeneracy-ordering","kind":"node","lean":"SimpleGraph.IsDegeneracyOrdering","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#is-degeneracy-ordering","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $k$ be a natural number. A map $e : V \\to \\mathbb{N}$ is a degeneracy ordering of $G$ with bound $k$ if $e$ is injective and every vertex $v$ has at most $k$ neighbors $w$ with $e(v) < e(w)$.","title":"Degeneracy ordering"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/exists-inj-on-ordering","kind":"node","lean":"SimpleGraph.IsDegenerateOn.exists_injOn_ordering","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#exists-inj-on-ordering","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $s \\subseteq V$. If $G$ is $k$-degenerate on $s$, then there is a map $e : V \\to \\mathbb{N}$, injective on $s$, such that every $v \\in s$ has at most $k$ neighbors $w \\in s$ with $e(v) < e(w)$.","title":"Degeneracy orderings of a vertex set"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/exists-degeneracy-ordering","kind":"node","lean":"SimpleGraph.IsDegenerate.exists_degeneracyOrdering","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#exists-degeneracy-ordering","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"If $G$ is a $k$-degenerate simple graph on a finite vertex set $V$, then $G$ has a degeneracy ordering with bound $k$: an injective map $e : V \\to \\mathbb{N}$ such that every vertex $v$ has at most $k$ neighbors $w$ with $e(v) < e(w)$.","title":"Degenerate graphs have degeneracy orderings"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/exists-coloring-fun","kind":"node","lean":"SimpleGraph.IsDegenerateOn.exists_coloringFun","parent":29449,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coloring a "],["math","\\(k\\)"],["text","-degenerate vertex set with "],["math","\\(k + 1\\)"],["text"," colors"]],"source":"Coloring a $k$-degenerate vertex set with $k + 1$ colors"},"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#exists-coloring-fun","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, let $s \\subseteq V$ and let $k$ be a natural number. If $G$ is $k$-degenerate on $s$, then there is a map $f : V \\to \\{0, 1, \\dots, k\\}$ such that $f(a) \\ne f(b)$ for all adjacent vertices $a, b \\in s$.","title":"Coloring a $k$-degenerate vertex set with $k + 1$ colors"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/colorable","kind":"node","lean":"SimpleGraph.IsDegenerate.colorable","parent":29449,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-degenerate graphs are "],["math","\\((k + 1)\\)"],["text","-colorable"]],"source":"$k$-degenerate graphs are $(k + 1)$-colorable"},"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#colorable","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $k$ be a natural number. Every $k$-degenerate simple graph on a finite vertex set is $(k + 1)$-colorable.","title":"$k$-degenerate graphs are $(k + 1)$-colorable"},{"id":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/chromatic-number-le","kind":"node","lean":"SimpleGraph.IsDegenerate.chromaticNumber_le","parent":29449,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring/index.html#chromatic-number-le","scope":"GraphTheory/graph-coloring/part-01/degeneracy-and-greedy-coloring","status":"fully_proved","summary":"Let $k$ be a natural number. If $G$ is a $k$-degenerate simple graph on a finite vertex set, then the chromatic number of $G$ is at most $k + 1$.","title":"Chromatic number of a degenerate graph"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/edge-coloring.json","id":"GraphTheory/graph-coloring/part-01/edge-coloring","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","title":"Edge coloring"},{"id":"GraphTheory/graph-coloring/part-01/edge-coloring/is-proper-edge-coloring","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.EdgeColoringWanted.IsProperEdgeColoring","parent":29465,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html#is-proper-edge-coloring","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on $V$. An edge labeling $C$ of $G$ with colors in $K$ is a proper edge coloring when any two distinct edges of $G$ that share a vertex receive different colors: whenever a vertex $v$ is adjacent to $w_1$ and to $w_2$ with $w_1 \\ne w_2$, the edges $v w_1$ and $v w_2$ have different colors under $C$.","title":"Proper edge coloring"},{"id":"GraphTheory/graph-coloring/part-01/edge-coloring/k-nig-s-edge-coloring-theorem-for-regula","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.EdgeColoringWanted.multigraph_regular_color","parent":29465,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html#k-nig-s-edge-coloring-theorem-for-regula","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","summary":"K\u00f6nig's line coloring, regular multigraph case: a `d`-regular bipartite multigraph (given by endpoint projections) is `d`-edge-colorable. The proof extracts a perfect matching by Hall's theorem and inducts. Shared API also used by the multigraph Galvin development.","title":"K\u00f6nig's edge coloring theorem for regular bipartite multigraphs"},{"id":"GraphTheory/graph-coloring/part-01/edge-coloring/k-nig-s-edge-coloring-theorem-for-bounde","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.EdgeColoringWanted.multigraph_bounded_color","parent":29465,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html#k-nig-s-edge-coloring-theorem-for-bounde","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","summary":"K\u00f6nig's line coloring, bounded-degree multigraph case: a bipartite multigraph of maximum degree at most `d` is `d`-edge-colorable. The proof pads both sides with dummy edges to reach the regular case. Shared API also used by the multigraph Galvin development.","title":"K\u00f6nig's edge coloring theorem for bounded-degree bipartite multigraphs"},{"id":"GraphTheory/graph-coloring/part-01/edge-coloring/konig-line-coloring","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.EdgeColoringWanted.konig_line_coloring","parent":29465,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html#konig-line-coloring","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","summary":"Every finite bipartite simple graph $G$ has a proper edge coloring with $\\Delta(G)$ colors, where $\\Delta(G)$ is the maximum degree of $G$.","title":"K\u00f6nig's line coloring theorem"},{"id":"GraphTheory/graph-coloring/part-01/edge-coloring/is-proper-sym2-coloring","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.EdgeColoringWanted.IsProperSym2Coloring","parent":29465,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html#is-proper-sym2-coloring","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on $V$. A map $c$ that assigns a color in $K$ to every unordered pair of elements of $V$ is a proper coloring of unordered pairs for $G$ when, for all vertices $x, y, z$ such that $x$ is adjacent to both $y$ and $z$ and $y \\ne z$, the pairs $\\{x, y\\}$ and $\\{x, z\\}$ receive different colors under $c$.","title":"Proper coloring of unordered pairs"},{"id":"GraphTheory/graph-coloring/part-01/edge-coloring/is-proper-edge-coloring-of-is-proper-sym","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.EdgeColoringWanted.isProperEdgeColoring_of_isProperSym2Coloring","parent":29465,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html#is-proper-edge-coloring-of-is-proper-sym","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on $V$ and let $c$ assign a color in $K$ to every unordered pair of elements of $V$. If $c$ is a proper coloring of unordered pairs for $G$, then the edge labeling of $G$ that gives each edge $e$ the color $c(e)$ is a proper edge coloring of $G$.","title":"Proper colorings of unordered pairs restrict to proper edge colorings"},{"id":"GraphTheory/graph-coloring/part-01/edge-coloring/mono","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.EdgeColoringWanted.IsProperSym2Coloring.mono","parent":29465,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html#mono","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","summary":"Let $G$ and $H$ be simple graphs on $V$ such that every edge of $H$ is an edge of $G$. If a map $c$ from unordered pairs of elements of $V$ to $K$ is a proper coloring of unordered pairs for $G$, then it is also a proper coloring of unordered pairs for $H$.","title":"Proper colorings of unordered pairs pass to subgraphs"},{"id":"GraphTheory/graph-coloring/part-01/edge-coloring/vizing","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.EdgeColoringWanted.vizing","parent":29465,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/edge-coloring/index.html#vizing","scope":"GraphTheory/graph-coloring/part-01/edge-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $\\Delta$ be its maximum degree. Then $G$ has a proper edge coloring with $\\Delta + 1$ colors: there is an assignment of colors from $\\{0, 1, \\dots, \\Delta\\}$ to the edges of $G$ such that any two distinct edges sharing a vertex receive different colors.","title":"Vizing's theorem"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/four-color-theorem.json","id":"GraphTheory/graph-coloring/part-01/four-color-theorem","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/four-color-theorem/index.html","scope":"GraphTheory/graph-coloring/part-01/four-color-theorem","status":"can_prove","title":"Four color theorem"},{"id":"GraphTheory/graph-coloring/part-01/four-color-theorem/has-k5-minor","kind":"node","lean":"MetaMathlibExt.HasK5Minor","parent":29474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Graphs with a "],["math","\\(K_5\\)"],["text"," minor"]],"source":"Graphs with a $K_5$ minor"},"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/four-color-theorem/index.html#has-k5-minor","scope":"GraphTheory/graph-coloring/part-01/four-color-theorem","status":"fully_proved","summary":"A simple graph $G$ has $K_5$ as a minor if there are five nonempty, pairwise disjoint sets of vertices $S_1, \\dots, S_5$ such that any two vertices in the same set are joined by a walk in $G$ all of whose vertices lie in that set, and for all distinct $i, j$ some vertex of $S_i$ is adjacent in $G$ to some vertex of $S_j$.","title":"Graphs with a $K_5$ minor"},{"id":"GraphTheory/graph-coloring/part-01/four-color-theorem/has-k33-minor","kind":"node","lean":"MetaMathlibExt.HasK33Minor","parent":29474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Graphs with a "],["math","\\(K_{3,3}\\)"],["text"," minor"]],"source":"Graphs with a $K_{3,3}$ minor"},"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/four-color-theorem/index.html#has-k33-minor","scope":"GraphTheory/graph-coloring/part-01/four-color-theorem","status":"fully_proved","summary":"A simple graph $G$ has $K_{3,3}$ as a minor if there are six nonempty, pairwise disjoint sets of vertices $L_1, L_2, L_3, R_1, R_2, R_3$ such that any two vertices in the same set are joined by a walk in $G$ all of whose vertices lie in that set, and for all $i, j \\in \\{1, 2, 3\\}$ some vertex of $L_i$ is adjacent in $G$ to some vertex of $R_j$.","title":"Graphs with a $K_{3,3}$ minor"},{"id":"GraphTheory/graph-coloring/part-01/four-color-theorem/is-planar","kind":"node","lean":"MetaMathlibExt.IsPlanar","parent":29474,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/four-color-theorem/index.html#is-planar","scope":"GraphTheory/graph-coloring/part-01/four-color-theorem","status":"fully_proved","summary":"A simple graph $G$ is called planar here if it has neither $K_5$ nor $K_{3,3}$ as a minor.","title":"Planarity by forbidden minors"},{"id":"GraphTheory/graph-coloring/part-01/four-color-theorem/four-color-theorem","kind":"node","lean":"MetaMathlibExt.fourColorTheorem","parent":29474,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/four-color-theorem/index.html#four-color-theorem","scope":"GraphTheory/graph-coloring/part-01/four-color-theorem","status":"can_prove","summary":"Let $G$ be a simple graph on a finite vertex set $V$ that has neither $K_5$ nor $K_{3,3}$ as a minor, and let $\\ell$ assign a natural number $\\ell(u, v)$ to every ordered pair of vertices $u, v$ such that $u$ and $v$ are adjacent in $G$ if and only if $u \\ne v$ and $\\ell(u, v) > 0$. Then $G$ has a proper vertex coloring with four colors.","title":"Four color theorem"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/fractional-coloring.json","id":"GraphTheory/graph-coloring/part-01/fractional-coloring","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/fractional-coloring/index.html","scope":"GraphTheory/graph-coloring/part-01/fractional-coloring","status":"fully_proved","title":"Fractional coloring"},{"id":"GraphTheory/graph-coloring/part-01/fractional-coloring/frac-clique-number","kind":"node","lean":"SimpleGraph.fracCliqueNumber","parent":29479,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/fractional-coloring/index.html#frac-clique-number","scope":"GraphTheory/graph-coloring/part-01/fractional-coloring","status":"fully_proved","summary":"Let $X$ be a graph with finite vertex set $V(X)$. A *fractional clique* of $X$ is a function $f \\colon V(X) \\to \\mathbb{R}$ with $f(v) \\ge 0$ for every vertex $v$, such that $\\sum_{v \\in S} f(v) \\le 1$ for every independent set $S$ of $X$. The *weight* of $f$ is $\\sum_{v \\in V(X)} f(v)$. The *fractional clique number* $\\omega^*(X)$ of $X$ is the supremum of the weights of all fractional cliques of $X$.","title":"Fractional clique number"},{"id":"GraphTheory/graph-coloring/part-01/fractional-coloring/frac-chromatic-number","kind":"node","lean":"SimpleGraph.fracChromaticNumber","parent":29479,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/fractional-coloring/index.html#frac-chromatic-number","scope":"GraphTheory/graph-coloring/part-01/fractional-coloring","status":"fully_proved","summary":"Let $X$ be a graph with finite vertex set $V(X)$. A *fractional colouring* of $X$ assigns to each set $S$ of vertices of $X$ a real weight $f(S) \\ge 0$, with $f(S) = 0$ unless $S$ is independent, such that $\\sum_{S \\ni x} f(S) \\ge 1$ for every vertex $x$ of $X$, the sum running over all sets of vertices containing $x$. The *weight* of $f$ is the sum of $f(S)$ over all sets $S$ of vertices of $X$. The *fractional\u2026","title":"Fractional chromatic number"},{"id":"GraphTheory/graph-coloring/part-01/fractional-coloring/is-regular","kind":"node","lean":"MetaMathlibExt.FracColouring.IsRegular","parent":29479,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/fractional-coloring/index.html#is-regular","scope":"GraphTheory/graph-coloring/part-01/fractional-coloring","status":"fully_proved","summary":"Let $X$ be a finite simple graph. A fractional colouring of $X$ assigns to each set $S$ of vertices of $X$ a real weight $f(S) \\ge 0$, with $f(S) = 0$ unless $S$ is independent, such that $\\sum_{S \\ni x} f(S) \\ge 1$ for every vertex $x$, the sum running over all sets of vertices containing $x$. The fractional colouring $f$ is *regular* if $\\sum_{S \\ni x} f(S) = 1$ for every vertex $x$ of $X$. Since $f$ vanishes off\u2026","title":"Regular fractional colouring"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem.json","id":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/index.html","scope":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem","status":"can_prove","title":"Hajnal\u2013Szemer\u00e9di theorem"},{"id":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/exists-equitable-coloring-of-max-degree","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.HajnalSzemerediWanted.exists_equitable_coloring_of_maxDegree_lt","parent":29483,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/index.html#exists-equitable-coloring-of-max-degree","scope":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set with maximum degree $\\Delta(G)$, and let $k$ be a natural number with $\\Delta(G) < k$. Then $G$ has a proper coloring with the $k$ colors $0, 1, \\dots, k-1$ that is equitable: for any two colors $i$ and $j$, the number of vertices of color $i$ is at most one more than the number of vertices of color $j$.","title":"Equitable coloring with more colors than the maximum degree"},{"id":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/hajnal-szemeredi-equitable-coloring","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.HajnalSzemerediWanted.hajnal_szemeredi_equitable_coloring","parent":29483,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/index.html#hajnal-szemeredi-equitable-coloring","scope":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set with maximum degree $\\Delta(G)$, and let $k$ be a natural number with $\\Delta(G) < k$. Then $G$ has a proper coloring with the $k$ colors $0, 1, \\dots, k-1$ that is equitable: for any two colors $i$ and $j$, the number of vertices of color $i$ is at most one more than the number of vertices of color $j$.","title":"Hajnal\u2013Szemer\u00e9di theorem"},{"id":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem914Wanted.conjecture","parent":29483,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem","status":"fully_proved","summary":"The proposition asserts that for all integers $r \\ge 2$ and $m \\ge 1$ and every finite simple graph $G$ with exactly $rm$ vertices, (a) if every vertex of $G$ has degree at least $m(r - 1)$, then $G$ contains $m$ pairwise disjoint cliques of $r$ vertices each, and (b) if every vertex of $G$ has degree at most $m - 1$, then $G$ has a proper colouring with $m$ colours in which every colour class has exactly $r$\u2026","title":"Erd\u0151s Problem 914 on clique factors and equitable colourings"},{"id":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem914Wanted.conjecture_holds","parent":29483,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem/index.html#conjecture-holds","scope":"GraphTheory/graph-coloring/part-01/hajnal-szemer-di-theorem","status":"can_prove","summary":"For all integers $r \\ge 2$ and $m \\ge 1$ and every finite simple graph $G$ with exactly $rm$ vertices, (a) if every vertex of $G$ has degree at least $m(r - 1)$, then $G$ contains $m$ pairwise disjoint cliques of $r$ vertices each, and (b) if every vertex of $G$ has degree at most $m - 1$, then $G$ has a proper colouring with $m$ colours in which every colour class has exactly $r$ vertices.","title":"Hajnal\u2013Szemer\u00e9di theorem in clique-factor and equitable-colouring form"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic.json","id":"GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic/index.html","scope":"GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic","status":"can_prove","title":"Independent sets in graphs of infinite chromatic number"},{"id":"GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem750Wanted.conjecture","parent":29488,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic","status":"fully_proved","summary":"The statement that the following three assertions all hold. (i) For every function $f : \\mathbb{N} \\to \\mathbb{R}_{\\ge 0}$ with $f(m) \\to \\infty$ as $m \\to \\infty$ there is a simple graph $G$ with infinite chromatic number such that for every $m > 0$, every set $S$ of exactly $m$ vertices of $G$ contains an independent set $I$ of $G$ with $|I| \\ge m/2 - f(m)$. (ii) For every real $\\varepsilon > 0$ there is a simple\u2026","title":"Erd\u0151s Problem 750"},{"id":"GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem750Wanted.conjecture_holds","parent":29488,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic/index.html#conjecture-holds","scope":"GraphTheory/graph-coloring/part-01/independent-sets-in-graphs-of-infinite-chromatic","status":"can_prove","summary":"The following three statements hold. (i) For every function $f : \\mathbb{N} \\to \\mathbb{R}_{\\ge 0}$ with $f(m) \\to \\infty$ as $m \\to \\infty$ there is a simple graph $G$ with infinite chromatic number such that for every $m > 0$, every set $S$ of exactly $m$ vertices of $G$ contains an independent set $I$ of $G$ with $|I| \\ge m/2 - f(m)$. (ii) For every real $\\varepsilon > 0$ there is a simple graph $G$ with\u2026","title":"Erd\u0151s Problem 750 is true"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/kneser-graphs.json","id":"GraphTheory/graph-coloring/part-01/kneser-graphs","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/kneser-graphs/index.html","scope":"GraphTheory/graph-coloring/part-01/kneser-graphs","status":"fully_proved","title":"Kneser graphs"},{"id":"GraphTheory/graph-coloring/part-01/kneser-graphs/kneser-graph","kind":"node","lean":"MetaMathlibExt.kneserGraph","parent":29491,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/kneser-graphs/index.html#kneser-graph","scope":"GraphTheory/graph-coloring/part-01/kneser-graphs","status":"fully_proved","summary":"For natural numbers $v$ and $k$, the Kneser graph $K_{v:k}$ has as vertices the $k$-element subsets of the $v$-element set $\\{0, 1, \\dots, v-1\\}$, with two distinct vertices adjacent when they are disjoint.","title":"Kneser graph"},{"id":"GraphTheory/graph-coloring/part-01/kneser-graphs/kneser-graph-adj-iff","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.KneserLovaszWanted.kneserGraph_adj_iff","parent":29491,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/kneser-graphs/index.html#kneser-graph-adj-iff","scope":"GraphTheory/graph-coloring/part-01/kneser-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and $k$-element subsets $x$ and $y$ of $\\{0, 1, \\dots, n-1\\}$, the vertices $x$ and $y$ of the Kneser graph $K(n, k)$ are adjacent if and only if $x$ and $y$ are disjoint and $x \\ne y$.","title":"Adjacency in the Kneser graph"},{"id":"GraphTheory/graph-coloring/part-01/kneser-graphs/kneser-lovasz","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.KneserLovaszWanted.kneser_lovasz","parent":29491,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/kneser-graphs/index.html#kneser-lovasz","scope":"GraphTheory/graph-coloring/part-01/kneser-graphs","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers with $0 < k$ and $2k \\le n$. The Kneser graph $K(n, k)$, whose vertices are the $k$-element subsets of $\\{0, 1, \\dots, n-1\\}$ and in which two such subsets are adjacent when they are disjoint and distinct, has chromatic number $n - 2k + 2$.","title":"Kneser\u2013Lov\u00e1sz theorem"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/list-coloring.json","id":"GraphTheory/graph-coloring/part-01/list-coloring","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"can_prove","title":"List coloring"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/is-list-colorable","kind":"node","lean":"SimpleGraph.IsListColorable","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#is-list-colorable","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"A simple graph $G$ is $q$-list-colorable if, for every set of colors and every assignment of a finite list $L(v)$ of at least $q$ colors to each vertex $v$, there is a proper coloring $c$ of $G$ with $c(v) \\in L(v)$ for every vertex $v$.","title":"List colorability"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/is-choice-number","kind":"node","lean":"SimpleGraph.IsChoiceNumber","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#is-choice-number","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"A natural number $q$ is the choice number of a simple graph $G$ if $G$ is $q$-list-colorable but is not $r$-list-colorable for any $r < q$.","title":"Choice number"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/mono","kind":"node","lean":"SimpleGraph.IsListColorable.mono","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#mono","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"If a simple graph $G$ is $q$-list-colorable and $q \\le r$, then $G$ is $r$-list-colorable.","title":"Monotonicity of list colorability"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/colorable","kind":"node","lean":"SimpleGraph.IsListColorable.colorable","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#colorable","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"If a simple graph $G$ is $q$-list-colorable, then $G$ is $q$-colorable, that is, it has a proper coloring with $q$ colors.","title":"List-colorable graphs are colorable"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/is-partial-list-coloring","kind":"node","lean":"MathlibExt.Combinatorics.PartialListColoringProportionalBoundWanted.IsPartialListColoring","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#is-partial-list-coloring","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"Let $G$ be a simple graph and assign to each vertex $v$ a finite list $L(v)$ of colors. A finite set $S$ of vertices is partially list-colorable from $L$ when the subgraph of $G$ induced by $S$ has a proper coloring $c$ with $c(v) \\in L(v)$ for every $v \\in S$. No colors are assigned to vertices outside $S$.","title":"Partial list colorability"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.PartialListColoringProportionalBoundWanted.conjecture","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"Let $G$ be a finite simple graph on $n$ vertices whose choice number $q$ is positive, let $t \\le q$ be a natural number, and assign to each vertex $v$ a list $L(v)$ of exactly $t$ colors. Then there is a set $S$ of vertices of $G$ that is partially list-colorable from $L$ (the subgraph of $G$ induced by $S$ has a proper coloring giving each vertex a color from its list) and satisfies $tn \\le q|S|$, that is,\u2026","title":"Albertson\u2013Grossman\u2013Haas partial list coloring conjecture"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.PartialListColoringProportionalBoundWanted.conjecture_refuted","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#conjecture-refuted","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"can_prove","summary":"There exist a finite simple graph $G$ on $n$ vertices whose choice number $q$ is positive, a natural number $t \\le q$, and an assignment to each vertex $v$ of a list $L(v)$ of exactly $t$ colors, such that every set $S$ of vertices of $G$ whose induced subgraph has a proper coloring giving each vertex a color from its list satisfies $q|S| < tn$.","title":"The partial list coloring conjecture is false"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/is-list-coloring","kind":"node","lean":"MathlibExt.GraphTheory.PartialListColoringWanted.IsListColoring","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#is-list-coloring","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"Let $G$ be a graph on a finite vertex set $V$, let $L$ assign to each vertex $v$ a finite list $L(v)$ of colours (natural numbers), and let $S \\subseteq V$. A map $f : V \\to \\mathbb{N}$ is a proper $L$-colouring of $S$ when $f(v) \\in L(v)$ for every $v \\in S$ and $f(v) \\ne f(w)$ for all adjacent $v, w \\in S$.","title":"Proper list colouring of a vertex subset"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/lambda-max","kind":"node","lean":"MathlibExt.GraphTheory.PartialListColoringWanted.lambdaMax","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#lambda-max","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"For a graph $G$ on a finite vertex set and an assignment $L$ of a finite list $L(v)$ of natural-number colours to each vertex $v$, $\\lambda_L$ is the largest $k$ such that some set of $k$ vertices of $G$ has a proper $L$-colouring.","title":"Maximum number of list-colourable vertices"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/lambda-min","kind":"node","lean":"MathlibExt.GraphTheory.PartialListColoringWanted.lambdaMin","parent":29495,"reading_title":{"format":"inline-parts-v1","parts":[["text","The partial list colouring number "],["math","\\(\\lambda_t\\)"]],"source":"The partial list colouring number $\\lambda_t$"},"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#lambda-min","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"For a graph $G$ on a finite vertex set and a natural number $t$, $\\lambda_t$ is the minimum of $\\lambda_L$ over all assignments $L$ of a list $L(v)$ of exactly $t$ natural-number colours to each vertex $v$, where $\\lambda_L$ is the largest $k$ such that some set of $k$ vertices of $G$ has a proper $L$-colouring.","title":"The partial list colouring number $\\lambda_t$"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/conjecture-2","kind":"node","lean":"MathlibExt.GraphTheory.PartialListColoringWanted.conjecture","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#conjecture-2","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"fully_proved","summary":"For every finite graph $G$ with choice number $\\chi_\\ell$ and all natural numbers $r$, $s$ with $1 \\le r \\le s \\le \\chi_\\ell$, one has $\\lambda_r / r \\ge \\lambda_s / s$, where $\\lambda_t$ is the minimum, over all assignments of lists of exactly $t$ colours to the vertices, of the largest number of vertices of $G$ that can be properly coloured from their lists.","title":"Iradmusa's partial list colouring conjecture"},{"id":"GraphTheory/graph-coloring/part-01/list-coloring/conjecture-refuted-2","kind":"node","lean":"MathlibExt.GraphTheory.PartialListColoringWanted.conjecture_refuted","parent":29495,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/list-coloring/index.html#conjecture-refuted-2","scope":"GraphTheory/graph-coloring/part-01/list-coloring","status":"can_prove","summary":"There exist a finite graph $G$ with choice number $\\chi_\\ell$ and natural numbers $r$, $s$ with $1 \\le r \\le s \\le \\chi_\\ell$ such that $\\lambda_r / r < \\lambda_s / s$, where $\\lambda_t$ is the minimum, over all assignments of lists of exactly $t$ colours to the vertices, of the largest number of vertices of $G$ that can be properly coloured from their lists. So Iradmusa's partial list colouring conjecture fails.","title":"The partial list colouring conjecture is false"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/odd-distance-graph.json","id":"GraphTheory/graph-coloring/part-01/odd-distance-graph","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/odd-distance-graph/index.html","scope":"GraphTheory/graph-coloring/part-01/odd-distance-graph","status":"can_prove","title":"Odd distance graph"},{"id":"GraphTheory/graph-coloring/part-01/odd-distance-graph/odd-distance-graph","kind":"node","lean":"MathlibExt.GraphTheory.OddDistanceGraphInfiniteChromaticWanted.ODGIC.OddDistanceGraph","parent":29508,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/odd-distance-graph/index.html#odd-distance-graph","scope":"GraphTheory/graph-coloring/part-01/odd-distance-graph","status":"fully_proved","summary":"The odd distance graph is the simple graph on the Euclidean plane $\\mathbb{R}^2$ in which two points $x$ and $y$ are adjacent when their Euclidean distance is an odd integer.","title":"Odd distance graph"},{"id":"GraphTheory/graph-coloring/part-01/odd-distance-graph/odd-distance-chromatic-infinite","kind":"node","lean":"MathlibExt.GraphTheory.OddDistanceGraphInfiniteChromaticWanted.ODGIC.OddDistanceChromaticInfinite","parent":29508,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/odd-distance-graph/index.html#odd-distance-chromatic-infinite","scope":"GraphTheory/graph-coloring/part-01/odd-distance-graph","status":"fully_proved","summary":"The proposition that the odd distance graph on $\\mathbb{R}^2$ has infinite chromatic number, that is, has no proper colouring with finitely many colours.","title":"Infinite chromatic number of the odd distance graph"},{"id":"GraphTheory/graph-coloring/part-01/odd-distance-graph/odd-distance-chromatic-infinite-holds","kind":"node","lean":"MathlibExt.GraphTheory.OddDistanceGraphInfiniteChromaticWanted.OddDistanceChromaticInfinite_holds","parent":29508,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/odd-distance-graph/index.html#odd-distance-chromatic-infinite-holds","scope":"GraphTheory/graph-coloring/part-01/odd-distance-graph","status":"can_prove","summary":"The odd distance graph on the Euclidean plane $\\mathbb{R}^2$, in which two points are adjacent when their distance is an odd integer, has infinite chromatic number: it has no proper colouring with finitely many colours.","title":"Davies' odd distance theorem"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/perfect-graphs.json","id":"GraphTheory/graph-coloring/part-01/perfect-graphs","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"can_prove","title":"Perfect graphs"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/is-perfect","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.PerfectGraphWanted.IsPerfect","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#is-perfect","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"A finite simple graph $G$ is perfect if, for every set $S$ of its vertices, the induced subgraph $G[S]$ has chromatic number equal to its clique number, the largest size of a clique in $G[S]$.","title":"Perfect graph"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/perf-ineq","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.PerfectGraphWanted.PerfIneq","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#perf-ineq","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"A finite simple graph $G$ satisfies condition (P) if, for every set $S$ of its vertices, the induced subgraph $G[S]$, with independence number $\\alpha(G[S])$ and clique number $\\omega(G[S])$, satisfies $|S| \\le \\alpha(G[S]) \\cdot \\omega(G[S])$.","title":"Gasparian's condition (P)"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/perfect-implies-ineq","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.PerfectGraphWanted.perfect_implies_ineq","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#perfect-implies-ineq","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"Every finite perfect graph $G$ satisfies Gasparian's condition (P): for every set $S$ of its vertices, the induced subgraph $G[S]$, with independence number $\\alpha(G[S])$ and clique number $\\omega(G[S])$, satisfies $|S| \\le \\alpha(G[S]) \\cdot \\omega(G[S])$.","title":"Perfect graphs satisfy condition (P)"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/lovasz-weak-perfect-graph-general","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.PerfectGraphWanted.lovasz_weak_perfect_graph_general","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#lovasz-weak-perfect-graph-general","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"A finite simple graph $G$ is perfect if and only if its complement $\\overline{G}$ is perfect.","title":"Weak perfect graph theorem"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/lovasz-weak-perfect-graph","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.PerfectGraphWanted.lovasz_weak_perfect_graph","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#lovasz-weak-perfect-graph","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"A finite simple graph $G$ is perfect if and only if its complement $\\overline{G}$ is perfect.","title":"Lov\u00e1sz's weak perfect graph theorem"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/perfect-graph-instance","kind":"node","lean":"MathlibExt.Combinatorics.PerfectGraphNPWanted.PerfectGraphInstance","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#perfect-graph-instance","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"An instance consists of a natural number $n$ and a Boolean $n \\times n$ matrix $A$ with rows and columns indexed by $\\{0, \\dots, n-1\\}$, read as the adjacency matrix of a graph on the vertices $0, \\dots, n-1$. The matrix is arbitrary; symmetry and the absence of loops are imposed separately by the validity condition.","title":"Adjacency-matrix graph instance"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/valid","kind":"node","lean":"MathlibExt.Combinatorics.PerfectGraphNPWanted.PerfectGraphInstance.Valid","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#valid","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"An instance with $n$ vertices and adjacency matrix $A$ is valid when $A$ is symmetric, that is, $A_{yx}$ is true whenever $A_{xy}$ is true, and has no loops, that is, $A_{xx}$ is false for every vertex $x$.","title":"Valid adjacency-matrix instance"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/to-simple-graph","kind":"node","lean":"MathlibExt.Combinatorics.PerfectGraphNPWanted.PerfectGraphInstance.toSimpleGraph","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#to-simple-graph","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"A valid instance with $n$ vertices and adjacency matrix $A$ determines the simple graph on the vertex set $\\{0, \\dots, n-1\\}$ in which vertices $x$ and $y$ are adjacent exactly when $A_{xy}$ is true.","title":"Graph of a valid instance"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/perfect-graph-problem","kind":"node","lean":"MathlibExt.Combinatorics.PerfectGraphNPWanted.perfectGraphProblem","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#perfect-graph-problem","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"The decision problem on bitstrings that accepts a bitstring $x$ exactly when $x$ decodes, under the bitstring encoding of adjacency-matrix instances, to a valid instance whose graph is perfect. Bitstrings that do not decode to an instance, or decode to an invalid one, are rejected.","title":"The perfect-graph decision problem"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.PerfectGraphNPWanted.conjecture","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"fully_proved","summary":"The perfect-graph decision problem, which accepts exactly the bitstrings that decode to a valid adjacency-matrix instance whose graph is perfect, lies in the complexity class NP: there are a polynomial $p$ with natural-number coefficients and a polynomial-time verifier $R$ on pairs of bitstrings such that a bitstring $x$ is accepted if and only if $R$ accepts $(x, w)$ for some bitstring $w$ of length at most\u2026","title":"Edmonds's question: an NP description of perfect graphs"},{"id":"GraphTheory/graph-coloring/part-01/perfect-graphs/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.PerfectGraphNPWanted.conjecture_holds","parent":29512,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/perfect-graphs/index.html#conjecture-holds","scope":"GraphTheory/graph-coloring/part-01/perfect-graphs","status":"can_prove","summary":"The perfect-graph decision problem, which accepts exactly the bitstrings that decode to a valid adjacency-matrix instance whose graph is perfect, lies in the complexity class NP.","title":"Perfect graphs have an NP description"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture.json","id":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/index.html","scope":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture","status":"can_prove","title":"Petersen coloring conjecture"},{"id":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/pvertex","kind":"node","lean":"MathlibExt.GraphTheory.PetersenColoringConjectureWanted.PVertex","parent":29524,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/index.html#pvertex","scope":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture","status":"fully_proved","summary":"The vertex set of the Petersen graph is $\\mathbb{Z}/5\\mathbb{Z} \\times \\{\\mathrm{false}, \\mathrm{true}\\}$; the vertices $(i, \\mathrm{false})$ with $i \\in \\mathbb{Z}/5\\mathbb{Z}$ form the outer 5-cycle and the vertices $(i, \\mathrm{true})$ form the inner 5-star.","title":"Vertices of the Petersen graph"},{"id":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/petersen-edge-set","kind":"node","lean":"MathlibExt.GraphTheory.PetersenColoringConjectureWanted.petersenEdgeSet","parent":29524,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/index.html#petersen-edge-set","scope":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture","status":"fully_proved","summary":"The edge set of the Petersen graph on $\\mathbb{Z}/5\\mathbb{Z} \\times \\{\\mathrm{false}, \\mathrm{true}\\}$ consists of the unordered pairs $\\{(i, \\mathrm{false}), (i + 1, \\mathrm{false})\\}$ (the outer 5-cycle), $\\{(i, \\mathrm{false}), (i, \\mathrm{true})\\}$ (the spokes) and $\\{(i, \\mathrm{true}), (i + 2, \\mathrm{true})\\}$ (the inner 5-star), for $i \\in \\mathbb{Z}/5\\mathbb{Z}$.","title":"Edges of the Petersen graph"},{"id":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/petersen-graph","kind":"node","lean":"MathlibExt.GraphTheory.PetersenColoringConjectureWanted.petersenGraph","parent":29524,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/index.html#petersen-graph","scope":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture","status":"fully_proved","summary":"The Petersen graph is the simple graph on $\\mathbb{Z}/5\\mathbb{Z} \\times \\{\\mathrm{false}, \\mathrm{true}\\}$ whose edges are the outer 5-cycle edges $\\{(i, \\mathrm{false}), (i + 1, \\mathrm{false})\\}$, the spokes $\\{(i, \\mathrm{false}), (i, \\mathrm{true})\\}$ and the inner 5-star edges $\\{(i, \\mathrm{true}), (i + 2, \\mathrm{true})\\}$ for $i \\in \\mathbb{Z}/5\\mathbb{Z}$.","title":"Petersen graph"},{"id":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/is-petersen-coloring","kind":"node","lean":"MathlibExt.GraphTheory.PetersenColoringConjectureWanted.IsPetersenColoring","parent":29524,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/index.html#is-petersen-coloring","scope":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex type $V$ and let $P$ be the Petersen graph. A map $f$ from unordered pairs of vertices of $G$ to unordered pairs of vertices of $P$ is a Petersen coloring of $G$ when $f$ sends every edge of $G$ to an edge of $P$ and, for every vertex $v$ of $G$, there is a vertex $w$ of $P$ such that $f$ maps the set of edges of $G$ incident to $v$ bijectively onto the set of edges of $P$\u2026","title":"Petersen coloring"},{"id":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/conjecture","kind":"node","lean":"MathlibExt.GraphTheory.PetersenColoringConjectureWanted.conjecture","parent":29524,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture","status":"fully_proved","summary":"For every finite vertex type $V$ and every simple graph $G$ on $V$, if every vertex of $G$ has exactly three neighbors and $G$ is bridgeless, then $G$ admits a Petersen coloring.","title":"Petersen coloring conjecture"},{"id":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/conjecture-refuted","kind":"node","lean":"MathlibExt.GraphTheory.PetersenColoringConjectureWanted.conjecture_refuted","parent":29524,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture/index.html#conjecture-refuted","scope":"GraphTheory/graph-coloring/part-01/petersen-coloring-conjecture","status":"can_prove","summary":"There are a finite vertex type $V$ and a simple graph $G$ on $V$ in which every vertex has exactly three neighbors and no edge is a bridge, such that $G$ has no Petersen coloring: no map $f$ from unordered pairs of vertices of $G$ to unordered pairs of vertices of the Petersen graph $P$ sends every edge of $G$ to an edge of $P$ and, for every vertex $v$ of $G$, maps the edges of $G$ at $v$ bijectively onto the\u2026","title":"The Petersen coloring conjecture is false"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number.json","id":"GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number/index.html","scope":"GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number","status":"can_prove","title":"Subgraphs of large cochromatic number"},{"id":"GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem760Wanted.conjecture","parent":29531,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number","status":"fully_proved","summary":"The proposition asserts two statements. First, there is a real number $c > 0$ such that for every simple graph $G$ on a finite vertex set with chromatic number $m$ there are a subgraph $H$ of $G$ and a natural number $k$ with $k \\ge c\\,m / \\log m$ such that the cochromatic number of $H$ is at least $k$. Second, there is a real number $c > 0$ with the same property for the bound $k \\ge c\\sqrt{m / \\log m}$.","title":"Erd\u0151s Problem 760: subgraphs of large cochromatic number"},{"id":"GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem760Wanted.conjecture_holds","parent":29531,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number/index.html#conjecture-holds","scope":"GraphTheory/graph-coloring/part-01/subgraphs-of-large-cochromatic-number","status":"can_prove","summary":"There is a real number $c > 0$ such that for every simple graph $G$ on a finite vertex set with chromatic number $m$ there are a subgraph $H$ of $G$ and a natural number $k$ with $k \\ge c\\,m / \\log m$ such that the cochromatic number of $H$ is at least $k$. Moreover, there is a real number $c > 0$ with the same property for the bound $k \\ge c\\sqrt{m / \\log m}$.","title":"Alon\u2013Krivelevich\u2013Sudakov theorem on subgraphs of large cochromatic number"},{"data_url":"scopes/GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe.json","id":"GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe","kind":"scope","navigation":"module","parent":29406,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe/index.html","scope":"GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe","status":"can_prove","title":"Triangle-free subgraphs of large chromatic number"},{"id":"GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem923Wanted.conjecture","parent":29534,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe/index.html#conjecture","scope":"GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe","status":"fully_proved","summary":"The proposition asserts that for every natural number $n$ there is a natural number $k$ such that every simple graph $G$, on any vertex set, whose chromatic number is at least $k$ has a triangle-free subgraph $H$ on the same vertex set whose chromatic number is at least $n$. Chromatic numbers may be infinite.","title":"Erd\u0151s Problem 923 on triangle-free subgraphs of large chromatic number"},{"id":"GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem923Wanted.conjecture_holds","parent":29534,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe/index.html#conjecture-holds","scope":"GraphTheory/graph-coloring/part-01/triangle-free-subgraphs-of-large-chromatic-numbe","status":"can_prove","summary":"For every natural number $n$ there is a natural number $k$ such that every simple graph $G$, on any vertex set, whose chromatic number is at least $k$ has a triangle-free subgraph $H$ on the same vertex set whose chromatic number is at least $n$. Chromatic numbers may be infinite.","title":"R\u00f6dl's theorem on triangle-free subgraphs"},{"data_url":"scopes/GraphTheory/graph-coloring/part-02.json","id":"GraphTheory/graph-coloring/part-02","kind":"scope","navigation":"transparent","parent":29405,"result":false,"route":"../roadmap/GraphTheory/graph-coloring/part-02/index.html","scope":"GraphTheory/graph-coloring/part-02","status":"can_prove","title":"Part 2 of 2"},{"id":"GraphTheory/graph-coloring/part-02/five-color-theorem","kind":"node","lean":"MetaMathlibExt.five_color_theorem","parent":29537,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-02/index.html#five-color-theorem","scope":"GraphTheory/graph-coloring/part-02","status":"can_prove","summary":"Every straight-line planar simple graph $G$ on a finite vertex set has a proper vertex coloring with five colors.","title":"Five color theorem"},{"id":"GraphTheory/graph-coloring/part-02/mycielski-theorem","kind":"node","lean":"MetaMathlibExt.mycielski_theorem","parent":29537,"result":true,"route":"../roadmap/GraphTheory/graph-coloring/part-02/index.html#mycielski-theorem","scope":"GraphTheory/graph-coloring/part-02","status":"fully_proved","summary":"Write $\\chi(G)$ for the chromatic number of a simple graph $G$. For every simple graph $G$ on a finite vertex set there is a simple graph $H$ on a finite vertex set with strictly more vertices than $G$ that contains a subgraph isomorphic to $G$, such that $H$ is triangle-free whenever $G$ is, and such that $\\chi(H) = \\chi(G) + 1$ whenever $\\chi(G)$ is finite. Moreover, for every natural number $k$ there is a\u2026","title":"Mycielski's theorem"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory.json","id":"GraphTheory/algebraic-graph-theory","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/index.html","scope":"GraphTheory/algebraic-graph-theory","status":"can_prove","summary":"Graphs studied through groups and matrices: Cayley and vertex-transitive graphs, strongly regular and Moore graphs, graph homomorphisms, Laplacians and the matrix-tree theorem, and named graphs such as the Petersen and Coxeter graphs.","title":"Algebraic graph theory"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01.json","id":"GraphTheory/algebraic-graph-theory/part-01","kind":"scope","navigation":"transparent","parent":29540,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"can_prove","title":"Part 1 of 2"},{"id":"GraphTheory/algebraic-graph-theory/part-01/andrasfai-graph","kind":"node","lean":"MetaMathlibExt.andrasfaiGraph","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#andrasfai-graph","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"For a natural number $k$, the Andr\u00e1sfai graph is the Cayley graph of the cyclic group $\\mathbb{Z}/(3k+2)\\mathbb{Z}$ whose connection set $C$ consists of the residues whose representative in $\\{0, 1, \\dots, 3k+1\\}$ is congruent to $1$ modulo $3$: two distinct residues $x$ and $y$ are adjacent when $y - x \\in C$ or $x - y \\in C$.","title":"Andr\u00e1sfai graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/directed-cayley-graph","kind":"node","lean":"MetaMathlibExt.directedCayleyGraph","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#directed-cayley-graph","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"Let $G$ be a set with a binary operation written as multiplication, and let $C \\subseteq G$. The *Cayley digraph* $X(G, C)$ is the directed graph whose vertices are the elements of $G$, with an arc $u \\to v$ exactly when $v = uc$ for some $c \\in C$.","title":"Cayley digraph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/cayley-formula","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.cayley_formula","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#cayley-formula","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"For every natural number $n \\ge 2$, the complete graph on $n$ vertices has exactly $n^{n-2}$ spanning trees.","title":"Cayley's formula"},{"id":"GraphTheory/algebraic-graph-theory/part-01/coxeter-graph","kind":"node","lean":"MetaMathlibExt.coxeterGraph","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#coxeter-graph","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"The Coxeter graph is the simple graph whose $28$ vertices are the pairs $(i, j)$ with $i \\in \\{0, 1, 2, 3\\}$ and $j \\in \\mathbb{Z}/7\\mathbb{Z}$, and whose edges are the following: for each $i \\in \\{1, 2, 3\\}$, the vertices $(i, j)$ and $(i, j')$ are adjacent when $j - j' = i$ or $j' - j = i$ in $\\mathbb{Z}/7\\mathbb{Z}$, and for each $j$ the vertex $(0, j)$ is adjacent to $(1, j)$, $(2, j)$ and $(3, j)$. There are\u2026","title":"Coxeter graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/edge-transposition-graph","kind":"node","lean":"MetaMathlibExt.edgeTranspositionGraph","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#edge-transposition-graph","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"Let $n$ be a natural number and let $T$ be a set of permutations of $\\{0, 1, \\dots, n - 1\\}$, typically a set of transpositions in the symmetric group $S_n$. The edge-transposition graph of $T$ is the simple graph with vertex set $\\{0, 1, \\dots, n - 1\\}$ in which two distinct vertices $i$ and $j$ are adjacent if and only if the transposition swapping $i$ and $j$ belongs to $T$.","title":"Edge-transposition graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/frucht","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.FruchtWanted.frucht","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#frucht","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"Every finite group is isomorphic to the automorphism group of a finite simple graph.","title":"Frucht's theorem"},{"id":"GraphTheory/algebraic-graph-theory/part-01/is-moore-graph","kind":"node","lean":"MetaMathlibExt.IsMooreGraph","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#is-moore-graph","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $d$ be a natural number. Then $G$ is a Moore graph of diameter $d$ when $G$ has diameter $d$ and girth $2d + 1$.","title":"Moore graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/petersen-graph","kind":"node","lean":"MetaMathlibExt.petersenGraph","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#petersen-graph","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"The Petersen graph is the generalized Johnson graph $J(5,2,0)$: its vertices are the $2$-element subsets of $\\{0, 1, 2, 3, 4\\}$, and two of them are adjacent exactly when they are disjoint.","title":"Petersen graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/adj-matrix-pow-weighted-walk-sum","kind":"node","lean":"MetaMathlibExt.adjMatrix_pow_weighted_walk_sum","parent":29541,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/index.html#adj-matrix-pow-weighted-walk-sum","scope":"GraphTheory/algebraic-graph-theory/part-01","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $k$ and $n$ be natural numbers, and let $A$ be a $k \\times k$ matrix over $R$ with rows and columns indexed by $\\{0, \\dots, k-1\\}$, viewed as the weighted adjacency matrix of a weighted digraph on these $k$ vertices. Then for all vertices $i$ and $j$,","title":"Powers of a weighted adjacency matrix sum the weights of walks"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs.json","id":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs","status":"fully_proved","title":"Arc-transitive graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/sarc","kind":"node","lean":"SimpleGraph.SArc","parent":29551,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(s\\)"],["text","-arc"]],"source":"$s$-arc"},"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/index.html#sarc","scope":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs","status":"fully_proved","summary":"Let $X$ be a finite graph and $s$ a natural number. An *$s$-arc* in $X$ is a sequence $(v_0, \\dots, v_s)$ of vertices of $X$ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \\le i \\le s$, and $v_{i-1} \\ne v_{i+1}$ for $0 < i < s$.","title":"$s$-arc"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/is-sarc-transitive","kind":"node","lean":"MetaMathlibExt.SimpleGraph.IsSArcTransitive","parent":29551,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(s\\)"],["text","-arc transitive graph"]],"source":"$s$-arc transitive graph"},"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/index.html#is-sarc-transitive","scope":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs","status":"fully_proved","summary":"A finite graph $G$ is $s$-arc transitive if its automorphism group acts transitively on its $s$-arcs: for any two $s$-arcs $(u_0, \\dots, u_s)$ and $(v_0, \\dots, v_s)$ of $G$ there is an automorphism $\\varphi$ of $G$ with $\\varphi(u_i) = v_i$ for every $i \\le s$.","title":"$s$-arc transitive graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/is-sarc-regular","kind":"node","lean":"MetaMathlibExt.SimpleGraph.IsSArcRegular","parent":29551,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(s\\)"],["text","-arc regular graph"]],"source":"$s$-arc regular graph"},"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/index.html#is-sarc-regular","scope":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs","status":"fully_proved","summary":"A finite graph $G$ is $s$-arc regular if it has at least one $s$-arc and, for any two $s$-arcs $(u_0, \\dots, u_s)$ and $(v_0, \\dots, v_s)$ of $G$, there is exactly one automorphism $\\varphi$ of $G$ with $\\varphi(u_i) = v_i$ for every $i \\le s$.","title":"$s$-arc regular graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/arc-digraph","kind":"node","lean":"SimpleGraph.arcDigraph","parent":29551,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(s\\)"],["text","-arc digraph"]],"source":"$s$-arc digraph"},"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs/index.html#arc-digraph","scope":"GraphTheory/algebraic-graph-theory/part-01/arc-transitive-graphs","status":"fully_proved","summary":"Let $X$ be a finite graph and $s$ a natural number. For an $(s+1)$-arc $\\gamma = (x_0, \\dots, x_{s+1})$ of $X$, its *head* is the $s$-arc $(x_1, \\dots, x_{s+1})$ and its *tail* is the $s$-arc $(x_0, \\dots, x_s)$. If $\\alpha$ and $\\beta$ are $s$-arcs of $X$, then $\\beta$ *follows* $\\alpha$ if there is an $(s+1)$-arc of $X$ whose tail is $\\alpha$ and whose head is $\\beta$. The *$s$-arc digraph* $X^{(s)}$ is the\u2026","title":"$s$-arc digraph"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs.json","id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","title":"Arithmetical structures on graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/is-arithmetical-structure","kind":"node","lean":"MetaMathlibExt.IsArithmeticalStructure","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#is-arithmetical-structure","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"An arithmetical structure on a graph $G$ with vertex set $V$ is a pair of labelings $d, r : V \\to \\mathbb{N}$ such that $d(v) > 0$ and $r(v) > 0$ for every vertex $v$, $d(v)\\, r(v) = \\sum_{w \\sim v} r(w)$ for every vertex $v$, where the sum runs over the neighbours $w$ of $v$, and the only natural number dividing every $r(v)$ is $1$. For a finite graph with adjacency matrix $A$ the middle condition reads\u2026","title":"Arithmetical structure"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/is-arithmetical-structure-top-fin-two","kind":"node","lean":"MetaMathlibExt.isArithmeticalStructure_top_fin_two","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#is-arithmetical-structure-top-fin-two","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"On the graph with two vertices joined by an edge, the labelings $d = (1, 1)$ and $r = (1, 1)$ form an arithmetical structure.","title":"The all-ones structure on a single edge"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-tree","kind":"node","lean":"MetaMathlibExt.coconutTree","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-tree","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For natural numbers $p$ and $s$, the coconut tree $\\mathrm{CT}(p, s)$ is the graph whose vertices are the path vertices $v_1, \\dots, v_p$ and the leaves $v_{\\ell_1}, \\dots, v_{\\ell_s}$, in which $v_i$ and $v_{i+1}$ are adjacent for $1 \\le i < p$, every leaf is adjacent to $v_p$, and there are no other edges. For $p = 0$ it consists of $s$ isolated vertices.","title":"Coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-tree-adj-inl-inl","kind":"node","lean":"MetaMathlibExt.coconutTree_adj_inl_inl","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-tree-adj-inl-inl","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"Let the coconut tree $CT(p, s)$ have path vertices $v_1, \\dots, v_p$. Two path vertices $v_a$ and $v_b$ are adjacent if and only if $|a - b| = 1$.","title":"Adjacent path vertices of a coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-tree-adj-inl-inr","kind":"node","lean":"MetaMathlibExt.coconutTree_adj_inl_inr","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-tree-adj-inl-inr","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"Let the coconut tree $CT(p, s)$ have path vertices $v_1, \\dots, v_p$, with central vertex $v_p$. A leaf of $CT(p, s)$ is adjacent to a path vertex $v_k$ if and only if $k = p$.","title":"Adjacency of leaves and path vertices in a coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-tree-not-adj-inr-inr","kind":"node","lean":"MetaMathlibExt.coconutTree_not_adj_inr_inr","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-tree-not-adj-inr-inr","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"No leaf of the coconut tree $CT(p, s)$ is adjacent to a leaf.","title":"Leaves of a coconut tree are pairwise nonadjacent"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/is-smooth-coconut","kind":"node","lean":"MetaMathlibExt.IsSmoothCoconut","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#is-smooth-coconut","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"Let $CT(p, s)$ be the coconut tree with path vertices $v_1, \\dots, v_p$ and leaves $\\ell_1, \\dots, \\ell_s$ adjacent to $v_p$. An arithmetical structure $(d, r)$ on $CT(p, s)$ is smooth if $d(v_i) \\ge 2$ for $1 \\le i \\le p - 1$ and $d(\\ell_j) \\ge 2$ for $1 \\le j \\le s$; there is no condition on $d(v_p)$.","title":"Smooth arithmetical structure on a coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/two-le-leaf","kind":"node","lean":"MetaMathlibExt.IsSmoothCoconut.two_le_leaf","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#two-le-leaf","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"If $(d, r)$ is a smooth arithmetical structure on the coconut tree $\\mathrm{CT}(p, s)$, then $d_{\\ell_j} \\ge 2$ for every leaf $\\ell_j$, $1 \\le j \\le s$.","title":"Leaves of a smooth coconut structure have label at least 2"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ballot-number","kind":"node","lean":"MetaMathlibExt.ballotNumber","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ballot-number","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the ballot number $B(n, k)$ is the integer part of $\\max(n + 1 - k, 0) \\binom{n+k}{n} / (n + 1)$. For $k \\le n + 1$ the division is exact, so $B(n, k) = \\frac{n - k + 1}{n + 1} \\binom{n+k}{n}$ (OEIS A009766), and $B(n, k) = 0$ for $k > n + 1$.","title":"Ballot numbers"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ballot-number-zero-right","kind":"node","lean":"MetaMathlibExt.ballotNumber_zero_right","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ballot-number-zero-right","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For every natural number $n$, the ballot number $B(n, 0)$ equals $1$.","title":"Ballot numbers with second index zero"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ballot-number-self","kind":"node","lean":"MetaMathlibExt.ballotNumber_self","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ballot-number-self","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For every natural number $n$, the ballot number $B(n, n)$ equals the $n$-th Catalan number $C_n$.","title":"Diagonal ballot numbers are Catalan numbers"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-count-term","kind":"node","lean":"MetaMathlibExt.coconutCountTerm","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-count-term","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For natural numbers $x$ and $y$, the term $A(x, y)$ is defined by $A(x, 0) = C_{x-1}$, $A(x, 1) = C_{x-1} - C_{x-2}$ and, for $y \\ge 2$,","title":"Terms of the coconut tree count"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-count-term-zero","kind":"node","lean":"MetaMathlibExt.coconutCountTerm_zero","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-count-term-zero","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For every natural number $x$, the term $A(x, 0)$ of the coconut tree count equals the Catalan number $C_{x-1}$, read as $C_0$ when $x = 0$.","title":"The coconut count term with second argument zero"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/euclidean-chain-length","kind":"node","lean":"MetaMathlibExt.euclideanChainLength","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#euclidean-chain-length","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For natural numbers $x_1$ and $x_2$, the Euclidean chain $x_1, x_2, x_3, \\dots$ is given by $x_{i+1} = (-x_{i-1}) \\bmod x_i$ (the least nonnegative residue) while $x_i \\ne 0$, and by $x_{i+1} = 0$ once $x_i = 0$. The Euclidean chain function is defined by the recursion $F(x, 0) = 1$ and $F(x, y) = 1 + F(y, (-x) \\bmod y)$ for $y \\ne 0$. For $x_1 > 0$, $F(x_1, x_2)$ is the number of positive terms of the Euclidean\u2026","title":"Euclidean chain function"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/euclidean-chain-length-zero","kind":"node","lean":"MetaMathlibExt.euclideanChainLength_zero","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#euclidean-chain-length-zero","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For every natural number $x$, the Euclidean chain function $F$ satisfies $F(x, 0) = 1$.","title":"Euclidean chain function with second argument zero"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/euclidean-chain-length-of-ne-zero","kind":"node","lean":"MetaMathlibExt.euclideanChainLength_of_ne_zero","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#euclidean-chain-length-of-ne-zero","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"For natural numbers $x$ and $y$ with $y \\ne 0$, the Euclidean chain function $F$ satisfies $F(x, y) = 1 + F(y, (-x) \\bmod y)$, where $(-x) \\bmod y$ is the least nonnegative residue of $-x$ modulo $y$.","title":"Recursion for the Euclidean chain function"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/euclidean-chain-length-thirteen-sixty","kind":"node","lean":"MetaMathlibExt.euclideanChainLength_thirteen_sixty","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#euclidean-chain-length-thirteen-sixty","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"The Euclidean chain function $F$ satisfies $F(13, 60) = 8$.","title":"Euclidean chain function of 13 and 60"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-tfae","kind":"node","lean":"MetaMathlibExt.IsArithmeticalStructure.coconut_tfae","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-tfae","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $(d, r)$ be an arithmetical structure on the coconut tree $CT(p, s)$ with path vertices $v_1, \\dots, v_p$, and write $d_k = d(v_k)$ and $r_k = r(v_k)$. The following are equivalent: (1) $d_k \\ge 2$ for $1 \\le k \\le p - 1$; (2) $0 < r_2 - r_1 \\le r_3 - r_2 \\le \\cdots \\le r_p - r_{p-1}$; (3) $r_1 < r_2 < \\cdots < r_p$.","title":"Monotonicity criteria for arithmetical structures on coconut trees"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/is-smooth-coconut-iff","kind":"node","lean":"MetaMathlibExt.IsArithmeticalStructure.isSmoothCoconut_iff","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#is-smooth-coconut-iff","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $p \\ge 1$ and let $(d, r)$ be an arithmetical structure on the coconut tree $CT(p, s)$ with path vertices $v_1, \\dots, v_p$ and leaves $\\ell_1, \\dots, \\ell_s$ adjacent to $v_p$. Then $(d, r)$ is smooth if and only if $r(\\ell_j) < r(v_p)$ for every $j$ and $r(v_1) < r(v_2) < \\dots < r(v_p)$.","title":"Characterization of smooth coconut structures"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/eq-one-of-dvd-leaves","kind":"node","lean":"MetaMathlibExt.IsSmoothCoconut.eq_one_of_dvd_leaves","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#eq-one-of-dvd-leaves","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $(d, r)$ be a smooth arithmetical structure on the coconut tree $\\mathrm{CT}(p, s)$ with leaves $v_{\\ell_1}, \\dots, v_{\\ell_s}$. If a natural number $g$ divides $r(v_{\\ell_j})$ for every $j = 1, \\dots, s$, then $g = 1$.","title":"The leaf labels of a smooth arithmetical structure on a coconut tree are coprime"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ncard-arithmetical-structure-coconut-tre","kind":"node","lean":"MetaMathlibExt.ncard_arithmeticalStructure_coconutTree_one_leaf_ne_one","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ncard-arithmetical-structure-coconut-tre","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"For every integer $p \\ge 1$, the number of arithmetical structures $(d, r)$ on the coconut tree $\\mathrm{CT}(p, 1)$, which is the path on $p + 1$ vertices, whose leaf $\\ell_1$ has $d_{\\ell_1} \\ne 1$ (so that the leaf cannot be smoothed) is $C_p - C_{p-1}$, where $C_n$ denotes the $n$-th Catalan number.","title":"Arithmetical structures on a one-leaf coconut tree with a non-smoothable leaf"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ncard-arithmetical-structure-coconut-tre-2","kind":"node","lean":"MetaMathlibExt.ncard_arithmeticalStructure_coconutTree","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ncard-arithmetical-structure-coconut-tre-2","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"For $p \\ge 1$ and $s \\ge 2$, the number of arithmetical structures on the coconut tree $CT(p, s)$ is","title":"Number of arithmetical structures on coconut trees"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ncard-arithmetical-structure-coconut-tre-3","kind":"node","lean":"MetaMathlibExt.ncard_arithmeticalStructure_coconutTree_two","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ncard-arithmetical-structure-coconut-tre-3","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"For $p \\ge 2$, the number of arithmetical structures on the coconut tree $CT(p, 2)$ is","title":"Number of arithmetical structures on coconut trees with two leaves"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ncard-arithmetical-structure-coconut-tre-4","kind":"node","lean":"MetaMathlibExt.ncard_arithmeticalStructure_coconutTree_star","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ncard-arithmetical-structure-coconut-tre-4","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"For $s \\ge 2$, the number of arithmetical structures on the star $CT(1, s)$, a centre joined to $s$ leaves, is","title":"Number of arithmetical structures on stars"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/euclidean-chain-length-add-mul","kind":"node","lean":"MetaMathlibExt.euclideanChainLength_add_mul","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#euclidean-chain-length-add-mul","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"For natural numbers $x \\ge 1$, $y$ and $k$, the Euclidean chain function $F$ satisfies $F(x + k y, y) = F(x, y)$ and $F(x, k x + y) = F(x, y) + k$.","title":"Euclidean chain function under adding multiples"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/euclidean-chain-length-of-dvd","kind":"node","lean":"MetaMathlibExt.euclideanChainLength_of_dvd","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#euclidean-chain-length-of-dvd","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"For natural numbers $x, y \\ge 1$ with $x \\mid y$, the Euclidean chain function $F$ satisfies $F(x, y) = 1 + y / x$.","title":"Euclidean chain function when the first argument divides the second"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/euclidean-chain-length-one-left-and-one","kind":"node","lean":"MetaMathlibExt.euclideanChainLength_one_left_and_one_right","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#euclidean-chain-length-one-left-and-one","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"For every natural number $y$, the Euclidean chain function $F$ satisfies $F(1, y) = y + 1$, and for every natural number $x \\ge 1$ it satisfies $F(x, 1) = 2$.","title":"Euclidean chain function at one"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/exists-unique-is-smooth-coconut","kind":"node","lean":"MetaMathlibExt.existsUnique_isSmoothCoconut","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#exists-unique-is-smooth-coconut","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $s \\ge 1$ and let $c$ and $a_1, \\dots, a_s$ be natural numbers with $a_i > 0$, $a_i \\mid c$ and $a_i < c$ for every $i$, such that the only natural number dividing $c$ and all of $a_1, \\dots, a_s$ is $1$. Write $CT(p, s)$ for the coconut tree with path vertices $v_1, \\dots, v_p$ and leaves $\\ell_1, \\dots, \\ell_s$ adjacent to $v_p$. Then there is exactly one $p \\ge 1$ such that $CT(p, s)$ carries a smooth\u2026","title":"Smooth coconut structures with prescribed labels"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/eq-euclidean-chain-length-sub-one","kind":"node","lean":"MetaMathlibExt.IsSmoothCoconut.eq_euclideanChainLength_sub_one","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#eq-euclidean-chain-length-sub-one","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $F$ be the Euclidean chain function, let $p \\ge 1$, and let $(d, r)$ be a smooth arithmetical structure on the coconut tree $\\mathrm{CT}(p, s)$, with path vertices $v_1, \\dots, v_p$ and leaves $v_{\\ell_1}, \\dots, v_{\\ell_s}$. Then $p = F(r(v_{\\ell_1}) + \\dots + r(v_{\\ell_s}), r(v_p)) - 1$.","title":"Path length of a smooth arithmetical structure on a coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/one-lt-center-iff","kind":"node","lean":"MetaMathlibExt.IsSmoothCoconut.one_lt_center_iff","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#one-lt-center-iff","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $p \\ge 1$ and let $(d, r)$ be a smooth arithmetical structure on the coconut tree $\\mathrm{CT}(p, s)$, with central vertex $v_p$ and leaves $\\ell_1, \\ldots, \\ell_s$. Then $d_{v_p} > 1$ if and only if $r_{v_p} < \\sum_{j=1}^{s} r_{\\ell_j}$.","title":"Smooth coconut structures with central label above one"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/center-le-one-of-two","kind":"node","lean":"MetaMathlibExt.IsSmoothCoconut.center_le_one_of_two","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#center-le-one-of-two","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $p \\ge 1$ and let $(d, r)$ be a smooth arithmetical structure on the coconut tree $CT(p, 2)$, whose path is $v_1, \\dots, v_p$ and whose two leaves are attached to the central vertex $v_p$. Then $d(v_p) \\le 1$.","title":"Smooth structures on coconut trees with two leaves"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-smooth-path","kind":"node","lean":"MetaMathlibExt.coconutSmoothPath","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-smooth-path","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"Let the coconut tree $CT(p + 1, s)$ have path vertices $v_1, \\dots, v_{p+1}$, central vertex $v_{p+1}$ and leaves $\\ell_1, \\dots, \\ell_s$, let $d$ and $r$ be labelings of its vertices by natural numbers, and let $1 \\le k \\le p + 1$. Smoothing $(d, r)$ at $v_k$ gives the labelings $(d', r')$ of $CT(p, s)$ obtained by deleting $v_k$ and renumbering the remaining path vertices\u2026","title":"Smoothing at a path vertex of a coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-smooth-leaf","kind":"node","lean":"MetaMathlibExt.coconutSmoothLeaf","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-smooth-leaf","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"Let the coconut tree $CT(p, s + 1)$ have path vertices $v_1, \\dots, v_p$, central vertex $v_p$ and leaves $\\ell_1, \\dots, \\ell_{s+1}$, let $d$ and $r$ be labelings of its vertices by natural numbers, and let $1 \\le k \\le s + 1$. Smoothing $(d, r)$ at the leaf $\\ell_k$ gives the labelings $(d', r')$ of $CT(p, s)$ obtained by deleting $\\ell_k$ and renumbering the remaining leaves\u2026","title":"Smoothing at a leaf of a coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-subdivide","kind":"node","lean":"MetaMathlibExt.coconutSubdivide","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-subdivide","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"Let the coconut tree $CT(p, s)$ have path vertices $v_1, \\dots, v_p$ and leaves $\\ell_1, \\dots, \\ell_s$, let $d$ and $r$ be labelings of its vertices by natural numbers, and let $1 \\le k \\le p$. Subdividing $(d, r)$ at position $k$ gives the labelings $(d', r')$ of $CT(p + 1, s)$ obtained by inserting a new path vertex $w$ just before $v_k$, so that the path becomes $v_1, \\dots, v_{k-1}, w, v_k, \\dots, v_p$, with\u2026","title":"Subdivision at a path vertex of a coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-smooth-path-example","kind":"node","lean":"MetaMathlibExt.coconutSmoothPath_example","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-smooth-path-example","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"List a labeling of a coconut tree as its values on the path vertices $v_1, v_2, \\dots$, then $\\mid$, then its values on the leaves. Smoothing the labelings $d = (3, 1, 3, 2, 2 \\mid 2, 2, 4)$ and $r = (1, 3, 2, 3, 4 \\mid 2, 2, 1)$ of $CT(5, 3)$ at $v_2$ gives the labelings $d' = (2, 2, 2, 2 \\mid 2, 2, 4)$ and $r' = (1, 2, 3, 4 \\mid 2, 2, 1)$ of $CT(4, 3)$.","title":"Example of smoothing a coconut tree at a path vertex"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-smooth-leaf-example","kind":"node","lean":"MetaMathlibExt.coconutSmoothLeaf_example","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-smooth-leaf-example","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"On the coconut tree $CT(4, 4)$, write a labeling as (values at $v_1, \\dots, v_4$ | values at the leaves $\\ell_1, \\dots, \\ell_4$). Take $d = (1, 3, 2, 3 \\mid 2, 2, 3, 1)$ and $r = (2, 2, 4, 6 \\mid 3, 3, 2, 6)$. Smoothing at the path vertex $v_1$ and then at the leaf $\\ell_4$ gives the labelings $d' = (2, 2, 2 \\mid 2, 2, 3)$ and $r' = (2, 4, 6 \\mid 3, 3, 2)$ on $CT(3, 3)$.","title":"An example of smoothing at a path vertex and at a leaf"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-subdivide-example","kind":"node","lean":"MetaMathlibExt.coconutSubdivide_example","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-subdivide-example","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"List a labeling of a coconut tree as its values on the path vertices $v_1, v_2, \\dots$, then $\\mid$, then its values on the leaves. Subdividing the labelings $d = (2, 2, 2, 2 \\mid 2, 2, 4)$ and $r = (1, 2, 3, 4 \\mid 2, 2, 1)$ of $CT(4, 3)$ at position $2$ gives the labelings $d' = (3, 1, 3, 2, 2 \\mid 2, 2, 4)$ and $r' = (1, 3, 2, 3, 4 \\mid 2, 2, 1)$ of $CT(5, 3)$. These are the labelings of Example 3.2, whose\u2026","title":"Example of subdividing a coconut tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-ancestor","kind":"node","lean":"MetaMathlibExt.CoconutAncestor","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-ancestor","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"fully_proved","summary":"Let $x$ be a pair of vertex labelings $(d, r)$ of the coconut tree $CT(p, s)$ and $y$ a pair of vertex labelings of $CT(q, t)$. Then $y$ is an ancestor of $x$, and $x$ a descendant of $y$, when $y$ is obtained from $x$ by a finite, possibly empty, sequence of smoothings, each performed at a vertex where the current $d$-label equals $1$: either at a path vertex other than the central vertex, or at a leaf.","title":"Ancestors of labelings of coconut trees"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-smooth-path-2","kind":"node","lean":"MetaMathlibExt.IsArithmeticalStructure.coconutSmoothPath","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-smooth-path-2","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $s \\ge 1$ and let $(d, r)$ be an arithmetical structure on the coconut tree $CT(p + 1, s)$, whose path vertices are $v_1, \\dots, v_{p+1}$ with central vertex $v_{p+1}$. If $d(v_k) = 1$ for some non-central path vertex $v_k$ (that is, $1 \\le k \\le p$), then smoothing $(d, r)$ at $v_k$ gives an arithmetical structure on $CT(p, s)$.","title":"Path smoothing preserves arithmetical structures on coconut trees"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-smooth-leaf-2","kind":"node","lean":"MetaMathlibExt.IsArithmeticalStructure.coconutSmoothLeaf","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-smooth-leaf-2","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $p$ and $s$ be natural numbers with $p + s \\ge 2$, let $(d, r)$ be an arithmetical structure on the coconut tree $CT(p, s + 1)$, and let $\\ell$ be a leaf with $d(\\ell) = 1$. Then the labelings $(d', r')$ obtained by smoothing $(d, r)$ at $\\ell$ form an arithmetical structure on $CT(p, s)$.","title":"Smoothing a leaf preserves arithmetical structures"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/coconut-subdivide-2","kind":"node","lean":"MetaMathlibExt.IsArithmeticalStructure.coconutSubdivide","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#coconut-subdivide-2","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"If $(d, r)$ is an arithmetical structure on the coconut tree $CT(p, s)$, whose path vertices are $v_1, \\dots, v_p$, then for every $1 \\le k \\le p$, subdividing $(d, r)$ at position $k$ (inserting a new path vertex just before $v_k$) gives an arithmetical structure on $CT(p + 1, s)$.","title":"Subdivision preserves arithmetical structures on coconut trees"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/exists-unique-smooth-coconut-ancestor","kind":"node","lean":"MetaMathlibExt.IsArithmeticalStructure.existsUnique_smooth_coconutAncestor","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#exists-unique-smooth-coconut-ancestor","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $s \\ge 1$ and let $(d, r)$ be an arithmetical structure on the coconut tree $CT(p, s)$ with $d(\\ell_j) \\ge 2$ at every leaf $\\ell_j$. Then there is exactly one pair consisting of a number $q$ with $1 \\le q \\le p$ and a smooth arithmetical structure on $CT(q, s)$ that is a coconut ancestor of $(d, r)$, that is, one obtained from $(d, r)$ by a possibly empty sequence of smoothings, each performed at a vertex\u2026","title":"Unique smooth ancestor"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ncard-coconut-descendants","kind":"node","lean":"MetaMathlibExt.ncard_coconutDescendants","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ncard-coconut-descendants","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"Let $s \\ge 2$ and $1 \\le i \\le p$, and let $(d, r)$ be a smooth arithmetical structure on the coconut tree $\\mathrm{CT}(i, s)$. Then exactly $B(p - 1, p - i)$ arithmetical structures on $\\mathrm{CT}(p, s)$ are descendants of $(d, r)$, where $B(n, k)$ is the ballot number.","title":"Number of descendants of a smooth coconut structure"},{"id":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/ncard-arithmetical-structure-coconut-tre-5","kind":"node","lean":"MetaMathlibExt.ncard_arithmeticalStructure_coconutTree_leaves_ge_two","parent":29556,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs/index.html#ncard-arithmetical-structure-coconut-tre-5","scope":"GraphTheory/algebraic-graph-theory/part-01/arithmetical-structures-on-graphs","status":"can_prove","summary":"For $p \\ge 1$ and $s \\ge 2$, the number of arithmetical structures $(d, r)$ on the coconut tree $CT(p, s)$ with $d(\\ell_j) \\ge 2$ at every leaf $\\ell_j$ is","title":"Counting coconut structures whose leaves cannot be smoothed"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps.json","id":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps","status":"can_prove","title":"Cycle-continuous maps"},{"id":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/fgraph","kind":"node","lean":"MathlibExt.GraphTheory.CycleContinuousAntichainWanted.FGraph","parent":29601,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/index.html#fgraph","scope":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps","status":"fully_proved","summary":"A multigraph $G$ consists of a type $V$ of vertices, a type $E$ of edges, and a map assigning to each edge an ordered pair of vertices, its two endpoints. Distinct edges may have the same endpoints (parallel edges), and the two endpoints of an edge may coincide (a loop). Neither $V$ nor $E$ is required to be finite.","title":"Multigraphs with loops and parallel edges"},{"id":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/is-finite-nontrivial","kind":"node","lean":"MathlibExt.GraphTheory.CycleContinuousAntichainWanted.IsFiniteNontrivial","parent":29601,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/index.html#is-finite-nontrivial","scope":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps","status":"fully_proved","summary":"A multigraph $G$ is finite and nontrivial when its set of vertices and its set of edges are both finite and it has at least one edge. These are the graphs on which the cycle-continuous order is considered.","title":"Finite multigraphs with at least one edge"},{"id":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/cycle-space","kind":"node","lean":"MathlibExt.GraphTheory.CycleContinuousAntichainWanted.cycleSpace","parent":29601,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/index.html#cycle-space","scope":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps","status":"fully_proved","summary":"Let $G$ be a multigraph. A set $S$ of edges of $G$ is a binary cycle when $S$ is finite and every vertex $v$ is an endpoint of an even number of edges of $S$, where only edges with exactly one endpoint equal to $v$ are counted; loops are not counted, which agrees modulo $2$ with counting each loop twice. The cycle space of $G$ is the set of all binary cycles of $G$.","title":"Binary cycle space"},{"id":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/is-cycle-continuous","kind":"node","lean":"MathlibExt.GraphTheory.CycleContinuousAntichainWanted.IsCycleContinuous","parent":29601,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/index.html#is-cycle-continuous","scope":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps","status":"fully_proved","summary":"Let $G$ and $H$ be multigraphs. A map $f$ from the edges of $G$ to the edges of $H$ is cycle-continuous when, for every binary cycle $C$ of $H$, the preimage $f^{-1}(C)$ is a binary cycle of $G$.","title":"Cycle-continuous maps"},{"id":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/conjecture","kind":"node","lean":"MathlibExt.GraphTheory.CycleContinuousAntichainWanted.conjecture","parent":29601,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/index.html#conjecture","scope":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps","status":"fully_proved","summary":"There is a sequence $G_0, G_1, G_2, \\dots$ of pairwise distinct finite multigraphs, each with at least one edge, such that for all $i \\ne j$ there is no cycle-continuous map from the edges of $G_i$ to the edges of $G_j$.","title":"Infinite antichains in the cycle-continuous order"},{"id":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/conjecture-holds","kind":"node","lean":"MathlibExt.GraphTheory.CycleContinuousAntichainWanted.conjecture_holds","parent":29601,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps/index.html#conjecture-holds","scope":"GraphTheory/algebraic-graph-theory/part-01/cycle-continuous-maps","status":"can_prove","summary":"There is a sequence $G_0, G_1, G_2, \\dots$ of pairwise distinct finite multigraphs, each with at least one edge, such that for all $i \\ne j$ there is no cycle-continuous map from the edges of $G_i$ to the edges of $G_j$.","title":"Samal's theorem: infinite antichains in the cycle-continuous order exist"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/deza-graphs.json","id":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/deza-graphs/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs","status":"fully_proved","title":"Deza graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs/is-deza-with","kind":"node","lean":"SimpleGraph.IsDezaWith","parent":29608,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/deza-graphs/index.html#is-deza-with","scope":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, and let $n, k, b, a$ be natural numbers. Then $G$ is a Deza graph with parameters $(n, k, b, a)$ when $a \\le b \\le k < n$, $V$ has exactly $n$ elements, every vertex of $G$ has degree $k$, and any two distinct vertices of $G$ have exactly $a$ or exactly $b$ common neighbours, whether or not they are adjacent.","title":"Deza graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs/a-le-k","kind":"node","lean":"SimpleGraph.IsDezaWith.a_le_k","parent":29608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parameter bound "],["math","\\(a \\le k\\)"],["text"," for Deza graphs"]],"source":"Parameter bound $a \\le k$ for Deza graphs"},"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/deza-graphs/index.html#a-le-k","scope":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs","status":"fully_proved","summary":"If a finite simple graph $G$ is a Deza graph with parameters $(n, k, b, a)$, then $a \\le k$.","title":"Parameter bound $a \\le k$ for Deza graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs/b-lt-n","kind":"node","lean":"SimpleGraph.IsDezaWith.b_lt_n","parent":29608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parameter bound "],["math","\\(b < n\\)"],["text"," for Deza graphs"]],"source":"Parameter bound $b < n$ for Deza graphs"},"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/deza-graphs/index.html#b-lt-n","scope":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs","status":"fully_proved","summary":"If a finite simple graph $G$ is a Deza graph with parameters $(n, k, b, a)$, then $b < n$.","title":"Parameter bound $b < n$ for Deza graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs/a-lt-n","kind":"node","lean":"SimpleGraph.IsDezaWith.a_lt_n","parent":29608,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parameter bound "],["math","\\(a < n\\)"],["text"," for Deza graphs"]],"source":"Parameter bound $a < n$ for Deza graphs"},"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/deza-graphs/index.html#a-lt-n","scope":"GraphTheory/algebraic-graph-theory/part-01/deza-graphs","status":"fully_proved","summary":"If a finite simple graph $G$ is a Deza graph with parameters $(n, k, b, a)$, then $a < n$.","title":"Parameter bound $a < n$ for Deza graphs"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers.json","id":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers","status":"fully_proved","title":"Graph homomorphisms, cores and covers"},{"id":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/is-retract","kind":"node","lean":"SimpleGraph.Subgraph.IsRetract","parent":29613,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/index.html#is-retract","scope":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers","status":"fully_proved","summary":"Let $Y$ be a subgraph of a graph $X$. A *retraction* from $X$ to $Y$ is a homomorphism $f$ from $X$ to $Y$, regarded as a graph in its own right, whose restriction to the vertex set of $Y$ is the identity map. We say that $Y$ is a *retract* of $X$ if there is a retraction from $X$ to $Y$.","title":"Retract"},{"id":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/has-core","kind":"node","lean":"SimpleGraph.HasCore","parent":29613,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/index.html#has-core","scope":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers","status":"fully_proved","summary":"A finite graph $X$ is a *core* if every homomorphism from $X$ to itself is a bijection. For finite graphs $X$ and $Y$, $Y$ is a *core of* $X$ if $Y$ is a core, there is a homomorphism from $X$ to $Y$, and there is an injective homomorphism from $Y$ to $X$; that is, $Y$ is isomorphic to a subgraph of $X$.","title":"Core of a graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/is-local-injection","kind":"node","lean":"SimpleGraph.IsLocalInjection","parent":29613,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/index.html#is-local-injection","scope":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers","status":"fully_proved","summary":"A homomorphism $f$ from a finite graph $X$ to a finite graph $Y$ is a *local injection* if any two distinct vertices of $X$ with the same image under $f$ are at distance at least $3$ in $X$.","title":"Local injection"},{"id":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/is-local-isomorphism","kind":"node","lean":"SimpleGraph.IsLocalIsomorphism","parent":29613,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/index.html#is-local-isomorphism","scope":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers","status":"fully_proved","summary":"A homomorphism $f$ from a finite graph $X$ to a finite graph $Y$ is a *local isomorphism* if for every vertex $x$ of $X$, $f$ maps the set of neighbours of $x$ bijectively onto the set of neighbours of $f(x)$.","title":"Local isomorphism"},{"id":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/covers","kind":"node","lean":"SimpleGraph.Covers","parent":29613,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/index.html#covers","scope":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers","status":"fully_proved","summary":"A finite graph $X$ *covers* a finite graph $Y$ if there is a surjective local isomorphism from $X$ to $Y$.","title":"Graph cover"},{"id":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/folding-of-a-graph-homomorphism","kind":"node","lean":"SimpleGraph.IsFolding","parent":29613,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers/index.html#folding-of-a-graph-homomorphism","scope":"GraphTheory/algebraic-graph-theory/part-01/graph-homomorphisms-cores-and-covers","status":"fully_proved","summary":"A homomorphism from X to Y is a *simple folding* if it has one fibre consisting of two vertices at distance two, and all other fibres are singletons. A homomorphism is a *folding* if it is the composition of a non-negative number of simple foldings.","title":"Folding of a graph homomorphism"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs.json","id":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs","status":"can_prove","title":"Grassmann graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/gr-vertex","kind":"node","lean":"MathlibExt.Combinatorics.PseudorandomGrassmannWanted.PseudorandomGrassmann.GrVertex","parent":29620,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/index.html#gr-vertex","scope":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs","status":"fully_proved","summary":"For natural numbers $k$ and $\\ell$, the vertices of the Grassmann graph $\\mathrm{Gr}(k, \\ell)$ are the $\\ell$-dimensional linear subspaces of $\\mathbb{F}_2^k$.","title":"Vertices of the Grassmann graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/gr","kind":"node","lean":"MathlibExt.Combinatorics.PseudorandomGrassmannWanted.PseudorandomGrassmann.Gr","parent":29620,"reading_title":{"format":"inline-parts-v1","parts":[["text","Grassmann graph over "],["math","\\(\\mathbb{F}_2\\)"]],"source":"Grassmann graph over $\\mathbb{F}_2$"},"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/index.html#gr","scope":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs","status":"fully_proved","summary":"For natural numbers $k$ and $\\ell$, the Grassmann graph $\\mathrm{Gr}(k, \\ell)$ has as vertices the $\\ell$-dimensional linear subspaces of $\\mathbb{F}_2^k$, and two distinct such subspaces $L$ and $L'$ are adjacent when $\\dim(L \\cap L') = \\ell - 1$.","title":"Grassmann graph over $\\mathbb{F}_2$"},{"id":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/sub-gr","kind":"node","lean":"MathlibExt.Combinatorics.PseudorandomGrassmannWanted.PseudorandomGrassmann.SubGr","parent":29620,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/index.html#sub-gr","scope":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs","status":"fully_proved","summary":"For natural numbers $k$ and $\\ell$ and linear subspaces $A$ and $B$ of $\\mathbb{F}_2^k$, $\\mathrm{Gr}(k, \\ell)[A, B]$ is the finite set of vertices $L$ of the Grassmann graph $\\mathrm{Gr}(k, \\ell)$, that is of $\\ell$-dimensional subspaces of $\\mathbb{F}_2^k$, with $A \\subseteq L \\subseteq B$. If $A \\not\\subseteq B$ the set is empty.","title":"Grassmann vertices between two subspaces"},{"id":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/untitled","kind":"node","lean":"MathlibExt.Combinatorics.PseudorandomGrassmannWanted.PseudorandomGrassmann.\u03a6","parent":29620,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/index.html#untitled","scope":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $S \\subseteq V$ be nonempty. Let $d$ be the degree in $G$ of a fixed vertex of $S$, chosen once for each $S$. The expansion of $S$ is the rational number $\\Phi_G(S) = |E(S, V \\setminus S)| / (d \\cdot |S|)$, where $E(S, V \\setminus S)$ is the set of edges of $G$ with one endpoint in $S$ and the other in $V \\setminus S$; it is $0$ when $d = 0$.","title":"Edge expansion of a vertex set"},{"id":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/theorem-1-12","kind":"node","lean":"MathlibExt.Combinatorics.PseudorandomGrassmannWanted.PseudorandomGrassmann.theorem_1_12","parent":29620,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs/index.html#theorem-1-12","scope":"GraphTheory/algebraic-graph-theory/part-01/grassmann-graphs","status":"can_prove","summary":"For every real $\\alpha$ with $0 < \\alpha < 1$ there are a real $\\varepsilon > 0$ and a natural number $r$ such that for all sufficiently large natural numbers $\\ell$ and, for each such $\\ell$, all sufficiently large natural numbers $k$, the following holds. For subspaces $A$ and $B$ of $\\mathbb{F}_2^k$, let $\\mathrm{Gr}(k, \\ell)[A, B]$ be the set of vertices $L$ of the Grassmann graph $\\mathrm{Gr}(k, \\ell)$ with\u2026","title":"Non-expanding sets in the Grassmann graph (Khot\u2013Minzer\u2013Safra)"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs.json","id":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs","status":"can_prove","title":"Integral Cayley graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/has-integral-spectrum","kind":"node","lean":"MathlibExt.GroupTheory.IntegralCayleyGraphsKourovka19Problem50Wanted.HasIntegralSpectrum","parent":29626,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/index.html#has-integral-spectrum","scope":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs","status":"fully_proved","summary":"A square integer matrix is integral: every complex root of its characteristic polynomial (every eigenvalue) is an integer.","title":"Integer matrix with integral spectrum"},{"id":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/involution-cayley-graph-integral","kind":"node","lean":"MathlibExt.GroupTheory.IntegralCayleyGraphsKourovka19Problem50Wanted.involutionCayleyGraphIntegral","parent":29626,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/index.html#involution-cayley-graph-integral","scope":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs","status":"fully_proved","summary":"This is the proposition that for every finite group $G$ and every subset $R \\subseteq G$ such that every element of $R$ is an involution ($r^2 = 1$ and $r \\ne 1$), $R$ is closed under conjugation ($g r g^{-1} \\in R$ for all $r \\in R$ and $g \\in G$) and $R$ generates $G$, the Cayley graph $\\mathrm{Cay}(G, R)$ is integral. Here $\\mathrm{Cay}(G, R)$ has vertex set $G$, distinct $x$ and $y$ being adjacent when\u2026","title":"The Kourovka 19.50(a) integrality claim"},{"id":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/three-cycles-one-two","kind":"node","lean":"MathlibExt.GroupTheory.IntegralCayleyGraphsKourovka19Problem50Wanted.threeCyclesOneTwo","parent":29626,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/index.html#three-cycles-one-two","scope":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs","status":"fully_proved","summary":"For a natural number $n$, this is the set of $3$-cycles $(1\\,2\\,k)$ with $3 \\le k \\le n$ in the symmetric group $S_n$. With the points numbered $0, 1, \\dots, n - 1$, these are the permutations $(0\\,1\\,k) = (0\\,1)(1\\,k)$ with $2 \\le k < n$, products composed right to left; the set is empty when $n < 3$.","title":"The 3-cycles (1 2 k)"},{"id":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/alternating-three-cycle-cayley-graph-int","kind":"node","lean":"MathlibExt.GroupTheory.IntegralCayleyGraphsKourovka19Problem50Wanted.alternatingThreeCycleCayleyGraphIntegral","parent":29626,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/index.html#alternating-three-cycle-cayley-graph-int","scope":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs","status":"fully_proved","summary":"For $n \\ge 3$ let $S$ be the set of $3$-cycles $(1\\,2\\,k)$ with $3 \\le k \\le n$ on the points $1, \\dots, n$, and let $R = S \\cup S^{-1}$, a subset of the alternating group $A_n$. This is the proposition that for every $n \\ge 3$ the Cayley graph $\\mathrm{Cay}(A_n, R)$ is integral. Here $\\mathrm{Cay}(A_n, R)$ has vertex set $A_n$, distinct $x$ and $y$ being adjacent when $x^{-1} y \\in R$ or $y^{-1} x \\in R$, and a\u2026","title":"The Kourovka 19.50(b) integrality claim"},{"id":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/involution-cayley-graph-integral-holds","kind":"node","lean":"MathlibExt.GroupTheory.IntegralCayleyGraphsKourovka19Problem50Wanted.involutionCayleyGraphIntegral_holds","parent":29626,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/index.html#involution-cayley-graph-integral-holds","scope":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs","status":"can_prove","summary":"Let $G$ be a finite group and let $R \\subseteq G$ be a set of involutions ($r^2 = 1$ and $r \\ne 1$ for every $r \\in R$) that is closed under conjugation ($g r g^{-1} \\in R$ for all $r \\in R$ and $g \\in G$) and generates $G$. Then the Cayley graph $\\mathrm{Cay}(G, R)$, whose vertices are the elements of $G$ with distinct $x$ and $y$ adjacent when $x^{-1} y \\in R$ or $y^{-1} x \\in R$, is integral: every eigenvalue of\u2026","title":"Cayley graphs on normal sets of involutions are integral"},{"id":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/alternating-three-cycle-cayley-graph-int-2","kind":"node","lean":"MathlibExt.GroupTheory.IntegralCayleyGraphsKourovka19Problem50Wanted.alternatingThreeCycleCayleyGraphIntegral_holds","parent":29626,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs/index.html#alternating-three-cycle-cayley-graph-int-2","scope":"GraphTheory/algebraic-graph-theory/part-01/integral-cayley-graphs","status":"can_prove","summary":"For every $n \\ge 3$, let $S$ be the set of $3$-cycles $(1\\,2\\,k)$ with $3 \\le k \\le n$ in the alternating group $A_n$, and let $R = S \\cup S^{-1}$. Then the Cayley graph $\\mathrm{Cay}(A_n, R)$, whose vertices are the elements of $A_n$ with distinct $x$ and $y$ adjacent when $x^{-1} y \\in R$ or $y^{-1} x \\in R$, is integral: every eigenvalue of its integer adjacency matrix (every complex root of the characteristic\u2026","title":"The 3-cycle Cayley graph of the alternating group is integral"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/johnson-graphs.json","id":"GraphTheory/algebraic-graph-theory/part-01/johnson-graphs","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/johnson-graphs/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/johnson-graphs","status":"fully_proved","title":"Johnson graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/johnson-graphs/johnson-graph","kind":"node","lean":"MetaMathlibExt.johnsonGraph","parent":29633,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/johnson-graphs/index.html#johnson-graph","scope":"GraphTheory/algebraic-graph-theory/part-01/johnson-graphs","status":"fully_proved","summary":"For natural numbers $v$, $k$ and $i$, the generalized Johnson graph $J(v,k,i)$ is the graph whose vertices are the $k$-element subsets of $\\{0, 1, \\dots, v-1\\}$, in which two distinct vertices are adjacent if and only if their intersection has exactly $i$ elements.","title":"Generalized Johnson graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/johnson-graphs/johnson-graph-2","kind":"node","lean":"SimpleGraph.johnsonGraph","parent":29633,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/johnson-graphs/index.html#johnson-graph-2","scope":"GraphTheory/algebraic-graph-theory/part-01/johnson-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the Johnson graph $J(n, k)$ has as vertices the $k$-element subsets of $\\{0, 1, \\dots, n-1\\}$, and two vertices $s$ and $t$ are adjacent when $|s \\cap t| + 1 = k$, that is, when $s$ and $t$ share exactly $k - 1$ elements.","title":"Johnson graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/johnson-graphs/johnson-graph-adj-iff","kind":"node","lean":"SimpleGraph.johnsonGraph_adj_iff","parent":29633,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/johnson-graphs/index.html#johnson-graph-adj-iff","scope":"GraphTheory/algebraic-graph-theory/part-01/johnson-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and $k$-element subsets $s$ and $t$ of $\\{0, 1, \\dots, n-1\\}$, the vertices $s$ and $t$ of the Johnson graph $J(n, k)$ are adjacent if and only if $|s \\cap t| + 1 = k$.","title":"Adjacency in the Johnson graph"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem.json","id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","title":"Kirchhoff's matrix-tree theorem"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/laplacian-int","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.laplacianInt","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#laplacian-int","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set $V$, the integer Laplacian of $G$ is the $V \\times V$ integer matrix $L$ with $L_{uu} = \\deg u$ for every vertex $u$, $L_{uv} = -1$ when $u \\ne v$ are adjacent, and $L_{uv} = 0$ when $u \\ne v$ are not adjacent.","title":"Integer Laplacian of a graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/reduced-laplacian","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.reducedLaplacian","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#reduced-laplacian","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set $V$ and a vertex $r \\in V$, the reduced Laplacian of $G$ at $r$ is the integer Laplacian of $G$ with the row and the column of $r$ deleted, a matrix indexed by the vertices of $G$ other than $r$.","title":"Reduced Laplacian"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/is-spanning-tree","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.IsSpanningTree","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#is-spanning-tree","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"A subgraph $T$ of a simple graph $G$ is a spanning tree of $G$ if it contains every vertex of $G$ and, regarded as a graph on its own vertex set, it is a tree (connected and acyclic).","title":"Spanning tree"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/num-spanning-trees","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.numSpanningTrees","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#num-spanning-trees","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set, the number of spanning trees of $G$ is the number of subgraphs of $G$ that are spanning trees, that is, subgraphs that contain every vertex of $G$ and are trees.","title":"Number of spanning trees"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/laplacian-int-eq-lap-matrix","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.laplacianInt_eq_lapMatrix","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#laplacian-int-eq-lap-matrix","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set, the integer Laplacian of $G$ equals $D - A$, where $D$ is the diagonal matrix of vertex degrees and $A$ is the adjacency matrix of $G$, both with integer entries.","title":"The integer Laplacian is degree minus adjacency"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/reduced-laplacian-eq-lap-matrix-submatri","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.reducedLaplacian_eq_lapMatrix_submatrix","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#reduced-laplacian-eq-lap-matrix-submatri","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $L = D - A$ be its Laplacian matrix over $\\mathbb{Z}$, where $D$ is the diagonal matrix of vertex degrees and $A$ is the adjacency matrix of $G$. For every vertex $r \\in V$, the reduced Laplacian of $G$ at $r$ equals the submatrix of $L$ obtained by deleting the row and the column of $r$.","title":"The reduced Laplacian is a principal submatrix"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/spanning-tree-subgraph-equiv","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.spanningTreeSubgraphEquiv","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#spanning-tree-subgraph-equiv","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"For a simple graph $G$ on a vertex set $V$, there is an equivalence between the spanning trees of $G$ (subgraphs of $G$ that contain every vertex and are trees) and the simple graphs $H$ on $V$ that are trees with $H \\le G$, that is, every edge of $H$ is an edge of $G$. It sends a spanning tree to the graph on $V$ with the same edges.","title":"Spanning trees as trees contained in the graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/num-spanning-trees-eq-card-trees","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.numSpanningTrees_eq_card_trees","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#num-spanning-trees-eq-card-trees","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set $V$, the number of spanning trees of $G$ equals the number of simple graphs $H$ on $V$ such that every edge of $H$ is an edge of $G$ and $H$ is a tree.","title":"Spanning trees as trees on the whole vertex set"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/kirchhoff-matrix-tree-of-root","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.kirchhoff_matrix_tree_of_root","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#kirchhoff-matrix-tree-of-root","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $r \\in V$. Then the determinant of the reduced Laplacian of $G$ at $r$, that is, the integer Laplacian of $G$ with the row and the column of $r$ deleted, equals the number of spanning trees of $G$.","title":"Matrix-tree theorem for a chosen root"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/kirchhoff-matrix-tree-lap-matrix","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.kirchhoff_matrix_tree_lapMatrix","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#kirchhoff-matrix-tree-lap-matrix","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, let $L = D - A$ be its Laplacian matrix over $\\mathbb{Z}$, where $D$ is the diagonal matrix of vertex degrees and $A$ is the adjacency matrix of $G$, and let $r \\in V$. Then the determinant of the submatrix of $L$ obtained by deleting the row and the column of $r$ equals the number of spanning trees of $G$.","title":"Matrix-tree theorem for principal minors of the Laplacian"},{"id":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/kirchhoff-matrix-tree","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MatrixTreeWanted.kirchhoff_matrix_tree","parent":29637,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem/index.html#kirchhoff-matrix-tree","scope":"GraphTheory/algebraic-graph-theory/part-01/kirchhoff-s-matrix-tree-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite nonempty vertex set $V$ and let $r \\in V$. Then the determinant of the reduced Laplacian of $G$ at $r$, that is, the integer Laplacian of $G$ with the row and the column of $r$ deleted, equals the number of spanning trees of $G$.","title":"Kirchhoff's matrix-tree theorem"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs.json","id":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs","status":"fully_proved","title":"Strongly regular graphs"},{"id":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/adj-of-adj-of-adj-of-mu-zero","kind":"node","lean":"SimpleGraph.IsSRGWith.adj_of_adj_of_adj_of_mu_zero","parent":29649,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/index.html#adj-of-adj-of-adj-of-mu-zero","scope":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs","status":"fully_proved","summary":"Let $G$ be a finite strongly regular graph with parameters $(n, k, \\ell, \\mu)$ where $\\mu = 0$: $G$ has $n$ vertices, every vertex has degree $k$, any two adjacent vertices have exactly $\\ell$ common neighbors, and any two distinct nonadjacent vertices have no common neighbor. If $u$, $v$, $w$ are vertices of $G$ such that $u$ is adjacent to $v$, $v$ is adjacent to $w$, and $u \\neq w$, then $u$ is adjacent to $w$.","title":"Two-edge paths close into triangles when \u03bc = 0"},{"id":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/nonprincipal-eigenvalue-quadratic","kind":"node","lean":"SimpleGraph.IsSRGWith.nonprincipal_eigenvalue_quadratic","parent":29649,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/index.html#nonprincipal-eigenvalue-quadratic","scope":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs","status":"fully_proved","summary":"Let $G$ be a finite strongly regular graph with parameters $(n, k, \\ell, \\mu)$: $G$ has $n$ vertices, every vertex has degree $k$, any two adjacent vertices have exactly $\\ell$ common neighbors, and any two distinct nonadjacent vertices have exactly $\\mu$ common neighbors. Let $A$ be the adjacency matrix of $G$ over $\\mathbb{C}$. Then every eigenvalue $x \\in \\mathbb{C}$ of $A$ with $x \\neq k$ satisfies\u2026","title":"Non-principal eigenvalues of a strongly regular graph"},{"id":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/srg-connected-mu-zero-complete","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.StronglyRegularWanted.srg_connected_mu_zero_complete","parent":29649,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/index.html#srg-connected-mu-zero-complete","scope":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs","status":"fully_proved","summary":"Let $G$ be a finite strongly regular graph with parameters $(n, k, \\ell, \\mu)$ where $\\mu = 0$: $G$ has $n$ vertices, every vertex has degree $k$, any two adjacent vertices have exactly $\\ell$ common neighbors, and any two distinct nonadjacent vertices have no common neighbor. If $G$ is connected, then $G$ is complete, that is, any two distinct vertices of $G$ are adjacent.","title":"Connected strongly regular graphs with \u03bc = 0 are complete"},{"id":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/srg-reachable-eq-or-adj-mu-zero","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.StronglyRegularWanted.srg_reachable_eq_or_adj_mu_zero","parent":29649,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/index.html#srg-reachable-eq-or-adj-mu-zero","scope":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs","status":"fully_proved","summary":"Let $G$ be a finite strongly regular graph with parameters $(n, k, \\ell, \\mu)$ where $\\mu = 0$: $G$ has $n$ vertices, every vertex has degree $k$, any two adjacent vertices have exactly $\\ell$ common neighbors, and any two distinct nonadjacent vertices have no common neighbor. If $v$ and $w$ are vertices of $G$ joined by a walk, then $v = w$ or $v$ and $w$ are adjacent. Thus each connected component of $G$ is a\u2026","title":"Components of a strongly regular graph with \u03bc = 0 are cliques"},{"id":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/srg-nonprincipal-eigenvalue-quadratic","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.StronglyRegularWanted.srg_nonprincipal_eigenvalue_quadratic","parent":29649,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs/index.html#srg-nonprincipal-eigenvalue-quadratic","scope":"GraphTheory/algebraic-graph-theory/part-01/strongly-regular-graphs","status":"fully_proved","summary":"Let $G$ be a finite strongly regular graph with parameters $(n, k, \\ell, \\mu)$: $G$ has $n$ vertices, every vertex has degree $k$, any two adjacent vertices have exactly $\\ell$ common neighbors, and any two distinct nonadjacent vertices have exactly $\\mu$ common neighbors. Suppose that $G$ is connected and not complete, and let $A$ be its adjacency matrix over $\\mathbb{C}$. Then every eigenvalue $x \\in \\mathbb{C}$\u2026","title":"Non-principal eigenvalues of a connected non-complete strongly regular graph"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets.json","id":"GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets","kind":"scope","navigation":"module","parent":29541,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets/index.html","scope":"GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets","status":"can_prove","title":"\u03b5-light vertex sets"},{"id":"GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets/is-epsilon-light","kind":"node","lean":"MathlibExt.Combinatorics.EpsilonLightSubsetWanted.IsEpsilonLight","parent":29655,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets/index.html#is-epsilon-light","scope":"GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, let $\\varepsilon$ be a real number and let $S \\subseteq V$. Let $L$ be the real Laplacian matrix of $G$ and $L_S$ the real Laplacian matrix of the graph on $V$ whose edges are the edges of $G$ with both endpoints in $S$. The set $S$ is $\\varepsilon$-light in $G$ if $\\varepsilon L - L_S$ is positive semidefinite.","title":"\u03b5-light vertex sets"},{"id":"GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets/epsilon-light-subset-exists","kind":"node","lean":"MathlibExt.Combinatorics.EpsilonLightSubsetWanted.epsilonLightSubsetExists","parent":29655,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets/index.html#epsilon-light-subset-exists","scope":"GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets","status":"fully_proved","summary":"The statement that there is a real constant $c > 0$ such that for every natural number $n$, every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ and every real number $\\varepsilon$ with $0 < \\varepsilon < 1$, some set $S$ of vertices that is $\\varepsilon$-light in $G$ has at least $c \\varepsilon n$ elements.","title":"The large \u03b5-light subset property"},{"id":"GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets/epsilon-light-subset-exists-holds","kind":"node","lean":"MathlibExt.Combinatorics.EpsilonLightSubsetWanted.epsilonLightSubsetExists_holds","parent":29655,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets/index.html#epsilon-light-subset-exists-holds","scope":"GraphTheory/algebraic-graph-theory/part-01/light-vertex-sets","status":"can_prove","summary":"There is a real constant $c > 0$ such that for every natural number $n$, every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ and every real number $\\varepsilon$ with $0 < \\varepsilon < 1$, there is a set $S$ of vertices with at least $c \\varepsilon n$ elements that is $\\varepsilon$-light in $G$: the matrix $\\varepsilon L - L_S$ is positive semidefinite, where $L$ is the real Laplacian matrix of $G$ and\u2026","title":"Every graph has a large \u03b5-light vertex set"},{"data_url":"scopes/GraphTheory/algebraic-graph-theory/part-02.json","id":"GraphTheory/algebraic-graph-theory/part-02","kind":"scope","navigation":"transparent","parent":29540,"result":false,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-02/index.html","scope":"GraphTheory/algebraic-graph-theory/part-02","status":"fully_proved","title":"Part 2 of 2"},{"id":"GraphTheory/algebraic-graph-theory/part-02/is-vertex-transitive","kind":"node","lean":"SimpleGraph.IsVertexTransitive","parent":29659,"result":true,"route":"../roadmap/GraphTheory/algebraic-graph-theory/part-02/index.html#is-vertex-transitive","scope":"GraphTheory/algebraic-graph-theory/part-02","status":"fully_proved","summary":"A graph is vertex-transitive when its automorphism group acts transitively on vertices.","title":"Vertex-transitive graph"},{"data_url":"scopes/GraphTheory/matchings-and-flows.json","id":"GraphTheory/matchings-and-flows","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/matchings-and-flows/index.html","scope":"GraphTheory/matchings-and-flows","status":"can_prove","summary":"Matchings and factors in graphs, stable matchings, and flows in networks, including K\u0151nig's theorem, Berge's lemma and the max-flow min-cut theorem.","title":"Matchings and flows"},{"data_url":"scopes/GraphTheory/matchings-and-flows/berge-s-lemma.json","id":"GraphTheory/matchings-and-flows/berge-s-lemma","kind":"scope","navigation":"module","parent":29661,"result":false,"route":"../roadmap/GraphTheory/matchings-and-flows/berge-s-lemma/index.html","scope":"GraphTheory/matchings-and-flows/berge-s-lemma","status":"fully_proved","title":"Berge's lemma"},{"id":"GraphTheory/matchings-and-flows/berge-s-lemma/berge-general","kind":"node","lean":"MetaMathlibExt.berge_general","parent":29662,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/berge-s-lemma/index.html#berge-general","scope":"GraphTheory/matchings-and-flows/berge-s-lemma","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $M$ be a matching subgraph of $G$, so that every vertex of $M$ is incident to exactly one edge of $M$. Then $M$ has the maximum number of edges among all matching subgraphs of $G$ if and only if there is no $M$-augmenting path: a path in $G$ of positive length between two distinct vertices $u$ and $v$, neither of which is a vertex of $M$, whose consecutive\u2026","title":"Berge's lemma for matching subgraphs"},{"id":"GraphTheory/matchings-and-flows/berge-s-lemma/berge","kind":"node","lean":"MetaMathlibExt.Berge","parent":29662,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/berge-s-lemma/index.html#berge","scope":"GraphTheory/matchings-and-flows/berge-s-lemma","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $M$ be a matching of $G$, that is, a finite set of edges of $G$ no two distinct members of which share a vertex. Then $M$ has maximum cardinality among all matchings of $G$ if and only if there is no $M$-augmenting path: a path in $G$ of positive length between two distinct vertices $u$ and $v$, neither of which lies on an edge of $M$, whose consecutive edges\u2026","title":"Berge's lemma"},{"data_url":"scopes/GraphTheory/matchings-and-flows/k-nig-s-theorem.json","id":"GraphTheory/matchings-and-flows/k-nig-s-theorem","kind":"scope","navigation":"module","parent":29661,"result":false,"route":"../roadmap/GraphTheory/matchings-and-flows/k-nig-s-theorem/index.html","scope":"GraphTheory/matchings-and-flows/k-nig-s-theorem","status":"fully_proved","title":"K\u0151nig's theorem"},{"id":"GraphTheory/matchings-and-flows/k-nig-s-theorem/konig-matching-cover-general","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.KonigMatchingCoverWanted.konig_matching_cover_general","parent":29665,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/k-nig-s-theorem/index.html#konig-matching-cover-general","scope":"GraphTheory/matchings-and-flows/k-nig-s-theorem","status":"fully_proved","summary":"Let $G$ be a bipartite simple graph on a finite vertex type $V$. Then there are a matching $M$ in $G$ and a vertex cover $C \\subseteq V$ of $G$ such that the number of edges of $M$ equals the number of vertices in $C$.","title":"K\u0151nig's theorem on a finite vertex type"},{"id":"GraphTheory/matchings-and-flows/k-nig-s-theorem/konig-matching-cover","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.KonigMatchingCoverWanted.konig_matching_cover","parent":29665,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/k-nig-s-theorem/index.html#konig-matching-cover","scope":"GraphTheory/matchings-and-flows/k-nig-s-theorem","status":"fully_proved","summary":"Every bipartite simple graph $G$ on a finite vertex type $V$ has a matching $M$ and a vertex cover $C \\subseteq V$ such that the number of edges of $M$ equals the number of vertices in $C$.","title":"K\u0151nig's theorem for finite bipartite graphs"},{"data_url":"scopes/GraphTheory/matchings-and-flows/matching-cuts.json","id":"GraphTheory/matchings-and-flows/matching-cuts","kind":"scope","navigation":"module","parent":29661,"result":false,"route":"../roadmap/GraphTheory/matchings-and-flows/matching-cuts/index.html","scope":"GraphTheory/matchings-and-flows/matching-cuts","status":"can_prove","title":"Matching cuts"},{"id":"GraphTheory/matchings-and-flows/matching-cuts/avg-degree","kind":"node","lean":"MathlibExt.Combinatorics.MatchingCutGirthWanted.avgDegree","parent":29668,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/matching-cuts/index.html#avg-degree","scope":"GraphTheory/matchings-and-flows/matching-cuts","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set $V$ with edge set $E$, the average degree of $G$ is the rational number $2|E|/|V|$; it equals $0$ when $V$ is empty.","title":"Average degree of a finite graph"},{"id":"GraphTheory/matchings-and-flows/matching-cuts/has-matching-cut","kind":"node","lean":"MathlibExt.Combinatorics.MatchingCutGirthWanted.HasMatchingCut","parent":29668,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/matching-cuts/index.html#has-matching-cut","scope":"GraphTheory/matchings-and-flows/matching-cuts","status":"fully_proved","summary":"A simple graph $G$ has a matching cut if some matching $M$ of $G$, that is, a set of pairwise disjoint edges of $G$, is such that deleting the edges of $M$ from $G$ leaves two vertices $u$ and $v$ that are not joined by any path. The matching may be empty, in which case this says that $G$ itself has two vertices not joined by any path.","title":"Matching cut"},{"id":"GraphTheory/matchings-and-flows/matching-cuts/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.MatchingCutGirthWanted.conjecture","parent":29668,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/matching-cuts/index.html#conjecture","scope":"GraphTheory/matchings-and-flows/matching-cuts","status":"fully_proved","summary":"For every real number $d$ there is a natural number $g$ such that every simple graph $G$ on a finite vertex set with at least two vertices, whose average degree is less than $d$ and whose girth is at least $g$, has a matching cut. An acyclic graph has infinite girth, so it meets the girth bound for every $g$.","title":"Matching cut and girth question, affirmative form"},{"id":"GraphTheory/matchings-and-flows/matching-cuts/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.MatchingCutGirthWanted.conjecture_refuted","parent":29668,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/matching-cuts/index.html#conjecture-refuted","scope":"GraphTheory/matchings-and-flows/matching-cuts","status":"can_prove","summary":"It is not the case that for every real number $d$ there is a natural number $g$ such that every simple graph on a finite vertex set with at least two vertices, average degree less than $d$ and girth at least $g$ has a matching cut. That is, there is a real number $d$ such that for every natural number $g$ some simple graph on a finite vertex set with at least two vertices, average degree less than $d$ and girth at\u2026","title":"Negative answer to the matching cut and girth question"},{"data_url":"scopes/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem.json","id":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","kind":"scope","navigation":"module","parent":29661,"result":false,"route":"../roadmap/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/index.html","scope":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","status":"fully_proved","title":"Max-flow min-cut theorem"},{"id":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/is-feasible-flow","kind":"node","lean":"MathlibExt.Combinatorics.Graph.MaxFlowMinCutWanted.IsFeasibleFlow","parent":29673,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/index.html#is-feasible-flow","scope":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","status":"fully_proved","summary":"Let $V$ be a finite set, $c : V \\times V \\to \\mathbb{R}$ a capacity function and $s, t \\in V$. A function $f : V \\times V \\to \\mathbb{R}$ is a feasible $s$-$t$ flow for $c$ if $0 \\le f(u, v) \\le c(u, v)$ for all $u, v \\in V$ and $f$ is conserved at every vertex $v \\ne s, t$, that is, $\\sum_{u \\in V} f(u, v) = \\sum_{w \\in V} f(v, w)$.","title":"Feasible flow"},{"id":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/flow-value","kind":"node","lean":"MathlibExt.Combinatorics.Graph.MaxFlowMinCutWanted.flowValue","parent":29673,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/index.html#flow-value","scope":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","status":"fully_proved","summary":"For a finite set $V$, a vertex $s \\in V$ and a function $f : V \\times V \\to \\mathbb{R}$, the value of $f$ is its net outflow from the source $s$, $\\sum_{v \\in V} f(s, v) - \\sum_{u \\in V} f(u, s)$.","title":"Flow value"},{"id":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/is-stcut","kind":"node","lean":"MathlibExt.Combinatorics.Graph.MaxFlowMinCutWanted.IsSTCut","parent":29673,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/index.html#is-stcut","scope":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","status":"fully_proved","summary":"For a finite set $V$ and vertices $s, t \\in V$, an $s$-$t$ cut is a subset $S \\subseteq V$ with $s \\in S$ and $t \\notin S$.","title":"Source\u2013sink cut"},{"id":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/cut-capacity","kind":"node","lean":"MathlibExt.Combinatorics.Graph.MaxFlowMinCutWanted.cutCapacity","parent":29673,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/index.html#cut-capacity","scope":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","status":"fully_proved","summary":"Let $V$ be a finite set and $c : V \\times V \\to \\mathbb{R}$ a capacity function. The capacity of a subset $S \\subseteq V$ is the total capacity of the arcs leaving $S$, $\\sum_{u \\in S} \\sum_{v \\in V \\setminus S} c(u, v)$.","title":"Cut capacity"},{"id":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/cut-flow-eq","kind":"node","lean":"MathlibExt.Combinatorics.Graph.MaxFlowMinCutWanted.cutFlow_eq","parent":29673,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/index.html#cut-flow-eq","scope":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","status":"fully_proved","summary":"Let $V$ be a finite set, $s, t \\in V$, and $f : V \\times V \\to \\mathbb{R}$ a function conserved at every vertex $v \\ne s, t$, that is, $\\sum_{u \\in V} f(u, v) = \\sum_{w \\in V} f(v, w)$; no sign or capacity condition is imposed. Then for every subset $S \\subseteq V$ with $s \\in S$ and $t \\notin S$, the flow value $\\sum_{v \\in V} f(s, v) - \\sum_{u \\in V} f(u, s)$ of $f$ equals\u2026","title":"Flow across a cut"},{"id":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/flow-value-le-cut-capacity","kind":"node","lean":"MathlibExt.Combinatorics.Graph.MaxFlowMinCutWanted.flowValue_le_cutCapacity","parent":29673,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/index.html#flow-value-le-cut-capacity","scope":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","status":"fully_proved","summary":"Let $V$ be a finite set, $c : V \\times V \\to \\mathbb{R}$ a capacity function and $s, t \\in V$. If $f$ is a feasible $s$-$t$ flow for $c$ and $S$ is an $s$-$t$ cut, then the flow value of $f$ is at most the capacity of $S$ for $c$.","title":"Weak duality for flows and cuts"},{"id":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/max-flow-min-cut","kind":"node","lean":"MathlibExt.Combinatorics.Graph.MaxFlowMinCutWanted.max_flow_min_cut","parent":29673,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/max-flow-min-cut-theorem/index.html#max-flow-min-cut","scope":"GraphTheory/matchings-and-flows/max-flow-min-cut-theorem","status":"fully_proved","summary":"Let $V$ be a finite set, $c : V \\times V \\to \\mathbb{R}$ a capacity function with $c(u, v) \\ge 0$ for all $u, v \\in V$, and $s, t \\in V$ with $s \\ne t$. Then there is a feasible $s$-$t$ flow $f$ for $c$ whose value is at least the value of every feasible $s$-$t$ flow for $c$ and at most the capacity of every $s$-$t$ cut, and there is an $s$-$t$ cut whose capacity equals the value of $f$. In other words, the maximum\u2026","title":"Max-flow min-cut theorem"},{"data_url":"scopes/GraphTheory/matchings-and-flows/petersen-s-theorem.json","id":"GraphTheory/matchings-and-flows/petersen-s-theorem","kind":"scope","navigation":"module","parent":29661,"result":false,"route":"../roadmap/GraphTheory/matchings-and-flows/petersen-s-theorem/index.html","scope":"GraphTheory/matchings-and-flows/petersen-s-theorem","status":"fully_proved","title":"Petersen's theorem"},{"id":"GraphTheory/matchings-and-flows/petersen-s-theorem/petersen-bridgeless-cubic-has-perfect-ma","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.PetersenMatchingWanted.petersen_bridgeless_cubic_has_perfect_matching_general","parent":29681,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/petersen-s-theorem/index.html#petersen-bridgeless-cubic-has-perfect-ma","scope":"GraphTheory/matchings-and-flows/petersen-s-theorem","status":"fully_proved","summary":"Every finite simple graph that is $3$-regular and has no bridge, that is, no edge whose removal disconnects its endpoints, has a perfect matching: a set of edges of the graph such that every vertex lies on exactly one of them.","title":"Petersen's theorem on cubic graphs"},{"id":"GraphTheory/matchings-and-flows/petersen-s-theorem/petersen-bridgeless-cubic-has-perfect-ma-2","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.PetersenMatchingWanted.petersen_bridgeless_cubic_has_perfect_matching","parent":29681,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/petersen-s-theorem/index.html#petersen-bridgeless-cubic-has-perfect-ma-2","scope":"GraphTheory/matchings-and-flows/petersen-s-theorem","status":"fully_proved","summary":"Every finite simple graph that is $3$-regular and has no bridge, that is, no edge whose removal disconnects its endpoints, has a perfect matching: a set of edges of the graph such that every vertex lies on exactly one of them.","title":"Petersen's perfect matching theorem"},{"data_url":"scopes/GraphTheory/matchings-and-flows/stable-marriage-problem.json","id":"GraphTheory/matchings-and-flows/stable-marriage-problem","kind":"scope","navigation":"module","parent":29661,"result":false,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","title":"Stable marriage problem"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/strict-total-order","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.StrictTotalOrder","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#strict-total-order","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"A strict total order on a type $\\alpha$ is a binary relation $<$ on $\\alpha$ that is irreflexive ($a < a$ holds for no $a$), transitive ($a < b$ and $b < c$ imply $a < c$) and total on distinct elements ($a \\ne b$ implies $a < b$ or $b < a$). In the Gale\u2013Shapley development it models strict preference.","title":"Strict preference order"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/mk","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.StrictTotalOrder.mk","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#mk","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"A binary relation $<$ on a set $\\alpha$ that is irreflexive ($a \\not< a$ for all $a$), transitive ($a < b$ and $b < c$ imply $a < c$) and total on distinct elements ($a \\ne b$ implies $a < b$ or $b < a$) defines a strict total order on $\\alpha$ whose relation is $<$.","title":"Building a strict total order from a relation"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/lt","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.StrictTotalOrder.lt","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#lt","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"The relation of a strict total order $T$ on a set $\\alpha$ is its underlying binary relation $\\lt _T$ on $\\alpha$, which is irreflexive, transitive, and relates any two distinct elements in one direction or the other.","title":"Relation of a strict total order"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/irrefl","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.StrictTotalOrder.irrefl","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#irrefl","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $T$ be a strict total order on a set $\\alpha$, with relation $\\lt _T$. Then $a \\lt _T a$ fails for every $a \\in \\alpha$.","title":"Irreflexivity of a strict total order"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/trans","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.StrictTotalOrder.trans","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#trans","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $<$ be the relation of a strict total order on a type $\\alpha$. If $a, b, c \\in \\alpha$ satisfy $a < b$ and $b < c$, then $a < c$.","title":"Transitivity of a strict total order"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/total","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.StrictTotalOrder.total","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#total","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $<$ be the relation of a strict total order on a type $\\alpha$. If $a, b \\in \\alpha$ and $a \\ne b$, then $a < b$ or $b < a$.","title":"Strict total orders are total on distinct elements"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/ext","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.StrictTotalOrder.ext","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#ext","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $T$ and $T'$ be strict total orders on a set $\\alpha$, with relations $\\lt _T$ and $\\lt _{T'}$. If $x \\lt _T y \\iff x \\lt _{T'} y$ for all $x, y \\in \\alpha$, then $T = T'$.","title":"Strict total orders are determined by their relation"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/mprefs","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.MPrefs","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#mprefs","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For sets $M$ (men) and $W$ (women), a preference profile of the men assigns to each $m \\in M$ a strict total order on $W$, the preferences of $m$.","title":"Men's preferences"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/wprefs","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.WPrefs","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#wprefs","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For types $M$ and $W$, a preference profile of $W$ over $M$ assigns to each $w \\in W$ a strict total order on $M$, read as the strict preference of $w$ among the elements of $M$.","title":"Women's preference profile"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/perfect-matching","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.PerfectMatching","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#perfect-matching","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"A perfect matching between sets $M$ and $W$ is a bijection $\\mu \\colon M \\to W$, which matches each $m \\in M$ with $\\mu(m) \\in W$.","title":"Perfect matching"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/is-blocking-pair","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.IsBlockingPair","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#is-blocking-pair","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $M$ and $W$ be sets. Suppose that each $m \\in M$ has a strict total order $\\lt _m$ of preference on $W$ and each $w \\in W$ has a strict total order $\\lt _w$ of preference on $M$, where $x \\lt _m y$ means that $m$ prefers $x$ to $y$, and likewise for $\\lt _w$. For a perfect matching $\\mu$ between $M$ and $W$, a pair $(m, w) \\in M \\times W$ is a blocking pair when $w \\lt _m \\mu(m)$ and $m \\lt _w \\mu^{-1}(w)$,\u2026","title":"Blocking pair"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/is-stable","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.IsStable","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#is-stable","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $M$ and $W$ be sets. Suppose that each $m \\in M$ has a strict total order of preference on $W$ and each $w \\in W$ has a strict total order of preference on $M$. A perfect matching $\\mu$ between $M$ and $W$ is stable for these preferences when no pair $(m, w) \\in M \\times W$ is a blocking pair for $\\mu$.","title":"Stable perfect matching"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gale-shapley-exists-stable-matching-gene","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.gale_shapley_exists_stable_matching_general","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gale-shapley-exists-stable-matching-gene","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $M$ and $W$ be finite sets with the same number of elements. Suppose that each $m \\in M$ has a strict total order of preference on $W$ and each $w \\in W$ has a strict total order of preference on $M$. Then some perfect matching between $M$ and $W$ is stable for these preferences.","title":"Existence of stable perfect matchings"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gale-shapley-exists-stable-matching","kind":"node","lean":"MathlibExt.Combinatorics.Matching.GaleShapley.gale_shapley_exists_stable_matching","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gale-shapley-exists-stable-matching","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $M$ and $W$ be finite sets with the same number of elements. Suppose that each $m \\in M$ has a strict total order of preference on $W$ and each $w \\in W$ has a strict total order of preference on $M$. Then some perfect matching between $M$ and $W$ is stable for these preferences.","title":"Gale\u2013Shapley theorem"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/preference-profile","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.PreferenceProfile","parent":29684,"reading_title":{"format":"inline-parts-v1","parts":[["text","Preference profile for "],["math","\\(n\\)"],["text"," men and "],["math","\\(n\\)"],["text"," women"]],"source":"Preference profile for $n$ men and $n$ women"},"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#preference-profile","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a natural number $n$, number $n$ men and $n$ women by $0, \\dots, n - 1$. A preference profile $P = (M, W)$ for them consists of a permutation $M_i$ of $\\{0, \\dots, n - 1\\}$ for each man $i$ and a permutation $W_j$ of $\\{0, \\dots, n - 1\\}$ for each woman $j$: $M_i(r)$ is the woman man $i$ ranks at zero-based position $r$, and $W_j(r)$ is the man woman $j$ ranks at position $r$.","title":"Preference profile for $n$ men and $n$ women"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/man-rank","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.manRank","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#man-rank","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a preference profile of $n$ men and $n$ women, the rank of woman $j$ for man $i$ is the zero-based position $r \\in \\{0, \\dots, n-1\\}$ at which $i$ ranks $j$, so his favorite has rank $0$ and his least favorite has rank $n - 1$.","title":"Rank of a woman for a man"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/woman-rank","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.womanRank","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#woman-rank","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a preference profile for $n$ men and $n$ women, the rank that woman $j$ gives man $i$ is the zero-based position at which she ranks him, that is, the paper's ranking minus one.","title":"A woman's rank of a man"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/is-soulmate-pair","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.IsSoulmatePair","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#is-soulmate-pair","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a preference profile of $n$ men and $n$ women, man $i$ and woman $j$ are soulmates when each ranks the other first: $j$ is the first choice of $i$ and $i$ is the first choice of $j$.","title":"Soulmate pair"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/soulmate-pair-count","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.soulmatePairCount","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#soulmate-pair-count","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"The number of soulmate pairs of a preference profile $P$ for $n$ men and $n$ women is the number of pairs $(i, j)$ of a man $i$ and a woman $j$ that are soulmates in $P$, that is, each ranks the other first.","title":"Number of soulmate pairs"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/man-prefs","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.manPrefs","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#man-prefs","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a preference profile of $n$ men and $n$ women, each man $i$ gets the strict total order on the women in which $i$ prefers woman $a$ to woman $b$ exactly when he ranks $a$ at an earlier position than $b$.","title":"Men's preferences induced by rankings"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/woman-prefs","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.womanPrefs","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#woman-prefs","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a preference profile $P$ for $n$ men and $n$ women, each woman $j$ gets the strict total order on the men in which man $a$ comes before (is preferred to) man $b$ exactly when woman $j$ ranks $a$ at an earlier position than $b$.","title":"Women's preference orders of a profile"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/is-stable-matching","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.IsStableMatching","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#is-stable-matching","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a preference profile of $n$ men and $n$ women, a matching $\\mu$, in which man $i$ marries woman $\\mu(i)$, is stable when no man and woman both strictly prefer each other to their own spouses. Equivalently, $\\mu$ is a stable perfect matching for the strict preferences induced by the men's and the women's rankings.","title":"Stable matching"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/is-stable-matching-iff","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.isStableMatching_iff","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#is-stable-matching-iff","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a preference profile of $n$ men and $n$ women, a matching $\\mu$, in which man $i$ marries woman $\\mu(i)$, is stable if and only if there are no man $i$ and woman $j$ such that $i$ ranks $j$ strictly above his wife $\\mu(i)$ and $j$ ranks $i$ strictly above her husband $\\mu^{-1}(j)$.","title":"Stability in terms of ranks"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/is-hell-couple","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.IsHellCouple","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#is-hell-couple","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $P$ be a preference profile of $n$ men and $n$ women, and let $\\mu$ be a matching in which man $i$ marries woman $\\mu(i)$. Man $i$ and his wife $\\mu(i)$ form a hell-couple when each ranks the other last: $i$ ranks $\\mu(i)$ in his last position and $\\mu(i)$ ranks $i$ in her last position.","title":"Hell-couple"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/latin-square-count","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.latinSquareCount","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#latin-square-count","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, $L_n$ is the number of Latin squares of order $n$: families $(M_0, \\dots, M_{n-1})$ of permutations of $\\{0, \\dots, n-1\\}$ (the rows) such that for every position $r$ the entries $M_0(r), \\dots, M_{n-1}(r)$ (column $r$) are pairwise distinct.","title":"Number of Latin squares"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gsengaged","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.GSEngaged","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gsengaged","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $P$ be a preference profile for $n$ men and $n$ women, with positions in each ranking numbered $0, \\dots, n - 1$, and consider the state of the men-proposing Gale\u2013Shapley algorithm in which each man $i'$ has proposed to the women at the positions $r' < p(i')$ of his ranking. Man $i$ is provisionally engaged in this state if $1 \\le p(i) \\le n$ and the woman $w$ at his position $p(i) - 1$ (the last woman he\u2026","title":"Provisional engagement in the Gale\u2013Shapley algorithm"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gs-proposals","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.gsProposals","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gs-proposals","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $P$ be a preference profile for $n$ men and $n$ women. The number $p_t(i)$ of women man $i$ has proposed to after $t$ rounds of the men-proposing Gale\u2013Shapley algorithm on $P$ is defined by $p_0(i) = 0$ and, for every $t$, $p_{t+1}(i) = p_t(i)$ if man $i$ is provisionally engaged in the state $p_t$ and $p_{t+1}(i) = p_t(i) + 1$ otherwise: in each round every man who is not provisionally engaged proposes to his\u2026","title":"Proposal counts in the Gale\u2013Shapley algorithm"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gale-shapley-rounds","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.galeShapleyRounds","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gale-shapley-rounds","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For a preference profile $P$ for $n$ men and $n$ women, the number of rounds of the men-proposing Gale\u2013Shapley algorithm on $P$ is the least $t$ such that every man is provisionally engaged after $t$ rounds (or $0$ if there is no such $t$).","title":"Number of Gale\u2013Shapley rounds"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gs-proposals-zero","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.gsProposals_zero","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gs-proposals-zero","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For every preference profile, after zero rounds of the men-proposing Gale\u2013Shapley algorithm every man has made zero proposals.","title":"No proposals before the first round"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gs-proposals-succ","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.gsProposals_succ","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gs-proposals-succ","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Let $P$ be a preference profile for $n$ men and $n$ women and let $t$ be a natural number. After $t + 1$ rounds of the men-proposing Gale\u2013Shapley algorithm on $P$, a man who is provisionally engaged after $t$ rounds has made as many proposals as after $t$ rounds, and every other man has made one proposal more than after $t$ rounds.","title":"One round of the Gale\u2013Shapley algorithm"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gale-shapley-rounds-of-zero","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.galeShapleyRounds_of_zero","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gale-shapley-rounds-of-zero","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"For the preference profile with no men and no women, the men-proposing Gale\u2013Shapley algorithm takes zero rounds.","title":"Gale\u2013Shapley rounds with no participants"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/soulmate-pair-count-le","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.soulmatePairCount_le","parent":29684,"reading_title":{"format":"inline-parts-v1","parts":[["text","A profile has at most "],["math","\\(n\\)"],["text"," soulmate pairs"]],"source":"A profile has at most $n$ soulmate pairs"},"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#soulmate-pair-count-le","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"Every preference profile for $n$ men and $n$ women has at most $n$ soulmate pairs, since a man's soulmate is his first choice.","title":"A profile has at most $n$ soulmate pairs"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/latin-square-count-zero","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.latinSquareCount_zero","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#latin-square-count-zero","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"fully_proved","summary":"There is exactly one Latin square of order $0$, that is, $L_0 = 1$.","title":"Latin squares of order zero"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gale-shapley-rounds-of-same-taste","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.galeShapleyRounds_of_same_taste","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gale-shapley-rounds-of-same-taste","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"Let $P$ be a preference profile for $n$ men and $n$ women in which all men rank the women in the same order. Then the men-proposing Gale\u2013Shapley algorithm on $P$ ends in exactly $n$ rounds.","title":"Gale\u2013Shapley rounds when all men have the same preferences"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/gale-shapley-rounds-eq-one-iff","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.galeShapleyRounds_eq_one_iff","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#gale-shapley-rounds-eq-one-iff","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"Let $n \\ge 1$ and let $P$ be a preference profile for $n$ men and $n$ women. The men-proposing Gale\u2013Shapley algorithm on $P$ ends in exactly one round if and only if every woman is the first choice of exactly one man.","title":"When the Gale\u2013Shapley algorithm ends in one round"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/disjoint-profiles-card-le-and-dvd","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.disjoint_profiles_card_le_and_dvd","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#disjoint-profiles-card-le-and-dvd","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"Call a preference profile for $n$ men and $n$ women disjoint if for every pair of positions $(r, s)$ there is exactly one man\u2013woman pair in which the man ranks the woman at position $r$ and the woman ranks the man at position $s$. For every natural number $n$, the number of disjoint profiles is at most $(n^2)!$ and is divisible by $(n!)^2$.","title":"Counting disjoint preference profiles"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/outside-no-soulmates-card","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.outside_no_soulmates_card","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#outside-no-soulmates-card","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers, where $n - k$ is read as $0$ when $k > n$. Consider the $n - k$ men $k, \\dots, n - 1$ and the $n - k$ women $k, \\dots, n - 1$ among $n$ men and $n$ women numbered from $0$, where each of these men ranks all $n$ women and each of these women ranks all $n$ men. The number of ways to choose these rankings so that no man and woman among them each rank the other first is","title":"Outsiders without soulmate pairs"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/soulmate-profiles-card","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.soulmate_profiles_card","parent":29684,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of profiles with exactly "],["math","\\(k\\)"],["text"," soulmate pairs"]],"source":"Number of profiles with exactly $k$ soulmate pairs"},"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#soulmate-profiles-card","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"For natural numbers $n$ and $k$, the number $F(k, n)$ of preference profiles for $n$ men and $n$ women with exactly $k$ soulmate pairs is","title":"Number of profiles with exactly $k$ soulmate pairs"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/unique-stable-matching-of-soulmate-pair","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.unique_stable_matching_of_soulmatePairCount_eq_pred","parent":29684,"reading_title":{"format":"inline-parts-v1","parts":[["text","Profiles with "],["math","\\(n - 1\\)"],["text"," soulmate pairs"]],"source":"Profiles with $n - 1$ soulmate pairs"},"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#unique-stable-matching-of-soulmate-pair","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"Let $P$ be a preference profile for $n$ men and $n$ women with exactly $n - 1$ soulmate pairs (read as $0$ when $n = 0$). Then $P$ has exactly one stable matching. Moreover, for every stable matching $\\mu$ of $P$ and every man $i$ who has no soulmate, the men-proposing Gale\u2013Shapley algorithm on $P$ ends after $r + 1$ rounds, where $r$ is the zero-based position at which man $i$ ranks his wife $\\mu(i)$.","title":"Profiles with $n - 1$ soulmate pairs"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/men-latin-soulmate-profiles-card","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.men_latin_soulmate_profiles_card","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#men-latin-soulmate-profiles-card","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"For all $n, k \\in \\mathbb{N}$, the number of preference profiles of $n$ men and $n$ women in which, for every position $r$, different men rank different women at position $r$ (the men's rankings form a Latin square), and which have exactly $k$ soulmate pairs, is","title":"Profiles with Latin-square men's preferences, counted by soulmate pairs"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/mutually-latin-soulmate-profiles-card","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.mutually_latin_soulmate_profiles_card","parent":29684,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mutually Latin profiles with "],["math","\\(k\\)"],["text"," soulmate pairs"]],"source":"Mutually Latin profiles with $k$ soulmate pairs"},"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#mutually-latin-soulmate-profiles-card","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"Let $n$ and $k$ be natural numbers and let $L_n$ be the number of Latin squares of order $n$. Call the men's preferences of a preference profile for $n$ men and $n$ women Latin when, for every position $r$, distinct men rank distinct women at position $r$, and likewise for the women's preferences. The number of preference profiles whose men's and women's preferences are both Latin and that have exactly $k$ soulmate\u2026","title":"Mutually Latin profiles with $k$ soulmate pairs"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/hell-couple-subsingleton","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.hellCouple_subsingleton","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#hell-couple-subsingleton","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"Let $P$ be a preference profile of $n$ men and $n$ women, and let $\\mu$ be a stable matching for $P$, in which man $i$ marries woman $\\mu(i)$. Then at most one man $i$ forms a hell-couple with his wife $\\mu(i)$.","title":"At most one hell-couple in a stable matching"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/hell-couple-mem-all-stable","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.hellCouple_mem_all_stable","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#hell-couple-mem-all-stable","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"Let $P$ be a preference profile for $n$ men and $n$ women and let $\\mu$ and $\\nu$ be stable matchings for $P$. If man $i$ and his wife $\\mu(i)$ form a hell-couple in $\\mu$ (each ranks the other last), then $\\nu(i) = \\mu(i)$.","title":"A hell-couple belongs to every stable matching"},{"id":"GraphTheory/matchings-and-flows/stable-marriage-problem/outcast-profiles-card","kind":"node","lean":"MetaMathlibExt.StableMatchingSequences.outcast_profiles_card","parent":29684,"result":true,"route":"../roadmap/GraphTheory/matchings-and-flows/stable-marriage-problem/index.html#outcast-profiles-card","scope":"GraphTheory/matchings-and-flows/stable-marriage-problem","status":"can_prove","summary":"In a preference profile for $n$ men and $n$ women, a man $i$ and a woman $j$ form a pair of outcasts when every man other than $i$ ranks $j$ last and every woman other than $j$ ranks $i$ last. The number of preference profiles with at least one pair of outcasts is $14$ if $n = 2$ and $n^4 ((n-1)!)^{2n}$ if $n \\ne 2$ (this is $0$ for $n = 0$).","title":"Profiles with a pair of outcasts"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity.json","id":"GraphTheory/paths-cycles-connectivity","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/index.html","scope":"GraphTheory/paths-cycles-connectivity","status":"can_prove","summary":"Walks, paths and cycles in graphs, Eulerian and Hamiltonian cycles, and vertex and edge connectivity, including Menger's theorem and Fleischner's theorem on squares of 2-connected graphs.","title":"Paths, cycles and connectivity"},{"id":"GraphTheory/paths-cycles-connectivity/is-path-append-of-is-path","kind":"node","lean":"SimpleGraph.Walk.isPath_append_of_isPath","parent":29729,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/index.html#is-path-append-of-is-path","scope":"GraphTheory/paths-cycles-connectivity","status":"fully_proved","summary":"Two paths that meet only at their common endpoint `m` concatenate to a path.","title":"Concatenation of paths meeting at one endpoint is a path"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs.json","id":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/index.html","scope":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs","status":"fully_proved","title":"Bridges and bridgeless graphs"},{"id":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/is-bridgeless","kind":"node","lean":"SimpleGraph.IsBridgeless","parent":29731,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/index.html#is-bridgeless","scope":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs","status":"fully_proved","summary":"A simple graph is bridgeless when no edge is a bridge.","title":"Bridgeless graph"},{"id":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/is-bridgeless-iff","kind":"node","lean":"SimpleGraph.isBridgeless_iff","parent":29731,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/index.html#is-bridgeless-iff","scope":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs","status":"fully_proved","summary":"A simple graph $G$ is bridgeless if and only if no edge of $G$ is a bridge.","title":"Characterization of bridgeless graphs"},{"id":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/mk","kind":"node","lean":"SimpleGraph.IsBridgeless.mk","parent":29731,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/index.html#mk","scope":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs","status":"fully_proved","summary":"If no edge of a simple graph $G$ is a bridge, then $G$ is bridgeless.","title":"Bridgeless from non-bridge edges"},{"id":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/not-is-bridge","kind":"node","lean":"SimpleGraph.IsBridgeless.not_isBridge","parent":29731,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/index.html#not-is-bridge","scope":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs","status":"fully_proved","summary":"Every edge of a bridgeless graph is not a bridge.","title":"Edges of a bridgeless graph are not bridges"},{"id":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/is-bridgeless-bot","kind":"node","lean":"SimpleGraph.isBridgeless_bot","parent":29731,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs/index.html#is-bridgeless-bot","scope":"GraphTheory/paths-cycles-connectivity/bridges-and-bridgeless-graphs","status":"fully_proved","summary":"The edgeless graph on any vertex set is vacuously bridgeless.","title":"The edgeless graph is bridgeless"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/cycle-covers.json","id":"GraphTheory/paths-cycles-connectivity/cycle-covers","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-covers/index.html","scope":"GraphTheory/paths-cycles-connectivity/cycle-covers","status":"fully_proved","title":"Cycle covers"},{"id":"GraphTheory/paths-cycles-connectivity/cycle-covers/cycle","kind":"node","lean":"SimpleGraph.Cycle","parent":29737,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-covers/index.html#cycle","scope":"GraphTheory/paths-cycles-connectivity/cycle-covers","status":"fully_proved","summary":"A cycle in a simple graph $G$ consists of a base vertex $v$ of $G$ together with a closed walk $v = v_0, v_1, \\dots, v_n = v$ in $G$ such that $n \\ge 1$, the $n$ edges $v_{k-1} v_k$ of the walk are pairwise distinct, and the vertices $v_1, \\dots, v_n$ are pairwise distinct. In a simple graph this forces $n \\ge 3$.","title":"Cycles in a graph"},{"id":"GraphTheory/paths-cycles-connectivity/cycle-covers/edges","kind":"node","lean":"SimpleGraph.Cycle.edges","parent":29737,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-covers/index.html#edges","scope":"GraphTheory/paths-cycles-connectivity/cycle-covers","status":"fully_proved","summary":"The edges of a cycle $c$ in a simple graph form the list of the edges of its closed walk, in the order in which $c$ traverses them starting from its base vertex; each edge is the unordered pair of its two endpoints.","title":"Edges of a cycle"},{"id":"GraphTheory/paths-cycles-connectivity/cycle-covers/length","kind":"node","lean":"SimpleGraph.Cycle.length","parent":29737,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-covers/index.html#length","scope":"GraphTheory/paths-cycles-connectivity/cycle-covers","status":"fully_proved","summary":"The length of a cycle in a simple graph is the number of edges of its closed walk.","title":"Length of a cycle"},{"id":"GraphTheory/paths-cycles-connectivity/cycle-covers/is-cycle-cover","kind":"node","lean":"SimpleGraph.IsCycleCover","parent":29737,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-covers/index.html#is-cycle-cover","scope":"GraphTheory/paths-cycles-connectivity/cycle-covers","status":"fully_proved","summary":"Let $G$ be a finite simple graph. A finite multiset $C$ of cycles in $G$ is a cycle cover of $G$ when every edge of $G$ is an edge of at least one cycle in $C$.","title":"Cycle covers"},{"id":"GraphTheory/paths-cycles-connectivity/cycle-covers/total-length","kind":"node","lean":"SimpleGraph.totalLength","parent":29737,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-covers/index.html#total-length","scope":"GraphTheory/paths-cycles-connectivity/cycle-covers","status":"fully_proved","summary":"The total length of a finite multiset $C$ of cycles in a simple graph is the sum of the lengths of the cycles in $C$, each counted with its multiplicity.","title":"Total length of a multiset of cycles"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference.json","id":"GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference/index.html","scope":"GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference","status":"fully_proved","title":"Cycle lengths and circumference"},{"id":"GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference/cycle-lengths","kind":"node","lean":"SimpleGraph.cycleLengths","parent":29743,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference/index.html#cycle-lengths","scope":"GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference","status":"fully_proved","summary":"For a simple graph $G$, the set of cycle lengths of $G$ is the set of natural numbers $m$ such that $G$ contains a cycle of length $m$.","title":"Cycle lengths"},{"id":"GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference/odd-cycle-lengths","kind":"node","lean":"SimpleGraph.oddCycleLengths","parent":29743,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference/index.html#odd-cycle-lengths","scope":"GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference","status":"fully_proved","summary":"For a simple graph $G$, the set of odd cycle lengths of $G$ is the set of those cycle lengths of $G$ that are odd, that is, the odd natural numbers $m$ such that $G$ contains a cycle of length $m$.","title":"Odd cycle lengths"},{"id":"GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference/is-longest-cycle","kind":"node","lean":"SimpleGraph.Walk.IsLongestCycle","parent":29743,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference/index.html#is-longest-cycle","scope":"GraphTheory/paths-cycles-connectivity/cycle-lengths-and-circumference","status":"fully_proved","summary":"A cycle is globally longest if no cycle in the graph has more edges.","title":"Longest cycle in a graph"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/eulerian-circuits.json","id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","title":"Eulerian circuits"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/count-p-edges-eq-degree","kind":"node","lean":"SimpleGraph.Walk.IsEulerian.countP_edges_eq_degree","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#count-p-edges-eq-degree","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $G$ be a finite simple graph and let $p$ be an Eulerian trail in $G$, that is, a walk between any two vertices that uses every edge of $G$ exactly once. Then for every vertex $x$, the number of edges of $p$ that contain $x$, counted with multiplicity along $p$, equals the degree of $x$ in $G$.","title":"Edges at a vertex of an Eulerian trail"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/exists-is-eulerian","kind":"node","lean":"SimpleGraph.Connected.exists_isEulerian","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#exists-is-eulerian","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Every finite connected simple graph in which every vertex has even degree has a closed Eulerian trail, that is, a closed walk that uses every edge of the graph exactly once.","title":"Euler circuits in connected even graphs"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/induce-support","kind":"node","lean":"SimpleGraph.Reachable.induce_support","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#induce-support","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $G$ be a simple graph and let $u$ and $v$ be non-isolated vertices of $G$, that is, vertices with at least one neighbour. If $v$ is reachable from $u$ in $G$, then $v$ is reachable from $u$ in the subgraph of $G$ induced on the set of non-isolated vertices.","title":"Reachability among non-isolated vertices"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/exists-is-eulerian-2","kind":"node","lean":"SimpleGraph.ConnectedSupport.exists_isEulerian","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#exists-is-eulerian-2","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $G$ be a finite simple graph in which every vertex has even degree, and suppose that the subgraph of $G$ induced on its non-isolated vertices is connected (in particular nonempty, so $G$ has at least one edge). Then $G$ has a closed Eulerian trail, that is, a closed walk that uses every edge of $G$ exactly once.","title":"Euler circuits in simple graphs whose non-isolated vertices are connected"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/edge-indexed-multigraph","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#edge-indexed-multigraph","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"An edge-indexed multigraph with vertex type $V$ and edge type $E$ assigns to each edge $e \\in E$ an unordered pair of endpoints in $V$, such that no edge has two equal endpoints. Distinct edges may have the same endpoints, so parallel edges are distinguished by their elements of $E$.","title":"Edge-indexed multigraph"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/indexed-walk","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#indexed-walk","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"In an edge-indexed multigraph, an indexed walk is a walk that records which edge it traverses at each step. Indexed walks are defined inductively: for every vertex $v$ there is the empty indexed walk from $v$ to $v$, and if $e$ is an edge with endpoints $u$ and $v$ and $p$ is an indexed walk from $v$ to $w$, then traversing $e$ from $u$ to $v$ and then following $p$ is an indexed walk from $u$ to $w$.","title":"Indexed walk"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/support","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.support","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#support","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"The support of an indexed walk $p$ from $u$ to $v$ in an edge-indexed multigraph is the list of the vertices that $p$ visits, in order and with repetitions, including both endpoints: the empty walk at $w$ has support $[w]$, and a walk that starts at $u$ with an edge and continues with an indexed walk $q$ has support $u$ followed by the support of $q$.","title":"Vertex list of an indexed walk"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/edges","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.edges","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#edges","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"The edge list of an indexed walk is the list of the edge identities it traverses, in order and with repetitions: the empty walk has the empty edge list, and the walk that traverses an edge $e$ and then follows an indexed walk $p$ has edge list $e$ followed by the edge list of $p$.","title":"Edge list of an indexed walk"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/is-trail","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.IsTrail","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#is-trail","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"An indexed walk $p$ in an edge-indexed multigraph is a trail when its edge list has no repeated entries, that is, $p$ traverses each edge of the multigraph at most once.","title":"Indexed trail"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/is-trail-iff","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.isTrail_iff","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#is-trail-iff","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"An indexed walk is a trail exactly when its edge list has no duplicates.","title":"Characterization of indexed trails"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/is-eulerian","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.IsEulerian","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#is-eulerian","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"An indexed walk is Eulerian when it is a trail, that is, it uses no edge identity twice, and every edge identity of the multigraph occurs in its edge list.","title":"Eulerian indexed walk"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/is-eulerian-iff","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.isEulerian_iff","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#is-eulerian-iff","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"An indexed walk is Eulerian exactly when it is a trail whose edge list contains every edge identity of the multigraph.","title":"Characterization of Eulerian indexed walks"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/support-nil","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.support_nil","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#support-nil","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"The support of the empty indexed walk at a vertex $v$ is the one-element list $[v]$.","title":"Support of the empty indexed walk"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/support-cons","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.support_cons","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#support-cons","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"If $e$ is an edge with endpoints $u$ and $v$ and $p$ is an indexed walk starting at $v$, then the support of the walk that traverses $e$ from $u$ to $v$ and then follows $p$ is $u$ followed by the support of $p$.","title":"Support of a walk extended at the front"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/edges-nil","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.edges_nil","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#edges-nil","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"For every vertex $v$ of an edge-indexed multigraph, the edge list of the empty indexed walk at $v$ is the empty list.","title":"The empty indexed walk has no edges"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/edges-cons","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.edges_cons","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#edges-cons","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $p$ be an indexed walk from $v$ to $w$ in an edge-indexed multigraph $M$, and let $e$ be an edge of $M$ with endpoints $u$ and $v$. The edge list of the walk from $u$ to $w$ that traverses $e$ and then follows $p$ is $e$ followed by the edge list of $p$.","title":"Edge list of an extended indexed walk"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/inc","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.Inc","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#inc","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"In an edge-indexed multigraph, a vertex $v$ is incident with an edge $e$ when $v$ belongs to the unordered pair of endpoints of $e$.","title":"Incidence"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/inc-iff","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.inc_iff","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#inc-iff","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"In an edge-indexed multigraph, a vertex $v$ is incident with an edge $e$ if and only if $v$ belongs to the unordered pair of endpoints of $e$.","title":"Incidence is membership in the endpoint pair"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/underlying","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.underlying","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#underlying","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"The underlying simple graph of an edge-indexed multigraph $M$ with vertex set $V$ is the simple graph on $V$ whose edges are the unordered pairs $\\{u, v\\}$ that are the endpoint pair of some edge of $M$; such pairs have $u \\ne v$ because $M$ is loopless. It forgets edge multiplicities.","title":"Underlying simple graph of an edge-indexed multigraph"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/underlying-adj","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.underlying_adj","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#underlying-adj","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ be an edge-indexed multigraph. Two vertices $u$ and $v$ are adjacent in the underlying simple graph of $M$ if and only if some edge $e$ of $M$ has endpoint pair $\\{u, v\\}$.","title":"Adjacency in the underlying graph"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/to-walk","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.toWalk","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#to-walk","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ be an edge-indexed multigraph. An indexed walk $p$ in $M$ from $u$ to $v$ yields a walk from $u$ to $v$ in the underlying simple graph of $M$ by forgetting edge identities: the empty indexed walk at $w$ goes to the empty walk at $w$, and an indexed walk that first traverses an edge $e$ with endpoints $\\{u, x\\}$ and then continues with an indexed walk $q$ goes to the walk that first steps from $u$ to $x$, an\u2026","title":"From an indexed walk to a walk in the underlying graph"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/support-to-walk","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.support_toWalk","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#support-to-walk","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ be an edge-indexed multigraph and $p$ an indexed walk in $M$ from $u$ to $v$. The walk from $u$ to $v$ in the underlying simple graph of $M$ obtained from $p$ by forgetting edge identities visits the same vertices as $p$, in the same order: the two supports are equal as lists.","title":"Forgetting edge identities preserves the vertex sequence"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/edges-to-walk","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.edges_toWalk","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#edges-to-walk","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"For every indexed walk $p$ from $u$ to $v$ in an edge-indexed multigraph, the edge list of the walk in the underlying simple graph obtained from $p$ by forgetting edge identities is the list of the endpoint pairs of the edges of $p$, taken in the same order.","title":"Edges of the forgetful walk"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/length-to-walk","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.IndexedWalk.length_toWalk","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#length-to-walk","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"For an indexed walk $p$, the walk in the underlying simple graph obtained by forgetting edge identities has the same length as $p$, namely the number of entries in the edge list of $p$.","title":"Forgetting edge identities preserves length"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/degree","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.degree","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#degree","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"The degree of a vertex $v$ in an edge-indexed multigraph $M$ is the number of edges of $M$ incident with $v$, that is, the cardinality of the set of edges whose endpoint pair contains $v$; it is defined to be $0$ when this set is infinite.","title":"Degree in an edge-indexed multigraph"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/degree-eq-ncard-inc","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.degree_eq_ncard_inc","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#degree-eq-ncard-inc","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"For every vertex $v$ of an edge-indexed multigraph $M$, the degree of $v$ equals the cardinality of the set of edges of $M$ whose endpoint pair contains $v$, where this cardinality is taken to be $0$ when the set is infinite.","title":"Degree as the number of incident edges"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/disjoint-sum","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.disjointSum","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#disjoint-sum","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Given edge-indexed multigraphs $M$ and $N$ on the same vertex type $V$, with edge types $E$ and $F$, their disjoint sum is the edge-indexed multigraph on $V$ whose edge type is the disjoint union $E \\sqcup F$, where an edge coming from $E$ has its endpoints in $M$ and an edge coming from $F$ has its endpoints in $N$.","title":"Disjoint sum of edge-indexed multigraphs"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/disjoint-sum-ends-inl","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.disjointSum_ends_inl","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#disjoint-sum-ends-inl","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ and $N$ be edge-indexed multigraphs on the same vertex type. Every edge $e$ of $M$, viewed as an edge of the disjoint sum of $M$ and $N$, has the same endpoints as in $M$.","title":"Endpoints of edges from the first summand"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/disjoint-sum-ends-inr","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.disjointSum_ends_inr","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#disjoint-sum-ends-inr","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ and $N$ be edge-indexed multigraphs on the same vertex type. Every edge $f$ of $N$, viewed as an edge of the disjoint sum of $M$ and $N$, has the same endpoints as in $N$.","title":"Endpoints of edges from the second summand"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/disjoint-sum-degree","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.disjointSum_degree","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#disjoint-sum-degree","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ and $N$ be edge-indexed multigraphs on the same vertex type, each with finitely many edges. Then for every vertex $v$, the degree of $v$ in the disjoint sum of $M$ and $N$ equals the degree of $v$ in $M$ plus the degree of $v$ in $N$.","title":"Degrees add under disjoint sums"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/disjoint-sum-connected-left","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.disjointSum_connected_left","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#disjoint-sum-connected-left","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ and $N$ be edge-indexed multigraphs on the same vertex type. If the underlying simple graph of $M$ is connected, then the underlying simple graph of the disjoint sum of $M$ and $N$ is connected.","title":"Adding edges preserves connectedness"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/even-sum-degree","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.even_sum_degree","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#even-sum-degree","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"In an edge-indexed multigraph with finitely many vertices and finitely many edges, the sum of the degrees of all vertices is even.","title":"Handshake lemma for edge-indexed multigraphs"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/exists-indexed-eulerian","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.exists_indexedEulerian","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#exists-indexed-eulerian","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ be an edge-indexed multigraph with finitely many vertices and finitely many edges. If the underlying simple graph of $M$ is connected and every vertex of $M$ has even degree, then there are a vertex $u$ and a closed indexed walk from $u$ to $u$ that is Eulerian, that is, it traverses every edge of $M$ exactly once.","title":"Euler circuits in edge-indexed multigraphs"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/exists-indexed-eulerian-connected-suppor","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.exists_indexedEulerian_connectedSupport","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#exists-indexed-eulerian-connected-suppor","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ be an edge-indexed multigraph with finitely many vertices and finitely many edges. Suppose that the subgraph of the underlying simple graph of $M$ induced on its non-isolated vertices is connected, and that every vertex of $M$ has even degree. Then there are a vertex $u$ and a closed indexed walk from $u$ to $u$ that is Eulerian, that is, it traverses every edge of $M$ exactly once.","title":"Euler circuits in edge-indexed multigraphs whose non-isolated vertices are connected"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/line-graph","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.lineGraph","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#line-graph","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"The line graph of an edge-indexed multigraph is the simple graph whose vertices are the edge identities, two of them being adjacent when they are distinct and share an endpoint. In particular, parallel edges are adjacent.","title":"Line graph of an edge-indexed multigraph"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/line-graph-adj","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.lineGraph_adj","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#line-graph-adj","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Two edge identities $e_1$ and $e_2$ are adjacent in the line graph of an edge-indexed multigraph exactly when $e_1 \\ne e_2$ and some vertex is incident with both of them.","title":"Adjacency in the line graph of a multigraph"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/max-degree","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.maxDegree","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#max-degree","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"For an edge-indexed multigraph with finitely many vertices and finitely many edge identities, the maximum degree is the largest degree of a vertex, and $0$ when there are no vertices.","title":"Maximum degree of a multigraph"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/degree-le-max-degree","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.degree_le_maxDegree","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#degree-le-max-degree","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"Let $M$ be an edge-indexed multigraph with finitely many vertices and finitely many edges. Then every vertex of $M$ has degree at most the maximum degree of $M$.","title":"Degrees are bounded by the maximum degree"},{"id":"GraphTheory/paths-cycles-connectivity/eulerian-circuits/filter-inc-card","kind":"node","lean":"SimpleGraph.EdgeIndexedMultigraph.filter_inc_card","parent":29747,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/eulerian-circuits/index.html#filter-inc-card","scope":"GraphTheory/paths-cycles-connectivity/eulerian-circuits","status":"fully_proved","summary":"In an edge-indexed multigraph with finitely many edges, the degree of every vertex $v$ equals the number of edges incident with $v$.","title":"Degree counts incident edges"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/menger-s-theorem.json","id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","title":"Menger's theorem"},{"id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem/menger-vertex","kind":"node","lean":"MetaMathlibExt.menger_vertex","parent":29787,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html#menger-vertex","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $s$ and $t$ be distinct, nonadjacent vertices of $G$. Then there is a natural number $n$ with the following two properties. First, there is a set $C$ of exactly $n$ vertices with $s \\notin C$ and $t \\notin C$ such that every walk from $s$ to $t$ passes through a vertex of $C$, and every set of vertices with these three properties has at least $n$ elements.\u2026","title":"Menger's theorem, vertex form"},{"id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem/is-internally-vertex-disjoint-stpaths","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MengerVertexWanted.IsInternallyVertexDisjointSTPaths","parent":29787,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html#is-internally-vertex-disjoint-stpaths","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph with vertices $s$ and $t$, and let $P_0, \\dots, P_{n-1}$ be a family of paths from $s$ to $t$ in $G$, indexed by $\\{0, \\dots, n-1\\}$. The family is internally vertex-disjoint when for all indices $i \\ne j$, every vertex lying on both $P_i$ and $P_j$ is $s$ or $t$.","title":"Internally vertex-disjoint paths"},{"id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem/is-stvertex-cut","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MengerVertexWanted.IsSTVertexCut","parent":29787,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html#is-stvertex-cut","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set, let $s$ and $t$ be vertices of $G$, and let $C$ be a set of vertices of $G$. Then $C$ is an $s$-$t$ vertex cut when $s \\notin C$, $t \\notin C$, and $t$ is not reachable from $s$ in the graph obtained from $G$ by deleting every edge that has an endpoint in $C$.","title":"Vertex cuts separating two vertices"},{"id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem/is-stvertex-cut-iff-walk-separator","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MengerVertexWanted.isSTVertexCut_iff_walkSeparator","parent":29787,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html#is-stvertex-cut-iff-walk-separator","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set, let $s$ and $t$ be vertices of $G$, and let $C$ be a set of vertices of $G$. Then $C$ is an $s$-$t$ vertex cut if and only if $s \\notin C$, $t \\notin C$, and every walk from $s$ to $t$ in $G$ passes through a vertex of $C$.","title":"Vertex cuts are walk separators"},{"id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem/menger-vertex-max-min","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MengerVertexWanted.menger_vertex_max_min","parent":29787,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html#menger-vertex-max-min","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $s$ and $t$ be distinct, nonadjacent vertices of $G$. Then there is a natural number $k$ such that there exist $k$ internally vertex-disjoint paths from $s$ to $t$ and an $s$-$t$ vertex cut with exactly $k$ vertices, every family of internally vertex-disjoint paths from $s$ to $t$ has at most $k$ members, and every $s$-$t$ vertex cut has at least $k$\u2026","title":"Menger's theorem for vertex cuts and internally disjoint paths"},{"id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem/is-edge-disjoint-stpaths","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MengerEdgeWanted.IsEdgeDisjointSTPaths","parent":29787,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html#is-edge-disjoint-stpaths","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph with vertices $s$ and $t$. A family $P_0, \\dots, P_{n-1}$ of paths from $s$ to $t$ in $G$ is edge-disjoint when, for all indices $i \\ne j$, no edge of $P_i$ is an edge of $P_j$.","title":"Edge-disjoint paths"},{"id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem/is-stedge-cut","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MengerEdgeWanted.IsSTEdgeCut","parent":29787,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html#is-stedge-cut","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $s$ and $t$ be vertices of $G$. A finite set $C$ of unordered pairs of vertices is an $s$-$t$ edge cut when every element of $C$ is an edge of $G$ and $t$ is not reachable from $s$ in the graph obtained from $G$ by deleting the edges in $C$.","title":"Edge cuts separating two vertices"},{"id":"GraphTheory/paths-cycles-connectivity/menger-s-theorem/menger-edge-max-min","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.MengerEdgeWanted.menger_edge_max_min","parent":29787,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/menger-s-theorem/index.html#menger-edge-max-min","scope":"GraphTheory/paths-cycles-connectivity/menger-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $s$ and $t$ be distinct vertices of $G$. Then there is a natural number $k$ such that there exist $k$ edge-disjoint paths from $s$ to $t$ and an $s$-$t$ edge cut with exactly $k$ edges, every family of edge-disjoint paths from $s$ to $t$ has at most $k$ members, and every $s$-$t$ edge cut has at least $k$ edges. In words, the maximum number of edge-disjoint\u2026","title":"Menger's theorem for edge cuts and edge-disjoint paths"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/reachability-and-connectedness.json","id":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/index.html","scope":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness","status":"fully_proved","title":"Reachability and connectedness"},{"id":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/exists-parent","kind":"node","lean":"SimpleGraph.exists_parent","parent":29796,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/index.html#exists-parent","scope":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness","status":"fully_proved","summary":"In a connected graph every vertex `v \u2260 s` has a neighbor `p` one step closer to `s` (a parent of `v` in a breadth-first tree rooted at `s`).","title":"Parent vertex towards a root in a connected graph"},{"id":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/internally-disjoint","kind":"node","lean":"SimpleGraph.InternallyDisjoint","parent":29796,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/index.html#internally-disjoint","scope":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness","status":"fully_proved","summary":"Two walks $p$ and $q$ in a simple graph $G$ are internally disjoint if no vertex is an interior vertex of both, where the interior vertices of a walk are the entries of its vertex sequence other than the first and the last.","title":"Internally disjoint walks"},{"id":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/infinitely-connected","kind":"node","lean":"SimpleGraph.InfinitelyConnected","parent":29796,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/index.html#infinitely-connected","scope":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness","status":"fully_proved","summary":"A simple graph $G$ is infinitely connected if it has at least two vertices and, for any two distinct vertices $u$ and $v$, there is an infinite set of paths from $u$ to $v$ in $G$ that are pairwise internally disjoint.","title":"Infinitely connected graphs"},{"id":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/bfs-step","kind":"node","lean":"SimpleGraph.bfsStep","parent":29796,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/index.html#bfs-step","scope":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set and a set $s$ of vertices, this is the set of vertices that lie in $s$ or are adjacent to some vertex of $s$, that is, $s$ together with all neighbours of its vertices.","title":"One round of breadth-first search"},{"id":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/bfs-iterate","kind":"node","lean":"SimpleGraph.bfsIterate","parent":29796,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/index.html#bfs-iterate","scope":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set, a natural number $n$ and a set $s$ of vertices, this applies rounds of breadth-first search to $s$ (each round adds all neighbours of the current vertices), stopping after $n$ rounds or as soon as a round adds no new vertex, whichever comes first, and returns the resulting set.","title":"Breadth-first search with a round limit"},{"id":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/reachable-finset","kind":"node","lean":"SimpleGraph.reachableFinset","parent":29796,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/reachability-and-connectedness/index.html#reachable-finset","scope":"GraphTheory/paths-cycles-connectivity/reachability-and-connectedness","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set $V$ and a vertex $u$, this is the set obtained from $\\{u\\}$ by breadth-first search with at most $|V|$ rounds, stopping early as soon as a round adds no new vertex.","title":"Vertices reachable from a vertex"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/vertex-connectivity.json","id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","title":"Vertex connectivity"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/is-vertex-connected","kind":"node","lean":"SimpleGraph.IsVertexConnected","parent":29803,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(k\\)"],["text","-vertex-connected graph"]],"source":"$k$-vertex-connected graph"},"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#is-vertex-connected","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"Let $k$ be a natural number. A simple graph $G$ on a finite vertex set $V$ is $k$-vertex-connected if $V$ has more than $k$ elements and, for every set $S$ of fewer than $k$ vertices, the subgraph of $G$ induced on $V \\setminus S$ is connected.","title":"$k$-vertex-connected graph"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/is-vertex-connected-iff","kind":"node","lean":"SimpleGraph.isVertexConnected_iff","parent":29803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of "],["math","\\(k\\)"],["text","-vertex-connectivity by vertex deletion"]],"source":"Characterization of $k$-vertex-connectivity by vertex deletion"},"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#is-vertex-connected-iff","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"Let $k$ be a natural number and let $G$ be a simple graph on a finite vertex set $V$. Then $G$ is $k$-vertex-connected if and only if $V$ has more than $k$ elements and, for every set $S$ of fewer than $k$ vertices, the subgraph of $G$ induced on $V \\setminus S$ is connected.","title":"Characterization of $k$-vertex-connectivity by vertex deletion"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/card-gt","kind":"node","lean":"SimpleGraph.IsVertexConnected.card_gt","parent":29803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vertex count of a "],["math","\\(k\\)"],["text","-vertex-connected graph"]],"source":"Vertex count of a $k$-vertex-connected graph"},"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#card-gt","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"Let $k$ be a natural number. If a simple graph $G$ on a finite vertex set $V$ is $k$-vertex-connected, then $V$ has more than $k$ elements.","title":"Vertex count of a $k$-vertex-connected graph"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/connected-induce-compl","kind":"node","lean":"SimpleGraph.IsVertexConnected.connected_induce_compl","parent":29803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Deleting fewer than "],["math","\\(k\\)"],["text"," vertices from a "],["math","\\(k\\)"],["text","-vertex-connected graph"]],"source":"Deleting fewer than $k$ vertices from a $k$-vertex-connected graph"},"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#connected-induce-compl","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"Let $k$ be a natural number and let $G$ be a $k$-vertex-connected simple graph on a finite vertex set $V$. Then for every set $S$ of fewer than $k$ vertices, the subgraph of $G$ induced on $V \\setminus S$ is connected.","title":"Deleting fewer than $k$ vertices from a $k$-vertex-connected graph"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/connected","kind":"node","lean":"SimpleGraph.IsVertexConnected.connected","parent":29803,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(k\\)"],["text","-vertex-connected graph with "],["math","\\(k \\ge 1\\)"],["text"," is connected"]],"source":"A $k$-vertex-connected graph with $k \\ge 1$ is connected"},"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#connected","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"Let $k \\ge 1$ be a natural number. Every $k$-vertex-connected simple graph on a finite vertex set is connected.","title":"A $k$-vertex-connected graph with $k \\ge 1$ is connected"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/connected-compl-singleton","kind":"node","lean":"SimpleGraph.IsVertexConnected.connected_compl_singleton","parent":29803,"reading_title":{"format":"inline-parts-v1","parts":[["text","Deleting one vertex from a "],["math","\\(k\\)"],["text","-vertex-connected graph with "],["math","\\(k \\ge 2\\)"]],"source":"Deleting one vertex from a $k$-vertex-connected graph with $k \\ge 2$"},"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#connected-compl-singleton","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"Let $k \\ge 2$ be a natural number and let $G$ be a $k$-vertex-connected simple graph on a finite vertex set $V$. Then for every vertex $v$, the subgraph of $G$ induced on $V \\setminus \\{v\\}$ is connected.","title":"Deleting one vertex from a $k$-vertex-connected graph with $k \\ge 2$"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/is-vertex-connected-two-iff","kind":"node","lean":"SimpleGraph.isVertexConnected_two_iff","parent":29803,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#is-vertex-connected-two-iff","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"A simple graph $G$ on a finite vertex set $V$ is 2-vertex-connected if and only if $V$ has more than two elements, $G$ is connected, and for every vertex $v$ the subgraph of $G$ induced on $V \\setminus \\{v\\}$ is connected.","title":"Characterization of 2-vertex-connected graphs"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/two-le-degree","kind":"node","lean":"SimpleGraph.IsVertexConnected.two_le_degree","parent":29803,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#two-le-degree","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"Every vertex of a 2-vertex-connected simple graph on a finite vertex set has degree at least two.","title":"Degrees in a 2-vertex-connected graph are at least two"},{"id":"GraphTheory/paths-cycles-connectivity/vertex-connectivity/exists-cycle-through","kind":"node","lean":"SimpleGraph.IsVertexConnected.exists_cycle_through","parent":29803,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/vertex-connectivity/index.html#exists-cycle-through","scope":"GraphTheory/paths-cycles-connectivity/vertex-connectivity","status":"fully_proved","summary":"Let $G$ be a 2-vertex-connected simple graph on a finite vertex set. Then for every vertex $x$ of $G$ there is a cycle in $G$ that starts and ends at $x$.","title":"Cycles through each vertex of a 2-vertex-connected graph"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/hamiltonian-cycles.json","id":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/index.html","scope":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles","status":"can_prove","title":"Hamiltonian cycles"},{"id":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/dirac","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.DiracWanted.dirac","parent":29813,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/index.html#dirac","scope":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles","status":"fully_proved","summary":"Let $G$ be a finite simple graph with $n \\ge 3$ vertices in which every vertex has degree at least $n/2$. Then $G$ is Hamiltonian, that is, it has a cycle that visits every vertex exactly once.","title":"Dirac's theorem"},{"id":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/of-cyclic-list","kind":"node","lean":"SimpleGraph.IsHamiltonian.of_cyclic_list","parent":29813,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/index.html#of-cyclic-list","scope":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, and let $v_0, v_1, \\dots, v_n$ be a list of vertices with $v_0 = v_n$ in which $v_{i-1}$ and $v_i$ are adjacent in $G$ for each $1 \\le i \\le n$. If $v_1, \\dots, v_n$ are pairwise distinct, every vertex of $V$ is among them, and $n \\ge 3$, then $G$ is Hamiltonian: it has a cycle that visits every vertex exactly once.","title":"A cyclic listing of the vertices gives a Hamiltonian cycle"},{"id":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/ore","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.OreWanted.ore","parent":29813,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/index.html#ore","scope":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ with $|V| \\ge 3$. If $\\deg u + \\deg v \\ge |V|$ for every pair of distinct nonadjacent vertices $u$ and $v$, then $G$ is Hamiltonian.","title":"Ore's theorem"},{"id":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.PlanarGraphPrismHamiltonicityWanted.conjecture","parent":29813,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/index.html#conjecture","scope":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles","status":"fully_proved","summary":"For every finite simple graph $G$ that is $3$-vertex-connected and straight-line planar, the prism $G \\mathbin{\\square} K_2$, the Cartesian product of $G$ with the complete graph on two vertices, is Hamiltonian.","title":"Prism-Hamiltonicity conjecture for 3-connected planar graphs"},{"id":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.PlanarGraphPrismHamiltonicityWanted.conjecture_refuted","parent":29813,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/hamiltonian-cycles/index.html#conjecture-refuted","scope":"GraphTheory/paths-cycles-connectivity/hamiltonian-cycles","status":"can_prove","summary":"The prism-Hamiltonicity conjecture for 3-connected planar graphs is false: there is a finite simple graph $G$ that is $3$-vertex-connected and straight-line planar whose prism $G \\mathbin{\\square} K_2$ is not Hamiltonian.","title":"\u0160pacapan's counterexample to prism-Hamiltonicity"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/square-of-a-graph.json","id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","title":"Square of a graph"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/square","kind":"node","lean":"SimpleGraph.square","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#square","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"The square of a simple graph $G$ is the simple graph on the same vertex set in which two distinct vertices $u$ and $v$ are adjacent when they are adjacent in $G$ or have a common neighbor in $G$, that is, a vertex $w$ adjacent in $G$ to both $u$ and $v$.","title":"Square of a graph"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/square-adj","kind":"node","lean":"SimpleGraph.square_adj","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#square-adj","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"For a simple graph $G$ and vertices $u$ and $v$, the vertices $u$ and $v$ are adjacent in the square of $G$ if and only if $u \\ne v$ and either $u$ and $v$ are adjacent in $G$ or there is a vertex $w$ with $u$ adjacent to $w$ and $w$ adjacent to $v$ in $G$.","title":"Adjacency in the graph square"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/square-mono","kind":"node","lean":"SimpleGraph.square_mono","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#square-mono","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"Let $G$ and $H$ be simple graphs on the same vertex set such that every edge of $G$ is an edge of $H$. Then every edge of the square of $G$ is an edge of the square of $H$.","title":"Monotonicity of the graph square"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/square-2","kind":"node","lean":"SimpleGraph.Connected.square","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#square-2","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"If a simple graph $G$ is connected, then the square of $G$ is connected.","title":"The square of a connected graph is connected"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/square-mono-2","kind":"node","lean":"SimpleGraph.IsHamiltonian.square_mono","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#square-mono-2","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"Let $G$ and $H$ be simple graphs on the same finite vertex set such that every edge of $G$ is an edge of $H$. If the square of $G$ is Hamiltonian, then the square of $H$ is Hamiltonian.","title":"Adding edges preserves a Hamiltonian square"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/eq-square-iff","kind":"node","lean":"SimpleGraph.eq_square_iff","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#eq-square-iff","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"Let $G$ and $H$ be simple graphs on the same vertex set. Then $H$ equals the square of $G$ if and only if, for all vertices $u$ and $v$, the vertices $u$ and $v$ are adjacent in $H$ exactly when $u \\ne v$ and either $u$ and $v$ are adjacent in $G$ or there is a vertex $w$ with $u$ adjacent to $w$ and $w$ adjacent to $v$ in $G$.","title":"Characterization of the graph square"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/is-chain-square-of-is-chain-cons-cons","kind":"node","lean":"SimpleGraph.isChain_square_of_isChain_cons_cons","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#is-chain-square-of-is-chain-cons-cons","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"Let $G$ be a simple graph and let $u, w, v, x_1, \\dots, x_k$ (with $k \\ge 0$) be a sequence of vertices in which consecutive vertices are adjacent in $G$. If $u \\ne v$, then in the sequence $u, v, x_1, \\dots, x_k$ consecutive vertices are adjacent in the square of $G$.","title":"Skipping a vertex of a walk in the graph square"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/square-of-cyclic-list","kind":"node","lean":"SimpleGraph.IsHamiltonian.square_of_cyclic_list","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#square-of-cyclic-list","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, and let $v_0, v_1, \\dots, v_n$ be vertices with $n \\ge 3$ and $v_0 = v_n$, such that $v_1, \\dots, v_n$ are pairwise distinct and every vertex of $V$ is among them. If for each $i < n$ the vertices $v_i$ and $v_{i+1}$ are distinct and either adjacent in $G$ or have a common neighbor in $G$, then the square of $G$ is Hamiltonian.","title":"A cyclic listing by square adjacencies gives a Hamiltonian square"},{"id":"GraphTheory/paths-cycles-connectivity/square-of-a-graph/four-le-square-degree","kind":"node","lean":"SimpleGraph.IsVertexConnected.four_le_square_degree","parent":29819,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/square-of-a-graph/index.html#four-le-square-degree","scope":"GraphTheory/paths-cycles-connectivity/square-of-a-graph","status":"fully_proved","summary":"In a 2-vertex-connected simple graph on a finite vertex set with at least five vertices, every vertex has at least four neighbors in the square of the graph.","title":"Minimum degree in the square of a 2-connected graph"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma.json","id":"GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma/index.html","scope":"GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma","status":"fully_proved","title":"\u0158\u00edha's cycle lemma"},{"id":"GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma/is-bound-vertex","kind":"node","lean":"SimpleGraph.Walk.IsBoundVertex","parent":29829,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma/index.html#is-bound-vertex","scope":"GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma","status":"fully_proved","summary":"Let $c$ be a closed walk in a simple graph $G$. A vertex $y$ is bound to $c$ when $y$ lies on $c$ and every neighbor of $y$ in $G$ lies on $c$.","title":"Bound vertices of a closed walk"},{"id":"GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma/is-bound-vertex-iff","kind":"node","lean":"SimpleGraph.Walk.isBoundVertex_iff","parent":29829,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma/index.html#is-bound-vertex-iff","scope":"GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma","status":"fully_proved","summary":"Let $c$ be a closed walk in a simple graph $G$. A vertex $y$ is bound to $c$ if and only if $y$ lies on $c$ and every neighbor of $y$ in $G$ lies on $c$.","title":"Characterization of bound vertices"},{"id":"GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma/exists-cycle-bound-vertex","kind":"node","lean":"SimpleGraph.IsVertexConnected.exists_cycle_bound_vertex","parent":29829,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma/index.html#exists-cycle-bound-vertex","scope":"GraphTheory/paths-cycles-connectivity/ha-s-cycle-lemma","status":"fully_proved","summary":"Let $G$ be a finite $2$-vertex-connected simple graph. Then for every vertex $x$ of $G$ there is a cycle $c$ through $x$ and a vertex $y \\ne x$ that is bound to $c$: $y$ lies on $c$ and every neighbor of $y$ in $G$ lies on $c$.","title":"\u0158\u00edha's cycle lemma"},{"data_url":"scopes/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem.json","id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","kind":"scope","navigation":"module","parent":29729,"result":false,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","title":"Fleischner's theorem"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/collapse-outside","kind":"node","lean":"SimpleGraph.collapseOutside","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#collapse-outside","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"For a set $D \\subseteq V$, the collapse map sends $V$ to $D$ together with one new vertex, the contraction vertex: each $v \\in D$ is sent to itself, and each $v \\notin D$ is sent to the contraction vertex.","title":"Collapsing the complement of a set"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/collapse-outside-of-mem","kind":"node","lean":"SimpleGraph.collapseOutside_of_mem","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#collapse-outside-of-mem","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"For a set $D \\subseteq V$, the collapse map from $V$ to $D$ together with the contraction vertex sends each $v \\in D$ to $v$ itself.","title":"The collapse map fixes the retained set"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/collapse-outside-of-not-mem","kind":"node","lean":"SimpleGraph.collapseOutside_of_notMem","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#collapse-outside-of-not-mem","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"For a set $D \\subseteq V$, the collapse map from $V$ to $D$ together with the contraction vertex sends each vertex $v \\notin D$ to the contraction vertex.","title":"The collapse map sends the complement to the contraction vertex"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/collapse-outside-eq-some-iff","kind":"node","lean":"SimpleGraph.collapseOutside_eq_some_iff","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#collapse-outside-eq-some-iff","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"For a set $D \\subseteq V$, a vertex $v \\in V$ and an element $a \\in D$, the collapse map from $V$ to $D$ together with the contraction vertex sends $v$ to $a$ if and only if $v = a$.","title":"Preimages of retained vertices under the collapse map"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/collapse-outside-eq-none-iff","kind":"node","lean":"SimpleGraph.collapseOutside_eq_none_iff","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#collapse-outside-eq-none-iff","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"For a set $D \\subseteq V$ and a vertex $v \\in V$, the collapse map from $V$ to $D$ together with the contraction vertex sends $v$ to the contraction vertex if and only if $v \\notin D$.","title":"Preimage of the contraction vertex under the collapse map"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/collapse-outside-surjective","kind":"node","lean":"SimpleGraph.collapseOutside_surjective","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#collapse-outside-surjective","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"For a set $D \\subseteq V$ whose complement $V \\setminus D$ is nonempty, the collapse map from $V$ to $D$ together with the contraction vertex is surjective.","title":"The collapse map is surjective"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/contract-outside","kind":"node","lean":"SimpleGraph.contractOutside","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#contract-outside","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex set $V$ and let $D \\subseteq V$. Contracting the complement of $D$ to a single vertex gives the simple graph whose vertices are the elements of $D$ together with one new vertex $\\ast$, in which two vertices $a, b \\in D$ are adjacent exactly when they are adjacent in $G$, and a vertex $a \\in D$ is adjacent to $\\ast$ exactly when $a$ has a neighbour in $G$ outside $D$. The new\u2026","title":"Contracting the complement of a vertex set"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/contract-outside-adj-some","kind":"node","lean":"SimpleGraph.contractOutside_adj_some","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#contract-outside-adj-some","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $G$ be a graph on $V$, let $D \\subseteq V$, and contract $V \\setminus D$ to a single vertex. Two vertices $a, b \\in D$ are adjacent in the contracted graph if and only if they are adjacent in $G$.","title":"Contraction preserves adjacency among retained vertices"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/contract-outside-adj-none","kind":"node","lean":"SimpleGraph.contractOutside_adj_none","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#contract-outside-adj-none","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $G$ be a graph on $V$, let $D \\subseteq V$, and contract $V \\setminus D$ to a single vertex. A vertex $a \\in D$ is adjacent to the contraction vertex in the contracted graph if and only if $a$ is adjacent in $G$ to some vertex $v \\notin D$.","title":"Neighbors of the contraction vertex"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/contract-outside-2","kind":"node","lean":"SimpleGraph.IsVertexConnected.contractOutside","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#contract-outside-2","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $G$ be a 2-vertex-connected graph on a finite vertex set $V$, and let $D \\subseteq V$ be a set with at least two elements that induces a connected subgraph of $G$ and whose complement $V \\setminus D$ is nonempty. Then the graph obtained from $G$ by contracting $V \\setminus D$ to a single vertex is 2-vertex-connected.","title":"Contracting the complement preserves 2-connectivity"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/card-contract-outside-lt","kind":"node","lean":"SimpleGraph.card_contractOutside_lt","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#card-contract-outside-lt","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $V$ be a finite set and $D \\subseteq V$. If the complement $V \\setminus D$ has at least two elements, then $D$ together with one new vertex has fewer elements than $V$, that is, $|D| + 1 < |V|$.","title":"Cardinality bound for contracting the complement"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/has-strong-square-cycle-2","kind":"node","lean":"SimpleGraph.HasStrongSquareCycle","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#has-strong-square-cycle-2","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $G$ be a graph on a finite vertex set $V$ and let $x \\in V$. Then $G$ has a strong square cycle at $x$ if the square $G^2$ of $G$ has a Hamiltonian cycle that starts and ends at $x$ and whose two edges at $x$, its first and its last edge, are edges of $G$.","title":"Strong square cycle at a vertex"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/has-strong-square-cycle-iff","kind":"node","lean":"SimpleGraph.hasStrongSquareCycle_iff","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#has-strong-square-cycle-iff","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $G$ be a graph on a finite vertex set $V$ and let $x \\in V$. Then $G$ has a strong square cycle at $x$ if and only if the square $G^2$ of $G$ has a Hamiltonian cycle $p$ that starts and ends at $x$ such that $x$ is adjacent in $G$ to the second vertex of $p$ and the second-to-last vertex of $p$ is adjacent in $G$ to $x$.","title":"Characterization of strong square cycles"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/has-strong-square-cycle","kind":"node","lean":"SimpleGraph.IsVertexConnected.hasStrongSquareCycle","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#has-strong-square-cycle","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $G$ be a finite simple graph that is 2-vertex-connected, that is, $G$ has at least three vertices and stays connected after deleting at most one vertex. Let $G^2$ be the square of $G$: the graph on the same vertices in which two distinct vertices are adjacent if and only if they are adjacent in $G$ or have a common neighbour in $G$. Then for every vertex $x$ of $G$, the graph $G^2$ has a Hamiltonian cycle whose\u2026","title":"Strengthened Fleischner theorem"},{"id":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/fleischner","kind":"node","lean":"MetaMathlibExt.fleischner","parent":29833,"result":true,"route":"../roadmap/GraphTheory/paths-cycles-connectivity/fleischner-s-theorem/index.html#fleischner","scope":"GraphTheory/paths-cycles-connectivity/fleischner-s-theorem","status":"fully_proved","summary":"Let $G$ be a finite simple graph that is 2-vertex-connected, that is, $G$ has at least three vertices and stays connected after deleting at most one vertex. Let $H$ be the square of $G$: the graph on the same vertices in which two distinct vertices are adjacent if and only if they are adjacent in $G$ or have a common neighbour in $G$. Then $H$ is Hamiltonian: it has a cycle that passes through every vertex exactly\u2026","title":"Fleischner's theorem"},{"data_url":"scopes/GraphTheory/independence-and-domination.json","id":"GraphTheory/independence-and-domination","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/index.html","scope":"GraphTheory/independence-and-domination","status":"can_prove","summary":"Independent sets, cliques and dominating sets in graphs, with bounds on their sizes and polynomials counting them.","title":"Independence, cliques and domination"},{"data_url":"scopes/GraphTheory/independence-and-domination/caro-wei-theorem.json","id":"GraphTheory/independence-and-domination/caro-wei-theorem","kind":"scope","navigation":"module","parent":29849,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/caro-wei-theorem/index.html","scope":"GraphTheory/independence-and-domination/caro-wei-theorem","status":"fully_proved","title":"Caro\u2013Wei theorem"},{"id":"GraphTheory/independence-and-domination/caro-wei-theorem/exists-is-indep-set-caro-wei","kind":"node","lean":"SimpleGraph.exists_isIndepSet_caroWei","parent":29850,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/caro-wei-theorem/index.html#exists-is-indep-set-caro-wei","scope":"GraphTheory/independence-and-domination/caro-wei-theorem","status":"fully_proved","summary":"Every finite graph has an independent set of size at least the sum over its vertices of the reciprocal of one more than the degree.","title":"Caro\u2013Wei bound"},{"id":"GraphTheory/independence-and-domination/caro-wei-theorem/caro-wei-le-indep-num","kind":"node","lean":"SimpleGraph.caroWei_le_indepNum","parent":29850,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/caro-wei-theorem/index.html#caro-wei-le-indep-num","scope":"GraphTheory/independence-and-domination/caro-wei-theorem","status":"fully_proved","summary":"For every finite simple graph $G$, the independence number of $G$ is at least $\\sum_{v} \\frac{1}{\\deg(v) + 1}$, the sum running over all vertices $v$ of $G$.","title":"Caro\u2013Wei bound for the independence number"},{"data_url":"scopes/GraphTheory/independence-and-domination/dominating-sets.json","id":"GraphTheory/independence-and-domination/dominating-sets","kind":"scope","navigation":"module","parent":29849,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","title":"Dominating sets"},{"id":"GraphTheory/independence-and-domination/dominating-sets/is-dominating","kind":"node","lean":"SimpleGraph.IsDominating","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#is-dominating","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"A set $S$ of vertices of a simple graph $G$ dominates $G$ when every vertex of $G$ lies in $S$ or is adjacent to some vertex of $S$.","title":"Dominating set"},{"id":"GraphTheory/independence-and-domination/dominating-sets/is-indep-dominating","kind":"node","lean":"SimpleGraph.IsIndepDominating","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#is-indep-dominating","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"A set $S$ of vertices of a simple graph $G$ is an independent dominating set of $G$ when no two vertices of $S$ are adjacent and $S$ dominates $G$.","title":"Independent dominating set"},{"id":"GraphTheory/independence-and-domination/dominating-sets/domination-number","kind":"node","lean":"SimpleGraph.dominationNumber","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#domination-number","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set, the domination number of $G$ is the least number of elements of a dominating set of $G$.","title":"Domination number"},{"id":"GraphTheory/independence-and-domination/dominating-sets/indep-domination-number","kind":"node","lean":"SimpleGraph.indepDominationNumber","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#indep-domination-number","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set, the independent domination number of $G$ is the least number of elements of an independent dominating set of $G$.","title":"Independent domination number"},{"id":"GraphTheory/independence-and-domination/dominating-sets/exists-is-indep-dominating","kind":"node","lean":"SimpleGraph.exists_isIndepDominating","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#exists-is-indep-dominating","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"Every finite graph has an independent dominating set.","title":"Existence of independent dominating sets"},{"id":"GraphTheory/independence-and-domination/dominating-sets/exists-indep-domination-number-eq-card","kind":"node","lean":"SimpleGraph.exists_indepDominationNumber_eq_card","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#exists-indep-domination-number-eq-card","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"Every simple graph $G$ on a finite vertex set has an independent dominating set whose number of elements equals the independent domination number of $G$.","title":"A minimum independent dominating set exists"},{"id":"GraphTheory/independence-and-domination/dominating-sets/indep-domination-number-le-card","kind":"node","lean":"SimpleGraph.indepDominationNumber_le_card","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#indep-domination-number-le-card","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set. If $S$ is an independent dominating set of $G$, then the independent domination number of $G$ is at most the number of elements of $S$.","title":"Independent dominating sets are at least as large as the independent domination number"},{"id":"GraphTheory/independence-and-domination/dominating-sets/exists-domination-number-eq-card","kind":"node","lean":"SimpleGraph.exists_dominationNumber_eq_card","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#exists-domination-number-eq-card","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"Every simple graph $G$ on a finite vertex set has a dominating set whose number of elements equals the domination number of $G$.","title":"A minimum dominating set exists"},{"id":"GraphTheory/independence-and-domination/dominating-sets/domination-number-le-card","kind":"node","lean":"SimpleGraph.dominationNumber_le_card","parent":29853,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/dominating-sets/index.html#domination-number-le-card","scope":"GraphTheory/independence-and-domination/dominating-sets","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set. If $S$ is a dominating set of $G$, then the domination number of $G$ is at most the number of elements of $S$.","title":"Dominating sets are at least as large as the domination number"},{"data_url":"scopes/GraphTheory/independence-and-domination/domination-in-rook-graphs.json","id":"GraphTheory/independence-and-domination/domination-in-rook-graphs","kind":"scope","navigation":"module","parent":29849,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"can_prove","title":"Domination in rook graphs"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-graph","kind":"node","lean":"SimpleGraph.rookGraph","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-graph","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the rook graph on the $n \\times m$ board is the simple graph whose vertices are the cells $(i, j)$ with $0 \\le i < n$ and $0 \\le j < m$, in which two cells are adjacent if and only if they are distinct and lie in the same row (same $i$) or the same column (same $j$).","title":"Rook graph"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-graph-is-dominating-iff","kind":"node","lean":"SimpleGraph.rookGraph_isDominating_iff","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-graph-is-dominating-iff","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $m$, a finite set $S$ of cells of the $n \\times m$ board dominates the rook graph on that board if and only if for every cell $(i, j)$, some cell of $S$ lies in row $i$ or some cell of $S$ lies in column $j$.","title":"Dominating sets of the rook graph"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-cover-count","kind":"node","lean":"MathlibExt.GraphTheory.RookDominationPolynomialWanted.rookCoverCount","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-cover-count","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"fully_proved","summary":"For natural numbers $n$, $m$ and $k$, $E_{n,m}(k)$ is the number of $k$-element sets of cells of the $n \\times m$ board that contain a cell in every row and a cell in every column (Mertens' $E_{n,m}(k)$).","title":"Rook placements covering all rows and columns"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-dominating-count","kind":"node","lean":"MathlibExt.GraphTheory.RookDominationPolynomialWanted.rookDominatingCount","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-dominating-count","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"fully_proved","summary":"For natural numbers $n$, $m$ and $k$, $d_{n,m}(k)$ is the number of $k$-element sets of cells of the $n \\times m$ board that are dominating sets of the rook graph on that board (Mertens' $d_{R_{n,m}}(k)$).","title":"Dominating rook placements"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-domination-polynomial","kind":"node","lean":"MathlibExt.GraphTheory.RookDominationPolynomialWanted.rookDominationPolynomial","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-domination-polynomial","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the rook domination polynomial of the $n \\times m$ board is the integer polynomial $D_{n,m}(x) = \\sum_{k=0}^{nm} d_{n,m}(k)\\, x^k$, where $d_{n,m}(k)$ is the number of $k$-element sets of cells of the $n \\times m$ board that are dominating sets of the rook graph on that board.","title":"Domination polynomial of the rook graph"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-dominating-count-eq-choose-sub-sum","kind":"node","lean":"MathlibExt.GraphTheory.RookDominationPolynomialWanted.rookDominatingCount_eq_choose_sub_sum_rookCoverCount","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-dominating-count-eq-choose-sub-sum","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"can_prove","summary":"For natural numbers $n, m \\ge 1$ and $k$, let $d_{n,m}(k)$ be the number of $k$-element sets of cells of the $n \\times m$ board that are dominating sets of the rook graph on that board, and let $E_{a,b}(k)$ be the number of $k$-element sets of cells of the $a \\times b$ board that contain a cell in every row and a cell in every column. Then","title":"Dominating rook placements via covering placements"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-cover-count-eq-choose-sub-sum-rook","kind":"node","lean":"MathlibExt.GraphTheory.RookDominationPolynomialWanted.rookCoverCount_eq_choose_sub_sum_rookCoverCount","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-cover-count-eq-choose-sub-sum-rook","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"can_prove","summary":"For natural numbers $n, m \\ge 1$ and $k$, let $E_{a,b}(k)$ denote the number of $k$-element sets of cells of the $a \\times b$ board that contain a cell in every row and a cell in every column. Then","title":"Recursion for rook placements covering all rows and columns"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-dominating-count-eq-choose-sub-mul","kind":"node","lean":"MathlibExt.GraphTheory.RookDominationPolynomialWanted.rookDominatingCount_eq_choose_sub_mul_choose","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-dominating-count-eq-choose-sub-mul","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"can_prove","summary":"For natural numbers $n, m \\ge 1$ and $k$ with $k > nm - n - m - \\min(n, m) + 2$, let $d_{n,m}(k)$ be the number of $k$-element sets of cells of the $n \\times m$ board that are dominating sets of the rook graph on that board. Then","title":"Dominating rook placements at high density"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-domination-polynomial-eq","kind":"node","lean":"MathlibExt.GraphTheory.RookDominationPolynomialWanted.rookDominationPolynomial_eq","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-domination-polynomial-eq","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"can_prove","summary":"For natural numbers $n, m \\ge 1$, the rook domination polynomial $D_{n,m}(x)$ of the $n \\times m$ board, an integer polynomial, satisfies","title":"Single-sum formula for the rook domination polynomial"},{"id":"GraphTheory/independence-and-domination/domination-in-rook-graphs/rook-domination-polynomial-eq-symmetric","kind":"node","lean":"MathlibExt.GraphTheory.RookDominationPolynomialWanted.rookDominationPolynomial_eq_symmetric","parent":29863,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-in-rook-graphs/index.html#rook-domination-polynomial-eq-symmetric","scope":"GraphTheory/independence-and-domination/domination-in-rook-graphs","status":"can_prove","summary":"For natural numbers $n, m \\ge 1$, the rook domination polynomial $D_{n,m}(x)$ of the $n \\times m$ board, an integer polynomial, satisfies","title":"Symmetric formula for the rook domination polynomial"},{"data_url":"scopes/GraphTheory/independence-and-domination/domination-polynomials-of-grids.json","id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","kind":"scope","navigation":"module","parent":29849,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"can_prove","title":"Domination polynomials of grids"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-state","kind":"node","lean":"MetaMathlibExt.DominationState","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-state","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"A domination state is one of three labels: uncovered, covered or occupied.","title":"Domination states"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-polynomial","kind":"node","lean":"MetaMathlibExt.dominationPolynomial","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-polynomial","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"For a finite graph $G$, the domination polynomial of $G$ is the polynomial $\\sum_S z^{|S|}$ with integer coefficients, the sum running over the dominating sets $S$ of $G$.","title":"Domination polynomial"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/vertex-domination-state","kind":"node","lean":"MetaMathlibExt.vertexDominationState","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#vertex-domination-state","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $G$ be a simple graph, $S$ a finite set of vertices of $G$ and $v$ a vertex of $G$. The domination state of $v$ relative to $S$ in $G$ is occupied if $v \\in S$, covered if $v \\notin S$ and $v$ is adjacent in $G$ to some vertex of $S$, and uncovered otherwise.","title":"Domination state of a vertex"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/is-domination-signature","kind":"node","lean":"MetaMathlibExt.IsDominationSignature","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#is-domination-signature","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $R$ be a simple graph on a vertex set $V$ (the row). A map $\\sigma$ assigning to each vertex of $V$ a domination state (uncovered, covered or occupied) is a signature for $R$ if no uncovered vertex is adjacent in $R$ to an occupied vertex: for all vertices $i, j$ adjacent in $R$, it is not the case that $\\sigma(i)$ is uncovered and $\\sigma(j)$ is occupied.","title":"Domination signature"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-signature","kind":"node","lean":"MetaMathlibExt.DominationSignature","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-signature","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"For a simple graph $R$ on a vertex set $V$ (the row), a signature of $R$ is a map $\\sigma$ assigning to each vertex of $V$ a domination state (uncovered, covered or occupied) such that $\\sigma$ is a domination signature for $R$, that is, no uncovered vertex is adjacent in $R$ to an occupied vertex. This is the type of all signatures of $R$; when $R$ is a cycle they are the paper's cyclic signatures.","title":"Signatures of a row graph"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/all-covered-signature","kind":"node","lean":"MetaMathlibExt.allCoveredSignature","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#all-covered-signature","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"For a graph $R$ (the row), the all-covered signature $\\sigma_\\bullet$ of $R$ is the signature that gives every vertex of $R$ the state covered; it is a signature because it has no uncovered vertex.","title":"All-covered signature"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-compatible","kind":"node","lean":"MetaMathlibExt.DominationCompatible","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-compatible","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $R$ be a graph (the row) and let $\\sigma$ and $\\tau$ assign a domination state (uncovered, covered or occupied) to each vertex of $R$. The new row $\\tau$ is compatible with the previous row $\\sigma$ when, for every vertex $i$ of $R$: if $\\sigma(i)$ is uncovered then $\\tau(i)$ is occupied; if $\\sigma(i)$ is occupied then $\\tau(i)$ is not uncovered; and if $\\tau(i)$ is covered then $\\sigma(i)$ is occupied or some\u2026","title":"Compatible row signatures"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/occupied-count","kind":"node","lean":"MetaMathlibExt.occupiedCount","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#occupied-count","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $V$ be a finite set and let $\\sigma$ assign a domination state (uncovered, covered or occupied) to each element of $V$. The occupied count $\\operatorname{occ}(\\sigma)$ is the number of elements $i \\in V$ such that $\\sigma(i)$ is occupied.","title":"Number of occupied vertices"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-transfer-matrix","kind":"node","lean":"MetaMathlibExt.dominationTransferMatrix","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-transfer-matrix","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $R$ be a simple graph on a finite vertex set (the row). The transfer matrix $A$ of $R$ is the matrix with entries in $\\mathbb{Z}[z]$, rows indexed by the new signature $\\tau$ and columns by the old signature $\\sigma$, both ranging over the signatures of $R$, whose entry $A_{\\tau\\sigma}$ is $z^{\\operatorname{occ}(\\tau)}$ if $\\tau$ is compatible with $\\sigma$ and $0$ otherwise; here $\\operatorname{occ}(\\tau)$ is\u2026","title":"Domination transfer matrix"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/almost-dominating-polynomial","kind":"node","lean":"MetaMathlibExt.almostDominatingPolynomial","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#almost-dominating-polynomial","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $R$ be a finite graph (the row), $n$ a natural number and $\\sigma$ an assignment of a domination state to each vertex of $R$. Let $H$ be the box product of $R$ with the path on the vertices $0, \\dots, n$, so that the last row of $H$ consists of the vertices $(i, n)$. Relative to a vertex set $S$ of $H$, a vertex is occupied if it lies in $S$, covered if it lies outside $S$ but is adjacent to a vertex of $S$,\u2026","title":"Almost dominating polynomial"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/vertex-domination-state-eq-occupied-iff","kind":"node","lean":"MetaMathlibExt.vertexDominationState_eq_occupied_iff","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#vertex-domination-state-eq-occupied-iff","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $G$ be a simple graph, $S$ a finite set of vertices of $G$ and $v$ a vertex of $G$. Then the domination state of $v$ relative to $S$ in $G$ is occupied if and only if $v \\in S$.","title":"A vertex is occupied exactly when it lies in the set"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/is-dominating-iff-forall-vertex-dominati","kind":"node","lean":"MetaMathlibExt.isDominating_iff_forall_vertexDominationState_ne_uncovered","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#is-dominating-iff-forall-vertex-dominati","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $G$ be a simple graph and $S$ a finite set of vertices of $G$. Then $S$ dominates $G$ if and only if no vertex of $G$ has domination state uncovered relative to $S$ in $G$.","title":"Dominating sets have no uncovered vertex"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-polynomial-eval-one","kind":"node","lean":"MetaMathlibExt.dominationPolynomial_eval_one","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-polynomial-eval-one","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"For every finite graph $G$, the value at $1$ of the domination polynomial of $G$ equals the number of dominating sets of $G$.","title":"The domination polynomial at one counts dominating sets"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/occupied-count-all-covered-signature","kind":"node","lean":"MetaMathlibExt.occupiedCount_allCoveredSignature","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#occupied-count-all-covered-signature","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"For every simple graph $R$ on a finite vertex set, the number of occupied vertices of the all-covered signature of $R$ is $0$.","title":"The all-covered signature has no occupied vertex"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-transfer-matrix-apply-all-cov","kind":"node","lean":"MetaMathlibExt.dominationTransferMatrix_apply_allCoveredSignature","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-transfer-matrix-apply-all-cov","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"fully_proved","summary":"Let $R$ be a simple graph on a finite vertex set (the row), let $A$ be its domination transfer matrix, with entries in $\\mathbb{Z}[z]$, and let $\\sigma_\\bullet$ be its all-covered signature. For every signature $\\tau$ of $R$, the entry $A_{\\tau\\sigma_\\bullet}$ equals $z^{\\operatorname{occ}(\\tau)}$, where $\\operatorname{occ}(\\tau)$ is the number of occupied vertices of $\\tau$, if every covered vertex of $\\tau$ is\u2026","title":"Transfer matrix column of the all-covered signature"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-polynomial-grid-eq-sum-almost","kind":"node","lean":"MetaMathlibExt.dominationPolynomial_grid_eq_sum_almostDominatingPolynomial","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-polynomial-grid-eq-sum-almost","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"can_prove","summary":"For all natural numbers $m$ and $n$, let $P_k$ denote the path on the vertices $0, \\dots, k - 1$. Then the domination polynomial of the grid $P_m \\mathbin{\\square} P_{n+1}$ equals $\\sum_\\sigma G^\\sigma(z)$, the sum running over the signatures $\\sigma$ of the row graph $P_m$ with no uncovered vertex, where $G^\\sigma(z)$ is the almost dominating polynomial of $P_m$ with $n + 1$ rows and last-row state $\\sigma$.","title":"Grid domination polynomial as a sum of almost dominating polynomials"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-polynomial-cylinder-eq-sum-al","kind":"node","lean":"MetaMathlibExt.dominationPolynomial_cylinder_eq_sum_almostDominatingPolynomial","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-polynomial-cylinder-eq-sum-al","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"can_prove","summary":"For all natural numbers $m$ and $n$, let $C_m$ be the cycle graph on the vertices $0, \\dots, m - 1$, in which distinct $i$ and $j$ are adjacent when $i - j \\equiv \\pm 1 \\pmod m$, and let $P_{n+1}$ be the path on the vertices $0, \\dots, n$. Then the domination polynomial of the cylinder $C_m \\mathbin{\\square} P_{n+1}$ equals $\\sum_\\sigma G^\\sigma(z)$, the sum running over the signatures $\\sigma$ of the row graph\u2026","title":"Cylinder domination polynomial as a sum of almost dominating polynomials"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-polynomial-grid-eq-sum-transf","kind":"node","lean":"MetaMathlibExt.dominationPolynomial_grid_eq_sum_transferMatrix_pow","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-polynomial-grid-eq-sum-transf","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"can_prove","summary":"For all natural numbers $m$ and $n$, let $P_k$ denote the path on the vertices $0, \\dots, k - 1$, let $A$ be the domination transfer matrix of the row graph $P_m$, indexed by the signatures of $P_m$, and let $\\sigma_\\bullet$ be the all-covered signature. Then the domination polynomial of the grid $P_m \\mathbin{\\square} P_n$ is","title":"Transfer-matrix formula for the domination polynomial of the grid"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-polynomial-cylinder-eq-sum-tr","kind":"node","lean":"MetaMathlibExt.dominationPolynomial_cylinder_eq_sum_transferMatrix_pow","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-polynomial-cylinder-eq-sum-tr","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"can_prove","summary":"For all natural numbers $m$ and $n$, let $C_m$ be the cycle graph on the vertices $0, \\dots, m - 1$, in which distinct $i$ and $j$ are adjacent when $i - j \\equiv \\pm 1 \\pmod m$, and let $P_n$ be the path on the vertices $0, \\dots, n - 1$. Let $\\bar A$ be the domination transfer matrix of the row graph $C_m$, indexed by the signatures of $C_m$, and let $\\sigma_\\bullet$ be the all-covered signature. Then the\u2026","title":"Transfer-matrix formula for the domination polynomial of the cylinder"},{"id":"GraphTheory/independence-and-domination/domination-polynomials-of-grids/domination-polynomial-torus-eq-trace-tra","kind":"node","lean":"MetaMathlibExt.dominationPolynomial_torus_eq_trace_transferMatrix_pow","parent":29874,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/domination-polynomials-of-grids/index.html#domination-polynomial-torus-eq-trace-tra","scope":"GraphTheory/independence-and-domination/domination-polynomials-of-grids","status":"can_prove","summary":"For all natural numbers $m$ and $n$ with $n \\ge 1$, let $C_k$ denote the cycle graph on the vertices $0, \\dots, k - 1$, in which distinct $i$ and $j$ are adjacent when $i - j \\equiv \\pm 1 \\pmod k$, and let $\\bar A$ be the domination transfer matrix of the row graph $C_m$. Then the domination polynomial of the torus $C_m \\mathbin{\\square} C_n$ equals the trace of $\\bar A^n$.","title":"Transfer-matrix formula for the domination polynomial of the torus"},{"data_url":"scopes/GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs.json","id":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs","kind":"scope","navigation":"module","parent":29849,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/index.html","scope":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs","status":"fully_proved","title":"Fibonacci numbers of graphs"},{"id":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/zykov-zero-indicator","kind":"node","lean":"MetaMathlibExt.zykovZeroIndicator","parent":29895,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/index.html#zykov-zero-indicator","scope":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs","status":"fully_proved","summary":"For a natural number $z$, the zero indicator of $z$ is the Boolean value that is true if $z = 0$ and false otherwise.","title":"Zero indicator of a natural number"},{"id":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/independent-set-characteristic","kind":"node","lean":"MetaMathlibExt.independentSetCharacteristic","parent":29895,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/index.html#independent-set-characteristic","scope":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex type $V$ and let $A$ be a finite independent set of $G$. The characteristic vector of $A$ is the Boolean function on $V$ that sends a vertex $v$ to true if $v \\in A$ and to false otherwise.","title":"Characteristic vector of an independent set"},{"id":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/relation-pairing","kind":"node","lean":"MetaMathlibExt.relationPairing","parent":29895,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/index.html#relation-pairing","scope":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs","status":"fully_proved","summary":"Let $V$ and $W$ be finite types, let $R$ be a relation between $V$ and $W$, and let $a$ and $b$ be Boolean functions on $V$ and on $W$. The pairing of $a$ and $b$ along $R$ is the natural number of pairs $(v, w) \\in V \\times W$ such that $R(v, w)$ holds and both $a(v)$ and $b(w)$ are true.","title":"Pairing of Boolean vectors along a relation"},{"id":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/zykov-transfer-matrix","kind":"node","lean":"MetaMathlibExt.zykovTransferMatrix","parent":29895,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs/index.html#zykov-transfer-matrix","scope":"GraphTheory/independence-and-domination/fibonacci-numbers-of-graphs","status":"fully_proved","summary":"Let $G$ and $H$ be simple graphs on finite vertex types $V$ and $W$, and let $R$ be a relation between $V$ and $W$. The transfer matrix of $G$ and $H$ along $R$ is indexed by pairs $(A, B)$ where $A$ is a finite independent set of $G$ and $B$ is a finite independent set of $H$. Its entry at $(A, B)$ is the Boolean value that is true exactly when the pairing along $R$ of the characteristic vectors of $A$ and $B$ is\u2026","title":"Transfer matrix of a generalized Zykov sum"},{"data_url":"scopes/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs.json","id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","kind":"scope","navigation":"module","parent":29849,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","title":"Independence polynomial of claw-free graphs"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/claw-free","kind":"node","lean":"MetaMathlibExt.clawFree","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#claw-free","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"A simple graph $G$ on a finite vertex set $V$ is claw-free if it has no induced claw $K_{1,3}$: there are no vertices $c, \\ell_1, \\ell_2, \\ell_3 \\in V$ such that $\\ell_1, \\ell_2, \\ell_3$ are pairwise distinct, each is adjacent to $c$ and different from $c$, and no two of them are adjacent.","title":"Claw-free finite graph"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/independence-poly-of-claw-free-splits","kind":"node","lean":"MetaMathlibExt.independence_poly_of_claw_free_splits","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#independence-poly-of-claw-free-splits","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"Let $G$ be a claw-free simple graph on a finite vertex set $V$, and let $I(G) = \\sum_{k=0}^{|V|} i_k X^k \\in \\mathbb{Z}[X]$ be its independence polynomial, where $i_k$ is the number of independent sets of $G$ with $k$ vertices. Then $I(G)$, viewed as a polynomial with real coefficients, splits over $\\mathbb{R}$: it is a constant times a product of linear factors $X - r$ with $r \\in \\mathbb{R}$, so all its roots are\u2026","title":"Chudnovsky\u2013Seymour theorem"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/claw-free-iff-not-is-ind-contained","kind":"node","lean":"MetaMathlibExt.clawFree_iff_not_isIndContained","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#claw-free-iff-not-is-ind-contained","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set. Then $G$ is claw-free if and only if the claw $K_{1,3}$, the star graph on four vertices whose center is adjacent to the other three, is not an induced subgraph of $G$, that is, there is no injective map from its vertices to those of $G$ that preserves both adjacency and nonadjacency.","title":"Claw-free finite graphs have no induced claw"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/independence-poly-coeff","kind":"node","lean":"MetaMathlibExt.independencePoly_coeff","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#independence-poly-coeff","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, and let $I(G) = \\sum_{k=0}^{|V|} i_k X^k \\in \\mathbb{Z}[X]$ be its independence polynomial, where $i_k$ is the number of independent sets of $G$ with $k$ vertices. Then for every natural number $k$, the coefficient of $X^k$ in $I(G)$ is the number of independent sets of $G$ with exactly $k$ vertices.","title":"Coefficients of the independence polynomial"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/aeval-independence-poly","kind":"node","lean":"MetaMathlibExt.aeval_independencePoly","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#aeval-independence-poly","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, and let $I(G) = \\sum_{k=0}^{|V|} i_k X^k \\in \\mathbb{Z}[X]$ be its independence polynomial, where $i_k$ is the number of independent sets of $G$ with $k$ vertices. Then for every element $x$ of a commutative ring $R$, the value of $I(G)$ at $x$ is $\\sum_S x^{|S|}$, the sum running over all independent sets $S \\subseteq V$ of $G$, the empty set included.","title":"Evaluating the independence polynomial"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/independence-poly-bot","kind":"node","lean":"MetaMathlibExt.independencePoly_bot","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#independence-poly-bot","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"Let $V$ be a finite set. For a simple graph $G$ on $V$, let $I(G) = \\sum_{k=0}^{|V|} i_k X^k \\in \\mathbb{Z}[X]$ be its independence polynomial, where $i_k$ is the number of independent sets of $G$ with $k$ vertices. If $G$ is the graph on $V$ with no edges, then $I(G) = (1 + X)^{|V|}$.","title":"Independence polynomial of the edgeless graph"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/independence-poly-top","kind":"node","lean":"MetaMathlibExt.independencePoly_top","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#independence-poly-top","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"Let $V$ be a finite set. For a simple graph $G$ on $V$, let $I(G) = \\sum_{k=0}^{|V|} i_k X^k \\in \\mathbb{Z}[X]$ be its independence polynomial, where $i_k$ is the number of independent sets of $G$ with $k$ vertices. If $G$ is the complete graph on $V$, then $I(G) = 1 + |V|\\,X$.","title":"Independence polynomial of the complete graph"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/is-claw-free","kind":"node","lean":"SimpleGraph.IsClawFree","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#is-claw-free","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"A simple graph $G$ on a vertex set $V$, not necessarily finite, is claw-free if it has no induced claw $K_{1,3}$: there are no vertices $c, \\ell_1, \\ell_2, \\ell_3 \\in V$ such that $\\ell_1, \\ell_2, \\ell_3$ are pairwise distinct, each is adjacent to $c$ and different from $c$, and no two of them are adjacent.","title":"Claw-free graph"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/is-claw-free-iff-not-is-ind-contained","kind":"node","lean":"SimpleGraph.isClawFree_iff_not_isIndContained","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#is-claw-free-iff-not-is-ind-contained","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on an arbitrary vertex set. Then $G$ is claw-free if and only if the claw $K_{1,3}$, the star graph on four vertices whose center is adjacent to the other three, is not an induced subgraph of $G$, that is, there is no injective map from its vertices to those of $G$ that preserves both adjacency and nonadjacency.","title":"Claw-free graphs have no induced claw"},{"id":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/claw-free-iff-is-claw-free","kind":"node","lean":"MetaMathlibExt.clawFree_iff_isClawFree","parent":29900,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs/index.html#claw-free-iff-is-claw-free","scope":"GraphTheory/independence-and-domination/independence-polynomial-of-claw-free-graphs","status":"fully_proved","summary":"A simple graph on a finite vertex set is a claw-free finite graph if and only if it is a claw-free graph in the sense defined for graphs on arbitrary vertex sets.","title":"The two definitions of claw-free agree"},{"data_url":"scopes/GraphTheory/independence-and-domination/maximal-cliques-and-triangles.json","id":"GraphTheory/independence-and-domination/maximal-cliques-and-triangles","kind":"scope","navigation":"module","parent":29849,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/maximal-cliques-and-triangles/index.html","scope":"GraphTheory/independence-and-domination/maximal-cliques-and-triangles","status":"fully_proved","title":"Maximal cliques and triangles"},{"id":"GraphTheory/independence-and-domination/maximal-cliques-and-triangles/maximal-clique-sizes","kind":"node","lean":"SimpleGraph.maximalCliqueSizes","parent":29911,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/maximal-cliques-and-triangles/index.html#maximal-clique-sizes","scope":"GraphTheory/independence-and-domination/maximal-cliques-and-triangles","status":"fully_proved","summary":"For a simple graph $G$, this is the set of natural numbers $k$ such that some finite clique of $G$ with exactly $k$ vertices is maximal under inclusion among the finite cliques of $G$, that is, no finite clique of $G$ strictly contains it.","title":"Sizes of maximal cliques"},{"id":"GraphTheory/independence-and-domination/maximal-cliques-and-triangles/triangles-containing","kind":"node","lean":"SimpleGraph.trianglesContaining","parent":29911,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/maximal-cliques-and-triangles/index.html#triangles-containing","scope":"GraphTheory/independence-and-domination/maximal-cliques-and-triangles","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set and an edge $uv$ of $G$, this is the finite set of triangles of $G$ that contain the edge $uv$, where a triangle is the vertex set of a $3$-clique of $G$; that is, the $3$-cliques of $G$ containing both $u$ and $v$.","title":"Triangles containing an edge"},{"data_url":"scopes/GraphTheory/independence-and-domination/odd-independent-sets.json","id":"GraphTheory/independence-and-domination/odd-independent-sets","kind":"scope","navigation":"module","parent":29849,"result":false,"route":"../roadmap/GraphTheory/independence-and-domination/odd-independent-sets/index.html","scope":"GraphTheory/independence-and-domination/odd-independent-sets","status":"fully_proved","title":"Odd independent sets"},{"id":"GraphTheory/independence-and-domination/odd-independent-sets/is-odd-independent","kind":"node","lean":"SimpleGraph.IsOddIndependent","parent":29914,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/odd-independent-sets/index.html#is-odd-independent","scope":"GraphTheory/independence-and-domination/odd-independent-sets","status":"fully_proved","summary":"A set $S$ of vertices of a simple graph $G$ is odd independent if $S$ is an independent set of $G$ and, for every vertex $v \\notin S$, the set of neighbours of $v$ that lie in $S$ is finite and is either empty or has an odd number of elements.","title":"Odd independent set"},{"id":"GraphTheory/independence-and-domination/odd-independent-sets/is-indep-set","kind":"node","lean":"SimpleGraph.IsOddIndependent.isIndepSet","parent":29914,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/odd-independent-sets/index.html#is-indep-set","scope":"GraphTheory/independence-and-domination/odd-independent-sets","status":"fully_proved","summary":"If $S$ is an odd independent set of a simple graph $G$, then $S$ is an independent set of $G$: no two distinct vertices of $S$ are adjacent.","title":"Odd independent sets are independent"},{"id":"GraphTheory/independence-and-domination/odd-independent-sets/finite-inter","kind":"node","lean":"SimpleGraph.IsOddIndependent.finite_inter","parent":29914,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/odd-independent-sets/index.html#finite-inter","scope":"GraphTheory/independence-and-domination/odd-independent-sets","status":"fully_proved","summary":"Let $S$ be an odd independent set of a simple graph $G$ and let $v$ be a vertex with $v \\notin S$. Then only finitely many neighbours of $v$ lie in $S$.","title":"Outside vertices have finitely many neighbours in an odd independent set"},{"id":"GraphTheory/independence-and-domination/odd-independent-sets/empty-or-odd","kind":"node","lean":"SimpleGraph.IsOddIndependent.empty_or_odd","parent":29914,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/odd-independent-sets/index.html#empty-or-odd","scope":"GraphTheory/independence-and-domination/odd-independent-sets","status":"fully_proved","summary":"Let $S$ be an odd independent set of a simple graph $G$ and let $v$ be a vertex with $v \\notin S$. Then the set of neighbours of $v$ that lie in $S$ is either empty or has an odd number of elements.","title":"Outside vertices see an empty or odd part of an odd independent set"},{"id":"GraphTheory/independence-and-domination/odd-independent-sets/is-odd-independent-empty","kind":"node","lean":"SimpleGraph.isOddIndependent_empty","parent":29914,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/odd-independent-sets/index.html#is-odd-independent-empty","scope":"GraphTheory/independence-and-domination/odd-independent-sets","status":"fully_proved","summary":"The empty set is odd independent in any simple graph.","title":"The empty set is odd independent"},{"id":"GraphTheory/independence-and-domination/odd-independent-sets/is-odd-independent-singleton","kind":"node","lean":"SimpleGraph.isOddIndependent_singleton","parent":29914,"result":true,"route":"../roadmap/GraphTheory/independence-and-domination/odd-independent-sets/index.html#is-odd-independent-singleton","scope":"GraphTheory/independence-and-domination/odd-independent-sets","status":"fully_proved","summary":"For every vertex $a$ of a simple graph $G$, the singleton $\\{a\\}$ is odd independent in $G$.","title":"Singletons are odd independent"},{"data_url":"scopes/GraphTheory/extremal-graph-theory.json","id":"GraphTheory/extremal-graph-theory","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/index.html","scope":"GraphTheory/extremal-graph-theory","status":"can_prove","summary":"How many edges or copies of a subgraph a graph can have while avoiding a forbidden structure: Tur\u00e1n and Zarankiewicz problems, saturation, and extremal graphs of given girth.","title":"Extremal graph theory"},{"id":"GraphTheory/extremal-graph-theory/kovari-sos-turan","kind":"node","lean":"MetaMathlibExt.kovari_sos_turan","parent":29921,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/index.html#kovari-sos-turan","scope":"GraphTheory/extremal-graph-theory","status":"fully_proved","summary":"Let $m, n, s, t$ be natural numbers with $s \\ge 1$ and $1 \\le t \\le n$. Let $z(m, n; s, t)$ be the Zarankiewicz number, the largest number of edges in a bipartite graph with parts of sizes $m$ and $n$ that contains no copy of the complete bipartite graph $K_{s,t}$. Then","title":"K\u0151v\u00e1ri\u2013S\u00f3s\u2013Tur\u00e1n bound"},{"id":"GraphTheory/extremal-graph-theory/monomial-graph-girth-eight-classificatio","kind":"node","lean":"MetaMathlibExt.monomialGraph_girth_eight_classification","parent":29921,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/index.html#monomial-graph-girth-eight-classificatio","scope":"GraphTheory/extremal-graph-theory","status":"can_prove","summary":"Let $p$ be an odd prime, $e$ a natural number, $q = p^e$, and $\\mathbb{F}$ a field with $q$ elements, and suppose that either $p \\ge 5$ and $e = 2^a 3^b$ for some natural numbers $a, b$, or $3 \\le q \\le 10^{10}$. For natural numbers $m_2, n_2, m_3, n_3$, let $G$ be the bipartite graph with two copies of $\\mathbb{F}^3$ as its parts, in which $(x_1, x_2, x_3)$ in the first part is adjacent to $(y_1, y_2, y_3)$ in the\u2026","title":"Monomial graphs of girth eight"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/balanced-subgraphs.json","id":"GraphTheory/extremal-graph-theory/balanced-subgraphs","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/balanced-subgraphs/index.html","scope":"GraphTheory/extremal-graph-theory/balanced-subgraphs","status":"can_prove","title":"Balanced subgraphs"},{"id":"GraphTheory/extremal-graph-theory/balanced-subgraphs/is-balanced","kind":"node","lean":"SimpleGraph.IsBalanced","parent":29924,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/balanced-subgraphs/index.html#is-balanced","scope":"GraphTheory/extremal-graph-theory/balanced-subgraphs","status":"fully_proved","summary":"Let $G$ be a finite simple graph and $D$ a real number. The graph $G$ is $D$-balanced if its maximum degree $\\Delta(G)$ is at most $D$ times its minimum degree $\\delta(G)$, that is, $\\Delta(G) \\le D \\cdot \\delta(G)$.","title":"Balanced graph"},{"id":"GraphTheory/extremal-graph-theory/balanced-subgraphs/main-positive","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1077Wanted.mainPositive","parent":29924,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/balanced-subgraphs/index.html#main-positive","scope":"GraphTheory/extremal-graph-theory/balanced-subgraphs","status":"fully_proved","summary":"For all reals $\\varepsilon$ and $\\alpha$ with $0 < \\varepsilon < 1$ and $0 < \\alpha < 1$, for all sufficiently large real $D$ and then all sufficiently large $n$, every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ with more than $n^{1+\\alpha}$ edges has a (not necessarily induced) subgraph $H$ with some number $m$ of vertices such that $H$ is $D$-balanced (its maximum degree is at most $D$ times its\u2026","title":"Erd\u0151s\u2013Simonovits balanced subgraph claim"},{"id":"GraphTheory/extremal-graph-theory/balanced-subgraphs/disproved-main","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1077Wanted.disprovedMain","parent":29924,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/balanced-subgraphs/index.html#disproved-main","scope":"GraphTheory/extremal-graph-theory/balanced-subgraphs","status":"fully_proved","summary":"The proposition that the Erd\u0151s\u2013Simonovits balanced subgraph claim fails, that is, that it is not the case that for all reals $\\varepsilon$ and $\\alpha$ with $0 < \\varepsilon < 1$ and $0 < \\alpha < 1$, for all sufficiently large real $D$ and then all sufficiently large $n$, every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ with more than $n^{1+\\alpha}$ edges has a (not necessarily induced) subgraph $H$\u2026","title":"Negation of the Erd\u0151s\u2013Simonovits balanced subgraph claim"},{"id":"GraphTheory/extremal-graph-theory/balanced-subgraphs/main-positive-refuted","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1077Wanted.mainPositive_refuted","parent":29924,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/balanced-subgraphs/index.html#main-positive-refuted","scope":"GraphTheory/extremal-graph-theory/balanced-subgraphs","status":"can_prove","summary":"It is not the case that for all reals $\\varepsilon$ and $\\alpha$ with $0 < \\varepsilon < 1$ and $0 < \\alpha < 1$, for all sufficiently large real $D$ and then all sufficiently large $n$, every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ with more than $n^{1+\\alpha}$ edges has a (not necessarily induced) subgraph $H$ with some number $m$ of vertices such that $H$ is $D$-balanced (its maximum degree is at\u2026","title":"The Erd\u0151s\u2013Simonovits balanced subgraph claim is false"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/books-in-graphs.json","id":"GraphTheory/extremal-graph-theory/books-in-graphs","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/books-in-graphs/index.html","scope":"GraphTheory/extremal-graph-theory/books-in-graphs","status":"can_prove","title":"Books in graphs"},{"id":"GraphTheory/extremal-graph-theory/books-in-graphs/book-number","kind":"node","lean":"SimpleGraph.bookNumber","parent":29929,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/books-in-graphs/index.html#book-number","scope":"GraphTheory/extremal-graph-theory/books-in-graphs","status":"fully_proved","summary":"Let $G$ be a finite simple graph. The book number of $G$ is the maximum, over ordered pairs $(u, v)$ of adjacent vertices, of the number of common neighbors of $u$ and $v$ (the number of triangles containing the edge $uv$); it is $0$ if $G$ has no edges.","title":"Book number"},{"id":"GraphTheory/extremal-graph-theory/books-in-graphs/le-book-number","kind":"node","lean":"SimpleGraph.le_bookNumber","parent":29929,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/books-in-graphs/index.html#le-book-number","scope":"GraphTheory/extremal-graph-theory/books-in-graphs","status":"fully_proved","summary":"Let $G$ be a finite simple graph. If $u$ and $v$ are adjacent vertices of $G$, then the number of common neighbors of $u$ and $v$ is at most the book number of $G$.","title":"Common neighbors of an edge are bounded by the book number"},{"id":"GraphTheory/extremal-graph-theory/books-in-graphs/book-number-le","kind":"node","lean":"SimpleGraph.bookNumber_le","parent":29929,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/books-in-graphs/index.html#book-number-le","scope":"GraphTheory/extremal-graph-theory/books-in-graphs","status":"fully_proved","summary":"Let $G$ be a finite simple graph and $b$ a natural number. If every pair of adjacent vertices $u$, $v$ of $G$ has at most $b$ common neighbors, then the book number of $G$ is at most $b$.","title":"Bounding the book number"},{"id":"GraphTheory/extremal-graph-theory/books-in-graphs/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem905Wanted.conjecture","parent":29929,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/books-in-graphs/index.html#conjecture","scope":"GraphTheory/extremal-graph-theory/books-in-graphs","status":"fully_proved","summary":"The proposition asserts that for every natural number $n$ and every simple graph $G$ on $n$ vertices with more than $n^2/4$ edges, some edge of $G$ lies in at least $n/6$ triangles of $G$.","title":"Erd\u0151s Problem 905 on edges in many triangles"},{"id":"GraphTheory/extremal-graph-theory/books-in-graphs/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem905Wanted.conjecture_holds","parent":29929,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/books-in-graphs/index.html#conjecture-holds","scope":"GraphTheory/extremal-graph-theory/books-in-graphs","status":"can_prove","summary":"For every natural number $n$, every simple graph $G$ on $n$ vertices with more than $n^2/4$ edges has an edge that lies in at least $n/6$ triangles of $G$.","title":"Edwards\u2013Khadzhiivanov\u2013Nikiforov theorem on edges in many triangles"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture.json","id":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/index.html","scope":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","status":"can_prove","title":"Erd\u0151s\u2013Faudree triangle conjecture"},{"id":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/joined-to-two","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1034Wanted.JoinedToTwo","parent":29935,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/index.html#joined-to-two","scope":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex type $V$ and let $T$ and $Y$ be finite sets of vertices. The set $Y$ is joined to two vertices of $T$ in $G$ if every vertex $y \\in Y$ is adjacent in $G$ to two distinct vertices $u, v \\in T$.","title":"Vertices joined to two vertices of a set"},{"id":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/conjecture","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1034Wanted.conjecture","parent":29935,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/index.html#conjecture","scope":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","status":"fully_proved","summary":"For every real $\\varepsilon > 0$ the following holds for all sufficiently large $n$: every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ with more than $n^2/4$ edges contains a triangle $T$ and a set $Y$ of vertices, each adjacent in $G$ to two distinct vertices of $T$, with $|Y| > (1/2 - \\varepsilon)n$.","title":"Erd\u0151s\u2013Faudree triangle conjecture"},{"id":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/disproved-form","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1034Wanted.disprovedForm","parent":29935,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/index.html#disproved-form","scope":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","status":"fully_proved","summary":"The proposition that the Erd\u0151s\u2013Faudree triangle conjecture fails, that is, that it is not the case that for every real $\\varepsilon > 0$ and all sufficiently large $n$, every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ with more than $n^2/4$ edges contains a triangle $T$ and a set $Y$ of vertices, each adjacent in $G$ to two distinct vertices of $T$, with $|Y| > (1/2 - \\varepsilon)n$.","title":"Negation of the Erd\u0151s\u2013Faudree triangle conjecture"},{"id":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/book-lower-bound","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1034Wanted.bookLowerBound","parent":29935,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/index.html#book-lower-bound","scope":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","status":"fully_proved","summary":"For every real $\\varepsilon > 0$ the following holds for all sufficiently large $n$: every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ with more than $n^2/4$ edges contains a triangle $T$ and a set $Y$ of vertices, each adjacent in $G$ to two distinct vertices of $T$, with $|Y| \\ge (1/6 - \\varepsilon)n$.","title":"Book lower bound for Erd\u0151s Problem 1034"},{"id":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/ma-tang-upper-bound","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1034Wanted.maTangUpperBound","parent":29935,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/index.html#ma-tang-upper-bound","scope":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","status":"fully_proved","summary":"For every real $\\varepsilon > 0$ the following holds for all sufficiently large $n$: there is a simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ with more than $n^2/4$ edges such that for every triangle $T$ of $G$, every set $Y$ of vertices each adjacent in $G$ to two distinct vertices of $T$ satisfies $|Y| \\le (2 - \\sqrt{5/2} + \\varepsilon)n$.","title":"Ma\u2013Tang upper bound for Erd\u0151s Problem 1034"},{"id":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/k4-free-upper-bound","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1034Wanted.k4FreeUpperBound","parent":29935,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(K_4\\)"],["text","-free upper bound for Erd\u0151s Problem 1034"]],"source":"$K_4$-free upper bound for Erd\u0151s Problem 1034"},"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/index.html#k4-free-upper-bound","scope":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","status":"fully_proved","summary":"For every real $\\varepsilon > 0$ the following holds for all sufficiently large $n$: there is a simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ that contains no $K_4$ and has more than $n^2/4$ edges, such that for every triangle $T$ of $G$, every set $Y$ of vertices each adjacent in $G$ to two distinct vertices of $T$ satisfies $|Y| \\le (2\\sqrt{3} - 3 + \\varepsilon)n$.","title":"$K_4$-free upper bound for Erd\u0151s Problem 1034"},{"id":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/conjecture-refuted","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem1034Wanted.conjecture_refuted","parent":29935,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture/index.html#conjecture-refuted","scope":"GraphTheory/extremal-graph-theory/erd-s-faudree-triangle-conjecture","status":"can_prove","summary":"It is not true that for every real $\\varepsilon > 0$ and all sufficiently large $n$, every simple graph on the vertex set $\\{0, \\dots, n-1\\}$ with more than $n^2/4$ edges contains a triangle $T$ and a set of more than $(1/2 - \\varepsilon)n$ vertices, each adjacent to two distinct vertices of $T$.","title":"The Erd\u0151s\u2013Faudree triangle conjecture is false"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five.json","id":"GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five/index.html","scope":"GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five","status":"can_prove","title":"Extremal graphs of girth five"},{"id":"GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five/girth-five-extremal-number","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.GirthFiveExtremalValuesWanted.girthFiveExtremalNumber","parent":29943,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five/index.html#girth-five-extremal-number","scope":"GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five","status":"fully_proved","summary":"For a natural number $n$, $\\mathrm{ex}(n; \\{C_3, C_4\\})$ is the largest number of edges of a simple graph on $n$ vertices whose girth is at least $5$, that is, which contains no triangle and no four-cycle (OEIS A006856). An acyclic graph has infinite girth, so the empty graph always qualifies.","title":"Girth-five extremal number"},{"id":"GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five/girth-five-extremal-number-one-to-twenty","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.GirthFiveExtremalValuesWanted.girthFiveExtremalNumber_one_to_twenty","parent":29943,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five/index.html#girth-five-extremal-number-one-to-twenty","scope":"GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five","status":"can_prove","summary":"For $n = 1, 2, \\dots, 20$ the girth-five extremal numbers $\\mathrm{ex}(n; \\{C_3, C_4\\})$, the largest numbers of edges of a simple graph on $n$ vertices that contains no triangle and no four-cycle, are, in order, $0, 1, 2, 3, 5, 6, 8, 10, 12, 15, 16, 18, 21, 23, 26, 28, 31, 34, 38, 41$.","title":"Girth-five extremal numbers up to twenty vertices"},{"id":"GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five/girth-five-extremal-number-fifty","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.GirthFiveExtremalValuesWanted.girthFiveExtremalNumber_fifty","parent":29943,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five/index.html#girth-five-extremal-number-fifty","scope":"GraphTheory/extremal-graph-theory/extremal-graphs-of-girth-five","status":"can_prove","summary":"The girth-five extremal number $\\mathrm{ex}(50; \\{C_3, C_4\\})$, the largest number of edges of a simple graph on $50$ vertices that contains no triangle and no four-cycle, equals $175$.","title":"Girth-five extremal number on fifty vertices"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/four-cycle-free-graphs.json","id":"GraphTheory/extremal-graph-theory/four-cycle-free-graphs","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/four-cycle-free-graphs/index.html","scope":"GraphTheory/extremal-graph-theory/four-cycle-free-graphs","status":"can_prove","title":"Four-cycle-free graphs"},{"id":"GraphTheory/extremal-graph-theory/four-cycle-free-graphs/extremal-number-cycle-graph-four-one-to","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.FourCycleExtremalValuesWanted.extremalNumber_cycleGraph_four_one_to_twenty","parent":29947,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/four-cycle-free-graphs/index.html#extremal-number-cycle-graph-four-one-to","scope":"GraphTheory/extremal-graph-theory/four-cycle-free-graphs","status":"can_prove","summary":"For a natural number $n$, let $\\mathrm{ex}(n; C_4)$ be the largest number of edges of a simple graph on $n$ vertices that contains no four-cycle $C_4$ as a (not necessarily induced) subgraph. For $n = 1, 2, \\dots, 20$ the values of $\\mathrm{ex}(n; C_4)$ are, in order, $0, 1, 3, 4, 6, 7, 9, 11, 13, 16, 18, 21, 24, 27, 30, 33, 36, 39, 42, 46$.","title":"Extremal numbers of the four-cycle up to twenty vertices"},{"id":"GraphTheory/extremal-graph-theory/four-cycle-free-graphs/extremal-number-cycle-graph-four-project","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.FourCycleExtremalValuesWanted.extremalNumber_cycleGraph_four_projectivePlane_order","parent":29947,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/four-cycle-free-graphs/index.html#extremal-number-cycle-graph-four-project","scope":"GraphTheory/extremal-graph-theory/four-cycle-free-graphs","status":"can_prove","summary":"For a natural number $n$, let $\\mathrm{ex}(n; C_4)$ be the largest number of edges of a simple graph on $n$ vertices that contains no four-cycle $C_4$ as a (not necessarily induced) subgraph. For every prime power $q > 13$, $\\mathrm{ex}(q^2 + q + 1; C_4) = q (q + 1)^2 / 2$; the division is exact because $q (q + 1)$ is even.","title":"F\u00fcredi's theorem on four-cycle-free graphs"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/generalized-tur-n-numbers.json","id":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/index.html","scope":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers","status":"fully_proved","title":"Generalized Tur\u00e1n numbers"},{"id":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/generalized-extremal-number","kind":"node","lean":"SimpleGraph.generalizedExtremalNumber","parent":29950,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/index.html#generalized-extremal-number","scope":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers","status":"fully_proved","summary":"Let $n$ be a natural number and let $T$ and $H$ be simple graphs, where $T$ has finitely many vertices. A copy of a graph in a graph $G$ is a subgraph of $G$ isomorphic to it, and $G$ is $H$-free if it contains no copy of $H$. The generalized Tur\u00e1n number $\\mathrm{ex}(n, T, H)$ is the largest number of copies of $T$ in an $H$-free simple graph on the vertex set $\\{0, 1, \\dots, n-1\\}$, and is $0$ if there is no such\u2026","title":"Generalized Tur\u00e1n number"},{"id":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/generalized-extremal-number-eq-sup","kind":"node","lean":"SimpleGraph.generalizedExtremalNumber_eq_sup","parent":29950,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/index.html#generalized-extremal-number-eq-sup","scope":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers","status":"fully_proved","summary":"Let $n$ be a natural number and let $T$ and $H$ be simple graphs, where $T$ has finitely many vertices. Then the generalized Tur\u00e1n number $\\mathrm{ex}(n, T, H)$ is the supremum in $\\mathbb{N}$, with the supremum of the empty family taken to be $0$, of the number of copies of $T$ in $G$ over all simple graphs $G$ on the vertex set $\\{0, 1, \\dots, n-1\\}$ that contain no copy of $H$ (a subgraph isomorphic to $H$).","title":"The generalized Tur\u00e1n number as a supremum"},{"id":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/copy-count-le-generalized-extremal-numbe","kind":"node","lean":"SimpleGraph.copyCount_le_generalizedExtremalNumber","parent":29950,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/index.html#copy-count-le-generalized-extremal-numbe","scope":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers","status":"fully_proved","summary":"Let $n$ be a natural number and let $T$ and $H$ be simple graphs, where $T$ has finitely many vertices. If $G$ is a simple graph on the vertex set $\\{0, 1, \\dots, n-1\\}$ that contains no copy of $H$ (a subgraph isomorphic to $H$), then the number of copies of $T$ in $G$ is at most the generalized Tur\u00e1n number $\\mathrm{ex}(n, T, H)$.","title":"The generalized Tur\u00e1n number is an upper bound"},{"id":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/generalized-extremal-number-le-iff","kind":"node","lean":"SimpleGraph.generalizedExtremalNumber_le_iff","parent":29950,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/index.html#generalized-extremal-number-le-iff","scope":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers","status":"fully_proved","summary":"Let $n$ and $m$ be natural numbers and let $T$ and $H$ be simple graphs, where $T$ has finitely many vertices. Then the generalized Tur\u00e1n number $\\mathrm{ex}(n, T, H)$ is at most $m$ if and only if every simple graph on the vertex set $\\{0, 1, \\dots, n-1\\}$ that contains no copy of $H$ (a subgraph isomorphic to $H$) has at most $m$ copies of $T$.","title":"Upper bounds for the generalized Tur\u00e1n number"},{"id":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/exists-free-copy-count-eq-generalized-ex","kind":"node","lean":"SimpleGraph.exists_free_copyCount_eq_generalizedExtremalNumber","parent":29950,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/generalized-tur-n-numbers/index.html#exists-free-copy-count-eq-generalized-ex","scope":"GraphTheory/extremal-graph-theory/generalized-tur-n-numbers","status":"fully_proved","summary":"Let $n$ be a natural number and let $T$ and $H$ be simple graphs, where $T$ has finitely many vertices. If some simple graph on the vertex set $\\{0, 1, \\dots, n-1\\}$ contains no copy of $H$ (a subgraph isomorphic to $H$), then some such graph $G$ has exactly $\\mathrm{ex}(n, T, H)$ copies of $T$, where $\\mathrm{ex}(n, T, H)$ is the generalized Tur\u00e1n number.","title":"The generalized Tur\u00e1n number is attained"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/graph-saturation.json","id":"GraphTheory/extremal-graph-theory/graph-saturation","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","title":"Graph saturation"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/is-saturating-nonedge","kind":"node","lean":"SimpleGraph.IsSaturatingNonedge","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#is-saturating-nonedge","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $F$ and $G$ be simple graphs. A pair of vertices $x, y$ of $G$ is an $F$-saturating non-edge of $G$ if $x \\ne y$, $x$ and $y$ are not adjacent in $G$, and the graph $G + xy$ obtained from $G$ by adding the edge $xy$ contains a copy of $F$ (a subgraph isomorphic to $F$, not necessarily induced).","title":"Saturating non-edge"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/new-copy-of-free","kind":"node","lean":"SimpleGraph.IsSaturatingNonedge.newCopy_of_free","parent":29956,"reading_title":{"format":"inline-parts-v1","parts":[["text","A saturating non-edge of an "],["math","\\(F\\)"],["text","-free graph creates a copy of "],["math","\\(F\\)"]],"source":"A saturating non-edge of an $F$-free graph creates a copy of $F$"},"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#new-copy-of-free","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $F$ and $G$ be simple graphs, and let $xy$ be an $F$-saturating non-edge of $G$. If $G$ is $F$-free, that is, $G$ contains no copy of $F$, then the graph $G + xy$ obtained from $G$ by adding the edge $xy$ contains a copy of $F$, and $G$ does not.","title":"A saturating non-edge of an $F$-free graph creates a copy of $F$"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/saturating-nonedge-graph","kind":"node","lean":"SimpleGraph.saturatingNonedgeGraph","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#saturating-nonedge-graph","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $F$ and $G$ be simple graphs. The graph of $F$-saturating non-edges of $G$ is the simple graph on the vertices of $G$ whose edges are exactly the $F$-saturating non-edges of $G$.","title":"Graph of saturating non-edges"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/num-saturating-nonedges","kind":"node","lean":"SimpleGraph.numSaturatingNonedges","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#num-saturating-nonedges","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $F$ and $G$ be simple graphs such that $G$ has only finitely many $F$-saturating non-edges. Then $f_F(G)$ is the number of $F$-saturating non-edges of $G$, counted as unordered pairs $\\{x, y\\}$, that is, the number of edges of the graph of $F$-saturating non-edges of $G$.","title":"Number of saturating non-edges"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/is-saturated","kind":"node","lean":"SimpleGraph.IsSaturated","parent":29956,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(H\\)"],["text","-saturated graphs"]],"source":"$H$-saturated graphs"},"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#is-saturated","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $H$ be a simple graph. A simple graph $G$ is $H$-saturated if $G$ is $H$-free (it contains no copy of $H$ as a subgraph) and is maximal with this property: every $H$-free simple graph $G'$ on the same vertex set as $G$ with $G \\subseteq G'$ equals $G$.","title":"$H$-saturated graphs"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/is-saturated-iff-forall-gt","kind":"node","lean":"SimpleGraph.isSaturated_iff_forall_gt","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#is-saturated-iff-forall-gt","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $H$ be a simple graph. A simple graph $G$ is $H$-saturated if and only if $G$ is $H$-free and every simple graph $G'$ on the same vertex set with $G \\subsetneq G'$ contains a copy of $H$.","title":"Saturation via strict supergraphs"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/is-saturated-iff-missing-edge","kind":"node","lean":"SimpleGraph.isSaturated_iff_missing_edge","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#is-saturated-iff-missing-edge","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $H$ be a simple graph. A simple graph $G$ is $H$-saturated if and only if $G$ is $H$-free and, for all distinct vertices $s$ and $t$ of $G$ that are not adjacent in $G$, the graph $G + st$ obtained from $G$ by adding the edge $st$ contains a copy of $H$.","title":"Saturation via missing edges"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/is-saturated-top","kind":"node","lean":"SimpleGraph.isSaturated_top","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#is-saturated-top","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $H$ be a simple graph. The complete graph on any vertex set is $H$-saturated if and only if it is $H$-free.","title":"Saturation of complete graphs"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/sat-counts","kind":"node","lean":"SimpleGraph.satCounts","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#sat-counts","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $H$ be a simple graph and $n$ a natural number. The set of saturated edge counts of $H$ on $n$ vertices is the set of natural numbers $m$ such that some $H$-saturated simple graph on the vertex set $\\{0, 1, \\dots, n - 1\\}$ has exactly $m$ edges.","title":"Edge counts of saturated graphs"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/sat","kind":"node","lean":"SimpleGraph.sat","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#sat","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $H$ be a simple graph and $n$ a natural number. The saturation number $\\operatorname{sat}(n, H)$ is the minimum number of edges of an $H$-saturated simple graph on the vertex set $\\{0, 1, \\dots, n - 1\\}$, and is $0$ when no such graph exists.","title":"Saturation number"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/sat-eq-zero-of-counts-empty","kind":"node","lean":"SimpleGraph.sat_eq_zero_of_counts_empty","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#sat-eq-zero-of-counts-empty","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $H$ be a simple graph, $n$ a natural number, and $\\operatorname{sat}(n, H)$ the saturation number of $H$ on $n$ vertices. If the set of saturated edge counts of $H$ on $n$ vertices is empty, that is, no $H$-saturated simple graph on the vertex set $\\{0, 1, \\dots, n - 1\\}$ exists, then $\\operatorname{sat}(n, H) = 0$.","title":"The saturation number is zero when no saturated graph exists"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/sat-le-card-of-is-saturated","kind":"node","lean":"SimpleGraph.sat_le_card_of_isSaturated","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#sat-le-card-of-is-saturated","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Let $H$ be a simple graph, $n$ a natural number, and $\\operatorname{sat}(n, H)$ the saturation number of $H$ on $n$ vertices. Every $H$-saturated simple graph on the vertex set $\\{0, 1, \\dots, n - 1\\}$ has at least $\\operatorname{sat}(n, H)$ edges.","title":"Saturated graphs bound the saturation number"},{"id":"GraphTheory/extremal-graph-theory/graph-saturation/exists-is-saturated-card-eq-sat","kind":"node","lean":"SimpleGraph.exists_isSaturated_card_eq_sat","parent":29956,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/graph-saturation/index.html#exists-is-saturated-card-eq-sat","scope":"GraphTheory/extremal-graph-theory/graph-saturation","status":"fully_proved","summary":"Attainment: when a saturated graph exists, some saturated graph has exactly `sat(n, H)` edges.","title":"Saturation number is attained by a saturated graph"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs.json","id":"GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs/index.html","scope":"GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs","status":"can_prove","title":"Pentagons in triangle-free graphs"},{"id":"GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs/conjecture","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem24Wanted.conjecture","parent":29970,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs/index.html#conjecture","scope":"GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs","status":"fully_proved","summary":"For every natural number $n$, every graph on $5n$ vertices with no three pairwise adjacent vertices has at most $n^5$ subgraphs isomorphic to the $5$-cycle $C_5$.","title":"Pentagons in triangle-free graphs (Erd\u0151s Problem 24)"},{"id":"GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs/conjecture-holds","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem24Wanted.conjecture_holds","parent":29970,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs/index.html#conjecture-holds","scope":"GraphTheory/extremal-graph-theory/pentagons-in-triangle-free-graphs","status":"can_prove","summary":"For every natural number $n$, every graph on $5n$ vertices with no three pairwise adjacent vertices has at most $n^5$ subgraphs isomorphic to the $5$-cycle $C_5$.","title":"Pentagon bound for triangle-free graphs (five-cycle case of Erd\u0151s Problem 24)"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs.json","id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/index.html","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","status":"can_prove","title":"Tur\u00e1n numbers of bipartite graphs"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/subdivision","kind":"node","lean":"MathlibExt.Combinatorics.ConlonLeeSubdivisionConjectureWanted.subdivision","parent":29973,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\((k-1)\\)"],["text","-subdivision of a graph"]],"source":"The $(k-1)$-subdivision of a graph"},"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/index.html#subdivision","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","status":"fully_proved","summary":"For a simple graph $H$ on the vertex set $\\{0, \\dots, m - 1\\}$ and a natural number $k \\ge 2$, the $(k-1)$-subdivision $H^{k-1}$ is the graph obtained from $H$ by replacing each edge $ij$ with $i < j$ by a path $i, (ij, 0), (ij, 1), \\dots, (ij, k - 2), j$ of length $k$ through $k - 1$ new vertices, the paths of different edges sharing no new vertices. For $k \\le 1$ there are no new vertices and the resulting graph\u2026","title":"The $(k-1)$-subdivision of a graph"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/contains-copy","kind":"node","lean":"MathlibExt.Combinatorics.ConlonLeeSubdivisionConjectureWanted.ContainsCopy","parent":29973,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/index.html#contains-copy","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","status":"fully_proved","summary":"A simple graph $G$ contains a copy of a simple graph $F$ if there is an injective map $f$ from the vertices of $F$ to the vertices of $G$ such that $f(u)$ and $f(v)$ are adjacent in $G$ whenever $u$ and $v$ are adjacent in $F$; the copy need not be induced.","title":"Containing a copy of a graph"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/ex","kind":"node","lean":"MathlibExt.Combinatorics.ConlonLeeSubdivisionConjectureWanted.ex","parent":29973,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/index.html#ex","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","status":"fully_proved","summary":"For a natural number $n$ and a simple graph $F$, the extremal number $\\mathrm{ex}(n, F)$ is the maximum number of edges of a simple graph on the vertex set $\\{0, \\dots, n - 1\\}$ that contains no copy of $F$. It is $0$ when every simple graph on these $n$ vertices contains a copy of $F$.","title":"The extremal number of a graph"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ConlonLeeSubdivisionConjectureWanted.conjecture","parent":29973,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/index.html#conjecture","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","status":"fully_proved","summary":"For a simple graph $H$ and a natural number $k$, write $H^{k-1}$ for the $(k-1)$-subdivision of $H$, and for a simple graph $F$ and a natural number $n$ write $\\mathrm{ex}(n, F)$ for the extremal number of $F$ on $n$ vertices. The conjecture is the statement that for every even natural number $k \\ge 2$, every natural number $m$ and every simple graph $H$ on the vertex set $\\{0, \\dots, m - 1\\}$, there are real\u2026","title":"The Conlon\u2013Lee subdivision conjecture"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ConlonLeeSubdivisionConjectureWanted.conjecture_holds","parent":29973,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/index.html#conjecture-holds","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","status":"can_prove","summary":"For a simple graph $F$ and a natural number $n$, let $\\mathrm{ex}(n, F)$ be the maximum number of edges of a simple graph on $n$ vertices that contains no copy of $F$ (not necessarily induced), or $0$ if there is no such graph. For a simple graph $H$ and a natural number $k \\ge 2$, let $H^{k-1}$ be the graph obtained from $H$ by replacing each edge with a path of length $k$ through $k - 1$ new vertices (internally\u2026","title":"The Conlon\u2013Lee subdivision conjecture holds"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/degenerate-extremal-claim","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem146Wanted.EP146.DegenerateExtremalClaim","parent":29973,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/index.html#degenerate-extremal-claim","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","status":"fully_proved","summary":"The statement that for every finite simple graph $H$ and every natural number $r \\ge 1$, if $H$ is bipartite and $r$-degenerate (every nonempty set of vertices contains a vertex with at most $r$ neighbours in that set), then $\\mathrm{ex}(n; H) = O(n^{2 - 1/r})$ as $n \\to \\infty$, where $\\mathrm{ex}(n; H)$ is the largest number of edges of a simple graph on $n$ vertices containing no copy of $H$.","title":"Erd\u0151s\u2013Simonovits degeneracy conjecture (Erd\u0151s Problem 146)"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/degenerate-extremal-claim-refuted","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem146Wanted.DegenerateExtremalClaim_refuted","parent":29973,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs/index.html#degenerate-extremal-claim-refuted","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-bipartite-graphs","status":"can_prove","summary":"It is not true that for every finite simple graph $H$ and every natural number $r \\ge 1$ such that $H$ is bipartite and $r$-degenerate (every nonempty set of vertices contains a vertex with at most $r$ neighbours in that set), the extremal number $\\mathrm{ex}(n; H)$ (the largest number of edges of a simple graph on $n$ vertices containing no copy of $H$) is $O(n^{2 - 1/r})$ as $n \\to \\infty$.","title":"Counterexample to the Erd\u0151s\u2013Simonovits degeneracy conjecture"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles.json","id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/index.html","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles","status":"can_prove","title":"Tur\u00e1n numbers of cycles"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/is-free-of574-cycles","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem574Wanted.IsFreeOf574Cycles","parent":29981,"reading_title":{"format":"inline-parts-v1","parts":[["text","Graphs without cycles of lengths "],["math","\\(2k-1\\)"],["text"," and "],["math","\\(2k\\)"]],"source":"Graphs without cycles of lengths $2k-1$ and $2k$"},"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/index.html#is-free-of574-cycles","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles","status":"fully_proved","summary":"For a natural number $k$, a simple graph $G$ is free of the cycle lengths $2k-1$ and $2k$ when it contains no copy of the cycle graph $C_{2k-1}$ and no copy of the cycle graph $C_{2k}$, where a copy is an injective graph homomorphism, that is, a subgraph that need not be induced.","title":"Graphs without cycles of lengths $2k-1$ and $2k$"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/extremal-number","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem574Wanted.extremalNumber","parent":29981,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extremal number for cycles of lengths "],["math","\\(2k-1\\)"],["text"," and "],["math","\\(2k\\)"]],"source":"Extremal number for cycles of lengths $2k-1$ and $2k$"},"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/index.html#extremal-number","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles","status":"fully_proved","summary":"For natural numbers $k$ and $n$, $\\mathrm{ex}(n; \\{C_{2k-1}, C_{2k}\\})$ is the supremum of the numbers of edges of the simple graphs on the vertex set $\\{0, \\dots, n-1\\}$ that contain no copy of the cycle graph $C_{2k-1}$ and no copy of the cycle graph $C_{2k}$, where the supremum of the empty set is $0$. Here $2k-1$ is computed in the natural numbers, so it is $0$ when $k = 0$.","title":"Extremal number for cycles of lengths $2k-1$ and $2k$"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem574Wanted.conjecture","parent":29981,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/index.html#conjecture","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles","status":"fully_proved","summary":"The statement that for every natural number $k \\ge 2$ the ratio $\\mathrm{ex}(n; \\{C_{2k-1}, C_{2k}\\}) / (n/2)^{1 + 1/k}$ tends to $1$ as $n \\to \\infty$, that is, $\\mathrm{ex}(n; \\{C_{2k-1}, C_{2k}\\}) = (1 + o(1)) (n/2)^{1 + 1/k}$. Here $\\mathrm{ex}(n; \\{C_{2k-1}, C_{2k}\\})$ is the extremal number: the supremum of the numbers of edges of simple graphs on $n$ vertices that contain no copy of the cycle graph\u2026","title":"Erd\u0151s Problem 574"},{"id":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem574Wanted.conjecture_refuted","parent":29981,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles/index.html#conjecture-refuted","scope":"GraphTheory/extremal-graph-theory/tur-n-numbers-of-cycles","status":"can_prove","summary":"It is not the case that for every natural number $k \\ge 2$ the ratio $\\mathrm{ex}(n; \\{C_{2k-1}, C_{2k}\\}) / (n/2)^{1 + 1/k}$ tends to $1$ as $n \\to \\infty$. Here $\\mathrm{ex}(n; \\{C_{2k-1}, C_{2k}\\})$ is the extremal number: the supremum of the numbers of edges of simple graphs on $n$ vertices that contain no copy of the cycle graph $C_{2k-1}$ and no copy of the cycle graph $C_{2k}$.","title":"Erd\u0151s Problem 574 is false"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube.json","id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"can_prove","title":"Tur\u00e1n problems in the hypercube"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-graph","kind":"node","lean":"SimpleGraph.hypercubeGraph","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-graph","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"For a natural number $n$, the $n$-dimensional hypercube $Q_n$ is the simple graph whose vertices are the functions from $\\{0, \\dots, n-1\\}$ to the Booleans $\\{\\mathrm{true}, \\mathrm{false}\\}$, with an edge between two vertices exactly when they differ in exactly one coordinate.","title":"Hypercube graph"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-graph-adj","kind":"node","lean":"SimpleGraph.hypercubeGraph_adj","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-graph-adj","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"For every natural number $n$ and all vertices $x, y$ of the $n$-dimensional hypercube $Q_n$ (functions from $\\{0, \\dots, n-1\\}$ to the Booleans), $x$ and $y$ are adjacent in $Q_n$ if and only if there is exactly one coordinate $i$ with $x_i \\ne y_i$.","title":"Adjacency in the hypercube"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-graph-zero-not-adj","kind":"node","lean":"SimpleGraph.hypercubeGraph_zero_not_adj","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-graph-zero-not-adj","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"For all vertices $x, y$ of the $0$-dimensional hypercube $Q_0$, $x$ and $y$ are not adjacent in $Q_0$.","title":"No two vertices of the zero-dimensional hypercube are adjacent"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-graph-one-adj","kind":"node","lean":"SimpleGraph.hypercubeGraph_one_adj","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-graph-one-adj","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"In the $1$-dimensional hypercube $Q_1$, the constant vertex $\\mathrm{true}$ and the constant vertex $\\mathrm{false}$ are adjacent.","title":"An edge of the one-dimensional hypercube"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-graph-zero-edge-set","kind":"node","lean":"SimpleGraph.hypercubeGraph_zero_edgeSet","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-graph-zero-edge-set","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"The edge set of the $0$-dimensional hypercube $Q_0$ is empty.","title":"The zero-dimensional hypercube has no edges"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-graph-subgraph-edge-set-ncard","kind":"node","lean":"SimpleGraph.hypercubeGraph_subgraph_edgeSet_ncard_le","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-graph-subgraph-edge-set-ncard","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"For every natural number $n$, every subgraph of the $n$-dimensional hypercube $Q_n$ has at most as many edges as $Q_n$.","title":"Subgraphs of the hypercube have at most as many edges"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-turan-number","kind":"node","lean":"SimpleGraph.hypercubeTuranNumber","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-turan-number","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"For a natural number $n$ and a simple graph $H$, let $Q_n$ be the $n$-dimensional hypercube. The hypercube Tur\u00e1n number $\\mathrm{ex}(Q_n, H)$ is the largest number of edges of a subgraph (not necessarily spanning) of $Q_n$ that contains no copy of $H$, that is, no subgraph isomorphic to $H$; it is $0$ when no such subgraph exists.","title":"Hypercube Tur\u00e1n number"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/le-hypercube-turan-number","kind":"node","lean":"SimpleGraph.le_hypercubeTuranNumber","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#le-hypercube-turan-number","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"For every natural number $n$ and every simple graph $H$, let $Q_n$ be the $n$-dimensional hypercube and let $\\mathrm{ex}(Q_n, H)$ be the hypercube Tur\u00e1n number of $H$ in $Q_n$. If $S$ is a subgraph of $Q_n$ that contains no copy of $H$, then $S$ has at most $\\mathrm{ex}(Q_n, H)$ edges.","title":"The hypercube Tur\u00e1n number bounds every free subgraph"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-turan-number-le-edge-set-ncard","kind":"node","lean":"SimpleGraph.hypercubeTuranNumber_le_edgeSet_ncard","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-turan-number-le-edge-set-ncard","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"For every natural number $n$ and every simple graph $H$, writing $Q_n$ for the $n$-dimensional hypercube, the hypercube Tur\u00e1n number $\\mathrm{ex}(Q_n, H)$ is at most the number of edges of $Q_n$.","title":"Upper bound for the hypercube Tur\u00e1n number"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/exists-attains-hypercube-turan-number","kind":"node","lean":"SimpleGraph.exists_attains_hypercubeTuranNumber","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#exists-attains-hypercube-turan-number","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"Let $n$ be a natural number, $H$ a simple graph, $Q_n$ the $n$-dimensional hypercube, and $\\mathrm{ex}(Q_n, H)$ the hypercube Tur\u00e1n number. If $Q_n$ has at least one subgraph (not necessarily spanning) that contains no copy of $H$, then some subgraph $S$ of $Q_n$ contains no copy of $H$ and has exactly $\\mathrm{ex}(Q_n, H)$ edges.","title":"Attainment of the hypercube Tur\u00e1n number"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-turan-number-zero","kind":"node","lean":"SimpleGraph.hypercubeTuranNumber_zero","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-turan-number-zero","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"fully_proved","summary":"For every simple graph $H$, writing $Q_0$ for the $0$-dimensional hypercube, the hypercube Tur\u00e1n number $\\mathrm{ex}(Q_0, H)$ is $0$.","title":"The hypercube Tur\u00e1n number in dimension zero"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-turan-number-cycle-graph-four","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.HypercubeFourCycleFreeWanted.hypercubeTuranNumber_cycleGraph_four_one_to_six","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-turan-number-cycle-graph-four","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"can_prove","summary":"For a natural number $n$, let $Q_n$ be the $n$-dimensional hypercube, let $C_4$ be the cycle on four vertices, and let $\\mathrm{ex}(Q_n, C_4)$ be the largest number of edges of a subgraph of $Q_n$ that contains no copy of $C_4$. Then for $n = 1, 2, 3, 4, 5, 6$ the value of $\\mathrm{ex}(Q_n, C_4)$ is $1$, $3$, $9$, $24$, $56$ and $132$ respectively.","title":"Four-cycle-free subgraphs of the hypercubes of dimensions one to six"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/le-hypercube-turan-number-seven-cycle-gr","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.HypercubeFourCycleFreeWanted.le_hypercubeTuranNumber_seven_cycleGraph_four","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#le-hypercube-turan-number-seven-cycle-gr","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"can_prove","summary":"Let $Q_7$ be the $7$-dimensional hypercube and $C_4$ the cycle on four vertices. The largest number $\\mathrm{ex}(Q_7, C_4)$ of edges of a subgraph of $Q_7$ that contains no copy of $C_4$ is at least $304$.","title":"Lower bound for four-cycle-free subgraphs of the seven-dimensional hypercube"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/hypercube-turan-number-seven-cycle-graph","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.HypercubeFourCycleFreeWanted.hypercubeTuranNumber_seven_cycleGraph_four_le","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#hypercube-turan-number-seven-cycle-graph","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"can_prove","summary":"Let $Q_7$ be the $7$-dimensional hypercube and $C_4$ the cycle on four vertices. The largest number $\\mathrm{ex}(Q_7, C_4)$ of edges of a subgraph of $Q_7$ that contains no copy of $C_4$ is at most $308$.","title":"Upper bound for four-cycle-free subgraphs of the seven-dimensional hypercube"},{"id":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/le-hypercube-turan-number-eight-cycle-gr","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.HypercubeFourCycleFreeWanted.le_hypercubeTuranNumber_eight_cycleGraph_four","parent":29986,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube/index.html#le-hypercube-turan-number-eight-cycle-gr","scope":"GraphTheory/extremal-graph-theory/tur-n-problems-in-the-hypercube","status":"can_prove","summary":"Let $Q_8$ be the $8$-dimensional hypercube and $C_4$ the cycle on four vertices. The largest number $\\mathrm{ex}(Q_8, C_4)$ of edges of a subgraph of $Q_8$ that contains no copy of $C_4$ is at least $682$.","title":"Lower bound for four-cycle-free subgraphs of the eight-dimensional hypercube"},{"data_url":"scopes/GraphTheory/extremal-graph-theory/unique-subgraphs.json","id":"GraphTheory/extremal-graph-theory/unique-subgraphs","kind":"scope","navigation":"module","parent":29921,"result":false,"route":"../roadmap/GraphTheory/extremal-graph-theory/unique-subgraphs/index.html","scope":"GraphTheory/extremal-graph-theory/unique-subgraphs","status":"can_prove","title":"Unique subgraphs"},{"id":"GraphTheory/extremal-graph-theory/unique-subgraphs/is-unique-subgraph","kind":"node","lean":"SimpleGraph.IsUniqueSubgraph","parent":30002,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/unique-subgraphs/index.html#is-unique-subgraph","scope":"GraphTheory/extremal-graph-theory/unique-subgraphs","status":"fully_proved","summary":"Let $G$ and $H$ be graphs on the same vertex set $V$. Then $G$ is a unique subgraph of $H$ if exactly one subgraph of $H$ is isomorphic to $G$. Here a subgraph of $H$ means a spanning subgraph: a graph on $V$ all of whose edges are edges of $H$ (not necessarily induced).","title":"Unique subgraph"},{"id":"GraphTheory/extremal-graph-theory/unique-subgraphs/unique-subgraph-count","kind":"node","lean":"SimpleGraph.uniqueSubgraphCount","parent":30002,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/unique-subgraphs/index.html#unique-subgraph-count","scope":"GraphTheory/extremal-graph-theory/unique-subgraphs","status":"fully_proved","summary":"Let $H$ be a graph on a finite vertex set $V$. The number of unique subgraphs of $H$ is the number of spanning subgraphs $G$ of $H$ (graphs on $V$ all of whose edges are edges of $H$) that are unique subgraphs of $H$, that is, isomorphic to no other spanning subgraph of $H$.","title":"Number of unique subgraphs"},{"id":"GraphTheory/extremal-graph-theory/unique-subgraphs/conjecture","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem426Wanted.conjecture","parent":30002,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/unique-subgraphs/index.html#conjecture","scope":"GraphTheory/extremal-graph-theory/unique-subgraphs","status":"fully_proved","summary":"For a graph $H$ on a finite vertex set, let $u(H)$ be the number of spanning subgraphs of $H$ that are isomorphic to no other spanning subgraph of $H$ (its unique subgraphs). The statement is that there is a real constant $c > 0$ such that for infinitely many $n$ some graph $H$ on $n$ vertices has $u(H) \\ge c \\cdot 2^{\\binom{n}{2}} / n!$.","title":"Many unique subgraphs (Erd\u0151s Problem 426)"},{"id":"GraphTheory/extremal-graph-theory/unique-subgraphs/disproved-form","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem426Wanted.disprovedForm","parent":30002,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/unique-subgraphs/index.html#disproved-form","scope":"GraphTheory/extremal-graph-theory/unique-subgraphs","status":"fully_proved","summary":"For a graph $H$ on a finite vertex set, let $u(H)$ be the number of spanning subgraphs of $H$ that are isomorphic to no other spanning subgraph of $H$. This is the proposition that there is no real constant $c > 0$ such that for infinitely many $n$ some graph $H$ on $n$ vertices has $u(H) \\ge c \\cdot 2^{\\binom{n}{2}} / n!$.","title":"Negation of the Erd\u0151s Problem 426 statement"},{"id":"GraphTheory/extremal-graph-theory/unique-subgraphs/conjecture-refuted","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem426Wanted.conjecture_refuted","parent":30002,"result":true,"route":"../roadmap/GraphTheory/extremal-graph-theory/unique-subgraphs/index.html#conjecture-refuted","scope":"GraphTheory/extremal-graph-theory/unique-subgraphs","status":"can_prove","summary":"For a graph $H$ on a finite vertex set, let $u(H)$ be the number of spanning subgraphs of $H$ that are isomorphic to no other spanning subgraph of $H$ (its unique subgraphs). There is no real constant $c > 0$ such that for infinitely many $n$ some graph $H$ on $n$ vertices has $u(H) \\ge c \\cdot 2^{\\binom{n}{2}} / n!$.","title":"Erd\u0151s Problem 426 is false (Brada\u010d\u2013Christoph)"},{"data_url":"scopes/GraphTheory/graph-ramsey-theory.json","id":"GraphTheory/graph-ramsey-theory","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/graph-ramsey-theory/index.html","scope":"GraphTheory/graph-ramsey-theory","status":"can_prove","summary":"Ramsey numbers of graphs, their known small values, and Ramsey\u2013Tur\u00e1n problems.","title":"Ramsey theory for graphs"},{"data_url":"scopes/GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers.json","id":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers","kind":"scope","navigation":"module","parent":30008,"result":false,"route":"../roadmap/GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/index.html","scope":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers","status":"can_prove","title":"Generalized Ramsey numbers"},{"id":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/has-four-five-colouring","kind":"node","lean":"MathlibExt.Combinatorics.FourCliqueFiveColoursErdosProblem136Wanted.HasFourFiveColouring","parent":30009,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\((4, 5)\\)"],["text","-colourings of "],["math","\\(K_n\\)"]],"source":"$(4, 5)$-colourings of $K_n$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/index.html#has-four-five-colouring","scope":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the complete graph $K_n$ on the vertices $0, 1, \\dots, n - 1$ has a $(4, 5)$-colouring with $k$ colours if there is a map $c$ assigning to every ordered pair $(x, y)$ of vertices one of $k$ colours, with $c(x, y) = c(y, x)$ for all $x, y$, such that for any four distinct vertices the six edges between them receive at least five distinct colours.","title":"$(4, 5)$-colourings of $K_n$"},{"id":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/four-five-colour-number","kind":"node","lean":"MathlibExt.Combinatorics.FourCliqueFiveColoursErdosProblem136Wanted.fourFiveColourNumber","parent":30009,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\((4, 5)\\)"],["text","-colouring number "],["math","\\(f(n)\\)"]],"source":"The $(4, 5)$-colouring number $f(n)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/index.html#four-five-colour-number","scope":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers","status":"fully_proved","summary":"For a natural number $n$, $f(n)$ is the least natural number $k$ such that $K_n$ has a $(4, 5)$-colouring with $k$ colours; such a $k$ always exists.","title":"The $(4, 5)$-colouring number $f(n)$"},{"id":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/four-five-colour-number-asymptotic","kind":"node","lean":"MathlibExt.Combinatorics.FourCliqueFiveColoursErdosProblem136Wanted.fourFiveColourNumberAsymptotic","parent":30009,"reading_title":{"format":"inline-parts-v1","parts":[["text","The asymptotic of the "],["math","\\((4, 5)\\)"],["text","-colouring number (Erd\u0151s Problem 136)"]],"source":"The asymptotic of the $(4, 5)$-colouring number (Erd\u0151s Problem 136)"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/index.html#four-five-colour-number-asymptotic","scope":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers","status":"fully_proved","summary":"Let $f(n)$ be the least number of colours in a $(4, 5)$-colouring of $K_n$. This is the proposition that $f(n)/n$ tends to $5/6$ as $n \\to \\infty$; equivalently, $f(n) = (5/6 + o(1))\\, n$.","title":"The asymptotic of the $(4, 5)$-colouring number (Erd\u0151s Problem 136)"},{"id":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/four-five-colour-number-asymptotic-holds","kind":"node","lean":"MathlibExt.Combinatorics.FourCliqueFiveColoursErdosProblem136Wanted.fourFiveColourNumberAsymptotic_holds","parent":30009,"reading_title":{"format":"inline-parts-v1","parts":[["text","Erd\u0151s Problem 136: the "],["math","\\((4, 5)\\)"],["text","-colouring number of "],["math","\\(K_n\\)"],["text"," is asymptotically "],["math","\\(5n/6\\)"]],"source":"Erd\u0151s Problem 136: the $(4, 5)$-colouring number of $K_n$ is asymptotically $5n/6$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers/index.html#four-five-colour-number-asymptotic-holds","scope":"GraphTheory/graph-ramsey-theory/generalized-ramsey-numbers","status":"can_prove","summary":"Let $f(n)$ be the least number of colours in a $(4, 5)$-colouring of $K_n$, that is, a symmetric colouring of the pairs of vertices of $K_n$ in which the six edges of every $K_4$ receive at least five distinct colours. Then $f(n)/n \\to 5/6$ as $n \\to \\infty$.","title":"Erd\u0151s Problem 136: the $(4, 5)$-colouring number of $K_n$ is asymptotically $5n/6$"},{"data_url":"scopes/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers.json","id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","kind":"scope","navigation":"module","parent":30008,"result":false,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"can_prove","title":"Graph Ramsey numbers"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/classical-ramsey","kind":"node","lean":"SimpleGraph.classicalRamsey","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#classical-ramsey","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"For natural numbers $s$, $t$ and $N$, the classical Ramsey property for $(s, t)$ holds at $N$ when every simple graph $G$ on $N$ vertices contains the complete graph $K_s$ as a subgraph, or its complement $\\overline{G}$ contains $K_t$ as a subgraph.","title":"The Ramsey property for cliques"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/graph-ramsey","kind":"node","lean":"SimpleGraph.graphRamsey","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#graph-ramsey","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"For natural numbers $s$ and $t$, the Ramsey number $R(s, t)$ is the least $N$ such that every simple graph on $N$ vertices contains the complete graph $K_s$ as a subgraph or has a complement containing $K_t$ as a subgraph, and $0$ if there is no such $N$.","title":"Classical Ramsey numbers"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/graph-ramsey-eq-s-inf","kind":"node","lean":"SimpleGraph.graphRamsey_eq_sInf","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#graph-ramsey-eq-s-inf","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"For all natural numbers $s$ and $t$, the Ramsey number $R(s, t)$ equals the infimum in $\\mathbb{N}$ (where the empty set has infimum $0$) of the set of $N$ such that every simple graph on $N$ vertices contains the complete graph $K_s$ as a subgraph or has a complement containing $K_t$ as a subgraph.","title":"Classical Ramsey numbers as an infimum"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/graph-pair-ramsey","kind":"node","lean":"SimpleGraph.graphPairRamsey","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#graph-pair-ramsey","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"For finite simple graphs $G$ and $H$, the Ramsey number $R(G, H)$ is the least $n$ such that every simple graph $C$ on $n$ vertices contains a copy of $G$ as a subgraph or has a complement containing a copy of $H$ as a subgraph, and $0$ if there is no such $n$.","title":"Ramsey number of a pair of graphs"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/graph-pair-ramsey-eq-s-inf","kind":"node","lean":"SimpleGraph.graphPairRamsey_eq_sInf","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#graph-pair-ramsey-eq-s-inf","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"For finite simple graphs $G$ and $H$, the Ramsey number $R(G, H)$ equals the infimum in $\\mathbb{N}$ (where the empty set has infimum $0$) of the set of $n$ such that every simple graph $C$ on $n$ vertices contains a copy of $G$ as a subgraph or has a complement containing a copy of $H$ as a subgraph.","title":"The pair Ramsey number as an infimum"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/diagonal-graph-ramsey","kind":"node","lean":"SimpleGraph.diagonalGraphRamsey","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#diagonal-graph-ramsey","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"For a finite simple graph $G$, the diagonal Ramsey number of $G$ is $R(G, G)$: the least $n$ such that for every simple graph $C$ on $n$ vertices, $C$ or its complement contains a copy of $G$ as a subgraph, and $0$ if there is no such $n$.","title":"Diagonal Ramsey number of a graph"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/is-ramsey-size-linear","kind":"node","lean":"SimpleGraph.IsRamseySizeLinear","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#is-ramsey-size-linear","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"Let $G$ be a finite simple graph. For a finite simple graph $H$, write $R(G, H)$ for the least $m$ such that every simple graph on $m$ vertices contains a copy of $G$ or has a complement containing a copy of $H$ ($0$ if there is no such $m$), and write $e(H)$ for the number of edges of $H$. Then $G$ is Ramsey size linear if there is a real constant $c > 0$ such that for every $n$ and every simple graph $H$ on $n$\u2026","title":"Ramsey size linear graphs"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem570Wanted.conjecture","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#conjecture","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"The statement that for every natural number $k \\ge 3$ there is an $M$ such that for every $m \\ge M$ and every finite simple graph $H$ with exactly $m$ edges and no isolated vertices, the Ramsey number $R(C_k, H)$ of the cycle $C_k$ on $k$ vertices and $H$ is at most $2m + \\lfloor (k-1)/2 \\rfloor$.","title":"Erd\u0151s Problem 570 (cycle versus graph Ramsey bound)"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem570Wanted.conjecture_holds","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#conjecture-holds","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"can_prove","summary":"For every natural number $k \\ge 3$ there is an $M$ such that for every $m \\ge M$ and every finite simple graph $H$ with exactly $m$ edges and no isolated vertices, the Ramsey number $R(C_k, H)$ of the cycle $C_k$ on $k$ vertices and $H$ is at most $2m + \\lfloor (k-1)/2 \\rfloor$.","title":"Resolution of Erd\u0151s Problem 570"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/conjecture-2","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem800Wanted.conjecture","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#conjecture-2","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"The proposition asserts that there is a real number $C > 0$ such that for every natural number $n$ and every simple graph $G$ on $n$ vertices in which no two adjacent vertices both have degree at least $3$, the diagonal Ramsey number $R(G, G)$ is at most $Cn$.","title":"Erd\u0151s Problem 800 on linear Ramsey numbers"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/conjecture-holds-2","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem800Wanted.conjecture_holds","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#conjecture-holds-2","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"can_prove","summary":"There is a real number $C > 0$ such that for every natural number $n$ and every simple graph $G$ on $n$ vertices in which no two adjacent vertices both have degree at least $3$, the diagonal Ramsey number $R(G, G)$ is at most $Cn$.","title":"Alon's theorem on linear Ramsey numbers"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/triangle-goodness-superlinear-threshold","kind":"node","lean":"MathlibExt.Combinatorics.TriangleGoodSparseConnectedGraphsErdosProblem1182Wanted.triangleGoodnessSuperlinearThreshold","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#triangle-goodness-superlinear-threshold","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"fully_proved","summary":"The statement that for every real number $C$, for all sufficiently large natural numbers $n$, every connected graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ with at most $Cn$ edges has $R(K_3, G) = 2n - 1$. Here $R(K_3, G)$ is the Ramsey number of the pair: the least $N$ such that every graph on $N$ vertices contains a triangle or has a complement containing a copy of $G$.","title":"Superlinear threshold for triangle-good graphs (Erd\u0151s Problem 1182)"},{"id":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/triangle-goodness-superlinear-threshold-2","kind":"node","lean":"MathlibExt.Combinatorics.TriangleGoodSparseConnectedGraphsErdosProblem1182Wanted.triangleGoodnessSuperlinearThreshold_refuted","parent":30014,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/graph-ramsey-numbers/index.html#triangle-goodness-superlinear-threshold-2","scope":"GraphTheory/graph-ramsey-theory/graph-ramsey-numbers","status":"can_prove","summary":"It is not true that for every real number $C$, for all sufficiently large natural numbers $n$, every connected graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ with at most $Cn$ edges has $R(K_3, G) = 2n - 1$. Equivalently, there is a real number $C$ such that for infinitely many $n$ some connected graph $G$ on $\\{0, \\dots, n - 1\\}$ with at most $Cn$ edges has $R(K_3, G) \\ne 2n - 1$. Here $R(K_3, G)$ is the Ramsey\u2026","title":"No superlinear threshold for triangle-good graphs (Erd\u0151s Problem 1182)"},{"data_url":"scopes/GraphTheory/graph-ramsey-theory/monochromatic-clique-packings.json","id":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings","kind":"scope","navigation":"module","parent":30008,"result":false,"route":"../roadmap/GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/index.html","scope":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings","status":"can_prove","title":"Monochromatic clique packings"},{"id":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/is-monochromatic-clique-packing","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticCliquePartitionLeftoverErdosProblem1015Wanted.IsMonochromaticCliquePacking","parent":30028,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/index.html#is-monochromatic-clique-packing","scope":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers, and view a simple graph $G$ on the vertex set $\\{0, 1, \\dots, n - 1\\}$ as a two-colouring of the edges of the complete graph $K_n$: the edges of $G$ are red and the edges of its complement $G^c$ are blue. A finite family $P$ of sets of vertices is a monochromatic $k$-clique packing for $G$ when the sets in $P$ are pairwise disjoint and each of them is a $k$-clique of $G$ or a\u2026","title":"Monochromatic clique packings"},{"id":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/leftover-value","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticCliquePartitionLeftoverErdosProblem1015Wanted.leftoverValue","parent":30028,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/index.html#leftover-value","scope":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings","status":"fully_proved","summary":"For natural numbers $k$ and $n$, let $r = R(k, k - 1)$ be the classical two-colour Ramsey number. The leftover value is $r - 1 + \\big((n + 1 - r) \\bmod k\\big)$, where subtraction is truncated at $0$ (in particular $n + 1 - r$ is read as $0$ when $n + 1 \\le r$) and $a \\bmod 0 = a$.","title":"Burr\u2013Erd\u0151s\u2013Spencer leftover value"},{"id":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/monochromatic-clique-partition-leftover","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticCliquePartitionLeftoverErdosProblem1015Wanted.monochromaticCliquePartitionLeftover","parent":30028,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/index.html#monochromatic-clique-partition-leftover","scope":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings","status":"fully_proved","summary":"This is the statement that for every natural number $k \\ge 2$ there is $n_0$ such that for every $n \\ge n_0$ the following holds. Let $r = R(k, k - 1)$ be the classical two-colour Ramsey number and let $L = r - 1 + \\big((n + 1 - r) \\bmod k\\big)$, where $n + 1 - r$ is read as $0$ if it is negative. View a simple graph $G$ on $n$ vertices as a two-colouring of the edges of $K_n$, with the edges of $G$ red and those\u2026","title":"Erd\u0151s Problem 1015 in the Burr\u2013Erd\u0151s\u2013Spencer form"},{"id":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/monochromatic-clique-partition-leftover-2","kind":"node","lean":"MathlibExt.Combinatorics.MonochromaticCliquePartitionLeftoverErdosProblem1015Wanted.monochromaticCliquePartitionLeftover_holds","parent":30028,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/monochromatic-clique-packings/index.html#monochromatic-clique-partition-leftover-2","scope":"GraphTheory/graph-ramsey-theory/monochromatic-clique-packings","status":"can_prove","summary":"For every natural number $k \\ge 2$ there is $n_0$ such that for every $n \\ge n_0$ the following holds. Let $r = R(k, k - 1)$ be the classical two-colour Ramsey number and let $L = r - 1 + \\big((n + 1 - r) \\bmod k\\big)$, where $n + 1 - r$ is read as $0$ if it is negative. View a simple graph $G$ on $n$ vertices as a two-colouring of the edges of $K_n$, with the edges of $G$ red and those of its complement $G^c$\u2026","title":"Burr\u2013Erd\u0151s\u2013Spencer theorem on monochromatic clique packings"},{"data_url":"scopes/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers.json","id":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","kind":"scope","navigation":"module","parent":30008,"result":false,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/index.html","scope":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","status":"can_prove","title":"Ordered Ramsey numbers"},{"id":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/is-ordered-embedding","kind":"node","lean":"MathlibExt.Combinatorics.OrderRobustOrderedRamseyWanted.IsOrderedEmbedding","parent":30033,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/index.html#is-ordered-embedding","scope":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","status":"fully_proved","summary":"Let $\\lt _H$ be a strict total order on $\\{0, \\dots, n - 1\\}$. A map $f \\colon \\{0, \\dots, n - 1\\} \\to \\{0, \\dots, N - 1\\}$ is an ordered embedding when it is injective and $a \\lt _H b$ implies $f(a) < f(b)$ for all $a$ and $b$, where $\\{0, \\dots, N - 1\\}$ carries its natural order.","title":"Ordered embedding"},{"id":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/has-red-ordered-copy","kind":"node","lean":"MathlibExt.Combinatorics.OrderRobustOrderedRamseyWanted.HasRedOrderedCopy","parent":30033,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/index.html#has-red-ordered-copy","scope":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","status":"fully_proved","summary":"Let $H$ be a simple graph on $\\{0, \\dots, n - 1\\}$ with a strict total order $\\lt _H$ on its vertices, and let $G$ be a simple graph on $\\{0, \\dots, N - 1\\}$, the red graph. Then $G$ contains a red ordered copy of $(H, \\lt _H)$ when there is an ordered embedding $f$ of $(\\{0, \\dots, n - 1\\}, \\lt _H)$ into $\\{0, \\dots, N - 1\\}$ such that $f(a)$ and $f(b)$ are adjacent in $G$ whenever $a$ and $b$ are adjacent in $H$.\u2026","title":"Red ordered copy"},{"id":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/has-blue-ordered-copy","kind":"node","lean":"MathlibExt.Combinatorics.OrderRobustOrderedRamseyWanted.HasBlueOrderedCopy","parent":30033,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/index.html#has-blue-ordered-copy","scope":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","status":"fully_proved","summary":"Let $H$ be a simple graph on $\\{0, \\dots, n - 1\\}$ with a strict total order $\\lt _H$ on its vertices, and let $G$ be a simple graph on $\\{0, \\dots, N - 1\\}$, the red graph. Then $G$ admits a blue ordered copy of $(H, \\lt _H)$ when there is an ordered embedding $f$ of $(\\{0, \\dots, n - 1\\}, \\lt _H)$ into $\\{0, \\dots, N - 1\\}$ such that $f(a)$ and $f(b)$ are not adjacent in $G$ whenever $a$ and $b$ are adjacent in\u2026","title":"Blue ordered copy"},{"id":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/ordered-ramsey-prop","kind":"node","lean":"MathlibExt.Combinatorics.OrderRobustOrderedRamseyWanted.OrderedRamseyProp","parent":30033,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/index.html#ordered-ramsey-prop","scope":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","status":"fully_proved","summary":"Let $H$ be a simple graph on $\\{0, \\dots, n - 1\\}$ with a strict total order $\\lt _H$ on its vertices. A natural number $N$ has the ordered Ramsey property for $(H, \\lt _H)$ when every simple graph $G$ on $\\{0, \\dots, N - 1\\}$ contains a red ordered copy of $(H, \\lt _H)$ or admits a blue ordered copy of $(H, \\lt _H)$. Equivalently, every red/blue colouring of the edges of the complete graph on $\\{0, \\dots, N - 1\\}$\u2026","title":"Ordered Ramsey property"},{"id":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/ordered-ramsey-number","kind":"node","lean":"MathlibExt.Combinatorics.OrderRobustOrderedRamseyWanted.orderedRamseyNumber","parent":30033,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/index.html#ordered-ramsey-number","scope":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","status":"fully_proved","summary":"For a simple graph $H$ on $\\{0, \\dots, n - 1\\}$ with a strict total order $\\lt _H$ on its vertices, the ordered Ramsey number $r_<(H, \\lt _H)$ is the least natural number $N$ with the ordered Ramsey property for $(H, \\lt _H)$, that is, such that every red/blue colouring of the edges of the complete graph on $\\{0, \\dots, N - 1\\}$ with its natural order contains a monochromatic ordered copy of $(H, \\lt _H)$. It is\u2026","title":"Ordered Ramsey number"},{"id":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.OrderRobustOrderedRamseyWanted.conjecture","parent":30033,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/index.html#conjecture","scope":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","status":"fully_proved","summary":"There are a real number $c > 0$ and a natural number $N_0$ such that for every $n \\ge N_0$ there is a $100$-regular simple graph $H$ on $\\{0, \\dots, n - 1\\}$ such that $r_<(H, \\lt _H) \\ge n^{1+c}$ for every strict total order $\\lt _H$ on its vertices, where $r_<(H, \\lt _H)$ is the ordered Ramsey number of $H$ with the order $\\lt _H$.","title":"Order-robust ordered Ramsey conjecture for 100-regular graphs"},{"id":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.OrderRobustOrderedRamseyWanted.conjecture_holds","parent":30033,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers/index.html#conjecture-holds","scope":"GraphTheory/graph-ramsey-theory/ordered-ramsey-numbers","status":"can_prove","summary":"There are a real number $c > 0$ and a natural number $N_0$ such that for every $n \\ge N_0$ there is a $100$-regular simple graph $H$ on $\\{0, \\dots, n - 1\\}$ such that $r_<(H, \\lt _H) \\ge n^{1+c}$ for every strict total order $\\lt _H$ on its vertices. Here $r_<(H, \\lt _H)$ is the least $N$ such that every red/blue colouring of the edges of the complete graph on $\\{0, \\dots, N - 1\\}$ with its natural order contains\u2026","title":"The order-robust ordered Ramsey conjecture holds"},{"data_url":"scopes/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle.json","id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","kind":"scope","navigation":"module","parent":30008,"result":false,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"can_prove","title":"Ramsey graphs for the triangle"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/is-triangle-ramsey","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRamseyMinimalThetaWanted.IsTriangleRamsey","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#is-triangle-ramsey","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"A graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ is $r$-Ramsey for the triangle when every colouring $c$ of the ordered pairs of vertices with $r$ colours that is symmetric ($c(a, b) = c(b, a)$) admits three distinct vertices $u, v, w$ that form a triangle in $G$ with $c(u, v) = c(v, w) = c(w, u)$.","title":"Ramsey graphs for the triangle"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/is-triangle-ramsey-delete-vertex","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRamseyMinimalThetaWanted.IsTriangleRamseyDeleteVertex","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#is-triangle-ramsey-delete-vertex","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"Let $G$ be a graph on the vertex set $\\{0, \\dots, n - 1\\}$, $r$ a natural number and $x$ a vertex. The property holds when every colouring $c$ of the ordered pairs of vertices with $r$ colours that is symmetric ($c(a, b) = c(b, a)$) admits three distinct vertices $u, v, w$, all different from $x$, that form a triangle in $G$ with $c(u, v) = c(v, w) = c(w, u)$.","title":"Ramsey property for the triangle after deleting a vertex"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/is-ramsey-minimal","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRamseyMinimalThetaWanted.IsRamseyMinimal","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#is-ramsey-minimal","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"A graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ is $r$-Ramsey-minimal for the triangle when $G$ is $r$-Ramsey for the triangle, no proper spanning subgraph of $G$ (a graph on the same vertices whose edges are edges of $G$, missing at least one of them) is $r$-Ramsey for the triangle, and for every vertex $x$ it is false that every symmetric colouring of the ordered pairs of vertices with $r$ colours admits a\u2026","title":"Ramsey-minimal graphs for the triangle"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/has-min-degree","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRamseyMinimalThetaWanted.HasMinDegree","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#has-min-degree","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"A graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ has minimum degree $d$ when every vertex of $G$ has at least $d$ neighbours and some vertex of $G$ has exactly $d$ neighbours; in particular $n \\ge 1$.","title":"Minimum degree of a finite graph"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/s-ramsey","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRamseyMinimalThetaWanted.sRamsey","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#s-ramsey","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"For a natural number $r$, $s_r(K_3)$ is the least natural number $d$ such that, for some $n$, some graph on the vertex set $\\{0, \\dots, n - 1\\}$ is $r$-Ramsey-minimal for the triangle and has minimum degree $d$; if no such graph exists, $s_r(K_3) = 0$.","title":"Least minimum degree of Ramsey-minimal graphs for the triangle"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRamseyMinimalThetaWanted.conjecture","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#conjecture","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"The statement that $s_r(K_3) = \\Theta(r^2 \\log r)$ as the natural number $r$ tends to infinity, where $s_r(K_3)$ is the least minimum degree of a graph that is $r$-Ramsey-minimal for the triangle ($0$ if there is none) and $\\log$ is the natural logarithm: there are constants $c, C > 0$ such that $c \\, r^2 \\log r \\le s_r(K_3) \\le C \\, r^2 \\log r$ for all sufficiently large $r$.","title":"Minimum-degree conjecture for minimal Ramsey graphs of the triangle"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.TriangleRamseyMinimalThetaWanted.conjecture_holds","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#conjecture-holds","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"can_prove","summary":"Let $s_r(K_3)$ be the least minimum degree of a graph that is $r$-Ramsey-minimal for the triangle, with $s_r(K_3) = 0$ if there is no such graph, as defined for the conjecture. Then $s_r(K_3) = \\Theta(r^2 \\log r)$ as the natural number $r$ tends to infinity, where $\\log$ is the natural logarithm: there are constants $c, C > 0$ such that $c \\, r^2 \\log r \\le s_r(K_3) \\le C \\, r^2 \\log r$ for all sufficiently large\u2026","title":"Minimum degree of minimal Ramsey graphs for the triangle (Guo\u2013Warnke)"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/is-triangle-tree","kind":"node","lean":"MathlibExt.Combinatorics.TriangleTreeLocalRamseyWanted.IsTriangleTree","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#is-triangle-tree","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"The trees of triangles on a vertex set $V$ are the graphs on $V$ built recursively as follows. A graph whose edges are exactly the three edges of a triangle on distinct vertices $a, b, c$ is a tree of triangles. If $H_0$ is a tree of triangles, $xy$ is an edge of $H_0$, and $z$ is a vertex different from $x$ and $y$ with no neighbours in $H_0$, then the graph obtained from $H_0$ by joining $z$ to $x$ and to $y$ is\u2026","title":"Trees of triangles"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/ramseys2-triangle","kind":"node","lean":"MathlibExt.Combinatorics.TriangleTreeLocalRamseyWanted.Ramseys2Triangle","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#ramseys2-triangle","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"A graph $G$ arrows the triangle in two colours, written $G \\to_2 K_3$, if every colouring of the edges of $G$ with two colours contains a monochromatic triangle: three distinct pairwise adjacent vertices whose three edges all have the same colour.","title":"Graphs that arrow the triangle"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/is-locally-triangle-tree","kind":"node","lean":"MathlibExt.Combinatorics.TriangleTreeLocalRamseyWanted.IsLocallyTriangleTree","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#is-locally-triangle-tree","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"A graph $G$ on a vertex set $V$ is locally a tree of triangles on $\\ell$ vertices if for every set $S$ of exactly $\\ell$ vertices there is a tree of triangles on $V$ that contains every edge of $G$ with both endpoints in $S$.","title":"Graphs that are locally trees of triangles"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/conjecture-2","kind":"node","lean":"MathlibExt.Combinatorics.TriangleTreeLocalRamseyWanted.conjecture","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#conjecture-2","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"fully_proved","summary":"This is the statement that for every natural number $\\ell$ there is a finite graph $G$ with at least $\\ell$ vertices such that $G \\to_2 K_3$ (every colouring of the edges of $G$ with two colours contains a monochromatic triangle) and $G$ is locally a tree of triangles on $\\ell$ vertices: for every set $S$ of exactly $\\ell$ vertices of $G$, some tree of triangles on the vertex set of $G$ contains every edge of $G$\u2026","title":"R\u00f6dl's question on Ramsey graphs that are locally trees of triangles"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/conjecture-holds-2","kind":"node","lean":"MathlibExt.Combinatorics.TriangleTreeLocalRamseyWanted.conjecture_holds","parent":30041,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle/index.html#conjecture-holds-2","scope":"GraphTheory/graph-ramsey-theory/ramsey-graphs-for-the-triangle","status":"can_prove","summary":"For every natural number $\\ell$ there is a finite graph $G$ with at least $\\ell$ vertices such that $G \\to_2 K_3$, that is, every colouring of the edges of $G$ with two colours contains a monochromatic triangle, and such that for every set $S$ of exactly $\\ell$ vertices of $G$, some tree of triangles on the vertex set of $G$ contains every edge of $G$ with both endpoints in $S$.","title":"Ramsey graphs for the triangle that are locally trees of triangles"},{"data_url":"scopes/GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory.json","id":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory","kind":"scope","navigation":"module","parent":30008,"result":false,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/index.html","scope":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory","status":"can_prove","title":"Ramsey\u2013Tur\u00e1n theory"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.K5FreeTriangleFreeSubsetWanted.conjecture","parent":30054,"reading_title":{"format":"inline-parts-v1","parts":[["text","EP-533: linear triangle-free sets in dense "],["math","\\(K_5\\)"],["text","-free graphs"]],"source":"EP-533: linear triangle-free sets in dense $K_5$-free graphs"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/index.html#conjecture","scope":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory","status":"fully_proved","summary":"The conjecture states that for every real $\\delta > 0$ there is a real $c > 0$ such that, for all sufficiently large $n$, every $K_5$-free simple graph on $n$ vertices with at least $\\delta n^2$ edges has a triangle-free set of at least $c n$ vertices (a set of vertices no three of which are pairwise adjacent).","title":"EP-533: linear triangle-free sets in dense $K_5$-free graphs"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.K5FreeTriangleFreeSubsetWanted.conjecture_refuted","parent":30054,"reading_title":{"format":"inline-parts-v1","parts":[["text","EP-533 is false: dense "],["math","\\(K_5\\)"],["text","-free graphs without linear triangle-free sets"]],"source":"EP-533 is false: dense $K_5$-free graphs without linear triangle-free sets"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/index.html#conjecture-refuted","scope":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory","status":"can_prove","summary":"There is a real $\\delta > 0$ such that for every real $c > 0$ there are infinitely many $n$ for which some $K_5$-free simple graph on $n$ vertices has at least $\\delta n^2$ edges and yet has no triangle-free set of at least $c n$ vertices (a set of vertices no three of which are pairwise adjacent). In other words, the EP-533 conjecture fails.","title":"EP-533 is false: dense $K_5$-free graphs without linear triangle-free sets"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/conjecture-2","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem22Wanted.conjecture","parent":30054,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ramsey\u2013Tur\u00e1n problem for "],["math","\\(K_4\\)"],["text"," (Erd\u0151s Problem 22)"]],"source":"Ramsey\u2013Tur\u00e1n problem for $K_4$ (Erd\u0151s Problem 22)"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/index.html#conjecture-2","scope":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory","status":"fully_proved","summary":"For a graph $G$ write $\\alpha(G)$ for its independence number and $e(G)$ for its number of edges, and call $G$ $K_4$-free if it has no four pairwise adjacent vertices. The statement is the conjunction of the following four assertions.","title":"Ramsey\u2013Tur\u00e1n problem for $K_4$ (Erd\u0151s Problem 22)"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/test-bot","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem22Wanted.test_bot","parent":30054,"reading_title":{"format":"inline-parts-v1","parts":[["text","The empty graph meets the Ramsey\u2013Tur\u00e1n edge bound for "],["math","\\(K_4\\)"]],"source":"The empty graph meets the Ramsey\u2013Tur\u00e1n edge bound for $K_4$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/index.html#test-bot","scope":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory","status":"fully_proved","summary":"For every natural number $n$ and every real $\\varepsilon > 0$, the empty graph on $n$ vertices has no four pairwise adjacent vertices, and its number of edges (namely $0$) is at most $(1/8 + \\varepsilon) n^2$.","title":"The empty graph meets the Ramsey\u2013Tur\u00e1n edge bound for $K_4$"},{"id":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/conjecture-holds","kind":"node","lean":"MathlibExt.GraphTheory.ErdosProblem22Wanted.conjecture_holds","parent":30054,"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory/index.html#conjecture-holds","scope":"GraphTheory/graph-ramsey-theory/ramsey-tur-n-theory","status":"can_prove","summary":"For a graph $G$ write $\\alpha(G)$ for its independence number and $e(G)$ for its number of edges, and call $G$ $K_4$-free if it has no four pairwise adjacent vertices. All four of the following assertions hold.","title":"Resolution of Erd\u0151s Problem 22 (Fox\u2013Loh\u2013Zhao)"},{"data_url":"scopes/GraphTheory/graph-ramsey-theory/small-ramsey-numbers.json","id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","kind":"scope","navigation":"module","parent":30008,"result":false,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","title":"Small Ramsey numbers"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/graph-ramsey-four-four","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.graphRamsey_four_four","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value of "],["math","\\(R(4, 4)\\)"]],"source":"The value of $R(4, 4)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#graph-ramsey-four-four","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(4, 4)$ equals $18$.","title":"The value of $R(4, 4)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/forty-three-le-graph-ramsey-five-five","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.forty_three_le_graphRamsey_five_five","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exoo's lower bound for "],["math","\\(R(5, 5)\\)"]],"source":"Exoo's lower bound for $R(5, 5)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#forty-three-le-graph-ramsey-five-five","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(5, 5)$ is at least $43$.","title":"Exoo's lower bound for $R(5, 5)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/graph-ramsey-five-five-le-forty-six","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.graphRamsey_five_five_le_forty_six","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Angeltveit\u2013McKay upper bound for "],["math","\\(R(5, 5)\\)"]],"source":"The Angeltveit\u2013McKay upper bound for $R(5, 5)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#graph-ramsey-five-five-le-forty-six","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(5, 5)$ is at most $46$.","title":"The Angeltveit\u2013McKay upper bound for $R(5, 5)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/graph-ramsey-three-five","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.graphRamsey_three_five","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value of "],["math","\\(R(3, 5)\\)"]],"source":"The value of $R(3, 5)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#graph-ramsey-three-five","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(3, 5)$ equals $14$.","title":"The value of $R(3, 5)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/graph-ramsey-three-six","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.graphRamsey_three_six","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value of "],["math","\\(R(3, 6)\\)"]],"source":"The value of $R(3, 6)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#graph-ramsey-three-six","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(3, 6)$ equals $18$.","title":"The value of $R(3, 6)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/graph-ramsey-three-seven","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.graphRamsey_three_seven","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value of "],["math","\\(R(3, 7)\\)"]],"source":"The value of $R(3, 7)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#graph-ramsey-three-seven","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(3, 7)$ equals $23$.","title":"The value of $R(3, 7)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/graph-ramsey-three-eight","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.graphRamsey_three_eight","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value of "],["math","\\(R(3, 8)\\)"]],"source":"The value of $R(3, 8)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#graph-ramsey-three-eight","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(3, 8)$ equals $28$.","title":"The value of $R(3, 8)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/graph-ramsey-three-nine","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.graphRamsey_three_nine","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value of "],["math","\\(R(3, 9)\\)"]],"source":"The value of $R(3, 9)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#graph-ramsey-three-nine","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(3, 9)$ equals $36$.","title":"The value of $R(3, 9)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/forty-le-graph-ramsey-three-ten","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.forty_le_graphRamsey_three_ten","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exoo's lower bound for "],["math","\\(R(3, 10)\\)"]],"source":"Exoo's lower bound for $R(3, 10)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#forty-le-graph-ramsey-three-ten","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(3, 10)$ is at least $40$.","title":"Exoo's lower bound for $R(3, 10)$"},{"id":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers/graph-ramsey-three-ten-le-forty-one","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.SmallRamseyNumbersWanted.graphRamsey_three_ten_le_forty_one","parent":30060,"reading_title":{"format":"inline-parts-v1","parts":[["text","Angeltveit's upper bound for "],["math","\\(R(3, 10)\\)"]],"source":"Angeltveit's upper bound for $R(3, 10)$"},"result":true,"route":"../roadmap/GraphTheory/graph-ramsey-theory/small-ramsey-numbers/index.html#graph-ramsey-three-ten-le-forty-one","scope":"GraphTheory/graph-ramsey-theory/small-ramsey-numbers","status":"can_prove","summary":"The classical Ramsey number $R(3, 10)$ is at most $41$.","title":"Angeltveit's upper bound for $R(3, 10)$"},{"data_url":"scopes/GraphTheory/graph-invariants-and-labelings.json","id":"GraphTheory/graph-invariants-and-labelings","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/index.html","scope":"GraphTheory/graph-invariants-and-labelings","status":"can_prove","summary":"Numerical and polynomial invariants of graphs such as degree sequences, arboricity, induced forest number, Shannon capacity and the chromatic symmetric function, and graph labelings.","title":"Graph invariants and labelings"},{"id":"GraphTheory/graph-invariants-and-labelings/erdos-gallai","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.ErdosGallaiWanted.erdos_gallai","parent":30071,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/index.html#erdos-gallai","scope":"GraphTheory/graph-invariants-and-labelings","status":"fully_proved","summary":"Let $V$ be a finite set and $d \\colon V \\to \\mathbb{N}$. There is a simple graph on $V$ in which every vertex $v$ has degree $d(v)$ if and only if the sum $\\sum_{v \\in V} d(v)$ is even and, for every subset $S \\subseteq V$,","title":"Erd\u0151s\u2013Gallai theorem"},{"data_url":"scopes/GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function.json","id":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function","kind":"scope","navigation":"module","parent":30071,"result":false,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/index.html","scope":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function","status":"fully_proved","title":"Chromatic symmetric function"},{"id":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/chromatic-profile-count","kind":"node","lean":"SimpleGraph.chromaticProfileCount","parent":30073,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/index.html#chromatic-profile-count","scope":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, let $k$ be a natural number and let $m : \\{0, \\dots, k-1\\} \\to \\mathbb{N}$. The chromatic profile count of $G$ at $k$ and $m$ is the number of proper colorings $c : V \\to \\{0, \\dots, k-1\\}$ of $G$ such that for every color $i$, exactly $m(i)$ vertices $v$ satisfy $c(v) = i$.","title":"Chromatic profile count"},{"id":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/same-chromatic-symmetric-function","kind":"node","lean":"SimpleGraph.SameChromaticSymmetricFunction","parent":30073,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/index.html#same-chromatic-symmetric-function","scope":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function","status":"fully_proved","summary":"Let $G$ and $H$ be simple graphs on finite vertex sets. We say that $G$ and $H$ have the same chromatic symmetric function when, for every natural number $k$ and every $m : \\{0, \\dots, k-1\\} \\to \\mathbb{N}$, the chromatic profile count of $G$ at $k$ and $m$ equals that of $H$: both graphs have equally many proper colorings with colors $\\{0, \\dots, k-1\\}$ in which each color $i$ is used on exactly $m(i)$ vertices.","title":"Graphs with the same chromatic symmetric function"},{"id":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/refl","kind":"node","lean":"SimpleGraph.SameChromaticSymmetricFunction.refl","parent":30073,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/index.html#refl","scope":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function","status":"fully_proved","summary":"Every simple graph on a finite vertex set has the same chromatic symmetric function as itself.","title":"Reflexivity of having the same chromatic symmetric function"},{"id":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/symm","kind":"node","lean":"SimpleGraph.SameChromaticSymmetricFunction.symm","parent":30073,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/index.html#symm","scope":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function","status":"fully_proved","summary":"Let $G$ and $H$ be simple graphs on finite vertex sets. If $G$ has the same chromatic symmetric function as $H$, then $H$ has the same chromatic symmetric function as $G$.","title":"Symmetry of having the same chromatic symmetric function"},{"id":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/trans","kind":"node","lean":"SimpleGraph.SameChromaticSymmetricFunction.trans","parent":30073,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/index.html#trans","scope":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function","status":"fully_proved","summary":"Let $G$, $H$ and $K$ be simple graphs on finite vertex sets. If $G$ has the same chromatic symmetric function as $H$, and $H$ has the same chromatic symmetric function as $K$, then $G$ has the same chromatic symmetric function as $K$.","title":"Transitivity of having the same chromatic symmetric function"},{"id":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/of-iso","kind":"node","lean":"SimpleGraph.SameChromaticSymmetricFunction.of_iso","parent":30073,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function/index.html#of-iso","scope":"GraphTheory/graph-invariants-and-labelings/chromatic-symmetric-function","status":"fully_proved","summary":"If $G$ and $H$ are isomorphic simple graphs on finite vertex sets, then $G$ and $H$ have the same chromatic symmetric function.","title":"Isomorphic graphs have the same chromatic symmetric function"},{"data_url":"scopes/GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings.json","id":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings","kind":"scope","navigation":"module","parent":30071,"result":false,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/index.html","scope":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings","status":"fully_proved","title":"Edge-graceful labelings"},{"id":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/incident-sum","kind":"node","lean":"SimpleGraph.incidentSum","parent":30080,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/index.html#incident-sum","scope":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, let $q$ be a natural number and let $\\ell$ be a bijection from the edge set of $G$ to $\\{0, 1, \\dots, q-1\\}$. The incident-edge sum at a vertex $v$ is $\\sum_{e \\ni v} (\\ell(e) + 1)$, the sum over the edges $e$ of $G$ that contain $v$; in other words, the edges carry the labels $1, \\dots, q$.","title":"Incident-edge sum"},{"id":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/induced-residue","kind":"node","lean":"SimpleGraph.inducedResidue","parent":30080,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/index.html#induced-residue","scope":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$, let $q$ be a natural number and let $\\ell$ be a bijection from the edge set of $G$ to $\\{0, \\dots, q-1\\}$. The induced residue of a vertex $v$ is the incident-edge sum at $v$ (with edge labels $\\ell(e) + 1$) reduced modulo $|V|$, viewed as an element of $\\{0, \\dots, |V| - 1\\}$. When $|V| = 0$ there are no vertices, so there is nothing to define.","title":"Induced vertex residue"},{"id":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/induced-residue-val","kind":"node","lean":"SimpleGraph.inducedResidue_val","parent":30080,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/index.html#induced-residue-val","scope":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite nonempty vertex set $V$, let $q$ be a natural number and let $\\ell$ be a bijection from the edge set of $G$ to $\\{0, \\dots, q-1\\}$. For every vertex $v$, the induced residue of $v$ with respect to $\\ell$, read as a natural number, equals the incident-edge sum at $v$ with respect to $\\ell$ modulo $|V|$.","title":"Value of the induced residue"},{"id":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/is-edge-graceful","kind":"node","lean":"SimpleGraph.IsEdgeGraceful","parent":30080,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/index.html#is-edge-graceful","scope":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings","status":"fully_proved","summary":"A simple graph $G$ on a finite vertex set $V$ is edge-graceful when there are a natural number $q$ and a bijection $\\ell$ from the edge set of $G$ to $\\{0, \\dots, q-1\\}$ such that the induced residue map, sending each vertex $v$ to the sum of $\\ell(e) + 1$ over the edges $e$ containing $v$ taken modulo $|V|$, is a bijection from $V$ to $\\{0, \\dots, |V| - 1\\}$.","title":"Edge-graceful graph"},{"id":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/is-edge-graceful-iff","kind":"node","lean":"SimpleGraph.isEdgeGraceful_iff","parent":30080,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings/index.html#is-edge-graceful-iff","scope":"GraphTheory/graph-invariants-and-labelings/edge-graceful-labelings","status":"fully_proved","summary":"A simple graph $G$ on a finite vertex set $V$ is edge-graceful if and only if there are a natural number $q$, a bijection $\\ell$ from the edge set of $G$ to $\\{0, \\dots, q-1\\}$ and a bijection $\\lambda$ from $V$ to $\\{0, \\dots, |V| - 1\\}$ such that for every vertex $v$, $\\lambda(v)$ is the induced residue of $v$ with respect to $\\ell$ and, if $V$ is nonempty, $\\lambda(v)$ equals the incident-edge sum at $v$ with\u2026","title":"Edge-graceful graphs via a vertex bijection"},{"data_url":"scopes/GraphTheory/graph-invariants-and-labelings/induced-forest-number.json","id":"GraphTheory/graph-invariants-and-labelings/induced-forest-number","kind":"scope","navigation":"module","parent":30071,"result":false,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/induced-forest-number/index.html","scope":"GraphTheory/graph-invariants-and-labelings/induced-forest-number","status":"fully_proved","title":"Induced forest number"},{"id":"GraphTheory/graph-invariants-and-labelings/induced-forest-number/induced-forest-number","kind":"node","lean":"SimpleGraph.inducedForestNumber","parent":30086,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/induced-forest-number/index.html#induced-forest-number","scope":"GraphTheory/graph-invariants-and-labelings/induced-forest-number","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set $V$, the induced forest number $a(G)$ is the maximum number of vertices of an induced forest in $G$, that is, the maximum cardinality of a set $S \\subseteq V$ such that the subgraph of $G$ induced on $S$ is acyclic.","title":"Induced forest number"},{"id":"GraphTheory/graph-invariants-and-labelings/induced-forest-number/card-le-induced-forest-number","kind":"node","lean":"SimpleGraph.card_le_inducedForestNumber","parent":30086,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/induced-forest-number/index.html#card-le-induced-forest-number","scope":"GraphTheory/graph-invariants-and-labelings/induced-forest-number","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$. For every set $S \\subseteq V$ such that the subgraph of $G$ induced on $S$ is acyclic, $|S|$ is at most the induced forest number of $G$.","title":"Induced forests are no larger than the induced forest number"},{"id":"GraphTheory/graph-invariants-and-labelings/induced-forest-number/induced-forest-number-le-card","kind":"node","lean":"SimpleGraph.inducedForestNumber_le_card","parent":30086,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/induced-forest-number/index.html#induced-forest-number-le-card","scope":"GraphTheory/graph-invariants-and-labelings/induced-forest-number","status":"fully_proved","summary":"For every simple graph $G$ on a finite vertex set $V$, the induced forest number of $G$ is at most $|V|$.","title":"The induced forest number is at most the number of vertices"},{"id":"GraphTheory/graph-invariants-and-labelings/induced-forest-number/exists-is-acyclic-card-eq-induced-forest","kind":"node","lean":"SimpleGraph.exists_isAcyclic_card_eq_inducedForestNumber","parent":30086,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/induced-forest-number/index.html#exists-is-acyclic-card-eq-induced-forest","scope":"GraphTheory/graph-invariants-and-labelings/induced-forest-number","status":"fully_proved","summary":"For every simple graph $G$ on a finite vertex set $V$, there is a set $S \\subseteq V$ such that the subgraph of $G$ induced on $S$ is acyclic and $|S|$ equals the induced forest number of $G$.","title":"The induced forest number is attained"},{"data_url":"scopes/GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem.json","id":"GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem","kind":"scope","navigation":"module","parent":30071,"result":false,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem/index.html","scope":"GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem","status":"fully_proved","title":"Nash-Williams arboricity theorem"},{"id":"GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem/is-arboricity-coloring","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.NashWilliamsWanted.IsArboricityColoring","parent":30091,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem/index.html#is-arboricity-coloring","scope":"GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph and $k$ a natural number. A labeling of the edges of $G$ with labels in $\\{0, 1, \\dots, k - 1\\}$ is an arboricity coloring when, for each label $i$, the graph on the vertices of $G$ whose edges are the edges of $G$ labeled $i$ is acyclic, that is, a forest.","title":"Arboricity colorings"},{"id":"GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem/nash-williams-arboricity","kind":"node","lean":"MathlibExt.Combinatorics.SimpleGraph.NashWilliamsWanted.nash_williams_arboricity","parent":30091,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem/index.html#nash-williams-arboricity","scope":"GraphTheory/graph-invariants-and-labelings/nash-williams-arboricity-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $k$ be a natural number. Then $G$ has an arboricity coloring with $k$ labels, that is, its edges can be labeled with $k$ labels so that the edges of each label form a forest, if and only if for every nonempty set $U \\subseteq V$, the number of edges of $G$ with both endpoints in $U$ is at most $k(|U| - 1)$.","title":"Nash\u2013Williams arboricity theorem"},{"data_url":"scopes/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs.json","id":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","kind":"scope","navigation":"module","parent":30071,"result":false,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/index.html","scope":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","status":"fully_proved","title":"Prime labelings of ladder graphs"},{"id":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/cyclic-label","kind":"node","lean":"MetaMathlibExt.cyclicLabel","parent":30094,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/index.html#cyclic-label","scope":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","status":"fully_proved","summary":"Let $n$ and $s$ be natural numbers. Number the vertices of the ladder $P_n \\times P_2$ along its perimeter by $0, 1, \\dots, 2n - 1$, with $v_k$ at position $k - 1$ and $u_k$ at position $2n - k$ for $1 \\le k \\le n$. The cyclic label with offset $s$ of the position $i \\in \\{0, \\dots, 2n - 1\\}$ is $((i + s) \\bmod 2n) + 1$.","title":"Cyclic labeling of the ladder perimeter"},{"id":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/ladder-adj","kind":"node","lean":"MetaMathlibExt.LadderAdj","parent":30094,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/index.html#ladder-adj","scope":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","status":"fully_proved","summary":"Number the vertices of the ladder $P_n \\times P_2$ along its perimeter by $0, 1, \\dots, 2n-1$, with $v_k$ at position $k - 1$ and $u_k$ at position $2n - k$ for $1 \\le k \\le n$. Two positions $i, j \\in \\{0, \\dots, 2n-1\\}$ are adjacent when $j \\equiv i + 1 \\pmod{2n}$ or $i \\equiv j + 1 \\pmod{2n}$ (neighbours along the perimeter cycle), or when $i + j = 2n - 1$ (the rung joining $v_k$ and $u_k$).","title":"Adjacency in the ladder graph"},{"id":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/is-consecutive-cyclic-prime-labeling","kind":"node","lean":"MetaMathlibExt.IsConsecutiveCyclicPrimeLabeling","parent":30094,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/index.html#is-consecutive-cyclic-prime-labeling","scope":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","status":"fully_proved","summary":"Let $n$ and $s$ be natural numbers. Number the vertices of the ladder $P_n \\times P_2$ along its perimeter by $0, 1, \\dots, 2n - 1$, with $v_k$ at position $k - 1$ and $u_k$ at position $2n - k$ for $1 \\le k \\le n$. The offset $s$ gives a consecutive cyclic prime labeling of the ladder when $n > 0$ and, for all positions $i, j \\in \\{0, \\dots, 2n - 1\\}$ that are adjacent in the ladder graph, the cyclic labels of $i$\u2026","title":"Consecutive cyclic prime labeling of the ladder"},{"id":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/cyclic-label-pos","kind":"node","lean":"MetaMathlibExt.cyclicLabel_pos","parent":30094,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/index.html#cyclic-label-pos","scope":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","status":"fully_proved","summary":"For all natural numbers $n$ and $s$ and every position $i \\in \\{0, \\dots, 2n-1\\}$, the cyclic label of $i$ with offset $s$ is at least $1$.","title":"Cyclic labels are positive"},{"id":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/cyclic-label-le-size","kind":"node","lean":"MetaMathlibExt.cyclicLabel_le_size","parent":30094,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/index.html#cyclic-label-le-size","scope":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","status":"fully_proved","summary":"For all natural numbers $n > 0$ and $s$ and every position $i \\in \\{0, \\dots, 2n-1\\}$, the cyclic label of $i$ with offset $s$ is at most $2n$.","title":"Cyclic labels do not exceed the number of vertices"},{"id":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/cyclic-label-mem-range","kind":"node","lean":"MetaMathlibExt.cyclicLabel_mem_range","parent":30094,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/index.html#cyclic-label-mem-range","scope":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","status":"fully_proved","summary":"For all natural numbers $n > 0$ and $s$ and every position $i \\in \\{0, \\dots, 2n-1\\}$, the cyclic label $c$ of $i$ with offset $s$ satisfies $1 \\le c \\le 2n$.","title":"Range of the cyclic labels"},{"id":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/cyclic-label-zero-offset-zero","kind":"node","lean":"MetaMathlibExt.cyclicLabel_zero_offset_zero","parent":30094,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs/index.html#cyclic-label-zero-offset-zero","scope":"GraphTheory/graph-invariants-and-labelings/prime-labelings-of-ladder-graphs","status":"fully_proved","summary":"Let $n > 0$ be a natural number and number the vertices of the ladder $P_n \\times P_2$ along its perimeter by $0, 1, \\dots, 2n - 1$, with $v_k$ at position $k - 1$ and $u_k$ at position $2n - k$ for $1 \\le k \\le n$. Then the cyclic label with offset $0$ of the position $0$ (the vertex $v_1$) is $1$.","title":"The cyclic label of the first position with offset zero"},{"data_url":"scopes/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph.json","id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","kind":"scope","navigation":"module","parent":30071,"result":false,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","title":"Shannon capacity of a graph"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/positive-nat","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleGraphCode.PositiveNat","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#positive-nat","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The type of positive integers, that is, of natural numbers $n$ with $1 \\le n$, used as the block lengths of codes.","title":"Positive block lengths"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/is-cycle-code","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleGraphCode.IsCycleCode","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#is-cycle-code","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $q$ and $n$ be natural numbers. A finite set $A$ of words of length $n$ over $\\mathbb{Z}/q\\mathbb{Z}$ is an independent code for the $q$-cycle when every difference $x = a_1 - a_2$ with $a_1, a_2 \\in A$ whose coordinates all lie in $\\{-1, 0, 1\\}$ is the zero word. For $q \\ge 3$ this says exactly that $A$ is an independent set in the $n$-th strong power of the cycle graph $C_q$.","title":"Independent codes for the modular cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/cycle-code","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleGraphCode.CycleCode","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#cycle-code","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"For natural numbers $q$ and $n$, a cycle code of length $n$ modulo $q$ is a finite set of words of length $n$ over $\\mathbb{Z}/q\\mathbb{Z}$ that is an independent code for the $q$-cycle.","title":"Cycle codes"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/cycle-code-finite","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleGraphCode.cycleCode_finite","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#cycle-code-finite","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"For every nonzero natural number $q$ and every natural number $n$, there are only finitely many cycle codes of length $n$ modulo $q$.","title":"Finitely many cycle codes of each length"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/cycle-code-rate","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleGraphCode.cycleCodeRate","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#cycle-code-rate","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"For a natural number $q$, a positive integer $n$ and a cycle code $A$ of length $n$ modulo $q$, the rate of $A$ is $|A|^{1/n}$, the $n$-th root of the number of codewords of $A$, taken in the extended nonnegative reals $[0, \\infty]$.","title":"Rate of a cycle code"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/shannon-capacity-cycle","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleGraphCode.shannonCapacityCycle","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#shannon-capacity-cycle","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"For a natural number $q$, the Shannon capacity of the $q$-cycle is the supremum, in $[0, \\infty]$, of the rates $|A|^{1/n}$ over all positive integers $n$ and all cycle codes $A$ of length $n$ modulo $q$.","title":"Shannon capacity of the modular cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/cycle-code-rate-le-shannon-capacity","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleGraphCode.cycleCodeRate_le_shannonCapacity","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#cycle-code-rate-le-shannon-capacity","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"For every natural number $q$, every positive integer $n$ and every cycle code $A$ of length $n$ modulo $q$, the rate of $A$ is at most the Shannon capacity of the $q$-cycle.","title":"Code rates are at most the cycle capacity"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/cycle-construction-implies-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleGraphCode.cycleConstruction_implies_lowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#cycle-construction-implies-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $q$, $n$ and $M$ be natural numbers with $n \\ge 1$. If some finite set of words of length $n$ over $\\mathbb{Z}/q\\mathbb{Z}$ is an independent code for the $q$-cycle and has exactly $M$ elements, then $M^{1/n}$ is at most the Shannon capacity of the $q$-cycle, as an inequality in $[0, \\infty]$.","title":"Codes give lower bounds on the cycle capacity"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/polak-schrijver-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.polakSchrijverLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#polak-schrijver-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"There is a finite set $A$ of words of length $5$ over $\\mathbb{Z}/7\\mathbb{Z}$ that is an independent code for the $7$-cycle and has exactly $367$ elements.","title":"Polak\u2013Schrijver code of size 367 for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/shannon1956-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.shannon1956UpperBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#shannon1956-upper-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"The Shannon capacity of the $7$-cycle is at most $7/2$.","title":"Shannon's upper bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/lovasz-theta-cycle-seven","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.lovaszThetaCycleSeven","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#lovasz-theta-cycle-seven","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"This is the real number $\\frac{7 \\cos(\\pi/7)}{1 + \\cos(\\pi/7)}$, Lov\u00e1sz's closed form for the theta number of the $7$-cycle, viewed as an element of $[0, \\infty]$.","title":"Lov\u00e1sz's theta value for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/lovasz1979-upper-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.lovasz1979UpperBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#lovasz1979-upper-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"The Shannon capacity of the $7$-cycle is at most $\\frac{7 \\cos(\\pi/7)}{1 + \\cos(\\pi/7)}$, as an inequality in $[0, \\infty]$.","title":"Lov\u00e1sz's upper bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bmrrst1971-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bmrrst1971LowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bmrrst1971-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"The Shannon capacity of the $7$-cycle is at least $343^{1/5}$, as an inequality in $[0, \\infty]$.","title":"Baumert et al. lower bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/vesel-zerovnik2002-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.veselZerovnik2002LowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#vesel-zerovnik2002-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"The Shannon capacity of the $7$-cycle is at least $108^{1/4}$.","title":"Vesel\u2013\u017derovnik lower bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/mathew-ostergard2017-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.mathewOstergard2017LowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#mathew-ostergard2017-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"The Shannon capacity of the $7$-cycle is at least $350^{1/5}$, as an inequality in $[0, \\infty]$.","title":"Mathew\u2013\u00d6sterg\u00e5rd lower bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/polak-schrijver-rate-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.polakSchrijverRateLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#polak-schrijver-rate-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"The Shannon capacity of the $7$-cycle is at least $367^{1/5}$, as an inequality in $[0, \\infty]$.","title":"Polak\u2013Schrijver lower bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/itty-rosin-carstensen-reichman2026-lower","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.ittyRosinCarstensenReichman2026LowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#itty-rosin-carstensen-reichman2026-lower","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"The Shannon capacity of the $7$-cycle is at least $134753^{1/10}$, as an inequality in $[0, \\infty]$.","title":"Itty\u2013Rosin\u2013Carstensen\u2013Reichman lower bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/itty-rosin-carstensen-reichman2026-const","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.ittyRosinCarstensenReichman2026Construction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#itty-rosin-carstensen-reichman2026-const","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"There is a finite set $A$ of words of length $10$ over $\\mathbb{Z}/7\\mathbb{Z}$ that is an independent code for the $7$-cycle and has exactly $134753$ elements.","title":"Itty\u2013Rosin\u2013Carstensen\u2013Reichman code of dimension 10 for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpzprofile","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.BPZProfile","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpzprofile","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"A BPZ profile is an arbitrary quadruple of integers, called its code, pairs, auxiliary and neutral coordinates: the four integer coordinates propagated by the valid-tuple product of Buys, Polak and Zuiddam. No validity condition is imposed on the quadruple.","title":"BPZ profiles"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/star","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.BPZProfile.star","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#star","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"For $i = 1, 2$ let $p_i = (c_i, \\pi_i, a_i, z_i)$ be a BPZ profile, with integer code, pairs, auxiliary and neutral coordinates. The product $p_1 \\star p_2$ is the BPZ profile with code coordinate $(c_1 - \\pi_1)(c_2 - \\pi_2) + a_1 \\pi_2 + \\pi_1 a_2$, pairs coordinate $z_1 \\pi_2 + \\pi_1 z_2$, auxiliary coordinate $a_1 a_2$ and neutral coordinate $z_1 z_2 + (a_1 - z_1)(a_2 - z_2)$.","title":"Product of BPZ profiles"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-profile-square-iter","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzProfileSquareIter","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-profile-square-iter","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Write $\\star$ for the BPZ profile product. For a natural number $k$ and a BPZ profile $p$, the $k$-fold iterated square $S_k(p)$ is defined by $S_0(p) = p$ and $S_{k+1}(p) = S_k(p \\star p)$; that is, $S_k(p)$ is obtained from $p$ by squaring $k$ times under $\\star$.","title":"Iterated squaring of BPZ profiles"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/gao2026-m40","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.gao2026M40","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#gao2026-m40","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"$M_{40}$ is the natural number $4085567963379990282748597333793399773399910167372462 112610203714626938509773125898252337545387866277249$, the cardinality Gao prints for a code of length $200$ for the $7$-cycle.","title":"Gao's code size in dimension 200"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/gao2026-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.gao2026Construction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#gao2026-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $M_{40}$ be Gao's printed dimension-200 cardinality, a natural number with 103 decimal digits. There is a finite set $A$ of words of length $200$ over $\\mathbb{Z}/7\\mathbb{Z}$ that is an independent code for the $7$-cycle and has exactly $M_{40}$ elements.","title":"Gao's code of dimension 200 for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/gao2026-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.gao2026LowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#gao2026-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $M_{40}$ be Gao's printed dimension-200 cardinality, a natural number with 103 decimal digits. The Shannon capacity of the $7$-cycle is at least $M_{40}^{1/200}$, as an inequality in $[0, \\infty]$.","title":"Gao's lower bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c7-pi1","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC7Pi1","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c7-pi1","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"This is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile whose code, pairs, auxiliary and neutral coordinates are $367$, $8$, $367$ and $322$.","title":"Buys\u2013Polak\u2013Zuiddam one-block profile for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c7-pi2","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC7Pi2","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c7-pi2","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"This is $p_1 \\star p_1$, where $p_1$ is the one-block 7-cycle profile, whose code, pairs, auxiliary and neutral coordinates are $367$, $8$, $367$ and $322$, and $\\star$ is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile product.","title":"Buys\u2013Polak\u2013Zuiddam two-block profile for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c7-pi3","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC7Pi3","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c7-pi3","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"A BPZ profile is a quadruple of integers (code, pairs, auxiliary, neutral), and $\\star$ denotes the BPZ profile product. The three-block seven-cycle profile of Buys\u2013Polak\u2013Zuiddam is $\\pi_3 = \\pi_1 \\star \\pi_2$, where $\\pi_1 = (367, 8, 367, 322)$ is the one-block profile and $\\pi_2 = \\pi_1 \\star \\pi_1$ is the two-block profile.","title":"Three-block BPZ seven-cycle profile"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c7-pi5","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC7Pi5","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c7-pi5","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"A BPZ profile is a quadruple of integers (code, pairs, auxiliary, neutral), and $\\star$ denotes the BPZ profile product. The five-block seven-cycle profile of Buys\u2013Polak\u2013Zuiddam is $\\pi_5 = \\pi_2 \\star \\pi_3$, where $\\pi_1 = (367, 8, 367, 322)$, $\\pi_2 = \\pi_1 \\star \\pi_1$ and $\\pi_3 = \\pi_1 \\star \\pi_2$ are the one-, two- and three-block profiles.","title":"Five-block BPZ seven-cycle profile"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c7-pi10","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC7Pi10","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c7-pi10","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"This is $p_5 \\star p_5$, where $p_5$ is the five-block 7-cycle profile and $\\star$ is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile product.","title":"Buys\u2013Polak\u2013Zuiddam ten-block profile for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c7-pi20","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC7Pi20","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c7-pi20","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"This is $p_{10} \\star p_{10}$, where $p_{10}$ is the ten-block 7-cycle profile and $\\star$ is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile product.","title":"Buys\u2013Polak\u2013Zuiddam twenty-block profile for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c7-pi40","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC7Pi40","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c7-pi40","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"A BPZ profile is a quadruple of integers (code, pairs, auxiliary, neutral), and $\\star$ denotes the BPZ profile product. The final forty-block seven-cycle profile of Buys\u2013Polak\u2013Zuiddam is $\\pi_{40} = \\pi_{20} \\star \\pi_{20}$, where $\\pi_1 = (367, 8, 367, 322)$, $\\pi_2 = \\pi_1 \\star \\pi_1$, $\\pi_3 = \\pi_1 \\star \\pi_2$, $\\pi_5 = \\pi_2 \\star \\pi_3$, $\\pi_{10} = \\pi_5 \\star \\pi_5$ and\u2026","title":"Forty-block BPZ seven-cycle profile"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c7-pi40-code","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC7Pi40_code","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c7-pi40-code","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The code coordinate (the first of the four integer coordinates) of the forty-block seven-cycle profile $\\pi_{40}$ of Buys\u2013Polak\u2013Zuiddam equals $4089635995108338053456272492117641008738901835089431264388780413340766441486608 118807897325023935704961$.","title":"Code coordinate of the forty-block BPZ seven-cycle profile"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-seven-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleSevenConstruction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-seven-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final forty-block seven-cycle profile, and let $M = \\max(c, 0)$. There is a finite set of words of length $200$ over $\\mathbb{Z}/7\\mathbb{Z}$ that is an independent code for the $7$-cycle and has exactly $M$ elements.","title":"Buys\u2013Polak\u2013Zuiddam code for the 7-cycle in length 200"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-seven-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleSevenLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-seven-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final forty-block seven-cycle profile, and let $M = \\max(c, 0)$. Then $M^{1/200}$ is at most the Shannon capacity of the $7$-cycle, as an inequality in $[0, \\infty]$.","title":"Buys\u2013Polak\u2013Zuiddam lower bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/tandon2026-mstar","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.tandon2026Mstar","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#tandon2026-mstar","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Tandon's dimension-500 code size is the natural number $M^*$ with $257$ decimal digits given by","title":"Tandon's dimension-500 code size"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/tandon2026-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.tandon2026Construction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#tandon2026-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $M^*$ be Tandon's dimension-500 code size. There is an independent code for the $7$-cycle consisting of words of length $500$ over $\\mathbb{Z}/7\\mathbb{Z}$ with exactly $M^*$ codewords.","title":"Tandon's dimension-500 code for the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/tandon2026-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.tandon2026LowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#tandon2026-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $M^*$ be Tandon's dimension-500 code size. The Shannon capacity of the $7$-cycle is at least $(M^*)^{1/500}$.","title":"Tandon's lower bound for the capacity of the 7-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi1","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi1","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi1","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The one-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile with code coordinate $148$, pairs coordinate $3$, auxiliary coordinate $148$ and neutral coordinate $142$.","title":"One-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi2","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi2","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi2","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The two-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile product $\\pi_1 \\star \\pi_1$, where $\\pi_1$ is their one-block eleven-cycle profile.","title":"Two-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi3","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi3","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi3","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The three-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile product $\\pi_1 \\star \\pi_2$, where $\\pi_1$ and $\\pi_2$ are their one-block and two-block eleven-cycle profiles.","title":"Three-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi4","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi4","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi4","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The four-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile product $\\pi_2 \\star \\pi_2$, where $\\pi_2$ is their two-block eleven-cycle profile.","title":"Four-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi7","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi7","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi7","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The seven-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile product $\\pi_3 \\star \\pi_4$, where $\\pi_3$ and $\\pi_4$ are their three-block and four-block eleven-cycle profiles.","title":"Seven-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi8","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi8","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi8","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The eight-block BPZ profile of the eleven-cycle is $\\pi_4 \\star \\pi_4$, where $\\pi_4$ is the four-block BPZ profile of the eleven-cycle and $\\star$ is the BPZ profile product.","title":"Eight-block BPZ profile of the eleven-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi14","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi14","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi14","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The fourteen-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile product $\\pi_7 \\star \\pi_7$, where $\\pi_7$ is their seven-block eleven-cycle profile.","title":"Fourteen-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi15","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi15","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi15","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The fifteen-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile product $\\pi_7 \\star \\pi_8$, where $\\pi_7$ and $\\pi_8$ are their seven-block and eight-block eleven-cycle profiles.","title":"Fifteen-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi29","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi29","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi29","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The twenty-nine-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile product $\\pi_{14} \\star \\pi_{15}$, where $\\pi_{14}$ and $\\pi_{15}$ are their fourteen-block and fifteen-block eleven-cycle profiles.","title":"Twenty-nine-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c11-pi58","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC11Pi58","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c11-pi58","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The final fifty-eight-block eleven-cycle profile of Buys, Polak and Zuiddam is the profile product $\\pi_{29} \\star \\pi_{29}$, where $\\pi_{29}$ is their twenty-nine-block eleven-cycle profile.","title":"Fifty-eight-block BPZ profile for the 11-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c13-pi1","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC13Pi1","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c13-pi1","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The one-block BPZ profile of the thirteen-cycle is the BPZ profile with code coordinate $62530$, pairs coordinate $1014$, auxiliary coordinate $62530$ and neutral coordinate $60502$.","title":"One-block BPZ profile of the thirteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c13-pi2","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC13Pi2","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c13-pi2","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The two-block BPZ profile of the thirteen-cycle is $\\pi_1 \\star \\pi_1$, where $\\pi_1$ is the one-block BPZ profile of the thirteen-cycle and $\\star$ is the BPZ profile product.","title":"Two-block BPZ profile of the thirteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c13-pi3","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC13Pi3","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c13-pi3","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The three-block BPZ profile of the thirteen-cycle is $\\pi_1 \\star \\pi_2$, where $\\pi_1$ and $\\pi_2$ are the one-block and two-block BPZ profiles of the thirteen-cycle and $\\star$ is the BPZ profile product.","title":"Three-block BPZ profile of the thirteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c13-pi4","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC13Pi4","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c13-pi4","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The four-block BPZ profile of the thirteen-cycle is $\\pi_2 \\star \\pi_2$, where $\\pi_2$ is the two-block BPZ profile of the thirteen-cycle and $\\star$ is the BPZ profile product.","title":"Four-block BPZ profile of the thirteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c13-pi5","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC13Pi5","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c13-pi5","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The five-block BPZ profile of the thirteen-cycle is $\\pi_2 \\star \\pi_3$, where $\\pi_2$ and $\\pi_3$ are the two-block and three-block BPZ profiles of the thirteen-cycle and $\\star$ is the BPZ profile product.","title":"Five-block BPZ profile of the thirteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c13-pi9","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC13Pi9","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c13-pi9","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The nine-block BPZ profile of the thirteen-cycle is $\\pi_4 \\star \\pi_5$, where $\\pi_4$ and $\\pi_5$ are the four-block and five-block BPZ profiles of the thirteen-cycle and $\\star$ is the BPZ profile product.","title":"Nine-block BPZ profile of the thirteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c13-pi72","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC13Pi72","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c13-pi72","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $\\pi_9$ be the nine-block BPZ profile of the thirteen-cycle, and let $\\star$ be the BPZ profile product. Set $\\sigma_0 = \\pi_9$ and $\\sigma_{k+1} = \\sigma_k \\star \\sigma_k$. The seventy-two-block BPZ profile of the thirteen-cycle is $\\sigma_3$, the result of squaring $\\pi_9$ three times.","title":"Seventy-two-block BPZ profile of the thirteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c15-pi1","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC15Pi1","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c15-pi1","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"The one-block BPZ profile of the fifteen-cycle is the BPZ profile with code coordinate $2842$, pairs coordinate $3$, auxiliary coordinate $2842$ and neutral coordinate $2833$.","title":"One-block BPZ profile of the fifteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c15-pi1024","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC15Pi1024","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c15-pi1024","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $\\pi_1$ be the one-block BPZ profile of the fifteen-cycle, and let $\\star$ be the BPZ profile product. Set $\\sigma_0 = \\pi_1$ and $\\sigma_{k+1} = \\sigma_k \\star \\sigma_k$. This profile is $\\sigma_{10}$, the result of squaring $\\pi_1$ ten times.","title":"BPZ profile of the fifteen-cycle after ten squarings"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c19-pi1","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC19Pi1","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c19-pi1","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"This is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile whose code, pairs, auxiliary and neutral coordinates are $7666$, $2$, $7666$ and $7661$.","title":"Buys\u2013Polak\u2013Zuiddam one-block profile for the 19-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c19-pi4096","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC19Pi4096","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c19-pi4096","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $p_0$ be the one-block 19-cycle profile, whose code, pairs, auxiliary and neutral coordinates are $7666$, $2$, $7666$ and $7661$, and let $p_{j+1} = p_j \\star p_j$, where $\\star$ is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile product. This profile is $p_{12}$, the result of squaring $p_0$ twelve times.","title":"Buys\u2013Polak\u2013Zuiddam 4096-block profile for the 19-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c21-pi1","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC21Pi1","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c21-pi1","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"This is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile whose code, pairs, auxiliary and neutral coordinates are $11441$, $10$, $11441$ and $11398$.","title":"Buys\u2013Polak\u2013Zuiddam one-block profile for the 21-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c21-pi1024","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC21Pi1024","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c21-pi1024","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $p_0$ be the one-block 21-cycle profile, whose code, pairs, auxiliary and neutral coordinates are $11441$, $10$, $11441$ and $11398$, and let $p_{j+1} = p_j \\star p_j$, where $\\star$ is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile product. This profile is $p_{10}$, the result of squaring $p_0$ ten times.","title":"Buys\u2013Polak\u2013Zuiddam 1024-block profile for the 21-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c23-pi1","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC23Pi1","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c23-pi1","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"This is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile whose code, pairs, auxiliary and neutral coordinates are $16466$, $30$, $16466$ and $16323$.","title":"Buys\u2013Polak\u2013Zuiddam one-block profile for the 23-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz-c23-pi512","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpzC23Pi512","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz-c23-pi512","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $p_0$ be the one-block 23-cycle profile, whose code, pairs, auxiliary and neutral coordinates are $16466$, $30$, $16466$ and $16323$, and let $p_{j+1} = p_j \\star p_j$, where $\\star$ is the Buys\u2013Polak\u2013Zuiddam (BPZ) profile product. This profile is $p_9$, the result of squaring $p_0$ nine times.","title":"Buys\u2013Polak\u2013Zuiddam 512-block profile for the 23-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-eleven-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleElevenConstruction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-eleven-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of the fifty-eight-block eleven-cycle profile $\\pi_{58}$ of Buys\u2013Polak\u2013Zuiddam. There is a finite set of words of length $174$ over $\\mathbb{Z}/11\\mathbb{Z}$ that is an independent code for the $11$-cycle and has exactly $\\max(c, 0)$ elements.","title":"Buys\u2013Polak\u2013Zuiddam code for the eleven-cycle in dimension 174"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-thirteen-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleThirteenConstruction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-thirteen-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final seventy-two-block thirteen-cycle profile, and let $M = \\max(c, 0)$. There is a finite set of words of length $432$ over $\\mathbb{Z}/13\\mathbb{Z}$ that is an independent code for the $13$-cycle and has exactly $M$ elements.","title":"Buys\u2013Polak\u2013Zuiddam code for the 13-cycle in length 432"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-fifteen-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleFifteenConstruction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-fifteen-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of the $1024$-block fifteen-cycle profile $\\pi_{1024}$ of Buys\u2013Polak\u2013Zuiddam. There is a finite set of words of length $4096$ over $\\mathbb{Z}/15\\mathbb{Z}$ that is an independent code for the $15$-cycle and has exactly $\\max(c, 0)$ elements.","title":"Buys\u2013Polak\u2013Zuiddam code for the fifteen-cycle in dimension 4096"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-nineteen-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleNineteenConstruction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-nineteen-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of the $4096$-block nineteen-cycle profile $\\pi_{4096}$ of Buys\u2013Polak\u2013Zuiddam. There is a finite set of words of length $16384$ over $\\mathbb{Z}/19\\mathbb{Z}$ that is an independent code for the $19$-cycle and has exactly $\\max(c, 0)$ elements.","title":"Buys\u2013Polak\u2013Zuiddam code for the nineteen-cycle in dimension 16384"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-twenty-one-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleTwentyOneConstruction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-twenty-one-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final twenty-one-cycle profile after ten squarings, and let $M = \\max(c, 0)$. There is a finite set of words of length $4096$ over $\\mathbb{Z}/21\\mathbb{Z}$ that is an independent code for the $21$-cycle and has exactly $M$ elements.","title":"Buys\u2013Polak\u2013Zuiddam code for the 21-cycle in length 4096"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-twenty-three-construction","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleTwentyThreeConstruction","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-twenty-three-construction","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final twenty-three-cycle profile after nine squarings, and let $M = \\max(c, 0)$. There is a finite set of words of length $2048$ over $\\mathbb{Z}/23\\mathbb{Z}$ that is an independent code for the $23$-cycle and has exactly $M$ elements.","title":"Buys\u2013Polak\u2013Zuiddam code for the 23-cycle in length 2048"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-eleven-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleElevenLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-eleven-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of the fifty-eight-block eleven-cycle profile $\\pi_{58}$ of Buys\u2013Polak\u2013Zuiddam. Then $\\max(c, 0)^{1/174}$ is at most the Shannon capacity of the $11$-cycle, as an inequality in $[0, \\infty]$.","title":"Buys\u2013Polak\u2013Zuiddam lower bound for the capacity of the eleven-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-thirteen-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleThirteenLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-thirteen-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final seventy-two-block thirteen-cycle profile, and let $M = \\max(c, 0)$. Then $M^{1/432}$ is at most the Shannon capacity of the $13$-cycle, as an inequality in $[0, \\infty]$.","title":"Buys\u2013Polak\u2013Zuiddam lower bound for the capacity of the 13-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-fifteen-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleFifteenLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-fifteen-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of the $1024$-block fifteen-cycle profile $\\pi_{1024}$ of Buys\u2013Polak\u2013Zuiddam. Then $\\max(c, 0)^{1/4096}$ is at most the Shannon capacity of the $15$-cycle, as an inequality in $[0, \\infty]$.","title":"Buys\u2013Polak\u2013Zuiddam lower bound for the capacity of the fifteen-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-nineteen-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleNineteenLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-nineteen-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final nineteen-cycle profile after twelve squarings, and let $M = \\max(c, 0)$. Then $M^{1/16384}$ is at most the Shannon capacity of the $19$-cycle, as an inequality in $[0, \\infty]$.","title":"Buys\u2013Polak\u2013Zuiddam lower bound for the capacity of the 19-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-twenty-one-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleTwentyOneLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-twenty-one-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final twenty-one-cycle profile after ten squarings, and let $M = \\max(c, 0)$. Then $M^{1/4096}$ is at most the Shannon capacity of the $21$-cycle, as an inequality in $[0, \\infty]$.","title":"Buys\u2013Polak\u2013Zuiddam lower bound for the capacity of the 21-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/bpz2026-cycle-twenty-three-lower-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.CycleShannonBoundsWanted.bpz2026CycleTwentyThreeLowerBound","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#bpz2026-cycle-twenty-three-lower-bound","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"Let $c$ be the code coordinate of BPZ's final twenty-three-cycle profile after nine squarings, and let $M = \\max(c, 0)$. Then $M^{1/2048}$ is at most the Shannon capacity of the $23$-cycle, as an inequality in $[0, \\infty]$.","title":"Buys\u2013Polak\u2013Zuiddam lower bound for the capacity of the 23-cycle"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/strong-power","kind":"node","lean":"MathlibExt.Combinatorics.ShannonCapacityPentagonWanted.Combinatorics.ShannonCapacityPentagon.strongPower","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#strong-power","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $G$ be a simple graph with vertex set $V$ and let $k$ be a natural number. The $k$-th strong power $G^{\\boxtimes k}$ is the simple graph whose vertices are the words $x = (x_1, \\dots, x_k) \\in V^k$ of length $k$, in which two words $x$ and $y$ are adjacent (confusable) when $x \\neq y$ and for every $i$ either $x_i = y_i$ or $x_i$ and $y_i$ are adjacent in $G$.","title":"Strong power of a graph"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/has-shannon-capacity","kind":"node","lean":"MathlibExt.Combinatorics.ShannonCapacityPentagonWanted.Combinatorics.ShannonCapacityPentagon.HasShannonCapacity","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#has-shannon-capacity","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $\\theta$ be a real number. For $k \\ge 1$ let $G^{\\boxtimes k}$ be the $k$-th strong power of $G$ and $\\alpha(G^{\\boxtimes k})$ its independence number. The graph $G$ has Shannon capacity $\\theta$ when $\\alpha(G^{\\boxtimes k})^{1/k} \\to \\theta$ as $k \\to \\infty$.","title":"Shannon capacity of a graph"},{"id":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/shannon-capacity-pentagon","kind":"node","lean":"MathlibExt.Combinatorics.ShannonCapacityPentagonWanted.Combinatorics.ShannonCapacityPentagon.shannon_capacity_pentagon","parent":30102,"result":true,"route":"../roadmap/GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph/index.html#shannon-capacity-pentagon","scope":"GraphTheory/graph-invariants-and-labelings/shannon-capacity-of-a-graph","status":"can_prove","summary":"The cycle graph $C_5$ on five vertices has Shannon capacity $\\sqrt{5}$: writing $\\alpha(C_5^{\\boxtimes k})$ for the independence number of the $k$-th strong power of $C_5$, we have $\\alpha(C_5^{\\boxtimes k})^{1/k} \\to \\sqrt{5}$ as $k \\to \\infty$.","title":"Lov\u00e1sz's theorem: the Shannon capacity of the pentagon"},{"data_url":"scopes/GraphTheory/directed-graphs.json","id":"GraphTheory/directed-graphs","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/index.html","scope":"GraphTheory/directed-graphs","status":"can_prove","summary":"Directed graphs, orientations of undirected graphs, digraph colorings and tournaments.","title":"Directed graphs and tournaments"},{"data_url":"scopes/GraphTheory/directed-graphs/connectivity-of-directed-graphs.json","id":"GraphTheory/directed-graphs/connectivity-of-directed-graphs","kind":"scope","navigation":"module","parent":30180,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/connectivity-of-directed-graphs/index.html","scope":"GraphTheory/directed-graphs/connectivity-of-directed-graphs","status":"can_prove","title":"Connectivity of directed graphs"},{"id":"GraphTheory/directed-graphs/connectivity-of-directed-graphs/is-directed-path","kind":"node","lean":"MathlibExt.Combinatorics.HamidouneReverseConnectivityAIMCombinatoricsProblem191Wanted.IsDirectedPath","parent":30181,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/connectivity-of-directed-graphs/index.html#is-directed-path","scope":"GraphTheory/directed-graphs/connectivity-of-directed-graphs","status":"fully_proved","summary":"A directed path in a digraph $D$ is a finite, possibly empty, sequence of vertices of $D$ with no vertex repeated in which each vertex is joined to the next by an arc of $D$.","title":"Directed paths"},{"id":"GraphTheory/directed-graphs/connectivity-of-directed-graphs/internal-vertices","kind":"node","lean":"MathlibExt.Combinatorics.HamidouneReverseConnectivityAIMCombinatoricsProblem191Wanted.internalVertices","parent":30181,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/connectivity-of-directed-graphs/index.html#internal-vertices","scope":"GraphTheory/directed-graphs/connectivity-of-directed-graphs","status":"fully_proved","summary":"The internal vertices of a path given as a vertex list (all but the first and the last).","title":"Internal vertices of a path"},{"id":"GraphTheory/directed-graphs/connectivity-of-directed-graphs/has-internally-disjoint-paths","kind":"node","lean":"MathlibExt.Combinatorics.HamidouneReverseConnectivityAIMCombinatoricsProblem191Wanted.HasInternallyDisjointPaths","parent":30181,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/connectivity-of-directed-graphs/index.html#has-internally-disjoint-paths","scope":"GraphTheory/directed-graphs/connectivity-of-directed-graphs","status":"fully_proved","summary":"Let $D$ be a digraph, $r$ a natural number and $y$, $x$ vertices of $D$. We write $\\kappa(y, x) \\ge r$ when there are $r$ pairwise distinct directed paths in $D$ from $y$ to $x$, no two of which share an internal vertex. Here a directed path from $y$ to $x$ is a sequence of distinct vertices starting at $y$ and ending at $x$ in which each vertex is joined to the next by an arc of $D$, and its internal vertices are\u2026","title":"Internally disjoint directed paths"},{"id":"GraphTheory/directed-graphs/connectivity-of-directed-graphs/hamidoune-some-rat-least-three","kind":"node","lean":"MathlibExt.Combinatorics.HamidouneReverseConnectivityAIMCombinatoricsProblem191Wanted.hamidouneSomeRAtLeastThree","parent":30181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hamidoune's reverse-connectivity conjecture for some "],["math","\\(r \\ge 3\\)"]],"source":"Hamidoune's reverse-connectivity conjecture for some $r \\ge 3$"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/connectivity-of-directed-graphs/index.html#hamidoune-some-rat-least-three","scope":"GraphTheory/directed-graphs/connectivity-of-directed-graphs","status":"fully_proved","summary":"The statement that there is a natural number $r \\ge 3$ such that every digraph $D$ without loops or parallel arcs on a finite nonempty vertex set, in which every vertex has at least $r$ out-neighbours and at least $r$ in-neighbours, has an arc $(x, y)$ from $x$ to $y$ with $\\kappa(y, x) \\ge r$, that is, with $r$ pairwise distinct directed paths from $y$ to $x$, no two of which share an internal vertex. Here a\u2026","title":"Hamidoune's reverse-connectivity conjecture for some $r \\ge 3$"},{"id":"GraphTheory/directed-graphs/connectivity-of-directed-graphs/hamidoune-some-rat-least-three-refuted","kind":"node","lean":"MathlibExt.Combinatorics.HamidouneReverseConnectivityAIMCombinatoricsProblem191Wanted.hamidouneSomeRAtLeastThree_refuted","parent":30181,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hamidoune's reverse-connectivity conjecture fails for every "],["math","\\(r \\ge 3\\)"]],"source":"Hamidoune's reverse-connectivity conjecture fails for every $r \\ge 3$"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/connectivity-of-directed-graphs/index.html#hamidoune-some-rat-least-three-refuted","scope":"GraphTheory/directed-graphs/connectivity-of-directed-graphs","status":"can_prove","summary":"For no natural number $r \\ge 3$ does the following hold: every digraph $D$ without loops or parallel arcs on a finite nonempty vertex set, in which every vertex has at least $r$ out-neighbours and at least $r$ in-neighbours, has an arc $(x, y)$ from $x$ to $y$ with $\\kappa(y, x) \\ge r$, that is, with $r$ pairwise distinct directed paths from $y$ to $x$, no two of which share an internal vertex. Here a directed path\u2026","title":"Hamidoune's reverse-connectivity conjecture fails for every $r \\ge 3$"},{"data_url":"scopes/GraphTheory/directed-graphs/dichromatic-number.json","id":"GraphTheory/directed-graphs/dichromatic-number","kind":"scope","navigation":"module","parent":30180,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/dichromatic-number/index.html","scope":"GraphTheory/directed-graphs/dichromatic-number","status":"fully_proved","title":"Dichromatic number"},{"id":"GraphTheory/directed-graphs/dichromatic-number/is-orientation","kind":"node","lean":"SimpleGraph.IsOrientation","parent":30187,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/dichromatic-number/index.html#is-orientation","scope":"GraphTheory/directed-graphs/dichromatic-number","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex set $V$. A binary relation $D$ on $V$ is an orientation of $G$ if $D(u, v)$ holds only when $u$ and $v$ are adjacent in $G$, and for every pair of adjacent vertices $u$ and $v$ exactly one of $D(u, v)$ and $D(v, u)$ holds.","title":"Orientations of a graph"},{"id":"GraphTheory/directed-graphs/dichromatic-number/is-directed-cycle","kind":"node","lean":"SimpleGraph.IsDirectedCycle","parent":30187,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/dichromatic-number/index.html#is-directed-cycle","scope":"GraphTheory/directed-graphs/dichromatic-number","status":"fully_proved","summary":"Let $D$ be a binary relation on a set $V$, viewed as a digraph, let $n$ be a natural number, and let $f_0, f_1, \\dots, f_{n+1}$ be elements of $V$. The sequence $(f_0, \\dots, f_{n+1})$ is a closed directed walk of length $n + 1$ in $D$ if $f_0 = f_{n+1}$ and $D(f_i, f_{i+1})$ holds for every $i$ with $0 \\le i \\le n$.","title":"Closed directed walks"},{"id":"GraphTheory/directed-graphs/dichromatic-number/dichromatic-colorable","kind":"node","lean":"SimpleGraph.DichromaticColorable","parent":30187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dichromatically "],["math","\\(k\\)"],["text","-colourable graphs"]],"source":"Dichromatically $k$-colourable graphs"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/dichromatic-number/index.html#dichromatic-colorable","scope":"GraphTheory/directed-graphs/dichromatic-number","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex set $V$ and let $k$ be a natural number. Then $G$ is dichromatically $k$-colourable if for every orientation $D$ of $G$ there is a map $c \\colon V \\to \\{0, \\dots, k - 1\\}$, not required to be a proper colouring, such that no closed directed walk of positive length in $D$ has all of its vertices of the same colour under $c$.","title":"Dichromatically $k$-colourable graphs"},{"id":"GraphTheory/directed-graphs/dichromatic-number/dichromatic-number","kind":"node","lean":"SimpleGraph.dichromaticNumber","parent":30187,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/dichromatic-number/index.html#dichromatic-number","scope":"GraphTheory/directed-graphs/dichromatic-number","status":"fully_proved","summary":"The dichromatic number of a simple graph $G$ is the least natural number $k$ such that $G$ is dichromatically $k$-colourable, taken in $\\mathbb{N} \\cup \\{\\infty\\}$; it is $\\infty$ if $G$ is dichromatically $k$-colourable for no natural number $k$.","title":"Dichromatic number"},{"id":"GraphTheory/directed-graphs/dichromatic-number/dichromatic-number-eq-s-inf","kind":"node","lean":"SimpleGraph.DichromaticColorable.dichromaticNumber_eq_sInf","parent":30187,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/dichromatic-number/index.html#dichromatic-number-eq-s-inf","scope":"GraphTheory/directed-graphs/dichromatic-number","status":"fully_proved","summary":"Let $G$ be a simple graph. If $G$ is dichromatically $n$-colourable for some natural number $n$, then the dichromatic number of $G$ equals the least natural number $k$ such that $G$ is dichromatically $k$-colourable.","title":"The dichromatic number as a least natural number"},{"id":"GraphTheory/directed-graphs/dichromatic-number/of-edgeless","kind":"node","lean":"SimpleGraph.IsOrientation.of_edgeless","parent":30187,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/dichromatic-number/index.html#of-edgeless","scope":"GraphTheory/directed-graphs/dichromatic-number","status":"fully_proved","summary":"If a simple graph $G$ has no edges, then the empty relation on its vertex set, the digraph with no arcs, is an orientation of $G$.","title":"The empty orientation of an edgeless graph"},{"id":"GraphTheory/directed-graphs/dichromatic-number/of-edgeless-2","kind":"node","lean":"SimpleGraph.DichromaticColorable.of_edgeless","parent":30187,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edgeless graphs are dichromatically "],["math","\\(1\\)"],["text","-colourable"]],"source":"Edgeless graphs are dichromatically $1$-colourable"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/dichromatic-number/index.html#of-edgeless-2","scope":"GraphTheory/directed-graphs/dichromatic-number","status":"fully_proved","summary":"If a simple graph $G$ has no edges, then $G$ is dichromatically $1$-colourable.","title":"Edgeless graphs are dichromatically $1$-colourable"},{"data_url":"scopes/GraphTheory/directed-graphs/labeled-directed-graphs.json","id":"GraphTheory/directed-graphs/labeled-directed-graphs","kind":"scope","navigation":"module","parent":30180,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/labeled-directed-graphs/index.html","scope":"GraphTheory/directed-graphs/labeled-directed-graphs","status":"fully_proved","title":"Labeled directed graphs"},{"id":"GraphTheory/directed-graphs/labeled-directed-graphs/labeled-digraph","kind":"node","lean":"MetaMathlibExt.LabeledDigraph","parent":30195,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/labeled-directed-graphs/index.html#labeled-digraph","scope":"GraphTheory/directed-graphs/labeled-directed-graphs","status":"fully_proved","summary":"A labeled directed graph with vertex set $V$, edge set $E$ and label set $L$ consists of a source map $s : E \\to V$, a target map $t : E \\to V$ and a label map $\\ell : E \\to L$; the edge $e$ goes from $s(e)$ to $t(e)$ and carries the label $\\ell(e)$. Loops and parallel edges are allowed.","title":"Labeled directed graph"},{"id":"GraphTheory/directed-graphs/labeled-directed-graphs/ext","kind":"node","lean":"MetaMathlibExt.LabeledDigraph.ext","parent":30195,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/labeled-directed-graphs/index.html#ext","scope":"GraphTheory/directed-graphs/labeled-directed-graphs","status":"fully_proved","summary":"Two labeled directed graphs $G$ and $H$ with the same vertex set, edge set and label set are equal if they have the same source map, the same target map and the same label map.","title":"Extensionality for labeled directed graphs"},{"id":"GraphTheory/directed-graphs/labeled-directed-graphs/restrict","kind":"node","lean":"MetaMathlibExt.LabeledDigraph.restrict","parent":30195,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/labeled-directed-graphs/index.html#restrict","scope":"GraphTheory/directed-graphs/labeled-directed-graphs","status":"fully_proved","summary":"Let $G$ be a labeled directed graph with vertex set $V$ and label set $L$, and let $p$ be a predicate on $V$. The restriction of $G$ to $p$ is the labeled directed graph with label set $L$ whose vertices are the vertices $v$ of $G$ with $p(v)$, whose edges are the edges of $G$ whose source and target both satisfy $p$, and in which each edge has the same source, target and label as in $G$.","title":"Restriction of a labeled directed graph"},{"id":"GraphTheory/directed-graphs/labeled-directed-graphs/restrict-src","kind":"node","lean":"MetaMathlibExt.LabeledDigraph.restrict_src","parent":30195,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/labeled-directed-graphs/index.html#restrict-src","scope":"GraphTheory/directed-graphs/labeled-directed-graphs","status":"fully_proved","summary":"Let $G$ be a labeled directed graph, $p$ a predicate on its vertices, and $e$ an edge of $G$ whose source and target satisfy $p$, viewed as an edge of the restriction of $G$ to $p$. The source of $e$ in the restriction, viewed as a vertex of $G$, is the source of $e$ in $G$.","title":"Source of an edge in a restriction"},{"id":"GraphTheory/directed-graphs/labeled-directed-graphs/restrict-tgt","kind":"node","lean":"MetaMathlibExt.LabeledDigraph.restrict_tgt","parent":30195,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/labeled-directed-graphs/index.html#restrict-tgt","scope":"GraphTheory/directed-graphs/labeled-directed-graphs","status":"fully_proved","summary":"Let $G$ be a labeled directed graph, $p$ a predicate on its vertices, and $e$ an edge of $G$ whose source and target satisfy $p$, viewed as an edge of the restriction of $G$ to $p$. The target of $e$ in the restriction, viewed as a vertex of $G$, is the target of $e$ in $G$.","title":"Target of an edge in a restriction"},{"id":"GraphTheory/directed-graphs/labeled-directed-graphs/restrict-label","kind":"node","lean":"MetaMathlibExt.LabeledDigraph.restrict_label","parent":30195,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/labeled-directed-graphs/index.html#restrict-label","scope":"GraphTheory/directed-graphs/labeled-directed-graphs","status":"fully_proved","summary":"Let $G$ be a labeled directed graph, $p$ a predicate on its vertices, and $e$ an edge of $G$ whose source and target satisfy $p$, viewed as an edge of the restriction of $G$ to $p$. The label of $e$ in the restriction is its label in $G$.","title":"Label of an edge in a restriction"},{"data_url":"scopes/GraphTheory/directed-graphs/modular-orientations.json","id":"GraphTheory/directed-graphs/modular-orientations","kind":"scope","navigation":"module","parent":30180,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html","scope":"GraphTheory/directed-graphs/modular-orientations","status":"can_prove","title":"Modular orientations"},{"id":"GraphTheory/directed-graphs/modular-orientations/crosses-cut","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.CrossesCut","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#crosses-cut","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"Consider a multigraph with vertex set $V$ and edge set $E$ in which each edge has a first and a second endpoint. For a set $S \\subseteq V$, an edge crosses the cut at $S$ when exactly one of its two endpoints lies in $S$.","title":"Edges crossing a cut"},{"id":"GraphTheory/directed-graphs/modular-orientations/cut-size","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.cutSize","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#cut-size","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"Consider a multigraph with finitely many edges in which each edge has a first and a second endpoint. For a set $S$ of vertices, the size of the cut at $S$ is the number of edges crossing it, that is, having exactly one endpoint in $S$. Distinct parallel edges are counted separately.","title":"Size of a cut"},{"id":"GraphTheory/directed-graphs/modular-orientations/is-four-kedge-connected","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.IsFourKEdgeConnected","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#is-four-kedge-connected","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"For a natural number $k$, a finite multigraph is $4k$-edge-connected when it has at least two vertices and, for every set $S$ of vertices such that both $S$ and its complement are nonempty, at least $4k$ edges cross the cut at $S$, that is, have exactly one endpoint in $S$. Parallel edges are counted separately. For $k \\ge 1$ the condition implies that the multigraph is connected.","title":"4k-edge-connected multigraphs"},{"id":"GraphTheory/directed-graphs/modular-orientations/oriented-tail","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.orientedTail","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#oriented-tail","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"Each edge $e$ of the multigraph has a reference direction from its first endpoint to its second endpoint, and an orientation chooses for each edge whether to reverse it. The tail of $e$ is its first endpoint if $e$ is not reversed and its second endpoint if $e$ is reversed.","title":"Tail of an oriented edge"},{"id":"GraphTheory/directed-graphs/modular-orientations/oriented-head","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.orientedHead","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#oriented-head","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"Each edge $e$ of the multigraph has a reference direction from its first endpoint to its second endpoint, and an orientation chooses for each edge whether to reverse it. The head of $e$ is its second endpoint if $e$ is not reversed and its first endpoint if $e$ is reversed.","title":"Head of an oriented edge"},{"id":"GraphTheory/directed-graphs/modular-orientations/indegree","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.indegree","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#indegree","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"In a multigraph with finitely many edges and a chosen orientation, the indegree of a vertex $v$ is the number of edges whose head is $v$, parallel edges counted separately.","title":"Indegree"},{"id":"GraphTheory/directed-graphs/modular-orientations/outdegree","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.outdegree","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#outdegree","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"In a multigraph with finitely many edges and a chosen orientation, the outdegree of a vertex $v$ is the number of edges whose tail is $v$, parallel edges counted separately.","title":"Outdegree"},{"id":"GraphTheory/directed-graphs/modular-orientations/is-modular-orientation","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.IsModularOrientation","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#is-modular-orientation","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"For a natural number $k$, an orientation of a multigraph with finitely many edges is a modulo $(2k+1)$ orientation when at every vertex the indegree minus the outdegree is congruent to $0$ modulo $2k + 1$.","title":"Modular orientations"},{"id":"GraphTheory/directed-graphs/modular-orientations/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.conjecture","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#conjecture","scope":"GraphTheory/directed-graphs/modular-orientations","status":"fully_proved","summary":"This is the statement that for every integer $k \\ge 1$, every finite multigraph (loops and parallel edges allowed) that has at least two vertices, and in which every partition of the vertices into two nonempty sets is crossed by at least $4k$ edges, admits an orientation of its edges in which, at every vertex, the indegree minus the outdegree is divisible by $2k + 1$.","title":"Jaeger's modular orientation conjecture"},{"id":"GraphTheory/directed-graphs/modular-orientations/conjecture-refuted","kind":"node","lean":"MathlibExt.Combinatorics.JaegerModularOrientationWanted.conjecture_refuted","parent":30202,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/modular-orientations/index.html#conjecture-refuted","scope":"GraphTheory/directed-graphs/modular-orientations","status":"can_prove","summary":"There exist an integer $k \\ge 1$ and a finite multigraph (loops and parallel edges allowed) with at least two vertices, in which every partition of the vertices into two nonempty sets is crossed by at least $4k$ edges, such that no orientation of the edges makes the indegree minus the outdegree of every vertex divisible by $2k + 1$.","title":"Jaeger's modular orientation conjecture is false"},{"data_url":"scopes/GraphTheory/directed-graphs/road-coloring-theorem.json","id":"GraphTheory/directed-graphs/road-coloring-theorem","kind":"scope","navigation":"module","parent":30180,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/road-coloring-theorem/index.html","scope":"GraphTheory/directed-graphs/road-coloring-theorem","status":"fully_proved","title":"Road coloring theorem"},{"id":"GraphTheory/directed-graphs/road-coloring-theorem/road-coloring","kind":"node","lean":"MetaMathlibExt.roadColoring'","parent":30213,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/road-coloring-theorem/index.html#road-coloring","scope":"GraphTheory/directed-graphs/road-coloring-theorem","status":"fully_proved","summary":"Let $V$ be a finite nonempty set and let $\\to$ be a relation on $V$, viewed as a directed graph on $V$ without parallel edges (loops allowed). Suppose that the graph is strongly connected (for all $u, v \\in V$ there is a directed walk, possibly of length $0$, from $u$ to $v$), aperiodic (for every integer $d > 1$ there is a closed directed walk of positive length not divisible by $d$), and of constant positive\u2026","title":"Road coloring theorem for a finite vertex type"},{"id":"GraphTheory/directed-graphs/road-coloring-theorem/road-coloring-2","kind":"node","lean":"MetaMathlibExt.roadColoring","parent":30213,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/road-coloring-theorem/index.html#road-coloring-2","scope":"GraphTheory/directed-graphs/road-coloring-theorem","status":"fully_proved","summary":"Let $V$ be a finite nonempty set and let $\\to$ be a relation on $V$, viewed as a directed graph on $V$ without parallel edges (loops allowed). Suppose that the graph is strongly connected (for all $u, v \\in V$ there is a directed walk, possibly of length $0$, from $u$ to $v$), aperiodic (for every integer $d > 1$ there is a closed directed walk of positive length not divisible by $d$), and of constant positive\u2026","title":"Road coloring theorem"},{"data_url":"scopes/GraphTheory/directed-graphs/robbins-theorem.json","id":"GraphTheory/directed-graphs/robbins-theorem","kind":"scope","navigation":"module","parent":30180,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/robbins-theorem/index.html","scope":"GraphTheory/directed-graphs/robbins-theorem","status":"fully_proved","title":"Robbins' theorem"},{"id":"GraphTheory/directed-graphs/robbins-theorem/robbins-theorem-connected","kind":"node","lean":"MetaMathlibExt.robbins_theorem_connected","parent":30216,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/robbins-theorem/index.html#robbins-theorem-connected","scope":"GraphTheory/directed-graphs/robbins-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite nonempty vertex set $V$. Then $G$ is connected and, after deleting any single edge of $G$, every vertex is still reachable from every vertex (that is, $G$ is 2-edge-connected) if and only if $G$ has a strongly connected orientation: a relation $D$ on $V$ such that $D(u, v)$ holds only when $u$ and $v$ are adjacent, for every edge $uv$ exactly one of $D(u, v)$ and $D(v, u)$\u2026","title":"Robbins' theorem"},{"id":"GraphTheory/directed-graphs/robbins-theorem/robbins-theorem","kind":"node","lean":"MetaMathlibExt.robbins_theorem","parent":30216,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/robbins-theorem/index.html#robbins-theorem","scope":"GraphTheory/directed-graphs/robbins-theorem","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite, possibly empty, vertex set $V$. Then every vertex of $G$ is reachable from every vertex and, after deleting any single edge of $G$, every vertex is still reachable from every vertex, if and only if $G$ has a strongly connected orientation: a relation $D$ on $V$ such that $D(u, v)$ holds only when $u$ and $v$ are adjacent, for every edge $uv$ exactly one of $D(u, v)$ and\u2026","title":"Robbins' theorem, allowing the empty graph"},{"data_url":"scopes/GraphTheory/directed-graphs/tournaments-and-score-sequences.json","id":"GraphTheory/directed-graphs/tournaments-and-score-sequences","kind":"scope","navigation":"module","parent":30180,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"can_prove","title":"Tournaments and score sequences"},{"id":"GraphTheory/directed-graphs/tournaments-and-score-sequences/is-tournament","kind":"node","lean":"MathlibExt.GraphTheory.TournamentScoreSequenceWanted.IsTournament","parent":30219,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html#is-tournament","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"fully_proved","summary":"A tournament on the players $1, \\dots, n$ is a binary relation $T$, where $T(i, j)$ means that $i$ beats $j$, such that no player beats itself and for any two distinct players $i$ and $j$ exactly one of $T(i, j)$ and $T(j, i)$ holds.","title":"Tournament"},{"id":"GraphTheory/directed-graphs/tournaments-and-score-sequences/score","kind":"node","lean":"MathlibExt.GraphTheory.TournamentScoreSequenceWanted.score","parent":30219,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html#score","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"fully_proved","summary":"Let $T$ be a binary relation on the players $1, \\dots, n$, where $T(i, j)$ means that $i$ beats $j$. The score of a player $i$ is the number of players $j$ such that $T(i, j)$ holds, that is, the number of players that $i$ beats.","title":"Score of a player"},{"id":"GraphTheory/directed-graphs/tournaments-and-score-sequences/is-reducible","kind":"node","lean":"MathlibExt.GraphTheory.TournamentScoreSequenceWanted.IsReducible","parent":30219,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html#is-reducible","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"fully_proved","summary":"A tournament is reducible if some nonempty proper set of players loses all games against the complement.","title":"Reducible tournament"},{"id":"GraphTheory/directed-graphs/tournaments-and-score-sequences/is-circular-triad","kind":"node","lean":"MathlibExt.GraphTheory.TournamentScoreSequenceWanted.IsCircularTriad","parent":30219,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html#is-circular-triad","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"fully_proved","summary":"Let $T$ be a binary relation on the players $1, \\dots, n$, where $T(i, j)$ means that $i$ beats $j$. A set $S$ of players is a circular triad of $T$ if $S$ has exactly three elements and every $i \\in S$ beats exactly one $j \\in S$.","title":"Circular triad"},{"id":"GraphTheory/directed-graphs/tournaments-and-score-sequences/egf-factorial-choose","kind":"node","lean":"MathlibExt.GraphTheory.TournamentScoreSequenceWanted.egf_factorial_choose","parent":30219,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html#egf-factorial-choose","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"can_prove","summary":"For natural numbers $a$ and $b$, let $g(n) = n! \\binom{n-a+b}{b}$ for $n \\ge a$ and $g(n) = 0$ for $n < a$, and let $A(x) = \\sum_{n \\ge 0} g(n) x^n / n!$ be its exponential generating function, a formal power series over $\\mathbb{Q}$. Then $(1 - x)^{b+1} A(x) = x^a$; equivalently, $A(x) = x^a (1 - x)^{-(b+1)}$.","title":"Exponential generating function of a shifted binomial sequence"},{"id":"GraphTheory/directed-graphs/tournaments-and-score-sequences/landau-score-sequence","kind":"node","lean":"MathlibExt.GraphTheory.TournamentScoreSequenceWanted.landau_score_sequence","parent":30219,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html#landau-score-sequence","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"can_prove","summary":"Let $s_1 \\le s_2 \\le \\dots \\le s_n$ be a nondecreasing sequence of natural numbers. There is a tournament on $n$ players in which player $i$ has score $s_i$ for every $i$ if and only if $\\sum_{i=1}^{k} s_i \\ge \\binom{k}{2}$ for every $k$ with $1 \\le k \\le n$ and $\\sum_{i=1}^{n} s_i = \\binom{n}{2}$.","title":"Landau's theorem on score sequences"},{"id":"GraphTheory/directed-graphs/tournaments-and-score-sequences/is-reducible-iff","kind":"node","lean":"MathlibExt.GraphTheory.TournamentScoreSequenceWanted.isReducible_iff","parent":30219,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html#is-reducible-iff","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"can_prove","summary":"Let $T$ be a tournament on $n$ players in which the score $s_i$ of player $i$ satisfies $s_1 \\le s_2 \\le \\dots \\le s_n$. Then $T$ is reducible if and only if $\\sum_{i=1}^{k} s_i = \\binom{k}{2}$ for some integer $k$ with $1 \\le k < n$.","title":"Moon's reducibility criterion"},{"id":"GraphTheory/directed-graphs/tournaments-and-score-sequences/card-circular-triads","kind":"node","lean":"MathlibExt.GraphTheory.TournamentScoreSequenceWanted.card_circularTriads","parent":30219,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/tournaments-and-score-sequences/index.html#card-circular-triads","scope":"GraphTheory/directed-graphs/tournaments-and-score-sequences","status":"can_prove","summary":"In a tournament on $n$ players with scores $s_1, \\dots, s_n$, the number of circular triads is $\\binom{n}{3} - \\sum_{i=1}^{n} \\binom{s_i}{2}$.","title":"Moon's count of circular triads"},{"data_url":"scopes/GraphTheory/directed-graphs/transitive-subtournaments.json","id":"GraphTheory/directed-graphs/transitive-subtournaments","kind":"scope","navigation":"module","parent":30180,"result":false,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"can_prove","title":"Transitive subtournaments"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/is-tournament","kind":"node","lean":"MathlibExt.Combinatorics.TransitiveSubtournamentLogErdosProblem1216Wanted.IsTournament","parent":30228,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#is-tournament","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$. A relation $T$ on $\\{0, \\dots, n-1\\}$ is a tournament when $T(x, x)$ holds for no $x$ and, for all distinct $x$ and $y$, exactly one of $T(x, y)$ and $T(y, x)$ holds.","title":"Tournaments"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/is-transitive-on","kind":"node","lean":"MathlibExt.Combinatorics.TransitiveSubtournamentLogErdosProblem1216Wanted.IsTransitiveOn","parent":30228,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#is-transitive-on","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $T$ be a relation on $\\{0, \\dots, n-1\\}$ and let $S$ be a set of its elements. Then $T$ is transitive on $S$ when, for all $x, y, z \\in S$, $T(x, y)$ and $T(y, z)$ imply $T(x, z)$. For a tournament $T$ this says that the subtournament of $T$ on $S$ is transitive.","title":"Transitivity on a vertex set"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/transitive-subtournament-number","kind":"node","lean":"MathlibExt.Combinatorics.TransitiveSubtournamentLogErdosProblem1216Wanted.transitiveSubtournamentNumber","parent":30228,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#transitive-subtournament-number","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, $f(n)$ is the largest $m$ such that every tournament $T$ on $\\{0, \\dots, n-1\\}$ is transitive on some set of exactly $m$ vertices, that is, contains a transitive subtournament on $m$ vertices.","title":"Transitive subtournament number"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/transitive-subtournament-number-eq-log-t","kind":"node","lean":"MathlibExt.Combinatorics.TransitiveSubtournamentLogErdosProblem1216Wanted.transitiveSubtournamentNumberEqLogTwo","parent":30228,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#transitive-subtournament-number-eq-log-t","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"fully_proved","summary":"For every $n \\ge 1$, $f(n) = \\lfloor \\log_2 n \\rfloor + 1$, where $f(n)$ is the largest $m$ such that every tournament on $n$ vertices contains a transitive subtournament on $m$ vertices.","title":"Erd\u0151s Problem 1216 (the Erd\u0151s\u2013Moser formula)"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/transitive-subtournament-number-eq-log-t-2","kind":"node","lean":"MathlibExt.Combinatorics.TransitiveSubtournamentLogErdosProblem1216Wanted.transitiveSubtournamentNumberEqLogTwo_refuted","parent":30228,"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#transitive-subtournament-number-eq-log-t-2","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"can_prove","summary":"It is not the case that $f(n) = \\lfloor \\log_2 n \\rfloor + 1$ for every $n \\ge 1$, where $f(n)$ is the largest $m$ such that every tournament on $n$ vertices contains a transitive subtournament on $m$ vertices.","title":"Erd\u0151s Problem 1216 is false"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/t-star-back-arcs","kind":"node","lean":"MathlibExt.Combinatorics.TournamentErdosHajnalSixVertexAIMCombinatoricsProblem55Wanted.tStarBackArcs","parent":30228,"reading_title":{"format":"inline-parts-v1","parts":[["text","The backward arcs of "],["math","\\(T^*\\)"]],"source":"The backward arcs of $T^*$"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#t-star-back-arcs","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"fully_proved","summary":"The backward arcs of $T^*$. The list of ordered pairs $(5, 0)$, $(5, 2)$, $(4, 1)$, $(3, 0)$ of vertices in $\\{0, \\dots, 5\\}$. These are the arcs of the six-vertex tournament $T^*$ that go from a larger to a smaller vertex: $6 \\to 1$, $6 \\to 3$, $5 \\to 2$ and $4 \\to 1$ in the source's labelling $1, \\dots, 6$, with every vertex lowered by one.","title":"The backward arcs of $T^*$"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/t-star","kind":"node","lean":"MathlibExt.Combinatorics.TournamentErdosHajnalSixVertexAIMCombinatoricsProblem55Wanted.tStar","parent":30228,"reading_title":{"format":"inline-parts-v1","parts":[["text","The six-vertex tournament "],["math","\\(T^*\\)"]],"source":"The six-vertex tournament $T^*$"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#t-star","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"fully_proved","summary":"The six-vertex tournament $T^*$. The tournament $T^*$ is the relation on the vertex set $\\{0, \\dots, 5\\}$ with an arc $i \\to j$ if and only if either $(i, j)$ is one of the backward arcs $(5, 0)$, $(5, 2)$, $(4, 1)$, $(3, 0)$, or $i < j$ and $(j, i)$ is not a backward arc.","title":"The six-vertex tournament $T^*$"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/contains-tstar","kind":"node","lean":"MathlibExt.Combinatorics.TournamentErdosHajnalSixVertexAIMCombinatoricsProblem55Wanted.ContainsTStar","parent":30228,"reading_title":{"format":"inline-parts-v1","parts":[["text","Containing "],["math","\\(T^*\\)"],["text"," as a subtournament"]],"source":"Containing $T^*$ as a subtournament"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#contains-tstar","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"fully_proved","summary":"Containing $T^*$ as a subtournament. A relation $T$ on the vertex set $\\{0, \\dots, n-1\\}$ contains $T^*$ when there is an injective map $f$ from the vertex set $\\{0, \\dots, 5\\}$ of $T^*$ to $\\{0, \\dots, n-1\\}$ such that for all vertices $i, j$ of $T^*$, $i \\to j$ in $T^*$ if and only if $T(f(i), f(j))$ holds.","title":"Containing $T^*$ as a subtournament"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/t-star-erdos-hajnal","kind":"node","lean":"MathlibExt.Combinatorics.TournamentErdosHajnalSixVertexAIMCombinatoricsProblem55Wanted.tStarErdosHajnal","parent":30228,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Erd\u0151s\u2013Hajnal problem for the tournament "],["math","\\(T^*\\)"]],"source":"The Erd\u0151s\u2013Hajnal problem for the tournament $T^*$"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#t-star-erdos-hajnal","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"fully_proved","summary":"The Erd\u0151s\u2013Hajnal problem for the tournament $T^*$. There is a real number $\\varepsilon > 0$ such that for every natural number $n$ and every tournament $T$ on the vertex set $\\{0, \\dots, n-1\\}$ that does not contain $T^*$ as a subtournament, there is a set $S$ of vertices on which $T$ is transitive with $|S| \\ge n^{\\varepsilon}$.","title":"The Erd\u0151s\u2013Hajnal problem for the tournament $T^*$"},{"id":"GraphTheory/directed-graphs/transitive-subtournaments/t-star-erdos-hajnal-holds","kind":"node","lean":"MathlibExt.Combinatorics.TournamentErdosHajnalSixVertexAIMCombinatoricsProblem55Wanted.tStarErdosHajnal_holds","parent":30228,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nguyen\u2013Scott\u2013Seymour: "],["math","\\(T^*\\)"],["text"," has the Erd\u0151s\u2013Hajnal property"]],"source":"Nguyen\u2013Scott\u2013Seymour: $T^*$ has the Erd\u0151s\u2013Hajnal property"},"result":true,"route":"../roadmap/GraphTheory/directed-graphs/transitive-subtournaments/index.html#t-star-erdos-hajnal-holds","scope":"GraphTheory/directed-graphs/transitive-subtournaments","status":"can_prove","summary":"Nguyen\u2013Scott\u2013Seymour: $T^*$ has the Erd\u0151s\u2013Hajnal property. There is a real number $\\varepsilon > 0$ such that for every natural number $n$ and every tournament $T$ on the vertex set $\\{0, \\dots, n-1\\}$ that does not contain $T^*$ as a subtournament, there is a set $S$ of vertices on which $T$ is transitive with $|S| \\ge n^{\\varepsilon}$.","title":"Nguyen\u2013Scott\u2013Seymour: $T^*$ has the Erd\u0151s\u2013Hajnal property"},{"data_url":"scopes/GraphTheory/graph-families-and-operations.json","id":"GraphTheory/graph-families-and-operations","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/graph-families-and-operations/index.html","scope":"GraphTheory/graph-families-and-operations","status":"fully_proved","summary":"Graph products, line graphs and other constructions, together with special families such as expanders and the hypercubic lattice.","title":"Graph families and operations"},{"id":"GraphTheory/graph-families-and-operations/is-line-graph","kind":"node","lean":"MetaMathlibExt.SimpleGraph.IsLineGraph","parent":30239,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/index.html#is-line-graph","scope":"GraphTheory/graph-families-and-operations","status":"fully_proved","summary":"A simple graph $G$ on a finite vertex type is a line graph if there are a natural number $k$ and a simple graph $H$ on the vertex set $\\{0, \\dots, k-1\\}$ such that $G$ is isomorphic to the line graph $L(H)$. Here $L(H)$ has the edges of $H$ as its vertices, and two distinct edges of $H$ are adjacent in $L(H)$ when they share an endpoint.","title":"Graphs that are line graphs"},{"id":"GraphTheory/graph-families-and-operations/lex-product","kind":"node","lean":"SimpleGraph.lexProduct","parent":30239,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/index.html#lex-product","scope":"GraphTheory/graph-families-and-operations","status":"fully_proved","summary":"Given simple graphs $X$ and $Y$, the *lexicographic product* $X[Y]$ of $X$ and $Y$ is the simple graph with vertex set $V(X) \\times V(Y)$ in which two vertices $(x_1, y_1)$ and $(x_2, y_2)$ are adjacent exactly when $x_1$ and $x_2$ are adjacent in $X$, or $x_1 = x_2$ and $y_1$ and $y_2$ are adjacent in $Y$.","title":"Lexicographic product of graphs"},{"data_url":"scopes/GraphTheory/graph-families-and-operations/expander-graphs.json","id":"GraphTheory/graph-families-and-operations/expander-graphs","kind":"scope","navigation":"module","parent":30239,"result":false,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","title":"Expander graphs"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/external-neighbor-set","kind":"node","lean":"SimpleGraph.externalNeighborSet","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#external-neighbor-set","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"For a simple graph $G$ on a vertex set $V$ and a set $A \\subseteq V$, the external neighbourhood of $A$ in $G$ is the set of vertices $v \\notin A$ that are adjacent in $G$ to some vertex of $A$.","title":"External neighbourhood"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/mem-external-neighbor-set","kind":"node","lean":"SimpleGraph.mem_externalNeighborSet","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#mem-external-neighbor-set","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex set $V$, let $A \\subseteq V$ and let $v \\in V$. Then $v$ lies in the external neighbourhood of $A$ in $G$ if and only if $v \\notin A$ and $v$ is adjacent in $G$ to some vertex $a \\in A$.","title":"Membership in the external neighbourhood"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/external-neighbor-set-disjoint","kind":"node","lean":"SimpleGraph.externalNeighborSet_disjoint","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#external-neighbor-set-disjoint","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"For every simple graph $G$ on a vertex set $V$ and every set $A \\subseteq V$, the external neighbourhood of $A$ in $G$ is disjoint from $A$.","title":"The external neighbourhood is disjoint from the set"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/exists-adj-of-mem-external-neighbor-set","kind":"node","lean":"SimpleGraph.exists_adj_of_mem_externalNeighborSet","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#exists-adj-of-mem-external-neighbor-set","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex set $V$ and let $A \\subseteq V$. If $v$ is an external neighbour of $A$ in $G$, then $v$ is adjacent in $G$ to some vertex $a \\in A$.","title":"External neighbours have a neighbour in the set"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/is-cexpander","kind":"node","lean":"SimpleGraph.IsCExpander","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#is-cexpander","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ with $n$ vertices, and let $C$ be a real number. Then $G$ is a $C$-expander if $C > 0$, every set $A \\subseteq V$ with $|A| < n/(2C)$ has an external neighbourhood in $G$ with at least $C|A|$ vertices, and for any two disjoint sets $A, B \\subseteq V$ with $|A| \\ge n/(2C)$ and $|B| \\ge n/(2C)$ some vertex of $A$ is adjacent in $G$ to some vertex of $B$.","title":"C-expander"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/pos","kind":"node","lean":"SimpleGraph.IsCExpander.pos","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#pos","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $C$ be a real number. If $G$ is a $C$-expander, then $C > 0$.","title":"The parameter of a C-expander is positive"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/small-expansion","kind":"node","lean":"SimpleGraph.IsCExpander.small_expansion","parent":30242,"reading_title":{"format":"inline-parts-v1","parts":[["text","Small sets expand in a "],["math","\\(C\\)"],["text","-expander"]],"source":"Small sets expand in a $C$-expander"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#small-expansion","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $C$ be a real number and let $G$ be a $C$-expander on a finite vertex set $V$. Then every set $A \\subseteq V$ with $|A| < |V|/(2C)$ has at least $C|A|$ external neighbours, that is, at least $C|A|$ vertices outside $A$ are adjacent to some vertex of $A$.","title":"Small sets expand in a $C$-expander"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/large-edge","kind":"node","lean":"SimpleGraph.IsCExpander.large_edge","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#large-edge","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ with $n$ vertices, let $C$ be a real number, and suppose $G$ is a $C$-expander. If $A, B \\subseteq V$ are disjoint sets with $|A| \\ge n/(2C)$ and $|B| \\ge n/(2C)$, then some vertex $a \\in A$ is adjacent in $G$ to some vertex $b \\in B$.","title":"Large disjoint sets in a C-expander span an edge"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/is-uniform-vertex-expander","kind":"node","lean":"SimpleGraph.IsUniformVertexExpander","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#is-uniform-vertex-expander","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $h$ be a real number. Then $G$ is a uniform vertex expander with parameter $h$ if $h > 0$ and every set $S \\subseteq V$ with $|S| \\le |V|/2$ has an external neighbourhood in $G$ with at least $h|S|$ vertices.","title":"Uniform vertex expander"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/pos-2","kind":"node","lean":"SimpleGraph.IsUniformVertexExpander.pos","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#pos-2","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $h$ be a real number. If $G$ is a uniform vertex expander with parameter $h$, then $h > 0$.","title":"The parameter of a uniform vertex expander is positive"},{"id":"GraphTheory/graph-families-and-operations/expander-graphs/small-expansion-2","kind":"node","lean":"SimpleGraph.IsUniformVertexExpander.small_expansion","parent":30242,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/expander-graphs/index.html#small-expansion-2","scope":"GraphTheory/graph-families-and-operations/expander-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set $V$ and let $h$ be a real number. If $G$ is a uniform vertex expander with parameter $h$ and $S \\subseteq V$ satisfies $|S| \\le |V|/2$, then the external neighbourhood of $S$ in $G$ has at least $h|S|$ vertices.","title":"Small sets expand in a uniform vertex expander"},{"data_url":"scopes/GraphTheory/graph-families-and-operations/finite-multigraphs.json","id":"GraphTheory/graph-families-and-operations/finite-multigraphs","kind":"scope","navigation":"module","parent":30239,"result":false,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","title":"Finite multigraphs"},{"id":"GraphTheory/graph-families-and-operations/finite-multigraphs/incident","kind":"node","lean":"EndpointMapMultigraph.incident","parent":30254,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html#incident","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","summary":"Let $\\mathrm{end}_1, \\mathrm{end}_2 : E \\to V$ be maps, viewed as an undirected multigraph on $V$ in which each edge $e \\in E$ has endpoints $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$. For a vertex $v \\in V$ and a finite set $F \\subseteq E$, the edges of $F$ incident to $v$ form the finite set of all $e \\in F$ with $\\mathrm{end}_1(e) = v$ or $\\mathrm{end}_2(e) = v$. An edge appears once even when it is a loop.","title":"Edges incident to a vertex"},{"id":"GraphTheory/graph-families-and-operations/finite-multigraphs/degree-within","kind":"node","lean":"EndpointMapMultigraph.degreeWithin","parent":30254,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html#degree-within","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","summary":"Let $\\mathrm{end}_1, \\mathrm{end}_2 : E \\to V$ be maps, viewed as an undirected multigraph on $V$ in which each edge $e \\in E$ has endpoints $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$. For a finite set $F \\subseteq E$ and a vertex $v \\in V$, the degree of $v$ in $F$ is the number of $e \\in F$ with $\\mathrm{end}_1(e) = v$ plus the number of $e \\in F$ with $\\mathrm{end}_2(e) = v$. Each endpoint incidence counts\u2026","title":"Degree within an edge set"},{"id":"GraphTheory/graph-families-and-operations/finite-multigraphs/support","kind":"node","lean":"EndpointMapMultigraph.support","parent":30254,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html#support","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","summary":"Let $\\mathrm{end}_1, \\mathrm{end}_2 : E \\to V$ be maps, viewed as an undirected multigraph on $V$ in which each edge $e \\in E$ has endpoints $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$. The support of a finite set $F \\subseteq E$ is the set of vertices $v \\in V$ such that $\\mathrm{end}_1(e) = v$ or $\\mathrm{end}_2(e) = v$ for some $e \\in F$.","title":"Support of an edge set"},{"id":"GraphTheory/graph-families-and-operations/finite-multigraphs/adjacent-using","kind":"node","lean":"EndpointMapMultigraph.AdjacentUsing","parent":30254,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html#adjacent-using","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","summary":"Let $\\mathrm{end}_1, \\mathrm{end}_2 : E \\to V$ be maps, viewed as an undirected multigraph on $V$ in which each edge $e \\in E$ has endpoints $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$. For a finite set $F \\subseteq E$, vertices $u, v \\in V$ are adjacent using $F$ when some $e \\in F$ has $\\mathrm{end}_1(e) = u$ and $\\mathrm{end}_2(e) = v$, or $\\mathrm{end}_1(e) = v$ and $\\mathrm{end}_2(e) = u$.","title":"Adjacency through an edge set"},{"id":"GraphTheory/graph-families-and-operations/finite-multigraphs/is-connected-on-support","kind":"node","lean":"EndpointMapMultigraph.IsConnectedOnSupport","parent":30254,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html#is-connected-on-support","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","summary":"Let $\\mathrm{end}_1, \\mathrm{end}_2 : E \\to V$ be maps, viewed as an undirected multigraph on $V$ in which each edge $e \\in E$ has endpoints $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$. A finite set $F \\subseteq E$ is connected on its support when for all vertices $u$ and $v$ in the support of $F$ there are vertices $u = w_0, w_1, \\dots, w_k = v$, with $k \\ge 0$, such that $w_{i-1}$ and $w_i$ are adjacent using $F$\u2026","title":"Edge sets connected on their support"},{"id":"GraphTheory/graph-families-and-operations/finite-multigraphs/is-circuit","kind":"node","lean":"EndpointMapMultigraph.IsCircuit","parent":30254,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html#is-circuit","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","summary":"Let $\\mathrm{end}_1, \\mathrm{end}_2 : E \\to V$ be maps, viewed as an undirected multigraph on $V$ in which each edge $e \\in E$ has endpoints $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$. A finite set $F \\subseteq E$ is a circuit when $F$ is nonempty, $F$ is connected on its support, and every vertex in the support of $F$ has degree exactly $2$ in $F$, where degree counts endpoint incidences, so that a loop\u2026","title":"Circuits in an edge set"},{"id":"GraphTheory/graph-families-and-operations/finite-multigraphs/cut","kind":"node","lean":"EndpointMapMultigraph.cut","parent":30254,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html#cut","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","summary":"Let $E$ be a finite set and $\\mathrm{end}_1, \\mathrm{end}_2 : E \\to V$ be maps, viewed as an undirected multigraph on $V$ in which each edge $e \\in E$ has endpoints $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$. For a finite set $X \\subseteq V$, the cut of $X$ is the set of edges $e \\in E$ such that exactly one of $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$ lies in $X$. Loops never cross a cut.","title":"Cut of a vertex set"},{"id":"GraphTheory/graph-families-and-operations/finite-multigraphs/degree-within-eq-card-incident-of-loople","kind":"node","lean":"EndpointMapMultigraph.degreeWithin_eq_card_incident_of_loopless","parent":30254,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/finite-multigraphs/index.html#degree-within-eq-card-incident-of-loople","scope":"GraphTheory/graph-families-and-operations/finite-multigraphs","status":"fully_proved","summary":"Let $\\mathrm{end}_1, \\mathrm{end}_2 : E \\to V$ be maps, viewed as an undirected multigraph on $V$ in which each edge $e \\in E$ has endpoints $\\mathrm{end}_1(e)$ and $\\mathrm{end}_2(e)$. Let $F \\subseteq E$ be a finite set of edges with no loops, that is, $\\mathrm{end}_1(e) \\ne \\mathrm{end}_2(e)$ for every $e \\in F$. Then for every vertex $v \\in V$, the degree of $v$ in $F$ equals the number of edges of $F$ incident\u2026","title":"Degree counts incident edges in a loopless edge set"},{"data_url":"scopes/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs.json","id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","kind":"scope","navigation":"module","parent":30239,"result":false,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","title":"Hypercubic lattice graphs"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/site","kind":"node","lean":"StatMech.Lattice.Site","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#site","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"A site of the `d`-dimensional hypercubic lattice `\u2124^d`, encoded as an integer vector.","title":"Site of the hypercubic lattice"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/nearest-neighbour","kind":"node","lean":"StatMech.Lattice.NearestNeighbour","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nearest neighbours in "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Nearest neighbours in $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#nearest-neighbour","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"Two sites are nearest neighbours when their `\u2113\u00b9` distance `\u2211 i, |x i - y i|` equals one.","title":"Nearest neighbours in $\\mathbb{Z}^d$"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/hypercubic-lattice","kind":"node","lean":"StatMech.Lattice.hypercubicLattice","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hypercubic lattice "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Hypercubic lattice $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#hypercubic-lattice","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The hypercubic lattice `\u2124^d`: the simple graph on `Site d` whose edges join nearest neighbours.","title":"Hypercubic lattice $\\mathbb{Z}^d$"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/box","kind":"node","lean":"StatMech.Lattice.box","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box "],["math","\\((-n, n)^d\\)"],["text"," in the hypercubic lattice"]],"source":"Box $(-n, n)^d$ in the hypercubic lattice"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#box","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The box `[-n, n]^d` of sites whose coordinates all have absolute value at most `n`.","title":"Box $(-n, n)^d$ in the hypercubic lattice"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/vertex-boundary","kind":"node","lean":"StatMech.Lattice.vertexBoundary","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#vertex-boundary","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The inner vertex boundary of `box d n`: sites of `box d n` that are not in `box d (n - 1)`.","title":"Inner vertex boundary of a box"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/edge-boundary","kind":"node","lean":"StatMech.Lattice.edgeBoundary","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#edge-boundary","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The edge boundary of a set `S` of sites: ordered nearest-neighbour pairs with exactly one endpoint in `S`.","title":"Edge boundary of a set of sites"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/finite-subset-box","kind":"node","lean":"StatMech.Lattice.finite_subset_box","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#finite-subset-box","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"Every finite set of sites lies in some box `box d R`.","title":"Every finite set of sites lies in a box"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/reachable-update-add","kind":"node","lean":"StatMech.Lattice.reachable_update_add","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#reachable-update-add","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"In `\u2124\u00b2`, increasing one coordinate by `n` stays in the same component.","title":"Increasing one coordinate stays in the same component"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/reachable-update","kind":"node","lean":"StatMech.Lattice.reachable_update","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#reachable-update","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"In `\u2124\u00b2`, changing one coordinate stays in the same component.","title":"Changing one coordinate stays in the same component"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/hypercubic-lattice-two-reachable","kind":"node","lean":"StatMech.Lattice.hypercubicLattice_two_reachable","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#hypercubic-lattice-two-reachable","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The square lattice `\u2124\u00b2` is connected.","title":"Square lattice is connected"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/box-sv-box-f","kind":"node","lean":"StatMech.Lattice.boxSV_boxF","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box "],["math","\\((-n, n)^d\\)"],["text"," as a finite set of sites"]],"source":"Box $(-n, n)^d$ as a finite set of sites"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#box-sv-box-f","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The box `[-n, n]^d` as a finite set of sites.","title":"Box $(-n, n)^d$ as a finite set of sites"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/box-sv-vb-f","kind":"node","lean":"StatMech.Lattice.boxSV_vbF","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vertex boundary of the box "],["math","\\((-n, n)^d\\)"],["text"," as a finite set"]],"source":"Vertex boundary of the box $(-n, n)^d$ as a finite set"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#box-sv-vb-f","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The vertex boundary of `[-n, n]^d` as a finite set of sites.","title":"Vertex boundary of the box $(-n, n)^d$ as a finite set"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/box-sv-boundary-card","kind":"node","lean":"StatMech.Lattice.boxSV_boundaryCard","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Size of the vertex boundary of the box "],["math","\\((-n, n)^d\\)"]],"source":"Size of the vertex boundary of the box $(-n, n)^d$"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#box-sv-boundary-card","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The number of sites in the vertex boundary of `[-n, n]^d`.","title":"Size of the vertex boundary of the box $(-n, n)^d$"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/box-sv-edge-f","kind":"node","lean":"StatMech.Lattice.boxSV_edgeF","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nearest-neighbour pairs inside the box "],["math","\\((-n, n)^d\\)"]],"source":"Nearest-neighbour pairs inside the box $(-n, n)^d$"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#box-sv-edge-f","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The ordered pairs of nearest-neighbour sites that both lie in `[-n, n]^d`.","title":"Nearest-neighbour pairs inside the box $(-n, n)^d$"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/box-sv-edge-card","kind":"node","lean":"StatMech.Lattice.boxSV_edgeCard","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of nearest-neighbour pairs inside the box "],["math","\\((-n, n)^d\\)"]],"source":"Number of nearest-neighbour pairs inside the box $(-n, n)^d$"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#box-sv-edge-card","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The number of ordered nearest-neighbour pairs inside `[-n, n]^d`.","title":"Number of nearest-neighbour pairs inside the box $(-n, n)^d$"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/box-sv-fwd-dom","kind":"node","lean":"StatMech.Lattice.boxSV_fwdDom","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#box-sv-fwd-dom","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"Sites of `[-n, n]^d` whose successor in the first coordinate direction is still in the box.","title":"Box sites with a forward neighbour in the box"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/pvr-first-idx-at-coord","kind":"node","lean":"StatMech.Lattice.pvr_firstIdxAtCoord","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#pvr-first-idx-at-coord","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The first index at which the walk `w` has `k`-th coordinate equal to `r`.","title":"First index where a walk reaches a coordinate value"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/shift","kind":"node","lean":"StatMech.Lattice.shift","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#shift","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The site obtained from `x` by adding `s` to its `j`-th coordinate.","title":"Shift of a site along a coordinate"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/cand-map","kind":"node","lean":"StatMech.Lattice.candMap","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#cand-map","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The neighbour of `x` in direction `j`, positive if the flag is `true` and negative otherwise.","title":"Neighbour of a site in a coordinate direction"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/cand-finset","kind":"node","lean":"StatMech.Lattice.candFinset","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#cand-finset","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The finite set of the `2d` nearest neighbours of `x`.","title":"Nearest neighbours of a lattice site"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/l1dist","kind":"node","lean":"StatMech.Lattice.l1dist","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\ell^1\\)"],["text"," distance on "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"$\\ell^1$ distance on $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#l1dist","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The `\u2113\u00b9` (graph) distance `\u2211 i, |x i - y i|` between two sites.","title":"$\\ell^1$ distance on $\\mathbb{Z}^d$"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/exterior","kind":"node","lean":"StatMech.Lattice.exterior","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Exterior of a box in "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Exterior of a box in $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#exterior","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The exterior of the box `[-R, R]^d`: sites with some coordinate of absolute value greater than `R`.","title":"Exterior of a box in $\\mathbb{Z}^d$"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/set-on","kind":"node","lean":"StatMech.Lattice.setOn","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#set-on","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The site obtained from `p` by setting every coordinate in `T` to `v`.","title":"Setting a set of coordinates of a lattice site"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/beacon","kind":"node","lean":"StatMech.Lattice.beacon","parent":30263,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#beacon","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"The site `(R + 1, \u2026, R + 1)`, a fixed base point of the exterior of `box d R`.","title":"Base point of the exterior of a box"},{"id":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/unique-infinite-component","kind":"node","lean":"StatMech.Lattice.unique_infinite_component","parent":30263,"reading_title":{"format":"inline-parts-v1","parts":[["text","Complement of a finite set in "],["math","\\(\\mathbb{Z}^d\\)"],["text"," has one infinite component"]],"source":"Complement of a finite set in $\\mathbb{Z}^d$ has one infinite component"},"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs/index.html#unique-infinite-component","scope":"GraphTheory/graph-families-and-operations/hypercubic-lattice-graphs","status":"fully_proved","summary":"For `d \u2265 2`, the complement of a finite set of sites of `\u2124^d` has exactly one infinite connected component.","title":"Complement of a finite set in $\\mathbb{Z}^d$ has one infinite component"},{"data_url":"scopes/GraphTheory/graph-families-and-operations/simplex-graphs.json","id":"GraphTheory/graph-families-and-operations/simplex-graphs","kind":"scope","navigation":"module","parent":30239,"result":false,"route":"../roadmap/GraphTheory/graph-families-and-operations/simplex-graphs/index.html","scope":"GraphTheory/graph-families-and-operations/simplex-graphs","status":"fully_proved","title":"Simplex graphs"},{"id":"GraphTheory/graph-families-and-operations/simplex-graphs/simplex-vertex","kind":"node","lean":"SimpleGraph.SimplexVertex","parent":30289,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/simplex-graphs/index.html#simplex-vertex","scope":"GraphTheory/graph-families-and-operations/simplex-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph on a vertex type $V$. A simplex vertex of $G$ is a finite set $s$ of vertices that is a clique of $G$, meaning that any two distinct elements of $s$ are adjacent in $G$.","title":"Simplex vertices"},{"id":"GraphTheory/graph-families-and-operations/simplex-graphs/empty-vertex","kind":"node","lean":"SimpleGraph.emptyVertex","parent":30289,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/simplex-graphs/index.html#empty-vertex","scope":"GraphTheory/graph-families-and-operations/simplex-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph. The empty simplex vertex of $G$ is the empty set of vertices, which is a clique of $G$.","title":"Empty simplex vertex"},{"id":"GraphTheory/graph-families-and-operations/simplex-graphs/singleton-vertex","kind":"node","lean":"SimpleGraph.singletonVertex","parent":30289,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/simplex-graphs/index.html#singleton-vertex","scope":"GraphTheory/graph-families-and-operations/simplex-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph. For a vertex $v$ of $G$, the singleton simplex vertex of $v$ is the one-element set containing $v$, which is a clique of $G$.","title":"Singleton simplex vertex"},{"id":"GraphTheory/graph-families-and-operations/simplex-graphs/simplex-graph","kind":"node","lean":"SimpleGraph.simplexGraph","parent":30289,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/simplex-graphs/index.html#simplex-graph","scope":"GraphTheory/graph-families-and-operations/simplex-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph. The simplex graph of $G$ is the simple graph whose vertices are the simplex vertices of $G$ (the finite cliques of $G$), in which two cliques $s$ and $t$ are adjacent exactly when their symmetric difference $s \\triangle t$ has exactly one element.","title":"Simplex graph"},{"id":"GraphTheory/graph-families-and-operations/simplex-graphs/adj-iff","kind":"node","lean":"SimpleGraph.adj_iff","parent":30289,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/simplex-graphs/index.html#adj-iff","scope":"GraphTheory/graph-families-and-operations/simplex-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph. Two simplex vertices $s$ and $t$ of $G$ are adjacent in the simplex graph of $G$ if and only if their symmetric difference $s \\triangle t$ has exactly one element.","title":"Adjacency in the simplex graph"},{"id":"GraphTheory/graph-families-and-operations/simplex-graphs/empty-adj-singleton","kind":"node","lean":"SimpleGraph.empty_adj_singleton","parent":30289,"result":true,"route":"../roadmap/GraphTheory/graph-families-and-operations/simplex-graphs/index.html#empty-adj-singleton","scope":"GraphTheory/graph-families-and-operations/simplex-graphs","status":"fully_proved","summary":"Let $G$ be a simple graph. For every vertex $v$ of $G$, the empty simplex vertex of $G$ and the singleton simplex vertex of $v$ are adjacent in the simplex graph of $G$.","title":"The empty clique is adjacent to every singleton"},{"data_url":"scopes/GraphTheory/random-graphs.json","id":"GraphTheory/random-graphs","kind":"scope","parent":28786,"result":false,"route":"../roadmap/GraphTheory/random-graphs/index.html","scope":"GraphTheory/random-graphs","status":"can_prove","summary":"Random graph models such as Erd\u0151s\u2013R\u00e9nyi, random regular and random k-out graphs, the typical values of their invariants (chromatic number, independence ratio, connectivity, spectra), and graphons as limits of dense graph sequences.","title":"Random graphs and graph limits"},{"data_url":"scopes/GraphTheory/random-graphs/chromatic-number-of-random-graphs.json","id":"GraphTheory/random-graphs/chromatic-number-of-random-graphs","kind":"scope","navigation":"module","parent":30296,"result":false,"route":"../roadmap/GraphTheory/random-graphs/chromatic-number-of-random-graphs/index.html","scope":"GraphTheory/random-graphs/chromatic-number-of-random-graphs","status":"can_prove","title":"Chromatic number of random graphs"},{"id":"GraphTheory/random-graphs/chromatic-number-of-random-graphs/prob-diff-gt","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem625Wanted.probDiffGT","parent":30297,"result":true,"route":"../roadmap/GraphTheory/random-graphs/chromatic-number-of-random-graphs/index.html#prob-diff-gt","scope":"GraphTheory/random-graphs/chromatic-number-of-random-graphs","status":"fully_proved","summary":"For natural numbers $C$ and $n$, the proportion of simple graphs $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ with $\\chi(G) - \\zeta(G) > C$, where $\\chi$ is the chromatic number and $\\zeta$ the cochromatic number; equivalently, the probability of this event for a uniformly random labelled graph, $G \\sim G(n, 1/2)$.","title":"Probability of a large chromatic\u2013cochromatic gap"},{"id":"GraphTheory/random-graphs/chromatic-number-of-random-graphs/conjecture","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem625Wanted.conjecture","parent":30297,"result":true,"route":"../roadmap/GraphTheory/random-graphs/chromatic-number-of-random-graphs/index.html#conjecture","scope":"GraphTheory/random-graphs/chromatic-number-of-random-graphs","status":"fully_proved","summary":"The statement that for every natural number $C$, the probability that a uniformly random simple graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ (that is, $G \\sim G(n, 1/2)$) satisfies $\\chi(G) - \\zeta(G) > C$ tends to $1$ as $n \\to \\infty$, where $\\chi$ is the chromatic number and $\\zeta$ the cochromatic number. In other words, $\\chi(G) - \\zeta(G) \\to \\infty$ with high probability.","title":"Erd\u0151s Problem 625"},{"id":"GraphTheory/random-graphs/chromatic-number-of-random-graphs/conjecture-holds","kind":"node","lean":"MathlibExt.Combinatorics.ErdosProblem625Wanted.conjecture_holds","parent":30297,"result":true,"route":"../roadmap/GraphTheory/random-graphs/chromatic-number-of-random-graphs/index.html#conjecture-holds","scope":"GraphTheory/random-graphs/chromatic-number-of-random-graphs","status":"can_prove","summary":"For every natural number $C$, the probability that a uniformly random simple graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ (that is, $G \\sim G(n, 1/2)$) satisfies $\\chi(G) - \\zeta(G) > C$ tends to $1$ as $n \\to \\infty$, where $\\chi$ is the chromatic number and $\\zeta$ the cochromatic number.","title":"Chromatic minus cochromatic number of a random graph tends to infinity"},{"data_url":"scopes/GraphTheory/random-graphs/connectivity-of-random-k-out-graphs.json","id":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs","kind":"scope","navigation":"module","parent":30296,"result":false,"route":"../roadmap/GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/index.html","scope":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs","status":"can_prove","title":"Connectivity of random k-out graphs"},{"id":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/kout-reachable","kind":"node","lean":"MathlibExt.Probability.ScottishBook38Wanted.KOutReachable","parent":30301,"result":true,"route":"../roadmap/GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/index.html#kout-reachable","scope":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs","status":"fully_proved","summary":"Let $c$ assign to each person $i$ among $N$ people a set $c(i)$ of people, the friends of $i$. A person $b$ is reachable from a person $a$ if there are $m \\ge 0$ and people $p_0 = a, p_1, \\dots, p_m = b$ such that for each $j < m$, $p_j$ is a friend of $p_{j+1}$ or $p_{j+1}$ is a friend of $p_j$.","title":"Reachability by a chain of friends"},{"id":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/kout-connected","kind":"node","lean":"MathlibExt.Probability.ScottishBook38Wanted.KOutConnected","parent":30301,"result":true,"route":"../roadmap/GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/index.html#kout-connected","scope":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs","status":"fully_proved","summary":"Let $c$ assign to each person $i$ among $N$ people a set $c(i)$ of people, the friends of $i$. The assignment $c$ connects all $N$ people when, for all people $a$ and $b$, there are $m \\ge 0$ and people $p_0 = a, p_1, \\dots, p_m = b$ such that for each $j < m$, $p_j$ is a friend of $p_{j+1}$ or $p_{j+1}$ is a friend of $p_j$.","title":"Assignments that connect everyone"},{"id":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/kout-connected-prob","kind":"node","lean":"MathlibExt.Probability.ScottishBook38Wanted.KOutConnectedProb","parent":30301,"reading_title":{"format":"inline-parts-v1","parts":[["text","Probability that a random "],["math","\\(k\\)"],["text","-out assignment connects everyone"]],"source":"Probability that a random $k$-out assignment connects everyone"},"result":true,"route":"../roadmap/GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/index.html#kout-connected-prob","scope":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs","status":"fully_proved","summary":"For natural numbers $k$ and $N$, $P_{kN}$ is the number of assignments that give each of $N$ people a set of exactly $k$ friends other than themselves and connect all $N$ people, divided by the number of all assignments that give each person a set of exactly $k$ friends other than themselves. When there is no such assignment, $P_{kN} = 0$.","title":"Probability that a random $k$-out assignment connects everyone"},{"id":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/k-out-connectivity-threshold","kind":"node","lean":"MathlibExt.Probability.ScottishBook38Wanted.k_out_connectivity_threshold","parent":30301,"reading_title":{"format":"inline-parts-v1","parts":[["text","Connectivity threshold for random "],["math","\\(k\\)"],["text","-out friendship graphs"]],"source":"Connectivity threshold for random $k$-out friendship graphs"},"result":true,"route":"../roadmap/GraphTheory/random-graphs/connectivity-of-random-k-out-graphs/index.html#k-out-connectivity-threshold","scope":"GraphTheory/random-graphs/connectivity-of-random-k-out-graphs","status":"can_prove","summary":"For natural numbers $k$ and $N$, let $P_{kN}$ be the proportion of the assignments giving each of $N$ people exactly $k$ friends other than themselves that connect all $N$ people. Then $P_{kN} \\to 1$ as $N \\to \\infty$ for every $k \\ge 2$, and $P_{1N} \\to 0$ as $N \\to \\infty$.","title":"Connectivity threshold for random $k$-out friendship graphs"},{"data_url":"scopes/GraphTheory/random-graphs/finitely-forcible-graphons.json","id":"GraphTheory/random-graphs/finitely-forcible-graphons","kind":"scope","navigation":"module","parent":30296,"result":false,"route":"../roadmap/GraphTheory/random-graphs/finitely-forcible-graphons/index.html","scope":"GraphTheory/random-graphs/finitely-forcible-graphons","status":"can_prove","title":"Finitely forcible graphons"},{"id":"GraphTheory/random-graphs/finitely-forcible-graphons/is-graphon","kind":"node","lean":"MathlibExt.Combinatorics.FinitelyForcibleOptimumOWR4212Problem011Wanted.IsGraphon","parent":30306,"result":true,"route":"../roadmap/GraphTheory/random-graphs/finitely-forcible-graphons/index.html#is-graphon","scope":"GraphTheory/random-graphs/finitely-forcible-graphons","status":"fully_proved","summary":"A graphon is a function $W$ of two real variables that is jointly measurable (measurable as a function on $\\mathbb{R}^2$), symmetric ($W(x, y) = W(y, x)$ for all real $x, y$) and takes values in $[0, 1]$ at every point of $\\mathbb{R}^2$.","title":"Graphon"},{"id":"GraphTheory/random-graphs/finitely-forcible-graphons/labelled-density","kind":"node","lean":"MathlibExt.Combinatorics.FinitelyForcibleOptimumOWR4212Problem011Wanted.labelledDensity","parent":30306,"result":true,"route":"../roadmap/GraphTheory/random-graphs/finitely-forcible-graphons/index.html#labelled-density","scope":"GraphTheory/random-graphs/finitely-forcible-graphons","status":"fully_proved","summary":"For a graph $G$ on the vertex set $\\{0, \\dots, k-1\\}$ and a function $W$ of two real variables, the labelled density of $G$ in $W$ is","title":"Labelled density"},{"id":"GraphTheory/random-graphs/finitely-forcible-graphons/density","kind":"node","lean":"MathlibExt.Combinatorics.FinitelyForcibleOptimumOWR4212Problem011Wanted.density","parent":30306,"result":true,"route":"../roadmap/GraphTheory/random-graphs/finitely-forcible-graphons/index.html#density","scope":"GraphTheory/random-graphs/finitely-forcible-graphons","status":"fully_proved","summary":"For a graph $H$ on the vertex set $\\{0, \\dots, k-1\\}$ and a function $W$ of two real variables, the density $d(H, W)$ is the sum of the labelled densities $\\tau(G, W)$ over all graphs $G$ on $\\{0, \\dots, k-1\\}$ that are isomorphic to $H$.","title":"Density of a graph in a graphon"},{"id":"GraphTheory/random-graphs/finitely-forcible-graphons/weakly-isomorphic","kind":"node","lean":"MathlibExt.Combinatorics.FinitelyForcibleOptimumOWR4212Problem011Wanted.WeaklyIsomorphic","parent":30306,"result":true,"route":"../roadmap/GraphTheory/random-graphs/finitely-forcible-graphons/index.html#weakly-isomorphic","scope":"GraphTheory/random-graphs/finitely-forcible-graphons","status":"fully_proved","summary":"Two functions $W_1$ and $W_2$ of two real variables are weakly isomorphic if $d(H, W_1) = d(H, W_2)$ for every natural number $k$ and every graph $H$ on the vertex set $\\{0, \\dots, k-1\\}$, where $d(H, W)$ is the density of $H$ in $W$.","title":"Weak isomorphism"},{"id":"GraphTheory/random-graphs/finitely-forcible-graphons/is-finitely-forcible","kind":"node","lean":"MathlibExt.Combinatorics.FinitelyForcibleOptimumOWR4212Problem011Wanted.IsFinitelyForcible","parent":30306,"result":true,"route":"../roadmap/GraphTheory/random-graphs/finitely-forcible-graphons/index.html#is-finitely-forcible","scope":"GraphTheory/random-graphs/finitely-forcible-graphons","status":"fully_proved","summary":"A function $W$ of two real variables is finitely forcible if there are finitely many graphs $H_1, \\dots, H_\\ell$ such that every graphon $W'$ with $d(H_i, W') = d(H_i, W)$ for all $i$ is weakly isomorphic to $W$.","title":"Finitely forcible graphon"},{"id":"GraphTheory/random-graphs/finitely-forcible-graphons/finitely-forcible-optimum","kind":"node","lean":"MathlibExt.Combinatorics.FinitelyForcibleOptimumOWR4212Problem011Wanted.finitelyForcibleOptimum","parent":30306,"result":true,"route":"../roadmap/GraphTheory/random-graphs/finitely-forcible-graphons/index.html#finitely-forcible-optimum","scope":"GraphTheory/random-graphs/finitely-forcible-graphons","status":"fully_proved","summary":"The proposition that for all graphs $H_1, \\dots, H_\\ell$ and reals $d_1, \\dots, d_\\ell$, if some graphon $W$ satisfies $d(H_i, W) = d_i$ for every $i$, then some finitely forcible graphon $W_0$ satisfies $d(H_i, W_0) = d_i$ for every $i$.","title":"Lov\u00e1sz's finitely forcible optimum conjecture"},{"id":"GraphTheory/random-graphs/finitely-forcible-graphons/finitely-forcible-optimum-refuted","kind":"node","lean":"MathlibExt.Combinatorics.FinitelyForcibleOptimumOWR4212Problem011Wanted.finitelyForcibleOptimum_refuted","parent":30306,"result":true,"route":"../roadmap/GraphTheory/random-graphs/finitely-forcible-graphons/index.html#finitely-forcible-optimum-refuted","scope":"GraphTheory/random-graphs/finitely-forcible-graphons","status":"can_prove","summary":"There are graphs $H_1, \\dots, H_\\ell$ and reals $d_1, \\dots, d_\\ell$ such that some graphon $W$ satisfies $d(H_i, W) = d_i$ for every $i$, but no finitely forcible graphon $W_0$ satisfies $d(H_i, W_0) = d_i$ for every $i$.","title":"Lov\u00e1sz's finitely forcible optimum conjecture is false"},{"data_url":"scopes/GraphTheory/random-graphs/random-regular-graphs.json","id":"GraphTheory/random-graphs/random-regular-graphs","kind":"scope","navigation":"module","parent":30296,"result":false,"route":"../roadmap/GraphTheory/random-graphs/random-regular-graphs/index.html","scope":"GraphTheory/random-graphs/random-regular-graphs","status":"can_prove","title":"Random regular graphs"},{"id":"GraphTheory/random-graphs/random-regular-graphs/regular-graphs","kind":"node","lean":"MathlibExt.Combinatorics.RandomRegularIndependenceRatioAMR11Problem0018Wanted.regularGraphs","parent":30314,"result":true,"route":"../roadmap/GraphTheory/random-graphs/random-regular-graphs/index.html#regular-graphs","scope":"GraphTheory/random-graphs/random-regular-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $d$, the finite set of all $d$-regular simple graphs on the vertex set $\\{0, \\dots, n - 1\\}$; the uniformly random $d$-regular graph on $n$ vertices is uniform on this set. It is empty when $nd$ is odd or $1 \\le n \\le d$.","title":"Regular simple graphs on a labelled vertex set"},{"id":"GraphTheory/random-graphs/random-regular-graphs/ratio-deviation-prob","kind":"node","lean":"MathlibExt.Combinatorics.RandomRegularIndependenceRatioAMR11Problem0018Wanted.ratioDeviationProb","parent":30314,"result":true,"route":"../roadmap/GraphTheory/random-graphs/random-regular-graphs/index.html#ratio-deviation-prob","scope":"GraphTheory/random-graphs/random-regular-graphs","status":"fully_proved","summary":"For natural numbers $n$ and $d$ and real numbers $c$ and $\\varepsilon$, the probability that the independence ratio $\\alpha(G)/n$ of a uniformly random $d$-regular simple graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ differs from $c$ by more than $\\varepsilon$, where $\\alpha(G)$ is the independence number of $G$: the number of $d$-regular simple graphs $G$ on this vertex set with\u2026","title":"Deviation probability of the independence ratio of a random regular graph"},{"id":"GraphTheory/random-graphs/random-regular-graphs/independence-ratio-converges","kind":"node","lean":"MathlibExt.Combinatorics.RandomRegularIndependenceRatioAMR11Problem0018Wanted.independenceRatioConverges","parent":30314,"result":true,"route":"../roadmap/GraphTheory/random-graphs/random-regular-graphs/index.html#independence-ratio-converges","scope":"GraphTheory/random-graphs/random-regular-graphs","status":"fully_proved","summary":"The statement that for every integer $d \\ge 1$ there is a real number $c$ such that, for every $\\varepsilon > 0$, the probability that the independence ratio $\\alpha(G)/n$ of a uniformly random $d$-regular simple graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ differs from $c$ by more than $\\varepsilon$ tends to $0$ as $n \\to \\infty$. Here $\\alpha(G)$ is the independence number of $G$, the largest size of an\u2026","title":"Bollob\u00e1s's question on the independence ratio of random regular graphs"},{"id":"GraphTheory/random-graphs/random-regular-graphs/independence-ratio-converges-holds","kind":"node","lean":"MathlibExt.Combinatorics.RandomRegularIndependenceRatioAMR11Problem0018Wanted.independenceRatioConverges_holds","parent":30314,"result":true,"route":"../roadmap/GraphTheory/random-graphs/random-regular-graphs/index.html#independence-ratio-converges-holds","scope":"GraphTheory/random-graphs/random-regular-graphs","status":"can_prove","summary":"For every integer $d \\ge 1$ there is a real number $c$ such that, for every $\\varepsilon > 0$, the probability that the independence ratio $\\alpha(G)/n$ of a uniformly random $d$-regular simple graph $G$ on the vertex set $\\{0, \\dots, n - 1\\}$ differs from $c$ by more than $\\varepsilon$ tends to $0$ as $n \\to \\infty$. Here $\\alpha(G)$ is the independence number of $G$, the largest size of an independent set of\u2026","title":"The independence ratio of random regular graphs converges"},{"id":"GraphTheory/random-graphs/random-regular-graphs/nonsingular-fraction","kind":"node","lean":"MathlibExt.Combinatorics.RegularGraphAdjacencyNonsingularOWR12861Problem026Wanted.nonsingularFraction","parent":30314,"result":true,"route":"../roadmap/GraphTheory/random-graphs/random-regular-graphs/index.html#nonsingular-fraction","scope":"GraphTheory/random-graphs/random-regular-graphs","status":"fully_proved","summary":"For natural numbers $d$ and $n$, this is the real number obtained by dividing the number of $d$-regular simple graphs on the vertex set $\\{0, 1, \\ldots, n-1\\}$ whose adjacency matrix has nonzero determinant over $\\mathbb{Q}$ by the number of all $d$-regular simple graphs on that vertex set; it is $0$ when there is no $d$-regular simple graph on that vertex set. It is the probability that the adjacency matrix of the\u2026","title":"Proportion of regular graphs with nonsingular adjacency matrix"},{"id":"GraphTheory/random-graphs/random-regular-graphs/regular-graph-adjacency-nonsingular","kind":"node","lean":"MathlibExt.Combinatorics.RegularGraphAdjacencyNonsingularOWR12861Problem026Wanted.regularGraphAdjacencyNonsingular","parent":30314,"result":true,"route":"../roadmap/GraphTheory/random-graphs/random-regular-graphs/index.html#regular-graph-adjacency-nonsingular","scope":"GraphTheory/random-graphs/random-regular-graphs","status":"fully_proved","summary":"This is the statement that for every natural number $d \\ge 3$, the probability that the adjacency matrix of the uniformly random $d$-regular simple graph on $n$ labeled vertices is nonsingular tends to $1$ as $n \\to \\infty$, where $n$ runs over the natural numbers with $nd$ even. Here the probability is the proportion of the $d$-regular simple graphs on $\\{0, 1, \\ldots, n-1\\}$ whose adjacency matrix has nonzero\u2026","title":"Vu's problem on nonsingular adjacency matrices of random regular graphs"},{"id":"GraphTheory/random-graphs/random-regular-graphs/regular-graph-adjacency-nonsingular-hold","kind":"node","lean":"MathlibExt.Combinatorics.RegularGraphAdjacencyNonsingularOWR12861Problem026Wanted.regularGraphAdjacencyNonsingular_holds","parent":30314,"result":true,"route":"../roadmap/GraphTheory/random-graphs/random-regular-graphs/index.html#regular-graph-adjacency-nonsingular-hold","scope":"GraphTheory/random-graphs/random-regular-graphs","status":"can_prove","summary":"For every natural number $d \\ge 3$, the probability that the adjacency matrix of the uniformly random $d$-regular simple graph on $n$ labeled vertices has nonzero determinant (over $\\mathbb{Q}$, equivalently over $\\mathbb{R}$) tends to $1$ as $n \\to \\infty$ along the natural numbers $n$ with $nd$ even.","title":"Nonsingularity of adjacency matrices of random regular graphs"},{"area":"Discrete Mathematics & Computer Science","data_url":"chapters/TheoryOfComputation.json","id":"TheoryOfComputation","kind":"scope","parent":0,"result":false,"route":"../roadmap/TheoryOfComputation/index.html","scope":"TheoryOfComputation","status":"can_prove","summary":"The mathematical study of what can be computed and at what cost: machine models, computability and undecidability, automata and formal languages, and computational complexity.","title":"Theory of Computation"},{"data_url":"scopes/TheoryOfComputation/np-completeness.json","id":"TheoryOfComputation/np-completeness","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/index.html","scope":"TheoryOfComputation/np-completeness","status":"can_prove","summary":"The Cook\u2013Levin theorem and NP-completeness of specific decision problems, by the problem's domain; includes both machine-level Karp reductions and stated NP-completeness results.","title":"NP-completeness"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin.json","id":"TheoryOfComputation/np-completeness/cook-levin","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin","status":"can_prove","summary":"The Cook\u2013Levin theorem and its machinery: SAT verifiers, dense and binary CNF encodings and converters between them, tableau formulas and the generator machines that emit them.","title":"Cook\u2013Levin theorem"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-sat-np-complete","kind":"node","lean":"MetaMathlibExt.cook_levin_SAT_np_complete","parent":30324,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/index.html#cook-levin-sat-np-complete","scope":"TheoryOfComputation/np-completeness/cook-levin","status":"can_prove","summary":"Let SAT be the decision problem on bitstrings that accepts exactly the encodings of satisfiable CNF formulas. Then SAT belongs to NP, and every decision problem $L$ in NP is polynomial-time many-one reducible to SAT: there is a function $f$ from bitstrings to bitstrings, computable in polynomial time, such that $L(x) = \\mathrm{SAT}(f(x))$ for every bitstring $x$.","title":"Cook\u2013Levin theorem: SAT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","title":"Cook-Levin: the counter-machine view shared by the clause generators"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/ms","kind":"node","lean":"Cslib.Complexity.CookLevin.CounterView.MS","parent":30326,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html#ms","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","summary":"The stages of the counter-machine multiplication routine form the four-element type with elements head, body, l2 and rest.","title":"Stages of the multiplication routine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/sk","kind":"node","lean":"Cslib.Complexity.CookLevin.CounterView.SK","parent":30326,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html#sk","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","summary":"For a type $C$, the stacks of a counter machine with counters indexed by $C$ are an output stack $\\mathrm{out}$ and one counter stack $c_x$ for each $x \\in C$.","title":"Stacks of a counter machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/s","kind":"node","lean":"Cslib.Complexity.CookLevin.CounterView.S\u0393","parent":30326,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html#s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","summary":"For a type $C$, the output stack $\\mathrm{out}$ of a counter machine with counters indexed by $C$ has the Booleans as its alphabet, and each counter stack $c_x$ has a one-element alphabet.","title":"Alphabets of a counter machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/st","kind":"node","lean":"Cslib.Complexity.CookLevin.CounterView.st","parent":30326,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","summary":"For a type $C$, counter values $f : C \\to \\mathbb{N}$ and a list $o$ of Booleans, $\\mathrm{st}(f, o)$ is the stack contents of a counter machine whose output stack $\\mathrm{out}$ holds $o$ and whose counter stack $c_x$ holds exactly $f(x)$ copies of the unit symbol, for each $x \\in C$.","title":"Stacks holding given counter values"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/mul-s","kind":"node","lean":"Cslib.Complexity.CookLevin.CounterView.mulS","parent":30326,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html#mul-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","summary":"Let $a, a_d, b, c, t$ be counters, let $\\mathrm{mk}$ assign a label to each stage of the multiplication routine, and let $\\mathrm{next}$ be a label. The routine assigns to each stage the following counter-machine instruction. At head: pop a unit from $a$, then push a unit onto $a_d$ and go to $\\mathrm{mk}(\\mathrm{body})$, or go to $\\mathrm{mk}(\\mathrm{rest})$ if $a$ was empty. At body: pop a unit from $b$, then\u2026","title":"The multiplication routine of a counter machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/gap-s","kind":"node","lean":"Cslib.Complexity.CookLevin.CounterView.gapS","parent":30326,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html#gap-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","summary":"Let $M$ be a counter machine, let $P, Q, T_1, T_2$ be pairwise distinct counters, let $y$ be a list of Booleans, let $\\mathrm{dec}$, $\\mathrm{cmp}$, $\\mathrm{emit}$, $z$, $\\mathrm{next}$ be labels, and let $r$, $r'$, $\\mathrm{out}$ assign a label to each comparison outcome $\\omega$, which is one of $\\mathrm{lt}$, $\\mathrm{eq}$, $\\mathrm{gt}$. In the instructions below, every pop sets the flag to true if a unit was\u2026","title":"Gap emission in the counter view"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/bud-st","kind":"node","lean":"Cslib.Complexity.CookLevin.CounterView.bud_st","parent":30326,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html#bud-st","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","summary":"Let $M$ be a counter machine, let $K, X \\in \\mathbb{N}$, let $F$ and $F'$ be counter values with $F = F'$, let $l$ be a label or the halting state, $v$ a flag, $o$ a list of Booleans and $d$ a configuration. If there is $n$ with $K + n \\le X$ such that $M$ goes in exactly $n$ steps from the configuration with label $l$, flag $v$ and stacks $\\mathrm{st}(F', o)$ to $d$, then the same holds from the configuration with\u2026","title":"Budgeted runs transport along equal counter values"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/st-loop","kind":"node","lean":"Cslib.Complexity.CookLevin.CounterView.stLoop","parent":30326,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th/index.html#st-loop","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-counter-machine-view-shared-by-th","status":"fully_proved","summary":"Let $M$ be a counter machine, let $c \\ne c_d$ be counters, and let $\\mathrm{head}$, $\\mathrm{body}$, $\\mathrm{rest}$, $\\mathrm{next}$ be labels such that at $\\mathrm{head}$, $M$ pops a unit from $c$, then pushes a unit onto $c_d$ and goes to $\\mathrm{body}$, or goes to $\\mathrm{rest}$ if $c$ was empty; and at $\\mathrm{rest}$, $M$ pops a unit from $c_d$, then pushes a unit onto $c$ and stays at $\\mathrm{rest}$, or\u2026","title":"A runtime `for` loop in the counter view"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas.json","id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","title":"Binary encoding of CNF formulas"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/read-unary-length","kind":"node","lean":"Cslib.CookLevin.readUnaryLength","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#read-unary-length","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"On a list of Booleans that begins with $k$ copies of $\\mathrm{true}$ followed by $\\mathrm{false}$, the reader returns $k$ together with the rest of the list after that $\\mathrm{false}$. On a list containing no $\\mathrm{false}$, including the empty list, it fails.","title":"Reading a unary length prefix"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/encode-nat-prefix","kind":"node","lean":"Cslib.CookLevin.encodeNatPrefix","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#encode-nat-prefix","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"The code of $n \\in \\mathbb{N}$ is the self-delimiting block of the binary representation of $n$ (least significant bit first, with $0$ represented by the empty list): each bit $c$ is written as the two bits $\\mathrm{true}, c$, and the block ends with a terminating $\\mathrm{false}$.","title":"Prefix-free code of a natural number"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/decode-nat-prefix","kind":"node","lean":"Cslib.CookLevin.decodeNatPrefix","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#decode-nat-prefix","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"Given a bit string $w$, writing $1$ for true and $0$ for false, the decoder reads one self-delimiting block $1\\,b_1\\,1\\,b_2 \\cdots 1\\,b_k\\,0$ from the front of $w$, with payload $b_1 \\cdots b_k$ and remaining bits $r$. If the payload is the binary expansion of some natural number $n$, least significant bit first and without trailing zeros (empty for $n = 0$), the result is $(n, r)$. Decoding fails if $w$ does not\u2026","title":"Decoding a prefix-coded natural number"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/encode-lit","kind":"node","lean":"Cslib.CookLevin.encodeLit","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#encode-lit","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"A literal $(n, b)$, with variable index $n \\in \\mathbb{N}$ and polarity $b \\in \\{\\mathrm{false}, \\mathrm{true}\\}$, is encoded as the prefix-free code of $n$ followed by the single bit $b$.","title":"Binary code of a literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/decode-lit","kind":"node","lean":"Cslib.CookLevin.decodeLit","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#decode-lit","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"Given a bit string $w$, the decoder reads a prefix-coded natural number $n$ from the front of $w$ and then one more bit $b$. The result is the literal $(n, b)$, with variable index $n$ and polarity $b$, together with the bits left after $b$. Decoding fails if no prefix-coded natural number can be read or no bit follows it.","title":"Decoding a literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/decode-lits","kind":"node","lean":"Cslib.CookLevin.decodeLits","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#decode-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"For a natural number $n$ and a bit string $w$, the decoder reads $n$ literals one after another from the front of $w$ with the literal decoder. The result is the clause made of these literals, in order, together with the bits left after them; for $n = 0$ it is the empty clause together with $w$. Decoding fails if one of the $n$ literals cannot be read.","title":"Decoding a fixed number of literals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/encode-clause","kind":"node","lean":"Cslib.CookLevin.encodeClause","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#encode-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, the code of a clause $C = (\\ell_1, \\dots, \\ell_k)$ is $1^k 0$ ($k$ ones followed by a zero) followed by the binary codes of $\\ell_1, \\dots, \\ell_k$ in order, where a literal $(x, b)$ with variable index $x$ and polarity $b$ is coded by the prefix-free code of $x$ followed by the bit $b$.","title":"Length-prefixed binary code of a clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/decode-clause","kind":"node","lean":"Cslib.CookLevin.decodeClause","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#decode-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"Given a bit string $w$, writing $1$ for true and $0$ for false, the decoder reads a unary length prefix $1^n 0$ ($n$ ones followed by a zero) from the front of $w$, then reads exactly $n$ literals from the remaining bits with the literal decoder. The result is the clause made of these literals, in order, together with the bits left after them. Decoding fails if $w$ does not begin with such a prefix or one of the\u2026","title":"Decoding a length-prefixed clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/decode-clauses","kind":"node","lean":"Cslib.CookLevin.decodeClauses","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#decode-clauses","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"For a natural number $n$ and a bit string $w$, the decoder reads $n$ clauses one after another from the front of $w$ with the length-prefixed clause decoder. The result is the CNF formula made of these clauses, in order, together with the bits left after them; for $n = 0$ it is the empty formula together with $w$. Decoding fails if one of the $n$ clauses cannot be read.","title":"Decoding a fixed number of clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/encode-cnf","kind":"node","lean":"Cslib.CookLevin.encodeCNF","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#encode-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, the code of a CNF formula $\\varphi = (C_1, \\dots, C_m)$ is $1^m 0$ ($m$ ones followed by a zero) followed by the length-prefixed binary codes of the clauses $C_1, \\dots, C_m$ in order.","title":"Length-prefixed binary code of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/decode-cnfaux","kind":"node","lean":"Cslib.CookLevin.decodeCNFAux","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#decode-cnfaux","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"Given a bit string $w$, writing $1$ for true and $0$ for false, the decoder reads a unary length prefix $1^n 0$ ($n$ ones followed by a zero) from the front of $w$, then reads exactly $n$ clauses from the remaining bits with the length-prefixed clause decoder. The result is the CNF formula made of these clauses, in order, together with the bits left after them. Decoding fails if $w$ does not begin with such a\u2026","title":"Decoding a length-prefixed CNF formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/decode-cnf","kind":"node","lean":"Cslib.CookLevin.decodeCNF","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#decode-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"Given a bit string $w$, the decoder reads one CNF formula $\\varphi$ from the front of $w$ with the length-prefixed CNF decoder, which also returns the bits left after $\\varphi$. The result is $\\varphi$ if no bits are left; decoding fails if no formula can be read or if bits remain after it.","title":"Decoding a complete length-prefixed CNF formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/undelimit-eq-some-iff","kind":"node","lean":"Cslib.CookLevin.undelimit_eq_some_iff","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#undelimit-eq-some-iff","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"For all lists of Booleans $u$, $p$ and $r$, parsing one self-delimiting block from the front of $u$ succeeds with payload $p$ and remainder $r$ if and only if $u$ is the self-delimiting block of $p$ followed by $r$.","title":"Parsing a self-delimiting block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/decode-cnf-eq-some-iff","kind":"node","lean":"Cslib.CookLevin.decodeCNF_eq_some_iff","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#decode-cnf-eq-some-iff","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"For every bit string $w$ and every CNF formula $\\varphi$, the complete CNF decoder returns $\\varphi$ on input $w$ if and only if $w$ is the length-prefixed binary code of $\\varphi$.","title":"The CNF decoder inverts the length-prefixed CNF code"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/is-satencoded","kind":"node","lean":"Cslib.CookLevin.IsSATEncoded","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#is-satencoded","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"A list $x$ of Booleans encodes a satisfiable CNF formula when $x$ is the binary code of some CNF formula $\\varphi$ and $\\varphi$ is satisfiable.","title":"Encodings of satisfiable CNF formulas"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/satlang","kind":"node","lean":"Cslib.CookLevin.SATLang","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#satlang","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"SAT is the set of lists of Booleans that encode a satisfiable CNF formula.","title":"The SAT language"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/sat-dec","kind":"node","lean":"Cslib.CookLevin.satDec","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#sat-dec","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"The SAT decision problem returns true on a list $x$ of Booleans when $x$ lies in the SAT language, that is, when $x$ encodes a satisfiable CNF formula, and false otherwise.","title":"SAT as a decision problem"},{"id":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/poly-many-one-red","kind":"node","lean":"Cslib.CookLevin.PolyManyOneRed","parent":30335,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas/index.html#poly-many-one-red","scope":"TheoryOfComputation/np-completeness/cook-levin/binary-encoding-of-cnf-formulas","status":"fully_proved","summary":"A decision problem $L_1$ reduces to a decision problem $L_2$ in polynomial time when there is a polynomial-time computable function $f$ from lists of Booleans to lists of Booleans such that $L_1(x) = L_2(f(x))$ for every list $x$ of Booleans.","title":"Polynomial-time many-one reduction"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat.json","id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","title":"Dense CNF encoding of SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/clause","kind":"node","lean":"Cslib.Complexity.DenseSAT.Clause","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#clause","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"A clause is a finite list of slots, where slot $j$ records how the variable $x_j$ occurs: the slot is either empty (the variable is absent) or holds a Boolean $b$, meaning the literal $x_j = b$; so $b$ true gives $x_j$ and $b$ false gives $\\lnot x_j$.","title":"Dense clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/cnf","kind":"node","lean":"Cslib.Complexity.DenseSAT.CNF","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"A CNF formula is a finite list of dense clauses, read as the conjunction of its clauses.","title":"Dense CNF formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/clause-sat","kind":"node","lean":"Cslib.Complexity.DenseSAT.clauseSat","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#clause-sat","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"A finite bit sequence $a = (a_0, \\dots, a_{n-1})$ satisfies a clause $c$ when there are an index $j$ and a Boolean $b$ such that slot $j$ of $c$ holds the literal $x_j = b$, $j < n$, and $a_j = b$. Variables $x_j$ with $j \\ge n$ are unassigned and satisfy no literal.","title":"Satisfaction of a dense clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/cnf-sat","kind":"node","lean":"Cslib.Complexity.DenseSAT.cnfSat","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#cnf-sat","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"A finite bit sequence $a$ satisfies a CNF formula $\\varphi$ when $a$ satisfies every clause of $\\varphi$.","title":"Satisfaction of a dense CNF formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/decode-aux","kind":"node","lean":"Cslib.Complexity.DenseSAT.decodeAux","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#decode-aux","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Given a bit string $w$ and the list $c$ of slots read so far for the current clause, stored in reverse order, the result is the CNF formula obtained by reading $w$ two bits at a time: $00$ outputs the current clause in reading order and continues with no slots read; $01$ adds an empty slot to the current clause; and $1s$, for a bit $s$, adds a slot holding the Boolean $\\lnot s$, so $10$ adds a positive and $11$ a\u2026","title":"Dense decoding with an accumulator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/decode-cnf","kind":"node","lean":"Cslib.Complexity.DenseSAT.decodeCNF","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#decode-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The CNF formula decoded from a bit string $w$ is the result of the accumulator decoding of $w$ started with no slots read for the current clause: $w$ is read two bits at a time, $00$ ends a clause, $01$ is an empty slot, $10$ a positive and $11$ a negative literal, and a trailing unterminated clause and a dangling odd bit are ignored.","title":"Dense decoding of CNF formulas"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/encode-slot","kind":"node","lean":"Cslib.Complexity.DenseSAT.encodeSlot","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#encode-slot","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Slot $j$ of a dense clause is empty (variable $x_j$ is absent), holds the positive literal $x_j$, or holds the negative literal $\\neg x_j$. Its code is $01$ when empty, $10$ for the positive literal and $11$ for the negative literal.","title":"Two-bit code of a slot"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/encode-clause","kind":"node","lean":"Cslib.Complexity.DenseSAT.encodeClause","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#encode-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The bit string encoding a clause is the concatenation of the two-bit codes of its slots, in order (empty slot $01$, positive literal $10$, negative literal $11$), followed by the terminator $00$.","title":"Dense encoding of a clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/encode-cnf","kind":"node","lean":"Cslib.Complexity.DenseSAT.encodeCNF","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#encode-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The dense encoding of a CNF formula $\\varphi$ is the bit string obtained by concatenating, in order, the encodings of the clauses of $\\varphi$, where a clause is encoded as the two-bit codes of its slots followed by the terminator $00$.","title":"Dense encoding of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/decode-cnf-encode-cnf","kind":"node","lean":"Cslib.Complexity.DenseSAT.decodeCNF_encodeCNF","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#decode-cnf-encode-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For every CNF formula $\\varphi$, decoding the bit string that encodes $\\varphi$ gives back $\\varphi$. In particular, every CNF formula is the decoding of some bit string.","title":"Decoding inverts the dense encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/sat","kind":"node","lean":"Cslib.Complexity.DenseSAT.SAT","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#sat","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"A bit string $w$ belongs to dense CNF-SAT when the CNF formula obtained by decoding $w$ in the dense clause encoding is satisfiable, that is, some assignment (a list of booleans) satisfies every clause of the decoded formula.","title":"Dense CNF-SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/eval-cnf","kind":"node","lean":"Cslib.Complexity.DenseSAT.evalCNF","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#eval-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The scan $E(r, u, v, f)$ takes the formula bits $r$ still to be read, the assignment bits $u$ not yet used by the current clause, the bits $v$ already used by it (most recent first) and a boolean $f$ recording whether the current clause is already satisfied; write $x u'$ for the list with first bit $x$ followed by the list $u'$. It reads $r$ two bits at a time; let $r'$ be the bits after the pair read. On $00$ (end\u2026","title":"Single-pass evaluation of a dense CNF encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/eval-idx","kind":"node","lean":"Cslib.Complexity.DenseSAT.evalIdx","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#eval-idx","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The scan $I(r, a, j, f)$ takes the formula bits $r$ still to be read, an assignment $a$ (a list of booleans), the position $j$ of the next slot in the current clause and a boolean $f$ recording whether the current clause is already satisfied. It reads $r$ two bits at a time; let $r'$ be the bits after the pair read. On $00$ (end of clause) the result is $f \\wedge I(r', a, 0, \\mathrm{false})$. On $01$ (an empty\u2026","title":"Indexed evaluation of a dense CNF encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/check","kind":"node","lean":"Cslib.Complexity.DenseSAT.check","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#check","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For bit strings $w$ (the encoded formula) and $y$ (the certificate), the check of $y$ against $w$ is the Boolean result of the verifier's left-to-right scan of the formula bits $w$, started with $y$ as the assignment bits not yet used by the current clause, no used bits, and the current clause not yet satisfied.","title":"Dense CNF-SAT certificate check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/length-le-of-mem-decode-aux","kind":"node","lean":"Cslib.Complexity.DenseSAT.length_le_of_mem_decodeAux","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#length-le-of-mem-decode-aux","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For every bit string $r$ and every clause $c_0$, every clause produced by the auxiliary decoder run on $r$ with accumulated partial clause $c_0$ has length at most $|c_0| + |r|$.","title":"Decoded clauses are no longer than the input"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/cnf-sat-take","kind":"node","lean":"Cslib.Complexity.DenseSAT.cnfSat_take","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#cnf-sat-take","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Let $w$ be a bit string and $\\varphi$ the CNF formula decoded from $w$. If a finite bit sequence $a$ satisfies $\\varphi$, then so does the prefix of $a$ of length $|w|$ (all of $a$ when $a$ has at most $|w|$ bits).","title":"Truncating a satisfying assignment"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/sat-iff-check","kind":"node","lean":"Cslib.Complexity.DenseSAT.sat_iff_check","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#sat-iff-check","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"A bit string $w$ belongs to dense CNF-SAT if and only if there is a bit string $y$ with $|y| \\le |w|$ that the certificate check accepts on formula bits $w$ and assignment bits $y$.","title":"Certificate characterization of dense CNF-SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/w","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.W","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#w","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The verifier has four work stacks: the input stack, a stack $F$ that stashes the formula, and two stacks $A$ and $B$ for the assignment.","title":"Work stacks of the dense CNF-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/g","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.G","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#g","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The work alphabet $G$ is the disjoint union of the Booleans and the optional Booleans. Its elements are the left symbols $\\mathrm{inl}(b)$ and the right symbols $\\mathrm{inr}(o)$, where $b$ is a Boolean and $o$ is either $\\mathrm{some}\\ b$ or $\\mathrm{none}$. This is the alphabet of the pair encoding, in which $\\mathrm{inr}(\\mathrm{none})$ is the separator $\\#$.","title":"Work alphabet of the dense-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/pair-encode","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.pairEncode","parent":30354,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pair encoding "],["math","\\(w \\# y\\)"]],"source":"Pair encoding $w \\# y$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#pair-encode","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For bit strings $w$ and $y$, the word over the pair alphabet $G = \\mathrm{Bool} \\oplus \\mathrm{Option}\\,\\mathrm{Bool}$ consisting of the bits $b$ of $w$ in order, each tagged on the left as $\\mathrm{inl}\\,b$, then the separator $\\# = \\mathrm{inr}\\,\\mathrm{none}$, then the bits $b$ of $y$ in order, each tagged on the right as $\\mathrm{inr}(\\mathrm{some}\\,b)$. This is the input of the dense CNF-SAT verifier.","title":"Pair encoding $w \\# y$"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/untitled","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.\u0393","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#untitled","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The stacks of the verifier are the output stack and one stack for each work stack name (input, formula stash $F$, assignment stacks $A$ and $B$). The output stack has the Booleans as alphabet, and every work stack has the alphabet $G$, the disjoint union of the Booleans and the Booleans extended by a separator $\\#$.","title":"Stack alphabets of the dense CNF-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/lab","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.Lab","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The labels of the verifier's program are $\\mathrm{read1}$, $\\mathrm{read2}$, $\\mathrm{back1}$ and $\\mathrm{back2}$; $\\mathrm{c0}(f)$ and $\\mathrm{c1}(f, b)$ for Booleans $f$ and $b$; $\\mathrm{adv}(f, o)$ for a Boolean $f$ and an optional Boolean $o$; $\\mathrm{restore}(f)$ for a Boolean $f$; and $\\mathrm{drainI}(a)$, $\\mathrm{drainA}(a)$, $\\mathrm{drainB}(a)$ and $\\mathrm{finish}(a)$ for a Boolean $a$. There are\u2026","title":"Labels of the dense-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/st","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.St","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The machine state of the dense CNF-SAT verifier is an element of $\\mathrm{Option}\\,G$: either nothing or a work symbol in $G$.","title":"Machine states of the verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/stmt","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.Stmt","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#stmt","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The statements of the verifier are the statements of Mathlib's multi-stack Turing machine model with stack alphabets $\\Gamma$, labels in the verifier's label type and machine states in $\\mathrm{Option}\\,G$.","title":"Statements of the dense CNF-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/cfg","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.Cfg","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#cfg","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"A configuration of the verifier is a configuration of the multi-stack Turing machine model with the verifier's labels, states and stack alphabets. It consists of an optional current label (absent once the machine has halted), a state, which is either a symbol of the work alphabet $G$ or nothing, and for each stack the list of symbols on it: Booleans on the output stack and symbols of $G$ on each of the four work\u2026","title":"Configurations of the dense-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/loop","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.loop","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#loop","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For a work stack $s$, a partial map $g$ on the work alphabet $G$, an optional target work stack $t$, and labels $\\ell$ (the loop's own label) and $\\ell'$ (its exit), the generic loop is the program statement that does the following. It pops the top of $s$ and stores the popped symbol $x$ as the machine state (nothing if $s$ is empty). If $x$ exists and $g(x) = y$ is defined, it pushes $y$ onto $t$ (or pushes\u2026","title":"Generic copy loop of the dense-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/g-read1","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.gRead1","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#g-read1","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The partial map on the work alphabet $G$ that sends each left symbol $\\mathrm{inl}(b)$ to itself and is undefined on every right symbol $\\mathrm{inr}(o)$, including the separator $\\mathrm{inr}(\\mathrm{none})$, written $\\#$.","title":"Formula filter of the dense-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/g-read2","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.gRead2","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#g-read2","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The partial map on the work alphabet $G$ that sends each right symbol $\\mathrm{inr}(\\mathrm{some}\\ b)$ to the left symbol $\\mathrm{inl}(b)$ and is undefined on every other symbol, that is on the left symbols and on the separator $\\mathrm{inr}(\\mathrm{none})$.","title":"Certificate filter of the dense-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/prog","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.prog","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The program assigns a statement to each label. Every pop stores the popped symbol, or nothing if the stack was empty, as the machine state.","title":"The dense CNF-SAT verifier program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/tm","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.tm","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#tm","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The finite multi-stack Turing machine with an output stack over the Booleans and four work stacks (the input stack, a formula stash $F$ and two assignment stacks $A$ and $B$) over the alphabet $G$, the disjoint union of the Booleans and the Booleans extended by a separator $\\#$. It reads its input from the input stack, writes its result to the output stack, starts at the first label of the verifier program with\u2026","title":"The dense CNF-SAT verifier machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/stk","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.stk","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#stk","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Given work-stack contents $V$ (a list $V(s)$ of work symbols for each work stack $s$) and a list $o$ of bits, the stack contents of the verifier that give the output stack $o$ and each work stack $s$ the list $V(s)$.","title":"All stacks of the verifier from work stacks and output"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/mk-v","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.mkV","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#mk-v","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Given four lists $i$, $f$, $a$, $b$ of work symbols in $G$, the contents of the verifier's work stacks in which the input stack holds $i$, the formula stash $F$ holds $f$, and the assignment stacks $A$ and $B$ hold $a$ and $b$.","title":"Work stacks from four lists"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/push-all","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.pushAll","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#push-all","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Given an optional work stack $t$, a list $ys$ of work symbols and work-stack contents $V$ (a list $V(s)$ of symbols for each work stack $s$), the contents obtained by pushing the elements of $ys$ one at a time onto $t$: if $t$ is a work stack, $V(t)$ is replaced by the reverse of $ys$ followed by $V(t)$ and every other stack is unchanged; if $t$ is none, the result is $V$.","title":"Pushing a list onto a verifier work stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/loop-run","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.loop_run","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#loop-run","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Let $s$ be a work stack, $g$ a partial map on the work alphabet $G$, $t$ an optional target work stack other than $s$, and $\\ell$, $\\ell'$ labels such that the verifier's program at $\\ell$ is the generic loop with source $s$, map $g$, target $t$, own label $\\ell$ and exit $\\ell'$. Let $o$ be a list of output bits. Let $x = x_1 \\dots x_n$, $y = y_1 \\dots y_n$ and $r$ be lists of symbols of $G$ with $g(x_j) = y_j$\u2026","title":"Running the generic copy loop of the dense-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/drain-run","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.drain_run","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#drain-run","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Let $\\mathit{ans}$ be a Boolean, let $i$, $a$, $b$ be lists of symbols of the work alphabet $G$, and let $v$ be any state. Start the verifier's program at the label $\\mathrm{drainI}(\\mathit{ans})$ in state $v$, with the input stack holding $i$, the formula stash empty, the assignment stacks $A$ and $B$ holding $a$ and $b$, and the output stack empty. It then runs for exactly $|i| + |a| + |b| + 3$ steps and reaches\u2026","title":"Draining the work stacks of the dense-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/enc","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.enc","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#enc","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For a list $l$ of bits, its encoding on a work stack of the dense-SAT verifier is the list of symbols of the work alphabet $G$ obtained by replacing each bit $b$ of $l$ with the left symbol $\\mathrm{inl}(b)$.","title":"Left-tagged encoding of a bit list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/t","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.T","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#t","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The number $T(r, bs, as, fl) \\in \\mathbb{N}$, for bit strings $r$, $bs$, $as$ and a flag $fl$, is defined by recursion on $r$, where $0$ is false and $1$ is true, juxtaposition denotes concatenation (so $b\\,r$ is $r$ with the bit $b$ prepended), $[\\,]$ is the empty string, $\\mathrm{rev}$ reverses a string and $|\\cdot|$ is length:","title":"Step count of the verifier's main scan"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/scan","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.scan","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#scan","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For all bit strings $r$, $bs$, $as$, every flag $fl$ and every machine state $v$, the verifier program leads in exactly $T(r, bs, as, fl)$ steps from the configuration at label $\\mathsf{c0}(fl)$ with machine state $v$, the input stack holding the bits of $r$, the stack $A$ holding the bits of $as$ and the stack $B$ holding the bits of $bs$ (each bit $b$ stored as $\\mathrm{inl}\\,b$, from the top in order), and the\u2026","title":"The verifier's main scan"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/enc-r","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.encR","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#enc-r","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For a list $y$ of bits, its right-tagged encoding is the list of symbols of the work alphabet $G$ obtained by replacing each bit $b$ of $y$ with the right symbol $\\mathrm{inr}(\\mathrm{some}\\ b)$. This is how the certificate appears after the separator in the pair encoding of a formula and a certificate.","title":"Right-tagged encoding of a certificate"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/prefix-run","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.prefix_run","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#prefix-run","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For all bit strings $w$ and $y$, the verifier program leads in exactly $(|w| + 1) + (|y| + 1) + (|w| + 1) + (|y| + 1)$ steps from the configuration at label $\\mathsf{read1}$ with empty machine state, the input stack holding (from the top) the bits $b$ of $w$ as $\\mathrm{inl}\\,b$, the separator $\\mathrm{inr}\\,\\mathrm{none}$ and the bits $b$ of $y$ as $\\mathrm{inr}(\\mathrm{some}\\,b)$, and all other stacks empty, to\u2026","title":"Splitting the verifier input"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/total-steps","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.totalSteps","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#total-steps","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For bit lists $w$ (a formula) and $y$ (an assignment), the number $(|w| + 1) + (|y| + 1) + (|w| + 1) + (|y| + 1) + T + 1$, where $T$ is the step count of the main scan on $w$ started with all of $y$ unused, no assignment bits used and the clause flag false.","title":"Step count of the dense CNF-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/time","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.time","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#time","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The polynomial $2(n + 5)^2$ over $\\mathbb{N}$ in one variable $n$. It bounds the running time of the verifier on inputs of length $n$; for a formula $w$ and an assignment $y$, the input $w \\# y$ has length $n = |w| + 1 + |y|$.","title":"Time bound of the dense CNF-SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/verifier","kind":"node","lean":"Cslib.Complexity.DenseSAT.Verifier.verifier","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#verifier","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The verifier machine computes, in polynomial time, the map sending a pair $(w, y)$ of bit lists to the Boolean $\\mathrm{check}(w, y)$, which is true exactly when $y$ satisfies the CNF formula decoded from $w$. The pair is given as the input $w \\# y$ (the bits of $w$, a separator $\\#$, then the bits of $y$), the output is the single bit $\\mathrm{check}(w, y)$, and the machine halts within $2(n + 5)^2$ steps on an\u2026","title":"Polynomial-time verifier for dense CNF-SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/dense-sat-dec","kind":"node","lean":"Cslib.Complexity.DenseSAT.denseSatDec","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#dense-sat-dec","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The decision problem of dense CNF-SAT accepts a bit string $w$ exactly when $w$ belongs to dense CNF-SAT, that is, when the formula decoded from $w$ in the dense clause encoding has a satisfying assignment.","title":"Dense CNF-SAT as a decision problem"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/is-poly-time-check","kind":"node","lean":"Cslib.Complexity.DenseSAT.isPolyTime_check","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#is-poly-time-check","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The map sending a pair $(w, y)$ of bit strings to the result of the certificate check on formula bits $w$ and assignment bits $y$ is polynomial-time computable, where pairs of bit strings carry their canonical bitstring encoding.","title":"The dense certificate check runs in polynomial time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/dense-sat-dec-mem-np","kind":"node","lean":"Cslib.Complexity.DenseSAT.denseSatDec_mem_NP","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#dense-sat-dec-mem-np","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Dense CNF-SAT is in NP.","title":"Dense CNF-SAT is in NP"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/encode-nat-two-pow-sub-one","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.encodeNat_two_pow_sub_one","parent":30354,"reading_title":{"format":"inline-parts-v1","parts":[["text","The binary code of "],["math","\\(2^k - 1\\)"]],"source":"The binary code of $2^k - 1$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#encode-nat-two-pow-sub-one","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For every natural number $k$, the binary code of $2^k - 1$ (Mathlib's encoding of natural numbers as bit lists, least significant bit first) is the list of $k$ ones.","title":"The binary code of $2^k - 1$"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/delimit-replicate","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.delimit_replicate","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#delimit-replicate","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For every $k \\in \\mathbb{N}$, the self-delimiting form of the bit string of $k$ ones is $k$ copies of $11$ followed by a single $0$.","title":"Self-delimiting code of a block of ones"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/encode-lit-two-pow","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.encodeLit_two_pow","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#encode-lit-two-pow","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For every $k \\in \\mathbb{N}$ and every truth value $b$, the binary code of the literal of polarity $b$ on variable $2^k - 1$ is $k$ copies of $11$, followed by $0$, followed by the bit $b$.","title":"Binary codes of literals on variables one less than a power of two"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/bin-clause","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.binClause","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#bin-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Let $k_0 \\in \\mathbb{N}$ and let $c$ be a dense clause with slots $s_0, \\dots, s_{L-1}$. Its binary-index translation with offset $k_0$ is the clause that has, for each slot $j$ holding a literal of polarity $b$, the literal of polarity $b$ on variable $2^{k_0 + L - 1 - j} - 1$, listed in increasing order of $j$; absent slots contribute no literal.","title":"Binary-index translation of a dense clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/bin-cnf","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.binCNF","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#bin-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The translation of a dense CNF formula $\\varphi$ replaces each clause $c$ of $\\varphi$, keeping their order, by its binary-index translation with offset $0$: if $c$ has length $L$, each slot $j$ of $c$ holding a literal of polarity $b$ becomes the literal of polarity $b$ on variable $2^{L - 1 - j} - 1$, and absent slots are dropped.","title":"Binary-index translation of a dense formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/cnf-sat-bin-cnf-iff","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.cnfSat_binCNF_iff","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#cnf-sat-bin-cnf-iff","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Let $\\varphi$ be a dense CNF formula all of whose clauses have the same length $V$. Then the binary-index translation of $\\varphi$ is satisfiable (some assignment of truth values to the variables satisfies every clause) if and only if some finite list of truth values satisfies $\\varphi$.","title":"The binary-index translation preserves satisfiability when all clauses have equal length"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/end-clause","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.endClause","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#end-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For a natural number $n$ and a bit string $o$, closing a clause with $n$ literals in front of $o$ gives the bit string consisting of $n$ ones, then a zero, then $o$; that is, the unary count of $n$ is prepended to $o$.","title":"Closing a clause in the translated output"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/conv","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.conv","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#conv","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The function $\\mathrm{conv}(w, \\iota, n, k, m, o)$ consumes the bit string $w$ two bits at a time and prepends to the output bit string $o$. The truth value $\\iota$ records whether a clause is open, $n$ counts the literals of the open clause, $k$ counts slots and $m$ counts clauses. Juxtaposition of bit strings is concatenation, $1^n$ is $n$ ones, $(11)^k$ is $k$ copies of $11$, $\\bar s$ is the negation of a bit\u2026","title":"The dense-to-binary translation on the reversed bit string"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/conv-slots","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.conv_slots","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#conv-slots","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Write $\\mathrm{conv}(w, \\iota, n, k, m, o)$ for the reversed-string translation run on the bit string $w$ with open-clause flag $\\iota$, literal count $n$, slot count $k$, clause count $m$ and output $o$; juxtaposition of bit strings is concatenation, $w^R$ is the reversal of $w$ and $|x|$ is the length of a list $x$. Let $c$ be a dense clause, $r$ and $o$ bit strings and $n, k, m \\in \\mathbb{N}$. Let $s(c)$ be the\u2026","title":"The translation on the slots of a clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/conv-clause","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.conv_clause","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#conv-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Write $\\mathrm{conv}(w, \\iota, n, k, m, o)$ for the reversed-string translation run on the bit string $w$ with open-clause flag $\\iota$, literal count $n$, slot count $k$, clause count $m$ and output $o$; juxtaposition of bit strings is concatenation, $w^R$ is the reversal of $w$, $1^n$ is $n$ ones and $|x|$ is the length of a list $x$. Let $c$ be a dense clause, $r$ and $o$ bit strings, $\\iota$ a truth value and\u2026","title":"The translation on a whole clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/conv-clauses","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.conv_clauses","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#conv-clauses","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Write $\\mathrm{conv}(w, \\iota, n, k, m, o)$ for the reversed-string translation run on the bit string $w$ with open-clause flag $\\iota$, literal count $n$, slot count $k$, clause count $m$ and output $o$; juxtaposition of bit strings is concatenation, $w^R$ is the reversal of $w$, $1^n$ is $n$ ones and $|x|$ is the length of a list $x$. For a dense clause $c$, let $E(c)$ be its dense encoding, $c'$ its binary-index\u2026","title":"The translation on the remaining clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/conv-encode-cnf","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.conv_encodeCNF","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#conv-encode-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Let $\\varphi$ be a dense CNF formula. Run the reversed-string translation on the reversal of the dense encoding of $\\varphi$, starting with no clause open, literal count $0$, slot count $0$, clause count $0$ and empty output. The result is the binary encoding of the binary-index translation of $\\varphi$.","title":"The translation computes the binary encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/sat-dec-encode-cnf","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.satDec_encodeCNF","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#sat-dec-encode-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For every CNF formula $\\psi$, the SAT decision problem returns true on the length-prefixed binary code of $\\psi$ if and only if $\\psi$ is satisfiable.","title":"SAT accepts the code of a formula exactly when it is satisfiable"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/sat-dec-of-decode-cnf-eq-none","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.satDec_of_decodeCNF_eq_none","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#sat-dec-of-decode-cnf-eq-none","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"If the decoder for binary-encoded CNF formulas fails on a bit string $x$, then the SAT decision problem rejects $x$.","title":"Unparsable input is not in SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/bin-conv","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.binConv","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#bin-conv","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The translation of a bit string $w$ is the backwards dense-to-binary translation applied to the reversal of $w$, started with no clause open, with its literal, slot and clause counters all equal to $0$, and with empty output.","title":"The dense-to-binary translation on bit strings"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/stk-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.Stk","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#stk-2","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The stack names of the Turing machine for the dense-to-binary translation form a type with exactly seven elements: the input, a stash, the output, the three counters $n$, $k$ and $m$, and a scratch stack $t$.","title":"Stacks of the dense-to-binary machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/lab-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.Lab","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#lab-2","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The labels of the dense-to-binary converter program are the moving label $\\mathrm{mv}$; the reading label $\\mathrm{rd}(\\iota)$ for a Boolean $\\iota$; the second-read label $\\mathrm{rd2}(\\iota, s)$ for Booleans $\\iota$ and $s$; the two literal labels $\\mathrm{litA}$ and $\\mathrm{litB}$; the two closing labels $\\mathrm{eN}$ and $\\mathrm{eK}$; the clearing label $\\mathrm{fK}(\\iota)$ for a Boolean $\\iota$; the\u2026","title":"Labels of the dense-to-binary converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/stk-3","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.stk","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#stk-3","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For bit strings $i$, $a$, $o$ and natural numbers $n, k, m, t$, this stack assignment gives the input stack $i$, the stash $a$, the output stack $o$, and the counter stacks $N$, $K$, $M$ and the scratch stack $T$ respectively $n$, $k$, $m$ and $t$ copies of $1$ (true).","title":"Stack contents of the converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/prog-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.prog","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#prog-2","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The program uses an input stack, a stash, an output stack, unary counter stacks $N$, $K$, $M$ and a scratch stack $T$; each label first pops one stack and keeps the popped bit (or nothing, if the stack was empty) as its internal state. Below, $\\iota$ says whether a clause is open, $0$ is false and $1$ is true.","title":"The dense-to-binary converter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/run","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.Run","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For a natural number $n$ and configurations $c$ and $d$ of the converter program, $c$ leads to $d$ in exactly $n$ steps when the program, started at $c$, is at configuration $d$ after exactly $n$ steps.","title":"Exact runs of the dense-to-binary converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/mv-run","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.mv_run","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#mv-run","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Let $l$, $a$, $o$ be bit strings and $v$ an optional bit, and list stacks top first. Started at the moving label with internal state $v$, input stack $l$, stash $a$, output stack $o$, empty counter stacks $N$, $K$, $M$ and empty scratch stack $T$, the dense-to-binary converter reaches, in exactly $|l| + 1$ steps, the label for reading with no clause open, with no internal state, an empty input stack, the reversal\u2026","title":"Moving the input onto the stash"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/fin-run","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.fin_run","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#fin-run","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Write $0$ for false, $1$ for true and $1^j$ for $j$ copies of $1$. Let $o$ be a bit string, $\\iota$ a flag saying whether a clause is open, and $n, k, m$ natural numbers with $n = 0$ if no clause is open; let $c = n + 1$ if a clause is open and $c = 0$ otherwise. Started at the label that clears the slot counter at the end, for the flag $\\iota$, with empty internal state, empty input and stash, output $o$, literal,\u2026","title":"The converter's final phase"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/conv-t","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.convT","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#conv-t","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"Write $0$ for false, $1$ for true, and $xyr$ for the bit string whose first two bits are $x, y$ followed by $r$. For a bit string $l$, a flag $\\iota$ (whether a clause is open) and natural numbers $n, k, m$, let $c = n + 1$ if $\\iota$ is true and $c = 0$ otherwise. The step count $T(l, \\iota, n, k, m)$ is defined by recursion on $l$: $T(00r, \\iota, n, k, m) = 2 + c + (k + 1) + T(r, 1, 0, 0, m + 1)$;\u2026","title":"Step count of the main loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/conv-t-le","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.convT_le","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#conv-t-le","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"For every bit string $l$, whether or not a clause is open, and all natural numbers $n, k, m, Q$ with $|l| + n + k + m \\le Q$, the exact step count of the main loop on stash $l$ with literal, slot and clause counters $n, k, m$ is at most $(|l| + 1)(2Q + 6) + Q + 4$.","title":"Step bound for the main loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/tm-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.tm","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#tm-2","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"The multi-stack Turing machine whose stacks are indexed by the stack names (input, stash, output, the counters $n$, $k$, $m$ and the scratch $t$), each with the Booleans as alphabet. It reads its input from the input stack and leaves its output on the output stack; its labels are the program labels, starting at the label that moves the input onto the stash; its internal state is an optional Boolean, initially\u2026","title":"The dense-to-binary machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/computable","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToBinary.computable","parent":30354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat/index.html#computable","scope":"TheoryOfComputation/np-completeness/cook-levin/dense-cnf-encoding-of-sat","status":"fully_proved","summary":"There is a finite multi-stack Turing machine whose input and output alphabets are the bits such that, for every bit string $w$ of length $L$, the machine started on input $w$ halts within $2L^2 + 10L + 11$ steps with the dense-to-binary translation of $w$ on its output stack and all other stacks empty.","title":"The dense-to-binary translation on raw bits is polynomial-time"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","title":"Cook-Levin: generator machine for the transition clause families"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/layout","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Layout","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#layout","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"A layout consists of four natural numbers derived from the verifier: the size $A$ of the label/state block of a row, the number $K$ of stacks, the number $g$ of symbol codes (code $0$ is the empty cell), and the window depth $d$.","title":"Variable layout of the transition clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/row-w","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Layout.rowW","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#row-w","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, and a height $H$, the row width is $W = A + K \\cdot (H \\cdot g)$: the label/state block of $A$ variables, followed by $K \\cdot H \\cdot g$ cell variables.","title":"Variables per time step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/num-vars","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Layout.numVars","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#num-vars","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ and bounds $T$ and $H$, the number of variables is $V = (T + 1) \\cdot W$, where $W$ is the number of variables per time step at height $H$.","title":"Number of variables of the transition-clause formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/x-idx","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Layout.xIdx","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#x-idx","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For height $H$, time $t$ and $a \\in \\mathbb{N}$, the index of label/state variable $a$ at time $t$ is $t \\cdot W + a$, where $W$ is the row width (variables per time step) at height $H$. It is used for $a < A$, the size of the label/state block.","title":"Index of a label/state variable"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/c-idx","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Layout.cIdx","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#c-idx","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ with label/state block size $A$, $K$ stacks and $g$ symbol codes, and natural numbers $H$, $t$, $k$, $j$, $x$, the variable for stack $k$, depth $j$ from the top and symbol code $x$ at time $t$ has index $tW + A + (kH + j)g + x$, where $W = A + KHg$ is the number of variables per time step. It is meant for $k < K$, $j < H$ and $x < g$ but is defined for all values.","title":"Index of a cell variable in the transition-family layout"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/n-lits","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Layout.nLits","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#n-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ with $KK$ stacks and window depth $d$, a label-transition clause has $KK \\cdot d + 2$ literals: the current label, the $KK \\cdot d$ window cells and the next label.","title":"Number of literals of a transition clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/lit-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.litPos","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#lit-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Fix a layout with $K$ stacks and window depth $d$, a height bound $H$, and tables giving for each situation $i$ a current label $a(i)$, a next label $a'(i)$ and window symbol codes $w(i, k, r)$. The position of literal $j$ of clause $(t, i)$ is: for $j = 0$, the index of the label/state variable $a(i)$ at time $t$; for $1 \\le j \\le Kd$, writing $j - 1 = kd + r$ with $r < d$, the index of the cell variable for stack\u2026","title":"Position of a literal in a transition clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/lits","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.lits","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ with $KK$ stacks and window depth $d$, tables $la$, $na$, $wa$, a height $H$, a time $t$ and a situation $i$, the literals of clause $(t, i)$ form the list of pairs $(p_j, b_j)$ for $j = 0, 1, \\dots, n - 1$, where $n = KK \\cdot d + 2$ is the number of literals, $p_j$ is the position of literal $j$ of clause $(t, i)$, and the polarity $b_j$ is true exactly when $j = n - 1$. So every literal is\u2026","title":"Literals of a transition clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/dense-clause","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.denseClause","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#dense-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Given $V \\in \\mathbb{N}$ and a list $\\ell$ of pairs $(p, b)$ of a variable index $p$ and a sign $b$, the dense clause over $V$ variables has, for each $v < V$, slot $v$ equal to the literal $x_v$ if the first pair in $\\ell$ with index $v$ is $(v, \\mathrm{true})$, the literal $\\neg x_v$ if it is $(v, \\mathrm{false})$, and empty if $\\ell$ has no pair with index $v$. Pairs with index at least $V$ are ignored.","title":"Dense clause from a list of literals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/family","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.family","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#family","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Fix a layout with label block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, tables $\\mathrm{la}, \\mathrm{na} : \\mathbb{N} \\to \\mathbb{N}$ and $\\mathrm{wa} : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$, and $T, H, N \\in \\mathbb{N}$. Let $W = A + KHg$, let $X[t, a]$ be the variable $tW + a$, and let $C[t, k, j, x]$ be the variable $tW + A + (kH + j)g + x$. The family is the CNF formula\u2026","title":"The transition clause family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gap-enc","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.gapEnc","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gap-enc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a starting slot $m$, a list $\\ell$ of literals $(p, b)$ (variable index $p$, sign $b$) and a number of variables $V$, the gap encoding of $\\ell$ from slot $m$ to $V$ is defined by recursion on $\\ell$. For the empty list it is $V - m$ copies of the absent-slot block $01$. For a list whose first literal is $(p, b)$ and whose remaining literals form the list $r$, it is $p - m$ copies of $01$, then the two-bit slot\u2026","title":"Gap encoding of a dense clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/inc","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Inc","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#inc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For $\\mathit{lo}, V \\in \\mathbb{N}$, a list of literals $(p_1, b_1), \\dots, (p_n, b_n)$ with positions $p_k \\in \\mathbb{N}$ and signs $b_k$ has increasing literal positions from $\\mathit{lo}$ below $V$ when $\\mathit{lo} \\le p_1 < p_2 < \\dots < p_n < V$; for the empty list ($n = 0$) the condition is $\\mathit{lo} \\le V$.","title":"Increasing literal positions"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/mono-lo","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Inc.mono_lo","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#mono-lo","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For natural numbers $lo' \\le lo$ and $V$ and a list $ls$ of literals, if the positions in $ls$ are strictly increasing, at least $lo$ and below $V$ (with $lo \\le V$ if $ls$ is empty), then the same holds with $lo'$ in place of $lo$.","title":"Lowering the lower bound keeps a literal list increasing"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/encode-clause-dense-clause","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.encodeClause_denseClause","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#encode-clause-dense-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $\\ell = [(p_1, b_1), \\dots, (p_n, b_n)]$ be a list of literals with $p_1 < p_2 < \\dots < p_n < V$. Then the bit encoding of the dense clause over $V$ variables with literals $\\ell$ is the gap encoding of $\\ell$ from slot $0$ to $V$, followed by the clause terminator $00$.","title":"Encoding a dense clause block by block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gap-enc-shift","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.gapEnc_shift","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gap-enc-shift","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $m \\le m'$ and let $\\ell = [(p_1, b_1), \\dots, (p_n, b_n)]$ be a list of literals with $m' \\le p_1 < \\dots < p_n < V$ (if $\\ell$ is empty, require $m' \\le V$). Then the gap encoding of $\\ell$ from slot $m$ to $V$ is $m' - m$ copies of the absent-slot block $01$ followed by the gap encoding of $\\ell$ from slot $m'$ to $V$.","title":"Shifting the start of a gap encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/div-mod-succ-of-lt","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.div_mod_succ_of_lt","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#div-mod-succ-of-lt","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $c$ and $d$ be natural numbers with $0 < d$ and $(c \\bmod d) + 1 < d$. Then $\\lfloor (c + 1) / d \\rfloor = \\lfloor c / d \\rfloor$ and $(c + 1) \\bmod d = (c \\bmod d) + 1$.","title":"Successor of a number with a non-maximal remainder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/inc-of-mono","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.inc_of_mono","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#inc-of-mono","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $f$ assign to each $j \\in \\mathbb{N}$ a literal $f(j) = (p_j, b_j)$ with position $p_j \\in \\mathbb{N}$, and let $m, V \\in \\mathbb{N}$. Suppose $p_j < p_{j+1}$ whenever $j + 1 < m$, and $p_j < V$ when $j + 1 = m$. Then for all $n, a, \\mathit{lo} \\in \\mathbb{N}$ with $a + n = m$ such that $\\mathit{lo} \\le V$ if $n = 0$ and $\\mathit{lo} \\le p_a$ if $n > 0$, the list $f(a), f(a + 1), \\dots, f(a + n - 1)$ has\u2026","title":"Increasing literal sequences"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/wf","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.WF","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#wf","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Given a layout with label/state block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, a current-code table $\\ell$, a next-code table $\\nu$ and a window table $w$, the tables are well formed for situation $i$ at height $H$ when $\\ell(i) < A$, $\\nu(i) < A$, $w(i, k, j) < g$ for all $k < K$ and $j < d$, and $d \\le H$.","title":"Well-formed situation tables for the transition clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/ck","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.CK","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#ck","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"The generator for the transition clause families keeps fifteen unit counters, each on its own stack: $\\mathrm{Tn}$, $\\mathrm{Ti}$, $\\mathrm{Hs}$, $\\mathrm{Gc}$, $\\mathrm{HG}$, $\\mathrm{Ka}$, $\\mathrm{Wc}$, $\\mathrm{Vc}$, $\\mathrm{TW}$, $\\mathrm{P}$, $\\mathrm{Q}$, $\\mathrm{tmp}$, $\\mathrm{md}$, $\\mathrm{c1}$ and $\\mathrm{c2}$.","title":"Counters of the transition-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gk","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.GK","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gk","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"The stacks of the counter machine over the generator's counters: one stack of Booleans for the output and one unit counter for each of the generator's fifteen counters.","title":"Stacks of the clause-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/g","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.G\u0393","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#g","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"The generator has an output stack and one counter stack for each of its counter names; the output stack has the alphabet $\\{\\mathrm{false}, \\mathrm{true}\\}$ and every counter stack has a one-element alphabet.","title":"Stack alphabets of the generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/ps","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.PS","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#ps","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"The control stages of the precomputation of the transition-clause generator. There are eight plain stages $eg$, $ew$, $ek$, $dk$, $cp_1$, $cp_2$, $inc$ and $dk_2$, and three families $hg(s)$, $w(s)$ and $v(s)$ indexed by the four stages $s$ of a multiplication site.","title":"Stages of the transition-clause generator's precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/ls","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.LS","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#ls","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"The control stages of the routine of the transition-clause generator that emits one literal. There are thirteen plain stages: two copy stages $cp_1$, $cp_2$ and the stages $ea$, $da$, $eb$, $db$, $ec$, $dec$, $cmp$, $emL$, $bad$, $slot$ and $dq$; two families $ma(s)$ and $mb(s)$ indexed by the four stages $s$ of a multiplication site; and two families $r(o)$ and $rB(o)$ indexed by the three comparison outcomes $o$\u2026","title":"Stages of the transition-clause generator's literal routine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gl","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.GL","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For $N, m \\in \\mathbb{N}$, a label is one of: a precomputation stage; the head or the tail of the time loop; a stage of the multiplication routine of a time step; for a situation $i < N$, one of three labels that start clause $i$, a stage of the routine emitting literal $j$ of clause $i$ for some $j < m$, or the label that finishes clause $i$; the label that ends a time step; or the final label $\\mathsf{done}$,\u2026","title":"Labels of the clause-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/lit-l","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.GL.litL","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#lit-l","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Among the labels of the generator with $N$ situations and $m$ literal numbers, the entry label of literal $j \\in \\mathbb{N}$ of situation $i < N$ is the first stage of the routine for literal $j$ of situation $i$ when $j < m$, and the halting label $\\mathrm{done}$ when $j \\ge m$.","title":"Entry label of a literal routine (transition-clause generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/after","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.GL.after","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#after","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $N$ be the number of situations and $m$ the number of literals per clause, let $i < N$ be a situation and $j$ a natural number. After the routine for literal $j$ of the clause of situation $i$, the generator moves to the label that emits the leading gap of that clause if $j = 0$. Otherwise it moves to the first label of the routine for literal $j - 1$ if $j - 1 < m$, and to the halting label if $j - 1 \\ge m$.","title":"Label after a literal routine in the transition-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/sit-l","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.GL.sitL","parent":30427,"reading_title":{"format":"inline-parts-v1","parts":[["text","Entry label of the clause routine for situation "],["math","\\(i\\)"]],"source":"Entry label of the clause routine for situation $i$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#sit-l","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For $i < N$, the generator label $\\mathrm{e00}_i$ at which the routine emitting the clause of situation $i$ starts; for $i \\ge N$, the end-of-time-step label $\\mathrm{dtw}$.","title":"Entry label of the clause routine for situation $i$"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/after-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.GL.afterS","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#after-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $N$ be the number of situations. For a natural number $i$, the label the generator moves to after the clause of situation $i$ is the end-of-time-step label if $i = 0$. Otherwise it is the first label of the clause routine for situation $i - 1$ if $i - 1 < N$, and the end-of-time-step label again if $i - 1 \\ge N$.","title":"Label after a situation clause in the transition-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/k-a","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.kA","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#k-a","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout with $K$ stacks and window depth $d$, the stack index $k_A(j)$ of literal $j$ is $\\lfloor (j - 1)/d \\rfloor$ when $1 \\le j \\le Kd$, and $0$ when $j = 0$ or $j > Kd$.","title":"Stack index of a literal in a transition clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/k-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.kB","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#k-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout with $K$ stacks and window depth $d$, the row offset $k_B(j)$ of literal $j$ is $0$ when $j \\le Kd$ and $1$ when $j > Kd$.","title":"Row offset of a literal in a transition clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/cst","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.cst","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#cst","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout with label block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, and tables $\\mathrm{la}, \\mathrm{na} : \\mathbb{N} \\to \\mathbb{N}$ and $\\mathrm{wa} : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$, the constant part of the position of literal $j$ of situation $i$ is $\\mathrm{la}(i)$ if $j = 0$, is\u2026","title":"Constant offset of a literal position"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/sgn","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.sgn","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#sgn","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout with $K$ stacks and window depth $d$, a label-transition clause has $n = K d + 2$ literals, and literal $j$ is positive (polarity $\\mathrm{true}$) exactly when $j = n - 1$: only the last (next-code) literal is positive.","title":"Polarity of a transition-clause literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/lit-p","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.litP","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#lit-p","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Fix a layout with label/state block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, tables $\\mathrm{la}$, $\\mathrm{na}$, $\\mathrm{wa}$ and a number $N$ of situations, and let $n = Kd + 2$ be the number of literals of a clause. For a situation $i < N$ and a literal number $j < n$, this assigns a generator instruction to each stage of the routine that emits literal $j$ of the clause of situation $i$; all\u2026","title":"Literal routine of the transition-clause generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/prog","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.prog","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Fix a layout $L$ with label/state block size $A$, $K$ stacks and $g$ symbol codes, tables $\\mathrm{la}$, $\\mathrm{na}$, $\\mathrm{wa}$, and a number $N$ of situations, and let $n = Kd + 2$ be the number of literals of a clause, where $d$ is the window depth. The generator is a multi-stack machine with a Boolean output stack, the unit counters of the generator, and a Boolean internal state; below, moving a counter\u2026","title":"Generator machine for the transition clause families"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gen-tm","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.genTM","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gen-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$, tables $\\mathit{la}$, $\\mathit{na}$, $\\mathit{wa}$ and $N$ situations, the generator program for these data, bundled as a multi-stack Turing machine with finitely many stacks, labels and states: its stacks are the generator's stacks (the Boolean output stack and the counters), its input stack is the time counter and its output stack is the Boolean output stack, its alphabets are the generator's\u2026","title":"The clause-family generator as a finite stack machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/rest","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Rest","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#rest","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"A counter assignment $F$ of the generator (a natural number for each of its counters) is at rest when its six scratch counters $Q$, $K_a$, $\\mathit{tmp}$, $\\mathit{md}$, $c_1$ and $c_2$ all hold $0$.","title":"Scratch counters at rest in the clause-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/lit-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.lit_run","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#lit-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $i < N$ be a situation, $j$ less than the number of literals, and $F$ a counter assignment whose scratch counters $Q$, $\\mathrm{Ka}$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ are $0$. Let $q = F(\\mathrm{TW}) + k_A(j) F(\\mathrm{HG}) + k_B(j) F(\\mathrm{Wc}) + c_i(j)$, where $k_A$, $k_B$ and $c_i(j)$ are the stack index, the row offset and the constant part of the position of literal $j$, and suppose\u2026","title":"Run of one literal routine (transition-clause generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/bnd","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.bnd","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#bnd","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $L$ be a layout with $K$ stacks and window depth $d$, let $la$, $na$, $wa$ be the tables and $T$, $H$, $t$, $i$, $j$ natural numbers. If $j$ is less than the number $m = Kd + 2$ of literals of a transition clause, this is the position of literal $j$ of clause $(t, i)$; otherwise it is the number $V$ of variables of $L$ for $T$ and $H$.","title":"Boundary before a literal of a transition clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/base","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Base","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#base","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $L$ be a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, let $T$, $H$, $t$ be natural numbers, and write $W = A + KHg$ for the number of variables per time step and $V = (T + 1)W$ for the total number of variables. Counter values $F$ of the generator are at rest at time $t$ when the scratch counters $\\mathrm{Q}$, $\\mathrm{Ka}$, $\\mathrm{tmp}$, $\\mathrm{md}$, $\\mathrm{c1}$, $\\mathrm{c2}$\u2026","title":"Resting counter values in the transition-family time loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/lit-e","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.litE","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#lit-e","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, and for $T, H \\in \\mathbb{N}$, let $W = A + KHg$ be the number of variables per time step and $V = (T + 1)W$ the number of variables. The per-literal time bound is $6V + K(2Hg + 3) + 2K + 2W + 22$.","title":"Per-literal time bound of the transition-clause generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/clause-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.clauseB","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#clause-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $L$ be a layout with label/state block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, and let $T$, $H$ be natural numbers. With $W = A + KHg$, $V = (T + 1)W$, $m = Kd + 2$ and $E = 6V + K(2Hg + 3) + 2K + 2W + 22$, the bound is $V(3V + 6) + mE + 3V + 4$.","title":"Step bound for one clause of the transition-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/lits-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.lits_run","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $t < T$, let the tables be well formed for situation $i < N$ at height $H$, and let $F$ be a counter assignment resting at time $t$: its scratch counters $Q$, $\\mathrm{Ka}$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ are $0$, $F(\\mathrm{TW}) = tW$, $F(\\mathrm{HG}) = Hg$, $F(\\mathrm{Wc}) = W$ and $F(\\mathrm{Vc}) = V$, where $g$ is the number of symbol codes of the layout, $W$ is the number of variables per time\u2026","title":"Run of the literal routines of a clause (transition-clause generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/clause-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.clause_run","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $L$ be a layout with tables $la$, $na$, $wa$, let $N$ be the number of situations, let $T$, $H$, $t$ be natural numbers with $t < T$, and let $i < N$ be a situation whose tables are well formed at height $H$. Let $F$ be counter values at rest in the time loop at time $t$ (for $T$ and $H$) with $F(\\mathrm{P}) = 0$, let $o$ be a bit string and $v$ a Boolean machine state. Started at the first label of the clause\u2026","title":"Run of the transition-family generator on one clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/blk","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.blk","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#blk","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Given a layout $L$, tables $la, na : \\mathbb{N} \\to \\mathbb{N}$ and $wa : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$, and natural numbers $N$, $T$, $H$, $t$, this is the bit string obtained by concatenating, for $i = 0, 1, \\dots, N - 1$ in this order, the encoding of the dense clause over $V$ variables whose literals are the literals of clause $(t, i)$, where $V$ is the number of variables of $L$ for\u2026","title":"Encoded transition clauses of one time step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/loop-p","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.LoopP","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#loop-p","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $L$ be a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, let $T, H \\in \\mathbb{N}$, and write $W = A + KHg$ for the number of variables per time step and $V = (T + 1)W$ for the total number of variables. Counter values $f$ of the generator satisfy the time-loop invariant when the scratch counters $\\mathrm{Q}$, $\\mathrm{Ka}$, $\\mathrm{tmp}$, $\\mathrm{md}$, $\\mathrm{c1}$, $\\mathrm{c2}$\u2026","title":"Time-loop invariant of the transition-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/body-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.bodyB","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#body-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ and natural numbers $N$, $T$, $H$, let $W$ be the number of variables per time step of $L$ at height $H$ and $C$ the step bound for one whole clause of $L$ for $T$ and $H$. The bound is $T(2W + 3) + TW + T + 3 + NC$.","title":"Step bound for one time-loop iteration of the transition-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/pre-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.preB","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#pre-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ with $KK$ stacks and $g$ symbol codes and bounds $T$ and $H$, the bound is $H (2g + 3) + KK (2Hg + 3) + (T + 1)(2W + 3) + H + 2 KK + 4T + 16$, where $W$ is the number of variables per time step at height $H$.","title":"Step bound for the transition-clause generator's precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gen-bound","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.genBound","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gen-bound","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ and $N, T, H \\in \\mathbb{N}$, the time bound of the whole generator is $P + T(B + 1) + T + 2$, where $P$ is the time bound of the precomputation for $L$, $T$ and $H$, and $B$ is the time bound of one iteration of the time loop for $L$, $N$, $T$ and $H$.","title":"Time bound of the generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/time-body","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.time_body","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#time-body","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $t < T$, suppose the tables are well formed at height $H$ for every situation $i < N$, and let $f$ be a counter assignment that satisfies the time-loop invariant for $T$ and $H$ and has counter $T_n$ equal to $t$. Then for every output word $o$, the generator started at the head label of the time-step multiplication, with state $\\mathrm{true}$, counters $f$ and output $o$, reaches the time-loop head label with\u2026","title":"One iteration of the time loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/out-after","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.outAfter","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#out-after","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$, tables $la$, $na$, $wa$, sizes $N$, $T$, $H$, a number $m$ of iterations and a Boolean list $o$, the output after $m$ iterations is the concatenation, over the $m$ consecutive time steps $s$ starting at $T - m$ in increasing order, of the encoded clauses of time step $s$, followed by $o$. Here $T - m$ is truncated subtraction, so for $m \\le T$ these are the time steps $T - m, \\dots, T - 1$.","title":"Generator output after part of the time loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/init-f","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.initF","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#init-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For natural numbers $T$ and $H$, the initial counter assignment gives the counter $\\mathrm{Tn}$ the value $T$, the counter $\\mathrm{Hs}$ the value $H$, and every other counter of the generator the value $0$.","title":"Initial counters of the transition-clause generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gen-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.gen_run","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gen-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $L$ be a layout, $\\mathit{la}, \\mathit{na} : \\mathbb{N} \\to \\mathbb{N}$ and $\\mathit{wa} : \\mathbb{N}^3 \\to \\mathbb{N}$ tables, and $N, T, H \\in \\mathbb{N}$, and suppose the tables are well formed at height $H$ for every situation $i < N$. Start the generator for $L$, $\\mathit{la}$, $\\mathit{na}$, $\\mathit{wa}$ and $N$ at its first precomputation stage, in state false, with $T$ on the time counter and $H$ on\u2026","title":"Correctness of the clause-family generator, with its explicit time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gen-correct","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.gen_correct","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gen-correct","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $L$ be a layout, $\\mathit{la}, \\mathit{na} : \\mathbb{N} \\to \\mathbb{N}$ and $\\mathit{wa} : \\mathbb{N}^3 \\to \\mathbb{N}$ tables, and $N, T, H \\in \\mathbb{N}$, and suppose the tables are well formed at height $H$ for every situation $i < N$. Start the generator for $L$, $\\mathit{la}$, $\\mathit{na}$, $\\mathit{wa}$ and $N$ at its first precomputation stage, in state false, with $T$ on the time counter and $H$ on\u2026","title":"Correctness of the clause-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/uni","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.Layout.uni","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#uni","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, the layout whose label/state block size, number of stacks, number of symbol codes and window depth are all equal to $k$.","title":"Uniform layout"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gen-bound-uni","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.genBound_uni","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gen-bound-uni","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For natural numbers $k \\ge 1$ and $y \\ge 1$, the explicit time bound of the generator for the layout whose label block size, number of stacks, number of symbol codes and window depth all equal $k$, with $N = k$ situations and $T = H = y$, is at most $323\\,k^5 y^5$.","title":"Uniform bound on the generator's time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gen-c","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.genC","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gen-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ with label/state block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, and for $N$ situations, the constant is $323\\,(A + K + g + d + N + 1)^5$.","title":"Constant of the generator's time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gen-poly","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.genPoly","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gen-poly","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"For a layout $L$ and a number $N$ of situations, this is the polynomial $c\\,(X + 1)^5$ with natural-number coefficients, where $c$ is the constant of the generator's time bound for $L$ and $N$.","title":"Time-bound polynomial of the clause-family generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/gen-time","kind":"node","lean":"Cslib.Complexity.CookLevin.Family.gen_time","parent":30427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition/index.html#gen-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-transition","status":"fully_proved","summary":"Let $L$ be a layout, $\\mathit{la}, \\mathit{na} : \\mathbb{N} \\to \\mathbb{N}$ and $\\mathit{wa} : \\mathbb{N}^3 \\to \\mathbb{N}$ tables, and $N, T, H \\in \\mathbb{N}$, and suppose the tables are well formed at height $H$ for every situation $i < N$. Start the generator for $L$, $\\mathit{la}$, $\\mathit{na}$, $\\mathit{wa}$ and $N$ at its first precomputation stage, in state false, with $T$ on the time counter and $H$ on\u2026","title":"Polynomial time bound for the clause-family generator"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","title":"Cook-Levin: the cell one-hot family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/alo-lits","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.aloLits","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#alo-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For $g, B \\in \\mathbb{N}$, the list of the $g$ positive literals on the variables $v_B, v_{B+1}, \\dots, v_{B+g-1}$, in this order, each given as the pair of its variable index and the sign true; these are the literals of the clause $v_B \\lor \\dots \\lor v_{B+g-1}$.","title":"At-least-one literals of a cell"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/amo-lits","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.amoLits","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#amo-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For $B, x, y \\in \\mathbb{N}$, the two-element list of the negative literals on the variables $v_{B+x}$ and $v_{B+y}$, in this order, each given as the pair of its variable index and the sign false; these are the literals of the clause $\\neg v_{B+x} \\lor \\neg v_{B+y}$. It is used for pairs $x < y$.","title":"At-most-one literals for a pair"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/rows-cnf","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.rowsCNF","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#rows-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For natural numbers $V$, $B$ and $y$, write $p_i$ for the variable with index $i$. This is the CNF formula listing, for $y' = 0, 1, \\dots, y - 1$ in turn and within each for $x = 0, 1, \\dots, y' - 1$ in turn, the dense clause over $V$ variables built from the literals $\\lnot p_{B + x}$ and $\\lnot p_{B + y'}$: its slot $i < V$ holds the literal on $p_i$ if there is one and is empty otherwise.","title":"At-most-one clauses of the first rows of a cell"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/cell-cnf","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.cellCNF","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#cell-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For $g, V, B \\in \\mathbb{N}$, the CNF formula whose first clause is the at-least-one clause $v_B \\lor \\dots \\lor v_{B+g-1}$ and whose remaining clauses are the at-most-one clauses $\\neg v_{B+x} \\lor \\neg v_{B+y}$ for $x < y < g$, ordered by $y$ and then by $x$. Each clause is written densely over the $V$ variables $v_0, \\dots, v_{V-1}$; literals on variables of index at least $V$ are dropped.","title":"One-hot clauses of a cell"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/oh-family","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.ohFamily","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#oh-family","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Fix a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, and bounds $T$ and $H$. Let $W = A + KHg$ be the row width and $V = (T+1)W$ the number of variables $v_0, \\dots, v_{V-1}$. For every time $t \\le T$, stack $k < K$ and depth $j < H$, the cell with base $B = tW + A + (kH + j)g$ contributes the at-least-one clause $v_B \\lor \\dots \\lor v_{B+g-1}$, followed by the at-most-one clauses\u2026","title":"The one-hot cell family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/ck","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.CK","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#ck","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"The finite type of the nineteen unit counters of the one-hot clause generator: Tn, Ti, Kc, Kd, Hs, Hi, Gc, HG, Wc, Vc, TW, KB, CB, P, Q, tmp, md, c1 and c2.","title":"Counters of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/cl","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.CL","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#cl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a natural number $g$, a clause kind of a cell with $g$ symbol codes is either the at-least-one kind, or the at-most-one kind of a pair $(x, y)$ of symbol codes $x, y < g$ (not necessarily ordered).","title":"Clause kinds of a one-hot cell"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/nl","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.CL.nl","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#nl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"The number of literals of the at-least-one clause kind of a cell with $g$ symbol codes is $g$, and that of every at-most-one clause kind $\\mathrm{amo}(x, y)$ is $2$.","title":"Number of literals of a one-hot clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/off","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.CL.off","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#off","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"The offset from the cell base of literal $l$ of a clause kind is $l$ for the at-least-one clause, and for the at-most-one clause $\\mathrm{amo}(x, y)$ it is $x$ when $l = 0$ and $y$ otherwise.","title":"Offset of a literal in a one-hot clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/sg","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.CL.sg","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#sg","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a clause kind $c$ of a cell, $\\mathrm{sg}(c)$ is the sign shared by all literals of $c$: true (positive) for the at-least-one clause and false (negative) for every at-most-one clause $\\mathrm{amo}(x, y)$.","title":"Literal sign of a clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/wf","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.CL.WF","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#wf","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"A clause kind of a cell is well formed when it is the at-least-one clause, or an at-most-one clause $\\mathrm{amo}(x, y)$ with $x < y$.","title":"Well-formed clause kinds"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/ps","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.PS","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#ps","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"The control labels of the generator's precomputation, sixteen in all: one each for incrementing the time counter $T_n$, writing the number $g$ of symbol codes on the counter $G_c$, writing the label/state block size $A$ on the counter $W_c$ and writing the number $K$ of stacks on the counter $K_c$, and one for each of the four stages of a multiplication routine in each of the three multiplications\u2026","title":"Stages of the one-hot precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/ls","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.LS","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#ls","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"The stages of the routine of the one-hot generator that emits one literal are: two copy stages, the offset stage, the decrement stage, the comparison stage, for each ordering $o$ (less, equal, greater) two restore stages $r(o)$ and $r_B(o)$, the gap stage, the failure stage, the slot stage and the drain stage.","title":"Stages of the one-hot literal routine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/gl","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.GL","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#gl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a natural number $g$, the labels are: a precomputation label for each precomputation stage; for the time loop, a head, a rest label, a multiplication label for each multiplication stage, and an end label; for the stack loop, a head, a rest label, an emission label, two copy labels, a multiplication label for each multiplication stage, and an end label; for the depth loop, a head, a rest label, two copy labels,\u2026","title":"Control labels of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/lit-l","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.GL.litL","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#lit-l","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a clause kind $c$ of a cell with $g$ symbol codes and a natural number $l$, the entry label of literal $l$ of $c$ is the first-stage label of the routine emitting literal $l$ of $c$ if $l < g$, and the halting label otherwise.","title":"Entry label of a one-hot literal routine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/after","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.GL.after","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#after","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a clause kind $c$ of a cell with $g$ symbol codes and an index $l$, the label the generator moves to once literals $l, l+1, \\dots$ of $c$ have been emitted (literals are emitted from the last one down to the first): if $l = 0$, the final stage of clause $c$, which emits the leading run of empty slots; otherwise the routine for literal $l - 1$ of $c$, or the halting label when $l - 1 \\ge g$.","title":"Label after a literal routine of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/pr","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.GL.pr","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#pr","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Fix a number $g$ of symbol codes. For natural numbers $x$ and $y$ with $x < y < g$, the generator label at which the emission of the at-most-one clause for the pair $(x, y)$ of the current cell begins; for any other pair it is the halting label, a case the generator never uses.","title":"Entry label of an at-most-one clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/y-done","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.GL.yDone","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#y-done","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $g$ be the number of symbol codes of the one-hot generator. For a natural number $y$, this is the label of the generator at which the at-least-one clause starts when $y \\le 1$; when $2 \\le y \\le g$, the label starting the at-most-one clause for the pair $(y - 2, y - 1)$; and when $y \\ge 2$ and $y > g$, the final label $\\mathrm{done}$.","title":"Entry label for the first rows of a cell"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/next-c","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.GL.nextC","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#next-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $c$ be a clause kind of a cell with $g$ symbol codes: the at-least-one clause, or the at-most-one clause $\\mathrm{amo}(x, y)$ for symbol codes $x, y < g$. The label the generator goes to after clause $c$ is the end of the cell if $c$ is the at-least-one clause. If $c = \\mathrm{amo}(x, y)$ with $x > 0$, it is the start of clause $\\mathrm{amo}(x - 1, y)$ when $x - 1 < y < g$, and the halting label otherwise. If\u2026","title":"One-hot generator: the label after a clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/lit-p","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.litP","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#lit-p","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, $c$ a clause kind of a cell of $L$ and $l < g$ a literal index, and write $\\mathrm{off}_c(l)$ for the offset of literal $l$ of $c$ from the cell base. The routine for literal $l$ of $c$ is the following fragment of the one-hot generator for $L$, which acts on unit counters and a Boolean output stack. It copies the cell base counter $\\mathrm{CB}$ onto counter $Q$ (restoring\u2026","title":"One-hot generator: the routine for one literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/prog","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.prog","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, the program of a counter machine with an output bit stack and the unary counters of the generator. Started with $T$ on the time counter $T_n$, $H$ on the height counter $H_s$ and every other counter empty, it first sets $T_n = T + 1$, $G_c = g$, $HG = Hg$, $W_c = A + K \\cdot Hg$, $K_c = K$ and $V_c = (T + 1)W_c$. It then loops over times\u2026","title":"The one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/oh-tm","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.ohTM","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#oh-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"The one-hot generator for a layout, bundled as a Mathlib multi-stack Turing machine with finitely many stacks, labels and internal states: its stacks are the output bit stack and one unary counter for each counter of the generator, its input stack is the time counter and its output stack is the output stack, it starts at its first precomputation label with internal state false, and it runs the generator's program.","title":"The one-hot generator as a finite multi-stack machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/clits","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.clits","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#clits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a cell base $B$ and a clause kind $c$ of a cell with $g$ symbol codes, the literals of $c$ form a list of pairs (variable, sign): for the at-least-one kind, the pairs $(B + x, \\mathrm{true})$ for $x = 0, \\dots, g - 1$ in increasing order; for the at-most-one kind of the pair $(x, y)$, the two pairs $(B + x, \\mathrm{false})$ and $(B + y, \\mathrm{false})$.","title":"Literals of a one-hot clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/bnd","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.bnd","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#bnd","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For $V, B \\in \\mathbb{N}$, a clause kind $c$ and an index $l$, the number $B + o_l$ if $l$ is less than the number of literals of $c$, where $o_l$ is the offset of literal $l$ of $c$ from the cell base, and $V$ otherwise.","title":"Boundary before a literal of a cell clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/rest","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.Rest","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#rest","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For an assignment $F$ of a natural number to each counter of the one-hot generator, $\\mathrm{Rest}(F)$ holds when the five scratch counters $Q$, $\\mathit{tmp}$, $\\mathit{md}$, $c_1$ and $c_2$ all have value $0$ under $F$.","title":"Empty scratch counters of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/lit-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.lit_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#lit-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $c$ be a clause kind of a cell with $g$ symbol codes, let $l < g$ be a literal index with offset $\\mathrm{off}_c(l)$, and let $F$ be counter values of the one-hot generator whose scratch counters $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ are $0$. Let $q, \\gamma \\in \\mathbb{N}$ with $q = F(\\mathrm{CB}) + \\mathrm{off}_c(l)$ and $F(P) = q + \\gamma + 1$. Then for\u2026","title":"Run of one literal routine (one-hot generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/lit-e","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.litE","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#lit-e","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a natural number $V$, the cost of emitting one literal, apart from the gap loop, is $6V + 9$.","title":"Per-literal step cost of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/clause-b","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.clauseB","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#clause-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For natural numbers $g$ and $V$, the step budget for one clause is $V(3V + 6) + g(6V + 9) + 3V + 4$, where $6V + 9$ is the per-literal cost apart from the gap loop.","title":"Step budget for one one-hot clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/lits-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.lits_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $c$ be a well-formed clause kind of a cell with $g$ symbol codes (the at-least-one clause, or the at-most-one clause $\\mathrm{amo}(x, y)$ with $x < y$) with $n$ literals, write $\\mathrm{off}_c(l)$ for the offset of literal $l$ of $c$ from the cell base, and let $B, V \\in \\mathbb{N}$ with $B + g \\le V$. For $l \\le n$ let $b_l$ be $B + \\mathrm{off}_c(l)$ if $l < n$ and\u2026","title":"Run of the literal routines of a clause (one-hot generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/clause-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.clause_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $c$ be a well-formed clause kind of a cell (the pair of an at-most-one kind is increasing), and let $B$ and $V$ be natural numbers with $B + g \\le V$. Let $F$ be counter values whose scratch counters $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ are zero, whose cell-base counter $\\mathrm{CB}$ holds $B$, whose variable counter $\\mathrm{Vc}$ holds $V$, and whose\u2026","title":"One clause of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/row-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.row_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#row-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $V$ and $B$ be natural numbers with $B + g \\le V$, write $p_i$ for the variable with index $i$, and let $F$ assign a value to each counter of the one-hot generator such that the scratch counters are empty ($\\mathrm{Rest}(F)$), $F(CB) = B$, $F(Vc) = V$ and $F(P) = 0$. Let $y < g$, let $\\beta = V(3V + 6) + g(6V + 9) + 3V + 4$ be the step budget of one clause, and for\u2026","title":"Generator run over one row of at-most-one clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/rows-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.rows_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#rows-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $V$ and $B$ be natural numbers with $B + g \\le V$, write $p_i$ for the variable with index $i$, and let $F$ assign a value to each counter of the one-hot generator such that the scratch counters are empty ($\\mathrm{Rest}(F)$), $F(CB) = B$, $F(Vc) = V$ and $F(P) = 0$. Let $\\beta = V(3V + 6) + g(6V + 9) + 3V + 4$ be the step budget of one clause, and for $y \\le g$ let\u2026","title":"Generator run over the first rows and the at-least-one clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/cell-b","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.cellB","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#cell-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For $g, H, V \\in \\mathbb{N}$, the number $2V + 2 + (H(2g + 3) + H + 2) + (1 + g^2)\\, b + V + 1$, where $b$ is the step bound for one clause for $g$ and $V$. It bounds the steps of one iteration of the one-hot generator's loop over cell depths.","title":"Step bound for one cell"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/cell-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.cell_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#cell-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes and let $j < H$, $V$ and $B$ be natural numbers. Let $F$ be counter values of the one-hot generator with the scratch counters $Q$, tmp, md, $c_1$, $c_2$ empty, the counters $CB$ and $P$ zero, $Vc = V$, $Gc = g$, $Hs = j$, $j g + KB = B$, and suppose $B + g \\le V$. Then for every output $o$ and flag $v$, the generator goes, in at most the one-cell step bound for $g$, $H$,\u2026","title":"Generator run for one cell"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/blk-k","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.blkK","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#blk-k","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a layout $L$ with $g$ symbol codes, natural numbers $T, H$, a time step $t$ and a stack $k$, the concatenation over $j = 0, \\dots, H - 1$ of the bit encodings of the one-hot clauses of the cell whose base is the cell variable of $L$ for stack $k$, depth $j$, symbol code $0$ at time $t$, written densely over the $V$ variables of $L$ for $T$ and $H$.","title":"Encoded one-hot clauses of one stack at one time step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/blk-t","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.blkT","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#blk-t","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a layout $L$ with $K$ stacks, natural numbers $T, H$ and a time step $t$, the concatenation over the stacks $k = 0, \\dots, K - 1$ of the encoded one-hot clauses of stack $k$ at time step $t$ (for $T$ and $H$).","title":"Encoded one-hot clauses of one time step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/k-body-b","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.kBodyB","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#k-body-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For natural numbers $g$, $K$, $H$ and $V$, let $c = V(3V + 6) + g(6V + 9) + 3V + 4$ be the step budget for one clause and $C = 2V + 2 + (H(2g + 3) + H + 2) + (1 + g^2)c + V + 1$ the step budget for one cell. The budget is $1 + (2V + 2) + (K(2Hg + 3) + K + 2) + (H(C + 1) + H + 2) + V + 1$.","title":"Step budget for one stack iteration of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/kbody-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.kbody_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#kbody-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, $K$ stacks and a label/state block of size $A$, let $T$, $H$, $t$, $k$ be natural numbers with $t \\le T$ and $k < K$, let $W = A + KHg$ be the row width of $L$ at height $H$, and let $V = (T + 1)W$ be the number of variables. For $j < H$ let $B_j = tW + A + (kH + j)g$ be the base of the cell at time $t$, stack $k$ and depth $j$, and let $w_j$ be the encoding of the dense\u2026","title":"One stack iteration of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/t-body-b","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.tBodyB","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#t-body-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $A$ label/state variables per row, $K$ stacks and $g$ symbol codes, let $T$ and $H$ be natural numbers, $W = A + K H g$ the row width and $V = (T + 1) W$ the number of variables, and let $\\kappa$ be the step budget of one iteration of the $k$ loop for $g$, $K$, $H$ and $V$. The budget is $T(2W + 3) + T + 2 + \\bigl(K(\\kappa + 1) + K + 2\\bigr) + V + 1$.","title":"Step budget for one time step of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/tbody-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.tbody_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#tbody-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Let $L$ be a layout with $A$ label/state variables per row, $K$ stacks and $g$ symbol codes, let $t \\le T$ and $H$ be natural numbers, and write $W = A + K H g$ for the row width, $V = (T + 1) W$ for the number of variables and $p_i$ for the variable with index $i$. Let $F$ assign a value to each counter of the one-hot generator such that the scratch counters are empty ($\\mathrm{Rest}(F)$),\u2026","title":"Generator run over one time step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/init-f","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.initF","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#init-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For natural numbers $T$ and $H$, the initial counter values are $T$ on the counter $\\mathrm{Tn}$, $H$ on the counter $\\mathrm{Hs}$, and $0$ on every other counter.","title":"Initial counters of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/pre-f","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.preF","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#pre-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, and bounds $T$ and $H$, let $W = A + KHg$. After the precomputation the generator's counters hold $T_n = T + 1$, $H_s = H$, $G_c = g$, $HG = Hg$, $W_c = W$, $K_c = K$ and $V_c = (T + 1)W$, and every other counter holds $0$.","title":"Counter values after the one-hot precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/pre-b","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.preB","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#pre-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, bounds $T$ and $H$, and row width $W = A + KHg$, the number of steps allotted to the generator's precomputation is $4 + (H(2g + 3) + H + 2) + (K(2Hg + 3) + K + 2) + ((T + 1)(2W + 3) + (T + 1) + 2)$.","title":"Step budget of the one-hot precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/oh-bound","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.ohBound","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#oh-bound","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a layout $L$ and natural numbers $T$ and $H$, the explicit time bound of the one-hot generator is $p + (T + 1)(b + 1) + (T + 1) + 2$, where $p$ is the cost of its precomputation and $b$ the cost of one iteration of its time loop, both for $L$, $T$ and $H$.","title":"Explicit time bound of the one-hot generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/oh-run","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.oh_run","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#oh-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Fix a layout, natural numbers $T$ and $H$, and a bit string $o$. Start the one-hot generator at its first precomputation label, with internal state false, the output stack holding $o$, the time counter holding $T$ and the height counter holding $H$ in unary, and every other counter empty. Then within $\\beta(T, H)$ steps, where $\\beta$ is the generator's explicit step bound, it reaches its halting label, in some\u2026","title":"Correctness of the one-hot generator, with its explicit time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/oh-bound-uni","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.ohBound_uni","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#oh-bound-uni","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For all natural numbers $k \\ge 1$ and $y \\ge 1$, the explicit time bound of the one-hot generator for the layout whose label block size, number of stacks, number of symbol codes and window depth all equal $k$, with $T = H = y$, is at most $668\\, k^7 y^6$.","title":"Time bound of the one-hot generator on a uniform layout"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/oh-c","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.ohC","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#oh-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, the constant of the one-hot generator's time bound is $668\\,(A + K + g + 1)^7$; it does not depend on the window depth of $L$.","title":"Constant of the one-hot time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/oh-poly","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.ohPoly","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#oh-poly","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"For a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, let $c = 668\\,(A + K + g + 1)^7$; this is the polynomial $c\\,(X + 1)^6$ with natural-number coefficients. For natural numbers $T$ and $H$, its value at $X = T + H$ is $c\\,(T + H + 1)^6$, the bound on the number of steps of the one-hot generator started on $T$ and $H$.","title":"The one-hot time bound as a polynomial"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/oh-time","kind":"node","lean":"Cslib.Complexity.CookLevin.OneHot.oh_time","parent":30486,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family/index.html#oh-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-cell-one-hot-family","status":"fully_proved","summary":"Fix a layout with label/state block size $A$, $K$ stacks and $g$ symbol codes, natural numbers $T$ and $H$, and a bit string $o$, and let $c = 668\\,(A + K + g + 1)^7$. Start the one-hot generator at its first precomputation label, with internal state false, the output stack holding $o$, the time counter holding $T$ and the height counter holding $H$ in unary, and every other counter empty. Then within\u2026","title":"Polynomial time bound for the one-hot generator"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","title":"Cook-Levin: generator machine for the cell-shift family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-lits","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shLits","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Fix a layout $L$ and tables $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$. For a height $H$, write $C[t,k,j,x]$ for the cell variable of stack $k$, depth $j$ and symbol code $x$ at time $t$, and $X[t,a]$ for the label/state variable $a$ at time $t$. With $c = \\mathrm{cn}(i,k)$ and $p = \\mathrm{pl}(i,k)$, the literals of the clause $(t, i, k, u, x)$ are, as (variable, sign) pairs in this order, $\\neg C[t,k,u+c,x]$,\u2026","title":"Literals of an in-range shift clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ov-lits","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.ovLits","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ov-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, row width $W = A + K H g$, situation offsets $s_a$, push counts $p$, consumed counts $c$ and numbers $H, t, i, k, r$, write $S[t, i] = (t + 1) W + s_a(i)$ for the situation variable (the label/state variable $s_a(i)$ of row $t + 1$), and $C[t', k', j, x] = t' W + A + (k' H + j) g + x$ for the cell variable of row $t'$, stack $k'$, depth\u2026","title":"Literals of an overflow clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/pf-lits","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.pfLits","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#pf-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, row width $W = A + K H g$, situation offsets $s_a$, pushed codes $p_u$ and numbers $H, t, i, k, j$, write $S[t, i] = (t + 1) W + s_a(i)$ for the situation variable (the label/state variable $s_a(i)$ of row $t + 1$), and $C[t', k', j', x] = t' W + A + (k' H + j') g + x$ for the cell variable of row $t'$, stack $k'$, depth $j'$ and symbol\u2026","title":"Literals of a pushed-fill clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-blk","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shBlk","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-blk","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Fix a layout with $A$ label variables, $K$ stacks and $g$ symbol codes, the generator's parameters $sa$, $pl$, $cn$, $pu$, and $T, H, t, i, k \\in \\mathbb{N}$. Write $p = pl(i, k)$, $c = cn(i, k)$, $W = A + K H g$ for the row width and $V = (T + 1) W$ for the number of variables. The block is the CNF formula consisting, in this order, of: for each $u < H - \\max(p, c)$ and each $x < g$ (ordered by $u$, then $x$), the\u2026","title":"The shift block of a situation and a stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-family","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shFamily","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-family","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Fix a layout $L$ with $K$ stacks and situation tables $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$. For $N, T, H \\in \\mathbb{N}$, the shift family is the CNF formula obtained by concatenating the shift blocks of situation $i$ and stack $k$ at time $t$ for all $t < T$, $i < N$ and $k < K$, ordered lexicographically by $(t, i, k)$.","title":"The cell-shift clause family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/wf","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.WF","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#wf","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Given a layout with label/state block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, a situation-offset table $\\mathrm{sa}$, push-count and consumed-count tables $\\mathrm{pl}$ and $\\mathrm{cn}$ and a pushed-code table $\\mathrm{pu}$, the tables are well formed for $N$ situations at height $H$ when $g > 0$, $d \\le H$, $\\mathrm{sa}(i) < A$ for every $i < N$, and for all $i < N$ and $k < K$:\u2026","title":"Well-formed situation tables for the cell-shift family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/depth-cover","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.depth_cover","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#depth-cover","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $p, c, H$ be natural numbers with $p \\le H$ and $c \\le H$, and let $m = \\max(p, c)$. Every natural number $j < H$ satisfies $j < p$, or $j = u + p$ for some natural number $u < H - m$, or $j = H - c + p + r$ for some natural number $r < c - p$, where $c - p$ is truncated subtraction (zero when $c \\le p$).","title":"Coverage of target depths"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/depth-disjoint","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.depth_disjoint","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#depth-disjoint","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $p, c, H$ be natural numbers with $p \\le H$ and $c \\le H$, and let $m = \\max(p, c)$. For every natural number $u < H - m$ we have $p \\le u + p$, $u + p < H - c + p$ and $u + p < H$. For every natural number $r < c - p$, where $c - p$ is truncated subtraction, we have $H - m + p = H - c + p$ and $H - c + p + r < H$.","title":"Bounds on in-range and overflow target depths"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ck","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.CK","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ck","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"The type with 23 elements, one for each unit counter of the counter machine that emits the cell-shift clause family: $\\mathrm{Tn}$, $\\mathrm{Ti}$, $\\mathrm{Hs}$, $\\mathrm{Gc}$, $\\mathrm{HG}$, $\\mathrm{Wc}$, $\\mathrm{Vc}$, $\\mathrm{TW}$, $\\mathrm{TW1}$, $\\mathrm{KB}$, $\\mathrm{U}$, $\\mathrm{Ui}$, $\\mathrm{R}$, $\\mathrm{Ri}$, $\\mathrm{CB}$, $\\mathrm{CW}$, $\\mathrm{P}$, $\\mathrm{Q}$, $\\mathrm{tmp}$, $\\mathrm{md}$,\u2026","title":"Counters of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/bs","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.BS","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#bs","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"The three bases $s$, $cb$ and $cw$ from which a literal position is computed; at time step $t$, with $W$ the row width, $s$ stands for the counter $TW1 = (t + 1) \\cdot W$ and serves the situation variable.","title":"Base counters of literal positions"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/k","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.BS.k","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#k","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"The map from the three kinds of literal base to counters of the generator. With $t$ the current time step and $W$ the number of variables per time step, the situation base goes to the counter holding $(t + 1) W$, the start of row $t + 1$; the current-row cell base goes to the cell-base counter; and the next-row cell base goes to the counter holding the cell base plus $W$.","title":"Counter holding a literal base"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/cl","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.CL","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#cl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For natural numbers $g$ and $d$, a clause kind is either an in-range shift clause $\\mathrm{sh}(x)$ for a symbol code $x < g$, the overflow-fill clause $\\mathrm{ov}$, or a pushed-fill clause $\\mathrm{pf}(j)$ for a pushed depth $j < d$. These are the kinds of clause in the block of a situation $i$ and a stack $k$.","title":"Clause kinds of the cell-shift family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/nl","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.CL.nl","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#nl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"An in-range shift clause has $3$ literals, and an overflow clause and a pushed-fill clause have $2$ each.","title":"Number of literals of a cell-shift clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/bs-2","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.CL.bs","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#bs-2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a clause kind $c$ and $l \\in \\mathbb{N}$: if $c$ is an in-range shift clause, literal $l$ has base $cb$ when $l = 0$, $s$ when $l = 1$ and $cw$ when $l \\ge 2$; if $c$ is an overflow or pushed-fill clause, it has base $s$ when $l = 0$ and $cw$ otherwise.","title":"Base counter of a literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/cs","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.CL.cs","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#cs","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $g$ be the number of symbol codes of the clause kinds, and let $s, p, c \\in \\mathbb{N}$ and $q : \\mathbb{N} \\to \\mathbb{N}$ (situation offset, push count, consumed count and pushed codes). The constant offset of literal $l$ is: for an in-range shift clause $\\mathrm{sh}(x)$, $cg + x$ if $l = 0$, $s$ if $l = 1$ and $pg + x$ otherwise; for the overflow-fill clause $\\mathrm{ov}$, $s$ if $l = 0$ and $pg$ otherwise;\u2026","title":"Constant offsets of the literals of a clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sg","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.CL.sg","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sg","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a clause kind $c$ of a cell-shift block and $l \\in \\mathbb{N}$, the sign of literal $l$ of $c$ is positive exactly when $l + 1$ equals the number of literals of $c$; so only the last literal is positive.","title":"Literal signs of a shift clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ps","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.PS","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ps","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"The labels of the precomputation that the cell-shift generator runs before its time loop: six plain stages and three multiplication sites, each of the latter tagged with a stage of the multiplication routine. For a layout with $A$ label variables, $K$ stacks and $g$ symbol codes, and the inputs $T$ (number of time steps) and $H$ (height), in order of execution they set a counter $G$ to $g$; set a counter $HG$ to\u2026","title":"Precomputation stages of the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ls","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.LS","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ls","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"The routine emitting one literal has fifteen stages: two copy stages, an offset stage, a decrement stage, a compare stage, two restore stages for each comparison outcome (less, equal, greater), an absent-slot emission stage, a failure stage, a slot emission stage and a drain stage.","title":"Stages of the cell-shift literal routine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/st","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.ST","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"The finite type of stages of the setup and loops of an $(i, k)$ block of the cell-shift generator: the base stages $\\mathrm{kbE}$, $\\mathrm{kbC1}$, $\\mathrm{kbC2}$, $\\mathrm{kbK}$, $\\mathrm{kbM}(s)$ for each stage $s$ of the multiplication routine, $\\mathrm{kbD}$, $\\mathrm{uC1}$ and $\\mathrm{uC2}$; the overflow-loop stages $\\mathrm{oE}$, $\\mathrm{oHead}$, $\\mathrm{oRest}$, $\\mathrm{oC1}$, $\\mathrm{oC2}$,\u2026","title":"Setup and loop stages of a cell-shift block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/bd","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.BD","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#bd","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a symbol count $g$ and a window depth $d$, the finite type of stages of the loop body of an $(i, k)$ block: for each flag $b$, two stages adding the block base, four stages building the second base and two drain stages; one stage for each stage of the multiplication routine; and for each clause kind $c$ (over $g$ and $d$), a terminator stage, two stages loading the variable count, one stage for each literal\u2026","title":"Loop-body stages of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/gl","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#gl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For natural numbers $N$, $K$, $g$ and $d$, a label is one of: a precomputation stage $\\mathrm{pre}(s)$; a time-loop label $\\mathrm{tHead}$, $\\mathrm{tRest}$, $\\mathrm{tw}(s)$ for a multiplication stage $s$, $\\mathrm{t1a}$, $\\mathrm{t1b}$, $\\mathrm{t1c}$, $\\mathrm{t1d}$, $\\mathrm{tEnd}$ or $\\mathrm{tEnd2}$; a setup or loop stage $\\mathrm{s}(i, k, x)$ of the block of situation $i < N$ and stack $k < K$; a decrement\u2026","title":"Labels of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/lit-l","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL.litL","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#lit-l","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a situation $i < N$, a stack $k < K$, a clause kind $c$ and $l \\in \\mathbb{N}$, the entry label of literal $l$ is the first stage of the routine emitting literal $l$ of a clause of kind $c$ in block $(i, k)$ when $l < 3$, and the generator's final label when $l \\ge 3$ (that case is never used).","title":"Entry label of a literal routine of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/after","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL.after","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#after","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For block $(i, k)$ of the cell-shift generator, a clause kind $c$ and $l \\in \\mathbb{N}$, with the literals of a clause emitted from the last down to the first: the label is the finishing stage of clause $c$ in block $(i, k)$ when $l = 0$; otherwise it is the first stage of the routine for literal $l - 1$ of $c$ in block $(i, k)$ when $l - 1 < 3$, and the final label $\\mathrm{done}$, at which the generator halts,\u2026","title":"Label after a literal routine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/x-from","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL.xFrom","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#x-from","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a situation $i < N$, a stack $k < K$ and a number $n$, the label of the generator with $N$ situations, $K$ stacks, $g$ symbol codes and window depth $d$ from which the in-range shift clauses for the symbol codes $x < n$ of the current cell pair of block $(i, k)$ are emitted before the in-range drains: the in-range drain stage $\\mathrm{dCB}$ of block $(i, k)$ if $n = 0$, the first stage of the routine emitting\u2026","title":"Entry label of the in-range chain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/pf-from","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL.pfFrom","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#pf-from","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a situation $i < N$, a stack $k < K$ and a number $n$, the label of the generator with $N$ situations, $K$ stacks, $g$ symbol codes and window depth $d$ from which the pushed-fill clauses $j < n$ of block $(i, k)$ are emitted before the fill drains: the fill-drain stage $\\mathrm{fD1}$ of block $(i, k)$ if $n = 0$, the first stage of the routine emitting pushed-fill clause $n - 1$ of block $(i, k)$ if\u2026","title":"Entry label of the pushed-fill chain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/next-c","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL.nextC","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#next-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a situation $i < N$ and a stack $k < K$, in block $(i, k)$: after the in-range clause for symbol $x$ the generator goes to the start of the in-range clause for symbol $x - 1$, or to the in-range drains when $x = 0$; after the overflow clause it goes to the overflow drains; after the pushed-fill clause for depth $j$ it goes to the start of the pushed-fill clause for depth $j - 1$, or to the pushed-fill drains\u2026","title":"Label after a clause of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sb-l","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL.sbL","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sb-l","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Among the generator labels for $N$ situations, $K$ stacks, $g$ symbol codes and window depth $d$, take a situation $i < N$, a stack $k < K$ and $n \\in \\mathbb{N}$. The label is the pushed-fill setup label of block $(i, k)$ if $n = 0$, the decrement label of block $(i, k)$ with index $n - 1$ if $1 \\le n \\le d + 1$, and the terminal label otherwise. It marks the point where $n$ decrements of the counter $U$ remain\u2026","title":"Label of the decrement chain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/i-from","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL.iFrom","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#i-from","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a natural number $n$, with $N$ the number of situations and $K$ the number of stacks, this is the end-of-time-step label when $n = 0$ or $K = 0$; the first setup stage of block $(n - 1, K - 1)$ when $1 \\le n \\le N$ and $K \\ge 1$; and the terminal (halting) label when $n > N$ and $K \\ge 1$. For $n \\le N$ it is the label from which the generator emits the blocks of situations $i < n$ and then ends the time step.","title":"Entry label of the situation chain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ik","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.GL.ik","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ik","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For natural numbers $i$ and $n$, with $N$ the number of situations and $K$ the number of stacks, this is the entry label of the situation chain for $i$ when $n = 0$; the first setup stage of block $(i, n - 1)$ when $1 \\le n \\le K$ and $i < N$; and the terminal (halting) label otherwise. For $i < N$ and $n \\le K$ it is the label from which the generator emits the blocks of stacks $k < n$ of situation $i$ and then\u2026","title":"Entry label of the stack chain of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/lab","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.Lab","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout with $K$ stacks, $g$ symbol codes and window depth $d$, and a number $N$ of situations, the labels of the generator are the finite label type with parameters $N$, $K$, $g$ and $d$.","title":"Labels of the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/csv","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.csv","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#csv","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout with $K$ stacks, $g$ symbol codes and window depth $d$, let $N \\in \\mathbb{N}$ be the number of situations, and let $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$ and $\\mathrm{pu}$ give the situation offsets, push counts, consumed counts and pushed codes. For a situation $i < N$, a stack $k < K$, a clause kind $c$ with parameters $g$ and $d$, and $l \\in \\mathbb{N}$, the constant offset of literal\u2026","title":"Constant offset of a literal in a block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/lit-p","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.litP","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#lit-p","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a situation $i < N$, a stack $k < K$, a clause kind $c$ and a literal index $l < 3$, this is the program fragment, indexed by the stages of the literal routine, that emits literal $l$ of clause $c$ in block $(i, k)$. Write $B$ for the base counter of literal $l$ and $a$ for its constant offset in block $(i, k)$. Started with $Q$ and the scratch counters empty, the routine copies the value $b$ of $B$ into the\u2026","title":"Literal routine of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/st-p","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.stP","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#st-p","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$ and $K$ stacks, push-count and consumed-count tables $\\mathrm{pl}$ and $\\mathrm{cn}$, a number $N$ of situations, a situation $i < N$ and a stack $k < K$, write $p = \\mathrm{pl}(i, k)$ and $c = \\mathrm{cn}(i, k)$. This assigns a statement of the generator to each setup or loop stage of block $(i, k)$; below, \"add $X$ to $Y$\" copies counter $X$ onto $Y$ through the scratch\u2026","title":"Block stages of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/bd-p","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.bdP","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#bd-p","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with $g$ symbol codes and a block $(i, k)$, the generator's statement at each loop-body stage, with flag $b$ true in the in-range loop and false in the overflow loop: add the counter $KB$ to the counter $CB$; when $b$ is true, add the product of the counters $U$ and $Gc$ to $CB$; add $CB$ and then the counter $Wc$ to the counter $CW$; then start the in-range clauses for the symbols\u2026","title":"Loop body of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/prog","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.prog","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, tables $s_a, p, c, p_u$ and $N$ situations, the counter machine with the unit counters of the generator and a Boolean output stack, meant to start with $T$ units on $\\mathrm{Tn}$, $H$ on $\\mathrm{Hs}$ and everything else empty. Its precomputation puts $g$ units on $\\mathrm{Gc}$, adds $\\mathrm{Hs} \\cdot \\mathrm{Gc}$ to $\\mathrm{HG}$, puts\u2026","title":"The cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-tm","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shTM","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$, situation tables $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$ and $N$ situations, the cell-shift generator for these data, bundled as a multi-stack Turing machine with finitely many stacks, labels and states: its stacks are the Boolean output stack and one counter stack for each counter of the generator, its input stack is the counter $\\mathrm{Tn}$ and its output stack is the Boolean\u2026","title":"The cell-shift generator as a finite stack machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/clits","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.clits","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#clits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $c$ be a clause kind with $n$ literals ($n = 3$ for an in-range shift clause, $n = 2$ for an overflow-fill or pushed-fill clause) and let $\\mathrm{pos} : \\mathbb{N} \\to \\mathbb{N}$. The literals of $c$ at positions $\\mathrm{pos}$ are the list $(\\mathrm{pos}(0), b_0), \\dots, (\\mathrm{pos}(n - 1), b_{n - 1})$ of variable-sign pairs, where the sign $b_l$ is true (a positive literal) exactly when $l = n - 1$.","title":"Literals of a clause kind at given positions"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/mono","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.Mono","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#mono","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For $p : \\mathbb{N} \\to \\mathbb{N}$ and $n, V \\in \\mathbb{N}$, the positions $p$ are increasing for $n$ literals below $V$ when $p(l) < p(l + 1)$ for every $l$ with $l + 1 < n$, and $p(n - 1) < V$ if $n \\ge 1$.","title":"Increasing literal positions below a bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/bnd","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.bnd","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#bnd","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For $V, n, l \\in \\mathbb{N}$ and literal positions $pos : \\mathbb{N} \\to \\mathbb{N}$, the boundary before literal $l$ is $pos(l)$ if $l < n$, and $V$ otherwise.","title":"Boundary before a literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/rest","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.Rest","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#rest","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"A valuation $F$ of the generator's counters satisfies this predicate when the five scratch counters $Q$, $\\mathit{tmp}$, $\\mathit{md}$, $c_1$ and $c_2$ all have value $0$ under $F$. The generator keeps them empty whenever a clause starts.","title":"Empty scratch counters of the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/lit-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.lit_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#lit-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $i < N$ be a situation, $k < K$ a stack, $c$ a clause kind and $l < 3$ a literal index. Let $F$ be counter values whose scratch counters ($Q$, the copy buffer, the multiplication buffer and the two comparison buffers) are $0$, let $b$ be the value under $F$ of the base counter of literal $l$, and let $a$ be the constant offset of literal $l$ of $c$ in block $(i, k)$. Let $q, m \\in \\mathbb{N}$ satisfy\u2026","title":"One literal routine of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/lit-e","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.litE","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#lit-e","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For $V \\in \\mathbb{N}$, the per-literal overhead is $6V + 9$.","title":"Per-literal step overhead of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/clause-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.clauseB","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#clause-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a natural number $V$ (the number of variables), the step bound for one clause of at most three literals is $V(3V + 6) + 3(6V + 9) + 3V + 4$, where $6V + 9$ is the per-literal cost apart from the gap loop.","title":"Step bound for one clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/lits-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.lits_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Fix $V \\in \\mathbb{N}$, a situation $i < N$, a stack $k < K$, a clause kind $c$ with $n$ literals, positions $p : \\mathbb{N} \\to \\mathbb{N}$, counter values $F$ whose scratch counters are $0$, and an output $o$. Suppose that for every $l$ the position $p(l)$ is the value under $F$ of the base counter of literal $l$ plus its constant offset in block $(i, k)$, that this base value is at most $V$, that\u2026","title":"The literal routines of one clause of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/clause-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.clause_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $M$ be the cell-shift generator for a layout $L$ with $K$ stacks, $g$ symbol codes and window depth $d$, situation offsets $\\mathrm{sa}$, push counts $\\mathrm{pl}$, consumed counts $\\mathrm{cn}$, pushed codes $\\mathrm{pu}$ and $N$ situations. Fix a situation $i < N$, a stack $k < K$, a clause kind $c$ with parameters $g$ and $d$ and with $n$ literals, positions $\\mathrm{pos} : \\mathbb{N} \\to \\mathbb{N}$,\u2026","title":"One clause of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/kbv","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.kbv","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#kbv","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout with label block size $A$, $K$ stacks and $g$ symbol codes, a stack height $H$, a time $t$ and a stack $k$, this is the number $k \\cdot (H g) + (t W + A)$, where $W = A + K H g$ is the number of variables per time step. It is the index of the variable for depth $0$ and symbol code $0$ of stack $k$ in row $t$.","title":"First cell variable of a block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shPos","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, row width $W = A + K H g$, situation offsets $s_a$, push counts $p$, consumed counts $c$ and numbers $H, t, i, k, u, x, l$, the position of literal $l$ of the in-range shift clause $(t, i, k, u, x)$ is: if $l = 0$, the cell variable of row $t$, stack $k$, depth $u + c(i, k)$ and symbol code $x$, namely\u2026","title":"Literal positions of an in-range shift clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ov-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.ovPos","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ov-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, row width $W = A + K H g$, situation offsets $s_a$, push counts $p$, consumed counts $c$ and numbers $H, t, i, k, r, l$, the position of literal $l$ of the overflow clause $(t, i, k, r)$ is the situation variable $(t + 1) W + s_a(i)$ if $l = 0$, and otherwise the cell variable of row $t + 1$, stack $k$, depth $H - c(i, k) + p(i, k) + r$\u2026","title":"Literal positions of an overflow clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/pf-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.pfPos","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#pf-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, row width $W = A + K H g$, situation offsets $s_a$, pushed codes $p_u$ and numbers $H, t, i, k, j, l$, the position of literal $l$ of the pushed-fill clause $(t, i, k, j)$ is the situation variable $(t + 1) W + s_a(i)$ if $l = 0$, and otherwise the cell variable of row $t + 1$, stack $k$, depth $j$ and symbol code $p_u(i, k, j)$, namely\u2026","title":"Literal positions of a pushed-fill clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/body","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.Body","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#body","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $KK$ stacks and $g$ symbol codes, $T, H, t \\in \\mathbb{N}$ and counter contents $F$, let $W = A + KK \\cdot H g$ be the row width. The condition holds when the scratch counters $Q$, $tmp$, $md$, $c_1$, $c_2$ and the counter $P$ are empty, $Vc$ holds the variable count $(T + 1) W$, $Gc$ holds $g$, $Wc$ holds $W$, $TW$ holds $t W$ and $TW1$ holds $(t + 1) W$.","title":"Counter state of a loop body"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/xs-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.xs_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#xs-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, let $H, T \\in \\mathbb{N}$, let $W = A + K H g$ be the row width and $V = (T + 1) W$ the number of variables, and let the tables $\\mathrm{sa}, \\mathrm{pl}, \\mathrm{cn}, \\mathrm{pu}$ be well formed for $N$ situations and height $H$: $g > 0$, $d \\le H$, and for all $i < N$ and $k < K$, $\\mathrm{sa}(i) < A$,\u2026","title":"The in-range chain of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/pf-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.pf_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#pf-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks and $g$ symbol codes, let $W = A + K H g$ be the row width and $V = (T + 1) W$ the number of variables, and let the tables $s_a, p, c, p_u$ be well formed for $N$ situations and height $H$. Let $t < T$, $i < N$, $k < K$, and let $F$ be counter values in which $\\mathrm{Q}$, $\\mathrm{tmp}$, $\\mathrm{md}$, $\\mathrm{c1}$, $\\mathrm{c2}$ and $\\mathrm{P}$ are $0$,\u2026","title":"The pushed-fill chain of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sb-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.sb_run","parent":30536,"reading_title":{"format":"inline-parts-v1","parts":[["text","The decrement chain takes "],["math","\\(n\\)"],["text"," steps"]],"source":"The decrement chain takes $n$ steps"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sb-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Fix a layout with $K$ stacks and window depth $d$, the generator's remaining parameters with $N$ situations, a situation $i < N$ and a stack $k < K$. For every $n \\le d + 1$, every counter valuation $F$ and every $h$ with $F(U) = h + n$, every flag $v$ and every output $o$, there is a flag $v'$ such that the generator, started at the decrement-chain label for $n$ remaining decrements of block $(i, k)$ with flag\u2026","title":"The decrement chain takes $n$ steps"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/s-body-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.sBodyB","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#s-body-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For natural numbers $g$, $H$ and $V$, with $B(V)$ the cost bound of one clause over $V$ variables, the bound is $(2V + 2) + (H(2g + 3) + H + 2) + (2V + 2) + (2V + 2) + g \\cdot B(V) + (V + 1) + (V + 1)$.","title":"Step bound for one in-range loop iteration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/o-body-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.oBodyB","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#o-body-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For $g, H, d, V \\in \\mathbb{N}$, the budget is","title":"Step budget of one overflow iteration of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sbody-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.sbody_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sbody-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, let $T, H \\in \\mathbb{N}$, let $W = A + K H g$ be the row width and $V = (T + 1) W$ the number of variables, and let the situation-offset, push-count, consumed-count and pushed-code tables $s_a, p, c, p_u$ be well formed for $N$ situations and height $H$. Let $t < T$, $i < N$, $k < K$ and $u \\in \\mathbb{N}$ with\u2026","title":"One iteration of the in-range loop of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/obody-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.obody_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#obody-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, let $W = A + K H g$ be the row width and $V = (T + 1) W$ the number of variables, and let the tables $s_a, p, c, p_u$ be well formed for $N$ situations and height $H$. Let $t < T$, $i < N$, $k < K$ and $r < c(i, k) - p(i, k)$. Let $F$ be counter values in which $\\mathrm{Q}$, $\\mathrm{tmp}$, $\\mathrm{md}$,\u2026","title":"One iteration of the overflow loop of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/blk","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.Blk","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#blk","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with $g$ symbol codes, $T, H, t \\in \\mathbb{N}$ and counter contents $F$: $F$ is a loop-body counter state for $L, T, H, t$, the counters $KB$, $U$, $Ui$, $R$, $Ri$, $Ka$, $CB$ and $CW$ are empty, $HG$ holds $H \\cdot g$ and $Hs$ holds $H$.","title":"Counter state at the start of a block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ik-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.ikB","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ik-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For natural numbers $g$, $K$, $H$, $d$ and $V$ (in use: the number of symbol codes, the number of stacks, the stack height, the window depth and the number of variables), the budget is","title":"Step budget for one block of the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ik-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.ik_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ik-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $t < T$, assume the well-formedness conditions for $N$ situations and stack height $H$, let $i < N$ and $k < K$, where $K$ is the number of stacks, and let $F$ be counter values satisfying the block-start conditions at time $t$. Then for every output $o$ and every Boolean state $v$, the generator, started at the first setup stage of block $(i, k)$ with state $v$, counters $F$ and output $o$, reaches in at most\u2026","title":"Correctness of the shift generator on one block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/blk-i","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.blkI","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#blk-i","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with $KK$ stacks, family data $sa, pl, cn, pu$ and $T, H, t, i \\in \\mathbb{N}$, the bit string obtained by concatenating, for $k = 0, \\dots, KK - 1$ in order, the encoding of the clause block of situation $i$, stack $k$ at time $t$.","title":"Encoded blocks of one situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/blk-t","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.blkT","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#blk-t","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$, family data $sa, pl, cn, pu$ and $N, T, H, t \\in \\mathbb{N}$, the bit string obtained by concatenating, for situations $i = 0, \\dots, N - 1$ in order, the encoded blocks of situation $i$ at time $t$.","title":"Encoded blocks of one time step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/ks-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.ks_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#ks-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $t < T$, assume the well-formedness conditions for $N$ situations and stack height $H$, let $i < N$, and let $F$ be counter values satisfying the block-start conditions at time $t$. Then for every $n \\le K$, where $K$ is the number of stacks, every output $o$ and every Boolean state $v$, there is a Boolean state $v'$ such that the generator, started at the entry label of the stacks $< n$ of situation $i$ with\u2026","title":"Correctness of the shift generator on the stack chain of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/is-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.is_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#is-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $t < T$, assume the well-formedness conditions for $N$ situations and stack height $H$, and let $F$ be counter values satisfying the block-start conditions at time $t$. Then for every $n \\le N$, every output $o$ and every Boolean state $v$, there is a Boolean state $v'$ such that the generator, started at the entry label of the situation chain for $n$ with state $v$, counters $F$ and output $o$, reaches in at\u2026","title":"Correctness of the shift generator on the situation chain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/t-body-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.tBodyB","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#t-body-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout with $K$ stacks, $g$ symbol codes and window depth $d$, let $N, T, H \\in \\mathbb{N}$, let $V = (T + 1) W$ be the number of variables, $W$ being the number of variables per time step at height $H$, and let $B$ be the step budget for one block of the shift generator for $g$, $K$, $H$, $d$ and $V$. The budget is","title":"Step budget for one time step of the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/tbody-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.tbody_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#tbody-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$ be situation tables, let $N, T, H, t \\in \\mathbb{N}$ with $t < T$, and suppose the tables are well formed for $N$ situations at height $H$. Write $W$ for the number of variables per time step at height $H$ and $V = (T + 1) W$ for the number of variables. Let $F$ be counter values of the cell-shift generator\u2026","title":"One iteration of the time loop of the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/init-f","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.initF","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#init-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For natural numbers $T$ and $H$, the assignment of counter values that puts $T$ on the time-bound input counter, $H$ on the stack-height input counter, and $0$ on every other counter of the generator.","title":"Initial counter values of the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/pre-f","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.preF","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#pre-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes and numbers $T, H$, the counter values with $T$ on $\\mathrm{Tn}$, $H$ on $\\mathrm{Hs}$, $g$ on $\\mathrm{Gc}$, $H g$ on $\\mathrm{HG}$, the row width $W = A + K H g$ on $\\mathrm{Wc}$, the number of variables $(T + 1) W$ on $\\mathrm{Vc}$, and $0$ on every other counter.","title":"Counters after the precomputation of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/pre-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.preB","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#pre-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, numbers $T$ and $H$, and the row width $W = A + K H g$, this is the step bound","title":"Cost of the precomputation of the cell-shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-bound","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shBound","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-bound","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ and $N, T, H \\in \\mathbb{N}$, the bound is $P + (T(B + 1) + T + 2)$, where $P$ is the cost of the precomputation for $L$, $T$, $H$ and $B$ is the cost of one iteration of the time loop for $L$, $N$, $T$, $H$.","title":"Explicit time bound of the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.sh_run","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout, let $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$ be situation tables and let $N, T, H \\in \\mathbb{N}$, and suppose the tables are well formed for $N$ situations at height $H$. Start the cell-shift generator for $L$, $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$ and $N$ at its first precomputation stage, in state false, with $T$ on the counter $\\mathrm{Tn}$ and $H$ on\u2026","title":"Correctness of the shift generator, with its explicit time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-bound-uni","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shBound_uni","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-bound-uni","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $k \\ge 1$ and $y \\ge 1$ be natural numbers and let $L_k$ be the layout with every constant equal to $k$. Then the explicit time bound of the shift generator for $L_k$ with $N = k$ situations and $T = H = y$ is at most $828\\, k^7 y^6$.","title":"Time bound of the shift generator on uniform layouts"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-c","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shC","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label/state block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, and a natural number $N$, the constant is $828 \\cdot (A + K + g + d + N + 1)^7$.","title":"Constant of the shift-generator time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-poly","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.shPoly","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-poly","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"For a layout $L$ with label/state block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, and a natural number $N$, let $c = 828\\,(A + K + g + d + N + 1)^7$ be the constant of the shift-generator time bound; this is the polynomial $c\\,(X + 1)^6$ with natural-number coefficients.","title":"The shift-generator time bound as a polynomial"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/sh-time","kind":"node","lean":"Cslib.Complexity.CookLevin.Shift.sh_time","parent":30536,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift/index.html#sh-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-cell-shift","status":"fully_proved","summary":"Let $L$ be a layout with label/state block size $A$, $K$ stacks, $g$ symbol codes and window depth $d$, let $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$ be situation tables and let $N, T, H \\in \\mathbb{N}$, and suppose the tables are well formed for $N$ situations at height $H$. Start the cell-shift generator for $L$, $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$ and $N$ at its first\u2026","title":"Polynomial time bound for the shift generator"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac","status":"fully_proved","title":"Cook-Levin: situation tables of a multi-stack machine and the correctness of the formula"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","title":"Top of a stack as read from the windows to Next label/state code of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/read-top","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.readTop","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#read-top","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $W$ assign to each stack $k$ a window, a list of entries that are symbols of $k$ or the empty cell $\\bot$; let $P$ assign to each stack $k$ a list of symbols of $k$; and let $c$ assign a natural number to each stack. The top of stack $k$ is the first element of $P(k)$ when $P(k)$ is nonempty. Otherwise it is the entry of $W(k)$ at position $c(k)$, counting from $0$, or $\\bot$ when $W(k)$ has fewer than\u2026","title":"Top of a stack as read from the windows"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/eff","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.eff","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#eff","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $W$ assign to each stack $k$ a window $W_k$, a list of cells each holding a symbol or $\\bot$. The window top of stack $k$ for pushed lists $P$ and consumption counts $c$ is the head of $P_k$ if $P_k$ is nonempty, and otherwise the entry of $W_k$ at position $c_k$ (counting from $0$), or $\\bot$ if there is none. The effect of a statement tree $q$ from a state $v$, pushed lists $P$ and consumption counts $c$ is a\u2026","title":"Symbolic effect of a statement tree"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/ops","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.ops","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#ops","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For a statement tree $q$ of a multi-stack Turing machine, $\\operatorname{ops}(q)$ is the largest number of push, peek and pop nodes on a path from the root of $q$ to a leaf. Load and branch nodes do not count, and goto and halt leaves contribute $0$.","title":"Stack operations on a path of a statement tree"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/push-sym","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.PushSym","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#push-sym","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $q$ be a statement tree of a multi-stack Turing machine with state type $\\sigma$, let $k$ be a stack and $x$ a symbol of the alphabet of $k$. Then $x$ can be pushed onto $k$ by $q$ when $q$ contains a push node onto stack $k$ with symbol function $f$ and some state $v \\in \\sigma$ satisfies $x = f(v)$. Every push node of $q$ counts, whichever branches are taken.","title":"Symbols a statement tree can push"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/eff-bound","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.eff_bound","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#eff-bound","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $q$ be a statement tree, let $W$ assign to each stack a window (a list of cells, each a symbol or $\\bot$), and let $n(q)$ be the largest number of push, peek and pop nodes on a root-to-leaf path of $q$. For every state $v$, pushed lists $P$ and consumption counts $c$, the pushed lists $P'$ and consumption counts $c'$ of the symbolic effect of $q$ on $W$ from $v$, $P$, $c$ satisfy $|P'_k| \\le |P_k| + n(q)$ and\u2026","title":"The symbolic effect grows by at most the number of stack operations"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/eff-pushed","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.eff_pushed","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#eff-pushed","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $q$ be a statement tree, let $W$ assign to each stack a window, and for each stack $k$ let $Q_k$ be a property of symbols of stack $k$ such that every symbol that some push node of $q$ can push onto stack $k$, for some state, satisfies $Q_k$. Then for every state $v$, pushed lists $P$ and consumption counts $c$ such that every symbol of $P_k$ satisfies $Q_k$ for every stack $k$, every symbol of the pushed list\u2026","title":"Pushed symbols come from the start or from push nodes"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/enc","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#enc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"An encoding of a finite TM2 machine $M$ consists of a number $K$, a map $\\kappa$ from the stacks of $M$ to the naturals below $K$ and a map $\\kappa'$ from naturals to stacks with $\\kappa'(\\kappa(k)) = k$ for every stack $k$ and $\\kappa(\\kappa'(n)) = n$ for every $n < K$; a number $A_0$, a map $\\lambda$ from pairs (label or halted, state) to the naturals below $A_0$ and a map $\\lambda'$ back with\u2026","title":"Encoding data for a finite TM2 machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/depth","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.depth","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#depth","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For a finite TM2 machine $M$, the window depth is the maximum, over all labels $l$ of $M$, of the largest number of push, peek and pop nodes on a root-to-leaf path of the statement tree of $l$.","title":"Window depth of a machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/sit-step","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.sitStep","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#sit-step","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine, let $l$ be either a label of $M$ or $\\mathrm{none}$ (the halted label), let $v$ be a state of $M$, and let $W$ assign to each stack $k$ a window, a list of entries that are symbols of $k$ or the empty cell. If $l = \\mathrm{none}$, the step returns the label $\\mathrm{none}$, the state $v$, no pushed symbols on any stack and no consumed symbols on any stack. If $l$ is a\u2026","title":"The situation step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/g","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.g","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#g","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For an encoding $E$ of a finite TM2 machine $M$, the symbol-code bound is $g = 1 + \\max_k s_k$, where the maximum runs over all stacks $k$ of $M$ and $s_k$ is the size of the effective alphabet of stack $k$.","title":"Symbol-code bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/g-2","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.G","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#g-2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For an encoding $E$ of a finite TM2 machine $M$ with $K$ numbered stacks and symbol-code bound $g$, the number of window combinations is $G = g^{K d}$, where $d$ is the window depth of $M$.","title":"Number of window combinations"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/n","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.N","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#n","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For an encoding $E$ of a finite multi-stack Turing machine $M$, the number of situations is $N = A_0 \\cdot G$, where $A_0$ is the number of (label, state) codes of $E$ and $G$ is the number of window combinations.","title":"Number of situations"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/layout","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.layout","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#layout","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For an encoding $E$ of a finite TM2 machine $M$, the layout of the transition clauses has label/state block size $A = A_0 + N$, where $A_0$ is the number of label/state codes and $N = A_0 G$ is the number of situations ($G$ the number of window combinations), together with the number $K$ of numbered stacks, the symbol-code bound $g$ and the window depth $d$ of $M$. The block is a one-hot part on $[0, A_0)$ followed\u2026","title":"Clause layout of a machine encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/la","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.la","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#la","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For an encoding $E$ of a finite TM2 machine with $A_0$ label/state codes and $G$ window combinations, the label/state code of a situation $i \\in \\mathbb{N}$ is $\\lfloor i / G \\rfloor \\bmod A_0$, the top digit of $i$ in the mixed radix whose lower digits are the window symbol codes.","title":"Label/state code of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/wa","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.wa","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#wa","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $E$ be an encoding of a finite multi-stack Turing machine $M$, let $s_\\kappa$ be the size of the effective alphabet of stack $\\kappa$ in $E$, let $g = 1 + \\max_\\kappa s_\\kappa$ (so the empty-cell code $0$ and every effective symbol code are below $g$), and let $d$ be the window depth of $M$. For natural numbers $i$, $k$ and $j$, the window symbol code of situation $i$ on stack number $k$ at depth $j$ is the\u2026","title":"Window symbol code of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/sa","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sa","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#sa","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For an encoding $E$ of a finite multi-stack Turing machine $M$ with $A_0$ (label, state) codes and a natural number $i$, the variable of situation $i$ sits at position $A_0 + i$ of the label/state block of a row.","title":"Position of a situation's variable"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/dec-o","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.decO","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#dec-o","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For an encoding $E$ of a finite TM2 machine $M$, a stack $k$ and a natural number $c$, the decoded symbol is the effective symbol of stack $k$ with index $c - 1$ when $1 \\le c$ and $c - 1 < s_k$, where $s_k$ is the size of the effective alphabet of stack $k$; otherwise, in particular for $c = 0$, it is $\\bot$ (no symbol).","title":"Decoding a symbol code"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/win","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.win","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#win","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $E$ be an encoding of a finite multi-stack Turing machine $M$ with window depth $d$, and let $i$ be a natural number. The window of situation $i$ on stack $k$ is the list of length $d$ whose entry at position $j < d$ decodes the window symbol code $c$ of situation $i$ on the stack number of $k$ at depth $j$. A code with $1 \\le c \\le s_k$, where $s_k$ is the size of the effective alphabet of $k$, decodes to the\u2026","title":"Decoded windows of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/res","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.res","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#res","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"Let $E$ be an encoding of a finite multi-stack Turing machine $M$ and $i$ a natural number. Let $(l, v)$, an optional label and a state, be the result of decoding the label/state code of situation $i$ with the inverse (label, state) numbering of $E$, and let $W$ be the decoded windows of situation $i$. The effect of situation $i$ is the situation step of $M$ from $l$, $v$ and $W$: a new optional label, a new state,\u2026","title":"Effect of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/na","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.na","parent":30613,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01/index.html#na","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-01","status":"fully_proved","summary":"For an encoding $E$ of a finite multi-stack Turing machine $M$ and a natural number $i$, the next label/state code of situation $i$ is the code, in the (label, state) numbering of $E$, of the pair formed by the label and the state that the effect of situation $i$ produces.","title":"Next label/state code of a situation"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","title":"Push-count table of a situation to Top window of a stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/pl","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.pl","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#pl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Fix an encoding record $E$ of a finite multi-stack machine. For a situation $i$ and a stack number $k$, $\\mathrm{pl}(i, k)$ is the number of symbols that the effect of situation $i$ has pushed onto the stack that the inverse stack numbering of $E$ assigns to $k$ and that are still on top of that stack at the end of the step.","title":"Push-count table of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/cn","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.cn","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#cn","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Fix an encoding record $E$ of a finite multi-stack machine. For a situation $i$ and a stack number $k$, $\\mathrm{cn}(i, k)$ is the number of original symbols that the effect of situation $i$ consumes from the stack that the inverse stack numbering of $E$ assigns to $k$; a symbol pushed and then popped within the same step is not counted.","title":"Consumption table of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/pu","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.pu","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#pu","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Fix an encoding record $E$ of a finite multi-stack machine. For a situation $i$, a stack number $k$ and a depth $j$, $\\mathrm{pu}(i, k, j)$ is the code of the symbol at position $j$ from the top (counting from $0$) among the symbols that the effect of situation $i$ leaves pushed on the stack that the inverse stack numbering of $E$ assigns to $k$, and $0$ if fewer than $j + 1$ symbols are left pushed there.","title":"Pushed-symbol table of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/g-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.g_pos","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#g-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"For an encoding $E$ of a finite TM2 machine, the symbol-code bound $g$ of $E$ satisfies $0 < g$.","title":"The symbol-code bound is positive"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/res-pushed","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.res_pushed","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#res-pushed","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Let $E$ be an encoding of a finite multi-stack Turing machine $M$, and for each stack $k$ of $M$ let $s_k$ be the size of the effective alphabet of stack $k$ in $E$ and $\\mathrm{dec}_k(n)$, for $n < s_k$, its $n$-th effective symbol. For every $i \\in \\mathbb{N}$, every stack $k$ of $M$ and every symbol $x$ of the alphabet of stack $k$: if $x$ is among the symbols that the effect of situation $i$ leaves pushed on\u2026","title":"Symbols left pushed by a situation are effective"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/pu-valid","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.pu_valid","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#pu-valid","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Fix an encoding record $E$ of a finite multi-stack machine. Let $i$ be a situation, $k$ a stack number, $s$ the stack that the inverse stack numbering of $E$ assigns to $k$, $n_s$ the size of the effective alphabet of $s$, and $j < \\mathrm{pl}(i, k)$. Then there is a symbol $x$ at position $j$ from the top (counting from $0$) among the symbols the effect of situation $i$ leaves pushed on $s$, such that\u2026","title":"Pushed codes are valid symbol codes"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/wf-upd","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.wf_upd","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#wf-upd","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with window depth $d$, let $E$ be an encoding of $M$ with $A_0$ label/state codes, $K$ stacks, $g$ symbol codes and $N$ situations, and let $H \\ge d$. For every $i \\in \\mathbb{N}$, the current-code table $\\mathrm{la}$, the next-code table $\\mathrm{na}$ and the window table $\\mathrm{wa}$ of $E$ are well formed for the transition clauses of the layout of $E$ at situation\u2026","title":"The transition tables of a machine are well formed"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/wf-sdef","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.wf_sdef","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#wf-sdef","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with window depth $d$, let $E$ be an encoding of $M$ with $A_0$ label/state codes, $K$ stacks, $g$ symbol codes and $N$ situations, and let $H \\ge d$. For every situation $i < N$, the current-code table $\\mathrm{la}$ of $E$, the table $\\mathrm{sa}(i) = A_0 + i$ as next-code table and the window table $\\mathrm{wa}$ of $E$ are well formed for the transition clauses of\u2026","title":"The situation-definition tables of a machine are well formed"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/wf-sh","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.wf_sh","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#wf-sh","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with window depth $d$, let $E$ be an encoding of $M$ with $A_0$ label/state codes, $K$ stacks, $g$ symbol codes and $N$ situations, and let $H \\ge d$. Then the table $\\mathrm{sa}(i) = A_0 + i$, the push counts $\\mathrm{pl}$, the consumption counts $\\mathrm{cn}$ and the pushed codes $\\mathrm{pu}$ of $E$ are well formed for the cell-shift clauses of the layout of $E$\u2026","title":"The cell-shift tables of a machine are well formed"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/halted","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.halted","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#halted","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"If the (label, state) pair decoded from the code $\\mathrm{la}(i)$ of situation $i$ has no label, so the machine has halted, then for every stack number $k$ the next code equals the current one, $\\mathrm{na}(i) = \\mathrm{la}(i)$, and $\\mathrm{pl}(i, k) = 0$ and $\\mathrm{cn}(i, k) = 0$: nothing is left pushed and nothing is consumed.","title":"Halted situations stay fixed"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/mix","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.mix","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#mix","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"For a base $b$, a top digit $c$, a length $m$ and digits $D : \\mathbb{N} \\to \\mathbb{N}$, $\\mathrm{mix}(b, c, 0, D) = c$ and $\\mathrm{mix}(b, c, m + 1, D) = D(0) + b \\cdot \\mathrm{mix}(b, c, m, n \\mapsto D(n + 1))$; that is, $D(0) + b\\,(D(1) + b\\,(\\cdots + b\\,(D(m - 1) + b\\,c)))$, least significant digit first.","title":"Mixed-radix number"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/mix-sit","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.mix_sit","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#mix-sit","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Fix an encoding record $E$ of a finite multi-stack machine $M$. Let $K$ be the number of stacks of $E$, $d$ the window depth of $M$, $A_0$ the number of (label, state) codes of $E$, $g$ the symbol-code bound of $E$ and $N$ the number of situations. For a code $c < A_0$ and window contents $w(k, j)$ with $w(k, j) < g$ for all $k < K$ and $j < d$, let $i = \\mathrm{mix}(g, c, K d, D)$ with digits\u2026","title":"Mixed-radix index of a situation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/sit-surj","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sit_surj","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#sit-surj","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Let $E$ be an encoding of a finite multi-stack Turing machine $M$ with $A_0$ label/state codes, $K$ stacks, $g$ symbol codes, window depth $d$ and $N$ situations. For every $c < A_0$ and every $w : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$ with $w(k, j) < g$ for all $k < K$ and $j < d$, there is a situation $i < N$ whose label/state code $\\mathrm{la}(i)$ equals $c$ and whose window codes satisfy\u2026","title":"Situation surjectivity"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/push-fs","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.pushFS","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#push-fs","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"For a stack $k$ and a statement tree $q$ of a multi-stack machine with a finite set of states, the finite set of symbols of stack $k$ that some push instruction of $q$ on stack $k$ pushes for some state: the union, over all push instructions on $k$ in all branches of $q$, of the images of their push functions over all states.","title":"Finite set of pushable symbols"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/sym-set","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.symSet","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#sym-set","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"For a finite multi-stack Turing machine $M$ and a stack $k$ of $M$, this is the finite set of symbols of stack $k$ consisting of every symbol that some push node on stack $k$ in the statement of some label of $M$ pushes in some state, together with, when $k$ is the input stack $k_0$, every symbol of the alphabet of $k_0$.","title":"Pushable and input symbols of a stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/of-fin-tm2","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.ofFinTM2","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#of-fin-tm2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"For every finite multi-stack Turing machine $M$, an encoding record of $M$, built with classical choice. If $M$ has $K$ stacks, they are numbered $0, \\dots, K - 1$ by a bijection, and numbers $n \\ge K$ decode to the input stack. The pairs of an optional label and a state are numbered $0, \\dots, A_0 - 1$ by a bijection, where $A_0$ is the number of such pairs, and numbers $c \\ge A_0$ decode to the pair of no label\u2026","title":"The classical encoding record of a finite multi-stack machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/upd-time","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.upd_time","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#upd-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with window depth $d$, let $E$ be an encoding of $M$ with $N$ situations, and let $H \\ge d$ and $T \\in \\mathbb{N}$. Run the transition-clause generator for the layout of $E$, the current-code table $\\mathrm{la}$, the next-code table $\\mathrm{na}$ and the window table $\\mathrm{wa}$ of $E$ and $N$ situations, started at its first precomputation stage, in state false,\u2026","title":"Time bound of the transition-clause generator on the tables of a machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/sdef-time","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.sdef_time","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#sdef-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with window depth $d$, let $E$ be an encoding of $M$ with $A_0$ label/state codes and $N$ situations, and let $H \\ge d$ and $T \\in \\mathbb{N}$. Run the transition-clause generator for the layout of $E$, the current-code table $\\mathrm{la}$ of $E$, the table $\\mathrm{sa}(i) = A_0 + i$ (the position of the variable of situation $i$) in place of the next-code table, the\u2026","title":"Time bound of the situation-definition generator on the tables of a machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/shift-time","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.shift_time","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#shift-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with window depth $d$, let $E$ be an encoding of $M$ with $A_0$ label/state codes and $N$ situations, and let $H \\ge d$ and $T \\in \\mathbb{N}$. Run the cell-shift generator for the layout of $E$, the table $\\mathrm{sa}(i) = A_0 + i$ (the position of the variable of situation $i$), the push counts $\\mathrm{pl}$, the consumption counts $\\mathrm{cn}$ and the pushed codes\u2026","title":"Time bound of the shift generator on the tables of a machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/window","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.window","parent":30634,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02/index.html#window","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-02","status":"fully_proved","summary":"For $d \\in \\mathbb{N}$ and a list $L = [x_0, x_1, \\dots]$ over a type $\\alpha$ (a stack, top first), the window of depth $d$ of $L$ is the list of length $d$ over $\\alpha$ extended by the empty cell $\\bot$ whose entry $j$ is $x_j$ if $j$ is less than the length of $L$ and $\\bot$ otherwise.","title":"Top window of a stack"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","title":"Applying an effect to the original stacks to Reading back a list from its cells"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/apply-eff","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.applyEff","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#apply-eff","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Let $S_0$ assign a list $S_0(k)$ to each stack $k$, and let $r = (\\ell, v, P, c)$ consist of a label $\\ell$ (possibly the halted label), a state $v$, a list $P(k)$ for each stack $k$ and a natural number $c(k)$ for each stack $k$. This is the configuration with label $\\ell$, state $v$, and stack $k$ equal to $P(k)$ followed by $S_0(k)$ with its first $c(k)$ symbols (from the top) removed.","title":"Applying an effect to the original stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/read-top-eq","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.readTop_eq","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#read-top-eq","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$, let $S_0$ and $P$ assign a list to each stack, let $c$ assign a natural number to each stack, and let $k$ be a stack with $c(k) < d$. Reading the top of stack $k$ from the depth-$d$ windows of the lists $S_0$, with pushed symbols $P$ and consumption counts $c$, gives the first symbol of $P(k)$ followed by $S_0(k)$ with its first $c(k)$ symbols removed, or \u22a5 if that list is empty.","title":"Reading the top through the windows"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/eff-agree","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.eff_agree","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#eff-agree","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Let $q$ be a statement tree of a multi-stack machine, $d \\in \\mathbb{N}$, and $S_0$ an assignment of a list $S_0(k)$ to each stack $k$. Let $v$ be a state, $P$ an assignment of a list $P(k)$ to each stack, and $c$ an assignment of a natural number $c(k)$ to each stack with $c(k) + \\operatorname{ops}(q) \\le d$ for every $k$. Then running $q$ from state $v$ on the stacks whose stack $k$ is $P(k)$ followed by $S_0(k)$\u2026","title":"The symbolic effect of a statement agrees with the real step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/step-i","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.stepI","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#step-i","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"For a finite TM2 machine $M$ and a configuration $c$, the idle-extended step of $M$ at $c$ is the configuration that one step of $M$ produces from $c$ when the label of $c$ is not the halted label, and is $c$ itself when it is (a halted configuration stays put).","title":"Idle-extended step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/run-i","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.runI","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#run-i","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"For a finite TM2 machine $M$ and a configuration $c$, the idle-extended run of $M$ from $c$ is the sequence of configurations whose term at time $0$ is $c$ and whose term at time $t + 1$ is the idle-extended step of $M$ applied to its term at time $t$.","title":"Idle-extended run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/win-c","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.winC","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#win-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Let $M$ be a finite TM2 machine with window depth $d$ and let $c$ be a configuration. The window of $c$ on stack $k$ is the list of length $d$ whose entry at position $j < d$ is the symbol at depth $j$ of stack $k$ (counted from the top, starting at $0$) when the stack has more than $j$ symbols, and the empty cell $\\bot$ otherwise.","title":"Windows of a configuration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/step-agree","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.step_agree","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#step-agree","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Let $M$ be a finite TM2 machine and $c$ a configuration with label $l$, state $v$ and stacks $S$, and let $(l', v', P, n)$ be the situation step of $M$ at label $l$ and state $v$ on the windows of $c$. Then the idle-extended step of $M$ at $c$ is the configuration with label $l'$, state $v'$ and, on each stack $k$, the list $P_k$ placed on top of what remains of $S_k$ after removing its top $n_k$ symbols.","title":"The idle-extended step is the situation step applied to the stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/code","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.code","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#code","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"For an encoding $E$ of a machine $M$ and a configuration $c$ of $M$, this is the number that $E$'s (label, state) numbering assigns to the pair formed by the label of $c$ (possibly the halted label) and the state of $c$.","title":"Label-and-state code of a configuration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/cell","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.cell","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#cell","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"For an encoding $E$ of a machine $M$, a configuration $c$ of $M$ and natural numbers $k$ and $j$, this is the code under $E$ of the symbol at position $j$ (counting from $0$ at the top) of the stack that $E$'s inverse stack numbering assigns to $k$, in $c$; it is $0$ (the code of \u22a5) if that stack has at most $j$ symbols.","title":"Cell code of a configuration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/sit-idx","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sitIdx","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#sit-idx","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Fix an encoding $E$ of a finite TM2 machine $M$, with $K$ stacks, symbol-code bound $g$ and window depth $d$. The situation index of a configuration $c$ is the number","title":"Situation index of a configuration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/tab-row","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.tabRow","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#tab-row","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Fix an encoding $E$ of a finite TM2 machine, a row height $H$, a situation $i$ and an old row $\\mathrm{old}$ giving a code for each stack number $k$ and depth $j$. For a stack number $k$, write $p$ for the number of symbols situation $i$ leaves pushed on stack $k$ and $n$ for the number of original symbols it consumes from stack $k$. The updated row at $(k, j)$ is the code of the symbol situation $i$ leaves at\u2026","title":"The table-driven row update"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/good","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.Good","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#good","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"For an encoding $E$ of a machine $M$, a configuration $c$ of $M$ is good when every symbol on every stack $k$ of $c$ is one of the effective symbols of stack $k$ listed by $E$ (so that it has a code).","title":"Good configurations"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/sit-idx-spec","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sitIdx_spec","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#sit-idx-spec","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Fix an encoding $E$ of a finite TM2 machine $M$, with $K$ stacks, $N$ situations and window depth $d$. If every symbol on every stack of a configuration $c$ lies in the effective alphabet of $E$, then the situation index $i$ of $c$ satisfies $i < N$, the label/state code of situation $i$ equals the combined (label, state) code of $c$, and for all $k < K$ and $j < d$ the window symbol code of situation $i$ on stack\u2026","title":"The situation index matches the code and windows"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/row-cell","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.row_cell","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#row-cell","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Let $e$ map elements of $\\alpha$ to natural numbers, and for a list $L$ over $\\alpha$ and $i \\in \\mathbb{N}$ let $L\\langle i\\rangle$ be $e$ of the entry of $L$ at position $i$ (counting from $0$), or $0$ if $L$ has at most $i$ entries. Subtraction of natural numbers is truncated at $0$. Let $P$ and $S$ be lists over $\\alpha$ and $n, H, d, j \\in \\mathbb{N}$ with $|P| \\le d$, $n \\le d$ and $|S| \\le H - d$, and let\u2026","title":"Cells of a pushed prefix over a truncated stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/table-agree","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.table_agree","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#table-agree","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Fix an encoding $E$ of a finite TM2 machine $M$, with $K$ stacks and window depth $d$, and a natural number $H$. Let $c$ be a configuration such that every symbol on every stack of $c$ lies in the effective alphabet of $E$ and every stack of $c$ has length at most $H \\mathbin{\\dot{-}} d$ (truncated subtraction). Let $c'$ be the idle-extended step of $M$ at $c$ and $i$ the situation index of $c$. Then the combined\u2026","title":"The table-driven step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/tab-row-reads-lt","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.tabRow_reads_lt","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#tab-row-reads-lt","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Fix an encoding $E$ of a finite TM2 machine and natural numbers $H$, $i$, $k$ and $j$. Write $p$ for the number of symbols situation $i$ leaves pushed on stack number $k$ and $n$ for the number of original symbols it consumes from stack number $k$. If $p \\le j$ and $j < (H \\mathbin{\\dot{-}} \\max(p, n)) + p$, where $\\dot{-}$ is truncated subtraction, then $j - p + n < H$.","title":"Reads of the in-range shift stay in the row"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/good-step","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.good_step","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#good-step","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Let $E$ be an encoding of a finite TM2 machine $M$ and let $c$ be a configuration of $M$. If $c$ is good for $E$, then so is the idle-extended step of $M$ at $c$.","title":"The idle-extended step preserves good configurations"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/table-agree-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.table_agree_run","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#table-agree-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Fix an encoding $E$ of a finite TM2 machine $M$, with $K$ stacks, $N$ situations and window depth $d$, a natural number $H$, an input list $s$ and a time $t$. For each time $u$ let $c_u$ be the configuration at time $u$ of the idle-extended run of $M$ from the initial configuration of $M$ on input $s$, and let $i$ be the situation index of $c_t$. If every stack of $c_t$ has length at most $H \\mathbin{\\dot{-}} d$\u2026","title":"The table-driven step along a run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/dec-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.decS","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#dec-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"For $f$ mapping each natural number to an element of $\\alpha$ or to \u22a5, and $j, m \\in \\mathbb{N}$, this is the list $[x_0, \\dots, x_{r-1}]$ with $f(j + i) = x_i$ for each $i < r$, where $r = m$ if none of $f(j), \\dots, f(j + m - 1)$ is \u22a5, and otherwise $r$ is the least $i < m$ with $f(j + i) = \\bot$.","title":"Reading a list from a sequence of cells"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/dec-s-of-list","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.decS_of_list","parent":30655,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03/index.html#dec-s-of-list","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-03","status":"fully_proved","summary":"Let $f$ map each natural number to an element of $\\alpha$ or to \u22a5, let $S$ be a list over $\\alpha$, and let $j, m \\in \\mathbb{N}$. Suppose that for every $i \\in \\mathbb{N}$, $f(j + i)$ is the entry of $S$ at position $i$ (counting from $0$) when $i$ is less than the length of $S$, and \u22a5 otherwise. If $S$ has length at most $m$, then reading $f$ from position $j$ for at most $m$ positions, stopping at the first \u22a5,\u2026","title":"Reading back a list from its cells"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","title":"Entries of a decoded stack on prefix-form input to Positive shift step on a real run (L3\u207a)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/get-elem-dec-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.getElem?_decS","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#get-elem-dec-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathrm{Option}\\,\\alpha$ and write $\\bot$ for the absent value. For $j, m \\in \\mathbb{N}$ let $\\mathrm{decS}(f, j, m)$ be the list of the values $f(j), f(j + 1), \\dots$, read for at most $m$ positions and stopped at the first $\\bot$. Suppose $f$ is in prefix form on the window from $j$ to $j + m$: whenever $j \\le a \\le b < j + m$ and $f(a) = \\bot$, also $f(b) = \\bot$. Then for every $i < m$,\u2026","title":"Entries of a decoded stack on prefix-form input"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/step-i-height","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.stepI_height","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#step-i-height","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine (finitely many stacks, labels and states, and a finite input alphabet), and let $d$ be its window depth, the largest number of push, peek and pop nodes on a root-to-leaf path of any of its statement trees. For every configuration $c$ of $M$ and every stack $k$, the length of stack $k$ after one idle-extended step of $M$ from $c$ is at most the length of stack $k$ of\u2026","title":"Stack heights grow by at most the window depth per step"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/dec-row","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.decRow","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#dec-row","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $H, n \\in \\mathbb{N}$, and let $w : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$. The decoded configuration has the label and the state that the inverse (label, state) numbering of $E$ assigns to $n$. For each stack $k$ of $M$, with stack number $\\kappa$, the decoded stack $k$ lists, from the top, the symbols of stack $k$ with codes $w(\\kappa, 0), w(\\kappa, 1), \\dots$,\u2026","title":"C1 decoder (decoding a row)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/row-ok","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.RowOK","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#row-ok","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine and $E$ an encoding of $M$ with $K$ numbered stacks, let $H \\in \\mathbb{N}$, and let $w : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$ assign a code $w(k, j)$ to each stack number $k$ and depth $j$. The row $w$ is well formed at height $H$ when for every $k < K$ and every $j < H$: $w(k, j)$ is at most the size of the effective alphabet of the stack numbered $k$, and if\u2026","title":"Well-formed rows of cells"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/dec-row-enc","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.decRow_enc","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#dec-row-enc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $H \\in \\mathbb{N}$, and let $c$ be a good configuration of $M$ (every symbol on every stack of $c$ lies in the effective alphabet of $E$) all of whose stacks have length at most $H$. Then the row decoder of $E$ at height $H$, applied to the (label, state) code of $c$ and the cell array of $c$, returns $c$.","title":"C1a (decoding recovers a good configuration)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/enc-dec-row","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.enc_decRow","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#enc-dec-row","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with $K$ numbered stacks and $A_0$ (label, state) codes, let $H \\in \\mathbb{N}$, let $n < A_0$, and let $w : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$ be a well-formed row of height $H$: for all $k < K$ and $j < H$, $w(k, j)$ is at most the size of the effective alphabet of stack number $k$, and if $w(k, j) = 0$ then $w(k, b) = 0$ for every $b$ with $j \\le b < H$. Let $c$ be\u2026","title":"C1b (decoding a well-formed row loses nothing)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/cell-row-ok","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.cell_rowOK","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#cell-row-ok","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with $K$ numbered stacks, and let $c$ be a good configuration of $M$, that is, every symbol on every stack of $c$ lies in the effective alphabet of $E$. Let $w$ be the cell array of $c$: $w(k, j)$ is the code of the symbol at depth $j$ from the top of stack number $k$, or $0$ if there is none. Then for every $H \\in \\mathbb{N}$, $w$ is a well-formed row of height $H$: for all\u2026","title":"C1c (cell arrays of good configurations are well-formed rows)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/dec-row-good","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.decRow_good","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#dec-row-good","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$. For all $H, n \\in \\mathbb{N}$ and every $w : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$, the configuration that the row decoder of $E$ at height $H$ returns on code $n$ and array $w$ is good: every symbol on each of its stacks lies in the effective alphabet of $E$.","title":"C1d (decoded configurations are good)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/dec-row-congr","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.decRow_congr","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#dec-row-congr","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with $K$ numbered stacks, let $H, n \\in \\mathbb{N}$, and let $w, w' : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$. If $w(k, j) = w'(k, j)$ for all $k < K$ and $j < H$, then the row decoder of $E$ at height $H$ returns the same configuration on code $n$ and array $w$ as on code $n$ and array $w'$.","title":"C1e (the row decoder reads only the window)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/row-inj","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.row_inj","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#row-inj","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine and $E$ an encoding of $M$ with $K$ numbered stacks, let $H \\in \\mathbb{N}$, and for a configuration $c$ write $\\mathrm{cell}(c, k, j)$ for the code of the symbol at depth $j$ of the stack numbered $k$ ($0$ if there is none). Let $c$ and $c'$ be configurations that are good for $E$ (every stack symbol lies in the effective alphabet of its stack) and all of whose stacks\u2026","title":"Good configurations are determined by their row (C1f)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/dec-row-tab-row","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.decRow_tabRow","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#dec-row-tab-row","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with window depth $d$, let $H \\ge d$ be a natural number, and let $c$ be a good configuration of $M$ (every symbol on every stack of $c$ lies in the effective alphabet of $E$) all of whose stacks have length at most $H - d$. Let $i$ be the situation index of $c$. Then the row decoder of $E$ at height $H$, applied to the next (label, state) code of situation $i$ and to the\u2026","title":"C2' (decoding the table-driven row gives the next configuration)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/tr","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Tr","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#tr","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"For an assignment $a$, given as a list of booleans, and a variable index $v \\in \\mathbb{N}$, the variable $v$ is true under $a$ when $v$ is less than the length of $a$ and the entry of $a$ at position $v$ is true. In particular, variables with index at least the length of $a$ are not true.","title":"Truth of a variable under an assignment"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/clause-sat-dense","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.clauseSat_dense","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#clause-sat-dense","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $V \\in \\mathbb{N}$, let $\\ell$ be a list of pairs $(p, b)$ of a variable index $p \\in \\mathbb{N}$ and a sign $b$ (true or false), and let $a$ be a list of booleans. If $a$ satisfies the dense clause over $V$ variables built from $\\ell$, then some pair $(p, b) \\in \\ell$ has $p$ less than the length of $a$ and $a_p = b$.","title":"L0 (a satisfied dense clause has a satisfied listed literal)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/sh-force","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.sh_force","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#sh-force","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $L$ be a layout with $K$ stacks and $g$ symbol codes, let $sa$, $pl$, $cn$, $pu$ be situation tables and $N, T, H \\in \\mathbb{N}$, and let $a$ be an assignment satisfying the cell-shift clause family of $L$ with these tables and bounds $N$, $T$, $H$. Call a variable true when its index is less than the length of $a$ and $a$ is true there. Write $S[t, i]$ for the label/state variable $sa(i)$ of row $t + 1$ and\u2026","title":"In-range shift forcing (L1)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/cells-are","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.CellsAre","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#cells-are","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with $K$ numbered stacks and $g$ symbol codes, let $a$ be a list of booleans, let $H, t \\in \\mathbb{N}$, and let $w : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$. The cell block of row $t$ is exactly $w$ under $a$ when, for all $k < K$, $j < H$ and $x < g$, the cell variable for time $t$, stack $k$, depth $j$ and symbol code $x$ in the layout of $E$ at height $H$ is true under\u2026","title":"Cell block of a row equal to an array"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/cells-hold","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.CellsHold","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#cells-hold","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with $K$ numbered stacks, let $a$ be a list of booleans, let $H, t \\in \\mathbb{N}$, and let $w : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$. The cell literals of $w$ hold in row $t$ under $a$ when, for all $k < K$ and $j < H$, the cell variable for time $t$, stack $k$, depth $j$ and symbol code $w(k, j)$ in the layout of $E$ at height $H$ is true under $a$, that is, its index\u2026","title":"Cell literals of an array hold in a row"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/sh-hold","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sh_hold","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#sh-hold","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine and $E$ an encoding of $M$ with $K$ numbered stacks, tables $sa$, $pl$, $cn$, $pu$ and $N$ situations, let $T, H, t \\in \\mathbb{N}$ with $t < T$, and let $a$ be an assignment satisfying the cell-shift clause family of the layout of $E$ with these tables and bounds $N$, $T$, $H$. Call a variable true when its index is less than the length of $a$ and $a$ is true there.\u2026","title":"Positive core of the shift step (L2\u207a)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/sh-step-cell","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sh_step_cell","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#sh-step-cell","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine and $E$ an encoding of $M$ with $K$ numbered stacks, $g$ symbol codes, tables $sa$, $pl$, $cn$, $pu$ and $N$ situations, let $T, H, t \\in \\mathbb{N}$ with $t < T$, and let $a$ be an assignment satisfying the cell-shift clause family of the layout of $E$ with these tables and bounds $N$, $T$, $H$. Call a variable true when its index is less than the length of $a$ and\u2026","title":"Exact shift step (L2)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/sh-step-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sh_step_run","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#sh-step-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine and $E$ an encoding of $M$ with $K$ numbered stacks, $g$ symbol codes, tables $sa$, $pl$, $cn$, $pu$ and $N$ situations, let $T, H, t \\in \\mathbb{N}$ with $t < T$, and let $a$ be an assignment satisfying the cell-shift clause family of the layout of $E$ with these tables and bounds $N$, $T$, $H$. Call a variable true when its index is less than the length of $a$ and\u2026","title":"Exact shift step on a real run (L3)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/sh-step-run-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sh_step_run_pos","parent":30676,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04/index.html#sh-step-run-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-04","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine and $E$ an encoding of $M$ with $K$ numbered stacks, tables $sa$, $pl$, $cn$, $pu$ and $N$ situations, let $T, H, t \\in \\mathbb{N}$ with $t < T$, and let $a$ be an assignment satisfying the cell-shift clause family of the layout of $E$ with these tables and bounds $N$, $T$, $H$. Call a variable true when its index is less than the length of $a$ and $a$ is true there.\u2026","title":"Positive shift step on a real run (L3\u207a)"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","title":"Transition clauses force the next label to The three depth regions cover every depth exactly once"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/fam-force","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.fam_force","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#fam-force","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $L$ be a layout with $K$ stacks and window depth $d$, let $\\mathrm{la}, \\mathrm{na} : \\mathbb{N} \\to \\mathbb{N}$ and $\\mathrm{wa} : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$ be tables, and let $a$ be an assignment satisfying the transition clause family of $L$, $\\mathrm{la}$, $\\mathrm{na}$ and $\\mathrm{wa}$ for bounds $T$, $H$ and $N$. Write $X[t, b]$ and $C[t, k, j, x]$ for the label/state and\u2026","title":"Transition clauses force the next label"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/fam-force-kj","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.fam_force_kj","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#fam-force-kj","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $L$ be a layout with $K$ stacks and window depth $d$, let $\\mathrm{la}, \\mathrm{na} : \\mathbb{N} \\to \\mathbb{N}$ and $\\mathrm{wa} : \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N} \\to \\mathbb{N}$ be tables, and let $a$ be an assignment satisfying the transition clause family of $L$, $\\mathrm{la}$, $\\mathrm{na}$ and $\\mathrm{wa}$ for bounds $T$, $H$ and $N$. Write $X[t, b]$ and $C[t, k, j, x]$ for the label/state and\u2026","title":"Transition clauses force the next label, stack-and-depth form"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/code-holds","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.CodeHolds","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#code-holds","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Given an encoding of a machine $M$, an assignment $a$, a height $H$, a time $t$ and a configuration $c$ of $M$, the code of $c$ holds in row $t$ when the label/state variable of row $t$ (at height $H$) whose index is the combined (label, state) code of $c$ is true under $a$.","title":"The code of a configuration holds in a row"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/sd-hold","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sd_hold","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#sd-hold","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with $K$ stacks and layout $L$, let $d$ be the window depth of $M$, let $H \\in \\mathbb{N}$ with $d \\le H$, let $W$ be the row width of $L$ at height $H$, and write $X[t, x] = tW + x$ for the label/state variables and $C[t, k, j, x]$ for the cell variable of row $t$, stack $k$, depth $j$ and symbol code $x$. For a configuration $c$ write $\\kappa(c)$ for its (label, state)\u2026","title":"L2 (S-definition, positive)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/sd-step-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sd_step_run","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#sd-step-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with $K$ stacks and layout $L$, let $d$ be the window depth of $M$, let $H \\in \\mathbb{N}$ with $d \\le H$, let $W$ be the row width of $L$ at height $H$, and write $X[t, x] = tW + x$ for the label/state variables and $C[t, k, j, x]$ for the cell variable of row $t$, stack $k$, depth $j$ and symbol code $x$. For a configuration $c$ write $\\kappa(c)$ for its (label, state)\u2026","title":"L3 (S-definition, real run)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/sdsh-step-run-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sdsh_step_run_pos","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#sdsh-step-run-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with $K$ stacks, layout $L$, $N$ situations and situation tables $\\mathrm{la}$, $\\mathrm{wa}$, $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$ and $\\mathrm{pu}$, let $d$ be the window depth of $M$, let $H \\in \\mathbb{N}$ with $d \\le H$, and let $T, t \\in \\mathbb{N}$ with $t < T$. Call a variable true under an assignment $a$ when its index is less than the length of $a$ and $a$\u2026","title":"Composition with the shift family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/upd-hold","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.upd_hold","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#upd-hold","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with $K$ stacks and layout $L$, let $d$ be the window depth of $M$, let $H \\in \\mathbb{N}$ with $d \\le H$, let $W$ be the row width of $L$ at height $H$, and write $X[t, x] = tW + x$ for the label/state variables and $C[t, k, j, x]$ for the cell variable of row $t$, stack $k$, depth $j$ and symbol code $x$. For a configuration $c$ write $\\kappa(c)$ for its (label, state)\u2026","title":"Update forcing (positive)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/upd-step-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.upd_step_run","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#upd-step-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with $K$ stacks and layout $L$, let $d$ be the window depth of $M$, let $H \\in \\mathbb{N}$ with $d \\le H$, let $W$ be the row width of $L$ at height $H$, and write $X[t, x] = tW + x$ for the label/state variables and $C[t, k, j, x]$ for the cell variable of row $t$, stack $k$, depth $j$ and symbol code $x$. For a configuration $c$ write $\\kappa(c)$ for its (label, state)\u2026","title":"Update forcing, real run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/pf-force","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.pf_force","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#pf-force","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $L$ be a layout with $A$ label/state variables, $K$ stacks and $g$ symbol codes, let $\\mathrm{sa}, \\mathrm{pl}, \\mathrm{cn}, \\mathrm{pu}$ be situation tables and $N, T, H \\in \\mathbb{N}$, and write $W = A + KHg$, $X[t, x] = tW + x$ and $C[t, k, j, x] = tW + A + (kH + j)g + x$. If the assignment $a$ satisfies the cell-shift family of these data, $t < T$, $i < N$, $k < K$, $j < \\mathrm{pl}(i, k)$ and the variable\u2026","title":"Pushed-fill forcing"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/ov-force","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.ov_force","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#ov-force","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $L$ be a layout with $K$ stacks, let $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$ and $\\mathrm{pu}$ be situation tables, and let $a$ be an assignment satisfying the cell-shift clause family of $L$ and these tables for bounds $N$, $T$ and $H$. Write $X[t, b]$ and $C[t, k, j, x]$ for the label/state and cell variables at height $H$. If $t < T$, $i < N$, $k < K$, $r < \\mathrm{cn}(i, k) - \\mathrm{pl}(i, k)$ and the\u2026","title":"Overflow forcing"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/pf-hold","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.pf_hold","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#pf-hold","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with $K$ stacks and layout $L$, let $H \\in \\mathbb{N}$, let $W$ be the row width of $L$ at height $H$, and write $X[t, x] = tW + x$ for the label/state variables and $C[t, k, j, x]$ for the cell variable of row $t$, stack $k$, depth $j$ and symbol code $x$. Let $T, t \\in \\mathbb{N}$ with $t < T$, let the assignment $a$ satisfy the cell-shift family of $E$ (tables\u2026","title":"Pushed-fill forcing for a good configuration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/ov-hold","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.ov_hold","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#ov-hold","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with $K$ stacks and window depth $d$, and let $a$ be an assignment satisfying the cell-shift clause family of $E$'s layout and tables $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$ for $E$'s number of situations and bounds $T$ and $H$. Write $X[t, b]$ and $C[t, k, j, x]$ for the label/state and cell variables at height $H$. Let $t < T$, let $c$ be a configuration\u2026","title":"Overflow cells of the next row hold the table value"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/pf-step-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.pf_step_run","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#pf-step-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with $K$ stacks and layout $L$, let $d$ be the window depth of $M$, let $H \\in \\mathbb{N}$ with $d \\le H$, let $W$ be the row width of $L$ at height $H$, and write $X[t, x] = tW + x$ for the label/state variables and $C[t, k, j, x]$ for the cell variable of row $t$, stack $k$, depth $j$ and symbol code $x$. Fix an input list $s$, let $c_n$ be the configuration after $n$\u2026","title":"Pushed-fill forcing on a real run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/ov-step-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.ov_step_run","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#ov-step-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with $K$ stacks and window depth $d$, let $s$ be an input, and let $c_t$ be the configuration at step $t$ of the idle-extended run of $M$ from the initial configuration on input $s$. Let $a$ be an assignment satisfying the cell-shift clause family of $E$'s layout and tables $\\mathrm{sa}$, $\\mathrm{pl}$, $\\mathrm{cn}$, $\\mathrm{pu}$ for $E$'s number of situations and bounds\u2026","title":"Overflow cells of the next row on a real run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/cap-h","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.capH","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#cap-h","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"For natural numbers $m$, $T$ and $d$ (an input bound, a run length and a per-step growth), the stack cap is $H = m + (T + 1) d + 1$.","title":"Stack cap"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/run-i-height","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.runI_height","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#run-i-height","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $M$ be a finite TM2 machine with window depth $d$, let $c$ be a configuration and $k$ a stack. For every $t \\in \\mathbb{N}$, stack $k$ of the configuration reached from $c$ after $t$ steps of the idle-extended run has length at most the length of stack $k$ of $c$ plus $t \\cdot d$.","title":"Stack heights along a run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/cap-h-height","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.capH_height","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#cap-h-height","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with window depth $d$, let $m$, $T$ and $t$ be natural numbers with $t \\le T$, and let $s$ be an input of length at most $m$. In the idle-extended run of $M$ from the initial configuration on input $s$, every stack of the configuration at step $t$ has length at most $H - d$, where $H = m + (T + 1) d + 1$ is the stack cap for input bound $m$, run length $T$ and per-step\u2026","title":"Stack heights are at most the cap minus the window depth"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/cap-h-c1","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.capH_c1","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#cap-h-c1","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"For natural numbers $\\ell$, $m$, $T$ and $d$ with $\\ell \\le m$, we have $\\ell + (T + 1) d \\le H$, where $H = m + (T + 1) d + 1$ is the stack cap for input bound $m$, run length $T$ and per-step growth $d$.","title":"Stack-cap constraint 1"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/cap-h-c2","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.capH_c2","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#cap-h-c2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"For natural numbers $m$, $T$ and $d$ with $T \\ge 1$, the stack cap $H = m + (T + 1) d + 1$ for input bound $m$, run length $T$ and per-step growth $d$ satisfies $2d \\le H$.","title":"Stack-cap constraint 2"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/region-cover","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.region_cover","parent":30697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05/index.html#region-cover","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-05","status":"fully_proved","summary":"Let $p, q, H, j$ be natural numbers with $j < H$, and let $m = H - \\max(p, q) + p$, where the subtraction is truncated at $0$. Then $j$ lies in exactly one of the ranges $j < p$ (pushed fill), $p \\le j < m$ (in-range shift) and $m \\le j < H$ (overflow).","title":"The three depth regions cover every depth exactly once"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","title":"Cell literals propagate one step along the real run to The init family discharges the init obligation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/cells-step-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.cells_step_run","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#cells-step-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with window depth $d$, let $s$ be a list of symbols of the input stack of $M$, and for each $n$ let $c_n$ be the configuration after $n$ steps of the idle-extended run of $M$ from its initial configuration on $s$. Let $T, H, t$ be natural numbers with $t < T$ and $d \\le H$. Suppose an assignment $a$ satisfies the transition clause family of $E$ with the situation-variable\u2026","title":"Cell literals propagate one step along the real run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/row-step-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.row_step_run","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#row-step-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Fix an encoding $E$ of a machine $M$ with $K$ stacks, window depth $d$ and $N$ situations, natural numbers $T$, $H$, $t$ with $t < T$ and $d \\le H$, an assignment $a$ and an input list $s$, and let $c_t$ be the configuration at time $t$ of the idle-extended run of $M$ from the start configuration on $s$. Suppose $a$ satisfies the transition clause families of the layout of $E$ with tables\u2026","title":"One-step invariant"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/row-step-cap-h","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.row_step_capH","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#row-step-cap-h","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Fix an encoding $E$ of a machine $M$ with $K$ stacks, window depth $d$ and $N$ situations, natural numbers $m$, $T$, $t$ with $t < T$, an assignment $a$ and an input list $s$ of length at most $m$. Let $H = m + (T + 1)d + 1$ be the stack cap and let $c_t$ be the configuration at time $t$ of the idle-extended run of $M$ from the start configuration on $s$. Suppose $a$ satisfies the transition clause families of the\u2026","title":"One-step invariant at the stack cap"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/row0","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.Row0","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#row0","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"For an encoding $E$ of a machine $M$ with $K$ stacks, an assignment $a$, a height $H$ and an input list $s$, let $c$ be the start configuration of $M$ on $s$. The row-0 obligation says that, in the layout of $E$ at height $H$, the literal of the (label, state) code of $c$ in row $0$ is true under $a$ and, for every stack number $k < K$ and depth $j < H$, the literal of the cell code of $c$ at stack $k$ and depth\u2026","title":"Row-0 obligation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-obl","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.InitObl","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-obl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with window depth $d$, let $a$ be an assignment, let $x$ and $Y$ be lists of symbols of the input stack of $M$, and let $m, T$ be natural numbers, and put $H = m + (T + 1)d + 1$. The init obligation for $a$, $x$, $Y$, $m$ and $T$ holds when there is a list $y$ of symbols of $Y$ with $|x| + |y| \\le m$ such that the row-0 obligation holds for $a$ at height $H$ and the input $x$\u2026","title":"Init obligation, input pinned"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/rows-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.rows_run","parent":30718,"reading_title":{"format":"inline-parts-v1","parts":[["text","Row-"],["math","\\(t\\)"],["text"," induction"]],"source":"Row-$t$ induction"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#rows-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Fix an encoding $E$ of a machine $M$ with $K$ stacks, window depth $d$ and $N$ situations, natural numbers $T$ and $H$ with $d \\le H$, an assignment $a$ and an input list $s$, and let $c_t$ be the configuration at time $t$ of the idle-extended run of $M$ from the start configuration on $s$. Suppose $a$ satisfies the transition clause families of the layout of $E$ with tables\u2026","title":"Row-$t$ induction"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/rows-cap-h","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.rows_capH","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#rows-cap-h","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Fix an encoding $E$ of a machine $M$ with $K$ stacks, window depth $d$ and $N$ situations, natural numbers $m$ and $T$, an assignment $a$ and an input list $s$ of length at most $m$. Let $H = m + (T + 1)d + 1$ be the stack cap and let $c_t$ be the configuration at time $t$ of the idle-extended run of $M$ from the start configuration on $s$. Suppose $a$ satisfies the transition clause families of the layout of $E$\u2026","title":"Row induction at the stack cap"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/rows-of-init","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.rows_of_init","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#rows-of-init","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Fix an encoding $E$ of a machine $M$ with $K$ stacks, window depth $d$ and $N$ situations, lists $x$ and $Y$ of input symbols, natural numbers $m$ and $T$, and an assignment $a$, and let $H = m + (T + 1)d + 1$ be the stack cap. Suppose $a$ satisfies the transition clause families of the layout of $E$ with tables $(\\mathrm{la}, \\mathrm{na}, \\mathrm{wa})$ and $(\\mathrm{la}, \\mathrm{sa}, \\mathrm{wa})$ and the\u2026","title":"Run tracking from the init obligation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/cnf-sat-append","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.cnfSat_append","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#cnf-sat-append","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"For dense CNF formulas $\\varphi$ and $\\psi$ and a finite bit sequence $a$, $a$ satisfies the concatenation of $\\varphi$ and $\\psi$ (the clauses of $\\varphi$ followed by those of $\\psi$) if and only if $a$ satisfies both $\\varphi$ and $\\psi$.","title":"Satisfying a concatenation of CNF formulas"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/unit-force","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.unit_force","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#unit-force","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"If an assignment $a$ satisfies a CNF formula $\\varphi$ containing the dense clause over $V$ variables built from the literal list whose only entry is the pair $(v, \\mathrm{true})$, then variable $v$ is true under $a$.","title":"Positive dense unit clauses force their variable"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/z-idx","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.zIdx","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#z-idx","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"For an encoding $E$ and natural numbers $H$ and $j$, this is the index, in the layout of $E$ at height $H$, of the cell variable for row $0$, the stack number of the input stack, depth $j$ from the top and symbol code $0$ (the empty cell).","title":"Index of the empty-cell variable of the input stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-code","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.initCode","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-code","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ and let $T, H$ be natural numbers. The start-code clause is the CNF formula with a single clause: the dense clause over the $(T + 1)W$ variables of the layout of $E$, where $W$ is its row width at height $H$, whose only literal is the positive literal of the label/state variable of row $0$ for the code under $E$ of the pair formed by the main label of $M$ and its initial state.","title":"Start-code clause of the init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-other","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.initOther","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-other","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with $K$ numbered stacks and let $T, H$ be natural numbers. The empty-stack clauses form the CNF formula that lists, for each stack number $k < K$ other than the number of the input stack in increasing order and, within it, each depth $j < H$ in increasing order, the dense clause over the $(T + 1)W$ variables of the layout of $E$, where $W$ is its row width at height $H$,\u2026","title":"Empty-stack clauses of the init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-in","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.initIn","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-in","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x = x_0 x_1 \\cdots x_{n-1}$ be a list of symbols of the input stack of $M$, and let $T, H$ be natural numbers. The input clauses form the CNF formula that lists, for $j = 0, 1, \\dots, n - 1$ in order, the dense clause over the $(T + 1)W$ variables of the layout of $E$, where $W$ is its row width at height $H$, whose only literal is the positive literal of the cell\u2026","title":"Input clauses of the init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-cert","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.initCert","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-cert","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, and let $m, T, H$ be natural numbers. For a depth $j$ and a symbol code $c$, write $C_0[j, c]$ for the cell variable of row $0$, input stack, depth $j$ and code $c$ in the layout of $E$ at height $H$, and let $e$ be the symbol code of $E$ on the input stack. The certificate clauses form the CNF formula that\u2026","title":"Certificate clauses of the init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-pre","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.initPre","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-pre","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x$ be a list of symbols of the input stack of $M$, and let $m, T, H$ be natural numbers. For a depth $j$, write $Z_j$ for the cell variable of row $0$, input stack, depth $j$ and code $0$ (the empty cell) in the layout of $E$ at height $H$. The prefix clauses form the CNF formula that lists, for $j = |x|, |x| + 1, \\dots, m - 1$ in order (none if $m \\le |x|$), the dense\u2026","title":"Prefix clauses of the init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-tail","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.initTail","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-tail","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"For an encoding $E$ and natural numbers $m$, $T$ and $H$, this is the CNF formula with one clause for each depth $j$ with $m \\le j < H$, in increasing order of $j$: the dense clause over the variables of the transition-clause formula of the layout of $E$ at bounds $T$ and $H$, built from the single positive literal of the empty-cell variable of the input stack at depth $j$ in row $0$.","title":"Length-bound clauses of the init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-family","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.initFamily","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-family","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, and let $m, T, H$ be natural numbers. The init family is the CNF formula whose clauses are, in this order, those of the start-code clause at $(T, H)$, the empty-stack clauses at $(T, H)$, the input clauses of $x$ at $(T, H)$, the certificate clauses of $x$ and $Y$ at $(m, T, H)$, the prefix clauses of $x$ at\u2026","title":"The row-0 init family of an encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-row0","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.init_row0","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-row0","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Fix an encoding $E$ of a machine $M$, lists $x$ and $Y$ of symbols of the input alphabet, natural numbers $m$, $T$, $H$ with $|x| \\le m < H$, and an assignment $a$ that satisfies the init family of $E$ for $x$, $Y$, $m$, $T$ and $H$. Then there is a list $y$ of input symbols, each occurring in $Y$, such that the concatenation $xy$ has length at most $m$ and the row-0 obligation holds for $a$, height $H$ and the\u2026","title":"Init semantics"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/init-obl-2","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.init_obl","parent":30718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06/index.html#init-obl-2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-06","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with window depth $d$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, and let $m, T$ be natural numbers with $|x| \\le m$, and put $H = m + (T + 1)d + 1$. If an assignment $a$ satisfies the init family of $E$ for $x$, $Y$, $m$, $T$ and $H$, then the init obligation holds for $a$, $x$, $Y$, $m$ and $T$: there is a list $y$ of symbols of $Y$ with\u2026","title":"The init family discharges the init obligation"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","title":"A satisfying assignment tracks a certified run to Unfolding the idle-extended run from the front"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/rows-of-init-family","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.rows_of_initFamily","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#rows-of-init-family","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with window depth $d$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, let $m, T \\in \\mathbb{N}$ with $|x| \\le m$, and let $H = m + (T + 1)d + 1$ be the corrected stack cap. Write $\\mathrm{la}, \\mathrm{na}, \\mathrm{sa}, \\mathrm{wa}, \\mathrm{pl}, \\mathrm{cn}, \\mathrm{pu}$ for the situation tables of $E$ and $N$ for its number of situations. Suppose an assignment\u2026","title":"A satisfying assignment tracks a certified run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/faithful","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Faithful","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#faithful","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"A list $\\ell$ of literals $(p, b)$, each a variable index $p \\in \\mathbb{N}$ with a sign $b$, is faithful for $V \\in \\mathbb{N}$ when every literal $(p, b)$ of $\\ell$ has $p < V$ and the first literal of $\\ell$ with index $p$ is $(p, b)$.","title":"Faithful literal list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/fam-ok","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.FamOK","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#fam-ok","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"A CNF formula $\\varphi$ is a faithful family over $V$ variables when every clause of $\\varphi$ is the dense clause over $V$ variables built from some list of literals that is faithful for $V$.","title":"Faithful dense clause family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/fam-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.FamPos","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#fam-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"A CNF formula $\\varphi$ is a faithful family with positive literals over $V$ variables when every clause of $\\varphi$ is the dense clause over $V$ variables built from some list $\\ell$ of literals that is faithful for $V$ and contains a positive literal $(p, \\mathrm{true})$ for some $p$.","title":"Faithful dense clause family with positive literals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/clause-sat-iff","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.clauseSat_iff","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#clause-sat-iff","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $V \\in \\mathbb{N}$, let $\\ell$ be a list of pairs $(p, b)$ of a variable index $p$ and a sign $b$ that is faithful for $V$, and let $a$ be a list of Booleans. Then $a$ satisfies the dense clause over $V$ variables built from $\\ell$ if and only if some $(p, b) \\in \\ell$ has $p < |a|$ and $a_p = b$.","title":"Exact clause semantics"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/sat-all-true","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.sat_allTrue","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#sat-all-true","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $V \\in \\mathbb{N}$ and let $\\ell$ be a list of literals that is faithful for $V$ and contains a positive literal $(p, \\mathrm{true})$. Then the all-true assignment of length $V$ satisfies the dense clause over $V$ variables built from $\\ell$.","title":"The all-true assignment satisfies a faithful clause with a positive literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/lt-num-vars","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.lt_numVars","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#lt-num-vars","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $L$ be a layout, let $T$, $H$, $t$ and $p$ be natural numbers with $t \\le T$, and let $W$ be the number of variables per time step of $L$ at height $H$. If $p < (t + 1) W$, then $p$ is less than the number of variables of $L$ for bounds $T$ and $H$.","title":"Variables of rows up to the time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/init-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.init_pos","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#init-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, and let $m, T, H \\in \\mathbb{N}$ with $|x| \\le m < H$. Let $V$ be the number of variables of the layout of $E$ at bounds $T$ and $H$. Then every clause of the init family of $E$ for $x$, $Y$, $m$, $T$, $H$ is the dense clause over $V$ variables built from a list of literals that is faithful for $V$ and contains a\u2026","title":"The init family is faithful with positive literals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/acode","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.acode","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#acode","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"For an encoding $E$ of a finite stack machine $M$, the accepting code is the (label, state) code that $E$ assigns to the pair of the halted label and the initial state of $M$, the label and state of a halted configuration as acceptance reads it.","title":"Accepting code"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/acc-code","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.accCode","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#acc-code","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a finite stack machine $M$ with $A_0$ (label, state) codes, let $c^\\ast$ be the code that $E$ assigns to the halted label together with the initial state of $M$, and let $T, H \\in \\mathbb{N}$. In the layout of $E$ at height $H$, let $W$ be the row width, $V = (T + 1)W$ the number of variables, and $X[t, u] = tW + u$ the label/state variable $u$ at time $t$. The accept-code clauses are the\u2026","title":"Accept-code clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/acc-cell","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.accCell","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#acc-cell","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a finite stack machine $M$ with output stack $k_1$, $g$ symbol codes and stack numbering $\\kappa$, let $\\alpha$ be a symbol of stack $k_1$ with code $e$, and let $T, H \\in \\mathbb{N}$. In the layout of $E$ at height $H$, let $W$ be the row width, $V = (T + 1)W$ the number of variables, and $C[t, k, j, x]$ the variable for stack $k$, depth $j$ and symbol code $x$ at time $t$. The\u2026","title":"Accept-cell clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/acc-family","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.accFamily","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#acc-family","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a finite stack machine $M$, let $\\alpha$ be a symbol of the output stack of $M$, and let $T, H \\in \\mathbb{N}$. The accept family for $\\alpha$ at bounds $T$ and $H$ is the CNF formula formed by the accept-code clauses at $T$ and $H$ followed by the accept-cell clauses for $\\alpha$ at $T$ and $H$.","title":"The accept family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/acc-code-2","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.acc_code","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#acc-code-2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a finite stack machine $M$ with $A_0$ (label, state) codes, let $\\alpha$ be a symbol of the output stack of $M$, and let $T, H, u \\in \\mathbb{N}$ with $u < A_0$. In the layout of $E$ at height $H$, let $X[t, w]$ be the label/state variable $w$ at time $t$, and call a variable $v$ true under an assignment $a$ when $v < |a|$ and $a_v$ is true. If an assignment $a$ satisfies the accept family\u2026","title":"Accept-code clauses pin the (label, state) code"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/acc-cell-2","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.acc_cell","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#acc-cell-2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a finite stack machine $M$ with output stack $k_1$, $g$ symbol codes and stack numbering $\\kappa$, let $\\alpha$ be a symbol of stack $k_1$, and let $T, H, x \\in \\mathbb{N}$ with $x < g$. In the layout of $E$ at height $H$, let $C[t, k, j, z]$ be the variable for stack $k$, depth $j$ and symbol code $z$ at time $t$, and call a variable $v$ true under an assignment $a$ when $v < |a|$ and\u2026","title":"Accept-cell clauses pin the top output symbol"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/accepts","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Accepts","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#accepts","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"A configuration $c$ of a finite stack machine $M$ is accepting for a symbol $\\alpha$ of the output stack of $M$ when $c$ is halted (it has no label), its state is the initial state of $M$, and the output stack of $c$ is nonempty with top symbol $\\alpha$.","title":"Accepting configuration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/accept-sound","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.accept_sound","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#accept-sound","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a finite stack machine $M$ with input stack $k_0$ and output stack $k_1$, let $x$ and $Y$ be lists of symbols of $k_0$, let $m, T \\in \\mathbb{N}$ with $|x| \\le m$, and let $\\alpha$ be a symbol of $k_1$ that is one of the effective symbols of $E$ for $k_1$. Put $H = m + (T + 1)d + 1$, where $d$ is the window depth of $M$, and let $N$ be the number of situations of $E$. Suppose an assignment\u2026","title":"Satisfying assignments yield accepting runs"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/unsat-of-no-accept","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.unsat_of_noAccept","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#unsat-of-no-accept","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with window depth $d$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, let $m, T \\in \\mathbb{N}$ with $|x| \\le m$, let $H = m + (T + 1)d + 1$, and let $\\alpha$ be a symbol of the output stack of $M$ in the effective alphabet of $E$. Write $\\mathrm{la}, \\mathrm{na}, \\mathrm{sa}, \\mathrm{wa}, \\mathrm{pl}, \\mathrm{cn}, \\mathrm{pu}$ for the situation tables of $E$\u2026","title":"No accepting certificate, no satisfying assignment"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/trans-all-true","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.trans_allTrue","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#trans-all-true","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with window depth $d$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, and let $m, T, H \\in \\mathbb{N}$ with $|x| \\le m < H$ and $d \\le H$. Write $\\mathrm{la}, \\mathrm{na}, \\mathrm{sa}, \\mathrm{wa}, \\mathrm{pl}, \\mathrm{cn}, \\mathrm{pu}$ for the situation tables of $E$, $N$ for its number of situations and $V$ for the number of variables of its layout at bounds\u2026","title":"The all-true assignment satisfies the transition, shift and init families"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/acc-family-all-true","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.accFamily_allTrue","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#acc-family-all-true","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"Let $E$ be an encoding of a finite stack machine $M$, let $\\alpha$ be a symbol of the output stack of $M$, and let $T, H \\in \\mathbb{N}$. The assignment that sets all $V$ variables to true, where $V$ is the number of variables of the layout of $E$ at bounds $T$ and $H$, does not satisfy the accept family for $\\alpha$ at bounds $T$ and $H$.","title":"The all-true assignment violates the accept family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/run-i-succ","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.runI_succ'","parent":30739,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07/index.html#run-i-succ","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-07","status":"fully_proved","summary":"For a finite TM2 machine $M$, a configuration $c$ and every $n \\in \\mathbb{N}$, the configuration at time $n + 1$ of the idle-extended run of $M$ from $c$ equals the configuration at time $n$ of the idle-extended run of $M$ from the idle-extended step of $M$ at $c$.","title":"Unfolding the idle-extended run from the front"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","title":"The idle-extended run follows the real run to Completeness of the five clause families"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/run-i-of-iterate","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.runI_of_iterate","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#run-i-of-iterate","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine, let $n \\in \\mathbb{N}$, and let $c, c'$ be configurations of $M$. If iterating the step function of $M$ $n$ times as a partial function, starting from $c$, yields $c'$, then the idle-extended run of $M$ from $c$ is at $c'$ after $n$ steps.","title":"The idle-extended run follows the real run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/run-i-halted","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.runI_halted","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#run-i-halted","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine and $c$ a configuration of $M$ whose label is empty (that is, $c$ has halted). Then for every $t \\in \\mathbb{N}$, the idle-extended run of $M$ from $c$ is at $c$ after $t$ steps.","title":"A halted configuration is fixed by the idle-extended run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/run-i-halt-list","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.runI_haltList","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#run-i-halt-list","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a machine, $s$ a word over its input alphabet and $l'$ a list over its output alphabet. Suppose that $M$, started in the initial configuration on $s$, reaches the halting configuration with output $l'$ (halted, in the initial state, with $l'$ on the output stack and every other stack empty) in at most $p$ steps. Then for every $T \\ge p$, the configuration reached after $T$ steps of the idle-extended run\u2026","title":"The idle-extended run stays at the halting configuration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/accepts-halt-list","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.accepts_haltList","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#accepts-halt-list","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a machine, $l'$ a list over its output alphabet and $\\mathrm{acc}$ an output symbol. The halting configuration with output $l'$ (halted, in the initial state, with $l'$ on the output stack and every other stack empty) is accepting for $\\mathrm{acc}$ if and only if $l'$ is nonempty and its first element is $\\mathrm{acc}$.","title":"Acceptance of a halting configuration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/row-pos-inj","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.row_pos_inj","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#row-pos-inj","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $W$, $t$, $t'$, $r$ and $r'$ be natural numbers with $r < W$ and $r' < W$. If $tW + r = t'W + r'$, then $t = t'$ and $r = r'$.","title":"Uniqueness of quotient and remainder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/mod-pow-eq-of-digits","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.mod_pow_eq_of_digits","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#mod-pow-eq-of-digits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $b$, $i$, $i'$ and $n$ be natural numbers. If $\\lfloor i / b^e \\rfloor \\bmod b = \\lfloor i' / b^e \\rfloor \\bmod b$ for every $e < n$, that is, $i$ and $i'$ have the same first $n$ base-$b$ digits, then $i \\bmod b^n = i' \\bmod b^n$.","title":"Equal low-order digits give equal residues"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/sit-inj","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sit_inj","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#sit-inj","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine with encoding data $E$, and write $N$ for the number of situations of $E$, $K$ for the number of stacks of $E$ and $d$ for the window depth of $M$. Let $c$ be a configuration of $M$ all of whose stack symbols lie in the effective alphabets of $E$, and let $i < N$. If the label/state code of situation $i$ equals the (label, state) code of $c$, and for every stack number $k < K$ and\u2026","title":"Situation injectivity"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/run-lit","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.RunLit","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#run-lit","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine with encoding data $E$, let $s$ be an input of $M$ and $H \\in \\mathbb{N}$, and for $t \\in \\mathbb{N}$ let $c_t$ be the configuration after $t$ steps of the idle-extended run of $M$ from its start configuration on $s$. Variable indices are those of the layout of $E$ at height $H$, $K$ is the number of stacks of $E$, and $A_0$ is the number of (label, state) codes of $E$. An index\u2026","title":"The real run's literals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/run-asg","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.runAsg","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#run-asg","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$, let $s$ be a word over the input alphabet, let $T$ and $H$ be natural numbers, and let $V$ be the number of variables of the transition-clause formula for $T$ and $H$. The assignment is the list of $V$ truth values whose entry $p$, for each $p < V$, is true exactly when $p$ is one of the real run's literals of $s$ at height $H$.","title":"The assignment of the real run's literals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/sat-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sat_run","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#sat-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine with encoding data $E$, let $s$ be an input of $M$, let $T, H \\in \\mathbb{N}$, and let $V$ be the number of variables of the layout of $E$ for $T$ and $H$. Let $\\ell$ be a list of literals that is faithful for $V$, and let $(p, b) \\in \\ell$ be a literal such that $b$ is true exactly when $p$ is a run literal of $s$ at height $H$. Then the dense clause over $V$ variables built from\u2026","title":"Faithful clauses with a literal agreeing with the real run are satisfied"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/sa-lt","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sa_lt","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#sa-lt","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $E$ be an encoding of a finite TM2 machine with $N$ situations. For every natural number $i < N$, the position of the variable of situation $i$ is less than the label/state block size $A$ of the clause layout of $E$.","title":"Situation variables lie in the label/state block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/fam-complete","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.fam_complete","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#fam-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with layout $L$, $N$ situations, label/state table $\\mathrm{la}$ and window table $\\mathrm{wa}$, let $d$ be the window depth of $M$, and let $T, H \\in \\mathbb{N}$ with $d \\le H$. Write $X[r, x]$ for the index of the label/state variable $x$ of row $r$ at height $H$. Fix an input list $s$, let $c_t$ be the configuration after $t$ steps of the idle-extended run of $M$ from\u2026","title":"Transition families with a run-literal head are satisfied by the run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/upd-complete","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.upd_complete","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#upd-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine with encoding data $E$, let $N$ be the number of situations of $E$, let $s$ be an input of $M$, and let $T, H \\in \\mathbb{N}$ with $d \\le H$, where $d$ is the window depth of $M$. For $t \\in \\mathbb{N}$ let $c_t$ be the configuration after $t$ steps of the idle-extended run of $M$ from its start configuration on $s$, and suppose that for every $t < T$ every stack of $c_t$ has\u2026","title":"The run assignment satisfies the update family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/sdef-complete","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sdef_complete","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#sdef-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine with encoding data $E$, let $N$ be the number of situations of $E$, let $s$ be an input of $M$, and let $T, H \\in \\mathbb{N}$ with $d \\le H$, where $d$ is the window depth of $M$. Then the run assignment of $s$ for $T$ and $H$ (true exactly on the run literals) satisfies the transition clause family of the layout of $E$ at height $H$, for times $t < T$ and situations $i < N$, with\u2026","title":"The run assignment satisfies the situation-definition family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/shift-complete","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.shift_complete","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#shift-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine with encoding data $E$, let $N$ be the number of situations of $E$, let $s$ be an input of $M$, and let $T, H \\in \\mathbb{N}$ with $d \\le H$, where $d$ is the window depth of $M$. For $t \\in \\mathbb{N}$ let $c_t$ be the configuration after $t$ steps of the idle-extended run of $M$ from its start configuration on $s$, and suppose that for every $t < T$ every stack of $c_t$ has\u2026","title":"The run assignment satisfies the cell-shift family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/sat-unit","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.sat_unit","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#sat-unit","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $M$ be a finite stack machine with encoding data $E$, let $s$ be an input of $M$, let $T, H \\in \\mathbb{N}$, and let $V$ be the number of variables of the layout of $E$ for $T$ and $H$. If $p < V$ is a run literal of $s$ at height $H$, then the dense clause over $V$ variables whose only literal is $x_p$ is satisfied by the run assignment of $s$ for $T$ and $H$, which is true exactly on the run literals.","title":"Unit clauses on run literals are satisfied"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/lit-cell0","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.lit_cell0","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#lit-cell0","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$ with $K$ stacks, let $s$ be a word over the input alphabet, and let $H$, $k$ and $j$ be natural numbers with $k < K$ and $j < H$. Let $x$ be the code of the cell of stack $k$ at depth $j$ in the initial configuration on $s$. Then the cell variable of time $0$, stack $k$, depth $j$ and symbol code $x$ at height $H$ is one of the real run's literals of $s$ at height $H$.","title":"Row-0 cells of the start configuration are run literals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/init-complete","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.init_complete","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#init-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$, let $x$, $y$ and $Y$ be words over the input alphabet such that every letter of $y$ occurs in $Y$, and let $m$, $T$ and $H$ be natural numbers with $|xy| \\le m < H$, where $xy$ is the concatenation. Then the assignment of the real run's literals of $xy$ for $T$ steps at height $H$ satisfies the init family (clauses I1 to I6) for $x$, $Y$, $m$, $T$ and $H$.","title":"Completeness of the init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/acc-complete","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.acc_complete","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#acc-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$, let $s$ be a word over the input alphabet, let $T$ and $H$ be natural numbers with $H > 0$, and let $\\mathrm{acc}$ be a symbol of the output stack. Suppose that the configuration reached after $T$ steps of the idle-extended run of $M$ from the initial configuration on $s$ is accepting for $\\mathrm{acc}$: it is halted, its state is the initial state of $M$, and\u2026","title":"Completeness of the accept family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/accept-complete","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.accept_complete","parent":30760,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08/index.html#accept-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-08","status":"fully_proved","summary":"Let $E$ be an encoding of the machine $M$, let $x$, $y$ and $Y$ be words over the input alphabet such that every letter of $y$ occurs in $Y$, and let $m$ and $T$ be natural numbers with $|xy| \\le m$, where $xy$ is the concatenation. Let $d$ be the window depth of $M$, let $H = m + (T + 1)d + 1$ be the stack cap, and let $a$ be the assignment of the real run's literals of $xy$ for $T$ steps at height $H$. Suppose\u2026","title":"Completeness of the five clause families"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-09.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-09","kind":"scope","navigation":"transparent","parent":30612,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-09/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-09","status":"fully_proved","title":"Part 9 of 9"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-09/five-iff","kind":"node","lean":"Cslib.Complexity.CookLevin.Situation.Enc.five_iff","parent":30781,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-09/index.html#five-iff","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-situation-tables-of-a-multi-stack-mac/part-09","status":"fully_proved","summary":"Let $E$ be an encoding of the multi-stack machine $M$, with input stack $k_0$, output stack $k_1$ and window depth $d$. Let $x$ and $Y$ be lists of input symbols, let $m, T \\in \\mathbb{N}$, let $\\mathrm{acc}$ be an output symbol, and let $H = m + (T + 1)d + 1$ be the stack cap. Suppose $|x| \\le m$ and $\\mathrm{acc}$ is one of the effective symbols that $E$ fixes for stack $k_1$. Then some list $a$ of Booleans\u2026","title":"The five Cook\u2013Levin clause families are satisfiable iff some certificate is accepted"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","title":"Cook-Levin: generator machine for the accept family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/code-lits","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.codeLits","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#code-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a layout $L$, a code $a_c$, a time $T$, a height $H$ and $u \\in \\mathbb{N}$, the literal list of code clause $u$ is the single literal whose variable is the label/state variable $X[T, u]$ (entry $u$ of the label/state block at time $T$, for height $H$) and which is positive exactly when $u = a_c$.","title":"Literals of an accept code clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/cell-lits","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.cellLits","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#cell-lits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a layout $L$, a stack $k_1$, a symbol code $e_a$, a time $T$, a height $H$ and $x \\in \\mathbb{N}$, the literal list of cell clause $x$ is the single literal whose variable is the cell variable $C[T, k_1, 0, x]$ (time $T$, stack $k_1$, depth $0$ from the top, symbol code $x$, for height $H$) and which is positive exactly when $x = e_a$.","title":"Literals of an accept cell clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-fam","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.accFam","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-fam","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks and $g$ symbol codes, let $n_1, a, k_1, e, T, H$ be natural numbers, $W = A + K H g$ and $V = (T + 1) W$. The accept family is the CNF formula over $V$ variables made of, first, for each $u < n_1$ in increasing order, the clause whose only literal is the label variable $X[T, u]$ (variable $T W + u$), positive if $u = a$ and negative otherwise; then, for each\u2026","title":"The accept family of a layout"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/ck","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.CK","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#ck","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"The accept-family generator has sixteen distinct unit counters, named $\\mathsf{Tn}$, $\\mathsf{Hs}$, $\\mathsf{Gc}$, $\\mathsf{HG}$, $\\mathsf{Wc}$, $\\mathsf{Kc}$, $\\mathsf{K1}$, $\\mathsf{Vc}$, $\\mathsf{TW}$, $\\mathsf{CB}$, $\\mathsf{P}$, $\\mathsf{Q}$, $\\mathsf{tmp}$, $\\mathsf{md}$, $\\mathsf{c1}$ and $\\mathsf{c2}$.","title":"Counters of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/ak","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.AK","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#ak","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For natural numbers $n_1$ and $g$, a clause kind is either a code clause $u$ for some $u < n_1$ or a cell clause $x$ for some $x < g$.","title":"Clause kinds of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/bc","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.AK.bc","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#bc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a clause kind $c$ of the accept generator, the base counter of its single literal is the counter $\\mathsf{TW}$ when $c$ is a code clause and the counter $\\mathsf{CB}$ when $c$ is a cell clause.","title":"Base counter of an accept clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/off","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.AK.off","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#off","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For $n_1, g \\in \\mathbb{N}$, the offset of a clause kind of the accept generator is its index: $u$ for the code clause $u < n_1$ and $x$ for the cell clause $x < g$. It is the distance of the clause's single literal from its base counter.","title":"Offset of a clause's literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/sg","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.AK.sg","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#sg","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Fix naturals $n_1$ and $g$, and let $a$ and $e$ be naturals. The polarity of the code clause of $u < n_1$ is true exactly when $u = a$, and the polarity of the cell clause of $x < g$ is true exactly when $x = e$.","title":"Polarity of an accept clause's literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/ps","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.PS","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#ps","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Write $A$, $K$ and $g$ for the label/state block size, the number of stacks and the number of symbol codes of the generator's layout, and $k_1$ for the index of the stack whose top cell it checks. A precomputation stage is one of eight plain stages or one of five multiplication stages, each of which carries a stage $s$ of the multiplication routine. Named by the step the generator takes there, in the order it runs\u2026","title":"Precomputation stages of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/ls","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.LS","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#ls","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"The stages of the accept generator's routine emitting a clause's literal are $\\mathrm{cp1}$, $\\mathrm{cp2}$, $\\mathrm{eo}$, $\\mathrm{dec}$, $\\mathrm{cmp}$, $r(o)$ and $r_B(o)$ for each ordering $o \\in \\{<, =, >\\}$, $\\mathrm{emL}$, $\\mathrm{bad}$, $\\mathrm{slot}$ and $\\mathrm{dq}$. They form a finite type with decidable equality.","title":"Stages of the accept generator's literal routine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/gl","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.GL","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#gl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For $n_1, g \\in \\mathbb{N}$, a label is one of: a precomputation stage; $\\mathrm{e00}(c)$, $\\mathrm{cv1}(c)$, $\\mathrm{cv2}(c)$ or $\\mathrm{fin}(c)$ for a clause kind $c$ (a code clause $u < n_1$ or a cell clause $x < g$); $\\mathrm{lit}(c, s)$ for a clause kind $c$ and a stage $s$ of the literal routine; or the halting label $\\mathrm{done}$. The labels form a finite type with decidable equality.","title":"Labels of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/code-from","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.GL.codeFrom","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#code-from","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a generator with $n_1$ code clauses and $g$ cell clauses and $u \\in \\mathbb{N}$, the code-chain label at $u$ is the entry label of code clause $u$ when $u < n_1$, and the halting label $\\mathrm{done}$ when $u \\ge n_1$.","title":"Entry label of an accept code clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/code-start","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.GL.codeStart","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#code-start","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a generator with $n_1$ code clauses, the code chain starts at the entry label of code clause $n_1 - 1$ (the last one, as clauses are emitted last first), or at the halting label $\\mathrm{done}$ when $n_1 = 0$.","title":"Start of the accept code chain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/cell-from","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.GL.cellFrom","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#cell-from","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a generator with $n_1$ code clauses and $g$ cell clauses and $x \\in \\mathbb{N}$, the cell-chain label at $x$ is the entry label of cell clause $x$ when $x < g$, and the start of the code chain when $x \\ge g$.","title":"Entry label of an accept cell clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/start","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.GL.start","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#start","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Fix naturals $n_1$ and $g$, the numbers of code clauses and of cell clauses. Clauses are emitted last first, so the label at which clause emission starts is the start label of the cell clause of $g - 1$ when $g > 0$. When $g = 0$ it is the start label of the code clause of $n_1 - 1$ if $n_1 > 0$, and the final label if $n_1 = 0$.","title":"First clause label of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/next-c","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.GL.nextC","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#next-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $n_1, g \\in \\mathbb{N}$; clauses are emitted last first. After the code clause $u < n_1$ the generator goes to the halting label $\\mathrm{done}$ if $u = 0$, and otherwise to the first label $\\mathrm{e00}$ of code clause $u - 1$. After the cell clause $x < g$ it goes, if $x = 0$, to the start of the code chain (the first label of code clause $n_1 - 1$, or $\\mathrm{done}$ if $n_1 = 0$), and otherwise to the first\u2026","title":"Next label after a clause of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/lit-p","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.litP","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#lit-p","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Fix a layout $L$ with $g$ symbol codes and parameters $n_1$, $\\mathrm{ac}$, $\\mathrm{ea}$, and a clause kind $c$ with base counter $b(c)$, offset $\\mathrm{off}(c)$ and polarity $\\sigma(c)$. The literal routine of $c$ gives the generator's statement at each label $\\mathrm{lit}(c, s)$; below, a stage $s$ stands for the label $\\mathrm{lit}(c, s)$, and moving a counter onto targets means popping it until it is empty\u2026","title":"Literal routine of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/prog","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.prog","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Fix a layout $L$ with label/state block size $A$, $K$ stacks and $g$ symbol codes, and naturals $n_1$, $a$, $k_1$, $e$. The generator is a multi-stack machine with a Boolean output stack, the unit counters of the generator, and a Boolean internal state. Its precomputation pushes $g$ units on $\\mathsf{Gc}$, adds $\\mathsf{Hs} \\cdot \\mathsf{Gc}$ to $\\mathsf{HG}$, pushes $A$ units on $\\mathsf{Wc}$, pushes $K$ units on\u2026","title":"Generator machine for the accept family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-tm","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.accTM","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a layout $L$ with $g$ symbol codes and natural numbers $n_1, a, k_1, e$, this is the finite multi-stack Turing machine whose stacks are an output stack of bits and one unary counter for each of the generator's sixteen counter names, whose input stack is the counter $T_n$ and output stack the bit stack, whose labels are the generator's labels for $n_1$ code clauses and $g$ cell clauses with start label the first\u2026","title":"The accept generator as a finite multi-stack machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/rest","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.Rest","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#rest","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"An assignment $F$ of a natural number to each counter of the generator satisfies this predicate when $F$ is zero on the scratch counters $\\mathsf{Q}$, $\\mathsf{tmp}$, $\\mathsf{md}$, $\\mathsf{c1}$ and $\\mathsf{c2}$.","title":"Empty scratch counters of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/lit-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.lit_run","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#lit-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $c$ be a clause kind with base counter $b(c)$, offset $\\mathrm{off}(c)$ and polarity $\\sigma(c)$, and let $F$ assign values to the counters of the accept generator (for layout $L$ and parameters $n_1$, $\\mathrm{ac}$, $k_1$, $\\mathrm{ea}$) such that the scratch counters $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ are $0$. Let $q, m \\in \\mathbb{N}$ with $q = F(b(c)) + \\mathrm{off}(c)$ and $F(P) = q + m + 1$.\u2026","title":"Run of the literal routine of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/clause-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.clauseB","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#clause-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For $V \\in \\mathbb{N}$, the step budget for one clause of the accept generator is $B(V) = V(3V + 6) + 10V + 13$.","title":"Step bound for one accept clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/clause-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.clause_run","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $n_1, a, k_1, e, V \\in \\mathbb{N}$, and let $c$ be a clause kind of the accept generator (a code clause $u < n_1$ or a cell clause $x < g$) with base counter $b(c)$, offset $\\mathrm{off}(c)$ and polarity $\\sigma(c)$ for $a$ and $e$. Let $F$ be counter values of the generator whose scratch counters $\\mathsf{Q}$, $\\mathsf{tmp}$, $\\mathsf{md}$, $\\mathsf{c1}$,\u2026","title":"One clause of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/codes-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.codes_run","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#codes-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $F$ assign values to the counters of the accept generator (for layout $L$ and parameters $n_1$, $\\mathrm{ac}$, $k_1$, $\\mathrm{ea}$) such that the scratch counters $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ are $0$, $F(\\mathrm{Vc}) = V$, $F(P) = 0$ and $F(\\mathrm{TW}) + n_1 \\le V$. Then for every $u < n_1$, every output word $o$ and every Boolean state $v$, the generator runs in at most\u2026","title":"Run of the code-clause chain of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/cells-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.cells_run","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#cells-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $n_1, a_c, k_1, e_a \\in \\mathbb{N}$ be the parameters of the generator, and let $F$ give a value to every counter of the generator such that the scratch counters $\\mathsf{Q}$, $\\mathsf{tmp}$, $\\mathsf{md}$, $\\mathsf{c1}$, $\\mathsf{c2}$ are $0$, $F(\\mathsf{Vc}) = V$, $F(\\mathsf{P}) = 0$, $F(\\mathsf{TW}) + n_1 \\le V$ and $F(\\mathsf{CB}) + g \\le V$. Write\u2026","title":"Run of the accept cell chain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/init-f","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.initF","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#init-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For $T, H \\in \\mathbb{N}$, the initial counter values are $T$ on the counter $\\mathrm{Tn}$, $H$ on the counter $\\mathrm{Hs}$, and $0$ on every other counter.","title":"Initial counters of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/pre-f","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.preF","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#pre-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a layout $L$ with $A$ label/state variables per row, $K$ stacks and $g$ symbol codes, and for $k_1, T, H \\in \\mathbb{N}$, let $W = A + KHg$ be the row width. The counter values after the precomputation are $T + 1$ on $\\mathrm{Tn}$, $H$ on $\\mathrm{Hs}$, $g$ on $\\mathrm{Gc}$, $Hg$ on $\\mathrm{HG}$, $W$ on $\\mathrm{Wc}$, $K$ on $\\mathrm{Kc}$, $k_1$ on $\\mathrm{K1}$, $(T + 1)W$ on $\\mathrm{Vc}$, $TW$ on\u2026","title":"Counters after the precomputation of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/pre-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.preB","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#pre-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a layout $L$ with $A$ label/state variables per row, $K$ stacks and $g$ symbol codes, and for $k_1, T, H \\in \\mathbb{N}$, let $W = A + KHg$ be the row width. The step budget of the precomputation is $3 + (H(2g + 3) + H + 2) + (K(2Hg + 3) + K + 2) + (T(2W + 3) + T + 2) + 1 + ((T + 1)(2W + 3) + (T + 1) + 2) + (2TW + 2) + 2 + (k_1(2Hg + 3) + k_1 + 2)$.","title":"Cost of the precomputation of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-bound","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.accBound","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-bound","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, and natural numbers $n_1$, $k_1$, $T$, $H$, let $W = A + K H g$ and $V = (T + 1) W$ (the number of variables). The bound is $P + (g + n_1)\\,C$, where $C = V(3V + 6) + 10V + 13$ is the cost of one clause and the precomputation cost is\u2026","title":"Explicit time bound of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/pre-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.pre_run","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#pre-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Fix a layout $L$ with label/state block size $A$, $K$ stacks and $g$ symbol codes, and naturals $n_1$, $a$, $k_1$, $e$. For all $T, H \\in \\mathbb{N}$ and every list $o$ of Booleans, write $W = A + K H g$ for the row width. The accept generator, started at its first precomputation label with internal state false, output $o$, $T$ units on counter $\\mathsf{Tn}$, $H$ units on $\\mathsf{Hs}$ and every other counter\u2026","title":"Run of the accept generator's precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.acc_run","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$ and $K$ stacks, and let $n_1, a, k_1, e, T, H$ be natural numbers with $n_1 \\le A$, $k_1 < K$ and $H > 0$. Start the accept generator for $L$, $n_1$, $a$, $k_1$, $e$ at its start label, with internal state false, $T$ and $H$ in unary on its counters $T_n$ and $H_s$, all other counters empty and an arbitrary bit list $o$ on its output stack. Within a number of steps at\u2026","title":"Correctness of the accept generator, with its explicit time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-bound-uni","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.accBound_uni","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-bound-uni","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For all natural numbers $k \\ge 1$ and $y \\ge 1$, the explicit time bound of the accept generator for the layout with all four constants equal to $k$, with $n_1 = k_1 = k$ and $T = H = y$, is at most $314\\,k^5 y^4$.","title":"The accept generator's time bound on the uniform layout"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-c","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.accC","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"For a layout $L$ with label block size $A$, $K$ stacks and $g$ symbol codes, and natural numbers $n_1$, $k_1$, the constant is $314\\,(A + K + g + n_1 + k_1 + 1)^5$.","title":"Constant of the accept generator's time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-time","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.acc_time","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks and $g$ symbol codes, let $n_1, a, k_1, e, T, H$ be natural numbers with $n_1 \\le A$, $k_1 < K$ and $H > 0$, and let $c = 314\\,(A + K + g + n_1 + k_1 + 1)^5$. Start the accept generator for $L$, $n_1$, $a$, $k_1$, $e$ at its start label, with internal state false, $T$ and $H$ in unary on its counters $T_n$ and $H_s$, all other counters empty and an arbitrary\u2026","title":"Polynomial time bound of the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-family-eq","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.accFamily_eq","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-family-eq","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with encoding $E$, let $\\mathit{acc}$ be a symbol of the output stack of $M$, and let $T$, $H$ be natural numbers. The accept family of $E$ for $\\mathit{acc}$ at $T$, $H$ equals, by definition, the accept family of the layout of $E$ at $T$, $H$ with parameters $n_1 = A_0$, the number of (label, state) codes of $E$; $a$, the (label, state) code of the halted\u2026","title":"The concrete accept family is a layout-level accept family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/acc-gen-time","kind":"node","lean":"Cslib.Complexity.CookLevin.Accept.acc_gen_time","parent":30783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami/index.html#acc-gen-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-accept-fami","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with encoding $E$, let $\\mathit{acc}$ be a symbol of the output stack of $M$, and let $T$, $H$ be natural numbers with $H > 0$. Let $L$ be the layout of $E$, with label block size $A$, $K$ stacks and $g$ symbol codes, let $A_0$ be the number of (label, state) codes of $E$, let $k_1$ be the number $E$ gives the output stack, and let\u2026","title":"Time bound of the accept generator for a concrete machine"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","title":"Cook-Levin: generator machine for the initial-row family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/ik","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.IK","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#ik","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a counter type $C$, the stacks are those of the counter machine with counters $C$ (its output stack and one counter stack for each element of $C$), plus one additional input stack.","title":"Stacks of a counter machine with an input stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.I\u0393","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a counter type $C$ and an input alphabet $G$, each stack of the counter machine with counters $C$ keeps its alphabet there (Booleans on the output stack, a single symbol on each counter stack), and the input stack has alphabet $G$.","title":"Alphabets of a counter machine with an input stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/emb","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.emb","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#emb","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Given stack contents $S$ of a counter machine with counters indexed by $C$ (a list over the alphabet of each of its stacks) and a list $x$ of symbols from $G$, the combined stack contents give each stack $k$ of the counter machine the list $S(k)$ and give the extra input stack the list $x$.","title":"Stacks of a counter machine with an extra input stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/lift-e","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.liftE","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#lift-e","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a counter index type $C$ and an input alphabet $G$, the stack embedding of a counter machine over $C$ into the same machine with one extra input stack over $G$: each stack (the output stack or a counter) goes to its copy among the extended machine's stacks, with the identity map on its alphabet.","title":"The counter stacks inside the generator's stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/run-lift","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.run_lift","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#run-lift","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $M$ be a counter machine (unit counters indexed by $C$ and a Boolean output stack) and $M'$ a machine on the same stacks plus an input stack over $G$, such that $M'$ embeds $M$ along the inclusion of the counter stacks with the same labels (at every label whose statement is not the halt statement, $M'$ runs $M$'s statement). If $M$ runs in $n$ steps from optional label $l$, flag $v$ and stacks $S$ to a label\u2026","title":"Run lifting"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/rd-s","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.rdS","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#rd-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Given labels $\\mathrm{hit}(i)$ and $\\mathrm{miss}(i)$ for the codes $i < g$, the read test for code $i$ is the statement, on a counter machine's stacks plus an input stack of codes, that peeks at the top of the input stack and sets the flag to whether it equals $i$. If it does, the test pops it and goes to $\\mathrm{hit}(i)$; otherwise (including when the input stack is empty) it goes to $\\mathrm{miss}(i)$.","title":"Read test for one code"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/read-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.read_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#read-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $M'$ be a machine on a counter machine's stacks plus an input stack of codes $< g$, with labels $\\mathrm{rd}(i)$, $\\mathrm{hit}(i)$ and $\\mathrm{miss}(i)$ for $i < g$, such that at $\\mathrm{rd}(i)$ it runs the read test for code $i$ with these hit and miss labels, and $\\mathrm{miss}(i) = \\mathrm{rd}(i + 1)$ whenever $i + 1 < g$. Then for all counter stacks $S$, codes $c$, input lists $xs$, all $k$ and codes $i$\u2026","title":"The read chain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/z-b","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.zB","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#z-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a variable layout $L$, a stack number $k_0$, a number $H$ of tracked depths per stack and a depth $j$, this is the index of the cell variable of $L$ at time $0$ for stack $k_0$, depth $j$ and symbol code $0$ (the empty cell $\\bot$), namely $A + (k_0 H + j)\\,g$, where $A$ is the size of the label/state block of $L$ and $g$ is its number of symbol codes.","title":"Empty-cell variable of the initial row"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i-code","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.iCode","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i-code","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout $L$, a label/state code $i_c$ and natural numbers $T$ and $H$, the start-code formula consists of a single clause: the dense clause over the variables of $L$ for $T$ time steps and depth $H$ whose only literal is the positive literal on the time-$0$ label/state variable with code $i_c$.","title":"Start-code clause of the initial row (I1)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i-other","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.iOther","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i-other","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks and $g$ symbol codes, let $k_0$ be a stack, and let $T, H$ be natural numbers. This is the CNF over $V = (T + 1)(A + KHg)$ variables that has, for every stack $k < K$ with $k \\ne k_0$ and every depth $j < H$ (stacks in increasing order, then depths in increasing order), the dense clause over $V$ variables whose only literal is the positive literal of\u2026","title":"Row-0 clauses for the other stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i-in","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.iIn","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i-in","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout $L$ with $g$ symbol codes, a stack number $k_0$, a list $x = (x_0, \\dots, x_{\\ell - 1})$ of codes below $g$ and natural numbers $T$ and $H$, the input formula has, for each $j = 0, 1, \\dots, \\ell - 1$ in order, one dense clause over the variables of $L$ for $T$ time steps and depth $H$, whose only literal is the positive literal on the time-$0$ cell variable of stack $k_0$ at depth $j$ with code $x_j$.","title":"Input clauses of the initial row (I3)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i-cert","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.iCert","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i-cert","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks and $g$ symbol codes, let $k_0$ be a stack, let $y = (y_1, \\dots, y_r)$ be a list of natural numbers and let $n, m, T, H$ be natural numbers. For a depth $j$ and a code $x$ write $C_{j,x} = A + (k_0 H + j)g + x$ for the row-$0$ variable saying that stack $k_0$ holds code $x$ at depth $j$. This is the CNF over $V = (T + 1)(A + KHg)$ variables that has, for\u2026","title":"Certificate clauses of the initial row (I4)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i-pre","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.iPre","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i-pre","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks and $g$ symbol codes, let $k_0$ be a stack, let $n, m, T, H$ be natural numbers, and write $z_j = A + (k_0 H + j)g$ for the row-$0$ variable saying that stack $k_0$ holds code $0$ (the empty cell) at depth $j$. This is the CNF over $V = (T + 1)(A + KHg)$ variables that has, for $i = 0, 1, \\dots, m - n - 1$ in order (no clauses if $m \\le n$), the dense clause\u2026","title":"Row-0 prefix clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i-tail","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.iTail","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i-tail","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks and $g$ symbol codes, let $k_0$ be a stack, let $m, T, H$ be natural numbers, and write $z_j = A + (k_0 H + j)g$ for the row-$0$ variable saying that stack $k_0$ holds code $0$ (the empty cell) at depth $j$. This is the CNF over $V = (T + 1)(A + KHg)$ variables that has, for $i = 0, 1, \\dots, H - m - 1$ in order (no clauses if $H \\le m$), the dense clause\u2026","title":"Row-0 tail clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-fam","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.initFam","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-fam","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout, let $i_c$ be a label/state code, let $k_0$ be a stack, let $y$ be a list of natural numbers, let $x$ be a list of symbol codes of $L$ of length $\\ell$, and let $m, T, H$ be natural numbers. The init family is the CNF formula obtained by concatenating, in this order: the start-code clause of the initial row for $i_c$, $T$, $H$ (I1); the row-$0$ clauses for the stacks other than $k_0$ for $T$,\u2026","title":"The init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/ck","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.CK","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#ck","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"The unit-counter stacks of the generator are indexed by a finite type with 27 distinct elements: Tn, Xn, En, Fn, Gc, Mm, Hs, HG, Wc, Vc, KA, ZA, Z45, Z6, Z1, Z2, N1, N2, CB, Zr, P, Q, tmp, md, c1, c2 and Id. Equality on it is decidable.","title":"Counters of the initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/ph","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.PH","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#ph","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"The five phase loops of the generator for the initial-row clauses, in the order the generator runs them (its input loop runs between the certificate phase and the two stack phases): the tail phase (the row-$0$ cells of the stack $k_0$ holding the input, past the certificate block, are empty), the prefix phase (an empty cell in the certificate block is followed by an empty cell), the certificate phase (each\u2026","title":"Loop phases of the initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/c-n","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.PH.cN","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#c-n","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $k_0$ be the index of the input stack in the tableau layout. For a loop phase $p$, the generator counter whose units count the remaining iterations of the loop of $p$: Fn for the I6 phase, En for both the I5 and the I4 phase, N2 for the I2 phase over the stacks $k > k_0$, and N1 for the I2 phase over the stacks $k < k_0$.","title":"Loop-count counter of a phase (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/c-z","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.PH.cZ","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#c-z","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $k_0$ be the index of the input stack in the tableau layout. For a loop phase $p$, the generator counter holding the base of the loop of $p$, the cell base of its iteration $0$: Z6 for the I6 phase, Z45 for both the I5 and the I4 phase, Z2 for the I2 phase over the stacks $k > k_0$, and Z1 for the I2 phase over the stacks $k < k_0$.","title":"Base counter of a phase (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/cl","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.CL","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#cl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For $g, n \\in \\mathbb{N}$, a clause kind is either the start-code clause (I1), an input clause (I3) with a symbol code $c < g$, or the clause of one of the five loop phases $p$. The parameter $n$ only indexes the type. There are finitely many clause kinds and their equality is decidable.","title":"Clause kinds of the initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/nl","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.CL.nl","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#nl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For clause kinds over $g$ codes and $n$ certificate codes, a clause of the certificate phase has $n + 1$ literals, a clause of the prefix phase has $2$, and every other clause kind has $1$.","title":"Number of literals of a clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/off","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.CL.off","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#off","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Fix numbers $i_c$ and $g'$ and a list $y = (y_1, \\dots, y_r)$ of numbers. The offset from the clause base of literal $l$ in a clause of kind $c$ is $i_c$ for the start-code clause; $x$ for the input clause with code $x$; for the certificate phase, $y_l$ if $1 \\le l \\le r$ and $0$ otherwise; for the prefix phase, $0$ if $l = 0$ and $g'$ otherwise; and $0$ for the other three phases.","title":"Literal offsets"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/sg","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.CL.sg","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#sg","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a clause kind $c$ of the init-family generator and a literal index $l \\in \\mathbb{N}$, the sign of literal $l$ of $c$ is positive in every case except one: for the prefix-form phase (clause group I5), literal $0$ is negative and every literal $l \\ne 0$ is positive.","title":"Literal signs of an init-family clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/bc","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.CL.bc","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#bc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a clause kind $c$, the generator counter whose value is added to the literal offsets of $c$ to give the literal positions: the counter Zr for the start-code clause (I1), and the counter CB for every other clause kind.","title":"Base counter of a clause kind (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/ps1","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.PS1","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#ps1","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"The finite type of labels with which the generator sets the counters $G_c = g$, $M_m = m$, $H_s = H$, $HG = H \\cdot g$, $W_c = W$ and $V_c = V$, where $g$ is the number of symbol codes, $m$ the end of the certificate block, $H$ the stack height, $W$ the row width and $V$ the number of variables. It consists of thirteen single labels and three multiplication loops, each with one label per multiplication stage.","title":"Labels of the first precomputation part"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/ps2","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.PS2","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#ps2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"The finite type of labels with which the generator computes the row-0 cell bases $Z_A, Z_{45}, Z_6, Z_1, Z_2$ and the loop counts $N_1, N_2$. It consists of sixteen single labels and five multiplication loops, each with one label per multiplication stage.","title":"Labels of the second precomputation part"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/ls","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.LS","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#ls","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"The stages of the generator's routine that emits one literal of a clause: two stages copying the clause base into a target counter, a stage adding the literal's offset to the target, a stage decrementing the position counter, a stage comparing the position with the target, two restore stages for each comparison outcome (less, equal or greater), a stage emitting one empty slot, a halting error stage, a stage\u2026","title":"Stages of the literal routine (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/gl","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.GL","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#gl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For natural numbers $g$ and $n$, the labels are: a label for each stage of the first precomputation part and one for each stage of the second; for each loop phase $p$, a loop-head label, a counter-reset label, two base-copy labels, a multiplication label for each multiplication stage and a base-drain label; for the input loop, a head label, two base-copy labels, a multiplication label for each multiplication stage,\u2026","title":"Labels of the initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/lit-l","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.GL.litL","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#lit-l","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a clause kind $c$ and $l \\in \\mathbb{N}$, among the generator labels with parameters $g$ and $n$: the first stage of the routine for literal $l$ of $c$ if $l < n + 2$, and the label $\\mathrm{done}$ otherwise (that case is never used).","title":"Entry label of a literal routine (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/after","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.GL.after","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#after","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $g, n \\in \\mathbb{N}$, let $c$ be a clause kind of the generator with parameters $g$ and $n$, and let $l \\in \\mathbb{N}$. If $l = 0$, the label is the final routine of $c$, which emits the leading gap. If $l \\ge 1$, it is the start of the routine for literal $l - 1$ of $c$ when $l - 1 < n + 2$, and the halting label otherwise. The generator goes here after the routine for literal $l$, and with $l$ the number of\u2026","title":"Label after a literal routine (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/next","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.GL.next","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#next","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Write $k_0$ for the stack holding the input. The label the generator moves to when the loop of phase $p$ finishes: the loop head of the prefix phase after the tail phase, the loop head of the certificate phase after the prefix phase, the head of the input loop after the certificate phase, the loop head of the phase for the stacks $k < k_0$ after the phase for the stacks $k > k_0$, and the first step of the\u2026","title":"Label after a phase loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/next-c","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.GL.nextC","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#next-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"The label the generator moves to after emitting a clause of kind $c$: the halting label after the start-code clause, the end of the input-loop iteration (which drains the clause base) after an input clause, and the end of the loop iteration of phase $p$ (which drains the clause base) after a clause of phase $p$.","title":"Label after a clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/rd0","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.GL.rd0","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#rd0","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For $g, n \\in \\mathbb{N}$, the generator label that tests for input code $0$ when $g > 0$, and the error label when $g = 0$.","title":"The first read test"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/miss","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.GL.miss","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#miss","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a code $i < g$, the label reached when the read test for $i$ fails is the read test for $i + 1$ if $i + 1 < g$, and the halting error label otherwise.","title":"Label after a failed read test"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/lb","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.Lb","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#lb","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout $L$ and a list $y$ of certificate codes, the label type of the generator is the type of labels of the initial-row generator with parameters $g$, the number of symbol codes of $L$, and $n$, the length of $y$.","title":"Labels of the initial-row generator for a layout"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/lit-p","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.litP","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#lit-p","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a clause kind $c$ and a literal index $l < n + 2$, where $n$ is the number of certificate codes, the counter program at the stages of the routine for literal $l$ of $c$. It copies the base counter of $c$ into the counter $Q$ (restoring the base counter through a scratch counter) and adds the offset of literal $l$ to $Q$. It then repeats: pop a unit from the counter $P$ (halting if $P$ is empty) and compare $P$\u2026","title":"The literal routine of the initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/prog","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.prog","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with label block size $A$, $K$ stacks and $g$ symbol codes, let $i_c$ and $k_0$ be natural numbers and let $y$ be a list of $n$ certificate codes. The counter part assigns to each label of the initial-row generator for $g$ and $n$ a statement of the counter machine with the generator's counters and a Boolean flag, as follows. Every pop sets the flag to whether a unit was popped, and pushes leave\u2026","title":"The counter part of the initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/iprog","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.iprog","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#iprog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $i_c$ and $k_0$ be natural numbers and let $y$ be a list of certificate codes. The generator assigns to each label of the initial-row generator for $L$ and $y$ a machine statement with a Boolean flag on the stacks of the counter machine with the generator's counters plus an input stack of codes below $g$, as follows. At the read test label $\\mathrm{rd}(i)$ for a code\u2026","title":"The initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-tm","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.initTM","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout $L$ with $g$ symbol codes, a start code $i_c$, a stack $k_0$ and a code list $Y$, this is the finite multi-stack Turing machine whose stacks are the generator's counters, its Boolean output stack and an extra input stack, with the extra stack as input stack and the Boolean stack as output stack; whose stack alphabets are unit symbols on the counters, bits on the output stack and codes below $g$ on the\u2026","title":"The init generator as a finite multi-stack machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/clits","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.clits","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#clits","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Given natural numbers $i_c$ and $g'$, a list $y$ of natural numbers, a base position $B$ and a clause kind $c$, the literal list of $c$ at base $B$ is the list of the pairs $(B + \\mathrm{off}(c, l), \\mathrm{sg}(c, l))$ for $l = 0, 1, \\dots, \\mathrm{nl}(c) - 1$ in order, where $\\mathrm{nl}(c)$ is the number of literals of $c$, $\\mathrm{off}(c, l)$ is the offset of literal $l$ (computed from $i_c$, $g'$ and $y$) and\u2026","title":"Literals of a clause kind"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/wf","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.WF","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#wf","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Fix natural numbers $i_c$ and $g'$ and a list $Y$ of natural numbers, which determine the offsets of the literals of each clause kind from the base. A clause kind $c$ is well formed when these offsets are strictly increasing: for every $l \\in \\mathbb{N}$ with $l + 1$ less than the number of literals of $c$, the offset of literal $l$ of $c$ is less than the offset of literal $l + 1$ of $c$.","title":"Well-formed init-family clause kinds"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/bnd","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.bnd","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#bnd","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a bound $V$, a base $B$, a clause kind $c$ and $l \\in \\mathbb{N}$, the boundary before literal $l$ is $B + \\mathrm{off}_c(l)$, the position of literal $l$, when $l$ is less than the number of literals of $c$, and $V$ otherwise. Here $\\mathrm{off}_c(l)$ is the offset of literal $l$ of $c$ determined by the start code $ic$, the symbol count $g'$ and the certificate codes $ycs$.","title":"Boundary before a literal (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/in-r","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.InR","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#in-r","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Given the start code $i_c$, a code count $g'$, the certificate codes $y$, a bound $V$, a base $B$ and a clause kind $c$ of the init generator, all literals of $c$ are in range when $B + \\mathrm{off}_c(l) < V$ for every literal index $l$ less than the number of literals of $c$, where $\\mathrm{off}_c(l)$ is the offset of literal $l$ of $c$ determined by $i_c$, $g'$ and $y$.","title":"Literals of a clause kind in range"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/rest","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.Rest","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#rest","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"An assignment $F$ of natural numbers to the generator's counters has empty scratch counters when $F(Q) = F(\\mathrm{tmp}) = F(\\mathrm{md}) = F(c_1) = F(c_2) = 0$.","title":"Empty scratch counters"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/lit-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.lit_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#lit-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout and a list of $n$ certificate codes, let $c$ be a clause kind with base counter $b$, let $l < n + 2$ be a literal index with offset $\\mathrm{off}(l)$, and let $F$ be counter values whose scratch counters $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ are $0$. Let $\\gamma \\in \\mathbb{N}$, let $q = F(b) + \\mathrm{off}(l)$ and suppose $F(P) = q + \\gamma + 1$. Then for every output $o$ and flag $v$, the\u2026","title":"Run of one literal routine (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/lit-e","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.litE","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#lit-e","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For $V \\in \\mathbb{N}$, the per-literal cost is $6V + 9$.","title":"Per-literal step cost of the initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/clause-b","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.clauseB","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#clause-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For natural numbers $m$ and $V$,","title":"Step bound for one clause (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/lits-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.lits_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $c$ be a clause kind with $k$ literals at base $B$, the $j$-th at position $B + \\mathrm{off}(j)$, such that the offsets are strictly increasing and every position is below $V$. For $l \\le k$ let $b_l$ be $B + \\mathrm{off}(l)$ if $l < k$ and $V$ otherwise, let $\\lambda_l$ be the list of literals of $c$ (position and sign) from index $l$ on, and let the label before literal $l$ be the first stage of the routine\u2026","title":"Run of the literal routines of a clause (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/clause-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.clause_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $i_c$ and $k_0$ be natural numbers and let $y$ be a list of $n$ certificate codes. Let $c$ be a clause kind for $g$ and $n$ that is well formed for $i_c$, $g$ and $y$, and let $V, B$ be natural numbers such that all literals of $c$ at base $B$ (for $i_c$, $g$ and $y$) are in range below $V$. Let $F$ be counter values of the generator with empty scratch counters such\u2026","title":"Run of one clause (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/body-b","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.bodyB","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#body-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For natural numbers $m, g, N, V$,","title":"Step bound for one loop iteration (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/body-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.body_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#body-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, let $ic, k_0 \\in \\mathbb{N}$, let $ycs$ be a list of $n$ certificate codes, and let $p$ be a loop phase with base counter $c_Z(p)$ and loop-count counter $c_N(p)$. Let $F$ give the value of each generator counter, with the scratch counters empty ($\\mathrm{Rest}(F)$), $F(\\mathrm{CB}) = F(\\mathrm{P}) = 0$, $F(\\mathrm{Vc}) = V$, $F(\\mathrm{Gc}) = g$, $F(c_Z(p)) = Z$ and\u2026","title":"One phase-loop iteration of the initial-row generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/phase-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.phase_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#phase-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Fix a layout $L$ with $g$ symbol codes, a start code $i_c$, a stack index $k_0$ and a list $y$ of $n$ certificate codes, and let $M$ be the counter part of the initial-row generator for these parameters. Let $p$ be a loop phase with loop counter $N_p$ and base counter $Z_p$, and let $F$ be counter values with the scratch counters $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ empty,\u2026","title":"One phase loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i3pre-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.i3pre_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i3pre-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes. Suppose the counters $F$ of the init generator's counter part have $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$ and $\\mathrm{CB}$ equal to $0$, $\\mathrm{Gc} = g$, $\\mathrm{ZA} = Z$ and $\\mathrm{Xn} = r + 1$. Then for every output $o$ and flag $v$, the counter part, started at the head of the I3 loop with flag $v$, counters $F$ and output $o$, reaches the first read\u2026","title":"Counter part of one I3 iteration"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i3-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.i3_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i3-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, consider the init generator for a start code, a stack $k_0$ and a list of $\\ell$ certificate codes, and let $x = (x_0, \\dots, x_{s-1})$ be a list of codes $x_j < g$. Suppose its counters $F$ have $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$, $\\mathrm{CB}$ and $P$ equal to $0$, $\\mathrm{Vc} = V$, $\\mathrm{Gc} = g$ and $\\mathrm{ZA} = Z$, that $jg + Z + g \\le V$ for every\u2026","title":"The I3 loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/qpos","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.qpos","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#qpos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout $L$ with $A$ label/state variables, $KK$ stacks and $g$ symbol codes, and all $T, H, q \\in \\mathbb{N}$ with $q < KK \\cdot H$, we have $A + q g + g \\le V$, where $V$ is the number of variables of $L$ for time bound $T$ and stack height $H$.","title":"Row-0 cells fit in the variable range"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/flat-blocks","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.flat_blocks","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#flat-blocks","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $\\varphi$ be a function from the natural numbers to a type $\\alpha$ and let $H$, $s$, $r$ be natural numbers. Concatenating, for $k = s, s + 1, \\dots, s + r - 1$ in order, the lists $[\\varphi(kH), \\varphi(kH + 1), \\dots, \\varphi(kH + H - 1)]$ gives the list $[\\varphi(sH), \\varphi(sH + 1), \\dots, \\varphi(sH + rH - 1)]$.","title":"Concatenating consecutive blocks of indices"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-f","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.initF","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For natural numbers $T, n, e, f$, the counters $\\mathrm{Tn}$, $\\mathrm{Xn}$, $\\mathrm{En}$ and $\\mathrm{Fn}$ of the generator hold $T$, $n$, $e$ and $f$ respectively, and every other counter holds $0$.","title":"Initial counters of the init generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/pre-f","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.preF","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#pre-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout $L$ with $A$ label/state variables, $K$ stacks and $g$ symbol codes, a stack index $k_0$ and numbers $T, n, e, f$, write $H = n + e + f$. The counter values after the precomputation are $\\mathrm{Tn} = T + 1$, $\\mathrm{Xn} = n$, $\\mathrm{En} = e$, $\\mathrm{Fn} = f$, $\\mathrm{Gc} = g$, $\\mathrm{Mm} = n + e$, $\\mathrm{Hs} = H$, $\\mathrm{HG} = Hg$, $\\mathrm{Wc} = A + KHg$ (the row width),\u2026","title":"Counters after the precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/pre-b","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.preB","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#pre-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout $L$ with $A$ label/state variables, $K$ stacks and $g$ symbol codes, and numbers $k_0, q, T, n, e, f$, write $h = n + e + f$. The bound is","title":"Step bound for the precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/pre-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.pre_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#pre-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $K$ stacks, let $i_c$ and $k_0$ be natural numbers and let $y$ be a list of certificate codes. For all natural numbers $T, n, e, f$ and every output $o$, the counter part of the generator for $L$, $i_c$, $k_0$, $y$ runs within $\\mathrm{preB}(L, k_0, K - k_0 - 1, T, n, e, f)$ steps, the step bound for the precomputation (with truncated subtraction), from the label $\\mathrm{eg}$ of the first\u2026","title":"Run of the precomputation (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/i3-full","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.i3_full","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#i3-full","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $g$ symbol codes, consider the init generator for a start code, a stack $k_0$ and a list of $\\ell$ certificate codes, and let $x = (x_0, \\dots, x_{s-1})$ be a list of codes $x_j < g$. Suppose its counters $F$ have $Q$, $\\mathrm{tmp}$, $\\mathrm{md}$, $c_1$, $c_2$, $\\mathrm{CB}$ and $P$ equal to $0$, $\\mathrm{Vc} = V$, $\\mathrm{Gc} = g$, $\\mathrm{ZA} = Z$ and $\\mathrm{Xn} = s$, that\u2026","title":"The I3 loop on the whole input"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/loop-b","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.loopB","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#loop-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For $m, g, N, V \\in \\mathbb{N}$, the cost of one phase loop with count bound $N$ is $N(\\beta + 1) + N + 2$, where $\\beta$ is the cost of one loop iteration with clauses of at most $m$ literals, $g$ symbol codes, count bound $N$ and $V$ variables.","title":"Step bound of one phase loop (initial-row generator)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-bound","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.initBound","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-bound","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $A$ label/state variables, $K$ stacks and $g$ symbol codes, and let $n_Y, k_0, q, T, n, e, f$ be natural numbers. Put $H = n + e + f$ and let $V = (T + 1)(A + K H g)$ be the number of variables. The bound is the precomputation cost $\\mathrm{preB}(L, k_0, q, T, n, e, f)$, plus $6$ times the phase-loop cost $\\mathrm{loopB}(n_Y + 2, g, K H + H, V)$, plus the clause cost\u2026","title":"Time bound of the init generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-run","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.init_run","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $A$ label/state variables, $K$ stacks and $g > 0$ symbol codes, let $i_c < A$ and $k_0 < K$, and let $Y$ be a strictly increasing list of codes with $0 < y < g$ for every $y \\in Y$. Let $xs$ be a list of codes below $g$ with $n = |xs|$, let $T, e, f$ be natural numbers with $f > 0$, and let $o$ be a bit string. Start the generator for $L, i_c, k_0, Y$ at its start label with state false,\u2026","title":"Correctness of the init generator, with its explicit time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-bound-uni","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.initBound_uni","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-bound-uni","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For all natural numbers $k \\ge 1$ and $y \\ge 1$, let $L$ be the uniform layout with all four parameters equal to $k$. Then the time bound of the init generator for $L$ with $n_Y = k_0 = q = k$ and $T = n = e = f = y$ is at most $18928\\,k^5 y^5$.","title":"Time bound of the init generator at uniform parameters"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-c","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.initC","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"For a layout $L$ with $A$ label/state variables, $K$ stacks and $g$ symbol codes, and natural numbers $n_Y$ and $k_0$, the constant is $18928\\,(A + K + g + n_Y + k_0 + 1)^5$.","title":"Constant of the init generator's time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-time","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.init_time","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $L$ be a layout with $A$ label/state variables, $K$ stacks and $g > 0$ symbol codes, let $i_c < A$ and $k_0 < K$, and let $Y$ be a strictly increasing list of codes with $0 < y < g$ for every $y \\in Y$. Let $xs$ be a list of codes below $g$ with $n = |xs|$, let $T, e, f$ be natural numbers with $f > 0$, and let $o$ be a bit string. Start the generator for $L, i_c, k_0, Y$ at its start label with state false,\u2026","title":"Polynomial time bound for the init generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/lookup-pos","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.lookup_pos","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#lookup-pos","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $\\lambda$ be a list of pairs of a natural number and a boolean in which every boolean is true. For every $v \\in \\mathbb{N}$, looking up $v$ in $\\lambda$ (the boolean of the first pair with first entry $v$, if any) gives true if $v$ is the first entry of some pair in $\\lambda$, and no value otherwise.","title":"Lookup in an all-positive literal list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/dense-clause-congr","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.denseClause_congr","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#dense-clause-congr","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $\\ell$ and $\\ell'$ be lists of literals (pairs of a position and a sign) in which every sign is positive. If every natural number $v$ is a position of $\\ell$ exactly when it is a position of $\\ell'$, then for every $V$ the dense clause over $V$ variables with literals $\\ell$ equals the dense clause over $V$ variables with literals $\\ell'$: order and repetitions in the lists are irrelevant.","title":"Dense clauses of positive literals depend only on their positions"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/ycs","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.ycs","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#ycs","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Given an encoding $E$ of a machine $M$ and a list $Y$ of symbols of the input stack of $M$, this is the list of the distinct codes that $E$ assigns to the symbols of $Y$ on the input stack, sorted in strictly increasing order.","title":"Sorted certificate code list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/enc-f","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.encF","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#enc-f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $E$ be an encoding of a finite multi-stack Turing machine $M$, and let $g$ be the number of symbol codes of the layout of $E$. A symbol $z$ of the input stack of $M$ is sent to its symbol code under $E$, viewed as an element of $\\{0, \\dots, g - 1\\}$; the code is below $g$ because every input symbol is an effective symbol of the input stack.","title":"Codes of input symbols"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-family-eq","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.initFamily_eq","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-family-eq","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, and let $m, T, H$ be natural numbers. The init family of $E$ for $x, Y, m, T, H$ equals the layout-level init family over the layout of $E$ with start code the code of the pair of the main label and the initial state of $M$, stack $k_0$ the number of the input stack, code list the codes of the symbols of $Y$ in\u2026","title":"The init family of an encoding is a layout-level init family"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-gen-time","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.init_gen_time","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-gen-time","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, let $m, T, H$ be natural numbers with $|x| \\le m < H$, and let $o$ be a bit string. Let $C$ be the time constant for the layout of $E$, the number of distinct codes of $Y$ and the number $k_0$ of the input stack. Start the generator for $E$ (start code: the code of the pair of the main label and the initial\u2026","title":"Time and output of the init generator for an encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/init-gen-cap-h","kind":"node","lean":"Cslib.Complexity.CookLevin.InitFamily.init_gen_capH","parent":30820,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row/index.html#init-gen-cap-h","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-generator-machine-for-the-initial-row","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$ with window depth $d$, let $x$ and $Y$ be lists of symbols of the input stack of $M$, let $m, T$ be natural numbers with $|x| \\le m$, let $H = m + (T + 1)d + 1$ be the stack cap, and let $o$ be a bit string. Let $C$ be the time constant for the layout of $E$, the number of distinct codes of $Y$ and the number $k_0$ of the input stack. Start the generator for $E$ (start code:\u2026","title":"The init generator at the stack cap"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","title":"Cook-Levin: the five generators as sub-machines of one host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/hk","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.HK","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#hk","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"The host has a shared output stack, an input-code stack, and five private counter blocks: an update block and a situation-definition block, each with one stack per counter of the family generator, and a shift block, an accept block and an init block, with one stack per counter of the shift, accept and init generator respectively.","title":"Stacks of the Cook\u2013Levin host machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/h","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.H\u0393","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#h","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"For $g \\in \\mathbb{N}$, the host stacks carry the following alphabets: the shared output stack holds Booleans, the input-code stack holds codes in $\\{0, \\dots, g - 1\\}$, and every stack of every private counter block has a one-element alphabet.","title":"Alphabets of the Cook\u2013Levin host machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/e-u","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.eU","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#e-u","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"The map from the stacks of the clause-family generator, used as the update generator, to the host stacks that sends the output stack to the shared output stack and the counter stack of each counter $x$ to the counter $x$ of the host's update block.","title":"Host stack renaming for the update generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/e-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.eS","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#e-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"The map from the stacks of the clause-family generator, used as the situation-definition generator, to the host stacks that sends the output stack to the shared output stack and the counter stack of each counter $x$ to the counter $x$ of the host's situation-definition block.","title":"Host stack renaming for the situation-definition generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/e-h","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.eH","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#e-h","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"The map from the stacks of the shift generator to the host stacks that sends the output stack to the shared output stack and the counter stack of each shift counter $x$ to the counter $x$ of the host's shift block.","title":"Host stack renaming for the shift generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/e-a","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.eA","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#e-a","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"The map from the stacks of the accept generator to the host stacks that sends the output stack to the shared output stack and the counter stack of each accept counter $x$ to the counter $x$ of the host's accept block.","title":"Host stack renaming for the accept generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/e-i","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.eI","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#e-i","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"The map from the stacks of the init generator to the host stacks that sends the output stack to the shared output stack, the input stack to the host's input-code stack, and the counter stack of each init counter $x$ to the counter $x$ of the host's init block.","title":"Host stack renaming for the init generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/emb-u","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.embU","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#emb-u","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"For $g \\in \\mathbb{N}$, this is the stack embedding of the family generator's stacks (a Boolean output stack and one unit counter stack per counter) into the host stacks with $g$ input codes, used for the update generator: the output stack goes to the host's shared output stack, and the counter stack $x$ goes to the stack $x$ of the host's private update block. Distinct stacks go to distinct host stacks, and each\u2026","title":"Embedding of the update generator's stacks into the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/emb-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.embS","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#emb-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"For $g$ input codes, the stack embedding of the clause-family generator's stacks, used as the situation-definition generator, into the host stacks that sends the output stack to the shared output stack and the counter stack of each counter $x$ to the counter $x$ of the host's situation-definition block, identifying each stack's alphabet with itself.","title":"Embedding of the situation-definition generator into the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/emb-h","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.embH","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#emb-h","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"For $g$ input codes, the stack embedding of the shift generator's stacks into the host stacks that sends the output stack to the shared output stack and the counter stack of each shift counter $x$ to the counter $x$ of the host's shift block, identifying each stack's alphabet with itself.","title":"Embedding of the shift generator into the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/emb-a","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.embA","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#emb-a","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"For $g$ input codes, the stack embedding of the accept generator's stacks into the host stacks that sends the output stack to the shared output stack and the counter stack of each accept counter $x$ to the counter $x$ of the host's accept block, identifying each stack's alphabet with itself.","title":"Embedding of the accept generator into the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/emb-i","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.embI","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#emb-i","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"For $g$ input codes, the stack embedding of the init generator's stacks (with input alphabet the $g$ codes) into the host stacks that sends the output stack to the shared output stack, the input stack to the host's input-code stack, and the counter stack of each init counter $x$ to the counter $x$ of the host's init block, identifying each stack's alphabet with itself.","title":"Embedding of the init generator into the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/hl","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.HL","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#hl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"For parameters $N, n_L, K, g, d, n_1, n_Y$, the host labels consist of a copy of the family generator's labels (for $N$ situations and $n_L$ literals) for the update phase, a second copy of them for the situation-definition phase, a copy of the shift generator's labels (for $N$ situations, $K$ stacks, $g$ symbols and depth $d$), a copy of the accept generator's labels (for $n_1$ code clauses and $g$ symbols), a\u2026","title":"Labels of the Cook\u2013Levin host machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/chain","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.chain","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#chain","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"For a host label $l$, the chaining step is the host statement that sets the Boolean internal state to false (the state in which every generator starts) and then jumps to $l$.","title":"Chaining step of the host program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/hprog","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.hprog","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#hprog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"Given the five generator programs (update, situation definition, shift, accept and init), the host program runs init, then update, then situation definition, then shift, then accept. At each label of a generator other than its final label, the host executes that generator's statement with the generator's stacks embedded into the host stacks (output onto the shared output stack, counters onto the generator's own\u2026","title":"The host program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/host-tm","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.hostTM","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#host-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"Given the five generator programs (update and situation-definition programs over the family generator's stacks, a shift program, an accept program, and an init program whose input alphabet has $g$ codes), the host machine is the finite multi-stack Turing machine whose stacks are the host stacks, with the input-code stack as input stack and the shared output stack as output stack, whose alphabets are the host\u2026","title":"The host as a finite multi-stack Turing machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/seq-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.seq_run","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#seq-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"Let $P_u$, $P_s$ (update and situation definition), $P_h$ (shift), $P_a$ (accept) and $P_i$ (init) be generator programs that halt at their label `done`, and let $\\xi$ be a list of input codes. Suppose that for each generator $j \\in \\{u, s, h, a, i\\}$ there are counter values $F_j$, a word $w_j$ and a bound $B_j$ such that, for every output word $o$, the run of $P_j$ from its start label with state bit false, on\u2026","title":"Sequencing"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/real-prog","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.realProg","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#real-prog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with encoding data $E$, let $Y$ be a list of input symbols, and let $\\mathit{acc}$ be an output symbol. This is the host program whose five generators are the real ones for $M$. The update generator is the family generator of $E$ for the (label, state) codes, window codes and next (label, state) codes of the situations of $E$. The situation-definition generator is the\u2026","title":"The host on the real generators of a machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/phi","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.Phi","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#phi","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with encoding data $E$, let $x$ and $Y$ be lists of input symbols, let $\\mathit{acc}$ be an output symbol, and let $m, T \\in \\mathbb{N}$. Let $H = m + (T + 1) d + 1$ be the stack cap, where $d$ is the window depth of $M$. The CNF formula $\\Phi(x, Y, \\mathit{acc}, m, T)$ is the concatenation, in this order, of the accept family for $\\mathit{acc}$, the shift family, the\u2026","title":"The formula emitted by the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/real-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.realB","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#real-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with encoding data $E$, let $Y$ be a list of input symbols, and let $m, T \\in \\mathbb{N}$. Let $H = m + (T + 1) d + 1$, where $d$ is the window depth of $M$, and let $R = T + H + 1$. Let $L$ be the layout of $E$ and $N$ its number of situations. Let $c_I$ be the init generator's constant for $L$, the number of distinct codes of symbols of $Y$, and the number of the\u2026","title":"Step bound of the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/real-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.real_run","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#real-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x$ and $Y$ be lists of input symbols of $M$, let $a$ be a symbol of its output stack, and let $m, T \\in \\mathbb{N}$ with $|x| \\le m$. Put $H = m + (T + 1)d + 1$, where $d$ is the window depth of $M$. Let $S_0$ be host stacks whose output stack holds a word $o$, whose input stack holds the symbol codes of $x$ in reverse order, whose update, situation-definition, shift\u2026","title":"The assembled generator, after the precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/phi-iff","kind":"node","lean":"Cslib.Complexity.CookLevin.Assembly.phi_iff","parent":30895,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o/index.html#phi-iff","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-five-generators-as-sub-machines-o","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with encoding data $E$, let $x$ and $Y$ be lists of input symbols of $M$, let $\\mathit{acc}$ be an output symbol of $M$, and let $m, T \\in \\mathbb{N}$. Suppose that $|x| \\le m$ and that $\\mathit{acc}$ is one of the effective symbols of the output stack under $E$. Then the formula $\\Phi(x, Y, \\mathit{acc}, m, T)$ emitted by the host is satisfiable if and only if there\u2026","title":"Correctness of the assembled formula"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","title":"Cook-Levin: the precomputation, composed with the assembled generator"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/tg","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.tg","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#tg","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a list $y_1, \\dots, y_j$ of counters, the list of transfer targets that pushes one unit on each counter $y_i$, in the same order.","title":"Unit targets of a multi-target transfer"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/xf","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.xf","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#xf","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Given counter values $F : C \\to \\mathbb{N}$, a counter $x \\in C$ and a list $Y$ of counters, the counter values after transferring $x$ to $Y$ form the function that sends $x$ to $0$, sends every $z \\ne x$ occurring in $Y$ to $F(z) + F(x)$, and sends every other $z$ to $F(z)$.","title":"Counters after a multi-target transfer"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/xfer-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.xferS","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#xfer-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $M$ be a counter machine with counters indexed by $C$, a Boolean output stack and a Boolean state, let $x \\in C$, and let $Y$ be a list of distinct counters not containing $x$. Suppose the instruction at label $\\ell$ is the transfer loop that pops counter $x$ and pushes one unit on every counter of $Y$, returning to $\\ell$ while $x$ is nonempty and continuing at label $\\ell'$ once it is empty. Then for all\u2026","title":"Multi-target transfer"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/pw","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.PW","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#pw","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"The labels of one power-loop site: a head stage, a multiplication stage $\\mathrm{mul}(s)$ for each stage $s$ of a multiplication site, a drain stage and a move stage.","title":"Stages of the power loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/pow-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.powS","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#pow-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Given six counters $b$, $k$, $p$, $q$, $a$, $t$, a label for each power-loop stage and an exit label, the power loop is the following program on the stages. At the head stage it pops one unit from $k$; if $k$ was empty it goes to the exit label, otherwise to the first multiplication stage. The multiplication stages add $p \\cdot b$ to $q$ (a multiplication site with index counter $a$ and scratch counter $t$) and\u2026","title":"Power-loop program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/pow-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.powB","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#pow-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For natural numbers $j$ (the number of rounds), $p$ (the start value) and $b$ (the base), the step budget of the power loop is $j \\, (p \\, (b + 1)^j \\, (3 b + 5) + 5) + 1$.","title":"Step budget of the power loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/pow-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.pow_run","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#pow-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $b, k, p, q, a, t$ be pairwise distinct counters of a counter machine $M$, let $\\mu$ assign a label to each power-loop stage and let $\\nu$ be a label. Suppose that for every power-loop stage $s$ the label $\\mu(s)$ runs the power-loop program for $b, k, p, q, a, t$ with stage labels $\\mu$ and exit label $\\nu$ at stage $s$. Let $j \\in \\mathbb{N}$, let $F$ be counter values with $F(k) = j$ and\u2026","title":"The power loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/hw","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.HW","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#hw","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"The Horner-round stages are $\\mathsf{mul}(s)$, one for each stage $s$ of a multiplication site, and the three stages $\\mathsf{drn}$, $\\mathsf{mv}$ and $\\mathsf{add}$. In a Horner round they label the multiplication, draining the accumulator, moving the product into the accumulator, and adding the coefficient.","title":"Stages of a Horner round"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horn-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.hornS","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horn-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For counters $b, \\mathit{acc}, q, d, t$, a number $a \\in \\mathbb{N}$, stage labels $\\mu$ and an exit label $\\nu$: the statements performing $\\mathit{acc} \\leftarrow \\mathit{acc} \\cdot b + a$ in four stages, each stage reaching the next through $\\mu$. The multiplication stages add $\\mathit{acc} \\cdot b$ to $q$ (with index counter $d$ and scratch counter $t$), then the drain stage empties $\\mathit{acc}$, the move\u2026","title":"The statements of one Horner round"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horn-step","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.hornStep","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horn-step","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For an accumulator value $A$ and a base $b$, the number $A(2b + 3) + A + 2 + (A + 1) + (Ab + 1) + 1$: the exact number of steps one Horner round takes from $\\mathit{acc} = A$ with base $b$.","title":"Step count of one Horner round"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horn-round","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.horn_round","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horn-round","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $b, \\mathit{acc}, q, d, t$ be pairwise distinct counters of a counter machine $M$, let $a \\in \\mathbb{N}$, let $\\mu$ assign a label to each Horner-round stage and let $\\nu$ be a label. Suppose that for every Horner-round stage $s$ the label $\\mu(s)$ runs the Horner-round statement for $b, \\mathit{acc}, q, d, t, a$ with stage labels $\\mu$ and exit label $\\nu$ at stage $s$. Let $F$ be counter values with\u2026","title":"One Horner round"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horn-val","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.hornVal","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horn-val","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a list $c_1, \\dots, c_\\ell$ of natural numbers and natural numbers $b$ and $A$, the Horner value of $c_1, \\dots, c_\\ell$ with base $b$ from accumulator $A$ is the number obtained from the starting value $x = A$ by replacing $x$ by $x b + c_i$ for $i = 1, \\dots, \\ell$ in turn; for the empty list it is $A$.","title":"Horner value"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horn-t","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.hornT","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horn-t","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a list $cs$ of coefficients, a base $b$ and an accumulator value $A$: $0$ if $cs$ is empty, and if $cs$ is $a$ followed by $cs'$, the step count $h(A, b)$ of one Horner round from $A$ with base $b$ plus the step count of $cs'$ with base $b$ from accumulator $A b + a$. This is the exact number of steps of the Horner rounds.","title":"Step count of the Horner rounds"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horner-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.horner_run","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horner-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $b, \\mathit{acc}, q, d, t$ be pairwise distinct counters of a counter machine $M$, let $c_0, \\dots, c_{\\ell-1}$ be natural numbers, let $\\mu$ assign a label $\\mu(i, s)$ to each $i \\in \\mathbb{N}$ and each Horner-round stage $s$, and let $\\nu$ be a label. Suppose that for every $i < \\ell$ and every stage $s$ the label $\\mu(i, s)$ runs, at stage $s$, the statement of one Horner round for\u2026","title":"Horner rounds"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horn-val-eq","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.hornVal_eq","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horn-val-eq","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a list $c_1, \\dots, c_\\ell$ of natural numbers and natural numbers $b$ and $A$, write $h(x)$ for the Horner value of $c_1, \\dots, c_\\ell$ with base $b$ from accumulator $x$: the number obtained from the starting value $x$ by replacing the current value $y$ by $y b + c_i$ for $i = 1, \\dots, \\ell$ in turn. Then $h(A) = A b^{\\ell} + h(0)$.","title":"Splitting off the starting accumulator in Horner evaluation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/coeffs","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.coeffs","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#coeffs","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $p$ with natural-number coefficients and natural degree $d$ (where $d = 0$ when $p = 0$), the coefficient list of $p$ is $[p_d, p_{d-1}, \\dots, p_1, p_0]$, where $p_i$ is the coefficient of $X^i$ in $p$.","title":"Coefficient list of a polynomial"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horn-val-coeffs","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.hornVal_coeffs","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horn-val-coeffs","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For every polynomial $p$ with natural coefficients and every $b \\in \\mathbb{N}$, running Horner rounds with base $b$ on the coefficients of $p$ (highest degree first), starting from accumulator $0$, yields $p(b)$.","title":"Horner's rule"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/ps","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.PS","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#ps","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"The type of the twelve private counters of the precomputation's counter program, with constructors $N$, $P$, $Q$, $AD$, $TMP$, $KC$, $Mc$, $B$, $Tc$, $Fx$, $Dd$, $Hc$.","title":"Scratch counters of the counter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/pt","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.PT","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#pt","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"The type of the twelve counters that the precomputation's counter program loads for the host, with constructors $uT$, $uH$, $sT$, $sH$, $hT$, $hH$, $aT$, $aH$, $iT$, $iX$, $iE$, $iF$.","title":"Target counters of the counter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/cc","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.CC","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#cc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"A counter of the counter program is either a scratch counter $s(x)$, for one of the twelve scratch counter names $x$, or a target counter $t(y)$, for one of the twelve target counter names $y$: the counters are the disjoint union of the scratch counters and the target counters.","title":"Counters of the counter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/pl","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.PL","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#pl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For $D \\in \\mathbb{N}$, the number of Horner rounds, the labels of the counter program are $p_1$; a label $\\mathrm{hr}(i, s)$ for each round $i < D$ and each Horner-round stage $s$; the labels $x_1, x_2, x_3, x_4, c_1, c_2, x_6, x_7, x_8$; a label $w_2(s)$ for each power-loop stage $s$; $x_9$; a label $\\mathrm{ml}(s)$ for each stage $s$ of a multiplication site; the labels $x_{10}, x_{11}, x_{12}, x_{13}, x_{14}$;\u2026","title":"Labels of the counter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/hlab","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.hlab","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#hlab","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a number $D$ of Horner rounds, $i \\in \\mathbb{N}$ and a Horner-round stage $s$: the counter-program label for stage $s$ of Horner round $i$ if $i < D$, and otherwise the label $x_1$ that follows the Horner rounds.","title":"Horner-round labels indexed by natural numbers"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/cprog","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.cprog","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#cprog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural-number coefficients, whose coefficient list $\\kappa_0, \\dots, \\kappa_{D-1}$ has length $D$, and $c, e, d \\in \\mathbb{N}$, the counter program is the program with the labels of the counter program for $D$ Horner rounds that acts on a Boolean output stack and on the scratch and target counters of the counter program as follows. It runs $D$ Horner rounds, round $i$ with coefficient\u2026","title":"The counter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/m-of","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.mOf","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#m-of","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural-number coefficients and $n \\in \\mathbb{N}$, the total length bound is $m = n + 1 + k(n)$.","title":"Total length bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/tof","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.TOf","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#tof","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural coefficients and $c, e, n \\in \\mathbb{N}$, the time bound is $T = (m + c)^e$, where $m = n + 1 + k(n)$.","title":"The precomputed time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/f0","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.F0","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#f0","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a natural number $n$, the start counter values of the counter program give the scratch counter $N$ the value $n$ and every other counter the value $0$.","title":"Start counter values"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/f1","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.F1","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#f1","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural-number coefficients and $n \\in \\mathbb{N}$, this is the assignment of values to the counters of the counter program that gives the scratch counter $N$ the value $n$, the scratch counter $P$ the value $k(n)$, and every other counter the value $0$.","title":"Counter values after the Horner loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/f2","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.F2","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#f2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural-number coefficients and $c, e, n \\in \\mathbb{N}$, write $m = n + 1 + k(n)$. This is the assignment of values to the counters of the counter program that gives the scratch counters $\\mathit{Mc}$, $B$, $\\mathit{KC}$ and $\\mathit{Tc}$ the values $m$, $m + c$, $e$ and $1$, the target counters $\\mathit{iX}$ and $\\mathit{iE}$ the values $n + 1$ and $k(n)$, and every other counter the\u2026","title":"Counter values before the power loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/f3","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.F3","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#f3","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural-number coefficients and $c, e, n \\in \\mathbb{N}$, write $m = n + 1 + k(n)$. This is the assignment of values to the counters of the counter program that gives the scratch counters $\\mathit{Mc}$, $B$ and $\\mathit{Tc}$ the values $m$, $m + c$ and $(m + c)^e$, the target counters $\\mathit{iX}$ and $\\mathit{iE}$ the values $n + 1$ and $k(n)$, and every other counter the value $0$.","title":"Counter values after the power loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/tv","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.tv","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#tv","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural-number coefficients and natural numbers $c, e, d, n$, let $m = n + 1 + k(n)$, $T = (m + c)^e$ and let $H = m + (T + 1)d + 1$ be the stack cap. The target values on the twelve target counters assign $T$ to each of $u_T, s_T, h_T, a_T, i_T$, assign $H$ to each of $u_H, s_H, h_H, a_H$, and assign $n + 1$ to $i_X$, $k(n)$ to $i_E$ and $(T + 1)d + 1$ to $i_F$.","title":"Target counter values"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/f4","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.F4","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#f4","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural-number coefficients and $c, e, d, n \\in \\mathbb{N}$, write $m = n + 1 + k(n)$ and $T = (m + c)^e$. This is the assignment of values to the counters of the counter program that gives the scratch counters $B$ and $\\mathit{Dd}$ the values $m + c$ and $d$, gives each target counter its target value ($T$ for $\\mathit{uT}$, $\\mathit{sT}$, $\\mathit{hT}$, $\\mathit{aT}$ and $\\mathit{iT}$;\u2026","title":"Counter values at the end of the counter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/c-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.cB","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#c-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a polynomial $k$ with natural-number coefficients and natural numbers $c, e, d, n$, let $m = n + 1 + k(n)$, $T = (m + c)^e$ and $H = m + (T + 1)d + 1$. The step budget of the counter program at $(k, c, e, d, n)$ is","title":"Step budget of the counter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/cprog-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.cprog_run","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#cprog-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $k$ be a polynomial with natural-number coefficients, let $c, e, d, n$ be natural numbers, and put $m = n + 1 + k(n)$ and $T = (m + c)^e$. Start the counter program for $k, c, e, d$ at its first label with any flag, any output stack $o$, the scratch counter $N$ equal to $n$ and every other counter equal to $0$. Then, in a number of steps at most the step budget of the counter program at $(k, c, e, d, n)$, it\u2026","title":"The counter program's run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/fk","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.FK","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#fk","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"The full machine has one stack for each host stack, one stack for the raw input, and one stack for each scratch counter of the counter program.","title":"Stacks of the full machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/f","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.F\u0393","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#f","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For natural numbers $g, r$, the alphabet of a stack of the full machine is the host's alphabet with $g$ input codes on each host stack, $\\{0, 1, \\dots, r - 1\\}$ on the raw input stack, and the one-point type on each private counter.","title":"Alphabets of the full machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/fl","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.FL","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#fl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For $D \\in \\mathbb{N}$ and a type $\\Lambda$ of host labels, the labels of the full machine are the read-loop label, one label for each label of the counter program with $D$ Horner rounds, and one label for each element of $\\Lambda$.","title":"Labels of the full machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/tgt","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.tgt","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#tgt","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"The map from the twelve targets to host stacks: $uT$ and $uH$ go to the counters $T_n$ and $H_s$ of the update generator's block, $sT$ and $sH$ to those of the situation-definition generator's block, $hT$ and $hH$ to those of the shift generator's block, $aT$ and $aH$ to those of the accept generator's block, and $iT$, $iX$, $iE$, $iF$ to the counters $T_n$, $X_n$, $E_n$, $F_n$ of the init generator's block.","title":"Host counter of each target"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/e-c","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.eC","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#e-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"The stack renaming of the counter program into the full machine sends the output stack to the host's output stack, each scratch counter $x$ to the full machine's private counter for $x$, and each target counter $y$ to the host counter that receives $y$.","title":"Stack renaming of the counter program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/emb-c","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.embC","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#emb-c","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For natural numbers $g, r$, the stack embedding of the counter program into the full machine with $g$ input codes and $r$ raw input symbols renames stacks by the stack renaming of the counter program and identifies the alphabet of each stack with that of its image by the identity (Booleans on the output stack, the one-point type on each counter).","title":"Embedding of the counter program's stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/emb-h","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.embH","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#emb-h","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For natural numbers $g, r$, the stack embedding of the host into the full machine with $g$ input codes and $r$ raw input symbols sends each host stack to the same host stack of the full machine and identifies their alphabets by the identity.","title":"Embedding of the host's stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/rd-t","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.rdT","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#rd-t","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Given a code map $c_f$ from the raw alphabet $\\mathrm{Fin}\\,r$ to the code alphabet $\\mathrm{Fin}\\,g$, a separator code $c_s$ and a list $L$ of raw symbols, the read-loop statement tests the symbols $i$ of $L$ in order against the top of the raw input stack. At the first $i$ found on top, it pops it, pushes $c_f(i)$ on the host input stack and one unit on the scratch counter $N$, and goes back to the read-loop\u2026","title":"The read loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/fst","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.fst","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#fst","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For a list $s$ of raw symbols in $\\mathrm{Fin}\\,r$, a list $w$ of input codes in $\\mathrm{Fin}\\,g$, $n \\in \\mathbb{N}$ and a list $o$ of booleans, these are the stack contents of the full machine in which the raw-input stack holds $s$, the host input stack holds $w$, the host output stack holds $o$, the scratch counter $N$ holds $n$ units, and every other stack is empty.","title":"Stacks during the read loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/read-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.read_run","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#read-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $M$ be a program of the full machine whose read-loop label carries the read loop over all of $\\mathrm{Fin}\\,r$, with code map $c_f$ and separator code $c_s$, and let $o$ be a list of booleans. For every raw input $s = s_1 \\cdots s_\\ell$, list of codes $w$, $n \\in \\mathbb{N}$ and flag $v$: started at the read-loop label with flag $v$, raw input stack $s$, input stack $w$, scratch counter $N = n$, output stack\u2026","title":"The read loop's run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/fprog","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.fprog","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#fprog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $\\gamma : \\mathrm{Fin}\\,r \\to \\mathrm{Fin}\\,g$ code the raw symbols, let $\\sigma \\in \\mathrm{Fin}\\,g$ be a separator code, let $k$ be a polynomial with natural-number coefficients, let $c, e, d \\in \\mathbb{N}$, and let $h$ be a host program with labels $\\Lambda$ and start label $\\lambda_0 \\in \\Lambda$. The full program, whose counter-program labels have $D$ Horner rounds for $D$ the length of the coefficient\u2026","title":"The full program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/pre-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.pre_run","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#pre-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Fix the full program built from a code map $c_f$ from the raw alphabet $\\mathrm{Fin}\\,r$ to the code alphabet $\\mathrm{Fin}\\,g$, a separator code $c_s$, a polynomial $k$ with natural coefficients, constants $c, e, d \\in \\mathbb{N}$, a host program and its start label. Let $s = s_1 \\cdots s_n$ be a raw input word and $o$ a list of booleans. Started at the read-loop label with flag false, raw input stack $s$, output\u2026","title":"The precomputation"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/full-prog","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.fullProg","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#full-prog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For an encoding $E$ of a machine $M$, a list $Y$ of symbols of the input stack alphabet of $M$, a symbol $a$ of the output stack alphabet of $M$, a map $\\iota$ from the raw symbols $\\mathrm{Fin}\\,r$ to input stack symbols of $M$, an input stack symbol $\\sigma$ of $M$ (the separator), a polynomial $k$ with natural-number coefficients and $c, e \\in \\mathbb{N}$, this is the full program in which each raw symbol $i$ is\u2026","title":"The full program on the real generators"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/full-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.fullB","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#full-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"For an encoding $E$ of a machine $M$, a list $Y$ of symbols of the input stack alphabet of $M$, a polynomial $k$ with natural-number coefficients and $c, e, n \\in \\mathbb{N}$, write $m = n + 1 + k(n)$ and $T = (m + c)^e$. The bound is $n + 1 + C + 1 + R$, where $C$ is the step budget of the counter program for $k$, $c$, $e$, $d = \\operatorname{depth} M$ and $n$, and $R$ is the host's step bound for $E$ and $Y$ at\u2026","title":"Step bound of the full machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/full-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.full_run","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#full-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"From the raw input `s` (all other stacks empty, output `o`), the full machine halts its host part at `fin` within `fullB` steps, having prepended exactly `encodeCNF (Phi \u2026)` for `x = s.map \u03b9 ++ [sep]`, `m = |s| + 1 + k(|s|)`, `T = (m + c)^e`. No hypotheses (`|x| \u2264 m` is discharged here).","title":"The composed machine, raw input to formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/full-tm","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.fullTM","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#full-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Given $r \\in \\mathbb{N}$, an encoding $E$ of the machine $M$, a list $Y_s$ of input symbols of $M$, an output symbol $a$ of $M$, a map $\\iota$ from $\\{0, \\dots, r - 1\\}$ to input symbols of $M$, an input symbol $\\sigma$ of $M$ (the separator), a polynomial $k$ with natural coefficients and $c, e \\in \\mathbb{N}$: the finite multi-stack machine whose stacks are the full stacks (the host's stacks, the raw input stack\u2026","title":"The full machine as a finite multi-stack Turing machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/init-list-full","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.initList_full","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#init-list-full","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $E$ be encoding data for a finite multi-stack Turing machine $M$, and let $r$ be a natural number. Fix a list $Y$ of input-stack symbols of $M$, an output-stack symbol $a$ of $M$, a map $\\iota$ from $\\{0, \\dots, r-1\\}$ to input-stack symbols of $M$, a separator input-stack symbol $\\sigma$ of $M$, a polynomial $k$ with natural-number coefficients and natural numbers $c$ and $e$. On every raw input $s$, a word\u2026","title":"Initial configuration of the full machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/is-poly","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.IsPoly","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#is-poly","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"A function $f : \\mathbb{N} \\to \\mathbb{N}$ is polynomial when there is a polynomial $p$ with natural-number coefficients such that $f(n) = p(n)$ for every $n \\in \\mathbb{N}$.","title":"Polynomial functions on the naturals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/horn-t-2","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.IsPoly.hornT","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#horn-t-2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $cs$ be a list of natural numbers and let $A : \\mathbb{N} \\to \\mathbb{N}$ be a polynomial function, that is, there is a polynomial $q$ with natural-number coefficients such that $A(n) = q(n)$ for all $n$. Then $n \\mapsto h(n, A(n))$ is a polynomial function in the same sense, where $h(b, a)$ is the step count of the Horner rounds over $cs$ with base $b$ from accumulator $a$.","title":"The Horner step count is polynomial"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/full-time-poly","kind":"node","lean":"Cslib.Complexity.CookLevin.Precompute.full_time_poly","parent":30918,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the/index.html#full-time-poly","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-the-precomputation-composed-with-the","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine with encoding data $E$, let $Y$ be a list of symbols of the input stack of $M$, $a$ a symbol of its output stack, $\\iota$ a map from the raw symbols $\\mathrm{Fin}\\,r$ to symbols of the input stack, $\\sigma$ a symbol of the input stack (the separator), $k$ a polynomial with natural-number coefficients and $c, e \\in \\mathbb{N}$. Then there is a polynomial $p$ with\u2026","title":"Polynomial time bound of the composed machine"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","title":"Cook-Levin: halting cleanup and packaging of the reduction machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/xfer-drain","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.xfer_drain","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#xfer-drain","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Let $M$ be a TM2 machine whose state type carries a flag bit, and suppose that the statement of $M$ at label $\\ell$ is the transfer loop that pops stack $k$ with no target stacks, returning to $\\ell$ while the pop succeeds and going to label $\\ell'$ once stack $k$ is empty. Then for every state $v$ and every stack assignment $S$, writing $S_k$ for the contents of stack $k$ under $S$ and $|S_k|$ for its length, in\u2026","title":"Draining a stack with the transfer loop"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/drain-s","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.drainS","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#drain-s","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Given a list $k_0, \\dots, k_{m-1}$ of stacks, a type $\\Lambda$ of labels and labels $\\lambda : \\mathbb{N} \\to \\Lambda$, stage $i$ of the drain program is as follows. For $i < m$ it pops stack $k_i$, recording in the state whether a symbol was popped, and jumps to $\\lambda(i)$ if one was and to $\\lambda(i + 1)$ if the stack was empty; repeated, this empties $k_i$. For $i \\ge m$ it sets the state to store the flag\u2026","title":"The drain program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/drain-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.drainB","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#drain-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"For a list $k_1, \\dots, k_m$ of stacks and contents $S$ assigning a list $S_k$ to each stack $k$, the drain step count is $1 + \\sum_{i=1}^{m} (|S_{k_i}| + 1)$.","title":"Drain step count"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/drain-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.drain_run","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#drain-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Let $M$ be a multi-stack machine program with labels in a type $\\Lambda$, let $k_0, \\dots, k_{m-1}$ be a list of distinct stacks, and let $\\lambda : \\mathbb{N} \\to \\Lambda$ be labels such that for every $i \\le m$ the instruction of $M$ at $\\lambda(i)$ is stage $i$ of the drain program for this list and these labels. Then for every state $v$ and stack contents $S$, the machine runs in exactly\u2026","title":"The drain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/dl","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.DL","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#dl","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"For a type $\\Lambda$ of labels and $n \\in \\mathbb{N}$, a label of the composed machine is either a label $l \\in \\Lambda$ of the original machine or a drain stage $i \\in \\{0, \\dots, n\\}$.","title":"Labels of the composed machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/dlab","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.dlab","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#dlab","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"For $n, j \\in \\mathbb{N}$, the label of the composed machine with drain stages $0, \\dots, n$ that names drain stage $\\min(j, n)$.","title":"Drain-stage label"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/id-e","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.idE","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#id-e","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"For a family of stack alphabets $\\Gamma$ indexed by $K$, the identity stack embedding of $\\Gamma$ into itself sends every stack to itself and identifies every alphabet with itself by the identity bijection: a pure relabelling.","title":"The identity stack embedding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/dks","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.dks","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#dks","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"The list of all stacks of the full machine other than its output stack, each appearing exactly once, in an unspecified order. These are the stacks that the drain empties before the composed machine halts.","title":"Stacks to drain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/drained","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.drained","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#drained","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"For any alphabets and any assignment $T$ of contents to the stacks of the full machine, the contents that are empty on every stack to drain and agree with $T$ elsewhere are the contents that agree with $T$ on the output stack and are empty on every other stack.","title":"Only the output stack survives the drain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/gprog","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.gprog","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#gprog","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Let $g, r \\in \\mathbb{N}$, let $P$ be a program with labels $\\Lambda$ and Boolean state over the stacks of the full machine with alphabets $F\\Gamma_{g,r}$, and let $\\mathit{fin} \\in \\Lambda$. The composed program has as labels the labels of $P$ and the drain stages $0, \\dots, n$, where $n$ is the number of stacks other than the output stack. At a label $\\ell \\ne \\mathit{fin}$ it runs $P(\\ell)$ with every jump\u2026","title":"The composed program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/g-run-gen","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.g_run_gen","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#g-run-gen","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Let $g, r \\in \\mathbb{N}$, let $P$ be a program with labels $\\Lambda$ and Boolean state over the stacks of the full machine with alphabets $F\\Gamma_{g,r}$, and let $\\mathit{fin}$ be a label with $P(\\mathit{fin}) = \\mathrm{halt}$. Suppose $P$ goes in at most $B$ steps from the configuration with label $\\ell$, state $v$ and stacks $S$ to the configuration with label $\\mathit{fin}$, state $v'$ and stacks $T$. Then the\u2026","title":"Running a program, then the drain"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/drain-b-le","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.drainB_le","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#drain-b-le","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Let $g, r, h \\in \\mathbb{N}$ and let $T$ assign to every stack $k$ of the full machine (with alphabets $\\Gamma_{g,r}$) a list $T_k$ of length at most $h$. Let $L$ be the number of stacks other than the output stack. Then the drain step count $1 + \\sum_k (|T_k| + 1)$, summed over the stacks $k$ other than the output stack, is at most $L(h + 1) + 1$.","title":"Step bound for draining bounded stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/g-tm","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.gTM","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#g-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Fix $r \\in \\mathbb{N}$, a finite TM2 machine $M$ with an encoding $E$, a list $Y$ of input symbols of $M$, an output symbol $a$, a map $\\iota$ from $\\{0, \\dots, r-1\\}$ to input symbols, an input symbol $\\mathit{sep}$, a polynomial $k$ with coefficients in $\\mathbb{N}$ and $c, e \\in \\mathbb{N}$. The reduction machine is the finite TM2 machine with the stacks and alphabets of the full machine, input on the raw-input\u2026","title":"The reduction machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/phi-of","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.phiOf","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#phi-of","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Fix $r \\in \\mathbb{N}$, a finite TM2 machine $M$ with an encoding $E$, a list $Y$ of input symbols of $M$, an output symbol $a$, a map $\\iota$ from $\\{0, \\dots, r-1\\}$ to input symbols, an input symbol $\\mathit{sep}$, a polynomial $k$ with coefficients in $\\mathbb{N}$ and $c, e \\in \\mathbb{N}$. For a raw input $s$, a list over $\\{0, \\dots, r-1\\}$, let $x$ be $s$ with each symbol mapped by $\\iota$, followed by\u2026","title":"The formula emitted on a raw input"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/g-b","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.gB","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#g-b","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Fix the verifier data (an encoding $E$ of a stack machine $M$, a list $Y$ of input symbols, an output symbol $\\mathrm{acc}$, a map $\\iota$ into the input alphabet, a separator $\\mathrm{sep}$, a polynomial $k$ and $c, e \\in \\mathbb{N}$) and an input length $n$. Let $B(n)$ be the step bound of the full machine on inputs of length $n$, let $D$ be the depth of the full machine (the most push, peek and pop operations on\u2026","title":"Step bound of the composed machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/g-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.g_run","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#g-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Fix $r \\in \\mathbb{N}$, a finite TM2 machine $M$ with an encoding $E$, a list $Y$ of input symbols of $M$, an output symbol $a$, a map $\\iota$ from $\\{0, \\dots, r-1\\}$ to input symbols, an input symbol $\\mathit{sep}$, a polynomial $k$ with coefficients in $\\mathbb{N}$ and $c, e \\in \\mathbb{N}$. For every raw input $s$, a list over $\\{0, \\dots, r-1\\}$, the program of the reduction machine goes in at most $g_B(|s|)$\u2026","title":"The reduction machine halts with the emitted formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/outputs-of-run-le","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.outputsOfRunLe","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#outputs-of-run-le","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Let $M$ be a finite TM2 machine, $l$ a list over its input alphabet, $l'$ a list over its output alphabet and $B \\in \\mathbb{N}$. If the program of $M$ goes in at most $B$ steps from the initial configuration on input $l$ (its start label, its initial state, $l$ on the input stack and every other stack empty) to the halting configuration with output $l'$ (no label, the initial state, $l'$ on the output stack and\u2026","title":"A bounded halting run certifies the output"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/outputs-of-run-le-2","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.outputsOfRunLe'","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#outputs-of-run-le-2","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Let $M$ be a finite TM2 machine, $l$ a list over its input alphabet, $l'$ and $l''$ lists over its output alphabet and $B \\in \\mathbb{N}$. If the program of $M$ goes in at most $B$ steps from the initial configuration on input $l$ to the halting configuration with output $l''$, and $l'' = l'$, then $M$ outputs $l'$ on input $l$ within $B$ steps.","title":"A bounded halting run certifies the output, up to an equal output"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/g-outputs","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.g_outputs","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#g-outputs","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Fix $r \\in \\mathbb{N}$, a finite TM2 machine $M$ with an encoding $E$, a list $Y$ of input symbols of $M$, an output symbol $a$, a map $\\iota$ from $\\{0, \\dots, r-1\\}$ to input symbols, an input symbol $\\mathit{sep}$, a polynomial $k$ with coefficients in $\\mathbb{N}$ and $c, e \\in \\mathbb{N}$. There is a polynomial $p$ with coefficients in $\\mathbb{N}$ such that for every raw input $s$, a list over\u2026","title":"One polynomial bounds the reduction machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/f-r","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.fR","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#f-r","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Fix $r \\in \\mathbb{N}$, a finite TM2 machine $M$ with an encoding $E$, a list $Y$ of input symbols of $M$, an output symbol $\\mathrm{acc}$ of $M$, a map $\\iota$ from $\\{0, \\dots, r-1\\}$ to input symbols of $M$, an input symbol $\\mathit{sep}$ of $M$, a polynomial $k$ with coefficients in $\\mathbb{N}$ and $c, e \\in \\mathbb{N}$. Let $\\mathit{ea}$ encode the elements of a type $\\alpha$ as lists over an alphabet\u2026","title":"The Cook\u2013Levin reduction function"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/g-comp","kind":"node","lean":"Cslib.Complexity.CookLevin.Package.gComp","parent":30972,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the/index.html#g-comp","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-halting-cleanup-and-packaging-of-the","status":"fully_proved","summary":"Fix $r \\in \\mathbb{N}$, a finite TM2 machine $M$ with an encoding $E$, a list $Y$ of input symbols of $M$, an output symbol $a$, a map $\\iota$ from $\\{0, \\dots, r-1\\}$ to input symbols, an input symbol $\\mathit{sep}$, a polynomial $k$ with coefficients in $\\mathbb{N}$ and $c, e \\in \\mathbb{N}$. Let $\\mathit{ea}$ encode the elements of a type $\\alpha$ as lists over an alphabet $\\Sigma$, and let $\\mathit{eqv}$ be a\u2026","title":"The reduction machine as a polynomial-time computation"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat","status":"fully_proved","title":"Polynomial-time verifier for SAT"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","kind":"scope","navigation":"transparent","parent":30994,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","title":"Occurrence list of a CNF formula to SAT is in NP if its verifier is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/vars-of","kind":"node","lean":"Cslib.CookLevin.varsOf","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#vars-of","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"For a CNF formula $\\varphi = (C_1, \\dots, C_m)$, the occurrence list of $\\varphi$ is the list of the variable indices of its literals, in order: the variables of the literals of $C_1$ in order, then those of $C_2$, and so on up to $C_m$. Duplicates are kept, so a variable appears once for each literal in which it occurs.","title":"Occurrence list of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/lookup-var","kind":"node","lean":"Cslib.CookLevin.lookupVar","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#lookup-var","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Given a list $N = (n_1, \\dots, n_k)$ of natural numbers, a list $w = (b_1, \\dots, b_m)$ of booleans and a natural number $x$, the lookup of $x$ in $N$ and $w$ is the timed computation (a returned boolean together with a natural-number cost) that scans the pairs $(n_1, b_1), (n_2, b_2), \\dots$ in order, charging one tick for each pair it examines, and returns $b_i$ for the first $i$ with $n_i = x$; if either list\u2026","title":"Witness lookup"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/check-clause","kind":"node","lean":"Cslib.CookLevin.checkClause","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#check-clause","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Given a list $N$ of natural numbers, a list $w$ of booleans and a CNF clause $C$, the check of $C$ against $N$ and $w$ is the timed computation (a returned boolean together with a natural-number cost) that scans the literals $(x, b)$ of $C$ in order: for each it charges one tick, then looks up $x$ in $N$ and $w$, paying the cost of the lookup, and returns true if the value found equals the polarity $b$; otherwise\u2026","title":"Clause check against a witness"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/check-cnf","kind":"node","lean":"Cslib.CookLevin.checkCNF","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#check-cnf","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Given a list $N$ of natural numbers, a list $w$ of booleans and a CNF formula $\\varphi$, the check of $\\varphi$ against $N$ and $w$ is the timed computation that runs the clause check against $N$ and $w$ on the clauses of $\\varphi$ in order, adding up their costs, and returns false as soon as one of them returns false. If every clause check returns true, in particular if $\\varphi$ has no clauses, it returns true.","title":"Formula check against a witness"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/sat-verifier","kind":"node","lean":"Cslib.CookLevin.satVerifier","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#sat-verifier","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"The SAT verifier takes a pair $(x, w)$ of lists of Booleans, an input $x$ and a witness $w$, and returns a Boolean together with a declared time cost. If the complete CNF decoder fails on $x$, it returns false at cost $0$. If $x$ decodes to a CNF formula $\\varphi$, it returns the result and the cost of the CNF check of $\\varphi$ against the witness $w$ and the occurrence list of $\\varphi$, so that $w$ is read as\u2026","title":"The SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/lookup-var-time-le","kind":"node","lean":"Cslib.CookLevin.lookupVar_time_le","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#lookup-var-time-le","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"For every list $N$ of natural numbers, every list $w$ of booleans and every natural number $x$, the cost of looking up $x$ in $N$ and $w$ is at most the length of $N$.","title":"Cost bound for witness lookup"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/lookup-var-ret-map","kind":"node","lean":"Cslib.CookLevin.lookupVar_ret_map","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#lookup-var-ret-map","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Let $v$ be a function from the natural numbers to the booleans, $N = (n_1, \\dots, n_k)$ a list of natural numbers and $x$ a natural number. Looking up $x$ in $N$ and the list $(v(n_1), \\dots, v(n_k))$ returns $v(x)$ if $x$ occurs in $N$, and false otherwise.","title":"Lookup in the witness of an assignment"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/check-clause-ret","kind":"node","lean":"Cslib.CookLevin.checkClause_ret","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#check-clause-ret","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Let $N$ be a list of natural numbers, $w$ a list of booleans and $C$ a CNF clause. The check of $C$ against $N$ and $w$ returns true if and only if $C$ is satisfied by the assignment that sends each natural number $x$ to the value returned by looking up $x$ in $N$ and $w$.","title":"Correctness of the witness clause check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/check-clause-time-le","kind":"node","lean":"Cslib.CookLevin.checkClause_time_le","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#check-clause-time-le","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Let $N$ be a list of natural numbers, $w$ a list of booleans and $C$ a CNF clause, viewed as a list of $\\ell$ literals (repetitions counted). The cost of checking $C$ against $N$ and $w$ is at most $\\ell \\cdot (|N| + 1)$, where $|N|$ is the length of $N$.","title":"Cost bound for the witness clause check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/check-cnf-ret","kind":"node","lean":"Cslib.CookLevin.checkCNF_ret","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#check-cnf-ret","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Let $N$ be a list of natural numbers, $w$ a list of booleans and $\\varphi$ a CNF formula. The check of $\\varphi$ against $N$ and $w$ returns true if and only if $\\varphi$ is satisfied by the assignment that sends each natural number $x$ to the value returned by looking up $x$ in $N$ and $w$.","title":"Correctness of the witness formula check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/check-cnf-time-le","kind":"node","lean":"Cslib.CookLevin.checkCNF_time_le","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#check-cnf-time-le","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Let $N$ be a list of natural numbers, $w$ a list of booleans and $\\varphi$ a CNF formula, and let $L$ be the sum of the lengths of the clauses of $\\varphi$ (its total number of literals). The cost of checking $\\varphi$ against $N$ and $w$ is at most $L \\cdot (|N| + 1)$, where $|N|$ is the length of $N$.","title":"Cost bound for the witness formula check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/vars-of-length","kind":"node","lean":"Cslib.CookLevin.varsOf_length","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#vars-of-length","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"For every CNF formula $\\varphi$, the length of the occurrence list of $\\varphi$ equals the total number of literals of $\\varphi$, that is, the sum over the clauses of $\\varphi$ of their lengths.","title":"Length of the occurrence list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/mem-vars-of","kind":"node","lean":"Cslib.CookLevin.mem_varsOf","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#mem-vars-of","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula, $C$ a clause of $\\varphi$ and $(x, b)$ a literal of $C$. Then $x$ belongs to the list of variables occurring in $\\varphi$.","title":"Variables of a formula occur in its occurrence list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/sum-length-le-encode-cnf-length","kind":"node","lean":"Cslib.CookLevin.sum_length_le_encodeCNF_length","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#sum-length-le-encode-cnf-length","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"For every CNF formula $\\varphi$, the sum over the clauses of $\\varphi$ of their lengths (the total number of literals of $\\varphi$) is at most the length of the length-prefixed binary code of $\\varphi$.","title":"The literal count is at most the code length"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/sat-verifier-none","kind":"node","lean":"Cslib.CookLevin.satVerifier_none","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#sat-verifier-none","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"For all lists of Booleans $x$ and $w$, if the complete CNF decoder fails on $x$, then the SAT verifier on $(x, w)$ returns false at declared time cost $0$.","title":"The SAT verifier rejects undecodable inputs"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/sat-verifier-some","kind":"node","lean":"Cslib.CookLevin.satVerifier_some","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#sat-verifier-some","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"For all lists of Booleans $x$ and $w$ and every CNF formula $\\varphi$, if the complete CNF decoder returns $\\varphi$ on $x$, then the SAT verifier on $(x, w)$ coincides with the CNF check of $\\varphi$ against the witness $w$ and the occurrence list of $\\varphi$: both the returned Boolean and the declared time cost are the same.","title":"The SAT verifier on a decodable input"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/sat-verifier-time-le","kind":"node","lean":"Cslib.CookLevin.satVerifier_time_le","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#sat-verifier-time-le","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"For all lists of Booleans $x$ (the input) and $w$ (the witness), the declared time cost of the SAT verifier on $(x, w)$ is at most $N^2 + N$, where $N = |x| + |w|$ is the sum of their lengths.","title":"Declared time bound of the SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/sat-verifier-verifier-in-declared-poly-t","kind":"node","lean":"Cslib.CookLevin.satVerifier_verifierInDeclaredPolyTime","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#sat-verifier-verifier-in-declared-poly-t","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"There is a polynomial $q$ with natural number coefficients such that for all lists of Booleans $x$ (the input) and $w$ (the witness), the declared time cost of the SAT verifier on $(x, w)$ is at most $q(|x| + |w|)$.","title":"The SAT verifier has polynomial declared time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/sat-verifier-verifies","kind":"node","lean":"Cslib.CookLevin.satVerifier_verifies","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#sat-verifier-verifies","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"The SAT verifier verifies the SAT decision problem with witness bound the polynomial $X$: for every list $x$ of Booleans, $x$ encodes a satisfiable CNF formula if and only if there is a list $w$ of Booleans, of length at most the length of $x$, such that the SAT verifier on $(x, w)$ returns true.","title":"The SAT verifier verifies SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/sat-dec-mem-np-of-is-poly-time","kind":"node","lean":"Cslib.CookLevin.satDec_mem_NP_of_isPolyTime","parent":30995,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01/index.html#sat-dec-mem-np-of-is-poly-time","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-01","status":"fully_proved","summary":"If the function sending a pair $(x, w)$ of lists of Booleans to the Boolean returned by the SAT verifier on $(x, w)$ is polynomial-time computable, by a multi-stack Turing machine under the canonical bitstring encodings, then the SAT decision problem belongs to NP.","title":"SAT is in NP if its verifier is polynomial-time"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","kind":"scope","navigation":"transparent","parent":30994,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","title":"Stack alphabet of the SAT verifier to A clause has at most as many tokens as code bits"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/sym","kind":"node","lean":"Cslib.Complexity.SatVerifier.Sym","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#sym","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"The symbols of the token and table stacks are a name bit $b$ for each boolean $b$, a polarity marker for each polarity $p$, a clause end marker, a value marker for each boolean $b$, and a dead marker. Equality of symbols is decidable.","title":"Stack alphabet of the SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/bit","kind":"node","lean":"Cslib.Complexity.SatVerifier.Sym.bit?","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#bit","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"A symbol of the token and table stacks that is the name bit $b$ reads as the bit $b$; every other symbol reads as no bit, and so ends a name block.","title":"Reading a symbol as a name bit"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/lit-toks","kind":"node","lean":"Cslib.Complexity.SatVerifier.litToks","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#lit-toks","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"For a literal $(n, p)$ with variable index $n$ and polarity $p$, the token list is one bit symbol for each bit of the canonical binary representation of $n$ (least significant bit first, empty for $n = 0$), in order, followed by the polarity marker symbol for $p$.","title":"Tokens of a literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/clause-toks","kind":"node","lean":"Cslib.Complexity.SatVerifier.clauseToks","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#clause-toks","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"The tokens of a CNF clause are the tokens of its literals, in order, followed by the clause-end symbol.","title":"Tokens of a clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/toks","kind":"node","lean":"Cslib.Complexity.SatVerifier.toks","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#toks","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"For a CNF formula $\\varphi$ with clauses $C_1, \\dots, C_k$ (in order), the token sequence of $\\varphi$ is the concatenation of the token sequences of $C_1, \\dots, C_k$. The token sequence of a clause is the concatenation of the token sequences of its literals, in order, followed by a clause-end symbol; the token sequence of a literal with variable index $n \\in \\mathbb{N}$ and polarity $p$ is the bits of the binary\u2026","title":"Token sequence of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/ent","kind":"node","lean":"Cslib.Complexity.SatVerifier.Ent","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#ent","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"A table entry is a pair of a name, which is a list of bits, and an optional bit, its value; the value is absent for occurrences past the end of the witness.","title":"Table entries"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/mark-of","kind":"node","lean":"Cslib.Complexity.SatVerifier.markOf","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#mark-of","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"The marker closing a table entry whose value is the boolean $b$ is the value symbol for $b$; for an entry with no value it is the dead symbol.","title":"Closing marker of a table entry"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/ent-toks","kind":"node","lean":"Cslib.Complexity.SatVerifier.entToks","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#ent-toks","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"The symbols of a table entry with name $q$ and value $v$ are the bits of $q$ as name-bit symbols, in order, followed by one closing marker: the value marker of $b$ if $v$ is a bit $b$, and the dead marker if $v$ is absent.","title":"Symbols of a table entry"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/tbl-stk","kind":"node","lean":"Cslib.Complexity.SatVerifier.tblStk","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#tbl-stk","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"Write each table entry as one bit symbol per bit of its name followed by its closing marker (the value symbol for its value, or the dead symbol if it has no value), and concatenate these in the order of the list. The table stack is the reverse of this sequence, with the head as the top of the stack, so the first entry lies at the bottom.","title":"Table stack of an entry list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/ents-of","kind":"node","lean":"Cslib.Complexity.SatVerifier.entsOf","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#ents-of","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"Given natural numbers $n_0, \\dots, n_{k-1}$ and a list $w$ of bits, the occurrence table is the list of $k$ table entries whose $i$-th entry has as name the binary code of $n_i$ (Mathlib's encoding of natural numbers as bit lists) and as value the bit $w_i$ if $i$ is less than the length of $w$, and no value otherwise.","title":"Occurrence table of a list of variables"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/ents","kind":"node","lean":"Cslib.Complexity.SatVerifier.ents","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#ents","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula, $w$ a list of booleans, and $n_0, \\dots, n_{k-1}$ the variable indices of the literals of $\\varphi$ in reading order, with repetitions. The occurrence table is the list of $k$ entries whose $i$-th entry pairs the binary name of $n_i$ (its canonical binary representation, least significant bit first, empty for $0$) with the value $w_i$ if $i$ is less than the length of $w$, and with\u2026","title":"Occurrence table of a formula and a witness"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/live-val","kind":"node","lean":"Cslib.Complexity.SatVerifier.liveVal","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#live-val","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"Given a list of table entries, each pairing a name (a bit string) with either a boolean value or no value, and a bit string $q$, the live value of $q$ is the value of the first entry whose name is $q$ and which carries a value, or false if there is no such entry.","title":"Live value of a name in a table"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/check-spec","kind":"node","lean":"Cslib.Complexity.SatVerifier.checkSpec","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#check-spec","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and a list $es$ of table entries, the clause check is true exactly when every clause of $\\varphi$ contains a literal $(x, p)$ whose live value equals its polarity $p$. Here the live value of $x$ is the value of the first entry of $es$ that is named by the binary code of $x$ and has a value, and false if there is no such entry.","title":"Clause check against an occurrence table"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/tokenize","kind":"node","lean":"Cslib.Complexity.SatVerifier.tokenize","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#tokenize","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"Given bit strings $x$ and $w$, the tokenizer rejects (returns no result) if $x$ does not decode to a CNF formula. If $x$ decodes to a formula $\\varphi$, it returns the pair consisting of the token stack, the reverse of the tokens of $\\varphi$ in reading order, and the table stack of the occurrence table of $\\varphi$ and $w$.","title":"Tokenizer of the SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/encode-nat-inj","kind":"node","lean":"Cslib.Complexity.SatVerifier.encodeNat_inj","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#encode-nat-inj","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"For natural numbers $m$ and $n$, the binary codes of $m$ and $n$ (Mathlib's encoding of natural numbers as bit lists) are equal if and only if $m = n$.","title":"The binary code of natural numbers is injective"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/encode-nat-decode-nat-eq-iff","kind":"node","lean":"Cslib.Complexity.SatVerifier.encodeNat_decodeNat_eq_iff","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#encode-nat-decode-nat-eq-iff","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"For every list $l$ of bits, encoding the natural number that $l$ decodes to (Mathlib's binary code of natural numbers, least significant bit first) gives back $l$ if and only if $l$ is empty or its last bit is true.","title":"Canonical binary names"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/live-val-ents-of","kind":"node","lean":"Cslib.Complexity.SatVerifier.liveVal_entsOf","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#live-val-ents-of","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"Let $n = (n_0, \\dots, n_{k-1})$ be a list of natural numbers, $w$ a list of booleans and $x$ a natural number. Write $[m]$ for the binary name of a natural number $m$: its canonical binary representation, least significant bit first, empty for $m = 0$. Form the table whose $i$-th entry, for $i < k$, pairs $[n_i]$ with $w_i$ if $i$ is less than the length of $w$, and with no value otherwise. Then the live value of\u2026","title":"Table lookup agrees with the verifier's variable lookup"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/check-spec-ents","kind":"node","lean":"Cslib.Complexity.SatVerifier.checkSpec_ents","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#check-spec-ents","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"For every CNF formula $\\varphi$ and every list $w$ of bits, the clause check of $\\varphi$ against the occurrence table of $\\varphi$ and $w$ equals the value returned (ignoring the time count) by the formula check of $\\varphi$ against $w$ that uses the variable occurrences of $\\varphi$, in order, as its variable list.","title":"The clause check on the occurrence table is the formula check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/sat-verifier-ret-eq","kind":"node","lean":"Cslib.Complexity.SatVerifier.satVerifier_ret_eq","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#sat-verifier-ret-eq","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"For all lists $x$ and $w$ of bits, the value returned by the SAT verifier on $(x, w)$, ignoring the time count, is false if the complete length-prefixed CNF decoder fails on $x$, and is the clause check of $\\varphi$ against the occurrence table of $\\varphi$ and $w$ if $x$ decodes to the CNF formula $\\varphi$.","title":"The SAT verifier's answer through the occurrence table"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/clause-toks-length-le","kind":"node","lean":"Cslib.Complexity.SatVerifier.clauseToks_length_le","parent":31016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02/index.html#clause-toks-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-02","status":"fully_proved","summary":"For every CNF clause $C$, the number of tokens of $C$ is at most the length of the binary code of $C$.","title":"A clause has at most as many tokens as code bits"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","kind":"scope","navigation":"transparent","parent":30994,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","title":"Token count is bounded by the encoding length to The lookup program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/toks-length-le","kind":"node","lean":"Cslib.Complexity.SatVerifier.toks_length_le","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#toks-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"For every CNF formula $\\varphi$, the number of tokens of $\\varphi$ is at most the length of the binary encoding of $\\varphi$.","title":"Token count is bounded by the encoding length"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/k","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.K","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#k","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"The eight stacks of the clause-check machine: $T$ (tokens), $B$ (table), $S$ (the lookup's stash), $A$ and $C$ (the lookup's scratch stacks), $V$ (looked-up value), $P$ (polarity) and $F$ (clause flag).","title":"Stacks of the clause check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/dlab","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.DLab","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#dlab","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"The seven labels of the clause-check driver: start (begin the next clause), lits (read the next literal), ret (return from the lookup), drop (discard a literal's name bits), endc (end of a clause), yes (accept) and no (reject).","title":"Driver labels of the clause check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/stmt","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.Stmt","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#stmt","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"For a label set $L$, the statements of a multi-stack Turing machine with the eight driver stacks, each over the symbols of the token and table stacks, with labels in $D \\sqcup L$, where $D$ is the set of driver labels, and with a Boolean internal state.","title":"Clause-check statements"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/cfg","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.Cfg","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#cfg","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"For a type $LL$ of lookup labels, a configuration of the multi-stack Turing machine of the clause check: a current label (absent once the machine halts) that is either a driver label or a label in $LL$, a Boolean internal state, and the contents of the eight stacks $T$, $B$, $S$, $A$, $C$, $V$, $P$, $F$ of the clause check, each a list of symbols of the token and table stacks (name bits, literal and clause markers,\u2026","title":"Configurations of the clause-check machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/call-s","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.callS","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#call-s","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"For the lookup's start label $l_0$ and a polarity $p$, the statement that pops the top of the token stack $T$, pushes the bit $p$ onto the polarity stack $P$, and then goes to label $l_0$.","title":"Calling the lookup"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/drv","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.drv","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#drv","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"Given the lookup's start label $l_0$, the statement at each driver label. At start: if the token stack $T$ is empty, go to yes; if its top is a clause marker, pop it, push the bit false onto the flag stack $F$ and go to lits; otherwise go to no. At lits: if the top of $T$ is a literal marker of polarity $p$, pop it, push the bit $p$ onto the polarity stack $P$ and go to $l_0$; otherwise go to endc. At ret: if the\u2026","title":"The clause-check driver"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/prog","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.prog","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"Given a lookup sub-program $lk$ on a label set $L$ and an entry label $l_0 \\in L$, this is the program on the labels $D \\sqcup L$, where $D$ is the set of driver labels: at a driver label $d$ it runs the driver's statement for $d$ (the driver calls the lookup by jumping to $l_0$), and at a label $l \\in L$ it runs $lk(l)$.","title":"The clause-check driver with a lookup sub-program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/lookup-spec","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.LookupSpec","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#lookup-spec","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"A lookup sub-program $lk$ with start label $l_0$ meets the contract with constants $a$ and $b$ when the following holds. For every name $q$ (a list of bits), every table $es$, with table stack $B_{es}$, and every list of symbols $R$ whose first symbol, if any, is not a bit, start the clause-check machine with lookup $lk$ at label $l_0$, with any internal state and with stack contents that have the bits of $q$ in\u2026","title":"The lookup contract"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/lit-rev","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.litRev","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#lit-rev","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"For a literal $(n, p)$ with variable index $n$ and polarity $p$, the list of symbols consisting of the literal marker of polarity $p$ followed by the bits of the binary encoding of $n$ (least significant bit first, empty for $n = 0$) in reverse order.","title":"A literal's tokens on the token stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/cl-rev","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.clRev","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#cl-rev","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"For a clause with literals $l_1, \\dots, l_k$, the list of symbols consisting of the clause marker followed by the stack tokens of $l_k$, then of $l_{k-1}$, down to $l_1$, where the stack tokens of a literal $(n, p)$ are the literal marker of polarity $p$ followed by the bits of the binary encoding of $n$ (least significant bit first, empty for $n = 0$) in reverse order.","title":"A clause's tokens on the token stack"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/lits-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.lits_run","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"Suppose the lookup sub-program $lk$ with start label $l_0$ meets the lookup contract with constants $a$ and $b$. Let $es$ be a table with table stack $B_{es}$, let $R$ be a list of symbols that is empty or starts with a clause marker, and let $s$ be a bit. The stack tokens of a literal $(n, p)$ are the literal marker of polarity $p$ followed by the bits of the binary encoding of $n$ (least significant bit first,\u2026","title":"Literal loop of the clause check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/clauses-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.clauses_run","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#clauses-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"Suppose the lookup sub-program $lk$ with start label $l_0$ meets the lookup contract with constants $a$ and $b$. Let $es$ be a table with table stack $B_{es}$, and let $\\varphi$ be a CNF formula with clauses $D_1, \\dots, D_m$. The stack tokens of a clause are its clause marker followed by the stack tokens of its literals, last literal first, where the stack tokens of a literal $(n, p)$ are the literal marker of\u2026","title":"Clause loop of the clause check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Check.run","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"Suppose the lookup sub-program $lk$ with entry label $l_0$ satisfies the lookup contract with constants $a, b \\in \\mathbb{N}$. Let $\\varphi$ be a CNF formula, $es$ a list of table entries, $t$ the token list of $\\varphi$ and $m$ the length of the table stack of $es$. Start the driver with lookup $lk$ at its start label, with any internal state, with $t$ reversed on the token stack, the table stack of $es$ on the\u2026","title":"The clause check"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/k-2","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.K","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#k-2","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"The lookup machine has six stacks: the query stack $T$, the table stack $B$, the stash $S$, the two scratch stacks $A$ and $C$ used for comparisons, and the value stack $V$.","title":"Stacks of the lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/lab","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.Lab","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"The lookup machine has the labels $\\mathsf{init}$, $\\mathsf{scan}$, $\\mathsf{cmp}(b)$ and $\\mathsf{put}(b)$ for each bit $b$, $\\mathsf{rstT}$, $\\mathsf{rstB}$, $\\mathsf{mvN}$, $\\mathsf{back}$, $\\mathsf{mark}$ and $\\mathsf{done}$. The labels $\\mathsf{cmp}(b)$ and $\\mathsf{put}(b)$ serve a live table entry with value $b$.","title":"Labels of the lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/mv-sym","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.mvSym","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#mv-sym","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"For stacks $a$ and $b$, a symbol $x$ and a map $f$ from symbols to labels, this machine instruction pops the top of $a$ (if any), pushes $x$ onto $b$ and goes to the label $f(x)$.","title":"Moving a symbol between stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/mv-mark","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.mvMark","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#mv-mark","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"For stacks $a$ and $b$, a label $e$ and a map $f$ from symbols to labels, this machine instruction does the following: if the top of $a$ is a table marker $y$ (that is, $\\mathsf{dead}$ or $\\mathsf{val}(c)$ for a bit $c$), it pops $y$ from $a$, pushes it onto $b$ and goes to $f(y)$; if $a$ is empty or has another symbol on top, it goes to $e$ and leaves the stacks unchanged.","title":"Moving a table marker between stacks"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/scan-next","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.scanNext","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#scan-next","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"The label at which the lookup scan continues after moving a marker symbol $x$: the comparison label $\\mathsf{cmp}(b)$ if $x$ is the value marker $\\mathsf{val}(b)$, and the name-moving label $\\mathsf{mvN}$ for every other symbol.","title":"Next label of the lookup scan"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/prog-2","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.prog","parent":31037,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03/index.html#prog-2","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-03","status":"fully_proved","summary":"The lookup machine works on the stacks $T$, $B$, $S$, $A$, $C$ and $V$ and acts at each label as follows. At $\\mathsf{init}$ it pushes the bit symbol of false onto $V$ and goes to $\\mathsf{scan}$. At $\\mathsf{scan}$, if the top of $B$ is a table marker it moves that marker onto $S$ and goes to $\\mathsf{cmp}(b)$ if the marker is $\\mathsf{val}(b)$ and to $\\mathsf{mvN}$ if it is $\\mathsf{dead}$; if $B$ is empty or has\u2026","title":"The lookup program"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","kind":"scope","navigation":"transparent","parent":30994,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","title":"Start label of the lookup machine to Reading a unary count"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/start","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.start","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#start","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"The lookup program starts at its initialization label $\\mathsf{init}$.","title":"Start label of the lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/stk","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.stk","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#stk","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"For lists of symbols $t$, $b$, $s$, $a$, $c$, $v$, the assignment of contents to the six stacks of the lookup machine that gives $T$ the list $t$, $B$ the list $b$, $S$ the list $s$, $A$ the list $a$, $C$ the list $c$ and $V$ the list $v$.","title":"Stack contents of the lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/bit-bit","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.bit?_bit","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#bit-bit","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"For every bit $b$, the name-bit symbol for $b$ reads as the bit $b$.","title":"A name bit reads as itself"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/init-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.init_run","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#init-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Let $t$ and $b$ be lists of symbols and $v$ a bit. In exactly one step, the lookup program goes from the label $\\mathsf{init}$ with internal bit $v$, stack $T$ holding $t$, stack $B$ holding $b$ and the stacks $S$, $A$, $C$, $V$ empty, to the label $\\mathsf{scan}$ with internal bit $v$, the same contents of $T$, $B$, $S$, $A$ and $C$, and stack $V$ holding the single symbol name bit false.","title":"Initialization step of the lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/tail-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.tail_run","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#tail-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Let $q$, $n$, $c$ be lists of bits such that $c$ is a prefix of both the reverse of $q$ and the reverse of $n$; let $R$, $r$, $s$, $w$ be lists of symbols with $r$ empty or starting with a symbol other than a name bit; and for a list of bits $x$ write $\\bar x$ for the list of its bits as name bits. Suppose the lookup program is at label $\\mathsf{rstT}$, with any internal bit, and its stacks are: $T$ holds the\u2026","title":"Restoring and stashing after a comparison in the lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/cmp-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.cmp_run","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#cmp-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Write stack contents top first, and for a list of bits $u$ let $\\overrightarrow{u}$ be its bit symbols in order and $\\overleftarrow{u}$ the same symbols in reverse order. Let $q$ and $n$ be lists of bits, $b$ and $x$ bits, $s$ a list of symbols, and $R$, $r$ lists of symbols each empty or topped by a symbol that is not a bit. From label $\\mathsf{cmp}(b)$ with any internal state, $\\overleftarrow{q}R$ on $T$,\u2026","title":"Serving a live table entry"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/entry-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.entry_run","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#entry-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Write stack contents top first; for a list of bits $u$ let $\\overleftarrow{u}$ be its bit symbols in reverse order; for a list $E$ of table entries let $\\tau(E)$ be the symbols of its entries in reading order (each entry's name bits, then its marker) and $\\mathrm{val}_E(q)$ the value of the first live entry of $E$ named $q$, or false if there is none. Let $q$ be a list of bits, $e = (n, m)$ a table entry, $E$ a\u2026","title":"One scan step of the lookup"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/scan-loop","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.scan_loop","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#scan-loop","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Let $q$ be a list of bits, let $R$ be a list of symbols that is empty or starts with a symbol other than a name bit, and let $pre$ and $post$ be lists of table entries; the live value of $q$ in an entry list is the value of its first entry named $q$ that has a value, or false if there is none. Suppose the lookup program is at label $\\mathsf{scan}$, with any internal bit, and its stacks are: $T$ holds the bits of\u2026","title":"The scan loop of the lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/back-loop","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.back_loop","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#back-loop","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Let $t$ and $w$ be lists of symbols, let $pre$ and $post$ be lists of table entries, and let $m$ be the total number of symbols of the entries of $post$. Suppose the lookup program is at label $\\mathsf{back}$, with any internal bit, and its stacks are: $T$ holds $t$; $B$ holds the table stack of $pre$; $S$ holds the symbols of the entries of $post$ in reading order, first entry on top; $A$ and $C$ are empty; and\u2026","title":"The return loop of the lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Lookup.run","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Let $q$ be a list of bits, $es$ a list of table entries and $R$ a list of symbols that is empty or starts with a symbol other than a name bit. Suppose the stacks of the lookup machine are as follows: $T$ holds the bits of $q$ as name bits, last bit on top, above $R$; $B$ holds the table stack of $es$, of length $\\ell$; and $S$, $A$, $C$ and $V$ are empty. Then the lookup program, started at its start label with any\u2026","title":"Correctness of the SAT-verifier lookup machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/k","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.K","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#k","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"The SAT tokenizer has six stacks: $X$ for the formula bits, $W$ for the witness bits, $T$ for the token output, $B$ for the table output, and the clause counter $\\mathsf{CC}$ and the literal counter $\\mathsf{LC}$.","title":"Stacks of the SAT tokenizer"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/lab","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.Lab","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"The labels of the SAT tokenizer are: $\\mathsf{un}(c)$ for a bit $c$, which reads a unary count onto the clause counter $\\mathsf{CC}$ if $c$ is true and onto the literal counter $\\mathsf{LC}$ otherwise; $\\mathsf{cls}$, which starts a clause; $\\mathsf{lits}$, which starts a literal; $\\mathsf{nm}(\\ell)$, where $\\ell$ is a bit or none, which reads the bit pairs of a name block, $\\ell$ being the last name bit read (none\u2026","title":"Labels of the SAT tokenizer"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/top-is-bit","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.topIsBit","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#top-is-bit","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"For a bit $b$ and the result of peeking at a stack (a symbol, or nothing when the stack is empty), this is true exactly when the peeked symbol is the bit symbol carrying $b$; it is false for an empty stack and for every other symbol.","title":"Testing whether the top symbol is a given bit"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/unary","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.unary","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#unary","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"For a stack $k$ of the tokenizer and labels $s$ and $t$, this is the tokenizer instruction that looks at the top of the formula stack $X$. If it is the bit true, the instruction pops it, pushes the clause-end symbol $\\mathsf{cl}$ onto $k$ and goes to $s$; if it is the bit false, it pops it and goes to $t$; otherwise, when $X$ is empty or its top is not a bit, it goes to $\\mathsf{rej}$. The internal bit is set to\u2026","title":"Instruction for reading a unary count"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/prog","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.prog","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"The tokenizer is a multi-stack machine with stacks $X$ (formula bits), $W$ (witness bits), $T$ (tokens), $B$ (table) and the unary counters $CC$, $LC$ (over the unit symbol $\\mathsf{cl}$), and one bit of internal state. Its instruction at each label is:","title":"The tokenizer program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/start-2","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.start","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#start-2","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"The tokenizer starts at the label $\\mathrm{un}(1)$, which reads a unary count from the input stack $X$ onto the clause counter $CC$.","title":"Start label of the SAT tokenizer"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/st","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.st","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"For symbol lists $x, w, t, b$ and natural numbers $c, \\ell$, this assigns to the tokenizer's stacks $X$, $W$, $T$, $B$ the lists $x$, $w$, $t$, $b$, and to the counter stacks $CC$ and $LC$ the lists of $c$ and $\\ell$ copies of the clause-end symbol $\\mathrm{cl}$ (the counters in unary).","title":"Tokenizer stack contents"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/rej","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.Rej","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#rej","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"For a natural number $n$ and a configuration $c$ of the tokenizer, the tokenizer rejects from $c$ within $n$ steps when there are an internal bit $v$ and stacks $S$ such that it runs in at most $n$ steps from $c$ to the configuration with label $\\mathsf{rej}$, internal bit $v$ and stacks $S$.","title":"Rejection of the SAT tokenizer within a step bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/rej-one","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.rej_one","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#rej-one","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Let $\\ell$ be a label, $v$ a bit and $S$ an assignment of symbol stacks to the tokenizer's stacks. If executing the tokenizer's instruction at $\\ell$ from internal bit $v$ and stacks $S$ yields a configuration with label $\\mathsf{rej}$, then the tokenizer reaches $\\mathsf{rej}$ within one step from the configuration with label $\\ell$, internal bit $v$ and stacks $S$.","title":"One step to rejection"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/un-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.un_run","parent":31058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04/index.html#un-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-04","status":"fully_proved","summary":"Let $c$ be a bit, $n$ a natural number, $r$ a bit string, $w, t, b$ symbol lists and $m, \\ell$ counter values. Started at the label $\\mathrm{un}(c)$ with any value of its Boolean variable, with $X$ holding the bits of $1^n 0 r$, with $W$, $T$, $B$ holding $w$, $t$, $b$, and with the counters $CC$ and $LC$ holding $m$ and $\\ell$ in unary, the tokenizer takes exactly $n + 1$ steps to reach the configuration whose\u2026","title":"Reading a unary count"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","kind":"scope","navigation":"transparent","parent":30994,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","title":"An unterminated unary count is rejected to Stack alphabets of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/un-fail","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.un_fail","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#un-fail","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $r$ be a bit string on which reading a unary length prefix fails. For every bit $c$, symbol lists $w, t, b$, counter values $m, \\ell$ and value of the Boolean variable, the tokenizer started at the label $\\mathrm{un}(c)$ with $X$ holding the bits of $r$, with $W$, $T$, $B$ holding $w$, $t$, $b$, and with the counters $CC$ and $LC$ holding $m$ and $\\ell$ in unary reaches the rejecting label within $|r| + 1$ steps.","title":"An unterminated unary count is rejected"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/nm-end","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.nmEnd","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#nm-end","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"For an optional bit $\\ell$ and a bit string $u$, the name-loop outcome is the tokenizer label defined by recursion on $u$: for empty $u$ it is $\\mathsf{rej}$ if $\\ell$ is the bit false and $\\mathsf{pol}$ otherwise, and for $u = c\\,u'$ it is the outcome for the bit $c$ and the string $u'$. So it is $\\mathsf{rej}$ exactly when the string formed by $\\ell$ (if present) followed by $u$ is non-empty and ends in false,\u2026","title":"Name-loop outcome"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/nm-end-none","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.nmEnd_none","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#nm-end-none","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"For every bit string $u$, the name-loop outcome with no previous bit after $u$ is the label $\\mathsf{pol}$ if $u$ is a canonical name, that is, encoding the natural number that $u$ decodes to gives back $u$, and the label $\\mathsf{rej}$ otherwise.","title":"Name-loop outcome from the start of a name"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/nm-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.nm_run","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#nm-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $u$ and $r$ be bit strings, $\\ell$ an optional bit, $v$ a bit, $w$, $t$, $b$ symbol stacks and $c$, $l$ natural numbers. Started at label $\\mathsf{nm}(\\ell)$ with internal bit $v$, the bits of the self-delimiting block of $u$ followed by $r$ on the formula stack $X$, $w$ on the witness stack $W$, $t$ on the token stack $T$, $b$ on the table stack $B$, clause counter $c$ and literal counter $l$, the tokenizer\u2026","title":"Reading a name block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/nm-fail","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.nm_fail","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#nm-fail","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $r$ be a bit string from whose front no self-delimiting block can be parsed. For every optional last bit $\\ell$, internal bit $v$, symbol stacks $w$, $t$, $b$ and natural numbers $c$, $l$, the tokenizer started at label $\\mathsf{nm}(\\ell)$ with internal bit $v$, the bits of $r$ on the formula stack $X$, $w$ on the witness stack $W$, $t$ on the token stack $T$, $b$ on the table stack $B$, clause counter $c$ and\u2026","title":"Rejecting an unterminated name block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/pol-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.pol_run","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#pol-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $p$, $v$ be bits, $r$, $w$ bit strings, $t$, $b$ symbol stacks and $c$, $l$ natural numbers. Started at label $\\mathsf{pol}$ with internal bit $v$, the bit $p$ followed by the bits of $r$ on the formula stack $X$, the bits of $w$ on the witness stack $W$, $t$ on the token stack $T$, $b$ on the table stack $B$, clause counter $c$ and literal counter $l$, the tokenizer moves in exactly one step to label\u2026","title":"Reading a polarity"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/lit-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.lit_run","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#lit-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"A configuration $\\langle \\ell, v \\mid x, w, t, b, i, j\\rangle$ of the SAT tokenizer has label $\\ell$, internal bit $v$, stacks $X$, $W$, $T$, $B$ holding $x$, $w$, $t$, $b$ (top first; a bit string stands for its list of bit symbols), and $i$ and $j$ marks on the clause and literal counters; $\\mathbin{+\\!\\!+}$ is concatenation. Let $n, i, j \\in \\mathbb{N}$, let $p$ and $v$ be bits, $r$ and $w$ bit strings and $t$\u2026","title":"Tokenizer run on one literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/lit-fail","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.lit_fail","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#lit-fail","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"A configuration $\\langle \\ell, v \\mid x, w, t, b, i, j\\rangle$ of the SAT tokenizer has label $\\ell$, internal bit $v$, stacks $X$, $W$, $T$, $B$ holding $x$, $w$, $t$, $b$ (top first; a bit string stands for its list of bit symbols), and $i$ and $j$ marks on the clause and literal counters. Let $r$ be a bit string from whose front no literal can be decoded (a literal being a self-delimited canonical variable code\u2026","title":"Tokenizer rejects an undecodable literal"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/ents-of-cons","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.entsOf_cons","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#ents-of-cons","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $n$ be a natural number, $L$ a list of natural numbers and $w$ a list of bits. The occurrence table of the list $n$ followed by $L$, with $w$, is the entry whose name is the binary code of $n$ and whose value is the first bit of $w$ (no value if $w$ is empty), followed by the occurrence table of $L$ with $w$ minus its first bit (with the empty list if $w$ is empty).","title":"Occurrence table of a nonempty list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/lits-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.lits_run","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $C$ be a clause with literals $(n_1, p_1), \\dots, (n_k, p_k)$, let $u$ be the concatenation of their literal codes, and let $r$, $w$ be bit strings, $t$, $b$ stacks of symbols, $c$ a natural number and $v$ a bit. Started at label $\\mathsf{lits}$ with internal bit $v$, the bits of $u$ followed by $r$ on the formula stack $X$, the bits of $w$ on the witness stack $W$, $t$ on the token stack $T$, $b$ on the table\u2026","title":"Reading the literals of a clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/lits-fail","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.lits_fail","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#lits-fail","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"A configuration $\\langle \\ell, v \\mid x, w, t, b, i, j\\rangle$ of the SAT tokenizer has label $\\ell$, internal bit $v$, stacks $X$, $W$, $T$, $B$ holding $x$, $w$, $t$, $b$ (top first; a bit string stands for its list of bit symbols), and $i$ and $j$ marks on the clause and literal counters. Let $c, k \\in \\mathbb{N}$, let $r$ and $w$ be bit strings, $t$ and $b$ symbol lists and $v$ a bit, and suppose that decoding\u2026","title":"Tokenizer rejects undecodable literals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/clause-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.clause_run","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"A configuration $\\langle \\ell, v \\mid x, w, t, b, i, j\\rangle$ of the SAT tokenizer has label $\\ell$, internal bit $v$, stacks $X$, $W$, $T$, $B$ holding $x$, $w$, $t$, $b$ (top first; a bit string stands for its list of bit symbols), and $i$ and $j$ marks on the clause and literal counters; $\\mathbin{+\\!\\!+}$ is concatenation. Let $C$ be a clause, $r$ and $w$ bit strings, $t$ and $b$ symbol lists,\u2026","title":"Tokenizer run on one clause"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/clauses-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.clauses_run","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#clauses-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"A configuration $\\langle \\ell, v \\mid x, w, t, b, i, j\\rangle$ of the SAT tokenizer has label $\\ell$, internal bit $v$, stacks $X$, $W$, $T$, $B$ holding $x$, $w$, $t$, $b$ (top first; a bit string stands for its list of bit symbols), and $i$ and $j$ marks on the clause and literal counters; $\\mathbin{+\\!\\!+}$ is concatenation. Let $r$ and $w$ be bit strings, $\\varphi$ a CNF formula with $m$ clauses, $t$ and $b$\u2026","title":"Tokenizer run on all clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/clauses-fail","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.clauses_fail","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#clauses-fail","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"A configuration $\\langle \\ell, v \\mid x, w, t, b, i, j\\rangle$ of the SAT tokenizer has label $\\ell$, internal bit $v$, stacks $X$, $W$, $T$, $B$ holding $x$, $w$, $t$, $b$ (top first; a bit string stands for its list of bit symbols), and $i$ and $j$ marks on the clause and literal counters. Let $n \\in \\mathbb{N}$, let $r$ and $w$ be bit strings, $t$ and $b$ symbol lists and $v$ a bit, and suppose that decoding $n$\u2026","title":"Tokenizer rejects undecodable clauses"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/fin-acc","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.fin_acc","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#fin-acc","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $w$, $t$, $b$ be lists of symbols, $i, j \\in \\mathbb{N}$ and $v$ a bit. In exactly one step, the SAT tokenizer goes from the label $\\mathsf{fin}$ with internal bit $v$, empty formula stack $X$, stacks $W$, $T$, $B$ holding $w$, $t$, $b$, and $i$ and $j$ marks on the clause and literal counters, to the label $\\mathsf{acc}$ with internal bit false and all stacks unchanged.","title":"The tokenizer accepts when the formula stack is empty"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/run-some","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.run_some","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#run-some","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $x$ and $w$ be bit strings such that $x$ decodes to the CNF formula $\\varphi$, and let $S$ be contents of the tokenizer's stacks in which $X$ holds the bits of $x$, $W$ holds the bits of $w$, and the counters $CC$ and $LC$ are empty. Started at the start label with stacks $S$ and any value of its Boolean variable, the tokenizer reaches the accepting label within $2|x| + 2$ steps, with final stacks $S'$ such\u2026","title":"Accepting run of the SAT tokenizer"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/run-none","kind":"node","lean":"Cslib.Complexity.SatVerifier.Tokenizer.run_none","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#run-none","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Let $x$ and $w$ be bit strings such that $x$ does not decode as a CNF formula. For every bit $v$ and stacks $S$ with the bits of $x$ on the formula stack $X$, the bits of $w$ on the witness stack $W$, and both counter stacks $CC$ and $LC$ empty, the tokenizer started at its start label with internal bit $v$ and stacks $S$ reaches a configuration with label $\\mathsf{rej}$ within $2|x| + 2$ steps.","title":"Rejecting run"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/g","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.G","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#g","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"The alphabet $G$ of the host's input stack is the pair alphabet $\\mathrm{Bool} \\oplus \\mathrm{Option}\\ \\mathrm{Bool}$ of the dense SAT verifier.","title":"Input alphabet of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/hk","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.HK","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#hk","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"The SAT verifier host has seventeen stacks: the input stack, the output stack, and fifteen work stacks, namely two holding the formula bits and the certificate bits, two scratch stacks used to move them in phase 0, the token and table stacks, six further stacks of the clause-check driver, two further stacks of the tokenizer, and one further stack of the 3-clause filter.","title":"Stacks of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/h","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.H\u0393","parent":31079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05/index.html#h","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-05","status":"fully_proved","summary":"Each stack of the SAT verifier host has an alphabet: the input stack has the input alphabet $G$, the output stack has alphabet $\\mathrm{Bool}$, and every other stack has the token alphabet $\\mathrm{Sym}$.","title":"Stack alphabets of the SAT verifier host"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","kind":"scope","navigation":"transparent","parent":30994,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","title":"Phase-0 labels of the SAT verifier host to Program of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/p0","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.P0","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#p0","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The five labels of phase 0 of the SAT verifier host: $m_1$ and $m_2$ move the formula bits $x$ off the input (in two passes), $\\mathsf{sep}$ drops the separator, and $m_3$ and $m_4$ move the witness bits $w$ (in two passes).","title":"Phase-0 labels of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/hlab","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.HLab","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#hlab","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"A label of the SAT verifier host is one of the following: a phase-0 label; a label of the tokenizer; a label of the 3-clause filter; a label of the clause-check driver or of the lookup embedded in it; a cleanup label carrying an answer bit; or an output label carrying an answer bit.","title":"Labels of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/equiv","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.HLab.equiv","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#equiv","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"Write $P_0$ for the phase-0 labels, $\\Lambda_T$ for the tokenizer labels, $\\Lambda_F$ for the labels of the clause-length filter, and $\\Lambda_D$ and $\\Lambda_L$ for the labels of the clause-check driver and of the lookup. The host labels are in bijection with $P_0 \\sqcup \\Lambda_T \\sqcup \\Lambda_F \\sqcup (\\Lambda_D \\sqcup \\Lambda_L) \\sqcup \\mathrm{Bool} \\sqcup \\mathrm{Bool}$: a phase-0 label $p$, tokenizer label,\u2026","title":"Host labels as a sum type"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/e-t","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.eT","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#e-t","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The stacks of the tokenizer (the formula bits $X$, the witness bits $W$, the token output $T$, the table output $B$, and the clause and literal counters $CC$ and $LC$) are placed in the host machine by the map that sends each of them to the host stack of the same name.","title":"Placement of the tokenizer's stacks in the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/emb-t","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.embT","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#emb-t","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"Every stack of the tokenizer has alphabet $\\mathrm{Sym}$. The tokenizer is embedded in the host machine by sending each of its stacks $X$, $W$, $T$, $B$, $CC$, $LC$ to the host stack of the same name (an injective map) and identifying the alphabet of each with that of its image, which is again $\\mathrm{Sym}$, by the identity.","title":"Stack embedding of the tokenizer into the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/e-l","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.eL","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#e-l","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The stacks of the lookup (the query $T$, the table $B$, the stash $S$, the scratch stacks $A$ and $C$, and the value $V$) are placed in the clause-check driver by the map that sends each of them to the driver stack of the same name.","title":"Placement of the lookup's stacks in the clause-check driver"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/emb-l","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.embL","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#emb-l","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"Every stack of the lookup and of the clause-check driver has alphabet $\\mathrm{Sym}$. The lookup is embedded in the driver by sending each of its stacks $T$, $B$, $S$, $A$, $C$, $V$ to the driver stack of the same name (an injective map) and identifying the alphabets by the identity of $\\mathrm{Sym}$.","title":"Stack embedding of the lookup into the clause-check driver"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/l","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.\u03c6L","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#l","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The relabelling that places the lookup's labels among the driver's labels, whose label set consists of the driver's own labels together with the lookup's labels: the lookup's final label $\\mathsf{done}$ goes to the driver's return label $\\mathsf{ret}$, and every other lookup label $l$ goes to the copy of $l$ among the lookup's labels.","title":"Lookup labels in the clause-check driver"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/lk-h","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.lkH","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#lk-h","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The lookup sub-program of the clause-check driver: at each lookup label $l$, it is the lookup program's statement at $l$ with every stack operation moved to the driver's stack of the same name (the lookup's stacks $T$, $B$, $S$, $A$, $C$, $V$) and every jump target relabelled by the lookup-label map, which sends $\\mathsf{done}$ to the driver's return label $\\mathsf{ret}$ and keeps every other label.","title":"The embedded lookup program"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/e-c","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.eC","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#e-c","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The map from the stacks of the clause-check driver into the host stacks sends each driver stack to the host stack of the same name.","title":"Driver stacks in the SAT host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/emb-c","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.embC","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#emb-c","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"Every stack of the clause-check driver has alphabet $\\mathrm{Sym}$. The driver is embedded in the host machine by sending each of its stacks $T$, $B$, $S$, $A$, $C$, $V$, $P$, $F$ to the host stack of the same name (an injective map) and identifying the alphabet of each with that of its image, which is again $\\mathrm{Sym}$, by the identity.","title":"Stack embedding of the clause-check driver into the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/e-f","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.eF","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#e-f","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The two stacks of the clause-length filter, the token stack $T$ and the side stack $U$, are placed in the host machine by the map that sends $T$ to the host stack $T$ and $U$ to the host stack $U$.","title":"Placement of the filter's stacks in the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/emb-f","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.embF","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#emb-f","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"Both stacks of the clause-length filter have alphabet $\\mathrm{Sym}$. The filter is embedded in the host machine by sending its stacks $T$ and $U$ to the host stacks of the same names (an injective map) and identifying the alphabet of each with that of its image, which is again $\\mathrm{Sym}$, by the identity.","title":"Stack embedding of the clause-length filter into the host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/f","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.\u03c6F","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#f","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The relabelling of the 3-clause filter's labels into the labels of the SAT verifier host sends the filter's exit label *pass* to the start label of the clause-check driver, its exit label *fail* to the cleanup label with answer false, and every other filter label $l$ to the host's copy of $l$ among its filter labels.","title":"Filter exits in the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/t","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.\u03c6T","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#t","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The map from labels of the tokenizer to labels of the host SAT verifier sends the tokenizer's accepting label to the start label of the embedded clause-length filter, its rejecting label to the cleanup label with answer false, and every other label $l$ to the host label that runs the tokenizer at $l$.","title":"Tokenizer labels in the host machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/c","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.\u03c6C","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#c","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The map from the labels of the clause-check driver (its own labels and those of the lookup embedded in it) to the host labels sends the driver's label for the answer yes to the cleanup label with answer true, its label for the answer no to the cleanup label with answer false, and every other label $l$ to the host label running the driver at $l$.","title":"Driver labels in the SAT host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/d-x","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.dX","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#d-x","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The input alphabet of the SAT host consists of the formula bits, the separator and the witness bits. This partial map sends a formula bit $b$ to $b$ and is undefined on the separator and on witness bits.","title":"Reading a formula bit in the SAT host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/d-w","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.dW","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#d-w","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The input alphabet of the SAT host consists of the formula bits, the separator and the witness bits. This partial map sends a witness bit $b$ to $b$ and is undefined on formula bits and on the separator.","title":"Reading a witness bit in the SAT host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/work-ks","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.workKs","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#work-ks","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"The cleanup loop of the host SAT verifier empties the list of stacks made of the input stack followed by the work stacks $X$, $W$, $T$, $B$, $S$, $A$, $C$, $V$, $P$, $F$, $CC$, $LC$, $X1$, $W1$, $U$, in that order: every host stack except the output stack.","title":"Stacks emptied by the SAT verifier's cleanup"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/hprog","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.hprog","parent":31100,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06/index.html#hprog","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-06","status":"fully_proved","summary":"For a value $c$ of the 3SAT flag, the host program acts at each host label as follows. Here moving a bit block from one stack to another means repeatedly popping the top symbol of the first stack while it is a bit and pushing it, as a bit token, onto the second, and continuing at the next label once the top is not a bit or the stack is empty.","title":"Program of the SAT verifier host"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","kind":"scope","navigation":"transparent","parent":30994,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","title":"The SAT verifier machine to SAT is in NP"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/tm","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.tm","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#tm","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For a flag $\\mathit{cap}$, the host machine is the finite multi-stack Turing machine whose stacks are the host stacks, with the input stack as input (its alphabet is that of the pair encoding: tagged formula bits, a separator and tagged witness bits), the output stack (alphabet the booleans) as output, the host alphabets on all stacks, the host labels as labels, the first phase-0 label as start label, a boolean\u2026","title":"The SAT verifier machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/phase0-run","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.phase0_run","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#phase0-run","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"Let $c$ be a flag, $x$ and $w$ bit strings of lengths $m$ and $n$, $v$ an internal state, and $S$ host stacks whose input stack holds the pair encoding $x \\# w$ and whose other stacks are all empty. Then there are an internal state $v'$ and host stacks $S'$ such that the host program with flag $c$ runs in exactly $2m + 2n + 5$ steps from the configuration at the phase-0 label $m_1$ with internal state $v$ and\u2026","title":"Phase 0 of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/embeds-tok","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.embeds_tok","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#embeds-tok","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For every flag $c$, the tokenizer program embeds in the host program with flag $c$ along the stack embedding of the tokenizer into the host and the map of tokenizer labels into host labels, which sends the tokenizer's accepting label to the start label of the clause-length filter, its rejecting label to the cleanup label with answer false, and every other label $\\ell$ to the host label running the tokenizer at\u2026","title":"The tokenizer embeds in the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/lookup-spec","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.lookupSpec","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#lookup-spec","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"The clause-check driver, with the embedded lookup program as its lookup sub-program and the lookup's start label as the label it calls, meets the driver's lookup contract with constants $a = 8$ and $b = 4$: from the lookup's start label, with a reversed name block $q$ on the token stack $T$ above a rest that does not begin with a name bit, the table stack of an entry list $es$ on $B$, and the stacks $S$, $A$, $C$,\u2026","title":"The embedded lookup meets the lookup contract"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/work-ks-nodup","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.workKs_nodup","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#work-ks-nodup","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"The list of stacks emptied by the cleanup of the SAT verifier host has no repeated entries.","title":"The cleanup stacks are distinct"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/clean-host","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.clean_host","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#clean-host","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"Let $m$ be a natural number and let $S$ assign contents to the host stacks so that every stack has length at most $m$ and the output stack is empty. Then for every flag $c$, answer bit $a$ and internal state $v$, the host program with flag $c$ goes in at most $m + 1$ steps from the cleanup label with answer $a$, internal state $v$ and stacks $S$ to the final label with answer $a$, internal state false and every\u2026","title":"Cleanup of the SAT host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/ents-of-map-length","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.entsOf_map_length","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#ents-of-map-length","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For every list $n_0, \\dots, n_{k-1}$ of natural numbers and every list $w$ of bits, the names of the entries of the occurrence table of $n_0, \\dots, n_{k-1}$ and $w$, in order, have the lengths of the binary codes of $n_0, \\dots, n_{k-1}$ (Mathlib's encoding of natural numbers as bit lists).","title":"Name lengths in an occurrence table"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/init-s","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.initS","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#init-s","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For a word $l$ over the input alphabet $G$, the initial host stacks on $l$ hold $l$ on the input stack and leave every other stack empty.","title":"Initial stacks of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/tok-s","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.tokS","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#tok-s","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For bit lists $x$ and $w$, these tokenizer stack contents put the bits of $x$ on the formula stack $X$ and the bits of $w$ on the witness stack $W$ (each bit as a bit symbol), and leave every other tokenizer stack (token, table and the two counters) empty.","title":"Tokenizer stacks after phase 0"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/chk-s","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.chkS","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#chk-s","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and a witness $w$, these are the contents of the clause-check driver's stacks once the tokenizer has accepted $\\varphi$: the token stack holds the token list of $\\varphi$ reversed, the table stack holds the table-stack encoding of the occurrence table of $\\varphi$ and $w$, and every other stack is empty.","title":"Driver stacks after tokenization"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/pre-steps","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.preSteps","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#pre-steps","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For an input length $N$, the number $(2N + 5) + (2N + 2) + (2N + 2) + (N(8N + 9) + 1)$.","title":"Step budget before the cleanup"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/finish","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.finish","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#finish","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"Fix either value of the 3SAT flag, an answer bit $b$, a natural number $m$, an internal state $v$, and host stacks $S$ such that every stack of $S$ has length at most $m$ and the output stack of $S$ is empty. Then for some $n \\le m + 2$, the host program runs in exactly $n$ steps from the configuration at the cleanup label with answer $b$, internal state $v$ and stacks $S$ to the halted configuration with internal\u2026","title":"Cleanup phase of the SAT verifier host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/ans","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.ans","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#ans","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For a flag $c$ and a pair $(x, w)$ of bit strings, the host's answer is false if $x$ does not decode to a CNF formula. If $x$ decodes to a formula $\\varphi$, give each variable $n$ the value $w_i$ (counting from $0$), where $i$ is the first position of $n$ in the list of variable occurrences of $\\varphi$ read clause by clause, and the value false if $i \\ge |w|$. Then the answer is true exactly when both (a) $c$ is\u2026","title":"Answer of the SAT host"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/total-steps","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.totalSteps","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#total-steps","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For a flag $\\mathit{cap}$ and an input length $N$, let $s(N) = (2N + 5) + (2N + 2) + (2N + 2) + (N(8N + 9) + 1)$ be the number of steps allowed before the cleanup, and let $p$ be the per-step push budget of the host machine with that flag. The total step bound is $s(N) + (N + p \\cdot s(N)) + 2$.","title":"Explicit step bound of the SAT verifier machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/init-list-tm","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.initList_tm","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#init-list-tm","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For every flag $c$ and every word $l$ over the input alphabet $G$ of the host, the initial configuration of the SAT verifier machine with flag $c$ on $l$ (its start label, its initial internal state, $l$ on the input stack and every other stack empty) is the configuration at the phase-0 label $m_1$ with internal state false and the initial host stacks on $l$.","title":"Initial configuration of the SAT verifier machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/time","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.time","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#time","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For a flag $\\mathit{cap}$, let $p$ be the per-step push budget of the host machine with that flag (the sum over its labels of the number of push instructions in each label's statement). The step bound is the polynomial $(p + 1)(8X^2 + 15X + 10) + X + 2$ with natural-number coefficients, in the input length $X$.","title":"Step bound of the SAT verifier machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/verifier","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.verifier","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#verifier","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For a flag $\\mathit{cap}$, the host machine with that flag computes the host's answer in polynomial time. For every pair $(x, w)$ of bit lists, started on the input $x \\# w$ (the bits of $x$, a separator, then the bits of $w$), it halts with the single bit $\\mathrm{ans}_{\\mathit{cap}}(x, w)$ on its output stack and every other stack empty within $T(n)$ steps, where $n$ is the length of $x \\# w$ and $T$ is the\u2026","title":"The `TM2` SAT verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/is-poly-time-ans","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.isPolyTime_ans","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#is-poly-time-ans","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"For each value of the flag $\\mathit{cap}$ (with or without the three-literal clause filter), the host's answer, as a function from pairs $(x, w)$ of bit strings to Booleans, is polynomial-time computable under the canonical bitstring encodings.","title":"The SAT host's answer is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/is-poly-time-sat-verifier","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.isPolyTime_satVerifier","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#is-poly-time-sat-verifier","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"The function that maps a pair $(x, w)$ of bit strings to the Boolean answer of the SAT verifier on $(x, w)$ (its return value, ignoring its declared time cost) is polynomial-time computable under the canonical bitstring encodings.","title":"The SAT verifier is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/sat-dec-mem-np","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.satDec_mem_NP","parent":31121,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07/index.html#sat-dec-mem-np","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-07","status":"fully_proved","summary":"The decision problem SAT, which accepts exactly the bit strings that encode a satisfiable CNF formula, belongs to the class NP.","title":"SAT is in NP"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08.json","id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08","kind":"scope","navigation":"transparent","parent":30994,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08","status":"fully_proved","title":"Cook\u2013Levin theorem to Finiteness of the SAT verifier alphabet (moved alias)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/cook-levin-sat-np-complete","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.cook_levin_SAT_np_complete","parent":31142,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/index.html#cook-levin-sat-np-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08","status":"fully_proved","summary":"CNF satisfiability is NP-complete: the decision problem accepting exactly the bit strings that encode a satisfiable CNF formula is in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"Cook\u2013Levin theorem"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/ans-true","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.ans_true","parent":31142,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/index.html#ans-true","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08","status":"fully_proved","summary":"For all bit strings $x$ and $w$, the answer of the SAT host with flag true on $(x, w)$ is the conjunction of two bits. The first is false if the complete length-prefixed CNF decoder fails on $x$; if $x$ decodes to a CNF formula $\\varphi$, it is true exactly when every clause of $\\varphi$ has at most three literals. The second is the return value of the SAT verifier on $(x, w)$.","title":"Answer of the SAT host with the three-literal filter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/sat3-dec-mem-np","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.sat3Dec_mem_NP","parent":31142,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/index.html#sat3-dec-mem-np","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08","status":"fully_proved","summary":"The decision problem 3SAT, which accepts exactly the bit strings that encode a satisfiable CNF formula in which every clause has at most three literals, belongs to the class NP.","title":"3SAT is in NP"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/toks-length-le","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.toks_length_le","parent":31142,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/index.html#toks-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08","status":"fully_proved","summary":"Moved to `SatVerifier.toks_length_le` (`SatVerifierSpec`).","title":"Token length bound for the SAT verifier (moved alias)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/clause-toks-length-le","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.clauseToks_length_le","parent":31142,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/index.html#clause-toks-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08","status":"fully_proved","summary":"Moved to `SatVerifier.clauseToks_length_le` (`SatVerifierSpec`).","title":"Clause token length bound for the SAT verifier (moved alias)"},{"id":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/inst-fintype","kind":"node","lean":"Cslib.Complexity.SatVerifier.Machine.Sym.instFintype","parent":31142,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08/index.html#inst-fintype","scope":"TheoryOfComputation/np-completeness/cook-levin/polynomial-time-verifier-for-sat/part-08","status":"fully_proved","summary":"Moved to the instance `SatVerifier.Sym.instFintype` (`SatVerifierSpec`); kept as a definition, not an instance, so instance search sees one `Fintype Sym`.","title":"Finiteness of the SAT verifier alphabet (moved alias)"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings","kind":"scope","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings","status":"fully_proved","title":"Conversions between bit-string encodings"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalize-nat-run-frame","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.normalizeNat_run_frame","parent":31149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/index.html#normalize-nat-run-frame","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings","status":"fully_proved","summary":"The canonical normalizer restores both work stacks and prepends its answer to an arbitrary output suffix, within a linear bit bound.","title":"Number normalizer restores its work stacks and prepends its answer"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block","status":"fully_proved","title":"Reading a framed bit block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block/read","kind":"node","lean":"Cslib.Complexity.FramedCNF.Block.read","parent":31151,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block/index.html#read","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block","status":"fully_proved","summary":"Consume one framed-payload pair or its terminator.","title":"Machine step reading one framed payload pair"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block/run-delimit","kind":"node","lean":"Cslib.Complexity.FramedCNF.Block.run_delimit","parent":31151,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block/index.html#run-delimit","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block","status":"fully_proved","summary":"Read a complete block, preserving an arbitrary suffix and all stacks other than the input and reversed-payload destination.","title":"Reading a complete framed block"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block/run-none","kind":"node","lean":"Cslib.Complexity.FramedCNF.Block.run_none","parent":31151,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block/index.html#run-none","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reading-a-framed-bit-block","status":"fully_proved","summary":"Truncated framing reaches the failure branch within input length plus one, without assuming that the input is an encoder image. The destination may contain a partial payload on failure; the caller must discard or clear it before using that stack again.","title":"Framed block reader fails on truncated framing"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco","status":"fully_proved","title":"Normalizing the total binary natural-number decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/normalize-nat","kind":"node","lean":"Cslib.Complexity.FramedCNF.normalizeNat","parent":31155,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/index.html#normalize-nat","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco","status":"fully_proved","summary":"Canonicalize a raw payload without enumerating its decoded value.","title":"Normalization of a raw binary natural-number payload"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/normalize-nat-eq-encode-decode","kind":"node","lean":"Cslib.Complexity.FramedCNF.normalizeNat_eq_encode_decode","parent":31155,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/index.html#normalize-nat-eq-encode-decode","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco","status":"fully_proved","summary":"Normalization agrees with the existing total decoder on every payload, not just canonical encodings.","title":"Payload normalization agrees with decode-then-encode"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/length-normalize-nat-le","kind":"node","lean":"Cslib.Complexity.FramedCNF.length_normalizeNat_le","parent":31155,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/index.html#length-normalize-nat-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco","status":"fully_proved","summary":"Normalization adds at most one bit.","title":"Payload normalization adds at most one bit"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/normalize-nat-encode-nat","kind":"node","lean":"Cslib.Complexity.FramedCNF.normalizeNat_encodeNat","parent":31155,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco/index.html#normalize-nat-encode-nat","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/normalizing-the-total-binary-natural-number-deco","status":"fully_proved","summary":"Canonical encodings are unchanged.","title":"Payload normalization fixes canonical encodings"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","title":"A linear-time machine for total binary-payload normalization"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/k","kind":"node","lean":"Cslib.Complexity.FramedCNF.NormalizeMachine.K","parent":31160,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html#k","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","summary":"Input, reverse buffer, and output.","title":"Stacks of the payload normalizer machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/lab","kind":"node","lean":"Cslib.Complexity.FramedCNF.NormalizeMachine.Lab","parent":31160,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","summary":"The reader records the last payload bit; `none` means empty input.","title":"Control labels of the payload normalizer machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/prog","kind":"node","lean":"Cslib.Complexity.FramedCNF.NormalizeMachine.prog","parent":31160,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","summary":"Normalization using only Boolean stack symbols and finite control.","title":"Program of the payload normalizer machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/st","kind":"node","lean":"Cslib.Complexity.FramedCNF.NormalizeMachine.st","parent":31160,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","summary":"The three stack contents used in the run invariant.","title":"Stack contents for the payload normalizer invariant"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/read-run","kind":"node","lean":"Cslib.Complexity.FramedCNF.NormalizeMachine.read_run","parent":31160,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html#read-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","summary":"The read phase consumes each input bit once and optionally adds the implicit terminal one. The prefix records already-consumed input.","title":"Read phase of the payload normalizer consumes each bit once"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/run","kind":"node","lean":"Cslib.Complexity.FramedCNF.NormalizeMachine.run","parent":31160,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","summary":"The whole component runs in at most `2 |x| + 3` steps.","title":"Linear step bound for the payload normalizer machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/component","kind":"node","lean":"Cslib.Complexity.FramedCNF.NormalizeMachine.component","parent":31160,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html#component","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","summary":"The normalizer in the canonical component interface.","title":"Payload normalizer as a host component"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/is-poly-time-normalize-nat","kind":"node","lean":"Cslib.Complexity.FramedCNF.NormalizeMachine.isPolyTime_normalizeNat","parent":31160,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n/index.html#is-poly-time-normalize-nat","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/a-linear-time-machine-for-total-binary-payload-n","status":"fully_proved","summary":"Total binary-payload normalization is polynomial-time in the canonical encoded machine model.","title":"Payload normalization is polynomial-time"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","title":"From the canonical pair encoding to the w # y form"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/enc-fst","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.encFst","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#enc-fst","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"For a bit list $x$, the bit list obtained by replacing every bit $b$ of $x$, in order, with the six bits true, true, true, $b$, true, false.","title":"Converter encoding of the first component"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/enc-snd","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.encSnd","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#enc-snd","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"For a bit list $w$, the bit list obtained by replacing every bit $b$ of $w$, in order, with the three bits true, $b$, false.","title":"Converter encoding of the second component"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/bit-encode-pair-eq","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.bitEncode_pair_eq","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#bit-encode-pair-eq","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"For bit lists $x$ and $w$, the canonical bitstring encoding of the pair $(x, w)$ is the concatenation of three parts: every bit $b$ of $x$, in order, written as the six bits true, true, true, $b$, true, false; then a single false; then every bit $b$ of $w$, in order, written as the three bits true, $b$, false.","title":"Canonical encoding of a pair of bit lists"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/stk","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.Stk","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#stk","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"The machine that rewrites the canonical encoding of a pair of bit lists into the verifier's input form has four stacks: the input stack, the output stack, and two stash stacks, one for each component of the pair.","title":"Stacks of the pair-encoding converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/untitled","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.\u0393","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#untitled","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"The output stack of the converter has as its alphabet the verifier's work alphabet $G = \\{0,1\\} \\sqcup (\\{0,1\\} \\cup \\{\\#\\})$; the input stack and the two stash stacks have the bits $\\{0,1\\}$ as their alphabet.","title":"Stack alphabets of the pair-encoding converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/lab","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.Lab","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"The five labels of the converter program: read the first component, read the second component, emit the second component, emit the separator, and emit the first component.","title":"Labels of the pair-encoding converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/st","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.St","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"The internal state of the converter is an optional bit: either a bit or nothing. It records the last bit the program stored.","title":"State of the pair-encoding converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/prog","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.prog","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"The program over the converter's four stacks, five labels and states (a bit or nothing). Each pop either stores the popped bit in the state (nothing if the stack is empty) or leaves the state unchanged, and each push of the stored bit uses false when the state holds nothing.","title":"Program of the pair-encoding converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/tm","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.tm","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#tm","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"The converter is the finite multi-stack Turing machine whose stacks are the input stack, the output stack and the two stash stacks, with bits as the alphabet of every stack except the output stack, which holds the verifier's symbols. Its labels are the five phases (read the first component, read the second, emit the second, emit the separator, emit the first), and it starts in the phase that reads the first\u2026","title":"The pair-encoding converter machine"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/stk-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.stk","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#stk-2","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"For bit lists $i$, $a$, $b$ and a list $o$ of verifier symbols, the contents of the converter's four stacks with $i$ on the input stack, $o$ on the output stack, $a$ on the first stash and $b$ on the second stash.","title":"Stack contents of the pair-encoding converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/run","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.Run","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and configurations $c$ and $d$ of the converter, the statement that $n$ steps of the converter program lead from $c$ to $d$.","title":"Runs of the pair-encoding converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/convert-run","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.convert_run","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#convert-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"For bit lists $x$ and $w$, the converter program goes in exactly $2|x| + 2|w| + 5$ steps from the configuration at the label *read first*, with no bit in the state, the input stack holding every bit $b$ of $x$ in order as the six bits true, true, true, $b$, true, false, then a single false, then every bit $b$ of $w$ in order as the three bits true, $b$, false, and the other three stacks empty, to the halted\u2026","title":"Full run of the pair-encoding converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/computable","kind":"node","lean":"Cslib.Complexity.DenseSAT.Converter.computable","parent":31169,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form/index.html#computable","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/from-the-canonical-pair-encoding-to-the-w-y-form","status":"fully_proved","summary":"A polynomial-time computation of the identity map on pairs $(x, w)$ of bit lists, from the canonical bitstring encoding of $(x, w)$ to the verifier's pair encoding of $(x, w)$ (the bits of $x$, the separator, then the bits of $w$), by the converter machine, with the bits and the verifier's symbols themselves as input and output alphabets and time bound $n + 5$: started on the canonical encoding of $(x, w)$, a list\u2026","title":"Polynomial-time conversion to the verifier's pair encoding"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","title":"Encoding converters for the Cook-Levin reduction"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/stk","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.Stk","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#stk","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The converter has four stacks: the input stack, whose symbols are left-tagged bits, right-tagged bits and the separator (the input is the bits of a first bit list tagged left, a separator, then the bits of a second bit list tagged right); the output stack, holding bits; and two stashes, holding the bits of the first and of the second list.","title":"Stacks of the converter to the canonical pair encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/untitled","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.\u0393","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#untitled","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The input stack of the converter from the separator form of a pair to its canonical encoding has the verifier's alphabet, and each of its other three stacks (the output and the two stashes) has the alphabet of bits.","title":"Stack alphabets of the separator-to-canonical converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/lab","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.Lab","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The machine that turns the separated form of a pair $(x, w)$ of bit lists (the bits of $x$ tagged left, a separator, then the bits of $w$ tagged right) into the canonical encoding of the pair has five labels: reading the first part, reading the second part, emitting the second part, writing the separator, and emitting the first part.","title":"Labels of the converter to the canonical pair encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/st","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.St","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"A state of the converter is either empty or one symbol of the verifier's input alphabet, that is, a left-tagged bit, a right-tagged bit or the separator.","title":"States of the converter to the canonical pair encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/is-fst","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.isFst","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#is-fst","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"For a state $s$ of the pair-to-canonical converter (either empty or a symbol of the verifier alphabet), the test is true exactly when $s$ is a left-tagged bit $\\mathrm{inl}\\,b$ for some bit $b$, and false when $s$ is empty, the separator $\\mathrm{inr}(\\mathrm{none})$ or a right-tagged bit.","title":"Left-tagged state test"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/is-snd","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.isSnd","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#is-snd","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"For a state $s$ of the pair-to-canonical converter (either empty or a symbol of the verifier alphabet), the test is true exactly when $s$ is a right-tagged bit $\\mathrm{inr}(\\mathrm{some}\\,b)$ for some bit $b$, and false when $s$ is empty, the separator $\\mathrm{inr}(\\mathrm{none})$ or a left-tagged bit.","title":"Right-tagged state test"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/bit","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.bit","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#bit","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"For a state $s$ of the pair-to-canonical converter (either empty or a symbol of the verifier alphabet), the bit of $s$ is $b$ when $s$ is a left-tagged bit $\\mathrm{inl}\\,b$ or a right-tagged bit $\\mathrm{inr}(\\mathrm{some}\\,b)$, and $\\mathrm{false}$ when $s$ is empty or the separator $\\mathrm{inr}(\\mathrm{none})$.","title":"Bit carried by a converter state"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/prog","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.prog","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The machine has an input stack of symbols, each a left-tagged bit, a right-tagged bit or the separator, and three stacks of bits: the output and two stashes. Its program first pops input symbols and pushes the bit of each left-tagged one onto the first stash, until the input is empty or the symbol popped is not a left-tagged bit. It then pops input symbols and pushes the bit of each right-tagged one onto the second\u2026","title":"Program of the converter to the canonical pair encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/tm","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.tm","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#tm","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The multi-stack Turing machine with the four stacks of the converter to the canonical pair encoding (input, output and two stashes), whose input stack holds symbols of the verifier's alphabet (left-tagged bits, right-tagged bits and the separator) and whose other stacks hold bits, reading its input from the input stack and leaving its output on the output stack, with the five labels of that converter and start\u2026","title":"Machine converting the separator form of a pair to its canonical encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/stk-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.stk","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#stk-2","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"Given a list $i$ of input symbols and bit lists $o$, $a$ and $b$, this is the contents of the four stacks of the converter with $i$ on the input stack, $o$ on the output stack, $a$ on the first stash and $b$ on the second stash.","title":"Stack contents of the converter to the canonical pair encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/run","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.Run","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"For a natural number $n$ and configurations $c$ and $d$ of the converter, this is the proposition that $n$ steps of its program lead from $c$ to $d$: applying the step function $n$ times to $c$ is defined at every step (no configuration before the last one is halted) and gives $d$.","title":"Runs of the converter to the canonical pair encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/convert-run","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.convert_run","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#convert-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"Let $x$ and $w$ be bit lists, write $x \\# w$ for the verifier input of the pair $(x, w)$, and read $1$ for true and $0$ for false. Let $\\mathrm{encFst}(x)$ replace each bit $b$ of $x$ by $1, 1, 1, b, 1, 0$ and $\\mathrm{encSnd}(w)$ replace each bit $b$ of $w$ by $1, b, 0$. The program of the pair-to-canonical converter goes in exactly $2|x| + 2|w| + 5$ steps from the configuration at label $\\mathrm{readFst}$ with\u2026","title":"Run of the pair-to-canonical converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/computable","kind":"node","lean":"Cslib.Complexity.DenseSAT.ToCanonical.computable","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#computable","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"This is a multi-stack Turing machine over the alphabet $\\mathrm{Bool}$, with time bound $n + 2$, such that for every bit list $x$, started on the canonical bitstring encoding of $x$, it halts within $n + 2$ steps, where $n$ is the length of that encoding, with $x$ on its output stack and all other stacks empty. So the identity on bit lists is computable in polynomial time from the canonical encoding to the raw bits.","title":"Canonical bit-list decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/bstk","kind":"node","lean":"Cslib.Complexity.DenseSAT.BStk","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#bstk","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The two bit-list converters use three stacks: the input stack, the output stack and one stash.","title":"Stacks of the bit-list converters"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/blab","kind":"node","lean":"Cslib.Complexity.DenseSAT.BLab","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#blab","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The two bit-list converters have two program labels, $\\mathrm{read}$ and $\\mathrm{emit}$.","title":"Labels of the bit-list converters"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/bstk-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.bstk","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#bstk-2","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"For bit lists $i$, $o$ and $a$, this assigns contents to the three stacks of the bit-list converters: $i$ to the input stack, $o$ to the output stack and $a$ to the stash. Every stack alphabet is $\\mathrm{Bool}$.","title":"Stack contents of the bit-list converters"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/prog-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.DecodeBits.prog","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#prog-2","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The program of the machine that decodes the canonical encoding of a bit list acts on the three stacks of the bit-list converters, all over bits, with an optional bit as internal state. At label $\\mathrm{read}$ it pops a symbol from the input stack into the state (which becomes empty if the input is empty). If that symbol is true, it pops a second symbol $b$ into the state, pops and discards a third, pushes $b$ onto\u2026","title":"Program of the bit-list decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/tm-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.DecodeBits.tm","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#tm-2","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The multi-stack Turing machine with the three stacks of the bit-list converters (input, output and a stash), all over bits, reading its input from the input stack and leaving its output on the output stack, with the two labels $\\mathrm{read}$ and $\\mathrm{emit}$ and start label $\\mathrm{read}$, an optional bit as internal state, initially empty, and the program of the bit-list decoder. It repeatedly pops an input\u2026","title":"Machine decoding the canonical encoding of a bit list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/run-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.DecodeBits.Run","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#run-2","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"For a natural number $n$ and configurations $c$ and $d$ of the machine decoding the canonical encoding of a bit list (each an optional label, an optional bit as internal state and the contents of the three bit stacks), this is the proposition that $n$ steps of its program lead from $c$ to $d$: applying the step function $n$ times to $c$ is defined at every step (no configuration before the last one is halted) and\u2026","title":"Runs of the bit-list decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/computable-2","kind":"node","lean":"Cslib.Complexity.DenseSAT.DecodeBits.computable","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#computable-2","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"This is a multi-stack Turing machine whose input and output alphabets are the bits, together with the polynomial time bound $t(n) = n + 2$, such that for every bit list $x$, whose canonical bitstring encoding has length $n$, the machine started with that encoding on its input stack and all other stacks empty halts within $t(n)$ steps with $x$ on its output stack, all other stacks empty and its internal state back\u2026","title":"Polynomial-time decoder of the canonical bit-list encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/prog-3","kind":"node","lean":"Cslib.Complexity.DenseSAT.EncodeBits.prog","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#prog-3","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The program of the machine that writes the canonical encoding of a bit list acts on the three stacks of the bit-list converters, all over bits, with an optional bit as internal state. At label $\\mathrm{read}$ it pops a bit from the input stack, pushes it onto the stash and returns to $\\mathrm{read}$; when the input is empty it goes to $\\mathrm{emit}$. At label $\\mathrm{emit}$ it pops a bit $b$ from the stash,\u2026","title":"Program of the bit-list encoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/tm-3","kind":"node","lean":"Cslib.Complexity.DenseSAT.EncodeBits.tm","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#tm-3","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"The multi-stack Turing machine with the three stacks of the bit-list converters (input, output and a stash), all over bits, reading its input from the input stack and leaving its output on the output stack, with the two labels $\\mathrm{read}$ and $\\mathrm{emit}$ and start label $\\mathrm{read}$, an optional bit as internal state, initially empty, and the program of the bit-list encoder. It moves the input, bit by\u2026","title":"Machine writing the canonical encoding of a bit list"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/run-3","kind":"node","lean":"Cslib.Complexity.DenseSAT.EncodeBits.Run","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#run-3","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"For a natural number $n$ and configurations $c$ and $d$ of the machine writing the canonical encoding of a bit list (each an optional label, an optional bit as internal state and the contents of the three bit stacks), this is the proposition that $n$ steps of its program lead from $c$ to $d$: applying the step function $n$ times to $c$ is defined at every step (no configuration before the last one is halted) and\u2026","title":"Runs of the bit-list encoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/computable-3","kind":"node","lean":"Cslib.Complexity.DenseSAT.EncodeBits.computable","parent":31183,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction/index.html#computable-3","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/encoding-converters-for-the-cook-levin-reduction","status":"fully_proved","summary":"This is a multi-stack Turing machine whose input and output alphabets are the bits, together with the polynomial time bound $t(n) = 2n + 2$, such that for every bit list $x$ of length $n$, the machine started with $x$ on its input stack and all other stacks empty halts within $t(n)$ steps with the canonical bitstring encoding of $x$ on its output stack, all other stacks empty and its internal state back to the\u2026","title":"Polynomial-time encoder to the canonical bit-list encoding"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","title":"Size bounds for the canonical framed codecs"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/blocks-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.blocks_length_le","parent":31208,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html#blocks-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","summary":"Each decoded block consumes at least its delimiter bit.","title":"Decoded block count is at most the input length"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/decoded-list-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.decoded_list_length_le","parent":31208,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html#decoded-list-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","summary":"A successfully decoded list has at most one element per input bit. No canonicality assumption on the element payloads is needed.","title":"Decoded list has at most one element per input bit"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/decoded-pair-parts","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.decoded_pair_parts","parent":31208,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html#decoded-pair-parts","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","summary":"Successful pair decoding exposes the exact raw component payloads. This includes payloads which are not canonical encodings.","title":"Decoded pair exposes its raw component payloads"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/decoded-fst-list-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.decoded_fst_list_length_le","parent":31208,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html#decoded-fst-list-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","summary":"A list in the first field of a decoded pair is raw-input bounded.","title":"Length bound for a list in the first field of a decoded pair"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/decoded-snd-list-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.decoded_snd_list_length_le","parent":31208,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html#decoded-snd-list-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","summary":"A list in the second field of a decoded pair is raw-input bounded.","title":"Length bound for a list in the second field of a decoded pair"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/decoded-middle-list-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.decoded_middle_list_length_le","parent":31208,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html#decoded-middle-list-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","summary":"The middle list of a right-associated triple is raw-input bounded. Used for TSP matrix rows and SET COVER set occurrences.","title":"Length bound for the middle list of a decoded triple"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/list-nat-encode-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.list_nat_encode_length_le","parent":31208,"reading_title":{"format":"inline-parts-v1","parts":[["text","Framed length bound for lists of "],["math","\\(m\\)"],["text","-bit naturals"]],"source":"Framed length bound for lists of $m$-bit naturals"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html#list-nat-encode-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","summary":"A list of binary naturals with at most `m` bits per entry has the stated framed length bound.","title":"Framed length bound for lists of $m$-bit naturals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/bounded-nat-list-encode-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.bounded_nat_list_encode_length_le","parent":31208,"reading_title":{"format":"inline-parts-v1","parts":[["text","Quadratic encoding bound for vertex lists below "],["math","\\(n\\)"]],"source":"Quadratic encoding bound for vertex lists below $n$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs/index.html#bounded-nat-list-encode-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/size-bounds-for-the-canonical-framed-codecs","status":"fully_proved","summary":"Lists of vertices below `n` have a quadratic certificate bound when their length is at most `n`; no unary Nat encoding is used.","title":"Quadratic encoding bound for vertex lists below $n$"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source","status":"fully_proved","title":"Copying a bit block without consuming the source"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source/split-b","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.splitB","parent":31217,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source/index.html#split-b","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source","status":"fully_proved","summary":"Pop a bit block onto two distinct destinations, reversing both copies.","title":"Two-destination bit-block transfer"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source/split-b-run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.splitB_run","parent":31217,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source/index.html#split-b-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source","status":"fully_proved","summary":"Two-destination bit transfer takes one step per bit plus one, with exact suffix and unrelated-stack preservation.","title":"Two-destination bit transfer takes one step per bit plus one"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source/copy-b-run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.copyB_run","parent":31217,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source/index.html#copy-b-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/copying-a-bit-block-without-consuming-the-source","status":"fully_proved","summary":"Copy a bit block in its original order, restoring both the source and scratch, in exactly `2 * p.length + 2` steps.","title":"Bit-block copy keeps the order and restores source and scratch"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse","status":"fully_proved","title":"Emitting a canonical frame in reverse"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse/emit-frame","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.emitFrame","parent":31221,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse/index.html#emit-frame","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse","status":"fully_proved","summary":"Emit one raw payload as a reversed self-delimiting frame.","title":"Reversed self-delimiting frame emitter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse/emit-frame-run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.emitFrame_run","parent":31221,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse/index.html#emit-frame-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/emitting-a-canonical-frame-in-reverse","status":"fully_proved","summary":"Emit a complete frame in `payload.length + 1` steps, restoring the input suffix and leaving every unrelated stack unchanged.","title":"Frame emitter takes one step per payload bit plus one"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonicalization-of-the-concrete-binary-list-nat.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonicalization-of-the-concrete-binary-list-nat","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonicalization-of-the-concrete-binary-list-nat/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonicalization-of-the-concrete-binary-list-nat","status":"fully_proved","title":"Canonicalization of the concrete binary List Nat 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bit"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/counting-frames-of-a-concrete-list-nat-input/blocks-run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListCount.blocks_run","parent":31231,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/counting-frames-of-a-concrete-list-nat-input/index.html#blocks-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/counting-frames-of-a-concrete-list-nat-input","status":"fully_proved","summary":"Counting a sequence of raw frames yields its length in unary, with the caller's counter suffix preserved and scratch empty.","title":"Frame counter yields the number of raw frames in 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naturals"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/counting-frames-of-a-concrete-list-nat-input/run-none","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListCount.run_none","parent":31231,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/counting-frames-of-a-concrete-list-nat-input/index.html#run-none","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/counting-frames-of-a-concrete-list-nat-input","status":"fully_proved","summary":"Malformed List Nat framing reaches the failure endpoint. 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After `bad`, clear partial workspace data before another invocation.","title":"Embedded list canonicalizer preserves outside stacks"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/structural-laws-for-the-existing-framed-list-dec.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/structural-laws-for-the-existing-framed-list-dec","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/structural-laws-for-the-existing-framed-list-dec/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/structural-laws-for-the-existing-framed-list-dec","status":"fully_proved","title":"Structural laws for the existing framed list decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/structural-laws-for-the-existing-framed-list-dec/blocks-delimit","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.blocks_delimit","parent":31257,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/structural-laws-for-the-existing-framed-list-dec/index.html#blocks-delimit","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/structural-laws-for-the-existing-framed-list-dec","status":"fully_proved","summary":"Parsing a framed head leaves exactly the raw remainder, including any later framing failure. 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Empty input is excluded because it is the valid empty list.","title":"Malformed first frame makes list decoding fail"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","title":"Raw projection of the canonical pair frame"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/k","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.PairSplit.K","parent":31261,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html#k","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","summary":"The original input becomes the second field; `left` holds the first.","title":"Stacks of the raw pair-frame splitter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/lab","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.PairSplit.Lab","parent":31261,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","summary":"Read one frame, restore its order, then report success or failure.","title":"Control labels of the raw pair-frame splitter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/prog","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.PairSplit.prog","parent":31261,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","summary":"Pair framing is split using the existing delimiter and move kernels.","title":"Program of the raw pair-frame splitter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/st","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.PairSplit.st","parent":31261,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","summary":"Explicit stack layout of the projection invariant.","title":"Stack layout of the raw pair-frame splitter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/run-delimit","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.PairSplit.run_delimit","parent":31261,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html#run-delimit","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","summary":"Extract both raw fields, restore first-field order, and clear scratch in exactly two steps per first-field bit plus two.","title":"Pair-frame splitter extracts both raw fields"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/run-some","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.PairSplit.run_some","parent":31261,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html#run-some","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","summary":"Every successful raw pair frame is projected within input length plus one; its two payloads are left unnormalized.","title":"Pair-frame splitter projects every well-framed input in linear time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/run-none","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.PairSplit.run_none","parent":31261,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html#run-none","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","summary":"Malformed first-field framing reaches the actual failure label. 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This is not a runtime assumption or implementation of either element codec.","title":"Raw field projection agrees with the pair decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.PairSplit.run","parent":31261,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/raw-projection-of-the-canonical-pair-frame","status":"fully_proved","summary":"Total framing outcome, without prematurely accepting either field as a value of its eventual element type.","title":"Pair-frame splitter reaches the framing 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stacks"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","title":"Projecting and normalizing a pair of binary 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projector"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/run-delimit","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.NatPair.run_delimit","parent":31273,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html#run-delimit","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","summary":"Every raw pair frame is projected and normalized without changing the total Nat decoding semantics. Previous output suffixes survive.","title":"Number-pair projector normalizes both fields of a raw pair frame"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/decode-pair-nat","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.NatPair.decode_pair_nat","parent":31273,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html#decode-pair-nat","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","summary":"Pair Nat decoding fails exactly when the first frame is malformed; both raw field payloads use Mathlib's total Nat decoder.","title":"Decoding a pair of naturals fails exactly on a malformed first frame"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/run-some","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.NatPair.run_some","parent":31273,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html#run-some","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","summary":"A successful pair decode produces both canonical binary payloads, with all scratch empty and arbitrary previous output suffixes preserved.","title":"Number-pair projector yields both canonical payloads on success"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/run-none","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.NatPair.run_none","parent":31273,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html#run-none","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","summary":"Malformed pair framing fails in linear time. Private partial workspace contents must be cleared before another invocation.","title":"Number-pair projector fails on malformed framing in linear time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/component","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.NatPair.component","parent":31273,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html#component","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","summary":"The concrete two-Nat projector in the existing Host interface.","title":"Number-pair projector as a host component"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/decoded-payload-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.NatPair.decoded_payload_length_le","parent":31273,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html#decoded-payload-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","summary":"Canonical output payload sizes are bounded by the raw input size, even when either field had a noncanonical final zero bit.","title":"Canonical payloads of a decoded pair are bounded by the raw input"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.NatPair.run","parent":31273,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","summary":"Total decoder outcome with exact successful output and empty private scratch. Failure does not promise reusable workspace contents.","title":"Number-pair projector reaches the decoding outcome"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/run-embedded","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.NatPair.run_embedded","parent":31273,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu/index.html#run-embedded","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/projecting-and-normalizing-a-pair-of-binary-natu","status":"fully_proved","summary":"Callers may embed this concrete projector without disturbing outside stacks. A failed workspace must be cleared before reuse.","title":"Embedded number-pair projector preserves outside stacks"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","title":"Canonical projection of lists of binary Nat pairs"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/projection-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.projection_length_le","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#projection-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Both projected canonical lists together are bounded by four times the original raw list length, including noncanonical input payloads.","title":"Projected pair lists total at most four times the input length"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/k","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.K","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#k","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Raw outer input, two reversed accumulators, two outputs, and the existing Nat-pair component's workspace.","title":"Stacks of the pair-list projector"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/lab","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.Lab","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Read one outer frame, normalize its pair, emit both fields, repeat.","title":"Control labels of the pair-list projector"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/pair-emb","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.pairEmb","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#pair-emb","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"The Nat-pair component is embedded without changing its alphabets.","title":"Stack embedding of the pair normalizer in the pair-list projector"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/pair-label","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.pairLabel","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#pair-label","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Only the pair component's actual halts are redirected.","title":"Pair-normalizer halt redirection in the pair-list projector"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/prog","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.prog","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Concrete outer parsing, pair normalization, and two framed streams.","title":"Program of the pair-list projector"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/st","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.st","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"The exact eleven-stack invariant, with caller output suffixes.","title":"Eleven-stack invariant of the pair-list projector"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/element-run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.element_run","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#element-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"One successfully decoded item contributes one canonical weight frame and one canonical value frame, with pair scratch emptied again.","title":"Pair-list projector emits one weight and one value frame per item"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/blocks-run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.blocks_run","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#blocks-run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"The main loop emits both lists in reverse, ready for the two final stack reversals. Order and multiplicity follow the decoded input.","title":"Pair-list projector main loop emits both lists in reverse"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/run-some","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.run_some","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#run-some","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Successful generic decoding yields the canonical weight and value lists on separate stacks. All input and scratch stacks are emptied.","title":"Pair-list projector yields canonical weight and value lists"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/run-none","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.run_none","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#run-none","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Malformed outer framing or any malformed pair reaches `bad` in linear time. Failed workspaces may contain partial data and need clearing.","title":"Pair-list projector fails on malformed input in linear time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/component","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.component","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#component","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"The concrete eleven-stack projector uses the existing Host model.","title":"Pair-list projector as a host component"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.run","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Total decoder outcome. Success projects canonical lists with empty scratch; failure promises termination, not reusable workspace contents.","title":"Pair-list projector always terminates at the decoding outcome"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/run-embedded","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPair.run_embedded","parent":31290,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair/index.html#run-embedded","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/canonical-projection-of-lists-of-binary-nat-pair","status":"fully_proved","summary":"Any stack embedding preserves the external frame on both outcomes. Only mapped-label reachability is promised: callers establish their own continuation-label separation. Failed private workspace needs clearing.","title":"Embedded pair-list projector preserves outside stacks"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","title":"Four canonical outputs from a Nat-pair list and a Nat pair"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/k","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.K","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#k","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"Two disjoint existing workspaces; the pair buffer also serves the initial outer-frame split and is cleared before normalization starts.","title":"Stacks of the four-output pair-list 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decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/split-label","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.splitLabel","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#split-label","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"Successful outer framing enters the item-list projector.","title":"Label map from the outer splitter to the list 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decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/pair-label","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.pairLabel","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#pair-label","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"The final pair's actual halts are the whole decoder's outcomes.","title":"Label map from final pair halts to the decoder outcomes"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/prog","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.prog","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"The whole finite program is a composition of existing components.","title":"Program of the four-output pair-list decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/st","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.st","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#st","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"Exact raw-field and four-output invariant, with empty private scratch. At entry the item field is empty and the entire raw input is `pairRaw`.","title":"Stack invariant of the four-output pair-list decoder"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/run-some","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.run_some","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#run-some","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"The entire raw decoder emits four canonical outputs, preserving caller suffixes and emptying all input and private scratch on success.","title":"Four-output decoder emits four canonical outputs on success"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/output-length-le","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.output_length_le","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#output-length-le","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"The combined newly emitted output size is at most twice raw input length, independently of decoded magnitudes and caller output suffixes.","title":"Four-output decoder output is at most twice the input length"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/run-none","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.run_none","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#run-none","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"Every failed raw decode reaches `bad` within the same whole-input bound. Failed owned workspace can be partial and must be discarded or cleared.","title":"Four-output decoder fails on malformed input"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/component","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.component","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#component","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"The fixed seventeen-stack decoder in the existing Host interface.","title":"Four-output pair-list decoder as a host component"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/run","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.run","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#run","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"Total raw decoder contract with four output suffixes. Exactly the canonical decoder's success/failure label is reached, with empty scratch on success.","title":"Four-output decoder reaches exactly the canonical decoding outcome"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/run-embedded","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.ListNatPairPair.run_embedded","parent":31305,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and/index.html#run-embedded","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/four-canonical-outputs-from-a-nat-pair-list-and","status":"fully_proved","summary":"Embedded execution preserves every outside stack on both outcomes. Successful output suffixes survive and private scratch is empty. Failed workspace may be partial. Callers establish continuation-label separation.","title":"Embedded four-output decoder preserves outside stacks"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con.json","id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","kind":"scope","navigation":"module","parent":31149,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","title":"Reusable unary-count to canonical binary Nat conversion"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/lab","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.UnaryNat.Lab","parent":31322,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html#lab","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","summary":"Existing emitter labels and the cleanup/unframing continuation.","title":"Control labels of the unary-to-binary converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/lab-equiv","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.UnaryNat.labEquiv","parent":31322,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html#lab-equiv","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","summary":"The finite label enumeration uses the existing emitter enumeration and two Boolean choices, without a noncomputable list conversion.","title":"Label enumeration of the unary-to-binary converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/embedding","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.UnaryNat.embedding","parent":31322,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html#embedding","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","summary":"The component retains the existing emitter's heterogeneous alphabets.","title":"Emitter stack embedding in the unary-to-binary converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/emit-label","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.UnaryNat.emitLabel","parent":31322,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html#emit-label","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","summary":"The actual emitter halt is followed by charged scratch cleanup.","title":"Label map from the emitter halt to scratch cleanup"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/prog","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.UnaryNat.prog","parent":31322,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html#prog","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","summary":"`framed = true` keeps the new frame; otherwise only that frame is removed, preserving any earlier output below it.","title":"Program of the unary-to-binary converter"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/run-framed","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.UnaryNat.run_framed","parent":31322,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html#run-framed","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","summary":"Emit one canonical Nat frame above any earlier output and restore all conversion scratch. The old counter is consumed; unrelated stacks, including `X`, are preserved. Cleanup is part of the charged run.","title":"Converter turns a unary count into a canonical binary frame"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/run-raw","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.UnaryNat.run_raw","parent":31322,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html#run-raw","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","summary":"Convert a unary count to the canonical unframed Nat payload, preserving an earlier output suffix and clearing every used work stack.","title":"Converter turns a unary count into a raw canonical binary payload"},{"id":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/component","kind":"node","lean":"Cslib.Complexity.PolynomialEncoding.UnaryNat.component","parent":31322,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con/index.html#component","scope":"TheoryOfComputation/np-completeness/cook-levin/conversions-between-bit-string-encodings/reusable-unary-count-to-canonical-binary-nat-con","status":"fully_proved","summary":"The adapter is a finite component of the existing Host. Its input counter has `n` physical symbols; the bound is linear in that length, not in the bit length of an arbitrary binary-encoded numeric input.","title":"Unary-to-binary converter as a host component"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl.json","id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","title":"Cook-Levin for dense CNF-SAT on the canonical classes"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/sym-set-x","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.symSetX","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#sym-set-x","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"For a finite multi-stack Turing machine $M$, a symbol $a$ of its output-stack alphabet and a stack $k$, this is the finite set of symbols that some statement of $M$ can push on stack $k$ (together with the whole input alphabet when $k$ is the input stack), with $a$ added when $k$ is the output stack.","title":"Effective alphabets with an adjoined accept symbol"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/enc-x","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.encX","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#enc-x","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Given a finite multi-stack Turing machine $M$ and a symbol $a$ of its output-stack alphabet, this is the encoding of $M$ that numbers the stacks and the pairs (optional label, state) through their finite enumerations, and whose effective alphabet on each stack $k$ is the set of symbols some statement of $M$ can push on $k$ (plus the whole input alphabet if $k$ is the input stack), with $a$ added if $k$ is the\u2026","title":"Tableau encoding with a chosen accept symbol"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/enc-x-acc","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.encX_acc","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#enc-x-acc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine and let $a$ be a symbol of its output-stack alphabet. In the tableau encoding of $M$ with $a$ adjoined to the output alphabet, some index of the effective output-stack alphabet decodes to $a$.","title":"The chosen accept symbol is encodable"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/vacc","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.vacc","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#vacc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time machine computing a function of pairs of bit lists from their pair encoding, whose output alphabet is identified with the Booleans. The accept symbol is the output symbol of $V$ that corresponds to $\\mathrm{true}$.","title":"Accept symbol of the verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/v-e","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.vE","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#v-e","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"For a polynomial-time Turing machine $V$ that computes a Boolean function on pairs of bit strings, reading a pair $(a, y)$ as the bits of $a$, a separator, then the bits of $y$, and writing its answer as one bit, this is the tableau encoding of $V$ whose effective output alphabet is enlarged by the accept symbol: the output symbol of $V$ for true.","title":"Tableau encoding of the verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/veqv","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.veqv","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#veqv","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"The bijection from $\\{0, 1\\}$, the indices below the number $2$ of Booleans, onto the Booleans that is inverse to the fixed enumeration of the Booleans.","title":"Bits as indices"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/v","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.v\u03b9","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#v","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time machine computing a function of pairs of bit lists from their pair encoding, whose input alphabet is identified with the pair alphabet $\\mathrm{Bool} \\oplus \\mathrm{Option}\\ \\mathrm{Bool}$, and let $\\beta$ be the fixed bijection from $\\{0, 1\\}$ onto the Booleans. For an index $i \\in \\{0, 1\\}$, this is the symbol of $V$'s input alphabet that corresponds to the left-tagged bit\u2026","title":"Input-bit symbol of the verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/vsep","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.vsep","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#vsep","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time machine computing a function of pairs of bit lists from their pair encoding, whose input alphabet is identified with the pair alphabet $\\mathrm{Bool} \\oplus \\mathrm{Option}\\ \\mathrm{Bool}$. This is the symbol of $V$'s input alphabet that corresponds to the separator $\\#$, the element $\\mathrm{inr}(\\mathrm{none})$.","title":"Separator symbol of the verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/v-y","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.vY","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#v-y","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time machine computing a function of pairs of bit lists from their pair encoding, whose input alphabet is identified with the pair alphabet $\\mathrm{Bool} \\oplus \\mathrm{Option}\\ \\mathrm{Bool}$. For a bit $g$, this is the symbol of $V$'s input alphabet that corresponds to the right-tagged bit $\\mathrm{inr}(\\mathrm{some}\\ g)$.","title":"Certificate symbol of the verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/v-ys","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.vYs","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#v-ys","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time machine computing a function of pairs of bit lists from their pair encoding, whose input alphabet is identified with the pair alphabet $\\mathrm{Bool} \\oplus \\mathrm{Option}\\ \\mathrm{Bool}$. This is the list of the symbols of $V$'s input alphabet corresponding to the right-tagged bits $\\mathrm{inr}(\\mathrm{some}\\ g)$ for $g = \\mathrm{true}$ and $g = \\mathrm{false}$, each once, in the\u2026","title":"Certificate symbols of the verifier"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/vc","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.vc","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#vc","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time machine computing a function of pairs of bit lists from their pair encoding, with time polynomial $t_V$. This is a natural number $c \\ge 1$, fixed by choice, such that for some exponent $e$ we have $t_V(j) \\le (j + c)^e$ for every natural number $j$.","title":"Offset of the verifier's time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/ve","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.ve","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#ve","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time machine computing a function of pairs of bit lists from their pair encoding, with time polynomial $t_V$, and let $c \\ge 1$ be the offset of its time bound, fixed by choice so that some exponent works. This is an exponent $e$, fixed by choice, with $t_V(j) \\le (j + c)^e$ for every natural number $j$.","title":"Exponent of the verifier's time bound"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/v-input","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.v_input","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#v-input","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time machine computing a function of pairs of bit lists from their pair encoding, whose input alphabet is identified with the pair alphabet $\\mathrm{Bool} \\oplus \\mathrm{Option}\\ \\mathrm{Bool}$, and let $\\beta$ be the fixed bijection from $\\{0, 1\\}$ onto the Booleans. For bit lists $a$ and $y$, translating the pair encoding $a \\# y$ symbol by symbol into $V$'s input alphabet gives the\u2026","title":"Input of the verifier on a pair"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/v-run","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.v_run","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#v-run","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a machine that computes a function $f_v$ of pairs of bit lists from their pair encoding, within $t_V(n)$ steps on an encoding of length $n$ for a polynomial $t_V$ with natural-number coefficients, with input alphabet identified with the pair alphabet $\\mathrm{Bool} \\oplus \\mathrm{Option}\\ \\mathrm{Bool}$ and output alphabet identified with the Booleans. Let $a$, $y$ be bit lists of lengths $|a|$, $|y|$,\u2026","title":"The verifier's run settles on its output"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/red","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.red","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#red","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time Turing machine that computes a Boolean function on pairs of bit strings, reading a pair $(a, y)$ as the bits of $a$, a separator, then the bits of $y$, and writing its answer as one bit; let $t$ be its time bound and fix constants $c \\ge 1$ and $e$ with $t(j) \\le (j + c)^e$ for all $j$. For a polynomial $p$ with natural-number coefficients, the reduction sends a bit string $a$ of length\u2026","title":"The Cook\u2013Levin reduction"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/v-correct","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.v_correct","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#v-correct","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time Turing machine that computes a Boolean function $f$ on pairs of bit strings, reading a pair $(a, y)$ as the bits of $a$, a separator, then the bits of $y$, and writing its answer as one bit. Let $p$ be a polynomial with natural-number coefficients, $R$ a relation on bit strings with $R(a, y)$ iff $f(a, y)$ is true, and $L'$ a predicate on bit strings such that $L'(a)$ holds iff some bit\u2026","title":"Correctness of the reduction"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/is-poly-time-red","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.isPolyTime_red","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#is-poly-time-red","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Let $V$ be a polynomial-time Turing machine that computes a Boolean function on pairs of bit strings, reading a pair $(a, y)$ as the bits of $a$, a separator, then the bits of $y$, and writing its answer as one bit. For every polynomial $p$ with natural-number coefficients, the reduction built from $V$ and $p$ is polynomial-time computable with respect to the canonical bit-string encodings.","title":"The Cook\u2013Levin reduction is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/dense-sat-dec-np-hard","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.denseSatDec_npHard","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#dense-sat-dec-np-hard","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Every decision problem $L$ in NP has a polynomial-time many-one reduction to dense CNF-SAT: a polynomial-time computable map $f$ on bit strings such that, for every bit string $x$, $L$ accepts $x$ exactly when dense CNF-SAT accepts $f(x)$.","title":"Every NP problem reduces to dense CNF-SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/dense-sat-dec-np-complete","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.denseSatDec_npComplete","parent":31331,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl/index.html#dense-sat-dec-np-complete","scope":"TheoryOfComputation/np-completeness/cook-levin/cook-levin-for-dense-cnf-sat-on-the-canonical-cl","status":"fully_proved","summary":"Dense CNF-SAT is NP-complete: it lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to dense CNF-SAT.","title":"Cook-Levin for dense CNF-SAT"},{"data_url":"scopes/TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat.json","id":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat","kind":"scope","navigation":"module","parent":30324,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/index.html","scope":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat","status":"fully_proved","title":"NP-hardness of binary-index SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/phi-length","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.Phi_length","parent":31351,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/index.html#phi-length","scope":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat","status":"fully_proved","summary":"Let $E$ be an encoding of a machine $M$, let $x$ and $Y$ be lists of input symbols of $M$, let $a$ be a symbol of its output stack, and let $m, T \\in \\mathbb{N}$. Put $H = m + (T + 1)d + 1$, where $d$ is the window depth of $M$. Then every clause of the Cook\u2013Levin formula $\\Phi(E, x, Y, a, m, T)$ is a dense clause of length $(T + 1)(A + K H g)$, the number of variables of the layout of $E$ at time bound $T$ and\u2026","title":"Every clause of the Cook\u2013Levin formula has one slot per variable"},{"id":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/is-poly-time-bin-conv","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.isPolyTime_binConv","parent":31351,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/index.html#is-poly-time-bin-conv","scope":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat","status":"fully_proved","summary":"The translation of bit strings from the dense CNF encoding to the binary-index CNF encoding, as a function on all bit strings, is polynomial-time computable with respect to the canonical bitstring encoding of bit lists.","title":"The dense-to-binary translation is polynomial-time in the canonical encoding"},{"id":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/sat-dec-np-hard","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.satDec_npHard","parent":31351,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/index.html#sat-dec-np-hard","scope":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat","status":"fully_proved","summary":"Every decision problem $L$ in NP has a polynomial-time many-one reduction to the language of bit strings that encode a satisfiable CNF formula in the binary-index encoding.","title":"Every NP problem reduces to binary-index CNF-SAT"},{"id":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/cook-levin-sat-np-complete-of-is-poly-ti","kind":"node","lean":"Cslib.Complexity.CookLevin.Hardness.cook_levin_SAT_np_complete_of_isPolyTime","parent":31351,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat/index.html#cook-levin-sat-np-complete-of-is-poly-ti","scope":"TheoryOfComputation/np-completeness/cook-levin/np-hardness-of-binary-index-sat","status":"fully_proved","summary":"If the Boolean result of the SAT verifier, as a function of pairs (formula code, assignment), is computable in polynomial time, then the language of bit strings that encode a satisfiable CNF formula in the binary-index encoding is in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"Cook\u2013Levin for binary-index CNF-SAT, given a polynomial-time verifier"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability.json","id":"TheoryOfComputation/np-completeness/satisfiability","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability","status":"can_prove","summary":"Variants of SAT and CSP: 3SAT and the SAT-to-3SAT reduction, bounded-occurrence, monotone, not-all-equal and 1-in-3 SAT, MAX-SAT, MAX-2SAT, MIN-ONES, circuit SAT, binary CSP and betweenness.","title":"Satisfiability and constraint satisfaction"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat.json","id":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat","status":"can_prove","title":"1-in-3 SAT"},{"id":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/one-in-three-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OneInThreeSat.OneInThreeSatInstance","parent":31357,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/index.html#one-in-three-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat","status":"fully_proved","summary":"A 1-in-3-SAT instance is a pair $(n, \\varphi)$ of a natural number $n$, the number of variables, and a CNF formula $\\varphi$.","title":"1-in-3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OneInThreeSat.WellFormed","parent":31357,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat","status":"fully_proved","summary":"A 1-in-3-SAT instance $(n, \\varphi)$ is well formed when every clause of $\\varphi$ has exactly three literals and every literal $(x, b)$ in it has variable index $x < n$.","title":"Well-formed 1-in-3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/one-in-three-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OneInThreeSat.OneInThreeSatFeasible","parent":31357,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/index.html#one-in-three-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat","status":"fully_proved","summary":"A 1-in-3-SAT instance $(n, \\varphi)$ is feasible when there is an assignment $v$ of truth values to the variables $0, \\dots, n - 1$, with variables $y \\ge n$ read as false, such that every clause of $\\varphi$ has exactly one position holding a literal $(y, b)$ with $v(y) = b$.","title":"Feasible 1-in-3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/one-in-three-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OneInThreeSat.oneInThreeSatDec","parent":31357,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/index.html#one-in-three-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat","status":"fully_proved","summary":"The 1-in-3-SAT decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a pair $(n, \\varphi)$ of a natural number $n$ and a CNF formula $\\varphi$ such that every clause of $\\varphi$ has exactly three literals, every literal $(y, b)$ of $\\varphi$ has variable index $y < n$, and some assignment $v$ of truth values to the variables $0, \\dots, n - 1$ makes exactly\u2026","title":"The 1-in-3-SAT decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/one-in-three-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OneInThreeSat.oneInThreeSat_npComplete","parent":31357,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat/index.html#one-in-three-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/1-in-3-sat","status":"can_prove","summary":"The 1-in-3-SAT decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"1-IN-3-SAT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences.json","id":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences","status":"can_prove","title":"3-SAT with bounded occurrences"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/bounded-occurrence-three-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BoundedOccurrenceThreeSat.BoundedOccurrenceThreeSatInstance","parent":31363,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/index.html#bounded-occurrence-three-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences","status":"fully_proved","summary":"A bounded-occurrence 3-SAT instance is a pair $(n, \\varphi)$ of a natural number $n$, the number of variables, and a CNF formula $\\varphi$.","title":"Bounded-occurrence 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BoundedOccurrenceThreeSat.WellFormed","parent":31363,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences","status":"fully_proved","summary":"A bounded-occurrence 3-SAT instance $(n, \\varphi)$ is well formed when every clause of $\\varphi$ is a list of exactly three literals, not necessarily distinct, every literal of $\\varphi$ has a variable less than $n$, and for every variable $v < n$ at most four of the literals in the clauses of $\\varphi$, counted with multiplicity and with either polarity, have variable $v$.","title":"Well-formed bounded-occurrence 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/bounded-occurrence-three-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BoundedOccurrenceThreeSat.BoundedOccurrenceThreeSatFeasible","parent":31363,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/index.html#bounded-occurrence-three-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences","status":"fully_proved","summary":"A bounded-occurrence 3-SAT instance $(n, \\varphi)$ is satisfiable when there is a list $(a_0, \\dots, a_{n-1})$ of booleans such that $\\varphi$ is satisfied by the assignment sending each variable $x < n$ to $a_x$ and every variable $x \\ge n$ to false.","title":"Satisfiable bounded-occurrence 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/bounded-occurrence-three-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BoundedOccurrenceThreeSat.boundedOccurrenceThreeSatDec","parent":31363,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/index.html#bounded-occurrence-three-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences","status":"fully_proved","summary":"The decision problem 3-SAT WITH BOUNDED OCCURRENCES accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of bounded-occurrence 3-SAT instances, to an instance that is well formed and satisfiable; it rejects every other bitstring, including every bitstring that does not decode.","title":"3-SAT with bounded occurrences as a decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/bounded-occurrence-three-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BoundedOccurrenceThreeSat.boundedOccurrenceThreeSat_npComplete","parent":31363,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences/index.html#bounded-occurrence-three-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/3-sat-with-bounded-occurrences","status":"can_prove","summary":"The decision problem 3-SAT WITH BOUNDED OCCURRENCES over bitstrings belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"3-SAT WITH BOUNDED OCCURRENCES is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability.json","id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","title":"3-satisfiability"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/is3","kind":"node","lean":"Cslib.Complexity.Karp.is3","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#is3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a CNF formula $\\varphi$, this Boolean test holds exactly when every clause of $\\varphi$ has at most three literals.","title":"Three-literal clause test"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/sat3-dec","kind":"node","lean":"Cslib.Complexity.Karp.sat3Dec","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#sat3-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"3SAT accepts a bit string $x$ exactly when $x$ is the binary code of a satisfiable CNF formula in which every clause has at most three literals.","title":"The 3SAT decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/ren","kind":"node","lean":"Cslib.Complexity.Karp.ren","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#ren","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Variable index $n$ is renamed to $2n$, so the variables of the original formula get even indices.","title":"Renaming of the original variables"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/fresh","kind":"node","lean":"Cslib.Complexity.Karp.fresh","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#fresh","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For $u \\in \\mathbb{N}$, the $u$-th fresh variable has index $2^{u+1} + 1$, an odd number.","title":"Fresh variable indices"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/ren-lit","kind":"node","lean":"Cslib.Complexity.Karp.renLit","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#ren-lit","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The literal with variable index $n$ and polarity $b$ is renamed to the literal with variable index $2n$ and the same polarity $b$.","title":"Renaming a literal"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/chain-tail","kind":"node","lean":"Cslib.Complexity.Karp.chainTail","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#chain-tail","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Write $y_v$ for the variable with index $2^{v+1} + 1$ (the $v$-th fresh variable), $\\ell'$ for a literal $\\ell$ whose variable index $n$ is replaced by $2n$ with the same polarity, and $z \\ominus j$ for $\\max(z - j, 0)$. Given literals $\\ell_1, \\dots, \\ell_m$ and a natural number $z$, the result is the empty formula if $m = 0$, and otherwise the $m$ clauses\u2026","title":"The tail of a clause chain"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/chain","kind":"node","lean":"Cslib.Complexity.Karp.chain","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#chain","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Write $y_v$ for the variable with index $2^{v+1} + 1$ (the $v$-th fresh variable) and $\\ell'$ for a literal $\\ell$ whose variable index $n$ is replaced by $2n$, with the same polarity. Given a clause $C$ and a natural number $u$, the empty clause is replaced by the empty clause alone, a one-literal clause $\\ell_1$ by the clause $\\ell_1'$, and a clause $\\ell_1 \\lor \\dots \\lor \\ell_k$ with $k \\ge 2$ by the $k$\u2026","title":"The chain of clauses replacing one clause"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/used","kind":"node","lean":"Cslib.Complexity.Karp.used","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#used","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The number of fresh variables reserved for a CNF clause $C$ in the SAT-to-3SAT translation: $k - 1$ if $C$ has $k \\ge 1$ literals, and $0$ if $C$ is empty.","title":"Fresh variables reserved for a clause"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/to3-aux","kind":"node","lean":"Cslib.Complexity.Karp.to3Aux","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#to3-aux","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a CNF formula this returns a pair $(N, \\psi)$ of a natural number $N$ and a CNF formula $\\psi$. The empty formula gives $N = 0$ and $\\psi$ empty. A formula with first clause $C$ and remaining clauses $\\varphi'$, whose pair is $(N', \\psi')$, gives $N = N' + \\max(|C| - 1, 0)$ and, as $\\psi$, the chain of clauses replacing $C$ with fresh numbers from $N'$ on, followed by the clauses of $\\psi'$.","title":"Clause-by-clause translation with a fresh-number counter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/to3","kind":"node","lean":"Cslib.Complexity.Karp.to3","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#to3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a CNF formula with clauses $C_1, \\dots, C_m$, the translation is the concatenation, in order, of the chains of clauses replacing $C_1, \\dots, C_m$, where the chain for $C_i$ uses fresh numbers from $u_i = \\sum_{j > i} \\max(|C_j| - 1, 0)$ on.","title":"The SAT-to-3SAT translation"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/reduce3","kind":"node","lean":"Cslib.Complexity.Karp.reduce3","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#reduce3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"A bit string that decodes, with no bits left over, to a CNF formula $\\varphi$ is sent to the binary code of the SAT-to-3SAT translation of $\\varphi$; any other bit string is sent to the empty string.","title":"The SAT-to-3SAT reduction on bit strings"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/length-le-three-of-mem-chain-tail","kind":"node","lean":"Cslib.Complexity.Karp.length_le_three_of_mem_chainTail","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#length-le-three-of-mem-chain-tail","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every clause $C$ and natural number $z$, every clause of $\\mathrm{chainTail}(C, z)$ has at most three literals.","title":"Chain-tail clauses have at most three literals"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/length-le-three-of-mem-chain","kind":"node","lean":"Cslib.Complexity.Karp.length_le_three_of_mem_chain","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#length-le-three-of-mem-chain","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every clause $C$ and natural number $u$, every clause of $\\mathrm{chain}(C, u)$ has at most three literals.","title":"Chain clauses have at most three literals"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/length-le-three-of-mem-to3-aux","kind":"node","lean":"Cslib.Complexity.Karp.length_le_three_of_mem_to3Aux","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#length-le-three-of-mem-to3-aux","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every CNF formula $\\varphi$, every clause of the clause list produced by the auxiliary SAT-to-3SAT translation of $\\varphi$ (which replaces each clause by its chain of clauses with fresh variables) has at most three literals.","title":"Clauses of the auxiliary 3SAT translation have at most three literals"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/is3-to3","kind":"node","lean":"Cslib.Complexity.Karp.is3_to3","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#is3-to3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every CNF formula $\\varphi$, every clause of the translation $\\mathrm{to3}(\\varphi)$ has at most three literals.","title":"The SAT-to-3SAT translation yields clauses of length at most three"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/chain-tail-sound","kind":"node","lean":"Cslib.Complexity.Karp.chainTail_sound","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#chain-tail-sound","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $C$ be a nonempty clause and $z$ a natural number. If an assignment $v'$ satisfies every clause of $\\mathrm{chainTail}(C, z)$ and gives the fresh variable $\\mathrm{fresh}(z)$ the value true, then the assignment $n \\mapsto v'(2n)$ satisfies $C$.","title":"Soundness of the tail of a chain"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/chain-sound","kind":"node","lean":"Cslib.Complexity.Karp.chain_sound","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#chain-sound","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $C$ be a clause and $u$ a natural number. If an assignment $v'$ satisfies every clause of $\\mathrm{chain}(C, u)$, then the assignment $n \\mapsto v'(2n)$ satisfies $C$.","title":"Soundness of the chain of one clause"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/to3-aux-sound","kind":"node","lean":"Cslib.Complexity.Karp.to3Aux_sound","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#to3-aux-sound","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every assignment $v' : \\mathbb{N} \\to \\mathrm{Bool}$ and every CNF formula $\\varphi$, if $v'$ satisfies the clause list produced by the auxiliary SAT-to-3SAT translation of $\\varphi$, then the assignment $n \\mapsto v'(2n)$ satisfies $\\varphi$.","title":"Soundness of the auxiliary 3SAT translation"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/chain-tail-complete","kind":"node","lean":"Cslib.Complexity.Karp.chainTail_complete","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#chain-tail-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $v$ and $v'$ be assignments with $v'(2n) = v(n)$ for every $n$, let $C$ be a nonempty clause with $m + 1$ literals and let $z$ be a natural number. Suppose that for every $j \\le m$, $v'$ gives the fresh variable $\\mathrm{fresh}(z - j)$ the value true exactly when $v$ satisfies some literal of the clause obtained from $C$ by dropping its first $j$ literals, where $z - j$ is truncated subtraction ($0$ when\u2026","title":"Completeness of the tail of a chain"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/chain-complete","kind":"node","lean":"Cslib.Complexity.Karp.chain_complete","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#chain-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $v$ and $v'$ be assignments with $v'(2n) = v(n)$ for every $n$, let $C$ be a clause and $u$ a natural number, and write $k = \\mathrm{used}(C)$. Suppose that $v$ satisfies $C$, and that for every $w$ with $u \\le w < u + k$, $v'$ gives the fresh variable $\\mathrm{fresh}(w)$ the value true exactly when $v$ satisfies some literal of the clause obtained from $C$ by dropping its first $u + k - w$ literals. Then $v'$\u2026","title":"Completeness of the chain of one clause"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/fresh-val","kind":"node","lean":"Cslib.Complexity.Karp.freshVal","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#fresh-val","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For an assignment $v : \\mathbb{N} \\to \\mathrm{Bool}$, a CNF formula $\\varphi$ and a natural number $w$, the value $f_v(\\varphi, w)$ is defined by recursion on $\\varphi$. For the empty formula it is false. For a formula with first clause $C$ and remaining clauses $\\varphi'$, let $N'$ be the next free fresh number of the clause-by-clause translation of $\\varphi'$, and let $k = \\max(|C| - 1, 0)$ be the number of fresh\u2026","title":"Values of the fresh variables in the 3SAT translation"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/to3-aux-complete","kind":"node","lean":"Cslib.Complexity.Karp.to3Aux_complete","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#to3-aux-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $v, v' : \\mathbb{N} \\to \\mathrm{Bool}$ be assignments with $v'(2n) = v(n)$ for every $n$, let $\\varphi$ be a CNF formula satisfied by $v$, and let $N$ be the next free fresh number of the auxiliary SAT-to-3SAT translation of $\\varphi$. If for every $w < N$ the assignment $v'$ gives fresh variable number $w$, the variable $2^{w+1} + 1$, the fresh-variable value of $w$ built from $v$ and $\\varphi$, then $v'$\u2026","title":"Completeness of the auxiliary 3SAT translation"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lift","kind":"node","lean":"Cslib.Complexity.Karp.lift","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lift","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Given an assignment $v : \\mathbb{N} \\to \\mathrm{Bool}$ and a CNF formula $\\varphi$, the lifted assignment sends an even variable $n$ to $v(n/2)$ and an odd variable $n$ to the fresh-variable value built from $v$ and $\\varphi$ at the number $\\lfloor \\log_2 (n - 1) \\rfloor - 1$, where $\\lfloor \\log_2 0 \\rfloor$ is taken to be $0$ and subtraction is truncated at $0$.","title":"Lifting an assignment to the 3SAT translation"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lift-ren","kind":"node","lean":"Cslib.Complexity.Karp.lift_ren","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lift-ren","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every assignment $v : \\mathbb{N} \\to \\mathrm{Bool}$, every CNF formula $\\varphi$ and every $n \\in \\mathbb{N}$, the assignment lifted from $v$ for the 3SAT translation of $\\varphi$ gives the renamed variable $2n$ the value $v(n)$.","title":"The lifted assignment on renamed variables"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lift-fresh","kind":"node","lean":"Cslib.Complexity.Karp.lift_fresh","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lift-fresh","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every assignment $v : \\mathbb{N} \\to \\mathrm{Bool}$, every CNF formula $\\varphi$ and every $w \\in \\mathbb{N}$, the assignment lifted from $v$ for the 3SAT translation of $\\varphi$ gives fresh variable number $w$, that is variable $2^{w+1} + 1$, the fresh-variable value of $w$ built from $v$ and $\\varphi$.","title":"The lifted assignment on fresh variables"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/cnf-sat-to3-iff","kind":"node","lean":"Cslib.Complexity.Karp.cnfSat_to3_iff","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#cnf-sat-to3-iff","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every CNF formula $\\varphi$, the translation $\\mathrm{to3}(\\varphi)$ is satisfiable if and only if $\\varphi$ is satisfiable.","title":"The SAT-to-3SAT translation preserves satisfiability"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/encode-cnf-ne-nil","kind":"node","lean":"Cslib.Complexity.Karp.encodeCNF_ne_nil","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#encode-cnf-ne-nil","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every CNF formula $\\varphi$, the binary encoding of $\\varphi$ is not the empty bit string.","title":"CNF encodings are nonempty"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/sat-dec-eq-sat3-dec-reduce3","kind":"node","lean":"Cslib.Complexity.Karp.satDec_eq_sat3Dec_reduce3","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#sat-dec-eq-sat3-dec-reduce3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every bit string $x$, SAT accepts $x$ (that is, $x$ encodes a satisfiable CNF formula) exactly when 3SAT accepts the image of $x$ under the SAT-to-3SAT reduction (that is, the image encodes a satisfiable CNF formula in which every clause has at most three literals).","title":"Correctness of the SAT-to-3SAT reduction on bit strings"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/encode-nat-fresh","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.encodeNat_fresh","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#encode-nat-fresh","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every natural number $z$, the binary representation (least significant bit first) of $\\mathrm{fresh}(z) = 2^{z+1} + 1$ is $\\mathrm{true}$, then $z$ copies of $\\mathrm{false}$, then $\\mathrm{true}$.","title":"Binary representation of a fresh name"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/encode-lit-fresh","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.encodeLit_fresh","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#encode-lit-fresh","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every natural number $z$ and every polarity $b$, the binary code of the literal $(\\mathrm{fresh}(z), b)$, where $\\mathrm{fresh}(z) = 2^{z+1} + 1$, is $\\mathrm{true}, \\mathrm{true}$, then $z$ copies of the pair $\\mathrm{true}, \\mathrm{false}$, then $\\mathrm{true}, \\mathrm{true}, \\mathrm{false}, b$.","title":"Code of a fresh literal"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/encode-lit-ren-lit","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.encodeLit_renLit","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#encode-lit-ren-lit","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every literal $(n, p)$, the binary code of the renamed literal $(2n, p)$ is: the pair $\\mathrm{true}, \\mathrm{false}$ if $n \\neq 0$ and nothing if $n = 0$, then the pair $\\mathrm{true}, b$ for each bit $b$ of the binary representation of $n$ (least significant bit first, no bits for $n = 0$), then $\\mathrm{false}, p$.","title":"Code of a renamed literal"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/k","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.K","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#k","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The 3SAT emitter machine has six stacks: the token stack $T$, the output stack $O$, the fresh counter $U$, two scratch counters $Z$ and $W$, and the output clause count $C$.","title":"Stacks of the 3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/untitled","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.\u0393","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#untitled","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The output stack of the emitter machine holds Booleans; each of its other stacks (the token stack, the fresh counter, the two scratch counters and the output clause count) holds symbols of the SAT verifier's token alphabet.","title":"Stack alphabets of the SAT-to-3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lab","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.Lab","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lab","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The emitter machine has the labels $\\mathrm{start}$, $\\mathrm{clause}$, $\\mathrm{fin}$ and $\\mathrm{done}$, and, for Booleans $f, p, s, x, e$, the labels $\\mathrm{lit}(f)$, $\\mathrm{fa}(s, x)$, $\\mathrm{fb}(s, x)$, $\\mathrm{fc}(s, x)$, $\\mathrm{fd}(s, x)$, $\\mathrm{fe}(s, x)$, $\\mathrm{body}(f, p)$, $\\mathrm{cp}(f, e)$, $\\mathrm{aft}(f)$ and $\\mathrm{sfx}(f)$, all distinct; equality of labels is decidable.","title":"Control labels of the SAT-to-3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/is-lit","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.isLit","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#is-lit","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For an optional token $s$ (the top of a stack, absent when the stack is empty), the test is true exactly when $s$ is present and is a literal marker, of either polarity.","title":"Literal-marker test of the 3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/pre","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.pre","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#pre","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For Booleans $f$ and $p$, the label where the emitter continues after popping the marker of a literal of polarity $p$ is $\\mathrm{body}(\\text{true}, p)$, which copies the name directly, if $f$ holds (the literal is the first walked in its clause), and $\\mathrm{fa}(\\text{true}, p)$, which first prepends the fresh literal of the current counter with polarity true, otherwise.","title":"Entry label of a literal"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/ret","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.ret","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#ret","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For Booleans $s$ and $x$, the label where the emitter continues after prepending the fresh literal of polarity $s$ is $\\mathrm{body}(\\text{false}, x)$ if $s$ is true, which copies the name of the pending literal of polarity $x$, and $\\mathrm{sfx}(x)$ if $s$ is false, which closes the output clause, $x$ being the first-literal flag.","title":"Return label of the fresh-literal subroutine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/prog","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.prog","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#prog","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The emitter is the multi-stack machine on the stacks $T, O, U, Z, W, C$ of the emitter ($O$ holds bits, the others token symbols), with the labels of the emitter and one internal Boolean variable. A unit is the clause symbol, and prepending bits $b_1, \\dots, b_k$ to $O$ pushes them so that $O$ reads $b_1, \\dots, b_k$ from the top. The labels act as follows.","title":"The SAT-to-3SAT emitter machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/start","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.start","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#start","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The start label of the emitter machine is its label $\\mathrm{start}$, at which the machine begins each input clause or, once the token stack is empty, moves on to its final phase, which writes the output clause count.","title":"Start label of the SAT-to-3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/st","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.st","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#st","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a list $t$ of tokens, a list $o$ of bits and $z, c \\in \\mathbb{N}$, $\\mathrm{st}(t, o, z, c)$ is the stack contents of the emitter machine in which the token stack holds $t$, the output stack holds $o$, the fresh counter holds exactly $z$ copies of the clause token $\\mathrm{cl}$, the output clause count holds exactly $c$ copies of $\\mathrm{cl}$, and the two scratch counters are empty.","title":"Stack contents of the SAT-to-3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/fresh-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.fresh_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#fresh-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For all booleans $s$ and $x$, every token list $t$, output bit list $o$, natural numbers $z$ and $c$, and flag $v$, there is a flag $v'$ such that the 3SAT emitter machine, started at the fresh-literal label for $(s, x)$ with flag $v$, token stack $t$, output stack $o$, fresh counter $z$, clause count $c$ and empty scratch counters, reaches in exactly $3z + 5$ steps the return label for $(s, x)$ with flag $v'$,\u2026","title":"Fresh-literal subroutine of the 3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/cp-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.cp_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#cp-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $f$ and $e$ be booleans, $R$ a token list whose first token, if there is one, is not a name bit, $z$ and $c$ natural numbers, $b_1, \\dots, b_m$ bits, $v$ a flag and $o$ an output bit list. Then there is a flag $v'$ such that the 3SAT emitter machine, started at the copy label for $(f, e)$ with flag $v$, token stack the name-bit tokens $b_1, \\dots, b_m$ followed by $R$, output stack $o$, fresh counter $z$,\u2026","title":"Name copy loop of the 3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/name-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.name_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#name-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Write $\\langle \\ell, w, t, o, z, c\\rangle$ for the configuration of the emitter machine at label $\\ell$ with internal variable $w$, token stack $t$, output stack $o$, fresh counter $z$, output-clause count $c$ and empty scratch stacks. Let $l = (n, b)$ be a literal, $f$ and $v$ Booleans, $R$ a token stack whose top, if any, is not a bit symbol, $o$ an output stack and $z$, $c$ natural numbers. Let $B$ be the $k$\u2026","title":"Copying a renamed name"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/head-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.head_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#head-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For all booleans $f$ and $p$, every token list $t$, output bit list $o$, natural numbers $z$ and $c$, and flag $v$, there is a flag $v'$ such that the 3SAT emitter machine, started at the literal label for $f$ with flag $v$, token stack the literal marker of polarity $p$ followed by $t$, output stack $o$, fresh counter $z$, clause count $c$ and empty scratch counters, reaches the renamed-literal label for $(f, p)$\u2026","title":"The 3SAT emitter reads a literal marker"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/tail-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.tail_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#tail-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $f$ be a Boolean, $R$ a list of tokens, $o$ a list of bits and $z, c \\in \\mathbb{N}$, and let $m = 1$ if $f$ is true and $m = 2$ otherwise. Start the emitter machine at its clause-closing label $\\mathrm{aft}(f)$, with any internal state, with stack contents $\\mathrm{st}(R, o, z, c)$: token stack $R$, output stack $o$, fresh counter $z$, output clause count $c$ and empty scratch counters.","title":"Closing an output clause in the SAT-to-3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lit-cl","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.litCl","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lit-cl","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For Booleans $f$ and $g$, a counter $z$ and a literal $l = (n, b)$, let $z' = z$ if $f$ holds and $z' = z + 1$ otherwise. The output clause of $l$ is the clause consisting of, in this order: the literal $(2^{z'+1} + 1, \\text{false})$ unless $g$ holds, then the renamed literal $(2n, b)$, then the literal $(2^{z+1} + 1, \\text{true})$ unless $f$ holds. Here $f$ tells whether $l$ is the first literal walked in its\u2026","title":"Output clause of one literal"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/nz","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.nz","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#nz","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a list $L$ of literals and a Boolean $f$, the number of fresh numbers used up by walking $L$ is $|L| - 1$ (truncated at $0$) if $f$ holds and $|L|$ otherwise; $f$ tells whether the last literal of $L$ is the first literal walked in its clause.","title":"Fresh numbers used by a run of literals"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/walk-f","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.walkF","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#walk-f","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a list $L$ of literals of a clause, Booleans $f$ and $g$, and $z \\in \\mathbb{N}$, $\\mathrm{walkF}(L, f, g, z)$ is the list of output clauses of the literals of $L$, in reading order, when $L$ is walked from its last literal with counter $z$; $f$ tells whether the last literal of $L$ is the first walked in its clause, and $g$ whether the first literal of $L$ is the last walked. It is defined by recursion on $L$.\u2026","title":"Output clauses of a walked list of literals"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/walk-f-chain","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.walkF_chain","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#walk-f-chain","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every non-empty clause $C$ and every $z \\in \\mathbb{N}$, the output clauses of the literals of $C$, walked from its last literal with counter $z$, with both the first-walked flag and the last-walked flag true, that is, $\\mathrm{walkF}(C, \\mathrm{true}, \\mathrm{true}, z)$, are the chain of clauses replacing $C$ with fresh numbers from $z$ on.","title":"Walking a whole clause gives its chain"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lit-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.lit_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lit-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Write $\\langle \\ell, w, t, o, z, c\\rangle$ for the configuration of the emitter machine at label $\\ell$ with internal variable $w$, token stack $t$, output stack $o$, fresh counter $z$, output-clause count $c$ and empty scratch stacks. Let $l = (n, b)$ be a literal, $f$ and $v$ Booleans, $R$ a token stack whose top, if any, is not a bit symbol, $o$ an output stack, and $z$, $c$, $Z_b$ natural numbers with\u2026","title":"One literal of the emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lits-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.lits_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Write $\\langle \\ell, w, t, o, z, c\\rangle$ for the configuration of the emitter machine at label $\\ell$ with internal variable $w$, token stack $t$, output stack $o$, fresh counter $z$, output-clause count $c$ and empty scratch stacks. Let $Z_b$ be a natural number, $L = (l_1, \\dots, l_k)$ a list of literals, $R$ a token stack whose top is a literal marker, $f$ and $v$ Booleans, $o$ an output stack and $z$, $c$\u2026","title":"The literals of a clause in the emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/clause-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.clause_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $C$ be a clause, $Z_b$, $z$ and $c$ natural numbers with $z + \\mathrm{used}(C) \\le Z_b$, where $\\mathrm{used}(C) = \\max(|C| - 1, 0)$, let $R$ be a token list that is empty or starts with a clause token, $o$ an output bit list and $v$ a flag. Let $\\mathrm{clRev}(C)$ be the tokens of $C$ as they lie on the token stack (a clause token, then the tokens of its literals, last literal first), and let $D_1, \\dots, D_k$\u2026","title":"The 3SAT emitter processes one clause"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/formula-run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.formula_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#formula-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $\\varphi = C_1 \\cdots C_m$ be a CNF formula, $u = \\sum_{i=1}^{m} \\max(|C_i| - 1, 0)$ the number of fresh variables its 3SAT translation uses, $Z_b \\ge u$ a natural number, $R$ a token list that is empty or starts with a clause token, $o$ an output bit list and $v$ a flag. Let $T$ be the token list $\\mathrm{clRev}(C_m) \\cdots \\mathrm{clRev}(C_1)$, where $\\mathrm{clRev}(C)$ is the list of tokens of $C$ as they\u2026","title":"The 3SAT emitter processes a whole formula"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/rep-single","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.rep_single","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#rep-single","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For every bit $x$ and every $n \\in \\mathbb{N}$, the repeated block $\\mathrm{rep}([x], n)$ is the list consisting of $n$ copies of $x$.","title":"Repeating a single bit"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/run","kind":"node","lean":"Cslib.Complexity.Karp.Emit3.run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula, $o$ a list of bits and $T$ the token sequence of $\\varphi$. Start the emitter machine at its start label, with any internal state, with the token stack holding $T$ reversed (so that the first token of $T$ is at the bottom), the output stack holding $o$, and every other stack empty. Then after at most $8(|T| + 1)^2$ steps the machine reaches its label $\\mathrm{done}$, with some\u2026","title":"The SAT-to-3SAT emitter writes the encoded translation in quadratic time"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/k-2","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.K","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#k-2","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The filter has exactly two stacks: the token stack $T$ and the side stack $U$.","title":"Stacks of the 3SAT clause-length filter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lab-2","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.Lab","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lab-2","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The filter has exactly seven labels: the walking labels $w_0, w_1, w_2, w_3$, at which it walks the token stack having seen that many literals of the current clause ($w_3$: three or more); the return label $\\mathsf{back}$, at which it moves the tokens back; and the exits $\\mathsf{pass}$ and $\\mathsf{fail}$.","title":"Labels of the 3SAT clause-length filter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/stmt","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.Stmt","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#stmt","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The statements of a multi-stack Turing machine (Mathlib's TM2 model) with the two stacks $T$ and $U$, each over the symbol alphabet of the SAT verifier, the labels of the filter, and a Boolean internal state.","title":"Statements of the filter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/wl","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.wl","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#wl","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The walking label after $n$ literals of the current clause: $w_0$, $w_1$, $w_2$ for $n = 0, 1, 2$, and $w_3$ for every $n \\ge 3$.","title":"Walking label for a literal count"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/nxt","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.nxt","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#nxt","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Write $w_i$ for the walking label after $i$ literals of the current clause ($w_3$: three or more). For a Boolean $c$ (whether the filter is capped), the label after one more literal sends $w_0$ to $w_1$, $w_1$ to $w_2$ and $w_2$ to $w_3$; it sends $w_3$ to the failure label when $c$ is true and to $w_3$ when $c$ is false; every other label is unchanged.","title":"Next walking label"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/mv-if","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.mvIf","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#mv-if","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For stacks $a$ and $b$ of the filter, a symbol $s$, a label $\\ell$ and a statement $q$: the statement that tests whether the top of stack $a$ is $s$ and sets the internal state to the outcome; if it is, it pops $a$, pushes $s$ onto $b$ and jumps to $\\ell$, and otherwise it runs $q$.","title":"Conditional move between stacks"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/walk","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.walk","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#walk","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Write $w_0, w_1, w_2, w_3$ for the walking labels after zero, one, two, and three or more literals of the current clause. For a Boolean cap flag $\\mathrm{cap}$ and a label $\\ell$, the filter statement run at $\\ell$: if the token stack is empty, go to the label $\\mathrm{back}$; if its top is the clause marker, move it onto the side stack and go to $w_0$; if its top is a polarity marker, move it onto the side stack\u2026","title":"One step of the clause-length walk"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/prog-2","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.prog","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#prog-2","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a Boolean $c$ (whether the filter is capped), the program on the labels of the filter: at a walking label it runs one walking step with cap $c$; at the label back it moves the top of the side stack $U$ onto the token stack $T$ and stays at back when that top is a token (a clause marker, a literal marker or a name bit), goes to the label pass when $U$ is empty, and goes to the label fail otherwise; at pass and\u2026","title":"The clause-length filter program"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/start-2","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.start","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#start-2","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The filter starts at the walking label $w_0$, the label for having seen no literal of the current clause.","title":"Start label of the filter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/is-tok","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.IsTok","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#is-tok","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"A symbol of the SAT verifier's alphabet is a token symbol when it is the clause marker, a literal marker of either polarity, or a name bit. The value markers and the dead marker are not token symbols.","title":"Token symbols of the 3SAT filter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/tgt","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.tgt","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#tgt","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Write $w_i$ for the walking label after $i$ literals of the current clause ($w_3$: three or more). For a Boolean $c$ (whether the filter is capped) and a label $\\ell$, the label reached from $\\ell$ on a symbol $s$ is: $w_0$ if $s$ is a clause marker; if $s$ is a literal marker, $w_1$, $w_2$ or $w_3$ when $\\ell$ is $w_0$, $w_1$ or $w_2$ respectively, the failure label when $\\ell = w_3$ and $c$ is true, and $\\ell$\u2026","title":"Label after a token"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/st-2","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.st","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#st-2","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For lists of symbols $t$ and $u$, the stack contents that put $t$ on the token stack $T$ and $u$ on the side stack $U$.","title":"Filter stacks from two lists"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/walk-step","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.walk_step","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#walk-step","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $\\mathrm{cap}$ be a Boolean, $n \\in \\mathbb{N}$, $s$ a token (a name bit, a polarity marker or the clause marker), $R$ and $U$ lists of symbols and $v$ a Boolean; stacks are lists written top first, and $w_m$ is the walking label after $m$ literals. Let $t$ be $w_0$ if $s$ is the clause marker, $w_n$ if $s$ is a name bit, and, if $s$ is a polarity marker, $\\mathrm{fail}$ when $\\mathrm{cap}$ is true and\u2026","title":"One step of the clause-length walk on a token"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/walk-bits","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.walk_bits","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#walk-bits","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $\\mathrm{cap}$ be a Boolean, $n \\in \\mathbb{N}$ and $R$ a list of symbols; stacks are lists written top first, $w_n$ is the walking label after $n$ literals and $\\mathrm{bit}(b)$ is the name-bit symbol carrying the Boolean $b$. For all Booleans $b_1, \\dots, b_k$, every list of symbols $U$ and every Boolean $v$ there is a Boolean $v'$ such that the clause-length filter with cap flag $\\mathrm{cap}$ goes in\u2026","title":"Walking over name bits keeps the label"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/walk-lit","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.walk_lit","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#walk-lit","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $\\mathrm{cap}$ be a Boolean and $n \\in \\mathbb{N}$ with $\\mathrm{cap}$ false or $n < 3$; stacks are lists written top first, $w_m$ is the walking label after $m$ literals, $\\mathrm{bit}(b)$ is the name-bit symbol carrying the Boolean $b$ and $\\mathrm{lit}(p)$ is the polarity marker of polarity $p$. Let $l = (i, p)$ be a literal with variable index $i \\in \\mathbb{N}$ and polarity $p$, let $e_1, \\dots, e_k$ be\u2026","title":"Walking over a literal advances the literal count"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/walk-lit-fail","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.walk_lit_fail","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#walk-lit-fail","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ with $n \\ge 3$ and let $w_n$ be the walking label after $n$ literals; stacks are lists written top first, $\\mathrm{bit}(b)$ is the name-bit symbol carrying the Boolean $b$ and $\\mathrm{lit}(p)$ is the polarity marker of polarity $p$. Let $l = (i, p)$ be a literal with variable index $i \\in \\mathbb{N}$ and polarity $p$, let $e_1, \\dots, e_m$ be the binary encoding $\\mathrm{encodeNat}(i)$ of\u2026","title":"A capped walk fails on a literal past the third"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lits-ok","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.lits_ok","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lits-ok","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $c$ be a Boolean (whether the filter is capped), let $\\ell_1, \\dots, \\ell_k$ be literals (pairs of a variable index and a polarity), let $n$ be a natural number with $c$ false or $n + k \\le 3$, and let $w$ be the concatenation of the stack tokens of $\\ell_1, \\dots, \\ell_k$. For all lists of symbols $R$ and $U$ and every internal state $v$, the filter program with cap $c$, started at the walking label for $n$\u2026","title":"The filter walks a clause's literals within the cap"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/lits-fail","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.lits_fail","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#lits-fail","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $n \\le 3$, let $\\ell_1, \\dots, \\ell_k$ be literals (pairs of a variable index and a polarity) with $3 < n + k$, and let $w$ be the concatenation of the stack tokens of $\\ell_1, \\dots, \\ell_k$. For all lists of symbols $R$ and $U$ and every internal state $v$, the capped filter program, started at the walking label for $n$ literals with internal state $v$, token stack $wR$ (top first) and side stack $U$, reaches\u2026","title":"The capped filter fails on a clause with too many literals"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/clause-ok","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.clause_ok","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#clause-ok","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $C$ be a clause, and suppose the cap flag is off or $C$ has at most three literals. Let $\\overline{C}$ be the tokens of $C$ as they lie on the token stack (the clause marker on top, then the literals' tokens, last literal first), and let $w_n$ be the walking label after $n$ literals. For every $n$, all lists of symbols $R$ and $U$ and every value $v$ of the Boolean variable, there is a value $v'$ such that the\u2026","title":"The 3SAT filter walks one clause within the cap"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/clause-fail","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.clause_fail","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#clause-fail","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $C$ be a clause with more than three literals, let $\\overline{C}$ be its tokens as they lie on the token stack (the clause marker on top, then the literals' tokens, last literal first), and let $w_n$ be the walking label after $n$ literals. For every $n$, all lists of symbols $R$ and $U$ and every value $v$ of the Boolean variable, the 3SAT clause-length filter with the cap on reaches label $\\mathsf{fail}$,\u2026","title":"A clause with more than three literals makes the capped 3SAT filter fail"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/cls-ok","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.cls_ok","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#cls-ok","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a clause $C$ let $\\overline{C}$ be its tokens as they lie on the token stack (the clause marker on top, then the literals' tokens, last literal first). Let $C_1, \\dots, C_k$ be clauses such that, for each $i$, the cap flag is off or $C_i$ has at most three literals. Let $X$ be the concatenation $\\overline{C_1} \\cdots \\overline{C_k}$ with $\\overline{C_1}$ on top, and let $w_n$ be the walking label after $n$\u2026","title":"The 3SAT filter walks a list of clauses within the cap"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/cls-fail","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.cls_fail","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#cls-fail","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a clause $C$ let $\\overline{C}$ be its tokens as they lie on the token stack (the clause marker on top, then the literals' tokens, last literal first). For a list of clauses $C_1, \\dots, C_k$, let $X$ be the concatenation $\\overline{C_1} \\cdots \\overline{C_k}$ with $\\overline{C_1}$ on top, and let $w_n$ be the walking label after $n$ literals. If some $C_i$ has more than three literals, then for every list of\u2026","title":"A long clause makes the capped 3SAT filter fail on a list of clauses"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/walk-end","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.walk_end","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#walk-end","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $\\mathrm{cap}$ be a Boolean, $n \\in \\mathbb{N}$, $U$ a list of symbols and $v$ a Boolean, and let $w_n$ be the walking label after $n$ literals. The clause-length filter with cap flag $\\mathrm{cap}$ goes in exactly one step from label $w_n$, local variable $v$, empty token stack and side stack $U$ to the label $\\mathrm{back}$, local variable true, empty token stack and side stack $U$.","title":"An empty token stack ends the walk"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/back-step","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.back_step","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#back-step","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For either value of the cap flag, every token symbol $s$, all lists of symbols $R$ and $M$ and every value $v$ of the Boolean variable, the 3SAT clause-length filter takes exactly one step from label $\\mathsf{back}$ with variable $v$, token stack $R$ and side stack $s$ on top of $M$ to label $\\mathsf{back}$ with variable true, token stack $s$ on top of $R$, and side stack $M$.","title":"One step of the 3SAT filter's return phase"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/back-run","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.back_run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#back-run","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For either value of the cap flag, all lists of symbols $M$ and $R$ such that every symbol of $M$ is a token symbol, and every value $v$ of the Boolean variable, there is a value $v'$ such that the 3SAT clause-length filter takes exactly $|M|$ steps from label $\\mathsf{back}$ with variable $v$, token stack $R$ and side stack $M$ to label $\\mathsf{back}$ with variable $v'$, token stack the reversal of $M$ on top of\u2026","title":"The 3SAT filter's return phase moves the side stack back"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/back-end","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.back_end","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#back-end","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For either value of the cap flag, every list of symbols $R$ and every value $v$ of the Boolean variable, the 3SAT clause-length filter takes exactly one step from label $\\mathsf{back}$ with variable $v$, token stack $R$ and empty side stack to label $\\mathsf{pass}$ with variable true, token stack $R$ and empty side stack.","title":"The return phase ends on an empty side stack"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/run-2","kind":"node","lean":"Cslib.Complexity.Karp.Filter3.run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#run-2","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $c$ be a Boolean (whether the filter is capped), $\\varphi$ a CNF formula and $v$ an internal state, and let $S$ be stacks holding the reversed token list of $\\varphi$ on the token stack $T$ and nothing on the side stack $U$. Let $b$ be the condition that $c$ is false or every clause of $\\varphi$ has at most three literals, and let $N$ be the number of tokens of $\\varphi$. Then the filter program with cap $c$,\u2026","title":"Correctness of the clause-length filter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/emitter","kind":"node","lean":"Cslib.Complexity.Karp.Emitter","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#emitter","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"An emitter is a multi-stack machine, given by a type of stacks and a type of labels (both with decidable equality), an alphabet for each stack and a program with boolean state, together with a start label and a different exit label at which the program halts, a token stack and a different output stack, a bijection from the SAT token symbols to the token stack's alphabet, a bijection from the output stack's alphabet\u2026","title":"Emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/out","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.out","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#out","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For an emitter $E$ with output map $g$ and a bit string $x$: if the length-prefixed CNF decoder reads a CNF formula $\\varphi$ from $x$ with no bits left over, the output is $g(\\varphi)$; otherwise it is the bit string that $E$ reserves as the output for undecodable inputs.","title":"Output of an emitter on a bit string"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/k-3","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.K","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#k-3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The stacks of the tokenize-then-emit machine for an emitter $E$ are the disjoint union of the SAT tokenizer's stacks and $E$'s stacks.","title":"Stacks of the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/untitled-2","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.\u0393","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#untitled-2","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For an emitter $E$, every stack of the combined machine that comes from the SAT tokenizer has the tokenizer's symbol alphabet, and every stack $k$ that comes from $E$ has the alphabet of $k$ in $E$.","title":"Alphabets of the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/l","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.L","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#l","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For an emitter $E$, the labels of the combined machine form the disjoint union of the SAT tokenizer's labels, the labels of $E$, and the two booleans: false labels the step that writes the output $E$ fixes for undecodable input, and true is the exit label.","title":"Labels of the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/fin","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.fin","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#fin","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The labels of the tokenize-then-emit machine for an emitter $E$ are the SAT tokenizer's labels, $E$'s labels and two extra labels indexed by the booleans; the exit label is the extra label indexed by true.","title":"Exit label of the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/e-t","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.eT","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#e-t","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The map from the SAT tokenizer's stacks to the stacks of the tokenize-then-emit machine for an emitter $E$ that sends the tokenizer's token stack $T$ to $E$'s token stack, and each other tokenizer stack ($X$, $W$, $B$, $CC$, $LC$) to its own copy among the tokenizer stacks of the combined machine.","title":"Tokenizer stacks in the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/emb-t","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.embT","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#emb-t","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The stack embedding of the SAT tokenizer into the tokenize-then-emit machine for an emitter $E$: the tokenizer's token stack $T$ goes to $E$'s token stack, with token symbols translated by $E$'s bijection from token symbols to that stack's alphabet, and each other tokenizer stack goes to its own copy among the tokenizer stacks of the combined machine, with symbols left unchanged.","title":"Embedding the tokenizer's stacks into tokenize-then-emit"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/emb-e","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.embE","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#emb-e","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The stack embedding of an emitter $E$ into the tokenize-then-emit machine for $E$: each stack of $E$ goes to its own copy among the emitter stacks of the combined machine, which carries the same alphabet, and symbols are left unchanged.","title":"Embedding the emitter's stacks into tokenize-then-emit"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/t","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.\u03c6T","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#t","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For an emitter $E$, the tokenizer's accepting label goes to the start label of $E$, its rejecting label goes to the label of the combined machine that writes the output $E$ fixes for undecodable input, and every other tokenizer label goes to its own copy among the combined machine's labels.","title":"Renaming the tokenizer's labels"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/e","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.\u03c6E","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#e","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The map from the labels of an emitter $E$ to the labels of the tokenize-then-emit machine for $E$ that sends $E$'s exit label to the machine's exit label, and every other label of $E$ to its own copy among the emitter labels.","title":"Emitter labels in the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/prog-3","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.prog","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#prog-3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For an emitter $E$, the combined machine acts as follows. At a label of the SAT tokenizer it runs the tokenizer's statement for that label, with the tokenizer's token stack replaced by the token stack of $E$ (token symbols identified with those of $E$), its other stacks kept as copies, and its labels renamed: acceptance goes to the start label of $E$, rejection goes to the step that writes the undecodable-input\u2026","title":"Program of the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/start-3","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.start","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#start-3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For an emitter $E$, the combined machine starts at the start label of the SAT tokenizer.","title":"Start label of the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/init","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.init","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#init","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a bit string $x$, the initial stack contents of the tokenize-then-emit machine for an emitter $E$: the tokenizer's input stack $X$ holds the bits of $x$ in order, each written as a bit symbol, and every other stack is empty.","title":"Initial stacks of the tokenize-then-emit machine"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/tok-s","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.tokS","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#tok-s","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For a bit string $x$, the SAT tokenizer's initial stacks hold $x$, as bit symbols, on the formula-bit stack $X$ and leave every other stack empty.","title":"Initial stacks of the tokenizer"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/run-3","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.run","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#run-3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $E$ be an emitter with step bound $B$, and let $x$ be a bit string. The tokenize-then-emit machine for $E$, started at its start label with internal state false, with $x$ as bit symbols on the tokenizer's input stack and every other stack empty, reaches its exit label, with some internal state, in at most $2|x| + 3 + B(|x|)$ steps, and the output stack of $E$ then holds the output on $x$, written in the output\u2026","title":"Tokenize, then emit"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/tok-ks","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.tokKs","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#tok-ks","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The list of the SAT tokenizer's six stacks in the order: the formula-bit stack $X$, the witness-bit stack $W$, the token stack $T$, the table stack $B$, the clause counter $CC$ and the literal counter $LC$.","title":"List of the tokenizer's stacks"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/comp","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.comp","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#comp","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Given an emitter $E$ with finitely many stacks, finite stack alphabets and finitely many labels, and a list of $E$'s stacks that contains every stack exactly once, this is the reduction-host component whose stacks, alphabets, labels and program are those of the tokenize-then-emit machine for $E$, with its stacks listed as the SAT tokenizer's six stacks followed by the given list of $E$'s stacks.","title":"The tokenize-then-emit machine as a host component"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/is-poly-time-out","kind":"node","lean":"Cslib.Complexity.Karp.TokThen.isPolyTime_out","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#is-poly-time-out","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $E$ be an emitter with finitely many stacks, finite stack alphabets and finitely many labels, given together with a list of its stacks that contains every stack exactly once. Write $B$ for the step bound of $E$, $g$ for its output map and $o$ for its output on undecodable inputs, and let $p$ be a polynomial with natural-number coefficients such that $2n + 3 + B(n) \\le p(n)$ for every $n$. Then the map sending a\u2026","title":"Emitter reductions out of SAT are polynomial-time"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/poly-many-one-red-trans","kind":"node","lean":"Cslib.Complexity.Karp.polyManyOneRed_trans","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#poly-many-one-red-trans","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"For decision problems $L_1$, $L_2$ and $L_3$, if $L_1$ has a polynomial-time many-one reduction to $L_2$ and $L_2$ has one to $L_3$, then $L_1$ has a polynomial-time many-one reduction to $L_3$.","title":"Transitivity of polynomial-time many-one reductions"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/np-hard-of-poly-many-one-red","kind":"node","lean":"Cslib.Complexity.Karp.npHard_of_polyManyOneRed","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#np-hard-of-poly-many-one-red","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Let $L$ and $L'$ be decision problems. If every decision problem in NP has a polynomial-time many-one reduction to $L$, and $L$ has a polynomial-time many-one reduction to $L'$, then every decision problem in NP has a polynomial-time many-one reduction to $L'$.","title":"NP-hardness transfers along reductions"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/emit3","kind":"node","lean":"Cslib.Complexity.Karp.emit3","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#emit3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The emitter whose stacks, stack alphabets, labels, program, start label and exit label are those of the 3SAT emitter machine, with that machine's token stack as its token stack and its bit-valued output stack as its output stack. Its output on a CNF formula $\\varphi$ is the binary encoding of the SAT-to-3SAT translation of $\\varphi$, its output on a bit string that encodes no CNF formula is the empty bit string,\u2026","title":"The SAT-to-3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/emit3-ks","kind":"node","lean":"Cslib.Complexity.Karp.emit3Ks","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#emit3-ks","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The list of all six stacks of the SAT-to-3SAT emitter machine, each listed once, in the order: token stack, output stack, fresh-variable counter, two scratch counters, output clause counter.","title":"The stacks of the 3SAT emitter"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/is-poly-time-reduce3","kind":"node","lean":"Cslib.Complexity.Karp.isPolyTime_reduce3","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#is-poly-time-reduce3","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The map on bit strings that sends the binary encoding of a CNF formula $\\varphi$ to the binary encoding of the SAT-to-3SAT translation of $\\varphi$, and sends every bit string that encodes no CNF formula to the empty bit string, is computable in polynomial time.","title":"The SAT-to-3SAT reduction is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/sat-dec-poly-many-one-red-sat3-dec","kind":"node","lean":"Cslib.Complexity.Karp.satDec_polyManyOneRed_sat3Dec","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#sat-dec-poly-many-one-red-sat3-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The decision problem SAT, which accepts exactly the bit strings that encode a satisfiable CNF formula, has a polynomial-time many-one reduction to the decision problem 3SAT, which accepts exactly the bit strings that encode a satisfiable CNF formula with at most three literals per clause.","title":"SAT reduces to 3SAT"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/sat3-dec-np-hard","kind":"node","lean":"Cslib.Complexity.Karp.sat3Dec_npHard","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#sat3-dec-np-hard","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"Every decision problem in NP has a polynomial-time many-one reduction to the decision problem 3SAT, which accepts exactly the bit strings that encode a satisfiable CNF formula with at most three literals per clause.","title":"3SAT is NP-hard"},{"id":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/sat3-dec-np-complete","kind":"node","lean":"Cslib.Complexity.Karp.sat3Dec_npComplete","parent":31369,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/3-satisfiability/index.html#sat3-dec-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/3-satisfiability","status":"fully_proved","summary":"The decision problem 3SAT, which accepts exactly the bit strings that encode a satisfiable CNF formula with at most three literals per clause, is in NP, and every decision problem in NP has a polynomial-time many-one reduction to 3SAT.","title":"3SAT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/betweenness.json","id":"TheoryOfComputation/np-completeness/satisfiability/betweenness","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/betweenness/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/betweenness","status":"can_prove","title":"Betweenness"},{"id":"TheoryOfComputation/np-completeness/satisfiability/betweenness/betweenness-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Betweenness.BetweennessInstance","parent":31479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/betweenness/index.html#betweenness-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/betweenness","status":"fully_proved","summary":"An instance of BETWEENNESS is a pair $(n, T)$ of a natural number $n$ (the number of elements) and a list $T$ of triples of natural numbers.","title":"BETWEENNESS instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/betweenness/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Betweenness.WellFormed","parent":31479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/betweenness/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/betweenness","status":"fully_proved","summary":"A BETWEENNESS instance $(n, T)$ is well formed when every triple $(a, b, c)$ in $T$ satisfies $a < n$, $b < n$ and $c < n$, and $a$, $b$, $c$ are pairwise distinct.","title":"Well-formed BETWEENNESS instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/betweenness/betweenness-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Betweenness.BetweennessFeasible","parent":31479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/betweenness/index.html#betweenness-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/betweenness","status":"fully_proved","summary":"A BETWEENNESS instance $(n, T)$ is feasible when there is an ordering of the numbers $0, \\dots, n-1$ such that for every triple $(a, b, c)$ in $T$, either $p(a) < p(b) < p(c)$ or $p(c) < p(b) < p(a)$, where $p(m)$ is the position of $m$ in the ordering, counting from $0$, and $p(m) = n$ when $m$ is not among $0, \\dots, n-1$.","title":"BETWEENNESS feasibility"},{"id":"TheoryOfComputation/np-completeness/satisfiability/betweenness/betweenness-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Betweenness.betweennessDec","parent":31479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/betweenness/index.html#betweenness-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/betweenness","status":"fully_proved","summary":"BETWEENNESS accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs $(n, T)$ of a natural number and a list of triples of natural numbers, to a BETWEENNESS instance that is both well formed and feasible. Bitstrings that do not decode to such a pair are rejected.","title":"The BETWEENNESS decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/betweenness/betweenness-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Betweenness.betweenness_npComplete","parent":31479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/betweenness/index.html#betweenness-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/betweenness","status":"can_prove","summary":"The BETWEENNESS decision problem belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"BETWEENNESS is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction.json","id":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction","status":"can_prove","title":"Binary constraint satisfaction"},{"id":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/binary-csp-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinaryCsp.BinaryCspInstance","parent":31485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/index.html#binary-csp-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction","status":"fully_proved","summary":"A binary CSP instance is a pair consisting of a list $(D_0, \\dots, D_{n-1})$ of lists of natural numbers, where $D_j$ is the domain of variable $j$, and a list of constraints, each a triple $(j, k, P)$ of two variable indices $j, k \\in \\mathbb{N}$ and a list $P$ of allowed value pairs in $\\mathbb{N} \\times \\mathbb{N}$.","title":"Binary CSP instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinaryCsp.WellFormed","parent":31485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction","status":"fully_proved","summary":"A binary CSP instance with domain list $(D_0, \\dots, D_{n-1})$ and constraint list $C$ is well formed when every constraint $(j, k, P) \\in C$ has $j < n$ and $k < n$.","title":"Well-formed binary CSP instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/binary-csp-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinaryCsp.BinaryCspFeasible","parent":31485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/index.html#binary-csp-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction","status":"fully_proved","summary":"A binary CSP instance with domain list $(D_0, \\dots, D_{n-1})$ and constraint list $C$ is feasible when there is a list $(a_0, \\dots, a_{n-1})$ of natural numbers with $a_j \\in D_j$ for every $j < n$ and $(a_j, a_k) \\in P$ for every constraint $(j, k, P) \\in C$, where $a_i$ is read as $0$ for every index $i \\ge n$.","title":"Feasible binary CSP instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/binary-csp-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinaryCsp.binaryCspDec","parent":31485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/index.html#binary-csp-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction","status":"fully_proved","summary":"The decision problem BINARY CONSTRAINT SATISFACTION accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of binary CSP instances, to an instance that is well formed and feasible; it rejects every other bitstring, including every bitstring that does not decode.","title":"Binary constraint satisfaction as a decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/binary-csp-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinaryCsp.binaryCsp_npComplete","parent":31485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction/index.html#binary-csp-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/binary-constraint-satisfaction","status":"can_prove","summary":"The decision problem BINARY CONSTRAINT SATISFACTION over bitstrings belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"BINARY CONSTRAINT SATISFACTION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability.json","id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"can_prove","title":"Circuit satisfiability"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/to-fin","kind":"node","lean":"Cslib.Circuits.Wire.toFin","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#to-fin","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"For a wire that refers to one of $n$ inputs or one of $g$ gates, its position in $\\{0, \\dots, n + g - 1\\}$ is $k$ if it is input $k$ and $n + j$ if it is gate $j$, so the inputs come first and the gates follow.","title":"Certificate position of a wire"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/verify-boolean-certificate","kind":"node","lean":"Cslib.Circuits.Circuit.verifyBooleanCertificate","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#verify-boolean-certificate","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Let $c$ be a circuit over the Boolean signature with $n$ inputs, $s$ gates and one output, and let $b = (b_0, \\dots, b_{m-1})$ be an array of truth values, read as a value for each wire: input $k$ at position $k$ and gate $j$ at position $n + j$. The verifier is a computation with a counted cost. It first checks whether $m = n + s$, at cost $1$; if not, it returns false. Otherwise, for each gate $j < s$ in order,\u2026","title":"Boolean certificate verifier"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/verify-boolean-certificate-iff","kind":"node","lean":"Cslib.Circuits.Circuit.verifyBooleanCertificate_iff","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#verify-boolean-certificate-iff","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Let $c$ be a circuit over the Boolean signature with $n$ inputs, $s$ gates and one output, and let $b = (b_0, \\dots, b_{m-1})$ be an array of truth values, where an entry at a position at or beyond $m$ is read as false and the position of a wire is $k$ for input $k$ and $n + j$ for gate $j$. The Boolean certificate verifier accepts $b$ if and only if $m = n + s$, for every gate $j < s$ the Boolean interpretation of\u2026","title":"Characterization of accepted certificates"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/verify-boolean-certificate-sound","kind":"node","lean":"Cslib.Circuits.Circuit.verifyBooleanCertificate_sound","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#verify-boolean-certificate-sound","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Let $c$ be a circuit over the Boolean signature with $n$ inputs, $s$ gates and one output, and let $b = (b_0, \\dots, b_{m-1})$ be an array of truth values, where an entry at a position at or beyond $m$ is read as false and the position of a wire is $k$ for input $k$ and $n + j$ for gate $j$. If the Boolean certificate verifier accepts $b$, then, writing $x = (b_0, \\dots, b_{n-1})$, the value of every wire of $c$ on\u2026","title":"Soundness of the Boolean certificate verifier"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/verify-boolean-certificate-complete","kind":"node","lean":"Cslib.Circuits.Circuit.verifyBooleanCertificate_complete","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#verify-boolean-certificate-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Let $c$ be a circuit over the Boolean signature with $n$ inputs, $s$ gates and one output, and let $x = (x_0, \\dots, x_{n-1})$ be truth values on which the output of $c$ under the Boolean interpretation is true. Let $b$ be the array $(x_0, \\dots, x_{n-1}, v_0, \\dots, v_{s-1})$, where $v_j$ is the value of gate $j$ of $c$ on input $x$. Then $b$ has length $n + s$ and the Boolean certificate verifier accepts $b$.","title":"Completeness of the Boolean certificate verifier"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/verify-boolean-certificate-satisfiable-i","kind":"node","lean":"Cslib.Circuits.Circuit.verifyBooleanCertificate_satisfiable_iff","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#verify-boolean-certificate-satisfiable-i","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Let $c$ be a circuit over the Boolean signature with $n$ inputs, $s$ gates and one output. The Boolean certificate verifier accepts some array of at most $n + 2s$ truth values if and only if there are truth values $x_0, \\dots, x_{n-1}$ on which the output of $c$ under the Boolean interpretation is true.","title":"Short accepted certificates exist exactly for satisfiable circuits"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/verify-boolean-certificate-time","kind":"node","lean":"Cslib.Circuits.Circuit.verifyBooleanCertificate_time","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#verify-boolean-certificate-time","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Let $c$ be a circuit over the Boolean signature with $n$ inputs, $s$ gates and one output, and let $b$ be an array of $m$ truth values. The cost counted by the Boolean certificate verifier on $b$ is $2 + s + \\sum_{j < s} a_j$ if $m = n + s$, where $a_j$ is the number of arguments of gate $j$, and $1$ otherwise.","title":"Cost of the Boolean certificate verifier"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/verify-boolean-certificate-time-le-of-fa","kind":"node","lean":"Cslib.Circuits.Circuit.verifyBooleanCertificate_time_le_of_fanInAtMost","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#verify-boolean-certificate-time-le-of-fa","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Let $K$ be a natural number and let $c$ be a circuit over the Boolean signature with $n$ inputs, $s$ gates and one output in which every gate has at most $K$ arguments. Then for every array $b$ of truth values, the cost counted by the Boolean certificate verifier on $b$ is at most $2 + (K + 1)s$.","title":"Linear cost of the Boolean certificate verifier"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/circuit-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CircuitSat.CircuitSatInstance","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#circuit-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"An instance of circuit satisfiability is a pair $(n, G)$ of a natural number $n$, the number of variables, and a finite list $G$ of gates, each gate a triple $(o, a, b)$ of natural numbers: an operation code $o$, a first input $a$ and a second input $b$.","title":"Circuit-satisfiability instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/circuit-ref","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CircuitSat.CircuitRef","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#circuit-ref","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Given an assignment $x = (x_0, \\dots, x_{n-1})$ of booleans to the variables and the list $v = (v_0, \\dots, v_{k-1})$ of the values of the gates evaluated so far, an input index $r$ resolves to $x_r$ if $r < n$, to $v_{r-n}$ if $n \\le r < n + k$, and to false if $r \\ge n + k$.","title":"Resolving a circuit input"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/gate-value","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CircuitSat.GateValue","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#gate-value","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Let $x = (x_0, \\dots, x_{n-1})$ be an assignment of booleans to the variables and $v$ the list of values of the earlier gates. An input index $r$ resolves to $x_r$ if $r < n$, and otherwise to $v_{r-n}$, or to false if that entry does not exist. The value of a gate $(o, a, b)$ is the conjunction of the resolved inputs $a$ and $b$ if $o = 0$, their disjunction if $o = 1$, the negation of the resolved input $a$ if\u2026","title":"Value of a gate"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/eval-circuit","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CircuitSat.EvalCircuit","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#eval-circuit","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"Given an assignment of booleans to the variables and a list of gates $g_1, \\dots, g_m$, circuit evaluation returns the list $(v_1, \\dots, v_m)$ of gate values, where $v_k$ is the value of the gate $g_k$ under the assignment and the earlier values $(v_1, \\dots, v_{k-1})$.","title":"Evaluating a circuit"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CircuitSat.WellFormed","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"An instance $(n, G)$ with gates $G = (g_0, \\dots, g_{m-1})$ is well formed when every gate $g_j = (o, a, b)$ satisfies $o \\le 5$, and moreover: if $o = 0$ or $o = 1$ then $a < n + j$ and $b < n + j$; if $o = 2$ then $a < n + j$; and if $o = 3$ then $a < n$.","title":"Well-formed circuit-satisfiability instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/circuit-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CircuitSat.CircuitSatFeasible","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#circuit-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"An instance $(n, G)$ is satisfiable when there is an assignment $x \\in \\{\\mathrm{true}, \\mathrm{false}\\}^n$ of booleans to the $n$ variables such that the evaluation of the circuit $G$ under $x$ has a last gate value and that value is true. In particular an instance with no gates is not satisfiable.","title":"Satisfiable circuit instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/circuit-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CircuitSat.circuitSatDec","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#circuit-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"fully_proved","summary":"CIRCUIT SAT is the decision problem that accepts a bitstring $s$ exactly when $s$ decodes, under the canonical bitstring encoding of circuit-satisfiability instances, to an instance that is well formed and satisfiable. Bitstrings that fail to decode are rejected.","title":"The CIRCUIT SAT decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/circuit-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CircuitSat.circuitSat_npComplete","parent":31491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability/index.html#circuit-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/circuit-satisfiability","status":"can_prove","summary":"The CIRCUIT SAT decision problem on bitstrings belongs to the class NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"CIRCUIT SAT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability.json","id":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability","status":"can_prove","title":"Maximum 2-satisfiability"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/max-two-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxTwoSat.MaxTwoSatInstance","parent":31508,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/index.html#max-two-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability","status":"fully_proved","summary":"A MAX-2SAT instance is a triple $(n, \\varphi, K)$ of a natural number $n$ (the number of variables), a CNF formula $\\varphi$ and a natural number $K$ (the bound). The type does not restrict clause lengths; that clauses have at most two literals is part of well-formedness.","title":"MAX-2SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxTwoSat.WellFormed","parent":31508,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability","status":"fully_proved","summary":"A MAX-2SAT instance $(n, \\varphi, K)$ is well formed when every clause of $\\varphi$ has at most two literals and every literal $(x, b)$ in it has variable index $x < n$.","title":"Well-formed MAX-2SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/max-two-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxTwoSat.MaxTwoSatFeasible","parent":31508,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/index.html#max-two-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability","status":"fully_proved","summary":"A MAX-2SAT instance $(n, \\varphi, K)$ is feasible when there are truth values $a_0, \\dots, a_{n-1}$ such that at least $K$ of the clauses in the list $\\varphi$, counted with multiplicity, are satisfied by the assignment $v$ with $v(x) = a_x$ for $x < n$ and $v(x) = \\mathrm{false}$ for $x \\ge n$.","title":"Feasible MAX-2SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/max-two-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxTwoSat.maxTwoSatDec","parent":31508,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/index.html#max-two-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability","status":"fully_proved","summary":"The decision problem MAXIMUM 2-SATISFIABILITY accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of MAX-2SAT instances, to an instance $(n, \\varphi, K)$ that is well formed (every clause of $\\varphi$ has at most two literals, and every literal has variable index less than $n$) and feasible (some assignment of truth values to the $n$ variables satisfies at least $K$ of the\u2026","title":"MAXIMUM 2-SATISFIABILITY as a decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/max-two-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxTwoSat.maxTwoSat_npComplete","parent":31508,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability/index.html#max-two-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-2-satisfiability","status":"can_prove","summary":"The decision problem MAXIMUM 2-SATISFIABILITY over bitstrings lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"MAXIMUM 2-SATISFIABILITY is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability.json","id":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability","status":"can_prove","title":"Maximum satisfiability"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/max-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxSat.MaxSatInstance","parent":31514,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/index.html#max-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability","status":"fully_proved","summary":"A MAX-SAT instance is a triple $(n, \\varphi, K)$ of a natural number $n$ (the number of variables), a CNF formula $\\varphi$ and a natural number $K$ (the bound).","title":"MAX-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxSat.WellFormed","parent":31514,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability","status":"fully_proved","summary":"A MAX-SAT instance $(n, \\varphi, K)$ is well formed when every literal $(x, b)$ in every clause of $\\varphi$ has variable index $x < n$.","title":"Well-formed MAX-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/max-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxSat.MaxSatFeasible","parent":31514,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/index.html#max-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability","status":"fully_proved","summary":"A MAX-SAT instance $(n, \\varphi, K)$ is feasible when there are truth values $a_0, \\dots, a_{n-1}$ such that at least $K$ of the clauses in the list $\\varphi$, counted with multiplicity, are satisfied by the assignment $v$ with $v(x) = a_x$ for $x < n$ and $v(x) = \\mathrm{false}$ for $x \\ge n$.","title":"Feasible MAX-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/max-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxSat.maxSatDec","parent":31514,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/index.html#max-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability","status":"fully_proved","summary":"The decision problem MAXIMUM SATISFIABILITY accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of MAX-SAT instances, to an instance $(n, \\varphi, K)$ that is well formed (every literal of $\\varphi$ has variable index less than $n$) and feasible (some assignment of truth values to the $n$ variables satisfies at least $K$ of the clauses of $\\varphi$, counted with multiplicity).\u2026","title":"MAXIMUM SATISFIABILITY as a decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/max-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxSat.maxSat_npComplete","parent":31514,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability/index.html#max-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/maximum-satisfiability","status":"can_prove","summary":"The decision problem MAXIMUM SATISFIABILITY over bitstrings lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"MAXIMUM SATISFIABILITY is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability.json","id":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability","status":"can_prove","title":"Minimum 2-satisfiability"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/min-two-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinTwoSat.MinTwoSatInstance","parent":31520,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/index.html#min-two-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability","status":"fully_proved","summary":"An instance of MINIMUM 2-SATISFIABILITY is a triple $(n, \\varphi, K)$ of a natural number $n$ (the number of variables), a CNF formula $\\varphi$ and a natural number $K$ (the bound). No condition is placed on $\\varphi$ here; clauses of at most two literals are required separately, by well-formedness.","title":"MINIMUM 2-SATISFIABILITY instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinTwoSat.WellFormed","parent":31520,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability","status":"fully_proved","summary":"A triple $(n, \\varphi, K)$ of a natural number $n$, a CNF formula $\\varphi$ and a natural number $K$ is well formed when every clause of $\\varphi$ has at most two literals and every literal of every clause has variable index less than $n$.","title":"Well-formed MINIMUM 2-SATISFIABILITY instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/min-two-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinTwoSat.MinTwoSatFeasible","parent":31520,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/index.html#min-two-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability","status":"fully_proved","summary":"A triple $(n, \\varphi, K)$ of a natural number $n$, a CNF formula $\\varphi$ with clauses $C_0, \\dots, C_{m-1}$ and a natural number $K$ is feasible when there is an assignment of truth values to the variables $0, \\dots, n-1$ that, with every variable $x \\ge n$ set to false, satisfies $C_j$ for at most $K$ indices $j < m$.","title":"Feasible MINIMUM 2-SATISFIABILITY instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/min-two-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinTwoSat.minTwoSatDec","parent":31520,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/index.html#min-two-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability","status":"fully_proved","summary":"MINIMUM 2-SATISFIABILITY is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, \\varphi, K)$ of a natural number $n$, a CNF formula $\\varphi$ with clauses $C_0, \\dots, C_{m-1}$ and a natural number $K$ such that every clause has at most two literals, all with variables less than $n$, and some assignment of truth values to the variables\u2026","title":"The MINIMUM 2-SATISFIABILITY decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/min-two-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinTwoSat.minTwoSat_npComplete","parent":31520,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability/index.html#min-two-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-2-satisfiability","status":"can_prove","summary":"Let MINIMUM 2-SATISFIABILITY be the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, \\varphi, K)$ of a natural number $n$, a CNF formula $\\varphi$ with clauses $C_0, \\dots, C_{m-1}$ and a natural number $K$ such that every clause has at most two literals, all with variables less than $n$, and some assignment of truth values to the\u2026","title":"MINIMUM 2-SATISFIABILITY is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat.json","id":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat","status":"can_prove","title":"Minimum-ones 3-SAT"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/min-ones-three-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinOnesThreeSat.MinOnesThreeSatInstance","parent":31526,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/index.html#min-ones-three-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat","status":"fully_proved","summary":"An instance of MINIMUM ONES 3-SAT is a triple $(n, \\varphi, K)$ of a natural number $n$ (the number of variables), a CNF formula $\\varphi$ and a natural number $K$ (the bound). No condition is placed on $\\varphi$ here; clauses of exactly three literals are required separately, by well-formedness.","title":"MINIMUM ONES 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinOnesThreeSat.WellFormed","parent":31526,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat","status":"fully_proved","summary":"A triple $(n, \\varphi, K)$ of a natural number $n$, a CNF formula $\\varphi$ and a natural number $K$ is well formed when every clause of $\\varphi$ is a list of exactly three literals and every literal of every clause has variable index less than $n$.","title":"Well-formed MINIMUM ONES 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/min-ones-three-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinOnesThreeSat.MinOnesThreeSatFeasible","parent":31526,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/index.html#min-ones-three-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat","status":"fully_proved","summary":"A triple $(n, \\varphi, K)$ of a natural number $n$, a CNF formula $\\varphi$ and a natural number $K$ is feasible when there is an assignment of truth values to the variables $0, \\dots, n-1$ that, with every variable $x \\ge n$ set to false, satisfies $\\varphi$ and sets at most $K$ of the variables $0, \\dots, n-1$ to true.","title":"Feasible MINIMUM ONES 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/min-ones-three-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinOnesThreeSat.minOnesThreeSatDec","parent":31526,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/index.html#min-ones-three-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat","status":"fully_proved","summary":"MINIMUM ONES 3-SAT is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, \\varphi, K)$ of a natural number $n$, a CNF formula $\\varphi$ and a natural number $K$ such that every clause of $\\varphi$ is a list of exactly three literals whose variables are all less than $n$, and some assignment of truth values to the variables\u2026","title":"The MINIMUM ONES 3-SAT decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/min-ones-three-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinOnesThreeSat.minOnesThreeSat_npComplete","parent":31526,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat/index.html#min-ones-three-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/minimum-ones-3-sat","status":"can_prove","summary":"Let MINIMUM ONES 3-SAT be the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, \\varphi, K)$ of a natural number $n$, a CNF formula $\\varphi$ and a natural number $K$ such that every clause of $\\varphi$ is a list of exactly three literals whose variables are all less than $n$, and some assignment of truth values to the variables\u2026","title":"MINIMUM ONES 3-SAT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat.json","id":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat","status":"can_prove","title":"Monotone 3-SAT"},{"id":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/monotone-three-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MonotoneThreeSat.MonotoneThreeSatInstance","parent":31532,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/index.html#monotone-three-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat","status":"fully_proved","summary":"A monotone 3-SAT instance is a pair $(n, \\varphi)$ of a natural number $n$, the number of variables, and a CNF formula $\\varphi$.","title":"Monotone 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MonotoneThreeSat.WellFormed","parent":31532,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat","status":"fully_proved","summary":"A monotone 3-SAT instance $(n, \\varphi)$ is well formed when every clause of the CNF formula $\\varphi$ is a list of exactly three literals (repetitions allowed), each with a variable below $n$, and the literals of each clause are all positive or all negative.","title":"Well-formed monotone 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/monotone-three-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MonotoneThreeSat.MonotoneThreeSatFeasible","parent":31532,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/index.html#monotone-three-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat","status":"fully_proved","summary":"A monotone 3-SAT instance $(n, \\varphi)$ is feasible when there is a list of $n$ truth values $a_0, \\dots, a_{n-1}$ such that the assignment sending each variable $x < n$ to $a_x$ and every variable $x \\ge n$ to false satisfies the CNF formula $\\varphi$.","title":"Satisfiable monotone 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/monotone-three-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MonotoneThreeSat.monotoneThreeSatDec","parent":31532,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/index.html#monotone-three-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat","status":"fully_proved","summary":"The decision problem MONOTONE 3-SAT accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a pair $(n, \\varphi)$ of a natural number $n$ and a CNF formula $\\varphi$ such that every clause of $\\varphi$ is a list of exactly three literals (repetitions allowed), every literal of $\\varphi$ has a variable below $n$, the literals of each clause are all positive or all negative, and\u2026","title":"MONOTONE 3-SAT as a decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/monotone-three-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MonotoneThreeSat.monotoneThreeSat_npComplete","parent":31532,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat/index.html#monotone-three-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/monotone-3-sat","status":"can_prove","summary":"The decision problem MONOTONE 3-SAT belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to MONOTONE 3-SAT.","title":"MONOTONE 3-SAT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat.json","id":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat","kind":"scope","navigation":"module","parent":31356,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/index.html","scope":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat","status":"can_prove","title":"Not-all-equal 3-SAT"},{"id":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/nae-three-sat-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.NaeThreeSat.NaeThreeSatInstance","parent":31538,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/index.html#nae-three-sat-instance","scope":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat","status":"fully_proved","summary":"A NOT-ALL-EQUAL 3-SAT instance is a pair $(n, \\varphi)$ of a natural number $n$, the number of variables, and a CNF formula $\\varphi$. The formula is not required to have three literals per clause; that is part of well-formedness.","title":"NOT-ALL-EQUAL 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.NaeThreeSat.WellFormed","parent":31538,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat","status":"fully_proved","summary":"A NOT-ALL-EQUAL 3-SAT instance $(n, \\varphi)$ is well-formed when every clause of $\\varphi$ is a list of exactly three literals, repetitions allowed, and every literal $(x, b)$ in it has variable index $x < n$.","title":"Well-formed NOT-ALL-EQUAL 3-SAT instances"},{"id":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/nae-three-sat-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.NaeThreeSat.NaeThreeSatFeasible","parent":31538,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/index.html#nae-three-sat-feasible","scope":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat","status":"fully_proved","summary":"A NOT-ALL-EQUAL 3-SAT instance $(n, \\varphi)$, with $n \\in \\mathbb{N}$ and $\\varphi$ a CNF formula, is not-all-equal satisfiable when there are truth values $a_0, \\dots, a_{n-1}$ such that, for the assignment $v$ with $v(x) = a_x$ for $x < n$ and $v(x) = \\mathrm{false}$ for $x \\ge n$, every clause of $\\varphi$ contains a literal satisfied by $v$ and a literal not satisfied by $v$.","title":"Not-all-equal satisfiability of an instance"},{"id":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/nae-three-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.NaeThreeSat.naeThreeSatDec","parent":31538,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/index.html#nae-three-sat-dec","scope":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat","status":"fully_proved","summary":"The decision problem NOT-ALL-EQUAL 3-SAT accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a NOT-ALL-EQUAL 3-SAT instance $(n, \\varphi)$ that is well-formed and not-all-equal satisfiable. Every bitstring that decodes to no instance is rejected.","title":"NOT-ALL-EQUAL 3-SAT as a decision problem"},{"id":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/nae-three-sat-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.NaeThreeSat.naeThreeSat_npComplete","parent":31538,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat/index.html#nae-three-sat-np-complete","scope":"TheoryOfComputation/np-completeness/satisfiability/not-all-equal-3-sat","status":"can_prove","summary":"The decision problem NOT-ALL-EQUAL 3-SAT over bitstrings belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"NOT-ALL-EQUAL 3-SAT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination","status":"can_prove","summary":"Clique, independent set, vertex cover, domination variants, feedback vertex and arc sets, odd cycle transversal, bicliques, densest subgraph, induced matching and minimum fill-in.","title":"Cliques, covers and domination in graphs"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique","status":"can_prove","title":"Balanced biclique"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/balanced-biclique-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BalancedBiclique.BalancedBicliqueInstance","parent":31545,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/index.html#balanced-biclique-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique","status":"fully_proved","summary":"An instance of BALANCED BICLIQUE is a triple $(n, es, K)$ consisting of a natural number $n$ (the number of vertices), a list $es$ of pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"Balanced biclique instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BalancedBiclique.Graph","parent":31545,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique","status":"fully_proved","summary":"The graph of a balanced biclique instance $(n, es, K)$ is the simple graph on $\\{0, \\dots, n-1\\}$ in which distinct vertices $u$ and $v$ are adjacent when $(u, v)$ or $(v, u)$ belongs to $es$.","title":"Graph of a balanced biclique instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BalancedBiclique.WellFormed","parent":31545,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique","status":"fully_proved","summary":"A balanced biclique instance $(n, es, K)$ is well formed when the entries of the list $es$ are distinct and every pair $(u, v)$ in $es$ satisfies $u < n$, $v < n$, $u \\neq v$ and $(v, u) \\notin es$.","title":"Well-formed balanced biclique instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/balanced-biclique-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BalancedBiclique.BalancedBicliqueFeasible","parent":31545,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/index.html#balanced-biclique-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique","status":"fully_proved","summary":"A balanced biclique instance $(n, es, K)$ is feasible when there are disjoint sets $A$ and $B$ of exactly $K$ vertices each in $\\{0, \\dots, n-1\\}$ such that every vertex of $A$ is adjacent to every vertex of $B$, where distinct vertices $u$ and $v$ are adjacent when $(u, v)$ or $(v, u)$ belongs to $es$. In other words, the graph contains a complete bipartite subgraph $K_{K,K}$, not necessarily induced.","title":"Feasible balanced biclique instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/balanced-biclique-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BalancedBiclique.balancedBicliqueDec","parent":31545,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/index.html#balanced-biclique-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique","status":"fully_proved","summary":"BALANCED BICLIQUE is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a balanced biclique instance $(n, es, K)$ that is well formed (the entries of $es$ are distinct, and every pair $(u, v)$ in $es$ has $u < n$, $v < n$, $u \\neq v$ and $(v, u) \\notin es$) and for which there are disjoint sets $A$ and $B$ of exactly $K$ vertices each in\u2026","title":"The decision problem BALANCED BICLIQUE"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/balanced-biclique-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BalancedBiclique.balancedBiclique_npComplete","parent":31545,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique/index.html#balanced-biclique-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/balanced-biclique","status":"can_prove","summary":"The decision problem BALANCED BICLIQUE belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"BALANCED BICLIQUE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover","status":"can_prove","title":"Biclique edge cover"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/biclique-edge-cover-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BicliqueEdgeCover.BicliqueEdgeCoverInstance","parent":31552,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/index.html#biclique-edge-cover-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover","status":"fully_proved","summary":"An instance of BICLIQUE EDGE COVER is a tuple $(L, R, E, K)$ of natural numbers $L$, $R$ and $K$ and a finite list $E$ of pairs of natural numbers. Here $L$ counts the left vertices, $R$ counts the right vertices, each pair $(a, b)$ in $E$ is an edge from left vertex $a$ to right vertex $b$, and $K$ is the bound on the number of bicliques.","title":"Biclique edge cover instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BicliqueEdgeCover.WellFormed","parent":31552,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover","status":"fully_proved","summary":"An instance $(L, R, E, K)$ of BICLIQUE EDGE COVER is well formed when the list $E$ has no repeated entries and every pair $(a, b)$ in $E$ satisfies $a < L$ and $b < R$.","title":"Well-formed biclique edge cover instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/biclique-edge-cover-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BicliqueEdgeCover.BicliqueEdgeCoverFeasible","parent":31552,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/index.html#biclique-edge-cover-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover","status":"fully_proved","summary":"An instance $(L, R, E, K)$ of BICLIQUE EDGE COVER is feasible when there is a list of at most $K$ pairs $(A, B)$ of finite lists of natural numbers, with every element of $A$ less than $L$ and every element of $B$ less than $R$, such that for every pair $(u, v)$ in $E$ some listed pair $(A, B)$ has $u \\in A$, $v \\in B$ and $(a, b)$ in $E$ for all $a \\in A$ and $b \\in B$.","title":"Feasible biclique edge cover instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/biclique-edge-cover-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BicliqueEdgeCover.bicliqueEdgeCoverDec","parent":31552,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/index.html#biclique-edge-cover-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover","status":"fully_proved","summary":"BICLIQUE EDGE COVER is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of biclique edge cover instances, to an instance that is well formed and feasible; it rejects every bitstring that fails to decode.","title":"The biclique edge cover decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/biclique-edge-cover-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BicliqueEdgeCover.bicliqueEdgeCover_npComplete","parent":31552,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover/index.html#biclique-edge-cover-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/biclique-edge-cover","status":"can_prove","summary":"The biclique edge cover decision problem belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"BICLIQUE EDGE COVER is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover","status":"fully_proved","title":"Clique, independent set and vertex cover"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","kind":"scope","navigation":"transparent","parent":31558,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","title":"Vertex pairs in encoding order to The vertex pairs are listed without repetition"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/pairs","kind":"node","lean":"Cslib.Complexity.Karp.pairs","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#pairs","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"For a natural number $n$, the list of all pairs $(i, j)$ of natural numbers with $i < j < n$, ordered by $i$ ascending and, for each $i$, by $j$ descending from $n - 1$ to $i + 1$.","title":"Vertex pairs in encoding order"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/adj-bits","kind":"node","lean":"Cslib.Complexity.Karp.adjBits","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#adj-bits","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"Let $n$, $i$, $j$ be natural numbers and $b$ a list of bits. List the pairs $(i', j')$ with $i' < j' < n$ in encoding order ($i'$ ascending and, for each $i'$, $j'$ descending) and match the $t$-th pair with the $t$-th bit of $b$, for as many positions as both lists have. The adjacency bit of $\\{i, j\\}$ is the bit matched with $(\\min(i, j), \\max(i, j))$, and false when that pair has no matched bit (for instance\u2026","title":"Adjacency bit of a vertex pair"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/to-graph","kind":"node","lean":"Cslib.Complexity.Karp.toGraph","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#to-graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"List the pairs $(i, j)$ of natural numbers with $i < j < n$ by $i$ ascending and, for each $i$, by $j$ descending, and give the $t$-th pair the $t$-th bit of a list of bits. The simple graph on the vertex set $\\{0, \\dots, n - 1\\}$ given by the list makes two vertices $i < j$ adjacent when the bit of $(i, j)$ is true; a pair left without a bit because the list is too short is not an edge, and bits beyond the last\u2026","title":"The graph encoded by pair bits"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/encode-kg","kind":"node","lean":"Cslib.Complexity.Karp.encodeKG","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#encode-kg","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"For natural numbers $k$, $n$ and a list of bits $b$, the encoding is $1^k\\,0\\,1^n\\,0\\,b$, writing $1$ for true and $0$ for false: $k$ ones, a zero, $n$ ones, a zero, and then $b$. Here $k$ is the target size and $b$ lists the pair bits of a graph on $n$ vertices.","title":"Encoding of a graph instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/decode-kg","kind":"node","lean":"Cslib.Complexity.Karp.decodeKG","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#decode-kg","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, a bit string of the form $1^k\\,0\\,1^n\\,0\\,b$ with natural numbers $k$, $n$ and a list of bits $b$ decodes to the triple $(k, n, b)$. A bit string with fewer than two zeros does not decode.","title":"Decoding a graph instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/no-kg","kind":"node","lean":"Cslib.Complexity.Karp.noKG","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#no-kg","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"The bit string $1\\,0\\,0$ (writing $1$ for true and $0$ for false), which encodes $k = 1$ together with the graph with no vertices ($n = 0$ and no pair bits).","title":"Fixed no-instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/kg-dec","kind":"node","lean":"Cslib.Complexity.Karp.kgDec","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#kg-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"Let $P$ be a property of a natural number $n$, a natural number $k$ and a simple graph on $\\{0, \\dots, n-1\\}$. The decision problem accepts a bit string $x$ exactly when, writing $1$ for true and $0$ for false, $x = 1^k\\,0\\,1^n\\,0\\,b$ for some natural numbers $k$, $n$ and a list of bits $b$ with one entry for each pair $i < j < n$, such that $P$ holds of $n$, $k$ and the graph on $\\{0, \\dots, n-1\\}$ given by $b$.\u2026","title":"Language of graph instances with a property"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/clique-dec","kind":"node","lean":"Cslib.Complexity.Karp.cliqueDec","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#clique-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"The decision problem CLIQUE accepts a bit string $x$ exactly when, writing $1$ for true and $0$ for false, $x = 1^k\\,0\\,1^n\\,0\\,b$ for some natural numbers $k$, $n$ and a list of bits $b$ with one entry for each pair $i < j < n$, such that the graph on $\\{0, \\dots, n-1\\}$ given by $b$ has a clique of exactly $k$ vertices. In that graph, distinct $i$ and $j$ are adjacent when the bit of the pair $\\{i, j\\}$ is true,\u2026","title":"CLIQUE"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/ind-set-dec","kind":"node","lean":"Cslib.Complexity.Karp.indSetDec","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#ind-set-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"The decision problem INDEPENDENT-SET accepts a bit string $x$ exactly when, writing $1$ for true and $0$ for false, $x = 1^k\\,0\\,1^n\\,0\\,b$ for some natural numbers $k$, $n$ and a list of bits $b$ with one entry for each pair $i < j < n$, such that the graph on $\\{0, \\dots, n-1\\}$ given by $b$ has an independent set of exactly $k$ vertices. In that graph, distinct $i$ and $j$ are adjacent when the bit of the pair\u2026","title":"INDEPENDENT-SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/vertex-cover-dec","kind":"node","lean":"Cslib.Complexity.Karp.vertexCoverDec","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#vertex-cover-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"The decision problem that accepts a bit string exactly when it encodes an instance $(k, G)$, where $G$ is a simple graph on $n$ vertices, and $G$ has a vertex cover of exactly $k$ vertices. The encoding is $k$ ones, a zero, $n$ ones, a zero, then one bit for each pair $(i, j)$ with $i < j < n$, listed by $i$ ascending and, for each $i$, by $j$ descending, a true bit meaning that $i$ and $j$ are adjacent; a string\u2026","title":"The VERTEX-COVER problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/occs","kind":"node","lean":"Cslib.Complexity.Karp.occs","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#occs","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"For a CNF formula $\\varphi$ with clauses $C_0, \\dots, C_{m-1}$, the list of triples $(i, x, b)$, one for each literal $(x, b)$ of each clause $C_i$, where $x$ is the variable index and $b$ the polarity. The triples are listed clause by clause in order of $i$, and within a clause in the order of its literals.","title":"Literal occurrences of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/occ-adj","kind":"node","lean":"Cslib.Complexity.Karp.occAdj","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#occ-adj","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"A literal occurrence is a triple $(c, x, b)$ of a clause index $c$, a variable index $x$ and a polarity $b$. Two occurrences $(c, x, b)$ and $(c', x', b')$ are adjacent when $c \\ne c'$ and they are not complementary, that is, it is not the case that $x = x'$ and $b \\ne b'$.","title":"Adjacency of literal occurrences in Karp's clique graph"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/clique-bits","kind":"node","lean":"Cslib.Complexity.Karp.cliqueBits","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#clique-bits","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"For a CNF formula $\\varphi$, list its literal occurrences $o_0, \\dots, o_{m-1}$ in formula order, each recorded by its clause index, its variable and its polarity. The result has one bit for each pair $(i, j)$ with $i < j < m$, the pairs taken in encoding order ($i$ ascending and, for each $i$, $j$ descending). The bit of $(i, j)$ is true exactly when $o_i$ and $o_j$ lie in different clauses and are not\u2026","title":"Adjacency bits of Karp's clique graph"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/to-clique","kind":"node","lean":"Cslib.Complexity.Karp.toClique","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#to-clique","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"For a CNF formula $\\varphi$ with $m$ clauses and $N$ literal occurrences, the bit string encoding the instance $(m, G)$: $m$ ones, a zero, $N$ ones, a zero, then one adjacency bit for each pair $(i, j)$ with $i < j < N$, listed by $i$ ascending and, for each $i$, by $j$ descending, a true bit meaning that $i$ and $j$ are adjacent. The graph $G$ has the literal occurrences of $\\varphi$, in order, as its vertices\u2026","title":"Karp's CLIQUE instance of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/reduce-clique","kind":"node","lean":"Cslib.Complexity.Karp.reduceClique","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#reduce-clique","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"The map on bit strings that decodes its input as a CNF formula $\\varphi$ (rejecting missing or trailing bits) and returns Karp's CLIQUE instance of $\\varphi$; an input that does not decode is sent to the fixed no-instance $k = 1$ on the graph with no vertices.","title":"Karp's SAT-to-CLIQUE map on bit strings"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/flip-kg","kind":"node","lean":"Cslib.Complexity.Karp.flipKG","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#flip-kg","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, the map sends a bit string $1^k\\,0\\,1^n\\,0\\,b$ to $1^k\\,0\\,1^n\\,0\\,\\bar b$, where $\\bar b$ is $b$ with every bit complemented. A bit string with fewer than two zeros goes to the fixed no-instance $1\\,0\\,0$.","title":"Complementing map from CLIQUE to INDEPENDENT-SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/is-to-vc","kind":"node","lean":"Cslib.Complexity.Karp.isToVc","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#is-to-vc","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, the map sends a bit string $1^k\\,0\\,1^n\\,0\\,b$ with $k \\le n$ to $1^{n-k}\\,0\\,1^n\\,0\\,b$. Every other bit string, namely one of this form with $k > n$ or one with fewer than two zeros, goes to the fixed no-instance $1\\,0\\,0$.","title":"Map from INDEPENDENT-SET to VERTEX-COVER instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/kg-dec-of-decode-kg-eq-none","kind":"node","lean":"Cslib.Complexity.Karp.kgDec_of_decodeKG_eq_none","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#kg-dec-of-decode-kg-eq-none","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"Let $P$ be any property of a triple $(n, k, G)$, where $n$ and $k$ are natural numbers and $G$ is a simple graph on the vertices $0, 1, \\dots, n - 1$. A bit string that does not begin with two unary numbers (each a run of $1$s ended by a $0$) is rejected by the language of valid instances $(k, G)$ satisfying $P$.","title":"Unparsable input is rejected"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/mem-pairs","kind":"node","lean":"Cslib.Complexity.Karp.mem_pairs","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#mem-pairs","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"For natural numbers $n$, $i$ and $j$, the pair $(i, j)$ belongs to the list of vertex pairs in encoding order for $n$ if and only if $i < j$ and $j < n$.","title":"Membership in the vertex pairs"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/pairs-nodup","kind":"node","lean":"Cslib.Complexity.Karp.pairs_nodup","parent":31559,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01/index.html#pairs-nodup","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-01","status":"fully_proved","summary":"For every natural number $n$, the list of pairs $(i, j)$ of natural numbers with $i < j < n$, in encoding order ($i$ ascending and, for each $i$, $j$ descending from $n - 1$ to $i + 1$), contains no pair twice.","title":"The vertex pairs are listed without repetition"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","kind":"scope","navigation":"transparent","parent":31558,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","title":"Pair bits are read off by position to The row-by-row check of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/adj-bits-eq-get-elem","kind":"node","lean":"Cslib.Complexity.Karp.adjBits_eq_getElem","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#adj-bits-eq-get-elem","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"Let $n$ be a natural number, and let the pair list of $n$ be the list of pairs $(i, j)$ with $i < j < n$ in encoding order. Let $b$ be a list of bits whose length equals the length of the pair list of $n$. For all $i < j < n$, the bit of the pair $(i, j)$ in $b$ is the entry of $b$ at the position of $(i, j)$ in the pair list of $n$.","title":"Pair bits are read off by position"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/adj-bits-map-pairs","kind":"node","lean":"Cslib.Complexity.Karp.adjBits_map_pairs","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#adj-bits-map-pairs","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"Let $n$ be a natural number, let $f$ be a function from pairs of natural numbers to bits, and let $b$ be the list obtained by applying $f$ to each entry of the pair list of $n$ (the pairs $(i, j)$ with $i < j < n$ in encoding order). Then for all $i < j < n$, the bit of the pair $(i, j)$ in $b$ is $f(i, j)$.","title":"Pair bits computed by a function"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/to-graph-map-not","kind":"node","lean":"Cslib.Complexity.Karp.toGraph_map_not","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#to-graph-map-not","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and let $b$ be a list of bits whose length is the number of pairs $(i, j)$ with $i < j < n$. Then the graph on $\\{0, \\dots, n-1\\}$ read off the list obtained by negating every bit of $b$ is the complement of the graph on $\\{0, \\dots, n-1\\}$ read off $b$.","title":"Complementing the pair bits complements the graph"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/clique-dec-eq-ind-set-dec-flip-kg","kind":"node","lean":"Cslib.Complexity.Karp.cliqueDec_eq_indSetDec_flipKG","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#clique-dec-eq-ind-set-dec-flip-kg","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For every bit string $x$, CLIQUE accepts $x$ if and only if INDEPENDENT-SET accepts the image of $x$ under the complementing map from CLIQUE to INDEPENDENT-SET instances.","title":"Correctness of the complementing map from CLIQUE to INDEPENDENT-SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/exists-is-nindep-set-iff-exists-is-verte","kind":"node","lean":"Cslib.Complexity.Karp.exists_isNIndepSet_iff_exists_isVertexCover","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#exists-is-nindep-set-iff-exists-is-verte","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"Let $G$ be a simple graph whose vertices are $0, 1, \\dots, n - 1$, and let $k \\le n$. Then $G$ has an independent set of exactly $k$ vertices if and only if $G$ has a vertex cover of exactly $n - k$ vertices.","title":"Independent sets and vertex covers of complementary sizes"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/mem-occs","kind":"node","lean":"Cslib.Complexity.Karp.mem_occs","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#mem-occs","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with clauses $C_0, \\dots, C_{m-1}$, let $i$ and $x$ be natural numbers and let $b$ be a Boolean. The triple $(i, x, b)$ belongs to the list of literal occurrences of $\\varphi$ if and only if $i < m$ and the literal $(x, b)$ belongs to $C_i$.","title":"Membership in the literal occurrences of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/to-graph-clique-bits-adj","kind":"node","lean":"Cslib.Complexity.Karp.toGraph_cliqueBits_adj","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#to-graph-clique-bits-adj","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula and let $o_0, \\dots, o_{m-1}$ be its literal occurrences (clause index, variable, polarity) in formula order. In Karp's graph of $\\varphi$, the graph on $\\{0, \\dots, m-1\\}$ read off the clique pair bits of $\\varphi$, vertices $i$ and $j$ are adjacent if and only if $i \\ne j$ and the occurrences $o_i$ and $o_j$ are adjacent, that is, they lie in different clauses and are not\u2026","title":"Adjacency in Karp's clique graph"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/cnf-sat-iff-exists-is-nclique","kind":"node","lean":"Cslib.Complexity.Karp.cnfSat_iff_exists_isNClique","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#cnf-sat-iff-exists-is-nclique","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"A CNF formula $\\varphi$ with $m$ clauses is satisfiable if and only if Karp's graph of $\\varphi$ has a clique of exactly $m$ vertices. Karp's graph has one vertex for each literal occurrence of $\\varphi$, and two occurrences are adjacent when they lie in different clauses and are not complementary.","title":"Karp's reduction from SAT to CLIQUE is correct"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/sat-dec-of-decode-cnf-eq-none","kind":"node","lean":"Cslib.Complexity.Karp.satDec_of_decodeCNF_eq_none","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#sat-dec-of-decode-cnf-eq-none","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"Moved to `DenseSAT.ToBinary.satDec_of_decodeCNF_eq_none` (`DenseToBinary`).","title":"SAT decision on undecodable input (moved alias)"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/mode","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.Mode","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#mode","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"A mode is one of three values, clique, independent set and vertex cover, naming the three graph problems CLIQUE, INDEPENDENT-SET and VERTEX-COVER that share one NP verifier.","title":"Modes of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/ok","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.Mode.ok","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#ok","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a mode $m$, the adjacency bit $b$ of a vertex pair $\\{i, j\\}$, and flags $s_i$, $s_j$ saying whether $i$ and $j$ are chosen, the pair condition $\\mathrm{ok}_m(b, s_i, s_j)$ is $\\lnot(s_i \\land s_j) \\lor b$ for clique, $\\lnot(s_i \\land s_j) \\lor \\lnot b$ for independent set, and $\\lnot b \\lor s_i \\lor s_j$ for vertex cover.","title":"Pair condition of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/kg-check","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.kgCheck","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#kg-check","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a mode $m$, an input $x$ and a witness $w$ (bit lists), the verifier rejects if $x$ does not decode to a triple $(k, n, \\mathit{bits})$. Otherwise it accepts if and only if $\\mathit{bits}$ has one bit per pair $(i, j)$ with $i < j < n$, $w$ has at least $n$ bits, exactly $k$ of the first $n$ bits of $w$ are set, and for every pair $(i, j)$, matched with its bit $b$ in the encoding order of the pairs, the pair\u2026","title":"The verifier for CLIQUE, INDEPENDENT-SET and VERTEX-COVER"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/free","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.Mode.free","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#free","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a mode and the Boolean saying whether a vertex $i$ is chosen, the result is true when $i$ alone settles the condition on every pair $\\{i, j\\}$: in the clique and independent-set modes it is true exactly when $i$ is not chosen, and in the vertex-cover mode exactly when $i$ is chosen.","title":"Vertices that settle their pairs alone"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/sat","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.Mode.sat","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#sat","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a mode $m$, a pair bit $b$ and a flag $s_j$ saying whether vertex $j$ is chosen, $\\mathrm{sat}_m(b, s_j)$ is $\\lnot s_j \\lor b$ for clique, $\\lnot s_j \\lor \\lnot b$ for independent set, and $\\lnot b \\lor s_j$ for vertex cover.","title":"The pair condition left to the later vertex"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/ok-eq","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.Mode.ok_eq","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#ok-eq","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a mode $m$ and Booleans $b$, $s_i$, $s_j$, let $\\mathrm{ok}_m(b, s_i, s_j)$ be the pair condition of $m$: $\\lnot(s_i \\land s_j) \\lor b$ for clique, $\\lnot(s_i \\land s_j) \\lor \\lnot b$ for independent set and $\\lnot b \\lor s_i \\lor s_j$ for vertex cover. Let $\\mathrm{free}_m(s_i)$, the part that vertex $i$ alone settles, be $\\lnot s_i$ for clique and independent set and $s_i$ for vertex cover, and let\u2026","title":"The pair condition splits by vertex"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/pairs-succ","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.pairs_succ","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#pairs-succ","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a natural number $m$, write $P(m)$ for the list of pairs $(i, j)$ of natural numbers with $i < j < m$, ordered by $i$ ascending and, for each $i$, by $j$ descending. Then for every natural number $n$, $P(n + 1)$ is the list $(0, n), (0, n - 1), \\dots, (0, 1)$ (empty when $n = 0$) followed by the pairs $(i + 1, j + 1)$ for the pairs $(i, j)$ of $P(n)$, in the order of $P(n)$.","title":"Vertex pairs in encoding order for one more vertex"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/pair-vals","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.pairVals","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#pair-vals","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a list $s = (s_0, \\dots, s_{\\ell - 1})$ of Booleans, the chosen-flag pairs of $s$ are the pairs $(s_i, s_j)$, one for each vertex pair $(i, j)$ with $i < j < \\ell$, listed in the encoding order of the vertex pairs of $\\ell$ vertices (rows $i$ ascending, within a row $j$ descending).","title":"Chosen-flag pairs"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/map-get-d-range","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.map_getD_range","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#map-get-d-range","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For every list $s$ of Booleans of length $\\ell$, reading $s$ at the indices $0, 1, \\dots, \\ell - 1$ in turn (with default value false) gives back $s$.","title":"Reading a list at every index"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/pair-vals-cons","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.pairVals_cons","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#pair-vals-cons","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a Boolean $a$ and a list $s = (s_0, \\dots, s_{\\ell - 1})$ of Booleans, the chosen-flag pairs of the list $(a, s_0, \\dots, s_{\\ell - 1})$ are $(a, s_{\\ell - 1}), (a, s_{\\ell - 2}), \\dots, (a, s_0)$, followed by the chosen-flag pairs of $s$.","title":"Chosen-flag pairs of a list with a new head"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/rows-ok","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.rowsOk","parent":31580,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02/index.html#rows-ok","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-02","status":"fully_proved","summary":"For a mode $m$, the row-by-row check of a list of chosen-flags against a list $b$ of pair bits is defined by recursion on the flags. For the empty list it holds iff $b$ is empty. For a flag $a$ followed by the flags $r$ of the later vertices, write $\\rho$ for the reverse of $r$: the check holds iff $|r| \\le |b|$, either $a$ is free for $m$ or $\\mathrm{sat}_m(b_t, \\rho_t)$ holds for every $t < |r|$ (indices from\u2026","title":"The row-by-row check of the clique verifier"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","kind":"scope","navigation":"transparent","parent":31558,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","title":"The row-by-row check is the pairwise check to Labels of the CLIQUE verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/rows-ok-eq","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.rowsOk_eq","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#rows-ok-eq","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For every mode $m$ and all lists of bits $s$ and $b$, the row-by-row check of $s$ against $b$ holds if and only if $b$ has exactly one bit per pair $i < j < |s|$ and, listing these pairs in encoding order (rows $i$ ascending, within a row $j$ descending), the $t$-th pair $(i, j)$ and the $t$-th bit $b_t$ satisfy the pair condition $\\mathrm{ok}_m(b_t, s_i, s_j)$ for every $t$.","title":"The row-by-row check is the pairwise check"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/kg-check-eq-rows-ok","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.kgCheck_eq_rowsOk","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#kg-check-eq-rows-ok","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"Let $m$ be a mode and $x$, $w$ bit lists. Suppose that $x$ decodes to $(k, n, \\mathit{bits})$, that $w$ has at least $n$ bits, and that exactly $k$ of the first $n$ bits of $w$ are set. Then the verifier for mode $m$ on input $x$ and witness $w$ returns the same value as the row-by-row check for mode $m$ of the first $n$ bits of $w$ against the pair bits $\\mathit{bits}$.","title":"The verifier as a row-by-row check"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/lookup-of-mem","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.lookup_of_mem","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#lookup-of-mem","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"Let $l$ be a list of pairs whose keys lie in a type with a lawful Boolean equality and are pairwise distinct. If $(a, b)$ occurs in $l$, then looking up the key $a$ in $l$ returns $b$.","title":"Lookup in a list of pairs with distinct keys"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/all-zip-pairs","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.all_zip_pairs","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#all-zip-pairs","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"Let $n$ be a natural number and $b$ a list of bits with one bit per vertex pair, that is, with as many entries as the list of pairs $(i, j)$ with $i < j < n$. Then for every Boolean predicate $P$ on a pair of natural numbers and a bit, $P$ holds on every entry of the pair list zipped with $b$ (the $t$-th pair together with the $t$-th bit of $b$) if and only if, for all natural numbers $i < j < n$, $P$ holds on the\u2026","title":"Checking each listed vertex pair against its bit"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/kg-check-iff","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.kgCheck_iff","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#kg-check-iff","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For a mode $m$ and bit lists $x$ and $w$, the verifier for mode $m$ accepts input $x$ with witness $w$ if and only if there are $k$, $n$ and a bit list $\\mathit{bits}$ such that $x$ is the encoding of the instance $(k, n, \\mathit{bits})$, $\\mathit{bits}$ has one bit per pair $(i, j)$ with $i < j < n$, $w$ has at least $n$ bits, exactly $k$ of the first $n$ bits of $w$ are set, and for all $i < j < n$ the pair\u2026","title":"Characterization of the CLIQUE verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/count-of-fn","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.count_ofFn","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#count-of-fn","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For every natural number $n$ and every function $f$ from $\\{0, \\dots, n-1\\}$ to the Booleans, the number of entries equal to true in the list $f(0), f(1), \\dots, f(n-1)$ equals the number of indices $i < n$ with $f(i)$ true.","title":"Counting the set bits of a list built from a function"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/take-eq-of-fn","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.take_eq_ofFn","parent":31601,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first "],["math","\\(n\\)"],["text"," bits of a list, read index by index"]],"source":"The first $n$ bits of a list, read index by index"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#take-eq-of-fn","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"If a bit list $w$ has length at least $n$, then the first $n$ entries of $w$ form the list $(w_0, \\dots, w_{n-1})$, where $w_i$ is the entry of $w$ at index $i$ (taken as false past the end of $w$).","title":"The first $n$ bits of a list, read index by index"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/set-ok","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.SetOk","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#set-ok","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For a mode $m$, a number $n$, a list of pair bits and a set $s$ of vertices in $\\{0, \\dots, n - 1\\}$, the set $s$ meets the condition of $m$ when for all vertices $i < j$ the pair condition $\\mathrm{ok}_m(b_{ij}, [i \\in s], [j \\in s])$ holds, where $[i \\in s]$ says whether $i$ is chosen and $b_{ij}$ is the bit of the pair $\\{i, j\\}$: listing the pairs $i' < j' < n$ in encoding order (rows $i'$ ascending, within a\u2026","title":"Vertex sets meeting a mode's pair condition"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/exists-set-ok","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.exists_setOk","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#exists-set-ok","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"Fix a mode (clique, independent set or vertex cover) and bit strings $x$ and $w$. If the verifier in that mode accepts the input $x$ with witness $w$, then there are natural numbers $k$ and $n$ and a list of bits $b$ such that $x$ is the encoding of the instance $(k, n, b)$, $b$ has exactly one bit per vertex pair $(i, j)$ with $i < j < n$, and some set of exactly $k$ of the vertices $0, \\dots, n-1$ meets the\u2026","title":"An accepted input encodes an instance with a good vertex set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/kg-check-of-fn","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.kgCheck_ofFn","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#kg-check-of-fn","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"Let $m$ be a mode, $k$ and $n$ natural numbers, let $\\mathit{bits}$ be a bit list with one bit per pair $(i, j)$, $i < j < n$, and let $s$ be a set of $k$ vertices of $\\{0, \\dots, n - 1\\}$ that meets the pair condition of $m$ on every pair $i < j$. Then the verifier for mode $m$ accepts the encoding of $(k, n, \\mathit{bits})$ with witness the characteristic vector of $s$: the list of $n$ bits whose $i$-th bit is\u2026","title":"The characteristic vector of a good vertex set is accepted"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/kg-dec-iff-exists","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.kgDec_iff_exists","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#kg-dec-iff-exists","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"Let $m$ be a mode and $P$ a property of a number $k$ and a graph on $n$ vertices. Suppose that for all $n$, $k$ and every bit list $\\mathit{bits}$ with one bit per pair $(i, j)$, $i < j < n$, the property $P$ holds for $k$ and the graph given by $\\mathit{bits}$ exactly when some set of $k$ vertices meets the pair condition of $m$ on every pair $i < j$. Then for every bit list $x$, $x$ belongs to the language of\u2026","title":"Verifying a graph language defined by a vertex-set condition"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/forall-ne-iff-lt","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.forall_ne_iff_lt","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#forall-ne-iff-lt","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"Let $n$ be a natural number and $R$ a relation on $\\{0, \\dots, n-1\\}$ such that $R(i, j)$ implies $R(j, i)$. Then $R(i, j)$ holds for all $i \\ne j$ if and only if $R(i, j)$ holds for all $i < j$.","title":"Symmetric relations on increasing pairs"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/clique-iff","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.clique_iff","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#clique-iff","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and a list of bits $b$, the graph on the vertices $0, \\dots, n-1$ read off $b$ has a clique of exactly $k$ vertices if and only if there is a set $s$ of exactly $k$ of these vertices meeting the clique-mode pair condition on every pair $i < j$: whenever $i$ and $j$ both lie in $s$, the bit of the pair $\\{i, j\\}$ in $b$ is set.","title":"Cliques of the graph given by pair bits"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/indep-iff","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.indep_iff","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#indep-iff","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and a list of bits $b$, the graph on the vertices $0, \\dots, n-1$ read off $b$ has an independent set of exactly $k$ vertices if and only if there is a set $s$ of exactly $k$ of these vertices meeting the independent-set-mode pair condition on every pair $i < j$: whenever $i$ and $j$ both lie in $s$, the bit of the pair $\\{i, j\\}$ in $b$ is not set.","title":"Independent sets of the graph given by pair bits"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/cover-iff","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.cover_iff","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#cover-iff","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and a list of bits $b$, the graph on the vertices $0, \\dots, n-1$ read off $b$ has a vertex cover of exactly $k$ vertices if and only if there is a set $s$ of exactly $k$ of these vertices meeting the cover-mode pair condition on every pair $i < j$: whenever the bit of the pair $\\{i, j\\}$ in $b$ is set, $i$ or $j$ lies in $s$.","title":"Vertex covers of the graph given by pair bits"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/clique-dec-iff","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.cliqueDec_iff","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#clique-dec-iff","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For every bit string $x$, $x$ is in CLIQUE if and only if there is a bit string $w$ of length at most the length of $x$ such that the shared CLIQUE / INDEPENDENT-SET / VERTEX-COVER verifier, run in clique mode, accepts the input $x$ with witness $w$.","title":"CLIQUE is verified with witnesses no longer than the input"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/ind-set-dec-iff","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.indSetDec_iff","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#ind-set-dec-iff","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"For every bit string $x$, $x$ is in INDEPENDENT-SET if and only if there is a bit string $w$ of length at most the length of $x$ such that the shared CLIQUE / INDEPENDENT-SET / VERTEX-COVER verifier, run in independent-set mode, accepts the input $x$ with witness $w$.","title":"INDEPENDENT-SET is verified with witnesses no longer than the input"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/vertex-cover-dec-iff","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.vertexCoverDec_iff","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#vertex-cover-dec-iff","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"A bit string $x$ is in VERTEX-COVER if and only if there is a bit string $w$, no longer than $x$, such that the clique-family verifier in vertex-cover mode accepts $x$ with witness $w$.","title":"VERTEX-COVER is verified by the clique verifier in cover mode"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/k","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.K","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#k","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"The verifier machine for CLIQUE, INDEPENDENT-SET and VERTEX-COVER, run on an instance with target size $k$ and $n$ vertices, has six stacks: the input $X$, the witness $W$, the counter $C$ (holding $k$), the counter $N$ (holding $n$), the vertex stack $V$ and its scratch stack $T$.","title":"Stacks of the CLIQUE verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/lab","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.Lab","parent":31601,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03/index.html#lab","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-03","status":"fully_proved","summary":"On an instance with target size $k$ and $n$ vertices, the verifier machine has the labels: read $k$, read $n$, load the witness, walk a row (carrying a bit $c$, the conjunction of the row's pair conditions so far), put the row back, accept, and reject.","title":"Labels of the CLIQUE verifier"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","kind":"scope","navigation":"transparent","parent":31558,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","title":"Statements of the clique verifier to Row-by-row pair bits see only adjacency"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/stmt","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.Stmt","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#stmt","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"The statements of the verifier are those of a multi-stack Turing machine (Mathlib's TM2 model) with the six stacks $X, W, C, N, V, T$, each over the symbol alphabet of the SAT verifier, the verifier's labels, and one Boolean of local state.","title":"Statements of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/read-u","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.readU","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#read-u","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"For a stack $c$ and labels $\\ell$ (its own) and $\\ell'$ (the next), the unary reader onto $c$ is the verifier statement that inspects the input stack $X$ and sets the state to whether $X$ is empty: if it is, it goes to the rejecting label; otherwise it pops the top symbol of $X$ and sets the state to whether that symbol is the bit $1$; if so, it pushes the bit $1$ onto $c$ and goes to $\\ell$, while any other symbol\u2026","title":"The clique verifier's unary reader"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/load-s","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.loadS","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#load-s","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"The statement run at the load label. If the counter $N$ is nonempty, it takes a unit off $N$, rejects if the witness stack $W$ is empty, and otherwise pops the top of $W$ and pushes its bit (false for a non-bit symbol) onto the vertex stack $V$; if that bit is set it also takes a unit off the counter $C$, rejecting if $C$ is empty; then it returns to the load label. If $N$ is empty, it goes to the walk label with\u2026","title":"Load step of the CLIQUE verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/walk-z","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.walkZ","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#walk-z","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Write $\\mathrm{free}_m(s)$ for the condition that, in mode $m$, a vertex with chosen-flag $s$ settles every pair of its row, and $\\mathrm{sat}_m(b', s')$ for the pair condition on pair bit $b'$ and later vertex flag $s'$. In mode $m$ with row flag $c$, after a bit $z$ has been popped off the vertex stack $V$: if $V$ is now empty, the machine moves to the put-back label when $\\mathrm{free}_m(z)$ or $c$ holds, and to\u2026","title":"Rest of a walk step of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/walk-s","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.walkS","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#walk-s","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"In mode $m$ with row flag $c$, the walk step looks at the vertex stack $V$. If $V$ is empty, it moves to the accept label if the input stack $X$ is empty and to the reject label otherwise. If $V$ is not empty, it pops the top symbol of $V$, reads it as a bit $z$ (a non-bit symbol reads as false), and continues with the rest of the walk step for $m$, $c$ and $z$.","title":"Walk step of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/prog","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.prog","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#prog","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"For a mode $m$ (clique, independent set or vertex cover), the verifier program of mode $m$ assigns a statement to each label of the verifier: to the label that reads the instance's target size $k$, the unary reader onto the counter $C$ with own label this label and next label the label that reads the instance's number $n$ of vertices; to the label that reads $n$, the unary reader onto the counter $N$ with own label\u2026","title":"Program of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/start","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.start","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#start","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"The verifier for CLIQUE, INDEPENDENT-SET and VERTEX-COVER starts at the label that reads the instance's size parameter $k$ (the number of vertices to choose) in unary off the input stack $X$ onto the counter $C$.","title":"Start label of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/run-one","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.run_one","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#run-one","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $M$ be a program that assigns a verifier statement to each label, $\\ell$ and $\\ell'$ labels, $v$ a state and $S$ stack contents. If executing the statement $M(\\ell)$ from state $v$ and stacks $S$ ends at label $\\ell'$, then there are a state $v'$ and stack contents $S'$ such that $M$ runs in exactly one step from label $\\ell$ with state $v$ and stacks $S$ to label $\\ell'$ with state $v'$ and stacks $S'$.","title":"One step of a clique-verifier program to a known label"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/read-u-some","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.readU_some","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#read-u-some","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $m$ be a mode, $c$ a stack other than the input stack $X$, and $\\ell$, $\\ell'$ labels such that the verifier program of mode $m$ runs, at label $\\ell$, the unary reader onto $c$ with own label $\\ell$ and next label $\\ell'$. Let $k, j \\in \\mathbb{N}$, let $r$ be a list of bits, $v$ a state and $S$ stack contents in which $X$ holds, from the top, $k$ ones, then a zero, then the bits of $r$, and $c$ holds exactly\u2026","title":"The clique verifier's unary reader succeeds"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/read-u-none","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.readU_none","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#read-u-none","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $m$ be a mode, $c$ a stack other than the input stack $X$, and $\\lambda$, $\\mu$ labels such that the verifier for $m$ runs at $\\lambda$ the unary reader onto $c$ that loops at $\\lambda$ and exits to $\\mu$. Let $l$ be a list of Booleans from which no unary length can be read (it contains no false). Then for every machine variable $v$ and all stack contents in which $X$ holds exactly the bits of $l$, the verifier\u2026","title":"Reading an unterminated unary number rejects"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/load-run","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.load_run","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#load-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $m$ be a mode, $n$ and $j$ natural numbers and $w$ and $\\mathit{acc}$ bit lists. Start the verifier for mode $m$ at the load label, in any internal state, with $n$ in unary on the counter $N$, $j$ in unary on the counter $C$, $w$ on the witness stack $W$ and $\\mathit{acc}$ on the vertex stack $V$. Then within at most $n + 1$ steps the machine reaches the walk label with its flag $c$ (the conjunction of the\u2026","title":"Run of the load phase"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/walk-row-run","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.walkRow_run","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#walk-row-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $m$ be a mode and $s$ a bit. Let $z = (z_0, \\dots, z_{r-1})$ and $b = (b_0, \\dots, b_{q-1})$ be lists of bits, $c$ a bit, $v$ a state and $S$ stack contents in which the vertex stack $V$ holds, from the top, the name bits $z_0, \\dots, z_{r-1}$ followed by the name bit $s$, and the input stack $X$ holds, from the top, the name bits $b_0, \\dots, b_{q-1}$. Then there are a state $v'$ and stack contents $S'$ such\u2026","title":"Walking the pairs of one row of the clique verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/row-run","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.row_run","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#row-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $m$ be a mode, $s_i$ a bit, $r$ and $b$ lists of bits, $v$ a state and $S$ stack contents in which the vertex stack $V$ holds, from the top, the bits of the reverse $\\rho$ of $r$ followed by $s_i$, the scratch stack $T$ is empty, and the input stack $X$ holds the bits $b$. Call the row condition: $|r| \\le |b|$, and either $s_i$ is free for $m$ or $\\mathrm{sat}_m(b_t, \\rho_t)$ holds for every $t < |r|$ (indices\u2026","title":"One row of the clique verifier's walk"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/rows-run","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.rows_run","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#rows-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $m$ be a mode, $r$ and $b$ lists of bits, $v$ a state and $S$ stack contents in which the vertex stack $V$ holds, from the top, the bits of the reverse of $r$ (so the last entry of $r$ is on top), the scratch stack $T$ is empty, and the input stack $X$ holds the bits $b$. Then there are a state $v'$ and stack contents $S'$ such that within at most $|r|(2|r| + 2) + 1$ steps the verifier program of mode $m$ goes\u2026","title":"All rows of the clique verifier's walk"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/run","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.run","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $m$ be a mode, $x$ and $w$ lists of bits, $v$ a state and $S$ stack contents in which the input stack $X$ holds the bits of $x$, the witness stack $W$ holds the bits of $w$, and every other stack is empty. Then there are a state $v'$ and stack contents $S'$ such that within at most $8(|x| + |w| + 1)^3$ steps the verifier program of mode $m$ goes from the start label with state $v$ and stacks $S$ to state $v'$\u2026","title":"The verifier machine for clique, independent set and vertex cover"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/pairs-nodup","kind":"node","lean":"Cslib.Complexity.Karp.KGVerify.pairs_nodup","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#pairs-nodup","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Moved to `Cslib.Complexity.Karp.pairs_nodup` (`Karp.CliqueCorrect`).","title":"Distinctness of CLIQUE verifier pairs (moved alias)"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/adj-l","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.adjL","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#adj-l","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"For literals $l = (x, p)$ and $l' = (x', q)$, each a variable $x, x' \\in \\mathbb{N}$ with a polarity $p, q \\in \\{\\mathrm{true}, \\mathrm{false}\\}$, the adjacency bit of $l$ and $l'$ is false when they are complementary, that is $x = x'$ and $p \\ne q$, and true otherwise. It is the adjacency of two occurrences in different clauses in Karp's graph.","title":"Karp adjacency of two literals"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/bits-rec","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.bitsRec","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#bits-rec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"For a list of literal occurrences (clause index, variable, polarity), the row-by-row pair bits are defined recursively: the empty list has none, and for a list $o, o_1, \\dots, o_k$ they are the adjacency bits of $o$ with $o_k, \\dots, o_1$, in that order, followed by the row-by-row pair bits of $o_1, \\dots, o_k$. Two occurrences are adjacent when they lie in different clauses and are not complementary.","title":"Row-by-row pair bits of an occurrence list"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/pair-bits-eq","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.pairBits_eq","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#pair-bits-eq","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Let $o_0, \\dots, o_{m-1}$ be a list of literal occurrences (clause index, variable, polarity). The list of adjacency bits of $o_i$ and $o_j$ over the pairs $(i, j)$ with $i < j < m$ in encoding order (rows $i$ ascending, and $j$ descending within a row) equals the recursive pair bits of the list, which are empty for the empty list and otherwise are the adjacency bits of $o_0$ with $o_{m-1}, o_{m-2}, \\dots, o_1$, in\u2026","title":"Pair bits of an occurrence list, row by row"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/bits-rec-map","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.bitsRec_map","parent":31622,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04/index.html#bits-rec-map","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-04","status":"fully_proved","summary":"Call two literal occurrences (clause index, variable, polarity) adjacent when they lie in different clauses and are not complementary. Let $f$ be a map on occurrences such that $f(a)$ and $f(b)$ are adjacent exactly when $a$ and $b$ are, for all occurrences $a$ and $b$. Then for every list of occurrences $o_1, \\dots, o_k$, the row-by-row pair bits of $f(o_1), \\dots, f(o_k)$ equal those of $o_1, \\dots, o_k$.","title":"Row-by-row pair bits see only adjacency"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","kind":"scope","navigation":"transparent","parent":31558,"reading_title":{"format":"inline-parts-v1","parts":[["text","Occurrences of a formula with a first clause to Moving a walked name from "],["math","\\(D\\)"],["text"," to "],["math","\\(E\\)"]],"source":"Occurrences of a formula with a first clause to Moving a walked name from $D$ to $E$"},"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","title":"Occurrences of a formula with a first clause to Moving a walked name from $D$ to $E$"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/occs-cons","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.occs_cons","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#occs-cons","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"Let $C$ be a clause and $\\varphi$ a CNF formula, and write a literal occurrence as a pair $(i, \\ell)$ of a clause index $i$ and a literal $\\ell$. The list of literal occurrences of the formula whose first clause is $C$ and whose remaining clauses are those of $\\varphi$ is the list of $(0, \\ell)$ for the literals $\\ell$ of $C$ in order, followed by the list of $(i + 1, \\ell)$ for the occurrences $(i, \\ell)$ of\u2026","title":"Occurrences of a formula with a first clause"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/d-tok","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.dTok","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#d-tok","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"For a literal $(x, p)$ with variable index $x \\in \\mathbb{N}$ and polarity $p$, the token list is the polarity marker symbol for $p$ followed by one name-bit symbol for each bit of the canonical binary representation of $x$ (least significant bit first, empty for $x = 0$), in order.","title":"Polarity-first tokens of a literal"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/item-tok","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.itemTok","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#item-tok","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"An item of the stack $D$ is either $\\mathrm{none}$, standing for a clause marker, or an occurrence $(x, p)$ of the variable $x$ with polarity $p$. The tokens of $\\mathrm{none}$ are the single clause-marker symbol; the tokens of $(x, p)$ are the polarity marker of $p$ followed by the binary encoding of $x$ as name-bit symbols.","title":"Tokens of an item on the walked stack"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/walk-bits","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.walkBits","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#walk-bits","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"Let $l = (x, p)$ be a literal occurrence with name $x \\in \\mathbb{N}$ and polarity $p$, let $f$ be a Boolean, and let $ds$ be a list whose items are literal occurrences or clause markers. The bit list $\\mathrm{walkBits}(l, f, ds)$ is defined by recursion on $ds$: it is empty if $ds$ is empty; if $ds$ is a clause marker followed by $ds'$, it is $\\mathrm{walkBits}(l, \\mathrm{true}, ds')$; and if $ds$ is an occurrence\u2026","title":"Bits pushed by a walk of the CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/items","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.items","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#items","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"For a CNF formula $\\varphi$ with clauses $C_1, \\dots, C_m$ in order, the items of $\\varphi$ are the list made of, for each $i = 1, \\dots, m$ in turn, the literals of $C_i$ in order, each as an occurrence item, followed by one clause-marker item $\\mathrm{none}$. These are the items on the stack $D$ once the clauses of $\\varphi$ are walked.","title":"Walked items of a formula"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/lrows","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.lrows","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#lrows","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"For a list $I$ of items and literals $l_1, \\dots, l_k$, the rows are the concatenation, for $i = 1, \\dots, k$ in turn, of the reverse of the walk bits of $l_i$ with flag false over the items $l_{i+1}, \\dots, l_k$ (as occurrence items) followed by $I$. These are the bits pushed when the literals of a clause are walked above the items $I$.","title":"Rows pushed for a clause's literals"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/rows-f","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.rowsF","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#rows-f","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"A literal is a pair $(x, s)$ of a variable $x \\in \\mathbb{N}$ and a polarity $s$, and an item is a literal or a clause marker. For a literal $\\ell = (x, s)$, a Boolean $f$ and a list of items, the walk bits are defined by recursion on the list: there are none for the empty list; for a list starting with a clause marker they are the walk bits of the rest with $f$ replaced by true; for a list starting with a literal\u2026","title":"Row bits of a formula for the CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/bits-rec-clause","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.bitsRec_clause","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#bits-rec-clause","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"Let $C$ be a clause and $\\varphi$ a CNF formula. Let $O_\\varphi$ be the list of literal occurrences of $\\varphi$ (clause index, variable, polarity), with every clause index increased by $1$. Then the row-by-row pair bits of the list made of the literals of $C$, each with clause index $0$, followed by $O_\\varphi$, are the rows pushed for the literals of $C$ walked above a clause marker and the items of $\\varphi$,\u2026","title":"Pair bits of a leading clause"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/rows-f-eq","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.rowsF_eq","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#rows-f-eq","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"For every CNF formula $\\varphi$, the row bits of $\\varphi$ equal the pair bits of Karp's CLIQUE graph of $\\varphi$: with $o_0, \\dots, o_{m-1}$ the literal occurrences of $\\varphi$, these are, for the pairs $(i, j)$ with $i < j < m$ in encoding order (rows $i$ ascending, and $j$ descending within a row), the bits telling whether $o_i$ and $o_j$ lie in different clauses and are not complementary.","title":"The CLIQUE emitter's row bits are Karp's pair bits"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/k","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.K","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#k","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"The emitter has nine stacks: $T$ (tokens), $O$ (output), $A$ (the current name), $B$ (scratch), $D$ (walked occurrences), $E$ (the walk's other side), $X$ (scratch), $C$ (the clause count) and $N$ (the occurrence count).","title":"Stacks of the SAT-to-CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/untitled","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.\u0393","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#untitled","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"In the CLIQUE emitter, the output stack $O$ has the Booleans as its alphabet, and every other stack has the token symbols as its alphabet.","title":"Stack alphabets of the CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/lab","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.Lab","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#lab","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"For bits $p, q, f, b$ the labels are $\\mathsf{start}$, $\\mathsf{lits}$, $\\mathsf{nm}(p)$, $\\mathsf{walk}(p, f)$, $\\mathsf{cmp}(p, q)$, $\\mathsf{res}(p, b)$, $\\mathsf{rsA}(p)$, $\\mathsf{rsD}(p)$, $\\mathsf{mvN}(p, f)$, $\\mathsf{back}(p)$, $\\mathsf{put}(p)$, $\\mathsf{put2}(p)$, $\\mathsf{put3}(p)$, $\\mathsf{fin}$, $\\mathsf{finN}$, $\\mathsf{fin2}$, $\\mathsf{finC}$ and $\\mathsf{done}$. Here $p$ is the polarity of the\u2026","title":"Labels of the SAT-to-CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/prog","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.prog","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#prog","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"The CLIQUE emitter is the multi-stack machine with stacks $T, O, A, B, D, E, X, C, N$, a Boolean flag and the labels named below. The stack $O$ holds Booleans and the others hold symbols, among them the bit symbols $\\mathrm{bit}(b)$, the literal symbols $\\mathrm{lit}(p)$ and the clause marker $\\mathrm{cl}$. Each test sets the flag to its outcome, a pop that reads a polarity or a bit sets the flag to it, and other\u2026","title":"The CLIQUE emitter machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/start","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.start","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#start","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"The start label of the CLIQUE emitter machine is its label $\\mathsf{start}$.","title":"Start label of the CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/st","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.st","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#st","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"For a token stack $t$, an output stack $o$ of bits, stacks $a$, $d$, $e$ of token symbols and natural numbers $c$ and $n$, this assigns to each stack of the CLIQUE emitter its contents: the token stack $T$ holds $t$, the output stack $O$ holds $o$, the name stack $A$ holds $a$, the walked-occurrence stack $D$ holds $d$, the walk's other stack $E$ holds $e$, the clause counter $C$ holds $c$ copies of the clause\u2026","title":"Configuration stacks of the CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/st-eq","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.st_eq","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#st-eq","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"Let $S$ assign contents to the stacks of the CLIQUE emitter and let $c, n \\in \\mathbb{N}$. If the scratch stacks $B$ and $X$ are empty and the counters $C$ and $N$ hold exactly $c$ and $n$ clause markers, then $S$ is the standard state whose stacks $T$, $O$, $A$, $D$, $E$ hold what they hold in $S$, whose stacks $B$ and $X$ are empty and whose counters $C$ and $N$ hold $c$ and $n$ clause markers.","title":"CLIQUE emitter: recognizing a standard state"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/is-tok","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.isTok","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#is-tok","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"A symbol is a token of the walk between the stacks $D$ and $E$ when it is a name bit, a literal marker or a clause marker; the value symbols and the dead symbol are not.","title":"Tokens moved by the walk"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/name","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.name","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#name","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"For $x \\in \\mathbb{N}$, the name of $x$ is the list of symbols obtained from the binary encoding of $x$ (least significant bit first, empty for $x = 0$) by writing each bit $b$ as the bit symbol $\\mathrm{bit}(b)$. This is the form in which the stacks $A$ and $D$ of the CLIQUE emitter hold variable names.","title":"Name bits of a variable in the CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/nm-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.nm_run","parent":31643,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#nm-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"Stacks are listed top first. Let $p$ be a Boolean, $x \\in \\mathbb{N}$ with name $w$ (its binary encoding written as bit symbols) of length $\\ell$, and $R$ a list of symbols that is empty or whose top symbol is not a bit symbol. Let $v$ be a flag, $o$ a list of Booleans, $d$ and $e$ lists of symbols, and $c, n \\in \\mathbb{N}$. Started at label $\\mathrm{nm}(p)$ with flag $v$, the token stack $T$ holding $w$ reversed\u2026","title":"CLIQUE emitter: reading a variable name"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/mv-n-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.mvN_run","parent":31643,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moving a walked name from "],["math","\\(D\\)"],["text"," to "],["math","\\(E\\)"]],"source":"Moving a walked name from $D$ to $E$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05/index.html#mv-n-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-05","status":"fully_proved","summary":"Write $\\langle \\ell, v \\mid t, o, a, d, e, c, n\\rangle$ for the configuration of the SAT-to-CLIQUE emitter at label $\\ell$ with local bit $v$, stacks $T = t$, $O = o$, $A = a$, $D = d$, $E = e$, empty scratch stacks $B$ and $X$, and $c$ and $n$ marks on the counters $C$ and $N$; juxtaposition of lists is concatenation. Let $p, f$ be bits, $w$ a list of bits with $\\bar w$ the corresponding list of name-bit symbols,\u2026","title":"Moving a walked name from $D$ to $E$"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","kind":"scope","navigation":"transparent","parent":31558,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","title":"Clique emitter: comparing two names to Pair-bit transform of each mode"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/cmp-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.cmp_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#cmp-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Let $p$, $q$ and $v$ be Booleans, $u$ and $w$ lists of bits, $t$, $e$ and $R$ stack-symbol lists with $R$ empty or not starting with a name bit, $o$ a list of bits and $c, n \\in \\mathbb{N}$. Let $b$ be $(p = q)$ if $u = w$ and true otherwise. For some flag $v'$, the emitter machine goes in at most $3|w| + 4$ steps from label $\\mathrm{cmp}(p, q)$ with flag $v$ and the state with token stack $t$, output $o$, name\u2026","title":"Clique emitter: comparing two names"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/put-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.put_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#put-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Stacks are listed top first. Let $p$ be a Boolean, $u$ a list of Booleans and $\\bar u$ the list of bit symbols $\\mathrm{bit}(b)$ for $b$ in $u$, let $v$ be a flag, $t$, $d$, $e$ lists of symbols, $o$ a list of Booleans and $c, n \\in \\mathbb{N}$. Started at label $\\mathrm{put}(p)$ with flag $v$, stacks $T$, $O$, $A$, $D$, $E$ holding $t$, $o$, $\\bar u$, $d$, $e$, empty scratch stacks $B$ and $X$, and $c$ and $n$\u2026","title":"CLIQUE emitter: putting the current occurrence on the walked stack"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/back-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.back_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#back-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Let $p$ and $v$ be Booleans, $t$, $a$ and $d$ stack-symbol lists, $o$ a list of bits, $c, n \\in \\mathbb{N}$, and let $L$ be a list of stack symbols each of which is a name bit, a polarity marker or a clause marker. From label $\\mathrm{back}(p)$ with flag $v$ and the state with token stack $t$, output $o$, name stack $a$, walked stack $d$, walk side $L$, clause count $c$, occurrence count $n$ and empty scratch\u2026","title":"Clique emitter: moving the walk side back"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/fin-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.fin_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#fin-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Let $v$ be a Boolean, $d$ and $e$ stack-symbol lists, $o$ a list of bits and $c, n \\in \\mathbb{N}$. From label fin with flag $v$ and the state with empty token stack, output $o$, empty name stack, walked stack $d$, walk side $e$, clause count $c$, occurrence count $n$ and empty scratch stacks, the emitter machine reaches in exactly $n + c + 4$ steps label done with some flag and some state whose output, read\u2026","title":"Clique emitter: the instance prefix"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/flat-map-reverse-length","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.flatMap_reverse_length","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#flat-map-reverse-length","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types, $l$ a list of elements of $\\alpha$ and $f$ a function from $\\alpha$ to lists of elements of $\\beta$. Concatenating the lists $f(a)$ for $a$ running through the reverse of $l$ gives a list of the same length as concatenating them for $a$ running through $l$.","title":"Reversing a list preserves the length of a flat map"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/walk-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.walk_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#walk-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Let $l = (x, p)$ be a literal occurrence, $ds$ a list of items (literal occurrences or clause markers), $f$ and $v$ Booleans, $t$ a token stack, $o$ an output stack of bits, $e$ a stack of token symbols and $c, n \\in \\mathbb{N}$. Let $D$ be the concatenation of the tokens of the items of $ds$: a clause marker for a clause marker, and the polarity marker of $l'$ above the name bits of $l'$ (the binary encoding of\u2026","title":"Walk over the occurrence stack in the CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/lit-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.lit_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#lit-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Write $\\langle \\ell, v \\mid t, o, a, d, e, c, n\\rangle$ for the configuration of the SAT-to-CLIQUE emitter at label $\\ell$ with local bit $v$, stacks $T = t$, $O = o$, $A = a$, $D = d$, $E = e$, empty scratch stacks $B$ and $X$, and $c$ and $n$ marks on the counters $C$ and $N$; juxtaposition of lists is concatenation. Let $l = (x, p)$ be a literal, $R$ a list of symbols that is empty or does not start with a\u2026","title":"Emitter run over one literal occurrence"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/lits-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.lits_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Write $\\langle \\ell, v \\mid t, o, a, d, e, c, n\\rangle$ for the configuration of the SAT-to-CLIQUE emitter at label $\\ell$ with local bit $v$, stacks $T = t$, $O = o$, $A = a$, $D = d$, $E = e$, empty scratch stacks $B$ and $X$, and $c$ and $n$ marks on the counters $C$ and $N$; juxtaposition of lists is concatenation. Let $L, c \\in \\mathbb{N}$, let $I$ be a list of items and $\\delta$ the concatenation of their\u2026","title":"Emitter run over a clause's literals"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/clause-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.clause_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Let $C$ be a clause, $\\varphi$ a CNF formula, $R$ a token list that is empty or starts with a clause marker, $o$ a list of bits, $c, n, B \\in \\mathbb{N}$ and $v$ a Boolean. Write $T_C$ for the tokens of $C$ as they lie on the token stack (a clause marker, then the literals last-first) and $D_\\psi$ for the tokens of the items of a formula $\\psi$. If $|T_C| + |D_\\varphi| \\le B$, then for some flag $v'$ the emitter\u2026","title":"Clique emitter: one clause"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/formula-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.formula_run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#formula-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Write $\\langle \\ell, v \\mid t, o, a, d, e, c, n\\rangle$ for the configuration of the SAT-to-CLIQUE emitter at label $\\ell$ with local bit $v$, stacks $T = t$, $O = o$, $A = a$, $D = d$, $E = e$, empty scratch stacks $B$ and $X$, and $c$ and $n$ marks on the counters $C$ and $N$; juxtaposition of lists is concatenation. Let $L \\in \\mathbb{N}$, let $\\varphi$ be a CNF formula with clauses $C_1, \\dots, C_m$ in order,\u2026","title":"Emitter run over a formula's clauses"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/run","kind":"node","lean":"Cslib.Complexity.Karp.EmitClique.run","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with token list $\\tau$ (in reading order), $o$ a list of Booleans, $v$ a flag, and $S$ stack contents in which the token stack $T$ holds $\\tau$ reversed (stacks are listed top first), the output stack $O$ holds $o$ and every other stack is empty. Started at its start label with flag $v$ and stacks $S$, the CLIQUE emitter reaches the label $\\mathrm{done}$ in at most $8(|\\tau| + 1)^3$\u2026","title":"The CLIQUE emitter computes Karp's instance in cubic time"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/mode","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.Mode","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#mode","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"A mode is one of two values, the flip mode and the vertex-cover mode. They select the reduction that the machine of this module computes: CLIQUE to INDEPENDENT-SET in the flip mode, INDEPENDENT-SET to VERTEX-COVER in the vertex-cover mode.","title":"Modes of the CLIQUE/VERTEX-COVER reduction machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/f","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.Mode.f","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#f","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"The map on bit strings that a mode of the CLIQUE/VERTEX-COVER reduction machine computes: in the flip mode it is the reduction from CLIQUE to INDEPENDENT-SET on bit strings, and in the vertex-cover mode it is the reduction from INDEPENDENT-SET to VERTEX-COVER on bit strings.","title":"Reduction computed in each mode"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/k","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.K","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#k","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"Stacks: input `X`, output `O`, counters `A` (`k`), `B` (`n`), `D` (`n`, then `n - k`), and the reversed pair bits `Y`.","title":"Stacks of the CLIQUE to VERTEX-COVER reduction machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/untitled","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.\u0393","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#untitled","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"The alphabet $\\Gamma(k)$ of a stack $k$ is the Booleans if $k$ is the output stack $O$, and the token alphabet $\\mathrm{Sym}$ for every other stack ($X$, $A$, $B$, $D$ and $Y$).","title":"Stack alphabets of the clique reduction machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/lab","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.Lab","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#lab","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"The type of the ten instruction labels of the KG-map machine: read $k$, read $n$, move the pair bits, subtract, output the pair bits, output $n$, write the separator, output the last number, the failure label and the halting label.","title":"Labels of the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/u","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.U","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#u","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"$U$ is the symbol $\\mathsf{cl}$ of the token alphabet $\\mathrm{Sym}$.","title":"Counter unit symbol of the clique reduction machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/is-t","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.isT","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#is-t","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"A test on the result of popping a stack: it is true exactly when a symbol was popped and that symbol is $\\mathrm{bit}(\\mathrm{true})$, and false for every other symbol and for an empty stack.","title":"Test for the input bit true"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/is-v","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.isV","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#is-v","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"A test on the result of popping a stack: it is true exactly when a symbol was popped and that symbol is $\\mathrm{val}(\\mathrm{true})$, and false for every other symbol and for an empty stack.","title":"Test for the moved bit true"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/fb","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.Mode.fb","parent":31664,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06/index.html#fb","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-06","status":"fully_proved","summary":"For a mode of the CLIQUE/VERTEX-COVER reduction machine, the map on booleans applied to each pair bit as it is written to the output: negation in the flip mode and the identity in the vertex-cover mode.","title":"Pair-bit transform of each mode"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","kind":"scope","navigation":"transparent","parent":31558,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","title":"Label after the move phase of the KG-map machine to SAT \u2264 CLIQUE is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/next","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.Mode.next","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#next","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"The map from the modes of the KG-map machine to its labels that sends the flip mode to the label output the pair bits and the vertex-cover mode to the subtraction label. The machine goes to this label after moving the pair bits.","title":"Label after the move phase of the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/ctr","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.Mode.ctr","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#ctr","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"The map from the modes of the KG-map machine to its stacks that sends the flip mode to the counter $A$ and the vertex-cover mode to the counter $D$.","title":"Counter of the first output number of the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/prog","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.prog","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#prog","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"For a mode $m$ (flip or vc), the KG-map machine $M_m$ is the multi-stack machine with stacks $X$, $A$, $B$, $D$, $Y$ of symbols and an output stack $O$ of bits, whose labels act as follows; the counters $A$, $B$, $D$ use a fixed unit symbol.","title":"The KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/start","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.start","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#start","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"The KG-map machine starts at the label read $k$.","title":"Start label of the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/rd-k-some","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.rdK_some","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#rd-k-some","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode, $k$ and $i$ natural numbers and $r$ a list of bits. Consider a configuration of the KG-map machine $M_m$ at the label read $k$, with any internal bit, in which stack $X$ holds the input symbols $\\mathrm{bit}(b)$ for the bits $b$ of the list of $k$ trues, a false and then $r$, top first, and stack $A$ holds $i$ copies of the unit symbol. Then in exactly $k + 1$ steps $M_m$ reaches the label read\u2026","title":"Reading the first unary number in the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/rd-k-none","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.rdK_none","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#rd-k-none","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode and $r$ a list of bits on which reading a unary length prefix fails (this happens exactly when $r$ contains no false). From any configuration of the KG-map machine $M_m$ at the label read $k$, with any internal bit, in which stack $X$ holds the input symbols $\\mathrm{bit}(b)$ for the bits $b$ of $r$, top first, $M_m$ reaches the failure label in exactly $|r| + 1$ steps, with every stack other than\u2026","title":"Unterminated first unary number in the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/rd-n-some","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.rdN_some","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#rd-n-some","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode, $n$ and $i$ natural numbers and $r$ a list of bits. Consider a configuration of the KG-map machine $M_m$ at the label read $n$, with any internal bit, in which stack $X$ holds the input symbols $\\mathrm{bit}(b)$ for the bits $b$ of the list of $n$ trues, a false and then $r$, top first, and stacks $B$ and $D$ each hold $i$ copies of the unit symbol. Then in exactly $n + 1$ steps $M_m$ reaches the\u2026","title":"Reading the second unary number in the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/rd-n-none","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.rdN_none","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#rd-n-none","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode and $r$ a list of bits on which reading a unary length prefix fails (this happens exactly when $r$ contains no false). From any configuration of the KG-map machine $M_m$ at the label read $n$, with any internal bit, in which stack $X$ holds the input symbols $\\mathrm{bit}(b)$ for the bits $b$ of $r$, top first, $M_m$ reaches the failure label in exactly $|r| + 1$ steps, with every stack other than\u2026","title":"Unterminated second unary number in the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/mv-run","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.mv_run","parent":31685,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moving the pair bits onto stack "],["math","\\(Y\\)"]],"source":"Moving the pair bits onto stack $Y$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#mv-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Fix a mode $m$ of the CLIQUE/VERTEX-COVER reduction machine, a list of bits $r$ and a list of symbols $y$. Start the machine for $m$ at the label for moving the pair bits, with any internal state and any stacks such that stack $X$ holds $r$ as the symbols $\\mathrm{bit}(b)$ and stack $Y$ holds $y$. Then after exactly $|r| + 1$ steps it is at the label that follows moving the pair bits in mode $m$, with some internal\u2026","title":"Moving the pair bits onto stack $Y$"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/sub-le","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.sub_le","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#sub-le","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode of the machine, let $k \\le n$ be natural numbers, and let $S$ be stack contents in which the counter stack $A$ holds $U^k$ ($k$ copies of the unit symbol $U$) and the counter stack $D$ holds $U^n$. Then for every Boolean local state $v$ there are a local state $v'$ and stack contents $S'$ such that the machine of mode $m$ runs in exactly $k + 1$ steps from label $\\mathsf{sub}$ with local state $v$\u2026","title":"Subtraction loop of the clique reduction machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/sub-gt","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.sub_gt","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#sub-gt","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode and $n < k$ natural numbers. From any configuration of the KG-map machine $M_m$ at the label subtract, with any internal bit, in which stack $A$ holds $k$ copies of the unit symbol and stack $D$ holds $n$ copies, $M_m$ reaches the failure label in exactly $n + 1$ steps, with every stack other than $A$ and $D$ unchanged.","title":"Failed subtraction in the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/out-y-run","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.outY_run","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#out-y-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode and $y$ and $o$ lists of bits. Consider a configuration of the KG-map machine $M_m$ at the label output pair bits, with any internal bit, in which stack $Y$ holds the symbols $\\mathrm{val}(b)$ for the bits $b$ of $y$, the first bit of $y$ on top, and the output stack $O$ holds $o$. Then in exactly $|y| + 1$ steps $M_m$ reaches the label output $n$ with $O$ equal, top first, to false, then the\u2026","title":"Output loop for the pair bits of the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/tail-run","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.tail_run","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#tail-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode of the machine and $C$ its final counter stack ($A$ for the flip mode, $D$ for the vertex-cover mode). Let $n, c$ be natural numbers, $u$ a symbol of the alphabet of $C$, and $S$ stack contents in which the stack $B$ holds $U^n$ ($n$ copies of the unit symbol $U$) and $C$ holds $u^c$. Then for every Boolean local state $v$ there are a local state $v'$ and stack contents $S'$ such that the machine\u2026","title":"Number output of the clique reduction machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/out-run","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.out_run","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#out-run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode, $r$ a list of bits and $n$ and $c$ natural numbers. Consider a configuration of the KG-map machine $M_m$ at the label output pair bits, with any internal bit, in which stack $Y$ holds the symbols $\\mathrm{val}(b)$ for the bits $b$ of $r$ in reverse order (the last bit of $r$ on top), stack $B$ holds $n$ copies of the unit symbol, the counter of $m$ ($A$ for flip, $D$ for vc) holds $c$ copies of\u2026","title":"Output phase of the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/run","kind":"node","lean":"Cslib.Complexity.Karp.KGMap.run","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#run","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $m$ be a mode and $x$ a list of bits, and let $F_m$ be the clique-to-independent-set map on bit strings for flip and the independent-set-to-vertex-cover map for vc. From any configuration of the KG-map machine $M_m$ at its start label, with any internal bit, in which stack $X$ holds the input symbols $\\mathrm{bit}(b)$ for the bits $b$ of $x$, top first, and every other stack is empty, $M_m$ reaches the halting\u2026","title":"Correctness and running time of the KG-map machine"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/kg-map-comp","kind":"node","lean":"Cslib.Complexity.Karp.kgMapComp","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#kg-map-comp","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"The CLIQUE-to-INDEPENDENT-SET and INDEPENDENT-SET-to-VERTEX-COVER map as a host component. For a mode $m$, standing for the reduction from CLIQUE to INDEPENDENT-SET or the one from INDEPENDENT-SET to VERTEX-COVER, this is the host component whose stacks (listed as $X, O, A, B, D, Y$), stack alphabets, labels and program are those of the Turing machine that computes the reduction of mode $m$.","title":"The CLIQUE-to-INDEPENDENT-SET and INDEPENDENT-SET-to-VERTEX-COVER map as a host component"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/is-poly-time-flip-kg","kind":"node","lean":"Cslib.Complexity.Karp.isPolyTime_flipKG","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#is-poly-time-flip-kg","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"The map on bit strings that sends $1^k 0 1^n 0 b$, for natural numbers $k$ and $n$ and a bit string $b$, to $1^k 0 1^n 0 \\bar b$, where $\\bar b$ is $b$ with every bit complemented, and sends every string not of this form to the string $100$, is computable in polynomial time.","title":"CLIQUE \u2264 INDEPENDENT-SET is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/is-poly-time-is-to-vc","kind":"node","lean":"Cslib.Complexity.Karp.isPolyTime_isToVc","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#is-poly-time-is-to-vc","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"The map on bit strings that sends $1^k 0 1^n 0 b$, for natural numbers $k \\le n$ and a bit string $b$, to $1^{n-k} 0 1^n 0 b$, and sends every other string (including $1^k 0 1^n 0 b$ with $k > n$) to the string $100$, is computable in polynomial time.","title":"INDEPENDENT-SET \u2264 VERTEX-COVER is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/emit-clique","kind":"node","lean":"Cslib.Complexity.Karp.emitClique","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#emit-clique","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"The emitter given by the CLIQUE emitter machine, run from its start label to its exit label, with token stack $T$ and output stack $O$. Its output on a CNF formula $\\varphi$ is Karp's CLIQUE instance of $\\varphi$, its output on undecodable input is the string $100$, and its step bound on $m$ tokens is $8(m + 1)^3$. Let $\\varphi$ be a CNF formula with $m$ tokens. Started with the tokens of $\\varphi$ on $T$, last\u2026","title":"The CLIQUE emitter"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/is-poly-time-reduce-clique","kind":"node","lean":"Cslib.Complexity.Karp.isPolyTime_reduceClique","parent":31685,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07/index.html#is-poly-time-reduce-clique","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-07","status":"fully_proved","summary":"Let $f$ be the map on bit strings that sends the binary code of a CNF formula $\\varphi$ to the encoding of Karp's CLIQUE instance $(k, G)$ of $\\varphi$, and sends every string that is not the code of a CNF formula to the string $100$. Here $G$ has one vertex per literal occurrence in $\\varphi$, two occurrences are adjacent when they lie in different clauses and are not complementary, and $k$ is the number of\u2026","title":"SAT \u2264 CLIQUE is polynomial-time"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","kind":"scope","navigation":"transparent","parent":31558,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","title":"CLIQUE is NP-hard to VERTEX-COVER (exact size `k`) is NP-complete"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/clique-dec-np-hard","kind":"node","lean":"Cslib.Complexity.Karp.cliqueDec_npHard","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#clique-dec-np-hard","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"Every decision problem $L$ in NP has a polynomial-time many-one reduction to CLIQUE: a polynomial-time computable map $f$ on bit strings such that, for every bit string $x$, $L$ accepts $x$ exactly when CLIQUE accepts $f(x)$.","title":"CLIQUE is NP-hard"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/ind-set-dec-np-hard","kind":"node","lean":"Cslib.Complexity.Karp.indSetDec_npHard","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#ind-set-dec-np-hard","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"Every decision problem $L$ in NP has a polynomial-time many-one reduction to INDEPENDENT-SET: a polynomial-time computable map $f$ on bit strings such that, for every bit string $x$, $L$ accepts $x$ exactly when INDEPENDENT-SET accepts $f(x)$.","title":"INDEPENDENT-SET is NP-hard"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/vertex-cover-dec-np-hard","kind":"node","lean":"Cslib.Complexity.Karp.vertexCoverDec_npHard","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#vertex-cover-dec-np-hard","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"Every decision problem in NP reduces by a polynomial-time many-one reduction to the decision problem VERTEX-COVER, which accepts exactly the valid encodings of a number $k$ and a graph $G$ such that $G$ has a vertex cover of exactly $k$ vertices.","title":"VERTEX-COVER (exact size `k`) is NP-hard"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/kg-verify-comp","kind":"node","lean":"Cslib.Complexity.Karp.kgVerifyComp","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#kg-verify-comp","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"For a mode $m$ (clique, independent set or vertex cover), this is the host component whose stacks are $X, W, C, N, V, T$, each over the symbol alphabet of the SAT verifier, and whose labels and program are those of the verifier machine of mode $m$.","title":"The CLIQUE, INDEPENDENT-SET and VERTEX-COVER verifier as a host component"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/mem-np-of-kg-check","kind":"node","lean":"Cslib.Complexity.Karp.mem_NP_of_kgCheck","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#mem-np-of-kg-check","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"Let $L$ be a decision problem, let $m$ be a mode (clique, independent set or vertex cover), and let $V_m$ be the verifier of mode $m$. If for every bit string $x$, $L$ accepts $x$ exactly when there is a bit string $w$ with $|w| \\le |x|$ such that $V_m$ accepts $(x, w)$, then $L$ is in NP.","title":"NP membership from the CLIQUE, INDEPENDENT-SET and VERTEX-COVER verifier"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/clique-dec-mem-np","kind":"node","lean":"Cslib.Complexity.Karp.cliqueDec_mem_NP","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#clique-dec-mem-np","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"CLIQUE is in NP.","title":"CLIQUE is in NP"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/ind-set-dec-mem-np","kind":"node","lean":"Cslib.Complexity.Karp.indSetDec_mem_NP","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#ind-set-dec-mem-np","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"INDEPENDENT-SET is in NP.","title":"INDEPENDENT-SET is in NP"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/vertex-cover-dec-mem-np","kind":"node","lean":"Cslib.Complexity.Karp.vertexCoverDec_mem_NP","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#vertex-cover-dec-mem-np","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"The decision problem VERTEX-COVER, which accepts exactly the valid encodings of a number $k$ and a graph $G$ such that $G$ has a vertex cover of exactly $k$ vertices, is in NP.","title":"VERTEX-COVER (exact size `k`) is in NP"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/clique-dec-np-complete","kind":"node","lean":"Cslib.Complexity.Karp.cliqueDec_npComplete","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#clique-dec-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"CLIQUE lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to CLIQUE.","title":"CLIQUE is NP-complete"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/ind-set-dec-np-complete","kind":"node","lean":"Cslib.Complexity.Karp.indSetDec_npComplete","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#ind-set-dec-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"INDEPENDENT-SET lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to INDEPENDENT-SET.","title":"INDEPENDENT-SET is NP-complete"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/vertex-cover-dec-np-complete","kind":"node","lean":"Cslib.Complexity.Karp.vertexCoverDec_npComplete","parent":31706,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08/index.html#vertex-cover-dec-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/clique-independent-set-and-vertex-cover/part-08","status":"fully_proved","summary":"The decision problem VERTEX-COVER, which accepts exactly the valid encodings of a number $k$ and a graph $G$ such that $G$ has a vertex cover of exactly $k$ vertices, is in NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"VERTEX-COVER (exact size `k`) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set","status":"can_prove","title":"Connected dominating set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/connected-dominating-set-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ConnectedDominatingSet.ConnectedDominatingSetInstance","parent":31718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/index.html#connected-dominating-set-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set","status":"fully_proved","summary":"An instance of CONNECTED DOMINATING SET is a triple $(n, E, K)$ of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"Instances of CONNECTED DOMINATING SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ConnectedDominatingSet.Graph","parent":31718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set","status":"fully_proved","summary":"For an instance $(n, E, K)$ of CONNECTED DOMINATING SET, its graph is the simple graph on the vertex set $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Graph of a CONNECTED DOMINATING SET instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ConnectedDominatingSet.WellFormed","parent":31718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set","status":"fully_proved","summary":"An instance $(n, E, K)$ of CONNECTED DOMINATING SET is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ has $u < n$, $v < n$ and $u \\ne v$, with the reversed pair $(v, u)$ not in $E$.","title":"Well-formed CONNECTED DOMINATING SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/connected-dominating-set-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ConnectedDominatingSet.ConnectedDominatingSetFeasible","parent":31718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/index.html#connected-dominating-set-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set","status":"fully_proved","summary":"An instance $(n, E, K)$ of CONNECTED DOMINATING SET is feasible when there is a set $S$ of vertices of its graph $G$ with at most $K$ elements such that every vertex of $G$ lies in $S$ or is adjacent in $G$ to a vertex of $S$, and the subgraph of $G$ induced by $S$ is connected (in particular, $S$ is nonempty).","title":"Feasible CONNECTED DOMINATING SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/connected-dominating-set-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ConnectedDominatingSet.connectedDominatingSetDec","parent":31718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/index.html#connected-dominating-set-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set","status":"fully_proved","summary":"CONNECTED DOMINATING SET accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance of CONNECTED DOMINATING SET that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"The decision problem CONNECTED DOMINATING SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/connected-dominating-set-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ConnectedDominatingSet.connectedDominatingSet_npComplete","parent":31718,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set/index.html#connected-dominating-set-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/connected-dominating-set","status":"can_prove","summary":"The decision problem CONNECTED DOMINATING SET belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"CONNECTED DOMINATING SET is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph","status":"can_prove","title":"Densest k-subgraph"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/densest-ksubgraph-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DensestKSubgraph.DensestKSubgraphInstance","parent":31725,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/index.html#densest-ksubgraph-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph","status":"fully_proved","summary":"An instance of DENSEST k-SUBGRAPH is a quadruple $(n, E, k, m)$ consisting of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edges), a natural number $k$ (the size of the vertex set sought) and a natural number $m$ (the edge threshold).","title":"Instances of DENSEST k-SUBGRAPH"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DensestKSubgraph.WellFormed","parent":31725,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph","status":"fully_proved","summary":"An instance $(n, E, k, m)$ of DENSEST k-SUBGRAPH is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, with the reversed pair $(v, u)$ not in $E$.","title":"Well-formed DENSEST k-SUBGRAPH instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/densest-ksubgraph-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DensestKSubgraph.DensestKSubgraphFeasible","parent":31725,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/index.html#densest-ksubgraph-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph","status":"fully_proved","summary":"An instance $(n, E, k, m)$ of DENSEST k-SUBGRAPH is feasible when there are a finite set $S$ of exactly $k$ natural numbers, each less than $n$, and a sublist $E'$ of $E$ (obtained by deleting entries and keeping the order) with at least $m$ entries such that both components of every pair in $E'$ lie in $S$.","title":"Feasibility for DENSEST k-SUBGRAPH"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/densest-ksubgraph-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DensestKSubgraph.densestKSubgraphDec","parent":31725,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/index.html#densest-ksubgraph-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph","status":"fully_proved","summary":"The decision problem DENSEST k-SUBGRAPH accepts a bitstring $x$ exactly when bitstring decoding of $x$ as a quadruple $(n, E, k, m)$ succeeds and returns an instance of DENSEST k-SUBGRAPH that is both well formed and feasible; every bitstring that does not decode is rejected.","title":"DENSEST k-SUBGRAPH as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/densest-ksubgraph-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DensestKSubgraph.densestKSubgraph_npComplete","parent":31725,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph/index.html#densest-ksubgraph-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/densest-k-subgraph","status":"can_prove","summary":"The decision problem DENSEST k-SUBGRAPH belongs to the class NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"DENSEST k-SUBGRAPH is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set","status":"can_prove","title":"Dominating set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/dominating-set-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DominatingSet.DominatingSetInstance","parent":31731,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/index.html#dominating-set-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set","status":"fully_proved","summary":"An instance of DOMINATING SET is a triple $(n, E, K)$ of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"Instances of DOMINATING SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DominatingSet.Graph","parent":31731,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set","status":"fully_proved","summary":"For an instance $(n, E, K)$ of DOMINATING SET, its graph is the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Graph of a DOMINATING SET instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DominatingSet.WellFormed","parent":31731,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set","status":"fully_proved","summary":"An instance $(n, E, K)$ of DOMINATING SET is well formed when the list $E$ has no repeated entries and every pair $(u, v) \\in E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$.","title":"Well-formed DOMINATING SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/dominating-set-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DominatingSet.DominatingSetFeasible","parent":31731,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/index.html#dominating-set-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set","status":"fully_proved","summary":"An instance $(n, E, K)$ of DOMINATING SET is feasible when there is a set $S$ of at most $K$ vertices of its graph such that every vertex lies in $S$ or is adjacent to a vertex of $S$, where the graph has vertex set $\\{0, \\dots, n - 1\\}$ and distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Feasibility of a DOMINATING SET instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/dominating-set-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DominatingSet.dominatingSetDec","parent":31731,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/index.html#dominating-set-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set","status":"fully_proved","summary":"The decision problem DOMINATING SET accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, E, K)$ of a natural number $n$, a list $E$ of pairs of natural numbers and a natural number $K$ such that $E$ has no repeated entries, every pair $(u, v) \\in E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$, and there is a set $S$ of at most $K$ vertices with\u2026","title":"DOMINATING SET as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/dominating-set-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DominatingSet.dominatingSet_npComplete","parent":31731,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set/index.html#dominating-set-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/dominating-set","status":"can_prove","summary":"The decision problem DOMINATING SET belongs to NP, and every decision problem in NP reduces to DOMINATING SET by a polynomial-time many-one reduction.","title":"DOMINATING SET is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set","status":"can_prove","title":"Edge dominating set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/edge-dominating-set-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeDominatingSet.EdgeDominatingSetInstance","parent":31738,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/index.html#edge-dominating-set-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set","status":"fully_proved","summary":"An instance of EDGE DOMINATING SET is a triple $(n, E, K)$ of a natural number $n$ (the vertex count), a finite list $E$ of ordered pairs of natural numbers (the edge list) and a natural number $K$ (the bound).","title":"EDGE DOMINATING SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/adj","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeDominatingSet.Adj","parent":31738,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/index.html#adj","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set","status":"fully_proved","summary":"For a list $E$ of ordered pairs of natural numbers and natural numbers $u$ and $v$, the Boolean adjacency of $u$ and $v$ in $E$ is true exactly when $(u, v)$ or $(v, u)$ occurs in $E$.","title":"Edge-list adjacency for EDGE DOMINATING SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeDominatingSet.WellFormed","parent":31738,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set","status":"fully_proved","summary":"An instance with vertex count $n$, edge list $E$ and bound $K$ is well formed when $E$ has no repeated entries and every pair $(a, b)$ in $E$ satisfies $a < n$, $b < n$ and $a \\ne b$, and $(b, a)$ does not occur in $E$. Thus $E$ lists the edges of a simple graph on the vertices $0, \\dots, n - 1$, each once and in one orientation.","title":"Well-formed EDGE DOMINATING SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/edge-dominating-set-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeDominatingSet.EdgeDominatingSetFeasible","parent":31738,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/index.html#edge-dominating-set-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set","status":"fully_proved","summary":"An instance with vertex count $n$, edge list $E$ and bound $K$ is feasible when some sublist $D$ of $E$ (its entries taken in order) has at most $K$ entries and dominates every edge of $E$: for every $e \\in E$ there is $d \\in D$ such that one of the two entries of $d$ equals one of the two entries of $e$.","title":"Feasible EDGE DOMINATING SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/edge-dominating-set-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeDominatingSet.edgeDominatingSetDec","parent":31738,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/index.html#edge-dominating-set-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set","status":"fully_proved","summary":"EDGE DOMINATING SET is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an EDGE DOMINATING SET instance $(n, E, K)$ that is well formed and feasible, that is, some sublist of $E$ with at most $K$ entries contains, for every edge of $E$, an edge sharing an endpoint with it. Bitstrings that do not decode to an instance are rejected.","title":"The EDGE DOMINATING SET decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/edge-dominating-set-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeDominatingSet.edgeDominatingSet_npComplete","parent":31738,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set/index.html#edge-dominating-set-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/edge-dominating-set","status":"can_prove","summary":"The EDGE DOMINATING SET decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"EDGE DOMINATING SET is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination","status":"can_prove","title":"Efficient domination"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/efficient-domination-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EfficientDomination.EfficientDominationInstance","parent":31745,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/index.html#efficient-domination-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination","status":"fully_proved","summary":"An instance of EFFICIENT DOMINATION is a pair $(n, E)$ of a natural number $n$, the number of vertices, and a list $E$ of pairs of natural numbers, the edges.","title":"Efficient Domination instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EfficientDomination.Graph","parent":31745,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination","status":"fully_proved","summary":"For an instance $(n, E)$, its graph is the simple graph on the vertex set $\\{0, \\dots, n - 1\\}$ in which distinct vertices $u$ and $v$ are adjacent when $(u, v)$ or $(v, u)$ belongs to $E$.","title":"Graph of an Efficient Domination instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EfficientDomination.WellFormed","parent":31745,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination","status":"fully_proved","summary":"An instance $(n, E)$ is well formed when the list $E$ has no repeated entries and every pair $(a, b)$ in $E$ satisfies $a < n$, $b < n$ and $a \\ne b$, and $(b, a)$ does not belong to $E$.","title":"Well-formed Efficient Domination instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/efficient-domination-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EfficientDomination.EfficientDominationFeasible","parent":31745,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/index.html#efficient-domination-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination","status":"fully_proved","summary":"An instance $(n, E)$ of EFFICIENT DOMINATION is feasible when its graph has an efficient dominating set (a perfect code): a set $S$ of vertices such that every vertex $v$ has exactly one vertex $u \\in S$ with $u = v$ or $u$ adjacent to $v$.","title":"Instances with an efficient dominating set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/efficient-domination-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EfficientDomination.efficientDominationDec","parent":31745,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/index.html#efficient-domination-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination","status":"fully_proved","summary":"The decision problem EFFICIENT DOMINATION accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of instances, to an instance $(n, E)$ that is well formed and whose graph has an efficient dominating set. Bitstrings that do not decode to an instance are rejected.","title":"EFFICIENT DOMINATION (PERFECT CODE) as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/efficient-domination-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EfficientDomination.efficientDomination_npComplete","parent":31745,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination/index.html#efficient-domination-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/efficient-domination","status":"can_prove","summary":"The decision problem EFFICIENT DOMINATION belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"EFFICIENT DOMINATION (PERFECT CODE) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set","status":"can_prove","title":"Feedback arc set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/feedback-arc-set-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackArcSet.FeedbackArcSetInstance","parent":31752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/index.html#feedback-arc-set-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set","status":"fully_proved","summary":"An instance of FEEDBACK ARC SET is a triple $(n, E, K)$ consisting of a natural number $n$ (the number of vertices), a finite list $E$ of ordered pairs of natural numbers (the arcs) and a natural number $K$ (the bound).","title":"Instances of FEEDBACK ARC SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/dadj","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackArcSet.DAdj","parent":31752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/index.html#dadj","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set","status":"fully_proved","summary":"For a list $E$ of ordered pairs of natural numbers and natural numbers $u$ and $v$, the directed adjacency test returns true when the pair $(u, v)$ occurs in $E$, that is, when $E$ contains an arc from $u$ to $v$, and false otherwise.","title":"Directed adjacency for FEEDBACK ARC SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackArcSet.WellFormed","parent":31752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set","status":"fully_proved","summary":"An instance $(n, E, K)$ of FEEDBACK ARC SET is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$; that is, $E$ lists distinct arcs, without loops, of a directed graph on the vertices $0, \\dots, n-1$.","title":"Well-formed FEEDBACK ARC SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/feedback-arc-set-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackArcSet.FeedbackArcSetFeasible","parent":31752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/index.html#feedback-arc-set-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set","status":"fully_proved","summary":"An instance $(n, E, K)$ of FEEDBACK ARC SET is feasible when there is an ordering $v_0, v_1, \\dots, v_{n-1}$ of the vertices $0, \\dots, n-1$, in which each of them occurs exactly once, such that at most $K$ entries $(u, v)$ of the list $E$, counted with repetition, are backward, meaning that the position of $v$ in the ordering is strictly smaller than the position of $u$. Here $v_j$ has position $j$, and a number\u2026","title":"Feasible FEEDBACK ARC SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/feedback-arc-set-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackArcSet.feedbackArcSetDec","parent":31752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/index.html#feedback-arc-set-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set","status":"fully_proved","summary":"The decision problem FEEDBACK ARC SET accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of instances, to an instance $(n, E, K)$ of FEEDBACK ARC SET that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"FEEDBACK ARC SET as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/feedback-arc-set-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackArcSet.feedbackArcSet_npComplete","parent":31752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set/index.html#feedback-arc-set-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-arc-set","status":"can_prove","summary":"The decision problem FEEDBACK ARC SET belongs to NP, and every decision problem in NP reduces to FEEDBACK ARC SET by a polynomial-time many-one reduction.","title":"FEEDBACK ARC SET is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set","status":"can_prove","title":"Feedback vertex set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/feedback-vertex-set-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackVertexSet.FeedbackVertexSetInstance","parent":31759,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/index.html#feedback-vertex-set-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set","status":"fully_proved","summary":"An instance of FEEDBACK VERTEX SET is a triple $(n, es, K)$ of a natural number $n$ (the number of vertices), a list $es$ of pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"Instances of FEEDBACK VERTEX SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackVertexSet.Graph","parent":31759,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set","status":"fully_proved","summary":"For an instance $(n, es, K)$ of FEEDBACK VERTEX SET, its graph is the simple graph on $\\{0, \\dots, n-1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $es$.","title":"Graph of a FEEDBACK VERTEX SET instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackVertexSet.WellFormed","parent":31759,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set","status":"fully_proved","summary":"An instance $(n, es, K)$ of FEEDBACK VERTEX SET is well formed when the list $es$ has no repeated entries and every pair $(u, v)$ in $es$ satisfies $u < n$, $v < n$ and $u \\ne v$, and $(v, u)$ does not belong to $es$.","title":"Well-formed FEEDBACK VERTEX SET instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/feedback-vertex-set-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackVertexSet.FeedbackVertexSetFeasible","parent":31759,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/index.html#feedback-vertex-set-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set","status":"fully_proved","summary":"An instance $(n, es, K)$ of FEEDBACK VERTEX SET is feasible when there is a set $S$ of at most $K$ vertices of its graph such that the subgraph of its graph induced on the vertices outside $S$ is acyclic.","title":"Feedback vertex sets within the bound"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/feedback-vertex-set-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackVertexSet.feedbackVertexSetDec","parent":31759,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/index.html#feedback-vertex-set-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set","status":"fully_proved","summary":"The decision problem FEEDBACK VERTEX SET accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of instances, to an instance of FEEDBACK VERTEX SET that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"FEEDBACK VERTEX SET as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/feedback-vertex-set-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FeedbackVertexSet.feedbackVertexSet_npComplete","parent":31759,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set/index.html#feedback-vertex-set-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/feedback-vertex-set","status":"can_prove","summary":"The decision problem FEEDBACK VERTEX SET belongs to NP, and every decision problem in NP reduces to FEEDBACK VERTEX SET by a polynomial-time many-one reduction.","title":"FEEDBACK VERTEX SET is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set","status":"can_prove","title":"Independent dominating set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/independent-dominating-set-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.IndependentDominatingSet.IndependentDominatingSetInstance","parent":31766,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/index.html#independent-dominating-set-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set","status":"fully_proved","summary":"An instance of INDEPENDENT DOMINATING SET is a triple $(n, E, K)$ of a natural number $n$, the number of vertices, a list $E$ of pairs of natural numbers, the edges, and a natural number $K$, the bound on the size of the set.","title":"Independent dominating set instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.IndependentDominatingSet.Graph","parent":31766,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set","status":"fully_proved","summary":"For an independent dominating set instance with $n$ vertices and edge list $E$, the graph of the instance is the simple graph on $\\{0, \\dots, n-1\\}$ in which distinct vertices $u$ and $v$ are adjacent when $(u, v)$ or $(v, u)$ belongs to $E$.","title":"Graph of an independent dominating set instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.IndependentDominatingSet.WellFormed","parent":31766,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set","status":"fully_proved","summary":"An independent dominating set instance $(n, E, K)$ is well formed when the list $E$ has no repeated entries and every pair $(a, b)$ in $E$ satisfies $a < n$, $b < n$, $a \\ne b$, and $(b, a)$ does not belong to $E$.","title":"Well-formed independent dominating set instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/independent-dominating-set-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.IndependentDominatingSet.IndependentDominatingSetFeasible","parent":31766,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/index.html#independent-dominating-set-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set","status":"fully_proved","summary":"An independent dominating set instance $(n, E, K)$ has a small independent dominating set when there is a set $S$ of vertices of its graph with at most $K$ elements such that every vertex lies in $S$ or is adjacent to a vertex of $S$, and no two distinct vertices of $S$ are adjacent.","title":"Instances with a small independent dominating set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/independent-dominating-set-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.IndependentDominatingSet.independentDominatingSetDec","parent":31766,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/index.html#independent-dominating-set-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set","status":"fully_proved","summary":"The decision problem INDEPENDENT DOMINATING SET accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an independent dominating set instance $(n, E, K)$ that is well formed and whose graph has an independent dominating set of at most $K$ vertices. It rejects every bitstring that does not decode to an instance.","title":"INDEPENDENT DOMINATING SET as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/independent-dominating-set-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.IndependentDominatingSet.independentDominatingSet_npComplete","parent":31766,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set/index.html#independent-dominating-set-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/independent-dominating-set","status":"can_prove","summary":"The decision problem INDEPENDENT DOMINATING SET belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"INDEPENDENT DOMINATING SET is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching","status":"can_prove","title":"Maximum induced matching"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/maximum-induced-matching-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumInducedMatching.MaximumInducedMatchingInstance","parent":31773,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/index.html#maximum-induced-matching-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching","status":"fully_proved","summary":"A maximum-induced-matching instance is a triple $(n, E, K)$ of a natural number $n$ (the vertex count), a list $E$ of pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"Maximum-induced-matching instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumInducedMatching.WellFormed","parent":31773,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching","status":"fully_proved","summary":"A maximum-induced-matching instance $(n, E, K)$ is well formed when the edge list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, with the reverse pair $(v, u)$ not in $E$.","title":"Well-formed maximum-induced-matching instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/maximum-induced-matching-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumInducedMatching.MaximumInducedMatchingFeasible","parent":31773,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/index.html#maximum-induced-matching-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching","status":"fully_proved","summary":"A maximum-induced-matching instance $(n, E, K)$, with vertex count $n$, edge list $E$ of pairs of natural numbers and bound $K$, is feasible when there is a sublist $M$ of $E$ of length at least $K$ such that any two distinct pairs in $M$ share no endpoint, and no pair in $E$ joins an endpoint of one pair in $M$ to an endpoint of a different pair in $M$.","title":"Feasible maximum-induced-matching instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/maximum-induced-matching-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumInducedMatching.maximumInducedMatchingDec","parent":31773,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/index.html#maximum-induced-matching-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching","status":"fully_proved","summary":"MAXIMUM INDUCED MATCHING is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a maximum-induced-matching instance $(n, E, K)$ that is well formed and feasible: the edge list $E$ has no repeated entries, every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$ with $(v, u)$ not in $E$, and some sublist $M$ of $E$ of length at least $K$\u2026","title":"The MAXIMUM INDUCED MATCHING decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/maximum-induced-matching-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumInducedMatching.maximumInducedMatching_npComplete","parent":31773,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching/index.html#maximum-induced-matching-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/maximum-induced-matching","status":"can_prove","summary":"The MAXIMUM INDUCED MATCHING decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"MAXIMUM INDUCED MATCHING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in","status":"can_prove","title":"Minimum fill-in"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/minimum-fill-in-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumFillIn.MinimumFillInInstance","parent":31779,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/index.html#minimum-fill-in-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in","status":"fully_proved","summary":"An instance of MINIMUM FILL-IN is a triple $(n, E, K)$ consisting of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edge list) and a natural number $K$ (the bound).","title":"Instances of MINIMUM FILL-IN"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/graph-aug","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumFillIn.GraphAug","parent":31779,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/index.html#graph-aug","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in","status":"fully_proved","summary":"For an instance $(n, E, K)$ of MINIMUM FILL-IN and a list $A$ of pairs of natural numbers, the augmented graph is the graph of the edge list $E$ followed by $A$ on $n$ vertices: the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$ or to $A$.","title":"Graph augmented by added edges"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumFillIn.WellFormed","parent":31779,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in","status":"fully_proved","summary":"An instance $(n, E, K)$ of MINIMUM FILL-IN is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and its reverse $(v, u)$ does not belong to $E$.","title":"Well-formed MINIMUM FILL-IN instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/minimum-fill-in-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumFillIn.MinimumFillInFeasible","parent":31779,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/index.html#minimum-fill-in-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in","status":"fully_proved","summary":"An instance $(n, E, K)$ of MINIMUM FILL-IN is feasible when there are an ordering $\\sigma$ of the vertices $0, \\dots, n - 1$ (a list that contains each of them exactly once) and a list $A$ of pairs of natural numbers, with no repeated entries and at most $K$ entries, such that every pair $(u, v)$ in $A$ satisfies $u < n$, $v < n$ and $u \\ne v$, neither $(u, v)$ nor $(v, u)$ belongs to $E$, and $(v, u)$ does not\u2026","title":"Feasible MINIMUM FILL-IN instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/minimum-fill-in-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumFillIn.minimumFillInDec","parent":31779,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/index.html#minimum-fill-in-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in","status":"fully_proved","summary":"The decision problem MINIMUM FILL-IN accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance $(n, E, K)$ of MINIMUM FILL-IN that is well formed and feasible.","title":"The MINIMUM FILL-IN (CHORDAL COMPLETION) problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/minimum-fill-in-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumFillIn.minimumFillIn_npComplete","parent":31779,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in/index.html#minimum-fill-in-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/minimum-fill-in","status":"can_prove","summary":"The decision problem MINIMUM FILL-IN lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to MINIMUM FILL-IN.","title":"MINIMUM FILL-IN (CHORDAL COMPLETION) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal","status":"can_prove","title":"Odd cycle transversal"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/odd-cycle-transversal-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OddCycleTransversal.OddCycleTransversalInstance","parent":31786,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/index.html#odd-cycle-transversal-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal","status":"fully_proved","summary":"An odd cycle transversal instance is a triple $(n, es, K)$ of a natural number $n$ (the number of vertices), a list $es$ of pairs of natural numbers (the edges) and a natural number $K$ (the bound on the number of removed vertices).","title":"Odd cycle transversal instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OddCycleTransversal.Graph","parent":31786,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal","status":"fully_proved","summary":"For an odd cycle transversal instance $(n, es, K)$, its graph is the graph of the edge list $es$ on $n$ vertices: the simple graph on $\\{0, \\dots, n-1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $es$.","title":"Graph of an odd cycle transversal instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OddCycleTransversal.WellFormed","parent":31786,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal","status":"fully_proved","summary":"An odd cycle transversal instance $(n, es, K)$ is well formed when the list $es$ has no repeated entries and every pair $(u, v)$ in $es$ satisfies $u < n$, $v < n$ and $u \\ne v$, and $(v, u)$ does not belong to $es$.","title":"Well-formed odd cycle transversal instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/odd-cycle-transversal-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OddCycleTransversal.OddCycleTransversalFeasible","parent":31786,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/index.html#odd-cycle-transversal-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal","status":"fully_proved","summary":"An odd cycle transversal instance $(n, es, K)$ is feasible when there is a set $S$ of at most $K$ vertices of its graph such that the subgraph induced on the vertices outside $S$ is bipartite.","title":"Feasible odd cycle transversal instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/odd-cycle-transversal-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OddCycleTransversal.oddCycleTransversalDec","parent":31786,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/index.html#odd-cycle-transversal-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal","status":"fully_proved","summary":"The decision problem ODD CYCLE TRANSVERSAL accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an odd cycle transversal instance $(n, es, K)$ that is well formed and feasible. Well formed means that $es$ has no repeated entries and every pair $(u, v)$ in $es$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin es$; feasible means that some set of at most $K$ vertices\u2026","title":"ODD CYCLE TRANSVERSAL as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/odd-cycle-transversal-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.OddCycleTransversal.oddCycleTransversal_npComplete","parent":31786,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal/index.html#odd-cycle-transversal-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/odd-cycle-transversal","status":"can_prove","summary":"The decision problem ODD CYCLE TRANSVERSAL, which accepts a bitstring exactly when it decodes to a well-formed instance $(n, es, K)$ whose graph becomes bipartite after removing some set of at most $K$ vertices, belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"ODD CYCLE TRANSVERSAL is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination","status":"can_prove","title":"Roman domination"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/roman-domination-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RomanDomination.RomanDominationInstance","parent":31793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/index.html#roman-domination-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination","status":"fully_proved","summary":"An instance of ROMAN DOMINATION is a triple $(n, E, K)$ of a natural number $n$, the number of vertices, a list $E$ of pairs of natural numbers, the edges, and a natural number $K$, the weight bound.","title":"Roman domination instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RomanDomination.Graph","parent":31793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination","status":"fully_proved","summary":"For an instance $(n, E, K)$ of ROMAN DOMINATION, its graph is the graph of the edge list $E$ on $n$ vertices: the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Graph of a Roman domination instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RomanDomination.WellFormed","parent":31793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination","status":"fully_proved","summary":"An instance $(n, E, K)$ of ROMAN DOMINATION is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ has $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$.","title":"Well-formed Roman domination instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/roman-domination-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RomanDomination.RomanDominationFeasible","parent":31793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/index.html#roman-domination-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination","status":"fully_proved","summary":"An instance $(n, E, K)$ of ROMAN DOMINATION is feasible when there is a function $f$ from the vertices $\\{0, \\dots, n - 1\\}$ of its graph to $\\{0, 1, 2\\}$ such that every vertex $v$ with $f(v) = 0$ is adjacent to some vertex $u$ with $f(u) = 2$, and the total weight $\\sum_v f(v)$ is at most $K$.","title":"Feasible Roman domination instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/roman-domination-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RomanDomination.romanDominationDec","parent":31793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/index.html#roman-domination-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination","status":"fully_proved","summary":"The decision problem ROMAN DOMINATION accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance $(n, E, K)$ of ROMAN DOMINATION that is well formed and feasible; it rejects every other bitstring.","title":"ROMAN DOMINATION as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/roman-domination-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RomanDomination.romanDomination_npComplete","parent":31793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination/index.html#roman-domination-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/roman-domination","status":"can_prove","summary":"The decision problem ROMAN DOMINATION belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"ROMAN DOMINATION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set.json","id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set","kind":"scope","navigation":"module","parent":31544,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/index.html","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set","status":"can_prove","title":"Total dominating set"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/total-dominating-set-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalDominatingSet.TotalDominatingSetInstance","parent":31800,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/index.html#total-dominating-set-instance","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set","status":"fully_proved","summary":"An instance of TOTAL DOMINATING SET is a triple $(n, E, K)$ consisting of a natural number $n$, the number of vertices, a list $E$ of pairs of natural numbers, the edges, and a natural number $K$, the bound.","title":"Instances of TOTAL DOMINATING SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalDominatingSet.Graph","parent":31800,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set","status":"fully_proved","summary":"The graph of an instance $(n, E, K)$ of TOTAL DOMINATING SET is the simple graph on $\\{0, \\dots, n-1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$. Loops are dropped, and pairs with an endpoint at least $n$ contribute no edge.","title":"Graph of a total dominating set instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalDominatingSet.WellFormed","parent":31800,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set","status":"fully_proved","summary":"An instance $(n, E, K)$ of TOTAL DOMINATING SET is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and its reversal $(v, u)$ does not belong to $E$.","title":"Well-formed total dominating set instances"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/total-dominating-set-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalDominatingSet.TotalDominatingSetFeasible","parent":31800,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/index.html#total-dominating-set-feasible","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set","status":"fully_proved","summary":"An instance $(n, E, K)$ of TOTAL DOMINATING SET is feasible when there is a set $S$ of at most $K$ vertices of its graph such that every vertex $v$ of the graph, including the vertices of $S$, is adjacent to some vertex $u \\in S$.","title":"Feasibility of a total dominating set instance"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/total-dominating-set-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalDominatingSet.totalDominatingSetDec","parent":31800,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/index.html#total-dominating-set-dec","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set","status":"fully_proved","summary":"TOTAL DOMINATING SET accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of triples of a natural number, a list of pairs of natural numbers and a natural number, to an instance $(n, E, K)$ that is well formed and feasible: some set of at most $K$ vertices of the graph of the instance contains a neighbor of every vertex. Bitstrings that do not decode to an instance are rejected.","title":"The decision problem TOTAL DOMINATING SET"},{"id":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/total-dominating-set-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalDominatingSet.totalDominatingSet_npComplete","parent":31800,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set/index.html#total-dominating-set-np-complete","scope":"TheoryOfComputation/np-completeness/graph-covering-and-domination/total-dominating-set","status":"can_prove","summary":"The decision problem TOTAL DOMINATING SET belongs to NP, and every decision problem in NP reduces to TOTAL DOMINATING SET by a polynomial-time many-one reduction.","title":"TOTAL DOMINATING SET is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning","status":"can_prove","summary":"Vertex, edge, list and precoloring-extension coloring, achromatic number, graph homomorphism, maximum and k-cuts, bisection, multiway cut, partition into triangles or paths, bandwidth, linear arrangement and linear ordering.","title":"Graph coloring, cuts and layouts"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number","status":"can_prove","title":"Achromatic number"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/achromatic-number-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AchromaticNumber.AchromaticNumberInstance","parent":31808,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/index.html#achromatic-number-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number","status":"fully_proved","summary":"An instance of ACHROMATIC NUMBER is a triple $(n, E, K)$ of a natural number $n$ (the number of vertices), a finite list $E$ of ordered pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"Instances of ACHROMATIC NUMBER"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AchromaticNumber.WellFormed","parent":31808,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number","status":"fully_proved","summary":"An instance $(n, E, K)$ of ACHROMATIC NUMBER is well-formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$.","title":"Well-formed ACHROMATIC NUMBER instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/achromatic-number-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AchromaticNumber.AchromaticNumberFeasible","parent":31808,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/index.html#achromatic-number-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number","status":"fully_proved","summary":"An instance $(n, E, K)$ of ACHROMATIC NUMBER is feasible when there are a natural number $q \\ge K$ and a list $c = (c_0, \\dots, c_{n-1})$ of $n$ natural numbers, with $c_u$ read as $0$ for $u \\ge n$, such that $c_v < q$ for every $v < n$, every $a < q$ equals $c_v$ for some $v < n$, $c_u \\ne c_v$ for every pair $(u, v)$ in $E$, and for all distinct $a, b < q$ some pair $(u, v)$ in $E$ has $(c_u, c_v) = (a, b)$ or\u2026","title":"Feasible ACHROMATIC NUMBER instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/achromatic-number-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AchromaticNumber.achromaticNumberDec","parent":31808,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/index.html#achromatic-number-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number","status":"fully_proved","summary":"The ACHROMATIC NUMBER decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of instances $(n, E, K)$, to an instance of ACHROMATIC NUMBER that is well-formed and feasible; it rejects every bitstring that does not decode to an instance.","title":"ACHROMATIC NUMBER as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/achromatic-number-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AchromaticNumber.achromaticNumber_npComplete","parent":31808,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number/index.html#achromatic-number-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/achromatic-number","status":"can_prove","summary":"The ACHROMATIC NUMBER decision problem belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"ACHROMATIC NUMBER is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring","status":"can_prove","title":"Edge coloring"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/edge-coloring-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeColoring.EdgeColoringInstance","parent":31814,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/index.html#edge-coloring-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring","status":"fully_proved","summary":"An instance of EDGE COLORING is a triple $(n, E, K)$ consisting of a natural number $n$, the number of vertices, a finite list $E$ of ordered pairs of natural numbers, the edges, and a natural number $K$, the bound on the number of colors.","title":"Edge coloring instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/adj","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeColoring.Adj","parent":31814,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/index.html#adj","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring","status":"fully_proved","summary":"Given a list $E$ of ordered pairs of natural numbers and natural numbers $u$ and $v$, the boolean adjacency of $u$ and $v$ in $E$ is true exactly when $(u, v)$ or $(v, u)$ occurs in $E$.","title":"Adjacency in an edge coloring edge list"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeColoring.WellFormed","parent":31814,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring","status":"fully_proved","summary":"An edge coloring instance $(n, E, K)$ is well formed when the list $E$ has no repeated entries and every pair $(a, b)$ in $E$ satisfies $a < n$, $b < n$ and $a \\ne b$, and $(b, a)$ does not occur in $E$.","title":"Well-formed edge coloring instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/edge-coloring-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeColoring.EdgeColoringFeasible","parent":31814,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/index.html#edge-coloring-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring","status":"fully_proved","summary":"Let $(n, E, K)$ be an edge coloring instance with edge list $E = (e_0, \\dots, e_{m-1})$, where the endpoints of a pair $(x, y)$ are $x$ and $y$. The instance is feasible when there is a list $(c_0, \\dots, c_{m-1})$ of natural numbers with $c_j < K$ for every $j$ such that $c_a \\ne c_b$ for all distinct $a, b < m$ for which $e_a$ and $e_b$ share an endpoint.","title":"Feasible edge coloring instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/edge-coloring-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeColoring.edgeColoringDec","parent":31814,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/index.html#edge-coloring-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring","status":"fully_proved","summary":"The EDGE COLORING decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of edge coloring instances, to an instance that is well formed and feasible. Bitstrings that do not decode are rejected.","title":"EDGE COLORING as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/edge-coloring-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.EdgeColoring.edgeColoring_npComplete","parent":31814,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring/index.html#edge-coloring-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/edge-coloring","status":"can_prove","summary":"The EDGE COLORING decision problem belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"EDGE COLORING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring","status":"fully_proved","title":"Graph 3-coloring"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","kind":"scope","navigation":"transparent","parent":31821,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","title":"Bit-string encoding of a graph to Decoding fails on a dangling half-pair"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/encode-e","kind":"node","lean":"Cslib.Complexity.Karp.encodeE","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#encode-e","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"The graph encoding of a number $n$ and a list $E$ of pairs of natural numbers is the bit string that writes $n$ in unary and then, for each pair $(u, v)$ of $E$ in order, writes $u$ and then $v$ in unary, where a natural number $k$ is written in unary as $k$ ones followed by one zero.","title":"Bit-string encoding of a graph"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/edge-graph","kind":"node","lean":"Cslib.Complexity.Karp.edgeGraph","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#edge-graph","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"For a natural number $n$ and a list $es$ of pairs of natural numbers, the graph of $es$ on $n$ vertices is the simple graph on $\\{0, \\dots, n-1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $es$. Loops are dropped, and pairs with an endpoint at least $n$ contribute no edge.","title":"Graph of an edge list"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/col3-dec","kind":"node","lean":"Cslib.Complexity.Karp.col3Dec","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#col3-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"The decision problem 3-COLOR accepts a bit string $x$ exactly when there are a natural number $n$ and a list $es$ of pairs of natural numbers such that $x$ is the encoding of $n$ and $es$ ($n$ in unary, then each edge $(u, v)$ as $u$ and $v$ in unary), every edge $(u, v)$ in $es$ has $u < n$ and $v < n$, and the graph on $\\{0, \\dots, n - 1\\}$ with edge list $es$ (symmetrized, loops dropped) is $3$-colorable.","title":"3-COLOR"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/col-st","kind":"node","lean":"Cslib.Complexity.Karp.ColSt","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#col-st","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"A state of the graph construction reducing CNF satisfiability to 3-colorability consists of a natural number, the next free vertex; a list of quadruples $(x, b, L, M)$ of a variable index $x$, a polarity $b$, the vertex $L$ of the literal and the vertex $M$ of its negation, recording the earlier literal occurrences with the latest first; and a list of pairs of natural numbers, the edges emitted so far, in emission\u2026","title":"State of the SAT-to-3-COLOR construction"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/cons-edges","kind":"node","lean":"Cslib.Complexity.Karp.consEdges","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#cons-edges","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"Let $l = (x, b)$ be a literal with variable index $x$ and polarity $b$, let $L$, $M$ be the vertices of a new occurrence of $l$, and let $(x', b', L', M')$ be an earlier occurrence. If $x' \\ne x$ the list of consistency edges is empty. If $x' = x$ and $b' = b$ it is $[(L', M), (M', L)]$; if $x' = x$ and $b' \\ne b$ it is $[(L', L), (M', M)]$.","title":"Consistency edges between two occurrences of a variable"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/col-occ","kind":"node","lean":"Cslib.Complexity.Karp.colOcc","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#col-occ","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"Given a literal $l = (x, b)$ with variable index $x$ and polarity $b$, and a state whose next free vertex is $n$, let $L = n$ and $M = n + 1$. The new state has next free vertex $n + 2$, records the occurrence $(x, b, L, M)$ at the front of the list of earlier occurrences, and appends to the emitted edges first $(L, M)$, $(L, 2)$, $(M, 2)$ (a triangle with the palette vertex $2$), then the consistency edges of the\u2026","title":"Adding a literal occurrence in the 3-COLOR construction"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/col-or","kind":"node","lean":"Cslib.Complexity.Karp.colOr","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#col-or","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"Given vertices $a$ and $o$ and a state whose next free vertex is $n$, let $u = n$, $v = n + 1$ and $w = n + 2$. The new state has next free vertex $n + 3$, the same earlier occurrences, and the emitted edges followed by $(a, u)$, $(o, v)$, $(u, v)$, $(u, w)$, $(v, w)$, $(w, 2)$. The result is this new state together with the output vertex $w$.","title":"OR gadget of the 3-COLOR construction"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/col-lits","kind":"node","lean":"Cslib.Complexity.Karp.colLits","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#col-lits","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"Given a list of literals, a vertex $o$ (the output so far) and a state, the result is defined by recursion on the list. For the empty list, the emitted edges of the state are followed by $(o, 1)$ and $(o, 2)$, joining $o$ to the palette vertices $1$ and $2$, and the other components are unchanged. For a list with head $l$ and tail $ls$, first add an occurrence of $l$, obtaining a state and the literal vertex $L$;\u2026","title":"Processing the remaining literals of a clause"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/col-clause","kind":"node","lean":"Cslib.Complexity.Karp.colClause","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#col-clause","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"Given a clause $C$ and a state whose next free vertex is $n$, the literals of $C$ are read from the last one. If $C$ is empty, the new state has next free vertex $n + 1$, the same earlier occurrences, and the emitted edges followed by $(n, 0)$, $(n, 1)$, $(n, 2)$, joining the fresh vertex $n$ to the palette vertices $0$, $1$, $2$. Otherwise let $l$ be the last literal of $C$ and $ls$ the other literals in reverse\u2026","title":"Processing one clause in the 3-COLOR construction"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/col-st-2","kind":"node","lean":"Cslib.Complexity.Karp.colSt","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#col-st-2","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"For a CNF formula $\\varphi$, start from the state whose next free vertex is $3$, with no earlier occurrences and with emitted edges $(0, 1)$, $(0, 2)$, $(1, 2)$ (the palette triangle on the vertices $0$, $1$, $2$), and process the clauses of $\\varphi$ one at a time, from the last clause to the first. The result is the final state.","title":"The SAT-to-3-COLOR graph construction"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/to-col","kind":"node","lean":"Cslib.Complexity.Karp.toCol","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#to-col","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"Run the gadget construction on a CNF formula $\\varphi$; it ends with a next free vertex $N$ and a list of emitted edges. The 3-COLOR instance of $\\varphi$ is the graph encoding of the number $N$ and the emitted edges in reverse order.","title":"The 3-COLOR instance of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/no-col","kind":"node","lean":"Cslib.Complexity.Karp.noCol","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#no-col","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"This is the graph encoding of the number $4$ and the edge list $(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)$, in that order, that is, of the complete graph on the vertices $0, 1, 2, 3$.","title":"The complete graph on four vertices as a bit string"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/reduce-col","kind":"node","lean":"Cslib.Complexity.Karp.reduceCol","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#reduce-col","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"The map sends a bit string $x$ that decodes to a CNF formula $\\varphi$ to the 3-COLOR instance of $\\varphi$, and sends every bit string that does not decode to a CNF formula to the graph encoding of the number $4$ and the edge list $(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)$, in that order, that is, of the complete graph on four vertices.","title":"The SAT to 3-COLOR reduction map on bit strings"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/length-lt-of-read-unary-length","kind":"node","lean":"Cslib.Complexity.Karp.length_lt_of_readUnaryLength","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#length-lt-of-read-unary-length","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"If reading a unary length prefix from a bit string $x$ succeeds with value $u$ and unconsumed suffix $r$, then $r$ is strictly shorter than $x$.","title":"Reading a unary number shortens the input"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/decode-es","kind":"node","lean":"Cslib.Complexity.Karp.decodeEs","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#decode-es","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"The edge-list decoder reads a bit string as a list of pairs of natural numbers, each number written in unary as that many $1$s followed by a $0$. On the empty string it returns the empty list. Otherwise it reads a number $u$, then a number $v$, decodes the rest recursively, and returns $(u, v)$ followed by the result; it fails if the input ends inside a number or after an unpaired number, or if decoding the rest\u2026","title":"Edge-list decoder"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/decode-e","kind":"node","lean":"Cslib.Complexity.Karp.decodeE","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#decode-e","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"To decode a bit string $x$, read a unary number $n$ from the front of $x$ ($n$ copies of true ended by false); if there is none, decoding fails. Otherwise decode the rest of $x$ as an edge list $es$ (pairs of unary numbers until the input ends); if that fails, decoding fails, and otherwise the result is the pair $(n, es)$.","title":"Decoding a graph"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/edge-bits","kind":"node","lean":"Cslib.Complexity.Karp.edgeBits","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#edge-bits","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"The edge bits of a list of pairs of natural numbers are the concatenation, over its pairs $(u, v)$ in order, of $u$ ones, a zero, $v$ ones and a zero.","title":"Bit encoding of an edge list"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/decode-es-nil","kind":"node","lean":"Cslib.Complexity.Karp.decodeEs_nil","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#decode-es-nil","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"The edge-list decoder, applied to the empty bit string, succeeds and returns the empty edge list.","title":"Decoding the empty input"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/decode-es-of-none","kind":"node","lean":"Cslib.Complexity.Karp.decodeEs_of_none","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#decode-es-of-none","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"If $x$ is a nonempty bit string and reading a unary length prefix from $x$ fails, then the edge-list decoder fails on $x$.","title":"The edge-list decoder fails on a cut-off first number"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/decode-es-of-some-none","kind":"node","lean":"Cslib.Complexity.Karp.decodeEs_of_some_none","parent":31822,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01/index.html#decode-es-of-some-none","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-01","status":"fully_proved","summary":"If reading a unary length prefix from a bit string $x$ returns a number $u$ and the remaining suffix $r$, and reading a unary length prefix from $r$ fails, then the edge-list decoder fails on $x$.","title":"Decoding fails on a dangling half-pair"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","kind":"scope","navigation":"transparent","parent":31821,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","title":"Decoding one pair, then the rest to Correctness of the reduction from SAT to 3-coloring"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/decode-es-of-some-some","kind":"node","lean":"Cslib.Complexity.Karp.decodeEs_of_some_some","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#decode-es-of-some-some","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"If reading a unary length prefix from a bit string $x$ returns $u$ with remaining suffix $r$, and reading one from $r$ returns $v$ with remaining suffix $r'$, then the edge-list decoder succeeds on $x$ exactly when it succeeds on $r'$, and in that case its result on $x$ is its result on $r'$ with the pair $(u, v)$ put in front.","title":"Decoding one pair, then the rest"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/decode-es-edge-bits","kind":"node","lean":"Cslib.Complexity.Karp.decodeEs_edgeBits","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#decode-es-edge-bits","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"For every list $es$ of pairs of natural numbers, the edge-list decoder applied to the edge bits of $es$ succeeds and returns $es$.","title":"Decoding the bits of an edge list"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/edge-bits-of-decode-es","kind":"node","lean":"Cslib.Complexity.Karp.edgeBits_of_decodeEs","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#edge-bits-of-decode-es","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"For every bit string $x$ and every list $es$ of pairs of natural numbers, if the edge-list decoder returns $es$ on $x$, then $x$ is the edge bits of $es$.","title":"The edge-list decoder succeeds only on edge bits"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/encode-e-eq","kind":"node","lean":"Cslib.Complexity.Karp.encodeE_eq","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#encode-e-eq","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"For every natural number $n$ and every list $es$ of pairs of natural numbers, the graph encoding of $n$ and $es$ is $n$ ones, then a zero, then the edge bits of $es$.","title":"Graph encoding as vertex count plus edge bits"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/decode-e-encode-e","kind":"node","lean":"Cslib.Complexity.Karp.decodeE_encodeE","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#decode-e-encode-e","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"For every natural number $n$ and every list $es$ of pairs of natural numbers, decoding the graph encoding of $n$ and $es$ succeeds and returns $(n, es)$.","title":"Graph decoding inverts graph encoding"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/encode-e-of-decode-e","kind":"node","lean":"Cslib.Complexity.Karp.encodeE_of_decodeE","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#encode-e-of-decode-e","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"Let $x$ be a bit string. If decoding $x$ as a graph yields a number $n$ and a list $E$ of pairs of natural numbers, then $x$ is the graph encoding of $n$ and $E$.","title":"Graph decoding succeeds only on encodings"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/not-colorable-no-col","kind":"node","lean":"Cslib.Complexity.Karp.not_colorable_noCol","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#not-colorable-no-col","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"The graph on the vertices $0, 1, 2, 3$ with an edge between every two distinct vertices has no proper coloring with $3$ colors.","title":"The complete graph on four vertices is not 3-colorable"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/col-proper","kind":"node","lean":"Cslib.Complexity.Karp.ColProper","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#col-proper","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"A map $c : \\mathbb{N} \\to \\{0, 1, 2\\}$ properly colors a list $E$ of pairs of natural numbers when $c(a) \\ne c(b)$ for every pair $(a, b)$ in $E$.","title":"Proper 3-colorings of an edge list"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/colorable-iff-exists-col-proper","kind":"node","lean":"Cslib.Complexity.Karp.colorable_iff_exists_colProper","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#colorable-iff-exists-col-proper","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"Let $n$ be a natural number and $E$ a list of pairs $(a, b)$ of natural numbers with $a < n$, $b < n$ and $a \\ne b$. The graph on $\\{0, \\dots, n - 1\\}$ with edge list $E$ is 3-colorable if and only if some map $c : \\mathbb{N} \\to \\{0, 1, 2\\}$ satisfies $c(a) \\ne c(b)$ for every pair $(a, b)$ in $E$.","title":"3-colorability through colorings of the natural numbers"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/or-gadget-colors","kind":"node","lean":"Cslib.Complexity.Karp.or_gadget_colors","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#or-gadget-colors","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"Read the colors $0, 1, 2$ as $T$, $F$ and $B$. For all colors $c_a, c_o \\in \\{0, 1, 2\\}$ with $c_a \\ne 2$ and $c_o \\ne 2$, there are colors $c_u, c_v, c_w \\in \\{0, 1, 2\\}$ with $c_a \\ne c_u$, $c_o \\ne c_v$, $c_u \\ne c_v$, $c_u \\ne c_w$, $c_v \\ne c_w$ and $c_w \\ne 2$, such that $c_w = 0$ if and only if $c_a = 0$ or $c_o = 0$.","title":"Coloring an OR gadget"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/or-gadget-sound","kind":"node","lean":"Cslib.Complexity.Karp.or_gadget_sound","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#or-gadget-sound","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"Read the colors $0, 1, 2$ as $T$, $F$ and $B$. If colors $c_a, c_o, c_u, c_v, c_w \\in \\{0, 1, 2\\}$ satisfy $c_a \\ne 2$, $c_o \\ne 2$, $c_a \\ne c_u$, $c_o \\ne c_v$, $c_u \\ne c_v$, $c_u \\ne c_w$, $c_v \\ne c_w$ and $c_w = 0$, then $c_a = 0$ or $c_o = 0$.","title":"Soundness of the OR gadget"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/col-occ-next","kind":"node","lean":"Cslib.Complexity.Karp.colOcc_next","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#col-occ-next","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"For a literal $l$ and a state $s$ of the SAT-to-3-COLOR construction whose next free vertex is $n$, the state obtained by adding the occurrence $l$ to $s$ has next free vertex $n + 2$.","title":"Adding a literal occurrence takes two new vertices"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/col-grows","kind":"node","lean":"Cslib.Complexity.Karp.ColGrows","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#col-grows","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"A construction state $s'$ extends a construction state $s$ when the next free vertex of $s'$ is at least that of $s$, every edge emitted in $s$ is emitted in $s'$, and every occurrence recorded in $s$ is recorded in $s'$.","title":"Growth of construction states"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/col-grows-extend","kind":"node","lean":"Cslib.Complexity.Karp.colGrows_extend","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#col-grows-extend","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"Let $s$ be a construction state with next free vertex $m$, let $n \\ge m$, and let $E$ be a list of pairs of natural numbers. Then the state with next free vertex $n$, the same recorded occurrences as $s$, and emitted edges those of $s$ followed by $E$ extends $s$: its next free vertex is at least that of $s$, and its emitted edges and recorded occurrences include those of $s$.","title":"Extending a state makes it grow"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/col-good","kind":"node","lean":"Cslib.Complexity.Karp.ColGood","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#col-good","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"A construction state $s$ with next free vertex $n$ is well formed when $3 \\le n$; every emitted edge $(a, b)$ has $a < n$, $b < n$ and $a \\ne b$; the literal vertex and the negation vertex of every recorded occurrence are smaller than $n$; the pairs $(0, 1)$, $(0, 2)$ and $(1, 2)$ are emitted edges (the palette triangle); for every recorded occurrence with literal vertex $L$ and negation vertex $M$, the pairs\u2026","title":"Well-formed construction states"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/extend","kind":"node","lean":"Cslib.Complexity.Karp.ColGood.extend","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#extend","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"Let $s$ be a well-formed construction state with next free vertex $m$, let $n \\ge m$, and let $E$ be a list of pairs $(a, b)$ of natural numbers with $a < n$, $b < n$ and $a \\ne b$. Then the state with next free vertex $n$, the same recorded occurrences as $s$, and emitted edges those of $s$ followed by $E$ is well formed.","title":"Extending a well-formed state keeps it well formed"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/col-st-em-lt","kind":"node","lean":"Cslib.Complexity.Karp.colSt_em_lt","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#col-st-em-lt","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"For every CNF formula $\\varphi$, both endpoints of every edge emitted by the construction for $\\varphi$ are smaller than the number of vertices the construction allocates (its next free vertex).","title":"The construction's edges lie below its vertex count"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/col-colored","kind":"node","lean":"Cslib.Complexity.Karp.ColColored","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#col-colored","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"Let $s$ be a construction state, let $v : \\mathbb{N} \\to \\mathrm{Bool}$ be an assignment, and let $c : \\mathbb{N} \\to \\{0, 1, 2\\}$ be a coloring. Then $c$ colors $s$ according to $v$ when $c(0) = 0$, $c(1) = 1$ and $c(2) = 2$ (the palette vertices $T$, $F$, $B$), $c(a) \\ne c(b)$ for every edge $(a, b)$ emitted in $s$, and for every occurrence recorded in $s$ with variable $x$, polarity $b$, literal vertex $L$ and\u2026","title":"Colorings that follow an assignment"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/sound-col-or","kind":"node","lean":"Cslib.Complexity.Karp.sound_colOr","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#sound-col-or","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"Let $c : \\mathbb{N} \\to \\{0, 1, 2\\}$ with $c(2) = 2$, let $a$ and $o$ be natural numbers, let $s$ be a state of the SAT-to-3-COLOR construction and let $E$ be a list of pairs of natural numbers such that $c(u) \\ne c(v)$ for every pair $(u, v)$ in $E$. Apply the OR gadget to $a$, $o$ and $s$, giving a new state and an output vertex $w$, and suppose every edge in the emitted-edge list of the new state belongs to $E$.\u2026","title":"Output of the OR gadget under a proper 3-coloring"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/cnf-sat-iff-colorable","kind":"node","lean":"Cslib.Complexity.Karp.cnfSat_iff_colorable","parent":31843,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02/index.html#cnf-sat-iff-colorable","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-02","status":"fully_proved","summary":"A CNF formula $\\varphi$ is satisfiable if and only if the graph built from $\\varphi$ by the Garey\u2013Johnson\u2013Stockmeyer construction is 3-colorable. Here, writing $n$ for the number of vertices the construction allocates, the graph has vertex set $\\{0, \\dots, n - 1\\}$, and two distinct vertices are adjacent when the construction emits an edge between them in either order.","title":"Correctness of the reduction from SAT to 3-coloring"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","kind":"scope","navigation":"transparent","parent":31821,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","title":"Correctness of the SAT to 3-COLOR map on bit strings to Quiescent states of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/sat-dec-eq-col3-dec-reduce-col","kind":"node","lean":"Cslib.Complexity.Karp.satDec_eq_col3Dec_reduceCol","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#sat-dec-eq-col3-dec-reduce-col","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"For every bit string $x$, SAT accepts $x$ if and only if 3-COLOR accepts the image of $x$ under the SAT to 3-COLOR reduction map on bit strings.","title":"Correctness of the SAT to 3-COLOR map on bit strings"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/e-b","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.eB","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#e-b","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The bits of an edge $(u, v)$ in the edge-list encoding of a graph: $u$ copies of true, then false, then $v$ copies of true, then false.","title":"Bit encoding of an edge"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/out","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.out","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#out","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"For a list of edges $e_1, \\dots, e_k$ (pairs of natural numbers), the output is the concatenation $b(e_k) \\cdots b(e_1)$, where $b(u, v)$ is $u$ ones, a zero, $v$ ones and a zero.","title":"Bits pushed for a list of edges"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/k","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.K","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#k","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The emitter has thirteen stacks: $T$ (tokens), $O$ (output), $N$ (the next free vertex), $R$ (the current OR output), $D$ (earlier occurrences), $E$ (the walk's other side), $A$ (the current name), $P$ (its polarity), $F$ (the first-literal flag), $U$ (an earlier literal vertex), and the scratch stacks $X$ and $B$ for comparisons and $W$ for copies.","title":"Stacks of the 3-coloring emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/untitled","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.\u0393","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#untitled","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The output stack has the booleans (bits) as its alphabet; every other stack has the token symbols as its alphabet.","title":"Stack alphabets of the 3-coloring emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/src","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Src","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#src","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The type with the three elements $n, r, u$, naming the counters of the 3-coloring emitter that a copy site can read: the next free vertex $n$, the current OR output $r$ and an earlier literal vertex $u$.","title":"Counters read by copy sites of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/k-2","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Src.k","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#k-2","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The counter $n$ (the next vertex) is held on stack $N$, the counter $r$ (the OR output) on stack $R$, and the counter $u$ (an earlier literal vertex) on stack $U$.","title":"Stack of a counter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/unit","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Src.unit","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#unit","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"For each of the three counters read by copy sites of the 3-coloring emitter, the unit symbol in the alphabet of the counter's stack is the clause-end token symbol.","title":"Unit symbol of a counter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/site","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Site","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#site","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The type of copy sites has 29 distinct elements: $o_1, \\dots, o_4$ for the occurrence triangle; $c_1, \\dots, c_4$ and $d_1, \\dots, d_4$ for the consistency edges with an earlier occurrence of equal and of opposite polarity; $r_1, \\dots, r_{11}$ for the OR gadget; $e_1, e_2$ for the clause end; $z_1, z_2, z_3$ for the empty clause; and one site for the vertex count. Equality of sites is decidable.","title":"Copy sites of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/all","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Site.all","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#all","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The list of all copy sites of the 3-COLOR emitter, each exactly once, in the order $o_1, \\dots, o_4$, $c_1, \\dots, c_4$, $d_1, \\dots, d_4$, $r_1, \\dots, r_{11}$, $e_1, e_2$, $z_1, z_2, z_3$ and the vertex-count site.","title":"The copy sites of the 3-COLOR emitter, listed"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/src-2","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Site.src","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#src-2","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"In the 3-coloring emitter, the sites $c_2$, $c_4$, $d_2$ and $d_4$ copy the earlier-literal counter $U$, the sites $r_4$, $e_1$ and $e_2$ copy the OR-output counter $R$, and every other site copies the next-vertex counter $N$.","title":"Counter copied by a copy site"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/top","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Site.top","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#top","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The list of bits a copy site of the 3-coloring emitter pushes onto the output stack below its counter copy, read top-first and written with $1$ for true and $0$ for false: $o_1$: 10, $o_2$: 0, $o_3$: 0110, $o_4$: 10110; $c_1$: 10, $c_2$: 0, $c_3$: 0, $c_4$: 10; $d_1$: 0, $d_2$: 0, $d_3$: 10, $d_4$: 10; $r_1$: 110, $r_2$: 0, $r_3$: 1110, $r_4$: 0, $r_5$: 1110, $r_6$: 110, $r_7$: 11110, $r_8$: 110, $r_9$: 11110,\u2026","title":"Constant block of a copy site"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/lab","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Lab","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#lab","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"`init`: palette; `cls`: next clause; `lits`: next literal; `lp`, `nm`: pop the polarity and the name; `walk` to `dr`: walk `D` (compare, restore, select the polarity case, move the name and the vertex to `E`, or the vertex to `U` and push the consistency edges); `back`: move `E` back to `D`; `putN` to `putP`: put the occurrence on `D`; `aft`, `fA` to `fC`: set `R` and advance `N`; `clr`: clear `R`; `zN`: advance\u2026","title":"Control labels of the 3-coloring emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/nxt","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Site.nxt","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#nxt","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The label at which the 3-COLOR emitter continues after a copy site. In each of the chains $o_1, \\dots, o_4$; $c_1, \\dots, c_4$; $d_1, \\dots, d_4$; $r_1, \\dots, r_{11}$; $e_1, e_2$; $z_1, z_2, z_3$, every site but the last continues with $\\mathtt{pre}$ of the next site of its chain. The last sites continue as follows: $o_4$ with $\\mathtt{lp}$, $c_4$ and $d_4$ with $\\mathtt{dr}$, $r_{11}$ with\u2026","title":"Successor label of a copy site"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/is-lit","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.isLit","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#is-lit","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"For an optional symbol (the top of a stack, absent when the stack is empty), the test is true exactly when the symbol is present and is a literal marker of some polarity; it is false for every other symbol and for an empty stack.","title":"Literal-marker test"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/unit-of","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.unitOf","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#unit-of","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The partial reading of token symbols as unary units used by the 3-coloring emitter: the clause-end symbol is read as the bit true, and every other symbol has no reading, so it ends a run of units.","title":"Reading unary units"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/pal","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.pal","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#pal","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The bits pushed for the edges $(0, 1)$, $(0, 2)$, $(1, 2)$ in this order, that is, for the triangle on the palette vertices $0$, $1$, $2$.","title":"Bits of the palette triangle"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/prog","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.prog","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#prog","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The TM2 program, with the stacks, stack alphabets and labels of the emitter and a Boolean local state, that builds the 3-COLOR instance of a CNF formula from its token stack. It first pushes the palette triangle on the output stack and adds $3$ to the next free vertex $N$. It then reads the clauses one at a time. For each literal occurrence it pushes the occurrence triangle, walks the earlier occurrences comparing\u2026","title":"The 3-COLOR emitter machine"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/start","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.start","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#start","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"The emitter machine starts at its palette label, the label whose instruction pushes the palette triangle onto the output stack.","title":"Start label of the 3-coloring emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/ms","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.MS","parent":31864,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03/index.html#ms","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-03","status":"fully_proved","summary":"A quiescent state records the contents of the token stack $T$ and of the stacks $D$, $E$, $A$, $P$ (lists of token symbols), the content of the output stack $O$ (a list of bits), and four natural numbers $n$, $r$, $f$, $u$: the values of the counters $N$ (next free vertex), $R$ (current OR output), $F$ (first-literal flag) and $U$ (an earlier literal vertex).","title":"Quiescent states of the 3-COLOR emitter"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","kind":"scope","navigation":"transparent","parent":31821,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","title":"Stacks of a quiescent state to Adding the vertex counter to the OR output"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/stk","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.MS.stk","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#stk","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"The stack contents of a quiescent state $m$ of the 3-coloring emitter: the token, output, earlier-occurrence, walk, name and polarity stacks hold the corresponding lists of $m$; the counters $N$, $R$, $F$ and $U$ hold $n$, $r$, $f$ and $u$ copies of the clause-end symbol, where $n$, $r$, $f$, $u$ are the counter values of $m$; and the scratch stacks $X$, $B$ and $W$ are empty.","title":"Stacks of a quiescent state"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/val","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.MS.val","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#val","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"For a quiescent state $m$ of the 3-coloring emitter, the value of the next-vertex counter is the next free vertex $n$ of $m$, that of the OR-output counter is the current OR output $r$ of $m$, and that of the earlier-literal counter is the earlier literal vertex $u$ of $m$.","title":"Counter values of a quiescent state"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/k-ne-o","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Src.k_ne_O","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#k-ne-o","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"For each of the three counters read by copy sites of the 3-coloring emitter, the stack of the counter is not the output stack $O$.","title":"Counters are not held on the output stack"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/dup-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.dup_run","parent":31885,"reading_title":{"format":"inline-parts-v1","parts":[["text","Copying a counter through "],["math","\\(W\\)"]],"source":"Copying a counter through $W$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#dup-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $s$ be a counter, $c$ a stack different from the stack of $s$ and from $W$, and $x$ a symbol of $c$. Suppose the instruction at label $l_1$ pops the stack of $s$ until it is empty, pushing $x$ onto $c$ and the unit symbol onto $W$ for each popped symbol, then goes to $l_2$; and the instruction at $l_2$ pops $W$ until it is empty, pushing the unit symbol of $s$ onto the stack of $s$ for each popped symbol, then\u2026","title":"Copying a counter through $W$"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/atom","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.atom","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#atom","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"For a quiescent state $m$ of the 3-COLOR emitter and a copy site $j$, the bit list of $j$ at $m$ is $v$ ones followed by the constant block of $j$, where $v$ is the value in $m$ of the counter that $j$ copies. Read top-first, the ones lie above the constant block.","title":"Bit list of a copy site of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/site-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.site_run","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#site-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $j$ be a copy site, $m$ a quiescent state and $v$ a local state, and let $c$ be the value in $m$ of the counter that $j$ copies. From the label that starts site $j$, with local state $v$, the machine runs in exactly $2c + 3$ steps to the label where $j$ continues, in some local state and in the quiescent state obtained from $m$ by pushing onto the output stack $c$ ones above the constant block of $j$.","title":"Running a copy site of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/chain","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Chain","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#chain","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Copy sites $j_1, \\dots, j_k$ run in sequence from label $\\ell_0$ to label $\\ell$ when $\\ell_0$ is the entry label of $j_1$, the continuation label of each $j_i$ with $i < k$ is the entry label of $j_{i+1}$, and the continuation label of $j_k$ is $\\ell$; for $k = 0$ this means $\\ell_0 = \\ell$. The condition is a Boolean test.","title":"Copy sites of the 3-COLOR emitter running in sequence"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/chain-out","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.chainOut","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#chain-out","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"For a quiescent state $m$ of the 3-COLOR emitter and copy sites $j_1, \\dots, j_k$, the bits of the sequence at $m$ are the bit lists at $m$ of $j_k, j_{k-1}, \\dots, j_1$, concatenated in that order. Read top-first, the bits of the last site lie on top.","title":"Bits pushed by a sequence of copy sites of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/chain-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.chain_run","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#chain-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let copy sites $j_1, \\dots, j_k$ run in sequence from label $\\ell_0$ to label $\\ell$, and let $m$ be a quiescent state of the 3-COLOR emitter whose counters $N$, $R$, $U$ have values $n$, $r$, $u$. Then for every Boolean $v$ there is a Boolean $v'$ such that the emitter goes in at most $k (2 (n + r + u) + 3)$ steps from label $\\ell_0$ with internal state $v$ and the stacks of $m$ to label $\\ell$ with internal state\u2026","title":"Running a sequence of copy sites of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/is-tok","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.isTok","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#is-tok","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"A symbol is a walk token, one the walk moves between the stacks $D$ and $E$, when it is a name bit, a literal marker or the clause symbol (which also serves as the unary unit); the value symbols and the dead symbol are not walk tokens.","title":"Walk tokens"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/name","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.name","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#name","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"The name of a variable $x \\in \\mathbb{N}$ is the list of name-bit tokens $\\mathtt{bit}(b)$, one for each bit $b$ of Mathlib's binary encoding of $x$ (least significant bit first, empty for $x = 0$), in that order; this is how the stacks $A$ and $D$ hold it.","title":"Name of a variable"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/d-tok","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.dTok","parent":31885,"reading_title":{"format":"inline-parts-v1","parts":[["text","Encoding of an earlier occurrence on stack "],["math","\\(D\\)"]],"source":"Encoding of an earlier occurrence on stack $D$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#d-tok","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"An occurrence $q = (x, p, L, M)$ of the literal $(x, p)$ with vertices $L$ and $M$ is written, top first, as the polarity marker of $p$, then the name bits of the variable $x$, then $L$ unit symbols (its literal vertex in unary); the vertex $M$ is not recorded.","title":"Encoding of an earlier occurrence on stack $D$"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/nm-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.nm_run","parent":31885,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moving the current name from "],["math","\\(T\\)"],["text"," to "],["math","\\(A\\)"]],"source":"Moving the current name from $T$ to $A$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#nm-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $m$ be a quiescent state of the 3-COLOR emitter whose stack $A$ is empty, $v$ a bit, $x \\in \\mathbb{N}$, and $R$ a list of token symbols whose first element, if any, is not a name bit. If the token stack $T$ of $m$ is the name of $x$ in reverse order followed by $R$, then for some bit $v'$ the emitter machine runs in exactly $k + 1$ steps, where $k$ is the length of the binary encoding of $x$, from label\u2026","title":"Moving the current name from $T$ to $A$"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/cmp-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.cmp_run","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#cmp-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $m$ be a quiescent state, $u$ and $w$ lists of bits, and $R$ a list of symbols that is empty or does not start with a name bit. If stack $A$ of $m$ holds $u$ as name bits and stack $D$ holds $w$ as name bits followed by $R$, then from label $\\mathsf{cmp}$ with the stacks of $m$ and any flag, the emitter reaches, in at most $3|w| + 3$ steps, label $\\mathsf{sel}$ if $u = w$ and label $\\mathsf{mvN}(\\mathrm{none})$\u2026","title":"Name comparison in the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/mv-n-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.mvN_run","parent":31885,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moving an earlier name from "],["math","\\(D\\)"],["text"," to "],["math","\\(E\\)"]],"source":"Moving an earlier name from $D$ to $E$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#mv-n-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $m$ be a quiescent state of the 3-COLOR emitter, $v$ a bit, $r$ either empty or a bit $b$, $w$ a list of bits, and $R$ a list of token symbols whose first element, if any, is not a name bit. If the stack $D$ of $m$ is the name bits of $w$ followed by $R$, then for some bit $v'$ the emitter machine runs in exactly $|w| + 1$ steps from label $\\mathtt{mvN}(r)$ with internal bit $v$ and the stacks of $m$ to\u2026","title":"Moving an earlier name from $D$ to $E$"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/sk-u-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.skU_run","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#sk-u-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $m$ be a quiescent state of the 3-COLOR emitter whose occurrence stack consists of $j$ units above a stack $R$ that is empty or does not have a unit on top. From the label $\\mathsf{skU}$, with any local state, the machine runs in exactly $j + 1$ steps to the label $\\mathsf{walk}$, in some local state and in the quiescent state obtained from $m$ by replacing the occurrence stack with $R$ and pushing $j$ units\u2026","title":"Skipping an earlier literal vertex"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/mv-u-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.mvU_run","parent":31885,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moving an earlier literal vertex from "],["math","\\(D\\)"],["text"," to "],["math","\\(U\\)"]],"source":"Moving an earlier literal vertex from $D$ to $U$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#mv-u-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $m$ be a quiescent state of the 3-COLOR emitter whose counter $U$ is $0$, let $v$ and $b$ be bits, $j \\in \\mathbb{N}$, and $R$ a list of token symbols whose first element, if any, is not the clause token $\\mathtt{cl}$. If the stack $D$ of $m$ is $j$ copies of $\\mathtt{cl}$ followed by $R$, then for some bit $v'$ the emitter machine runs in exactly $j + 1$ steps from label $\\mathtt{mvU}(b)$ with internal bit $v$\u2026","title":"Moving an earlier literal vertex from $D$ to $U$"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/dr-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.dr_run","parent":31885,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moving "],["math","\\(U\\)"],["text"," onto "],["math","\\(E\\)"]],"source":"Moving $U$ onto $E$"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#dr-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $m$ be a quiescent state and $u$ the value of its counter $U$. From label $\\mathsf{dr}$ with the stacks of $m$ and any flag, the emitter reaches label $\\mathsf{walk}$ in exactly $u + 1$ steps, with some flag and the stacks of $m$ except that $U$ is empty (value $0$) and $u$ unit symbols are pushed onto $E$.","title":"Moving $U$ onto $E$"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/clr-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.clr_run","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#clr-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"Let $m$ be a quiescent state of the 3-COLOR emitter whose OR-output counter $R$ has value $r$, let $v$ be a Boolean and $b$ a Boolean. Then there is a Boolean $v'$ such that the emitter runs in exactly $r + 1$ steps from label $\\mathsf{clr}(b)$ with internal state $v$ and the stacks of $m$ to the stacks of $m$ with $R$ set to $0$ and internal state $v'$, at label $\\mathsf{fA}(\\mathrm{false})$ if $b$ is true and at\u2026","title":"The 3-COLOR emitter clears its OR-output counter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/f-ab-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.fAB_run","parent":31885,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04/index.html#f-ab-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-04","status":"fully_proved","summary":"For every quiescent state $m$ with counters $n$ on stack $N$ and $r$ on stack $R$, every local state $v$ and every boolean $b$, the emitter goes from label $\\mathrm{fA}(b)$ with state $(v, m)$ to label $\\mathrm{fC}(b)$, with some local state, in exactly $2n + 2$ steps, ending in the state $m$ with the counter on $R$ replaced by $n + r$; the counter $N$ is restored through the scratch stack $W$.","title":"Adding the vertex counter to the OR output"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","kind":"scope","navigation":"transparent","parent":31821,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","title":"3-COLOR emitter: moving the walk side back to The emitter on a list of clauses"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/back-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.back_run","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#back-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $L$ be a list of token symbols each of which is a walk token (a name bit, a literal marker or the clause symbol), let $m$ be a quiescent state of the 3-COLOR emitter whose stack $E$ (the walk's other side) holds $L$, and let $v$ be a Boolean. Then there is a Boolean $v'$ such that the emitter runs in exactly $|L| + 1$ steps from label $\\mathsf{back}$ with internal state $v$ and the stacks of $m$ to label\u2026","title":"3-COLOR emitter: moving the walk side back"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/put-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.put_run","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#put-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $m$ be a quiescent state of the 3-COLOR emitter with next free vertex $n$, whose name stack holds the name bits of a variable $x$ and whose polarity stack holds only the marker for polarity $b$. From the label $\\mathsf{putN}$, with any local state, the machine runs in exactly $2n + 2|w| + 5$ steps, where $w$ is the binary encoding of $x$, to the label $\\mathsf{aft}$, in some local state and in the quiescent\u2026","title":"Recording the current occurrence"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/occ-sites","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.occSites","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#occ-sites","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"The list of copy sites $o_1, o_2, o_3, o_4$, in this order.","title":"Copy sites of the occurrence triangle"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/cons-sites","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.consSites","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#cons-sites","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"For a flag $b$, the list of copy sites that push the consistency edges with an earlier occurrence whose literal vertex $L'$ is on stack $U$: the sites $c_1, c_2, c_3, c_4$ (equal polarity) when $b$ is true, and the sites $d_1, d_2, d_3, d_4$ (opposite polarity) when $b$ is false.","title":"Copy sites of the consistency edges"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/or-sites","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.orSites","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#or-sites","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"The list of the eleven copy sites $r_1, \\dots, r_{11}$ of the 3-COLOR emitter, in this order.","title":"The OR-gadget copy sites"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/end-sites","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.endSites","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#end-sites","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"The list of copy sites $e_1, e_2$, which push the edges joining the current OR output $R$ to the palette vertices $F = 1$ and $B = 2$.","title":"Copy sites of the clause end"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/emp-sites","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.empSites","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#emp-sites","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"The list of copy sites $z_1, z_2, z_3$, which push the edges joining the next free vertex $N$ to the palette vertices $0$, $1$ and $2$.","title":"Copy sites of the empty clause"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/cons-edges-same","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.consEdges_same","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#cons-edges-same","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $x$, $L'$ and $n$ be natural numbers and let $p$, $b$ be Booleans. The consistency edges between a new occurrence of the literal $(x, p)$ with vertices $L = n$, $M = n + 1$ and an earlier occurrence $(x, b, L', L' + 1)$ are $[(L', n + 1), (L' + 1, n)]$ if $b = p$, and $[(L', n), (L' + 1, n + 1)]$ otherwise.","title":"Consistency edges of two occurrences of the same variable"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/ent-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.ent_run","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#ent-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $l = (x, p)$ be a new occurrence and $q = (x', b', L', M')$ an earlier one with $M' = L' + 1$. Let $m$ be a quiescent state whose stack $D$ holds the tokens of $q$ (the marker of polarity $b'$, the name bits of $x'$, then $L'$ unary unit symbols) above a list $S$ whose first symbol, if any, is neither a unary unit nor a name bit; whose stack $A$ holds the name bits of $x$; whose stack $P$ holds only the marker\u2026","title":"One step of the walk over earlier occurrences"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/walk-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.walk_run","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#walk-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $l = (x, b)$ be a literal and let $q_1, \\dots, q_k$ be earlier occurrences, each a tuple $(y, c, L, M)$ of a variable, a polarity, a literal vertex and a negation vertex with $M = L + 1$. Let $m$ be a quiescent state of the 3-coloring emitter whose earlier-occurrence stack $D$ holds the concatenation $\\delta$ of the stack encodings of $q_1, \\dots, q_k$, whose name stack holds the name bits of $x$, whose\u2026","title":"Walking the earlier occurrences"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/occ-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.occ_run","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#occ-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $\\ell = (x, p)$ be a literal, $R$ a list of token symbols whose first element, if any, is not a name bit, and $qs$ a list of earlier occurrences $(x_i, p_i, L_i, M_i)$ with $M_i = L_i + 1$ for each. Let $m$ be a quiescent state of the 3-COLOR emitter with counter values $n$ for $N$ and $r$ for the OR output, and $v$ a bit. Suppose the token stack of $m$ is the tokens of $\\ell$ (its polarity marker above its\u2026","title":"Emitting one literal occurrence"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/inv","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Inv","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#inv","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"A construction state $s$ satisfies the size invariant after $\\tau$ tokens when its next free vertex $n$ is at most $5\\tau + 3$, it has at most $\\tau$ earlier occurrences, the total number of tokens of its earlier occurrences as they lie on stack $D$ is at most $\\tau (n + 1)$, and every earlier occurrence $(x, b, L, M)$ has $M = L + 1$.","title":"Size invariant of the 3-coloring emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/bump","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Inv.bump","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#bump","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $s$ and $s'$ be construction states with the same earlier occurrences, such that the next free vertex of $s'$ is at least that of $s$ and at most that of $s$ plus $5$. If $s$ satisfies the size invariant after $\\tau$ tokens, then $s'$ satisfies the size invariant after $\\tau + 1$ tokens.","title":"One more token preserves the 3-COLOR emitter's size invariant"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/occ","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.Inv.occ","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#occ","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $s$ and $s'$ be states of the 3-COLOR construction with next free vertices $n$ and $n'$, and let $l = (x, p)$ be a literal. Suppose that $s$ satisfies the size invariant after $\\tau$ tokens, that the earlier occurrences of $s'$ are those of $s$ preceded by the occurrence of $l$ with literal vertex $n$ and negation vertex $n + 1$, and that $n \\le n' \\le n + 5$. Then $s'$ satisfies the size invariant after\u2026","title":"The size invariant survives a new literal occurrence"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/budget","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.budget","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#budget","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"For a bound $L$ on the number of tokens of a formula, the step budget per token is $400 (L + 1)^2$.","title":"Per-token step budget of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/enc","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.enc","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#enc","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"For a construction state $s$, a token list $t$, a bit list $o_0$ and natural numbers $r$ and $f$: the quiescent state whose token stack is $t$, whose output stack is the bits of the edges emitted by $s$ (the last one on top) above $o_0$, whose counter $N$ is the next free vertex of $s$, whose counter $R$ is $r$, whose stack $D$ holds the encodings of the earlier occurrences of $s$ (latest on top), whose stacks $E$,\u2026","title":"The emitter state of a construction state"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/lit-cost","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.lit_cost","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#lit-cost","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $L_b, \\tau, n, r, P, D, w$ be natural numbers with $\\tau + w + 1 \\le L_b$, $n \\le 5\\tau + 3$, $r \\le n$, $P \\le \\tau$ and $D \\le \\tau (n + 1)$. Then $13 D + P (8 (n + r) + 21) + 40 (n + r) + 3 w + 100 \\le \\mathrm{budget}(L_b)$, where $\\mathrm{budget}(L_b) = 400 (L_b + 1)^2$ is the per-token step budget of the 3-COLOR emitter.","title":"Step budget for one literal"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/lits-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.lits_run","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#lits-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $L_b \\in \\mathbb{N}$, let $R$ be a list of token symbols that is empty or starts with the clause token $\\mathtt{cl}$, and let $o_0$ be a list of bits. Let $ls$ be a list of literals, $o, \\tau \\in \\mathbb{N}$, $s$ a construction state and $v$ a bit, and write $T$ for the concatenation of the token-stack tokens of the literals of $ls$ and $TR$ for $T$ followed by $R$. Suppose $s$ satisfies the size invariant\u2026","title":"The emitter processes the remaining literals of a clause"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/clause-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.clause_run","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#clause-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $C$ be a CNF clause, $T_C$ its tokens as they lie on the token stack, $s$ a construction state that satisfies the size invariant after $\\tau$ tokens, and $L$ a bound with $\\tau + |T_C| \\le L$. Let $Y$ be a list of symbols that is empty or starts with the clause marker, and $o_0$ a list of bits. For a construction state $t$ and a symbol list $X$, write $q(t, X)$ for the quiescent state encoding $t$ with token\u2026","title":"The 3-COLOR emitter processes one clause"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/formula-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.formula_run","parent":31906,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05/index.html#formula-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-05","status":"fully_proved","summary":"Let $L_b$ be a natural number, $o_0$ a list of bits, $C_s$ a list of clauses, $s$ a construction state and $\\tau$ a natural number, and let $T$ be the concatenation over the clauses $C$ of $C_s$, in order, of the tokens of $C$ as they lie on the token stack. If the size invariant holds for $\\tau$ and $s$ and $\\tau + |T| \\le L_b$, then for every local state $v$ the emitter goes from label cls with the quiescent\u2026","title":"The emitter on a list of clauses"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","kind":"scope","navigation":"transparent","parent":31821,"reading_title":{"format":"inline-parts-v1","parts":[["text","Correctness and step bound of the 3-COLOR emitter to Witness colors of the first "],["math","\\(n\\)"],["text"," vertices"]],"source":"Correctness and step bound of the 3-COLOR emitter to Witness colors of the first $n$ vertices"},"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","title":"Correctness and step bound of the 3-COLOR emitter to Witness colors of the first $n$ vertices"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/run","kind":"node","lean":"Cslib.Complexity.Karp.EmitColor.run","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with token list $t(\\varphi)$, let $o$ be a bit list and $v$ a Boolean. Started at its start label with local state $v$, with the token stack holding $t(\\varphi)$ reversed, the output stack holding $o$ and every other stack empty, the emitter machine reaches the label $\\mathsf{done}$ within $500\\,(|t(\\varphi)| + 1)^4$ steps, in some local state and with some stacks whose output stack\u2026","title":"Correctness and step bound of the 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/col","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.col","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#col","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"The color code of vertex $i$ in a bit string $w$ is the pair of bits at positions $2i$ and $2i + 1$ of $w$ (counting from $0$), where a position past the end of $w$ reads as $0$. The codes $00$, $01$, $10$ stand for the colors $0$, $1$, $2$; $11$ is invalid.","title":"Color code of a vertex in a witness"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/ok-col","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.okCol","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#ok-col","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"A color code, a pair of bits $(c_1, c_2)$, is valid when it is not $11$, that is, when $c_1$ and $c_2$ are not both $1$.","title":"Valid color code"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/col-check","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.colCheck","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#col-check","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"The verifier accepts an input $x$ and a witness $w$ (bit strings) iff $x$ decodes to a number $n$ and an edge list $E$, every endpoint of an edge in $E$ is less than $n$, $w$ has at least $2n$ bits, every vertex $i < n$ has a valid color code in $w$ (bits $2i$ and $2i + 1$, not $11$), and the two ends of every edge $(u, v) \\in E$ with $u \\ne v$ have different color codes.","title":"The 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/col-check-iff","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.colCheck_iff","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#col-check-iff","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For all bit strings $x$ and $w$, the verifier accepts $(x, w)$ iff there are a natural number $n$ and a list $E$ of pairs of natural numbers such that $x$ is the encoding of the graph with $n$ vertices and edge list $E$, both entries of every pair in $E$ are less than $n$, $w$ has at least $2n$ bits, the color code of every vertex $i < n$ in $w$ is valid, and for every $(u, v) \\in E$ with $u \\ne v$ the color codes\u2026","title":"Acceptance condition of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/code","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.code","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#code","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"The code of a color $c \\in \\{0, 1, 2\\}$ is the pair of bits whose first bit is $1$ iff $c = 2$ and whose second bit is $1$ iff $c = 1$: $0 \\mapsto 00$, $1 \\mapsto 01$ and $2 \\mapsto 10$.","title":"Two-bit code of a color"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/uncode","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.uncode","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#uncode","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"The color in $\\{0, 1, 2\\}$ of a pair of bits $(a, b)$, writing $0$ for false and $1$ for true: $(0, 1)$ decodes to $1$, $(1, 0)$ decodes to $2$, and both $(0, 0)$ and the invalid code $(1, 1)$ decode to $0$.","title":"Decoding a color code"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/uncode-code","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.uncode_code","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#uncode-code","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For every color $c \\in \\{0, 1, 2\\}$, decoding the two-bit code of $c$ (where $0 \\mapsto 00$, $1 \\mapsto 01$, $2 \\mapsto 10$) gives back $c$.","title":"Decoding a color code recovers the color"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/ok-col-code","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.okCol_code","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#ok-col-code","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For each color $c \\in \\{0, 1, 2\\}$, the code of $c$ (which is $00$, $01$ or $10$) is a valid color code, that is, it is not $11$.","title":"Color codes are valid"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/code-uncode","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.code_uncode","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#code-uncode","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For every valid color code $c$ (a pair of bits other than $11$), the code of the color that $c$ decodes to is $c$.","title":"Re-encoding a decoded valid color"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/flat","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.flat","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#flat","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"The flattening of a list $[(a_1, b_1), \\dots, (a_k, b_k)]$ of pairs of bits is the list of bits $[a_1, b_1, \\dots, a_k, b_k]$.","title":"Flattening pairs of bits"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/flat-nil","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.flat_nil","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#flat-nil","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"The flattening of the empty list of pairs of bits is the empty list of bits.","title":"Flattening the empty list"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/flat-cons","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.flat_cons","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#flat-cons","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For a pair of bits $c = (a, b)$ and a list $cs$ of pairs of bits, the flattening of the list with head $c$ and tail $cs$ is the list with $a$ first, then $b$, then the flattening of $cs$.","title":"Flattening a nonempty list"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/flat-append","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.flat_append","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#flat-append","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For lists $cs$ and $ds$ of pairs of bits, the flattening of the concatenation of $cs$ and $ds$ is the concatenation of the flattening of $cs$ and the flattening of $ds$.","title":"Flattening a concatenation"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/length-flat","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.length_flat","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#length-flat","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For every list $cs$ of pairs of bits, the flattening of $cs$ has length $2|cs|$, two bits per pair.","title":"Length of a flattening"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/col-flat","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.col_flat","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#col-flat","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For every list $cs$ of pairs of bits and every natural number $i$, the color code of vertex $i$ in the flattening of $cs$ is the $i$-th pair of $cs$ (counting from $0$) if $i$ is less than the length of $cs$, and $00$ otherwise.","title":"Color codes of flattened pairs"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/col-flat-of-fn","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.col_flat_ofFn","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#col-flat-of-fn","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For every $n$, every map $f$ from $\\{0, \\dots, n - 1\\}$ to pairs of bits and every $i < n$, the color code of vertex $i$ in the flattening of the list $f(0), \\dots, f(n - 1)$ is $f(i)$.","title":"Color codes of a flattened coloring"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/colorable-iff","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.colorable_iff","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#colorable-iff","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$ and let $es$ be a list of pairs of natural numbers whose entries are all less than $n$. The graph on the vertices $0, \\dots, n-1$ with edge list $es$ (symmetrized, loops dropped) is 3-colorable iff there is a list $w$ of exactly $2n$ bits such that, writing the color code of vertex $i$ for the pair of bits $2i$ and $2i+1$ of $w$, the color code of every vertex $i < n$ is valid (not $11$), and\u2026","title":"3-colorability by witness bits"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/col3-dec-iff","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.col3Dec_iff","parent":31927,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#col3-dec-iff","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"A bit string $x$ is in 3-COLOR iff some bit string $w$ of length at most $2|x|$ is accepted together with $x$ by the 3-COLOR verifier.","title":"3-COLOR is verified by the witness check"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/cols","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.cols","parent":31927,"reading_title":{"format":"inline-parts-v1","parts":[["text","Witness colors of the first "],["math","\\(n\\)"],["text"," vertices"]],"source":"Witness colors of the first $n$ vertices"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06/index.html#cols","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-06","status":"fully_proved","summary":"For a bit list $w$ and $n \\in \\mathbb{N}$, the list of the color codes of the vertices $0, 1, \\dots, n-1$ in $w$, in that order, where the color code of vertex $i$ is the pair of bits $2i$ and $2i+1$ of $w$ (a missing bit reads as false).","title":"Witness colors of the first $n$ vertices"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","kind":"scope","navigation":"transparent","parent":31821,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","title":"Edge verdict to Interleaved updates of two stacks of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/edge-ok","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.edgeOk","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#edge-ok","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"For a list $cs$ of color codes (pairs of bits), one per vertex, and $u, v \\in \\mathbb{N}$, the edge $(u, v)$ passes iff $u$ and $v$ are both less than the length of $cs$ and either $u = v$ or entries $u$ and $v$ of $cs$ differ.","title":"Edge verdict"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/edges-ok","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.edgesOk","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#edges-ok","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"For a list $cs$ of color codes and a bit list $r$, the verdict is true iff $r$ parses as a list of edges, each written as two unary numbers $u$ and $v$ (a number $k$ as $k$ ones, then a zero), and every edge $(u, v)$ of that list passes the edge verdict with vertex colors $cs$.","title":"Edge-loop verdict"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/length-cols","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.length_cols","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#length-cols","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"For every list of bits $w$ and every $n \\in \\mathbb{N}$, the list of the witness color codes of the vertices $0, \\dots, n - 1$ read from $w$ has length $n$.","title":"Number of witness colors"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/get-d-cols","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.getD_cols","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#get-d-cols","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"For a list of bits $w$ and natural numbers $i$ and $n$, write $c_w(j)$ for the color code of vertex $j$ in $w$: the pair of bits $2j$ and $2j + 1$ of $w$ (counting from $0$), each read as false past the end of $w$. If $i < n$, then entry $i$ (counting from $0$) of the list $c_w(0), \\dots, c_w(n - 1)$ of witness colors is $c_w(i)$.","title":"Reading a witness color"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/cols-cons-cons","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.cols_cons_cons","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#cols-cons-cons","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"For bits $a, b$, a bit list $w$ and $n \\in \\mathbb{N}$, the color codes of the vertices $0, \\dots, n$ read from the bit list $a, b, w$ (the list $w$ with $a$ and $b$ prepended) are $(a, b)$ followed by the color codes of the vertices $0, \\dots, n-1$ read from $w$.","title":"Peeling a vertex off the witness colors"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/col-check-eq","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.colCheck_eq","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#col-check-eq","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"For all bit strings $x$ and $w$, the verifier accepts $(x, w)$ iff reading a unary prefix $1^n 0$ off $x$ succeeds, giving $n$ and the rest $r$ of $x$, and then $w$ has at least $2n$ bits, each of the witness colors $c_0, \\dots, c_{n-1}$ of $w$ is valid (not $11$), and the edge loop accepts $r$ with the vertex colors $c_0, \\dots, c_{n-1}$.","title":"The 3-COLOR verifier as an edge loop"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/edge-ok-cons-succ","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.edgeOk_cons_succ","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#edge-ok-cons-succ","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"For a color code $c$, a list $cs$ of color codes and $u, v \\in \\mathbb{N}$, the edge verdict for $(u + 1, v + 1)$ with vertex colors $c$ followed by $cs$ equals the edge verdict for $(u, v)$ with vertex colors $cs$.","title":"Shifting the edge verdict past the first color"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/k","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.K","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#k","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"The verifier has seven stacks: the input $X$ and the witness $W$; the counter $N$ (for the number $n$ of vertices); the color stack $P$ (the $n$ color codes, vertex $0$ on top) and its scratch stack $T$; and the endpoint counters $U$ and $V$.","title":"Stacks of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/lab","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.Lab","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#lab","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"The labels are: read the number $n$ of vertices onto the counter stack $N$; load the witness colors onto the scratch stack $T$; move them onto the color stack $P$; start an edge; read its first endpoint onto the counter stack $U$; read its second endpoint onto the counter stack $V$; walk the colors towards both endpoints; for each pair of bits $(a, b)$, walk on towards the farther endpoint holding the nearer one's\u2026","title":"Labels of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/stmt","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.Stmt","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#stmt","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"The type of statements of the verifier as a multi-stack Turing machine in Mathlib's model: each of the seven stacks (the input, the witness, the counter for the number of vertices, the color stack and its scratch stack, and the two endpoint counters) holds symbols of the SAT verifier's alphabet, the labels are the verifier's labels, and the internal state is a single bit.","title":"Statements of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/read-u","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.readU","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#read-u","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"Given a counter stack $c$ and labels $s$ (its own) and $t$ (the next): if the input stack $X$ is empty, reject; otherwise pop a bit off $X$; if it is $1$, push a unit onto $c$ and go back to $s$, and if it is $0$, go to $t$. A symbol of $X$ that is not a bit is read as $0$.","title":"Unary reader of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/load-s","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.loadS","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#load-s","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"If the counter stack $N$ is empty, go to the label that moves the codes onto the color stack $P$. Otherwise pop one symbol off $N$ and two bits $a$, $b$ off the witness stack $W$, pushing each onto the scratch stack $T$ as it is popped (so the codes end up reversed on $T$), and go back to the load label; reject instead if $W$ is empty when a bit is due, or if $a$ and $b$ are both $1$ (the invalid code $11$). A\u2026","title":"Load step of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/edge-s","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.edgeS","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#edge-s","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"The verifier statement that begins checking the next edge: if the input stack $X$ is empty it goes to the accepting label, and otherwise it goes to the label that reads the first endpoint of the next edge onto the counter $U$.","title":"Edge-loop start of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/mv-p","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.mvP","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#mv-p","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"Given a map $f$ from pairs of bits to labels: if the color stack $P$ is empty, reject (an endpoint lies past the last vertex); otherwise pop two bits $a$, $b$ off $P$, push them onto the scratch stack $T$ in that order (so $b$ ends on top), and go to the label $f(a, b)$. A missing or non-bit symbol is read as $0$.","title":"Moving one color code in the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/walk-s","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.walkS","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#walk-s","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"The statement pops the top symbol of each of the endpoint counters $U$ and $V$ (a pop from an empty counter leaves it empty) and then moves the next color code $(a, b)$ from the color stack $P$ onto the scratch stack $T$, rejecting if $P$ is empty. It continues at the joint walk again if both counters were nonempty; at the walk towards the endpoint of $U$, holding the color $(a, b)$, if only $U$ was nonempty (so\u2026","title":"One step of the joint walk of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/walk1-s","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.walk1S","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#walk1-s","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"Given a stack $k$ of the 3-COLOR verifier (used as a counter), a label $\\ell$ and a pair of bits $(a, b)$ (the color of the nearer endpoint), the statement pops the top symbol of $k$ and then moves the next color code $(a', b')$ from the color stack $P$ onto the scratch stack $T$, rejecting if $P$ is empty. If $k$ was nonempty it continues at $\\ell$. Otherwise $(a', b')$ is the color of the farther endpoint, and\u2026","title":"One step of the walk towards the farther endpoint"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/prog","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.prog","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#prog","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"The verifier is the program on the seven stacks $X$ (input), $W$ (witness), $N$, $P$, $T$, $U$, $V$, with internal state a bit, that runs as follows. It reads the unary number $n$ off $X$ onto the counter $N$; loads the first $n$ color codes of $W$ onto $T$, rejecting if $W$ runs out or one of these $n$ codes is $11$; moves the bits on top of $T$ onto the color stack $P$; and then repeatedly starts an edge,\u2026","title":"The 3-COLOR verifier machine"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/start","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.start","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#start","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"The 3-coloring verifier machine starts at the label that reads the vertex count $n$, written in unary on the input stack $X$, onto the counter stack $N$.","title":"Start label of the 3-coloring verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/run-one","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.run_one","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#run-one","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"Let $l$ and $l'$ be labels, $v$ an internal state and $S$ an assignment of stacks. If executing the verifier's statement at $l$ from state $v$ and stacks $S$ ends at label $l'$, then for some internal state $v'$ and stacks $S'$ the verifier goes in one step from label $l$ with state $v$ and stacks $S$ to label $l'$ with state $v'$ and stacks $S'$.","title":"One step of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/update-update-update-update","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.update_update_update_update","parent":31948,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07/index.html#update-update-update-update","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-07","status":"fully_proved","summary":"Let $a \\ne b$ be two stacks of the verifier, $S$ an assignment of a list of symbols to every stack, and $p, p', t, t'$ lists of symbols. Setting stack $a$ to $p$, then $b$ to $t$, then $a$ to $p'$, then $b$ to $t'$ gives the same assignment as setting $a$ to $p'$ and then $b$ to $t'$.","title":"Interleaved updates of two stacks of the 3-COLOR verifier"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","kind":"scope","navigation":"transparent","parent":31821,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","title":"Moving one color code from the color stack to The 3-COLOR verifier as a host component"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/step-aux-mv-p-cons","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.stepAux_mvP_cons","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#step-aux-mv-p-cons","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $f$ assign a label to each pair of bits, and suppose the color stack $P$ of stacks $S$ holds, from the top, the bit symbols $c_1$ and $c_2$ followed by a rest $r$. Then executing the color-moving block with continuation $f$, from any internal value and the stacks $S$, ends at label $f(c_1, c_2)$ with internal value $c_2$ and with the stacks $S$ changed only in that $P$ becomes $r$ and the scratch stack $T$ gets\u2026","title":"Moving one color code from the color stack"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/step-aux-mv-p-nil","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.stepAux_mvP_nil","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#step-aux-mv-p-nil","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"If the color stack $P$ of stacks $S$ is empty, then for every continuation $f$ (a label for each pair of bits), executing the color-moving block with continuation $f$ from any internal value and the stacks $S$ ends at the rejecting label with internal value true and the stacks $S$ unchanged.","title":"The color-moving block rejects on an empty color stack"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/step-aux-walk1-s-nil","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.stepAux_walk1S_nil","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#step-aux-walk1-s-nil","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $k$ be a stack, $\\ell$ a label and $a, b$ bits. If stack $k$ of stacks $S$ is empty, then executing the walk-on block for counter $k$, label $\\ell$ and held color $(a, b)$, from any internal value and the stacks $S$, has the same outcome as executing, from internal value false and the stacks $S$, the color-moving block that continues to the rejecting label if the color it moves is $(a, b)$ and to the label that\u2026","title":"Walk-on step with the counter exhausted"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/step-aux-walk1-s-cons","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.stepAux_walk1S_cons","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#step-aux-walk1-s-cons","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Write $1^j$ for the stack of $j$ copies of the bit symbol for true. Let $k$ be a stack, $\\ell$ a label and $a, b$ bits. If stack $k$ of stacks $S$ holds $1^{j+1}$, then executing the walk-on block for counter $k$, label $\\ell$ and held color $(a, b)$, from any internal value and the stacks $S$, has the same outcome as executing, from internal value true and the stacks $S$ with stack $k$ replaced by $1^j$, the\u2026","title":"Walk-on step with a unit on the counter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/walk1-run","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.walk1_run","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#walk1-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $k$ be a stack of the 3-COLOR verifier other than the color stack $P$ and the scratch stack $T$, $(a, b)$ a pair of bits, and $\\ell$ a label whose statement is the step of the walk towards the farther endpoint on the counter $k$, holding the color $(a, b)$ and continuing at $\\ell$. Let $j$ be a natural number, $\\mathit{pre}$ and $\\mathit{post}$ lists of color codes, $s$ a bit and $S$ an assignment of a list of\u2026","title":"The walk towards the farther endpoint decides the edge"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/step-aux-walk-s-cons-cons","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.stepAux_walkS_cons_cons","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#step-aux-walk-s-cons-cons","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Write $1^j$ for the stack of $j$ copies of the bit symbol for true. If the endpoint counters of stacks $S$ hold $U = 1^{u+1}$ and $V = 1^{v+1}$, then executing the joint-walk block from any internal value and the stacks $S$ has the same outcome as executing, from internal value true and the stacks $S$ with $U$ replaced by $1^u$ and $V$ by $1^v$, the color-moving block that continues to the joint-walk label whatever\u2026","title":"Joint-walk step with both counters nonempty"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/step-aux-walk-s-cons-nil","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.stepAux_walkS_cons_nil","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#step-aux-walk-s-cons-nil","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Write $1^j$ for the stack of $j$ copies of the bit symbol for true. If the endpoint counters of stacks $S$ hold $U = 1^{u+1}$ and $V$ empty, then executing the joint-walk block from any internal value and the stacks $S$ has the same outcome as executing, from internal value false and the stacks $S$ with $U$ replaced by $1^u$, the color-moving block that, after moving a color $(a, b)$, continues to the label that\u2026","title":"Joint-walk step when the second endpoint counter runs out first"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/step-aux-walk-s-nil-cons","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.stepAux_walkS_nil_cons","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#step-aux-walk-s-nil-cons","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Write $1^j$ for the stack of $j$ copies of the bit symbol for true. If the endpoint counters of stacks $S$ hold $U$ empty and $V = 1^{v+1}$, then executing the joint-walk block from any internal value and the stacks $S$ has the same outcome as executing, from internal value true and the stacks $S$ with $V$ replaced by $1^v$, the color-moving block that, after moving a color $(a, b)$, continues to the label that\u2026","title":"Joint-walk step when the first endpoint counter runs out first"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/step-aux-walk-s-nil-nil","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.stepAux_walkS_nil_nil","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#step-aux-walk-s-nil-nil","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"If the endpoint counters $U$ and $V$ of stacks $S$ are both empty, then executing the joint-walk block from any internal value and the stacks $S$ has the same outcome as executing, from internal value false and the same stacks $S$, the color-moving block that continues to the label restoring the color stack whatever color it moves.","title":"Joint-walk step with both counters exhausted"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/edge-ok-zero-succ","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.edgeOk_zero_succ","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#edge-ok-zero-succ","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"For a color code $c$, a list $cs$ of color codes and $v \\in \\mathbb{N}$, the edge $(0, v + 1)$ passes the edge verdict with vertex colors $c$ followed by $cs$ iff $v$ is less than the length of $cs$ and entry $v$ of $cs$ differs from $c$.","title":"Edge verdict for an edge starting at vertex 0"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/edge-ok-succ-zero","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.edgeOk_succ_zero","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#edge-ok-succ-zero","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"For a color code $c$, a list $cs$ of color codes and $u \\in \\mathbb{N}$, the edge $(u + 1, 0)$ passes the edge verdict with vertex colors $c$ followed by $cs$ iff $u$ is less than the length of $cs$ and entry $u$ of $cs$ differs from $c$.","title":"Edge verdict for an edge ending at vertex 0"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/walk-run","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.walk_run","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#walk-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $u, v$ be natural numbers, $\\mathit{pre}$ and $\\mathit{post}$ lists of color codes (pairs of bits), $s$ a bit and $S$ an assignment of a list of symbols to every stack. Suppose that $S$ holds $u$ unit symbols on the counter $U$ and $v$ on the counter $V$, the bits of $\\mathit{post}$ on the color stack $P$ (each code $(a, b)$ contributing $a$ then $b$, the first code on top), and the bits of $\\mathit{pre}$,\u2026","title":"The joint walk of the 3-COLOR verifier decides one edge"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/read-u-some","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.readU_some","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#read-u-some","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $c \\ne X$ be a stack and $s$, $t$ labels such that the verifier runs, at $s$, the unary reader from $X$ onto the counter $c$ that loops at $s$ and exits to $t$. Let $k, j$ be natural numbers and $r$ a list of bits. If $X$ holds $k$ ones, then a zero, then $r$, and $c$ holds exactly $j$ units, then from label $s$, in any internal state, the verifier reaches label $t$ in exactly $k + 1$ steps, with $X$ holding\u2026","title":"The unary reader on a terminated number"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/read-u-none","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.readU_none","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#read-u-none","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $c \\ne X$ be a stack and $s$, $t$ labels such that the verifier runs, at $s$, the unary reader from $X$ onto the counter $c$ that loops at $s$ and exits to $t$. If $X$ holds a bit list $l$ on which reading a unary number fails, that is, $l$ consists of ones only, then from label $s$, in any internal state and with any other stacks, the verifier reaches the rejecting label within at most $|l| + 1$ steps.","title":"The unary reader rejects an unterminated number"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/load-run","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.load_run","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#load-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $n$ be a natural number and $w$, $a$ lists of bits. Start the verifier at its load label, in any internal state, with stacks in which the counter $N$ holds exactly $n$ units, the witness stack $W$ holds $w$ and the scratch stack $T$ holds $a$. Then within at most $n + 1$ steps the verifier reaches the label that moves the codes onto the color stack $P$ if $w$ has at least $2n$ bits and the color codes of the\u2026","title":"Run of the load phase of the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/edges-ok-of-some-some","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.edgesOk_of_some_some","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#edges-ok-of-some-some","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $cs$ be a list of color codes and $r$ a bit list. If $r$ begins with $u$ in unary ($u$ ones, then a zero) followed by the rest $r_1$, and $r_1$ begins with $v$ in unary followed by the rest $r_2$, then the edge-loop verdict on $r$ with vertex colors $cs$ holds iff the edge verdict for $(u, v)$ holds and the edge-loop verdict on $r_2$ holds.","title":"Edge-loop verdict after one parsed edge"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/back-run","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.back_run","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#back-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $\\mathit{pre}$ and $\\mathit{post}$ be lists of pairs of bits, let $s$ be a Boolean state, and let $S$ assign to each stack a list of symbols. Suppose that in $S$ the stack $P$ holds the bits of the flattening of $\\mathit{post}$ and the stack $T$ holds the bits of the flattening of $\\mathit{pre}$ in reverse order (as bit symbols, top first). Then the verifier program runs in exactly $2 |\\mathit{pre}| + 1$ steps\u2026","title":"Restoring the color stack in the 3-COLOR verifier"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/edges-run","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.edges_run","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#edges-run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $cs$ be a list of color codes (pairs of bits) and $r$ a list of bits, and encode an edge $(u, v)$ of natural numbers as $u$ ones, a zero, $v$ ones and a zero. Call $r$ good for $cs$ when $r$ is a concatenation of encoded edges such that, for every edge $(u, v)$ among them, $u$ and $v$ are below the length of $cs$ and, if $u \\ne v$, entries $u$ and $v$ of $cs$ (counting from $0$) differ. Suppose the 3-COLOR\u2026","title":"Correctness and running time of the edge loop"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/run","kind":"node","lean":"Cslib.Complexity.Karp.ColVerify.run","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#run","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"Let $x$ and $w$ be bit strings. Start the verifier machine at its start label, with any internal value, with $x$ written as bit symbols on the input stack $X$, $w$ written as bit symbols on the witness stack $W$, and all other stacks empty. Then within at most $8(|x| + |w| + 1)^3$ steps it reaches the accepting label if the 3-coloring check accepts $(x, w)$, and the rejecting label otherwise.","title":"Correctness and time bound of the 3-coloring verifier machine"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/col-verify-comp","kind":"node","lean":"Cslib.Complexity.Karp.colVerifyComp","parent":31969,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08/index.html#col-verify-comp","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-08","status":"fully_proved","summary":"The component machine whose stacks are the seven stacks $X, W, N, P, T, U, V$ of the 3-COLOR verifier machine, listed in that order and each over the token alphabet of the SAT verifier, and whose labels and program are those of the 3-COLOR verifier machine.","title":"The 3-COLOR verifier as a host component"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09","kind":"scope","navigation":"transparent","parent":31821,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09","status":"fully_proved","title":"3-COLOR is in NP to 3-COLOR is NP-complete"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/col3-dec-mem-np","kind":"node","lean":"Cslib.Complexity.Karp.col3Dec_mem_NP","parent":31990,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/index.html#col3-dec-mem-np","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09","status":"fully_proved","summary":"3-COLOR is in NP.","title":"3-COLOR is in NP"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/emit-color","kind":"node","lean":"Cslib.Complexity.Karp.emitColor","parent":31990,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/index.html#emit-color","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09","status":"fully_proved","summary":"The emitter whose stacks, alphabets, labels, program and start label are those of the 3-COLOR emitter machine, which exits at the machine's final label, reads the tokens of a CNF formula from stack $T$ and writes its output bits on stack $O$ (both symbol maps being the identity), and has step bound $B(n) = 500 (n + 1)^4$ for $n$ tokens. Its output for a CNF formula $\\varphi$ is the encoded 3-COLOR instance of\u2026","title":"The 3-COLOR emitter"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/is-poly-time-reduce-col","kind":"node","lean":"Cslib.Complexity.Karp.isPolyTime_reduceCol","parent":31990,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/index.html#is-poly-time-reduce-col","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09","status":"fully_proved","summary":"The map on bit strings that sends a string decoding to a CNF formula $\\varphi$ to the encoding of the 3-COLOR instance of $\\varphi$, and every other string to the encoding of the complete graph $K_4$, is computable in polynomial time.","title":"The SAT-to-3-COLOR reduction is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/col3-dec-np-hard","kind":"node","lean":"Cslib.Complexity.Karp.col3Dec_npHard","parent":31990,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/index.html#col3-dec-np-hard","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09","status":"fully_proved","summary":"Every decision problem in NP reduces to 3-COLOR by a polynomial-time many-one reduction.","title":"3-COLOR is NP-hard"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/col3-dec-np-complete","kind":"node","lean":"Cslib.Complexity.Karp.col3Dec_npComplete","parent":31990,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09/index.html#col3-dec-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-3-coloring/part-09","status":"fully_proved","summary":"3-COLOR is in NP, and every decision problem in NP reduces to 3-COLOR by a polynomial-time many-one reduction.","title":"3-COLOR is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth","status":"can_prove","title":"Graph bandwidth"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/bandwidth-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Bandwidth.BandwidthInstance","parent":31996,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/index.html#bandwidth-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth","status":"fully_proved","summary":"An instance of BANDWIDTH is a triple $(n, E, K)$ of a natural number $n$ (the number of vertices), a list $E$ of ordered pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"BANDWIDTH instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Bandwidth.WellFormed","parent":31996,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth","status":"fully_proved","summary":"A BANDWIDTH instance $(n, E, K)$ is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and its reverse $(v, u)$ does not occur in $E$. Thus $E$ lists the edges of a loopless undirected graph on the vertices $0, \\dots, n-1$, each edge exactly once.","title":"Well-formed BANDWIDTH instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/bandwidth-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Bandwidth.BandwidthFeasible","parent":31996,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/index.html#bandwidth-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth","status":"fully_proved","summary":"A BANDWIDTH instance $(n, E, K)$ is feasible when there is an ordering of the numbers $0, \\dots, n-1$ such that $|p(u) - p(v)| \\le K$ for every pair $(u, v)$ in $E$, where $p(m)$ is the position of $m$ in the ordering, counting from $0$, and $p(m) = n$ when $m$ is not among $0, \\dots, n-1$.","title":"BANDWIDTH feasibility"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/bandwidth-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Bandwidth.bandwidthDec","parent":31996,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/index.html#bandwidth-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth","status":"fully_proved","summary":"BANDWIDTH accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of triples $(n, E, K)$ of a natural number, a list of pairs of natural numbers and a natural number, to a BANDWIDTH instance that is both well formed and feasible. Bitstrings that do not decode to such a triple are rejected.","title":"The BANDWIDTH decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/bandwidth-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Bandwidth.bandwidth_npComplete","parent":31996,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth/index.html#bandwidth-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-bandwidth","status":"can_prove","summary":"The BANDWIDTH decision problem belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"BANDWIDTH is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring","status":"can_prove","title":"Graph coloring"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/graph-coloring-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphColoring.GraphColoringInstance","parent":32002,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/index.html#graph-coloring-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring","status":"fully_proved","summary":"An instance of GRAPH COLORING is a triple $(n, E, K)$ consisting of a natural number $n$, the number of vertices, a list $E$ of pairs of natural numbers, the edges, and a natural number $K$, the bound on the number of colors.","title":"Instances of GRAPH COLORING"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphColoring.Graph","parent":32002,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring","status":"fully_proved","summary":"The graph of an instance $(n, E, K)$ of GRAPH COLORING is the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Graph of a GRAPH COLORING instance"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphColoring.WellFormed","parent":32002,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring","status":"fully_proved","summary":"An instance $(n, E, K)$ of GRAPH COLORING is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$.","title":"Well-formed GRAPH COLORING instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/graph-coloring-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphColoring.GraphColoringFeasible","parent":32002,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/index.html#graph-coloring-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring","status":"fully_proved","summary":"An instance $(n, E, K)$ of GRAPH COLORING is feasible when its graph is $K$-colorable, that is, when its vertices can be assigned colors from $\\{0, \\dots, K - 1\\}$ so that adjacent vertices receive different colors.","title":"Feasible GRAPH COLORING instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/graph-coloring-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphColoring.graphColoringDec","parent":32002,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/index.html#graph-coloring-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring","status":"fully_proved","summary":"The decision problem GRAPH COLORING accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance of GRAPH COLORING that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"GRAPH COLORING as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/graph-coloring-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphColoring.graphColoring_npComplete","parent":32002,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring/index.html#graph-coloring-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-coloring","status":"can_prove","summary":"The decision problem GRAPH COLORING belongs to NP, and every decision problem $L$ in NP has a polynomial-time many-one reduction to GRAPH COLORING.","title":"GRAPH COLORING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism","status":"can_prove","title":"Graph homomorphism"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/graph-homomorphism-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphHomomorphism.GraphHomomorphismInstance","parent":32009,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/index.html#graph-homomorphism-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism","status":"fully_proved","summary":"A graph-homomorphism instance is a quadruple $(n, E, m, F)$ where $n$ is a natural number (the number of source vertices), $E$ is a list of pairs of natural numbers (the source edges), $m$ is a natural number (the number of target vertices) and $F$ is a list of pairs of natural numbers (the target edges).","title":"Graph-homomorphism instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/graph-target","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphHomomorphism.GraphTarget","parent":32009,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/index.html#graph-target","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism","status":"fully_proved","summary":"The target graph of a graph-homomorphism instance $(n, E, m, F)$ is the simple graph on $\\{0, \\dots, m - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $F$. Pairs of $F$ with an endpoint at least $m$ contribute no edge.","title":"Target graph of a graph-homomorphism instance"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphHomomorphism.WellFormed","parent":32009,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism","status":"fully_proved","summary":"A graph-homomorphism instance $(n, E, m, F)$ is well formed when $E$ has no repeated entries, every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$, $F$ has no repeated entries, and every pair $(u, v)$ in $F$ satisfies $u < m$, $v < m$, $u \\ne v$ and $(v, u) \\notin F$.","title":"Well-formed graph-homomorphism instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/graph-homomorphism-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphHomomorphism.GraphHomomorphismFeasible","parent":32009,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/index.html#graph-homomorphism-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism","status":"fully_proved","summary":"A graph-homomorphism instance $(n, E, m, F)$ is feasible when there is a map $f : \\{0, \\dots, n - 1\\} \\to \\{0, \\dots, m - 1\\}$ such that, for every pair $(u, v)$ in $E$ with $u < n$ and $v < n$, the vertices $f(u)$ and $f(v)$ are adjacent in the target graph of the instance, that is, $f(u) \\ne f(v)$ and $(f(u), f(v))$ or $(f(v), f(u))$ belongs to $F$. Pairs of $E$ with an endpoint at least $n$ impose no condition.","title":"Feasible graph-homomorphism instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/graph-homomorphism-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphHomomorphism.graphHomomorphismDec","parent":32009,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/index.html#graph-homomorphism-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism","status":"fully_proved","summary":"The decision problem GRAPH HOMOMORPHISM accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a graph-homomorphism instance $(n, E, m, F)$ that is well formed (neither $E$ nor $F$ has repeated entries, every pair $(u, v)$ in $E$ has $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$, and every pair $(u, v)$ in $F$ has $u < m$, $v < m$, $u \\ne v$ and $(v, u) \\notin F$) and for\u2026","title":"GRAPH HOMOMORPHISM as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/graph-homomorphism-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GraphHomomorphism.graphHomomorphism_npComplete","parent":32009,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism/index.html#graph-homomorphism-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/graph-homomorphism","status":"can_prove","summary":"The decision problem GRAPH HOMOMORPHISM belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"GRAPH HOMOMORPHISM is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments","status":"can_prove","title":"Linear ordering of tournaments"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/linear-ordering-tournament-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LinearOrderingTournament.LinearOrderingTournamentInstance","parent":32016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/index.html#linear-ordering-tournament-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments","status":"fully_proved","summary":"An instance of LINEAR ORDERING (TOURNAMENTS) is a triple $(n, E, K)$ of a natural number $n$, the number of vertices, a list $E$ of ordered pairs of natural numbers, the arcs, and a natural number $K$, the bound.","title":"Tournament linear ordering instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LinearOrderingTournament.WellFormed","parent":32016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments","status":"fully_proved","summary":"An instance $(n, E, K)$ of LINEAR ORDERING (TOURNAMENTS) is well formed when $n > 0$, $E$ has no repeated entries, every pair $(a, b)$ in $E$ satisfies $a < n$, $b < n$, $a \\ne b$ and $(b, a) \\notin E$, and for all distinct $a, b < n$ at least one of $(a, b)$ and $(b, a)$ lies in $E$. That is, $E$ is a tournament on the vertices $0, \\dots, n - 1$.","title":"Well-formed tournament linear ordering instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/linear-ordering-tournament-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LinearOrderingTournament.LinearOrderingTournamentFeasible","parent":32016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/index.html#linear-ordering-tournament-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments","status":"fully_proved","summary":"An instance $(n, E, K)$ of LINEAR ORDERING (TOURNAMENTS) is feasible when there are an ordering of the vertices $0, \\dots, n - 1$ and a subsequence $F$ of the list $E$ with at least $K$ entries such that for every pair $(a, b)$ in $F$, the position of $a$ in the ordering is less than the position of $b$, where positions are counted from $0$ and a number that does not occur in the ordering is given position $n$.","title":"Orderings with many forward arcs"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/linear-ordering-tournament-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LinearOrderingTournament.linearOrderingTournamentDec","parent":32016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/index.html#linear-ordering-tournament-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments","status":"fully_proved","summary":"The decision problem LINEAR ORDERING (TOURNAMENTS) accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of triples of a natural number, a list of pairs of natural numbers and a natural number, to an instance $(n, E, K)$ that is well formed ($n > 0$ and $E$ is a tournament on the vertices $0, \\dots, n - 1$ without repeated arcs) and feasible (some ordering of the vertices has at\u2026","title":"LINEAR ORDERING (TOURNAMENTS) as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/linear-ordering-tournament-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LinearOrderingTournament.linearOrderingTournament_npComplete","parent":32016,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments/index.html#linear-ordering-tournament-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/linear-ordering-of-tournaments","status":"can_prove","summary":"The decision problem LINEAR ORDERING (TOURNAMENTS) on bitstrings lies in NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"LINEAR ORDERING (TOURNAMENTS) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring","status":"can_prove","title":"List coloring"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/list-coloring-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ListColoring.ListColoringInstance","parent":32022,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/index.html#list-coloring-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring","status":"fully_proved","summary":"A list-coloring instance is a triple $(n, E, L)$ of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edges), and a list $L$ of lists of natural numbers (the allowed colors of each vertex).","title":"List-coloring instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ListColoring.WellFormed","parent":32022,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring","status":"fully_proved","summary":"A list-coloring instance $(n, E, L)$ is well formed when the edge list $E$ has no repeated entries, every edge $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$ and its reversal $(v, u)$ does not occur in $E$, and the list $L$ of color lists has length $n$.","title":"Well-formed list-coloring instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/list-coloring-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ListColoring.ListColoringFeasible","parent":32022,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/index.html#list-coloring-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring","status":"fully_proved","summary":"A list-coloring instance $(n, E, L)$ is feasible when there is a list $c = (c_0, \\dots, c_{n-1})$ of $n$ natural numbers such that for every $v < n$ the number $c_v$ belongs to the $v$-th entry $L_v$ of $L$ (taken to be the empty list when $L$ has at most $v$ entries), and $c_u \\ne c_v$ for every edge $(u, v)$ in $E$, where $c_w$ is read as $0$ when $w \\ge n$.","title":"Feasible list-coloring instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/list-coloring-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ListColoring.listColoringDec","parent":32022,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/index.html#list-coloring-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring","status":"fully_proved","summary":"LIST COLORING is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a list-coloring instance that is well formed and feasible. Bitstrings that do not decode are rejected.","title":"The LIST COLORING decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/list-coloring-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ListColoring.listColoring_npComplete","parent":32022,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring/index.html#list-coloring-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/list-coloring","status":"can_prove","summary":"The LIST COLORING decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"LIST COLORING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut","status":"can_prove","title":"Maximum cut"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/max-cut-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxCut.MaxCutInstance","parent":32028,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/index.html#max-cut-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut","status":"fully_proved","summary":"An instance of MAXIMUM CUT is a triple $(n, E, K)$ of a natural number $n$ (the number of vertices), a finite list $E$ of ordered pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"Instances of MAXIMUM CUT"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxCut.WellFormed","parent":32028,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut","status":"fully_proved","summary":"An instance $(n, E, K)$ of MAXIMUM CUT is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and the reversed pair $(v, u)$ does not occur in $E$.","title":"Well-formed MAXIMUM CUT instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/max-cut-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxCut.MaxCutFeasible","parent":32028,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/index.html#max-cut-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut","status":"fully_proved","summary":"An instance $(n, E, K)$ of MAXIMUM CUT is feasible when there are a finite set $S$ of natural numbers, all less than $n$, and a list $C$ of length at least $K$ obtained from $E$ by deleting zero or more of its entries, such that every pair $(u, v)$ in $C$ has exactly one of $u$ and $v$ in $S$.","title":"Feasible MAXIMUM CUT instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/max-cut-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxCut.maxCutDec","parent":32028,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/index.html#max-cut-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut","status":"fully_proved","summary":"The decision problem MAXIMUM CUT accepts a bitstring $x$ exactly when decoding $x$ with the canonical bitstring encoding of instances of MAXIMUM CUT succeeds and yields an instance that is well formed and feasible. All other bitstrings, including those that fail to decode, are rejected.","title":"MAXIMUM CUT as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/max-cut-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxCut.maxCut_npComplete","parent":32028,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut/index.html#max-cut-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-cut","status":"can_prove","summary":"The decision problem MAXIMUM CUT lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to MAXIMUM CUT.","title":"MAXIMUM CUT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut","status":"can_prove","title":"Maximum k-cut"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/max-kcut-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxKCut.MaxKCutInstance","parent":32034,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/index.html#max-kcut-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut","status":"fully_proved","summary":"An instance of MAXIMUM k-CUT is a quadruple $(n, E, k, K)$ of a natural number $n$ (the number of vertices), a finite list $E$ of ordered pairs of natural numbers (the edges), a natural number $k$ (the number of parts) and a natural number $K$ (the bound).","title":"Instances of MAXIMUM k-CUT"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxKCut.WellFormed","parent":32034,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut","status":"fully_proved","summary":"An instance $(n, E, k, K)$ of MAXIMUM k-CUT is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and the reversed pair $(v, u)$ does not occur in $E$.","title":"Well-formed MAXIMUM k-CUT instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/max-kcut-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxKCut.MaxKCutFeasible","parent":32034,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/index.html#max-kcut-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut","status":"fully_proved","summary":"An instance $(n, E, k, K)$ of MAXIMUM k-CUT is feasible when there are a list $(a_0, \\dots, a_{n-1})$ of $n$ natural numbers, each less than $k$, and a list $C$ of length at least $K$ obtained from $E$ by deleting zero or more of its entries, such that $a_u \\ne a_v$ for every pair $(u, v)$ in $C$, where $a_w$ is read as $0$ when $w \\ge n$.","title":"Feasible MAXIMUM k-CUT instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/max-kcut-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxKCut.maxKCutDec","parent":32034,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/index.html#max-kcut-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut","status":"fully_proved","summary":"The decision problem MAXIMUM k-CUT accepts a bitstring $x$ exactly when decoding $x$ with the canonical bitstring encoding of instances of MAXIMUM k-CUT succeeds and yields an instance that is well formed and feasible. All other bitstrings, including those that fail to decode, are rejected.","title":"MAXIMUM k-CUT as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/max-kcut-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxKCut.maxKCut_npComplete","parent":32034,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut/index.html#max-kcut-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/maximum-k-cut","status":"can_prove","summary":"The decision problem MAXIMUM k-CUT lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to MAXIMUM k-CUT.","title":"MAXIMUM k-CUT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection","status":"can_prove","title":"Minimum bisection"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/min-bisection-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinBisection.MinBisectionInstance","parent":32040,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/index.html#min-bisection-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection","status":"fully_proved","summary":"An instance of MINIMUM BISECTION is a triple $(n, es, K)$ of a natural number $n$ (the number of vertices), a list $es$ of pairs of natural numbers (the edge list) and a natural number $K$ (the bound on the number of crossing edges).","title":"Instances of MINIMUM BISECTION"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinBisection.WellFormed","parent":32040,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection","status":"fully_proved","summary":"An instance $(n, es, K)$ of MINIMUM BISECTION is well formed when $n$ is even, the list $es$ has no repeated entries, and every entry $(u, v)$ of $es$ satisfies $u < n$, $v < n$ and $u \\ne v$, and $(v, u)$ is not an entry of $es$.","title":"Well-formed MINIMUM BISECTION instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/min-bisection-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinBisection.MinBisectionFeasible","parent":32040,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/index.html#min-bisection-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection","status":"fully_proved","summary":"An instance $(n, es, K)$ of MINIMUM BISECTION is feasible when there is a set $S \\subseteq \\{0, \\dots, n - 1\\}$ with $\\lfloor n / 2 \\rfloor$ elements such that no sublist of $es$ (a list obtained from $es$ by deleting entries) with $K + 1$ entries consists only of entries $(u, v)$ with exactly one of $u$ and $v$ in $S$; that is, at most $K$ entries of $es$ have exactly one endpoint in $S$.","title":"Feasible MINIMUM BISECTION instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/min-bisection-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinBisection.minBisectionDec","parent":32040,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/index.html#min-bisection-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection","status":"fully_proved","summary":"The decision problem MINIMUM BISECTION accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of instances, to an instance $(n, es, K)$ of MINIMUM BISECTION that is well formed and feasible (its vertices split into two sets of $n / 2$ vertices with at most $K$ entries of $es$ between them). Bitstrings that do not decode are rejected.","title":"The decision problem MINIMUM BISECTION"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/min-bisection-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinBisection.minBisection_npComplete","parent":32040,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection/index.html#min-bisection-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-bisection","status":"can_prove","summary":"The decision problem MINIMUM BISECTION belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"MINIMUM BISECTION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement","status":"can_prove","title":"Minimum linear arrangement"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/minimum-linear-arrangement-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumLinearArrangement.MinimumLinearArrangementInstance","parent":32046,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/index.html#minimum-linear-arrangement-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement","status":"fully_proved","summary":"An instance of MINIMUM LINEAR ARRANGEMENT is a triple $(n, E, K)$ of a natural number $n$ (the number of vertices), a finite list $E$ of ordered pairs of natural numbers (the edges) and a natural number $K$ (the bound).","title":"Minimum linear arrangement instance"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumLinearArrangement.WellFormed","parent":32046,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement","status":"fully_proved","summary":"An instance $(n, E, K)$ of MINIMUM LINEAR ARRANGEMENT is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, while the reversed pair $(v, u)$ does not occur in $E$. Thus $E$ lists the edges of a simple graph on the vertices $0, \\dots, n-1$, each edge once and in one orientation.","title":"Well-formed minimum linear arrangement instance"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/minimum-linear-arrangement-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumLinearArrangement.MinimumLinearArrangementFeasible","parent":32046,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/index.html#minimum-linear-arrangement-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement","status":"fully_proved","summary":"An instance $(n, E, K)$ of MINIMUM LINEAR ARRANGEMENT is feasible when there is an ordering $\\pi$ of $0, 1, \\dots, n-1$ (a list containing each of these numbers exactly once) such that $\\sum_{(u, v) \\in E} |\\pi(u) - \\pi(v)| \\le K$. Here $\\pi(v)$ is the position of $v$ in $\\pi$, counted from $0$, and is $n$ when $v \\ge n$; the sum runs over the list $E$, so a repeated pair is counted once for each occurrence.","title":"Feasible minimum linear arrangement instance"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/minimum-linear-arrangement-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumLinearArrangement.minimumLinearArrangementDec","parent":32046,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/index.html#minimum-linear-arrangement-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement","status":"fully_proved","summary":"It accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, E, K)$ of a natural number $n$, a list $E$ of pairs of natural numbers and a natural number $K$ with the following properties: every entry of $E$ is a pair $(u, v)$ of distinct numbers below $n$, the list $E$ has no repeated entries and never contains both $(u, v)$ and $(v, u)$, and there is an ordering of\u2026","title":"The MINIMUM LINEAR ARRANGEMENT decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/minimum-linear-arrangement-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumLinearArrangement.minimumLinearArrangement_npComplete","parent":32046,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement/index.html#minimum-linear-arrangement-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/minimum-linear-arrangement","status":"can_prove","summary":"The MINIMUM LINEAR ARRANGEMENT decision problem lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"MINIMUM LINEAR ARRANGEMENT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut","status":"can_prove","title":"Multiway cut"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/multiway-cut-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayCut.MultiwayCutInstance","parent":32052,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/index.html#multiway-cut-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut","status":"fully_proved","summary":"A multiway cut instance is a tuple $(n, E, T, K)$ consisting of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edges), a list $T$ of natural numbers (the terminals) and a natural number $K$ (the bound on the number of cut edges).","title":"Multiway cut instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayCut.WellFormed","parent":32052,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut","status":"fully_proved","summary":"A multiway cut instance $(n, E, T, K)$ is well-formed when the entries of $E$ are pairwise distinct; every entry $(u, v)$ of $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and $(v, u)$ is not an entry of $E$; the entries of $T$ are pairwise distinct; and every entry $t$ of $T$ satisfies $t < n$. Thus $E$ lists each edge of a simple undirected graph on the vertices $0, \\dots, n-1$ exactly once, in one orientation,\u2026","title":"Well-formed multiway cut instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/multiway-cut-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayCut.MultiwayCutFeasible","parent":32052,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/index.html#multiway-cut-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut","status":"fully_proved","summary":"Let $(n, E, T, K)$ be a multiway cut instance with terminal list $T = (t_0, \\dots, t_{k-1})$. It is feasible when there is a labelling $f$ of the vertices $0, \\dots, n-1$ by labels in $\\{0, \\dots, \\max(k, 1) - 1\\}$ such that $f(t_j) = j$ for every $j < k$, and at most $K$ entries $(u, v)$ of $E$, counted with repetition, satisfy $f(u) \\ne f(v)$. Here every number $v \\ge n$ counts as a vertex with label $0$.","title":"Feasible multiway cut instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/multiway-cut-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayCut.multiwayCutDec","parent":32052,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/index.html#multiway-cut-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut","status":"fully_proved","summary":"MULTIWAY CUT is the decision problem on bitstrings that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a multiway cut instance $(n, E, T, K)$ that is well-formed and feasible. Bitstrings that do not decode to a multiway cut instance are rejected.","title":"The MULTIWAY CUT decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/multiway-cut-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayCut.multiwayCut_npComplete","parent":32052,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut/index.html#multiway-cut-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/multiway-cut","status":"can_prove","summary":"Let MULTIWAY CUT be the decision problem that accepts exactly the bitstrings decoding to a well-formed, feasible multiway cut instance: a simple undirected graph on the vertices $0, \\dots, n-1$, a list of distinct terminal vertices and a bound $K$ such that deleting at most $K$ edges of the graph leaves no two terminals connected. Then MULTIWAY CUT is in NP, and every decision problem in NP reduces to MULTIWAY CUT\u2026","title":"MULTIWAY CUT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2","status":"can_prove","title":"Partition into paths of length 2"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/partition-into-p3s-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoP3s.PartitionIntoP3sInstance","parent":32058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/index.html#partition-into-p3s-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2","status":"fully_proved","summary":"An instance of PARTITION INTO PATHS OF LENGTH 2 is a pair $(n, E)$ of a natural number $n$, the number of vertices, and a finite list $E$ of pairs of natural numbers, the edge list.","title":"Instances of PARTITION INTO PATHS OF LENGTH 2"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoP3s.Graph","parent":32058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2","status":"fully_proved","summary":"For an instance $(n, E)$, the graph of the instance is the simple graph on the vertex set $\\{0, \\dots, n - 1\\}$ in which distinct vertices $u$ and $v$ are adjacent exactly when $(u, v)$ or $(v, u)$ belongs to $E$. Pairs in $E$ with an entry at least $n$, and pairs $(u, u)$, contribute no edge.","title":"Graph of a PARTITION INTO PATHS OF LENGTH 2 instance"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoP3s.WellFormed","parent":32058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2","status":"fully_proved","summary":"An instance $(n, E)$ is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and the reversed pair $(v, u)$ does not belong to $E$.","title":"Well-formed PARTITION INTO PATHS OF LENGTH 2 instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/partition-into-p3s-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoP3s.PartitionIntoP3sFeasible","parent":32058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/index.html#partition-into-p3s-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2","status":"fully_proved","summary":"An instance $(n, E)$ admits a partition into paths of length 2 when there is a list of triples, each a list of three distinct vertices from $\\{0, \\dots, n - 1\\}$, such that every vertex occurs exactly once among all the triples taken together, and every triple contains pairwise distinct vertices $x$, $y$, $z$ such that $x$ is adjacent to $y$ and $y$ is adjacent to $z$ in the graph of the instance.","title":"Partitions into paths of length 2"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/partition-into-p3s-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoP3s.partitionIntoP3sDec","parent":32058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/index.html#partition-into-p3s-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2","status":"fully_proved","summary":"The decision problem PARTITION INTO PATHS OF LENGTH 2 accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of instances, to an instance $(n, E)$ that is well formed and admits a partition into paths of length 2. Bitstrings that do not decode to an instance are rejected.","title":"PARTITION INTO PATHS OF LENGTH 2"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/partition-into-p3s-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoP3s.partitionIntoP3s_npComplete","parent":32058,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2/index.html#partition-into-p3s-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-paths-of-length-2","status":"can_prove","summary":"The decision problem PARTITION INTO PATHS OF LENGTH 2 belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"PARTITION INTO PATHS OF LENGTH 2 is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles","status":"can_prove","title":"Partition into triangles"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/partition-into-triangles-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoTriangles.PartitionIntoTrianglesInstance","parent":32065,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/index.html#partition-into-triangles-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles","status":"fully_proved","summary":"An instance of PARTITION INTO TRIANGLES is a pair consisting of a natural number $n$, the number of vertices, and a list of pairs of natural numbers, the edges.","title":"Instances of PARTITION INTO TRIANGLES"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoTriangles.Graph","parent":32065,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/index.html#graph","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles","status":"fully_proved","summary":"For an instance consisting of a natural number $n$ and a list $E$ of pairs of natural numbers, its graph is the graph of the edge list $E$ on $n$ vertices: the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$. Pairs with an endpoint at least $n$ contribute no edge.","title":"Graph of a PARTITION INTO TRIANGLES instance"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoTriangles.WellFormed","parent":32065,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles","status":"fully_proved","summary":"An instance consisting of a natural number $n$ and a list $E$ of pairs of natural numbers is well formed when $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$.","title":"Well-formed PARTITION INTO TRIANGLES instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/partition-into-triangles-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoTriangles.PartitionIntoTrianglesFeasible","parent":32065,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/index.html#partition-into-triangles-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles","status":"fully_proved","summary":"An instance consisting of a natural number $n$ and a list $E$ of pairs of natural numbers admits a partition into triangles when the vertex set $\\{0, \\dots, n - 1\\}$ can be partitioned into sets of three vertices, any two of which are adjacent in the graph of $E$ on $n$ vertices.","title":"Partition into triangles"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/partition-into-triangles-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoTriangles.partitionIntoTrianglesDec","parent":32065,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/index.html#partition-into-triangles-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles","status":"fully_proved","summary":"The decision problem PARTITION INTO TRIANGLES accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance consisting of a natural number $n$ and a list $E$ of pairs of natural numbers such that $E$ has no repeated entries, every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$, and the vertex set $\\{0, \\dots, n - 1\\}$ can be partitioned\u2026","title":"PARTITION INTO TRIANGLES as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/partition-into-triangles-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PartitionIntoTriangles.partitionIntoTriangles_npComplete","parent":32065,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles/index.html#partition-into-triangles-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/partition-into-triangles","status":"can_prove","summary":"The decision problem PARTITION INTO TRIANGLES lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to PARTITION INTO TRIANGLES.","title":"PARTITION INTO TRIANGLES is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension.json","id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension","kind":"scope","navigation":"module","parent":31807,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/index.html","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension","status":"can_prove","title":"Precoloring extension"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/precoloring-extension-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecoloringExtension.PrecoloringExtensionInstance","parent":32072,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/index.html#precoloring-extension-instance","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension","status":"fully_proved","summary":"An instance of precoloring extension is a tuple $(n, E, c, K)$ consisting of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edges), a list $c$ of natural numbers (the partial coloring, where an entry $0$ means uncolored and an entry $k + 1$ means color $k$), and a natural number $K$ (the number of colors).","title":"Instances of PRECOLORING EXTENSION"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecoloringExtension.WellFormed","parent":32072,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension","status":"fully_proved","summary":"An instance $(n, E, c, K)$ of precoloring extension is well formed when the edge list $E$ has no repeated entries, every edge $(u, v)$ in $E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$, and the partial coloring $c$ has length $n$.","title":"Well-formed precoloring extension instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/precoloring-extension-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecoloringExtension.PrecoloringExtensionFeasible","parent":32072,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/index.html#precoloring-extension-feasible","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension","status":"fully_proved","summary":"An instance $(n, E, c, K)$ of precoloring extension is feasible when there is a list $\\chi = (\\chi_0, \\dots, \\chi_{n-1})$ of $n$ natural numbers such that $\\chi_v < K$ for every vertex $v < n$, $\\chi_v = c_v - 1$ for every vertex $v < n$ with $c_v \\ne 0$, and $\\chi_u \\ne \\chi_v$ for every edge $(u, v)$ in $E$. Here $c_v$ is the entry of $c$ at position $v$, and an entry of a list at a position beyond its length is\u2026","title":"Feasible precoloring extension instances"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/precoloring-extension-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecoloringExtension.precoloringExtensionDec","parent":32072,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/index.html#precoloring-extension-dec","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension","status":"fully_proved","summary":"The decision problem PRECOLORING EXTENSION accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a precoloring extension instance that is well formed and feasible; bitstrings that do not decode are rejected.","title":"PRECOLORING EXTENSION as a decision problem"},{"id":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/precoloring-extension-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecoloringExtension.precoloringExtension_npComplete","parent":32072,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension/index.html#precoloring-extension-np-complete","scope":"TheoryOfComputation/np-completeness/graph-coloring-and-partitioning/precoloring-extension","status":"can_prove","summary":"The decision problem PRECOLORING EXTENSION lies in NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"PRECOLORING EXTENSION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs","status":"can_prove","summary":"Hamiltonian paths and cycles, longest paths and cycles, disjoint paths and path covers, constrained spanning and Steiner trees, minimum equivalent digraph, traveling salesperson and vehicle routing, and (induced) subgraph isomorphism.","title":"Paths, cycles, trees and subgraph problems"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem","status":"can_prove","title":"Asymmetric traveling salesperson problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/asymmetric-tsp-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AsymmetricTsp.AsymmetricTspInstance","parent":32079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/index.html#asymmetric-tsp-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem","status":"fully_proved","summary":"An instance of the asymmetric traveling salesperson problem is a triple $(n, w, B)$ consisting of a natural number $n$ (the number of cities), a weight matrix $w$ given as a list of rows, each row a list of natural numbers, and a natural number $B$ (the bound).","title":"Asymmetric TSP instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/tour-length","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AsymmetricTsp.TourLength","parent":32079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/index.html#tour-length","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem","status":"fully_proved","summary":"Let $w$ be a weight matrix, given as a list of rows of natural numbers, and let $t = (t_0, \\dots, t_{L-1})$ be a list of natural numbers. The length of the tour $t$ under $w$ is $\\sum_{j=0}^{L-1} w_{t_j, t_{(j+1) \\bmod L}}$, the sum of the weights along the closed walk including the return edge from $t_{L-1}$ to $t_0$, where $w_{a,b}$ denotes the entry in row $a$ and column $b$ of $w$, taken to be $0$ when that\u2026","title":"Length of a tour in the asymmetric TSP"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AsymmetricTsp.WellFormed","parent":32079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem","status":"fully_proved","summary":"An asymmetric TSP instance $(n, w, B)$, with $n$ the number of cities, $w$ the weight matrix and $B$ the bound, is well formed when $n > 0$, $w$ has exactly $n$ rows, every row of $w$ has length $n$, and the diagonal entry $w_{a,a}$ in row $a$ and column $a$ is $0$ for every $a < n$.","title":"Well-formed asymmetric TSP instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/asymmetric-tsp-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AsymmetricTsp.AsymmetricTspFeasible","parent":32079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/index.html#asymmetric-tsp-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem","status":"fully_proved","summary":"An asymmetric TSP instance $(n, w, B)$ is feasible when there is a list $t$ of natural numbers with no repeated entries, of length $n$ and containing every $v < n$ (so $t$ lists the cities $0, \\dots, n - 1$ in some order), whose tour length under $w$ is at most $B$.","title":"Feasible asymmetric TSP instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/asymmetric-tsp-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AsymmetricTsp.asymmetricTspDec","parent":32079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/index.html#asymmetric-tsp-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem","status":"fully_proved","summary":"The asymmetric TSP decision problem accepts a bitstring $x$ when $x$ decodes, under the canonical bitstring encoding of asymmetric TSP instances, to an instance $(n, w, B)$ that is well formed and feasible, and rejects $x$ otherwise; in particular it rejects every bitstring that does not decode to an instance.","title":"The asymmetric traveling salesperson decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/asymmetric-tsp-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.AsymmetricTsp.asymmetricTsp_npComplete","parent":32079,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem/index.html#asymmetric-tsp-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/asymmetric-traveling-salesperson-problem","status":"can_prove","summary":"The asymmetric traveling salesperson decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"TRAVELING SALESPERSON (ASYMMETRIC) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing","status":"can_prove","title":"Capacitated vehicle routing"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/cvrpinstance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CVRP.CVRPInstance","parent":32086,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/index.html#cvrpinstance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing","status":"fully_proved","summary":"A capacitated vehicle routing instance is a tuple $(n, d, D, m, C, K)$ consisting of a natural number $n$ (the number of customers), a list $d$ of natural numbers (the demands), a list $D$ of lists of natural numbers (the distance matrix, whose rows and columns include the depot), and natural numbers $m$ (the number of vehicles), $C$ (the capacity) and $K$ (the bound on the total length).","title":"Capacitated vehicle routing instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/route-length","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CVRP.RouteLength","parent":32086,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/index.html#route-length","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing","status":"fully_proved","summary":"Let $D$ be a list of lists of natural numbers, read as a distance matrix whose entry $D_{u,v}$ is the entry at position $v$ of the row at position $u$ (positions counted from $0$), or $0$ if there is no such entry, and let $o$ be a natural number, the depot. The length of the empty route is $0$, and the length of a route given by a list $(r_1, \\dots, r_k)$ of natural numbers with $k \\ge 1$ is\u2026","title":"Length of a route"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CVRP.WellFormed","parent":32086,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing","status":"fully_proved","summary":"A capacitated vehicle routing instance $(n, d, D, m, C, K)$ is well formed when the demand list $d$ has length $n$, the distance matrix $D$ has $n + 1$ rows, and every row of $D$ has length $n + 1$.","title":"Well-formed capacitated vehicle routing instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/cvrpfeasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CVRP.CVRPFeasible","parent":32086,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/index.html#cvrpfeasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing","status":"fully_proved","summary":"Let $(n, d, D, m, C, K)$ be a capacitated vehicle routing instance, with customers $0, \\dots, n - 1$ and depot $n$, and let the demand $d_c$ of $c$ be the entry of the list $d$ at position $c$ (counting from $0$), or $0$ if $d$ has no such entry. The instance is feasible when there is a list of at most $m$ routes, each a list of natural numbers, such that every route has no repeated entries and only entries less\u2026","title":"Feasibility of a capacitated vehicle routing instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/cvrp-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CVRP.cvrpDec","parent":32086,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/index.html#cvrp-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing","status":"fully_proved","summary":"The decision problem CAPACITATED VEHICLE ROUTING accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a capacitated vehicle routing instance that is well formed and feasible; it rejects every bitstring that does not decode.","title":"Capacitated vehicle routing as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/cvrp-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CVRP.cvrp_npComplete","parent":32086,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing/index.html#cvrp-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/capacitated-vehicle-routing","status":"can_prove","summary":"The decision problem CAPACITATED VEHICLE ROUTING lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"CAPACITATED VEHICLE ROUTING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree","status":"can_prove","title":"Degree-constrained spanning tree"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/degree-constrained-spanning-tree-instanc","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DegreeConstrainedSpanningTree.DegreeConstrainedSpanningTreeInstance","parent":32093,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/index.html#degree-constrained-spanning-tree-instanc","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree","status":"fully_proved","summary":"An instance of DEGREE-CONSTRAINED SPANNING TREE is a triple $(n, E, K)$ consisting of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edges) and a natural number $K$ (the degree bound).","title":"Instances of DEGREE-CONSTRAINED SPANNING TREE"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DegreeConstrainedSpanningTree.WellFormed","parent":32093,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree","status":"fully_proved","summary":"An instance $(n, E, K)$ of DEGREE-CONSTRAINED SPANNING TREE is well formed when $n > 0$, the list $E$ has no repeated entries, and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, with the reversed pair $(v, u)$ not in $E$.","title":"Well-formed DEGREE-CONSTRAINED SPANNING TREE instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/degree-constrained-spanning-tree-feasibl","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DegreeConstrainedSpanningTree.DegreeConstrainedSpanningTreeFeasible","parent":32093,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/index.html#degree-constrained-spanning-tree-feasibl","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree","status":"fully_proved","summary":"An instance $(n, E, K)$ of DEGREE-CONSTRAINED SPANNING TREE is feasible when there is a list $T$ of pairs of natural numbers such that $T$ is a sublist of $E$ (obtained by deleting entries and keeping the order), $T$ has no repeated entries, $T$ has exactly $n - 1$ entries, the graph of $T$ on $n$ vertices is connected, and for every vertex $v < n$ there is no sublist of $T$ with $K + 1$ entries each of which has\u2026","title":"Feasibility for DEGREE-CONSTRAINED SPANNING TREE"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/degree-constrained-spanning-tree-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DegreeConstrainedSpanningTree.degreeConstrainedSpanningTreeDec","parent":32093,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/index.html#degree-constrained-spanning-tree-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree","status":"fully_proved","summary":"The decision problem DEGREE-CONSTRAINED SPANNING TREE accepts a bitstring $x$ exactly when bitstring decoding of $x$ as a triple $(n, E, K)$ succeeds and returns an instance of DEGREE-CONSTRAINED SPANNING TREE that is both well formed and feasible; every bitstring that does not decode is rejected.","title":"DEGREE-CONSTRAINED SPANNING TREE as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/degree-constrained-spanning-tree-np-comp","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DegreeConstrainedSpanningTree.degreeConstrainedSpanningTree_npComplete","parent":32093,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree/index.html#degree-constrained-spanning-tree-np-comp","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/degree-constrained-spanning-tree","status":"can_prove","summary":"The decision problem DEGREE-CONSTRAINED SPANNING TREE belongs to the class NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"DEGREE-CONSTRAINED SPANNING TREE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle","status":"can_prove","title":"Directed Hamiltonian cycle"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/hamiltonian-cycle-directed-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleDirected.HamiltonianCycleDirectedInstance","parent":32099,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/index.html#hamiltonian-cycle-directed-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle","status":"fully_proved","summary":"An instance of HAMILTONIAN CYCLE (DIRECTED) is a pair $(n, E)$ of a natural number $n$, the number of vertices, and a finite list $E$ of ordered pairs of natural numbers, the arcs.","title":"Instances of HAMILTONIAN CYCLE (DIRECTED)"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/dadj","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleDirected.DAdj","parent":32099,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/index.html#dadj","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle","status":"fully_proved","summary":"For a finite list $E$ of ordered pairs of natural numbers and natural numbers $u$ and $v$, the directed adjacency of $u$ to $v$ in $E$ is the boolean value that is true exactly when the pair $(u, v)$ occurs in $E$.","title":"Directed adjacency for HAMILTONIAN CYCLE (DIRECTED)"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleDirected.WellFormed","parent":32099,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle","status":"fully_proved","summary":"An instance $(n, E)$ of HAMILTONIAN CYCLE (DIRECTED), with $n$ a natural number and $E$ a finite list of ordered pairs of natural numbers, is well formed when $n > 0$, the list $E$ has no repeated entries, and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$.","title":"Well-formed instances of HAMILTONIAN CYCLE (DIRECTED)"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/hamiltonian-cycle-directed-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleDirected.HamiltonianCycleDirectedFeasible","parent":32099,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/index.html#hamiltonian-cycle-directed-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle","status":"fully_proved","summary":"An instance $(n, E)$ of HAMILTONIAN CYCLE (DIRECTED), with $n$ a natural number and $E$ a finite list of ordered pairs of natural numbers, has a directed Hamiltonian cycle when there is a list $(v_0, \\dots, v_{n-1})$ of $n$ distinct natural numbers containing every natural number below $n$ such that the pair $(v_j, v_{(j+1) \\bmod n})$ occurs in $E$ for every $j < n$.","title":"Directed Hamiltonian cycles of an instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/hamiltonian-cycle-directed-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleDirected.hamiltonianCycleDirectedDec","parent":32099,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/index.html#hamiltonian-cycle-directed-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle","status":"fully_proved","summary":"The decision problem HAMILTONIAN CYCLE (DIRECTED) accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance $(n, E)$, consisting of a natural number $n$ and a finite list $E$ of ordered pairs of natural numbers, such that $(n, E)$ is well formed ($n > 0$, $E$ has no repeated entries, and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$) and has a\u2026","title":"The directed Hamiltonian cycle decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/hamiltonian-cycle-directed-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleDirected.hamiltonianCycleDirected_npComplete","parent":32099,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle/index.html#hamiltonian-cycle-directed-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/directed-hamiltonian-cycle","status":"can_prove","summary":"The decision problem HAMILTONIAN CYCLE (DIRECTED) belongs to NP, and every decision problem $L$ in NP has a polynomial-time many-one reduction to it.","title":"HAMILTONIAN CYCLE (DIRECTED) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs","status":"fully_proved","title":"Hamiltonian path in directed graphs"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","title":"Hamiltonian paths to Witness encoding of a nonempty list"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/is-ham-path","kind":"node","lean":"Cslib.Complexity.Karp.IsHamPath","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#is-ham-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"Let $E$ be a list of pairs of natural numbers, viewed as the digraph on $0, \\dots, n - 1$ whose arcs are the pairs in $E$ with both ends below $n$. A list $p$ is a Hamiltonian path from $s$ to $t$ when it lists each of the vertices $0, \\dots, n - 1$ exactly once, starts at $s$, ends at $t$, and each two consecutive entries of $p$ form an arc.","title":"Hamiltonian paths"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/ham-path-dec","kind":"node","lean":"Cslib.Complexity.Karp.hamPathDec","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#ham-path-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"The decision problem that accepts a bit string $x$ exactly when, for some $s$, $t$, $n$ and list of pairs $E$, $x$ is the PATH encoding of $s$, $t$ and the digraph on $0, \\dots, n - 1$ with edge list $E$, and this digraph has a Hamiltonian path from $s$ to $t$.","title":"The HAMPATH problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/ham-vars","kind":"node","lean":"Cslib.Complexity.Karp.hamVars","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#ham-vars","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"The list of the distinct variables of a CNF formula $\\varphi$, in order of occurrence in $\\varphi$, with each variable kept at its last occurrence.","title":"Distinct variables of a formula"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/hp-t","kind":"node","lean":"Cslib.Complexity.Karp.hpT","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#hp-t","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"In the HAMPATH construction with $m$ clauses, the top of diamond $i$ is the vertex $m + i(3m + 2)$; it is also the bottom of diamond $i - 1$.","title":"Tops of the diamonds"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/hp-r","kind":"node","lean":"Cslib.Complexity.Karp.hpR","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#hp-r","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"In the HAMPATH construction with $m$ clauses, write $T(i)$ for the top of diamond $i$. Row vertex $l$ of diamond $i$ is the number $T(i) + 1 + l$; the row of diamond $i$ consists of the vertices with $l \\le 3m$.","title":"Row vertices of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/diamond-edges","kind":"node","lean":"Cslib.Complexity.Karp.diamondEdges","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#diamond-edges","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, and write $T(i)$ for the top of diamond $i$ and $R(i, l)$ for its row vertex $l$. The arcs of diamond $i$ for variable $x$ are $T(i) \\to R(i, 0)$, $T(i) \\to R(i, 3m)$, $R(i, 0) \\to T(i + 1)$ and $R(i, 3m) \\to T(i + 1)$; for every $l < 3m$, both $R(i, l) \\to R(i, l + 1)$ and $R(i, l + 1) \\to R(i, l)$; and for every clause $j < m$, the arcs $R(i, 3j + 1) \\to j$ and\u2026","title":"Arcs of a variable diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/ham-edges","kind":"node","lean":"Cslib.Complexity.Karp.hamEdges","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#ham-edges","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula and let $x_0, \\dots, x_{k-1}$ be its distinct variables, in the order of the distinct-variable list of $\\varphi$. The arcs of the construction for $\\varphi$ are the arcs of diamond $i$ for variable $x_i$, for $i = 0, \\dots, k - 1$, concatenated in this order.","title":"Arcs of the HAMPATH construction"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/to-ham","kind":"node","lean":"Cslib.Complexity.Karp.toHam","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#to-ham","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, numbered $0, \\dots, m - 1$, and let $x_0, \\dots, x_{k-1}$ be its distinct variables, in the order of their last occurrences in $\\varphi$ (clauses read in order, and the literals of each clause in order). Put $T_i = m + i(3m + 2)$ and $R_{i,l} = T_i + 1 + l$. The result is the encoded PATH instance with vertices $0, \\dots, T_k$, source $T_0$, target $T_k$, and, for\u2026","title":"Sipser's HAMPATH instance of a CNF formula"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/no-ham","kind":"node","lean":"Cslib.Complexity.Karp.noHam","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#no-ham","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"The encoded PATH instance on the two vertices $0$ and $1$, with no arcs, whose source and target are both $0$.","title":"A fixed HAMPATH no-instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/reduce-ham","kind":"node","lean":"Cslib.Complexity.Karp.reduceHam","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#reduce-ham","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"A bit string $x$ that decodes to a CNF formula $\\varphi$ is sent to Sipser's HAMPATH instance of $\\varphi$, and a bit string that does not decode to a CNF formula is sent to the fixed HAMPATH no-instance (two vertices, no arcs, source and target $0$).","title":"The SAT-to-HAMPATH reduction map"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/detours-normal","kind":"node","lean":"Cslib.Complexity.Karp.DetoursNormal","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#detours-normal","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, and write $R(i, l)$ for row vertex $l$ of diamond $i$ in the HAMPATH construction for $\\varphi$. A list of vertices $p$ has normal detours when for every clause $j < m$ there is a diamond $i < k$ such that $p$ contains $R(i, 3j + 1), j, R(i, 3j + 2)$ or $R(i, 3j + 2), j, R(i, 3j + 1)$ as consecutive entries.","title":"Normal clause detours"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/hp-t-mono","kind":"node","lean":"Cslib.Complexity.Karp.hpT_mono","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#hp-t-mono","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"In the HAMPATH construction with $m$ clauses, if $i \\le i'$ then the top of diamond $i$ is at most the top of diamond $i'$.","title":"Tops are monotone in the diamond index"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/hp-r-lt-hp-t-succ","kind":"node","lean":"Cslib.Complexity.Karp.hpR_lt_hpT_succ","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#hp-r-lt-hp-t-succ","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"In the HAMPATH construction with $m$ clauses, for every diamond $i$ and every $l \\le 3m$, row vertex $l$ of diamond $i$ is a smaller number than the top of diamond $i + 1$.","title":"Row vertices lie below the next top"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/edge-bits-cons","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.edgeBits_cons","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#edge-bits-cons","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"For natural numbers $u, v$ and a list $E$ of pairs of natural numbers, the bit encoding of the edge list whose first edge is $(u, v)$ and whose remaining edges are $E$ is: $u$ copies of true, then false, then $v$ copies of true, then false, then the bit encoding of $E$.","title":"Bits of an edge list, first edge first"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/edge-bits-nil","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.edgeBits_nil","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#edge-bits-nil","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"The edge-list bit encoding of the empty list of edges is the empty bit string.","title":"Edge bits of the empty edge list"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/edge-bits-append","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.edgeBits_append","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#edge-bits-append","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"For all lists $E$ and $E'$ of pairs of natural numbers, the bit encoding of the concatenation of $E$ and $E'$ is the bit encoding of $E$ followed by the bit encoding of $E'$.","title":"Bits of concatenated edge lists"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/enc-w","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.encW","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#enc-w","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"The witness encoding of a list $p$ of natural numbers (the vertices of a candidate path) is the bit string obtained by writing each entry $v$ of $p$, in order, as $v$ copies of true followed by one false, and concatenating the results.","title":"Witness encoding"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/decode-w","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.decodeW","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#decode-w","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"The witness decoder reads a list $w$ of bits as a list of natural numbers. On the empty list it returns the empty list. Otherwise it reads a unary number $v$ from the front of $w$, a run of $v$ copies of true ended by false, and if decoding the remaining bits gives a list $p$, it returns $v$ followed by $p$. It fails if no unary number can be read from a nonempty list, or if decoding the remaining bits fails.","title":"Witness decoder"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/enc-w-nil","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.encW_nil","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#enc-w-nil","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"The witness encoding of the empty list of vertices is the empty bit string.","title":"Witness encoding of the empty list"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/enc-w-cons","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.encW_cons","parent":32107,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01/index.html#enc-w-cons","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-01","status":"fully_proved","summary":"For a natural number $v$ and a list $p$ of natural numbers, the witness encoding of the list with first entry $v$ followed by $p$ is the bit string made of $v$ copies of true, then one false, then the witness encoding of $p$.","title":"Witness encoding of a nonempty list"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","title":"The witness encoding respects concatenation to Length of an encoded PATH instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/enc-w-append","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.encW_append","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#enc-w-append","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For lists $p$ and $p'$ of natural numbers, the witness encoding of the concatenation of $p$ and $p'$ is the witness encoding of $p$ followed by the witness encoding of $p'$.","title":"The witness encoding respects concatenation"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/decode-w-nil","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.decodeW_nil","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#decode-w-nil","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"The witness decoder succeeds on the empty list of bits and returns the empty list of vertices.","title":"Decoding the empty witness"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/decode-w-of-none","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.decodeW_of_none","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#decode-w-of-none","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"If a list $w$ of bits is nonempty and reading a unary number (a run of true ended by false) from the front of $w$ fails, that is, every bit of $w$ is true, then the witness decoder fails on $w$.","title":"Witness decoding fails on a cut-off first number"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/decode-w-of-some","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.decodeW_of_some","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#decode-w-of-some","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"If reading a unary number from the front of a list $w$ of bits returns $v$ with remaining bits $r$, that is, $w$ is $v$ copies of true, then false, then $r$, then the witness decoder succeeds on $w$ exactly when it succeeds on $r$, and its result on $w$ is $v$ followed by its result on $r$.","title":"Decoding peels off the first number"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/decode-w-enc-w","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.decodeW_encW","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#decode-w-enc-w","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For every list $p$ of natural numbers, the witness decoder applied to the witness encoding of $p$ (each entry $v$ written as $v$ copies of true followed by false) succeeds and returns $p$.","title":"Decoding inverts the witness encoding"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/enc-w-of-decode-w","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.encW_of_decodeW","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#enc-w-of-decode-w","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For every bit string $w$ and every list $p$ of natural numbers, if the witness decoder returns $p$ on $w$, then $w$ is the witness encoding of $p$.","title":"The witness decoder succeeds only on encodings"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/decode-w-eq-some","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.decodeW_eq_some","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#decode-w-eq-some","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For every list $w$ of bits and every list $p$ of natural numbers, the witness decoder returns $p$ on $w$ if and only if $w$ is the witness encoding of $p$, which writes each entry $v$ of $p$ as $v$ copies of true followed by false.","title":"The witness codec"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/length-enc-w","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.length_encW","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#length-enc-w","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For every list $p = (v_1, \\dots, v_k)$ of natural numbers, the witness encoding of $p$ has length $\\sum_{i=1}^{k} (v_i + 1)$.","title":"Witness length"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/length-enc-w-le","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.length_encW_le","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#length-enc-w-le","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For every natural number $n$ and every list $p$ of natural numbers all below $n$, the witness encoding of $p$ has length at most $|p| \\cdot n$.","title":"Witness length bound"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/length-le-length-enc-w","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.length_le_length_encW","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#length-le-length-enc-w","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For every list $p$ of natural numbers, the witness encoding of $p$ has length at least $|p|$.","title":"A witness is at least as long as its vertex list"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/chain-ok","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.chainOk","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#chain-ok","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"Given $n \\in \\mathbb{N}$ and a list $E$ of pairs of natural numbers, the arc check of a list $p$ of natural numbers returns true when every two consecutive entries $u, v$ of $p$ satisfy $(u, v) \\in E$, $u < n$ and $v < n$, and false otherwise; in particular every list with fewer than two entries passes.","title":"Arc check of a vertex list"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/chain-ok-iff","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.chainOk_iff","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#chain-ok-iff","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, every list $E$ of pairs of natural numbers and every list $p$ of natural numbers, the arc check of $p$ against $n$ and $E$ returns true if and only if every two consecutive entries $u, v$ of $p$ form an arc of the digraph on $0, \\dots, n - 1$ listed by $E$, that is, $(u, v) \\in E$, $u < n$ and $v < n$.","title":"Checking a vertex list against the arcs"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/ham-ok","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hamOk","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#ham-ok","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"Given a natural number $n$, an edge list $E$, vertices $s$ and $t$ and a list $p$ of natural numbers, the Boolean test that $p$ has no repeated entries, has length $n$, has all entries below $n$, has first entry $s$ and last entry $t$, and that every two consecutive entries $u, v$ of $p$ satisfy $(u, v) \\in E$, $u < n$ and $v < n$.","title":"The Hamiltonian path check"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/ham-ok-iff","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hamOk_iff","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#ham-ok-iff","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For all natural numbers $n$, $s$, $t$, every edge list $E$ and every list $p$ of natural numbers, the Hamiltonian path check on $(n, E, s, t, p)$ returns true if and only if $p$ is a Hamiltonian path from $s$ to $t$ in the digraph on $\\{0, \\dots, n-1\\}$ with edge list $E$.","title":"The Hamiltonian path check is correct"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/ham-check","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hamCheck","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#ham-check","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"Given bit strings $x$ and $w$, the verifier decodes $x$ as a PATH instance, consisting of a start vertex $s$, a target vertex $t$, a vertex count $n$ and an edge list $E$, and decodes $w$ as a witness, a list $p$ of natural numbers each written in unary. If both decodings succeed, it accepts exactly when the Hamiltonian path check passes, that is, when $p$ is a Hamiltonian path from $s$ to $t$ in the digraph on\u2026","title":"The HAMPATH verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/ham-check-iff","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hamCheck_iff","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#ham-check-iff","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For bit strings $x$ and $w$, the HAMPATH verifier accepts $(x, w)$ if and only if there are natural numbers $s$, $t$, $n$, a list $es$ of pairs of natural numbers and a list $p$ of natural numbers such that $x$ is the PATH encoding of $(s, t, n, es)$, $w$ is the witness encoding of $p$, and $p$ is a Hamiltonian path from $s$ to $t$ in the digraph on $0, \\dots, n - 1$ with edge list $es$.","title":"What the HAMPATH verifier accepts"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/ham-check-encode","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hamCheck_encode","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#ham-check-encode","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For natural numbers $s$, $t$, $n$, a list $es$ of pairs of natural numbers and a list $p$ of natural numbers, the HAMPATH verifier applied to the PATH encoding of $(s, t, n, es)$ and the witness encoding of $p$ returns the same boolean as the Hamiltonian path check on $n$, $es$, $s$, $t$ and $p$.","title":"The verifier on an encoded instance and witness"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/ham-check-of-decode-p","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hamCheck_of_decodeP","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#ham-check-of-decode-p","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"If the PATH instance decoder fails on a bit string $x$, then for every bit string $w$ the HAMPATH verifier rejects $(x, w)$, that is, returns false.","title":"Instances that do not decode are rejected"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/ham-check-of-decode-w","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hamCheck_of_decodeW","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#ham-check-of-decode-w","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"If the witness decoder fails on a bit string $w$, then for every bit string $x$ the HAMPATH verifier rejects $(x, w)$, that is, returns false.","title":"Witnesses that do not decode are rejected"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/length-encode-p","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.length_encodeP","parent":32128,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02/index.html#length-encode-p","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-02","status":"fully_proved","summary":"For all natural numbers $s$, $t$, $n$ and every edge list $E$, the encoding of the PATH instance $(s, t, n, E)$ has length $s + t + n + 3 + \\ell$, where $\\ell$ is the length of the bit encoding of the edge list $E$.","title":"Length of an encoded PATH instance"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","title":"HAMPATH is verified with quadratic-size witnesses to The frame of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-path-dec-iff","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hamPathDec_iff","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-path-dec-iff","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"For every bit string $x$, $x$ is a yes-instance of HAMPATH if and only if there is a bit string $w$ of length at most $(|x| + 1)^2$ such that the HAMPATH verifier accepts $(x, w)$.","title":"HAMPATH is verified with quadratic-size witnesses"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/step-ok","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.stepOk","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#step-ok","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Given natural numbers $s$ and $n$, an edge list $E$, a count $c$ of vertices already seen, the previous vertex $b$, the current vertex $a$ and the list $q$ of remaining vertices, the step check is true exactly when $a < n$, $a$ does not occur in $q$, and either $c = 0$ and $a = s$, or $c \\neq 0$ and $(b, a) \\in E$.","title":"One step of the HAMPATH walk"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/walk-ok","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.walkOk","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#walk-ok","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Fix natural numbers $s, t, n$ and a list $E$ of pairs of natural numbers. The walk check accepts a list $q$ of natural numbers after $c$ vertices, the last of which was $b$, by recursion on $q$. On the empty list it accepts if and only if $c = n$, $c \\neq 0$ and $b = t$. On a list with first entry $a$ and remaining entries $q'$ it accepts if and only if the step check passes and the walk check accepts $q'$ after\u2026","title":"The walk check of the HAMPATH verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/walk-ok-iff","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.walkOk_iff","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#walk-ok-iff","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $s, t, n$ be natural numbers and $E$ a list of pairs of natural numbers. For every list $q = (q_1, \\dots, q_k)$ of natural numbers and all natural numbers $c \\neq 0$ and $b$, the walk check for $s, t, n, E$ accepts $q$ after $c$ vertices the last of which was $b$ if and only if every $q_i$ is less than $n$, every pair $(u, v)$ of consecutive entries of the list $(b, q_1, \\dots, q_k)$ is an entry of $E$, the\u2026","title":"The walk check after the first vertex"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/walk-ok-eq","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.walkOk_eq","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#walk-ok-eq","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"For all natural numbers $s$, $t$, $n$, every edge list $E$ and every list $p$ of natural numbers, the walk over $p$ started with count $0$ and previous vertex $0$ returns the same Boolean value as the Hamiltonian path check on $(n, E, s, t, p)$.","title":"The machine's walk is the check"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/hp-t-sep","kind":"node","lean":"Cslib.Complexity.Karp.hpT_sep","parent":32149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tops of distinct diamonds are at least "],["math","\\(3m + 2\\)"],["text"," apart"]],"source":"Tops of distinct diamonds are at least $3m + 2$ apart"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#hp-t-sep","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"In the vertex numbering of the Hamiltonian path construction with $m$ clauses, where the top of diamond $i$ is the vertex $T(i) = m + i(3m + 2)$, any two indices $i \\ne i'$ satisfy $T(i) + 3m + 2 \\le T(i')$ or $T(i') + 3m + 2 \\le T(i)$.","title":"Tops of distinct diamonds are at least $3m + 2$ apart"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/mem-ham-edges","kind":"node","lean":"Cslib.Complexity.Karp.mem_hamEdges","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#mem-ham-edges","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with clauses $C_0, \\dots, C_{m-1}$ and distinct variables $x_0, \\dots, x_{k-1}$ (in the order the construction lists them), and write $T(i) = m + i(3m + 2)$ for the top of diamond $i$ and $R(i, l) = T(i) + 1 + l$ for its row vertex $l$. Then $(u, v)$ is an arc of the Hamiltonian path instance of $\\varphi$ if and only if there is an $i < k$ such that $(u, v)$ is one of the frame arcs\u2026","title":"Arcs of the Hamiltonian path construction"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-edges-cases","kind":"node","lean":"Cslib.Complexity.Karp.hamEdges_cases","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-edges-cases","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, and write $T_i = m + i(3m + 2)$ and $R_{i,l} = T_i + 1 + l$. If $(u, v)$ is an arc of Sipser's construction for $\\varphi$, then there is $i < k$ such that one of the following holds: $u = T_i$ and $v \\in \\{R_{i,0}, R_{i,3m}\\}$; or $u \\in \\{R_{i,0}, R_{i,3m}\\}$ and $v = T_{i+1}$; or there is $l < 3m$ with $(u, v) = (R_{i,l}, R_{i,l+1})$ or\u2026","title":"Classification of the arcs of Sipser's graph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-edges-to-hp-r","kind":"node","lean":"Cslib.Complexity.Karp.hamEdges_to_hpR","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-edges-to-hp-r","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, and write $T_i = m + i(3m + 2)$ and $R_{i,l} = T_i + 1 + l$. Let $i$ and $l \\le 3m$ be natural numbers, and let $(v, R_{i,l})$ be an arc of Sipser's construction for $\\varphi$. Then $l < 3m$ and $v = R_{i,l+1}$; or $0 < l$ and $v = R_{i,l-1}$; or $l \\in \\{0, 3m\\}$ and $v = T_i$; or $v < m$ and $l \\in \\{3v + 1, 3v + 2\\}$.","title":"In-neighbours of a row vertex in Sipser's graph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-edges-from-hp-r","kind":"node","lean":"Cslib.Complexity.Karp.hamEdges_from_hpR","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-edges-from-hp-r","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, and write $T_i = m + i(3m + 2)$ and $R_{i,l} = T_i + 1 + l$. Let $i$ and $l \\le 3m$ be natural numbers, and let $(R_{i,l}, v)$ be an arc of Sipser's construction for $\\varphi$. Then $l < 3m$ and $v = R_{i,l+1}$; or $0 < l$ and $v = R_{i,l-1}$; or $l \\in \\{0, 3m\\}$ and $v = T_{i+1}$; or $v < m$ and $l \\in \\{3v + 1, 3v + 2\\}$.","title":"Out-neighbours of a row vertex in Sipser's graph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-edges-to-clause","kind":"node","lean":"Cslib.Complexity.Karp.hamEdges_to_clause","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-edges-to-clause","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, and write $R_{i,l} = m + i(3m + 2) + 1 + l$ for row vertex $l$ of diamond $i$. If $j < m$ and $(v, j)$ is an arc of Sipser's construction for $\\varphi$, then there is $i < k$ with $v = R_{i,3j+1}$ or $v = R_{i,3j+2}$.","title":"In-neighbours of a clause vertex in Sipser's graph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-edges-from-clause","kind":"node","lean":"Cslib.Complexity.Karp.hamEdges_from_clause","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-edges-from-clause","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, and write $R_{i,l} = m + i(3m + 2) + 1 + l$ for row vertex $l$ of diamond $i$. If $j < m$ and $(j, v)$ is an arc of Sipser's construction for $\\varphi$, then there is $i < k$ with $v = R_{i,3j+1}$ or $v = R_{i,3j+2}$.","title":"Out-neighbours of a clause vertex in Sipser's graph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/mem-of-is-ham-path","kind":"node","lean":"Cslib.Complexity.Karp.mem_of_isHamPath","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#mem-of-is-ham-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"If $p$ is a Hamiltonian path from $s$ to $t$ in the digraph on the vertices $0, \\dots, n - 1$ with arc list $E$, then every vertex $v < n$ occurs in $p$.","title":"A Hamiltonian path visits every vertex"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-edges-of-infix","kind":"node","lean":"Cslib.Complexity.Karp.hamEdges_of_infix","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-edges-of-infix","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, let $T_i = m + i(3m + 2)$, and let $p$ be a Hamiltonian path from $T_0$ to $T_k$ in the digraph on the vertices $0, \\dots, T_k$ whose arcs are those of Sipser's construction for $\\varphi$. If $v$ comes immediately after $u$ in $p$, then $(u, v)$ is an arc of the construction.","title":"Consecutive vertices of a Hamiltonian path of Sipser's graph form arcs"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-path-row-end-false","kind":"node","lean":"Cslib.Complexity.Karp.hamPath_rowEnd_false","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-path-row-end-false","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m \\ge 1$ clauses and $k$ distinct variables, write $T_i = m + i(3m + 2)$ and $R_{i,l} = T_i + 1 + l$, and let $p$ be a Hamiltonian path from $T_0$ to $T_k$ in the digraph on the vertices $0, \\dots, T_k$ whose arcs are those of Sipser's construction for $\\varphi$. For $i < k$ and $l \\in \\{0, 3m\\}$, it is impossible that $R_{i,l}$ comes immediately after $T_i$ in $p$ and $T_{i+1}$\u2026","title":"A Hamiltonian path of Sipser's graph does not cross a diamond through one row end"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-path-separator-false","kind":"node","lean":"Cslib.Complexity.Karp.hamPath_separator_false","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-path-separator-false","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, write $T_i = m + i(3m + 2)$ and $R_{i,l} = T_i + 1 + l$, and let $p$ be a Hamiltonian path from $T_0$ to $T_k$ in the digraph on the vertices $0, \\dots, T_k$ whose arcs are those of Sipser's construction for $\\varphi$. Let $i < k$, let $l \\le 3m$ be a multiple of $3$, and let $l' \\le 3m$ satisfy $l' = l + 1$ or $l' + 1 = l$. Then $R_{i,l'}$\u2026","title":"A separator is next to each of its row neighbours on a Hamiltonian path"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/ham-path-pair-false","kind":"node","lean":"Cslib.Complexity.Karp.hamPath_pair_false","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#ham-path-pair-false","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, write $T_i = m + i(3m + 2)$ and $R_{i,l} = T_i + 1 + l$, and let $p$ be a Hamiltonian path from $T_0$ to $T_k$ in the digraph on the vertices $0, \\dots, T_k$ whose arcs are those of Sipser's construction for $\\varphi$. Let $i < k$ and $j < m$, and let $(l_a, l_c)$ be $(3j + 1, 3j + 2)$ or $(3j + 2, 3j + 1)$. If the clause vertex $j$ comes\u2026","title":"A clause detour returns to its own pair (Sipser's separator argument)"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/detours-normal-of-is-ham-path","kind":"node","lean":"Cslib.Complexity.Karp.detoursNormal_of_isHamPath","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#detours-normal-of-is-ham-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, and write $T_i = m + i(3m + 2)$ and $R_{i,l} = T_i + 1 + l$ for the top and the row vertices of diamond $i$. Every Hamiltonian path from $T_0$ to $T_k$ in the digraph on the vertices $0, \\dots, T_k$ whose arcs are those of Sipser's construction for $\\varphi$ has normal detours: for every clause $j < m$ there is $i < k$ such that\u2026","title":"Hamiltonian paths of Sipser's graph have normal detours"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/to-ham-eq","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.toHam_eq","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#to-ham-eq","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, and let $t = m + k(3m + 2)$ be the top of diamond $k$. Then the HAMPATH instance of $\\varphi$ is the bit string made of $m$ ones and a zero (the start vertex $s = m$), then $t$ ones and a zero (the target vertex $t$), then $t + 1$ ones and a zero (the vertex count $n = t + 1$), followed by the arcs of the construction, each arc $(u, v)$\u2026","title":"The HAMPATH instance as unary blocks and arc bits"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/frame-e","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.frameE","parent":32149,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03/index.html#frame-e","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-03","status":"fully_proved","summary":"For natural numbers $m$ (the number of clauses) and $i$, write $t_i = m + i(3m + 2)$ for the top of diamond $i$ and $r_{i,l} = t_i + 1 + l$ for its row vertex $l$. The frame of diamond $i$ is the list of the four arcs $(t_i, r_{i,0})$, $(t_i, r_{i,3m})$, $(r_{i,0}, t_{i+1})$ and $(r_{i,3m}, t_{i+1})$: the top points to both row ends, and both row ends point to the top of the next diamond.","title":"The frame of a diamond"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","title":"The row arcs of a diamond to Label after a HAMPATH copy site"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/row-e","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.rowE","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#row-e","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"For natural numbers $m$ (the number of clauses), $i$ and $l$, let $r_{i,l} = m + i(3m + 2) + 1 + l$ be row vertex $l$ of diamond $i$. The row arcs between row vertices $l$ and $l + 1$ of diamond $i$ are the list of the two arcs $(r_{i,l}, r_{i,l+1})$ and $(r_{i,l+1}, r_{i,l})$.","title":"The row arcs of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/det-e","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.detE","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#det-e","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"For natural numbers $m$ (the number of clauses), $i$, $x$ and $j$, let $r_{i,l} = m + i(3m + 2) + 1 + l$ be row vertex $l$ of diamond $i$. For a clause $C$, a list of literals, the detours of clause $j$ with contents $C$ in diamond $i$ of variable $x$ are the list of the arcs $(r_{i,3j+1}, j)$ and $(j, r_{i,3j+2})$ if the positive literal $x$ is in $C$, followed by the arcs $(r_{i,3j+2}, j)$ and $(j, r_{i,3j+1})$\u2026","title":"Clause detours in a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/hp-t-succ-eq-hp-r","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.hpT_succ_eq_hpR","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#hp-t-succ-eq-hp-r","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"For natural numbers $m$ (the number of clauses) and $i$, write $t_i = m + i(3m + 2)$ for the top of diamond $i$ and $r_{i,l} = t_i + 1 + l$ for its row vertex $l$. Then $t_{i+1} = r_{i,3m} + 1$: the top of diamond $i + 1$ comes right after the last row vertex of diamond $i$.","title":"The next diamond top follows the row"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/range-flat-map-get-d","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.range_flatMap_getD","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#range-flat-map-get-d","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"Let $P$ and $Q$ be CNF formulas, $C$ a clause, and $\\varphi$ the formula whose clauses are those of $P$, then $C$, then those of $Q$. Write $|P|$ for the number of clauses of $P$ and $\\varphi_j$ for clause $j$ of $\\varphi$, counting from $0$ (the empty clause if $j$ is out of range). Let $f$ map each natural number $j$ and clause to a list of elements of a type $\\alpha$. Then concatenating the lists\u2026","title":"Splitting off the middle clause of a concatenation"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/ins-v","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.insV","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#ins-v","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"For a natural number $x$ and a list $V$ of natural numbers, inserting $x$ into $V$ gives $V$ itself if $x$ occurs in $V$, and otherwise the list obtained by prepending $x$ to $V$.","title":"Insert a variable if new"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/foldr-ins-v","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.foldr_insV","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#foldr-ins-v","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"For every list $l$ of natural numbers, folding $l$ from the right into the empty list, prepending each element unless it is already present, gives $l$ with duplicates removed: each element is kept at its last occurrence in $l$, in order.","title":"Removing duplicates by insertion"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/hp-r-succ","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.hpR_succ","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#hp-r-succ","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"In the HAMPATH construction with $m$ clauses, for every diamond $i$ and every $l \\in \\mathbb{N}$, row vertex $l + 1$ of diamond $i$ is one more than row vertex $l$ of diamond $i$.","title":"Consecutive row vertices"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/edge-bits-length-le","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.edgeBits_length_le","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#edge-bits-length-le","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"Let $B$ be a natural number and $E$ a list of pairs of natural numbers such that every pair $(u, v)$ in $E$ has $u \\le B$ and $v \\le B$. Then the bit encoding of $E$, which writes each pair $(u, v)$ as $u$ ones, a zero, $v$ ones and a zero, has length at most $|E| \\cdot (2B + 2)$, where $|E|$ is the length of $E$.","title":"Length bound for the bit encoding of an edge list"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/k","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.K","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#k","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The emitter has eighteen distinct stacks: $T$ (the tokens), $O$ (the output), $F$ (the tokens in reading order), $V$ (the distinct variables), $W$ (the other side of the walk over $V$), $Q$ (the edges in progress), $G$ (garbage), $X$ and $Y$ (comparison scratch), $C$ (the clause count), $M$ (three times the clause count), $P$ (the diamond top), $N$ (the row end), $R$ (the row vertex), $L$ (the row loop index), $J$\u2026","title":"Stacks of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/untitled","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.\u0393","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#untitled","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The alphabet of the output stack $O$ is the set of bits (booleans); every other stack has the token symbols as its alphabet.","title":"Stack alphabets of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/src","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Src","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#src","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The type with the five elements $p, n, r, j, d$, naming the counters of the HAMPATH emitter that a copy site can read: the diamond top $p$, the row end $n$, the row vertex $r$, the clause index $j$ and the detour vertex $d$.","title":"Counters read by HAMPATH copy sites"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/k-2","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Src.k","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#k-2","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The counters $p$, $n$, $r$, $j$, $d$ that a copy site of the HAMPATH emitter reads are held on the emitter's stacks $P$, $N$, $R$, $J$, $D$ respectively (the diamond top, the row end, the row vertex, the clause index and the detour vertex).","title":"Stack of a HAMPATH copy-site counter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/unit","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Src.unit","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#unit","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"For every counter $s$ read by copy sites, the unit symbol of $s$ is the token symbol $\\mathrm{cl}$, as a letter of the alphabet of the stack that holds $s$.","title":"Unit symbol of an emitter counter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/site","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Site","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#site","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The type of the twenty copy sites of the emitter machine. Each site writes one endpoint of an edge of the emitted HAMPATH instance, $u$ for the source and $v$ for the target: the sites $f_iu, f_iv$ for $i = 1, \\dots, 4$ belong to the four frame edges, $w_iu, w_iv$ for $i = 1, 2$ to the two row edges, and $d_iu, d_iv$ for $i = 1, \\dots, 4$ to the detour edges, positive for $i = 1, 2$ and negative for $i = 3, 4$.","title":"Copy sites of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/all","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Site.all","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#all","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The list of the twenty copy sites of the HAMPATH emitter, in this order: the frame sites $f_1u, f_1v, f_2u, f_2v, f_3u, f_3v, f_4u, f_4v$, the row sites $w_1u, w_1v, w_2u, w_2v$, and the detour sites $d_1u, d_1v, d_2u, d_2v, d_3u, d_3v, d_4u, d_4v$.","title":"Copy sites of the HAMPATH emitter, listed"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/src-2","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Site.src","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#src-2","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The map from copy sites to counters sending $f_1u, f_1v, f_2u, f_3u$ to the diamond top $p$; $f_2v, f_3v, f_4u, f_4v$ to the row end $n$; $w_1u, w_1v, w_2u, w_2v$ to the row vertex $r$; $d_1u, d_2v, d_3u, d_4v$ to the detour vertex $d$; and $d_1v, d_2u, d_3v, d_4u$ to the clause index $j$.","title":"Counter read by a copy site"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/ext","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Site.ext","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#ext","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The number of extra ones a copy site adds to its counter's value is $1$ for the sites $f_{1v}$, $f_{3u}$, $f_{3v}$, $f_{4v}$, $w_{1v}$, $w_{2u}$, $d_{2v}$ and $d_{3u}$, and $0$ for every other site.","title":"Extra ones of a copy site"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/lab","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Lab","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#lab","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"Let $f$, $p_p$, $p_n$ and $b$ range over bits and $j$ over copy sites. A label of the emitter is one of $\\mathrm{a0}$, $\\mathrm{anm}$, $\\mathrm{aw}(f)$, $\\mathrm{ac}(f)$, $\\mathrm{ar1}(f)$, $\\mathrm{ar2}(f)$, $\\mathrm{amv}(f)$, $\\mathrm{asep}(f)$, $\\mathrm{ab}(f)$, $\\mathrm{ad2}$, $\\mathrm{ad3}$, $\\mathrm{b0}$, $\\mathrm{b1}$, $\\mathrm{h0}, \\dots, \\mathrm{h5}$, $\\mathrm{r0}, \\dots, \\mathrm{r4}$, $\\mathrm{wi}$,\u2026","title":"Labels of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/bb","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.bb","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#bb","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"The list of all pairs of Booleans in the order $(\\mathrm{true}, \\mathrm{true})$, $(\\mathrm{true}, \\mathrm{false})$, $(\\mathrm{false}, \\mathrm{true})$, $(\\mathrm{false}, \\mathrm{false})$.","title":"The four pairs of Booleans"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/nxt","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Site.nxt","parent":32170,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04/index.html#nxt","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-04","status":"fully_proved","summary":"For a copy site $j$ and a bit $b$, the label that follows $j$ is $\\mathrm{cpA}(j', b)$, where $j'$ is the site after $j$ in one of the chains $f_{1u}, f_{1v}, f_{2u}, f_{2v}, f_{3u}, f_{3v}, f_{4u}, f_{4v}$, or $w_{1u}, w_{1v}, w_{2u}, w_{2v}$, or $d_{1u}, d_{1v}, d_{2u}, d_{2v}$, or $d_{3u}, d_{3v}, d_{4u}, d_{4v}$. At the ends of the chains it is $\\mathrm{r0}$ after $f_{4v}$, $\\mathrm{wi}$ after $w_{2v}$ and\u2026","title":"Label after a HAMPATH copy site"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","title":"End of a clause in the detour walk to HAMPATH emitter, phase A: copying a reversed name onto two stacks"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/cl-end","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.clEnd","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#cl-end","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"For booleans $p_p$ and $p_n$ (the flags recording whether the current clause contains the current variable positively, respectively negatively), the detour walk continues at the end of the clause at the label where the first positive detour site $d_{1u}$ starts with passed bit $p_n$ if $p_p$ is true; otherwise, if $p_n$ is true, at the label where the first negative detour site $d_{3u}$ starts with passed bit\u2026","title":"End of a clause in the detour walk"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/prog","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.prog","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#prog","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"The emitter is the multi-stack machine with the emitter's stacks (the output stack $O$ holds bits, every other stack holds token symbols), the emitter's labels, and a boolean internal state. Below, *moving the bit block* of a stack $A$ onto a stack $B$ means popping $A$ while its top is a name bit $\\mathrm{bit}(b)$ and pushing each popped bit onto $B$ (as $\\mathrm{bit}(b)$, or as the bit $b$ when $B = O$).\u2026","title":"The HAMPATH emitter machine"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/start","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.start","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#start","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"The emitter machine starts at the label $a_0$, the first phase A label, which reads the next token.","title":"Start label of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/ms","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.MS","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#ms","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"A quiescent state consists of lists of token symbols $t$, $f$, $v$, $w$, $g$, $x$ (meant as the token stack, the tokens in reading order, the distinct variables, the other side of the variable walk, garbage, and the first comparison scratch stack), a list of bits $o$ (the output stack), a list of bits $q$ (the edge bits so far, in reading order), and natural numbers $c$, $c_3$, $p$, $n$, $r$, $l$, $j$, $d$ (meant\u2026","title":"Quiescent states of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/stk","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.MS.stk","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#stk","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"In a quiescent state $s$, the stacks $T$, $O$, $F$, $V$, $W$, $G$ and $X$ hold the lists that $s$ records for them; the stack $Q$ holds the edge bits of $s$ as bit symbols, reversed so that the last bit is on top; each counter stack $C$, $M$, $P$, $N$, $R$, $L$, $J$, $D$ holds as many copies of the symbol $\\mathrm{cl}$ as the value that $s$ records for the corresponding counter (the clause count, the counter\u2026","title":"Stacks of a quiescent emitter state"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/val","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.MS.val","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#val","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"For a quiescent state $s$, the value of the counters $p$, $n$, $r$, $j$, $d$ is respectively the diamond top, the row end, the row vertex, the clause index and the detour vertex recorded in $s$.","title":"Counter values of a quiescent emitter state"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/cp-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.cp_run","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#cp-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"Let $a$, $c$ and $Z$ be three distinct stacks of the emitter machine (with $Z$ its copy scratch stack), $u$ a symbol of $a$ and $x$ a symbol of $c$. Suppose the machine at label $l_1$ pops $a$ until it is empty, pushing $x$ onto $c$ and the clause separator $\\mathrm{cl}$ onto $Z$ for each popped symbol, and then continues at $l_2$; and at $l_2$ pops $Z$ until it is empty, pushing $u$ onto $a$ for each popped\u2026","title":"Copying a counter in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/atom","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.atom","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#atom","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"For a quiescent state $s$ of the HAMPATH emitter and a copy site $j$, the atom of $j$ in $s$ is the list of $c + e$ copies of true followed by one false, where $c$ is the value in $s$ of the counter that $j$ copies and $e \\in \\{0, 1\\}$ is the number of extra ones that $j$ adds.","title":"Bits pushed by a copy site of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/site-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.site_run","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#site-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"Let $j$ be a copy site, $b$ and $v$ bits and $s$ a quiescent state of the HAMPATH emitter; let $c$ be the value in $s$ of the counter that $j$ reads and $e$ the number of ones that $j$ adds to its counter. Started at label $\\mathrm{cpA}(j, b)$ with internal state $v$ and the stacks of $s$, the emitter runs in exactly $2c + 3$ steps to the label that follows $j$ for $b$, with some internal state and the stacks of\u2026","title":"Run of a HAMPATH copy site"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/chain","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.Chain","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#chain","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"Copy sites $j_1, \\dots, j_n$ are chained from label $l_0$ to label $l$ with passed bit $b$ when $l_0$ is the label at which $j_1$ starts with passed bit $b$, the label where each $j_i$ with $i < n$ continues with passed bit $b$ is the label at which $j_{i+1}$ starts with passed bit $b$, and $j_n$ continues at $l$ with passed bit $b$. The empty list of sites is chained from $l_0$ to $l$ exactly when $l_0 = l$.","title":"Copy sites chained in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/chain-cost","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.chainCost","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#chain-cost","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"For a quiescent state $s$ and copy sites $j_1, \\dots, j_n$, the step count is $\\sum_{i=1}^{n} (2 v_i + 3)$, where $v_i$ is the value in $s$ of the counter that $j_i$ copies.","title":"Step count of a list of copy sites in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/chain-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.chain_run","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#chain-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"Let $j_1, \\dots, j_n$ be copy sites chained from label $l_0$ to label $l$ with passed bit $b$. Then for every quiescent state $s$ and every flag $v$ there is a flag $v'$ such that the emitter machine goes, in exactly as many steps as the step count of $j_1, \\dots, j_n$ at $s$, from label $l_0$ with flag $v$ and the stacks of $s$ to label $l$ with flag $v'$ and the stacks of the state that differs from $s$ only in\u2026","title":"Running a chain of copy sites in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/bit-inv","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.bit_inv","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#bit-inv","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"For every symbol $x$ of the stack alphabet of the SAT verifier and every bit $b$, if $x$ reads as the name bit $b$, then $x$ is the name-bit symbol for $b$.","title":"A symbol that reads as a name bit is that bit"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/cmp-r-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.cmpR_run","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#cmp-r-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"Let $l_c$ be a label at which the emitter machine compares the bit blocks of the stacks $F$ and $V$, with scratch stacks $X$ and $Y$, going to $r_1(b_T)$ if the blocks are equal and to $r_1(b_F)$ otherwise. Suppose that for each bit $b$ the machine at $r_1(b)$ moves the bit block of $X$ onto $F$ and then continues at $r_2(b)$, and at $r_2(b)$ moves the bit block of $Y$ onto $V$ and then continues at $n_x(b)$. Let\u2026","title":"Comparing a name with the top entry of the variable stack"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/name","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.name","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#name","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"The name of a variable $x \\in \\mathbb{N}$ is the list of name-bit symbols $\\mathrm{bit}(b_0), \\dots, \\mathrm{bit}(b_{k-1})$, where $b_0 \\cdots b_{k-1}$ is the binary encoding of $x$, least significant bit first (the empty list for $x = 0$).","title":"Name of a variable in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/vstk","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.vstk","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#vstk","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"For a list $x_1, \\dots, x_k$ of variables (natural numbers), the variable stack is the list of symbols obtained by concatenating, for $i = 1, \\dots, k$ in order, the name bits of $x_i$ followed by the separator symbol $\\mathrm{cl}$. With the head of the list on top, $x_1$ is the top entry and each name sits above its separator.","title":"Variable stack of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/is-vs","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.isVS","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#is-vs","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"A token symbol is a variable-stack symbol when it is a name bit $\\mathrm{bit}(b)$ for some bit $b$ or the separator $\\mathrm{cl}$; the polarity markers, the value markers and the dead marker are not.","title":"Variable-stack symbols of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/ab-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.ab_run","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#ab-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"Let $f$ be a Boolean, let $L$ be a list of symbols each of which is a name bit or the separator $\\mathrm{cl}$, and let $s$ be a quiescent state of the HAMPATH emitter whose stack $W$ is $L$. Then for every value $v$ of the machine's Boolean variable there is a value $v'$ such that the emitter goes in exactly $|L|$ steps from label $\\mathrm{ab}(f)$, variable $v$ and the stacks of $s$ to label $\\mathrm{ab}(f)$,\u2026","title":"HAMPATH emitter, phase A: moving the walk stack back onto the variable stack"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/aw-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.aw_run","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#aw-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"For $y \\in \\mathbb{N}$ write $\\mathrm{name}(y)$ for the canonical binary encoding of $y$ as name-bit symbols, and for a list $\\mathit{vs}$ of naturals write $\\mathrm{vstk}(\\mathit{vs})$ for the variable stack: the names of the entries of $\\mathit{vs}$, each followed by the separator $\\mathrm{cl}$. Let $x \\in \\mathbb{N}$ and let $A$ be a list of symbols that is empty or does not start with a name bit. Then for every\u2026","title":"HAMPATH emitter, phase A: searching the variable stack for a name"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/ad3-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.ad3_run","parent":32191,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05/index.html#ad3-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-05","status":"fully_proved","summary":"Let $u$ be a list of bits and let $s$ be a quiescent state of the HAMPATH emitter whose stack $X$ holds $u$ as name-bit symbols. Then for every value $v$ of the machine's Boolean variable there is a value $v'$ such that the emitter goes in exactly $|u| + 1$ steps from label $\\mathrm{ad3}$, variable $v$ and the stacks of $s$ to label $\\mathrm{a0}$, variable $v'$ and the stacks of the state obtained from $s$ by\u2026","title":"HAMPATH emitter, phase A: copying a reversed name onto two stacks"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","title":"HAMPATH emitter, phase A: one literal to Moving the tokens back in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/alit-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.alit_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#alit-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"For $x \\in \\mathbb{N}$ write $\\mathrm{name}(x)$ for the canonical binary encoding of $x$ as name-bit symbols. Let $\\ell = (x, p)$ be a literal with variable $x$ and polarity $p$: its tokens are $\\mathrm{name}(x)$ followed by the polarity marker $\\mathrm{lit}(p)$, and its reversed tokens $r_\\ell$ are $\\mathrm{lit}(p)$ followed by the reverse of $\\mathrm{name}(x)$. For a list $\\mathit{vs}$ of naturals write\u2026","title":"HAMPATH emitter, phase A: one literal"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/alits-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.alits_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#alits-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Notation: $\\mathrm{name}(x)$ is the canonical binary encoding of $x \\in \\mathbb{N}$ as name-bit symbols; a literal $\\ell = (x, p)$ has tokens $\\mathrm{name}(x)$ followed by the polarity marker $\\mathrm{lit}(p)$ and reversed tokens $r_\\ell$, namely $\\mathrm{lit}(p)$ followed by the reverse of $\\mathrm{name}(x)$; $\\mathrm{vstk}(\\mathit{vs})$ is the list of the names of the entries of a list $\\mathit{vs}$ of naturals,\u2026","title":"HAMPATH emitter, phase A: the literals of a clause"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/aclause-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.aclause_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#aclause-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Notation: $\\mathrm{name}(x)$ is the canonical binary encoding of $x \\in \\mathbb{N}$ as name-bit symbols; a literal $\\ell = (x, p)$ has tokens $\\mathrm{name}(x)$ followed by the polarity marker $\\mathrm{lit}(p)$ and reversed tokens $r_\\ell$, namely $\\mathrm{lit}(p)$ followed by the reverse of $\\mathrm{name}(x)$; $\\mathrm{vstk}(\\mathit{vs})$ is the list of the names of the entries of a list $\\mathit{vs}$ of naturals,\u2026","title":"HAMPATH emitter, phase A: one clause"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/aformula-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.aformula_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#aformula-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Notation: $\\mathrm{name}(x)$ is the canonical binary encoding of $x \\in \\mathbb{N}$ as name-bit symbols; a literal $\\ell = (x, p)$ has tokens $\\mathrm{name}(x)$ followed by the polarity marker $\\mathrm{lit}(p)$; a clause $C$ with literals $\\ell_1, \\dots, \\ell_k$ has tokens the concatenated tokens of $\\ell_1, \\dots, \\ell_k$ followed by the separator $\\mathrm{cl}$, and reversed tokens $r_C$, namely $\\mathrm{cl}$\u2026","title":"HAMPATH emitter, phase A: the clauses of a formula"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/init-ms","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.initMS","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#init-ms","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"For a CNF formula $\\varphi$ and an output list $o$, the initial state is the quiescent state whose token stack holds the tokens of $\\varphi$ in reverse reading order, whose output stack is $o$, and all of whose other stacks are empty and counters zero.","title":"Initial state of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/dia-ms","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.diaMS","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#dia-ms","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"For a CNF formula $\\varphi$ with $m$ clauses and an output list $o$, this is the quiescent state whose output stack is $o$, whose stack $F$ holds the tokens of $\\varphi$ in reading order, whose stack $V$ is the variable stack of the distinct variables of $\\varphi$ (each kept at its last occurrence), whose clause count and diamond top are both $m$, whose count of three times the clauses is $3m$, and all of whose\u2026","title":"The emitter state before the diamonds"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/phase-a-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.phaseA_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#phase-a-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses whose token list has length $L$, let $o$ be a list of bits and let $v$ be a bit. Started at label $\\mathrm{a0}$ with internal state $v$ and the stacks of the initial emitter state of $\\varphi$ and $o$ (the token stack holding the tokens of $\\varphi$ reversed, the output stack holding $o$, and everything else empty), the emitter reaches, within at most\u2026","title":"Phase A of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/is-tok","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.isTok","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#is-tok","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"A symbol is a token when it is a name bit, a polarity marker or the clause separator $\\mathsf{cl}$; the value markers and the dead marker are not tokens. The emitter moves tokens between its stacks $T$ and $F$.","title":"Token symbols of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/lit-toks-head","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.litToks_head","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#lit-toks-head","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"For every literal $(n, p)$ with variable index $n \\in \\mathbb{N}$ and polarity $p$, the token list of $(n, p)$ is nonempty and its first symbol is not the clause-end marker $\\mathsf{cl}$.","title":"The tokens of a literal do not start with a clause end"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/dlit-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.dlit_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#dlit-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $y \\in \\mathbb{N}$ have binary encoding $\\mathrm{bin}(y)$, let $l = (z, b)$ be a literal with token list $\\tau$, let $\\rho$ and $\\nu$ be lists of symbols, let $p_+$, $p_-$ and $v$ be bits, and let $s$ be a quiescent state of the HAMPATH emitter such that $F$ holds $\\tau$ followed by $\\rho$, $V$ holds the name of $y$, then the clause symbol $\\mathrm{cl}$, then $\\nu$, and the scratch stack $X$ is empty. Then\u2026","title":"One literal of the detour walk"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/dlits-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.dlits_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#dlits-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $y \\in \\mathbb{N}$ have binary encoding $\\mathrm{bin}(y)$ and let $\\nu$ and $\\rho$ be lists of symbols. For every list $C$ of literals with token list $\\tau$ (the tokens of its literals in order, without a clause separator), all bits $p_+, p_-, b$, and every quiescent state $s$ of the emitter such that $F$ holds $\\tau$ followed by $\\rho$, $V$ holds the name of $y$, then a clause separator, then $\\nu$, and the\u2026","title":"The literals of a clause in the detour walk"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/dclause-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.dclause_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#dclause-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $B, m, i, y, j \\in \\mathbb{N}$, let $C$ be a clause with token list $\\mathrm{toks}(C)$, let $\\rho$ and $\\nu$ be lists of symbols and $b$ a state bit, and let $s$ be a quiescent state of the emitter such that $F$ holds $\\mathrm{toks}(C)$ followed by $\\rho$, $V$ holds the name of $y$, then a clause separator, then $\\nu$, the scratch stack $X$ is empty, the counter $J$ equals $j$, the counter $D$ equals row vertex\u2026","title":"One clause of the detour walk"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/dwalk-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.dwalk_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#dwalk-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $B$, $m$, $i$ and $y$ be natural numbers and $\\varphi = C_0 \\cdots C_{m-1}$ a CNF formula with $m$ clauses. Write $\\tau = m + i(3m+2)$ for the top vertex of diamond $i$ and $\\rho_l = \\tau + 1 + l$ for its row vertices, encode an arc $(a, b)$ by the bits $1^a 0 1^b 0$, write $\\mathrm{toks}(\\psi)$ for the token sequence of a list of clauses $\\psi$, and call the bits of the binary code $\\mathrm{enc}(y)$ of $y$, as\u2026","title":"The detour walk of a diamond in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/frame-sites","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.frameSites","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#frame-sites","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"The list of the eight copy sites of a diamond's frame arcs, in order: the source site and then the target site of the first, second, third and fourth frame arc.","title":"Frame copy sites of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/row-sites","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.rowSites","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#row-sites","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"The list of the four copy sites of the two row arcs between consecutive row vertices of a diamond, in order: the source site and then the target site of the first and of the second row arc.","title":"Row copy sites of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/dfront-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.dfront_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#dfront-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $B, m, i \\in \\mathbb{N}$, let $b$ be a state bit, and let $s$ be a quiescent state of the emitter whose stack $V$ is nonempty, whose counter $P$ equals the top of diamond $i$ for $m$ clauses, whose counter $M$ equals $3m$, whose counter $N$ is $0$, and whose counters $R$, $J$, $D$ are at most $B$, and suppose row vertex $3m$ of diamond $i$ is at most $B$. Then for some state bit $b'$ the emitter goes in at most\u2026","title":"Front of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/drow-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.drow_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#drow-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $B$, $m$ and $i$ be natural numbers, write $\\tau = m + i(3m+2)$ for the top vertex of diamond $i$ and $\\rho_l = \\tau + 1 + l$ for its row vertices, and encode an arc $(a, b)$ by the bits $1^a 0 1^b 0$. Let $s$ be a quiescent state of the emitter whose counters satisfy $M = 3m$, $L = 0$, $R = \\rho_0$, $P \\le B$, $N \\le B$, $J \\le B$ and $D \\le B$, and suppose $\\rho_{3m} \\le B$ and $3m \\le B$. Then for every\u2026","title":"The row loop of a diamond in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/add-u-nil","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.addU_nil","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#add-u-nil","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"For every $i \\in \\mathbb{N}$ and all stack contents $S$ of the HAMPATH emitter, adding $i$ units to the stacks named in the empty list of (stack, symbol) pairs leaves $S$ unchanged: $\\mathrm{addU}([\\,], i, S) = S$.","title":"Adding units along no stacks"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/drain-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.drain_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#drain-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $k$ be a stack of the emitter and $\\ell, \\ell'$ labels such that the instruction at $\\ell$ is the transfer loop that pops stack $k$ until it is empty, pushing nothing, and then goes to $\\ell'$. If $S$ is an assignment of stack contents in which stack $k$ consists of $n$ copies of a symbol $u$, then for every state bit $b$ the emitter goes in exactly $n + 1$ steps from label $\\ell$ with state bit $b$ and stacks\u2026","title":"Draining a counter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/t0-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.t0_run","parent":32212,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06/index.html#t0-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-06","status":"fully_proved","summary":"Let $L$ be a list of symbols that are all tokens, $s$ a quiescent state of the emitter whose stack $T$ holds $L$, and $v$ an internal bit. Then there is an internal bit $v'$ such that the emitter goes in exactly $|L| + 1$ steps from label $\\mathsf{t0}$ with bit $v$ and the stacks of $s$ to label $\\mathsf{t1}$ with bit $v'$ and the stacks of the state obtained from $s$ by emptying $T$ and replacing $F$ by $L$\u2026","title":"Moving the tokens back in the HAMPATH emitter"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","title":"The row of a diamond in the HAMPATH emitter to A row starts at its left end"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/drow-full-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.drowFull_run","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#drow-full-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $B$, $m$ and $i$ be natural numbers, write $\\tau = m + i(3m+2)$ for the top vertex of diamond $i$ and $\\rho_l = \\tau + 1 + l$ for its row vertices, and encode an arc $(a, b)$ by the bits $1^a 0 1^b 0$. Let $s$ be a quiescent state of the emitter whose counters satisfy $P = \\tau$, $M = 3m$, $L = 0$, $R = 0$, $N \\le B$, $J \\le B$ and $D \\le B$, and suppose $\\rho_{3m} \\le B$. Then for every internal bit $v$ there\u2026","title":"The row of a diamond in the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/ddet-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.ddet_run","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#ddet-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $B, i, y \\in \\mathbb{N}$, let $\\varphi$ be a CNF formula with $m$ clauses and token list $\\mathrm{toks}(\\varphi)$, let $\\nu$ be a list of symbols and $b$ a state bit, and let $s$ be a quiescent state of the emitter such that $F$ holds $\\mathrm{toks}(\\varphi)$, $T$ and the scratch stack $X$ are empty, $V$ holds the name of $y$, then a clause separator, then $\\nu$, the counter $P$ equals the top of diamond $i$\u2026","title":"The detours of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/dback-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.dback_run","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#dback-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with token list $\\mathrm{toks}(\\varphi)$, let $y \\in \\mathbb{N}$ have binary encoding $\\mathrm{bin}(y)$, let $\\nu$ be a list of symbols and $b$ a state bit, and let $s$ be a quiescent state of the emitter whose stack $T$ holds $\\mathrm{toks}(\\varphi)$ reversed, whose stack $F$ is empty, and whose stack $V$ holds the name of $y$, then a clause separator, then $\\nu$. Let $p, n, r, j, d$\u2026","title":"End of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/diamond-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.diamond_run","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#diamond-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $B, i, y \\in \\mathbb{N}$, let $u$ be a list of natural numbers, let $\\varphi$ be a CNF formula with $m$ clauses and token list $\\mathrm{toks}(\\varphi)$, let $b$ be a state bit, and let $s$ be a quiescent state of the emitter such that $T$ and the scratch stack $X$ are empty, $F$ holds $\\mathrm{toks}(\\varphi)$, $V$ is the variable stack of the list $y$ followed by $u$, the counter $P$ equals the top of diamond\u2026","title":"One diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/diamonds-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.diamonds_run","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#diamonds-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $B \\in \\mathbb{N}$ and let $\\varphi$ be a CNF formula with $m$ clauses, token list $\\mathrm{toks}(\\varphi)$ and list of distinct variables $\\mathrm{vars}(\\varphi)$ of length $k$. Suppose the top of diamond $k$ for $m$ clauses, the length of $\\mathrm{toks}(\\varphi)$, and the length of the binary encoding of every $y \\in \\mathrm{vars}(\\varphi)$ are at most $B$. Let $\\mathrm{vars}(\\varphi)$ split as a list $\\pi$\u2026","title":"The diamonds"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/final-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.final_run","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#final-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $s$ be a quiescent state of the emitter whose variable stack $V$ is empty, with clause count $C = c$, diamond-top counter $P = p$, edge bits $q$ and output stack $o$. Then for every internal bit $v$ there are an internal bit $v'$ and stacks $S'$ such that the emitter goes in exactly $|q| + 4p + 2c + 11$ steps from label $\\mathsf{h0}$ with bit $v$ and the stacks of $s$ to label $\\mathsf{done}$ with bit $v'$ and\u2026","title":"The output phase of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/run","kind":"node","lean":"Cslib.Complexity.Karp.EmitHam.run","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with token sequence $\\mathrm{toks}(\\varphi)$, let $o$ be a list of bits, and let $S$ be stacks in which the token stack $T$ holds $\\mathrm{toks}(\\varphi)$ reversed, the output stack $O$ holds $o$, and every other stack is empty. Then for every internal bit $v$ there are an internal bit $v'$ and stacks $S'$ such that the emitter goes, in at most $2000(|\\mathrm{toks}(\\varphi)| + 1)^5$\u2026","title":"Correctness and running time of the HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/hf-row","kind":"node","lean":"Cslib.Complexity.Karp.hfRow","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#hf-row","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $m$ and $i$ be natural numbers and $c : \\mathbb{N} \\to \\mathbb{N}$ a map, and write $r_l = m + i(3m + 2) + 1 + l$ for the row vertices of diamond $i$. The row of diamond $i$ for $c$ is the list $r_0, r_1, \\dots, r_{3m}$ in which, for every $j < m$ with $c(j) = i$, the clause vertex $j$ is inserted right after $r_{3j+1}$.","title":"The row of a diamond with its clauses inserted"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/hf-picks","kind":"node","lean":"Cslib.Complexity.Karp.hfPicks","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#hf-picks","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"For natural numbers $m$ and $i$ and a map $c : \\mathbb{N} \\to \\mathbb{N}$, the clauses picked up by diamond $i$ are the list of all $j < m$ with $c(j) = i$, in increasing order.","title":"Clauses picked up by a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/hf-diamond","kind":"node","lean":"Cslib.Complexity.Karp.hfDiamond","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#hf-diamond","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"For $m$ clauses, a diamond $i$, a clause-to-diamond map $c$ and a direction $b$, the visit of diamond $i$ is the list consisting of its top $T_i = m + i(3m+2)$ followed by its row with the clauses sent to $i$ by $c$ inserted, read left to right if $b$ is true and right to left otherwise.","title":"The visit of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/hf-path","kind":"node","lean":"Cslib.Complexity.Karp.hfPath","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#hf-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $m$ and $k$ be natural numbers, let $c : \\mathbb{N} \\to \\mathbb{N}$ be a map and let $d : \\mathbb{N} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ give each diamond $i$ a direction $d(i)$. The forward path through the first $k$ diamonds is the list obtained by concatenating, for $i = 0, 1, \\dots, k - 1$ in this order, the traversal of diamond $i$ for $c$ in direction $d(i)$, and then appending the vertex\u2026","title":"The forward path through the diamonds"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/mem-hf-picks","kind":"node","lean":"Cslib.Complexity.Karp.mem_hfPicks","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#mem-hf-picks","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $m$ and $i$ be natural numbers and let $c : \\mathbb{N} \\to \\mathbb{N}$. A natural number $v$ is one of the clause vertices picked up by the row of diamond $i$ under $c$, for $m$ clauses, if and only if $v < m$ and $c(v) = i$.","title":"Clause vertices picked up by a row"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/nodup-hf-picks","kind":"node","lean":"Cslib.Complexity.Karp.nodup_hfPicks","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#nodup-hf-picks","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"For all natural numbers $m$, $i$ and every function $c : \\mathbb{N} \\to \\mathbb{N}$, the list of clause vertices picked up by the row of diamond $i$ under $c$, for $m$ clauses, has no repeated entries.","title":"The clause vertices of a row are distinct"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/flat-map-hf-picks-perm","kind":"node","lean":"Cslib.Complexity.Karp.flatMap_hfPicks_perm","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#flat-map-hf-picks-perm","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $m, k$ be natural numbers and let $c$ send every clause $j < m$ to a diamond $c(j) < k$. For each diamond $i$, list the clauses $j < m$ with $c(j) = i$ in increasing order (the clause vertices picked up by the row of diamond $i$). Concatenating these lists for $i = 0, \\dots, k - 1$ gives a permutation of $0, 1, \\dots, m - 1$.","title":"Each clause is picked up by exactly one row"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/hf-row-perm","kind":"node","lean":"Cslib.Complexity.Karp.hfRow_perm","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#hf-row-perm","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"For natural numbers $m$ and $i$ and a map $c : \\mathbb{N} \\to \\mathbb{N}$, let $r = m + i(3m + 2) + 1$ be the left end of the row of diamond $i$. The row of diamond $i$ for $c$ (with its clause vertices inserted) is a permutation of the concatenation of the list $r, r + 1, \\dots, r + 3m$ and the list of clauses picked up by diamond $i$ for $c$, that is, of all $j < m$ with $c(j) = i$ in increasing order.","title":"The vertices of a diamond row"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/hf-diamond-perm","kind":"node","lean":"Cslib.Complexity.Karp.hfDiamond_perm","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#hf-diamond-perm","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"For $m$ clauses, a diamond $i$, a clause-to-diamond map $c$ and a direction $b$, let $T_i = m + i(3m+2)$ be the top of diamond $i$. The visit of diamond $i$ in direction $b$ is a rearrangement of the list that consists of $T_i, T_i + 1, \\dots, T_i + 3m + 1$ and then the clauses $j < m$ with $c(j) = i$ in increasing order.","title":"A diamond visit is a block of consecutive vertices plus its clauses"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/flat-map-range-hp-t","kind":"node","lean":"Cslib.Complexity.Karp.flatMap_range'_hpT","parent":32233,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first "],["math","\\(t\\)"],["text"," diamonds occupy consecutive vertices"]],"source":"The first $t$ diamonds occupy consecutive vertices"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#flat-map-range-hp-t","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"For natural numbers $m$ and $t$, let $T_i = m + i(3m+2)$. Concatenating, for $i = 0, \\dots, t - 1$, the lists $T_i, T_i + 1, \\dots, T_i + 3m + 1$ of $3m + 2$ consecutive numbers (the top and the row of diamond $i$) gives the list $m, m + 1, \\dots, m + t(3m+2) - 1$, that is, the numbers from $m$ to $T_t - 1$ in increasing order.","title":"The first $t$ diamonds occupy consecutive vertices"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/hf-path-perm","kind":"node","lean":"Cslib.Complexity.Karp.hfPath_perm","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#hf-path-perm","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $m, k$ be natural numbers, let $c$ send every clause $j < m$ to a diamond $c(j) < k$, let $d$ be any directions, and let $T_k = m + k(3m+2)$. The forward path through the first $k$ diamonds in directions $d$ with clause choice $c$ is a permutation of the list $0, 1, \\dots, T_k$.","title":"The forward path visits every vertex once"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/is-chain-hf-row","kind":"node","lean":"Cslib.Complexity.Karp.isChain_hfRow","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#is-chain-hf-row","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"Let $R$ be a relation on $\\mathbb{N}$, let $m$ and $i$ be natural numbers and $c : \\mathbb{N} \\to \\mathbb{N}$ a map, and write $r_l = m + i(3m + 2) + 1 + l$ for the row vertices of diamond $i$. If $R(r_l, r_{l+1})$ holds for every $l < 3m$, and $R(r_{3j+1}, j)$ and $R(j, r_{3j+2})$ hold for every $j < m$ with $c(j) = i$, then the row of diamond $i$ for $c$ is a chain for $R$: every entry is related by $R$ to the\u2026","title":"Chain condition for a diamond row"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/head-hf-row","kind":"node","lean":"Cslib.Complexity.Karp.head?_hfRow","parent":32233,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07/index.html#head-hf-row","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-07","status":"fully_proved","summary":"For $m$ clauses, a diamond $i$ and any clause-to-diamond map $c$, the row of diamond $i$ read left to right, with the clauses sent to $i$ by $c$ inserted, starts at its left end, the row vertex $T_i + 1$, where $T_i = m + i(3m+2)$ is the top of diamond $i$.","title":"A row starts at its left end"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","title":"A row ends at its right end to Consecutive vertices of the contracted path are joined by arcs"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/get-last-hf-row","kind":"node","lean":"Cslib.Complexity.Karp.getLast?_hfRow","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#get-last-hf-row","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"For $m$ clauses, a diamond $i$ and any clause-to-diamond map $c$, the row of diamond $i$ read left to right, with the clauses sent to $i$ by $c$ inserted, ends at the row vertex $T_i + 1 + 3m$, where $T_i = m + i(3m+2)$ is the top of diamond $i$.","title":"A row ends at its right end"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/is-chain-hf-diamond","kind":"node","lean":"Cslib.Complexity.Karp.isChain_hfDiamond","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#is-chain-hf-diamond","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $R$ be a relation on $\\mathbb{N}$, let $m$ and $i$ be natural numbers, $c : \\mathbb{N} \\to \\mathbb{N}$ a map and $b$ a boolean, and write $t_a = m + a(3m + 2)$ for the top of diamond $a$ and $r_l = t_i + 1 + l$ for the row vertices of diamond $i$. Suppose that $R(t_i, r_0)$, $R(t_i, r_{3m})$, $R(r_0, t_{i+1})$ and $R(r_{3m}, t_{i+1})$ hold; that $R(r_l, r_{l+1})$ and $R(r_{l+1}, r_l)$ hold for every $l < 3m$;\u2026","title":"Chain condition for a diamond traversal"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/is-chain-hf-path","kind":"node","lean":"Cslib.Complexity.Karp.isChain_hfPath","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#is-chain-hf-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $R$ be a relation on $\\mathbb{N}$, let $m$ and $k$ be natural numbers, $c : \\mathbb{N} \\to \\mathbb{N}$ a map and $d : \\mathbb{N} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ a choice of directions, and write $t_a = m + a(3m + 2)$ for the top of diamond $a$. If for every $i < k$ the traversal of diamond $i$ for $c$ in direction $d(i)$, followed by $t_{i+1}$, is a chain for $R$ (every entry is related by $R$ to the\u2026","title":"Chain condition for the forward path"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/head-hf-path","kind":"node","lean":"Cslib.Complexity.Karp.head?_hfPath","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#head-hf-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"For $m$ clauses, $k$ diamonds, any clause-to-diamond map $c$ and any directions $d$, the forward path through the first $k$ diamonds in directions $d$ with clause choice $c$ starts at the top $T_0 = m$ of diamond $0$.","title":"The forward path starts at the first top"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/get-last-hf-path","kind":"node","lean":"Cslib.Complexity.Karp.getLast?_hfPath","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#get-last-hf-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"For $m$ clauses, $k$ diamonds, any clause-to-diamond map $c$ and any directions $d$, the forward path through the first $k$ diamonds in directions $d$ with clause choice $c$ ends at the top $T_k = m + k(3m+2)$.","title":"The forward path ends at the last top"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/mem-ham-edges-of-mem-diamond-edges","kind":"node","lean":"Cslib.Complexity.Karp.mem_hamEdges_of_mem_diamondEdges","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#mem-ham-edges-of-mem-diamond-edges","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula, let $x_0, \\dots, x_{k-1}$ be its distinct variables in the order used by the construction, and let $i < k$. Every arc of diamond $i$ for variable $x_i$ is an arc of the HAMPATH construction for $\\varphi$.","title":"Diamond arcs are arcs of the construction"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/mem-diamond-edges-frame","kind":"node","lean":"Cslib.Complexity.Karp.mem_diamondEdges_frame","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#mem-diamond-edges-frame","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, let $i$ and $x$ be natural numbers, and write $t_a = m + a(3m + 2)$ for the top of diamond $a$ and $r_l = t_i + 1 + l$ for the row vertices of diamond $i$. The four arcs $(t_i, r_0)$, $(t_i, r_{3m})$, $(r_0, t_{i+1})$ and $(r_{3m}, t_{i+1})$ are arcs of diamond $i$ for the variable $x$ in the construction for $\\varphi$.","title":"The frame arcs of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/mem-diamond-edges-row","kind":"node","lean":"Cslib.Complexity.Karp.mem_diamondEdges_row","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#mem-diamond-edges-row","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, let $i$ and $x$ be natural numbers, and write $r_i(l)$ for row vertex $l$ of diamond $i$. For every $l < 3m$, the arcs of diamond $i$ for variable $x$ include both the arc from $r_i(l)$ to $r_i(l+1)$ and the arc from $r_i(l+1)$ to $r_i(l)$.","title":"Row arcs of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/mem-diamond-edges-pos","kind":"node","lean":"Cslib.Complexity.Karp.mem_diamondEdges_pos","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#mem-diamond-edges-pos","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, let $i$, $x$, $j$ be natural numbers with $j < m$, and write $r_i(l)$ for row vertex $l$ of diamond $i$. If the variable $x$ occurs positively in clause $j$ of $\\varphi$ (counting clauses from $0$), then the arcs of diamond $i$ for variable $x$ include both the arc from $r_i(3j+1)$ to clause vertex $j$ and the arc from clause vertex $j$ to $r_i(3j+2)$.","title":"Positive detour arcs of a diamond"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/mem-diamond-edges-neg","kind":"node","lean":"Cslib.Complexity.Karp.mem_diamondEdges_neg","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#mem-diamond-edges-neg","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, let $i$, $x$ and $j$ be natural numbers with $j < m$, and write $r_l = m + i(3m + 2) + 1 + l$ for the row vertices of diamond $i$. If the negative literal $\\neg x$ occurs in clause $j$ of $\\varphi$, then $(r_{3j+2}, j)$ and $(j, r_{3j+1})$ are arcs of diamond $i$ for the variable $x$ in the construction for $\\varphi$.","title":"The detour arcs of a negative occurrence"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/is-ham-path-hf-path","kind":"node","lean":"Cslib.Complexity.Karp.isHamPath_hfPath","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#is-ham-path-hf-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, let $v_0, \\dots, v_{k-1}$ be its distinct variables in the order used by the construction, let $\\sigma$ assign a truth value to every variable, and write $t_a = m + a(3m + 2)$ for the top of diamond $a$. Let $c : \\mathbb{N} \\to \\mathbb{N}$ be such that for every $j < m$ we have $c(j) < k$ and clause $j$ of $\\varphi$ contains the literal over $v_{c(j)}$ that $\\sigma$\u2026","title":"A satisfying choice gives a Hamiltonian path"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/exists-is-ham-path-of-cnf-sat","kind":"node","lean":"Cslib.Complexity.Karp.exists_isHamPath_of_cnfSat","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#exists-is-ham-path-of-cnf-sat","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a satisfiable CNF formula with $m$ clauses and $k$ distinct variables, and let $T_i = m + i(3m+2)$ be the top of diamond $i$. Then the digraph on the vertices $0, \\dots, T_k$ whose arcs are the arcs of the HAMPATH construction for $\\varphi$ has a Hamiltonian path from $T_0$ to $T_k$.","title":"Satisfiable formulas give Hamiltonian paths (SAT \u2264 HAMPATH, forward direction)"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/hp-t-arc","kind":"node","lean":"Cslib.Complexity.Karp.hpT_arc","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#hp-t-arc","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses. In Sipser's HAMPATH graph for $\\varphi$, every arc out of the top $t_i$ of diamond $i$ ends at one of the two ends $r_{i,0}$ and $r_{i,3m}$ of the row of diamond $i$.","title":"Out-neighbours of a top"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/hp-r-arc-clause","kind":"node","lean":"Cslib.Complexity.Karp.hpR_arc_clause","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#hp-r-arc-clause","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses $C_0, \\dots, C_{m-1}$, and let $x_i$ be the $i$-th of its distinct variables (variable $0$ if $\\varphi$ has at most $i$ distinct variables). In Sipser's HAMPATH graph for $\\varphi$, clause $j$ is vertex $j$, and $r_{i,l}$ denotes vertex $l$ of the row of diamond $i$. If $l \\le 3m$, $j < m$ and $(r_{i,l}, j)$ is an arc of the graph, then either $l = 3j + 1$ and the\u2026","title":"Arcs from a row into a clause vertex"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/hp-ro","kind":"node","lean":"Cslib.Complexity.Karp.hpRo","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#hp-ro","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"For $m$ clauses, let $r_{i,l} = m + i(3m + 2) + 1 + l$ be vertex $l$ of the row of diamond $i$. For a Boolean $b$ and a natural number $l$, this is $r_{i,l}$ if $b$ is true and $r_{i,3m - l}$ if $b$ is false, where $3m - l$ is truncated subtraction ($0$ when $l > 3m$). For $l \\le 3m$ it is vertex $l$ of the row counted from the left end ($b$ true) or from the right end ($b$ false).","title":"Row vertices counted from either end"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/hp-ro-arc","kind":"node","lean":"Cslib.Complexity.Karp.hpRo_arc","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#hp-ro-arc","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses. In Sipser's HAMPATH graph for $\\varphi$, clause $j$ is vertex $j < m$, the top of diamond $i$ is $t_i = m + i(3m + 2)$ (and $t_{i+1}$ is its bottom), and $\\rho_b(i, l)$ denotes vertex $l$ of the row of diamond $i$ counted from the left end if $b$ is true and from the right end if $b$ is false. Let $l \\le 3m$ and $v \\ge m$. If $(\\rho_b(i, l), v)$ is an arc of the\u2026","title":"Out-neighbours of a row vertex outside the clause vertices"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/arc-hp-ro","kind":"node","lean":"Cslib.Complexity.Karp.arc_hpRo","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#arc-hp-ro","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses. In Sipser's HAMPATH graph for $\\varphi$, clause $j$ is vertex $j < m$, the top of diamond $i$ is $t_i = m + i(3m + 2)$, and $\\rho_b(i, l)$ denotes vertex $l$ of the row of diamond $i$ counted from the left end if $b$ is true and from the right end if $b$ is false. Let $l \\le 3m$ and $u \\ge m$. If $(u, \\rho_b(i, l))$ is an arc of the graph, then $l > 0$ and\u2026","title":"In-neighbours of a row vertex outside the clause vertices"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/drop-clauses","kind":"node","lean":"Cslib.Complexity.Karp.dropClauses","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#drop-clauses","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"For a CNF formula $\\varphi$ with $m$ clauses (the clause vertices of Sipser's HAMPATH graph are $0, \\dots, m - 1$) and a list $p$ of natural numbers, the list obtained from $p$ by deleting every entry $v < m$, keeping the other entries in their order.","title":"Deleting the clause vertices from a path"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/infix-drop-clauses","kind":"node","lean":"Cslib.Complexity.Karp.infix_dropClauses","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#infix-drop-clauses","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses, let $p$ be a list of natural numbers, and let $q$ be the list obtained from $p$ by deleting every entry less than $m$. If $a, j, c$ occur consecutively in $p$ with $j < m$, $a \\ge m$ and $c \\ge m$, then $a$ is immediately followed by $c$ in $q$.","title":"Contracting a clause detour"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/arc-of-infix-drop-clauses","kind":"node","lean":"Cslib.Complexity.Karp.arc_of_infix_dropClauses","parent":32254,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08/index.html#arc-of-infix-drop-clauses","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-08","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables. In Sipser's HAMPATH graph for $\\varphi$, clause $j$ is vertex $j < m$, the top of diamond $i$ is $t_i = m + i(3m + 2)$, and vertex $l \\le 3m$ of its row is $r_{i,l} = t_i + 1 + l$. Let $p$ be a Hamiltonian path of this graph (on the vertices $0, \\dots, t_k$) from $t_0$ to $t_k$ whose detours are normal: for every $j < m$ there is $i < k$\u2026","title":"Consecutive vertices of the contracted path are joined by arcs"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","title":"Every vertex of the contracted path except its end has a successor to Second counter of a HAMPATH-verifier comparison"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/exists-infix-drop-clauses","kind":"node","lean":"Cslib.Complexity.Karp.exists_infix_dropClauses","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#exists-infix-drop-clauses","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables. In Sipser's HAMPATH graph for $\\varphi$, clause $j$ is vertex $j < m$ and the top of diamond $i$ is $t_i = m + i(3m + 2)$. Let $p$ be a Hamiltonian path of this graph (on the vertices $0, \\dots, t_k$) from $t_0$ to $t_k$, and let $p'$ be $p$ with the clause vertices deleted. Then every vertex $v \\ne t_k$ of $p'$ is immediately followed in\u2026","title":"Every vertex of the contracted path except its end has a successor"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/no-shortcut","kind":"node","lean":"Cslib.Complexity.Karp.no_shortcut","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#no-shortcut","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m > 0$ clauses and $n$ distinct variables. For $k \\le n$ let $t_k = m + k(3m + 2)$ be the top of diamond $k$, and for $k < n$ let row $k$ consist of the vertices $t_k + 1 + l$ with $0 \\le l \\le 3m$; its left end is $t_k + 1$ and its right end is $t_k + 1 + 3m$. Let $p$ be a Hamiltonian path from $t_0$ to $t_n$ in Sipser's digraph for $\\varphi$ on the vertices $0, \\dots, t_n$\u2026","title":"Entering a row and leaving it at once is impossible"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/row-run","kind":"node","lean":"Cslib.Complexity.Karp.row_run","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#row-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $n$ distinct variables. For $k \\le n$ let $t_k = m + k(3m + 2)$ be the top of diamond $k$, and for $k < n$ let row $k$ consist of the vertices $t_k + 1 + l$ with $0 \\le l \\le 3m$; its left end is $t_k + 1$ and its right end is $t_k + 1 + 3m$. Let $p$ be a Hamiltonian path from $t_0$ to $t_n$ in Sipser's digraph for $\\varphi$ on the vertices $0, \\dots, t_n$ whose\u2026","title":"A row entered at one end is run to the other end"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/exists-top-succ","kind":"node","lean":"Cslib.Complexity.Karp.exists_top_succ","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#exists-top-succ","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables. In Sipser's HAMPATH graph for $\\varphi$, clause $j$ is vertex $j < m$, the top of diamond $i$ is $t_i = m + i(3m + 2)$, and vertex $l \\le 3m$ of its row is $r_{i,l} = t_i + 1 + l$. Let $p$ be a Hamiltonian path of this graph (on the vertices $0, \\dots, t_k$) from $t_0$ to $t_k$ whose detours are normal: for every $j < m$ there is $i < k$\u2026","title":"Every top but the last is followed by an end of its row"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/idx-of-get-d-ham-vars","kind":"node","lean":"Cslib.Complexity.Karp.idxOf_getD_hamVars","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#idx-of-get-d-ham-vars","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula, let $v_0, \\dots, v_{k-1}$ be the list of distinct variables of $\\varphi$, and let $i < k$. Then the first position at which $v_i$ occurs in this list is $i$.","title":"Positions in the list of distinct variables"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/cnf-sat-of-is-ham-path","kind":"node","lean":"Cslib.Complexity.Karp.cnfSat_of_isHamPath","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#cnf-sat-of-is-ham-path","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables. In Sipser's HAMPATH graph for $\\varphi$, clause $j$ is vertex $j < m$, the top of diamond $i$ is $t_i = m + i(3m + 2)$, and vertex $l \\le 3m$ of its row is $r_{i,l} = t_i + 1 + l$. If $p$ is a Hamiltonian path of this graph (on the vertices $0, \\dots, t_k$) from $t_0$ to $t_k$ whose detours are normal, that is, for every $j < m$ there is\u2026","title":"Backward direction of SAT \u2264 HAMPATH for paths with normal detours"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/exists-is-ham-path-iff-cnf-sat","kind":"node","lean":"Cslib.Complexity.Karp.exists_isHamPath_iff_cnfSat","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#exists-is-ham-path-iff-cnf-sat","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula with $m$ clauses and $k$ distinct variables, and let $t_i = m + i(3m + 2)$ be the top vertex of diamond $i$ in Sipser's construction. The digraph on the vertices $0, \\dots, t_k$ whose arcs are the arcs of the construction for $\\varphi$ has a Hamiltonian path from $t_0$ to $t_k$ if and only if $\\varphi$ is satisfiable.","title":"The construction has a Hamiltonian path iff the formula is satisfiable"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/ham-path-dec-encode-p","kind":"node","lean":"Cslib.Complexity.Karp.hamPathDec_encodeP","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#ham-path-dec-encode-p","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"For all natural numbers $s$, $t$, $n$ and every list $E$ of pairs of natural numbers, the HAMPATH decision problem accepts the PATH encoding of $(s, t, n, E)$ if and only if the digraph on $0, \\dots, n - 1$ with edge list $E$ has a Hamiltonian path from $s$ to $t$.","title":"HAMPATH on an encoded instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/ham-path-dec-no-ham","kind":"node","lean":"Cslib.Complexity.Karp.hamPathDec_noHam","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#ham-path-dec-no-ham","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"The HAMPATH decision problem rejects the PATH encoding of the digraph on the two vertices $0, 1$ with no arcs and $s = t = 0$: a path from $0$ to $0$ that visits both vertices would repeat $0$.","title":"HAMPATH rejects the fixed no-instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/sat-dec-eq-ham-path-dec-reduce-ham","kind":"node","lean":"Cslib.Complexity.Karp.satDec_eq_hamPathDec_reduceHam","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#sat-dec-eq-ham-path-dec-reduce-ham","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"For every bit string $x$, SAT accepts $x$ if and only if HAMPATH accepts the image of $x$ under the SAT-to-HAMPATH reduction map.","title":"SAT \u2264 HAMPATH preserves membership"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/k","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.K","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#k","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"The verifier has twelve stacks: the input $X$ and the witness $W$; the counters $S$, $T$, $N$ (holding $s$, $t$, $n$), $C$ (the number of vertices seen), $A$ (the current vertex) and $B$ (the previous one); the save stacks $E$ (for $X$) and $Z$ (for $W$); and the scratch stacks $R$ and $Q$.","title":"Stacks of the Hamiltonian path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/ci","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.CI","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#ci","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"The four-element type of the comparisons the verifier makes between its counters: the current vertex $a$ with the vertex count $n$, the current vertex $a$ with the start vertex $s$, the number $c$ of vertices seen with $n$, and the previous vertex $b$ with the end vertex $t$.","title":"Counter comparisons of the HAMPATH verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/lab","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.Lab","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#lab","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"The instruction labels of the verifier: reading $s$, $t$, $n$; validating the edge list and putting it back; validating the witness and putting it back; the vertex loop, reading a vertex, and the head-or-edge dispatch; for each of the four counter comparisons, the comparison itself and, for each of its three outcomes, the two loops restoring the compared counters; the edge scan (comparing an edge's ends, with a\u2026","title":"Labels of the Hamiltonian path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/stmt","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.Stmt","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#stmt","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"A statement of the verifier is a statement of a multi-stack Turing machine whose stacks are indexed by the verifier's stacks $K$, each stack over the alphabet $\\mathrm{Sym}$, with labels $\\mathrm{Lab}$ and a Boolean internal variable.","title":"Statements of the Hamiltonian-path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/read-u","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.readU","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#read-u","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"For stacks $\\mathsf{src}$ and $c$ and labels $\\ell$ (its own) and $m$ (the next one), the statement that goes to label $\\mathrm{no}$ if $\\mathsf{src}$ is empty; otherwise it pops the top symbol of $\\mathsf{src}$ and, if that symbol is the bit `true`, pushes the bit `true` (one unit) onto $c$ and goes to $\\ell$, while if it is anything else it goes to $m$. Each of the two tests also stores its outcome in the Boolean\u2026","title":"The unary reader of the Hamiltonian path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/skip-u","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.skipU","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#skip-u","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"For stacks $\\mathsf{src}$ and $\\mathsf{dst}$ and labels $\\ell$ (its own) and $m$ (the next one), the statement that goes to label $\\mathrm{no}$ if $\\mathsf{src}$ is empty; otherwise it pops the top symbol of $\\mathsf{src}$, reads it as the bit `true` if it is the bit `true` and as `false` otherwise, pushes that bit onto $\\mathsf{dst}$, and goes to $\\ell$ if the bit is `true` and to $m$ if it is `false`. Each of the\u2026","title":"The unary mover of the Hamiltonian path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/if-e","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.ifE","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#if-e","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"For a stack $k$ and labels $e$ and $ne$, the verifier instruction that looks at the top of stack $k$ without changing any stack, then goes to $e$ if $k$ is empty and to $ne$ otherwise.","title":"Emptiness test"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/cb-s","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.cbS","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#cb-s","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"For stacks $\\mathit{src}$, $\\mathit{sv}$, $c$ of the HAMPATH verifier, label families $\\mathit{self}$ and $\\mathit{fin}$ indexed by booleans, and a boolean verdict $\\mathit{ok}$, this is the instruction that pops the top of $\\mathit{src}$, reads it as a bit $b$ (false if $\\mathit{src}$ is empty or its top is not a bit symbol), and pushes the bit symbol of $b$ onto $\\mathit{sv}$. If $b$ is true, it pops $c$; if $c$\u2026","title":"Step of the HAMPATH verifier's unary comparison loop"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/ca","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.ca","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#ca","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"The function from the four counter comparisons of the HAMPATH verifier to its stacks that sends the comparisons of the current vertex with the vertex count and of the current vertex with the start vertex to the current-vertex counter $A$, the comparison of the number of vertices seen with the vertex count to the vertices-seen counter $C$, and the comparison of the previous vertex with the end vertex to the\u2026","title":"First counter of a HAMPATH-verifier comparison"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/cb","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.cb","parent":32275,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09/index.html#cb","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-09","status":"fully_proved","summary":"The function from the four counter comparisons of the HAMPATH verifier to its stacks that sends the comparisons of the current vertex with the vertex count and of the number of vertices seen with the vertex count to the vertex-count counter $N$, the comparison of the current vertex with the start vertex to the start-vertex counter $S$, and the comparison of the previous vertex with the end vertex to the end-vertex\u2026","title":"Second counter of a HAMPATH-verifier comparison"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","title":"Exit labels of the counter comparisons to The edge scan of the Hamiltonian path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/cout","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.cout","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#cout","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"The label the verifier goes to after comparing two counters, where $a$ is the current vertex, $b$ the previous one, $c$ the number of vertices seen, $n$ the number of vertices and $s$, $t$ the two ends: after comparing $a$ with $n$ it goes to the head-or-edge dispatch if $a < n$; after comparing $a$ with $s$ it goes to the repeat scan if $a = s$; after comparing $c$ with $n$ it goes to the nonzero-count check if\u2026","title":"Exit labels of the counter comparisons"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/prog","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.prog","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#prog","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"The verifier is the program of the multi-stack machine with the stacks and labels of the Hamiltonian path verifier, every stack over the alphabet $\\mathrm{Sym}$, and a Boolean internal variable, given below label by label. Write $\\mathrm{bit}(x)$ for the bit symbol of a boolean $x$. Write $(a, n)$, $(a, s)$, $(c, n)$ and $(b, t)$ for the four counter comparisons, and $\\alpha(i)$ and $\\beta(i)$ for the first and the\u2026","title":"The Hamiltonian path verifier machine"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/start","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.start","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#start","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"The verifier starts at the label $\\mathsf{rdS}$, where it reads the first unary number of the input stack $X$ (the start vertex $s$) onto the counter stack $S$.","title":"Start label of the Hamiltonian-path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/update-update-update-update","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.update_update_update_update","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#update-update-update-update","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $a \\ne b$ be two stacks of the verifier, $S$ an assignment of a list of symbols to each stack, and $p, p', t, t'$ lists of symbols. Setting $a$ to $p$, then $b$ to $t$, then $a$ to $p'$, then $b$ to $t'$ in $S$ gives the same assignment as setting $a$ to $p'$ and then $b$ to $t'$.","title":"Interleaved stack updates in the Hamiltonian-path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/step1","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.step1","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#step1","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $l, l'$ be labels, $v$ a flag value and $S, S'$ assignments of a list of symbols to each stack. If executing the verifier's statement at $l$ with flag $v$ and stacks $S$ jumps to the label $l'$ and leaves the stacks $S'$, then the verifier goes in exactly one step from label $l$, flag $v$ and stacks $S$ to label $l'$ and stacks $S'$, with some flag value.","title":"A single step of the Hamiltonian-path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/if-e-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.ifE_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#if-e-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"If the verifier's instruction at label $l$ is the emptiness test of stack $k$ with targets $e$ and $ne$, then from label $l$, in any internal state and on any stacks, exactly one step of the verifier leads to label $e$ if stack $k$ is empty and to label $ne$ otherwise, in some internal state and with the stacks unchanged.","title":"One step of the emptiness test"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/read-u-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.readU_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#read-u-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $\\mathsf{src} \\ne c$ be stacks and $\\ell$, $m$ labels such that the verifier's statement at $\\ell$ is the unary reader from $\\mathsf{src}$ onto $c$ with own label $\\ell$ and next label $m$. Let $k, j$ be natural numbers and $r$ a bit list, and suppose that $\\mathsf{src}$ holds $k$ copies of `true`, then `false`, then $r$ (as bit symbols, first bit on top) and that $c$ holds $j$ copies of the bit `true`. Then\u2026","title":"The unary reader of the Hamiltonian path verifier reads a terminated number"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/read-u-none","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.readU_none","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#read-u-none","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $\\mathsf{src} \\ne c$ be stacks and $\\ell$, $m$ labels such that the verifier's statement at $\\ell$ is the unary reader from $\\mathsf{src}$ onto $c$ with own label $\\ell$ and next label $m$. Let $l$ be a bit list from which no unary length can be read (that is, $l$ contains no `false`), and suppose that $\\mathsf{src}$ holds the bits of $l$ (as bit symbols, first bit on top). Then from label $\\ell$, with either\u2026","title":"The unary reader of the Hamiltonian path verifier rejects an unterminated number"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/blk-succ","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.blk_succ","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#blk-succ","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"For $a \\in \\mathbb{N}$, let $\\beta_a$ be the list of symbols obtained by mapping the bit symbol over the reversal of the unary block of $a$ ($a$ copies of true followed by one false), that is, the bit symbol of false followed by $a$ bit symbols of true. Then for every list $L$ of symbols, $\\beta_a$ followed by the bit symbol of true and then $L$ equals $\\beta_{a+1}$ followed by $L$.","title":"Extending a reversed unary block"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/skip-u-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.skipU_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#skip-u-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $\\mathsf{src} \\ne \\mathsf{dst}$ be stacks and $\\ell$, $m$ labels such that the verifier's statement at $\\ell$ is the unary mover from $\\mathsf{src}$ to $\\mathsf{dst}$ with own label $\\ell$ and next label $m$. Let $k$ be a natural number and $r$ a bit list, and suppose that $\\mathsf{src}$ holds $k$ copies of `true`, then `false`, then $r$ (as bit symbols, first bit on top). Then from label $\\ell$, with either\u2026","title":"The unary mover of the Hamiltonian path verifier moves a terminated number"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/skip-u-none","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.skipU_none","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#skip-u-none","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $\\mathsf{src} \\ne \\mathsf{dst}$ be stacks and $\\ell$, $m$ labels such that the verifier's statement at $\\ell$ is the unary mover from $\\mathsf{src}$ to $\\mathsf{dst}$ with own label $\\ell$ and next label $m$. Let $l$ be a bit list from which no unary length can be read (that is, $l$ contains no `false`), and suppose that $\\mathsf{src}$ holds the bits of $l$ (as bit symbols, first bit on top). Then from label\u2026","title":"The unary mover of the Hamiltonian path verifier rejects an unterminated number"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/mv-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.mv_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#mv-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Suppose the verifier's instruction at label $p$ is the bit-moving loop from a stack $a$ to a different stack $b$, continuing at label $p'$ when done. If stack $a$ holds exactly the bits of a list $l$, then from label $p$ in any internal state the verifier takes exactly $|l| + 1$ steps to reach label $p'$, in some internal state, with stack $a$ empty, stack $b$ holding the bits of $l$ in reverse order followed by\u2026","title":"Moving a stack of bits"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/cmp-i-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.cmpI_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#cmp-i-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $i$ be one of the four counter comparisons of the HAMPATH verifier, with first counter $\\mathrm{ca}(i)$ and second counter $\\mathrm{cb}(i)$, and let $m, n \\in \\mathbb{N}$. Suppose the stacks $\\sigma$ hold $m$ units (bit symbols of true) on $\\mathrm{ca}(i)$, $n$ units on $\\mathrm{cb}(i)$, and nothing on the scratch stacks $R$ and $Q$. Then for every flag $v$ the verifier goes, in at most $3 \\min(m, n) + 4$\u2026","title":"Run of a HAMPATH-verifier counter comparison"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/cout-an","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.cout_an","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#cout-an","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"For all $m, n \\in \\mathbb{N}$, the label that the comparison of the current vertex with the vertex count picks on the ordering $\\mathrm{compare}(m, n)$ of $m$ and $n$ is the head-or-edge dispatch label $\\mathrm{hd}$ if $m < n$, and the reject label $\\mathrm{no}$ otherwise.","title":"Outcome of the HAMPATH verifier's vertex-range comparison"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/cout-eq","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.cout_eq","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#cout-eq","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Write $\\mathrm{cout}(i, o)$ for the label that counter comparison $i$ of the HAMPATH verifier picks on the ordering $o \\in \\{\\mathrm{lt}, \\mathrm{eq}, \\mathrm{gt}\\}$. Let $i$ be a counter comparison and $l$ a label such that $\\mathrm{cout}(i, \\mathrm{eq}) = l$ and $\\mathrm{cout}(i, o)$ is the reject label $\\mathrm{no}$ for every ordering $o \\ne \\mathrm{eq}$. Then for all $m, n \\in \\mathbb{N}$, the label\u2026","title":"Outcome of a HAMPATH-verifier equality comparison"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/cb-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.cb_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#cb-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $\\mathit{src}$, $\\mathit{sv}$, $c$ and the scratch stack $R$ be four distinct stacks of the HAMPATH verifier, and let $\\mathrm{cb}$ and $\\mathrm{fin}$ be families of labels indexed by booleans such that for every boolean $\\mathit{ok}$ the instruction at $\\mathrm{cb}(\\mathit{ok})$ is the unary comparison step on $\\mathit{src}$, $\\mathit{sv}$, $c$ with loop labels $\\mathrm{cb}$, exit labels $\\mathrm{fin}$ and\u2026","title":"Run of the HAMPATH verifier's unary comparison loop"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/cmp-n-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.cmpN_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#cmp-n-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $\\mathit{src}$, $\\mathit{sv}$, $c$ and the scratch stack $R$ be four distinct stacks of the HAMPATH verifier, and let $\\mathrm{cb}$, $\\mathrm{rs}$ and $\\mathrm{nx}$ be families of labels indexed by booleans such that for every boolean $\\mathit{ok}$ the instruction at $\\mathrm{cb}(\\mathit{ok})$ is the unary comparison step on $\\mathit{src}$, $\\mathit{sv}$, $c$ with loop labels $\\mathrm{cb}$, exit labels\u2026","title":"HAMPATH-verifier unary comparison with the counter restored"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/val-e-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.valE_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#val-e-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $r$ and $acc$ be bit lists, and let the stacks $St$ hold $r$ on the input stack $X$ and the reverse of $acc$ on the save stack $E$. Started at the label $\\mathsf{vE}$ with any flag value, the verifier reaches within at most $2|r| + 1$ steps the label $\\mathsf{vEb}$ if $r$ decodes as an edge list (a sequence of pairs of unary numbers, each a block of ones ended by a zero) and the label $\\mathsf{no}$ otherwise.\u2026","title":"The edge-list check of the Hamiltonian-path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/val-w-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.valW_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#val-w-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"Let $w$ and $acc$ be bit lists, and let the stacks $St$ hold $w$ on the witness stack $W$ and the reverse of $acc$ on the save stack $Z$. Started at the label $\\mathsf{vW}$ with any flag value, the verifier reaches within at most $2|w| + 1$ steps the label $\\mathsf{vWb}$ if $w$ decodes as a vertex list (a sequence of unary numbers, each a block of ones ended by a zero) and the label $\\mathsf{no}$ otherwise. When\u2026","title":"The witness check of the Hamiltonian-path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/scan-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.scan_run","parent":32296,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10/index.html#scan-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-10","status":"fully_proved","summary":"For a list $e$ of pairs of natural numbers write $\\mathrm{bits}(e)$ for its edge-list encoding, in which each pair $(u, u')$ contributes $u$ copies of `true`, then `false`, then $u'$ copies of `true`, then `false`. Let $a, b$ be natural numbers and $r$, $d$ lists of pairs of natural numbers. Suppose that the verifier's stacks hold, as bit symbols with the first bit on top, $\\mathrm{bits}(r)$ on $\\mathsf{X}$,\u2026","title":"The edge scan of the Hamiltonian path verifier"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","kind":"scope","navigation":"transparent","parent":32106,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","title":"The repeat scan of the Hamiltonian path verifier to HAMPATH is NP-complete"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/nd-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.nd_run","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#nd-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"For a list $p$ of natural numbers write $\\mathrm{enc}(p)$ for its witness encoding: each entry in unary (that many copies of `true`) followed by `false`. Let $a$ be a natural number and $q$, $d$ lists of natural numbers. Suppose that the verifier's stacks hold, as bit symbols with the first bit on top, the bits of $\\mathrm{enc}(q)$ on $\\mathsf{W}$, the bits of $\\mathrm{enc}(d)$ in reverse order on $\\mathsf{Z}$, $a$\u2026","title":"The repeat scan of the Hamiltonian path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/lay","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.lay","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#lay","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"For an edge list $es$, numbers $s, t, n, c, b$ and a list $q$ of vertices, the stacks of the verifier between two vertex steps: $X$ holds the bits of the edge list $es$, $W$ holds the bits of the witness encoding of $q$, the counters $S$, $T$, $N$, $C$ and $B$ hold $s$, $t$, $n$, $c$ and $b$ in unary, and the stacks $A$, $E$, $Z$, $R$ and $Q$ are empty.","title":"Stack layout between vertex steps"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/mid","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.mid","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#mid","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"For an edge list $es$, numbers $s, t, n, c, b, a$ and a list $q$ of vertices, the stacks of the verifier during the step on vertex $a$: the layout between vertex steps for $es, s, t, n, c, b, q$, except that stack $A$ holds $a$ in unary.","title":"Stack layout during a vertex step"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/hd-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.hd_run","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#hd-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"Let $es$ be an edge list, $s, t, n, c, b, a$ numbers and $q$ a list of vertices, and say the check passes when $a = s$ in case $c = 0$, and $(b, a) \\in es$ in case $c \\neq 0$. Started at the head-or-edge dispatch, in any internal state, on the stack layout during the step on vertex $a$ for $es, s, t, n, c, b, a, q$, the verifier reaches within $3a + 4\\ell + 7$ steps, where $\\ell$ is the number of bits of $es$, the\u2026","title":"Running the head-or-edge check"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/tail-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.tail_run","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#tail-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"Let $es$ be an edge list, $s, t, n, c, b, a$ natural numbers and $q$ a list of natural numbers. Start the verifier, with any flag value, at the label $\\mathsf{nd}$ (the repeat scan) with its stacks in the mid-step layout: the between-steps layout for $es$, $s$, $t$, $n$, count $c$, previous vertex $b$ and remaining vertices $q$, with $a$ in unary on the stack $A$. Let $w$ be the unary encoding of $q$. Then within\u2026","title":"The rest of a vertex step in the Hamiltonian-path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/step-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.step_run","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#step-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"Let $es$ be an edge list, $s, t, n, c, b, a$ natural numbers and $q$ a list of natural numbers. Start the verifier, with any flag value, at the label $\\mathsf{rdA}$ (read the next vertex) with its stacks in the between-steps layout for $es$, $s$, $t$, $n$, count $c$, previous vertex $b$ and remaining vertices $a, q$. Let $e$ be the bit encoding of $es$ and $w$ the unary encoding of $q$. Then within at most\u2026","title":"One vertex step of the Hamiltonian-path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/loop-run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.loop_run","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#loop-run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"Let $es$ be an edge list, $s, t, n$ numbers and $L$ a bound with $n \\le L$, $t \\le L$ and at most $L$ bits in the encoding of $es$. For every list $q$ of vertices whose witness encoding has at most $L$ bits, every count $c$, every previous vertex $b \\le L$ and every internal state, the verifier started at the vertex loop on the stack layout between vertex steps for $es, s, t, n, c, b, q$ reaches within\u2026","title":"Running the vertex loop"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/run","kind":"node","lean":"Cslib.Complexity.Karp.HamVerify.run","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#run","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"Let $x$ and $w$ be bit lists. Start the verifier at its start label, with either value of its Boolean variable, with $x$ on stack $\\mathsf{X}$ and $w$ on stack $\\mathsf{W}$ (as bit symbols, first bit on top) and all other stacks empty. Then within $100(|x| + |w| + 1)^4$ steps it reaches the label $\\mathrm{yes}$ if the Hamiltonian path check accepts $x$ and $w$, and the label $\\mathrm{no}$ otherwise.","title":"Correctness and running time of the Hamiltonian path verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/ham-verify-comp","kind":"node","lean":"Cslib.Complexity.Karp.hamVerifyComp","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#ham-verify-comp","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"The component machine whose program and labels are those of the HAMPATH verifier machine, whose stacks are the twelve stacks of that machine (the input $X$, the witness $W$, the counters $S, T, N, C, A, B$, the save stacks $E, Z$ and the scratch stacks $R, Q$), each over the symbol alphabet of the SAT verifier.","title":"The HAMPATH verifier as a host component"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/ham-path-dec-mem-np","kind":"node","lean":"Cslib.Complexity.Karp.hamPathDec_mem_NP","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#ham-path-dec-mem-np","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"HAMPATH is in NP.","title":"HAMPATH is in NP"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/emit-ham","kind":"node","lean":"Cslib.Complexity.Karp.emitHam","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#emit-ham","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"The emitter whose stacks, alphabets, labels, program and start label are those of the HAMPATH emitter machine, which exits at the machine's label $\\mathsf{done}$, reads the tokens of a CNF formula from stack $T$ and writes its output bits on stack $O$ (both symbol maps being the identity), and has step bound $B(n) = 2000 (n + 1)^5$ for $n$ tokens. Its output for a CNF formula $\\varphi$ is Sipser's HAMPATH instance\u2026","title":"The HAMPATH emitter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/is-poly-time-reduce-ham","kind":"node","lean":"Cslib.Complexity.Karp.isPolyTime_reduceHam","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#is-poly-time-reduce-ham","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"The map on bit strings that sends a bit string decoding to a CNF formula $\\varphi$ to Sipser's HAMPATH instance for $\\varphi$, and every other bit string to the fixed no-instance with two vertices, no arcs and $s = t = 0$, is computable in polynomial time.","title":"SAT \u2264 HAMPATH is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/ham-path-dec-np-hard","kind":"node","lean":"Cslib.Complexity.Karp.hamPathDec_npHard","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#ham-path-dec-np-hard","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"Every decision problem $L$ in NP has a polynomial-time many-one reduction to HAMPATH: a polynomial-time computable map $f$ on bit strings such that, for every bit string $x$, $L$ accepts $x$ exactly when HAMPATH accepts $f(x)$.","title":"HAMPATH is NP-hard"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/ham-path-dec-np-complete","kind":"node","lean":"Cslib.Complexity.Karp.hamPathDec_npComplete","parent":32317,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11/index.html#ham-path-dec-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/hamiltonian-path-in-directed-graphs/part-11","status":"fully_proved","summary":"HAMPATH lies in NP, and every decision problem $L$ in NP has a polynomial-time many-one reduction to HAMPATH: a polynomial-time computable map $f$ on bit strings such that, for every bit string $x$, $L$ accepts $x$ exactly when HAMPATH accepts $f(x)$.","title":"HAMPATH is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","status":"can_prove","title":"Induced subgraph isomorphism"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/induced-subgraph-isomorphism-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.InducedSubgraphIsomorphism.InducedSubgraphIsomorphismInstance","parent":32332,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/index.html#induced-subgraph-isomorphism-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","status":"fully_proved","summary":"An instance of INDUCED SUBGRAPH ISOMORPHISM is a quadruple $(n, E, k, F)$ consisting of a natural number $n$ (the number of host vertices), a list $E$ of pairs of natural numbers (the host edges), a natural number $k$ (the number of pattern vertices) and a list $F$ of pairs of natural numbers (the pattern edges).","title":"Induced-subgraph-isomorphism instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/graph-host","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.InducedSubgraphIsomorphism.GraphHost","parent":32332,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/index.html#graph-host","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","status":"fully_proved","summary":"For an instance $(n, E, k, F)$, the host graph is the graph of the edge list $E$ on $n$ vertices: the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Host graph of an induced-subgraph-isomorphism instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/graph-pattern","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.InducedSubgraphIsomorphism.GraphPattern","parent":32332,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/index.html#graph-pattern","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","status":"fully_proved","summary":"For an instance $(n, E, k, F)$, the pattern graph is the graph of the edge list $F$ on $k$ vertices: the simple graph on $\\{0, \\dots, k - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $F$.","title":"Pattern graph of an induced-subgraph-isomorphism instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.InducedSubgraphIsomorphism.WellFormed","parent":32332,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","status":"fully_proved","summary":"An instance $(n, E, k, F)$ is well-formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$, and likewise the list $F$ has no repeated entries and every pair $(u, v)$ in $F$ satisfies $u < k$, $v < k$, $u \\ne v$ and $(v, u) \\notin F$.","title":"Well-formed induced-subgraph-isomorphism instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/induced-subgraph-isomorphism-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.InducedSubgraphIsomorphism.InducedSubgraphIsomorphismFeasible","parent":32332,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/index.html#induced-subgraph-isomorphism-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","status":"fully_proved","summary":"An instance $(n, E, k, F)$ is feasible when there is an injective map $f$ from $\\{0, \\dots, k - 1\\}$ to $\\{0, \\dots, n - 1\\}$ such that, for all $a, b \\in \\{0, \\dots, k - 1\\}$, the vertices $a$ and $b$ are adjacent in the pattern graph if and only if $f(a)$ and $f(b)$ are adjacent in the host graph.","title":"Induced embedding of the pattern into the host"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/induced-subgraph-isomorphism-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.InducedSubgraphIsomorphism.inducedSubgraphIsomorphismDec","parent":32332,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/index.html#induced-subgraph-isomorphism-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","status":"fully_proved","summary":"The decision problem INDUCED SUBGRAPH ISOMORPHISM accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance $(n, E, k, F)$ (host vertex count, host edge list, pattern vertex count, pattern edge list) that is well-formed and for which there is an injective map $f$ from $\\{0, \\dots, k - 1\\}$ to $\\{0, \\dots, n - 1\\}$ such that, for all $a$ and $b$, the vertices $a$ and\u2026","title":"INDUCED SUBGRAPH ISOMORPHISM as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/induced-subgraph-isomorphism-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.InducedSubgraphIsomorphism.inducedSubgraphIsomorphism_npComplete","parent":32332,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism/index.html#induced-subgraph-isomorphism-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/induced-subgraph-isomorphism","status":"can_prove","summary":"The decision problem INDUCED SUBGRAPH ISOMORPHISM is in NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"INDUCED SUBGRAPH ISOMORPHISM is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle","status":"can_prove","title":"Longest cycle"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/longest-cycle-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCycle.LongestCycleInstance","parent":32340,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/index.html#longest-cycle-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle","status":"fully_proved","summary":"An instance of LONGEST CYCLE is a triple $(n, E, K)$ of a natural number $n$ (the number of vertices), a list $E$ of pairs of natural numbers (the edges), and a natural number $K$ (the bound).","title":"Instances of LONGEST CYCLE"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCycle.Graph","parent":32340,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/index.html#graph","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle","status":"fully_proved","summary":"For an instance $(n, E, K)$ of LONGEST CYCLE, its graph is the simple graph on $\\{0, \\dots, n-1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$. Loops are dropped, and pairs with an endpoint at least $n$ contribute no edge.","title":"Graph of a LONGEST CYCLE instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCycle.WellFormed","parent":32340,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle","status":"fully_proved","summary":"An instance $(n, E, K)$ of LONGEST CYCLE is well formed when the list $E$ has no repeated entries and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E$.","title":"Well-formed LONGEST CYCLE instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/longest-cycle-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCycle.LongestCycleFeasible","parent":32340,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/index.html#longest-cycle-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle","status":"fully_proved","summary":"An instance $(n, E, K)$ of LONGEST CYCLE is feasible when there are pairwise distinct vertices $v_1, \\dots, v_m$ of the graph of the instance, with $m \\ge K$ and $m \\ge 3$, such that $v_j$ is adjacent to $v_{j+1}$ for every $1 \\le j < m$ and $v_m$ is adjacent to $v_1$. That is, the graph has a simple cycle with at least $K$ edges and at least three vertices.","title":"Feasible LONGEST CYCLE instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/longest-cycle-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCycle.longestCycleDec","parent":32340,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/index.html#longest-cycle-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle","status":"fully_proved","summary":"LONGEST CYCLE accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of instances, to an instance of LONGEST CYCLE that is well formed and feasible. Bitstrings that do not decode, and bitstrings that decode to instances that are not well formed, are rejected.","title":"The decision problem LONGEST CYCLE"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/longest-cycle-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCycle.longestCycle_npComplete","parent":32340,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle/index.html#longest-cycle-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-cycle","status":"can_prove","summary":"The decision problem LONGEST CYCLE belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to LONGEST CYCLE.","title":"LONGEST CYCLE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path","status":"can_prove","title":"Longest path"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/longest-path-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestPath.LongestPathInstance","parent":32347,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/index.html#longest-path-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path","status":"fully_proved","summary":"A longest-path instance is a tuple $(n, E, K, s, t)$ consisting of a natural number $n$, the number of vertices; a list $E$ of pairs of natural numbers, the edges; and natural numbers $K$, the length bound, $s$, the start vertex, and $t$, the target vertex.","title":"Longest-path instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestPath.Graph","parent":32347,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/index.html#graph","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path","status":"fully_proved","summary":"For a longest-path instance with vertex count $n$ and edge list $E$, its graph is the graph of the edge list $E$ on $n$ vertices: the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Graph of a longest-path instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestPath.WellFormed","parent":32347,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path","status":"fully_proved","summary":"A longest-path instance $(n, E, K, s, t)$ is well formed when $s < n$, $t < n$, the list $E$ has no repeated entries, and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and its reverse $(v, u)$ does not belong to $E$.","title":"Well-formed longest-path instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/longest-path-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestPath.LongestPathFeasible","parent":32347,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/index.html#longest-path-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path","status":"fully_proved","summary":"A longest-path instance $(n, E, K, s, t)$ is feasible when the graph of $E$ on $n$ vertices has a simple path from $s$ to $t$ with at least $K$ edges, that is, a list $v_0, \\dots, v_m$ of pairwise distinct vertices with $m \\ge K$, $v_0 = s$ and $v_m = t$, in which $v_j$ and $v_{j+1}$ are adjacent for every $j < m$.","title":"Feasibility of a longest-path instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/longest-path-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestPath.longestPathDec","parent":32347,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/index.html#longest-path-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path","status":"fully_proved","summary":"LONGEST PATH is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a longest-path instance $(n, E, K, s, t)$ that is well formed and in which the graph of $E$ on $n$ vertices has a simple path from $s$ to $t$ with at least $K$ edges. Bitstrings that do not decode to a longest-path instance are rejected.","title":"The LONGEST PATH decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/longest-path-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestPath.longestPath_npComplete","parent":32347,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path/index.html#longest-path-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/longest-path","status":"can_prove","summary":"The LONGEST PATH decision problem belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to LONGEST PATH.","title":"LONGEST PATH is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","status":"can_prove","title":"Maximum common induced subgraph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/maximum-common-induced-subgraph-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCommonInducedSubgraph.MaximumCommonInducedSubgraphInstance","parent":32354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/index.html#maximum-common-induced-subgraph-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","status":"fully_proved","summary":"An instance of MAXIMUM COMMON INDUCED SUBGRAPH is a tuple $(n_1, E_1, n_2, E_2, K)$ in which $n_1$, $n_2$ and $K$ are natural numbers and $E_1$, $E_2$ are finite lists of pairs of natural numbers: $n_1$ is the vertex count of the first graph, $E_1$ its edge list, $n_2$ the vertex count of the second graph, $E_2$ its edge list, and $K$ the bound.","title":"Instances of MAXIMUM COMMON INDUCED SUBGRAPH"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/graph-first","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCommonInducedSubgraph.GraphFirst","parent":32354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/index.html#graph-first","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","status":"fully_proved","summary":"For an instance $(n_1, E_1, n_2, E_2, K)$ of MAXIMUM COMMON INDUCED SUBGRAPH, the first graph is the graph of the edge list $E_1$ on $n_1$ vertices: the simple graph on $\\{0, \\dots, n_1 - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E_1$.","title":"First graph of an instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/graph-second","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCommonInducedSubgraph.GraphSecond","parent":32354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/index.html#graph-second","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","status":"fully_proved","summary":"For an instance $(n_1, E_1, n_2, E_2, K)$ of MAXIMUM COMMON INDUCED SUBGRAPH, the second graph is the graph of the edge list $E_2$ on $n_2$ vertices: the simple graph on $\\{0, \\dots, n_2 - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E_2$.","title":"Second graph of an instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCommonInducedSubgraph.WellFormed","parent":32354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","status":"fully_proved","summary":"An instance $(n_1, E_1, n_2, E_2, K)$ of MAXIMUM COMMON INDUCED SUBGRAPH is well formed when, for each $k \\in \\{1, 2\\}$, the list $E_k$ has no repeated entries and every pair $(u, v)$ in $E_k$ has $u < n_k$, $v < n_k$, $u \\ne v$ and $(v, u) \\notin E_k$.","title":"Well-formed instances of MAXIMUM COMMON INDUCED SUBGRAPH"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/maximum-common-induced-subgraph-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCommonInducedSubgraph.MaximumCommonInducedSubgraphFeasible","parent":32354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/index.html#maximum-common-induced-subgraph-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","status":"fully_proved","summary":"An instance $(n_1, E_1, n_2, E_2, K)$ of MAXIMUM COMMON INDUCED SUBGRAPH is feasible when, writing $G_1$ and $G_2$ for its first and second graphs, there are a set $A$ of vertices of $G_1$ and a set $B$ of vertices of $G_2$ with $|A| \\ge K$ and $|B| = |A|$, and a bijection $f : A \\to B$ such that for all $a, b \\in A$, the vertices $a$ and $b$ are adjacent in $G_1$ if and only if $f(a)$ and $f(b)$ are adjacent in\u2026","title":"Feasible instances of MAXIMUM COMMON INDUCED SUBGRAPH"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/maximum-common-induced-subgraph-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCommonInducedSubgraph.maximumCommonInducedSubgraphDec","parent":32354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/index.html#maximum-common-induced-subgraph-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","status":"fully_proved","summary":"The decision problem MAXIMUM COMMON INDUCED SUBGRAPH accepts a bitstring $x$ exactly when the canonical bitstring decoding of $x$ yields an instance of MAXIMUM COMMON INDUCED SUBGRAPH that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"MAXIMUM COMMON INDUCED SUBGRAPH as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/maximum-common-induced-subgraph-np-compl","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCommonInducedSubgraph.maximumCommonInducedSubgraph_npComplete","parent":32354,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph/index.html#maximum-common-induced-subgraph-np-compl","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-common-induced-subgraph","status":"can_prove","summary":"The decision problem MAXIMUM COMMON INDUCED SUBGRAPH belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"MAXIMUM COMMON INDUCED SUBGRAPH is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree","status":"can_prove","title":"Maximum leaf spanning tree"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/maximum-leaf-spanning-tree-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumLeafSpanningTree.MaximumLeafSpanningTreeInstance","parent":32362,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/index.html#maximum-leaf-spanning-tree-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree","status":"fully_proved","summary":"An instance of MAXIMUM LEAF SPANNING TREE is a triple $(n, es, K)$ of a natural number $n$ (the number of vertices), a list $es$ of pairs of natural numbers (the edge list) and a natural number $K$ (the bound on the number of leaves).","title":"Instances of MAXIMUM LEAF SPANNING TREE"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumLeafSpanningTree.WellFormed","parent":32362,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree","status":"fully_proved","summary":"An instance $(n, es, K)$ of MAXIMUM LEAF SPANNING TREE is well formed when $n > 0$, the list $es$ has no repeated entries, and every entry $(u, v)$ of $es$ satisfies $u < n$, $v < n$ and $u \\ne v$, and $(v, u)$ is not an entry of $es$.","title":"Well-formed MAXIMUM LEAF SPANNING TREE instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/maximum-leaf-spanning-tree-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumLeafSpanningTree.MaximumLeafSpanningTreeFeasible","parent":32362,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/index.html#maximum-leaf-spanning-tree-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree","status":"fully_proved","summary":"An instance $(n, es, K)$ of MAXIMUM LEAF SPANNING TREE is feasible when there are a sublist $T$ of $es$ (a list obtained from $es$ by deleting entries) with no repeated entries and exactly $n - 1$ entries, and a set $L$ of at least $K$ vertices in $\\{0, \\dots, n - 1\\}$, such that the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ is an entry of\u2026","title":"Feasible MAXIMUM LEAF SPANNING TREE instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/maximum-leaf-spanning-tree-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumLeafSpanningTree.maximumLeafSpanningTreeDec","parent":32362,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/index.html#maximum-leaf-spanning-tree-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree","status":"fully_proved","summary":"The decision problem MAXIMUM LEAF SPANNING TREE accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of instances, to an instance $(n, es, K)$ of MAXIMUM LEAF SPANNING TREE that is well formed and feasible (its graph has a spanning tree with at least $K$ leaves). Bitstrings that do not decode are rejected.","title":"The decision problem MAXIMUM LEAF SPANNING TREE"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/maximum-leaf-spanning-tree-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumLeafSpanningTree.maximumLeafSpanningTree_npComplete","parent":32362,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree/index.html#maximum-leaf-spanning-tree-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/maximum-leaf-spanning-tree","status":"can_prove","summary":"The decision problem MAXIMUM LEAF SPANNING TREE belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"MAXIMUM LEAF SPANNING TREE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","status":"can_prove","title":"Minimum equivalent digraph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/minimum-equivalent-digraph-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumEquivalentDigraph.MinimumEquivalentDigraphInstance","parent":32368,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/index.html#minimum-equivalent-digraph-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","status":"fully_proved","summary":"An instance of MINIMUM EQUIVALENT DIGRAPH is a triple $(n, E, K)$ consisting of a natural number $n$ (the number of vertices), a finite list $E$ of ordered pairs of natural numbers (the arcs) and a natural number $K$ (the bound on the number of arcs).","title":"Instances of MINIMUM EQUIVALENT DIGRAPH"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/dadj","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumEquivalentDigraph.DAdj","parent":32368,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/index.html#dadj","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","status":"fully_proved","summary":"For a finite list $E$ of ordered pairs of natural numbers and natural numbers $u$ and $v$, the directed adjacency test of $E$ at $(u, v)$ returns true when the pair $(u, v)$ occurs in $E$, and false otherwise.","title":"Directed adjacency for MINIMUM EQUIVALENT DIGRAPH"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/reachable","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumEquivalentDigraph.Reachable","parent":32368,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/index.html#reachable","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","status":"fully_proved","summary":"Let $E$ be a finite list of ordered pairs of natural numbers, read as the arcs of a directed graph, and let $n$, $u$ and $v$ be natural numbers. Then $v$ is reachable from $u$ within $n$ steps along $E$ when there are natural numbers $w_0, w_1, \\dots, w_k$ with $k \\le n$, $w_0 = u$ and $w_k = v$ such that $(w_j, w_{j+1})$ occurs in $E$ for every $j < k$.","title":"Bounded reachability in a digraph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumEquivalentDigraph.WellFormed","parent":32368,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","status":"fully_proved","summary":"An instance $(n, E, K)$ of MINIMUM EQUIVALENT DIGRAPH is well formed when the list $E$ has no repeated entries and every pair $(a, b)$ in $E$ satisfies $a < n$, $b < n$ and $a \\ne b$; that is, $E$ lists distinct arcs without loops between vertices among $0, \\dots, n - 1$.","title":"Well-formed MINIMUM EQUIVALENT DIGRAPH instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/minimum-equivalent-digraph-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumEquivalentDigraph.MinimumEquivalentDigraphFeasible","parent":32368,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/index.html#minimum-equivalent-digraph-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","status":"fully_proved","summary":"An instance $(n, E, K)$ of MINIMUM EQUIVALENT DIGRAPH is feasible when some sublist $F$ of $E$ (obtained by deleting entries of $E$) has at most $K$ entries and, for all vertices $u < n$ and $v < n$, $v$ is reachable from $u$ within $n$ steps along $E$ if and only if $v$ is reachable from $u$ within $n$ steps along $F$.","title":"Feasible MINIMUM EQUIVALENT DIGRAPH instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/minimum-equivalent-digraph-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumEquivalentDigraph.minimumEquivalentDigraphDec","parent":32368,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/index.html#minimum-equivalent-digraph-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","status":"fully_proved","summary":"The decision problem MINIMUM EQUIVALENT DIGRAPH accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance of MINIMUM EQUIVALENT DIGRAPH that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"MINIMUM EQUIVALENT DIGRAPH as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/minimum-equivalent-digraph-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumEquivalentDigraph.minimumEquivalentDigraph_npComplete","parent":32368,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph/index.html#minimum-equivalent-digraph-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/minimum-equivalent-digraph","status":"can_prove","summary":"The decision problem MINIMUM EQUIVALENT DIGRAPH belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"MINIMUM EQUIVALENT DIGRAPH is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph","status":"can_prove","title":"Path cover of a digraph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/directed-path-cover-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedPathCover.DirectedPathCoverInstance","parent":32376,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/index.html#directed-path-cover-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph","status":"fully_proved","summary":"An instance of DIRECTED PATH COVER is a triple $(n, E, K)$, where the natural number $n$ is the number of vertices, the list $E$ of ordered pairs of natural numbers lists the directed edges, and the natural number $K$ bounds the number of paths.","title":"Instances of DIRECTED PATH COVER"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/dadj","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedPathCover.DAdj","parent":32376,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/index.html#dadj","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph","status":"fully_proved","summary":"For a list $E$ of ordered pairs of natural numbers and natural numbers $u$ and $v$, the directed adjacency test of $E$ at $(u, v)$ is the boolean that is true exactly when the pair $(u, v)$ occurs in $E$, that is, when $E$ contains an edge from $u$ to $v$.","title":"Directed adjacency in an edge list"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedPathCover.WellFormed","parent":32376,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph","status":"fully_proved","summary":"An instance $(n, E, K)$ of DIRECTED PATH COVER is well formed when the edge list $E$ has no repeated entries and every edge $(a, b)$ in $E$ satisfies $a < n$, $b < n$ and $a \\ne b$.","title":"Well-formed DIRECTED PATH COVER instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/directed-path-cover-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedPathCover.DirectedPathCoverFeasible","parent":32376,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/index.html#directed-path-cover-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph","status":"fully_proved","summary":"An instance $(n, E, K)$ of DIRECTED PATH COVER is feasible when there is a list of at most $K$ paths, each a list of natural numbers, such that every path has no repeated vertex, every vertex on a path is less than $n$, and whenever $v$ immediately follows $u$ on a path the pair $(u, v)$ occurs in $E$; and such that every vertex $v < n$ lies on some path, and all paths containing $v$ are equal.","title":"Feasible DIRECTED PATH COVER instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/directed-path-cover-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedPathCover.directedPathCoverDec","parent":32376,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/index.html#directed-path-cover-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph","status":"fully_proved","summary":"DIRECTED PATH COVER is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance $(n, E, K)$ of DIRECTED PATH COVER that is well formed and feasible; bitstrings that do not decode to an instance are rejected.","title":"DIRECTED PATH COVER as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/directed-path-cover-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedPathCover.directedPathCover_npComplete","parent":32376,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph/index.html#directed-path-cover-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/path-cover-of-a-digraph","status":"can_prove","summary":"The decision problem DIRECTED PATH COVER is in NP, and every decision problem in NP has a polynomial-time many-one reduction to DIRECTED PATH COVER.","title":"DIRECTED PATH COVER is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree","status":"can_prove","title":"Steiner tree"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/steiner-tree-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SteinerTree.SteinerTreeInstance","parent":32383,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/index.html#steiner-tree-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree","status":"fully_proved","summary":"A Steiner tree instance is a quadruple $(n, E, R, K)$ of a natural number $n$ (the number of vertices), a list $E$ of triples $(u, v, w)$ of natural numbers (an edge between $u$ and $v$ of weight $w$), a list $R$ of natural numbers (the terminals) and a natural number $K$ (the budget).","title":"Steiner tree instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SteinerTree.Graph","parent":32383,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/index.html#graph","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree","status":"fully_proved","summary":"Given a Steiner tree instance with vertex count $n$ and a list $T$ of triples $(u, v, w)$ of natural numbers, the graph of $T$ is the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j, w)$ or $(j, i, w)$ belongs to $T$ for some natural number $w$. Weights are ignored, and triples with an endpoint at least $n$ contribute no edge.","title":"Graph of a candidate edge set"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SteinerTree.WellFormed","parent":32383,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree","status":"fully_proved","summary":"A Steiner tree instance with vertex count $n$, edge list $E$ of triples $(u, v, w)$ and terminal list $R$ is well formed when $E$ has no repeated entries; every $(u, v, w)$ in $E$ has $u < n$, $v < n$, $u \\ne v$ and $w > 0$; any two entries of $E$ with the same endpoints, in the same or the reverse order, are equal; $R$ has no repeated entries; and every terminal in $R$ is less than $n$.","title":"Well-formed Steiner tree instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/steiner-tree-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SteinerTree.SteinerTreeFeasible","parent":32383,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/index.html#steiner-tree-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree","status":"fully_proved","summary":"A Steiner tree instance with vertex count $n$, edge list $E$ of triples $(u, v, w)$, terminal list $R$ and budget $K$ is feasible when every terminal in $R$ is less than $n$ and there is a sublist $T$ of $E$ (a not necessarily contiguous subsequence) whose weights $w$ sum to at most $K$ such that any two terminals in $R$ are joined by a path in the graph of $T$, the simple graph on $\\{0, \\dots, n - 1\\}$ in which\u2026","title":"Feasible Steiner tree instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/steiner-tree-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SteinerTree.steinerTreeDec","parent":32383,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/index.html#steiner-tree-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree","status":"fully_proved","summary":"The decision problem STEINER TREE accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of Steiner tree instances, to a Steiner tree instance that is both well formed and feasible. Bitstrings that do not decode are rejected.","title":"The STEINER TREE decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/steiner-tree-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SteinerTree.steinerTree_npComplete","parent":32383,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree/index.html#steiner-tree-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/steiner-tree","status":"can_prove","summary":"The decision problem STEINER TREE belongs to NP, and every decision problem in NP reduces to STEINER TREE by a polynomial-time many-one reduction.","title":"STEINER TREE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism","status":"can_prove","title":"Subgraph isomorphism"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/subgraph-isomorphism-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubgraphIsomorphism.SubgraphIsomorphismInstance","parent":32390,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/index.html#subgraph-isomorphism-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism","status":"fully_proved","summary":"A subgraph-isomorphism instance is a tuple $(n, E_G, m, E_H)$ of a natural number $n$ (the number of host vertices), a list $E_G$ of pairs of natural numbers (the host edges), a natural number $m$ (the number of pattern vertices) and a list $E_H$ of pairs of natural numbers (the pattern edges).","title":"Subgraph-isomorphism instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/graph-host","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubgraphIsomorphism.GraphHost","parent":32390,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/index.html#graph-host","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism","status":"fully_proved","summary":"For a subgraph-isomorphism instance $(n, E_G, m, E_H)$, the host graph is the graph of the edge list $E_G$ on $n$ vertices: the simple graph on $\\{0, \\dots, n - 1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E_G$.","title":"Host graph of a subgraph-isomorphism instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubgraphIsomorphism.WellFormed","parent":32390,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism","status":"fully_proved","summary":"A subgraph-isomorphism instance $(n, E_G, m, E_H)$ is well formed when the list $E_G$ has no repeated entries and every $(u, v) \\in E_G$ satisfies $u < n$, $v < n$, $u \\ne v$ and $(v, u) \\notin E_G$, and likewise the list $E_H$ has no repeated entries and every $(u, v) \\in E_H$ satisfies $u < m$, $v < m$, $u \\ne v$ and $(v, u) \\notin E_H$.","title":"Well-formed subgraph-isomorphism instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/subgraph-isomorphism-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubgraphIsomorphism.SubgraphIsomorphismFeasible","parent":32390,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/index.html#subgraph-isomorphism-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism","status":"fully_proved","summary":"A subgraph-isomorphism instance $(n, E_G, m, E_H)$ is feasible when there is an injective map $f : \\{0, \\dots, m - 1\\} \\to \\{0, \\dots, n - 1\\}$ such that for every $(u, v) \\in E_H$ with $u < m$ and $v < m$, the vertices $f(u)$ and $f(v)$ are adjacent in the host graph of the instance.","title":"Feasible subgraph-isomorphism instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/subgraph-isomorphism-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubgraphIsomorphism.subgraphIsomorphismDec","parent":32390,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/index.html#subgraph-isomorphism-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism","status":"fully_proved","summary":"The decision problem SUBGRAPH ISOMORPHISM accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of subgraph-isomorphism instances, to an instance that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"SUBGRAPH ISOMORPHISM as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/subgraph-isomorphism-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubgraphIsomorphism.subgraphIsomorphism_npComplete","parent":32390,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism/index.html#subgraph-isomorphism-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/subgraph-isomorphism","status":"can_prove","summary":"The decision problem SUBGRAPH ISOMORPHISM belongs to NP, and every decision problem $L$ in NP has a polynomial-time many-one reduction to SUBGRAPH ISOMORPHISM.","title":"SUBGRAPH ISOMORPHISM is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"can_prove","title":"Traveling salesperson problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/hamiltonian-cycle-to-tsp","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.hamiltonianCycleToTsp","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#hamiltonian-cycle-to-tsp","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"The Hamiltonian-cycle to TSP converter takes a Boolean array $x$ and, starting from the empty array, scans $x$ from left to right and appends the negation of each cell, charging one time unit before and one time unit after each append (one read and one write per cell). It returns the resulting array of negated cells together with the total time charged.","title":"Hamiltonian-cycle to TSP converter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/hamiltonian-cycle-to-tsp-size","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.hamiltonianCycleToTsp_size","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#hamiltonian-cycle-to-tsp-size","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"For every Boolean array $x$, the array output by the Hamiltonian-cycle to TSP converter on $x$ has the same length as $x$.","title":"The Hamiltonian-cycle to TSP converter preserves length"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/hamiltonian-cycle-to-tsp-get-elem","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.hamiltonianCycleToTsp_getElem?","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#hamiltonian-cycle-to-tsp-get-elem","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"For every Boolean array $x$ and every index $k$, the array output by the Hamiltonian-cycle to TSP converter on $x$ has a cell at index $k$ exactly when $x$ does, and that cell is the negation of the cell of $x$ at index $k$.","title":"The Hamiltonian-cycle to TSP converter negates every cell"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/hamiltonian-cycle-to-tsp-time","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.hamiltonianCycleToTsp_time","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#hamiltonian-cycle-to-tsp-time","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"For every Boolean array $x$, the time charged by the Hamiltonian-cycle to TSP converter on $x$ is $2m$, where $m$ is the length of $x$: one unit for reading and one unit for writing each cell.","title":"Running time of the Hamiltonian-cycle to TSP converter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/hamiltonian-cycle-to-tsp-costs","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.hamiltonianCycleToTsp_costs","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#hamiltonian-cycle-to-tsp-costs","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertex set $\\{0, \\dots, n-1\\}$, let $x$ be a Boolean array of length $n^2$ such that for all $i, j < n$ the cell of $x$ at index $in + j$ exists and is true exactly when $i$ and $j$ are adjacent in $G$, and let $y$ be the array output by the Hamiltonian-cycle to TSP converter on $x$, with a Boolean cell read as $1$ if true and $0$ if false. Then for all $i, j < n$ the cell of $y$ at\u2026","title":"Costs produced by the Hamiltonian-cycle to TSP converter"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/hamiltonian-cycle-to-tsp-forward","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.hamiltonianCycleToTsp_forward","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#hamiltonian-cycle-to-tsp-forward","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertex set $\\{0, \\dots, n-1\\}$, let $x$ be a Boolean array of length $n^2$ such that for all $i, j < n$ the cell of $x$ at index $in + j$ exists and is true exactly when $i$ and $j$ are adjacent in $G$, and let $y$ be the array output by the Hamiltonian-cycle to TSP converter on $x$, with a Boolean cell read as $1$ if true and $0$ if false. Let $p$ be a Hamiltonian cycle in $G$\u2026","title":"Hamiltonian cycles of the source graph give zero-cost tours"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/hamiltonian-cycle-to-tsp-reverse","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.hamiltonianCycleToTsp_reverse","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#hamiltonian-cycle-to-tsp-reverse","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertex set $\\{0, \\dots, n-1\\}$, let $x$ be a Boolean array of length $n^2$ such that for all $i, j < n$ the cell of $x$ at index $in + j$ exists and is true exactly when $i$ and $j$ are adjacent in $G$, and let $y$ be the array output by the Hamiltonian-cycle to TSP converter on $x$, with a Boolean cell read as $1$ if true and $0$ if false. Let $p$ be a Hamiltonian cycle based at a\u2026","title":"Tours of cost zero give Hamiltonian cycles of the source graph"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/hamiltonian-cycle-to-tsp-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.hamiltonianCycleToTsp_iff","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#hamiltonian-cycle-to-tsp-iff","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertex set $V = \\{0, \\dots, n-1\\}$, let $x$ be a Boolean array of length $n^2$ such that for all $i, j < n$ the cell of $x$ at index $in + j$ exists and is true exactly when $i$ and $j$ are adjacent in $G$, and let $y$ be the array output by the Hamiltonian-cycle to TSP converter on $x$, with a Boolean cell read as $1$ if true and $0$ if false. Then $G$ has a Hamiltonian cycle if\u2026","title":"Correctness of the Hamiltonian-cycle to TSP reduction"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/tsp-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Tsp.TspInstance","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#tsp-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"An instance of the symmetric traveling salesperson problem is a triple $(n, w, B)$ of a natural number $n$, the number of cities, a weight matrix $w$ given as a finite list of rows, each a finite list of natural numbers, and a natural number $B$, the bound.","title":"Symmetric TSP instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/tour-length","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Tsp.TourLength","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#tour-length","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"Let $w$ be a matrix of natural numbers given as a list of rows, where an entry $w_{a,b}$ outside the list (no row $a$, or row $a$ too short) is read as $0$, and rows and columns are numbered from $0$. The length of a list $(t_0, \\dots, t_{k-1})$ of natural numbers with respect to $w$ is $\\sum_{j\\lt k} w_{t_j, t_{(j+1) \\bmod k}}$, the total weight of the edges along the cycle $t_0, t_1, \\dots, t_{k-1}, t_0$,\u2026","title":"Length of a tour in the symmetric TSP"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Tsp.WellFormed","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"A symmetric TSP instance $(n, w, B)$ is well formed when $n > 0$, the weight matrix $w$ has exactly $n$ rows and each row has exactly $n$ entries, $w$ is symmetric, that is $w_{a,b} = w_{b,a}$ for all $a, b < n$, and $w$ has zero diagonal, that is $w_{a,a} = 0$ for all $a < n$. Rows and columns are numbered from $0$.","title":"Well-formed symmetric TSP instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/tsp-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Tsp.TspFeasible","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#tsp-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"A symmetric TSP instance $(n, w, B)$ is feasible when there is a list of $n$ pairwise distinct natural numbers containing every $v < n$, that is, an ordering of the cities $0, 1, \\dots, n-1$ visiting each exactly once, whose tour length with respect to $w$ (the sum of the weights of consecutive edges, including the return edge) is at most $B$.","title":"Feasible symmetric TSP instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/tsp-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Tsp.tspDec","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#tsp-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"fully_proved","summary":"The symmetric traveling salesperson problem is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of symmetric TSP instances, to an instance $(n, w, B)$ that is well formed and feasible. Bitstrings that fail to decode, and bitstrings that decode to instances that are not well formed, are rejected.","title":"Symmetric TSP as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/tsp-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Tsp.tsp_npComplete","parent":32397,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem/index.html#tsp-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/traveling-salesperson-problem","status":"can_prove","summary":"The symmetric traveling salesperson decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"TRAVELING SALESPERSON (SYMMETRIC) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"can_prove","title":"Undirected Hamiltonian cycle"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/verify-hamiltonian-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.verifyHamiltonianCycle","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#verify-hamiltonian-cycle","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"Given bit strings $x$ and $y$, let $m = \\lfloor \\sqrt{|x|} \\rfloor$ and let $w$ be the least natural number with $m \\le 2^w$. The verifier reads $x$ as the adjacency matrix of a graph on the vertices $0, \\dots, m - 1$, stored row by row, and $y$ as $m$ words of $w$ bits, the $i$-th word giving a number $v_i$ in binary, least significant bit first. It accepts exactly when $|x| = m^2$, every diagonal entry of the\u2026","title":"Hamiltonian-cycle verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/verify-hamiltonian-cycle-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.verifyHamiltonianCycle_iff","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#verify-hamiltonian-cycle-iff","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"Let $x$ and $y$ be bit strings, $m = \\lfloor \\sqrt{|x|} \\rfloor$, and $w$ the least natural number with $m \\le 2^w$. For each $i$ let $d_i$ be the number whose binary digits, least significant first, are the bits of $y$ at positions $iw, \\dots, iw + w - 1$ (counting from $0$; positions past the end of $y$ are omitted). The Hamiltonian-cycle verifier accepts $x$ and $y$ if and only if $|x| = m^2$, $m \\ge 3$,\u2026","title":"Acceptance condition of the Hamiltonian-cycle verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/verify-hamiltonian-cycle-sound","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.verifyHamiltonianCycle_sound","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#verify-hamiltonian-cycle-sound","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"If the Hamiltonian-cycle verifier accepts the bit strings $x$ and $y$, then, with $m = \\lfloor \\sqrt{|x|} \\rfloor$, we have $|x| = m^2$ and $m \\ge 3$, and there is a simple graph $G$ on the vertices $0, \\dots, m - 1$ in which $i$ and $j$ are adjacent exactly when entry $im + j$ of $x$ (counting from $0$) is true, and $G$ has a Hamiltonian cycle.","title":"Soundness of the Hamiltonian-cycle verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/verify-hamiltonian-cycle-complete","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.verifyHamiltonianCycle_complete","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#verify-hamiltonian-cycle-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"Let $x$ be a bit string and $m = \\lfloor \\sqrt{|x|} \\rfloor$, suppose $|x| = m^2$, and let $G$ be a simple graph on the vertices $0, \\dots, m - 1$ in which $i$ and $j$ are adjacent exactly when entry $im + j$ of $x$ (counting from $0$) is true. If $G$ has a Hamiltonian cycle, then there is a bit string $y$ of length $mw$, where $w$ is the least natural number with $m \\le 2^w$, such that the Hamiltonian-cycle\u2026","title":"Completeness of the Hamiltonian-cycle verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/verify-hamiltonian-cycle-language-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.verifyHamiltonianCycle_language_iff","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#verify-hamiltonian-cycle-language-iff","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"For every bit string $x$, with $m = \\lfloor \\sqrt{|x|} \\rfloor$, some bit string $y$ makes the Hamiltonian-cycle verifier accept $x$ and $y$ if and only if $|x| = m^2$, $m \\ge 3$, and there is a simple graph $G$ on the vertices $0, \\dots, m - 1$ in which $i$ and $j$ are adjacent exactly when entry $im + j$ of $x$ (counting from $0$) is true, and $G$ has a Hamiltonian cycle.","title":"Certificates exist exactly for Hamiltonian graphs"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/verify-hamiltonian-cycle-certificate-siz","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.verifyHamiltonianCycle_certificate_size","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#verify-hamiltonian-cycle-certificate-siz","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"If the Hamiltonian-cycle verifier accepts the bit strings $x$ and $y$, then $|y| = mw$, where $m = \\lfloor \\sqrt{|x|} \\rfloor$ and $w$ is the least natural number with $m \\le 2^w$, and moreover $|y| \\le |x|$.","title":"Size of an accepted Hamiltonian-cycle certificate"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/verify-hamiltonian-cycle-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.verifyHamiltonianCycle_time_le","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#verify-hamiltonian-cycle-time-le","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"There is a constant $c > 0$ such that for all bit strings $x$ and $y$, the step count of the Hamiltonian-cycle verifier on $x$ and $y$ is at most $c\\,(|x| + |y| + 1)^2$.","title":"Quadratic step bound for the Hamiltonian-cycle verifier"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/verify-hamiltonian-cycle-time-le-of-acce","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.verifyHamiltonianCycle_time_le_of_accept","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#verify-hamiltonian-cycle-time-le-of-acce","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"There is a constant $c > 0$ such that for all bit strings $x$ and $y$ that the Hamiltonian-cycle verifier accepts, its step count on $x$ and $y$ is at most $c\\,(|x| + 1)^2$.","title":"Step bound for accepted Hamiltonian-cycle certificates"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/hamiltonian-cycle-undirected-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleUndirected.HamiltonianCycleUndirectedInstance","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#hamiltonian-cycle-undirected-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"An instance of HAMILTONIAN CYCLE (UNDIRECTED) is a pair $(n, E)$ of a natural number $n$, the number of vertices, and a finite list $E$ of pairs of natural numbers, the edges.","title":"Instances of HAMILTONIAN CYCLE (UNDIRECTED)"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleUndirected.Graph","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#graph","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"For an instance $(n, E)$, its graph is the simple graph on the vertices $\\{0, \\dots, n-1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Graph of a HAMILTONIAN CYCLE (UNDIRECTED) instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleUndirected.WellFormed","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"An instance $(n, E)$ is well formed when $0 < n$, the list $E$ has no repeated entries, and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and its reversal $(v, u)$ does not belong to $E$.","title":"Well-formed HAMILTONIAN CYCLE (UNDIRECTED) instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/hamiltonian-cycle-undirected-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleUndirected.HamiltonianCycleUndirectedFeasible","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#hamiltonian-cycle-undirected-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"An instance $(n, E)$ is feasible when $n \\ge 3$ and its graph has a Hamiltonian cycle, that is, a cycle that visits every vertex exactly once.","title":"Feasible HAMILTONIAN CYCLE (UNDIRECTED) instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/hamiltonian-cycle-undirected-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleUndirected.hamiltonianCycleUndirectedDec","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#hamiltonian-cycle-undirected-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"fully_proved","summary":"The decision problem HAMILTONIAN CYCLE (UNDIRECTED) accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a well-formed instance $(n, E)$ with $n \\ge 3$ whose graph has a Hamiltonian cycle. It rejects every bitstring that does not decode to an instance.","title":"HAMILTONIAN CYCLE (UNDIRECTED) as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/hamiltonian-cycle-undirected-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianCycleUndirected.hamiltonianCycleUndirected_npComplete","parent":32412,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle/index.html#hamiltonian-cycle-undirected-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-cycle","status":"can_prove","summary":"The decision problem HAMILTONIAN CYCLE (UNDIRECTED) belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"HAMILTONIAN CYCLE (UNDIRECTED) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path","status":"can_prove","title":"Undirected Hamiltonian path"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/hamiltonian-path-undirected-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianPathUndirected.HamiltonianPathUndirectedInstance","parent":32427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/index.html#hamiltonian-path-undirected-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path","status":"fully_proved","summary":"An instance of undirected Hamiltonian path is a pair $(n, E)$ of a natural number $n$, the number of vertices, and a finite list $E$ of pairs of natural numbers, the edge list.","title":"Instances of undirected Hamiltonian path"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/graph","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianPathUndirected.Graph","parent":32427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/index.html#graph","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path","status":"fully_proved","summary":"For an instance $(n, E)$ of undirected Hamiltonian path, its graph is the simple graph on the vertex set $\\{0, \\dots, n-1\\}$ in which distinct vertices $i$ and $j$ are adjacent when $(i, j)$ or $(j, i)$ belongs to $E$.","title":"Graph of an undirected Hamiltonian path instance"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianPathUndirected.WellFormed","parent":32427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path","status":"fully_proved","summary":"An instance $(n, E)$ of undirected Hamiltonian path is well formed when $n > 0$, the list $E$ has no repeated entries, and every pair $(u, v)$ in $E$ satisfies $u < n$, $v < n$ and $u \\ne v$, and the reversed pair $(v, u)$ does not belong to $E$.","title":"Well-formed undirected Hamiltonian path instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/hamiltonian-path-undirected-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianPathUndirected.HamiltonianPathUndirectedFeasible","parent":32427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/index.html#hamiltonian-path-undirected-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path","status":"fully_proved","summary":"An instance $(n, E)$ of undirected Hamiltonian path is feasible when its graph has a Hamiltonian path: a list of length $n$ of vertices of the graph, with no repeated entries and containing every vertex, in which any two consecutive entries are adjacent.","title":"Feasible undirected Hamiltonian path instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/hamiltonian-path-undirected-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianPathUndirected.hamiltonianPathUndirectedDec","parent":32427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/index.html#hamiltonian-path-undirected-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path","status":"fully_proved","summary":"The decision problem HAMILTONIAN PATH (UNDIRECTED) accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance $(n, E)$ of undirected Hamiltonian path that is well formed and feasible; bitstrings that do not decode to an instance are rejected.","title":"HAMILTONIAN PATH (UNDIRECTED) as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/hamiltonian-path-undirected-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.HamiltonianPathUndirected.hamiltonianPathUndirected_npComplete","parent":32427,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path/index.html#hamiltonian-path-undirected-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/undirected-hamiltonian-path","status":"can_prove","summary":"The decision problem HAMILTONIAN PATH (UNDIRECTED) belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"HAMILTONIAN PATH (UNDIRECTED) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs.json","id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs","kind":"scope","navigation":"module","parent":32078,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/index.html","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs","status":"can_prove","title":"Vertex-disjoint paths in digraphs"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/directed-disjoint-paths-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedDisjointPaths.DirectedDisjointPathsInstance","parent":32434,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/index.html#directed-disjoint-paths-instance","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs","status":"fully_proved","summary":"An instance of DIRECTED DISJOINT PATHS is a triple $(n, A, P)$ consisting of a natural number $n$ (the number of vertices), a finite list $A$ of ordered pairs of natural numbers (the arcs) and a finite list $P$ of ordered pairs of natural numbers (the source-target pairs).","title":"Directed disjoint paths instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/dadj","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedDisjointPaths.DAdj","parent":32434,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/index.html#dadj","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs","status":"fully_proved","summary":"For a finite list $E$ of ordered pairs of natural numbers and natural numbers $u$ and $v$, the directed adjacency of $u$ to $v$ in $E$ is the boolean value that is true exactly when the pair $(u, v)$ occurs in $E$.","title":"Directed adjacency for DIRECTED DISJOINT PATHS"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedDisjointPaths.WellFormed","parent":32434,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs","status":"fully_proved","summary":"An instance $(n, A, P)$ of DIRECTED DISJOINT PATHS is well formed when the arc list $A$ has no repeated entries, every arc $(u, v)$ in $A$ satisfies $u < n$, $v < n$ and $u \\ne v$, and every source-target pair $(s, t)$ in $P$ satisfies $s < n$ and $t < n$.","title":"Well-formed directed disjoint paths instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/directed-disjoint-paths-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedDisjointPaths.DirectedDisjointPathsFeasible","parent":32434,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/index.html#directed-disjoint-paths-feasible","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs","status":"fully_proved","summary":"Let $(n, A, P)$ be an instance of DIRECTED DISJOINT PATHS whose list of source-target pairs is $P = ((s_0, t_0), \\dots, (s_{k-1}, t_{k-1}))$. The instance is feasible when there are lists $\\pi_0, \\dots, \\pi_{k-1}$ of natural numbers such that, for every $p < k$, the list $\\pi_p$ is nonempty, has no repeated entries, has first entry $s_p$ and last entry $t_p$, has all its entries less than $n$, and every two\u2026","title":"Feasible directed disjoint paths instances"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/directed-disjoint-paths-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedDisjointPaths.directedDisjointPathsDec","parent":32434,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/index.html#directed-disjoint-paths-dec","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs","status":"fully_proved","summary":"The decision problem DIRECTED DISJOINT PATHS accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance of DIRECTED DISJOINT PATHS that is well formed and feasible; bitstrings that do not decode to an instance are rejected.","title":"DIRECTED DISJOINT PATHS as a decision problem"},{"id":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/directed-disjoint-paths-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.DirectedDisjointPaths.directedDisjointPaths_npComplete","parent":32434,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs/index.html#directed-disjoint-paths-np-complete","scope":"TheoryOfComputation/np-completeness/paths-cycles-and-subgraphs/vertex-disjoint-paths-in-digraphs","status":"can_prove","summary":"The decision problem DIRECTED DISJOINT PATHS belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"DIRECTED DISJOINT PATHS is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems","status":"can_prove","summary":"Subset sum, subset product, partition, 3-partition, multiway partition, knapsack variants, bin and strip packing, numerical 3-dimensional matching and 0-1 integer programming.","title":"Number partitioning, knapsack and packing"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming","status":"can_prove","title":"0-1 integer programming"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/zero-one-ilpinstance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ZeroOneILP.ZeroOneILPInstance","parent":32442,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/index.html#zero-one-ilpinstance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming","status":"fully_proved","summary":"A 0-1 integer programming instance is a triple $(n, A, b)$ consisting of a natural number $n$ (the number of variables), a coefficient matrix $A$ given as a list of rows, each row a list of pairs of natural numbers, and a right-hand side $b$ given as a list of pairs of natural numbers. A pair $(p, q)$ is read as the integer $p - q$, so coefficients and right-hand side entries are stored by positive and negative\u2026","title":"0-1 integer programming instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ZeroOneILP.WellFormed","parent":32442,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming","status":"fully_proved","summary":"A 0-1 integer programming instance $(n, A, b)$ is well formed when the right-hand side list $b$ has as many entries as there are rows of the coefficient matrix $A$, and every row of $A$ has exactly $n$ entries.","title":"Well-formed 0-1 integer programming instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/zero-one-ilpfeasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ZeroOneILP.ZeroOneILPFeasible","parent":32442,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/index.html#zero-one-ilpfeasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming","status":"fully_proved","summary":"Let $(n, A, b)$ be a 0-1 integer programming instance whose coefficient matrix has the rows $A_0, \\dots, A_{m-1}$, with list entries indexed from $0$. For $r < m$ and $j < n$, let $a_{rj}$ be the integer $p - q$ if row $A_r$ has a $j$-th entry $(p, q)$, and $a_{rj} = 0$ otherwise; let $b_r$ be the integer $p' - q'$ if $b$ has an $r$-th entry $(p', q')$, and $b_r = 0$ otherwise. The instance is feasible when there\u2026","title":"Feasible 0-1 integer programs"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/zero-one-ilpdec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ZeroOneILP.zeroOneILPDec","parent":32442,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/index.html#zero-one-ilpdec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming","status":"fully_proved","summary":"The decision problem 0-1 INTEGER PROGRAMMING FEASIBILITY accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a 0-1 integer programming instance that is well formed and feasible; it rejects every bitstring that does not decode to such an instance.","title":"0-1 integer programming feasibility as a decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/zero-one-ilp-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ZeroOneILP.zeroOneILP_npComplete","parent":32442,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming/index.html#zero-one-ilp-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/0-1-integer-programming","status":"can_prove","summary":"The decision problem 0-1 INTEGER PROGRAMMING FEASIBILITY belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"0-1 INTEGER PROGRAMMING FEASIBILITY is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem","status":"can_prove","title":"3-partition problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/three-partition-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreePartition.ThreePartitionInstance","parent":32448,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/index.html#three-partition-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem","status":"fully_proved","summary":"An instance of 3-PARTITION is a pair $(a, B)$ of a finite list $a$ of natural numbers and a natural number $B$, the target sum.","title":"3-partition instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreePartition.WellFormed","parent":32448,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem","status":"fully_proved","summary":"A 3-partition instance $(a, B)$ is well formed when the length of the list $a$ is a multiple of $3$.","title":"Well-formed 3-partition instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/three-partition-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreePartition.ThreePartitionFeasible","parent":32448,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/index.html#three-partition-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem","status":"fully_proved","summary":"Let $(a, B)$ be a 3-partition instance with $a = (a_0, \\dots, a_{N-1})$, and let $m = \\lfloor N/3 \\rfloor$. The instance is feasible when there is a map $g : \\{0, \\dots, N-1\\} \\to \\{0, \\dots, m-1\\}$ such that for every $t < m$ exactly three indices $j$ satisfy $g(j) = t$, and the sum of $a_j$ over these indices equals $B$.","title":"Feasible 3-partition instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/three-partition-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreePartition.threePartitionDec","parent":32448,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/index.html#three-partition-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem","status":"fully_proved","summary":"The decision problem 3-PARTITION accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a 3-partition instance $(a, B)$ that is well formed (the length of the list $a$ is a multiple of $3$) and feasible (the entries of $a$ can be split into triples each summing to $B$). Bitstrings that do not decode to an instance are rejected.","title":"3-PARTITION as a decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/three-partition-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreePartition.threePartition_npComplete","parent":32448,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem/index.html#three-partition-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/3-partition-problem","status":"can_prove","summary":"The decision problem 3-PARTITION on bitstrings is in NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"3-PARTITION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing","status":"can_prove","title":"Bin packing"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/bin-packing-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinPacking.BinPackingInstance","parent":32454,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/index.html#bin-packing-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing","status":"fully_proved","summary":"An instance of BIN PACKING is a triple $(s, B, K)$ of a finite list $s$ of natural numbers (the item sizes), a natural number $B$ (the bin capacity) and a natural number $K$ (the number of bins).","title":"Bin packing instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/bin-packing-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinPacking.BinPackingFeasible","parent":32454,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/index.html#bin-packing-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing","status":"fully_proved","summary":"An instance $(s, B, K)$ of BIN PACKING is feasible when there is a list of exactly $K$ bins, each a finite list of natural numbers with sum at most $B$, such that the concatenation of the bins is a permutation of $s$.","title":"Feasible bin packing instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/bin-packing-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinPacking.binPackingDec","parent":32454,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/index.html#bin-packing-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing","status":"fully_proved","summary":"BIN PACKING is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of bin packing instances, to a feasible instance; it rejects every bitstring that fails to decode.","title":"The bin packing decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/bin-packing-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.BinPacking.binPacking_npComplete","parent":32454,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing/index.html#bin-packing-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/bin-packing","status":"can_prove","summary":"The bin packing decision problem belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"BIN PACKING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum","status":"can_prove","title":"Exact-cardinality subset sum"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/cardinality-subset-sum-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CardinalitySubsetSum.CardinalitySubsetSumInstance","parent":32459,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/index.html#cardinality-subset-sum-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum","status":"fully_proved","summary":"An exact-cardinality subset-sum instance is a triple $(a, K, T)$ consisting of a list $a$ of natural numbers, a natural number $K$ (the cardinality) and a natural number $T$ (the target).","title":"Exact-cardinality subset-sum instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/cardinality-subset-sum-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CardinalitySubsetSum.CardinalitySubsetSumFeasible","parent":32459,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/index.html#cardinality-subset-sum-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum","status":"fully_proved","summary":"An exact-cardinality subset-sum instance $(a, K, T)$ is feasible when some subsequence of the list $a$ (entries kept in order, not necessarily contiguous) has exactly $K$ entries and sum $T$.","title":"Feasibility of an exact-cardinality subset-sum instance"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/cardinality-subset-sum-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CardinalitySubsetSum.cardinalitySubsetSumDec","parent":32459,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/index.html#cardinality-subset-sum-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum","status":"fully_proved","summary":"The decision problem EXACT-CARDINALITY SUBSET SUM accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an exact-cardinality subset-sum instance that is feasible; it rejects every bitstring that does not decode.","title":"Exact-cardinality subset sum as a decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/cardinality-subset-sum-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.CardinalitySubsetSum.cardinalitySubsetSum_npComplete","parent":32459,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum/index.html#cardinality-subset-sum-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/exact-cardinality-subset-sum","status":"can_prove","summary":"The decision problem EXACT-CARDINALITY SUBSET SUM lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"EXACT-CARDINALITY SUBSET SUM is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem","status":"can_prove","title":"Knapsack problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/knapsack-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Knapsack.KnapsackInstance","parent":32464,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/index.html#knapsack-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem","status":"fully_proved","summary":"A 0/1 knapsack instance is a triple $(I, C, T)$ of a list $I$ of items, each a pair $(w, v)$ of natural numbers giving its weight $w$ and value $v$, a natural-number capacity $C$, and a natural-number target value $T$.","title":"Knapsack instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/knapsack-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Knapsack.KnapsackFeasible","parent":32464,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/index.html#knapsack-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem","status":"fully_proved","summary":"A knapsack instance $(I, C, T)$, with item list $I$, capacity $C$ and target value $T$, is feasible when some subsequence of $I$, possibly empty, has total weight at most $C$ and total value at least $T$. Repeated items in $I$ are distinct occurrences that can be selected separately.","title":"Feasible knapsack instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/knapsack-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Knapsack.knapsackDec","parent":32464,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/index.html#knapsack-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem","status":"fully_proved","summary":"KNAPSACK is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a knapsack instance $(I, C, T)$ for which some subsequence of the item list $I$, possibly empty, has total weight at most $C$ and total value at least $T$. Bitstrings that do not decode are rejected.","title":"The KNAPSACK decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/knapsack-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Knapsack.knapsack_npComplete","parent":32464,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem/index.html#knapsack-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/knapsack-problem","status":"can_prove","summary":"The KNAPSACK decision problem is in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"KNAPSACK is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem","status":"can_prove","title":"Multiple knapsack problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/multiple-knapsack-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultipleKnapsack.MultipleKnapsackInstance","parent":32469,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/index.html#multiple-knapsack-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem","status":"fully_proved","summary":"A multiple-knapsack instance consists of a finite list of items, each a pair $(w, v)$ of natural numbers giving its weight $w$ and value $v$, a finite list of natural-number knapsack capacities, and a natural-number target value.","title":"Multiple-knapsack instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/multiple-knapsack-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultipleKnapsack.MultipleKnapsackFeasible","parent":32469,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/index.html#multiple-knapsack-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem","status":"fully_proved","summary":"Let a multiple-knapsack instance have items $(w_0, v_0), \\dots, (w_{k-1}, v_{k-1})$ (weight, value), knapsack capacities $c_0, \\dots, c_{m-1}$ and target $V$. The instance is feasible when there are $a_0, \\dots, a_{k-1} \\in \\{0, \\dots, m\\}$, where $a_j = b < m$ puts item $j$ in knapsack $b$ and $a_j = m$ leaves item $j$ out, such that for every $b < m$ the total weight $\\sum_{j : a_j = b} w_j$ is at most $c_b$, and\u2026","title":"Feasible multiple-knapsack instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/multiple-knapsack-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultipleKnapsack.multipleKnapsackDec","parent":32469,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/index.html#multiple-knapsack-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem","status":"fully_proved","summary":"The decision problem MULTIPLE KNAPSACK accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a feasible multiple-knapsack instance. Bitstrings that do not decode are rejected.","title":"MULTIPLE KNAPSACK as a decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/multiple-knapsack-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultipleKnapsack.multipleKnapsack_npComplete","parent":32469,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem/index.html#multiple-knapsack-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiple-knapsack-problem","status":"can_prove","summary":"The decision problem MULTIPLE KNAPSACK belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to MULTIPLE KNAPSACK.","title":"MULTIPLE KNAPSACK is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning","status":"can_prove","title":"Multiway number partitioning"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/multiway-partition-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayPartition.MultiwayPartitionInstance","parent":32474,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/index.html#multiway-partition-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning","status":"fully_proved","summary":"A multiway number partition instance is a pair $(a, K)$ consisting of a list $a$ of natural numbers and a natural number $K$ (the number of parts).","title":"Multiway number partition instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/multiway-partition-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayPartition.MultiwayPartitionFeasible","parent":32474,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/index.html#multiway-partition-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning","status":"fully_proved","summary":"Let $(a, K)$ be a multiway number partition instance with $a = (a_0, \\dots, a_{m-1})$. It is feasible when $K \\ge 1$ and there is a map $f$ from $\\{0, \\dots, m-1\\}$ onto $\\{0, \\dots, K-1\\}$ such that the sums $\\sum_{j < m,\\, f(j) = p} a_j$ for $p = 0, \\dots, K-1$ are all equal. In other words, the entries of $a$, counted with repetition, can be split into $K$ nonempty parts with equal sums.","title":"Feasible multiway number partition instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/multiway-partition-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayPartition.multiwayPartitionDec","parent":32474,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/index.html#multiway-partition-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning","status":"fully_proved","summary":"MULTIWAY NUMBER PARTITION is the decision problem on bitstrings that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a multiway number partition instance $(a, K)$ that is feasible. Bitstrings that do not decode to a multiway number partition instance are rejected.","title":"The MULTIWAY NUMBER PARTITION decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/multiway-partition-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MultiwayPartition.multiwayPartition_npComplete","parent":32474,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning/index.html#multiway-partition-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/multiway-number-partitioning","status":"can_prove","summary":"Let MULTIWAY NUMBER PARTITION be the decision problem that accepts exactly the bitstrings decoding to a feasible multiway number partition instance: a list of natural numbers and a number $K \\ge 1$ such that the entries of the list, counted with repetition, can be split into $K$ nonempty parts with equal sums. Then MULTIWAY NUMBER PARTITION is in NP, and every decision problem in NP reduces to it by a\u2026","title":"MULTIWAY NUMBER PARTITION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching","status":"can_prove","title":"Numerical 3-dimensional matching"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/numerical3-dminstance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Numerical3DM.Numerical3DMInstance","parent":32479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/index.html#numerical3-dminstance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching","status":"fully_proved","summary":"An instance of NUMERICAL 3-DIMENSIONAL MATCHING is a quadruple $(A, B, C, t)$ of three finite lists $A$, $B$, $C$ of natural numbers and a natural number $t$, the target.","title":"Numerical 3-dimensional matching instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Numerical3DM.WellFormed","parent":32479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching","status":"fully_proved","summary":"An instance $(A, B, C, t)$ of NUMERICAL 3-DIMENSIONAL MATCHING is well formed when the lists $B$ and $C$ have the same length as the list $A$.","title":"Well-formed numerical 3-dimensional matching instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/numerical3-dmfeasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Numerical3DM.Numerical3DMFeasible","parent":32479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/index.html#numerical3-dmfeasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching","status":"fully_proved","summary":"Let $(A, B, C, t)$ be an instance and let $n$ be the length of $A$, with lists indexed from $0$. The instance is feasible when there are permutations $p$ and $q$ of the numbers $0$ through $n-1$ such that $A_j + B_{p(j)} + C_{q(j)} = t$ for every $j < n$, where an entry of $B$ or $C$ at an index past the end of its list counts as $0$.","title":"Feasible numerical 3-dimensional matching instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/numerical3-dmdec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Numerical3DM.numerical3DMDec","parent":32479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/index.html#numerical3-dmdec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching","status":"fully_proved","summary":"The decision problem NUMERICAL 3-DIMENSIONAL MATCHING accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding (natural numbers in binary), to an instance $(A, B, C, t)$ that is well formed and feasible: the lists $B$ and $C$ have the same length $n$ as the list $A$, and there are permutations $p$ and $q$ of the numbers $0$ through $n-1$ such that $A_j + B_{p(j)} + C_{q(j)} = t$ for\u2026","title":"NUMERICAL 3-DIMENSIONAL MATCHING as a decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/numerical3-dm-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Numerical3DM.numerical3DM_npComplete","parent":32479,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching/index.html#numerical3-dm-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/numerical-3-dimensional-matching","status":"can_prove","summary":"The decision problem NUMERICAL 3-DIMENSIONAL MATCHING belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"NUMERICAL 3-DIMENSIONAL MATCHING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing","status":"can_prove","title":"Strip packing"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/strip-packing-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.StripPacking.StripPackingInstance","parent":32485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/index.html#strip-packing-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing","status":"fully_proved","summary":"A strip packing instance is a triple $(R, W, H)$ consisting of a finite list $R = ((w_1, h_1), \\dots, (w_n, h_n))$ of pairs of natural numbers, the widths and heights of the rectangles, a natural number $W$, the width of the strip, and a natural number $H$, the height bound.","title":"Strip packing instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.StripPacking.WellFormed","parent":32485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing","status":"fully_proved","summary":"A strip packing instance $(R, W, H)$ is well formed when every rectangle $(w, h)$ in the list $R$ has $w > 0$ and $h > 0$.","title":"Well-formed strip packing instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/strip-packing-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.StripPacking.StripPackingFeasible","parent":32485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/index.html#strip-packing-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing","status":"fully_proved","summary":"A strip packing instance $(R, W, H)$ with $R = ((w_1, h_1), \\dots, (w_n, h_n))$ is feasible when there are natural numbers $x_1, y_1, \\dots, x_n, y_n$ such that $x_j + w_j \\le W$ and $y_j + h_j \\le H$ for every $1 \\le j \\le n$, and for all $1 \\le a < b \\le n$ at least one of $x_a + w_a \\le x_b$, $x_b + w_b \\le x_a$, $y_a + h_a \\le y_b$ and $y_b + h_b \\le y_a$ holds.","title":"Feasible strip packings"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/strip-packing-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.StripPacking.stripPackingDec","parent":32485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/index.html#strip-packing-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing","status":"fully_proved","summary":"STRIP PACKING is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of strip packing instances, to a strip packing instance that is well formed and feasible. Bitstrings that do not decode to a strip packing instance are rejected.","title":"The STRIP PACKING decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/strip-packing-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.StripPacking.stripPacking_npComplete","parent":32485,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing/index.html#strip-packing-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/strip-packing","status":"can_prove","summary":"The STRIP PACKING decision problem belongs to NP, and every decision problem $L$ in NP reduces to STRIP PACKING by a polynomial-time many-one reduction.","title":"STRIP PACKING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product","status":"can_prove","title":"Subset product"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/subset-product-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubsetProduct.SubsetProductInstance","parent":32491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/index.html#subset-product-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product","status":"fully_proved","summary":"An instance of SUBSET PRODUCT is a pair $(a, T)$ of a finite list $a$ of natural numbers and a target natural number $T$.","title":"SUBSET PRODUCT instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubsetProduct.WellFormed","parent":32491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product","status":"fully_proved","summary":"An instance $(a, T)$ of SUBSET PRODUCT, with $a$ a finite list of natural numbers and $T$ a natural number, is well formed when every entry of $a$ is positive.","title":"Well-formed SUBSET PRODUCT instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/subset-product-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubsetProduct.SubsetProductFeasible","parent":32491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/index.html#subset-product-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product","status":"fully_proved","summary":"An instance $(a, T)$ of SUBSET PRODUCT, with $a$ a finite list of natural numbers and $T$ a natural number, is feasible when some sublist of $a$ (a list obtained from $a$ by deleting entries and keeping the order of the rest) has product exactly $T$.","title":"Feasible SUBSET PRODUCT instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/subset-product-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubsetProduct.subsetProductDec","parent":32491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/index.html#subset-product-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product","status":"fully_proved","summary":"The decision problem SUBSET PRODUCT accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs of a finite list of natural numbers and a natural number, to an instance $(a, T)$ that is well formed (every entry of $a$ is positive) and feasible (some sublist of $a$ has product exactly $T$). It rejects every bitstring that does not decode to an instance.","title":"SUBSET PRODUCT as a decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/subset-product-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SubsetProduct.subsetProduct_npComplete","parent":32491,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product/index.html#subset-product-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-product","status":"can_prove","summary":"The decision problem SUBSET PRODUCT lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to SUBSET PRODUCT.","title":"SUBSET PRODUCT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum","status":"fully_proved","title":"Subset sum"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","kind":"scope","navigation":"transparent","parent":32497,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","title":"Bit-string encoding of SUBSET-SUM instances to Value of the empty bit list"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/encode-ss","kind":"node","lean":"Cslib.Complexity.Karp.encodeSS","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#encode-ss","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"The instance with numbers $a_1, \\dots, a_r$ and target $t$ is encoded as the block of $t$ followed by the blocks of $a_1, \\dots, a_r$ in order. The block of a number is its binary expansion, least significant bit first and empty for the number $0$, made self-delimiting by writing each bit $c$ as $1c$ and appending a final $0$.","title":"Bit-string encoding of SUBSET-SUM instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/subset-sum-dec","kind":"node","lean":"Cslib.Complexity.Karp.subsetSumDec","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#subset-sum-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"The decision problem that accepts a bit string $x$ exactly when $x$ encodes a SUBSET-SUM instance with a list of numbers $a_1, \\dots, a_r$ and a target $t$ such that some sublist of $a_1, \\dots, a_r$ sums to $t$. A sublist keeps some of the entries in their order, so a number may be used as often as it occurs in the list.","title":"The SUBSET-SUM problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/pair-digit","kind":"node","lean":"Cslib.Complexity.Karp.pairDigit","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#pair-digit","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"The place value of pair number $q$ is $4^q$. The numbers of the reduction from VERTEX-COVER to SUBSET-SUM are written in base $4$, with the digit at position $q$ belonging to pair $q$.","title":"Place value of a vertex pair in the SUBSET-SUM reduction"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/touches","kind":"node","lean":"Cslib.Complexity.Karp.touches","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#touches","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $p_0, \\dots, p_{m-1}$ be the vertex pairs $(i, j)$ with $i < j < n$ in encoding order (rows $i$ ascending, within a row $j$ descending). For a list $b$ of pair bits, a vertex $v$ and an index $q$, the test is true when bit $b_q$ is set and $v$ is one of the two entries of $p_q$. An index past the end of $b$ counts as an unset bit, and for $q \\ge m$ the pair $p_q$ is taken to be $(0, 0)$.","title":"Pairs touching a vertex"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/vtx-num","kind":"node","lean":"Cslib.Complexity.Karp.vtxNum","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#vtx-num","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $p_0, \\dots, p_{m-1}$ be the vertex pairs $(i, j)$ with $i < j < n$ in encoding order (rows $i$ ascending, within a row $j$ descending), and let $b$ be a list of pair bits. The number of a vertex $v$ is $4^m + \\sum 4^q$, the sum over the indices $q < m$ such that bit $b_q$ is set and $v$ is an entry of $p_q$, where an index past the end of $b$ counts as unset. In base $4$ it has digit $1$ at position $m$ and at\u2026","title":"Vertex numbers of the SUBSET-SUM reduction"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/edge-num","kind":"node","lean":"Cslib.Complexity.Karp.edgeNum","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#edge-num","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Given a list $b$ of pair bits and a pair index $q$, the number of pair $q$ is $4^q$ if bit $b_q$ is set and $0$ otherwise; an index past the end of $b$ counts as an unset bit.","title":"Edge numbers of the SUBSET-SUM reduction"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/ss-target","kind":"node","lean":"Cslib.Complexity.Karp.ssTarget","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#ss-target","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $m = n(n-1)/2$ be the number of vertex pairs $(i, j)$ with $i < j < n$, and let $b$ be a list of pair bits. For $k \\in \\mathbb{N}$ the target is $k \\cdot 4^m + \\sum 2 \\cdot 4^q$, the sum over the indices $q < m$ such that bit $b_q$ is set (an index past the end of $b$ counts as unset). So the quotient of the target by $4^m$ is $k$, and its base-$4$ digit at position $q < m$ is $2$ when $b_q$ is set and $0$\u2026","title":"Target of the SUBSET-SUM reduction"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/ss-nums","kind":"node","lean":"Cslib.Complexity.Karp.ssNums","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#ss-nums","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"For $n$ vertices and a list $b$ of pair bits, with $m = n(n-1)/2$ vertex pairs, the numbers of the instance are the vertex numbers of $v = 0, \\dots, n - 1$, in this order, followed by the edge numbers of the pairs $q = 0, \\dots, m - 1$, in this order.","title":"Numbers of the SUBSET-SUM reduction"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/no-ss","kind":"node","lean":"Cslib.Complexity.Karp.noSS","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#no-ss","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"The encoding of the SUBSET-SUM instance with no numbers and target $1$. It is a no-instance, because the only sublist of the empty list sums to $0$.","title":"Fixed no-instance of SUBSET-SUM"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/reduce-ss","kind":"node","lean":"Cslib.Complexity.Karp.reduceSS","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#reduce-ss","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"The map on bit strings reads $x$ as $k$ ones, a zero, $n$ ones, a zero and a remaining list $b$ of pair bits. If this parse succeeds and $b$ has exactly $n(n-1)/2$ bits, one per vertex pair, then $x$ is sent to the encoding of the SUBSET-SUM instance whose numbers are the numbers of the reduction for $n$ and $b$ and whose target is the target for $k$, $n$ and $b$. Otherwise $x$ is sent to the fixed no-instance with\u2026","title":"Reduction from VERTEX-COVER to SUBSET-SUM"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/subset-sum-dec-no-ss","kind":"node","lean":"Cslib.Complexity.Karp.subsetSumDec_noSS","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#subset-sum-dec-no-ss","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"The encoded SUBSET-SUM instance with no numbers and target $1$ is a no-instance: the SUBSET-SUM decision problem returns false on it.","title":"The fixed no-instance of SUBSET-SUM"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/add-sum-digits-eq-iff","kind":"node","lean":"Cslib.Complexity.Karp.add_sum_digits_eq_iff","parent":32498,"reading_title":{"format":"inline-parts-v1","parts":[["text","Base-4 expansions with digits below "],["math","\\(4\\)"],["text"," are unique"]],"source":"Base-4 expansions with digits below $4$ are unique"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#add-sum-digits-eq-iff","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $m, A, B \\in \\mathbb{N}$ and let $d, e : \\mathbb{N} \\to \\mathbb{N}$ satisfy $d(q) < 4$ and $e(q) < 4$ for every $q < m$. Then $A \\cdot 4^m + \\sum_{q < m} d(q) \\, 4^q = B \\cdot 4^m + \\sum_{q < m} e(q) \\, 4^q$ holds if and only if $A = B$ and $d(q) = e(q)$ for every $q < m$.","title":"Base-4 expansions with digits below $4$ are unique"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/exists-sublist-ss-nums-iff","kind":"node","lean":"Cslib.Complexity.Karp.exists_sublist_ssNums_iff","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#exists-sublist-ss-nums-iff","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $n, t \\in \\mathbb{N}$, let $b$ be a list of booleans, and let $m$ be the length of the list of vertex pairs $(i, j)$, $i < j < n$. Write $x_v$ for the vertex number of $v$ and $y_q$ for the pair number of $q$ defined from $n$ and $b$. Some sublist (a subsequence, not necessarily contiguous) of the reduction's numbers for $n$ and $b$ sums to $t$ if and only if there are finite sets $S$ of vertices $v < n$ and\u2026","title":"Sublists of the reduction's numbers as vertex and pair sets"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/vtx-num-eq","kind":"node","lean":"Cslib.Complexity.Karp.vtxNum_eq","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#vtx-num-eq","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $n, v \\in \\mathbb{N}$, let $b$ be a list of booleans, and let $m$ be the length of the list of vertex pairs $(i, j)$, $i < j < n$. The vertex number of $v$ defined from $n$ and $b$ equals $4^m + \\sum_{q < m} t_q 4^q$, where $t_q = 1$ if entry $q$ of $b$ (counting from $0$) is true and $v$ is an endpoint of the $q$-th pair, and $t_q = 0$ otherwise.","title":"A vertex number in base 4"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/ss-target-eq","kind":"node","lean":"Cslib.Complexity.Karp.ssTarget_eq","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#ss-target-eq","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $k, n \\in \\mathbb{N}$, let $b$ be a list of booleans, and let $m$ be the length of the list of vertex pairs $(i, j)$, $i < j < n$. The reduction's target for $k$, $n$ and $b$ equals $k \\cdot 4^m + \\sum_{q < m} c_q 4^q$, where $c_q = 2$ if entry $q$ of $b$ (counting from $0$) is true and $c_q = 0$ otherwise, in particular when $b$ has no entry $q$.","title":"The target in base 4"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/sum-vtx-num-add-sum-edge-num","kind":"node","lean":"Cslib.Complexity.Karp.sum_vtxNum_add_sum_edgeNum","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#sum-vtx-num-add-sum-edge-num","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $b$ be a list of booleans, let $m$ be the length of the list of vertex pairs $(i, j)$, $i < j < n$, and let $S$ and $F$ be finite sets of natural numbers with every element of $F$ below $m$. Write $x_v$ for the vertex number of $v$ and $y_q$ for the pair number of $q$ defined from $n$ and $b$. For $q < m$ let $t_{v,q} = 1$ if entry $q$ of $b$ (counting from $0$) is true and $v$ is an\u2026","title":"A subset sum of the reduction's numbers in base 4"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/sum-touches-eq","kind":"node","lean":"Cslib.Complexity.Karp.sum_touches_eq","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#sum-touches-eq","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $n$ be a natural number, let $p_0, \\dots, p_{m-1}$ be the vertex pairs in encoding order for $n$, let $b$ be a list of pair bits, let $q < m$ with $p_q = (i, j)$, and let $S$ be a finite set of natural numbers. Write $[P]$ for $1$ if $P$ holds and $0$ otherwise. Then the number of $v \\in S$ for which the test whether pair $q$ touches $v$ under $b$ is true equals $[i \\in S] + [j \\in S]$ if the bit $b_q$ is set,\u2026","title":"Vertices of a set touching a vertex pair"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/is-vertex-cover-to-graph-iff","kind":"node","lean":"Cslib.Complexity.Karp.isVertexCover_toGraph_iff","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#is-vertex-cover-to-graph-iff","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $b$ be a list of booleans with one entry for each pair in the list of vertex pairs $(i, j)$, $i < j < n$, and let $G$ be the graph on the vertices $0, \\dots, n - 1$ encoded by the pair bits $b$. Let $s$ be a finite set of vertices of $G$ and $S$ a finite set of natural numbers such that each vertex $i$ lies in $s$ exactly when $i \\in S$. Then $s$ is a vertex cover of $G$ if and only if,\u2026","title":"Vertex covers in terms of the pair bits"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/exists-cover-iff-exists-sublist","kind":"node","lean":"Cslib.Complexity.Karp.exists_cover_iff_exists_sublist","parent":32498,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vertex covers of size "],["math","\\(k\\)"],["text"," match subset sums hitting the target"]],"source":"Vertex covers of size $k$ match subset sums hitting the target"},"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#exists-cover-iff-exists-sublist","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"Let $k, n \\in \\mathbb{N}$ and let $b$ be a list of booleans with one entry for each pair in the list of vertex pairs $(i, j)$, $i < j < n$. Let $G$ be the graph on the vertices $0, \\dots, n - 1$ encoded by the pair bits $b$. Then $G$ has a vertex cover with exactly $k$ vertices if and only if some sublist (a subsequence, not necessarily contiguous) of the reduction's numbers for $n$ and $b$ sums to the reduction's\u2026","title":"Vertex covers of size $k$ match subset sums hitting the target"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/bits-val-nil","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.bitsVal_nil","parent":32498,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01/index.html#bits-val-nil","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-01","status":"fully_proved","summary":"The empty bit list, read as a binary number least significant bit first, has value $0$.","title":"Value of the empty bit list"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","kind":"scope","navigation":"transparent","parent":32497,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","title":"Value of a bit list with a leading bit to Reading a pair bit or vertex mark"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/bits-val-cons","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.bitsVal_cons","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#bits-val-cons","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"Write $\\mathrm{val}(l)$ for the value of a bit list $l$ read as a binary number least significant bit first, and for a bit $b$ write $[b]$ for $1$ if $b$ is true and $0$ if $b$ is false. For every bit $b$ and every bit list $l$, the list with first entry $b$ followed by $l$ has value $[b] + 2 \\cdot \\mathrm{val}(l)$.","title":"Value of a bit list with a leading bit"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/encode-nat-bits-val","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.encodeNat_bitsVal","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#encode-nat-bits-val","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"For every bit list $l$ whose last entry is not false (the empty list included), the binary code, least significant bit first, of the value of $l$ read least significant bit first is $l$ itself.","title":"A bit list without trailing false is the binary code of its value"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/bits-val-flat-map2","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.bitsVal_flatMap2","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#bits-val-flat-map2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"Write $\\mathrm{val}(l)$ for the value of a bit list $l$ read as a binary number least significant bit first. Let $f$ assign to each bit a list of exactly two bits. Then for every bit list $b_0, b_1, \\dots, b_{m-1}$, the concatenation $f(b_0) f(b_1) \\cdots f(b_{m-1})$ has value $\\sum_{q < m} 4^q \\, \\mathrm{val}(f(b_q))$.","title":"Value of a list of two-bit digits"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/encode-nat-bits-val-rev","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.encodeNat_bitsVal_rev","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#encode-nat-bits-val-rev","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"Read a bit list $l$ as a binary number, least significant bit first. The binary code of its value (least significant bit first, with no trailing false bits) is $l$ with all of its trailing false bits removed.","title":"Binary code of the value of a bit list"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/delimit-append-true","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.delimit_append_true","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#delimit-append-true","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"For every bit list $u = (u_1, \\dots, u_k)$, the self-delimiting form of $u$ followed by the bit true is $(\\mathrm{true}, u_1, \\mathrm{true}, u_2, \\dots, \\mathrm{true}, u_k, \\mathrm{true}, \\mathrm{true}, \\mathrm{false})$.","title":"Delimiting a bit list that ends in true"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/blk","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.blk","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#blk","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"For a bit $t$, the block is the four-bit list $(\\mathrm{true}, t, \\mathrm{true}, \\mathrm{false})$.","title":"Output block of a vertex-number digit"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/edge-enc","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.edgeEnc","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#edge-enc","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"For $q \\in \\mathbb{N}$ and a bit $b$: if $b$ is true, the block is $q$ copies of $(\\mathrm{true}, \\mathrm{false}, \\mathrm{true}, \\mathrm{false})$ followed by $(\\mathrm{true}, \\mathrm{true}, \\mathrm{false})$; if $b$ is false, it is the one-bit list $(\\mathrm{false})$.","title":"Output block of an edge number"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/edge-out","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.edgeOut","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#edge-out","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"Write $E(q, b)$ for the output block of the edge number of pair $q$ with bit $b$. For a bit list $(b_{m-1}, \\dots, b_1, b_0)$, listing the bit of the last pair first, the output is the concatenation $E(0, b_0) \\, E(1, b_1) \\cdots E(m-1, b_{m-1})$.","title":"Output of all edge numbers"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/walk-m","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.walkM","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#walk-m","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"For lists $L$ and $R$ of booleans, the pair walk $W(L, R)$ is the list of pairs of booleans defined by recursion on $L$. If $L$ is empty, then $W(L, R)$ is empty. If $L$ has first entry $a$ and remaining entries $L'$, then $W(L, R)$ consists of the pairs $(a, r)$ for the entries $r$ of $R$ in order, followed by $W(L', R')$, where $R'$ is the list $R$ with $a$ put in front.","title":"The pair walk over lists of marks"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/tri","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.tri","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#tri","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"For a natural number $n$, the triangular number $T(n)$ is the sum $0 + 1 + \\cdots + (n - 1)$ of the natural numbers below $n$. It is defined by recursion: $T(0) = 0$ and $T(i + 1) = T(i) + i$.","title":"Triangular numbers"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/walk-m-pairs","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.walkM_pairs","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#walk-m-pairs","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"Let $f$ be a function from the natural numbers to the booleans, let $n$ be a natural number, and let $W$ be the pair walk. For every $i \\le n$, let $L$ be the list $f(i - 1), \\dots, f(1), f(0)$ (empty when $i = 0$) and $R$ the list $f(i), f(i + 1), \\dots, f(n - 1)$ (empty when $i = n$). Then $W(L, R)$ is obtained as follows: list the pairs $(a, j)$ of natural numbers with $a < i$ and $a < j < n$, with $a$ ascending\u2026","title":"The pair walk lists the vertex pairs in reverse"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/mk-l","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.mkL","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#mk-l","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"For natural numbers $n$ and $v$, the list of $n$ bits whose entries, from first to last, record for $i = n-1, n-2, \\dots, 0$ whether $i = v$.","title":"Vertex marks for a vertex"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/walk-touches","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.walk_touches","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#walk-touches","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"Let $n$ and $v$ be natural numbers, let $m$ be the number of vertex pairs in encoding order for $n$, and let $b$ be a list of $m$ pair bits. Let $L$ be the list of vertex marks for $v$ among $n$, let $(x_0, y_0), \\dots, (x_{k-1}, y_{k-1})$ be the pair walk $W(L, [\\,])$, and let $c_0, \\dots, c_{m-1}$ be the list $b$ reversed. Then the list of the entries $c_l \\wedge (y_l \\vee x_l)$ for\u2026","title":"Touch bits from the pair walk of a vertex"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/k","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.K","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#k","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"The machine has fifteen stacks: the input $X$ and the output $O$; $A$, which holds the cover size $k$, and $H$, used to halve it; $N$, which holds the number $n$ of vertices, and $M$, the pair count; $C$, which counts the pair bits; the scratch stacks $S_1$ and $S_2$; $Y$ and $Z$, which hold the pair bits; $T$, which holds the target's bits; and $L$, $R$ and $R_t$, which hold the vertex marks.","title":"Stacks of the SUBSET-SUM reduction machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/untitled","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.\u0393","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#untitled","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"In the VERTEX-COVER to SUBSET-SUM machine the output stack $\\mathsf{O}$ holds Booleans, and every other stack holds token symbols.","title":"Stack alphabets of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/lab","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.Lab","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#lab","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"The control labels of the subset-sum emitter machine are: $\\mathrm{rdK}$, $\\mathrm{rdN}$ and $\\mathrm{mv}$ (parsing the two unary numbers $k$ and $n$ that open the input, and moving the pair bits); $\\mathrm{tri}$, $\\mathrm{triC}$, $\\mathrm{triC2}$, $\\mathrm{cmp}$ and, for every ordering $o$, the labels $\\mathrm{rc}(o)$ and $\\mathrm{rcB}(o)$ (counting the pairs and comparing that count with the number of pair bits);\u2026","title":"Labels of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/u","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.U","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#u","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"The unit symbol of the counters of the VERTEX-COVER to SUBSET-SUM machine is the token symbol $\\mathrm{cl}$; a counter with value $n$ holds exactly $n$ copies of it.","title":"Counter unit of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/is-t","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.isT","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#is-t","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"On the result of a pop, which is either a symbol or nothing (when the stack was empty), the test returns true exactly when the result is the input-bit symbol $\\mathrm{bit}(\\mathrm{true})$, and false otherwise.","title":"Test for the input bit true (SUBSET-SUM reduction machine)"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/is-v","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.isV","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#is-v","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"On the result of a pop, which is either a symbol or nothing (when the stack was empty), the test returns true exactly when the result is the value symbol $\\mathrm{val}(\\mathrm{true})$, which this machine uses for a pair bit or vertex mark that is true, and false otherwise.","title":"Test for the value symbol true (SUBSET-SUM reduction machine)"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/val-of","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.valOf","parent":32519,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02/index.html#val-of","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-02","status":"fully_proved","summary":"The partial map from token symbols to bits that sends the symbol $\\mathrm{val}(b)$ to $b$ and gives no value on every other symbol.","title":"Reading a pair bit or vertex mark"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","kind":"scope","navigation":"transparent","parent":32497,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","title":"Branch after the count comparison (SUBSET-SUM reduction machine) to One halving step of the binary conversion (SUBSET-SUM reduction machine)"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/cmp-out","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.cmpOut","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#cmp-out","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"The label the machine takes after comparing the pair-bit count with the pair count: the result \"equal\" leads to the label $\\mathrm{eg}$, which starts the edge-number phase, and the results \"less than\" and \"greater than\" lead to the failure label $\\mathrm{bad}$, which writes the fixed no-instance of SUBSET-SUM.","title":"Branch after the count comparison (SUBSET-SUM reduction machine)"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/prog","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.prog","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#prog","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"A multi-stack machine with the labels of the subset-sum emitter as control labels, a Boolean internal state, and the stacks of the emitter: the output stack $O$ holds bits, every other stack holds symbols. Stacks are read top first, $U$ is the counter unit, and to prepend a list to $O$ is to push it so that it reads top first. The labels act as follows.","title":"The subset-sum emitter machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/start","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.start","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#start","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"The VERTEX-COVER to SUBSET-SUM machine starts at the label $\\mathsf{rdK}$, which begins parsing the input by reading its leading unary number.","title":"Start label of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/is-t-bit","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.isT_bit","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#is-t-bit","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"For every bit $b$, the test for the input bit true of the SUBSET-SUM reduction machine, applied to the popped symbol $\\mathrm{bit}(b)$, returns $b$.","title":"Testing a popped input bit"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/is-v-val","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.isV_val","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#is-v-val","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"For every bit $b$, the test for the input bit true of the SUBSET-SUM reduction machine, applied to the popped symbol $\\mathrm{bit}(b)$, returns $b$.","title":"Testing a popped input bit"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/is-bit-block-val","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.isBitBlock_val","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#is-bit-block-val","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"For every bit $b$, the test for the input bit true of the SUBSET-SUM reduction machine, applied to the popped symbol $\\mathrm{bit}(b)$, returns $b$.","title":"Testing a popped input bit"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/rd-k-some","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.rdK_some","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#rd-k-some","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $k$ and $i$ be natural numbers, $v$ a Boolean state, $r = (r_1, \\dots, r_m)$ a list of bits and $S$ stack contents such that stack $X$ holds, top first, $k$ copies of $\\mathrm{bit}(\\mathrm{true})$, then $\\mathrm{bit}(\\mathrm{false})$, then $\\mathrm{bit}(r_1), \\dots, \\mathrm{bit}(r_m)$, and stack $A$ holds exactly $i$ copies of the counter unit $U$. Then $k + 1$ steps of the subset-sum emitter machine lead from\u2026","title":"Subset-sum emitter: reading the unary k"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/rd-k-none","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.rdK_none","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#rd-k-none","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $r = (r_1, \\dots, r_m)$ be a list of bits on which reading a unary length prefix fails (equivalently, no $r_i$ is $\\mathrm{false}$). Then for every Boolean state $v$ and all stack contents $S$ whose stack $X$ holds $\\mathrm{bit}(r_1), \\dots, \\mathrm{bit}(r_m)$, top first, $m + 1$ steps of the subset-sum emitter machine lead from label $\\mathrm{rdK}$ with state $v$ and stacks $S$ to label $\\mathrm{bad}$ with\u2026","title":"Subset-sum emitter: unterminated unary prefix at rdK"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/rd-n-some","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.rdN_some","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#rd-n-some","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $n$ and $i$ be natural numbers, $v$ a Boolean state, $r = (r_1, \\dots, r_m)$ a list of bits and $S$ stack contents such that stack $X$ holds, top first, $n$ copies of $\\mathrm{bit}(\\mathrm{true})$, then $\\mathrm{bit}(\\mathrm{false})$, then $\\mathrm{bit}(r_1), \\dots, \\mathrm{bit}(r_m)$, stack $N$ holds exactly $i$ copies of the counter unit $U$, and stack $L$ holds exactly $i$ copies of\u2026","title":"Subset-sum emitter: reading the unary n"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/rd-n-none","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.rdN_none","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#rd-n-none","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $r = (r_1, \\dots, r_m)$ be a list of bits on which reading a unary length prefix fails (equivalently, no $r_i$ is $\\mathrm{false}$). Then for every Boolean state $v$ and all stack contents $S$ whose stack $X$ holds $\\mathrm{bit}(r_1), \\dots, \\mathrm{bit}(r_m)$, top first, $m + 1$ steps of the subset-sum emitter machine lead from label $\\mathrm{rdN}$ with state $v$ and stacks $S$ to label $\\mathrm{bad}$ with\u2026","title":"Subset-sum emitter: unterminated unary prefix at rdN"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/mv-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.mv_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#mv-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $r = (b_1, \\dots, b_m)$ be a list of bits, $v$ a Boolean state, $c$ a natural number and $S$ stack contents such that stack $X$ holds $\\mathrm{bit}(b_1), \\dots, \\mathrm{bit}(b_m)$, top first, and stack $C$ holds exactly $c$ copies of the counter unit $U$. Then $m + 1$ steps of the subset-sum emitter machine lead from label $\\mathrm{mv}$ with state $v$ and stacks $S$ to label $\\mathrm{tri}$ with some state and\u2026","title":"Subset-sum emitter: moving the pair bits"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/tri-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.tri_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#tri-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $n, p \\in \\mathbb{N}$ and let $T_n = 0 + 1 + \\dots + (n - 1)$. Start the VERTEX-COVER to SUBSET-SUM machine at label $\\mathsf{tri}$ with any state bit and with stacks $S$ in which the counter stacks $\\mathsf{N}$ and $\\mathsf{M}$ hold exactly $n$ and $p$ unit symbols and the scratch stack $\\mathsf{S1}$ is empty. Then after exactly $(n + 1)^2$ steps the machine is at label $\\mathsf{cmp}$ with some state bit and\u2026","title":"Run of the pair-counting loop of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/eg-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.eg_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#eg-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $d$ be a list of bits (in the machine, the pair bits listed from the last pair down), $v$ a value of the machine's Boolean flag and $S$ an assignment of contents to the stacks, and write $\\mathrm{val}(d)$ for the list of the symbols $\\mathrm{val}(b)$ for the entries $b$ of $d$, in order. Suppose stack $Y$ holds $\\mathrm{val}(d)$ (first element on top), stacks $C$ and $M$ each hold $|d|$ copies of the unit\u2026","title":"Edge-number phase (SUBSET-SUM reduction machine)"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/pr-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.pr_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#pr-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Stacks are listed top first. Let $a$ be a bit, $\\ell$ a list of symbols, $(m_1, \\dots, m_p)$ and $(b_1, \\dots, b_p)$ two lists of bits of the same length $p$, $y$ a list of symbols, $v$ a Boolean state and $S$ stack contents such that $S(R)$ is $\\mathrm{val}(m_1), \\dots, \\mathrm{val}(m_p)$, $S(L)$ is $\\mathrm{val}(a)$ followed by $\\ell$, and $S(Y)$ is $\\mathrm{val}(b_1), \\dots, \\mathrm{val}(b_p)$ followed by $y$.\u2026","title":"Subset-sum emitter: one row of the pair walk"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/walk-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.walk_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#walk-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $L_m \\ne []$, $r_m$ and $w$ be lists of booleans, and for a list $\\ell$ of booleans write $\\bar\\ell$ for the list of mark symbols $\\mathrm{val}(x)$, $x \\in \\ell$. Let $(a_1, c_1), \\dots, (a_m, c_m)$ be the mark pairs $\\mathrm{walkM}(L_m, r_m)$, and suppose $w = (b_1, \\dots, b_m)$ has the same length $m$. Start the emitter machine at label $\\mathrm{pr}$ with any internal state $v$ and stacks $S$ such that\u2026","title":"Run of the subset-sum emitter's pair walk"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/round-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.round_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#round-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Stacks are listed top first. Let $n$ be a natural number, $p_0, \\dots, p_{P-1}$ the pairs $(i, j)$ with $i < j < n$ in encoding order (rows $i$ ascending, within a row $j$ descending), $(\\beta_0, \\dots, \\beta_{P-1})$ a list of $P$ pair bits and $w < n$ a vertex. Let $N_w = 4^P + \\sum_q 4^q$, the sum over the $q < P$ such that $\\beta_q$ is true and $w$ is an endpoint of $p_q$, and let $m_u$ be true if $u = w$ and\u2026","title":"Subset-sum emitter: one vertex number"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/rs-f-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.rsF_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#rs-f-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $c$ be a natural number, $\\rho = (\\rho_1, \\dots, \\rho_k)$ a list of bits, $v$ a Boolean state and $S$ stack contents whose stack $R$ holds, top first, $c$ copies of $\\mathrm{val}(\\mathrm{false})$ followed by $\\mathrm{val}(\\rho_1), \\dots, \\mathrm{val}(\\rho_k)$. Then $c$ steps of the subset-sum emitter machine lead from label $\\mathrm{rs}$ with state $v$ and stacks $S$ back to label $\\mathrm{rs}$ with some state\u2026","title":"Subset-sum emitter: moving unmarked vertices back to L"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/rs-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.rs_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#rs-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $n, w \\in \\mathbb{N}$ with $w + 1 < n$. Start the VERTEX-COVER to SUBSET-SUM machine at label $\\mathsf{rs}$ with any state bit and with stacks $S$ in which stack $\\mathsf{L}$ is empty and stack $\\mathsf{R}$ holds, top first, the symbols $\\mathrm{val}(b_0), \\dots, \\mathrm{val}(b_{n-1})$, where $b_i$ is true exactly when $i = w + 1$. Then after exactly $n + 1$ steps the machine is at label $\\mathsf{vtx}$ with\u2026","title":"Run of the mark-return loop of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/vtx-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.vtx_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#vtx-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $p_0, \\dots, p_{P-1}$ be the vertex pairs $(i, j)$ with $i < j < n$, listed with $i$ ascending and, for each $i$, with $j$ descending, let $\\beta_0, \\dots, \\beta_{P-1}$ be bits, and let $w < n$. For a vertex $u$ let $a_u$ be $4^P$ plus the sum of $4^q$ over the indices $q < P$ such that $\\beta_q$ is true and $p_q$ contains $u$. Start the VERTEX-COVER to SUBSET-SUM machine at label\u2026","title":"Run of the vertex-number phase of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/hv0-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.hv0_run","parent":32540,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03/index.html#hv0-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-03","status":"fully_proved","summary":"Let $c$ and $p$ be natural numbers, $r$ a bit (read as $0$ or $1$ in arithmetic), $v$ a value of the machine's Boolean flag and $S$ an assignment of contents to the stacks. Suppose stack $A$ holds $2c + r$ copies of the unit symbol and stack $H$ holds $p$ copies. Then, starting at label $\\mathrm{hv0}$ with flag $v$ and stacks $S$, the machine reaches label $\\mathrm{hvB}$ in exactly $2c + r + 1$ steps, with some\u2026","title":"One halving step of the binary conversion (SUBSET-SUM reduction machine)"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","kind":"scope","navigation":"transparent","parent":32497,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","title":"Binary conversion (SUBSET-SUM reduction machine) to Statements of the SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/hv-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.hv_run","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#hv-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"Let $k$ be a natural number, $v$ a value of the machine's Boolean flag and $S$ an assignment of contents to the stacks, and list the binary digits of $k$ least significant first, without leading zeros (so there are none for $k = 0$). Suppose stack $A$ holds $k$ copies of the unit symbol and stack $H$ is empty. Then, starting at label $\\mathrm{hv}$ with flag $v$ and stacks $S$, the machine reaches label\u2026","title":"Binary conversion (SUBSET-SUM reduction machine)"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/tem-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.tem_run","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#tem-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"Let $l = (b_1, \\dots, b_m)$ be a list of bits. Start the VERTEX-COVER to SUBSET-SUM machine at label $\\mathsf{tem}$ with any state bit and with stacks $S$ in which stack $\\mathsf{T}$ holds exactly the bit symbols of $b_1, \\dots, b_m$, top first. Then after exactly $m + 1$ steps the machine is at label $\\mathsf{done}$ with some state bit and stacks $S'$ in which the output stack $\\mathsf{O}$ holds, top first,\u2026","title":"Run of the target-bit loop of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/tsk-run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.tsk_run","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#tsk-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"Let $l$ be a list of bits, let $l'$ be $l$ with its leading $\\mathrm{false}$ bits removed, and let $o$ be a list of bits. Start the VERTEX-COVER to SUBSET-SUM machine at label $\\mathsf{tsk}$ with any state bit and with stacks $S$ in which stack $\\mathsf{T}$ holds exactly the bit symbols of $l$, first bit on top, and the output stack $\\mathsf{O}$ holds $\\mathrm{false}$ followed by $o$. Then after exactly $|l| + 1$\u2026","title":"Run of the target-output phase of the subset-sum emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/run","kind":"node","lean":"Cslib.Complexity.Karp.EmitSS.run","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"Let $x$ be a bit string. Start the VERTEX-COVER to SUBSET-SUM machine at its start label with any state bit and with stacks in which the input stack $\\mathsf{X}$ holds the bit symbols of $x$, first bit on top, and every other stack is empty. Then within at most $40(|x| + 1)^4$ steps the machine reaches the label $\\mathsf{done}$ with some state bit and stacks in which the output stack $\\mathsf{O}$ holds, top first,\u2026","title":"Run of the VERTEX-COVER to SUBSET-SUM machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/last-f","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.lastF","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#last-f","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For a Boolean $z$ and a list of bits $p$, the value is $z$ when $p$ is empty and the negation of the last bit of $p$ otherwise. So if $z$ records whether the bits read so far end in false, the value records whether those bits followed by $p$ end in false.","title":"Whether a bit string ends in false"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/encode-nat-decode-nat-iff","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.encodeNat_decodeNat_iff","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#encode-nat-decode-nat-iff","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For every bit string $p$, encoding in binary (least significant bit first) the natural number that $p$ decodes to gives back $p$ if and only if the trailing-zero test, started with no trailing zero, reports that $p$ does not end in $0$; that is, $p$ is empty or its last bit is $1$.","title":"Canonical binary strings are those not ending in zero"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/nums-aux","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.numsAux","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#nums-aux","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For a fuel $n$ and a list of bits $r$, the result is either a list of numbers or a failure. If $r$ is empty, the result is the empty list. If $r$ is non-empty and $n = 0$, parsing fails. If $r$ is non-empty and $n = m + 1$, one prefix-coded number is decoded off the front of $r$: if this gives a number $a$ with remainder $r'$, the result is $a$ followed by the result for $r'$ with fuel $m$ (failing if that fails),\u2026","title":"Parsing number blocks, with fuel"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/decode-ss","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.decodeSS","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#decode-ss","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"A block holding a bit string $p$ writes each bit $b$ of $p$ as the two bits $1, b$ and ends with a single $0$. The decoder reads a bit string $x$ as one block holding the target $t$, followed by blocks holding numbers $a_1, \\dots, a_n$ that run exactly to the end of $x$, where each block must hold the canonical binary representation of its number (least significant bit first, no trailing $0$). It returns the list\u2026","title":"Decoding a SUBSET-SUM instance"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/decode-ss-eq-some","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.decodeSS_eq_some","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#decode-ss-eq-some","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For every bit string $x$, every list $a$ of natural numbers and every natural number $t$, the SUBSET-SUM decoder returns the instance $(a, t)$ on $x$ if and only if $x$ is the encoding of the SUBSET-SUM instance with numbers $a$ and target $t$.","title":"Decoding a SUBSET-SUM instance inverts its encoding"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/pick","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.pick","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#pick","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For a list of numbers $a_0, \\dots, a_{k-1}$ and a list of bits $w$, the list of those $a_i$ with $w_i$ true, in their original order, where positions $i$ beyond the end of $w$ count as false.","title":"Numbers chosen by a mask"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/ss-check","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.ssCheck","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#ss-check","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For bit strings $x$ and $w$, the verifier accepts $(x, w)$ exactly when $x$ is the encoding of a SUBSET-SUM instance with numbers $a_0, \\dots, a_{n-1}$ and target $t$, and the sum of the $a_i$ for which bit $i$ of $w$ is true equals $t$; bits beyond the end of $w$ count as false, and bits of $w$ past position $n - 1$ are ignored. If $x$ is not the encoding of an instance, the verifier rejects.","title":"The SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/exists-pick","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.exists_pick","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#exists-pick","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"If $l$ is a sublist of a list $a$ of natural numbers, then there is a list $w$ of Booleans of the same length as $a$ such that the entries $a_i$ of $a$ with $w_i$ true, taken in order, form exactly $l$.","title":"Every sublist is selected by a mask"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/subset-sum-dec-iff","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.subsetSumDec_iff","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#subset-sum-dec-iff","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"A bit string $x$ is a yes-instance of SUBSET-SUM if and only if there is a bit string $w$ with $|w| \\le |x|$ such that the SUBSET-SUM verifier accepts $(x, w)$.","title":"SUBSET-SUM is characterized by its verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/carry","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.carry","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#carry","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For bits $a$, $b$ and a carry-in $c$, the carry out of adding $a$, $b$ and $c$ is $(a \\wedge b) \\vee (c \\wedge (a \\oplus b))$, which is true iff at least two of $a$, $b$, $c$ are true.","title":"Carry bit of a full adder"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/inc-b","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.incB","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#inc-b","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For a carry bit $c$ and a bit string $u$ (least significant bit first), $\\mathrm{inc}(c, u)$ adds $c$ to $u$ by ripple carry: on the empty string it is $[1]$ if $c$ is true and empty otherwise, and on a string with first bit $a$ and rest $u'$ it is the bit $a \\oplus c$ followed by $\\mathrm{inc}(a \\wedge c, u')$.","title":"Adding a carry bit"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/add-b","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.addB","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#add-b","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For a carry-in bit $c$ and bit lists $u$ and $p$, both least significant bit first, the ripple-carry sum walks $p$. If $p$ is empty, the result is $u$ with the carry $c$ added (a final carry becomes a new last bit). Otherwise, with $a$ the first bit of $u$ ($a$ is false when $u$ is empty) and $b$ the first bit of $p$, the result is the bit $a \\oplus b \\oplus c$ followed by the ripple-carry sum of the rest of $u$\u2026","title":"Ripple-carry addition of bit lists"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/add-b-encode-nat","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.addB_encodeNat","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#add-b-encode-nat","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"For every bit $c$ and all natural numbers $m$ and $n$, the ripple-carry sum, with carry-in $c$, of the canonical binary expansions of $m$ and $n$ (least significant bit first, no trailing false bit) is the canonical binary expansion of $m + n + c$, where $c$ counts as $1$ if true and $0$ if false. In particular the sum stays canonical.","title":"Addition is correct on canonical numbers"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/k","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.K","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#k","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"The verifier has seven stacks: the input $X$, the witness $W$, the target $T$ (most significant bit on top), the running sum $A$ (least significant bit on top), the new sum $B$ (most significant bit on top), and the scratch stacks $P$ and $Q$ of the final comparison.","title":"Stacks of the SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/lab","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.Lab","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#lab","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"The verifier has the labels $\\mathrm{tgt}(z)$, reading the target; $\\mathrm{next}$, starting the next number; $\\mathrm{add}(c, z)$, adding a chosen number with carry $c$; $\\mathrm{skip}(z)$, skipping an unchosen number; $\\mathrm{fin}(c)$, carrying $c$ through the rest of the sum; $\\mathrm{back}$, moving the new sum back; $\\mathrm{fmove}$, moving the final sum for the comparison; $\\mathrm{cmp}$, comparing;\u2026","title":"Labels of the SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/stmt","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.Stmt","parent":32561,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04/index.html#stmt","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-04","status":"fully_proved","summary":"The statements of the SUBSET-SUM verifier machine are the statements of a multi-stack Turing machine in Mathlib's TM2 model whose stacks are the seven stacks $X$, $W$, $T$, $A$, $B$, $P$, $Q$, each over the symbol alphabet of the SAT verifier, whose labels are the labels of the SUBSET-SUM verifier, and whose local state is a Boolean.","title":"Statements of the SUBSET-SUM verifier"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","kind":"scope","navigation":"transparent","parent":32497,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","title":"Reading one pair of a block to Comparing the sum with the target"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/rd-s","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.rdS","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#rd-s","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"For a flag $z$ (the bits read so far end in false), a statement $\\mathit{done}$ and statements $\\mathit{body}(b)$ for each bit $b$: if the input stack $X$ is empty, go to the label $\\mathsf{no}$. Otherwise pop the top of $X$ as a bit, a symbol that is not a bit counting as false. If it is false (the end marker), go to $\\mathsf{no}$ when $z$ is true and run $\\mathit{done}$ otherwise. If it is true, go to\u2026","title":"Reading one pair of a block"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/add-s","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.addS","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#add-s","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"The machine statement for adding the payload bit $b$ with carry $c$: pop the top symbol of the sum stack $A$ and read it as a bit $a$ (false if $A$ is empty or the symbol is not a bit), push the bit $a \\oplus b \\oplus c$ onto $B$, and go to the label that adds a chosen number with carry the carry out of $a$, $b$ and $c$ and flag $\\lnot b$ (the flag records whether the bits of the block read so far end in false).","title":"Adding one payload bit to the sum"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/fin-s","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.finS","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#fin-s","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"For a carry bit $c$, the verifier statement that carries $c$ through the rest of the sum on stack $A$: if $A$ is empty, it sets the local variable to true, pushes the bit $1$ onto $B$ when $c$ is true and goes to label $\\mathrm{back}$; otherwise it pops the top symbol of $A$, read as a bit $a$ (a symbol that is not a bit reads as $0$), sets the local variable to $a$, pushes the bit $a \\oplus c$ onto $B$ and goes to\u2026","title":"Carry step of the SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/next-s","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.nextS","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#next-s","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"The verifier's statement at the label $\\mathsf{next}$. If the input stack $X$ is empty, it goes to the label $\\mathsf{fmove}$ (move the sum, then compare). Otherwise it pops the top of the witness stack $W$ and reads it as a bit $c$, taking $c$ false when $W$ is empty or its top is not a bit; $c$ becomes the internal state. It then goes to the label $\\mathsf{add}$ with carry false if $c$ is true, and to the label\u2026","title":"Starting the next number"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/prog","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.prog","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#prog","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"The program on the stacks $X, W, T, A, B, P, Q$ over the verifier's symbols, with Boolean internal state, given label by label. A popped symbol that is not a bit counts as false, as does a pop from an empty stack. Reading a pair of a block off $X$ with flag $z$ (\"the bits read so far end in false\") goes to $\\mathsf{no}$ if $X$ is empty; on the end marker false it goes to $\\mathsf{no}$ if $z$ holds and otherwise\u2026","title":"The SUBSET-SUM verifier program"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/start","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.start","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#start","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"The SUBSET-SUM verifier machine starts in its target-reading state, with the flag recording whether the bits read so far end in false set to false.","title":"Start state of the SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/st","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.st","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#st","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"For lists of symbols $x$, $w$, $t$, $a$, $b$, this assigns contents to the seven stacks of the SUBSET-SUM verifier machine: the input stack $X$ holds $x$, the witness stack $W$ holds $w$, the target stack $T$ holds $t$, the sum stacks $A$ and $B$ hold $a$ and $b$, and the scratch stacks $P$ and $Q$ are empty.","title":"Stack contents of the SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/rej","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.Rej","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#rej","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"For a natural number $n$ and a configuration $c$ of the SUBSET-SUM verifier, $\\mathrm{Rej}(n, c)$ holds when there are an internal state $v$ and stack contents $S$ such that the verifier runs from $c$ in at most $n$ steps to the configuration with label $\\mathsf{no}$, internal state $v$ and stacks $S$.","title":"Rejection by the SUBSET-SUM verifier within a step bound"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/rej-one","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.rej_one","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#rej-one","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $l$ be a label, $v$ a Boolean and $S$ an assignment of stack contents. If executing the verifier's statement at label $l$ from internal state $v$ and stacks $S$ yields the next label $\\mathsf{no}$, then the verifier reaches a configuration with label $\\mathsf{no}$ from the configuration $(l, v, S)$ within one step.","title":"Rejecting in one step"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/tgt-run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.tgt_run","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#tgt-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $p$ and $s$ be lists of bits, and write $\\mathrm{delimit}(p)$ for the self-delimiting encoding of $p$ (each bit $b$ written as true followed by $b$, then a terminating false). Suppose the SUBSET-SUM verifier machine is in its target-reading state, with flag $z$ recording whether the bits read so far end in false and any value of its local Boolean variable, with the bits of $\\mathrm{delimit}(p)$ followed by the\u2026","title":"Reading the target block"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/skip-run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.skip_run","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#skip-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $p$ and $s$ be lists of bits, and write $\\mathrm{delimit}(p)$ for the self-delimiting encoding of $p$ (each bit $b$ written as true followed by $b$, then a terminating false). Suppose the SUBSET-SUM verifier machine is in the state that skips an unchosen number block, with flag $z$ recording whether the bits read so far end in false and any value of its local Boolean variable, with the bits of\u2026","title":"Skipping a number block"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/fin-run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.fin_run","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#fin-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $x$, $w$, $t$ and $\\mathit{acc}$ be lists of stack symbols, $u$ a list of bits, $c$ a carry bit and $v$ a Boolean, and write $\\mathrm{inc}(c, u)$ for the result of adding the carry $c$ to $u$ (least significant bit first). Started at label $\\mathrm{fin}(c)$ with variable $v$, with $x$, $w$, $t$ on the stacks $X$, $W$, $T$, the bits of $u$ on $A$ (first bit on top), $\\mathit{acc}$ on $B$ and $P$, $Q$ empty, the\u2026","title":"Carry phase of the SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/add-run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.add_run","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#add-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $p$ and $\\mathit{rest}$ be bit lists and $z$ a bit such that $p$ is nonempty and ends in true, or $p$ is empty and $z$ is false (the flag $z$ records whether the bits of the block read so far end in false), and let $c$ be a bit. Start the SUBSET-SUM verifier machine at the label that adds a chosen number with carry $c$ and flag $z$, with any value of the local variable, the self-delimiting encoding of $p$\u2026","title":"Adding a canonical chosen block"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/add-rej","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.add_rej","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#add-rej","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $p$ and $\\mathit{rest}$ be bit lists and $z$ a bit such that $p$ is nonempty and ends in false, or $p$ is empty and $z$ is true (the flag $z$ records whether the bits of the block read so far end in false). Let $c$ be a bit and $u$ a bit list. Start the SUBSET-SUM verifier machine at the label that adds a chosen number with carry $c$ and flag $z$, with any value of the local variable, the self-delimiting\u2026","title":"The verifier rejects a chosen block that ends in false"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/tgt-fail","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.tgt_fail","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#tgt-fail","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $r$ be a list of bits that does not begin with a self-delimiting block, that is, parsing one self-delimiting block off the front of $r$ fails. Suppose the SUBSET-SUM verifier machine is in its target-reading state, with either value of the flag $z$ recording whether the bits read so far end in false and either value of its local Boolean variable, with the bits of $r$ on the input stack $X$, arbitrary contents\u2026","title":"The verifier rejects an unterminated target block"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/skip-fail","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.skip_fail","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#skip-fail","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $r$ be a list of bits that does not begin with a self-delimiting block, that is, parsing one self-delimiting block off the front of $r$ fails. Suppose the SUBSET-SUM verifier machine is in the state that skips an unchosen number block, with either value of the flag $z$ recording whether the bits read so far end in false and either value of its local Boolean variable, with the bits of $r$ on the input stack $X$,\u2026","title":"The verifier rejects an unterminated skipped block"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/add-fail","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.add_fail","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#add-fail","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $r$ be a bit list from whose front no complete self-delimiting block can be parsed, let $c$ and $z$ be bits, and let $u$ be a bit list. Start the SUBSET-SUM verifier machine at the label that adds a chosen number with carry $c$ and flag $z$ (the flag records whether the bits of the block read so far end in false), with any value of the local variable, $r$ (as bit symbols) on the input stack $X$, any symbol\u2026","title":"The verifier rejects an unterminated chosen block"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/back-run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.back_run","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#back-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $\\mathit{res}$ be a bit list. Start the SUBSET-SUM verifier machine at the label that moves the new sum back, with any value of the local variable, symbol lists $x$, $w$, $t$ on $X$, $W$, $T$, the sum stack $A$ empty, $\\mathit{res}$ (as bit symbols, last bit on top) on $B$, and the scratch stacks $P$ and $Q$ empty. Then after exactly $|\\mathit{res}| + 1$ steps the machine is at the label that starts the next\u2026","title":"Moving the new sum back"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/fmove-run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.fmove_run","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#fmove-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $x$, $w$, $t$ be lists of stack symbols, $s$ a list of bits and $v$ a Boolean. Started at label $\\mathrm{fmove}$ with variable $v$, with $x$, $w$, $t$ on the stacks $X$, $W$, $T$, the bits of $s$ on $A$ (first bit on top) and $B$, $P$, $Q$ empty, the verifier takes exactly $|s| + 1$ steps to reach label $\\mathrm{cmp}$, with some value of the variable, where $X$, $W$, $T$ are unchanged, $A$, $P$, $Q$ are empty,\u2026","title":"Moving the final sum for the comparison"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/cmp-run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.cmp_run","parent":32582,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05/index.html#cmp-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-05","status":"fully_proved","summary":"Let $s$ and $\\tau$ be bit lists. Start the SUBSET-SUM verifier machine at the comparison label, with any value of the local variable, symbol lists $x$ and $w$ on $X$ and $W$, the target $\\tau$ on $T$ and the sum $s$ on $B$ (both as bit symbols, last bit on top), and the stacks $A$, $P$ and $Q$ empty. Then within $|s| + 1$ steps the machine reaches the accepting label if $s = \\tau$ and the rejecting label otherwise.","title":"Comparing the sum with the target"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","kind":"scope","navigation":"transparent","parent":32497,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","title":"The summing loop of the SUBSET-SUM verifier to Binary addition of framed numbers in quadratic time"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/loop-run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.loop_run","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#loop-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Let $N$ be a natural number and $t$ a list of symbols. Write $\\langle m \\rangle$ for the canonical binary code of a natural number $m$, least significant bit first, and put a bit list on a stack as the list of its name-bit symbols. Let $n$ and $s$ be natural numbers, $r$ and $w$ bit lists and $v$ a Boolean, with $|r| \\le n$ and $|\\langle s \\rangle| + |r| \\le N$. Start the SUBSET-SUM verifier at the label\u2026","title":"The summing loop of the SUBSET-SUM verifier"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/cost-le","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.cost_le","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#cost-le","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"For natural numbers $X$, $W$, $P$, $R$ and $S$ with $P + R \\le X$ and $S \\le X$, $P + 1 + (R(2X + 6) + 1) + (S + 1) + (S + 1) \\le 16 (X + W + 1)^3$.","title":"Cubic bound on the verifier's phase costs"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/run","kind":"node","lean":"Cslib.Complexity.Karp.SSVerify.run","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Let $x$ and $w$ be lists of bits and $v$ a Boolean, and let $S$ assign stack contents with the bits of $x$ on $X$, the bits of $w$ on $W$ and every other stack empty. Started at the start label with internal state $v$ and stacks $S$, the verifier program reaches, within $16(|x| + |w| + 1)^3$ steps, a configuration with label $\\mathsf{yes}$ if the SUBSET-SUM check accepts $(x, w)$, and with label $\\mathsf{no}$\u2026","title":"The SUBSET-SUM verifier machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/ss-verify-comp","kind":"node","lean":"Cslib.Complexity.Karp.ssVerifyComp","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#ss-verify-comp","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"The multi-stack Turing machine that checks SUBSET-SUM certificates, packaged as a host component: it has seven stacks, each holding symbols of the SAT verifier alphabet, its labels and its program are those of the machine, and the list of its stacks names each of the seven stacks exactly once.","title":"The SUBSET-SUM verifier machine as a host component"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/subset-sum-dec-mem-np","kind":"node","lean":"Cslib.Complexity.Karp.subsetSumDec_mem_NP","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#subset-sum-dec-mem-np","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"SUBSET-SUM is the decision problem on bit strings that accepts $x$ exactly when $x$ encodes a target $t$ followed by natural numbers $a_1, \\dots, a_n$ (each written in binary and delimited) such that some, possibly empty, sublist of $a_1, \\dots, a_n$ sums to $t$. This decision problem belongs to NP: there are a polynomial $p$ with natural-number coefficients and a polynomial-time computable verifier $R$ such that\u2026","title":"SUBSET-SUM is in NP"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/emit-sscomp","kind":"node","lean":"Cslib.Complexity.Karp.emitSSComp","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#emit-sscomp","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"The multi-stack Turing machine that computes the VERTEX-COVER to SUBSET-SUM reduction on bit strings, packaged as a host component: its fifteen stacks (the output stack holds booleans, every other stack holds symbols of the SAT verifier alphabet), its labels and its program are those of the machine, and the list of its stacks names each of the fifteen stacks exactly once.","title":"The VERTEX-COVER to SUBSET-SUM machine as a host component"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/is-poly-time-reduce-ss","kind":"node","lean":"Cslib.Complexity.Karp.isPolyTime_reduceSS","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#is-poly-time-reduce-ss","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"The reduction map on bit strings, which sends the encoding of a VERTEX-COVER instance $(k, G)$ to the encoding of the SUBSET-SUM instance built from it (one base-4 number for each vertex and for each pair of vertices of $G$, and a target) and every other bit string to the instance with no numbers and target $1$, is computable in polynomial time.","title":"The VERTEX-COVER to SUBSET-SUM reduction is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/subset-sum-dec-np-hard","kind":"node","lean":"Cslib.Complexity.Karp.subsetSumDec_npHard","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#subset-sum-dec-np-hard","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Let SUBSET-SUM be the decision problem on bit strings that accepts exactly the strings encoding a target $t$ followed by natural numbers $a_1, \\dots, a_n$ (each written in binary and delimited) such that some, possibly empty, sublist of $a_1, \\dots, a_n$ sums to $t$. Every decision problem $L$ in NP reduces to SUBSET-SUM by a polynomial-time many-one reduction: there is a polynomial-time computable map $f$ on bit\u2026","title":"SUBSET-SUM is NP-hard"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/subset-sum-dec-np-complete","kind":"node","lean":"Cslib.Complexity.Karp.subsetSumDec_npComplete","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#subset-sum-dec-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Let SUBSET-SUM be the decision problem on bit strings that accepts exactly the strings encoding a target $t$ followed by natural numbers $a_1, \\dots, a_n$ (each written in binary and delimited) such that some, possibly empty, sublist of $a_1, \\dots, a_n$ sums to $t$. Then SUBSET-SUM belongs to NP, and every decision problem $L$ in NP reduces to SUBSET-SUM by a polynomial-time many-one reduction: there is a\u2026","title":"SUBSET-SUM is NP-complete"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/sum-component-run","kind":"node","lean":"Cslib.Complexity.SSVerifierSum.sum_component_run","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#sum-component-run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"On canonical delimited `encodeNat` blocks, the existing component leaves the selected sum at a halt label. Normalize raw payloads first: malformed blocks can instead reject early, leaving only a partial sum. The cost uses framed bit length, never numeric magnitudes. The mask is consumed; copy it before reuse in another component.","title":"SUBSET-SUM verifier computes the mask-selected sum"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/pick-all","kind":"node","lean":"Cslib.Complexity.SSVerifierSum.pick_all","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#pick-all","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"A mask long enough to select every item gives the original list.","title":"A long enough all-true mask selects the whole list"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/sum-embed","kind":"node","lean":"Cslib.Complexity.SSVerifierSum.sum_embed","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#sum-embed","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Reach the common mapped label after summing canonical `encodeNat` blocks; unembedded stacks remain intact. The input agreement requires prior normalization. A raw parse-failure exit can expose a partial sum. This is only a reachability guarantee: `h\u03c6` does not make the endpoint fresh, distinct from mapped live labels, or a halt of the host. Callers must establish their intended label separation and the code run\u2026","title":"Embedded SUBSET-SUM summing component reaches the mapped label"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/pair-input","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.pairInput","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#pair-input","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Two canonical Nat blocks in the existing list framing.","title":"Framed input of two natural numbers"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/lab","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.Lab","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#lab","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Original verifier labels plus initialization, clear, reverse, and halt.","title":"Labels of the binary addition machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/identity","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.identity","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#identity","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Reuse all original verifier stacks without re-encoding symbols.","title":"Identity embedding of the verifier stacks"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/redirect","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.redirect","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#redirect","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Redirect the two halt labels, leaving every live label distinct.","title":"Redirection of the verifier's halt labels"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/redirect-eq-clear-iff","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.redirect_eq_clear_iff","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#redirect-eq-clear-iff","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"The cleanup continuation cannot alias any live verifier label.","title":"Only halt labels are redirected to the cleanup label"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/prog","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.prog","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#prog","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Run the existing sum machine, then clear the mask and restore LSB order. Only the original halt labels are redirected to a continuation.","title":"Binary addition program built on the SUBSET-SUM machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/length-sum-le-input","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.length_sum_le_input","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#length-sum-le-input","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"The canonical sum fits within the two framed operands' bit length.","title":"Bit length of the sum is at most the input length"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/run-2","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.run","parent":32603,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06/index.html#run-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-06","status":"fully_proved","summary":"Add two canonically framed numbers, leaving only the LSB-first result. The bound is quadratic in encoded input length; no loop enumerates values.","title":"Binary addition of framed numbers in quadratic time"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-07.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-07","kind":"scope","navigation":"transparent","parent":32497,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-07/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-07","status":"fully_proved","title":"Part 7 of 7"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-07/run-embedded","kind":"node","lean":"Cslib.Complexity.SSVerifierArithmetic.run_embedded","parent":32624,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-07/index.html#run-embedded","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/subset-sum/part-07","status":"fully_proved","summary":"The addition component runs in a host, with every external stack unchanged and all embedded workspace cleared.","title":"Embedded run of the binary addition component"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","title":"Partition problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/partition-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.PartitionInstance","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#partition-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"A partition instance: a list of natural numbers.","title":"PARTITION instance as a list of naturals"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/partition-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.PartitionFeasible","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#partition-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"A sublist summing to half the total.","title":"Feasibility of a PARTITION instance"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/partition-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.partitionDec","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#partition-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"PARTITION using the canonical binary list decoder; bad framing rejects.","title":"Binary-coded PARTITION decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/partition-dec-bit-encode","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.partitionDec_bitEncode","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#partition-dec-bit-encode","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"A canonical encoding has exactly the intended feasibility semantics.","title":"Binary PARTITION decider is correct on canonical encodings"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/from-subset-sum","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.fromSubsetSum","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#from-subset-sum","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The two padding entries in the SUBSET-SUM reduction. No subtraction or range check is needed, even when `t > as.sum`.","title":"Padding map from SUBSET-SUM to PARTITION"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/from-subset-sum-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.fromSubsetSum_feasible","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#from-subset-sum-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"SUBSET-SUM reduces to PARTITION by adding two nonnegative entries.","title":"Correctness of the SUBSET-SUM to PARTITION padding map"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/feasible-iff-doubled-sublist","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.feasible_iff_doubled_sublist","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#feasible-iff-doubled-sublist","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Doubling each entry reduces PARTITION membership to SUBSET-SUM without truncating division on odd totals.","title":"PARTITION as SUBSET-SUM on the doubled list"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/phase","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.Phase","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#phase","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Reading the first marker, a later marker, or a payload bit.","title":"Scan phases of the PARTITION doubling pass"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/next","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.next","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#next","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Phase after reading one input bit.","title":"Phase transition of the PARTITION doubling pass"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/chunk","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.chunk","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#chunk","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Output bits for one input bit; at a nonzero block prepend its low zero.","title":"Output bits per input bit in the doubling pass"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/transform","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.transform","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#transform","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The raw finite-state pass. Malformed strings have a total output too; validation belongs to the preceding canonical decoder.","title":"Finite-state doubling pass on raw bitstrings"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/transform-block","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.transform_block","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#transform-block","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"A canonical number block becomes the canonical block of its double.","title":"Doubling pass maps a number block to the block of its double"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/transform-nums","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.transform_nums","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#transform-nums","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The stream pass doubles every number and preserves list multiplicity.","title":"Doubling pass doubles every number in a list"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/k","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.K","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#k","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Input, reversed output, and output stacks of the doubling pass.","title":"Stacks of the PARTITION doubling machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/lab","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.Lab","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#lab","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Three scan phases followed by reversal and a halting continuation.","title":"Control labels of the PARTITION doubling machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/prog","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.prog","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#prog","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Finite-stack implementation of `transform`.","title":"Program of the PARTITION doubling machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/run","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.run","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#run","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The complete pass takes at most `4 * input.length + 2` steps.","title":"Linear step bound for the PARTITION doubling machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/comp","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.comp","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#comp","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The doubling transducer as a finite component for the canonical host.","title":"PARTITION doubling machine as a host component"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/is-poly-time-transform","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.DoubleStream.isPolyTime_transform","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#is-poly-time-transform","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Polynomial time in the canonical bitstring encoding, from the actual finite-state machine, including malformed raw strings.","title":"PARTITION doubling pass is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/reduce-from-subset-sum","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.reduceFromSubsetSum","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#reduce-from-subset-sum","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"SUBSET-SUM to PARTITION, with malformed inputs sent to `[1]`.","title":"Encoded reduction from SUBSET-SUM to PARTITION"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/reduce-to-subset-sum","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.reduceToSubsetSum","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#reduce-to-subset-sum","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"PARTITION to SUBSET-SUM, preserving the generic decoder's semantics.","title":"Encoded reduction from PARTITION to SUBSET-SUM"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/reduce-from-subset-sum-correct","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.reduceFromSubsetSum_correct","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#reduce-from-subset-sum-correct","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The encoded reduction preserves every yes/no answer, including malformed source strings. This is the semantic, not the time, theorem.","title":"Correctness of the encoded SUBSET-SUM to PARTITION reduction"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/reduce-to-subset-sum-correct","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.reduceToSubsetSum_correct","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#reduce-to-subset-sum-correct","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The membership map preserves the generic decoder, including accepted noncanonical payloads and rejected malformed framing.","title":"Correctness of the encoded PARTITION to SUBSET-SUM reduction"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/sum-encoding-length-le","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.sum_encoding_length_le","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#sum-encoding-length-le","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Summing a binary PARTITION instance cannot exceed its encoded length.","title":"Bitlength of a PARTITION sum is at most the encoded length"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/from-subset-sum-length","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.fromSubsetSum_length","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#from-subset-sum-length","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The padding reduction has linear output length in its encoded input.","title":"Padding reduction has linear output length"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/reduce-from-subset-sum-length","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.reduceFromSubsetSum_length","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#reduce-from-subset-sum-length","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Linear output bound on all strings, including malformed encodings.","title":"Encoded SUBSET-SUM to PARTITION map has linear output length"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/to-subset-sum-length","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.toSubsetSum_length","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#to-subset-sum-length","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Linear bitlength bound for the membership instance.","title":"PARTITION membership instance has linear bitlength"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/k-3","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessCore.K","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#k-3","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Input, reverse buffer, target, output, and unchanged sum-machine stacks.","title":"Stacks of the SUBSET-SUM to PARTITION padding machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/lab-3","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessCore.Lab","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#lab-3","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Parse target, preserve numbers, sum, then emit the two padding blocks.","title":"Control labels of the SUBSET-SUM to PARTITION padding machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/sum-emb","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessCore.sumEmb","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#sum-emb","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The original sum component's stacks are not reimplemented.","title":"Summation-machine stack embedding in the padding machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/sum-label","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessCore.sumLabel","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#sum-label","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Both genuine sum-machine halts resume at target emission.","title":"Label map from summation halts to target emission"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/prog-3","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessCore.prog","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#prog-3","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Padding construction by binary stack operations.","title":"Program of the SUBSET-SUM to PARTITION padding 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machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/run-3","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessCore.run","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#run-3","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The subtraction-free padding map is computed by a quadratic binary machine on every canonical SUBSET-SUM instance, including `t > as.sum`.","title":"Quadratic run of the padding machine on canonical instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/comp-3","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessCore.comp","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#comp-3","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Finite padding component for the strict source-decoding wrapper.","title":"SUBSET-SUM to PARTITION padding machine as a host component"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/k-2","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessMachine.K","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#k-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Disjoint workspaces, a preserved input, and comparison scratch stacks.","title":"Stacks of the raw-input SUBSET-SUM to PARTITION machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/lab-2","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessMachine.Lab","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#lab-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Copy, exact normalization, canonicality check, and numeric computation.","title":"Control labels of the raw-input SUBSET-SUM to PARTITION machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/prog-2","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessMachine.prog","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#prog-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"A total finite-stack transducer for the exact source codec.","title":"Program of the raw-input SUBSET-SUM to PARTITION machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/run-2","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessMachine.run","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#run-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Total raw-input hardness reduction, with exact source rejection and an explicit bound polynomial in encoded length, not numeric magnitude.","title":"Quadratic run of the raw-input SUBSET-SUM to PARTITION machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/comp-2","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessMachine.comp","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#comp-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The all-string hardness reduction in the existing finite host.","title":"Raw-input SUBSET-SUM to PARTITION machine as a host component"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/is-poly-time-reduce-from-subset-sum","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.HardnessMachine.isPolyTime_reduceFromSubsetSum","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#is-poly-time-reduce-from-subset-sum","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The total SUBSET-SUM-to-PARTITION map is polynomial-time.","title":"SUBSET-SUM to PARTITION map is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/k-5","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.K","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#k-5","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Input, reversal buffer, and the two embedded components' stacks.","title":"Stacks of the canonical PARTITION membership transducer"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/untitled","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.\u0393","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#untitled","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The summation stacks use the existing verifier alphabet.","title":"Stack alphabets of the canonical PARTITION membership transducer"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/lab-5","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.Lab","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#lab-5","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Component phases and the final sum-prefix emitter.","title":"Control labels of the canonical PARTITION membership transducer"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/sum-emb-2","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.sumEmb","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#sum-emb-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Stack embedding of the existing sum component.","title":"Summation-machine stack embedding in the membership transducer"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/double-emb","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.doubleEmb","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#double-emb","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Stack embedding of the doubling component.","title":"Doubling-machine stack embedding in the membership transducer"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/sum-label-2","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.sumLabel","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#sum-label-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Both sum halts resume at the doubling pass.","title":"Label map from summation halts to the doubling pass"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/double-label","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.doubleLabel","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#double-label","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The doubling halt resumes at the prefix emitter.","title":"Label map from the doubling halt to the prefix emitter"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/prog-5","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.prog","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#prog-5","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Copy, sum, double, and prepend the sum, using finite-stack operations.","title":"Program of the canonical PARTITION membership transducer"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/embeds-sum-2","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.embeds_sum","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#embeds-sum-2","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Only genuine halts are replaced when embedding the sum machine.","title":"Membership transducer embeds the summation machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/embeds-double","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.embeds_double","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#embeds-double","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Only the doubling halt is replaced by the prefix-emitter continuation.","title":"Membership transducer embeds the doubling machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/run-5","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.run","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#run-5","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The canonical PARTITION membership map is computed in quadratic time by summing, doubling, and prefixing, with no value-sized loop.","title":"Canonical PARTITION membership transducer runs in quadratic time"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/comp-5","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipCore.comp","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#comp-5","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Finite component for normalization followed by the membership map.","title":"Canonical PARTITION membership transducer as a host component"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/k-4","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipMachine.K","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#k-4","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Disjoint normalizer and numeric-core workspaces and a transfer buffer.","title":"Stacks of the raw-input PARTITION membership machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/lab-4","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipMachine.Lab","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#lab-4","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Normalization, transfer, canonical computation, and rejection.","title":"Control labels of the raw-input PARTITION membership machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/norm-emb","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipMachine.normEmb","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#norm-emb","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The exact canonicalizer's workspace.","title":"Normalizer workspace of the raw-input PARTITION membership machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/core-emb","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipMachine.coreEmb","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#core-emb","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The canonical numeric core's workspace.","title":"Numeric-core workspace of the raw-input PARTITION membership machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/prog-4","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipMachine.prog","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#prog-4","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Normalize exactly, then run the polynomial numeric core.","title":"Program of the raw-input PARTITION membership machine"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/run-4","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipMachine.run","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#run-4","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The exact all-string membership reduction has a quadratic machine bound, including malformed inputs and noncanonical binary payloads.","title":"Raw-input PARTITION membership machine runs in quadratic time"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/comp-4","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipMachine.comp","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#comp-4","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The total membership reduction in the existing finite host.","title":"Raw-input PARTITION membership machine as a host component"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/is-poly-time-reduce-to-subset-sum","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.MembershipMachine.isPolyTime_reduceToSubsetSum","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#is-poly-time-reduce-to-subset-sum","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"The total PARTITION-to-SUBSET-SUM map is polynomial-time.","title":"PARTITION to SUBSET-SUM map is polynomial-time"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/partition-dec-mem-np","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.partitionDec_mem_NP","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#partition-dec-mem-np","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Exact binary-coded PARTITION belongs to canonical NP.","title":"Binary-coded PARTITION is in NP"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/subset-sum-poly-many-one-red-partition","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.subsetSum_polyManyOneRed_partition","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#subset-sum-poly-many-one-red-partition","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Binary-coded SUBSET-SUM reduces to the exact PARTITION decision problem.","title":"Polynomial-time reduction from SUBSET-SUM to PARTITION"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/partition-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Partition.partition_npComplete","parent":32626,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem/index.html#partition-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/partition-problem","status":"fully_proved","summary":"Source: M. R. Garey and D. S. Johnson, *Computers and Intractability* (1979), PARTITION; R. M. Karp, \"Reducibility among combinatorial problems\" (1972).","title":"PARTITION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem.json","id":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem","kind":"scope","navigation":"module","parent":32441,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/index.html","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem","status":"can_prove","title":"Unbounded knapsack problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/unbounded-knapsack-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.UnboundedKnapsack.UnboundedKnapsackInstance","parent":32691,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/index.html#unbounded-knapsack-instance","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem","status":"fully_proved","summary":"An instance of the unbounded knapsack problem is a triple $(I, C, K)$ of a finite list $I = ((s_1, v_1), \\dots, (s_m, v_m))$ of items, where item $j$ has weight $s_j \\in \\mathbb{N}$ and value $v_j \\in \\mathbb{N}$, a capacity $C \\in \\mathbb{N}$ and a target value $K \\in \\mathbb{N}$.","title":"Unbounded knapsack instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/unbounded-knapsack-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.UnboundedKnapsack.UnboundedKnapsackFeasible","parent":32691,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/index.html#unbounded-knapsack-feasible","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem","status":"fully_proved","summary":"An unbounded knapsack instance with items $(s_1, v_1), \\dots, (s_m, v_m)$, capacity $C$ and target value $K$ is feasible when there are multiplicities $c_1, \\dots, c_m \\in \\mathbb{N}$ with $\\sum_{j=1}^{m} c_j s_j \\le C$ and $\\sum_{j=1}^{m} c_j v_j \\ge K$.","title":"Feasible unbounded knapsack instances"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/unbounded-knapsack-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.UnboundedKnapsack.unboundedKnapsackDec","parent":32691,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/index.html#unbounded-knapsack-dec","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem","status":"fully_proved","summary":"The unbounded knapsack problem is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of unbounded knapsack instances, to a feasible instance. Bitstrings that fail to decode are rejected.","title":"Unbounded knapsack as a decision problem"},{"id":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/unbounded-knapsack-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.UnboundedKnapsack.unboundedKnapsack_npComplete","parent":32691,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem/index.html#unbounded-knapsack-np-complete","scope":"TheoryOfComputation/np-completeness/number-and-packing-problems/unbounded-knapsack-problem","status":"can_prove","summary":"The unbounded knapsack decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"UNBOUNDED KNAPSACK is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location","status":"can_prove","summary":"3-dimensional matching, exact cover, set cover, set packing, set splitting, maximum coverage, minimum test set, generalized and quadratic assignment, facility location, k-center and k-median.","title":"Set systems, assignment and location"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching","status":"can_prove","title":"3-dimensional matching"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/three-dimensional-matching-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreeDimensionalMatching.ThreeDimensionalMatchingInstance","parent":32697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/index.html#three-dimensional-matching-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching","status":"fully_proved","summary":"An instance of 3-DIMENSIONAL MATCHING is a pair $(n, T)$ of a natural number $n$, the part size, and a finite list $T$ of triples of natural numbers.","title":"3-dimensional matching instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreeDimensionalMatching.WellFormed","parent":32697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching","status":"fully_proved","summary":"A 3-dimensional matching instance $(n, T)$ is well formed when each of the three coordinates of every triple in $T$ is less than $n$.","title":"Well-formed 3-dimensional matching instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/three-dimensional-matching-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreeDimensionalMatching.ThreeDimensionalMatchingFeasible","parent":32697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/index.html#three-dimensional-matching-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching","status":"fully_proved","summary":"A 3-dimensional matching instance $(n, T)$ is feasible, or has a perfect matching, when there is a list $M$ of length $n$, obtained from the list of triples $T$ by deleting some of its entries, such that for each of the three coordinate positions, every natural number $a < n$ occurs in that position in exactly one triple of $M$.","title":"Feasible 3-dimensional matching instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/three-dimensional-matching-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreeDimensionalMatching.threeDimensionalMatchingDec","parent":32697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/index.html#three-dimensional-matching-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching","status":"fully_proved","summary":"The decision problem 3-DIMENSIONAL MATCHING accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a 3-dimensional matching instance $(n, T)$ that is well formed (every coordinate of every triple in $T$ is less than $n$) and feasible (it has a perfect matching). Bitstrings that do not decode to an instance are rejected.","title":"3-DIMENSIONAL MATCHING as a decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/three-dimensional-matching-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ThreeDimensionalMatching.threeDimensionalMatching_npComplete","parent":32697,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching/index.html#three-dimensional-matching-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/3-dimensional-matching","status":"can_prove","summary":"The decision problem 3-DIMENSIONAL MATCHING on bitstrings is in NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"3-DIMENSIONAL MATCHING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover","status":"can_prove","title":"Exact cover"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/exact-cover-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCover.ExactCoverInstance","parent":32703,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/index.html#exact-cover-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover","status":"fully_proved","summary":"An instance of EXACT COVER is a pair $(n, C)$ of a natural number $n$, the size of the universe $\\{0, \\dots, n - 1\\}$, and a finite list $C$ of finite lists of natural numbers, the candidate subsets. That each entry of $C$ lists distinct elements of the universe is required by well-formedness, not by the type of instances.","title":"Instances of EXACT COVER"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCover.WellFormed","parent":32703,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover","status":"fully_proved","summary":"An instance $(n, C)$ of EXACT COVER is well formed when every entry of $C$ is a list of pairwise distinct natural numbers, each less than $n$.","title":"Well-formed EXACT COVER instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/exact-cover-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCover.ExactCoverFeasible","parent":32703,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/index.html#exact-cover-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover","status":"fully_proved","summary":"An instance $(n, C)$ of EXACT COVER is feasible when $C$ has a sublist $S$, obtained by deleting some entries of $C$, such that for every natural number $x < n$ there is exactly one list that occurs in $S$ and contains $x$.","title":"Feasible EXACT COVER instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/exact-cover-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCover.exactCoverDec","parent":32703,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/index.html#exact-cover-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover","status":"fully_proved","summary":"The decision problem EXACT COVER accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs of a natural number and a list of lists of natural numbers, to an instance of EXACT COVER that is well formed and feasible; bitstrings that do not decode are rejected.","title":"EXACT COVER as a decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/exact-cover-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCover.exactCover_npComplete","parent":32703,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover/index.html#exact-cover-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover","status":"can_prove","summary":"The decision problem EXACT COVER is in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"EXACT COVER is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets","status":"can_prove","title":"Exact cover by 3-sets"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/exact-cover-by3-sets-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCoverBy3Sets.ExactCoverBy3SetsInstance","parent":32709,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/index.html#exact-cover-by3-sets-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets","status":"fully_proved","summary":"An instance of EXACT COVER BY 3-SETS is a pair $(q, C)$ of a natural number $q$ and a finite list $C$ of finite lists of natural numbers. The intended universe is $\\{0, \\dots, 3q - 1\\}$ and the entries of $C$ are the candidate three-element subsets; that each entry lists three distinct elements of the universe is required by well-formedness, not by the type of instances.","title":"Instances of EXACT COVER BY 3-SETS"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCoverBy3Sets.WellFormed","parent":32709,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets","status":"fully_proved","summary":"An instance $(q, C)$ of EXACT COVER BY 3-SETS is well formed when every entry of $C$ is a list of exactly three pairwise distinct natural numbers, each less than $3q$.","title":"Well-formed EXACT COVER BY 3-SETS instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/exact-cover-by3-sets-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCoverBy3Sets.ExactCoverBy3SetsFeasible","parent":32709,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/index.html#exact-cover-by3-sets-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets","status":"fully_proved","summary":"An instance $(q, C)$ of EXACT COVER BY 3-SETS is feasible when $C$ has a sublist $S$, obtained by deleting some entries of $C$, such that for every natural number $x < 3q$ there is exactly one list that occurs in $S$ and contains $x$.","title":"Feasible EXACT COVER BY 3-SETS instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/exact-cover-by3-sets-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCoverBy3Sets.exactCoverBy3SetsDec","parent":32709,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/index.html#exact-cover-by3-sets-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets","status":"fully_proved","summary":"The decision problem EXACT COVER BY 3-SETS accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs of a natural number and a list of lists of natural numbers, to an instance of EXACT COVER BY 3-SETS that is well formed and feasible; bitstrings that do not decode are rejected.","title":"EXACT COVER BY 3-SETS as a decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/exact-cover-by3-sets-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ExactCoverBy3Sets.exactCoverBy3Sets_npComplete","parent":32709,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets/index.html#exact-cover-by3-sets-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/exact-cover-by-3-sets","status":"can_prove","summary":"The decision problem EXACT COVER BY 3-SETS is in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"EXACT COVER BY 3-SETS is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/facility-location.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location","status":"can_prove","title":"Facility location"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/facility-location-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FacilityLocation.FacilityLocationInstance","parent":32715,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/index.html#facility-location-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location","status":"fully_proved","summary":"A facility location instance is a triple $(o, D, B)$ made of a finite list $o$ of natural numbers (the opening costs of the facilities), a finite list $D$ of finite lists of natural numbers (the $c$-th list holds the costs of assigning client $c$ to each facility), and a natural number $B$ (the budget).","title":"Facility location instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FacilityLocation.WellFormed","parent":32715,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location","status":"fully_proved","summary":"A facility location instance $(o, D, B)$, with opening costs $o$ and assignment-cost rows $D$, is well-formed when every row of $D$ has the same length as $o$.","title":"Well-formed facility location instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/facility-location-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FacilityLocation.FacilityLocationFeasible","parent":32715,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/index.html#facility-location-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location","status":"fully_proved","summary":"Let $(o, D, B)$ be a facility location instance with opening costs $o = (o_0, \\dots, o_{n-1})$, assignment-cost rows $D = (D_0, \\dots, D_{m-1})$ and budget $B$, and write $D_{c,j}$ for the $j$-th entry of the row $D_c$, read as $0$ when $D_c$ has no $j$-th entry. The instance is feasible when there is a map $\\sigma$ from the clients $\\{0, \\dots, m-1\\}$ to the facilities $\\{0, \\dots, n-1\\}$ such that the opening\u2026","title":"Feasible facility location instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/facility-location-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FacilityLocation.facilityLocationDec","parent":32715,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/index.html#facility-location-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location","status":"fully_proved","summary":"FACILITY LOCATION is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a facility location instance that is well-formed and feasible; bitstrings that do not decode to an instance are rejected.","title":"The FACILITY LOCATION decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/facility-location-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FacilityLocation.facilityLocation_npComplete","parent":32715,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/facility-location/index.html#facility-location-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/facility-location","status":"can_prove","summary":"FACILITY LOCATION belongs to NP, and every decision problem $L$ in NP reduces to FACILITY LOCATION by a polynomial-time many-one reduction: there is a polynomial-time computable map $f$ on bitstrings such that, for every bitstring $x$, $L$ accepts $x$ exactly when FACILITY LOCATION accepts $f(x)$.","title":"FACILITY LOCATION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem","status":"can_prove","title":"Generalized assignment problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/generalized-assignment-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GeneralizedAssignment.GeneralizedAssignmentInstance","parent":32721,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/index.html#generalized-assignment-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem","status":"fully_proved","summary":"A generalized-assignment instance is a tuple $(R, C, c, K)$ where $R$ and $C$ are finite lists of finite lists of natural numbers, read as matrices indexed by job and machine ($R_{j,i}$ is the resource job $j$ consumes on machine $i$ and $C_{j,i}$ is the cost of assigning job $j$ to machine $i$), $c$ is a finite list of natural numbers (the machine capacities $c_i$), and $K$ is a natural number (the cost bound).","title":"Generalized-assignment instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GeneralizedAssignment.WellFormed","parent":32721,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem","status":"fully_proved","summary":"A generalized-assignment instance $(R, C, c, K)$ is well formed when $R$ and $C$ have the same number of rows and every row of $R$ and every row of $C$ has as many entries as $c$ (one per machine).","title":"Well-formed generalized-assignment instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/generalized-assignment-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GeneralizedAssignment.GeneralizedAssignmentFeasible","parent":32721,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/index.html#generalized-assignment-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem","status":"fully_proved","summary":"Let $(R, C, c, K)$ be a generalized-assignment instance, let $n$ be the number of rows of $R$ (the jobs) and $m$ the length of $c$ (the machines), and read entries missing from $R$ or $C$ as $0$. The instance is feasible when there are machines $a_1, \\dots, a_n \\in \\{1, \\dots, m\\}$ such that for every machine $i$ the sum of $R_{j,i}$ over the jobs $j$ with $a_j = i$ is at most $c_i$, and\u2026","title":"Feasible generalized assignments"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/generalized-assignment-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GeneralizedAssignment.generalizedAssignmentDec","parent":32721,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/index.html#generalized-assignment-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem","status":"fully_proved","summary":"The decision problem GENERALIZED ASSIGNMENT accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of generalized-assignment instances, to an instance that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"The generalized assignment decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/generalized-assignment-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.GeneralizedAssignment.generalizedAssignment_npComplete","parent":32721,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem/index.html#generalized-assignment-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/generalized-assignment-problem","status":"can_prove","summary":"The generalized assignment decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"GENERALIZED ASSIGNMENT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage","status":"can_prove","title":"Maximum coverage"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/maximum-coverage-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCoverage.MaximumCoverageInstance","parent":32727,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/index.html#maximum-coverage-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage","status":"fully_proved","summary":"A maximum-coverage instance is a tuple $(n, S, K, M)$ of a natural number $n$ (the universe size), a list $S$ of lists of natural numbers (the subsets), a natural number $K$ (the set budget) and a natural number $M$ (the coverage target).","title":"Maximum-coverage instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCoverage.WellFormed","parent":32727,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage","status":"fully_proved","summary":"A maximum-coverage instance $(n, S, K, M)$ is well formed when every subset in the list $S$ has no repeated elements and all of its elements are less than $n$.","title":"Well-formed maximum-coverage instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/maximum-coverage-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCoverage.MaximumCoverageFeasible","parent":32727,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/index.html#maximum-coverage-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage","status":"fully_proved","summary":"A maximum-coverage instance $(n, S, K, M)$, with universe size $n$, list of subsets $S$, set budget $K$ and coverage target $M$, is feasible when there are a sublist $T$ of $S$ with at most $K$ entries and a list of at least $M$ pairwise distinct natural numbers, each less than $n$ and each belonging to some subset in $T$.","title":"Feasible maximum-coverage instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/maximum-coverage-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCoverage.maximumCoverageDec","parent":32727,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/index.html#maximum-coverage-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage","status":"fully_proved","summary":"MAXIMUM COVERAGE is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a maximum-coverage instance $(n, S, K, M)$ that is well formed and feasible: every subset in the list $S$ has no repeated elements and only elements less than $n$, and some sublist of $S$ with at most $K$ entries covers at least $M$ distinct elements less than $n$. Bitstrings\u2026","title":"The MAXIMUM COVERAGE decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/maximum-coverage-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaximumCoverage.maximumCoverage_npComplete","parent":32727,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage/index.html#maximum-coverage-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-coverage","status":"can_prove","summary":"The MAXIMUM COVERAGE decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"MAXIMUM COVERAGE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage","status":"can_prove","title":"Maximum n-times coverage"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxNTimesCoverage.Instance","parent":32733,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/index.html#instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage","status":"fully_proved","summary":"An instance: `n`, the coverage rows of the overlays, element weights, `k`, `\u03c4`.","title":"Instance of maximum n-times coverage"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxNTimesCoverage.WellFormed","parent":32733,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage","status":"fully_proved","summary":"Well-formedness: every row has one entry per element.","title":"Well-formed maximum n-times coverage instance"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/covered-weight","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxNTimesCoverage.coveredWeight","parent":32733,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/index.html#covered-weight","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage","status":"fully_proved","summary":"The weight of the elements covered at least `n` times by the rows `sub`.","title":"Weight covered at least n times"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxNTimesCoverage.Feasible","parent":32733,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/index.html#feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage","status":"fully_proved","summary":"At most `k` overlays cover weight at least `\u03c4` at least `n` times.","title":"Feasibility for maximum n-times coverage"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/max-ntimes-coverage-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxNTimesCoverage.maxNTimesCoverageDec","parent":32733,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/index.html#max-ntimes-coverage-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage","status":"fully_proved","summary":"MAXIMUM n-TIMES COVERAGE as a decision problem over bitstrings.","title":"Maximum n-times coverage decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/max-ntimes-coverage-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MaxNTimesCoverage.maxNTimesCoverage_npComplete","parent":32733,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage/index.html#max-ntimes-coverage-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/maximum-n-times-coverage","status":"can_prove","summary":"MAXIMUM n-TIMES COVERAGE is NP-complete (Liu, Dimitrakakis, Carter and Gifford 2022).","title":"MAXIMUM n-TIMES COVERAGE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set","status":"can_prove","title":"Minimum test set"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/minimum-test-set-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumTestSet.MinimumTestSetInstance","parent":32740,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/index.html#minimum-test-set-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set","status":"fully_proved","summary":"An instance of MINIMUM TEST SET is a triple $(n, C, K)$ of a natural number $n$ (the number of elements), a finite list $C$ of finite lists of natural numbers (the tests) and a natural number $K$ (the bound).","title":"Minimum test set instance"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumTestSet.WellFormed","parent":32740,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set","status":"fully_proved","summary":"An instance $(n, C, K)$ of MINIMUM TEST SET is well formed when every test in the list $C$ is a list of numbers below $n$ without repeated entries, so that each test lists a subset of $\\{0, \\dots, n-1\\}$, each element once.","title":"Well-formed minimum test set instance"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/minimum-test-set-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumTestSet.MinimumTestSetFeasible","parent":32740,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/index.html#minimum-test-set-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set","status":"fully_proved","summary":"An instance $(n, C, K)$ of MINIMUM TEST SET is feasible when some sublist $T$ of $C$ (obtained by deleting entries of $C$) has length at most $K$ and distinguishes every pair: for all distinct $a, b < n$ some test in $T$ contains exactly one of $a$ and $b$.","title":"Feasible minimum test set instance"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/minimum-test-set-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumTestSet.minimumTestSetDec","parent":32740,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/index.html#minimum-test-set-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set","status":"fully_proved","summary":"It accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, C, K)$ of a natural number $n$, a list $C$ of lists of natural numbers (the tests) and a natural number $K$ with the following properties: every test is a list of numbers below $n$ without repeated entries, and some sublist $T$ of $C$, obtained by deleting entries of $C$, has length at most $K$ and\u2026","title":"The MINIMUM TEST SET decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/minimum-test-set-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.MinimumTestSet.minimumTestSet_npComplete","parent":32740,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set/index.html#minimum-test-set-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/minimum-test-set","status":"can_prove","summary":"The MINIMUM TEST SET decision problem lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"MINIMUM TEST SET is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center","status":"can_prove","title":"Nonmetric k-center"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/kcenter-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KCenter.KCenterInstance","parent":32746,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/index.html#kcenter-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center","status":"fully_proved","summary":"An instance of generalized nonmetric k-center is a triple $(D, K, B)$ of a finite list $D$ of rows, each a finite list of natural numbers (row $c$ lists the costs from client $c$ to the locations), and natural numbers $K$, the bound on the number of locations, and $B$, the radius.","title":"k-center instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KCenter.WellFormed","parent":32746,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center","status":"fully_proved","summary":"A k-center instance $(D, K, B)$ is well formed when all rows of $D$ have the same length.","title":"Well-formed k-center instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/kcenter-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KCenter.KCenterFeasible","parent":32746,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/index.html#kcenter-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center","status":"fully_proved","summary":"Let $(D, K, B)$ be a k-center instance with rows $D_0, \\dots, D_{m-1}$, and let $n$ be the length of $D_0$, with $n = 0$ if $m = 0$. For $c < m$ and a natural number $f$, write $D_c(f)$ for the entry of $D_c$ at position $f$, counting from $0$, and set $D_c(f) = 0$ if $D_c$ has at most $f$ entries. The instance is feasible when there is a set $S$ of at most $K$ locations $f < n$ such that every client $c < m$ has\u2026","title":"Feasible k-center instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/k-center-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KCenter.kCenterDec","parent":32746,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/index.html#k-center-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center","status":"fully_proved","summary":"This decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a k-center instance that is well formed and feasible; bitstrings that do not decode to an instance are rejected.","title":"Generalized nonmetric k-center as a decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/k-center-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KCenter.kCenter_npComplete","parent":32746,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center/index.html#k-center-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-center","status":"can_prove","summary":"The decision problem of generalized nonmetric k-center on bitstrings is NP-complete: it belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"k-CENTER (GENERALIZED, NONMETRIC) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median","status":"can_prove","title":"Nonmetric k-median"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/kmedian-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KMedian.KMedianInstance","parent":32752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/index.html#kmedian-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median","status":"fully_proved","summary":"A k-median instance is a triple $(d, K, B)$ of a cost matrix $d$, given as a list of rows of natural numbers (one row per client, one entry per location; the rows need not have equal length), a natural number $K$ bounding the number of open locations, and a natural-number budget $B$.","title":"k-median instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KMedian.WellFormed","parent":32752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median","status":"fully_proved","summary":"A k-median instance $(d, K, B)$ is well formed when all rows of its cost matrix $d$ have the same length.","title":"Well-formed k-median instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/kmedian-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KMedian.KMedianFeasible","parent":32752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/index.html#kmedian-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median","status":"fully_proved","summary":"Let $(d, K, B)$ be a k-median instance whose cost matrix $d$ has rows $d_0, \\dots, d_{n-1}$, one per client, and let $m$ be the length of $d_0$, or $m = 0$ when $n = 0$. The instance is feasible when there are a set $S \\subseteq \\{0, \\dots, m-1\\}$ of at most $K$ locations and a choice of $a(c) \\in S$ for each client $c < n$ such that $\\sum_{c < n} d_c(a(c)) \\le B$, where $d_c(j)$ is the entry of $d_c$ at index $j$\u2026","title":"Feasible k-median instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/k-median-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KMedian.kMedianDec","parent":32752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/index.html#k-median-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median","status":"fully_proved","summary":"The k-MEDIAN decision problem. k-MEDIAN (generalized, nonmetric) is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a k-median instance $(d, K, B)$ that is well formed and feasible. That is, $d$ has rows $d_0, \\dots, d_{n-1}$ (the clients) of a common length $m$ (the locations), and there are a set $S \\subseteq \\{0, \\dots, m-1\\}$ of at most $K$\u2026","title":"The k-MEDIAN decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/k-median-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.KMedian.kMedian_npComplete","parent":32752,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median/index.html#k-median-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/nonmetric-k-median","status":"can_prove","summary":"The k-MEDIAN decision problem is in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"k-MEDIAN (GENERALIZED, NONMETRIC) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem","status":"can_prove","title":"Quadratic assignment problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/quadratic-assignment-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.QuadraticAssignment.QuadraticAssignmentInstance","parent":32758,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/index.html#quadratic-assignment-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem","status":"fully_proved","summary":"An instance of quadratic assignment is a tuple $(n, F, D, K)$ consisting of a natural number $n$ (the size), a flow matrix $F$ and a distance matrix $D$, each given as a list of rows that are lists of natural numbers, and a natural number $K$ (the cost bound).","title":"Instances of QUADRATIC ASSIGNMENT"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.QuadraticAssignment.WellFormed","parent":32758,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem","status":"fully_proved","summary":"An instance $(n, F, D, K)$ of quadratic assignment is well formed when the flow matrix $F$ and the distance matrix $D$ each have exactly $n$ rows and every row of each has length $n$.","title":"Well-formed quadratic assignment instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/quadratic-assignment-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.QuadraticAssignment.QuadraticAssignmentFeasible","parent":32758,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/index.html#quadratic-assignment-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem","status":"fully_proved","summary":"An instance $(n, F, D, K)$ of quadratic assignment is feasible when there is a permutation $\\pi$ of the numbers $0, \\dots, n - 1$ such that","title":"Feasible quadratic assignment instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/quadratic-assignment-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.QuadraticAssignment.quadraticAssignmentDec","parent":32758,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/index.html#quadratic-assignment-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem","status":"fully_proved","summary":"The decision problem QUADRATIC ASSIGNMENT accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a quadratic assignment instance that is well formed and feasible; bitstrings that do not decode are rejected.","title":"QUADRATIC ASSIGNMENT as a decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/quadratic-assignment-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.QuadraticAssignment.quadraticAssignment_npComplete","parent":32758,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem/index.html#quadratic-assignment-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/quadratic-assignment-problem","status":"can_prove","summary":"The decision problem QUADRATIC ASSIGNMENT lies in NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"QUADRATIC ASSIGNMENT is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/set-cover.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover","status":"can_prove","title":"Set cover"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/set-cover-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetCover.SetCoverInstance","parent":32764,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/index.html#set-cover-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover","status":"fully_proved","summary":"A set-cover instance is a triple $(n, C, K)$ consisting of a natural number $n$, the size of the universe, a finite list $C$ of finite lists of natural numbers, the subsets, and a natural number $K$, the bound.","title":"Set-cover instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetCover.WellFormed","parent":32764,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover","status":"fully_proved","summary":"A set-cover instance $(n, C, K)$ is well formed when every list $s$ in $C$ has no repeated entries and every entry of $s$ is less than $n$, that is, each member of $C$ lists the elements of a subset of $\\{0, \\dots, n - 1\\}$ without repetition.","title":"Well-formed set-cover instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/set-cover-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetCover.SetCoverFeasible","parent":32764,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/index.html#set-cover-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover","status":"fully_proved","summary":"A set-cover instance $(n, C, K)$ is feasible when there is a sublist $D$ of $C$ (a list obtained from $C$ by deleting entries while keeping their order) with at most $K$ entries such that every natural number $x < n$ belongs to some list in $D$.","title":"Feasible set-cover instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/set-cover-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetCover.setCoverDec","parent":32764,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/index.html#set-cover-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover","status":"fully_proved","summary":"SET COVER is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a set-cover instance $(n, C, K)$ that is both well formed and feasible: every list in $C$ has no repeated entries and only entries less than $n$, and some sublist of $C$ with at most $K$ entries contains every natural number $y < n$ in one of its lists. Bitstrings that do not decode to\u2026","title":"The SET COVER decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/set-cover-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetCover.setCover_npComplete","parent":32764,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-cover/index.html#set-cover-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-cover","status":"can_prove","summary":"The SET COVER decision problem lies in NP, and every decision problem $L$ in NP has a polynomial-time many-one reduction to SET COVER.","title":"SET COVER is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/set-packing.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing","status":"can_prove","title":"Set packing"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/set-packing-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetPacking.SetPackingInstance","parent":32770,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/index.html#set-packing-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing","status":"fully_proved","summary":"A set-packing instance is a triple $(n, C, K)$ consisting of a natural number $n$, the size of the universe, a finite list $C$ of finite lists of natural numbers, the subsets, and a natural number $K$, the bound.","title":"Set-packing instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetPacking.WellFormed","parent":32770,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing","status":"fully_proved","summary":"A set-packing instance $(n, C, K)$ is well formed when every list $s$ in $C$ has no repeated entries and every entry of $s$ is less than $n$, that is, each member of $C$ lists the elements of a subset of $\\{0, \\dots, n - 1\\}$ without repetition.","title":"Well-formed set-packing instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/set-packing-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetPacking.SetPackingFeasible","parent":32770,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/index.html#set-packing-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing","status":"fully_proved","summary":"A set-packing instance $(n, C, K)$ is feasible when there is a sublist $D$ of $C$ (a list obtained from $C$ by deleting entries while keeping their order) with no repeated entries and at least $K$ entries such that any two distinct entries of $D$ have no element in common.","title":"Feasible set-packing instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/set-packing-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetPacking.setPackingDec","parent":32770,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/index.html#set-packing-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing","status":"fully_proved","summary":"SET PACKING is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a set-packing instance $(n, C, K)$ that is both well formed and feasible: every list in $C$ has no repeated entries and only entries less than $n$, and some sublist of $C$ with no repeated entries and at least $K$ entries has no element in common between any two of its distinct\u2026","title":"The SET PACKING decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/set-packing-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetPacking.setPacking_npComplete","parent":32770,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-packing/index.html#set-packing-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-packing","status":"can_prove","summary":"The SET PACKING decision problem lies in NP, and every decision problem $L$ in NP has a polynomial-time many-one reduction to SET PACKING.","title":"SET PACKING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting.json","id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting","kind":"scope","navigation":"module","parent":32696,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/index.html","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting","status":"can_prove","title":"Set splitting"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/set-splitting-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetSplitting.SetSplittingInstance","parent":32776,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/index.html#set-splitting-instance","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting","status":"fully_proved","summary":"A set-splitting instance is a pair $(n, C)$ of a natural number $n$, the size of the universe, and a list $C$ of lists of natural numbers, the subsets to be split.","title":"Set-splitting instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetSplitting.WellFormed","parent":32776,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting","status":"fully_proved","summary":"A set-splitting instance $(n, C)$, a natural number $n$ together with a list $C$ of lists of natural numbers, is well formed when every list in $C$ has no repeated entries and all its entries are less than $n$.","title":"Well-formed set-splitting instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/set-splitting-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetSplitting.SetSplittingFeasible","parent":32776,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/index.html#set-splitting-feasible","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting","status":"fully_proved","summary":"A set-splitting instance $(n, C)$, a natural number $n$ together with a list $C$ of lists of natural numbers, is feasible when there is a finite set $S$ of natural numbers, all less than $n$, such that every list in $C$ has an entry in $S$ and an entry not in $S$.","title":"Feasible set-splitting instances"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/set-splitting-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetSplitting.setSplittingDec","parent":32776,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/index.html#set-splitting-dec","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting","status":"fully_proved","summary":"SET SPLITTING is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a pair $(n, C)$ of a natural number $n$ and a list $C$ of lists of natural numbers such that every list in $C$ has no repeated entries and only entries less than $n$, and some finite set $S$ of natural numbers less than $n$ contains an entry of every list in $C$ and misses an entry\u2026","title":"The SET SPLITTING decision problem"},{"id":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/set-splitting-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.SetSplitting.setSplitting_npComplete","parent":32776,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting/index.html#set-splitting-np-complete","scope":"TheoryOfComputation/np-completeness/set-systems-and-location/set-splitting","status":"can_prove","summary":"The SET SPLITTING decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"SET SPLITTING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling.json","id":"TheoryOfComputation/np-completeness/scheduling","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/index.html","scope":"TheoryOfComputation/np-completeness/scheduling","status":"can_prove","summary":"Single-machine, parallel-machine, flow-shop, precedence-constrained and resource-constrained scheduling with deadlines, tardiness or weighted completion objectives.","title":"Scheduling problems"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits.json","id":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits","status":"can_prove","title":"Job sequencing with deadlines and profits"},{"id":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/job-sequencing-with-processing-times-ins","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.JobSequencingWithProcessingTimes.JobSequencingWithProcessingTimesInstance","parent":32783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/index.html#job-sequencing-with-processing-times-ins","scope":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits","status":"fully_proved","summary":"An instance of job sequencing with deadlines, processing times and profits is a pair $(J, K)$ of a finite list $J$ of jobs, each a triple $(p, d, w)$ of natural numbers giving its processing time $p$, deadline $d$ and profit $w$, and a natural number $K$, the profit bound.","title":"Job-sequencing instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/job-sequencing-with-processing-times-fea","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.JobSequencingWithProcessingTimes.JobSequencingWithProcessingTimesFeasible","parent":32783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/index.html#job-sequencing-with-processing-times-fea","scope":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits","status":"fully_proved","summary":"Let $(J, K)$ be a job-sequencing instance with jobs $J = (p_0, d_0, w_0), \\dots, (p_{n-1}, d_{n-1}, w_{n-1})$. The instance is feasible when there are distinct indices $j_1, \\dots, j_m < n$, for some $m \\ge 0$, such that $w_{j_1} + \\dots + w_{j_m} \\ge K$ and, for every $t$ with $1 \\le t \\le m$, $p_{j_1} + \\dots + p_{j_t} \\le d_{j_t}$: the selected jobs, run one after another in this order from time $0$, each finish\u2026","title":"Feasible job-sequencing instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/job-sequencing-with-processing-times-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.JobSequencingWithProcessingTimes.jobSequencingWithProcessingTimesDec","parent":32783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/index.html#job-sequencing-with-processing-times-dec","scope":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits","status":"fully_proved","summary":"This decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a job-sequencing instance that is feasible; bitstrings that do not decode to an instance are rejected.","title":"Job sequencing with deadlines, processing times and profits as a decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/job-sequencing-with-processing-times-np","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.JobSequencingWithProcessingTimes.jobSequencingWithProcessingTimes_npComplete","parent":32783,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits/index.html#job-sequencing-with-processing-times-np","scope":"TheoryOfComputation/np-completeness/scheduling/job-sequencing-with-deadlines-and-profits","status":"can_prove","summary":"The decision problem of job sequencing with deadlines, processing times and profits on bitstrings is NP-complete: it belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"JOB SEQUENCING WITH DEADLINES, PROCESSING TIMES, AND PROFITS is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines.json","id":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines","status":"can_prove","title":"Parallel machine scheduling with deadlines"},{"id":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/parallel-machines-deadlines-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ParallelMachinesDeadlines.ParallelMachinesDeadlinesInstance","parent":32788,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/index.html#parallel-machines-deadlines-instance","scope":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines","status":"fully_proved","summary":"An instance of parallel machine scheduling with deadlines is a pair consisting of a finite list of jobs, each a pair $(p, d)$ of natural numbers giving its processing time $p$ and deadline $d$, and a natural number $m$, the number of machines.","title":"Parallel machine scheduling instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/parallel-machines-deadlines-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ParallelMachinesDeadlines.ParallelMachinesDeadlinesFeasible","parent":32788,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/index.html#parallel-machines-deadlines-feasible","scope":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines","status":"fully_proved","summary":"An instance consisting of jobs $(p_1, d_1), \\dots, (p_k, d_k)$, where job $j$ has processing time $p_j$ and deadline $d_j$, and a number $m$ of machines is feasible when there are start times $s_1, \\dots, s_k \\in \\mathbb{N}$ and machines $\\mu_1, \\dots, \\mu_k \\in \\{0, \\dots, m - 1\\}$ such that $s_j + p_j \\le d_j$ for every $j$, and for all $a \\ne b$ with $\\mu_a = \\mu_b$, either $s_a + p_a \\le s_b$ or\u2026","title":"Feasible schedules on parallel machines with deadlines"},{"id":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/parallel-machines-deadlines-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ParallelMachinesDeadlines.parallelMachinesDeadlinesDec","parent":32788,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/index.html#parallel-machines-deadlines-dec","scope":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines","status":"fully_proved","summary":"This decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to jobs $(p_1, d_1), \\dots, (p_k, d_k)$ of natural numbers (processing times and deadlines) and a number $m$ of machines for which there are start times $s_1, \\dots, s_k \\in \\mathbb{N}$ and machines $\\mu_1, \\dots, \\mu_k \\in \\{0, \\dots, m - 1\\}$ such that $s_j + p_j \\le d_j$ for every $j$, and for all\u2026","title":"The PARALLEL MACHINES WITH DEADLINES decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/parallel-machines-deadlines-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ParallelMachinesDeadlines.parallelMachinesDeadlines_npComplete","parent":32788,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines/index.html#parallel-machines-deadlines-np-complete","scope":"TheoryOfComputation/np-completeness/scheduling/parallel-machine-scheduling-with-deadlines","status":"can_prove","summary":"The PARALLEL MACHINES WITH DEADLINES decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"PARALLEL MACHINES WITH DEADLINES is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling.json","id":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling","status":"can_prove","title":"Precedence-constrained multiprocessor scheduling"},{"id":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/precedence-multiprocessor-scheduling-ins","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceMultiprocessorScheduling.PrecedenceMultiprocessorSchedulingInstance","parent":32793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/index.html#precedence-multiprocessor-scheduling-ins","scope":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling","status":"fully_proved","summary":"An instance is a tuple $(d, P, m, D)$ in which $d = (d_0, \\dots, d_{n-1})$ is a list of natural numbers, $d_j$ being the duration of job $j$; $P$ is a list of pairs $(a, b)$ of natural numbers, each read as the precedence constraint that job $a$ precedes job $b$; $m$ is a natural number, the number of machines; and $D$ is a natural number, the deadline.","title":"Instances of precedence-constrained multiprocessor scheduling"},{"id":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceMultiprocessorScheduling.WellFormed","parent":32793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling","status":"fully_proved","summary":"An instance $(d, P, m, D)$ whose list of job durations $d$ has length $n$ is well formed when every precedence pair $(a, b) \\in P$ satisfies $a < n$, $b < n$ and $a \\ne b$, and there is an ordering of the jobs $0, \\dots, n-1$ (a permutation of this list) in which $a$ comes before $b$ for every $(a, b) \\in P$.","title":"Well-formed precedence-multiprocessor scheduling instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/precedence-multiprocessor-scheduling-fea","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceMultiprocessorScheduling.PrecedenceMultiprocessorSchedulingFeasible","parent":32793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/index.html#precedence-multiprocessor-scheduling-fea","scope":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling","status":"fully_proved","summary":"An instance $(d, P, m, D)$ with job durations $d = (d_0, \\dots, d_{n-1})$ is feasible when there are start times $s_0, \\dots, s_{n-1} \\in \\mathbb{N}$ and machines $\\mu_0, \\dots, \\mu_{n-1} \\in \\mathbb{N}$ such that $\\mu_j < m$ and $s_j + d_j \\le D$ for every $j < n$, $s_a + d_a \\le s_b$ for every precedence pair $(a, b) \\in P$, and for all jobs $a < b < n$ with $\\mu_a = \\mu_b$, either $s_a + d_a \\le s_b$ or\u2026","title":"Feasible precedence-multiprocessor scheduling instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/precedence-multiprocessor-scheduling-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceMultiprocessorScheduling.precedenceMultiprocessorSchedulingDec","parent":32793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/index.html#precedence-multiprocessor-scheduling-dec","scope":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling","status":"fully_proved","summary":"The decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a precedence-multiprocessor scheduling instance that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"The precedence-constrained multiprocessor scheduling decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/precedence-multiprocessor-scheduling-np","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceMultiprocessorScheduling.precedenceMultiprocessorScheduling_npComplete","parent":32793,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling/index.html#precedence-multiprocessor-scheduling-np","scope":"TheoryOfComputation/np-completeness/scheduling/precedence-constrained-multiprocessor-scheduling","status":"can_prove","summary":"The precedence-constrained multiprocessor scheduling decision problem is in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"PRECEDENCE-CONSTRAINED MULTIPROCESSOR SCHEDULING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling.json","id":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling","status":"can_prove","title":"Resource-constrained project scheduling"},{"id":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/rcpspinstance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RCPSP.RCPSPInstance","parent":32799,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/index.html#rcpspinstance","scope":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling","status":"fully_proved","summary":"An instance of resource-constrained project scheduling (an RCPSP instance) is a quadruple $(J, P, C, D)$: a list $J$ of jobs, each a pair $(p, r) \\in \\mathbb{N} \\times \\mathbb{N}$ of a duration $p$ and a resource demand $r$; a list $P$ of precedence pairs $(a, b) \\in \\mathbb{N} \\times \\mathbb{N}$ of job indices; a resource capacity $C \\in \\mathbb{N}$; and a deadline $D \\in \\mathbb{N}$.","title":"Instances of resource-constrained project scheduling"},{"id":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RCPSP.WellFormed","parent":32799,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling","status":"fully_proved","summary":"An RCPSP instance $(J, P, C, D)$ whose job list $J$ has $n$ jobs is well formed when every precedence pair $(a, b) \\in P$ satisfies $a < n$, $b < n$ and $a \\ne b$, and there is a listing of $0, 1, \\dots, n-1$, each exactly once, in which $a$ comes before $b$ for every $(a, b) \\in P$.","title":"Well-formed RCPSP instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/rcpspfeasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RCPSP.RCPSPFeasible","parent":32799,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/index.html#rcpspfeasible","scope":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling","status":"fully_proved","summary":"Let $(J, P, C, D)$ be an RCPSP instance with $n$ jobs, where job $j < n$ has duration $p_j$ and resource demand $r_j$, and set $p_j = 0$ for $j \\ge n$. The instance is feasible when there are start times $s_0, \\dots, s_{n-1} \\in \\mathbb{N}$, with $s_j = 0$ for $j \\ge n$, such that every job finishes by the deadline ($s_j + p_j \\le D$ for all $j < n$), every precedence pair $(a, b) \\in P$ satisfies\u2026","title":"Feasible RCPSP instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/rcpsp-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RCPSP.rcpspDec","parent":32799,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/index.html#rcpsp-dec","scope":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling","status":"fully_proved","summary":"RESOURCE-CONSTRAINED PROJECT SCHEDULING is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of RCPSP instances, to an RCPSP instance that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"Resource-constrained project scheduling as a decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/rcpsp-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.RCPSP.rcpsp_npComplete","parent":32799,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling/index.html#rcpsp-np-complete","scope":"TheoryOfComputation/np-completeness/scheduling/resource-constrained-project-scheduling","status":"can_prove","summary":"The decision problem RESOURCE-CONSTRAINED PROJECT SCHEDULING belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"RESOURCE-CONSTRAINED PROJECT SCHEDULING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines.json","id":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines","status":"can_prove","title":"Scheduling with release times and deadlines"},{"id":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/release-deadlines-scheduling-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ReleaseDeadlinesScheduling.ReleaseDeadlinesSchedulingInstance","parent":32805,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/index.html#release-deadlines-scheduling-instance","scope":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines","status":"fully_proved","summary":"A release-deadlines instance is a list of jobs $(p_0, r_0, d_0), \\dots, (p_{n-1}, r_{n-1}, d_{n-1})$ of triples of natural numbers, where $p_j$ is the processing time, $r_j$ the release time and $d_j$ the deadline of job $j$.","title":"Instances of single-machine scheduling with release times and deadlines"},{"id":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/release-deadlines-scheduling-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ReleaseDeadlinesScheduling.ReleaseDeadlinesSchedulingFeasible","parent":32805,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/index.html#release-deadlines-scheduling-feasible","scope":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines","status":"fully_proved","summary":"A release-deadlines instance with jobs $(p_j, r_j, d_j)$ for $j < n$ is feasible when there are start times $s_0, \\dots, s_{n-1} \\in \\mathbb{N}$ such that $r_j \\le s_j$ and $s_j + p_j \\le d_j$ for every $j < n$, and for all $a < b < n$ either $s_a + p_a \\le s_b$ or $s_b + p_b \\le s_a$.","title":"Feasible single-machine schedules with release times and deadlines"},{"id":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/release-deadlines-scheduling-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ReleaseDeadlinesScheduling.releaseDeadlinesSchedulingDec","parent":32805,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/index.html#release-deadlines-scheduling-dec","scope":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines","status":"fully_proved","summary":"SINGLE-MACHINE SCHEDULING WITH RELEASE TIMES AND DEADLINES is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of release-deadlines instances, to a feasible instance. Bitstrings that do not decode to an instance are rejected.","title":"Single-machine scheduling with release times and deadlines as a decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/release-deadlines-scheduling-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ReleaseDeadlinesScheduling.releaseDeadlinesScheduling_npComplete","parent":32805,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines/index.html#release-deadlines-scheduling-np-complete","scope":"TheoryOfComputation/np-completeness/scheduling/scheduling-with-release-times-and-deadlines","status":"can_prove","summary":"The decision problem SINGLE-MACHINE SCHEDULING WITH RELEASE TIMES AND DEADLINES belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"SINGLE-MACHINE SCHEDULING WITH RELEASE TIMES AND DEADLINES is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling.json","id":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling","status":"can_prove","title":"Three-machine flow shop scheduling"},{"id":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/flow-shop-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FlowShop.FlowShopInstance","parent":32810,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/index.html#flow-shop-instance","scope":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling","status":"fully_proved","summary":"A flow-shop instance is a pair consisting of a finite list of jobs, each a finite list of natural numbers (its processing times on machines 1, 2 and 3, in that order), and a natural number $D$, the deadline. The type does not force a job to have exactly three entries; that is part of well-formedness.","title":"Flow-shop instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FlowShop.WellFormed","parent":32810,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling","status":"fully_proved","summary":"A flow-shop instance is well formed when every job in it is a list of exactly three natural numbers.","title":"Well-formed flow-shop instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/flow-shop-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FlowShop.FlowShopFeasible","parent":32810,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/index.html#flow-shop-feasible","scope":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling","status":"fully_proved","summary":"Let an instance consist of jobs $1, \\dots, n$, where job $j$ is a list of natural numbers whose $k$-th entry $p_{j,k}$ is its processing time on machine $k$ (read as $0$ if the list has fewer than $k$ entries), and a deadline $D$. The instance is feasible when there are natural numbers $s_{j,k}$, for each job $j$ and each $k \\in \\{1, 2, 3\\}$, such that for every job $j$ we have $s_{j,1} + p_{j,1} \\le s_{j,2}$,\u2026","title":"Feasible flow-shop schedules"},{"id":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/flow-shop-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FlowShop.flowShopDec","parent":32810,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/index.html#flow-shop-dec","scope":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling","status":"fully_proved","summary":"The decision problem 3-MACHINE FLOW SHOP SCHEDULING accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of flow-shop instances, to an instance that is well formed and feasible. Bitstrings that do not decode to an instance are rejected.","title":"The 3-machine flow shop scheduling decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/flow-shop-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.FlowShop.flowShop_npComplete","parent":32810,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling/index.html#flow-shop-np-complete","scope":"TheoryOfComputation/np-completeness/scheduling/three-machine-flow-shop-scheduling","status":"can_prove","summary":"The 3-machine flow shop scheduling decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"3-MACHINE FLOW SHOP SCHEDULING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling.json","id":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling","status":"can_prove","title":"Total tardiness scheduling"},{"id":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/total-tardiness-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalTardiness.TotalTardinessInstance","parent":32816,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/index.html#total-tardiness-instance","scope":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling","status":"fully_proved","summary":"An instance of TOTAL TARDINESS is a pair $(J, K)$ consisting of a finite list $J = ((p_0, d_0), \\dots, (p_{n-1}, d_{n-1}))$ of jobs, where job $j$ has processing time $p_j \\in \\mathbb{N}$ and due date $d_j \\in \\mathbb{N}$, and a bound $K \\in \\mathbb{N}$.","title":"Instances of TOTAL TARDINESS"},{"id":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/tardiness-step","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalTardiness.TardinessStep","parent":32816,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/index.html#tardiness-step","scope":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling","status":"fully_proved","summary":"Let $J = ((p_0, d_0), \\dots, (p_{n-1}, d_{n-1}))$ be a list of jobs with processing times $p_i \\in \\mathbb{N}$ and due dates $d_i \\in \\mathbb{N}$, let $j \\in \\mathbb{N}$, and let $(p, d) = (p_j, d_j)$ if $j < n$ and $(p, d) = (0, 0)$ otherwise. Running job $j$ from the state $(t, s) \\in \\mathbb{N} \\times \\mathbb{N}$, where $t$ is the elapsed time and $s$ the accumulated tardiness, gives the state\u2026","title":"One step of the tardiness computation"},{"id":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/total-tardiness-value","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalTardiness.TotalTardinessValue","parent":32816,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/index.html#total-tardiness-value","scope":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling","status":"fully_proved","summary":"Let $J = ((p_0, d_0), \\dots, (p_{n-1}, d_{n-1}))$ be a list of jobs with processing times $p_i \\in \\mathbb{N}$ and due dates $d_i \\in \\mathbb{N}$, and set $(p_i, d_i) = (0, 0)$ for $i \\ge n$. For a list $(j_1, \\dots, j_m)$ of natural numbers (job indices), let $C_k = p_{j_1} + \\dots + p_{j_k}$ be the completion time of the $k$-th job when the jobs are run in this order. The total tardiness of the order is\u2026","title":"Total tardiness of a job order"},{"id":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/total-tardiness-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalTardiness.TotalTardinessFeasible","parent":32816,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/index.html#total-tardiness-feasible","scope":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling","status":"fully_proved","summary":"An instance $(J, K)$ of TOTAL TARDINESS with $n$ jobs is feasible when some ordering of its jobs, that is, some permutation $(j_1, \\dots, j_n)$ of $(0, 1, \\dots, n - 1)$, has total tardiness at most $K$.","title":"Feasible TOTAL TARDINESS instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/total-tardiness-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalTardiness.totalTardinessDec","parent":32816,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/index.html#total-tardiness-dec","scope":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling","status":"fully_proved","summary":"The decision problem TOTAL TARDINESS accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding (natural numbers in binary), to an instance of TOTAL TARDINESS that is feasible, that is, whose jobs admit an ordering of total tardiness at most its bound. It rejects every bitstring that does not decode to an instance.","title":"The TOTAL TARDINESS decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/total-tardiness-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.TotalTardiness.totalTardiness_npComplete","parent":32816,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling/index.html#total-tardiness-np-complete","scope":"TheoryOfComputation/np-completeness/scheduling/total-tardiness-scheduling","status":"can_prove","summary":"The TOTAL TARDINESS decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"TOTAL TARDINESS is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra.json","id":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","status":"can_prove","title":"Weighted completion time with precedence constraints"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/precedence-weighted-completion-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceWeightedCompletion.PrecedenceWeightedCompletionInstance","parent":32823,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/index.html#precedence-weighted-completion-instance","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","status":"fully_proved","summary":"An instance of the scheduling problem $1 \\mid \\mathrm{prec} \\mid \\sum w_j C_j$ (one machine, precedence constraints, objective the sum over jobs $j$ of the weight $w_j$ times the completion time $C_j$) is a triple $(J, P, K)$: a list $J$ of jobs, each a pair $(p, w)$ of natural numbers giving its processing time and weight; a list $P$ of pairs of natural numbers, read as precedence constraints between job indices;\u2026","title":"Instances of single-machine weighted completion with precedence"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/weighted-completion-step","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceWeightedCompletion.WeightedCompletionStep","parent":32823,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/index.html#weighted-completion-step","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","status":"fully_proved","summary":"Let $J$ be a list of jobs, each a pair (processing time, weight), let $(t, s)$ be a state consisting of the elapsed time $t$ and the accumulated weighted completion time $s$, and let $j$ be a job index. With $(p, w)$ the job at position $j$ of $J$ counting from $0$, or $(0, 0)$ if $J$ has no such position, the step sends $(t, s)$ to $(t + p, s + w (t + p))$.","title":"One step of the weighted completion time computation"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/weighted-completion","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceWeightedCompletion.WeightedCompletion","parent":32823,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/index.html#weighted-completion","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","status":"fully_proved","summary":"Let $J$ be a list of jobs, each a pair $(p_i, w_i)$ of processing time and weight indexed from $0$, with $(p_i, w_i) = (0, 0)$ for $i$ outside the list. For a list $j_1, \\dots, j_m$ of job indices, the total weighted completion time is $\\sum_{k=1}^{m} w_{j_k} C_k$, where $C_k = p_{j_1} + \\dots + p_{j_k}$ is the completion time of the $k$-th job in the order.","title":"Total weighted completion time of a job order"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceWeightedCompletion.WellFormed","parent":32823,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","status":"fully_proved","summary":"An instance $(J, P, K)$ whose job list $J$ has $n$ jobs is well formed when every precedence pair $(a, b)$ in $P$ satisfies $a < n$, $b < n$ and $a \\ne b$.","title":"Well-formed weighted completion instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/precedence-weighted-completion-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceWeightedCompletion.PrecedenceWeightedCompletionFeasible","parent":32823,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/index.html#precedence-weighted-completion-feasible","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","status":"fully_proved","summary":"Let $(J, P, K)$ be an instance whose job list $J$ consists of $n$ jobs $(p_0, w_0), \\dots, (p_{n-1}, w_{n-1})$ (processing time, weight). It is feasible when there is an ordering $j_1, \\dots, j_n$ of the indices $0, \\dots, n-1$, each listed exactly once, such that for every pair $(a, b)$ in $P$ the position of $a$ in the ordering is smaller than the position of $b$, where an index that does not occur gets position\u2026","title":"Feasible weighted completion instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/precedence-weighted-completion-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceWeightedCompletion.precedenceWeightedCompletionDec","parent":32823,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/index.html#precedence-weighted-completion-dec","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","status":"fully_proved","summary":"The decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to an instance $(J, P, K)$ that is well formed and feasible: every precedence pair in $P$ consists of two distinct job indices of $J$, and some ordering of all the jobs that puts the first job of every precedence pair before the second has total weighted completion time at most $K$. Bitstrings that do not\u2026","title":"SINGLE-MACHINE WEIGHTED COMPLETION WITH PRECEDENCE as a decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/precedence-weighted-completion-np-comple","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.PrecedenceWeightedCompletion.precedenceWeightedCompletion_npComplete","parent":32823,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra/index.html#precedence-weighted-completion-np-comple","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-completion-time-with-precedence-constra","status":"can_prove","summary":"The decision problem SINGLE-MACHINE WEIGHTED COMPLETION WITH PRECEDENCE on bitstrings belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"SINGLE-MACHINE WEIGHTED COMPLETION WITH PRECEDENCE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs.json","id":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs","kind":"scope","navigation":"module","parent":32782,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/index.html","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs","status":"can_prove","title":"Weighted number of tardy jobs"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/weighted-tardy-jobs-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.WeightedTardyJobs.WeightedTardyJobsInstance","parent":32831,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/index.html#weighted-tardy-jobs-instance","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs","status":"fully_proved","summary":"An instance of WEIGHTED NUMBER OF TARDY JOBS is a pair consisting of a finite list of jobs, each a triple $(p, d, w)$ of natural numbers giving its processing time $p$, due date $d$ and weight $w$, and a natural number $K$, the bound.","title":"Weighted tardy jobs instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/weighted-tardy-step","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.WeightedTardyJobs.WeightedTardyStep","parent":32831,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/index.html#weighted-tardy-step","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs","status":"fully_proved","summary":"Let $J$ be a list of jobs, each a triple $(p, d, w)$ of natural numbers (processing time, due date, weight). Given a state $(t, s)$ and a natural number $j$, let $(p, d, w)$ be the job at position $j$ of $J$, counting from $0$, or $(0, 0, 0)$ if $J$ has no position $j$, and let $t' = t + p$. The step returns $(t', s + w)$ if $d < t'$, and $(t', s)$ otherwise.","title":"One step of the weighted tardy jobs computation"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/weighted-tardy","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.WeightedTardyJobs.WeightedTardy","parent":32831,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/index.html#weighted-tardy","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs","status":"fully_proved","summary":"Let $J$ be a list of jobs, each a triple of natural numbers (processing time, due date, weight), and write $(p_i, d_i, w_i)$ for the job at position $i$ of $J$, counting from $0$, with $(p_i, d_i, w_i) = (0, 0, 0)$ when $J$ has no position $i$. For a list $(j_1, \\dots, j_m)$ of natural numbers, the weighted number of tardy jobs of this order is the sum of $w_{j_k}$ over all $k$ with $d_{j_k} < C_k$, where\u2026","title":"Weighted number of tardy jobs of an ordering"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/weighted-tardy-jobs-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.WeightedTardyJobs.WeightedTardyJobsFeasible","parent":32831,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/index.html#weighted-tardy-jobs-feasible","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs","status":"fully_proved","summary":"A weighted tardy jobs instance, consisting of a list of $n$ jobs and a bound $K$, is feasible when there is an ordering of its jobs, that is, a list of the job indices $0, 1, \\dots, n - 1$ in which each index occurs exactly once, whose weighted number of tardy jobs is at most $K$.","title":"Feasible weighted tardy jobs instances"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/weighted-tardy-jobs-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.WeightedTardyJobs.weightedTardyJobsDec","parent":32831,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/index.html#weighted-tardy-jobs-dec","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs","status":"fully_proved","summary":"WEIGHTED NUMBER OF TARDY JOBS is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of weighted tardy jobs instances, to a feasible instance, that is, an instance having an ordering of its jobs whose weighted number of tardy jobs is at most its bound. A bitstring that decodes to no instance is rejected.","title":"The weighted number of tardy jobs decision problem"},{"id":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/weighted-tardy-jobs-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.WeightedTardyJobs.weightedTardyJobs_npComplete","parent":32831,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs/index.html#weighted-tardy-jobs-np-complete","scope":"TheoryOfComputation/np-completeness/scheduling/weighted-number-of-tardy-jobs","status":"can_prove","summary":"The weighted number of tardy jobs decision problem lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"WEIGHTED NUMBER OF TARDY JOBS is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles","kind":"scope","parent":30323,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles","status":"can_prove","summary":"String problems (closest string, longest common subsequence, shortest common supersequence) and puzzles (Sudoku, Nonogram, Akari, Minesweeper, Latin square completion).","title":"Strings, puzzles and Latin squares"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles/akari.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari","kind":"scope","navigation":"module","parent":32838,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/akari/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari","status":"can_prove","title":"Akari"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari/akari-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Akari.AkariInstance","parent":32839,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/akari/index.html#akari-instance","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari","status":"fully_proved","summary":"An Akari instance is a board given as a list of rows, each row a list of natural numbers: the entry $0$ is an empty cell, $1$ is a wall, and $2 + k$ is a wall numbered $k$. The rows need not have the same length; that is part of well-formedness.","title":"Akari boards"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari/illuminates","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Akari.Illuminates","parent":32839,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/akari/index.html#illuminates","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari","status":"fully_proved","summary":"Let $B$ be a board, a list of rows of natural numbers, and write $B_{r,c}$ for the entry in row $r$ and column $c$, read as $1$ when there is no such entry. A bulb at $(b_r, b_c)$ illuminates the cell $(r, c)$ when either $b_r = r$ and $B_{r,c'} = 0$ for every column $c'$ strictly between $b_c$ and $c$, or $b_c = c$ and $B_{r',c} = 0$ for every row $r'$ strictly between $b_r$ and $r$. In particular, a bulb\u2026","title":"Illumination on an Akari board"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Akari.WellFormed","parent":32839,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/akari/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari","status":"fully_proved","summary":"An Akari board is well formed when every entry is at most $6$ and all its rows have the same length.","title":"Well-formed Akari boards"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari/akari-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Akari.AkariFeasible","parent":32839,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/akari/index.html#akari-feasible","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari","status":"fully_proved","summary":"Let $B$ be an Akari board with $m$ rows, let $n$ be the length of its first row ($n = 0$ when there are no rows), and write $B_{r,c}$ for the entry in row $r$ and column $c$, read as $1$ when there is no such entry. The board $B$ is feasible when there is a finite set $S$ of pairs of natural numbers (the bulbs) such that every bulb $(r, c) \\in S$ has $r < m$, $c < n$ and $B_{r,c} = 0$; every cell $(r, c)$ with\u2026","title":"Feasible Akari boards"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari/akari-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Akari.akariDec","parent":32839,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/akari/index.html#akari-dec","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari","status":"fully_proved","summary":"The Akari (Light Up) decision problem accepts a list of booleans $x$ exactly when $x$ decodes, under the canonical bitstring encoding of lists of lists of natural numbers, to an Akari board that is well formed and feasible; it rejects every list that does not decode.","title":"The Akari decision problem"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari/akari-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Akari.akari_npComplete","parent":32839,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/akari/index.html#akari-np-complete","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/akari","status":"can_prove","summary":"The Akari decision problem belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"AKARI (LIGHT UP) is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string","kind":"scope","navigation":"module","parent":32838,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string","status":"can_prove","title":"Closest string"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/closest-string-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ClosestString.ClosestStringInstance","parent":32846,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/index.html#closest-string-instance","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string","status":"fully_proved","summary":"A closest-string instance is a pair $(S, d)$ of a finite list $S$ of bitstrings (finite lists of booleans) and a natural number $d$, the distance bound.","title":"Closest-string instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ClosestString.WellFormed","parent":32846,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string","status":"fully_proved","summary":"A closest-string instance $(S, d)$ is well formed when all bitstrings in the list $S$ have the same length.","title":"Well-formed closest-string instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/closest-string-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ClosestString.ClosestStringFeasible","parent":32846,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/index.html#closest-string-feasible","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string","status":"fully_proved","summary":"Write $\\ell(x)$ for the length of a bitstring $x$. A closest-string instance $(S, d)$ is feasible when there is a bitstring $c$ with $\\ell(c) = \\ell(s_1)$, where $s_1$ is the first bitstring of $S$ (and $\\ell(c) = 0$ if $S$ is empty), such that for every $s$ in $S$ the number of positions $j < \\min(\\ell(c), \\ell(s))$ at which $c$ and $s$ differ is at most $d$.","title":"Feasible closest-string instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/closest-string-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ClosestString.closestStringDec","parent":32846,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/index.html#closest-string-dec","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string","status":"fully_proved","summary":"The decision problem CLOSEST STRING accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of closest-string instances, to an instance that is well formed and feasible. Bitstrings that do not decode to a closest-string instance are rejected.","title":"The CLOSEST STRING decision problem"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/closest-string-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ClosestString.closestString_npComplete","parent":32846,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string/index.html#closest-string-np-complete","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/closest-string","status":"can_prove","summary":"The CLOSEST STRING decision problem belongs to NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"CLOSEST STRING is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion","kind":"scope","navigation":"module","parent":32838,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion","status":"can_prove","title":"Latin square completion"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/latin-square-completion-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LatinSquareCompletion.LatinSquareCompletionInstance","parent":32852,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/index.html#latin-square-completion-instance","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion","status":"fully_proved","summary":"An instance of LATIN SQUARE COMPLETION is a pair $(n, G)$ of a natural number $n$, the intended size, and a list $G$ of lists of natural numbers, read as the rows of a partial grid in which the entry $0$ marks a blank cell.","title":"Latin square completion instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LatinSquareCompletion.WellFormed","parent":32852,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion","status":"fully_proved","summary":"An instance $(n, G)$ of LATIN SQUARE COMPLETION is well formed when $n > 0$, $G$ has exactly $n$ rows, every row of $G$ has length $n$, and every entry of $G$ is at most $n$.","title":"Well-formed Latin square completion instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/latin-square-completion-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LatinSquareCompletion.LatinSquareCompletionFeasible","parent":32852,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/index.html#latin-square-completion-feasible","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion","status":"fully_proved","summary":"An instance $(n, G)$ of LATIN SQUARE COMPLETION is feasible when there is a grid $C$ consisting of $n$ rows of length $n$ such that for all $r, c < n$ the entry $G_{rc}$ is either $0$ or equal to $C_{rc}$, every row of $C$ is a permutation of $1, \\dots, n$, and every column of $C$ is a permutation of $1, \\dots, n$. Here $G_{rc}$ is taken to be $0$ when $G$ has no entry in row $r$ and column $c$.","title":"Completable partial Latin squares"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/latin-square-completion-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LatinSquareCompletion.latinSquareCompletionDec","parent":32852,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/index.html#latin-square-completion-dec","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion","status":"fully_proved","summary":"The decision problem LATIN SQUARE COMPLETION accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs of a natural number and a list of lists of natural numbers, to an instance $(n, G)$ that is well formed ($n > 0$ and $G$ consists of $n$ rows of length $n$ with entries at most $n$) and feasible ($G$ can be completed to an $n \\times n$ grid whose rows and columns are\u2026","title":"LATIN SQUARE COMPLETION as a decision problem"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/latin-square-completion-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LatinSquareCompletion.latinSquareCompletion_npComplete","parent":32852,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion/index.html#latin-square-completion-np-complete","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/latin-square-completion","status":"can_prove","summary":"The decision problem LATIN SQUARE COMPLETION on bitstrings lies in NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"LATIN SQUARE COMPLETION is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings","kind":"scope","navigation":"module","parent":32838,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings","status":"can_prove","title":"Longest common subsequence of many strings"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/longest-common-subsequence-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCommonSubsequence.LongestCommonSubsequenceInstance","parent":32858,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/index.html#longest-common-subsequence-instance","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings","status":"fully_proved","summary":"A longest-common-subsequence instance is a pair $(R, K)$ of a finite list $R$ of binary strings (finite lists of booleans) and a natural number $K$.","title":"Longest-common-subsequence instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/longest-common-subsequence-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCommonSubsequence.LongestCommonSubsequenceFeasible","parent":32858,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/index.html#longest-common-subsequence-feasible","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings","status":"fully_proved","summary":"A longest-common-subsequence instance $(R, K)$ is feasible when there is a binary string $w$ of length at least $K$ that is a subsequence of every string in $R$, that is, $w$ is obtained from each string in $R$ by deleting some of its entries.","title":"Feasible longest-common-subsequence instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/longest-common-subsequence-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCommonSubsequence.longestCommonSubsequenceDec","parent":32858,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/index.html#longest-common-subsequence-dec","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings","status":"fully_proved","summary":"LONGEST COMMON SUBSEQUENCE is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a feasible longest-common-subsequence instance. Bitstrings that do not decode are rejected.","title":"The LONGEST COMMON SUBSEQUENCE decision problem"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/longest-common-subsequence-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.LongestCommonSubsequence.longestCommonSubsequence_npComplete","parent":32858,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings/index.html#longest-common-subsequence-np-complete","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/longest-common-subsequence-of-many-strings","status":"can_prove","summary":"The LONGEST COMMON SUBSEQUENCE decision problem, over binary strings, belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"LONGEST COMMON SUBSEQUENCE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency","kind":"scope","navigation":"module","parent":32838,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency","status":"can_prove","title":"Minesweeper consistency"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/minesweeper-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Minesweeper.MinesweeperInstance","parent":32863,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/index.html#minesweeper-instance","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency","status":"fully_proved","summary":"A Minesweeper instance consists of a number of rows $R$, a number of columns $C$ and a finite list of clues, where $R$ and $C$ are natural numbers and each clue is a triple $(r, c, k)$ of natural numbers giving a row $r$, a column $c$ and the number $k$ shown.","title":"Minesweeper instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Minesweeper.WellFormed","parent":32863,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency","status":"fully_proved","summary":"A Minesweeper instance with $R$ rows, $C$ columns and a list of clues is well formed when every clue $(r, c, k)$ in the list satisfies $r < R$, $c < C$ and $k \\le 8$.","title":"Well-formed Minesweeper instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/minesweeper-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Minesweeper.MinesweeperFeasible","parent":32863,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/index.html#minesweeper-feasible","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency","status":"fully_proved","summary":"Index rows and columns by natural numbers, and call a cell $(r', c') \\ne (r, c)$ a neighbour of $(r, c)$ when $|r' - r| \\le 1$ and $|c' - c| \\le 1$. A Minesweeper instance with $R$ rows, $C$ columns and a list of clues is feasible when there is a finite set $M$ of cells $(r, c)$ with $r < R$ and $c < C$ (the mines) such that for every clue $(r, c, k)$ in the list the cell $(r, c)$ is not in $M$ and exactly $k$\u2026","title":"Feasible Minesweeper instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/minesweeper-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Minesweeper.minesweeperDec","parent":32863,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/index.html#minesweeper-dec","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency","status":"fully_proved","summary":"The Minesweeper consistency decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a Minesweeper instance that is well formed and feasible, that is, admits a mine placement consistent with all of its clues. It rejects every bitstring that does not decode to a Minesweeper instance.","title":"Minesweeper consistency as a decision problem"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/minesweeper-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Minesweeper.minesweeper_npComplete","parent":32863,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency/index.html#minesweeper-np-complete","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/minesweeper-consistency","status":"can_prove","summary":"The Minesweeper consistency decision problem lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to it.","title":"MINESWEEPER CONSISTENCY is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram","kind":"scope","navigation":"module","parent":32838,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram","status":"can_prove","title":"Nonogram"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/nonogram-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Nonogram.NonogramInstance","parent":32869,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/index.html#nonogram-instance","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram","status":"fully_proved","summary":"A nonogram instance is a pair $(R, C)$ of finite lists, the row clues $R$ and the column clues $C$, where each clue is a finite list of natural numbers.","title":"Nonogram instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/runs-step","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Nonogram.RunsStep","parent":32869,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/index.html#runs-step","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram","status":"fully_proved","summary":"For a Boolean $b$ and a pair $(k, R)$ of a natural number $k$ and a finite list $R$ of natural numbers, the step returns $(k + 1, R)$ if $b$ is true. If $b$ is false it returns $(0, R)$ when $k = 0$, and $(0, R')$ when $k > 0$, where $R'$ is the list $R$ with $k$ prepended.","title":"One step of the run-length computation"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/runs","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Nonogram.Runs","parent":32869,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/index.html#runs","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram","status":"fully_proved","summary":"For a finite list $l$ of Booleans, the runs of $l$ are its maximal blocks of consecutive true entries. The run list of $l$ is the list of the lengths of these blocks, in order from left to right; it is empty when $l$ has no true entry.","title":"Run lengths of a row"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/nonogram-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Nonogram.NonogramFeasible","parent":32869,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/index.html#nonogram-feasible","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram","status":"fully_proved","summary":"A nonogram instance with row clues $R_0, \\dots, R_{m-1}$ and column clues $C_0, \\dots, C_{n-1}$ is feasible when there is a picture, a list of $m$ rows of Booleans each of length $n$, such that for every $r < m$ the run list of row $r$ equals $R_r$, and for every $c < n$ the run list of column $c$, read from row $0$ to row $m - 1$, equals $C_c$.","title":"Feasible nonogram instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/nonogram-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Nonogram.nonogramDec","parent":32869,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/index.html#nonogram-dec","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram","status":"fully_proved","summary":"NONOGRAM is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of nonogram instances, to a feasible nonogram instance. Bitstrings that do not decode to an instance are rejected.","title":"The NONOGRAM decision problem"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/nonogram-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Nonogram.nonogram_npComplete","parent":32869,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram/index.html#nonogram-np-complete","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/nonogram","status":"can_prove","summary":"The NONOGRAM decision problem belongs to NP, and every decision problem in NP reduces to NONOGRAM by a polynomial-time many-one reduction.","title":"NONOGRAM is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence","kind":"scope","navigation":"module","parent":32838,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence","status":"can_prove","title":"Shortest common supersequence"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/shortest-common-supersequence-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ShortestCommonSupersequence.ShortestCommonSupersequenceInstance","parent":32876,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/index.html#shortest-common-supersequence-instance","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence","status":"fully_proved","summary":"An instance of SHORTEST COMMON SUPERSEQUENCE is a pair $(R, K)$ of a list $R$ of binary strings (lists of booleans) and a natural number $K$, the length bound.","title":"Shortest common supersequence instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/shortest-common-supersequence-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ShortestCommonSupersequence.ShortestCommonSupersequenceFeasible","parent":32876,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/index.html#shortest-common-supersequence-feasible","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence","status":"fully_proved","summary":"An instance $(R, K)$ of SHORTEST COMMON SUPERSEQUENCE, a list $R$ of lists of booleans together with a natural number $K$, is feasible when there is a list $w$ of booleans of length at most $K$ such that every list in $R$ is a subsequence of $w$ (its entries occur in $w$ in the same order, not necessarily consecutively).","title":"Feasible shortest common supersequence instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/shortest-common-supersequence-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ShortestCommonSupersequence.shortestCommonSupersequenceDec","parent":32876,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/index.html#shortest-common-supersequence-dec","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence","status":"fully_proved","summary":"SHORTEST COMMON SUPERSEQUENCE is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a pair $(R, K)$ of a list $R$ of lists of booleans and a natural number $K$ for which some list of booleans of length at most $K$ has every list in $R$ as a subsequence. Bitstrings that do not decode to such a pair are rejected.","title":"The SHORTEST COMMON SUPERSEQUENCE decision problem"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/shortest-common-supersequence-np-complet","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.ShortestCommonSupersequence.shortestCommonSupersequence_npComplete","parent":32876,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence/index.html#shortest-common-supersequence-np-complet","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/shortest-common-supersequence","status":"can_prove","summary":"The SHORTEST COMMON SUPERSEQUENCE decision problem over binary strings belongs to NP, and every decision problem in NP reduces to it by a polynomial-time many-one reduction.","title":"SHORTEST COMMON SUPERSEQUENCE is NP-complete"},{"data_url":"scopes/TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku.json","id":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku","kind":"scope","navigation":"module","parent":32838,"result":false,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/index.html","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku","status":"can_prove","title":"Sudoku"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/sudoku-instance","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Sudoku.SudokuInstance","parent":32881,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/index.html#sudoku-instance","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku","status":"fully_proved","summary":"A Sudoku instance is a pair $(n, G)$ of a natural number $n$, the block size, and a grid $G$ given as a finite list of rows, each row a finite list of natural numbers; an entry $0$ stands for a blank cell.","title":"Sudoku instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Sudoku.WellFormed","parent":32881,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/index.html#well-formed","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku","status":"fully_proved","summary":"A Sudoku instance $(n, G)$, with block size $n$ and grid $G$ a list of rows of natural numbers, is well formed when $n > 0$, $G$ has exactly $n^2$ rows, every row of $G$ has exactly $n^2$ entries, and every entry of $G$ is at most $n^2$.","title":"Well-formed Sudoku instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/sudoku-feasible","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Sudoku.SudokuFeasible","parent":32881,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/index.html#sudoku-feasible","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku","status":"fully_proved","summary":"A Sudoku instance $(n, G)$, with block size $n$ and grid $G$ a list of rows of natural numbers, is feasible when there is a grid $C$ of natural numbers with exactly $n^2$ rows, each of exactly $n^2$ entries, such that, indexing rows and columns from $0$: for all $r, c < n^2$ the entry $G_{rc}$ is $0$ or equals $C_{rc}$, where an entry missing from $G$ counts as $0$; every row of $C$ and every column of $C$ lists\u2026","title":"Feasible Sudoku instances"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/sudoku-dec","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Sudoku.sudokuDec","parent":32881,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/index.html#sudoku-dec","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku","status":"fully_proved","summary":"The decision problem SUDOKU accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs of a natural number and a finite list of finite lists of natural numbers, to a Sudoku instance that is well formed and feasible. It rejects every bitstring that does not decode to an instance.","title":"SUDOKU as a decision problem"},{"id":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/sudoku-np-complete","kind":"node","lean":"MetaMathlibExt.Complexity.NPComplete.Sudoku.sudoku_npComplete","parent":32881,"result":true,"route":"../roadmap/TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku/index.html#sudoku-np-complete","scope":"TheoryOfComputation/np-completeness/strings-and-puzzles/sudoku","status":"can_prove","summary":"The decision problem SUDOKU lies in NP, and every decision problem in NP has a polynomial-time many-one reduction to SUDOKU.","title":"SUDOKU is NP-complete"},{"data_url":"scopes/TheoryOfComputation/complexity-classes.json","id":"TheoryOfComputation/complexity-classes","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/index.html","scope":"TheoryOfComputation/complexity-classes","status":"can_prove","summary":"Definitions of time and space complexity on machine models (P, NP, PSPACE, EXP, L, NL, the polynomial hierarchy, FPT and W[1]), polynomial-time encodings and reductions, simulation overheads, Blum measures, and inclusions and collapses among deterministic and nondeterministic classes (Savitch, Immerman\u2013Szelepcs\u00e9nyi, Karp\u2013Lipton, Ladner, Mahaney). Randomized, counting, circuit and quantum classes have their own\u2026","title":"Complexity classes"},{"id":"TheoryOfComputation/complexity-classes/immerman-szelepcsenyi","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Immerman.immerman_szelepcsenyi","parent":32887,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/index.html#immerman-szelepcsenyi","scope":"TheoryOfComputation/complexity-classes","status":"can_prove","summary":"The complexity class NL is closed under complementation: a decision problem $L$ belongs to NL if and only if its complement, the decision problem that is true exactly on the bitstrings where $L$ is false, belongs to NL.","title":"Immerman-Szelepcsenyi theorem: NL = coNL"},{"id":"TheoryOfComputation/complexity-classes/karp-lipton","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.KarpLipton.karp_lipton","parent":32887,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/index.html#karp-lipton","scope":"TheoryOfComputation/complexity-classes","status":"can_prove","summary":"If every decision problem in NP belongs to P/poly (polynomial time with polynomial advice), then the polynomial hierarchy PH equals its second level $\\Sigma_2^p$.","title":"Karp-Lipton theorem: NP in P/poly collapses PH"},{"id":"TheoryOfComputation/complexity-classes/ladner","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Ladner.ladner","parent":32887,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/index.html#ladner","scope":"TheoryOfComputation/complexity-classes","status":"can_prove","summary":"If P $\\ne$ NP, then there is a decision problem $L$ in NP that is not in P and to which not every decision problem in NP reduces by a polynomial-time many-one reduction; that is, $L$ is in NP but is neither in P nor NP-complete.","title":"Ladner's theorem: NP-intermediate languages exist"},{"id":"TheoryOfComputation/complexity-classes/savitch","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Savitch.savitch","parent":32887,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/index.html#savitch","scope":"TheoryOfComputation/complexity-classes","status":"can_prove","summary":"The complexity class NPSPACE of decision problems decided in polynomial space by nondeterministic multi-tape Turing machines equals the class PSPACE of decision problems decided in polynomial space by deterministic multi-tape Turing machines.","title":"Savitch's theorem: NPSPACE = PSPACE"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/blum-complexity-measures.json","id":"TheoryOfComputation/complexity-classes/blum-complexity-measures","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/blum-complexity-measures/index.html","scope":"TheoryOfComputation/complexity-classes/blum-complexity-measures","status":"fully_proved","title":"Blum complexity measures"},{"id":"TheoryOfComputation/complexity-classes/blum-complexity-measures/blum-measure","kind":"node","lean":"Cslib.Decidability.BlumMeasure","parent":32892,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/blum-complexity-measures/index.html#blum-measure","scope":"TheoryOfComputation/complexity-classes/blum-complexity-measures","status":"fully_proved","summary":"A Blum measure $\\Phi$ assigns to every program $c$ and input $n \\in \\mathbb{N}$ a partially defined cost $\\Phi_c(n) \\in \\mathbb{N}$ satisfying Blum's two axioms: $\\Phi_c(n)$ is defined exactly when program $c$ halts on input $n$, and the relation $\\Phi_c(n) = m$ between $c$, $n$ and $m$ is decidable.","title":"Blum complexity measure"},{"id":"TheoryOfComputation/complexity-classes/blum-complexity-measures/blum-class","kind":"node","lean":"Cslib.Decidability.BlumClass","parent":32892,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/blum-complexity-measures/index.html#blum-class","scope":"TheoryOfComputation/complexity-classes/blum-complexity-measures","status":"fully_proved","summary":"Given a Blum measure $\\Phi$ and a function $t$ from $\\mathbb{N}$ to $\\mathbb{N}$, the Blum class of $t$ consists of the partial functions $f$ from $\\mathbb{N}$ to $\\mathbb{N}$ that are computed by some program $c$ whose cost $\\Phi_c(n)$ is defined and at most $t(n)$ for all but finitely many inputs $n$.","title":"Blum complexity class"},{"id":"TheoryOfComputation/complexity-classes/blum-complexity-measures/mono","kind":"node","lean":"Cslib.Decidability.BlumClass.mono","parent":32892,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/blum-complexity-measures/index.html#mono","scope":"TheoryOfComputation/complexity-classes/blum-complexity-measures","status":"fully_proved","summary":"If `t` is eventually dominated by `t'`, every function in the `t`-class is in the `t'`-class.","title":"Blum classes are monotone in the bound"},{"id":"TheoryOfComputation/complexity-classes/blum-complexity-measures/blum-class-iff","kind":"node","lean":"Cslib.Decidability.blumClass_iff","parent":32892,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/blum-complexity-measures/index.html#blum-class-iff","scope":"TheoryOfComputation/complexity-classes/blum-complexity-measures","status":"fully_proved","summary":"Let $\\Phi$ be a Blum measure, $t$ a function from $\\mathbb{N}$ to $\\mathbb{N}$ and $f$ a partial function from $\\mathbb{N}$ to $\\mathbb{N}$. Then $f$ lies in the Blum class of $t$ with respect to $\\Phi$ if and only if there are a program $c$ that computes $f$ and a threshold $N$ such that for every input $n \\ge N$ the cost $\\Phi_c(n)$ is defined and at most $t(n)$.","title":"Blum classes via an explicit threshold"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/polynomially-bounded-functions.json","id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","title":"Polynomially bounded functions"},{"id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions/polynomially-bounded-iff-le-mul-pow-self","kind":"node","lean":"Cslib.polynomiallyBounded_iff_le_mul_pow_self","parent":32897,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html#polynomially-bounded-iff-le-mul-pow-self","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","summary":"Polynomial growth is equivalent to a global bound whose coefficient equals its exponent.","title":"Polynomial bounds with equal coefficient and exponent"},{"id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions/const","kind":"node","lean":"Cslib.PolynomiallyBounded.const","parent":32897,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html#const","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","summary":"Every constant natural-valued function is polynomially bounded.","title":"Constant functions are polynomially bounded"},{"id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions/id","kind":"node","lean":"Cslib.PolynomiallyBounded.id","parent":32897,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html#id","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","summary":"The identity function on natural numbers is polynomially bounded.","title":"The identity function is polynomially bounded"},{"id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions/add","kind":"node","lean":"Cslib.PolynomiallyBounded.add","parent":32897,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html#add","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","summary":"A pointwise sum of polynomially bounded functions is polynomially bounded.","title":"Sums of polynomially bounded functions are polynomially bounded"},{"id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions/mul","kind":"node","lean":"Cslib.PolynomiallyBounded.mul","parent":32897,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html#mul","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","summary":"A pointwise product of polynomially bounded functions is polynomially bounded.","title":"Products of polynomially bounded functions are polynomially bounded"},{"id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions/pow","kind":"node","lean":"Cslib.PolynomiallyBounded.pow","parent":32897,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html#pow","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","summary":"A fixed pointwise power of a polynomially bounded function is polynomially bounded.","title":"Powers of polynomially bounded functions are polynomially bounded"},{"id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions/comp","kind":"node","lean":"Cslib.PolynomiallyBounded.comp","parent":32897,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html#comp","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","summary":"A composition of polynomially bounded functions is polynomially bounded.","title":"Compositions of polynomially bounded functions are polynomially bounded"},{"id":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions/mono-bound","kind":"node","lean":"Cslib.PolynomiallyBounded.mono_bound","parent":32897,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomially-bounded-functions/index.html#mono-bound","scope":"TheoryOfComputation/complexity-classes/polynomially-bounded-functions","status":"fully_proved","summary":"The normalized bound is itself polynomially bounded and monotone in its input.","title":"The normalized polynomial bound is polynomially bounded and monotone"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation.json","id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","title":"Quadratic overhead of single-tape simulation"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/relates-in-steps-go","kind":"node","lean":"Cslib.Turing.TM2Time.relatesInSteps_go","parent":32906,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#relates-in-steps-go","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"Let $M$ be a multi-stack machine, $k$ a stack, $L$ tape contents whose $k$-th track lists a stack $S$ from bottom to top, and $n \\le |S|$. In the $\\mathrm{TM1}$ emulator of $M$, the configuration in the state that walks to stack $k$ (for a given stack action and continuation), with internal state $v$ and head at the bottom marker of $L$, reaches in exactly $n$ steps the same configuration with the head moved $n$\u2026","title":"The emulator walks right to the top of a stack"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/relates-in-steps-ret","kind":"node","lean":"Cslib.Turing.TM2Time.relatesInSteps_ret","parent":32906,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#relates-in-steps-ret","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"Let $M$ be a multi-stack machine and $L$ tape contents. For every $n$, in the $\\mathrm{TM1}$ emulator of $M$, the configuration in the return state (for a given continuation), with internal state $v$ and head $n$ cells to the right of the bottom marker of $L$, reaches in exactly $n$ steps the same configuration with the head at the bottom marker.","title":"The emulator walks back to the bottom"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/exists-relates-in-steps-step-aux","kind":"node","lean":"Cslib.Turing.TM2Time.exists_relatesInSteps_stepAux","parent":32906,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#exists-relates-in-steps-step-aux","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"Let $M$ be a multi-stack machine and $N$ a statement over the same stack alphabets, labels and internal states, and write $|N|$ for the structural size of $N$. Let $H \\in \\mathbb{N}$, let $v$ be an internal state, let $S$ be a family of stacks $S_k$, each of height at most $H$, and let $L$ be tape contents whose $k$-th track lists $S_k$ from bottom to top, for every $k$. Running the translation of $N$ in the\u2026","title":"Running one statement on the emulator"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/exists-relates-within-steps-tm1-of-step","kind":"node","lean":"Cslib.Turing.TM2Time.exists_relatesWithinSteps_tm1_of_step","parent":32906,"reading_title":{"format":"inline-parts-v1","parts":[["text","One multi-stack step costs at most "],["math","\\(1 + B(2(H + B) + 2)\\)"],["text"," emulator steps"]],"source":"One multi-stack step costs at most $1 + B(2(H + B) + 2)$ emulator steps"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#exists-relates-within-steps-tm1-of-step","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"Let $M$ be a multi-stack machine in which the statement of every label has size at most $B$, and consider its emulation by a single-tape machine with an internal store that keeps the stacks side by side on its tape. Let $c$ be a configuration of $M$ in which every stack has height at most $H$, and let $c_1$ be a configuration of the emulator that represents $c$. If $M$ moves from $c$ to $c'$ in one step, then there\u2026","title":"One multi-stack step costs at most $1 + B(2(H + B) + 2)$ emulator steps"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/tm0-step-eq-none-of-step-eq-none","kind":"node","lean":"Cslib.Turing.TM2Time.tm0_step_eq_none_of_step_eq_none","parent":32906,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#tm0-step-eq-none-of-step-eq-none","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"A halted configuration is related only to configurations of the single-tape emulation that do not step.","title":"The emulation halts where the multi-stack machine halts"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/relates-within-steps-tm0-of-relates-in-s","kind":"node","lean":"Cslib.Turing.TM2Time.relatesWithinSteps_tm0_of_relatesInSteps_tm1","parent":32906,"reading_title":{"format":"inline-parts-v1","parts":[["text","Each step of the "],["math","\\(\\mathrm{TM1}\\)"],["text"," emulator is at most three single-tape steps"]],"source":"Each step of the $\\mathrm{TM1}$ emulator is at most three single-tape steps"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#relates-within-steps-tm0-of-relates-in-s","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"Let $M$ be a multi-stack machine with a default label and a default internal state. If the $\\mathrm{TM1}$ emulator of $M$ goes from $c$ to $c'$ in exactly $m$ steps, then the single-tape emulation of $M$ goes from the single-tape form of $c$ to the single-tape form of $c'$ in at most $3m$ steps.","title":"Each step of the $\\mathrm{TM1}$ emulator is at most three single-tape steps"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/emulation-constant","kind":"node","lean":"Cslib.Turing.TM2Time.emulationConstant","parent":32906,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#emulation-constant","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"For a bundled multi-stack machine $M$, the emulation constant is $C = 3(2B + 1) + P$, where $B = \\sum_l |M(l)|$ is the total structural size of the statements $M(l)$ over all labels $l$ (so each statement has size at most $B$), and $P$ is the push bound of $M$, the total number of push instructions in its statements, which bounds the number of pushes in one step.","title":"Emulation constant"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/exists-tm0-run-of-tm2-run","kind":"node","lean":"Cslib.Turing.TM2Time.exists_tm0_run_of_tm2_run","parent":32906,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#exists-tm0-run-of-tm2-run","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"Let $M$ be a bundled multi-stack machine with emulation constant $C$, and $L$ an input for its input stack. If $M$, started on $L$, reaches a configuration $c$ after $t$ steps, then the single-tape emulation of $M$, started on the encoding of $L$, reaches in some number $n \\le C t (|L| + C t + 1)$ of steps the single-tape form of a configuration of the $\\mathrm{TM1}$ emulator of $M$ that represents $c$: it has the\u2026","title":"Quadratic overhead of the single-tape emulation"},{"id":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/exists-tm0-halt-of-tm2-halt","kind":"node","lean":"Cslib.Turing.TM2Time.exists_tm0_halt_of_tm2_halt","parent":32906,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation/index.html#exists-tm0-halt-of-tm2-halt","scope":"TheoryOfComputation/complexity-classes/quadratic-overhead-of-single-tape-simulation","status":"fully_proved","summary":"Let $M$ be a bundled multi-stack machine with emulation constant $C$, and $L$ an input for its input stack. If $M$, started on $L$, reaches after $t$ steps a configuration $c$ from which it does not step, then the single-tape emulation of $M$, started on the encoding of $L$, reaches in some number $n \\le C t (|L| + C t + 1)$ of steps a configuration from which it does not step. That configuration is the single-tape\u2026","title":"Single-tape emulation of halting multi-stack runs"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp.json","id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","title":"The classes P, NP and coNP"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/bitstring-encoding","kind":"node","lean":"MetaMathlibExt.BitstringEncoding","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#bitstring-encoding","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"A canonical bitstring encoding of a type $\\alpha$ consists of an encoding map from $\\alpha$ to finite lists of booleans and a partial decoding map from finite lists of booleans to $\\alpha$, such that decoding the encoding of any $a \\in \\alpha$ succeeds and returns $a$.","title":"Bitstring encodings"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/bit-encode","kind":"node","lean":"MetaMathlibExt.BitstringEncoding.bitEncode","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#bit-encode","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"For a type $\\alpha$ with a canonical bitstring encoding, the bitstring encoding of $a \\in \\alpha$ is the finite list of booleans that this encoding assigns to $a$.","title":"Encoding function of a bitstring encoding"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/bit-decode","kind":"node","lean":"MetaMathlibExt.BitstringEncoding.bitDecode","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#bit-decode","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"For a type $\\alpha$ with a canonical bitstring encoding, bitstring decoding is the decoding function of that encoding: it sends each finite list of booleans either to an element of $\\alpha$ or to failure.","title":"Decoding function of a bitstring encoding"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/bit-decode-bit-encode","kind":"node","lean":"MetaMathlibExt.BitstringEncoding.bitDecode_bitEncode","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#bit-decode-bit-encode","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"Let $\\alpha$ be a type with a canonical bitstring encoding. For every $a \\in \\alpha$, decoding the bitstring encoding of $a$ succeeds and returns $a$.","title":"Decoding inverts bitstring encoding"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/delimit","kind":"node","lean":"MetaMathlibExt.BitstringEncoding.delimit","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#delimit","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"The self-delimiting form of a bitstring $(b_1, \\dots, b_n)$ is $(\\mathrm{true}, b_1, \\dots, \\mathrm{true}, b_n, \\mathrm{false})$: each bit is preceded by $\\mathrm{true}$ and the block is terminated by $\\mathrm{false}$. In particular the empty bitstring becomes $(\\mathrm{false})$.","title":"Self-delimiting form of a bitstring"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/undelimit","kind":"node","lean":"MetaMathlibExt.BitstringEncoding.undelimit","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#undelimit","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"Write booleans as bits, with true as $1$ and false as $0$. Reading a list of bits from the front, a $0$ ends the block, and a $1$ followed by a bit $b$ contributes $b$ to the payload. The result is the payload together with the rest of the list after the terminating $0$. Parsing fails if the list runs out before a terminating $0$ is read.","title":"Parsing one self-delimiting block"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/undelimit-blocks-aux","kind":"node","lean":"MetaMathlibExt.BitstringEncoding.undelimitBlocksAux","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#undelimit-blocks-aux","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"Given a bound $n$ and a list of booleans $s$, the parse returns the empty list of payloads if $s$ is empty. If $s$ is nonempty and $n = 0$ it fails. If $s$ is nonempty and $n = m + 1$, it parses one self-delimiting block from the front of $s$, then parses the remaining list with bound $m$, and returns the first payload followed by the payloads of the rest. It fails if either step fails. So the parse succeeds\u2026","title":"Bounded parsing of self-delimiting blocks"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/undelimit-blocks","kind":"node","lean":"MetaMathlibExt.BitstringEncoding.undelimitBlocks","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#undelimit-blocks","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"Given a list of booleans $s$, this repeatedly parses one self-delimiting block from the front of $s$ until $s$ is used up, allowing at most as many blocks as the length of $s$, and returns the list of payloads in order. It fails if some block fails to parse. The bound never binds, since every block uses at least one bit. So the parse succeeds exactly when $s$ is a concatenation of self-delimiting blocks.","title":"Parsing a list of booleans into self-delimiting blocks"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/decision-problem","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.DecisionProblem","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#decision-problem","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"A decision problem is a function from finite lists of booleans (bitstrings) to booleans.","title":"Decision problems"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/decision-complexity-class","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.DecisionComplexityClass","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#decision-complexity-class","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"A complexity class is a set of decision problems.","title":"Complexity classes"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/is-poly-time-with-encoding","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.IsPolyTimeWithEncoding","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#is-poly-time-with-encoding","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"Let $e_\\alpha$ encode $\\alpha$ as words over an alphabet $\\Gamma_\\alpha$ and $e_\\beta$ encode $\\beta$ as words over $\\Gamma_\\beta$. A function $f : \\alpha \\to \\beta$ is polynomial-time computable with respect to $e_\\alpha$ and $e_\\beta$ when there are a multi-stack Turing machine with finitely many stacks, function labels and internal states and a finite input stack alphabet, identifications of its input and output\u2026","title":"Polynomial-time computability under given encodings"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/is-poly-time","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.IsPolyTime","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#is-poly-time","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types with canonical bitstring encodings. A function $f : \\alpha \\to \\beta$ is polynomial-time computable when it is polynomial-time computable with respect to the canonical bitstring encodings of $\\alpha$ and $\\beta$.","title":"Polynomial-time computable functions"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/p","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.P","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#p","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"P is the set of decision problems (functions from lists of booleans to booleans) that are computable in polynomial time by a deterministic multi-stack Turing machine, with respect to the canonical bitstring encodings of inputs and outputs.","title":"The complexity class P"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/np","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.NP","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#np","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"NP is the set of decision problems $L$ (functions from lists of booleans to booleans) for which there exist a polynomial $p$ with natural-number coefficients and a polynomial-time computable function $R$ from pairs of lists of booleans to booleans such that, for every list of booleans $x$, $L(x)$ is true if and only if there is a list of booleans $w$ whose length is at most $p$ evaluated at the length of $x$ and\u2026","title":"The complexity class NP"},{"id":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/co-np","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.coNP","parent":32916,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp/index.html#co-np","scope":"TheoryOfComputation/complexity-classes/the-classes-p-np-and-conp","status":"fully_proved","summary":"The class coNP. coNP is the set of decision problems $L$ whose complement, the decision problem that accepts exactly the bitstrings that $L$ rejects, belongs to NP.","title":"The class coNP"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/mahaney-s-theorem.json","id":"TheoryOfComputation/complexity-classes/mahaney-s-theorem","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/mahaney-s-theorem/index.html","scope":"TheoryOfComputation/complexity-classes/mahaney-s-theorem","status":"can_prove","title":"Mahaney's theorem"},{"id":"TheoryOfComputation/complexity-classes/mahaney-s-theorem/sparse","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Mahaney.Sparse","parent":32932,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/mahaney-s-theorem/index.html#sparse","scope":"TheoryOfComputation/complexity-classes/mahaney-s-theorem","status":"fully_proved","summary":"A decision problem $L$ is sparse when there is a polynomial $p$ with natural-number coefficients such that, for every natural number $n$, the set of bitstrings of length at most $n$ accepted by $L$ is finite and has at most $p(n)$ elements.","title":"Sparse decision problems"},{"id":"TheoryOfComputation/complexity-classes/mahaney-s-theorem/npcomplete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Mahaney.NPComplete","parent":32932,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/mahaney-s-theorem/index.html#npcomplete","scope":"TheoryOfComputation/complexity-classes/mahaney-s-theorem","status":"fully_proved","summary":"A decision problem $L$ is NP-complete when $L$ is in NP and every decision problem in NP has a polynomial-time many-one reduction to $L$.","title":"NP-complete decision problems"},{"id":"TheoryOfComputation/complexity-classes/mahaney-s-theorem/mahaney","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Mahaney.mahaney","parent":32932,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/mahaney-s-theorem/index.html#mahaney","scope":"TheoryOfComputation/complexity-classes/mahaney-s-theorem","status":"can_prove","summary":"If some decision problem is both sparse and NP-complete, then the complexity classes $\\mathsf{P}$ and $\\mathsf{NP}$ are equal.","title":"Mahaney's theorem: sparse NP-complete sets imply P = NP"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming.json","id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","title":"Polynomial time in a cost-annotated programming model"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/decides-in-declared-poly-time","kind":"node","lean":"Cslib.TimeMComplexity.DecidesInDeclaredPolyTime","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#decides-in-declared-poly-time","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $f$ be a timed function that maps each list $x$ of Booleans to a Boolean $f(x)$ together with a declared time $t_f(x) \\in \\mathbb{N}$. Then $f$ decides in declared polynomial time when there is a polynomial $p$ with natural-number coefficients such that $t_f(x) \\le p(|x|)$ for every list $x$ of Booleans, where $|x|$ is the length of $x$.","title":"Declared polynomial time for timed deciders"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/computes-in-declared-poly-time","kind":"node","lean":"Cslib.TimeMComplexity.ComputesInDeclaredPolyTime","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#computes-in-declared-poly-time","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $f$ be a timed function that maps each list $x$ of Booleans to an output list $f(x)$ of Booleans together with a declared time $t_f(x) \\in \\mathbb{N}$. Then $f$ computes in declared polynomial time when there is a polynomial $p$ with natural-number coefficients such that, for every list $x$ of Booleans, both $t_f(x)$ and the length of $f(x)$ are at most $p(|x|)$, where $|x|$ is the length of $x$.","title":"Declared polynomial time for timed functions on bitstrings"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/verifier-in-declared-poly-time","kind":"node","lean":"Cslib.TimeMComplexity.VerifierInDeclaredPolyTime","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#verifier-in-declared-poly-time","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"A timed verifier $V$, which assigns to each pair of bit strings $(x, w)$ (an input and a witness) a timed computation returning a Boolean, runs in declared polynomial time when there is a polynomial $q$ with natural coefficients such that the declared time of $V(x, w)$ is at most $q(|x| + |w|)$ for all $x$ and $w$.","title":"Verifiers with polynomial declared time"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/verifies","kind":"node","lean":"Cslib.TimeMComplexity.Verifies","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#verifies","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"A timed verifier $V$, which assigns to each pair of bit strings $(x, w)$ (an input and a witness) a timed computation returning a Boolean, verifies a decision problem $L$ with witness bound $p$, a polynomial with natural coefficients, when for every bit string $x$, $L(x)$ holds if and only if there is a witness $w$ with $|w| \\le p(|x|)$ such that $V(x, w)$ returns true.","title":"Verifying a decision problem with a witness bound"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/declared-poly-many-one-red-via","kind":"node","lean":"Cslib.TimeMComplexity.DeclaredPolyManyOneRedVia","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#declared-poly-many-one-red-via","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be decision problems and let $f$ be a timed function that maps each list $x$ of Booleans to a list $f(x)$ of Booleans together with a declared time. Then $f$ is a declared-time many-one reduction from $L_1$ to $L_2$ when $f$ computes in declared polynomial time and $L_1(x) = L_2(f(x))$ for every list $x$ of Booleans.","title":"Declared-time many-one reduction by a timed function"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/exists-decides-in-declared-poly-time","kind":"node","lean":"Cslib.TimeMComplexity.exists_decidesInDeclaredPolyTime","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#exists-decides-in-declared-poly-time","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"For every decision problem $L$ there is a timed function $f$ from lists of Booleans to Booleans that decides in declared polynomial time and whose output $f(x)$ equals $L(x)$ for every list $x$ of Booleans.","title":"Every decision problem has a timed decider in declared polynomial time"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/verifier-in-declared-poly-time-of-decide","kind":"node","lean":"Cslib.TimeMComplexity.verifierInDeclaredPolyTime_of_decidesInDeclaredPolyTime","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#verifier-in-declared-poly-time-of-decide","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"If a timed decider $f$ on bit strings has declared time bounded by a polynomial in the input length, then the verifier that sends an input $x$ and a witness $w$ to $f(x)$, ignoring $w$, has declared time bounded by a polynomial in $|x| + |w|$.","title":"A decider with polynomial declared time gives a verifier with polynomial declared time"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/verifies-of-decides","kind":"node","lean":"Cslib.TimeMComplexity.verifies_of_decides","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#verifies-of-decides","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L$ be a decision problem and let $f$ be a timed decider whose return value on every bit string $x$ equals $L(x)$. Then the verifier that sends an input $x$ and a witness $w$ to $f(x)$, ignoring $w$, verifies $L$ with the zero polynomial as witness bound, so only the empty witness is allowed.","title":"A decider is a verifier with witness bound zero"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/declared-poly-many-one-red-via-refl","kind":"node","lean":"Cslib.TimeMComplexity.declaredPolyManyOneRedVia_refl","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#declared-poly-many-one-red-via-refl","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"For every decision problem $L$, the timed identity function, which returns its input with declared time $0$, is a declared-time many-one reduction from $L$ to $L$.","title":"Declared-time many-one reduction is reflexive"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/declared-poly-many-one-red-via-trans","kind":"node","lean":"Cslib.TimeMComplexity.declaredPolyManyOneRedVia_trans","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#declared-poly-many-one-red-via-trans","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L_1$, $L_2$ and $L_3$ be decision problems. If $f_1$ is a declared-time many-one reduction from $L_1$ to $L_2$ and $f_2$ is a declared-time many-one reduction from $L_2$ to $L_3$, then the timed function that on input $x$ returns $f_2(f_1(x))$, with declared time the declared time of $f_1$ on $x$ plus that of $f_2$ on $f_1(x)$, is a declared-time many-one reduction from $L_1$ to $L_3$.","title":"Declared-time many-one reductions compose"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/decides-in-declared-poly-time-comp-reduc","kind":"node","lean":"Cslib.TimeMComplexity.decidesInDeclaredPolyTime_comp_reduction","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#decides-in-declared-poly-time-comp-reduc","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be decision problems, let $f$ be a declared-time many-one reduction from $L_1$ to $L_2$, and let $g$ be a timed function from lists of Booleans to Booleans that decides in declared polynomial time. Then the timed function that on input $x$ returns $g(f(x))$, with declared time the declared time of $f$ on $x$ plus that of $g$ on $f(x)$, decides in declared polynomial time.","title":"Declared polynomial time is preserved under precomposition with a reduction"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/decides-comp-reduction","kind":"node","lean":"Cslib.TimeMComplexity.decides_comp_reduction","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#decides-comp-reduction","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be decision problems, let $f$ be a declared-time many-one reduction from $L_1$ to $L_2$, and let $g$ be a timed function from lists of Booleans to Booleans whose output on every list $y$ is $L_2(y)$. Then for every list $x$ of Booleans, the timed function that runs $f$ on $x$ and then $g$ on its output returns $L_1(x)$.","title":"A declared-time reduction followed by a decider for the target decides the source"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/verifies-comp-reduction","kind":"node","lean":"Cslib.TimeMComplexity.verifies_comp_reduction","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#verifies-comp-reduction","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be decision problems, let $f$ be a declared polynomial-time many-one reduction from $L_1$ to $L_2$, and let $V$ be a verifier with polynomial declared time that verifies $L_2$ with witness bound $p$. Let $V'$ be the verifier that on input $x$ and witness $w$ first runs $f$ on $x$, obtaining $y$, then runs $V$ on $(y, w)$ and accepts when $V$ accepts and $|w| \\le p(|y|)$; its declared time is the\u2026","title":"Pulling a verifier back along a declared reduction"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/mem-p-of-decides","kind":"node","lean":"Cslib.TimeMComplexity.mem_P_of_decides","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#mem-p-of-decides","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L$ be a decision problem and let $f$ assign to each bit string $x$ a timed computation, that is, a Boolean return value together with a declared time in $\\mathbb{N}$, whose return value equals $L(x)$ for every $x$. If the function sending $x$ to the return value of $f(x)$ is polynomial-time computable, then $L$ belongs to the class P.","title":"A timed decider with polynomial-time output function gives membership in P"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/mem-np-of-verifies","kind":"node","lean":"Cslib.TimeMComplexity.mem_NP_of_verifies","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#mem-np-of-verifies","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L$ be a decision problem, $p$ a polynomial with natural-number coefficients, and $V$ a timed function from pairs of lists of Booleans to Booleans that verifies $L$ with witness bound $p$. If the function sending $(x, w)$ to the output of $V$ on $(x, w)$ is polynomial-time computable by a Turing machine for the canonical bitstring encodings, then $L$ is in NP.","title":"A timed verifier with polynomial-time output function puts its language in NP"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/cook-levin-poly-many-one-red-of-declared","kind":"node","lean":"Cslib.TimeMComplexity.cookLevinPolyManyOneRed_of_declared","parent":32936,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming/index.html#cook-levin-poly-many-one-red-of-declared","scope":"TheoryOfComputation/complexity-classes/polynomial-time-in-a-cost-annotated-programming","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be decision problems and let $f$ be a declared-time many-one reduction from $L_1$ to $L_2$. If the function sending $x$ to the output of $f$ on $x$ is polynomial-time computable by a Turing machine for the canonical bitstring encodings, then $L_1$ reduces to $L_2$ by a polynomial-time many-one reduction.","title":"From declared-time to polynomial-time many-one reductions"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines.json","id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","title":"Polynomial time on single-tape machines"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/k","kind":"node","lean":"Cslib.Turing.RevTail.K","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#k","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"The two stack indices of the reversal component: the input stack $X$ and the output stack $O$.","title":"Stacks of the reversal component"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/l","kind":"node","lean":"Cslib.Turing.RevTail.L","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#l","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"The three labels of the reversal component: mv (move the input onto the output), drop (drop one symbol) and done.","title":"Labels of the reversal component"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/prog","kind":"node","lean":"Cslib.Turing.RevTail.prog","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#prog","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"The multi-stack program over the stacks $X$ and $O$, both with alphabet $\\mathrm{Bool}$, with a Boolean internal state: at label mv it moves the bits of $X$ one at a time onto $O$ (reversing them) and then goes to drop; at drop it pops $O$ once and goes to done; at done it halts.","title":"Program of the reversal component"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/comp","kind":"node","lean":"Cslib.Turing.RevTail.comp","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#comp","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"The component machine with two stacks, the input stack $X$ and the output stack $O$, both over the alphabet $\\mathrm{Bool}$; the three labels mv, drop and done; the program that moves the bits of $X$ onto $O$ (reversing them), pops $O$ once and halts; and the stack list $[X, O]$.","title":"The reversal component"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/run","kind":"node","lean":"Cslib.Turing.RevTail.run","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#run","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"For every bit list $x$, the program of the reversal component, started at label mv with internal state false, with the bits of $x$ on the input stack $X$ and every other stack empty, reaches label done within at most $|x| + 2$ steps, with some internal state and some stacks whose output stack $O$ holds the reversal of $x$ with its first entry removed.","title":"Running the reversal component"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/computable","kind":"node","lean":"Cslib.Turing.RevTail.computable","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#computable","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"A finite multi-stack Turing machine whose input and output alphabets are both identified with the bits, together with a polynomial $p$ with natural-number coefficients, such that for every bit list $x$, given bit by bit as its own encoding, the machine outputs the reversal of $x$ with its first entry removed (that is, $x$ without its last bit, in reverse order) within $p(|x|)$ steps.","title":"Reversing and dropping the first bit is polynomial-time"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/answer-tm-outputs-within-time","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.answerTM_outputsWithinTime","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#answer-tm-outputs-within-time","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine with finite stack alphabets, whose input alphabet is identified with the bits by $e$ and whose output alphabet by $e_1$. For every non-blank cell $x$ of the single-tape emulation alphabet of $M$, the answer machine of $M$, started on the one-cell tape $x$, halts within at most one step on the one-cell tape holding the input-bit cell of the answer of $x$ (the bit telling\u2026","title":"The answer machine runs in one step"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/decider-tm-outputs-within-time","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.deciderTM_outputsWithinTime","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#decider-tm-outputs-within-time","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine with finite stack alphabets, whose input alphabet is identified with the bits by $e$ and whose output alphabet by $e_1$, and let $C$ be the emulation constant of $M$. Let $w$ be a bit list, $b$ a bit and $t$ a natural number. Suppose $M$, started with its input stack holding the bits of $w$ in reverse order followed by $\\mathrm{true}$ (read through $e^{-1}$), is after exactly\u2026","title":"The decider within time"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/single-tape-p","kind":"node","lean":"Cslib.Complexity.SingleTapeP","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#single-tape-p","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"The class of decision problems $L$ for which there are a finite alphabet $\\Sigma$ with a designated default element, an injective map $\\iota$ from the bits to $\\Sigma$, a single-tape Turing machine $M$ over $\\Sigma$ and a polynomial $p$ with natural-number coefficients such that, for every bit list $w$, the machine $M$ started on the word $\\iota(w)$ (the bits of $w$ mapped by $\\iota$) halts on the one-cell tape\u2026","title":"Polynomial time on single-tape machines"},{"id":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/p-subset-single-tape-p","kind":"node","lean":"Cslib.Complexity.P_subset_singleTapeP","parent":32953,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines/index.html#p-subset-single-tape-p","scope":"TheoryOfComputation/complexity-classes/polynomial-time-on-single-tape-machines","status":"fully_proved","summary":"Every decision problem in the class $\\mathrm{P}$ (defined with multi-stack machines) belongs to the class of decision problems decided in polynomial time by single-tape machines.","title":"Problems in P are decided in polynomial time by single-tape machines"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines.json","id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","title":"Time and space classes on multi-tape machines"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/time-bounded-by","kind":"node","lean":"Cslib.Turing.MultiTapeTM.TimeBoundedBy","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#time-bounded-by","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"A deterministic Turing machine $M$ with $k$ work tapes is time bounded by $t : \\mathbb{N} \\to \\mathbb{N}$ when, for every input word $w$, every computation path of $M$ on $w$ with at least $t(|w|)$ steps ends in a halted configuration, where $|w|$ is the length of $w$.","title":"Time-bounded multi-tape machine"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/space-bounded-by","kind":"node","lean":"Cslib.Turing.MultiTapeTM.SpaceBoundedBy","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#space-bounded-by","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"A deterministic Turing machine $M$ with $k$ work tapes is space bounded by $s : \\mathbb{N} \\to \\mathbb{N}$ when, for every input word $w$, every computation path of $M$ on $w$ visits at most $s(|w|)$ work-tape cells, counting the cells visited on each work tape and summing over the tapes, where $|w|$ is the length of $w$.","title":"Space-bounded multi-tape machine"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/runs-in-space-of-runs-in-time","kind":"node","lean":"Cslib.Turing.MultiTapeTM.runsInSpace_of_runsInTime","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#runs-in-space-of-runs-in-time","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"Let $M$ be a deterministic Turing machine with $k$ work tapes, let $w$ be an input word and let $T \\in \\mathbb{N}$. If $M$ runs in time $T$ on $w$, that is, every computation path of $M$ on $w$ with at least $T$ steps ends in a halted configuration, then $M$ runs in space $k(T + 1)$ on $w$: every computation path of $M$ on $w$ visits at most $k(T + 1)$ work-tape cells, counting the cells visited on each work tape\u2026","title":"Time bounds space on one input"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/space-bounded-by-2","kind":"node","lean":"Cslib.Turing.MultiTapeTM.TimeBoundedBy.spaceBoundedBy","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#space-bounded-by-2","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"Let $M$ be a deterministic Turing machine with $k$ work tapes and let $t : \\mathbb{N} \\to \\mathbb{N}$. If $M$ is time bounded by $t$, then $M$ is space bounded by the function $n \\mapsto k(t(n) + 1)$.","title":"Uniform time bounds give uniform space bounds"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/decides","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Decides","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#decides","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"A deterministic multi-tape Turing machine over the Boolean alphabet decides a language $L$ of Boolean strings when, on every input $w$, it eventually halts with output the one-bit string $[\\mathrm{true}]$ if $w \\in L$ and $[\\mathrm{false}]$ if $w \\notin L$.","title":"A multi-tape machine deciding a language"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/decidable-in-time-and-space-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.decidableInTimeAndSpace_iff","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#decidable-in-time-and-space-iff","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"Let $L$ be a language of Boolean strings and let $t$ and $s$ assign a natural number to every Boolean string. Then $L$ is decidable within time $t$ and space $s$ in CSLib's sense, with the identity encoding of inputs, if and only if there are a number $k$, a finite type of states and a deterministic Turing machine with $k$ work tapes over the Boolean alphabet that decides $L$ and, on every input $w$, runs in time\u2026","title":"Decidability within time and space bounds, unfolded"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/dtime","kind":"node","lean":"Cslib.Turing.MultiTapeTM.DTIME","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#dtime","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"For $t : \\mathbb{N} \\to \\mathbb{N}$, $\\mathrm{DTIME}(t)$ is the set of languages $L$ of Boolean strings for which there are constants $c_1, c_2 \\in \\mathbb{N}$, a function $s : \\mathbb{N} \\to \\mathbb{N}$ and a deterministic multi-tape Turing machine with Boolean tape alphabet and finitely many states such that, on every input $w$ of length $n$, the machine halts with the one-bit output $[\\mathrm{true}]$ if\u2026","title":"Deterministic time classes on multi-tape machines"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/dspace","kind":"node","lean":"Cslib.Turing.MultiTapeTM.DSPACE","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#dspace","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"For $s : \\mathbb{N} \\to \\mathbb{N}$, $\\mathrm{DSPACE}(s)$ is the set of languages $L$ of Boolean strings for which there are constants $c_1, c_2 \\in \\mathbb{N}$, a function $t : \\mathbb{N} \\to \\mathbb{N}$ and a deterministic multi-tape Turing machine with Boolean tape alphabet and finitely many states such that, on every input $w$ of length $n$, the machine halts with the one-bit output $[\\mathrm{true}]$ if\u2026","title":"Deterministic space classes on multi-tape machines"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/dtime-mono","kind":"node","lean":"Cslib.Turing.MultiTapeTM.DTIME_mono","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#dtime-mono","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"Let $t_1, t_2 : \\mathbb{N} \\to \\mathbb{N}$ and $a, b \\in \\mathbb{N}$ satisfy $t_1(n) \\le a \\, t_2(n) + b$ for every $n \\in \\mathbb{N}$. Then $\\mathrm{DTIME}(t_1) \\subseteq \\mathrm{DTIME}(t_2)$.","title":"Deterministic time classes are monotone up to affine changes of the bound"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/dspace-mono","kind":"node","lean":"Cslib.Turing.MultiTapeTM.DSPACE_mono","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#dspace-mono","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"Let $s_1, s_2 : \\mathbb{N} \\to \\mathbb{N}$ and $a, b \\in \\mathbb{N}$ satisfy $s_1(n) \\le a \\, s_2(n) + b$ for every $n \\in \\mathbb{N}$. Then $\\mathrm{DSPACE}(s_1) \\subseteq \\mathrm{DSPACE}(s_2)$.","title":"Deterministic space classes are monotone up to affine changes of the bound"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/dtime-subset-dspace","kind":"node","lean":"Cslib.Turing.MultiTapeTM.DTIME_subset_DSPACE","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#dtime-subset-dspace","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"For every $t : \\mathbb{N} \\to \\mathbb{N}$, $\\mathrm{DTIME}(t) \\subseteq \\mathrm{DSPACE}(t)$.","title":"Time classes are contained in space classes"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/p","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Classes.P","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#p","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"$\\mathrm{P}$ is the set of languages of Boolean strings that lie in $\\mathrm{DTIME}(n \\mapsto n^d)$ for some $d \\in \\mathbb{N}$.","title":"Polynomial time on multi-tape machines"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/pspace","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Classes.PSPACE","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#pspace","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"$\\mathrm{PSPACE}$ is the set of languages of Boolean strings that lie in $\\mathrm{DSPACE}(n \\mapsto n^d)$ for some $d \\in \\mathbb{N}$.","title":"Polynomial space on multi-tape machines"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/p-subset-pspace","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Classes.P_subset_PSPACE","parent":32964,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines/index.html#p-subset-pspace","scope":"TheoryOfComputation/complexity-classes/time-and-space-classes-on-multi-tape-machines","status":"fully_proved","summary":"On the deterministic multi-tape model, $\\mathrm{P} \\subseteq \\mathrm{PSPACE}$.","title":"P is contained in PSPACE"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes.json","id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","title":"Time and space complexity classes"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/lang-of","kind":"node","lean":"Cslib.Complexity.Classes.langOf","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#lang-of","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The language of a decision problem $L$ is the set of bitstrings $x$ with $L(x) = \\mathrm{true}$.","title":"Language of a decision problem"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/pspace","kind":"node","lean":"Cslib.Complexity.Classes.PSPACE","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#pspace","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class PSPACE of decision problems consists of the decision problems $L$ whose language $\\{x : L(x) = \\mathrm{true}\\}$ lies in the multi-tape class PSPACE, that is, is decided by a multi-tape Turing machine that halts on every input and uses space $O(n^d)$ for some natural number $d$.","title":"Polynomial space for decision problems"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/exp","kind":"node","lean":"Cslib.Complexity.Classes.EXP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#exp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"EXP, deterministic exponential time, is the class of decision problems $L$ for which there is a natural number $d$ such that the language of $L$, the set of bitstrings $x$ with $L(x)$ true, belongs to $\\mathrm{DTIME}(2^{n^d})$: it is decided by a multi-tape Turing machine over the binary alphabet with finitely many states in time at most $c_1 2^{n^d} + c_2$ on inputs of length $n$, for some natural numbers $c_1$\u2026","title":"The class EXP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/expspace","kind":"node","lean":"Cslib.Complexity.Classes.EXPSPACE","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#expspace","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"EXPSPACE, deterministic exponential space, is the class of decision problems $L$ for which there is a natural number $d$ such that the language of $L$, the set of bitstrings $x$ with $L(x)$ true, belongs to $\\mathrm{DSPACE}(2^{n^d})$: it is decided by a multi-tape Turing machine over the binary alphabet with finitely many states that halts on every input and visits at most $c_1 2^{n^d} + c_2$ work-tape cells on\u2026","title":"The class EXPSPACE"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/e","kind":"node","lean":"Cslib.Complexity.Classes.E","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#e","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"E is the class of decision problems $L$ for which there is a natural number $c$ such that the language of $L$, the set of bitstrings $x$ with $L(x)$ true, belongs to $\\mathrm{DTIME}(2^{cn})$: it is decided by a multi-tape Turing machine over the binary alphabet with finitely many states in time at most $c_1 2^{cn} + c_2$ on inputs of length $n$, for some natural numbers $c_1$ and $c_2$.","title":"The class E"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/log-space","kind":"node","lean":"Cslib.Complexity.Classes.LogSpace","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#log-space","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class $\\mathsf{L}$ consists of the decision problems $L$ for which there is a constant $c \\in \\mathbb{N}$ such that the language of $L$ lies in $\\mathrm{DSPACE}(s)$ for $s(n) = c \\lfloor \\log_2(n + 1) \\rfloor + c$, the class of languages decided by a multi-tape Turing machine with binary alphabet and finitely many states, within some time bound, visiting at most $c_1 s(n) + c_2$ work-tape cells on inputs of\u2026","title":"Deterministic logarithmic space"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/log-space-computable","kind":"node","lean":"Cslib.Complexity.Classes.LogSpaceComputable","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#log-space-computable","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A function $f$ from bitstrings to bitstrings is log-space computable when there are a constant $c \\in \\mathbb{N}$ and a time bound $t : \\mathbb{N} \\to \\mathbb{N}$ such that some multi-tape Turing machine with binary alphabet and finitely many states, on every bitstring input $x$, halts with output $f(x)$ within $t(|x|)$ steps and visits at most $c \\lfloor \\log_2(|x| + 1) \\rfloor + c$ work-tape cells in total.","title":"Log-space computable functions"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/log-space-red","kind":"node","lean":"Cslib.Complexity.Classes.LogSpaceRed","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#log-space-red","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L_1$ is log-space many-one reducible to a decision problem $L_2$ when there is a log-space computable function $f$ from bitstrings to bitstrings such that, for every bitstring $x$, $L_1(x)$ is true if and only if $L_2(f(x))$ is true.","title":"Log-space many-one reductions"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ntmdecides","kind":"node","lean":"Cslib.Complexity.Classes.NTMDecides","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ntmdecides","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A nondeterministic multi-tape Turing machine $M$ over the boolean alphabet decides a set $L$ of bitstrings if, for every bitstring $x$, $x \\in L$ if and only if some computation path of $M$ on $x$ ends in a halted configuration whose output is the one-element list $[\\mathrm{true}]$.","title":"Nondeterministic decision"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ntmdecides-in-time","kind":"node","lean":"Cslib.Complexity.Classes.NTMDecidesInTime","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ntmdecides-in-time","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $M$ be a nondeterministic multi-tape Turing machine over the boolean alphabet, $t : \\mathbb{N} \\to \\mathbb{N}$ and $L$ a set of bitstrings. Then $M$ decides $L$ in time $t$ if, for every input $x$, every computation path of $M$ on $x$ with at least $t(|x|)$ steps ends in a halted configuration, where $|x|$ is the length of $x$, and for every bitstring $x$, $x \\in L$ if and only if some computation path of $M$\u2026","title":"Nondeterministic decision in bounded time"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ntmdecides-in-space","kind":"node","lean":"Cslib.Complexity.Classes.NTMDecidesInSpace","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ntmdecides-in-space","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $M$ be a nondeterministic multi-tape Turing machine over the boolean alphabet, $s : \\mathbb{N} \\to \\mathbb{N}$ and $L$ a set of bitstrings. Then $M$ decides $L$ in space $s$ if, for every input $x$, every computation path of $M$ on $x$ visits at most $s(|x|)$ work-tape cells in total, where $|x|$ is the length of $x$, and for every bitstring $x$, $x \\in L$ if and only if some computation path of $M$ on $x$ ends\u2026","title":"Nondeterministic decision in bounded space"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ntime","kind":"node","lean":"Cslib.Complexity.Classes.NTIME","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ntime","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For $t : \\mathbb{N} \\to \\mathbb{N}$, $\\mathrm{NTIME}(t)$ is the set of languages $L$ of bitstrings for which there are natural numbers $c_1$, $c_2$, $k$ and a nondeterministic multi-tape Turing machine $M$ over the boolean alphabet with finitely many states and $k$ work tapes that decides $L$ in time $c_1 t(n) + c_2$: on every input $x$, every computation path of $M$ with at least $c_1 t(|x|) + c_2$ steps ends in a\u2026","title":"Nondeterministic time classes"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/nspace","kind":"node","lean":"Cslib.Complexity.Classes.NSPACE","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#nspace","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For $s : \\mathbb{N} \\to \\mathbb{N}$, $\\mathrm{NSPACE}(s)$ is the set of languages $L$ of bitstrings for which there are natural numbers $c_1$, $c_2$, $k$ and a nondeterministic multi-tape Turing machine $M$ over the boolean alphabet with finitely many states and $k$ work tapes that decides $L$ in space $c_1 s(n) + c_2$: on every input $x$, every computation path of $M$ visits at most $c_1 s(|x|) + c_2$ work-tape\u2026","title":"Nondeterministic space classes"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/nl","kind":"node","lean":"Cslib.Complexity.Classes.NL","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#nl","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"NL is the set of decision problems $L$ for which there is a natural number $c$ such that the language of $L$ (the set of bitstrings on which $L$ is true) belongs to $\\mathrm{NSPACE}(s)$ for $s(n) = c \\lfloor \\log_2 (n + 1) \\rfloor + c$, where $\\mathrm{NSPACE}(s)$ is the class of languages decided by a nondeterministic multi-tape Turing machine with finitely many states in space $c_1 s(n) + c_2$ for some natural\u2026","title":"The complexity class NL"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/npspace","kind":"node","lean":"Cslib.Complexity.Classes.NPSPACE","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#npspace","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"NPSPACE is the set of decision problems $L$ for which there is a natural number $d$ such that the language of $L$ (the set of bitstrings on which $L$ is true) belongs to $\\mathrm{NSPACE}(s)$ for $s(n) = n^d$, where $\\mathrm{NSPACE}(s)$ is the class of languages decided by a nondeterministic multi-tape Turing machine with finitely many states in space $c_1 s(n) + c_2$ for some natural numbers $c_1, c_2$.","title":"The complexity class NPSPACE"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/nexp","kind":"node","lean":"Cslib.Complexity.Classes.NEXP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#nexp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"NEXP is the set of decision problems $L$ for which there is a polynomial $p$ with natural-number coefficients such that the language of $L$ (the set of bitstrings on which $L$ is true) belongs to $\\mathrm{NTIME}(t)$ for $t(n) = 2^{p(n)}$, where $\\mathrm{NTIME}(t)$ is the class of languages decided by a nondeterministic multi-tape Turing machine with finitely many states in time $c_1 t(n) + c_2$ for some natural\u2026","title":"The complexity class NEXP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/bpp","kind":"node","lean":"Cslib.Complexity.Classes.BPP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#bpp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class BPP of bounded-error probabilistic polynomial-time decision problems consists of the decision problems $L$ for which there are a polynomial $p$ with natural number coefficients and a polynomial-time computable function $R$ from pairs of bitstrings to booleans such that, for every bitstring $x$, writing $m = p(|x|)$ and $c_x$ for the number of strings $r \\in \\{0,1\\}^m$ with $R(x, r)$ true: if $L(x)$ is\u2026","title":"BPP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/pp","kind":"node","lean":"Cslib.Complexity.Classes.PP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#pp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class PP consists of the decision problems $L$ for which there are a polynomial $p$ with natural-number coefficients and a polynomial-time computable function $R$ on pairs of bitstrings such that, for every bitstring $x$ and $r$ uniformly random among the bitstrings of length $p(|x|)$, the probability that $R(x, r)$ is true is greater than $1/2$ if $L(x)$ is true, and at most $1/2$ if $L(x)$ is false.","title":"Probabilistic polynomial time with unbounded error"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ppoly","kind":"node","lean":"Cslib.Complexity.Classes.PPoly","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ppoly","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class P/poly consists of the decision problems $L$ for which there are a polynomial $p$ with natural-number coefficients, bitstrings $a_0, a_1, a_2, \\ldots$ with $|a_n| \\le p(n)$ for every $n$, and a polynomial-time computable function $R$ on pairs of bitstrings such that for every bitstring $x$, $L(x)$ is true if and only if $R(x, a_{|x|})$ is true.","title":"Polynomial time with polynomial advice"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/sharp-p","kind":"node","lean":"Cslib.Complexity.Classes.SharpP","parent":32979,"reading_title":{"format":"inline-parts-v1","parts":[["text","The counting class "],["math","\\(\\#P\\)"]],"source":"The counting class $\\#P$"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#sharp-p","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A function $f$ from bitstrings to natural numbers is in $\\#P$ when there are a polynomial $p$ with natural-number coefficients and a polynomial-time computable Boolean function $R$ on pairs of bitstrings such that for every bitstring $x$, $f(x)$ is the number of bitstrings $w$ of length exactly $p(|x|)$ for which $R(x, w)$ is true.","title":"The counting class $\\#P$"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/eval-oracle","kind":"node","lean":"Cslib.Complexity.Classes.evalOracle","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#eval-oracle","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $\\alpha$ be a type of oracle answers and $\\beta$ a type of outputs, and call a list of pairs of a bitstring and an element of $\\alpha$ a history. Fix a length cap $c \\in \\mathbb{N}$, a next-query function $N$ sending an input bitstring and a history to a bitstring, a decision function $F$ sending an input bitstring and a history to an element of $\\beta$, an oracle $O$ from bitstrings to $\\alpha$ and an input\u2026","title":"Evaluation of an oracle machine with adaptive queries"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/pwith-oracle","kind":"node","lean":"Cslib.Complexity.Classes.PWithOracle","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#pwith-oracle","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a Boolean oracle $O$ on bitstrings, the class $\\mathsf{P}^O$ consists of the decision problems $L$ for which there are polynomials $p$ and $q$ and polynomial-time functions $\\mathrm{next}$ and $\\mathrm{final}$ such that, for every bitstring $x$, $L(x)$ is true exactly when the following oracle machine outputs true on $x$. Starting from the empty history $h$, it runs $p(|x|)$ rounds; in each round it computes\u2026","title":"Polynomial time with an oracle"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/npwith-oracle","kind":"node","lean":"Cslib.Complexity.Classes.NPWithOracle","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#npwith-oracle","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $O$ be a function from bitstrings to booleans. The class $\\mathrm{NP}^O$ is the set of decision problems $L$ for which there are polynomials $p$ and $q$ with natural-number coefficients and polynomial-time computable functions $\\mathrm{nextQ}$ and $\\mathrm{final}$, each taking a bitstring and a list of query/answer pairs, such that for every bitstring $x$, $L(x)$ is true if and only if there is a bitstring $w$\u2026","title":"NP relative to an oracle"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/pwith-sharp-p","kind":"node","lean":"Cslib.Complexity.Classes.PWithSharpP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#pwith-sharp-p","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class $\\mathsf{P}^{\\#\\mathsf{P}}$ consists of the decision problems $L$ for which there are a function $f$ in $\\#\\mathsf{P}$, polynomials $p$ and $q$, and polynomial-time functions $\\mathrm{next}$ and $\\mathrm{final}$ such that, for every bitstring $x$, $L(x)$ is true exactly when the following oracle machine outputs true on $x$. Starting from the empty history $h$, it runs $p(|x|)$ rounds; in each round it\u2026","title":"Polynomial time with a #P oracle"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/sharp-pcomplete","kind":"node","lean":"Cslib.Complexity.Classes.SharpPComplete","parent":32979,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\#P\\)"],["text","-complete functions"]],"source":"$\\#P$-complete functions"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#sharp-pcomplete","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A function $f$ from bitstrings to natural numbers is $\\#P$-complete when $f$ is in $\\#P$ and every function $g$ in $\\#P$ is computed from $f$ by a polynomial-time oracle machine. That is, there are polynomials $p$ and $q$ with natural-number coefficients and polynomial-time computable functions choosing the next query and the final natural-number answer from the input and the history of queries and oracle answers,\u2026","title":"$\\#P$-complete functions"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/alt-quant","kind":"node","lean":"Cslib.Complexity.Classes.AltQuant","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#alt-quant","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $b : \\mathbb{N} \\to \\mathbb{N}$ be a length bound, $k$ a natural number, $R$ a Boolean function of a bitstring and a list of bitstrings, and $x$ a bitstring. The statement with $k$ blocks, beginning with $\\exists$ when the given flag is true and with $\\forall$ when it is false, is","title":"Alternating bounded quantifiers"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/sigma-k","kind":"node","lean":"Cslib.Complexity.Classes.SigmaK","parent":32979,"reading_title":{"format":"inline-parts-v1","parts":[["text","The class "],["math","\\(\\Sigma_k^p\\)"]],"source":"The class $\\Sigma_k^p$"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#sigma-k","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a natural number $k$, a decision problem $L$ is in $\\Sigma_k^p$ when there are a polynomial $p$ with natural-number coefficients and a polynomial-time computable Boolean function $R$ of a bitstring and a list of bitstrings such that for every bitstring $x$, $L(x)$ is true if and only if $Q_1 w_1\\, Q_2 w_2 \\cdots Q_k w_k$ with $R(x, [w_1, \\dots, w_k])$ true, where $Q_i$ is $\\exists$ for odd $i$ and $\\forall$ for\u2026","title":"The class $\\Sigma_k^p$"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/pi-k","kind":"node","lean":"Cslib.Complexity.Classes.PiK","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#pi-k","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a natural number $k$, the class $\\Pi_k^p$ consists of the decision problems $L$ for which there are a polynomial $p$ with natural-number coefficients and a polynomial-time computable function $R$, taking a bitstring and a list of bitstrings to a boolean, such that for every bitstring $x$, $L(x)$ is true if and only if","title":"The universal levels of the polynomial hierarchy"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ph","kind":"node","lean":"Cslib.Complexity.Classes.PH","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ph","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class PH is the union, over all natural numbers $k$, of the classes $\\Sigma_k^p$ of the polynomial hierarchy.","title":"The polynomial hierarchy"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/am","kind":"node","lean":"Cslib.Complexity.Classes.AM","parent":32979,"reading_title":{"format":"inline-parts-v1","parts":[["text","The class "],["math","\\(\\mathsf{AM}\\)"],["text"," (Arthur\u2013Merlin)"]],"source":"The class $\\mathsf{AM}$ (Arthur\u2013Merlin)"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#am","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L$ on bitstrings is in $\\mathsf{AM}$ when there are polynomials $p$ and $q$ with natural-number coefficients and a polynomial-time computable Boolean function $R(x, r, w)$ of three bitstrings such that, for every bitstring $x$, writing $A_x$ for the set of strings $r \\in \\{0,1\\}^{p(|x|)}$ for which some bitstring $w$ of length at most $q(|x|)$ makes $R(x, r, w)$ true: if $L(x)$ is true then\u2026","title":"The class $\\mathsf{AM}$ (Arthur\u2013Merlin)"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/co-am","kind":"node","lean":"Cslib.Complexity.Classes.coAM","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#co-am","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class coAM. coAM is the class of decision problems $L$ whose complement, the decision problem that accepts exactly the bitstrings that $L$ rejects, belongs to AM.","title":"The class coAM"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ipgame","kind":"node","lean":"Cslib.Complexity.Classes.IPGame","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ipgame","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The game takes a length cap $c$, a verifier message function $V_{\\mathrm{msg}}$, a verifier decision function $V_{\\mathrm{dec}}$, a prover $P$ mapping transcripts (lists of bitstrings) to bitstrings, an input $x$, a coin string $\\rho$, a number $k$ of rounds and a transcript $t$, and returns a Boolean outcome. If $k = 0$, the outcome is $V_{\\mathrm{dec}}(x, \\rho, t)$. Otherwise the verifier sends\u2026","title":"Interactive-proof game"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ip","kind":"node","lean":"Cslib.Complexity.Classes.IP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ip","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L$ is in $\\mathsf{IP}$ when there are polynomials $s$, $r$ and $q$ with natural-number coefficients and polynomial-time functions $V_{\\mathrm{msg}}$ (the verifier's message function) and $V_{\\mathrm{dec}}$ (its decision function) with the following property. For a bitstring $x$, a prover $P$ (an arbitrary function from transcripts to bitstrings) and a coin string $c$, let $G(P, c)$ be the\u2026","title":"The class IP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/pcp","kind":"node","lean":"Cslib.Complexity.Classes.PCP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#pcp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For functions $r, q : \\mathbb{N} \\to \\mathbb{N}$, the class $\\mathrm{PCP}(r, q)$ consists of the decision problems $L$ for which there are polynomial-time computable functions $Q$ and $D$ such that $Q(x, \\rho)$ is a list of at most $q(|x|)$ natural numbers for all bitstrings $x$ and $\\rho$, and such that for every bitstring $x$ the following holds. For a proof\u2026","title":"Probabilistically checkable proofs"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/acgate","kind":"node","lean":"Cslib.Complexity.Classes.ACGate","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#acgate","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A gate of a flat Boolean circuit on $n$ inputs is one of: an input variable $x_i$ for some $i < n$; a Boolean constant; a negation gate with one predecessor index $a \\in \\mathbb{N}$; a conjunction gate or a disjunction gate with a list of predecessor indices; or a mod gate with a modulus $m \\in \\mathbb{N}$ and a list of predecessor indices.","title":"Gates of a flat Boolean circuit"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/gate-inputs","kind":"node","lean":"Cslib.Complexity.Classes.gateInputs","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#gate-inputs","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a gate $g$ of a flat Boolean circuit on $n$ inputs, this is the list of gate indices that $g$ reads: the empty list for an input variable or a constant, the one-element list $[a]$ for the negation of gate $a$, and the argument list $l$ for an and, or or mod gate with argument list $l$ (the modulus of a mod gate is not included).","title":"Predecessors of a gate"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/well-formed-circuit","kind":"node","lean":"Cslib.Complexity.Classes.WellFormedCircuit","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#well-formed-circuit","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A flat Boolean circuit on $n$ inputs, given as a list of gates $(g_0, \\dots, g_{m-1})$, is well-formed when every gate references only strictly earlier gates: for every $j < m$, every gate index $a$ among the inputs of $g_j$ satisfies $a < j$.","title":"Well-formed flat circuits"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/gate-value","kind":"node","lean":"Cslib.Complexity.Classes.gateValue","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#gate-value","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $v$ be a list of Boolean values of earlier gates, where an index outside $v$ reads as false, and let $\\sigma$ assign a Boolean value to each of the $n$ inputs. Given $v$ and $\\sigma$, the value of the input variable $x_i$ is $\\sigma(i)$; the value of the constant $b$ is $b$; the value of the negation of gate $a$ is the negation of $v_a$; the value of an and gate (or gate) with argument list $l$ is the\u2026","title":"Value of a gate"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/circuit-eval","kind":"node","lean":"Cslib.Complexity.Classes.circuitEval","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#circuit-eval","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a circuit on $n$ inputs given as a list of gates $g_0, \\dots, g_{m-1}$ and an assignment $\\alpha$ of booleans to the inputs, the list of gate values is $(v_0, \\dots, v_{m-1})$, where $v_j$ is computed from $v_0, \\dots, v_{j-1}$, a reference to a gate $a \\ge j$ reading as false: an input gate $i$ has value $\\alpha(i)$, a constant gate has its constant, a negation gate on $a$ has value $\\lnot v_a$, an and gate\u2026","title":"Evaluation of a flat Boolean circuit"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ac-gate-to-pair","kind":"node","lean":"Cslib.Complexity.Classes.acGateToPair","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ac-gate-to-pair","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A gate of a flat Boolean circuit on $n$ inputs is encoded as a pair of a natural number (its kind) and a list of natural numbers (its references), the shared encoding used by decoding-based problems such as Circuit Value and succinct circuits: the input variable $x_i$ is encoded as $(0, [i])$; the constant true as $(1, [1])$ and the constant false as $(1, [0])$; the negation of gate $a$ as $(2, [a])$; the\u2026","title":"Encoding a gate as a (kind, references) pair"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ac-gate-of-pair","kind":"node","lean":"Cslib.Complexity.Classes.acGateOfPair","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ac-gate-of-pair","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A pair $(k, r)$ of a natural number $k$ and a list $r$ of natural numbers decodes to a gate of a flat Boolean circuit on $n$ inputs as follows: $(0, [i])$ decodes to the input variable $x_i$ when $i < n$ and to the constant false when $i \\ge n$; $(1, [b])$ decodes to the constant that is true if and only if $b = 1$; $(2, [a])$ decodes to the negation of gate $a$; $(3, r)$ decodes to the conjunction of the gates\u2026","title":"Decoding a gate from its (kind, references) pair"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/serial-gate-eval","kind":"node","lean":"Cslib.Complexity.Classes.serialGateEval","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#serial-gate-eval","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $v$ be a list of Boolean values of earlier gates, let $a$ assign a Boolean to each of the $n$ inputs $0, \\dots, n - 1$, and let $(k, \\ell)$ be a serialized gate. Write $v_c$ for entry $c$ of $v$, read as false when $c$ is out of range. The value of the gate is: $a(i)$ if $(k, \\ell) = (0, [i])$ with $i < n$, and false if $(k, \\ell) = (0, [i])$ with $i \\ge n$; true exactly when $b = 1$ if $(k, \\ell) = (1, [b])$;\u2026","title":"Evaluating a serialized gate"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/gate-value-of-pair","kind":"node","lean":"Cslib.Complexity.Classes.gateValue_ofPair","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#gate-value-of-pair","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For every list $v$ of Boolean values of earlier gates, every assignment of Boolean values to the $n$ inputs and every pair $g = (k, r)$ of a natural number $k$ (the gate kind) and a list $r$ of natural numbers (its references), evaluating $g$ with the serial $(\\mathrm{kind}, \\mathrm{refs})$ gate evaluator gives the same value as evaluating the gate decoded from $g$ with the gate-value function.","title":"Serial gate evaluation agrees with typed gate evaluation"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/gate-depth-of","kind":"node","lean":"Cslib.Complexity.Classes.gateDepthOf","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#gate-depth-of","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $d$ be the list of depths of the earlier gates of a flat Boolean circuit on $n$ inputs, and for a gate index $a$ let $d_a$ be the entry of $d$ at (zero-based) position $a$, or $0$ when $a$ is not a position of $d$. The depth of an input-variable gate or a constant gate is $0$; the depth of a negation, AND, OR or modular gate is one more than the maximum of $d_a$ over the gates $a$ it references, where the\u2026","title":"Depth of a gate"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/circuit-depths","kind":"node","lean":"Cslib.Complexity.Classes.circuitDepths","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#circuit-depths","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a circuit given as a list of gates $g_0, \\dots, g_{m-1}$, the list of gate depths is $(d_0, \\dots, d_{m-1})$, where $d_j = 0$ if $g_j$ is an input or constant gate, and otherwise $d_j$ is one more than the maximum of the depths $d_a$ of the gates $a$ that $g_j$ references (the maximum over no references is $0$), a reference $a \\ge j$ counting as depth $0$.","title":"Gate depths of a flat Boolean circuit"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/circuit-depth","kind":"node","lean":"Cslib.Complexity.Classes.circuitDepth","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#circuit-depth","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The depth of a circuit, given as a list of gates, is the maximum of the depths of its gates, and $0$ for the empty circuit. Input and constant gates have depth $0$; a negation, and, or or mod gate has depth one more than the maximum depth of the gates it references (the maximum over no references is $0$), where a reference to a gate that does not come earlier in the list counts as depth $0$.","title":"Depth of a flat Boolean circuit"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-mod-free","kind":"node","lean":"Cslib.Complexity.Classes.IsModFree","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-mod-free","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A flat Boolean circuit $C$ on $n$ inputs is mod-free when none of its gates is a mod gate.","title":"Circuits without mod gates"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/mod-only","kind":"node","lean":"Cslib.Complexity.Classes.ModOnly","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#mod-only","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a natural number $m$, a flat Boolean circuit on $n$ inputs satisfies the modulus-$m$ condition when every $\\mathrm{mod}$ gate in it has modulus exactly $m$; gates of the other kinds are unrestricted.","title":"Circuits whose modular gates have a fixed modulus"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-monotone-circuit","kind":"node","lean":"Cslib.Complexity.Classes.IsMonotoneCircuit","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-monotone-circuit","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A flat circuit $C$ on $n$ input variables, given as a list of gates, is monotone when every gate of $C$ is an input variable, an AND gate or an OR gate. Constant, NOT and mod gates are not allowed.","title":"Monotone circuits"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ac0","kind":"node","lean":"Cslib.Complexity.Classes.AC0","parent":32979,"reading_title":{"format":"inline-parts-v1","parts":[["text","The class "],["math","\\(\\mathsf{AC}^0\\)"]],"source":"The class $\\mathsf{AC}^0$"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ac0","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L$ on bitstrings is in $\\mathsf{AC}^0$ when there are a depth bound $d$, a polynomial $p$ with natural-number coefficients and, for every $n$, a flat Boolean circuit $C_n$ on $n$ inputs such that for every $n$: every gate of $C_n$ refers only to earlier gates; $C_n$ has no mod gates; $C_n$ has at most $p(n)$ gates and depth at most $d$; and for every bitstring $x$ of length $n$, $L(x)$ is true\u2026","title":"The class $\\mathsf{AC}^0$"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ac0-mod","kind":"node","lean":"Cslib.Complexity.Classes.AC0Mod","parent":32979,"reading_title":{"format":"inline-parts-v1","parts":[["text","The class "],["math","\\(\\mathsf{AC}^0(m)\\)"]],"source":"The class $\\mathsf{AC}^0(m)$"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ac0-mod","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a natural number $m$, a decision problem $L$ on bitstrings is in $\\mathsf{AC}^0[m]$ when there are a depth bound $d$, a polynomial $p$ with natural-number coefficients and, for every $n$, a flat Boolean circuit $C_n$ on $n$ inputs such that for every $n$: every gate of $C_n$ refers only to earlier gates; every mod gate of $C_n$ has modulus $m$; $C_n$ has at most $p(n)$ gates and depth at most $d$; and for every\u2026","title":"The class $\\mathsf{AC}^0(m)$"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/acc0","kind":"node","lean":"Cslib.Complexity.Classes.ACC0","parent":32979,"reading_title":{"format":"inline-parts-v1","parts":[["text","The class "],["math","\\(\\mathsf{ACC}^0\\)"]],"source":"The class $\\mathsf{ACC}^0$"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#acc0","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"$\\mathsf{ACC}^0$ is the union of the classes $\\mathsf{AC}^0[m]$ over all natural numbers $m$: a decision problem is in $\\mathsf{ACC}^0$ when it is in $\\mathsf{AC}^0[m]$ for some $m$. A circuit family witnessing membership uses one fixed modulus $m$ throughout.","title":"The class $\\mathsf{ACC}^0$"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/parity-dec","kind":"node","lean":"Cslib.Complexity.Classes.parityDec","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#parity-dec","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The parity decision problem accepts a bitstring exactly when an odd number of its entries are true.","title":"The parity language"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/majority-dec","kind":"node","lean":"Cslib.Complexity.Classes.majorityDec","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#majority-dec","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The majority language is the decision problem that accepts a bitstring $x$ exactly when $x$ has more true bits than false bits, that is, when $2k > |x|$, where $k$ is the number of true bits of $x$ and $|x|$ is its length.","title":"The majority language"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/rp","kind":"node","lean":"Cslib.Complexity.Classes.RP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#rp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L$ is in $\\mathsf{RP}$ when there are a polynomial $p$ with natural-number coefficients and a polynomial-time computable Boolean function $R$ on pairs of bitstrings such that for every bitstring $x$: if $L(x)$ is true, then $R(x, r)$ is true for at least half of the $2^{p(|x|)}$ bitstrings $r$ of length $p(|x|)$; and if $L(x)$ is false, then $R(x, r)$ is false for every bitstring $r$ of length\u2026","title":"The class RP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/co-rp","kind":"node","lean":"Cslib.Complexity.Classes.coRP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#co-rp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class coRP. coRP is the class of decision problems $L$ whose complement, the decision problem that accepts exactly the bitstrings that $L$ rejects, belongs to RP.","title":"The class coRP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/mem-co-rp","kind":"node","lean":"Cslib.Complexity.Classes.mem_coRP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#mem-co-rp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L$ belongs to $\\mathsf{coRP}$ if and only if its complement, the decision problem that accepts exactly the bitstrings $L$ rejects, belongs to $\\mathsf{RP}$.","title":"Membership in coRP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/zpp","kind":"node","lean":"Cslib.Complexity.Classes.ZPP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#zpp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class $\\mathsf{ZPP}$ (zero-error probabilistic polynomial time) is defined as the set of decision problems that lie in both $\\mathsf{RP}$ and $\\mathsf{coRP}$, that is, $\\mathsf{ZPP} = \\mathsf{RP} \\cap \\mathsf{coRP}$.","title":"ZPP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/zpp-eq","kind":"node","lean":"Cslib.Complexity.Classes.zpp_eq","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#zpp-eq","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class $\\mathsf{ZPP}$ equals $\\mathsf{RP} \\cap \\mathsf{coRP}$; here this holds by definition.","title":"ZPP is the intersection of RP and coRP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/zpp-subset-rp","kind":"node","lean":"Cslib.Complexity.Classes.zpp_subset_rp","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#zpp-subset-rp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Every decision problem in $\\mathsf{ZPP}$ lies in $\\mathsf{RP}$, that is, $\\mathsf{ZPP} \\subseteq \\mathsf{RP}$.","title":"ZPP is contained in RP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/zpp-subset-corp","kind":"node","lean":"Cslib.Complexity.Classes.zpp_subset_corp","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#zpp-subset-corp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Every decision problem in $\\mathsf{ZPP}$ lies in $\\mathsf{coRP}$, that is, $\\mathsf{ZPP} \\subseteq \\mathsf{coRP}$.","title":"ZPP is contained in coRP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ma","kind":"node","lean":"Cslib.Complexity.Classes.MA","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ma","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L$ is in $\\mathsf{MA}$ when there are polynomials $p$ and $q$ with natural-number coefficients and a polynomial-time computable predicate $R$ on triples $(x, w, r)$ of bitstrings such that, for every bitstring $x$, with $N_x(w)$ the number of strings $r$ of length $p(|x|)$ for which $R(x, w, r)$ is true: if $L(x)$ is true, then some $w$ with $|w| \\le q(|x|)$ satisfies\u2026","title":"The class MA (Merlin\u2013Arthur)"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/co-ma","kind":"node","lean":"Cslib.Complexity.Classes.coMA","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#co-ma","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class coMA. coMA is the class of decision problems $L$ whose complement, the decision problem that accepts exactly the bitstrings that $L$ rejects, belongs to MA.","title":"The class coMA"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/mem-co-ma","kind":"node","lean":"Cslib.Complexity.Classes.mem_coMA","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#mem-co-ma","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L$ belongs to $\\mathsf{coMA}$ if and only if its complement, the decision problem that accepts exactly the bitstrings $L$ rejects, belongs to $\\mathsf{MA}$.","title":"Membership in coMA"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/serial-circuit","kind":"node","lean":"Cslib.Complexity.Classes.SerialCircuit","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#serial-circuit","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A serial circuit is a list of pairs $(k, \\ell)$, where $k$ is a natural number (the gate kind) and $\\ell$ is a list of natural numbers (the gate's references).","title":"Serial circuits"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/uniform-circuit-family","kind":"node","lean":"Cslib.Complexity.Classes.UniformCircuitFamily","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#uniform-circuit-family","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A family $(C_n)_{n \\in \\mathbb{N}}$ of flat Boolean circuits, $C_n$ having $n$ inputs, is log-space uniform when there is a log-space computable function $D$ on bitstrings such that, for every $n$, decoding $D(1^n)$ (where $1^n$ is the string of $n$ ones) as a list of (gate kind, references) pairs succeeds and returns the serial encoding of $C_n$, that is, the list of the serial encodings of its gates in order.\u2026","title":"Log-space uniform circuit families"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-fan-in-le-two","kind":"node","lean":"Cslib.Complexity.Classes.IsFanInLeTwo","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-fan-in-le-two","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A flat Boolean circuit $C$ on $n$ inputs has fan-in at most two when every and gate, or gate and mod gate of $C$ has an argument list of length at most $2$.","title":"Circuits of fan-in at most two"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/nck","kind":"node","lean":"Cslib.Complexity.Classes.NCK","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#nck","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a natural number $k$, a decision problem $L$ is in $\\mathsf{NC}^k$ when there are a constant $c$, a polynomial $p$ with natural number coefficients and a log-space uniform family of flat Boolean circuits $(C_n)_{n \\in \\mathbb{N}}$, where $C_n$ has $n$ inputs, such that for every $n$ the circuit $C_n$ is well formed, has no $\\mathrm{mod}$ gates, has at most two inputs at every and gate and every or gate, has at\u2026","title":"The class NC^k"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/nc","kind":"node","lean":"Cslib.Complexity.Classes.NC","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#nc","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"$\\mathsf{NC}$ is the complexity class of decision problems that belong to $\\mathsf{NC}^k$ for some natural number $k$, that is, $\\mathsf{NC} = \\bigcup_{k} \\mathsf{NC}^k$.","title":"The class NC"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/nc-eq","kind":"node","lean":"Cslib.Complexity.Classes.nc_eq","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#nc-eq","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class $\\mathsf{NC}$ equals the union $\\bigcup_{k \\in \\mathbb{N}} \\mathsf{NC}^k$ of the classes $\\mathsf{NC}^k$ over all natural numbers $k$; this holds by definition.","title":"NC is the union of its levels"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/mem-nc","kind":"node","lean":"Cslib.Complexity.Classes.mem_nc","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#mem-nc","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A decision problem $L$ belongs to $\\mathsf{NC}$ if and only if $L$ belongs to $\\mathsf{NC}^k$ for some natural number $k$.","title":"Membership in NC"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/nck-mono","kind":"node","lean":"Cslib.Complexity.Classes.nck_mono","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#nck-mono","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For every natural number $k$, every decision problem in $\\mathsf{NC}^k$ is in $\\mathsf{NC}^{k+1}$, that is, $\\mathsf{NC}^k \\subseteq \\mathsf{NC}^{k+1}$.","title":"The NC hierarchy is increasing"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/rat-enc","kind":"node","lean":"Cslib.Complexity.Classes.RatEnc","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#rat-enc","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A rational number is encoded as a triple $(s, a, b)$ of a boolean $s$ and natural numbers $a$ and $b$, read as the sign, the numerator and the denominator.","title":"Encoded rational number"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/to-rat","kind":"node","lean":"Cslib.Complexity.Classes.toRat","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#to-rat","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A triple $(s, p, q)$, with $s$ a boolean and $p, q$ natural numbers, encodes the rational number $-p/q$ when $s$ is true and $p/q$ when $s$ is false; when $q = 0$ the value is $0$, since division by zero gives $0$.","title":"Rational number of an encoding"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/to-complex","kind":"node","lean":"Cslib.Complexity.Classes.toComplex","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#to-complex","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A pair $(e_1, e_2)$ of rational encodings encodes the complex number $q_1 + q_2 i$, where $q_1$ and $q_2$ are the rational numbers encoded by $e_1$ and $e_2$.","title":"Complex number of a pair of rational encodings"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/q-bit","kind":"node","lean":"Cslib.Complexity.Classes.qBit","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#q-bit","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For natural numbers $k$ and $i$, this is the digit of $i$ at position $k$ in binary, counting from the least significant digit at position $0$, that is, $\\lfloor i / 2^k \\rfloor \\bmod 2$, taken as an element of $\\{0, 1\\}$.","title":"Binary digit of a natural number"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/q-idx2","kind":"node","lean":"Cslib.Complexity.Classes.qIdx2","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#q-idx2","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A pair $(a, b)$ of bits $a, b \\in \\{0, 1\\}$ has index $2a + b$, its position in the enumeration $(0, 0), (0, 1), (1, 0), (1, 1)$ of the four pairs.","title":"Index of a pair of bits"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/matrix-of-entries","kind":"node","lean":"Cslib.Complexity.Classes.matrixOfEntries","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#matrix-of-entries","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Number the pairs $(a, b)$ with $a, b \\in \\{0, 1\\}$ by $2a + b$, and index the rows and columns of a $4 \\times 4$ complex matrix by these pairs. Given a list $e$ of pairs of encoded rational numbers, the matrix of $e$ has in row $p$ and column $q$ the complex number whose real part is the first and whose imaginary part is the second rational of the entry of $e$ at position $4p + q$ (counting from $0$, with $p$ and\u2026","title":"Matrix encoded by a list of entries"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/set-bits","kind":"node","lean":"Cslib.Complexity.Classes.setBits","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#set-bits","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For natural numbers $N$, $s$, $i$, $j$ and a pair $b = (b_1, b_2)$ of bits, this is the index in $\\{0, \\dots, 2^N - 1\\}$ given by $(s - s_i 2^i - s_j 2^j + b_1 2^i + b_2 2^j) \\bmod 2^N$, where $s_k$ is bit $k$ of $s$ and each subtraction of natural numbers is truncated at $0$. When $i \\ne j$ this is $s$ with bit $i$ set to $b_1$ and bit $j$ set to $b_2$, reduced modulo $2^N$.","title":"Setting two bits of an index"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/qgate","kind":"node","lean":"Cslib.Complexity.Classes.QGate","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#qgate","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A 2-local quantum gate is encoded as a triple $(i, j, e)$ of natural numbers $i$ and $j$, the qubits it acts on, and a list $e$ of pairs of encoded rational numbers, the real and imaginary parts of the entries of its $4 \\times 4$ matrix in row-major order.","title":"Encoded 2-local quantum gate"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/qgate-well-formed","kind":"node","lean":"Cslib.Complexity.Classes.QGateWellFormed","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#qgate-well-formed","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A 2-local gate acting on qubits $i$ and $j$ with entry list $e$ is well-formed on $N$ qubits when $i < j < N$ and the $4 \\times 4$ complex matrix $U$ encoded by $e$ is unitary, that is, $U^* U = I$.","title":"Well-formed quantum gate"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/apply-gate","kind":"node","lean":"Cslib.Complexity.Classes.applyGate","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#apply-gate","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $N$ be a number of qubits, and let a gate consist of two qubit positions $i$ and $j$ together with a list of entries encoding a $4 \\times 4$ complex matrix $U$ whose rows and columns are indexed by pairs of bits. For a state vector $\\psi$ on $N$ qubits, that is, a function from $\\{0, \\dots, 2^N - 1\\}$ to $\\mathbb{C}$, applying the gate gives the state whose value at a basis index $s$ is\u2026","title":"Applying a 2-local gate to a state"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/apply-circuit","kind":"node","lean":"Cslib.Complexity.Classes.applyCircuit","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#apply-circuit","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a state vector $\\psi \\in \\mathbb{C}^{2^N}$ on $N$ qubits and a list of 2-local gates $g_1, \\dots, g_k$, the result of applying the circuit to $\\psi$ is obtained by applying the gate $g_1$ to $\\psi$, then $g_2$ to the result, and so on up to $g_k$. For the empty list it is $\\psi$ itself.","title":"Applying a quantum circuit to a state"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/q-num-of","kind":"node","lean":"Cslib.Complexity.Classes.qNumOf","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#q-num-of","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A list $b_0, b_1, \\ldots, b_{n-1}$ of booleans, read as bits with true as $1$ and false as $0$, denotes the natural number $\\sum_{i < n} b_i 2^i$; the first entry is the least significant bit.","title":"Number with a given binary expansion"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/basis-state","kind":"node","lean":"Cslib.Complexity.Classes.basisState","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#basis-state","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a bitstring $x = (x_0, \\dots, x_{n-1})$, reading true as $1$ and false as $0$, the computational basis state of $x$ is the vector $\\psi : \\{0, \\dots, 2^n - 1\\} \\to \\mathbb{C}$ with $\\psi(s) = 1$ if $s = \\sum_{k < n} x_k 2^k$ (the number whose binary expansion is $x$, least significant bit first) and $\\psi(s) = 0$ otherwise.","title":"Computational basis state of a bitstring"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/tensor-prod","kind":"node","lean":"Cslib.Complexity.Classes.tensorProd","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#tensor-prod","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For vectors $\\varphi \\in \\mathbb{C}^{2^a}$ and $\\psi \\in \\mathbb{C}^{2^b}$, indexed by $\\{0, \\dots, 2^a - 1\\}$ and $\\{0, \\dots, 2^b - 1\\}$, their tensor product is the vector in $\\mathbb{C}^{2^{a+b}}$ whose entry at $s$ is $\\varphi_{s \\bmod 2^a}\\, \\psi_{\\lfloor s / 2^a \\rfloor}$. Thus $\\varphi$ occupies the low-order $a$ bits of the index and $\\psi$ the high-order $b$ bits.","title":"Tensor product of state vectors"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/zero-state","kind":"node","lean":"Cslib.Complexity.Classes.zeroState","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#zero-state","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The all-zero state on $a$ qubits is the vector in $\\mathbb{C}^{2^a}$, indexed by $s \\in \\{0, \\dots, 2^a - 1\\}$, whose entry at $s = 0$ is $1$ and whose other entries are $0$; that is, the computational basis state $|0 \\cdots 0\\rangle$.","title":"All-zero state"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/sq-norm","kind":"node","lean":"Cslib.Complexity.Classes.sqNorm","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#sq-norm","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a vector $\\psi \\in \\mathbb{C}^{2^m}$, indexed by $\\{0, \\dots, 2^m - 1\\}$, its squared norm is the real number $\\sum_{s} \\overline{\\psi_s}\\, \\psi_s = \\sum_{s} |\\psi_s|^2$.","title":"Squared norm of a state vector"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/accept-prob","kind":"node","lean":"Cslib.Complexity.Classes.acceptProb","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#accept-prob","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a state vector $\\psi$ on $N$ qubits, given by its amplitudes $\\psi(s) \\in \\mathbb{C}$ at the basis states $s \\in \\{0, \\dots, 2^N - 1\\}$, the acceptance probability is the sum of $|\\psi(s)|^2$ over the basis states $s$ whose qubit $0$ (the output qubit, the least significant bit of $s$) equals $1$. By the Born rule, for a unit vector $\\psi$ this is the probability that measuring the output qubit gives $1$.","title":"Acceptance probability of a quantum state"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/bqp-init-state","kind":"node","lean":"Cslib.Complexity.Classes.bqpInitState","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#bqp-init-state","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"For a bitstring $x = (x_0, \\dots, x_{n-1})$, with true read as $1$ and false as $0$, and a number $a$ of ancilla qubits, the initial state is the tensor product of the computational basis state of $x$ and the all-zero state on $a$ qubits: the vector on $\\{0, \\dots, 2^{n+a} - 1\\}$ whose value at $s$ is the value of the basis state of $x$ at $s \\bmod 2^n$ times the value of the all-zero state at\u2026","title":"Initial state of a BQP computation"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/bqp","kind":"node","lean":"Cslib.Complexity.Classes.BQP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#bqp","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The class BQP of bounded-error quantum polynomial-time decision problems consists of the decision problems $L$ for which there are polynomials $a$ and $q$ with natural number coefficients and a polynomial-time computable map sending each bitstring $x$ to a list of 2-local quantum gates such that, for every bitstring $x$, with $n = |x|$ and $N = n + a(n)$: the list has at most $q(n)$ gates; each gate acts on qubits\u2026","title":"BQP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/opt-problem","kind":"node","lean":"Cslib.Complexity.Classes.OptProblem","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#opt-problem","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"An optimization problem consists of a function $v$ assigning a rational value $v(x, s)$ to every instance $x$ and every candidate solution $s$, both bitstrings; a feasibility predicate saying whether $s$ is a feasible solution for $x$; and a polynomial $p$ with natural-number coefficients such that every feasible solution $s$ for $x$ has length at most $p(|x|)$. It is required that $v(x, s) \\ge 0$ for all $x$ and\u2026","title":"Optimization problems"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/approx-at","kind":"node","lean":"Cslib.Complexity.Classes.ApproxAt","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#approx-at","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $P$ be a maximization problem with objective value $\\mathrm{val}(x, s)$ for instance $x$ and solution $s$, let $A$ be an algorithm taking a pair of bitstrings, let $k$ be a natural number and let $\\rho$ be a rational number. Then $A$ achieves ratio $\\rho$ on $P$ at precision $k$ when, for every instance $x$ and every feasible solution $s$ of $x$, the output $y = A(x, 1^k)$ of $A$ on $x$ and the unary string\u2026","title":"Approximation ratio at a precision"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-npo","kind":"node","lean":"Cslib.Complexity.Classes.IsNPO","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-npo","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $P$ be an optimization problem whose instances $x$ and solutions $s$ are bitstrings, with rational value $\\mathrm{val}(x, s)$ and feasibility predicate $\\mathrm{feas}(x, s)$. Then $P$ is an NPO problem when the map $(x, s) \\mapsto \\mathrm{feas}(x, s)$ is polynomial-time computable, and there is a polynomial-time computable map $\\mathrm{enc}$ sending each pair $(x, s)$ to a triple $(b, a, d)$ of a boolean and\u2026","title":"NPO optimization problems"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-apx","kind":"node","lean":"Cslib.Complexity.Classes.IsAPX","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-apx","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A maximization problem $P$ is in $\\mathsf{APX}$ when $P$ is an NPO problem and there are a rational number $\\rho$ with $0 < \\rho \\le 1$ and a polynomial-time algorithm $A$ such that, for every instance $x$ and every feasible solution $s$ for $x$, the output $A(x, 1^2)$ of $A$ on $x$ with the unary precision tape of length $2$ is a feasible solution for $x$ whose value is at least $\\rho$ times the value of $s$.","title":"Constant-factor approximable problems"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/apx","kind":"node","lean":"Cslib.Complexity.Classes.APX","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#apx","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"APX is the set of maximization problems $P$ that are NP optimization problems (feasibility is decidable in polynomial time, and a polynomial-time computable map sends every instance-solution pair to a signed fraction with nonzero denominator that equals the objective value whenever the solution is feasible) and for which there are a rational number $\\rho$ with $0 < \\rho \\le 1$ and a polynomial-time algorithm $A$\u2026","title":"The class APX"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-ptas","kind":"node","lean":"Cslib.Complexity.Classes.IsPTAS","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-ptas","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"An optimization problem $P$, with value $\\mathrm{val}(x, s)$ of a solution $s$ for an instance $x$, admits a polynomial-time approximation scheme when $P$ is an NPO problem and for every integer $k \\ge 1$ there is a polynomial-time computable algorithm $A$ on pairs of bitstrings with the following property: for every instance $x$ and every feasible solution $s$ for $x$, the output $A(x, 1^k)$ on $x$ and the string\u2026","title":"Polynomial-time approximation schemes"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ptas","kind":"node","lean":"Cslib.Complexity.Classes.PTAS","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ptas","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"PTAS is the set of optimization problems $P$ that are NPO problems (feasibility is decidable in polynomial time and the objective value is given by a polynomial-time computable rational encoding) and that have a polynomial-time approximation scheme: for every natural number $k \\ge 1$ there is a polynomial-time algorithm $A$ such that, for every instance $x$ and every feasible solution $s$ of $x$, the output of $A$\u2026","title":"The class PTAS"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-fptas","kind":"node","lean":"Cslib.Complexity.Classes.IsFPTAS","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-fptas","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A maximization problem $P$ has an FPTAS when $P$ is an NPO problem and there is one polynomial-time algorithm $A$ such that, for every integer $k \\ge 1$, every instance $x$ and every feasible solution $s$ for $x$, the output $A(x, 1^k)$ of $A$ on $x$ with the unary precision tape of length $k$ is a feasible solution for $x$ whose value is at least $1 - 1/k$ times the value of $s$.","title":"Fully polynomial-time approximation schemes"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/fptas","kind":"node","lean":"Cslib.Complexity.Classes.FPTAS","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#fptas","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"FPTAS is the class of optimization problems that have a fully polynomial-time approximation scheme.","title":"The class FPTAS"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/fptas-subset-ptas","kind":"node","lean":"Cslib.Complexity.Classes.fptas_subset_ptas","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#fptas-subset-ptas","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Every optimization problem in the class FPTAS of problems with a fully polynomial-time approximation scheme belongs to the class PTAS of problems with a polynomial-time approximation scheme.","title":"FPTAS is contained in PTAS"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/ptas-subset-apx","kind":"node","lean":"Cslib.Complexity.Classes.ptas_subset_apx","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#ptas-subset-apx","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Every optimization problem in the class PTAS belongs to the class APX.","title":"PTAS is contained in APX"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/negligible","kind":"node","lean":"Cslib.Complexity.Classes.Negligible","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#negligible","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A function $e : \\mathbb{N} \\to \\mathbb{Q}$ is negligible if for every natural number $c$ there is a natural number $N$ such that $e(n) \\le 1 / n^c$ for all natural numbers $n \\ge N$.","title":"Negligible functions"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/invert-succ","kind":"node","lean":"Cslib.Complexity.Classes.invertSucc","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#invert-succ","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $f$ be a family of functions, $f_n$ mapping $n$-bit strings to bitstrings, and let $A$ be an inverter mapping a triple $(n, y, r)$ of a length $n$, an output $y$ and a coin string $r$ to a bitstring. For natural numbers $n$ and $m$, the inversion success probability of $A$ against $f$ at length $n$ with $m$ coins is the fraction of the $2^{n+m}$ pairs $(x, r)$ of an $n$-bit string $x$ and an $m$-bit string $r$\u2026","title":"Inversion success probability"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-one-way-function","kind":"node","lean":"Cslib.Complexity.Classes.IsOneWayFunction","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-one-way-function","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $f$ assign to each $n \\in \\mathbb{N}$ and each $n$-bit string $x$ a bitstring $f_n(x)$. The family $f$ is one-way when three conditions hold. First, its output lengths are polynomially bounded: some polynomial $p$ satisfies $|f_n(x)| \\le p(n)$ for all $n$ and $x$. Second, it is uniformly computable: some polynomial-time computable function maps the bitstring encoding of the pair $(n, x)$ to $f_n(x)$ for all $n$\u2026","title":"One-way function families"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/perm-invert-succ","kind":"node","lean":"Cslib.Complexity.Classes.permInvertSucc","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#perm-invert-succ","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $\\pi$ be a family of maps $\\pi_n$ from $n$-bit strings to $n$-bit strings, let $A$ be a function sending a natural number and two bitstrings to a bitstring, and let $n$ and $m$ be natural numbers. For a bitstring $w$, write $\\mathrm{trunc}_n(w)$ for the $n$-bit string whose bit at each position $i < n$ is the entry of $w$ at position $i$, or false if $w$ has at most $i$ entries. The success probability of $A$\u2026","title":"Inversion success probability against a permutation family"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/is-one-way-permutation","kind":"node","lean":"Cslib.Complexity.Classes.IsOneWayPermutation","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#is-one-way-permutation","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $\\pi$ assign to each $n \\in \\mathbb{N}$ a map $\\pi_n$ from $n$-bit strings to $n$-bit strings. The family $\\pi$ is a one-way permutation when three conditions hold. First, each $\\pi_n$ is injective, hence a permutation of the $n$-bit strings. Second, it is uniformly computable: some polynomial-time computable function maps the bitstring encoding of the pair $(n, x)$ to $\\pi_n(x)$ for all $n$ and all $n$-bit\u2026","title":"One-way permutation families"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/one-way-permutation-implies-one-way-func","kind":"node","lean":"Cslib.Complexity.Classes.one_way_permutation_implies_one_way_function","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#one-way-permutation-implies-one-way-func","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Let $\\pi$ be a one-way permutation family, so that each $\\pi_n$ maps $n$-bit strings to $n$-bit strings. Then the family that sends an $n$-bit string $x$ to the bitstring $\\pi_n(x)$ is a one-way function family.","title":"One-way permutations are one-way functions"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/output-length-le","kind":"node","lean":"Cslib.Complexity.Preimage.output_length_le","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#output-length-le","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"A polynomial-time machine has polynomially bounded encoded output.","title":"Polynomial-time machines have polynomially bounded output"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/raw-computable","kind":"node","lean":"Cslib.Complexity.Preimage.raw_computable","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#raw-computable","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Canonical polynomial-time maps can also be run on unframed bit lists.","title":"Polynomial-time maps run on unframed bit lists"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/map-fst","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.IsPolyTime.map_fst","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#map-fst","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Mapping the first component of an encoded pair is polynomial-time; the second component is preserved, not recomputed or guessed.","title":"Mapping the first component of a pair is polynomial-time"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/length-bound","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.IsPolyTime.length_bound","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#length-bound","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Encoded output length is polynomial in the unframed input length.","title":"Output of a polynomial-time map is polynomial in raw input length"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/sat-dec-comp-mem-np","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.satDec_comp_mem_NP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#sat-dec-comp-mem-np","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Pulling SAT back along a polynomial map preserves NP membership. SAT verifier soundness holds even for certificates above its length bound.","title":"Preimage of SAT under a polynomial-time map is in NP"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/comp-mem-np","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.comp_mem_NP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#comp-mem-np","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Canonical NP is closed under polynomial-time preimages. Cook\u2013Levin supplies a verifier sound for arbitrarily long certificates, so enlarging the transformed witness bound introduces no false positives.","title":"NP is closed under polynomial-time preimages"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/mem-np-of-poly-many-one-red","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.mem_NP_of_polyManyOneRed","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#mem-np-of-poly-many-one-red","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"NP membership transfers backwards along a polynomial many-one reduction.","title":"NP membership pulls back along polynomial many-one reductions"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/comp-mem-co-np","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.comp_mem_coNP","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#comp-mem-co-np","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"Canonical coNP is closed under polynomial-time preimages.","title":"coNP is closed under polynomial-time preimages"},{"id":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/compl","kind":"node","lean":"Cslib.CookLevin.PolyManyOneRed.compl","parent":32979,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-space-complexity-classes/index.html#compl","scope":"TheoryOfComputation/complexity-classes/time-and-space-complexity-classes","status":"fully_proved","summary":"The same reduction function reduces the complements.","title":"Polynomial many-one reductions also reduce complements"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes.json","id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"can_prove","title":"Containments between time and space classes"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/halt-time-lt-core-bound","kind":"node","lean":"Cslib.Turing.MultiTapeTM.haltTime_lt_coreBound","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#halt-time-lt-core-bound","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"fully_proved","summary":"The first halting time is smaller than the canonical bound on reachable cores.","title":"First halting time is below the configuration-core bound"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/runs-in-time-core-bound","kind":"node","lean":"Cslib.Turing.MultiTapeTM.runsInTime_coreBound","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#runs-in-time-core-bound","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"fully_proved","summary":"A halting, space-bounded run has an actual step bound.","title":"Halting space-bounded runs obey the core-count time bound"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/runs-in-time-exponential-of-space","kind":"node","lean":"Cslib.Turing.MultiTapeTM.runsInTime_exponential_of_space","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#runs-in-time-exponential-of-space","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"fully_proved","summary":"Exponential time from a visited-cell space bound and a halting run.","title":"Exponential time bound from a space bound and halting"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/exists-exponential-time-bound","kind":"node","lean":"Cslib.Turing.MultiTapeTM.exists_exponential_time_bound","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#exists-exponential-time-bound","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"fully_proved","summary":"One pair of constants works for every input of the fixed machine.","title":"Uniform constants for the exponential time-from-space bound"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/exists-exponential-time-bounded-by","kind":"node","lean":"Cslib.Turing.MultiTapeTM.SpaceBoundedBy.exists_exponential_timeBoundedBy","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#exists-exponential-time-bounded-by","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"fully_proved","summary":"The same finite polynomial-space decider has an exponential time bound. All constants precede every input. Halting comes from `Decides`, not space.","title":"Polynomial-space decider runs in exponential time"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/pspace-subset-exp","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PSPACESubsetEXP.pspace_subset_exp","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#pspace-subset-exp","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"fully_proved","summary":"Every canonical polynomial-space language is in exponential time, using its original finite multi-tape decider.","title":"PSPACE is contained in EXP"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/nexp-subset-expspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NEXPSubsetEXPSPACE.nexp_subset_expspace","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#nexp-subset-expspace","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"can_prove","summary":"Every decision problem in NEXP, the class of nondeterministic exponential time, also belongs to EXPSPACE, the class of deterministic exponential space.","title":"NEXP is contained in EXPSPACE"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/nl-subset-p","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NLSubsetP.nl_subset_p","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#nl-subset-p","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"can_prove","summary":"Every decision problem in NL, the class of nondeterministic logarithmic space, also belongs to P, the class of decision problems computable in deterministic polynomial time.","title":"NL is contained in P"},{"id":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/np-subset-pspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NPSubsetPSPACE.np_subset_pspace","parent":33097,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes/index.html#np-subset-pspace","scope":"TheoryOfComputation/complexity-classes/containments-between-time-and-space-classes","status":"can_prove","summary":"Every decision problem in NP also belongs to PSPACE, the class of decision problems decidable in polynomial space.","title":"NP is contained in PSPACE"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/parameterized-complexity.json","id":"TheoryOfComputation/complexity-classes/parameterized-complexity","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"can_prove","title":"Parameterized complexity"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/param-problem","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.ParamProblem","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#param-problem","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"A parameterized problem is a function that assigns a boolean answer to every pair of a bitstring (a finite list of booleans) $x$ and a natural-number parameter $k$.","title":"Parameterized problems"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/fpttime","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.FPTTime","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#fpttime","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"Let $\\beta$ be a type with a canonical bitstring encoding. A function $R$ from pairs $(x, k)$ of a bitstring $x$ and a natural number $k$ to $\\beta$ is computable in FPT time when there are a computable function $f : \\mathbb{N} \\to \\mathbb{N}$ and a polynomial-time computable function $A$ from pairs of bitstrings to $\\beta$ such that $R(x, k) = A(x, 1^{f(k)})$ for every bitstring $x$ and every $k \\in \\mathbb{N}$,\u2026","title":"FPT time"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/fpt","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.FPT","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#fpt","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"A parameterized problem $P$ is fixed-parameter tractable when there is a function $D$, from pairs $(x, k)$ of a bitstring $x$ and a natural number $k$ to booleans, that is computable in FPT time and satisfies $P(x, k) = D(x, k)$ for every bitstring $x$ and every $k \\in \\mathbb{N}$.","title":"Fixed-parameter tractable problems"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/fptreduces","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.FPTReduces","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#fptreduces","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"A parameterized problem $P$ FPT-reduces to a parameterized problem $Q$ when there are a map $R$, sending each pair $(x, k)$ of a bitstring $x$ and a natural number $k$ to such a pair $R(x, k) = (x', k')$, that is computable in FPT time, and a computable function $g : \\mathbb{N} \\to \\mathbb{N}$, such that for every bitstring $x$ and every $k \\in \\mathbb{N}$, $k' \\le g(k)$ and $P(x, k)$ is true if and only if\u2026","title":"FPT-reductions"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wcgate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.WCGate","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wcgate","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"An encoded gate is a pair $(k, r)$ of a natural number $k$, its kind, and a list $r$ of natural numbers, its references.","title":"Encoded gates for weighted circuit satisfiability"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wc-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.wcCircuitEval","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wc-circuit-eval","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"Let $C = (g_0, \\dots, g_{m-1})$ be a list of gates encoded as (kind, references) pairs and let $a = (a_0, \\dots, a_{n-1})$ be a list of booleans. The gate values of $C$ under $a$ are the values $(v_0, \\dots, v_{m-1})$ of the flat Boolean circuit on $n$ inputs obtained by decoding each $g_j$ to a gate on $n$ inputs, evaluated at the assignment that gives input $i$ the value $a_i$.","title":"Gate values of an encoded circuit"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wc-gate-preds","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.wcGatePreds","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wc-gate-preds","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"The predecessor list of an encoded gate $(k, r)$ is $r$ when $k$ is $2$, $3$ or $4$ (a NOT, AND or OR gate), and the empty list for every other kind.","title":"Predecessors of an encoded gate"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wc-gate-weft","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.wcGateWeft","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wc-gate-weft","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"Given a list $w$ of wefts of earlier gates, where $w_a$ reads as $0$ when $a$ is not a position of $w$, the weft of an encoded gate $(k, r)$ is $1$ if it is a large AND or OR gate ($k \\in \\{3, 4\\}$ and $r$ has more than $2$ entries) and $0$ otherwise, plus the maximum of $w_a$ over the entries $a$ of its predecessor list (the maximum over an empty list is $0$). In particular, input and constant gates have weft $0$.","title":"Weft of an encoded gate"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wc-gate-depth","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.wcGateDepth","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wc-gate-depth","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"Let $d = (d_0, d_1, \\dots)$ be a list of natural numbers (the depths of earlier gates), where $d_a$ is read as $0$ when $a$ is out of range, and let $(c, r)$ be a gate encoded as a (kind, references) pair. Its depth is $0$ if $c = 0$ or $c = 1$ (an input or constant gate), and otherwise one plus the maximum of $d_a$ over the entries $a$ of $r$, the maximum over no entries being $0$.","title":"Depth of one encoded gate"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wc-circuit-wefts","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.wcCircuitWefts","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wc-circuit-wefts","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"For a list $C = (g_0, \\dots, g_{m-1})$ of gates encoded as (kind, references) pairs, the list of wefts is $(w_0, \\dots, w_{m-1})$, where $w_j$ is computed from $w_0, \\dots, w_{j-1}$ as follows. Write $g_j = (c, r)$. The weft $w_j$ is $1$ if $c = 3$ or $c = 4$ (an AND or OR gate) and $r$ has more than $2$ entries, and $0$ otherwise, plus, when $c = 2$, $c = 3$ or $c = 4$, the maximum of $w_a$ over the entries $a$ of\u2026","title":"Wefts of the gates of an encoded circuit"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wc-circuit-depths","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.wcCircuitDepths","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wc-circuit-depths","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"For a list $C = (g_0, \\dots, g_{m-1})$ of gates encoded as (kind, references) pairs, the list of depths is $(d_0, \\dots, d_{m-1})$, where $d_j$ is computed from $d_0, \\dots, d_{j-1}$ as follows. Write $g_j = (c, r)$. If $c = 0$ or $c = 1$ (an input or constant gate), then $d_j = 0$; otherwise $d_j$ is one plus the maximum of $d_a$ over the entries $a$ of $r$, where $d_a$ is read as $0$ for $a \\ge j$ and the maximum\u2026","title":"Depths of the gates of an encoded circuit"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wcwell-formed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.WCWellFormed","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wcwell-formed","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"A list $C = (g_0, \\dots, g_{m-1})$ of encoded gates is well-formed when every NOT, AND or OR gate references only strictly earlier gates: for every $j < m$ such that $g_j = (k, r)$ has kind $k \\in \\{2, 3, 4\\}$, every entry $a$ of $r$ satisfies $a < j$.","title":"Well-formed encoded circuits"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/weighted-circuit-sat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.WeightedCircuitSat","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#weighted-circuit-sat","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"For natural numbers $t$ and $d$, weighted circuit satisfiability is the parameterized problem that, on a bitstring $x$ and a parameter $k$, answers true exactly when $x$ decodes to a pair $(n, C)$ of a natural number $n$ and a list $C$ of encoded gates such that: every NOT, AND or OR gate of $C$ references only strictly earlier gates; every gate of $C$ has weft at most $t$ and depth at most $d$; and there is an\u2026","title":"Weighted circuit satisfiability with bounded weft and depth"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/clique-param","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.CliqueParam","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#clique-param","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"The parameterized $k$-Clique problem answers a bitstring $x$ and a parameter $k \\in \\mathbb{N}$ as follows. If $x$ decodes, under the canonical bitstring encoding, to a pair $(n, E)$ of a natural number $n$ and a list $E$ of pairs of natural numbers, the answer is true exactly when there is a list of $k$ distinct natural numbers, each less than $n$, such that for any two distinct entries $u$ and $v$ of the list,\u2026","title":"Parameterized Clique"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/wlevel","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.WLevel","parent":33107,"reading_title":{"format":"inline-parts-v1","parts":[["text","The level "],["math","\\(\\mathrm{W}(t)\\)"],["text"," of the W-hierarchy"]],"source":"The level $\\mathrm{W}(t)$ of the W-hierarchy"},"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#wlevel","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"fully_proved","summary":"For a natural number $t$, $\\mathrm{W}[t]$ is the set of parameterized problems $P$ for which there is a natural number $d$ such that $P$ FPT-reduces to weighted circuit satisfiability for circuits of weft at most $t$ and depth at most $d$.","title":"The level $\\mathrm{W}(t)$ of the W-hierarchy"},{"id":"TheoryOfComputation/complexity-classes/parameterized-complexity/parameterized-clique-w1-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Parameterized.parameterized_clique_w1_complete","parent":33107,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/parameterized-complexity/index.html#parameterized-clique-w1-complete","scope":"TheoryOfComputation/complexity-classes/parameterized-complexity","status":"can_prove","summary":"Write $W[t]$ for the $t$-th level of the W-hierarchy. The parameterized $k$-Clique problem lies in $W[1]$, and every parameterized problem in $W[1]$ has an FPT-reduction to it. Every fixed-parameter tractable parameterized problem lies in $W[1]$. For every $t \\in \\mathbb{N}$, $W[t] \\subseteq W[t + 1]$.","title":"Parameterized complexity: FPT and W(1)-completeness of Clique"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/polynomial-hierarchy.json","id":"TheoryOfComputation/complexity-classes/polynomial-hierarchy","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-hierarchy/index.html","scope":"TheoryOfComputation/complexity-classes/polynomial-hierarchy","status":"can_prove","title":"Polynomial hierarchy"},{"id":"TheoryOfComputation/complexity-classes/polynomial-hierarchy/ph-collapses","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PHCollapse.ph_collapses","parent":33124,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-hierarchy/index.html#ph-collapses","scope":"TheoryOfComputation/complexity-classes/polynomial-hierarchy","status":"can_prove","summary":"If the complexity classes P and NP are equal, then the polynomial hierarchy PH, the union of the classes $\\Sigma_k^p$ over all natural numbers $k$, is equal to P.","title":"If P = NP then the polynomial hierarchy collapses to P"},{"id":"TheoryOfComputation/complexity-classes/polynomial-hierarchy/ph-subset-pspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PHSubsetPSPACE.ph_subset_pspace","parent":33124,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/polynomial-hierarchy/index.html#ph-subset-pspace","scope":"TheoryOfComputation/complexity-classes/polynomial-hierarchy","status":"can_prove","summary":"Every decision problem in the polynomial hierarchy PH belongs to PSPACE.","title":"The polynomial hierarchy is contained in PSPACE"},{"data_url":"scopes/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi.json","id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","kind":"scope","navigation":"module","parent":32887,"result":false,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","title":"Time- and resource-bounded runs of abstract machines"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/next","kind":"node","lean":"Cslib.OutputReachability.next","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#next","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"One iteration, preserving a state once its output is available.","title":"One iteration step with absorbing halted states"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/bounded-step","kind":"node","lean":"Cslib.OutputReachability.boundedStep","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#bounded-step","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"An active step with a resource bound on its source.","title":"Resource-bounded active step"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/reaches-output","kind":"node","lean":"Cslib.OutputReachability.ReachesOutput","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#reaches-output","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Reach an output in at most `T` active steps, with every state bounded by `S`.","title":"Output reachability within time and resource bounds"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/runs","kind":"node","lean":"Cslib.OutputReachability.Runs","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#runs","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Reach an output starting from an input's initial state.","title":"Bounded run from an input's initial state"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/iterate-of-some","kind":"node","lean":"Cslib.OutputReachability.iterate_of_some","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#iterate-of-some","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Every iterate of an already halted state is that state.","title":"Iterates of a halted state are constant"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/halt","kind":"node","lean":"Cslib.OutputReachability.ReachesOutput.halt","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#halt","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"A halted state reaches its output with any remaining time budget.","title":"A halted state reaches its output with any time budget"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/prepend","kind":"node","lean":"Cslib.OutputReachability.ReachesOutput.prepend","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#prepend","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Prepend an active bounded step to a successful computation.","title":"Prepending a bounded step to a successful computation"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/relates-in-steps-iterate","kind":"node","lean":"Cslib.OutputReachability.relatesInSteps_iterate","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#relates-in-steps-iterate","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"A canonical finite path ending in an output has the source's all-iterate bounds.","title":"Canonical finite path to an output bounds all iterates"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/iterate","kind":"node","lean":"Cslib.OutputReachability.ReachesOutput.iterate","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#iterate","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Reachability gives stable outputs after the budget and bounds every iterate.","title":"Output reachability gives stable outputs and bounded iterates"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/reaches-output-of-iterate","kind":"node","lean":"Cslib.OutputReachability.reachesOutput_of_iterate","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#reaches-output-of-iterate","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"The source iteration condition constructs a canonical bounded path.","title":"Output reachability from the iteration condition"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/reaches-output-iff-iterate","kind":"node","lean":"Cslib.OutputReachability.reachesOutput_iff_iterate","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#reaches-output-iff-iterate","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Exact source characterization, including bounds on all natural-number iterates.","title":"Characterization of output reachability by iterates"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/iterate-stable","kind":"node","lean":"Cslib.OutputReachability.iterate_stable","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#iterate-stable","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"A terminal iterate remains unchanged at every later iteration count.","title":"Terminal iterates are stable"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/reaches-output-iff-bounded-iterate","kind":"node","lean":"Cslib.OutputReachability.reachesOutput_iff_bounded_iterate","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#reaches-output-iff-bounded-iterate","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Finite visited-state bounds are equivalent to all-iterate bounds after halting.","title":"Output reachability via bounded iterates after halting"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/mono","kind":"node","lean":"Cslib.OutputReachability.ReachesOutput.mono","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#mono","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Increasing either bound preserves reachability.","title":"Monotonicity of output reachability in the bounds"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/unique","kind":"node","lean":"Cslib.OutputReachability.ReachesOutput.unique","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#unique","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Output values reached from the same state are unique, regardless of bounds.","title":"Uniqueness of reached outputs"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/run","kind":"node","lean":"Cslib.OutputReachability.run","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#run","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Iteration initialized by an input, with absorbing halting.","title":"Input-initialized iteration with absorbing halting"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/runs-iff-iterate","kind":"node","lean":"Cslib.OutputReachability.runs_iff_iterate","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#runs-iff-iterate","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"The source `Runs` interface, stated using initialized iteration.","title":"Characterization of bounded runs by initialized iteration"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/out-run","kind":"node","lean":"Cslib.OutputReachability.Runs.out_run","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#out-run","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"The output of a successful initialized run is present at every later time.","title":"The output of a successful run persists"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/mono-2","kind":"node","lean":"Cslib.OutputReachability.Runs.mono","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#mono-2","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Increasing time and resource bounds preserves initialized runs.","title":"Monotonicity of bounded runs in time and resources"},{"id":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/unique-2","kind":"node","lean":"Cslib.OutputReachability.Runs.unique","parent":33127,"result":true,"route":"../roadmap/TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi/index.html#unique-2","scope":"TheoryOfComputation/complexity-classes/time-and-resource-bounded-runs-of-abstract-machi","status":"fully_proved","summary":"Initialized runs have unique outputs.","title":"Uniqueness of run outputs"},{"data_url":"scopes/TheoryOfComputation/hierarchy-theorems-and-barriers.json","id":"TheoryOfComputation/hierarchy-theorems-and-barriers","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/index.html","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers","status":"can_prove","summary":"Time and space hierarchy theorems and diagonalization against time-bounded machines, separations such as P \u2260 EXP, and barriers to separating classes: relativization (Baker\u2013Gill\u2013Solovay, oracle separations), natural proofs and algebrization.","title":"Hierarchy theorems and proof barriers"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/nondet-time-hierarchy","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NondetTimeHierarchy.nondet_time_hierarchy","parent":33148,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/index.html#nondet-time-hierarchy","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers","status":"can_prove","summary":"Source: S. A. Cook, \"A hierarchy for nondeterministic time complexity\", Journal of Computer and System Sciences 7 (1973), 343-353; S. Zak, \"A Turing machine time hierarchy\", Theoretical Computer Science 26 (1983), 327-333.","title":"Nondeterministic time hierarchy"},{"data_url":"scopes/TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier.json","id":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier","kind":"scope","navigation":"module","parent":33148,"result":false,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/index.html","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier","status":"can_prove","title":"Algebrization barrier"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/multilinear-ext","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Algebrization.multilinearExt","parent":33150,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/index.html#multilinear-ext","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier","status":"fully_proved","summary":"Identify the booleans true and false with $1$ and $0$. Let $F$ be a field, $n$ a natural number and $f : \\{0,1\\}^n \\to \\{0,1\\}$. The multilinear extension of $f$ at a point $x = (x_0, \\dots, x_{n-1}) \\in F^n$ is","title":"Multilinear extension of a Boolean function"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/base-at","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Algebrization.baseAt","parent":33150,"reading_title":{"format":"inline-parts-v1","parts":[["text","Restriction of an oracle to inputs of length "],["math","\\(n\\)"]],"source":"Restriction of an oracle to inputs of length $n$"},"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/index.html#base-at","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier","status":"fully_proved","summary":"For a function $B$ from bitstrings to booleans and a natural number $n$, the restriction of $B$ to inputs of length $n$ is the function on $\\{0,1\\}^n$ sending $a = (a_0, \\dots, a_{n-1})$ to $B(a_0 a_1 \\cdots a_{n-1})$, the value of $B$ at the bitstring $a$.","title":"Restriction of an oracle to inputs of length $n$"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/poly-bits","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Algebrization.polyBits","parent":33150,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coefficient bitstring of a polynomial over "],["math","\\(\\mathbb{F}_2\\)"]],"source":"Coefficient bitstring of a polynomial over $\\mathbb{F}_2$"},"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/index.html#poly-bits","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier","status":"fully_proved","summary":"For a polynomial $q$ over $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$ and a natural number $n$, the coefficient bitstring of $q$ with parameter $n$ is the bitstring $b_0 b_1 \\cdots b_{n+1}$ of length $n + 2$ in which $b_i$ is true exactly when the coefficient of $X^i$ in $q$ equals $1$.","title":"Coefficient bitstring of a polynomial over $\\mathbb{F}_2$"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/ext-answer","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Algebrization.extAnswer","parent":33150,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/index.html#ext-answer","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier","status":"fully_proved","summary":"Let $n$ be a natural number, let $p$ be a monic irreducible polynomial of degree $n + 1$ over $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$, let $F = \\mathbb{F}_2[X]/(p)$ with $\\alpha$ the class of $X$, let $B$ be a function from bitstrings to booleans, and let $\\tilde B_n$ be the multilinear extension over $F$ of the restriction of $B$ to inputs of length $n$. Given a list $(e_0, e_1, \\ldots)$ of bitstrings and an index\u2026","title":"Answer bits of the algebraic extension oracle"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/is-algebraic-extension","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Algebrization.IsAlgebraicExtension","parent":33150,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/index.html#is-algebraic-extension","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier","status":"fully_proved","summary":"Let $B$ and $O$ be oracles, that is, functions from bitstrings to booleans. Then $O$ is an algebraic extension of $B$ when there are polynomials $\\pi_0, \\pi_1, \\pi_2, \\ldots$ over $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$, each $\\pi_n$ monic and irreducible of degree $n + 1$, with the following two properties, where $F_n$ is the field $\\mathbb{F}_2[X]/(\\pi_n)$, $\\alpha_n$ is the class of $X$ in $F_n$, and\u2026","title":"Algebraic extension oracles"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/aaronson-wigderson-algebrization","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Algebrization.aaronson_wigderson_algebrization","parent":33150,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier/index.html#aaronson-wigderson-algebrization","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/algebrization-barrier","status":"can_prove","summary":"Call an oracle a function from bitstrings to booleans. There are oracles $B$ and $\\tilde B$ such that $\\tilde B$ is an algebraic extension of $B$ and $\\mathsf{P}^B = \\mathsf{NP}^{\\tilde B}$. There are also oracles $B$ and $\\tilde B$ such that $\\tilde B$ is an algebraic extension of $B$ and $\\mathsf{NP}^B$ is not contained in $\\mathsf{P}^{\\tilde B}$.","title":"Aaronson-Wigderson: the algebrization barrier"},{"data_url":"scopes/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier.json","id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","kind":"scope","navigation":"module","parent":33148,"result":false,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"can_prove","title":"Natural proofs barrier"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/npgate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.NPGate","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#npgate","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"fully_proved","summary":"An encoded gate of a flat Boolean circuit is a pair $(k, r)$ of a natural number $k$ (its kind) and a list $r$ of natural numbers (its references), in the shared encoding of gates as (kind, references) pairs.","title":"Encoded gates"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/np-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.npCircuitEval","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#np-circuit-eval","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"fully_proved","summary":"Given a list of encoded gates $g_0, \\dots, g_{m-1}$, each a (kind, references) pair, and a list of Booleans $a = (a_0, \\dots, a_{n-1})$, this is the list of the values of all $m$ gates, in order: each pair is decoded to a gate of a flat Boolean circuit on $n$ inputs, and the resulting circuit is evaluated with input $i$ set to $a_i$. An input pair $(0, [i])$ with $i \\ge n$ decodes to the constant false, so it reads\u2026","title":"Evaluation of an encoded circuit"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/truth-table","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.truthTable","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#truth-table","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"fully_proved","summary":"The truth table of a Boolean function $g$ on $n$ bits is the list of the $2^n$ values $g(x^{(0)}), g(x^{(1)}), \\dots, g(x^{(2^n - 1)})$, where for $i < 2^n$ the input $x^{(i)}$ has bit $j$ equal to the $j$-th binary digit $\\lfloor i / 2^j \\rfloor \\bmod 2$ of $i$ (bit $0$ being the least significant), for every $j < n$.","title":"Truth table"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/natural-property","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.NaturalProperty","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#natural-property","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"fully_proved","summary":"Let $C = (C_n)_{n \\in \\mathbb{N}}$, where $C_n$ is a set of Boolean functions on $n$ bits. Then $C$ is natural when it is constructive and large. Constructive: there are a polynomial $q$ with natural-number coefficients and, for every $n$, a circuit given as a list of at most $q(2^n)$ encoded gates, such that a Boolean function $g$ on $n$ bits lies in $C_n$ if and only if the evaluation of the circuit on the truth\u2026","title":"Natural property"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/lang-of-family","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.langOfFamily","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#lang-of-family","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"fully_proved","summary":"For a family $f = (f_n)_{n \\in \\mathbb{N}}$, where $f_n$ is a Boolean function on $n$ bits, this is the decision problem that maps a bitstring $x = x_0 x_1 \\cdots x_{n-1}$ of length $n$ to $f_n(x_0, \\dots, x_{n-1})$.","title":"The decision problem of a family of Boolean functions"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/useful-against-ppoly","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.UsefulAgainstPPoly","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#useful-against-ppoly","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"fully_proved","summary":"Let $C = (C_n)_{n \\in \\mathbb{N}}$, where $C_n$ is a set of Boolean functions on $n$ bits. Then $C$ is useful against P/poly when, for every family $f = (f_n)_{n \\in \\mathbb{N}}$ of Boolean functions $f_n$ on $n$ bits such that $f_n \\in C_n$ for infinitely many $n$, the decision problem of $f$ is not in P/poly. Equivalently, if the decision problem of a family $f$ is in P/poly, then $f_n \\in C_n$ for only finitely\u2026","title":"Usefulness against P/poly"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/prfin-ppoly","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.PRFInPPoly","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#prfin-ppoly","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"fully_proved","summary":"Let $F = (F_n)_{n \\in \\mathbb{N}}$ be a keyed family of Boolean functions: $F_n(k, x)$ is a Boolean value for every key $k \\in \\{0,1\\}^{n^2}$ and every input $x \\in \\{0,1\\}^n$. Then $F$ is in P/poly when there is a polynomial $p$ with natural-number coefficients such that for every $n$ some circuit, given as a list of at most $p(n)$ encoded gates, computes $F_n$: for all $k$ and $x$, the evaluation of the circuit\u2026","title":"Keyed function families in P/poly"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/prfsecure","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.PRFSecure","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#prfsecure","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"fully_proved","summary":"Let $F = (F_n)_{n \\in \\mathbb{N}}$ be a keyed family of Boolean functions: $F_n(k, x)$ is a Boolean value for every key $k \\in \\{0,1\\}^{n^2}$ and every input $x \\in \\{0,1\\}^n$. Write $F_{n,k}$ for the Boolean function $x \\mapsto F_n(k, x)$ on $n$ bits and $T(g)$ for the truth table of a Boolean function $g$. Then $F$ is secure when the following holds for every polynomial $q$ with natural-number coefficients and\u2026","title":"Security of a keyed function family against truth-table distinguishers"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/razborov-rudich-natural-proofs","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NaturalProofs.razborov_rudich_natural_proofs","parent":33157,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier/index.html#razborov-rudich-natural-proofs","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/natural-proofs-barrier","status":"can_prove","summary":"Let $F = (F_n)_{n \\in \\mathbb{N}}$ be a keyed family of Boolean functions, where $F_n(k, x)$ is a Boolean value for every key $k \\in \\{0,1\\}^{n^2}$ and every input $x \\in \\{0,1\\}^n$. If $F$ is in P/poly (computed by circuits of size polynomial in $n$) and is secure against truth-table distinguishers, then there is no property $C = (C_n)_{n \\in \\mathbb{N}}$, with $C_n$ a set of Boolean functions on $n$ bits, that is\u2026","title":"Razborov-Rudich: the natural proofs barrier"},{"data_url":"scopes/TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations.json","id":"TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations","kind":"scope","navigation":"module","parent":33148,"result":false,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations/index.html","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations","status":"can_prove","title":"Relativization and oracle separations"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations/baker-gill-solovay","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.BakerGillSolovay.baker_gill_solovay","parent":33167,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations/index.html#baker-gill-solovay","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations","status":"can_prove","summary":"For an oracle $O$, a function from bitstrings to booleans, write $\\mathsf{P}^O$ and $\\mathsf{NP}^O$ for the classes of decision problems decidable in polynomial time, respectively in nondeterministic polynomial time, by machines that may query $O$. There is an oracle $O$ such that $\\mathsf{P}^O \\ne \\mathsf{NP}^O$, and there is an oracle $O$ such that $\\mathsf{P}^O = \\mathsf{NP}^O$.","title":"Baker-Gill-Solovay: oracles separate and collapse P vs NP"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations/oracle-np-co-np-separation","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.OracleNPCoNPSeparation.oracle_np_co_np_separation","parent":33167,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations/index.html#oracle-np-co-np-separation","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/relativization-and-oracle-separations","status":"can_prove","summary":"There is an oracle $O$, a function from bitstrings to booleans, such that $\\mathrm{NP}^O$ differs from the class $\\mathrm{coNP}^O$ of decision problems $L$ whose complement, the decision problem that accepts exactly the bitstrings that $L$ rejects, belongs to $\\mathrm{NP}^O$.","title":"An oracle separates NP from coNP"},{"data_url":"scopes/TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem.json","id":"TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem","kind":"scope","navigation":"module","parent":33148,"result":false,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem/index.html","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem","status":"can_prove","title":"Space hierarchy theorem"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem/l-ne-pspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PSPACEL.l_ne_pspace","parent":33170,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem/index.html#l-ne-pspace","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem","status":"can_prove","summary":"There is a decision problem that belongs to PSPACE but not to the class $\\mathsf{L}$ of decision problems decidable in deterministic logarithmic space.","title":"There is a PSPACE language outside L (space-hierarchy separation)"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem/space-hierarchy","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SpaceHierarchy.space_hierarchy","parent":33170,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem/index.html#space-hierarchy","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/space-hierarchy-theorem","status":"can_prove","summary":"There is a language $L$ of bitstrings that lies in $\\mathrm{DSPACE}(n^2)$ but not in $\\mathrm{DSPACE}(n)$, the deterministic space classes of multi-tape Turing machines for the space bounds $n \\mapsto n^2$ and $n \\mapsto n$.","title":"Deterministic space hierarchy: quadratic versus linear space"},{"data_url":"scopes/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem.json","id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","kind":"scope","navigation":"module","parent":33148,"result":false,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"can_prove","title":"Time hierarchy theorem"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/decides-within","kind":"node","lean":"Cslib.Turing.Universal.Code.DecidesWithin","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#decides-within","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"A code $c$ decides a decision problem $L$ within the time bound $t : \\mathbb{N} \\to \\mathbb{N}$ when, on every bit list $x$, the machine of $c$ started on the input word of $x$ halts within $t(|x|)$ steps, and its tape from the head rightwards, without trailing blanks, is the single symbol of the answer $L(x)$: $2$ for true and $1$ for false.","title":"Deciding within a time bound"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/code-time","kind":"node","lean":"Cslib.Turing.Universal.CodeTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#code-time","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For $t : \\mathbb{N} \\to \\mathbb{N}$, $\\mathrm{CodeTime}(t)$ is the class of decision problems $L$ such that some code decides $L$ within $t(|x|)$ steps on every input $x$.","title":"Time classes on codes"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/mono","kind":"node","lean":"Cslib.Turing.Universal.Code.DecidesWithin.mono","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#mono","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"If a code $c$ decides a decision problem $L$ within a time bound $t : \\mathbb{N} \\to \\mathbb{N}$, and $t' : \\mathbb{N} \\to \\mathbb{N}$ satisfies $t(n) \\le t'(n)$ for every $n$, then $c$ decides $L$ within $t'$.","title":"Deciding within a larger time bound"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/code-time-mono","kind":"node","lean":"Cslib.Turing.Universal.codeTime_mono","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#code-time-mono","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $t, t' : \\mathbb{N} \\to \\mathbb{N}$ with $t(n) \\le t'(n)$ for every $n$. Then every decision problem that some code decides within $t(|x|)$ steps on every input $x$ is decided by some code within $t'(|x|)$ steps on every input $x$: $\\mathrm{CodeTime}(t) \\subseteq \\mathrm{CodeTime}(t')$.","title":"Monotonicity of time classes on codes"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/const-false","kind":"node","lean":"Cslib.Turing.Universal.Code.constFalse","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#const-false","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The code whose table has a single row, for state $0$: on the blank $0$ it writes the symbol $1$, the symbol of the bit false, and enters state $1$; on the input symbols $1$ and $2$ it moves right and stays in state $0$; on every other symbol it halts. State $1$ has no row, so the machine halts there. This code answers false on every input.","title":"The constant-false code"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/const-false-decides-within","kind":"node","lean":"Cslib.Turing.Universal.Code.constFalse_decidesWithin","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#const-false-decides-within","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $c$ be the code whose table has one row, for state $0$: on the input symbols $1$ and $2$ it moves right and stays in state $0$, and on the blank $0$ it writes $1$, the symbol of the bit $\\mathrm{false}$, and enters state $1$, which has no row, so that $c$ halts. Then $c$ decides the empty language within $n + 1$ steps on every input of length $n$.","title":"The constant-false code decides the empty language"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/const-false-mem-code-time","kind":"node","lean":"Cslib.Turing.Universal.const_false_mem_codeTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#const-false-mem-code-time","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $n + 1 \\le f(n)$ for every $n$. Then the empty language, the decision problem that answers false on every bit list, belongs to the time class of $f$: some code decides it within $f(|x|)$ steps on every input $x$, where $|x|$ is the length of $x$.","title":"Time classes are not empty once the bound exceeds the input length"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/code-time-eq-empty","kind":"node","lean":"Cslib.Turing.Universal.codeTime_eq_empty","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#code-time-eq-empty","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ with $f(0) = 0$. Then no decision problem is decided by a code within $f(|x|)$ steps on every input $x$: the time class $\\mathrm{CodeTime}(f)$ is empty. With no step on the empty input, the tape stays blank and holds no answer bit.","title":"Some step is needed"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/lookup-eq-of-code-cells-eq","kind":"node","lean":"Cslib.Turing.Universal.Code.lookup_eq_of_codeCells_eq","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#lookup-eq-of-code-cells-eq","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"If two codes $c$ and $c'$ have the same code stack, then for every state $q$ and every symbol $a$, the table entry of $c$ at $(q, a)$ equals the table entry of $c'$ at $(q, a)$.","title":"The code stack determines the table lookups"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/eval-steps-eq-of-code-cells-eq","kind":"node","lean":"Cslib.Turing.Universal.Code.evalSteps_eq_of_codeCells_eq","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#eval-steps-eq-of-code-cells-eq","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Codes with the same code stack have the same clocked evaluator.","title":"The code stack determines the clocked semantics"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/read-unary-length-eq-some-iff-cell","kind":"node","lean":"Cslib.Turing.Universal.Diag.readUnaryLength_eq_some_iff_cell","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#read-unary-length-eq-some-iff-cell","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For bit lists $x$ and $r$ and a natural number $n$, reading a unary length prefix of $x$ returns $n$ together with the rest $r$ if and only if $x$ is the cell of $n$ (that is, $n$ copies of $\\mathrm{true}$ followed by $\\mathrm{false}$) followed by $r$.","title":"Reading a unary length prefix reads one cell"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/cells-decode","kind":"node","lean":"Cslib.Turing.Universal.Diag.cellsDecode","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#cells-decode","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For a bit list $x$, the list of natural numbers $l$ whose cells are $x$, if there is one, and nothing otherwise. Here the cells of $l$ are its numbers written in order, each number $n$ as $n$ copies of $\\mathrm{true}$ followed by one $\\mathrm{false}$; distinct lists have distinct cells. The decoder reads one cell at a time with the unary length reader.","title":"Decoding a list of cells"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/entries-of-nums","kind":"node","lean":"Cslib.Turing.Universal.Diag.entriesOfNums","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#entries-of-nums","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"A list of natural numbers is read as a list of entries $(q, a, q', s)$, four consecutive numbers per entry, if possible: the empty list gives no entries; a list that starts with $q, a, q', s$ and continues with $l$ gives the entry $(q, a, q', s)$ followed by the entries read from $l$, if that succeeds; and the reading fails when the length of the list is not a multiple of $4$.","title":"Reading entries from numbers"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/keys-ascend-from","kind":"node","lean":"Cslib.Turing.Universal.Diag.keysAscendFrom","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#keys-ascend-from","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For natural numbers $p_q$, $p_a$ and a list of entries $(q, a, q', s) \\in \\mathbb{N}^4$, this Boolean test is true exactly when the keys $(q, a)$ of the entries strictly increase in lexicographic order and the first key, if there is one, is at least $(p_q, p_a)$ in that order.","title":"Ascending keys of a list of entries"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/code-of-entries","kind":"node","lean":"Cslib.Turing.Universal.Diag.codeOfEntries","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#code-of-entries","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $E$ be a list of entries $(q, a, q', s)$ of natural numbers, and let $Q$ and $M$ be the sums, over the entries of $E$ counted with multiplicity, of $q + 1$ and of $a + 1$ respectively. The code of $E$ has a table with $Q$ rows and $M$ columns, and for $q < Q$ and $a < M$ its position $(q, a)$ holds $(q', s)$ for the first entry $(q, a, q', s)$ of $E$ with key $(q, a)$, and is empty (halting) if $E$ has no entry\u2026","title":"The code of a list of entries"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/decode-code-cells","kind":"node","lean":"Cslib.Turing.Universal.decodeCodeCells","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#decode-code-cells","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For a bit list $x$, the decoding of $x$ as a code is defined as follows. Write the cell of a number $n$ for the $n$ bits true followed by one bit false, and the cells of a list of numbers for the cells of its numbers in order. If $x$ is the cells of a list $l$ of natural numbers (such an $l$ is unique), the length of $l$ is a multiple of $4$, and the keys $(q, a)$ of the entries $(q, a, q', s)$ formed by the\u2026","title":"Decoding a code stack"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag","kind":"node","lean":"Cslib.Turing.Universal.diag","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For $f : \\mathbb{N} \\to \\mathbb{N}$, the diagonal language of $f$ holds at a bit list $x$ iff $x$ encodes a code $c$ and the clocked evaluator, running $c$ on $x$ for at most $f(|x|)$ steps, does not return the output $[2]$, the symbol of the answer true: either $c$ does not halt on $x$ within $f(|x|)$ steps, or it halts with another output. Bit lists that encode no code are not in the language. This is the\u2026","title":"The diagonal language"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/code-cells-of-decode-code-cells-eq-some","kind":"node","lean":"Cslib.Turing.Universal.codeCells_of_decodeCodeCells_eq_some","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#code-cells-of-decode-code-cells-eq-some","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every bit list $x$ and every code $c$, if $x$ decodes to $c$, then the code stack of $c$ is $x$.","title":"Decoding inverts the code stack"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/is-some-decode-code-cells-iff","kind":"node","lean":"Cslib.Turing.Universal.isSome_decodeCodeCells_iff","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#is-some-decode-code-cells-iff","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every bit list $x$, the decoding of $x$ as a code succeeds iff $x$ is the code stack of some code.","title":"Code stacks are recognized"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/is-some-decode-code-cells-iff-entries","kind":"node","lean":"Cslib.Turing.Universal.isSome_decodeCodeCells_iff_entries","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#is-some-decode-code-cells-iff-entries","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"A bit list $x$ encodes a code (its decoding succeeds) iff there is a list of entries $(q, a, q', s)$ of natural numbers whose keys $(q, a)$ strictly increase in lexicographic order, such that $x$ is the list of cells of the numbers of these entries, four numbers per entry.","title":"Code stacks, entry by entry"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/decode-code-cells-code-cells","kind":"node","lean":"Cslib.Turing.Universal.decodeCodeCells_codeCells","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#decode-code-cells-code-cells","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every code $c$, decoding the code stack of $c$ succeeds and returns a code $c'$ whose code stack equals the code stack of $c$.","title":"Code stacks decode to codes with the same code stack"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-code-cells","kind":"node","lean":"Cslib.Turing.Universal.diag_codeCells","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-code-cells","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every time bound $f : \\mathbb{N} \\to \\mathbb{N}$ and every code $c$ with code stack $x$, the bit list $x$ belongs to the diagonal language for $f$ iff it is not the case that $c$, run on $x$, halts within $f(|x|)$ steps with output the single symbol $2$ of the bit true.","title":"The diagonal language on a code stack"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-eq-false","kind":"node","lean":"Cslib.Turing.Universal.diag_eq_false","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-eq-false","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every $f : \\mathbb{N} \\to \\mathbb{N}$ and every bit list $x$ that is the code stack of no code, $x$ is not in the diagonal language of $f$.","title":"Bit lists that encode no code are outside the diagonal language"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-code-cells-eq-enc-out","kind":"node","lean":"Cslib.Turing.Universal.diag_codeCells_eq_encOut","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-code-cells-eq-enc-out","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every time bound $f : \\mathbb{N} \\to \\mathbb{N}$ and every code $c$ with code stack $x$, the bit list $x$ belongs to the diagonal language for $f$ iff the output encoding of the result of running $c$ on $x$ for at most $f(|x|)$ steps differs from the output encoding of the result whose output is the single symbol $2$ of the bit true.","title":"The diagonal language through the universal machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-not-mem-code-time","kind":"node","lean":"Cslib.Turing.Universal.diag_not_mem_codeTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-not-mem-code-time","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every $f : \\mathbb{N} \\to \\mathbb{N}$, the diagonal language of $f$ is not in the time class of $f$ on codes: no code decides it on every input $x$ within $f(|x|)$ steps. This is the argument from the proof of Sipser's Theorem 9.10: a code $c$ deciding it would, on its own code stack, answer the opposite of the diagonal language of $f$.","title":"The diagonal argument"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/code-time-ssubset-of-diag-mem","kind":"node","lean":"Cslib.Turing.Universal.codeTime_ssubset_of_diag_mem","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#code-time-ssubset-of-diag-mem","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $f, g : \\mathbb{N} \\to \\mathbb{N}$ with $f(n) \\le g(n)$ for every $n$, and for a time bound $t$ let $\\mathrm{CodeTime}(t)$ be the class of decision problems that some code decides within $t(|x|)$ steps on every input $x$. If the diagonal language $\\mathrm{diag}(f)$ belongs to $\\mathrm{CodeTime}(g)$, then $\\mathrm{CodeTime}(f)$ is a proper subclass of $\\mathrm{CodeTime}(g)$.","title":"A hierarchy within codes, given an upper bound"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-arg","kind":"node","lean":"Cslib.Turing.Universal.diagArg","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-arg","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For a time bound $f : \\mathbb{N} \\to \\mathbb{N}$ and a bit list $x$, the argument is a triple of a code, an input and a clock. If $x$ decodes to a code $c$, it is $(c, x, f(|x|))$. Otherwise it is the code with the empty table, the input consisting of the single bit true, and the clock $0$; that code halts at once with the answer true, so that the diagonal answer is false there.","title":"The universal machine's argument for the diagonal language"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/enc-in-diag-arg","kind":"node","lean":"Cslib.Turing.Universal.encIn_diagArg","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#enc-in-diag-arg","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For $f : \\mathbb{N} \\to \\mathbb{N}$ and a bit list $x$, the input bits of the clocked universal machine on its argument for the diagonal language of $f$ on $x$ are, when $x$ encodes a code, the cell of $f(|x|)$, then the cells of the input symbols of $x$, then the bit false, then $x$ reversed; when $x$ encodes no code, they are the fixed list\u2026","title":"The reformatting function in bits"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-answer","kind":"node","lean":"Cslib.Turing.Universal.diagAnswer","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-answer","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For a clocked result $o$, either a timeout or the output of a halted run, the diagonal answer of $o$ is false if $o$ is the output consisting of the single symbol $2$, the symbol of the bit true, and true otherwise, including on a timeout.","title":"The diagonal answer"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-answer-eq-decide-enc-out","kind":"node","lean":"Cslib.Turing.Universal.diagAnswer_eq_decide_encOut","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-answer-eq-decide-enc-out","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every clocked result $o$, either a timeout or the output of a halted run, the diagonal answer of $o$ is true exactly when the output encoding of $o$ differs from the output encoding of the result whose output is the single symbol $2$ of the bit true, which is the bit list true, true, true, false.","title":"The diagonal answer read from the encoded output"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-eq-comp","kind":"node","lean":"Cslib.Turing.Universal.diag_eq_comp","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-eq-comp","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every $f : \\mathbb{N} \\to \\mathbb{N}$ and every bit list $x$, let $(c, y, T)$ be the universal machine's argument for the diagonal language of $f$ on $x$. Then the diagonal language of $f$ holds at $x$ iff the clocked evaluator, running $c$ on $y$ for at most $T$ steps, does not return the output $[2]$, the symbol of the answer true.","title":"The diagonal language through the clocked evaluator"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/k-3","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.K","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#k-3","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The readout machine, which reads the answer of the diagonal language off the output of the clocked universal machine, has exactly two stacks: the input stack $X$ and the output stack $O$.","title":"Stacks of the readout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/lab-3","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.Lab","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#lab-3","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The readout machine has eight labels: $r_0$, $r_1$, $r_2$ and $r_3$ read in turn the bits $\\mathrm{true}, \\mathrm{true}, \\mathrm{true}, \\mathrm{false}$, which encode, at the output of the clocked universal machine, a halt whose output is the single symbol of the bit $\\mathrm{true}$; $r_4$ checks that no bit follows; $\\mathrm{hit}$ and $\\mathrm{miss}$ write the answer; and $\\mathrm{done}$ stops.","title":"Labels of the readout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/untitled-3","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.\u0393","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#untitled-3","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Every stack of the readout machine, namely the input stack $X$ and the output stack $O$, has the alphabet of bits.","title":"Stack alphabet of the readout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/expect","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.expect","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#expect","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For a bit $b$ and a label $\\ell$, the statement that pops the top of the input stack $X$ and goes to $\\ell$ if the popped symbol is $b$, and to the label miss otherwise, including when $X$ is empty.","title":"Expecting a bit in the readout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/prog-3","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.prog","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#prog-3","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The readout machine pops the bits of its input stack $X$ one at a time and compares them with $\\mathrm{true}, \\mathrm{true}, \\mathrm{true}, \\mathrm{false}$, which encode, at the output of the clocked universal machine, a halt whose output is the single symbol of the bit $\\mathrm{true}$. If the input is exactly this list, it pushes $\\mathrm{false}$ onto its output stack $O$; otherwise (at the first bit that differs,\u2026","title":"Program of the readout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/comp-3","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.comp","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#comp-3","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The component machine whose stacks are the input $X$ and the output $O$, each over the alphabet of bits, whose labels and program are those of the readout machine, which checks whether $X$ holds exactly $(\\mathrm{true}, \\mathrm{true}, \\mathrm{true}, \\mathrm{false})$ and pushes $\\mathrm{false}$ on $O$ if so and $\\mathrm{true}$ otherwise, and whose list of stacks is $(X, O)$.","title":"The readout machine as a host component"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/run-3","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.run","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#run-3","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $l$ be a bit list, and start the readout machine at the label $r_0$ with its internal Boolean variable set to $\\mathrm{false}$, $l$ on the input stack $X$ and the output stack $O$ empty. Then the machine reaches the label $\\mathrm{done}$ within $6$ steps, and $O$ then holds exactly one bit, which is $\\mathrm{true}$ if and only if $l \\ne [\\mathrm{true}, \\mathrm{true}, \\mathrm{true}, \\mathrm{false}]$, the list\u2026","title":"The readout run"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/bits-poly-time","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.bitsPolyTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#bits-poly-time","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"A finite multi-stack Turing machine, whose input and output alphabets are identified with the bits, together with a polynomial time bound, such that for every bit list $l$, given $l$ as its input, it halts within the bound, evaluated at $|l|$, with output the one-bit list $[b]$, where $b$ is true if and only if $l$ differs from the output encoding of the result whose output is the single symbol $2$ of the bit true,\u2026","title":"The readout on raw bits in polynomial time"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/poly-time-3","kind":"node","lean":"Cslib.Turing.Universal.Diag.Readout.polyTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#poly-time-3","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"A finite multi-stack Turing machine, given the output encoding $\\mathrm{encOut}(o)$ of a result $o$ of the clocked universal function (a list of symbols, or a timeout), outputs the one-bit list $[\\mathrm{diagAnswer}(o)]$ within a number of steps polynomial in the length of $\\mathrm{encOut}(o)$. Here $\\mathrm{diagAnswer}(o)$ is $\\mathrm{true}$ unless $o$ is a halt whose output is the single symbol of the bit\u2026","title":"The readout in polynomial time"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/nonempty-diag-poly-time","kind":"node","lean":"Cslib.Turing.Universal.nonempty_diag_polyTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#nonempty-diag-poly-time","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$. Suppose that a finite multi-stack Turing machine, the reformatting machine, computes in polynomial time, from a bit list $x$ given as itself, the input $\\mathrm{encIn}(c, y, T)$ of the clocked universal machine, where $(c, y, T) = \\mathrm{diagArg}_f(x)$ is the argument of the clocked universal function for the diagonal language $\\mathrm{diag}_f$ on $x$. Then a finite multi-stack\u2026","title":"The diagonal language in polynomial time, given the reformatting machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/chk-from","kind":"node","lean":"Cslib.Turing.Universal.Diag.chkFrom","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#chk-from","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The check of a bit list $x$ from a key $(p_q, p_a)$ of natural numbers is defined as follows. The empty list passes. Otherwise, calling the cell of a number $n$ the $n$ bits $\\mathrm{true}$ followed by one bit $\\mathrm{false}$, read four cells off the front of $x$, those of $q$, $a$, $q'$ and $s$ in this order, leaving the rest $r$. The check fails if one of these cells cannot be read because what remains contains\u2026","title":"Cell-by-cell check of a code stack"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/chk-from-iff","kind":"node","lean":"Cslib.Turing.Universal.Diag.chkFrom_iff","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#chk-from-iff","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For all natural numbers $p_q, p_a$ and every bit list $x$, the cell-by-cell check of $x$ from the key $(p_q, p_a)$ succeeds if and only if there is a list of $m \\ge 0$ entries $(q_1, a_1, q'_1, s_1), \\dots, (q_m, a_m, q'_m, s_m)$ of natural numbers such that $x$ is the concatenation of the cells of $q_1, a_1, q'_1, s_1, \\dots, q_m, a_m, q'_m, s_m$, in this order, and the keys $(q_i, a_i)$ strictly increase in\u2026","title":"The cell-by-cell check recognizes code stacks"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/k","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.K","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#k","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The checking machine has twelve stacks, named $X, Y, W, V, Q, A, P_Q, P_A, T_1, T_2, J, O$.","title":"Stacks of the checking machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/lab","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.Lab","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#lab","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The labels of the checking machine: 23 labels without arguments and four families of labels indexed by the outcome of a comparison ($<$, $=$ or $>$), 35 distinct labels in all.","title":"Labels of the checking machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/untitled","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.\u0393","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#untitled","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Every stack of the layout machine holds bits: the alphabet of each stack is the type of Booleans.","title":"Stack alphabet of the layout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/out-q","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.outQ","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#out-q","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The label the checking machine goes to after comparing the bound $p$ for the state (the state of the previous key, or $0$ at the first entry) with the state $q$ of the current key: if $p < q$, the label $\\mathsf{sk0}$, where it starts skipping the rest of the entry; if $p = q$, the label $\\mathsf{ca}$, where it compares the symbols; if $p > q$, the label $\\mathsf{bad}$, where it rejects.","title":"Branch after comparing states"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/out-a","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.outA","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#out-a","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The label the checking machine goes to after comparing the bound $p$ for the symbol with the symbol $a$ of the current key: if $p > a$, the label $\\mathsf{bad}$, where it rejects; if $p < a$ or $p = a$, the label $\\mathsf{sk0}$, where it starts skipping the rest of the entry.","title":"Branch after comparing symbols"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/prog","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.prog","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#prog","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, this is the program of a multi-stack machine whose stacks $X, Y, W, V, Q, A, P_Q, P_A, T_1, T_2, J, O$ all hold bits. To read a cell from $V$ onto a stack $S$ means to pop the leading $1$s of $V$, pushing a $1$ on $S$ for each, and then to pop the $0$ that follows; if $V$ is empty at that point, the machine rejects. From its first label the program pops the bits of $X$ one at\u2026","title":"Program of the checking machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/answer","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.answer","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#answer","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For a bit list $x$, the answer is the list $x$ with the bit $\\mathrm{true}$ put in front if the cell-by-cell check of $x$ from the key $(0, 0)$ succeeds, and the one-bit list $[\\mathrm{false}]$ otherwise.","title":"Answer of the checking machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/run","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.run","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#run","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every bit list $x$, the checking machine, started at the label $\\mathsf{dupP}$ with internal state false, with $x$ on its input stack $X$ and every other stack empty, reaches the label $\\mathsf{done}$ within $33(|x| + 1)^2$ steps, with some internal state and stacks in which the output stack $O$ holds the answer of the checking machine on $x$.","title":"The checking run"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/comp","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.comp","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#comp","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The component machine with the twelve stacks $X, Y, W, V, Q, A, P_Q, P_A, T_1, T_2, J, O$ of the checking machine, each holding bits, with the labels and the program of the checking machine, and with its stacks listed in the order $X, Y, W, V, Q, A, P_Q, P_A, T_1, T_2, J, O$.","title":"The checking machine as a host component"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/poly-time","kind":"node","lean":"Cslib.Turing.Universal.Diag.Check.polyTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#poly-time","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"A finite multi-stack Turing machine with a polynomial time bound that computes the checking answer: given a bit list $x$ as its input, it halts within the bound, evaluated at $|x|$, with the checking answer on $x$ as its output.","title":"The checking machine in polynomial time"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/k-2","kind":"node","lean":"Cslib.Turing.Universal.Diag.Layout.K","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#k-2","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The layout machine has twelve stacks, named $X, Z, P, P_2, U, N, A_2, B_2, D, C, T, O$.","title":"Stacks of the layout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/lab-2","kind":"node","lean":"Cslib.Turing.Universal.Diag.Layout.Lab","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#lab-2","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The layout machine has 17 labels, named $\\mathsf{s0}$, $\\mathsf{mvZ}$, $\\mathsf{cp0}$, $\\mathsf{cp1}$, $\\mathsf{lo0}$, $\\mathsf{lo1}$, $\\mathsf{lo2}$, $\\mathsf{lo3}$, $\\mathsf{lo4}$, $\\mathsf{mh}$, $\\mathsf{mb}$, $\\mathsf{mk}$, $\\mathsf{mr}$, $\\mathsf{lo5}$, $\\mathsf{lo6}$, $\\mathsf{bad}$ and $\\mathsf{done}$.","title":"Labels of the layout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/untitled-2","kind":"node","lean":"Cslib.Turing.Universal.Diag.Layout.\u0393","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#untitled-2","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Every stack of the layout machine holds bits: the alphabet of each of its stacks is the type of Booleans.","title":"Stack symbols of the layout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/prog-2","kind":"node","lean":"Cslib.Turing.Universal.Diag.Layout.prog","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#prog-2","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, this is the program of a multi-stack machine whose stacks $X, Z, P, P_2, U, N, A_2, B_2, D, C, T, O$ all hold bits. To move a stack $S$ onto a stack $S'$ means to pop the bits of $S$ one at a time, pushing each onto $S'$, until $S$ is empty; to transfer $S$ to some stacks means to pop $S$ until it is empty, pushing a $1$ on each of these stacks for every bit popped. From the\u2026","title":"Program of the layout machine"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/layout","kind":"node","lean":"Cslib.Turing.Universal.Diag.Layout.layout","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#layout","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Write $\\mathrm{cell}(n)$ for the bit list of $n$ bits $\\mathrm{true}$ followed by one bit $\\mathrm{false}$. On a bit list $\\mathrm{true} :: x$, the layout function returns $\\mathrm{cell}(|x|^2 + 1)$, then the cells $\\mathrm{cell}(s)$ of the symbols $s$ of the input word of $x$ (the symbol $2$ for each bit $\\mathrm{true}$ and $1$ for each bit $\\mathrm{false}$), then the bit $\\mathrm{false}$, then the reverse of $x$.\u2026","title":"The layout function"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/run-2","kind":"node","lean":"Cslib.Turing.Universal.Diag.Layout.run","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#run-2","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every bit list $y$, the layout machine, started at the label $\\mathsf{s0}$ with internal state false, with $y$ on its input stack $X$ and every other stack empty, reaches the label $\\mathsf{done}$ within $18(|y| + 1)^2$ steps, with some internal state and stacks in which the output stack $O$ holds the layout of $y$.","title":"The layout run"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/chk-from-zero","kind":"node","lean":"Cslib.Turing.Universal.Diag.chkFrom_zero","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#chk-from-zero","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every bit list $x$, the cell-by-cell check of $x$ from the key $(0, 0)$ succeeds if and only if the decoding of $x$ as a code stack returns a code.","title":"The cell-by-cell check agrees with code-stack decoding"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/comp-2","kind":"node","lean":"Cslib.Turing.Universal.Diag.Layout.comp","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#comp-2","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The component machine with the twelve stacks $X, Z, P, P_2, U, N, A_2, B_2, D, C, T, O$ of the layout machine, each holding bits, with the labels and the program of the layout machine, and with its stacks listed in the order $X, Z, P, P_2, U, N, A_2, B_2, D, C, T, O$.","title":"The layout machine as a host component"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/poly-time-2","kind":"node","lean":"Cslib.Turing.Universal.Diag.Layout.polyTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#poly-time-2","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"A finite multi-stack Turing machine, whose input and output alphabets are identified with the bits, together with a polynomial time bound, that computes the layout function: given a bit list $y$ as its input, it halts within the bound, evaluated at $|y|$, with the layout of $y$ as its output.","title":"The layout in polynomial time"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/layout-answer","kind":"node","lean":"Cslib.Turing.Universal.Diag.layout_answer","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#layout-answer","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"For every bit list $x$, let $(c, y, T)$ be the universal machine's argument for the diagonal language of $f(n) = n^2 + 1$ on $x$. Then the layout function applied to the checking machine's answer on $x$ equals the input bit list of the clocked universal machine for the code $c$, the input $y$ and the clock $T$.","title":"Layout after checking gives the universal machine's input"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/reformat","kind":"node","lean":"Cslib.Turing.Universal.Diag.reformat","parent":33173,"reading_title":{"format":"inline-parts-v1","parts":[["text","The reformatting machine for "],["math","\\(n \\mapsto n^2 + 1\\)"]],"source":"The reformatting machine for $n \\mapsto n^2 + 1$"},"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#reformat","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $f(n) = n^2 + 1$. This is a finite multi-stack Turing machine whose input and output alphabets are identified with the bits, together with a polynomial $p$, such that for every bit list $x$, started with $x$ on its input stack it halts within $p(|x|)$ steps with output the clocked universal machine's input bits for the universal machine's argument $(c, w, T)$ for the diagonal language of $f$ at $x$: the clock\u2026","title":"The reformatting machine for $n \\mapsto n^2 + 1$"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/nonempty-diag-sq-add-one-poly-time","kind":"node","lean":"Cslib.Turing.Universal.nonempty_diag_sq_add_one_polyTime","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#nonempty-diag-sq-add-one-poly-time","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Let $D$ be the diagonal language for the time bound $f(n) = n^2 + 1$. There is a finite multi-stack Turing machine that, given a bit list $x$ as its input, halts within a number of steps polynomial in $|x|$ with the single bit $D(x)$ as its output.","title":"Deciding the diagonal language on raw bits in polynomial time"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/diag-sq-add-one-mem-p","kind":"node","lean":"Cslib.Turing.Universal.diag_sq_add_one_mem_P","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#diag-sq-add-one-mem-p","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"The diagonal language for the time bound $f(n) = n^2 + 1$, the set of bit lists $x$ that encode a code which does not halt on $x$ within $|x|^2 + 1$ steps with the answer $\\mathrm{true}$, is in the class P of decision problems decidable in polynomial time.","title":"The quadratic diagonal language is in P"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/exists-mem-p-not-mem-code-time-sq-add-on","kind":"node","lean":"Cslib.Turing.Universal.exists_mem_P_not_mem_codeTime_sq_add_one","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#exists-mem-p-not-mem-code-time-sq-add-on","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"There is a decision problem in the class P that no code decides within $n^2 + 1$ steps: no code halts on every input $x$ within $|x|^2 + 1$ steps with the answer for $x$ as its output.","title":"Diagonalization across models"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/not-p-subset-code-time-sq-add-one","kind":"node","lean":"Cslib.Turing.Universal.not_P_subset_codeTime_sq_add_one","parent":33173,"reading_title":{"format":"inline-parts-v1","parts":[["text","Polynomial time is not contained in code time "],["math","\\(n^2 + 1\\)"]],"source":"Polynomial time is not contained in code time $n^2 + 1$"},"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#not-p-subset-code-time-sq-add-one","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"fully_proved","summary":"Some decision problem in $\\mathsf{P}$ is decided by no code within $n^2 + 1$ steps on every input of length $n$; that is, $\\mathsf{P}$ is not a subset of the class $\\mathrm{CodeTime}(n \\mapsto n^2 + 1)$.","title":"Polynomial time is not contained in code time $n^2 + 1$"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/p-ne-exp","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PneqEXP.p_ne_exp","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#p-ne-exp","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"can_prove","summary":"There is a decision problem that belongs to the class EXP but not to the complexity class P.","title":"There is an EXP language outside P (time-hierarchy separation)"},{"id":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/time-hierarchy","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.TimeHierarchy.time_hierarchy","parent":33173,"result":true,"route":"../roadmap/TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem/index.html#time-hierarchy","scope":"TheoryOfComputation/hierarchy-theorems-and-barriers/time-hierarchy-theorem","status":"can_prove","summary":"There is a language $L$ of bitstrings that lies in the multi-tape deterministic time class $\\mathrm{DTIME}(n^2)$ but not in $\\mathrm{DTIME}(n)$.","title":"Deterministic time hierarchy: quadratic versus linear time"},{"data_url":"scopes/TheoryOfComputation/regular-languages.json","id":"TheoryOfComputation/regular-languages","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/index.html","scope":"TheoryOfComputation/regular-languages","status":"fully_proved","summary":"Deterministic, nondeterministic, two-way and \u03b5-automata, transducers, regular expressions and Kleene's theorem, closure properties of regular languages, automaton minimization, and the algebraic theory of recognition by monoids.","title":"Finite automata and regular languages"},{"data_url":"scopes/TheoryOfComputation/regular-languages/automata-and-expressions.json","id":"TheoryOfComputation/regular-languages/automata-and-expressions","kind":"scope","parent":33244,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/index.html","scope":"TheoryOfComputation/regular-languages/automata-and-expressions","status":"fully_proved","summary":"DFA, NFA, \u03b5-NFA and two-way automata, sequential transducers and Mealy and Moore machines, regular expressions, Thompson's construction and Kleene's theorem, star height, and closure of regular languages under homomorphism, quotient, shuffle and substitution.","title":"Automata and regular expressions"},{"data_url":"scopes/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages.json","id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","kind":"scope","navigation":"module","parent":33245,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","title":"Closure properties of regular languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/bound","kind":"node","lean":"Cslib.Language.HomImage.bound","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#bound","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"For a map $\\varphi$ from a finite alphabet $T$ to words over $T'$, the bound is the maximum of the lengths $|\\varphi(a)|$ over all letters $a \\in T$ (and $0$ when $T$ is empty).","title":"Maximal length of the image of a letter"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/state","kind":"node","lean":"Cslib.Language.HomImage.State","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#state","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $T$ be a finite alphabet, let $\\varphi$ map letters of $T$ to words over $T'$, let $\\sigma$ be a set (the states of a DFA), and let $N$ be the maximum of the lengths $|\\varphi(a)|$ over $a \\in T$. The states of the \u03b5-NFA form the disjoint union of $\\sigma$ and the set of triples $(q, a, i)$ with $q \\in \\sigma$, $a \\in T$ and $0 \\le i < N$.","title":"States of the homomorphic-image \u03b5-NFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/advance","kind":"node","lean":"Cslib.Language.HomImage.advance","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#advance","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $T$ with transition function $\\delta$, and let $\\varphi$ map each letter of $T$ to a word over $T'$. For a state $q$ of $M$, a letter $a$ and a natural number $i$, the state reached after emitting position $i$ (counted from $0$) of $\\varphi(a)$ from $q$ is the intermediate state $(q, a, i + 1)$ of the \u03b5-NFA if $i + 1 < |\\varphi(a)|$, and the state $\\delta(q, a)$ of $M$\u2026","title":"Next state while spelling out a letter's image"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/nfa","kind":"node","lean":"Cslib.Language.HomImage.\u03b5nfa","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#nfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $T$ with start state $q_0$, transition function $\\delta$ and accepting states $F$, let $\\varphi$ map letters of $T$ to words over $T'$, and let $N$ be the maximum of the lengths $|\\varphi(a)|$ over $a \\in T$. The \u03b5-NFA over $T'$ has as states the states of $M$ together with the triples $(q, a, i)$ with $q$ a state of $M$, $a \\in T$ and $0 \\le i < N$. Write\u2026","title":"The \u03b5-NFA for the homomorphic image"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/code","kind":"node","lean":"Cslib.Language.HomImage.code","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#code","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"For a map $\\varphi$ from letters of $T$ to words over $T'$ and a letter $a$, the label sequence of $a$ is a single \u03b5-label when $\\varphi(a)$ is empty, and otherwise the letters of $\\varphi(a)$ in order, each as a non-\u03b5 label.","title":"Edge labels for a letter's image"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/is-path-advance","kind":"node","lean":"Cslib.Language.HomImage.isPath_advance","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#is-path-advance","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $T$ with transition function $\\delta$, and let $\\varphi$ map letters of $T$ to words over $T'$. For every state $q$ of $M$, letter $a$ and natural number $i$, the \u03b5-NFA of $M$ and $\\varphi$ has a path to the state $\\delta(q, a)$ of $M$ from the state $(q, a, i + 1)$ if $i + 1 < |\\varphi(a)|$, and from $\\delta(q, a)$ otherwise, whose labels are the letters of $\\varphi(a)$\u2026","title":"Finishing a letter's image in the image \u03b5-NFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/is-path-code","kind":"node","lean":"Cslib.Language.HomImage.isPath_code","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#is-path-code","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $T$ with transition function $\\delta$, and let $\\varphi$ map letters of $T$ to words over $T'$. For every state $q$ of $M$ and letter $a$, the \u03b5-NFA of $M$ and $\\varphi$ has a path from the state $q$ to the state $\\delta(q, a)$ whose label sequence is a single \u03b5-label if $\\varphi(a)$ is empty, and the letters of $\\varphi(a)$ in order, all non-\u03b5, otherwise.","title":"Spelling out a letter's image in the image \u03b5-NFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/exists-of-is-path","kind":"node","lean":"Cslib.Language.HomImage.exists_of_isPath","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#exists-of-is-path","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $T$ with start state $q_0$, and let $\\varphi$ map letters of $T$ to words over $T'$; for a word $w$ over $T$, write $\\varphi(w)$ for the concatenation of the images of its letters. Suppose the \u03b5-NFA of $M$ and $\\varphi$ has a path from the state $q_0$ of $M$ to a state $u$, with label sequence $x$, and let $\\bar x$ be the word formed by the non-\u03b5 labels of $x$. If $u$ is a\u2026","title":"Soundness of the homomorphic-image \u03b5-NFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/hom-image","kind":"node","lean":"Cslib.Language.IsRegular.homImage","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#hom-image","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $T$ be a finite alphabet, $T'$ an arbitrary (possibly infinite) alphabet, and $\\varphi$ a map from $T$ to words over $T'$, extended to words by $\\varphi(a_1 \\cdots a_n) = \\varphi(a_1) \\cdots \\varphi(a_n)$. If $L$ is a regular language over $T$, then its image $\\{\\varphi(w) : w \\in L\\}$ is a regular language over $T'$.","title":"Homomorphic images of regular languages over finite alphabets are regular"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/right-dfa","kind":"node","lean":"Cslib.Language.RegularQuotient.rightDFA","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#right-dfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Given a DFA $M$ over an alphabet $T$ with state set $\\sigma$ and a language $R$ over $T$, this is the DFA with the same states, transition function and start state as $M$, whose accepting states are the states $q$ from which some word of $R$ leads $M$ to an accepting state of $M$.","title":"DFA for the right quotient by a language"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/left-nfa","kind":"node","lean":"Cslib.Language.RegularQuotient.leftNFA","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#left-nfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Given a DFA $M$ over an alphabet $T$ with state set $\\sigma$ and a language $R$ over $T$, this is the NFA with state set $\\sigma$ and the same accepting states as $M$, in which from a state $q$ on a letter $a$ the only possible move is to the state $M$ reaches from $q$ on $a$, and whose start states are the states that $M$ reaches from its start state on the words of $R$.","title":"NFA for the left quotient by a language"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/right-quotient-set","kind":"node","lean":"Cslib.Language.IsRegular.rightQuotientSet","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#right-quotient-set","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"If $L$ is a regular language over an alphabet $T$ and $R$ is any language over $T$, then the right quotient $\\{x \\mid \\exists y \\in R,\\ xy \\in L\\}$ of $L$ by $R$ is regular.","title":"Regular languages are closed under right quotients by arbitrary languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/left-quotient-set","kind":"node","lean":"Cslib.Language.IsRegular.leftQuotientSet","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#left-quotient-set","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"If $L$ is a regular language over an alphabet $T$ and $R$ is any language over $T$, then the left quotient $\\{y \\mid \\exists x \\in R,\\ xy \\in L\\}$ of $L$ by $R$ is regular.","title":"Regular languages are closed under left quotients by arbitrary languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/is-shuffle","kind":"node","lean":"Cslib.Language.IsShuffle","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#is-shuffle","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"The relation \"$w$ is a shuffle of $u$ and $v$\" on words over $\\alpha$ is defined inductively: the empty word is a shuffle of the empty word and the empty word, and if $w$ is a shuffle of $u$ and $v$, then for every letter $a$ the word $aw$ is a shuffle of $au$ and $v$, and a shuffle of $u$ and $av$. Thus $w$ interleaves the letters of $u$ and of $v$, keeping their order within each word.","title":"Shuffle of two words"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/shuffle","kind":"node","lean":"Cslib.Language.shuffle","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#shuffle","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"For languages $L$ and $M$ over $\\alpha$, the shuffle of $L$ and $M$ is the language of all words $w$ over $\\alpha$ such that $w$ is a shuffle of some word $u \\in L$ and some word $v \\in M$, that is, $w$ interleaves $u$ and $v$.","title":"Shuffle of two languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/cyclic-shift","kind":"node","lean":"Cslib.Language.cyclicShift","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#cyclic-shift","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"For a language $L$ over $\\alpha$, the cyclic shift of $L$ is the language of all words $yx$ over $\\alpha$ such that $xy \\in L$ for some words $x$ and $y$; that is, the language of all rotations of the words of $L$.","title":"Cyclic shift of a language"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/shuffle-nfa","kind":"node","lean":"Cslib.Language.RegularShuffle.shuffleNFA","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#shuffle-nfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ and $N$ be DFAs over $\\alpha$ with state types $\\sigma$ and $\\tau$, start states $s_M$ and $s_N$, sets of accepting states $F_M$ and $F_N$, and transition functions $\\delta_M$ and $\\delta_N$. Their shuffle automaton is the NFA over $\\alpha$ with state type $\\sigma \\times \\tau$, the single start state $(s_M, s_N)$ and accepting states $F_M \\times F_N$, which on a letter $a$ moves from a state $(p, q)$ to the\u2026","title":"Shuffle automaton of two DFAs"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/from-to","kind":"node","lean":"Cslib.Language.RegularShuffle.fromTo","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#from-to","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"For a DFA $M$ over $\\alpha$ with state type $\\sigma$, a state $p \\in \\sigma$ and a set $S \\subseteq \\sigma$ of states, this is the language of all words $w$ over $\\alpha$ such that the run of $M$ started in $p$ on $w$ ends in a state of $S$.","title":"Words leading a DFA from a state into a set of states"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/shuffle-2","kind":"node","lean":"Cslib.Language.IsRegular.shuffle","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#shuffle-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"The shuffle of two regular languages is regular.","title":"The shuffle of two regular languages is regular"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/cyclic-shift-2","kind":"node","lean":"Cslib.Language.IsRegular.cyclicShift","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#cyclic-shift-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"The cyclic shifts of the words of a regular language form a regular language.","title":"The cyclic shifts of the words of a regular language form a regular language"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/substitution-2","kind":"node","lean":"Cslib.Language.substitution","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#substitution-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $T$, and let $s$ assign to each letter $a \\in T$ a language $s(a)$ over an alphabet $T'$. The substitution of $s$ into $L$ is the language over $T'$ of all words $y_1 y_2 \\cdots y_n$ for which there is a word $a_1 a_2 \\cdots a_n \\in L$ with $y_i \\in s(a_i)$ for every $i$.","title":"Substitution of languages for letters"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/nfa-2","kind":"node","lean":"Cslib.Language.RegularSubstitution.\u03b5nfa","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#nfa-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ be a DFA over $T$ with states $\\sigma$, and for each letter $a \\in T$ let $N_a$ be a DFA over $T'$ with states $\\tau_a$. This \u03b5-NFA over $T'$ has as states the states $q$ of $M$ and the triples $(q, a, p)$ with $q$ a state of $M$, $a \\in T$ and $p$ a state of $N_a$, read as the state $p$ of the copy of $N_a$ entered from $q$. A state $q$ of $M$ has an \u03b5-move to $(q, a, s_a)$ for every $a \\in T$, where $s_a$\u2026","title":"The \u03b5-NFA for a substitution"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/is-path-run","kind":"node","lean":"Cslib.Language.RegularSubstitution.isPath_run","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#is-path-run","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ be a DFA over $T$, and for each letter $a \\in T$ let $N_a$ be a DFA over $T'$. For every state $q$ of $M$, letter $a \\in T$, state $p$ of $N_a$ and word $y$ over $T'$, the \u03b5-NFA built from $M$ and the $N_a$ has a path reading the letters of $y$, with no \u03b5-moves, from the state $p$ of the copy of $N_a$ entered from $q$ to the state of the same copy that $N_a$ reaches from $p$ on $y$.","title":"Paths inside one copy of a letter automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/detour","kind":"node","lean":"Cslib.Language.RegularSubstitution.detour","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#detour","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"For a word $y = b_1 \\cdots b_n$ over $T'$, the detour labels of $y$ are the sequence of \u03b5-NFA labels $\\varepsilon, b_1, \\ldots, b_n, \\varepsilon$: an \u03b5-move, the letters of $y$ in order, and a final \u03b5-move.","title":"Detour labels"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/exists-of-is-path-2","kind":"node","lean":"Cslib.Language.RegularSubstitution.exists_of_isPath","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#exists-of-is-path-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $M$ be a DFA over $T$ with start state $q_0$, and for each letter $a \\in T$ let $N_a$ be a DFA over $T'$, with accepted language $L(N_a)$; for a word $w = a_1 \\cdots a_n$ over $T$ write $L_w = L(N_{a_1}) \\cdots L(N_{a_n})$, which is $\\{\\varepsilon\\}$ when $w$ is empty. Let $x$ be a sequence of labels, each a letter of $T'$ or $\\varepsilon$, let $\\bar x$ be the word over $T'$ obtained from $x$ by deleting the\u2026","title":"Soundness of the paths of the substitution \u03b5-NFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/substitution","kind":"node","lean":"Cslib.Language.IsRegular.substitution","parent":33246,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages/index.html#substitution","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/closure-properties-of-regular-languages","status":"fully_proved","summary":"Let $T$ be a finite alphabet and $T'$ any alphabet. If $L$ is a regular language over $T$ and $s(a)$ is a regular language over $T'$ for every letter $a \\in T$, then the substitution of $s$ into $L$ is a regular language over $T'$.","title":"Regular languages are closed under substitution over a finite alphabet"},{"data_url":"scopes/TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages.json","id":"TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages","kind":"scope","navigation":"module","parent":33245,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages/index.html","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages","status":"fully_proved","title":"Finite languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages/singleton-word","kind":"node","lean":"Cslib.Language.IsRegular.singleton_word","parent":33273,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages/index.html#singleton-word","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages","status":"fully_proved","summary":"For every word $w$ over any alphabet, the language whose only word is $w$ is regular.","title":"Single-word languages are regular"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages/of-finite","kind":"node","lean":"Cslib.Language.IsRegular.of_finite","parent":33273,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages/index.html#of-finite","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages","status":"fully_proved","summary":"Every finite language over any alphabet is regular.","title":"Finite languages are regular"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages/of-length-le","kind":"node","lean":"Cslib.Language.IsRegular.of_length_le","parent":33273,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages/index.html#of-length-le","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/finite-languages","status":"fully_proved","summary":"Over a finite alphabet, if every word of a language $L$ has length at most $n$, for some natural number $n$, then $L$ is regular.","title":"Languages of bounded word length are regular"},{"data_url":"scopes/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem.json","id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","kind":"scope","navigation":"module","parent":33245,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","title":"Kleene's theorem"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/is-regex-denoted","kind":"node","lean":"Cslib.Language.KleeneConverse.IsRegexDenoted","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#is-regex-denoted","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"A language is denoted by a regular expression when it is `r.matches'` for some `r`.","title":"Language denoted by a regular expression"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/zero","kind":"node","lean":"Cslib.Language.KleeneConverse.IsRegexDenoted.zero","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#zero","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The empty language is denoted by `0`.","title":"The empty language is regex-denoted"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/one","kind":"node","lean":"Cslib.Language.KleeneConverse.IsRegexDenoted.one","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#one","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The language `{[]}` is denoted by `1`.","title":"The language of the empty word is regex-denoted"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/char","kind":"node","lean":"Cslib.Language.KleeneConverse.IsRegexDenoted.char","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#char","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"A singleton `{[a]}` is denoted by `char a`.","title":"Single-letter languages are regex-denoted"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/add","kind":"node","lean":"Cslib.Language.KleeneConverse.IsRegexDenoted.add","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#add","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Denoted languages are closed under union.","title":"Regex-denoted languages are closed under union"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/mul","kind":"node","lean":"Cslib.Language.KleeneConverse.IsRegexDenoted.mul","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#mul","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Denoted languages are closed under concatenation.","title":"Regex-denoted languages are closed under concatenation"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/kstar","kind":"node","lean":"Cslib.Language.KleeneConverse.IsRegexDenoted.kstar","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#kstar","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Denoted languages are closed under Kleene star.","title":"Regex-denoted languages are closed under Kleene star"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/sum","kind":"node","lean":"Cslib.Language.KleeneConverse.IsRegexDenoted.sum","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#sum","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Denoted languages are closed under finite unions.","title":"Regex-denoted languages are closed under finite unions"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/mem-sum","kind":"node","lean":"Cslib.Language.KleeneConverse.mem_sum","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#mem-sum","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Membership in a finite union of languages.","title":"Membership in a finite union of languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/nil-not-mem-mul-of-nil-not-mem-right","kind":"node","lean":"Cslib.Language.KleeneConverse.nil_notMem_mul_of_nil_notMem_right","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#nil-not-mem-mul-of-nil-not-mem-right","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The empty word is not in a concatenation whose right factor excludes it.","title":"Empty word outside a concatenation"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/coeff","kind":"node","lean":"Cslib.Language.KleeneConverse.coeff","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#coeff","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The system of equations of a DFA at a state: the coefficient of `X j` in the equation for `X q`, the finite union of the singletons `{[a]}` over the letters taking `q` to `j`.","title":"Coefficient in the equation system of a DFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/const-term","kind":"node","lean":"Cslib.Language.KleeneConverse.constTerm","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#const-term","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The constant term of the equation for `X q`: `1` on accepting states, `0` otherwise.","title":"Constant term in the equation system of a DFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/from-state","kind":"node","lean":"Cslib.Language.KleeneConverse.fromState","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#from-state","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The language accepted from a state: the unknown `X q` of the system.","title":"Language accepted from a DFA state"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/mem-coeff","kind":"node","lean":"Cslib.Language.KleeneConverse.mem_coeff","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#mem-coeff","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Membership in a coefficient: the word is a single letter taking `q` to `j`.","title":"Membership in a DFA equation coefficient"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/nil-not-mem-coeff","kind":"node","lean":"Cslib.Language.KleeneConverse.nil_notMem_coeff","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#nil-not-mem-coeff","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"No coefficient contains the empty word.","title":"DFA equation coefficients exclude the empty word"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/denoted-coeff","kind":"node","lean":"Cslib.Language.KleeneConverse.denoted_coeff","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#denoted-coeff","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Every coefficient is denoted by a regular expression.","title":"DFA equation coefficients are regex-denoted"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/denoted-const-term","kind":"node","lean":"Cslib.Language.KleeneConverse.denoted_constTerm","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#denoted-const-term","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Every constant term is denoted by a regular expression.","title":"DFA equation constant terms are regex-denoted"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/from-state-eq","kind":"node","lean":"Cslib.Language.KleeneConverse.fromState_eq","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#from-state-eq","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The DFA equations: from `q`, either step on a letter and continue, or stop if `q` accepts.","title":"Equation system satisfied by DFA state languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/elim-coeff","kind":"node","lean":"Cslib.Language.KleeneConverse.elimCoeff","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#elim-coeff","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The coefficient of the reduced system obtained by eliminating the last state: the old coefficient, plus the detour through the last state, which may loop there any number of times.","title":"Coefficient after eliminating the last state"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/elim-const","kind":"node","lean":"Cslib.Language.KleeneConverse.elimConst","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#elim-const","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The constant term of the reduced system obtained by eliminating the last state.","title":"Constant term after eliminating the last state"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/exists-regex-of-system","kind":"node","lean":"Cslib.Language.KleeneConverse.exists_regex_of_system","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#exists-regex-of-system","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"If the languages `X i` satisfy `X i = (\u2211 j, A i j * X j) + B i` with every coefficient `A i j` excluding the empty word and every coefficient and constant term denoted by a regular expression, then every `X i` is denoted by a regular expression. The proof eliminates the last state with Arden's lemma and applies the induction hypothesis to the reduced system.","title":"Solving a system of language equations"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/exists-regex-of-dfa-fin","kind":"node","lean":"Cslib.Language.KleeneConverse.exists_regex_of_dfa_fin","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#exists-regex-of-dfa-fin","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The language of a DFA with `Fin n` states is denoted by a regular expression.","title":"Languages of DFAs on finitely many numbered states are regex-denoted"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/exists-regex-of-dfa","kind":"node","lean":"Cslib.Language.KleeneConverse.exists_regex_of_dfa","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#exists-regex-of-dfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The language of a DFA over a finite state type is denoted by a regular expression.","title":"Kleene's theorem: DFA languages are regex-denoted"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/regex-letters","kind":"node","lean":"Cslib.Language.KleeneConverse.regexLetters","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#regex-letters","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The letters occurring in a regular expression.","title":"Letters occurring in a regular expression"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/mem-matches-letters","kind":"node","lean":"Cslib.Language.KleeneConverse.mem_matches'_letters","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#mem-matches-letters","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Every letter of a word matched by `r` occurs in `r`.","title":"Letters of matched words occur in the regular expression"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/singleton-dfa","kind":"node","lean":"Cslib.Language.KleeneConverse.singletonDFA","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#singleton-dfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The three-state DFA accepting exactly the one-letter words.","title":"DFA accepting the one-letter words"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/singleton-dfa-accepts","kind":"node","lean":"Cslib.Language.KleeneConverse.singletonDFA_accepts","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#singleton-dfa-accepts","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"The one-letter-word DFA accepts exactly the words of length one.","title":"The one-letter-word DFA accepts exactly words of length one"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/exists-regex","kind":"node","lean":"Cslib.Language.IsRegular.exists_regex","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#exists-regex","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Kleene's theorem, converse direction (Sipser, *Introduction to the Theory of Computation*, 3rd ed., Theorem 1.54): every regular language over a finite alphabet is denoted by a regular expression. Source: M. Sipser, *Introduction to the Theory of Computation*, 3rd ed., Theorem 1.54.","title":"Kleene's theorem, converse direction"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/is-regular-iff-exists-regex","kind":"node","lean":"Cslib.Language.isRegular_iff_exists_regex","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#is-regular-iff-exists-regex","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Kleene's theorem (Sipser, 3rd ed., Theorem 1.54): over a finite alphabet, a language is regular if and only if it is denoted by a regular expression. Source: M. Sipser, *Introduction to the Theory of Computation*, 3rd ed., Theorem 1.54.","title":"Kleene's theorem"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/exists-regular-not-regex","kind":"node","lean":"Cslib.Language.exists_regular_not_regex","parent":33277,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem/index.html#exists-regular-not-regex","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/kleene-s-theorem","status":"fully_proved","summary":"Over an infinite alphabet, the language of one-letter words is regular but is denoted by no regular expression, since a regular expression mentions only finitely many letters. Source: M. Sipser, *Introduction to the Theory of Computation*, 3rd ed., Theorem 1.54 (finite alphabet).","title":"Sharpness of the finite-alphabet hypothesis"},{"data_url":"scopes/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines.json","id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","kind":"scope","navigation":"module","parent":33245,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","title":"Sequential transducers, Mealy and Moore machines"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/sequential-transducer","kind":"node","lean":"Cslib.Automata.SequentialTransducer","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#sequential-transducer","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"A sequential transducer with states in a type $Q$, input letters in a type $\\Sigma$ and output letters in a type $\\Gamma$ consists of a transition function assigning to each state $q \\in Q$ and input letter $a \\in \\Sigma$ a successor state in $Q$ and an output word over $\\Gamma$, a start state in $Q$, a set of accepting states $F \\subseteq Q$, an initial output word over $\\Gamma$, and, for each state in $Q$, a\u2026","title":"Sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-from","kind":"node","lean":"Cslib.Automata.SequentialTransducer.runFrom","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-from","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"The run of a sequential transducer $S$ from a state $q$ on an input word $a_1 \\cdots a_n$ is the pair $(q_n, w_1 w_2 \\cdots w_n)$, where $q_0 = q$ and, for each $1 \\le i \\le n$, $S$ moves from $q_{i-1}$ on $a_i$ to $q_i$ emitting the word $w_i$. The initial output and the final outputs of $S$ are not included.","title":"Run of a sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-state","kind":"node","lean":"Cslib.Automata.SequentialTransducer.runState","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-state","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a sequential transducer $S$, a state $q$ and an input word $w$, the final state of the run of $S$ from $q$ on $w$ is obtained by starting in $q$, reading the letters of $w$ in order, moving at each letter to the successor state given by the transition function of $S$, and taking the state reached at the end.","title":"Final state of a sequential transducer run"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-out","kind":"node","lean":"Cslib.Automata.SequentialTransducer.runOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-out","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a sequential transducer $S$, a state $q$ and an input word $x$, the output word of the run from $q$ on $x$ is the concatenation of the output words that $S$ emits on the successive letters of $x$, starting in $q$ and moving to the successor state after each letter; the initial and final outputs are not included.","title":"Output word of a sequential transducer run"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-state-cons","kind":"node","lean":"Cslib.Automata.SequentialTransducer.runState_cons","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-state-cons","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S$ be a sequential transducer, $q$ a state, $a$ an input letter and $w$ an input word, and let $q'$ be the successor state of $q$ on $a$ under the transition function of $S$. Then the final state of the run of $S$ from $q$ on the word $aw$ equals the final state of the run of $S$ from $q'$ on $w$.","title":"Final state of a sequential transducer run on a nonempty word"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-out-cons","kind":"node","lean":"Cslib.Automata.SequentialTransducer.runOut_cons","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-out-cons","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S$ be a sequential transducer, $q$ a state, $a$ an input letter and $x$ an input word, and let $q'$ and $w$ be the successor state and the output word of $S$ in state $q$ on letter $a$. The output word of the run of $S$ from $q$ on $ax$ is $w$ followed by the output word of the run from $q'$ on $x$.","title":"Output of a sequential transducer on a nonempty word"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-state-append","kind":"node","lean":"Cslib.Automata.SequentialTransducer.runState_append","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-state-append","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a sequential transducer $S$, a state $q$ and input words $x$ and $y$, the final state of the run of $S$ from $q$ on $xy$ is the final state of the run on $y$ from the final state of the run from $q$ on $x$.","title":"Final state of a sequential transducer on a concatenation"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-out-append","kind":"node","lean":"Cslib.Automata.SequentialTransducer.runOut_append","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-out-append","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a sequential transducer $S$, a state $q$ and input words $x$ and $y$, the output word of the run of $S$ from $q$ on $xy$ is the output word of the run from $q$ on $x$ followed by the output word of the run on $y$ from the final state of the run from $q$ on $x$.","title":"Output of a sequential transducer on a concatenation"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/translates","kind":"node","lean":"Cslib.Automata.SequentialTransducer.Translates","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#translates","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"A sequential transducer $S$ translates an input word $x$ to an output word $y$ when the run of $S$ on $x$ from its start state ends in an accepting state $q$ and $y$ is the concatenation of the initial output of $S$, the output words emitted by the transitions along the run, and the final output of $S$ at $q$.","title":"Translation by a sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/comp","kind":"node","lean":"Cslib.Automata.SequentialTransducer.comp","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#comp","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from $I$ to $M$ with start state $s_1$, initial output $\\iota_1$ and final output map $\\phi_1$, and let $S_2$ be one from $M$ to $O$ with start state $s_2$, initial output $\\iota_2$ and final output map $\\phi_2$. For a transducer $S$, a state $q$ and a word $w$, write $\\delta_S(q, w)$ for the state $S$ reaches by reading $w$ from $q$ and $\\lambda_S(q, w)$ for the concatenation\u2026","title":"Composition of sequential transducers"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/comp-step-fst","kind":"node","lean":"Cslib.Automata.SequentialTransducer.comp_step_fst","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#comp-step-fst","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from $I$ to $M$, $S_2$ one from $M$ to $O$, and $C$ their composition. Let $q_1$ be a state of $S_1$, $q_2$ a state of $S_2$ and $a \\in I$ a letter, and suppose that $S_1$, reading $a$ in state $q_1$, moves to $q_1'$ and emits the word $w$. Then $C$, reading $a$ in state $(q_1, q_2)$, moves to $(q_1', q_2')$, where $q_2'$ is the state $S_2$ reaches by reading $w$ from $q_2$.","title":"One step of a composite transducer: the successor state"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/comp-step-snd","kind":"node","lean":"Cslib.Automata.SequentialTransducer.comp_step_snd","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#comp-step-snd","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from $I$ to $M$, $S_2$ one from $M$ to $O$, and $C$ their composition. Let $q_1$ be a state of $S_1$, $q_2$ a state of $S_2$ and $a \\in I$ a letter, and let $w$ be the word $S_1$ emits when it reads $a$ in state $q_1$. Then the word $C$ emits when it reads $a$ in state $(q_1, q_2)$ is the concatenation of the words $S_2$ emits while reading $w$ from $q_2$.","title":"One step of a composite transducer: the output"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/comp-run-state","kind":"node","lean":"Cslib.Automata.SequentialTransducer.comp_runState","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#comp-run-state","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from $I$ to $M$, $S_2$ one from $M$ to $O$, and $C$ their composition. For a transducer $S$, a state $q$ and a word $w$, write $\\delta_S(q, w)$ for the state $S$ reaches by reading $w$ from $q$ and $\\lambda_S(q, w)$ for the concatenation of the words it emits along the way (without initial or final output). Then for all states $q_1$ of $S_1$ and $q_2$ of $S_2$ and every word $x$\u2026","title":"States reached by a composite sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/comp-run-out","kind":"node","lean":"Cslib.Automata.SequentialTransducer.comp_runOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#comp-run-out","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from $I$ to $M$, $S_2$ one from $M$ to $O$, and $C$ their composition. For a transducer $S$, a state $q$ and a word $w$, write $\\lambda_S(q, w)$ for the concatenation of the words $S$ emits while reading $w$ from $q$ (without initial or final output). Then for all states $q_1$ of $S_1$ and $q_2$ of $S_2$ and every word $x$ over $I$,\u2026","title":"Output of a composite sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/comp-run-state-init-out","kind":"node","lean":"Cslib.Automata.SequentialTransducer.comp_runState_initOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#comp-run-state-init-out","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from $I$ to $M$ and $S_2$ one from $M$ to $O$, with start states $s_1$ and $s_2$. For a transducer $S$, a state $q$ and a word $w$, write $\\delta_S(q, w)$ for the state $S$ reaches by reading $w$ from $q$ and $\\lambda_S(q, w)$ for the concatenation of the words it emits along the way (without initial or final output); juxtaposition of words is concatenation. For a word $x$ over\u2026","title":"State of the second transducer after the first one's whole output"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/comp-run-out-init-out","kind":"node","lean":"Cslib.Automata.SequentialTransducer.comp_runOut_initOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#comp-run-out-init-out","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from $I$ to $M$ and $S_2$ one from $M$ to $O$, with start states $s_1$ and $s_2$. For a transducer $S$, a state $q$ and a word $w$, write $\\delta_S(q, w)$ for the state $S$ reaches by reading $w$ from $q$ and $\\lambda_S(q, w)$ for the concatenation of the words it emits along the way (without initial or final output); juxtaposition of words is concatenation. For a word $x$ over\u2026","title":"Output of the second transducer on the first one's whole output"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/comp-output-eq","kind":"node","lean":"Cslib.Automata.SequentialTransducer.comp_output_eq","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#comp-output-eq","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from $I$ to $M$, $S_2$ one from $M$ to $O$, and $C$ their composition. For a sequential transducer $S$ with start state $s$, initial output $\\iota$ and final output map $\\phi$, and a word $x$, let $\\delta_S(s, x)$ be the state $S$ reaches by reading $x$ from $s$ and $\\lambda_S(s, x)$ the concatenation of the words it emits along the way, and call the concatenation\u2026","title":"Total output of a composite sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/translates-comp-iff","kind":"node","lean":"Cslib.Automata.SequentialTransducer.translates_comp_iff","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#translates-comp-iff","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S_1$ be a sequential transducer from an alphabet $A$ to an alphabet $B$, and $S_2$ one from $B$ to an alphabet $C$. For every word $x$ over $A$ and every word $y$ over $C$, the composition of $S_1$ and $S_2$ translates $x$ to $y$ if and only if there is a word $z$ over $B$ such that $S_1$ translates $x$ to $z$ and $S_2$ translates $z$ to $y$.","title":"Translations of a composite sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/preimage","kind":"node","lean":"Cslib.Automata.SequentialTransducer.preimage","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#preimage","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S$ be a sequential transducer with input alphabet $\\Sigma$ and output alphabet $\\Gamma$, and let $L$ be a language over $\\Gamma$. The preimage of $L$ under $S$ is the language over $\\Sigma$ consisting of the words $x$ for which there is a word $y \\in L$ such that $S$ translates $x$ to $y$.","title":"Preimage of a language under a sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/preimage-dfa","kind":"node","lean":"Cslib.Automata.SequentialTransducer.preimageDFA","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#preimage-dfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S$ be a sequential transducer with state type $Q$, input alphabet $\\Sigma$ and output alphabet $\\Gamma$, and let $M$ be a DFA over $\\Gamma$ with state type $P$. The product DFA of $S$ and $M$ is the DFA over $\\Sigma$ with state type $Q \\times P$ defined as follows. In state $(q, r)$ on a letter $a$, if $S$ moves from $q$ to $q'$ emitting the word $w$, it moves to $(q', r')$, where $r'$ is the state $M$ reaches\u2026","title":"Product DFA of a sequential transducer and a DFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/preimage-dfa-eval-from","kind":"node","lean":"Cslib.Automata.SequentialTransducer.preimageDFA_evalFrom","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#preimage-dfa-eval-from","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S$ be a sequential transducer and $M$ a DFA over the output alphabet of $S$. For every input word $x$, state $q$ of $S$ and state $r$ of $M$, the product DFA of $S$ and $M$, started in $(q, r)$ and run on $x$, ends in the state $(q', r')$, where $q'$ is the final state of the run of $S$ from $q$ on $x$ and $r'$ is the state that $M$ reaches from $r$ on the output word of that run.","title":"Runs of the product DFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/preimage-dfa-accepts","kind":"node","lean":"Cslib.Automata.SequentialTransducer.preimageDFA_accepts","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#preimage-dfa-accepts","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S$ be a sequential transducer with output alphabet $\\Gamma$ and let $M$ be a DFA over $\\Gamma$. The language accepted by the product DFA of $S$ and $M$ equals the preimage under $S$ of the language accepted by $M$, that is, the set of input words that $S$ translates to some word accepted by $M$.","title":"The product DFA recognizes the preimage"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/is-regular-preimage","kind":"node","lean":"Cslib.Automata.SequentialTransducer.isRegular_preimage","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#is-regular-preimage","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $S$ be a sequential transducer with finitely many states, input alphabet $I$ and output alphabet $O$, and let $L$ be a regular language over $O$. Then the preimage of $L$ under $S$, the set of words over $I$ that $S$ translates to some word of $L$, is regular.","title":"Inverse sequential transduction preserves regular languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/mealy-machine","kind":"node","lean":"Cslib.Automata.MealyMachine","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#mealy-machine","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"A Mealy machine with state set $Q$, input alphabet $I$ and output alphabet $O$ consists of a start state in $Q$ and a transition function that, given a state $q \\in Q$ and an input letter $a \\in I$, returns a successor state in $Q$ and a single output letter in $O$; thus the output of a transition depends on the state and the input letter.","title":"Mealy machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-from-2","kind":"node","lean":"Cslib.Automata.MealyMachine.runFrom","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-from-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"The run of a Mealy machine $M$ from a state $q$ on an input word $a_1 \\cdots a_n$ is the pair $(q_n, o_1 \\cdots o_n)$, where $q_0 = q$ and, for each $1 \\le i \\le n$, $M$ moves from $q_{i-1}$ on $a_i$ to $q_i$ emitting the output letter $o_i$.","title":"Run of a Mealy machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-state-2","kind":"node","lean":"Cslib.Automata.MealyMachine.runState","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-state-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Mealy machine $M$, a state $q$ and an input word $x$, the final state of the run from $q$ on $x$ is the state reached by starting in $q$ and moving, on each successive letter of $x$, to the successor state given by the transition function of $M$; on the empty word it is $q$.","title":"Final state of a Mealy machine run"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-out-2","kind":"node","lean":"Cslib.Automata.MealyMachine.runOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-out-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Mealy machine $M$, a state $q$ and an input word $x$, the output word of $M$ from $q$ on $x$ is the second component of the run of $M$ from $q$ on $x$: the word of output letters emitted along that run, one per input letter.","title":"Output word of a Mealy machine run"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-state-cons-2","kind":"node","lean":"Cslib.Automata.MealyMachine.runState_cons","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-state-cons-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $M$ be a Mealy machine, $q$ a state, $a$ an input letter and $x$ an input word, and let $q'$ be the successor state of $M$ in state $q$ on letter $a$. The final state of the run of $M$ from $q$ on $ax$ is the final state of the run from $q'$ on $x$.","title":"Final state of a Mealy machine on a nonempty word"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-out-cons-2","kind":"node","lean":"Cslib.Automata.MealyMachine.runOut_cons","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-out-cons-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $M$ be a Mealy machine, $q$ a state, $a$ an input letter and $x$ an input word, and let $q'$ and $o$ be the successor state and the output letter of $M$ in state $q$ on letter $a$. The output word of the run of $M$ from $q$ on $ax$ is $o$ followed by the output word of the run from $q'$ on $x$.","title":"Output of a Mealy machine on a nonempty word"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/transduce","kind":"node","lean":"Cslib.Automata.MealyMachine.transduce","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#transduce","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $M$ be a Mealy machine with start state $q_0$. On an input word $a_1 a_2 \\cdots a_n$, for each $i$ let $q_i$ and $b_i$ be the state that $M$ moves to and the letter that it outputs when it reads $a_i$ in state $q_{i-1}$. The transduction of $M$ on this word is the output word $b_1 b_2 \\cdots b_n$.","title":"Transduction of a Mealy machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/to-sequential","kind":"node","lean":"Cslib.Automata.MealyMachine.toSequential","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#to-sequential","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"The sequential transducer of a Mealy machine $M$ has the states and the start state of $M$; on an input letter $a$ in a state $q$ it moves to the state that $M$ moves to and emits the one-letter word consisting of the letter that $M$ outputs on this transition. Every state accepts, and the initial and final outputs are empty.","title":"Mealy machine as a sequential transducer"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/seq-run-state-eq","kind":"node","lean":"Cslib.Automata.MealyMachine.seq_runState_eq","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#seq-run-state-eq","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Mealy machine $M$, a state $q$ and an input word $w$, the final state of the run from $q$ on $w$ of the sequential transducer associated with $M$ equals the final state of the run of $M$ from $q$ on $w$.","title":"The sequential transducer of a Mealy machine has the same final states"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/seq-run-out-eq","kind":"node","lean":"Cslib.Automata.MealyMachine.seq_runOut_eq","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#seq-run-out-eq","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Mealy machine $M$, a state $q$ and an input word $w$, the output word of the run from $q$ on $w$ of the sequential transducer associated with $M$ (excluding its initial and final outputs) equals the output word of the run of $M$ from $q$ on $w$.","title":"The sequential transducer of a Mealy machine has the same run outputs"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/to-sequential-accept","kind":"node","lean":"Cslib.Automata.MealyMachine.toSequential_accept","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#to-sequential-accept","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Mealy machine $M$, the set of accepting states of the sequential transducer associated with $M$ is the set of all states.","title":"Every state of the sequential transducer of a Mealy machine is accepting"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/to-sequential-init-out","kind":"node","lean":"Cslib.Automata.MealyMachine.toSequential_initOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#to-sequential-init-out","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For every Mealy machine $M$, the initial output of the sequential transducer of $M$ is the empty word.","title":"Initial output of the sequential transducer of a Mealy machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/to-sequential-final-out","kind":"node","lean":"Cslib.Automata.MealyMachine.toSequential_finalOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#to-sequential-final-out","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For every Mealy machine $M$ and every state $q$, the final output of the sequential transducer of $M$ at $q$ is the empty word.","title":"Final outputs of the sequential transducer of a Mealy machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/translates-to-sequential-iff","kind":"node","lean":"Cslib.Automata.MealyMachine.translates_toSequential_iff","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#translates-to-sequential-iff","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For every Mealy machine $M$, input word $x$ and output word $y$, the sequential transducer of $M$ translates $x$ to $y$ if and only if the transduction of $M$ on $x$ is $y$.","title":"The sequential transducer of a Mealy machine computes its transduction"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/moore-machine","kind":"node","lean":"Cslib.Automata.MooreMachine","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#moore-machine","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"A Moore machine with state type $Q$, input alphabet $\\Sigma$ and output alphabet $\\Gamma$ consists of a transition function, assigning to each state $q$ and input letter $a$ a successor state $\\delta(q, a) \\in Q$, an output function $\\lambda : Q \\to \\Gamma$, and a start state $q_0 \\in Q$. The output depends only on the state.","title":"Moore machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-from-3","kind":"node","lean":"Cslib.Automata.MooreMachine.runFrom","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-from-3","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"The run of a Moore machine $M$ with output function $\\lambda$ from a state $q$ on an input word $a_1 \\cdots a_n$ is the pair $(q_n, \\lambda(q_0) \\lambda(q_1) \\cdots \\lambda(q_n))$, where $q_0 = q$ and, for each $1 \\le i \\le n$, $q_i$ is the successor of $q_{i-1}$ on $a_i$. The output word has $n + 1$ letters and begins with the output of the starting state.","title":"Run of a Moore machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-state-3","kind":"node","lean":"Cslib.Automata.MooreMachine.runState","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-state-3","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Moore machine $M$, a state $q$ and an input word $x$, the final state of the run from $q$ on $x$ is the state reached by starting in $q$ and moving, on each successive letter of $x$, to the successor state given by the transition function of $M$; on the empty word it is $q$.","title":"Final state of a Moore machine run"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-out-3","kind":"node","lean":"Cslib.Automata.MooreMachine.runOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-out-3","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Moore machine $M$, a state $q$ and an input word $x$, the output word of $M$ from $q$ on $x$ is the second component of the run of $M$ from $q$ on $x$: the outputs of the states visited along that run, starting with the output of $q$; it has one more letter than $x$.","title":"Output word of a Moore machine run"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-state-cons-3","kind":"node","lean":"Cslib.Automata.MooreMachine.runState_cons","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-state-cons-3","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Moore machine $M$ with transition function $\\delta$, a state $q$, an input letter $a$ and an input word $w$, the final state of the run of $M$ from $q$ on the word $aw$ equals the final state of the run of $M$ from $\\delta(q, a)$ on $w$.","title":"Final state of a Moore machine run on a nonempty word"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-out-cons-3","kind":"node","lean":"Cslib.Automata.MooreMachine.runOut_cons","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-out-cons-3","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $M$ be a Moore machine, $q$ a state, $a$ an input letter and $x$ an input word, and let $q'$ be the successor state of $M$ in state $q$ on letter $a$. The output word of the run of $M$ from $q$ on $ax$ is the output of $q$ followed by the output word of the run from $q'$ on $x$.","title":"Output of a Moore machine on a nonempty word"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/transduce-2","kind":"node","lean":"Cslib.Automata.MooreMachine.transduce","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#transduce-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $M$ be a Moore machine with start state $q_0$, and write $o(q)$ for the output of a state $q$. On an input word $a_1 a_2 \\cdots a_n$, let $q_i$ be the state that $M$ moves to from $q_{i-1}$ on $a_i$. The transduction of $M$ on this word is the output word $o(q_0)\\, o(q_1) \\cdots o(q_n)$, of length $n + 1$.","title":"Transduction of a Moore machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/to-sequential-2","kind":"node","lean":"Cslib.Automata.MooreMachine.toSequential","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#to-sequential-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"Let $M$ be a Moore machine with transition function $\\delta$, output function $\\lambda$ and start state $q_0$. Its sequential transducer has the same states and the start state $q_0$; reading a letter $a$ in state $q$ it moves to $\\delta(q, a)$ and emits the one-letter word $\\lambda(\\delta(q, a))$; every state is accepting; the initial output is the one-letter word $\\lambda(q_0)$; and every final output is empty.","title":"The sequential transducer of a Moore machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/run-out-eq-cons","kind":"node","lean":"Cslib.Automata.MooreMachine.runOut_eq_cons","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#run-out-eq-cons","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Moore machine $M$, a state $q$ and an input word $x$, the output word of the run of $M$ from $q$ on $x$ is the output of $q$ followed by the output word of the run of the sequential rendering of $M$ from $q$ on $x$.","title":"Moore machine output via its sequential rendering"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/to-sequential-accept-2","kind":"node","lean":"Cslib.Automata.MooreMachine.toSequential_accept","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#to-sequential-accept-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Moore machine $M$, the set of accepting states of the sequential transducer associated with $M$ is the set of all states.","title":"Every state of the sequential transducer of a Moore machine is accepting"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/to-sequential-init-out-2","kind":"node","lean":"Cslib.Automata.MooreMachine.toSequential_initOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#to-sequential-init-out-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For every Moore machine $M$, the initial output of the sequential transducer of $M$ is the one-letter word consisting of the output of the start state of $M$.","title":"Initial output of the sequential transducer of a Moore machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/to-sequential-final-out-2","kind":"node","lean":"Cslib.Automata.MooreMachine.toSequential_finalOut","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#to-sequential-final-out-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For a Moore machine $M$ and every state $q$, the final output word at $q$ of the sequential transducer associated with $M$ is empty.","title":"Final outputs of the sequential transducer of a Moore machine"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/translates-to-sequential-iff-2","kind":"node","lean":"Cslib.Automata.MooreMachine.translates_toSequential_iff","parent":33308,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines/index.html#translates-to-sequential-iff-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/sequential-transducers-mealy-and-moore-machines","status":"fully_proved","summary":"For every Moore machine $M$, input word $x$ and output word $y$, the sequential transducer of $M$ translates $x$ to $y$ if and only if the transduction of $M$ on $x$ is $y$.","title":"The sequential transducer of a Moore machine computes its transduction"},{"data_url":"scopes/TheoryOfComputation/regular-languages/automata-and-expressions/star-height.json","id":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height","kind":"scope","navigation":"module","parent":33245,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/star-height/index.html","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height","status":"fully_proved","title":"Star height"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height/regular-star-height","kind":"node","lean":"Cslib.Language.regularStarHeight","parent":33359,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/star-height/index.html#regular-star-height","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height","status":"fully_proved","summary":"The star height of an ordinary regular expression is the nesting depth of the Kleene star in it: the expressions for the empty language and the empty word and the single letters have star height $0$; a union $e_1 + e_2$ and a concatenation $e_1 e_2$ have the larger of the star heights of $e_1$ and $e_2$; and $e^*$ has star height one more than $e$.","title":"Star height of a regular expression"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height/gen-reg-exp","kind":"node","lean":"Cslib.Language.GenRegExp","parent":33359,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/star-height/index.html#gen-reg-exp","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height","status":"fully_proved","summary":"A generalized regular expression over an alphabet $\\alpha$ is built from ordinary regular expressions over $\\alpha$ by complement, union, concatenation and Kleene star: every ordinary regular expression over $\\alpha$ is one, and if $e$, $e_1$, $e_2$ are generalized regular expressions, then so are the complement $\\neg e$, the union $e_1 + e_2$, the concatenation $e_1 e_2$ and the star $e^*$.","title":"Generalized regular expressions"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height/denote","kind":"node","lean":"Cslib.Language.GenRegExp.denote","parent":33359,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/star-height/index.html#denote","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height","status":"fully_proved","summary":"For an alphabet $\\alpha$, the language over $\\alpha$ denoted by a generalized regular expression over $\\alpha$ is defined recursively, for generalized regular expressions $e$, $e_1$, $e_2$: an ordinary regular expression denotes the language it matches; the complement $\\neg e$ denotes the complement, in the set of all words over $\\alpha$, of the language of $e$; the union $e_1 + e_2$ denotes the union and the\u2026","title":"Language of a generalized regular expression"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height/star-height","kind":"node","lean":"Cslib.Language.GenRegExp.starHeight","parent":33359,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/star-height/index.html#star-height","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height","status":"fully_proved","summary":"The generalized star height of a generalized regular expression is the nesting depth of the Kleene star in it, counting the stars inside its ordinary regular subexpressions: for generalized regular expressions $e$, $e_1$, $e_2$, an ordinary regular expression has its usual star height; the complement $\\neg e$ has the star height of $e$; the union $e_1 + e_2$ and the concatenation $e_1 e_2$ have the larger of the\u2026","title":"Generalized star height"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height/has-gen-height-le","kind":"node","lean":"Cslib.Language.HasGenHeightLe","parent":33359,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/star-height/index.html#has-gen-height-le","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/star-height","status":"fully_proved","summary":"A language $L$ over an alphabet $\\alpha$ has generalized star height at most $n$, for a natural number $n$, if some generalized regular expression over $\\alpha$ whose generalized star height is at most $n$ denotes $L$.","title":"Bounded generalized star height"},{"data_url":"scopes/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction.json","id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","kind":"scope","navigation":"module","parent":33245,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","title":"Thompson's construction"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/zero-aut","kind":"node","lean":"Cslib.Automata.Thompson.zeroAut","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#zero-aut","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"Over an alphabet $\\alpha$, this is the \u03b5-NFA whose set of states is empty; consequently it has no start states, no accepting states and no transitions.","title":"The \u03b5-NFA with no states"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/one-aut","kind":"node","lean":"Cslib.Automata.Thompson.oneAut","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#one-aut","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"The \u03b5-NFA over the alphabet $\\alpha$ with a single state, which is both its start state and its accepting state, and no moves at all (neither letter moves nor \u03b5-moves).","title":"One-state automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/char-aut","kind":"node","lean":"Cslib.Automata.Thompson.charAut","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#char-aut","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"For a letter $a$ of the alphabet $\\alpha$, this is the \u03b5-NFA over $\\alpha$ with two states, a start state and an accepting state, and a single move, which reads $a$ and goes from the start state to the accepting state; it has no other moves and no \u03b5-moves.","title":"Single-letter automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/union","kind":"node","lean":"Cslib.Automata.Thompson.union","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#union","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"Given \u03b5-NFAs $M$ with state set $\\sigma$ and $N$ with state set $\\tau$ over the same alphabet, their union is the \u03b5-NFA on the disjoint union $\\sigma \\sqcup \\tau$ whose moves are the moves of $M$ and the moves of $N$, whose start states are the start states of $M$ and of $N$, and whose accepting states are the accepting states of $M$ and of $N$.","title":"Union of \u03b5-NFAs"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/concat","kind":"node","lean":"Cslib.Automata.Thompson.concat","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#concat","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"Given \u03b5-NFAs $M$ with state set $\\sigma$ and $N$ with state set $\\tau$ over the same alphabet, their concatenation is the \u03b5-NFA on the disjoint union $\\sigma \\sqcup \\tau$ whose moves are the moves of $M$, the moves of $N$, and an \u03b5-move from every accepting state of $M$ to every start state of $N$. Its start states are the start states of $M$ and its accepting states are the accepting states of $N$.","title":"Concatenation of \u03b5-NFAs"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/star","kind":"node","lean":"Cslib.Automata.Thompson.star","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#star","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"Given an \u03b5-NFA $M$ with state set $\\sigma$, its star automaton has the states of $M$ together with one new state $q_0$, which is its only start state and its only accepting state. Its moves are the moves of $M$, an \u03b5-move from $q_0$ to every start state of $M$, and an \u03b5-move from every accepting state of $M$ to $q_0$.","title":"Kleene star of an \u03b5-NFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/state","kind":"node","lean":"Cslib.Automata.Thompson.State","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#state","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"The state set $S(r)$ of Thompson's automaton of a regular expression $r$ is defined by structural recursion: $S(0)$ is empty, $S(\\varepsilon)$ has one element, $S(a)$ has two elements for a letter $a$, both $S(P + Q)$ and $S(PQ)$ are the disjoint union $S(P) \\sqcup S(Q)$, and $S(P^*)$ is $S(P)$ together with one new element.","title":"States of Thompson's automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/to-nfa","kind":"node","lean":"Cslib.Automata.Thompson.to\u03b5NFA","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#to-nfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"To each regular expression $r$ over an alphabet $\\alpha$ it assigns an \u03b5-NFA over $\\alpha$ with state set $S(r)$, the states of Thompson's automaton of $r$, by structural recursion: $0$ goes to the automaton with no states, $\\varepsilon$ to the one-state automaton, a letter $a$ to the single-letter automaton of $a$, $P + Q$ to the union of the automata of $P$ and $Q$, $PQ$ to their concatenation, and $P^*$ to the\u2026","title":"Thompson's construction"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/accepts-to-nfa","kind":"node","lean":"Cslib.Automata.Thompson.accepts_to\u03b5NFA","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#accepts-to-nfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"The automaton accepts exactly the words the regular expression matches.","title":"Correctness of Thompson's construction"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/num-states","kind":"node","lean":"Cslib.Automata.Thompson.numStates","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#num-states","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"The state count $n(r)$ of a regular expression $r$ is defined by structural recursion: $n(0) = 0$, $n(\\varepsilon) = 1$, $n(a) = 2$ for a letter $a$, $n(P + Q) = n(P) + n(Q)$, $n(PQ) = n(P) + n(Q)$ and $n(P^*) = n(P) + 1$.","title":"Thompson state count"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/card-state","kind":"node","lean":"Cslib.Automata.Thompson.card_state","parent":33365,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction/index.html#card-state","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/thompson-s-construction","status":"fully_proved","summary":"For every regular expression $r$, the number of states of Thompson's automaton of $r$ is $n(r)$, the Thompson state count of $r$: $n(0) = 0$, $n(\\varepsilon) = 1$, $n(a) = 2$ for a letter $a$, $n(P + Q) = n(P) + n(Q)$, $n(PQ) = n(P) + n(Q)$ and $n(P^*) = n(P) + 1$.","title":"Number of states of Thompson's automaton"},{"data_url":"scopes/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata.json","id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","kind":"scope","navigation":"module","parent":33245,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","title":"Two-way finite automata"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/two-way-dfa","kind":"node","lean":"Cslib.Automata.TwoWayDFA","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#two-way-dfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"A deterministic two-way automaton with state set $Q$ over the alphabet $\\Sigma$ consists of a partial transition function, which assigns to each state $q \\in Q$ and input symbol $a \\in \\Sigma$ at most one pair of a successor state and a head move (left, stay or right), together with a single start state and a set of accepting states.","title":"Deterministic two-way automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/to-two-way-na","kind":"node","lean":"Cslib.Automata.TwoWayDFA.toTwoWayNA","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#to-two-way-na","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"A deterministic two-way automaton $A$ is viewed as the nondeterministic two-way automaton with the same states and accepting states, whose only start state is the start state of $A$, and which can go from state $q$ reading a symbol $a$ to state $q'$ with head move $m$ exactly when the step function of $A$ at $q$ and $a$ returns $q'$ and $m$.","title":"A deterministic two-way automaton as a nondeterministic one"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/tr-iff","kind":"node","lean":"Cslib.Automata.TwoWayDFA.tr_iff","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#tr-iff","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"Let $A$ be a deterministic two-way automaton, viewed as a nondeterministic two-way automaton. On reading the symbol $x$ in state $q$, it has a transition to state $q'$ with head move $m$ if and only if the transition function of $A$ sends $(q, x)$ to the pair $(q', m)$.","title":"Transition relation of a deterministic two-way automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/tr-functional","kind":"node","lean":"Cslib.Automata.TwoWayDFA.tr_functional","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#tr-functional","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"Let $A$ be a deterministic two-way automaton, viewed as a nondeterministic two-way automaton. If, on reading the symbol $x$ in state $q$, it has a transition to state $q'$ with head move $m$ and a transition to state $q''$ with head move $m'$, then $m = m'$ and $q' = q''$.","title":"Deterministic two-way automata have unique transitions"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/language","kind":"node","lean":"Cslib.Automata.TwoWayDFA.language","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#language","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"The language of a deterministic two-way automaton $A$ is the set of words accepted by $A$ viewed as a nondeterministic two-way automaton: a word $w$ belongs to it when, starting in the start state with the head at the first position of $w$ and following the step function of $A$ (which moves the head one position left, keeps it in place, or moves it one position right, and gets stuck when it is undefined or would\u2026","title":"Language of a deterministic two-way automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/mem-language","kind":"node","lean":"Cslib.Automata.TwoWayDFA.mem_language","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#mem-language","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"For every deterministic two-way automaton $A$ and every word $w$, $w$ belongs to the language of $A$ if and only if $w$ is accepted by $A$ viewed as a nondeterministic two-way automaton.","title":"Membership in the language of a two-way deterministic automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/language-is-regular","kind":"node","lean":"Cslib.Automata.TwoWayDFA.language_isRegular","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#language-is-regular","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"The language of every deterministic two-way automaton with finitely many states is regular.","title":"Finite two-way deterministic automata accept regular languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/of-dfa","kind":"node","lean":"Cslib.Automata.TwoWayDFA.ofDFA","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#of-dfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"For a DFA $M$ with transition function $\\delta$, this is the deterministic two-way automaton with the same states, start state and accepting states as $M$ which, in state $q$ reading a symbol $a$, goes to state $\\delta(q, a)$ and moves its head one symbol to the right.","title":"A DFA as a two-way deterministic automaton"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/to-two-way-na-of-dfa","kind":"node","lean":"Cslib.Automata.TwoWayDFA.toTwoWayNA_ofDFA","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#to-two-way-na-of-dfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"For every DFA $M$, the deterministic two-way automaton that simulates $M$, viewed as a nondeterministic two-way automaton, equals the nondeterministic two-way automaton obtained by viewing $M$ as an NFA and letting that NFA always move its head one symbol to the right.","title":"The two-way simulation of a DFA agrees with that of its NFA"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/language-of-dfa","kind":"node","lean":"Cslib.Automata.TwoWayDFA.language_ofDFA","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#language-of-dfa","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"For every DFA $M$, the deterministic two-way automaton that simulates $M$ by always moving its head one symbol to the right has exactly the language accepted by $M$.","title":"The two-way simulation of a DFA accepts the same language"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/exists-of-dfa-of-is-regular","kind":"node","lean":"Cslib.Automata.TwoWayDFA.exists_ofDFA_of_isRegular","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#exists-of-dfa-of-is-regular","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"For every regular language $L$ over an alphabet, there is a deterministic two-way automaton with finitely many states whose language is exactly $L$.","title":"Regular languages are accepted by two-way deterministic automata"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/language-is-regular-2","kind":"node","lean":"Cslib.Automata.TwoWayNA.language_isRegular","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#language-is-regular-2","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"Every nondeterministic two-way automaton with finitely many states accepts a regular language.","title":"Finite two-way nondeterministic automata accept regular languages"},{"id":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/exists-two-way-na-of-is-regular","kind":"node","lean":"Cslib.Automata.exists_twoWayNA_of_isRegular","parent":33377,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata/index.html#exists-two-way-na-of-is-regular","scope":"TheoryOfComputation/regular-languages/automata-and-expressions/two-way-finite-automata","status":"fully_proved","summary":"For every regular language $L$ over an alphabet, there is a nondeterministic two-way automaton with finitely many states that accepts exactly the words of $L$.","title":"Regular languages are accepted by two-way nondeterministic automata"},{"data_url":"scopes/TheoryOfComputation/regular-languages/minimization.json","id":"TheoryOfComputation/regular-languages/minimization","kind":"scope","parent":33244,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/index.html","scope":"TheoryOfComputation/regular-languages/minimization","status":"fully_proved","summary":"Nerode equivalence, the minimal automaton and its universal property, and the minimization algorithms of Moore, Hopcroft and Brzozowski with their correctness and cost.","title":"Automaton minimization"},{"data_url":"scopes/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem.json","id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","kind":"scope","navigation":"module","parent":33391,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","title":"Myhill\u2013Nerode theorem"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-rel","kind":"node","lean":"Cslib.Language.nerodeRel","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-rel","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. Two words $x, y \\in \\Sigma^*$ are Nerode equivalent for $L$ when they have the same left quotient, that is, $\\{z \\in \\Sigma^* \\mid xz \\in L\\} = \\{z \\in \\Sigma^* \\mid yz \\in L\\}$.","title":"Nerode equivalence"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-setoid","kind":"node","lean":"Cslib.Language.nerodeSetoid","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-setoid","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For a language $L$ over an alphabet $\\Sigma$, the Nerode setoid of $L$ is the set $\\Sigma^*$ of words equipped with Nerode equivalence for $L$, which is reflexive, symmetric and transitive.","title":"The Nerode setoid"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-class","kind":"node","lean":"Cslib.Language.NerodeClass","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-class","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For a language $L$ over an alphabet $\\Sigma$, the Nerode classes of $L$ are the elements of the quotient of the set of words over $\\Sigma$ by Nerode equivalence for $L$, under which two words $x$ and $y$ are equivalent when they have the same left quotient, that is, when for every word $z$ we have $xz \\in L$ if and only if $yz \\in L$.","title":"Nerode classes"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-rel-iff","kind":"node","lean":"Cslib.Language.nerodeRel_iff","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-rel-iff","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. Two words $x, y \\in \\Sigma^*$ are Nerode equivalent for $L$ if and only if, for every word $z \\in \\Sigma^*$, $xz \\in L$ holds exactly when $yz \\in L$.","title":"Continuation characterization of Nerode equivalence"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/mem-iff","kind":"node","lean":"Cslib.Language.nerodeRel.mem_iff","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#mem-iff","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language. If two words $x$ and $y$ are Nerode equivalent for $L$, then $x \\in L$ if and only if $y \\in L$.","title":"Nerode equivalent words agree on membership"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-rel-append-right","kind":"node","lean":"Cslib.Language.nerodeRel_append_right","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-rel-append-right","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. If two words $x, y \\in \\Sigma^*$ are Nerode equivalent for $L$, then for every word $z \\in \\Sigma^*$ the words $xz$ and $yz$ are Nerode equivalent for $L$.","title":"Nerode equivalence is a right congruence"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-rel-append-singleton","kind":"node","lean":"Cslib.Language.nerodeRel_append_singleton","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-rel-append-singleton","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. If two words $x, y \\in \\Sigma^*$ are Nerode equivalent for $L$, then for every symbol $a \\in \\Sigma$ the words $xa$ and $ya$ are Nerode equivalent for $L$.","title":"Appending a symbol preserves Nerode equivalence"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-rel-append-nil","kind":"node","lean":"Cslib.Language.nerodeRel_append_nil","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-rel-append-nil","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. For every word $x \\in \\Sigma^*$, the word $x\\varepsilon$ obtained by appending the empty word $\\varepsilon$ is Nerode equivalent to $x$ for $L$.","title":"Appending the empty word preserves the Nerode class"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-class-2","kind":"node","lean":"Cslib.Language.nerodeClass","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-class-2","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For a language $L$ and a word $x$, the Nerode class $[x]$ of $x$ is the equivalence class of $x$ under Nerode equivalence for $L$.","title":"Nerode class of a word"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/class-residual","kind":"node","lean":"Cslib.Language.classResidual","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#class-residual","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language. The residual map sends the Nerode class $[x]$ of a word $x$ to the left quotient $x^{-1}L = \\{ y \\mid xy \\in L \\}$, viewed as an element of the set $\\{ w^{-1}L \\mid w \\text{ a word} \\}$ of left quotients of $L$. It is well-defined because Nerode equivalent words have the same left quotient.","title":"Residual map on Nerode classes"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-class-equiv-residual","kind":"node","lean":"Cslib.Language.nerodeClassEquivResidual","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-class-equiv-residual","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For every language $L$, the map sending the Nerode class $[x]$ of a word $x$ to the left quotient $x^{-1}L = \\{ y \\mid xy \\in L \\}$ is a bijection from the set of Nerode classes of $L$ onto the set $\\{ w^{-1}L \\mid w \\text{ a word} \\}$ of left quotients of $L$.","title":"Bijection between Nerode classes and left quotients"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/step","kind":"node","lean":"Cslib.Language.NerodeClass.step","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#step","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$ and $a \\in \\Sigma$ a symbol. Appending $a$ sends the Nerode class of a word $x$ to the Nerode class of $xa$; this is well defined because Nerode equivalence is preserved by appending a symbol.","title":"Transition on Nerode classes"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/accepting","kind":"node","lean":"Cslib.Language.NerodeClass.Accepting","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#accepting","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language. The accepting Nerode classes of $L$ are those whose words belong to $L$: the Nerode class $[x]$ of a word $x$ is accepting if and only if $x \\in L$. This is well-defined because Nerode equivalent words are either both in $L$ or both outside $L$.","title":"Accepting Nerode classes"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-dfa","kind":"node","lean":"Cslib.Language.nerodeDFA","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-dfa","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. The Nerode automaton of $L$ is the deterministic automaton over $\\Sigma$, not necessarily finite, whose states are the Nerode classes of words for $L$, whose start state is the class of the empty word, which on reading a symbol $a \\in \\Sigma$ moves from the class of a word $x$ to the class of $xa$, and whose accepting states are the classes of the words in $L$.","title":"The Nerode automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/accepts-nerode-dfa","kind":"node","lean":"Cslib.Language.accepts_nerodeDFA","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#accepts-nerode-dfa","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For every language $L$, the language accepted by the Nerode automaton of $L$ is exactly $L$.","title":"The Nerode automaton recognizes its language"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-dfa-state-reachable","kind":"node","lean":"Cslib.Language.nerodeDFA_state_reachable","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-dfa-state-reachable","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For every language $L$ and every Nerode class $q$ of $L$, there is a word $x$ such that the Nerode automaton of $L$, started in its start state and reading $x$, ends in the state $q$.","title":"Every state of the Nerode automaton is reachable"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/finite-nerode-class-iff","kind":"node","lean":"Cslib.Language.finite_nerodeClass_iff","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#finite-nerode-class-iff","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For every language $L$, writing $x^{-1}L = \\{ y \\mid xy \\in L \\}$ for the left quotient of $L$ by a word $x$, there are finitely many Nerode classes of $L$ if and only if the set $\\{ x^{-1}L \\mid x \\text{ a word} \\}$ of left quotients of $L$ is finite.","title":"Finitely many Nerode classes iff finitely many left quotients"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/is-regular-iff-finite-nerode-class","kind":"node","lean":"Cslib.Language.isRegular_iff_finite_nerodeClass","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#is-regular-iff-finite-nerode-class","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"A language $L$ is regular, that is, accepted by some deterministic finite automaton, if and only if $L$ has finitely many Nerode classes.","title":"Myhill\u2013Nerode theorem, quotient form"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/distinguishable","kind":"node","lean":"Cslib.Language.Distinguishable","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#distinguishable","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. Two words $x$ and $y$ over $\\Sigma$ are distinguishable by $L$ when there is a word $z$ over $\\Sigma$ such that exactly one of $xz$ and $yz$ belongs to $L$.","title":"Distinguishable words"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-rel-of-eval-eq","kind":"node","lean":"Cslib.Language.nerodeRel_of_eval_eq","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-rel-of-eval-eq","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $M$ be a DFA and let $L$ be the language it accepts. If $M$ reaches the same state from its start state on reading the words $x$ and $y$, then $x$ and $y$ are Nerode equivalent for $L$.","title":"Words reaching the same DFA state are Nerode equivalent"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/eval-ne-of-distinguishable","kind":"node","lean":"Cslib.Language.eval_ne_of_distinguishable","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#eval-ne-of-distinguishable","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language and let $M$ be a DFA that accepts exactly $L$. If two words $x$ and $y$ are distinguishable by $L$, then the states that $M$ reaches from its start state on reading $x$ and on reading $y$ are different.","title":"Distinguishable words reach different DFA states"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/state-equivalent","kind":"node","lean":"Cslib.Language.StateEquivalent","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#state-equivalent","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Two states $p$ and $q$ of a DFA $M$ are equivalent when they accept the same continuation language: for every word $z$, running $M$ from $p$ on $z$ ends in an accepting state if and only if running $M$ from $q$ on $z$ does.","title":"Equivalent states of a DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-rel-iff-state-equivalent","kind":"node","lean":"Cslib.Language.nerodeRel_iff_stateEquivalent","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-rel-iff-state-equivalent","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $M$ be a DFA and let $L$ be the language it accepts. Two words $x$ and $y$ are Nerode equivalent for $L$ if and only if the states that $M$ reaches from its start state on reading $x$ and on reading $y$ are observationally equivalent, that is, $M$ accepts the same language from both states.","title":"Nerode equivalence is equivalence of the reached states"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/reachable-residuals","kind":"node","lean":"Cslib.Language.reachableResiduals","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#reachable-residuals","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For a DFA $M$ and a state $q$ of $M$, let $L_q$ be the language of words $z$ such that running $M$ from $q$ on $z$ ends in an accepting state. The reachable residuals of $M$ form the set of the languages $L_q$ over all states $q$ that $M$ reaches from its start state on some input word.","title":"Residual languages of reachable states"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nerode-class-equiv-reachable-residuals","kind":"node","lean":"Cslib.Language.nerodeClassEquivReachableResiduals","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nerode-class-equiv-reachable-residuals","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language and let $M$ be a DFA that accepts exactly $L$. This is the bijection from the Nerode classes of $L$ to the reachable residuals of $M$, the languages accepted by $M$ from states it reaches from its start state. It sends the Nerode class of a word $x$ to the left quotient $\\{z \\mid xz \\in L\\}$, which is the language $M$ accepts from the state it reaches on reading $x$.","title":"Nerode classes correspond to reachable residuals"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/representative","kind":"node","lean":"Cslib.Language.NerodeClass.representative","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#representative","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For a language $L$ and a Nerode class $q$ of $L$, the representative of $q$ is a word whose Nerode class is $q$, chosen once and for all.","title":"Representative of a Nerode class"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/to-dfastate","kind":"node","lean":"Cslib.Language.NerodeClass.toDFAState","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#to-dfastate","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"For a language $L$ and a DFA $M$ over the same alphabet, this map sends each Nerode class $q$ of $L$ to the state that $M$ reaches from its start state on the chosen representative of $q$.","title":"Nerode classes to DFA states"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/to-dfastate-injective","kind":"node","lean":"Cslib.Language.NerodeClass.toDFAState_injective","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#to-dfastate-injective","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $M$ be a DFA that accepts exactly the language $L$. Then the map sending each Nerode class of $L$ to the state that $M$ reaches from its start state on the chosen representative of the class is injective. In particular, $M$ has at least one distinct state for each Nerode class.","title":"Distinct Nerode classes reach distinct states"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/card-nerode-class-le-card-state","kind":"node","lean":"Cslib.Language.card_nerodeClass_le_card_state","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#card-nerode-class-le-card-state","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language with finitely many Nerode classes, and let $M$ be a DFA with a finite set of states $Q$ that accepts exactly $L$. Then the number of Nerode classes of $L$ is at most $|Q|$.","title":"Nerode classes bound the number of DFA states"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/nat-card-range-left-quotient-le-nat-card","kind":"node","lean":"Cslib.Language.natCard_range_leftQuotient_le_natCard_state","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#nat-card-range-left-quotient-le-nat-card","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language and let $M$ be a DFA with a finite set of states $Q$ that accepts exactly $L$. For a word $x$, the left quotient of $L$ by $x$ is the language $\\{z \\mid xz \\in L\\}$. Then the number of distinct left quotients of $L$ is at most $|Q|$.","title":"Myhill\u2013Nerode lower bound via left quotients"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/card-nerode-class-eq-card-state-iff","kind":"node","lean":"Cslib.Language.card_nerodeClass_eq_card_state_iff","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#card-nerode-class-eq-card-state-iff","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language with finitely many Nerode classes, and let $M$ be a DFA with a finite set of states $Q$ that accepts exactly $L$. Send each Nerode class of $L$ to the state that $M$ reaches from its start state on the chosen representative word of the class. Then the number of Nerode classes of $L$ equals $|Q|$ if and only if every state in $Q$ is the image of some Nerode class under this map.","title":"Equality case of the Nerode state bound"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/card-nerode-class-lt-card-state","kind":"node","lean":"Cslib.Language.card_nerodeClass_lt_card_state","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#card-nerode-class-lt-card-state","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"Let $L$ be a language with finitely many Nerode classes, and let $M$ be a DFA with a finite set of states $Q$ that accepts exactly $L$. Send each Nerode class of $L$ to the state that $M$ reaches from its start state on the chosen representative word of the class. If some state in $Q$ is not the image of any Nerode class under this map, then the number of Nerode classes of $L$ is strictly less than $|Q|$.","title":"Strict Nerode state bound"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/reachable-state","kind":"node","lean":"Cslib.Language.ReachableState","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#reachable-state","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"A state is reachable when some input word reaches it from the start state.","title":"Reachable state of a DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/accessible","kind":"node","lean":"Cslib.Language.Accessible","parent":33392,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem/index.html#accessible","scope":"TheoryOfComputation/regular-languages/minimization/myhill-nerode-theorem","status":"fully_proved","summary":"A DFA is accessible when every state is reachable.","title":"Accessible DFA"},{"data_url":"scopes/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton.json","id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","kind":"scope","navigation":"module","parent":33391,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","title":"Minimal deterministic automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/left-quotient-accepts-from","kind":"node","lean":"Cslib.Language.leftQuotient_acceptsFrom","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#left-quotient-accepts-from","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA, write $\\delta^*(q, x)$ for the state $M$ reaches from the state $q$ by reading the word $x$, and let $L_M(q)$ be the language of words accepted by $M$ when started in $q$. For every state $q$ and every word $x$, the left quotient $\\{z : xz \\in L_M(q)\\}$ of $L_M(q)$ by $x$ equals $L_M(\\delta^*(q, x))$.","title":"Left quotients of continuation languages"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-state","kind":"node","lean":"Cslib.Language.minimizedState","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-state","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For a DFA $M$ and a word $x$, the minimized state of $x$ is the language accepted by $M$ starting from the state that $M$ reaches on reading $x$ from its start state, viewed as an element of the reachable residuals of $M$.","title":"Residual reached by a word"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/left-quotient-equiv-reachable-residuals","kind":"node","lean":"Cslib.Language.leftQuotientEquivReachableResiduals","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#left-quotient-equiv-reachable-residuals","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA accepting the language $L$. The set of left quotients $\\{z : xz \\in L\\}$ of $L$ by words $x$ equals the set of reachable residuals of $M$ (the languages accepted by $M$ from the states it reaches from its start state). The bijection from the first set to the second sends each language to itself.","title":"Left quotients as reachable residuals"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-dfa","kind":"node","lean":"Cslib.Language.minimizedDFA","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-dfa","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA accepting the language $L$. The minimized DFA of $M$ has as states the reachable residuals of $M$, that is, the languages accepted by $M$ from the states it reaches from its start state. Its start state is $L$; on a letter $a$ it moves from a state $K$ to the left quotient $\\{z : az \\in K\\}$; and its accepting states are those that contain the empty word.","title":"Minimized DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/to-dfa-eval-val","kind":"node","lean":"Cslib.Language.toDFA_eval_val","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#to-dfa-eval-val","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For every language $L$ and every word $x$, the state that the residual-language DFA of $L$ reaches on reading $x$ is the left quotient $x^{-1}L = \\{z : xz \\in L\\}$. This automaton has the left quotients of $L$ as states, start state $L$, transition from $K$ on a letter $a$ to $a^{-1}K$, and accepts $K$ when $K$ contains the empty word.","title":"States reached in the residual-language DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-dfa-eval-equiv","kind":"node","lean":"Cslib.Language.minimizedDFA_eval_equiv","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-dfa-eval-equiv","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA accepting the language $L$, and consider the automaton whose states are the left quotients of $L$, with start state $L$, moving from $K$ on a letter $a$ to $\\{z : az \\in K\\}$. For every word $x$, the state that the minimized DFA of $M$ reaches on $x$, carried to the left quotients of $L$ by the bijection that sends each language to itself, is the state that this automaton reaches on $x$.","title":"Evaluation in the minimized DFA agrees with the left-quotient automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-dfa-eval","kind":"node","lean":"Cslib.Language.minimizedDFA_eval","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-dfa-eval","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For every DFA $M$ and every word $x$, the state that the minimized DFA of $M$ reaches on $x$ is the language accepted by $M$ from the state that $M$ reaches on $x$.","title":"States reached in the minimized DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-dfa-step-val","kind":"node","lean":"Cslib.Language.minimizedDFA_step_val","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-dfa-step-val","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA. For every reachable residual $K$ of $M$ (a state of the minimized DFA of $M$) and every letter $a$, the minimized DFA of $M$ moves from $K$ on input $a$ to the left quotient $\\{z : az \\in K\\}$ of $K$ by $a$.","title":"Transitions of the minimized DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-dfa-step-minimized-state","kind":"node","lean":"Cslib.Language.minimizedDFA_step_minimizedState","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-dfa-step-minimized-state","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA, and for a word $x$ let $[x]$ be the language accepted by $M$ from the state that $M$ reaches on $x$. For every word $x$ and every letter $a$, the minimized DFA of $M$ moves from the state $[x]$ on input $a$ to the state $[xa]$.","title":"Transitions of the minimized DFA on reached states"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-dfa-accepts","kind":"node","lean":"Cslib.Language.minimizedDFA_accepts","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-dfa-accepts","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For every DFA $M$, the minimized DFA of $M$ accepts exactly the language accepted by $M$.","title":"The minimized DFA accepts the same language"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/to-dfa-accessible","kind":"node","lean":"Cslib.Language.toDFA_accessible","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#to-dfa-accessible","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For every language $L$, the residual-language DFA of $L$ is accessible. This automaton has as states the left quotients $x^{-1}L = \\{z : xz \\in L\\}$ for words $x$, start state $L$, transition from $K$ on a letter $a$ to $a^{-1}K$, and accepts $K$ when $K$ contains the empty word.","title":"The residual-language DFA is accessible"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-dfa-accessible","kind":"node","lean":"Cslib.Language.minimizedDFA_accessible","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-dfa-accessible","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For every DFA $M$, every state of the minimized DFA of $M$ is reached from its start state by reading some word.","title":"The minimized DFA is accessible"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/nerode-class-equiv-minimized-state","kind":"node","lean":"Cslib.Language.nerodeClassEquivMinimizedState","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#nerode-class-equiv-minimized-state","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For a DFA $M$ accepting the language $L$, there is a bijection from the Nerode classes of $L$ to the reachable residuals of $M$; it sends the class of a word $x$ to the left quotient $x^{-1}L = \\{z : xz \\in L\\}$.","title":"Nerode classes and reachable residuals"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-state-equiv-iterated","kind":"node","lean":"Cslib.Language.minimizedStateEquivIterated","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-state-equiv-iterated","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA accepting the language $L$. The reachable residuals of the minimized DFA of $M$ and the reachable residuals of $M$ both form the set of left quotients $x^{-1}L = \\{z : xz \\in L\\}$ of $L$ by words $x$, and this is the bijection from the first set to the second that sends each language to itself.","title":"Reachable residuals of the minimized DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/minimized-dfa-card-eq-nerode","kind":"node","lean":"Cslib.Language.minimizedDFA_card_eq_nerode","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#minimized-dfa-card-eq-nerode","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA accepting the language $L$. If the set of reachable residuals of $M$ (the states of the minimized DFA of $M$) and the set of Nerode classes of $L$ are both finite, then they have the same number of elements.","title":"When finite, the minimized DFA has as many states as there are Nerode classes"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/to-dfa","kind":"node","lean":"Cslib.Automata.DA.FinAcc.toDFA","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#to-dfa","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"A CSLib deterministic automaton $M$ on finite words is viewed as a Mathlib DFA with the same states, the same transition function, the same start state and the same set of accepting states as $M$.","title":"CSLib automata as Mathlib DFAs"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/of-dfa","kind":"node","lean":"Cslib.Automata.DA.FinAcc.ofDFA","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#of-dfa","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"A Mathlib DFA $M$ is viewed as a CSLib deterministic automaton on finite words with the same states, the same transition function, the same start state and the same set of accepting states as $M$.","title":"Mathlib DFAs as CSLib automata"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/language-to-dfa","kind":"node","lean":"Cslib.Automata.DA.FinAcc.language_toDFA","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#language-to-dfa","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For every deterministic automaton $M$ on finite words, the DFA with the same states, transition function, start state and accepting states as $M$ accepts exactly the language of $M$.","title":"Converting a CSLib automaton to a Mathlib DFA preserves its language"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/nerode-congruence-r-iff","kind":"node","lean":"Cslib.Automata.DA.FinAcc.nerodeCongruence_r_iff","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#nerode-congruence-r-iff","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$ and let $x, y \\in \\Sigma^*$. Then $x$ and $y$ are related by the Nerode congruence of $L$, that is, $xz \\in L \\iff yz \\in L$ for every word $z$, if and only if they are Nerode equivalent for $L$, that is, their left quotients $x^{-1}L = \\{z : xz \\in L\\}$ and $y^{-1}L$ are equal.","title":"The Nerode congruence relates words with equal left quotients"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/nerode-residual","kind":"node","lean":"Cslib.Automata.DA.FinAcc.nerodeResidual","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#nerode-residual","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$, and write $x^{-1}L = \\{z : xz \\in L\\}$ for the left quotient of $L$ by a word $x$. This is the map from the classes of the Nerode congruence of $L$ to the set $\\{w^{-1}L : w \\in \\Sigma^*\\}$ of residual languages of $L$ that sends the class of each word $x$ to $x^{-1}L$; it is well defined because Nerode-congruent words have the same left quotient.","title":"The residual language of a Nerode class"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/nerode-quotient-equiv-residual","kind":"node","lean":"Cslib.Automata.DA.FinAcc.nerodeQuotientEquivResidual","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#nerode-quotient-equiv-residual","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$, and write $x^{-1}L = \\{z : xz \\in L\\}$ for the left quotient of $L$ by a word $x$. This is the bijection from the classes of the Nerode congruence of $L$ onto the set $\\{x^{-1}L : x \\in \\Sigma^*\\}$ of residual languages of $L$ that sends the class of each word $x$ to $x^{-1}L$.","title":"Nerode classes are the residual languages"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/reindex-nerode-congruence-da","kind":"node","lean":"Cslib.Automata.DA.FinAcc.reindex_nerodeCongruenceDA","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#reindex-nerode-congruence-da","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$, and write $x^{-1}L = \\{z : xz \\in L\\}$ for the left quotient of $L$ by a word $x$. Consider the canonical automaton of $L$ as a DFA: its states are the classes $[x]$ of the Nerode congruence of $L$, its start state is the class of the empty word, on a letter $a$ it goes from $[x]$ to $[xa]$, and its accepting states are the classes of words of $L$. Renaming each\u2026","title":"The Nerode automaton is the residual-language automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/reindex-nerode-congruence-da-eq-minimize","kind":"node","lean":"Cslib.Automata.DA.FinAcc.reindex_nerodeCongruenceDA_eq_minimizedDFA","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#reindex-nerode-congruence-da-eq-minimize","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA over an alphabet $\\Sigma$ accepting the language $L$, and write $x^{-1}L = \\{z : xz \\in L\\}$ for the left quotient of $L$ by a word $x$. Consider the canonical automaton of $L$ as a DFA: its states are the classes $[x]$ of the Nerode congruence of $L$, its start state is the class of the empty word, on a letter $a$ it goes from $[x]$ to $[xa]$, and its accepting states are the classes of words of\u2026","title":"The canonical automaton of a DFA's language is its minimized DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/dfahom","kind":"node","lean":"Cslib.Language.DFAHom","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#dfahom","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ and $N$ be DFAs over the same alphabet with transition functions $\\delta_M$ and $\\delta_N$. A morphism from $M$ to $N$ is a map $f$ from the states of $M$ to the states of $N$ that sends the start state of $M$ to the start state of $N$, satisfies $f(\\delta_M(q, a)) = \\delta_N(f(q), a)$ for every state $q$ of $M$ and every symbol $a$, and such that each state $q$ of $M$ is accepting if and only if $f(q)$ is\u2026","title":"Morphisms of DFAs"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/map-eval-from","kind":"node","lean":"Cslib.Language.DFAHom.map_evalFrom","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#map-eval-from","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"If $f$ is a morphism from a DFA $M$ to a DFA $N$, then for every state $q$ of $M$ and every word $x$, running $N$ on $x$ from $f(q)$ ends in the image under $f$ of the state reached by running $M$ on $x$ from $q$.","title":"DFA morphisms commute with evaluation from a state"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/map-eval","kind":"node","lean":"Cslib.Language.DFAHom.map_eval","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#map-eval","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"If $f$ is a morphism from a DFA $M$ to a DFA $N$, then for every word $x$, the state that $N$ reaches on input $x$ from its start state is the image under $f$ of the state that $M$ reaches on input $x$ from its start state.","title":"DFA morphisms commute with evaluation"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/accepts-eq","kind":"node","lean":"Cslib.Language.DFAHom.accepts_eq","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#accepts-eq","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"If $f$ is a morphism from a DFA $M$ to a DFA $N$ over the same alphabet, then $M$ and $N$ accept the same language.","title":"DFA morphisms preserve the accepted language"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/refl","kind":"node","lean":"Cslib.Language.DFAHom.refl","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#refl","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For every DFA $M$, the identity map on the states of $M$ is a DFA morphism from $M$ to itself.","title":"Identity DFA morphism"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/comp","kind":"node","lean":"Cslib.Language.DFAHom.comp","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#comp","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"If $f$ is a DFA morphism from $M$ to $N$ and $g$ is a DFA morphism from $N$ to $P$, where $M$, $N$, $P$ are DFAs over the same alphabet, then the composite of $g$ and $f$ is the DFA morphism from $M$ to $P$ whose state map is $g \\circ f$.","title":"Composition of DFA morphisms"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/state-separated","kind":"node","lean":"Cslib.Language.StateSeparated","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#state-separated","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"A DFA $M$ is state-separated when any two equivalent states of $M$ are equal: if $p \\ne q$, then the languages of words accepted by $M$ starting from $p$ and from $q$ are different.","title":"State-separated DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/canonical-residual-hom","kind":"node","lean":"Cslib.Language.canonicalResidualHom","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#canonical-residual-hom","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be an accessible DFA accepting the language $L$. The residual-language DFA of $L$ has as states the left quotients $x^{-1}L = \\{y \\mid xy \\in L\\}$ for words $x$, start state $L$, transitions $K \\mapsto a^{-1}K$ on a symbol $a$, and a state $K$ is accepting when it contains the empty word. The canonical residual map is the DFA morphism from $M$ to this DFA that sends each state $q$ of $M$ to its continuation\u2026","title":"The canonical residual map"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/canonical-residual-hom-surjective","kind":"node","lean":"Cslib.Language.canonicalResidualHom_surjective","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#canonical-residual-hom-surjective","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For an accessible DFA $M$, every state of the residual-language DFA of the language accepted by $M$ is the image of some state of $M$ under the canonical residual map.","title":"The canonical residual map is surjective"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/canonical-residual-hom-injective-iff","kind":"node","lean":"Cslib.Language.canonicalResidualHom_injective_iff","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#canonical-residual-hom-injective-iff","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For an accessible DFA $M$, the canonical residual map from $M$ to the residual-language DFA of its accepted language is injective if and only if $M$ is state-separated, that is, any two states of $M$ with the same continuation language are equal.","title":"Injectivity of the canonical residual map"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/state-equiv-residuals-of-separated","kind":"node","lean":"Cslib.Language.stateEquivResidualsOfSeparated","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#state-equiv-residuals-of-separated","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"Let $M$ be a DFA that is accessible and state-separated, and let $L$ be the language it accepts. The map sending each state $q$ of $M$ to the language of words accepted by $M$ starting from $q$ is a bijection from the states of $M$ onto the set of left quotients $\\{z : xz \\in L\\}$ of $L$ by words $x$.","title":"States of an accessible, state-separated DFA correspond to residuals"},{"id":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/canonical-residual-hom-unique","kind":"node","lean":"Cslib.Language.canonicalResidualHom_unique","parent":33427,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton/index.html#canonical-residual-hom-unique","scope":"TheoryOfComputation/regular-languages/minimization/minimal-deterministic-automaton","status":"fully_proved","summary":"For an accessible DFA $M$ accepting the language $L$, every DFA morphism from $M$ to the residual-language DFA of $L$ equals the canonical residual map.","title":"Uniqueness of the canonical residual map"},{"data_url":"scopes/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm.json","id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","kind":"scope","navigation":"module","parent":33391,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","title":"Brzozowski's algorithm"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/rev-det","kind":"node","lean":"DFA.revDet","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#rev-det","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ with finitely many states, the determinized reversal of $M$ is the subset construction applied to the reversal of $M$: its states are the sets of states of $M$; reading a letter $a$ from a set $S$ leads to the set of states $q$ whose $a$-successor in $M$ lies in $S$; its start state is the set of accepting states of $M$; and it accepts exactly the sets that contain the start state of $M$.","title":"Determinized reversal of a DFA"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/rev-det-eval","kind":"node","lean":"DFA.revDet_eval","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#rev-det-eval","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with finitely many states and $w$ a word. On input $w$, the determinized reversal of $M$ reaches the set of states $q$ of $M$ from which $M$, reading $w$ backwards starting at $q$, ends in an accepting state.","title":"States reached in the determinized reversal"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/rev-det-accepts","kind":"node","lean":"DFA.revDet_accepts","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#rev-det-accepts","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with finitely many states and $L$ its language, and write $w^R$ for the word $w$ read backwards. The determinized reversal of $M$ accepts exactly the reversed language $L^R = \\{w \\mid w^R \\in L\\}$.","title":"The determinized reversal accepts the reversed language"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/rev-det-eval-eq-iff","kind":"node","lean":"DFA.revDet_eval_eq_iff","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#rev-det-eval-eq-iff","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with finitely many states in which every state is reached from the start state by some word, and let $L$ be its language. Write $u^R$ for the word $u$ read backwards, $L^R = \\{u \\mid u^R \\in L\\}$ for the reversed language, and $w^{-1}K = \\{v \\mid wv \\in K\\}$ for the left quotient of a language $K$ by a word $w$. Then two words $w$ and $w'$ lead the determinized reversal of $M$ from its start state\u2026","title":"Brzozowski's lemma"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/mid-state","kind":"node","lean":"DFA.MidState","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#mid-state","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finite state set $Q$, the subsets of $Q$ reachable from the start state in the subset construction of the reversal of $M$.","title":"States of the intermediate Brzozowski automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski-mid","kind":"node","lean":"DFA.brzozowskiMid","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski-mid","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states, the subset construction of the reversal of $M$, restricted to the subsets reachable from its start state.","title":"The intermediate Brzozowski automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski-state","kind":"node","lean":"DFA.BrzozowskiState","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski-state","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states, the subsets of states of the intermediate Brzozowski automaton of $M$ that are reachable from the start state in the subset construction of its reversal.","title":"States of Brzozowski's automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski","kind":"node","lean":"DFA.brzozowski","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states, the subset construction of the reversal of the intermediate Brzozowski automaton of $M$, restricted to the subsets reachable from its start state. Equivalently: reverse, determinize by the subset construction and keep the reachable subsets, twice.","title":"Brzozowski's automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski-mid-surjective","kind":"node","lean":"DFA.brzozowskiMid_surjective","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski-mid-surjective","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states, every state of the intermediate Brzozowski automaton of $M$ (the subset construction of the reversal of $M$ restricted to its reachable subsets) is reached from its start state by some word.","title":"Every state of the intermediate Brzozowski automaton is reachable"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski-surjective","kind":"node","lean":"DFA.brzozowski_surjective","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski-surjective","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states, every state of Brzozowski's automaton of $M$ is reached from its start state by some word.","title":"Every state of Brzozowski's automaton is reachable"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski-accepts","kind":"node","lean":"DFA.brzozowski_accepts","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski-accepts","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states, Brzozowski's automaton of $M$ accepts exactly the language of $M$.","title":"Brzozowski's automaton accepts the language of the original automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski-eval-eq-iff","kind":"node","lean":"DFA.brzozowski_eval_eq_iff","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski-eval-eq-iff","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states and words $w$ and $w'$, Brzozowski's automaton of $M$ reaches the same state on $w$ as on $w'$ if and only if $w$ and $w'$ are Nerode equivalent for the language of $M$, that is, have the same left quotient.","title":"Brzozowski's automaton identifies exactly the Nerode equivalent words"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski-residual","kind":"node","lean":"DFA.brzozowskiResidual","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski-residual","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states, the map that sends a state $p$ of Brzozowski's automaton of $M$ to the language accepted by $M$ from the state that $M$ reaches on $w$, where $w$ is a chosen word leading Brzozowski's automaton from its start state to $p$; this language is one of the residual languages of the reachable states of $M$.","title":"Residual language of a state of Brzozowski's automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/brzozowski-equiv","kind":"node","lean":"DFA.brzozowskiEquiv","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#brzozowski-equiv","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"For a deterministic finite automaton $M$ over a finite alphabet with finitely many states, the bijection from the states of Brzozowski's automaton of $M$ to the residual languages of the reachable states of $M$ that sends each state $p$ to the language accepted by $M$ from the state that $M$ reaches on $w$, where $w$ is a chosen word leading Brzozowski's automaton from its start state to $p$.","title":"States of Brzozowski's automaton as residual languages"},{"id":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/reindex-brzozowski","kind":"node","lean":"DFA.reindex_brzozowski","parent":33463,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm/index.html#reindex-brzozowski","scope":"TheoryOfComputation/regular-languages/minimization/brzozowski-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states, and let $L$ be its language. For a language $K$ and a word $w$, write $w^{-1}K = \\{v \\mid wv \\in K\\}$ for the left quotient of $K$ by $w$. Every state of Brzozowski's automaton of $M$ is reached by some word $w$; rename that state to $w^{-1}L$, which does not depend on the choice of $w$. The renamed automaton is exactly the minimized DFA of $M$: its\u2026","title":"Brzozowski's automaton is the minimal automaton"},{"data_url":"scopes/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm.json","id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","kind":"scope","navigation":"module","parent":33391,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","title":"Moore's algorithm"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-step","kind":"node","lean":"DFA.mooreStep","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-step","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with state set $S$ and transition function $\\delta$. One round of Moore's refinement sends a set $E \\subseteq S \\times S$ of pairs of states to the set of pairs $(p, q) \\in S \\times S$ such that $p$ is accepting if and only if $q$ is accepting and, for every letter $a$, the pair $(\\delta(p, a), \\delta(q, a))$ lies in $E$.","title":"Moore refinement step"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-iter","kind":"node","lean":"DFA.mooreIter","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-iter","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states, and write $R$ for one round of Moore's refinement. For a set $E$ of pairs of states of $M$ and a budget $n \\in \\mathbb{N}$, the stopping Moore iteration is defined by recursion on $n$: with budget $0$ it returns $E$; with budget $n + 1$ it returns $E$ if $R(E) = E$, and otherwise the stopping Moore iteration from $R(E)$ with budget $n$.","title":"Stopping Moore iteration"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-iter-spec","kind":"node","lean":"DFA.mooreIter_spec","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-iter-spec","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states, and write $R$ for one round of Moore's refinement. For every $n \\in \\mathbb{N}$ and every set $E$ of pairs of states of $M$, there is $k \\le n$ such that the stopping Moore iteration from $E$ with budget $n$ returns $R^k(E)$, and if $k < n$ then $R^k(E)$ is a fixed point of $R$, that is, $R(R^k(E)) = R^k(E)$.","title":"Output of the stopping Moore iteration"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-rel","kind":"node","lean":"DFA.mooreRel","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-rel","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ over a finite alphabet with finite set of states $Q$, Moore's relation is the set of pairs of states obtained from $Q \\times Q$ by repeating rounds of Moore's refinement until a round changes nothing, for at most $|Q \\times Q|$ rounds.","title":"Moore's relation"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/subset-moore-rel","kind":"node","lean":"DFA.subset_mooreRel","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#subset-moore-rel","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA. If a set $Z$ of pairs of states of $M$ is contained in its image under one round of Moore's refinement, then $Z$ is contained in Moore's relation of $M$.","title":"Moore's relation contains every post-fixed point"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/mem-moore-rel-iff","kind":"node","lean":"DFA.mem_mooreRel_iff","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#mem-moore-rel-iff","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ over a finite alphabet with finitely many states, two states $p$ and $q$ of $M$ are related by Moore's relation if and only if they have the same continuation language: for every word $w$, $M$ started at $p$ accepts $w$ exactly when $M$ started at $q$ accepts $w$.","title":"Correctness of Moore's refinement"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-setoid","kind":"node","lean":"DFA.mooreSetoid","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-setoid","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"For a DFA $M$, two reachable states $p$ and $q$ of $M$ are related when the pair $(p, q)$ lies in Moore's relation of $M$; this relation is an equivalence relation on the reachable states of $M$.","title":"Moore equivalence on reachable states"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-state","kind":"node","lean":"DFA.MooreState","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-state","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"For a DFA $M$, these are the equivalence classes of the reachable states of $M$ under Moore equivalence, that is, under Moore's relation of $M$ restricted to reachable states. They are the states of Moore's minimal automaton.","title":"States of Moore's automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-class","kind":"node","lean":"DFA.mooreClass","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-class","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states. For a reachable state $s$ of $M$, its Moore class is the equivalence class of $s$ among the reachable states of $M$ under Moore's relation; it is a state of Moore's minimal automaton of $M$.","title":"Moore class of a reachable state"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-dfa","kind":"node","lean":"DFA.mooreDFA","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-dfa","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ over a finite alphabet with finitely many states, Moore's minimal automaton of $M$ has as states the classes of reachable states of $M$ under Moore's relation. Its start state is the class of the start state of $M$; reading a letter $a$ sends the class of a reachable state $s$ to the class of the state that $M$ reaches from $s$ on $a$; and the class of $s$ is accepting exactly when $s$ is an accepting\u2026","title":"Moore's minimal automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-dfa-accepts","kind":"node","lean":"DFA.mooreDFA_accepts","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-dfa-accepts","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ over a finite alphabet with finitely many states, Moore's minimal automaton of $M$ accepts exactly the language accepted by $M$.","title":"Moore's automaton accepts the same language"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-residual","kind":"node","lean":"DFA.mooreResidual","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-residual","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states. Each state of Moore's minimal automaton of $M$, a class of reachable states of $M$, is sent to the continuation language of any member $s$, the set of words that $M$ accepts when started at $s$; this is a reachable residual language of $M$. It does not depend on the choice of $s$, because Moore-related states have the same continuation language.","title":"Continuation language of a Moore class"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/moore-equiv","kind":"node","lean":"DFA.mooreEquiv","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#moore-equiv","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states. The states of Moore's minimal automaton of $M$ are in bijection with the reachable residual languages of $M$: the bijection sends the class of a reachable state $s$ to the continuation language of $s$, the set of words that $M$ accepts when started at $s$.","title":"Moore classes are the reachable residuals"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/reindex-moore-dfa","kind":"node","lean":"DFA.reindex_mooreDFA","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#reindex-moore-dfa","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA. Renaming each state of Moore's minimal automaton of $M$, a class of reachable states, to the continuation language accepted from its members turns Moore's automaton into exactly the minimized DFA of $M$, whose states are the reachable continuation languages of $M$.","title":"Moore's automaton is the minimal automaton"},{"id":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/card-moore-state","kind":"node","lean":"DFA.card_mooreState","parent":33479,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/moore-s-algorithm/index.html#card-moore-state","scope":"TheoryOfComputation/regular-languages/minimization/moore-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ over a finite alphabet with finitely many states, Moore's minimal automaton of $M$ has exactly as many states as there are Nerode classes of the language accepted by $M$.","title":"Moore's automaton has one state per Nerode class"},{"data_url":"scopes/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm.json","id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","kind":"scope","navigation":"module","parent":33391,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","title":"Hopcroft's algorithm"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/preimage","kind":"node","lean":"DFA.Hopcroft.preimage","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#preimage","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ with transition function $\\delta$, a set $C$ of states and a letter $a$, the preimage of $C$ under $a$ is the set of states $q$ with $\\delta(q, a) \\in C$.","title":"Preimage of a set under a letter"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splits","kind":"node","lean":"DFA.Hopcroft.Splits","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splits","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"A set $D$ of states is split by a set $P$ of states when $D$ contains both a state in $P$ and a state not in $P$, that is, when $D \\cap P$ and $D \\setminus P$ are both nonempty.","title":"A set of states split by another"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/split-block","kind":"node","lean":"DFA.Hopcroft.splitBlock","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#split-block","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For sets $P$ and $D$ of states, splitting $D$ by $P$ gives the two-element list $[D \\cap P,\\ D \\setminus P]$ if both of these sets are nonempty, and the one-element list $[D]$ otherwise.","title":"Splitting a block by a set"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/smaller","kind":"node","lean":"DFA.Hopcroft.smaller","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#smaller","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For finite sets $A$ and $B$ of states, the smaller one is $A$ if $|A| \\le |B|$ and $B$ otherwise; on a tie in cardinality it is $A$.","title":"Smaller of two finite sets of states"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/new-splitters","kind":"node","lean":"DFA.Hopcroft.newSplitters","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#new-splitters","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Given a waiting list $W$ of pairs of a set of states and a letter and sets $S$ and $D$ of states, write $D_1 = D \\cap S$ and $D_2 = D \\setminus S$. The new waiting entries for $D$ consist, for each letter $b$ in the enumeration order of the alphabet, of both $(D_1, b)$ and $(D_2, b)$ if $(D, b)$ is in $W$, and otherwise of the single entry $(E, b)$, where $E$ is the smaller of $D_1$ and $D_2$ by cardinality ($D_1$\u2026","title":"New splitters for a split block"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/round-p","kind":"node","lean":"DFA.Hopcroft.roundP","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#round-p","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ with transition function $\\delta$, a list $P$ of blocks of states, a set $C$ of states and a letter $a$, let $S = \\{q : \\delta(q, a) \\in C\\}$ be the preimage of $C$ under $a$. The partition after the splitter $(C, a)$ replaces each block $D$ of $P$, in order, by the two blocks $D \\cap S$ and $D \\setminus S$ when both are nonempty, and keeps $D$ otherwise.","title":"One refinement round of the partition"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/round-w","kind":"node","lean":"DFA.Hopcroft.roundW","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#round-w","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ with transition function $\\delta$, a list $P$ of blocks of states, a list $W$ of pairs of a set of states and a letter, a set $C$ of states and a letter $a$, let $S = \\{q : \\delta(q, a) \\in C\\}$ be the preimage of $C$ under $a$, and say that $S$ splits a set $D$ when $D$ meets both $S$ and its complement. The waiting list after the splitter $(C, a)$, from the rest $W$ of the waiting list, consists of\u2026","title":"One update of the waiting list"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/acc-block","kind":"node","lean":"DFA.Hopcroft.accBlock","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#acc-block","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ with finite state set $Q$, this is the set $F \\subseteq Q$ of accepting states of $M$, as a finite set.","title":"Block of accepting states"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/rej-block","kind":"node","lean":"DFA.Hopcroft.rejBlock","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#rej-block","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ with state set $Q$ and set of accepting states $F$, the rejecting block is the set $Q \\setminus F$ of states of $M$ that are not accepting.","title":"Rejecting states"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/init-p","kind":"node","lean":"DFA.Hopcroft.initP","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#init-p","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ with finite state set $Q$ and set of accepting states $F$, the initial partition is the list of blocks $F$, $Q \\setminus F$, in this order, with any empty block removed.","title":"Hopcroft's initial partition"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/init-w","kind":"node","lean":"DFA.Hopcroft.initW","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#init-w","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with finite state set $Q$, set of accepting states $F$ and finite alphabet $\\Sigma$. If $F$ and $Q \\setminus F$ are both nonempty, the initial waiting list consists of the splitters $(S, b)$ for every letter $b \\in \\Sigma$, where $S = F$ if $|F| \\le |Q \\setminus F|$ and $S = Q \\setminus F$ otherwise; if one of $F$ and $Q \\setminus F$ is empty, the initial waiting list is empty.","title":"Hopcroft's initial waiting list"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop","kind":"node","lean":"DFA.hopcroftLoop","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA. Given a round budget $k$, a partition $P$ of the states of $M$ (a list of blocks) and a waiting list $W$ of splitters $(C, a)$, each a block $C$ with a letter $a$, the loop returns the pair $(P, W)$ unchanged if $k = 0$ or $W$ is empty. Otherwise it removes the first splitter $(C, a)$ from $W$, replaces $P$ by the partition after the splitter $(C, a)$ and the remaining waiting list by the waiting\u2026","title":"Hopcroft's refinement loop"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-partition","kind":"node","lean":"DFA.hopcroftPartition","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-partition","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with finite state set $Q$ over a finite alphabet $\\Sigma$. Hopcroft's partition of $M$ is the partition returned by Hopcroft's loop run from the initial partition and the initial waiting list with a budget of $|\\Sigma| \\cdot (|Q| + 1)$ rounds.","title":"Hopcroft's partition"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-rel","kind":"node","lean":"DFA.hopcroftRel","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-rel","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ with finitely many states, Hopcroft's relation is the set of pairs $(p, q)$ of states of $M$ such that $p$ and $q$ lie in a common block of Hopcroft's partition.","title":"Hopcroft's relation"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/is-partition","kind":"node","lean":"DFA.Hopcroft.IsPartition","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#is-partition","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"A list $P$ of sets of states is a partition into nonempty blocks when entries of $P$ at different positions are disjoint, every entry of $P$ is nonempty, and every state lies in some entry of $P$.","title":"Partitions of the states"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/rel","kind":"node","lean":"DFA.Hopcroft.Rel","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#rel","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a list $P$ of sets of states, two states $p$ and $q$ are related by $P$ when some block $D$ of $P$ contains both $p$ and $q$.","title":"Lying in a common block"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/mem-iff-of-rel","kind":"node","lean":"DFA.Hopcroft.IsPartition.mem_iff_of_rel","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#mem-iff-of-rel","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $P$ be a list of sets of states that is a partition into nonempty blocks, let $C$ be an entry of $P$, and let $p$ and $q$ be states lying in a common entry of $P$. Then $p \\in C$ if and only if $q \\in C$.","title":"A block contains both or neither of two related states"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/length-le","kind":"node","lean":"DFA.Hopcroft.IsPartition.length_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#length-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"If a list $P$ of sets of states is a partition into nonempty blocks, then the length of $P$ is at most the number $|Q|$ of states.","title":"A partition has at most as many blocks as there are states"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/side-of-mem-split-block","kind":"node","lean":"DFA.Hopcroft.side_of_mem_splitBlock","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#side-of-mem-split-block","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $S$ and $D$ be sets of states. Every piece of $D$ split by $S$ (the two sets $D \\cap S$ and $D \\setminus S$ if $D$ meets both $S$ and its complement, and $D$ itself otherwise) is contained in $S$ or disjoint from $S$: any two states $x$ and $y$ of the piece satisfy $x \\in S \\iff y \\in S$.","title":"Pieces of a split block lie on one side"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/rel-round-p-iff","kind":"node","lean":"DFA.Hopcroft.rel_roundP_iff","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#rel-round-p-iff","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with transition function $\\delta$, $P$ a list of blocks of states, $C$ a set of states and $a$ a letter. Two states $p$ and $q$ lie in a common block after the splitter $(C, a)$ is applied to $P$ if and only if they lie in a common block of $P$ and $\\delta(p, a) \\in C \\iff \\delta(q, a) \\in C$.","title":"Effect of one splitter on the common-block relation"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/length-round-p","kind":"node","lean":"DFA.Hopcroft.length_roundP","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#length-round-p","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with transition function $\\delta$, let $P$ be a list of sets of states, $C$ a set of states and $a$ a letter, and let $\\mathrm{pre} = \\{q : \\delta(q, a) \\in C\\}$. The partition after the splitter $(C, a)$, which replaces each entry $D$ of $P$ meeting both $\\mathrm{pre}$ and its complement by $D \\cap \\mathrm{pre}$ and $D \\setminus \\mathrm{pre}$, has length equal to the length of $P$ plus the number\u2026","title":"Number of blocks after one round"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/length-round-w-le","kind":"node","lean":"DFA.Hopcroft.length_roundW_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#length-round-w-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with transition function $\\delta$ over a finite alphabet $\\Sigma$, let $P$ be a list of sets of states with no repeated entries, let $W$ be a list of pairs of a set of states and a letter, let $C$ be a set of states and $a$ a letter, and let $S = \\{q : \\delta(q, a) \\in C\\}$ be the preimage of $C$ under $a$. Then the waiting list after the splitter $(C, a)$, computed from the blocks $P$ and the rest\u2026","title":"Growth of the waiting list in one round"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/inv","kind":"node","lean":"DFA.Hopcroft.Inv","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#inv","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with transition function $\\delta$. A list $P$ of sets of states and a waiting list $W$ of pairs $(C, b)$ of a set of states $C$ and a letter $b$ satisfy the invariants when: (i) $P$ is a partition of the states into nonempty blocks; (ii) for every $(C, b)$ in $W$, $C$ is an entry of $P$; (iii) for every letter $b$ and all states $p$, $q$ lying in a common entry of $P$, if for every $C$ with\u2026","title":"Invariants of Hopcroft's loop"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/round","kind":"node","lean":"DFA.Hopcroft.Inv.round","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#round","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA, $P$ a list of blocks of states, $W$ a list of pairs of a set of states and a letter, $C$ a set of states and $a$ a letter. If $P$ and the waiting list obtained by putting $(C, a)$ in front of $W$ satisfy the invariants of Hopcroft's loop, then so do the partition after the splitter $(C, a)$ and the waiting list after the splitter $(C, a)$ from the rest $W$.","title":"One round preserves the invariants"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/rel-iff","kind":"node","lean":"DFA.Hopcroft.Inv.rel_iff","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#rel-iff","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA and $P$ a list of blocks of states such that $P$ with the empty waiting list satisfies the invariants of Hopcroft's loop. Then two states $p$ and $q$ lie in a common block of $P$ if and only if they accept the same continuation languages, that is, $M$ accepts a word from $p$ exactly when it accepts it from $q$.","title":"With an empty waiting list, blocks are the classes of equal continuation languages"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/inv-init","kind":"node","lean":"DFA.Hopcroft.inv_init","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#inv-init","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For every DFA $M$ with finitely many states over a finite alphabet, Hopcroft's initial partition and initial waiting list satisfy the invariants of Hopcroft's loop.","title":"The invariants hold initially"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop-spec","kind":"node","lean":"DFA.hopcroftLoop_spec","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop-spec","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with finite state set $Q$ over a finite alphabet $\\Sigma$, let $P$ be a partition of the states given as a list of blocks, let $W$ be a waiting list of splitters, and let $k$ be a natural number. If $P$ and $W$ satisfy the invariants of Hopcroft's loop and $|W| + |\\Sigma| \\cdot (|Q| - |P|) \\le k$, where $|P|$ and $|W|$ are the lengths of the lists, then Hopcroft's loop run from $(P, W)$ with a\u2026","title":"Termination of Hopcroft's loop"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop-waiting-eq-nil","kind":"node","lean":"DFA.hopcroftLoop_waiting_eq_nil","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop-waiting-eq-nil","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA with finite state set $Q$ over a finite alphabet $\\Sigma$. Hopcroft's loop, run from the initial partition and the initial waiting list with a budget of $|\\Sigma| \\cdot (|Q| + 1)$ rounds, ends with an empty waiting list.","title":"Hopcroft's waiting list is exhausted"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/is-partition-hopcroft-partition","kind":"node","lean":"DFA.isPartition_hopcroftPartition","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#is-partition-hopcroft-partition","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$, Hopcroft's final partition of $M$ is a partition of the states of $M$ into nonempty blocks.","title":"Hopcroft's final partition is a partition"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/mem-hopcroft-rel-iff","kind":"node","lean":"DFA.mem_hopcroftRel_iff","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#mem-hopcroft-rel-iff","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$, two states $p$ and $q$ share a block of Hopcroft's final partition of $M$ exactly when the language accepted by $M$ from $p$ equals the language accepted by $M$ from $q$.","title":"Correctness of Hopcroft's algorithm"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-rel-eq-moore-rel","kind":"node","lean":"DFA.hopcroftRel_eq_mooreRel","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-rel-eq-moore-rel","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For every DFA $M$ with finitely many states over a finite alphabet, Hopcroft's relation of $M$, the set of pairs of states lying in a common block of Hopcroft's partition, equals Moore's state equivalence of $M$ as a set of pairs of states.","title":"Hopcroft's partition is Moore's relation"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/moore-class-eq-iff-mem-hopcroft-rel","kind":"node","lean":"DFA.mooreClass_eq_iff_mem_hopcroftRel","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#moore-class-eq-iff-mem-hopcroft-rel","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"For a DFA $M$ and reachable states $p$ and $q$ of $M$, the classes of $p$ and $q$ under Moore's equivalence on reachable states (the states of Moore's minimal automaton) are equal if and only if $p$ and $q$ share a block of Hopcroft's final partition of $M$.","title":"Moore classes are Hopcroft blocks"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/mem-block-iff","kind":"node","lean":"DFA.mem_block_iff","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#mem-block-iff","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Let $M$ be a DFA, $D$ a block of Hopcroft's final partition of $M$ and $p$ a state in $D$. Then a state $q$ lies in $D$ if and only if the language accepted by $M$ from $q$ equals the language accepted by $M$ from $p$.","title":"Blocks of Hopcroft's partition are language classes"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/two-mul-smaller-card-le","kind":"node","lean":"DFA.Hopcroft.two_mul_smaller_card_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#two-mul-smaller-card-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The smaller of two sets has at most half their combined size: it is one of the two sets, so twice its size is at most the sum of the two sizes.","title":"The smaller of two sets has at most half their total size"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/two-mul-smaller-card-le-of-split","kind":"node","lean":"DFA.Hopcroft.two_mul_smaller_card_le_of_split","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#two-mul-smaller-card-le-of-split","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"If two disjoint sets union to `D`, the smaller of them has at most half the size of `D`.","title":"The smaller part of a split has at most half the size"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/mul-pow-le-card-of-halving-chain","kind":"node","lean":"DFA.Hopcroft.mul_pow_le_card_of_halving_chain","parent":33495,"reading_title":{"format":"inline-parts-v1","parts":[["text","Halving chains shrink blocks by a factor of at least "],["math","\\(2^k\\)"]],"source":"Halving chains shrink blocks by a factor of at least $2^k$"},"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#mul-pow-le-card-of-halving-chain","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"After `k` halving steps, the initial block is at least `2 ^ k` times the final block. The halving hypothesis is stated for the cardinalities only; no subset hypothesis is needed.","title":"Halving chains shrink blocks by a factor of at least $2^k$"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/smaller-half-chain-le-log","kind":"node","lean":"DFA.Hopcroft.smaller_half_chain_le_log","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#smaller-half-chain-le-log","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"In a chain of blocks where each step at least halves the size, a block containing a given state can occur at most `Nat.log 2 n` times past the start, where `n` is the number of states. In Hopcroft's algorithm, the successive smaller halves containing one fixed state form such a chain: each of them is the smaller half of a split of the previous block containing that state.","title":"The smaller-half counting lemma"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/mem-new-splitters","kind":"node","lean":"DFA.Hopcroft.mem_newSplitters","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#mem-new-splitters","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Every entry `newSplitters` adds for a block `D` is a half of `D` (one of the two sides of the split by `pre`), or the smaller half.","title":"New splitters are halves of the split block"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/two-mul-card-le-of-mem-new-splitters-sma","kind":"node","lean":"DFA.Hopcroft.two_mul_card_le_of_mem_newSplitters_smaller","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#two-mul-card-le-of-mem-new-splitters-sma","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"A smaller-half entry added by `newSplitters` for a block `D` has at most half the size of `D`: the two halves are disjoint and union to `D`. No split hypothesis is needed: the halves of any block partition it.","title":"Smaller-half splitters have at most half the block size"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-splitters","kind":"node","lean":"DFA.Hopcroft.hopcroftSplitters","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-splitters","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The splitters popped by Hopcroft's loop, in order: the same recursion as `DFA.hopcroftLoop`, collecting each popped splitter. The cost of processing one popped splitter `(C, a)` is `(preimage M C a).card + 1`: one predecessor visit per state whose `a`-successor lies in `C`, plus `O(1)` per-splitter overhead. See `hopcroftCost`. `hopcroftLoop_eq_hopcroftLoop_of_length` proves this trace records exactly the splitters\u2026","title":"Trace of splitters popped by Hopcroft's algorithm"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-cost","kind":"node","lean":"DFA.Hopcroft.hopcroftCost","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-cost","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The predecessor cost of a splitter trace: for each popped splitter `(C, a)`, one predecessor visit per state in the preimage `preimage M C a`, plus `1` per-splitter overhead. This is the quantity the total-work bound constrains to `O(|\u03a3| n log n)`. Charging splitter size `C.card` instead would undercount: if all `n` states transition on `a` to one state `q`, processing the singleton splitter `({q}, a)` costs `n`\u2026","title":"Predecessor cost of a Hopcroft splitter trace"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-splitters-length-le","kind":"node","lean":"DFA.Hopcroft.hopcroftSplitters_length_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-splitters-length-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The loop and the trace recurse on the same state, so they take the same number of steps. The two theorems below make the correspondence usable: the trace length is at most `k`, and running the loop for exactly the trace's length gives the full loop result, so the trace contains every splitter the loop pops and no others.","title":"The splitter trace records exactly the splitters the loop pops"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop-eq-hopcroft-loop-of-length","kind":"node","lean":"DFA.Hopcroft.hopcroftLoop_eq_hopcroftLoop_of_length","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop-eq-hopcroft-loop-of-length","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The trace and the loop take the same number of steps: the trace length is the fuel the loop consumed, i.e. the loop would stop after exactly that many rounds if run with only that much fuel.","title":"Hopcroft splitter trace length equals loop fuel"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splitter-count","kind":"node","lean":"DFA.Hopcroft.splitterCount","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splitter-count","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Number of splitters with letter `b` containing `q`, with multiplicity. This applies both to the waiting list and to the actual popped trace.","title":"Splitter count of a state and letter"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splitter-count-new-splitters","kind":"node","lean":"DFA.Hopcroft.splitterCount_newSplitters","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splitter-count-new-splitters","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Replacement contributes one token at a state in the parent block; otherwise only membership in the smaller half contributes a token.","title":"Splitter count of newly added splitters"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splitter-count-round-w","kind":"node","lean":"DFA.Hopcroft.splitterCount_roundW","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splitter-count-round-w","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The exact token update at a state. Only its unique parent block can contribute a new token, even when other blocks split in the same round.","title":"Splitter count update after one Hopcroft round"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splitter-count-pos-of-mem","kind":"node","lean":"DFA.Hopcroft.splitterCount_pos_of_mem","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splitter-count-pos-of-mem","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"A waiting entry supplies a token; duplicate entries remain counted.","title":"Waiting entries give positive splitter count"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/subset-of-mem-round-p","kind":"node","lean":"DFA.Hopcroft.IsPartition.subset_of_mem_roundP","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#subset-of-mem-round-p","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"A refined block containing `q` is a subset of the old block containing `q`. Both partitions are the canonical Hopcroft partitions.","title":"Refined Hopcroft blocks lie inside their old blocks"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splitter-count-round-potential","kind":"node","lean":"DFA.Hopcroft.splitterCount_round_potential","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splitter-count-round-potential","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"One refinement cannot increase logarithmic block-size credit plus pending-token credit. Replacing a pending splitter spends no new credit; adding a nonpending smaller half spends one logarithmic unit.","title":"Potential inequality for one Hopcroft refinement round"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splitter-count-hopcroft-splitters-le","kind":"node","lean":"DFA.Hopcroft.splitterCount_hopcroftSplitters_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splitter-count-hopcroft-splitters-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Total state/letter charging along the actual execution. The bound includes initial tokens and tokens still pending through replacements. No amortized bound is assumed in the execution invariant.","title":"Splitter count bound along a Hopcroft execution"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splitter-count-init-w-le","kind":"node","lean":"DFA.Hopcroft.splitterCount_initW_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splitter-count-init-w-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Initially there is at most one token for each state and letter.","title":"Initial splitter count is at most one"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/splitter-count-init-trace-le","kind":"node","lean":"DFA.Hopcroft.splitterCount_init_trace_le","parent":33495,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hopcroft visits each of "],["math","\\(n\\)"],["text"," states at most "],["math","\\(\\log_2 n + 1\\)"],["text"," times per letter"]],"source":"Hopcroft visits each of $n$ states at most $\\log_2 n + 1$ times per letter"},"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#splitter-count-init-trace-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Each state is processed with a fixed letter at most `log\u2082 n + 1` times during any prefix of the actual initialized Hopcroft execution.","title":"Hopcroft visits each of $n$ states at most $\\log_2 n + 1$ times per letter"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/nonempty-of-mem-hopcroft-splitters","kind":"node","lean":"DFA.Hopcroft.nonempty_of_mem_hopcroftSplitters","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#nonempty-of-mem-hopcroft-splitters","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Every popped splitter has a nonempty block, including splitters whose predecessor set is empty.","title":"Popped Hopcroft splitters are nonempty"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-cost-eq-sum-splitter-count","kind":"node","lean":"DFA.Hopcroft.hopcroftCost_eq_sum_splitterCount","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-cost-eq-sum-splitter-count","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Double counting visits by source state and letter, retaining every incoming transition even when many sources have the same successor.","title":"Hopcroft predecessor-work as splitter counts plus pops"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/length-le-sum-splitter-count","kind":"node","lean":"DFA.Hopcroft.length_le_sum_splitterCount","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#length-le-sum-splitter-count","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"A nonempty popped block pays for its pop, even if its preimage is empty. This bounds the `+1` term independently of predecessor visits.","title":"Number of pops bounded by the total splitter count"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-cost-init-trace-le","kind":"node","lean":"DFA.Hopcroft.hopcroftCost_init_trace_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-cost-init-trace-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Total predecessor-work bound, including one unit for every pop, for every prefix of the initialized execution. No runtime claim is made for the scanning implementation or for initialization.","title":"Total predecessor-work bound"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-cost-init-trace-le-log","kind":"node","lean":"DFA.Hopcroft.hopcroftCost_init_trace_le_log","parent":33495,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(O(|\\Sigma| n \\log n)\\)"],["text"," bound on Hopcroft predecessor-work"]],"source":"$O(|\\Sigma| n \\log n)$ bound on Hopcroft predecessor-work"},"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-cost-init-trace-le-log","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"A uniform `O(|\u03a3| n log n)` bound, with the explicit constant `4`. The condition `n \u2265 2` is essential when absorbing the additive unit into the logarithm. This concerns predecessor-work, not scan runtime.","title":"$O(|\\Sigma| n \\log n)$ bound on Hopcroft predecessor-work"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-cost-complete-le","kind":"node","lean":"DFA.Hopcroft.hopcroftCost_complete_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-cost-complete-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The bound for the complete terminating minimization execution. `DFA.hopcroftLoop_waiting_eq_nil` proves this fuel exhausts the queue; `hopcroftLoop_eq_hopcroftLoop_of_length` ties the trace to its result.","title":"Predecessor-work bound for the complete Hopcroft run"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/init-w-eq-nil-of-card-le-one","kind":"node","lean":"DFA.Hopcroft.initW_eq_nil_of_card_le_one","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#init-w-eq-nil-of-card-le-one","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"With at most one state, the accepting/rejecting split has an empty side, so the algorithm has no initial splitters.","title":"No initial Hopcroft splitters with at most one state"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-cost-eq-zero-of-card-le-one","kind":"node","lean":"DFA.Hopcroft.hopcroftCost_eq_zero_of_card_le_one","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-cost-eq-zero-of-card-le-one","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Zero predecessor-work for zero/one-state boundary parameters. This does not assert that constructing the initial partition costs zero.","title":"Zero Hopcroft predecessor-work with at most one state"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-cost-eq-zero-of-alphabet-card-e","kind":"node","lean":"DFA.Hopcroft.hopcroftCost_eq_zero_of_alphabet_card_eq_zero","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-cost-eq-zero-of-alphabet-card-e","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"An empty alphabet causes no predecessor-work, for any state count.","title":"Zero Hopcroft predecessor-work over an empty alphabet"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop-work","kind":"node","lean":"DFA.Hopcroft.hopcroftLoopWork","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop-work","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The actual loop with predecessor-visit and pop events recorded in CSLib's `TimeM`. Pure scans are not charged in this event model.","title":"Hopcroft loop with recorded predecessor-work"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop-work-ret","kind":"node","lean":"DFA.Hopcroft.hopcroftLoopWork_ret","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop-work-ret","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"Instrumentation preserves the exact loop result.","title":"Instrumented Hopcroft loop returns the original result"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop-work-time","kind":"node","lean":"DFA.Hopcroft.hopcroftLoopWork_time","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop-work-time","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The recorded cost is exactly the work of the actual popped trace.","title":"Recorded Hopcroft cost equals the work of the popped trace"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop-work-init-le","kind":"node","lean":"DFA.Hopcroft.hopcroftLoopWork_init_le","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop-work-init-le","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"The canonical cost monad records the proved total predecessor-work bound for every initialized execution prefix.","title":"Recorded Hopcroft cost obeys the predecessor-work bound"},{"id":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/hopcroft-loop-work-complete-ret","kind":"node","lean":"DFA.Hopcroft.hopcroftLoopWork_complete_ret","parent":33495,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm/index.html#hopcroft-loop-work-complete-ret","scope":"TheoryOfComputation/regular-languages/minimization/hopcroft-s-algorithm","status":"fully_proved","summary":"At the proved terminating fuel, instrumentation returns the original correct minimization partition and an empty waiting list.","title":"Instrumented complete Hopcroft run returns the minimization partition"},{"data_url":"scopes/TheoryOfComputation/regular-languages/syntactic-monoids.json","id":"TheoryOfComputation/regular-languages/syntactic-monoids","kind":"scope","parent":33244,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/index.html","scope":"TheoryOfComputation/regular-languages/syntactic-monoids","status":"fully_proved","summary":"Recognition of languages by monoid morphisms, syntactic congruences and monoids, transition monoids, language contexts and quotients, and inverse images under word morphisms.","title":"Syntactic monoids and algebraic recognition"},{"data_url":"scopes/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid.json","id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","kind":"scope","navigation":"module","parent":33562,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","title":"Syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel","kind":"node","lean":"Cslib.Language.syntacticRel","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. Two words $x, y \\in \\Sigma^*$ are syntactically equivalent for $L$ when, for all words $u, v \\in \\Sigma^*$, $uxv \\in L$ if and only if $uyv \\in L$.","title":"Syntactic equivalence"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel-prefix","kind":"node","lean":"Cslib.Language.syntacticRel_prefix","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel-prefix","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. If two words $x, y \\in \\Sigma^*$ are syntactically equivalent for $L$, then for every word $w \\in \\Sigma^*$ the words $wx$ and $wy$ are syntactically equivalent for $L$.","title":"Syntactic equivalence is stable under left multiplication"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel-suffix","kind":"node","lean":"Cslib.Language.syntacticRel_suffix","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel-suffix","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. If two words $x, y \\in \\Sigma^*$ are syntactically equivalent for $L$, then for every word $w \\in \\Sigma^*$ the words $xw$ and $yw$ are syntactically equivalent for $L$.","title":"Syntactic equivalence is stable under right multiplication"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel-append","kind":"node","lean":"Cslib.Language.syntacticRel_append","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel-append","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$ and let $w, x, y, z \\in \\Sigma^*$ be words. If $w$ and $x$ are syntactically equivalent for $L$, and $y$ and $z$ are syntactically equivalent for $L$, then $wy$ and $xz$ are syntactically equivalent for $L$.","title":"Syntactic equivalence is compatible with concatenation"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-con","kind":"node","lean":"Cslib.Language.syntacticCon","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-con","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a language $L$ over an alphabet $\\Sigma$, the syntactic congruence of $L$ is the monoid congruence on the free monoid $\\Sigma^*$ that relates words $x$ and $y$ when, for all words $u$ and $v$, $uxv \\in L$ if and only if $uyv \\in L$.","title":"Syntactic congruence"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-monoid","kind":"node","lean":"Cslib.Language.SyntacticMonoid","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-monoid","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a language $L$ over an alphabet $\\Sigma$, the syntactic monoid of $L$ is the quotient of the free monoid $\\Sigma^*$ of words by the syntactic congruence of $L$, which relates words $x$ and $y$ when $uxv \\in L \\iff uyv \\in L$ for all words $u$ and $v$.","title":"Syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-morphism","kind":"node","lean":"Cslib.Language.syntacticMorphism","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-morphism","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a language $L$ over an alphabet $\\Sigma$, the syntactic morphism of $L$ is the monoid homomorphism from the free monoid $\\Sigma^*$ of words to the syntactic monoid of $L$ that sends each word to its class under the syntactic congruence of $L$.","title":"Syntactic morphism"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel-mem-iff","kind":"node","lean":"Cslib.Language.syntacticRel_mem_iff","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel-mem-iff","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. If two words $x, y \\in \\Sigma^*$ are syntactically equivalent for $L$, then $x \\in L$ if and only if $y \\in L$.","title":"Syntactically equivalent words agree on membership"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-accept","kind":"node","lean":"Cslib.Language.syntacticAccept","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-accept","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a language $L$, the accepting set of the syntactic monoid of $L$ is the set of syntactic classes of words of $L$: a class belongs to it when its representatives lie in $L$, which does not depend on the representative chosen.","title":"Accepting set of the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/mem-iff-syntactic-morphism-mem","kind":"node","lean":"Cslib.Language.mem_iff_syntacticMorphism_mem","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#mem-iff-syntactic-morphism-mem","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. For every word $x \\in \\Sigma^*$, $x \\in L$ if and only if the image of $x$ under the syntactic morphism of $L$ lies in the accepting set of the syntactic monoid of $L$, the set of syntactic classes of words of $L$.","title":"The syntactic morphism recognizes the language"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel-nerode-rel","kind":"node","lean":"Cslib.Language.syntacticRel_nerodeRel","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel-nerode-rel","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. If two words $x, y \\in \\Sigma^*$ are syntactically equivalent for $L$, then they are Nerode equivalent for $L$: the sets $\\{z : xz \\in L\\}$ and $\\{z : yz \\in L\\}$ are equal.","title":"Syntactic equivalence refines Nerode equivalence"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/nerode-class-eq-of-syntactic-morphism-eq","kind":"node","lean":"Cslib.Language.nerodeClass_eq_of_syntacticMorphism_eq","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#nerode-class-eq-of-syntactic-morphism-eq","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language. For all words $x$ and $y$, if the syntactic morphism of $L$ sends $x$ and $y$ to the same element of the syntactic monoid of $L$, then $x$ and $y$ have the same Nerode class with respect to $L$.","title":"Equal syntactic classes give equal Nerode classes"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/transition-action","kind":"node","lean":"Cslib.Language.transitionAction","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#transition-action","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a DFA $M$ and a word $x$, the transition action of $x$ is the map sending each state $q$ of $M$ to the state that $M$ reaches by reading $x$ starting from $q$.","title":"Transition action of a word"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel-of-transition-action-eq","kind":"node","lean":"Cslib.Language.syntacticRel_of_transitionAction_eq","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel-of-transition-action-eq","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $M$ be a DFA over an alphabet $\\Sigma$ and let $L$ be the language accepted by $M$. If two words $x, y \\in \\Sigma^*$ have the same transition action on $M$, that is, for every state $q$, running $M$ from $q$ on $x$ ends in the same state as running $M$ from $q$ on $y$, then $x$ and $y$ are syntactically equivalent for $L$.","title":"Words with the same transition action are syntactically equivalent"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/transition-action-eq-of-syntactic-rel","kind":"node","lean":"Cslib.Language.transitionAction_eq_of_syntacticRel","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#transition-action-eq-of-syntactic-rel","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $M$ be a DFA that is accessible (every state is reachable from the start state) and state-separated (two states with the same continuation language are equal). If two words $x$ and $y$ are syntactically equivalent for the language accepted by $M$, then $x$ and $y$ have the same transition action on $M$: for every state $q$, reading $x$ from $q$ and reading $y$ from $q$ lead to the same state.","title":"Syntactically equivalent words act identically"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel-iff-transition-action-eq","kind":"node","lean":"Cslib.Language.syntacticRel_iff_transitionAction_eq","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel-iff-transition-action-eq","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $M$ be a DFA over an alphabet $\\Sigma$ that is accessible (every state is reached from the start state by some word) and state-separated (distinct states have distinct continuation languages), and let $L$ be the language accepted by $M$. Then two words $x, y \\in \\Sigma^*$ are syntactically equivalent for $L$ if and only if they have the same transition action on $M$: for every state $q$, running $M$ from $q$ on\u2026","title":"Syntactic equivalence via transition actions"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/state-transformation","kind":"node","lean":"Cslib.Language.StateTransformation","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#state-transformation","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a type $Q$ of states, the state transformation monoid of $Q$ consists of all functions $Q \\to Q$, multiplied by composition in the opposite order: the product $fg$ first applies $f$ and then $g$, so $(fg)(q) = g(f(q))$, and the unit is the identity function.","title":"State transformation monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/state-transformation-equiv","kind":"node","lean":"Cslib.Language.stateTransformationEquiv","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#state-transformation-equiv","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a type $Q$ of states, the bijection from the state transformations of $Q$ to the functions $Q \\to Q$ that sends each transformation to its underlying function; it forgets the multiplication of state transformations.","title":"State transformations as functions"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/transition-transformation","kind":"node","lean":"Cslib.Language.transitionTransformation","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#transition-transformation","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a DFA $M$ and a word $x$, the transition transformation of $x$ is the transition action of $x$, the map sending each state $q$ to the state that $M$ reaches by reading $x$ from $q$, regarded as an element of the monoid of state transformations of $M$ (multiplied in reading order: $fg$ applies $f$ first, then $g$).","title":"Transition transformation of a word"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/transition-morphism","kind":"node","lean":"Cslib.Language.transitionMorphism","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#transition-morphism","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For a DFA $M$, the transition morphism is the monoid homomorphism from the free monoid of words to the state transformations of $M$ that sends each word $x$ to its transition action, the map taking each state $q$ to the state that $M$ reaches by reading $x$ from $q$. State transformations are multiplied in reading order, so the product $fg$ applies $f$ first and then $g$; with this order, the word $xy$ is sent to\u2026","title":"Transition morphism of a DFA"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/transition-monoid","kind":"node","lean":"Cslib.Language.TransitionMonoid","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#transition-monoid","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"The transition monoid of a DFA $M$ is the image of the transition morphism of $M$: the submonoid of state transformations of $M$ consisting of the transition actions of all words, where a word $x$ acts by sending each state $q$ to the state that $M$ reaches by reading $x$ from $q$. State transformations are multiplied in reading order, so the product $fg$ applies $f$ first and then $g$.","title":"Transition monoid of a DFA"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/to-transition-monoid","kind":"node","lean":"Cslib.Language.toTransitionMonoid","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#to-transition-monoid","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $M$ be a DFA over an alphabet $\\Sigma$. Each word $x \\in \\Sigma^*$ determines the element of the transition monoid of $M$ given by the state transformation that sends each state $q$ to the state $M$ reaches from $q$ after reading $x$.","title":"Transition monoid element of a word"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-con-eq-transition-morphism-ker","kind":"node","lean":"Cslib.Language.syntacticCon_eq_transitionMorphism_ker","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-con-eq-transition-morphism-ker","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $M$ be an accessible, state-separated DFA and let $L$ be the language $M$ accepts. The syntactic congruence of $L$ equals the kernel of the transition morphism of $M$: two words are syntactically equivalent for $L$ if and only if they induce the same transformation of the states of $M$.","title":"Syntactic congruence as a kernel"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-action-hom","kind":"node","lean":"Cslib.Language.syntacticActionHom","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-action-hom","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $M$ be an accessible, state-separated DFA with state type $Q$, and let $L$ be the language $M$ accepts. This is the monoid homomorphism from the syntactic monoid of $L$ to the state transformation monoid of $Q$ that sends the syntactic class of each word $x$ to the map taking each state $q$ to the state $M$ reaches from $q$ by reading $x$. It is well defined because syntactically equivalent words act\u2026","title":"Action of the syntactic monoid on states"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-action-hom-range","kind":"node","lean":"Cslib.Language.syntacticActionHom_range","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-action-hom-range","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $M$ be an accessible, state-separated DFA and let $L$ be the language $M$ accepts. The image of the action of the syntactic monoid of $L$ on the states of $M$, the monoid homomorphism sending the syntactic class of each word $x$ to the state transformation induced by reading $x$, is exactly the transition monoid of $M$.","title":"Image of the syntactic action"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizes","kind":"node","lean":"Cslib.Language.Recognizes","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizes","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$, let $\\varphi : \\Sigma^* \\to M$ be a monoid morphism from the free monoid on $\\Sigma$ into a monoid $M$, and let $A \\subseteq M$. Then $\\varphi$ recognizes $L$ through $A$ when, for every word $w \\in \\Sigma^*$, $w \\in L$ if and only if $\\varphi(w) \\in A$.","title":"Recognition of a language by a monoid morphism"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizes-syntactic-morphism","kind":"node","lean":"Cslib.Language.recognizes_syntacticMorphism","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizes-syntactic-morphism","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For every language $L$ over an alphabet $\\Sigma$, the syntactic morphism of $L$, from the free monoid $\\Sigma^*$ to the syntactic monoid of $L$, recognizes $L$ through the accepting subset of the syntactic monoid, namely the set of syntactic classes of words in $L$.","title":"The syntactic morphism recognizes its language"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/comp","kind":"node","lean":"Cslib.Language.Recognizes.comp","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#comp","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$ and let $\\varphi : \\Sigma^* \\to C$ be a monoid morphism from the free monoid $\\Sigma^*$ that recognizes $L$ through a subset $A \\subseteq C$. If $\\varphi = f \\circ \\psi$ for monoid morphisms $\\psi : \\Sigma^* \\to T$ and $f : T \\to C$, then $\\psi$ recognizes $L$ through the preimage $f^{-1}(A)$.","title":"Recognition pulls back along a factorization"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-rel-of-recognizer-eq","kind":"node","lean":"Cslib.Language.syntacticRel_of_recognizer_eq","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-rel-of-recognizer-eq","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $\\varphi : \\Sigma^* \\to M$ be a monoid morphism from the free monoid on an alphabet $\\Sigma$ into a monoid $M$ that recognizes a language $L$ over $\\Sigma$ through a set $A \\subseteq M$. If two words $x, y \\in \\Sigma^*$ satisfy $\\varphi(x) = \\varphi(y)$, then $x$ and $y$ are syntactically equivalent for $L$: for all words $u, v \\in \\Sigma^*$, $uxv \\in L$ if and only if $uyv \\in L$.","title":"Words with equal recognizer images are syntactically equivalent"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizer-ker-le-syntactic-con","kind":"node","lean":"Cslib.Language.recognizer_ker_le_syntacticCon","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizer-ker-le-syntactic-con","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"If a monoid morphism $\\varphi$ from the free monoid $\\Sigma^*$ recognizes a language $L$ over $\\Sigma$ through a subset $A$ of its codomain, then the kernel congruence of $\\varphi$ is contained in the syntactic congruence of $L$: any words $x, y$ with $\\varphi(x) = \\varphi(y)$ are syntactically equivalent for $L$.","title":"Recognizer kernels refine the syntactic congruence"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizer-quotient-to-syntactic","kind":"node","lean":"Cslib.Language.recognizerQuotientToSyntactic","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizer-quotient-to-syntactic","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$, let $\\varphi : \\Sigma^* \\to M$ be a monoid morphism from the free monoid on $\\Sigma$ into a monoid $M$, and let $A \\subseteq M$ be such that $\\varphi$ recognizes $L$ through $A$. Since the kernel congruence of $\\varphi$ (words $x, y$ with $\\varphi(x) = \\varphi(y)$) is contained in the syntactic congruence of $L$, there is a monoid morphism from the quotient of\u2026","title":"Canonical map from a recognizer's kernel quotient to the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizer-quotient-to-syntactic-surject","kind":"node","lean":"Cslib.Language.recognizerQuotientToSyntactic_surjective","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizer-quotient-to-syntactic-surject","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"If a monoid morphism $\\varphi$ from the free monoid $\\Sigma^*$ recognizes a language $L$ over $\\Sigma$ through a subset $A$ of its codomain, then the canonical monoid morphism from the quotient of $\\Sigma^*$ by the kernel congruence of $\\varphi$ to the syntactic monoid of $L$, which sends the class of a word $x$ to its syntactic class, is surjective.","title":"The kernel quotient of a recognizer maps onto the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizer-image-to-syntactic","kind":"node","lean":"Cslib.Language.recognizerImageToSyntactic","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizer-image-to-syntactic","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $\\varphi : \\Sigma^* \\to M$ be a monoid morphism from the free monoid $\\Sigma^*$ that recognizes a language $L$ over $\\Sigma$ through a subset $A \\subseteq M$. This is the monoid morphism from the image $\\varphi(\\Sigma^*)$ to the syntactic monoid of $L$ that sends $\\varphi(x)$ to the syntactic class of $x$ for every word $x$. It is well defined because words with the same image under $\\varphi$ are syntactically\u2026","title":"Map from a recognizer's image to the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizer-image-to-syntactic-surjective","kind":"node","lean":"Cslib.Language.recognizerImageToSyntactic_surjective","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizer-image-to-syntactic-surjective","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $\\varphi : \\Sigma^* \\to M$ be a monoid morphism from the free monoid $\\Sigma^*$ that recognizes a language $L$ over $\\Sigma$ through a subset $A \\subseteq M$. Then the canonical monoid morphism from the image $\\varphi(\\Sigma^*)$ to the syntactic monoid of $L$, which sends $\\varphi(x)$ to the syntactic class of $x$, is surjective.","title":"The image of a recognizer maps onto the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/finite-syntactic-monoid-of-recognizer","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_of_recognizer","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#finite-syntactic-monoid-of-recognizer","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"If a monoid morphism from the free monoid $\\Sigma^*$ to a finite monoid $M$ recognizes a language $L$ over $\\Sigma$ through some subset of $M$, then the syntactic monoid of $L$ is finite.","title":"Finite recognizers give finite syntactic monoids"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizes-iff-preimage","kind":"node","lean":"Cslib.Language.recognizes_iff_preimage","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizes-iff-preimage","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$, let $\\varphi : \\Sigma^* \\to M$ be a monoid morphism from the free monoid on $\\Sigma$ into a monoid $M$, and let $A \\subseteq M$. Then $\\varphi$ recognizes $L$ through $A$ if and only if the set of words of $\\Sigma^*$ that belong to $L$ equals $\\varphi^{-1}(A)$.","title":"Recognition as an inverse image"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/congr-accept","kind":"node","lean":"Cslib.Language.Recognizes.congr_accept","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#congr-accept","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $\\varphi : \\Sigma^* \\to M$ be a monoid morphism from the free monoid $\\Sigma^*$ that recognizes a language $L$ over $\\Sigma$ through a subset $A_1 \\subseteq M$. If $A_2 \\subseteq M$ satisfies $\\varphi(x) \\in A_1 \\iff \\varphi(x) \\in A_2$ for every word $x$, then $\\varphi$ also recognizes $L$ through $A_2$.","title":"Changing the accepting set off the image of a recognizer"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-dfa","kind":"node","lean":"Cslib.Language.syntacticDFA","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-dfa","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. The syntactic DFA of $L$ has the syntactic monoid of $L$ as its set of states, the identity $1$ as start state, and the syntactic classes of the words of $L$ as accepting states; from a state $q$, reading a letter $a \\in \\Sigma$ leads to $q \\cdot [a]$, where $[a]$ is the syntactic class of the one-letter word $a$.","title":"The syntactic DFA"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-dfa-eval-from","kind":"node","lean":"Cslib.Language.syntacticDFA_evalFrom","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-dfa-eval-from","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\Sigma$. For every state $q$ of the syntactic DFA of $L$ (an element of the syntactic monoid of $L$) and every word $x \\in \\Sigma^*$, the state reached from $q$ by reading $x$ is $q \\cdot [x]$, where $[x]$ is the image of $x$ under the syntactic morphism of $L$.","title":"Runs of the syntactic DFA"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-dfa-accepts","kind":"node","lean":"Cslib.Language.syntacticDFA_accepts","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-dfa-accepts","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For every language $L$, the syntactic DFA of $L$ accepts exactly $L$.","title":"The syntactic DFA accepts its language"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/syntactic-dfa-accessible","kind":"node","lean":"Cslib.Language.syntacticDFA_accessible","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#syntactic-dfa-accessible","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"For every language $L$, every state of the syntactic DFA of $L$ is reachable: some word leads to it from the start state.","title":"The syntactic DFA is accessible"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/is-regular-of-finite-syntactic-monoid","kind":"node","lean":"Cslib.Language.isRegular_of_finite_syntacticMonoid","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#is-regular-of-finite-syntactic-monoid","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Every language whose syntactic monoid is finite is regular.","title":"Finite syntactic monoid implies regularity"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/exists-finite-syntactic-dfa","kind":"node","lean":"Cslib.Language.exists_finite_syntacticDFA","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#exists-finite-syntactic-dfa","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"If the syntactic monoid of a language $L$ is finite, then the syntactic DFA of $L$ has finitely many states and accepts exactly $L$.","title":"The syntactic DFA of a language with finite syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/recognizes-transition-morphism","kind":"node","lean":"Cslib.Language.recognizes_transitionMorphism","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#recognizes-transition-morphism","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Let $M$ be a DFA over an alphabet $\\Sigma$ with state set $Q$, start state $q_0$ and set of accepting states $F$. The transition morphism of $M$, from the free monoid $\\Sigma^*$ to the monoid of state transformations of $Q$, recognizes the language accepted by $M$ through the set of transformations $f$ with $f(q_0) \\in F$.","title":"The transition morphism of a DFA recognizes its language"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/finite-syntactic-monoid-of-finite-dfa","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_of_finite_dfa","parent":33563,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid/index.html#finite-syntactic-monoid-of-finite-dfa","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/syntactic-monoid","status":"fully_proved","summary":"Every language accepted by a DFA with finitely many states has a finite syntactic monoid.","title":"Languages of finite automata have finite syntactic monoids"},{"data_url":"scopes/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity.json","id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","kind":"scope","navigation":"module","parent":33562,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","title":"Transition monoids and regularity"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/transition-accept","kind":"node","lean":"Cslib.Language.transitionAccept","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#transition-accept","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"For a DFA $M$ with set of states $Q$, start state $q_0$ and accepting states $F$, the accepting transformations of $M$ are the state transformations $f$ of $Q$ with $f(q_0) \\in F$.","title":"Accepting transformations of a DFA"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/recognizes-transition-accept","kind":"node","lean":"Cslib.Language.recognizes_transitionAccept","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#recognizes-transition-accept","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"Let $M$ be a DFA with start state $q_0$ and accepting states $F$. The transition morphism of $M$, from words to state transformations, recognizes the language accepted by $M$ through the accepting transformations of $M$: a word is accepted by $M$ exactly when the transformation it induces sends $q_0$ into $F$.","title":"The transition morphism recognizes through accepting transformations"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/recognizes-to-transition-monoid","kind":"node","lean":"Cslib.Language.recognizes_toTransitionMonoid","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#recognizes-to-transition-monoid","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"Let $M$ be a DFA with start state $q_0$ and accepting states $F$. The map sending each word to the element of the transition monoid of $M$ that it induces recognizes the language accepted by $M$, with accepting set the elements $f$ of the transition monoid such that $f(q_0) \\in F$: a word is accepted by $M$ exactly when the transformation it induces sends $q_0$ into $F$.","title":"The transition monoid recognizes the accepted language"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/finite-state-transformation","kind":"node","lean":"Cslib.Language.finite_stateTransformation","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#finite-state-transformation","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"If $Q$ is a finite set, then the monoid of state transformations of $Q$, consisting of all functions $Q \\to Q$ multiplied in diagrammatic order, is finite.","title":"Finitely many state transformations"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/finite-transition-monoid","kind":"node","lean":"Cslib.Language.finite_transitionMonoid","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#finite-transition-monoid","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"For every DFA $M$ with a finite set of states, the transition monoid of $M$ is finite.","title":"Transition monoids of finite automata are finite"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/finite-syntactic-monoid-of-is-regular","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_of_isRegular","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#finite-syntactic-monoid-of-is-regular","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"If $L$ is a regular language, then the syntactic monoid of $L$ is finite.","title":"Regular languages have finite syntactic monoids"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/is-regular-iff-finite-syntactic-monoid","kind":"node","lean":"Cslib.Language.isRegular_iff_finite_syntacticMonoid","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#is-regular-iff-finite-syntactic-monoid","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"A language $L$ is regular if and only if the syntactic monoid of $L$ is finite.","title":"Regular languages are those with finite syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/transition-monoid-to-syntactic","kind":"node","lean":"Cslib.Language.transitionMonoidToSyntactic","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#transition-monoid-to-syntactic","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"For a DFA $M$ accepting the language $L$, this is the monoid homomorphism from the transition monoid of $M$ to the syntactic monoid of $L$ that sends the transformation induced by each word $w$ to the syntactic class of $w$.","title":"The transition monoid maps to the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/card-syntactic-monoid-le-card-transition","kind":"node","lean":"Cslib.Language.card_syntacticMonoid_le_card_transitionMonoid","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#card-syntactic-monoid-le-card-transition","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"Let $M$ be a DFA accepting the language $L$, and suppose that the transition monoid of $M$ and the syntactic monoid of $L$ are both finite. Then the syntactic monoid of $L$ has at most as many elements as the transition monoid of $M$.","title":"The syntactic monoid is no larger than the transition monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/card-syntactic-monoid-le-card-endofuncti","kind":"node","lean":"Cslib.Language.card_syntacticMonoid_le_card_endofunction","parent":33609,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity/index.html#card-syntactic-monoid-le-card-endofuncti","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/transition-monoids-and-regularity","status":"fully_proved","summary":"Let $M$ be a DFA with a finite set of states $Q$, and suppose the syntactic monoid of the language accepted by $M$ is finite. Then that syntactic monoid has at most $|Q|^{|Q|}$ elements.","title":"Endofunction bound for the syntactic monoid"},{"data_url":"scopes/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages.json","id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","kind":"scope","navigation":"module","parent":33562,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","title":"Inverse homomorphic images of regular languages"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/alphabet-preimage","kind":"node","lean":"Cslib.Language.alphabetPreimage","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#alphabet-preimage","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets and $L$ a language over $\\Gamma$. The alphabet preimage of $L$ under $f$ is the language over $\\Sigma$ consisting of the words $a_1 a_2 \\cdots a_n$ such that $f(a_1) f(a_2) \\cdots f(a_n) \\in L$.","title":"Alphabet preimage"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/alphabet-preimage-morphism","kind":"node","lean":"Cslib.Language.alphabetPreimageMorphism","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#alphabet-preimage-morphism","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets and $L$ a language over $\\Gamma$. The alphabet preimage morphism of $f$ and $L$ is the monoid morphism from the free monoid $\\Sigma^*$ to the syntactic monoid of $L$ that sends a word $w$ to the syntactic class of $f^*(w)$, the word obtained by applying $f$ to each letter of $w$. It is the composite of the letterwise map $f^* \\colon \\Sigma^* \\to \\Gamma^*$\u2026","title":"Alphabet preimage morphism"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/recognizes-alphabet-preimage-morphism","kind":"node","lean":"Cslib.Language.recognizes_alphabetPreimageMorphism","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#recognizes-alphabet-preimage-morphism","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets and $L$ a language over $\\Gamma$. Write $f^*(w)$ for the word obtained by applying $f$ to each letter of a word $w$, $[u]_L$ for the class of a word $u$ in the syntactic monoid $M(L)$ of $L$, $A_L \\subseteq M(L)$ for the accepting set of classes of words in $L$, and $\\varphi \\colon \\Sigma^* \\to M(L)$ for the alphabet preimage morphism\u2026","title":"The alphabet preimage morphism recognizes the alphabet preimage"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/syntactic-rel-alphabet-preimage-of-map","kind":"node","lean":"Cslib.Language.syntacticRel_alphabetPreimage_of_map","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#syntactic-rel-alphabet-preimage-of-map","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets, $L$ a language over $\\Gamma$, and $K$ the alphabet preimage $\\{w \\in \\Sigma^* \\mid f^*(w) \\in L\\}$, where $f^*(w)$ is the word obtained by applying $f$ to each letter of $w$. If $x, y \\in \\Sigma^*$ are words such that $f^*(x)$ and $f^*(y)$ are syntactically equivalent for $L$, then $x$ and $y$ are syntactically equivalent for $K$.","title":"Syntactic equivalence pulls back along an alphabet map"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/alphabet-preimage-morphism-ker-le","kind":"node","lean":"Cslib.Language.alphabetPreimageMorphism_ker_le","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#alphabet-preimage-morphism-ker-le","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets, $L$ a language over $\\Gamma$ with syntactic monoid $M(L)$, and $\\varphi \\colon \\Sigma^* \\to M(L)$ the alphabet preimage morphism, which sends a word $w$ to the class in $M(L)$ of the word $f^*(w)$ obtained by applying $f$ to each letter of $w$. If two words $x, y \\in \\Sigma^*$ satisfy $\\varphi(x) = \\varphi(y)$, then $x$ and $y$ are syntactically congruent\u2026","title":"The alphabet preimage morphism refines the preimage's syntactic congruence"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/alphabet-preimage-to-syntactic","kind":"node","lean":"Cslib.Language.alphabetPreimageToSyntactic","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#alphabet-preimage-to-syntactic","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets and $L$ a language over $\\Gamma$. Write $f^*(w)$ for the word obtained by applying $f$ to each letter of a word $w$, $[u]_L$ for the class of a word $u$ in the syntactic monoid $M(L)$ of $L$, $K$ for the alphabet preimage $\\{w \\in \\Sigma^* \\mid f^*(w) \\in L\\}$, and $\\varphi \\colon \\Sigma^* \\to M(L)$ for the alphabet preimage morphism\u2026","title":"Division map for the syntactic monoid of an alphabet preimage"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/finite-syntactic-monoid-alphabet-preimag","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_alphabetPreimage","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#finite-syntactic-monoid-alphabet-preimag","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets and $L$ a language over $\\Gamma$ whose syntactic monoid is finite. Then the syntactic monoid of the alphabet preimage $\\{w \\in \\Sigma^* \\mid f^*(w) \\in L\\}$ is finite, where $f^*(w)$ is the word obtained by applying $f$ to each letter of $w$.","title":"Alphabet preimages preserve finiteness of the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/alphabet-preimage-2","kind":"node","lean":"Cslib.Language.IsRegular.alphabetPreimage","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#alphabet-preimage-2","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets. If $L$ is a regular language over $\\Gamma$, then its alphabet preimage $\\{w \\in \\Sigma^* \\mid f^*(w) \\in L\\}$ is regular, where $f^*(w)$ is the word obtained by applying $f$ to each letter of $w$.","title":"Regular languages are closed under alphabet preimages"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/is-regular-alphabet-preimage-equiv-iff","kind":"node","lean":"Cslib.Language.isRegular_alphabetPreimage_equiv_iff","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#is-regular-alphabet-preimage-equiv-iff","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $e \\colon \\Sigma \\to \\Gamma$ be a bijection between alphabets and $L$ a language over $\\Gamma$. The alphabet preimage $\\{w \\in \\Sigma^* \\mid e^*(w) \\in L\\}$, where $e^*(w)$ is the word obtained by applying $e$ to each letter of $w$, is regular if and only if $L$ is regular.","title":"Renaming letters by a bijection preserves and reflects regularity"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/finite-syntactic-monoid-alphabet-preimag-2","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_alphabetPreimage_equiv_iff","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#finite-syntactic-monoid-alphabet-preimag-2","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $e \\colon \\Sigma \\to \\Gamma$ be a bijection between alphabets and $L$ a language over $\\Gamma$. The syntactic monoid of the alphabet preimage $\\{w \\in \\Sigma^* \\mid e^*(w) \\in L\\}$, where $e^*(w)$ is the word obtained by applying $e$ to each letter of $w$, is finite if and only if the syntactic monoid of $L$ is finite.","title":"Finiteness of the syntactic monoid is invariant under renaming letters"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/word-morphism-preimage","kind":"node","lean":"Cslib.Language.wordMorphismPreimage","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#word-morphism-preimage","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $\\Sigma$ and $\\Gamma$ be alphabets, $h \\colon \\Sigma^* \\to \\Gamma^*$ a monoid morphism between their free monoids of words, and $L$ a language over $\\Gamma$. The inverse image of $L$ under $h$ is the language $h^{-1}(L) = \\{w \\in \\Sigma^* \\mid h(w) \\in L\\}$ over $\\Sigma$.","title":"Inverse image under a word morphism"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/to-list-word-morphism-of-list","kind":"node","lean":"Cslib.Language.toList_wordMorphism_ofList","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#to-list-word-morphism-of-list","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $h \\colon \\Sigma^* \\to \\Gamma^*$ be a monoid morphism between free monoids. For every word $a_1 a_2 \\cdots a_n \\in \\Sigma^*$ (including the empty word, $n = 0$), we have $h(a_1 a_2 \\cdots a_n) = h(a_1) h(a_2) \\cdots h(a_n)$, the concatenation of the images of its letters.","title":"Word morphisms act letter by letter"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/word-morphism-preimage-eq-flatten","kind":"node","lean":"Cslib.Language.wordMorphismPreimage_eq_flatten","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#word-morphism-preimage-eq-flatten","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $h \\colon \\Sigma^* \\to \\Gamma^*$ be a monoid morphism between free monoids and $L$ a language over $\\Gamma$. The inverse image $h^{-1}(L) = \\{w \\in \\Sigma^* \\mid h(w) \\in L\\}$ consists exactly of the words $a_1 a_2 \\cdots a_n \\in \\Sigma^*$ such that the concatenation $h(a_1) h(a_2) \\cdots h(a_n)$ of the images of their letters lies in $L$.","title":"Letterwise description of the inverse image under a word morphism"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/word-morphism-recognizer","kind":"node","lean":"Cslib.Language.wordMorphismRecognizer","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#word-morphism-recognizer","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $h \\colon \\Sigma^* \\to \\Gamma^*$ be a monoid morphism between free monoids and $L$ a language over $\\Gamma$ with syntactic monoid $M(L)$. The recognizing morphism of $h$ and $L$ is the monoid morphism $\\Sigma^* \\to M(L)$ that sends a word $w$ to the syntactic class $[h(w)]_L$ of $h(w)$; it is the composite of $h$ with the syntactic morphism of $L$.","title":"Recognizing morphism of an inverse image"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/syntactic-rel-word-morphism-preimage-of","kind":"node","lean":"Cslib.Language.syntacticRel_wordMorphismPreimage_of_map","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#syntactic-rel-word-morphism-preimage-of","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $h \\colon \\Sigma^* \\to \\Gamma^*$ be a monoid morphism between free monoids, $L$ a language over $\\Gamma$, and $K = h^{-1}(L) = \\{w \\in \\Sigma^* \\mid h(w) \\in L\\}$ its inverse image. If $x, y \\in \\Sigma^*$ are words such that $h(x)$ and $h(y)$ are syntactically equivalent for $L$, then $x$ and $y$ are syntactically equivalent for $K$.","title":"Syntactic equivalence pulls back along a word morphism"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/word-morphism-preimage-to-syntactic","kind":"node","lean":"Cslib.Language.wordMorphismPreimageToSyntactic","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#word-morphism-preimage-to-syntactic","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $h \\colon \\Sigma^* \\to \\Gamma^*$ be a monoid morphism between free monoids and $L$ a language over $\\Gamma$. Write $[u]_L$ for the class of a word $u$ in the syntactic monoid $M(L)$ of $L$, $K$ for the inverse image $h^{-1}(L) = \\{w \\in \\Sigma^* \\mid h(w) \\in L\\}$, and $\\varphi \\colon \\Sigma^* \\to M(L)$ for the recognizing morphism $w \\mapsto [h(w)]_L$. This is the monoid morphism from the submonoid\u2026","title":"Division map for the syntactic monoid of an inverse image"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/finite-syntactic-monoid-word-morphism-pr","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_wordMorphismPreimage","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#finite-syntactic-monoid-word-morphism-pr","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $h \\colon \\Sigma^* \\to \\Gamma^*$ be a monoid morphism between free monoids and $L$ a language over $\\Gamma$ whose syntactic monoid is finite. Then the syntactic monoid of the inverse image $h^{-1}(L) = \\{w \\in \\Sigma^* \\mid h(w) \\in L\\}$ is finite.","title":"Inverse images under word morphisms preserve finiteness of the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/word-morphism-preimage-2","kind":"node","lean":"Cslib.Language.IsRegular.wordMorphismPreimage","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#word-morphism-preimage-2","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $h \\colon \\Sigma^* \\to \\Gamma^*$ be a monoid morphism between free monoids. If $L$ is a regular language over $\\Gamma$, then its inverse image $h^{-1}(L) = \\{w \\in \\Sigma^* \\mid h(w) \\in L\\}$ is regular.","title":"Regular languages are closed under inverse word morphisms"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/alphabet-preimage-eq-word-morphism-preim","kind":"node","lean":"Cslib.Language.alphabetPreimage_eq_wordMorphismPreimage","parent":33620,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages/index.html#alphabet-preimage-eq-word-morphism-preim","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/inverse-homomorphic-images-of-regular-languages","status":"fully_proved","summary":"Let $f \\colon \\Sigma \\to \\Gamma$ be a map between alphabets, $f^* \\colon \\Sigma^* \\to \\Gamma^*$ the monoid morphism that applies $f$ to each letter of a word, and $L$ a language over $\\Gamma$. Then the alphabet preimage of $L$ under $f$ equals the inverse image $(f^*)^{-1}(L) = \\{w \\in \\Sigma^* \\mid f^*(w) \\in L\\}$ of $L$ under $f^*$.","title":"Alphabet preimages are inverse images under word morphisms"},{"data_url":"scopes/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids.json","id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","kind":"scope","navigation":"module","parent":33562,"result":false,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","title":"Language quotients and syntactic monoids"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/context-quotient","kind":"node","lean":"Cslib.Language.contextQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#context-quotient","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$ and let $u, v$ be words over $A$. The context quotient of $L$ by $(u, v)$ is the language $u^{-1} L v^{-1} = \\{x \\in A^* \\mid uxv \\in L\\}$ of words that land in $L$ when placed between $u$ and $v$.","title":"Context quotient"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/context-quotient-context-quotient","kind":"node","lean":"Cslib.Language.contextQuotient_contextQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#context-quotient-context-quotient","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$, and for words $u, v$ over $A$ write $u^{-1} L v^{-1} = \\{x \\mid uxv \\in L\\}$. Then for all words $u_1, v_1, u_2, v_2$ over $A$, $u_2^{-1} (u_1^{-1} L v_1^{-1}) v_2^{-1} = (u_1 u_2)^{-1} L (v_2 v_1)^{-1}$.","title":"Composing context quotients"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/context-accept","kind":"node","lean":"Cslib.Language.contextAccept","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#context-accept","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $\\varphi \\colon A^* \\to M$ be a monoid morphism from the free monoid on an alphabet $A$ to a monoid $M$, let $P \\subseteq M$, and let $u, v$ be words over $A$. The accepting set for the context $(u, v)$ is $\\{m \\in M \\mid \\varphi(u)\\, m\\, \\varphi(v) \\in P\\}$.","title":"Accepting set for a context quotient"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/context-quotient-2","kind":"node","lean":"Cslib.Language.Recognizes.contextQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#context-quotient-2","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $\\varphi \\colon A^* \\to M$ be a monoid morphism from the free monoid on an alphabet $A$ to a monoid $M$ that recognizes a language $L$ over $A$ through $P \\subseteq M$, that is, a word $x$ lies in $L$ exactly when $\\varphi(x) \\in P$. Then for all words $u, v$ over $A$, $\\varphi$ recognizes the context quotient $\\{x \\mid uxv \\in L\\}$ through the set $\\{m \\in M \\mid \\varphi(u)\\, m\\, \\varphi(v) \\in P\\}$.","title":"Recognizers recognize context quotients"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/context-quotient-to-syntactic","kind":"node","lean":"Cslib.Language.contextQuotientToSyntactic","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#context-quotient-to-syntactic","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$, let $u, v$ be words over $A$, and write $u^{-1} L v^{-1} = \\{x \\mid uxv \\in L\\}$. Let $\\eta$ be the syntactic morphism of $L$ and $\\eta'$ the syntactic morphism of $u^{-1} L v^{-1}$. This is the monoid morphism from the image of $\\eta$ to the syntactic monoid of $u^{-1} L v^{-1}$ that sends $\\eta(x)$ to $\\eta'(x)$ for every word $x$ over $A$.","title":"The canonical map to the syntactic monoid of a context quotient"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/card-syntactic-monoid-context-quotient-l","kind":"node","lean":"Cslib.Language.card_syntacticMonoid_contextQuotient_le_mrange","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#card-syntactic-monoid-context-quotient-l","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$, let $u, v$ be words over $A$, and write $u^{-1} L v^{-1} = \\{x \\mid uxv \\in L\\}$ for the context quotient. If the image of the syntactic morphism of $L$ and the syntactic monoid of $u^{-1} L v^{-1}$ are both finite, then the syntactic monoid of $u^{-1} L v^{-1}$ has at most as many elements as the image of the syntactic morphism of $L$.","title":"Size bound for the syntactic monoid of a context quotient"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/finite-syntactic-monoid-context-quotient","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_contextQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#finite-syntactic-monoid-context-quotient","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$ whose syntactic monoid is finite. Then for all words $u, v$ over $A$, the syntactic monoid of the context quotient $u^{-1} L v^{-1} = \\{x \\mid uxv \\in L\\}$ is finite.","title":"Context quotients preserve finiteness of the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/context-quotient-3","kind":"node","lean":"Cslib.Language.IsRegular.contextQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#context-quotient-3","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"If $L$ is a regular language over an alphabet $A$, then for all words $u, v$ over $A$, the context quotient $u^{-1} L v^{-1} = \\{x \\mid uxv \\in L\\}$ is regular.","title":"Regular languages are closed under context quotients"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/syntactic-rel-iff-mem-context-quotient","kind":"node","lean":"Cslib.Language.syntacticRel_iff_mem_contextQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#syntactic-rel-iff-mem-context-quotient","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$. Two words $x, y$ over $A$ are syntactically equivalent for $L$ if and only if, for all words $u, v$ over $A$, $x$ lies in the context quotient $u^{-1} L v^{-1} = \\{w \\mid uwv \\in L\\}$ exactly when $y$ does.","title":"Syntactic equivalence via context quotients"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/right-quotient","kind":"node","lean":"Cslib.Language.rightQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#right-quotient","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"For a language $L$ over an alphabet $\\Sigma$ and a word $w \\in \\Sigma^*$, the right quotient of $L$ by $w$ is the language $\\{x \\in \\Sigma^* : xw \\in L\\}$ of all words $x$ whose concatenation $xw$ with $w$ lies in $L$.","title":"Right quotient of a language by a word"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/left-quotient-accept","kind":"node","lean":"Cslib.Language.leftQuotientAccept","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#left-quotient-accept","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $\\varphi \\colon A^* \\to M$ be a monoid morphism from the free monoid on an alphabet $A$ to a monoid $M$, let $P \\subseteq M$, and let $u$ be a word over $A$. The accepting set for the left quotient by $u$ is $\\{m \\in M \\mid \\varphi(u)\\, m \\in P\\}$.","title":"Accepting set for a left quotient"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/right-quotient-accept","kind":"node","lean":"Cslib.Language.rightQuotientAccept","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#right-quotient-accept","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $\\varphi \\colon A^* \\to M$ be a monoid morphism from the free monoid on an alphabet $A$ to a monoid $M$, let $P \\subseteq M$, and let $u$ be a word over $A$. The accepting set for the right quotient by $u$ is $\\{m \\in M \\mid m\\, \\varphi(u) \\in P\\}$.","title":"Accepting set for a right quotient"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/left-quotient","kind":"node","lean":"Cslib.Language.Recognizes.leftQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#left-quotient","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $\\varphi \\colon A^* \\to M$ be a monoid morphism from the free monoid on an alphabet $A$ to a monoid $M$ that recognizes a language $L$ over $A$ through $P \\subseteq M$, that is, a word $x$ lies in $L$ exactly when $\\varphi(x) \\in P$. Then for every word $u$ over $A$, $\\varphi$ recognizes the left quotient $\\{x \\mid ux \\in L\\}$ through the set $\\{m \\in M \\mid \\varphi(u)\\, m \\in P\\}$.","title":"Recognizers recognize left quotients"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/right-quotient-2","kind":"node","lean":"Cslib.Language.Recognizes.rightQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#right-quotient-2","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $\\varphi \\colon A^* \\to M$ be a monoid morphism from the free monoid on an alphabet $A$ to a monoid $M$ that recognizes a language $L$ over $A$ through $P \\subseteq M$, that is, a word $x$ lies in $L$ exactly when $\\varphi(x) \\in P$. Then for every word $u$ over $A$, $\\varphi$ recognizes the right quotient $\\{x \\mid xu \\in L\\}$ through the set $\\{m \\in M \\mid m\\, \\varphi(u) \\in P\\}$.","title":"Recognizers recognize right quotients"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/left-quotient-to-syntactic","kind":"node","lean":"Cslib.Language.leftQuotientToSyntactic","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#left-quotient-to-syntactic","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$, let $u$ be a word over $A$, and write $u^{-1} L = \\{x \\mid ux \\in L\\}$. Let $\\eta$ be the syntactic morphism of $L$ and $\\eta'$ the syntactic morphism of $u^{-1} L$. This is the monoid morphism from the image of $\\eta$ to the syntactic monoid of $u^{-1} L$ that sends $\\eta(x)$ to $\\eta'(x)$ for every word $x$ over $A$.","title":"The canonical map to the syntactic monoid of a left quotient"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/right-quotient-to-syntactic","kind":"node","lean":"Cslib.Language.rightQuotientToSyntactic","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#right-quotient-to-syntactic","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$, let $u$ be a word over $A$, and write $L u^{-1} = \\{x \\mid xu \\in L\\}$. Let $\\eta$ be the syntactic morphism of $L$ and $\\eta'$ the syntactic morphism of $L u^{-1}$. This is the monoid morphism from the image of $\\eta$ to the syntactic monoid of $L u^{-1}$ that sends $\\eta(x)$ to $\\eta'(x)$ for every word $x$ over $A$.","title":"The canonical map to the syntactic monoid of a right quotient"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/finite-syntactic-monoid-left-quotient","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_leftQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#finite-syntactic-monoid-left-quotient","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$ whose syntactic monoid is finite. Then for every word $u$ over $A$, the syntactic monoid of the left quotient $u^{-1} L = \\{x \\mid ux \\in L\\}$ is finite.","title":"Left quotients preserve finiteness of the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/finite-syntactic-monoid-right-quotient","kind":"node","lean":"Cslib.Language.finite_syntacticMonoid_rightQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#finite-syntactic-monoid-right-quotient","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $A$ whose syntactic monoid is finite. Then for every word $u$ over $A$, the syntactic monoid of the right quotient $L u^{-1} = \\{x \\mid xu \\in L\\}$ is finite.","title":"Right quotients preserve finiteness of the syntactic monoid"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/right-quotient-3","kind":"node","lean":"Cslib.Language.IsRegular.rightQuotient'","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#right-quotient-3","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"If $L$ is a regular language over an alphabet $A$, then for every word $u$ over $A$, the right quotient $L u^{-1} = \\{x \\mid xu \\in L\\}$ is regular.","title":"Regular languages are closed under right quotients by fixed words"},{"id":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/left-quotient-2","kind":"node","lean":"Cslib.Language.IsRegular.leftQuotient","parent":33640,"result":true,"route":"../roadmap/TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids/index.html#left-quotient-2","scope":"TheoryOfComputation/regular-languages/syntactic-monoids/language-quotients-and-syntactic-monoids","status":"fully_proved","summary":"If $L$ is a regular language over an alphabet $A$, then for every word $u$ over $A$, the left quotient $u^{-1} L = \\{x \\mid ux \\in L\\}$ is regular.","title":"Regular languages are closed under left quotients by fixed words"},{"data_url":"scopes/TheoryOfComputation/context-free-languages.json","id":"TheoryOfComputation/context-free-languages","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/index.html","scope":"TheoryOfComputation/context-free-languages","status":"fully_proved","summary":"Context-free grammars and derivations, normal forms, parsing algorithms, pushdown automata, deterministic and ambiguous languages, closure and non-closure properties, pumping and Ogden's lemmas, and Parikh images.","title":"Context-free languages"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata","kind":"scope","parent":33661,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata","status":"fully_proved","summary":"Grammar basics and derivation forests, Chomsky and Greibach normal forms, \u03b5- and unit-production elimination, CYK and Earley recognizers, leftmost derivations and (inherent) ambiguity, pushdown automata, deterministic context-free languages, and the grammar\u2013PDA equivalence.","title":"Grammars, parsing and pushdown automata"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/is-context-free","kind":"node","lean":"Cslib.Language.IsRegular.isContextFree","parent":33662,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/index.html#is-context-free","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata","status":"fully_proved","summary":"Over a finite alphabet, every regular language is context-free.","title":"Regular languages over a finite alphabet are context-free"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","title":"Context-free grammars"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/eval-symbol","kind":"node","lean":"Cslib.Language.evalSymbol","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#eval-symbol","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Under an assignment $V$ of languages to nonterminals, a terminal $a$ denotes the language containing only the one-letter word $a$, and a nonterminal $A$ denotes its assigned language $V(A)$.","title":"Interpretation of a grammar symbol"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/eval-form","kind":"node","lean":"Cslib.Language.evalForm","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#eval-form","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Under an assignment $V$ of languages to nonterminals, a sentential form denotes the concatenation, in order, of the languages its symbols denote: a terminal $a$ denotes the language containing only the one-letter word $a$, and a nonterminal $A$ denotes $V(A)$. The empty sentential form denotes the language containing only the empty word.","title":"Interpretation of a sentential form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/eval-form-append","kind":"node","lean":"Cslib.Language.evalForm_append","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#eval-form-append","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"For every assignment $V$ of languages to nonterminals and all sentential forms $u$ and $v$, the language that $uv$ denotes under $V$ is the concatenation of the languages that $u$ and $v$ denote under $V$.","title":"Interpretation of a concatenation"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/eval-form-map-terminal","kind":"node","lean":"Cslib.Language.evalForm_map_terminal","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#eval-form-map-terminal","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Under every assignment of languages to nonterminals, a terminal word $w$, read as a sentential form, denotes the singleton language $\\{w\\}$.","title":"A terminal word denotes its singleton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/is-valuation","kind":"node","lean":"Cslib.Language.IsValuation","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#is-valuation","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar with terminals $T$, and let $V$ assign a language over $T$ to each nonterminal of $g$. Interpret a sentential form under $V$ by sending each terminal $a$ to the language $\\{a\\}$ of the one-letter word $a$ and each nonterminal $B$ to $V(B)$, and concatenating these languages in order (the empty form is interpreted as $\\{\\varepsilon\\}$). Then $V$ is a *valuation* of $g$ when, for\u2026","title":"Valuation of a grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/eval-form-le-of-produces","kind":"node","lean":"Cslib.Language.evalForm_le_of_produces","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#eval-form-le-of-produces","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar and $V$ a valuation of $g$. If a sentential form $u$ rewrites to $v$ by one production of $g$, then the language that $v$ denotes under $V$ is contained in the language that $u$ denotes under $V$.","title":"A production step shrinks interpretations under a valuation"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/eval-form-le-of-derives","kind":"node","lean":"Cslib.Language.evalForm_le_of_derives","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#eval-form-le-of-derives","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar and $V$ a valuation of $g$. If a sentential form $u$ derives $v$ in $g$ (in zero or more steps), then the language that $v$ denotes under $V$ is contained in the language that $u$ denotes under $V$.","title":"Derivations shrink interpretations under a valuation"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/language-le-of-valuation","kind":"node","lean":"Cslib.Language.language_le_of_valuation","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#language-le-of-valuation","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"For every context-free grammar $g$ and every valuation $V$ of $g$, the language generated by $g$ is contained in $V(S)$, where $S$ is the initial nonterminal of $g$.","title":"Valuations bound the language of a grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/derives-append","kind":"node","lean":"Cslib.Language.derives_append","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#derives-append","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar and let $u$, $u'$, $v$, $v'$ be sentential forms of $g$. If $u$ derives $u'$ and $v$ derives $v'$ in $g$ (each in zero or more steps), then the concatenation $uv$ derives $u'v'$ in $g$.","title":"Concatenating derivations"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/map-symbol","kind":"node","lean":"Cslib.Language.mapSymbol","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#map-symbol","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Given a map $f$ from nonterminals $N$ to nonterminals $N'$, renaming a grammar symbol by $f$ leaves a terminal $a$ unchanged and sends a nonterminal $A$ to the nonterminal $f(A)$.","title":"Renaming the nonterminals of a symbol"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/map-rule","kind":"node","lean":"Cslib.Language.mapRule","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#map-rule","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Given a map $f$ from nonterminals $N$ to nonterminals $N'$, the renaming by $f$ of a production $A \\to u$ is the production $f(A) \\to u'$, where $u'$ is obtained from $u$ by replacing each nonterminal $B$ with $f(B)$ and leaving each terminal unchanged.","title":"Renaming the nonterminals of a production"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/eval-form-map","kind":"node","lean":"Cslib.Language.evalForm_map","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#eval-form-map","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $f \\colon N \\to N'$ be a map of nonterminals, extended to sentential forms by renaming each nonterminal and keeping each terminal, and let $V$ assign a language to each nonterminal in $N'$. For every sentential form $u$ over $N$, the language that $f(u)$ denotes under $V$ equals the language that $u$ denotes under the assignment $V \\circ f$.","title":"Interpretation after renaming nonterminals"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/derives-map-of-forall-mem","kind":"node","lean":"Cslib.Language.derives_map_of_forall_mem","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#derives-map-of-forall-mem","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ and $G$ be context-free grammars over the same terminal alphabet, and let $f$ map the nonterminals of $g$ to those of $G$. Extend $f$ to symbols and sentential forms by renaming each nonterminal and keeping each terminal. Suppose that for every production $A \\to \\alpha$ of $g$, the renamed production $f(A) \\to f(\\alpha)$ is a production of $G$. Then whenever a sentential form $u$ derives $v$ in $g$ (in zero\u2026","title":"Derivations transfer along a renaming of nonterminals"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/language-from","kind":"node","lean":"Cslib.Language.languageFrom","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#language-from","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"For a context-free grammar $g$ and a nonterminal $A$ of $g$, the language generated by $g$ from $A$ is the set of words $w$ over the terminals of $g$ such that $g$ derives $w$ from the one-symbol sentential form $A$. It is the language of $g$ with $A$ in place of the initial nonterminal.","title":"Language generated from a nonterminal"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/derives-of-mem-eval-form-language-from","kind":"node","lean":"Cslib.Language.derives_of_mem_evalForm_languageFrom","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#derives-of-mem-eval-form-language-from","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar and assign to each nonterminal $A$ of $g$ the language of terminal words that $g$ derives from $A$. If a terminal word $w$ lies in the language that a sentential form $u$ of $g$ denotes under this assignment, then $u$ derives $w$ in $g$.","title":"Words of the canonical interpretation are derivable"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/is-valuation-language-from","kind":"node","lean":"Cslib.Language.isValuation_languageFrom","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#is-valuation-language-from","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"For every context-free grammar $g$, the assignment that sends each nonterminal $A$ of $g$ to the language generated by $g$ from $A$ is a valuation of $g$.","title":"The languages generated from each nonterminal form a valuation"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/map-symbol-injective","kind":"node","lean":"Cslib.Language.mapSymbol_injective","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#map-symbol-injective","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"If $f$ is an injective map from nonterminals $N$ to nonterminals $N'$, then renaming grammar symbols by $f$, which fixes each terminal and sends each nonterminal $A$ to $f(A)$, is injective.","title":"Injective renamings are injective on symbols"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/map-rule-injective","kind":"node","lean":"Cslib.Language.mapRule_injective","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#map-rule-injective","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"If $f$ is an injective map from nonterminals $N$ to nonterminals $N'$, then renaming the nonterminals of productions over $N$ by $f$ is injective: two productions with the same renaming are equal.","title":"Injective renamings are injective on productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/map-map-symbol-terminal","kind":"node","lean":"Cslib.Language.map_mapSymbol_terminal","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#map-map-symbol-terminal","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $f$ map nonterminals $N$ to nonterminals $N'$, and let $w$ be a word over the terminals. Reading $w$ as a sentential form over $N$ and renaming its nonterminals by $f$ gives $w$ read as a sentential form over $N'$.","title":"Renaming nonterminals fixes terminal words"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/zero","kind":"node","lean":"Cslib.Language.IsContextFree.zero","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#zero","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"The empty language over any alphabet is context-free: take a grammar with no productions.","title":"The empty language is context-free"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/one","kind":"node","lean":"Cslib.Language.IsContextFree.one","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#one","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"For every alphabet $T$, the language over $T$ whose only word is the empty word is context-free: a grammar with one nonterminal and a single production erasing it generates this language.","title":"The language of the empty word is context-free"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/singleton-word","kind":"node","lean":"Cslib.Language.IsContextFree.singleton_word","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#singleton-word","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"For every word $w$ over an alphabet $T$, the language $\\{w\\}$ consisting of $w$ alone is context-free: one production spells the word out.","title":"Singleton languages are context-free"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/yields","kind":"node","lean":"Cslib.Language.Yields","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#yields","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"For a context-free grammar $g$ with terminal alphabet $T$, the relation \"the sentential form $u$ yields the terminal word $w$\" is the smallest relation such that: the empty form yields the empty word; if $u$ yields $w$, then for every terminal $a$ the form $a u$ yields $a w$; and if $A \\to \\alpha$ is a production of $g$, $\\alpha$ yields $w_1$ and $u$ yields $w_2$, then the form $A u$ yields $w_1 w_2$.","title":"Yield of a sentential form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/append","kind":"node","lean":"Cslib.Language.Yields.append","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#append","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If the sentential form $u$ yields the terminal word $w_1$ and the sentential form $v$ yields the terminal word $w_2$, then the concatenated form $u v$ yields $w_1 w_2$.","title":"Concatenation of yields"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/append-inv","kind":"node","lean":"Cslib.Language.Yields.append_inv","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#append-inv","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If the concatenated sentential form $u v$ yields the terminal word $w$, then there are terminal words $w_1$ and $w_2$ with $w = w_1 w_2$ such that $u$ yields $w_1$ and $v$ yields $w_2$.","title":"Splitting a yield over a concatenation"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/yields-of-rule","kind":"node","lean":"Cslib.Language.yields_of_rule","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#yields-of-rule","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar and $A \\to \\alpha$ a production of $g$. If $\\alpha$ yields the terminal word $w$, then the one-symbol sentential form $A$ yields $w$.","title":"Yield through one production"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/derives-of-yields","kind":"node","lean":"Cslib.Language.derives_of_yields","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#derives-of-yields","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If the sentential form $u$ yields the terminal word $w$, then $u$ derives $w$, read as a string of terminal symbols, in $g$: that is, $u \\Rightarrow^* w$ in zero or more rewriting steps.","title":"Yields are derivable"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/yields-of-derives","kind":"node","lean":"Cslib.Language.yields_of_derives","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#yields-of-derives","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If the sentential form $u$ derives the terminal word $w$, read as a string of terminal symbols, in $g$, then $u$ yields $w$.","title":"Derivable terminal words are yields"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/yields-iff-derives","kind":"node","lean":"Cslib.Language.yields_iff_derives","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#yields-iff-derives","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar, $u$ a sentential form of $g$ and $w$ a word of terminals. Then $u$ yields $w$ in $g$ if and only if $u$ derives in $g$, by zero or more rewriting steps, the sentential form consisting of the letters of $w$ read as terminals.","title":"Yields coincide with derivations of terminal words"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/mem-language-iff-yields","kind":"node","lean":"Cslib.Language.mem_language_iff_yields","parent":33664,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars/index.html#mem-language-iff-yields","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar with initial nonterminal $S$. A terminal word $w$ belongs to the language of $g$ if and only if the one-symbol sentential form $S$ yields $w$.","title":"Language membership through yields"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","title":"Ambiguity of context-free grammars"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/left-derivation","kind":"node","lean":"Cslib.Language.LeftDerivation","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#left-derivation","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar, $u$ a sentential form of $g$, $w$ a word of terminals and $r_1, \\dots, r_n$ a list of rules over the terminals and nonterminals of $g$. The list $r_1, \\dots, r_n$ is a leftmost derivation of $w$ from $u$ in $g$ when every $r_i$ is a production of $g$, and, starting from $u$ and for $i = 1, \\dots, n$ in turn, the leftmost nonterminal of the current sentential form is the left-hand\u2026","title":"Leftmost derivation"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/is-ambiguous","kind":"node","lean":"Cslib.Language.IsAmbiguous","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#is-ambiguous","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"A context-free grammar $g$ is ambiguous if some word of terminals $w$ has two leftmost derivations from the start symbol of $g$ that use different lists of rules.","title":"Ambiguous grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/derives","kind":"node","lean":"Cslib.Language.LeftDerivation.derives","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#derives","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a sentential form $u$ has a leftmost derivation in $g$ to a word of terminals $w$, then $u$ derives $w$ in $g$ (by zero or more rewriting steps).","title":"Leftmost derivations are derivations"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/mem-language","kind":"node","lean":"Cslib.Language.LeftDerivation.mem_language","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#mem-language","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a word of terminals $w$ has a leftmost derivation in $g$ from the start symbol of $g$, then $w$ belongs to the language of $g$.","title":"Words with a leftmost derivation lie in the language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/eq-of-terminal","kind":"node","lean":"Cslib.Language.LeftDerivation.eq_of_terminal","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#eq-of-terminal","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar and $u$ a word of terminals. If $u$ has a leftmost derivation in $g$ with list of rules $\\rho$ to a word of terminals $w$, then $\\rho$ is the empty list and $w = u$.","title":"Leftmost derivations from a word of terminals are empty"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/inherently-ambiguous-lang","kind":"node","lean":"Cslib.Language.inherentlyAmbiguousLang","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#inherently-ambiguous-lang","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"The language `{a ^ i b ^ j c ^ k | i = j or j = k}` over the alphabet `Fin 3` (`0, 1, 2` for `a, b, c`): the standard inherently ambiguous context-free language.","title":"Canonical inherently ambiguous context-free language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/is-inherently-ambiguous","kind":"node","lean":"Cslib.Language.IsInherentlyAmbiguous","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#is-inherently-ambiguous","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"A language is inherently ambiguous when it is context-free and every grammar generating it is ambiguous.","title":"Inherently ambiguous language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/inherently-ambiguous-lang-eq","kind":"node","lean":"Cslib.Language.inherentlyAmbiguousLang_eq","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#inherently-ambiguous-lang-eq","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"The example language is the union of `a ^ n b ^ n c ^ *` and `a ^ * b ^ n c ^ n`.","title":"Inherently ambiguous example as a union of two languages"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/inherently-ambiguous-lang-is-context-fre","kind":"node","lean":"Cslib.Language.inherentlyAmbiguousLang_isContextFree","parent":33695,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars/index.html#inherently-ambiguous-lang-is-context-fre","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/ambiguity-of-context-free-grammars","status":"fully_proved","summary":"The example language is context-free.","title":"The example language is context-free"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","title":"Earley parser"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/earley-item","kind":"node","lean":"Cslib.Language.EarleyItem","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#earley-item","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"An Earley item: a production with a dot after `dot` symbols of its output, recognizing input that starts at the end of `pre` and whose recognized prefix is `consumed`. The full input is `pre ++ consumed ++ rest` for some `rest`.","title":"Earley item"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/earley-chart","kind":"node","lean":"Cslib.Language.EarleyChart","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#earley-chart","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"The Earley chart for input `w`: the least set of items closed under initialization, prediction, scanning, and completion.","title":"Earley chart"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/earley-chart-item","kind":"node","lean":"Cslib.Language.EarleyChartItem","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#earley-chart-item","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"The chart as a predicate on `EarleyItem`s.","title":"Earley chart as a predicate on items"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/earley-accepts","kind":"node","lean":"Cslib.Language.earleyAccepts","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#earley-accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"Acceptance by the Earley recognizer: some completed item for a production of the initial nonterminal consumes the whole input.","title":"Acceptance by the Earley recognizer"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/dot-le-of-earley-chart","kind":"node","lean":"Cslib.Language.dot_le_of_earleyChart","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#dot-le-of-earley-chart","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"The dot of a chart item never passes the end of its production.","title":"Earley item dot stays within its production"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/yields-take-of-earley-chart","kind":"node","lean":"Cslib.Language.yields_take_of_earleyChart","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#yields-take-of-earley-chart","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"Soundness: the recognized prefix of a chart item yields the input it consumed.","title":"Soundness of the Earley chart"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/earley-chart-complete-of-yields","kind":"node","lean":"Cslib.Language.earleyChart_complete_of_yields","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#earley-chart-complete-of-yields","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"Completeness: if a chart item waits on a sentential form and that form yields a word fitting in the remaining input, the chart contains the item completed by that word.","title":"Completeness of the Earley chart"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/exists-rule-of-yields-singleton","kind":"node","lean":"Cslib.Language.exists_rule_of_yields_singleton","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#exists-rule-of-yields-singleton","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"Inversion of a yield from a single nonterminal: it begins with one production of that nonterminal.","title":"A yield from one nonterminal starts with one of its productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/earley-accepts-iff","kind":"node","lean":"Cslib.Language.earleyAccepts_iff","parent":33705,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser/index.html#earley-accepts-iff","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/earley-parser","status":"fully_proved","summary":"The Earley recognizer accepts exactly the grammar's language.","title":"Correctness of the Earley recognizer"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","title":"Elimination of empty and unit productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/ne-nil-of-forall-output-ne-nil","kind":"node","lean":"Cslib.Language.Yields.ne_nil_of_forall_output_ne_nil","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#ne-nil-of-forall-output-ne-nil","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"In a context-free grammar in which no production has an empty right-hand side, every nonempty sentential form yields only nonempty words.","title":"No empty words without empty productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/nullable","kind":"node","lean":"Cslib.Language.Nullable","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#nullable","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"A nonterminal $A$ of a context-free grammar $g$ is nullable when $g$ derives the empty word from $A$.","title":"Nullable nonterminal"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/nullable-deletions","kind":"node","lean":"Cslib.Language.nullableDeletions","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#nullable-deletions","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"For a context-free grammar $g$ and a sentential form $u$, the nullable deletions of $u$ are the forms obtained from $u$ by deleting any selection of occurrences of nullable nonterminals, possibly none and possibly all, keeping the remaining symbols in order. They form a finite set.","title":"Nullable deletions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/eval-form-le-of-mem-nullable-deletions","kind":"node","lean":"Cslib.Language.evalForm_le_of_mem_nullableDeletions","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#eval-form-le-of-mem-nullable-deletions","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar and $V$ an assignment of a language $V(A)$ to each nonterminal $A$ of $g$ such that the empty word belongs to $V(A)$ whenever $A$ is nullable (derives the empty word in $g$). If a sentential form $\\alpha'$ is obtained from a sentential form $u$ by deleting some occurrences of nullable nonterminals, then the language denoted by $\\alpha'$ under $V$ is contained in the language\u2026","title":"Deleting nullable nonterminals shrinks the denoted language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/eps-grammar","kind":"node","lean":"Cslib.Language.epsGrammar","parent":33715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Elimination of "],["math","\\(\\varepsilon\\)"],["text","-productions"]],"source":"Elimination of $\\varepsilon$-productions"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#eps-grammar","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"For a context-free grammar $g$, the grammar obtained from $g$ by eliminating $\\varepsilon$-productions has the same nonterminals and the same initial nonterminal as $g$. Its productions are the productions $A \\to \\alpha'$ such that $g$ has a production $A \\to \\alpha$, the form $\\alpha'$ is obtained from $\\alpha$ by deleting some (possibly no) occurrences of nullable nonterminals (those deriving the empty word in\u2026","title":"Elimination of $\\varepsilon$-productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/mem-eps-grammar-rules","kind":"node","lean":"Cslib.Language.mem_epsGrammar_rules","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#mem-eps-grammar-rules","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar with a production $A \\to u$, and let $\\alpha$ be a nonempty form obtained from $u$ by deleting some (possibly no) occurrences of nullable nonterminals. Then $A \\to \\alpha$ is a production of the grammar obtained from $g$ by eliminating \u03b5-productions.","title":"Productions of the \u03b5-free grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/cases-of-mem-eps-grammar-rules","kind":"node","lean":"Cslib.Language.cases_of_mem_epsGrammar_rules","parent":33715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Productions of the "],["math","\\(\\varepsilon\\)"],["text","-eliminated grammar come from productions of "],["math","\\(g\\)"]],"source":"Productions of the $\\varepsilon$-eliminated grammar come from productions of $g$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#cases-of-mem-eps-grammar-rules","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar. Every production $A \\to \\alpha'$ of the grammar obtained from $g$ by eliminating $\\varepsilon$-productions arises from a production $A \\to \\alpha$ of $g$ with the same left-hand side $A$, such that $\\alpha'$ is obtained from $\\alpha$ by deleting some occurrences of nullable nonterminals and $\\alpha'$ is nonempty.","title":"Productions of the $\\varepsilon$-eliminated grammar come from productions of $g$"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/eps-grammar-output-ne-nil","kind":"node","lean":"Cslib.Language.epsGrammar_output_ne_nil","parent":33715,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\varepsilon\\)"],["text","-eliminated grammar has no empty productions"]],"source":"The $\\varepsilon$-eliminated grammar has no empty productions"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#eps-grammar-output-ne-nil","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"For a context-free grammar $g$, no production of the grammar obtained from $g$ by eliminating $\\varepsilon$-productions has empty right-hand side.","title":"The $\\varepsilon$-eliminated grammar has no empty productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/is-valuation-eps-grammar","kind":"node","lean":"Cslib.Language.isValuation_epsGrammar","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#is-valuation-eps-grammar","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar and, for each nonterminal $A$, let $L_g(A)$ be the language of words $g$ derives from $A$. Then $L_g$ is a valuation of the grammar obtained from $g$ by eliminating \u03b5-productions: for every production $A \\to \\alpha$ of that grammar, the language of $\\alpha$ under $L_g$ (the concatenation of the languages of its symbols, a terminal $a$ denoting $\\{a\\}$) is contained in $L_g(A)$.","title":"Soundness of \u03b5-production elimination"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/yields-eps-grammar-of-yields","kind":"node","lean":"Cslib.Language.yields_epsGrammar_of_yields","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#yields-eps-grammar-of-yields","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a sentential form $\\alpha$ of $g$ yields a terminal word $w$ in $g$, then some form $\\alpha'$ obtained from $\\alpha$ by deleting some occurrences of nullable nonterminals yields $w$ in the grammar obtained from $g$ by eliminating $\\varepsilon$-productions, and $w$ is empty if $\\alpha'$ is empty.","title":"Completeness of eliminating empty productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/eps-grammar-language","kind":"node","lean":"Cslib.Language.epsGrammar_language","parent":33715,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(\\varepsilon\\)"],["text","-eliminated grammar generates the nonempty words of "],["math","\\(g\\)"]],"source":"The $\\varepsilon$-eliminated grammar generates the nonempty words of $g$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#eps-grammar-language","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"For a context-free grammar $g$ and every word $w$ over its terminals, $w$ is generated by the grammar obtained from $g$ by eliminating $\\varepsilon$-productions if and only if $w$ is generated by $g$ and $w$ is nonempty.","title":"The $\\varepsilon$-eliminated grammar generates the nonempty words of $g$"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/unit-steps","kind":"node","lean":"Cslib.Language.UnitSteps","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#unit-steps","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar. A nonterminal $A$ of $g$ reaches a nonterminal $B$ by unit productions when there is a chain $A = A_0, A_1, \\dots, A_n = B$ with $n \\ge 0$ such that each $A_{i-1} \\to A_i$ is a production of $g$. In particular, every nonterminal reaches itself.","title":"Reachability by unit productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/is-unit-output","kind":"node","lean":"Cslib.Language.IsUnitOutput","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#is-unit-output","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"A sentential form is a unit output when it consists of a single nonterminal.","title":"Unit output"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/form-nts","kind":"node","lean":"Cslib.Language.formNTs","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#form-nts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"For a sentential form $u$, a finite word of terminal and nonterminal symbols, the nonterminals of $u$ form the finite set of all nonterminals that occur in $u$.","title":"Nonterminals of a sentential form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/grammar-nts","kind":"node","lean":"Cslib.Language.grammarNTs","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#grammar-nts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"The nonterminals mentioned by a context-free grammar $g$ form the finite set consisting of the initial nonterminal of $g$, the left-hand side of every production of $g$, and every nonterminal occurring in the right-hand side of a production of $g$.","title":"Set of nonterminals mentioned by a grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/mem-grammar-nts-of-unit-steps","kind":"node","lean":"Cslib.Language.mem_grammarNTs_of_unitSteps","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#mem-grammar-nts-of-unit-steps","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar and let $B$ be reachable from $A$ by a chain of zero or more unit productions of $g$. If $B$ is the left-hand side of some production of $g$, then $A$ is among the nonterminals mentioned by $g$ (its initial nonterminal, the left-hand sides of its productions, and the nonterminals in their right-hand sides).","title":"Unit chains start at mentioned nonterminals"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/unit-grammar","kind":"node","lean":"Cslib.Language.unitGrammar","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#unit-grammar","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"A unit production is one whose output is a single nonterminal. For a context-free grammar $g$, the grammar obtained by eliminating unit productions has the same nonterminals and the same start symbol as $g$. Its productions are the $A \\to \\alpha$ such that $A$ is a nonterminal mentioned by $g$ (its start symbol or one occurring in a production of $g$), $g$ has a production $B \\to \\alpha$ that is not a unit\u2026","title":"Eliminating unit productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/mem-unit-grammar-rules","kind":"node","lean":"Cslib.Language.mem_unitGrammar_rules","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#mem-unit-grammar-rules","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar, let $B$ be reachable from $A$ by a chain of zero or more unit productions of $g$, and let $B \\to \\beta$ be a production of $g$ whose right-hand side $\\beta$ is not a single nonterminal. If $A$ is among the nonterminals mentioned by $g$, then $A \\to \\beta$ is a production of the grammar obtained from $g$ by eliminating unit productions.","title":"Productions of the unit-free grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/cases-of-mem-unit-grammar-rules","kind":"node","lean":"Cslib.Language.cases_of_mem_unitGrammar_rules","parent":33715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Productions of the unit-eliminated grammar come from non-unit productions of "],["math","\\(g\\)"]],"source":"Productions of the unit-eliminated grammar come from non-unit productions of $g$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#cases-of-mem-unit-grammar-rules","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar. For every production $A \\to \\beta$ of the grammar obtained from $g$ by eliminating unit productions, there are a nonterminal $B$ and a production $B \\to \\beta$ of $g$ such that $\\beta$ is not a single nonterminal and $B$ is reachable from $A$ by a possibly empty chain of unit productions of $g$.","title":"Productions of the unit-eliminated grammar come from non-unit productions of $g$"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/unit-grammar-not-is-unit-output","kind":"node","lean":"Cslib.Language.unitGrammar_not_isUnitOutput","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#unit-grammar-not-is-unit-output","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"For every context-free grammar $g$, no production of the grammar obtained from $g$ by eliminating unit productions has an output consisting of a single nonterminal.","title":"Eliminating unit productions leaves no unit production"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/unit-grammar-output-of-mem","kind":"node","lean":"Cslib.Language.unitGrammar_output_of_mem","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#unit-grammar-output-of-mem","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar. For every production of the grammar obtained from $g$ by eliminating unit productions, some production of $g$ has the same output.","title":"Eliminating unit productions creates no new outputs"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/language-from-le-of-unit-steps","kind":"node","lean":"Cslib.Language.languageFrom_le_of_unitSteps","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#language-from-le-of-unit-steps","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar and, for each nonterminal $A$, let $L_g(A)$ be the language of words $g$ derives from $A$. If $B$ is reachable from $A$ by a chain of zero or more unit productions $A \\to A_1 \\to \\dots \\to B$ of $g$, then $L_g(B) \\subseteq L_g(A)$.","title":"Languages along unit chains"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/is-valuation-unit-grammar","kind":"node","lean":"Cslib.Language.isValuation_unitGrammar","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#is-valuation-unit-grammar","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar and, for each nonterminal $A$, let $L_g(A)$ be the language of words $g$ derives from $A$. Then $L_g$ is a valuation of the grammar obtained from $g$ by eliminating unit productions: for every production $A \\to \\alpha$ of that grammar, the language of $\\alpha$ under $L_g$ (the concatenation of the languages of its symbols, a terminal $a$ denoting $\\{a\\}$) is contained in $L_g(A)$.","title":"Soundness of unit-production elimination"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/yields-unit-grammar-of-yields","kind":"node","lean":"Cslib.Language.yields_unitGrammar_of_yields","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#yields-unit-grammar-of-yields","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a sentential form $\\alpha$ of $g$ yields a terminal word $w$ in $g$, then $\\alpha$ yields $w$ in the grammar obtained from $g$ by eliminating unit productions.","title":"Eliminating unit productions keeps every yield"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/unit-grammar-language","kind":"node","lean":"Cslib.Language.unitGrammar_language","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#unit-grammar-language","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"For every context-free grammar $g$, the grammar obtained from $g$ by eliminating unit productions generates the same language as $g$.","title":"Eliminating unit productions preserves the language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/is-reduced","kind":"node","lean":"Cslib.Language.IsReduced","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#is-reduced","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"A context-free grammar is reduced when no production has empty right-hand side and no production has a single nonterminal as its right-hand side.","title":"Reduced grammars"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/exists-reduced-grammar","kind":"node","lean":"Cslib.Language.exists_reduced_grammar","parent":33715,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions/index.html#exists-reduced-grammar","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/elimination-of-empty-and-unit-productions","status":"fully_proved","summary":"For every context-free grammar $g$ over an alphabet $T$ there is a context-free grammar $g'$ over $T$ that generates exactly the nonempty words generated by $g$, such that no production of $g'$ has empty right-hand side and no production of $g'$ has a single nonterminal as its right-hand side.","title":"Existence of reduced grammars"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","title":"Chomsky normal form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/is-chomsky-normal-form","kind":"node","lean":"Cslib.Language.IsChomskyNormalForm","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#is-chomsky-normal-form","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"A grammar is in Chomsky normal form when every production has output either two nonterminals, a single terminal, or the empty word with the initial nonterminal on the left; and either the initial nonterminal occurs in no output (so an empty production cannot leak the empty word to other nonterminals) or there are no empty outputs at all.","title":"Chomsky normal form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/grammar-terminals","kind":"node","lean":"Cslib.Language.grammarTerminals","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#grammar-terminals","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"For a context-free grammar $G$, the list of terminals occurring in the outputs of its productions: going through the productions of $G$ in a fixed but unspecified order, list the terminals of each output from left to right. A terminal may appear more than once.","title":"Terminals of a grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/to-cnf-nt","kind":"node","lean":"Cslib.Language.toCnfNT","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#to-cnf-nt","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"Let $G$ be a context-free grammar with nonterminals $N$, and let $t_0, \\dots, t_{k-1}$ be the list of terminals occurring in the production outputs of $G$. This map sends a symbol of $G$ to an element of $N \\sqcup \\mathbb{N}$: a nonterminal $A$ goes to $A$, and a terminal $a$ goes to the least index $i$ with $t_i = a$, or to $k$ if $a$ does not occur in the list.","title":"Encoding symbols as basic nonterminals of the binarized grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/from-cnf-nt","kind":"node","lean":"Cslib.Language.fromCnfNT","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#from-cnf-nt","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"Let $G$ be a context-free grammar with nonterminals $N$ and initial nonterminal $S$, and let $t_0, \\dots, t_{k-1}$ be the list of terminals occurring in the production outputs of $G$. This map sends an element of $N \\sqcup \\mathbb{N}$ (a basic nonterminal of the binarized grammar) to a symbol of $G$: a nonterminal $A \\in N$ goes to $A$, an index $i < k$ goes to the terminal $t_i$, and an index $i \\ge k$ goes to the\u2026","title":"Decoding basic nonterminals of the binarized grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/cnf-head","kind":"node","lean":"Cslib.Language.cnfHead","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#cnf-head","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"Let $g$ be a context-free grammar and let $a_0, a_1, \\dots$ be its list of terminals occurring in production outputs. For a list $\\sigma$ of basic symbols (each a nonterminal of $g$ or a natural number, a terminal wrapper), the head $h(\\sigma)$ is the following form over the nonterminals of the binarized grammar (basic symbols and chain nonterminals $\\langle\\tau\\rangle$ for lists $\\tau$ of basic symbols):\u2026","title":"Head of a binarized output"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/cnf-grammar","kind":"node","lean":"Cslib.Language.cnfGrammar","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#cnf-grammar","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"Let $g$ be a context-free grammar with nonterminals $N$ and initial nonterminal $S$, and let $t_0, \\dots, t_{k-1}$ be the list of terminals occurring in the production outputs of $g$. Encode each symbol of a production output of $g$ as a basic symbol in $N \\sqcup \\mathbb{N}$: a nonterminal $A$ as $A$, and a terminal $a$ as the least index $i$ with $t_i = a$. The binarized grammar of $g$ has as nonterminals the\u2026","title":"Binarization of a grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/cnf-valuation","kind":"node","lean":"Cslib.Language.cnfValuation","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#cnf-valuation","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"Let $g$ be a context-free grammar and let $a_0, a_1, \\dots$ be its list of terminals occurring in production outputs. A basic symbol is a nonterminal of $g$ or a natural number (a terminal wrapper); the nonterminals of the binarized grammar are the basic symbols together with a chain nonterminal $\\langle\\sigma\\rangle$ for every list $\\sigma$ of basic symbols. Decode a basic symbol as a symbol of $g$: a nonterminal\u2026","title":"Valuation of the binarized grammar by yields"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/cnf-grammar-language-le-of-reduced","kind":"node","lean":"Cslib.Language.cnfGrammar_language_le_of_reduced","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#cnf-grammar-language-le-of-reduced","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"If a context-free grammar $g$ is reduced, then every word generated by the binarization of $g$ is generated by $g$.","title":"Binarization adds no words"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/cnf-grammar-output-ne-nil","kind":"node","lean":"Cslib.Language.cnfGrammar_output_ne_nil","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#cnf-grammar-output-ne-nil","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"If a context-free grammar $g$ is reduced, then no production of the binarization of $g$ has the empty word as output.","title":"Binarization has no empty productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/cnf-grammar-is-chomsky-normal-form","kind":"node","lean":"Cslib.Language.cnfGrammar_isChomskyNormalForm","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#cnf-grammar-is-chomsky-normal-form","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"If a context-free grammar $g$ is reduced, then the binarization of $g$ is in Chomsky normal form.","title":"Binarization of a reduced grammar is in Chomsky normal form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/cnf-grammar-language-of-reduced","kind":"node","lean":"Cslib.Language.cnfGrammar_language_of_reduced","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#cnf-grammar-language-of-reduced","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"If a context-free grammar $g$ is reduced, then the binarization of $g$ generates exactly the language of $g$.","title":"Binarization preserves the language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/eps-start-grammar","kind":"node","lean":"Cslib.Language.epsStartGrammar","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#eps-start-grammar","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"Given a context-free grammar $G$ with nonterminals $N$ and initial nonterminal $S$, this is the grammar with nonterminals $N \\cup \\{S_0\\}$ for a fresh symbol $S_0$, initial nonterminal $S_0$, and the following productions: every production of $G$; the empty production $S_0 \\to \\varepsilon$; and $S_0 \\to \\alpha$ for every production $S \\to \\alpha$ of $G$.","title":"Grammar with a fresh nullable start symbol"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/eps-start-grammar-language","kind":"node","lean":"Cslib.Language.epsStartGrammar_language","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#eps-start-grammar-language","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"For every context-free grammar $G$, the grammar obtained from $G$ by adding a fresh initial nonterminal (with an empty production and copies of the productions of the old initial nonterminal) generates exactly $L(G) \\cup \\{\\varepsilon\\}$, where $L(G)$ is the language of $G$ and $\\varepsilon$ is the empty word.","title":"Language of the grammar with a fresh nullable start symbol"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/eps-start-grammar-initial-not-mem-output","kind":"node","lean":"Cslib.Language.epsStartGrammar_initial_not_mem_output","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#eps-start-grammar-initial-not-mem-output","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"For every context-free grammar $g$, the fresh initial nonterminal of the fresh-start grammar of $g$ occurs in no production output of the fresh-start grammar.","title":"The fresh start symbol occurs in no output"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/eps-start-grammar-language-of-mem-nil","kind":"node","lean":"Cslib.Language.epsStartGrammar_language_of_mem_nil","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#eps-start-grammar-language-of-mem-nil","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"If the empty word belongs to the language of a context-free grammar $g$, then the fresh-start grammar of $g$ generates exactly the language of $g$.","title":"A fresh start symbol is harmless when the empty word is generated"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/eps-start-grammar-is-chomsky-normal-form","kind":"node","lean":"Cslib.Language.epsStartGrammar_isChomskyNormalForm","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#eps-start-grammar-is-chomsky-normal-form","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"If a context-free grammar $g$ is in Chomsky normal form and no production of $g$ has empty output, then the fresh-start grammar of $g$ is in Chomsky normal form.","title":"Adding a fresh start symbol preserves Chomsky normal form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/exists-chomsky-normal-form","kind":"node","lean":"Cslib.Language.exists_chomskyNormalForm","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#exists-chomsky-normal-form","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"For every context-free grammar $G$ there is a context-free grammar in Chomsky normal form that generates the same language as $G$.","title":"Chomsky normal form theorem"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/exists-chomsky-normal-form-2","kind":"node","lean":"Cslib.Language.IsContextFree.exists_chomskyNormalForm","parent":33743,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form/index.html#exists-chomsky-normal-form-2","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/chomsky-normal-form","status":"fully_proved","summary":"Every context-free language $L$ is generated by some context-free grammar in Chomsky normal form.","title":"Chomsky normal form for context-free languages"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm","kind":"scope","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm","status":"fully_proved","title":"CYK algorithm"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","kind":"scope","navigation":"module","parent":33762,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","title":"The Cocke\u2013Younger\u2013Kasami recognizer"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/no-empty-rule-of-mem-output","kind":"node","lean":"Cslib.Language.no_empty_rule_of_mem_output","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#no-empty-rule-of-mem-output","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Let $g$ be a context-free grammar in Chomsky normal form. If a nonterminal $B$ occurs in the output of some production of $g$, then $g$ has no production $B \\to \\varepsilon$.","title":"Nonterminals in outputs have no empty production"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/empty-rule-of-yields-nil","kind":"node","lean":"Cslib.Language.empty_rule_of_yields_nil","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#empty-rule-of-yields-nil","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"If $g$ is in Chomsky normal form, then every symbol of a sentential form that yields the empty word is a nonterminal $B$ for which $g$ has a production $B \\to \\varepsilon$.","title":"Forms yielding the empty word under Chomsky normal form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/yields-nonterminal-iff-of-ne-nil","kind":"node","lean":"Cslib.Language.yields_nonterminal_iff_of_ne_nil","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#yields-nonterminal-iff-of-ne-nil","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Let $g$ be a context-free grammar in Chomsky normal form, $A$ a nonterminal and $w$ a nonempty word. Then $A$ yields $w$ if and only if $g$ has a production $A \\to a$ with $a$ a terminal and $w = a$, or a production $A \\to BC$ with nonterminals $B$ and $C$ and nonempty words $w_1$ and $w_2$ such that $w = w_1 w_2$, $B$ yields $w_1$ and $C$ yields $w_2$.","title":"Shape of yields under Chomsky normal form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-fits","kind":"node","lean":"Cslib.Language.cykFits","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-fits","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Let $f$ assign a cell (a finite set of nonterminals) to every word. A production output fits a word $w$ with respect to $f$ exactly in two cases: the output is a single terminal $a$ and $w$ is the one-letter word $a$; or the output is a pair of nonterminals $B C$ and there is some $i$ with $0 < i < |w|$ such that $B$ lies in the cell of the first $i$ letters of $w$ and $C$ lies in the cell of the remaining letters.\u2026","title":"When a production output fits a word"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-cell-from","kind":"node","lean":"Cslib.Language.cykCellFrom","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-cell-from","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Given an assignment $f$ of a cell (a finite set of nonterminals of $g$) to every word, the cell of a word $w$ built from $f$ is the set of left-hand sides of those productions of $g$ whose outputs fit $w$ with respect to $f$.","title":"The CYK cell built from given cells"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/mem-cyk-cell-from-iff","kind":"node","lean":"Cslib.Language.mem_cykCellFrom_iff","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#mem-cyk-cell-from-iff","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Let $g$ be a context-free grammar in Chomsky normal form, let $f$ assign to each word over the terminals a finite set of nonterminals, and let $w$ be a nonempty word. For $0 < i < |w|$ write $u_i$ for the prefix of $w$ of length $i$ and $v_i$ for the remaining suffix, so that $w = u_i v_i$. Suppose that for every such $i$ and every nonterminal $B$, $B \\in f(u_i)$ if and only if $B$ yields $u_i$, and $B \\in f(v_i)$\u2026","title":"The CYK step"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-cell","kind":"node","lean":"Cslib.Language.cykCell","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-cell","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"The depth-$n$ CYK cell of a word $w$ is defined by recursion on $n$: at depth $0$ every cell is empty, and the depth-$(n + 1)$ cell of $w$ is the CYK cell of $w$ built from the depth-$n$ cells of all words. Cells are recomputed without sharing.","title":"The unshared CYK recurrence"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/mem-cyk-cell-iff","kind":"node","lean":"Cslib.Language.mem_cykCell_iff","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#mem-cyk-cell-iff","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Let $g$ be a context-free grammar in Chomsky normal form. For every natural number $n$, every nonterminal $A$ and every nonempty word $w$ of length at most $n$, $A$ lies in the depth-$n$ CYK cell of $w$ if and only if $A$ yields $w$.","title":"Correctness of the CYK recurrence"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-lookup","kind":"node","lean":"Cslib.Language.cykLookup","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-lookup","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"In a table listing pairs of a word and a cell, the cell recorded for a word $u$ is the cell of the first entry whose word is $u$, or the empty set if no entry has word $u$.","title":"Looking up a word in a CYK table"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-table","kind":"node","lean":"Cslib.Language.cykTable","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-table","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"The CYK table of a word $w$ up to length $l$ is a list of pairs of a word and a cell, defined by recursion on $l$. Up to length $0$ it is empty. The table up to length $l + 1$ is the table $t$ up to length $l$ followed by one entry for each position $i$ (counted from $0$) with $i + l + 1 \\le |w|$, in increasing order of $i$: the factor $u$ of $w$ of length $l + 1$ starting at position $i$, together with the CYK\u2026","title":"The CYK table"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-table-correct","kind":"node","lean":"Cslib.Language.CykTableCorrect","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-table-correct","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"A table listing pairs $(u, X)$ of a word $u$ and a set $X$ of nonterminals of $g$ is correct when, for every entry $(u, X)$ with $u$ nonempty, $X$ consists exactly of the nonterminals $B$ such that the one-symbol form $B$ yields $u$.","title":"Correct CYK tables"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-table-spec","kind":"node","lean":"Cslib.Language.cykTable_spec","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-table-spec","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Let $g$ be a context-free grammar in Chomsky normal form and $w$ a word. For every $l$, the CYK table of $w$ up to length $l$ is correct, and for all $j$ and $m$ with $0 < m \\le l$ and $j + m \\le |w|$, some entry of the table has as its word the factor of $w$ of length $m$ starting at position $j$ (positions counted from $0$).","title":"Correctness of the CYK table"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-accepts","kind":"node","lean":"Cslib.Language.cykAccepts","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Let $g$ be a context-free grammar with initial nonterminal $S$. The CYK recognizer accepts the empty word exactly when $g$ has a production $S \\to \\varepsilon$, and accepts a nonempty word $w$ exactly when $S$ lies in the cell recorded for $w$ in the CYK table of $w$ up to length $|w|$.","title":"The CYK recognizer"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/cyk-accepts-iff","kind":"node","lean":"Cslib.Language.cykAccepts_iff","parent":33763,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer/index.html#cyk-accepts-iff","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/the-cocke-younger-kasami-recognizer","status":"fully_proved","summary":"Let $g$ be a context-free grammar in Chomsky normal form. For every word $w$ over the terminals of $g$, the CYK recognizer accepts $w$ if and only if $w$ belongs to the language of $g$.","title":"Correctness of the CYK recognizer"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","kind":"scope","navigation":"module","parent":33762,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","title":"Indexed CYK primitives"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/member","kind":"node","lean":"Cslib.Language.CYKIndexed.member","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#member","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Full membership scan, charging equality and list/control operations.","title":"Charged list-membership scan"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/member-ret","kind":"node","lean":"Cslib.Language.CYKIndexed.member_ret","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#member-ret","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"The charged scan has ordinary list-membership semantics.","title":"Charged membership scan agrees with list membership"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/lookup","kind":"node","lean":"Cslib.Language.CYKIndexed.lookup","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#lookup","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Two bounded array reads; invalid indices denote an empty cell.","title":"CYK table lookup with empty default cells"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/splits","kind":"node","lean":"Cslib.Language.CYKIndexed.splits","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#splits","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Scan split positions `1,...,count`, with two table reads per split. The 16 units cover four array reads, bounds/default tests, index arithmetic, Boolean combinations and recursion control.","title":"Charged scan of CYK split positions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/splits-ret","kind":"node","lean":"Cslib.Language.CYKIndexed.splits_ret","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#splits-ret","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Every successful scan names a split in exactly the scanned interval.","title":"Successful CYK split scans find a split in range"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/fits","kind":"node","lean":"Cslib.Language.CYKIndexed.fits","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#fits","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Test a production output without constructing or comparing substrings. Pattern inspection is bounded (at most three cons and two symbol tags).","title":"Production-output fit test for indexed CYK"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/cell","kind":"node","lean":"Cslib.Language.CYKIndexed.cell","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#cell","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Scan every production once, retaining its input exactly when it fits. Duplicates are allowed; no set insertion or uncharged deduplication occurs.","title":"Charged computation of one CYK cell"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/mem-cell","kind":"node","lean":"Cslib.Language.CYKIndexed.mem_cell","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#mem-cell","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Membership in a computed cell names an actual scanned production.","title":"CYK cell members come from scanned productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/row","kind":"node","lean":"Cslib.Language.CYKIndexed.row","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#row","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Build one row in increasing start-position order. Each cell is evaluated once against the already completed shorter rows.","title":"Charged construction of one CYK row"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/table","kind":"node","lean":"Cslib.Language.CYKIndexed.table","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#table","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Fill rows `1,...,depth` in a fixed outer array of size `w.size + 1`. Row conversion is linear; the outer update is an indexed primitive.","title":"Charged construction of the indexed CYK table"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/epsilon","kind":"node","lean":"Cslib.Language.CYKIndexed.epsilon","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#epsilon","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"Scan start epsilon rules; output emptiness is a single list-tag test.","title":"Charged scan for start epsilon productions"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/recognizes","kind":"node","lean":"Cslib.Language.CYKIndexed.recognizes","parent":33778,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives/index.html#recognizes","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-primitives","status":"fully_proved","summary":"The charged indexed recognizer. The nonempty branch pays for the input length scan, conversion, completed table, final lookup and membership.","title":"Charged indexed CYK recognizer"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","kind":"scope","navigation":"module","parent":33762,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","title":"Correctness of indexed CYK"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/cell-correct","kind":"node","lean":"Cslib.Language.CYKIndexed.cell_correct","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#cell-correct","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"A cell is correct once all strictly shorter, valid factors are correct. The rule enumeration certificate constrains data only, not computation.","title":"Correctness of one indexed CYK cell"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/row-get","kind":"node","lean":"Cslib.Language.CYKIndexed.row_get","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#row-get","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"Row construction preserves the exact start-position indexing.","title":"Indexing of a constructed CYK row"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/table-size","kind":"node","lean":"Cslib.Language.CYKIndexed.table_size","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#table-size","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"The outer table retains its allocated size through every update.","title":"Indexed CYK table keeps its allocated size"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/lookup-table-other","kind":"node","lean":"Cslib.Language.CYKIndexed.lookup_table_other","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#lookup-table-other","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"Updating one length never changes another length's cells.","title":"CYK table update leaves other lengths' cells unchanged"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/lookup-table-new","kind":"node","lean":"Cslib.Language.CYKIndexed.lookup_table_new","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#lookup-table-new","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"The new length's row contains exactly the cells just computed.","title":"New CYK table row holds the computed cells"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/table-correct","kind":"node","lean":"Cslib.Language.CYKIndexed.table_correct","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#table-correct","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"Every completed valid cell agrees with canonical grammar derivations.","title":"Correctness of the completed indexed CYK table"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/epsilon-ret","kind":"node","lean":"Cslib.Language.CYKIndexed.epsilon_ret","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#epsilon-ret","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"Epsilon scanning agrees with existence of a start epsilon production.","title":"Correctness of the CYK epsilon scan"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/recognizes-iff","kind":"node","lean":"Cslib.Language.CYKIndexed.recognizes_iff","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#recognizes-iff","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"The indexed recognizer decides the existing canonical language.","title":"Correctness of the indexed CYK recognizer"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/recognizes-eq-cyk-accepts","kind":"node","lean":"Cslib.Language.CYKIndexed.recognizes_eq_cykAccepts","parent":33791,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk/index.html#recognizes-eq-cyk-accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/correctness-of-indexed-cyk","status":"fully_proved","summary":"Exact Boolean correspondence with the existing list-table recognizer.","title":"Indexed CYK agrees with the list-table CYK recognizer"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","kind":"scope","navigation":"module","parent":33762,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","title":"Cubic cost of indexed CYK"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/member-time","kind":"node","lean":"Cslib.Language.CYKIndexed.member_time","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#member-time","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"Exact charge for a full membership scan, including a failed scan.","title":"Exact cost of a membership scan"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/cell-length","kind":"node","lean":"Cslib.Language.CYKIndexed.cell_length","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#cell-length","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"Even with duplicates, a cell never exceeds the rule enumeration length.","title":"Computed CYK cell length bounded by the rule count"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/lookup-length","kind":"node","lean":"Cslib.Language.CYKIndexed.lookup_length","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#lookup-length","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"All table cells, including invalid/default positions, have bounded membership-scan length. This invariant is proved from construction.","title":"Every CYK table lookup is bounded by the rule count"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/splits-time-le","kind":"node","lean":"Cslib.Language.CYKIndexed.splits_time_le","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#splits-time-le","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"Each split pays for both indexed cell reads and both membership scans.","title":"Cost bound for the CYK split scan"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/fits-time-le","kind":"node","lean":"Cslib.Language.CYKIndexed.fits_time_le","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#fits-time-le","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"Output inspection is bounded; only binary productions scan splits.","title":"Cost bound for the CYK production fit test"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/cell-time-le","kind":"node","lean":"Cslib.Language.CYKIndexed.cell_time_le","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#cell-time-le","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"A cell charges every rule, its test, and its possible output cons.","title":"Cost bound for computing one CYK cell"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/row-time-le","kind":"node","lean":"Cslib.Language.CYKIndexed.row_time_le","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#row-time-le","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"Row cost counts all starts, not only successful or distinct factors.","title":"Cost bound for building one CYK row"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/table-time-le","kind":"node","lean":"Cslib.Language.CYKIndexed.table_time_le","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#table-time-le","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"Table cost includes allocation, every row conversion and every update. The memory-size hypothesis needed for split costs is discharged internally.","title":"Cost bound for filling the indexed CYK table"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/row-length","kind":"node","lean":"Cslib.Language.CYKIndexed.row_length","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#row-length","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"The linear row-conversion charge uses the actual number of entries.","title":"Length of a computed CYK row"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/epsilon-time","kind":"node","lean":"Cslib.Language.CYKIndexed.epsilon_time","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#epsilon-time","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"Exact cost of the empty-input production scan.","title":"Exact cost of the CYK epsilon scan"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/recognizes-time-le","kind":"node","lean":"Cslib.Language.CYKIndexed.recognizes_time_le","parent":33801,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#recognizes-time-le","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"Explicit cubic bound, for every word including the empty word. The coefficient depends only on the rule enumeration and equality costs; no semantic or assumed recurrence hypothesis occurs in this cost theorem.","title":"Explicit cubic cost bound for indexed CYK"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/recognizes-time-le-cubic","kind":"node","lean":"Cslib.Language.CYKIndexed.recognizes_time_le_cubic","parent":33801,"reading_title":{"format":"inline-parts-v1","parts":[["text","Indexed CYK costs a constant times "],["math","\\(n^3\\)"],["text"," on nonempty words"]],"source":"Indexed CYK costs a constant times $n^3$ on nonempty words"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk/index.html#recognizes-time-le-cubic","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/cubic-cost-of-indexed-cyk","status":"fully_proved","summary":"In the usual nonempty-input form, the bound is a constant times `n\u00b3`.","title":"Indexed CYK costs a constant times $n^3$ on nonempty words"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","kind":"scope","navigation":"module","parent":33762,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","title":"Indexed CYK for the canonical encoded grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/decode-symbols","kind":"node","lean":"Cslib.Language.CYKIndexed.decodeSymbols","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#decode-symbols","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"Decode symbols with a charge for each constructor test and output cons.","title":"Charged decoding of encoded grammar symbols"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/decode-symbols-ret","kind":"node","lean":"Cslib.Language.CYKIndexed.decodeSymbols_ret","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#decode-symbols-ret","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"The charged conversion is exactly the canonical symbol decoding.","title":"Charged symbol decoding agrees with canonical decoding"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/decode-rules","kind":"node","lean":"Cslib.Language.CYKIndexed.decodeRules","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#decode-rules","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"Decode every rule, charging all right-hand-side traversals.","title":"Charged decoding of encoded grammar rules"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/decode-rules-ret","kind":"node","lean":"Cslib.Language.CYKIndexed.decodeRules_ret","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#decode-rules-ret","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"The charged rule conversion agrees with the existing decoded grammar.","title":"Charged rule decoding agrees with the decoded grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/recognizes-encoded","kind":"node","lean":"Cslib.Language.CYKIndexed.recognizesEncoded","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#recognizes-encoded","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"Run indexed CYK on the existing encoded grammar, including preparation.","title":"Indexed CYK recognizer for encoded grammars"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/recognizes-encoded-iff","kind":"node","lean":"Cslib.Language.CYKIndexed.recognizesEncoded_iff","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#recognizes-encoded-iff","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"Correctness for the canonical encoded language, including epsilon.","title":"Correctness of indexed CYK on encoded grammars"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/decode-symbols-time","kind":"node","lean":"Cslib.Language.CYKIndexed.decodeSymbols_time","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#decode-symbols-time","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"Exact symbol-decoding traversal cost.","title":"Exact cost of grammar symbol decoding"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/decode-rules-time","kind":"node","lean":"Cslib.Language.CYKIndexed.decodeRules_time","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#decode-rules-time","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"Exact grammar-preparation cost, including all production symbols.","title":"Exact cost of grammar rule decoding"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/recognizes-encoded-time-le","kind":"node","lean":"Cslib.Language.CYKIndexed.recognizesEncoded_time_le","parent":33814,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar/index.html#recognizes-encoded-time-le","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/cyk-algorithm/indexed-cyk-for-the-canonical-encoded-grammar","status":"fully_proved","summary":"Cubic word-length bound including canonical grammar preparation. All dependence on the fixed grammar and symbol-equality model is explicit.","title":"Cubic cost bound for indexed CYK on encoded grammars"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form","status":"fully_proved","title":"Greibach normal form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form/is-greibach-normal-form","kind":"node","lean":"Cslib.Language.IsGreibachNormalForm","parent":33824,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form/index.html#is-greibach-normal-form","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form","status":"fully_proved","summary":"A grammar is in Greibach normal form when every production has output either a single terminal followed by nonterminals, or the empty word with the initial nonterminal on the left; and either the initial nonterminal occurs in no output (so an empty production cannot leak the empty word to other nonterminals) or there are no empty outputs at all.","title":"Greibach normal form"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form/exists-terminal-head-of-output-ne-nil","kind":"node","lean":"Cslib.Language.exists_terminal_head_of_output_ne_nil","parent":33824,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form/index.html#exists-terminal-head-of-output-ne-nil","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/greibach-normal-form","status":"fully_proved","summary":"In Greibach normal form, every nonempty production output begins with a terminal followed by nonterminals.","title":"Nonempty Greibach productions begin with a terminal"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","title":"Pushdown automata"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/pda","kind":"node","lean":"Cslib.Automata.PDA","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#pda","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"A pushdown automaton over an input alphabet $T$, with states in $\\sigma$ and stack alphabet $\\Gamma$, consists of a transition function, a start state in $\\sigma$ and an initial stack symbol in $\\Gamma$. The transition function assigns to each state $q$, each input option (a letter of $T$, or none for an $\\varepsilon$-transition) and each popped stack symbol $\\gamma$ a finite set of pairs $(q', u)$ of a next state\u2026","title":"Pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/config","kind":"node","lean":"Cslib.Automata.PDA.Config","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#config","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"A configuration of a pushdown automaton with input alphabet $T$, states in $\\sigma$ and stack alphabet $\\Gamma$ consists of a current state in $\\sigma$, the remaining unread input (a word over $T$), and the stack contents (a list over $\\Gamma$ whose first element is the top of the stack).","title":"Pushdown automaton configuration"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/step","kind":"node","lean":"Cslib.Automata.PDA.Step","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#step","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Write a configuration of a pushdown automaton $M$ as $(q, w, s)$: state $q$, remaining input $w$, and stack $s$ listed from the top. A configuration $(q, w, \\gamma s)$ steps to $(q', w, u s)$ whenever $(q', u)$ is an $\\varepsilon$-transition of $M$ from $q$ with top symbol $\\gamma$, and a configuration $(q, a w, \\gamma s)$ steps to $(q', w, u s)$ whenever $(q', u)$ is a transition of $M$ from $q$ on the input\u2026","title":"One step of a pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/derives","kind":"node","lean":"Cslib.Automata.PDA.Derives","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#derives","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"For a pushdown automaton $M$, a configuration $c$ derives a configuration $c'$ when $M$ can reach $c'$ from $c$ in zero or more computation steps; that is, derivation is the reflexive transitive closure of the one-step computation relation of $M$.","title":"Multi-step computation of a pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/append-input-stack","kind":"node","lean":"Cslib.Automata.PDA.Derives.append_input_stack","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#append-input-stack","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Let $M$ be a pushdown automaton. If $M$ can compute from a configuration with state $q$, remaining input $w$ and stack $s$ to a configuration with state $q'$, remaining input $w'$ and stack $s'$, then for every word $v$ and every list $t$ of stack symbols, $M$ can compute from the configuration with state $q$, remaining input $wv$ and stack $st$ to the configuration with state $q'$, remaining input $w'v$ and stack\u2026","title":"Computations extend by unread input and stack bottom"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/finitely-supported","kind":"node","lean":"Cslib.Automata.PDA.FinitelySupported","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#finitely-supported","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"A pushdown automaton $M$ is finitely supported when there are finite sets $\\sigma_0$ of states, $\\Gamma_0$ of stack symbols and $T_0$ of input letters such that the start state lies in $\\sigma_0$, the initial stack symbol lies in $\\Gamma_0$, every transition from a state in $\\sigma_0$ that pops a symbol in $\\Gamma_0$ (on any input letter or on $\\varepsilon$) moves to a state in $\\sigma_0$ and pushes only symbols in\u2026","title":"Finitely supported pushdown automata"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/init-config","kind":"node","lean":"Cslib.Automata.PDA.initConfig","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#init-config","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"The initial configuration of a pushdown automaton $M$ on an input word $w$ has the start state of $M$ as its state, all of $w$ as its remaining input, and the stack consisting of the initial stack symbol of $M$ alone.","title":"Initial configuration of a pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/accepts","kind":"node","lean":"Cslib.Automata.PDA.accepts","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"The language accepted by a pushdown automaton $M$, by empty stack, consists of the words $w$ for which some computation of $M$ leads from the initial configuration on input $w$ to a configuration, in some state, with no remaining input and an empty stack.","title":"Language accepted by a pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/closed-under","kind":"node","lean":"Cslib.Automata.PDA.ClosedUnder","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#closed-under","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"A finite set $\\Gamma_0$ of stack symbols is closed for a pushdown automaton $M$ when every transition of $M$ that pops a symbol in $\\Gamma_0$, from any state and on any input letter or on $\\varepsilon$, pushes only symbols in $\\Gamma_0$. Computations starting with a stack inside a closed set keep their stack inside it.","title":"Closed sets of stack symbols"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/lift-stack","kind":"node","lean":"Cslib.Automata.PDA.liftStack","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#lift-stack","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Given a finite set $\\Gamma_0$ of stack symbols and a stack $s = \\gamma_1 \\cdots \\gamma_n$ all of whose symbols lie in $\\Gamma_0$, the lifted stack is the same list $\\gamma_1 \\cdots \\gamma_n$, in the same order, regarded as a stack over the alphabet $\\Gamma_0$.","title":"Lifting a stack to a set of stack symbols"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/restrict","kind":"node","lean":"Cslib.Automata.PDA.restrict","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#restrict","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Let $M$ be a pushdown automaton with stack alphabet $\\Gamma$, and let $\\Gamma_0 \\subseteq \\Gamma$ be a finite set of stack symbols that contains the initial stack symbol of $M$ and is closed for $M$. The restriction of $M$ to $\\Gamma_0$ is the pushdown automaton with the same input alphabet, states and start state as $M$, stack alphabet $\\Gamma_0$, and the initial stack symbol of $M$ as its initial stack symbol.\u2026","title":"Restricting the stack alphabet of a pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/mem-restrict-step","kind":"node","lean":"Cslib.Automata.PDA.mem_restrict_step","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#mem-restrict-step","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Let $M$ be a pushdown automaton with stack alphabet $\\Gamma$, and let $M|_{\\Gamma_0}$ be its restriction to a finite set $\\Gamma_0 \\subseteq \\Gamma$ of stack symbols that contains the initial stack symbol and is closed for $M$. Let $q, q'$ be states, $o$ an input option (an input letter, or none for an $\\varepsilon$-transition), $\\gamma \\in \\Gamma_0$ and $s$ a list of symbols of $\\Gamma_0$. Then $(q', s)$ is a\u2026","title":"Transitions of the restricted pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/derives-val-of-derives-restrict","kind":"node","lean":"Cslib.Automata.PDA.derives_val_of_derives_restrict","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#derives-val-of-derives-restrict","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Let $M$ be a pushdown automaton and let $\\Gamma_0$ be a finite set of stack symbols that contains the initial stack symbol of $M$ and is closed for $M$. If the restriction of $M$ to $\\Gamma_0$ can compute from a configuration with state $q$, remaining input $w$ and stack $s$ to a configuration with state $q'$, remaining input $w'$ and stack $s'$, then $M$ can compute from the configuration $(q, w, s)$ to the\u2026","title":"Computations of the restriction are computations of the automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/stack-subset-of-derives","kind":"node","lean":"Cslib.Automata.PDA.stack_subset_of_derives","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#stack-subset-of-derives","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Let $M$ be a pushdown automaton and $\\Gamma_0$ a finite set of stack symbols that is closed for $M$. If a configuration $c$ derives a configuration $c'$ in zero or more steps of $M$, and every symbol on the stack of $c$ lies in $\\Gamma_0$, then every symbol on the stack of $c'$ lies in $\\Gamma_0$.","title":"Computations stay inside a closed set of stack symbols"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/derives-restrict-of-derives","kind":"node","lean":"Cslib.Automata.PDA.derives_restrict_of_derives","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#derives-restrict-of-derives","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Let $M$ be a pushdown automaton and let $\\Gamma_0$ be a finite set of stack symbols that contains the initial stack symbol of $M$ and is closed for $M$. Suppose $M$ can compute from a configuration with state $q$, remaining input $w$ and stack $s$ to a configuration with state $q'$, remaining input $w'$ and stack $s'$, and every symbol of $s$ lies in $\\Gamma_0$. Then the symbols of $s'$ also lie in $\\Gamma_0$, and\u2026","title":"Computations inside a closed set lift to the restriction"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/accepts-restrict","kind":"node","lean":"Cslib.Automata.PDA.accepts_restrict","parent":33827,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata/index.html#accepts-restrict","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata","status":"fully_proved","summary":"Let $M$ be a pushdown automaton and let $\\Gamma_0$ be a finite set of stack symbols of $M$ that contains the initial stack symbol of $M$ and is closed for $M$. Then the restriction of $M$ to the stack alphabet $\\Gamma_0$ accepts, by empty stack, exactly the same language as $M$.","title":"Restricting the stack alphabet preserves the accepted language"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","title":"Pushdown automata and context-free grammars"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/cfg-step","kind":"node","lean":"Cslib.Language.cfgStep","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#cfg-step","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Let $g$ be a context-free grammar over an alphabet $T$. To the single state, an input option (a letter of $T$, or none for an $\\varepsilon$-transition) and a popped symbol of $g$ (a terminal or a nonterminal), this assigns a finite set of pairs of the state and a list of symbols of $g$ to push. Popping a nonterminal $A$ with no input gives, for each production of $g$ with left-hand side $A$, the pair pushing its\u2026","title":"Transitions of the grammar-simulating pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/cfg-pda","kind":"node","lean":"Cslib.Language.cfgPDA","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#cfg-pda","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"For a context-free grammar $g$ over an alphabet $T$, this is the pushdown automaton with input alphabet $T$, a single state, the terminals and nonterminals of $g$ as stack alphabet, and the initial nonterminal of $g$ as initial stack symbol. Popping a nonterminal $A$ on an $\\varepsilon$-transition, it may push the right-hand side of any production of $g$ with left-hand side $A$ (whose first symbol becomes the new\u2026","title":"The pushdown automaton simulating a context-free grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/cfg-pda-accepts","kind":"node","lean":"Cslib.Language.ContextFreeGrammar.cfgPDA_accepts","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#cfg-pda-accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"For every context-free grammar $g$ over an alphabet $T$, the language accepted by empty stack by the one-state pushdown automaton simulating $g$ equals the language generated by $g$.","title":"The grammar-simulating pushdown automaton accepts the grammar's language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/exists-pda-accepts","kind":"node","lean":"Cslib.Language.IsContextFree.exists_pda_accepts","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#exists-pda-accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"For every context-free language $L$ over an alphabet $T$ there are a context-free grammar $g$ over $T$ and a pushdown automaton with input alphabet $T$, a single state, and the terminals and nonterminals of $g$ as stack alphabet, whose language accepted by empty stack is $L$.","title":"Context-free languages are accepted by pushdown automata"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/cfg-pda-finitely-supported","kind":"node","lean":"Cslib.Language.ContextFreeGrammar.cfgPDA_finitelySupported","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#cfg-pda-finitely-supported","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"For every context-free grammar $g$ over an alphabet $T$, the one-state pushdown automaton simulating $g$ is finitely supported.","title":"The grammar-simulating pushdown automaton is finitely supported"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/exists-finitely-supported-pda-accepts","kind":"node","lean":"Cslib.Language.IsContextFree.exists_finitely_supported_pda_accepts","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#exists-finitely-supported-pda-accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"For every context-free language $L$ over an alphabet $T$ there are a context-free grammar $g$ over $T$ and a finitely supported pushdown automaton with input alphabet $T$, a single state, and the terminals and nonterminals of $g$ as stack alphabet, whose language accepted by empty stack is $L$.","title":"Context-free languages are accepted by finitely supported pushdown automata"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/chain-form","kind":"node","lean":"Cslib.Language.chainForm","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#chain-form","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Write $(p, \\gamma, q)$ for the nonterminal symbol given by states $p, q$ and a stack symbol $\\gamma$ (the nonterminals of the triple grammar are these triples and one extra initial nonterminal). The chain form of a list $\\gamma_1 \\cdots \\gamma_k$ of stack symbols along a nonempty list $s_0, s_1, \\dots, s_m$ of states is the word $(s_0, \\gamma_1, s_1)(s_1, \\gamma_2, s_2) \\cdots (s_{j-1}, \\gamma_j, s_j)$ with\u2026","title":"Chain of triple nonterminals"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/nonterminal-of-mem-chain-form","kind":"node","lean":"Cslib.Language.nonterminal_of_mem_chainForm","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#nonterminal-of-mem-chain-form","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"For every list of stack symbols and every list of states, every symbol of their chain form is a nonterminal.","title":"Chain forms consist of nonterminals"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/state-lists","kind":"node","lean":"Cslib.Language.stateLists","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#state-lists","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"For a finite set $\\sigma$ of states and a natural number $n$, this is the finite set of all lists of length $n$ with entries in $\\sigma$. It is built recursively: for $n = 0$ it contains only the empty list, and for $n + 1$ it consists of the lists obtained by putting a state $p \\in \\sigma$ in front of a list from the set for $n$.","title":"Lists of states of a given length"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/opt-terms","kind":"node","lean":"Cslib.Language.optTerms","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#opt-terms","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"As a word of grammar symbols (terminals and triple nonterminals), an input that is a letter $a$ contributes the one-symbol word consisting of the terminal $a$, and the empty input $\\varepsilon$ contributes the empty word.","title":"Terminal word of an optional input letter"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/step-rule","kind":"node","lean":"Cslib.Language.stepRule","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#step-rule","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Let $p$ and $r$ be states, $\\gamma$ a stack symbol, $a$ an optional input letter, $\\varepsilon_1 \\cdots \\varepsilon_k$ a list of stack symbols and $s_1, \\dots, s_m$ a list of states. Write $s_0 = r$ and let $q = s_m$ be the last state of the list $s_0, s_1, \\dots, s_m$ (so $q = r$ when $m = 0$). The production has left-hand side the nonterminal $(p, \\gamma, q)$, and its right-hand side is the letter $a$ (nothing if\u2026","title":"Production of the triple grammar for a transition"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/pda-grammar","kind":"node","lean":"Cslib.Language.pdaGrammar","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#pda-grammar","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Let $M$ be a pushdown automaton over the alphabet $T$ with a finite set $\\sigma$ of states, a finite stack alphabet $\\Gamma$, start state $q_0$ and initial stack symbol $Z_0$, and let $T_0$ be a finite set of letters. The triple grammar of $M$ and $T_0$ is the context-free grammar over $T$ whose nonterminals are an initial nonterminal $S$ and the triples $(p, \\gamma, q)$ with $p, q \\in \\sigma$ and\u2026","title":"Triple grammar of a pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/mem-rules-start","kind":"node","lean":"Cslib.Language.mem_rules_start","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#mem-rules-start","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Let $M$ be a pushdown automaton with a finite set of states, a finite stack alphabet, start state $q_0$ and initial stack symbol $Z_0$, and let $T_0$ be a finite set of input letters. For every state $q$, the triple grammar of $M$ and $T_0$ contains the production $S \\to (q_0, Z_0, q)$, where $S$ is its initial nonterminal.","title":"Start productions of the triple grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/mem-rules-step","kind":"node","lean":"Cslib.Language.mem_rules_step","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#mem-rules-step","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Let $M$ be a pushdown automaton with a finite set of states and a finite stack alphabet, and let $T_0$ be a finite set of input letters. Let $a$ be either $\\varepsilon$ or a letter in $T_0$, and suppose that $M$ has a transition from state $p$ on $a$ popping the stack symbol $\\gamma$ that moves to state $r$ and pushes $\\gamma_1 \\cdots \\gamma_k$. Then for all states $s_1, \\dots, s_k$, writing $s_0 = r$, the triple\u2026","title":"Transition productions of the triple grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/cases-of-mem-rules","kind":"node","lean":"Cslib.Language.cases_of_mem_rules","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#cases-of-mem-rules","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Let $M$ be a pushdown automaton with a finite set of states, a finite stack alphabet, start state $q_0$ and initial stack symbol $Z_0$, and let $T_0$ be a finite set of input letters. Every production of the triple grammar of $M$ and $T_0$ is of one of two kinds. Either it is a start production $S \\to (q_0, Z_0, q)$ for some state $q$, where $S$ is the initial nonterminal; or there are states $p$ and $r$, a stack\u2026","title":"Productions of the triple grammar"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/pda-grammar-language","kind":"node","lean":"Cslib.Language.PDA.pdaGrammar_language","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#pda-grammar-language","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Let $M$ be a pushdown automaton with a finite set of states and a finite stack alphabet, and let $T_0$ be a finite set of input letters such that $M$ has no transition, from any state and popping any stack symbol, that consumes a letter outside $T_0$. Then the language generated by the triple grammar of $M$ and $T_0$ equals the language accepted by $M$ by empty stack.","title":"The triple grammar generates the accepted language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/is-context-free-accepts","kind":"node","lean":"Cslib.Language.PDA.isContextFree_accepts","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#is-context-free-accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"Let $M$ be a pushdown automaton with a finite set of states and a finite stack alphabet, and suppose that there is a finite set $T_0$ of input letters such that $M$ has no transition, from any state and popping any stack symbol, that consumes a letter outside $T_0$. Then the language accepted by $M$ by empty stack is context-free.","title":"Finite pushdown automata accept context-free languages"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/is-context-free-iff-exists-pda-accepts","kind":"node","lean":"Cslib.Language.isContextFree_iff_exists_pda_accepts","parent":33844,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars/index.html#is-context-free-iff-exists-pda-accepts","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/pushdown-automata-and-context-free-grammars","status":"fully_proved","summary":"A language $L$ over an alphabet $T$ is context-free if and only if there are a pushdown automaton $M$ over $T$ with a finite set of states and a finite stack alphabet, and a finite set $T_0$ of input letters such that $M$ has no transition, from any state and popping any stack symbol, that consumes a letter outside $T_0$, for which the language accepted by $M$ by empty stack is $L$.","title":"Context-free languages are exactly the languages of finite pushdown automata"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages.json","id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","kind":"scope","navigation":"module","parent":33662,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","title":"Deterministic context-free languages"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/is-deterministic","kind":"node","lean":"Cslib.Automata.PDA.IsDeterministic","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#is-deterministic","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"A pushdown automaton is deterministic when every transition set has at most one element, and an available `\u03b5`-transition precludes every consuming transition from the same state and stack symbol.","title":"Deterministic pushdown automaton"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-encode","kind":"node","lean":"Cslib.Language.endMarkerEncode","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-encode","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"Encode a word by copying it to the left summand and appending the end marker `Sum.inr ()`.","title":"End-marker encoding of a word"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-erase","kind":"node","lean":"Cslib.Language.endMarkerErase","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-erase","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"Erase the end marker: a left letter maps to its one-letter word, the end marker maps to the empty word.","title":"End-marker erasure"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/encoded-words","kind":"node","lean":"Cslib.Language.encodedWords","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#encoded-words","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"The encodings of words over `T`: the words over `T \u2295 Unit` consisting of left letters followed by a single end marker.","title":"Language of end-marker encoded words"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-erase-end-marker-encode","kind":"node","lean":"Cslib.Language.endMarkerErase_endMarkerEncode","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-erase-end-marker-encode","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"Erasing an encoding recovers the word.","title":"Erasing the end-marker encoding recovers the word"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-encode-injective","kind":"node","lean":"Cslib.Language.endMarkerEncode_injective","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-encode-injective","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"Encoding is injective.","title":"Injectivity of the end-marker encoding"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-dfa","kind":"node","lean":"Cslib.Language.endMarkerDFA","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-dfa","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"The DFA for the encodings: state `inl false` while reading left letters, `inl true` after the end marker (the accepting state), and `inr ()` dead once anything follows the end marker.","title":"DFA recognizing end-marker encodings"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-dfa-eval-from-dead","kind":"node","lean":"Cslib.Language.endMarkerDFA_evalFrom_dead","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-dfa-eval-from-dead","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"The dead state of the encoding DFA stays dead.","title":"Dead state of the encoding DFA is absorbing"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-dfa-eval-from-inl","kind":"node","lean":"Cslib.Language.endMarkerDFA_evalFrom_inl","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-dfa-eval-from-inl","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"The encoding DFA stays in its start state while reading left letters.","title":"Encoding DFA stays in its start state on left letters"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-dfa-eval-from-after-marker","kind":"node","lean":"Cslib.Language.endMarkerDFA_evalFrom_afterMarker","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-dfa-eval-from-after-marker","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"After the end marker, the encoding DFA can only be in its accepting state on empty input.","title":"Encoding DFA after the end marker accepts only the empty input"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/end-marker-dfa-eval-from-encode","kind":"node","lean":"Cslib.Language.endMarkerDFA_evalFrom_encode","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#end-marker-dfa-eval-from-encode","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"The encoding DFA accepts exactly the encodings: it reaches the accepting state on an encoding.","title":"Encoding DFA accepts every end-marker encoding"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/exists-end-marker-encode-of-end-marker-d","kind":"node","lean":"Cslib.Language.exists_endMarkerEncode_of_endMarkerDFA_evalFrom","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#exists-end-marker-encode-of-end-marker-d","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"The encoding DFA reaches the accepting state only on encodings.","title":"Encoding DFA accepts only end-marker encodings"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/is-regular-encoded-words","kind":"node","lean":"Cslib.Language.isRegular_encodedWords","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#is-regular-encoded-words","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"The encodings of words form a regular language.","title":"End-marker encodings form a regular language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/is-deterministic-context-free","kind":"node","lean":"Cslib.Language.IsDeterministicContextFree","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#is-deterministic-context-free","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"A language is deterministic context-free when some deterministic pushdown automaton with finite state and stack alphabets, whose transitions consume letters from a finite set, accepts by empty stack exactly the end-marker encodings of its words.","title":"Deterministic context-free language"},{"id":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/is-context-free","kind":"node","lean":"Cslib.Language.IsDeterministicContextFree.isContextFree","parent":33863,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages/index.html#is-context-free","scope":"TheoryOfComputation/context-free-languages/grammars-and-pushdown-automata/deterministic-context-free-languages","status":"fully_proved","summary":"Every deterministic context-free language is context-free: the encoded language is accepted by a pushdown automaton, hence context-free; intersecting with the regular language of encodings and erasing the end marker recovers the original language.","title":"Deterministic context-free languages are context-free"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/closure-and-pumping.json","id":"TheoryOfComputation/context-free-languages/closure-and-pumping","kind":"scope","parent":33661,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/index.html","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping","status":"fully_proved","summary":"Closure of context-free languages under union, concatenation, star, homomorphism, inverse homomorphism, substitution, regular intersection and quotients, non-closure under intersection and complement, the pumping and Ogden lemmas, and Parikh's theorem.","title":"Closure properties and pumping lemmas"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages.json","id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","kind":"scope","navigation":"module","parent":33879,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","title":"Closure properties of context-free languages"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/add","kind":"node","lean":"Cslib.Language.IsContextFree.add","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#add","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"If $L$ and $M$ are context-free languages over the same alphabet, then their union $L \\cup M$ is context-free.","title":"Closure of context-free languages under union"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/mul","kind":"node","lean":"Cslib.Language.IsContextFree.mul","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#mul","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"If $L$ and $M$ are context-free languages over the same alphabet, then their concatenation $LM = \\{uv : u \\in L,\\ v \\in M\\}$ is context-free.","title":"Closure of context-free languages under concatenation"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/kstar","kind":"node","lean":"Cslib.Language.IsContextFree.kstar","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#kstar","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"If $L$ is a context-free language, then its Kleene star $L^*$ is context-free.","title":"Closure of context-free languages under Kleene star"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/pow","kind":"node","lean":"Cslib.Language.IsContextFree.pow","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#pow","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"If $L$ is a context-free language, then for every natural number $n$ the power $L^n$, the concatenation of $n$ copies of $L$ (with $L^0$ the language containing only the empty word), is context-free.","title":"Powers of a context-free language are context-free"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/of-finite","kind":"node","lean":"Cslib.Language.IsContextFree.of_finite","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#of-finite","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Every finite language is context-free.","title":"Finite languages are context-free"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/of-length-le","kind":"node","lean":"Cslib.Language.IsContextFree.of_length_le","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#of-length-le","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Over a finite alphabet, if $n$ is a natural number and every word of a language $L$ has length at most $n$, then $L$ is context-free.","title":"Languages of bounded word length are context-free"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/hom-image","kind":"node","lean":"Cslib.Language.IsContextFree.homImage","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#hom-image","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $T$ and $T'$ be alphabets, $L$ a context-free language over $T$, and $\\varphi$ a map sending each letter of $T$ to a word over $T'$. Then the language of all words $\\varphi(a_1) \\cdots \\varphi(a_n)$ with $a_1 \\cdots a_n \\in L$ is context-free.","title":"Context-free languages are closed under homomorphic images"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/inf-dfa-accepts","kind":"node","lean":"Cslib.Language.IsContextFree.inf_dfa_accepts","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#inf-dfa-accepts","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"If $L$ is a context-free language over an alphabet $T$ and $M$ is a deterministic finite automaton over $T$ with finitely many states, then the intersection of $L$ with the language accepted by $M$ is context-free.","title":"Intersection of a context-free language with the language of a DFA"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/inf-is-regular","kind":"node","lean":"Cslib.Language.IsContextFree.inf_isRegular","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#inf-is-regular","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"If $L$ is a context-free language and $R$ a regular language over the same alphabet, then $L \\cap R$ is context-free.","title":"Intersection of a context-free language with a regular language"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/map","kind":"node","lean":"Cslib.Language.IsContextFree.map","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#map","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $L$ be a context-free language over an alphabet $T$ and let $f \\colon T \\to T'$ be a map of alphabets. Then the image of $L$ under $f$, the language $\\{f(a_1) \\cdots f(a_n) \\mid a_1 \\cdots a_n \\in L\\}$ over $T'$, is context-free.","title":"Context-free languages are closed under letter maps"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/code","kind":"node","lean":"Cslib.Language.Substitution.code","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#code","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"For a context-free grammar $g$ and a letter $a$, the number of $a$ is the position (counted from $0$) of the first occurrence of $a$ in the list of terminals occurring in the productions of $g$, or the length of that list when $a$ does not occur in it.","title":"Letter numbering for the substitution grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/decode","kind":"node","lean":"Cslib.Language.Substitution.decode","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#decode","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"For a context-free grammar $g$ and a natural number $k$, the letter with number $k$ is the entry at position $k$ (counted from $0$) of the list of terminals occurring in the productions of $g$; there is no such letter when $k$ is at least the length of that list.","title":"Letter decoding for the substitution grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/subst-symbol","kind":"node","lean":"Cslib.Language.Substitution.substSymbol","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#subst-symbol","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T$ with nonterminals $N_g$, and let $e$ assign to each letter $a \\in T$ an encoded grammar $e(a)$ over $T'$. Each symbol of $g$ becomes a nonterminal in $N_g \\sqcup (\\mathbb{N} \\times \\mathbb{N})$: a nonterminal $B$ of $g$ becomes $B$ in the left part, and a terminal $a$ becomes the pair $(k, n)$, where $k$ is the number of $a$ (the position, counted from $0$, of the first\u2026","title":"Substituting the symbols of a grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/main-rule","kind":"node","lean":"Cslib.Language.Substitution.mainRule","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#main-rule","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T$ with nonterminals $N_g$, and let $e$ assign to each letter $a \\in T$ an encoded grammar $e(a)$ over $T'$. A production $A \\to \\alpha$ over $T$ with nonterminals in $N_g$ gives the production $A \\to \\alpha'$ over $T'$ with nonterminals in $N_g \\sqcup (\\mathbb{N} \\times \\mathbb{N})$, where $A$ is taken in the left part and $\\alpha'$ is obtained from $\\alpha$ by replacing\u2026","title":"Main production of the substitution grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/sub-rule","kind":"node","lean":"Cslib.Language.Substitution.subRule","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#sub-rule","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T$ with nonterminals $N_g$, let $a \\in T$ have number $k$ (the position, counted from $0$, of the first occurrence of $a$ in the list of terminals occurring in the productions of $g$, or the length of that list when $a$ does not occur in it), and let $n \\to w$ be a production over $T'$ whose nonterminals are natural numbers. It gives the production $(k, n) \\to w'$ with\u2026","title":"Letter productions of the substitution grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/rules","kind":"node","lean":"Cslib.Language.Substitution.rules","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#rules","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T$, and let $e$ assign to each letter $a \\in T$ an encoded grammar $e(a)$ over $T'$. The productions of the substitution grammar are the main productions obtained from the productions of $g$, together with, for each terminal $a$ occurring in a production of $g$, the letter productions for $a$ obtained from the productions of $e(a)$.","title":"Productions of the substitution grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/grammar","kind":"node","lean":"Cslib.Language.Substitution.grammar","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#grammar","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T$ with nonterminals $N_g$ and start symbol $S$, and let $e$ assign to each letter $a \\in T$ an encoded grammar $e(a)$ over $T'$. The substitution grammar is the context-free grammar over $T'$ with nonterminals $N_g \\sqcup (\\mathbb{N} \\times \\mathbb{N})$, start symbol $S$ taken in the left part, and the productions of the substitution grammar for $g$ and $e$.","title":"The substitution grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/val","kind":"node","lean":"Cslib.Language.Substitution.val","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#val","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T$ with nonterminals $N_g$, let $e$ assign to each letter $a \\in T$ an encoded grammar $e(a)$ over $T'$, and let $s$ assign to each letter $a \\in T$ a language $s(a)$ over $T'$. Each element of $N_g \\sqcup (\\mathbb{N} \\times \\mathbb{N})$, the nonterminals of the substitution grammar, is assigned a language over $T'$. A nonterminal $B \\in N_g$ is assigned the substitution of\u2026","title":"Intended languages of the substitution grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/mem-grammar-terminals-of-derives","kind":"node","lean":"Cslib.Language.Substitution.mem_grammarTerminals_of_derives","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#mem-grammar-terminals-of-derives","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar and let the sentential form $u$ derive $v$ in $g$ in zero or more steps. If every terminal occurring in $u$ occurs in some production of $g$, then every terminal occurring in $v$ occurs in some production of $g$.","title":"Derivations keep the terminals among those of the grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/substitution","kind":"node","lean":"Cslib.Language.IsContextFree.substitution","parent":33880,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages/index.html#substitution","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/closure-properties-of-context-free-languages","status":"fully_proved","summary":"Context-free languages are closed under substitution by context-free languages.","title":"Context-free languages are closed under substitution by context-free languages"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu.json","id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","kind":"scope","navigation":"module","parent":33879,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","title":"Inverse homomorphic images of context-free languages"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/proj-right","kind":"node","lean":"Cslib.Language.projRight","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#proj-right","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"For a word $u$ over the disjoint union $T \\oplus T'$ of two alphabets, its right projection is the word over $T'$ obtained from $u$ by deleting every letter from $T$ and reading each remaining letter as a letter of $T'$, keeping their order.","title":"Right projection of a word"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/ins-symbol","kind":"node","lean":"Cslib.Language.insSymbol","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#ins-symbol","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T'$ and let $T$ be an alphabet. Each symbol of $g$ is translated to a string of symbols with terminals in $T \\sqcup T'$ and nonterminals those of $g$ together with two new nonterminals $S$ and $D$: a terminal $b \\in T'$ becomes the marker $D$ followed by $b$, and a nonterminal $A$ of $g$ becomes $A$.","title":"Marker translation of a symbol"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/marked-form","kind":"node","lean":"Cslib.Language.markedForm","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#marked-form","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T'$, let $T$ be an alphabet, and let $u$ be a word over $T \\sqcup T'$. The marked form of $u$ is the sentential form, with terminals in $T \\sqcup T'$ and nonterminals those of $g$ together with two new nonterminals $S$ and $D$, that lists the letters of $u$ from $T'$ in order, each preceded by the marker $D$, followed by one more $D$.","title":"Marked form of a word"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/ins-grammar","kind":"node","lean":"Cslib.Language.insGrammar","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#ins-grammar","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T'$ with start symbol $A_0$, and let $T$ be a finite alphabet. The marker-insertion grammar of $g$ is the context-free grammar over $T \\sqcup T'$ whose nonterminals are those of $g$ together with a new start symbol $S$ and a marker $D$, and whose rules are: for each rule $A \\to \\alpha$ of $g$, the rule $A \\to \\alpha'$, where $\\alpha'$ is $\\alpha$ with $D$ inserted in front of\u2026","title":"Marker-insertion grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/ins-grammar-derives-flat-map","kind":"node","lean":"Cslib.Language.insGrammar_derives_flatMap","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#ins-grammar-derives-flat-map","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T'$ and let $T$ be a finite alphabet. If $g$ derives a sentential form $y$ from a sentential form $x$, then the marker-insertion grammar of $g$ over $T \\sqcup T'$ derives the translation of $y$ from the translation of $x$, where a sentential form is translated by replacing each of its symbols by its marker translation.","title":"The marker-insertion grammar simulates the original grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/ins-grammar-derives-marked-form","kind":"node","lean":"Cslib.Language.insGrammar_derives_markedForm","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#ins-grammar-derives-marked-form","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T'$ and let $T$ be a finite alphabet. For every word $u$ over $T \\sqcup T'$, the marker-insertion grammar of $g$ derives $u$ from the marked form of $u$, the sentential form that lists the letters of $u$ from $T'$ in order, each preceded by the marker nonterminal $D$, followed by one more $D$.","title":"Marker loops fill in the left letters"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/ins-valuation","kind":"node","lean":"Cslib.Language.insValuation","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#ins-valuation","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T'$ and let $T$ be an alphabet. This valuation assigns a language over $T \\sqcup T'$ to each nonterminal of $g$ and to the two new nonterminals $S$ and $D$ of the marker-insertion grammar of $g$: to the marker $D$ the words with no letter from $T'$; to the new start symbol $S$ the words whose letters from $T'$ form a word of the language of $g$; and to a nonterminal $A$ of\u2026","title":"Valuation for the marker-insertion grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/ins-grammar-language","kind":"node","lean":"Cslib.Language.insGrammar_language","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#ins-grammar-language","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $g$ be a context-free grammar over $T'$ and let $T$ be a finite alphabet. The marker-insertion grammar of $g$ generates exactly the words $u$ over $T \\sqcup T'$ whose letters from $T'$, read in order, form a word of the language of $g$.","title":"Language of the marker-insertion grammar"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/preimage-proj-right","kind":"node","lean":"Cslib.Language.IsContextFree.preimage_projRight","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#preimage-proj-right","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $T$ be a finite alphabet and let $L$ be a context-free language over an alphabet $T'$. Then the language of words $u$ over the disjoint union $T \\oplus T'$ whose letters from $T'$, read in order, form a word of $L$ is context-free.","title":"Context-free languages are closed under inserting letters"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/block","kind":"node","lean":"Cslib.Language.block","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#block","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"For a map $\\varphi$ from $T$ to words over $T'$ and a letter $a \\in T$, the block of $a$ is the word over the disjoint union $T \\sqcup T'$ that consists of $a$, as a letter of $T$, followed by the letters of $\\varphi(a)$, as letters of $T'$.","title":"Block of a letter"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/erase-right","kind":"node","lean":"Cslib.Language.eraseRight","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#erase-right","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Erasing right letters is the map from letters of the disjoint union $T \\sqcup T'$ to words over $T$ that sends a letter $a$ of $T$ to the one-letter word $a$ and every letter of $T'$ to the empty word.","title":"Erasing right letters"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/preimage-hom","kind":"node","lean":"Cslib.Language.IsContextFree.preimage_hom","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#preimage-hom","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Context-free languages are closed under inverse homomorphisms from a finite alphabet.","title":"Context-free languages are closed under inverse homomorphisms"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/preimage-word-morphism","kind":"node","lean":"Cslib.Language.IsContextFree.preimage_wordMorphism","parent":33901,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu/index.html#preimage-word-morphism","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/inverse-homomorphic-images-of-context-free-langu","status":"fully_proved","summary":"Let $T$ be a finite alphabet, $T'$ an alphabet, $\\sigma \\colon T^* \\to {T'}^*$ a morphism of free monoids and $L$ a context-free language over $T'$. Then the inverse image $\\sigma^{-1}(L) = \\{x \\in T^* \\mid \\sigma(x) \\in L\\}$ is a context-free language over $T$.","title":"Context-free languages are closed under inverse free-monoid morphisms"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets.json","id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","kind":"scope","navigation":"module","parent":33879,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","title":"Parikh images and semilinear sets"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/parikh-vector","kind":"node","lean":"Cslib.Language.parikhVector","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#parikh-vector","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The Parikh vector of a word: the function counting, for each letter, how many times it occurs in the word.","title":"Parikh vector of a word"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/parikh-image","kind":"node","lean":"Cslib.Language.parikhImage","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#parikh-image","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The Parikh image of a language: the set of Parikh vectors of its words.","title":"Parikh image of a language"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/parikh-vector-nil","kind":"node","lean":"Cslib.Language.parikhVector_nil","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#parikh-vector-nil","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The Parikh vector of the empty word is zero.","title":"Parikh vector of the empty word is zero"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/parikh-vector-append","kind":"node","lean":"Cslib.Language.parikhVector_append","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#parikh-vector-append","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"Parikh vectors add over concatenation.","title":"Parikh vectors add over concatenation"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/parikh-image-union","kind":"node","lean":"Cslib.Language.parikhImage_union","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#parikh-image-union","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The Parikh image of a union is the union of the Parikh images.","title":"Parikh image of a union of languages"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/parikh-image-empty","kind":"node","lean":"Cslib.Language.parikhImage_empty","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#parikh-image-empty","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The Parikh image of the empty language is empty.","title":"Parikh image of the empty language is empty"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/parikh-image-singleton","kind":"node","lean":"Cslib.Language.parikhImage_singleton","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#parikh-image-singleton","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The Parikh image of a singleton language is the singleton of the Parikh vector.","title":"Parikh image of a singleton language"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/lin-comb","kind":"node","lean":"Cslib.Language.linComb","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#lin-comb","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The linear combination of a list of period vectors with a coefficient list `c`: pair the two lists position by position and sum the scaled periods, and reject mismatched lengths by returning `0` (the length guard is checked up front, before the recursion, so a shorter coefficient list cannot silently truncate and a longer one cannot silently drop coefficients). Coefficients are indexed by *position* in `P`, not by\u2026","title":"Linear combination of period vectors"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/lin-comb-eq-aux-of-length-eq","kind":"node","lean":"Cslib.Language.linComb_eq_aux_of_length_eq","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#lin-comb-eq-aux-of-length-eq","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The unguarded recursion agrees with `linComb` on equal-length inputs.","title":"Unguarded linear combination agrees on equal lengths"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/lin-comb-eq-zero-of-length-lt","kind":"node","lean":"Cslib.Language.linComb_eq_zero_of_length_lt","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#lin-comb-eq-zero-of-length-lt","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"Mismatched lengths are rejected: a shorter coefficient list does not silently truncate.","title":"Linear combination with too few coefficients is zero"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/lin-comb-eq-zero-of-length-gt","kind":"node","lean":"Cslib.Language.linComb_eq_zero_of_length_gt","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#lin-comb-eq-zero-of-length-gt","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"Mismatched lengths are rejected: a longer coefficient list does not silently drop coefficients.","title":"Linear combination with too many coefficients is zero"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/lin-comb-dup","kind":"node","lean":"Cslib.Language.linComb_dup","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#lin-comb-dup","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"Duplicate periods get independent coefficients: with `P = [p, p]`, the coefficients `(1, 0)` give `p`.","title":"Duplicate periods get independent coefficients"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/is-linear","kind":"node","lean":"Cslib.Language.IsLinear","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#is-linear","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"A set of vectors is linear when it consists of a base vector plus arbitrary nonnegative combinations of finitely many period vectors, with one coefficient per position in the period list.","title":"Linear set of vectors"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/is-semilinear","kind":"node","lean":"Cslib.Language.IsSemilinear","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#is-semilinear","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"A set of vectors is semilinear when it is a finite union of linear sets.","title":"Semilinear set of vectors"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/is-linear-singleton","kind":"node","lean":"Cslib.Language.isLinear_singleton","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#is-linear-singleton","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"A singleton set of vectors is linear, with no periods.","title":"A singleton set of vectors is linear"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/is-semilinear-2","kind":"node","lean":"Cslib.Language.IsLinear.isSemilinear","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#is-semilinear-2","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"Linear sets are semilinear.","title":"Linear sets are semilinear"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/is-semilinear-empty","kind":"node","lean":"Cslib.Language.isSemilinear_empty","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#is-semilinear-empty","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"The empty set of vectors is semilinear.","title":"The empty set is semilinear"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/is-semilinear-singleton","kind":"node","lean":"Cslib.Language.isSemilinear_singleton","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#is-semilinear-singleton","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"A singleton set of vectors is semilinear.","title":"A singleton set of vectors is semilinear"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/union","kind":"node","lean":"Cslib.Language.IsSemilinear.union","parent":33915,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets/index.html#union","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/parikh-images-and-semilinear-sets","status":"fully_proved","summary":"Semilinear sets are closed under union.","title":"Semilinear sets are closed under union"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages.json","id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","kind":"scope","navigation":"module","parent":33879,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","title":"Pumping lemma for context-free languages"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/yields-h","kind":"node","lean":"Cslib.Language.YieldsH","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#yields-h","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar. For a sentential form $\\alpha$ (a list of terminals and nonterminals of $g$), a terminal word $w$ and a natural number $n$, say that $\\alpha$ yields $w$ at height $n$ when this follows from three rules: the empty form yields the empty word at every height $n$; if $\\alpha$ yields $w$ at height $n$, then for every terminal $a$ the form $a\\alpha$ yields $aw$ at height $n$; and if\u2026","title":"Height-bounded derivations"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/mono","kind":"node","lean":"Cslib.Language.YieldsH.mono","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#mono","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a sentential form $\\alpha$ yields the terminal word $w$ through a derivation forest of height at most $n$, and $n \\le m$, then $\\alpha$ yields $w$ through a derivation forest of height at most $m$.","title":"Raising the height bound of a yield"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/of-yields","kind":"node","lean":"Cslib.Language.YieldsH.of_yields","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#of-yields","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a sentential form $\\alpha$ yields the terminal word $w$, then there is a natural number $n$ such that $\\alpha$ yields $w$ through a derivation forest of height at most $n$.","title":"Every yield has a height bound"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/of-yields-h","kind":"node","lean":"Cslib.Language.Yields.of_yieldsH","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#of-yields-h","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a sentential form $\\alpha$ yields the terminal word $w$ through a derivation forest of height at most $n$, then $\\alpha$ yields $w$.","title":"Height-bounded yields are yields"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/append","kind":"node","lean":"Cslib.Language.YieldsH.append","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#append","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a sentential form $u$ yields the terminal word $w_1$ through a derivation forest of height at most $n$, and a sentential form $v$ yields the terminal word $w_2$ through one of height at most $m$, then the concatenation $uv$ yields $w_1 w_2$ through a derivation forest of height at most $\\max(n, m)$.","title":"Concatenating height-bounded yields"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/append-inv","kind":"node","lean":"Cslib.Language.YieldsH.append_inv","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#append-inv","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar and $u$, $v$ sentential forms of $g$. If the concatenation $uv$ yields the terminal word $w$ through a derivation forest of height at most $n$, then there are terminal words $w_1$, $w_2$ with $w = w_1 w_2$ such that $u$ yields $w_1$ and $v$ yields $w_2$, each through a derivation forest of height at most $n$.","title":"Splitting a height-bounded yield at a concatenation"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/length-le-of-yields-h","kind":"node","lean":"Cslib.Language.List.length_le_of_yieldsH","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#length-le-of-yields-h","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar and $R$ a natural number such that every production of $g$ has an output of length at most $R$. If a sentential form $\\alpha$ yields the terminal word $w$ through a derivation forest of height at most $n$, then $|w| \\le |\\alpha| \\cdot \\max(2, R)^n$.","title":"Length bound for height-bounded yields"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/exists-minimal-yields-h","kind":"node","lean":"Cslib.Language.exists_minimal_yieldsH","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#exists-minimal-yields-h","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar. If a sentential form $\\alpha$ yields the terminal word $w$, then there is a natural number $n$ such that $\\alpha$ yields $w$ through a derivation forest of height at most $n$, but through no derivation forest of height at most $m$ for any $m < n$.","title":"Minimal-height derivation forests exist"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/yields-context","kind":"node","lean":"Cslib.Language.YieldsContext","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#yields-context","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar, let $A$ and $B$ be nonterminals of $g$, and let $l$ and $r$ be words over the terminal alphabet. Then $(l, r)$ is a derivation context from $A$ to $B$ when, for every terminal word $w$ that the nonterminal $B$ yields in $g$, the nonterminal $A$ yields the word $l w r$ in $g$.","title":"Derivation contexts"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/of-rule","kind":"node","lean":"Cslib.Language.YieldsContext.of_rule","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#of-rule","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar and $A \\to \\alpha_1 B \\alpha_2$ a production of $g$, where $B$ is a nonterminal and $\\alpha_1$, $\\alpha_2$ are sentential forms. If $\\alpha_1$ yields the terminal word $l$ and $\\alpha_2$ yields the terminal word $r$, then for every terminal word $w$ yielded by $B$ the word $l w r$ is yielded by $A$; that is, $(l, r)$ is a derivation context from $A$ to $B$.","title":"A production gives derivation contexts"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/iterate","kind":"node","lean":"Cslib.Language.YieldsContext.iterate","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#iterate","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar, $A$ a nonterminal and $l$, $r$, $x$ terminal words. Suppose that for every terminal word $w$ yielded by $A$ the word $l w r$ is also yielded by $A$, and that $A$ yields $x$. Then for every natural number $i$ the nonterminal $A$ yields $l^i x r^i$, where $l^i$ and $r^i$ denote the concatenations of $i$ copies of $l$ and of $r$.","title":"Pumping through a self-context"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/pumping","kind":"node","lean":"Cslib.Language.IsContextFree.pumping","parent":33935,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages/index.html#pumping","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/pumping-lemma-for-context-free-languages","status":"fully_proved","summary":"For every context-free language $L$ over an alphabet $T$ there is a natural number $n > 0$ such that every word $w \\in L$ with $|w| \\ge n$ can be written as $w = uvxyz$ with words $u, v, x, y, z$ over $T$ such that $vy$ is nonempty, $|vxy| \\le n$, and $u v^i x y^i z \\in L$ for every natural number $i$, where $v^i$ and $y^i$ denote the concatenations of $i$ copies of $v$ and of $y$.","title":"Pumping lemma for context-free languages"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages.json","id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","kind":"scope","navigation":"module","parent":33879,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","title":"Non-closure of context-free languages"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/mem-kstar-singleton","kind":"node","lean":"Cslib.Language.mem_kstar_singleton","parent":33948,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#mem-kstar-singleton","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"For a letter $a$ of an alphabet, a word $w$ belongs to the Kleene star of the language $\\{a\\}$ whose only word is the one-letter word $a$ if and only if $w = a^k$ for some $k \\in \\mathbb{N}$.","title":"The Kleene star of a one-letter word"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/eq-pow-lang","kind":"node","lean":"Cslib.Language.eqPowLang","parent":33948,"reading_title":{"format":"inline-parts-v1","parts":[["text","The language "],["math","\\(a^n b^n\\)"]],"source":"The language $a^n b^n$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#eq-pow-lang","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"For letters $a$ and $b$ of an alphabet, this is the language of the words $a^n b^n$ for $n \\in \\mathbb{N}$: $n$ copies of $a$ followed by $n$ copies of $b$.","title":"The language $a^n b^n$"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/eq-pow-grammar","kind":"node","lean":"Cslib.Language.eqPowGrammar","parent":33948,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grammar "],["math","\\(S \\to aSb \\mid \\varepsilon\\)"]],"source":"The grammar $S \\to aSb \\mid \\varepsilon$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#eq-pow-grammar","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"For letters $a$ and $b$ of an alphabet with decidable equality, this is the context-free grammar with a single nonterminal $S$, which is its start symbol, and the two rules $S \\to aSb$ and $S \\to \\varepsilon$.","title":"The grammar $S \\to aSb \\mid \\varepsilon$"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/eq-pow-grammar-language","kind":"node","lean":"Cslib.Language.eqPowGrammar_language","parent":33948,"reading_title":{"format":"inline-parts-v1","parts":[["text","The grammar "],["math","\\(S \\to aSb \\mid \\varepsilon\\)"],["text"," generates "],["math","\\(a^n b^n\\)"]],"source":"The grammar $S \\to aSb \\mid \\varepsilon$ generates $a^n b^n$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#eq-pow-grammar-language","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"For letters $a$ and $b$ of an alphabet with decidable equality, the language of the context-free grammar with a single nonterminal $S$, its start symbol, and the two rules $S \\to aSb$ and $S \\to \\varepsilon$ is the language of the words $a^n b^n$ for $n \\in \\mathbb{N}$.","title":"The grammar $S \\to aSb \\mid \\varepsilon$ generates $a^n b^n$"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/eq-pow-lang-is-context-free","kind":"node","lean":"Cslib.Language.eqPowLang_isContextFree","parent":33948,"reading_title":{"format":"inline-parts-v1","parts":[["text","The language "],["math","\\(a^n b^n\\)"],["text"," is context-free"]],"source":"The language $a^n b^n$ is context-free"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#eq-pow-lang-is-context-free","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"For any alphabet and any letters $a$ and $b$ of it, the language of the words $a^n b^n$ for $n \\in \\mathbb{N}$ is context-free.","title":"The language $a^n b^n$ is context-free"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/anbncn-lang","kind":"node","lean":"Cslib.Language.anbncnLang","parent":33948,"reading_title":{"format":"inline-parts-v1","parts":[["text","The language "],["math","\\(a^n b^n c^n\\)"]],"source":"The language $a^n b^n c^n$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#anbncn-lang","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"Over the three-letter alphabet $\\{a, b, c\\}$ (the letters $0, 1, 2$), this is the language of the words $a^n b^n c^n$ for $n \\in \\mathbb{N}$: $n$ copies of $a$, then $n$ copies of $b$, then $n$ copies of $c$.","title":"The language $a^n b^n c^n$"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/anbncn-lang-not-is-context-free","kind":"node","lean":"Cslib.Language.anbncnLang_not_isContextFree","parent":33948,"reading_title":{"format":"inline-parts-v1","parts":[["text","The language "],["math","\\(a^n b^n c^n\\)"],["text"," is not context-free"]],"source":"The language $a^n b^n c^n$ is not context-free"},"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#anbncn-lang-not-is-context-free","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"Over the three-letter alphabet $\\{a, b, c\\}$ (the letters $0, 1, 2$), the language of the words $a^n b^n c^n$ for $n \\in \\mathbb{N}$ is not context-free.","title":"The language $a^n b^n c^n$ is not context-free"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/exists-is-context-free-inf-not-is-contex","kind":"node","lean":"Cslib.Language.exists_isContextFree_inf_not_isContextFree","parent":33948,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#exists-is-context-free-inf-not-is-contex","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"There are context-free languages $L$ and $M$ over the three-letter alphabet $\\{0, 1, 2\\}$ whose intersection $L \\cap M$ is not context-free.","title":"Context-free languages are not closed under intersection"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/exists-is-context-free-compl-not-is-cont","kind":"node","lean":"Cslib.Language.exists_isContextFree_compl_not_isContextFree","parent":33948,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#exists-is-context-free-compl-not-is-cont","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"There is a context-free language $L$ over the three-letter alphabet $\\{0, 1, 2\\}$ whose complement is not context-free.","title":"Context-free languages are not closed under complement"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/not-forall-is-context-free-inf","kind":"node","lean":"Cslib.Language.not_forall_isContextFree_inf","parent":33948,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#not-forall-is-context-free-inf","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"Over the three-letter alphabet $\\{0, 1, 2\\}$, it is not the case that for all context-free languages $L$ and $M$ the intersection $L \\cap M$ is context-free.","title":"Intersections of context-free languages need not be context-free"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/not-forall-is-context-free-compl","kind":"node","lean":"Cslib.Language.not_forall_isContextFree_compl","parent":33948,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages/index.html#not-forall-is-context-free-compl","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/non-closure-of-context-free-languages","status":"fully_proved","summary":"It is not the case that for every context-free language $L$ over the three-letter alphabet $\\{0, 1, 2\\}$, the complement of $L$ is context-free.","title":"Complements of context-free languages need not be context-free"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma.json","id":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma","kind":"scope","navigation":"module","parent":33879,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/index.html","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma","status":"fully_proved","title":"Ogden's lemma"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/marks","kind":"node","lean":"Cslib.Language.marks","parent":33960,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/index.html#marks","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma","status":"fully_proved","summary":"A marked word over an alphabet $T$ is a word whose letters are pairs $(a, b)$ with $a \\in T$ and $b$ a Boolean; the letter $(a, b)$ is marked when $b$ is true. The number of marked letters of a marked word $w$ is the number of letters of $w$ that are marked, counted with multiplicity.","title":"Number of marked letters"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/ogden-of-bound","kind":"node","lean":"Cslib.Language.ogden_of_bound","parent":33960,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/index.html#ogden-of-bound","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma","status":"fully_proved","summary":"Let $g$ be a context-free grammar over an alphabet $T$ and let $R$, $K$, $n$ be natural numbers with $R \\ge 2$ such that the right-hand side of every production of $g$ has length at most $R$, some list of at most $K$ nonterminals contains the left-hand side of every production of $g$, and $R^{K+1} \\le n$. A marked word is a word whose letters are pairs $(a, b)$ with $a \\in T$ and $b$ a Boolean, the letter being\u2026","title":"Ogden's lemma for a grammar, with an explicit bound"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/pumping-of-bound","kind":"node","lean":"Cslib.Language.pumping_of_bound","parent":33960,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/index.html#pumping-of-bound","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma","status":"fully_proved","summary":"Let $g$ be a context-free grammar over an alphabet $T$ and let $R$, $K$, $n$ be natural numbers with $R \\ge 2$ such that the right-hand side of every production of $g$ has length at most $R$, some list of at most $K$ nonterminals contains the left-hand side of every production of $g$, and $R^{K+1} \\le n$. Then every word $w$ in the language of $g$ of length at least $n$ can be written as $w = uvxyz$ with words\u2026","title":"The pumping lemma for a grammar, with an explicit bound"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/ogden","kind":"node","lean":"Cslib.Language.IsContextFree.ogden","parent":33960,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma/index.html#ogden","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/ogden-s-lemma","status":"fully_proved","summary":"Let $L$ be a context-free language over an alphabet $T$. A marked word is a word whose letters are pairs $(a, b)$ with $a \\in T$ and $b$ a Boolean, the letter being marked when $b$ is true; its underlying word is the word of first components. Then there is a natural number $n > 0$ such that every marked word $w$ whose underlying word lies in $L$ and which has at least $n$ marked letters can be written as\u2026","title":"Ogden's lemma"},{"data_url":"scopes/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages.json","id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","kind":"scope","navigation":"module","parent":33879,"result":false,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","title":"Quotients of context-free languages"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/right-quotient-lang","kind":"node","lean":"Cslib.Language.rightQuotientLang","parent":33965,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html#right-quotient-lang","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","summary":"For languages $L$ and $R$ over an alphabet $T$, the right quotient of $L$ by $R$ is the language of all words $x$ over $T$ such that the concatenation $xy$ lies in $L$ for some word $y \\in R$.","title":"Right quotient of a language by a language"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/left-words","kind":"node","lean":"Cslib.Language.RightQuotientLang.leftWords","parent":33965,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html#left-words","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","summary":"For an alphabet $T$, write $T \\sqcup T$ for the disjoint union of a left and a right copy of $T$. This is the language over $T \\sqcup T$ of the words obtained from a word $x$ over $T$ by replacing each letter with its left copy.","title":"Words over the left copy of a doubled alphabet"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/right-words","kind":"node","lean":"Cslib.Language.RightQuotientLang.rightWords","parent":33965,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html#right-words","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","summary":"For a language $R$ over an alphabet $T$, write $T \\sqcup T$ for the disjoint union of a left and a right copy of $T$. This is the language over $T \\sqcup T$ of the words obtained from a word $y \\in R$ by replacing each letter with its right copy.","title":"Words of a language marked by the right copy"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/left-dfa","kind":"node","lean":"Cslib.Language.RightQuotientLang.leftDFA","parent":33965,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html#left-dfa","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","summary":"For an alphabet $T$, write $T \\sqcup T$ for the disjoint union of a left and a right copy of $T$. This is the deterministic automaton over $T \\sqcup T$ with the two states true and false, whose start state and only accepting state are true, and which moves from state $b$ on a letter $c$ to true exactly when $b$ is true and $c$ lies in the left copy, and to false otherwise.","title":"Automaton for words over the left copy"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/right-dfa","kind":"node","lean":"Cslib.Language.RightQuotientLang.rightDFA","parent":33965,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html#right-dfa","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","summary":"Let $D$ be a deterministic automaton over an alphabet $T$, and write $T \\sqcup T$ for the disjoint union of a left and a right copy of $T$. This is the deterministic automaton over $T \\sqcup T$ whose states are the states of $D$ together with one extra dead state $\\bot$. From a state $p$ of $D$, on the right copy of a letter $a$ it moves to the state that $D$ reaches from $p$ on $a$; on a letter of the left copy,\u2026","title":"Automaton running a deterministic automaton on the right copy"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/unmark","kind":"node","lean":"Cslib.Language.RightQuotientLang.unmark","parent":33965,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html#unmark","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","summary":"For an alphabet $T$, write $T \\sqcup T$ for the disjoint union of a left and a right copy of $T$. This map sends the left copy and the right copy of a letter $a \\in T$ to the one-letter word $a$ over $T$.","title":"Forgetting the copy of a letter"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/right-quotient-lang-2","kind":"node","lean":"Cslib.Language.IsContextFree.rightQuotientLang","parent":33965,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html#right-quotient-lang-2","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","summary":"The right quotient of a context-free language by a regular language is context-free, over a finite alphabet.","title":"The right quotient of a context-free language by a regular language is context-free"},{"id":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/left-quotient-lang","kind":"node","lean":"Cslib.Language.IsContextFree.leftQuotientLang","parent":33965,"result":true,"route":"../roadmap/TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages/index.html#left-quotient-lang","scope":"TheoryOfComputation/context-free-languages/closure-and-pumping/quotients-of-context-free-languages","status":"fully_proved","summary":"The left quotient of a context-free language by a regular language is context-free, over a finite alphabet.","title":"The left quotient of a context-free language by a regular language is context-free"},{"data_url":"scopes/TheoryOfComputation/decision-procedures.json","id":"TheoryOfComputation/decision-procedures","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/index.html","scope":"TheoryOfComputation/decision-procedures","status":"fully_proved","summary":"Algorithms deciding questions about finite automata, regular expressions and context-free grammars given as data: emptiness, membership, universality, equivalence, and computable grammar reduction, together with the encodings of automata and grammars they operate on; and satisfiability of string constraints (word equations with regular membership and length bounds).","title":"Decision procedures for automata, grammars and strings"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars.json","id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","kind":"scope","navigation":"module","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","title":"Encodings of context-free grammars"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/map-symbol-map-symbol","kind":"node","lean":"Cslib.Language.mapSymbol_mapSymbol","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#map-symbol-map-symbol","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"For maps $f : N_1 \\to N_2$ and $b : N_2 \\to N_3$ and a grammar symbol $s$ with terminals in $T$ and nonterminals in $N_1$, renaming the nonterminals of $s$ by $f$ and then by $b$ gives the same symbol as renaming them by $b \\circ f$.","title":"Renaming the nonterminals of a symbol twice"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/language-eq-of-maps","kind":"node","lean":"Cslib.Language.language_eq_of_maps","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#language-eq-of-maps","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ and $G$ be context-free grammars over the same terminals. Suppose there are maps $f$ from the nonterminals of $g$ to those of $G$ and $b$ from the nonterminals of $G$ to those of $g$ such that renaming the nonterminals of any production of $g$ by $f$ gives a production of $G$, renaming the nonterminals of any production of $G$ by $b$ gives a production of $g$, $f$ sends the start nonterminal of $g$ to that\u2026","title":"Grammars with renamings in both directions generate the same language"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/encoded-cfg","kind":"node","lean":"Cslib.Language.EncodedCFG","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#encoded-cfg","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"An encoded context-free grammar over a terminal type $T$ consists of a start nonterminal, which is a natural number, and a list of productions. Each production is a pair of a left-hand side, a natural number, and a right-hand side, a list of elements of the disjoint union $\\mathbb{N} \\sqcup T$, in which a number $i$ from the left summand stands for nonterminal $i$ and an element $a$ of $T$ for terminal $a$.","title":"Encoded context-free grammars"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/decode-symbol","kind":"node","lean":"Cslib.Language.EncodedCFG.decodeSymbol","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#decode-symbol","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"Decoding sends an element of the disjoint union $\\mathbb{N} \\sqcup T$ to a grammar symbol with terminals $T$ and nonterminals $\\mathbb{N}$: a natural number $i$ from the left summand goes to the nonterminal $i$, and an element $a$ of $T$ from the right summand goes to the terminal $a$.","title":"Decoding a grammar symbol"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/encode-symbol","kind":"node","lean":"Cslib.Language.EncodedCFG.encodeSymbol","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#encode-symbol","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"Encoding sends a grammar symbol with terminals $T$ and nonterminals $\\mathbb{N}$ to an element of the disjoint union $\\mathbb{N} \\sqcup T$: the terminal $a$ goes to $a$ in the right summand $T$, and the nonterminal $i$ goes to $i$ in the left summand $\\mathbb{N}$.","title":"Encoding a grammar symbol"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/decode-rule","kind":"node","lean":"Cslib.Language.EncodedCFG.decodeRule","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#decode-rule","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"Decoding a pair $(A, s)$ of a natural number $A$ and a list $s$ of elements of the disjoint union $\\mathbb{N} \\sqcup T$ gives the production, with terminals $T$ and nonterminals $\\mathbb{N}$, whose left-hand side is the nonterminal $A$ and whose right-hand side is $s$ with each entry decoded as a symbol: a number $i$ from the left summand becomes the nonterminal $i$ and an element $a$ of $T$ becomes the terminal $a$.","title":"Decoding a production"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/encode-rule","kind":"node","lean":"Cslib.Language.EncodedCFG.encodeRule","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#encode-rule","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"Encoding a production with terminals $T$ and nonterminals $\\mathbb{N}$ gives the pair $(A, s)$ where $A \\in \\mathbb{N}$ is its left-hand side and $s$ is its right-hand side with each symbol encoded as an element of the disjoint union $\\mathbb{N} \\sqcup T$: the terminal $a$ becomes $a$ in the right summand and the nonterminal $i$ becomes $i$ in the left summand.","title":"Encoding a production"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/to-cfg","kind":"node","lean":"Cslib.Language.EncodedCFG.toCFG","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#to-cfg","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$ over $T$, the context-free grammar over $T$ whose nonterminals are the natural numbers, whose initial nonterminal is the start nonterminal of $g$, and whose productions are the decodings of the encoded productions of $g$: an encoded production $(i, u)$ becomes the production rewriting the nonterminal $i$ to the word obtained from $u$ by reading each entry $\\mathrm{inl}\\, j$\u2026","title":"Grammar denoted by an encoding"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/language","kind":"node","lean":"Cslib.Language.EncodedCFG.language","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#language","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"The language of an encoded context-free grammar $g$ over $T$ is the language generated by the context-free grammar that $g$ denotes, whose nonterminals are the natural numbers, whose initial nonterminal is the start nonterminal of $g$ and whose productions are the decoded productions of $g$.","title":"Language of an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/language-is-context-free","kind":"node","lean":"Cslib.Language.EncodedCFG.language_isContextFree","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#language-is-context-free","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"The language of an encoded grammar is context-free.","title":"Encoded grammars generate context-free languages"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/nt-of","kind":"node","lean":"Cslib.Language.EncodedCFG.ntOf","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#nt-of","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"The partial map from grammar symbols with terminals in $T$ and nonterminals in $N$ to $N$ that sends a nonterminal $A$ to $A$ and has no value on terminals.","title":"Nonterminal of a symbol"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/used-nt","kind":"node","lean":"Cslib.Language.EncodedCFG.usedNT","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#used-nt","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"For a context-free grammar $G$, the list consisting of the initial nonterminal of $G$ followed, for each production of $G$ in turn (in some fixed order of its finite set of productions), by the left-hand side of the production and then the nonterminals of its right-hand side in order. Nonterminals may occur more than once.","title":"List of nonterminals mentioned by a grammar"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/nt-index","kind":"node","lean":"Cslib.Language.EncodedCFG.ntIndex","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#nt-index","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"For a context-free grammar $G$, the number of a nonterminal $A$ of $G$ is the position, counting from $0$, of the first occurrence of $A$ in the list of nonterminals that $G$ mentions (its initial nonterminal first); if $A$ does not occur in that list, its number is the length of the list.","title":"Numbering of nonterminals"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/nt-of-index","kind":"node","lean":"Cslib.Language.EncodedCFG.ntOfIndex","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#nt-of-index","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"For a context-free grammar $G$ and a natural number $n$, the nonterminal of $G$ with number $n$ is the entry at position $n$, counting from $0$, of the list of nonterminals that $G$ mentions (its initial nonterminal first); if the list has no entry at position $n$, it is the initial nonterminal of $G$.","title":"Nonterminal with a given number"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/map-rule-nt-of-index-nt-index","kind":"node","lean":"Cslib.Language.EncodedCFG.mapRule_ntOfIndex_ntIndex","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#map-rule-nt-of-index-nt-index","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"Let $G$ be a context-free grammar over $T$ and $r$ a production of $G$. Replacing each nonterminal $A$ of $r$ by its number, the position, counting from $0$, of the first occurrence of $A$ in the list of nonterminals that $G$ mentions, and then replacing each number $n$ by the nonterminal of $G$ with number $n$ (the entry at position $n$ of that list, or the initial nonterminal of $G$ if the list has no such entry)\u2026","title":"Numbering the nonterminals of a production and back recovers it"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/of-cfg","kind":"node","lean":"Cslib.Language.EncodedCFG.ofCFG","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#of-cfg","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"For a context-free grammar $G$ over $T$, the encoded grammar whose start nonterminal is the number of the initial nonterminal of $G$ and whose productions are the productions of $G$, listed in some fixed order, with every nonterminal $A$ replaced by its number, the position, counting from $0$, of the first occurrence of $A$ in the list of nonterminals that $G$ mentions (its initial nonterminal first).","title":"Encoding of a context-free grammar"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/language-of-cfg","kind":"node","lean":"Cslib.Language.EncodedCFG.language_ofCFG","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#language-of-cfg","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"The encoding of a grammar generates the same language.","title":"The encoding of a grammar generates the same language"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/is-context-free-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.isContextFree_iff","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#is-context-free-iff","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"A language is context-free exactly when an encoded grammar generates it.","title":"A language is context-free exactly when an encoded grammar generates it"},{"id":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/equiv-prod","kind":"node","lean":"Cslib.Language.EncodedCFG.equivProd","parent":33975,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars/index.html#equiv-prod","scope":"TheoryOfComputation/decision-procedures/encodings-of-context-free-grammars","status":"fully_proved","summary":"Sending an encoded grammar over $T$ to the pair of its start nonterminal and its list of productions is a bijection from encoded grammars over $T$ onto pairs $(s, R)$ of a natural number $s$ and a list $R$ of pairs of a natural number and a list of elements of the disjoint union $\\mathbb{N} \\sqcup T$.","title":"Encoded grammars as pairs"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/primitive-recursive-list-operations.json","id":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations","kind":"scope","navigation":"module","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/index.html","scope":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations","status":"fully_proved","title":"Primitive recursive list operations"},{"id":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/mem-b","kind":"node","lean":"Cslib.Decidability.memB","parent":33995,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/index.html#mem-b","scope":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations","status":"fully_proved","summary":"For a list $G$ of natural numbers and a natural number $j$, the Boolean value that is true exactly when $j$ is an entry of $G$.","title":"Boolean membership test for lists of natural numbers"},{"id":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/dedup-b","kind":"node","lean":"Cslib.Decidability.dedupB","parent":33995,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/index.html#dedup-b","scope":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations","status":"fully_proved","summary":"For a list $l$ of natural numbers, the list obtained from $l$ by deleting every entry that occurs again later in $l$: each element of $l$ is kept once, at its last occurrence, and the kept entries stay in their order in $l$.","title":"Duplicate removal for lists of natural numbers"},{"id":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/iterate-closed","kind":"node","lean":"Cslib.Decidability.iterate_closed","parent":33995,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/index.html#iterate-closed","scope":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations","status":"fully_proved","summary":"Let $f$ map lists of natural numbers to lists of natural numbers and be monotone for list inclusion: $f(G) \\subseteq f(G')$ whenever $G \\subseteq G'$, where $G \\subseteq G'$ means that every entry of $G$ is an entry of $G'$. If $B$ is a list with $f(G) \\subseteq B$ for every list $G$, and $n$ is the length of $B$, then $f(f^n([\\,])) \\subseteq f^n([\\,])$, where $f^n$ is the $n$-fold iterate of $f$ and $[\\,]$ is the\u2026","title":"Bounded monotone rounds on lists close off"},{"id":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/computable-pred-of-primrec-bool","kind":"node","lean":"Cslib.Decidability.computablePred_of_primrec_bool","parent":33995,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/primitive-recursive-list-operations/index.html#computable-pred-of-primrec-bool","scope":"TheoryOfComputation/decision-procedures/primitive-recursive-list-operations","status":"fully_proved","summary":"Let $\\beta$ be a primcodable type, $p$ a predicate on $\\beta$, and $f : \\beta \\to \\mathrm{Bool}$ a primitive recursive function such that $f(b) = \\mathrm{true}$ exactly when $p(b)$ holds, for every $b \\in \\beta$. Then $p$ is a computable predicate.","title":"Predicates with a primitive recursive decision procedure are computable"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars.json","id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","kind":"scope","navigation":"module","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","title":"Emptiness of context-free grammars"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/sym-ok","kind":"node","lean":"Cslib.Language.EncodedCFG.symOk","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#sym-ok","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Given a list $G$ of natural numbers and a symbol $s$ of an encoded grammar over an alphabet $T$, which is either a nonterminal $i \\in \\mathbb{N}$ or a terminal in $T$, $s$ is acceptable when it is a terminal or a nonterminal that occurs in $G$. The test returns a Boolean.","title":"Acceptable symbols"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/rule-ok","kind":"node","lean":"Cslib.Language.EncodedCFG.ruleOk","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#rule-ok","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Given a list $G$ of nonterminals (natural numbers), the test accepts an encoded production $(A, \\rho)$ when every symbol of $\\rho$ is acceptable: a terminal always is, and a nonterminal is when it belongs to $G$. So the production is accepted exactly when every nonterminal occurring in $\\rho$ belongs to $G$.","title":"Acceptable productions"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/gen-step","kind":"node","lean":"Cslib.Language.EncodedCFG.genStep","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#gen-step","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let a list of productions be given, each a pair of a nonterminal $A \\in \\mathbb{N}$ and a right-hand side made of nonterminals and terminals, together with a list $G$ of nonterminals. One round returns, in order, the left-hand sides of those productions in whose right-hand side every nonterminal belongs to $G$.","title":"One round of the generating-nonterminal computation"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/generating","kind":"node","lean":"Cslib.Language.EncodedCFG.generating","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#generating","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For an encoded grammar $g$, start from the empty list of nonterminals and, in each round, replace the current list $G$ by the list of left-hand sides of those productions of $g$ whose right-hand sides contain no nonterminal outside $G$. The generating nonterminals of $g$ are the list obtained after as many rounds as $g$ has productions.","title":"Generating nonterminals"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/is-empty-b","kind":"node","lean":"Cslib.Language.EncodedCFG.isEmptyB","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#is-empty-b","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For an encoded grammar $g$, the test returns true exactly when the start nonterminal of $g$ is not among the generating nonterminals of $g$.","title":"The emptiness test for context-free grammars"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/gen-step-generating-subset","kind":"node","lean":"Cslib.Language.EncodedCFG.genStep_generating_subset","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#gen-step-generating-subset","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For an encoded grammar $g$, one more round of the generating-nonterminal computation, applied to the productions of $g$ and the generating nonterminals of $g$, yields only generating nonterminals of $g$: every left-hand side of a production of $g$ whose right-hand side has all its nonterminals among the generating nonterminals of $g$ is itself a generating nonterminal of $g$.","title":"The generating nonterminals are closed under one more round"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/exists-derives-of-forall","kind":"node","lean":"Cslib.Language.EncodedCFG.exists_derives_of_forall","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#exists-derives-of-forall","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"In a context-free grammar $G$, if every nonterminal occurring in a sentential form $u$ derives some terminal word, then $u$ derives some terminal word.","title":"Sentential forms of productive nonterminals derive terminal words"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/exists-derives-of-mem-iterate","kind":"node","lean":"Cslib.Language.EncodedCFG.exists_derives_of_mem_iterate","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#exists-derives-of-mem-iterate","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar. A round sends a list $H$ of nonterminals to the left-hand sides of those productions of $g$ in whose right-hand side every nonterminal belongs to $H$. For every $k \\ge 0$, every nonterminal obtained after $k$ rounds starting from the empty list derives some terminal word in the context-free grammar that $g$ denotes.","title":"Soundness of the productive-nonterminal iteration"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/eval-form-eq-zero-of-mem","kind":"node","lean":"Cslib.Language.EncodedCFG.evalForm_eq_zero_of_mem","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#eval-form-eq-zero-of-mem","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $T$ be an alphabet and $N$ a set of nonterminals, and let $V$ assign a language over $T$ to each nonterminal. A sentential form $X_1 \\cdots X_m$ over $T$ and $N$ denotes under $V$ the concatenation $L_1 \\cdots L_m$, where $L_t = V(X_t)$ if $X_t$ is a nonterminal and $L_t = \\{X_t\\}$ if $X_t$ is a terminal; the empty form denotes $\\{\\varepsilon\\}$. Let $A \\in N$ with $V(A) = \\emptyset$. Then every sentential form\u2026","title":"A sentential form containing an empty nonterminal denotes the empty language"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/start-mem-generating-of-mem-language","kind":"node","lean":"Cslib.Language.EncodedCFG.start_mem_generating_of_mem_language","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#start-mem-generating-of-mem-language","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"If an encoded grammar $g$ generates some word $w$, then the start nonterminal of $g$ is among the generating nonterminals of $g$ computed by the emptiness procedure.","title":"Completeness of the generating-nonterminal computation"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/is-empty-b-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.isEmptyB_iff","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#is-empty-b-iff","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$, the emptiness test returns true on $g$ if and only if the language of $g$ is empty.","title":"Correctness of the emptiness test"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/computable-pred-language-eq-zero","kind":"node","lean":"Cslib.Language.EncodedCFG.computablePred_language_eq_zero","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#computable-pred-language-eq-zero","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $T$ be an alphabet with decidable equality and a primitive recursive encoding into the natural numbers. The predicate on encoded context-free grammars $g$ over $T$ that holds exactly when the language of $g$ is empty is a computable predicate.","title":"Emptiness of context-free grammars is computable"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/tm2-decidable-language-eq-zero","kind":"node","lean":"Cslib.Language.EncodedCFG.tm2Decidable_language_eq_zero","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#tm2-decidable-language-eq-zero","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For every alphabet $T$ with decidable equality and a primitive recursive encoding, the property of an encoded context-free grammar $g$ over $T$ that the language of $g$ is empty is decidable by a multi-stack machine. This is Sipser, Theorem 4.8: the problem $E_{\\mathrm{CFG}}$ is decidable.","title":"Emptiness of context-free grammars is decidable"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/factor","kind":"node","lean":"Cslib.Language.EncodedCFG.factor","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#factor","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For a word $w = w_0 w_1 \\cdots w_{n-1}$ and natural numbers $i$ and $j$, the factor of $w$ from position $i$ to position $j$ is obtained by dropping the first $i$ letters of $w$ and keeping at most the first $j - i$ letters of the rest, where $j - i$ is $0$ when $j \\le i$. For $i \\le j \\le n$ it is $w_i w_{i+1} \\cdots w_{j-1}$.","title":"Factor of a word"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/factor-eq-append","kind":"node","lean":"Cslib.Language.EncodedCFG.factor_eq_append","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#factor-eq-append","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $w$ be a word and $i \\le j \\le |w|$, and write $w[i, j)$ for the factor of $w$ from position $i$ to position $j$. If $w[i, j) = uv$, then $i + |u| \\le j$, $w[i, i + |u|) = u$ and $w[i + |u|, j) = v$.","title":"Splitting a factor"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/sym-fits","kind":"node","lean":"Cslib.Language.EncodedCFG.symFits","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#sym-fits","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Write $\\langle s, t \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ given by $\\langle s, t \\rangle = t^2 + s$ if $s < t$ and $\\langle s, t \\rangle = s^2 + s + t$ otherwise. Given a word $w$, a list $R$ of items, a list $S$ of positions and an encoded symbol $X$, the test at a position $i$ succeeds when either $X$ is a terminal $a$, $i + 1 \\in S$ and the letter of $w$ at position $i$\u2026","title":"Fitting a symbol at a position"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/starts","kind":"node","lean":"Cslib.Language.EncodedCFG.starts","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#starts","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Write $\\langle s, t \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ given by $\\langle s, t \\rangle = t^2 + s$ if $s < t$ and $\\langle s, t \\rangle = s^2 + s + t$ otherwise. Given a word $w$, a list $R$ of items (natural numbers), a position $j$ and an encoded right-hand side $X_1 \\cdots X_m$, a position $i$ is a start when there are positions $i = p_0, p_1, \\ldots, p_m = j$ with\u2026","title":"Start positions of a right-hand side"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/mem-step","kind":"node","lean":"Cslib.Language.EncodedCFG.memStep","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#mem-step","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Write $\\langle s, t \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ given by $\\langle s, t \\rangle = t^2 + s$ if $s < t$ and $\\langle s, t \\rangle = s^2 + s + t$ otherwise. Given an encoded grammar $g$, a word $w$ and a list $R$ of items found so far, the round lists, for every production $A \\to \\rho$ of $g$, every position $j \\le |w|$ and every start $i$ of $\\rho$ for the factor of\u2026","title":"One round of the membership procedure"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/items","kind":"node","lean":"Cslib.Language.EncodedCFG.items","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#items","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For an encoded grammar $g$ and a word $w$, the candidate items are the triples $(A, i, j)$, encoded as natural numbers by pairing, where $A$ is the left-hand side of a production of $g$ and $i, j \\le |w|$; each triple is listed once for every production whose left-hand side is $A$.","title":"Candidate items for membership"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/found","kind":"node","lean":"Cslib.Language.EncodedCFG.found","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#found","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar and $w = w_0 \\cdots w_{n-1}$ a word. An item is a triple $(A, i, j)$ of natural numbers, coded as a natural number by nested pairing. A round sends a list $R$ of items to the items $(A, i, j)$ such that $A \\to X_1 \\cdots X_m$ is a production of $g$, $j \\le n$, and there are positions $i = k_0, k_1, \\ldots, k_m = j$, each at most $n$, such that for every $t$ with\u2026","title":"Items found by the membership procedure"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/mem-language-b","kind":"node","lean":"Cslib.Language.EncodedCFG.memLanguageB","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#mem-language-b","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For an encoded grammar $g$ with start nonterminal $S$ and a word $w$, the test returns true exactly when the item $(S, 0, |w|)$ is among the items found for $g$ and $w$.","title":"The membership test for context-free grammars"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/sound-item","kind":"node","lean":"Cslib.Language.EncodedCFG.SoundItem","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#sound-item","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Write $\\langle s, t \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ given by $\\langle s, t \\rangle = t^2 + s$ if $s < t$ and $\\langle s, t \\rangle = s^2 + s + t$ otherwise. Given an encoded grammar $g$ and a word $w$, a natural number $x$ is a sound item when $x = \\langle A, \\langle i, j \\rangle \\rangle$ for some $A$, $i$, $j$ with $i \\le j \\le |w|$ such that the nonterminal $A$\u2026","title":"Sound items"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/mem-starts-of-yields","kind":"node","lean":"Cslib.Language.EncodedCFG.mem_starts_of_yields","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#mem-starts-of-yields","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over an alphabet $T$ with decidable equality, let $w$ be a word over $T$, and let $\\rho$ be an encoded right-hand side whose decoding yields a word $u$ in the grammar that $g$ denotes. If $i \\le j \\le |w|$ and $u$ is the factor of $w$ from position $i$ to position $j$, then $i$ is a start position of $\\rho$ at $j$ for the word $w$ and the list of items found by the\u2026","title":"Completeness of the start positions"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/mem-language-b-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.memLanguageB_iff","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#mem-language-b-iff","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$ and every word $w$, the membership test returns true on $g$ and $w$ if and only if $w$ belongs to the language of $g$.","title":"Correctness of the membership test"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/tm2-decidable-mem-language","kind":"node","lean":"Cslib.Language.EncodedCFG.tm2Decidable_mem_language","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#tm2-decidable-mem-language","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For every alphabet $T$ with decidable equality and a primitive recursive encoding, the property of a pair $(g, w)$, where $g$ is an encoded context-free grammar over $T$ and $w$ is a word over $T$, that $w$ is in the language of $g$ is decidable by a multi-stack machine. This is Sipser, Theorem 4.7: the problem $A_{\\mathrm{CFG}}$ is decidable.","title":"Membership for context-free grammars is decidable"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/len-bound","kind":"node","lean":"Cslib.Language.EncodedCFG.lenBound","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#len-bound","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For an encoded grammar $g$, the length bound is the maximum of $2$ and the lengths of the right-hand sides of the productions of $g$.","title":"Right-hand side length bound"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/pump-bound","kind":"node","lean":"Cslib.Language.EncodedCFG.pumpBound","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#pump-bound","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For an encoded grammar $g$, let $L$ be the length bound of $g$, the larger of $2$ and the largest length of a right-hand side of a production of $g$, and let $K$ be the number of distinct left-hand sides of the productions of $g$. The pumping bound of $g$ is $L^{K+1}$.","title":"Pumping bound of an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/words-over","kind":"node","lean":"Cslib.Language.EncodedCFG.wordsOver","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#words-over","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For a list $A$ of letters and a natural number $n$, the list obtained from the list whose only entry is the empty word by $n$ rounds, each of which replaces every word $w$, in order, by the words $a w$ for the letters $a$ of $A$, in order. Its entries are exactly the words of length $n$ all of whose letters occur in $A$.","title":"The words of a given length over a list of letters"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/terminals","kind":"node","lean":"Cslib.Language.EncodedCFG.terminals","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#terminals","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$ over an alphabet $T$ with decidable equality, the list of the terminals that occur in the right-hand sides of the productions of $g$, each listed once.","title":"The terminals of an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/infinite-b","kind":"node","lean":"Cslib.Language.EncodedCFG.infiniteB","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#infinite-b","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded grammar and $N$ its pumping bound. The test returns true exactly when some word $w$ whose letters all occur as terminals in the productions of $g$, with $N \\le |w| < 2N$, is accepted by the membership test for $g$.","title":"The infiniteness test"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/mem-terminals-of-mem-language","kind":"node","lean":"Cslib.Language.EncodedCFG.mem_terminals_of_mem_language","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#mem-terminals-of-mem-language","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"If a word $w$ is generated by an encoded grammar $g$, then every letter of $w$ is among the terminals of $g$, that is, it occurs in the right-hand side of some production of $g$.","title":"Letters of generated words occur in the productions"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/pumping-pump-bound","kind":"node","lean":"Cslib.Language.EncodedCFG.pumping_pumpBound","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#pumping-pump-bound","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded grammar with pumping bound $N$. Every word $w$ generated by $g$ with $|w| \\ge N$ can be written as $w = uvxyz$ with $vy$ nonempty and $|vxy| \\le N$, such that $u v^i x y^i z$ is generated by $g$ for every natural number $i$, including $i = 0$.","title":"Pumping lemma for encoded grammars"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/infinite-of-mem-of-le","kind":"node","lean":"Cslib.Language.EncodedCFG.infinite_of_mem_of_le","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#infinite-of-mem-of-le","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"If the language of an encoded grammar $g$ contains a word whose length is at least the pumping bound of $g$, then the language of $g$ is infinite.","title":"Long words make a context-free language infinite"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/exists-mem-le-lt","kind":"node","lean":"Cslib.Language.EncodedCFG.exists_mem_le_lt","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#exists-mem-le-lt","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar, let $L$ be the largest of $2$ and the lengths of the right-hand sides of the productions of $g$, let $K$ be the number of distinct left-hand sides of these productions, and let $N = L^{K+1}$ be the pumping bound of $g$. For every word $w$ in the language of $g$ with $|w| \\ge N$, there is a word $v$ in the language of $g$ with $N \\le |v| < 2N$.","title":"Context-free pumping down below twice the pumping bound"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/infinite-b-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.infiniteB_iff","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#infinite-b-iff","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$, the infiniteness test returns true on $g$ if and only if the language of $g$ is infinite.","title":"Correctness of the infiniteness test"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/tm2-decidable-language-infinite","kind":"node","lean":"Cslib.Language.EncodedCFG.tm2Decidable_language_infinite","parent":34000,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars/index.html#tm2-decidable-language-infinite","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-grammars","status":"fully_proved","summary":"For every alphabet $T$ with decidable equality and a primitive recursive encoding, the property of an encoded context-free grammar $g$ over $T$ that the language of $g$ is infinite is decidable by a multi-stack machine. This is the problem $\\mathrm{INFINITE}_{\\mathrm{CFG}}$.","title":"Infiniteness of context-free languages is decidable"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars.json","id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","kind":"scope","navigation":"module","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","title":"Computable reduction of context-free grammars"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/terminal-not-mem","kind":"node","lean":"Cslib.Language.Yields.terminal_not_mem","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#terminal-not-mem","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $G$ be a context-free grammar over terminals $T$ and let $\\alpha$ be a sentential form of $G$. If $\\alpha$ yields the empty word in $G$, then no terminal symbol $a \\in T$ occurs in $\\alpha$.","title":"A form yielding the empty word contains no terminal"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/no-terminals","kind":"node","lean":"Cslib.Language.EncodedCFG.noTerminals","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#no-terminals","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$, the encoded grammar with the same start nonterminal whose productions are the productions of $g$, in their order, whose right-hand sides contain no terminal.","title":"Productions without terminals of an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/nullable","kind":"node","lean":"Cslib.Language.EncodedCFG.nullable","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#nullable","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$, the list of generating nonterminals of the encoded grammar that has the start nonterminal of $g$ and, as productions, the productions of $g$ whose right-hand sides contain no terminal.","title":"Computed nullable nonterminals of an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/yields-no-terminals","kind":"node","lean":"Cslib.Language.EncodedCFG.yields_noTerminals","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#yields-no-terminals","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar and $\\alpha$ a sentential form of the grammar $g$ denotes. If $\\alpha$ yields the empty word in that grammar, then $\\alpha$ also yields the empty word in the grammar denoted by the encoded grammar consisting of the productions of $g$ that contain no terminal.","title":"Yields of the empty word need only terminal-free productions"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/eq-nil-of-yields-no-terminals","kind":"node","lean":"Cslib.Language.EncodedCFG.eq_nil_of_yields_noTerminals","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#eq-nil-of-yields-no-terminals","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar and consider the grammar of its productions without terminals. If a sentential form $\\alpha$ that contains no terminal yields a word $w$ in that grammar, then $w$ is the empty word.","title":"Terminal-free productions yield only the empty word"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/yields-of-yields-no-terminals","kind":"node","lean":"Cslib.Language.EncodedCFG.yields_of_yields_noTerminals","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#yields-of-yields-no-terminals","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar, $\\alpha$ a sentential form and $w$ a word. If $\\alpha$ yields $w$ in the grammar denoted by the encoded grammar consisting of the productions of $g$ that contain no terminal, then $\\alpha$ yields $w$ in the grammar $g$ denotes.","title":"Yields in the terminal-free subgrammar are yields of the grammar"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/mem-nullable-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.mem_nullable_iff","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#mem-nullable-iff","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$ and every natural number $A$, $A$ belongs to the computed list of nullable nonterminals of $g$ (the generating nonterminals of the productions of $g$ without terminals) if and only if $A$ is nullable in the grammar denoted by $g$, that is, $A$ derives the empty word.","title":"The nullable nonterminals are computed"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/deletions","kind":"node","lean":"Cslib.Language.EncodedCFG.deletions","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#deletions","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $N$ be a list of natural numbers and $u$ an encoded right-hand side, that is, a word whose letters are nonterminals $i \\in \\mathbb{N}$ or terminals. The result is the list of all words obtained from $u$ by deleting some of the occurrences of nonterminals that belong to $N$ (possibly none, possibly all) and keeping every other letter in order; each choice of deleted occurrences contributes one entry of the list.","title":"Deleting nonterminals from an encoded right-hand side"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/mem-nullable-deletions-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.mem_nullableDeletions_iff","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#mem-nullable-deletions-iff","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar. For every encoded word $u$ and every sentential form $\\alpha$ of the grammar denoted by $g$, $\\alpha$ is obtained from the decoding of $u$ by deleting some occurrences of nullable nonterminals of the grammar denoted by $g$ if and only if $\\alpha$ is the decoding of a word obtained from $u$ by deleting some occurrences of nonterminals in the computed list of nullable\u2026","title":"Computed deletions agree with nullable deletions"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/eps-elim","kind":"node","lean":"Cslib.Language.EncodedCFG.epsElim","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#eps-elim","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$, the encoded grammar with the same start nonterminal whose productions are, for each production $A \\to u$ of $g$, the productions $A \\to v$ for every nonempty word $v$ obtained from $u$ by deleting some occurrences of nonterminals in the computed list of nullable nonterminals of $g$.","title":"Removing empty productions from an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/mem-eps-elim-rules","kind":"node","lean":"Cslib.Language.EncodedCFG.mem_epsElim_rules","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#mem-eps-elim-rules","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$, a production is a production of the grammar denoted by the encoded grammar obtained from $g$ by removing empty productions if and only if it is a production of the grammar obtained from the grammar denoted by $g$ by eliminating empty productions.","title":"Productions of the computable removal of empty productions"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/eps-elim-to-cfg","kind":"node","lean":"Cslib.Language.EncodedCFG.epsElim_toCFG","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#eps-elim-to-cfg","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$, the grammar denoted by the encoded grammar obtained from $g$ by removing empty productions is equal to the grammar obtained from the grammar denoted by $g$ by eliminating empty productions: from each production, all productions obtained by deleting some occurrences of nullable nonterminals from its output, except those with empty output.","title":"Computable removal of empty productions agrees with the classical one"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/language-eps-elim","kind":"node","lean":"Cslib.Language.EncodedCFG.language_epsElim","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#language-eps-elim","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$, the language of the encoded grammar obtained from $g$ by removing empty productions consists of the nonempty words of the language of $g$.","title":"Removing empty productions drops only the empty word"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/nt-list","kind":"node","lean":"Cslib.Language.EncodedCFG.ntList","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#nt-list","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$, the list obtained by duplicate removal from the list made of the start nonterminal of $g$ followed, for each production of $g$ in order, by its left-hand side and then the nonterminals of its right-hand side in order. It lists without repetitions the nonterminals that $g$ mentions.","title":"Nonterminals of an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/unit-target","kind":"node","lean":"Cslib.Language.EncodedCFG.unitTarget","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#unit-target","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded right-hand side $u$, a list whose entries are nonterminals $i \\in \\mathbb{N}$ or terminals, the optional value is $B$ when $u$ consists of the single nonterminal $B$, and there is no value otherwise.","title":"Target of a unit right-hand side"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/unit-step","kind":"node","lean":"Cslib.Language.EncodedCFG.unitStep","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#unit-step","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$, a natural number $A$ and a list $S$ of natural numbers, the list consisting of $A$ followed, for each production of $g$ in order whose left-hand side lies in $S$ and whose right-hand side is a single nonterminal $B$, by that nonterminal $B$.","title":"One round of unit reachability"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/unit-reach","kind":"node","lean":"Cslib.Language.EncodedCFG.unitReach","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#unit-reach","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$ and a natural number $A$, let $n$ be the number of distinct nonterminals mentioned in $g$ (the start, the left-hand sides, and the nonterminals on right-hand sides). The list is obtained from the empty list by applying one round of unit reachability from $A$ exactly $n + 1$ times.","title":"Nonterminals reachable by unit productions in an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/mem-unit-reach-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.mem_unitReach_iff","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#mem-unit-reach-iff","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar and let $A$ and $B$ be natural numbers. Then $B$ belongs to the computed list of nonterminals reachable from $A$ by unit productions of $g$ if and only if there are $k \\ge 0$ and nonterminals $A = C_0, C_1, \\dots, C_k = B$ such that $C_{i-1} \\to C_i$ is a production of the grammar denoted by $g$ for every $i$ with $1 \\le i \\le k$.","title":"Unit reachability is computed"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/unit-elim","kind":"node","lean":"Cslib.Language.EncodedCFG.unitElim","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#unit-elim","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$, the encoded grammar with the start nonterminal of $g$ whose productions are listed as follows: for each nonterminal $A$ in the list of nonterminals of $g$, and for each production $B \\to u$ of $g$ in order such that $B$ is in the computed list of nonterminals reachable from $A$ by unit productions and $u$ is not a single nonterminal, the production $A \\to u$.","title":"Unit elimination for encoded grammars"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/mem-unit-elim-rules","kind":"node","lean":"Cslib.Language.EncodedCFG.mem_unitElim_rules","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#mem-unit-elim-rules","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over terminals $T$ and let $r$ be a production with terminals in $T$ and nonterminals in $\\mathbb{N}$. Then $r$ is a production of the grammar denoted by the encoded unit elimination of $g$ if and only if $r$ is a production of the grammar obtained from the grammar denoted by $g$ by eliminating unit productions.","title":"Productions of the encoded unit elimination"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/unit-elim-to-cfg","kind":"node","lean":"Cslib.Language.EncodedCFG.unitElim_toCFG","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#unit-elim-to-cfg","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$, the grammar denoted by the encoded unit elimination of $g$ is equal to the grammar obtained from the grammar denoted by $g$ by eliminating unit productions.","title":"The encoded unit elimination denotes the unit-eliminated grammar"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/reduce","kind":"node","lean":"Cslib.Language.EncodedCFG.reduce","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#reduce","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For an encoded context-free grammar $g$, the encoded grammar obtained by first removing the empty productions of $g$ (encoded $\\varepsilon$-elimination) and then removing the unit productions of the result (encoded unit elimination).","title":"Reduction of an encoded grammar"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/language-reduce","kind":"node","lean":"Cslib.Language.EncodedCFG.language_reduce","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#language-reduce","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$, the language of the reduction of $g$ (empty productions removed, then unit productions removed) consists of the nonempty words of the language of $g$.","title":"The reduced grammar generates the language minus the empty word"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/is-reduced-reduce","kind":"node","lean":"Cslib.Language.EncodedCFG.isReduced_reduce","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#is-reduced-reduce","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"For every encoded context-free grammar $g$, every production of the grammar denoted by the reduction of $g$ (empty productions removed, then unit productions removed) has a nonempty output, and that output is not a single nonterminal.","title":"The reduced grammar has no empty and no unit productions"},{"id":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/primrec-reduce","kind":"node","lean":"Cslib.Language.EncodedCFG.primrec_reduce","parent":34037,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars/index.html#primrec-reduce","scope":"TheoryOfComputation/decision-procedures/computable-reduction-of-context-free-grammars","status":"fully_proved","summary":"Let $T$ be a type of terminals with a primitive recursive encoding. The map sending an encoded context-free grammar $g$ over $T$ to its reduction, the encoded grammar obtained from $g$ by removing empty productions and then unit productions, is primitive recursive, where encoded grammars are encoded as pairs of a start nonterminal and a list of productions.","title":"Reduction is primitive recursive"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/encoded-finite-automata.json","id":"TheoryOfComputation/decision-procedures/encoded-finite-automata","kind":"scope","navigation":"module","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","title":"Encoded finite automata"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/encoded-dfa","kind":"node","lean":"Cslib.Automata.EncodedDFA","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#encoded-dfa","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"An encoded deterministic finite automaton over an alphabet $\\alpha$ is a deterministic finite automaton given as data with natural-number states: a start state in $\\mathbb{N}$, a list of accepting states in $\\mathbb{N}$, and a list of transitions $((s, a), t)$ with $s, t \\in \\mathbb{N}$ and $a \\in \\alpha$, read as \"on letter $a$, state $s$ moves to $t$\".","title":"Encoded DFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/step","kind":"node","lean":"Cslib.Automata.EncodedDFA.step","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#step","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded DFA $D$, a state $s \\in \\mathbb{N}$ and a letter $a$, the step of $D$ from $s$ on $a$ is the target of the first transition that $D$ lists from $s$ on $a$, or $s$ itself when $D$ lists no such transition.","title":"Transition function of an encoded DFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/to-dfa","kind":"node","lean":"Cslib.Automata.EncodedDFA.toDFA","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#to-dfa","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"An encoded DFA $D$ over an alphabet $\\alpha$ denotes the deterministic automaton over $\\alpha$ with state set $\\mathbb{N}$ whose start state is the start state of $D$, whose transition function is the transition function of $D$, and whose accepting states are the states in the accepting list of $D$.","title":"The automaton denoted by an encoded DFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/language","kind":"node","lean":"Cslib.Automata.EncodedDFA.language","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#language","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"The language of an encoded DFA $D$ is the set of words $w$ such that the state reached by running $D$ on $w$ from its start state, following its transition function, is one of its accepting states.","title":"Language of an encoded DFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/letters","kind":"node","lean":"Cslib.Automata.EncodedDFA.letters","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#letters","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"The letters of an encoded DFA $D$ form the list of the letters $a$ of its listed transitions $((s, a), t)$, in the order the transitions are listed.","title":"Letters of an encoded DFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/states","kind":"node","lean":"Cslib.Automata.EncodedDFA.states","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#states","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"The states of an encoded DFA $D$ form the list consisting of its start state followed by the targets of its listed transitions, in the order they are listed.","title":"States of an encoded DFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/step-eq-self","kind":"node","lean":"Cslib.Automata.EncodedDFA.step_eq_self","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#step-eq-self","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Let $D$ be an encoded DFA, $s \\in \\mathbb{N}$ a state and $a$ a letter. If $D$ lists no transition from $s$ on $a$, then the transition function of $D$ takes $s$ on $a$ to $s$.","title":"Unlisted transitions fix the state"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/step-eq-of-forall","kind":"node","lean":"Cslib.Automata.EncodedDFA.step_eq_of_forall","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#step-eq-of-forall","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Let $D$ be an encoded DFA, let $s, t \\in \\mathbb{N}$ be states and let $a$ be a letter. If $D$ lists at least one transition from $s$ on $a$, and every transition listed from $s$ on $a$ has target $t$, then the transition function of $D$ takes $s$ on $a$ to $t$.","title":"A step determined by its listed transitions"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/step-of-not-mem-letters","kind":"node","lean":"Cslib.Automata.EncodedDFA.step_of_not_mem_letters","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#step-of-not-mem-letters","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Let $D$ be an encoded DFA and let $a$ be a letter that is not among the letters of $D$, that is, no listed transition of $D$ is labelled $a$. Then for every state $s \\in \\mathbb{N}$ the transition function of $D$ takes $s$ on $a$ to $s$.","title":"Letters without listed transitions fix every state"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/step-mem-or-eq","kind":"node","lean":"Cslib.Automata.EncodedDFA.step_mem_or_eq","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#step-mem-or-eq","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For every encoded DFA $D$, state $s \\in \\mathbb{N}$ and letter $a$, the transition function of $D$ takes $s$ on $a$ either to the target of some listed transition of $D$ or to $s$ itself.","title":"Where a step can lead"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/step-mem-states","kind":"node","lean":"Cslib.Automata.EncodedDFA.step_mem_states","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#step-mem-states","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Let $D$ be an encoded DFA and let $s$ be one of its states, that is, its start state or the target of one of its listed transitions. Then for every letter $a$ the transition function of $D$ takes $s$ on $a$ to one of the states of $D$.","title":"The listed states are closed under transitions"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/compl","kind":"node","lean":"Cslib.Automata.EncodedDFA.compl","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#compl","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"The complement of an encoded DFA $D$ has the same start state and the same transitions as $D$, and its accepting states are those states of $D$ (its start state and the targets of its listed transitions) that $D$ does not accept.","title":"Complement of an encoded DFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/mem-language-compl","kind":"node","lean":"Cslib.Automata.EncodedDFA.mem_language_compl","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#mem-language-compl","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For every encoded DFA $D$ and every word $w$, the complement of $D$ accepts $w$ if and only if $D$ does not accept $w$.","title":"The complement DFA accepts the complement language"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/exists-language-eq","kind":"node","lean":"Cslib.Automata.EncodedDFA.exists_language_eq","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#exists-language-eq","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"A DFA over a finite alphabet with finitely many states has an encoded DFA with the same language.","title":"Every finite DFA has an encoding"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/equiv-prod","kind":"node","lean":"Cslib.Automata.EncodedDFA.equivProd","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#equiv-prod","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Sending an encoded DFA over $\\alpha$ to the triple of its start state, its list of accepting states and its list of transitions is an equivalence (a bijection with explicit inverse) between encoded DFAs over $\\alpha$ and the set $\\mathbb{N} \\times \\mathrm{List}\\,\\mathbb{N} \\times \\mathrm{List}((\\mathbb{N} \\times \\alpha) \\times \\mathbb{N})$.","title":"Encoded DFAs as triples"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/primrec-mk","kind":"node","lean":"Cslib.Automata.EncodedDFA.primrec_mk","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#primrec-mk","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with a primitive recursive encoding, and encode encoded DFAs over $\\alpha$ through their start state, accepting states and transitions. The map sending a triple $(s, F, T)$, where $s \\in \\mathbb{N}$, $F$ is a list of natural numbers and $T$ is a list of pairs $((q, a), t)$ with $q, t \\in \\mathbb{N}$ and $a \\in \\alpha$, to the encoded DFA over $\\alpha$ with start state $s$, accepting\u2026","title":"Assembling an encoded DFA is primitive recursive"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/primrec-step","kind":"node","lean":"Cslib.Automata.EncodedDFA.primrec_step","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#primrec-step","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with a primitive recursive encoding and decidable equality, and encode encoded DFAs over $\\alpha$ through their start state, accepting states and transitions. The map sending an encoded DFA $D$ over $\\alpha$, a state $s \\in \\mathbb{N}$ and a letter $a \\in \\alpha$ to the state that $D$ moves to from $s$ on $a$ (the target of the first transition listed for $(s, a)$, or $s$ if there is\u2026","title":"The transition function of encoded DFAs is primitive recursive"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/primrec-eval","kind":"node","lean":"Cslib.Automata.EncodedDFA.primrec\u2082_eval","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#primrec-eval","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with a primitive recursive encoding and decidable equality, and encode encoded DFAs over $\\alpha$ through their start state, accepting states and transitions. The map sending an encoded DFA $D$ and a word $w$ over $\\alpha$ to the state reached by running $D$ on $w$ from its start state is primitive recursive.","title":"Running an encoded DFA is primitive recursive"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/encoded-nfa","kind":"node","lean":"Cslib.Automata.EncodedNFA","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#encoded-nfa","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"An encoded NFA over an alphabet $\\alpha$ is a nondeterministic finite automaton with \u03b5-moves given as data, with natural numbers as states: a list of start states, a list of accepting states, a list of transitions $((s, a), t)$, meaning that on the letter $a$ the state $s$ may move to $t$, and a list of \u03b5-moves $(s, t)$, meaning that the state $s$ may move to $t$ without reading a letter.","title":"Encoded NFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/to-nfa","kind":"node","lean":"Cslib.Automata.EncodedNFA.to\u03b5NFA","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#to-nfa","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"An encoded NFA $N$ over $\\alpha$ denotes the \u03b5-NFA over $\\alpha$ with state set $\\mathbb{N}$ whose start states are the start states listed in $N$, whose accepting states are the accepting states listed in $N$, and in which a state $s$ may move to $t$ on a letter $a$ exactly when $((s, a), t)$ is a transition of $N$, and without reading a letter exactly when $(s, t)$ is an \u03b5-move of $N$.","title":"The \u03b5-NFA of an encoded NFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/language-2","kind":"node","lean":"Cslib.Automata.EncodedNFA.language","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#language-2","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"The language of an encoded NFA $N$ over an alphabet $\\alpha$ is the language accepted by the \u03b5-NFA that $N$ denotes, whose states are the natural numbers: the words $w$ over $\\alpha$ such that some accepting state of $N$ is among the states reached on $w$ from the start states of $N$, with \u03b5-moves allowed before, between and after the letters.","title":"Language of an encoded NFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/states-2","kind":"node","lean":"Cslib.Automata.EncodedNFA.states","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#states-2","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$, this is the list of the start states of $N$, followed by the targets $t$ of its transitions $((s, a), t)$, followed by the targets $t$ of its \u03b5-moves $(s, t)$.","title":"States an encoded NFA mentions as start states or targets"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/eps-rel","kind":"node","lean":"Cslib.Automata.EncodedNFA.EpsRel","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#eps-rel","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$, the \u03b5-move relation holds between states $s$ and $t$ exactly when $(s, t)$ is one of the \u03b5-moves of $N$.","title":"\u03b5-move relation"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/mem-closure-iff","kind":"node","lean":"Cslib.Automata.EncodedNFA.mem_\u03b5Closure_iff","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#mem-closure-iff","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$, a set $X$ of natural numbers and a natural number $t$, $t$ lies in the \u03b5-closure of $X$ in the \u03b5-NFA that $N$ denotes if and only if $t$ is reachable from some $u \\in X$ by a sequence of zero or more \u03b5-moves of $N$.","title":"The \u03b5-closure is reachability along \u03b5-moves"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/eps-next","kind":"node","lean":"Cslib.Automata.EncodedNFA.epsNext","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#eps-next","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$ and a state $s$, the \u03b5-successors of $s$ are the list of the targets $t$ of the \u03b5-moves $(s, t)$ of $N$, in the order in which these \u03b5-moves are listed.","title":"\u03b5-successors of a state"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/closure","kind":"node","lean":"Cslib.Automata.EncodedNFA.closure","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#closure","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$ and a list $S$ of states, the \u03b5-closure of $S$ is a list whose members are exactly the states reachable from some state of $S$ by finitely many \u03b5-moves of $N$ (zero moves included), computed by a search from each state of $S$.","title":"\u03b5-closure of a list of states"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/letter-step","kind":"node","lean":"Cslib.Automata.EncodedNFA.letterStep","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#letter-step","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$, a list $S$ of states and a letter $a$, the letter step lists the target $t$ of every transition $((s, a), t)$ of $N$ with $s \\in S$, in the order of the transition list of $N$, without taking the \u03b5-closure.","title":"Letter step of an encoded NFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/step-list","kind":"node","lean":"Cslib.Automata.EncodedNFA.stepList","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#step-list","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$, a list $S$ of states and a letter $a$, the list of states reached from $S$ on $a$: the \u03b5-closure, computed as a list, of the list of states $t$ such that $((s, a), t)$ is a transition of $N$ for some $s$ in $S$.","title":"One step of an encoded NFA on a list of states"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/run","kind":"node","lean":"Cslib.Automata.EncodedNFA.run","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#run","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$ and a word $w = a_1 \\cdots a_n$, the run of $N$ on $w$ is the list of states obtained by starting from the \u03b5-closure of the start states of $N$ and, for each letter $a_i$ in turn, taking the letter step on $a_i$ and then the \u03b5-closure.","title":"Run of an encoded NFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/accepts-b","kind":"node","lean":"Cslib.Automata.EncodedNFA.acceptsB","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#accepts-b","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$ and a word $w$, the membership test is the Boolean that is true exactly when some state reached on $w$ is an accepting state of $N$. The states reached on $w$ are computed from the \u03b5-closure of the start states by, for each letter of $w$ in turn, following the transitions on that letter and taking the \u03b5-closure.","title":"Membership test for encoded NFAs"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/mem-run-iff","kind":"node","lean":"Cslib.Automata.EncodedNFA.mem_run_iff","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#mem-run-iff","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$, a word $w$ and a natural number $t$, $t$ is among the states in the run of $N$ on $w$ if and only if $t$ is among the states that the \u03b5-NFA denoted by $N$ reaches on $w$ from its start states, with \u03b5-moves allowed before, between and after the letters.","title":"The run of an encoded NFA agrees with \u03b5-NFA evaluation"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/accepts-b-iff","kind":"node","lean":"Cslib.Automata.EncodedNFA.acceptsB_iff","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#accepts-b-iff","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$ and a word $w$, the membership test returns true on $w$ if and only if $w$ belongs to the language of $N$.","title":"Correctness of the NFA membership test"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/eval-eps-rel","kind":"node","lean":"Cslib.Automata.EncodedNFA.eval_epsRel","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#eval-eps-rel","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"Let $N$ be an encoded NFA and $w$ a word. If $u$ is one of the states that the \u03b5-NFA denoted by $N$ can be in after reading $w$, and $N$ has an \u03b5-move from $u$ to $v$, then $v$ is also one of these states.","title":"The states reached on a word are closed under \u03b5-moves"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/next","kind":"node","lean":"Cslib.Automata.EncodedNFA.next","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#next","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$ and a state $s$, this is the list of the targets $t$ of the transitions $((s, a), t)$ of $N$ on any letter $a$, followed by the targets $t$ of the \u03b5-moves $(s, t)$ of $N$, each in the order of the corresponding list of $N$.","title":"Successors of a state in an encoded NFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/reachable","kind":"node","lean":"Cslib.Automata.EncodedNFA.reachable","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#reachable","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$, this is the list, found by a search from each start state of $N$ in turn through the start states of $N$ and the targets of its transitions and \u03b5-moves, of the states reachable from a start state of $N$ by zero or more moves, each a transition of $N$ on some letter or an \u03b5-move of $N$.","title":"Reachable states of an encoded NFA"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/mem-reachable-iff","kind":"node","lean":"Cslib.Automata.EncodedNFA.mem_reachable_iff","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#mem-reachable-iff","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$ and a natural number $t$, $t$ is among the reachable states of $N$ if and only if there is a word $w$ such that $t$ is among the states that the \u03b5-NFA denoted by $N$ reaches on $w$ from its start states, with \u03b5-moves allowed before, between and after the letters.","title":"The reachable states are those some word can lead to"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/is-empty-b","kind":"node","lean":"Cslib.Automata.EncodedNFA.isEmptyB","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#is-empty-b","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For an encoded NFA $N$, the emptiness test is the Boolean that is true exactly when no state in the list of reachable states of $N$ is an accepting state of $N$; the reachable states are those reachable from a start state of $N$ by zero or more moves, each a transition on some letter or an \u03b5-move.","title":"Emptiness test for encoded NFAs"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/exists-language-eq-nfa","kind":"node","lean":"Cslib.Automata.EncodedNFA.exists_language_eq_\u03b5NFA","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#exists-language-eq-nfa","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"An `\u03b5NFA` over a finite alphabet with finitely many states has an encoded NFA with the same language.","title":"Every finite \u03b5-NFA has an encoding"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/exists-language-eq-2","kind":"node","lean":"Cslib.Automata.EncodedNFA.exists_language_eq","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#exists-language-eq-2","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"An NFA over a finite alphabet with finitely many states has an encoded NFA with the same language.","title":"Every finite NFA has an encoding"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/equiv-prod-2","kind":"node","lean":"Cslib.Automata.EncodedNFA.equivProd","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#equiv-prod-2","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"The bijection between encoded NFAs over $\\alpha$ and quadruples consisting of a list of natural numbers, a list of natural numbers, a list of elements of $(\\mathbb{N} \\times \\alpha) \\times \\mathbb{N}$ and a list of pairs of natural numbers, which sends an encoded NFA to its start states, accepting states, transitions and \u03b5-moves.","title":"Encoded NFAs as tuples"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/tm2-decidable-mem-language","kind":"node","lean":"Cslib.Automata.EncodedNFA.tm2Decidable_mem_language","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#tm2-decidable-mem-language","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For every alphabet $\\alpha$ with a primitive recursive encoding, the property of a pair $(N, w)$, where $N$ is an encoded NFA (with \u03b5-moves) over $\\alpha$ and $w$ is a word over $\\alpha$, that $w$ is in the language of $N$ is decidable by a multi-stack machine. This is Sipser, Theorem 4.2: the problem $A_{\\mathrm{NFA}}$ is decidable.","title":"Membership for NFAs is decidable"},{"id":"TheoryOfComputation/decision-procedures/encoded-finite-automata/tm2-decidable-language-eq-zero","kind":"node","lean":"Cslib.Automata.EncodedNFA.tm2Decidable_language_eq_zero","parent":34063,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/encoded-finite-automata/index.html#tm2-decidable-language-eq-zero","scope":"TheoryOfComputation/decision-procedures/encoded-finite-automata","status":"fully_proved","summary":"For every alphabet $\\alpha$ with a primitive recursive encoding, the property of an encoded NFA $N$ (with \u03b5-moves) over $\\alpha$ that its language is empty is decidable by a multi-stack machine. This is the problem $E_{\\mathrm{NFA}}$.","title":"Emptiness of NFAs is decidable"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata.json","id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","kind":"scope","navigation":"module","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","title":"Decision procedures for finite automata"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/symm-diff","kind":"node","lean":"DFA.symmDiff","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#symm-diff","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For DFAs $M$ and $N$ over the same alphabet, with sets of states $\\sigma$ and $\\sigma'$, this is the product DFA with set of states $\\sigma \\times \\sigma'$ whose start state is the pair of the start states of $M$ and $N$, whose transition takes $(p, q)$ on a letter $a$ to the pair of the state to which $M$ moves from $p$ on $a$ and the state to which $N$ moves from $q$ on $a$, and whose accepting states are the\u2026","title":"Symmetric difference of two DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/reach-step","kind":"node","lean":"DFA.reachStep","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#reach-step","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $\\alpha$. The round sends a finite set $S$ of states of $M$ to the set consisting of the start state of $M$ and of every state that the transition function of $M$ takes some $s \\in S$ to on some letter $a \\in \\alpha$.","title":"One round of the DFA reachability search"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/reachable","kind":"node","lean":"DFA.reachable","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#reachable","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet whose set of states $\\sigma$ is finite and has decidable equality. The reachable states of $M$ form the finite set obtained from the empty set by applying one round of the DFA reachability search for $M$ exactly $|\\sigma|$ times, where $|\\sigma|$ is the number of states of $M$.","title":"Reachable states of a DFA"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/mem-reachable-iff","kind":"node","lean":"DFA.mem_reachable_iff","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#mem-reachable-iff","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states. A state $s$ of $M$ is in the set of reachable states of $M$ if and only if there is a word $w$ such that running $M$ on $w$ from its start state ends in $s$.","title":"A state is reachable exactly when some word leads to it"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/start-mem-reachable","kind":"node","lean":"DFA.start_mem_reachable","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#start-mem-reachable","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For a DFA $M$ over a finite alphabet with finitely many states, the start state of $M$ belongs to the reachable states of $M$.","title":"The start state is reachable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/step-mem-reachable","kind":"node","lean":"DFA.step_mem_reachable","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#step-mem-reachable","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet whose set of states is finite and has decidable equality. If a state $s$ of $M$ is reachable from the start state, then for every letter $a$ the state to which $M$ moves from $s$ on $a$ is also reachable from the start state.","title":"Reachable states are closed under transitions"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/reachable-within","kind":"node","lean":"DFA.reachableWithin","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#reachable-within","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet whose set of states has decidable equality. For a natural number $n$ and a finite set $S$ of states of $M$, the reachability search from $S$ with at most $n$ rounds returns $S$ when $n = 0$. When $n = k + 1$, it returns $S$ if one round of the reachability search for $M$ sends $S$ to $S$ itself, and otherwise it returns the search from the image of $S$ under that round with\u2026","title":"Early-stopping DFA reachability search"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/reachable-within-eq-reachable","kind":"node","lean":"DFA.reachableWithin_eq_reachable","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#reachable-within-eq-reachable","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states. For every natural number $n$ at least the number of states of $M$, the early-stopping reachability search for $M$ from the empty set with at most $n$ rounds returns the set of reachable states of $M$.","title":"The early-stopping search finds exactly the reachable states"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/accessible-part-within","kind":"node","lean":"DFA.accessiblePartWithin","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#accessible-part-within","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $\\alpha$ with a finite state set $\\sigma$ with decidable equality, and let $n \\ge |\\sigma|$. The accessible part of $M$ found with $n$ rounds is the DFA over $\\alpha$ whose states are the states found by the reachability search from the empty set with at most $n$ rounds, whose start state is the start state of $M$, whose transition function is that of $M$ restricted to these\u2026","title":"Accessible part of a DFA, by a bounded search"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/accessible-part","kind":"node","lean":"DFA.accessiblePart","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#accessible-part","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $\\alpha$ with a finite state set $\\sigma$ with decidable equality. The accessible part of $M$ is its accessible part found by the reachability search with $|\\sigma|$ rounds: the DFA over $\\alpha$ whose states are the states found by the reachability search from the empty set with at most $|\\sigma|$ rounds (exactly the states that some word leads to from the start state of\u2026","title":"Accessible part of a DFA"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/accessible-part-within-surjective","kind":"node","lean":"DFA.accessiblePartWithin_surjective","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#accessible-part-within-surjective","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet $\\alpha$ with a finite state set $\\sigma$ with decidable equality, and let $n \\ge |\\sigma|$. For every state $p$ of the accessible part of $M$ found with $n$ rounds, some word $w$ leads the accessible part from its start state to $p$.","title":"Every state of the accessible part is reached"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/is-empty-b","kind":"node","lean":"DFA.isEmptyB","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#is-empty-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For a DFA $M$ over a finite alphabet with finitely many states, the emptiness test returns true exactly when none of the reachable states of $M$ is an accepting state.","title":"Emptiness test for DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/is-empty-b-iff","kind":"node","lean":"DFA.isEmptyB_iff","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#is-empty-b-iff","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states. The emptiness test applied to $M$ returns true if and only if the language accepted by $M$ is empty.","title":"The emptiness test decides emptiness of DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/is-univ-b","kind":"node","lean":"DFA.isUnivB","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#is-univ-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For a DFA $M$ over a finite alphabet with finitely many states, the universality test applied to $M$ is the emptiness test applied to the complement of $M$: the DFA with the same states, start state and transitions whose accepting states are the states that $M$ does not accept.","title":"Universality test for DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/is-univ-b-iff","kind":"node","lean":"DFA.isUnivB_iff","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#is-univ-b-iff","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ be a DFA over a finite alphabet with finitely many states. The universality test applied to $M$ returns true if and only if $M$ accepts every word over the alphabet.","title":"The universality test decides universality of DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/equiv-b","kind":"node","lean":"DFA.equivB","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#equiv-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ and $N$ be DFAs over the same finite alphabet, with finite state sets with decidable equality and decidable sets of accepting states. Their symmetric difference is the product DFA on pairs of states that starts at the pair of start states, moves on a letter by moving both components, and accepts a pair $(p, q)$ when exactly one of the statements \"$M$ accepts $p$\" and \"$N$ accepts $q$\" holds. The equivalence\u2026","title":"Equivalence test for DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/equiv-b-iff","kind":"node","lean":"DFA.equivB_iff","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#equiv-b-iff","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $M$ and $N$ be DFAs over the same finite alphabet, with finite state sets with decidable equality and decidable sets of accepting states. The equivalence test of $M$ and $N$ returns true if and only if $M$ and $N$ accept the same language.","title":"Correctness of the equivalence test for DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/accepts-b","kind":"node","lean":"Cslib.Automata.EncodedDFA.acceptsB","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#accepts-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For an encoded DFA $D$ over an alphabet $\\alpha$ with decidable equality and a word $w$ over $\\alpha$, the acceptance test is the Boolean that is true exactly when the state reached by running $D$ on $w$ from its start state is in the list of accepting states of $D$.","title":"Acceptance test for encoded DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/reach-round","kind":"node","lean":"Cslib.Automata.EncodedDFA.reachRound","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#reach-round","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $s_0 \\in \\mathbb{N}$, let $\\mathrm{next}$ assign to each $s \\in \\mathbb{N}$ a list $\\mathrm{next}(s)$ of natural numbers, and let $B$ and $R$ be lists of natural numbers. The round from $s_0$ with successors $\\mathrm{next}$ kept inside $B$ sends the list $R$ of states found so far to the following list: write $s_0$ followed, for each $s$ in $R$ in order, by $s$ and then the list $\\mathrm{next}(s)$; keep the\u2026","title":"One round of the list reachability search"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/reach-n","kind":"node","lean":"Cslib.Automata.EncodedDFA.reachN","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#reach-n","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $s_0 \\in \\mathbb{N}$, let $\\mathrm{next}$ assign to each $s \\in \\mathbb{N}$ a list $\\mathrm{next}(s)$ of natural numbers, and let $B$ be a list of natural numbers. One search round sends a list $R$ to the list obtained by writing $s_0$ followed, for each $s$ in $R$ in order, by $s$ and then the list $\\mathrm{next}(s)$, keeping the entries that lie in $B$, and deleting every entry that occurs again later; its\u2026","title":"Bounded reachability search on lists of states"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/mem-reach-n-iff","kind":"node","lean":"Cslib.Automata.EncodedDFA.mem_reachN_iff","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#mem-reach-n-iff","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $s_0 \\in \\mathbb{N}$, let $\\mathrm{next}$ assign to every natural number a list of natural numbers, its successors, and let $B$ be a list of natural numbers that contains $s_0$ and is closed under successors: whenever $s$ is in $B$ and $t$ is a successor of $s$, then $t$ is in $B$. Then a natural number $t$ is in the list computed by the reachability search from $s_0$ through $\\mathrm{next}$ within $B$ if and\u2026","title":"Correctness of the bounded reachability search"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/reachable-2","kind":"node","lean":"Cslib.Automata.EncodedDFA.reachable","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#reachable-2","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For an encoded DFA $D$ over an alphabet $\\alpha$ with decidable equality, the reachable states of $D$ form the list returned by the bounded reachability search on lists of states started at the start state of $D$, kept inside the states of $D$, in which the successors of a state $s$ are the states that the transition function of $D$ takes $s$ to on the letters of $D$, in the order of the letters.","title":"Reachable states of an encoded DFA"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/is-empty-b-2","kind":"node","lean":"Cslib.Automata.EncodedDFA.isEmptyB","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#is-empty-b-2","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For an encoded DFA $D$ over an alphabet $\\alpha$ with decidable equality, the emptiness test is the Boolean that is true exactly when no entry of the list of reachable states of $D$ is in the accepting list of $D$.","title":"Emptiness test for encoded DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/pair-states","kind":"node","lean":"Cslib.Automata.EncodedDFA.pairStates","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#pair-states","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For encoded DFAs $D$ and $E$ over the same alphabet, code a pair $(s, t)$ of natural numbers by its image $\\langle s, t \\rangle \\in \\mathbb{N}$ under Mathlib's pairing function. This is the list of the numbers $\\langle s, t \\rangle$ with $s$ running over the states of $D$ in their listed order and, for each $s$, $t$ running over the states of $E$ in their listed order.","title":"Coded pairs of states of two encoded DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/pair-next","kind":"node","lean":"Cslib.Automata.EncodedDFA.pairNext","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#pair-next","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $D$ and $E$ be encoded DFAs over the same alphabet, and let $p \\in \\mathbb{N}$ code the pair of states $(s, t)$ under the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$. The successors of $p$ form the list that has, for each letter $a$ in the letters of $D$ followed by the letters of $E$, the code of the pair made of the step of $D$ from $s$ on $a$ and the step of $E$ from $t$ on $a$.","title":"Successors of a pair of states"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/pair-reachable","kind":"node","lean":"Cslib.Automata.EncodedDFA.pairReachable","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#pair-reachable","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For encoded DFAs $D$ and $E$ over the same alphabet, code a pair $(s, t)$ of natural numbers by its image $\\langle s, t \\rangle \\in \\mathbb{N}$ under Mathlib's pairing function. The reachable pairs of $D$ and $E$ form the list returned by the bounded reachability search on lists of states started at $\\langle s_D, s_E \\rangle$, where $s_D$ and $s_E$ are the start states of $D$ and $E$, kept inside the list of coded\u2026","title":"Reachable pairs of states of two encoded DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/equiv-b-2","kind":"node","lean":"Cslib.Automata.EncodedDFA.equivB","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#equiv-b-2","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $D$ and $E$ be encoded DFAs over an alphabet $\\alpha$ with decidable equality. The equivalence test of $D$ and $E$ is the Boolean that is true exactly when, for every entry $p$ of the list of reachable pairs of states of $D$ and $E$, writing $p = \\langle s, t \\rangle$ with Mathlib's pairing function, $s$ is in the accepting list of $D$ if and only if $t$ is in the accepting list of $E$.","title":"Equivalence test for encoded DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/is-univ-b-2","kind":"node","lean":"Cslib.Automata.EncodedDFA.isUnivB","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#is-univ-b-2","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For an encoded DFA $D$ over an alphabet $\\alpha$ with decidable equality, the universality test is the Boolean that is true exactly when every entry of the list of reachable states of $D$ is in the accepting list of $D$.","title":"Universality test for encoded DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/to-fin-dfa","kind":"node","lean":"Cslib.Automata.EncodedDFA.toFinDFA","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#to-fin-dfa","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For an encoded DFA $D$, this is the DFA whose set of states is the finite set of states of $D$ (its start state and the targets of its listed transitions), whose start state is the start state of $D$, whose transition function is that of $D$ restricted to these states, and whose accepting states are those in the accepting list of $D$.","title":"An encoded DFA on its listed states"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/infinite-of-mem-of-le","kind":"node","lean":"Cslib.Automata.EncodedDFA.infinite_of_mem_of_le","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#infinite-of-mem-of-le","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $D$ be an encoded DFA and let $N$ be the length of the list of states of $D$. If $D$ accepts a word of length at least $N$, then the language of $D$ is infinite.","title":"A long accepted word makes the language of an encoded DFA infinite"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/exists-mem-le-lt","kind":"node","lean":"Cslib.Automata.EncodedDFA.exists_mem_le_lt","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#exists-mem-le-lt","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $D$ be an encoded DFA over an alphabet $\\alpha$ with decidable equality, and let $N$ be the length of its list of states. For every natural number $n$ and every word $w$ of length $n$ in the language of $D$ with $N \\le n$, the language of $D$ contains a word $v$ whose length $|v|$ satisfies $N \\le |v| < 2N$.","title":"Accepted words of length between the number of listed states and twice it"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/words-of-length","kind":"node","lean":"Cslib.Automata.EncodedDFA.wordsOfLength","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#words-of-length","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For a finite alphabet $\\alpha$ given with an enumeration of its letters and for $n \\in \\mathbb{N}$, the list of the words of length $n$ over $\\alpha$: starting from the list whose only entry is the empty word, it replaces $n$ times each word $w$ of the current list by the words $a w$ for the letters $a$ of $\\alpha$, in their enumeration order.","title":"Words of a given length"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/infinite-b","kind":"node","lean":"Cslib.Automata.EncodedDFA.infiniteB","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#infinite-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"Let $D$ be an encoded DFA over a finite alphabet and let $N$ be the length of its list of states. The infiniteness test is the Boolean that is true exactly when the language of $D$ contains a word $w$ with $N \\le |w| < 2N$; it checks every word of each length $N + i$ with $0 \\le i < N$.","title":"Infiniteness test for encoded DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/infinite-b-iff","kind":"node","lean":"Cslib.Automata.EncodedDFA.infiniteB_iff","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#infinite-b-iff","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For every encoded DFA $D$ over a finite alphabet, the infiniteness test returns true on $D$ if and only if the language of $D$ is infinite.","title":"Correctness of the infiniteness test for encoded DFAs"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/tm2-decidable-mem-language","kind":"node","lean":"Cslib.Automata.EncodedDFA.tm2Decidable_mem_language","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#tm2-decidable-mem-language","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For every alphabet $\\alpha$ with decidable equality and a primitive recursive encoding, the property of a pair $(D, w)$ of an encoded DFA $D$ over $\\alpha$ and a word $w$ over $\\alpha$ that $w$ belongs to the language of $D$ is decidable by a multi-stack machine. This is the problem $A_{\\mathrm{DFA}}$.","title":"Acceptance for DFAs is decidable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/tm2-decidable-language-eq-zero","kind":"node","lean":"Cslib.Automata.EncodedDFA.tm2Decidable_language_eq_zero","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#tm2-decidable-language-eq-zero","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For every alphabet $\\alpha$ with decidable equality and a primitive recursive encoding, the property of an encoded DFA over $\\alpha$ that its language is empty is decidable by a multi-stack machine. This is the problem $E_{\\mathrm{DFA}}$.","title":"Emptiness of DFAs is decidable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/tm2-decidable-language-eq","kind":"node","lean":"Cslib.Automata.EncodedDFA.tm2Decidable_language_eq","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#tm2-decidable-language-eq","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For every alphabet $\\alpha$ with decidable equality and a primitive recursive encoding, the property of a pair of encoded DFAs over $\\alpha$ that their languages are equal is decidable by a multi-stack machine. This is the problem $\\mathrm{EQ}_{\\mathrm{DFA}}$.","title":"Equivalence of DFAs is decidable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/tm2-decidable-language-eq-top","kind":"node","lean":"Cslib.Automata.EncodedDFA.tm2Decidable_language_eq_top","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#tm2-decidable-language-eq-top","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For every alphabet $\\alpha$ with decidable equality and a primitive recursive encoding, the property of an encoded DFA over $\\alpha$ that its language is the set of all words over $\\alpha$ is decidable by a multi-stack machine. This is the problem $\\mathrm{ALL}_{\\mathrm{DFA}}$.","title":"Universality of DFAs is decidable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/tm2-decidable-language-infinite","kind":"node","lean":"Cslib.Automata.EncodedDFA.tm2Decidable_language_infinite","parent":34106,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata/index.html#tm2-decidable-language-infinite","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-finite-automata","status":"fully_proved","summary":"For every finite alphabet $\\alpha$, given with an enumeration of its letters, decidable equality and a primitive recursive encoding, the property of an encoded DFA over $\\alpha$ that its language is infinite is decidable by a multi-stack machine. This is the problem $\\mathrm{INFINITE}_{\\mathrm{DFA}}$.","title":"Infiniteness of DFAs is decidable"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti.json","id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","kind":"scope","navigation":"module","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","title":"Emptiness of context-free and regular intersections"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/dfa-sym-fits","kind":"node","lean":"Cslib.Language.EncodedCFG.dfaSymFits","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#dfa-sym-fits","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"Let $D$ be an encoded DFA, $R$ and $S$ lists of natural numbers (the items found so far and the states where the rest of a right-hand side can start), $X$ a grammar symbol and $p \\in \\mathbb{N}$. The test is true exactly when either $X$ is a terminal $a$ and the step of $D$ from $p$ on $a$ lies in $S$, or $X$ is a nonterminal $B$ and some $k$ in $S$ has $\\langle B, \\langle p, k \\rangle \\rangle$ in $R$, where\u2026","title":"Whether a symbol fits at a DFA state"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/dfa-starts","kind":"node","lean":"Cslib.Language.EncodedCFG.dfaStarts","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#dfa-starts","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"Let $D$ be an encoded DFA, $R$ a list of natural numbers (the items found so far) and $q \\in \\mathbb{N}$. The start states of a right-hand side $\\beta$, a list of grammar symbols, for $q$ given $R$ are defined by recursion on $\\beta$: for the empty right-hand side they are the one-element list $[q]$, and for a symbol $X$ followed by $\\beta'$ they are the states of $D$, in order, at which $X$ fits given $R$ and the\u2026","title":"Start states of a right-hand side in a DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/dfa-step","kind":"node","lean":"Cslib.Language.EncodedCFG.dfaStep","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#dfa-step","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For an encoded grammar $g$, an encoded DFA $D$ and a list $R$ of natural numbers (the items found so far), one round produces the list of the codes $\\langle A, \\langle p, q \\rangle \\rangle$, where $\\langle \\cdot, \\cdot \\rangle$ is the natural-number pairing, for every production $A \\to \\beta$ of $g$, every state $q$ of $D$ and every start state $p$ of $\\beta$ for $q$ given $R$.","title":"One round of the item search against a DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/dfa-items","kind":"node","lean":"Cslib.Language.EncodedCFG.dfaItems","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#dfa-items","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For an encoded grammar $g$ and an encoded DFA $D$, the candidate items form the list of the codes $\\langle A, \\langle p, q \\rangle \\rangle$, where $\\langle \\cdot, \\cdot \\rangle$ is the natural-number pairing, for every left-hand side $A$ of a production of $g$ (listed once per production) and all states $p$ and $q$ of $D$.","title":"Candidate items for a grammar and a DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/dfa-found","kind":"node","lean":"Cslib.Language.EncodedCFG.dfaFound","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#dfa-found","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For an encoded grammar $g$ and an encoded DFA $D$, the found items are the list obtained from the empty list by applying one round of the item search of $g$ and $D$ as many times as the length of the list of candidate items of $g$ and $D$.","title":"Items found for a grammar and a DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/inter-empty-b","kind":"node","lean":"Cslib.Language.EncodedCFG.interEmptyB","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#inter-empty-b","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"Let $g$ be an encoded grammar with start nonterminal $S$, let $D$ be an encoded DFA with start state $s$, and write $\\langle \\cdot, \\cdot \\rangle$ for the natural-number pairing. The test of $g$ and $D$ is true exactly when there is no accepting state $f$ of $D$ that is also one of the states of $D$ and for which the code $\\langle S, \\langle s, f \\rangle \\rangle$ is among the found items of $g$ and $D$.","title":"The intersection-emptiness test for a grammar and a DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/dfa-starts-subset","kind":"node","lean":"Cslib.Language.EncodedCFG.dfaStarts_subset","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#dfa-starts-subset","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"Let $D$ be an encoded DFA, $R$ a list of natural numbers and $q$ one of the states of $D$. Then for every right-hand side $\\beta$, every start state of $\\beta$ for $q$ given $R$ is one of the states of $D$.","title":"Start states of a right-hand side are states of the DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/dfa-sound-item","kind":"node","lean":"Cslib.Language.EncodedCFG.DfaSoundItem","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#dfa-sound-item","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For an encoded grammar $g$ and an encoded DFA $D$, a natural number $x$ is a sound item when there are natural numbers $A$, $p$, $q$ and a word $w$ such that $x = \\langle A, \\langle p, q \\rangle \\rangle$, where $\\langle \\cdot, \\cdot \\rangle$ is the natural-number pairing, the nonterminal $A$ yields $w$ in the grammar denoted by $g$, and the automaton denoted by $D$, run on $w$ from state $p$, ends in state $q$.","title":"Sound items for a grammar and a DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/mem-dfa-starts-of-yields","kind":"node","lean":"Cslib.Language.EncodedCFG.mem_dfaStarts_of_yields","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#mem-dfa-starts-of-yields","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar and $D$ an encoded DFA over an alphabet $T$ with decidable equality, and let $\\rho$ be an encoded right-hand side whose decoding yields a word $u$ in the grammar that $g$ denotes. If $p$ is one of the states of $D$ and the automaton that $D$ denotes, run on $u$ from state $p$, ends in state $q$, then $p$ is a start state of $\\rho$ for $q$ given the items found for $g$ and\u2026","title":"Completeness of the start states in a DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/inter-empty-b-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.interEmptyB_iff","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#inter-empty-b-iff","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For every encoded grammar $g$ and every encoded DFA $D$, the intersection-emptiness test of $g$ and $D$ returns true if and only if the intersection of the language of $g$ and the language of $D$ is empty.","title":"Correctness of the intersection-emptiness test"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/subset-b","kind":"node","lean":"Cslib.Language.EncodedCFG.subsetB","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#subset-b","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For an encoded context-free grammar $g$ and an encoded DFA $D$ over the same alphabet $T$, where $T$ has decidable equality, the inclusion test on $g$ and $D$ is the Boolean value of the emptiness test on $g$ and the complement of $D$.","title":"The inclusion test for a grammar and a DFA"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/subset-b-iff","kind":"node","lean":"Cslib.Language.EncodedCFG.subsetB_iff","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#subset-b-iff","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For every alphabet $T$ with decidable equality, every encoded context-free grammar $g$ over $T$ and every encoded DFA $D$ over $T$, the inclusion test on $g$ and $D$ returns true if and only if the language of $g$ is contained in the language of $D$.","title":"Correctness of the inclusion test"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/tm2-decidable-inter-eq-zero","kind":"node","lean":"Cslib.Language.EncodedCFG.tm2Decidable_inter_eq_zero","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#tm2-decidable-inter-eq-zero","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For every alphabet $T$ with decidable equality and a primitive recursive encoding, the property of a pair $(g, D)$, where $g$ is an encoded context-free grammar over $T$ and $D$ is an encoded DFA over $T$, that the intersection of the language of $g$ and the language of $D$ is empty is decidable by a multi-stack machine.","title":"Emptiness of the intersection of a context-free and a regular language is decidable"},{"id":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/tm2-decidable-language-le","kind":"node","lean":"Cslib.Language.EncodedCFG.tm2Decidable_language_le","parent":34146,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti/index.html#tm2-decidable-language-le","scope":"TheoryOfComputation/decision-procedures/emptiness-of-context-free-and-regular-intersecti","status":"fully_proved","summary":"For every alphabet $T$ with decidable equality and a primitive recursive encoding, the property of a pair $(g, D)$, where $g$ is an encoded context-free grammar over $T$ and $D$ is an encoded DFA over $T$, that the language of $g$ is contained in the language of $D$ is decidable by a multi-stack machine.","title":"Inclusion of a context-free language in a regular language is decidable"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions.json","id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","kind":"scope","navigation":"module","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","title":"Decision procedures for regular expressions"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/regex-node","kind":"node","lean":"Cslib.Language.RegexNode","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#regex-node","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"A node of an encoded regular expression over an alphabet $\\alpha$ is one of six kinds: zero, epsilon, a letter $a \\in \\alpha$, a sum with arguments $i, j \\in \\mathbb{N}$, a product (concatenation) with arguments $i, j \\in \\mathbb{N}$, or a star with argument $i \\in \\mathbb{N}$, where the arguments are indices of nodes.","title":"Regular-expression node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/encoded-regex","kind":"node","lean":"Cslib.Language.EncodedRegex","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#encoded-regex","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"An encoded regular expression over an alphabet $\\alpha$ is a list of regular-expression nodes, each a constant ($0$, $1$ or a letter) or an operator (sum, concatenation or star) whose arguments are node indices; the last node is the root. Any list is allowed: in the semantics, an argument that is not the index of an earlier node reads as $0$, and the empty list denotes $0$.","title":"Encoded regular expressions"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/node","kind":"node","lean":"Cslib.Language.EncodedRegex.node","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#node","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an encoded regular expression $e$ and a natural number $k$, the node of $e$ at index $k$ is the entry at position $k$ (counting from $0$) of its list of nodes, or the zero node when $k$ is at least the length of the list.","title":"Node at an index"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/regex-at","kind":"node","lean":"Cslib.Language.EncodedRegex.regexAt","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#regex-at","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an encoded regular expression $e$ and $k \\in \\mathbb{N}$, the regular expression at node $k$ is defined by recursion on $k$ from the node of $e$ at index $k$ (a zero node when $k$ is past the end): a zero node gives $0$, an epsilon node gives $1$, a node for a letter $a$ gives $a$, and a sum, product or star node gives the sum, product or star of the regular expressions at its arguments, where an argument that\u2026","title":"Regular expression at a node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/to-regex","kind":"node","lean":"Cslib.Language.EncodedRegex.toRegex","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#to-regex","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"An encoded regular expression $e$ with $n$ nodes denotes the regular expression at its node $n - 1$, the last node. For $n = 0$ this is the expression at node $0$, which lies past the end of the list and is the empty expression $0$.","title":"The regular expression denoted by an encoding"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/regex-at-append","kind":"node","lean":"Cslib.Language.EncodedRegex.regexAt_append","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#regex-at-append","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $l$ and $m$ be lists of regular-expression nodes and let $k$ be smaller than the length of $l$. Then the regular expression at node $k$ of the encoding with nodes $l$ followed by $m$ equals the regular expression at node $k$ of the encoding with nodes $l$.","title":"Appending nodes does not change earlier expressions"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/flatten-aux","kind":"node","lean":"Cslib.Language.EncodedRegex.flattenAux","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#flatten-aux","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For a regular expression $r$ and a list $l$ of nodes, the nodes of $r$ are appended after $l$ in postorder, by recursion on $r$. For $0$, $1$ and a letter $a$, append the single node $0$, $1$ or $a$. For a sum $s + t$ (respectively a concatenation $st$), append the nodes of $s$ after $l$, giving $l_1$, then the nodes of $t$ after $l_1$, giving $l_2$, then a sum (respectively concatenation) node with arguments\u2026","title":"Postorder encoding after a prefix"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/flatten","kind":"node","lean":"Cslib.Language.EncodedRegex.flatten","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#flatten","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"The encoding of a regular expression $r$ is the list of nodes obtained by appending the nodes of $r$ in postorder after the empty list, so that the root of $r$ is the last node.","title":"Encoding of a regular expression"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/to-regex-flatten","kind":"node","lean":"Cslib.Language.EncodedRegex.toRegex_flatten","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#to-regex-flatten","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For every regular expression $r$ over an alphabet $\\alpha$, the encoded regular expression obtained by flattening $r$ (listing the nodes of its subexpressions in postorder, so that the root of $r$ is the last node) denotes $r$.","title":"Every regular expression has an encoding"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/zero-flags","kind":"node","lean":"Cslib.Language.EncodedRegex.zeroFlags","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#zero-flags","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"The flag triple of the empty language is $(\\mathrm{false}, \\mathrm{false}, \\mathrm{false})$; the three flags of a language say whether it is nonempty, whether it has a nonempty word, and whether it is infinite. This triple is also read for an argument of a node that is not an earlier node.","title":"Flags of the empty language"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/plus-flags","kind":"node","lean":"Cslib.Language.EncodedRegex.plusFlags","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#plus-flags","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For triples of booleans $b = (b_1, b_2, b_3)$ and $c = (c_1, c_2, c_3)$, the flags of the sum are $(b_1 \\lor c_1, b_2 \\lor c_2, b_3 \\lor c_3)$.","title":"Flags of a sum"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/comp-flags","kind":"node","lean":"Cslib.Language.EncodedRegex.compFlags","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#comp-flags","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For triples of Booleans $b = (b_1, b_2, b_3)$ and $c = (c_1, c_2, c_3)$, the flags of the product are $(b_1 \\land c_1,\\ (b_2 \\land c_1) \\lor (b_1 \\land c_2),\\ (b_3 \\land c_1) \\lor (b_1 \\land c_3))$.","title":"Flags of a concatenation"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/star-flags","kind":"node","lean":"Cslib.Language.EncodedRegex.starFlags","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#star-flags","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Given a triple of Booleans $(b_1, b_2, b_3)$, the flags of its star are $(\\mathrm{true}, b_2, b_2)$. Read as the flags (nonempty, has a nonempty word, infinite) of a language, a star is always nonempty, and it has a nonempty word, and is infinite, exactly when the argument has a nonempty word.","title":"Flags of a star"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/step-flags","kind":"node","lean":"Cslib.Language.EncodedRegex.stepFlags","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#step-flags","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Given the list $A$ of flag triples of the nodes before it, the flag triple of a node is $(\\mathrm{false}, \\mathrm{false}, \\mathrm{false})$ for a zero node, $(\\mathrm{true}, \\mathrm{false}, \\mathrm{false})$ for an epsilon node and $(\\mathrm{true}, \\mathrm{true}, \\mathrm{false})$ for a letter node. For a sum, product or star node it is the flags of a sum, product or star applied to the entries of $A$ at its\u2026","title":"Flags of the next node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/flags","kind":"node","lean":"Cslib.Language.EncodedRegex.flags","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#flags","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For a list of nodes $n_0, \\dots, n_{m-1}$, the flags $f_0, \\dots, f_{m-1}$ are computed in one left-to-right pass: $f_k$ is the step flags of node $n_k$ computed from the flags $f_0, \\dots, f_{k-1}$ of the nodes before it.","title":"Flags of all nodes"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/flags-ok","kind":"node","lean":"Cslib.Language.EncodedRegex.FlagsOK","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#flags-ok","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"A triple of Booleans $(b_1, b_2, b_3)$ describes a language $L$ when $b_1$ is true exactly when $L$ is nonempty, $b_2$ is true exactly when $L$ contains a nonempty word, and $b_3$ is true exactly when $L$ is infinite.","title":"Flags of a language"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/flags-ok-flags","kind":"node","lean":"Cslib.Language.EncodedRegex.flagsOK_flags","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#flags-ok-flags","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For a list $l$ of nodes and a natural number $k$, let $f_k$ be the entry at position $k$ of the flags of $l$, or the zero flags $(\\mathrm{false}, \\mathrm{false}, \\mathrm{false})$ if $k$ is at least the length of $l$. Then $f_k$ describes the language of the regular expression at node $k$ of the encoding $l$ when $k$ is less than the length of $l$, and describes the empty language otherwise.","title":"Correctness of the flag pass"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/infinite-b","kind":"node","lean":"Cslib.Language.EncodedRegex.infiniteB","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#infinite-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $e$ be an encoded regular expression with nodes $n_0, \\dots, n_{m-1}$, and let $f_0, \\dots, f_{m-1}$ be the flags of all nodes of $e$, the triples of Booleans computed in one left-to-right pass over the nodes. The infiniteness test of $e$ is the third component of $f_{m-1}$, the flag triple of the last node (the root); when $e$ has no nodes it is false.","title":"Infiniteness test for encoded regular expressions"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/plus-rules","kind":"node","lean":"Cslib.Language.EncodedRegex.plusRules","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#plus-rules","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For natural numbers $k$, $i$ and $j$, the productions of a sum node $k$ with arguments $i$ and $j$ are the unit production $k \\to i$ if $i < k$, followed by the unit production $k \\to j$ if $j < k$ (and nothing else). Nonterminals are node indices.","title":"Productions of a sum node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/comp-rules","kind":"node","lean":"Cslib.Language.EncodedRegex.compRules","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#comp-rules","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For natural numbers $k$, $i$ and $j$, the productions of a concatenation node $k$ with arguments $i$ and $j$ are the single production $k \\to i\\,j$ (nonterminal $i$ followed by nonterminal $j$) when $i < k$ and $j < k$, and there are none otherwise.","title":"Productions of a concatenation node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/star-rules","kind":"node","lean":"Cslib.Language.EncodedRegex.starRules","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#star-rules","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For natural numbers $k$ and $i$, the productions of a star node $k$ with argument $i$ are $k \\to \\varepsilon$, followed by $k \\to i\\,k$ if $i < k$ (and nothing else). Nonterminals are node indices.","title":"Productions of a star node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/node-rules","kind":"node","lean":"Cslib.Language.EncodedRegex.nodeRules","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#node-rules","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For a natural number $k$ and a node $n$, the productions of node $k$ are the following encoded productions, all with left-hand side $k$, where nonterminal $i$ stands for node $i$: none if $n$ is the zero node; $k \\to \\varepsilon$ if $n$ is the empty-word node; $k \\to a$ if $n$ is the character $a$; if $n$ is a sum with arguments $i$ and $j$, the unit production $k \\to i$ when $i < k$ and the unit production\u2026","title":"Productions of a node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/to-grammar","kind":"node","lean":"Cslib.Language.EncodedRegex.toGrammar","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#to-grammar","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an encoded regular expression $e$ with $n$ nodes, its grammar is the encoded context-free grammar with start nonterminal $n - 1$ (truncated subtraction, so $0$ when $n = 0$), in which nonterminal $k$ stands for node $k$. Its productions are, for each node $k < n$: none if node $k$ is $0$; $k \\to \\varepsilon$ if it is $\\varepsilon$; if it is the letter $a$, the production $k \\to a$; if it is the sum of nodes $i$\u2026","title":"The grammar of an encoded regular expression"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/form-lang","kind":"node","lean":"Cslib.Language.EncodedRegex.formLang","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#form-lang","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an encoded regular expression $e$, the language of a sentential form $X_1 \\cdots X_n$ (terminals in $\\alpha$, nonterminals in $\\mathbb{N}$) is the product $L_1 \\cdots L_n$, where $L_i$ is $\\{a\\}$, the language of the one-letter word $a$, when $X_i$ is the terminal $a$, and the language of the regular expression at node $k$ of $e$ when $X_i$ is the nonterminal $k$. The empty form has language $\\{\\varepsilon\\}$.","title":"Language of a sentential form"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/mem-matches-of-mem-node-rules","kind":"node","lean":"Cslib.Language.EncodedRegex.mem_matches_of_mem_nodeRules","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#mem-matches-of-mem-node-rules","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $e$ be an encoded regular expression, $k$ a natural number and $x$ one of the productions of node $k$ built from the node of $e$ at index $k$. If a word $w$ belongs to the language of the right-hand side of $x$ (the concatenation, over its symbols, of $\\{a\\}$ for a terminal $a$ and of the language of the regular expression at node $i$ for a nonterminal $i$), then $w$ belongs to the language of the regular\u2026","title":"A production of a node only produces words of its language"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/mem-form-lang-of-yields","kind":"node","lean":"Cslib.Language.EncodedRegex.mem_formLang_of_yields","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#mem-form-lang-of-yields","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $e$ be an encoded regular expression. If a sentential form $u$ of the context-free grammar of $e$ yields a terminal word $w$, then $w$ belongs to the language of $u$: the concatenation, over the symbols of $u$, of $\\{a\\}$ for a terminal $a$ and of the language of the regular expression at node $k$ for a nonterminal $k$.","title":"Soundness of the grammar of a regular expression"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/yields-of-mem-matches","kind":"node","lean":"Cslib.Language.EncodedRegex.yields_of_mem_matches","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#yields-of-mem-matches","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with decidable equality, $e$ an encoded regular expression over $\\alpha$ and $k$ a natural number. Every word in the language of the regular expression at node $k$ of $e$ is yielded by the sentential form consisting of the single nonterminal $k$ in the context-free grammar of $e$.","title":"Completeness of the grammar of an encoded regular expression"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/language-to-grammar","kind":"node","lean":"Cslib.Language.EncodedRegex.language_toGrammar","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#language-to-grammar","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For every encoded regular expression $e$, the language of the context-free grammar of $e$ (one nonterminal per node, the root as start symbol) equals the language of the regular expression that $e$ denotes.","title":"The grammar generates the language of the expression"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/mem-b","kind":"node","lean":"Cslib.Language.EncodedRegex.memB","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#mem-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an encoded regular expression $e$ and a word $w$, the test returns the answer of the membership test for encoded context-free grammars on the word $w$ and the grammar of $e$, which has one nonterminal per node and the root as start symbol.","title":"Membership test for regular expressions"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/is-empty-b","kind":"node","lean":"Cslib.Language.EncodedRegex.isEmptyB","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#is-empty-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an encoded regular expression $e$, the test returns the answer of the emptiness test for encoded context-free grammars on the grammar of $e$, which has one nonterminal per node and the root as start symbol.","title":"Emptiness test for regular expressions"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/subset-b","kind":"node","lean":"Cslib.Language.EncodedRegex.subsetB","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#subset-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an encoded regular expression $e$ and an encoded DFA $D$, the inclusion test returns the Boolean computed by the CFG \u2286 DFA inclusion test on the grammar of $e$ and the automaton $D$.","title":"Inclusion test"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/node-equiv","kind":"node","lean":"Cslib.Language.EncodedRegex.nodeEquiv","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#node-equiv","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"The bijection between the nodes of encoded regular expressions over $\\alpha$ and $1 \\oplus 1 \\oplus \\alpha \\oplus (\\mathbb{N} \\times \\mathbb{N}) \\oplus (\\mathbb{N} \\times \\mathbb{N}) \\oplus \\mathbb{N}$, where $1$ is the one-point type, that sends the zero node and the empty-word node to the first and second copies of $1$, a character node $a$ to $a$, a sum node with arguments $i, j$ to $(i, j)$ in the first copy of\u2026","title":"Regex nodes as a sum type"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/equiv-list","kind":"node","lean":"Cslib.Language.EncodedRegex.equivList","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#equiv-list","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"The map sending an encoded regular expression over $\\alpha$ to its list of nodes is a bijection onto the lists of regular-expression nodes over $\\alpha$; its inverse wraps a list of nodes as an encoding.","title":"Encodings are lists of nodes"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/tm2-decidable-mem-matches","kind":"node","lean":"Cslib.Language.EncodedRegex.tm2Decidable_mem_matches","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#tm2-decidable-mem-matches","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with a primitive recursive coding by natural numbers. The set of pairs $(e, w)$ of an encoded regular expression $e$ over $\\alpha$ and a word $w$ over $\\alpha$ such that $w$ belongs to the language of the regular expression denoted by $e$ is decidable by a multi-stack machine.","title":"Membership for regular expressions is decidable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/tm2-decidable-matches-eq-zero","kind":"node","lean":"Cslib.Language.EncodedRegex.tm2Decidable_matches_eq_zero","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#tm2-decidable-matches-eq-zero","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an alphabet $\\alpha$ with a primitive recursive encoding, the property of an encoded regular expression over $\\alpha$ that the language it denotes is empty (E_REX) is decidable by a multi-stack machine.","title":"Emptiness of regular expressions is decidable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/tm2-decidable-matches-infinite","kind":"node","lean":"Cslib.Language.EncodedRegex.tm2Decidable_matches_infinite","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#tm2-decidable-matches-infinite","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an alphabet $\\alpha$ with a primitive recursive encoding, the property of an encoded regular expression over $\\alpha$ that the language it denotes is infinite is decidable by a multi-stack machine.","title":"Infiniteness of regular expressions is decidable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/tm2-decidable-matches-le","kind":"node","lean":"Cslib.Language.EncodedRegex.tm2Decidable_matches_le","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#tm2-decidable-matches-le","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with decidable equality and a primitive recursive coding by natural numbers. The set of pairs $(e, D)$ of an encoded regular expression $e$ over $\\alpha$ and an encoded DFA $D$ over $\\alpha$ such that the language of the regular expression denoted by $e$ is contained in the language of $D$ is decidable by a multi-stack machine.","title":"Inclusion of a regular language in a DFA language is decidable"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/matches-eq-iff-forall-length-lt","kind":"node","lean":"Cslib.Automata.Thompson.matches'_eq_iff_forall_length_lt","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#matches-eq-iff-forall-length-lt","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Two regular expressions $r$ and $s$ denote the same language if and only if every word of length less than $2^{n(r) + n(s)}$, where $n$ denotes the Thompson state count, is matched by $r$ exactly when it is matched by $s$.","title":"Short distinguishing words"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/step-states","kind":"node","lean":"Cslib.Language.EncodedRegex.stepStates","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#step-states","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $c = (c_0, \\dots, c_{m-1})$ be a list of natural numbers, and read $c_i$ as $0$ when $i \\ge m$. The Thompson state count of a regular-expression node from $c$ is $0$ for the node $0$, $1$ for the node $\\varepsilon$, $2$ for a letter, $c_i + c_j$ for a sum node and for a concatenation node with arguments $i$ and $j$, and $c_i + 1$ for a star node with argument $i$.","title":"One step of the Thompson state count"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/state-counts","kind":"node","lean":"Cslib.Language.EncodedRegex.stateCounts","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#state-counts","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For a list $l$ of regular-expression nodes, the list of Thompson state counts of $l$ is computed in one left-to-right pass: starting from the empty list, each node $n$ of $l$ in turn appends the count of $n$ computed from the list of counts built so far.","title":"Thompson state counts of a node list"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/num-thompson-states","kind":"node","lean":"Cslib.Language.EncodedRegex.numThompsonStates","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#num-thompson-states","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"The number of Thompson states of an encoded regular expression $e$ is the entry of the state counts of the nodes of $e$ at the last node, the root (read as $0$ when $e$ has no nodes).","title":"Thompson state count of an encoding"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/length-state-counts","kind":"node","lean":"Cslib.Language.EncodedRegex.length_stateCounts","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#length-state-counts","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For every list $l$ of nodes of encoded regular expressions, the list of state counts of $l$ has the same length as $l$.","title":"One state count per node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/state-counts-append-singleton","kind":"node","lean":"Cslib.Language.EncodedRegex.stateCounts_append_singleton","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#state-counts-append-singleton","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For every list $l$ of regular-expression nodes and every node $n$, let $l'$ be the list $l$ with $n$ appended at the end. Then the list of Thompson state counts of $l'$ is the list of counts of $l$ followed by one more entry, the count of $n$ computed from the counts of $l$.","title":"Appending a node appends its count"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/get-d-state-counts","kind":"node","lean":"Cslib.Language.EncodedRegex.getD_stateCounts","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#get-d-state-counts","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For every list $l$ of nodes and every index $k$, the entry at index $k$ of the state counts of $l$ (read as $0$ past the end of the list) equals the Thompson state count of the regular expression at node $k$ of the encoding with nodes $l$ when $k$ is less than the length of $l$, and $0$ otherwise.","title":"The pass computes the Thompson state count of every node"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/num-thompson-states-eq","kind":"node","lean":"Cslib.Language.EncodedRegex.numThompsonStates_eq","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#num-thompson-states-eq","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For every encoded regular expression $e$, the number of Thompson states of $e$ computed by the pass over its nodes equals the Thompson state count of the regular expression that $e$ denotes.","title":"The pass counts the states of Thompson's automaton"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/eq-bound","kind":"node","lean":"Cslib.Language.EncodedRegex.eqBound","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#eq-bound","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For encoded regular expressions $e_1$ and $e_2$, the search bound is $2^{m + n}$, where $m$ and $n$ are the numbers of Thompson states of $e_1$ and $e_2$, each computed by one pass over the nodes of the encoding.","title":"Search bound for equivalence"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/matches-eq-iff-forall-length-lt-2","kind":"node","lean":"Cslib.Language.EncodedRegex.matches'_eq_iff_forall_length_lt","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#matches-eq-iff-forall-length-lt-2","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $e_1$ and $e_2$ be encoded regular expressions over an alphabet $\\alpha$, and let $B$ be their search bound for equivalence. The regular expressions denoted by $e_1$ and $e_2$ match the same language if and only if every word $w$ over $\\alpha$ of length less than $B$ is matched by the expression denoted by $e_1$ exactly when it is matched by the expression denoted by $e_2$.","title":"Short distinguishing words for encoded regular expressions"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/eq-b","kind":"node","lean":"Cslib.Language.EncodedRegex.eqB","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#eq-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Over a finite alphabet with decidable equality and a fixed enumeration of its letters, the equivalence test of two encoded regular expressions $e_1$ and $e_2$ returns true exactly when, for every length $n$ less than the search bound of $e_1$ and $e_2$ and every word $w$ of length $n$, the membership test gives the same answer for $e_1$ on $w$ as for $e_2$ on $w$.","title":"Equivalence test"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/eq-b-iff","kind":"node","lean":"Cslib.Language.EncodedRegex.eqB_iff","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#eq-b-iff","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Over a finite alphabet with decidable equality and a fixed enumeration of its letters, for all encoded regular expressions $e_1$ and $e_2$, the equivalence test returns true on $e_1$ and $e_2$ if and only if the regular expressions that $e_1$ and $e_2$ denote have the same language.","title":"The equivalence test is correct"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/primrec-step-states","kind":"node","lean":"Cslib.Language.EncodedRegex.primrec_stepStates","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#primrec-step-states","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an alphabet $\\alpha$ with a primitive recursive encoding, the map sending a list $c$ of natural numbers and a regular-expression node over $\\alpha$ to the Thompson state count of the node computed from $c$ is primitive recursive.","title":"One step of the Thompson state count is primitive recursive"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/primrec-state-counts","kind":"node","lean":"Cslib.Language.EncodedRegex.primrec_stateCounts","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#primrec-state-counts","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an alphabet $\\alpha$ with a primitive recursive encoding, the map sending a list of regular-expression nodes over $\\alpha$ to the list of their Thompson state counts, computed in one left-to-right pass, is primitive recursive.","title":"The pass of Thompson state counts is primitive recursive"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/primrec-num-thompson-states","kind":"node","lean":"Cslib.Language.EncodedRegex.primrec_numThompsonStates","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#primrec-num-thompson-states","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an alphabet $\\alpha$ with a primitive recursive encoding, the number of states of Thompson's automaton of an encoded regular expression over $\\alpha$, computed by one pass over its nodes, is a primitive recursive function of the encoding.","title":"The Thompson state count of an encoding is primitive recursive"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/primrec-eq-bound","kind":"node","lean":"Cslib.Language.EncodedRegex.primrec_eqBound","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#primrec-eq-bound","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"For an alphabet $\\alpha$ with a primitive recursive encoding, the search bound $2^{m+n}$ for equivalence of two encoded regular expressions over $\\alpha$, where $m$ and $n$ are the numbers of states of Thompson's automata of the two expressions, is a primitive recursive function of the pair of encoded expressions.","title":"The equivalence search bound is primitive recursive"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/primrec-eq-b","kind":"node","lean":"Cslib.Language.EncodedRegex.primrec_eqB","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#primrec-eq-b","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Over a primitively encodable finite alphabet with decidable equality and a fixed enumeration of its letters, the equivalence test, as a function of two encoded regular expressions, is primitive recursive.","title":"The equivalence test is primitive recursive"},{"id":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/tm2-decidable-matches-eq","kind":"node","lean":"Cslib.Language.EncodedRegex.tm2Decidable_matches_eq","parent":34161,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions/index.html#tm2-decidable-matches-eq","scope":"TheoryOfComputation/decision-procedures/decision-procedures-for-regular-expressions","status":"fully_proved","summary":"Let $\\alpha$ be a finite alphabet with a primitive recursive encoding. Then equivalence of encoded regular expressions over $\\alpha$ is decidable by a multi-stack machine: some multi-stack machine decides, for a pair $(e_1, e_2)$ of encoded regular expressions over $\\alpha$, whether the regular expressions that $e_1$ and $e_2$ denote have the same language.","title":"EQ_REX is decidable"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations.json","id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations","kind":"scope","parent":33974,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/index.html","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations","status":"fully_proved","title":"String constraints and word equations"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/levi-of-le","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.levi_of_le","parent":34217,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/index.html#levi-of-le","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations","status":"fully_proved","summary":"Length-directed Levi's lemma: if `u` is no longer than `x`, then `x` splits as `u ++ w` and the remainders satisfy `v = w ++ y`. The general form is Mathlib's `List.append_eq_append_iff`, used directly.","title":"Length-directed Levi's lemma"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics.json","id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","kind":"scope","navigation":"module","parent":34217,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","title":"String constraints: syntax and semantics"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/term","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.Term","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#term","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"A string term over variables `V`: a constant word, a variable, the concatenation of two terms, or a letter-to-word morphism applied to a term.","title":"String term over variables"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/eval","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.eval","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#eval","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Evaluation of a term under an assignment.","title":"Evaluation of a string term under an assignment"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/vars-of","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.varsOf","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#vars-of","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"The variables occurring in a term, as a list (duplicates allowed).","title":"Variables occurring in a string term"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/eval-eq-of-agree-on-vars","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.eval_eq_of_agree_on_vars","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#eval-eq-of-agree-on-vars","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Evaluation depends only on the values of the variables in the term.","title":"String term evaluation depends only on its variables"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/atom","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.Atom","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#atom","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"A string atom: a word equation, a regular-membership test, or a length bound. Regular languages are represented by the existing `Cslib.Language.EncodedRegex`.","title":"String constraint atom"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/vars-of-atom","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.varsOfAtom","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#vars-of-atom","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"The variables occurring in an atom.","title":"Variables occurring in a string atom"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/formula","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.Formula","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#formula","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Positive Boolean combinations of atoms (conjunction, disjunction, `true`, `false`; no negation), as in the positive fragment of SMT-LIB assertions.","title":"String formulas"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/and-chain","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.andChain","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#and-chain","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Conjunction of a list of formulas.","title":"Conjunction of a list of string formulas"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/or-chain","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.orChain","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#or-chain","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Disjunction of a list of formulas.","title":"Disjunction of a list of string formulas"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/vars-of-formula","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.varsOfFormula","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#vars-of-formula","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"The variables occurring in a formula.","title":"Variables occurring in a string formula"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/sat-atom","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.SatAtom","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#sat-atom","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Satisfaction of an atom under an assignment.","title":"Satisfaction of a string atom"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/sat","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.Sat","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#sat","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Satisfaction of a formula under an assignment.","title":"Satisfaction of a string formula"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/sat-and-chain","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.sat_andChain","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#sat-and-chain","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Satisfaction of a conjunction chain: every member is satisfied.","title":"Satisfaction of a conjunction chain"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/sat-or-chain","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.sat_orChain","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#sat-or-chain","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Satisfaction of a disjunction chain: one member is satisfied.","title":"Satisfaction of a disjunction 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variables"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/sat-congr","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.sat_congr","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#sat-congr","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Satisfaction depends only on the values of the variables in the formula.","title":"Formula satisfaction depends only on its variables"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/sat-atom-b","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.satAtomB","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#sat-atom-b","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"Boolean checker for atom satisfaction. Regular membership uses `EncodedRegex.memB`.","title":"Boolean checker for atom satisfaction"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/sat-atom-b-iff","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.satAtomB_iff","parent":34219,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics/index.html#sat-atom-b-iff","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/string-constraints-syntax-and-semantics","status":"fully_proved","summary":"The Boolean checker is correct.","title":"Correctness of the Boolean atom 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enumeration"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/mem-words-up-to","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.mem_wordsUpTo","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#mem-words-up-to","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"Membership in the enumeration is exactly the length bound.","title":"Membership in the bounded-length word 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enumeration"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/get-val","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.getVal","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#get-val","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"Lookup in an association list; unbound variables read as `[]`.","title":"Association-list lookup defaulting to the empty word"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/get-val-length-le","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.getVal_length_le","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#get-val-length-le","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"A lookup returns one of the values in the list, so it inherits the length bound on values.","title":"Lookup inherits the length bound on values"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/get-val-map","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.getVal_map","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#get-val-map","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"Lookup in the association list built from an assignment returns the assignment.","title":"Lookup recovers the assignment from its association list"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/assign-lists","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.assignLists","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#assign-lists","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"All assignments of the variables in `vs` to words of length at most `B`, as association lists, each occurring once (as in `wordsUpTo`).","title":"Enumeration of bounded assignments as association lists"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/mem-assign-lists","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.mem_assignLists","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#mem-assign-lists","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"Membership in the assignment enumeration: the keys are exactly `vs` and every value has length at most `B`.","title":"Membership in the bounded assignment enumeration"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/assign-lists-length","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.assignLists_length","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#assign-lists-length","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"Counting the assignment enumeration: the total (worst-case) size of the candidate search space is `(wordsUpToCount B) ^ |vs|`. This counts enumerated association lists; see the cost model in the module header for when it coincides with the number of distinct assignments.","title":"Size of the bounded assignment enumeration"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/assign-lists-eq-map-get-val","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.assignLists_eq_map_getVal","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#assign-lists-eq-map-get-val","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"Under `vs.Nodup`, an enumerated association list is recovered from its lookup function: there is no duplicate key for `getVal` to shadow, so the list is exactly the association list built from its own lookups.","title":"Enumerated lists over distinct variables are recovered from lookups"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/assign-lists-get-val-injective","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.assignLists_getVal_injective","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#assign-lists-get-val-injective","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"Under `vs.Nodup`, distinct enumerated association lists denote distinct assignments: if two enumerated lists agree as lookup functions on `vs`, they are the same list. Without `Nodup` this fails, since a duplicate key lets a later entry be shadowed by an earlier one.","title":"Enumerated lists over distinct variables give distinct assignments"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/bounded-sat-spec","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.boundedSatSpec","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#bounded-sat-spec","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"The bounded-search specification: enumerate bounded assignments and check `\u03c6` on each one. This is a classical specification, not an executable decision procedure: `assignLists` is `noncomputable`, so this Boolean cannot be evaluated by `decide`.","title":"Bounded-search satisfiability specification"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/bounded-sat-spec-iff","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.boundedSatSpec_iff","parent":34240,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi/index.html#bounded-sat-spec-iff","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bounded-string-constraints-finite-search-specifi","status":"fully_proved","summary":"Soundness and completeness of the bounded-search specification: it reports `true` exactly when `\u03c6` has a solution whose values on `vs` have length at most `B`. The hypothesis is that `vs` covers the variables of `\u03c6`.","title":"Soundness and completeness of the bounded-search specification"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor.json","id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor","kind":"scope","navigation":"module","parent":34217,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor/index.html","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor","status":"fully_proved","title":"SAT and bounded string constraints: reduction correctness"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor/encode-cnf","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.encodeCNF","parent":34261,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor/index.html#encode-cnf","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor","status":"fully_proved","summary":"SAT as string constraints: each clause is a disjunction of word equations `x = [b]`, one per literal.","title":"Encoding of CNF formulas as string constraints"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor/cnf-sat-iff-string-sat","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.cnfSat_iff_stringSat","parent":34261,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor/index.html#cnf-sat-iff-string-sat","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor","status":"fully_proved","summary":"Correctness of the SAT-to-string-constraints translation at length bound `1`: Boolean assignments become length-`\u2264 1` string assignments and back.","title":"Correctness of the SAT-to-string-constraints translation"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor/cnf-sat-iff-bounded-sat-spec","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.cnfSat_iff_boundedSatSpec","parent":34261,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor/index.html#cnf-sat-iff-bounded-sat-spec","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/sat-and-bounded-string-constraints-reduction-cor","status":"fully_proved","summary":"SAT iff the bounded-search specification holds of the encoding at length bound `1`, for any variable list covering the variables of the encoding. This is the bridge from the reduction to the bounded-search specification: `boundedSatSpec_iff` (soundness / completeness of the finite search) plus the reduction correctness.","title":"SAT iff the bounded-search specification holds of the encoding"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami.json","id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","kind":"scope","navigation":"module","parent":34217,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","title":"PCP characterization of a string-constraint family"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/block-sym","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.BlockSym","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#block-sym","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The alphabet for the PCP encoding: dominoes (as index letters) and plain letters.","title":"Alphabet for the PCP encoding"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/domino-letter","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.dominoLetter","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#domino-letter","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"A domino as an index letter.","title":"Domino as an index letter"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/top-fun","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.topFun","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#top-fun","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The top morphism: a domino letter maps to its top word as plain letters; plain letters map to `[]` (they never occur in an index word).","title":"Top morphism of the PCP encoding"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/bot-fun","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.botFun","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#bot-fun","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The bottom morphism.","title":"Bottom morphism of the PCP encoding"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/domino-regex","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.dominoRegex","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#domino-regex","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The regular language of index words: concatenations of domino letters.","title":"Regular language of domino index words"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/pcp-constraint","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.pcpConstraint","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#pcp-constraint","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The PCP encoding as a string constraint over one variable.","title":"PCP encoding as a one-variable string constraint"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/block-domino","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.blockDomino","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#block-domino","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The domino named by a one-letter block; junk `([], [])` otherwise.","title":"Domino named by a one-letter block"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/mem-sum-matches","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.mem_sum_matches","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#mem-sum-matches","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"Membership in a sum of regular expressions.","title":"Membership in a sum of regular expressions"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/mem-domino-regex","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.mem_dominoRegex","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#mem-domino-regex","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"Membership in the index-word language: the word is a concatenation of domino letters.","title":"Membership in the domino index-word language"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/sat-mem-domino-regex","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.sat_mem_dominoRegex","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#sat-mem-domino-regex","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The membership atom of the encoding checks index-word membership.","title":"Membership atom of the PCP encoding checks index words"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/flat-map-top-fun","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.flatMap_topFun","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#flat-map-top-fun","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The top morphism applied to an index word.","title":"Top morphism applied to an index word"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/flat-map-bot-fun","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.flatMap_botFun","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#flat-map-bot-fun","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"The bottom morphism applied to an index word.","title":"Bottom morphism applied to an index word"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/map-inl-flatten","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.map_inl_flatten","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#map-inl-flatten","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"Flattening a list of one-letter blocks.","title":"Flattening a list of one-letter blocks"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/pcp-iff-string-sat","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.pcp_iff_stringSat","parent":34265,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami/index.html#pcp-iff-string-sat","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/pcp-characterization-of-a-string-constraint-fami","status":"fully_proved","summary":"For each PCP instance `P`, `PCP P` holds iff the encoding constraint is satisfiable. This is a semantic characterization, not a computable reduction to a formula-indexed satisfiability predicate (see the module header).","title":"PCP is characterized by string-constraint satisfiability:"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable.json","id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable","kind":"scope","navigation":"module","parent":34217,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/index.html","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable","status":"fully_proved","title":"The closed straight-line fragment is decidable"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/slsys","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.SLSys","parent":34280,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/index.html#slsys","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable","status":"fully_proved","summary":"A closed straight-line system: one defining term per variable, plus a query formula. The definition at position `k` is the definition of variable `k`. There are no free input variables: every variable the system talks about must be defined by the system itself.","title":"Closed straight-line system"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/acyclic","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.Acyclic","parent":34280,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/index.html#acyclic","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable","status":"fully_proved","summary":"A closed straight-line system is acyclic when every defining term mentions only earlier variables, and the query mentions only defined variables. Hence every variable occurring anywhere in the system is defined, and unfolding the definitions terminates at ground terms: there is no free input variable.","title":"Acyclic closed straight-line system"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/slsat","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.SLSat","parent":34280,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/index.html#slsat","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable","status":"fully_proved","summary":"Satisfaction of a closed straight-line system: the assignment obeys every definition and satisfies the query.","title":"Satisfaction of a closed straight-line 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The variable condition says the term only looks at positions below `acc.length + k`.","title":"Defining equation for the folded values"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/sol-satisfies-defs","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.sol_satisfies_defs","parent":34280,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/index.html#sol-satisfies-defs","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable","status":"fully_proved","summary":"The candidate assignment satisfies every definition.","title":"Candidate assignment satisfies every 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variables"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/exists-sat-iff-sat-sol","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.exists_sat_iff_sat_sol","parent":34280,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/index.html#exists-sat-iff-sat-sol","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable","status":"fully_proved","summary":"Satisfiability of a closed straight-line system is equivalent to the query holding on the candidate assignment.","title":"Straight-line satisfiability iff the query holds on the candidate"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/decidable-exists-sat-of-acyclic","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.decidableExistsSatOfAcyclic","parent":34280,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable/index.html#decidable-exists-sat-of-acyclic","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/the-closed-straight-line-fragment-is-decidable","status":"fully_proved","summary":"The decision procedure checks the query on the unique ground assignment; it does not search over free inputs, because the closed fragment has none.","title":"Satisfiability of a closed acyclic straight-line system is decidable"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations.json","id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","kind":"scope","navigation":"module","parent":34217,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","title":"Ordinary word 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word"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/word-term","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.wordTerm","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#word-term","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Embed an ordinary word in the larger string-constraint term language.","title":"Embedding of ordinary words into string terms"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/eval-word-term","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.eval_wordTerm","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#eval-word-term","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"The embedding preserves substitution semantics.","title":"Word embedding preserves substitution 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satisfaction"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/eval-word-eq-of-agree-on-vars","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.evalWord_eq_of_agree_on_vars","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#eval-word-eq-of-agree-on-vars","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Evaluation depends only on variables that occur in the word.","title":"Word evaluation depends only on occurring variables"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/length-le-eval-word-of-mem","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.length_le_evalWord_of_mem","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#length-le-eval-word-of-mem","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Each occurring variable contributes a contiguous block, so its length cannot exceed the length of the evaluated word, even when substitutions may be empty.","title":"Occurring variable's image is no longer than the evaluated word"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/normalize-word-subst","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.normalizeWordSubst","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#normalize-word-subst","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Retain the images of occurring variables and replace every unused image by `[]`. Only equality of variables, not equality of alphabet symbols, is needed.","title":"Word substitution normalized by emptying unused images"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/normalize-word-subst-of-mem","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.normalizeWordSubst_of_mem","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#normalize-word-subst-of-mem","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Normalization preserves the image of every occurring variable.","title":"Normalization keeps images of occurring variables"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/normalize-word-subst-of-not-mem","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.normalizeWordSubst_of_not_mem","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#normalize-word-subst-of-not-mem","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Every variable absent from the word has empty normalized image.","title":"Normalization empties images of absent variables"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/eval-word-normalize-word-subst","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.evalWord_normalizeWordSubst","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#eval-word-normalize-word-subst","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Removing unused images does not change the evaluated word.","title":"Normalization does not change the evaluated word"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/length-normalize-word-subst-le","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.length_normalizeWordSubst_le","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#length-normalize-word-subst-le","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"After normalization, every variable image, including unused ones, is bounded by the length of the original evaluation.","title":"Normalized images are bounded by the evaluation length"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/pattern-match-iff-bounded","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.patternMatch_iff_bounded","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#pattern-match-iff-bounded","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Matching a pattern against a fixed target loses no solutions by bounding ALL variable images by the target length. Unused variables are normalized to `[]`; an arbitrary unnormalized solution need not obey this bound.","title":"Bounding variable images by the target loses no pattern matches"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/ground-sat-b","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.WordEq.groundSatB","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#ground-sat-b","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"Executable satisfaction test for a ground equation. Since the variable type is `Empty`, no substitution search is required.","title":"Executable satisfaction test for ground word equations"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/ground-sat-b-eq-true-iff","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.WordEq.groundSatB_eq_true_iff","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#ground-sat-b-eq-true-iff","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"The ground test agrees with satisfaction under any substitution.","title":"Ground test agrees with satisfaction under any substitution"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/ground-sat-b-iff","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.WordEq.groundSatB_iff","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#ground-sat-b-iff","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"The executable ground test decides existence of a solution, not just satisfaction of one guessed substitution.","title":"Ground test decides solvability of ground word equations"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/word-system-satisfiable-iff-exists-bound","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.wordSystem_satisfiable_iff_exists_bound","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#word-system-satisfiable-iff-exists-bound","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"A system over finitely many variables has a solution iff one substitution solves every equation at some finite global length bound. The witness bound is obtained from the solution; this supplies no terminating general UNSAT test.","title":"Solvability of word equation systems at a finite length bound"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/satisfiable-iff-exists-bound","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.WordEq.satisfiable_iff_exists_bound","parent":34299,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations/index.html#satisfiable-iff-exists-bound","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/ordinary-word-equations","status":"fully_proved","summary":"An equation over `Fin n` is satisfiable iff it is satisfiable at some finite global length bound. There is no equation-computable bound claimed.","title":"Satisfiability of a single word equation at a finite length bound"},{"data_url":"scopes/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll.json","id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","kind":"scope","navigation":"module","parent":34217,"result":false,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","title":"Bit-blasting word equations to the verified DPLL solver"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/bit-var","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.bitVar","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#bit-var","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"A positive SAT index for position `p` of variable `i`; `0` is reserved.","title":"SAT variable index for a bit of a string variable"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/blast-word","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.blastWord","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#blast-word","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Expand a word into literals at a fixed length for each string variable.","title":"Bit-blasting a word into literals"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/iff-clauses","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.iffClauses","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#iff-clauses","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Two implication clauses forcing equal truth values of two literals.","title":"Clauses forcing two literals to be equal"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/equal-word-clauses","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.equalWordClauses","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#equal-word-clauses","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Compare literal vectors position by position; a length mismatch emits an empty clause.","title":"Position-wise equality clauses for literal vectors"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/equation-clauses","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.equationClauses","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#equation-clauses","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Short-circuit unequal lengths before emitting position-wise equality clauses.","title":"Clauses for a word equation with a length check"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/blast-system","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.blastSystem","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#blast-system","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"CNF for a system at fixed lengths, with shared variables and one constant bit.","title":"CNF encoding of a word-equation system at fixed lengths"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/decode-assignment","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.decodeAssignment","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#decode-assignment","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Read each string variable's bits from a SAT assignment.","title":"Decoding string values from a SAT assignment"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/encode-assignment","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.encodeAssignment","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#encode-assignment","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Encode a substitution as a total SAT assignment, using inverse pairing.","title":"Encoding a substitution as a SAT assignment"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/cnf-sat-blast-system","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.cnfSat_blastSystem","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#cnf-sat-blast-system","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Fixed-length bit-blasting is satisfiability-preserving in both directions.","title":"Fixed-length bit-blasting is equisatisfiable"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/fixed-length-sat-b","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.fixedLengthSatB","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#fixed-length-sat-b","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Solve the fixed-length system by bit-blasting and the verified DPLL solver.","title":"Fixed-length word-equation solver via DPLL"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/fixed-length-sat-b-iff","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.fixedLengthSatB_iff","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#fixed-length-sat-b-iff","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"The executable SAT-backed procedure accepts exactly the fixed-length solutions.","title":"Correctness of the fixed-length SAT-backed solver"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/length-profiles","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.lengthProfiles","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#length-profiles","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"All bounded length profiles, using Mathlib's finite Cartesian-product enumeration.","title":"Enumeration of bounded length profiles"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/mem-length-profiles","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.mem_lengthProfiles","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#mem-length-profiles","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"The profile enumeration contains exactly the pointwise-bounded length assignments.","title":"Membership in the length-profile enumeration"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/bounded-word-sat-b","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.boundedWordSatB","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#bounded-word-sat-b","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Solve bounded word equations by trying finite length profiles and calling DPLL.","title":"Bounded word-equation solver via DPLL"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/bounded-word-sat-b-iff","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.boundedWordSatB_iff","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#bounded-word-sat-b-iff","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"The bounded SAT-backed solver is sound and complete, including empty strings.","title":"Soundness and completeness of the bounded word-equation solver"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/decidable-bounded-word-eq","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.decidableBoundedWordEq","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#decidable-bounded-word-eq","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"An executable decision procedure for bounded Boolean-alphabet word equations.","title":"Decision procedure for bounded Boolean word equations"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/length-profiles-length","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.lengthProfiles_length","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#length-profiles-length","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"The complete length-profile search has `(B + 1)^n` entries.","title":"Size of the length-profile search"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/equation-clauses-length","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.equationClauses_length","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#equation-clauses-length","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"One empty clause for unequal lengths; otherwise two clauses per bit.","title":"Clause count for one word equation"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/blast-system-length","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.blastSystem_length","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#blast-system-length","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Exact CNF size: one constant-bit clause plus the position clauses for every equation. This is an output-size count, not a `TimeM` or DPLL runtime bound.","title":"Exact clause count of the CNF encoding"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/blast-system-width","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.blastSystem_width","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#blast-system-width","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Every emitted clause has at most two literals, including the constant unit clause.","title":"CNF encoding uses clauses of width at most two"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/word-system-satisfiable-iff-exists-bound","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.wordSystem_satisfiable_iff_exists_boundedSatB","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#word-system-satisfiable-iff-exists-bound","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"A source system is satisfiable iff some bounded SAT search accepts. This does not make failure at one bound a decision of general UNSAT.","title":"Satisfiability as acceptance by some bounded SAT search"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/pattern-match-b","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.patternMatchB","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#pattern-match-b","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Decide matching against a concrete target, with no bound supplied by the caller.","title":"Pattern-matching decision against a concrete target"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/pattern-match-b-iff","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.patternMatchB_iff","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#pattern-match-b-iff","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"The target length is a sufficient search bound for unrestricted pattern matching.","title":"Target length bounds the pattern-matching search"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/decidable-pattern-match","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.decidablePatternMatch","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#decidable-pattern-match","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Executable decidability for unrestricted Boolean-word pattern matching.","title":"Decidability of Boolean-word pattern matching"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/blast-word-length-le","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.blastWord_length_le","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#blast-word-length-le","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"Expanded words are at most linear in source-word length and the supplied bound. Together with `blastSystem_length`, this bounds CNF output size, not solver time.","title":"Linear size bound for bit-blasted words"},{"id":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/bounded-word-sat-b-formula-iff","kind":"node","lean":"Cslib.AutomatedReasoning.StringConstraints.boundedWordSatB_formula_iff","parent":34320,"result":true,"route":"../roadmap/TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll/index.html#bounded-word-sat-b-formula-iff","scope":"TheoryOfComputation/decision-procedures/string-constraints-and-word-equations/bit-blasting-word-equations-to-the-verified-dpll","status":"fully_proved","summary":"The SAT backend decides the same bounded semantics as the existing formula language.","title":"SAT backend agrees with the bounded formula semantics"},{"data_url":"scopes/TheoryOfComputation/turing-machines.json","id":"TheoryOfComputation/turing-machines","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/index.html","scope":"TheoryOfComputation/turing-machines","status":"fully_proved","summary":"Definitions, semantics and programming of Turing machines (single-tape, multi-tape and multi-stack), simulations between tape models, and universal machines with their machine codes. Results showing that a model computes exactly the recursive functions go to Equivalence of models of computation.","title":"Turing machines"},{"data_url":"scopes/TheoryOfComputation/turing-machines/single-tape.json","id":"TheoryOfComputation/turing-machines/single-tape","kind":"scope","parent":34347,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/index.html","scope":"TheoryOfComputation/turing-machines/single-tape","status":"fully_proved","summary":"Single-tape machines: configurations, steps, halting, composition of machines, tape views, state restriction, and translation of multi-stack machines to one tape (simulation and its halting behaviour, not its time cost).","title":"Single-tape Turing machines"},{"data_url":"scopes/TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin.json","id":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin","kind":"scope","navigation":"module","parent":34348,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/index.html","scope":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin","status":"fully_proved","title":"Restricting a single-tape machine to a supporting set of states"},{"id":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/restrict","kind":"node","lean":"MetaMathlibExt.TM0Restrict.restrict","parent":34349,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/index.html#restrict","scope":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with tape alphabet $\\Gamma$ and states $\\Lambda$, and let $S \\subseteq \\Lambda$ support $M$: $S$ contains the default state and every transition of $M$ from a state in $S$ leads to a state in $S$. The restriction of $M$ to $S$ is the machine with states $S$ that, in state $q \\in S$ reading $a$, does what $M$ does in state $q$ reading $a$: it halts if $M$ halts, and otherwise\u2026","title":"Restriction of a single-tape machine to a supporting set of states"},{"id":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/cfg-val","kind":"node","lean":"MetaMathlibExt.TM0Restrict.cfgVal","parent":34349,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/index.html#cfg-val","scope":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin","status":"fully_proved","summary":"Let $S$ be a set of states of a single-tape Turing machine with tape alphabet $\\Gamma$ and states $\\Lambda$. A configuration with state $q \\in S$ and tape $T$ is sent to the configuration with state $q$, viewed as an element of $\\Lambda$, and the same tape $T$.","title":"Forgetting the restriction on a configuration"},{"id":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/respects","kind":"node","lean":"MetaMathlibExt.TM0Restrict.respects","parent":34349,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/index.html#respects","scope":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin","status":"fully_proved","summary":"Let $S$ be a set of states supporting a single-tape Turing machine $M$, and relate a configuration of the restriction of $M$ to $S$ to the configuration of $M$ with the same state, viewed as a state of $M$, and the same tape. Suppose the configuration $c$ of the restricted machine is related to the configuration $d$ of $M$. If the restricted machine steps from $c$ to $c'$, then $M$ reaches, in one or more steps\u2026","title":"The original machine simulates the restricted one step for step"},{"id":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/eval-restrict","kind":"node","lean":"MetaMathlibExt.TM0Restrict.eval_restrict","parent":34349,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin/index.html#eval-restrict","scope":"TheoryOfComputation/turing-machines/single-tape/restricting-a-single-tape-machine-to-a-supportin","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with tape alphabet $\\Gamma$ and states $\\Lambda$, and let $S \\subseteq \\Lambda$ be a set of states that supports $M$: it contains the default state and is closed under the transitions of $M$. If the default state of the restriction of $M$ to $S$ is the default state of $M$, then for every input list $l$ over $\\Gamma$ the evaluation of the restricted machine on $l$ equals that\u2026","title":"The restricted machine evaluates like the original"},{"data_url":"scopes/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single.json","id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","kind":"scope","navigation":"module","parent":34348,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","title":"Decidable and recognizable predicates on single-tape Turing machines"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/halts-on","kind":"node","lean":"Cslib.Turing.SingleTapeTM.HaltsOn","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#halts-on","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"A single-tape Turing machine $M$ halts on an input word $w$ when its run from the initial configuration on $w$ (the start state, with $w$ written from the head position rightwards on an otherwise blank tape) reaches, in finitely many steps (possibly none), a configuration in the halting state.","title":"Halting on an input"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/halts-from","kind":"node","lean":"Cslib.Turing.SingleTapeTM.HaltsFrom","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#halts-from","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"A single-tape Turing machine $M$ halts from a configuration $c$ when its run from $c$ reaches, in finitely many steps (possibly none), a configuration in the halting state.","title":"Halting from a configuration"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/halts-on-iff-halts-from","kind":"node","lean":"Cslib.Turing.SingleTapeTM.haltsOn_iff_haltsFrom","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#halts-on-iff-halts-from","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"A single-tape Turing machine $M$ halts on an input word $w$ if and only if $M$ halts from the initial configuration on $w$.","title":"Halting on an input is halting from the initial configuration"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/step-some","kind":"node","lean":"Cslib.Turing.SingleTapeTM.step_some","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#step-some","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine, $q$ a state of $M$ other than the halting state and $t$ a tape, and let the transition function of $M$ at $q$ and the symbol under the head of $t$ (possibly blank) give the symbol $s$ to write (possibly blank), the head movement $d$ (left, right or none) and the next state $q'$ (possibly the halting state). Then the configuration in state $q$ with tape $t$ has a successor,\u2026","title":"One step of a single-tape Turing machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/halts-on-2","kind":"node","lean":"Cslib.Turing.SingleTapeTM.Outputs.haltsOn","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#halts-on-2","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"If a single-tape Turing machine $M$ outputs a word $w'$ on an input word $w$, then $M$ halts on $w$.","title":"A machine that produces an output halts"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/right-unique-transition-relation","kind":"node","lean":"Cslib.Turing.SingleTapeTM.rightUnique_transitionRelation","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#right-unique-transition-relation","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"For a single-tape Turing machine $M$, every configuration of $M$ has at most one successor under the transition relation of $M$.","title":"Single-tape Turing machines are deterministic"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/step-eq-none-iff","kind":"node","lean":"Cslib.Turing.SingleTapeTM.step_eq_none_iff","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#step-eq-none-iff","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"For a single-tape Turing machine $M$ and a configuration $c$ of $M$, one step of $M$ from $c$ gives no configuration if and only if $c$ is in the halting state.","title":"Halting configurations are exactly those without a successor"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/halts-from-iff-normalizable","kind":"node","lean":"Cslib.Turing.SingleTapeTM.haltsFrom_iff_normalizable","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#halts-from-iff-normalizable","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine and $c$ a configuration of $M$. Then $M$ halts from $c$ if and only if $c$ is normalizable for the transition relation of $M$, that is, the run of $M$ from $c$ reaches, in finitely many steps (possibly none), a configuration that has no successor.","title":"Halting from a configuration is normalizability"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/halts-from-iff-of-refl-trans-gen","kind":"node","lean":"Cslib.Turing.SingleTapeTM.haltsFrom_iff_of_reflTransGen","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#halts-from-iff-of-refl-trans-gen","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine and let $a$ and $b$ be configurations of $M$ such that the run of $M$ from $a$ reaches $b$ in finitely many steps (possibly none). Then $M$ halts from $a$ if and only if $M$ halts from $b$.","title":"Halting from a configuration is invariant along a run"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/halts-from-iff-of-step-eq","kind":"node","lean":"Cslib.Turing.SingleTapeTM.haltsFrom_iff_of_step_eq","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#halts-from-iff-of-step-eq","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine and let $a$ and $b$ be configurations of $M$ that have the same successor, or that both have no successor. Then $M$ halts from $a$ if and only if $M$ halts from $b$.","title":"Configurations with the same successor halt together"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/mk-injective","kind":"node","lean":"Cslib.Turing.SingleTapeTM.mk\u2081_injective","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#mk-injective","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"The map sending a word $w$ to the tape that holds $w$ from the head position rightwards, every other cell blank, is injective.","title":"Distinct words give distinct tapes"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/unique","kind":"node","lean":"Cslib.Turing.SingleTapeTM.Outputs.unique","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#unique","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"If a single-tape Turing machine $M$ outputs the word $l_1$ on the input $l$ and also outputs the word $l_2$ on the input $l$, then $l_1 = l_2$. Here $M$ outputs a word $w$ on $l$ when its run from the initial configuration on $l$ halts with exactly $w$ on the tape, every other cell blank and the head on the first symbol of $w$ (an all-blank tape if $w$ is empty).","title":"Outputs are unique"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/single-tape-decidable","kind":"node","lean":"Cslib.Decidability.SingleTapeDecidable","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#single-tape-decidable","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$. A predicate $p$ on $\\alpha$ is decidable by a single-tape machine when there are a finite, nonempty tape alphabet $\\Sigma$, an injective map $\\iota$ from the bits $\\{0, 1\\}$ to $\\Sigma$ and a single-tape Turing machine $M$ over $\\Sigma$ such that for every $a \\in \\alpha$ there is a bit $b$ for which $M$ outputs the one-symbol word\u2026","title":"Predicates decidable by a single-tape Turing machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/single-tape-recognizable","kind":"node","lean":"Cslib.Decidability.SingleTapeRecognizable","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#single-tape-recognizable","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$. A predicate $p$ on $\\alpha$ is recognizable by a single-tape machine when there are a finite, nonempty tape alphabet $\\Sigma$, an injective map $\\iota$ from the bits $\\{0, 1\\}$ to $\\Sigma$ and a single-tape Turing machine $M$ over $\\Sigma$ such that for every $a \\in \\alpha$, the run of $M$ on the input word obtained by writing the binary\u2026","title":"Predicates recognizable by a single-tape Turing machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/single-tape-co-recognizable","kind":"node","lean":"Cslib.Decidability.SingleTapeCoRecognizable","parent":34354,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single/index.html#single-tape-co-recognizable","scope":"TheoryOfComputation/turing-machines/single-tape/decidable-and-recognizable-predicates-on-single","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type. A predicate $p$ on $\\alpha$ is co-recognizable by a single-tape Turing machine when its complement $\\neg p$ is recognizable by a single-tape Turing machine.","title":"Co-recognizability by single-tape machines"},{"data_url":"scopes/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o.json","id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","kind":"scope","navigation":"module","parent":34348,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","title":"Running two single-tape machines one after the other"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/comp-cfg-left","kind":"node","lean":"Cslib.Turing.SingleTapeTM.compCfgLeft","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#comp-cfg-left","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. A configuration of $M_1$ with tape $t$ is sent to the configuration\u2026","title":"Embedding the first machine's configurations into a composite"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/comp-cfg-right","kind":"node","lean":"Cslib.Turing.SingleTapeTM.compCfgRight","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#comp-cfg-right","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. A configuration of $M_2$ with tape $t$ is sent to the configuration\u2026","title":"Embedding the second machine's configurations into a composite"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/transition-relation-comp-cfg-left","kind":"node","lean":"Cslib.Turing.SingleTapeTM.transitionRelation_compCfgLeft","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#transition-relation-comp-cfg-left","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. Let $\\iota_1$ send a configuration of $M_1$ to the configuration of\u2026","title":"The composite simulates steps of the first machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/step-comp-cfg-right","kind":"node","lean":"Cslib.Turing.SingleTapeTM.step_compCfgRight","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#step-comp-cfg-right","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. Let $\\iota_2$ send a configuration of $M_2$ to the configuration of\u2026","title":"Step function of a composite on configurations of the second machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/transition-relation-comp-cfg-right","kind":"node","lean":"Cslib.Turing.SingleTapeTM.transitionRelation_compCfgRight","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#transition-relation-comp-cfg-right","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. Let $\\iota_2$ send a configuration of $M_2$ to the configuration of\u2026","title":"The composite simulates steps of the second machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/exists-of-refl-trans-gen-comp-cfg-right","kind":"node","lean":"Cslib.Turing.SingleTapeTM.exists_of_reflTransGen_compCfgRight","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#exists-of-refl-trans-gen-comp-cfg-right","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. Let $\\iota_2$ send a configuration of $M_2$ to the configuration of\u2026","title":"Runs of a composite from configurations of the second machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/comp-computer","kind":"node","lean":"Cslib.Turing.SingleTapeTM.Outputs.compComputer","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#comp-computer","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. Say that a machine outputs a word $w$ on a word $v$ when, started in\u2026","title":"Outputs compose"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/comp-computer-2","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OutputsWithinTime.compComputer","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#comp-computer-2","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. Say that a machine outputs a word $w$ on a word $v$ within $t$ steps\u2026","title":"Timed outputs compose"},{"id":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/halts-on-comp-computer-iff","kind":"node","lean":"Cslib.Turing.SingleTapeTM.haltsOn_compComputer_iff","parent":34370,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o/index.html#halts-on-comp-computer-iff","scope":"TheoryOfComputation/turing-machines/single-tape/running-two-single-tape-machines-one-after-the-o","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be single-tape Turing machines over the same alphabet, and let $M$ be their composite, whose states are the disjoint union of those of $M_1$ and of $M_2$: $M$ starts in the start state of $M_1$ and runs $M_1$, where $M_1$ would halt it enters the start state of $M_2$ instead, and it then runs $M_2$ and halts where $M_2$ halts. Say that a machine outputs a word $w$ on a word $v$ when, started in\u2026","title":"Halting of a composite"},{"data_url":"scopes/TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape.json","id":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape","kind":"scope","navigation":"module","parent":34348,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/index.html","scope":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape","status":"fully_proved","title":"Simulating multi-stack machines on a single tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/tr","kind":"node","lean":"MetaMathlibExt.TM2ToTM0.tr","parent":34380,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/index.html#tr","scope":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape","status":"fully_proved","summary":"For a multi-stack Turing machine $M$ with stacks indexed by $K$ and stack alphabets $\\Gamma_k$, this is the single-tape machine obtained by translating $M$ first into a single-tape machine with a store and then into a plain single-tape machine. Its tape alphabet consists of a bottom-of-stack flag together with one optional symbol of $\\Gamma_k$ for each stack $k$. Its states are pairs of an optional statement of the\u2026","title":"Two-step single-tape translation of a multi-stack machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/tr-eval-dom","kind":"node","lean":"MetaMathlibExt.TM2ToTM0.tr_eval_dom","parent":34380,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/index.html#tr-eval-dom","scope":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape","status":"fully_proved","summary":"Let $M$ be a multi-stack Turing machine with stacks indexed by $K$, let $k \\in K$, and let $L$ be a list over the alphabet of stack $k$. The single-tape emulation of $M$, started on the tape encoding of the stacks with $L$ on stack $k$ and all other stacks empty, halts if and only if $M$, started with $L$ on stack $k$ and all other stacks empty, halts.","title":"The emulation halts iff the multi-stack machine halts"},{"id":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/tr-supports","kind":"node","lean":"MetaMathlibExt.TM2ToTM0.tr_supports","parent":34380,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/index.html#tr-supports","scope":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape","status":"fully_proved","summary":"Let $M$ be a multi-stack Turing machine whose internal state type is finite, and let $S$ be a finite set of labels that supports $M$. Then the single-tape emulation of $M$ is supported by an explicit finite set of states built from $S$ through the two translations: this set contains the initial state and is closed under the transitions of the emulation.","title":"The emulation visits finitely many states"},{"id":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/supports-stmt-univ","kind":"node","lean":"MetaMathlibExt.TM2ToTM0.supportsStmt_univ","parent":34380,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/index.html#supports-stmt-univ","scope":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape","status":"fully_proved","summary":"If the label type $\\Lambda$ is finite, then every statement of a multi-stack Turing machine with labels $\\Lambda$ is supported by the finite set of all labels: every label it can jump to lies in this set.","title":"Every statement is supported by all labels"},{"id":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/supports-univ","kind":"node","lean":"MetaMathlibExt.TM2ToTM0.supports_univ","parent":34380,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/index.html#supports-univ","scope":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape","status":"fully_proved","summary":"A multi-stack Turing machine with finitely many labels is supported by the set of all its labels: this set contains the default label, and every statement of the machine is supported by it.","title":"A multi-stack machine is supported by all its labels"},{"id":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/init-list-eq-init","kind":"node","lean":"MetaMathlibExt.TM2ToTM0.initList_eq_init","parent":34380,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape/index.html#init-list-eq-init","scope":"TheoryOfComputation/turing-machines/single-tape/simulating-multi-stack-machines-on-a-single-tape","status":"fully_proved","summary":"For a bundled multi-stack Turing machine and every list $l$ over the alphabet of its input stack, the initial configuration of the bundled machine on $l$ equals the generic multi-stack initial configuration with $l$ on the input stack, when the main label is taken as the default label.","title":"The initial configuration of a bundled multi-stack machine"},{"data_url":"scopes/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations.json","id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","kind":"scope","navigation":"module","parent":34348,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","title":"Equivalent tape representations"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-2","kind":"node","lean":"Cslib.Turing.StackTape.toListBlank","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"A stack tape $s$ over an alphabet $\\Gamma$ is a finite list of cells, each a symbol of $\\Gamma$ or the blank cell, that does not end in a blank cell. Its half-tape view is the half-tape of cells (a sequence of cells taken up to trailing blanks, the blank cell being the default) whose cells are those of $s$ in order, followed by blank cells.","title":"Half-tape view of a stack tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-nil-2","kind":"node","lean":"Cslib.Turing.StackTape.toListBlank_nil","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-nil-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"The half-tape view of the empty stack tape over $\\Gamma$ is the half-tape all of whose cells are blank.","title":"Half-tape view of the empty stack tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-cons-2","kind":"node","lean":"Cslib.Turing.StackTape.toListBlank_cons","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-cons-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every cell $c$ (a symbol of $\\Gamma$ or the blank cell) and every stack tape $s$ over $\\Gamma$, the half-tape view of the stack tape obtained by pushing $c$ onto $s$ is the half-tape view of $s$ with $c$ pushed onto its front.","title":"Pushing a cell onto a stack tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/head-to-list-blank-2","kind":"node","lean":"Cslib.Turing.StackTape.head_toListBlank","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#head-to-list-blank-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every stack tape $s$ over $\\Gamma$ (a finite list of cells, each a symbol of $\\Gamma$ or the blank, that does not end in a blank), the first cell of the half-tape view of $s$ (its cells followed by infinitely many blanks) is the first cell of $s$, which is the blank if $s$ is empty.","title":"First cell of the half-tape view of a stack tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/tail-to-list-blank-2","kind":"node","lean":"Cslib.Turing.StackTape.tail_toListBlank","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#tail-to-list-blank-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every stack tape $s$ over $\\Gamma$ (a list of cells, each a symbol of $\\Gamma$ or the blank cell, not ending in a blank cell), the tail of the half-tape view of $s$ is the half-tape view of the tail of $s$: dropping the first cell of $s$ drops the first cell of its half-tape.","title":"Tail of the half-tape view of a stack tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/nth-to-list-blank","kind":"node","lean":"Cslib.Turing.StackTape.nth_toListBlank","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#nth-to-list-blank","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every stack tape $s$ over $\\Gamma$ (a finite list of cells, each a symbol of $\\Gamma$ or the blank, that does not end in a blank) and every $n \\in \\mathbb{N}$, the cell at position $n$ of the half-tape view of $s$ (its cells followed by infinitely many blanks) is the $n$-th cell of $s$, counting from $0$, if $n$ is less than the length of $s$, and the blank otherwise.","title":"Cells of the half-tape view of a stack tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-map-some-2","kind":"node","lean":"Cslib.Turing.StackTape.toListBlank_mapSome","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-map-some-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every word $a_1 \\cdots a_n$ over $\\Gamma$, the half-tape view of the stack tape whose cells are the non-blank cells $a_1, \\dots, a_n$ is the half-tape whose first $n$ cells are $a_1, \\dots, a_n$ and whose later cells are blank.","title":"Half-tape view of a word"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-injective-2","kind":"node","lean":"Cslib.Turing.StackTape.toListBlank_injective","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-injective-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"If two stack tapes over $\\Gamma$ have the same half-tape view, then they are equal.","title":"The half-tape view of stack tapes is injective"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-2","kind":"node","lean":"Cslib.Turing.BiTape.toTape","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"A CSLib tape $t$ over an alphabet $\\Gamma$ has a head cell and two halves, each a finite list of cells not ending in a blank, where a cell is a symbol of $\\Gamma$ or the blank. Its cell view is the Mathlib tape over cells, with the blank as default symbol, whose head is the head cell of $t$ and whose left and right halves are the halves of $t$, blank past their ends. So the cell at position $i \\in \\mathbb{Z}$ of\u2026","title":"Cell view of a two-way tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/head-to-tape-2","kind":"node","lean":"Cslib.Turing.BiTape.head_toTape","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#head-to-tape-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every two-way tape $t$ over $\\Gamma$ (a head cell with a left and a right stack tape), the head cell of the view of $t$ as a tape of cells is the head cell of $t$.","title":"Head cell of the view of a two-way tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/left-to-tape","kind":"node","lean":"Cslib.Turing.BiTape.left_toTape","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#left-to-tape","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every two-way tape $t$ over $\\Gamma$ (a head cell with a left and a right stack tape), the left half of the view of $t$ as a tape of cells is the half-tape view of the left stack tape of $t$.","title":"Left half of the view of a two-way tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/right-to-tape","kind":"node","lean":"Cslib.Turing.BiTape.right_toTape","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#right-to-tape","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every two-sided tape $t$ over $\\Gamma$ (a head cell together with a left and a right stack tape), the right half of the tape view of $t$ is the half-tape view of the right half of $t$.","title":"Right half of the tape view"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/nth-to-tape","kind":"node","lean":"Cslib.Turing.BiTape.nth_toTape","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#nth-to-tape","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $t$ be a two-way tape over $\\Gamma$ with head cell $h$, left stack tape $(\\ell_0, \\dots, \\ell_{p-1})$ and right stack tape $(r_0, \\dots, r_{q-1})$. For every integer $i$, the cell at position $i$ of the view of $t$ as a tape of cells is $h$ if $i = 0$, $r_{i-1}$ if $0 < i \\le q$, $\\ell_{-i-1}$ if $-p \\le i < 0$, and the blank otherwise.","title":"The cells of the view"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-write-2","kind":"node","lean":"Cslib.Turing.BiTape.toTape_write","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-write-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every tape $t$ and every cell $a$ (a symbol or the blank), the cell view of the tape obtained from $t$ by writing $a$ under the head equals the tape obtained from the cell view of $t$ by writing $a$ under its head.","title":"The cell view commutes with writing"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-move-2","kind":"node","lean":"Cslib.Turing.BiTape.toTape_move","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-move-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every tape $t$ and each direction $d$ (left or right), the cell view of the tape obtained from $t$ by moving the head one cell in direction $d$ equals the tape obtained from the cell view of $t$ by moving its head one cell in direction $d$.","title":"The cell view commutes with head moves"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-option-move-2","kind":"node","lean":"Cslib.Turing.BiTape.toTape_optionMove","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-option-move-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every two-way tape $t$ over $\\Gamma$ and every optional head move $d$, which is either no move or a move one cell in a direction, the cell view of the tape obtained from $t$ by the optional move $d$ is the cell view of $t$ when $d$ is no move, and the cell view of $t$ with its head moved one cell in direction $d$ when $d$ is a direction.","title":"The cell view commutes with optional head moves"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/option-move-offset","kind":"node","lean":"Cslib.Turing.BiTape.optionMoveOffset","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#option-move-offset","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"The offset of an optional head move $d$ is $0$ if $d$ is no move, $-1$ if $d$ is a move to the left and $1$ if $d$ is a move to the right.","title":"Offset of an optional head move"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/option-move-offset-bound","kind":"node","lean":"Cslib.Turing.BiTape.optionMoveOffset_bound","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#option-move-offset-bound","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every optional head move $d$ (no move, a move to the left, or a move to the right), the offset of $d$ lies between $-1$ and $1$, that is, $-1 \\le \\mathrm{offset}(d) \\le 1$.","title":"The offset of an optional move is at most one"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/nth-to-tape-write","kind":"node","lean":"Cslib.Turing.BiTape.nth_toTape_write","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#nth-to-tape-write","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every two-way tape $t$ over $\\Gamma$, every cell $a$ (a symbol of $\\Gamma$ or the blank) and every integer $i$, the cell at position $i$ of the view of the tape obtained from $t$ by writing $a$ under the head is $a$ if $i = 0$, and the cell at position $i$ of the view of $t$ otherwise.","title":"Cells after a write"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/nth-to-tape-option-move","kind":"node","lean":"Cslib.Turing.BiTape.nth_toTape_optionMove","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#nth-to-tape-option-move","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For an optional direction $d$, let $\\delta(d)$ be $0$ if $d$ is no move, $-1$ if $d$ is a move to the left and $1$ if $d$ is a move to the right. For every two-way tape $t$ over $\\Gamma$ and every integer $i$, the cell at position $i$ of the view of the tape obtained from $t$ by the optional move $d$ is the cell at position $i + \\delta(d)$ of the view of $t$.","title":"Cells after an optional move"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/nth-to-tape-write-option-move","kind":"node","lean":"Cslib.Turing.BiTape.nth_toTape_write_optionMove","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#nth-to-tape-write-option-move","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For an optional direction $d$, let $\\delta(d)$ be $0$ if $d$ is no move, $-1$ if $d$ is a move to the left and $1$ if $d$ is a move to the right. For every two-way tape $t$ over $\\Gamma$, every cell $a$ (a symbol of $\\Gamma$ or the blank) and every integer $i$, the cell at position $i$ of the view of the tape obtained from $t$ by writing $a$ under the head and then performing the optional move $d$ is $a$ if\u2026","title":"Cells after a write and an optional move"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/nth-to-tape-move-right-iterate","kind":"node","lean":"Cslib.Turing.BiTape.nth_toTape_moveRight_iterate","parent":34387,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cells after "],["math","\\(n\\)"],["text"," moves to the right"]],"source":"Cells after $n$ moves to the right"},"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#nth-to-tape-move-right-iterate","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every two-way tape $t$ over $\\Gamma$, every $n \\in \\mathbb{N}$ and every integer $i$, the cell at position $i$ of the view of the tape obtained from $t$ by moving the head $n$ times to the right is the cell at position $i + n$ of the view of $t$.","title":"Cells after $n$ moves to the right"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-mk-2","kind":"node","lean":"Cslib.Turing.BiTape.toTape_mk\u2081","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-mk-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every word $w = w_0 w_1 \\cdots w_{n-1}$ over $\\Gamma$, the cell view of the input tape of $w$ is the Mathlib input tape of the list of non-blank cells $w_0, \\dots, w_{n-1}$: the head is on $w_0$, the letters $w_1, \\dots, w_{n-1}$ lie to its right in order, and every other cell is blank (for the empty word, every cell is blank).","title":"The cell view of an input tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/nth-to-tape-mk","kind":"node","lean":"Cslib.Turing.BiTape.nth_toTape_mk\u2081","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#nth-to-tape-mk","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every word $w = w_0 \\cdots w_{m-1}$ over $\\Gamma$ and every integer $i$, the cell at position $i$ of the view of the input tape of $w$ is $w_i$ if $0 \\le i < m$, and the blank otherwise.","title":"Cells of an input tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-injective-2","kind":"node","lean":"Cslib.Turing.BiTape.toTape_injective","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-injective-2","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"If two two-sided tapes $t$ and $t'$ over $\\Gamma$ (each a head cell together with a left and a right stack tape) have the same tape view as a tape of cells, then $t = t'$.","title":"The tape view is injective"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/non-blank","kind":"node","lean":"Cslib.Turing.SingleTapeTM.NonBlank","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#non-blank","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For an alphabet $\\Gamma_0$ with a distinguished blank symbol $b$, the non-blank symbols of $\\Gamma_0$ are the elements $x \\in \\Gamma_0$ with $x \\ne b$. They form the tape alphabet of the single-tape machine that simulates a Post\u2013Turing machine over $\\Gamma_0$.","title":"Non-blank symbols"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/cell-symbol","kind":"node","lean":"Cslib.Turing.SingleTapeTM.cellSymbol","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#cell-symbol","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet with blank symbol $b$. A tape cell over the non-blank symbols of $\\Gamma_0$ is either blank or holds a symbol $x \\ne b$ of $\\Gamma_0$. Its cell symbol is $b$ for the blank cell and $x$ for the cell holding $x$. This map from cells to $\\Gamma_0$ sends the blank cell to the blank symbol, so it is a pointed map.","title":"Cell symbol"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/cell-symbol-apply","kind":"node","lean":"Cslib.Turing.SingleTapeTM.cellSymbol_apply","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#cell-symbol-apply","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every cell $o$ over the non-blank symbols of an alphabet $\\Gamma_0$ with blank symbol $b$, the cell symbol of $o$ is $b$ when $o$ is the blank cell and is $x$ when $o$ holds the non-blank symbol $x$.","title":"Value of the cell-symbol map"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/cell-symbol-injective","kind":"node","lean":"Cslib.Turing.SingleTapeTM.cellSymbol_injective","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#cell-symbol-injective","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Distinct cells over the non-blank symbols of an alphabet $\\Gamma_0$ have distinct cell symbols in $\\Gamma_0$.","title":"The cell-symbol map is injective"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toListBlank","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet with blank symbol $b$ and $N = \\Gamma_0 \\setminus \\{b\\}$ its non-blank symbols. A stack tape $s$ over $N$, a finite list of cells each blank or holding a symbol of $N$ and not ending in a blank cell, is read as the TM0 half-tape over $\\Gamma_0$ whose $i$-th symbol is that of the $i$-th cell of $s$: $b$ for a blank cell, $x$ for a cell holding $x$, and $b$ past the end of $s$.","title":"Half-tape of a stack tape of non-blank symbols"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toTape","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet whose default symbol is the blank. The Post\u2013Turing reading of a two-way tape $t$ whose cells are blank or hold non-blank symbols of $\\Gamma_0$ is the Post\u2013Turing machine tape over $\\Gamma_0$ obtained cell by cell: take the view of $t$ as a tape of cells, each cell at its position relative to the head and blank cells past the ends, and replace each cell by its symbol, a non-blank symbol\u2026","title":"Post\u2013Turing reading of a tape over the non-blank symbols"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-nil","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toListBlank_nil","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-nil","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For an alphabet $\\Gamma_0$ with blank symbol $b$, the empty stack tape over the non-blank symbols of $\\Gamma_0$ reads as the half-tape of the empty list, all of whose symbols are $b$.","title":"The empty stack tape reads as the blank half-tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-cons","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toListBlank_cons","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-cons","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet with blank symbol $b$ and $N = \\Gamma_0 \\setminus \\{b\\}$. Read a cell over $N$ as a symbol of $\\Gamma_0$: $b$ for the blank cell, $x$ for a cell holding $x$. For every cell $o$ and every stack tape $s$ over $N$, the half-tape read from the stack tape obtained by pushing $o$ onto $s$ is the half-tape read from $s$ with the symbol of $o$ pushed onto its front.","title":"Pushing a cell onto a stack tape of non-blank symbols"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/head-to-list-blank","kind":"node","lean":"Cslib.Turing.SingleTapeTM.head_toListBlank","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#head-to-list-blank","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet with blank symbol $b$. For every stack tape $s$ of cells over the non-blank symbols of $\\Gamma_0$ (a finite list of cells, blank beyond its end), the first symbol of the half-tape over $\\Gamma_0$ read from $s$ is the cell symbol of the first cell of $s$; when $s$ is empty this is $b$.","title":"First symbol of a half-tape read from non-blank cells"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/tail-to-list-blank","kind":"node","lean":"Cslib.Turing.SingleTapeTM.tail_toListBlank","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#tail-to-list-blank","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For an alphabet $\\Gamma_0$ with blank symbol $b$ and every stack tape $s$ over the non-blank symbols of $\\Gamma_0$, dropping the first symbol of the half-tape read from $s$ gives the half-tape read from $s$ with its first cell removed (the empty stack tape staying empty).","title":"Dropping the first cell of a stack tape of non-blank symbols"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-map-some","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toListBlank_mapSome","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-map-some","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For an alphabet $\\Gamma_0$ with blank symbol $b$ and a word $w$ of non-blank symbols of $\\Gamma_0$, the half-tape read from the stack tape whose cells hold the letters of $w$ in order is the half-tape of $w$, viewed as a word over $\\Gamma_0$.","title":"A word of non-blank symbols reads as itself"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-list-blank-injective","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toListBlank_injective","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-list-blank-injective","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For an alphabet $\\Gamma_0$ with blank symbol $b$, two stack tapes over the non-blank symbols of $\\Gamma_0$ that read as the same half-tape over $\\Gamma_0$ are equal.","title":"Distinct stack tapes of non-blank symbols read as distinct half-tapes"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/head-to-tape","kind":"node","lean":"Cslib.Turing.SingleTapeTM.head_toTape","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#head-to-tape","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For every two-way tape $t$ of cells over the non-blank symbols of an alphabet $\\Gamma_0$, the symbol under the head of the tape over $\\Gamma_0$ read from $t$ is the cell symbol of the cell under the head of $t$.","title":"Head symbol of a tape read from non-blank cells"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-write","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toTape_write","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-write","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet whose default symbol is the blank, and for a two-way tape $t$ whose cells are blank or hold non-blank symbols of $\\Gamma_0$, let $\\tau(t)$ be the Post\u2013Turing reading of $t$, a Post\u2013Turing machine tape over $\\Gamma_0$. For every such $t$ and every cell $o$, blank or holding a non-blank symbol, $\\tau$ of $t$ with $o$ written under the head is $\\tau(t)$ with the symbol of $o$ written\u2026","title":"Writing commutes with the Post\u2013Turing reading of a tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-move","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toTape_move","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-move","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet whose default symbol is the blank, and for a two-way tape $t$ whose cells are blank or hold non-blank symbols of $\\Gamma_0$, let $\\tau(t)$ be the Post\u2013Turing reading of $t$, a Post\u2013Turing machine tape over $\\Gamma_0$. For every such $t$ and every direction $d$, left or right, $\\tau$ of $t$ with its head moved in direction $d$ is $\\tau(t)$ with its head moved in direction $d$.","title":"Head moves commute with the Post\u2013Turing reading of a tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-option-move","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toTape_optionMove","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-option-move","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet whose default symbol is the blank, and for a two-way tape $t$ whose cells are blank or hold non-blank symbols of $\\Gamma_0$, let $\\tau(t)$ be the Post\u2013Turing reading of $t$, a Post\u2013Turing machine tape over $\\Gamma_0$. For every such $t$ and every optional move $d$, which is either no move or a direction, $\\tau$ of $t$ after the optional move $d$ is $\\tau(t)$ when $d$ is no move, and\u2026","title":"Optional head moves commute with the Post\u2013Turing reading of a tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-mk","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toTape_mk\u2081","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-mk","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be an alphabet whose default symbol is the blank. For every word $l$ of non-blank symbols of $\\Gamma_0$, the Post\u2013Turing reading of the two-way input tape of $l$ (the head on the first symbol of $l$, the rest of $l$ to its right, every other cell blank) is the Post\u2013Turing machine input tape of $l$ read as a word over $\\Gamma_0$, with the head on the first symbol of $l$ and the rest of $l$ to its right.","title":"The Post\u2013Turing reading of an input tape"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/to-tape-injective","kind":"node","lean":"Cslib.Turing.SingleTapeTM.toTape_injective","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#to-tape-injective","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"For an alphabet $\\Gamma_0$ with blank symbol $b$, two tapes over the non-blank symbols of $\\Gamma_0$ that read, cell by cell, as the same TM0 tape over $\\Gamma_0$ are equal.","title":"Distinct tapes of non-blank symbols read as distinct TM0 tapes"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/of-tm0","kind":"node","lean":"Cslib.Turing.SingleTapeTM.ofTM0","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#of-tm0","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be a finite alphabet with blank symbol $b$ and at least one non-blank symbol, and let $M$ be a Post\u2013Turing machine over $\\Gamma_0$ with start state $q_0$, supported by the finite set of states $S$ (so $q_0 \\in S$, and from a state in $S$ the machine $M$ only enters states in $S$). The single-tape machine simulating $M$ has the non-blank symbols of $\\Gamma_0$ as tape alphabet, $S$ as set of states and\u2026","title":"Single-tape simulation of a Post\u2013Turing machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/step-of-tm0-of-step-eq-none","kind":"node","lean":"Cslib.Turing.SingleTapeTM.step_ofTM0_of_step_eq_none","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#step-of-tm0-of-step-eq-none","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $M$ be a TM0 machine over a finite alphabet $\\Gamma_0$ with blank symbol $b$ and at least one non-blank symbol, supported by a finite set $S$ of states, and let $\\widehat{M}$ be the single-tape machine over the non-blank symbols $N = \\Gamma_0 \\setminus \\{b\\}$ simulating $M$. For a tape $t$ over $N$, write $\\tau(t)$ for the TM0 tape read from $t$ cell by cell. If $q \\in S$ and $M$ has no step from the\u2026","title":"The simulation halts one step after the TM0 machine"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/step-of-tm0-of-step-eq-some","kind":"node","lean":"Cslib.Turing.SingleTapeTM.step_ofTM0_of_step_eq_some","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#step-of-tm0-of-step-eq-some","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $M$ be a TM0 machine over a finite alphabet $\\Gamma_0$ with blank symbol $b$ and at least one non-blank symbol, supported by a finite set $S$ of states, and let $\\widehat{M}$ be the single-tape machine over the non-blank symbols $N = \\Gamma_0 \\setminus \\{b\\}$ simulating $M$. For a tape $t$ over $N$, write $\\tau(t)$ for the TM0 tape read from $t$ cell by cell. If $q \\in S$ and one step of $M$ takes the\u2026","title":"Each TM0 step is one step of the simulation"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/relates-in-steps-of-tm0","kind":"node","lean":"Cslib.Turing.SingleTapeTM.relatesInSteps_ofTM0","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#relates-in-steps-of-tm0","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $M$ be a TM0 machine over a finite alphabet $\\Gamma_0$ with blank symbol $b$ and at least one non-blank symbol, supported by a finite set $S$ of states, and let $\\widehat{M}$ be the single-tape machine over the non-blank symbols $N = \\Gamma_0 \\setminus \\{b\\}$ simulating $M$. For a tape $t$ over $N$, write $\\tau(t)$ for the TM0 tape read from $t$ cell by cell. If $M$ goes from a configuration $(q, T)$ to a\u2026","title":"Runs of a TM0 machine are runs of its single-tape simulation"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/halts-on-of-tm0-iff","kind":"node","lean":"Cslib.Turing.SingleTapeTM.haltsOn_ofTM0_iff","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#halts-on-of-tm0-iff","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be a finite alphabet with at least one non-blank symbol, let $M$ be a Post\u2013Turing machine over $\\Gamma_0$ supported by the finite set of states $S$, and let $l$ be a word of non-blank symbols of $\\Gamma_0$. The single-tape machine simulating $M$ halts on the input $l$ if and only if $M$, started on $l$, halts.","title":"The simulating machine halts exactly when the Post\u2013Turing machine halts"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/outputs-within-time-of-tm0","kind":"node","lean":"Cslib.Turing.SingleTapeTM.outputsWithinTime_ofTM0","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#outputs-within-time-of-tm0","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be a finite alphabet with at least one non-blank symbol, let $M$ be a Post\u2013Turing machine over $\\Gamma_0$ supported by the finite set of states $S$, and let $l$ and $l'$ be words of non-blank symbols of $\\Gamma_0$. If $M$, started on $l$, reaches in exactly $n$ steps a configuration $c$ in which it halts, and the tape of $c$ holds $l'$ starting at the head with blanks everywhere else, then the\u2026","title":"The single-tape simulation outputs within one extra step"},{"id":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/outputs-of-tm0","kind":"node","lean":"Cslib.Turing.SingleTapeTM.outputs_ofTM0","parent":34387,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations/index.html#outputs-of-tm0","scope":"TheoryOfComputation/turing-machines/single-tape/equivalent-tape-representations","status":"fully_proved","summary":"Let $\\Gamma_0$ be a finite alphabet with at least one non-blank symbol, let $M$ be a Post\u2013Turing machine over $\\Gamma_0$ supported by the finite set of states $S$, and let $l$ and $l'$ be words of non-blank symbols of $\\Gamma_0$. If $M$, started on $l$, halts in a configuration whose tape holds $l'$ starting at the head, with blanks everywhere else, then the single-tape machine simulating $M$ outputs $l'$ on the\u2026","title":"The single-tape simulation has the same output"},{"data_url":"scopes/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack.json","id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack","kind":"scope","parent":34347,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/index.html","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack","status":"fully_proved","summary":"Multi-tape and multi-stack (TM2) machines and their program library: bit primitives, cleanup, composition, stack embeddings and generic host machines that run component machines, and state relabelling.","title":"Multi-tape and multi-stack machines"},{"data_url":"scopes/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines.json","id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","kind":"scope","navigation":"module","parent":34438,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","title":"Multi-tape Turing machines"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/step-with","kind":"node","lean":"Cslib.Turing.MultiTapeTM.stepWith","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#step-with","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $\\delta$ be a transition function of a multi-tape Turing machine with $k$ work tapes, assigning an action to each state, input symbol and tuple of $k$ work tape symbols, where a symbol may be blank. The successor of a configuration $c$ under $\\delta$ is $c$ itself if $c$ is halted, and otherwise the configuration obtained by applying to $c$ the action $\\delta(q, a, b)$, where $q$ is the state of $c$, $a$ the\u2026","title":"Successor under a transition function"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/step-of-tr","kind":"node","lean":"Cslib.Turing.MultiTapeTM.step_ofTr","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#step-of-tr","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be the deterministic multi-tape Turing machine with initial state $q_0$ and transition function $\\delta$. For every configuration $c$, the step of $M$ from $c$ is the successor of $c$ under $\\delta$: $c$ itself if $c$ is halted, and otherwise the result of applying to $c$ the action that $\\delta$ selects for the state of $c$ and the symbols under its input and work tape heads.","title":"Step of a machine given by a transition function"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/run-from-of-tr","kind":"node","lean":"Cslib.Turing.MultiTapeTM.runFrom_ofTr","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#run-from-of-tr","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $\\delta$ be a transition function that assigns an action to every state, symbol or blank read on the input tape, and tuple of symbols or blanks read on the work tapes, and let $M$ be the deterministic multi-tape Turing machine with initial state $q_0$ and transition function $\\delta$. Let $S_\\delta$ be the map on configurations that fixes every halted configuration and sends a running configuration in state $q$\u2026","title":"Runs of a machine given by a transition function"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/step-deterministic","kind":"node","lean":"Cslib.Turing.MultiTapeTM.step_deterministic","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#step-deterministic","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a deterministic multi-tape Turing machine, with the one-step relation it inherits as a nondeterministic machine. If a configuration $c$ steps to both $c_1$ and $c_2$, then $c_1 = c_2$.","title":"Step determinism"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/run-from-succ","kind":"node","lean":"Cslib.Turing.MultiTapeTM.runFrom_succ'","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#run-from-succ","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"For a configuration $c$ of a deterministic multi-tape Turing machine and every $t \\in \\mathbb{N}$, the run from $c$ after $t + 1$ steps is in the same configuration as the run from the successor of $c$ after $t$ steps.","title":"Peeling off the first step of a run"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/run-from-of-halted","kind":"node","lean":"Cslib.Turing.MultiTapeTM.runFrom_of_halted","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#run-from-of-halted","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"If a configuration $c$ of a deterministic multi-tape Turing machine is halted, then for every $t \\in \\mathbb{N}$ the run from $c$ is in configuration $c$ after $t$ steps.","title":"Halted configurations are fixed"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/halted-run-from-mono","kind":"node","lean":"Cslib.Turing.MultiTapeTM.halted_runFrom_mono","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#halted-run-from-mono","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"For a configuration $c$ of a deterministic multi-tape Turing machine and steps $s \\le t$, if the run from $c$ is halted after $s$ steps, then it is halted after $t$ steps.","title":"Halting persists along a run"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/halts-at-zero-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.haltsAt_zero_iff","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#halts-at-zero-iff","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"A deterministic multi-tape Turing machine started in a configuration $c$ halts at step $0$ if and only if $c$ is halted.","title":"Halting at step zero"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/halts-at-add","kind":"node","lean":"Cslib.Turing.MultiTapeTM.haltsAt_add","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#halts-at-add","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a configuration of a deterministic multi-tape Turing machine and let $m, n \\in \\mathbb{N}$. If the run from $c$ is not halted after any $s < m$ steps, and the run from the configuration reached after $m$ steps halts at step $n$ (it is halted after $n$ steps and not halted after fewer), then the run from $c$ halts at step $m + n$.","title":"Run composition for halting"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/run-from","kind":"node","lean":"Cslib.Turing.MultiTapeTM.HaltsAt.runFrom","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#run-from","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a deterministic multi-tape Turing machine, $c$ a configuration of $M$ and $m, n$ natural numbers. If $M$ started in $c$ halts at step $m + n$, that is, it is halted after $m + n$ steps and not halted after any fewer, then $M$ started in the configuration reached from $c$ after $m$ steps halts at step $n$.","title":"Halting step after a run prefix"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/halted-run-from-of-runs-in-time","kind":"node","lean":"Cslib.Turing.MultiTapeTM.halted_runFrom_of_runsInTime","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#halted-run-from-of-runs-in-time","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"If a deterministic multi-tape Turing machine runs in time $t$ on input $w$, then the run from the initial configuration on $w$ is halted after $t$ steps.","title":"A time bound forces halting"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/runs-in-time-iff-halted","kind":"node","lean":"Cslib.Turing.MultiTapeTM.runsInTime_iff_halted","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#runs-in-time-iff-halted","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a deterministic multi-tape Turing machine, $w$ an input and $t$ a natural number. Then $M$ runs in time $t$ on $w$, that is, every computation path of $M$ on $w$ with at least $t$ steps ends in a halted configuration, if and only if $M$ started in its initial configuration on $w$ is halted after $t$ steps.","title":"Time bounds of deterministic machines"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/computes-unique","kind":"node","lean":"Cslib.Turing.MultiTapeTM.computes_unique","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#computes-unique","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"A deterministic machine computes at most one output on an input.","title":"Output determinism"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/computes-iff-exists-halts-at","kind":"node","lean":"Cslib.Turing.MultiTapeTM.computes_iff_exists_haltsAt","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#computes-iff-exists-halts-at","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"A deterministic multi-tape Turing machine computes the output $o$ on input $w$ if and only if there is a step $t$ at which the run from the initial configuration on $w$ halts (it is halted after $t$ steps and not halted after any $s < t$ steps) and the configuration it reaches after $t$ steps has output tape $o$.","title":"Computing an output means halting with it"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/computes-iff-output-of-runs-in-time","kind":"node","lean":"Cslib.Turing.MultiTapeTM.computes_iff_output_of_runsInTime","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#computes-iff-output-of-runs-in-time","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"If a deterministic multi-tape Turing machine runs in time $t$ on input $w$, then for every word $o$ the machine computes $o$ on $w$ if and only if the configuration reached from the initial configuration on $w$ after $t$ steps has output tape $o$.","title":"Output at a time bound"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/space-of-deterministic","kind":"node","lean":"Cslib.Turing.MultiTapeTM.space_ofDeterministic","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#space-of-deterministic","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a deterministic multi-tape Turing machine, $x$ an input and $t \\in \\mathbb{N}$. The number of work-tape cells visited along the computation path of $M$ on $x$ made of the first $t$ steps from the initial configuration equals the space used by the run of $M$ from the initial configuration on $x$ up to step $t$. Both count the visited cells on each work tape separately and sum over the work tapes.","title":"Space of the deterministic computation path"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/runs-in-space-iff-forall-space-used-le","kind":"node","lean":"Cslib.Turing.MultiTapeTM.runsInSpace_iff_forall_spaceUsed_le","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#runs-in-space-iff-forall-space-used-le","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a deterministic multi-tape Turing machine, $x$ an input and $s \\in \\mathbb{N}$. Then $M$ runs in space $s$ on $x$, that is, every computation path of $M$ on $x$ visits at most $s$ work-tape cells, if and only if for every $t \\in \\mathbb{N}$ the run of $M$ from the initial configuration on $x$ visits at most $s$ work-tape cells within its first $t$ steps. Visited cells are counted on each work tape\u2026","title":"Space bounds of deterministic machines"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/relabel-action","kind":"node","lean":"Cslib.Turing.MultiTapeTM.relabelAction","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#relabel-action","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"For a map $f : Q \\to Q'$ of states and an action $a$ of a machine with $k$ work tapes and states $Q$, the relabelled action is the action with states $Q'$ that moves the input head, writes and moves on the work tapes and outputs exactly as $a$ does, and whose successor state is $f(q)$ when $a$ goes to state $q$ and is halting when $a$ halts.","title":"Relabelling the successor state of an action"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/apply-relabel-action","kind":"node","lean":"Cslib.Turing.MultiTapeTM.apply_relabelAction","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#apply-relabel-action","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $f : Q \\to Q'$ be a map of states; relabelling by $f$ replaces a state $q$ by $f(q)$ and keeps the halting state. For every action $a$ of a machine with $k$ work tapes and states $Q$ and every configuration $c$ of such a machine, applying $a$ with its successor state relabelled by $f$ to $c$ with its state relabelled by $f$ gives the configuration obtained by applying $a$ to $c$ and then relabelling its state\u2026","title":"Applying a relabelled action"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/is-relabelling","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#is-relabelling","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be deterministic machines with $k$ work tapes over the same alphabet, with state sets $Q$ and $Q'$, and let $f : Q \\to Q'$. Then $M'$ simulates $M$ up to the relabelling $f$ when the initial state of $M'$ is $f(q_0)$, where $q_0$ is the initial state of $M$, and for every state $q \\in Q$, every possibly blank symbol $a$ under the input head and every tuple $w$ of possibly blank symbols under the\u2026","title":"Simulation up to a state relabelling"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/halted-map-state-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.halted_mapState_iff","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#halted-map-state-iff","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $f$ be a map of states. For every configuration $c$, the configuration obtained from $c$ by replacing its state $q$ by $f(q)$, and keeping the halting state, is halted if and only if $c$ is halted.","title":"Relabelling preserves halting"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/refl","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.refl","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#refl","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Every deterministic machine with $k$ work tapes simulates itself up to the identity relabelling of its states.","title":"Every machine is a relabelling of itself"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/trans","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.trans","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#trans","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$, $M'$ and $M''$ be multi-tape Turing machines with the same number of work tapes and the same alphabet, with state sets $Q$, $Q'$ and $Q''$. If $M'$ simulates $M$ up to the state relabelling $f : Q \\to Q'$ and $M''$ simulates $M'$ up to the state relabelling $g : Q' \\to Q''$, then $M''$ simulates $M$ up to the state relabelling $g \\circ f$.","title":"Relabellings compose"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/step","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.step","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#step","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be multi-tape Turing machines with the same number of work tapes and the same alphabet, with state sets $Q$ and $Q'$, such that $M'$ simulates $M$ up to the state relabelling $f : Q \\to Q'$. For a configuration $c$ of $M$, write $f_*(c)$ for the configuration of $M'$ obtained by replacing the state $q$ of $c$ by $f(q)$ and keeping the input head, the work tapes, the work heads and the output (a\u2026","title":"Relabelled machines step in lockstep"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/run-from-2","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.runFrom","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#run-from-2","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be multi-tape Turing machines with the same number of work tapes and the same alphabet, with state sets $Q$ and $Q'$, such that $M'$ simulates $M$ up to the state relabelling $f : Q \\to Q'$. For a configuration $c$ of $M$, write $f_*(c)$ for the configuration of $M'$ obtained by replacing the state $q$ of $c$ by $f(q)$ and keeping the input head, the work tapes, the work heads and the output (a\u2026","title":"Lockstep runs"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/init-cfg","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.initCfg","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#init-cfg","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be deterministic machines with $k$ work tapes over the same alphabet such that $M'$ simulates $M$ up to a state relabelling $f$. Then for every input word $x$, the initial configuration of $M'$ on $x$ is the initial configuration of $M$ on $x$ with its state $q$ replaced by $f(q)$.","title":"Initial configurations under relabelling"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/run-from-init-cfg","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.runFrom_initCfg","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#run-from-init-cfg","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be deterministic machines with $k$ work tapes over the same alphabet such that $M'$ simulates $M$ up to a state relabelling $f$. Then for every input word $x$ and every $t \\in \\mathbb{N}$, the configuration of $M'$ after $t$ steps on input $x$ is the configuration of $M$ after $t$ steps on input $x$ with its state relabelled by $f$: a state $q$ becomes $f(q)$ and the halting state stays halting.","title":"Lockstep runs from the initial configuration"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/computes-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.computes_iff","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#computes-iff","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be deterministic machines with $k$ work tapes over the same alphabet such that $M'$ simulates $M$ up to a state relabelling $f$. Then for all words $x$ and $o$ over the alphabet, $M'$ started on input $x$ eventually halts with output $o$ if and only if $M$ does.","title":"Relabelling preserves what a machine computes"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/runs-in-time-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.runsInTime_iff","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#runs-in-time-iff","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be multi-tape Turing machines with the same number of work tapes and the same alphabet such that $M'$ simulates $M$ up to a state relabelling $f$. Then for every input $x$ and every $t \\in \\mathbb{N}$, $M'$ runs in time $t$ on $x$ if and only if $M$ runs in time $t$ on $x$.","title":"Relabelling preserves time bounds"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/runs-in-space-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.runsInSpace_iff","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#runs-in-space-iff","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be multi-tape Turing machines with the same number of work tapes and the same alphabet such that $M'$ simulates $M$ up to a state relabelling $f$. Then for every input $x$ and every $s \\in \\mathbb{N}$, $M'$ runs in space $s$ on $x$ if and only if $M$ runs in space $s$ on $x$.","title":"Relabelling preserves space bounds"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/computes-fun-in-time-and-space-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.computesFunInTimeAndSpace_iff","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#computes-fun-in-time-and-space-iff","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be deterministic machines with $k$ work tapes over the same alphabet such that $M'$ simulates $M$ up to a state relabelling $f$. Let $e_{\\mathrm{in}}$ and $e_{\\mathrm{out}}$ be injective encodings of types $\\alpha$ and $\\beta$ as words over the alphabet, $g : \\alpha \\to \\beta$ a function and $t, s : \\alpha \\to \\mathbb{N}$ bounds. Then $M'$ computes $g$ with respect to these encodings in time $t$\u2026","title":"Relabelling preserves computing a function within time and space bounds"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/decides-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.IsRelabelling.decides_iff","parent":34439,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines/index.html#decides-iff","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ and $M'$ be deterministic machines with $k$ work tapes over the alphabet $\\{\\mathrm{false}, \\mathrm{true}\\}$ such that $M'$ simulates $M$ up to a state relabelling $f$. Then for every set $L$ of binary words, $M'$ decides $L$ if and only if $M$ decides $L$.","title":"Relabelling preserves deciding a language"},{"data_url":"scopes/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines.json","id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","kind":"scope","navigation":"module","parent":34438,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","title":"Programming multi-stack machines"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/flag","kind":"node","lean":"MetaMathlibExt.TM2Program.Flag","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#flag","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"A flag on a type $\\sigma$ of machine states consists of a map $\\mathrm{tag} : \\mathrm{Bool} \\to \\sigma$ that stores a bit in a state and a map $\\mathrm{untag} : \\sigma \\to \\mathrm{Bool}$ that reads it back, such that $\\mathrm{untag}(\\mathrm{tag}(\\beta)) = \\beta$ for every bit $\\beta$. The program primitives use it to branch on whether a pop found a symbol.","title":"Flag bit of a machine state"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/iter-none","kind":"node","lean":"MetaMathlibExt.TM2Program.iter_none","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#iter-none","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $f : \\sigma \\to \\mathrm{Option}\\,\\sigma$ be a partial step function and let $\\hat f$ be its lift to $\\mathrm{Option}\\,\\sigma$, which sends $\\mathrm{none}$ to $\\mathrm{none}$ and $\\mathrm{some}\\,x$ to $f(x)$. Then $\\hat f^{\\,n}(\\mathrm{none}) = \\mathrm{none}$ for every $n \\in \\mathbb{N}$.","title":"A halted run stays halted"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/run","kind":"node","lean":"MetaMathlibExt.TM2Program.Run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a TM2 machine, given by a statement for each label, and let $c$ and $d$ be configurations. For $n \\in \\mathbb{N}$, $M$ runs from $c$ to $d$ in exactly $n$ steps when applying the step function of $M$ $n$ times, starting at $c$, is defined at every step (no configuration before the last one has halted) and ends at $d$.","title":"Exact-length runs of a TM2 machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/single","kind":"node","lean":"MetaMathlibExt.TM2Program.Run.single","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#single","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a TM2 machine, $l$ a label, $v$ an internal state, $S$ an assignment of a list of symbols to each stack, and $d$ a configuration. If executing the statement $M(l)$ from state $v$ and stacks $S$ yields $d$, then $M$ runs in exactly one step from the configuration with label $l$, state $v$ and stacks $S$ to $d$.","title":"A single step executes the statement at the current label"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/cnt","kind":"node","lean":"MetaMathlibExt.TM2Program.cnt","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#cnt","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"The unary counter with unit symbol $u$ and value $n \\in \\mathbb{N}$ is the list $u^n$ consisting of exactly $n$ copies of $u$.","title":"Unary counter"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/incr","kind":"node","lean":"MetaMathlibExt.TM2Program.incr","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#incr","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a stack $k$, a symbol $u$ of stack $k$ (the unit of a unary counter) and a label $\\ell$, this statement pushes $u$ onto stack $k$ and then goes to label $\\ell$.","title":"Counter increment"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/decr","kind":"node","lean":"MetaMathlibExt.TM2Program.decr","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#decr","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a stack $k$ and labels $\\ell_0$ and $\\ell_+$, this machine statement pops stack $k$, stores in the state whether a symbol was popped, and goes to $\\ell_+$ if a symbol was popped and to $\\ell_0$ if stack $k$ was empty.","title":"Decrement"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/push-all","kind":"node","lean":"MetaMathlibExt.TM2Program.pushAll","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#push-all","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $ps$ be a list of pairs $(k, x)$ of a stack index $k$ and a symbol $x$ of the alphabet of stack $k$, and let $q$ be a statement of a TM2 machine. This is the statement that, for each pair $(k, x)$ of $ps$ in list order, pushes $x$ onto stack $k$, and then continues with $q$.","title":"Pushing symbols onto several stacks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/add-u","kind":"node","lean":"MetaMathlibExt.TM2Program.addU","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#add-u","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $ps$ be a list of pairs $(j, x)$ of a stack $j$ and a symbol $x$ of stack $j$, let $i \\in \\mathbb{N}$ and let $S$ be stacks. The resulting stacks are as follows: if $(k, x)$ is the first pair in $ps$ whose stack is $k$, stack $k$ holds $i$ copies of $x$ followed by $S_k$ (so the copies are on top); if no pair in $ps$ has stack $k$, stack $k$ holds $S_k$.","title":"Adding units to target stacks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/xfer","kind":"node","lean":"MetaMathlibExt.TM2Program.xfer","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#xfer","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a stack $k$, a list $ps$ of pairs $(t, x)$ of a stack $t$ and a symbol $x$ of stack $t$, and labels $\\ell$ and $\\ell'$, this machine statement pops stack $k$, stores in the state whether a symbol was popped, and branches on it: if a symbol was popped, it pushes $x$ onto $t$ for each pair $(t, x)$ of $ps$ in list order and goes to $\\ell$; if stack $k$ was empty, it goes to $\\ell'$.","title":"Counter transfer loop"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/xfer-run","kind":"node","lean":"MetaMathlibExt.TM2Program.xfer_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#xfer-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a multi-stack machine whose state carries a flag bit, and suppose that at label $\\ell$ it executes the transfer loop that empties stack $k$ into the targets $p_1, \\dots, p_m$ and then goes to label $\\ell'$. Here each target $p_i$ is a pair of a stack $t_i$ and a symbol $x_i$ of its alphabet, the stacks $t_1, \\dots, t_m$ are distinct, and $k$ is none of them. Let $u$ be a symbol of stack $k$. Then for\u2026","title":"Running a counter transfer"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/cmp-step","kind":"node","lean":"MetaMathlibExt.TM2Program.cmpStep","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#cmp-step","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For stacks $a, b, t_a, t_b$, symbols $u_{t_a}$ of stack $t_a$ and $u_{t_b}$ of stack $t_b$, and labels $\\mathit{self}$, $\\mathit{lt}$, $\\mathit{eq}$ and $\\mathit{gt}$, this machine statement pops stack $a$ and then stack $b$; after each pop it stores in the state whether the pop found a symbol and branches on it. If both pops found a symbol, it pushes $u_{t_a}$ onto $t_a$, then $u_{t_b}$ onto $t_b$, and goes to\u2026","title":"One round of counter comparison"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/cmp-loop-run","kind":"node","lean":"MetaMathlibExt.TM2Program.cmpLoop_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#cmp-loop-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $a, b, t_a, t_b$ be pairwise distinct stacks with unit symbols $u_a, u_b, u_{t_a}, u_{t_b}$, and let $r$ map each ordering to a label. Suppose the machine $M$ has at label $\\ell$ the comparison round of $a$ and $b$ with scratch stacks $t_a, t_b$ and units $u_{t_a}, u_{t_b}$ that repeats at $\\ell$ and exits to $r(\\mathrm{lt})$, $r(\\mathrm{eq})$ and $r(\\mathrm{gt})$: it pops $a$ and $b$, pushes $u_{t_a}$ onto\u2026","title":"The comparison loop"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/cmp-run","kind":"node","lean":"MetaMathlibExt.TM2Program.cmp_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#cmp-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a TM2 machine whose state type carries a flag bit, let $a, b, t_a, t_b$ be pairwise distinct stacks with symbols $u_a$, $u_b$, $u_{t_a}$, $u_{t_b}$ of their alphabets, let $\\ell$ be a label and let $r$, $r'$, $o$ map each ordering to a label. Suppose that $M$ has at $\\ell$ the comparison round of $a$ and $b$ with scratch stacks $t_a, t_b$ and units $u_{t_a}, u_{t_b}$ that goes to $\\ell$ again and exits\u2026","title":"Non-destructive comparison of two counters"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/sum-from","kind":"node","lean":"MetaMathlibExt.TM2Program.sumFrom","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#sum-from","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For $f : \\mathbb{N} \\to \\mathbb{N}$ and $i, j \\in \\mathbb{N}$, $\\mathrm{sumFrom}(f, i, j) = f(i) + f(i+1) + \\dots + f(i+j-1)$ is the sum of $f(t)$ over the $j$ indices $t$ with $i \\le t < i + j$; it is $0$ when $j = 0$.","title":"Sum of consecutive values"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/sum-from-le","kind":"node","lean":"MetaMathlibExt.TM2Program.sumFrom_le","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#sum-from-le","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ and $c, i, j \\in \\mathbb{N}$. If $f(t) \\le c$ for every $t$ with $i \\le t < i + j$, then $\\mathrm{sumFrom}(f, i, j) = f(i) + f(i+1) + \\dots + f(i+j-1)$ is at most $j \\cdot c$.","title":"Termwise bound on a sum of consecutive values"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/for-head","kind":"node","lean":"MetaMathlibExt.TM2Program.forHead","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#for-head","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For stacks $c$ and $c'$, a symbol $u'$ of stack $c'$ and labels $b$ and $r$, the loop head is the statement that pops stack $c$, replaces the machine state by the state whose flag bit records whether a symbol was popped, and branches on that bit: if a symbol was popped, it pushes $u'$ onto the index stack $c'$ and goes to label $b$ (the loop body); if $c$ was empty, it goes to label $r$.","title":"Loop head of a for loop over a counter"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/for-loop-run","kind":"node","lean":"MetaMathlibExt.TM2Program.forLoop_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#for-loop-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a multi-stack machine whose states carry a flag bit, with $\\mathrm{tag}(\\beta)$ the state storing the bit $\\beta$; write $(\\ell, v, S)$ for the configuration at label $\\ell$ with state $v$ and stack contents $S$, and $u^m$ for a stack holding exactly $m$ copies of the symbol $u$. Let $c \\ne c'$ be stacks with a symbol $u$ of stack $c$ and a symbol $u'$ of stack $c'$, and let $h, b, r, e$ be labels such\u2026","title":"Exact running time of a for loop over a counter"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/push-list","kind":"node","lean":"MetaMathlibExt.TM2Program.pushList","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#push-list","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $k$ be a stack index, $xs = [x_1, \\dots, x_m]$ a list of symbols of the alphabet of stack $k$, and $q$ a statement of a TM2 machine. This is the statement that pushes $x_1, \\dots, x_m$ onto stack $k$ in that order, so that $x_m$ ends on top when $m \\ge 1$, and then continues with $q$.","title":"Pushing a list of symbols onto one stack"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/emit","kind":"node","lean":"MetaMathlibExt.TM2Program.emit","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#emit","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a stack $k$, a list $(x_1, \\dots, x_m)$ of symbols of stack $k$ and a label $\\ell$, this statement pushes $x_1, \\dots, x_m$ onto stack $k$ in this order, so that $x_m$ ends on top, and then goes to label $\\ell$. The machine executes it in a single step.","title":"Pushing a constant list"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/copy-run","kind":"node","lean":"MetaMathlibExt.TM2Program.copy_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#copy-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $a, c, t$ be pairwise distinct stacks with unit symbols $u_a, u_c, u_t$. Write $\\mathrm{xfer}(k, ps, \\ell, \\ell')$ for the transfer loop at label $\\ell$ that pops stack $k$ until it is empty, pushing $x$ onto stack $j$ for each pair $(j, x)$ in $ps$ per popped symbol, and then goes to $\\ell'$. Suppose the machine $M$ has $M(l_1) = \\mathrm{xfer}(a, [(c, u_c), (t, u_t)], l_1, l_2)$ and\u2026","title":"Copying a counter through a scratch stack"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/add-run","kind":"node","lean":"MetaMathlibExt.TM2Program.add_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#add-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $a, b, c, t$ be stacks with $a \\ne c$, $a \\ne t$, $b \\ne c$, $b \\ne t$ and $c \\ne t$, with unit symbols $u_a, u_b, u_c, u_t$. Write $\\mathrm{xfer}(k, ps, \\ell, \\ell')$ for the transfer loop at label $\\ell$ that pops stack $k$ until it is empty, pushing $x$ onto stack $j$ for each pair $(j, x)$ in $ps$ per popped symbol, and then goes to $\\ell'$. Suppose the machine $M$ has\u2026","title":"Adding two counters into a third"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/mul-run","kind":"node","lean":"MetaMathlibExt.TM2Program.mul_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#mul-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a multi-stack machine whose states carry a flag bit, with $\\mathrm{tag}(\\mathrm{false})$ the state storing the bit false; write $(\\ell, v, S)$ for the configuration at label $\\ell$ with state $v$ and stack contents $S$, and $u^m$ for a stack holding exactly $m$ copies of the symbol $u$. Let $a, a', b, c, t$ be pairwise distinct stacks with symbols $u_a, u_{a'}, u_b, u_c, u_t$ of the respective stacks,\u2026","title":"Multiplying unary counters"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/rep","kind":"node","lean":"MetaMathlibExt.TM2Program.rep","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#rep","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a list $ys$ and $n \\in \\mathbb{N}$, $\\mathrm{rep}(ys, n)$ is the concatenation of $n$ copies of $ys$; it is the empty list when $n = 0$.","title":"Repeated block"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/emit-r","kind":"node","lean":"MetaMathlibExt.TM2Program.emitR","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#emit-r","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a stack $k$, a list $(y_1, \\dots, y_m)$ of symbols of stack $k$ and a label $\\ell$, this statement pushes $y_m, \\dots, y_1$ onto stack $k$ in this order and then goes to label $\\ell$; it is the constant-list push of the reversed list. Afterwards $y_1, \\dots, y_m$ sit on top of stack $k$, read from the top down.","title":"Prepending a constant list"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/xfer-e","kind":"node","lean":"MetaMathlibExt.TM2Program.xferE","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#xfer-e","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $k$ and $o$ be stack indices, $ys$ a list of symbols of the alphabet of stack $o$, and $\\ell$ and $\\ell'$ labels. This is the TM2 statement that pops stack $k$ and sets the internal state to the flag bit recording whether a symbol was popped. If one was, it prepends $ys$ to stack $o$ (so that $ys$ reads from the top down) and goes to $\\ell$; if stack $k$ was empty, it goes to $\\ell'$.","title":"Draining a counter while prepending a block"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/emit-diff-run","kind":"node","lean":"MetaMathlibExt.TM2Program.emitDiff_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#emit-diff-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $P, Q, T_1, T_2, o$ be pairwise distinct stacks, with unit symbols $u_P, u_Q, u_{T_1}, u_{T_2}$ for the first four, and let $ys$ be a list of symbols of stack $o$. Write $\\mathrm{xfer}(k, ps, \\ell, \\ell')$ for the transfer loop at label $\\ell$ that pops stack $k$ until it is empty, pushing $x$ onto stack $j$ for each pair $(j, x)$ in $ps$ per popped symbol, and then goes to $\\ell'$. Let $r, r_B, \\mathit{out}$\u2026","title":"Gap emission"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/reach","kind":"node","lean":"MetaMathlibExt.TM2Program.Reach","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#reach","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a TM2 machine $M$ and configurations $c$ and $d$, $d$ is reachable from $c$ when there is some $n \\in \\mathbb{N}$ such that $M$ runs from $c$ to $d$ in exactly $n$ steps.","title":"Reachability for a TM2 machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/for-loop-reach","kind":"node","lean":"MetaMathlibExt.TM2Program.forLoop_reach","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#for-loop-reach","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a multi-stack machine whose states carry a flag bit, with $\\mathrm{tag}(\\beta)$ the state storing the bit $\\beta$; write $(\\ell, v, S)$ for the configuration at label $\\ell$ with state $v$ and stack contents $S$, and $u^m$ for a stack holding exactly $m$ copies of the symbol $u$. Let $c \\ne c'$ be stacks with a symbol $u$ of stack $c$ and a symbol $u'$ of stack $c'$, and let $h, b, r, e$ be labels such\u2026","title":"A for loop over a counter reaches its exit"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/run-le","kind":"node","lean":"MetaMathlibExt.TM2Program.RunLe","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#run-le","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a TM2 machine $M$, a bound $B \\in \\mathbb{N}$ and configurations $c$ and $d$, $M$ runs from $c$ to $d$ within $B$ steps when there is some $n \\le B$ such that $M$ runs from $c$ to $d$ in exactly $n$ steps.","title":"Runs within a step budget"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/bud","kind":"node","lean":"MetaMathlibExt.TM2Program.Bud","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#bud","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a machine $M$, numbers $K, X \\in \\mathbb{N}$ and configurations $c$ and $d$, the budgeted run from $c$ to $d$ with $K$ steps already spent and budget $X$ holds when there is an $n \\in \\mathbb{N}$ with $K + n \\le X$ such that $n$ steps of $M$ lead from $c$ to $d$.","title":"Budgeted run"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/for-loop-le","kind":"node","lean":"MetaMathlibExt.TM2Program.forLoop_le","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#for-loop-le","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a multi-stack machine whose states carry a flag bit, with $\\mathrm{tag}(\\beta)$ the state storing the bit $\\beta$; write $(\\ell, v, S)$ for the configuration at label $\\ell$ with state $v$ and stack contents $S$, and $u^m$ for a stack holding exactly $m$ copies of the symbol $u$. Let $c \\ne c'$ be stacks with a symbol $u$ of stack $c$ and a symbol $u'$ of stack $c'$, and let $h, b, r, e$ be labels such\u2026","title":"Time bound for a for loop over a counter"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/is-bit","kind":"node","lean":"MetaMathlibExt.TM2Program.isBit","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#is-bit","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $d$ be a decoder that reads each symbol either as a bit or as a marker. For an optional symbol $o$ (the top of a stack, absent when the stack is empty), the test is true when $o$ is a symbol that $d$ reads as a bit, and false when $o$ is absent or $d$ reads it as a marker.","title":"Bit test for the top of a stack"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/bit-of","kind":"node","lean":"MetaMathlibExt.TM2Program.bitOf","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#bit-of","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $d$ be a decoder that reads each symbol either as a bit or as a marker. For an optional symbol $o$ (the top of a stack, absent when the stack is empty), the bit of $o$ under $d$ is $x$ when $o$ is a symbol that $d$ reads as the bit $x$, and false when $o$ is absent or $d$ reads it as a marker.","title":"Bit on top of a stack"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/is-bit-block","kind":"node","lean":"MetaMathlibExt.TM2Program.IsBitBlock","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#is-bit-block","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $d$ be a decoder that reads each symbol either as a bit or as a marker. A list of symbols $s_1, \\dots, s_n$ is the bit block of a bit string $p$ under $d$ when $p$ has the same length $n$ and, for each $i$, $d$ reads $s_i$ as the $i$-th bit of $p$.","title":"Bit blocks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/ends-bit-block","kind":"node","lean":"MetaMathlibExt.TM2Program.EndsBitBlock","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#ends-bit-block","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $d$ be a decoder that reads each symbol either as a bit or as a marker. A list of symbols $A$ (the rest of a stack below a block) ends a bit block under $d$ when $A$ is empty or $d$ reads its first symbol as a marker.","title":"End of a bit block"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/move-b","kind":"node","lean":"MetaMathlibExt.TM2Program.moveB","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#move-b","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For stacks $a$ and $b$, a decoder $d$ reading each symbol of $a$ as a bit or a marker, an encoder $g$ from bits to symbols of $b$, and labels $\\mathit{self}$ and $\\mathit{next}$, the bit-moving loop from $a$ to $b$ is the statement that looks at the top of $a$: if it is a symbol that $d$ reads as a bit $x$, it pops it, pushes $g(x)$ onto $b$ and goes to $\\mathit{self}$; if it is a marker or $a$ is empty, it goes to\u2026","title":"Bit-moving loop"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/move-b-run","kind":"node","lean":"MetaMathlibExt.TM2Program.moveB_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#move-b-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $a \\ne b$ be stacks of a program $M$ whose state carries a flag bit, $d$ a decoder reading each symbol of $a$ as a bit or a marker, $g$ an encoder from bits to symbols of $b$, and $\\mathit{self}$, $\\mathit{next}$ labels such that $M$ runs at $\\mathit{self}$ the bit-moving loop from $a$ to $b$ with decoder $d$ and encoder $g$, looping at $\\mathit{self}$ and exiting to $\\mathit{next}$. Let $p = p_1 \\cdots p_n$ be\u2026","title":"Running the bit-moving loop"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/cmp-b","kind":"node","lean":"MetaMathlibExt.TM2Program.cmpB","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#cmp-b","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $a$ and $b$ be stacks with decoders $d_a$ and $d_b$ that read each symbol either as a bit or as a marker, let $t_a$ and $t_b$ be scratch stacks with bit encoders $e_a$ and $e_b$, and let $\\ell$, $\\ell_=$ and $\\ell_{\\ne}$ be labels. The comparison statement with loop label $\\ell$ performs one round of comparison: if the tops of $a$ and $b$ are the same bit $x$, it pops both, pushes $e_a(x)$ onto $t_a$ and\u2026","title":"Bit-block comparison statement"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/distinct4","kind":"node","lean":"MetaMathlibExt.TM2Program.Distinct4","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#distinct4","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Four stack indices $a$, $b$, $t_a$, $t_b$ are pairwise distinct: $a \\ne b$, $a \\ne t_a$, $a \\ne t_b$, $b \\ne t_a$, $b \\ne t_b$ and $t_a \\ne t_b$.","title":"Four pairwise distinct stacks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/cmp-b-prefix","kind":"node","lean":"MetaMathlibExt.TM2Program.cmpB_prefix","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#cmp-b-prefix","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Suppose the program has at label $\\ell$ the bit-block comparison statement of stacks $a$ and $b$ (decoders $d_a$, $d_b$) with scratch stacks $t_a$ and $t_b$ (encoders $e_a$, $e_b$) and loop label $\\ell$, and that $a$, $b$, $t_a$, $t_b$ are pairwise distinct. Let $p = p_1 \\dots p_n$ be a bit string, and take a configuration at label $\\ell$, with any state, in which stack $a$ holds a list $X$ on top of $A$ and stack\u2026","title":"Comparison pops a common prefix"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/cmp-b-last","kind":"node","lean":"MetaMathlibExt.TM2Program.cmpB_last","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#cmp-b-last","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Suppose the program has at label $\\ell$ the bit-block comparison statement of stacks $a$ and $b$ (decoders $d_a$, $d_b$; scratch stacks arbitrary) with loop label $\\ell$, equality label $\\ell_=$ and mismatch label $\\ell_{\\ne}$. From a configuration at label $\\ell$, with any state, in which the tops of $a$ and $b$ are not the same bit, the program takes one step to a configuration with the same stacks and some\u2026","title":"Deciding step of the comparison"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/cmp-b-run","kind":"node","lean":"MetaMathlibExt.TM2Program.cmpB_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#cmp-b-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Suppose the program has at label $\\ell$ the bit-block comparison statement of stacks $a$ and $b$ (decoders $d_a$, $d_b$) with scratch stacks $t_a$ and $t_b$ (encoders $e_a$, $e_b$), loop label $\\ell$, equality label $\\ell_=$ and mismatch label $\\ell_{\\ne}$, and that $a$, $b$, $t_a$, $t_b$ are pairwise distinct. Let $u$ and $w$ be bit strings, and take a configuration at label $\\ell$, with any state, in which stack\u2026","title":"Correctness of the bit-block comparison"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/take-eq-map","kind":"node","lean":"MetaMathlibExt.TM2Program.IsBitBlock.take_eq_map","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#take-eq-map","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $d$ be a decoder reading symbols as bits or markers and $g$ an encoder from bits to symbols such that $g(b) = x$ whenever $d$ reads $x$ as the bit $b$. If the symbol list $xs$ is a bit block for the bit list $u$ under $d$ and $c = c_1 \\cdots c_m$ is a prefix of $u$, then the first $m$ symbols of $xs$ are $g(c_1), \\dots, g(c_m)$.","title":"Recovering a bit block from its bits"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/move-b-restore","kind":"node","lean":"MetaMathlibExt.TM2Program.moveB_restore","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#move-b-restore","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $t \\ne a$ be stacks with decoders $d_t$ and $d_a$ that read each symbol either as a bit or as a marker. Let $e_t$ be a bit encoder for $t$ that $d_t$ reads back ($d_t$ reads $e_t(x)$ as the bit $x$), and let $g_a$ be a bit encoder for $a$ that inverts $d_a$ ($g_a(x) = s$ whenever $d_a$ reads the symbol $s$ as the bit $x$). Suppose the program has at label $\\ell$ the bit-block move statement from $t$ (decoder\u2026","title":"Restoring a stack after a comparison"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/pop-all","kind":"node","lean":"MetaMathlibExt.TM2Program.popAll","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#pop-all","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a list $ks$ of stacks and a statement $q$, this is the statement that pops the stacks listed in $ks$ in order, once per entry (an empty stack stays empty), after each pop replacing the state by the one storing the old flag or-ed with whether that pop found a symbol, and then continues with $q$.","title":"Popping a list of stacks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/clean-s","kind":"node","lean":"MetaMathlibExt.TM2Program.cleanS","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#clean-s","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"For a list $ks$ of stacks and labels $\\mathit{self}$ and $\\mathit{next}$, the cleanup round is the statement that sets the state to the one storing the flag value false, pops the stacks listed in $ks$ in order, once per entry (an empty stack stays empty), while recording in the flag whether some pop found a symbol, and then goes to $\\mathit{self}$ if one did and to $\\mathit{next}$ otherwise.","title":"Cleanup round"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/clean-run","kind":"node","lean":"MetaMathlibExt.TM2Program.clean_run","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#clean-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $ks$ be a list of pairwise distinct stacks of a program $M$ whose state carries a flag bit, and $\\mathit{self}$, $\\mathit{next}$ labels such that $M$ runs at $\\mathit{self}$ the cleanup round for $ks$, looping at $\\mathit{self}$ and exiting to $\\mathit{next}$. Let $m$ be a natural number and $S$ stack contents in which every stack of $ks$ holds at most $m$ symbols. Then for every state $v$, $M$ runs in at most\u2026","title":"Cleanup empties its stacks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/semb","kind":"node","lean":"MetaMathlibExt.TM2Embedding.SEmb","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#semb","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Given a family of stack alphabets $\\Gamma$ indexed by the stacks $K$ of a sub-machine and a family $\\Gamma'$ indexed by the stacks $K'$ of a host, a stack embedding consists of an injective map $e : K \\to K'$ renaming the stacks together with, for each stack $k \\in K$, a bijection $\\iota_k : \\Gamma_k \\to \\Gamma'_{e(k)}$ from the alphabet of $k$ onto the alphabet of its image.","title":"Stack embedding"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/map-s","kind":"node","lean":"MetaMathlibExt.TM2Embedding.mapS","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#map-s","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $E$ be a stack embedding with stack renaming $e$ and alphabet bijections $\\iota_k$, and let $\\varphi$ map the sub-machine's labels to the host's labels. The translation of a sub-machine statement is the host statement obtained recursively, where $s$ denotes the current state. A push of the symbol $f(s)$ on stack $k$ becomes a push of $\\iota_k(f(s))$ on stack $e(k)$. A peek or pop on stack $k$ becomes a peek or\u2026","title":"Translating a statement along a stack embedding"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/embeds","kind":"node","lean":"MetaMathlibExt.TM2Embedding.Embeds","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#embeds","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $E$ be a stack embedding and $\\varphi$ a map from the labels of a machine $M$ to the labels of a host machine $M'$ over the same internal states. Then $M$ embeds in $M'$ along $E$ and $\\varphi$ when, for every label $\\ell$ of $M$, either the statement of $M'$ at $\\varphi(\\ell)$ is the statement of $M$ at $\\ell$ translated along $E$ and $\\varphi$, or the statement of $M$ at $\\ell$ is the halt statement (in which\u2026","title":"Embedding of a machine in a host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/agree","kind":"node","lean":"MetaMathlibExt.TM2Embedding.Agree","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#agree","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $E$ be a stack embedding with stack renaming $e$ and alphabet bijections $\\iota_k$. Stack contents $S$ of the sub-machine and $S'$ of the host agree when, for every sub-machine stack $k$, the host stack $e(k)$ holds the list obtained from $S(k)$ by applying $\\iota_k$ to each symbol.","title":"Agreement of host and sub-machine stacks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/frame","kind":"node","lean":"MetaMathlibExt.TM2Embedding.Frame","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#frame","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $E$ be a stack embedding with stack renaming $e$. Two contents $S'$ and $S''$ of the host's stacks satisfy the frame condition when every host stack $k'$ outside the image of $e$ has the same contents in both: $S''(k') = S'(k')$ whenever $e(k) \\ne k'$ for every sub-machine stack $k$.","title":"Frame condition for a stack embedding"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/step-aux-map-s","kind":"node","lean":"MetaMathlibExt.TM2Embedding.stepAux_mapS","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#step-aux-map-s","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $E$ be a stack embedding with stack renaming $e$, and let $\\varphi$ map sub-machine labels to host labels. Let $q$ be a sub-machine statement, $v$ a state, $S$ sub-machine stacks and $S'$ host stacks that agree with $S$. Let executing $q$ from state $v$ and stacks $S$ yield the next label $\\ell$ (possibly none), state $w$ and stacks $T$. Then there are host stacks $T'$ such that executing the translation of $q$\u2026","title":"Executing a translated statement"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/run-embed","kind":"node","lean":"MetaMathlibExt.TM2Embedding.run_embed","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#run-embed","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $E$ be a stack embedding with stack renaming $e$, let $\\varphi$ map the labels of a machine $M$ to those of a host machine $M'$, and suppose $M$ embeds in $M'$ along $E$ and $\\varphi$. Suppose $M$ goes in exactly $n$ steps from label $\\ell$, state $v$ and stacks $S$ to a live label $\\ell'$, state $v'$ and stacks $T$, where the start may also be the halted configuration with no label (whose image under $\\varphi$\u2026","title":"Run embedding"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/run-le-embed","kind":"node","lean":"MetaMathlibExt.TM2Embedding.runLe_embed","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#run-le-embed","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Let $E$ be a stack embedding with stack renaming $e$, let $\\varphi$ map the labels of a machine $M$ to those of a host machine $M'$, and suppose $M$ embeds in $M'$ along $E$ and $\\varphi$. Suppose $M$ goes in at most $B$ steps from label $\\ell$, state $v$ and stacks $S$ to label $\\ell'$, state $v'$ and stacks $T$, and host stacks $S'$ agree with $S$. Then there are host stacks $T'$ such that $M'$ goes in at most\u2026","title":"Run embedding for bounded runs"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/run-embed-exact","kind":"node","lean":"MetaMathlibExt.TM2Embedding.run_embed_exact","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#run-embed-exact","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Exact program matching transports a run, including a halted endpoint, while preserving the local state, embedded stacks, untouched host stacks, and exact number of steps.","title":"Exact program matching transports a run with its step count"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/run-embed-exact-of-agree-frame","kind":"node","lean":"MetaMathlibExt.TM2Embedding.run_embed_exact_of_agree_frame","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#run-embed-exact-of-agree-frame","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Source-shaped transport to supplied host final stacks: agreement fixes embedded slots and the frame condition fixes every other slot, so the host endpoint is exact.","title":"Exact run transport to supplied host final stacks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/run-le-embed-exact","kind":"node","lean":"MetaMathlibExt.TM2Embedding.runLe_embed_exact","parent":34472,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines/index.html#run-le-embed-exact","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/programming-multi-stack-machines","status":"fully_proved","summary":"Exact matching transports a bounded run with arbitrary optional endpoints, retaining the same step witness and budget as well as agreement and the host frame.","title":"Exact matching transports a bounded run"},{"data_url":"scopes/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines.json","id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","kind":"scope","navigation":"module","parent":34438,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","title":"Composition of multi-stack machines"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/ck","kind":"node","lean":"MetaMathlibExt.TM2Composition.CK","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#ck","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"For finite multi-stack Turing machines $M_1$ and $M_2$ with stack index sets $K_1$ and $K_2$, the stacks of their composite machine are indexed by the disjoint union $K_1 \\sqcup K_2 \\sqcup \\{\\ast\\}$: the stacks of $M_1$, the stacks of $M_2$, and one scratch stack $\\ast$.","title":"Stack indices of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/x","kind":"node","lean":"MetaMathlibExt.TM2Composition.X","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#x","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"For a multi-stack Turing machine (a TM2 machine) $M_2$ with input stack $k_0$, this is the alphabet of the symbols on stack $k_0$ of $M_2$.","title":"Intermediate alphabet of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/c","kind":"node","lean":"MetaMathlibExt.TM2Composition.C\u0393","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#c","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be finite multi-stack Turing machines and let $X$ be the alphabet of the input stack of $M_2$. The stacks of their composite machine are the stacks of $M_1$, the stacks of $M_2$ and one scratch stack. Each stack of $M_1$ keeps its alphabet from $M_1$, each stack of $M_2$ keeps its alphabet from $M_2$, and the scratch stack has alphabet $X$.","title":"Stack alphabets of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/c-2","kind":"node","lean":"MetaMathlibExt.TM2Composition.C\u039b","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#c-2","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be finite multi-stack Turing machines with label sets $\\Lambda_1$ and $\\Lambda_2$, and let $X$ be the alphabet of the input stack of $M_2$. The label set of their composite machine is the disjoint union of $\\Lambda_1$, $\\Lambda_2$ and two copies of $X \\sqcup \\{\\bot\\}$, one for each copy loop: in each copy, $\\bot$ labels the step that pops and $x \\in X$ labels the step that pushes $x$.","title":"Labels of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/c-3","kind":"node","lean":"MetaMathlibExt.TM2Composition.C\u03c3","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#c-3","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be finite multi-stack Turing machines with internal state sets $\\sigma_1$ and $\\sigma_2$, and let $X$ be the alphabet of the input stack of $M_2$. The internal states of their composite machine are the triples in $\\sigma_1 \\times \\sigma_2 \\times (X \\sqcup \\{\\bot\\})$: a state of $M_1$, a state of $M_2$, and either the symbol just popped by a copy loop or $\\bot$ when there is none.","title":"States of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/pop1","kind":"node","lean":"MetaMathlibExt.TM2Composition.pop1","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#pop1","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"In the composite machine of $M_1$ and $M_2$, this is the label $\\mathrm{none}$ in the first copy-loop part of the label set; the composite program uses it to pop a symbol from $M_1$'s output stack.","title":"Pop label of the first copy loop"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/push1","kind":"node","lean":"MetaMathlibExt.TM2Composition.push1","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#push1","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"In the composite machine of $M_1$ and $M_2$, for a symbol $x$ of $M_2$'s input alphabet, this is the label $\\mathrm{some}\\ x$ in the first copy-loop part of the label set; the composite program uses it to push $x$ on the scratch stack.","title":"Push labels of the first copy loop"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/pop2","kind":"node","lean":"MetaMathlibExt.TM2Composition.pop2","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#pop2","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"In the composite machine of $M_1$ and $M_2$, this is the label $\\mathrm{none}$ in the second copy-loop part of the label set; the composite program uses it to pop a symbol from the scratch stack.","title":"Pop label of the second copy loop"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/push2","kind":"node","lean":"MetaMathlibExt.TM2Composition.push2","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#push2","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"In the composite machine of $M_1$ and $M_2$, for a symbol $x$ of $M_2$'s input alphabet, this is the label $\\mathrm{some}\\ x$ in the second copy-loop part of the label set; the composite program uses it to push $x$ on $M_2$'s input stack.","title":"Push labels of the second copy loop"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/tmp","kind":"node","lean":"MetaMathlibExt.TM2Composition.tmp","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#tmp","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"For multi-stack Turing machines (TM2 machines) $M_1$ and $M_2$ with stack index types $K_1$ and $K_2$, the composite machine has stack indices $K_1 \\oplus K_2 \\oplus \\mathrm{Unit}$; its scratch stack is the index given by the unique element of the $\\mathrm{Unit}$ summand.","title":"Scratch stack of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/lift","kind":"node","lean":"MetaMathlibExt.TM2Composition.lift\u2081","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#lift","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Given finite multi-stack Turing machines $M_1$ and $M_2$, a statement of $M_1$ is translated into a statement of their composite machine, whose state is a triple $(v_1, v_2, o)$, by recursion on the statement: an operation on a stack $k$ of $M_1$ acts on the copy of $k$ among the composite stacks, every read or update of $M_1$'s state reads or updates the component $v_1$ and leaves $v_2$ and $o$ unchanged, a branch\u2026","title":"Lifting a statement of the first machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/lift-2","kind":"node","lean":"MetaMathlibExt.TM2Composition.lift\u2082","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#lift-2","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Given finite multi-stack Turing machines $M_1$ and $M_2$, a statement of $M_2$ is translated into a statement of their composite machine, whose state is a triple $(v_1, v_2, o)$, by recursion on the statement: an operation on a stack $k$ of $M_2$ acts on the copy of $k$ among the composite stacks, every read or update of $M_2$'s state reads or updates the component $v_2$ and leaves $v_1$ and $o$ unchanged, a branch\u2026","title":"Lifting a statement of the second machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/prog","kind":"node","lean":"MetaMathlibExt.TM2Composition.prog","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#prog","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be finite multi-stack Turing machines and $\\varphi$ a map from $M_1$'s output alphabet to $M_2$'s input alphabet. At a label of $M_1$ or of $M_2$ the composite program runs the lifted statement of that machine at that label. At the pop label of the first copy loop it pops $M_1$'s output stack, stores the image under $\\varphi$ of the popped symbol (or nothing) in the third state component, and\u2026","title":"The program of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/comp-tm","kind":"node","lean":"MetaMathlibExt.TM2Composition.compTM","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#comp-tm","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $M_1$ and $M_2$ be finite multi-stack Turing machines, let $X$ be the alphabet of the input stack of $M_2$, and let $\\varphi$ map the output alphabet of $M_1$ to $X$. The composite machine has the stacks of $M_1$, the stacks of $M_2$ and one scratch stack with alphabet $X$; its input stack is that of $M_1$ and its output stack is that of $M_2$. It starts at the main label of $M_1$, in the state made of the\u2026","title":"The composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/mk-stk","kind":"node","lean":"MetaMathlibExt.TM2Composition.mkStk","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#mk-stk","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Given finite multi-stack Turing machines $M_1$ and $M_2$, contents $S_1$ for the stacks of $M_1$, contents $S_2$ for the stacks of $M_2$, and a list $t$ of symbols of $M_2$'s input alphabet, the stack contents of their composite machine put $S_1(k)$ on the copy of each stack $k$ of $M_1$, $S_2(k)$ on the copy of each stack $k$ of $M_2$, and $t$ on the scratch stack.","title":"Assembling the stacks of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/lab","kind":"node","lean":"MetaMathlibExt.TM2Composition.lab\u2081","parent":34532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Composite label of an "],["math","\\(M_1\\)"],["text"," label"]],"source":"Composite label of an $M_1$ label"},"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#lab","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"A label $l$ of $M_1$ is sent to $l$ viewed as a label of the $M_1$ part of the composite of $M_1$ and $M_2$, and the halted label is sent to the label that pops $M_1$'s output stack, which starts the first copy loop.","title":"Composite label of an $M_1$ label"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/emb","kind":"node","lean":"MetaMathlibExt.TM2Composition.emb\u2081","parent":34532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Embedding of "],["math","\\(M_1\\)"],["text","'s configurations into the composite machine"]],"source":"Embedding of $M_1$'s configurations into the composite machine"},"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#emb","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"A configuration $c$ of $M_1$ is sent to the configuration of the composite of $M_1$ and $M_2$ whose label is the label of $c$ viewed as a label of the $M_1$ part, or the label that pops $M_1$'s output stack (the start of the first copy loop) if $c$ is halted, whose state is $(v, q_2, \\mathrm{none})$ with $v$ the state of $c$ and $q_2$ the initial state of $M_2$, and whose stacks are the stacks of $c$ on $M_1$'s\u2026","title":"Embedding of $M_1$'s configurations into the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/emb-2","kind":"node","lean":"MetaMathlibExt.TM2Composition.emb\u2082","parent":34532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Embedding of "],["math","\\(M_2\\)"],["text","'s configurations into the composite machine"]],"source":"Embedding of $M_2$'s configurations into the composite machine"},"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#emb-2","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"A configuration $c$ of $M_2$ is sent to the configuration of the composite of $M_1$ and $M_2$ whose label is the label of $c$ viewed as a label of the $M_2$ part (halted if $c$ is halted), whose state is $(q_1, v, \\mathrm{none})$ with $q_1$ the initial state of $M_1$ and $v$ the state of $c$, and whose stacks are the stacks of $c$ on $M_2$'s stacks, with $M_1$'s stacks and the scratch stack empty.","title":"Embedding of $M_2$'s configurations into the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/iter-sim","kind":"node","lean":"MetaMathlibExt.TM2Composition.iter_sim","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#iter-sim","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $f : \\alpha \\to \\mathrm{Option}\\,\\alpha$ and $g : \\beta \\to \\mathrm{Option}\\,\\beta$ be partial step functions (a value $\\mathrm{none}$ means halting) and $e : \\alpha \\to \\beta$ a map such that $f(c) = c'$ implies $g(e(c)) = e(c')$ for all $c, c'$. Then for every $n$ and all $c, d \\in \\alpha$, if $n$ steps of $f$ starting from $c$ end at $d$, then $n$ steps of $g$ starting from $e(c)$ end at $e(d)$.","title":"Simulation transports runs"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/copy","kind":"node","lean":"MetaMathlibExt.TM2Composition.copy\u2081","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#copy","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Consider the composite program of machines $M_1$ and $M_2$ with symbol translation $\\varphi$. Suppose it is at the label that pops $M_1$'s output stack, with state $(v_1, v_2, o)$, with $M_1$'s stacks holding $S_1$, $M_2$'s stacks holding $S_2$, the scratch stack holding $t$, and $M_1$'s output stack holding the list $y$. Then after exactly $2|y| + 1$ steps it is at the label that pops the scratch stack, with state\u2026","title":"First copy loop of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/copy-2","kind":"node","lean":"MetaMathlibExt.TM2Composition.copy\u2082","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#copy-2","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Consider the composite program of machines $M_1$ and $M_2$ with symbol translation $\\varphi$. Suppose it is at the label that pops the scratch stack, with state $(v_1, v_2, o)$, with $M_1$'s stacks holding $S_1$, $M_2$'s stacks holding $S_2$, and the scratch stack holding the list $t$. Then after exactly $2|t| + 1$ steps it is at the start label of $M_2$, with state $(v_1, v_2, \\mathrm{none})$, $M_1$'s stacks still\u2026","title":"Second copy loop of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/pushes","kind":"node","lean":"MetaMathlibExt.TM2Composition.pushes","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#pushes","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"For a statement $q$ of a multi-stack Turing machine (Mathlib's TM2 model), this is the number of push nodes in $q$: a push node counts $1$ plus the count of its continuation; peek, pop and load nodes count as their continuation; a branch counts the sum of its two branches; goto and halt count $0$.","title":"Number of push instructions in a TM2 statement"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/push-bound","kind":"node","lean":"MetaMathlibExt.TM2Composition.pushBound","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#push-bound","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"For a finite multi-stack Turing machine $M$ (Mathlib's TM2 model with finitely many labels), the push bound of $M$ is the sum, over all labels $l$ of $M$, of the number of push instructions in the statement that $M$ assigns to $l$.","title":"Push bound of a finite multi-stack Turing machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/eval-mono","kind":"node","lean":"MetaMathlibExt.TM2Composition.eval_mono","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#eval-mono","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"For every polynomial $p$ with coefficients in $\\mathbb{N}$ and all natural numbers $a \\le b$, we have $p(a) \\le p(b)$.","title":"Evaluation of a natural-number polynomial is monotone"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/init-list-stk-length-le","kind":"node","lean":"MetaMathlibExt.TM2Composition.initList_stk_length_le","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#init-list-stk-length-le","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine (Mathlib's TM2 model with finitely many stacks, function labels and internal states, and a finite input alphabet) with input stack $k_0$, and let $l$ be a list of symbols from the alphabet of stack $k_0$. Then for every stack $k$ of $M$, the stack $k$ in the initial configuration of $M$ on input $l$ has length at most the length of $l$.","title":"Stacks of the initial configuration are no longer than the input"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/glue","kind":"node","lean":"MetaMathlibExt.TM2Composition.glue","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#glue","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $h_1$ be a polynomial-time TM2 computation of $f : \\alpha \\to \\beta$ and $h_2$ one of $g : \\beta \\to \\gamma$, where $\\beta$ is encoded by lists over an alphabet $\\Gamma_\\beta$. A symbol $y$ of the output-stack alphabet of $h_1$'s machine is sent to the symbol of the input-stack alphabet of $h_2$'s machine that corresponds, under $h_2$'s input alphabet bijection with $\\Gamma_\\beta$, to the letter that $y$\u2026","title":"Symbol translation between composed machines"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/mid-poly","kind":"node","lean":"MetaMathlibExt.TM2Composition.midPoly","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#mid-poly","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $h_1$ be a polynomial-time TM2 computation of $f : \\alpha \\to \\beta$ with polynomial time bound $p_1 \\in \\mathbb{N}[X]$, and let $c$ be the push bound of the machine of $h_1$. This is the polynomial $X + c\\,p_1 \\in \\mathbb{N}[X]$.","title":"Intermediate length bound of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/comp-time","kind":"node","lean":"MetaMathlibExt.TM2Composition.compTime","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#comp-time","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $p_1, p_2 \\in \\mathbb{N}[X]$ be the polynomial time bounds of a machine $M_1$ computing $f$ and a machine $M_2$ computing $g$, and let $c$ be the sum, over the labels of $M_1$, of the number of push instructions in the statement at that label. The composite time bound is the polynomial $p_1 + 4(X + c\\,p_1) + 2 + p_2 \\circ (X + c\\,p_1)$.","title":"Time bound of the composite machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/comp-outputs","kind":"node","lean":"MetaMathlibExt.TM2Composition.compOutputs","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#comp-outputs","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $M_1$ compute $f : \\alpha \\to \\beta$ in polynomial time from the encoding $e_\\alpha$ to $e_\\beta$, and let $M_2$ compute $g : \\beta \\to \\gamma$ in polynomial time from $e_\\beta$ to $e_\\gamma$. For every $a \\in \\alpha$, let $n$ be the length of $e_\\alpha(a)$ and $T$ the composite time bound. Started on $e_\\alpha(a)$, written in the input alphabet of $M_1$, the composite machine of $M_1$ and $M_2$ (with the glue\u2026","title":"The composite machine computes the composite function in time"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/comp-computable","kind":"node","lean":"MetaMathlibExt.TM2Composition.compComputable","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#comp-computable","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $h_1$ be a polynomial-time TM2 computation of $f : \\alpha \\to \\beta$ with respect to encodings $e_\\alpha$ and $e_\\beta$, and let $h_2$ be one of $g : \\beta \\to \\gamma$ with respect to $e_\\beta$ and $e_\\gamma$. This is the polynomial-time TM2 computation of $g \\circ f$ with respect to $e_\\alpha$ and $e_\\gamma$ whose machine is the composite machine of the machines of $h_1$ and $h_2$ with the glue map between\u2026","title":"The composite polynomial-time TM2 computation"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/tm2-computable-in-poly-time-comp","kind":"node","lean":"MetaMathlibExt.tm2ComputableInPolyTime_comp","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#tm2-computable-in-poly-time-comp","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $\\alpha$, $\\beta$, $\\gamma$ be types encoded as lists over alphabets by $e_\\alpha$, $e_\\beta$, $e_\\gamma$. If $f : \\alpha \\to \\beta$ is computable in polynomial time by a multi-stack Turing machine (a TM2 machine) with respect to $e_\\alpha$ and $e_\\beta$, and $g : \\beta \\to \\gamma$ is computable in polynomial time by a TM2 machine with respect to $e_\\beta$ and $e_\\gamma$, then there is a TM2 machine computing\u2026","title":"Polynomial-time TM2-computable functions are closed under composition"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/comp","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.IsPolyTimeWithEncoding.comp","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#comp","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $e_\\alpha$, $e_\\beta$, $e_\\gamma$ be encodings of types $\\alpha$, $\\beta$, $\\gamma$ as lists over alphabets $\\Gamma_\\alpha$, $\\Gamma_\\beta$, $\\Gamma_\\gamma$. If $f : \\alpha \\to \\beta$ is polynomial-time computable with respect to $e_\\alpha$ and $e_\\beta$, and $g : \\beta \\to \\gamma$ is polynomial-time computable with respect to $e_\\beta$ and $e_\\gamma$, then $g \\circ f$ is polynomial-time computable with respect\u2026","title":"Composition of polynomial-time functions under encodings"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/comp-2","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.IsPolyTime.comp","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#comp-2","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $\\alpha$, $\\beta$, $\\gamma$ be types with canonical bitstring encodings. If $f : \\alpha \\to \\beta$ and $g : \\beta \\to \\gamma$ are polynomial-time computable, then so is $g \\circ f$.","title":"Polynomial-time functions are closed under composition"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/comp-mem-p","kind":"node","lean":"MetaMathlibExt.ComplexityTheory.comp_mem_P","parent":34532,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines/index.html#comp-mem-p","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/composition-of-multi-stack-machines","status":"fully_proved","summary":"Let $L$ be a decision problem on bitstrings and let $f$ be a polynomial-time computable function from bitstrings to bitstrings. If $L$ is in P, then $L \\circ f$ is in P. In particular P is closed under polynomial-time many-one reductions.","title":"P is closed under polynomial-time preimages"},{"data_url":"scopes/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine.json","id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","kind":"scope","navigation":"module","parent":34438,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","title":"Simulating component machines in a host machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/comp","kind":"node","lean":"Cslib.Complexity.Host.Comp","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#comp","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"A component machine consists of a finite type $K$ of stacks with decidable equality, a finite alphabet $\\Gamma_k$ for each stack $k$, a finite type $L$ of labels with decidable equality, a multi-stack Turing machine program assigning to each label in $L$ a statement over the alphabets $\\Gamma$ and the labels $L$ with Boolean internal state, and a list of the stacks in which every stack occurs exactly once, so that\u2026","title":"Component machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/hk","kind":"node","lean":"Cslib.Complexity.Host.HK","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#hk","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component machine $c$ with stack indices $K$, the host stacks are an input stack, an output stack, two buffers $X_1$ and $W_1$ used in phase 0 (which moves the input onto the component's stacks), and one copy of each component stack $k \\in K$.","title":"Stacks of the generic host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/all-ks","kind":"node","lean":"Cslib.Complexity.Host.allKs","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#all-ks","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component machine $c$, the host stacks are listed as the input stack, the output stack, the two buffer stacks $X_1$ and $W_1$, and then the host's copies of the stacks of $c$, in the order of the list of stacks that $c$ carries.","title":"The list of host stacks"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/h","kind":"node","lean":"Cslib.Complexity.Host.H\u0393","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#h","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component machine $c$ with stack alphabets $(\\Gamma_k)_k$ and a type $\\alpha$, the alphabet of a host stack is $\\alpha$ on the input stack, $\\Gamma_k$ on the host copy of the component's stack $k$, and the booleans on the output stack and on the two phase-0 buffers.","title":"Alphabets of the generic host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/p0","kind":"node","lean":"Cslib.Complexity.Host.P0","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#p0","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"The six labels of a host's phase 0, which moves the input onto the component's stacks: two for the two passes that move the first block $x$, one that drops the separator, two for the two passes that move the second block $w$, and one that resets the state and starts the component.","title":"Phase-0 labels of the generic host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/hlab","kind":"node","lean":"Cslib.Complexity.Host.HLab","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#hlab","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component machine $c$ with labels $L$, and with $P_0$ the set of phase-0 labels, the host labels are one label for each $p \\in P_0$, one copy of each component label $l \\in L$, and, for each boolean answer $a$, a cleanup label and an output label carrying $a$.","title":"Labels of the generic host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/equiv","kind":"node","lean":"Cslib.Complexity.Host.HLab.equiv","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#equiv","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component machine $c$ with labels $L$, let $P_0$ be the set of phase-0 labels and $B = \\{\\mathrm{false}, \\mathrm{true}\\}$. The host labels of $c$ are in bijection with the disjoint union $P_0 \\sqcup L \\sqcup B \\sqcup B$: the phase-0 label $p$ corresponds to $p$, the host copy of the component label $l$ to $l$, the cleanup label with answer $a$ to $a$ in the first copy of $B$, and the output label with answer\u2026","title":"Labels of the generic host as a sum type"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/emb","kind":"node","lean":"Cslib.Complexity.Host.emb","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#emb","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component machine $c$ and an input alphabet $\\alpha$, the stack embedding of $c$ into the host over $\\alpha$ sends each stack $k$ of $c$ to the host's copy of $k$, whose alphabet is the alphabet $\\Gamma_k$ of $k$, and identifies the two alphabets by the identity. Distinct stacks have distinct copies, so the embedding is injective.","title":"The component's stacks inside a host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/agree-emb","kind":"node","lean":"Cslib.Complexity.Host.agree_emb","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#agree-emb","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine and $\\alpha$ an input alphabet. For stack contents $S$ of $c$ and stack contents $T$ of the host over $\\alpha$, $T$ agrees with $S$ under the embedding of the component's stacks into the host if and only if, for every stack $k$ of $c$, the host's copy of $k$ holds exactly $S(k)$.","title":"Agreement with the embedded component"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/move2-run","kind":"node","lean":"Cslib.Complexity.Host.move2_run","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#move2-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $a$, $t$, $b$ be pairwise distinct stacks of a program $M$ with Boolean state, and let $l_1$, $l_2$, $l_3$ be labels such that $M$ runs at $l_1$ the bit-moving loop from $a$ to $t$ with decoder $d$ and encoder $g_t$, looping at $l_1$ and exiting to $l_2$, and at $l_2$ the bit-moving loop from $t$ to $b$ with decoder $d_t$ and encoder $g$, looping at $l_2$ and exiting to $l_3$, where $d_t$ decodes $g_t(y)$ to\u2026","title":"Moving a bit block through a buffer"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/clean-ks","kind":"node","lean":"Cslib.Complexity.Host.cleanKs","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#clean-ks","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component machine $c$ and a host stack $o$, the cleanup stacks are the list of all host stacks with $o$ removed, in the order of the list of host stacks. The cleanup empties these stacks and leaves the output stack $o$.","title":"Cleanup stacks of the generic host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/mem-clean-ks","kind":"node","lean":"Cslib.Complexity.Host.mem_cleanKs","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#mem-clean-ks","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component machine $c$ and host stacks $o$ and $k$, the stack $k$ belongs to the list of stacks that the cleanup for the output stack $o$ empties if and only if $k \\ne o$.","title":"Membership in the cleanup stacks of the generic host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/cleanup-run","kind":"node","lean":"Cslib.Complexity.Host.cleanup_run","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#cleanup-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine, $\\alpha$ an input alphabet, $o$ a host stack, $a$ a bit, and $M$ a host program with Boolean state whose statement at the cleanup label for $a$ is the cleanup round on the list of all host stacks with $o$ removed, kept in the order of the list of host stacks, repeating at the cleanup label for $a$ and exiting to the output label for $a$. If $S$ assigns to every host stack at most $m$\u2026","title":"Cleanup in a host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/stk-length-le","kind":"node","lean":"Cslib.Complexity.Host.stk_length_le","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#stk-length-le","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $M$ be a finite multi-stack Turing machine with push bound $B$. If every stack of $S$ has length at most $N$ and $M$ runs in $n$ steps from a configuration with some label, state $v$ and stacks $S$ to a configuration with some label, state $v'$ and stacks $S'$, then every stack of $S'$ has length at most $N + Bn$.","title":"Stack lengths grow by at most the push bound per step"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/init-list-eq","kind":"node","lean":"Cslib.Complexity.Host.initList_eq","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#init-list-eq","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a finite multi-stack Turing machine $M$ with input stack $k_0$, main label $\\lambda_0$ and initial internal state $\\sigma_0$, and a list $l$ over the alphabet of $k_0$, the initial configuration of $M$ on $l$ has current label $\\lambda_0$, internal state $\\sigma_0$, the list $l$ on $k_0$ and every other stack empty.","title":"Initial configuration of a finite multi-stack machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/halt-list-eq","kind":"node","lean":"Cslib.Complexity.Host.haltList_eq","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#halt-list-eq","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a finite multi-stack Turing machine $M$ with output stack $k_1$ and initial internal state $\\sigma_0$, and a list $l$ over the alphabet of $k_1$, the halting configuration of $M$ with output $l$ has no current label, internal state $\\sigma_0$, the list $l$ on $k_1$ and every other stack empty.","title":"Halting configuration of a finite multi-stack machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/update-length-le","kind":"node","lean":"Cslib.Complexity.Host.update_length_le","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#update-length-le","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine and consider its host with input alphabet $\\alpha$. If a host stack $o$ holds a list $l$ and every other host stack is empty, then every host stack has length at most the length of $l$.","title":"Stack lengths when only one stack is filled"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/r","kind":"node","lean":"Cslib.Complexity.Host.\u03c6R","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#r","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"For a component with exit label $\\mathit{done}$, the label $\\mathit{done}$ is sent to the host's cleanup label with answer false, and every other component label $l$ to the host label that runs the component at $l$.","title":"Component labels in the reduction host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/rprog","kind":"node","lean":"Cslib.Complexity.Host.rprog","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#rprog","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"The program of the reduction host for a component with stacks $c_X$ and $c_O$, an encoding $g_X$ of bits as symbols of $c_X$ and labels $\\mathit{start}$ and $\\mathit{done}$. Phase 0 moves the whole input stack onto the buffer $X_1$ with the bit-moving loop, then moves $X_1$ onto the host's copy of $c_X$ re-encoded by $g_X$ (so in the original order), then sets the state to false and goes to $\\mathit{start}$, or to\u2026","title":"Reduction host program"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/rtm","kind":"node","lean":"Cslib.Complexity.Host.rtm","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#rtm","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"The reduction host as a finite multi-stack Turing machine: its stacks are the host stacks with the host alphabets for bit input, its labels are the host labels, its state is a bit starting at false, its program is the reduction host program started at the first phase-0 label, its input stack is the input stack (raw bits) and its output stack is the host's copy of the component's stack $c_O$.","title":"Reduction host machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/rtime","kind":"node","lean":"Cslib.Complexity.Host.rtime","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#rtime","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with stacks $c_X$ and $c_O$, an encoding $g_X$ of bits as symbols of $c_X$ and labels $\\mathit{start}$ and $\\mathit{done}$, and let $B$ be the push bound of the reduction host machine built from these data. For a polynomial $p$ with natural-number coefficients, the step bound of the reduction host is the polynomial in the variable $X$","title":"Step bound of the reduction host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/embeds-rprog","kind":"node","lean":"Cslib.Complexity.Host.embeds_rprog","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#embeds-rprog","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with stacks $c_X$ and $c_O$, a map $g_X$ from bits to the alphabet of $c_X$, and labels $s$ and $d$, and suppose that $c$ halts at $d$. Rename the labels of $c$ by sending $d$ to the cleanup label with answer false and every other label $l$ to the host's copy of $l$, and embed the stacks of $c$ as the host's copies over the bit input alphabet. Then the reduction host built from $c_X$,\u2026","title":"The reduction host embeds the component"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/rprog-run-at","kind":"node","lean":"Cslib.Complexity.Host.rprog_run_at","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#rprog-run-at","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component with stacks $c_X$ and $c_O$, an encoding $g_X$ of bits as symbols of $c_X$, an identification $e_O$ of the alphabet of $c_O$ with the bits, and labels $\\mathit{start}$ and $\\mathit{done}$, where the component halts at $\\mathit{done}$. Let $p$ be a polynomial with natural-number coefficients and $x$, $o$ bit lists, and suppose that the component, started at $\\mathit{start}$ with state false,\u2026","title":"The reduction host's run on one input"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/rprog-run","kind":"node","lean":"Cslib.Complexity.Host.rprog_run","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#rprog-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component with stacks $c_X$ and $c_O$, an encoding $g_X$ of bits as symbols of $c_X$, an identification $e_O$ of the alphabet of $c_O$ with the bits, and labels $\\mathit{start}$ and $\\mathit{done}$, where the component halts at $\\mathit{done}$. Let $f$ be a function from bit lists to bit lists and $p$ a polynomial with natural-number coefficients, and suppose that for every bit list $y$ the component,\u2026","title":"The reduction host's run"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/reduction","kind":"node","lean":"Cslib.Complexity.Host.reduction","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#reduction","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component with stacks $c_X$ and $c_O$, an encoding $g_X$ of bits as symbols of $c_X$, an identification $e_O$ of the alphabet of $c_O$ with the bits, and labels $\\mathit{start}$ and $\\mathit{done}$, where the component halts at $\\mathit{done}$. Let $f$ be a function from bit lists to bit lists and $p$ a polynomial with natural-number coefficients. Suppose that for every bit list $x$ the component,\u2026","title":"Reduction host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/is-poly-time-reduction","kind":"node","lean":"Cslib.Complexity.Host.isPolyTime_reduction","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#is-poly-time-reduction","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with stacks $c_X$ and $c_O$ and labels $\\mathit{start}$ and $\\mathit{done}$, where the statement at $\\mathit{done}$ is the halting statement. Let $g_X$ map bits to symbols of $c_X$, let $e_O$ be a bijection from the alphabet of $c_O$ to the booleans, let $f$ map bit lists to bit lists, and let $p$ be a polynomial with coefficients in $\\mathbb{N}$. Suppose that for every bit list $x$,\u2026","title":"Reduction host, canonical form"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/is-poly-time-decider","kind":"node","lean":"Cslib.Complexity.Host.isPolyTime_decider","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#is-poly-time-decider","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with stacks $c_X$ and $c_O$ and labels $\\mathit{start}$ and $\\mathit{done}$, where the statement at $\\mathit{done}$ is the halting statement. Let $g_X$ map bits to symbols of $c_X$, let $e_O$ be a bijection from the alphabet of $c_O$ to the booleans, let $L$ map bit lists to booleans, and let $p$ be a polynomial with coefficients in $\\mathbb{N}$. Suppose that for every bit list $x$,\u2026","title":"Decider host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/v","kind":"node","lean":"Cslib.Complexity.Host.\u03c6V","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#v","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Given labels $\\mathit{yes}$ and $\\mathit{no}$ of a component machine $c$, a label $l$ of $c$ is sent to the host label that starts the cleanup with answer true if $l = \\mathit{yes}$, to the host label that starts the cleanup with answer false if $l \\ne \\mathit{yes}$ and $l = \\mathit{no}$, and to the host label that runs the component at $l$ otherwise.","title":"Relabelling in the verifier host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/d-x","kind":"node","lean":"Cslib.Complexity.Host.dX","parent":34567,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reading a bit of "],["math","\\(x\\)"]],"source":"Reading a bit of $x$"},"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#d-x","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"On the pair alphabet of the $x \\# w$ input, the symbol that encodes a bit $b$ of $x$ (the left injection of $b$) is read as the bit $b$; every other symbol, the separator $\\#$ or a bit of $w$ (both right injections), is read as no bit.","title":"Reading a bit of $x$"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/d-w","kind":"node","lean":"Cslib.Complexity.Host.dW","parent":34567,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reading a bit of "],["math","\\(w\\)"]],"source":"Reading a bit of $w$"},"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#d-w","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"On the pair alphabet of the $x \\# w$ input, the symbol that encodes a bit $b$ of $w$ (the right injection of $\\mathrm{some}\\,b$) is read as the bit $b$; every other symbol, a bit of $x$ (a left injection) or the separator $\\#$ (the right injection of $\\mathrm{none}$), is read as no bit.","title":"Reading a bit of $w$"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/vprog","kind":"node","lean":"Cslib.Complexity.Host.vprog","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#vprog","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"The program of the verifier host for a component machine $c$ with stacks $c_X$ and $c_W$, encodings $g_X$ and $g_W$ of bits as symbols of $c_X$ and $c_W$, and labels $\\mathit{start}$, $\\mathit{yes}$ and $\\mathit{no}$, on the host stacks with the work alphabet of the dense-SAT verifier on the input stack. On an input $x \\# w$ in the pair encoding, phase 0 moves the symbols on top of the input stack that encode bits\u2026","title":"Verifier host program"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/vtm","kind":"node","lean":"Cslib.Complexity.Host.vtm","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#vtm","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"The verifier host as a finite multi-stack Turing machine for a component machine $c$: its stacks are the host stacks, with the pair alphabet on the input stack, the alphabets of $c$ on the component's stacks and bits on the other stacks; its labels are the host labels and its program is the verifier host program; it starts at the first phase-0 label with a boolean internal state, initially false; it reads its input\u2026","title":"The verifier host machine"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/vtime","kind":"node","lean":"Cslib.Complexity.Host.vtime","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#vtime","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with stacks $c_X$ and $c_W$, encodings $g_X$ and $g_W$ of bits as symbols of $c_X$ and $c_W$, and labels $\\mathit{start}$, $\\mathit{yes}$ and $\\mathit{no}$, and let $B$ be the push bound of the verifier host machine built from these data. For a polynomial $p$ with natural-number coefficients, the step bound of the verifier host is the polynomial in the variable $X$","title":"Step bound of the verifier host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/embeds-vprog","kind":"node","lean":"Cslib.Complexity.Host.embeds_vprog","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#embeds-vprog","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with stacks $c_X$ and $c_W$, maps $g_X$ and $g_W$ from bits to the alphabets of $c_X$ and $c_W$, and labels $\\mathit{start}$, $\\mathit{yes}$ and $\\mathit{no}$, and suppose that the statement of $c$ at $\\mathit{yes}$ and the statement of $c$ at $\\mathit{no}$ are the halt statement. Rename the labels of $c$ by sending $\\mathit{yes}$ to the cleanup label with answer true, $\\mathit{no}$\u2026","title":"The verifier host embeds the component"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/vprog-run","kind":"node","lean":"Cslib.Complexity.Host.vprog_run","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#vprog-run","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with stacks $c_X \\ne c_W$, maps $g_X$ and $g_W$ from bits to the alphabets of $c_X$ and $c_W$, and labels $\\mathit{start}$, $\\mathit{yes}$ and $\\mathit{no}$ with $\\mathit{yes} \\ne \\mathit{no}$, where the program halts at $\\mathit{yes}$ and at $\\mathit{no}$. Let $f$ assign a boolean to each pair of bit lists and let $p$ be a polynomial with natural number coefficients. Suppose that for\u2026","title":"The verifier host's run"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/verifier","kind":"node","lean":"Cslib.Complexity.Host.verifier","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#verifier","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with stacks $c_X \\ne c_W$, maps $g_X$ and $g_W$ from bits to the alphabets of $c_X$ and $c_W$, and labels $\\mathit{start}$, $\\mathit{yes}$ and $\\mathit{no}$ with $\\mathit{yes} \\ne \\mathit{no}$, where the program halts at $\\mathit{yes}$ and at $\\mathit{no}$. Let $f$ assign a boolean to each pair of bit lists and let $p$ be a polynomial with natural number coefficients. Suppose that for\u2026","title":"Verifier host"},{"id":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/is-poly-time-verifier","kind":"node","lean":"Cslib.Complexity.Host.isPolyTime_verifier","parent":34567,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine/index.html#is-poly-time-verifier","scope":"TheoryOfComputation/turing-machines/multi-tape-and-multi-stack/simulating-component-machines-in-a-host-machine","status":"fully_proved","summary":"Let $c$ be a component machine with distinct stacks $c_X$ and $c_W$ and labels $\\mathit{start}$, $\\mathit{yes}$ and $\\mathit{no}$, where $\\mathit{yes} \\ne \\mathit{no}$ and the statements at $\\mathit{yes}$ and $\\mathit{no}$ are the halting statement. Let $g_X$ and $g_W$ map bits to symbols of $c_X$ and $c_W$, let $f$ assign a boolean $f(x, w)$ to all bit lists $x$ and $w$, and let $p$ be a polynomial with\u2026","title":"Verifier host, canonical form"},{"data_url":"scopes/TheoryOfComputation/turing-machines/universal-machines.json","id":"TheoryOfComputation/turing-machines/universal-machines","kind":"scope","parent":34347,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/index.html","scope":"TheoryOfComputation/turing-machines/universal-machines","status":"fully_proved","summary":"Codes for machines, clocked evaluators and universal machines, completeness of codes, and the cost of universal simulation. Diagonalization against time-bounded codes goes to Hierarchy theorems.","title":"Universal Turing machines"},{"data_url":"scopes/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines.json","id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","kind":"scope","navigation":"module","parent":34605,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","title":"Codes for multi-tape Turing machines"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/decode-move","kind":"node","lean":"Cslib.Turing.MultiTapeTM.decodeMove","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#decode-move","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Decode a head move: `0` moves left, `1` stays, anything else moves right.","title":"Decoding a Turing machine head move"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/code-action","kind":"node","lean":"Cslib.Turing.MultiTapeTM.CodeAction","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#code-action","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"A coded action for a machine with $k$ work tapes consists of a coded input-head move; for each of the $k$ work tapes, an optional symbol to write, the symbol being a bit or the blank (so a blank can be written), together with a coded head move; an optional output bit; and an optional next state in $\\mathbb{N}$, where no next state means halting. A head move is coded by a natural number: $0$ moves left, $1$ stays,\u2026","title":"Coded actions"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/equiv-prod","kind":"node","lean":"Cslib.Turing.MultiTapeTM.CodeAction.equivProd","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#equiv-prod","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The bijection between coded actions and tuples $(m, w, b, q)$ that sends an action to its coded input head move $m \\in \\mathbb{N}$, its work tape instructions $w$ (for each of the $k$ work tapes, an optional symbol to write and a coded head move), its optional output bit $b$ and its optional next state $q$.","title":"Coded actions as tuples"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/to-action","kind":"node","lean":"Cslib.Turing.MultiTapeTM.CodeAction.toAction","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#to-action","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"A coded action $a$ stands for the action of CSLib's multi-tape model that moves the input head by the decoded input move of $a$, on each work tape writes the optional symbol that $a$ gives for that tape and moves its head by the decoded move, emits the optional output bit of $a$, and enters the next state of $a$ (halting when there is none).","title":"Decoding a coded action"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/halt","kind":"node","lean":"Cslib.Turing.MultiTapeTM.CodeAction.halt","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#halt","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The default action of a missing table entry: halt without moving any head, writing any symbol or emitting an output bit.","title":"Halting action"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/code-key","kind":"node","lean":"Cslib.Turing.MultiTapeTM.CodeKey","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#code-key","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"A key of a table entry for a machine with $k$ work tapes is a triple $(q, a, w)$ of a state $q \\in \\mathbb{N}$, the symbol $a$ under the input head and the symbols $w = (w_i)_{i < k}$ under the $k$ work heads, where each symbol is a bit or the blank.","title":"Keys of a code's table"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/code","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#code","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"A code for a machine with $k$ work tapes over the binary alphabet, with states in $\\mathbb{N}$, is a finite list of table entries, each a key (a state together with the symbols under the input head and the $k$ work heads) paired with a coded action. The table is looked up by key, the first matching entry wins, and a missing key halts the machine.","title":"Codes for multi-tape machines"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/equiv-table","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.equivTable","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#equiv-table","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The bijection between codes for machines with $k$ work tapes and finite lists of table entries, each a pair of a key and a coded action, that sends a code to its table.","title":"Multi-tape codes as tables"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/encode","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.encode","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#encode","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The natural number coding a code $c$: its encoding under the standard primitive recursive coding of codes, which identifies a code with its table, a list of pairs of a key and a coded action.","title":"Numeric coding of codes"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/decode","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.decode","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#decode","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The code coded by a natural number $n$, if any: decoding under the standard primitive recursive coding of codes, which identifies a code with its table, a list of pairs of a key and a coded action. The result is empty when $n$ codes no code.","title":"Decoding numbers to codes"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/decode-encode","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.decode_encode","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#decode-encode","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For every code $c$ with $k$ work tapes, decoding the natural number $\\operatorname{encode}(c)$ returns $c$: $\\operatorname{decode}(\\operatorname{encode}(c))$ is defined and equals $c$.","title":"Code round-trip"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/encode-injective","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.encode_injective","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#encode-injective","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"If two codes $c$ and $c'$ with the same number of work tapes have the same numeric code, then $c = c'$.","title":"Injectivity of the numeric coding of codes"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/act","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.act","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#act","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a code $c$ with $k$ work tapes, a state $q \\in \\mathbb{N}$, a symbol $a$ under the input head and symbols $w = (w_i)_{i < k}$ under the work heads (each symbol a bit or the blank), the action of $c$ at $(q, a, w)$ is the coded action of the first table entry of $c$ whose key is $(q, a, w)$, or, if there is no such entry, the halting action, which moves no head, writes and emits nothing, and has no next state.","title":"Action of a code at a key"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/tr","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.tr","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#tr","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The transition function of a code $c$ sends a state $q$, the symbol $a$ under the input head and the symbols $w$ under the work heads (each a bit or blank) to the action of CSLib's multi-tape model that the coded action of $c$ at $(q, a, w)$ stands for. That coded action is the one of the first table entry of $c$ with key $(q, a, w)$, or, if there is none, the one that halts without moving, writing or emitting.","title":"Transition function of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/to-tm","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.toTM","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#to-tm","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The machine of a code $c$ for $k$ work tapes is the machine of CSLib's multi-tape model with $k$ work tapes, the binary alphabet and states in $\\mathbb{N}$ whose initial state is $0$ and whose transition function is the transition function of $c$.","title":"The multi-tape machine of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/step-to-tm","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.step_toTM","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#step-to-tm","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, a binary input $x$ and a configuration $C$ of the machine of $c$ on $x$, the step of the machine of $c$ sends $C$ to its successor under the transition function of $c$: $C$ itself if it has halted, and otherwise the result of applying to $C$ the action that the transition function of $c$ selects for the state of $C$ and the symbols under its heads.","title":"Step of the machine of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/states","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.states","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#states","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The states of a code $c$ for machines with $k$ work tapes form the finite set of natural numbers consisting of the initial state $0$ and the next state of every table entry of $c$ that does not halt.","title":"States of a multi-tape code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/next-act-mem-states","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.next_act_mem_states","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#next-act-mem-states","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, a state $q$, an input symbol $a$ and work tape symbols $w$, if the action of $c$ at $(q, a, w)$ has a next state $q'$ (rather than halting), then $q'$ belongs to the set of states of $c$.","title":"Next states of a code lie in its state set"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/state-run-from-mem-states","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.state_runFrom_mem_states","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#state-run-from-mem-states","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a code for a machine with $k$ work tapes. For every binary input $x$ and every number of steps $t$, if the machine of $c$, run for $t$ steps from its initial configuration on $x$, has not halted and is in state $q$, then $q$ lies in the finite set of states of $c$.","title":"Codes denote finite-state machines"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/tape","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Tape","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#tape","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"A work tape is a pair of finite lists of cells, each cell a bit or blank: the cells left of the head, nearest first, and the cells from the head rightwards; cells beyond both lists are blank.","title":"Work tapes as zippers"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/read","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Tape.read","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#read","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The symbol read from a work tape given as a zipper $(l, r)$ is the first cell of $r$, or the blank if $r$ is empty.","title":"Symbol under the head of a zipper tape"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/write","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Tape.write","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#write","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Writing an optional symbol on a work tape leaves the tape unchanged when no symbol is given; given a symbol $s$ (a bit or blank), it replaces the cell under the head by $s$ and keeps all other cells.","title":"Writing on a zipper tape"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/move","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Tape.move","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#move","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a coded move $m$ and a work tape given as a zipper $(l, r)$: if $m = 0$ the head moves one cell left, removing the first cell of $l$ (the blank if $l$ is empty) and putting it in front of $r$; if $m = 1$ the tape is unchanged; otherwise the head moves one cell right, removing the first cell of $r$ (the blank if $r$ is empty) and putting it in front of $l$.","title":"Moving the head of a zipper tape"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/cell","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Tape.cell","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#cell","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a work tape $t$ in zipper form and an integer $j$, the cell of $t$ at $j$ is the symbol at offset $j$ from the head: for $j = n \\ge 0$ it is entry $n$ (counting from $0$) of the list of cells from the head rightwards, and for $j = -(n + 1)$ it is entry $n$ of the list of cells left of the head, nearest first; in either case it is the blank when the list has no entry $n$.","title":"Cell of a zipper tape"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/config","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Config","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#config","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"An explicit configuration for a machine with $k$ work tapes consists of an optional state in $\\mathbb{N}$ (none when the machine has halted); the input head position, a natural number shifted by one, so that position $0$ is the cell left of the input; the $k$ work tapes, each a zipper (the cells left of the head, nearest first, and the cells from the head rightwards, all further cells blank); and the output written\u2026","title":"Explicit configurations of the code evaluator"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/equiv-prod-2","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Config.equivProd","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#equiv-prod-2","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The bijection between explicit configurations and tuples $(q, p, t, o)$ that sends a configuration to its optional state $q$ (empty when halted), its shifted input head position $p \\in \\mathbb{N}$, its family $t$ of $k$ work tapes (each a zipper) and its output $o$ so far, a list of bits.","title":"Explicit configurations as tuples"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/init","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Config.init","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#init","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The initial explicit configuration has state $0$, the input head on the first input cell (position $1$, positions being shifted by one), every work tape blank (both halves of its zipper empty), and empty output.","title":"Initial configuration of the clocked evaluator"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/input-sym","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Config.inputSym","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#input-sym","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For an input $x$ and a shifted input head position $p$, the input symbol is the blank if $p = 0$; otherwise it is the entry of $x$ at index $p - 1$ (counting from $0$) if $p - 1 < |x|$, and the blank if $p - 1 \\ge |x|$.","title":"Input symbol under the input head"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/move-input","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Config.moveInput","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#move-input","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For an input of length $n$, a shifted input head position $p$ and a coded move $m$, the new position is $p - 1$ (truncated at $0$) if $m = 0$, $p$ if $m = 1$, and $\\min(p + 1, n + 1)$ otherwise, so a move right never passes position $n + 1$, one cell beyond the input.","title":"Input head move of the clocked evaluator"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/apply","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Config.apply","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#apply","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Given an input word $x$, a coded action $a$ and an explicit configuration $e$, applying $a$ to $e$ gives the configuration whose state is the next state of $a$; whose input head position is that of $e$, say $p$, moved by the coded input move of $a$, where a left move gives $p - 1$ (truncated at $0$), staying gives $p$, and a right move gives $\\min(p + 1, |x| + 1)$; whose work tape $i$ is work tape $i$ of $e$ after\u2026","title":"Applying a coded action"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/step","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Config.step","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#step","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"One step of the evaluator for a code $c$ on a binary input $x$ leaves an explicit configuration $e$ unchanged if $e$ has halted; if $e$ is in state $q$, it applies to $e$ the coded action of $c$ at $q$, the input symbol at the input head position of $e$ and the symbols under the heads of the work tapes of $e$.","title":"Evaluator step"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/eval-steps","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.evalSteps","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#eval-steps","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The clocked evaluator runs a code $c$ on a binary input $x$ for $T$ steps: it applies the evaluator's step $T$ times to the initial explicit configuration. If the resulting configuration is halted, it returns that configuration's output; otherwise it returns no result.","title":"Clocked evaluator"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/represents","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Represents","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#represents","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"An explicit configuration $e$ represents a configuration $C$ of CSLib's multi-tape model on input $x$ when they have the same state, the same shifted input head position and the same output, and for every work tape $i$ and every cell $z \\in \\mathbb{Z}$, the content of cell $z$ of tape $i$ in $C$ equals the cell of tape $i$ of $e$ at offset $z - p_i$ from its head, where $p_i$ is the head position of tape $i$ in $C$.","title":"Explicit configurations representing machine configurations"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/step-2","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.Represents.step","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#step-2","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a code and $x$ a binary input. If an explicit configuration $e$ represents a configuration $C$ of the machine of $c$ on $x$, then the evaluator's step of $e$ for $c$ on $x$ represents the successor of $C$ under the machine of $c$.","title":"The evaluator simulates the machine step for step"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/represents-iterate","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.represents_iterate","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#represents-iterate","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, an input $x$ and every $t$, the explicit configuration obtained from the initial one by $t$ steps of the evaluator for $c$ on $x$ represents the configuration that the machine of $c$ reaches after $t$ steps from its initial configuration on $x$.","title":"The evaluator tracks the machine run"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/eval-steps-eq-some-iff","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.evalSteps_eq_some_iff","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#eval-steps-eq-some-iff","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For every code $c$, binary input $x$, clock $T \\in \\mathbb{N}$ and binary word $o$, the clocked evaluator of $c$ on $x$ with clock $T$ returns $o$ if and only if the machine of $c$, run for $T$ steps from its initial configuration on $x$, has halted and its output is $o$.","title":"Evaluator agreement"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/is-some-eval-steps-iff-runs-in-time","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.isSome_evalSteps_iff_runsInTime","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#is-some-eval-steps-iff-runs-in-time","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, an input $x$ and a clock $T$, the clocked evaluator of $c$ on $x$ with clock $T$ returns an output if and only if the machine of $c$ runs in time $T$ on $x$, that is, every computation path of that machine on $x$ with at least $T$ steps ends in a halted configuration.","title":"The evaluator returns within the clock exactly when the machine runs in time"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/eval-steps-mono","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.evalSteps_mono","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#eval-steps-mono","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a code $c$ for machines with $k$ work tapes, a binary input $x$, a binary word $o$ and clocks $T \\le T'$ in $\\mathbb{N}$, if the clocked evaluator of $c$ on $x$ returns $o$ with clock $T$, then it also returns $o$ with clock $T'$.","title":"Monotonicity of the clocked evaluator for multi-tape codes"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/computes-iff-exists-eval-steps","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.computes_iff_exists_evalSteps","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#computes-iff-exists-eval-steps","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a code $c$ with $k$ work tapes and binary words $x$ and $o$, the machine of $c$ computes $o$ on input $x$, that is, it eventually halts on $x$ with output $o$, if and only if there is a clock $T \\in \\mathbb{N}$ such that the clocked evaluator of $c$ run on $x$ for $T$ steps returns $o$.","title":"A code computes exactly what its clocked evaluator returns"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/primrec-eval-steps","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.primrec_evalSteps","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#primrec-eval-steps","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The evaluator is primitive recursive in the code, the input and the clock.","title":"The evaluator is primitive recursive"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/encode-move","kind":"node","lean":"Cslib.Turing.MultiTapeTM.encodeMove","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#encode-move","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"A head move is coded as a natural number: left is $0$, stay is $1$ and right is $2$. Decoding a move code recovers the move.","title":"Coding head moves"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/encode-action","kind":"node","lean":"Cslib.Turing.MultiTapeTM.encodeAction","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#encode-action","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $e : Q \\to \\mathbb{N}$ number the states $Q$ of a binary multi-tape machine with $k$ work tapes. The coded action of an action $a$ of the machine under $e$ records as its input-head move the code of the input-head move of $a$ (left $0$, stay $1$, right $2$); for each work tape it records what $a$ writes there and the code of the head move of $a$ there; it records the output bit of $a$, if any; and its next\u2026","title":"Coding an action"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/to-action-encode-action","kind":"node","lean":"Cslib.Turing.MultiTapeTM.toAction_encodeAction","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#to-action-encode-action","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $Q$ be a type, $e : Q \\to \\mathbb{N}$ a map and $a$ an action of a binary machine with $k$ work tapes and states in $Q$. The action that the coded action of $a$, with its successor state numbered by $e$, stands for is $a$ with its successor state relabelled by $e$.","title":"Decoding the coding of an action"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/state-bound","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.stateBound","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#state-bound","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"The states of a code $c$ are the initial state $0$ and every successor state named in its table. The state bound of $c$ is one more than the largest of them; it is a strict upper bound on the states of $c$.","title":"State bound of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/fin-state","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.finState","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#fin-state","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For natural numbers $N$ and $q$, this is $q$ as an element of $\\{0, \\dots, N - 1\\}$ if $q < N$, and no value otherwise.","title":"A number as a bounded state"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/fin-tr","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.finTr","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#fin-tr","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a code with $k$ work tapes and let $N$ be its state bound. For a state $q < N$, a symbol $a$ under the input head and symbols $w$ under the $k$ work heads (each a bit or blank), this is the action of $c$ at $(q, a, w)$ with the same input-head move, work-tape writes and moves, and output, and with its successor state read as a state below $N$; a successor state that is not below $N$ is replaced by halting.","title":"Transition function of a code on bounded states"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/to-fin-tm","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.toFinTM","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#to-fin-tm","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For a code $c$ with $k$ work tapes and state bound $N$, this is the binary machine with $k$ work tapes and states $0, 1, \\dots, N - 1$ that starts in state $0$ and, in a state $q < N$ reading $a$ under the input head and $w$ under the work heads, takes the action of $c$ at $(q, a, w)$, with a successor state not below $N$ replaced by halting. Since every state the machine of $c$ can enter is below $N$, it is the\u2026","title":"The finite-state machine of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/is-relabelling-to-fin-tm","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.isRelabelling_toFinTM","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#is-relabelling-to-fin-tm","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For every code $c$, the machine of $c$ simulates the finite-state machine of $c$ up to the relabelling that sends each state $q < N$, where $N$ is the state bound of $c$, to the natural number $q$.","title":"A code's machine simulates its finite-state machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/computable-in-time-and-space","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.computableInTimeAndSpace","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#computable-in-time-and-space","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a code with $k$ work tapes, let $g : \\alpha \\to \\beta$ be a function, let inputs in $\\alpha$ and outputs in $\\beta$ be encoded by injective maps into binary words, and let $t, s : \\alpha \\to \\mathbb{N}$. If the machine of $c$, started on the encoding of any input $x$, halts with the encoding of $g(x)$ as output, halts within $t(x)$ steps, and visits at most $s(x)$ work-tape cells, then some binary\u2026","title":"Computation by a code is finite-state computation"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/decidable-in-time-and-space","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.decidableInTimeAndSpace","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#decidable-in-time-and-space","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a code with $k$ work tapes, let $L$ be a set of binary words, and let $t$ and $s$ map binary words to natural numbers. If the machine of $c$ decides $L$, that is, started on any binary word $w$ it halts with the one-bit output $\\mathrm{true}$ if $w \\in L$ and $\\mathrm{false}$ otherwise, and if on every binary word $w$ it halts within $t(w)$ steps and visits at most $s(w)$ work-tape cells, then some\u2026","title":"Deciding by a code is finite-state decidability"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/of-tm","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.ofTM","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#of-tm","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a machine over the binary alphabet with $k$ work tapes and a finite set of states $Q$, and let $e : Q \\to \\mathbb{N}$. The code of $M$ with states numbered by $e$ has one table entry for each state $q \\in Q$, symbol $a$ under the input head and symbols $w$ under the $k$ work heads (each a bit or blank): its key is $(e(q), a, w)$ and its value is the coded action of $M$ in state $q$ reading $a$ and $w$,\u2026","title":"The code of a finite-state binary machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/act-of-tm","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.act_ofTM","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#act-of-tm","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a binary multi-tape machine with $k$ work tapes and a finite set of states $Q$, and let $e : Q \\to \\mathbb{N}$ be injective. For every state $q \\in Q$, every symbol $a$ under the input head and every tuple $w$ of symbols under the $k$ work heads (each a bit or blank), the coded action of the code of $M$ under $e$ at state $e(q)$ and symbols $a$, $w$ is the coding, with successor states numbered by $e$,\u2026","title":"The code of a machine takes its coded actions"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/is-relabelling-of-tm","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.isRelabelling_ofTM","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#is-relabelling-of-tm","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a machine over the binary alphabet with $k$ work tapes, a finite set of states $Q$ and initial state $q_0$, and let $e : Q \\to \\mathbb{N}$ be injective with $e(q_0) = 0$. Then the machine of the code of $M$ with states numbered by $e$ simulates $M$ up to the relabelling $e$.","title":"The code of a machine simulates it"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/exists-injective-nat-eq-zero","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.exists_injective_nat_eq_zero","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#exists-injective-nat-eq-zero","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For every finite type $Q$ and every $q_0 \\in Q$ there is an injective map $e : Q \\to \\mathbb{N}$ with $e(q_0) = 0$.","title":"Numbering a finite type from a chosen element"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/exists-code-is-relabelling","kind":"node","lean":"Cslib.Turing.MultiTapeTM.Code.exists_code_isRelabelling","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#exists-code-is-relabelling","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"A binary machine with finitely many states is simulated, up to an injective state relabelling, by the machine of some code.","title":"Every finite-state machine has a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/computable-in-time-and-space-iff-exists","kind":"node","lean":"Cslib.Turing.MultiTapeTM.computableInTimeAndSpace_iff_exists_code","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#computable-in-time-and-space-iff-exists","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $g : \\alpha \\to \\beta$ be a function, let inputs in $\\alpha$ and outputs in $\\beta$ be encoded by injective maps into binary words, and let $t, s : \\alpha \\to \\mathbb{N}$. Say a machine computes $g$ within $t$ and $s$ when, started on the encoding of any input $x$, it halts with the encoding of $g(x)$ as output, halts within $t(x)$ steps, and visits at most $s(x)$ work-tape cells. Then some binary multi-tape\u2026","title":"Computability over codes"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/decidable-in-time-and-space-iff-exists-c","kind":"node","lean":"Cslib.Turing.MultiTapeTM.decidableInTimeAndSpace_iff_exists_code","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#decidable-in-time-and-space-iff-exists-c","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"Let $L$ be a set of binary words and let $t$ and $s$ map binary words to natural numbers. Say a machine decides $L$ within $t$ and $s$ when, started on any binary word $w$, it halts with the one-bit output $\\mathrm{true}$ if $w \\in L$ and $\\mathrm{false}$ otherwise, halts within $t(w)$ steps, and visits at most $s(w)$ work-tape cells. Then some binary multi-tape machine with finitely many states decides $L$ within\u2026","title":"Decidability over codes"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/mem-dtime-iff-exists-code","kind":"node","lean":"Cslib.Turing.MultiTapeTM.mem_DTIME_iff_exists_code","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#mem-dtime-iff-exists-code","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For every $t : \\mathbb{N} \\to \\mathbb{N}$ and every language $L$ of binary words, $L \\in \\mathsf{DTIME}(t)$ if and only if there are natural numbers $c_1, c_2, k$ and a code $c$ with $k$ work tapes such that the machine of $c$ decides $L$, on every input $w$ runs in time $c_1 t(|w|) + c_2$, and for some $s : \\mathbb{N} \\to \\mathbb{N}$ on every input $w$ runs in space $s(|w|)$.","title":"DTIME over codes"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/mem-dspace-iff-exists-code","kind":"node","lean":"Cslib.Turing.MultiTapeTM.mem_DSPACE_iff_exists_code","parent":34606,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines/index.html#mem-dspace-iff-exists-code","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-multi-tape-turing-machines","status":"fully_proved","summary":"For every $s : \\mathbb{N} \\to \\mathbb{N}$ and every language $L$ of binary words, $L \\in \\mathsf{DSPACE}(s)$ if and only if there are natural numbers $c_1, c_2, k$ and a code $c$ with $k$ work tapes such that the machine of $c$ decides $L$, on every input $w$ runs in space $c_1 s(|w|) + c_2$, and for some $t : \\mathbb{N} \\to \\mathbb{N}$ on every input $w$ runs in time $t(|w|)$.","title":"DSPACE over codes"},{"data_url":"scopes/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines.json","id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","kind":"scope","navigation":"module","parent":34605,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","title":"Codes for single-tape Turing machines"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/halts-within","kind":"node","lean":"Cslib.Turing.Universal.HaltsWithin","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#halts-within","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"A single-tape Turing machine $M$ halts within $T$ steps on a word $l$ when, started in its initial state with $l$ on the tape and the head on the first letter of $l$ (on a blank if $l$ is empty), it reaches after some $j \\le T$ steps a configuration that has no successor.","title":"Bounded halting of a single-tape machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/mono","kind":"node","lean":"Cslib.Turing.Universal.HaltsWithin.mono","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#mono","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a single-tape Turing machine $M$, a list $l$ of symbols and numbers $T \\le T'$, if $M$ started on $l$ halts within $T$ steps, then $M$ started on $l$ halts within $T'$ steps.","title":"Halting within a bound is monotone in the bound"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/dom-eval-iff-exists-halts-within","kind":"node","lean":"Cslib.Turing.Universal.dom_eval_iff_exists_haltsWithin","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#dom-eval-iff-exists-halts-within","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a single-tape Turing machine $M$ and an input word $l$, the evaluation of $M$ on $l$ is defined (that is, $M$ halts on $l$) if and only if there is $T \\in \\mathbb{N}$ such that $M$ halts on $l$ within $T$ steps.","title":"Halting is halting within some number of steps"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/code","kind":"node","lean":"Cslib.Turing.Universal.Code","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#code","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"A code is a transition table: a finite list of rows, row $q$ belonging to the state $q \\in \\mathbb{N}$, each a finite list of entries, entry $a$ of a row belonging to the symbol $a \\in \\mathbb{N}$. The entry in row $q$ and column $a$ is either empty, meaning that the machine halts, or a pair $(q', s)$ of natural numbers, meaning that the machine enters the state $q'$ and does the action coded by $s$: $0$ moves\u2026","title":"Codes for single-tape machines"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/decode-stmt","kind":"node","lean":"Cslib.Turing.Universal.decodeStmt","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#decode-stmt","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"A natural number codes a statement of a single-tape Turing machine with symbols in $\\mathbb{N}$: $0$ moves the head left, $1$ moves the head right, and $a + 2$ writes the symbol $a$.","title":"Decoding an action"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/bit-sym","kind":"node","lean":"Cslib.Turing.Universal.bitSym","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#bit-sym","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The tape symbol of a bit $b$ is $1$ if $b$ is false and $2$ if $b$ is true; the symbol $0$ is the blank.","title":"Symbol of an input bit"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/input","kind":"node","lean":"Cslib.Turing.Universal.input","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#input","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The input word of a list of bits $x$ is the list of natural numbers obtained from $x$ by replacing each bit by its symbol: $1$ for false and $2$ for true.","title":"Input word of a list of bits"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/equiv-table","kind":"node","lean":"Cslib.Turing.Universal.Code.equivTable","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#equiv-table","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"Codes for single-tape machines correspond bijectively to transition tables, that is, to lists of rows, each a list of entries that are either empty or a pair of natural numbers: a code corresponds to its table.","title":"Single-tape codes as tables"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/lookup","kind":"node","lean":"Cslib.Turing.Universal.Code.lookup","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#lookup","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$ and numbers $q$ and $a$, the lookup of $c$ at state $q$ and symbol $a$ is the entry in row $q$, column $a$ of the table of $c$: either a pair $(q', s)$ of a next state and an action code, or no entry (halt). It is no entry when the table has no row $q$ or that row has no column $a$.","title":"Table lookup of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/machine","kind":"node","lean":"Cslib.Turing.Universal.Code.machine","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#machine","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The machine of a code $c$ is the single-tape Turing machine with states and symbols in $\\mathbb{N}$ that, in state $q$ reading symbol $a$, halts if the table lookup of $c$ at state $q$ and symbol $a$ has no entry, and otherwise, if the entry is $(q', s)$, enters state $q'$ and does the action coded by $s$: $0$ moves left, $1$ moves right and $a' + 2$ writes $a'$.","title":"The machine of a single-tape code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/states","kind":"node","lean":"Cslib.Turing.Universal.Code.states","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#states","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The set of states of a code $c$ is the finite set of natural numbers consisting of the initial state $0$ and every state $q'$ such that some entry of the table of $c$ is a pair $(q', s)$.","title":"States of a single-tape code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/supports-machine","kind":"node","lean":"Cslib.Turing.Universal.Code.supports_machine","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#supports-machine","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The decoded machine is supported by the finite set of states of its code.","title":"Codes denote finite machines"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/cfg","kind":"node","lean":"Cslib.Turing.Universal.Code.Cfg","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#cfg","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"A configuration of the simulation consists of a state $q \\in \\mathbb{N}$, the list of the cells left of the head (nearest first), the symbol in the cell under the head, and the list of the cells right of the head (nearest first); states and symbols are natural numbers.","title":"Explicit configurations"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/equiv-prod","kind":"node","lean":"Cslib.Turing.Universal.Code.Cfg.equivProd","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#equiv-prod","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"Configurations correspond bijectively to tuples in $\\mathbb{N} \\times \\mathrm{List}(\\mathbb{N}) \\times \\mathbb{N} \\times \\mathrm{List}(\\mathbb{N})$: a configuration corresponds to the tuple of its state, its cells left of the head, its cell under the head and its cells right of the head.","title":"Configurations as tuples"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/to-tm0","kind":"node","lean":"Cslib.Turing.Universal.Code.Cfg.toTM0","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#to-tm0","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The TM0 configuration of a configuration $k$ of the simulation has the state of $k$, the cell under the head of $k$ as its head cell, and as the halves of its tape left and right of the head the cells of $k$ left and right of the head, nearest first, followed by infinitely many blanks.","title":"TM0 configuration of a simulation configuration"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/act","kind":"node","lean":"Cslib.Turing.Universal.Code.Cfg.act","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#act","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"Let $k$ be a configuration with left cells $L$ (nearest first), head cell $h$ and right cells $R$ (nearest first), let $q'$ be a state and let $s$ be an action code. Acting on $k$ with $q'$ and $s$ gives a configuration with state $q'$, as follows. For $s = 0$ (move left), its head cell is the first entry of $L$, or the blank $0$ if $L$ is empty, its left cells are $L$ without its first entry, and its right cells\u2026","title":"Acting on a configuration"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/output","kind":"node","lean":"Cslib.Turing.Universal.Code.Cfg.output","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#output","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The output of a configuration $k$ is the list consisting of the symbol under the head of $k$ followed by the cells right of the head (nearest first), with all trailing blanks $0$ removed.","title":"Output of a configuration"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/init","kind":"node","lean":"Cslib.Turing.Universal.Code.init","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#init","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The initial configuration of the simulation on a list of bits $x$ has state $0$, an empty list of cells left of the head, the head on the first symbol of the input word of $x$ (the blank $0$ if $x$ is empty), and the remaining symbols of the input word, in order, right of the head.","title":"Initial configuration of the code simulation"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/step","kind":"node","lean":"Cslib.Turing.Universal.Code.step","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#step","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$ and a configuration $k$ of the simulation, look up the table entry of $c$ at the state of $k$ and the symbol under the head of $k$. If there is no entry, the step is undefined (the machine halts); if the entry is $(q', s)$, the step is the configuration obtained from $k$ by entering the state $q'$ and doing the action coded by $s$.","title":"One step of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/step-or-stay","kind":"node","lean":"Cslib.Turing.Universal.Code.stepOrStay","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#step-or-stay","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$ and a configuration $k$ of the simulation, the configuration after one step of $c$ from $k$ if that step is defined, and $k$ itself if $c$ halts at $k$.","title":"One step of a code, or stay when halted"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/cfg-after","kind":"node","lean":"Cslib.Turing.Universal.Code.cfgAfter","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#cfg-after","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, a list of bits $x$ and $T \\in \\mathbb{N}$, the configuration of $c$ on $x$ after $T$ steps is obtained from the initial configuration on $x$ by applying $T$ times the step of $c$, where a step leaves a halted configuration unchanged. It is the configuration after $T$ steps, or the halting configuration if the machine halts earlier.","title":"Clocked configuration of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/eval-steps","kind":"node","lean":"Cslib.Turing.Universal.Code.evalSteps","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#eval-steps","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"Given a code $c$, a list of bits $x$ and a clock $T \\in \\mathbb{N}$, run the simulation of $c$ on $x$ for at most $T$ steps. If it halts within $T$ steps, the result is the output of the halting configuration: the tape from the head rightwards, with trailing blanks dropped. Otherwise there is no result.","title":"The clocked evaluator"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/step-to-tm0","kind":"node","lean":"Cslib.Turing.Universal.Code.step_toTM0","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#step-to-tm0","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For every code $c$ and every configuration $k$ of the simulation, the decoded TM0 machine of $c$ halts at the TM0 configuration of $k$ exactly when the step of $c$ from $k$ is undefined, and otherwise its step from the TM0 configuration of $k$ is the TM0 configuration of the step of $c$ from $k$.","title":"Explicit steps agree with the decoded machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/init-to-tm0","kind":"node","lean":"Cslib.Turing.Universal.Code.init_toTM0","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#init-to-tm0","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a list of bits $x$, the initial configuration of the simulation on $x$, read as a configuration of a single-tape machine, is the initial configuration of a single-tape machine on the input word of $x$.","title":"The explicit initial configuration is the machine's"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/right-to-tm0","kind":"node","lean":"Cslib.Turing.Universal.Code.right\u2080_toTM0","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#right-to-tm0","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For every explicit configuration $k$, the half-tape of the Mathlib configuration of $k$ from the head rightwards (the cell under the head followed by the cells to its right) is the output of $k$, read with blanks beyond its end.","title":"The output of a configuration as a Mathlib half-tape"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/iterate-step-to-tm0","kind":"node","lean":"Cslib.Turing.Universal.Code.iterate_step_toTM0","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#iterate-step-to-tm0","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a code, $n \\in \\mathbb{N}$ and $k$ an explicit configuration. Run the machine of $c$ for $n$ steps from the Mathlib configuration of $k$, and run the explicit simulation of $c$ for $n$ steps from $k$, where a run that reaches a configuration without a successor before completing $n$ steps has no result. Then the first run has a result exactly when the second does, and its result is the Mathlib\u2026","title":"Iterated steps of the decoded machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/halts-within-machine-iff","kind":"node","lean":"Cslib.Turing.Universal.Code.haltsWithin_machine_iff","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#halts-within-machine-iff","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, a list of bits $x$ and $T \\in \\mathbb{N}$, the decoded machine of $c$ halts within $T$ steps on the input word of $x$ if and only if the explicit simulation of $c$, started from the initial configuration on $x$, reaches after some $j \\le T$ steps a configuration that has no successor step.","title":"Bounded halting of a decoded machine through its simulation"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/is-some-eval-steps-iff-step-eq-none","kind":"node","lean":"Cslib.Turing.Universal.Code.isSome_evalSteps_iff_step_eq_none","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#is-some-eval-steps-iff-step-eq-none","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, a list of bits $x$ and a clock $T \\in \\mathbb{N}$, the clocked evaluator of $c$ on $x$ with clock $T$ returns a value if and only if the simulation of $c$ has no step from the configuration after $T$ clocked steps on $x$ (the configuration after $T$ steps, or the halting configuration if the machine halts earlier).","title":"Definedness of the clocked evaluator"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/is-some-eval-steps-iff","kind":"node","lean":"Cslib.Turing.Universal.Code.isSome_evalSteps_iff","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#is-some-eval-steps-iff","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, a list of bits $x$ and a clock $T \\in \\mathbb{N}$, the clocked evaluator of $c$ on $x$ with clock $T$ returns a value if and only if the machine of $c$, started on the input word of $x$, halts within $T$ steps.","title":"Clocked halting"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/eval-steps-eq-some-iff-output","kind":"node","lean":"Cslib.Turing.Universal.Code.evalSteps_eq_some_iff_output","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#eval-steps-eq-some-iff-output","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$, a list of bits $x$, a clock $T \\in \\mathbb{N}$ and a list $l$ of natural numbers, the clocked evaluator returns $l$ on $c$, $x$ and $T$ if and only if the configuration of $c$ after $T$ clocked steps on $x$ has no successor step and its output is $l$.","title":"Characterization of clocked results"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/eval-steps-mono","kind":"node","lean":"Cslib.Turing.Universal.Code.evalSteps_mono","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#eval-steps-mono","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a code for a single-tape machine, $x$ a list of bits, $l$ a list of natural numbers and $T \\le T'$ natural numbers. If the clocked evaluator returns $l$ on $c$ and $x$ with clock $T$, then it also returns $l$ with clock $T'$.","title":"Monotonicity of the single-tape clocked evaluator"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/eval-eq-some-of-eval-steps-eq-some","kind":"node","lean":"Cslib.Turing.Universal.Code.eval_eq_some_of_evalSteps_eq_some","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#eval-eq-some-of-eval-steps-eq-some","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"Let $c$ be a code, $x$ a list of bits, $T$ a natural number and $l$ a list of natural numbers. If the clocked evaluator, running $c$ on $x$ for at most $T$ steps, returns $l$, then the decoded machine of $c$, started on the input word of $x$, halts, and the tape from its head rightwards in the halting configuration is $l$ followed by blanks.","title":"Clocked results agree with the machine's semantics"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/dom-eval-iff-exists-is-some-eval-steps","kind":"node","lean":"Cslib.Turing.Universal.Code.dom_eval_iff_exists_isSome_evalSteps","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#dom-eval-iff-exists-is-some-eval-steps","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"For a code $c$ and a list of bits $x$, the machine decoded from $c$ halts on the input word of $x$ if and only if there is a clock $T \\in \\mathbb{N}$ for which the clocked evaluator of $c$ on $x$ with clock $T$ returns a result.","title":"Unclocked halting"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/exists-code","kind":"node","lean":"Cslib.Turing.Universal.Code.exists_code","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#exists-code","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"Universality of codes, for machines that read bit inputs. Let $M$ be a single-tape Turing machine with a finite alphabet $\\Gamma$ and a finite set of states that contains the initial state and is closed under the transitions of $M$, and let $\\iota$ be an injective map from the bits to $\\Gamma$ whose values are not the blank. Then there are a code $c$ and an injective map $f$ from $\\Gamma$ to the natural numbers,\u2026","title":"Universality of codes"},{"id":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/primrec-eval-steps","kind":"node","lean":"Cslib.Turing.Universal.Code.primrec_evalSteps","parent":34666,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines/index.html#primrec-eval-steps","scope":"TheoryOfComputation/turing-machines/universal-machines/codes-for-single-tape-turing-machines","status":"fully_proved","summary":"The clocked evaluator is primitive recursive in the code, the input and the clock.","title":"The clocked evaluator is primitive recursive"},{"data_url":"scopes/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine.json","id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","kind":"scope","navigation":"module","parent":34605,"result":false,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","title":"Clocked universal Turing machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/cell","kind":"node","lean":"Cslib.Turing.Universal.Clocked.cell","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#cell","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The cell of a natural number $n$ is the list of bits consisting of $n$ copies of the bit true (units) followed by one bit false (a marker).","title":"The cell of a number"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/cells","kind":"node","lean":"Cslib.Turing.Universal.Clocked.cells","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#cells","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The cells of a list $(n_1, \\ldots, n_m)$ of natural numbers are the concatenation of the cells of $n_1, \\ldots, n_m$ in this order, so that the cell of $n_1$ comes first (on top of a stack).","title":"The cells of a list of numbers"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/k","kind":"node","lean":"Cslib.Turing.Universal.Clocked.K","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#k","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The clocked universal machine has fifteen stacks: the state $Q$, the next state $N$, the action $A$ and the head cell $H$ (these four hold counters of units), the cells left ($L$) and right ($R$) of the head, the code $C$, the scan buffer $B$, the scratch counters $T$, $U$, $V$, the clock $\\mathrm{Ck}$, and the parsing stacks: the encoded input $X$, the parse buffer $P$ and the output $O$.","title":"Stacks of the clocked universal machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/lab","kind":"node","lean":"Cslib.Turing.Universal.Clocked.Lab","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#lab","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The labels of the program of the clocked universal machine: the tape update (`act0`\u2013`act3`, `wr0`, `wr1`, `mvL0`\u2013`mvL3`, `mvR0`\u2013`mvR3`) and the clock tick `fin`; the scan, which starts at `sc0` and has labels for the state (`rq0`\u2013`rq2`; `cqA o` and `cqB o` for each comparison outcome `o`; `sq0 m` and `sq1 m` for each bit `m`), for the head cell (`ra0`\u2013`ra2`; `caA o`, `caB o`; `sa0 m`, `sa1 m`), for skipping the\u2026","title":"Labels of the clocked universal machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/untitled","kind":"node","lean":"Cslib.Turing.Universal.Clocked.\u0393","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#untitled","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Every stack $k$ of the clocked universal machine holds bits: its alphabet is $\\Gamma(k) = \\mathrm{Bool}$.","title":"Stack alphabet of the clocked universal machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/d-u","kind":"node","lean":"Cslib.Turing.Universal.Clocked.dU","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#d-u","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The unit decoder sends the bit true (a unit) to the bit true, and sends the bit false (a marker) to no bit, which ends a block.","title":"Decoding units"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/mark","kind":"node","lean":"Cslib.Turing.Universal.Clocked.mark","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#mark","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a label $\\ell$, this instruction pops the top symbol of the code stack $C$ and discards it (doing nothing if $C$ is empty), pushes a marker (the bit false) onto the scan buffer $B$, and goes to $\\ell$; the state bit is unchanged.","title":"Moving a marker onto the scan buffer"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/prog","kind":"node","lean":"Cslib.Turing.Universal.Clocked.prog","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#prog","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The program of the clocked universal machine gives an instruction for every label. Numbers sit on stacks as counters of units (the bit true), and a cell is a counter followed by a marker (the bit false).","title":"The clocked universal program"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/entries","kind":"node","lean":"Cslib.Turing.Universal.Clocked.entries","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#entries","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a code $c$, the list of quadruples $(q, a, q', s)$, one for each state $q$ and symbol $a$ inside the table of $c$ whose table entry is a transition $(q', s)$ (enter the state $q'$ and do the action coded by $s$); halting positions are skipped. The quadruples are listed row by row, by increasing $q$, and within a row by increasing $a$.","title":"Entries of a code"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/find-entry","kind":"node","lean":"Cslib.Turing.Universal.Clocked.findEntry","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#find-entry","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For natural numbers $q$ and $a$ and a list of quadruples $(q_i, a_i, q'_i, s_i)$ of natural numbers, the result is the pair $(q'_i, s_i)$ of the first quadruple in the list with $q_i = q$ and $a_i = a$, and no result if no quadruple has the key $(q, a)$. This is the order in which the machine scans the code stack.","title":"Looking up a key in a list of entries"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/find-entry-entries","kind":"node","lean":"Cslib.Turing.Universal.Clocked.findEntry_entries","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#find-entry-entries","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For every code $c$ and all natural numbers $q$ and $a$, looking up the key $(q, a)$ in the entries of $c$ gives the entry of the table of $c$ at state $q$ and symbol $a$: the pair $(q', s)$ if that position holds the transition $(q', s)$, and no result if it halts, including every position outside the table.","title":"Scanning the entries finds the table entry"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/entry-nums","kind":"node","lean":"Cslib.Turing.Universal.Clocked.entryNums","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#entry-nums","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a list of quadruples of natural numbers, the list of natural numbers obtained by replacing each quadruple $(q, a, q', s)$ by the four numbers $q, a, q', s$ in this order, keeping the order of the quadruples.","title":"Numbers of a list of entries"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/code-cells","kind":"node","lean":"Cslib.Turing.Universal.Clocked.codeCells","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#code-cells","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The code stack of a code $c$ is the list of bits formed by the cells of the numbers $q, a, q', s$ of every entry $(q, a, q', s)$ of $c$ (a table position $(q, a)$ holding the transition $(q', s)$), with the entries ordered by $q$ and then by $a$, and the four numbers of an entry in the order $q, a, q', s$.","title":"The code stack"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/stk","kind":"node","lean":"Cslib.Turing.Universal.Clocked.stk","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#stk","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a code $c$, a natural number $t$ (the clock), a configuration $k$ and natural numbers $q'$ and $s$, the stacks of $c$, $t$ and $k$ with next state $q'$ and action $s$ hold, as counters of units (the bit true): the state of $k$ on $Q$, $q'$ on $N$, $s$ on $A$, the head symbol of $k$ on $H$ and $t$ on $Ck$. The stacks $L$ and $R$ hold the cells of the symbols left and right of the head, nearest first, $C$ holds\u2026","title":"The stacks of a simulated configuration"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/act-time","kind":"node","lean":"Cslib.Turing.Universal.Clocked.actTime","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#act-time","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a configuration $k$ with state $q$, head cell $h$, cells $\\ell$ left of the head and cells $r$ right of the head (nearest first), a next state $q'$ and an action $s$, the step bound of the tape update is $q + q' + s + h + \\ell_0 + r_0 + 8$, where $\\ell_0$ and $r_0$ are the first entries of $\\ell$ and $r$, taken to be $0$ for an empty list.","title":"Step bound of the tape update"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/act-run","kind":"node","lean":"Cslib.Turing.Universal.Clocked.act_run","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#act-run","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, $k$ a configuration and $t$, $q'$, $s$ natural numbers. Started at the label $\\mathsf{act0}$, with any value of its internal variable, on the stacks holding the code $c$, the clock $t$ and the configuration $k$, with $q'$ on the next-state stack, $s$ on the action stack and all other stacks empty, the program of the clocked universal machine reaches the label $\\mathsf{fin}$, with some value of\u2026","title":"The tape update"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/entry-cost","kind":"node","lean":"Cslib.Turing.Universal.Clocked.entryCost","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#entry-cost","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The step budget for scanning past an entry $(q, a, q', s)$ of the code stack is $5q + 5a + 2q' + 2s + 30$.","title":"Cost of scanning past one entry"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/step-time","kind":"node","lean":"Cslib.Turing.Universal.Clocked.stepTime","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#step-time","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a code $c$ and a configuration $k$ with state $q$, head cell $h$, nearest left cell $l$ and nearest right cell $r$ (each read as the blank $0$ when there is no such cell), let $|C|$ be the length of the code stack of $c$. The step bound of $c$ at $k$ is $10|C| + q + h + l + r + 10$.","title":"Step bound of one simulated step"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/step-run-of-step-eq-some","kind":"node","lean":"Cslib.Turing.Universal.Clocked.step_run_of_step_eq_some","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#step-run-of-step-eq-some","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, $k$ and $k'$ configurations such that one step of $c$ from $k$ gives $k'$, $t$ a natural number, $v$ a bit, and $b$ the step bound of $c$ at $k$. Started at the label $\\mathsf{sc0}$ with state bit $v$ and the stacks of the code $c$, the clock $t$ and the configuration $k$ with next state and action $0$, the clocked universal program reaches the label $\\mathsf{fin}$ within $b$ steps, with some\u2026","title":"One simulated step, running case"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/step-run-of-step-eq-none","kind":"node","lean":"Cslib.Turing.Universal.Clocked.step_run_of_step_eq_none","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#step-run-of-step-eq-none","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, $k$ a configuration at which $c$ halts (one step of $c$ from $k$ gives nothing), $t$ a natural number, $v$ a bit, and $b$ the step bound of $c$ at $k$. Started at the label $\\mathsf{sc0}$ with state bit $v$ and the stacks of the code $c$, the clock $t$ and the configuration $k$ with next state and action $0$, the clocked universal program reaches the label $\\mathsf{hlt}$ within $b$ steps, with\u2026","title":"One simulated step, halting case"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/bounded","kind":"node","lean":"Cslib.Turing.Universal.Clocked.Bounded","parent":34702,"reading_title":{"format":"inline-parts-v1","parts":[["text","Configurations bounded by "],["math","\\(M\\)"]],"source":"Configurations bounded by $M$"},"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#bounded","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"A configuration $k$ is bounded by a natural number $M$ when its state, its head cell and every cell left and right of the head are at most $M$.","title":"Configurations bounded by $M$"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/loop-step-time","kind":"node","lean":"Cslib.Turing.Universal.Clocked.loopStepTime","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#loop-step-time","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a code $c$ and a number $M$, the loop step time is $10\\ell + 4M + 11$, where $\\ell$ is the length of the code stack of $c$. It serves as the cost bound of one round of the loop (a simulated step and a clock tick) on configurations whose numbers are at most $M$.","title":"Step bound of the clocked loop"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/loop-run","kind":"node","lean":"Cslib.Turing.Universal.Clocked.loop_run","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#loop-run","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, let $M$ be at least the length of the code stack of $c$, let $k$ be a configuration whose state and cells are all at most $M$, let $t$ be a natural number and $v$ a bit, and let $k_t$ be the result of $t$ steps of $c$ from $k$, where a halted configuration stays put. Start the program of the clocked universal machine at label `sc0` with variable $v$ and the stacks of $c$, the clock $t$ and $k$\u2026","title":"The clocked loop"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/clock-time","kind":"node","lean":"Cslib.Turing.Universal.Clocked.clockTime","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#clock-time","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a code $c$ and a clock $T$, let $|C|$ be the length of the code stack of $c$. The step bound of the clocked simulation on the clock $T$ is $T + 1$ times the cost of one round of the loop on configurations whose numbers are at most $|C| + 2$, that is, $(T + 1)(14|C| + 19)$.","title":"Step bound of the clocked simulation"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/clocked-run","kind":"node","lean":"Cslib.Turing.Universal.Clocked.clocked_run","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#clocked-run","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, $x$ a list of bits and $T$ a natural number, and let $|C|$ be the length of the code stack of $c$. Start the program of the clocked universal machine at the label $\\mathsf{sc0}$, with any value of its internal variable, on the stacks holding the code $c$, the clock $T$ and the initial configuration on $x$, all other stacks empty. If the clocked evaluation of $c$ on $x$ for $T$ steps is defined,\u2026","title":"The clocked universal machine on stacks"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/enc-in","kind":"node","lean":"Cslib.Turing.Universal.Clocked.encIn","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#enc-in","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a code $c$, a list of bits $x$ and a clock $T$, the input of the clocked universal machine is the list of bits formed by the cell of $T$, then the cells of the input word of $x$, then the cell of $0$ (a blank cell, the single bit false), then the code stack of $c$ in reverse order.","title":"Input encoding of the clocked universal machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/parse-time","kind":"node","lean":"Cslib.Turing.Universal.Clocked.parseTime","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#parse-time","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a code $c$, a list of bits $x$ and a natural number $T$, let $w_1, \\dots, w_n$ be the input word of $x$ (false as the symbol $1$, true as $2$), with $w_1$ read as $0$ when $x$ is empty, let $S_j = \\sum_{i=1}^{n} (w_i + j)$, and let $m$ be the length of the code stack of $c$. The cost of reading the input is $T + 2 + (S_3 + 2) + (S_1 + 1) + (w_1 + 2) + (m + 1)$.","title":"Cost of reading the input"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/parse-run","kind":"node","lean":"Cslib.Turing.Universal.Clocked.parse_run","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#parse-run","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, $x$ a list of bits and $T$ a natural number, and let $p$ be the cost of reading the input of $c$, $x$ and $T$. Started at the label $\\mathsf{pc0}$ with state bit false, the input encoding of $c$, $x$ and $T$ on the stack $X$ and every other stack empty, the clocked universal program reaches the label $\\mathsf{sc0}$ in exactly $p$ steps, with some state bit and the stacks of the code $c$, the\u2026","title":"Reading the input"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/enc-out","kind":"node","lean":"Cslib.Turing.Universal.Clocked.encOut","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#enc-out","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The output bits of an optional list of numbers: no result (a timeout) is encoded as the single bit false, and a result $l$ is encoded as the bit true followed by the cells of $l$, where a number $n$ is written as $n$ bits true followed by one bit false.","title":"Output encoding of the clocked universal machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/out-run","kind":"node","lean":"Cslib.Turing.Universal.Clocked.out_run","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#out-run","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, $t$ a natural number, $k$ a configuration and $v$ a bit, and let $N$ be the length of the cells of the head symbol of $k$ followed by the symbols to its right (a symbol $a$ takes $a + 1$ bits). Started at the label $\\mathsf{hlt}$ with state bit $v$ and the stacks of the code $c$, the clock $t$ and the configuration $k$ with next state and action $0$, the clocked universal program reaches the\u2026","title":"Writing the output"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/tmo-run","kind":"node","lean":"Cslib.Turing.Universal.Clocked.tmo_run","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#tmo-run","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, $t$ a natural number, $k$ a configuration and $v$ a bit. Started at the label $\\mathsf{tmo}$ with state bit $v$ and the stacks of the code $c$, the clock $t$ and the configuration $k$ with next state and action $0$, the clocked universal program reaches the label $\\mathsf{done}$ in exactly one step, with the same state bit and the same stacks except that the output stack $O$ holds the timeout\u2026","title":"Writing a timeout"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/univ-time","kind":"node","lean":"Cslib.Turing.Universal.Clocked.univTime","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#univ-time","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For a code $c$, a list of bits $x$ and a natural number $T$, the step bound of the whole machine on $(c, x, T)$ is the sum of three terms: the cost of reading the input on $(c, x, T)$, the step bound of the clocked simulation on the clock $T$, and the output bound $4\\,(|x| + T + 1)\\,(|C| + 3) + 8$, where $|x|$ is the length of $x$ and $|C|$ is the length of the code stack of $c$.","title":"Step bound of the clocked universal machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/univ-run","kind":"node","lean":"Cslib.Turing.Universal.Clocked.univ_run","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#univ-run","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"Let $c$ be a code, $x$ a list of bits and $T$ a natural number. Start the clocked universal stack program at the label $\\mathrm{pc0}$ with internal bit $\\mathrm{false}$, with every stack empty except $X$, which holds the input encoding of $(c, x, T)$: the cell of $T$, the cells of the input symbols of $x$ (the symbol $1$ for $\\mathrm{false}$ and $2$ for $\\mathrm{true}$), the bit $\\mathrm{false}$, and the code stack\u2026","title":"Correctness and running time of the clocked universal machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/univ-comp","kind":"node","lean":"Cslib.Turing.Universal.Clocked.univComp","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#univ-comp","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The component machine whose stacks are the fifteen stacks of the clocked universal machine, each holding bits, whose labels and program are those of the clocked universal machine, and whose list of stacks is, in order: the state $Q$, the next state $N$, the action $A$, the head cell $H$, the cells left ($L$) and right ($R$) of the head, the code $C$, the scan buffer $B$, the scratch counters $T$, $U$, $V$, the\u2026","title":"The clocked universal machine as a host component"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/univ-time-le","kind":"node","lean":"Cslib.Turing.Universal.Clocked.univTime_le","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#univ-time-le","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"For every code $c$, list of bits $x$ and clock $T \\in \\mathbb{N}$, the step bound of the whole clocked universal machine on $(c, x, T)$ is at most $18 (n + 1)^2$, where $n$ is the length of the input encoding of $(c, x, T)$.","title":"The step bound of the clocked universal machine is quadratic"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/univ-poly","kind":"node","lean":"Cslib.Turing.Universal.Clocked.univPoly","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#univ-poly","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"The polynomial $18 (X + 1)^2$ with natural-number coefficients, used as the step bound of the clocked universal machine as a host component.","title":"Step bound polynomial of the clocked universal machine"},{"id":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/nonempty-univ-poly-time","kind":"node","lean":"Cslib.Turing.Universal.Clocked.nonempty_univPolyTime","parent":34702,"result":true,"route":"../roadmap/TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine/index.html#nonempty-univ-poly-time","scope":"TheoryOfComputation/turing-machines/universal-machines/clocked-universal-turing-machine","status":"fully_proved","summary":"There are a multi-stack Turing machine with finitely many stacks, labels and internal states, whose input and output alphabets are in bijection with the bits, and a polynomial $p$ with natural-number coefficients, such that for every code $c$, list of bits $x$ and clock $T \\in \\mathbb{N}$ the following holds. Let $n$ be the length of the input encoding of $(c, x, T)$. Started at its main label in its initial\u2026","title":"The clocked universal function is polynomial-time computable"},{"data_url":"scopes/TheoryOfComputation/equivalence-of-models.json","id":"TheoryOfComputation/equivalence-of-models","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/index.html","scope":"TheoryOfComputation/equivalence-of-models","status":"can_prove","summary":"Register (URM), counter and RAM machines, and theorems that Turing machines and other models compute exactly the partial recursive functions or decide and recognize exactly the recursive and r.e. predicates (Church\u2013Turing equivalences).","title":"Equivalence of models of computation"},{"data_url":"scopes/TheoryOfComputation/equivalence-of-models/counter-machines.json","id":"TheoryOfComputation/equivalence-of-models/counter-machines","kind":"scope","navigation":"module","parent":34740,"result":false,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","title":"Counter machines"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-instr","kind":"node","lean":"Cslib.Decidability.CounterInstr","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-instr","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"A counter-machine instruction is one of: increment register $r$, decrement register $r$, or jump to instruction $q$ if register $r$ is zero, where the register index $r$ and the jump target $q$ are natural numbers.","title":"Counter-machine instructions"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-state","kind":"node","lean":"Cslib.Decidability.CounterState","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-state","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"A counter-machine state consists of a program counter, which is a natural number, and the register contents, a function assigning a natural number to every register index $i \\in \\mathbb{N}$.","title":"Counter-machine states"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-step","kind":"node","lean":"Cslib.Decidability.counterStep","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-step","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"Let $p$ be a counter-machine program, a list of instructions indexed from $0$, and let $s$ be a state with program counter $c$. If $c$ is at least the length of $p$ (the machine has halted), the step leaves $s$ unchanged. Otherwise it executes instruction $c$ of $p$: increment register $r$ adds $1$ to register $r$ and sets the program counter to $c + 1$; decrement register $r$ subtracts $1$ from register $r$ if it\u2026","title":"One step of a counter machine"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-init","kind":"node","lean":"Cslib.Decidability.counterInit","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-init","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"The initial counter-machine state on a list of inputs $x_0, \\dots, x_{k-1}$ has program counter $0$, register $i$ holding $x_i$ for each $i < k$, and every other register holding $0$.","title":"Initial state of a counter machine"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-exec-n","kind":"node","lean":"Cslib.Decidability.counterExecN","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-exec-n","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"For a counter-machine program $p$, a natural number $t$ and a state $s$, the execution of $p$ for $t$ steps from $s$ is the state obtained from $s$ by applying $t$ single steps of $p$.","title":"Bounded execution of a counter machine"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-eval","kind":"node","lean":"Cslib.Decidability.counterEval","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-eval","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"The output of a counter-machine program $p$ on a list of inputs is the partial natural number obtained as follows: run $p$ from the initial state on those inputs; if for some number $t$ of steps the program counter is at least the length of $p$ (the machine has halted), the output is the content of register $0$ after the least such $t$; if no such $t$ exists, the output is undefined.","title":"Output of a counter machine"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-step-sticky","kind":"node","lean":"Cslib.Decidability.counterStep_sticky","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-step-sticky","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"Past the end of the program the state does not change.","title":"A counter step is sticky at halting states"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-exec-n-stable","kind":"node","lean":"Cslib.Decidability.counterExecN_stable","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-exec-n-stable","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"Let $p$ be a counter-machine program, $s$ a state and $t$ a natural number. If the state reached from $s$ after $t$ steps of $p$ has program counter at least the length of $p$, then the state reached from $s$ after $t + 1$ steps equals the state reached after $t$ steps.","title":"Counter-machine execution is stable once halted"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-exec-n-eq-of-halted","kind":"node","lean":"Cslib.Decidability.counterExecN_eq_of_halted","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-exec-n-eq-of-halted","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"Let $p$ be a counter-machine program, $s$ a state, and $t_1 \\le t_2$ natural numbers. If the state reached from $s$ after $t_1$ steps of $p$ has program counter at least the length of $p$, then the state reached from $s$ after $t_2$ steps equals the state reached after $t_1$ steps.","title":"Counter-machine execution is constant after halting"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/mem-counter-eval-of-halted","kind":"node","lean":"Cslib.Decidability.mem_counterEval_of_halted","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#mem-counter-eval-of-halted","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"Let $P$ be a counter-machine program, let $x$ be a list of natural-number inputs, and let $t$ and $y$ be natural numbers. If the state reached after $t$ steps of $P$ from the initial state on $x$ has program counter at least the length of $P$ (so the machine has halted) and register $0$ holds $y$ in that state, then the output of $P$ on $x$ is defined and equals $y$.","title":"Output membership for counter machines"},{"id":"TheoryOfComputation/equivalence-of-models/counter-machines/counter-decidable","kind":"node","lean":"Cslib.Decidability.CounterDecidable","parent":34741,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/counter-machines/index.html#counter-decidable","scope":"TheoryOfComputation/equivalence-of-models/counter-machines","status":"fully_proved","summary":"A predicate $P$ on the natural numbers is counter-machine decidable when some counter-machine program computes its characteristic function: for every natural number $n$, the output of the program on the single input $n$ is defined and equals $1$ if $P(n)$ holds and $0$ otherwise.","title":"Counter-machine decidable predicates"},{"data_url":"scopes/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun.json","id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","kind":"scope","navigation":"module","parent":34740,"result":false,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"can_prove","title":"Equivalence of Turing machines and recursive functions"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/encode-nat-eq-cons","kind":"node","lean":"MetaMathlibExt.TM2Recursive.encodeNat_eq_cons","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#encode-nat-eq-cons","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For every natural number $n \\neq 0$, the binary encoding of $n$ is the list whose first bit is true exactly when $n$ is odd, followed by the binary encoding of $\\lfloor n/2 \\rfloor$.","title":"Binary encoding, least significant bit first"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/bits-val","kind":"node","lean":"MetaMathlibExt.TM2Recursive.bitsVal","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#bits-val","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"The value of a list of bits $b_0, b_1, \\dots, b_{m-1}$ is $\\sum_{i < m} [b_i]\\, 2^i$, where $[b_i]$ is $1$ if $b_i$ is true and $0$ otherwise. So the first bit is the least significant, and the empty list has value $0$.","title":"Value of a little-endian bit list"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/partrec-run","kind":"node","lean":"MetaMathlibExt.TM2Recursive.partrec_run","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#partrec-run","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $\\alpha$, $\\sigma$ and $\\beta$ be primitive recursively encodable types, and let $\\mathrm{init} : \\alpha \\to \\sigma$, $\\mathrm{step} : \\sigma \\to \\sigma$, $\\mathrm{halted} : \\sigma \\to \\mathrm{Bool}$ and $\\mathrm{out} : \\sigma \\to \\beta$ be primitive recursive. Then the partial function sending $a$ to $\\mathrm{out}(\\mathrm{step}^k(\\mathrm{init}(a)))$, where $k$ is the least natural number for which\u2026","title":"Running a primitive recursive machine is partial recursive"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/primrec-of-eventually-const","kind":"node","lean":"MetaMathlibExt.TM2Recursive.primrec_of_eventually_const","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#primrec-of-eventually-const","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $\\beta$ be a primitive recursively encodable type. If $g : \\mathbb{N} \\to \\beta$ and $N \\in \\mathbb{N}$ satisfy $g(c) = g(N)$ for all $c \\ge N$, then $g$ is primitive recursive.","title":"Eventually constant functions on \u2115 are primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/primrec-of-eventually-const-2","kind":"node","lean":"MetaMathlibExt.TM2Recursive.primrec\u2082_of_eventually_const","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#primrec-of-eventually-const-2","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $\\alpha$ be a finite primitive recursively encodable type and $\\beta$ a primitive recursively encodable type. If $g : \\alpha \\to \\mathbb{N} \\to \\beta$ and $N \\in \\mathbb{N}$ satisfy $g(a, c) = g(a, N)$ for all $a \\in \\alpha$ and all $c \\ge N$, then $g$ is primitive recursive as a function of two arguments.","title":"Eventually constant functions of a finite parameter are primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/primrec-of-finite-left","kind":"node","lean":"MetaMathlibExt.TM2Recursive.primrec\u2082_of_finite_left","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#primrec-of-finite-left","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $\\alpha$, $\\beta$ and $\\sigma$ be primitive recursively encodable types with $\\alpha$ finite, and let $g : \\alpha \\to \\beta \\to \\sigma$. If $b \\mapsto g(a, b)$ is primitive recursive for every $a \\in \\alpha$, then $g$ is primitive recursive as a function of two arguments.","title":"Joint primitive recursion over a finite first argument"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/primcodable-arrow","kind":"node","lean":"MetaMathlibExt.TM2Recursive.primcodableArrow","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#primcodable-arrow","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite type $K$ and a primitive recursively encodable type $\\beta$, the primitive recursive encoding of the functions $K \\to \\beta$ transported from that of the functions $\\mathrm{Fin}\\, n \\to \\beta$, where $n$ is the cardinality of $K$, along a chosen bijection between $K$ and $\\mathrm{Fin}\\, n$.","title":"Encoding of functions on a finite type"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/pushes","kind":"node","lean":"MetaMathlibExt.TM2Recursive.pushes","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#pushes","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, a stack $k$ and a statement $q$ over the machine's stack alphabets, labels and internal states, the set of symbols of stack $k$ consisting of every value, at every internal state, of the symbol function of every push instruction onto stack $k$ that occurs in $q$, in either branch of each conditional.","title":"Symbols a statement may push"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/relevant","kind":"node","lean":"MetaMathlibExt.TM2Recursive.relevant","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#relevant","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a stack index $k$, the relevant symbols of stack $k$ are the symbols $f(v)$ of $\\Gamma_k$, for every internal state $v$, over all statements that push $f$ onto stack $k$ and occur in the program of some label, together with, when $k$ is the input stack $k_0$, every symbol of $\\Gamma_{k_0}$.","title":"Relevant symbols of a stack"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/sym-list","kind":"node","lean":"MetaMathlibExt.TM2Recursive.symList","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#sym-list","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine and a stack index $k$, a list without repetitions whose entries are exactly the relevant symbols of stack $k$, in an order fixed by choice.","title":"Enumeration of the relevant symbols of a stack"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/code","kind":"node","lean":"MetaMathlibExt.TM2Recursive.code","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#code","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, a stack index $k$ and a symbol $x$ of stack $k$, the code of $x$ is the position of $x$ (counting from $0$) in the chosen enumeration of the relevant symbols of stack $k$, or the length of that enumeration if $x$ is not a relevant symbol.","title":"Code of a stack symbol"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/decode","kind":"node","lean":"MetaMathlibExt.TM2Recursive.decode","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#decode","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, a stack index $k$ and a natural number $c$, the decoding of $c$ is the entry at position $c$ (counting from $0$) of the chosen enumeration of the relevant symbols of stack $k$, and is undefined when $c$ is at least the length of that enumeration.","title":"Decoding a stack symbol"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/relevant-2","kind":"node","lean":"MetaMathlibExt.TM2Recursive.Relevant","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#relevant-2","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"A family of stacks $S$, consisting of a list $S_k$ of symbols of $\\Gamma_k$ for each stack index $k$, has only relevant symbols when for every stack index $k$, every symbol on $S_k$ is a relevant symbol of stack $k$.","title":"Stacks with only relevant symbols"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/primcodable","kind":"node","lean":"MetaMathlibExt.TM2Recursive.primcodable\u039b","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#primcodable","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, the primitive recursive encoding of its finite type of labels transported from that of $\\mathrm{Fin}\\, n$, where $n$ is the number of labels, along a chosen bijection between the two.","title":"Encoding of labels"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/primcodable-2","kind":"node","lean":"MetaMathlibExt.TM2Recursive.primcodable\u03c3","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#primcodable-2","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, the primitive recursive encoding of its finite type of internal states transported from that of $\\mathrm{Fin}\\, n$, where $n$ is the number of internal states, along a chosen bijection between the two.","title":"Encoding of internal states"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/coded-cfg","kind":"node","lean":"MetaMathlibExt.TM2Recursive.CodedCfg","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#coded-cfg","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine with label set $\\Lambda$, internal state set $\\sigma$ and stack index set $K$, a coded configuration is a triple consisting of an optional label in $\\Lambda$ (no label marks a halted configuration), an internal state in $\\sigma$, and an assignment of a list of natural numbers to each $k \\in K$.","title":"Coded configurations of a multi-stack machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/enc-stk","kind":"node","lean":"MetaMathlibExt.TM2Recursive.encStk","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#enc-stk","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, the coded stacks of a family $S$ of stacks assign to each stack index $k$ the list obtained from the stack $S_k$ by replacing each symbol with its code.","title":"Coded stacks"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/enc","kind":"node","lean":"MetaMathlibExt.TM2Recursive.enc","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#enc","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, the coded configuration of a configuration $c$ has the same optional label and internal state as $c$, and the coded stacks of the stacks of $c$.","title":"Coding a configuration"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/read-top","kind":"node","lean":"MetaMathlibExt.TM2Recursive.readTop","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#read-top","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $k$ be a stack index of the machine and let $S$ assign to each stack index $j$ a list $S_j$ of natural numbers (codes of symbols). The top of stack $k$ as read from $S$ is no symbol when $S_k$ is empty. Otherwise, if $c$ is the first entry of $S_k$, it is the decoding of $c$: the symbol of $\\Gamma_k$ at position $c$, counting from $0$, in the list of relevant symbols of stack $k$, or no symbol when $c$ is at\u2026","title":"Reading the top of a coded stack"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/step-aux-c","kind":"node","lean":"MetaMathlibExt.TM2Recursive.stepAuxC","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#step-aux-c","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"This mirrors the machine's execution of one statement on stacks of symbol codes. Given a statement $q$, an internal state $v$ and coded stacks $S$ (a list $S_k$ of natural numbers for each stack index $k$), it returns a coded configuration (an optional label, an internal state and coded stacks), by recursion on $q$. A push of $f$ onto stack $k$ followed by $q'$ replaces $S_k$ by the code of the symbol $f(v)$ of\u2026","title":"The coded execution of a statement"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/step-c","kind":"node","lean":"MetaMathlibExt.TM2Recursive.stepC","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#step-c","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a coded configuration $(l, v, S)$, consisting of an optional label $l$, an internal state $v$ and coded stacks $S$, the coded step is undefined when $l$ is absent (the configuration has halted). When $l$ is a label, it is the coded configuration obtained by the coded execution of the program statement of $l$ from the internal state $v$ on the coded stacks $S$.","title":"The coded step"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/primrec-read-top","kind":"node","lean":"MetaMathlibExt.TM2Recursive.primrec\u2082_readTop","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#primrec-read-top","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $k$ be a stack index of a finite multi-stack machine, write $\\Gamma_k$ for the alphabet of stack $k$, and let $f$ be any function sending an internal state and an optional symbol of $\\Gamma_k$ to an internal state. Then the map sending an internal state $v$ and coded stacks $S$ (a list $S_j$ of natural numbers for each stack index $j$) to $f(v, t)$, where $t$ is the top of stack $k$ as read from $S$, is\u2026","title":"Updating the state from the top of a coded stack is primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/enc-stk-update","kind":"node","lean":"MetaMathlibExt.TM2Recursive.encStk_update","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#enc-stk-update","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, a family $S$ of stacks, a stack index $k$ and a list $l$ of symbols of stack $k$, the coded stacks of the family obtained from $S$ by replacing stack $k$ with $l$ are the coded stacks of $S$ with the entry at $k$ replaced by the list obtained from $l$ by replacing each symbol with its code.","title":"Coded stacks after replacing one stack"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/opt-step-c","kind":"node","lean":"MetaMathlibExt.TM2Recursive.optStepC","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#opt-step-c","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine, the coded step extended to optional coded configurations: it sends the absent configuration to the absent configuration, and a coded configuration $c$ to the coded step of $c$, which is itself absent when $c$ has halted.","title":"The coded step on optional configurations"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/opt-step-c-iterate","kind":"node","lean":"MetaMathlibExt.TM2Recursive.optStepC_iterate","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#opt-step-c-iterate","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $o$ be either no configuration or a configuration of a finite multi-stack machine all of whose stack symbols are relevant. Let running one step send a running configuration to its successor, and a halted configuration or no configuration to no configuration. Then for every $j$, applying the coded step on optional configurations $j$ times to the coding of $o$ gives the coding of the result of running $j$ steps\u2026","title":"The coded run simulates the machine's run"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/relevant-init-list","kind":"node","lean":"MetaMathlibExt.TM2Recursive.relevant_initList","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#relevant-init-list","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a finite multi-stack machine and every list $s$ of symbols of the input alphabet $\\Gamma_{k_0}$, the stacks of the initial configuration with input $s$ (the list $s$ on the input stack $k_0$ and every other stack empty) have only relevant symbols.","title":"The initial stacks have only relevant symbols"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/running-before-halt","kind":"node","lean":"MetaMathlibExt.TM2Recursive.running_before_halt","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#running-before-halt","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Write $\\mathrm{step}$ for the machine's step map, which is undefined on halted configurations (those with no label), and iterate it as a partial map. Let $c_0$ and $c$ be configurations and $s$ a natural number such that $s$ steps from $c_0$ reach $c$. Then for every $j < s$, $j$ steps from $c_0$ reach some configuration $c'$, and $c'$ is running: its label is defined.","title":"Configurations before a reached one are running"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/computable-of-tm2-computable","kind":"node","lean":"MetaMathlibExt.TM2Recursive.computable_of_tm2Computable","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#computable-of-tm2-computable","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ and suppose that some finite multi-stack Turing machine (finitely many stacks, labels and internal states, and a finite input alphabet), together with identifications of its input and output stack alphabets with the booleans, computes $f$ under the binary encoding of natural numbers (least significant bit first, with $0$ encoded as the empty list): for every $n$, the machine\u2026","title":"Turing-computable functions are recursive"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/bool-to","kind":"node","lean":"MetaMathlibExt.TM2Recursive.boolTo\u0393'","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#bool-to","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $\\Gamma' = \\{\\mathtt{cons}_l, \\mathtt{cons}, \\mathtt{bit0}, \\mathtt{bit1}\\}$ be the stack alphabet of Mathlib's multi-stack machine $\\mathrm{tr}$ that evaluates partial recursive codes. This is the map from bits to $\\Gamma'$ that sends $\\mathrm{true}$ to $\\mathtt{bit1}$ and $\\mathrm{false}$ to $\\mathtt{bit0}$.","title":"Bits as stack symbols"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/to-bool","kind":"node","lean":"MetaMathlibExt.TM2Recursive.\u0393'ToBool","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#to-bool","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"The map from the stack alphabet $\\Gamma'$ of Mathlib's code evaluator (symbols $\\mathrm{cons}_l$, $\\mathrm{cons}$, $\\mathrm{bit0}$, $\\mathrm{bit1}$) to the booleans sends $\\mathrm{bit1}$ to true and every other symbol to false.","title":"Reading a stack symbol as a bit"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/tr-nat-eq-map","kind":"node","lean":"MetaMathlibExt.TM2Recursive.trNat_eq_map","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#tr-nat-eq-map","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For every natural number $n$, the list of stack symbols by which Mathlib's code evaluator encodes $n$ is obtained from the binary encoding of $n$ (least significant bit first) by writing each bit as $\\mathrm{bit1}$ if it is true and as $\\mathrm{bit0}$ if it is false.","title":"The evaluator encodes numbers in binary"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/fintype-k","kind":"node","lean":"MetaMathlibExt.TM2Recursive.fintypeK'","parent":34753,"reading_title":{"format":"inline-parts-v1","parts":[["text","The stack set "],["math","\\(K'\\)"],["text"," is finite"]],"source":"The stack set $K'$ is finite"},"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#fintype-k","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"The set $K' = \\{\\mathrm{main}, \\mathrm{rev}, \\mathrm{aux}, \\mathrm{stack}\\}$ of the four stacks of Mathlib's multi-stack machine that evaluates partial recursive codes is a finite type whose elements are exactly these four stacks.","title":"The stack set $K'$ is finite"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/phase","kind":"node","lean":"MetaMathlibExt.TM2Recursive.Phase","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#phase","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"The type with exactly the five elements pre0, pre1, pre2, post1 and post2, with decidable equality.","title":"Adapter labels"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/f","kind":"node","lean":"MetaMathlibExt.TM2Recursive.F\u0393","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#f","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $\\mathrm{tr}$ be Mathlib's multi-stack machine that evaluates partial recursive codes, with stacks $K' = \\{\\mathrm{main}, \\mathrm{rev}, \\mathrm{aux}, \\mathrm{stack}\\}$ and alphabet $\\Gamma' = \\{\\mathtt{cons}_l, \\mathtt{cons}, \\mathtt{bit0}, \\mathtt{bit1}\\}$. The stacks of the wrapped machine are indexed by $K' \\sqcup \\{\\mathrm{false}, \\mathrm{true}\\}$: each stack in $K'$ has alphabet $\\Gamma'$, and the two\u2026","title":"Stack alphabets of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/supp","kind":"node","lean":"MetaMathlibExt.TM2Recursive.supp","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#supp","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a code $c$, the support of $c$ is the finite set of labels of Mathlib's code evaluator $\\mathrm{tr}$ that Mathlib associates with evaluating $c$ under the halting continuation.","title":"Support of a code"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/f-2","kind":"node","lean":"MetaMathlibExt.TM2Recursive.F\u039b","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#f-2","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a partial recursive code and let $\\mathrm{tr}$ be Mathlib's multi-stack machine that evaluates such codes. The labels of the wrapped machine for $c$ form the disjoint union $\\mathrm{Phase} \\sqcup \\mathrm{supp}(c)$, where $\\mathrm{Phase} = \\{\\mathrm{pre}_0, \\mathrm{pre}_1, \\mathrm{pre}_2, \\mathrm{post}_1, \\mathrm{post}_2\\}$ is the set of five adapter labels and $\\mathrm{supp}(c)$ is Mathlib's support of\u2026","title":"Labels of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/fstmt","kind":"node","lean":"MetaMathlibExt.TM2Recursive.FStmt","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#fstmt","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a partial recursive code, and let $\\mathrm{tr}$ be Mathlib's multi-stack machine that evaluates such codes, with stacks $K' = \\{\\mathrm{main}, \\mathrm{rev}, \\mathrm{aux}, \\mathrm{stack}\\}$ and alphabet $\\Gamma' = \\{\\mathtt{cons}_l, \\mathtt{cons}, \\mathtt{bit0}, \\mathtt{bit1}\\}$. A statement of the wrapped machine for $c$ is a statement of Mathlib's multi-stack (TM2) Turing machine model whose stacks are\u2026","title":"Statements of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/restrict","kind":"node","lean":"MetaMathlibExt.TM2Recursive.restrict","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#restrict","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $\\mathrm{tr}$ be Mathlib's four-stack machine that evaluates partial recursive codes, and let $S$ be the support of a code $c$. A statement $q$ of $\\mathrm{tr}$ all of whose jump targets lie in $S$ becomes a statement of the wrapped machine of $c$ with the same structure: each push, peek and pop acts on the same stack of $\\mathrm{tr}$ inside the wrapped machine with the same function, loads and branches are\u2026","title":"Restricting a statement of the code evaluator to the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/start-mem","kind":"node","lean":"MetaMathlibExt.TM2Recursive.start_mem","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#start-mem","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For every code $c$, the label at which Mathlib's code evaluator $\\mathrm{tr}$ begins evaluating $c$ with the halting continuation belongs to the support of $c$.","title":"The start label lies in the support"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/fwd-prog","kind":"node","lean":"MetaMathlibExt.TM2Recursive.fwdProg","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#fwd-prog","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a partial recursive code and let $\\mathrm{tr}$ be Mathlib's multi-stack machine that evaluates such codes, with stacks $\\mathrm{main}, \\mathrm{rev}, \\mathrm{aux}, \\mathrm{stack}$ and alphabet $\\Gamma' = \\{\\mathtt{cons}_l, \\mathtt{cons}, \\mathtt{bit0}, \\mathtt{bit1}\\}$. The program of the wrapped machine for $c$ assigns a statement to each of its labels, namely the five adapter labels\u2026","title":"The program of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/fwd-tm","kind":"node","lean":"MetaMathlibExt.TM2Recursive.fwdTM","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#fwd-tm","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a code for a partial recursive function on lists of natural numbers, and let $\\mathrm{tr}$ be Mathlib's multi-stack machine that evaluates such codes, with stacks $K' = \\{\\mathrm{main}, \\mathrm{rev}, \\mathrm{aux}, \\mathrm{stack}\\}$ and alphabet $\\Gamma' = \\{\\mathtt{cons}_l, \\mathtt{cons}, \\mathtt{bit0}, \\mathtt{bit1}\\}$. The wrapped machine of $c$ is the finite multi-stack machine whose stacks are\u2026","title":"The wrapped machine of a code"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/mk-stk","kind":"node","lean":"MetaMathlibExt.TM2Recursive.mkStk","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#mk-stk","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Given lists $a, b, c, d$ of symbols of $\\Gamma'$ and lists $i, o$ of bits, the stack contents of the wrapped machine with $a$, $b$, $c$ and $d$ on the stacks main, rev, aux and stack, $i$ on the input stack and $o$ on the output stack.","title":"Stack contents of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/fcfg","kind":"node","lean":"MetaMathlibExt.TM2Recursive.FCfg","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#fcfg","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a partial recursive code, and let $\\Gamma' = \\{\\mathtt{cons}_l, \\mathtt{cons}, \\mathtt{bit0}, \\mathtt{bit1}\\}$ be the alphabet of Mathlib's multi-stack machine $\\mathrm{tr}$ that evaluates such codes. A configuration of the wrapped machine for $c$ is a configuration of Mathlib's multi-stack (TM2) Turing machine model with the stack alphabets and labels of the wrapped machine for $c$ and internal states\u2026","title":"Configurations of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/fstep","kind":"node","lean":"MetaMathlibExt.TM2Recursive.fstep","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#fstep","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For a partial recursive code $c$, the step function of the wrapped machine for $c$ is Mathlib's multi-stack (TM2) step function for the program of the wrapped machine for $c$. It sends a configuration without a current label (a halted one) to none, and a configuration with current label $l$ to the configuration obtained by executing the statement that the program assigns to $l$ on its internal state and stacks.","title":"One step of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/frun","kind":"node","lean":"MetaMathlibExt.TM2Recursive.frun","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#frun","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a partial recursive code and let $s$ be the map that sends none to none and a configuration $d$ of the wrapped machine for $c$ to the result of one step of the wrapped machine on $d$ (which is none if $d$ has halted). For $j \\in \\mathbb{N}$ and $x$ either a configuration of the wrapped machine for $c$ or none, the $j$-fold run from $x$ is $s^{j}(x)$, the result of applying $s$ to $x$ $j$ times.","title":"Iterated steps of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/pre1-loop","kind":"node","lean":"MetaMathlibExt.TM2Recursive.pre1_loop","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#pre1-loop","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Write $\\Gamma'$ for the stack alphabet of Mathlib's code evaluator. Let $c$ be a code, $a$, $r$, $x$ and $y$ lists of symbols of $\\Gamma'$, $b$ a list of bits of length $k$ and $v$ an internal state (a symbol of $\\Gamma'$ or none), and write $\\beta(b)$ for $b$ with each bit $\\mathrm{true}$ replaced by the symbol bit1 and each bit $\\mathrm{false}$ by the symbol bit0 of $\\Gamma'$. Started at the adapter label pre1\u2026","title":"The first pre-phase loop of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/pre-phase","kind":"node","lean":"MetaMathlibExt.TM2Recursive.pre_phase","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#pre-phase","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a code and $b$ a list of bits of length $k$, and write $\\beta(b)$ for $b$ with each bit $0$ replaced by the symbol bit0 and each bit $1$ by the symbol bit1 of $\\Gamma'$. Started at the adapter label pre0 with empty local state, $b$ on the input stack and all other stacks empty, the wrapped machine is after exactly $2(k + 1) + 1$ steps at the label $\\mathrm{trNormal}(c, \\mathrm{halt})$ at which Mathlib's\u2026","title":"Pre-phase of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/post1-loop","kind":"node","lean":"MetaMathlibExt.TM2Recursive.post1_loop","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#post1-loop","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Write $\\Gamma'$ for the stack alphabet of Mathlib's code evaluator. Let $c$ be a code, $b$ a list of bits of length $k$, $r$ a list of symbols of $\\Gamma'$, $o$ a list of bits and $v$ an internal state (a symbol of $\\Gamma'$ or none), and write $\\beta(b)$ for $b$ with each bit $\\mathrm{true}$ replaced by the symbol bit1 and each bit $\\mathrm{false}$ by the symbol bit0 of $\\Gamma'$. Started at the adapter label\u2026","title":"The first post-phase loop of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/post-phase","kind":"node","lean":"MetaMathlibExt.TM2Recursive.post_phase","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#post-phase","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a code and $b$ a list of bits of length $k$, and write $\\beta(b)$ for $b$ with each bit $0$ replaced by the symbol bit0 and each bit $1$ by the symbol bit1 of $\\Gamma'$. Started at the adapter label post1 with empty local state, $\\beta(b)$ followed by the symbol cons on the main stack and all other stacks empty, the wrapped machine is after exactly $2(k + 1)$ steps in the halting configuration (no label)\u2026","title":"Post-phase of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/lift-stk","kind":"node","lean":"MetaMathlibExt.TM2Recursive.liftStk","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#lift-stk","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Given contents $s$ of the four stacks main, rev, aux and stack of Mathlib's code-evaluating machine $\\mathrm{tr}$, the stack contents of the wrapped machine that agree with $s$ on these four stacks and have empty input and output stacks.","title":"Lifting stacks of tr"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/label-rel","kind":"node","lean":"MetaMathlibExt.TM2Recursive.LabelRel","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#label-rel","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a code for a partial recursive function on lists of natural numbers and let $\\mathrm{tr}$ be Mathlib's multi-stack machine that evaluates such codes. An optional label $o$ of $\\mathrm{tr}$ (a label, or none for halting) corresponds to an optional label $o'$ of the wrapped machine for $c$ when either $o$ is none and $o'$ is the first post-phase label $\\mathrm{post}_1$, or $o$ is a label $l$ of\u2026","title":"Correspondence of labels between the code evaluator and the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/tr-rel","kind":"node","lean":"MetaMathlibExt.TM2Recursive.TrRel","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#tr-rel","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"A configuration $x$ of Mathlib's code evaluator $\\mathrm{tr}$ is related to a configuration $d$ of the wrapped machine of a code $c$ when $d$ has the same internal state as $x$, the stacks of $d$ are the four stacks of $x$ together with empty input and output stacks, and the labels correspond: if $x$ has halted then $d$ is at the first post-phase label, and if $x$ is at a label $l$ then $l$ lies in the support of\u2026","title":"Simulation relation between the code evaluator and the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/exists-iterate-eq-of-reaches","kind":"node","lean":"MetaMathlibExt.TM2Recursive.exists_iterate_eq_of_reaches","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#exists-iterate-eq-of-reaches","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $\\alpha$ be a type, let $\\bot \\notin \\alpha$, and let $f : \\alpha \\to \\alpha \\cup \\{\\bot\\}$ be a partial step function, where $f(x) = \\bot$ means that no step is possible from $x$. Extend $f$ to $\\bar f : \\alpha \\cup \\{\\bot\\} \\to \\alpha \\cup \\{\\bot\\}$ by $\\bar f(\\bot) = \\bot$. If $b \\in \\alpha$ is reachable from $a \\in \\alpha$ under $f$, that is, $b$ is obtained from $a$ by finitely many (possibly zero) steps\u2026","title":"Reachability gives an iteration count"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/main-phase","kind":"node","lean":"MetaMathlibExt.TM2Recursive.main_phase","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#main-phase","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a code and $f : \\mathbb{N} \\to \\mathbb{N}$ such that $c$ evaluates to $[f(n)]$ on input $[n]$ for every $n$. Then for every $n$ there is $N$ such that the wrapped machine, started at the label $\\mathrm{trNormal}(c, \\mathrm{halt})$ with empty local state, the encoding $\\mathrm{trList}([n])$ on the main stack and all other stacks empty, is after exactly $N$ steps at the adapter label post1 with empty local\u2026","title":"Main phase of the wrapped machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/tr-list-singleton","kind":"node","lean":"MetaMathlibExt.TM2Recursive.trList_singleton","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#tr-list-singleton","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For every natural number $m$, the list of stack symbols by which Mathlib's code evaluator encodes the one-element list $[m]$ is the binary encoding of $m$ (least significant bit first) with each bit written as $\\mathrm{bit1}$ if it is true and as $\\mathrm{bit0}$ if it is false, followed by the symbol $\\mathrm{cons}$.","title":"The evaluator's encoding of a one-element list"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/tm2-computable-of-code","kind":"node","lean":"MetaMathlibExt.TM2Recursive.tm2ComputableOfCode","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#tm2-computable-of-code","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a code and $f : \\mathbb{N} \\to \\mathbb{N}$ a function such that, for every $n$, evaluating $c$ on the one-element list $[n]$ returns $[f(n)]$. Then the wrapped machine of $c$, with input and output alphabet the booleans, computes $f$ under the binary encoding of natural numbers (least significant bit first): for every $n$, started with the encoding of $n$ on its input stack and all other stacks empty, it\u2026","title":"The wrapped machine computes the function a code evaluates"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/tm2-computable-of-computable","kind":"node","lean":"MetaMathlibExt.TM2Recursive.tm2Computable_of_computable","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#tm2-computable-of-computable","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Every computable function $f : \\mathbb{N} \\to \\mathbb{N}$ is computed by a finite multi-stack Turing machine (finitely many stacks, labels and internal states) whose input and output alphabets are identified with the booleans, under the binary encoding of natural numbers (least significant bit first, with $0$ encoded as the empty list): for every $n$, the machine started with the encoding of $n$ on its input stack\u2026","title":"Recursive functions are Turing-computable"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/computable-iff-nonempty-tm2-computable","kind":"node","lean":"MetaMathlibExt.TM2Recursive.computable_iff_nonempty_tm2Computable","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#computable-iff-nonempty-tm2-computable","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"A function $f : \\mathbb{N} \\to \\mathbb{N}$ is computable (recursive) if and only if some bundled finite multi-stack Turing machine computes it in binary. Here the binary word of $m \\in \\mathbb{N}$ is its list of binary digits, least significant first, with $1$ read as true, $0$ as false and no leading zeros (so the word of $0$ is empty). That is, there are such a machine and identifications of its input and output\u2026","title":"Turing computable equals recursive"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/tm2-halts-on","kind":"node","lean":"MetaMathlibExt.TM2Recursive.TM2HaltsOn","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#tm2-halts-on","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"A bundled finite multi-stack Turing machine halts on a word $l$ over its input alphabet when its evaluation from the initial configuration on $l$ (the main label, the initial internal state, $l$ on the input stack and every other stack empty) is defined, that is, its run from that configuration reaches a configuration without successor.","title":"Halting of a multi-stack machine on an input"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/exists-iterate-eq-some-of-eq-none","kind":"node","lean":"MetaMathlibExt.TM2Recursive.exists_iterate_eq_some_of_eq_none","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#exists-iterate-eq-some-of-eq-none","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $f : \\sigma \\to \\mathrm{Option}\\ \\sigma$ be a step function and $a \\in \\sigma$, and write $f^{(k)}(a) \\in \\mathrm{Option}\\ \\sigma$ for the result of $k$ steps from $a$: the $k$-th iterate, applied to $\\mathrm{some}\\ a$, of the map sending $\\mathrm{none}$ to $\\mathrm{none}$ and $\\mathrm{some}\\ x$ to $f(x)$. If $f^{(k)}(a) = \\mathrm{none}$ for some $k$, then there are $j \\in \\mathbb{N}$ and $b \\in \\sigma$ with\u2026","title":"A run that stops passes through a configuration without successor"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/dom-eval-iff-exists-iterate","kind":"node","lean":"MetaMathlibExt.TM2Recursive.dom_eval_iff_exists_iterate","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#dom-eval-iff-exists-iterate","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $f : \\sigma \\to \\mathrm{Option}\\ \\sigma$ be a step function and $a \\in \\sigma$, and write $f^{(k)}(a) \\in \\mathrm{Option}\\ \\sigma$ for the result of $k$ steps from $a$: the $k$-th iterate, applied to $\\mathrm{some}\\ a$, of the map sending $\\mathrm{none}$ to $\\mathrm{none}$ and $\\mathrm{some}\\ x$ to $f(x)$. The evaluation of $f$ from $a$ (the configuration where the run from $a$ stops) is defined if and only if\u2026","title":"A run halts iff it reaches a configuration without successor"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/iterate-eq-none-of-lt","kind":"node","lean":"MetaMathlibExt.TM2Recursive.iterate_eq_none_of_lt","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#iterate-eq-none-of-lt","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $f : \\sigma \\to \\mathrm{Option}\\ \\sigma$ be a step function and $a, b \\in \\sigma$, and write $f^{(k)}(a) \\in \\mathrm{Option}\\ \\sigma$ for the result of $k$ steps from $a$: the $k$-th iterate, applied to $\\mathrm{some}\\ a$, of the map sending $\\mathrm{none}$ to $\\mathrm{none}$ and $\\mathrm{some}\\ x$ to $f(x)$. If $f^{(k)}(a) = \\mathrm{some}\\ b$ and $f(b) = \\mathrm{none}$, then $f^{(m)}(a) = \\mathrm{none}$ for\u2026","title":"A run stays stopped after a configuration without successor"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/re-pred-tm2-halts-on","kind":"node","lean":"MetaMathlibExt.TM2Recursive.rePred_tm2HaltsOn","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#re-pred-tm2-halts-on","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For every bundled finite multi-stack Turing machine and every identification of its input alphabet with the booleans, the predicate on $\\mathbb{N}$ that holds at $n$ exactly when the machine halts on the binary digits of $n$ (least significant first, with $1$ read as true, $0$ as false and no leading zeros, so that $0$ gives the empty word) is recursively enumerable.","title":"The halting set of a multi-stack machine is r.e"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/tm2-halts-on-fwd-tm-iff","kind":"node","lean":"MetaMathlibExt.TM2Recursive.tm2HaltsOn_fwdTM_iff","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#tm2-halts-on-fwd-tm-iff","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"Let $c$ be a code for a partial recursive function on lists of natural numbers and $n \\in \\mathbb{N}$, and suppose that every value of $c$ on the one-element list $[n]$ is itself a one-element list. Then the wrapped multi-stack machine of $c$ halts on the binary digits of $n$ (least significant first, with $1$ read as true, $0$ as false and no leading zeros, so that $0$ gives the empty word) if and only if the\u2026","title":"The wrapped machine of a code halts exactly where the code is defined"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/exists-finite-tm2-halts-on-iff-of-re-pre","kind":"node","lean":"MetaMathlibExt.TM2Recursive.exists_finite_tm2HaltsOn_iff_of_rePred","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#exists-finite-tm2-halts-on-iff-of-re-pre","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For every recursively enumerable predicate $p$ on $\\mathbb{N}$ there are a bundled finite multi-stack Turing machine whose stack alphabets are all finite and an identification of its input alphabet with the booleans such that, for every $n \\in \\mathbb{N}$, $p(n)$ holds if and only if the machine halts on the binary digits of $n$ (least significant first, with $1$ read as true, $0$ as false and no leading zeros, so\u2026","title":"Every r.e. predicate is the halting set of a machine with finite alphabets"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/exists-tm2-halts-on-iff-of-re-pred","kind":"node","lean":"MetaMathlibExt.TM2Recursive.exists_tm2HaltsOn_iff_of_rePred","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#exists-tm2-halts-on-iff-of-re-pred","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"For every recursively enumerable predicate $p$ on $\\mathbb{N}$ there are a bundled finite multi-stack Turing machine and an identification of its input alphabet with the booleans such that, for every $n \\in \\mathbb{N}$, $p(n)$ holds if and only if the machine halts on the binary digits of $n$ (least significant first, with $1$ read as true, $0$ as false and no leading zeros, so that $0$ gives the empty word).","title":"Every r.e. predicate is the halting set of a multi-stack machine"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/re-pred-iff-exists-tm2-halts-on","kind":"node","lean":"MetaMathlibExt.TM2Recursive.rePred_iff_exists_tm2HaltsOn","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#re-pred-iff-exists-tm2-halts-on","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"fully_proved","summary":"A predicate $p$ on $\\mathbb{N}$ is recursively enumerable if and only if there are a bundled finite multi-stack Turing machine and an identification of its input alphabet with the booleans such that, for every $n \\in \\mathbb{N}$, $p(n)$ holds if and only if the machine halts on the binary digits of $n$ (least significant first, with $1$ read as true, $0$ as false and no leading zeros, so that $0$ gives the empty\u2026","title":"R.e. predicates are the halting sets of multi-stack machines"},{"id":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/turing-recursive-equiv","kind":"node","lean":"MathlibExt.TheoryOfComputation.TuringRecursiveWanted.ModelTheory.TuringRecursive.turing_recursive_equiv","parent":34753,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun/index.html#turing-recursive-equiv","scope":"TheoryOfComputation/equivalence-of-models/equivalence-of-turing-machines-and-recursive-fun","status":"can_prove","summary":"A function $f : \\mathbb{N} \\to \\mathbb{N}$ is computable, that is, total recursive, if and only if some finite multi-stack Turing machine computes it on binary representations: there are a machine with finitely many stacks, internal states and program labels and bijections from its input and output stack alphabets to $\\{0, 1\\}$ such that, for every $n \\in \\mathbb{N}$, the machine started in its initial state with\u2026","title":"Recursive functions are the Turing-computable functions"},{"data_url":"scopes/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic.json","id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","kind":"scope","navigation":"module","parent":34740,"result":false,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","title":"Multi-tape machines decide only recursive predicates"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/multi-tape-decidable","kind":"node","lean":"Cslib.Decidability.MultiTapeDecidable","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#multi-tape-decidable","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is decidable by a multi-tape machine when there are a number $k$ and a deterministic Turing machine $M$ with finitely many states, a read-only input tape, $k$ work tapes and a write-only output tape, all over the Boolean alphabet, such that for every\u2026","title":"Decidability by multi-tape machines"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/multi-tape-recognizable","kind":"node","lean":"Cslib.Decidability.MultiTapeRecognizable","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#multi-tape-recognizable","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is recognizable by a multi-tape machine when there are a number $k$ and a deterministic Turing machine $M$ with finitely many states, a read-only input tape, $k$ work tapes and a write-only output tape, all over the Boolean alphabet, such that for every\u2026","title":"Recognizability by multi-tape machines"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/multi-tape-co-recognizable","kind":"node","lean":"Cslib.Decidability.MultiTapeCoRecognizable","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#multi-tape-co-recognizable","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is co-recognizable by a multi-tape machine when its complement $\\neg p$ is recognizable by a multi-tape machine.","title":"Co-recognizability by multi-tape machines"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/re-pred","kind":"node","lean":"Cslib.Decidability.MultiTapeRecognizable.rePred","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#re-pred","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is recognizable by a multi-tape machine, then $p$ is recursively enumerable, that is, the partial function on $\\alpha$ that is defined, with a trivial value, exactly where $p$ holds is partial recursive.","title":"A predicate recognizable by a multi-tape machine is recursively enumerable"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/re-pred-2","kind":"node","lean":"Cslib.Decidability.MultiTapeDecidable.rePred","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#re-pred-2","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a multi-tape machine, then $p$ is recursively enumerable, that is, the partial function on $\\alpha$ that is defined, with a trivial value, exactly where $p$ holds is partial recursive.","title":"A predicate decided by a multi-tape machine is recursively enumerable"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/re-pred-not","kind":"node","lean":"Cslib.Decidability.MultiTapeDecidable.rePred_not","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#re-pred-not","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a multi-tape machine, then its complement $\\neg p$ is recursively enumerable, that is, the partial function on $\\alpha$ that is defined, with a trivial value, exactly where $\\neg p$ holds is partial recursive.","title":"A predicate decided by a multi-tape machine has a recursively enumerable complement"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/computable-pred","kind":"node","lean":"Cslib.Decidability.MultiTapeDecidable.computablePred","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#computable-pred","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a multi-tape machine, then $p$ is computable, that is, $p$ is decidable and its Boolean characteristic function $\\alpha \\to \\mathrm{Bool}$ is computable.","title":"A predicate decided by a multi-tape machine is computable"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/multi-tape-decidable-implies-tm2-decidab","kind":"node","lean":"Cslib.Decidability.multiTapeDecidable_implies_tm2Decidable","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#multi-tape-decidable-implies-tm2-decidab","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). Every predicate on $\\alpha$ that is decidable by a multi-tape machine is decidable by a multi-stack machine.","title":"Multi-tape machines decide no more than multi-stack machines"},{"id":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/multi-tape-recognizable-implies-tm2-reco","kind":"node","lean":"Cslib.Decidability.multiTapeRecognizable_implies_tm2Recognizable","parent":34822,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic/index.html#multi-tape-recognizable-implies-tm2-reco","scope":"TheoryOfComputation/equivalence-of-models/multi-tape-machines-decide-only-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). Every predicate on $\\alpha$ that is recognizable by a multi-tape machine is recognizable by a multi-stack machine.","title":"Multi-tape machines recognize no more than multi-stack machines"},{"data_url":"scopes/TheoryOfComputation/equivalence-of-models/random-access-machines.json","id":"TheoryOfComputation/equivalence-of-models/random-access-machines","kind":"scope","navigation":"module","parent":34740,"result":false,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","title":"Random-access machines"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ramaddr","kind":"node","lean":"Cslib.Decidability.RAMAddr","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ramaddr","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"An addressing mode of a random-access machine specifies an operand by a natural number $n$ in one of three ways: immediate (the operand is the value $n$ itself), direct (the operand is the contents of register $n$), or indirect (the operand is the contents of the register whose index is the contents of register $n$).","title":"RAM addressing mode"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ramdest","kind":"node","lean":"Cslib.Decidability.RAMDest","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ramdest","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"A store destination of a random-access machine names, by a natural number $n$, the register into which the accumulator is stored: either register $n$ itself (direct) or the register whose index is the contents of register $n$ (indirect).","title":"RAM store destination"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/raminstr","kind":"node","lean":"Cslib.Decidability.RAMInstr","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#raminstr","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"An instruction of a random-access machine is one of the following: load an operand into the accumulator; store the accumulator into a store destination; add an operand to the accumulator; subtract an operand from the accumulator, truncating at $0$; jump to instruction $q$; jump to instruction $q$ if the accumulator is $0$; halt. Here an operand is given by a RAM addressing mode, a store destination is a RAM store\u2026","title":"RAM instruction"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ramstate","kind":"node","lean":"Cslib.Decidability.RAMState","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ramstate","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"A state of a random-access machine consists of a program counter in $\\mathbb{N}$, an accumulator value in $\\mathbb{N}$, and register contents, a function $\\mathbb{N} \\to \\mathbb{N}$ giving the contents of each register.","title":"RAM state"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-resolve","kind":"node","lean":"Cslib.Decidability.ramResolve","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-resolve","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"In a RAM state with register contents $r$, an operand resolves to a natural number: the immediate operand $n$ resolves to $n$ itself, the direct operand $n$ to the contents $r(n)$ of register $n$, and the indirect operand $n$ to $r(r(n))$.","title":"Value of a RAM operand"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-dest-index","kind":"node","lean":"Cslib.Decidability.ramDestIndex","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-dest-index","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"In a RAM state with register contents $r$, a store destination resolves to a register index: the direct destination $n$ resolves to $n$, and the indirect destination $n$ resolves to the contents $r(n)$ of register $n$.","title":"Register written by a RAM store"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-step","kind":"node","lean":"Cslib.Decidability.ramStep","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-step","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"Let $P = (P_0, \\dots, P_{\\ell-1})$ be a program of length $\\ell$ and $s$ a state with program counter $p$, accumulator $a$ and register contents $r$. If $p \\ge \\ell$, the step leaves $s$ unchanged. Otherwise it executes $P_p$, where $v$ denotes the value of the instruction's operand in $s$: a load sets the accumulator to $v$, an add sets it to $a + v$, a subtract sets it to $\\max(a - v, 0)$, and a store writes $a$\u2026","title":"One step of a RAM machine"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-init","kind":"node","lean":"Cslib.Decidability.ramInit","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-init","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"The initial state on inputs $x_0, \\dots, x_{k-1}$ has program counter $0$ and accumulator $0$; register $i$ holds $x_i$ for $i < k$, and every other register holds $0$.","title":"Initial RAM state"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-exec-n","kind":"node","lean":"Cslib.Decidability.ramExecN","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-exec-n","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"For a program $P$, a natural number $t$ and a state $s$, the state after $t$ steps of $P$ from $s$ is obtained by applying one step of $P$ to $s$ exactly $t$ times.","title":"Bounded execution of a RAM machine"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-eval","kind":"node","lean":"Cslib.Decidability.ramEval","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-eval","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"Let $P$ be a program of length $\\ell$ and $x_0, \\dots, x_{k-1}$ inputs. Run $P$ from the initial state on these inputs. If for some $t$ the program counter after $t$ steps is at least $\\ell$, the output of $P$ on these inputs is the accumulator after the least such number of steps; otherwise the output is undefined.","title":"Output of a RAM machine"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-step-sticky","kind":"node","lean":"Cslib.Decidability.ramStep_sticky","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-step-sticky","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"If the program counter of a state $s$ is at least the length of a program $P$, then one step of $P$ leaves $s$ unchanged.","title":"A RAM step is sticky at halting states"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-exec-n-stable","kind":"node","lean":"Cslib.Decidability.ramExecN_stable","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-exec-n-stable","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"Let $P$ be a program of length $\\ell$, $s$ a state and $t$ a natural number. If the program counter after $t$ steps of $P$ from $s$ is at least $\\ell$, then the state after $t + 1$ steps of $P$ from $s$ equals the state after $t$ steps.","title":"RAM execution is stable once halted"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ram-exec-n-eq-of-halted","kind":"node","lean":"Cslib.Decidability.ramExecN_eq_of_halted","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ram-exec-n-eq-of-halted","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"Let $P$ be a program of length $\\ell$, $s$ a state and $t_1 \\le t_2$ natural numbers. If the program counter after $t_1$ steps of $P$ from $s$ is at least $\\ell$, then the state after $t_2$ steps of $P$ from $s$ equals the state after $t_1$ steps.","title":"RAM execution stays at its halted state"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/mem-ram-eval-of-halted","kind":"node","lean":"Cslib.Decidability.mem_ramEval_of_halted","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#mem-ram-eval-of-halted","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"Let $P$ be a program of length $\\ell$, let $x_0, \\dots, x_{k-1}$ be inputs and let $t$ and $y$ be natural numbers. If, after $t$ steps of $P$ from the initial state on these inputs, the program counter is at least $\\ell$ and the accumulator equals $y$, then the output of $P$ on these inputs is defined and equals $y$.","title":"Output membership for RAM machines"},{"id":"TheoryOfComputation/equivalence-of-models/random-access-machines/ramdecidable","kind":"node","lean":"Cslib.Decidability.RAMDecidable","parent":34832,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/random-access-machines/index.html#ramdecidable","scope":"TheoryOfComputation/equivalence-of-models/random-access-machines","status":"fully_proved","summary":"A predicate $P$ on the natural numbers is RAM decidable when there is a RAM program, a finite list of RAM instructions, such that for every natural number $n$ the program, started with $n$ in register $0$ and with the accumulator, the program counter and every other register at $0$, halts (its program counter reaches or passes the end of the program) with $1$ in the accumulator if $P(n)$ holds and with $0$ in the\u2026","title":"RAM-decidable predicate"},{"data_url":"scopes/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic.json","id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","kind":"scope","navigation":"module","parent":34740,"result":false,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","title":"Single-tape machines decide the recursive predicates"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/cons-computer","kind":"node","lean":"Cslib.Turing.SingleTapeTM.consComputer","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#cons-computer","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For a symbol $c$, this is the single-tape machine with two states that, from its start state, writes back the symbol under the head, moves left and enters its second state, and from its second state writes $c$ and halts without moving.","title":"The machine that prepends a symbol"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/write-move-left-mk","kind":"node","lean":"Cslib.Turing.SingleTapeTM.write_moveLeft_mk\u2081","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#write-move-left-mk","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $c$ be a symbol and $l$ a list of symbols, and consider the tape holding $l$ from the head rightwards, all other cells blank. Writing back the symbol under the head, moving the head one cell left, and then writing $c$ gives the tape holding the list $c$ followed by $l$ from the head rightwards, all other cells blank.","title":"Prepending a symbol by stepping left"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/cons-computer-outputs-within-time","kind":"node","lean":"Cslib.Turing.SingleTapeTM.consComputer_outputsWithinTime","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#cons-computer-outputs-within-time","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For every symbol $c$ and every word $l$, the machine that prepends $c$, started on $l$, halts with output the word $c$ followed by $l$ within at most two steps.","title":"Prepending a symbol takes at most two steps"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/cons-computer-outputs","kind":"node","lean":"Cslib.Turing.SingleTapeTM.consComputer_outputs","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#cons-computer-outputs","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For every symbol $c$ and every word $l$, the machine that prepends $c$, started on $l$, halts with output the word $c$ followed by $l$.","title":"Correctness of the prepending machine"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/input-code","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.inputCode","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#input-code","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For a list of bits $w$, let $b_0, \\dots, b_n$ be the reversal of $w$ followed by $\\mathsf{true}$, reading $\\mathsf{true}$ as $1$ and $\\mathsf{false}$ as $0$. The input code of $w$ is the natural number $\\sum_{i=0}^{n} b_i 2^i$. Its binary expansion, least significant bit first, is $b_0, \\dots, b_n$: the input stack on which the single-tape emulation starts for the input word $w$.","title":"Input code of a bit word"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/reverse-tail-encode-nat-input-code","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.reverse_tail_encodeNat_inputCode","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#reverse-tail-encode-nat-input-code","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For every list of bits $w$, reversing the binary encoding (least significant bit first) of the input code of $w$ and then dropping the first bit of the result gives back $w$.","title":"Recovering a word from its input code"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/input-decode","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.inputDecode","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#input-decode","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For a natural number $m > 0$, let $b_0, \\dots, b_n$ be its binary expansion, least significant bit first, without trailing zeros. The input decoding of $m$ is $\\sum_{i=0}^{n-1} b_{n-1-i} 2^i$, the value, least significant bit first, of the reversed expansion with its first bit dropped. The input decoding of $0$, whose expansion is empty, is $0$.","title":"Input decoding"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/in-cell","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.inCell","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#in-cell","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a multi-stack machine with finitely many stacks, labels and internal states, and let $e$ be a bijection from the alphabet of its input stack to the bits. The input cell of a bit $b$ is the non-blank cell of the single-tape emulation of $M$ whose bottom-of-stack flag is off, whose entry for the input stack is $e^{-1}(b)$, and whose entries for all other stacks are empty.","title":"Input cell of a bit"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/bot-cell","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.botCell","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#bot-cell","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine whose input stack alphabet is identified with the bits. The bottom cell of the emulation's start tape is the cell flagged as the stack bottom that holds the symbol for $\\mathrm{true}$ on the input stack and nothing on every other stack.","title":"The bottom cell of the start tape"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/emu-support","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.emuSupport","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#emu-support","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a multi-stack machine with finitely many stacks, labels and internal states and a finite input alphabet. Its plain single-tape emulation is the translation of $M$ to a single-tape machine with a finite store, followed by the translation of that machine to a plain single-tape machine, whose states are pairs of a statement of the intermediate machine, or the halting state, and an internal state. This is\u2026","title":"States of the single-tape emulation"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/emu-tm","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.emuTM","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#emu-tm","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a multi-stack machine with finitely many stacks, labels and internal states, and with finite stack alphabets. Its finite single-tape emulation is the single-tape machine that runs the plain single-tape emulation of $M$ (through a single-tape machine with a finite store) step for step: its states are those in the finite set of states the emulation visits, and its tape alphabet is the set of non-blank\u2026","title":"Finite single-tape emulation of a multi-stack machine"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/recognizer-tm","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.recognizerTM","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#recognizer-tm","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine all of whose stack alphabets are finite, and let $e$ be an identification of the alphabet of its input stack with the booleans. The recognizer of $M$ is the single-tape machine, over the non-blank cells of the multi-stack-to-single-tape translation of $M$, that first runs the machine prepending the bottom cell determined by $M$ and $e$ (the bottom-flagged cell holding\u2026","title":"The recognizer of a multi-stack machine"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/map-val-bot-cell-cons","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.map_val_botCell_cons","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#map-val-bot-cell-cons","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a multi-stack machine with finitely many stacks, labels and internal states, and let $e$ be a bijection from the alphabet of its input stack to the bits. For every list of bits $w$, the bottom cell followed by the input cells of the bits of $w$, read as cells of the emulation, is the start tape that the single-tape emulation assigns to the input stack obtained by applying $e^{-1}$ to each entry of the\u2026","title":"The bottom cell followed by the input word is the emulation's start tape"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/halts-on-recognizer-tm-iff","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.haltsOn_recognizerTM_iff","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#halts-on-recognizer-tm-iff","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a multi-stack machine with finitely many stacks, labels and internal states and with finite stack alphabets, and let $e$ be a bijection from the alphabet of its input stack to the bits. For every list of bits $w$, the recognizer built from $M$ and $e$ halts on the word of input cells of the bits of $w$ if and only if $M$ halts when started on the input stack obtained by applying $e^{-1}$ to each entry of\u2026","title":"The recognizer halts iff the multi-stack machine does"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/exists-mem-eval-tr","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.exists_mem_eval_tr","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#exists-mem-eval-tr","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a multi-stack machine with finitely many stacks, labels and internal states and a finite input alphabet, let $L$ be a list of symbols of its input stack, and suppose that $M$, started with $L$ on its input stack and every other stack empty, halts at the configuration $c$. Then its plain single-tape emulation, started on the tape that holds the symbols of $L$ bottom first on the input stack with the first\u2026","title":"The emulation's final tape"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/add-bottom-eq-of-length-le-one","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.addBottom_eq_of_length_le_one","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#add-bottom-eq-of-length-le-one","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine, and let $S$ assign to every stack $k$ of $M$ a list $S_k$ of length at most $1$. Let $L'$ be a tape of stack columns whose $k$-th track, for every $k$, is $S_k$ laid out bottom first and followed by blanks. Then adding the bottom marker to $L'$, as in the translation of $M$ to a single-tape machine, gives the tape whose first cell is flagged as the bottom and holds the top\u2026","title":"Stacks of height at most one give a one-cell tape"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/cell-answer","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.cellAnswer","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#cell-answer","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine whose output stack alphabet is identified with the bits. The answer read from a non-blank cell of the single-tape emulation of $M$ is $\\mathrm{true}$ exactly when the cell holds the symbol for $\\mathrm{true}$ on the output stack.","title":"The answer bit of a cell"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/answer-tm","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.answerTM","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#answer-tm","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine whose stack alphabets are all finite, with its input and output stack alphabets identified with the bits. The answer machine of $M$ is the single-tape machine over the non-blank cells of the single-tape emulation of $M$ with a single state that, in one step, replaces the symbol under the head by the input-bit cell for the answer read from that cell (the answer\u2026","title":"The answer machine"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/answer-tm-outputs","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.answerTM_outputs","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#answer-tm-outputs","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine whose stack alphabets are all finite, with its input and output stack alphabets identified with the bits. For every non-blank cell $x$ of the single-tape emulation of $M$, the answer machine of $M$, started on the one-cell input $x$, halts with the one-cell output holding the input-bit cell for the answer read from $x$.","title":"The answer machine outputs the answer of a one-cell input"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/decider-tm","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.deciderTM","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#decider-tm","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine whose stack alphabets are all finite, with its input and output stack alphabets identified with the bits. The decider of $M$ is the single-tape machine over the non-blank cells of the single-tape emulation of $M$ that runs the recognizer of $M$ and, once the recognizer halts, runs the answer machine of $M$ on the resulting tape.","title":"The decider of a multi-stack machine"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/decider-tm-outputs","kind":"node","lean":"Cslib.Turing.SingleTapeTM.OfTM2.deciderTM_outputs","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#decider-tm-outputs","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a finite multi-stack machine whose stack alphabets are all finite, with its input and output stack alphabets identified with the bits, and let $w$ be a word of bits. Suppose that $M$, started with the input stack holding the reverse of $w$ followed by $\\mathrm{true}$, halts with the output stack holding a list $u$ of length at most $1$, every other stack empty and its variable back at its initial value.\u2026","title":"The decider outputs the answer bit"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/of-re-pred","kind":"node","lean":"Cslib.Decidability.SingleTapeRecognizable.of_rePred","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#of-re-pred","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"R.e. predicates are recognizable by single-tape machines.","title":"R.e. predicates are recognizable by single-tape machines"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/of-computable-pred","kind":"node","lean":"Cslib.Decidability.SingleTapeDecidable.of_computablePred","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#of-computable-pred","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$, and let $p$ be a computable predicate on $\\alpha$, that is, one whose indicator function $\\alpha \\to \\{0, 1\\}$ is computable. Then $p$ is decidable by a single-tape Turing machine.","title":"Computable predicates are decidable by single-tape machines"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/coded-cons","kind":"node","lean":"Cslib.Turing.SingleTapeTM.codedCons","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#coded-cons","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For any type $\\Sigma$, an element $x$ of $\\Sigma \\cup \\{\\bot\\}$ (with $\\bot$ the blank) and a finite list $l$ over $\\Sigma \\cup \\{\\bot\\}$, the push of $x$ onto $l$ is the empty list when $x = \\bot$ and $l$ is empty, and otherwise the list obtained by prepending $x$ to $l$.","title":"Pushing a cell onto a coded tape"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/head-eq-head-i","kind":"node","lean":"Cslib.Turing.StackTape.head_eq_headI","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#head-eq-head-i","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For every one-sided tape $s$ over an alphabet $\\Sigma$, stored as a finite list of cells in $\\Sigma \\cup \\{\\bot\\}$ (with $\\bot$ the blank) that does not end in $\\bot$, the symbol at the head of $s$ is the first entry of that list, and the blank $\\bot$ when the list is empty.","title":"The head of a stack tape is the first entry of its list"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/to-list-tail","kind":"node","lean":"Cslib.Turing.StackTape.toList_tail","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#to-list-tail","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $s$ be a stack tape, a one-sided tape stored as its list of cells without trailing blanks. Then the cell list of the tail of $s$ is the tail of the cell list of $s$, the tail of the empty list being empty.","title":"Tail of a stack tape, on cell lists"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/to-list-cons","kind":"node","lean":"Cslib.Turing.StackTape.toList_cons","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#to-list-cons","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $s$ be a stack tape, a one-sided tape stored as its list of cells without trailing blanks, and $x$ a cell content, either a symbol or blank. Then the cell list of the tape obtained by pushing $x$ onto $s$ is the coded push of $x$ applied to the cell list of $s$: $x$ prepended to it, except that pushing a blank onto the empty tape gives the empty list.","title":"Pushing onto a stack tape, on cell lists"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/primrec-coded-cons","kind":"node","lean":"Cslib.Turing.SingleTapeTM.primrec_codedCons","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#primrec-coded-cons","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\Sigma$ be a type with a primitive recursive coding, and let $x$ be a cell content, either a symbol of $\\Sigma$ or blank. Then the coded push of $x$, the map on lists of cell contents that prepends $x$ to a list, except that it sends the empty list to the empty list when $x$ is blank, is primitive recursive.","title":"Pushing a cell onto a coded tape is primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/primrec-map-encode-nat","kind":"node","lean":"Cslib.Turing.SingleTapeTM.primrec_map_encodeNat","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#primrec-map-encode-nat","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\Sigma$ be a type with a primitive recursive coding and $\\iota$ any map from the Booleans to $\\Sigma$. Then the map sending $n \\in \\mathbb{N}$ to the word over $\\Sigma$ obtained by applying $\\iota$ letterwise to the binary code of $n$ (least significant bit first, the empty word for $0$) is primitive recursive.","title":"Binary codes of natural numbers in a tape alphabet are primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/halts-on-iff-dom-eval","kind":"node","lean":"Cslib.Turing.SingleTapeTM.haltsOn_iff_dom_eval","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#halts-on-iff-dom-eval","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"A single-tape machine $M$ halts on an input word $l$ if and only if iterating the step function of $M$ from the initial configuration on $l$, until a configuration with no successor is reached, terminates.","title":"Halting is termination of the run"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/outputs-iff-mem-eval","kind":"node","lean":"Cslib.Turing.SingleTapeTM.outputs_iff_mem_eval","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#outputs-iff-mem-eval","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"A single-tape machine $M$ outputs the word $l'$ on the input word $l$ if and only if iterating the step function of $M$ from the initial configuration on $l$, until a configuration with no successor is reached, terminates at the halting configuration for $l'$: the halting state, with $l'$ written from the head rightwards and every other cell blank.","title":"Outputs are the ends of runs"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/coded-cfg","kind":"node","lean":"Cslib.Turing.SingleTapeTM.CodedCfg","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#coded-cfg","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For a single-tape Turing machine with state set $Q$ and tape alphabet $\\Sigma$, write $\\bot$ for an absent value. A coded configuration is a quadruple $(q, h, L, R)$ where $q \\in Q \\cup \\{\\bot\\}$ is the state ($\\bot$ meaning halted), $h \\in \\Sigma \\cup \\{\\bot\\}$ is the symbol under the head ($\\bot$ being the blank), and $L$ and $R$ are finite lists over $\\Sigma \\cup \\{\\bot\\}$, meant as the cells to the left and to\u2026","title":"Coded configurations of a single-tape machine"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/to-coded-cfg","kind":"node","lean":"Cslib.Turing.SingleTapeTM.Cfg.toCodedCfg","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#to-coded-cfg","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"The code of a configuration of a single-tape machine is the quadruple consisting of its state (or the halting state), the symbol under the head (or blank), and the lists of cells, without trailing blanks, of the tapes to the left and to the right of the head.","title":"Code of a single-tape configuration"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/coded-step","kind":"node","lean":"Cslib.Turing.SingleTapeTM.codedStep","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#coded-step","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with transition function $\\delta$, and write $\\bot$ both for the halting state and for the blank. For a finite list $l$ of cells, let $\\mathrm{hd}(l)$ be the first entry of $l$, the blank $\\bot$ when $l$ is empty; let $\\mathrm{tl}(l)$ be $l$ without its first entry, the empty list when $l$ is empty; and for a cell $w$ let $w \\cdot l$, the push of $w$ onto $l$, be $l$ when\u2026","title":"One step of a single-tape machine on coded configurations"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/coded-step-to-coded-cfg","kind":"node","lean":"Cslib.Turing.SingleTapeTM.codedStep_toCodedCfg","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#coded-step-to-coded-cfg","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For every configuration $c$ of a single-tape machine $M$, the code of $c$ has no successor under the coded step exactly when $c$ has no successor under the step function of $M$, and otherwise the coded step sends the code of $c$ to the code of the successor of $c$.","title":"The coded step commutes with coding"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/primrec-coded-step","kind":"node","lean":"Cslib.Turing.SingleTapeTM.primrec_codedStep","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#primrec-coded-step","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine over a finite alphabet $\\Sigma$, and fix primitive recursive codings of $\\Sigma$ and of the states of $M$. Then the coded step of $M$, which maps a coded configuration to the coded configuration after one step of $M$ and has no value when the state is the halting state, is primitive recursive.","title":"The coded step of a single-tape machine is primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/eval-coded-step","kind":"node","lean":"Cslib.Turing.SingleTapeTM.eval_codedStep","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#eval-coded-step","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For every configuration $c$ of a single-tape machine $M$, iterating the coded step from the code of $c$ until a coded configuration with no successor is reached terminates exactly when iterating the step function of $M$ from $c$ until a configuration with no successor is reached does, and then it ends at the code of the configuration where the run of $M$ from $c$ ends.","title":"Running the coded step mirrors running the machine"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/primrec-to-coded-cfg-init-cfg","kind":"node","lean":"Cslib.Turing.SingleTapeTM.primrec_toCodedCfg_initCfg","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#primrec-to-coded-cfg-init-cfg","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine over a finite alphabet $\\Sigma$, and fix primitive recursive codings of $\\Sigma$ and of the states of $M$. Then the map sending an input word $w$ over $\\Sigma$ to the code of the initial configuration of $M$ on $w$ is primitive recursive.","title":"The code of the initial configuration is primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/partrec-eval","kind":"node","lean":"Cslib.Turing.SingleTapeTM.partrec_eval","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#partrec-eval","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine whose finite alphabet $\\Sigma$ carries a primitive recursive coding. The partial function sending an input word $l$ over $\\Sigma$ to the final tape of the run of $M$ on $l$ (the symbol under the head, and the lists of cells to the left and to the right of the head), defined exactly when $M$ halts on $l$, is partial recursive.","title":"The final tape of a run is partial recursive in the input word"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/re-pred-halts-on","kind":"node","lean":"Cslib.Turing.SingleTapeTM.rePred_haltsOn","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#re-pred-halts-on","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine over a finite alphabet $\\Sigma$, and fix a primitive recursive coding of $\\Sigma$. Then the set of input words over $\\Sigma$ on which $M$ halts is recursively enumerable.","title":"The halting set of a single-tape machine is r.e"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/re-pred-halts-on-encode-nat","kind":"node","lean":"Cslib.Turing.SingleTapeTM.rePred_haltsOn_encodeNat","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#re-pred-halts-on-encode-nat","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine over a finite alphabet $\\Sigma$ and $\\iota$ any map from the Booleans to $\\Sigma$. Then the set of natural numbers $n$ such that $M$ halts on the word obtained by applying $\\iota$ letterwise to the binary code of $n$ (least significant bit first, the empty word for $0$) is recursively enumerable.","title":"The halting set of a single-tape machine on binary inputs is r.e"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/re-pred","kind":"node","lean":"Cslib.Decidability.SingleTapeRecognizable.rePred","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#re-pred","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"A predicate recognized by a single-tape machine is r.e.","title":"A predicate recognized by a single-tape machine is r.e"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/computable-pred","kind":"node","lean":"Cslib.Decidability.SingleTapeDecidable.computablePred","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#computable-pred","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"The decider's answer is the symbol under the head of its final tape.","title":"A predicate decided by a single-tape machine is computable"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-decidable-iff-computable-pre","kind":"node","lean":"Cslib.Decidability.singleTapeDecidable_iff_computablePred","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-decidable-iff-computable-pre","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is decidable by a single-tape machine if and only if it is computable, that is, $p$ is decidable and its Boolean characteristic function $\\alpha \\to \\mathrm{Bool}$ is computable.","title":"Single-tape decidable predicates are exactly the computable predicates"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-recognizable-iff-re-pred","kind":"node","lean":"Cslib.Decidability.singleTapeRecognizable_iff_rePred","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-recognizable-iff-re-pred","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is recognizable by a single-tape machine if and only if it is recursively enumerable, that is, the partial function on $\\alpha$ that is defined, with a trivial value, exactly where $p$ holds is partial recursive.","title":"Single-tape recognizable predicates are exactly the r.e. predicates"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-co-recognizable-iff-re-pred","kind":"node","lean":"Cslib.Decidability.singleTapeCoRecognizable_iff_rePred_not","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-co-recognizable-iff-re-pred","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is co-recognizable by a single-tape machine if and only if its complement $\\neg p$ is recursively enumerable, that is, the partial function on $\\alpha$ that is defined, with a trivial value, exactly where $\\neg p$ holds is partial recursive.","title":"Single-tape co-recognizable predicates are those with an r.e. complement"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-decidable-iff-tm2-decidable","kind":"node","lean":"Cslib.Decidability.singleTapeDecidable_iff_tm2Decidable","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-decidable-iff-tm2-decidable","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Single-tape and multi-stack machines decide the same predicates.","title":"Single-tape and multi-stack machines decide the same predicates"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-recognizable-iff-tm2-recogni","kind":"node","lean":"Cslib.Decidability.singleTapeRecognizable_iff_tm2Recognizable","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-recognizable-iff-tm2-recogni","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is recognizable by a single-tape machine if and only if it is recognizable by a multi-stack machine.","title":"Single-tape and multi-stack machines recognize the same predicates"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-decidable-iff-single-tape-re","kind":"node","lean":"Cslib.Decidability.singleTapeDecidable_iff_singleTapeRecognizable_and_singleTapeCoRecognizable","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-decidable-iff-single-tape-re","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Post's theorem for single-tape machines: a predicate is decidable iff it is recognizable and co-recognizable.","title":"Post's theorem"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-recognizable","kind":"node","lean":"Cslib.Decidability.SingleTapeDecidable.singleTapeRecognizable","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-recognizable","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a single-tape machine, then $p$ is recognizable by a single-tape machine.","title":"Decidable predicates are recognizable by single-tape machines"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-co-recognizable","kind":"node","lean":"Cslib.Decidability.SingleTapeDecidable.singleTapeCoRecognizable","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-co-recognizable","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a single-tape machine, then $p$ is co-recognizable by a single-tape machine, that is, its complement $\\neg p$ is recognizable by a single-tape machine.","title":"Single-tape decidable predicates are co-recognizable"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/not","kind":"node","lean":"Cslib.Decidability.SingleTapeDecidable.not","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#not","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a single-tape machine, then so is its complement $\\neg p$.","title":"Single-tape decidable predicates are closed under complement"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/not-single-tape-decidable-halting","kind":"node","lean":"Cslib.Decidability.not_singleTapeDecidable_halting","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#not-single-tape-decidable-halting","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, no single-tape machine decides whether a partial recursive code halts on input $n$.","title":"The halting problem for partial recursive codes is undecidable on single-tape machines"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/single-tape-recognizable-halting","kind":"node","lean":"Cslib.Decidability.singleTapeRecognizable_halting","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#single-tape-recognizable-halting","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the set of partial recursive codes whose program halts on input $n$ is recognizable by a single-tape machine.","title":"The halting problem for partial recursive codes is recognizable on single-tape machines"},{"id":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/not-single-tape-co-recognizable-halting","kind":"node","lean":"Cslib.Decidability.not_singleTapeCoRecognizable_halting","parent":34848,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic/index.html#not-single-tape-co-recognizable-halting","scope":"TheoryOfComputation/equivalence-of-models/single-tape-machines-decide-the-recursive-predic","status":"fully_proved","summary":"The halting problem for partial recursive codes is not co-recognizable on single-tape machines. For every $n \\in \\mathbb{N}$, the set of partial recursive codes whose program halts on input $n$ is not co-recognizable by a single-tape machine: no single-tape machine halts on exactly the codes whose program does not halt on input $n$.","title":"The halting problem for partial recursive codes is not co-recognizable on single-tape machines"},{"data_url":"scopes/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines.json","id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","kind":"scope","navigation":"module","parent":34740,"result":false,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","title":"Nondeterministic Turing machines"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/accepts-iff-exists-labels","kind":"node","lean":"Cslib.Computability.Turing.SingleTape.SingleTapeNTM.accepts_iff_exists_labels","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#accepts-iff-exists-labels","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"A single-tape nondeterministic Turing machine $m$ accepts a word $x$ if and only if there is a list $\\mu$ of transition labels (read a symbol, write a symbol, move, skip) such that applying the labels of $\\mu$ in order to the input tape of $x$ (the tape holding $x$ from the head cell rightward, every other cell blank) never fails, that is, every read label finds its symbol in the cell under the head, and $\\mu$\u2026","title":"Acceptance as a label list"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/accepts-append","kind":"node","lean":"Cslib.Computability.Turing.SingleTape.SingleTapeNTM.accepts_append","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#accepts-append","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"If a single-tape nondeterministic Turing machine $m$ accepts a word $x$ over its tape alphabet, then $m$ accepts $xy$ for every word $y$ over that alphabet.","title":"Acceptance is closed under extension on the right"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/accepts-of-accepts-nil","kind":"node","lean":"Cslib.Computability.Turing.SingleTape.SingleTapeNTM.accepts_of_accepts_nil","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#accepts-of-accepts-nil","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"If a single-tape nondeterministic Turing machine $m$ accepts the empty word, then $m$ accepts every word over its tape alphabet.","title":"An NTM that accepts the empty word accepts every word"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/not-forall-accepts-encode-nat-iff-eq-zer","kind":"node","lean":"Cslib.Computability.Turing.SingleTape.SingleTapeNTM.not_forall_accepts_encodeNat_iff_eq_zero","parent":34904,"reading_title":{"format":"inline-parts-v1","parts":[["text","No NTM accepts exactly the binary code of "],["math","\\(0\\)"]],"source":"No NTM accepts exactly the binary code of $0$"},"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#not-forall-accepts-encode-nat-iff-eq-zer","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"Let $m$ be a single-tape nondeterministic Turing machine with tape alphabet $\\Sigma$, and let $\\iota$ be any map from the bits to $\\Sigma$. For a natural number $n$, write $w_n$ for the word over $\\Sigma$ obtained by applying $\\iota$ letter by letter to the little-endian binary code of $n$, in which $0$ is coded by the empty word. Then it is not the case that, for every natural number $n$, $m$ accepts $w_n$ if and\u2026","title":"No NTM accepts exactly the binary code of $0$"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/re-pred-accepts","kind":"node","lean":"Cslib.Computability.Turing.SingleTape.SingleTapeNTM.rePred_accepts","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#re-pred-accepts","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"Acceptance by an NTM with finite state and tape alphabets is r.e. in the input word (Sipser, Theorem 3.16, through recursion theory): search for a label list that applies to the input tape, checked on coded tapes, and along which the image of the start states, a finite set, meets the accepting states.","title":"Acceptance by an NTM with finite state and tape alphabets is r.e"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/single-tape-recognizable-accepts","kind":"node","lean":"Cslib.Decidability.singleTapeRecognizable_accepts","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#single-tape-recognizable-accepts","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"Let $m$ be a single-tape nondeterministic Turing machine with a finite set of states and a finite tape alphabet $\\Sigma$ that has decidable equality and a primitive recursive encoding. Then the predicate on words over $\\Sigma$ that holds at a word exactly when $m$ accepts it is recognizable by a single-tape deterministic Turing machine.","title":"Deterministic machines recognize NTM acceptance"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/single-tape-recognizable-accepts-comp","kind":"node","lean":"Cslib.Decidability.singleTapeRecognizable_accepts_comp","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#single-tape-recognizable-accepts-comp","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"Let $m$ be a single-tape nondeterministic Turing machine with a finite set of states and a finite tape alphabet $\\Sigma$ that has decidable equality and a primitive recursive encoding. Let $\\alpha$ be a primitively encodable type and $f$ a computable map from $\\alpha$ to words over $\\Sigma$. Then the predicate on $\\alpha$ that holds at $a$ exactly when $m$ accepts $f(a)$ is recognizable by a single-tape\u2026","title":"Deterministic machines recognize NTM acceptance along a computable map"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/single-tape-ntmrecognizable","kind":"node","lean":"Cslib.Decidability.SingleTapeNTMRecognizable","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#single-tape-ntmrecognizable","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$. A predicate $p$ on $\\alpha$ is recognizable by a single-tape nondeterministic machine when there are a finite set of states, a finite tape alphabet $\\Sigma$ with decidable equality, an injective map $\\iota$ from the bits $\\{0, 1\\}$ to $\\Sigma$ and a single-tape nondeterministic Turing machine $M$ with these states over $\\Sigma$ such that\u2026","title":"Predicates recognizable by a single-tape nondeterministic machine"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/single-tape-recognizable","kind":"node","lean":"Cslib.Decidability.SingleTapeNTMRecognizable.singleTapeRecognizable","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#single-tape-recognizable","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"A predicate recognized by a single-tape NTM is recognized by a deterministic single-tape machine (Sipser, Theorem 3.16, through recursion theory).","title":"A predicate recognized by a single-tape NTM is recognized by a deterministic single-tape machine"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/not-single-tape-ntmrecognizable-eq-zero","kind":"node","lean":"Cslib.Decidability.not_singleTapeNTMRecognizable_eq_zero","parent":34904,"reading_title":{"format":"inline-parts-v1","parts":[["text","No single-tape NTM recognizes "],["math","\\(n = 0\\)"]],"source":"No single-tape NTM recognizes $n = 0$"},"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#not-single-tape-ntmrecognizable-eq-zero","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"The predicate $n = 0$ on the natural numbers is not recognizable by any single-tape nondeterministic Turing machine.","title":"No single-tape NTM recognizes $n = 0$"},{"id":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/exists-single-tape-decidable-not-single","kind":"node","lean":"Cslib.Decidability.exists_singleTapeDecidable_not_singleTapeNTMRecognizable","parent":34904,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines/index.html#exists-single-tape-decidable-not-single","scope":"TheoryOfComputation/equivalence-of-models/nondeterministic-turing-machines","status":"fully_proved","summary":"There is a predicate $p$ on the natural numbers that is decidable by a single-tape deterministic Turing machine but is not recognizable by any single-tape nondeterministic Turing machine.","title":"A decidable predicate that no single-tape NTM recognizes"},{"data_url":"scopes/TheoryOfComputation/equivalence-of-models/unlimited-register-machines.json","id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","kind":"scope","navigation":"module","parent":34740,"result":false,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","title":"Unlimited register machines"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/encode-instr","kind":"node","lean":"Cslib.URM.encodeInstr","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#encode-instr","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Code of a URM instruction as a natural number: the instruction tag paired with the paired arguments, `Z n \u21a6 pair 0 n`, `S n \u21a6 pair 1 n`, `T m n \u21a6 pair 2 (pair m n)`, `J m n q \u21a6 pair 3 (pair m (pair n q))`.","title":"Code of a URM instruction"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/encode-instr-unpair-fst","kind":"node","lean":"Cslib.URM.encodeInstr_unpair_fst","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#encode-instr-unpair-fst","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For every URM instruction $I$, unpairing the code of $I$ gives first component $0$ if $I$ is a zero instruction $Z\\,n$, $1$ if it is a successor instruction $S\\,n$, $2$ if it is a transfer $T\\,m\\,n$, and $3$ if it is a jump $J\\,m\\,n\\,q$.","title":"Tag of an instruction code"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/encode-instr-unpair-snd","kind":"node","lean":"Cslib.URM.encodeInstr_unpair_snd","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#encode-instr-unpair-snd","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Write $\\langle a, b \\rangle$ for the pairing of natural numbers $a$ and $b$. For every URM instruction $I$, unpairing the code of $I$ gives second component $n$ if $I$ is $Z\\,n$ or $S\\,n$, $\\langle m, n \\rangle$ if $I$ is $T\\,m\\,n$, and $\\langle m, \\langle n, q \\rangle \\rangle$ if $I$ is $J\\,m\\,n\\,q$.","title":"Payload of an instruction code"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/coded-step","kind":"node","lean":"Cslib.URM.codedStep","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#coded-step","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A coded state is a program counter $k$ together with a list $r$ of register contents (in use, the contents of registers $0, \\dots, R$, where $R$ is the register bound of the program), and a program is given by the list $c$ of its instruction codes. Write $c_k$ for entry $k$ of $c$, and $r_j$ for entry $j$ of $r$, read as $0$ when $j$ is past the end of $r$; setting an entry past the end of $r$ leaves $r$ unchanged.\u2026","title":"One coded URM step"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/primrec-coded-step","kind":"node","lean":"Cslib.URM.primrec_codedStep","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#primrec-coded-step","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"The coded step is primitive recursive, for any fixed list of instruction codes.","title":"The coded step is primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/reg-bound","kind":"node","lean":"Cslib.URM.regBound","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#reg-bound","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"The register bound $R$ of a URM program $p$ is the sum, over the instructions of $p$, of the largest register index each instruction refers to. Every register that an instruction of $p$ can read or write is at most $R$.","title":"Register bound"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/to-regs","kind":"node","lean":"Cslib.URM.toRegs","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#to-regs","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a URM program $p$ with register bound $R$, the register list of a register state $\\sigma$ is the list $(\\sigma(0), \\sigma(1), \\dots, \\sigma(R))$ of the contents of registers $0, \\dots, R$.","title":"Register list"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/to-coded","kind":"node","lean":"Cslib.URM.toCoded","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#to-coded","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a URM program $p$ with register bound $R$, the coded state of a machine state $s$ is the pair consisting of the program counter of $s$ and the register list $(r_0, \\dots, r_R)$ of the contents of registers $0, \\dots, R$ of $s$.","title":"Coded machine state"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/of-coded","kind":"node","lean":"Cslib.URM.ofCoded","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#of-coded","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A coded state is a pair $(c, \\ell)$ of a natural number $c$ and a list $\\ell$ of natural numbers. Its machine state has program counter $c$, and register $n$ holds the $n$-th entry of $\\ell$ (counting from $0$) when $n$ is less than the length of $\\ell$, and $0$ otherwise.","title":"Decoding a coded state"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/length-to-regs","kind":"node","lean":"Cslib.URM.length_toRegs","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#length-to-regs","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a URM program $p$ with register bound $R$ and a register state $\\sigma$, the register list of $\\sigma$ (the contents of registers $0, \\dots, R$) has length $R + 1$.","title":"Length of the register list"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/get-d-to-regs","kind":"node","lean":"Cslib.URM.getD_toRegs","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#get-d-to-regs","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program with register bound $R$ and $\\sigma$ a register state. For every $n$, the entry at position $n$ of the register list of $\\sigma$, read with default $0$, is $\\sigma(n)$ if $n \\le R$ and $0$ otherwise.","title":"Reading the register list"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/to-regs-update","kind":"node","lean":"Cslib.URM.toRegs_update","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#to-regs-update","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program with register bound $R$, let $\\sigma$ be a register state, and let $n \\le R$ and $v$ be natural numbers. The register list of the state obtained from $\\sigma$ by setting register $n$ to $v$ equals the register list of $\\sigma$ with its entry at position $n$ replaced by $v$.","title":"Updating a register updates the register list"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/max-register-le-reg-bound","kind":"node","lean":"Cslib.URM.maxRegister_le_regBound","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#max-register-le-reg-bound","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a URM program $p$ with register bound $R$ and every instruction $I$ of $p$, the largest register index referenced by $I$ is at most $R$.","title":"Instructions stay within the register bound"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/writes-to-le-reg-bound","kind":"node","lean":"Cslib.URM.writesTo_le_regBound","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#writes-to-le-reg-bound","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program with register bound $R$. If an instruction of $p$ writes to register $n$, then $n \\le R$.","title":"Written registers lie within the register bound"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/reads-le-reg-bound","kind":"node","lean":"Cslib.URM.reads_le_regBound","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#reads-le-reg-bound","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program. If $n$ is a register read by some instruction of $p$, then $n$ is at most the register bound of $p$.","title":"Registers read by a program lie within the register bound"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/to-coded-of-coded","kind":"node","lean":"Cslib.URM.toCoded_ofCoded","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#to-coded-of-coded","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program with register bound $R$, and let $c$ be a coded state: a program counter together with a list of register contents. If the list has length $R + 1$, then the coded state, with respect to $p$, of the machine state decoded from $c$ is $c$.","title":"Decoding and re-encoding a coded state gives it back"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/pc-lt-of-step","kind":"node","lean":"Cslib.URM.pc_lt_of_step","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#pc-lt-of-step","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"If a state $s$ of a URM program $p$ makes a step to some state, then the program counter of $s$ is less than the number of instructions of $p$.","title":"A state that can step is running"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/get-d-codes","kind":"node","lean":"Cslib.URM.getD_codes","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#get-d-codes","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"If the URM program $p$ has instruction $I$ at position $pc$, then the entry at position $pc$ of the list of instruction codes of $p$, read with default $0$, is the code of $I$.","title":"Instruction codes by position"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/coded-step-of-step","kind":"node","lean":"Cslib.URM.codedStep_of_step","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#coded-step-of-step","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program, let $R$ be the sum over the instructions of $p$ of the largest register index each one mentions, and let $s'$ be a state that $p$ reaches from the state $s$ in one step. Then the coded step, run on the list of codes of the instructions of $p$, sends the coded state of $s$ (its program counter together with the list of the contents of its registers $0, \\dots, R$) to the coded state of $s'$.","title":"A machine step is a coded step"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/step-of-coded-step","kind":"node","lean":"Cslib.URM.step_of_codedStep","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#step-of-coded-step","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program and $s$ a machine state. If one coded URM step, run on the list of codes of the instructions of $p$ from the coded state of $s$, yields $c'$, then there is a machine state $s'$ such that $p$ steps from $s$ to $s'$ in one step and the coded state of $s'$ is $c'$.","title":"A coded step is a machine step"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/decidable","kind":"node","lean":"Cslib.URM.Decidable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#decidable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A predicate $P$ on the natural numbers is URM-decidable when some URM program computes its characteristic function: started on any input $n$ (register $0$ holding $n$, all other registers $0$), the program halts, and register $0$ then holds $1$ if $P(n)$ holds and $0$ otherwise.","title":"URM-decidable predicate"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/recognizable","kind":"node","lean":"Cslib.URM.Recognizable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#recognizable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A predicate $P$ on the natural numbers is URM-recognizable when there is a URM program that, started with input $n$ in register $0$ and every other register $0$, halts if and only if $P(n)$ holds.","title":"URM-recognizable predicates"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/step-fn","kind":"node","lean":"Cslib.URM.stepFn","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#step-fn","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Given a list of instruction codes, the one-step function sends a coded state $s$ to the result of one coded URM step from $s$ when that step is defined, and leaves $s$ unchanged when it is not (in particular when $s$ is halted).","title":"One-step function on coded states"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/primrec-step-fn","kind":"node","lean":"Cslib.URM.primrec_stepFn","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#primrec-step-fn","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For every list of natural numbers, read as instruction codes, the function on coded states (pairs of a program counter and a register list) that sends a coded state to its successor under one coded step, and a coded state with no successor to itself, is primitive recursive.","title":"The one-step function is primitive recursive"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/exec-n","kind":"node","lean":"Cslib.URM.execN","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#exec-n","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a list of instruction codes, a number $t$ and a coded state $s$, bounded execution applies the one-step function for these codes $t$ times, starting from $s$.","title":"Bounded execution"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/exec-n-succ","kind":"node","lean":"Cslib.URM.execN_succ","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#exec-n-succ","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a list of instruction codes, a number $t$ and a coded state $s$, bounded execution for $t + 1$ steps from $s$ is the one-step function applied to the result of bounded execution for $t$ steps from $s$.","title":"Bounded execution unfolds by one step"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/init-coded","kind":"node","lean":"Cslib.URM.initCoded","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#init-coded","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a URM program $p$ with register bound $R$ and an input $x$, the initial coded state has program counter $0$ and register list $(x, 0, \\dots, 0)$ of length $R + 1$: register $0$ holds $x$ and registers $1, \\dots, R$ hold $0$.","title":"Initial coded state"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/primrec-init-coded","kind":"node","lean":"Cslib.URM.primrec_initCoded","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#primrec-init-coded","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"The initial coded state is primitive recursive in the input.","title":"The initial coded state is primitive recursive in the input"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/primrec-exec-n","kind":"node","lean":"Cslib.URM.primrec_execN","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#primrec-exec-n","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a URM program $p$, the bounded execution of $p$ for $t$ steps ($t$ iterations of the one-step function on the list of instruction codes of $p$), started from the initial coded state for input $x$, is a primitive recursive function of the pair $(x, t)$.","title":"Bounded execution is primitive recursive in the input and the step bound"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/init-coded-eq-to-coded","kind":"node","lean":"Cslib.URM.initCoded_eq_toCoded","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#init-coded-eq-to-coded","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a URM program $p$ and an input $x$, the initial coded state for $x$ equals the coded state of the initial machine state with input list $[x]$.","title":"The initial coded state codes the initial machine state"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/exec-n-to-coded-steps","kind":"node","lean":"Cslib.URM.execN_toCoded_steps","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#exec-n-to-coded-steps","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program, $x$ an input and $t$ a natural number. Then there is a machine state $s'$, reachable from the initial state with input list $[x]$ by steps of $p$, whose coded state equals the result of bounded execution for $t$ steps on the instruction codes of $p$ from the initial coded state for $x$.","title":"Bounded execution only reaches codes of reachable states"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/halts-of-exec-n-halted","kind":"node","lean":"Cslib.URM.halts_of_execN_halted","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#halts-of-exec-n-halted","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program, $x$ an input and $t$ a number. If bounded execution for $t$ steps on the instruction codes of $p$ from the initial coded state for $x$ reaches a program counter at least the length of the list of instruction codes, then $p$ halts on the input list $[x]$.","title":"A halted coded state gives a halting machine run"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/exists-exec-n-eq-of-steps","kind":"node","lean":"Cslib.URM.exists_execN_eq_of_steps","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#exists-exec-n-eq-of-steps","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $p$ be a URM program and $x$ an input. If a machine state $s$ is reachable from the initial state with input list $[x]$ by steps of $p$, then for some number $t$, bounded execution for $t$ steps on the instruction codes of $p$ from the initial coded state for $x$ gives the coded state of $s$.","title":"Every reachable state is reached by bounded execution"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/exec-n-succ-of-halted","kind":"node","lean":"Cslib.URM.execN_succ_of_halted","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#exec-n-succ-of-halted","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $c$ be a list of instruction codes, $s$ a coded state and $t$ a natural number. If after $t$ steps of bounded execution of $c$ from $s$ the program counter is at least the length of $c$, then $t + 1$ steps of bounded execution of $c$ from $s$ give the same coded state as $t$ steps.","title":"Bounded URM execution is stable once halted"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/exec-n-eq-of-le","kind":"node","lean":"Cslib.URM.execN_eq_of_le","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#exec-n-eq-of-le","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $c$ be a list of instruction codes, $s$ a coded state, and $t_1 \\le t_2$ natural numbers. If after $t_1$ steps of bounded execution of $c$ from $s$ the program counter is at least the length of $c$, then $t_2$ steps of bounded execution of $c$ from $s$ give the same coded state as $t_1$ steps.","title":"Bounded URM execution stays at its halted value"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/halted-output","kind":"node","lean":"Cslib.URM.haltedOutput","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#halted-output","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For a URM program $p$, an input $x$ and a number $t$, run bounded execution for $t$ steps on the instruction codes of $p$ from the initial coded state for $x$. If the resulting program counter is less than the length of the list of instruction codes, the program is still running and there is no output; otherwise the output is the entry at position $0$ of the resulting register list, read with default $0$.","title":"Output after a given number of steps"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/computable-halted-output","kind":"node","lean":"Cslib.URM.computable\u2082_haltedOutput","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#computable-halted-output","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"For every URM program $p$, the function sending an input $x$ and a step count $t$ to the output of $p$ on $x$ after $t$ steps (nothing while $p$ is still running, the contents of register $0$ once it has halted) is a computable function of the pair $(x, t)$.","title":"The URM output after a given number of steps is computable"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/partrec-of-computable","kind":"node","lean":"Cslib.URM.partrec_of_computable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#partrec-of-computable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $f$ be a partial function from $\\mathbb{N}$ to $\\mathbb{N}$. Suppose some URM program has the following property for every natural number $x$: started with $x$ in register $0$ and all other registers $0$, it halts exactly when $f(x)$ is defined, and then it holds $f(x)$ in register $0$. Then $f$ is partial recursive.","title":"A URM-computable partial function is partial recursive"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/computable-pred","kind":"node","lean":"Cslib.URM.Decidable.computablePred","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#computable-pred","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"If a predicate $P$ on the natural numbers is URM-decidable, then it is a computable predicate: its Boolean decision function, sending $n$ to whether $P(n)$ holds, is computable.","title":"A URM-decidable predicate is computable"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/re-pred","kind":"node","lean":"Cslib.URM.Recognizable.rePred","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#re-pred","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A URM-recognizable predicate is recursively enumerable.","title":"A URM-recognizable predicate is recursively enumerable"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/single-tape-decidable","kind":"node","lean":"Cslib.URM.Decidable.singleTapeDecidable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#single-tape-decidable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A URM-decidable predicate is decidable by a single-tape Turing machine.","title":"A URM-decidable predicate is decidable by a single-tape Turing machine"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/tm2-decidable","kind":"node","lean":"Cslib.URM.Decidable.tm2Decidable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#tm2-decidable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A URM-decidable predicate is decidable by a multi-stack Turing machine.","title":"A URM-decidable predicate is decidable by a multi-stack Turing machine"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/single-tape-recognizable","kind":"node","lean":"Cslib.URM.Recognizable.singleTapeRecognizable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#single-tape-recognizable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A URM-recognizable predicate is recognizable by a single-tape Turing machine.","title":"A URM-recognizable predicate is recognizable by a single-tape Turing machine"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/tm2-recognizable","kind":"node","lean":"Cslib.URM.Recognizable.tm2Recognizable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#tm2-recognizable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"A URM-recognizable predicate is recognizable by a multi-stack Turing machine.","title":"A URM-recognizable predicate is recognizable by a multi-stack Turing machine"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/urm-computable-to-tm2","kind":"node","lean":"Cslib.Decidability.urmComputable_to_tm2","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#urm-computable-to-tm2","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$. Suppose there is a URM program that, for every $n \\in \\mathbb{N}$, started with $n$ in register $0$ and all other registers $0$, halts with $f(n)$ in register $0$. Then there is a multi-stack Turing machine that computes $f$ with input and output numbers written in binary: started on the binary word of $n$, it halts with the binary word of $f(n)$.","title":"A URM-computable total function is computed by a multi-stack Turing machine"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/urm-decidable-to-tm2-decidable","kind":"node","lean":"Cslib.Decidability.urmDecidable_to_tm2Decidable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#urm-decidable-to-tm2-decidable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Every predicate $P$ on $\\mathbb{N}$ that is URM-decidable is decidable by a multi-stack Turing machine.","title":"URM-decidability implies multi-stack decidability"},{"id":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines/urm-recognizable-to-tm2-recognizable","kind":"node","lean":"Cslib.Decidability.urmRecognizable_to_tm2Recognizable","parent":34916,"result":true,"route":"../roadmap/TheoryOfComputation/equivalence-of-models/unlimited-register-machines/index.html#urm-recognizable-to-tm2-recognizable","scope":"TheoryOfComputation/equivalence-of-models/unlimited-register-machines","status":"fully_proved","summary":"Every predicate $P$ on $\\mathbb{N}$ that is URM-recognizable is recognizable by a multi-stack Turing machine.","title":"URM-recognizability implies multi-stack recognizability"},{"data_url":"scopes/TheoryOfComputation/computability.json","id":"TheoryOfComputation/computability","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/computability/index.html","scope":"TheoryOfComputation/computability","status":"can_prove","summary":"Recursive and recursively enumerable predicates and their closure properties, the halting problem, Rice's and Rice\u2013Shapiro theorems, the recursion and s-m-n theorems, enumerators, index sets such as totality, finiteness and minimal programs, and Busy Beaver functions.","title":"Computability and undecidability"},{"id":"TheoryOfComputation/computability/exists-computable-code","kind":"node","lean":"MetaMathlibExt.exists_computable_code","parent":34964,"result":true,"route":"../roadmap/TheoryOfComputation/computability/index.html#exists-computable-code","scope":"TheoryOfComputation/computability","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into the natural numbers, and for each $a \\in \\alpha$ let $F_a$ be a partial function from $\\mathbb{N}$ to $\\mathbb{N}$, such that $(a, x) \\mapsto F_a(x)$ is partial recursive. Then there is a computable map $g$ from $\\alpha$ to codes of partial recursive functions such that for all $a \\in \\alpha$ and $x \\in \\mathbb{N}$, the partial function with code\u2026","title":"Parametrized s-m-n theorem"},{"data_url":"scopes/TheoryOfComputation/computability/busy-beaver.json","id":"TheoryOfComputation/computability/busy-beaver","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html","scope":"TheoryOfComputation/computability/busy-beaver","status":"can_prove","title":"Busy beaver"},{"id":"TheoryOfComputation/computability/busy-beaver/move","kind":"node","lean":"Cslib.BusyBeaver.Move","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#move","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A head movement is one of exactly two directions: left or right.","title":"Head movement"},{"id":"TheoryOfComputation/computability/busy-beaver/shift","kind":"node","lean":"Cslib.BusyBeaver.Move.shift","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#shift","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"The displacement of a head movement is $-1$ for a move to the left and $1$ for a move to the right.","title":"Head displacement"},{"id":"TheoryOfComputation/computability/busy-beaver/machine","kind":"node","lean":"Cslib.BusyBeaver.Machine","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#machine","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A Busy Beaver machine with $n$ working states is a transition table that assigns to each of the $n$ working states and each read symbol in $\\{0, 1\\}$ a symbol in $\\{0, 1\\}$ to write, a head movement (left or right), and a next state, which is either one of the $n$ working states or the halt state.","title":"Busy Beaver machine"},{"id":"TheoryOfComputation/computability/busy-beaver/config","kind":"node","lean":"Cslib.BusyBeaver.Config","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#config","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A configuration of an $n$-state machine consists of a current state, which is either one of the $n$ working states or the halt state, an integer head position, and a tape assigning a symbol in $\\{0, 1\\}$ to every integer cell.","title":"Busy Beaver machine configuration"},{"id":"TheoryOfComputation/computability/busy-beaver/step","kind":"node","lean":"Cslib.BusyBeaver.step","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#step","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"Let $M$ be a machine with $n$ working states and let $c$ be a configuration with head position $h$. If $c$ is halted, the step leaves $c$ unchanged. Otherwise let $q$ be the current state, let $a$ be the symbol in cell $h$, and let $M(q, a) = (b, d, q')$. The step writes $b$ in cell $h$, leaving every other cell unchanged, moves the head to $h - 1$ if $d$ is left and to $h + 1$ if $d$ is right, and passes to the\u2026","title":"One step of a Busy Beaver machine"},{"id":"TheoryOfComputation/computability/busy-beaver/initial-state","kind":"node","lean":"Cslib.BusyBeaver.initialState","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#initial-state","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For a machine with $n$ working states, the initial state is state $0$ when $n > 0$, and the halt state when $n = 0$.","title":"Initial state"},{"id":"TheoryOfComputation/computability/busy-beaver/initial","kind":"node","lean":"Cslib.BusyBeaver.initial","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#initial","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"The initial configuration of a machine with $n$ working states is in the initial state (state $0$, or the halt state when $n = 0$), with the head at position $0$ and every tape cell blank, that is, holding the symbol $0$.","title":"Initial configuration of a Busy Beaver machine"},{"id":"TheoryOfComputation/computability/busy-beaver/run","kind":"node","lean":"Cslib.BusyBeaver.run","parent":34966,"reading_title":{"format":"inline-parts-v1","parts":[["text","Configuration after "],["math","\\(t\\)"],["text"," steps"]],"source":"Configuration after $t$ steps"},"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#run","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For a Busy Beaver machine $M$ and $t \\in \\mathbb{N}$, the configuration of $M$ after $t$ steps is the result of applying the one-step map of $M$ $t$ times to the initial configuration on the blank tape.","title":"Configuration after $t$ steps"},{"id":"TheoryOfComputation/computability/busy-beaver/halts-within","kind":"node","lean":"Cslib.BusyBeaver.HaltsWithin","parent":34966,"reading_title":{"format":"inline-parts-v1","parts":[["text","Halting within "],["math","\\(t\\)"],["text"," steps"]],"source":"Halting within $t$ steps"},"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#halts-within","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A Busy Beaver machine $M$ with $n$ working states halts within $t$ steps when the configuration reached after running $M$ for $t$ steps from the initial blank-tape configuration is in the halt state.","title":"Halting within $t$ steps"},{"id":"TheoryOfComputation/computability/busy-beaver/halts","kind":"node","lean":"Cslib.BusyBeaver.Halts","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#halts","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A Busy Beaver machine $M$ halts when there is a number of steps $t$ such that $M$ halts within $t$ steps, that is, when $M$ started from the initial blank-tape configuration eventually reaches the halt state.","title":"Halting on the blank tape"},{"id":"TheoryOfComputation/computability/busy-beaver/halt-time","kind":"node","lean":"Cslib.BusyBeaver.haltTime","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#halt-time","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For an $n$-state machine $M$ that halts on the blank tape, its halting time is the least $t$ such that $M$, started on the blank tape, is in the halt state after $t$ steps.","title":"Halting time"},{"id":"TheoryOfComputation/computability/busy-beaver/abs-head-le","kind":"node","lean":"Cslib.BusyBeaver.abs_head_le","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#abs-head-le","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For every $n$-state Busy Beaver machine $M$ and every $t \\in \\mathbb{N}$, if $h \\in \\mathbb{Z}$ is the head position after $M$ has run for $t$ steps from the initial configuration on the blank tape, then $|h| \\le t$.","title":"Head displacement bound"},{"id":"TheoryOfComputation/computability/busy-beaver/tape-eq-zero-of-abs-gt","kind":"node","lean":"Cslib.BusyBeaver.tape_eq_zero_of_abs_gt","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#tape-eq-zero-of-abs-gt","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"Let $M$ be a machine with $n$ working states, started in cell $0$ on the blank tape. For every $t \\in \\mathbb{N}$ and every cell $i \\in \\mathbb{Z}$ with $|i| > t$, the symbol in cell $i$ after $t$ steps is $0$.","title":"Cells far from the origin stay blank"},{"id":"TheoryOfComputation/computability/busy-beaver/ones-at","kind":"node","lean":"Cslib.BusyBeaver.onesAt","parent":34966,"reading_title":{"format":"inline-parts-v1","parts":[["text","Number of ones after "],["math","\\(t\\)"],["text"," steps"]],"source":"Number of ones after $t$ steps"},"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#ones-at","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For a Busy Beaver machine $M$ and $t \\in \\mathbb{N}$, the number of ones of $M$ after $t$ steps is the number of cells $i \\in \\mathbb{Z}$ with $-t \\le i \\le t$ that hold the symbol $1$ in the configuration of $M$ after $t$ steps, started on the blank tape.","title":"Number of ones after $t$ steps"},{"id":"TheoryOfComputation/computability/busy-beaver/mem-window-of-tape-eq-one","kind":"node","lean":"Cslib.BusyBeaver.mem_window_of_tape_eq_one","parent":34966,"reading_title":{"format":"inline-parts-v1","parts":[["text","After "],["math","\\(t\\)"],["text"," steps, every one lies within distance "],["math","\\(t\\)"],["text"," of the origin"]],"source":"After $t$ steps, every one lies within distance $t$ of the origin"},"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#mem-window-of-tape-eq-one","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For every Busy Beaver machine $M$, every $t \\in \\mathbb{N}$ and every cell $i \\in \\mathbb{Z}$, if cell $i$ holds the symbol $1$ after $M$ runs for $t$ steps from the initial blank-tape configuration, then $-t \\le i \\le t$.","title":"After $t$ steps, every one lies within distance $t$ of the origin"},{"id":"TheoryOfComputation/computability/busy-beaver/ones-at-succ-le","kind":"node","lean":"Cslib.BusyBeaver.onesAt_succ_le","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#ones-at-succ-le","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For every Busy Beaver machine $M$ and every $t \\in \\mathbb{N}$, the number of ones of $M$ after $t + 1$ steps is at most the number of ones of $M$ after $t$ steps plus $1$.","title":"The number of ones grows by at most one per step"},{"id":"TheoryOfComputation/computability/busy-beaver/ones-at-le-steps","kind":"node","lean":"Cslib.BusyBeaver.onesAt_le_steps","parent":34966,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most "],["math","\\(t\\)"],["text"," ones after "],["math","\\(t\\)"],["text"," steps"]],"source":"At most $t$ ones after $t$ steps"},"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#ones-at-le-steps","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For every Busy Beaver machine $M$ and every $t \\in \\mathbb{N}$, the number of ones of $M$ after $t$ steps is at most $t$.","title":"At most $t$ ones after $t$ steps"},{"id":"TheoryOfComputation/computability/busy-beaver/steps-of","kind":"node","lean":"Cslib.BusyBeaver.stepsOf","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#steps-of","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For a machine $M$ with $n$ working states, the runtime of $M$ is its halting time on the blank tape if $M$ halts, and $0$ if $M$ does not halt. This is the quantity whose maximum over all $n$-state machines is the shift function $S(n)$.","title":"Runtime of a Busy Beaver machine"},{"id":"TheoryOfComputation/computability/busy-beaver/score-of","kind":"node","lean":"Cslib.BusyBeaver.scoreOf","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#score-of","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For a Busy Beaver machine $M$: if $M$ halts on the blank tape, with halting time $T$, its score is the number of ones of $M$ after $T$ steps; if $M$ does not halt, its score is $0$.","title":"Score of a machine"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-steps","kind":"node","lean":"Cslib.BusyBeaver.busyBeaverSteps","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-steps","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, Rado's shift function $S(n)$ is the largest halting time of an $n$-state two-symbol machine that halts when started on the blank tape; machines that do not halt contribute $0$.","title":"Busy Beaver shift function"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-ones","kind":"node","lean":"Cslib.BusyBeaver.busyBeaverOnes","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-ones","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, Rado's score function $\\Sigma(n)$ is the largest number of ones left on the tape, at its halting time, by an $n$-state two-symbol machine that halts when started on the blank tape; machines that do not halt contribute $0$.","title":"Busy Beaver score function"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-ones-le-steps","kind":"node","lean":"Cslib.BusyBeaver.busyBeaverOnes_le_steps","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-ones-le-steps","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, Rado's score function $\\Sigma$ and Rado's shift function $S$ satisfy $\\Sigma(n) \\le S(n)$: the largest number of ones left on the tape by a halting $n$-state two-symbol machine started on the blank tape is at most the largest halting time of such a machine.","title":"The Busy Beaver score is at most the shift"},{"id":"TheoryOfComputation/computability/busy-beaver/exists-steps-of-eq","kind":"node","lean":"Cslib.BusyBeaver.exists_stepsOf_eq","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#exists-steps-of-eq","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"Let $S(n)$ be Rado's shift function, the largest runtime of an $n$-state machine, where the runtime of a machine is its halting time on the blank tape if it halts and $0$ otherwise. For every $n$, some $n$-state machine $M$ has runtime equal to $S(n)$. Thus $S(n)$ is the halting time of a halting $n$-state machine, or $S(n) = 0$.","title":"The shift function is attained"},{"id":"TheoryOfComputation/computability/busy-beaver/exists-score-of-eq","kind":"node","lean":"Cslib.BusyBeaver.exists_scoreOf_eq","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#exists-score-of-eq","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"Let $\\Sigma(n)$ be the busy beaver score function, the largest score of an $n$-state machine, where the score of a machine is the number of ones on its final tape if it halts on the blank tape and $0$ otherwise. For every $n$, some $n$-state machine $M$ has score equal to $\\Sigma(n)$. Thus $\\Sigma(n)$ is the number of ones left by a halting $n$-state machine, or $\\Sigma(n) = 0$.","title":"The score function is attained"},{"id":"TheoryOfComputation/computability/busy-beaver/embed-machine","kind":"node","lean":"Cslib.BusyBeaver.embedMachine","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#embed-machine","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"An $n$-state machine $M$ is embedded into $n + 1$ states as follows. In a working state $q < n$ reading a symbol $s$, the new machine writes the symbol, makes the move and enters the next state that $M$ prescribes in state $q$ on $s$, halting when $M$ halts. In the new state $n$ it writes $0$, moves left and halts, whatever it reads.","title":"Embedding a machine into one more state"},{"id":"TheoryOfComputation/computability/busy-beaver/embed-config","kind":"node","lean":"Cslib.BusyBeaver.embedConfig","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#embed-config","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For a configuration $c$ of an $n$-state machine, its embedding is the configuration for $n + 1$ states with the same head position and the same tape as $c$. Its state is the state of $c$, where a working state $q \\in \\{0, \\dots, n - 1\\}$ stays $q$ and the halt state stays the halt state.","title":"Embedding a configuration into one more state"},{"id":"TheoryOfComputation/computability/busy-beaver/not-halts-of-one-state-march","kind":"node","lean":"Cslib.BusyBeaver.not_halts_of_one_state_march","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#not-halts-of-one-state-march","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"Let $M$ be a Busy Beaver machine with a single working state $0$. If the transition of $M$ from state $0$ on reading the blank symbol $0$ has next state $0$, then $M$ does not halt.","title":"A one-state machine that stays in its state on a blank never halts"},{"id":"TheoryOfComputation/computability/busy-beaver/halts-within-one-of-halts","kind":"node","lean":"Cslib.BusyBeaver.haltsWithin_one_of_halts","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#halts-within-one-of-halts","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"Every halting one-state machine halts within one step.","title":"One-state machines halt within one step"},{"id":"TheoryOfComputation/computability/busy-beaver/machine-code","kind":"node","lean":"Cslib.BusyBeaver.machineCode","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#machine-code","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"The code of a machine $M$ with $n$ working states is the natural number that encodes the pair $(n, i)$ under a fixed pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$, where $i$ is the index of $M$ in a fixed enumeration of the finitely many machines with $n$ working states.","title":"Machine code"},{"id":"TheoryOfComputation/computability/busy-beaver/sentence-code","kind":"node","lean":"Cslib.BusyBeaver.sentenceCode","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#sentence-code","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"The sentence code with tag $a$ and payload $p$ is the natural number $4p + a$. The tags in use are $0$ (the coded machine halts), $1$ (the coded machine does not halt), $2$ (the false sentence), and $3$ with payload the pairing code of $n$ and $B$ (every halting $n$-state machine halts within $B$ steps).","title":"Sentence codes"},{"id":"TheoryOfComputation/computability/busy-beaver/halts-sentence","kind":"node","lean":"Cslib.BusyBeaver.haltsSentence","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#halts-sentence","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For an $n$-state machine $M$, the halting sentence of $M$ is the sentence code with tag $0$ and payload the code of $M$. It stands for \"$M$ halts on the blank tape\"; this meaning comes from soundness hypotheses on a theory, not from the code itself.","title":"Halting sentence of a machine"},{"id":"TheoryOfComputation/computability/busy-beaver/not-halts-sentence","kind":"node","lean":"Cslib.BusyBeaver.notHaltsSentence","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#not-halts-sentence","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For a Busy Beaver machine $M$, the non-halting sentence of $M$ is the sentence code with tag $1$ whose payload is the code of $M$; it stands for \"$M$ does not halt on the blank tape\".","title":"Non-halting sentence of a machine"},{"id":"TheoryOfComputation/computability/busy-beaver/false-sentence","kind":"node","lean":"Cslib.BusyBeaver.falseSentence","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#false-sentence","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"The false sentence is the sentence code with tag $2$ and payload $0$, that is, the natural number $2$.","title":"The false sentence"},{"id":"TheoryOfComputation/computability/busy-beaver/bound-sentence","kind":"node","lean":"Cslib.BusyBeaver.boundSentence","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#bound-sentence","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For natural numbers $n$ and $B$, the runtime-bound sentence is the sentence code with tag $3$ and payload $\\langle n, B \\rangle$, the pairing of $n$ and $B$; as a number it equals $4 \\langle n, B \\rangle + 3$.","title":"Runtime-bound sentence"},{"id":"TheoryOfComputation/computability/busy-beaver/formal-theory","kind":"node","lean":"Cslib.BusyBeaver.FormalTheory","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#formal-theory","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A formal theory is given by a provability predicate on sentence codes, which are natural numbers: for each natural number it specifies whether that sentence code is provable in the theory.","title":"Formal theory"},{"id":"TheoryOfComputation/computability/busy-beaver/consistent","kind":"node","lean":"Cslib.BusyBeaver.Consistent","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#consistent","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A formal theory $T$ is consistent when $T$ does not prove the false sentence.","title":"Consistent formal theory"},{"id":"TheoryOfComputation/computability/busy-beaver/sound-for-halting","kind":"node","lean":"Cslib.BusyBeaver.SoundForHalting","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#sound-for-halting","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A formal theory $T$ is sound for halting sentences when, for every $n$ and every Busy Beaver machine $M$ with $n$ states, if $T$ proves the halting sentence of $M$ then $M$ halts on the blank tape, and if $T$ proves the non-halting sentence of $M$ then $M$ does not halt on the blank tape.","title":"Soundness for halting sentences"},{"id":"TheoryOfComputation/computability/busy-beaver/decides-halting","kind":"node","lean":"Cslib.BusyBeaver.DecidesHalting","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#decides-halting","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A formal theory $T$ decides halting for $n$-state machines when, for every $n$-state machine $M$, $T$ proves the halting sentence of $M$ or $T$ proves the non-halting sentence of $M$.","title":"Theories deciding halting at a state count"},{"id":"TheoryOfComputation/computability/busy-beaver/first-undecidable","kind":"node","lean":"Cslib.BusyBeaver.firstUndecidable","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#first-undecidable","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For a formal theory $T$, the first undecidable number of states is the least $n$ such that $T$ does not decide halting for all $n$-state machines, that is, there is an $n$-state machine $M$ for which $T$ proves neither the halting sentence nor the non-halting sentence of $M$. It is $0$ when there is no such $n$.","title":"First undecidable number of states"},{"id":"TheoryOfComputation/computability/busy-beaver/six-le-first-undecidable-of-decides-le-f","kind":"node","lean":"Cslib.BusyBeaver.six_le_firstUndecidable_of_decides_le_five","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#six-le-first-undecidable-of-decides-le-f","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"If a formal theory $T$ decides halting for $n$-state machines for every $n \\le 5$, and fails to decide halting for $n$-state machines for some $n$, then the first undecidable size of $T$ is at least $6$.","title":"Decidability through five states and the first undecidable size"},{"id":"TheoryOfComputation/computability/busy-beaver/not-decides-of-consistency-machine","kind":"node","lean":"Cslib.BusyBeaver.not_decides_of_consistency_machine","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#not-decides-of-consistency-machine","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"Let $T$ be a formal theory and $M$ a Busy Beaver machine with $n$ working states. Suppose that $M$ halts if and only if $T$ proves the false sentence, that $M$ halts whenever $T$ proves the halting sentence of $M$, that $T$ is consistent, and that $T$ does not prove the non-halting sentence of $M$. Then $T$ does not decide halting for all machines with $n$ working states.","title":"A consistency-search machine blocks decidability at its size"},{"id":"TheoryOfComputation/computability/busy-beaver/decides-halting-of-succ","kind":"node","lean":"Cslib.BusyBeaver.decidesHalting_of_succ","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#decides-halting-of-succ","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"Let $T$ be a formal theory such that, for every $m$ and every Busy Beaver machine $M$ with $m$ working states, if $T$ proves the halting sentence of the embedding of $M$ into $m + 1$ states then $T$ proves the halting sentence of $M$, and if $T$ proves the non-halting sentence of the embedding of $M$ into $m + 1$ states then $T$ proves the non-halting sentence of $M$. Then for every $n$, if $T$ decides halting for\u2026","title":"Decidability of halting passes to fewer states"},{"id":"TheoryOfComputation/computability/busy-beaver/zfcpresentation","kind":"node","lean":"Cslib.BusyBeaver.ZFCPresentation","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#zfcpresentation","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"A ZFC-like presentation consists of a formal theory $T$, given by a provability predicate on sentence codes in $\\mathbb{N}$, together with a Boolean proof checker $\\mathrm{check}(p, s)$ on pairs of natural numbers, such that the following hold. A sentence code $s$ is provable in $T$ if and only if $\\mathrm{check}(p, s)$ is true for some $p \\in \\mathbb{N}$. $T$ is sound for halting: for every machine $M$, if $T$\u2026","title":"Proof-checker presentation of a ZFC-like theory"},{"id":"TheoryOfComputation/computability/busy-beaver/decides-halting-le-five","kind":"node","lean":"Cslib.BusyBeaver.ZFCPresentation.decidesHalting_le_five","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#decides-halting-le-five","scope":"TheoryOfComputation/computability/busy-beaver","status":"fully_proved","summary":"For every ZFC-like presentation $Z$ and every $n \\le 5$, the theory presented by $Z$ decides halting for $n$-state machines: for every $n$-state machine it proves the machine's halting sentence or its non-halting sentence.","title":"ZFC-like theories decide halting through five states"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-steps-three-wanted","kind":"node","lean":"Cslib.BusyBeaverWanted.busyBeaverSteps_three_wanted","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-steps-three-wanted","scope":"TheoryOfComputation/computability/busy-beaver","status":"can_prove","summary":"Rado's shift function $S$ satisfies $S(3) = 21$: among the two-symbol Busy Beaver machines with $3$ working states that halt when started on the blank tape, the largest halting time is $21$.","title":"Three-state Busy Beaver shift value"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-ones-three-wanted","kind":"node","lean":"Cslib.BusyBeaverWanted.busyBeaverOnes_three_wanted","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-ones-three-wanted","scope":"TheoryOfComputation/computability/busy-beaver","status":"can_prove","summary":"Rado's score function $\\Sigma$ satisfies $\\Sigma(3) = 6$: among the two-symbol Busy Beaver machines with $3$ working states that halt when started on the blank tape, the largest number of ones left on the tape at the halting time is $6$.","title":"Three-state Busy Beaver score value"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-steps-four-wanted","kind":"node","lean":"Cslib.BusyBeaverWanted.busyBeaverSteps_four_wanted","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-steps-four-wanted","scope":"TheoryOfComputation/computability/busy-beaver","status":"can_prove","summary":"Rado's shift function $S$ satisfies $S(4) = 107$: among the two-symbol Busy Beaver machines with $4$ working states that halt when started on the blank tape, the largest halting time is $107$.","title":"Four-state Busy Beaver shift value"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-ones-four-wanted","kind":"node","lean":"Cslib.BusyBeaverWanted.busyBeaverOnes_four_wanted","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-ones-four-wanted","scope":"TheoryOfComputation/computability/busy-beaver","status":"can_prove","summary":"Rado's score function $\\Sigma$ satisfies $\\Sigma(4) = 13$: among the two-symbol Busy Beaver machines with $4$ working states that halt when started on the blank tape, the largest number of ones left on the tape at the halting time is $13$.","title":"Four-state Busy Beaver score value"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-steps-five-wanted","kind":"node","lean":"Cslib.BusyBeaverWanted.busyBeaverSteps_five_wanted","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-steps-five-wanted","scope":"TheoryOfComputation/computability/busy-beaver","status":"can_prove","summary":"Rado's shift function $S$ satisfies $S(5) = 47176870$: among the two-symbol Busy Beaver machines with $5$ working states that halt when started on the blank tape, the largest halting time is $47176870$.","title":"Five-state Busy Beaver shift value"},{"id":"TheoryOfComputation/computability/busy-beaver/busy-beaver-ones-five-wanted","kind":"node","lean":"Cslib.BusyBeaverWanted.busyBeaverOnes_five_wanted","parent":34966,"result":true,"route":"../roadmap/TheoryOfComputation/computability/busy-beaver/index.html#busy-beaver-ones-five-wanted","scope":"TheoryOfComputation/computability/busy-beaver","status":"can_prove","summary":"Rado's score function $\\Sigma$ satisfies $\\Sigma(5) = 4098$: among the two-symbol Busy Beaver machines with $5$ working states that halt when started on the blank tape, the largest number of ones left on the tape at the halting time is $4098$.","title":"Five-state Busy Beaver score value"},{"data_url":"scopes/TheoryOfComputation/computability/decidable-and-recognizable-predicates.json","id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","title":"Decidable and recognizable predicates"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/and","kind":"node","lean":"MetaMathlibExt.ComputablePred.and","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#and","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$. If $p$ and $q$ are computable predicates on $\\alpha$, then the predicate $a \\mapsto p(a) \\land q(a)$ is computable.","title":"Computable predicates are closed under conjunction"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/or","kind":"node","lean":"MetaMathlibExt.ComputablePred.or","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#or","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$. If $p$ and $q$ are computable predicates on $\\alpha$, then the predicate $a \\mapsto p(a) \\lor q(a)$ is computable.","title":"Computable predicates are closed under disjunction"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/and-2","kind":"node","lean":"MetaMathlibExt.REPred.and","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#and-2","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$. If $p$ and $q$ are recursively enumerable predicates on $\\alpha$, then the predicate $a \\mapsto p(a) \\land q(a)$ is recursively enumerable.","title":"Recursively enumerable predicates are closed under conjunction"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/or-2","kind":"node","lean":"MetaMathlibExt.REPred.or","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#or-2","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$. If $p$ and $q$ are recursively enumerable predicates on $\\alpha$, then the predicate $a \\mapsto p(a) \\lor q(a)$ is recursively enumerable.","title":"Recursively enumerable predicates are closed under disjunction"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/exists-mem","kind":"node","lean":"MetaMathlibExt.REPred.exists_mem","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#exists-mem","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types with primitive recursive encodings into $\\mathbb{N}$, let $R$ be a relation between $\\alpha$ and $\\beta$ that is recursively enumerable as a predicate on pairs $(a, b)$, and let $c$ be a primitive recursive function sending each $a \\in \\alpha$ to a finite list $c(a)$ of elements of $\\beta$. Then the predicate $a \\mapsto \\exists b \\in c(a),\\ R(a, b)$ is recursively enumerable.","title":"Bounded existential quantification preserves recursive enumerability"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/forall-mem","kind":"node","lean":"MetaMathlibExt.REPred.forall_mem","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#forall-mem","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types with primitive recursive encodings into $\\mathbb{N}$, let $R$ be a relation between $\\alpha$ and $\\beta$ that is recursively enumerable as a predicate on pairs $(a, b)$, and let $c$ be a primitive recursive function sending each $a \\in \\alpha$ to a finite list $c(a)$ of elements of $\\beta$. Then the predicate $a \\mapsto \\forall b \\in c(a),\\ R(a, b)$ is recursively enumerable.","title":"Bounded universal quantification preserves recursive enumerability"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/computable-list-any","kind":"node","lean":"MetaMathlibExt.computable_list_any","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#computable-list-any","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\gamma$ and $\\beta$ be types with primitive recursive encodings into $\\mathbb{N}$, let $l$ be a computable function sending each $x \\in \\gamma$ to a finite list $l(x)$ of elements of $\\beta$, and let $g$ be a computable Boolean-valued function of $x \\in \\gamma$ and $b \\in \\beta$. Then the Boolean-valued function that sends $x$ to true exactly when $g(x, b)$ is true for some $b$ in $l(x)$ is computable.","title":"Searching a computable list with a computable test is computable"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/exists-partrec-choose","kind":"node","lean":"MetaMathlibExt.REPred.exists_partrec_choose","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#exists-partrec-choose","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types with primitive recursive encodings into $\\mathbb{N}$, and let $R$ be a relation between $\\alpha$ and $\\beta$ that is recursively enumerable as a predicate on pairs $(a, b)$. Then there is a partial recursive function $g$ from $\\alpha$ to $\\beta$ such that, for every $a \\in \\alpha$, if $g(a)$ is defined then its value $b$ satisfies $R(a, b)$, and $g(a)$ is defined whenever some\u2026","title":"Selection for r.e. relations"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/exists","kind":"node","lean":"MetaMathlibExt.REPred.exists","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#exists","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types with primitive recursive encodings into $\\mathbb{N}$, and let $R$ be a relation between $\\alpha$ and $\\beta$ that is recursively enumerable as a predicate on pairs $(a, b)$. Then the predicate $a \\mapsto \\exists b \\in \\beta,\\ R(a, b)$ is recursively enumerable.","title":"Existential projection preserves recursive enumerability"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-decidable","kind":"node","lean":"Cslib.Decidability.TM2Decidable","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-decidable","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is decidable by a multi-stack machine when there is a function $f : \\mathbb{N} \\to \\mathbb{N}$ computed by a multi-stack Turing machine, with input and output numbers written in binary, such that for every $a \\in \\alpha$, $f(\\mathrm{encode}(a)) = 1$ if\u2026","title":"Decidability by multi-stack machines"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-recognizable","kind":"node","lean":"Cslib.Decidability.TM2Recognizable","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-recognizable","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is recognizable by a multi-stack machine when there are a multi-stack Turing machine $M$ and an identification of its input alphabet with the Booleans such that for every $a \\in \\alpha$, $p(a)$ holds if and only if $M$ halts when started on the binary\u2026","title":"Recognizability by multi-stack machines"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-co-recognizable","kind":"node","lean":"Cslib.Decidability.TM2CoRecognizable","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-co-recognizable","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is co-recognizable by a multi-stack machine when its complement $\\neg p$ is recognizable by a multi-stack machine.","title":"Co-recognizability by multi-stack machines"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-decidable-iff-computable-pred","kind":"node","lean":"Cslib.Decidability.tm2Decidable_iff_computablePred","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-decidable-iff-computable-pred","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is decidable by a multi-stack machine if and only if it is computable, that is, $p$ is decidable and its Boolean characteristic function $\\alpha \\to \\mathrm{Bool}$ is computable.","title":"Multi-stack decidable predicates are exactly the computable predicates"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-recognizable-iff-re-pred","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_iff_rePred","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-recognizable-iff-re-pred","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is recognizable by a multi-stack machine if and only if it is recursively enumerable, that is, the partial function on $\\alpha$ that is defined, with a trivial value, exactly where $p$ holds is partial recursive.","title":"Multi-stack recognizable predicates are exactly the r.e. predicates"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-co-recognizable-iff-re-pred-not","kind":"node","lean":"Cslib.Decidability.tm2CoRecognizable_iff_rePred_not","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-co-recognizable-iff-re-pred-not","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is co-recognizable by a multi-stack machine if and only if its complement $\\neg p$ is recursively enumerable, that is, the partial function on $\\alpha$ that is defined, with a trivial value, exactly where $\\neg p$ holds is partial recursive.","title":"Multi-stack co-recognizable predicates are exactly the co-r.e. predicates"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-recognizable-2","kind":"node","lean":"Cslib.Decidability.TM2Decidable.tm2Recognizable","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-recognizable-2","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a multi-stack machine, then $p$ is recognizable by a multi-stack machine.","title":"Decidable predicates are recognizable by multi-stack machines"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/not","kind":"node","lean":"Cslib.Decidability.TM2Decidable.not","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#not","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a multi-stack machine, then so is its complement $\\neg p$.","title":"Decidability by multi-stack machines is closed under complement"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-co-recognizable-2","kind":"node","lean":"Cslib.Decidability.TM2Decidable.tm2CoRecognizable","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-co-recognizable-2","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). If a predicate $p$ on $\\alpha$ is decidable by a multi-stack machine, then $p$ is co-recognizable by a multi-stack machine, that is, its complement $\\neg p$ is recognizable by a multi-stack machine.","title":"Decidable predicates are co-recognizable by multi-stack machines"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/tm2-decidable-iff-tm2-recognizable-and-t","kind":"node","lean":"Cslib.Decidability.tm2Decidable_iff_tm2Recognizable_and_tm2CoRecognizable","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#tm2-decidable-iff-tm2-recognizable-and-t","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a primcodable type (it has an encoding $\\mathrm{encode} : \\alpha \\to \\mathbb{N}$ whose decoding is primitive recursive). A predicate $p$ on $\\alpha$ is decidable by a multi-stack machine if and only if it is both recognizable and co-recognizable by a multi-stack machine.","title":"Post's theorem for multi-stack machines"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/and-3","kind":"node","lean":"Cslib.Decidability.TM2Decidable.and","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#and-3","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $p$ and $q$ be predicates on a type $\\alpha$ with a primitive recursive encoding. If $p$ and $q$ are both decidable by multi-stack machines, then so is the predicate $a \\mapsto p(a) \\wedge q(a)$.","title":"Decidable predicates are closed under conjunction"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/or-3","kind":"node","lean":"Cslib.Decidability.TM2Decidable.or","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#or-3","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding. If $p$ and $q$ are predicates on $\\alpha$ that are decidable by a multi-stack Turing machine, then so is the predicate $a \\mapsto p(a) \\lor q(a)$.","title":"Decidable predicates are closed under disjunction"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/and-4","kind":"node","lean":"Cslib.Decidability.TM2Recognizable.and","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#and-4","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $p$ and $q$ be predicates on a type $\\alpha$ with a primitive recursive encoding. If $p$ and $q$ are both recognizable by multi-stack machines, then so is the predicate $a \\mapsto p(a) \\wedge q(a)$.","title":"Recognizable predicates are closed under conjunction"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/or-4","kind":"node","lean":"Cslib.Decidability.TM2Recognizable.or","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#or-4","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding. If $p$ and $q$ are predicates on $\\alpha$ that are recognizable by a multi-stack Turing machine, then so is the predicate $a \\mapsto p(a) \\lor q(a)$.","title":"Recognizable predicates are closed under disjunction"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/bool-lang","kind":"node","lean":"Cslib.Decidability.boolLang","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#bool-lang","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"For a Boolean-valued function $f$ on words over an alphabet $\\sigma$, the language of $f$ is the set of words $u$ over $\\sigma$ with $f(u) = \\mathrm{true}$.","title":"Language of a Boolean test"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/mem-mul-iff-exists-append","kind":"node","lean":"Cslib.Decidability.mem_mul_iff_exists_append","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#mem-mul-iff-exists-append","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"For languages $L_1$ and $L_2$ over an alphabet $\\sigma$ and a word $w$ over $\\sigma$, $w$ lies in the concatenation $L_1 L_2$ if and only if there are words $u$ and $v$ with $w = uv$, $u \\in L_1$ and $v \\in L_2$.","title":"Membership in a concatenation of languages"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/split-test","kind":"node","lean":"Cslib.Decidability.splitTest","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#split-test","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Given Boolean tests $f$ and $g$ on words over $\\sigma$, a word $w$ and a natural number $n$, the split test is the Boolean value that is true exactly when there is some $i < n$ such that $f$ is true on the prefix of $w$ of length $\\min(i, |w|)$ and $g$ is true on the remaining suffix of $w$.","title":"Bounded split test"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/exists-computable","kind":"node","lean":"Cslib.Decidability.TM2Decidable.exists_computable","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#exists-computable","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\sigma$ with a primitive recursive encoding. If membership in $L$ is decidable by a multi-stack machine, then there is a computable function $f$ from words over $\\sigma$ to Booleans such that $L$ is the language of $f$, that is, the set of words $u$ with $f(u) = \\mathrm{true}$.","title":"Decidable languages have computable membership tests"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/mul","kind":"node","lean":"Cslib.Decidability.TM2Decidable.mul","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#mul","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\sigma$ be an alphabet with a primitive recursive encoding. If $L_1$ and $L_2$ are languages over $\\sigma$ whose membership predicates are decidable by a multi-stack Turing machine, then the membership predicate of their concatenation $L_1 L_2 = \\{uv : u \\in L_1, v \\in L_2\\}$ is also decidable by a multi-stack Turing machine.","title":"Decidable languages are closed under concatenation"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/splits","kind":"node","lean":"Cslib.Decidability.splits","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#splits","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"For a word $w$ over $\\sigma$ of length $n$, this is the list of the $n + 1$ pairs $(u, v)$ of words with $uv = w$, ordered by the length of $u$: for each $i = 0, 1, \\dots, n$ it contains the prefix of $w$ of length $i$ paired with the remaining suffix.","title":"Splits of a word"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/mul-2","kind":"node","lean":"Cslib.Decidability.TM2Recognizable.mul","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#mul-2","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $L_1$ and $L_2$ be languages over an alphabet $\\sigma$ with a primitive recursive encoding. If membership in $L_1$ and membership in $L_2$ are recognizable by multi-stack machines, then so is membership in the concatenation $L_1 L_2$, the set of words $uv$ with $u \\in L_1$ and $v \\in L_2$.","title":"Recognizable languages are closed under concatenation"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/extend-partitions","kind":"node","lean":"Cslib.Decidability.extendPartitions","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#extend-partitions","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Given a letter $a$ and a list $\\pi = [u_1, \\dots, u_k]$ of words, the extensions of $\\pi$ by $a$ are listed as follows: first $[[a], u_1, \\dots, u_k]$, which starts a new piece $[a]$, and then, only when $k \\ge 1$, also $[a u_1, u_2, \\dots, u_k]$, which puts the letter $a$ in front of the first piece.","title":"Extending a list of pieces by a letter"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/partitions","kind":"node","lean":"Cslib.Decidability.partitions","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#partitions","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"For a word $w$ over $\\sigma$, this is the list of all ways to write $w$ as a concatenation $w = u_1 u_2 \\cdots u_k$ of nonempty words $u_1, \\dots, u_k$, each recorded as the list $[u_1, \\dots, u_k]$. The empty word has exactly one such partition, the empty list.","title":"Partitions of a word into nonempty factors"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/kstar","kind":"node","lean":"Cslib.Decidability.TM2Recognizable.kstar","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#kstar","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\sigma$ with a primitive recursive encoding. If membership in $L$ is recognizable by a multi-stack machine, then so is membership in the Kleene star $L^*$, the set of all concatenations $u_1 \\cdots u_k$ with $k \\ge 0$ and $u_1, \\dots, u_k \\in L$.","title":"Recognizable languages are closed under Kleene star"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/kstar-2","kind":"node","lean":"Cslib.Decidability.TM2Decidable.kstar","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#kstar-2","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $\\sigma$ with a primitive recursive encoding. If membership in $L$ is decidable by a multi-stack machine, then so is membership in the Kleene star $L^*$, the set of all concatenations $u_1 \\cdots u_k$ with $k \\ge 0$ and $u_1, \\dots, u_k \\in L$.","title":"Decidable languages are closed under Kleene star"},{"id":"TheoryOfComputation/computability/decidable-and-recognizable-predicates/map","kind":"node","lean":"Cslib.Decidability.TM2Recognizable.map","parent":35017,"result":true,"route":"../roadmap/TheoryOfComputation/computability/decidable-and-recognizable-predicates/index.html#map","scope":"TheoryOfComputation/computability/decidable-and-recognizable-predicates","status":"fully_proved","summary":"Let $\\sigma$ and $\\tau$ be alphabets with primitive recursive encodings, let $f : \\sigma \\to \\tau$ be primitive recursive, and let $L$ be a language over $\\sigma$. If membership in $L$ is recognizable by a multi-stack machine, then so is membership in the image language $f(L) = \\{f(a_1) \\cdots f(a_n) : a_1 \\cdots a_n \\in L\\}$, obtained by applying $f$ letter by letter to the words of $L$.","title":"Recognizable languages are closed under alphabet maps"},{"data_url":"scopes/TheoryOfComputation/computability/enumerators.json","id":"TheoryOfComputation/computability/enumerators","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/enumerators/index.html","scope":"TheoryOfComputation/computability/enumerators","status":"fully_proved","title":"Enumerators"},{"id":"TheoryOfComputation/computability/enumerators/enumerates","kind":"node","lean":"Cslib.Decidability.Enumerates","parent":35053,"result":true,"route":"../roadmap/TheoryOfComputation/computability/enumerators/index.html#enumerates","scope":"TheoryOfComputation/computability/enumerators","status":"fully_proved","summary":"For a type $\\alpha$ with a primitive recursive encoding, a function $e$ assigning to each natural number $n$ either an element $e(n)$ of $\\alpha$ or nothing enumerates a predicate $p$ on $\\alpha$ when, for every $a \\in \\alpha$, $p(a)$ holds if and only if $e(n) = a$ for some natural number $n$.","title":"Enumerating a predicate"},{"id":"TheoryOfComputation/computability/enumerators/exists-enumerator","kind":"node","lean":"Cslib.Decidability.TM2Recognizable.exists_enumerator","parent":35053,"result":true,"route":"../roadmap/TheoryOfComputation/computability/enumerators/index.html#exists-enumerator","scope":"TheoryOfComputation/computability/enumerators","status":"fully_proved","summary":"For every type $\\alpha$ with a primitive recursive encoding and every predicate $p$ on $\\alpha$ that is recognizable by a multi-stack machine, there is a computable function $e$, assigning to each natural number either an element of $\\alpha$ or nothing, that enumerates $p$.","title":"Recognizable languages are enumerable"},{"id":"TheoryOfComputation/computability/enumerators/tm2-recognizable-of-enumerates","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_of_enumerates","parent":35053,"result":true,"route":"../roadmap/TheoryOfComputation/computability/enumerators/index.html#tm2-recognizable-of-enumerates","scope":"TheoryOfComputation/computability/enumerators","status":"fully_proved","summary":"For every type $\\alpha$ with a primitive recursive encoding, every predicate $p$ on $\\alpha$ and every computable function $e$, assigning to each natural number either an element of $\\alpha$ or nothing, that enumerates $p$, the predicate $p$ is recognizable by a multi-stack machine.","title":"Enumerable languages are recognizable"},{"id":"TheoryOfComputation/computability/enumerators/tm2-recognizable-iff-exists-enumerator","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_iff_exists_enumerator","parent":35053,"result":true,"route":"../roadmap/TheoryOfComputation/computability/enumerators/index.html#tm2-recognizable-iff-exists-enumerator","scope":"TheoryOfComputation/computability/enumerators","status":"fully_proved","summary":"For every type $\\alpha$ with a primitive recursive encoding, a predicate $p$ on $\\alpha$ is recognizable by a multi-stack machine if and only if some computable function $e$, assigning to each natural number either an element of $\\alpha$ or nothing, enumerates $p$. This is Sipser, Theorem 3.21: a language is Turing-recognizable if and only if some enumerator enumerates it.","title":"Recognizable languages are exactly the enumerable ones"},{"data_url":"scopes/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem.json","id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","title":"Halting problem and Rice's theorem"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/not-tm2-decidable-halting","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_halting","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#not-tm2-decidable-halting","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, no multi-stack machine decides whether a partial recursive code halts on input $n$.","title":"The halting problem is undecidable by multi-stack machines"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/tm2-recognizable-halting","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_halting","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#tm2-recognizable-halting","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the set of partial recursive codes whose program halts on input $n$ is recognizable by a multi-stack machine.","title":"The halting problem is recognizable by multi-stack machines"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/not-tm2-co-recognizable-halting","kind":"node","lean":"Cslib.Decidability.not_tm2CoRecognizable_halting","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#not-tm2-co-recognizable-halting","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the set of partial recursive codes whose program halts on input $n$ is not co-recognizable by a multi-stack machine: no multi-stack machine halts on exactly the codes whose program does not halt on input $n$.","title":"The halting problem is not co-recognizable by multi-stack machines"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/exists-not-tm2-decidable-tm2-halts-on","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_tm2HaltsOn","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#exists-not-tm2-decidable-tm2-halts-on","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"There is a multi-stack machine $M$, with a bijection between its input alphabet and the Booleans, such that no multi-stack machine decides, for $n \\in \\mathbb{N}$, whether $M$ halts when started on the binary representation of $n$ (written in $M$'s input alphabet through the bijection).","title":"Some multi-stack machine has an undecidable halting set"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/rice-tm2","kind":"node","lean":"Cslib.Decidability.rice_tm2","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#rice-tm2","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"Let $C$ be a set of partial functions $\\mathbb{N} \\to \\mathbb{N}$. If a multi-stack machine decides, for each partial recursive code, whether the function the code computes lies in $C$, then for all partial recursive functions $f$ and $g$, $f \\in C$ implies $g \\in C$.","title":"Rice's theorem for multi-stack machines"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/many-one-reducible-iff-exists-tm2","kind":"node","lean":"Cslib.Decidability.manyOneReducible_iff_exists_tm2","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#many-one-reducible-iff-exists-tm2","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"Let $p$ and $q$ be predicates on primitive recursively encodable types $\\alpha$ and $\\beta$, and write $\\langle x \\rangle \\in \\mathbb{N}$ for the code of $x$. Then $p$ is many-one reducible to $q$ if and only if there is a function $f : \\alpha \\to \\beta$ with $p(a) \\iff q(f(a))$ for all $a \\in \\alpha$, together with a function $F : \\mathbb{N} \\to \\mathbb{N}$ that a multi-stack machine computes on binary\u2026","title":"Mapping reductions are computed by multi-stack machines"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/tm2-decidable-of-many-one-reducible","kind":"node","lean":"Cslib.Decidability.tm2Decidable_of_manyOneReducible","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#tm2-decidable-of-many-one-reducible","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"Let $p$ and $q$ be predicates on primitive recursively encodable types $\\alpha$ and $\\beta$. If $p$ is many-one reducible to $q$ and $q$ is decidable by a multi-stack machine, then $p$ is decidable by a multi-stack machine.","title":"Decidability transfers along mapping reductions"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/tm2-recognizable-of-many-one-reducible","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_of_manyOneReducible","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#tm2-recognizable-of-many-one-reducible","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"Let $p$ and $q$ be predicates on primitive recursively encodable types $\\alpha$ and $\\beta$. If $p$ is many-one reducible to $q$ and $q$ is recognizable by a multi-stack machine, then $p$ is recognizable by a multi-stack machine.","title":"Recognizability transfers along mapping reductions"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/not-tm2-decidable-of-nontrivial","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_of_nontrivial","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#not-tm2-decidable-of-nontrivial","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"Let $C$ be a set of partial functions $\\mathbb{N} \\rightharpoonup \\mathbb{N}$, and suppose there are partial recursive functions $f \\in C$ and $g \\notin C$. Then the property of a partial recursive code $c$ that the partial function computed by $c$ belongs to $C$ is not decidable by a multi-stack Turing machine.","title":"Rice's theorem for partial recursive codes"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/not-tm2-decidable-language","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_language","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#not-tm2-decidable-language","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"Let $P$ be a set of subsets of $\\mathbb{N}$, and suppose there are partial recursive codes $c_1$ and $c_2$ such that the domain of $c_1$ (the set of inputs on which it halts) belongs to $P$ and the domain of $c_2$ does not. Then the property of a partial recursive code $c$ that the domain of $c$ belongs to $P$ is not decidable by a multi-stack Turing machine.","title":"Rice's theorem for languages"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/accepts","kind":"node","lean":"Cslib.Decidability.Accepts","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#accepts","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"A partial recursive code $c$ accepts a natural number $n$ when $c$ halts on input $n$ with output $0$. Acceptance is a predicate on pairs $(c, n)$; it is the analogue of Sipser's $A_{\\mathrm{TM}}$, where a machine may halt without accepting.","title":"Acceptance by a code"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/halts","kind":"node","lean":"Cslib.Decidability.Halts","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#halts","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"A partial recursive code $c$ halts on a natural number $n$ when its evaluation at $n$ is defined. Halting is a predicate on pairs $(c, n)$; it is the analogue of Sipser's $\\mathrm{HALT}_{\\mathrm{TM}}$.","title":"Halting of a code on an input"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/exists-code-zero-or-loop","kind":"node","lean":"Cslib.Decidability.exists_code_zero_or_loop","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#exists-code-zero-or-loop","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"There is a partial recursive code $z$ that on input $0$ halts with output $0$ and on every input $x \\ne 0$ does not halt.","title":"A code that halts only on zero"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/many-one-reducible-accepts-halts","kind":"node","lean":"Cslib.Decidability.manyOneReducible_accepts_halts","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#many-one-reducible-accepts-halts","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"Say a partial recursive code $c$ accepts a natural number $n$ when $c$ halts on input $n$ with output $0$. Acceptance many-one reduces to halting: there is a computable map $f$ from pairs $(c, n)$ of a partial recursive code and a natural number to such pairs, such that for every pair $(c, n)$, the code $c$ accepts $n$ if and only if, writing $f(c, n) = (c', n')$, the code $c'$ halts on $n'$.","title":"Acceptance reduces to halting"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/many-one-reducible-halts-accepts","kind":"node","lean":"Cslib.Decidability.manyOneReducible_halts_accepts","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#many-one-reducible-halts-accepts","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"Say a partial recursive code $c$ accepts a natural number $n$ when $c$ halts on input $n$ with output $0$. Halting many-one reduces to acceptance: there is a computable map $f$ from pairs $(c, n)$ of a partial recursive code and a natural number to such pairs, such that for every pair $(c, n)$, the code $c$ halts on $n$ if and only if, writing $f(c, n) = (c', n')$, the code $c'$ accepts $n'$.","title":"Halting reduces to acceptance"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/not-tm2-decidable-halts","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_halts","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#not-tm2-decidable-halts","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"No multi-stack Turing machine decides, for pairs $(c, n)$ of a partial recursive code and a natural number, whether $c$ halts on $n$.","title":"Halting is undecidable"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/not-tm2-decidable-accepts","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_accepts","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#not-tm2-decidable-accepts","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"No multi-stack Turing machine decides, for pairs $(c, n)$ of a partial recursive code and a natural number, whether $c$ accepts $n$, that is, halts on $n$ with output $0$.","title":"Acceptance is undecidable"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/unary-lang","kind":"node","lean":"Cslib.Decidability.unaryLang","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#unary-lang","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"For a set $S \\subseteq \\mathbb{N}$, the unary language of $S$ is the language over the one-letter alphabet $\\{\\ast\\}$ consisting of the words whose length lies in $S$.","title":"Unary language of a set of naturals"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/regular-domain-code","kind":"node","lean":"Cslib.Decidability.RegularDomainCode","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#regular-domain-code","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"A partial recursive code $c$ has a regular halting set when the unary language of its domain (the set of inputs on which $c$ halts), that is, the language of the words over a one-letter alphabet whose length lies in the domain of $c$, is regular.","title":"Codes with a regular halting set"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/not-is-regular-unary-lang-squares","kind":"node","lean":"Cslib.Decidability.not_isRegular_unaryLang_squares","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#not-is-regular-unary-lang-squares","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"The unary language of the perfect squares, the set of words over a one-letter alphabet whose length is $m^2$ for some $m \\in \\mathbb{N}$, is not regular.","title":"The unary squares are not regular"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/regular-red-fun","kind":"node","lean":"Cslib.Decidability.regularRedFun","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#regular-red-fun","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"For a partial recursive code $c$ and natural numbers $n$ and $x$, the value at $((c, n), x)$ is $0$ if $x$ is a perfect square; otherwise it is defined exactly when $c$ halts on input $n$, and then it is $0$.","title":"Partial function of the halting-to-regularity reduction"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/exists-code-regular-red-fun","kind":"node","lean":"Cslib.Decidability.exists_code_regularRedFun","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#exists-code-regular-red-fun","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"For a pair $q = (c, n)$ of a partial recursive code and a natural number, let $g_q$ be the partial function on $\\mathbb{N}$ that on input $x$ halts with output $0$ if $x$ is a perfect square, and otherwise runs $c$ on $n$ and, if that halts, outputs $0$ (and is undefined if it does not). Write $\\ulcorner q \\urcorner$ for the standard encoding of $q$ as a natural number, and $\\langle a, b \\rangle$ for the standard\u2026","title":"A single code for the regularity reduction function"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/many-one-reducible-halts-regular-domain","kind":"node","lean":"Cslib.Decidability.manyOneReducible_halts_regularDomain","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#many-one-reducible-halts-regular-domain","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"There is a computable map $f$ from pairs $(c, n)$ of a partial recursive code and a natural number to partial recursive codes such that for every pair $(c, n)$, the code $c$ halts on $n$ if and only if the halting set of $f(c, n)$ (the set of inputs on which it halts), read as a unary language, is regular. That is, halting many-one reduces to regularity of halting sets.","title":"Halting reduces to regularity of halting sets"},{"id":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/not-tm2-decidable-regular-domain","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_regularDomain","parent":35058,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-and-rice-s-theorem/index.html#not-tm2-decidable-regular-domain","scope":"TheoryOfComputation/computability/halting-problem-and-rice-s-theorem","status":"fully_proved","summary":"For a partial recursive code $c$, let $\\operatorname{dom}(c) \\subseteq \\mathbb{N}$ be the set of inputs on which $c$ halts. The property of $c$ that the unary language $\\{w \\in \\{\\ast\\}^* : |w| \\in \\operatorname{dom}(c)\\}$ is regular is not decidable by a multi-stack Turing machine.","title":"Regularity of halting sets is undecidable"},{"data_url":"scopes/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs.json","id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","title":"Emptiness and equivalence problems for programs"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/empty-code","kind":"node","lean":"Cslib.Decidability.EmptyCode","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#empty-code","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set $E$ consists of the codes $c$ of partial recursive functions on $\\mathbb{N}$ that are defined on no input: for every $n \\in \\mathbb{N}$, the function computed by $c$ is undefined at $n$.","title":"Codes with empty domain"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/not-tm2-decidable-empty","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_empty","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#not-tm2-decidable-empty","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set $E$ of codes of partial recursive functions on $\\mathbb{N}$ that are defined on no input is not decidable by multi-stack machines.","title":"Emptiness is undecidable"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/tm2-recognizable-not-empty","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_not_empty","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#tm2-recognizable-not-empty","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set of codes of partial recursive functions $\\mathbb{N} \\to \\mathbb{N}$ that are defined on at least one input is recognizable by multi-stack machines: some multi-stack machine halts on the binary encoding of a code $c$ exactly when the function coded by $c$ is defined on some input.","title":"Nonemptiness is recognizable"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/tm2-co-recognizable-empty","kind":"node","lean":"Cslib.Decidability.tm2CoRecognizable_empty","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#tm2-co-recognizable-empty","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set of codes of partial recursive functions $\\mathbb{N} \\to \\mathbb{N}$ that are defined on no input is co-recognizable by multi-stack machines: some multi-stack machine halts on the binary encoding of a code $c$ exactly when the function coded by $c$ is defined on at least one input.","title":"Emptiness is co-recognizable"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/not-tm2-recognizable-empty","kind":"node","lean":"Cslib.Decidability.not_tm2Recognizable_empty","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#not-tm2-recognizable-empty","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set $E$ of codes of partial recursive functions on $\\mathbb{N}$ that are defined on no input is not recognizable by multi-stack machines: no multi-stack machine, started on the binary encoding of a code, halts exactly when the code is in $E$.","title":"Emptiness is not recognizable"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/eq-code","kind":"node","lean":"Cslib.Decidability.EqCode","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#eq-code","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set $EQ$ consists of the pairs $(c_1, c_2)$ of codes of partial recursive functions on $\\mathbb{N}$ that recognize the same language, that is, whose computed functions have the same domain: for every $n \\in \\mathbb{N}$, the function computed by $c_1$ is defined at $n$ if and only if the function computed by $c_2$ is.","title":"Equivalent codes"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/exists-code-empty","kind":"node","lean":"Cslib.Decidability.exists_code_empty","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#exists-code-empty","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"There is a code $e$ of a partial recursive function on $\\mathbb{N}$ whose computed function is the empty partial function, undefined at every input.","title":"A code defined nowhere"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/many-one-reducible-empty-eq","kind":"node","lean":"Cslib.Decidability.manyOneReducible_empty_eq","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#many-one-reducible-empty-eq","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set $E$ of codes of partial recursive functions defined on no input is many-one reducible to the set $EQ$ of pairs of codes with the same domain: there is a computable map $f$ from codes to pairs of codes such that, for every code $c$, $c \\in E$ if and only if $f(c) \\in EQ$.","title":"Emptiness reduces to equivalence"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/not-tm2-recognizable-eq","kind":"node","lean":"Cslib.Decidability.not_tm2Recognizable_eq","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#not-tm2-recognizable-eq","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set $EQ$ of pairs of codes of partial recursive functions on $\\mathbb{N}$ with the same domain is not recognizable by multi-stack machines: no multi-stack machine, started on the binary encoding of a pair of codes, halts exactly when the pair is in $EQ$.","title":"Equivalence is not recognizable"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/halts-on-zero-code","kind":"node","lean":"Cslib.Decidability.haltsOnZeroCode","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#halts-on-zero-code","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"For a code $c$ of a partial recursive function on $\\mathbb{N}$, the code $h(c)$ is the composition $k_0 \\circ c \\circ k_0$, where $k_0$ is the code of the constant function $0$: whatever its input, $h(c)$ runs $c$ on $0$ and, if this halts, outputs $0$.","title":"Halting-on-zero code"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/many-one-reducible-halting-eq","kind":"node","lean":"Cslib.Decidability.manyOneReducible_halting_eq","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#many-one-reducible-halting-eq","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set of codes $c$ of partial recursive functions on $\\mathbb{N}$ that halt on input $0$ (are defined at $0$) is many-one reducible to the set $EQ$ of pairs of codes with the same domain: there is a computable map $f$ from codes to pairs of codes such that, for every code $c$, $c$ halts on $0$ if and only if $f(c) \\in EQ$.","title":"Halting on zero reduces to equivalence"},{"id":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/not-tm2-co-recognizable-eq","kind":"node","lean":"Cslib.Decidability.not_tm2CoRecognizable_eq","parent":35083,"result":true,"route":"../roadmap/TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs/index.html#not-tm2-co-recognizable-eq","scope":"TheoryOfComputation/computability/emptiness-and-equivalence-problems-for-programs","status":"fully_proved","summary":"The set $EQ$ of pairs of codes of partial recursive functions on $\\mathbb{N}$ with the same domain is not co-recognizable by multi-stack machines: no multi-stack machine, started on the binary encoding of a pair of codes, halts exactly when the pair is not in $EQ$.","title":"Equivalence is not co-recognizable"},{"data_url":"scopes/TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines.json","id":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/index.html","scope":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines","status":"fully_proved","title":"Halting problem for single-tape Turing machines"},{"id":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/tm0-halts-on","kind":"node","lean":"Cslib.Decidability.TM0HaltsOn","parent":35096,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/index.html#tm0-halts-on","scope":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines","status":"fully_proved","summary":"A single-tape Turing machine $M$ halts on an input word $l$ when its run, started in its start state with $l$ on the tape, reaches a halting configuration.","title":"Halting of a single-tape machine"},{"id":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/tm2-halts-on-iff-eval-dom","kind":"node","lean":"Cslib.Decidability.tm2HaltsOn_iff_eval_dom","parent":35096,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/index.html#tm2-halts-on-iff-eval-dom","scope":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines","status":"fully_proved","summary":"A multi-stack machine halts on a word $l$ over the alphabet of its input stack if and only if the evaluation of its transition function, started at its main label and initial internal state with $l$ on the input stack and all other stacks empty, is defined.","title":"Halting agrees with evaluation for multi-stack machines"},{"id":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/primcodable-of-fintype","kind":"node","lean":"Cslib.Decidability.primcodableOfFintype","parent":35096,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/index.html#primcodable-of-fintype","scope":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines","status":"fully_proved","summary":"A finite type $\\alpha$ with $n$ elements gets the primitive recursive coding transported from $\\{0, \\dots, n - 1\\}$ along a fixed (noncanonically chosen) bijection between $\\alpha$ and $\\{0, \\dots, n - 1\\}$.","title":"Primitive recursive coding of a finite type"},{"id":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/primrec-tr-init-encode-nat","kind":"node","lean":"Cslib.Decidability.primrec_trInit_encodeNat","parent":35096,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/index.html#primrec-tr-init-encode-nat","scope":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines","status":"fully_proved","summary":"Let $M$ be a multi-stack machine together with a bijection $e$ from the alphabet of its input stack to the Booleans, and suppose the tape alphabet $\\Gamma'$ of the single-tape emulation of $M$ is finite and carries a primitive recursive coding. Then the map sending $n \\in \\mathbb{N}$ to the initial tape of the emulation on the input word $e^{-1}(b_1) \\cdots e^{-1}(b_m)$, where $b_1 \\cdots b_m$ is the binary code of\u2026","title":"The input encoding of the single-tape emulation is primitive recursive"},{"id":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/exists-not-tm2-decidable-tm0-halts-on","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_tm0HaltsOn","parent":35096,"result":true,"route":"../roadmap/TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines/index.html#exists-not-tm2-decidable-tm0-halts-on","scope":"TheoryOfComputation/computability/halting-problem-for-single-tape-turing-machines","status":"fully_proved","summary":"There are a finite tape alphabet $\\Gamma$ with a blank symbol, decidable equality and a primitive recursive coding, a type of states $\\Lambda$ with a start state, a single-tape Turing machine $M$ with alphabet $\\Gamma$ and states $\\Lambda$, and a finite set $S$ of states that contains the start state and is closed under the transitions of $M$, such that the set of input words on which $M$ halts is not decidable by\u2026","title":"The halting problem for single-tape machines is undecidable"},{"data_url":"scopes/TheoryOfComputation/computability/kleene-s-recursion-theorem.json","id":"TheoryOfComputation/computability/kleene-s-recursion-theorem","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/kleene-s-recursion-theorem/index.html","scope":"TheoryOfComputation/computability/kleene-s-recursion-theorem","status":"fully_proved","title":"Kleene's recursion theorem"},{"id":"TheoryOfComputation/computability/kleene-s-recursion-theorem/exists-quine","kind":"node","lean":"Cslib.Decidability.exists_quine","parent":35102,"result":true,"route":"../roadmap/TheoryOfComputation/computability/kleene-s-recursion-theorem/index.html#exists-quine","scope":"TheoryOfComputation/computability/kleene-s-recursion-theorem","status":"fully_proved","summary":"There is a code $c$ of a partial recursive function on $\\mathbb{N}$ that, on every input $n \\in \\mathbb{N}$, halts and outputs the number encoding $c$.","title":"A quine"},{"id":"TheoryOfComputation/computability/kleene-s-recursion-theorem/exists-dom-eq-singleton-encode","kind":"node","lean":"Cslib.Decidability.exists_dom_eq_singleton_encode","parent":35102,"result":true,"route":"../roadmap/TheoryOfComputation/computability/kleene-s-recursion-theorem/index.html#exists-dom-eq-singleton-encode","scope":"TheoryOfComputation/computability/kleene-s-recursion-theorem","status":"fully_proved","summary":"There is a code $c$ of a partial recursive function on $\\mathbb{N}$ such that, for every $n \\in \\mathbb{N}$, $c$ halts on input $n$ if and only if $n$ is the number encoding $c$.","title":"A code that halts exactly on its own number"},{"data_url":"scopes/TheoryOfComputation/computability/minimal-programs.json","id":"TheoryOfComputation/computability/minimal-programs","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/minimal-programs/index.html","scope":"TheoryOfComputation/computability/minimal-programs","status":"fully_proved","title":"Minimal programs"},{"id":"TheoryOfComputation/computability/minimal-programs/desc-length","kind":"node","lean":"Cslib.Decidability.descLength","parent":35105,"result":true,"route":"../roadmap/TheoryOfComputation/computability/minimal-programs/index.html#desc-length","scope":"TheoryOfComputation/computability/minimal-programs","status":"fully_proved","summary":"For a code $c$ of a partial recursive function, the description length of $c$ is the number of binary digits of the encoding of $c$ as a natural number, that is, the least $k$ such that this encoding is less than $2^k$ (so it is $0$ when the encoding is $0$).","title":"Description length of a code"},{"id":"TheoryOfComputation/computability/minimal-programs/minimal-code","kind":"node","lean":"Cslib.Decidability.MinimalCode","parent":35105,"result":true,"route":"../roadmap/TheoryOfComputation/computability/minimal-programs/index.html#minimal-code","scope":"TheoryOfComputation/computability/minimal-programs","status":"fully_proved","summary":"A code $c$ of a partial recursive function is minimal when no code with a shorter description has the same domain: for every code $d$ whose description length is strictly less than that of $c$, the domain of $d$ (the set of inputs on which its evaluation is defined) differs from the domain of $c$. The set of minimal codes is denoted $\\mathrm{MIN}$.","title":"Minimal codes"},{"id":"TheoryOfComputation/computability/minimal-programs/exists-code-dom-singleton","kind":"node","lean":"Cslib.Decidability.exists_code_dom_singleton","parent":35105,"result":true,"route":"../roadmap/TheoryOfComputation/computability/minimal-programs/index.html#exists-code-dom-singleton","scope":"TheoryOfComputation/computability/minimal-programs","status":"fully_proved","summary":"For every natural number $n$ there is a code $c$ of a partial recursive function whose evaluation is defined exactly at $n$, that is, the domain of the partial function computed by $c$ is $\\{n\\}$.","title":"Codes with singleton domain"},{"id":"TheoryOfComputation/computability/minimal-programs/exists-minimal-code-dom","kind":"node","lean":"Cslib.Decidability.exists_minimalCode_dom","parent":35105,"result":true,"route":"../roadmap/TheoryOfComputation/computability/minimal-programs/index.html#exists-minimal-code-dom","scope":"TheoryOfComputation/computability/minimal-programs","status":"fully_proved","summary":"For every code $c$ of a partial recursive function there is a minimal code $D$ whose domain equals the domain of $c$, where the domain of a code is the set of inputs on which its evaluation is defined.","title":"Every language of a code has a minimal code"},{"id":"TheoryOfComputation/computability/minimal-programs/exists-minimal-code-gt","kind":"node","lean":"Cslib.Decidability.exists_minimalCode_gt","parent":35105,"result":true,"route":"../roadmap/TheoryOfComputation/computability/minimal-programs/index.html#exists-minimal-code-gt","scope":"TheoryOfComputation/computability/minimal-programs","status":"fully_proved","summary":"For every natural number $N$ there is a minimal code $D$ of a partial recursive function whose encoding as a natural number is greater than $N$.","title":"Minimal codes are unbounded"},{"id":"TheoryOfComputation/computability/minimal-programs/not-tm2-recognizable-minimal-code","kind":"node","lean":"Cslib.Decidability.not_tm2Recognizable_minimalCode","parent":35105,"result":true,"route":"../roadmap/TheoryOfComputation/computability/minimal-programs/index.html#not-tm2-recognizable-minimal-code","scope":"TheoryOfComputation/computability/minimal-programs","status":"fully_proved","summary":"The set $\\mathrm{MIN}$ of minimal codes of partial recursive functions is not recognizable by a multi-stack Turing machine: there is no finite multi-stack machine that, for every code $c$, halts on the binary encoding of the natural-number encoding of $c$ exactly when $c$ is minimal.","title":"Minimal codes are not recognizable"},{"data_url":"scopes/TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings.json","id":"TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings/index.html","scope":"TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings","status":"fully_proved","title":"Rice's theorem for acceptable numberings"},{"id":"TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings/acceptable-numbering-fixed-point","kind":"node","lean":"MetaMathlibExt.acceptableNumbering_fixed_point","parent":35112,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings/index.html#acceptable-numbering-fixed-point","scope":"TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings","status":"fully_proved","summary":"Let $\\varphi$ assign to each natural number $e$ a partial function $\\varphi_e : \\mathbb{N} \\rightharpoonup \\mathbb{N}$, and suppose that the universal function $(e, n) \\mapsto \\varphi_e(n)$ is partial computable, that every partial computable function $f : \\mathbb{N} \\rightharpoonup \\mathbb{N}$ equals $\\varphi_e$ for some $e$, and that for every partial computable\u2026","title":"Rogers's fixed-point theorem for acceptable numberings"},{"id":"TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings/rice-theorem","kind":"node","lean":"MetaMathlibExt.rice_theorem","parent":35112,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings/index.html#rice-theorem","scope":"TheoryOfComputation/computability/rice-s-theorem-for-acceptable-numberings","status":"fully_proved","summary":"Let $\\varphi$ assign to each natural number $e$ a partial function $\\varphi_e : \\mathbb{N} \\rightharpoonup \\mathbb{N}$, and suppose that every $\\varphi_e$ is partial computable, that the universal function $(e, n) \\mapsto \\varphi_e(n)$ is partial computable, that every partial computable function $f : \\mathbb{N} \\rightharpoonup \\mathbb{N}$ equals $\\varphi_e$ for some $e$, and that for every partial computable\u2026","title":"Rice's theorem for acceptable numberings"},{"data_url":"scopes/TheoryOfComputation/computability/rice-shapiro-theorem.json","id":"TheoryOfComputation/computability/rice-shapiro-theorem","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","title":"Rice\u2013Shapiro theorem"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/rice-shapiro-mono","kind":"node","lean":"Cslib.Decidability.rice_shapiro_mono","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#rice-shapiro-mono","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"In a class with a recognizable index set, every partial recursive extension of a partial recursive member is a member.","title":"Rice\u2013Shapiro, upward closure"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/rice-shapiro-finite","kind":"node","lean":"Cslib.Decidability.rice_shapiro_finite","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#rice-shapiro-finite","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"Every partial recursive member of a class with a recognizable index set has a partial recursive finite subfunction in the class.","title":"Rice\u2013Shapiro, finite approximation"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/fin-fun","kind":"node","lean":"Cslib.Decidability.finFun","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#fin-fun","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"For a finite list $\\theta$ of pairs of natural numbers, the finite function of $\\theta$ is the partial function $\\mathbb{N} \\rightharpoonup \\mathbb{N}$ whose value at $x$ is the second component of the first pair in $\\theta$ with first component $x$, and which is undefined at $x$ when $\\theta$ has no pair with first component $x$.","title":"Finite function of a list of pairs"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/is-graph","kind":"node","lean":"Cslib.Decidability.IsGraph","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#is-graph","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"A finite list $\\theta$ of pairs of natural numbers is a graph when, for every pair $(x, y)$ in $\\theta$, the finite function of $\\theta$ takes the value $y$ at $x$; that is, the first pair in $\\theta$ with first component $x$ is $(x, y)$.","title":"Graphs of finite functions"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/re-pred-is-graph","kind":"node","lean":"Cslib.Decidability.rePred_isGraph","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#re-pred-is-graph","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"The set of finite lists of pairs of natural numbers that are graphs is recursively enumerable.","title":"Being a graph is recursively enumerable"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/exists-fin-fun-eq","kind":"node","lean":"Cslib.Decidability.exists_finFun_eq","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#exists-fin-fun-eq","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"For every partial function $g : \\mathbb{N} \\rightharpoonup \\mathbb{N}$ with finite domain there is a finite list $\\theta$ of pairs of natural numbers such that the finite function of $\\theta$ is $g$ and, for every pair $(x, y)$ in $\\theta$, $g(x)$ is defined and equals $y$.","title":"Finite-domain partial functions are finite functions"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/re-pred-subgraph","kind":"node","lean":"Cslib.Decidability.rePred_subgraph","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#re-pred-subgraph","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"The set of pairs $(c, \\theta)$, where $c$ is a code of a partial recursive function and $\\theta$ is a finite list of pairs of natural numbers, such that for every pair $(x, y)$ in $\\theta$ the function computed by $c$ is defined at $x$ with value $y$, is recursively enumerable.","title":"Extending a finite list of pairs is recursively enumerable"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/tm2-recognizable-of-basis","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_of_basis","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#tm2-recognizable-of-basis","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"Let $C$ be a class of partial functions $\\mathbb{N} \\rightharpoonup \\mathbb{N}$ and let $B$ be a recognizable set of finite lists of pairs of natural numbers. Suppose that a partial recursive function $f$ is in $C$ if and only if some $\\theta \\in B$ satisfies: for every pair $(x, y)$ in $\\theta$, $f(x)$ is defined and equals $y$. Then the index set of $C$, the set of codes $c$ of partial recursive functions such\u2026","title":"Effective Rice\u2013Shapiro, sufficiency"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/exists-basis-of-tm2-recognizable","kind":"node","lean":"Cslib.Decidability.exists_basis_of_tm2Recognizable","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#exists-basis-of-tm2-recognizable","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"Let $C$ be a class of partial functions $\\mathbb{N} \\rightharpoonup \\mathbb{N}$ whose index set, the set of codes $c$ of partial recursive functions such that the function computed by $c$ is in $C$, is recognizable. Then there is a recognizable set $B$ of finite lists of pairs of natural numbers, each of which is a graph, such that a partial recursive function $f$ is in $C$ if and only if some $\\theta \\in B$\u2026","title":"Effective Rice\u2013Shapiro, necessity"},{"id":"TheoryOfComputation/computability/rice-shapiro-theorem/tm2-recognizable-iff-exists-basis","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_iff_exists_basis","parent":35115,"result":true,"route":"../roadmap/TheoryOfComputation/computability/rice-shapiro-theorem/index.html#tm2-recognizable-iff-exists-basis","scope":"TheoryOfComputation/computability/rice-shapiro-theorem","status":"fully_proved","summary":"A class of partial functions has a recognizable index set iff, among partial recursive functions, its members are exactly the extensions of the finite functions in some recognizable set of graphs.","title":"The effective Rice\u2013Shapiro theorem"},{"data_url":"scopes/TheoryOfComputation/computability/totality-problem-for-programs.json","id":"TheoryOfComputation/computability/totality-problem-for-programs","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/totality-problem-for-programs/index.html","scope":"TheoryOfComputation/computability/totality-problem-for-programs","status":"fully_proved","title":"Totality problem for programs"},{"id":"TheoryOfComputation/computability/totality-problem-for-programs/total-code","kind":"node","lean":"Cslib.Decidability.TotalCode","parent":35126,"result":true,"route":"../roadmap/TheoryOfComputation/computability/totality-problem-for-programs/index.html#total-code","scope":"TheoryOfComputation/computability/totality-problem-for-programs","status":"fully_proved","summary":"A code $c$ of a partial recursive function is total when it halts on every input: writing $\\varphi_c$ for the partial function on $\\mathbb{N}$ that $c$ computes, $\\varphi_c(n)$ is defined for every $n \\in \\mathbb{N}$.","title":"Total codes"},{"id":"TheoryOfComputation/computability/totality-problem-for-programs/many-one-reducible-halting-total","kind":"node","lean":"Cslib.Decidability.manyOneReducible_halting_total","parent":35126,"result":true,"route":"../roadmap/TheoryOfComputation/computability/totality-problem-for-programs/index.html#many-one-reducible-halting-total","scope":"TheoryOfComputation/computability/totality-problem-for-programs","status":"fully_proved","summary":"Writing $\\varphi_c$ for the partial function on $\\mathbb{N}$ computed by a code $c$ of a partial recursive function, the set of codes $c$ with $\\varphi_c(0)$ defined is many-one reducible to the set of total codes, the codes that halt on every input: there is a computable function $f$ from codes to codes such that, for every code $c$, $\\varphi_c(0)$ is defined if and only if $f(c)$ is total.","title":"Halting on zero reduces to totality"},{"id":"TheoryOfComputation/computability/totality-problem-for-programs/not-tm2-co-recognizable-total","kind":"node","lean":"Cslib.Decidability.not_tm2CoRecognizable_total","parent":35126,"result":true,"route":"../roadmap/TheoryOfComputation/computability/totality-problem-for-programs/index.html#not-tm2-co-recognizable-total","scope":"TheoryOfComputation/computability/totality-problem-for-programs","status":"fully_proved","summary":"The set of total codes, the codes of partial recursive functions that halt on every input, is not co-recognizable by multi-stack machines: no multi-stack machine halts on the binary encoding of a code $c$ exactly when $c$ is not total.","title":"Totality is not co-recognizable"},{"id":"TheoryOfComputation/computability/totality-problem-for-programs/until-halts-fun","kind":"node","lean":"Cslib.Decidability.untilHaltsFun","parent":35126,"result":true,"route":"../roadmap/TheoryOfComputation/computability/totality-problem-for-programs/index.html#until-halts-fun","scope":"TheoryOfComputation/computability/totality-problem-for-programs","status":"fully_proved","summary":"For a code $c$ of a partial recursive function, this is the partial function $\\mathbb{N} \\to \\mathbb{N}$ that on input $k$ is undefined if $c$ halts on input $0$ within $k$ steps of bounded evaluation, and equals $0$ otherwise.","title":"Diverging once a code halts on zero"},{"id":"TheoryOfComputation/computability/totality-problem-for-programs/many-one-reducible-not-halting-total","kind":"node","lean":"Cslib.Decidability.manyOneReducible_not_halting_total","parent":35126,"result":true,"route":"../roadmap/TheoryOfComputation/computability/totality-problem-for-programs/index.html#many-one-reducible-not-halting-total","scope":"TheoryOfComputation/computability/totality-problem-for-programs","status":"fully_proved","summary":"Writing $\\varphi_c$ for the partial function on $\\mathbb{N}$ computed by a code $c$ of a partial recursive function, the set of codes $c$ with $\\varphi_c(0)$ undefined is many-one reducible to the set of total codes, the codes that halt on every input: there is a computable function $f$ from codes to codes such that, for every code $c$, $\\varphi_c(0)$ is undefined if and only if $f(c)$ is total.","title":"Non-halting on zero reduces to totality"},{"id":"TheoryOfComputation/computability/totality-problem-for-programs/not-tm2-recognizable-total","kind":"node","lean":"Cslib.Decidability.not_tm2Recognizable_total","parent":35126,"result":true,"route":"../roadmap/TheoryOfComputation/computability/totality-problem-for-programs/index.html#not-tm2-recognizable-total","scope":"TheoryOfComputation/computability/totality-problem-for-programs","status":"fully_proved","summary":"The set of total codes, the codes of partial recursive functions that halt on every input, is not recognizable by multi-stack machines: no multi-stack machine halts on the binary encoding of a code $c$ exactly when $c$ is total.","title":"Totality is not recognizable"},{"id":"TheoryOfComputation/computability/totality-problem-for-programs/not-tm2-decidable-total","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_total","parent":35126,"result":true,"route":"../roadmap/TheoryOfComputation/computability/totality-problem-for-programs/index.html#not-tm2-decidable-total","scope":"TheoryOfComputation/computability/totality-problem-for-programs","status":"fully_proved","summary":"The set of total codes, the codes of partial recursive functions that halt on every input, is not decidable by multi-stack machines: no multi-stack machine computes a total function $f : \\mathbb{N} \\to \\mathbb{N}$ that takes the value $1$ at the number encoding a code $c$ exactly when $c$ is total.","title":"Totality is undecidable"},{"data_url":"scopes/TheoryOfComputation/computability/finiteness-problem-for-programs.json","id":"TheoryOfComputation/computability/finiteness-problem-for-programs","kind":"scope","navigation":"module","parent":34964,"result":false,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","title":"Finiteness problem for programs"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/finite-code","kind":"node","lean":"Cslib.Decidability.FiniteCode","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#finite-code","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"A code $c$ of a partial recursive function $\\varphi_c$ on $\\mathbb{N}$ is finite when the domain of $\\varphi_c$, the set of inputs on which $c$ halts, is finite. The set of finite codes is denoted $\\mathrm{FIN}$.","title":"Codes with finite domain"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/until-halts-fun-dom","kind":"node","lean":"Cslib.Decidability.untilHaltsFun_dom","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#until-halts-fun-dom","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"For a code $c$ of a partial recursive function, let $u_c$ be the partial function on $\\mathbb{N}$ that on input $j$ is undefined if $c$ halts on input $0$ within $j$ steps (the step-bounded evaluation of $c$ on input $0$ with bound $j$ returns a value) and outputs $0$ otherwise. Then for every $k \\in \\mathbb{N}$, $u_c$ is defined at $k$ if and only if $c$ does not halt on input $0$ within $k$ steps.","title":"Domain of the until-halts function"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/many-one-reducible-halting-finite","kind":"node","lean":"Cslib.Decidability.manyOneReducible_halting_finite","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#many-one-reducible-halting-finite","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"The set of codes of partial recursive functions that halt on input $0$ is many-one reducible to the set of finite codes: there is a computable function $f$ from codes to codes such that, for every code $c$, $c$ halts on input $0$ if and only if the partial function of $f(c)$ has finite domain.","title":"Halting reduces to finiteness"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/not-tm2-co-recognizable-finite","kind":"node","lean":"Cslib.Decidability.not_tm2CoRecognizable_finite","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#not-tm2-co-recognizable-finite","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"The set of codes of partial recursive functions with finite domain is not co-recognizable by multi-stack Turing machines: no finite multi-stack machine whose input alphabet is identified with $\\{0, 1\\}$ halts on the binary encoding of the number of a code $c$ exactly when the partial function of $c$ has infinite domain.","title":"Finiteness is not co-recognizable"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/halts-within-fun","kind":"node","lean":"Cslib.Decidability.haltsWithinFun","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#halts-within-fun","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"For a code $c$ of a partial recursive function, $h_c$ is the partial function on $\\mathbb{N}$ that on input $k$ outputs $0$ if $c$ halts on input $0$ within $k$ steps (the step-bounded evaluation of $c$ on input $0$ with bound $k$ returns a value) and is undefined otherwise.","title":"Halts-within function"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/halts-within-fun-dom","kind":"node","lean":"Cslib.Decidability.haltsWithinFun_dom","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#halts-within-fun-dom","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"For a code $c$ of a partial recursive function, let $h_c$ be the partial function on $\\mathbb{N}$ that on input $j$ outputs $0$ if $c$ halts on input $0$ within $j$ steps (the step-bounded evaluation of $c$ on input $0$ with bound $j$ returns a value) and is undefined otherwise. Then for every $k \\in \\mathbb{N}$, $h_c$ is defined at $k$ if and only if $c$ halts on input $0$ within $k$ steps.","title":"Domain of the halts-within function"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/many-one-reducible-not-halting-finite","kind":"node","lean":"Cslib.Decidability.manyOneReducible_not_halting_finite","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#many-one-reducible-not-halting-finite","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"The set of codes of partial recursive functions that do not halt on input $0$ is many-one reducible to the set of finite codes: there is a computable function $f$ from codes to codes such that, for every code $c$, $c$ does not halt on input $0$ if and only if the partial function of $f(c)$ has finite domain.","title":"Non-halting reduces to finiteness"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/not-tm2-recognizable-finite","kind":"node","lean":"Cslib.Decidability.not_tm2Recognizable_finite","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#not-tm2-recognizable-finite","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"The set of codes of partial recursive functions with finite domain is not recognizable by multi-stack Turing machines: no finite multi-stack machine whose input alphabet is identified with $\\{0, 1\\}$ halts on the binary encoding of the number of a code $c$ exactly when the partial function of $c$ has finite domain.","title":"Finiteness is not recognizable"},{"id":"TheoryOfComputation/computability/finiteness-problem-for-programs/not-tm2-decidable-finite","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_finite","parent":35134,"result":true,"route":"../roadmap/TheoryOfComputation/computability/finiteness-problem-for-programs/index.html#not-tm2-decidable-finite","scope":"TheoryOfComputation/computability/finiteness-problem-for-programs","status":"fully_proved","summary":"The set of codes of partial recursive functions with finite domain is not decidable by multi-stack Turing machines: there is no total function $f : \\mathbb{N} \\to \\mathbb{N}$ computed by a finite multi-stack machine on binary encodings such that, for every code $c$ with number $n_c$, $f(n_c) = 1$ if and only if the partial function of $c$ has finite domain.","title":"Finiteness is undecidable"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems.json","id":"TheoryOfComputation/undecidable-problems","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/index.html","scope":"TheoryOfComputation/undecidable-problems","status":"fully_proved","summary":"Specific undecidable problems and the reductions proving them undecidable: the Post correspondence problem and its variants, string rewriting, undecidable questions about context-free grammars (ambiguity, intersection, universality, regularity), and emptiness of linear bounded automata.","title":"Undecidable problems"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem.json","id":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/index.html","scope":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem","status":"fully_proved","title":"Diophantine sets and Hilbert's tenth problem"},{"id":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/is-diophantine","kind":"node","lean":"MetaMathlibExt.IsDiophantine","parent":35145,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/index.html#is-diophantine","scope":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem","status":"fully_proved","summary":"Let $\\varphi : A \\to R$ be a ring homomorphism of commutative rings and let $k \\in \\mathbb{N}$. Polynomials with coefficients in $A$ are evaluated at points with coordinates in $R$ by first mapping their coefficients into $R$ by $\\varphi$. A subset $S \\subseteq R^k$ is Diophantine over $R$ with coefficients from $A$ through $\\varphi$ when there are $n \\in \\mathbb{N}$ and a finite set $P$ of polynomials with\u2026","title":"Diophantine sets"},{"id":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/is-diophantine-set","kind":"node","lean":"MetaMathlibExt.IsDiophantineSet","parent":35145,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/index.html#is-diophantine-set","scope":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem","status":"fully_proved","summary":"Let $\\varphi : A \\to R$ be a ring homomorphism of commutative rings. A subset $S \\subseteq R$ is Diophantine over $R$ with coefficients from $A$ through $\\varphi$ when the set $\\{(s) : s \\in S\\} \\subseteq R^1$ of one-coordinate points with coordinate in $S$ is a Diophantine subset of $R^1$ over $R$ with coefficients from $A$ through $\\varphi$.","title":"Diophantine subsets of a ring"},{"id":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/is-diophantine-univ","kind":"node","lean":"MetaMathlibExt.isDiophantine_univ","parent":35145,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/index.html#is-diophantine-univ","scope":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem","status":"fully_proved","summary":"Let $\\varphi : A \\to R$ be a ring homomorphism of commutative rings. For every $k \\in \\mathbb{N}$, the whole space $R^k$ is Diophantine over $R$ with coefficients from $A$ through $\\varphi$.","title":"The whole space is Diophantine"},{"id":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/is-diophantine-empty","kind":"node","lean":"MetaMathlibExt.isDiophantine_empty","parent":35145,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/index.html#is-diophantine-empty","scope":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem","status":"fully_proved","summary":"Let $\\varphi : A \\to R$ be a ring homomorphism of commutative rings, with $R$ nontrivial. For every $k \\in \\mathbb{N}$, the empty subset of $R^k$ is Diophantine over $R$ with coefficients from $A$ through $\\varphi$.","title":"The empty set is Diophantine"},{"id":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/dprm-theorem","kind":"node","lean":"MathlibExt.NumberTheory.DPRMWanted.dprm_theorem","parent":35145,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem/index.html#dprm-theorem","scope":"TheoryOfComputation/undecidable-problems/diophantine-sets-and-hilbert-s-tenth-problem","status":"fully_proved","summary":"A set $S$ of natural numbers is recursively enumerable if and only if it is Diophantine, that is, there is a polynomial $p$ with integer coefficients such that $n \\in S$ exactly when $p(n, t_1, \\dots, t_k) = 0$ for some natural numbers $t_1, \\dots, t_k$.","title":"Davis\u2013Putnam\u2013Robinson\u2013Matiyasevich theorem"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata.json","id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","title":"Emptiness problem for linear bounded automata"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/fence-symbol","kind":"node","lean":"Cslib.Turing.SingleTapeTM.FenceSymbol","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#fence-symbol","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For an alphabet $\\Sigma$, the tape alphabet of the fenced machine consists of a symbol for each letter of $\\Sigma$, two kinds of padding cells (left and right), which the simulated machine reads as blanks, and the wall, which marks the ends of the input.","title":"Fenced alphabet"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/fence-state","kind":"node","lean":"Cslib.Turing.SingleTapeTM.FenceState","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#fence-state","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For a set of states $Q$, the states of the fenced machine are two start states, a running state for each state $q \\in Q$ of the simulated machine, and a trap state.","title":"States of the fenced machine"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/read","kind":"node","lean":"Cslib.Turing.SingleTapeTM.FenceSymbol.read","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#read","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"The simulated cell of a cell of the fenced machine is the cell of the simulated machine that it stands for: a symbol of the simulated machine stands for itself, and every other cell (a padding cell, a wall or a blank) stands for a blank.","title":"Simulated cell of a fenced cell"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/of-cell","kind":"node","lean":"Cslib.Turing.SingleTapeTM.FenceSymbol.ofCell","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#of-cell","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For a cell $o$ of the simulated machine, a symbol or a blank, the fenced cell of $o$ is the cell the fenced machine writes for a write of $o$: $o$ itself if $o$ is a symbol, and the padding cell $\\mathrm{pad}$ if $o$ is a blank.","title":"Fenced cell of a write"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/read-of-cell","kind":"node","lean":"Cslib.Turing.SingleTapeTM.FenceSymbol.read_ofCell","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#read-of-cell","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For every cell $o$ of the simulated machine, a symbol or a blank, the simulated cell of the fenced cell of $o$ is $o$.","title":"Reading back a fenced cell of a write"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/read-map-sym","kind":"node","lean":"Cslib.Turing.SingleTapeTM.FenceSymbol.read_map_sym","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#read-map-sym","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"Let $o$ be a cell of the simulated machine, a symbol or a blank, and copy it to the fenced machine, a symbol as itself and a blank as a blank. Then the simulated cell of the copy is $o$.","title":"Reading back a copied cell"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/of-cell-ne-wall","kind":"node","lean":"Cslib.Turing.SingleTapeTM.FenceSymbol.ofCell_ne_wall","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#of-cell-ne-wall","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For every cell $o$ of the simulated machine, a symbol or a blank, the fenced cell of $o$ is not a wall.","title":"A fenced cell of a write is never a wall"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/fence-input","kind":"node","lean":"Cslib.Turing.SingleTapeTM.fenceInput","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#fence-input","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For a word $w$ over an alphabet $\\Sigma$ and $i, j \\in \\mathbb{N}$, the fenced input of $w$ with $i$ padding cells on the left and $j$ on the right is the word over the fenced alphabet of $\\Sigma$ made of a wall, $i$ left padding cells, the letters of $w$ as symbols, $j$ right padding cells and a wall, in this order.","title":"Fenced input"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/get-elem-fence-input","kind":"node","lean":"Cslib.Turing.SingleTapeTM.getElem?_fenceInput","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#get-elem-fence-input","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"Let $w$ be a word of length $\\ell$ and $i, j, n \\in \\mathbb{N}$. The cell at position $n$, counting from $0$, of the fenced input of $w$ with $i$ left and $j$ right padding cells is the wall if $n = 0$; a left padding cell if $1 \\le n \\le i$; the letter of $w$ at position $n - i - 1$, also counting from $0$, as a symbol, if $i < n \\le i + \\ell$; a right padding cell if $i + \\ell < n \\le i + \\ell + j$; the wall if\u2026","title":"Cells of a fenced input"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/length-fence-input","kind":"node","lean":"Cslib.Turing.SingleTapeTM.length_fenceInput","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#length-fence-input","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For a word $w$ of length $n$ and natural numbers $i$ and $j$, the fenced input of $w$ with $i$ padding cells on the left and $j$ on the right has length $i + n + j + 2$: the word, the two paddings and the two walls.","title":"Length of a fenced input"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/fence-lift","kind":"node","lean":"Cslib.Turing.SingleTapeTM.fenceLift","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#fence-lift","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"A transition of a single-tape machine $M$, made of an optional symbol to write, an optional move and an optional next state, becomes the transition of the fenced machine of $M$ that writes the symbol, or a right padding cell where $M$ writes a blank, makes the same move, and goes to the running state of the next state $q$ of $M$, or halts where $M$ halts.","title":"A transition lifted to the fenced machine"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/fence-run","kind":"node","lean":"Cslib.Turing.SingleTapeTM.fenceRun","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#fence-run","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"In the running state of a state $q$ of a single-tape machine $M$, the fenced machine of $M$ acts on the cell under its head as follows: on a symbol $a$ it makes the lifted transition of $M$ in state $q$ on $a$; on a left or right padding cell it makes the lifted transition of $M$ in state $q$ on a blank; on a wall or a blank cell it leaves the cell unchanged, does not move, and enters the trap.","title":"The running transition of the fenced machine"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/fence","kind":"node","lean":"Cslib.Turing.SingleTapeTM.fence","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#fence","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For a single-tape machine $M$ over a finite inhabited alphabet $\\Sigma$, the fenced machine of $M$ is the single-tape machine over the fenced alphabet of $\\Sigma$ whose states are the states of the fenced machine for the states of $M$, which starts in the first start state and acts on the cell under its head as follows. In the first start state, on a wall it moves right into the second start state, and on any other\u2026","title":"The fenced machine"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/is-lba-fence","kind":"node","lean":"Cslib.Turing.SingleTapeTM.isLBA_fence","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#is-lba-fence","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"For every single-tape Turing machine $M$, the fenced machine of $M$ is a linear bounded automaton.","title":"The fenced machine is a linear bounded automaton"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/exists-halts-on-fence-iff","kind":"node","lean":"Cslib.Turing.SingleTapeTM.exists_haltsOn_fence_iff","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#exists-halts-on-fence-iff","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"Let $M$ be a single-tape machine over a finite inhabited alphabet and $w$ a word over that alphabet. The fenced machine of $M$ halts on the fenced input of $w$ with $i$ padding cells on the left and $j$ on the right for some $i, j \\in \\mathbb{N}$ if and only if $M$ halts on $w$.","title":"The fenced machine halts on some padding of a word iff the machine halts on it"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/forall-not-halts-on-fence-iff","kind":"node","lean":"Cslib.Turing.SingleTapeTM.forall_not_haltsOn_fence_iff","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#forall-not-halts-on-fence-iff","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"Let $M$ be a single-tape machine over a finite inhabited alphabet and $w$ a word over that alphabet. The fenced machine of $M$ halts on none of the fenced inputs of $w$, with $i$ padding cells on the left and $j$ on the right for $i, j \\in \\mathbb{N}$, if and only if $M$ does not halt on $w$.","title":"The fenced machine halts on no padding of a word iff the machine does not halt on it"},{"id":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/exists-is-lba-not-computable-pred-forall","kind":"node","lean":"Cslib.Turing.SingleTapeTM.exists_isLBA_not_computablePred_forall_not_haltsOn","parent":35151,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata/index.html#exists-is-lba-not-computable-pred-forall","scope":"TheoryOfComputation/undecidable-problems/emptiness-problem-for-linear-bounded-automata","status":"fully_proved","summary":"There exist a finite inhabited alphabet $\\Gamma$ with a primitive recursive encoding, a linear bounded automaton $F$ over the fenced alphabet of $\\Gamma$, and a primitive recursive map $e$ from codes of partial recursive functions to words over $\\Gamma$, such that the set of codes $c$ for which $F$ halts on none of the fenced inputs of $e(c)$, with $i$ padding cells on the left and $j$ on the right for any\u2026","title":"Emptiness of a linear bounded automaton on computable input slices is undecidable"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/post-correspondence-problem.json","id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","title":"Post correspondence problem"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/domino","kind":"node","lean":"Cslib.Decidability.Post.Domino","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#domino","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"A domino over an alphabet $\\alpha$ is a pair $(t, b)$ of words over $\\alpha$: its top word $t$ and its bottom word $b$.","title":"Domino (Post correspondence problem)"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/top","kind":"node","lean":"Cslib.Decidability.Post.top","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#top","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a sequence of dominoes $(t_1, b_1), \\ldots, (t_n, b_n)$, its top word is the concatenation $t_1 t_2 \\cdots t_n$ of their top words; the empty sequence has the empty top word.","title":"Top word of a sequence of dominoes"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/bottom","kind":"node","lean":"Cslib.Decidability.Post.bottom","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#bottom","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a sequence $s = d_1, \\dots, d_k$ of dominoes, the bottom word of $s$ is the concatenation $b_1 b_2 \\cdots b_k$, where $b_i$ is the bottom word of $d_i$.","title":"Bottom word of a domino sequence"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/is-solution","kind":"node","lean":"Cslib.Decidability.Post.IsSolution","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#is-solution","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $P$ be a finite list of dominoes. A solution of $P$ is a nonempty sequence $d_1, \\dots, d_k$ of dominoes, each occurring in $P$ (repetitions allowed), such that the concatenation of the top words of $d_1, \\dots, d_k$ equals the concatenation of their bottom words.","title":"PCP solution"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/pcp","kind":"node","lean":"Cslib.Decidability.Post.PCP","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#pcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"A list $P$ of dominoes is a solvable instance of the Post correspondence problem when it has a PCP solution: a nonempty sequence of dominoes from $P$, repetitions allowed, whose top words and bottom words concatenate to the same word.","title":"The Post correspondence problem"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/mpcp","kind":"node","lean":"Cslib.Decidability.Post.MPCP","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#mpcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"A list $P$ of dominoes has an MPCP solution when $P$ is nonempty and some PCP solution of $P$ starts with the first domino of $P$.","title":"The modified Post correspondence problem"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/nonempty-words","kind":"node","lean":"Cslib.Decidability.Post.NonemptyWords","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#nonempty-words","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"A list $P$ of dominoes has nonempty words when every domino of $P$ has a nonempty top word and a nonempty bottom word.","title":"Instances with nonempty words"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/sym","kind":"node","lean":"Cslib.Decidability.Post.Sym","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#sym","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For an alphabet $\\alpha$, the extended alphabet consists of the letters of $\\alpha$ together with two new symbols: the star $\\star$ and the end marker $\\diamond$.","title":"The extended alphabet of the MPCP-to-PCP reduction"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/letter","kind":"node","lean":"Cslib.Decidability.Post.letter","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#letter","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Each letter $a$ of the alphabet $\\alpha$ is regarded as a symbol of the extended alphabet of the MPCP-to-PCP reduction, which consists of the letters of $\\alpha$, the star $\\star$ and the end marker $\\diamond$.","title":"The alphabet inside the extended alphabet"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/star","kind":"node","lean":"Cslib.Decidability.Post.star","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#star","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"The star $\\star$ is the new symbol of the extended alphabet that is neither a letter of $\\alpha$ nor the end marker $\\diamond$.","title":"The star symbol"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/end-mark","kind":"node","lean":"Cslib.Decidability.Post.endMark","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#end-mark","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"The end marker $\\diamond$ is the symbol of the extended alphabet of the MPCP-to-PCP reduction that is neither a letter of $\\alpha$ nor the star $\\star$.","title":"End marker"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/star-before","kind":"node","lean":"Cslib.Decidability.Post.starBefore","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#star-before","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\star$ be the star of the extended alphabet. For a word $u = a_1 a_2 \\cdots a_n$ over $\\alpha$, the word $\\star u$ over the extended alphabet is $\\star a_1 \\star a_2 \\cdots \\star a_n$, obtained by putting a star before every letter of $u$; the empty word gives the empty word.","title":"A star before every letter"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/star-after","kind":"node","lean":"Cslib.Decidability.Post.starAfter","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#star-after","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\star$ be the star of the extended alphabet. For a word $u = a_1 a_2 \\cdots a_n$ over $\\alpha$, the word $u\\star$ over the extended alphabet is $a_1 \\star a_2 \\star \\cdots a_n \\star$, obtained by putting a star after every letter of $u$; the empty word gives the empty word.","title":"A star after every letter"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/letters","kind":"node","lean":"Cslib.Decidability.Post.letters","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#letters","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a word $w$ over the extended alphabet of the MPCP-to-PCP reduction, this is the word over $\\alpha$ obtained from $w$ by deleting every star $\\star$ and every end marker $\\diamond$, keeping the letters of $\\alpha$ in their order.","title":"Erasing stars and end markers"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/map-domino","kind":"node","lean":"Cslib.Decidability.Post.mapDomino","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#map-domino","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a word $u$ over $\\alpha$, write $\\star u$ for the word over the extended alphabet obtained by putting a star $\\star$ before every letter of $u$, and $u\\star$ for the word obtained by putting a star after every letter. A domino $(t, b)$ over $\\alpha$ is sent to the domino $(\\star t, b\\star)$ over the extended alphabet.","title":"Starred image of a domino"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/start-domino","kind":"node","lean":"Cslib.Decidability.Post.startDomino","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#start-domino","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a domino $(t, b)$ over an alphabet $\\alpha$, its start domino is the domino $(\\star t, \\star b \\star)$ over the extended alphabet, where $\\star t$ puts the star $\\star$ before every letter of $t$ and $b \\star$ puts a star after every letter of $b$, so that the bottom word gets one extra star in front.","title":"The start domino of the MPCP-to-PCP reduction"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/end-domino","kind":"node","lean":"Cslib.Decidability.Post.endDomino","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#end-domino","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Over the alphabet $\\alpha$ extended with the star $\\star$ and the end marker $\\diamond$, the end domino is $(\\star\\diamond, \\diamond)$: its top word is $\\star$ followed by $\\diamond$, and its bottom word is $\\diamond$.","title":"End domino of the MPCP-to-PCP reduction"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/to-pcp","kind":"node","lean":"Cslib.Decidability.Post.toPCP","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#to-pcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Write $\\star$ for the star and $\\diamond$ for the end marker of the extended alphabet; for a word $u$ over $\\alpha$, $\\star u$ puts a star before every letter of $u$ and $u \\star$ puts a star after every letter of $u$. The empty list of dominoes is sent to the empty list. A list $P = [d_0, d_1, \\ldots, d_n]$ of dominoes over $\\alpha$, with $d_0 = (t_0, b_0)$, is sent to the following list of dominoes over the\u2026","title":"The reduction from MPCP to PCP"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/unmap","kind":"node","lean":"Cslib.Decidability.Post.unmap","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#unmap","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a domino $(t, b)$ over the extended alphabet, its unmapping is the domino over $\\alpha$ obtained by deleting every star $\\star$ and every end marker $\\diamond$ from both $t$ and $b$, keeping the letters of $\\alpha$ in order.","title":"Erasing the markers from a domino"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/pcp-to-pcp-of-mpcp","kind":"node","lean":"Cslib.Decidability.Post.pcp_toPCP_of_mpcp","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#pcp-to-pcp-of-mpcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"If a list $P$ of dominoes has an MPCP solution, then the image of $P$ under the reduction from MPCP to PCP has a PCP solution.","title":"An MPCP solution gives a PCP solution of the reduced instance"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/mpcp-of-pcp-to-pcp","kind":"node","lean":"Cslib.Decidability.Post.mpcp_of_pcp_toPCP","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#mpcp-of-pcp-to-pcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"On instances with nonempty words, a PCP solution of the reduced instance gives an MPCP solution. Let $P$ be a list of dominoes in which every domino has a nonempty top and a nonempty bottom. If the image of $P$ under the reduction from MPCP to PCP has a PCP solution, then $P$ has an MPCP solution.","title":"On instances with nonempty words, a PCP solution of the reduced instance gives an MPCP solution"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/mpcp-iff-pcp-to-pcp","kind":"node","lean":"Cslib.Decidability.Post.mpcp_iff_pcp_toPCP","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#mpcp-iff-pcp-to-pcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $P$ be a list of dominoes over an alphabet $\\alpha$ in which every domino has a nonempty top and a nonempty bottom. Then $P$ has an MPCP solution if and only if the image of $P$ under the reduction from MPCP to PCP, an instance over $\\alpha$ extended with a star and an end marker, has a PCP solution.","title":"MPCP reduces to PCP on instances with nonempty words"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/nonempty-words-b","kind":"node","lean":"Cslib.Decidability.Post.nonemptyWordsB","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#nonempty-words-b","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a list $P$ of dominoes, this Boolean test returns true exactly when every domino of $P$ has a nonempty top and a nonempty bottom.","title":"Boolean test for nonempty words"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/reduce","kind":"node","lean":"Cslib.Decidability.Post.reduce","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#reduce","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a list $P$ of dominoes, the total reduction sends $P$ to its image under the reduction from MPCP to PCP when every domino of $P$ has a nonempty top and a nonempty bottom, and to the empty instance, which has no PCP solution, otherwise.","title":"The total reduction from MPCP to PCP"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/primrec-nonempty-words-b","kind":"node","lean":"Cslib.Decidability.Post.primrec_nonemptyWordsB","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#primrec-nonempty-words-b","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For an alphabet $\\alpha$ with a primitive recursive coding, the Boolean test of whether every domino of a list over $\\alpha$ has a nonempty top and a nonempty bottom is a primitive recursive function of the list.","title":"The nonempty-words test is primitive recursive"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/many-one-reducible-mpcp-pcp","kind":"node","lean":"Cslib.Decidability.Post.manyOneReducible_mpcp_pcp","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#many-one-reducible-mpcp-pcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with a primitive recursive coding, and let $\\alpha'$ be $\\alpha$ extended with two new symbols, the star and the end marker. There is a computable function $f$ from finite lists of dominoes over $\\alpha$ to finite lists of dominoes over $\\alpha'$ such that, for every list $P$ of dominoes over $\\alpha$, $f(P)$ has a PCP solution if and only if both every domino of $P$ has a nonempty top\u2026","title":"MPCP many-one reduces to PCP"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/is-solution-b","kind":"node","lean":"Cslib.Decidability.Post.isSolutionB","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#is-solution-b","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with a primitive recursive coding and decidable equality, and let $P$ and $s$ be lists of dominoes over $\\alpha$. The Boolean solution check of $s$ for $P$ is true exactly when $s$ is nonempty, every domino of $s$ occurs in $P$, and the top word of $s$ equals the bottom word of $s$, and it is false otherwise.","title":"Boolean check for PCP solutions"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/re-pred-pcp","kind":"node","lean":"Cslib.Decidability.Post.rePred_pcp","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#re-pred-pcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For an alphabet $\\alpha$ with a primitive recursive coding, the set of lists of dominoes over $\\alpha$ that have a PCP solution is recursively enumerable.","title":"PCP is recursively enumerable"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/tm2-recognizable-pcp","kind":"node","lean":"Cslib.Decidability.Post.tm2Recognizable_pcp","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#tm2-recognizable-pcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For every alphabet $\\alpha$ with a primitive recursive encoding, there is a multi-stack Turing machine that, started on the binary encoding of a list $P$ of dominoes over $\\alpha$, halts exactly when $P$ has a PCP solution.","title":"PCP is Turing-recognizable"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/length-bottom-le-length-top","kind":"node","lean":"Cslib.Decidability.Post.length_bottom_le_length_top","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#length-bottom-le-length-top","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Write $|w|$ for the length of a word $w$. Let $s$ be a finite sequence of dominoes over an alphabet. If every domino $(t, b)$ in $s$ satisfies $|b| \\le |t|$, then the concatenation of the bottom words of $s$ is no longer than the concatenation of the top words of $s$.","title":"Total length of dominoes that are at least as long on top"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/length-bottom-lt-length-top","kind":"node","lean":"Cslib.Decidability.Post.length_bottom_lt_length_top","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#length-bottom-lt-length-top","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Write $|w|$ for the length of a word $w$. Let $s$ be a nonempty finite sequence of dominoes over an alphabet. If every domino $(t, b)$ in $s$ satisfies $|b| < |t|$, then the concatenation of the bottom words of $s$ is strictly shorter than the concatenation of the top words of $s$.","title":"Total length of dominoes that are strictly longer on top"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/swap","kind":"node","lean":"Cslib.Decidability.Post.swap","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#swap","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a finite sequence $s$ of dominoes, the swapped sequence is obtained from $s$ by exchanging the top and bottom word of every domino: each $(t, b)$ becomes $(b, t)$, and the order of the dominoes is unchanged.","title":"Swapping top and bottom"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/pcp-iff-of-subsingleton","kind":"node","lean":"Cslib.Decidability.Post.pcp_iff_of_subsingleton","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#pcp-iff-of-subsingleton","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with at most one letter, and write $|w|$ for the length of a word $w$. A finite list $P$ of dominoes over $\\alpha$ has a PCP solution if and only if either some domino $(t, b)$ in $P$ has $|t| = |b|$, or some domino $(t, b)$ in $P$ has $|t| < |b|$ and some domino $(t', b')$ in $P$ has $|b'| < |t'|$.","title":"The unary PCP criterion"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/unary-test","kind":"node","lean":"Cslib.Decidability.Post.unaryTest","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#unary-test","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with at most one letter, and write $|w|$ for the length of a word $w$. For a finite list $P$ of dominoes over $\\alpha$, the unary test of $P$ is the Boolean value that is true exactly when some domino $(t, b)$ in $P$ has $|t| = |b|$, or some domino $(t, b)$ in $P$ has $|t| < |b|$ and some domino $(t', b')$ in $P$ has $|b'| < |t'|$.","title":"The unary PCP test"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/computable-pred-pcp-of-subsingleton","kind":"node","lean":"Cslib.Decidability.Post.computablePred_pcp_of_subsingleton","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#computable-pred-pcp-of-subsingleton","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with at most one letter whose elements have a computable encoding. Then the Post correspondence problem over $\\alpha$ is a computable predicate: some computable function decides, for every finite list $P$ of dominoes over $\\alpha$, whether $P$ has a PCP solution.","title":"PCP over a one-letter alphabet is decidable"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/tm2-decidable-pcp-of-subsingleton","kind":"node","lean":"Cslib.Decidability.Post.tm2Decidable_pcp_of_subsingleton","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#tm2-decidable-pcp-of-subsingleton","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with at most one letter whose elements have a computable encoding. Then the Post correspondence problem over $\\alpha$ is decidable by a multi-stack machine: some multi-stack Turing machine decides, for every finite list $P$ of dominoes over $\\alpha$, whether $P$ has a PCP solution.","title":"Unary PCP is decidable by a multi-stack machine"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/tmsym","kind":"node","lean":"Cslib.Decidability.TMRewriting.TMSym","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#tmsym","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a tape alphabet $\\Gamma$ and a set of states $\\Lambda$, the symbols of the rewriting system simulating a single-tape machine are a tape symbol $a$ for each $a \\in \\Gamma$, a state symbol $q$ for each $q \\in \\Lambda$, the left end marker $\\vdash$, the right end marker $\\dashv$ and the halting symbol $H$.","title":"Symbols of the rewriting system of a single-tape machine"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/is-ctl","kind":"node","lean":"Cslib.Decidability.TMRewriting.IsCtl","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#is-ctl","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"A symbol of the rewriting system simulating a single-tape Turing machine is a control symbol if it is a state symbol or the halting symbol $H$; tape symbols and the end markers $\\vdash$ and $\\dashv$ are not control symbols.","title":"Control symbols"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/eq-of-append-cons-eq","kind":"node","lean":"Cslib.Decidability.TMRewriting.eq_of_append_cons_eq","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#eq-of-append-cons-eq","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $P$ be a property of the elements of a type $\\alpha$, let $A$ and $B$ be lists over $\\alpha$ none of whose elements satisfy $P$, and let $y \\in \\alpha$. If $C$ and $D$ are lists over $\\alpha$ and $x \\in \\alpha$ satisfies $P$ with $C\\,x\\,D = A\\,y\\,B$ (concatenation), then $C = A$, $x = y$ and $D = B$.","title":"Unique splitting at a marked letter"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/split-site","kind":"node","lean":"Cslib.Decidability.TMRewriting.split_site","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#split-site","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $A$, $B$, $u$, $v$, $p$, $s$ be words over the symbols of the rewriting system simulating a single-tape machine, and let $x$ and $y$ be symbols. If no symbol of $A$ or of $B$ is a control symbol (a state symbol or the halting symbol), $x$ is a control symbol, and $u\\,p\\,x\\,s\\,v = A\\,y\\,B$, then $u\\,p = A$, $x = y$ and $s\\,v = B$.","title":"Splitting a rewrite site at its control symbol"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/spell","kind":"node","lean":"Cslib.Decidability.TMRewriting.spell","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#spell","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a state $q$, a tape symbol $h$ and words $L$, $R$ of tape symbols, the word of the configuration in state $q$ with head symbol $h$, left part $L$ (in reading order, from left to right) and right part $R$ is $\\vdash L\\,q\\,h\\,R \\dashv$, where $\\vdash$ and $\\dashv$ are the end markers.","title":"The word of a configuration"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/halt-word","kind":"node","lean":"Cslib.Decidability.TMRewriting.haltWord","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#halt-word","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"The halting word is the word $\\vdash H \\dashv$ made of the left end marker, the halting symbol and the right end marker. It is the target word of the rewriting simulation of a single-tape machine.","title":"The halting word"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/spells","kind":"node","lean":"Cslib.Decidability.TMRewriting.Spells","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#spells","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"A word $w$ spells a configuration $c$ of a single-tape Turing machine if there are a state $q$, a tape symbol $h$ and words $L$, $R$ of tape symbols such that $c$ is in state $q$, its head reads $h$, its tape reads $L$ immediately to the left of the head and $R$ immediately to the right of it, with blanks beyond, and $w = \\vdash L\\,q\\,h\\,R \\dashv$.","title":"Words spelling a configuration"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/init-word","kind":"node","lean":"Cslib.Decidability.TMRewriting.initWord","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#init-word","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For an input $l$ of tape symbols, the start word is $\\vdash q_0\\, h\\, R \\dashv$, where $\\vdash$ and $\\dashv$ are the left and right end markers, $q_0$ is the default state, $h$ is the first symbol of $l$ (the default tape symbol if $l$ is empty) and $R$ is the rest of $l$. It spells the configuration in state $q_0$ with head symbol $h$, empty left part and right part $R$.","title":"The start word"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/rules-for","kind":"node","lean":"Cslib.Decidability.TMRewriting.rulesFor","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#rules-for","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, let $\\sqcup$ be its blank symbol, and write $\\vdash$, $\\dashv$ for the end markers and $H$ for the halting symbol. The rules for a state $q$ reading a tape symbol $a$ are: the rule $q\\,a \\to H$ if $M$ has no transition from $q$ reading $a$; the rule $q\\,a \\to q'\\,b$ if $M$ writes $b$ and enters $q'$; the rules\u2026","title":"The rules for one transition"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/erase-rules","kind":"node","lean":"Cslib.Decidability.TMRewriting.eraseRules","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#erase-rules","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"For a finite tape alphabet $\\Gamma$, let $H$ be the halting symbol of the rewriting system simulating a single-tape machine with tape alphabet $\\Gamma$. The erasing rules are the rewriting rules $c\\,H \\to H$ and $H\\,c \\to H$ for every tape symbol $c \\in \\Gamma$: the halting symbol deletes the tape symbol on either side of it.","title":"Erasing rules"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/rules","kind":"node","lean":"Cslib.Decidability.TMRewriting.rules","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#rules","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and write $H$ for the halting symbol. The rewriting system simulating $M$ is the finite list of rules consisting of the rules for every state $q$ reading every tape symbol $a$, followed by the erasing rules $c\\,H \\to H$ and $H\\,c \\to H$ for every tape symbol $c$.","title":"The rewriting system simulating a Turing machine"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/cases-of-mem-rules","kind":"node","lean":"Cslib.Decidability.TMRewriting.cases_of_mem_rules","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#cases-of-mem-rules","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with a finite tape alphabet $\\Gamma$, a default (blank) symbol $\\beta \\in \\Gamma$ and finitely many states. The symbols are the tape symbols, the states, the end markers $\\vdash$ and $\\dashv$, and the halting symbol $H$. Every rule $l \\to r$ of the rewriting system simulating $M$ has one of the following eight forms, where $q, q'$ are states and $a, b, c$ are tape symbols.\u2026","title":"The eight kinds of rules of the machine-simulating rewriting system"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/exists-step-of-spells","kind":"node","lean":"Cslib.Decidability.TMRewriting.exists_step_of_spells","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#exists-step-of-spells","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many tape symbols and states. If the word $w$ spells the configuration $c$ and $M$ steps from $c$ to $c'$, then $w$ rewrites in one step of the rewriting system simulating $M$ to a word $w'$ that spells $c'$.","title":"One machine step is one rewrite"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/derives-erase","kind":"node","lean":"Cslib.Decidability.TMRewriting.derives_erase","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#derives-erase","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and let $\\vdash$, $\\dashv$ be the end markers and $H$ the halting symbol of the rewriting system simulating $M$. For all words $X$ and $Y$ of tape symbols, the word $\\vdash X\\,H\\,Y \\dashv$ rewrites in zero or more steps of this rewriting system to the halting word $\\vdash H \\dashv$.","title":"The halting symbol erases the tape"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/derives-of-spells-of-step-eq-none","kind":"node","lean":"Cslib.Decidability.TMRewriting.derives_of_spells_of_step_eq_none","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#derives-of-spells-of-step-eq-none","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and let $\\vdash$, $\\dashv$ be the end markers and $H$ the halting symbol of the rewriting system simulating $M$. Let $c$ be a configuration of $M$ at which $M$ has no next step, and let $w$ be a word that spells $c$: for some state $q$, tape symbol $h$ and lists $L$, $R$ of tape symbols, $c$ is in state $q$ with $h$ under the head,\u2026","title":"A halted configuration rewrites to the halting word"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/inv","kind":"node","lean":"Cslib.Decidability.TMRewriting.Inv","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#inv","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape machine with finitely many tape symbols and states, and $l$ an input. A word $w$ satisfies the invariant for $l$ when either $w$ spells a configuration that $M$ reaches from its initial configuration on $l$, or $M$ halts on $l$ and $w = \\vdash X\\, H\\, Y \\dashv$ for some words $X$ and $Y$ of tape symbols, where $\\vdash$ and $\\dashv$ are the left and right end markers and $H$ is the halting\u2026","title":"Invariant of the rewriting simulation of a single-tape machine"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/step","kind":"node","lean":"Cslib.Decidability.TMRewriting.Inv.step","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#step","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape machine with finite tape alphabet and finite set of states, and let $l$ be an input word. Write $\\vdash$ and $\\dashv$ for the left and right end markers and $H$ for the halting symbol. A word $w$ satisfies the invariant for $M$ and $l$ when either $w$ spells a configuration that $M$ reaches from its initial configuration on $l$ (the configuration with state $q$, head symbol $h$, left part\u2026","title":"Every rewrite preserves the invariant of reachable words"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/tm0-halts-on-iff-derives","kind":"node","lean":"Cslib.Decidability.TMRewriting.tm0HaltsOn_iff_derives","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#tm0-halts-on-iff-derives","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape machine with finite tape alphabet and finite set of states, and write $\\vdash$ and $\\dashv$ for the left and right end markers and $H$ for the halting symbol. For every input word $l$, $M$ halts on $l$ if and only if the start word ${\\vdash}\\,q_0\\,h\\,R\\,{\\dashv}$ of $l$ rewrites to the halting word ${\\vdash}\\,H\\,{\\dashv}$ in zero or more steps under the rules of $M$, where $q_0$ is the\u2026","title":"A single-tape machine halts iff its start word rewrites to the halting word"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/many-one-reducible-tm0-halts-on-sr","kind":"node","lean":"Cslib.Decidability.TMRewriting.manyOneReducible_tm0HaltsOn_sr","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#many-one-reducible-tm0-halts-on-sr","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols. Then the set of inputs on which $M$ halts is many-one reducible to the set of instances $(R, x, y)$, where $R$ is a list of rules and $x$, $y$ are words over the symbols of the rewriting system simulating $M$, such that every rule of $R$ has a nonempty left side and a nonempty right side and $x$ rewrites to $y$ in zero or more steps\u2026","title":"Single-tape halting many-one reduces to string rewriting reachability"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/exists-not-tm2-decidable-sr","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_sr","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#exists-not-tm2-decidable-sr","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"There is a finite alphabet $\\alpha$, with a primitive recursive coding of its letters by natural numbers, such that no multi-stack Turing machine decides, for a finite list $R$ of rewriting rules over $\\alpha$ and words $x$ and $y$ over $\\alpha$, whether every rule of $R$ has a nonempty left side and a nonempty right side and $x$ rewrites to $y$ in zero or more steps of $R$.","title":"Reachability for string rewriting is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/exists-not-tm2-decidable-mpcp","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_mpcp","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#exists-not-tm2-decidable-mpcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"There is a finite alphabet $\\alpha$, with a primitive recursive coding of its letters by natural numbers, such that no multi-stack Turing machine decides, for a finite list $P$ of dominoes over $\\alpha$, whether every domino of $P$ has a nonempty top and a nonempty bottom and $P$ has a solution of the modified Post correspondence problem, that is, a nonempty sequence of dominoes of $P$, repetitions allowed, that\u2026","title":"MPCP is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/exists-not-tm2-decidable-pcp","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_pcp","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#exists-not-tm2-decidable-pcp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"There is a finite alphabet $\\alpha$, with a primitive recursive coding of its letters by natural numbers, such that no multi-stack Turing machine decides whether a finite list of dominoes over $\\alpha$ has a solution, that is, a nonempty sequence of dominoes of the list, repetitions allowed, whose tops and bottoms concatenate to the same word.","title":"The Post correspondence problem is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/is-ctl-b","kind":"node","lean":"Cslib.Decidability.TMRewriting.isCtlB","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#is-ctl-b","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"The Boolean function on the symbols of the rewriting system simulating a single-tape Turing machine that is true on every state symbol and on the halting symbol $H$, and false on every tape symbol and on the end markers $\\vdash$ and $\\dashv$.","title":"Boolean test for control symbols"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/ctl-count","kind":"node","lean":"Cslib.Decidability.TMRewriting.ctlCount","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#ctl-count","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"The number of control symbols of a word $w$ over the symbols of the rewriting system simulating a single-tape machine is the number of letters of $w$ that are states or the halting symbol $H$.","title":"Number of control symbols"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/exists-split-of-ctl-count-eq-one","kind":"node","lean":"Cslib.Decidability.TMRewriting.exists_split_of_ctlCount_eq_one","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#exists-split-of-ctl-count-eq-one","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"If a word $w$ over the symbols of the rewriting system simulating a single-tape machine contains exactly one control symbol (a state symbol or the halting symbol), then $w = p\\,x\\,s$ for a control symbol $x$ and words $p$ and $s$ that contain no control symbol.","title":"A word with one control symbol splits at it"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/next-state","kind":"node","lean":"Cslib.Decidability.TMRewriting.nextState","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#next-state","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and let $\\sqcup$ be its blank symbol. For words $u$, $v$ and a state $q$, the successor of $u\\,q\\,v$ is defined as follows. If $v$ does not begin with a tape symbol, there is none. If $v = a\\,v'$ for a tape symbol $a$: when $M$ has no transition from $q$ reading $a$, it is $u\\,H\\,v'$, where $H$ is the halting symbol; when $M$ writes\u2026","title":"Rewriting at a state symbol"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/step-result","kind":"node","lean":"Cslib.Decidability.TMRewriting.step_result","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#step-result","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape machine with finite tape alphabet and finite set of states. Let $x$ be a control symbol (a state symbol or the halting symbol $H$) and let $u$ and $v$ be words that contain no control symbol. If $u\\,x\\,v$ rewrites in one step to $w'$ under the rules of $M$, then either $x$ is a state symbol $q$ and the successor of $u\\,q\\,v$ under the rules for $q$ is defined and equals $w'$, or $x = H$ and\u2026","title":"A rewrite at the control symbol"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/ctl-count-eq-of-step","kind":"node","lean":"Cslib.Decidability.TMRewriting.ctlCount_eq_of_step","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#ctl-count-eq-of-step","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and call the states and the halting symbol $H$ control symbols. If a word $w$ rewrites to $w'$ in one step of the rewriting system simulating $M$, then $w$ and $w'$ contain the same number of control symbols.","title":"Rewriting preserves the number of control symbols"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/step-confluent","kind":"node","lean":"Cslib.Decidability.TMRewriting.step_confluent","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#step-confluent","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape machine with finite tape alphabet and finite set of states, and let $w$ be a word over the symbols of its rewriting system that contains exactly one control symbol (a state symbol or the halting symbol). If $w$ rewrites in one step to $w_1$ and in one step to $w_2$ under the rules of $M$, then there is a word $d$ such that $w_1$ rewrites to $d$ in at most one step and $w_2$ rewrites to $d$\u2026","title":"Local confluence on words with one control symbol"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/not-step-halt-word","kind":"node","lean":"Cslib.Decidability.TMRewriting.not_step_haltWord","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#not-step-halt-word","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols. There is no word $w$ such that the halting word $\\vdash H \\dashv$ rewrites to $w$ in one step of the rewriting system simulating $M$.","title":"The halting word is irreducible"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/ctl-step","kind":"node","lean":"Cslib.Decidability.TMRewriting.CtlStep","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#ctl-step","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and call the states and the halting symbol $H$ control symbols. A word $w$ rewrites to $w'$ by a controlled step when $w$ contains exactly one control symbol and $w$ rewrites to $w'$ in one step of the rewriting system simulating $M$.","title":"Rewrites from words with one control symbol"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/eqv-gen-ctl-step","kind":"node","lean":"Cslib.Decidability.TMRewriting.eqvGen_ctlStep","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#eqv-gen-ctl-step","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and call the states and the halting symbol $H$ control symbols. Suppose $w$ contains exactly one control symbol, and $w$ and $w'$ are equivalent under the equivalence relation generated by one-step rewriting in the rewriting system simulating $M$. Then $w$ and $w'$ are equivalent under the equivalence relation generated by the\u2026","title":"Thue equivalence from a word with one control symbol uses only controlled steps"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/confluent-ctl-step","kind":"node","lean":"Cslib.Decidability.TMRewriting.confluent_ctlStep","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#confluent-ctl-step","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and call the states and the halting symbol $H$ control symbols. Consider the relation in which a word $w$ with exactly one control symbol rewrites to $w'$ by one step of the rewriting system simulating $M$. This relation is confluent: if $w$ rewrites in zero or more such steps to both $w_1$ and $w_2$, then $w_1$ and $w_2$ both\u2026","title":"The rewrites from words with one control symbol are confluent"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/tm0-halts-on-iff-eqv-gen","kind":"node","lean":"Cslib.Decidability.TMRewriting.tm0HaltsOn_iff_eqvGen","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#tm0-halts-on-iff-eqv-gen","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, write $\\vdash$ and $\\dashv$ for the left and right end markers and $H$ for the halting symbol, and let $l$ be an input word of tape symbols. Then $M$ halts on $l$ if and only if the start word of $l$ and the halting word ${\\vdash}\\,H\\,{\\dashv}$ are equivalent under the reflexive, symmetric and transitive closure of one-step\u2026","title":"A single-tape machine halts iff its start word is Thue equivalent to the halting word"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/many-one-reducible-tm0-halts-on-eqv-gen","kind":"node","lean":"Cslib.Decidability.TMRewriting.manyOneReducible_tm0HaltsOn_eqvGen","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#many-one-reducible-tm0-halts-on-eqv-gen","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $M$ be a single-tape Turing machine with finitely many states and tape symbols, and let $R_M$ be the rewriting system simulating $M$. Then the set of inputs on which $M$ halts is many-one reducible to the set of pairs $(x, y)$ of words over the symbols of $R_M$ that are equivalent under the reflexive, symmetric and transitive closure of one-step rewriting by $R_M$.","title":"Single-tape halting many-one reduces to the word problem of a fixed Thue system"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/exists-not-tm2-decidable-eqv-gen","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_eqvGen","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#exists-not-tm2-decidable-eqv-gen","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"(Post, Markov) There are a finite alphabet $\\alpha$, with a primitive recursive coding of its letters by natural numbers, and a finite list $R$ of rewriting rules over $\\alpha$ such that no multi-stack Turing machine decides, for words $x$ and $y$ over $\\alpha$, whether $x$ and $y$ are equivalent under the reflexive, symmetric and transitive closure of one-step rewriting by $R$.","title":"Some finite Thue system has an undecidable word problem"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/exists-not-tm2-decidable-thue-wp","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_thueWP","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#exists-not-tm2-decidable-thue-wp","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"There is a finite alphabet $\\alpha$, with a primitive recursive coding of its letters by natural numbers, such that no multi-stack Turing machine decides, for a finite list $R$ of rewriting rules over $\\alpha$ and words $x$ and $y$ over $\\alpha$, whether $x$ and $y$ are equivalent under the reflexive, symmetric and transitive closure of one-step rewriting by $R$.","title":"The uniform word problem for Thue systems is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/map-instance","kind":"node","lean":"Cslib.Decidability.Post.mapInstance","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#map-instance","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Given a map $f$ from words over $\\alpha$ to words over $\\beta$ and a list $P$ of dominoes over $\\alpha$, the image of $P$ under $f$ is the list of dominoes over $\\beta$ obtained by replacing each domino $(u, v)$ of $P$ by $(f(u), f(v))$, in the same order.","title":"Image of a PCP instance under a word map"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/pcp-map-iff","kind":"node","lean":"Cslib.Decidability.Post.pcp_map_iff","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#pcp-map-iff","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $h$ map each letter of $\\alpha$ to a word over $\\beta$, and extend it to words by $h(a_1 \\cdots a_n) = h(a_1) \\cdots h(a_n)$. If this extension is injective on words over $\\alpha$, then for every list $P$ of dominoes over $\\alpha$, the list obtained from $P$ by replacing each domino $(u, v)$ by $(h(u), h(v))$ has a PCP solution if and only if $P$ has a PCP solution.","title":"An injective word homomorphism preserves and reflects PCP solvability"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/unary-code","kind":"node","lean":"Cslib.Decidability.Post.unaryCode","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#unary-code","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Given an indexing $e$ assigning a natural number to each letter of $\\alpha$, the unary code of a letter $a$ is the Boolean word consisting of $e(a)$ copies of $\\mathrm{true}$ followed by one $\\mathrm{false}$; writing $1$ for true and $0$ for false, it is $1^{e(a)} 0$.","title":"Unary code of a letter"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/unary-hom-injective","kind":"node","lean":"Cslib.Decidability.Post.unaryHom_injective","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#unary-hom-injective","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $e$ be an injective map from $\\alpha$ to the natural numbers, and let $c(a)$ be the unary code of a letter $a$: $e(a)$ copies of $\\mathrm{true}$ followed by one $\\mathrm{false}$. Then the map sending a word $a_1 \\cdots a_n$ over $\\alpha$ to the Boolean word $c(a_1) \\cdots c(a_n)$ is injective.","title":"The unary code of an injective indexing is uniquely decodable"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/many-one-reducible-pcp-bool","kind":"node","lean":"Cslib.Decidability.Post.manyOneReducible_pcp_bool","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#many-one-reducible-pcp-bool","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"Let $\\alpha$ be a finite alphabet with a primitive recursive encoding of its letters as natural numbers. Then the Post correspondence problem over $\\alpha$ is many-one reducible to the Post correspondence problem over the two-letter alphabet of Booleans $\\mathrm{true}$ and $\\mathrm{false}$: there is a computable map $f$ sending each finite list $P$ of dominoes over $\\alpha$ to a finite list $f(P)$ of dominoes over\u2026","title":"PCP over a finite alphabet many-one reduces to PCP over the binary alphabet"},{"id":"TheoryOfComputation/undecidable-problems/post-correspondence-problem/not-tm2-decidable-pcp-bool","kind":"node","lean":"Cslib.Decidability.not_tm2Decidable_pcp_bool","parent":35169,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/post-correspondence-problem/index.html#not-tm2-decidable-pcp-bool","scope":"TheoryOfComputation/undecidable-problems/post-correspondence-problem","status":"fully_proved","summary":"The Post correspondence problem over the two-letter alphabet of Booleans $\\mathrm{true}$ and $\\mathrm{false}$, that is, the property of a finite list $P$ of dominoes over the Booleans of having a PCP solution, is not decidable by a multi-stack Turing machine: no multi-stack machine computes a total function $g : \\mathbb{N} \\to \\mathbb{N}$ such that, for every such $P$ with natural-number encoding $n_P$,\u2026","title":"The Post correspondence problem is undecidable over the binary alphabet"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages.json","id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","title":"Intersection problem for context-free languages"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/letter-sym","kind":"node","lean":"Cslib.Decidability.Grammar.letterSym","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#letter-sym","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"For a letter $a \\in \\alpha$, this is $a$ as an encoded terminal over the alphabet $\\alpha \\oplus \\mathbb{N}$: the element $\\mathrm{inr}(\\mathrm{inl}\\, a)$ of $\\mathbb{N} \\oplus (\\alpha \\oplus \\mathbb{N})$, whose left summand $\\mathbb{N}$ holds the nonterminals and whose right summand holds the terminals.","title":"The encoded letter terminal"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index-sym","kind":"node","lean":"Cslib.Decidability.Grammar.indexSym","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#index-sym","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"For $i \\in \\mathbb{N}$, this is the index $i$ as an encoded terminal over the alphabet $\\alpha \\oplus \\mathbb{N}$: the element $\\mathrm{inr}(\\mathrm{inr}\\, i)$ of $\\mathbb{N} \\oplus (\\alpha \\oplus \\mathbb{N})$, whose left summand $\\mathbb{N}$ holds the nonterminals and whose right summand holds the terminals.","title":"The encoded index terminal"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/nt-sym","kind":"node","lean":"Cslib.Decidability.Grammar.ntSym","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#nt-sym","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"For a natural number $A$, the nonterminal $A$ as a symbol of the right-hand side of an encoded production over the terminals $\\alpha \\sqcup \\mathbb{N}$: the left injection of $A$ into $\\mathbb{N} \\sqcup (\\alpha \\sqcup \\mathbb{N})$.","title":"Encoded nonterminal symbol"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/domino","kind":"node","lean":"Cslib.Decidability.Grammar.domino","parent":35249,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(i\\)"],["text","-th domino of a list"]],"source":"The $i$-th domino of a list"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#domino","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"For a list $P$ of dominoes and $i \\in \\mathbb{N}$, this is the $i$-th domino of $P$ (counting from $0$) when $i$ is less than the length of $P$, and the domino whose top and bottom words are both empty otherwise.","title":"The $i$-th domino of a list"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/cont-rule","kind":"node","lean":"Cslib.Decidability.Grammar.contRule","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#cont-rule","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"For a list $P$ of dominoes, a map $\\sigma$ assigning a word over $\\alpha$ to each domino, a nonterminal $A \\in \\mathbb{N}$ and an index $i \\in \\mathbb{N}$, this is the production $A \\to w_i\\, A\\, i$ in encoded form, where $w_i$ is $\\sigma$ applied to the $i$-th domino of $P$ (the domino with both words empty when $i$ is past the end of $P$): its right-hand side is the letters of $w_i$ as terminals, then the\u2026","title":"The continuation production"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/term-rule","kind":"node","lean":"Cslib.Decidability.Grammar.termRule","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#term-rule","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $P = (d_0, \\dots, d_{k-1})$ be a list of dominoes over $\\alpha$, set $d_i = (\\varepsilon, \\varepsilon)$ for $i \\ge k$, let $\\sigma$ assign a word over $\\alpha$ to each domino, and write $w_i = \\sigma(d_i)$. For a nonterminal $A$ and a natural number $i$, this is the encoded production $A \\to w_i \\, i$, whose right-hand side is the letters of $w_i$ as terminals followed by the index terminal $i$.","title":"The terminating production of a side grammar"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/side-rules","kind":"node","lean":"Cslib.Decidability.Grammar.sideRules","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#side-rules","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $P = (d_0, \\dots, d_{k-1})$ be a list of dominoes over $\\alpha$, let $\\sigma$ assign a word over $\\alpha$ to each domino, write $w_i = \\sigma(d_i)$, and let $A$ be a nonterminal. The side productions are the list of encoded productions $A \\to w_i \\, A \\, i$ and $A \\to w_i \\, i$ for $i = 0, 1, \\dots, k - 1$, where the letters of $w_i$ are terminals and $i$ is an index terminal.","title":"The productions of a side grammar"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/side-grammar","kind":"node","lean":"Cslib.Decidability.Grammar.sideGrammar","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#side-grammar","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $P = (d_0, \\dots, d_{k-1})$ be a list of dominoes over $\\alpha$, let $\\sigma$ assign a word over $\\alpha$ to each domino, and write $w_i = \\sigma(d_i)$. The side grammar of $\\sigma$ and $P$ is the encoded context-free grammar whose terminals are the letters of $\\alpha$ and the index terminals $0, 1, 2, \\dots$, whose start nonterminal is $S = 0$, and whose productions are $S \\to w_i \\, S \\, i$ and\u2026","title":"The side grammar of a PCP instance"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/flat","kind":"node","lean":"Cslib.Decidability.Grammar.flat","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#flat","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"For a list $P$ of dominoes, a map $\\sigma$ assigning a word over $\\alpha$ to each domino, and a sequence $i_1, \\dots, i_n$ of natural numbers, this is the word $w_{i_1} w_{i_2} \\cdots w_{i_n}$ over $\\alpha$, where $w_i$ is $\\sigma$ applied to the $i$-th domino of $P$ (the domino with both words empty when $i$ is past the end of $P$).","title":"The letter part of an index sequence"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/side-word","kind":"node","lean":"Cslib.Decidability.Grammar.sideWord","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#side-word","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $P = (d_0, \\dots, d_{k-1})$ be a list of dominoes over $\\alpha$, set $d_i = (\\varepsilon, \\varepsilon)$ for $i \\ge k$, let $\\sigma$ assign a word over $\\alpha$ to each domino, and write $w_i = \\sigma(d_i)$. The side word of a sequence $i_1, \\dots, i_n$ of natural numbers is the word $w_{i_1} \\cdots w_{i_n} \\, i_n \\cdots i_1$ over the letters of $\\alpha$ and the index terminals $0, 1, 2, \\dots$: the letters of\u2026","title":"The side word of an index sequence"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/eq-of-derives-terminal","kind":"node","lean":"Cslib.Decidability.Grammar.eq_of_derives_terminal","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#eq-of-derives-terminal","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be a context-free grammar with terminal alphabet $\\alpha \\oplus \\mathbb{N}$ and let $u$ be a word over $\\alpha \\oplus \\mathbb{N}$. If $g$ derives a sentential form $v$ from $u$, read as a string of terminals, then $v = u$.","title":"A terminal word derives only itself"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/eq-of-terminal-append-nonterminal","kind":"node","lean":"Cslib.Decidability.Grammar.eq_of_terminal_append_nonterminal","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#eq-of-terminal-append-nonterminal","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $N$ be a type of nonterminals, $u, v$ words over $\\alpha \\oplus \\mathbb{N}$, $A, B \\in N$, and $p, q$ strings of terminals in $\\alpha \\oplus \\mathbb{N}$ and nonterminals in $N$. If $u\\, A\\, v = p\\, B\\, q$, where $u$ and $v$ are read as strings of terminals, then $p = u$, $B = A$ and $q = v$.","title":"Splitting a sentential form at its single nonterminal"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/eq-side-word-of-derives","kind":"node","lean":"Cslib.Decidability.Grammar.eq_sideWord_of_derives","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#eq-side-word-of-derives","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $P$ be a list of dominoes, $\\sigma$ a map assigning a word over $\\alpha$ to each domino, and $G$ the context-free grammar of the side grammar of $P$ for $\\sigma$, with start nonterminal $S$ and terminals $\\alpha \\oplus \\mathbb{N}$ (letters and indices). Let $u, v, x$ be words over $\\alpha \\oplus \\mathbb{N}$. If $G$ derives the terminal word $x$ from the sentential form $u\\, S\\, v$, then there is a nonempty\u2026","title":"The analysis of derivations"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/derives-side-word","kind":"node","lean":"Cslib.Decidability.Grammar.derives_sideWord","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#derives-side-word","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $P$ be a list of dominoes, $\\sigma$ a map assigning a word over $\\alpha$ to each domino, and $G$ the context-free grammar of the side grammar of $P$ for $\\sigma$, with start nonterminal $S$ and terminals $\\alpha \\oplus \\mathbb{N}$ (letters and indices). For every nonempty sequence $i_1, \\dots, i_n$ of indices, each less than the length of $P$, and all words $u, v$ over $\\alpha \\oplus \\mathbb{N}$, $G$ derives\u2026","title":"The construction of derivations"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/mem-language-side-grammar","kind":"node","lean":"Cslib.Decidability.Grammar.mem_language_sideGrammar","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#mem-language-side-grammar","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with decidable equality, let $P = (d_0, \\dots, d_{k-1})$ be a list of dominoes over $\\alpha$, let $\\sigma$ assign a word over $\\alpha$ to each domino, and write $w_i = \\sigma(d_i)$. A word $u$ over the letters of $\\alpha$ and the index terminals $0, 1, 2, \\dots$ lies in the language of the side grammar of $\\sigma$ and $P$ if and only if there is a nonempty sequence of indices\u2026","title":"The language of a side grammar"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/eq-of-map-inl-append-map-inr","kind":"node","lean":"Cslib.Decidability.Grammar.eq_of_map_inl_append_map_inr","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#eq-of-map-inl-append-map-inr","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $A, C$ be words over $\\alpha$ and $B, D$ words over $\\beta$, all viewed as words over the disjoint union $\\alpha \\oplus \\beta$. If $AB = CD$ as words over $\\alpha \\oplus \\beta$, then $A = C$ and $B = D$.","title":"Unique splitting of letters followed by markers"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/pcp-iff-exists-mem-inter","kind":"node","lean":"Cslib.Decidability.Grammar.pcp_iff_exists_mem_inter","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#pcp-iff-exists-mem-inter","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"Let $P$ be a list of dominoes over an alphabet $\\alpha$ with decidable equality. Then $P$ has a PCP solution if and only if some word lies both in the language of the side grammar of $P$ for the top words and in the language of the side grammar of $P$ for the bottom words.","title":"A PCP instance is solvable exactly when its top and bottom grammars share a word"},{"id":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/many-one-reducible-pcp-inter","kind":"node","lean":"Cslib.Decidability.Grammar.manyOneReducible_pcp_inter","parent":35249,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages/index.html#many-one-reducible-pcp-inter","scope":"TheoryOfComputation/undecidable-problems/intersection-problem-for-context-free-languages","status":"fully_proved","summary":"PCP many-one reduces to nonemptiness of the intersection of two context-free languages. Let $\\alpha$ be an alphabet with a primitive recursive encoding and decidable equality. The Post correspondence problem over $\\alpha$ is many-one reducible to the problem of deciding, for a pair $(G_1, G_2)$ of encoded context-free grammars whose terminals are the letters of $\\alpha$ together with the natural numbers, whether\u2026","title":"PCP many-one reduces to nonemptiness of the intersection of two context-free languages"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars.json","id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","title":"Ambiguity problem for context-free grammars"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/amb-grammar","kind":"node","lean":"Cslib.Decidability.Grammar.ambGrammar","parent":35268,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#amb-grammar","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P = [(t_0, b_0), \\dots, (t_{m-1}, b_{m-1})]$ be a list of dominoes over an alphabet $\\alpha$, with top words $t_i$ and bottom words $b_i$. The ambiguity grammar of $P$ is the encoded context-free grammar over the terminals $\\alpha \\sqcup \\mathbb{N}$ (letters and indices) with nonterminals $S = 0$, $A = 1$, $B = 2$, start symbol $S$, and the productions $S \\to A$ and $S \\to B$ together with, for every $i < m$,\u2026","title":"The ambiguity grammar of a PCP instance"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/canon","kind":"node","lean":"Cslib.Decidability.Grammar.canon","parent":35268,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#canon","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P$ be a list of dominoes over $\\alpha$, let $\\sigma$ assign a word over $\\alpha$ to each domino, let $A$ be a natural number, and for $i \\in \\mathbb{N}$ write $w_i$ for $\\sigma$ applied to the $i$-th domino of $P$ (the domino with both words empty when $i$ is past the end of $P$). For a sequence $i_1, \\dots, i_n$ of natural numbers, this is the list of productions, with terminals $\\alpha \\sqcup \\mathbb{N}$ and\u2026","title":"The canonical rule sequence along an index sequence"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/cases-of-mem-amb-grammar-of-input","kind":"node","lean":"Cslib.Decidability.Grammar.cases_of_mem_ambGrammar_of_input","parent":35268,"reading_title":{"format":"inline-parts-v1","parts":[["text","The productions of "],["math","\\(A\\)"],["text"," and "],["math","\\(B\\)"],["text"," in the ambiguity grammar"]],"source":"The productions of $A$ and $B$ in the ambiguity grammar"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#cases-of-mem-amb-grammar-of-input","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P = [(t_0, b_0), \\dots, (t_{m-1}, b_{m-1})]$ be a list of dominoes, and let $A$, $B$ be the nonterminals of the ambiguity grammar of $P$ for its top and bottom words. Every production of that grammar with left-hand side $A$ is $A \\to t_i\\, A\\, i$ or $A \\to t_i\\, i$ for some $i < m$, and every production with left-hand side $B$ is $B \\to b_i\\, B\\, i$ or $B \\to b_i\\, i$ for some $i < m$.","title":"The productions of $A$ and $B$ in the ambiguity grammar"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/eq-canon-of-left-derivation","kind":"node","lean":"Cslib.Decidability.Grammar.eq_canon_of_leftDerivation","parent":35268,"reading_title":{"format":"inline-parts-v1","parts":[["text","Leftmost derivations from "],["math","\\(A\\)"],["text"," and "],["math","\\(B\\)"],["text"," follow an index sequence"]],"source":"Leftmost derivations from $A$ and $B$ follow an index sequence"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#eq-canon-of-left-derivation","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P = [(t_0, b_0), \\dots, (t_{m-1}, b_{m-1})]$ be a list of dominoes, let $A$, $B$ be the nonterminals of the ambiguity grammar of $P$ for its top and bottom words, and let $u$, $v$ be words of terminals. Every leftmost derivation of a word $w$ from the sentential form $u\\,A\\,v$ is the canonical rule sequence from $A$ for the words $t_i$ along some nonempty sequence $i_1, \\dots, i_n$ of indices less than $m$,\u2026","title":"Leftmost derivations from $A$ and $B$ follow an index sequence"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/left-derivation-canon","kind":"node","lean":"Cslib.Decidability.Grammar.leftDerivation_canon","parent":35268,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#left-derivation-canon","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P = [(t_0, b_0), \\dots, (t_{m-1}, b_{m-1})]$ be a list of dominoes, and let $A$, $B$ be the nonterminals of the ambiguity grammar of $P$ for its top and bottom words. For every nonempty sequence $i_1, \\dots, i_n$ of indices less than $m$ and all words of terminals $u$, $v$, the canonical rule sequence from $A$ for the words $t_i$ along $i_1, \\dots, i_n$ is a leftmost derivation of\u2026","title":"The leftmost derivation along an index sequence"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/start-rule","kind":"node","lean":"Cslib.Decidability.Grammar.startRule","parent":35268,"reading_title":{"format":"inline-parts-v1","parts":[["text","The start production "],["math","\\(S \\to A\\)"]],"source":"The start production $S \\to A$"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#start-rule","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"For an alphabet $\\alpha$ and a natural number $A$, this is the production with terminals $\\alpha \\sqcup \\mathbb{N}$ and nonterminals $\\mathbb{N}$ whose left-hand side is the nonterminal $S = 0$ and whose right-hand side is the single nonterminal $A$.","title":"The start production $S \\to A$"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/cases-of-left-derivation-start","kind":"node","lean":"Cslib.Decidability.Grammar.cases_of_leftDerivation_start","parent":35268,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#cases-of-left-derivation-start","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P = [(t_0, b_0), \\dots, (t_{m-1}, b_{m-1})]$ be a list of dominoes, and let $S$, $A$, $B$ be the nonterminals of the ambiguity grammar of $P$. Every leftmost derivation of a word $w$ from $S$ in that grammar has one of two forms, for some nonempty sequence $i_1, \\dots, i_n$ of indices less than $m$: either it is $S \\to A$ followed by the canonical rule sequence from $A$ for the words $t_i$ along\u2026","title":"Leftmost derivations in the ambiguity grammar"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/left-derivation-start","kind":"node","lean":"Cslib.Decidability.Grammar.leftDerivation_start","parent":35268,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#left-derivation-start","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P = [(t_0, b_0), \\dots, (t_{m-1}, b_{m-1})]$ be a list of dominoes, and let $S$, $A$, $B$ be the nonterminals of the ambiguity grammar of $P$. For every nonempty sequence $i_1, \\dots, i_n$ of indices less than $m$, the production $S \\to A$ followed by the canonical rule sequence from $A$ for the words $t_i$ along $i_1, \\dots, i_n$ is a leftmost derivation of $t_{i_1} \\cdots t_{i_n}\\, i_n \\cdots i_1$ from $S$,\u2026","title":"Leftmost derivations from the start symbol along an index sequence"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/is-ambiguous-iff-pcp","kind":"node","lean":"Cslib.Decidability.Grammar.isAmbiguous_iff_pcp","parent":35268,"reading_title":{"format":"inline-parts-v1","parts":[["text","The ambiguity grammar of "],["math","\\(P\\)"],["text"," is ambiguous exactly when "],["math","\\(P\\)"],["text"," has a solution"]],"source":"The ambiguity grammar of $P$ is ambiguous exactly when $P$ has a solution"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#is-ambiguous-iff-pcp","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"For every list $P$ of dominoes, the ambiguity grammar of $P$ is ambiguous, that is, some word has two different leftmost derivations from its start symbol, if and only if $P$ has a PCP solution.","title":"The ambiguity grammar of $P$ is ambiguous exactly when $P$ has a solution"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/primrec-amb-grammar","kind":"node","lean":"Cslib.Decidability.Grammar.primrec_ambGrammar","parent":35268,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#primrec-amb-grammar","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"For every alphabet $\\alpha$ with a primitive recursive encoding by natural numbers, the map sending a list $P$ of dominoes over $\\alpha$ to the ambiguity grammar of $P$ is primitive recursive, where encoded grammars are encoded as pairs of a start nonterminal and a list of productions.","title":"The ambiguity grammar is primitive recursive"},{"id":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/many-one-reducible-pcp-is-ambiguous","kind":"node","lean":"Cslib.Decidability.Grammar.manyOneReducible_pcp_isAmbiguous","parent":35268,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars/index.html#many-one-reducible-pcp-is-ambiguous","scope":"TheoryOfComputation/undecidable-problems/ambiguity-problem-for-context-free-grammars","status":"fully_proved","summary":"For every alphabet $\\alpha$ with a primitive recursive encoding by natural numbers and decidable equality, the Post correspondence problem over $\\alpha$ many-one reduces to ambiguity of encoded context-free grammars over the terminals $\\alpha \\sqcup \\mathbb{N}$: there is a computable function sending each list $P$ of dominoes over $\\alpha$ to such a grammar, which is ambiguous if and only if $P$ has a PCP solution.","title":"PCP many-one reduces to ambiguity of context-free grammars"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/string-rewriting-systems.json","id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","title":"String rewriting systems"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/rule","kind":"node","lean":"Cslib.Decidability.Rewriting.Rule","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#rule","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"A rewriting rule over an alphabet $\\alpha$ is a pair $(l, r)$ of words over $\\alpha$; applying it replaces an occurrence of $l$ by $r$.","title":"Rewriting rules"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/step","kind":"node","lean":"Cslib.Decidability.Rewriting.Step","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#step","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $R$ be a string rewriting system, given as a list of rules. A word $w$ rewrites to a word $w'$ in one step of $R$ when there are a rule $(l, r)$ of $R$ and words $u$ and $v$ with $w = u l v$ and $w' = u r v$.","title":"One-step rewriting"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/derives","kind":"node","lean":"Cslib.Decidability.Rewriting.Derives","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#derives","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"For a list of rules $R$, a word $x$ derives a word $y$ when $x$ rewrites to $y$ in zero or more steps of $R$, that is, when $(x, y)$ lies in the reflexive-transitive closure of the one-step rewriting relation of $R$.","title":"Derivations in a string rewriting system"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/sr","kind":"node","lean":"Cslib.Decidability.Rewriting.SR","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#sr","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"An instance $(R, x, y)$ consists of a string rewriting system $R$, given as a list of rules, and two words $x$ and $y$; it is a yes-instance of the reachability problem when $x$ rewrites to $y$ in zero or more steps of $R$.","title":"The reachability problem for string rewriting"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/nonempty-rules","kind":"node","lean":"Cslib.Decidability.Rewriting.NonemptyRules","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#nonempty-rules","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"A string rewriting system $R$, given as a list of rules $(l, r)$, has nonempty sides when every rule of $R$ has a nonempty left side $l$ and a nonempty right side $r$.","title":"Rewriting systems with nonempty sides"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/lift","kind":"node","lean":"Cslib.Decidability.Rewriting.lift","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#lift","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"A word $a_1 \\cdots a_n$ over $\\alpha$ is sent to the word $a_1 \\cdots a_n$ over the extended alphabet of $\\alpha$ in which each $a_i$ is read as a letter.","title":"Lifting a word to the extended alphabet"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/start-card","kind":"node","lean":"Cslib.Decidability.Rewriting.startCard","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#start-card","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $x$ be a word over an alphabet $\\alpha$, and extend $\\alpha$ by a star $\\star$ and an end marker $\\diamond$. The start domino of $x$ has top the one-letter word $\\diamond$ and bottom the word $\\diamond x \\star$, with the letters of $x$ read as letters of the extended alphabet.","title":"The start domino of the reduction from reachability to MPCP"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/final-card","kind":"node","lean":"Cslib.Decidability.Rewriting.finalCard","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#final-card","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"For a word $y$ over $\\alpha$, the final domino of $y$ is the domino over the extended alphabet with top $y \\star \\diamond$ and bottom $\\diamond$, where $y$ is read as a word of letters, $\\star$ is the row separator and $\\diamond$ is the end marker.","title":"Final domino"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/sep-card","kind":"node","lean":"Cslib.Decidability.Rewriting.sepCard","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#sep-card","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Over the extended alphabet, that is, an alphabet $\\alpha$ together with a star $\\star$ and an end marker $\\diamond$, the separator domino has top and bottom both equal to the one-letter word $\\star$; the star separates the rows of a derivation.","title":"The separator domino"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/rule-card","kind":"node","lean":"Cslib.Decidability.Rewriting.ruleCard","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#rule-card","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"For a rule $(l, r)$ over an alphabet $\\alpha$, the domino of the rule has top $l$ and bottom $r$, read as words over the extended alphabet, that is, $\\alpha$ together with a star and an end marker.","title":"The domino of a rewriting rule"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/copy-card","kind":"node","lean":"Cslib.Decidability.Rewriting.copyCard","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#copy-card","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"For a letter $a$ of $\\alpha$, the copy domino of $a$ is the domino over the extended alphabet whose top and bottom are both the one-letter word $a$.","title":"Copy domino"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/symbols","kind":"node","lean":"Cslib.Decidability.Rewriting.symbols","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#symbols","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"For an instance $(R, x, y)$ of the reachability problem, where $R = [(l_1, r_1), \\dots, (l_k, r_k)]$ is a list of rewriting rules and $x$, $y$ are words, the letters of the instance form the word $l_1 r_1 l_2 r_2 \\cdots l_k r_k \\, x \\, y$, a list in which a letter may occur several times.","title":"Letters of a rewriting instance"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/to-mpcp","kind":"node","lean":"Cslib.Decidability.Rewriting.toMPCP","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#to-mpcp","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $(R, x, y)$ be an instance of the reachability problem over an alphabet $\\alpha$, where $R = [(l_1, r_1), \\dots, (l_k, r_k)]$, and write $\\#$ for the separator and $\\diamond$ for the end marker of the extended alphabet $\\mathrm{Sym}(\\alpha)$, in which letters of $\\alpha$ are read as letters. Let $a_1 \\cdots a_m$ be the word $l_1 r_1 l_2 r_2 \\cdots l_k r_k \\, x \\, y$, the letters of the instance. The MPCP\u2026","title":"The MPCP instance of a reachability instance"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/letter-ne-star","kind":"node","lean":"Cslib.Decidability.Rewriting.letter_ne_star","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#letter-ne-star","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"For every letter $a$ of the alphabet $\\alpha$, the symbol $a$ of the extended alphabet of the MPCP-to-PCP reduction is different from the star $\\star$.","title":"Letters differ from the star"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/exists-of-lift-append-eq","kind":"node","lean":"Cslib.Decidability.Rewriting.exists_of_lift_append_eq","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#exists-of-lift-append-eq","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $u, v$ be words over $\\alpha$, read as words of letters, and let $X, Y$ be words over the extended alphabet of $\\alpha$. If $u X = v \\star Y$, where $\\star$ is the row separator, then there is a word $w$ over $\\alpha$ with $v = u w$ and $X = w \\star Y$.","title":"Comparing words of letters before a separator"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/derives-of-top-eq","kind":"node","lean":"Cslib.Decidability.Rewriting.derives_of_top_eq","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#derives-of-top-eq","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $(R, x, y)$ be a reachability instance and read words over $\\alpha$ as words of letters. Let $A$ be a sequence of dominoes of the MPCP instance of $(R, x, y)$. For all words $v, s, t$ over $\\alpha$, if $s$ rewrites to $t$ under $R$ in zero or more steps and the top of $A$ equals $v \\star t$ followed by the bottom of $A$, where $\\star$ is the row separator, then $s v$ rewrites to $y$ under $R$ in zero or more\u2026","title":"The backward direction of the MPCP reduction"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/mem-symbols-of-step","kind":"node","lean":"Cslib.Decidability.Rewriting.mem_symbols_of_step","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#mem-symbols-of-step","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $(R, x, y)$ be an instance of the reachability problem over an alphabet $\\alpha$. If a word $w$ rewrites to a word $w'$ in one step of $R$ and every letter of $w$ is among the letters of the instance, then every letter of $w'$ is among the letters of the instance.","title":"Rewriting steps keep the letters of the instance"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/exists-cards-of-derives","kind":"node","lean":"Cslib.Decidability.Rewriting.exists_cards_of_derives","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#exists-cards-of-derives","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $(R, x, y)$ be a reachability instance and read words over $\\alpha$ as words of letters. If a word $w$ over $\\alpha$ rewrites to $y$ under $R$ in zero or more steps and every letter of $w$ occurs in a rule of $R$, in $x$ or in $y$, then there is a sequence $A$ of dominoes of the MPCP instance of $(R, x, y)$ whose top is $w \\star$ followed by the bottom of $A$, where $\\star$ is the row separator.","title":"The forward direction of the MPCP reduction"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/sr-iff-mpcp-to-mpcp","kind":"node","lean":"Cslib.Decidability.Rewriting.sr_iff_mpcp_toMPCP","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#sr-iff-mpcp-to-mpcp","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"For an instance $(R, x, y)$ of the reachability problem, $x$ rewrites to $y$ in zero or more steps of $R$ if and only if the MPCP instance of $(R, x, y)$ has an MPCP solution, that is, a solution that starts with its first domino.","title":"Reachability reduces to MPCP"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/reduce","kind":"node","lean":"Cslib.Decidability.Rewriting.reduce","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#reduce","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"On an instance $(R, x, y)$ of the reachability problem, the reduction returns the MPCP instance of $(R, x, y)$ when every rule of $R$ has a nonempty left side and a nonempty right side, and otherwise the empty list of dominoes, which has no solution.","title":"The total reduction of reachability with nonempty sides to MPCP"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/many-one-reducible-sr-mpcp","kind":"node","lean":"Cslib.Decidability.Rewriting.manyOneReducible_sr_mpcp","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#many-one-reducible-sr-mpcp","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with a primitive recursive coding, and let $\\alpha'$ be $\\alpha$ extended with two new symbols, the star and the end marker. There is a computable function $f$ from instances $(R, x, y)$ of the reachability problem over $\\alpha$, where $R$ is a list of rewriting rules over $\\alpha$ and $x$ and $y$ are words over $\\alpha$, to finite lists of dominoes over $\\alpha'$ such that, for every\u2026","title":"Reachability with nonempty sides many-one reduces to MPCP with nonempty words"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/many-one-reducible-sr-pcp","kind":"node","lean":"Cslib.Decidability.Rewriting.manyOneReducible_sr_pcp","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#many-one-reducible-sr-pcp","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"Let $\\alpha$ be an alphabet with a primitive recursive coding, let $\\alpha'$ be $\\alpha$ extended with two new symbols, the star and the end marker, and let $\\alpha''$ be $\\alpha'$ extended in the same way by two further new symbols. There is a computable function $f$ from instances $(R, x, y)$ of the reachability problem over $\\alpha$, where $R$ is a list of rewriting rules over $\\alpha$ and $x$ and $y$ are words\u2026","title":"Reachability for rules with nonempty sides many-one reduces to PCP"},{"id":"TheoryOfComputation/undecidable-problems/string-rewriting-systems/thue-wp","kind":"node","lean":"Cslib.Decidability.Rewriting.ThueWP","parent":35280,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/string-rewriting-systems/index.html#thue-wp","scope":"TheoryOfComputation/undecidable-problems/string-rewriting-systems","status":"fully_proved","summary":"On an instance $(R, x, y)$, where $R$ is a list of rewriting rules and $x$, $y$ are words, the word problem asks whether $x$ and $y$ are related by the equivalence relation generated by the one-step rewriting relation of $R$, that is, by its reflexive, symmetric and transitive closure.","title":"The word problem of a Thue system"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars.json","id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","title":"Universality problem for context-free grammars"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/bit-code","kind":"node","lean":"Cslib.Decidability.Grammar.bitCode","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#bit-code","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"The code of an index $i \\in \\mathbb{N}$ is the bit word $\\mathbf{t}\\,\\mathbf{f}^i$ of length $i + 1$: the bit $\\mathbf{t}$ (true) followed by $i$ copies of the bit $\\mathbf{f}$ (false); with $|$ for true and $1$ for false, it is the word $|\\,1^i$.","title":"Unary code of an index"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/bit-codes","kind":"node","lean":"Cslib.Decidability.Grammar.bitCodes","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#bit-codes","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a finite sequence $I = (i_1, \\dots, i_n)$ of natural numbers, its code is the bit word $c(i_1)\\,c(i_2) \\cdots c(i_n)$, the concatenation of the unary codes $c(i) = \\mathbf{t}\\,\\mathbf{f}^i$ of its entries; the empty sequence has the empty code.","title":"Concatenated code of an index sequence"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/list-word","kind":"node","lean":"Cslib.Decidability.Grammar.listWord","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#list-word","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P = (d_0, \\dots, d_{m-1})$ be a list of dominoes over $\\alpha$, let $s$ be a function sending each domino to a word over $\\alpha$, and set $w_i = s(d_i)$ for $i < m$ and $w_i = s(\\varepsilon, \\varepsilon)$ for $i \\ge m$, where $(\\varepsilon, \\varepsilon)$ is the domino with empty top and bottom. The list word of an index sequence $i_1, \\dots, i_n$ is the word $w_{i_1} \\cdots w_{i_n}\\, c(i_n) \\cdots c(i_1)$\u2026","title":"List word of an index sequence"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/list-lang","kind":"node","lean":"Cslib.Decidability.Grammar.listLang","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#list-lang","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P = (d_0, \\dots, d_{m-1})$ be a list of dominoes over $\\alpha$, let $s$ be a function sending each domino to a word over $\\alpha$ (for instance its top word), and write $w_i = s(d_i)$. The list language of $s$ and $P$ is the language over $\\alpha \\oplus \\mathrm{Bool}$ of all words $w_{i_1} \\cdots w_{i_n}\\, c(i_n) \\cdots c(i_1)$ with $n \\ge 1$ and $i_1, \\dots, i_n < m$, where $c(i)$ is the unary code of $i$:\u2026","title":"List language of a PCP instance"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/bit-codes-injective","kind":"node","lean":"Cslib.Decidability.Grammar.bitCodes_injective","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#bit-codes-injective","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a finite sequence $I = (i_1, \\dots, i_n)$ of natural numbers write $c(I) = c(i_1) \\cdots c(i_n)$, where $c(i) = \\mathbf{t}\\,\\mathbf{f}^i$ is the unary code of $i$. If $I$ and $J$ are finite sequences of natural numbers with $c(I) = c(J)$, then $I = J$.","title":"Concatenated index codes are uniquely decodable"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/pcp-iff-exists-mem-list-lang","kind":"node","lean":"Cslib.Decidability.Grammar.pcp_iff_exists_mem_listLang","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#pcp-iff-exists-mem-list-lang","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"A list $P$ of dominoes has a PCP solution if and only if some word belongs both to the list language of the top words of $P$ and to the list language of the bottom words of $P$.","title":"A PCP instance is solvable exactly when its top and bottom list languages meet"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/u-nt","kind":"node","lean":"Cslib.Decidability.Grammar.uNt","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#u-nt","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a type $\\alpha$ and a natural number $A$, the nonterminal $A$ as a symbol of an encoded grammar over the alphabet $\\alpha \\oplus \\mathrm{Bool}$, that is, the element $\\mathrm{inl}\\, A$ of $\\mathbb{N} \\oplus (\\alpha \\oplus \\mathrm{Bool})$.","title":"Encoded nonterminal"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/u-letter","kind":"node","lean":"Cslib.Decidability.Grammar.uLetter","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#u-letter","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a letter $a \\in \\alpha$, the terminal $a$ of the alphabet $\\alpha \\oplus \\mathrm{Bool}$ as a symbol of an encoded grammar, that is, the element $\\mathrm{inr}(\\mathrm{inl}\\, a)$ of $\\mathbb{N} \\oplus (\\alpha \\oplus \\mathrm{Bool})$.","title":"Encoded letter terminal"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/u-bit","kind":"node","lean":"Cslib.Decidability.Grammar.uBit","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#u-bit","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a type $\\alpha$ and a bit $b$, the terminal $b$ of the alphabet $\\alpha \\oplus \\mathrm{Bool}$ as a symbol of an encoded grammar, that is, the element $\\mathrm{inr}(\\mathrm{inr}\\, b)$ of $\\mathbb{N} \\oplus (\\alpha \\oplus \\mathrm{Bool})$.","title":"Encoded bit terminal"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/all-letters","kind":"node","lean":"Cslib.Decidability.Grammar.allLetters","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#all-letters","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a finite alphabet $\\alpha$, the list of all letters of $\\alpha$, each appearing exactly once, in some fixed but unspecified order.","title":"List of all letters"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/others","kind":"node","lean":"Cslib.Decidability.Grammar.others","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#others","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a letter $a$ of a finite alphabet with decidable equality, the list of all letters $b$ of the alphabet with $b \\ne a$, in the order of a fixed enumeration of the alphabet.","title":"The letters other than a given letter"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/base-rules","kind":"node","lean":"Cslib.Decidability.Grammar.baseRules","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#base-rules","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a finite alphabet $\\alpha$ and $k \\in \\mathbb{N}$, this is a list of productions whose nonterminals $N_0, N_1, \\dots$ are encoded as natural numbers and whose terminals are the letters $a \\in \\alpha$ and the two bits $\\mathbf{t}$ (true) and $\\mathbf{f}$ (false). Writing $c(i) = \\mathbf{t}\\,\\mathbf{f}^i$ for the unary code of an index $i$, the list consists of the productions $N_3 \\to \\varepsilon$,\u2026","title":"Domino-independent productions of the universality grammar"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/mismatch-rules","kind":"node","lean":"Cslib.Decidability.Grammar.mismatchRules","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#mismatch-rules","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"The mismatch productions of a nonterminal $A$ for one side of a list $P$ of dominoes over a finite alphabet. Let $s$ be a map from dominoes to words (in use, the top or the bottom). For each index $i < |P|$ write $w = s(d_i)$ for the $i$-th domino $d_i$ of $P$ and $\\mathrm{code}(i)$ for the unary code of $i$ as bits. With letters and bits as terminals and $N_3$, $N_4$, $N_7$ the nonterminals $3$, $4$, $7$ (meant to\u2026","title":"The mismatch productions"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/univ-grammar","kind":"node","lean":"Cslib.Decidability.Grammar.univGrammar","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#univ-grammar","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a finite alphabet $\\alpha$ with decidable equality and a list $P$ of dominoes over $\\alpha$, the encoded context-free grammar over $\\alpha \\oplus \\mathrm{Bool}$ with start nonterminal $0$ whose productions are the base productions for $k = |P|$, followed by the mismatch productions of the top words of $P$ with nonterminal $1$, followed by the mismatch productions of the bottom words of $P$ with nonterminal $2$.","title":"The universality grammar"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/letters-lang","kind":"node","lean":"Cslib.Decidability.Grammar.lettersLang","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#letters-lang","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For an alphabet $\\alpha$, the language of words of letters is the set of all words over $\\alpha \\oplus \\mathrm{Bool}$ whose symbols all lie in $\\alpha$, that is, the words over $\\alpha$ (including the empty word) viewed as words over $\\alpha \\oplus \\mathrm{Bool}$.","title":"Words of letters"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/letters1-lang","kind":"node","lean":"Cslib.Decidability.Grammar.letters1Lang","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#letters1-lang","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For an alphabet $\\alpha$, the language of nonempty words of letters is the set of all nonempty words over $\\alpha \\oplus \\mathrm{Bool}$ whose symbols all lie in $\\alpha$, that is, the nonempty words over $\\alpha$ viewed as words over $\\alpha \\oplus \\mathrm{Bool}$.","title":"Nonempty words of letters"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/bits-lang","kind":"node","lean":"Cslib.Decidability.Grammar.bitsLang","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#bits-lang","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For an alphabet $\\alpha$, the language over the disjoint union of $\\alpha$ and the bits $\\{\\mathbf{t}, \\mathbf{f}\\}$ consisting of the words in which every symbol is a bit, including the empty word.","title":"Words of bits"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/codes-lang","kind":"node","lean":"Cslib.Decidability.Grammar.codesLang","parent":35304,"reading_title":{"format":"inline-parts-v1","parts":[["text","Codes of index sequences below "],["math","\\(k\\)"]],"source":"Codes of index sequences below $k$"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#codes-lang","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For an alphabet $\\alpha$ and $k \\in \\mathbb{N}$, the language over the disjoint union of $\\alpha$ and the bits $\\{\\mathbf{t}, \\mathbf{f}\\}$ consisting of the bit words $c(j_1) \\cdots c(j_n)$ with $n \\ge 0$ and $j_1, \\dots, j_n < k$, where $c(j) = \\mathbf{t}\\,\\mathbf{f}^j$ is the unary code of $j$. In particular it contains the empty word.","title":"Codes of index sequences below $k$"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/mismatch-lang","kind":"node","lean":"Cslib.Decidability.Grammar.mismatchLang","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#mismatch-lang","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"The mismatch language of one side of a list $P$ of dominoes. Let $s$ be a map from dominoes to words (in use, the top or the bottom), write $w_i = s(d_i)$ for the $i$-th domino $d_i$ of $P$ and $\\mathrm{code}(i)$ for the unary code of $i$ as bits. The language, over letters and bits, consists of the words $x\\,\\mathrm{code}(i_n) \\cdots \\mathrm{code}(i_1)$ where $x$ is a word of letters and $i_1, \\dots, i_n$ with\u2026","title":"The mismatch language"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/malformed-lang","kind":"node","lean":"Cslib.Decidability.Grammar.malformedLang","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#malformed-lang","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Consider words over $\\alpha \\oplus \\mathrm{Bool}$, whose symbols are letters from $\\alpha$ and bits, and read the bit $\\mathrm{true}$ as the separator $|$ and the bit $\\mathrm{false}$ as $1$. For $k \\in \\mathbb{N}$, a word is malformed with bound $k$ when at least one of the following holds: it consists of letters only (possibly none); it is a word of letters followed by a nonempty word of bits whose first bit is\u2026","title":"Malformed words"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/univ-val","kind":"node","lean":"Cslib.Decidability.Grammar.univVal","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#univ-val","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"For a list $P$ of dominoes over $\\alpha$, the assignment of a language over $\\alpha \\oplus \\mathrm{Bool}$ to each natural number that sends $0$ to the union of the malformed language for $k = |P|$ and the mismatch languages of the top words and of the bottom words of $P$; $1$ to the mismatch language of the top words of $P$; $2$ to the mismatch language of the bottom words of $P$; $3$ to the words of letters; $4$\u2026","title":"Intended languages of the nonterminals of the universality grammar"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/mem-eval-form-one","kind":"node","lean":"Cslib.Decidability.Grammar.mem_evalForm_one","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#mem-eval-form-one","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $V$ assign a language over $T$ to each nonterminal. For terminal words $u$ and $v$, a nonterminal $A$ and a word $w$, the word $w$ lies in the language that $V$ assigns to the sentential form $u\\,A\\,v$ if and only if $w = u z v$ for some $z \\in V(A)$.","title":"Evaluating a form with one nonterminal"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/mem-eval-form-two","kind":"node","lean":"Cslib.Decidability.Grammar.mem_evalForm_two","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#mem-eval-form-two","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $V$ assign a language over $T$ to each nonterminal. For terminal words $u$, $v$ and $x$, nonterminals $A$ and $B$ and a word $w$, the word $w$ lies in the language that $V$ assigns to the sentential form $u\\,A\\,v\\,B\\,x$ if and only if $w = u z_1 v z_2 x$ for some $z_1 \\in V(A)$ and $z_2 \\in V(B)$.","title":"Evaluating a form with two nonterminals"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/is-valuation-univ-val","kind":"node","lean":"Cslib.Decidability.Grammar.isValuation_univVal","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#is-valuation-univ-val","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $\\alpha$ be a finite alphabet with decidable equality and let $P$ be a list of dominoes over $\\alpha$. The assignment of intended languages to the nonterminals of the universality grammar of $P$ is a valuation of that grammar: for every production $A \\to \\gamma$, the language that $\\gamma$ denotes under the assignment is contained in the language assigned to $A$.","title":"The intended languages form a valuation of the universality grammar"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/exists-mismatch-of-not-prefix","kind":"node","lean":"Cslib.Decidability.Grammar.exists_mismatch_of_not_prefix","parent":35304,"reading_title":{"format":"inline-parts-v1","parts":[["text","A word that does not start with "],["math","\\(w\\)"],["text"," leaves "],["math","\\(w\\)"],["text"," at some position"]],"source":"A word that does not start with $w$ leaves $w$ at some position"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#exists-mismatch-of-not-prefix","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $w$ and $x$ be words over an alphabet $\\alpha$ such that $w$ is not a prefix of $x$. Then there is a position $m < |w|$ with the following property: writing $w = u\\,c\\,r$ with $|u| = m$ and $c \\in \\alpha$, either $x = u$, or $x = u\\,a\\,y$ for some letter $a \\ne c$ and some word $y$.","title":"A word that does not start with $w$ leaves $w$ at some position"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/mem-mismatch-rules","kind":"node","lean":"Cslib.Decidability.Grammar.mem_mismatchRules","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#mem-mismatch-rules","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P$ be a list of dominoes over a finite alphabet with decidable equality, $A$ a nonterminal, $s$ a map from dominoes to words (in use, the top or the bottom), $i < |P|$ an index, $w = s(d_i)$ for the $i$-th domino $d_i$ of $P$, and $\\mathrm{code}(i)$ the unary code of $i$ as bits. Write letters and bits as terminals and $N_3$, $N_4$, $N_7$ for the nonterminals $3$, $4$ and $7$. Each of the following encoded\u2026","title":"Membership in the mismatch productions"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/mismatch-mem","kind":"node","lean":"Cslib.Decidability.Grammar.mismatch_mem","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#mismatch-mem","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P$ be a list of dominoes over a finite alphabet, $s$ a map from dominoes to words (in use, the top or the bottom), $w_i = s(d_i)$ for the $i$-th domino $d_i$ of $P$, and $\\mathrm{code}(i)$ the unary code of $i$ as bits. Let $A$ be a nonterminal such that every mismatch production of $A$ for $s$ and $P$ is a production of the universality grammar of $P$. Then for every nonempty sequence $i_1, \\dots, i_n$ of\u2026","title":"Every mismatched word is derivable from the mismatch nonterminal"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/language-univ-grammar","kind":"node","lean":"Cslib.Decidability.Grammar.language_univGrammar","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#language-univ-grammar","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $\\alpha$ be a finite alphabet with decidable equality, let $P$ be a list of dominoes over $\\alpha$ of length $m$, and for $i < m$ let $t_i$ and $b_i$ be the top and bottom words of the domino of $P$ at position $i$, counting from $0$. Read the bit $\\mathrm{true}$ as the separator $|$ and the bit $\\mathrm{false}$ as $1$, and let $c(i)$ be the unary code of $i$: the bit $|$ followed by $i$ bits $1$. The language\u2026","title":"The language of the universality grammar"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/append-inr-inl-ne","kind":"node","lean":"Cslib.Decidability.Grammar.append_inr_inl_ne","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#append-inr-inl-ne","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types and consider words over the disjoint union $\\alpha \\sqcup \\beta$. For all words $u, v$ over $\\alpha \\sqcup \\beta$, all $b \\in \\beta$ and $a \\in \\alpha$, all words $X$ over $\\alpha$ and all words $Y$ over $\\beta$, we have $u\\,b\\,a\\,v \\ne X\\,Y$: a word over $\\alpha$ followed by a word over $\\beta$ never has a symbol of $\\beta$ immediately followed by a symbol of $\\alpha$.","title":"No right symbol directly precedes a left symbol in a left-then-right word"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/exists-suffix-of-inr","kind":"node","lean":"Cslib.Decidability.Grammar.exists_suffix_of_inr","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#exists-suffix-of-inr","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types, call the elements of $\\alpha$ letters and those of $\\beta$ bits, and regard words over $\\alpha$ and over $\\beta$ as words over the disjoint union $\\alpha \\oplus \\beta$. If $u\\,b\\,z = X\\,Y$ for words $u, z$ over $\\alpha \\oplus \\beta$, a bit $b$, a word $X$ of letters and a word $Y$ of bits, then there are words $Y_1, Y_2$ of bits with $Y = Y_1 Y_2$ and $b\\,z = Y_2$.","title":"A bit inside a word of letters followed by bits starts a suffix of the bits"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/bit-codes-ne-append-bit-code","kind":"node","lean":"Cslib.Decidability.Grammar.bitCodes_ne_append_bitCode","parent":35304,"reading_title":{"format":"inline-parts-v1","parts":[["text","Codes of indices below "],["math","\\(k\\)"],["text"," contain no code of "],["math","\\(k\\)"]],"source":"Codes of indices below $k$ contain no code of $k$"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#bit-codes-ne-append-bit-code","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $k \\in \\mathbb{N}$ and let $J = (j_1, \\dots, j_n)$ be a finite sequence of natural numbers with $j_1, \\dots, j_n < k$. Write $c(i) = \\mathbf{t}\\,\\mathbf{f}^i$ for the unary code of $i$ and $c(J) = c(j_1) \\cdots c(j_n)$. Then there are no bit words $Y$ and $Z$ with $c(J) = Y\\,c(k)\\,Z$.","title":"Codes of indices below $k$ contain no code of $k$"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/not-mem-list-lang-of-malformed","kind":"node","lean":"Cslib.Decidability.Grammar.not_mem_listLang_of_malformed","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#not-mem-list-lang-of-malformed","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P$ be a list of dominoes and $s$ a map from dominoes to words (in use, the top or the bottom). Write the code of an index $i$ as the bits $|\\,1^i$, where $|$ stands for the bit true and $1$ for the bit false. A word over letters and bits that is malformed with respect to $|P|$ (it has no bits, its first bit is a $1$, a bit is followed by a letter, or it contains the factor $|\\,1^{|P|}$) does not belong to the\u2026","title":"Malformed words are in no list language"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/not-mem-list-lang-of-mismatch","kind":"node","lean":"Cslib.Decidability.Grammar.not_mem_listLang_of_mismatch","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#not-mem-list-lang-of-mismatch","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $P$ be a list of dominoes and $s$ a map from dominoes to words (in use, the top or the bottom). No word of the mismatch language of $s$ for $P$ belongs to the list language of $s$ for $P$.","title":"Mismatched words are not in the list language"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/malformed-or-well-formed","kind":"node","lean":"Cslib.Decidability.Grammar.malformed_or_wellFormed","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#malformed-or-well-formed","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $k \\in \\mathbb{N}$ and let $w$ be a word over $\\alpha \\oplus \\mathrm{Bool}$. Then $w$ is malformed with bound $k$, or $w = x\\, c(j_1) \\cdots c(j_n)$ for a word $x$ over $\\alpha$ and a nonempty sequence of indices $j_1, \\dots, j_n < k$, where $c(j)$ is the unary code of $j$: the bit $\\mathrm{true}$ followed by $j$ bits $\\mathrm{false}$.","title":"Every word is malformed or well formed"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/forall-mem-language-univ-grammar-iff","kind":"node","lean":"Cslib.Decidability.Grammar.forall_mem_language_univGrammar_iff","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#forall-mem-language-univ-grammar-iff","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $\\alpha$ be a finite alphabet with decidable equality and let $P$ be a list of dominoes over $\\alpha$. The universality grammar of $P$ generates every word over $\\alpha \\oplus \\mathrm{Bool}$ if and only if the Post correspondence problem $P$ has no solution.","title":"The universality grammar generates every word exactly when PCP has no solution"},{"id":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/many-one-reducible-not-pcp-univ","kind":"node","lean":"Cslib.Decidability.Grammar.manyOneReducible_not_pcp_univ","parent":35304,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars/index.html#many-one-reducible-not-pcp-univ","scope":"TheoryOfComputation/undecidable-problems/universality-problem-for-context-free-grammars","status":"fully_proved","summary":"Let $\\alpha$ be a finite alphabet with decidable equality and a primitive recursive encoding. There is a computable map sending each PCP instance $P$ (a list of dominoes over $\\alpha$) to an encoded context-free grammar $G_P$ over the alphabet $\\alpha \\oplus \\mathrm{Bool}$ (the letters of $\\alpha$ together with two bits) such that $P$ has no PCP solution if and only if $G_P$ generates every word over\u2026","title":"The complement of PCP many-one reduces to universality of context-free grammars"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars.json","id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","title":"Undecidable problems for context-free grammars"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-decidable-overlap","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_overlap","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-decidable-overlap","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is an alphabet $T$, with decidable equality and a primitive recursive encoding into $\\mathbb{N}$, such that no multi-stack machine decides (in the sense of TM2-decidability), for a pair $(G_1, G_2)$ of encoded context-free grammars over $T$, whether some word lies in both $L(G_1)$ and $L(G_2)$, where $L(G)$ is the language of $G$.","title":"Whether two context-free languages intersect is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-decidable-disjoint","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_disjoint","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-decidable-disjoint","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is an alphabet $T$, with decidable equality and a primitive recursive encoding into $\\mathbb{N}$, such that no multi-stack machine decides (in the sense of TM2-decidability), for a pair $(G_1, G_2)$ of encoded context-free grammars over $T$, whether no word lies in both $L(G_1)$ and $L(G_2)$, where $L(G)$ is the language of $G$.","title":"Whether two context-free languages are disjoint is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-decidable-is-ambiguous","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_isAmbiguous","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-decidable-is-ambiguous","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is an alphabet $T$, with decidable equality and a primitive recursive encoding into $\\mathbb{N}$, such that no multi-stack machine decides (in the sense of TM2-decidability), for an encoded context-free grammar $G$ over $T$, whether $G$ is ambiguous, that is, whether some word has two different leftmost derivations from the start nonterminal of $G$.","title":"Whether a context-free grammar is ambiguous is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-decidable-univ","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_univ","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-decidable-univ","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is a finite alphabet $T$, with decidable equality and a primitive recursive encoding into $\\mathbb{N}$, such that no multi-stack machine decides (in the sense of TM2-decidability), for an encoded context-free grammar $G$ over $T$, whether every word over $T$ lies in the language of $G$.","title":"Whether a context-free grammar generates every word is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/all-words-cfg","kind":"node","lean":"Cslib.Decidability.allWordsCFG","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#all-words-cfg","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"For a finite alphabet $T$, the encoded context-free grammar over $T$ whose start nonterminal is $S = 0$ and whose productions are $S \\to \\varepsilon$ and $S \\to a S$ for every letter $a \\in T$.","title":"Grammar of all words"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-decidable-language-eq","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_language_eq","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-decidable-language-eq","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is a finite alphabet $T$, with decidable equality and a primitive recursive encoding into $\\mathbb{N}$, such that no multi-stack machine decides (in the sense of TM2-decidability), for a pair $(G_1, G_2)$ of encoded context-free grammars over $T$, whether $L(G_1) = L(G_2)$, where $L(G)$ is the language of $G$.","title":"Equivalence of context-free grammars is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-decidable-language-le","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_language_le","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-decidable-language-le","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is a finite alphabet $T$, with decidable equality and a primitive recursive encoding into $\\mathbb{N}$, such that no multi-stack machine decides (in the sense of TM2-decidability), for a pair $(G_1, G_2)$ of encoded context-free grammars over $T$, whether $L(G_1) \\subseteq L(G_2)$, where $L(G)$ is the language of $G$.","title":"Inclusion of context-free languages is undecidable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/not-tm2-recognizable-not-of","kind":"node","lean":"Cslib.Decidability.not_tm2Recognizable_not_of","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#not-tm2-recognizable-not-of","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"A recognizable undecidable predicate has an unrecognizable complement.","title":"Post's theorem, contrapositive"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-recognizable-not-pcp","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Recognizable_not_pcp","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-recognizable-not-pcp","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"The complement of PCP is not recognizable, over some finite alphabet.","title":"The complement of PCP is not recognizable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/computable-pred-mem-fst","kind":"node","lean":"Cslib.Decidability.computablePred_mem_fst","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#computable-pred-mem-fst","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"Let $T$ be an alphabet with a primcodable encoding and decidable equality. The predicate that holds of a pair $((g_1, g_2), w)$, where $g_1, g_2$ are encoded context-free grammars over $T$ and $w$ is a word over $T$, exactly when $w$ lies in the language of $g_1$, is a computable predicate.","title":"Membership in the first grammar of a pair is computable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/tm2-recognizable-overlap","kind":"node","lean":"Cslib.Decidability.tm2Recognizable_overlap","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#tm2-recognizable-overlap","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"Let $T$ be an alphabet with a primitive recursive coding into the natural numbers and decidable equality. For an encoded context-free grammar $G$ over $T$, write $L(G)$ for the language it generates. The problem $\\mathrm{OVERLAP}_{\\mathrm{CFG}}$, the set of pairs $(G, H)$ of encoded context-free grammars over $T$ such that some word $w$ over $T$ lies in both $L(G)$ and $L(H)$, is recognizable by a multi-stack\u2026","title":"The overlap problem for context-free grammars is recognizable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/tm2-co-recognizable-univ","kind":"node","lean":"Cslib.Decidability.tm2CoRecognizable_univ","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#tm2-co-recognizable-univ","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"For every alphabet $T$ with a primcodable encoding and decidable equality, the set of encoded context-free grammars over $T$ whose language contains every word over $T$ is co-recognizable by a multi-stack Turing machine.","title":"Universality of context-free grammars is co-recognizable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/tm2-co-recognizable-language-eq","kind":"node","lean":"Cslib.Decidability.tm2CoRecognizable_language_eq","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#tm2-co-recognizable-language-eq","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"For every alphabet $T$ with a primcodable encoding and decidable equality, the set of pairs $(g_1, g_2)$ of encoded context-free grammars over $T$ whose languages are equal is co-recognizable by a multi-stack Turing machine.","title":"Equivalence of context-free grammars is co-recognizable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/tm2-co-recognizable-language-le","kind":"node","lean":"Cslib.Decidability.tm2CoRecognizable_language_le","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#tm2-co-recognizable-language-le","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"For every alphabet $T$ with a primcodable encoding and decidable equality, the set of pairs $(g_1, g_2)$ of encoded context-free grammars over $T$ for which the language of $g_1$ is contained in the language of $g_2$ is co-recognizable by a multi-stack Turing machine.","title":"Inclusion of context-free languages is co-recognizable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-recognizable-disjoint","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Recognizable_disjoint","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-recognizable-disjoint","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is an alphabet $T$, with a primcodable encoding and decidable equality, such that the set of pairs $(g_1, g_2)$ of encoded context-free grammars over $T$ whose languages are disjoint (no word lies in both) is not recognizable by a multi-stack Turing machine.","title":"Disjointness of context-free languages is not recognizable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-recognizable-univ","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Recognizable_univ","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-recognizable-univ","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is a finite alphabet $T$, with a primcodable encoding and decidable equality, such that the set of encoded context-free grammars over $T$ whose language contains every word over $T$ is not recognizable by a multi-stack Turing machine.","title":"Universality of context-free grammars is not recognizable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-recognizable-language-eq","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Recognizable_language_eq","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-recognizable-language-eq","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is a finite alphabet $T$, with a primcodable encoding and decidable equality, such that the set of pairs $(g_1, g_2)$ of encoded context-free grammars over $T$ whose languages are equal is not recognizable by a multi-stack Turing machine.","title":"Equivalence of context-free grammars is not recognizable"},{"id":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/exists-not-tm2-recognizable-language-le","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Recognizable_language_le","parent":35341,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars/index.html#exists-not-tm2-recognizable-language-le","scope":"TheoryOfComputation/undecidable-problems/undecidable-problems-for-context-free-grammars","status":"fully_proved","summary":"There is a finite alphabet $T$, with a primcodable encoding and decidable equality, such that the set of pairs $(g_1, g_2)$ of encoded context-free grammars over $T$ for which the language of $g_1$ is contained in the language of $g_2$ is not recognizable by a multi-stack Turing machine.","title":"Inclusion of context-free languages is not recognizable"},{"data_url":"scopes/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages.json","id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","kind":"scope","navigation":"module","parent":35144,"result":false,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","title":"Regularity problem for context-free languages"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/letter","kind":"node","lean":"Cslib.Decidability.Regularity.Letter","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#letter","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"For a type $T$, the alphabet $T \\sqcup \\{\\#, 0, 1\\}$ consists of the letters of $T$, a separator $\\#$ and two bits $0$ and $1$, all distinct.","title":"Alphabet of the reduction from universality to regularity"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/sep","kind":"node","lean":"Cslib.Decidability.Regularity.sep","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#sep","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"For an alphabet $T$, among the letters of the reduction (the letters of $T$, a separator and two bits), this is the separator, written $\\#$.","title":"The separator letter"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/tl","kind":"node","lean":"Cslib.Decidability.Regularity.tl","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#tl","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"For an alphabet $T$, each letter $t \\in T$ is sent to the corresponding letter among the letters of the reduction (the letters of $T$, a separator and two bits).","title":"Alphabet letters in the extended alphabet"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/bl","kind":"node","lean":"Cslib.Decidability.Regularity.bl","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#bl","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Each boolean $b$ (with false read as $0$ and true as $1$) is sent to the corresponding bit letter of the alphabet consisting of the letters of $T$, a separator $\\#$ and two bits $0$ and $1$.","title":"Bits as letters"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/zero-one","kind":"node","lean":"Cslib.Decidability.Regularity.zeroOne","parent":35360,"reading_title":{"format":"inline-parts-v1","parts":[["text","The language "],["math","\\(\\{0^n 1^n\\}\\)"]],"source":"The language $\\{0^n 1^n\\}$"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#zero-one","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"The language over the two bits $0$ and $1$ (the Booleans false and true) consisting of the words $0^n 1^n$ for $n \\in \\mathbb{N}$, that is, $n$ zeros followed by $n$ ones.","title":"The language $\\{0^n 1^n\\}$"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/reg-lang","kind":"node","lean":"Cslib.Decidability.Regularity.regLang","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#reg-lang","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"For a language $L$ over an alphabet $T$, this is the language over the letters of $T$, a separator $\\#$ and the two bits $0, 1$ consisting of the words $u \\# v$ with $u \\in T^*$ and $v \\in \\{0,1\\}^*$ such that $u \\in L$ or $v = 0^n 1^n$ for some $n \\in \\mathbb{N}$; that is, $L \\# \\{0,1\\}^* \\cup T^* \\# \\{0^n 1^n : n \\in \\mathbb{N}\\}$.","title":"The target language of the regularity reduction"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/eq-of-map-tl-append-sep","kind":"node","lean":"Cslib.Decidability.Regularity.eq_of_map_tl_append_sep","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#eq-of-map-tl-append-sep","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Over the alphabet consisting of the letters of $T$, a separator $\\#$ and two bits $0$ and $1$, let $u, u'$ be words over $T$ and $v, v'$ words over the bits. If $u \\# v = u' \\# v'$, then $u = u'$ and $v = v'$.","title":"Unique decomposition at the separator"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/not-is-regular-zero-one","kind":"node","lean":"Cslib.Decidability.Regularity.not_isRegular_zeroOne","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#not-is-regular-zero-one","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"The words `0\u2071` are pairwise inequivalent for the Nerode congruence.","title":"`{0\u207f1\u207f}` is not regular"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/letter-words","kind":"node","lean":"Cslib.Decidability.Regularity.letterWords","parent":35360,"reading_title":{"format":"inline-parts-v1","parts":[["text","Words over "],["math","\\(T\\)"],["text"," in the reduction alphabet"]],"source":"Words over $T$ in the reduction alphabet"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#letter-words","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"This is the language $T^*$ over the alphabet $T \\sqcup \\{\\#, 0, 1\\}$ of the reduction from universality to regularity: the words, including the empty word, all of whose letters lie in $T$.","title":"Words over $T$ in the reduction alphabet"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/bit-words","kind":"node","lean":"Cslib.Decidability.Regularity.bitWords","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#bit-words","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Over the alphabet consisting of the letters of $T$, a separator $\\#$ and two bits $0$ and $1$, the bit words form the language $\\{0, 1\\}^*$ of all finite words, including the empty word, all of whose letters are bits.","title":"Bit words"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/letter-set","kind":"node","lean":"Cslib.Decidability.Regularity.letterSet","parent":35360,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-letter words over "],["math","\\(T\\)"]],"source":"One-letter words over $T$"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#letter-set","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"This is the language over the alphabet $T \\sqcup \\{\\#, 0, 1\\}$ of the reduction from universality to regularity consisting of the one-letter words $t$ with $t \\in T$.","title":"One-letter words over $T$"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/bit-set","kind":"node","lean":"Cslib.Decidability.Regularity.bitSet","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#bit-set","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Over the alphabet consisting of the letters of $T$, a separator $\\#$ and two bits $0$ and $1$, the single-bit words form the language $\\{0, 1\\}$ whose only words are the one-letter words $0$ and $1$.","title":"Single-bit words"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/is-regular-reg-lang-top","kind":"node","lean":"Cslib.Decidability.Regularity.isRegular_regLang_top","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#is-regular-reg-lang-top","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $T$ be a finite alphabet and $L$ a language over $T$ that contains every word over $T$. Over the letters of $T$, a separator $\\#$ and two bits $0$ and $1$, the language $L \\# \\{0,1\\}^* \\cup T^* \\# \\{0^n 1^n : n \\ge 0\\}$, which here equals $T^* \\# \\{0,1\\}^*$, is regular.","title":"The reduction language of a universal language is regular"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/not-is-regular-reg-lang","kind":"node","lean":"Cslib.Decidability.Regularity.not_isRegular_regLang","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#not-is-regular-reg-lang","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $L$ be a language over an alphabet $T$, let $T \\sqcup \\{\\#, 0, 1\\}$ extend $T$ by a separator $\\#$ and two bits, and let $R(L) = L\\,\\#\\,\\{0,1\\}^* \\cup T^*\\,\\#\\,\\{0^n1^n : n \\in \\mathbb{N}\\}$ over this alphabet. If some word $u$ over $T$ is not in $L$, then $R(L)$ is not regular.","title":"Non-universality makes the reduction language non-regular"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/is-regular-reg-lang-iff","kind":"node","lean":"Cslib.Decidability.Regularity.isRegular_regLang_iff","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#is-regular-reg-lang-iff","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $T$ be a finite alphabet and $L$ a language over $T$. Over the letters of $T$, a separator $\\#$ and two bits $0$ and $1$, the language $L \\# \\{0,1\\}^* \\cup T^* \\# \\{0^n 1^n : n \\ge 0\\}$, of the words $u \\# v$ with $u$ a word over $T$, $v$ a word over the bits, and $u \\in L$ or $v = 0^n 1^n$ for some $n$, is regular if and only if $L$ contains every word over $T$.","title":"Regularity of the reduction language characterises universality"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/shift-sym","kind":"node","lean":"Cslib.Decidability.Regularity.shiftSym","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#shift-sym","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $T$ be a finite alphabet with decidable equality, and consider the alphabet $T \\sqcup \\{\\#, 0, 1\\}$ of the letters of $T$, a separator $\\#$ and two bits $0$ and $1$. A symbol of an encoded grammar over $T$, that is, a nonterminal $n \\in \\mathbb{N}$ or a letter $t \\in T$, is sent to a symbol of an encoded grammar over $T \\sqcup \\{\\#, 0, 1\\}$: the nonterminal $n$ goes to the nonterminal $n + 4$, and the letter\u2026","title":"Shifting a symbol into the regularity grammar"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/shift-rule","kind":"node","lean":"Cslib.Decidability.Regularity.shiftRule","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#shift-rule","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $T$ be a finite alphabet with decidable equality, and consider the alphabet $T \\sqcup \\{\\#, 0, 1\\}$ of the letters of $T$, a separator $\\#$ and two bits $0$ and $1$. An encoded production $(n, \\alpha)$ over $T$, with left-hand side the nonterminal $n$ and right-hand side $\\alpha$, is sent to the encoded production $(n + 4, \\alpha')$ over $T \\sqcup \\{\\#, 0, 1\\}$, where $\\alpha'$ is $\\alpha$ with each nonterminal\u2026","title":"Shifting a production into the regularity grammar"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/fixed-rules","kind":"node","lean":"Cslib.Decidability.Regularity.fixedRules","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#fixed-rules","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"For a natural number $s$, these are the following productions over the nonterminals $\\mathbb{N}$ and the alphabet consisting of the letters of the finite alphabet $T$, a separator $\\#$ and two bits $0$ and $1$, where $A_n$ denotes nonterminal $n$: $A_0 \\to A_{s+4} \\# A_1$, $A_0 \\to A_2 \\# A_3$, $A_1 \\to \\varepsilon$, $A_1 \\to 0 A_1$, $A_1 \\to 1 A_1$, $A_2 \\to \\varepsilon$, $A_3 \\to \\varepsilon$, $A_3 \\to 0 A_3 1$,\u2026","title":"The auxiliary productions of the regularity grammar"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/reg-grammar","kind":"node","lean":"Cslib.Decidability.Regularity.regGrammar","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#reg-grammar","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over a finite alphabet $T$ with start nonterminal $s$, and write $N_k$ for nonterminal $k \\in \\mathbb{N}$ and $\\varepsilon$ for the empty word. The reduction grammar of $g$ is the encoded grammar over $T \\sqcup \\{\\#, 0, 1\\}$ (the letters of $T$, a separator $\\#$ and two bits $0$ and $1$) with start nonterminal $N_0$ and the productions $N_0 \\to N_{s+4}\\,\\#\\,N_1$,\u2026","title":"Grammar of the reduction from universality to regularity"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/reg-val","kind":"node","lean":"Cslib.Decidability.Regularity.regVal","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#reg-val","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over a finite alphabet $T$ with decidable equality, let $L(g)$ be its language and $L_g(n)$ the language $g$ generates from nonterminal $n$. Over the alphabet $T \\sqcup \\{\\#, 0, 1\\}$ (the letters of $T$, a separator $\\#$ and two bits $0$ and $1$), write $\\bar u$ for a word $u$ over $T$ read in this alphabet. This assigns to each $m \\in \\mathbb{N}$ the language $V(m)$ over\u2026","title":"Languages assigned to the nonterminals of the regularity grammar"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/mem-eval-form-shift","kind":"node","lean":"Cslib.Decidability.Regularity.mem_evalForm_shift","parent":35360,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shifted right-hand sides denote images of words of "],["math","\\(g\\)"]],"source":"Shifted right-hand sides denote images of words of $g$"},"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#mem-eval-form-shift","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over $T$, let $L_g(n)$ be the language $g$ generates from nonterminal $n$ and $L(g)$ its language. For a word $u$ over $T$ write $\\bar u$ for the same word over $T \\sqcup \\{\\#, 0, 1\\}$. Let $V$ assign to each $m \\in \\mathbb{N}$ the language $V(m)$ over $T \\sqcup \\{\\#, 0, 1\\}$ given by $V(0) = L(g)\\,\\#\\,\\{0,1\\}^* \\cup T^*\\,\\#\\,\\{0^k1^k : k \\in \\mathbb{N}\\}$,\u2026","title":"Shifted right-hand sides denote images of words of $g$"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/is-valuation-reg-val","kind":"node","lean":"Cslib.Decidability.Regularity.isValuation_regVal","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#is-valuation-reg-val","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over a finite alphabet $T$ with decidable equality, and let $G$ be the regularity grammar of $g$, over the letters of $T$, a separator $\\#$ and two bits $0$ and $1$; words over $T$ are read as words over this alphabet. Assign to the nonterminals of $G$ the languages $V(0) = L(g) \\# \\{0,1\\}^* \\cup T^* \\# \\{0^n 1^n : n \\ge 0\\}$, where $L(g)$ is the language of $g$,\u2026","title":"Every production of the regularity grammar is sound for its intended languages"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/mem-language-from-of-mem-rules","kind":"node","lean":"Cslib.Decidability.Regularity.mem_languageFrom_of_mem_rules","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#mem-language-from-of-mem-rules","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over a finite alphabet $T$ with decidable equality, let $G$ be the regularity grammar of $g$, over the letters of $T$, a separator $\\#$ and two bits $0$ and $1$, and for a nonterminal $B$ let $L_G(B)$ be the language that the grammar denoted by $G$ generates from $B$. If $(A, \\alpha)$ is an encoded production of $G$ and $w$ is a word in the language that the decoded\u2026","title":"Productions of the regularity grammar preserve generated languages"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/mem-eval-form-shift-of","kind":"node","lean":"Cslib.Decidability.Regularity.mem_evalForm_shift_of","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#mem-eval-form-shift-of","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over a finite alphabet $T$, let $G$ be the grammar over $T \\sqcup \\{\\#, 0, 1\\}$ that the reduction from universality to regularity builds from $g$, and let $L_G(m)$ be the language $G$ generates from nonterminal $m$. For a word $u$ over $T$ write $\\bar u$ for the same word over $T \\sqcup \\{\\#, 0, 1\\}$. Let $W$ assign to each $n \\in \\mathbb{N}$ a language $W(n)$ over $T$\u2026","title":"Shifted right-hand sides generate the images of words"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/shift-mem","kind":"node","lean":"Cslib.Decidability.Regularity.shift_mem","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#shift-mem","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over a finite alphabet $T$, and let $n$ be a nonterminal of $g$. If $g$ derives a word $u$ over $T$ from $n$, then the regularity grammar of $g$ derives from its nonterminal $n + 4$ the word $u$ read over the extended alphabet, each letter $t$ of $T$ replaced by the corresponding letter.","title":"Shifted derivations in the regularity grammar"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/language-reg-grammar","kind":"node","lean":"Cslib.Decidability.Regularity.language_regGrammar","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#language-reg-grammar","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over a finite alphabet $T$ with decidable equality, with language $L(g)$. Over the letters of $T$, a separator $\\#$ and two bits $0$ and $1$, the regularity grammar of $g$ generates exactly $L(g) \\# \\{0,1\\}^* \\cup T^* \\# \\{0^n 1^n : n \\ge 0\\}$, the set of words $u \\# v$ with $u$ a word over $T$, $v$ a word over the bits, and $u \\in L(g)$ or $v = 0^n 1^n$ for some $n$.","title":"The language of the regularity grammar"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/univ-iff-is-regular","kind":"node","lean":"Cslib.Decidability.Regularity.univ_iff_isRegular","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#univ-iff-is-regular","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $g$ be an encoded context-free grammar over a finite alphabet $T$. Then $g$ generates every word over $T$ if and only if the regularity grammar of $g$ generates a regular language.","title":"A grammar is universal iff its regularity grammar generates a regular language"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/many-one-reducible-univ-is-regular","kind":"node","lean":"Cslib.Decidability.Regularity.manyOneReducible_univ_isRegular","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#many-one-reducible-univ-is-regular","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"Let $T$ be a finite alphabet with decidable equality and a primitive recursive coding, and let $T \\sqcup \\{\\#, 0, 1\\}$ extend $T$ by a separator $\\#$ and two bits. There is a computable map $f$ from encoded context-free grammars over $T$ to encoded context-free grammars over $T \\sqcup \\{\\#, 0, 1\\}$ such that a grammar $g$ generates every word over $T$ if and only if the language generated by $f(g)$ is regular.","title":"Universality many-one reduces to regularity of context-free languages"},{"id":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/exists-not-tm2-decidable-is-regular","kind":"node","lean":"Cslib.Decidability.exists_not_tm2Decidable_isRegular","parent":35360,"result":true,"route":"../roadmap/TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages/index.html#exists-not-tm2-decidable-is-regular","scope":"TheoryOfComputation/undecidable-problems/regularity-problem-for-context-free-languages","status":"fully_proved","summary":"There is a finite alphabet $T$, with a primitive recursive encoding and decidable equality, such that no multi-stack Turing machine decides, for an encoded context-free grammar $g$ over $T$, whether the language of $g$ is regular.","title":"Regularity of context-free languages is undecidable"},{"data_url":"scopes/TheoryOfComputation/degrees-and-hierarchies.json","id":"TheoryOfComputation/degrees-and-hierarchies","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/index.html","scope":"TheoryOfComputation/degrees-and-hierarchies","status":"fully_proved","summary":"Turing and many-one reducibility, Turing degrees and the jump, recursive isomorphism (Myhill), productive, creative, simple and immune sets, r.e.-completeness, and the arithmetical hierarchy.","title":"Degrees of unsolvability and the arithmetical hierarchy"},{"data_url":"scopes/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets.json","id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","kind":"scope","navigation":"module","parent":35390,"result":false,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","title":"Productive, creative, simple and immune sets"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/enum-re","kind":"node","lean":"Cslib.Decidability.enumRE","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#enum-re","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"For natural numbers $e$ and $n$, $n$ belongs to $W_e$, the $e$-th recursively enumerable set, when the partial recursive function $\\varphi_e$ computed by the program with index $e$ in the standard numbering of partial recursive programs is defined at $n$; thus $W_e$ is the domain of $\\varphi_e$.","title":"Standard enumeration of the recursively enumerable sets"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/productive","kind":"node","lean":"Cslib.Decidability.Productive","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#productive","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"A set $A$ of natural numbers is productive when there is a partial recursive function $f$ such that for every natural number $e$ with $W_e \\subseteq A$, where $W_e$ is the $e$-th recursively enumerable set in the standard enumeration, $f(e)$ is defined, $f(e) \\in A$ and $f(e) \\notin W_e$.","title":"Productive set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/creative","kind":"node","lean":"Cslib.Decidability.Creative","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#creative","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"A set $A$ of natural numbers is creative when it is recursively enumerable and its complement $\\mathbb{N} \\setminus A$ is productive.","title":"Creative set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/immune","kind":"node","lean":"Cslib.Decidability.Immune","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#immune","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"A set $A$ of natural numbers is immune when it is infinite and every recursively enumerable set $B \\subseteq A$ is finite.","title":"Immune set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/simple","kind":"node","lean":"Cslib.Decidability.Simple","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#simple","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"A set $P$ of natural numbers is simple when it is recursively enumerable and its complement $\\{n \\mid \\neg P(n)\\}$ is immune.","title":"Simple set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/productive-not-re","kind":"node","lean":"Cslib.Decidability.productive_not_re","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#productive-not-re","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"No productive set of natural numbers is recursively enumerable: if it were the $e$-th recursively enumerable set $W_e$, its productive function would produce a member of it outside $W_e$.","title":"A productive set is not recursively enumerable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/immune-not-re","kind":"node","lean":"Cslib.Decidability.immune_not_re","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#immune-not-re","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"It would be an infinite RE subset of itself.","title":"An immune set is not recursively enumerable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/simple-not-computable","kind":"node","lean":"Cslib.Decidability.simple_not_computable","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#simple-not-computable","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"Its complement would be an infinite RE set, contradicting immunity.","title":"A simple set is not computable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/simple-re","kind":"node","lean":"Cslib.Decidability.simple_re","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#simple-re","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"A simple set is recursively enumerable, by definition.","title":"A simple set is recursively enumerable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/simple-complement-immune","kind":"node","lean":"Cslib.Decidability.simple_complement_immune","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#simple-complement-immune","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"The complement of a simple set is immune, by definition.","title":"The complement of a simple set is immune"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/creative-re","kind":"node","lean":"Cslib.Decidability.creative_re","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#creative-re","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"A creative set is recursively enumerable, by definition.","title":"A creative set is recursively enumerable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/creative-not-computable","kind":"node","lean":"Cslib.Decidability.creative_not_computable","parent":35391,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets/index.html#creative-not-computable","scope":"TheoryOfComputation/degrees-and-hierarchies/productive-creative-simple-and-immune-sets","status":"fully_proved","summary":"Its complement is productive, hence not RE, but the complement of a computable set is RE.","title":"A creative set is not computable"},{"data_url":"scopes/TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence.json","id":"TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence","kind":"scope","navigation":"module","parent":35390,"result":false,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence/index.html","scope":"TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence","status":"fully_proved","title":"Recursive isomorphism and 1-equivalence"},{"id":"TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence/recursive-isomorphic","kind":"node","lean":"Cslib.Decidability.RecursiveIsomorphic","parent":35404,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence/index.html#recursive-isomorphic","scope":"TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence","status":"fully_proved","summary":"Two predicates $P$ and $Q$ on $\\mathbb{N}$ are recursively isomorphic when there is a computable bijection $f : \\mathbb{N} \\to \\mathbb{N}$ such that for every $n$, $P(n)$ holds if and only if $Q(f(n))$ holds.","title":"Recursively isomorphic predicates"},{"id":"TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence/computable-inv-fun-of-bijective","kind":"node","lean":"Cslib.Decidability.computable_invFun_of_bijective","parent":35404,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence/index.html#computable-inv-fun-of-bijective","scope":"TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ be a computable bijection. Then its inverse function $f^{-1} : \\mathbb{N} \\to \\mathbb{N}$ is computable.","title":"The inverse of a computable bijection is computable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence/recursive-isomorphic-to-one-one-equiv","kind":"node","lean":"Cslib.Decidability.recursiveIsomorphic_to_oneOneEquiv","parent":35404,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence/index.html#recursive-isomorphic-to-one-one-equiv","scope":"TheoryOfComputation/degrees-and-hierarchies/recursive-isomorphism-and-1-equivalence","status":"fully_proved","summary":"Let $P$ and $Q$ be predicates on $\\mathbb{N}$. If $P$ and $Q$ are recursively isomorphic, then they are 1-equivalent: $P$ is one-one reducible to $Q$ and $Q$ is one-one reducible to $P$. Here $P$ is one-one reducible to $Q$ when there is a computable injective function $g : \\mathbb{N} \\to \\mathbb{N}$ such that for every $n$, $P(n)$ holds if and only if $Q(g(n))$ holds.","title":"A recursive isomorphism gives a 1-equivalence"},{"data_url":"scopes/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility.json","id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","kind":"scope","navigation":"module","parent":35390,"result":false,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","title":"Turing reducibility"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/char-fun","kind":"node","lean":"Cslib.Decidability.charFun","parent":35408,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html#char-fun","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","summary":"The characteristic function of a set $p$ of natural numbers is the partial function $\\mathbb{N} \\rightharpoonup \\mathbb{N}$ that is defined at every $n$, with value $1$ if $n \\in p$ and value $0$ otherwise.","title":"Characteristic function of a set of naturals"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/computable-pred-iff-partrec-char-fun","kind":"node","lean":"Cslib.Decidability.computablePred_iff_partrec_charFun","parent":35408,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html#computable-pred-iff-partrec-char-fun","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","summary":"A set $p$ of natural numbers is decidable (computable as a predicate) if and only if its characteristic function is partial recursive.","title":"A set is decidable iff its characteristic function is partial recursive"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/of-turing-reducible","kind":"node","lean":"Cslib.Decidability.ComputablePred.of_turingReducible","parent":35408,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html#of-turing-reducible","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","summary":"Let $p$ and $q$ be sets of natural numbers such that the characteristic function of $p$ is Turing reducible to the characteristic function of $q$. If $q$ is decidable, then $p$ is decidable.","title":"Decidability transfers down Turing reducibility"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/halting-set","kind":"node","lean":"Cslib.Decidability.haltingSet","parent":35408,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html#halting-set","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","summary":"Fix the standard bijective numbering $n \\mapsto c_n$ of the codes of partial recursive functions $\\mathbb{N} \\rightharpoonup \\mathbb{N}$. The halting set is the set of natural numbers $n$ such that the partial recursive function with code $c_n$ is defined at $0$.","title":"Halting set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/not-turing-reducible-halting","kind":"node","lean":"Cslib.Decidability.not_turingReducible_halting","parent":35408,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html#not-turing-reducible-halting","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","summary":"For every decidable set $q$ of natural numbers, the characteristic function of the halting set is not Turing reducible to the characteristic function of $q$.","title":"The halting set is not Turing reducible to a decidable set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/turing-reducible-of-many-one-reducible","kind":"node","lean":"Cslib.Decidability.turingReducible_of_manyOneReducible","parent":35408,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html#turing-reducible-of-many-one-reducible","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","summary":"If $p$ and $q$ are sets of natural numbers and $p$ is many-one reducible to $q$, that is, there is a computable $f : \\mathbb{N} \\to \\mathbb{N}$ with $n \\in p \\iff f(n) \\in q$ for all $n$, then the characteristic function of $p$ is Turing reducible to the characteristic function of $q$.","title":"Many-one reducibility implies Turing reducibility"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/turing-equivalent-char-fun-not","kind":"node","lean":"Cslib.Decidability.turingEquivalent_charFun_not","parent":35408,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html#turing-equivalent-char-fun-not","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","summary":"For every set $p$ of natural numbers, the characteristic function of the complement $\\mathbb{N} \\setminus p$ and the characteristic function of $p$ are Turing equivalent: each is Turing reducible to the other.","title":"Every set is Turing equivalent to its complement"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/not-many-one-reducible-not-halting","kind":"node","lean":"Cslib.Decidability.not_manyOneReducible_not_halting","parent":35408,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-reducibility/index.html#not-many-one-reducible-not-halting","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-reducibility","status":"fully_proved","summary":"There is no computable function $f : \\mathbb{N} \\to \\mathbb{N}$ such that, for every $n$, $n$ is outside the halting set if and only if $f(n)$ is in the halting set.","title":"The complement of the halting set is not many-one reducible to it"},{"data_url":"scopes/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy.json","id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","kind":"scope","navigation":"module","parent":35390,"result":false,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","title":"Arithmetical hierarchy"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/is-delta","kind":"node","lean":"Cslib.Decidability.IsDelta","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#is-delta","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"Let $n$ be a natural number. A set $P$ of natural numbers is in $\\Delta_n$ when it is in both classes $\\Sigma_n$ and $\\Pi_n$ of the arithmetical hierarchy.","title":"The class \u0394\u2099 of the arithmetical hierarchy"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/computable-pred-comp","kind":"node","lean":"Cslib.Decidability.computablePred_comp","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#computable-pred-comp","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"If $P$ is a decidable set of natural numbers and $g : \\mathbb{N} \\to \\mathbb{N}$ is a computable function, then the set $\\{w \\in \\mathbb{N} : g(w) \\in P\\}$ is decidable.","title":"Computable predicates are closed under computable precomposition"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/computable-unpair-fst","kind":"node","lean":"Cslib.Decidability.computable_unpair_fst","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#computable-unpair-fst","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"Let $\\langle a, b \\rangle$ denote the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$, $\\langle a, b \\rangle = b^2 + a$ if $a < b$ and $\\langle a, b \\rangle = a^2 + a + b$ otherwise. The function sending $z \\in \\mathbb{N}$ to the first component $a$ of the unique pair $(a, b)$ with $\\langle a, b \\rangle = z$ is computable.","title":"The first projection of the pairing function is computable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/computable-unpair-snd","kind":"node","lean":"Cslib.Decidability.computable_unpair_snd","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#computable-unpair-snd","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"Let $\\langle a, b \\rangle$ denote the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$, $\\langle a, b \\rangle = b^2 + a$ if $a < b$ and $\\langle a, b \\rangle = a^2 + a + b$ otherwise. The function sending $z \\in \\mathbb{N}$ to the second component $b$ of the unique pair $(a, b)$ with $\\langle a, b \\rangle = z$ is computable.","title":"The second projection of the pairing function is computable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/computable-pair","kind":"node","lean":"Cslib.Decidability.computable_pair","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#computable-pair","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"The pairing bijection $\\langle \\cdot, \\cdot \\rangle : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$, given by $\\langle a, b \\rangle = b^2 + a$ if $a < b$ and $\\langle a, b \\rangle = a^2 + a + b$ otherwise, is a computable function of its two arguments.","title":"The pairing function is computable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/computable-pred-comp-prod","kind":"node","lean":"Cslib.Decidability.computablePred_comp_prod","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#computable-pred-comp-prod","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"Let $\\langle a, b \\rangle$ denote the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$, $\\langle a, b \\rangle = b^2 + a$ if $a < b$ and $\\langle a, b \\rangle = a^2 + a + b$ otherwise. If $Q$ is a decidable set of natural numbers, then the set of pairs $(a, b) \\in \\mathbb{N} \\times \\mathbb{N}$ with $\\langle a, b \\rangle \\in Q$ is decidable.","title":"Computable predicates are closed under precomposition with pairing"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/comp-closure","kind":"node","lean":"Cslib.Decidability.comp_closure","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#comp-closure","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"Let $n$ be a natural number and let $g : \\mathbb{N} \\to \\mathbb{N}$ be computable. If a set $P$ of natural numbers is in the class $\\Sigma_n$ of the arithmetical hierarchy, then so is $\\{w \\in \\mathbb{N} : g(w) \\in P\\}$; and if $P$ is in the class $\\Pi_n$, then so is $\\{w \\in \\mathbb{N} : g(w) \\in P\\}$.","title":"The hierarchy is closed under computable precomposition"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/sigma-subset-pi-succ","kind":"node","lean":"Cslib.Decidability.sigma_subset_pi_succ","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#sigma-subset-pi-succ","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"Let $\\Sigma_n$ and $\\Pi_n$ be the levels of the arithmetical hierarchy of predicates on $\\mathbb{N}$: $\\Sigma_0 = \\Pi_0$ is the class of computable predicates, $P \\in \\Sigma_{n+1}$ when for some $Q \\in \\Pi_n$ and all $x$, $P(x)$ holds iff $Q(\\langle x, y \\rangle)$ holds for some $y$, and $P \\in \\Pi_{n+1}$ when for some $Q \\in \\Sigma_n$ and all $x$, $P(x)$ holds iff $Q(\\langle x, y \\rangle)$ holds for all $y$; here\u2026","title":"Every \u03a3\u2099 predicate is \u03a0\u2099\u208a\u2081"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/pi-subset-sigma-succ","kind":"node","lean":"Cslib.Decidability.pi_subset_sigma_succ","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#pi-subset-sigma-succ","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"For every natural number $n$, every set of natural numbers in the class $\\Pi_n$ of the arithmetical hierarchy is also in $\\Sigma_{n+1}$; that is, $\\Pi_n \\subseteq \\Sigma_{n+1}$, by adding a dummy existential quantifier.","title":"Every \u03a0\u2099 set is \u03a3\u2099\u208a\u2081"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/mono-aux","kind":"node","lean":"Cslib.Decidability.mono_aux","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#mono-aux","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"For every natural number $n$, writing $\\Sigma_n$ and $\\Pi_n$ for the classes of the arithmetical hierarchy, every $\\Sigma_n$ set of natural numbers is in $\\Sigma_{n+1}$ and every $\\Pi_n$ set of natural numbers is in $\\Pi_{n+1}$.","title":"Monotonicity of the hierarchy"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/sigma-mono","kind":"node","lean":"Cslib.Decidability.sigma_mono","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#sigma-mono","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"Let $\\Sigma_n$ and $\\Pi_n$ be the levels of the arithmetical hierarchy of predicates on $\\mathbb{N}$: $\\Sigma_0 = \\Pi_0$ is the class of computable predicates, $P \\in \\Sigma_{n+1}$ when for some $Q \\in \\Pi_n$ and all $x$, $P(x)$ holds iff $Q(\\langle x, y \\rangle)$ holds for some $y$, and $P \\in \\Pi_{n+1}$ when for some $Q \\in \\Sigma_n$ and all $x$, $P(x)$ holds iff $Q(\\langle x, y \\rangle)$ holds for all $y$; here\u2026","title":"Every \u03a3\u2099 predicate is \u03a3\u2099\u208a\u2081"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/pi-mono","kind":"node","lean":"Cslib.Decidability.pi_mono","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#pi-mono","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"For every natural number $n$, every set of natural numbers in the class $\\Pi_n$ of the arithmetical hierarchy is also in $\\Pi_{n+1}$; that is, $\\Pi_n \\subseteq \\Pi_{n+1}$.","title":"Monotonicity of the \u03a0 hierarchy"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/sigma-pi-subset-delta-succ","kind":"node","lean":"Cslib.Decidability.sigma_pi_subset_delta_succ","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#sigma-pi-subset-delta-succ","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"Let $\\Sigma_n$ and $\\Pi_n$ be the levels of the arithmetical hierarchy of predicates on $\\mathbb{N}$: $\\Sigma_0 = \\Pi_0$ is the class of computable predicates, $P \\in \\Sigma_{n+1}$ when for some $Q \\in \\Pi_n$ and all $x$, $P(x)$ holds iff $Q(\\langle x, y \\rangle)$ holds for some $y$, and $P \\in \\Pi_{n+1}$ when for some $Q \\in \\Sigma_n$ and all $x$, $P(x)$ holds iff $Q(\\langle x, y \\rangle)$ holds for all $y$; here\u2026","title":"\u03a3\u2099 and \u03a0\u2099 predicates are \u0394\u2099\u208a\u2081"},{"id":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/sigma-one-eq-re","kind":"node","lean":"Cslib.Decidability.sigma_one_eq_re","parent":35417,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy/index.html#sigma-one-eq-re","scope":"TheoryOfComputation/degrees-and-hierarchies/arithmetical-hierarchy","status":"fully_proved","summary":"A predicate $P$ on $\\mathbb{N}$ is recursively enumerable if and only if it is in $\\Sigma_1$, that is, if and only if there is a computable predicate $Q$ on $\\mathbb{N}$ such that for all $x$, $P(x)$ holds iff $Q(\\langle x, y \\rangle)$ holds for some $y$, where $\\langle x, y \\rangle$ is the standard pairing function of $\\mathbb{N}$.","title":"\u03a3\u2081 is the class of recursively enumerable predicates"},{"data_url":"scopes/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set.json","id":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","kind":"scope","navigation":"module","parent":35390,"result":false,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/index.html","scope":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","status":"fully_proved","title":"Completeness of the halting set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/eval-comp-const","kind":"node","lean":"Cslib.Decidability.eval_comp_const","parent":35432,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/index.html#eval-comp-const","scope":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","status":"fully_proved","summary":"Let $c$ be a code of a partial recursive function and let $n, m \\in \\mathbb{N}$. The code that first computes the constant function with value $n$ and then applies $c$, evaluated at $m$, gives the same partial value as $c$ evaluated at $n$: one is defined exactly when the other is, and then they are equal.","title":"Composing a code with a constant code ignores the input"},{"id":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/of-many-one-reducible","kind":"node","lean":"Cslib.Decidability.REPred.of_manyOneReducible","parent":35432,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/index.html#of-many-one-reducible","scope":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be types with primitive recursive encodings into $\\mathbb{N}$, and let $p$ and $q$ be predicates on $\\alpha$ and on $\\beta$. Suppose that $p$ many-one reduces to $q$, that is, there is a computable function $f : \\alpha \\to \\beta$ such that for every $a \\in \\alpha$, $p(a)$ holds if and only if $q(f(a))$ holds. If $q$ is recognizable (recursively enumerable: the domain of a partial computable\u2026","title":"Recognizability transfers down many-one reductions"},{"id":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/re-pred-halting-set","kind":"node","lean":"Cslib.Decidability.rePred_haltingSet","parent":35432,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/index.html#re-pred-halting-set","scope":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","status":"fully_proved","summary":"The halting set, the set of numbers $n$ such that the code with number $n$ halts on input $0$, is recursively enumerable: it is the domain of a partial computable function on $\\mathbb{N}$.","title":"The halting set is recognizable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/one-one-reducible-halting-set","kind":"node","lean":"Cslib.Decidability.oneOneReducible_haltingSet","parent":35432,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/index.html#one-one-reducible-halting-set","scope":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$, and let $p$ be a recognizable (recursively enumerable) predicate on $\\alpha$, that is, the domain of a partial computable function on $\\alpha$. Here the halting set is the set of $n \\in \\mathbb{N}$ such that the partial recursive function with code number $n$ halts on input $0$. Then there is an injective computable function\u2026","title":"Every recognizable predicate one-one reduces to the halting set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/many-one-reducible-halting-set","kind":"node","lean":"Cslib.Decidability.manyOneReducible_haltingSet","parent":35432,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/index.html#many-one-reducible-halting-set","scope":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$, and let $p$ be a recognizable (recursively enumerable) predicate on $\\alpha$, that is, the domain of a partial computable function on $\\alpha$. Here the halting set is the set of $n \\in \\mathbb{N}$ such that the partial recursive function with code number $n$ halts on input $0$. Then there is a computable function\u2026","title":"Every recognizable predicate many-one reduces to the halting set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/re-pred-iff-many-one-reducible-halting-s","kind":"node","lean":"Cslib.Decidability.rePred_iff_manyOneReducible_haltingSet","parent":35432,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/index.html#re-pred-iff-many-one-reducible-halting-s","scope":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$ and let $p$ be a predicate on $\\alpha$. Here the halting set is the set of $n \\in \\mathbb{N}$ such that the partial recursive function with code number $n$ halts on input $0$. Then $p$ is recognizable (recursively enumerable: the domain of a partial computable function on $\\alpha$) if and only if there is a computable function\u2026","title":"A predicate is recognizable iff it many-one reduces to the halting set"},{"id":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/re-pred-iff-one-one-reducible-halting-se","kind":"node","lean":"Cslib.Decidability.rePred_iff_oneOneReducible_haltingSet","parent":35432,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set/index.html#re-pred-iff-one-one-reducible-halting-se","scope":"TheoryOfComputation/degrees-and-hierarchies/completeness-of-the-halting-set","status":"fully_proved","summary":"Let $\\alpha$ be a type with a primitive recursive encoding into $\\mathbb{N}$ and let $p$ be a predicate on $\\alpha$. Here the halting set is the set of $n \\in \\mathbb{N}$ such that the partial recursive function with code number $n$ halts on input $0$. Then $p$ is recognizable (recursively enumerable: the domain of a partial computable function on $\\alpha$) if and only if there is an injective computable function\u2026","title":"A predicate is recognizable iff it one-one reduces to the halting set"},{"data_url":"scopes/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump.json","id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","kind":"scope","navigation":"module","parent":35390,"result":false,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","title":"Turing degrees and the Turing jump"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-code","kind":"node","lean":"Cslib.Decidability.OracleCode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-code","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Oracle codes are the finite programs built from the basic codes $\\mathtt{zero}$, $\\mathtt{succ}$, $\\mathtt{left}$, $\\mathtt{right}$ and $\\mathtt{oracle}$ by the binary constructors $\\mathtt{pair}$, $\\mathtt{comp}$ and $\\mathtt{prec}$ and the unary constructor $\\mathtt{rfind}$. They are the codes for partial recursive functions (constant zero, successor, the two unpairing projections, pairing, composition, primitive\u2026","title":"Oracle codes"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/encode-oracle-code","kind":"node","lean":"Cslib.Decidability.encodeOracleCode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#encode-oracle-code","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Write $\\langle a, b \\rangle$ for the pairing of natural numbers equal to $b^2 + a$ if $a < b$ and to $a^2 + a + b$ otherwise. The code number $\\ulcorner c \\urcorner$ of an oracle code $c$ is defined by recursion on $c$: $\\mathsf{zero}$, $\\mathsf{succ}$, $\\mathsf{left}$, $\\mathsf{right}$ and $\\mathsf{oracle}$ have code numbers $\\langle 0, 0 \\rangle$, $\\langle 1, 0 \\rangle$, $\\langle 2, 0 \\rangle$,\u2026","title":"The G\u00f6del numbering of oracle codes"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/encode-oracle-code-injective","kind":"node","lean":"Cslib.Decidability.encodeOracleCode_injective","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#encode-oracle-code-injective","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"The G\u00f6del numbering of oracle codes is injective.","title":"The G\u00f6del numbering of oracle codes is injective"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/pair-ge-fst","kind":"node","lean":"Cslib.Decidability.pair_ge_fst","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#pair-ge-fst","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Let $\\langle a, b \\rangle$ denote the pairing of natural numbers, equal to $b^2 + a$ if $a < b$ and to $a^2 + a + b$ otherwise. For all natural numbers $a$ and $b$, $a \\le \\langle a, b \\rangle$.","title":"Pairing is at least its first argument"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/pair-ge-snd","kind":"node","lean":"Cslib.Decidability.pair_ge_snd","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#pair-ge-snd","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Let $\\langle a, b \\rangle$ denote the pairing of natural numbers, equal to $b^2 + a$ if $a < b$ and to $a^2 + a + b$ otherwise. For all natural numbers $a$ and $b$, $b \\le \\langle a, b \\rangle$.","title":"Pairing is at least its second argument"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/pair-four-gt","kind":"node","lean":"Cslib.Decidability.pair_four_gt","parent":35440,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pairing with "],["math","\\(4\\)"],["text"," strictly increases the second argument"]],"source":"Pairing with $4$ strictly increases the second argument"},"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#pair-four-gt","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Let $\\langle a, b \\rangle$ denote the pairing of natural numbers, equal to $b^2 + a$ if $a < b$ and to $a^2 + a + b$ otherwise. For every natural number $x$, $x < \\langle 4, x \\rangle$.","title":"Pairing with $4$ strictly increases the second argument"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/pair-gt-snd-of-pos","kind":"node","lean":"Cslib.Decidability.pair_gt_snd_of_pos","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#pair-gt-snd-of-pos","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Let $\\langle a, b \\rangle$ denote the pairing of natural numbers, equal to $b^2 + a$ if $a < b$ and to $a^2 + a + b$ otherwise. For all natural numbers $t$ and $x$ with $t \\ge 1$, $x < \\langle t, x \\rangle$.","title":"Pairing with a positive first argument strictly increases the second"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/unpair-fst-le","kind":"node","lean":"Cslib.Decidability.unpair_fst_le","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#unpair-fst-le","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Let $\\mathrm{pair} : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ be the pairing bijection with $\\mathrm{pair}(a, b) = b^2 + a$ if $a < b$ and $\\mathrm{pair}(a, b) = a^2 + a + b$ otherwise. For every natural number $n$, the first component of $\\mathrm{pair}^{-1}(n)$ is at most $n$.","title":"The first unpairing component is at most the number"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/unpair-snd-le","kind":"node","lean":"Cslib.Decidability.unpair_snd_le","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#unpair-snd-le","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Let $\\mathrm{pair} : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ be the pairing bijection with $\\mathrm{pair}(a, b) = b^2 + a$ if $a < b$ and $\\mathrm{pair}(a, b) = a^2 + a + b$ otherwise. For every natural number $n$, the second component of $\\mathrm{pair}^{-1}(n)$ is at most $n$.","title":"The second unpairing component is at most the number"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-code-depth","kind":"node","lean":"Cslib.Decidability.oracleCodeDepth","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-code-depth","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Leaves have depth `0`; each constructor adds one to the maximum depth of its children.","title":"Depth of an oracle code"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-oracle-code-fuel","kind":"node","lean":"Cslib.Decidability.decodeOracleCodeFuel","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-oracle-code-fuel","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Write $\\langle a, b \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ of the G\u00f6del numbering. Given a fuel $f$, a tag $t$ and a payload $p$, the decoder either returns an oracle code or fails; to decode a number $n = \\langle n_1, n_2 \\rangle$ with fuel $f$ means to run it with fuel $f$, tag $n_1$ and payload $n_2$. With fuel $0$ it fails. With fuel $f + 1$, writing\u2026","title":"Fuel-indexed decoder of oracle codes"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-oracle-code-fuel-complete","kind":"node","lean":"Cslib.Decidability.decodeOracleCodeFuel_complete","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-oracle-code-fuel-complete","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Write $\\langle a, b \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ of the G\u00f6del numbering. Let $c$ be an oracle code with G\u00f6del number $\\langle t, p \\rangle$. For every fuel $f$ strictly greater than the depth of $c$, the fuel-indexed decoder with fuel $f$, tag $t$ and payload $p$ returns $c$.","title":"The fuel-indexed decoder inverts the encoder"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-oracle-code-fuel-sound","kind":"node","lean":"Cslib.Decidability.decodeOracleCodeFuel_sound","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-oracle-code-fuel-sound","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Write $\\langle a, b \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ of the G\u00f6del numbering. For every fuel $f$, tag $t$, payload $p$ and oracle code $c$, if the fuel-indexed decoder with fuel $f$, tag $t$ and payload $p$ returns $c$, then the G\u00f6del number of $c$ is $\\langle t, p \\rangle$.","title":"Decoder soundness"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-code-depth-le-encode","kind":"node","lean":"Cslib.Decidability.oracleCodeDepth_le_encode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-code-depth-le-encode","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"For every oracle code $c$, the depth of $c$ is at most the code number of $c$ under the G\u00f6del numbering of oracle codes.","title":"Depth is bounded by the code number"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-oracle-code","kind":"node","lean":"Cslib.Decidability.decodeOracleCode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-oracle-code","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"The computable decoder for the G\u00f6del numbering of oracle codes sends a natural number $n$ to the result of the fuel-indexed decoder with fuel $n + 1$, tag $n_1$ and payload $n_2$, where $(n_1, n_2)$ is the unpairing of $n$; the result is either an oracle code or undefined. The fuel $n + 1$ is more than enough, since the depth of an oracle code is at most its code number.","title":"Decoding oracle codes"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-oracle-code-encode","kind":"node","lean":"Cslib.Decidability.decodeOracleCode_encode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-oracle-code-encode","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"For every oracle code $c$, the computable decoder applied to the G\u00f6del number of $c$ returns $c$.","title":"The computable decoder inverts the encoder"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-oracle-code-fuel-eq-decode","kind":"node","lean":"Cslib.Decidability.decodeOracleCodeFuel_eq_decode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-oracle-code-fuel-eq-decode","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"The fuel-indexed decoder agrees with the total decoder whenever the code number is below the fuel: decoding `(tag, payload)` at any fuel above the code number gives the same result as decoding the code number itself.","title":"The fuel-indexed decoder agrees with the total decoder"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-oracle-code-pair","kind":"node","lean":"Cslib.Decidability.decodeOracleCode_pair","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-oracle-code-pair","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Write $\\langle t, p \\rangle$ for the pairing of natural numbers used by the G\u00f6del numbering of oracle codes, $(p_1, p_2)$ for the unpairing of $p$ (so that $\\langle p_1, p_2 \\rangle = p$), and $\\mathrm{dec}(n)$ for the result of the computable decoder on $n$, which is either an oracle code or undefined. Then for all natural numbers $t$ and $p$, the value $\\mathrm{dec}\\langle t, p \\rangle$ is: $\\mathsf{zero}$,\u2026","title":"The decoder, by cases on the tag"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-is-some","kind":"node","lean":"Cslib.Decidability.decodeIsSome","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-is-some","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"For a natural number $n$, this is the Boolean value that is true exactly when the computable decoder of oracle codes returns an oracle code on $n$.","title":"Whether a code number decodes"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-is-some-pair","kind":"node","lean":"Cslib.Decidability.decodeIsSome_pair","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-is-some-pair","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Write $\\langle a, b \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ of the G\u00f6del numbering, and say that a natural number decodes when the computable decoder returns an oracle code on it. Let $t$ and $p$ be natural numbers and write $p = \\langle p_1, p_2 \\rangle$. Then $\\langle t, p \\rangle$ decodes if and only if one of the following holds: $t$ is $0$, $1$, $2$, $3$ or $8$ and\u2026","title":"The definedness predicate, by cases on the tag"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-is-some-step","kind":"node","lean":"Cslib.Decidability.decodeIsSomeStep","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-is-some-step","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Write $\\langle a, b \\rangle$ for the pairing bijection $\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ of the G\u00f6del numbering. The step takes an argument of the unit type, which it ignores, and a list $L$ of natural numbers; for $i \\in \\mathbb{N}$ write $L_i$ for the entry of $L$ at position $i$ (counting from $0$), with $L_i = 0$ when $i$ is at least the length of $L$. Let $m$ be the length of $L$, and write\u2026","title":"One step of the course-of-values recursion for definedness"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/primrec-decode-is-some-step","kind":"node","lean":"Cslib.Decidability.primrec\u2082_decodeIsSomeStep","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#primrec-decode-is-some-step","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"One step of the course-of-values recursion for definedness is primitive recursive as a function of its two arguments, an element of the unit type and a list of natural numbers.","title":"The course-of-values step for definedness is primitive recursive"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/decode-is-some-step-correct","kind":"node","lean":"Cslib.Decidability.decodeIsSomeStep_correct","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#decode-is-some-step-correct","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"For a natural number $n$, let $b_n$ be $1$ if the computable decoder returns an oracle code on $n$, and $0$ otherwise. For every natural number $m$, the one-step function of the course-of-values recursion for definedness, applied to the list $(b_0, b_1, \\dots, b_{m-1})$, returns $b_m$.","title":"The definedness step function is correct"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/nat-primrec-decode-is-some","kind":"node","lean":"Cslib.Decidability.natPrimrec_decodeIsSome","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#nat-primrec-decode-is-some","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"The function $\\mathbb{N} \\to \\mathbb{N}$ sending $n$ to $1$ if $n$ decodes to an oracle code under the computable decoder and to $0$ otherwise is primitive recursive.","title":"The definedness predicate is primitive recursive"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-code-of-nat","kind":"node","lean":"Cslib.Decidability.oracleCodeOfNat","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-code-of-nat","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"The $n$-th oracle code is the code that the computable decoder of the G\u00f6del numbering of oracle codes assigns to $n$, or the basic code $\\mathtt{zero}$ (the constant $0$ program) when $n$ is not the number of any code. Because the decoder inverts the G\u00f6del numbering, this is an effective enumeration tied to that numbering, not an arbitrary countable encoding.","title":"Effective enumeration of oracle codes"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-code-of-nat-encode","kind":"node","lean":"Cslib.Decidability.oracleCodeOfNat_encode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-code-of-nat-encode","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"For every oracle code $c$, the effective enumeration of oracle codes sends the code number of $c$ to $c$.","title":"The effective enumeration is a left inverse of the G\u00f6del numbering"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-code-of-nat-surjective","kind":"node","lean":"Cslib.Decidability.oracleCodeOfNat_surjective","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-code-of-nat-surjective","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"The enumeration of oracle codes is surjective.","title":"The enumeration of oracle codes is surjective"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/primrec-encode-oracle-code","kind":"node","lean":"Cslib.Decidability.primrec_encodeOracleCode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#primrec-encode-oracle-code","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Regard oracle codes as coded by their G\u00f6del numbers, with the computable decoder as decoding. Then the map sending each oracle code to its G\u00f6del number is primitive recursive.","title":"The G\u00f6del encoding of oracle codes is primitive recursive"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/primrec-decode-oracle-code","kind":"node","lean":"Cslib.Decidability.primrec_decodeOracleCode","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#primrec-decode-oracle-code","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Regard oracle codes as coded by their G\u00f6del numbers. Then the computable decoder, which sends a natural number $n$ to the oracle code with G\u00f6del number $n$ when there is one and to no value otherwise, is primitive recursive as a map from natural numbers to optional oracle codes.","title":"The G\u00f6del decoder of oracle codes is primitive recursive"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/primrec-oracle-code-of-nat","kind":"node","lean":"Cslib.Decidability.primrec_oracleCodeOfNat","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#primrec-oracle-code-of-nat","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Regard oracle codes as coded by their G\u00f6del numbers. Then the effective enumeration of oracle codes, which sends a natural number $n$ to the oracle code with G\u00f6del number $n$ when there is one and to the zero code otherwise, is primitive recursive.","title":"The effective enumeration of oracle codes is primitive recursive"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/computable-oracle-code-of-nat","kind":"node","lean":"Cslib.Decidability.computable_oracleCodeOfNat","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#computable-oracle-code-of-nat","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"The effective enumeration of oracle codes is computable.","title":"The effective enumeration of oracle codes is computable"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-evaln","kind":"node","lean":"Cslib.Decidability.oracleEvaln","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-evaln","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Write $\\langle a, b \\rangle$ for the pairing of natural numbers. For an oracle $o$ assigning to each natural number a natural number or no value, a fuel $k$, an oracle code $c$ and an input $n$, the evaluator $E_k(c, n)$ returns a natural number or fails. $E_0(c, n)$ always fails. $E_{k+1}(c, n)$ fails if $n > k$; otherwise, with $n = \\langle a, m \\rangle$:","title":"Fuel-indexed evaluation of oracle codes"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-evaln-bound","kind":"node","lean":"Cslib.Decidability.oracleEvaln_bound","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-evaln-bound","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"For every oracle $o$, fuel $k$, oracle code $c$ and input $n$, if the fuel-indexed evaluator with fuel $k$ returns a value on $c$ and $n$ relative to $o$, then $n < k$.","title":"Fuel bounds the inputs"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/oracle-eval","kind":"node","lean":"Cslib.Decidability.oracleEval","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#oracle-eval","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Let $o$ be an oracle that assigns to each natural number either a natural number or no value, $c$ an oracle code and $n$ a natural number. The value of $c$ on $n$ relative to $o$ is the value returned by the fuel-indexed evaluator of $c$ on $n$ relative to $o$ at the least fuel $k$ for which that evaluator returns a value; it is undefined when there is no such $k$.","title":"Evaluation of oracle codes"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/jump","kind":"node","lean":"Cslib.Decidability.Jump","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#jump","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"For a predicate $P$ on $\\mathbb{N}$, the Turing jump $P'$ is the predicate that holds at $n$ exactly when the $n$-th oracle code in the effective enumeration of oracle codes, evaluated with the oracle that answers $1$ at $m$ if $P(m)$ holds and $0$ otherwise, halts on input $n$. Thus $P'$ is the diagonal halting set of the universal oracle machine with oracle $P$.","title":"The Turing jump"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/turing-reducible-pred","kind":"node","lean":"Cslib.Decidability.TuringReduciblePred","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#turing-reducible-pred","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"A predicate $P$ on the natural numbers is Turing reducible to a predicate $Q$ on the natural numbers when the characteristic function of $P$ is Turing reducible to the characteristic function of $Q$, that is, partial recursive relative to an oracle for the characteristic function of $Q$.","title":"Turing reducibility of predicates"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/turing-equivalent-pred","kind":"node","lean":"Cslib.Decidability.TuringEquivalentPred","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#turing-equivalent-pred","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Two predicates $P$ and $Q$ on the natural numbers are Turing equivalent when the characteristic function of $P$ is Turing reducible to the characteristic function of $Q$ and the characteristic function of $Q$ is Turing reducible to the characteristic function of $P$.","title":"Turing equivalence of predicates"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/turing-degree-of","kind":"node","lean":"Cslib.Decidability.turingDegreeOf","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#turing-degree-of","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"The Turing degree of a predicate $P$ on the natural numbers is the Turing degree of its characteristic function, that is, the class of the characteristic function of $P$ under Turing equivalence of partial functions $\\mathbb{N} \\rightharpoonup \\mathbb{N}$.","title":"Turing degree of a predicate"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/turing-reducible-pred-refl","kind":"node","lean":"Cslib.Decidability.turingReduciblePred_refl","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#turing-reducible-pred-refl","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"Every predicate $P$ on the natural numbers is Turing reducible to itself.","title":"Turing reducibility is reflexive"},{"id":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/turing-reducible-pred-trans","kind":"node","lean":"Cslib.Decidability.turingReduciblePred_trans","parent":35440,"result":true,"route":"../roadmap/TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump/index.html#turing-reducible-pred-trans","scope":"TheoryOfComputation/degrees-and-hierarchies/turing-degrees-and-the-turing-jump","status":"fully_proved","summary":"For predicates $P$, $Q$ and $R$ on the natural numbers, if $P$ is Turing reducible to $Q$ and $Q$ is Turing reducible to $R$, then $P$ is Turing reducible to $R$.","title":"Turing reducibility is transitive"},{"data_url":"scopes/TheoryOfComputation/omega-automata-and-games.json","id":"TheoryOfComputation/omega-automata-and-games","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/index.html","scope":"TheoryOfComputation/omega-automata-and-games","status":"fully_proved","summary":"B\u00fcchi, generalized B\u00fcchi, parity, Rabin, Streett and Muller automata on infinite words, their emptiness and conversions, and parity games with attractors and traps.","title":"Automata on infinite words and infinite games"},{"data_url":"scopes/TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual.json","id":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual","kind":"scope","navigation":"module","parent":35481,"result":false,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/index.html","scope":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual","status":"fully_proved","title":"B\u00fcchi automata for infinitely often and eventually always"},{"id":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/infinitely-often","kind":"node","lean":"Cslib.\u03c9Language.Example.infinitelyOften","parent":35482,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/index.html#infinitely-often","scope":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual","status":"fully_proved","summary":"For a set $p$ of symbols, the \u03c9-language \u201cinfinitely often $p$\u201d consists of the infinite words $x_0 x_1 x_2 \\ldots$ with $x_k \\in p$ for infinitely many $k$.","title":"Infinitely often"},{"id":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/eventually-always","kind":"node","lean":"Cslib.\u03c9Language.Example.eventuallyAlways","parent":35482,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/index.html#eventually-always","scope":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual","status":"fully_proved","summary":"For a set $p$ of symbols, the \u03c9-language \u201ceventually always $p$\u201d consists of the infinite words $x_0 x_1 x_2 \\ldots$ for which there is an $N$ with $x_k \\in p$ for all $k \\ge N$.","title":"Eventually always"},{"id":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/compl-infinitely-often","kind":"node","lean":"Cslib.\u03c9Language.Example.compl_infinitelyOften","parent":35482,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/index.html#compl-infinitely-often","scope":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual","status":"fully_proved","summary":"For every set $p$ of symbols, the complement of the \u03c9-language of infinite words with infinitely many letters in $p$ is the \u03c9-language of infinite words whose letters are eventually all in the complement of $p$.","title":"Complement of \u201cinfinitely often\u201d"},{"id":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/infinitely-often-na","kind":"node","lean":"Cslib.\u03c9Language.Example.infinitelyOftenNa","parent":35482,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/index.html#infinitely-often-na","scope":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual","status":"fully_proved","summary":"For a set $p$ of symbols, this is the nondeterministic B\u00fcchi automaton over the alphabet of symbols with states $\\mathsf{true}$ and $\\mathsf{false}$, single start state $\\mathsf{true}$ and single accepting state $\\mathsf{true}$, in which reading a letter $x$ from either state leads to exactly one state: $\\mathsf{true}$ if $x \\in p$ and $\\mathsf{false}$ otherwise. So the state records whether the last letter read\u2026","title":"Two-state B\u00fcchi automaton for \u201cinfinitely often\u201d"},{"id":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/eventually-always-na","kind":"node","lean":"Cslib.\u03c9Language.Example.eventuallyAlwaysNa","parent":35482,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual/index.html#eventually-always-na","scope":"TheoryOfComputation/omega-automata-and-games/b-chi-automata-for-infinitely-often-and-eventual","status":"fully_proved","summary":"For a set $p$ of symbols, this is the nondeterministic B\u00fcchi automaton over the alphabet of symbols with states $\\mathsf{false}$ and $\\mathsf{true}$, single start state $\\mathsf{false}$ and single accepting state $\\mathsf{true}$. From $\\mathsf{false}$ it may read any letter and move to either state; from $\\mathsf{true}$ it may read only letters in $p$, and then it stays in $\\mathsf{true}$.","title":"Two-state B\u00fcchi automaton for \u201ceventually always\u201d"},{"data_url":"scopes/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata.json","id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","kind":"scope","navigation":"module","parent":35481,"result":false,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","title":"Emptiness of B\u00fcchi automata"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/step","kind":"node","lean":"Cslib.Automata.NA.Buchi.Step","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#step","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"For a nondeterministic B\u00fcchi automaton, the transition graph has an edge from a state $s$ to a state $t$ exactly when the automaton has a transition from $s$ to $t$ reading some symbol.","title":"Symbol-forgetting transition graph of a B\u00fcchi automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/has-lasso","kind":"node","lean":"Cslib.Automata.NA.Buchi.HasLasso","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#has-lasso","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"A nondeterministic B\u00fcchi automaton has a lasso if there are a start state $s_0$ and an accepting state $s$ such that the automaton can move from $s_0$ to $s$ by reading some finite word $u$, and from $s$ back to $s$ by reading some nonempty finite word $v$.","title":"Lasso of a B\u00fcchi automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/mem-language-iff","kind":"node","lean":"Cslib.Automata.NA.Buchi.mem_language_iff","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#mem-language-iff","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"A nondeterministic B\u00fcchi automaton accepts an infinite word $x = x_0 x_1 x_2 \\cdots$ if and only if it has a run on $x$, that is, a sequence of states $s_0 s_1 s_2 \\cdots$ with $s_0$ a start state and a transition from $s_k$ to $s_{k+1}$ reading $x_k$ for every $k$, such that $s_k$ is an accepting state for infinitely many $k$.","title":"B\u00fcchi acceptance condition"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/lasso-mem-language","kind":"node","lean":"Cslib.Automata.NA.Buchi.lasso_mem_language","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#lasso-mem-language","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"Let $\\mathcal{A}$ be a nondeterministic B\u00fcchi automaton over a nonempty alphabet. Suppose $s_0$ is a start state and $s$ an accepting state of $\\mathcal{A}$, that reading the finite word $u$ the automaton can move from $s_0$ to $s$, and that reading a nonempty finite word $v$ it can move from $s$ back to $s$. Then $\\mathcal{A}$ accepts the infinite word $u v^\\omega = u v v v \\cdots$.","title":"Lassos give accepted ultimately periodic words"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/has-lasso-of-mem","kind":"node","lean":"Cslib.Automata.NA.Buchi.hasLasso_of_mem","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#has-lasso-of-mem","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"If a nondeterministic B\u00fcchi automaton with finitely many states accepts some infinite word, then it has a lasso: some accepting state is visited twice along an accepting run, and the run between the two visits is the cycle.","title":"An accepted word yields a lasso"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/language-nonempty-iff-has-lasso","kind":"node","lean":"Cslib.Automata.NA.Buchi.language_nonempty_iff_hasLasso","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#language-nonempty-iff-has-lasso","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"For finitely many states, the language of a nondeterministic B\u00fcchi automaton is nonempty iff some accepting state is reachable from a start state and lies on a nonempty cycle.","title":"Lasso characterization of B\u00fcchi nonemptiness"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/exists-lasso-word-of-nonempty","kind":"node","lean":"Cslib.Automata.NA.Buchi.exists_lasso_word_of_nonempty","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#exists-lasso-word-of-nonempty","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"Let $A$ be a nondeterministic B\u00fcchi automaton with finitely many states over a nonempty alphabet. If $A$ accepts some infinite word, then there are finite words $u$ and $v$ with $v$ nonempty such that $A$ accepts the ultimately periodic word $u v^\\omega = u v v v \\cdots$.","title":"Nonempty B\u00fcchi languages contain ultimately periodic words"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/exists-m-tr-iff-refl-trans-gen","kind":"node","lean":"Cslib.Automata.NA.Buchi.exists_mTr_iff_reflTransGen","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#exists-m-tr-iff-refl-trans-gen","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"For states $s$ and $t$ of a nondeterministic B\u00fcchi automaton, the automaton can move from $s$ to $t$ by reading some finite word (possibly empty) if and only if $t$ is reachable from $s$ in zero or more steps of the transition graph, which has an edge from $p$ to $q$ when some symbol takes $p$ to $q$.","title":"Reachability by words is reachability in the transition graph"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/exists-ne-nil-m-tr-iff-trans-gen","kind":"node","lean":"Cslib.Automata.NA.Buchi.exists_ne_nil_mTr_iff_transGen","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#exists-ne-nil-m-tr-iff-trans-gen","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"For states $s$ and $t$ of a nondeterministic B\u00fcchi automaton, the automaton can move from $s$ to $t$ by reading some nonempty finite word if and only if $t$ is reachable from $s$ in one or more steps of the transition graph, which has an edge from $p$ to $q$ when some symbol takes $p$ to $q$.","title":"Nonempty words and paths of positive length"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/has-lasso-iff-reachable","kind":"node","lean":"Cslib.Automata.NA.Buchi.hasLasso_iff_reachable","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#has-lasso-iff-reachable","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"A nondeterministic B\u00fcchi automaton has a lasso if and only if there are a start state $s_0$ and an accepting state $s$ such that $s$ is reachable from $s_0$ in zero or more steps of the transition graph and $s$ is reachable from itself in one or more steps; the transition graph has an edge from $p$ to $q$ when some symbol takes $p$ to $q$.","title":"The lasso condition in the transition graph"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/start-fin","kind":"node","lean":"Cslib.Automata.NA.Buchi.startFin","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#start-fin","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"For a nondeterministic B\u00fcchi automaton with finitely many states and a decidable start set, this is the set of its start states, viewed as a finite set.","title":"Finite set of start states"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/lasso-states","kind":"node","lean":"Cslib.Automata.NA.Buchi.lassoStates","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#lasso-states","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"For a nondeterministic B\u00fcchi automaton with finitely many states, decidable equality of states, a finite alphabet, and decidable transitions, start set and accepting set, the lasso states form the finite set of accepting states $s$ that are reachable from some start state in zero or more transitions and lie on a nonempty cycle, that is, $s$ is reachable from itself in one or more transitions.","title":"Lasso states of a finite B\u00fcchi automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/has-lasso-iff-lasso-states-nonempty","kind":"node","lean":"Cslib.Automata.NA.Buchi.hasLasso_iff_lassoStates_nonempty","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#has-lasso-iff-lasso-states-nonempty","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"Let $A$ be a nondeterministic B\u00fcchi automaton with finitely many states and finitely many symbols, with decidable equality of states, and whose transitions, start states and accepting states are decidable. Then $A$ has a lasso if and only if its set of lasso states, computed by finite reachability closures as the accepting states that are reachable from a start state and lie on a nonempty cycle, is nonempty.","title":"Lasso detection by finite reachability"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/is-empty-b","kind":"node","lean":"Cslib.Automata.NA.Buchi.isEmptyB","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#is-empty-b","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"For a nondeterministic B\u00fcchi automaton with finitely many states and finitely many symbols, with decidable equality of states, and whose transitions, start states and accepting states are decidable, the emptiness test returns true exactly when the automaton has no lasso. It decides this by computing the set of lasso states; by the lasso characterization, it returns true iff no infinite word is accepted.","title":"B\u00fcchi emptiness test"},{"id":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/is-empty-b-iff","kind":"node","lean":"Cslib.Automata.NA.Buchi.isEmptyB_iff","parent":35488,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata/index.html#is-empty-b-iff","scope":"TheoryOfComputation/omega-automata-and-games/emptiness-of-b-chi-automata","status":"fully_proved","summary":"For a nondeterministic B\u00fcchi automaton with finitely many states and finitely many symbols, with decidable equality of states, and whose transitions, start states and accepting states are decidable, the emptiness test returns true if and only if the automaton accepts no infinite word.","title":"Correctness of the B\u00fcchi emptiness test"},{"data_url":"scopes/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata.json","id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","kind":"scope","navigation":"module","parent":35481,"result":false,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","title":"Generalized B\u00fcchi automata"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/gen-buchi","kind":"node","lean":"Cslib.Automata.NA.GenBuchi","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#gen-buchi","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"A generalized B\u00fcchi automaton with states $S$ and symbols $\\Sigma$ is a nondeterministic automaton (a labelled transition relation on $S$ together with a set of start states) equipped with a natural number $k$ and $k$ acceptance sets $F_0, \\dots, F_{k-1} \\subseteq S$. A run is accepting when it visits every acceptance set infinitely often.","title":"Generalized B\u00fcchi automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/mem-language-iff","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.mem_language_iff","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#mem-language-iff","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"An infinite word $x$ belongs to the language of a generalized B\u00fcchi automaton $g$ with acceptance sets $F_0, \\dots, F_{k-1}$ iff there is a run $s$ of $g$ on $x$ such that for every $j < k$ there are infinitely many times $t$ with $s_t \\in F_j$. Here a run of $g$ on $x$ is an infinite sequence of states $s_0, s_1, \\dots$ with $s_0$ a start state and a transition from $s_i$ to $s_{i+1}$ labelled $x_i$ for every $i$.","title":"Language of a generalized B\u00fcchi automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/next-count","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.nextCount","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#next-count","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with acceptance sets $F_0, \\dots, F_{k-1}$. The counter update takes a counter value $i \\in \\{0, \\dots, k\\}$ and a state $t$ of $g$ and returns a counter value in $\\{0, \\dots, k\\}$: if $i < k$, it returns $i + 1$ when $t \\in F_i$ and $i$ when $t \\notin F_i$; if $i = k$, it returns $0$.","title":"Counter update for degeneralization"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/next-count-eq-self","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.nextCount_eq_self","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#next-count-eq-self","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with acceptance sets $F_0, \\dots, F_{k-1}$ and counter update $\\mathrm{next}$. If $i \\in \\{0, \\dots, k\\}$ satisfies $i < k$ and the state $t$ is not in $F_i$, then $\\mathrm{next}(i, t) = i$.","title":"The degeneralization counter stays when the awaited set is missed"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/next-count-eq-succ","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.nextCount_eq_succ","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#next-count-eq-succ","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with acceptance sets $F_0, \\dots, F_{k-1}$, let $i \\in \\{0, \\dots, k\\}$ be a counter value and let $t$ be a state of $g$. If $i < k$ and $t \\in F_i$, then the counter update of $i$ at $t$ is $i + 1$.","title":"The degeneralization counter advances on a hit"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/next-count-eq-zero","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.nextCount_eq_zero","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#next-count-eq-zero","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with $k$ acceptance sets and let $i \\in \\{0, \\dots, k\\}$ be a counter value. If $i = k$, then for every state $t$ of $g$ the counter update of $i$ at $t$ is $0$.","title":"The degeneralization counter resets at the top"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/count-seq","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.countSeq","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#count-seq","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with $k$ acceptance sets and let $s = (s_0, s_1, \\dots)$ be an infinite sequence of states. The counter sequence $c_0, c_1, \\dots$ of $s$, with values in $\\{0, \\dots, k\\}$, is defined by $c_0 = 0$ and $c_{n+1} = \\mathrm{next}(c_n, s_{n+1})$, where $\\mathrm{next}$ is the counter update of $g$. So $c_n$ is the counter after reading the states $s_1, \\dots, s_n$; the state $s_0$\u2026","title":"Counter sequence of a state sequence"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/count-seq-zero","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.countSeq_zero","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#count-seq-zero","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"For every generalized B\u00fcchi automaton $g$ and every infinite sequence of states $s$, the counter sequence of $s$ has value $0$ at time $0$.","title":"The counter sequence starts at zero"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/count-seq-succ","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.countSeq_succ","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#count-seq-succ","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton, $s$ an infinite sequence of states, and $c_0, c_1, \\dots$ the counter sequence of $s$. For every $n$, $c_{n+1} = \\mathrm{next}(c_n, s_{n+1})$, where $\\mathrm{next}$ is the counter update of $g$.","title":"Recursion step of the counter sequence"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/count-seq-stay","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.countSeq_stay","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#count-seq-stay","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with acceptance sets $F_0, \\dots, F_{k-1}$, let $s$ be an infinite sequence of states with counter sequence $c_0, c_1, \\dots$, and let $i < k$ and $n$ satisfy $c_n = i$. If $m \\ge n$ and $s_{j+1} \\notin F_i$ for every $j$ with $n \\le j < m$, then $c_m = i$.","title":"The degeneralization counter waits for the awaited set"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/count-seq-reach","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.countSeq_reach","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#count-seq-reach","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with acceptance sets $F_0, \\dots, F_{k-1}$, and let $s$ be an infinite sequence of states that visits every $F_j$ infinitely often, with counter sequence $c_0, c_1, \\dots$. Then for every time $n$ and every $j$ with $c_n \\le j \\le k$ there is a time $m \\ge n$ with $c_m = j$.","title":"The degeneralization counter reaches every higher value"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/counter-cross","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.counter_cross","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#counter-cross","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with acceptance sets $F_0, \\dots, F_{k-1}$ and counter update $\\mathrm{next}$, let $s$ be an infinite sequence of states, and let $c_0, c_1, \\dots$ be any sequence in $\\{0, \\dots, k\\}$ with $c_{n+1} = \\mathrm{next}(c_n, s_{n+1})$ for every $n$. Let $a \\le b$ be times with $c_a = 0$ and $c_b = d \\le k$, and suppose $c_t \\ne k$ for every $t$ with $a < t < b$. Then for every\u2026","title":"The degeneralization counter crosses every value below its target"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/to-buchi","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.toBuchi","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#to-buchi","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"Let $g$ be a generalized B\u00fcchi automaton with $k$ acceptance sets. Its degeneralization is the B\u00fcchi automaton over the same alphabet whose states are the pairs $(s, i)$ of a state $s$ of $g$ and a counter value $i \\in \\{0, \\dots, k\\}$. On a symbol $x$ there is a transition from $(s, i)$ to $(s', i')$ iff $g$ has a transition from $s$ to $s'$ on $x$ and $i'$ is the counter update of $i$ at the state $s'$. The start\u2026","title":"Degeneralization of a generalized B\u00fcchi automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/language-to-buchi","kind":"node","lean":"Cslib.Automata.NA.GenBuchi.language_toBuchi","parent":35504,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata/index.html#language-to-buchi","scope":"TheoryOfComputation/omega-automata-and-games/generalized-b-chi-automata","status":"fully_proved","summary":"A word is accepted by the degeneralized B\u00fcchi automaton iff it is accepted by the generalized B\u00fcchi automaton.","title":"Degeneralization preserves the language"},{"data_url":"scopes/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences.json","id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","kind":"scope","navigation":"module","parent":35481,"result":false,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","title":"Parity condition on infinite sequences"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/is-max-inf-occ","kind":"node","lean":"Cslib.\u03c9Sequence.IsMaxInfOcc","parent":35519,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html#is-max-inf-occ","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","summary":"Let $(p_k)_{k \\in \\mathbb{N}}$ be a sequence of natural numbers and $n \\in \\mathbb{N}$. Then $n$ is the largest value occurring infinitely often in $(p_k)$ if $p_k = n$ for infinitely many $k$, and every $m \\in \\mathbb{N}$ with $p_k = m$ for infinitely many $k$ satisfies $m \\le n$.","title":"Largest infinitely occurring value"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/parity-even","kind":"node","lean":"Cslib.\u03c9Sequence.ParityEven","parent":35519,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html#parity-even","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","summary":"A sequence $(p_k)_{k \\in \\mathbb{N}}$ of natural numbers (priorities) satisfies the even max-parity condition if there is an even $n$ that is the largest value occurring infinitely often in $(p_k)$.","title":"Even max-parity condition"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/parity-odd","kind":"node","lean":"Cslib.\u03c9Sequence.ParityOdd","parent":35519,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html#parity-odd","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","summary":"A sequence $(p_k)_{k \\in \\mathbb{N}}$ of natural numbers (priorities) satisfies the odd max-parity condition if there is an odd $n$ that is the largest value occurring infinitely often in $(p_k)$.","title":"Odd max-parity condition"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/exists-is-max-inf-occ","kind":"node","lean":"Cslib.\u03c9Sequence.exists_isMaxInfOcc","parent":35519,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html#exists-is-max-inf-occ","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$. If a sequence $(p_k)_{k \\in \\mathbb{N}}$ of natural numbers satisfies $p_k \\le d$ for all $k$, then some natural number $n \\le d$ is the largest value occurring infinitely often in $(p_k)$.","title":"Bounded sequences have a largest infinitely occurring value"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/eventually-le","kind":"node","lean":"Cslib.\u03c9Sequence.IsMaxInfOcc.eventually_le","parent":35519,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html#eventually-le","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$, let $(p_k)_{k \\in \\mathbb{N}}$ be a sequence of natural numbers with $p_k \\le d$ for all $k$, and let $n$ be the largest value occurring infinitely often in $(p_k)$. Then $p_k \\le n$ for all sufficiently large $k$.","title":"The tail of a bounded sequence is bounded by its largest infinitely occurring value"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/parity-odd-iff-not-parity-even","kind":"node","lean":"Cslib.\u03c9Sequence.parityOdd_iff_not_parityEven","parent":35519,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html#parity-odd-iff-not-parity-even","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","summary":"Let $d \\in \\mathbb{N}$. If a sequence $(p_k)_{k \\in \\mathbb{N}}$ of natural numbers satisfies $p_k \\le d$ for all $k$, then it satisfies the odd max-parity condition if and only if it does not satisfy the even max-parity condition.","title":"For bounded sequences the odd parity condition is the negation of the even one"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/parity-even-iff-frequently-two","kind":"node","lean":"Cslib.\u03c9Sequence.parityEven_iff_frequently_two","parent":35519,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html#parity-even-iff-frequently-two","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","summary":"Let $(p_k)_{k \\in \\mathbb{N}}$ be a sequence of natural numbers with $p_k \\in \\{1, 2\\}$ for every $k$. Then $(p_k)$ satisfies the even max-parity condition if and only if $p_k = 2$ for infinitely many $k$.","title":"Max-parity with priorities 1 and 2 is the B\u00fcchi condition"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/is-max-inf-occ-drop-iff","kind":"node","lean":"Cslib.\u03c9Sequence.isMaxInfOcc_drop_iff","parent":35519,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences/index.html#is-max-inf-occ-drop-iff","scope":"TheoryOfComputation/omega-automata-and-games/parity-condition-on-infinite-sequences","status":"fully_proved","summary":"Let $(p_k)_{k \\in \\mathbb{N}}$ be a sequence of natural numbers and let $j, n \\in \\mathbb{N}$. Then $n$ is the largest value occurring infinitely often in the shifted sequence $(p_{j+k})_{k \\in \\mathbb{N}}$ if and only if $n$ is the largest value occurring infinitely often in $(p_k)$.","title":"Prefix independence of the largest infinitely occurring value"},{"data_url":"scopes/TheoryOfComputation/omega-automata-and-games/parity-automata.json","id":"TheoryOfComputation/omega-automata-and-games/parity-automata","kind":"scope","navigation":"module","parent":35481,"result":false,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-automata/index.html","scope":"TheoryOfComputation/omega-automata-and-games/parity-automata","status":"fully_proved","title":"Parity automata"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-automata/parity","kind":"node","lean":"Cslib.Automata.NA.Parity","parent":35528,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-automata/index.html#parity","scope":"TheoryOfComputation/omega-automata-and-games/parity-automata","status":"fully_proved","summary":"A nondeterministic parity automaton with states in $Q$ and symbols in $\\Sigma$ is a nondeterministic automaton (a transition relation labelled by symbols on $Q$, together with a set of start states) with a priority $p(q) \\in \\mathbb{N}$ assigned to each state $q \\in Q$.","title":"Nondeterministic parity automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-automata/run-parity-even-or-parity-odd","kind":"node","lean":"Cslib.Automata.NA.Parity.run_parityEven_or_parityOdd","parent":35528,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-automata/index.html#run-parity-even-or-parity-odd","scope":"TheoryOfComputation/omega-automata-and-games/parity-automata","status":"fully_proved","summary":"Let $\\mathcal{A}$ be a nondeterministic parity automaton with finitely many states and priority function $p$. For every infinite sequence of states $s_0, s_1, s_2, \\dots$, the priority sequence $p(s_0), p(s_1), p(s_2), \\dots$ satisfies the even or the odd max-parity condition: the largest priority occurring infinitely often exists and is even, or it exists and is odd.","title":"Every state sequence of a finite parity automaton has a parity winner"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-automata/to-parity","kind":"node","lean":"Cslib.Automata.NA.Buchi.toParity","parent":35528,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-automata/index.html#to-parity","scope":"TheoryOfComputation/omega-automata-and-games/parity-automata","status":"fully_proved","summary":"A nondeterministic B\u00fcchi automaton is turned into the nondeterministic parity automaton with the same states, transitions and start states, in which every accepting state has priority $2$ and every other state has priority $1$.","title":"B\u00fcchi automata as parity automata"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-automata/language-to-parity","kind":"node","lean":"Cslib.Automata.NA.Buchi.language_toParity","parent":35528,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-automata/index.html#language-to-parity","scope":"TheoryOfComputation/omega-automata-and-games/parity-automata","status":"fully_proved","summary":"Every nondeterministic B\u00fcchi automaton accepts the same language of infinite words as its presentation as a parity automaton (priority $2$ on accepting states, $1$ elsewhere).","title":"B\u00fcchi-to-parity conversion preserves the language"},{"data_url":"scopes/TheoryOfComputation/omega-automata-and-games/parity-games.json","id":"TheoryOfComputation/omega-automata-and-games/parity-games","kind":"scope","navigation":"module","parent":35481,"result":false,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","title":"Parity games"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/player","kind":"node","lean":"Cslib.Games.Player","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#player","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"A parity game has exactly two players, the even player and the odd player.","title":"Players of a parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/opponent","kind":"node","lean":"Cslib.Games.Player.opponent","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#opponent","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"The opponent of a player of a parity game is the other player: the opponent of the even player is the odd player, and the opponent of the odd player is the even player.","title":"Opponent"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/eq-or-eq-opponent","kind":"node","lean":"Cslib.Games.Player.eq_or_eq_opponent","parent":35533,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every player is "],["math","\\(p\\)"],["text"," or the opponent of "],["math","\\(p\\)"]],"source":"Every player is $p$ or the opponent of $p$"},"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#eq-or-eq-opponent","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"For all players $p$ and $q$ of a parity game, $q = p$ or $q$ is the opponent of $p$.","title":"Every player is $p$ or the opponent of $p$"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/wins","kind":"node","lean":"Cslib.Games.Player.Wins","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#wins","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"A player wins an infinite sequence of natural-number priorities when the sequence satisfies that player's parity condition: the even player wins it when the largest priority occurring infinitely often exists and is even, and the odd player wins it when the largest priority occurring infinitely often exists and is odd.","title":"Parity winning condition of a player"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/parity-game","kind":"node","lean":"Cslib.Games.ParityGame","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#parity-game","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"A parity game on a type of positions consists of an owner for each position, which is one of the two players, a move relation between positions, where the owner of $v$ may move from $v$ to $w$ when $v$ is related to $w$, and a natural-number priority for each position, such that every position has at least one move (there are no dead ends).","title":"Parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/is-play","kind":"node","lean":"Cslib.Games.ParityGame.IsPlay","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#is-play","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"An infinite sequence of positions $\\pi_0, \\pi_1, \\dots$ of a parity game is a play if for every $i$ there is a move from $\\pi_i$ to $\\pi_{i+1}$.","title":"Plays of a parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/won-by","kind":"node","lean":"Cslib.Games.ParityGame.WonBy","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#won-by","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, write $\\Omega(v)$ for the priority of a position $v$, and let $p$ be a player. An infinite sequence $\\pi = \\pi_0 \\pi_1 \\pi_2 \\dots$ of positions of $G$ is won by $p$ when its priority sequence $\\Omega(\\pi_0) \\Omega(\\pi_1) \\Omega(\\pi_2) \\dots$ satisfies the parity condition of $p$: among the priorities occurring infinitely often there is a largest one, and it is even if $p$ is the even\u2026","title":"Winner of a play in a parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/positional-strategy","kind":"node","lean":"Cslib.Games.ParityGame.PositionalStrategy","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#positional-strategy","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"A positional (memoryless) strategy for player `p`: a choice of successor at each position, legal at the positions `p` owns. Its values elsewhere are irrelevant.","title":"Positional strategy in a parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/nonempty-positional-strategy","kind":"node","lean":"Cslib.Games.ParityGame.nonempty_positionalStrategy","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#nonempty-positional-strategy","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"In a parity game, every player has a positional strategy, since every position has a move.","title":"Positional strategies exist"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/conforms","kind":"node","lean":"Cslib.Games.ParityGame.Conforms","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#conforms","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $\\sigma$ be a positional strategy of player $p$ in a parity game. An infinite sequence of positions $\\pi_0, \\pi_1, \\dots$ (not necessarily a play) conforms to $\\sigma$ if $\\pi_{i+1} = \\sigma(\\pi_i)$ for every $i$ such that $p$ owns $\\pi_i$.","title":"Plays conforming to a strategy"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/wins-from","kind":"node","lean":"Cslib.Games.ParityGame.WinsFrom","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#wins-from","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $p$ be a player of a parity game. A positional strategy $\\sigma$ of $p$ wins from a position $v$ when every play starting at $v$ that conforms to $\\sigma$ (at every position owned by $p$, the play moves to the successor chosen by $\\sigma$) is won by $p$.","title":"Winning strategy from a position"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/winning-region","kind":"node","lean":"Cslib.Games.ParityGame.WinningRegion","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#winning-region","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"The positions from which player `p` has a winning positional strategy.","title":"Winning region of a parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/won-by-or-won-by-opponent","kind":"node","lean":"Cslib.Games.ParityGame.wonBy_or_wonBy_opponent","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#won-by-or-won-by-opponent","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $p$ be a player of a parity game in which every position has priority at most $d$, for some natural number $d$. Then every infinite sequence of positions, in particular every play, is won by $p$ or by the opponent of $p$.","title":"Bounded priorities give every play a winner"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/not-won-by-and-won-by-opponent","kind":"node","lean":"Cslib.Games.ParityGame.not_wonBy_and_wonBy_opponent","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#not-won-by-and-won-by-opponent","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"In a parity game, for every player $p$, no infinite sequence of positions (play or not) is won both by $p$ and by the opponent of $p$.","title":"No play is won by both players"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/won-by-drop-iff","kind":"node","lean":"Cslib.Games.ParityGame.wonBy_drop_iff","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#won-by-drop-iff","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $p$ be a player of a parity game, $\\pi = (\\pi_0, \\pi_1, \\dots)$ an infinite sequence of positions and $j$ a natural number. Then the sequence $(\\pi_j, \\pi_{j+1}, \\dots)$, obtained by dropping the first $j$ positions of $\\pi$, is won by $p$ if and only if $\\pi$ is won by $p$.","title":"Winning is prefix independent"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/joint-next","kind":"node","lean":"Cslib.Games.ParityGame.jointNext","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#joint-next","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $\\sigma$ be a positional strategy of player $p$ and $\\tau$ a positional strategy of the opponent of $p$ in a parity game. The joint successor of a position $v$ is $\\sigma(v)$ if $p$ owns $v$, and $\\tau(v)$ otherwise.","title":"Joint successor of two strategies"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/outcome","kind":"node","lean":"Cslib.Games.ParityGame.outcome","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#outcome","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $p$ be a player of a parity game, $\\sigma$ a positional strategy of $p$ and $\\tau$ a positional strategy of the opponent of $p$. The outcome of $\\sigma$ and $\\tau$ from a position $v$ is the infinite sequence of positions $v_0, v_1, v_2, \\dots$ with $v_0 = v$ in which $v_{i+1}$ is the successor chosen by $\\sigma$ at $v_i$ if $p$ owns $v_i$, and the successor chosen by $\\tau$ at $v_i$ otherwise. It is the play\u2026","title":"Outcome of two positional strategies"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/disjoint-winning-region","kind":"node","lean":"Cslib.Games.ParityGame.disjoint_winningRegion","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#disjoint-winning-region","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"In a parity game, for every player $p$, no position lies both in the winning region of $p$ and in the winning region of the opponent of $p$.","title":"The winning regions of the two players are disjoint"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/of-move","kind":"node","lean":"Cslib.Games.ParityGame.WinsFrom.of_move","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#of-move","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"In a parity game, let $\\sigma$ be a positional strategy of player $p$ that wins from a position $v$, and let $w$ be a position with a move from $v$ to $w$ such that $w = \\sigma(v)$ if $p$ owns $v$. Then $\\sigma$ wins from $w$.","title":"Winning strategies keep winning along allowed moves"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/move-mem-winning-region","kind":"node","lean":"Cslib.Games.ParityGame.move_mem_winningRegion","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#move-mem-winning-region","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"In a parity game, let $v$ be a position in the winning region of player $p$ that is owned by the opponent of $p$. Then every position $w$ with a move from $v$ to $w$ is in the winning region of $p$.","title":"Opponent moves stay in the winning region"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/next-mem-winning-region","kind":"node","lean":"Cslib.Games.ParityGame.next_mem_winningRegion","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#next-mem-winning-region","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"In a parity game, let $\\sigma$ be a positional strategy of player $p$ that wins from a position $v$ owned by $p$. Then the successor $\\sigma(v)$ chosen by $\\sigma$ is in the winning region of $p$.","title":"A winning strategy's move stays in the winning region"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/cpre","kind":"node","lean":"Cslib.Games.ParityGame.cpre","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#cpre","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $X$ a set of positions. The controllable predecessor $\\mathrm{cpre}_p(X)$ of $X$ for $p$ is the set of positions $v$ such that either $v$ is owned by $p$ and some move from $v$ leads into $X$, or $v$ is not owned by $p$ and every move from $v$ leads into $X$.","title":"Controllable predecessor"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-step","kind":"node","lean":"Cslib.Games.ParityGame.attractorStep","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-step","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $T$ a set of positions. The attractor operator of $T$ for $p$ is the monotone map on sets of positions sending $X$ to $T \\cup \\mathrm{cpre}_p(X)$, where $\\mathrm{cpre}_p$ is the controllable predecessor for $p$.","title":"Attractor operator"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor","kind":"node","lean":"Cslib.Games.ParityGame.attractor","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $T$ a set of positions. The attractor of $T$ for $p$ is the least fixed point of the attractor operator $X \\mapsto T \\cup \\mathrm{cpre}_p(X)$ on sets of positions, where $\\mathrm{cpre}_p$ is the controllable predecessor for $p$.","title":"Attractor"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-stage","kind":"node","lean":"Cslib.Games.ParityGame.attractorStage","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-stage","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player, $T$ a set of positions and $n$ a natural number. The $n$-th attractor stage $S_n$ of $T$ for $p$ is the set obtained by applying the attractor operator $X \\mapsto T \\cup \\mathrm{cpre}_p(X)$ $n$ times to $\\emptyset$, where $\\mathrm{cpre}_p$ is the controllable predecessor for $p$; thus $S_0 = \\emptyset$ and $S_{n+1} = T \\cup \\mathrm{cpre}_p(S_n)$.","title":"Attractor stages"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/union-cpre-attractor","kind":"node","lean":"Cslib.Games.ParityGame.union_cpre_attractor","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#union-cpre-attractor","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $T$ a set of positions, and let $A$ be the attractor of $T$ for $p$. Then $T \\cup \\mathrm{cpre}_p(A) = A$, where $\\mathrm{cpre}_p(A)$ is the controllable predecessor of $A$ for $p$; that is, $A$ is a fixed point of the attractor operator $X \\mapsto T \\cup \\mathrm{cpre}_p(X)$.","title":"The attractor is a fixed point"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-subset-of","kind":"node","lean":"Cslib.Games.ParityGame.attractor_subset_of","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-subset-of","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $T$ a set of positions. If a set $X$ of positions satisfies $T \\cup \\mathrm{cpre}_p(X) \\subseteq X$, where $\\mathrm{cpre}_p$ is the controllable predecessor for $p$, then the attractor of $T$ for $p$ is contained in $X$.","title":"The attractor lies in every closed set"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-eq-attractor-stage-card","kind":"node","lean":"Cslib.Games.ParityGame.attractor_eq_attractorStage_card","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-eq-attractor-stage-card","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game with finitely many positions, $N$ of them, let $p$ be a player and $T$ a set of positions. Then the attractor of $T$ for $p$ equals the $N$-th attractor stage of $T$ for $p$, that is, the result of applying the attractor operator $X \\mapsto T \\cup \\mathrm{cpre}_p(X)$ $N$ times to $\\emptyset$, where $\\mathrm{cpre}_p$ is the controllable predecessor for $p$.","title":"The attractor stabilizes after as many stages as there are positions"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-strategy","kind":"node","lean":"Cslib.Games.ParityGame.attractorStrategy","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-strategy","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $T$ a set of positions. The attractor strategy of $p$ for $T$ is the positional strategy of $p$ that at a position $v$ moves to a successor of $v$ lying in the $n$-th attractor stage of $T$ for $p$, for the least $n$ for which $v$ has such a successor, and to an arbitrary successor of $v$ if no successor of $v$ lies in any attractor stage.","title":"Attractor strategy"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-strategy-next-mem","kind":"node","lean":"Cslib.Games.ParityGame.attractorStrategy_next_mem","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-strategy-next-mem","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player, $T$ a set of positions, $n$ a natural number and $S_n$ the $n$-th attractor stage of $T$ for $p$. If a position $v$ owned by $p$ lies in the controllable predecessor $\\mathrm{cpre}_p(S_n)$ of $S_n$ for $p$, then the successor that the attractor strategy of $p$ for $T$ chooses at $v$ lies in $S_n$.","title":"The attractor strategy moves into the stage from its controllable predecessor"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-strategy-reaches","kind":"node","lean":"Cslib.Games.ParityGame.attractorStrategy_reaches","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-strategy-reaches","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player, $T$ a set of positions and $n$ a natural number. If $\\pi$ is a play of $G$ that conforms to the attractor strategy of $p$ for $T$ and whose initial position $\\pi_0$ lies in the $n$-th attractor stage of $T$ for $p$, then $\\pi_i \\in T$ for some $i < n$.","title":"The attractor strategy reaches the target from an attractor stage"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/forces-reach","kind":"node","lean":"Cslib.Games.ParityGame.ForcesReach","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#forces-reach","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player, $\\sigma$ a positional strategy of $p$, $T$ a set of positions and $v$ a position. The strategy $\\sigma$ forces a visit to $T$ from $v$ when every play $\\pi$ of $G$ with $\\pi_0 = v$ that conforms to $\\sigma$ visits $T$, that is, $\\pi_i \\in T$ for some $i \\in \\mathbb{N}$.","title":"Forcing a visit to a set"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-strategy-reaches-of-mem-attrac","kind":"node","lean":"Cslib.Games.ParityGame.attractorStrategy_reaches_of_mem_attractor","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-strategy-reaches-of-mem-attrac","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game with finitely many positions, $N$ of them, let $p$ be a player and $T$ a set of positions. If $\\pi$ is a play of $G$ that conforms to the attractor strategy of $p$ for $T$ and whose initial position $\\pi_0$ lies in the attractor of $T$ for $p$, then $\\pi_i \\in T$ for some $i < N$.","title":"The attractor strategy reaches the target from the attractor"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/forces-reach-attractor-strategy","kind":"node","lean":"Cslib.Games.ParityGame.forcesReach_attractorStrategy","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#forces-reach-attractor-strategy","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game with finitely many positions, $p$ a player and $T$ a set of positions. From every position $v$ of the attractor of $T$ for $p$, the attractor strategy of $p$ for $T$ forces a visit to $T$: every play from $v$ conforming to it visits $T$.","title":"The attractor strategy forces a visit"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/is-trap","kind":"node","lean":"Cslib.Games.ParityGame.IsTrap","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#is-trap","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game and $p$ a player. A set $U$ of positions is a trap for $p$ when, for every position $v \\in U$, every move from $v$ leads into $U$ if $v$ belongs to $p$, and some move from $v$ leads into $U$ if $v$ does not belong to $p$. Thus $p$ cannot force the play out of $U$.","title":"Trap for a player"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/is-trap-compl-attractor","kind":"node","lean":"Cslib.Games.ParityGame.isTrap_compl_attractor","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#is-trap-compl-attractor","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $T$ a set of positions. The complement of the attractor of $T$ for $p$ is a trap for $p$.","title":"The complement of an attractor is a trap"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/disjoint-attractor","kind":"node","lean":"Cslib.Games.ParityGame.IsTrap.disjoint_attractor","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#disjoint-attractor","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $T$ a set of positions. If a set $U$ of positions is a trap for $p$ and is disjoint from $T$, then $U$ is disjoint from the attractor of $T$ for $p$.","title":"Traps disjoint from a set avoid its attractor"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/trap-strategy","kind":"node","lean":"Cslib.Games.ParityGame.trapStrategy","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#trap-strategy","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $U$ a set of positions. The trap strategy for $U$ is the positional strategy of the opponent of $p$ that, at each position $v$, moves to some position $w \\in U$ with a move from $v$ to $w$ if there is one, and to an arbitrary position reachable from $v$ in one move otherwise.","title":"Trap strategy"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/trap-strategy-stays","kind":"node","lean":"Cslib.Games.ParityGame.IsTrap.trapStrategy_stays","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#trap-strategy-stays","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $U$ a trap for $p$. If $\\pi$ is a play with $\\pi_0 \\in U$ that conforms to the trap strategy of the opponent of $p$ for $U$, then $\\pi_i \\in U$ for every $i \\in \\mathbb{N}$.","title":"The trap strategy stays in the trap"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/trap-strategy-avoids","kind":"node","lean":"Cslib.Games.ParityGame.trapStrategy_avoids","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#trap-strategy-avoids","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player, $T$ a set of positions and $A$ the attractor of $T$ for $p$. If $\\pi$ is a play starting at a position outside $A$ that conforms to the trap strategy of the opponent of $p$ for the complement of $A$, then $\\pi_i \\notin T$ for every $i \\in \\mathbb{N}$.","title":"The trap strategy avoids the target"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/not-forces-reach-of-not-mem-attractor","kind":"node","lean":"Cslib.Games.ParityGame.not_forcesReach_of_notMem_attractor","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#not-forces-reach-of-not-mem-attractor","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game, $p$ a player and $T$ a set of positions. If a position $v$ is not in the attractor of $T$ for $p$, then no positional strategy of $p$ forces a visit to $T$ from $v$.","title":"No strategy forces a visit from outside the attractor"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/mem-attractor-iff","kind":"node","lean":"Cslib.Games.ParityGame.mem_attractor_iff","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#mem-attractor-iff","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be a parity game with finitely many positions, $p$ a player and $T$ a set of positions. A position $v$ lies in the attractor of $T$ for $p$ if and only if some positional strategy of $p$ forces a visit to $T$ from $v$.","title":"Characterization of the attractor"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/is-trap-winning-region","kind":"node","lean":"Cslib.Games.ParityGame.isTrap_winningRegion","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#is-trap-winning-region","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"In a parity game $G$, the winning region of a player $p$, the set of positions from which $p$ has a winning positional strategy, is a trap for the opponent of $p$.","title":"Winning regions are traps for the opponent"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/encoded-parity-game","kind":"node","lean":"Cslib.Games.EncodedParityGame","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#encoded-parity-game","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"An encoded finite parity game on the $n$ positions $0, \\dots, n - 1$ consists of an owner for each position, given as a Boolean (false for the even player, true for the odd player), a move relation, given as a Boolean-valued $n \\times n$ matrix, and a priority for each position, given as a natural number.","title":"Encoded parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/to-parity-game","kind":"node","lean":"Cslib.Games.EncodedParityGame.toParityGame","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#to-parity-game","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be an encoded parity game on the positions $\\{0, \\dots, n-1\\}$ in which every position has a move to some position. The associated parity game has the same positions, moves and priorities, and a position is owned by the odd player if $G$ assigns it to the odd player and by the even player otherwise; every position has a move by the hypothesis on $G$.","title":"The parity game of a serial encoded game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/dual-game","kind":"node","lean":"Cslib.Games.EncodedParityGame.dualGame","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#dual-game","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"The dual of an encoded parity game $G$ is the encoded parity game with the same positions and moves in which every position is owned by the other player than in $G$ and has priority one more than in $G$.","title":"Dual parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/cpre-in","kind":"node","lean":"Cslib.Games.EncodedParityGame.cpreIn","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#cpre-in","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be an encoded parity game, $p$ a player (encoded as a Boolean: false for even, true for odd), and $U$, $X$ sets of positions. The controllable predecessor of $X$ for $p$ inside $U$ is the set of positions $v \\in U$ such that either $v$ is owned by $p$ and some move of $G$ goes from $v$ to a position of $U \\cap X$, or $v$ is not owned by $p$ and every move of $G$ from $v$ to a position of $U$ ends in $X$.","title":"Controllable predecessor inside a set of positions"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/attractor-in","kind":"node","lean":"Cslib.Games.EncodedParityGame.attractorIn","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#attractor-in","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be an encoded parity game on $n$ positions, $p$ a player (encoded as a Boolean: false for even, true for odd), and $T$, $U$ sets of positions. The attractor of $T$ for $p$ inside $U$ is the set obtained from $\\emptyset$ by applying $n$ times the map $X \\mapsto (T \\cap U) \\cup \\mathrm{CPre}_p^U(X)$, where $\\mathrm{CPre}_p^U(X)$ is the controllable predecessor of $X$ for $p$ inside $U$.","title":"Attractor inside a set of positions"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-rank","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityRank","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-rank","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"The rank of a priority `p` among the distinct priorities occurring in the game: the number of distinct priorities `q < p` that occur.","title":"Rank of a priority in a parity game"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-compressed","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityCompressed","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-compressed","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be an encoded parity game on the positions $\\{0, \\dots, n-1\\}$. The compressed priority of a natural number $p$ is $2r + (p \\bmod 2)$, where $r$ is the rank of $p$ in $G$: the number of distinct priorities $q < p$ that occur in $G$.","title":"Compressed priority"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-compress","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityCompress","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-compress","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"For an encoded parity game $G$ on $n$ positions, the compressed game has the same owners and moves as $G$ and gives each position $v$ the priority $2r + (q \\bmod 2)$, where $q$ is the priority of $v$ in $G$ and $r$ is the number of distinct priorities occurring in $G$ that are smaller than $q$.","title":"Priority compression"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-rank-lt-card","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityRank_lt_card","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-rank-lt-card","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be an encoded parity game, and for a natural number $p$ let the rank of $p$ be the number of distinct priorities $q < p$ that occur in $G$. For every position $v$ of $G$, the rank of the priority of $v$ is strictly less than the number of distinct priorities that occur in $G$.","title":"Priority ranks are below the number of priorities"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-rank-lt-of-lt","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityRank_lt_of_lt","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-rank-lt-of-lt","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be an encoded parity game, and for a natural number $p$ let the rank of $p$ be the number of distinct priorities $q < p$ that occur in $G$. If $v$ and $w$ are positions of $G$ and the priority of $w$ is strictly less than the priority of $v$, then the rank of the priority of $w$ is strictly less than the rank of the priority of $v$.","title":"Strict monotonicity of priority ranks"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-rank-le-of-le","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityRank_le_of_le","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-rank-le-of-le","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"Let $G$ be an encoded parity game, and for a natural number $p$ let the rank of $p$ be the number of distinct priorities $q < p$ that occur in $G$. If $v$ and $w$ are positions of $G$ and the priority of $w$ is at most the priority of $v$, then the rank of the priority of $w$ is at most the rank of the priority of $v$.","title":"Monotonicity of priority ranks"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-compressed-lt","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityCompressed_lt","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-compressed-lt","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"For an encoded parity game $G$ on $n$ positions, every priority of the priority-compressed game of $G$ is strictly less than $2n$.","title":"Bound on compressed priorities"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-compressed-lt-iff","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityCompressed_lt_iff","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-compressed-lt-iff","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"For an encoded parity game $G$ and positions $v$, $w$, the priority of $v$ is smaller than the priority of $w$ in the priority-compressed game of $G$ if and only if it is smaller in $G$.","title":"Priority compression preserves and reflects the order of priorities"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/priority-compressed-mod-two","kind":"node","lean":"Cslib.Games.EncodedParityGame.priorityCompressed_mod_two","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#priority-compressed-mod-two","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"For an encoded parity game $G$ and every position $v$, the priority of $v$ in the priority-compressed game of $G$ has the same parity as the priority of $v$ in $G$.","title":"Priority compression preserves parity"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/max-priority-in","kind":"node","lean":"Cslib.Games.EncodedParityGame.maxPriorityIn","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#max-priority-in","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"For an encoded parity game $G$ and a set $U$ of positions, the maximal priority on $U$ is the largest priority of $G$ at a position of $U$, or $0$ if $U$ is empty.","title":"Maximal priority on a set of positions"},{"id":"TheoryOfComputation/omega-automata-and-games/parity-games/max-positions-in","kind":"node","lean":"Cslib.Games.EncodedParityGame.maxPositionsIn","parent":35533,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/parity-games/index.html#max-positions-in","scope":"TheoryOfComputation/omega-automata-and-games/parity-games","status":"fully_proved","summary":"For an encoded parity game $G$ and a set $U$ of positions, the positions of maximal priority in $U$ are the positions $v \\in U$ whose priority equals the maximal priority of $G$ on $U$.","title":"Positions of maximal priority"},{"data_url":"scopes/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata.json","id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","kind":"scope","navigation":"module","parent":35481,"result":false,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","title":"Rabin and Streett automata"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/rabin","kind":"node","lean":"Cslib.Automata.NA.Rabin","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#rabin","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"A Rabin automaton is a nondeterministic automaton together with a set of Rabin pairs $(E, F)$, each a pair of sets of states. A run is accepting when some pair is hit: the run visits $E$ infinitely often and $F$ only finitely often.","title":"Rabin automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/mem-language-iff","kind":"node","lean":"Cslib.Automata.NA.Rabin.mem_language_iff","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#mem-language-iff","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"An infinite word $x$ is accepted by a Rabin automaton $r$ if and only if there is a run $s$ of $r$ on $x$ and a Rabin pair $(E, F)$ of $r$ such that $s$ visits $E$ infinitely often and visits $F$ only finitely often.","title":"Membership in the language of a Rabin automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/to-muller","kind":"node","lean":"Cslib.Automata.NA.Rabin.toMuller","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#to-muller","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"Given a Rabin automaton with set of Rabin pairs $\\mathcal{R}$, this is the Muller automaton with the same states, initial states and transitions whose accepting family consists of the sets $S$ of states for which some pair $(E, F) \\in \\mathcal{R}$ satisfies $S \\cap E \\neq \\emptyset$ and $S \\cap F = \\emptyset$. A Muller automaton accepts a run when the set of states that the run visits infinitely often belongs to\u2026","title":"Muller presentation of a Rabin automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/language-to-muller","kind":"node","lean":"Cslib.Automata.NA.Rabin.language_toMuller","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#language-to-muller","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"If the state space is finite, then a Rabin automaton $r$ accepts the same language as its Muller presentation: the Muller automaton with the same transitions and initial states as $r$ whose acceptance sets are the sets $S$ of states such that, for some Rabin pair $(E, F)$ of $r$, $S$ meets $E$ and is disjoint from $F$.","title":"Rabin-to-Muller conversion preserves the language"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/streett","kind":"node","lean":"Cslib.Automata.NA.Streett","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#streett","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"A Streett automaton is a nondeterministic automaton together with a set of Streett pairs $(E, F)$, each a pair of sets of states. A run is accepting when every pair is respected: if the run visits $E$ infinitely often then it also visits $F$ infinitely often.","title":"Streett automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/mem-language-iff-2","kind":"node","lean":"Cslib.Automata.NA.Streett.mem_language_iff","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#mem-language-iff-2","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"An infinite word $x$ is accepted by a Streett automaton $r$ if and only if there is a run $s$ of $r$ on $x$ such that for every Streett pair $(E, F)$ of $r$, if $s$ visits $E$ infinitely often then $s$ visits $F$ infinitely often.","title":"Membership in the language of a Streett automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/to-muller-2","kind":"node","lean":"Cslib.Automata.NA.Streett.toMuller","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#to-muller-2","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"Given a Streett automaton with set of Streett pairs $\\mathcal{P}$, this is the Muller automaton with the same states, initial states and transitions whose accepting family consists of the sets $S$ of states such that every pair $(E, F) \\in \\mathcal{P}$ with $S \\cap E \\neq \\emptyset$ also satisfies $S \\cap F \\neq \\emptyset$. A Muller automaton accepts a run when the set of states that the run visits infinitely often\u2026","title":"Muller presentation of a Streett automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/language-to-muller-2","kind":"node","lean":"Cslib.Automata.NA.Streett.language_toMuller","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#language-to-muller-2","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"If the state space is finite, then a Streett automaton $r$ accepts the same language as its Muller presentation: the Muller automaton with the same transitions and initial states as $r$ whose acceptance sets are the sets $S$ of states such that, for every Streett pair $(E, F)$ of $r$, if $S$ meets $E$ then $S$ meets $F$.","title":"Streett-to-Muller conversion preserves the language"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/to-rabin","kind":"node","lean":"Cslib.Automata.NA.Parity.toRabin","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#to-rabin","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"Given a parity automaton with priority function $\\Omega$ from states to natural numbers, this is the Rabin automaton with the same states, initial states and transitions whose Rabin pairs are the pairs $(E_i, F_i)$ for $i \\in \\mathbb{N}$, where $E_i = \\{s : \\Omega(s) = 2i\\}$ and $F_i = \\{s : \\Omega(s) > 2i\\}$.","title":"Rabin presentation of a parity automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/language-to-rabin","kind":"node","lean":"Cslib.Automata.NA.Parity.language_toRabin","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#language-to-rabin","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"If the state space is finite, then a parity automaton $a$ accepts the same language as its Rabin presentation: the Rabin automaton with the same transitions and initial states as $a$ whose Rabin pairs are the pairs $(E_i, F_i)$ for $i \\in \\mathbb{N}$, where $E_i$ is the set of states of priority $2i$ and $F_i$ the set of states of priority greater than $2i$.","title":"Parity-to-Rabin conversion preserves the language"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/to-streett","kind":"node","lean":"Cslib.Automata.NA.Parity.toStreett","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#to-streett","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"Given a parity automaton with priority function $\\Omega$ from states to natural numbers, this is the Streett automaton with the same states, initial states and transitions whose Streett pairs are the pairs $(E_i, F_i)$ for $i \\in \\mathbb{N}$, where $E_i = \\{s : \\Omega(s) = 2i + 1\\}$ and $F_i = \\{s : \\Omega(s) > 2i + 1\\}$.","title":"Streett presentation of a parity automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/language-to-streett","kind":"node","lean":"Cslib.Automata.NA.Parity.language_toStreett","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#language-to-streett","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"If the state space is finite, then a parity automaton $a$ accepts the same language as its Streett presentation: the Streett automaton with the same transitions and initial states as $a$ whose Streett pairs are the pairs $(E_i, F_i)$ for $i \\in \\mathbb{N}$, where $E_i$ is the set of states of priority $2i + 1$ and $F_i$ the set of states of priority greater than $2i + 1$.","title":"Parity-to-Streett conversion preserves the language"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/to-muller-3","kind":"node","lean":"Cslib.Automata.NA.Parity.toMuller","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#to-muller-3","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"The Muller automaton of a parity automaton is the Muller presentation of its Rabin presentation. It has the same states, initial states and transitions, and its accepting family consists of the sets $S$ of states for which there is a natural number $i$ such that $S$ contains a state of priority $2i$ and no state of priority greater than $2i$. A Muller automaton accepts a run when the set of states that the run\u2026","title":"Muller presentation of a parity automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/language-to-muller-3","kind":"node","lean":"Cslib.Automata.NA.Parity.language_toMuller","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#language-to-muller-3","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"If the state space is finite, then a parity automaton accepts the same language as its Muller presentation, the Muller automaton obtained by converting it to its Rabin presentation and that to a Muller automaton.","title":"Parity-to-Muller conversion preserves the language"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/to-streett-2","kind":"node","lean":"Cslib.Automata.NA.Rabin.toStreett","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#to-streett-2","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"Given a Rabin automaton, this is the Streett automaton with the same states, initial states and transitions and the same set of pairs.","title":"Streett automaton with the pairs of a Rabin automaton"},{"id":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/accepts-run-iff-not-streett","kind":"node","lean":"Cslib.Automata.NA.Rabin.acceptsRun_iff_not_streett","parent":35593,"result":true,"route":"../roadmap/TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata/index.html#accepts-run-iff-not-streett","scope":"TheoryOfComputation/omega-automata-and-games/rabin-and-streett-automata","status":"fully_proved","summary":"Let $r$ be a Rabin automaton and let $s$ be an infinite sequence of states. Then $s$ satisfies the Rabin condition of $r$, that is, for some Rabin pair $(E, F)$ of $r$ the sequence $s$ visits $E$ infinitely often and $F$ only finitely often, if and only if $s$ fails the Streett condition with the same pairs, that is, it is not the case that for every Rabin pair $(E, F)$ of $r$, if $s$ visits $E$ infinitely often\u2026","title":"Run-wise Rabin/Streett duality"},{"data_url":"scopes/TheoryOfComputation/context-sensitive-languages.json","id":"TheoryOfComputation/context-sensitive-languages","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/index.html","scope":"TheoryOfComputation/context-sensitive-languages","status":"fully_proved","summary":"Context-sensitive grammars, linear bounded automata and space-bounded single-tape machines, and the equivalence between them.","title":"Context-sensitive languages"},{"data_url":"scopes/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars.json","id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","kind":"scope","navigation":"module","parent":35610,"result":false,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","title":"Context-sensitive grammars"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/is-nonterminal","kind":"node","lean":"Symbol.isNonterminal","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#is-nonterminal","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Whether a grammar symbol (`Mathlib.Computability.Language.Symbol`, shared with the context-free development) is a nonterminal.","title":"Nonterminal grammar symbols"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/csgrammar","kind":"node","lean":"Cslib.Language.CSGrammar","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#csgrammar","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"An unrestricted grammar: a finite list of productions and an initial nonterminal.","title":"Unrestricted grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/rewrites","kind":"node","lean":"Cslib.Language.Rewrites","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#rewrites","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"One rewriting step: apply a production in a sentential form context. This is the generic `Cslib.Decidability.Rewriting.Step` at the grammar's rules, so grammar derivations reuse the string-rewriting API instead of a second, incompatible copy of it.","title":"One rewriting step of a grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/derives","kind":"node","lean":"Cslib.Language.Derives","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#derives","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Derivation: the reflexive-transitive closure of rewriting.","title":"Derivation in an unrestricted grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/language","kind":"node","lean":"Cslib.Language.language","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#language","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"The language of a grammar: terminal words derived from the initial nonterminal.","title":"Language of an unrestricted grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/is-context-sensitive","kind":"node","lean":"Cslib.Language.IsContextSensitive","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#is-context-sensitive","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"A grammar is context-sensitive when every production has a nonterminal on its left side (so the left side is nonempty and not terminal-only) and no production shrinks. The standard `S \u2192 \u03b5` exception is not used here: the `a\u207fb\u207fc\u207f` grammar has `n \u2265 1`.","title":"Context-sensitive grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/abc","kind":"node","lean":"Cslib.Language.ABC","parent":35611,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terminals of the "],["math","\\(a^n b^n c^n\\)"],["text"," grammar"]],"source":"Terminals of the $a^n b^n c^n$ grammar"},"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#abc","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Terminals of the `a\u207fb\u207fc\u207f` grammar.","title":"Terminals of the $a^n b^n c^n$ grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/abcnt","kind":"node","lean":"Cslib.Language.ABCNT","parent":35611,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonterminals of the "],["math","\\(a^n b^n c^n\\)"],["text"," grammar"]],"source":"Nonterminals of the $a^n b^n c^n$ grammar"},"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#abcnt","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Nonterminals of the `a\u207fb\u207fc\u207f` grammar.","title":"Nonterminals of the $a^n b^n c^n$ grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/abc-grammar","kind":"node","lean":"Cslib.Language.abcGrammar","parent":35611,"reading_title":{"format":"inline-parts-v1","parts":[["text","Context-sensitive grammar for "],["math","\\(a^n b^n c^n\\)"]],"source":"Context-sensitive grammar for $a^n b^n c^n$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#abc-grammar","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"The standard context-sensitive grammar for `{a\u207fb\u207fc\u207f | n \u2265 1}`.","title":"Context-sensitive grammar for $a^n b^n c^n$"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/step","kind":"node","lean":"Cslib.Language.step","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#step","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"One grammar step: rewrite `lhs` to `rhs` in context `p, q`.","title":"Applying a production in context is a grammar step"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/abc-grammar-context-sensitive","kind":"node","lean":"Cslib.Language.abcGrammar_contextSensitive","parent":35611,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(a^n b^n c^n\\)"],["text"," grammar is context-sensitive"]],"source":"The $a^n b^n c^n$ grammar is context-sensitive"},"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#abc-grammar-context-sensitive","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Every production of the `a\u207fb\u207fc\u207f` grammar is length-nondecreasing, so the grammar is context-sensitive.","title":"The $a^n b^n c^n$ grammar is context-sensitive"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/abc-mem","kind":"node","lean":"Cslib.Language.abc_mem","parent":35611,"reading_title":{"format":"inline-parts-v1","parts":[["text","The word abc is generated by the "],["math","\\(a^n b^n c^n\\)"],["text"," grammar"]],"source":"The word abc is generated by the $a^n b^n c^n$ grammar"},"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#abc-mem","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"`abc` is generated.","title":"The word abc is generated by the $a^n b^n c^n$ grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/aabbcc-mem","kind":"node","lean":"Cslib.Language.aabbcc_mem","parent":35611,"reading_title":{"format":"inline-parts-v1","parts":[["text","The word aabbcc is generated by the "],["math","\\(a^n b^n c^n\\)"],["text"," grammar"]],"source":"The word aabbcc is generated by the $a^n b^n c^n$ grammar"},"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#aabbcc-mem","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"`aabbcc` is generated: grow two `a`'s, shuttle the `C`'s right past the `B`'s, then convert `B`'s and `C`'s to terminals in order.","title":"The word aabbcc is generated by the $a^n b^n c^n$ grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/cnt","kind":"node","lean":"Cslib.Language.cnt","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#cnt","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Count of the symbol `s` in a sentential form (terminals `a`, `b`, `c`, and the \"unborn\" `b`/`c` nonterminals `B`, `C`).","title":"Symbol count of a sentential form"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/cnt-a","kind":"node","lean":"Cslib.Language.cntA","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#cnt-a","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"The `a`-count of a sentential form.","title":"Count of a's in a sentential form"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/cnt-b","kind":"node","lean":"Cslib.Language.cntB","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#cnt-b","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"The `b`-count: terminal `b`'s plus unborn `B`'s.","title":"Count of b's in a sentential form"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/cnt-c","kind":"node","lean":"Cslib.Language.cntC","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#cnt-c","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"The `c`-count: terminal `c`'s plus unborn `C`'s.","title":"Count of c's in a sentential form"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/balanced","kind":"node","lean":"Cslib.Language.Balanced","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#balanced","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"A sentential form is balanced when its `a`-, `b`- and `c`-counts agree.","title":"Balanced sentential form"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/cnt-concat3","kind":"node","lean":"Cslib.Language.cnt_concat3","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#cnt-concat3","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Counting over a concatenation of three forms splits additively.","title":"Symbol counts are additive over concatenation"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/rewrites-balanced","kind":"node","lean":"Cslib.Language.rewrites_balanced","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#rewrites-balanced","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"One rewriting step preserves balancedness: every production has the same imbalance on both sides, and the context contributes equally to all three counts.","title":"Rewriting steps preserve balancedness"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/derives-balanced","kind":"node","lean":"Cslib.Language.derives_balanced","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#derives-balanced","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Derivations preserve balancedness.","title":"Derivations preserve balancedness"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/balanced-initial","kind":"node","lean":"Cslib.Language.balanced_initial","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#balanced-initial","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"The initial form `[S]` is balanced, with all counts zero.","title":"The initial sentential form is balanced"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/language-subset-balanced","kind":"node","lean":"Cslib.Language.language_subset_balanced","parent":35611,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#language-subset-balanced","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"Every generated terminal word has equally many `a`'s, `b`'s and `c`'s.","title":"Generated words have equally many a's, b's and c's"},{"id":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/aabbc-not-mem","kind":"node","lean":"Cslib.Language.aabbc_not_mem","parent":35611,"reading_title":{"format":"inline-parts-v1","parts":[["text","The word aabbc is not generated by the "],["math","\\(a^n b^n c^n\\)"],["text"," grammar"]],"source":"The word aabbc is not generated by the $a^n b^n c^n$ grammar"},"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars/index.html#aabbc-not-mem","scope":"TheoryOfComputation/context-sensitive-languages/context-sensitive-grammars","status":"fully_proved","summary":"`aabbc` is not generated: its counts are `2 = 2 \u2260 1`.","title":"The word aabbc is not generated by the $a^n b^n c^n$ grammar"},{"data_url":"scopes/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata.json","id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","kind":"scope","navigation":"module","parent":35610,"result":false,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","title":"Linear bounded automata"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/lbaboundary","kind":"node","lean":"Cslib.Language.LBABoundary","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#lbaboundary","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Where the head sits relative to the tape boundaries: strictly inside the tape, on the first cell, on the last cell, or on the only cell of a one-cell tape. The transition function observes this tag, so the machine can tell the two ends of its tape apart from interior cells.","title":"Boundary tag of a linear bounded automaton head"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/of-head","kind":"node","lean":"Cslib.Language.LBABoundary.ofHead","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#of-head","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The boundary tag of head position `h` on a tape of length `n`.","title":"Boundary tag of a head position"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/of-head-eq-both","kind":"node","lean":"Cslib.Language.LBABoundary.ofHead_eq_both","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#of-head-eq-both","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The boundary tag at the head of a one-cell tape.","title":"Boundary tag on a one-cell tape"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/of-head-eq-left","kind":"node","lean":"Cslib.Language.LBABoundary.ofHead_eq_left","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#of-head-eq-left","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The boundary tag at the head of the first cell of a longer tape.","title":"Boundary tag on the first cell of a longer tape"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/of-head-eq-right","kind":"node","lean":"Cslib.Language.LBABoundary.ofHead_eq_right","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#of-head-eq-right","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The boundary tag at the head of the last cell of a longer tape.","title":"Boundary tag on the last cell of a longer tape"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/of-head-eq-middle","kind":"node","lean":"Cslib.Language.LBABoundary.ofHead_eq_middle","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#of-head-eq-middle","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The boundary tag at an interior head position.","title":"Boundary tag at an interior head position"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/lbamove","kind":"node","lean":"Cslib.Language.LBAMove","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#lbamove","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A head move of a linear bounded automaton: left, stay, or right.","title":"Head move of a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/move-head","kind":"node","lean":"Cslib.Language.LBAMove.moveHead","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#move-head","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The head position after a move: left decrements, stay keeps, right increments.","title":"Head position after a move"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/legal","kind":"node","lean":"Cslib.Language.LBAMove.legal","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#legal","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A move is legal from the head position `h` on a tape of length `n` when the source head is on the tape and the new head position is still on the tape. In particular a left move from an out-of-bounds head is illegal, even though the truncated new head `h - 1` would still be out of bounds: legality requires `h < n` for left moves as well.","title":"Legal head move of a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/linear-bounded-automaton","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#linear-bounded-automaton","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A linear bounded automaton: a nondeterministic Turing machine whose tape is the input word. The transition function gives, for a state, the symbol under the head (`none` for a blank cell), and the boundary tag of the head position, the finite set of possible next states, symbols to write (`none` for a blank), and head moves. The boundary tag is what lets the machine detect the ends of its tape: without it the model\u2026","title":"Linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/lbaconfig","kind":"node","lean":"Cslib.Language.LBAConfig","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#lbaconfig","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A configuration of a linear bounded automaton: the state, the tape (a list of cells, one per input cell; `none` is a blank), and the head position.","title":"Configuration of a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/init-cfg","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.initCfg","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#init-cfg","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The initial configuration on the input `w`: the initial state, the tape `w` itself (no blanks), the head on the first cell.","title":"Initial configuration of a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/step","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.Step","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#step","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"One step of the machine: choose a transition for the current state, the symbol under the head, and its boundary tag, write the symbol, move the head legally. A step from a configuration whose head is off the tape is impossible: legality requires the source head to be on the tape.","title":"One step of a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/accepts","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.Accepts","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#accepts","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"`M` accepts `w` when some run from the initial configuration on `w` reaches a configuration in the accepting state.","title":"Acceptance by a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/language","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.language","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#language","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The language of `M`: the words it accepts.","title":"Language of a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/lbant","kind":"node","lean":"Cslib.Language.LBANT","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#lbant","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The nonterminals of the grammar of an LBA: a cell, a generator, and the start symbol. A cell records the original input symbol, the current tape symbol, the head state if the head is on the cell, whether the cell is active (simulation acts only on active cells), and the boundary tag of its position (so simulation can apply the boundary-aware transitions).","title":"Nonterminals of the grammar of a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/cell-sym","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.cellSym","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#cell-sym","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A grammar symbol for the LBA grammar: a cell as a nonterminal.","title":"Cell symbol of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/start-sym","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.startSym","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#start-sym","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The start symbol as a grammar symbol.","title":"Start symbol of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/gen-sym","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.genSym","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#gen-sym","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The generator as a grammar symbol.","title":"Generator symbol of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/term-sym","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.termSym","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#term-sym","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A terminal symbol of the grammar.","title":"Terminal symbol of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/gen-rules","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.genRules","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#gen-rules","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The generation rules: the start symbol produces the first cell and the generator (or, for a one-cell word, the first cell alone, tagged `both`); the generator produces a middle non-head cell and itself, or a final right non-head cell. The first cell of a longer word is tagged `left`.","title":"Generation rules of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/activate-rules","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.activateRules","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#activate-rules","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The activation rules: an inactive cell becomes active, keeping its original, current, head state, and boundary tag.","title":"Activation rules of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/sim-stay-rules","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.simStayRules","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#sim-stay-rules","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The simulation rules for a stay move: the head cell rewrites its current symbol and state, staying active, keeping its boundary tag.","title":"Simulation rules for a stay move"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/sim-right-rules","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.simRightRules","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#sim-right-rules","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The simulation rules for a right move: the head cell rewrites its current symbol and loses the head; the neighbouring cell gains the head. Both cells keep their boundary tags.","title":"Simulation rules for a right move"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/sim-left-rules","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.simLeftRules","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#sim-left-rules","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The simulation rules for a left move: the neighbouring cell gains the head; the head cell rewrites its current symbol and loses the head. Both cells keep their boundary tags.","title":"Simulation rules for a left move"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/conv-rules","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.convRules","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#conv-rules","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The conversion rules: an active cell becomes its original terminal. The head cell converts only in the accepting state, so a terminal word is produced only from an accepting configuration. The boundary tag is discarded on conversion.","title":"Conversion rules of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/all-rules","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.allRules","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#all-rules","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"All the rules of the grammar.","title":"All rules of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/lba-grammar","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.lbaGrammar","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#lba-grammar","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The grammar whose rules are the generation, activation, simulation, and conversion rules above, starting from the start symbol.","title":"The grammar of an LBA"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-gen-rules-start","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_genRules_start","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-gen-rules-start","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The first generation rule is a rule of the grammar.","title":"The start rule is a generation rule of the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/lba-grammar-context-sensitive","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.lbaGrammar_contextSensitive","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#lba-grammar-context-sensitive","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Every production has a nonterminal on its left side and does not shrink.","title":"The grammar of an LBA is context-sensitive"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/encode-cells-aux","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.encodeCellsAux","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#encode-cells-aux","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The cells of a configuration, as a list of cell tuples: original symbol, current symbol, head state, active, boundary tag. The head is given as `some i` for the head position, or `none` once it has been passed. The `start` index is the position of the first cell in the whole word and `total` is the whole word's length, so the boundary tag of the cell at offset `j` is `ofHead (start + j) total`.","title":"Cell tuples of an LBA configuration from an offset"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/cell-tuple-sym","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.cellTupleSym","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#cell-tuple-sym","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A cell tuple as a grammar symbol.","title":"Cell tuple as a grammar symbol"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/encode-cfg","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.encodeCfg","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#encode-cfg","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A configuration as a sentential form: the cells of its tape, with the original symbols taken from the input word `w`.","title":"Configuration encoded as a sentential form"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/tail-cells-mid","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.tailCellsMid","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#tail-cells-mid","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The middle non-head generated cells of a word, inactive: the cells the generator produces while the generator remains.","title":"Middle generated cells of a word"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/tail-cells","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.tailCells","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#tail-cells","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The non-head generated cells of a word, inactive, as the tail of a generated word: middle cells and a final right cell.","title":"Tail generated cells of a word"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/gen-cells","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.genCells","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#gen-cells","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The generated cells of a word, inactive, with the head state on the first cell: the sentential form generation produces. The first cell is tagged `both` for a one-cell word and `left` otherwise; the tail is `tailCells`.","title":"Generated cells of a word"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/gen-encode","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.genEncode","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#gen-encode","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The generated sentential form of a word.","title":"Generated sentential form of a word"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/tail-encode","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.tailEncode","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#tail-encode","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The non-head generated sentential form of a word.","title":"Tail generated sentential form of a word"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/tail-encode-mid","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.tailEncodeMid","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#tail-encode-mid","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The middle non-head generated sentential form of a word: every cell tagged `middle`, for the generator's intermediate stage.","title":"Middle generated sentential form of a word"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/length-encode-cells-aux","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.length_encodeCellsAux","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#length-encode-cells-aux","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The encoding has as many cells as the input word, when the tape has the same length.","title":"Length of the cell-tuple encoding"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/get-elem-encode-cells-aux","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.getElem_encodeCellsAux","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#get-elem-encode-cells-aux","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The `i`-th cell of the encoding: original `w[i]`, current `tape[i]`, head state exactly at the head position, boundary tag of position `start + i` in a word of length `total`.","title":"Entries of the cell-tuple encoding"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-replace-list","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_replace_list","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-replace-list","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"One grammar step replacing a block: if `u` splits as `p ++ lhs ++ q` and `(lhs, rhs)` is a rule, then `u` derives `p ++ rhs ++ q`.","title":"Replacing a block by a rule is a derivation step"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-single-cell","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_single_cell","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-single-cell","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"One grammar step replacing a single cell by a single cell.","title":"Replacing a single cell is a derivation step"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-two-cells","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_two_cells","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-two-cells","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"One grammar step replacing two consecutive cells.","title":"Replacing two consecutive cells is a derivation step"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-singleton-cell","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_singleton_cell","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-singleton-cell","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A singleton-to-singleton production is a derivation step.","title":"Singleton-to-singleton productions are derivation steps"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-append-left","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_append_left","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-append-left","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A derivation may be appended on the left.","title":"Derivations extend by a left context"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-append-right","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_append_right","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-append-right","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A derivation may be appended on the right.","title":"Derivations extend by a right context"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-append","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_append","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-append","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Derivations compose under append.","title":"Derivations compose under concatenation"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-trans","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_trans","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-trans","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Derivations are transitive.","title":"Transitivity of grammar derivations"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-gen-start","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_gen_start","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-gen-start","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The first generation rule is a rule of the grammar.","title":"The first generation rule belongs to the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-gen-single","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_gen_single","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-gen-single","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The one-cell generation rule is a rule of the grammar.","title":"The one-cell generation rule belongs to the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-gen-step","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_gen_step","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-gen-step","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The generator step rule is a rule of the grammar.","title":"The generator step rule belongs to the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-gen-last","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_gen_last","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-gen-last","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The final generation rule is a rule of the grammar.","title":"The final generation rule belongs to the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-activate","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_activate","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-activate","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The activation rule is a rule of the grammar.","title":"The activation rule belongs to the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-sim-stay","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_sim_stay","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-sim-stay","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A stay transition gives a simulation rule.","title":"Stay transitions give simulation rules"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-sim-right","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_sim_right","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-sim-right","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A right transition gives a simulation rule.","title":"Right transitions give simulation rules"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-sim-left","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_sim_left","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-sim-left","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A left transition gives a simulation rule.","title":"Left transitions give simulation rules"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-conv-acc","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_conv_acc","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-conv-acc","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The accepting conversion rule is a rule of the grammar.","title":"The accepting conversion rule belongs to the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mem-rule-conv-none","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.mem_rule_conv_none","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mem-rule-conv-none","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The non-head conversion rule is a rule of the grammar.","title":"The non-head conversion rule belongs to the LBA grammar"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-gen-tail","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_gen_tail","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-gen-tail","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The generator derives the middle tail cells followed by itself.","title":"The generator derives the middle cells and itself"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-gen-tail-last","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_gen_tail_last","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-gen-tail-last","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The generator derives the tail cells, ending the generation.","title":"The generator derives the full tail of cells"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-start-gen-encode","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_start_genEncode","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-start-gen-encode","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The start symbol derives the generated sentential form of a word.","title":"The start symbol derives the generated form of a word"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-tail-activate","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_tail_activate","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-tail-activate","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The tail cells, sitting at offset `start \u2265 1` in a word of total length `start + w.length`, derive their activated encoding. The offset stays `\u2265 1` in the induction step, so the first tail cell is never tagged `left`: it is `right` when it is the last cell and `middle` otherwise.","title":"Tail cells derive their activated encoding"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-gen-activate","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_gen_activate","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-gen-activate","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The generated form derives the encoding of the initial configuration.","title":"Generated form derives the initial configuration encoding"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/take-append-two","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.take_append_two","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#take-append-two","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A list splits at two consecutive positions.","title":"Splitting a list at two consecutive positions"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-set-single","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_set_single","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-set-single","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A single-cell replacement that is a production is a derivation step.","title":"Setting one list position by a production is a derivation step"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/drop-set-of-gt","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.drop_set_of_gt","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#drop-set-of-gt","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Dropping past a set position is unaffected by the set.","title":"Dropping past an updated list position"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/take-set","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.take_set","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#take-set","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Taking past a set position keeps the new value at that position.","title":"Taking past an updated list position"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-set-two","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_set_two","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-set-two","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A two-cell replacement that is a production is a derivation step.","title":"Setting two adjacent list positions by a production is a derivation step"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-set-two-2","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_set_two'","parent":35636,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-cell replacement at indices "],["math","\\(i\\)"],["text"," and "],["math","\\(j = i + 1\\)"]],"source":"Two-cell replacement at indices $i$ and $j = i + 1$"},"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-set-two-2","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A two-cell replacement at positions `i` and `j = i + 1`.","title":"Two-cell replacement at indices $i$ and $j = i + 1$"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/length-encode-cfg","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.length_encodeCfg","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#length-encode-cfg","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The encoding of a configuration has as many cells as the input word.","title":"Length of a configuration encoding"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/get-elem-encode-cfg","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.getElem_encodeCfg","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#get-elem-encode-cfg","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The `i`-th cell of the encoding.","title":"Entries of a configuration encoding"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/step-tape-length","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.step_tape_length","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#step-tape-length","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"One step preserves the tape length.","title":"LBA steps preserve tape length"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/step-head-lt","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.step_head_lt","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#step-head-lt","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"One step keeps the head on the tape, when the source head is on the tape.","title":"LBA steps keep the head on the tape"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/reachable-tape-length","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.reachable_tape_length","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#reachable-tape-length","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Reachable configurations have the input tape length and the head on the tape, for a nonempty input word.","title":"Reachable LBA configurations keep the tape length and the head on the tape"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-encode-cfg-step","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_encodeCfg_step","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-encode-cfg-step","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"One machine step is one grammar derivation step on encodings.","title":"An LBA step is a grammar derivation step on encodings"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/encode-cells-aux-state","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.encodeCellsAux_state","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#encode-cells-aux-state","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Every cell of an encoding has head state `none` or the given state.","title":"Head states in a configuration encoding"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/encode-cells-aux-active","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.encodeCellsAux_active","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#encode-cells-aux-active","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Every cell of an encoding is active.","title":"Every cell of a configuration encoding is active"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/map-orig-encode-cells-aux","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.map_orig_encodeCellsAux","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#map-orig-encode-cells-aux","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"The original symbols of an encoding are the input word.","title":"Original symbols of an encoding are the input word"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-convert-cells","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_convert_cells","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-convert-cells","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A list of cells, each non-head or accepting, converts to its original terminals.","title":"Non-head or accepting cells convert to terminals"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/derives-encode-cfg-refl-trans-gen","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.derives_encodeCfg_reflTransGen","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#derives-encode-cfg-refl-trans-gen","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Derivations along a run, keeping the tape length and head on the tape.","title":"Runs of an LBA give grammar derivations"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/lba-accepts-implies-grammar-derives","kind":"node","lean":"Cslib.Language.LinearBoundedAutomaton.lba_accepts_implies_grammar_derives","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#lba-accepts-implies-grammar-derives","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"This is one inclusion of the LBA-to-grammar direction.","title":"If the LBA accepts a nonempty word, the grammar derives it"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/space-bounded-by","kind":"node","lean":"Cslib.Turing.SingleTapeTM.SpaceBoundedBy","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#space-bounded-by","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Let $s : \\mathbb{N} \\to \\mathbb{N}$. A single-tape Turing machine $M$ runs in space $s$ when, for every input word $l$ of length $|l|$, every configuration reachable from the initial configuration of $M$ on $l$ uses at most $s(|l|)$ cells, whether or not the run halts. The cells used by a configuration are the head cell together with the cells on each side of it up to the outermost non-blank cell.","title":"Space-bounded single-tape machines"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/is-lba","kind":"node","lean":"Cslib.Turing.SingleTapeTM.IsLBA","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#is-lba","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A single-tape machine is a linear bounded automaton when it runs in space $n + 1$ on inputs of length $n$: every configuration of its run on an input of length $n$ uses at most $n + 1$ cells, whether or not the run halts.","title":"Linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/mono","kind":"node","lean":"Cslib.Turing.SingleTapeTM.SpaceBoundedBy.mono","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#mono","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"If a single-tape machine runs in space $s : \\mathbb{N} \\to \\mathbb{N}$ and $s' : \\mathbb{N} \\to \\mathbb{N}$ satisfies $s(n) \\le s'(n)$ for every $n \\in \\mathbb{N}$, then it runs in space $s'$.","title":"Space bounds can be weakened"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/max-one-le","kind":"node","lean":"Cslib.Turing.SingleTapeTM.SpaceBoundedBy.max_one_le","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#max-one-le","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"If a single-tape machine runs in space $s : \\mathbb{N} \\to \\mathbb{N}$, then $\\max(1, n) \\le s(n)$ for the length $n$ of every input word, since the initial configuration on an input of length $n$ uses $\\max(1, n)$ cells.","title":"A space bound is at least the space of the input"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/is-lba-of-stays-on-blank","kind":"node","lean":"Cslib.Turing.SingleTapeTM.isLBA_of_stays_on_blank","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#is-lba-of-stays-on-blank","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"A machine that, on a blank cell, writes a blank and does not move is a linear bounded automaton. Its head never leaves a blank cell, so it stays on the input or on the two blank cells next to it, and it writes non-blank symbols only on the input. The machine may change state on a blank cell, or halt there.","title":"A machine that, on a blank cell, writes a blank and does not move is a linear bounded automaton"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/cfg-bound","kind":"node","lean":"Cslib.Turing.SingleTapeTM.cfgBound","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#cfg-bound","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"For a single-tape machine with state set $Q$ over a finite alphabet $\\Sigma$ and $k \\in \\mathbb{N}$, the configuration bound at $k$ is the number $(|Q| + 1)(|\\Sigma| + 1)(|\\Sigma| + 2)^{2k}$.","title":"Configuration bound"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/encard-le-cfg-bound","kind":"node","lean":"Cslib.Turing.SingleTapeTM.encard_le_cfgBound","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#encard-le-cfg-bound","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Let $M$ be a single-tape machine with state set $Q$ over a finite alphabet $\\Sigma$, where a configuration uses the cell under the head and the cells on either side of it up to the last non-blank one. For every $k \\in \\mathbb{N}$, at most $(|Q| + 1)(|\\Sigma| + 1)(|\\Sigma| + 2)^{2k}$ configurations of $M$ use at most $k$ cells.","title":"Configurations of bounded space are finitely many"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/exists-halted-relates-within-steps","kind":"node","lean":"Cslib.Turing.SingleTapeTM.SpaceBoundedBy.exists_halted_relatesWithinSteps","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#exists-halted-relates-within-steps","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Let $M$ be a single-tape machine with state set $Q$ over a finite alphabet $\\Sigma$ that runs in space $s : \\mathbb{N} \\to \\mathbb{N}$. If $M$ halts on an input word $w$ of length $n$, then from the initial configuration on $w$ it reaches a configuration in the halting state in at most $(|Q| + 1)(|\\Sigma| + 1)(|\\Sigma| + 2)^{2 s(n)}$ steps.","title":"Space-bounded machines halt within the configuration bound"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/computable-pred-halts-on","kind":"node","lean":"Cslib.Turing.SingleTapeTM.SpaceBoundedBy.computablePred_haltsOn","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#computable-pred-halts-on","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Let $M$ be a single-tape machine with state set $Q$ over a finite alphabet $\\Sigma$, fix any primitive recursive coding of $\\Sigma$, and let $s : \\mathbb{N} \\to \\mathbb{N}$ be computable. If $M$ runs in space $s$, then the set of input words over $\\Sigma$ on which $M$ halts is computable.","title":"Halting of a machine running in computable space is computable"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/computable-pred-halts-on-2","kind":"node","lean":"Cslib.Turing.SingleTapeTM.IsLBA.computablePred_haltsOn","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#computable-pred-halts-on-2","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Let $M$ be a single-tape machine over a finite alphabet $\\Sigma$, and fix any primitive recursive coding of $\\Sigma$. If $M$ is a linear bounded automaton, then the set of input words over $\\Sigma$ on which $M$ halts is computable.","title":"Halting of a linear bounded automaton is decidable"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/computable-pred-halts-on-encode-nat","kind":"node","lean":"Cslib.Turing.SingleTapeTM.IsLBA.computablePred_haltsOn_encodeNat","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#computable-pred-halts-on-encode-nat","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Let $M$ be a single-tape machine over a finite alphabet $\\Sigma$ that is a linear bounded automaton, let $\\alpha$ be a primitive recursively coded type, and let $\\iota : \\{0, 1\\} \\to \\Sigma$ be any map. For $a \\in \\alpha$, let $w_a$ be the word over $\\Sigma$ obtained by applying $\\iota$ to each bit of the binary expansion, least significant bit first and without leading zeros, of the natural-number code of $a$; so\u2026","title":"Halting of a linear bounded automaton on binary codes is decidable"},{"id":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/single-tape-decidable-halts-on-encode-na","kind":"node","lean":"Cslib.Turing.SingleTapeTM.IsLBA.singleTapeDecidable_haltsOn_encodeNat","parent":35636,"result":true,"route":"../roadmap/TheoryOfComputation/context-sensitive-languages/linear-bounded-automata/index.html#single-tape-decidable-halts-on-encode-na","scope":"TheoryOfComputation/context-sensitive-languages/linear-bounded-automata","status":"fully_proved","summary":"Let $M$ be a linear bounded automaton over a finite tape alphabet $\\Sigma$, let $\\iota$ be any map from the Booleans to $\\Sigma$, and let $\\alpha$ be a primitively encodable type. For $a \\in \\alpha$, let $w_a$ be the word obtained by writing, letter by letter through $\\iota$, the binary expansion (least significant bit first) of the natural-number code of $a$. Then the predicate on $\\alpha$ that holds at $a$\u2026","title":"A single-tape machine decides the halting problem of a linear bounded automaton"},{"data_url":"scopes/TheoryOfComputation/randomized-and-quantum-complexity.json","id":"TheoryOfComputation/randomized-and-quantum-complexity","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/index.html","scope":"TheoryOfComputation/randomized-and-quantum-complexity","status":"can_prove","summary":"Probabilistic classes (RP, BPP), Arthur\u2013Merlin and interactive proofs (AM, MA, IP = PSPACE, graph (non)isomorphism), BPP versus nonuniform and alternating classes, randomized reductions (Valiant\u2013Vazirani), and quantum classes BQP and QMA.","title":"Randomized, interactive and quantum complexity"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/adleman","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Adleman.adleman","parent":35731,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/index.html#adleman","scope":"TheoryOfComputation/randomized-and-quantum-complexity","status":"can_prove","summary":"Every decision problem in BPP lies in P/poly.","title":"Adleman's theorem: BPP is contained in P/poly"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/ip-eq-pspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.IPEqualsPSPACE.ip_eq_pspace","parent":35731,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/index.html#ip-eq-pspace","scope":"TheoryOfComputation/randomized-and-quantum-complexity","status":"can_prove","summary":"The class $\\mathsf{IP}$ of decision problems that have interactive proofs (a polynomial-time verifier using polynomially many rounds and random coins, with completeness at least $2/3$ and soundness at most $1/3$ against every prover) equals the class $\\mathrm{PSPACE}$ of decision problems whose languages are decided by multi-tape Turing machines in polynomial space.","title":"IP = PSPACE (Shamir)"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/sipser-gacs-lautemann","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SipserGacs.sipser_gacs_lautemann","parent":35731,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sipser-G\u00e1cs-Lautemann: BPP is in "],["math","\\(\\Sigma_2^p\\)"],["text"," and "],["math","\\(\\Pi_2^p\\)"]],"source":"Sipser-G\u00e1cs-Lautemann: BPP is in $\\Sigma_2^p$ and $\\Pi_2^p$"},"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/index.html#sipser-gacs-lautemann","scope":"TheoryOfComputation/randomized-and-quantum-complexity","status":"can_prove","summary":"Every decision problem in BPP lies in $\\Sigma_2^p$ and in $\\Pi_2^p$, the classes of the polynomial hierarchy defined by two alternating blocks of polynomially bounded quantifiers, beginning with $\\exists$ and with $\\forall$ respectively.","title":"Sipser-G\u00e1cs-Lautemann: BPP is in $\\Sigma_2^p$ and $\\Pi_2^p$"},{"data_url":"scopes/TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes.json","id":"TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes","kind":"scope","navigation":"module","parent":35731,"result":false,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes/index.html","scope":"TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes","status":"can_prove","title":"Arthur\u2013Merlin and Merlin\u2013Arthur classes"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes/am-subset-pi2","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.AMSubsetPi2.am_subset_pi2","parent":35735,"reading_title":{"format":"inline-parts-v1","parts":[["text","AM is contained in "],["math","\\(\\Pi_2^p\\)"],["text"," (Babai)"]],"source":"AM is contained in $\\Pi_2^p$ (Babai)"},"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes/index.html#am-subset-pi2","scope":"TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes","status":"can_prove","summary":"Every decision problem in the Arthur\u2013Merlin class $\\mathsf{AM}$ lies in $\\Pi_2^p$, the second universal level of the polynomial hierarchy.","title":"AM is contained in $\\Pi_2^p$ (Babai)"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes/ma-subset-am","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MASubsetAM.ma_subset_am","parent":35735,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes/index.html#ma-subset-am","scope":"TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes","status":"can_prove","summary":"Every decision problem in the class $\\mathsf{MA}$ is in the class $\\mathsf{AM}$.","title":"MA is contained in AM"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes/ma-subset-ph","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MASubsetPH.ma_subset_ph","parent":35735,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes/index.html#ma-subset-ph","scope":"TheoryOfComputation/randomized-and-quantum-complexity/arthur-merlin-and-merlin-arthur-classes","status":"can_prove","summary":"Every decision problem in the class $\\mathsf{MA}$ is in the polynomial hierarchy $\\mathsf{PH}$.","title":"MA is contained in PH"},{"data_url":"scopes/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp.json","id":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp","kind":"scope","navigation":"module","parent":35731,"result":false,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp/index.html","scope":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp","status":"can_prove","title":"Bounded-error probabilistic classes RP and BPP"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp/bpp-subset-pspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.BPPSubsetPSPACE.bpp_subset_pspace","parent":35739,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp/index.html#bpp-subset-pspace","scope":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp","status":"can_prove","summary":"Every decision problem in BPP lies in PSPACE.","title":"BPP is contained in PSPACE"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp/rp-subset-bpp","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RPSubsetBPP.rp_subset_bpp","parent":35739,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp/index.html#rp-subset-bpp","scope":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp","status":"can_prove","summary":"Every decision problem in the class $\\mathsf{RP}$ of one-sided-error randomized polynomial-time decision problems also belongs to the class $\\mathsf{BPP}$ of bounded-error probabilistic polynomial-time decision problems.","title":"RP is contained in BPP"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp/rp-subset-np","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RPSubsetNP.rp_subset_np","parent":35739,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp/index.html#rp-subset-np","scope":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-probabilistic-classes-rp-and-bpp","status":"can_prove","summary":"Every decision problem in the class $\\mathsf{RP}$ of one-sided-error randomized polynomial-time decision problems also belongs to the class $\\mathsf{NP}$.","title":"RP is contained in NP"},{"data_url":"scopes/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp.json","id":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp","kind":"scope","navigation":"module","parent":35731,"result":false,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp/index.html","scope":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp","status":"can_prove","title":"Bounded-error quantum polynomial time (BQP)"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp/bqp-subset-pp","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.BQPSubsetPP.bqp_subset_pp","parent":35743,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp/index.html#bqp-subset-pp","scope":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp","status":"can_prove","summary":"Every decision problem in BQP lies in PP.","title":"BQP is contained in PP"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp/bqp-subset-pspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.BQPSubsetPSPACE.bqp_subset_pspace","parent":35743,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp/index.html#bqp-subset-pspace","scope":"TheoryOfComputation/randomized-and-quantum-complexity/bounded-error-quantum-polynomial-time-bqp","status":"can_prove","summary":"Every decision problem in BQP, the class of bounded-error quantum polynomial-time decision problems, lies in PSPACE, the class of decision problems decidable in polynomial space.","title":"BQP is contained in PSPACE"},{"data_url":"scopes/TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem.json","id":"TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem","kind":"scope","navigation":"module","parent":35731,"result":false,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem/index.html","scope":"TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem","status":"can_prove","title":"Goldwasser\u2013Sipser theorem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem/pub-ipgame","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldwasserSipser.PubIPGame","parent":35746,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem/index.html#pub-ipgame","scope":"TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem","status":"fully_proved","summary":"The game takes a length cap $c$, a number $k$ of fresh coins per round, a verifier message function $V_{\\mathrm{msg}}$, a verifier decision function $V_{\\mathrm{dec}}$, a prover $P$ mapping pairs of a coin prefix and a transcript (a list of bitstrings) to bitstrings, an input $x$, a coin string $\\rho$, a number $j$ of remaining rounds, a total number $T$ of rounds and a transcript $t$, and returns a Boolean\u2026","title":"Public-coin interactive-proof game"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem/pub-ip","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldwasserSipser.PubIP","parent":35746,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem/index.html#pub-ip","scope":"TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem","status":"fully_proved","summary":"A decision problem $L$ is in $\\mathsf{PubIP}$ when there are polynomials $r$, $q$ and $k$ with natural-number coefficients and polynomial-time functions $V_{\\mathrm{msg}}$ (the verifier's message function) and $V_{\\mathrm{dec}}$ (its decision function) with the following property. For a bitstring $x$, a prover $P$ (an arbitrary function from pairs of a coin prefix and a transcript to bitstrings) and a coin string\u2026","title":"Public-coin interactive proofs"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem/goldwasser-sipser","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldwasserSipser.goldwasser_sipser","parent":35746,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem/index.html#goldwasser-sipser","scope":"TheoryOfComputation/randomized-and-quantum-complexity/goldwasser-sipser-theorem","status":"can_prove","summary":"The class $\\mathsf{IP}$ of decision problems with (private-coin) interactive proofs equals the class $\\mathsf{PubIP}$ of decision problems with public-coin interactive proofs with polynomially many rounds.","title":"Goldwasser-Sipser: private coins add no power to interactive proofs"},{"data_url":"scopes/TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs.json","id":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs","kind":"scope","navigation":"module","parent":35731,"result":false,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/index.html","scope":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs","status":"can_prove","title":"Graph isomorphism and Arthur\u2013Merlin proofs"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/graph-iso-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GIInNPCoAM.graphIsoDec","parent":35750,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/index.html#graph-iso-dec","scope":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs","status":"fully_proved","summary":"The decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, E_1, E_2)$ of a natural number $n$ and two lists $E_1$ and $E_2$ of pairs of natural numbers for which there is a bijection $f$ of $\\{0, \\dots, n - 1\\}$ such that for all $a, b < n$, $(a, b) \\in E_1$ if and only if $(f(a), f(b)) \\in E_2$. Bitstrings that do not decode to such a triple are\u2026","title":"Directed graph isomorphism as a decision problem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/graph-iso-in-np-coam","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GIInNPCoAM.graph_iso_in_np_coam","parent":35750,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/index.html#graph-iso-in-np-coam","scope":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs","status":"can_prove","summary":"The directed graph isomorphism decision problem belongs to NP and to coAM.","title":"Graph isomorphism is in NP and coAM"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/graph-non-iso-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GNIInAM.graphNonIsoDec","parent":35750,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/index.html#graph-non-iso-dec","scope":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs","status":"fully_proved","summary":"The decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of triples $(n, E_1, E_2)$ of a natural number and two lists of pairs of natural numbers, to a triple for which there is no bijection $f$ from $\\{0, \\ldots, n - 1\\}$ to itself such that, for all $a, b < n$, $(a, b) \\in E_1$ if and only if $(f(a), f(b)) \\in E_2$. Bitstrings that do not decode are rejected.","title":"Directed graph non-isomorphism as a decision problem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/graph-noniso-in-am","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GNIInAM.graph_noniso_in_am","parent":35750,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs/index.html#graph-noniso-in-am","scope":"TheoryOfComputation/randomized-and-quantum-complexity/graph-isomorphism-and-arthur-merlin-proofs","status":"can_prove","summary":"The directed graph non-isomorphism decision problem belongs to the class $\\mathsf{AM}$ of decision problems decided by Arthur\u2013Merlin protocols.","title":"Graph non-isomorphism is in AM"},{"data_url":"scopes/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem.json","id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","kind":"scope","navigation":"module","parent":35731,"result":false,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"can_prove","title":"Local Hamiltonian problem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/init-state","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.initState","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#init-state","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"For a bitstring $x = (x_0, \\dots, x_{n-1})$, with true read as $1$ and false as $0$, numbers $m$ and $a$, and a vector $\\psi \\in \\mathbb{C}^{2^m}$ (the witness on $m$ qubits), the initial state is the tensor product of the computational basis state of $x$, the vector $\\psi$ and the all-zero state on $a$ ancilla qubits, in this order. It is the vector on $\\{0, \\dots, 2^{n+m+a} - 1\\}$ whose value at $s$ is\u2026","title":"Initial state of a QMA verifier"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/qmapromise","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.QMAPromise","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#qmapromise","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"A pair $(\\Pi_{\\mathrm{yes}}, \\Pi_{\\mathrm{no}})$ of sets of bitstrings is in QMA when there are polynomials $m$, $a$ and $s$ with natural-number coefficients and a polynomial-time computable map sending each bitstring $x$ to a list of encoded 2-local quantum gates such that, for every bitstring $x$, writing $N = |x| + m(|x|) + a(|x|)$: the list has at most $s(|x|)$ gates, each well-formed on $N$ qubits; if\u2026","title":"QMA for promise problems"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/promise-reduces","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.PromiseReduces","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#promise-reduces","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"A promise problem $(Y_1, N_1)$, given by sets $Y_1$ and $N_1$ of bitstrings, reduces to another such promise problem $(Y_2, N_2)$ when there is a polynomial-time computable map $f$ from bitstrings to bitstrings with $f(x) \\in Y_2$ for every $x \\in Y_1$ and $f(x) \\in N_2$ for every $x \\in N_1$.","title":"Polynomial-time reduction between promise problems"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/local-term","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.LocalTerm","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#local-term","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"A 2-local Hamiltonian term is encoded as a triple $(i, j, e)$ of natural numbers $i$ and $j$, the qubits it acts on, and a list $e$ of pairs of encoded rational numbers, the real and imaginary parts of the entries of its $4 \\times 4$ matrix in row-major order.","title":"Encoded 2-local Hamiltonian term"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/embed-term","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.embedTerm","parent":35755,"reading_title":{"format":"inline-parts-v1","parts":[["text","Embedding of a 2-local term into "],["math","\\(n\\)"],["text"," qubits"]],"source":"Embedding of a 2-local term into $n$ qubits"},"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#embed-term","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"Let a 2-local term act on qubits $i$ and $j$, and let $M$ be the $4 \\times 4$ complex matrix encoded by its list of entries, with rows and columns indexed by pairs of bits. Its embedding into $n$ qubits is the $2^n \\times 2^n$ complex matrix whose entry at basis indices $s, t \\in \\{0, \\dots, 2^n - 1\\}$ is $M_{(s_i, s_j), (t_i, t_j)}$ if $s_k = t_k$ for every $k \\ne i, j$, and $0$ otherwise, where\u2026","title":"Embedding of a 2-local term into $n$ qubits"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/hamiltonian","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.hamiltonian","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#hamiltonian","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"For a number $n$ of qubits and a list $h_1, \\dots, h_m$ of 2-local terms, the Hamiltonian is the $2^n \\times 2^n$ complex matrix $\\tilde h_1 + \\dots + \\tilde h_m$, where $\\tilde h_l$ is the embedding of $h_l$ into $n$ qubits; for the empty list it is the zero matrix.","title":"Hamiltonian of a list of 2-local terms"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/ground-energy","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.groundEnergy","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#ground-energy","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"For a natural number $n$, the ground energy of a Hermitian $2^n \\times 2^n$ complex matrix $H$ is the smallest eigenvalue of $H$.","title":"Ground energy of a Hermitian matrix"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/local-term-valid","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.localTermValid","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#local-term-valid","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"A 2-local term acting on qubits $i$ and $j$ with entry list $e$ is valid on $n$ qubits when $i < j < n$, the list $e$ has exactly $16$ entries, each entry has nonzero denominators in the encodings of both its real and its imaginary part, the $4 \\times 4$ complex matrix $M$ encoded by $e$ is positive semidefinite, and $\\sum_{p, q} |M_{pq}|^2 \\le 1$, that is, $M$ has Frobenius norm at most $1$.","title":"Valid 2-local Hamiltonian term"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/local-ham-yes","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.LocalHamYes","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#local-ham-yes","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"A bitstring $x$ is a yes-instance when decoding it under the canonical bitstring encoding gives a tuple $(n, (h_1, \\dots, h_m), a, b)$ of a natural number $n$, a list of 2-local terms $h_1, \\dots, h_m$ and two encoded rational numbers, whose values we also write $a$ and $b$, such that every $h_l$ is valid on $n$ qubits, $b - a \\ge 1/(m + 1)$, and the Hamiltonian $H$ of $h_1, \\dots, h_m$ on $n$ qubits is Hermitian\u2026","title":"Yes-instances of the 2-local Hamiltonian problem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/local-ham-no","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.LocalHamNo","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#local-ham-no","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"fully_proved","summary":"A bitstring $x$ is a no-instance when decoding it under the canonical bitstring encoding gives a tuple $(n, (h_1, \\dots, h_m), a, b)$ of a natural number $n$, a list of 2-local terms $h_1, \\dots, h_m$ and two encoded rational numbers, whose values we also write $a$ and $b$, such that every $h_l$ is valid on $n$ qubits, $b - a \\ge 1/(m + 1)$, and the Hamiltonian $H$ of $h_1, \\dots, h_m$ on $n$ qubits is Hermitian\u2026","title":"No-instances of the 2-local Hamiltonian problem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/local-hamiltonian-qma-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LocalHamiltonianQMA.local_hamiltonian_qma_complete","parent":35755,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem/index.html#local-hamiltonian-qma-complete","scope":"TheoryOfComputation/randomized-and-quantum-complexity/local-hamiltonian-problem","status":"can_prove","summary":"The 2-local Hamiltonian promise problem, given by its yes-instances and no-instances, is in QMA, and every promise problem in QMA whose sets of yes-instances and no-instances are disjoint sets of bitstrings has a polynomial-time reduction to it.","title":"The 2-local Hamiltonian problem is QMA-complete"},{"data_url":"scopes/TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem.json","id":"TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem","kind":"scope","navigation":"module","parent":35731,"result":false,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem/index.html","scope":"TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem","status":"can_prove","title":"Valiant\u2013Vazirani theorem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem/sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ValiantVazirani.satDec'","parent":35767,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem/index.html#sat-dec","scope":"TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem","status":"fully_proved","summary":"A bitstring $x$ is a yes-instance exactly when bitstring decoding of $x$, as a pair of a list of booleans and a CNF formula, succeeds and returns a pair $(u, \\varphi)$ such that, writing $m$ for the length of $u$, every variable occurring in $\\varphi$ is less than $m$ and some assignment of booleans to the variables $0, \\dots, m - 1$ satisfies $\\varphi$ once it is extended by assigning false to every variable\u2026","title":"SAT with a unary variable count as a decision problem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem/unique-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ValiantVazirani.uniqueSatDec","parent":35767,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem/index.html#unique-sat-dec","scope":"TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem","status":"fully_proved","summary":"A bitstring $x$ is a yes-instance exactly when bitstring decoding of $x$, as a pair of a list of booleans and a CNF formula, succeeds and returns a pair $(u, \\varphi)$ such that, writing $m$ for the length of $u$, every variable occurring in $\\varphi$ is less than $m$ and exactly one assignment of booleans to the variables $0, \\dots, m - 1$ satisfies $\\varphi$ once it is extended by assigning false to every\u2026","title":"UniqueSAT with a unary variable count as a decision problem"},{"id":"TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem/valiant-vazirani","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ValiantVazirani.valiant_vazirani","parent":35767,"result":true,"route":"../roadmap/TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem/index.html#valiant-vazirani","scope":"TheoryOfComputation/randomized-and-quantum-complexity/valiant-vazirani-theorem","status":"can_prove","summary":"There are a polynomial $p$ with natural-number coefficients and a polynomial-time computable function $R$ from pairs of bitstrings to bitstrings such that, for every bitstring $x$ of length $n$: if $x$ is not in SAT, then for every bitstring $r$ of length $p(n)$ the bitstring $R(x, r)$ is in neither SAT nor UniqueSAT; and if $x$ is in SAT, then the number $N$ of bitstrings $r$ of length $p(n)$ for which $R(x, r)$\u2026","title":"Valiant-Vazirani: SAT reduces to UniqueSAT"},{"data_url":"scopes/TheoryOfComputation/counting-complexity.json","id":"TheoryOfComputation/counting-complexity","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/counting-complexity/index.html","scope":"TheoryOfComputation/counting-complexity","status":"can_prove","summary":"\\#P and PP: #P-completeness (counting SAT, permanent), PP-completeness, Toda's theorem, threshold languages, and approximate counting algorithms and schemes (Stockmeyer, Karp\u2013Luby, Jerrum\u2013Sinclair\u2013Vigoda).","title":"Counting complexity"},{"id":"TheoryOfComputation/counting-complexity/stockmeyer-approximate-counting","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Stockmeyer.stockmeyer_approximate_counting","parent":35771,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/index.html#stockmeyer-approximate-counting","scope":"TheoryOfComputation/counting-complexity","status":"can_prove","summary":"For every function $f$ in $\\#P$ there are a decision problem $O$ in NP, polynomials $p_r$, $p$ and $q$ with natural-number coefficients, and polynomial-time computable functions $\\mathrm{nextQ}$ and $\\mathrm{final}$, each taking a bitstring and a list of query/answer pairs (bitstring, boolean), with $\\mathrm{nextQ}$ returning a bitstring and $\\mathrm{final}$ returning a natural number, such that the following\u2026","title":"Stockmeyer: approximate counting of #P functions in BPP^NP"},{"id":"TheoryOfComputation/counting-complexity/toda","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Toda.toda","parent":35771,"reading_title":{"format":"inline-parts-v1","parts":[["text","Toda's theorem: PH is contained in "],["math","\\(\\mathsf{P}^{\\#\\mathsf{P}}\\)"]],"source":"Toda's theorem: PH is contained in $\\mathsf{P}^{\\#\\mathsf{P}}$"},"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/index.html#toda","scope":"TheoryOfComputation/counting-complexity","status":"can_prove","summary":"Every decision problem in the polynomial hierarchy PH lies in $\\mathsf{P}^{\\#\\mathsf{P}}$, the class of decision problems decidable in polynomial time with a $\\#\\mathsf{P}$ oracle.","title":"Toda's theorem: PH is contained in $\\mathsf{P}^{\\#\\mathsf{P}}$"},{"data_url":"scopes/TheoryOfComputation/counting-complexity/complexity-of-the-permanent.json","id":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent","kind":"scope","navigation":"module","parent":35771,"result":false,"route":"../roadmap/TheoryOfComputation/counting-complexity/complexity-of-the-permanent/index.html","scope":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent","status":"can_prove","title":"Complexity of the permanent"},{"id":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent/perm-of","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.JerrumSinclairVigoda.permOf","parent":35774,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/complexity-of-the-permanent/index.html#perm-of","scope":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent","status":"fully_proved","summary":"For a bitstring $x$: if $x$ decodes, under the canonical bitstring encoding, to a pair $(n, e)$ of a natural number $n$ and a list $e = (e_0, \\dots, e_{n^2 - 1})$ of $n^2$ natural numbers each at most $1$, the value at $x$ is the permanent $\\sum_{\\sigma} \\prod_{i=0}^{n-1} e_{in + \\sigma(i)}$ of the $n \\times n$ matrix with row-major entries $e$, where $\\sigma$ ranges over the permutations of $\\{0, \\dots, n - 1\\}$;\u2026","title":"Permanent of an encoded 0-1 matrix"},{"id":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent/jerrum-sinclair-vigoda-permanent","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.JerrumSinclairVigoda.jerrum_sinclair_vigoda_permanent","parent":35774,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/complexity-of-the-permanent/index.html#jerrum-sinclair-vigoda-permanent","scope":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent","status":"can_prove","summary":"For a bitstring $x$, let $\\operatorname{perm}(x)$ be the permanent of the 0-1 matrix encoded by $x$: if $x$ decodes, under the canonical bitstring encoding, to a pair $(k, e)$ of a natural number $k$ and a list $e$ of $k^2$ natural numbers each at most $1$, then $\\operatorname{perm}(x)$ is the permanent of the $k \\times k$ matrix with row-major entries $e$, and otherwise $\\operatorname{perm}(x) = 0$. There are a\u2026","title":"Jerrum-Sinclair-Vigoda: FPRAS for the permanent"},{"id":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent/perm-fun","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PermanentSharpPComplete.permFun","parent":35774,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/complexity-of-the-permanent/index.html#perm-fun","scope":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent","status":"fully_proved","summary":"For a bitstring $x$ that decodes to a pair $(n, e)$ of a natural number $n$ and a list $e = (e_0, \\dots, e_{m-1})$ of natural numbers with $m = n^2$ and every $e_j \\le 1$, the value at $x$ is the permanent $\\sum_{\\sigma} \\prod_{i < n} e_{in + \\sigma(i)}$ of the $n \\times n$ matrix with entries $e$ in row-major order, the sum running over all permutations $\\sigma$ of $\\{0, \\dots, n - 1\\}$. For every other bitstring\u2026","title":"Permanent of an encoded 0-1 matrix"},{"id":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent/permanent-sharp-p-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PermanentSharpPComplete.permanent_sharpP_complete","parent":35774,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/complexity-of-the-permanent/index.html#permanent-sharp-p-complete","scope":"TheoryOfComputation/counting-complexity/complexity-of-the-permanent","status":"can_prove","summary":"The function that sends a bitstring encoding a square 0-1 matrix to its permanent, and every other bitstring to $0$, is $\\#P$-complete.","title":"The permanent is #P-complete (Valiant)"},{"data_url":"scopes/TheoryOfComputation/counting-complexity/counting-satisfiability-sat.json","id":"TheoryOfComputation/counting-complexity/counting-satisfiability-sat","kind":"scope","navigation":"module","parent":35771,"result":false,"route":"../roadmap/TheoryOfComputation/counting-complexity/counting-satisfiability-sat/index.html","scope":"TheoryOfComputation/counting-complexity/counting-satisfiability-sat","status":"can_prove","title":"Counting satisfiability (#SAT)"},{"id":"TheoryOfComputation/counting-complexity/counting-satisfiability-sat/count-sat-fun","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.CountSATSharpPComplete.countSatFun","parent":35779,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/counting-satisfiability-sat/index.html#count-sat-fun","scope":"TheoryOfComputation/counting-complexity/counting-satisfiability-sat","status":"fully_proved","summary":"The counting function $f$ from bitstrings to natural numbers is defined as follows. If a bitstring $x$ decodes, under the canonical bitstring encoding of pairs, to a pair $(s, \\varphi)$ of a bitstring $s$ of length $n$ and a CNF formula $\\varphi$ every literal of which has variable index less than $n$, then $f(x)$ is the number of assignments $a : \\{0, \\dots, n - 1\\} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ such that\u2026","title":"Counting satisfying assignments of a CNF formula"},{"id":"TheoryOfComputation/counting-complexity/counting-satisfiability-sat/count-sat-sharp-p-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.CountSATSharpPComplete.count_sat_sharpP_complete","parent":35779,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/counting-satisfiability-sat/index.html#count-sat-sharp-p-complete","scope":"TheoryOfComputation/counting-complexity/counting-satisfiability-sat","status":"can_prove","summary":"The function counting the satisfying assignments of a CNF formula is $\\#P$-complete: it lies in $\\#P$, and every function in $\\#P$ is computed from it by a polynomial-time oracle machine.","title":"Counting SAT is #P-complete"},{"data_url":"scopes/TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting.json","id":"TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting","kind":"scope","navigation":"module","parent":35771,"result":false,"route":"../roadmap/TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting/index.html","scope":"TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting","status":"can_prove","title":"Karp\u2013Luby algorithm for DNF counting"},{"id":"TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting/dnf-sat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.KarpLuby.dnfSat","parent":35782,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting/index.html#dnf-sat","scope":"TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting","status":"fully_proved","summary":"A DNF formula is a list of terms, each a list of literals $(v, b)$ of a natural number $v$ (the variable) and a boolean $b$ (the polarity). An assignment $a$ of booleans to the natural numbers satisfies the formula when some term of it has $a(v) = b$ for every literal $(v, b)$ of that term.","title":"Satisfaction of a DNF formula"},{"id":"TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting/dnf-count-of","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.KarpLuby.dnfCountOf","parent":35782,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting/index.html#dnf-count-of","scope":"TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting","status":"fully_proved","summary":"For a bitstring $x$: if $x$ decodes, under the canonical bitstring encoding, to a pair $(u, \\varphi)$ of a bitstring $u$ of length $m$ and a DNF formula $\\varphi$ all of whose literals have variables less than $m$, the value at $x$ is the number of assignments of booleans to the variables $0, \\dots, m - 1$ that satisfy $\\varphi$; otherwise the value is $0$.","title":"Number of satisfying assignments of an encoded DNF formula"},{"id":"TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting/karp-luby-dnf-counting","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.KarpLuby.karp_luby_dnf_counting","parent":35782,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting/index.html#karp-luby-dnf-counting","scope":"TheoryOfComputation/counting-complexity/karp-luby-algorithm-for-dnf-counting","status":"can_prove","summary":"For a bitstring $x$, let $N(x)$ be the number of satisfying assignments of the DNF formula encoded by $x$: if $x$ decodes, under the canonical bitstring encoding, to a pair $(u, \\varphi)$ of a bitstring $u$ of length $m$ and a DNF formula $\\varphi$ whose variables are all less than $m$, then $N(x)$ is the number of assignments of booleans to the variables $0, \\dots, m - 1$ that satisfy $\\varphi$, and otherwise\u2026","title":"Karp-Luby: FPRAS for DNF counting"},{"data_url":"scopes/TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions.json","id":"TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions","kind":"scope","navigation":"module","parent":35771,"result":false,"route":"../roadmap/TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions/index.html","scope":"TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions","status":"can_prove","title":"Threshold languages of #P functions"},{"id":"TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions/threshold-lang","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SharpPThresholdPSPACE.thresholdLang","parent":35786,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions/index.html#threshold-lang","scope":"TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions","status":"fully_proved","summary":"For a function $f$ from bitstrings to natural numbers, the threshold language of $f$ is the decision problem that accepts a bitstring $y$ exactly when decoding $y$ with the canonical bitstring encoding of pairs of a bitstring and a natural number succeeds and gives a pair $(x, k)$ with $k \\le f(x)$; bitstrings that do not decode are rejected.","title":"Threshold language of a counting function"},{"id":"TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions/sharp-p-threshold-pspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SharpPThresholdPSPACE.sharpP_threshold_pspace","parent":35786,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions/index.html#sharp-p-threshold-pspace","scope":"TheoryOfComputation/counting-complexity/threshold-languages-of-p-functions","status":"can_prove","summary":"For every function $f$ in $\\#P$, the threshold language of $f$, which accepts a bitstring exactly when it decodes, under the canonical bitstring encoding of pairs, to a pair $(x, k)$ of a bitstring $x$ and a natural number $k$ with $k \\le f(x)$, is in PSPACE.","title":"Threshold languages of #P functions are in PSPACE"},{"data_url":"scopes/TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp.json","id":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp","kind":"scope","navigation":"module","parent":35771,"result":false,"route":"../roadmap/TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/index.html","scope":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp","status":"can_prove","title":"Unbounded-error probabilistic time (PP)"},{"id":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/maj-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MAJSATPPComplete.majSatDec","parent":35789,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/index.html#maj-sat-dec","scope":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp","status":"fully_proved","summary":"MAJSAT is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs, to a pair $(u, \\varphi)$ of a bitstring $u$ and a CNF formula $\\varphi$ such that, with $n = |u|$, every literal of $\\varphi$ has variable index less than $n$, and more than half of the $2^n$ assignments $a : \\{0, \\dots, n-1\\} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ satisfy\u2026","title":"The MAJSAT language"},{"id":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/majsat-pp-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MAJSATPPComplete.majsat_pp_complete","parent":35789,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/index.html#majsat-pp-complete","scope":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp","status":"can_prove","summary":"The MAJSAT language is in PP, and every decision problem in PP has a polynomial-time many-one reduction to MAJSAT.","title":"MAJSAT is PP-complete"},{"id":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/np-subset-pp","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NPSubsetPP.np_subset_pp","parent":35789,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/index.html#np-subset-pp","scope":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp","status":"can_prove","summary":"Every decision problem in NP also belongs to PP, the class of decision problems decided in probabilistic polynomial time with unbounded error.","title":"NP is contained in PP"},{"id":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/pp-subset-pspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PPSubsetPSPACE.pp_subset_pspace","parent":35789,"result":true,"route":"../roadmap/TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp/index.html#pp-subset-pspace","scope":"TheoryOfComputation/counting-complexity/unbounded-error-probabilistic-time-pp","status":"can_prove","summary":"Every decision problem in PP, the class of decision problems decided with unbounded error in probabilistic polynomial time, belongs to PSPACE.","title":"PP is contained in PSPACE"},{"data_url":"scopes/TheoryOfComputation/pseudorandomness-and-one-way-functions.json","id":"TheoryOfComputation/pseudorandomness-and-one-way-functions","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/index.html","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions","status":"can_prove","summary":"Complexity-theoretic foundations of cryptography and derandomization: one-way functions, hardcore predicates (Goldreich\u2013Levin), hardness amplification (Yao's XOR lemma), hardness-versus-randomness generators (Nisan\u2013Wigderson), and Impagliazzo's five worlds.","title":"Pseudorandomness and one-way functions"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/nisan-wigderson","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NisanWigderson.nisan_wigderson","parent":35794,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/index.html#nisan-wigderson","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions","status":"can_prove","summary":"Suppose there are a decision problem $f$ in E and a natural number $c > 0$ such that, for every $n$, every well-formed flat Boolean circuit $C$ on $n$ inputs without mod gates that computes $f$ on all bitstrings of length $n$ (for every bitstring $x$ of length $n$, the value of its last gate on input $x$ is $f(x)$) satisfies $|C|^c \\ge 2^n$, where $|C|$ is its number of gates; that is, $C$ has at least $2^{n/c}$\u2026","title":"Nisan-Wigderson: exponential hardness in E derandomizes BPP"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/owf-implies-p-ne-np","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.OWFImpliesPNeNP.owf_implies_p_ne_np","parent":35794,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/index.html#owf-implies-p-ne-np","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions","status":"can_prove","summary":"If some family $f$, assigning to each natural number $n$ and each $n$-bit string $x$ a bitstring $f_n(x)$, is a one-way function family, then $\\mathrm{P} \\neq \\mathrm{NP}$.","title":"One-way functions imply P \u2260 NP"},{"data_url":"scopes/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem.json","id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","kind":"scope","navigation":"module","parent":35794,"result":false,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"can_prove","title":"Goldreich\u2013Levin theorem"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/glgate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldreichLevin.GLGate","parent":35797,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html#glgate","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"fully_proved","summary":"A serialized gate is a pair $(k, r)$ of a natural number $k$, the kind of the gate, and a list $r$ of natural numbers, its references.","title":"Serialized circuit gates"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/gl-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldreichLevin.glCircuitEval","parent":35797,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html#gl-circuit-eval","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"fully_proved","summary":"For a list $C = (g_0, \\dots, g_{m-1})$ of serialized gates and a list $a = (a_0, \\dots, a_{n-1})$ of Booleans, the gate values of $C$ on $a$ are the list of gate values of the flat Boolean circuit on $n$ inputs whose $j$-th gate is decoded from the (kind, references) pair $g_j$, evaluated on the assignment that gives input $i$ the value $a_i$.","title":"Gate values of a serialized circuit"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/gl-circuit-outputs","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldreichLevin.glCircuitOutputs","parent":35797,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html#gl-circuit-outputs","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"fully_proved","summary":"For a list $C$ of serialized gates, a list $a$ of Booleans and a natural number $k$, the outputs of $C$ on $a$ are the last $k$ entries of the gate values of $C$ on $a$, or all of them when $C$ has fewer than $k$ gates.","title":"Outputs of a serialized circuit"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/invert-succ","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldreichLevin.invertSucc","parent":35797,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html#invert-succ","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"fully_proved","summary":"For a list $C$ of serialized gates and a map $\\pi$ from $n$-bit strings to $n$-bit strings, the inversion success probability of $C$ against $\\pi$ is the rational number $N / 2^n$, where $N$ is the number of $n$-bit strings $x$ such that the outputs of $C$ on input $\\pi(x)$, taken as its last $n$ gate values, are exactly $x$.","title":"Inversion success probability of a circuit"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/xor-dot","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldreichLevin.xorDot","parent":35797,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html#xor-dot","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"fully_proved","summary":"For $n$-bit strings $x$ and $r$, their inner product modulo 2 is true if and only if the number of indices $i < n$ with $x_i$ and $r_i$ both true is odd, that is, $\\langle x, r \\rangle = \\sum_{i < n} x_i r_i \\bmod 2$.","title":"Inner product modulo 2"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/gl-pair","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldreichLevin.glPair","parent":35797,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html#gl-pair","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"fully_proved","summary":"For $n$-bit strings $y$ and $r$, their concatenation is the $2n$-bit string whose bit $i$ is $y_i$ for $i < n$ and $r_{i - n}$ for $n \\le i < 2n$.","title":"Concatenation of two bitstrings"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/predict-succ","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldreichLevin.predictSucc","parent":35797,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html#predict-succ","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"fully_proved","summary":"For a list $C$ of serialized gates and a map $\\pi$ from $n$-bit strings to $n$-bit strings, the prediction success probability of $C$ against $\\pi$ is the rational number $N / 2^{2n}$, where $N$ is the number of pairs $(x, r)$ of $n$-bit strings such that $C$ has at least one gate and the value of its last gate on the concatenation of $\\pi(x)$ and $r$, a $2n$-bit input, equals the inner product of $x$ and $r$\u2026","title":"Prediction success probability for the inner-product bit"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/goldreich-levin","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GoldreichLevin.goldreich_levin","parent":35797,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem/index.html#goldreich-levin","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/goldreich-levin-theorem","status":"can_prove","summary":"Let $\\pi$ be a family of bijections $\\pi_n$ of the set of $n$-bit strings, one for each natural number $n$. Suppose that $\\pi$ is computable by polynomial-size circuits: there is a polynomial $q$ with natural number coefficients such that for every $n$ some list of at most $q(n)$ serialized gates has, on every $n$-bit string $x$, outputs (its last $n$ gate values) equal to $\\pi_n(x)$. Suppose also that $\\pi$ is\u2026","title":"Goldreich-Levin: hardcore predicate from one-way permutations"},{"data_url":"scopes/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds.json","id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","kind":"scope","navigation":"module","parent":35794,"result":false,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"can_prove","title":"Impagliazzo's five worlds"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/fwgate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.FWGate","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#fwgate","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"A serialized gate of a flat Boolean circuit is a pair $(k, r)$ of a natural number $k$ (the gate kind) and a list $r$ of natural numbers (its references, depending on the kind: an input index, a constant bit, or indices of other gates), in the shared encoding of gates as (kind, references) pairs.","title":"Serialized gates of five-worlds circuits"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/fw-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.fwCircuitEval","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#fw-circuit-eval","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Let $C$ be a flat Boolean circuit given as a list of gates, each encoded as a (kind, references) pair, and let $a = (a_0, \\dots, a_{k-1})$ be a list of booleans. The gate values of $C$ on $a$ are the gate values, in order, of the circuit on $k$ inputs obtained by decoding each gate of $C$, evaluated on the assignment that sends input $i$ to $a_i$.","title":"Evaluation of an encoded flat circuit"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/fw-circuit-outputs","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.fwCircuitOutputs","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#fw-circuit-outputs","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"For an encoded flat Boolean circuit $C$, a list of booleans $a$ and a natural number $k$, the outputs of $C$ on $a$ are the last $k$ entries of the list of gate values of $C$ on $a$, or the whole list if $C$ has fewer than $k$ gates.","title":"Outputs of an encoded flat circuit"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/fw-invert-succ","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.fwInvertSucc","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#fw-invert-succ","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Let $f$ map $n$-bit strings to bitstrings and let $C$ be an encoded flat Boolean circuit. For an $n$-bit string $x$, the candidate preimage $x'$ consists of the last $n$ gate values of $C$ on input $f(x)$ (all of them if $C$ has fewer than $n$ gates), padded with false to length $n$. The inversion success probability of $C$ against $f$ is the fraction of the $2^n$ strings $x$ for which $f(x') = f(x)$.","title":"Inversion success probability of a circuit"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/owfexists","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.OWFExists","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#owfexists","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"There are a family of functions $f_n$, each mapping $n$-bit strings to bitstrings, and a polynomial $q$ with natural-number coefficients such that the following hold: $|f_n(x)| \\le q(n)$ for every $n$ and every $n$-bit string $x$; some polynomial-time computable function on bitstrings maps the canonical bitstring encoding of every pair $(n, x)$ to $f_n(x)$; and for every family of flat Boolean circuits $C_n$, given\u2026","title":"One-way functions exist"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/samplable","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.Samplable","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#samplable","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Let $p$ be a polynomial with natural-number coefficients and let $S$ be a family of maps $S_n$ from $p(n)$-bit coin strings to bitstrings. The pair $(p, S)$ is samplable when every $S_n(r)$ has length exactly $n$ and some polynomial-time computable function on bitstrings maps the canonical bitstring encoding of every pair $(n, r)$ to $S_n(r)$.","title":"Samplable distributions"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/avg-easy","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.AvgEasy","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#avg-easy","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Let $p$ be a polynomial with natural-number coefficients, $L$ a decision problem, and $S$ a family of samplers, where $S_n$ sends each string $r$ of $p(n)$ random bits to a bitstring $S_n(r)$. The pair $(L, S)$ is easy on average when there is a polynomial-time computable function $A$ from bitstrings to booleans such that for every natural number $c$ there is $N$ such that for all $n \\ge N$","title":"Easy on average"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/avg-hard","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.AvgHard","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#avg-hard","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Let $p$ be a polynomial with natural-number coefficients, $L$ a decision problem, and $S$ a family of samplers, where $S_n$ sends each string $r$ of $p(n)$ random bits to a bitstring $S_n(r)$. The pair $(L, S)$ is hard on average when it is not easy on average: there is no polynomial-time computable function $A$ from bitstrings to booleans such that for every natural number $c$ and all sufficiently large $n$, $A$\u2026","title":"Hard on average"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/fw-key-pair","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.fwKeyPair","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#fw-key-pair","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"For a map $G$ from bitstrings to bitstrings (the key generator) and a bitstring $r$ of coins, the key pair of $G$ on $r$ is the pair of bitstrings obtained by decoding $G(r)$ with the canonical bitstring encoding of pairs of bitstrings, or the pair of empty bitstrings if this decoding fails.","title":"Key pair produced by a key generator"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/fw-pkadv","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.fwPKAdv","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#fw-pkadv","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Let $G$ (key generation) and $E$ (encryption) be maps from bitstrings to bitstrings, let $C$ be an encoded flat Boolean circuit, let $\\ell$ be a polynomial with natural-number coefficients and let $n$ be a natural number. For key coins $r$ and encryption coins $s$, each a string of $\\ell(n)$ bits, and a bit $b$, let $pk$ be the first component of the key pair of $G$ on $r$, and let the ciphertext $e$ be $E$ applied\u2026","title":"Eavesdropping advantage against a public-key scheme"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/pkexists","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.PKExists","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#pkexists","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"There are a polynomial $\\ell$ with natural-number coefficients and polynomial-time computable functions $G$ (key generation), $E$ (encryption) and $D$ (decryption) on bitstrings with the following two properties. Here tuples are passed to these functions and to circuits through their canonical bitstring encodings, and for a bitstring $r$ the keys $(\\mathit{pk}, \\mathit{sk})$ are the decoding of $G(r)$ as a pair of\u2026","title":"Public-key encryption exists"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/algorithmica","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.Algorithmica","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#algorithmica","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Algorithmica is the statement that the complexity classes $\\mathrm{P}$ and $\\mathrm{NP}$ are equal.","title":"Algorithmica"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/heuristica","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.Heuristica","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#heuristica","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Heuristica: `P \u2260 NP`, but every NP problem is easy on average for every samplable distribution.","title":"Heuristica"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/pessiland","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.Pessiland","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#pessiland","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Pessiland is the statement that $\\mathsf{P} \\ne \\mathsf{NP}$, that one-way functions do not exist, and that some decision problem $L$ in NP is hard on average for some samplable distribution: there are a polynomial $p$ with natural-number coefficients and a family $S$ of maps $S_n$ from $p(n)$-bit coin strings to bitstrings such that $(p, S)$ is samplable and $L$ is hard on average with respect to $S$.","title":"Pessiland"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/minicrypt","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.Minicrypt","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#minicrypt","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Minicrypt: one-way functions exist, but public-key cryptography does not.","title":"Minicrypt"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/cryptomania","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.Cryptomania","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#cryptomania","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Cryptomania is the statement that public-key cryptography exists: there are a polynomial $\\ell$ with natural-number coefficients and polynomial-time computable maps $G$ (key generation), $E$ (encryption) and $D$ (decryption) from bitstrings to bitstrings with the following two properties. First, decryption is correct: for every natural number $n$, all key coins $r$ and encryption coins $s$ that are bitstrings of\u2026","title":"Cryptomania"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/exactly-one-world","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.ExactlyOneWorld","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#exactly-one-world","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"fully_proved","summary":"Exactly one of the statements Algorithmica, Heuristica, Pessiland, Minicrypt and Cryptomania is true: one of them holds and the other four fail.","title":"Exactly one of Impagliazzo's five worlds holds"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/impagliazzo-five-worlds","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FiveWorlds.impagliazzo_five_worlds","parent":35806,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds/index.html#impagliazzo-five-worlds","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/impagliazzo-s-five-worlds","status":"can_prove","summary":"The following four statements hold. First, if public-key encryption exists, then one-way functions exist. Second, if one-way functions exist, then some decision problem $L$ in NP is hard on average for some samplable distribution: there are a polynomial $p$ with natural-number coefficients and a family $S$ of maps $S_n$ from $p(n)$-bit coin strings to bitstrings such that $(p, S)$ is samplable and $L$ is hard on\u2026","title":"Impagliazzo's five worlds"},{"data_url":"scopes/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma.json","id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","kind":"scope","navigation":"module","parent":35794,"result":false,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/index.html","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","status":"can_prove","title":"Yao's XOR lemma"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/xorgate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.YaoXOR.XORGate","parent":35825,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/index.html#xorgate","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","status":"fully_proved","summary":"An encoded gate is a pair $(k, r)$ of a natural number $k$ (the gate kind) and a list $r$ of natural numbers (its references), the shared (kind, references) encoding of gates of flat Boolean circuits.","title":"Encoded gates"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/xor-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.YaoXOR.xorCircuitEval","parent":35825,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/index.html#xor-circuit-eval","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","status":"fully_proved","summary":"For a list $C$ of encoded gates and a list $x$ of Booleans of length $\\ell$, the gate values of $C$ on $x$ are obtained by decoding each pair of $C$ to a gate of a flat Boolean circuit on $\\ell$ inputs and evaluating the decoded circuit with the assignment that gives input $i$ the value $x_i$; the result lists the values of all gates in order. A pair referring to an input index $i \\ge \\ell$ decodes to the constant\u2026","title":"Gate values of an encoded circuit"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/xor-flat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.YaoXOR.xorFlat","parent":35825,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/index.html#xor-flat","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","status":"fully_proved","summary":"For $n > 0$ and $k$ blocks $x_0, \\dots, x_{k-1}$ of $n$ bits each, the flattened string has $nk$ bits, and its bit $i$ is bit $i \\bmod n$ of the block $x_{\\lfloor i/n \\rfloor}$.","title":"Flattening blocks of bits"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/xor-fold","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.YaoXOR.xorFold","parent":35825,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/index.html#xor-fold","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","status":"fully_proved","summary":"For $k$ bits $b_0, \\dots, b_{k-1}$, their XOR is true if and only if an odd number of the $b_i$ are true.","title":"XOR of finitely many bits"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/xor-of","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.YaoXOR.xorOf","parent":35825,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/index.html#xor-of","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","status":"fully_proved","summary":"For natural numbers $n$ and $k$ and a Boolean function $f$ on $n$ bits, the function $f^{\\oplus k}$ takes $k$ blocks $x_0, \\dots, x_{k-1}$ of $n$ bits each to $f(x_0) \\oplus \\cdots \\oplus f(x_{k-1})$, which is true if and only if an odd number of the values $f(x_i)$ are true.","title":"XOR of independent copies of a function"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/agreement","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.YaoXOR.agreement","parent":35825,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/index.html#agreement","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","status":"fully_proved","summary":"Let $\\alpha$ be a finite type, let $C$ be a circuit given as a list of encoded gates, let $g$ assign a Boolean to each $a \\in \\alpha$, and let $\\mathrm{enc}$ encode each $a \\in \\alpha$ as a list of Booleans. The agreement of $C$ with $g$ is the rational number $N / |\\alpha|$ (which is $0$ when $\\alpha$ is empty), where $N$ is the number of $a \\in \\alpha$ such that $C$ has at least one gate and the value of its last\u2026","title":"Agreement of a circuit with a Boolean function"},{"id":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/yao-xor-lemma","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.YaoXOR.yao_xor_lemma","parent":35825,"result":true,"route":"../roadmap/TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma/index.html#yao-xor-lemma","scope":"TheoryOfComputation/pseudorandomness-and-one-way-functions/yao-s-xor-lemma","status":"can_prove","summary":"There is a natural number $c$ such that the following holds for all natural numbers $n \\ge 1$ and $k$, every rational $\\delta$ with $0 < \\delta \\le 1/2$, every Boolean function $f$ on $n$ bits and every natural number $s$. Suppose every circuit given as a list of at most $s$ encoded gates has agreement at most $1 - \\delta$ with $f$, where an input $x \\in \\{0,1\\}^n$ is given to the circuit as its $n$ bits. Then\u2026","title":"Yao's XOR lemma: hardness amplification by XOR"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity.json","id":"TheoryOfComputation/circuit-complexity","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/index.html","scope":"TheoryOfComputation/circuit-complexity","status":"can_prove","summary":"Boolean circuits as a model of computation, including arithmetic circuits built from gates (adders, two's complement), parallel classes AC0, NC1 and NC, and circuit lower bounds (parity and majority not in AC0, Smolensky, Razborov's monotone bound, Shannon counting, Williams).","title":"Boolean circuit complexity"},{"id":"TheoryOfComputation/circuit-complexity/smolensky","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Smolensky.smolensky","parent":35833,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/index.html#smolensky","scope":"TheoryOfComputation/circuit-complexity","status":"can_prove","summary":"The parity language, which accepts a bitstring exactly when an odd number of its entries are true, is not in $\\mathsf{AC}^0[3]$, the class $\\mathsf{AC}^0[m]$ for $m = 3$ of decision problems decided by polynomial-size, bounded-depth families of well-formed flat Boolean circuits in which every mod gate has modulus $3$.","title":"Smolensky: parity is not in AC0(mod 3)"},{"id":"TheoryOfComputation/circuit-complexity/williams-nexp-not-in-acc0","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Williams.williams_nexp_not_in_acc0","parent":35833,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/index.html#williams-nexp-not-in-acc0","scope":"TheoryOfComputation/circuit-complexity","status":"can_prove","summary":"Some decision problem lies in NEXP but not in $\\mathsf{ACC}^0$.","title":"Williams: NEXP is not contained in ACC0"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity/boolean-circuits.json","id":"TheoryOfComputation/circuit-complexity/boolean-circuits","kind":"scope","navigation":"module","parent":35833,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/boolean-circuits/index.html","scope":"TheoryOfComputation/circuit-complexity/boolean-circuits","status":"fully_proved","title":"Boolean circuits"},{"id":"TheoryOfComputation/circuit-complexity/boolean-circuits/boolean-gate","kind":"node","lean":"Cslib.Circuits.BooleanGate","parent":35836,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/boolean-circuits/index.html#boolean-gate","scope":"TheoryOfComputation/circuit-complexity/boolean-circuits","status":"fully_proved","summary":"The Boolean gates are a NOT gate and, for each natural number $a$, an AND gate and an OR gate indexed by $a$, which stand for gates with $a + 1$ inputs. The numbers of inputs are assigned by the Boolean signature, so there is no AND or OR gate with zero inputs.","title":"Boolean gates"},{"id":"TheoryOfComputation/circuit-complexity/boolean-circuits/boolean-signature","kind":"node","lean":"Cslib.Circuits.booleanSignature","parent":35836,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/boolean-circuits/index.html#boolean-signature","scope":"TheoryOfComputation/circuit-complexity/boolean-circuits","status":"fully_proved","summary":"The Boolean signature has the Boolean gates as its operation symbols, where NOT has arity $1$ and, for each natural number $a$, the AND gate and the OR gate indexed by $a$ have arity $a + 1$. In particular, no AND or OR gate takes zero inputs.","title":"Boolean signature"},{"id":"TheoryOfComputation/circuit-complexity/boolean-circuits/boolean-interpretation","kind":"node","lean":"Cslib.Circuits.booleanInterpretation","parent":35836,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/boolean-circuits/index.html#boolean-interpretation","scope":"TheoryOfComputation/circuit-complexity/boolean-circuits","status":"fully_proved","summary":"The Boolean interpretation evaluates the gates of the Boolean signature on the truth values $\\{\\mathrm{false}, \\mathrm{true}\\}$: NOT sends its input $x$ to $\\lnot x$, and for each natural number $a$, the AND gate and the OR gate with inputs $x_0, \\dots, x_a$ return $(\\cdots((\\mathrm{true} \\land x_0) \\land x_1) \\cdots) \\land x_a$ and $(\\cdots((\\mathrm{false} \\lor x_0) \\lor x_1) \\cdots) \\lor x_a$ respectively, that\u2026","title":"Boolean interpretation"},{"id":"TheoryOfComputation/circuit-complexity/boolean-circuits/boolean-interpretation-not","kind":"node","lean":"Cslib.Circuits.booleanInterpretation_not","parent":35836,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/boolean-circuits/index.html#boolean-interpretation-not","scope":"TheoryOfComputation/circuit-complexity/boolean-circuits","status":"fully_proved","summary":"Under the Boolean interpretation, the NOT gate applied to a truth value $x$ returns its negation $\\lnot x$.","title":"Boolean NOT"},{"id":"TheoryOfComputation/circuit-complexity/boolean-circuits/boolean-interpretation-and","kind":"node","lean":"Cslib.Circuits.booleanInterpretation_and","parent":35836,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/boolean-circuits/index.html#boolean-interpretation-and","scope":"TheoryOfComputation/circuit-complexity/boolean-circuits","status":"fully_proved","summary":"For every natural number $a$ and truth values $x_0, \\dots, x_a$, the Boolean interpretation of the AND gate with $a + 1$ inputs, applied to $x_0, \\dots, x_a$, returns $(\\cdots((\\mathrm{true} \\land x_0) \\land x_1) \\cdots) \\land x_a$, folding conjunction over the inputs from left to right starting from true.","title":"Boolean AND as a left fold"},{"id":"TheoryOfComputation/circuit-complexity/boolean-circuits/boolean-interpretation-or","kind":"node","lean":"Cslib.Circuits.booleanInterpretation_or","parent":35836,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/boolean-circuits/index.html#boolean-interpretation-or","scope":"TheoryOfComputation/circuit-complexity/boolean-circuits","status":"fully_proved","summary":"For every natural number $a$ and truth values $x_0, \\dots, x_a$, the Boolean interpretation of the OR gate with $a + 1$ inputs, applied to $x_0, \\dots, x_a$, returns $(\\cdots((\\mathrm{false} \\lor x_0) \\lor x_1) \\cdots) \\lor x_a$, folding disjunction over the inputs from left to right starting from false.","title":"Boolean OR as a left fold"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits.json","id":"TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits","kind":"scope","navigation":"module","parent":35833,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits/index.html","scope":"TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits","status":"can_prove","title":"Lower bounds for AC0 circuits"},{"id":"TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits/majority-not-in-ac0","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MajorityNotInAC0.majority_not_in_ac0","parent":35843,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits/index.html#majority-not-in-ac0","scope":"TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits","status":"can_prove","summary":"The majority language is not in the class $\\mathsf{AC}^0$.","title":"Majority is not in AC0"},{"id":"TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits/parity-not-in-ac0","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ParityNotInAC0.parity_not_in_ac0","parent":35843,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits/index.html#parity-not-in-ac0","scope":"TheoryOfComputation/circuit-complexity/lower-bounds-for-ac0-circuits","status":"can_prove","summary":"The parity language, which accepts a bitstring exactly when an odd number of its entries are true, is not in the class $\\mathsf{AC}^0$.","title":"Parity is not in AC0"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique.json","id":"TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique","kind":"scope","navigation":"module","parent":35833,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique/index.html","scope":"TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique","status":"can_prove","title":"Monotone circuit lower bound for clique"},{"id":"TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique/clique-holds","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RazborovMonotoneClique.cliqueHolds","parent":35846,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique/index.html#clique-holds","scope":"TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique","status":"fully_proved","summary":"For a natural number $n$ and a bitstring $x$, read $x$ as an $n \\times n$ adjacency matrix in row-major order: the entry for a pair $(a, b)$ is the bit of $x$ at position $an + b$, where a position beyond the end of $x$ reads as false. The clique predicate holds for $n$ and $x$ when there is a set $S$ of exactly $\\lfloor n/3 \\rfloor$ vertices among $0, \\dots, n - 1$ such that the entry for $(a, b)$ is true for all\u2026","title":"The clique function on adjacency-matrix bits"},{"id":"TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique/razborov-monotone-clique","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RazborovMonotoneClique.razborov_monotone_clique","parent":35846,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique/index.html#razborov-monotone-clique","scope":"TheoryOfComputation/circuit-complexity/monotone-circuit-lower-bound-for-clique","status":"can_prove","summary":"For every polynomial $p$ with natural-number coefficients there is a natural number $n$ with the following property. Let $C$ be a well-formed monotone flat circuit on $n^2$ inputs, that is, every gate of $C$ is an input variable, an and gate or an or gate, and refers only to earlier gates. Suppose that for every bitstring $x$ of length $n^2$, when input variable $i$ is given the $i$-th bit of $x$, $C$ has a last\u2026","title":"Razborov: monotone circuits for clique need superpolynomial size"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc.json","id":"TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc","kind":"scope","navigation":"module","parent":35833,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc/index.html","scope":"TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc","status":"can_prove","title":"Parallel complexity class NC"},{"id":"TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc/nc1-subset-logspace","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NC1SubsetL.nc1_subset_logspace","parent":35849,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc/index.html#nc1-subset-logspace","scope":"TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc","status":"can_prove","summary":"Every decision problem in $\\mathsf{NC}^1$ belongs to the class $\\mathsf{L}$ of decision problems decidable in deterministic logarithmic space.","title":"NC^1 is contained in L"},{"id":"TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc/nc-subset-p","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.NCSubsetP.nc_subset_p","parent":35849,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc/index.html#nc-subset-p","scope":"TheoryOfComputation/circuit-complexity/parallel-complexity-class-nc","status":"can_prove","summary":"Every decision problem in $\\mathsf{NC}$ belongs to $\\mathsf{P}$.","title":"NC is contained in P"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity/ripple-carry-adder.json","id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","kind":"scope","navigation":"module","parent":35833,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","title":"Ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/arith-gate","kind":"node","lean":"Cslib.Circuits.ArithGate","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#arith-gate","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"The gate symbols of arithmetic circuits are exactly three: XOR, AND and OR.","title":"Arithmetic gate symbols"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/arith-signature","kind":"node","lean":"Cslib.Circuits.arithSignature","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#arith-signature","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"The circuit signature whose operation symbols are the three arithmetic gate symbols XOR, AND and OR, each of arity $2$.","title":"Arithmetic basis"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/arith-interpretation","kind":"node","lean":"Cslib.Circuits.arithInterpretation","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#arith-interpretation","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"The interpretation of the arithmetic basis in the Booleans: on a first argument $a$ and a second argument $b$, the XOR gate computes $a \\oplus b$, the AND gate computes $a \\wedge b$, and the OR gate computes $a \\vee b$.","title":"Boolean interpretation of the arithmetic gates"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/bin-line","kind":"node","lean":"Cslib.Circuits.binLine","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#bin-line","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"For an arithmetic gate symbol $\\mathrm{op}$ and wires $u$ and $v$ of a circuit with $m$ inputs (each wire is an input or one of the $g$ preceding gates), the gate that applies $\\mathrm{op}$ to $u$ as its first argument and $v$ as its second.","title":"Binary gate line"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/carry-wire","kind":"node","lean":"Cslib.Circuits.RippleCarry.carryWire","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#carry-wire","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"For natural numbers $n$, $g$ and $i$ with $5i \\le g$, the wire carrying the carry into bit $i$ of the $n$-bit adder, in a circuit with $2n + 1$ inputs and $g$ gates: for $i = 0$ it is the carry-in input (input $2n$), and for $i \\ne 0$ it is gate $5i - 1$, the carry-out gate of bit $i - 1$.","title":"Carry wire"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/line","kind":"node","lean":"Cslib.Circuits.RippleCarry.line","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#line","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"For natural numbers $n$ and $g$ with $g < 5n$, gate $g$ of the $n$-bit ripple-carry adder is the following binary gate of the arithmetic basis, whose arguments are among the $2n + 1$ inputs and the $g$ gates before it. Write $g = 5i + r$ with $0 \\le r < 5$, let $a$ be input $i$ and $b$ input $n + i$, and let $c$ be the carry into bit $i$: input $2n$ if $i = 0$ and gate $5i - 1$ otherwise. Gate $g$ is the XOR gate\u2026","title":"Gates of the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/rca-program","kind":"node","lean":"Cslib.Circuits.rcaProgram","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#rca-program","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"For $g \\le 5n$, the straight-line program over the arithmetic basis with $2n + 1$ inputs whose gates are, in order, gates $0, 1, \\dots, g - 1$ of the $n$-bit ripple-carry adder; for $g = 0$ it is the empty program.","title":"Gate program of the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/ripple-carry-adder","kind":"node","lean":"Cslib.Circuits.rippleCarryAdder","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#ripple-carry-adder","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"The $n$-bit ripple-carry adder is the circuit with $2n + 1$ inputs, $5n$ binary gates, each an XOR, AND or OR gate, and $n + 1$ outputs, built as follows. Write $a_i$ for input $i$, $b_i$ for input $n + i$ and $c_0$ for input $2n$, where $i < n$. For each $i < n$, gates $g_{5i}, \\dots, g_{5i+4}$ form a full adder for position $i$: $g_{5i} = a_i \\oplus b_i$, $g_{5i+1} = g_{5i} \\oplus c_i$,\u2026","title":"Ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/rca-input","kind":"node","lean":"Cslib.Circuits.rcaInput","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#rca-input","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"For $n$-bit vectors $x$ and $y$ and a bit $c$, the assignment to the $2n + 1$ inputs of the $n$-bit adder: for $j < n$, input $j$ is bit $j$ of $x$; for $n \\le j < 2n$, input $j$ is bit $j - n$ of $y$; and input $2n$ is $c$. Bit $0$ is the least significant bit.","title":"Input of the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/rca-spec","kind":"node","lean":"Cslib.Circuits.rcaSpec","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#rca-spec","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"Let $x, y$ be $n$-bit vectors, $c$ a bit and $j$ a natural number, and write $j = 5i + r$ with $0 \\le r < 5$. Let $x_i$ and $y_i$ be bit $i$ of $x$ and $y$ (bit $0$ is the least significant, and bits at positions $i \\ge n$ are $0$), and let $c_i$ be the carry into position $i$ when $x$, $y$ and $c$ are added: reading $x$ and $y$ as natural numbers, $c_i = 1$ exactly when $(x \\bmod 2^i) + (y \\bmod 2^i) + c \\ge 2^i$.\u2026","title":"Gate-value specification of the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/carry-wire-value","kind":"node","lean":"Cslib.Circuits.carryWire_value","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#carry-wire-value","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit vectors, $c$ a bit, and $i$ and $g$ natural numbers with $5i \\le g$. Suppose the first $g$ gates are given values $v_0, \\dots, v_{g-1}$ with each $v_j$ equal to the specified value of gate $j$ of the adder on $x$, $y$, $c$. Then, on the adder input for $x$, $y$, $c$ and with these gate values, the carry wire into bit $i$ carries the carry into bit $i$ of $x + y + c$: it is true exactly\u2026","title":"Value of the carry wire"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/rca-program-eval","kind":"node","lean":"Cslib.Circuits.rcaProgram_eval","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#rca-program-eval","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit vectors and $c$ a bit. For every $g \\le 5n$ and every $j < g$, evaluating the first $g$ gates of the $n$-bit adder with the Boolean interpretation of the arithmetic gates on the adder input for $x$, $y$, $c$ gives gate $j$ its specified value on $x$, $y$, $c$.","title":"Gate values of the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/ripple-carry-adder-sum","kind":"node","lean":"Cslib.Circuits.rippleCarryAdder_sum","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#ripple-carry-adder-sum","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"Let $x, y$ be $n$-bit vectors, $c$ a bit and $i < n$, and evaluate the $n$-bit ripple-carry adder, with its gates read as Boolean XOR, AND and OR, on the input whose entry $j$ is bit $j$ of $x$ and entry $n + j$ is bit $j$ of $y$ for $j < n$ (bit $0$ being the least significant), and whose entry $2n$ is $c$. Its output number $i$ (outputs being numbered from $0$) is bit $i$ of the $n$-bit sum\u2026","title":"Sum bits of the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/ripple-carry-adder-carry","kind":"node","lean":"Cslib.Circuits.rippleCarryAdder_carry","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#ripple-carry-adder-carry","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"Let $x, y$ be $n$-bit vectors and $c$ a bit, and evaluate the $n$-bit ripple-carry adder, with its gates read as Boolean XOR, AND and OR, on the input whose entry $j$ is bit $j$ of $x$ and entry $n + j$ is bit $j$ of $y$ for $j < n$ (bit $0$ being the least significant), and whose entry $2n$ is $c$. Its output number $n$ (the last, outputs being numbered from $0$) is the carry out of $x + y + c$: reading $x$ and\u2026","title":"Carry out of the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/ripple-carry-adder/ripple-carry-adder-size","kind":"node","lean":"Cslib.Circuits.rippleCarryAdder_size","parent":35852,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/ripple-carry-adder/index.html#ripple-carry-adder-size","scope":"TheoryOfComputation/circuit-complexity/ripple-carry-adder","status":"fully_proved","summary":"The $n$-bit ripple-carry adder has exactly $5n$ gates.","title":"Size of the ripple-carry adder"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity/kogge-stone-adder.json","id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","kind":"scope","navigation":"module","parent":35833,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","title":"Kogge\u2013Stone adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/append-gates","kind":"node","lean":"Cslib.Circuits.appendGates","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#append-gates","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $p$ be a straight-line program with $g$ gates over $m$ inputs, let $c \\in \\mathbb{N}$, and for each $j < c$ let $f_j$ be a line (a gate operation with its argument wires) that reads the inputs and the first $g + j$ gates. Appending $f_0, \\dots, f_{c-1}$ to $p$ gives the program with $g + c$ gates whose first $g$ gates are those of $p$ and whose gate $g + j$ is $f_j$ for each $j < c$. It is defined by recursion\u2026","title":"Appending a block of gates"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/append-gates-eval-lt","kind":"node","lean":"Cslib.Circuits.appendGates_eval_lt","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#append-gates-eval-lt","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $p$ be a program with $g$ gates over $m$ inputs, let $c \\in \\mathbb{N}$, and for each $j < c$ let $f_j$ be a line that reads the inputs and the first $g + j$ gates. Under any interpretation of the gate symbols and any input values, for every $k < g$ the value of gate $k$ of the program obtained by appending $f_0, \\dots, f_{c-1}$ to $p$ equals the value of gate $k$ of $p$.","title":"Appending gates keeps the old gate values"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/append-gates-eval-le","kind":"node","lean":"Cslib.Circuits.appendGates_eval_le","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#append-gates-eval-le","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $p$ be a program with $g$ gates over $m$ inputs, let $c \\in \\mathbb{N}$, and for each $j \\le c$ let $f_j$ be a line that reads the inputs and the first $g + j$ gates. Under any interpretation of the gate symbols and any input values, for every $k < g + c$ the value of gate $k$ of the program obtained by appending $f_0, \\dots, f_c$ to $p$ equals the value of gate $k$ of the program obtained by appending\u2026","title":"Appending one more gate keeps the earlier gate values"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/append-gates-eval-ge","kind":"node","lean":"Cslib.Circuits.appendGates_eval_ge","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#append-gates-eval-ge","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $p$ be a program with $g$ gates over $m$ inputs, let $c \\in \\mathbb{N}$, and for each $j < c$ let $f_j$ be a line that reads the inputs and the first $g + j$ gates. Under any interpretation of the gate symbols and any input values $x$, for every $j < c$ the value of gate $g + j$ of the program obtained by appending $f_0, \\dots, f_{c-1}$ to $p$ is the value of the line $f_j$ on $x$ and the values of gates\u2026","title":"Values of appended gates"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/combine","kind":"node","lean":"Cslib.Circuits.CarryLookahead.combine","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#combine","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For pairs of Booleans $(G, P)$ and $(G', P')$, their combination is $(G \\lor (P \\land G'), P \\land P')$.","title":"Combining generate\u2013propagate pairs"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/gp-bit","kind":"node","lean":"Cslib.Circuits.CarryLookahead.gpBit","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#gp-bit","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For bit vectors $x, y$ of width $n$ and an index $i \\in \\mathbb{N}$, the generate\u2013propagate pair of bit $i$ is $(x_i \\wedge y_i,\\ x_i \\oplus y_i)$, where $x_i$ and $y_i$ are bit $i$ of $x$ and $y$, taken to be false when $i \\ge n$.","title":"Generate and propagate signals of a bit"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/gp-range","kind":"node","lean":"Cslib.Circuits.CarryLookahead.gpRange","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#gp-range","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For bit vectors $x, y$ of the same width, a start bit $\\ell$ and a length $m$, the generate\u2013propagate pair $\\mathrm{GP}(\\ell, m)$ of the block of bits $\\ell, \\dots, \\ell + m - 1$ is defined by recursion on $m$: the empty block has $\\mathrm{GP}(\\ell, 0) = (\\mathrm{false}, \\mathrm{true})$, and $\\mathrm{GP}(\\ell, m + 1) = \\mathrm{gp}(\\ell + m) \\circ \\mathrm{GP}(\\ell, m)$, where $\\mathrm{gp}(\\ell + m)$ is the\u2026","title":"Generate and propagate signals of a block"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/gp-range-add","kind":"node","lean":"Cslib.Circuits.CarryLookahead.gpRange_add","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#gp-range-add","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Fix bit vectors $x, y$ of the same width, write $\\mathrm{GP}(\\ell, m)$ for the generate\u2013propagate pair of the block of the $m$ bits $\\ell, \\dots, \\ell + m - 1$, and write $h \\circ l$ for the block operator with $h$ placed above $l$. Then for all natural numbers $\\ell, a, b$, $\\mathrm{GP}(\\ell, a + b) = \\mathrm{GP}(\\ell + a, b) \\circ \\mathrm{GP}(\\ell, a)$.","title":"Adjacent blocks compose"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-add","kind":"node","lean":"Cslib.Circuits.CarryLookahead.carry_add","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-add","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $x, y$ be $n$-bit vectors and $c$ a carry-in bit, and for $i \\in \\mathbb{N}$ let $c_i$ be the carry into bit $i$ when computing $x + y + c$, that is, $c_i$ holds exactly when $(x \\bmod 2^i) + (y \\bmod 2^i) + c \\ge 2^i$, where $x$, $y$ are read as natural numbers and $c$ counts as $1$ when true. For all $a, \\ell \\in \\mathbb{N}$, if $(G, P)$ are the generate and propagate signals of the block of bits\u2026","title":"The carry out of a block of bits"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/low-bit","kind":"node","lean":"Cslib.Circuits.CarryLookahead.lowBit","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#low-bit","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"The lowest bit of the block that bit $i$ holds after $k$ Kogge-Stone levels, $i + 1 - \\min(i + 1, 2^k)$, that is, $\\max(0, i + 1 - 2^k)$.","title":"Lowest bit of a Kogge-Stone block"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/gp-spec","kind":"node","lean":"Cslib.Circuits.CarryLookahead.gpSpec","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#gp-spec","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Fix bit vectors $x, y$ of the same width. For $k, i \\in \\mathbb{N}$, the pair $S_k(i)$ that bit $i$ holds after $k$ Kogge-Stone levels is the generate\u2013propagate pair of the block of the $\\min(i + 1, 2^k)$ bits ending at bit $i$, that is, of the bits $i + 1 - \\min(i + 1, 2^k), \\dots, i$.","title":"The block held after Kogge-Stone levels"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/gp-spec-succ","kind":"node","lean":"Cslib.Circuits.CarryLookahead.gpSpec_succ","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#gp-spec-succ","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Fix bit vectors $x, y$ of the same width and write $S_k(i)$ for the generate\u2013propagate pair of the block of the $\\min(i + 1, 2^k)$ bits ending at bit $i$, the pair bit $i$ holds after $k$ Kogge-Stone levels. For all $k, i \\in \\mathbb{N}$: if $2^k \\le i$, then $S_{k+1}(i) = S_k(i) \\circ S_k(i - 2^k)$, where $\\circ$ is the block operator with its left argument placed above its right; otherwise $S_{k+1}(i) = S_k(i)$.","title":"The Kogge-Stone step"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-of-gp-spec","kind":"node","lean":"Cslib.Circuits.CarryLookahead.carry_of_gpSpec","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-of-gp-spec","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $x, y$ be $n$-bit vectors, $c$ a carry-in bit, and $K$, $i$ natural numbers with $n \\le 2^K$ and $i < n$. Let $(G, P)$ be the generate and propagate signals that bit $i$ holds after $K$ Kogge\u2013Stone levels, those of the block of the at most $2^K$ bits ending at bit $i$. Then the carry into bit $i + 1$ of $x + y + c$, which is true exactly when $(x \\bmod 2^{i+1}) + (y \\bmod 2^{i+1}) + c \\ge 2^{i+1}$, equals\u2026","title":"Carries from the Kogge\u2013Stone prefixes"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/level-base","kind":"node","lean":"Cslib.Circuits.CarryLookahead.levelBase","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#level-base","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For an $n$-bit adder, the number $L_n(k)$ of gates in level $0$ together with the first $k$ Kogge-Stone levels is given by $L_n(0) = 2n$ and $L_n(k + 1) = L_n(k) + 3n$, so $L_n(k) = 2n + 3nk$.","title":"Gate count of the first Kogge-Stone levels"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/g-num","kind":"node","lean":"Cslib.Circuits.CarryLookahead.gNum","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#g-num","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For $n$-bit Kogge-Stone adders, the generate signal of bit $i$ after level $0$ is held by gate $2i$, and after Kogge-Stone level $k + 1$ by gate $2n + 3nk + 3i + 1$; here $2n + 3nk$ is the number of gates in levels $0, \\dots, k$.","title":"Index of the generate gate"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/p-num","kind":"node","lean":"Cslib.Circuits.CarryLookahead.pNum","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#p-num","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For natural numbers $n$, $k$ and $i$, $p_n(k, i)$ is the index of the gate holding the propagate signal of bit $i$ after level $k$ of the $n$-bit carry-lookahead adder: $p_n(0, i) = 2i + 1$, and $p_n(k + 1, i) = 2n + 3nk + 3i + 2$, where $2n + 3nk$ is the number of gates in levels $0, \\dots, k$.","title":"Propagate gate of a bit in the Kogge\u2013Stone adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/partner","kind":"node","lean":"Cslib.Circuits.CarryLookahead.partner","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#partner","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"The bit combined with bit $i$ at level $k + 1$ of the Kogge\u2013Stone adder is bit $i - 2^k$ if $2^k \\le i$, and bit $i$ itself otherwise.","title":"Kogge\u2013Stone partner bit"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/line0","kind":"node","lean":"Cslib.Circuits.CarryLookahead.line0","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#line0","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Gate $j < 2n$ of level $0$ of the $n$-bit adder of $x$ and $y$ reads the inputs $\\lfloor j/2 \\rfloor$ and $n + \\lfloor j/2 \\rfloor$, which carry bit $\\lfloor j/2 \\rfloor$ of $x$ and of $y$. For even $j$ it is an AND gate, the generate signal of that bit; for odd $j$ it is an XOR gate, its propagate signal.","title":"Gates of level 0"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/line-k","kind":"node","lean":"Cslib.Circuits.CarryLookahead.lineK","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#line-k","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Gate $j < 3n$ of Kogge-Stone level $k + 1$ of the $n$-bit adder serves bit $i = \\lfloor j/3 \\rfloor$ and its partner $q$, which is $i - 2^k$ if $2^k \\le i$ and $i$ otherwise. Write $G_b$ and $P_b$ for the generate and propagate gates of bit $b$ after level $k$. If $j \\equiv 0 \\pmod 3$ the gate computes $P_i \\wedge G_q$; if $j \\equiv 1 \\pmod 3$ it computes the new generate signal $G_i \\vee (P_i \\wedge G_q)$ from\u2026","title":"Gates of a Kogge-Stone level"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/levels","kind":"node","lean":"Cslib.Circuits.CarryLookahead.levels","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#levels","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For an $n$-bit adder, the program over the XOR, AND and OR gates on $2n + 1$ inputs obtained from the empty program by appending the $2n$ gates of level $0$ and then, in turn, the $3n$ gates of each Kogge-Stone level $1, \\dots, k$. It has $2n + 3nk$ gates.","title":"The first Kogge-Stone levels"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/line-f","kind":"node","lean":"Cslib.Circuits.CarryLookahead.lineF","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#line-f","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Gate $j < 3n$ of the final level of the $n$-bit adder with $K$ Kogge-Stone levels serves bit $i = \\lfloor j/3 \\rfloor$. Write $G_i$ and $P_i$ for the generate and propagate gates of bit $i$ after the $K$ levels, $p_i$ for the propagate gate of bit $i$ at level $0$, and $c$ for the carry input. If $j \\equiv 0 \\pmod 3$ the gate computes $P_i \\wedge c$; if $j \\equiv 1 \\pmod 3$ it computes the carry\u2026","title":"Gates of the final level"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/cla-program","kind":"node","lean":"Cslib.Circuits.CarryLookahead.claProgram","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#cla-program","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For natural numbers $n$ and $K$, the gate program of the $n$-bit adder with $K$ Kogge\u2013Stone levels is the straight-line program over the XOR/AND/OR basis on $2n + 1$ inputs (the $n$ bits of one summand, the $n$ bits of the other and a carry-in $c$) made of level $0$ and Kogge\u2013Stone levels $1, \\dots, K$, followed by a final level of $3n$ gates. For each bit $i < n$, let $G_i$ and $P_i$ be the generate and propagate\u2026","title":"Gates of the carry-lookahead adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-lookahead-adder","kind":"node","lean":"Cslib.Circuits.CarryLookahead.carryLookaheadAdder","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-lookahead-adder","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For natural numbers $n$ and $K$, the $n$-bit carry-lookahead adder with $K$ levels is the circuit over the XOR/AND/OR basis with $2n + 1$ inputs (the bits $x_0, \\dots, x_{n-1}$ of one summand, the bits $y_0, \\dots, y_{n-1}$ of the other, and a carry-in $c$) and $n + 1$ outputs, whose gates are those of the $n$-bit adder program with $K$ Kogge\u2013Stone levels. For $i < n$, output $i$ is the sum gate of bit $i$ in the\u2026","title":"Kogge\u2013Stone carry-lookahead adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/cla-levels","kind":"node","lean":"Cslib.Circuits.CarryLookahead.claLevels","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#cla-levels","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For a natural number $n$, the number of Kogge\u2013Stone levels used for $n$ bits is $\\lfloor \\log_2 n \\rfloor + 1$, with $\\lfloor \\log_2 0 \\rfloor = 0$.","title":"Number of carry-lookahead levels"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/levels-zero-eval","kind":"node","lean":"Cslib.Circuits.CarryLookahead.levels_zero_eval","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#levels-zero-eval","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit vectors and $c$ a carry bit, and evaluate level $0$ of the adder on the bits of $x$, the bits of $y$ and $c$. For every bit $i < n$, the gate holding the generate signal of bit $i$ carries the generate signal, and the gate holding its propagate signal carries the propagate signal, of the block consisting of bit $i$ alone, namely $x_i \\wedge y_i$ and $x_i \\oplus y_i$.","title":"Level 0 computes generate and propagate"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/levels-eval","kind":"node","lean":"Cslib.Circuits.CarryLookahead.levels_eval","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#levels-eval","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit vectors and $c$ a carry bit, and evaluate level $0$ and the first $k$ Kogge-Stone levels of the adder on the bits of $x$, the bits of $y$ and $c$. For every bit $i < n$, the gate holding the generate signal of bit $i$ carries the generate signal, and the gate holding its propagate signal carries the propagate signal, of the block of the at most $2^k$ bits ending at $i$, that is, of the\u2026","title":"Each Kogge-Stone level is correct"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/levels-eval-base","kind":"node","lean":"Cslib.Circuits.CarryLookahead.levels_eval_base","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#levels-eval-base","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For the $n$-bit adder, every interpretation of the XOR, AND and OR gates in a type $U$, every assignment of values in $U$ to the $2n + 1$ inputs, and every $K$, each of the $2n$ gates of level $0$ has the same value in the program with level $0$ and $K$ Kogge-Stone levels as in the program with level $0$ alone.","title":"Level-0 gates are unchanged by later levels"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/cla-program-carry","kind":"node","lean":"Cslib.Circuits.CarryLookahead.claProgram_carry","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#cla-program-carry","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $x, y$ be $n$-bit vectors, $c$ a carry-in bit and $K$ a natural number with $n \\le 2^K$. Evaluate the gate program of the $n$-bit adder with $K$ Kogge\u2013Stone levels in the Boolean interpretation of the XOR/AND/OR basis, on the input made of the bits of $x$, the bits of $y$ and $c$. Then for every $i < n$, gate $3i + 1$ of the final level holds the carry into bit $i + 1$ of $x + y + c$, which is true exactly when\u2026","title":"The carry gates"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/cla-program-sum","kind":"node","lean":"Cslib.Circuits.CarryLookahead.claProgram_sum","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#cla-program-sum","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $x, y$ be $n$-bit vectors with bits $x_i$, $y_i$, $c$ a carry-in bit and $K$ a natural number with $n \\le 2^K$. Evaluate the gate program of the $n$-bit adder with $K$ Kogge\u2013Stone levels in the Boolean interpretation of the XOR/AND/OR basis, on the input made of the bits of $x$, the bits of $y$ and $c$. Then for every $i < n$, gate $3i + 2$ of the final level is $x_i \\oplus y_i \\oplus c_i$, where $c_i$ is the\u2026","title":"The sum gates"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-lookahead-adder-sum","kind":"node","lean":"Cslib.Circuits.CarryLookahead.carryLookaheadAdder_sum","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-lookahead-adder-sum","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $x, y$ be $n$-bit vectors, $c$ a carry-in bit and $K$ a natural number with $n \\le 2^K$. Evaluate the $n$-bit Kogge\u2013Stone carry-lookahead adder with $K$ levels in the Boolean interpretation of the XOR/AND/OR basis, on the input made of the bits of $x$, the bits of $y$ and $c$. Then for every $i < n$, output $i$ of the adder is bit $i$ (the bit of weight $2^i$) of the sum $x + y + c$ taken modulo $2^n$.","title":"Sum bits of the carry-lookahead adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-lookahead-adder-carry","kind":"node","lean":"Cslib.Circuits.CarryLookahead.carryLookaheadAdder_carry","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-lookahead-adder-carry","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $n, K \\in \\mathbb{N}$ with $n \\le 2^K$, let $x, y$ be $n$-bit vectors and let $c$ be a carry-in bit. Evaluate the $n$-bit Kogge-Stone carry-lookahead adder with $K$ levels over the Boolean XOR/AND/OR basis on the input that lists the bits of $x$ from least to most significant, then those of $y$ in the same order, then $c$. Its output $n$, the last of its $n + 1$ outputs numbered from $0$, is the carry out of\u2026","title":"Carry out of the carry-lookahead adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-lookahead-adder-size","kind":"node","lean":"Cslib.Circuits.CarryLookahead.carryLookaheadAdder_size","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-lookahead-adder-size","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For all $n, K \\in \\mathbb{N}$, the $n$-bit Kogge-Stone carry-lookahead adder with $K$ levels has exactly $5n + 3nK$ gates: $2n$ in the first level, $3n$ in each of the $K$ Kogge-Stone levels, and $3n$ in the final level.","title":"Gate count of the carry-lookahead adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/depth-interpretation","kind":"node","lean":"Cslib.Circuits.depthInterpretation","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#depth-interpretation","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For a signature $\\sigma$, the depth interpretation is the interpretation of $\\sigma$ in $\\mathbb{N}$ that sends an operation symbol of arity $k$, applied to argument depths $d_0, \\dots, d_{k-1}$, to $\\max(0, d_0, \\dots, d_{k-1}) + 1$: a gate is one deeper than its deepest argument, and a gate without arguments has depth $1$.","title":"The depth interpretation"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/depths-eq-eval","kind":"node","lean":"Cslib.Circuits.Program.depths_eq_eval","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#depths-eq-eval","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For every program $p$ over a signature $\\sigma$, the depth of each gate of $p$ equals the value of that gate when $p$ is evaluated under the depth interpretation of $\\sigma$ with every input set to $0$.","title":"Gate depths are evaluation in the depth interpretation"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/depth-interpretation-arith","kind":"node","lean":"Cslib.Circuits.depthInterpretation_arith","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#depth-interpretation-arith","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For each gate of the arithmetic basis (XOR, AND or OR) and all argument depths $d_0, d_1 \\in \\mathbb{N}$, the depth interpretation of the arithmetic signature assigns the gate the depth $\\max(d_0, d_1) + 1$.","title":"Depth of an arithmetic gate"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/levels-depth","kind":"node","lean":"Cslib.Circuits.CarryLookahead.levels_depth","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#levels-depth","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"In the program made of level $0$ and the first $k$ Kogge-Stone levels of the $n$-bit adder, where inputs have depth $0$ and a gate is one deeper than its deepest argument, for every bit $i < n$ the gate holding the generate signal of bit $i$ has depth $2k + 1$ and the gate holding its propagate signal has depth $k + 1$.","title":"Depths after each level"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/cla-program-carry-depth","kind":"node","lean":"Cslib.Circuits.CarryLookahead.claProgram_carry_depth","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#cla-program-carry-depth","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For all natural numbers $n$, $K$ and $i < n$, in the gate program of the $n$-bit adder with $K$ Kogge\u2013Stone levels, the carry gate of bit $i$ in the final level (gate $3i + 1$ of that level) has depth $\\max(2K + 1, K + 2) + 1$, where inputs have depth $0$ and each gate is one deeper than its deepest argument.","title":"Depth of the carry gates"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/cla-program-sum-depth","kind":"node","lean":"Cslib.Circuits.CarryLookahead.claProgram_sum_depth","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#cla-program-sum-depth","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For all natural numbers $n$, $K$ and $i < n$, in the gate program of the $n$-bit adder with $K$ Kogge\u2013Stone levels, the sum gate of bit $i$ in the final level (gate $3i + 2$ of that level) has depth $2$ if $i = 0$ and depth $\\max(2K + 1, K + 2) + 2$ otherwise, where inputs have depth $0$ and each gate is one deeper than its deepest argument.","title":"Depth of the sum gates"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-lookahead-adder-depth","kind":"node","lean":"Cslib.Circuits.CarryLookahead.carryLookaheadAdder_depth","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-lookahead-adder-depth","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For all $K \\ge 1$ and $n \\ge 2$, the $n$-bit Kogge-Stone carry-lookahead adder with $K$ levels has depth exactly $2K + 3$, the depth being the largest depth of an output with the inputs at depth $0$.","title":"Exact depth of the carry-lookahead adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-lookahead-adder-depth-cla-levels","kind":"node","lean":"Cslib.Circuits.CarryLookahead.carryLookaheadAdder_depth_claLevels","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-lookahead-adder-depth-cla-levels","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For every $n \\ge 2$, the $n$-bit Kogge-Stone carry-lookahead adder with $\\lfloor \\log_2 n \\rfloor + 1$ levels has depth exactly $2 \\lfloor \\log_2 n \\rfloor + 5$, the depth being the largest depth of an output with the inputs at depth $0$.","title":"Depth of the carry-lookahead adder with logarithmically many levels"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/carry-depth","kind":"node","lean":"Cslib.Circuits.RippleCarry.carryDepth","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#carry-depth","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For $i \\in \\mathbb{N}$, the carry depth of bit $i$ is $0$ if $i = 0$ and $2i + 1$ if $i \\ge 1$. It is used as the depth of the carry into bit $i$ of the ripple-carry adder, where the carry into bit $0$ is the carry-in input.","title":"Carry depth in the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/gate-depth","kind":"node","lean":"Cslib.Circuits.RippleCarry.gateDepth","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#gate-depth","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"Let $c(i)$ be the carry depth of bit $i$: $c(0) = 0$ and $c(i) = 2i + 1$ for $i \\ge 1$. The formula assigns to gate $j$ of the ripple-carry adder, for every $j \\in \\mathbb{N}$, the depth $1$ when $j \\bmod 5$ is $0$ or $2$, the depth $\\max(1, c(\\lfloor j/5 \\rfloor)) + 1$ when $j \\bmod 5$ is $1$ or $3$, and the depth $c(\\lfloor j/5 \\rfloor + 1)$ when $j \\bmod 5 = 4$.","title":"Gate-depth formula for the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/rca-program-depth","kind":"node","lean":"Cslib.Circuits.RippleCarry.rcaProgram_depth","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#rca-program-depth","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For natural numbers $n$, $g \\le 5n$ and $j < g$, evaluating the first $g$ gates of the $n$-bit ripple-carry adder in $\\mathbb{N}$, with every input at depth $0$ and every gate one deeper than its deepest argument, gives gate $j$ the depth $d(j)$. Here, with $q = \\lfloor j / 5 \\rfloor$, $c(0) = 0$ and $c(i) = 2i + 1$ for $i \\ge 1$, $d(j) = 1$ if $j \\bmod 5 \\in \\{0, 2\\}$, $d(j) = \\max(1, c(q)) + 1$ if\u2026","title":"Gate depths of the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/kogge-stone-adder/ripple-carry-adder-depth","kind":"node","lean":"Cslib.Circuits.rippleCarryAdder_depth","parent":35868,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/kogge-stone-adder/index.html#ripple-carry-adder-depth","scope":"TheoryOfComputation/circuit-complexity/kogge-stone-adder","status":"fully_proved","summary":"For every $n \\ge 1$, the $n$-bit ripple-carry adder has depth $2n + 1$, the length of its carry chain.","title":"The ripple-carry adder has linear depth"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound.json","id":"TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound","kind":"scope","navigation":"module","parent":35833,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound/index.html","scope":"TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound","status":"can_prove","title":"Shannon's circuit-size lower bound"},{"id":"TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound/wire-count","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ShannonHardFunction.wireCount","parent":35913,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound/index.html#wire-count","scope":"TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound","status":"fully_proved","summary":"The wire count of a flat Boolean circuit on $n$ inputs, given as a list of gates, is the sum over its gates of the lengths of their predecessor lists: every reference of a gate to a gate index is counted, with multiplicity, while input variables and constants contribute nothing and the modulus of a mod gate is not counted.","title":"Wire count of a flat circuit"},{"id":"TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound/shannon-hard-function","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ShannonHardFunction.shannon_hard_function","parent":35913,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound/index.html#shannon-hard-function","scope":"TheoryOfComputation/circuit-complexity/shannon-s-circuit-size-lower-bound","status":"can_prove","summary":"For every natural number $n \\ge 10$, at least $2^{2^n}/2$ of the Boolean functions $f$ on $n$ input bits have the following property: for every well-formed flat Boolean circuit $C$ on $n$ inputs without mod gates whose wire count (the total length of the predecessor lists of its gates) is at most $\\lfloor 2^n / (10 n) \\rfloor$, there is an input $x \\in \\{0,1\\}^n$ such that $C$ has no gates or the value of the last\u2026","title":"Shannon: most Boolean functions need exponential circuits"},{"data_url":"scopes/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow.json","id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","kind":"scope","navigation":"module","parent":35833,"result":false,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","title":"Two's complement subtraction and overflow"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/adc-not-true","kind":"node","lean":"Cslib.Circuits.adc_not_true","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#adc-not-true","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit words and let $\\bar y$ be the bitwise complement of $y$. Adding $x$ and $\\bar y$ with carry in $1$ gives the $n$-bit result $x - y$ (the difference modulo $2^n$), and the carry out is $1$ exactly when there is no unsigned borrow, that is, when $y \\le x$ as unsigned integers.","title":"Subtraction by addition"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/ripple-carry-adder-sub","kind":"node","lean":"Cslib.Circuits.rippleCarryAdder_sub","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#ripple-carry-adder-sub","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit words and let $\\bar y$ be the bitwise complement of $y$. Evaluate the $n$-bit ripple-carry adder, with XOR, AND and OR gates, on the bits of $x$, the bits of $\\bar y$ and carry in $1$. Then for every $i < n$, output $i$ is bit $i$ (counting from the least significant bit $0$) of $x - y$ modulo $2^n$.","title":"The ripple-carry adder subtracts"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/ripple-carry-adder-no-borrow","kind":"node","lean":"Cslib.Circuits.rippleCarryAdder_noBorrow","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#ripple-carry-adder-no-borrow","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit words and let $\\bar y$ be the bitwise complement of $y$. Evaluate the $n$-bit ripple-carry adder, with XOR, AND and OR gates, on the bits of $x$, the bits of $\\bar y$ and carry in $1$. Its carry out (output $n$) is $1$ exactly when there is no unsigned borrow in $x - y$, that is, when $y \\le x$ as unsigned integers.","title":"Carry out of ripple-carry subtraction"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/carry-lookahead-adder-sub","kind":"node","lean":"Cslib.Circuits.carryLookaheadAdder_sub","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#carry-lookahead-adder-sub","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit words, let $\\bar y$ be the bitwise complement of $y$, and let $K$ satisfy $n \\le 2^K$. Evaluate the $n$-bit carry-lookahead adder with $K$ levels, with XOR, AND and OR gates, on the bits of $x$, the bits of $\\bar y$ and carry in $1$. Then for every $i < n$, output $i$ is bit $i$ (counting from the least significant bit $0$) of $x - y$ modulo $2^n$.","title":"The carry-lookahead adder subtracts"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/sadd-overflow-eq-carry-xor","kind":"node","lean":"Cslib.Circuits.saddOverflow_eq_carry_xor","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#sadd-overflow-eq-carry-xor","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"Let $n \\ge 1$ and let $x$ and $y$ be $n$-bit words. The two's complement addition $x + y$ overflows (the sum of the two's complement values of $x$ and $y$ is at least $2^{n-1}$ or less than $-2^{n-1}$) exactly when the carry into the sign bit $n - 1$ differs from the carry out of it, both carries taken when adding $x$ and $y$ with carry in $0$.","title":"Signed overflow is the XOR of the last two carries"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/overflow-adder","kind":"node","lean":"Cslib.Circuits.overflowAdder","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#overflow-adder","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"The $n$-bit overflow adder is the $n$-bit ripple-carry adder with one extra XOR gate. It has the same $2n + 1$ inputs (the bits of the two summands, then the carry in) and $5n + 1$ gates: the $5n$ gates of the ripple-carry adder, followed by an XOR of the carry into bit $n - 1$ and the carry into bit $n$ (the carry out). Here the carry into bit $0$ is the carry in, and for $n = 0$ the index $n - 1$ is read as $0$.\u2026","title":"The overflow adder"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/overflow-adder-eval-of-lt","kind":"node","lean":"Cslib.Circuits.overflowAdder_eval_of_lt","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#overflow-adder-eval-of-lt","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit words and let $c$ be a carry-in bit. Evaluate the $n$-bit overflow adder and the $n$-bit ripple-carry adder, with XOR, AND and OR gates, on the bits of $x$, the bits of $y$ and $c$. Then for every $o < n + 1$, output $o$ of the overflow adder equals output $o$ of the ripple-carry adder: the first $n + 1$ outputs (the $n$ sum bits and the carry out) agree.","title":"The overflow adder extends the ripple-carry adder"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/overflow-adder-overflow","kind":"node","lean":"Cslib.Circuits.overflowAdder_overflow","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#overflow-adder-overflow","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"Let $n \\ge 1$ and let $x$ and $y$ be $n$-bit words. Evaluate the $n$-bit overflow adder, with XOR, AND and OR gates, on the bits of $x$, the bits of $y$ and carry in $0$. Its last output (output $n + 1$) is $1$ exactly when the two's complement addition $x + y$ overflows, that is, when the sum of the two's complement values of $x$ and $y$ is at least $2^{n-1}$ or less than $-2^{n-1}$.","title":"The overflow flag"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/overflow-adder-sum","kind":"node","lean":"Cslib.Circuits.overflowAdder_sum","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#overflow-adder-sum","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"Let $x$ and $y$ be $n$-bit words. Evaluate the $n$-bit overflow adder, with XOR, AND and OR gates, on the bits of $x$, the bits of $y$ and carry in $0$. Then for every $i < n$, output $i$ is bit $i$ (counting from the least significant bit $0$) of $x + y$ modulo $2^n$.","title":"Sum bits of the overflow adder"},{"id":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/overflow-adder-size","kind":"node","lean":"Cslib.Circuits.overflowAdder_size","parent":35916,"result":true,"route":"../roadmap/TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow/index.html#overflow-adder-size","scope":"TheoryOfComputation/circuit-complexity/two-s-complement-subtraction-and-overflow","status":"fully_proved","summary":"The $n$-bit overflow adder has $5n + 1$ gates.","title":"Size of the overflow adder"},{"data_url":"scopes/TheoryOfComputation/algebraic-complexity.json","id":"TheoryOfComputation/algebraic-complexity","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/index.html","scope":"TheoryOfComputation/algebraic-complexity","status":"can_prove","summary":"Arithmetic circuits over a field and the complexity of algebraic problems: matrix multiplication exponents (square, rectangular, dual) and their bounds, tensor rank, and the bit complexity of integer multiplication.","title":"Algebraic complexity"},{"data_url":"scopes/TheoryOfComputation/algebraic-complexity/arithmetic-circuits.json","id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","kind":"scope","navigation":"module","parent":35927,"result":false,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","title":"Arithmetic circuits"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/arithmetic-operand","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticOperand","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#arithmetic-operand","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"An arithmetic operand over $F$ with inputs indexed by $A$ and $n$ nodes is a constant $x \\in F$, an input $a \\in A$, or a reference to a node $i < n$.","title":"Arithmetic operand"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/arithmetic-gate","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticGate","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#arithmetic-gate","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"An arithmetic gate over $F$ with inputs indexed by $A$ and $n$ prior nodes is an addition, a subtraction or a multiplication of two arithmetic operands, each of which is a constant in $F$, an input $a \\in A$ or a reference to a node $i < n$.","title":"Arithmetic gate"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/arithmetic-circuit","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticCircuit","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#arithmetic-circuit","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"An arithmetic circuit over $F$ with inputs indexed by $A$ and outputs indexed by $B$ consists of a size $s \\in \\mathbb{N}$, a gate $g_i$ for each $i < s$, and an output operand for each $b \\in B$. Each gate $g_i$ is an addition, subtraction or multiplication of two operands, each a constant in $F$, an input $a \\in A$ or an earlier gate $g_j$ with $j < i$; an output operand is a constant in $F$, an input or any gate\u2026","title":"Arithmetic circuit"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/evaluate","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticOperand.evaluate","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#evaluate","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"Let $F$ and $A$ be types and $n \\in \\mathbb{N}$. Given a value in $F$ for each input $a \\in A$ and a value in $F$ for each node $i < n$, an arithmetic operand over $F$ with inputs indexed by $A$ and $n$ nodes evaluates as follows: a constant $x$ evaluates to $x$, an input $a$ to its assigned value, and a reference to node $i$ to the value given for node $i$.","title":"Operand evaluation"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/evaluate-2","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticGate.evaluate","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#evaluate-2","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"Let $F$ be a type with addition, subtraction and multiplication, let $A$ be a type and let $n \\in \\mathbb{N}$. Given a value in $F$ for each input $a \\in A$ and a value in $F$ for each node $i < n$, write $v(x)$ for the value of an arithmetic operand $x$: a constant evaluates to itself, an input $a$ to its assigned value and a reference to node $i$ to the value given for node $i$. An arithmetic gate over $F$ with\u2026","title":"Gate evaluation"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/ext","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticCircuit.ext","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#ext","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"Two arithmetic circuits over $F$ with inputs indexed by $A$ and outputs indexed by $B$ are equal if they have the same size, the same gate at each position and the same output operand for each $b \\in B$.","title":"Extensionality for arithmetic circuits"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/evaluate-node","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticCircuit.evaluateNode","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#evaluate-node","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"Let $F$ be a type with addition, subtraction and multiplication and let $c$ be an arithmetic circuit over $F$ with inputs indexed by $A$. Given a value in $F$ for each input $a \\in A$, the value of gate $g_i$ of $c$ is the value of the operation of $g_i$ on its two operands, where a constant $x$ has value $x$, an input has its assigned value and a reference to an earlier gate $g_j$ with $j < i$ takes the value of\u2026","title":"Value of a gate in a circuit"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/evaluate-3","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticCircuit.evaluate","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#evaluate-3","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"Let $F$ be a type with addition, subtraction and multiplication and let $c$ be an arithmetic circuit over $F$ with inputs indexed by $A$ and outputs indexed by $B$. Given a value in $F$ for each input $a \\in A$, the value of $c$ at an output $b \\in B$ is the value of the output operand of $b$, where a constant $x$ has value $x$, an input has its assigned value and a reference to gate $g_j$ has the value of $g_j$,\u2026","title":"Circuit evaluation"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/outer-product","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticCircuit.outerProduct","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#outer-product","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"For a type $F$ of scalars and $m, n \\in \\mathbb{N}$, the outer-product circuit has inputs $a_0, \\dots, a_{m-1}$ and $b_0, \\dots, b_{n-1}$, one output for each pair $(i, j)$ with $i < m$ and $j < n$, and exactly $mn$ gates: the gate for $(i, j)$ multiplies $a_i$ by $b_j$, and the output at $(i, j)$ is that gate.","title":"Outer-product circuit"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/outer-product-evaluate","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticCircuit.outerProduct_evaluate","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#outer-product-evaluate","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"Let $F$ be a type with addition, subtraction and multiplication. For all $m, n \\in \\mathbb{N}$, $a \\in F^m$, $b \\in F^n$, $i < m$ and $j < n$, the outer-product circuit over $F$ evaluated on the inputs $a$ and $b$ has value $a_i b_j$ at the output $(i, j)$.","title":"Correctness of the outer-product circuit"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits/outer-product-size","kind":"node","lean":"Cslib.Complexity.Algebraic.ArithmeticCircuit.outerProduct_size","parent":35928,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits/index.html#outer-product-size","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits","status":"fully_proved","summary":"For every $F$ and all $m, n \\in \\mathbb{N}$, the outer-product circuit over $F$ for $m \\times n$ outer products has exactly $mn$ gates.","title":"Size of the outer-product circuit"},{"data_url":"scopes/TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n.json","id":"TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n","kind":"scope","navigation":"module","parent":35927,"result":false,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n/index.html","scope":"TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n","status":"can_prove","title":"Integer multiplication in time O(n log n)"},{"id":"TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n/encode-prod","kind":"node","lean":"MathlibExt.TheoryOfComputation.IntegerMultiplicationWanted.IntegerMultiplication.encodeProd","parent":35940,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n/index.html#encode-prod","scope":"TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n","status":"fully_proved","summary":"Given encodings $e_A : A \\to \\Gamma_A^*$ and $e_B : B \\to \\Gamma_B^*$ of types $A$ and $B$ as lists over alphabets $\\Gamma_A$ and $\\Gamma_B$, and a new symbol $\\bot$ in neither alphabet, a pair $(a, b)$ is encoded as a list over the alphabet $(\\Gamma_A \\cup \\{\\bot\\}) \\times (\\Gamma_B \\cup \\{\\bot\\})$ by zipping: its length is the larger of the lengths of $e_A(a)$ and $e_B(b)$, and its $i$-th entry is the pair of the\u2026","title":"Encoding of pairs"},{"id":"TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n/encode-nat-binary","kind":"node","lean":"MathlibExt.TheoryOfComputation.IntegerMultiplicationWanted.IntegerMultiplication.encodeNatBinary","parent":35940,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n/index.html#encode-nat-binary","scope":"TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n","status":"fully_proved","summary":"The binary encoding of a natural number $n$ is the list of its base-$2$ digits, least significant first, with the digit $1$ written as the Boolean value true and the digit $0$ as false. The list has no trailing zero digit, so $0$ is encoded by the empty list.","title":"Binary encoding of natural numbers"},{"id":"TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n/theorem-1-1","kind":"node","lean":"MathlibExt.TheoryOfComputation.IntegerMultiplicationWanted.IntegerMultiplication.theorem_1_1","parent":35940,"reading_title":{"format":"inline-parts-v1","parts":[["text","Harvey\u2013van der Hoeven: integer multiplication in time "],["math","\\(O(n \\log n)\\)"]],"source":"Harvey\u2013van der Hoeven: integer multiplication in time $O(n \\log n)$"},"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n/index.html#theorem-1-1","scope":"TheoryOfComputation/algebraic-complexity/integer-multiplication-in-time-o-n-log-n","status":"can_prove","summary":"There is a multi-stack Turing machine with finite alphabets and a function $t : \\mathbb{N} \\to \\mathbb{N}$ such that, for all natural numbers $x$ and $y$, the machine started on the encoding of the pair $(x, y)$ (the binary encodings of $x$ and $y$, zipped) halts within $t(\\ell)$ steps with the binary encoding of $xy$, where $\\ell$ is the length of the input, and $t(n) = O(n \\log n)$ as $n \\to \\infty$.","title":"Harvey\u2013van der Hoeven: integer multiplication in time $O(n \\log n)$"},{"data_url":"scopes/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication.json","id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","kind":"scope","navigation":"module","parent":35927,"result":false,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","title":"Arithmetic circuits for matrix multiplication"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/matrix-arithmetic-circuit","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixArithmeticCircuit","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#matrix-arithmetic-circuit","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For a type $F$ of scalars and $m, r, n \\in \\mathbb{N}$, a matrix arithmetic circuit is an arithmetic circuit over $F$ whose inputs are indexed by the entries $(i, k)$ of an $m \\times r$ matrix together with the entries $(k, j)$ of an $r \\times n$ matrix, and whose outputs are indexed by the entries $(i, j)$ of an $m \\times n$ matrix.","title":"Matrix arithmetic circuit"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/matrix-input","kind":"node","lean":"Cslib.Complexity.Algebraic.matrixInput","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#matrix-input","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For matrices $A \\in F^{m \\times r}$ and $B \\in F^{r \\times n}$, the input assignment of $(A, B)$ is the function on the entry positions $(i, k)$ of an $m \\times r$ matrix together with the entry positions $(k, j)$ of an $r \\times n$ matrix that sends $(i, k)$ to $A_{ik}$ and $(k, j)$ to $B_{kj}$. It supplies the inputs of a matrix arithmetic circuit.","title":"Matrix input assignment"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/is-matrix-multiplication","kind":"node","lean":"Cslib.Complexity.Algebraic.IsMatrixMultiplication","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#is-matrix-multiplication","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"A circuit $c$ over a field $F$ whose inputs are the entries of an $m \\times r$ matrix and an $r \\times n$ matrix computes matrix multiplication when, for all matrices $A \\in F^{m \\times r}$ and $B \\in F^{r \\times n}$ and all $i < m$, $j < n$, its output at $(i, j)$ on the entries of $A$ and $B$ equals $(AB)_{ij}$.","title":"Circuits that compute the matrix product"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/outer-matrix-circuit","kind":"node","lean":"Cslib.Complexity.Algebraic.outerMatrixCircuit","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#outer-matrix-circuit","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For a type $F$ of scalars and $m, n \\in \\mathbb{N}$, the outer-product matrix circuit is the matrix arithmetic circuit for an $m \\times 1$ matrix times a $1 \\times n$ matrix with exactly $mn$ gates: for $i < m$ and $j < n$, the gate for $(i, j)$ multiplies the input at entry $(i, 0)$ of the first matrix by the input at entry $(0, j)$ of the second, and the output at $(i, j)$ is that gate.","title":"Outer-product matrix circuit"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/outer-matrix-circuit-correct","kind":"node","lean":"Cslib.Complexity.Algebraic.outerMatrixCircuit_correct","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#outer-matrix-circuit-correct","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every field $F$ and all $m, n \\in \\mathbb{N}$, the outer-product matrix circuit over $F$ computes the product of an $m \\times 1$ matrix and a $1 \\times n$ matrix.","title":"The outer-product matrix circuit computes the matrix product"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/outer-matrix-circuit-size","kind":"node","lean":"Cslib.Complexity.Algebraic.outerMatrixCircuit_size","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#outer-matrix-circuit-size","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every type $F$ of scalars and all $m, n \\in \\mathbb{N}$, the outer-product matrix circuit over $F$ has exactly $mn$ gates.","title":"Size of the outer-product matrix circuit"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/output-is-node","kind":"node","lean":"Cslib.Complexity.Algebraic.IsMatrixMultiplication.output_is_node","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#output-is-node","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and let $c$ be a circuit over $F$ that computes the product of an $m \\times r$ matrix and an $r \\times n$ matrix, where $r \\ge 1$. Then for all $i < m$ and $j < n$, the output of $c$ at $(i, j)$ is one of its gates, not a constant or an input.","title":"Outputs of a matrix multiplication circuit are gates"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/output-cost-le","kind":"node","lean":"Cslib.Complexity.Algebraic.IsMatrixMultiplication.output_cost_le","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#output-cost-le","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and let $c$ be a circuit over $F$ that computes the product of an $m \\times r$ matrix and an $r \\times n$ matrix, where $r \\ge 1$. Then $c$ has at least $mn$ gates.","title":"Lower bound for matrix multiplication circuits"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/has-rectangular-circuit-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.HasRectangularCircuitExponent","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#has-rectangular-circuit-exponent","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and $k, w$ real numbers. The exponent $w$ is admissible at aspect $k$ over $F$ when $k \\ge 0$, $w \\ge 0$, and for every $\\varepsilon > 0$ there are a real $C > 0$ and a natural number $N \\ge 1$ such that for every $n \\ge N$ some arithmetic circuit over $F$ computes the product of an $n \\times \\lceil n^k \\rceil$ matrix and an $\\lceil n^k \\rceil \\times n$ matrix with at most $C n^{w + \\varepsilon}$\u2026","title":"Admissible rectangular circuit exponents"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/rectangular-circuit-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.rectangularCircuitExponent","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#rectangular-circuit-exponent","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For a field $F$ and a real $k$, the rectangular circuit exponent of $F$ at aspect $k$ is the infimum in $[0, \\infty]$ of the admissible rectangular circuit exponents for $F$ at aspect $k$: the reals $w \\ge 0$ such that $k \\ge 0$ and, for every $\\varepsilon > 0$, there are $C > 0$ and $N \\ge 1$ such that for every $n \\ge N$ some arithmetic circuit over $F$ of size at most $C n^{w + \\varepsilon}$ computes the product\u2026","title":"Rectangular circuit exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/dual-circuit-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.dualCircuitExponent","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#dual-circuit-exponent","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For a field $F$, the dual matrix multiplication exponent $\\alpha(F) \\in [0, \\infty]$ of the arithmetic-circuit model is the supremum of all real $k \\ge 0$ at which the rectangular matrix multiplication exponent of the arithmetic-circuit model over $F$ equals $2$.","title":"Dual matrix multiplication exponent (arithmetic circuits)"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/rectangular-circuit-exponent-le-of-admis","kind":"node","lean":"Cslib.Complexity.Algebraic.rectangularCircuitExponent_le_of_admissible","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#rectangular-circuit-exponent-le-of-admis","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and let $k$ and $w$ be reals. If $w$ is an admissible rectangular circuit exponent for $F$ at aspect $k$, then the rectangular circuit exponent of $F$ at aspect $k$ is at most $w$.","title":"Admissible exponents bound the rectangular circuit exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/le-rectangular-circuit-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.le_rectangularCircuitExponent","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#le-rectangular-circuit-exponent","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field, $k$ a real number and $W \\in [0, \\infty]$. If $W \\le w$ for every real $w$ that is an admissible exponent at aspect $k$ over $F$, then $W$ is at most the rectangular matrix multiplication exponent of the arithmetic-circuit model over $F$ at aspect $k$.","title":"Lower bounds for the rectangular circuit exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/le-dual-circuit-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.le_dualCircuitExponent","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#le-dual-circuit-exponent","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and $k \\ge 0$ a real number. If the rectangular matrix multiplication exponent of the arithmetic-circuit model over $F$ at aspect $k$ equals $2$, then $k \\le \\alpha(F)$, where $\\alpha(F)$ is the dual matrix multiplication exponent of the arithmetic-circuit model.","title":"Aspects with circuit exponent two bound the dual exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/dual-circuit-exponent-le","kind":"node","lean":"Cslib.Complexity.Algebraic.dualCircuitExponent_le","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#dual-circuit-exponent-le","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and $K \\in [0, \\infty]$. If $k \\le K$ for every real $k \\ge 0$ at which the rectangular matrix multiplication exponent of the arithmetic-circuit model over $F$ equals $2$, then the dual matrix multiplication exponent $\\alpha(F)$ of the arithmetic-circuit model satisfies $\\alpha(F) \\le K$.","title":"Upper bounds for the dual circuit exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/has-rectangular-circuit-exponent-zero-tw","kind":"node","lean":"Cslib.Complexity.Algebraic.hasRectangularCircuitExponent_zero_two","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#has-rectangular-circuit-exponent-zero-tw","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every field $F$, the exponent $2$ is an admissible rectangular circuit exponent at aspect $0$ over $F$: for every $\\varepsilon > 0$ there are a real $C > 0$ and a natural number $N \\ge 1$ such that for every $n \\ge N$ some arithmetic circuit over $F$ computes the product of an $n \\times \\lceil n^0 \\rceil$ matrix and a $\\lceil n^0 \\rceil \\times n$ matrix, that is of an $n \\times 1$ matrix and a $1 \\times n$\u2026","title":"Exponent two is admissible at aspect zero"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/rectangular-inner-dimension-pos","kind":"node","lean":"Cslib.Complexity.Algebraic.rectangularInnerDimension_pos","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#rectangular-inner-dimension-pos","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every natural number $n \\ge 1$ and every real $k$, the rounded inner dimension $\\lceil n^k \\rceil$ is positive.","title":"The rounded inner dimension is positive"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/two-le","kind":"node","lean":"Cslib.Complexity.Algebraic.HasRectangularCircuitExponent.two_le","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#two-le","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and let $k$ and $w$ be reals. If $w$ is an admissible rectangular circuit exponent for $F$ at aspect $k$, then $w \\ge 2$.","title":"Admissible rectangular circuit exponents are at least two"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/two-le-rectangular-circuit-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.two_le_rectangularCircuitExponent","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#two-le-rectangular-circuit-exponent","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every field $F$ and every real $k$, the rectangular circuit exponent of $F$ at aspect $k$ is at least $2$ (it may be $\\infty$).","title":"The rectangular circuit exponent is at least two"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/rectangular-circuit-exponent-zero","kind":"node","lean":"Cslib.Complexity.Algebraic.rectangularCircuitExponent_zero","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#rectangular-circuit-exponent-zero","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every field $F$, the rectangular circuit exponent of $F$ at aspect $0$ equals $2$.","title":"The rectangular circuit exponent at aspect zero is two"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/dual-circuit-exponent-family-nonempty","kind":"node","lean":"Cslib.Complexity.Algebraic.dualCircuitExponent_family_nonempty","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#dual-circuit-exponent-family-nonempty","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every field $F$, there is a real $k \\ge 0$ at which the rectangular matrix multiplication exponent of the arithmetic-circuit model over $F$ equals $2$. Hence the set of such $k$, viewed in $[0, \\infty]$, whose supremum is the dual matrix multiplication exponent, is nonempty.","title":"Some aspect has rectangular circuit exponent two"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/mono","kind":"node","lean":"Cslib.Complexity.Algebraic.HasRectangularCircuitExponent.mono","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#mono","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and $k, w, w'$ real numbers with $w \\le w'$. If $w$ is an admissible exponent at aspect $k$ over $F$, then so is $w'$.","title":"Admissible circuit exponents are upward closed"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/has-rectangular-circuit-exponent-of-inf","kind":"node","lean":"Cslib.Complexity.Algebraic.hasRectangularCircuitExponent_of_inf_lt","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#has-rectangular-circuit-exponent-of-inf","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and $k, w$ real numbers with $w > 0$. If the rectangular matrix multiplication exponent of the arithmetic-circuit model over $F$ at aspect $k$ is less than $w$, then $w$ is an admissible exponent at aspect $k$ over $F$.","title":"Exponents above the circuit exponent are admissible"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/rectangular-circuit-exponent-le-iff","kind":"node","lean":"Cslib.Complexity.Algebraic.rectangularCircuitExponent_le_iff","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#rectangular-circuit-exponent-le-iff","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and let $k \\ge 0$ and $w \\ge 0$ be reals. The rectangular circuit exponent of $F$ at aspect $k$ is at most $w$ if and only if $w$ is an admissible rectangular circuit exponent for $F$ at aspect $k$: for every $\\varepsilon > 0$ there are $C > 0$ and $N \\ge 1$ such that for every $n \\ge N$ some arithmetic circuit over $F$ of size at most $C n^{w + \\varepsilon}$ computes the product of an\u2026","title":"Upper bounds on the rectangular circuit exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/rectangular-circuit-exponent-endpoint","kind":"node","lean":"Cslib.Complexity.Algebraic.rectangularCircuitExponent_endpoint","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#rectangular-circuit-exponent-endpoint","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field and $k \\ge 0$ a real number. If the rectangular matrix multiplication exponent $\\omega$ of the arithmetic-circuit model over $F$ at aspect $k$ is finite, then $\\omega$ is itself an admissible exponent at aspect $k$ over $F$.","title":"A finite rectangular circuit exponent is attained"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/square-matrix-multiplication-circuit-exp","kind":"node","lean":"Cslib.Complexity.Algebraic.squareMatrixMultiplicationCircuitExponent","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#square-matrix-multiplication-circuit-exp","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For a field $F$, the square matrix multiplication circuit exponent of $F$ is the rectangular circuit exponent of $F$ at aspect $1$. Since $\\lceil n^1 \\rceil = n$, it is the infimum in $[0, \\infty]$ of the reals $w \\ge 0$ such that for every $\\varepsilon > 0$ there are $C > 0$ and $N \\ge 1$ such that for every $n \\ge N$ some arithmetic circuit over $F$ of size at most $C n^{w + \\varepsilon}$ computes the product of\u2026","title":"Square matrix multiplication circuit exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/square-matrix-multiplication-circuit-exp-2","kind":"node","lean":"Cslib.Complexity.Algebraic.squareMatrixMultiplicationCircuitExponent_def","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#square-matrix-multiplication-circuit-exp-2","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every field $F$, the square matrix multiplication circuit exponent of $F$ equals the rectangular circuit exponent of $F$ at aspect $1$.","title":"The square circuit exponent is the aspect-one rectangular exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/two-le-square-matrix-multiplication-circ","kind":"node","lean":"Cslib.Complexity.Algebraic.two_le_squareMatrixMultiplicationCircuitExponent","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#two-le-square-matrix-multiplication-circ","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"For every field $F$, the square matrix multiplication circuit exponent of $F$ is at least $2$.","title":"The square circuit exponent is at least two"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/square-matrix-multiplication-circuit-exp-3","kind":"node","lean":"Cslib.Complexity.Algebraic.squareMatrixMultiplicationCircuitExponent_le_iff","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#square-matrix-multiplication-circuit-exp-3","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field with square matrix multiplication circuit exponent $\\omega(F) \\in [0, \\infty]$, and let $w \\ge 0$ be real. Then $\\omega(F) \\le w$ if and only if $F$ has rectangular circuit exponent $w$ at aspect $1$: for every $\\varepsilon > 0$ there are a real $C > 0$ and a natural number $N \\ge 1$ such that for every $n \\ge N$ some arithmetic circuit over $F$ with at most $C n^{w + \\varepsilon}$ gates computes\u2026","title":"Upper bounds on the square matrix multiplication circuit exponent"},{"id":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/square-matrix-multiplication-circuit-exp-4","kind":"node","lean":"Cslib.Complexity.Algebraic.squareMatrixMultiplicationCircuitExponent_endpoint","parent":35944,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication/index.html#square-matrix-multiplication-circuit-exp-4","scope":"TheoryOfComputation/algebraic-complexity/arithmetic-circuits-for-matrix-multiplication","status":"fully_proved","summary":"Let $F$ be a field whose square matrix multiplication circuit exponent $\\omega(F)$ is finite. Then $F$ has rectangular circuit exponent $\\omega(F)$ at aspect $1$: for every $\\varepsilon > 0$ there are a real $C > 0$ and a natural number $N \\ge 1$ such that for every $n \\ge N$ some arithmetic circuit over $F$ with at most $C n^{\\omega(F) + \\varepsilon}$ gates computes the product of every pair of $n \\times n$\u2026","title":"The square matrix multiplication circuit exponent is attained"},{"data_url":"scopes/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent.json","id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","kind":"scope","navigation":"module","parent":35927,"result":false,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","title":"Matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/rank-one-trilinear-term","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.RankOneTrilinearTerm","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#rank-one-trilinear-term","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a type $F$ with multiplication and types $A$, $B$, $C$, a rank-one trilinear term is a triple of functions $u : A \\to F$, $v : B \\to F$, $w : C \\to F$ (its left, middle and right factors), representing one rank-one summand $u \\otimes v \\otimes w$ of a trilinear tensor decomposition.","title":"Rank-one trilinear term"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/eval","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.RankOneTrilinearTerm.eval","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#eval","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a rank-one trilinear term over a semigroup $F$ with factors $u : A \\to F$, $v : B \\to F$ and $w : C \\to F$, its value at $a \\in A$, $b \\in B$, $c \\in C$ is the product $u(a)\\,v(b)\\,w(c)$.","title":"Value of a rank-one trilinear term"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-trilinear-rank-at-most","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.HasTrilinearRankAtMost","parent":35975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Trilinear rank at most "],["math","\\(r\\)"]],"source":"Trilinear rank at most $r$"},"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-trilinear-rank-at-most","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a semiring and $A$, $B$, $C$ types. An array $T : A \\to B \\to C \\to F$ has trilinear rank at most $r$ when there are $s \\le r$ and functions $u_i : A \\to F$, $v_i : B \\to F$, $w_i : C \\to F$ for $i = 1, \\dots, s$ such that $T(a, b, c) = \\sum_{i=1}^{s} u_i(a)\\,v_i(b)\\,w_i(c)$ for all $a \\in A$, $b \\in B$, $c \\in C$.","title":"Trilinear rank at most $r$"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/mono","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.HasTrilinearRankAtMost.mono","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#mono","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a semiring and $T : A \\to B \\to C \\to F$ an array. If $T$ has trilinear rank at most $r$ and $r \\le s$, then $T$ has trilinear rank at most $s$.","title":"Monotonicity of trilinear rank bounds"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-trilinear-rank-at-most-card","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.hasTrilinearRankAtMost_card","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-trilinear-rank-at-most-card","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a semiring and $A$, $B$, $C$ finite types. Every array $T : A \\to B \\to C \\to F$ has trilinear rank at most $|A| \\cdot |B| \\cdot |C|$.","title":"Trivial upper bound on trilinear rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/trilinear-rank","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.trilinearRank","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#trilinear-rank","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a semiring and let $A$, $B$, $C$ be finite types. The trilinear rank of a coefficient array $T : A \\to B \\to C \\to F$ is the least $r \\in \\mathbb{N}$ for which there are $s \\le r$ functions $u_i : A \\to F$, $v_i : B \\to F$, $w_i : C \\to F$ with $T(a, b, c) = \\sum_{i=1}^{s} u_i(a)\\,v_i(b)\\,w_i(c)$ for all $a \\in A$, $b \\in B$, $c \\in C$.","title":"Trilinear rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/trilinear-rank-spec","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.trilinearRank_spec","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#trilinear-rank-spec","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a semiring, let $A$, $B$, $C$ be finite types, let $T : A \\to B \\to C \\to F$ be a coefficient array, and let $R(T)$ be its trilinear rank. Then there are $s \\le R(T)$ functions $u_i : A \\to F$, $v_i : B \\to F$, $w_i : C \\to F$ with $T(a, b, c) = \\sum_{i=1}^{s} u_i(a)\\,v_i(b)\\,w_i(c)$ for all $a \\in A$, $b \\in B$, $c \\in C$.","title":"The trilinear rank is attained"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/trilinear-rank-le","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.trilinearRank_le","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#trilinear-rank-le","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a semiring, let $A$, $B$, $C$ be finite types, and let $T : A \\to B \\to C \\to F$ be a coefficient array. If there are $s \\le r$ functions $u_i : A \\to F$, $v_i : B \\to F$, $w_i : C \\to F$ with $T(a, b, c) = \\sum_{i=1}^{s} u_i(a)\\,v_i(b)\\,w_i(c)$ for all $a \\in A$, $b \\in B$, $c \\in C$, then the trilinear rank of $T$ is at most $r$.","title":"Decompositions bound the trilinear rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-trilinear-rank-at-most-iff","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.hasTrilinearRankAtMost_iff","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-trilinear-rank-at-most-iff","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a semiring and $A$, $B$, $C$ finite types. For every array $T : A \\to B \\to C \\to F$ and every natural number $r$, $T$ has trilinear rank at most $r$ if and only if the trilinear rank of $T$ (the least length of a decomposition of $T$ into rank-one arrays) is at most $r$.","title":"Rank bounds via the trilinear rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/trilinear-tensor","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.TrilinearTensor","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#trilinear-tensor","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a field $F$ and types $A$, $B$, $C$, write $F^X$ for the $F$-vector space of functions $X \\to F$. The space of trilinear tensors over $F$ indexed by $A$, $B$, $C$ is the tensor product $(F^A \\otimes_F F^B) \\otimes_F F^C$ over $F$.","title":"Trilinear tensors"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/trilinear-tensor-coefficients","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.trilinearTensorCoefficients","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#trilinear-tensor-coefficients","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $A$ and $C$ be finite types, let $B$ be a type, and write $F^X$ for the space of functions $X \\to F$. The coefficient map is the $F$-linear isomorphism from $(F^A \\otimes_F F^B) \\otimes_F F^C$ to the space of arrays $A \\to B \\to C \\to F$ that sends each pure tensor $(u \\otimes v) \\otimes w$ to the array $(a, b, c) \\mapsto u(a)\\,v(b)\\,w(c)$.","title":"Coefficient array of a trilinear tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/trilinear-tensor-coefficients-tmul","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.trilinearTensorCoefficients_tmul","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#trilinear-tensor-coefficients-tmul","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $A$ and $C$ be finite types and let $B$ be a type. For $u : A \\to F$, $v : B \\to F$, $w : C \\to F$ and $a \\in A$, $b \\in B$, $c \\in C$, the coefficient array of the pure tensor $(u \\otimes v) \\otimes w$ takes the value $u(a)\\,v(b)\\,w(c)$ at $(a, b, c)$.","title":"Coefficients of a pure tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/tensor","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.RankOneTrilinearTerm.tensor","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#tensor","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and $A$, $B$, $C$ types. A rank-one trilinear term with factors $u : A \\to F$, $v : B \\to F$ and $w : C \\to F$ determines the pure tensor $(u \\otimes v) \\otimes w$ in the trilinear tensor space over $F$ indexed by $A$, $B$, $C$.","title":"The pure tensor of a rank-one term"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-tensor-rank-at-most","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.HasTensorRankAtMost","parent":35975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tensor rank at most "],["math","\\(r\\)"]],"source":"Tensor rank at most $r$"},"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-tensor-rank-at-most","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and $A$, $B$, $C$ types. An element $T$ of $(F^A \\otimes_F F^B) \\otimes_F F^C$ has tensor rank at most $r$ when there are $s \\le r$ and functions $u_i : A \\to F$, $v_i : B \\to F$, $w_i : C \\to F$ for $i = 1, \\dots, s$ such that $T = \\sum_{i=1}^{s} (u_i \\otimes v_i) \\otimes w_i$.","title":"Tensor rank at most $r$"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-tensor-rank-at-most-iff","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.hasTensorRankAtMost_iff","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-tensor-rank-at-most-iff","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and $A$, $B$, $C$ types with $A$ and $C$ finite, and let $T$ be an element of $(F^A \\otimes_F F^B) \\otimes_F F^C$. For every natural number $r$, $T$ has tensor rank at most $r$ if and only if its coefficient array $A \\to B \\to C \\to F$, obtained by the canonical linear equivalence, has trilinear rank at most $r$.","title":"Tensor rank bounds agree with coefficient rank bounds"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/tensor-rank","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.tensorRank","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#tensor-rank","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and let $A$, $B$, $C$ be finite types. The tensor rank of a trilinear tensor $T$ over $F$ indexed by $A$, $B$, $C$ is the least $r \\in \\mathbb{N}$ such that $T$ is a sum of at most $r$ pure tensors $(u \\otimes v) \\otimes w$ with $u : A \\to F$, $v : B \\to F$ and $w : C \\to F$.","title":"Tensor rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/tensor-rank-spec","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.tensorRank_spec","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#tensor-rank-spec","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $A$, $B$, $C$ be finite types, let $T$ be a trilinear tensor over $F$ indexed by $A$, $B$, $C$, and let $R(T)$ be its tensor rank. Then $T$ is a sum of at most $R(T)$ pure tensors $(u \\otimes v) \\otimes w$ with $u : A \\to F$, $v : B \\to F$ and $w : C \\to F$.","title":"The tensor rank is attained"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/tensor-rank-le","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.tensorRank_le","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#tensor-rank-le","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $A$, $B$, $C$ be finite types, and let $T$ be a trilinear tensor over $F$ indexed by $A$, $B$, $C$. If $T$ is a sum of at most $r$ pure tensors $(u \\otimes v) \\otimes w$ with $u : A \\to F$, $v : B \\to F$ and $w : C \\to F$, then the tensor rank of $T$ is at most $r$.","title":"Decompositions bound the tensor rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-coordinate","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.MatrixCoordinate","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-coordinate","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"The index set of an $m \\times n$ matrix: pairs $(i, j)$ of a row index $i \\in \\{0, \\dots, m-1\\}$ and a column index $j \\in \\{0, \\dots, n-1\\}$.","title":"Matrix coordinates"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-tensor","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationTensor","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-tensor","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a field $F$ and natural numbers $m$, $n$, $p$, the matrix multiplication tensor is the array over $F$ indexed by triples $((i, j), (j', k), (k', i'))$ of an $m \\times n$, an $n \\times p$ and a $p \\times m$ matrix coordinate, whose entry is $1$ if $j = j'$, $k = k'$ and $i' = i$, and $0$ otherwise.","title":"Matrix multiplication tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/square-matrix-multiplication-tensor","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.squareMatrixMultiplicationTensor","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#square-matrix-multiplication-tensor","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a field $F$ and a natural number $n$, the square matrix multiplication tensor over $F$ is the matrix multiplication tensor for multiplying an $n \\times n$ matrix by an $n \\times n$ matrix. Its value at a triple of matrix coordinates $((i, j), (j', k), (k', i'))$ is $1$ when $j = j'$, $k = k'$ and $i = i'$, and $0$ otherwise.","title":"Square matrix multiplication tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-structure-tensor","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationStructureTensor","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-structure-tensor","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a field $F$ and natural numbers $m, n, p$, the structure tensor of multiplying an $m \\times n$ matrix by an $n \\times p$ matrix over $F$ is the element $\\sum_{i < m} \\sum_{j < n} \\sum_{k < p} e_{(i,j)} \\otimes e_{(j,k)} \\otimes e_{(k,i)}$ of the trilinear tensor space $F^{[m] \\times [n]} \\otimes F^{[n] \\times [p]} \\otimes F^{[p] \\times [m]}$, where $[m] = \\{0, \\dots, m - 1\\}$ and $e_x$ is the standard basis\u2026","title":"Matrix multiplication structure tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-structure-tensor-c","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationStructureTensor_coefficients","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-structure-tensor-c","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $m, n, p$ be natural numbers, and write $e_x$ for the standard basis vector with index $x$. The coordinates of the structure tensor $\\sum_{i < m,\\, j < n,\\, k < p} e_{(i,j)} \\otimes e_{(j,k)} \\otimes e_{(k,i)}$ in $F^{m \\times n} \\otimes F^{n \\times p} \\otimes F^{p \\times m}$ are the matrix multiplication coefficients: the coordinate at $((i, j), (j', k), (k', i'))$ is $1$ if $j = j'$,\u2026","title":"Coefficients of the matrix multiplication structure tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/square-matrix-multiplication-rank","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.squareMatrixMultiplicationRank","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#square-matrix-multiplication-rank","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a field $F$ and a natural number $n$, the square matrix multiplication rank of $F$ at $n$ is the tensor rank of the structure tensor of multiplying two $n \\times n$ matrices over $F$: the least number of pure tensors whose sum is that structure tensor.","title":"Square matrix multiplication rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/square-matrix-multiplication-rank-eq","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.squareMatrixMultiplicationRank_eq","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#square-matrix-multiplication-rank-eq","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every field $F$ and natural number $n$, the square matrix multiplication rank of $F$ at $n$ equals the trilinear rank of the square $n \\times n$ matrix multiplication tensor over $F$.","title":"Square matrix multiplication rank equals the coefficient tensor rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-tensor-contraction","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationTensorContraction","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-tensor-contraction","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $m, n, p$ be natural numbers, let $A \\in F^{m \\times n}$ and $B \\in F^{n \\times p}$, and let $i < m$ and $k < p$. Write $T$ for the matrix multiplication tensor over $F$ for $m, n, p$ and, for a natural number $q$, $[q] = \\{0, \\dots, q - 1\\}$. The contraction of $T$ against $A$ and $B$ at output coordinate $(i, k)$ is","title":"Matrix multiplication tensor contraction"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-tensor-contraction-2","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationTensorContraction_eq_mul","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-tensor-contraction-2","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a field $F$, matrices $A \\in F^{m \\times n}$ and $B \\in F^{n \\times p}$, and indices $i < m$ and $k < p$, the contraction of the matrix multiplication tensor against $A$ and $B$ at output coordinate $(i, k)$ equals the entry $(AB)_{ik}$ of the matrix product.","title":"Contracting the matrix multiplication tensor gives the matrix product"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-matrix-multiplication-rank-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.HasMatrixMultiplicationRankExponent","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-matrix-multiplication-rank-exponent","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"A real number $w$ is a matrix multiplication rank exponent over a field $F$ when $w \\ge 0$ and there are a real $C > 0$ and a natural number $N_0 \\ge 1$ such that for every $n \\ge N_0$ the $n \\times n$ matrix multiplication coefficient tensor over $F$ is a sum of at most $C n^w$ rank-one trilinear terms.","title":"Matrix multiplication rank exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-matrix-multiplication-rank-exponent-2","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.hasMatrixMultiplicationRankExponent_iff","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-matrix-multiplication-rank-exponent-2","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and $w$ a real number. Then $w$ is a matrix multiplication rank exponent over $F$ if and only if $w \\ge 0$ and there are a real $C > 0$ and a natural number $N_0 \\ge 1$ such that for every $n \\ge N_0$ the tensor rank of the $n \\times n$ matrix multiplication structure tensor over $F$ is at most $C n^w$.","title":"Rank exponents via the minimum tensor rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/mono-2","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.HasMatrixMultiplicationRankExponent.mono","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#mono-2","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and let $w \\le w'$ be real numbers. If $w$ is a matrix multiplication rank exponent for $F$, then so is $w'$.","title":"Monotonicity of matrix multiplication rank exponents"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationExponent","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-exponent","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"The matrix multiplication exponent $\\omega(F)$ of a field $F$ is the infimum, taken in $[0, \\infty]$, of the real numbers $w$ that are matrix multiplication rank exponents over $F$; it is $\\infty$ when there is no such $w$.","title":"Matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-exponent-le-of-adm","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationExponent_le_of_admissible","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-exponent-le-of-adm","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"If a real number $w$ is a matrix multiplication rank exponent over a field $F$, then the matrix multiplication exponent $\\omega(F)$ of $F$ satisfies $\\omega(F) \\le w$.","title":"Rank exponents bound the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/le-matrix-multiplication-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.le_matrixMultiplicationExponent","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#le-matrix-multiplication-exponent","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and let $W$ be an element of $[0, \\infty]$. If $W \\le w$ for every real number $w$ that is a matrix multiplication rank exponent over $F$, then $W$ is at most the matrix multiplication exponent $\\omega(F)$.","title":"Lower bounds for the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-tensor-rank-le-pro","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationTensor_rank_le_product","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-tensor-rank-le-pro","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every field $F$ and natural numbers $m, n, p$, the coefficient tensor of multiplying an $m \\times n$ matrix by an $n \\times p$ matrix over $F$ is a sum of at most $mnp$ rank-one trilinear terms.","title":"Direct rank bound for the matrix multiplication tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-structure-tensor-r","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationStructureTensor_rank_le_product","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-structure-tensor-r","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $m, n, p$ be natural numbers, and write $e_x$ for the standard basis vector with index $x$. The matrix multiplication structure tensor $\\sum_{i < m,\\, j < n,\\, k < p} e_{(i,j)} \\otimes e_{(j,k)} \\otimes e_{(k,i)}$ in $F^{m \\times n} \\otimes F^{n \\times p} \\otimes F^{p \\times m}$ is a sum of at most $mnp$ pure tensors $u \\otimes v \\otimes w$.","title":"Direct decomposition of the structure tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/square-matrix-multiplication-tensor-rank","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.squareMatrixMultiplicationTensor_rank_le_seven","parent":35975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Strassen's rank bound for "],["math","\\(2 \\times 2\\)"],["text"," matrix multiplication"]],"source":"Strassen's rank bound for $2 \\times 2$ matrix multiplication"},"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#square-matrix-multiplication-tensor-rank","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every field $F$, the square $2 \\times 2$ matrix multiplication tensor over $F$ is a sum of at most $7$ rank-one trilinear terms.","title":"Strassen's rank bound for $2 \\times 2$ matrix multiplication"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-structure-tensor-r-2","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationStructureTensor_rank_le_seven","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-structure-tensor-r-2","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and write $e_x$ for the standard basis vector with index $x$. The $2 \\times 2$ matrix multiplication structure tensor $\\sum_{i, j, k < 2} e_{(i,j)} \\otimes e_{(j,k)} \\otimes e_{(k,i)}$ in $F^{2 \\times 2} \\otimes F^{2 \\times 2} \\otimes F^{2 \\times 2}$ is a sum of at most $7$ pure tensors $u \\otimes v \\otimes w$.","title":"Strassen's decomposition of the structure tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/square-matrix-multiplication-tensor-rank-2","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.squareMatrixMultiplicationTensor_rank_le_cube","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#square-matrix-multiplication-tensor-rank-2","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every field $F$ and natural number $n$, the square $n \\times n$ matrix multiplication tensor over $F$ is a sum of at most $n^3$ rank-one trilinear terms.","title":"Direct rank bound for the square matrix multiplication tensor"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-matrix-multiplication-rank-exponent-3","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.hasMatrixMultiplicationRankExponent_three","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-matrix-multiplication-rank-exponent-3","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every field $F$, the number $3$ is a matrix multiplication rank exponent over $F$.","title":"Exponent three for matrix multiplication rank"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-exponent-le-three","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationExponent_le_three","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-exponent-le-three","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every field $F$, the matrix multiplication exponent $\\omega(F)$ of $F$ satisfies $\\omega(F) \\le 3$.","title":"The matrix multiplication exponent is at most three"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-exponent-ne-top","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationExponent_ne_top","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-exponent-ne-top","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every field $F$, the matrix multiplication exponent of $F$, an element of $[0, \\infty]$, is not $\\infty$.","title":"The matrix multiplication exponent is finite"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-matrix-multiplication-rank-exponent-4","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.hasMatrixMultiplicationRankExponent_of_inf_lt","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-matrix-multiplication-rank-exponent-4","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $\\omega(F)$ be its matrix multiplication exponent and let $w > 0$ be real. If $\\omega(F) < w$, then there are $C > 0$ and $N_0 \\ge 1$ such that for every $n \\ge N_0$ the $n \\times n$ matrix multiplication tensor over $F$ has rank at most $C n^w$.","title":"Exponents above the matrix multiplication exponent admit rank bounds"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-exponent-le-iff","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationExponent_le_iff","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-exponent-le-iff","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field, let $\\omega(F)$ be its matrix multiplication exponent and let $w \\ge 0$ be real. Then $\\omega(F) \\le w$ if and only if for every real $\\varepsilon > 0$ there are $C > 0$ and $N_0 \\ge 1$ such that for every $n \\ge N_0$ the $n \\times n$ matrix multiplication tensor over $F$ has rank at most $C n^{w + \\varepsilon}$.","title":"Upper bounds on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/matrix-multiplication-exponent-endpoint","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.matrixMultiplicationExponent_endpoint","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#matrix-multiplication-exponent-endpoint","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and let $\\omega(F)$ be its matrix multiplication exponent, a finite real number. For every real $\\varepsilon > 0$ there are $C > 0$ and $N_0 \\ge 1$ such that for every $n \\ge N_0$ the $n \\times n$ matrix multiplication tensor over $F$ has rank at most $C n^{\\omega(F) + \\varepsilon}$.","title":"Rank bounds just above the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/rectangular-inner-dimension","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.rectangularInnerDimension","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#rectangular-inner-dimension","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a natural number $n$ and a real aspect $k$, the rectangular inner dimension is $\\lceil n^k \\rceil$, the least natural number that is at least the real power $n^k$.","title":"Rectangular inner dimension"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/rectangular-inner-dimension-zero","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.rectangularInnerDimension_zero","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#rectangular-inner-dimension-zero","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every natural number $n$, including $n = 0$, the rectangular inner dimension at dimension $n$ and aspect $0$, namely $\\lceil n^0 \\rceil$, equals $1$, so aspect $0$ is the outer product of an $n \\times 1$ matrix and a $1 \\times n$ matrix.","title":"Rectangular inner dimension at aspect zero"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/rectangular-inner-dimension-one","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.rectangularInnerDimension_one","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#rectangular-inner-dimension-one","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every natural number $n$, the rectangular inner dimension at dimension $n$ and aspect $1$, namely $\\lceil n^1 \\rceil$, equals $n$, so aspect $1$ is square matrix multiplication without rounding.","title":"Rectangular inner dimension at aspect one"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/rectangular-inner-dimension-one-base","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.rectangularInnerDimension_one_base","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#rectangular-inner-dimension-one-base","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For every real aspect $k$, the rectangular inner dimension at dimension $1$ and aspect $k$, namely $\\lceil 1^k \\rceil$, equals $1$.","title":"Rectangular inner dimension at dimension one"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/has-rectangular-matrix-multiplication-ra","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.HasRectangularMatrixMultiplicationRankExponent","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#has-rectangular-matrix-multiplication-ra","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"Let $F$ be a field and let $k, w$ be real. Then $w$ is a rank exponent for rectangular matrix multiplication over $F$ at aspect $k$ when $k \\ge 0$, $w \\ge 0$, and for every $\\varepsilon > 0$ there are $C > 0$ and $N_0 \\ge 1$ such that for every $n \\ge N_0$ the tensor of multiplying an $n \\times \\lceil n^k \\rceil$ matrix by a $\\lceil n^k \\rceil \\times n$ matrix over $F$ has rank at most $C n^{w + \\varepsilon}$.","title":"Rank exponents for rectangular matrix multiplication"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/rectangular-matrix-multiplication-expone","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.rectangularMatrixMultiplicationExponent","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#rectangular-matrix-multiplication-expone","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a field $F$ and a real number $k$, the rectangular matrix multiplication exponent $\\omega(k)$ of $F$ is the infimum in $[0, \\infty]$ of all real $w \\ge 0$ with the following property: for every $\\varepsilon > 0$ there are a real $C > 0$ and a natural number $N_0 \\ge 1$ such that for every $n \\ge N_0$, the coefficient tensor $\\sum e_{ij} \\otimes e_{jl} \\otimes e_{li}$ of multiplying an\u2026","title":"Rectangular matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/dual-matrix-multiplication-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.MatrixMultiplicationExponent.dualMatrixMultiplicationExponent","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#dual-matrix-multiplication-exponent","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"For a field $F$, the dual matrix multiplication exponent of $F$ is the supremum, taken in the extended nonnegative reals $[0, \\infty]$, of the set of real aspects $k \\ge 0$ for which the rectangular matrix multiplication exponent of $F$ at aspect $k$ equals $2$.","title":"Dual matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/upper-bound-one","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.upperBoundOne","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#upper-bound-one","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the dual matrix multiplication exponent $\\alpha(F)$ of the arithmetic-circuit model satisfies $\\alpha(F) \\le 1$.","title":"Dual exponent at most one"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/coppersmith1982-exponent","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.coppersmith1982Exponent","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#coppersmith1982-exponent","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the rectangular matrix multiplication exponent of the arithmetic-circuit model at aspect $0.172$ equals $2$: an $n \\times \\lceil n^{0.172} \\rceil$ matrix times an $\\lceil n^{0.172} \\rceil \\times n$ matrix costs $n^{2 + o(1)}$ arithmetic operations.","title":"Coppersmith's rectangular exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/coppersmith1997-strict-lower","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.coppersmith1997StrictLower","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#coppersmith1997-strict-lower","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the dual matrix multiplication exponent $\\alpha(F)$ of the arithmetic-circuit model satisfies $\\alpha(F) > 0.29462$.","title":"Coppersmith's 1997 bound on the dual exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/le-gall2012-strict-lower","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.leGall2012StrictLower","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#le-gall2012-strict-lower","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the dual matrix multiplication exponent $\\alpha(F)$ of the arithmetic-circuit model satisfies $\\alpha(F) > 0.30298$.","title":"Le Gall's 2012 bound on the dual exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/le-gall-urrutia2018-lower","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.leGallUrrutia2018Lower","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#le-gall-urrutia2018-lower","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the dual matrix multiplication exponent $\\alpha(F)$ of the arithmetic-circuit model satisfies $\\alpha(F) \\ge 0.31389$.","title":"Le Gall\u2013Urrutia bound on the dual exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/williams-xu-xu-zhou2024-strict-lower","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.williamsXuXuZhou2024StrictLower","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#williams-xu-xu-zhou2024-strict-lower","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the dual matrix multiplication exponent $\\alpha(F)$ of the arithmetic-circuit model satisfies $\\alpha(F) > 0.321334$.","title":"Williams\u2013Xu\u2013Xu\u2013Zhou bound on the dual exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/rectangular-naive-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.rectangularNaiveUpper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#rectangular-naive-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$ and every real $k \\ge 0$, the rectangular matrix multiplication exponent of the arithmetic-circuit model at aspect $k$ is at most $2 + k$.","title":"Naive bound for rectangular matrix multiplication"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/rectangular-operation-rank-bridge","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.rectangularOperationRankBridge","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#rectangular-operation-rank-bridge","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$ and every real $k \\ge 0$, the rectangular matrix multiplication exponent at aspect $k$ defined by arithmetic-circuit size equals the one defined by tensor rank.","title":"Circuit size and tensor rank give the same rectangular exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/characteristic-invariance","kind":"node","lean":"Cslib.Complexity.Algebraic.DualMatrixMultiplicationBoundsWanted.characteristicInvariance","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#characteristic-invariance","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"If two fields $F$ and $K$ have the same characteristic, then for every real $k \\ge 0$ the rectangular matrix multiplication exponents of $F$ and $K$ in the arithmetic-circuit model at aspect $k$ are equal.","title":"The rectangular exponent depends only on the characteristic"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/omega-equals-two-conjecture","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.omegaEqualsTwoConjecture","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#omega-equals-two-conjecture","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"fully_proved","summary":"The proposition that for every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model equals $2$.","title":"Conjecture that the square exponent is two"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/naive-upper-three","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.naiveUpperThree","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#naive-upper-three","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 3$.","title":"Naive bound for square matrix multiplication"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/strassen1969-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.strassen1969Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#strassen1969-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le \\log_2 7$.","title":"Strassen's 1969 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/pan1978-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.pan1978Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#pan1978-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le \\log_{70} 143640$.","title":"Pan's 1978 bound on the square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/bini-capovani-romani-lotti1979-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.biniCapovaniRomaniLotti1979Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#bini-capovani-romani-lotti1979-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 3 \\log_{12} 10$.","title":"Bini\u2013Capovani\u2013Romani\u2013Lotti bound on the square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/pan1979-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.pan1979Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#pan1979-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 3 \\log_{436} 196$.","title":"Pan's 1979 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/pan1980-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.pan1980Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#pan1980-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le \\log_{48} 47216$.","title":"Pan's 1980 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/schonhage1981-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.schonhage1981Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#schonhage1981-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le \\log_{110} 140600$.","title":"Sch\u00f6nhage's 1981 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/pan1981-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.pan1981Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#pan1981-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.5161$.","title":"Pan's 1981 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/romani1982-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.romani1982Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#romani1982-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.5166$.","title":"Romani's 1982 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/coppersmith-winograd1982-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.coppersmithWinograd1982Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#coppersmith-winograd1982-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.495548$.","title":"Coppersmith\u2013Winograd 1982 bound on the square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/strassen1987-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.strassen1987Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#strassen1987-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.4785$.","title":"Strassen's 1987 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/coppersmith-winograd1990-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.coppersmithWinograd1990Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#coppersmith-winograd1990-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.375477$.","title":"Coppersmith\u2013Winograd 1990 bound on the square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/cohn-kleinberg-szegedy-umans2005-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.cohnKleinbergSzegedyUmans2005Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#cohn-kleinberg-szegedy-umans2005-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.41$.","title":"Cohn\u2013Kleinberg\u2013Szegedy\u2013Umans bound on the square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/stothers-davie2013-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.stothersDavie2013Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#stothers-davie2013-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.373689703$.","title":"Stothers\u2013Davie bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/vassilevska-williams2012-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.vassilevskaWilliams2012Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#vassilevska-williams2012-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.372873$.","title":"Vassilevska Williams's 2012 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/zhdanovich2012-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.zhdanovich2012Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#zhdanovich2012-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.373$.","title":"Zhdanovich's 2012 bound on the matrix multiplication exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/le-gall2014-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.leGall2014Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#le-gall2014-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.3728639$.","title":"Le Gall's 2014 bound on the square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/alman-vassilevska-williams2020-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.almanVassilevskaWilliams2020Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#alman-vassilevska-williams2020-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.3728596$.","title":"Alman\u2013Vassilevska Williams 2020 bound on the square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/duan-wu-zhou2022-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.duanWuZhou2022Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#duan-wu-zhou2022-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent $\\omega(F)$ of the arithmetic-circuit model satisfies $\\omega(F) \\le 2.371866$.","title":"Duan\u2013Wu\u2013Zhou bound on the square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/vassilevska-williams-xu-xu-zhou2023-uppe","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.vassilevskaWilliamsXuXuZhou2023Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#vassilevska-williams-xu-xu-zhou2023-uppe","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent of $F$ in the arithmetic-circuit model is at most $2.371552$.","title":"Vassilevska Williams\u2013Xu\u2013Xu\u2013Zhou upper bound for matrix multiplication"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/alman-duan-vassilevska-williams-xu-xu-zh","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.almanDuanVassilevskaWilliamsXuXuZhou2024Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#alman-duan-vassilevska-williams-xu-xu-zh","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent of $F$ in the arithmetic-circuit model is at most $2.371339$.","title":"Alman\u2013Duan\u2013Vassilevska Williams\u2013Xu\u2013Xu\u2013Zhou upper bound for matrix multiplication"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/dupont-eisenberger-et-al2026-upper","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.dupontEisenbergerEtAl2026Upper","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#dupont-eisenberger-et-al2026-upper","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent of $F$ in the arithmetic-circuit model is at most $2.371177$.","title":"Dupont\u2013Eisenberger et al. upper bound for matrix multiplication"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/square-operation-rank-bridge","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.squareOperationRankBridge","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#square-operation-rank-bridge","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"For every field $F$, the square matrix multiplication exponent of $F$ defined by arithmetic-circuit size equals the matrix multiplication exponent of $F$ defined by tensor rank.","title":"Circuit size and tensor rank give the same square exponent"},{"id":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/characteristic-invariance-2","kind":"node","lean":"Cslib.Complexity.Algebraic.SquareMatrixMultiplicationBoundsWanted.characteristicInvariance","parent":35975,"result":true,"route":"../roadmap/TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent/index.html#characteristic-invariance-2","scope":"TheoryOfComputation/algebraic-complexity/matrix-multiplication-exponent","status":"can_prove","summary":"If two fields $F$ and $K$ have the same characteristic, then the square matrix multiplication exponents of $F$ and $K$ in the arithmetic-circuit model are equal.","title":"The square exponent depends only on the characteristic"},{"data_url":"scopes/TheoryOfComputation/descriptive-complexity.json","id":"TheoryOfComputation/descriptive-complexity","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/index.html","scope":"TheoryOfComputation/descriptive-complexity","status":"can_prove","summary":"Logical characterizations of complexity classes on finite structures: Fagin's theorem, Immerman\u2013Vardi (least fixed point and P), and transitive closure logic and NL.","title":"Descriptive complexity"},{"data_url":"scopes/TheoryOfComputation/descriptive-complexity/fagin-s-theorem.json","id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","kind":"scope","navigation":"module","parent":36061,"result":false,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"can_prove","title":"Fagin's theorem"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-graph","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.FaginGraph","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-graph","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"A finite graph is a pair $(n, t)$ of a natural number $n$, the number of vertices, and a finite list $t$ of booleans, read as a row-major edge table. The length of $t$ is not required to be $n^2$.","title":"Finite graphs as edge tables"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-edge","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.faginEdge","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-edge","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"Let $G$ be a graph with vertices $0, \\dots, n-1$ and edge table $t$, a finite list of booleans. For vertices $a, b < n$, there is an edge from $a$ to $b$ exactly when the entry of $t$ at position $an + b$ (counting from $0$) is true; a position beyond the end of $t$ counts as false.","title":"Edge relation of a graph"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-fo","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.FaginFO","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-fo","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"Variables and relation symbols are indexed by natural numbers. Formulas are built from equations $x_a = x_b$ between variables and atomic formulas applying a relation symbol $i$ to a finite list of variables, by negation, conjunction, disjunction and existential quantification $\\exists \\varphi$, which binds variable $0$ of $\\varphi$ (de Bruijn indexing).","title":"First-order formulas over graphs"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.faginEval","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-eval","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"Let $G$ be a graph, let $\\rho$ assign a vertex $\\rho(j)$ of $G$ to each variable $x_j$ ($j \\in \\mathbb{N}$), and let $S$ assign to each relation symbol $i \\in \\mathbb{N}$ a predicate $S_i$ on finite lists of vertices of $G$. Truth of a formula is defined by recursion: $x_a = x_b$ holds when $\\rho(a) = \\rho(b)$; relation symbol $i$ applied to the variables $x_{a_1}, \\dots, x_{a_m}$ holds when $S_i$ holds of the list\u2026","title":"Evaluation of a first-order formula in a graph"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-base","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.faginBase","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-base","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"Let $G$ be a graph and let $(E_j)_{j \\in \\mathbb{N}}$ be a family of relations, each a predicate on finite lists of vertices of $G$. Relation symbol $0$ is interpreted as the edge relation of $G$: it holds of a list exactly when the list is $[a, b]$ for vertices $a, b$ with an edge from $a$ to $b$. For $i \\ge 1$, relation symbol $i$ is interpreted as $E_{i-1}$.","title":"Interpretation of relation symbols in a graph"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-esosat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.faginESOSat","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-esosat","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"Let $G$ be a graph with $n > 0$ vertices, $k$ a natural number and $\\varphi$ a first-order formula. Then $G$ satisfies the existential second-order sentence $(k, \\varphi)$ if there are relations $R_0, \\dots, R_{k-1}$, each a predicate on finite lists of vertices of $G$, such that $\\varphi$ holds in $G$ when relation symbol $0$ is interpreted as the edge relation (holding exactly of the lists $[a, b]$ with an edge\u2026","title":"Satisfaction of an existential second-order sentence"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-encodings","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.faginEncodings","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-encodings","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"For a class $K$ of graphs, this is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs of a natural number and a list of booleans, to a graph $G \\in K$ with $n$ vertices whose edge table has length exactly $n^2$.","title":"Encodings of a graph class as a decision problem"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-iso","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.faginIso","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-iso","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"Graphs $G_1$ and $G_2$ are isomorphic if they have the same number of vertices and there is a bijection $\\sigma$ from the vertices of $G_1$ to the vertices of $G_2$ such that for all vertices $a, b$ of $G_1$, there is an edge from $a$ to $b$ in $G_1$ if and only if there is an edge from $\\sigma(a)$ to $\\sigma(b)$ in $G_2$.","title":"Isomorphic graphs"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-iso-closed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.faginIsoClosed","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-iso-closed","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"A class $K$ of graphs is isomorphism-closed if for every $G_1 \\in K$, every graph $G_2$ isomorphic to $G_1$ also belongs to $K$.","title":"Isomorphism-closed graph classes"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin-esodefinable","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.faginESODefinable","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin-esodefinable","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"fully_proved","summary":"A class $K$ of graphs is existential second-order definable if there are a natural number $k$, a first-order formula $\\varphi$ and a boolean $e$ such that every graph $G$ with $n$ vertices whose edge table has length exactly $n^2$ satisfies the following: if $n > 0$, then $G \\in K$ if and only if $G$ satisfies the existential second-order sentence $(k, \\varphi)$; if $n = 0$, then $G \\in K$ if and only if $e$ is true.","title":"Existential second-order definable graph classes"},{"id":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem/fagin","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Fagin.fagin","parent":36062,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/fagin-s-theorem/index.html#fagin","scope":"TheoryOfComputation/descriptive-complexity/fagin-s-theorem","status":"can_prove","summary":"Let $K$ be a class of finite graphs, each given as a pair $(n, t)$ of a number $n$ of vertices and a row-major edge table $t$ (a list of booleans), and suppose that $K$ is closed under graph isomorphism. Then the decision problem that accepts exactly the bitstrings decoding to a member $(n, t)$ of $K$ whose edge table has length $n^2$ is in NP if and only if $K$ is definable in existential second-order logic: there\u2026","title":"Fagin's theorem: NP is existential second-order logic"},{"data_url":"scopes/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem.json","id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","kind":"scope","navigation":"module","parent":36061,"result":false,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"can_prove","title":"Immerman\u2013Vardi theorem"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/ordered-graph","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.OrderedGraph","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#ordered-graph","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"A finite ordered graph is a pair $(n, t)$ of a natural number $n$ and a finite list $t$ of booleans of any length. Here $n$ is the size of the domain $\\{0, \\dots, n - 1\\}$, ordered numerically, and $t$ is read as a row-major edge table.","title":"Finite ordered graphs"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/og-edge","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.ogEdge","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#og-edge","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"Let $G$ be an ordered graph with vertices $0, \\dots, n - 1$ and edge table $t$, a list of booleans. There is an edge from $a$ to $b$ when the entry of $t$ at position $a n + b$ is true; positions beyond the end of $t$ count as false.","title":"Edge relation of an ordered graph"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/ogfo","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.OGFO","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#ogfo","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"Variables and relation symbols are natural numbers. Formulas are built from equalities $x_a = x_b$ and atoms $R_i(x_{a_1}, \\dots, x_{a_k})$ by negation, conjunction, disjunction, existential quantification and least-fixed-point formulas $\\mathrm{lfp}(R, k, \\varphi, (a_1, \\dots, a_m))$, the least fixed point of the operator defined by $\\varphi$ (with $R$ as the recursive relation, of arity $k$) applied to the tuple\u2026","title":"Formulas of first-order logic with least fixed points"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/positive-in","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.PositiveIn","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#positive-in","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"A relation symbol $R$ occurs only positively in a first-order formula with least fixed points over ordered graphs when every atomic formula with relation symbol $R$ in it, including those inside least-fixed-point subformulas whatever symbol they bind, lies under an even number of negations.","title":"Positive occurrence of a relation symbol"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/negative-in","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.NegativeIn","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#negative-in","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"A relation symbol $R$ occurs only negatively in a formula $\\varphi$ of first-order logic with least fixed points when every atomic formula $R(x_{a_1}, \\dots, x_{a_k})$ in $\\varphi$, including those inside least-fixed-point subformulas, lies under an odd number of negations.","title":"Negative occurrence of a relation symbol"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/all-lfppositive","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.AllLFPPositive","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#all-lfppositive","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"Let $\\varphi$ be a formula of first-order logic with least fixed points, whose relation symbols are natural numbers. All fixed points of $\\varphi$ are positive when every least-fixed-point subformula of $\\varphi$, nested ones included, binds a relation symbol $R \\ge 2$ (so neither the order symbol $0$ nor the edge symbol $1$) and $R$ occurs only positively in the body of that subformula.","title":"Formulas whose fixed points are all positive"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/og-base","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.ogBase","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#og-base","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"Let $G$ be an ordered graph with vertices $0, \\dots, n - 1$ and let $(E_i)_{i \\in \\mathbb{N}}$ be predicates on lists of vertices. Relation symbol $0$ holds of a list exactly when it is a pair $(a, b)$ with $a < b$, relation symbol $1$ holds of a list exactly when it is a pair $(a, b)$ with an edge from $a$ to $b$ in $G$, and relation symbol $i + 2$ is interpreted as $E_i$.","title":"Interpretation of relation symbols in an ordered graph"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/ogwell-formed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.OGWellFormed","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#ogwell-formed","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"Let $e$ assign to each relation symbol (a natural number) either an arity or nothing. Whether a first-order formula with least fixed points over ordered graphs is well-formed relative to $e$ is defined by recursion on the formula. An equality of two variables is well-formed. An atomic formula $R(x_1, \\dots, x_k)$ is well-formed when $k = a$ if $e(R) = a$; when $k = 2$ if $e$ assigns nothing to $R$ and $R$ is $0$\u2026","title":"Well-formed FO + LFP formulas"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/og-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.ogEval","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#og-eval","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"Let $G$ be an ordered graph with vertex set $V = \\{0, \\dots, n - 1\\}$, let $\\rho$ assign a vertex to each variable $0, 1, 2, \\dots$, and let $S$ assign to each relation symbol $i$ a predicate $S_i$ on lists of vertices. The truth of a formula is defined by recursion: $x_a = x_b$ holds when $\\rho(a) = \\rho(b)$; the atom $R_i(x_{a_1}, \\dots, x_{a_k})$ holds when $S_i$ holds of $(\\rho(a_1), \\dots, \\rho(a_k))$;\u2026","title":"Evaluation of FO + LFP formulas in an ordered graph"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/og-lfpsat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.ogLFPSat","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#og-lfpsat","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"An ordered graph $G$ with $n > 0$ vertices satisfies a formula $\\varphi$ of first-order logic with least fixed points when $\\varphi$ is well-formed in the environment where no relation symbol is bound, all fixed points of $\\varphi$ are positive, and $\\varphi$ is true in $G$ with every variable assigned the vertex $0$, relation symbol $0$ interpreted as the order $<$, relation symbol $1$ as the edge relation, and\u2026","title":"Satisfaction of an FO + LFP formula by an ordered graph"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/og-encodings","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.ogEncodings","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#og-encodings","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"For a class $K$ of ordered graphs, this is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs of a natural number and a list of booleans, to some $G \\in K$ whose edge table has length exactly $n^2$, where $n$ is the number of vertices of $G$.","title":"Encodings of a class of ordered graphs"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/og-lfpdefinable","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.ogLFPDefinable","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#og-lfpdefinable","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"fully_proved","summary":"A class $K$ of ordered graphs is definable in first-order logic with least fixed points when there are a formula $\\varphi$ and a boolean $b$ such that every ordered graph $G$ with $n$ vertices whose edge table has length $n^2$ lies in $K$ exactly when either $n > 0$ and $G$ satisfies $\\varphi$, or $n = 0$ and $b$ is true.","title":"Definability in FO + LFP"},{"id":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/immerman-vardi","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ImmermanVardi.immerman_vardi","parent":36074,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem/index.html#immerman-vardi","scope":"TheoryOfComputation/descriptive-complexity/immerman-vardi-theorem","status":"can_prove","summary":"Source: N. Immerman, \"Relational queries definable in polynomial time\", Information and Control 68 (1986), 193-243 (STOC 1983); M. Y. Vardi, \"The complexity of relational query languages\", Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing (STOC 1982), 137-146.","title":"Immerman-Vardi: FO + least fixed point is P on ordered structures"},{"data_url":"scopes/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl.json","id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","kind":"scope","navigation":"module","parent":36061,"result":false,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"can_prove","title":"Transitive closure logic and NL"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tcordered-graph","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.TCOrderedGraph","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tcordered-graph","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"An ordered graph is encoded as a pair $(n, e)$ of a natural number $n$, the domain size, and a finite list $e$ of booleans of arbitrary length, the edge table in row-major order. It is read as the graph on $\\{0, \\dots, n - 1\\}$, ordered numerically, with an edge from $a$ to $b$ when the entry of $e$ at position $an + b$ is true; a missing entry counts as false.","title":"Ordered graphs for first-order logic with transitive closure"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tc-edge","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.tcEdge","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tc-edge","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"For an ordered graph $(n, T)$ with edge table $T$ and elements $a, b < n$, there is an edge from $a$ to $b$ when the entry of $T$ at position $a n + b$ is true; positions beyond the end of $T$ count as false.","title":"Edge relation of an ordered graph for FO + TC"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tcfo","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.TCFO","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tcfo","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"FO + TC formulas, with variables and relation symbols indexed by natural numbers, are built from equations between two variables and relation symbols applied to lists of variables by negation, conjunction, disjunction, existential quantification (which binds variable $0$, de Bruijn style) and transitive closure $\\mathrm{TC}(\\varphi; \\bar u, \\bar v)$ of a formula $\\varphi$ applied to two lists $\\bar u$, $\\bar v$ of\u2026","title":"Formulas of first-order logic with transitive closure"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tc-base","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.tcBase","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tc-base","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"In an ordered graph with domain $\\{0, \\dots, n - 1\\}$, relation symbol $0$ holds of a list of elements exactly when the list is $[a, b]$ with $a < b$, relation symbol $1$ holds exactly when the list is $[a, b]$ and there is an edge from $a$ to $b$, and every other relation symbol holds of no list.","title":"Order and edge relations as FO + TC relation symbols"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tc-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.tcEval","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tc-eval","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"Fix an ordered graph with domain $D = \\{0, \\dots, n - 1\\}$, an assignment $\\rho$ of an element of $D$ to each variable index and an assignment $S$ of a relation on lists of elements of $D$ to each relation symbol. The truth value of a formula is defined by recursion. The equation between variables $i$ and $j$ holds when $\\rho(i) = \\rho(j)$. Relation symbol $i$ applied to variables $j_1, \\dots, j_k$ holds when $S_i$\u2026","title":"Evaluation of FO + TC formulas"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tc-param-example","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.tcParamExample","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tc-param-example","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"The FO + TC formula $\\mathrm{TC}(\\varphi; (1), (2))$ of arity $1$ whose body $\\varphi$ is $E(0, 1) \\lor (E(0, 2) \\land E(2, 1))$, where $E$ is relation symbol $1$ (the edge relation). In the body, variable $0$ is the source, variable $1$ the target, and variable $2$ is outer variable $0$, shifted by $2$; the endpoint arguments $1$ and $2$ are outer variables, not shifted. So the formula expresses that outer\u2026","title":"Example: parameterized transitive closure in FO + TC"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tc-pair-example","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.tcPairExample","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tc-pair-example","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"The FO + TC formula $\\mathrm{TC}(\\varphi; (0, 1), (2, 3))$ of arity $2$ whose body $\\varphi$ states that relation symbol $1$ (the edge relation) holds of variables $0, 2$ and of variables $1, 3$. In the body, variables $0, 1$ are the source pair and $2, 3$ the target pair, so a step moves both components along an edge at the same time, and the formula expresses that the pair of outer variables $(2, 3)$ is reachable\u2026","title":"Example: simultaneous reachability of pairs in FO + TC"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tc-sat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.tcSat","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tc-sat","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"An ordered graph with nonempty domain $\\{0, \\dots, n - 1\\}$ satisfies an FO + TC formula $\\varphi$ when $\\varphi$ evaluates to true with every variable assigned the element $0$, relation symbol $0$ interpreted as the order $<$ and relation symbol $1$ as the edge relation (both only on two-element lists), and every other relation symbol interpreted as the empty relation.","title":"Satisfaction of an FO + TC formula in an ordered graph"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tc-encodings","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.tcEncodings","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tc-encodings","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"For a set $K$ of ordered graphs, this is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of pairs of a natural number and a list of booleans, to some ordered graph $(n, T) \\in K$ whose edge table $T$ has length exactly $n^2$.","title":"Encodings of a class of ordered graphs"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/tc-definable","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.tcDefinable","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#tc-definable","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"fully_proved","summary":"A set $K$ of ordered graphs is FO + TC-definable when there are an FO + TC formula $\\varphi$ and a boolean $b$ such that for every ordered graph $(n, T)$ whose edge table $T$ has length exactly $n^2$: if $n > 0$, then $(n, T) \\in K$ exactly when $(n, T)$ satisfies $\\varphi$, and if $n = 0$, then $(n, T) \\in K$ exactly when $b$ is true.","title":"Definability in FO + TC"},{"id":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/fo-tc-eq-nl","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.FOTC.fo_tc_eq_nl","parent":36088,"result":true,"route":"../roadmap/TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl/index.html#fo-tc-eq-nl","scope":"TheoryOfComputation/descriptive-complexity/transitive-closure-logic-and-nl","status":"can_prove","summary":"For every set $K$ of finite ordered graphs, the decision problem of the encodings of $K$ (the bitstrings that decode to a member $(n, T)$ of $K$ whose edge table $T$ has length exactly $n^2$) belongs to $\\mathrm{NL}$ if and only if $K$ is definable in first-order logic with transitive closure.","title":"Immerman: FO + transitive closure is NL on ordered structures"},{"data_url":"scopes/TheoryOfComputation/kolmogorov-complexity.json","id":"TheoryOfComputation/kolmogorov-complexity","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"Algorithmic information content of individual strings: universal description methods, incompressible strings, non-computability and upper semicomputability of Kolmogorov complexity.","title":"Kolmogorov complexity"},{"id":"TheoryOfComputation/kolmogorov-complexity/run","kind":"node","lean":"Cslib.Kolmogorov.run","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#run","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"For a code $c$ of a partial recursive function on the natural numbers and a binary string $w$, running $c$ on $w$ evaluates $c$ at the number encoding $w$; if this halts with output $m$, the result is the binary string encoded by $m$. The result is undefined when the evaluation does not halt or $m$ encodes no binary string.","title":"Running a program on a binary string"},{"id":"TheoryOfComputation/kolmogorov-complexity/describe","kind":"node","lean":"Cslib.Kolmogorov.describe","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#describe","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"Let $c$ be a code for a partial recursive function and let $n$ be its number in the standard encoding of codes as natural numbers. For a binary string $w$, the description of $c$ running on $w$ is the binary string $1^n 0 w$, that is, $n$ ones, then a zero, then $w$.","title":"Description of a code running on an input"},{"id":"TheoryOfComputation/kolmogorov-complexity/u","kind":"node","lean":"Cslib.Kolmogorov.U","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#u","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"A description `1\u207f 0 w` runs code number `n` on `w`; a string of `1`s describes nothing.","title":"The universal description method"},{"id":"TheoryOfComputation/kolmogorov-complexity/k","kind":"node","lean":"Cslib.Kolmogorov.K","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#k","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"The Kolmogorov complexity $K(x)$ of a binary string $x$ is the least length of a binary string $p$ such that $U(p)$ is defined and equals $x$, where $U$ is the universal description method.","title":"Kolmogorov complexity"},{"id":"TheoryOfComputation/kolmogorov-complexity/k-le","kind":"node","lean":"Cslib.Kolmogorov.K_le","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#k-le","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"Let $x$ and $p$ be binary strings such that $U(p)$ is defined and equals $x$, where $U$ is the universal description method. Then the Kolmogorov complexity of $x$ is at most the length of $p$: $K(x) \\le |p|$.","title":"A description bounds the complexity"},{"id":"TheoryOfComputation/kolmogorov-complexity/exists-length-eq-k","kind":"node","lean":"Cslib.Kolmogorov.exists_length_eq_K","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#exists-length-eq-k","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"For every binary string $x$ there is a binary string $p$ whose length is the Kolmogorov complexity $K(x)$ and such that $U(p)$ is defined and equals $x$, where $U$ is the universal description method.","title":"Shortest descriptions exist"},{"id":"TheoryOfComputation/kolmogorov-complexity/k-le-length-add","kind":"node","lean":"Cslib.Kolmogorov.K_le_length_add","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#k-le-length-add","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"There is a natural number $c$ such that every binary string $x$ satisfies $K(x) \\le |x| + c$, where $K(x)$ is the Kolmogorov complexity of $x$ and $|x|$ is its length.","title":"Complexity is at most the length plus a constant"},{"id":"TheoryOfComputation/kolmogorov-complexity/partrec-u","kind":"node","lean":"Cslib.Kolmogorov.partrec_U","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#partrec-u","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"The universal description method, as a partial function from binary strings to binary strings, is partial recursive.","title":"The universal description method is partial recursive"},{"id":"TheoryOfComputation/kolmogorov-complexity/exists-run-eq","kind":"node","lean":"Cslib.Kolmogorov.exists_run_eq","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#exists-run-eq","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"Let $D$ be a partial recursive function from binary strings to binary strings. Then there is a code $c$ for a partial recursive function such that running $c$ equals $D$ as a partial function: for every binary string $w$, running $c$ on $w$ is defined exactly when $D(w)$ is, and then has the same value.","title":"Every partial recursive description method has a code"},{"id":"TheoryOfComputation/kolmogorov-complexity/exists-k-le-of-partrec","kind":"node","lean":"Cslib.Kolmogorov.exists_K_le_of_partrec","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#exists-k-le-of-partrec","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"Let $D$ be a partial recursive function from binary strings to binary strings, viewed as a description method. Then there is a natural number $c$ such that for all binary strings $x$ and $p$ for which $D(p)$ is defined and equals $x$, the Kolmogorov complexity of $x$ satisfies $K(x) \\le |p| + c$, where $|p|$ is the length of $p$.","title":"Optimality (the invariance theorem)"},{"id":"TheoryOfComputation/kolmogorov-complexity/k-append-self-le","kind":"node","lean":"Cslib.Kolmogorov.K_append_self_le","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#k-append-self-le","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"There is a natural number $c$ such that for every binary string $x$, the Kolmogorov complexity of the concatenation $xx$ of $x$ with itself satisfies $K(xx) \\le K(x) + c$.","title":"Doubling costs a constant"},{"id":"TheoryOfComputation/kolmogorov-complexity/pair-describe","kind":"node","lean":"Cslib.Kolmogorov.pairDescribe","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#pair-describe","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"For binary strings $p$ and $q$, the pair description of $p$ and $q$ is the binary string $1^{|p|} 0 p q$, that is, $|p|$ ones, where $|p|$ is the length of $p$, then a zero, then $p$, then $q$.","title":"Pair description"},{"id":"TheoryOfComputation/kolmogorov-complexity/pair-u","kind":"node","lean":"Cslib.Kolmogorov.pairU","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#pair-u","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"Write a binary string $s$ as $1^n s'$, where $n$ is the number of leading $1$s, and let $r$ be $s'$ without its first letter ($r$ is empty when $s'$ is). Let $r_1$ be the first $\\min(n, |r|)$ letters of $r$ and $r_2$ the remaining ones. The pair reader maps $s$ to the concatenation $U(r_1)\\,U(r_2)$, where $U$ is the universal description method, and is undefined when $U(r_1)$ or $U(r_2)$ is undefined. On a pair\u2026","title":"Reading pair descriptions"},{"id":"TheoryOfComputation/kolmogorov-complexity/k-append-le","kind":"node","lean":"Cslib.Kolmogorov.K_append_le","parent":36100,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/index.html#k-append-le","scope":"TheoryOfComputation/kolmogorov-complexity","status":"fully_proved","summary":"There is a natural number $c$ such that for all binary strings $x$ and $y$, the Kolmogorov complexity of their concatenation $xy$ satisfies $K(xy) \\le 2K(x) + K(y) + c$.","title":"Concatenation costs at most twice the first part"},{"data_url":"scopes/TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity.json","id":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity","kind":"scope","navigation":"module","parent":36100,"result":false,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/index.html","scope":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity","status":"fully_proved","title":"Incompressible strings, and Kolmogorov complexity is not computable"},{"id":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/pad","kind":"node","lean":"Cslib.Kolmogorov.pad","parent":36115,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/index.html#pad","scope":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity","status":"fully_proved","summary":"For a natural number $n$ and a binary string $p$, the padding of $p$ to length $n$ is the string $0^k 1 p$ with $k = \\max(n - 1 - |p|, 0)$; when $|p| < n$ it has length $n$.","title":"Padding a string to a given length"},{"id":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/card-lt-two-pow","kind":"node","lean":"Cslib.Kolmogorov.card_lt_two_pow","parent":36115,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/index.html#card-lt-two-pow","scope":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity","status":"fully_proved","summary":"Fewer than `2 ^ m` strings have complexity below `m`. Padding their shortest descriptions to length `m` maps them injectively to strings of length `m` other than `0\u1d50`.","title":"Few strings have small complexity"},{"id":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/strings-of-length","kind":"node","lean":"Cslib.Kolmogorov.stringsOfLength","parent":36115,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/index.html#strings-of-length","scope":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity","status":"fully_proved","summary":"For a natural number $n$, the finite set of all binary strings of length $n$.","title":"Binary strings of a given length"},{"id":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/card-compressible-lt","kind":"node","lean":"Cslib.Kolmogorov.card_compressible_lt","parent":36115,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/index.html#card-compressible-lt","scope":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity","status":"fully_proved","summary":"For all natural numbers $n$ and $c$, fewer than $2^{n-c}$ binary strings of length $n$ have Kolmogorov complexity less than $n - c$, where $n - c$ is read as $0$ when $c > n$.","title":"Most strings are incompressible"},{"id":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/exists-incompressible","kind":"node","lean":"Cslib.Kolmogorov.exists_incompressible","parent":36115,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/index.html#exists-incompressible","scope":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity","status":"fully_proved","summary":"Some string of length `n` has complexity at least `n`.","title":"Incompressible strings exist at every length"},{"id":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/not-computable-k","kind":"node","lean":"Cslib.Kolmogorov.not_computable_K","parent":36115,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity/index.html#not-computable-k","scope":"TheoryOfComputation/kolmogorov-complexity/incompressible-strings-and-kolmogorov-complexity","status":"fully_proved","summary":"The function sending each binary string $x$ to its Kolmogorov complexity $K(x)$ is not computable: no partial recursive function, acting on binary strings through their encodings as natural numbers, halts on every binary string $x$ with output $K(x)$.","title":"Kolmogorov complexity is not computable"},{"data_url":"scopes/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity.json","id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","kind":"scope","navigation":"module","parent":36100,"result":false,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","title":"Upper semicomputability of Kolmogorov complexity, and incompressible strings"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/re-pred-mem-u","kind":"node","lean":"Cslib.Kolmogorov.rePred_mem_U","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#re-pred-mem-u","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $U$ be the universal description method. The set of pairs $(x, p)$ of binary strings such that $U(p)$ is defined and equal to $x$ is recursively enumerable (recognizable).","title":"The description relation is recognizable"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/re-pred-k-le","kind":"node","lean":"Cslib.Kolmogorov.rePred_K_le","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#re-pred-k-le","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $K$ denote Kolmogorov complexity. The set of pairs $(x, n)$ of a binary string $x$ and a natural number $n$ with $K(x) \\le n$ is recursively enumerable (recognizable).","title":"Kolmogorov complexity is upper semicomputable"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/exists-k-map-le","kind":"node","lean":"Cslib.Kolmogorov.exists_K_map_le","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#exists-k-map-le","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $K$ denote Kolmogorov complexity and let $f$ be a partial recursive function from binary strings to binary strings. Then there is a natural number $c$ such that $K(y) \\le K(x) + c$ for all binary strings $x$ and $y$ such that $f(x)$ is defined and equal to $y$.","title":"A partial recursive map raises complexity by at most a constant"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/exists-shortest-incompressible","kind":"node","lean":"Cslib.Kolmogorov.exists_shortest_incompressible","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#exists-shortest-incompressible","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $U$ be the universal description method, $K$ Kolmogorov complexity, and $|p|$ the length of a binary string $p$. There is a natural number $b$ such that for all binary strings $x$ and $p$, if $p$ is a shortest description of $x$, that is, $U(p)$ is defined and equal to $x$ and $|p| = K(x)$, then $|p| \\le K(p) + b$.","title":"Shortest descriptions are incompressible"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/exists-bound-of-re-pred-le-k","kind":"node","lean":"Cslib.Kolmogorov.exists_bound_of_rePred_le_K","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#exists-bound-of-re-pred-le-k","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $K$ denote Kolmogorov complexity, and let $S$ be a recursively enumerable (recognizable) set of pairs $(x, n)$ of a binary string $x$ and a natural number $n$ such that $n \\le K(x)$ for every $(x, n) \\in S$. Then there is a natural number $B$ with $n \\le B$ for every $(x, n) \\in S$.","title":"Chaitin's incompleteness theorem"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/exists-bound-of-computable-le-k","kind":"node","lean":"Cslib.Kolmogorov.exists_bound_of_computable_le_K","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#exists-bound-of-computable-le-k","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $K$ denote Kolmogorov complexity. If $f$ is a computable function from binary strings to natural numbers with $f(x) \\le K(x)$ for every binary string $x$, then there is a natural number $B$ with $f(x) \\le B$ for every binary string $x$.","title":"A computable lower bound of Kolmogorov complexity is bounded"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/not-re-pred-le-k","kind":"node","lean":"Cslib.Kolmogorov.not_rePred_le_K","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#not-re-pred-le-k","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $K$ denote Kolmogorov complexity. The set of pairs $(x, n)$ of a binary string $x$ and a natural number $n$ with $n \\le K(x)$ is not recursively enumerable (recognizable).","title":"Kolmogorov complexity is not lower semicomputable"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/not-re-pred-of-forall-incompressible","kind":"node","lean":"Cslib.Kolmogorov.not_rePred_of_forall_incompressible","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#not-re-pred-of-forall-incompressible","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $K$ denote Kolmogorov complexity and $|x|$ the length of a binary string $x$. There is no recursively enumerable (recognizable) set $S$ of binary strings such that for every natural number $n$ some $x \\in S$ has $|x| \\ge n$, and every $x \\in S$ satisfies $|x| \\le K(x)$.","title":"No recognizable set with arbitrarily long members consists of incompressible strings"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/not-re-pred-incompressible","kind":"node","lean":"Cslib.Kolmogorov.not_rePred_incompressible","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#not-re-pred-incompressible","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $K$ denote Kolmogorov complexity and $|x|$ the length of a binary string $x$. The set of incompressible binary strings, that is, of binary strings $x$ with $|x| \\le K(x)$, is not recursively enumerable (recognizable).","title":"The incompressible strings are not recognizable"},{"id":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/not-computable-pred-incompressible","kind":"node","lean":"Cslib.Kolmogorov.not_computablePred_incompressible","parent":36122,"result":true,"route":"../roadmap/TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity/index.html#not-computable-pred-incompressible","scope":"TheoryOfComputation/kolmogorov-complexity/upper-semicomputability-of-kolmogorov-complexity","status":"fully_proved","summary":"Let $K$ denote Kolmogorov complexity and $|x|$ the length of a binary string $x$. The set of incompressible binary strings, that is, of binary strings $x$ with $|x| \\le K(x)$, is not decidable.","title":"The incompressible strings are not decidable"},{"data_url":"scopes/TheoryOfComputation/pcp-and-inapproximability.json","id":"TheoryOfComputation/pcp-and-inapproximability","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/index.html","scope":"TheoryOfComputation/pcp-and-inapproximability","status":"can_prove","summary":"The PCP theorem and Dinur's gap amplification, optimization classes (NPO, APX, MAXSNP), and hardness of approximation results such as H\u00e5stad's MAX-3SAT bound and the absence of a PTAS for MAX-CUT.","title":"Probabilistically checkable proofs and inapproximability"},{"id":"TheoryOfComputation/pcp-and-inapproximability/pcp-theorem","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PCPTheorem.pcp_theorem","parent":36133,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/index.html#pcp-theorem","scope":"TheoryOfComputation/pcp-and-inapproximability","status":"can_prove","summary":"A decision problem belongs to NP if and only if, for some natural numbers $c$ and $q$, it belongs to $\\mathrm{PCP}(r, s)$, where the randomness bound is $r(n) = c \\lfloor \\log_2(n + 1) \\rfloor$ and the query bound is the constant function $s(n) = q$.","title":"The PCP theorem: NP = PCP(O(log n), O(1))"},{"data_url":"scopes/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat.json","id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","kind":"scope","navigation":"module","parent":36133,"result":false,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"can_prove","title":"Approximability of MAX-3SAT"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/hastad-nondegenerate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Hastad.hastadNondegenerate","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#hastad-nondegenerate","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"A clause, given as a list of literals $(v, s)$ with variable index $v \\in \\mathbb{N}$ and polarity $s$, is nondegenerate if it has exactly three literals, its variable indices are pairwise distinct, and it contains no complementary pair, that is, no two literals $(v, s)$ and $(v, \\lnot s)$.","title":"Nondegenerate 3-clauses"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/hastad-val","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Hastad.hastadVal","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#hastad-val","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"For bitstrings $x$ and $a$, read $a$ as the assignment giving variable $v$ the value $a_v$ if $v < |a|$ and false otherwise. If $x$ decodes, under the canonical bitstring decoding, to a pair $(u, \\varphi)$ of a bitstring $u$ and a CNF formula $\\varphi$ in which every variable index is less than $|u|$ and every clause is nondegenerate, then the MAX-3SAT value of $a$ on $x$ is the number of clauses of $\\varphi$,\u2026","title":"Value of a MAX-3SAT assignment"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/hastad-opt","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Hastad.hastadOpt","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#hastad-opt","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"The MAX-3SAT optimum of a bitstring $x$ is the maximum, over all bitstrings $a$ of length $|x|$, of the MAX-3SAT value of $a$ on $x$.","title":"Optimum of a MAX-3SAT instance"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/hastad-max3sat-inapproximability","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.Hastad.hastad_max3sat_inapproximability","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#hastad-max3sat-inapproximability","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"can_prove","summary":"If $\\mathsf{P} \\ne \\mathsf{NP}$, then for every rational $\\varepsilon > 0$ there is no polynomial-time function $A$ from bitstrings to bitstrings such that, for every bitstring $x$, the bitstring $A(x)$ has the same length as $x$ and the MAX-3SAT value of the assignment $A(x)$ on $x$ is at least $(7/8 + \\varepsilon)$ times the MAX-3SAT optimum of $x$.","title":"Hastad: optimal inapproximability of MAX-3SAT"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/max3-satval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxThreeSATMAXSNPComplete.max3SATVal","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#max3-satval","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"The value of MAX-3SAT on an instance $x$ under an assignment $a$, both bitstrings, is defined as follows. Suppose $x$ decodes, under the canonical bitstring encoding, to a pair $(u, \\varphi)$ of a bitstring $u$ and a CNF formula $\\varphi$, and suppose moreover that $|a| \\le |x|$, that every literal of $\\varphi$ has variable index less than $|u|$ and that every clause of $\\varphi$ has at most $3$ literals. Then the\u2026","title":"The MAX-3SAT objective"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/npoproblem","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxThreeSATMAXSNPComplete.NPOProblem","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#npoproblem","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"An NPO (NP optimization) problem consists of a rational-valued value function $v$, a Boolean feasibility predicate $F$ and an objective encoding $e$, each taking an instance $x$ and a solution $s$ (both bitstrings), together with a polynomial $b$ with natural-number coefficients, such that the following hold. $F$ and $e$ are polynomial-time computable. For all $x$ and $s$, $e(x, s)$ is a triple $(\\sigma, p, q)$ of\u2026","title":"NPO problems"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/opt-val","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxThreeSATMAXSNPComplete.optVal","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#opt-val","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"The optimum of an NPO problem $M$, with value function $v$ and solution length bound $b$, on an instance $x$ is the maximum of $v(x, s)$ over all bitstrings $s$ with $|s| \\le b(|x|)$.","title":"Optimum of an NPO problem"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/apx","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxThreeSATMAXSNPComplete.APX","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#apx","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"An NPO problem $M$ is in APX when there are a rational number $\\rho \\ge 1$ and a polynomial-time computable map $A$ from bitstrings to bitstrings such that, for every instance $x$, the solution $A(x)$ is feasible for $x$ in $M$ and its value is at least $\\mathrm{opt}_M(x) / \\rho$, where $\\mathrm{opt}_M(x)$ is the optimum of $M$ on $x$.","title":"Membership in APX"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/lreduces","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxThreeSATMAXSNPComplete.LReduces","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#lreduces","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"Write $\\mathrm{opt}_M(x)$ for the optimum of an NPO problem $M$ on an instance $x$ and $v_M(x, s)$ for the value of a solution $s$ for $x$. An NPO problem $M_1$ L-reduces to an NPO problem $M_2$ when there are polynomial-time computable maps $f$, from bitstrings to bitstrings, and $g$, from pairs of bitstrings to bitstrings, and rational numbers $\\alpha > 0$ and $\\beta > 0$, such that for every instance $x$ the\u2026","title":"L-reductions"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/maxsnpproblem","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxThreeSATMAXSNPComplete.MAXSNPProblem","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#maxsnpproblem","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"A MAXSNP problem, in the normal form of Papadimitriou and Yannakakis, is an NPO problem $M$, with value function $v$ and solution length bound $b$, together with natural numbers $k$ and $N$, a polynomial-time computable map $W$ sending each instance $x$ to a list $W(x)$ of windows, each a pair $(w, j)$ of a list $w = (w_1, \\dots, w_m)$ of natural numbers (variable indices) and a natural number $j$ (a pattern\u2026","title":"MAXSNP problems"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/maxsnpcomplete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxThreeSATMAXSNPComplete.MAXSNPComplete","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#maxsnpcomplete","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"fully_proved","summary":"A MAXSNP problem $M$ is MAXSNP-complete when its underlying NPO problem is in APX and, for every MAXSNP problem $M'$, the underlying NPO problem of $M'$ L-reduces to the underlying NPO problem of $M$.","title":"MAXSNP-completeness"},{"id":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/max-three-sat-maxsnp-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxThreeSATMAXSNPComplete.max_three_sat_maxsnp_complete","parent":36135,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat/index.html#max-three-sat-maxsnp-complete","scope":"TheoryOfComputation/pcp-and-inapproximability/approximability-of-max-3sat","status":"can_prove","summary":"There is a MAXSNP problem whose value function is the MAX-3SAT value function and which is MAXSNP-complete: its underlying NPO problem is in APX, and the underlying NPO problem of every MAXSNP problem L-reduces to it.","title":"MAX-3SAT is MAXSNP-complete"},{"data_url":"scopes/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification.json","id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","kind":"scope","navigation":"module","parent":36133,"result":false,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"can_prove","title":"Dinur's gap amplification"},{"id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/cspinstance","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.DinurGapAmplification.CSPInstance","parent":36148,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html#cspinstance","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"fully_proved","summary":"A 2CSP instance is a triple $(n, \\Sigma, C)$ of a natural number $n$ (the number of variables), a natural number $\\Sigma$ (the alphabet size) and a finite list $C$ of constraints, where each constraint is a triple $(u, v, P)$ of natural numbers $u$ and $v$ (the variables it mentions) and a finite list $P$ of pairs of natural numbers (the allowed label pairs).","title":"2CSP instances"},{"id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/cspwell-formed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.DinurGapAmplification.CSPWellFormed","parent":36148,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html#cspwell-formed","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"fully_proved","summary":"A 2CSP instance $(n, \\Sigma, C)$ is well-formed when every constraint $(u, v, P)$ in $C$ satisfies $u < n$ and $v < n$, and every allowed pair $(a, b)$ in $P$ satisfies $a < \\Sigma$ and $b < \\Sigma$.","title":"Well-formed 2CSP instances"},{"id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/csp-constraint-sat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.DinurGapAmplification.cspConstraintSat","parent":36148,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html#csp-constraint-sat","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"fully_proved","summary":"Let $I = (n, \\Sigma, C)$ be a 2CSP instance and $\\sigma$ an assignment of a label less than $\\Sigma$ to each variable less than $n$. A constraint $(u, v, P)$ is satisfied by $\\sigma$ when $u < n$, $v < n$ and the pair $(\\sigma(u), \\sigma(v))$ belongs to $P$.","title":"Satisfied constraint of a 2CSP instance"},{"id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/csp-val","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.DinurGapAmplification.cspVal","parent":36148,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html#csp-val","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"fully_proved","summary":"Let $I = (n, \\Sigma, C)$ be a 2CSP instance and $\\sigma$ an assignment of a label less than $\\Sigma$ to each variable less than $n$. The value of $I$ under $\\sigma$ is the rational number $1$ if $C$ is empty, and otherwise the number of entries of the list $C$ that are constraints satisfied by $\\sigma$, divided by the length of $C$.","title":"Value of a 2CSP instance under an assignment"},{"id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/csp-degree","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.DinurGapAmplification.cspDegree","parent":36148,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html#csp-degree","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"fully_proved","summary":"For a 2CSP instance $(n, \\Sigma, C)$ and a natural number $v$, the degree of $v$ is the number of entries $(u, w, P)$ of the list $C$ with $u = v$ or $w = v$.","title":"Degree of a variable in a 2CSP instance"},{"id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/gap-amplification","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.DinurGapAmplification.gapAmplification","parent":36148,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html#gap-amplification","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"fully_proved","summary":"Let $c, \\alpha$ be rationals and $d, D, e$ natural numbers. A map $T$ from 2CSP instances to 2CSP instances is a gap amplification with constants $c, \\alpha, d, D, e$ when $c > 1$, $0 < \\alpha < 1$, $d \\ge 1$, $D \\ge 1$, and for every 2CSP instance $I$ with $n$ variables, alphabet size $\\Sigma$ and $m$ constraints: $T(I)$ has exactly $n$ variables, alphabet size at most $\\Sigma^e$ and at most $d m$ constraints; if\u2026","title":"Gap amplification with given constants"},{"id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/alphabet-reduction","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.DinurGapAmplification.alphabetReduction","parent":36148,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html#alphabet-reduction","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"fully_proved","summary":"Let $S_0, e, d', V$ be natural numbers and $c', \\alpha'$ rationals. A map $T$ from 2CSP instances to 2CSP instances is an alphabet reduction with constants $S_0, e, c', \\alpha', d', V$ when $S_0 \\ge 2$, $e \\ge 2$, $0 < c' \\le 1$, $0 < \\alpha' < 1$, $d' \\ge 1$, $V \\ge 1$, and for every 2CSP instance $I$ with $n$ variables, alphabet size at most $S_0^e$ and $m$ constraints: $T(I)$ has at most $V(n + 1)$ variables,\u2026","title":"Alphabet reduction with given constants"},{"id":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/dinur-gap-amplification","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.DinurGapAmplification.dinur_gap_amplification","parent":36148,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification/index.html#dinur-gap-amplification","scope":"TheoryOfComputation/pcp-and-inapproximability/dinur-s-gap-amplification","status":"can_prove","summary":"There exist natural numbers $S_0, e, d, D, d', V$ and rationals $c, \\alpha, c', \\alpha'$ with $S_0 \\ge 2$, $e \\ge 2$, $c > 1$, $0 < \\alpha < 1$, $d \\ge 1$, $D \\ge 1$, $0 < c' \\le 1$, $0 < \\alpha' < 1$, $d' \\ge 1$ and $V \\ge 1$, and maps $A$ and $R$ from 2CSP instances to 2CSP instances, with the following properties. Both maps are polynomial-time computable: for each $T \\in \\{A, R\\}$ there is a polynomial-time\u2026","title":"Dinur's gap amplification and alphabet reduction"},{"data_url":"scopes/TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut.json","id":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut","kind":"scope","navigation":"module","parent":36133,"result":false,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/index.html","scope":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut","status":"can_prove","title":"Inapproximability of MAX-CUT"},{"id":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/max-cut-n","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxCutNoPTAS.maxCutN","parent":36157,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/index.html#max-cut-n","scope":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut","status":"fully_proved","summary":"The vertex count of a MAX-CUT instance, a bitstring $x$ of length $|x|$, is $\\lfloor \\sqrt{|x|} \\rfloor$.","title":"Vertex count of a MAX-CUT instance"},{"id":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/max-cut-feas","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxCutNoPTAS.maxCutFeas","parent":36157,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/index.html#max-cut-feas","scope":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut","status":"fully_proved","summary":"A bitstring $s$ is a feasible solution of a MAX-CUT instance $x$ exactly when the length of $s$ equals the vertex count $\\lfloor \\sqrt{|x|} \\rfloor$ of $x$, so that $s$ has one bit per vertex.","title":"Feasible MAX-CUT solutions"},{"id":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/max-cut-val","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxCutNoPTAS.maxCutVal","parent":36157,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/index.html#max-cut-val","scope":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut","status":"fully_proved","summary":"Let $x$ be a MAX-CUT instance with vertex count $n = \\lfloor \\sqrt{|x|} \\rfloor$ and let $s$ be a bitstring, with bits indexed from $0$. If $s$ is a feasible solution of $x$ (its length is $n$), the value of $s$ is the number, as a rational number, of pairs $(i, j)$ with $i < j < n$ such that bit $i n + j$ of $x$ is true and bits $i$ and $j$ of $s$ differ; otherwise the value is $0$.","title":"Value of a MAX-CUT solution"},{"id":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/max-cut","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxCutNoPTAS.maxCut","parent":36157,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/index.html#max-cut","scope":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut","status":"fully_proved","summary":"MAX-CUT is the optimization problem whose instances are bitstrings $x$, read as an adjacency matrix on $n = \\lfloor \\sqrt{|x|} \\rfloor$ vertices in which, for $i < j < n$, bit $i n + j$ of $x$ (bits indexed from $0$) says whether $\\{i, j\\}$ is an edge. Its feasible solutions for $x$ are the bitstrings of length $n$, one bit per vertex; the value of a feasible solution $s$ is the number of pairs $i < j < n$ that are\u2026","title":"MAX-CUT as an optimization problem"},{"id":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/max-cut-no-ptas","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxCutNoPTAS.max_cut_no_ptas","parent":36157,"result":true,"route":"../roadmap/TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut/index.html#max-cut-no-ptas","scope":"TheoryOfComputation/pcp-and-inapproximability/inapproximability-of-max-cut","status":"can_prove","summary":"If $\\mathsf{P} \\neq \\mathsf{NP}$, then the optimization problem MAX-CUT does not admit a polynomial-time approximation scheme.","title":"MAX-CUT has no PTAS unless P = NP"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes.json","id":"TheoryOfComputation/completeness-other-classes","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/index.html","scope":"TheoryOfComputation/completeness-other-classes","status":"can_prove","summary":"Problems complete for classes other than NP: P (circuit value), NL (s-t connectivity), coNP (UNSAT), PSPACE (TQBF, Geography), levels of the polynomial hierarchy (QSAT_k), EXP, NEXP and EXPSPACE (succinct problems, regular expressions with squaring).","title":"Completeness for other complexity classes"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/circuit-value-problem.json","id":"TheoryOfComputation/completeness-other-classes/circuit-value-problem","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/circuit-value-problem/index.html","scope":"TheoryOfComputation/completeness-other-classes/circuit-value-problem","status":"can_prove","title":"Circuit value problem"},{"id":"TheoryOfComputation/completeness-other-classes/circuit-value-problem/cvgate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.CircuitValuePComplete.CVGate","parent":36164,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/circuit-value-problem/index.html#cvgate","scope":"TheoryOfComputation/completeness-other-classes/circuit-value-problem","status":"fully_proved","summary":"An encoded gate of a flat Boolean circuit is a pair $(k, r)$ of a natural number $k$, the kind of the gate, and a list $r$ of natural numbers, its references.","title":"Encoded circuit gates"},{"id":"TheoryOfComputation/completeness-other-classes/circuit-value-problem/cv-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.CircuitValuePComplete.cvCircuitEval","parent":36164,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/circuit-value-problem/index.html#cv-circuit-eval","scope":"TheoryOfComputation/completeness-other-classes/circuit-value-problem","status":"fully_proved","summary":"Let $n$ be a number of inputs, let $C$ be a list of gates, each encoded as a (kind, references) pair, and let $a = (a_0, a_1, \\dots)$ be a list of booleans. The evaluation of $C$ on $a$ is the list of values of all gates, in order, of the flat Boolean circuit on $n$ inputs obtained by decoding each pair of $C$, under the assignment that sends each input $i < n$ to $a_i$, or to false when $i$ is at least the length\u2026","title":"Evaluation of an encoded Boolean circuit"},{"id":"TheoryOfComputation/completeness-other-classes/circuit-value-problem/cvwell-formed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.CircuitValuePComplete.CVWellFormed","parent":36164,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/circuit-value-problem/index.html#cvwell-formed","scope":"TheoryOfComputation/completeness-other-classes/circuit-value-problem","status":"fully_proved","summary":"A list $C$ of encoded gates, each a (kind, references) pair, is well formed as a circuit on $n$ inputs when the flat Boolean circuit on $n$ inputs obtained by decoding each pair of $C$ is well formed: every decoded gate references only strictly earlier gates.","title":"Well-formed encoded circuits"},{"id":"TheoryOfComputation/completeness-other-classes/circuit-value-problem/cvp-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.CircuitValuePComplete.cvpDec","parent":36164,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/circuit-value-problem/index.html#cvp-dec","scope":"TheoryOfComputation/completeness-other-classes/circuit-value-problem","status":"fully_proved","summary":"The Circuit Value Problem is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of such triples, to a triple $(n, C, a)$ of a natural number $n$, a list $C$ of encoded gates and a list $a$ of booleans such that the flat Boolean circuit on $n$ inputs obtained by decoding the gates of $C$ is well formed, $a$ has length $n$, and the evaluation of $C$ on\u2026","title":"The Circuit Value Problem"},{"id":"TheoryOfComputation/completeness-other-classes/circuit-value-problem/cvp-p-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.CircuitValuePComplete.cvp_p_complete","parent":36164,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/circuit-value-problem/index.html#cvp-p-complete","scope":"TheoryOfComputation/completeness-other-classes/circuit-value-problem","status":"can_prove","summary":"The Circuit Value Problem is in P, and every decision problem in P is log-space many-one reducible to the Circuit Value Problem.","title":"The Circuit Value Problem is P-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity.json","id":"TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity/index.html","scope":"TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity","status":"can_prove","title":"Directed s-t connectivity"},{"id":"TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity/stcon-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.STCONNLComplete.stconDec","parent":36170,"reading_title":{"format":"inline-parts-v1","parts":[["text","Directed "],["math","\\(s\\)"],["text","-"],["math","\\(t\\)"],["text"," connectivity"]],"source":"Directed $s$-$t$ connectivity"},"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity/index.html#stcon-dec","scope":"TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity","status":"fully_proved","summary":"The decision problem of directed $s$-$t$ connectivity accepts a bitstring $x$ exactly when decoding $x$ with the canonical bitstring encoding of tuples $(n, E, s, t)$, where $n$, $s$ and $t$ are natural numbers and $E$ is a list of pairs of natural numbers, succeeds and gives a tuple such that $s < n$, $t < n$, and there are pairwise distinct natural numbers $v_0, \\dots, v_k$, all less than $n$, with $v_0 = s$,\u2026","title":"Directed $s$-$t$ connectivity"},{"id":"TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity/stcon-nl-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.STCONNLComplete.stcon_nl_complete","parent":36170,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity/index.html#stcon-nl-complete","scope":"TheoryOfComputation/completeness-other-classes/directed-s-t-connectivity","status":"can_prove","summary":"Directed $s$-$t$ connectivity is in NL, and every decision problem in NL is log-space many-one reducible to directed $s$-$t$ connectivity.","title":"ST-connectivity is NL-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/generalized-geography.json","id":"TheoryOfComputation/completeness-other-classes/generalized-geography","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/generalized-geography/index.html","scope":"TheoryOfComputation/completeness-other-classes/generalized-geography","status":"can_prove","title":"Generalized geography"},{"id":"TheoryOfComputation/completeness-other-classes/generalized-geography/geo-wins","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GeographyPSPACEComplete.GeoWins","parent":36173,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/generalized-geography/index.html#geo-wins","scope":"TheoryOfComputation/completeness-other-classes/generalized-geography","status":"fully_proved","summary":"Let $E$ be a list of pairs of natural numbers, read as directed edges, let $v$ be a natural number (the current vertex) and let $V$ be a list of natural numbers (the visited vertices). For a natural number $k$ (the fuel), the predicate $\\mathrm{GeoWins}(k, E, v, V)$ is defined by recursion on $k$: $\\mathrm{GeoWins}(0, E, v, V)$ is false, and $\\mathrm{GeoWins}(k + 1, E, v, V)$ holds exactly when there is a natural\u2026","title":"Winning positions in Geography"},{"id":"TheoryOfComputation/completeness-other-classes/generalized-geography/geography-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GeographyPSPACEComplete.geographyDec","parent":36173,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/generalized-geography/index.html#geography-dec","scope":"TheoryOfComputation/completeness-other-classes/generalized-geography","status":"fully_proved","summary":"The Geography decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of triples $(n, E, s)$ of a natural number, a list of pairs of natural numbers and a natural number, to a triple with $s < n$, with both entries of every pair in $E$ less than $n$, and with $\\mathrm{GeoWins}(n + 1, E, s, [s])$, where $[s]$ is the list whose only entry is $s$. Bitstrings that do\u2026","title":"Geography as a decision problem"},{"id":"TheoryOfComputation/completeness-other-classes/generalized-geography/geography-pspace-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.GeographyPSPACEComplete.geography_pspace_complete","parent":36173,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/generalized-geography/index.html#geography-pspace-complete","scope":"TheoryOfComputation/completeness-other-classes/generalized-geography","status":"can_prove","summary":"The Geography decision problem belongs to PSPACE, and every decision problem in PSPACE has a polynomial-time many-one reduction to it.","title":"Geography is PSPACE-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna.json","id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"can_prove","title":"Quantified Boolean formulas with bounded alternation"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/qbfmatrix-gate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.QBFMatrixGate","parent":36177,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#qbfmatrix-gate","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"A gate of a matrix circuit is a pair $(t, r)$ of a natural number $t$, its kind, and a list $r$ of natural numbers, its references. The kinds are read as follows: kind $0$ with $r = [v]$ is the variable $v$, kind $1$ with $r = [b]$ is the constant true if $b = 1$ and false otherwise, kind $2$ is a negation, kind $3$ a conjunction and kind $4$ a disjunction, where for kinds $2$, $3$ and $4$ the references are\u2026","title":"Gates of a quantified Boolean formula matrix"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/qbf-gate-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.qbfGateEval","parent":36177,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#qbf-gate-eval","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"Given a list $s$ of truth values (the values of the earlier gates) and an assignment $\\alpha$ of a truth value to every natural number, the value of a gate $(t, r)$ is computed as follows, where $s_a$ is the $a$-th entry of $s$ if $a$ is less than the length of $s$ and false otherwise: if $t = 0$ and $r = [v]$, it is $\\alpha(v)$; if $t = 1$ and $r = [b]$, it is true exactly when $b = 1$; if $t = 2$ and $r = [a]$,\u2026","title":"Evaluation of a QBF matrix gate"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/qbf-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.qbfCircuitEval","parent":36177,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#qbf-circuit-eval","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"Given a matrix circuit $C = (g_0, \\dots, g_{m-1})$, a list of gates, and an assignment $\\alpha$ of a truth value to every natural number, the list $(s_0, \\dots, s_{m-1})$ of the values of the gates of $C$, where $s_i$ is the value of the gate $g_i$ given the values $(s_0, \\dots, s_{i-1})$ of the earlier gates and the assignment $\\alpha$.","title":"Evaluation of a QBF matrix circuit"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/qbf-circuit-vars","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.qbfCircuitVars","parent":36177,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#qbf-circuit-vars","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"For a matrix circuit $C$, a list of gates, the list of the variables $v$ of the gates of $C$ of the form $(0, [v])$, in the order of the gates of $C$ and with repetitions.","title":"Variables of a QBF matrix circuit"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/qbf-circuit-well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.qbfCircuitWellFormed","parent":36177,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#qbf-circuit-well-formed","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"A matrix circuit $C = (g_0, \\dots, g_{m-1})$, a list of gates, is well formed when for every $j < m$ whose gate $g_j = (t, r)$ has kind $t \\in \\{2, 3, 4\\}$ (negation, conjunction or disjunction), every reference $a$ in $r$ satisfies $a < j$.","title":"Well-formed QBF matrix circuits"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/eval-qbfk","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.EvalQBFK","parent":36177,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#eval-qbfk","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"Let $P$ be a quantifier prefix, a list of pairs $(q, v)$ of a quantifier $q$, $\\exists$ (true) or $\\forall$ (false), and a variable $v \\in \\mathbb{N}$; let $\\varphi$ be a matrix circuit and $\\alpha$ an assignment of a truth value to every natural number. Truth of $(P, \\varphi)$ under $\\alpha$ is defined by recursion on $P$. If $P$ is empty, it means that $\\varphi$ is nonempty and the value of its last gate under\u2026","title":"Truth of a quantified Boolean formula with circuit matrix"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/block-count-aux","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.blockCountAux","parent":36177,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#block-count-aux","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"For a quantifier $p$ and a quantifier prefix $(q_1, v_1), \\dots, (q_n, v_n)$, the number of indices $1 \\le i \\le n$ with $q_i \\ne q_{i-1}$, where $q_0 = p$.","title":"Quantifier alternations after a given quantifier"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/block-count","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.blockCount","parent":36177,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#block-count","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"The number of maximal blocks of consecutive equal quantifiers in a quantifier prefix $(q_1, v_1), \\dots, (q_n, v_n)$: it is $0$ if $n = 0$, and otherwise $1$ plus the number of indices $2 \\le i \\le n$ with $q_i \\ne q_{i-1}$.","title":"Number of quantifier blocks"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/qsat-kdec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.qsatKDec","parent":36177,"reading_title":{"format":"inline-parts-v1","parts":[["text","The decision problem "],["math","\\(\\mathrm{QSAT}_k\\)"]],"source":"The decision problem $\\mathrm{QSAT}_k$"},"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#qsat-kdec","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"fully_proved","summary":"For a natural number $k$, $\\mathrm{QSAT}_k$ accepts a bitstring $x$ exactly when $x$ decodes, under the standard bitstring encoding, to a pair $(P, \\varphi)$ of a quantifier prefix $P = (q_1, v_1), \\dots, (q_n, v_n)$, where each $q_i$ is $\\exists$ (true) or $\\forall$ (false) and each $v_i$ is a natural number, and a matrix circuit $\\varphi$, such that: $P$ is nonempty and $q_1$ is $\\exists$; $(v_1, \\dots, v_n)$ is\u2026","title":"The decision problem $\\mathrm{QSAT}_k$"},{"id":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/qsat-k-sigma-k-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.QSATKComplete.qsat_k_sigma_k_complete","parent":36177,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\mathrm{QSAT}_k\\)"],["text"," is "],["math","\\(\\Sigma_k^p\\)"],["text","-complete, uniformly in "],["math","\\(k\\)"]],"source":"$\\mathrm{QSAT}_k$ is $\\Sigma_k^p$-complete, uniformly in $k$"},"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna/index.html#qsat-k-sigma-k-complete","scope":"TheoryOfComputation/completeness-other-classes/quantified-boolean-formulas-with-bounded-alterna","status":"can_prove","summary":"For every natural number $k \\ge 1$, the decision problem $\\mathrm{QSAT}_k$ lies in $\\Sigma_k^p$, and every decision problem $L$ in $\\Sigma_k^p$ reduces to $\\mathrm{QSAT}_k$ by a polynomial-time many-one reduction.","title":"$\\mathrm{QSAT}_k$ is $\\Sigma_k^p$-complete, uniformly in $k$"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring.json","id":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/index.html","scope":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring","status":"can_prove","title":"Regular expressions with squaring"},{"id":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/star-lang","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RegexSquaringEXPSPACEComplete.starLang","parent":36188,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/index.html#star-lang","scope":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring","status":"fully_proved","summary":"For a set $S$ of bitstrings, its Kleene star is the set of all concatenations $u_1 u_2 \\cdots u_k$, for $k \\ge 0$, of bitstrings $u_1, \\dots, u_k \\in S$; the empty concatenation ($k = 0$) is the empty bitstring.","title":"Kleene star of a bitstring language"},{"id":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/concat-lang","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RegexSquaringEXPSPACEComplete.concatLang","parent":36188,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/index.html#concat-lang","scope":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring","status":"fully_proved","summary":"For sets $S$ and $T$ of bitstrings, their concatenation is the set of all bitstrings $uv$ with $u \\in S$ and $v \\in T$.","title":"Concatenation of bitstring languages"},{"id":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/eval-postfix","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RegexSquaringEXPSPACEComplete.evalPostfix","parent":36188,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/index.html#eval-postfix","scope":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring","status":"fully_proved","summary":"A list of natural-number tokens is evaluated from left to right on a given stack of languages of bitstrings. Token $0$ pushes the empty language, token $1$ pushes the language containing only the empty bitstring, and tokens $2$ and $3$ push the languages containing only the one-letter bitstring false, respectively true. Token $4$ pops the top language $S$ and the language $T$ below it and pushes the concatenation\u2026","title":"Evaluation of postfix regular expressions with squaring"},{"id":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/regex-univ-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RegexSquaringEXPSPACEComplete.regexUnivDec","parent":36188,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/index.html#regex-univ-dec","scope":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring","status":"fully_proved","summary":"This is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding of lists of natural numbers, to a token list whose evaluation as a postfix regular expression with squaring, started on the empty stack, succeeds and returns the set of all bitstrings. A bitstring that does not decode to a list of natural numbers is rejected.","title":"Universality of postfix regular expressions with squaring"},{"id":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/regex-squaring-expspace-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.RegexSquaringEXPSPACEComplete.regex_squaring_expspace_complete","parent":36188,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring/index.html#regex-squaring-expspace-complete","scope":"TheoryOfComputation/completeness-other-classes/regular-expressions-with-squaring","status":"can_prove","summary":"The universality problem for postfix regular expressions with squaring belongs to $\\mathsf{EXPSPACE}$, and every decision problem in $\\mathsf{EXPSPACE}$ reduces to it by a polynomial-time many-one reduction. Here the universality problem accepts a bitstring exactly when it decodes to a list of natural-number tokens whose postfix evaluation, started on the empty stack, returns the set of all bitstrings.","title":"Regular expressions with squaring: universality is EXPSPACE-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear.json","id":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/index.html","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","status":"can_prove","title":"Satisfiability and model checking for the linear-time \u03bc-calculus"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/linear-mu-sat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LinearMuCalculusPSPACEComplete.linearMuSat","parent":36194,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/index.html#linear-mu-sat","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","status":"fully_proved","summary":"A closed linear \u03bc-calculus formula is satisfiable: some word and position satisfy it under the empty environment.","title":"Satisfiability of a closed linear \u03bc-calculus formula"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/linear-mu-valid","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LinearMuCalculusPSPACEComplete.linearMuValid","parent":36194,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/index.html#linear-mu-valid","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","status":"fully_proved","summary":"A closed linear \u03bc-calculus formula is valid: every word and position satisfy it under the empty environment.","title":"Validity of a closed linear \u03bc-calculus formula"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/linear-mu-model-check","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LinearMuCalculusPSPACEComplete.linearMuModelCheck","parent":36194,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/index.html#linear-mu-model-check","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","status":"fully_proved","summary":"Model checking a linear \u03bc-calculus formula against a finite Kripke structure: every infinite path `\u03c0` from `start` following `edges` satisfies `\u03c6` at position `0`, where the path turns the structure's valuation `worldVal` into a word valuation.","title":"Linear \u03bc-calculus model checking on a finite Kripke structure"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/linear-mu-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LinearMuCalculusPSPACEComplete.linearMuSatDec","parent":36194,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/index.html#linear-mu-sat-dec","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","status":"fully_proved","summary":"Satisfiability of the linear \u03bc-calculus as a decision problem over bitstrings: the input decodes to a formula token list, which decodes to a formula; the answer is whether that formula is closed and satisfiable.","title":"Linear \u03bc-calculus satisfiability as a decision problem"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/linear-mu-valid-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LinearMuCalculusPSPACEComplete.linearMuValidDec","parent":36194,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/index.html#linear-mu-valid-dec","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","status":"fully_proved","summary":"Validity of the linear \u03bc-calculus as a decision problem over bitstrings.","title":"Linear \u03bc-calculus validity as a decision problem"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/linear-mu-model-check-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LinearMuCalculusPSPACEComplete.linearMuModelCheckDec","parent":36194,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/index.html#linear-mu-model-check-dec","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","status":"fully_proved","summary":"Model checking of the linear \u03bc-calculus as a decision problem over bitstrings: the input decodes to `(n, edges, worldVals, start, tokens)`, where `worldVals` lists, per world, the atoms true at that world; the answer is whether the decoded formula is closed, the structure is well-formed and total (every world has a successor world, so the path quantification is not vacuous at deadlocks), and every path from `start`\u2026","title":"Linear \u03bc-calculus model checking as a decision problem"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/linear-mu-calculus-pspace-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LinearMuCalculusPSPACEComplete.linear_mu_calculus_pspace_complete","parent":36194,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear/index.html#linear-mu-calculus-pspace-complete","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-and-model-checking-for-the-linear","status":"can_prove","summary":"The model checking, satisfiability and validity problems of the linear modal \u03bc-calculus are PSPACE-complete, like for linear temporal logic. Source: M. Y. Vardi, \"A temporal fixpoint calculus\", POPL 1988, 250-259; cf. A. P. Sistla and E. M. Clarke, \"The complexity of propositional linear temporal logics\", JACM 32 (1985), 733-749.","title":"The model checking, satisfiability and validity problems of the linear modal \u03bc-calculus are PSPACE-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus.json","id":"TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus/index.html","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus","status":"can_prove","title":"Satisfiability of the modal \u03bc-calculus"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus/mu-sat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MuCalculusEXPTIMEComplete.muSat","parent":36202,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus/index.html#mu-sat","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus","status":"fully_proved","summary":"A closed \u03bc-calculus formula is satisfiable: some Kripke model has a world satisfying it under the empty environment.","title":"Satisfiability of a closed modal \u03bc-calculus formula"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus/mu-calculus-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MuCalculusEXPTIMEComplete.muCalculusSatDec","parent":36202,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus/index.html#mu-calculus-sat-dec","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus","status":"fully_proved","summary":"Satisfiability of the modal \u03bc-calculus as a decision problem over bitstrings: the input decodes to a formula token list, which decodes to a formula; the answer is whether that formula is closed and satisfiable.","title":"Modal \u03bc-calculus satisfiability as a decision problem"},{"id":"TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus/mu-calculus-sat-exp-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MuCalculusEXPTIMEComplete.mu_calculus_sat_exp_complete","parent":36202,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus/index.html#mu-calculus-sat-exp-complete","scope":"TheoryOfComputation/completeness-other-classes/satisfiability-of-the-modal-calculus","status":"can_prove","summary":"Sources: D. Kozen, \"Results on the propositional \u03bc-calculus\", Theoretical Computer Science 27 (1983), 333-354; E. A. Emerson and C. S. Jutla, \"Tree automata, \u03bc-calculus and determinacy\", FOCS 1991, 368-377.","title":"Satisfiability of a modal \u03bc-calculus formula is EXPTIME-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/succinct-3sat.json","id":"TheoryOfComputation/completeness-other-classes/succinct-3sat","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"can_prove","title":"Succinct 3SAT"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/s3-gate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.S3Gate","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#s3-gate","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"An encoded gate is a pair $(k, r)$ of a natural number $k$, the kind of the gate, and a list $r$ of natural numbers, its references. This is the (kind, references) encoding of the gates of a flat Boolean circuit.","title":"Encoded gates"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/s3-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.s3CircuitEval","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#s3-circuit-eval","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"Let $C = (g_0, \\dots, g_{m-1})$ be a list of gates, each encoded as a (kind, references) pair, let $\\alpha$ be a list of booleans and let $n$ be a natural number. The gate values of $C$ on $\\alpha$ are the values $(v_0, \\dots, v_{m-1})$ of the flat Boolean circuit on $n + 2$ inputs obtained by decoding every $g_j$, when input $i$ is given entry $i$ of $\\alpha$, read as false when $i$ is at least the length of\u2026","title":"Gate values of an encoded circuit"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/s3-circuit-outputs","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.s3CircuitOutputs","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#s3-circuit-outputs","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"Let $C$ be a list of $m$ gates encoded as (kind, references) pairs, let $\\alpha$ be a list of booleans and let $k$ and $n$ be natural numbers. The last $k$ outputs of $C$ on $\\alpha$ are the last $k$ of the gate values $(v_0, \\dots, v_{m-1})$ of $C$ on $\\alpha$ as a circuit on $n + 2$ inputs, that is $(v_{m-k}, \\dots, v_{m-1})$; when $k > m$ they are all $m$ gate values.","title":"Outputs of an encoded circuit"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/s3-valid-gate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.s3ValidGate","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#s3-valid-gate","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"Let $C$ be a list of gates encoded as (kind, references) pairs and let $n$ and $j$ be natural numbers. Gate $j$ of $C$ is valid when $C$ has more than $j$ entries and its entry $j$ (counting from $0$) is one of: $(0, [i])$ with $i < n + 2$ (an input gate); $(1, [b])$ with $b \\le 1$ (a constant gate); $(2, [a])$ with $a < j$ (a negation gate); or $(3, r)$ or $(4, r)$ where every entry $a$ of $r$ satisfies $a < j$ (a\u2026","title":"Valid gates of a succinct 3SAT descriptor"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/s3-well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.s3WellFormed","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#s3-well-formed","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"Let $C$ be a list of gates encoded as (kind, references) pairs and let $n$ be a natural number. The descriptor $C$ is well-formed for $n$ when $C$ has at least $n + 1$ gates, every gate of $C$ is valid for $n$, and the flat Boolean circuit on $n + 2$ inputs obtained by decoding every gate of $C$ is well-formed.","title":"Well-formed succinct 3SAT descriptors"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/s3-bits-of","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.s3BitsOf","parent":36206,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lowest "],["math","\\(n\\)"],["text"," bits of a number, least significant first"]],"source":"Lowest $n$ bits of a number, least significant first"},"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#s3-bits-of","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"For natural numbers $i$ and $n$, this is the list of booleans $(b_0, \\dots, b_{n-1})$ of length $n$ in which $b_k$ is true exactly when $\\lfloor i / 2^k \\rfloor$ is odd: the $n$ lowest binary digits of $i$, least significant bit first.","title":"Lowest $n$ bits of a number, least significant first"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/s3-bits-to-num","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.s3BitsToNum","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#s3-bits-to-num","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"A list of booleans $(b_0, \\dots, b_{m-1})$ represents the natural number $\\sum_{k < m} [b_k]\\, 2^k$, where $[b_k]$ is $1$ if $b_k$ is true and $0$ otherwise.","title":"Value of a list of bits, least significant first"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/succ-literal","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.succLiteral","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#succ-literal","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"Let $C$ be a list of gates encoded as (kind, references) pairs and let $n$, $i$ and $j$ be natural numbers. Give $C$, as a circuit on $n + 2$ inputs, the input consisting of the $n$ lowest bits of $i$ followed by the $2$ lowest bits of $j$, each least significant bit first, and let $(o_0, \\dots, o_{p-1})$ be its last $n + 1$ outputs. Literal $j$ of clause $i$ is the pair $(v, s)$ with\u2026","title":"Literals of a succinctly described formula"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/succ-clause-sat","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.succClauseSat","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#succ-clause-sat","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"Let $C$ be a list of gates encoded as (kind, references) pairs, let $n$ and $i$ be natural numbers, and let $\\sigma$ assign a boolean to every natural number. Clause $i$ of the formula described by $C$ for $n$ is satisfied by $\\sigma$ when, for some $j < 3$, literal $j$ of clause $i$ of that formula is a pair $(v, s)$ with $\\sigma(v) = s$.","title":"Satisfied clauses of a succinctly described formula"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/succinct-three-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.succinctThreeSatDec","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#succinct-three-sat-dec","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"fully_proved","summary":"Succinct 3SAT is the decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a pair $(n, C)$ of a natural number $n$ and a list $C$ of gates encoded as (kind, references) pairs such that $C$ is a well-formed descriptor for $n$ and some assignment $\\sigma$ of booleans to the natural numbers satisfies clause $c$ of the formula described by $C$ for every\u2026","title":"Succinct 3SAT"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-3sat/succinct-three-sat-nexp-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctThreeSATNEXPComplete.succinct_three_sat_nexp_complete","parent":36206,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-3sat/index.html#succinct-three-sat-nexp-complete","scope":"TheoryOfComputation/completeness-other-classes/succinct-3sat","status":"can_prove","summary":"The succinct 3SAT decision problem lies in NEXP, and every decision problem in NEXP has a polynomial-time many-one reduction to it.","title":"Succinct 3SAT is NEXP-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem.json","id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"can_prove","title":"Succinct circuit value problem"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/scvgate","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.SCVGate","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#scvgate","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"An encoded gate is a pair $(k, r)$ of a natural number $k$, the kind of the gate, and a list $r$ of natural numbers, its references.","title":"Encoded gates (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/scv-circuit-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.scvCircuitEval","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#scv-circuit-eval","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"Let $C$ be a list of encoded gates, $a$ a list of booleans and $n \\in \\mathbb{N}$. Decode each gate of $C$ into a gate of a flat Boolean circuit on $n$ inputs, and assign to input $j < n$ the entry $a_j$ of $a$, read as false when $j$ is at least the length of $a$. The result is the list of the values of all gates of the decoded circuit under this assignment, in order.","title":"Gate values of an encoded circuit (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/scv-circuit-outputs","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.scvCircuitOutputs","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#scv-circuit-outputs","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"Let $C$ be a list of encoded gates, $a$ a list of booleans and $k, n \\in \\mathbb{N}$. Decode each gate of $C$ into a gate of a flat Boolean circuit on $n$ inputs and evaluate the decoded circuit with input $j < n$ set to the entry $a_j$ of $a$, read as false when $j$ is at least the length of $a$. The outputs of $C$ on $a$ are the last $k$ of the resulting gate values, or all of them if $C$ has fewer than $k$ gates.","title":"Outputs of an encoded circuit (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/scv-bits-of","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.scvBitsOf","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#scv-bits-of","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"For natural numbers $i$ and $n$, this is the list of $n$ booleans $(b_0, \\dots, b_{n-1})$, least significant bit first, where $b_k$ is true exactly when $\\lfloor i / 2^k \\rfloor$ is odd. For $i < 2^n$ it is the $n$-bit binary expansion of $i$; in general it is the $n$-bit expansion of $i \\bmod 2^n$.","title":"Fixed-length binary expansion (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/scv-bits-to-num","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.scvBitsToNum","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#scv-bits-to-num","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"A list of booleans $(b_0, \\dots, b_{m-1})$, read least significant bit first, denotes the natural number $\\sum_{k < m} [b_k] \\, 2^k$, where $[b_k]$ is $1$ if $b_k$ is true and $0$ otherwise; the empty list denotes $0$.","title":"Number with a given binary expansion (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/succ-desc","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.succDesc","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#succ-desc","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"Let $C$ be a list of encoded gates and let $n, i \\in \\mathbb{N}$. Decode each gate of $C$ into a gate of a flat Boolean circuit on $n$ inputs and evaluate it with input $j < n$ set to bit $j$ of $i$, least significant bit first, that is, true exactly when $\\lfloor i / 2^j \\rfloor$ is odd. The description of gate $i$ of the circuit succinctly described by $C$ is the list of the last $2n + 3$ resulting gate values,\u2026","title":"Gate description in a succinct circuit (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/succ-gate-info","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.succGateInfo","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#succ-gate-info","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"Let $C$ be a list of encoded gates, let $n, i \\in \\mathbb{N}$, and let $d$ be the description of gate $i$ of the circuit succinctly described by $C$: decode each gate of $C$ into a gate of a flat Boolean circuit on $n$ inputs, evaluate the decoded circuit with input $j < n$ set to bit $j$ of $i$ (true exactly when $\\lfloor i / 2^j \\rfloor$ is odd), and take the last $2n + 3$ gate values, or all of them if $C$ has\u2026","title":"Decoded gate description (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/succ-gate-val","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.succGateVal","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#succ-gate-val","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"Let $C$ be a list of encoded gates, let $n, i \\in \\mathbb{N}$, let $x$ be a list of booleans (the input) and let $v$ be a list of booleans (the values of earlier gates), where an entry outside $x$ or $v$ reads as false. Let $(t, p_1, p_2)$ be the decoded description of gate $i$ of the circuit succinctly described by $C$. If $i < n$, gate $i$ is an input gate and its value is $x_i$. Otherwise its value is\u2026","title":"Value of a gate of a succinct circuit (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/succ-eval","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.succEval","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#succ-eval","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"Let $C$ be a list of encoded gates, $n \\in \\mathbb{N}$ and $x$ a list of booleans. The gate values of the circuit with $2^n$ gates succinctly described by $C$ on input $x$ form the list $(v_0, \\dots, v_{2^n - 1})$, where each $v_i$ is the value of gate $i$ given $x$ and the values $(v_0, \\dots, v_{i-1})$ of the earlier gates.","title":"Evaluation of a succinct circuit (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/succ-desc-well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.succDescWellFormed","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#succ-desc-well-formed","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"A list $C$ of encoded gates is a well-formed description circuit on $n$ inputs when $C$ has at least $2n + 3$ gates and the flat Boolean circuit on $n$ inputs obtained by decoding each gate of $C$ is well-formed, that is, every gate references only strictly earlier gates.","title":"Well-formed description circuits (Succinct Circuit Value)"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/succ-well-formed","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.succWellFormed","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#succ-well-formed","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"Let $C$ be a list of encoded gates, read as a flat Boolean circuit on $n$ inputs that describes a circuit $D$ with $2^n$ gates. For each gate $i$ of $D$, let $t_i$, $p_i$ and $q_i$ be its gate type, first predecessor and second predecessor, as decoded from the outputs of $C$ on the $n$-bit binary expansion of $i$. The description is well formed when, for every $i < 2^n$: if $i < n$, then $t_i = 0$ (gate $i$ is an\u2026","title":"Well-formed succinct circuit descriptions"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/succinct-circuit-value-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.succinctCircuitValueDec","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#succinct-circuit-value-dec","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"fully_proved","summary":"The decision problem Succinct Circuit Value accepts a bitstring exactly when it decodes to a triple $(n, C, x)$ of a natural number $n$, a list $C$ of encoded gates and a bitstring $x$ such that: $x$ has length $n$; $C$ has at least $2n + 3$ gates and, read as a flat Boolean circuit on $n$ inputs, is well formed; the circuit $D$ with $2^n$ gates succinctly described by $C$, whose gate $i$ is described by the\u2026","title":"Succinct Circuit Value"},{"id":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/succinct-circuit-value-exp-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.SuccinctCircuitValueEXPComplete.succinct_circuit_value_exp_complete","parent":36218,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem/index.html#succinct-circuit-value-exp-complete","scope":"TheoryOfComputation/completeness-other-classes/succinct-circuit-value-problem","status":"can_prove","summary":"The decision problem Succinct Circuit Value belongs to the class EXP, and every decision problem in EXP reduces to Succinct Circuit Value by a polynomial-time many-one reduction.","title":"Succinct Circuit Value is EXP-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas.json","id":"TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas/index.html","scope":"TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas","status":"can_prove","title":"True quantified Boolean formulas"},{"id":"TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas/eval-qbf","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.TQBFPSPACEComplete.EvalQBF","parent":36232,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas/index.html#eval-qbf","scope":"TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas","status":"fully_proved","summary":"Let $\\varphi$ be a CNF formula, let $a : \\mathbb{N} \\to \\{\\mathrm{true}, \\mathrm{false}\\}$ be an assignment, and let $Q$ be a quantifier prefix, a list of pairs $(e, v)$ of a flag $e$ (true for $\\exists$, false for $\\forall$) and a variable $v \\in \\mathbb{N}$. Write $a[v \\mapsto b]$ for the assignment that agrees with $a$ except that it sends $v$ to $b$. With the empty prefix, the formula holds under $a$ when $a$\u2026","title":"Evaluation of a quantified CNF formula"},{"id":"TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas/tqbf-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.TQBFPSPACEComplete.tqbfDec","parent":36232,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas/index.html#tqbf-dec","scope":"TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas","status":"fully_proved","summary":"The TQBF decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a pair $(Q, \\varphi)$, where $Q$ is a quantifier prefix, a list of $m$ pairs $(e_1, v_1), \\dots, (e_m, v_m)$ of a flag $e_i$ (true for $\\exists$, false for $\\forall$) and a variable $v_i \\in \\mathbb{N}$, and $\\varphi$ is a CNF formula, such that $v_1, \\dots, v_m$ are $0, 1, \\dots, m - 1$ in some\u2026","title":"TQBF as a decision problem"},{"id":"TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas/tqbf-pspace-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.TQBFPSPACEComplete.tqbf_pspace_complete","parent":36232,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas/index.html#tqbf-pspace-complete","scope":"TheoryOfComputation/completeness-other-classes/true-quantified-boolean-formulas","status":"can_prove","summary":"The TQBF decision problem, of true quantified Boolean formulas with a CNF matrix, lies in the class PSPACE of decision problems, and every decision problem in PSPACE has a polynomial-time many-one reduction to it.","title":"TQBF is PSPACE-complete"},{"data_url":"scopes/TheoryOfComputation/completeness-other-classes/unsatisfiability.json","id":"TheoryOfComputation/completeness-other-classes/unsatisfiability","kind":"scope","navigation":"module","parent":36163,"result":false,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/unsatisfiability/index.html","scope":"TheoryOfComputation/completeness-other-classes/unsatisfiability","status":"can_prove","title":"Unsatisfiability"},{"id":"TheoryOfComputation/completeness-other-classes/unsatisfiability/unsat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.UNSATCoNPComplete.unsatDec","parent":36236,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/unsatisfiability/index.html#unsat-dec","scope":"TheoryOfComputation/completeness-other-classes/unsatisfiability","status":"fully_proved","summary":"The UNSAT decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a pair $(n, \\varphi)$ of a natural number $n$ and a CNF formula $\\varphi$ such that every variable occurring in $\\varphi$ is less than $n$ and, for every assignment $a$ of booleans to the variables $0, \\dots, n - 1$, the assignment that agrees with $a$ below $n$ and sends every variable $v \\ge n$\u2026","title":"UNSAT as a decision problem"},{"id":"TheoryOfComputation/completeness-other-classes/unsatisfiability/unsat-conp-complete","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.UNSATCoNPComplete.unsat_conp_complete","parent":36236,"result":true,"route":"../roadmap/TheoryOfComputation/completeness-other-classes/unsatisfiability/index.html#unsat-conp-complete","scope":"TheoryOfComputation/completeness-other-classes/unsatisfiability","status":"can_prove","summary":"The UNSAT decision problem lies in coNP, and every decision problem in coNP has a polynomial-time many-one reduction to it.","title":"UNSAT is coNP-complete"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems.json","id":"TheoryOfComputation/polynomial-time-problems","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/index.html","scope":"TheoryOfComputation/polynomial-time-problems","status":"can_prove","summary":"Proofs or statements that specific problems lie in P or L: graph reachability (PATH) with its marking machine, undirected connectivity in L, 2-SAT, primality, matching, maximum flow, linear programming, integer programming in fixed dimension, and hypergraph matching and cut questions.","title":"Problems solvable in polynomial time"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/2-satisfiability.json","id":"TheoryOfComputation/polynomial-time-problems/2-satisfiability","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/2-satisfiability/index.html","scope":"TheoryOfComputation/polynomial-time-problems/2-satisfiability","status":"can_prove","title":"2-satisfiability"},{"id":"TheoryOfComputation/polynomial-time-problems/2-satisfiability/two-sat-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.TwoSATInP.twoSatDec","parent":36240,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/2-satisfiability/index.html#two-sat-dec","scope":"TheoryOfComputation/polynomial-time-problems/2-satisfiability","status":"fully_proved","summary":"The 2-SAT decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a pair $(n, \\varphi)$ of a natural number $n$ and a CNF formula $\\varphi$ such that every clause of $\\varphi$ has at most two literals, every variable occurring in $\\varphi$ is less than $n$, and there is a list $(a_0, \\dots, a_{n-1})$ of booleans such that $\\varphi$ is satisfied by the assignment\u2026","title":"2-SAT as a decision problem"},{"id":"TheoryOfComputation/polynomial-time-problems/2-satisfiability/two-sat-in-p","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.TwoSATInP.two_sat_in_p","parent":36240,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/2-satisfiability/index.html#two-sat-in-p","scope":"TheoryOfComputation/polynomial-time-problems/2-satisfiability","status":"can_prove","summary":"The 2-SAT decision problem lies in the complexity class P.","title":"2-SAT is in P"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/aks-primality-test.json","id":"TheoryOfComputation/polynomial-time-problems/aks-primality-test","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/aks-primality-test/index.html","scope":"TheoryOfComputation/polynomial-time-problems/aks-primality-test","status":"can_prove","title":"AKS primality test"},{"id":"TheoryOfComputation/polynomial-time-problems/aks-primality-test/primes-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PrimesInP.primesDec","parent":36243,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/aks-primality-test/index.html#primes-dec","scope":"TheoryOfComputation/polynomial-time-problems/aks-primality-test","status":"fully_proved","summary":"The primality decision problem is the decision problem that accepts a bitstring $x$ exactly when decoding $x$ with the canonical bitstring encoding of the natural numbers (Mathlib's binary encoding) yields a prime number, and rejects $x$ otherwise.","title":"The primality decision problem"},{"id":"TheoryOfComputation/polynomial-time-problems/aks-primality-test/primes-in-p","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.PrimesInP.primes_in_P","parent":36243,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/aks-primality-test/index.html#primes-in-p","scope":"TheoryOfComputation/polynomial-time-problems/aks-primality-test","status":"can_prove","summary":"The primality decision problem, which accepts a bitstring exactly when it decodes to a prime number under the canonical bitstring encoding of the natural numbers, belongs to the complexity class P.","title":"PRIMES is in P"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/graph-reachability.json","id":"TheoryOfComputation/polynomial-time-problems/graph-reachability","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","title":"Graph reachability"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/encode-p","kind":"node","lean":"Cslib.Complexity.Path.encodeP","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#encode-p","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, the instance with source $s$, target $t$ and the digraph given by $n$ and the edge list $E$ is encoded as $s$ ones, a zero, $t$ ones, a zero, then $n$ ones and a zero, followed, for each pair $(u, v)$ of $E$ in order, by $u$ ones, a zero, $v$ ones and a zero.","title":"Encoding of a PATH instance"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/decode-p","kind":"node","lean":"Cslib.Complexity.Path.decodeP","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#decode-p","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Writing $1$ for true and $0$ for false, a bit string $x$ is parsed by reading a unary number $s$ (a run of ones ended by a zero), then a unary number $t$ from the rest, then a graph from what remains: a unary number $n$ followed by pairs $(u, v)$ of unary numbers until the input ends. The result is $(s, t, n, E)$, with $E$ the list of these pairs in order; there is no result unless the input ends right after $n$ or\u2026","title":"Decoding a PATH instance"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/arc","kind":"node","lean":"Cslib.Complexity.Path.arc","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#arc","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and a list $E$ of pairs of natural numbers, there is an arc from $u$ to $v$ when the pair $(u, v)$ occurs in $E$, $u < n$ and $v < n$; listed pairs with an end at least $n$ give no arc.","title":"Arcs of the PATH digraph"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/path-dec","kind":"node","lean":"Cslib.Complexity.Path.pathDec","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#path-dec","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"PATH is the decision problem that accepts a bit string $x$ exactly when $x$ is the encoding of some instance $(s, t, n, E)$ with $s < n$, $t < n$ and a sequence $s = v_0, v_1, \\dots, v_k = t$ with $k \\ge 0$ such that each $(v_i, v_{i+1})$ is an arc, that is, a pair listed in $E$ with both ends below $n$.","title":"The PATH problem"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/init-marks","kind":"node","lean":"Cslib.Complexity.Path.initMarks","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#init-marks","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For $n, s \\in \\mathbb{N}$, the initial marks are the list of $n$ booleans whose entry at position $i < n$ is true exactly when $i = s$; all entries are false when $s \\ge n$.","title":"Initial marks"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/mark-pass","kind":"node","lean":"Cslib.Complexity.Path.markPass","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#mark-pass","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Given $n \\in \\mathbb{N}$, an edge list $E$ and a list $m$ of booleans (the marks), the pass scans the pairs $(u, v)$ of $E$ in order and, whenever $u < n$, $v < n$ and position $u$ of the current marks is true, sets position $v$ to true. A position beyond the end of the list counts as false and setting it changes nothing. Marks set earlier in the pass are seen by later pairs.","title":"One marking pass"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/reach-marks","kind":"node","lean":"Cslib.Complexity.Path.reachMarks","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#reach-marks","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For $n, s \\in \\mathbb{N}$ and an edge list $E$, the reach marks are obtained from the initial marks of $s$ among $0, \\dots, n - 1$ by applying the marking pass over $E$ with bound $n$ exactly $n$ times.","title":"Reach marks"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/path-check","kind":"node","lean":"Cslib.Complexity.Path.pathCheck","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#path-check","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"On a bit string $x$, the check decodes $x$ as a PATH instance. If decoding fails it returns false; if it yields $(s, t, n, E)$ it returns true exactly when $s < n$, $t < n$ and position $t$ of the marks after $n$ marking passes from $s$ over $E$ is true.","title":"The marking algorithm for PATH"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/decode-p-encode-p","kind":"node","lean":"Cslib.Complexity.Path.decodeP_encodeP","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#decode-p-encode-p","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For natural numbers $s, t, n$ and a list $E$ of pairs of natural numbers, the PATH instance parser applied to the PATH instance encoding of $(s, t, n, E)$ succeeds and returns $(s, t, n, E)$.","title":"Decoding inverts the PATH instance encoding"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/decode-p-eq-some","kind":"node","lean":"Cslib.Complexity.Path.decodeP_eq_some","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#decode-p-eq-some","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For a bit string $x$, natural numbers $s, t, n$ and a list $E$ of pairs of natural numbers, the PATH instance parser applied to $x$ succeeds with result $(s, t, n, E)$ if and only if $x$ is the PATH instance encoding of $(s, t, n, E)$.","title":"The PATH instance parser succeeds exactly on encodings"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/encode-p-inj","kind":"node","lean":"Cslib.Complexity.Path.encodeP_inj","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#encode-p-inj","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $s, t, n, s', t', n'$ be natural numbers and $E, E'$ lists of pairs of natural numbers. If the PATH instance encodings of $(s, t, n, E)$ and $(s', t', n', E')$ are equal, then $s = s'$, $t = t'$, $n = n'$ and $E = E'$.","title":"The PATH instance encoding is injective"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/get-d-set-true","kind":"node","lean":"Cslib.Complexity.Path.getD_set_true","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#get-d-set-true","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $m$ be a list of booleans and $v, w$ natural numbers, and read a position at or beyond the end of a list as $\\mathrm{false}$. Set the entry of $m$ at position $w$ to $\\mathrm{true}$, leaving $m$ unchanged if $w$ is not less than the length of $m$. Then position $v$ of the result holds $\\mathrm{true}$ if and only if position $v$ of $m$ holds $\\mathrm{true}$, or $v = w$ and $w$ is less than the length of $m$.","title":"Setting an entry of a boolean list to true"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/init-marks-get-d","kind":"node","lean":"Cslib.Complexity.Path.initMarks_getD","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#init-marks-get-d","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n, s, v$ be natural numbers, and read a position at or beyond the end of a list as $\\mathrm{false}$. Position $v$ of the initial marks for $n$ vertices and source $s$ holds $\\mathrm{true}$ if and only if $v < n$ and $v = s$.","title":"The initial marks mark exactly the source, if it is a vertex"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/length-mark-pass","kind":"node","lean":"Cslib.Complexity.Path.length_markPass","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#length-mark-pass","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n$ be a natural number, $E$ a list of pairs of natural numbers and $m$ a list of booleans. A marking pass over $E$ for $n$ vertices, applied to $m$, gives a list of the same length as $m$.","title":"A marking pass preserves the length"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/mark-pass-mono","kind":"node","lean":"Cslib.Complexity.Path.markPass_mono","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#mark-pass-mono","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n, v \\in \\mathbb{N}$, let $E$ be a list of pairs of natural numbers and let $m$ be a list of booleans. If $v < |m|$ and the $v$-th entry of $m$ (counting from $0$) is true, then the same holds for the result of the marking pass of $E$ with bound $n$ on $m$.","title":"Marks persist through a pass"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/mark-pass-sound","kind":"node","lean":"Cslib.Complexity.Path.markPass_sound","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#mark-pass-sound","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $P$ be a property of natural numbers and let $E$ be a list of pairs of natural numbers such that $P(u)$ implies $P(v)$ whenever $(u, v) \\in E$, $u < n$ and $v < n$. If $m$ is a list of booleans such that $P(v)$ holds for every $v < |m|$ whose entry in $m$ is true, then $P(v)$ holds for every $v$ whose entry is true in the result of the marking pass of $E$ with bound $n$ on $m$.","title":"Soundness of a pass"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/mark-pass-closed","kind":"node","lean":"Cslib.Complexity.Path.markPass_closed","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#mark-pass-closed","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$, let $E$ be a list of pairs of natural numbers and let $m$ be a list of booleans, where $m$ marks $w$ when $w < |m|$ and the $w$-th entry of $m$ (counting from $0$) is true. If $(u, v) \\in E$ with $u < n$ and $v < n$, $v < |m|$ and $m$ marks $u$, then the result of the marking pass of $E$ with bound $n$ on $m$ marks $v$.","title":"A pass marks the heads of arcs with marked tails"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/count-lt-of-get-d-imp","kind":"node","lean":"Cslib.Complexity.Path.count_lt_of_getD_imp","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#count-lt-of-get-d-imp","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $l_1$ and $l_2$ be lists of booleans of the same length such that every position holding $\\mathrm{true}$ in $l_1$ also holds $\\mathrm{true}$ in $l_2$. Then $l_1$ has at most as many $\\mathrm{true}$ entries as $l_2$, and strictly fewer if $l_1 \\ne l_2$.","title":"Counting true entries of pointwise dominated boolean lists"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/mark-pass-eq-or-count-lt","kind":"node","lean":"Cslib.Complexity.Path.markPass_eq_or_count_lt","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#mark-pass-eq-or-count-lt","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, every list $E$ of pairs of natural numbers and every list $m$ of booleans, let $m'$ be the result of the marking pass of $E$ with bound $n$ on $m$. Then $m' = m$, or $m'$ has strictly more entries equal to true than $m$.","title":"A pass either changes nothing or adds a mark"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/length-iterate-mark-pass","kind":"node","lean":"Cslib.Complexity.Path.length_iterate_markPass","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#length-iterate-mark-pass","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n$ be a natural number, $E$ a list of pairs of natural numbers and $m$ a list of booleans. For every natural number $k$, applying $k$ marking passes over $E$ for $n$ vertices to $m$ gives a list of the same length as $m$.","title":"Repeated marking passes preserve the length"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/iterate-mark-pass-mono","kind":"node","lean":"Cslib.Complexity.Path.iterate_markPass_mono","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#iterate-mark-pass-mono","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n$ and $v$ be natural numbers, $E$ a list of pairs of natural numbers and $m$ a list of booleans whose position $v$ holds $\\mathrm{true}$ (positions at or beyond the end read as $\\mathrm{false}$). Then for every natural number $k$, position $v$ still holds $\\mathrm{true}$ after $k$ marking passes over $E$ for $n$ vertices, starting from $m$.","title":"Marks survive repeated passes"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/length-reach-marks","kind":"node","lean":"Cslib.Complexity.Path.length_reachMarks","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#length-reach-marks","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For natural numbers $n, s$ and a list $E$ of pairs of natural numbers, the marks after $n$ marking passes over $E$ for $n$ vertices, starting from the initial marks for source $s$, form a list of length $n$.","title":"The final marks have one entry per vertex"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/mark-pass-reach-marks","kind":"node","lean":"Cslib.Complexity.Path.markPass_reachMarks","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#mark-pass-reach-marks","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n, s \\in \\mathbb{N}$ with $s < n$ and let $E$ be a list of pairs of natural numbers. Let $r$ be the list obtained from the initial marks (only $s$ marked among $0, \\dots, n - 1$) by $n$ marking passes of $E$ with bound $n$. Then one more marking pass of $E$ with bound $n$ leaves $r$ unchanged.","title":"Repeated passes reach a fixpoint"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/reach-marks-get-d-iff","kind":"node","lean":"Cslib.Complexity.Path.reachMarks_getD_iff","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#reach-marks-get-d-iff","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n, s, t \\in \\mathbb{N}$ with $s < n$ and $t < n$, and let $E$ be a list of pairs of natural numbers. Start from the marks on $0, \\dots, n - 1$ in which only $s$ is marked and apply $n$ marking passes of $E$ with bound $n$. Then $t$ is marked if and only if $t$ is reachable from $s$ in zero or more steps along the arcs of $E$ below $n$, the pairs $(u, v) \\in E$ with $u < n$ and $v < n$.","title":"The final marks are exactly the reachable vertices"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/path-dec-eq-path-check","kind":"node","lean":"Cslib.Complexity.Path.pathDec_eq_pathCheck","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#path-dec-eq-path-check","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For every bit string $x$, the marking algorithm returns the same answer on $x$ as the decision problem PATH.","title":"The marking algorithm decides PATH"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/k","kind":"node","lean":"Cslib.Complexity.Path.Machine.K","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#k","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"The PATH marking machine, run on an instance with source $s$, target $t$ and $n$ vertices, has eleven stacks: the input stack $X$, the output stack $O$, the stack $E$ of the edge bits (reversed between passes), the junk stack $J$, the counters $S$, $T$ and $N$ for $s$, $t$ and $n$ ($N$ counts the passes left), the counters $U$ and $V$ for the endpoints $u$ and $v$ of an arc, the mark stack $M$ (with the mark of\u2026","title":"Stacks of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/untitled","kind":"node","lean":"Cslib.Complexity.Path.Machine.\u0393","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#untitled","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"The output stack $O$ of the PATH marking machine holds Booleans, and every other stack holds symbols of the SAT verifier's symbol alphabet.","title":"Stack alphabets of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/lab","kind":"node","lean":"Cslib.Complexity.Path.Machine.Lab","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#lab","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For an instance with source $s$, target $t$, $n$ vertices and arcs $(u, v)$, the labels of the PATH marking machine are $\\mathsf{rdS}$, $\\mathsf{rdT}$ and $\\mathsf{rdN}$ (read $s$, $t$ and $n$); $\\mathsf{val0}$, $\\mathsf{valU}$ and $\\mathsf{valV}$ (validate the edge list: at a pair, inside $u$, inside $v$); $\\mathsf{goS}$ and $\\mathsf{backS}$ (walk to $s$ and set its mark, then restore); $\\mathsf{pass}$ (the pass\u2026","title":"Labels of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/stmt","kind":"node","lean":"Cslib.Complexity.Path.Machine.Stmt","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#stmt","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"The statements of the multi-stack Turing machine (TM2) model whose stacks carry the stack alphabets $\\Gamma$ of the marking machine, whose labels are the labels of the marking machine, and whose internal state is a boolean.","title":"Statements of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/one","kind":"node","lean":"Cslib.Complexity.Path.Machine.one","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#one","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"The unit symbol of the counters of the PATH marking machine is the clause-end symbol $\\mathrm{cl}$ of the SAT verifier's symbol alphabet.","title":"The counter unit of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/read-s","kind":"node","lean":"Cslib.Complexity.Path.Machine.readS","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#read-s","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Given a stack $d$, an encoding $g$ of bits as symbols of $d$, a list $ps$ of stack\u2013symbol pairs and labels $\\ell$ and $\\ell'$, the instruction of the marking machine that jumps to `bad` if $X$ is empty, and otherwise pops the top of $X$ as a bit $b$ (a symbol that is not a bit counts as false) and pushes $g(b)$ on $d$; then, if $b$ is true, it pushes $a$ on $q$ for each pair $(q, a)$ of $ps$, in order, and jumps to\u2026","title":"The unary read instruction"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/walk-s","kind":"node","lean":"Cslib.Complexity.Path.Machine.walkS","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#walk-s","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Given a counter stack $c$, a label $\\ell$ and a final statement $f$, the walk step pops the top of $c$. If $c$ was nonempty, it goes to $\\ell$ after moving the top mark of the marks stack $M$, if there is one, onto the parked stack $P$: the mark is popped, read as a bit (a symbol that is not a bit reads as false), and pushed on $P$ as that bit. If $c$ was empty, it runs $f$.","title":"Walk step of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/set-s","kind":"node","lean":"Cslib.Complexity.Path.Machine.setS","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#set-s","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For a label $\\ell$, the statement of the marking machine that pops the top cell of the mark stack $M$ and, if there was one, pushes the set mark (the bit true) onto $M$ in its place; in either case it then goes to $\\ell$.","title":"Mark-setting step"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/prog","kind":"node","lean":"Cslib.Complexity.Path.Machine.prog","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#prog","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"The multi-stack machine program whose input lies on the stack $X$ and whose answer is pushed on the Boolean stack $O$. To read a unary number off $X$ onto a stack $d$, it pops a bit off $X$ and pushes it on $d$, repeating while the bit is true and moving on after the first false bit; an empty $X$ jumps to `bad`. To walk along a counter $c$, it pops units off $c$ one at a time, each time moving the top mark of $M$\u2026","title":"The marking machine for PATH"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/start","kind":"node","lean":"Cslib.Complexity.Path.Machine.start","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#start","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"The marking machine for PATH starts at the label that reads the source vertex $s$, in unary, off the input.","title":"Start label of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/set-head","kind":"node","lean":"Cslib.Complexity.Path.Machine.setHead","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#set-head","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For a list of booleans, the list obtained by setting its first entry: the empty list is left unchanged, and a list $b :: r$ with first entry $b$ and remaining entries $r$ becomes $\\mathrm{true} :: r$.","title":"Setting the first mark"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/take-append-set-head","kind":"node","lean":"Cslib.Complexity.Path.Machine.take_append_setHead","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#take-append-set-head","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For every list $m$ of booleans and every natural number $j$, the first $j$ entries of $m$ followed by the remaining entries with their first entry set to true (the empty list stays empty) equal $m$ with entry $j$ set to true (which is $m$ itself when $j \\ge |m|$).","title":"Setting an entry by splitting the list"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/bit-of-drop","kind":"node","lean":"Cslib.Complexity.Path.Machine.bitOf_drop","parent":36246,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reading the mark of vertex "],["math","\\(j\\)"]],"source":"Reading the mark of vertex $j$"},"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#bit-of-drop","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $m$ be a list of Boolean marks and $j$ a natural number. Drop the first $j$ marks of $m$ and write the rest as bit symbols; reading the top symbol as a bit, with false when there is no top symbol, gives the entry of $m$ at position $j$ (counting from $0$), or false when $j$ is at least the length of $m$.","title":"Reading the mark of vertex $j$"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/arc-eq","kind":"node","lean":"Cslib.Complexity.Path.Machine.arc_eq","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#arc-eq","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $m$ be a list of $n$ Boolean marks and let $u$, $v$ be natural numbers; write $m_u$ for the entry of $m$ at position $u$ (counting from $0$), taken to be false when $u \\ge n$. If $u < n$, $v < n$ and $m_u$ is true, set position $v$ of $m$ to true, and otherwise keep $m$; the result is the same as setting position $v$ of $m$ to true when $m_u$ is true and keeping $m$ otherwise.","title":"Out-of-range arcs need no guard"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/set-replicate","kind":"node","lean":"Cslib.Complexity.Path.Machine.set_replicate","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#set-replicate","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For all natural numbers $n$ and $s$, setting entry $s$ of the list of $n$ copies of false to true (which changes nothing when $s \\ge n$) gives the initial marks of $n$ vertices from $s$: the list of length $n$ whose entry $i$ is true exactly when $i = s$.","title":"Initial marks by setting one entry"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/get-d-ite","kind":"node","lean":"Cslib.Complexity.Path.Machine.getD_ite","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#get-d-ite","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For every decidable proposition $p$, the optional mark that is a cleared mark (false) when $p$ holds and absent otherwise, read with default false, is false.","title":"A cleared mark or nothing reads as cleared"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/mark-pass-replicate","kind":"node","lean":"Cslib.Complexity.Path.Machine.markPass_replicate","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#mark-pass-replicate","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For all natural numbers $n$ and $k$ and every list $es$ of arcs of natural numbers, one pass of the marking algorithm with $n$ vertices over $es$ (Mark Pass) maps the list of $k$ cleared marks to itself.","title":"A pass sets no mark when none is set"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/path-check-eq","kind":"node","lean":"Cslib.Complexity.Path.Machine.pathCheck_eq","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#path-check-eq","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"If a bit string $x$ parses as a PATH instance with source $s$, target $t$, $n$ vertices and edge list $es$, then the marking algorithm on $x$ returns the mark of $t$ in the reach marks of $s$ (the marks after $n$ marking passes over $es$, starting from the marks on $0, \\dots, n - 1$ with only $s$ marked, and none if $s \\ge n$), where a vertex $t \\ge n$ counts as unmarked.","title":"The marking algorithm reads the mark of the target"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/fr","kind":"node","lean":"Cslib.Complexity.Path.Machine.Fr","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#fr","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For a list $A$ of stacks of the marking machine and two contents $S$, $S'$ of its stacks, $S'$ agrees with $S$ off $A$ when every stack not in $A$ holds the same list in $S'$ as in $S$.","title":"Agreement off a set of stacks"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/refl","kind":"node","lean":"Cslib.Complexity.Path.Machine.Fr.refl","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#refl","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For every list $A$ of stacks of the marking machine and every assignment $S$ of contents to its stacks, $S$ agrees with itself off $A$.","title":"Frames are reflexive"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/trans","kind":"node","lean":"Cslib.Complexity.Path.Machine.Fr.trans","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#trans","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $A$, $B$, $C$ be lists of stacks such that every stack in $A$ or in $B$ is in $C$. If stacks $S_2$ agree with $S_1$ on every stack outside $A$, and $S_3$ agree with $S_2$ on every stack outside $B$, then $S_3$ agree with $S_1$ on every stack outside $C$.","title":"Frames compose"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/upd","kind":"node","lean":"Cslib.Complexity.Path.Machine.Fr.upd","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#upd","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $A$ be a list of stacks of the PATH marking machine, and let $S$ and $S'$ be stack contents such that $S'$ agrees with $S$ on every stack outside $A$. Then for every stack $k \\in A$ and every list $l$ over the alphabet of $k$, the contents obtained from $S'$ by replacing stack $k$ with $l$ still agree with $S$ on every stack outside $A$.","title":"Updating a framed stack preserves the frame"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/fr-add-u","kind":"node","lean":"Cslib.Complexity.Path.Machine.fr_addU","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#fr-add-u","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $d$ be a stack of the marking machine, $ps$ a list of pairs of a stack and a symbol, $i$ a natural number, and $S$, $S'$ two contents of the stacks. Suppose every stack $q$ other than the input $X$ and $d$ holds in $S'$ the list it holds in $S$ with $i$ further copies of the symbol of the first entry for $q$ in $ps$ pushed on top (and unchanged if $ps$ has no entry for $q$). Then $S'$ agrees with $S$ on every\u2026","title":"The frame of a read"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/read-s-some","kind":"node","lean":"Cslib.Complexity.Path.Machine.readS_some","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#read-s-some","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $d$ be a stack other than $X$, let $g$ encode bits as symbols of $d$, and let $ps$ be a list of stack\u2013symbol pairs whose stacks are distinct and differ from $X$ and $d$. Suppose the instruction of the marking machine at a label $\\ell$ is the unary read off $X$ onto $d$ through $g$ that pushes $ps$ for each true bit, repeats at $\\ell$ and continues at $\\ell'$. If $X$ holds, as bit symbols, $k$ true bits, then a\u2026","title":"Reading a unary number"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/read-s-none","kind":"node","lean":"Cslib.Complexity.Path.Machine.readS_none","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#read-s-none","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $d$ be a stack other than $X$ and $O$, let $g$ encode bits as symbols of $d$, and let $ps$ be a list of stack\u2013symbol pairs whose stacks are distinct and differ from $X$ and $O$. Suppose the instruction of the marking machine at a label $\\ell$ is the unary read off $X$ onto $d$ through $g$ that pushes $ps$ for each true bit, repeats at $\\ell$ and continues at $\\ell'$. If a bit list $l$ has no unary length prefix\u2026","title":"Reading a cut-off unary number"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/restore-run","kind":"node","lean":"Cslib.Complexity.Path.Machine.restore_run","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#restore-run","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Suppose the instruction of the marking machine at a label $\\ell$ moves the bits on top of $P$ onto $M$ one at a time, repeating at $\\ell$ and continuing at $\\ell'$ once $P$ is empty or its top is not a bit. If $P$ holds the reverse of a bit list $p$ and $M$ holds a bit list $q$, both as bit symbols, then in exactly $|p| + 1$ steps the machine goes from $\\ell$, with any flag, to $\\ell'$, leaving $P$ empty, $M$\u2026","title":"Restoring the parked marks"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/step-aux-walk-s-nil","kind":"node","lean":"Cslib.Complexity.Path.Machine.stepAux_walkS_nil","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#step-aux-walk-s-nil","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $c$ be a stack, $\\mathit{self}$ a label, $\\mathit{fin}$ a statement and $S$ stacks in which $c$ is empty. For every internal state $v$, executing from $v$ and $S$ the walk step counted by $c$ that loops at $\\mathit{self}$ and finishes with $\\mathit{fin}$ gives the same result as executing $\\mathit{fin}$ from internal state false and the same stacks $S$.","title":"A walk step with an exhausted counter"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/walk-run","kind":"node","lean":"Cslib.Complexity.Path.Machine.walk_run","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#walk-run","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $c$ be a stack other than the marks stack $M$ and the parked stack $P$, let $u$ be a symbol of the alphabet of $c$, and let $\\ell$ be a label at which the program is the walk step counted by $c$ that returns to $\\ell$ (with some final statement). Let $j$ be a natural number, $p$ and $q$ lists of bits, $v$ a Boolean, and $S$ stack contents in which $c$ holds exactly $j$ copies of $u$, $M$ holds $q$ as bit\u2026","title":"Walking along the marks of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/step-aux-set-s","kind":"node","lean":"Cslib.Complexity.Path.Machine.stepAux_setS","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#step-aux-set-s","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $\\ell$ be a label, $v$ an internal state and $S$ stacks whose mark stack $M$ holds the bits of a list $l$ of booleans. Executing the mark-setting step to $\\ell$ from internal state $v$ and stacks $S$ ends at label $\\ell$ with some internal state and the stacks $S$ with $M$ replaced by the bits of $l$ with its first entry set to true (the empty list stays empty); every other stack is unchanged.","title":"Effect of the mark-setting step"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/set-w-run","kind":"node","lean":"Cslib.Complexity.Path.Machine.setW_run","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#set-w-run","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $c$ be a stack other than the mark stack $M$ and the parking stack $P$, and let $\\mathit{self}$, $\\mathit{back}$, $\\mathit{next}$ be labels such that the marking machine at $\\mathit{self}$ is the walk step counted by $c$ that loops at $\\mathit{self}$ and finishes with the mark-setting step to $\\mathit{back}$, and at $\\mathit{back}$ it moves the bit block of $P$ onto $M$, looping at $\\mathit{back}$ and then\u2026","title":"Walking to a vertex and setting its mark"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/read-w-run","kind":"node","lean":"Cslib.Complexity.Path.Machine.readW_run","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#read-w-run","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $u \\in \\mathbb{N}$ and let $m$ be a list of marks. Start the marking machine at the label `go1`, with any flag, from stacks on which $U$ holds $u$ counter units, $M$ holds $m$ as bit symbols and $P$ is empty. Then within $2u + 2$ steps it reaches `go2` if the mark of vertex $u$ in $m$ is set and `drV` otherwise (a vertex past the end of $m$ counts as unmarked), with $U$ and $P$ empty, $M$ holding $m$ again, and\u2026","title":"Reading the mark of a vertex"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/val-run","kind":"node","lean":"Cslib.Complexity.Path.Machine.val_run","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#val-run","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $w$ be a list of bits, $v$ a Boolean and $S$ stack contents whose input stack $X$ holds $w$ as bit symbols. Then there are a Boolean $v'$ and stack contents $S'$ such that, in at most $2|w| + 1$ steps, the machine goes from the validation label (at a pair) with state $v$ and stacks $S$ to state $v'$ and stacks $S'$ at the label that walks to the source vertex if $w$ decodes as an edge list, and at the label\u2026","title":"Validation phase of the PATH marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/edges-run","kind":"node","lean":"Cslib.Complexity.Path.Machine.edges_run","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#edges-run","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n$ be a natural number, $es$ a list of arcs $(u, w)$ of natural numbers and $m$ a list of $n$ Boolean marks. Encode an arc $(u, w)$ by $u$ ones, a zero, $w$ ones and a zero, and $es$ by the concatenation of the codes of its arcs; let $\\ell$ be the length of this code. Suppose the marking machine is at the edge-loop label, in any internal state, with stacks $S$ such that the input stack $X$ holds the code of\u2026","title":"The edge loop of a pass"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/passes-run","kind":"node","lean":"Cslib.Complexity.Path.Machine.passes_run","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#passes-run","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"Let $n, k \\in \\mathbb{N}$, let $es$ be a list of pairs of natural numbers with edge bits $b$, and let $m$ be a list of $n$ marks. Start the marking machine at the label `pass`, with any flag, from stacks on which $N$ holds $k$ counter units, $X$, $P$, $U$ and $V$ are empty, $E$ holds the reverse of $b$ and $M$ holds $m$, both as bit symbols. Then within $k(5|b| + 3) + 1$ steps the machine reaches the label `goT`,\u2026","title":"The pass loop of the marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/run","kind":"node","lean":"Cslib.Complexity.Path.Machine.run","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#run","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"For every bit string $x$, the marking machine started at its start label, with any flag, with $x$ on the input stack $X$ as bit symbols and every other stack empty, reaches the label `done` within $8(|x| + 1)^4$ steps, with the output stack $O$ holding exactly one entry: the answer of the marking algorithm on $x$.","title":"Running time and output of the marking machine"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/path-comp","kind":"node","lean":"Cslib.Complexity.Path.pathComp","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#path-comp","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"The component machine whose stack indices, stack alphabets, labels and program are those of the marking machine, with its eleven stacks listed once each, in the order: input, output, edge bits, junk, the counters for the source, the target and the number of vertices, the counters for the two ends of an arc, the marks and the parked marks.","title":"The marking machine as a host component"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/is-poly-time-path-check","kind":"node","lean":"Cslib.Complexity.Path.isPolyTime_pathCheck","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#is-poly-time-path-check","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"The marking algorithm for PATH, as a function from bit strings to booleans, is computable in polynomial time by a Turing machine with respect to the canonical bitstring encodings of lists of booleans and of booleans.","title":"The marking algorithm is polynomial-time"},{"id":"TheoryOfComputation/polynomial-time-problems/graph-reachability/path-dec-mem-p","kind":"node","lean":"Cslib.Complexity.Path.pathDec_mem_P","parent":36246,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/graph-reachability/index.html#path-dec-mem-p","scope":"TheoryOfComputation/polynomial-time-problems/graph-reachability","status":"fully_proved","summary":"PATH \u2208 P (Sipser, Theorem 7.14). The decision problem PATH belongs to the class P of decision problems decidable in polynomial time. Here PATH accepts a bit string exactly when it encodes natural numbers $s$, $t$, $n$ and a list $E$ of pairs of natural numbers such that $s < n$, $t < n$ and $t$ is reachable from $s$ in zero or more steps along the arcs of $E$ below $n$, the pairs $(u, v) \\in E$ with $u < n$ and\u2026","title":"PATH \u2208 P"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k.json","id":"TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k/index.html","scope":"TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k","status":"can_prove","title":"Hypergraph k-cut for fixed k"},{"id":"TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k/hypergraph-kcut-dec","kind":"node","lean":"MathlibExt.Complexity.HypergraphKCutFixedKAMR29Problem0056Wanted.hypergraphKCutDec","parent":36310,"reading_title":{"format":"inline-parts-v1","parts":[["text","Decision version of hypergraph "],["math","\\(k\\)"],["text","-cut"]],"source":"Decision version of hypergraph $k$-cut"},"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k/index.html#hypergraph-kcut-dec","scope":"TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k","status":"fully_proved","summary":"For a natural number $k$, this is the decision problem on bitstrings that decodes an input as a triple $(n, E, B)$: a number $n$ of vertices $0, \\dots, n-1$, a list $E$ of hyperedges, each a list of vertices with a natural-number cost, and a natural-number budget $B$. It accepts the input iff every hyperedge lists only vertices below $n$ and there is a partition of the vertices into $k$ non-empty parts, given by a\u2026","title":"Decision version of hypergraph $k$-cut"},{"id":"TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k/hypergraph-kcut-in-p","kind":"node","lean":"MathlibExt.Complexity.HypergraphKCutFixedKAMR29Problem0056Wanted.hypergraphKCutInP","parent":36310,"reading_title":{"format":"inline-parts-v1","parts":[["text","The EGRES question on hypergraph "],["math","\\(k\\)"],["text","-cut for fixed "],["math","\\(k\\)"]],"source":"The EGRES question on hypergraph $k$-cut for fixed $k$"},"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k/index.html#hypergraph-kcut-in-p","scope":"TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k","status":"fully_proved","summary":"The statement that for every natural number $k$, the decision version of hypergraph $k$-cut is in the complexity class P. That problem accepts a hypergraph on the vertices $0, \\dots, n-1$, given by hyperedges with natural-number costs, and a budget $B$ iff every hyperedge lists only vertices below $n$ and some partition of the vertices into $k$ non-empty parts has total cost of the hyperedges meeting two parts at\u2026","title":"The EGRES question on hypergraph $k$-cut for fixed $k$"},{"id":"TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k/hypergraph-kcut-in-p-holds","kind":"node","lean":"MathlibExt.Complexity.HypergraphKCutFixedKAMR29Problem0056Wanted.hypergraphKCutInP_holds","parent":36310,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hypergraph "],["math","\\(k\\)"],["text","-cut for fixed "],["math","\\(k\\)"],["text"," is in P"]],"source":"Hypergraph $k$-cut for fixed $k$ is in P"},"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k/index.html#hypergraph-kcut-in-p-holds","scope":"TheoryOfComputation/polynomial-time-problems/hypergraph-k-cut-for-fixed-k","status":"can_prove","summary":"For every natural number $k$, the decision version of hypergraph $k$-cut is in the complexity class P. That problem accepts a hypergraph on the vertices $0, \\dots, n-1$, given by hyperedges with natural-number costs, and a budget $B$ iff every hyperedge lists only vertices below $n$ and some partition of the vertices into $k$ non-empty parts has total cost of the hyperedges meeting two parts at most $B$; inputs\u2026","title":"Hypergraph $k$-cut for fixed $k$ is in P"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension.json","id":"TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension/index.html","scope":"TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension","status":"can_prove","title":"Integer programming in fixed dimension"},{"id":"TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension/decode-int-ilp","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ILPFixedDimInP.decodeIntILP","parent":36314,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension/index.html#decode-int-ilp","scope":"TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension","status":"fully_proved","summary":"A pair $(n, s)$ of a natural number $n$ and a Boolean sign $s$ decodes to the integer $n$ if $s$ is true and to $-n$ if $s$ is false.","title":"Integer from magnitude and sign"},{"id":"TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension/ilp-fixed-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ILPFixedDimInP.ilpFixedDec","parent":36314,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension/index.html#ilp-fixed-dec","scope":"TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension","status":"fully_proved","summary":"For a natural number $d$, this decision problem accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring decoding, to a pair $(A, b)$ such that $A$ is a list of rows, each a list of exactly $d$ integers, $b$ is a list of integers with as many entries as $A$ has rows, and there is an integer vector $y \\in \\mathbb{Z}^d$ with $\\sum_{j < d} A_{rj} y_j \\le b_r$ for every row index $r$. Each\u2026","title":"Integer linear-programming feasibility in fixed dimension"},{"id":"TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension/ilp-fixed-dim-in-p","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.ILPFixedDimInP.ilp_fixed_dim_in_p","parent":36314,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension/index.html#ilp-fixed-dim-in-p","scope":"TheoryOfComputation/polynomial-time-problems/integer-programming-in-fixed-dimension","status":"can_prove","summary":"For every natural number $d$, integer linear-programming feasibility in dimension $d$ is in the complexity class P.","title":"Integer linear programming in fixed dimension is in P (Lenstra)"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility.json","id":"TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility/index.html","scope":"TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility","status":"can_prove","title":"Linear programming feasibility"},{"id":"TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility/decode-int","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LPInPWanted.decodeInt","parent":36318,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility/index.html#decode-int","scope":"TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility","status":"fully_proved","summary":"The integer decoded from a pair $(k, s)$ of a natural number $k$ and a boolean $s$ is $k$ if $s$ is true and $-k$ if $s$ is false.","title":"Signed integer from a magnitude\u2013sign pair"},{"id":"TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility/lp-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LPInPWanted.lpDec","parent":36318,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility/index.html#lp-dec","scope":"TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility","status":"fully_proved","summary":"The decision problem of rational linear-program feasibility accepts a bitstring $w$ exactly when $w$ decodes, under the canonical bitstring encoding, to a pair $(A, b)$ of a list $A$ of $m$ rows, each a list of magnitude\u2013sign pairs, and a list $b$ of $m'$ magnitude\u2013sign pairs, such that every row of $A$ has the length $n$ of the first row ($n = 0$ when $m = 0$), $m' = m$, and some $x \\in \\mathbb{Q}^n$ satisfies\u2026","title":"Rational linear-program feasibility"},{"id":"TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility/lp-in-p","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.LPInPWanted.lp_in_p","parent":36318,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility/index.html#lp-in-p","scope":"TheoryOfComputation/polynomial-time-problems/linear-programming-feasibility","status":"can_prove","summary":"The decision problem of rational linear-program feasibility is in the complexity class P. This problem accepts exactly the bitstrings that decode, under the canonical bitstring encoding, to a list $A$ of $m$ rows of integers, all of the same length $n$, and a list $b$ of $m$ integers, for which some $x \\in \\mathbb{Q}^n$ satisfies $\\sum_{j\\lt n} A_{rj} x_j \\le b_r$ for every $r < m$, where $A_{rj}$ is the $j$-th\u2026","title":"Linear programming is in P (Khachiyan)"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs.json","id":"TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs/index.html","scope":"TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs","status":"can_prove","title":"Matching in general graphs"},{"id":"TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs/matching-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MatchingInP.matchingDec","parent":36322,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs/index.html#matching-dec","scope":"TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs","status":"fully_proved","summary":"The decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a triple $(n, E, K)$ of a natural number $n$, a list $E$ of pairs of natural numbers and a natural number $K$ such that every pair $(u, v)$ in $E$ has $u < n$, $v < n$ and $u \\neq v$, and some sublist $M$ of $E$ (a subsequence, keeping the order of $E$) with no repeated entries and length at least\u2026","title":"General matching decision problem"},{"id":"TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs/matching-in-p","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MatchingInP.matching_in_p","parent":36322,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs/index.html#matching-in-p","scope":"TheoryOfComputation/polynomial-time-problems/matching-in-general-graphs","status":"can_prove","summary":"The general matching decision problem belongs to the complexity class P. This is the problem of deciding, for a vertex count $n$, a list $E$ of edges joining distinct vertices below $n$ and a target $K$, encoded as a bitstring, whether $E$ contains at least $K$ distinct pairwise vertex-disjoint edges.","title":"General matching is in P (Edmonds' blossom algorithm)"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/maximum-flow.json","id":"TheoryOfComputation/polynomial-time-problems/maximum-flow","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/maximum-flow/index.html","scope":"TheoryOfComputation/polynomial-time-problems/maximum-flow","status":"can_prove","title":"Maximum flow"},{"id":"TheoryOfComputation/polynomial-time-problems/maximum-flow/max-flow-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxFlowInP.maxFlowDec","parent":36325,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/maximum-flow/index.html#max-flow-dec","scope":"TheoryOfComputation/polynomial-time-problems/maximum-flow","status":"fully_proved","summary":"The decision problem that accepts a bitstring $x$ exactly when $x$ decodes, under the canonical bitstring encoding, to a tuple $(n, E, s, t, K)$ of a natural number $n$, a list $E = (e_0, \\dots, e_{m-1})$ of triples $e_j = (u_j, v_j, c_j)$ of natural numbers and natural numbers $s$, $t$ and $K$ such that $u_j < n$, $v_j < n$ and $u_j \\neq v_j$ for every $j < m$, $s < n$, $t < n$ and $s \\neq t$, and there are\u2026","title":"Max-flow decision problem"},{"id":"TheoryOfComputation/polynomial-time-problems/maximum-flow/max-flow-in-p","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.MaxFlowInP.max_flow_in_p","parent":36325,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/maximum-flow/index.html#max-flow-in-p","scope":"TheoryOfComputation/polynomial-time-problems/maximum-flow","status":"can_prove","summary":"The max-flow decision problem belongs to the complexity class P. This is the problem of deciding, for a vertex count $n$, a list of edges joining distinct vertices below $n$ with natural-number capacities, distinct vertices $s, t < n$ and a target $K$, encoded as a bitstring, whether some flow in natural numbers that respects the capacities and is conserved at every vertex other than $s$ and $t$ has net outflow\u2026","title":"Max flow is in P (Edmonds-Karp)"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs.json","id":"TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs/index.html","scope":"TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs","status":"can_prove","title":"Perfect matchings in dense uniform hypergraphs"},{"id":"TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs/perfect-matching-problem","kind":"node","lean":"MathlibExt.Complexity.DenseHypergraphPerfectMatchingOWR1386Problem015Wanted.perfectMatchingProblem","parent":36328,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs/index.html#perfect-matching-problem","scope":"TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs","status":"fully_proved","summary":"For a natural number $k$ and a rational $c$, $\\mathrm{PM}(k, c)$ is the decision problem on bitstrings that decodes an input as a pair $(n, E)$ of a natural number $n$ and a list $E$ of lists of natural numbers, and accepts it iff every list in $E$ consists of $k$ distinct vertices below $n$, $k \\mid n$, every $(k-1)$-element subset of $\\{0, \\dots, n-1\\}$ is contained in at least $cn$ edges of the hypergraph $H$\u2026","title":"Perfect matching in uniform hypergraphs with a codegree condition"},{"id":"TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs/dense-hypergraph-perfect-matching-in-p","kind":"node","lean":"MathlibExt.Complexity.DenseHypergraphPerfectMatchingOWR1386Problem015Wanted.denseHypergraphPerfectMatchingInP","parent":36328,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs/index.html#dense-hypergraph-perfect-matching-in-p","scope":"TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs","status":"fully_proved","summary":"The statement that for every natural number $k \\ge 2$ and every rational $c$ with $1/k \\le c < 1/2$, the decision problem $\\mathrm{PM}(k, c)$ is in the complexity class P. Here $\\mathrm{PM}(k, c)$ accepts a list of edges of a $k$-uniform hypergraph $H$ on the vertices $0, \\dots, n-1$ iff every edge consists of $k$ distinct vertices below $n$, $k \\mid n$, every $(k-1)$-element set of vertices lies in at least $cn$\u2026","title":"Ruci\u0144ski's question on perfect matchings in dense hypergraphs"},{"id":"TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs/dense-hypergraph-perfect-matching-in-p-h","kind":"node","lean":"MathlibExt.Complexity.DenseHypergraphPerfectMatchingOWR1386Problem015Wanted.denseHypergraphPerfectMatchingInP_holds","parent":36328,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs/index.html#dense-hypergraph-perfect-matching-in-p-h","scope":"TheoryOfComputation/polynomial-time-problems/perfect-matchings-in-dense-uniform-hypergraphs","status":"can_prove","summary":"Resolved true: Keevash, Knox and Mycroft give a polynomial-time algorithm when the minimum codegree is at least `n/k + \u03b3n` for a fixed `\u03b3 > 0`, and Han extends it to minimum codegree at least `n/k`, so the problem is in `P` for every `1/k \u2264 c < 1/2`. Inputs whose edge count is below what the codegree condition forces are rejected first, so `n` is polynomial in the input size whenever the remaining checks run.\u2026","title":"Perfect matchings in dense hypergraphs are decidable in polynomial time"},{"data_url":"scopes/TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity.json","id":"TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity","kind":"scope","navigation":"module","parent":36239,"result":false,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity/index.html","scope":"TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity","status":"can_prove","title":"Undirected s-t connectivity"},{"id":"TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity/ustcon-dec","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.USTCONInL.ustconDec","parent":36332,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity/index.html#ustcon-dec","scope":"TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity","status":"fully_proved","summary":"A bitstring $x$ is a yes-instance exactly when bitstring decoding of $x$, as a tuple $(n, E, s, t)$ of natural numbers $n$, $s$, $t$ and a finite list $E$ of pairs of natural numbers, succeeds and returns $(n, E, s, t)$ such that $s < n$, $t < n$, and there are pairwise distinct natural numbers $v_0, \\dots, v_k$, all less than $n$, with $v_0 = s$, $v_k = t$ and, for every $i < k$, $(v_i, v_{i+1}) \\in E$ or\u2026","title":"Undirected s-t connectivity as a decision problem"},{"id":"TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity/ustcon-in-l","kind":"node","lean":"MetaMathlibExt.Complexity.KeyTheorems.USTCONInL.ustcon_in_l","parent":36332,"result":true,"route":"../roadmap/TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity/index.html#ustcon-in-l","scope":"TheoryOfComputation/polynomial-time-problems/undirected-s-t-connectivity","status":"can_prove","summary":"The undirected $s$-$t$ connectivity decision problem lies in the class $\\mathsf{L}$ of decision problems decidable in deterministic logarithmic space.","title":"Undirected ST-connectivity is in L (Reingold)"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity.json","id":"TheoryOfComputation/fine-grained-complexity","kind":"scope","parent":30322,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/index.html","scope":"TheoryOfComputation/fine-grained-complexity","status":"can_prove","summary":"Exact running times of hard problems beyond the polynomial/non-polynomial divide: exponential, subexponential and quasipolynomial algorithms for NP-hard or NP-intermediate problems, such as the tensor-network 2^o(n)-time algorithm for 3-SAT and Babai's quasipolynomial-time graph isomorphism theorem, and the exponential time hypothesis.","title":"Fine-grained and exponential-time complexity"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time.json","id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","kind":"scope","navigation":"module","parent":36335,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"can_prove","title":"Graph isomorphism in quasipolynomial time"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/graph-code","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.GraphCode","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#graph-code","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"fully_proved","summary":"A graph code consists of a vertex count $n \\in \\mathbb{N}$, with vertices $0, \\dots, n-1$, and a Boolean adjacency matrix $A$ on these vertices that is symmetric ($A_{ij} = A_{ji}$ for all $i, j$) and loop-free ($A_{ii}$ is false for every $i$).","title":"Graph codes"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/nat-bits","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.natBits","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#nat-bits","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"fully_proved","summary":"The binary digits of $n \\in \\mathbb{N}$, most significant first, as a list of bits with the digit $1$ read as true and $0$ as false; for $n = 0$ this is the single digit $0$.","title":"Binary digits of a natural number"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/nat-to-bits","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.natToBits","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#nat-to-bits","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"fully_proved","summary":"The prefix-free encoding of $n \\in \\mathbb{N}$ consists of $k$ true bits, where $k$ is the number of binary digits of $n$, then one false bit, then the binary digits of $n$, most significant first.","title":"Prefix-free encoding of a natural number"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/encode","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.encode","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#encode","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"fully_proved","summary":"The bit encoding of a graph code with $n$ vertices and adjacency matrix $A$ is the prefix-free encoding of $n$ followed by the $n^2$ entries of $A$, as bits, in row-major order: $A_{0,0}, A_{0,1}, \\dots, A_{0,n-1}, A_{1,0}, \\dots, A_{n-1,n-1}$.","title":"Bit encoding of a graph code"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/encode-pair","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.encodePair","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#encode-pair","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"fully_proved","summary":"The bit encoding of a pair $(G, H)$ of graph codes is the bit encoding of $G$ followed by the bit encoding of $H$.","title":"Bit encoding of a pair of graph codes"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/isomorphic","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.Isomorphic","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#isomorphic","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"fully_proved","summary":"Graph codes $G$ and $H$ with adjacency matrices $A$ and $B$ are isomorphic when they have the same vertex count $n$ and there is a permutation $\\sigma$ of the vertices $\\{0, \\dots, n-1\\}$ such that $A_{ij} = B_{\\sigma(i)\\sigma(j)}$ for all $i, j$.","title":"Isomorphism of graph codes"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/qp-time","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.qpTime","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#qp-time","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"fully_proved","summary":"For an exponent $c \\in \\mathbb{N}$ and an input length $n \\in \\mathbb{N}$, the quasipolynomial time bound is $q_c(n) = 2^{(\\lfloor \\log_2(n+2) \\rfloor + 2)^c}$.","title":"Quasipolynomial time bound"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/conjecture","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.conjecture","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#conjecture","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"fully_proved","summary":"The proposition that graph isomorphism is decidable in quasipolynomial time: there are a function $D$ from pairs of graph codes to Booleans, an exponent $c \\in \\mathbb{N}$, and a deterministic multi-stack Turing machine with a time function $t : \\mathbb{N} \\to \\mathbb{N}$ that, on the bit encoding of every pair $(G, H)$, outputs the single bit $D(G, H)$ within $t(\\ell)$ steps, where $\\ell$ is the length of the\u2026","title":"The quasipolynomial graph isomorphism statement"},{"id":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/conjecture-holds","kind":"node","lean":"MathlibExt.ComputerScience.GraphIsomorphismQuasipolynomialWanted.conjecture_holds","parent":36336,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time/index.html#conjecture-holds","scope":"TheoryOfComputation/fine-grained-complexity/graph-isomorphism-in-quasipolynomial-time","status":"can_prove","summary":"The quasipolynomial graph isomorphism statement holds: there are a function $D$ from pairs of graph codes to Booleans, an exponent $c \\in \\mathbb{N}$, and a deterministic multi-stack Turing machine with a time function $t : \\mathbb{N} \\to \\mathbb{N}$ that, on the bit encoding of every pair $(G, H)$, outputs the single bit $D(G, H)$ within $t(\\ell)$ steps, where $\\ell$ is the length of the encoding, such that\u2026","title":"Babai's theorem: graph isomorphism in quasipolynomial time"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction","kind":"scope","parent":36335,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction","status":"fully_proved","title":"Tensor-network contraction"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/grid-value-eq-sum-choices","kind":"node","lean":"TensorNetwork.gridValue_eq_sum_choices","parent":36346,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/index.html#grid-value-eq-sum-choices","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction","status":"fully_proved","summary":"If every active rectangle kernel is a finite sum, the grid value is the sum over one kernel choice for each active rectangle. Raw rectangle indices outside `gridRectangles` retain the original kernel, which does not affect `gridValue`.","title":"Grid value expanded over kernel choices"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/abs-to-instance-le","kind":"node","lean":"TensorNetwork.Factored.abs_toInstance_le","parent":36346,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/index.html#abs-to-instance-le","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction","status":"fully_proved","summary":"A converted tensor entry is a sum of at most `2 ^ P` terms, each containing one weight and `Fintype.card J - 1` off-diagonal factors.","title":"Entry bound for a converted tensor instance"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","title":"Integer-valued factored tensor networks"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/ordered-pairs","kind":"node","lean":"TensorNetwork.orderedPairs","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#ordered-pairs","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"Pairs `(a, b)` with `a < b`, representing unordered pairs once.","title":"Ordered pairs representing unordered pairs"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/mem-ordered-pairs","kind":"node","lean":"TensorNetwork.mem_orderedPairs","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#mem-ordered-pairs","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"Membership in `orderedPairs` is strict ordering of the two coordinates.","title":"Membership in the ordered pairs"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/pair-index","kind":"node","lean":"TensorNetwork.pairIndex","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#pair-index","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"The ordered representative `(min a b, max a b)` of the pair `{a, b}`.","title":"Ordered representative of an unordered pair"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/edge-labelings","kind":"node","lean":"TensorNetwork.edgeLabelings","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#edge-labelings","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"Complete-graph edge labelings, free on `orderedPairs` and default elsewhere.","title":"Edge labelings of the complete graph"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/mem-edge-labelings","kind":"node","lean":"TensorNetwork.mem_edgeLabelings","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#mem-edge-labelings","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"A labeling belongs to `edgeLabelings` exactly when active edges use allowed labels and all other coordinates contain the default.","title":"Characterization of complete-graph edge labelings"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/incident-labels","kind":"node","lean":"TensorNetwork.incidentLabels","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#incident-labels","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"Labels incident to `v`, indexed by the other endpoint.","title":"Edge labels incident to a vertex"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/instance","kind":"node","lean":"TensorNetwork.Instance","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#instance","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"An integer-valued local tensor at each vertex.","title":"Integer-valued complete-graph tensor instance"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/value","kind":"node","lean":"TensorNetwork.Instance.value","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#value","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"The sum, over complete-graph edge labelings, of the product of local tensors.","title":"Value of a complete-graph tensor instance"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/factored","kind":"node","lean":"TensorNetwork.Factored","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#factored","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"An integer-valued factored tensor network.","title":"Integer-valued factored tensor network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/label-set","kind":"node","lean":"TensorNetwork.Factored.labelSet","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#label-set","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"The label set of an unordered vertex pair.","title":"Label set of an unordered vertex pair"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/label-set-comm","kind":"node","lean":"TensorNetwork.Factored.labelSet_comm","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#label-set-comm","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"The label set is independent of endpoint order.","title":"Label set is symmetric in its endpoints"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/interaction","kind":"node","lean":"TensorNetwork.Factored.interaction","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#interaction","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"The interaction of two states, summed over their shared edge labels.","title":"Interaction of two states in a factored network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/interaction-comm","kind":"node","lean":"TensorNetwork.Factored.interaction_comm","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#interaction-comm","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"Reading an interaction from the other endpoint swaps its states.","title":"Interaction is symmetric under swapping endpoints"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/value-2","kind":"node","lean":"TensorNetwork.Factored.value","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#value-2","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"The network value: sum over states of vertex weights times all pair interactions.","title":"Value of a factored tensor network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/total-capacity","kind":"node","lean":"TensorNetwork.Factored.totalCapacity","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#total-capacity","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"The directed total capacity `\u03a3_{a \u2260 b} c a b`; both orientations are counted.","title":"Directed total capacity of a factored network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/shaped","kind":"node","lean":"TensorNetwork.Factored.Shaped","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#shaped","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"A network of shape parameter `P`: at most `2 ^ P` states per vertex, symmetric capacities summing to at most `P` at each vertex, and at most `2 ^ c` labels per edge.","title":"Factored tensor network with shape parameter"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/sized","kind":"node","lean":"TensorNetwork.Factored.Sized","parent":36349,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks/index.html#sized","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/integer-valued-factored-tensor-networks","status":"fully_proved","summary":"A shaped network whose weights, factors and off-diagonal interactions are bounded by `2 ^ B` in absolute value.","title":"Shaped tensor network with bounded entries"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks","status":"fully_proved","title":"Contraction of factored tensor networks"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/merge","kind":"node","lean":"TensorNetwork.Factored.merge","parent":36367,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/index.html#merge","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks","status":"fully_proved","summary":"Merge vertices along `\u03c7`, retaining source states and labels on their classes and padding all other coordinates with the supplied defaults.","title":"Merging vertices of a factored tensor network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/merge-interaction-lt","kind":"node","lean":"TensorNetwork.Factored.merge_interaction_lt","parent":36367,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/index.html#merge-interaction-lt","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks","status":"fully_proved","summary":"The interaction of two increasingly ordered classes is the product of all source interactions crossing between them.","title":"Interaction of merged classes in increasing order"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/merge-interaction","kind":"node","lean":"TensorNetwork.Factored.merge_interaction","parent":36367,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/index.html#merge-interaction","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks","status":"fully_proved","summary":"The interaction of two distinct classes is the product of all source interactions crossing between them, oriented from the first class to the second.","title":"Interaction of two distinct merged classes"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/value-merge","kind":"node","lean":"TensorNetwork.Factored.value_merge","parent":36367,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/index.html#value-merge","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks","status":"fully_proved","summary":"Merging along any map preserves the exact integer-valued network value.","title":"Merging vertices preserves network value"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/card-merge-s","kind":"node","lean":"TensorNetwork.Factored.card_merge_S","parent":36367,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks/index.html#card-merge-s","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/contraction-of-factored-tensor-networks","status":"fully_proved","summary":"The number of merged states is the product of the state-set sizes in the class.","title":"Number of states of a merged vertex"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ","status":"fully_proved","title":"Bounds for contraction of factored tensor networks"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ/merge-shaped","kind":"node","lean":"TensorNetwork.Factored.merge_shaped","parent":36373,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ/index.html#merge-shaped","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ","status":"fully_proved","summary":"If every contraction class has at most `m` vertices, contraction multiplies the network's shape parameter by `m`.","title":"Shape bound for contracted tensor networks"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ/merge-sized","kind":"node","lean":"TensorNetwork.Factored.merge_sized","parent":36373,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ/index.html#merge-sized","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-contraction-of-factored-tensor-networ","status":"fully_proved","summary":"If every contraction class has at most `m` vertices, contraction multiplies the shape parameter by `m` and changes the absolute-value exponent from `B` to `B * (m + m * m)`.","title":"Size bound for contracted tensor networks"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ","status":"fully_proved","title":"Equivalent presentations of finite tensor networks"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/to-instance","kind":"node","lean":"TensorNetwork.Factored.toInstance","parent":36376,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/index.html#to-instance","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ","status":"fully_proved","summary":"Turn a factored network into a complete-graph tensor instance.","title":"Complete-graph instance of a factored network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/value-to-instance","kind":"node","lean":"TensorNetwork.Factored.value_toInstance","parent":36376,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/index.html#value-to-instance","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ","status":"fully_proved","summary":"Naming every source edge alphabet bijectively inside `labels` preserves network value.","title":"Converting a factored network to an instance preserves value"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/to-factored","kind":"node","lean":"TensorNetwork.Instance.toFactored","parent":36376,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/index.html#to-factored","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ","status":"fully_proved","summary":"Present an instance as a half-edge network.","title":"Half-edge factored presentation of a tensor instance"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/value-to-factored","kind":"node","lean":"TensorNetwork.Instance.value_toFactored","parent":36376,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/index.html#value-to-factored","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ","status":"fully_proved","summary":"The half-edge presentation of an instance has the same value.","title":"Half-edge presentation preserves instance value"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/value-eq-half-edges","kind":"node","lean":"TensorNetwork.Instance.value_eq_halfEdges","parent":36376,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ/index.html#value-eq-half-edges","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/equivalent-presentations-of-finite-tensor-networ","status":"fully_proved","summary":"Expand an instance value as a sum over directed half-edge vectors with equality factors.","title":"Instance value as a sum over half-edge vectors"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding","status":"fully_proved","title":"Tensor-instance alphabet padding"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/pad","kind":"node","lean":"TensorNetwork.Instance.pad","parent":36382,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/index.html#pad","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding","status":"fully_proved","summary":"Rename the active edge alphabet of an instance through `dec`, returning zero when any off-diagonal target label is outside `D`. The diagonal coordinate is replaced by `sourceDefault`.","title":"Alphabet padding of a tensor instance"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/to-instance-to-factored-eq-pad","kind":"node","lean":"TensorNetwork.Instance.toInstance_toFactored_eq_pad","parent":36382,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/index.html#to-instance-to-factored-eq-pad","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding","status":"fully_proved","summary":"Factoring an instance and converting it back through constant edge names gives its padded version. The decoder must send every admitted target name into the explicit source alphabet.","title":"Factoring and unfactoring a tensor instance gives its padding"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/to-instance-to-factored-univ-eq-pad","kind":"node","lean":"TensorNetwork.Instance.toInstance_toFactored_univ_eq_pad","parent":36382,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/index.html#to-instance-to-factored-univ-eq-pad","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding","status":"fully_proved","summary":"The source conversion theorem with the full finite source alphabet.","title":"Factoring round trip with the full source alphabet"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/value-pad","kind":"node","lean":"TensorNetwork.Instance.value_pad","parent":36382,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding/index.html#value-pad","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/tensor-instance-alphabet-padding","status":"fully_proved","summary":"Padding through a bijective renaming of an explicit source alphabet preserves instance value.","title":"Bijective alphabet padding preserves instance value"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","title":"Finite rectangular-grid tensor evaluation"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/grid-rectangles","kind":"node","lean":"TensorNetwork.gridRectangles","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#grid-rectangles","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"Ordered grid rectangles, indexed by an increasing row pair and increasing column pair.","title":"Ordered rectangles of a finite grid"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/rectangle-apply","kind":"node","lean":"TensorNetwork.rectangleApply","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#rectangle-apply","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"Apply a rectangle kernel to its four directed diagonal ends `x`, `y`, `u`, `v`.","title":"Applying a rectangle kernel to its diagonal ends"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/grid-labelings","kind":"node","lean":"TensorNetwork.gridLabelings","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#grid-labelings","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"Directed grid labelings with independent finite choices on edges and diagonal entries.","title":"Directed labelings of a finite grid"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/mem-grid-labelings","kind":"node","lean":"TensorNetwork.mem_gridLabelings","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#mem-grid-labelings","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"Membership in `gridLabelings` means precisely that diagonal and edge coordinates use their respective finite choice sets.","title":"Characterization of directed grid labelings"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/grid-value","kind":"node","lean":"TensorNetwork.gridValue","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#grid-value","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"The finite grid evaluation: local tensors, row and column edge equalities, and one kernel per ordered rectangle, summed over all directed labelings.","title":"Finite grid tensor evaluation"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/on-grid","kind":"node","lean":"TensorNetwork.Instance.onGrid","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#on-grid","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"Transport an instance to a grid, returning zero unless the self-label is the chosen default.","title":"Transporting a tensor instance to a grid"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/crossing","kind":"node","lean":"TensorNetwork.crossing","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#crossing","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"The crossing kernel requires opposite directed copies on both rectangle diagonals to agree.","title":"Crossing kernel on grid rectangles"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/value-eq-grid-value-crossing","kind":"node","lean":"TensorNetwork.Instance.value_eq_gridValue_crossing","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#value-eq-grid-value-crossing","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"Complete-graph tensor evaluation is the crossing grid evaluation after transport. The transport gates diagonal entries at `default`; other allowed diagonal labels therefore contribute zero. The default need not belong to the edge-label set.","title":"Complete-graph tensor value as a crossing grid evaluation"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/card-grid-rectangles","kind":"node","lean":"TensorNetwork.card_gridRectangles","parent":36387,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation/index.html#card-grid-rectangles","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-rectangular-grid-tensor-evaluation","status":"fully_proved","summary":"The number of ordered grid rectangles is the product of the two pair counts.","title":"Number of ordered grid rectangles"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors","status":"fully_proved","title":"Finite row networks for grid tensors"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/grid-row-network","kind":"node","lean":"TensorNetwork.gridRowNetwork","parent":36397,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/index.html#grid-row-network","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors","status":"fully_proved","summary":"Merge every grid row into one vertex of a factored tensor network. The state and label carriers may be infinite; only the supplied alphabets are enumerated. Capacity metadata charges `L` per vertical edge and the supplied cut capacity for each active rectangle.","title":"Row network of a grid tensor"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/grid-row-network-interaction","kind":"node","lean":"TensorNetwork.gridRowNetwork_interaction","parent":36397,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/index.html#grid-row-network-interaction","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors","status":"fully_proved","summary":"The interaction between increasing rows is the rectangle-kernel product times vertical equality indicators. For finite alphabets, an equal vertical value contributes only when it is in `labels`.","title":"Interaction between rows of the grid row network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/grid-row-network-interaction-of-mem","kind":"node","lean":"TensorNetwork.gridRowNetwork_interaction_of_mem","parent":36397,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/index.html#grid-row-network-interaction-of-mem","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors","status":"fully_proved","summary":"When the left state is enumerated by the row network, the alphabet-membership condition is automatic, so the interaction recovers the source vertical-equality formula. The right state may be arbitrary because equality with an allowed left value already forces its vertical value.","title":"Row interaction for enumerated left states"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/grid-row-network-value","kind":"node","lean":"TensorNetwork.gridRowNetwork_value","parent":36397,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors/index.html#grid-row-network-value","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/finite-row-networks-for-grid-tensors","status":"fully_proved","summary":"If each rectangle kernel is the finite sum of its top and bottom factors, contracting the constructed row network gives the canonical finite grid value.","title":"Grid row network computes the grid value"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks","status":"fully_proved","title":"Bounds for finite grid row networks"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/grid-row-network-capacity-le","kind":"node","lean":"TensorNetwork.gridRowNetwork_capacity_le","parent":36402,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/index.html#grid-row-network-capacity-le","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks","status":"fully_proved","summary":"The capacity between two grid rows is at most `g\u00b2mL` when every active rectangle is charged at most `2mL` and there are `g` columns.","title":"Capacity bound between grid rows"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/grid-row-network-shaped","kind":"node","lean":"TensorNetwork.gridRowNetwork_shaped","parent":36402,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/index.html#grid-row-network-shaped","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks","status":"fully_proved","summary":"The grid row network has shape `m g\u00b3 L` when both raw-state alphabets have at most `2^L` elements and every rectangle label set and capacity obeys its declared bound.","title":"Shape bound for the grid row network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/grid-row-network-sized","kind":"node","lean":"TensorNetwork.gridRowNetwork_sized","parent":36402,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/index.html#grid-row-network-sized","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks","status":"fully_proved","summary":"Under uniform raw tensor and factor bounds, the grid row network has exact source exponents `m g\u00b3 L` for shape and `g B + g\u00b2 b` for integer magnitudes.","title":"Size bound for the grid row network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/grid-row-network-total-capacity","kind":"node","lean":"TensorNetwork.gridRowNetwork_totalCapacity","parent":36402,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks/index.html#grid-row-network-total-capacity","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/bounds-for-finite-grid-row-networks","status":"fully_proved","summary":"The directed total capacity consists of all vertical row edges plus twice the active rectangle capacity; diagonal row capacities are excluded.","title":"Total capacity of the grid row network"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations","status":"fully_proved","title":"Transposition of finite grid tensor evaluations"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/transpose","kind":"node","lean":"TensorNetwork.Instance.transpose","parent":36407,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/index.html#transpose","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations","status":"fully_proved","summary":"Swap the row and column coordinates of a grid tensor instance.","title":"Transpose of a grid tensor instance"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/transpose-transpose","kind":"node","lean":"TensorNetwork.Instance.transpose_transpose","parent":36407,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/index.html#transpose-transpose","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations","status":"fully_proved","summary":"Transposing a grid tensor instance twice recovers the original instance.","title":"Transposing a grid tensor instance is an involution"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/transpose-rectangle-kernel","kind":"node","lean":"TensorNetwork.transposeRectangleKernel","parent":36407,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/index.html#transpose-rectangle-kernel","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations","status":"fully_proved","summary":"Transpose a rectangle kernel, exchanging the `y` and `u` directed diagonal ends.","title":"Transpose of a rectangle kernel"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/transpose-rectangle-kernel-transpose-rec","kind":"node","lean":"TensorNetwork.transposeRectangleKernel_transposeRectangleKernel","parent":36407,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/index.html#transpose-rectangle-kernel-transpose-rec","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations","status":"fully_proved","summary":"Transposing a rectangle kernel twice recovers the original kernel.","title":"Transposing a rectangle kernel is an involution"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/grid-value-transpose","kind":"node","lean":"TensorNetwork.gridValue_transpose","parent":36407,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations/index.html#grid-value-transpose","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/transposition-of-finite-grid-tensor-evaluations","status":"fully_proved","summary":"Transposing the grid coordinates, local tensors, and rectangle kernels preserves the finite grid evaluation. Edge and diagonal label choices remain independent.","title":"Transposition preserves the grid evaluation"},{"data_url":"scopes/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values.json","id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values","kind":"scope","navigation":"module","parent":36346,"result":false,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values/index.html","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values","status":"fully_proved","title":"State padding preserves tensor-network values"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values/sum-state-padding","kind":"node","lean":"TensorNetwork.sum_statePadding","parent":36413,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values/index.html#sum-state-padding","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values","status":"fully_proved","summary":"Padding the finite state sets preserves the full sum of vertex weights and arbitrary pair kernels. No factorization, finiteness of the state carriers, or global injectivity is required.","title":"State padding preserves the weighted pair sum"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values/state-padding","kind":"node","lean":"TensorNetwork.Factored.StatePadding","parent":36413,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values/index.html#state-padding","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values","status":"fully_proved","summary":"The source's local padding protocol for factored networks. The factor clause is retained on source states and source labels, even though value preservation uses the interaction clause. Only capacities between distinct vertices must agree.","title":"State padding of a factored tensor network"},{"id":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values/value-total-capacity","kind":"node","lean":"TensorNetwork.Factored.StatePadding.value_totalCapacity","parent":36413,"result":true,"route":"../roadmap/TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values/index.html#value-total-capacity","scope":"TheoryOfComputation/fine-grained-complexity/tensor-network-contraction/state-padding-preserves-tensor-network-values","status":"fully_proved","summary":"A local state-padding certificate preserves both network value and directed total capacity.","title":"State padding preserves value and total capacity"},{"area":"Discrete Mathematics & Computer Science","data_url":"chapters/ComputerScience.json","id":"ComputerScience","kind":"scope","parent":0,"result":false,"route":"../roadmap/ComputerScience/index.html","scope":"ComputerScience","status":"can_prove","summary":"Computer science formalizes algorithms, programming-language structures, data abstractions, and correctness properties.","title":"Computer Science"},{"data_url":"scopes/ComputerScience/data-structures.json","id":"ComputerScience/data-structures","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/data-structures/index.html","scope":"ComputerScience/data-structures","status":"fully_proved","summary":"Representations of collections with their invariants, operations and costs (worst-case and amortized), whether the structure is developed on its own or as an augmented variant; an algorithm that merely uses a structure goes to the field of the problem it solves.","title":"Data structures"},{"data_url":"scopes/ComputerScience/data-structures/search-trees.json","id":"ComputerScience/data-structures/search-trees","kind":"scope","parent":36418,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/index.html","scope":"ComputerScience/data-structures/search-trees","status":"fully_proved","summary":"Ordered dictionaries: binary search trees and their balanced variants (AVL, red-black, 2-3, B-trees, splay, treaps, weight-balanced, Cartesian, order-statistic) and skip lists, including their insertion and deletion procedures.","title":"Search trees and skip lists"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/2-3-tree.json","id":"ComputerScience/data-structures/search-trees/2-3-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","title":"2\u20133 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/tree","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Tree","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#tree","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For a type $\\alpha$ of keys, the 2-3 trees over $\\alpha$ indexed by height are generated as follows. The leaf is a tree of height $0$ and stores no key. For a natural number $h$, a key $a$ and trees $l$, $r$ of height $h$, the 2-node with separator $a$ and children $l$, $r$ is a tree of height $h + 1$. For a natural number $h$, keys $a$, $b$ and trees $l$, $m$, $r$ of height $h$, the 3-node with separators $a$, $b$\u2026","title":"Height-indexed 2-3 trees"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/overflow","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Overflow","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#overflow","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For a type $\\alpha$ and a natural number $h$, an insertion result at height $h$ is either a 2-3 tree over $\\alpha$ of height $h$, or a split: two 2-3 trees $l$ and $r$ over $\\alpha$ of height $h$ together with a promoted key $k \\in \\alpha$.","title":"Insertion results for 2-3 trees"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/height","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.height","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#height","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"The height of a 2-3 tree is computed along its leftmost spine: a leaf has height $0$, and a 2-node or a 3-node has height one more than the height of its leftmost child.","title":"Height of a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/height-eq","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.height_eq","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#height-eq","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every 2-3 tree $t$ with height index $h$, the height of $t$ computed along its leftmost spine equals $h$.","title":"Computed height equals the height index"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/balanced","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Balanced","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#balanced","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"A leaf is balanced, and a node is balanced when all of its subtrees are balanced and have the same computed height, where the computed height of a tree is the length of its leftmost path from the root down to a leaf. Hence all leaves of a balanced tree lie at the same depth.","title":"Balance invariant for 2-3 trees"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/balanced-2","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.balanced","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#balanced-2","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every height $h$, every 2-3 tree of height $h$ satisfies the balance invariant: at every node, the subtrees are balanced and have equal computed heights.","title":"Every height-indexed 2-3 tree is balanced"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/mem","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Mem","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#mem","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"A key $x$ is a member of a 2-3 tree when it is one of the separator keys stored at its nodes: nothing is a member of a leaf; $x$ is a member of a 2-node with key $a$ and children $l, r$ when it is a member of $l$, equals $a$, or is a member of $r$; and $x$ is a member of a 3-node with keys $a, b$ and children $l, m, r$ when it is a member of $l$, equals $a$, is a member of $m$, equals $b$, or is a member of $r$.","title":"Membership in a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/all","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.All","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#all","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For a predicate $p$ on keys and a 2-3 tree $t$, $\\mathrm{All}_p(t)$ states that every separator key stored at a node of $t$ satisfies $p$: it is true at a leaf, and at a node it requires $p$ of each key of the node and $\\mathrm{All}_p$ of each of its subtrees.","title":"Predicate on all keys of a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/all-iff-forall-mem","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.all_iff_forall_mem","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#all-iff-forall-mem","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every predicate $p$ on keys and every 2-3 tree $t$ of height $h$, the recursive predicate requiring $p$ at every key stored at a node of $t$ holds if and only if $p(x)$ holds for every member $x$ of $t$, where $x$ is a member of $t$ when it equals one of the keys stored at some node of $t$.","title":"All-keys predicate agrees with membership on 2-3 trees"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ordered","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Ordered","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ordered","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $<$. The leaf is ordered. A 2-node with separator $a$ and children $l$, $r$ is ordered when $l$ and $r$ are ordered, $y < a$ for every key $y$ stored in $l$, and $a < y$ for every key $y$ stored in $r$. A 3-node with separators $a$, $b$ and children $l$, $m$, $r$ is ordered when $l$, $m$ and $r$ are ordered, $a < b$, $y < a$ for every key $y$ stored in $l$, $a < y$ and $y < b$\u2026","title":"Ordering invariant for 2-3 trees"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/flatten-aux","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.flattenAux","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#flatten-aux","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For a 2-3 tree $t$ and a list $s$, the accumulated traversal $F(t, s)$ is defined by recursion on $t$: for a leaf, $F(t, s) = s$; for a 2-node with key $a$ and children $l, r$, $F(t, s)$ is $F(l, s')$ where $s'$ is $a$ prepended to $F(r, s)$; for a 3-node with keys $a, b$ and children $l, m, r$, $F(t, s)$ is $F(l, s')$ where $s'$ is $a$ prepended to $F(m, s'')$ and $s''$ is $b$ prepended to $F(r, s)$.","title":"Accumulated inorder traversal of a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/flatten","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.flatten","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#flatten","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"The flattening of a 2-3 tree $t$ is the accumulated inorder traversal of $t$ onto the empty list. Equivalently, the flattening of a leaf is empty; that of a 2-node with key $a$ and children $l, r$ is the flattening of $l$, then $a$, then the flattening of $r$; and that of a 3-node with keys $a, b$ and children $l, m, r$ is the flattening of $l$, then $a$, then the flattening of $m$, then $b$, then the flattening of\u2026","title":"Inorder traversal of a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/flatten-aux-eq","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.flattenAux_eq","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#flatten-aux-eq","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every 2-3 tree $t$ (of any height) and every list $s$, the accumulated inorder traversal of $t$ onto $s$ equals the flattening of $t$ followed by $s$.","title":"The accumulated traversal appends the suffix"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/mem-flatten","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.mem_flatten","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#mem-flatten","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every 2-3 tree $t$ of height $h$ and every key $x$, $x$ occurs in the inorder traversal of $t$ if and only if $x$ is a member of $t$, that is, one of the separator keys stored at its nodes.","title":"The inorder traversal lists exactly the keys of a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/all-iff-forall-flatten","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.all_iff_forall_flatten","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#all-iff-forall-flatten","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every predicate $p$ on keys and every 2-3 tree $t$ of height $h$, every key stored at a node of $t$ satisfies $p$ if and only if every element of the inorder flattening of $t$ satisfies $p$.","title":"All-keys predicate agrees with the flattening on 2-3 trees"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ordered-flatten-pairwise","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.ordered_flatten_pairwise","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ordered-flatten-pairwise","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every natural number $h$ and every ordered 2-3 tree $t$ over $\\alpha$ of height $h$, the flattening (inorder traversal) of $t$ is strictly increasing: whenever a key $x$ occurs before a key $y$ in it, $x < y$.","title":"Inorder traversal of an ordered 2-3 tree is strictly increasing"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/flatten-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.flatten_nodup","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#flatten-nodup","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"If the keys are linearly ordered and a 2-3 tree $t$ (of any height) is ordered, then no key occurs twice in the flattening of $t$.","title":"The traversal of an ordered 2-3 tree has no duplicates"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/contains","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.contains","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#contains","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For a linearly ordered key type, searching a 2-3 tree for a key $x$ returns false at a leaf. At a 2-node with key $a$, it searches the left subtree if $x < a$, otherwise the right subtree if $a < x$, and otherwise returns true. At a 3-node with keys $a$ and $b$, it searches the left subtree if $x < a$; otherwise, if $a < x$, it searches the middle subtree if $x < b$, otherwise the right subtree if $b < x$, and\u2026","title":"Search in a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/contains-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.contains_eq_true_iff","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#contains-eq-true-iff","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let the keys be linearly ordered and let $t$ be a 2-3 tree of height $h$ that satisfies the ordering invariant. Then searching $t$ for a key $x$ returns true if and only if $x$ occurs in the inorder flattening of $t$.","title":"Search in an ordered 2-3 tree is correct"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ins","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.ins","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ins","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Inserting a key $x$ into a 2-3 tree of height $h$ yields either a single tree of height $h$ or a split into two trees of height $h$ around a promoted key. Inserting into a leaf gives the split of two leaves around $x$. At a 2-node with key $a$, insertion recurses into the left child if $x < a$ and into the right child if $a < x$; a returned tree replaces that child, and a returned split around $k$ is absorbed by\u2026","title":"Insertion into a height-indexed 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ins-spec","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.InsSpec","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ins-spec","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be linearly ordered, let $x \\in \\alpha$, and let $t$ be a 2-3 tree of height $h$. An insertion result of height $h$ meets the specification for inserting $x$ into $t$ when the following holds. If it is a single tree $t'$ of height $h$, then $t'$ is ordered and its inorder traversal equals the ordered insertion of $x$ into the inorder traversal of $t$. If it is a split into trees $l$ and $r$ of height\u2026","title":"Insertion specification for 2-3 trees"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ins-spec-2","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.ins_spec","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ins-spec-2","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be linearly ordered, let $x \\in \\alpha$, and let $t$ be an ordered 2-3 tree of height $h$. Then inserting $x$ into $t$ meets the insertion specification. If the insertion returns a single tree $t'$ of height $h$, then $t'$ is ordered and its inorder traversal is the ordered insertion of $x$ into the inorder traversal of $t$. If it returns a split into trees $l$ and $r$ of height $h$ around a promoted\u2026","title":"Correctness of 2-3 tree insertion"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/root","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#root","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For a type $\\alpha$, a 2-3 tree of unspecified height over $\\alpha$ is a pair $(h, t)$ of a natural number $h$ and a height-indexed 2-3 tree $t$ over $\\alpha$ of height $h$.","title":"2-3 trees of unspecified height"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/to-root","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Overflow.toRoot","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#to-root","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"An insertion result at height $h$ becomes a 2-3 tree of unspecified height as follows: a tree $t$ of height $h$ gives $t$ itself, of height $h$, and a split into trees $l$, $r$ of height $h$ around a key $k$ gives the 2-node with separator $k$, left child $l$ and right child $r$, of height $h + 1$.","title":"Turning an insertion result into a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/empty","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.empty","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#empty","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"The empty 2-3 tree is the root of height $0$ whose underlying tree is the leaf.","title":"Empty 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/flatten-2","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.flatten","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#flatten-2","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"The flattening of a 2-3 tree root is the flattening of its underlying tree, that is, the list of its keys in inorder.","title":"Inorder traversal of a 2-3 tree root"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/contains-2","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.contains","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#contains-2","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For a linearly ordered key type, searching a root (a 2-3 tree packaged with its height) for a key $x$ returns the Boolean result of the 2-3 tree search for $x$ in the underlying height-indexed tree.","title":"Search in a rooted 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ordered-2","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.Ordered","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ordered-2","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $<$. A 2-3 tree of unspecified height over $\\alpha$ is ordered when its underlying height-indexed 2-3 tree satisfies the ordering invariant.","title":"Ordering invariant for 2-3 trees of unspecified height"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.insert","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#insert","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"To insert a key $x$ into a root, that is, a 2-3 tree together with its height $h$, insert $x$ into the underlying tree. If this returns a single tree of height $h$, that tree is the new root, of height $h$. If it returns a split into trees $l$ and $r$ around a key $k$, the new root is the 2-node with key $k$ and children $l$ and $r$, of height $h + 1$.","title":"Insertion into a 2-3 tree root"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ordered-empty","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.ordered_empty","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ordered-empty","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every type $\\alpha$ with a relation $<$, the empty 2-3 tree of unspecified height over $\\alpha$ (the leaf, of height $0$) satisfies the ordering invariant.","title":"The empty 2-3 tree is ordered"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/height-tree","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.height_tree","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#height-tree","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every root $t$, that is, a 2-3 tree together with its height index, the height of the underlying tree computed along its leftmost spine equals the stored height of $t$.","title":"The stored height of a 2-3 tree root is its computed height"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/balanced-3","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.balanced","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#balanced-3","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For every root, that is, a 2-3 tree packaged with its height, the underlying height-indexed tree satisfies the balance invariant.","title":"Every rooted 2-3 tree is balanced"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.ordered_insert","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ordered-insert","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. If $t$ is an ordered 2-3 tree of unspecified height over $\\alpha$ and $x \\in \\alpha$, then the tree obtained by inserting $x$ into $t$ is ordered.","title":"Insertion preserves the ordering of 2-3 trees"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/flatten-insert","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.flatten_insert","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#flatten-insert","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let the keys be linearly ordered, let $t$ be an ordered 2-3 tree root and let $x$ be a key. Then the flattening of the root obtained by inserting $x$ into $t$ equals the ordered insertion of $x$ into the flattening of $t$.","title":"Insertion into a 2-3 tree is ordered insertion into its traversal"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/mem-flatten-insert","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.mem_flatten_insert","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#mem-flatten-insert","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be linearly ordered and let $t$ be an ordered 2-3 tree root over $\\alpha$. For all $x, y \\in \\alpha$, $y$ occurs in the inorder traversal of the root obtained by inserting $x$ into $t$ if and only if $y = x$ or $y$ occurs in the inorder traversal of $t$.","title":"Keys after insertion into a 2-3 tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/contains-eq-true-iff-2","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.contains_eq_true_iff","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#contains-eq-true-iff-2","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let the keys be linearly ordered and let $t$ be a root (a 2-3 tree packaged with its height) whose underlying tree satisfies the ordering invariant. Then searching $t$ for a key $x$ returns true if and only if $x$ occurs in the inorder flattening of $t$.","title":"Search in a rooted 2-3 tree is correct"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/height-insert-le","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.height_insert_le","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#height-insert-le","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For linearly ordered keys, every key $x$ and every 2-3 tree root $t$, the stored height of the root obtained by inserting $x$ into $t$ is at most the stored height of $t$ plus one.","title":"Insertion raises the height of a 2-3 tree by at most one"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/from-list","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.fromList","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#from-list","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"For linearly ordered keys, the 2-3 tree root built from a list is defined by recursion: the empty list gives the empty 2-3 tree, and a list with head $x$ and tail $xs$ gives the result of inserting $x$ into the root built from $xs$. Thus the elements are inserted into the empty tree from the last to the first.","title":"Building a 2-3 tree from a list"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/ordered-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.ordered_fromList","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#ordered-from-list","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every list $L$ of elements of $\\alpha$, the 2-3 tree of unspecified height built from $L$, by inserting the elements of $L$ from right to left into the empty tree, is ordered.","title":"Building a 2-3 tree from a list gives an ordered tree"},{"id":"ComputerScience/data-structures/search-trees/2-3-tree/mem-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.TwoThreeTree.Root.mem_fromList","parent":36420,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/2-3-tree/index.html#mem-from-list","scope":"ComputerScience/data-structures/search-trees/2-3-tree","status":"fully_proved","summary":"Let $\\alpha$ be linearly ordered. For every list $L$ over $\\alpha$ and every $x \\in \\alpha$, $x$ occurs in the inorder traversal of the 2-3 tree built from $L$ (by inserting its elements from right to left into the empty tree) if and only if $x$ occurs in $L$.","title":"Keys of a 2-3 tree built from a list"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/b-tree.json","id":"ComputerScience/data-structures/search-trees/b-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","title":"B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/btree","kind":"node","lean":"Cslib.Algorithms.Lean.BTree","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#btree","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"A raw B-tree with keys in a type $\\alpha$ is either a leaf, holding a list of keys, or an internal node, holding a list of separator keys and a list of child raw B-trees. The type imposes no invariant: that an internal node with $k$ keys has $k + 1$ children, the bounds on key counts, the balance of the tree and the ordering of the keys are imposed separately by the structural and ordering invariants of B-trees.","title":"Raw B-trees"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-aux","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flattenAux","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-aux","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Given a raw B-tree $T$ and a list $s$, this returns the inorder traversal of $T$ followed by $s$. For a leaf it is the leaf's key list followed by $s$. For an internal node with separator keys $k_1, \\dots, k_m$ and children $c_0, \\dots, c_n$ it is the sequence $c_0, k_1, c_1, k_2, \\dots$, in which children and keys alternate starting with a child until one of the two lists runs out, then the remaining keys or\u2026","title":"Accumulating flattening of a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"The flattening of a raw B-tree is the list of its keys defined recursively: for a leaf it is the leaf's key list, and for an internal node with keys $k_1, k_2, \\dots$ and children $c_1, c_2, \\dots$ it is the flattening of $c_1$, then $k_1$, then the flattening of $c_2$, then $k_2$, and so on, alternating while both lists last; whatever remains of the longer list is then appended in order, each surplus child\u2026","title":"Inorder flattening of a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-aux-eq","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flattenAux_eq","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-aux-eq","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For every raw B-tree $T$ and every list $s$, the accumulating flattening of $T$ with suffix $s$ equals the inorder flattening of $T$ followed by $s$.","title":"The accumulating B-tree flattening is correct"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-node-nil-cons","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_node_nil_cons","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-node-nil-cons","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a raw B-tree $c$ and a list $C$ of raw B-trees, the flattening of the node with no keys and child list $c$ followed by $C$ is the flattening of $c$ followed by the flattening of the node with no keys and child list $C$.","title":"Flattening a keyless node, one child at a time"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-node-cons-nil","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_node_cons_nil","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-node-cons-nil","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a key $k$ and a list of keys $K$, the flattening of the node with key list $k$ followed by $K$ and no children is $k$ followed by the flattening of the node with key list $K$ and no children.","title":"Flattening a childless node, one key at a time"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-node-nil-nil","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_node_nil_nil","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-node-nil-nil","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"The flattening of the internal node with no keys and no children is the empty list.","title":"Flattening of the empty node"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-node-append-cons","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_node_append_cons","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-node-append-cons","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $K_1$ and $K_2$ be lists of keys, $m$ a key, and $C_1$ and $C_2$ lists of raw B-trees such that $C_1$ has exactly one more element than $K_1$. Then the flattening of the node whose key list is $K_1$ followed by $m$ followed by $K_2$ and whose child list is $C_1$ followed by $C_2$ equals the flattening of the node with keys $K_1$ and children $C_1$, followed by $m$, followed by the flattening of the node with\u2026","title":"Splitting a node's flattening at a separator"},{"id":"ComputerScience/data-structures/search-trees/b-tree/mem","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.Mem","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#mem","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"A key $x$ belongs to a tree $T$ when $x$ occurs in the inorder flattening of $T$.","title":"Membership in a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/ordered","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.Ordered","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#ordered","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"A B-tree over a type with a relation $<$ is ordered when its inorder flattening is strictly increasing, that is, each key of the flattening is less than every key that comes after it. This single condition packages sortedness of the keys at each node and the separator bounds between a node's keys and the keys stored in its subtrees.","title":"Ordered B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/node-keys-pairwise","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.Ordered.node_keys_pairwise","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#node-keys-pairwise","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"If an internal node with separator list $K$ is ordered, then $K$ is strictly increasing: every key of $K$ is smaller than every key that comes after it in $K$.","title":"The separator keys of an ordered node are sorted"},{"id":"ComputerScience/data-structures/search-trees/b-tree/separator-bounds","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.Ordered.separator_bounds","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#separator-bounds","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $K_1$ and $K_2$ be lists of keys, $k$ a key, and $C_1$ and $C_2$ lists of B-trees such that $C_1$ has exactly one more tree than $K_1$ has keys. If the internal node whose key list is $K_1$ followed by $k$ followed by $K_2$, and whose child list is $C_1$ followed by $C_2$, is ordered, then every key in the flattening of the node with keys $K_1$ and children $C_1$ is less than $k$, and $k$ is less than every key\u2026","title":"Separator bounds in an ordered B-tree node"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.Ordered.flatten_nodup","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-nodup","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a linearly ordered key type, the flattening of an ordered raw B-tree has no duplicate entries.","title":"An ordered B-tree has no duplicate keys"},{"id":"ComputerScience/data-structures/search-trees/b-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.Valid","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#valid","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a minimum degree $t$ and a minimum key count $m$ for the top node, the invariant is defined inductively over the height: a leaf with $k$ keys satisfies it at height $0$ when $m \\le k \\le 2t - 1$, and for every natural number $h$, an internal node with $k$ keys satisfies it at height $h + 1$ when $m \\le k \\le 2t - 1$, the node has exactly $k + 1$ children, and every child satisfies the invariant with minimum key\u2026","title":"Structural invariant of a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/valid-root","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.ValidRoot","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#valid-root","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a minimum degree $t$ and a height $h$, a tree satisfies the root invariant when either $h = 0$ and the tree is a leaf with at most $2t - 1$ keys (possibly none, for the empty tree), or $h = h' + 1$ and the tree is an internal node with between $1$ and $2t - 1$ keys and exactly one more child than keys, every child satisfying the structural invariant for minimum degree $t$, minimum top-node key count $t - 1$ and\u2026","title":"Root invariant of a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/valid-2","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.ValidRoot.valid","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#valid-2","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"If a tree satisfies the root invariant for minimum degree $t$ at height $h$, then for some natural number $m$ it satisfies the structural invariant for minimum degree $t$, minimum top-node key count $m$ and height $h$.","title":"Root-valid B-trees are valid"},{"id":"ComputerScience/data-structures/search-trees/b-tree/contains","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.contains","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#contains","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a linearly ordered key type, the search for a key $x$ in a raw B-tree proceeds as follows. At a leaf, report whether $x$ is in the leaf's key list. At an internal node with separators $k_1, \\dots, k_m$ and children $c_0, c_1, \\dots$, scan the separators in order. On reaching $k_i$, report failure if the node has no child $c_{i-1}$; otherwise search $c_{i-1}$ if $x < k_i$, report success if $x = k_i$, and move\u2026","title":"B-tree search"},{"id":"ComputerScience/data-structures/search-trees/b-tree/ins-result","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.InsResult","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#ins-result","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"An insertion result over a key type $\\alpha$ is either a single tree over $\\alpha$, or a triple $(\\ell, m, r)$ of two trees $\\ell$ and $r$ over $\\alpha$ and a key $m$ of $\\alpha$. In insertion, the first form is a subtree that did not split, and the second form is an overflowing node split into two trees of the same height, with the median key $m$ promoted between them; the type itself does not record heights.","title":"Result of inserting into a B-tree subtree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/grown-flatten","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.grownFlatten","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#grown-flatten","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"The flattening of an insertion result that is a single tree is the flattening of that tree; for a result that splits into a left tree $\\ell$, a promoted median key $m$ and a right tree $r$, it is the flattening of $\\ell$, followed by $m$, followed by the flattening of $r$.","title":"Flattening of a B-tree insertion result"},{"id":"ComputerScience/data-structures/search-trees/b-tree/ins-leaf","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.insLeaf","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#ins-leaf","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Over a linearly ordered key type, let $t$ be a natural number, $x$ a key, $K$ the key list of a leaf, and $K'$ the result of ordered insertion of $x$ into $K$. If $K'$ has at most $2t - 1$ keys, the insertion result is the single leaf with keys $K'$. Otherwise the leaf is split: the result consists of the leaf holding the first $t - 1$ keys of $K'$, the next key of $K'$ promoted as the median, and the leaf holding\u2026","title":"Insertion into a B-tree leaf"},{"id":"ComputerScience/data-structures/search-trees/b-tree/split-node","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.splitNode","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#split-node","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t$ be a minimum degree, $s = t - 1$ (so $s = 0$ when $t = 0$), $k_1, \\dots, k_m$ a list of keys and $c_1, \\dots, c_n$ a list of B-trees. If $m > s$, splitting returns the pair of internal nodes formed by the keys $k_1, \\dots, k_s$ with the first $t$ children (all of them if $n < t$) and by the keys $k_{s+2}, \\dots, k_m$ with the children after the first $t$, together with the key $k_{s+1}$ promoted between\u2026","title":"Splitting a B-tree node"},{"id":"ComputerScience/data-structures/search-trees/b-tree/mk-node","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.mkNode","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#mk-node","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For minimum degree $t$, a separator list $K$ and a child list $C$, the result is the internal node with separators $K$ and children $C$, kept as a single tree when $K$ has at most $2t - 1$ keys, and otherwise split by node splitting into two nodes and a promoted median key.","title":"Rebuilding an internal node after insertion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/ins","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.ins","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#ins","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For minimum degree $t$ and a key $x$, insertion of $x$ into a tree returns an insertion result and is defined by recursion. At a leaf with key list $K$, it is leaf insertion of $x$ into $K$. At an internal node, the separator list and the child list are scanned together from the left. If $x$ is smaller than the current separator, or no separator is left but a child is, $x$ is inserted into the current child. If $x$\u2026","title":"Split-returning insertion into a B-tree subtree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.insert","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#insert","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For minimum degree $t$, the tree obtained by inserting a key $x$ into a tree $T$ is computed from the split-returning insertion of $x$ into $T$: if that returns a single tree, this tree is the result; if it returns two trees $\\ell$ and $r$ with a promoted median key $m$, the result is a new root node with the single separator $m$ and the two children $\\ell$ and $r$.","title":"B-tree insertion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/empty","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.empty","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#empty","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"The empty B-tree is the leaf with no keys.","title":"The empty B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/is-btree","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#is-btree","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"A tree $T$ is a B-tree of minimum degree $t$ when $t \\ge 2$ (the CLRS Chapter 18 minimum-degree requirement), $T$ is ordered, and $T$ is root-valid at some height. This is the invariant preserved by insertion and by building a tree from a list.","title":"B-trees of a given minimum degree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/two-le","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree.two_le","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#two-le","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"If a tree is a B-tree of minimum degree $t$, then $t \\ge 2$.","title":"The minimum degree of a B-tree is at least two"},{"id":"ComputerScience/data-structures/search-trees/b-tree/ordered-2","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree.ordered","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#ordered-2","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"If a tree is a B-tree of minimum degree $t$, then it is ordered: each key of its inorder flattening is less than every key that comes after it.","title":"B-trees are ordered"},{"id":"ComputerScience/data-structures/search-trees/b-tree/exists-valid-root","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree.exists_validRoot","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#exists-valid-root","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"If $T$ is a B-tree of minimum degree $t$, then there is a height $h$ such that $T$ satisfies the root structural invariant of minimum degree $t$ at height $h$.","title":"B-trees satisfy the root invariant"},{"id":"ComputerScience/data-structures/search-trees/b-tree/empty-2","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree.empty","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#empty-2","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For every key type equipped with a binary relation $<$ and every natural number $t \\ge 2$, the empty B-tree is a B-tree of minimum degree $t$.","title":"The empty tree is a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-insert","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_insert","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-insert","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $\\alpha$ be linearly ordered, let $t \\ge 2$, and let $T$ be a B-tree of minimum degree $t$ with keys in $\\alpha$. For every key $x \\in \\alpha$, the inorder flattening of the tree obtained by inserting $x$ into $T$ with minimum degree $t$ equals the sorted insertion of $x$ into the inorder flattening of $T$.","title":"B-tree insertion as sorted list insertion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/mem-insert","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.mem_insert","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#mem-insert","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For all keys $x$ and $y$, $y$ belongs to the tree obtained by inserting $x$ into $T$ if and only if $y = x$ or $y$ belongs to $T$.","title":"Membership after B-tree insertion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/insert-2","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree.insert","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#insert-2","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let the keys come from a linearly ordered type. If $T$ is a B-tree of minimum degree $t$, then for every key $x$ the tree obtained by inserting $x$ into $T$ is again a B-tree of minimum degree $t$: it is ordered and root-valid at some height.","title":"Insertion preserves the B-tree invariant"},{"id":"ComputerScience/data-structures/search-trees/b-tree/contains-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.contains_eq_true_iff","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#contains-eq-true-iff","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $\\alpha$ be linearly ordered and let $T$ be a B-tree of minimum degree $t$ with keys in $\\alpha$. For every key $x \\in \\alpha$, the search procedure returns true on $x$ and $T$ if and only if $x$ is a member of $T$, that is, $x$ occurs in the inorder flattening of $T$.","title":"Correctness of B-tree search"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-length-insert","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_length_insert","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-length-insert","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $\\alpha$ be linearly ordered, let $t \\ge 2$, let $T$ be a B-tree of minimum degree $t$ with keys in $\\alpha$, and let $x \\in \\alpha$. Let $T'$ be the tree obtained by inserting $x$ into $T$ with minimum degree $t$. If $x$ occurs in the inorder flattening of $T$, then the flattening of $T'$ has the same length as that of $T$; otherwise it is one longer.","title":"Size of a B-tree after insertion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/from-list","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.fromList","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#from-list","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a linearly ordered key type and a natural number $t$, the tree built from the empty list is the empty tree, and the tree built from a list with head $x$ and tail $L$ is the result of inserting $x$, with minimum degree $t$, into the tree built from $L$. Thus the elements are inserted from right to left.","title":"B-tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/b-tree/from-list-2","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree.fromList","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#from-list-2","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a linearly ordered key type, every natural number $t \\ge 2$ and every list $L$ of keys, the tree obtained by inserting the elements of $L$ from right to left into the empty tree, each insertion with minimum degree $t$, is a B-tree of minimum degree $t$.","title":"Building from a list yields a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/mem-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.mem_fromList","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#mem-from-list","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $xs$ be a list over a linearly ordered key type. A key $x$ belongs to the tree of minimum degree $t$ built from $xs$ (by inserting the elements of $xs$ into the empty tree, from right to left) if and only if $x$ is an element of $xs$.","title":"Membership in a B-tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-from-list-of-pairwise","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_fromList_of_pairwise","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-from-list-of-pairwise","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$. For every strictly increasing list $L$ of keys, the tree built from $L$ by inserting its keys from right to left into the empty tree with minimum degree $t$ has flattening exactly $L$.","title":"Building a B-tree from a strictly increasing list"},{"id":"ComputerScience/data-structures/search-trees/b-tree/delete-rebuild","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.deleteRebuild","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#delete-rebuild","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a minimum degree $t$, a key $x$ and a tree $T$, the rebuilding deletion of $x$ from $T$ is the tree obtained by inserting the keys of the list $L$, from right to left, into the empty tree with B-tree insertion of minimum degree $t$, where $L$ is the inorder flattening of $T$ with the first occurrence of $x$ removed ($L$ is the flattening itself when $x$ does not occur in it).","title":"B-tree deletion by rebuilding"},{"id":"ComputerScience/data-structures/search-trees/b-tree/delete-rebuild-eq-from-list-erase","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.deleteRebuild_eq_fromList_erase","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#delete-rebuild-eq-from-list-erase","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For every minimum degree $t$, key $x$ and tree $T$, the rebuilding deletion of $x$ from $T$ with minimum degree $t$ equals the tree built from the list obtained by removing the first occurrence of $x$ from the flattening of $T$, by inserting its keys from right to left into the empty tree with minimum degree $t$.","title":"Rebuilding deletion unfolds to erase and rebuild"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-delete-rebuild","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_deleteRebuild","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-delete-rebuild","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$, let $T$ be an ordered tree and let $x$ be a key. The flattening of the rebuilding deletion of $x$ from $T$ with minimum degree $t$ is the flattening of $T$ with the first occurrence of $x$ removed (unchanged if $x$ does not occur).","title":"Flattening after rebuilding deletion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/delete-rebuild-2","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree.deleteRebuild","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#delete-rebuild-2","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$. For every key $x$ and every tree $T$, whether or not $T$ is itself a B-tree, the rebuilding deletion of $x$ from $T$ with minimum degree $t$ is a B-tree of minimum degree $t$.","title":"Rebuilding deletion yields a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/ordered-delete-rebuild","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.ordered_deleteRebuild","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#ordered-delete-rebuild","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let keys come from a linearly ordered type. For every key $x$ and every tree $T$, not necessarily ordered, the tree obtained from $T$ by rebuild deletion of $x$ (removing the first occurrence of $x$, if any, from the inorder flattening of $T$ and rebuilding a B-tree of minimum degree $t$ from the remaining keys) is ordered: its inorder flattening is strictly increasing.","title":"Rebuild deletion yields an ordered tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/delete-rebuild-valid-root","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.deleteRebuild_validRoot","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#delete-rebuild-valid-root","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$. For every key $x$ and every tree $T$, there is a height $h$ such that the rebuilding deletion of $x$ from $T$ with minimum degree $t$ is root-valid of minimum degree $t$ and height $h$: all leaves sit at depth $h$, every node holds at most $2t - 1$ keys, every non-root node holds at least $t - 1$ keys, an internal root holds at least one key, and an internal node with $k$ keys has $k + 1$ children.","title":"Rebuilding deletion yields a root-valid tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/mem-delete-rebuild","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.mem_deleteRebuild","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#mem-delete-rebuild","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ with keys in a linearly ordered type. For all keys $x$ and $y$, the key $y$ lies in the tree obtained from $T$ by rebuild deletion of $x$ (erasing $x$ from the inorder flattening of $T$ and rebuilding a B-tree of minimum degree $t$ from the remaining keys) if and only if $y \\ne x$ and $y$ lies in $T$.","title":"Membership after rebuild deletion from a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/not-mem-delete-rebuild","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.not_mem_deleteRebuild","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#not-mem-delete-rebuild","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ with keys in a linearly ordered type. For every key $x$, the tree obtained from $T$ by rebuild deletion of $x$ (erasing $x$ from the inorder flattening of $T$ and rebuilding a B-tree of minimum degree $t$ from the remaining keys) does not contain $x$.","title":"The deleted key is absent after rebuild deletion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/contains-delete-rebuild-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.contains_deleteRebuild_iff","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#contains-delete-rebuild-iff","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$, let $T$ be a B-tree of minimum degree $t$, and let $x$ and $y$ be keys. The B-tree search for $y$ in the rebuilding deletion of $x$ from $T$ with minimum degree $t$ returns true if and only if $y \\ne x$ and $y$ is stored in $T$.","title":"Search after rebuilding deletion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-delete-rebuild-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_deleteRebuild_of_not_mem","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-delete-rebuild-of-not-mem","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$, let $T$ be an ordered tree and let $x$ be a key that does not occur in the flattening of $T$. Then the rebuilding deletion of $x$ from $T$ with minimum degree $t$ has the same flattening as $T$.","title":"Rebuilding deletion of an absent key"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-length-delete-rebuild","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_length_deleteRebuild","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-length-delete-rebuild","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$, let $T$ be an ordered tree with flattening of length $n$, and let $x$ be a key. The flattening of the rebuilding deletion of $x$ from $T$ with minimum degree $t$ has length $n - 1$ if $x$ occurs in the flattening of $T$, and length $n$ otherwise.","title":"Key count after rebuilding deletion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-delete-rebuild-delete-rebuild","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_deleteRebuild_deleteRebuild","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-delete-rebuild-delete-rebuild","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$, let $T$ be an ordered tree (its flattening is strictly increasing) and let $x$ be a key. Deleting $x$ twice by rebuilding with minimum degree $t$ gives a tree with the same flattening as deleting $x$ once.","title":"Rebuilding deletion of the same key twice"},{"id":"ComputerScience/data-structures/search-trees/b-tree/key-count","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.keyCount","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#key-count","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a raw B-tree $T$, this is the length of the key list stored at the root of $T$: the number of keys of $T$ if $T$ is a leaf, and the number of separator keys of $T$ if $T$ is an internal node. Keys in the children are not counted.","title":"Number of keys at the root of a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/split-last","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.splitLast","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#split-last","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a nonempty list $l$, the result is the pair consisting of $l$ with its last element removed and the last element of $l$; for the empty list there is no result.","title":"Splitting off the last element of a list"},{"id":"ComputerScience/data-structures/search-trees/b-tree/max-key","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.maxKey","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#max-key","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a tree $T$, the rightmost key of $T$ is an optional key defined by descending the rightmost spine: for a leaf it is the last of its keys (none if the leaf has no keys); for an internal node with at least one child it is the rightmost key of its last child; for an internal node with no children it is the last of its separator keys (none if there are none).","title":"Rightmost key of a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/min-key","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.minKey","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#min-key","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"For a tree $T$, the leftmost key of $T$ is an optional key defined by descending the leftmost spine: for a leaf it is the first of its keys (none if the leaf has no keys); for an internal node with at least one child it is the leftmost key of its first child; for an internal node with no children it is the first of its separator keys (none if there are none).","title":"Leftmost key of a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/merge","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.merge","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#merge","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Given a key $s$ and trees $L$ and $R$, the merge of $L$ and $R$ around $s$ is defined as follows. If $L$ and $R$ are leaves with key lists $k_1$ and $k_2$, it is the leaf with keys $k_1$, then $s$, then $k_2$. If $L$ and $R$ are internal nodes with key lists $k_1, k_2$ and child lists $c_1, c_2$, it is the internal node with keys $k_1$, then $s$, then $k_2$, and children $c_1$ followed by $c_2$. In every other case\u2026","title":"Merging two B-tree nodes around a separator"},{"id":"ComputerScience/data-structures/search-trees/b-tree/borrow-right","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.borrowRight","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#borrow-right","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Given a tree $c$ (the child), a separator key $k$ and a tree $s$ (the right sibling) such that $c$ and $s$ are both leaves or both internal nodes, $s$ has at least one key, and, when they are internal nodes, $s$ has at least one child: $k$ becomes the last key of $c$, the first key of $s$ is removed from $s$ and becomes the new separator, and, for internal nodes, the first child of $s$ is removed from $s$ and\u2026","title":"Borrowing a key from the right sibling"},{"id":"ComputerScience/data-structures/search-trees/b-tree/borrow-left","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.borrowLeft","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#borrow-left","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Given a tree $s$ (the left sibling), a separator key $k$ and a tree $c$ (the child) such that $s$ and $c$ are both leaves or both internal nodes, $s$ has at least one key, and, when they are internal nodes, $s$ has at least one child: the last key of $s$ is removed from $s$ and becomes the new separator, $k$ becomes the first key of $c$, and, for internal nodes, the last child of $s$ is removed from $s$ and becomes\u2026","title":"Borrowing a key from the left sibling"},{"id":"ComputerScience/data-structures/search-trees/b-tree/del-list","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.delList","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#del-list","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $D$ be an operation that deletes a key from a tree, let $t$ be a natural number, let $x$ be a key of a linearly ordered key type, and write $|c|$ for the number of keys stored at the root of a tree $c$. Deleting $x$ within a node with separator list $K$ and child list $C$, using $D$ on children, produces a new separator list and a new child list. Write $k$ for the first separator, $c$ for the first child, and\u2026","title":"Deletion within a B-tree node"},{"id":"ComputerScience/data-structures/search-trees/b-tree/del-h","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.delH","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#del-h","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Given a height $h$, a minimum degree $t$, a key $x$ of a linearly ordered key type and a tree $T$, the result is defined as follows. If $T$ is a leaf, the result is the leaf whose key list is that of $T$ with the first occurrence of $x$ erased, whatever $h$ is. If $T$ is an internal node and $h = 0$, the result is $T$. If $T$ is an internal node and $h = n + 1$, the result is the internal node whose separator and\u2026","title":"Height-indexed B-tree deletion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/tree-height","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.treeHeight","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#tree-height","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"The height of a B-tree is computed by following first children: a leaf has height $0$, an internal node with no children has height $0$, and an internal node whose first child is $c$ has height one more than the height of $c$.","title":"Height of a B-tree along first children"},{"id":"ComputerScience/data-structures/search-trees/b-tree/delete","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.delete","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#delete","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Deleting a key $x$ of a linearly ordered key type from a tree $T$ with minimum degree $t$ applies height-indexed deletion of $x$ with minimum degree $t$ to $T$ at the height of $T$, where the height of a leaf, or of an internal node without children, is $0$, and the height of any other internal node is one more than the height of its first child. If the outcome is an internal node with no keys and exactly one\u2026","title":"Top-down B-tree deletion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_delete","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-delete","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For every key $x$, the inorder flattening of the tree obtained from $T$ by top-down deletion of $x$ with minimum degree $t$ is the flattening of $T$ with the first occurrence of $x$ removed (unchanged if $x$ does not occur).","title":"Deletion from a B-tree erases the key from its flattening"},{"id":"ComputerScience/data-structures/search-trees/b-tree/delete-valid-root","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.delete_validRoot","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#delete-valid-root","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be an ordered tree over a linearly ordered key type (its inorder flattening is strictly increasing) that satisfies the root invariant of B-trees of minimum degree $t$ at height $h$. For every key $x$, the tree obtained by deleting $x$ from $T$ with minimum degree $t$ satisfies the root invariant of minimum degree $t$ at height $h$ or at height $h - 1$, where $h - 1$ is truncated\u2026","title":"Top-down deletion lowers the height by at most one"},{"id":"ComputerScience/data-structures/search-trees/b-tree/delete-2","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.IsBTree.delete","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#delete-2","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For every key $x$, the tree obtained by deleting $x$ from $T$ with minimum degree $t$ is again a B-tree of minimum degree $t$.","title":"Top-down deletion preserves the B-tree invariant"},{"id":"ComputerScience/data-structures/search-trees/b-tree/ordered-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.ordered_delete","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#ordered-delete","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For every key $x$, the tree obtained from $T$ by deleting $x$ (with minimum degree $t$) is ordered: its inorder flattening is strictly increasing.","title":"Deletion from a B-tree yields an ordered tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/delete-valid-root-exists","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.delete_validRoot_exists","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#delete-valid-root-exists","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For every key $x$, the tree obtained by deleting $x$ from $T$ with minimum degree $t$ satisfies the root invariant of B-trees of minimum degree $t$ at some height.","title":"Top-down deletion yields a root-valid tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/mem-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.mem_delete","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#mem-delete","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For all keys $x$ and $y$, $y$ belongs to the tree obtained from $T$ by deleting $x$ (with minimum degree $t$) if and only if $y \\ne x$ and $y$ belongs to $T$.","title":"Membership after deletion from a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/not-mem-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.not_mem_delete","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#not-mem-delete","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For every key $x$, $x$ does not belong to the tree obtained from $T$ by deleting $x$ (with minimum degree $t$).","title":"The deleted key is absent after deletion from a B-tree"},{"id":"ComputerScience/data-structures/search-trees/b-tree/contains-delete-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.contains_delete_iff","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#contains-delete-iff","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For all keys $x$ and $y$, searching for $y$ in the tree obtained by deleting $x$ from $T$ with minimum degree $t$ succeeds if and only if $y \\ne x$ and $y$ is a key of $T$.","title":"Search after top-down deletion"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-length-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_length_delete","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-length-delete","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$, let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type, and let $x$ be a key. If $x$ occurs in the inorder flattening of $T$, the flattening of the tree obtained from $T$ by deleting $x$ (with minimum degree $t$) has length one less than the flattening of $T$; otherwise the two flattenings have the same length.","title":"Deletion shortens the flattening by one exactly when the key is present"},{"id":"ComputerScience/data-structures/search-trees/b-tree/flatten-delete-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BTree.flatten_delete_delete","parent":36461,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/b-tree/index.html#flatten-delete-delete","scope":"ComputerScience/data-structures/search-trees/b-tree","status":"fully_proved","summary":"Let $t \\ge 2$ and let $T$ be a B-tree of minimum degree $t$ over a linearly ordered key type. For every key $x$, deleting $x$ from $T$ twice with minimum degree $t$ gives a tree with the same inorder flattening as deleting $x$ from $T$ once.","title":"Repeated top-down deletion"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/binary-search-tree.json","id":"ComputerScience/data-structures/search-trees/binary-search-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","title":"Binary search tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/mem","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.Mem","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#mem","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"A key $x$ belongs to a binary tree $T$ when it is the key of some node of $T$: no key belongs to the empty tree, and $x$ belongs to a node with key $k$, left subtree $L$ and right subtree $R$ exactly when $x$ belongs to $L$, $x = k$, or $x$ belongs to $R$.","title":"Membership in a binary tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/inorder-aux","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.inorderAux","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#inorder-aux","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a binary tree $t$ and a list $s$, the accumulated traversal of the empty tree onto $s$ is $s$, and the accumulated traversal of a node with key $k$, left subtree $L$ and right subtree $R$ onto $s$ is the accumulated traversal of $L$ onto the list consisting of $k$ followed by the accumulated traversal of $R$ onto $s$.","title":"Inorder traversal of a binary tree with an accumulator"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/inorder","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.inorder","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#inorder","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"The inorder traversal of a binary tree $t$ is the accumulated inorder traversal of $t$ onto the empty list. Equivalently, the empty tree gives the empty list, and a node with key $k$, left subtree $L$ and right subtree $R$ gives the inorder traversal of $L$, then $k$, then the inorder traversal of $R$.","title":"Inorder traversal of a binary tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/all","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.All","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#all","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a predicate $p$ on a type $\\alpha$, the empty binary tree satisfies \"every key satisfies $p$\", and a node with key $k$, left subtree $L$ and right subtree $R$ satisfies it exactly when $L$ satisfies it, $p(k)$ holds and $R$ satisfies it.","title":"A predicate holding at every key of a binary tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/ordered","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.Ordered","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#ordered","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $<$. The empty tree is ordered, and a binary tree with root key $k$, left subtree $L$ and right subtree $R$ is ordered when $L$ and $R$ are ordered, every key of $L$ is less than $k$, and $k$ is less than every key of $R$.","title":"Ordered binary trees"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/contains","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.contains","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#contains","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, a key $x \\in \\alpha$ and a binary tree $t$ with keys in $\\alpha$, the search for $x$ in $t$ returns a Boolean: false if $t$ is empty, and if $t$ is a node with key $k$, left subtree $L$ and right subtree $R$, the search for $x$ in $L$ when $x < k$, the search for $x$ in $R$ when $k < x$, and true otherwise.","title":"Search in a binary search tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.insert","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#insert","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, a key $x \\in \\alpha$ and a binary tree $t$ with keys in $\\alpha$, inserting $x$ into the empty tree gives the single node with key $x$ and empty subtrees; inserting $x$ into a node with key $k$, left subtree $L$ and right subtree $R$ inserts $x$ into $L$ when $x < k$, inserts $x$ into $R$ when $k < x$, and returns the node unchanged when $x = k$.","title":"Insertion into a binary search tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/rank","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.rank","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#rank","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, a key $x \\in \\alpha$ and a binary tree $t$ with keys in $\\alpha$, the rank of $x$ in $t$ is the number of positions of the inorder traversal of $t$ whose entry is strictly less than $x$.","title":"Rank of a key in a binary tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/select","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.select","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#select","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a binary tree $t$ and a natural number $i$, selecting index $i$ in $t$ returns the entry at zero-based position $i$ of the inorder traversal of $t$ when $i$ is less than the length of that traversal, and returns nothing otherwise.","title":"Inorder selection in a binary tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/from-list","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.fromList","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#from-list","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, the binary tree built from a list $[x_1, \\ldots, x_n]$ of elements of $\\alpha$ is obtained from the empty tree by inserting $x_n$, then $x_{n-1}$, and so on down to $x_1$, using binary search tree insertion; the empty list gives the empty tree.","title":"Binary search tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/all-iff-forall-mem","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.all_iff_forall_mem","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#all-iff-forall-mem","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a type $\\alpha$, a predicate $p$ on $\\alpha$ and a binary tree $t$ with keys in $\\alpha$, every key of $t$ satisfies $p$ in the recursive sense (the empty tree qualifies, and a node qualifies when its left subtree qualifies, its key satisfies $p$ and its right subtree qualifies) if and only if $p(x)$ holds for every $x$ that is a member of $t$.","title":"A predicate holds at all keys of a binary tree iff it holds at all its members"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/mem-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.mem_inorder","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#mem-inorder","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For every binary tree $T$ and every key $x$, $x$ occurs in the inorder traversal of $T$ if and only if $x$ lies in $T$.","title":"Membership in the inorder traversal of a binary tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/mem-insert","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.mem_insert","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#mem-insert","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let the keys be linearly ordered. For all keys $x, y$ and every binary tree $T$, $y$ lies in the tree obtained by inserting $x$ into $T$ if and only if $y = x$ or $y$ lies in $T$.","title":"Membership after insertion into a binary search tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.ordered_insert","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#ordered-insert","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every key $x \\in \\alpha$ and every ordered binary tree $T$ over $\\alpha$, the tree obtained by inserting $x$ into $T$ is ordered.","title":"Binary search tree insertion preserves order"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/contains-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.contains_eq_true_iff","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#contains-eq-true-iff","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, a key $x \\in \\alpha$ and a binary tree $t$ with keys in $\\alpha$ that is ordered (recursively, every key in the left subtree of a node is strictly smaller than the node's key and every key in its right subtree is strictly larger), the comparison search for $x$ in $t$ returns true if and only if $x$ is a member of $t$.","title":"Search in a binary search tree is correct"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/ordered-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.ordered_inorder","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#ordered-inorder","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. If a binary tree $T$ over $\\alpha$ is ordered, then in the inorder traversal of $T$ every key is less than every key that comes after it.","title":"The inorder traversal of an ordered binary tree is strictly increasing"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/inorder-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.inorder_nodup","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#inorder-nodup","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$ and a binary tree $t$ with keys in $\\alpha$ that is ordered (recursively, every key in the left subtree of a node is strictly smaller than the node's key and every key in its right subtree is strictly larger), the inorder traversal of $t$ is a list with no repeated entries.","title":"The inorder traversal of a binary search tree has no duplicates"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/ordered-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.ordered_fromList","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#ordered-from-list","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every list $xs$ of elements of $\\alpha$, the tree obtained from the empty tree by inserting the elements of $xs$ one at a time, from the last element to the first, is ordered.","title":"Binary search trees built from a list are ordered"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/mem-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.mem_fromList","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#mem-from-list","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, a key $x \\in \\alpha$ and a list $xs$ of elements of $\\alpha$, $x$ is a member of the binary tree built from $xs$ by successive binary search tree insertion if and only if $x$ occurs in $xs$.","title":"Members of the binary search tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/num-nodes-eq-length-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.numNodes_eq_length_inorder","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#num-nodes-eq-length-inorder","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For every binary tree $T$, the number of nodes of $T$ (each node carries one key, and the empty tree has none) equals the length of the inorder traversal of $T$.","title":"The length of the inorder traversal of a binary tree is its number of nodes"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/num-nodes-insert-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.numNodes_insert_of_mem","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#num-nodes-insert-of-mem","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let the keys be linearly ordered, and let $T$ be an ordered binary tree. If a key $x$ lies in $T$, then the tree obtained by inserting $x$ into $T$ has the same number of nodes as $T$.","title":"Inserting a present key into a binary search tree leaves its size unchanged"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/num-nodes-insert-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.numNodes_insert_of_not_mem","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#num-nodes-insert-of-not-mem","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let the keys be linearly ordered. For every binary tree $T$ and every key $x$ that does not lie in $T$, the tree obtained by inserting $x$ into $T$ has one more node than $T$.","title":"Inserting an absent key into a binary search tree adds one node"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/rank-eq-count-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.rank_eq_count_inorder","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#rank-eq-count-inorder","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, a key $x \\in \\alpha$ and a binary tree $t$ with keys in $\\alpha$, the rank of $x$ in $t$ equals the length of the list obtained from the inorder traversal of $t$ by keeping exactly the entries strictly less than $x$.","title":"Rank counts the smaller inorder entries"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/select-eq-get-elem-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.select_eq_getElem?_inorder","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#select-eq-get-elem-inorder","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For a binary tree $t$ and a natural number $i$, selecting index $i$ in $t$ gives the same result as looking up zero-based position $i$ of the inorder traversal of $t$: the entry at that position when $i$ is less than the length of the traversal, and nothing otherwise.","title":"Selection is indexing into the inorder traversal"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/select-rank","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.select_rank","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#select-rank","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and let $t$ be a binary search tree with keys in $\\alpha$, that is, at every node each key of the left subtree is strictly smaller and each key of the right subtree is strictly larger than the node's key. If $x$ is a key of $t$, then selecting, in $t$, the index equal to the rank of $x$ in $t$ returns $x$.","title":"Selecting the rank of a key returns the key"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/rank-select","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.rank_select","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#rank-select","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and let $t$ be a binary search tree with keys in $\\alpha$, that is, at every node each key of the left subtree is strictly smaller and each key of the right subtree is strictly larger than the node's key. If selecting index $i$ in $t$ returns a key $x$, then the rank of $x$ in $t$ equals $i$.","title":"The rank of a selected key is its index"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/join","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.join","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#join","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"The join of the empty tree and a tree $R$ is $R$. The join of a tree with root key $k$, left subtree $L_1$ and right subtree $L_2$ and a tree $R$ is the tree with root key $k$, left subtree $L_1$ and right subtree the join of $L_2$ and $R$. Thus, when the first tree is nonempty, $R$ becomes the right subtree of its rightmost node.","title":"Join of two binary trees"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/delete","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.delete","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#delete","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $x$ be a key of a linearly ordered type. Deleting $x$ from the empty tree gives the empty tree. Deleting $x$ from a tree with root key $k$, left subtree $L$ and right subtree $R$ gives the tree with root key $k$, left subtree $L$ with $x$ deleted and right subtree $R$ if $x < k$; the tree with root key $k$, left subtree $L$ and right subtree $R$ with $x$ deleted if $k < x$; and the join of $L$ and $R$\u2026","title":"Deletion from a binary search tree"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/mem-join","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.mem_join","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#mem-join","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For binary trees $L$ and $R$, a key $x$ occurs in the join of $L$ and $R$ if and only if $x$ occurs in $L$ or $x$ occurs in $R$.","title":"Membership in a join"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/inorder-join","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.inorder_join","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#inorder-join","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"For binary trees $L$ and $R$, the inorder traversal of the join of $L$ and $R$ is the inorder traversal of $L$ followed by the inorder traversal of $R$.","title":"Inorder traversal of a join"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/ordered-join","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.ordered_join","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#ordered-join","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $L$ and $R$ be ordered binary search trees over a type with a relation $<$. If $a < b$ for every key $a$ of $L$ and every key $b$ of $R$, then the join of $L$ and $R$ is ordered.","title":"Joining separated ordered trees"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/mem-delete-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.mem_delete_of_mem","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#mem-delete-of-mem","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"Let $t$ be a binary tree over a linearly ordered type, not necessarily ordered, and let $x$ and $y$ be keys with $y \\ne x$. If $y$ occurs in $t$, then $y$ occurs in the tree obtained by deleting $x$ from $t$.","title":"Deletion keeps the other keys"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/not-mem-delete-self","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.not_mem_delete_self","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#not-mem-delete-self","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"If $t$ is an ordered binary search tree over a linearly ordered type and $x$ is a key, then $x$ does not occur in the tree obtained by deleting $x$ from $t$.","title":"The deleted key is gone"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/mem-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.mem_delete","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#mem-delete","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"If $t$ is an ordered binary search tree over a linearly ordered type, then for all keys $x$ and $y$, the key $y$ occurs in the tree obtained by deleting $x$ from $t$ if and only if $y$ occurs in $t$ and $y \\ne x$.","title":"Membership after deletion"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/ordered-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.ordered_delete","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#ordered-delete","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"If $t$ is an ordered binary search tree over a linearly ordered type, then for every key $x$ the tree obtained by deleting $x$ from $t$ is ordered.","title":"Deletion preserves the search-tree order"},{"id":"ComputerScience/data-structures/search-trees/binary-search-tree/inorder-delete","kind":"node","lean":"Cslib.Algorithms.Lean.BinarySearchTree.inorder_delete","parent":36534,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/binary-search-tree/index.html#inorder-delete","scope":"ComputerScience/data-structures/search-trees/binary-search-tree","status":"fully_proved","summary":"If $t$ is an ordered binary search tree over a linearly ordered type and $x$ is a key, then the inorder traversal of the tree obtained by deleting $x$ from $t$ is the inorder traversal of $t$ with the first occurrence of $x$ removed (and is the inorder traversal of $t$ itself if $x$ does not occur in it).","title":"Inorder traversal after deletion"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/avl-tree.json","id":"ComputerScience/data-structures/search-trees/avl-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","title":"AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/tree","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.Tree","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#tree","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a type $\\alpha$, a tree with cached heights over $\\alpha$ (the underlying type of AVL trees) is either empty, or a node made of a natural number $h$ (its cached height), a left subtree, a key in $\\alpha$ and a right subtree.","title":"Binary trees with cached heights"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/height","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.height","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#height","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"The cached height of the empty tree is $0$, and the cached height of a node is the height value stored at that node, so it is read off the root in constant time.","title":"Cached height of an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mk-node","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.mkNode","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mk-node","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Write $h(T)$ for the cached height at the root of a tree $T$, with $h$ of the empty tree equal to $0$. Given trees $L$ and $R$ and a key $k$, this is the node with left subtree $L$, key $k$ and right subtree $R$ whose cached height is $\\max(h(L), h(R)) + 1$.","title":"AVL node with computed height"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.CacheCorrect","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"The empty tree has correct cached heights. A node storing the height $c$, with left subtree $\\ell$ and right subtree $r$, has correct cached heights when $\\ell$ and $r$ have correct cached heights and $c = \\max(h(\\ell), h(r)) + 1$, where $h$ denotes the height cached at the root of a tree (and $0$ for the empty tree).","title":"Correct cached heights"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.Balanced","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"The empty tree is balanced. A node with left subtree $\\ell$ and right subtree $r$ is balanced when $\\ell$ and $r$ are balanced and $h(\\ell) \\le h(r) + 1$ and $h(r) \\le h(\\ell) + 1$, where $h$ denotes the height cached at the root of a tree (and $0$ for the empty tree). Equivalently, at every node the balance factor $h(\\ell) - h(r)$ lies in $\\{-1, 0, 1\\}$.","title":"AVL balance invariant"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/to-binary-tree","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.toBinaryTree","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#to-binary-tree","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type. Erasing the cached heights sends a tree with cached heights over $\\alpha$ to a binary tree over $\\alpha$: the empty tree goes to the empty binary tree, and a node with height $h$, left subtree $L$, key $k$ and right subtree $R$ goes to the binary-tree node with key $k$ whose left and right subtrees are the erasures of $L$ and $R$.","title":"Erasing the cached heights of an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mem","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.Mem","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mem","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"A key $x$ belongs to a tree $T$ when $x$ belongs to the binary tree obtained from $T$ by erasing the cached heights. Equivalently, no key belongs to the empty tree, and $x$ belongs to a node with left subtree $L$, key $k$ and right subtree $R$ exactly when $x$ belongs to $L$, $x = k$, or $x$ belongs to $R$.","title":"Membership in an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/all","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.All","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#all","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a predicate $p$ on keys and a tree $t$ with cached heights, $\\mathrm{All}(p, t)$ holds when $p$ holds at the key of every node of $t$. It is defined as the corresponding predicate on the plain binary tree obtained from $t$ by erasing its cached heights.","title":"Every key of an AVL tree satisfies a predicate"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.Ordered","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $<$. A tree with cached heights over $\\alpha$ is ordered when the binary tree obtained by erasing its cached heights is an ordered binary search tree: the empty tree is ordered, and a node with key $k$ is ordered when its left and right subtrees are ordered, every key of the left subtree is less than $k$, and every key of the right subtree is greater than $k$.","title":"Ordered AVL trees"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/inorder-aux","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.inorderAux","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#inorder-aux","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Given a tree $t$ and a list $s$, this is the list of keys of $t$ in inorder (left subtree, root key, right subtree, recursively) followed by $s$. It is computed in linear time by the binary search tree traversal of $t$ with its cached heights erased.","title":"Inorder traversal of an AVL tree onto a suffix"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/inorder","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.inorder","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#inorder","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"The inorder traversal of a tree $t$ lists its keys with the left subtree first, then the root key, then the right subtree, recursively. It is the binary search tree inorder traversal of $t$ with its cached heights erased, computed in linear time.","title":"Inorder traversal of an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mem-to-binary-tree","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.mem_toBinaryTree","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mem-to-binary-tree","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"A key $x$ lies in a tree $T$ if and only if $x$ lies in the binary tree obtained from $T$ by erasing the cached heights.","title":"AVL membership is membership in the underlying binary tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/all-to-binary-tree","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.all_toBinaryTree","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#all-to-binary-tree","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a predicate $p$ on keys and a tree $t$ with cached heights, $\\mathrm{All}(p, t)$ holds if and only if $p$ holds at every key of the plain binary tree obtained from $t$ by erasing its cached heights. This holds by definition.","title":"The all-keys predicate is unchanged by erasing height caches"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-to-binary-tree","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_toBinaryTree","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-to-binary-tree","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $<$. A tree $t$ with cached heights over $\\alpha$ is ordered if and only if the binary tree obtained from $t$ by erasing the cached heights is an ordered binary search tree.","title":"Order of an AVL tree via its underlying binary tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/inorder-to-binary-tree","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.inorder_toBinaryTree","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#inorder-to-binary-tree","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For every AVL tree $t$, the inorder traversal of $t$ equals the binary search tree inorder traversal of the binary tree obtained from $t$ by erasing its cached heights; this holds by definition.","title":"AVL inorder traversal agrees with the erased binary search tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/contains","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.contains","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#contains","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$ and a key $x$, the search for $x$ returns false on the empty tree; at a node with key $k$ it searches the left subtree if $x < k$, searches the right subtree if $k < x$, and returns true otherwise. The cached heights are ignored.","title":"Membership search in an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balance","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balance","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balance","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For trees $l$, $r$ with cached heights and a key $x$, write $h(T)$ for the cached height at the root of a tree $T$ (zero for the empty tree) and $N(A, y, B)$ for the node with left subtree $A$, key $y$, right subtree $B$ and cached height $\\max(h(A), h(B)) + 1$. The node $\\mathrm{balance}(l, x, r)$ is defined by cases. If $h(l) > h(r) + 1$ and $l$ has left subtree $A$, key $y$ and right subtree $B$, it is the\u2026","title":"AVL rebalancing"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.insert","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#insert","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let the keys be linearly ordered, and write $\\mathrm{balance}(L, k, R)$ for the AVL rebalancing of the node with left subtree $L$, key $k$ and right subtree $R$. Inserting $x$ into the empty tree gives the node with key $x$, two empty children and cached height $1$. Inserting $x$ into a node with left subtree $L$, key $k$ and right subtree $R$ gives $\\mathrm{balance}(L', k, R)$, where $L'$ is the result of\u2026","title":"Insertion into an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/from-list","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.fromList","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#from-list","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, the tree built from the empty list is the empty tree, and the tree built from a list with head $x$ and tail $xs$ is obtained by AVL insertion of $x$ into the tree built from $xs$. Thus the list is inserted from right to left, starting from the empty tree.","title":"Building an AVL tree from a list"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/all-iff-forall-mem","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.all_iff_forall_mem","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#all-iff-forall-mem","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a predicate $p$ on keys and a tree $t$ with cached heights, $\\mathrm{All}(p, t)$ holds if and only if $p(x)$ holds for every $x \\in t$, where $x \\in t$ means that $x$ is the key of some node of $t$.","title":"The all-keys predicate in terms of membership (AVL trees)"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mem-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.mem_inorder","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mem-inorder","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"A key $x$ occurs in the inorder traversal of a tree $T$ if and only if $x$ lies in $T$.","title":"Membership in the inorder traversal of an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_inorder","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-inorder","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. If a tree $t$ with cached heights over $\\alpha$ is ordered, then in the inorder traversal of $t$ every key is less than every key that comes after it.","title":"The inorder traversal of an ordered AVL tree is strictly increasing"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/inorder-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.inorder_nodup","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#inorder-nodup","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, if an AVL tree is ordered, then its inorder traversal contains no key twice.","title":"The inorder traversal of an ordered AVL tree has no duplicates"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct-mk-node","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.cacheCorrect_mkNode","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct-mk-node","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"If two AVL trees $\\ell$ and $r$ have correct cached heights, then for every key $k$ the node built from $\\ell$, $k$ and $r$ by the smart constructor, which stores the height $\\max(h(\\ell), h(r)) + 1$ where $h$ denotes the height cached at the root (and $0$ for the empty tree), also has correct cached heights.","title":"The AVL node constructor preserves cached heights"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced-mk-node","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balanced_mkNode","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced-mk-node","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\ell$ and $r$ be AVL trees that satisfy the AVL balance invariant, and write $h(\\ell)$ and $h(r)$ for the heights cached at their roots (with height $0$ for the empty tree). If $h(\\ell) \\le h(r) + 1$ and $h(r) \\le h(\\ell) + 1$, then for every key $k$ the node built from $\\ell$, $k$ and $r$ by the smart constructor, which stores the height $\\max(h(\\ell), h(r)) + 1$, satisfies the AVL balance invariant.","title":"The AVL node constructor preserves balance"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balance-eq-mk-node-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balance_eq_mkNode_of_le","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balance-eq-mk-node-of-le","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $l$ and $r$ be trees with cached heights, let $x$ be a key, and write $h(T)$ for the cached height at the root of a tree $T$ (zero for the empty tree). If $h(l) \\le h(r) + 1$ and $h(r) \\le h(l) + 1$, then $\\mathrm{balance}(l, x, r)$ is the node with left subtree $l$, key $x$, right subtree $r$ and cached height $\\max(h(l), h(r)) + 1$; no rotation is performed.","title":"Rebalancing when the heights of the children differ by at most one"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balance-left-single","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balance_left_single","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balance-left-single","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Write $h(T)$ for the cached height at the root of a tree $T$ (zero for the empty tree), $\\langle c; A, y, B \\rangle$ for the node with cached height $c$, left subtree $A$, key $y$ and right subtree $B$, and $N(A, y, B)$ for the node $\\langle \\max(h(A), h(B)) + 1; A, y, B \\rangle$. Let $A$, $B$, $R$ be trees with cached heights, let $x$, $y$ be keys and let $c$ be a natural number. If $h(R) + 1 < c$ and\u2026","title":"Single right rotation in AVL rebalancing"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balance-left-double","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balance_left_double","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balance-left-double","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Write $h(T)$ for the cached height at the root of a tree $T$ (zero for the empty tree), $\\langle c; A, y, B \\rangle$ for the node with cached height $c$, left subtree $A$, key $y$ and right subtree $B$, and $N(A, y, B)$ for the node $\\langle \\max(h(A), h(B)) + 1; A, y, B \\rangle$. Let $A$, $B$, $C$, $R$ be trees with cached heights, let $x$, $y$, $z$ be keys and let $c$, $c'$ be natural numbers. If $h(R) + 1 < c$\u2026","title":"Left-right double rotation in AVL rebalancing"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balance-right-single","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balance_right_single","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balance-right-single","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Write $h(T)$ for the cached height at the root of a tree $T$ (zero for the empty tree), $\\langle c; A, y, B \\rangle$ for the node with cached height $c$, left subtree $A$, key $y$ and right subtree $B$, and $N(A, y, B)$ for the node $\\langle \\max(h(A), h(B)) + 1; A, y, B \\rangle$. Let $L$, $A$, $B$ be trees with cached heights, let $x$, $y$ be keys and let $c$ be a natural number. If $h(L) + 1 < c$ and\u2026","title":"Single left rotation in AVL rebalancing"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balance-right-double","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balance_right_double","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balance-right-double","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Write $h(T)$ for the cached height at the root of a tree $T$ (zero for the empty tree), $\\langle c; A, y, B \\rangle$ for the node with cached height $c$, left subtree $A$, key $y$ and right subtree $B$, and $N(A, y, B)$ for the node $\\langle \\max(h(A), h(B)) + 1; A, y, B \\rangle$. Let $L$, $A$, $B$, $C$ be trees with cached heights, let $x$, $y$, $z$ be keys and let $c$, $c'$ be natural numbers. If $h(L) + 1 < c$\u2026","title":"Right-left double rotation in AVL rebalancing"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct-balance","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.cacheCorrect_balance","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct-balance","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"If two AVL trees $\\ell$ and $r$ have correct cached heights, then for every key $k$ the tree produced by the rebalancing function from $\\ell$, $k$ and $r$ also has correct cached heights.","title":"AVL rebalancing preserves cached heights"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/height-balance-le","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.height_balance_le","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#height-balance-le","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $l$ and $r$ be trees whose cached heights are correct at every node, and write $h$ for the cached height. Then for every key $k$, the rebalanced node $b$ built from $l$, $k$ and $r$ satisfies $h(b) \\le \\max(h(l), h(r)) + 1$.","title":"Upper bound on height after AVL rebalancing"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/le-height-balance","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.le_height_balance","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#le-height-balance","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Write $h(T)$ for the cached height at the root of a tree $T$, with $h$ of the empty tree equal to $0$, and write $\\mathrm{balance}(L, k, R)$ for the AVL rebalancing of the node with left subtree $L$, key $k$ and right subtree $R$. If $L$ and $R$ are trees whose cached heights are correct at every node, then for every key $k$ we have $\\max(h(L), h(R)) \\le h(\\mathrm{balance}(L, k, R))$.","title":"Rebalancing does not drop below the taller child"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced-balance","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balanced_balance","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced-balance","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $l$ and $r$ be trees whose cached heights are correct at every node and which are balanced (at every node the cached heights of the two subtrees differ by at most one), let $x$ be a key, and write $h(T)$ for the cached height at the root of a tree $T$ (zero for the empty tree). If $h(l) \\le h(r) + 2$ and $h(r) \\le h(l) + 2$, then $\\mathrm{balance}(l, x, r)$ is balanced.","title":"AVL rebalancing yields a balanced tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/inorder-balance","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.inorder_balance","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#inorder-balance","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For all trees $l$ and $r$ and every key $k$, the inorder traversal of the rebalanced node built from $l$, $k$ and $r$ is the inorder traversal of $l$, followed by $k$, followed by the inorder traversal of $r$.","title":"AVL rebalancing preserves the inorder traversal"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mem-balance","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.mem_balance","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mem-balance","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Write $\\mathrm{balance}(L, k, R)$ for the AVL rebalancing of the node with left subtree $L$, key $k$ and right subtree $R$. For trees $L$ and $R$ and keys $k$ and $x$, the key $x$ lies in $\\mathrm{balance}(L, k, R)$ if and only if $x$ lies in $L$, $x = k$, or $x$ lies in $R$.","title":"Membership after AVL rebalancing"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-balance","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_balance","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-balance","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $k \\in \\alpha$, and let $L$ and $R$ be ordered trees with cached heights over $\\alpha$ such that every key of $L$ is less than $k$ and every key of $R$ is greater than $k$. Then the tree that rebalancing produces from left subtree $L$, key $k$ and right subtree $R$ is ordered.","title":"Rebalancing an AVL node preserves the search-tree order"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct-insert","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.cacheCorrect_insert","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct-insert","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. If an AVL tree $t$ over $\\alpha$ has correct cached heights, then for every $x \\in \\alpha$ the tree obtained by inserting $x$ into $t$ also has correct cached heights.","title":"AVL insertion preserves cached heights"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/height-insert-le","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.height_insert_le","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#height-insert-le","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, if a tree $t$ has correct cached heights at every node, then for every key $x$ the cached height of the tree obtained by inserting $x$ into $t$ is at most the cached height of $t$ plus one.","title":"Upper bound on height after AVL insertion"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/le-height-insert","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.le_height_insert","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#le-height-insert","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Write $h(T)$ for the cached height at the root of a tree $T$, with $h$ of the empty tree equal to $0$. Let the keys be linearly ordered, and let $T$ be a tree whose cached heights are correct at every node and which satisfies the AVL balance invariant at every node. Then for every key $x$, if $T'$ is the tree obtained by inserting $x$ into $T$, we have $h(T) \\le h(T')$.","title":"Insertion does not lower the height"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced-insert","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balanced_insert","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced-insert","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $t$ be an AVL tree over $\\alpha$ and let $x \\in \\alpha$. If $t$ has correct cached heights and satisfies the AVL balance invariant, then the tree obtained by inserting $x$ into $t$ also satisfies the AVL balance invariant.","title":"AVL insertion preserves balance"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mem-insert","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.mem_insert","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mem-insert","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let the keys be linearly ordered. A key $y$ lies in the tree obtained by inserting $x$ into a tree $T$ if and only if $y = x$ or $y$ lies in $T$.","title":"Membership after insertion into an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_insert","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-insert","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every key $x \\in \\alpha$ and every ordered tree $t$ with cached heights over $\\alpha$, the tree obtained by inserting $x$ into $t$ is ordered.","title":"AVL insertion preserves the search-tree order"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/contains-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.contains_eq_true_iff","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#contains-eq-true-iff","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and let $t$ be an ordered AVL tree over $\\alpha$. Then for every $x \\in \\alpha$, searching for $x$ in $t$ by following comparisons from the root returns true if and only if $x$ is a key of $t$.","title":"Correctness of AVL search"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.cacheCorrect_fromList","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct-from-list","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For every list $L$ over a linearly ordered type, the AVL tree built from $L$ by successive insertion has correct cached heights.","title":"AVL trees built from a list have correct cached heights"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balanced_fromList","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced-from-list","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For every list $L$ over a linearly ordered type, the AVL tree built from $L$ by successive insertion satisfies the AVL balance invariant.","title":"AVL trees built from a list are balanced"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_fromList","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-from-list","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every list $xs$ of elements of $\\alpha$, the tree obtained from the empty tree by inserting the elements of $xs$ one at a time, from the last element to the first, is ordered.","title":"AVL trees built from a list are ordered"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mem-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.mem_fromList","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mem-from-list","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let the keys be linearly ordered. A key lies in the tree obtained by inserting the entries of a list one at a time, from last to first, into the empty tree exactly when it is in the list.","title":"Membership in an AVL tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/min-d","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.minD","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#min-d","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Given a default value $d$ and a binary tree $T$ with cached heights, the leftmost key of $T$ with default $d$ is $d$ when $T$ is empty; when $T$ has left subtree $L$ and root key $k$, it is the leftmost key of $L$ with default $k$. Thus for a nonempty tree it is the key of the node reached from the root by following left children as far as possible, and it does not depend on $d$.","title":"Leftmost key of a tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/erase-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.eraseMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#erase-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a tree $t$ with cached heights, the tree obtained by removing the leftmost key of $t$ is defined by recursion on $t$. The empty tree stays empty. A node whose left subtree is empty is replaced by its right subtree. A node with nonempty left subtree $l$, key $k$ and right subtree $r$ is replaced by $\\mathrm{balance}(l', k, r)$, where $l'$ is $l$ with its leftmost key removed and $\\mathrm{balance}$ is the AVL\u2026","title":"Removing the leftmost key of an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/delete-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.deleteMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#delete-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Minimum deletion maps a binary tree with cached heights to the tree obtained by removing its leftmost key and rebalancing on the way back up. It is the operation of removing the minimum, under the name used for the successor step of deletion, where a deleted node with two children is replaced by the minimum of its right subtree.","title":"Minimum deletion in an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/erase","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.erase","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#erase","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $x \\in \\alpha$. Deleting $x$ from a tree with cached heights is defined by recursion on the tree, where $\\mathrm{balance}$ denotes the AVL rebalancing function. The empty tree stays empty. For a node with left subtree $l$, key $k$ and right subtree $r$: if $x < k$, the result is $\\mathrm{balance}(l', k, r)$, where $l'$ is $l$ with $x$ deleted; if $k < x$, the result is\u2026","title":"AVL tree deletion"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/delete","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.delete","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#delete","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, a key $x \\in \\alpha$ and a tree $t$ with cached heights, $\\mathrm{delete}(x, t)$ is defined to be $\\mathrm{erase}(x, t)$, the AVL deletion of $x$ from $t$.","title":"AVL tree deletion under its alternative name"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/min-d-mem","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.minD_mem","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#min-d-mem","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"For every nonempty binary tree $T$ with cached heights and every default value $d$, the leftmost key of $T$ computed with default $d$ occurs in $T$.","title":"The leftmost key lies in the tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct-erase-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.cacheCorrect_eraseMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct-erase-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with a cached height at each node whose cached heights are correct: each node stores one more than the larger of the cached heights of its two subtrees, the empty tree having height $0$. Then the cached heights of the tree obtained from $t$ by removing its leftmost key, rebalancing on the way back up, are again correct.","title":"Removing the minimum keeps cached heights correct"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/height-erase-min-le","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.height_eraseMin_le","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#height-erase-min-le","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"If $t$ is a tree whose cached heights are correct, then the cached height of the tree obtained from $t$ by removing its leftmost key is at most the cached height of $t$.","title":"Removing the leftmost key does not increase the height of an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/le-height-erase-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.le_height_eraseMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#le-height-erase-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"If $t$ is a tree whose cached heights are correct and which is balanced, then the cached height of $t$ is at most one more than the cached height of the tree obtained from $t$ by removing its leftmost key.","title":"Removing the leftmost key lowers the height of an AVL tree by at most one"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced-erase-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balanced_eraseMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced-erase-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with a cached height at each node whose cached heights are correct, and which is balanced: at every node the cached heights of the two subtrees differ by at most one. Then the tree obtained from $t$ by removing its leftmost key, rebalancing on the way back up, is again balanced.","title":"Removing the minimum preserves AVL balance"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-erase-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_eraseMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-erase-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $T$ be a binary tree with cached heights over a linearly ordered type. If $T$ is ordered as a search tree, then so is the tree obtained from $T$ by removing its leftmost key and rebalancing on the way back up.","title":"Removing the leftmost key preserves search-tree order"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/all-min-d-lt-erase-min-of-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.all_minD_lt_eraseMin_of_ordered","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#all-min-d-lt-erase-min-of-ordered","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys from a linearly ordered type and a cached height at each node, and suppose $t$ is ordered: at every node, all keys in the left subtree are strictly smaller and all keys in the right subtree are strictly larger than the key at that node. Let $m$ be the leftmost key of $t$, or an arbitrary default value if $t$ is empty. Then every key of the tree obtained from $t$ by removing its\u2026","title":"Keys left after removing the minimum exceed it"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/inorder-eq-min-d-cons-erase-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.inorder_eq_minD_cons_eraseMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#inorder-eq-min-d-cons-erase-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a nonempty tree with cached heights and $m$ its leftmost key. Then the inorder traversal of $t$ is $m$ followed by the inorder traversal of the tree obtained from $t$ by removing its leftmost key.","title":"The inorder traversal begins with the leftmost key"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct-erase","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.cacheCorrect_erase","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct-erase","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys from a linearly ordered type and a cached height at each node whose cached heights are correct: each node stores one more than the larger of the cached heights of its two subtrees, the empty tree having height $0$. Then for every key $x$, the cached heights of the tree obtained from $t$ by deleting $x$, rebalancing along the search path, are again correct.","title":"Deletion keeps cached heights correct"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/height-erase-le","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.height_erase_le","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#height-erase-le","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$, and $t$ a tree whose cached heights are correct. Then the cached height of the tree obtained by deleting $x$ from $t$ is at most the cached height of $t$.","title":"Deletion does not increase the height of an AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/le-height-erase","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.le_height_erase","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#le-height-erase","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$, and $t$ a balanced tree whose cached heights are correct. Then the cached height of $t$ is at most one more than the cached height of the tree obtained by deleting $x$ from $t$.","title":"Deletion lowers the height of an AVL tree by at most one"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced-erase","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balanced_erase","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced-erase","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys from a linearly ordered type and a cached height at each node, whose cached heights are correct, and which is balanced: at every node the cached heights of the two subtrees differ by at most one. Then for every key $x$, the tree obtained from $t$ by deleting $x$, rebalancing along the search path, is again balanced.","title":"Deletion preserves AVL balance"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-erase","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_erase","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-erase","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $T$ be a binary tree with cached heights over a linearly ordered type, ordered as a search tree (every key in a left subtree is smaller, and every key in a right subtree is larger, than the key at the node). Then for every key $x$, the tree obtained by deleting $x$ from $T$ with rebalancing is again ordered as a search tree.","title":"AVL deletion preserves search-tree order"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/inorder-erase","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.inorder_erase","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#inorder-erase","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$, and $t$ an ordered tree. Then the inorder traversal of the tree obtained by deleting $x$ from $t$ is the inorder traversal of $t$ with the first occurrence of $x$ removed; if $x$ does not occur, the traversal is unchanged.","title":"Inorder traversal after AVL deletion"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mem-erase-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.mem_erase_of_mem","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mem-erase-of-mem","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $T$ be a binary tree with cached heights whose keys come from a linearly ordered type, and let $x$ and $y$ be keys with $y \\ne x$. If $y$ occurs in $T$, then $y$ occurs in the tree obtained by deleting $x$ from $T$. No search-tree ordering of $T$ is assumed.","title":"AVL deletion keeps every other key"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/not-mem-erase-self","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.not_mem_erase_self","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#not-mem-erase-self","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $T$ be a binary tree with cached heights over a linearly ordered type, ordered as a search tree. Then for every key $x$, $x$ does not occur in the tree obtained by deleting $x$ from $T$.","title":"Deletion removes the deleted key from an ordered AVL tree"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/mem-erase","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.mem_erase","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#mem-erase","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $T$ be a binary tree with cached heights over a linearly ordered type, ordered as a search tree (every key in a left subtree is smaller, and every key in a right subtree is larger, than the key at the node). Then for all keys $x$ and $y$, $y$ occurs in the tree obtained by deleting $x$ from $T$ if and only if $y$ occurs in $T$ and $y \\ne x$.","title":"Keys of an ordered AVL tree after deletion"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct-delete","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.cacheCorrect_delete","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct-delete","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys from a linearly ordered type and a cached height at each node whose cached heights are correct: each node stores one more than the larger of the cached heights of its two subtrees, the empty tree having height $0$. Then for every key $x$, the cached heights of the tree obtained from $t$ by deleting $x$, rebalancing along the search path, are again correct.","title":"Deleting a key keeps cached heights correct"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced-delete","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balanced_delete","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced-delete","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys from a linearly ordered type and a cached height at each node, whose cached heights are correct, and which is balanced: at every node the cached heights of the two subtrees differ by at most one. Then for every key $x$, the tree obtained from $t$ by deleting $x$, rebalancing along the search path, is again balanced.","title":"Deleting a key preserves AVL balance"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-delete","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_delete","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-delete","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $T$ be a binary tree with cached heights over a linearly ordered type. If $T$ is ordered as a search tree, then for every key $x$ the tree obtained by deleting $x$ from $T$ with rebalancing is again ordered as a search tree.","title":"Deleting a key from an ordered AVL tree keeps it ordered"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/inorder-delete","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.inorder_delete","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#inorder-delete","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$, and $t$ an ordered tree. Then the inorder traversal of $\\mathrm{delete}(x, t)$, the AVL deletion of $x$ from $t$ under its alternative name, is the inorder traversal of $t$ with the first occurrence of $x$ removed; if $x$ does not occur, the traversal is unchanged.","title":"Inorder traversal after AVL deletion, alternative name"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/cache-correct-delete-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.cacheCorrect_deleteMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#cache-correct-delete-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with a cached height at each node whose cached heights are correct: each node stores one more than the larger of the cached heights of its two subtrees, the empty tree having height $0$. Then the cached heights of the tree obtained from $t$ by minimum deletion (removing its leftmost key, rebalancing on the way back up) are again correct.","title":"Minimum deletion keeps cached heights correct"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/balanced-delete-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.balanced_deleteMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#balanced-delete-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with a cached height at each node whose cached heights are correct, and which is balanced: at every node the cached heights of the two subtrees differ by at most one. Then the tree obtained from $t$ by minimum deletion (removing its leftmost key, rebalancing on the way back up) is again balanced.","title":"Minimum deletion preserves AVL balance"},{"id":"ComputerScience/data-structures/search-trees/avl-tree/ordered-delete-min","kind":"node","lean":"Cslib.Algorithms.Lean.AvlTree.ordered_deleteMin","parent":36571,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/avl-tree/index.html#ordered-delete-min","scope":"ComputerScience/data-structures/search-trees/avl-tree","status":"fully_proved","summary":"Let $T$ be a binary tree with cached heights over a linearly ordered type. If $T$ is ordered as a search tree, then so is the tree obtained by the minimum-removal step used for the successor in deletion, which removes the leftmost key of $T$ and rebalances on the way back up.","title":"Successor-step minimum removal preserves search-tree order"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/cartesian-tree.json","id":"ComputerScience/data-structures/search-trees/cartesian-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","title":"Cartesian tree"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/heap-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.HeapOrdered","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#heap-ordered","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $\\le$. A binary tree with keys in $\\alpha$ is min-heap ordered when, at every node, the key $k$ of that node satisfies $k \\le y$ for every key $y$ in its left subtree and every key $y$ in its right subtree. The empty tree is min-heap ordered.","title":"Min-heap ordered binary tree"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/root","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.root","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#root","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For a binary tree $T$ with keys in $\\alpha$, the root key of $T$ is an optional element of $\\alpha$: it is absent (none) when $T$ is empty, and it is $k$ when the top node of $T$ holds the key $k$.","title":"Optional root key of a binary tree"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/all-iff-forall-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.all_iff_forall_inorder","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#all-iff-forall-inorder","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For every predicate $p$ on $\\alpha$ and every binary tree $T$ with keys in $\\alpha$, every key of $T$ satisfies $p$ if and only if every element of the inorder traversal of $T$ satisfies $p$.","title":"A predicate holds on every key iff it holds along the inorder traversal"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/mem-of-root-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.mem_of_root_eq_some","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#mem-of-root-eq-some","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For every binary tree $T$ and every key $k$, if $T$ is nonempty with root key $k$, then $k$ is a key of $T$.","title":"The root of a binary tree is one of its keys"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/root-le-of-heap-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.root_le_of_heapOrdered","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#root-le-of-heap-ordered","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"Let $T$ be a binary tree over a preordered type that is heap-ordered (in min-heap order). If $T$ has root key $k$, then $k \\le x$ for every key $x$ occurring in $T$.","title":"The root of a heap-ordered tree is below every key"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/split-at-first","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.splitAtFirst","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#split-at-first","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For a value $m$ of a type with decidable equality and a list $xs$, the split of $xs$ at $m$ is the pair of the part of $xs$ before the first occurrence of $m$ and the part after it; that occurrence itself is dropped. If $m$ does not occur in $xs$, the split is the pair of $xs$ and the empty list.","title":"Splitting a list at the first occurrence of a value"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/split-at-first-not-mem-fst","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.splitAtFirst_not_mem_fst","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#split-at-first-not-mem-fst","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"Let $m$ be an element of a type with decidable equality and let $xs$ be a list in which $m$ occurs. When $xs$ is split at the first occurrence of $m$, the left part does not contain $m$.","title":"Splitting at the first occurrence leaves the value out of the left part"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/list-min","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.listMin","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#list-min","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For an element $a$ and a list $as$ over a linear order, the left-fold minimum of $a$ and $as$ is the minimum of the list with head $a$ and tail $as$. It is computed from the head: it equals $a$ when $as$ is empty, and when $as$ has head $b$ and tail $bs$ it equals the left-fold minimum of $\\min(a, b)$ and $bs$.","title":"Minimum of a nonempty list"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/build-fuel","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.buildFuel","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#build-fuel","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linear order. For a natural number $n$ (the fuel) and a list $xs$ over $\\alpha$, the tree built from $xs$ with fuel $n$ is the empty tree if $n = 0$ or $xs$ is empty. Otherwise $n = k + 1$ and $xs$ is nonempty; let $m$ be the minimum of $xs$, and let $L$ and $R$ be the parts of $xs$ before and after the first occurrence of $m$. Then the tree has root $m$, its left subtree is built from $L$ with\u2026","title":"Fuel-bounded Cartesian tree construction"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/build","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.build","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#build","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For a list $xs$ over a linear order, the Cartesian tree of $xs$ is the result of the fuel-bounded Cartesian tree construction on $xs$ with fuel equal to the length of $xs$. So the empty list gives the empty tree, and otherwise the root is the minimum $m$ of $xs$, the left subtree is built in the same way from the elements before the first occurrence of $m$, and the right subtree from the elements after that\u2026","title":"Cartesian tree of a list"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/inorder-build","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.inorder_build","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#inorder-build","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For every list $xs$ over a linear order, the inorder traversal of the Cartesian tree of $xs$ is $xs$.","title":"Inorder traversal of a Cartesian tree"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/perm-build","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.perm_build","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#perm-build","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For every list $xs$ over a linear order, the inorder traversal of the Cartesian tree of $xs$ is a permutation of $xs$.","title":"The keys of a Cartesian tree permute its input"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/mem-build","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.mem_build","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#mem-build","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For every list $xs$ over a linear order and every element $x$, $x$ is a key of the Cartesian tree of $xs$ if and only if $x$ occurs in $xs$.","title":"Keys of a Cartesian tree"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/heap-ordered-build","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.heapOrdered_build","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#heap-ordered-build","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For every list $xs$ over a linear order, the Cartesian tree of $xs$ is min-heap ordered.","title":"Cartesian trees are heap-ordered"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/root-build-cons-is-some","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.root_build_cons_isSome","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#root-build-cons-is-some","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"For every nonempty list $xs$ over a linearly ordered type, the Cartesian tree of $xs$ has a root key $m$.","title":"The Cartesian tree of a nonempty list has a root"},{"id":"ComputerScience/data-structures/search-trees/cartesian-tree/root-build-min","kind":"node","lean":"Cslib.Algorithms.Lean.CartesianTree.root_build_min","parent":36649,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/cartesian-tree/index.html#root-build-min","scope":"ComputerScience/data-structures/search-trees/cartesian-tree","status":"fully_proved","summary":"Let $xs$ be a list over a linearly ordered type. If the Cartesian tree of $xs$ has root key $m$, then $m$ occurs in $xs$ and $m \\le y$ for every element $y$ of $xs$.","title":"The root of a Cartesian tree is a minimum of the list"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/order-statistic-tree.json","id":"ComputerScience/data-structures/search-trees/order-statistic-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","title":"Order statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/entry","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.Entry","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#entry","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"A node entry of an order-statistic tree over a type $\\alpha$ consists of a key in $\\alpha$ and a natural number, the cached number of nodes in the subtree rooted at that node.","title":"Order-statistic tree node entry"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/tree","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.Tree","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#tree","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"An order-statistic tree with keys in a type $\\alpha$ is a binary tree, possibly empty, each of whose nodes carries a key in $\\alpha$ together with a natural number, meant as the cached number of nodes in the subtree rooted at that node. That these caches are correct is a separate condition.","title":"Order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/cached-size","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.cachedSize","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#cached-size","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"The cached root size of an order-statistic tree is the subtree size stored in the entry of its root node, and is $0$ for the empty tree.","title":"Cached root size"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/erase","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.erase","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#erase","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"The erasure of an order-statistic tree is the binary tree of the same shape whose node keys are the keys of the original nodes: the empty tree is sent to the empty tree, and a node is sent to the node with the same key whose left and right subtrees are the erasures of the original left and right subtrees, so every cached subtree size is dropped.","title":"Erasing the size caches of an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/cache-correct","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.CacheCorrect","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#cache-correct","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"An order-statistic tree has correct size caches when, at every node, the cached subtree size equals the number of nodes of the left subtree plus the number of nodes of the right subtree plus one, that is, the number of nodes of the subtree rooted at that node. The empty tree has correct size caches.","title":"Correct size caches"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/ordered","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.Ordered","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#ordered","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"An order-statistic tree over a type equipped with a relation $<$ is ordered when the binary tree obtained by erasing its cached sizes is an ordered binary search tree: at every node, each key of the left subtree is $<$ the node's key, the node's key is $<$ each key of the right subtree, and both subtrees are ordered.","title":"Ordered order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/inorder","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.inorder","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#inorder","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"The inorder traversal of an order-statistic tree is the inorder traversal of the binary tree obtained by erasing its cached subtree sizes: the traversal of the left subtree, then the root key, then the traversal of the right subtree, and the empty list for the empty tree.","title":"Inorder traversal of an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/node","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.node","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#node","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Given a key $k$ and order-statistic trees $L$ and $R$, the node built from them has key $k$, left subtree $L$, right subtree $R$, and cached subtree size equal to the cached root size of $L$ plus the cached root size of $R$ plus one, where the empty tree has cached root size zero.","title":"Order-statistic tree node with computed cache"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/contains","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.contains","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#contains","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $q \\in \\alpha$. Searching for $q$ in the empty tree returns false. At a node with key $k$, the search continues in the left subtree if $q < k$, continues in the right subtree if $k < q$, and returns true otherwise.","title":"Search in an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.insert","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#insert","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Inserting a key $k$ of a linearly ordered type into the empty tree gives a single node with key $k$ and cached size $1$. For a nonempty tree with root key $r$: if $k < r$, the key is inserted into the left subtree; if $r < k$, it is inserted into the right subtree; in both cases the root is rebuilt with key $r$ and cached size equal to the cached root sizes of its two subtrees plus one (an empty subtree counts as\u2026","title":"Insertion into an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/from-list","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.fromList","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#from-list","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"For a linearly ordered key type, the order-statistic tree built from a list of keys is obtained by inserting the keys into the empty tree from right to left: the empty list gives the empty tree, and a list with head $k$ and tail $l$ gives the insertion of $k$ into the tree built from $l$.","title":"Order-statistic tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/rank","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.rank","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#rank","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"For a key $q$ of a linearly ordered type, the rank of $q$ in an order-statistic tree is computed by descending from the root. It is $0$ for the empty tree. At a node with key $k$, left subtree $L$ and right subtree $R$, it is the rank of $q$ in $L$ if $q < k$; it is $s(L) + 1$ plus the rank of $q$ in $R$ if $k < q$; and it is $s(L)$ if $q = k$. Here $s(L)$ is the subtree size cached at the root of $L$, and $0$ if\u2026","title":"Rank in an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/select","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.select","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#select","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"For an order-statistic tree $T$ and a natural number $i$, select at index $i$ in $T$ is computed by descending from the root. It returns nothing for the empty tree. At a node with key $k$, left subtree $L$ and right subtree $R$, it is select at index $i$ in $L$ if $i < s(L)$; it returns $k$ if $i = s(L)$; and it is select at index $i - s(L) - 1$ in $R$ if $i > s(L)$. Here $s(L)$ is the subtree size cached at the\u2026","title":"Select in an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/cached-size-eq-num-nodes","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.cachedSize_eq_numNodes","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#cached-size-eq-num-nodes","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"If an order-statistic tree $T$ has correct cached subtree sizes at every node, then the cached root size of $T$ (zero for the empty tree) equals the number of nodes of $T$.","title":"Correct caches give the tree size"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/cache-correct-node","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.cacheCorrect_node","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#cache-correct-node","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $k$ be a key and let $L$ and $R$ be order-statistic trees whose cached subtree sizes are correct at every node. Then the node with key $k$, left subtree $L$ and right subtree $R$, whose root cache is the cached root size of $L$ plus the cached root size of $R$ plus one, also has correct cached subtree sizes at every node.","title":"The cache-computing node constructor preserves correct size caches"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/erase-insert","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.erase_insert","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#erase-insert","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $k \\in \\alpha$ and $T$ an order-statistic tree over $\\alpha$. Forgetting the cached sizes of the tree obtained by inserting $k$ into $T$ gives the same binary tree as inserting $k$, by ordinary binary-search-tree insertion, into $T$ with its cached sizes forgotten.","title":"Insertion commutes with forgetting the size caches"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/cache-correct-insert","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.cacheCorrect_insert","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#cache-correct-insert","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $k \\in \\alpha$, and let $T$ be an order-statistic tree over $\\alpha$ whose cached subtree sizes are correct at every node. Then the tree obtained by inserting $k$ into $T$ also has correct cached subtree sizes at every node.","title":"Insertion preserves correct size caches"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.ordered_insert","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#ordered-insert","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $T$ be an order-statistic tree over a linearly ordered key type that is ordered, meaning that after its cached sizes are erased, at every node each key of the left subtree is strictly smaller than the node's key and each key of the right subtree is strictly larger. Then for every key $k$, the tree obtained by inserting $k$ into $T$ is again ordered.","title":"Insertion preserves ordering of order-statistic trees"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/cache-correct-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.cacheCorrect_fromList","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#cache-correct-from-list","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every list $\\ell$ of elements of $\\alpha$, the order-statistic tree obtained by inserting the elements of $\\ell$ into the empty tree, from right to left, has correct cached subtree sizes at every node.","title":"Order-statistic trees built from a list have correct size caches"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/ordered-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.ordered_fromList","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#ordered-from-list","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"For every list $l$ of keys from a linearly ordered type, the order-statistic tree built from $l$ by successive insertion is ordered: after its cached sizes are erased, at every node each key of the left subtree is strictly smaller than the node's key and each key of the right subtree is strictly larger.","title":"Order-statistic trees built from lists are ordered"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/mem-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.mem_fromList","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#mem-from-list","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"For a linearly ordered key type, a key $k$ and a list $l$ of keys, $k$ is a key of some node of the order-statistic tree built from $l$ by successive insertion if and only if $k$ occurs in $l$.","title":"Keys of an order-statistic tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/contains-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.contains_eq_true_iff","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#contains-eq-true-iff","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $T$ an order-statistic tree over $\\alpha$ that is ordered: after forgetting the cached sizes, at every node all keys in the left subtree are smaller than the node's key and all keys in the right subtree are larger. Then for every $q \\in \\alpha$, comparison search for $q$ in $T$ returns true if and only if $q$ is the key of some node of $T$.","title":"Search in an ordered order-statistic tree is correct"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/rank-eq-bst-rank","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.rank_eq_bstRank","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#rank-eq-bst-rank","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $T$ be an order-statistic tree over a linearly ordered key type whose cached subtree sizes are correct at every node and which is ordered. Then for every query $q$, the rank of $q$ computed by descending $T$ with cached left-subtree sizes equals the number of entries strictly smaller than $q$ in the inorder traversal of the binary tree obtained by erasing the caches of $T$.","title":"Cached rank agrees with traversal rank"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/select-eq-bst-select","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.select_eq_bstSelect","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#select-eq-bst-select","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"For every order-statistic tree $T$ whose cached subtree sizes are correct and every natural number $i$, select at index $i$ in $T$ (descending with the cached sizes) returns the same result as taking the element at zero-based position $i$ of the inorder traversal of the binary tree obtained from $T$ by erasing the cached sizes; in particular both return nothing when that position does not exist.","title":"Cached select agrees with inorder select"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/select-rank","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.select_rank","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#select-rank","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $T$ be an order-statistic tree over a linearly ordered type whose cached subtree sizes are correct and which satisfies the binary-search-tree ordering. For every key $k$ in $T$, select in $T$ at the index equal to the rank of $k$ in $T$ returns $k$.","title":"Select inverts rank in an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/rank-select","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.rank_select","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#rank-select","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $T$ be an order-statistic tree over a linearly ordered type whose cached subtree sizes are correct and which satisfies the binary-search-tree ordering. If select at zero-based index $i$ in $T$ returns a key $k$, then the rank of $k$ in $T$ equals $i$.","title":"Rank inverts select in an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/merge","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.merge","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#merge","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"The merge of order-statistic trees $A$ and $B$ is defined by recursion on $A$. If $A$ is empty, it is $B$. If $A$ has root key $x$, left subtree $L$ and right subtree $R$, it is the node with key $x$, left subtree $L$ and right subtree the merge of $R$ and $B$, whose cached size is the sum of the cached root sizes of its children plus one, an empty child counting as zero. Thus $B$ is attached below the rightmost\u2026","title":"Merging order-statistic trees"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/delete","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.delete","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#delete","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Deleting a key $k$ from an order-statistic tree over a linearly ordered type follows the comparison path from the root. The empty tree stays empty. At a node with key $x$, left subtree $L$ and right subtree $R$: if $k < x$, the result is the node with key $x$, left subtree the deletion of $k$ from $L$ and right subtree $R$; if $x < k$, it is the node with key $x$, left subtree $L$ and right subtree the deletion of\u2026","title":"Deletion from an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/cache-correct-merge","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.cacheCorrect_merge","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#cache-correct-merge","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $A$ and $B$ be order-statistic trees whose cached subtree size at every node equals the number of nodes in that node's subtree. Then the merge of $A$ and $B$ has the same property. No ordering between the keys of $A$ and $B$ is assumed.","title":"Order-statistic merge preserves cache correctness"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/cache-correct-delete","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.cacheCorrect_delete","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#cache-correct-delete","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $k \\in \\alpha$ a key and $T$ an order-statistic tree over $\\alpha$. If the cached subtree size at every node of $T$ equals the number of nodes in that node's subtree, then the same holds for the tree obtained by deleting $k$ from $T$.","title":"Order-statistic deletion preserves cache correctness"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/ordered-merge","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.ordered_merge","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#ordered-merge","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and let $L$ and $R$ be ordered order-statistic trees over $\\alpha$ such that $x < y$ for every key $x$ of $L$ and every key $y$ of $R$. Then the merge of $L$ and $R$ is ordered.","title":"Merging separated ordered order-statistic trees preserves order"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/ordered-delete","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.ordered_delete","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#ordered-delete","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and let $T$ be an order-statistic tree over $\\alpha$ that is ordered, that is, after its cached sizes are removed, every key in a left subtree is strictly smaller than the key of its node and every key in a right subtree is strictly larger. Then for every key $k$, the tree obtained from $T$ by deleting $k$ is again ordered.","title":"Deletion preserves order in an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/inorder-merge","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.inorder_merge","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#inorder-merge","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"For order-statistic trees $A$ and $B$, the inorder traversal of the merge of $A$ and $B$ is the inorder traversal of $A$ followed by the inorder traversal of $B$. No ordering between the keys of $A$ and $B$ is assumed.","title":"Inorder traversal of an order-statistic merge"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/mem-merge","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.mem_merge","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#mem-merge","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"For order-statistic trees $A$ and $B$ over a type $\\alpha$ and any $x \\in \\alpha$, $x$ is a key of the merge of $A$ and $B$ if and only if $x$ is a key of $A$ or a key of $B$. No ordering between the keys of $A$ and $B$ is assumed.","title":"Keys of an order-statistic merge"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/inorder-delete","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.inorder_delete","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#inorder-delete","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $k \\in \\alpha$ and $T$ an ordered order-statistic tree over $\\alpha$ (at every node, all keys of the left subtree are smaller and all keys of the right subtree are larger than the node's key). The inorder traversal of the tree obtained by deleting $k$ from $T$ is the inorder traversal of $T$ with every occurrence of $k$ removed.","title":"Inorder traversal after order-statistic deletion"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/mem-delete-iff","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.mem_delete_iff","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#mem-delete-iff","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $k \\in \\alpha$ and $T$ an ordered order-statistic tree over $\\alpha$ (at every node, all keys of the left subtree are smaller and all keys of the right subtree are larger than the node's key). For every $y \\in \\alpha$, $y$ is a key of the tree obtained by deleting $k$ from $T$ if and only if $y$ is a key of $T$ and $y \\ne k$.","title":"Keys of an order-statistic tree after deletion"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/not-mem-delete","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.not_mem_delete","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#not-mem-delete","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $k \\in \\alpha$ and $T$ an ordered order-statistic tree over $\\alpha$ (at every node, all keys of the left subtree are smaller and all keys of the right subtree are larger than the node's key). Then $k$ is not a key of the tree obtained by deleting $k$ from $T$.","title":"Deleted key is absent from an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/contains-delete-self","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.contains_delete_self","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#contains-delete-self","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $k \\in \\alpha$ and $T$ an ordered order-statistic tree over $\\alpha$ (at every node, all keys of the left subtree are smaller and all keys of the right subtree are larger than the node's key). Comparison search for $k$ in the tree obtained by deleting $k$ from $T$ returns false.","title":"Search misses a key deleted from an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/rank-delete-eq-bst-rank","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.rank_delete_eq_bstRank","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#rank-delete-eq-bst-rank","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and let $T$ be an order-statistic tree over $\\alpha$ that is ordered and whose cached subtree sizes are correct. Let $k$ and $q$ be keys and let $T'$ be the tree obtained from $T$ by deleting $k$. Then the rank of $q$ in $T'$, computed by navigation with cached sizes, equals the number of entries strictly smaller than $q$ in the inorder traversal of $T'$ with its cached sizes\u2026","title":"Rank after deletion in an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/select-delete-eq-bst-select","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.select_delete_eq_bstSelect","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#select-delete-eq-bst-select","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and let $T$ be an order-statistic tree over $\\alpha$ whose cached subtree sizes are correct. Let $k$ be a key and let $T'$ be the tree obtained from $T$ by deleting $k$. Then for every natural number $i$, selecting position $i$ in $T'$ by navigation with cached sizes returns the entry at zero-based position $i$ of the inorder traversal of $T'$ with its cached sizes removed,\u2026","title":"Selection after deletion in an order-statistic tree"},{"id":"ComputerScience/data-structures/search-trees/order-statistic-tree/select-rank-delete","kind":"node","lean":"Cslib.Algorithms.Lean.OrderStatisticTree.select_rank_delete","parent":36666,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/order-statistic-tree/index.html#select-rank-delete","scope":"ComputerScience/data-structures/search-trees/order-statistic-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $T$ be an order-statistic tree over $\\alpha$ that is ordered and whose cached subtree sizes are correct, let $k$ be a key, and let $T'$ be the tree obtained from $T$ by deleting $k$. If $y$ is a key of $T'$, then selecting, with cached sizes, the position in $T'$ given by the cached rank of $y$ in $T'$ returns $y$.","title":"Selecting the rank of a surviving key after deletion"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/red-black-tree.json","id":"ComputerScience/data-structures/search-trees/red-black-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","title":"Red\u2013black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/color","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.Color","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#color","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The colors of red-black tree nodes are the two values red and black.","title":"Red-black node colors"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/tree","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.Tree","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#tree","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"A red-black tree with keys in a type $\\alpha$ is either empty or a node consisting of a color (red or black), a left subtree, a key in $\\alpha$, and a right subtree.","title":"Red-black tree data type"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/erase","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.erase","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#erase","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The color erasure of a red-black tree is the binary tree obtained by forgetting the node colors: the empty tree goes to the empty binary tree, and a node with key $k$ and subtrees $l$ and $r$ goes to the binary-tree node with key $k$ whose subtrees are the color erasures of $l$ and $r$.","title":"Color erasure"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/mem","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.Mem","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#mem","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"An element $x$ is a member of a red-black tree $t$ when $x$ is a member of the binary tree obtained from $t$ by erasing its colors, that is, when $x$ equals the key of some node of $t$.","title":"Membership in a red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/inorder-aux","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.inorderAux","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#inorder-aux","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a red-black tree $t$ and a list $s$, this is the accumulator form of the binary-search-tree inorder traversal applied to the color erasure of $t$: for the empty tree it returns $s$, and for a node with key $k$ and subtrees $l$ and $r$ it returns the result for $l$ with suffix the list $k$ followed by $u$, where $u$ is the result for $r$ with suffix $s$.","title":"Inorder traversal onto a suffix"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/inorder","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.inorder","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#inorder","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The inorder traversal of a red-black tree $t$ is the binary-search-tree inorder traversal of the color erasure of $t$, computed in linear time with an accumulator starting from the empty list. It lists the keys of $t$ from left to right: the empty tree gives the empty list, and a node with key $k$ and subtrees $l$ and $r$ gives the traversal of $l$, then $k$, then the traversal of $r$.","title":"Inorder traversal of a red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/all","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.All","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#all","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a predicate $p$ on $\\alpha$ and a red-black tree $t$ with keys in $\\alpha$, $p$ holds at every key of $t$ when $p$ holds at every key of the binary search tree obtained from $t$ by erasing colors.","title":"Predicate holding at every key of a red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/ordered","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.Ordered","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#ordered","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $<$. A red-black tree with keys in $\\alpha$ is ordered when, at every node with key $k$, every key $y$ in the left subtree satisfies $y < k$ and every key $z$ in the right subtree satisfies $k < z$.","title":"Ordered red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/mem-erase","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.mem_erase","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#mem-erase","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every red-black tree $t$ and every $x$, $x$ is a member of $t$ if and only if $x$ is a member of the binary tree obtained from $t$ by erasing its colors.","title":"Red-black membership is membership in the underlying binary tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/all-erase","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.all_erase","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#all-erase","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every predicate $p$ on $\\alpha$ and every red-black tree $t$ with keys in $\\alpha$, $p$ holds at every key of $t$ if and only if $p$ holds at every key of the binary search tree obtained from $t$ by erasing colors. This holds by definition.","title":"All keys of a red-black tree via its color erasure"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/ordered-erase","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.ordered_erase","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#ordered-erase","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For keys with a relation $<$, a red-black tree $t$ is ordered if and only if the binary tree obtained from $t$ by erasing its colors is an ordered binary search tree.","title":"A red-black tree is ordered exactly when its underlying binary tree is"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/inorder-erase","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.inorder_erase","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#inorder-erase","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every red-black tree $t$, the inorder traversal of $t$ equals the binary-search-tree inorder traversal of the color erasure of $t$; this holds by definition.","title":"Inorder traversal ignores colors"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/color-of","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.colorOf","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#color-of","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The root color of a red-black tree is the color of its root node; the empty tree has root color black.","title":"Root color"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/red-ok","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.RedOk","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#red-ok","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"A red-black tree satisfies the red property when no red node has a red child, where empty subtrees count as black.","title":"Red property of red-black trees"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/almost-red-ok","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.AlmostRedOk","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#almost-red-ok","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"A red-black tree satisfies the relaxed red invariant when it is empty, or when its left and right subtrees both satisfy the red invariant (no red node has a red child). The root itself is unconstrained, so a red root may have a red child.","title":"Red invariant away from the root"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/black-height","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.blackHeight","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#black-height","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The black height of a red-black tree is the number of black nodes on its leftmost root-to-leaf path, counting the empty leaf as one black node: the empty tree has black height $1$, a red node has the black height of its left subtree, and a black node has the black height of its left subtree plus $1$.","title":"Black height"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/balanced","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.Balanced","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#balanced","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"A red-black tree is balanced when every root-to-leaf path contains the same number of black nodes. Recursively: the empty tree is balanced, and a node is balanced when both of its subtrees are balanced and have the same black height (the number of black nodes on the leftmost root-to-leaf path, counting the leaf).","title":"Black-height balanced red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.Valid","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#valid","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $<$. A red-black tree with keys in $\\alpha$ is valid when it is ordered, satisfies the red property (no red node has a red child), is balanced (every root-to-leaf path contains the same number of black nodes), and its root is black, where the empty tree counts as black.","title":"Valid red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/contains","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.contains","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#contains","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Searching a red-black tree over a linearly ordered type for a key $x$ returns a Boolean, ignoring colors: the empty tree gives false, and at a node with key $k$ the search continues in the left subtree if $x < k$, continues in the right subtree if $k < x$, and returns true otherwise.","title":"Search in a red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/balance","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.balance","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#balance","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Given a color $c$, red-black trees $l$ and $r$, and a key $k$, rebalancing produces a tree as follows. Suppose $c$ is black and one of these configurations occurs, taking the first that applies: $l$ is a red node with a red left child; $l$ is a red node with a red right child; $r$ is a red node with a red left child; $r$ is a red node with a red right child. The black root and the two red nodes then hold three keys\u2026","title":"Okasaki's rebalancing"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/ins","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.ins","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#ins","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For keys from a linearly ordered type, the unchecked insertion of $x$ into a red-black tree is defined by recursion on the tree. Inserting $x$ into the empty tree gives a red node with key $x$ and two empty subtrees. Inserting $x$ into a node with color $c$, left subtree $l$, key $k$ and right subtree $r$ gives: if $x < k$, Okasaki's rebalancing of color $c$, left subtree the insertion of $x$ into $l$, key $k$ and\u2026","title":"Unchecked red-black insertion"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/make-black","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.makeBlack","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#make-black","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Recoloring the root of a red-black tree black leaves the empty tree unchanged and turns a node with any color, left subtree $l$, key $k$ and right subtree $r$ into the black node with left subtree $l$, key $k$ and right subtree $r$.","title":"Blackening the root"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/erase-make-black","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.erase_makeBlack","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#erase-make-black","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every red-black tree $t$, the color erasure of the tree obtained from $t$ by recoloring its root black (the empty tree is left unchanged) equals the color erasure of $t$.","title":"Recoloring the root black does not change the color erasure"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.insert","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#insert","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For keys from a linearly ordered type, inserting $x$ into a red-black tree $t$ gives the unchecked insertion of $x$ into $t$ with its root recolored black.","title":"Red-black tree insertion"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/from-list","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.fromList","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#from-list","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The red-black tree built from a list over a linearly ordered type inserts the elements of the list into the empty tree from right to left: the empty list gives the empty tree, and a list with head $x$ and tail $xs$ gives the insertion of $x$ into the tree built from $xs$.","title":"Red-black tree from a list"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/all-iff-forall-mem","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.all_iff_forall_mem","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#all-iff-forall-mem","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every predicate $p$ on $\\alpha$ and every red-black tree $t$ with keys in $\\alpha$, $p$ holds at every key of $t$ if and only if $p(x)$ holds for every $x \\in \\alpha$ that is a (structural) member of $t$.","title":"All keys versus members of a red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/mem-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.mem_inorder","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#mem-inorder","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every red-black tree $t$ and every $x$, $x$ occurs in the inorder traversal of $t$ if and only if $x$ is a member of $t$.","title":"Membership in the inorder traversal of a red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/ordered-inorder","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.ordered_inorder","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#ordered-inorder","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For keys from a linearly ordered type, if a red-black tree $t$ is ordered, then every entry of the inorder traversal of $t$ is strictly smaller than every later entry.","title":"The inorder traversal of an ordered red-black tree is strictly increasing"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/mem-insert","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.mem_insert","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#mem-insert","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For keys from a linearly ordered type, a key $y$ is a member of the red-black tree obtained by inserting $x$ into $t$ if and only if $y = x$ or $y$ is a member of $t$.","title":"Membership after insertion into a red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.ordered_insert","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#ordered-insert","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$ a key, and $t$ a red-black tree with keys in $\\alpha$. If $t$ is ordered (at every node with key $k$, every key in the left subtree is less than $k$ and every key in the right subtree is greater than $k$), then the tree obtained by inserting $x$ into $t$ is ordered.","title":"Red-black insertion preserves the search-tree order"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/red-ok-insert","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.redOk_insert","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#red-ok-insert","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$ a key, and $t$ a red-black tree with keys in $\\alpha$. If no red node of $t$ has a red child, then no red node of the tree obtained by inserting $x$ into $t$ has a red child.","title":"Red-black insertion preserves the red property"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/balanced-insert","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.balanced_insert","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#balanced-insert","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. If a red-black tree $t$ with keys in $\\alpha$ is balanced (every root-to-leaf path contains the same number of black nodes), then for every $x \\in \\alpha$ the tree obtained by inserting $x$ into $t$ is balanced.","title":"Red-black insertion preserves black-height balance"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/black-height-insert","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.blackHeight_insert","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#black-height-insert","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every red-black tree $t$ with keys in $\\alpha$ and every $x \\in \\alpha$, the black height of the tree obtained by inserting $x$ into $t$ equals either the black height of $t$ or the black height of $t$ plus one.","title":"Insertion raises the black height by at most one"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/valid-insert-of-invariants","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.valid_insert_of_invariants","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#valid-insert-of-invariants","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$ a key, and $t$ a red-black tree with keys in $\\alpha$ that is ordered, satisfies the red property (no red node has a red child), and is balanced (every root-to-leaf path contains the same number of black nodes). Then the tree obtained by inserting $x$ into $t$ is valid.","title":"Red-black insertion yields a valid tree from the invariants"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/valid-insert","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.valid_insert","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#valid-insert","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $x \\in \\alpha$. If $t$ is a valid red-black tree with keys in $\\alpha$, then the tree obtained by inserting $x$ into $t$ is also valid.","title":"Red-black insertion preserves validity"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/contains-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.contains_eq_true_iff","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#contains-eq-true-iff","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $t$ be a red-black tree over a linearly ordered type that is ordered (every left key is smaller and every right key larger, recursively). Then for every key $x$, the search for $x$ in $t$ returns true if and only if $x$ is a key of some node of $t$.","title":"Search in an ordered red-black tree is correct"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/inorder-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.inorder_nodup","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#inorder-nodup","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a red-black tree $t$ over a linearly ordered type that is ordered (every left key is smaller and every right key larger, recursively), the inorder traversal of $t$ contains no key twice.","title":"Ordered red-black trees have duplicate-free inorder traversals"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.valid_empty","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#valid-empty","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every type $\\alpha$ with a relation $<$, the empty red-black tree with keys in $\\alpha$ is a valid red-black tree.","title":"The empty red-black tree is valid"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/valid-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.valid_fromList","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#valid-from-list","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every list $l$ of elements of $\\alpha$, the red-black tree obtained from the empty tree by inserting the elements of $l$ one at a time, from the last to the first, is valid.","title":"Red-black trees built from a list are valid"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/mem-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.mem_fromList","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#mem-from-list","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For keys from a linearly ordered type and every list $xs$, a key $x$ is a member of the red-black tree built from $xs$ by successive insertion if and only if $x$ occurs in $xs$.","title":"Membership in a red-black tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/join","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.join","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#join","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The splice of a red-black tree $t$ with a red-black tree $s$ is defined by recursion on $t$. The splice of the empty tree with $s$ is $s$. The splice of the node with color $c$, left subtree $l$, key $k$ and right subtree $r$ with $s$ is the node with color $c$, left subtree $l$, key $k$ and right subtree the splice of $r$ with $s$.","title":"Splicing two red-black trees"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/del","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.del","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#del","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a key $x$ of a linearly ordered type, unbalanced deletion of $x$ from a red-black tree is defined by recursion on the tree. The empty tree stays empty. At a node with color $c$, left subtree $l$, key $k$ and right subtree $r$: if $x < k$, the result is the node with color $c$, left subtree the unbalanced deletion of $x$ from $l$, key $k$ and right subtree $r$; if $k < x$, it is the node with color $c$, left\u2026","title":"Unbalanced red-black tree deletion"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/rebalance","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.rebalance","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#rebalance","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a linearly ordered key type, the rebuild of a red-black tree $t$ is the tree obtained from the empty tree by inserting the keys of the inorder traversal of $t$ one at a time with Okasaki insertion, starting from the last key.","title":"Rebuilding a red-black tree from its keys"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/delete","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.delete","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#delete","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a key $x$ of a linearly ordered type and a red-black tree $t$, deleting $x$ from $t$ by rebuilding first forms the unbalanced deletion of $x$ from $t$, then rebalances it: starting from the empty tree, it inserts the keys of the inorder traversal of that tree one at a time, from the last key to the first.","title":"Red-black tree deletion by rebuilding"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/mem-join","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.mem_join","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#mem-join","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For red-black trees $t$ and $s$ with keys of any type, a key $x$ belongs to the splice of $t$ with $s$ if and only if $x$ belongs to $t$ or $x$ belongs to $s$.","title":"Membership in a splice"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/inorder-join","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.inorder_join","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#inorder-join","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For red-black trees $t$ and $s$ with keys of any type, the inorder traversal of the splice of $t$ with $s$ is the inorder traversal of $t$ followed by the inorder traversal of $s$.","title":"Inorder traversal of a splice"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/ordered-join","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.ordered_join","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#ordered-join","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $<$ be a relation on the keys, and let $L$ and $R$ be red-black trees that are ordered for $<$ such that $a < b$ for every key $a$ of $L$ and every key $b$ of $R$. Then the join of $L$ and $R$, obtained by putting $R$ in place of the empty right subtree of the rightmost node of $L$ (or $R$ itself if $L$ is empty) and keeping all colors, is ordered.","title":"Joining separated ordered trees preserves order"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/mem-delete","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.mem_delete","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#mem-delete","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $x$ and $y$ be keys of a linearly ordered type and let $t$ be an ordered red-black tree. Then $y$ belongs to the tree obtained from $t$ by deleting $x$ by rebuilding if and only if $y$ belongs to $t$ and $y \\ne x$.","title":"Membership after deletion by rebuilding"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/not-mem-delete-self","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.not_mem_delete_self","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#not-mem-delete-self","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $x$ be a key of a linearly ordered type and let $t$ be an ordered red-black tree. Then $x$ does not belong to the tree obtained from $t$ by deleting $x$ by rebuilding.","title":"The deleted key is absent after deletion by rebuilding"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/inorder-delete","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.inorder_delete","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#inorder-delete","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $x$ be a key of a linearly ordered type and let $t$ be an ordered red-black tree. Then the inorder traversal of the tree obtained from $t$ by deleting $x$ by rebuilding is the inorder traversal of $t$ with the first occurrence of $x$ removed (it is unchanged if $x$ does not occur).","title":"Inorder traversal after deletion by rebuilding"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/valid-delete","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.valid_delete","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#valid-delete","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every linearly ordered key type, every key $x$ and every red-black tree $t$, the tree obtained by deleting $x$ from $t$ is a valid red-black tree: it is ordered, no red node has a red child, every root-to-leaf path contains the same number of black nodes, and its root is black (the empty tree counts as black). No hypothesis on $t$ is needed.","title":"Red-black deletion yields a valid red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/ordered-delete","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.ordered_delete","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#ordered-delete","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every linearly ordered key type, every key $x$ and every red-black tree $t$, the tree obtained by deleting $x$ from $t$ is ordered: at every node, all keys in the left subtree are strictly smaller than the node's key and all keys in the right subtree are strictly larger. No hypothesis on $t$ is needed.","title":"Red-black deletion yields an ordered tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/red-ok-delete","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.redOk_delete","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#red-ok-delete","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every linearly ordered key type, every key $x$ and every red-black tree $t$, the tree obtained by deleting $x$ from $t$ satisfies the red invariant: no red node has a red child. No hypothesis on $t$ is needed.","title":"Red-black deletion yields a tree with no red-red edge"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/balanced-delete","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.balanced_delete","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#balanced-delete","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every linearly ordered type of keys, every key $x$ and every red-black tree $t$, with no hypothesis on $t$, the tree obtained from $t$ by deleting $x$ by rebuilding is black-height balanced.","title":"Deletion by rebuilding is black-height balanced"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/color-of-delete","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.colorOf_delete","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#color-of-delete","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every linearly ordered type of keys, every key $x$ and every red-black tree $t$, with no hypothesis on $t$, the root of the tree obtained from $t$ by deleting $x$ by rebuilding is black, where the empty tree counts as having a black root.","title":"Deletion by rebuilding gives a black root"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/bcolor","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.BColor","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#bcolor","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"An extended color is one of three values: red, black or double black.","title":"Extended root colors"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/bbtree","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.BBTree","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#bbtree","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"A tree with a possibly double-black root over a type $\\alpha$ is one of the following: a genuine red-black tree over $\\alpha$; the double-black empty tree; or a double-black node consisting of a left subtree, a key in $\\alpha$ and a right subtree, where both subtrees are genuine red-black trees. Hence a double black can occur only at the root.","title":"Trees with a possibly double-black root"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/inorder-bb","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.inorderBB","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#inorder-bb","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The inorder traversal of a red-black tree whose root may be double black ignores the double-black mark: a genuine tree gives its inorder traversal, the double-black empty tree gives the empty list, and a double-black node with left child $l$, key $k$ and right child $r$ gives the inorder traversal of $l$, followed by $k$, followed by the inorder traversal of $r$.","title":"Inorder traversal of a possibly double-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/bh-bb","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.bhBB","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#bh-bb","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The black height of a genuine red-black tree is its usual black height. The double-black empty tree has black height $2$, and a double-black node with left subtree $L$ has black height $h(L) + 2$, where $h(L)$ is the usual black height of $L$. Thus a double-black root counts as two black nodes, where the usual black height counts the empty tree as one.","title":"Black height of a possibly double-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/balanced-bb","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.BalancedBB","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#balanced-bb","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"A tree whose root may be double black is balanced when it is a genuine red-black tree that is balanced, when it is the double-black empty tree, or when it is a double-black node whose left and right subtrees are both balanced and have the same black height.","title":"Black-height balance of a possibly double-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/red-ok-bb","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.RedOkBB","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#red-ok-bb","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"A tree whose root may be double black satisfies the red invariant in the following cases: a genuine red-black tree satisfies it when it satisfies the usual red invariant (no red node has a red child); the double-black empty tree always satisfies it; and a double-black node with left subtree $l$ and right subtree $r$ satisfies it when both $l$ and $r$ satisfy the usual red invariant.","title":"Red invariant for trees with a possibly double-black root"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/color-bb","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.colorBB","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#color-bb","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"The extended color of a tree whose root may be double black is the color of its root, red or black, when it is a genuine red-black tree, where the empty tree counts as black; it is double black for the double-black empty tree and for every double-black node.","title":"Extended root color"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/incr-black","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.incrBlack","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#incr-black","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Add one black to the root of a red-black tree, producing a tree whose root may be double black: the empty tree becomes the double-black empty tree, a red root becomes black, and a black root becomes double black.","title":"Adding a black to the root"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/root-black","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.rootBlack","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#root-black","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a red-black tree $t$, this is the tree obtained from $t$ by coloring its root black: the empty tree stays empty, and a node with left subtree $l$, key $k$ and right subtree $r$, of either color, becomes the black node with left subtree $l$, key $k$ and right subtree $r$.","title":"Recoloring the root black"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/blacken","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.blacken","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#blacken","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Blackening sends a tree whose root may be double black to a genuine red-black tree by changing only its root: a genuine tree has its root recolored black (the empty tree stays empty), the double-black empty tree becomes the empty tree, and a double-black node becomes the black node with the same subtrees and key.","title":"Absorbing a double-black root"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/min-key","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.minKey","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#min-key","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a red-black tree $t$ over a type $\\alpha$, the leftmost key of $t$ is either a key in $\\alpha$ or undefined. It is undefined when $t$ is empty. When $t$ is a node, of either color, with left subtree $l$ and key $k$, it is the leftmost key of $l$ if that is defined, and $k$ otherwise. Hence the leftmost key is defined exactly when $t$ is nonempty.","title":"Leftmost key of a red-black tree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/fix-left","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.fixLeft","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#fix-left","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Write $\\mathrm{R}$ and $\\mathrm{B}$ for red and black, and $N(c, T, y, T')$ for the node of color $c$ with left subtree $T$, key $y$ and right subtree $T'$. Given a color $c$, a tree $D$, a key $k$ and a tree $S$, the left repair returns a tree whose root may be double black, by the first case that applies. If $S = N(\\mathrm{B}, T_1, y, N(\\mathrm{R}, T_2, z, T_3))$ or, failing that,\u2026","title":"Left repair after double-black deletion"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/fix-right","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.fixRight","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#fix-right","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Write $\\mathrm{R}$ and $\\mathrm{B}$ for red and black, and $N(c, T, y, T')$ for the node of color $c$ with left subtree $T$, key $y$ and right subtree $T'$. Given a color $c$, a tree $S$, a key $k$ and a tree $D$, the right repair returns a tree whose root may be double black, by the first case that applies. If $S = N(\\mathrm{B}, N(\\mathrm{R}, T_1, y, T_2), z, T_3)$ or, failing that,\u2026","title":"Right repair after double-black deletion"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/bal-left","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.balLeft","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#bal-left","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $c$ be a color, $L$ a tree whose root may be double black, $k$ a key and $R$ a red-black tree. If $L$ is a genuine red-black tree, the result is the genuine node with color $c$, left subtree $L$, key $k$ and right subtree $R$. Otherwise the double black of $L$ is absorbed first: a double-black empty tree becomes the empty tree, and a double-black node becomes the black node with the same children and key. The\u2026","title":"Rebalancing after deletion from the left subtree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/bal-right","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.balRight","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#bal-right","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $c$ be a color, $L$ a red-black tree, $k$ a key and $R$ a tree whose root may be double black. If $R$ is a genuine red-black tree, the result is the genuine node with color $c$, left subtree $L$, key $k$ and right subtree $R$. Otherwise the double black of $R$ is absorbed first: a double-black empty tree becomes the empty tree, and a double-black node becomes the black node with the same children and key. The\u2026","title":"Rebalancing after deletion from the right subtree"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/del-2","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.del","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#del-2","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a key $x$ of a linearly ordered type, deleting $x$ from a red-black tree returns a tree whose root may be double black, by recursion on the tree. The empty tree gives the empty tree. At a node of color $c$ with left subtree $l$, key $k$ and right subtree $r$: if $x < k$, the result is the left reassembly of color $c$, the deletion of $x$ from $l$, key $k$ and $r$; if $k < x$, it is the right reassembly of color\u2026","title":"Double-black deletion"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a key $x$ of a linearly ordered type and a red-black tree $t$, deleting $x$ from $t$ runs the double-black deletion of $x$ from $t$, which returns a tree whose root may be double black, and then absorbs a remaining double black at the root and recolors the root black.","title":"Red-black tree deletion by the double-black method"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/inorder-delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.inorder_deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#inorder-delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For a key $x$ of a linearly ordered type and an ordered red-black tree $t$, the inorder traversal of the tree obtained by deleting $x$ from $t$ by the double-black method is the inorder traversal of $t$ with the first occurrence of $x$ removed (unchanged if $x$ does not occur).","title":"Double-black deletion erases the key from the inorder traversal"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/mem-delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.mem_deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#mem-delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For keys $x$ and $y$ of a linearly ordered type and an ordered red-black tree $t$, $y$ is a key of the tree obtained by deleting $x$ from $t$ by the double-black method if and only if $y$ is a key of $t$ and $y \\ne x$.","title":"Keys after double-black deletion"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/not-mem-delete-bal-self","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.not_mem_deleteBal_self","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#not-mem-delete-bal-self","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $t$ a red-black tree with keys in $\\alpha$ that is ordered, that is, at every node all keys of the left subtree are strictly smaller and all keys of the right subtree are strictly larger than the node's key. Then for every $x \\in \\alpha$, the key $x$ does not occur in the tree obtained by deleting $x$ from $t$ by the double-black method.","title":"The deleted key is absent after double-black deletion"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/ordered-delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.ordered_deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#ordered-delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $t$ a red-black tree with keys in $\\alpha$ that is ordered, that is, at every node all keys of the left subtree are strictly smaller and all keys of the right subtree are strictly larger than the node's key. Then for every $x \\in \\alpha$, the tree obtained by deleting $x$ from $t$ by the double-black method is again ordered.","title":"Double-black deletion preserves the search-tree order"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/valid-delete-bal-of-invariants","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.valid_deleteBal_of_invariants","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#valid-delete-bal-of-invariants","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $t$ a red-black tree with keys in $\\alpha$ that is ordered, satisfies the red invariant (no red node has a red child), and is black-height balanced (all root-to-leaf paths contain the same number of black nodes). Then for every $x \\in \\alpha$, the tree obtained by deleting $x$ from $t$ by the double-black method is a valid red-black tree.","title":"Double-black deletion yields a valid tree from the structural invariants"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/valid-delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.valid_deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#valid-delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $t$ a valid red-black tree with keys in $\\alpha$: it is ordered, satisfies the red invariant, is black-height balanced, and has a black root. Then for every $x \\in \\alpha$, the tree obtained by deleting $x$ from $t$ by the double-black method is again a valid red-black tree.","title":"Double-black deletion preserves validity"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/red-ok-delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.redOk_deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#red-ok-delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $t$ a valid red-black tree with keys in $\\alpha$: it is ordered, satisfies the red invariant, is black-height balanced, and has a black root. Then for every $x \\in \\alpha$, the tree obtained by deleting $x$ from $t$ by the double-black method satisfies the red invariant: no red node has a red child.","title":"Double-black deletion from a valid tree keeps the red invariant"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/balanced-delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.balanced_deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#balanced-delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$, and $T$ a valid red-black tree over $\\alpha$. Then the tree obtained from $T$ by double-black deletion of $x$ is balanced.","title":"Double-black deletion keeps a valid tree balanced"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/color-of-delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.colorOf_deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#color-of-delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"For every key $x$ of a linearly ordered type and every red-black tree $t$, valid or not, the root of the tree obtained by deleting $x$ from $t$ by the double-black method is black, where the empty tree counts as black.","title":"Double-black deletion leaves a black root"},{"id":"ComputerScience/data-structures/search-trees/red-black-tree/black-height-delete-bal","kind":"node","lean":"Cslib.Algorithms.Lean.RedBlackTree.DeleteBalanced.blackHeight_deleteBal","parent":36708,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/red-black-tree/index.html#black-height-delete-bal","scope":"ComputerScience/data-structures/search-trees/red-black-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $x \\in \\alpha$, $T$ a valid red-black tree over $\\alpha$, and $T'$ the tree obtained from $T$ by double-black deletion of $x$. Then the black height of $T'$ either equals the black height of $T$ or is one less than it.","title":"Black height after double-black deletion"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/skip-list.json","id":"ComputerScience/data-structures/search-trees/skip-list","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","title":"Skip list"},{"id":"ComputerScience/data-structures/search-trees/skip-list/entry","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.Entry","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#entry","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"A skip-list node consists of a key, a value and a natural number, the height of its tower. The node itself does not require the height to be positive; positivity is part of the validity invariant of skip lists.","title":"Skip-list node"},{"id":"ComputerScience/data-structures/search-trees/skip-list/skip-list","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#skip-list","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"A deterministic skip list with keys in $\\alpha$ and values in $\\beta$ is represented by the list of its bottom-level nodes, each node an entry carrying a key in $\\alpha$, a value in $\\beta$ and a tower height, a natural number. No ordering or height condition is imposed by this structure.","title":"Skip list"},{"id":"ComputerScience/data-structures/search-trees/skip-list/empty","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.empty","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#empty","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"The empty skip list is the skip list with no nodes.","title":"Empty skip list"},{"id":"ComputerScience/data-structures/search-trees/skip-list/valid","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.Valid","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#valid","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. A skip list with keys in $\\alpha$ and values in $\\beta$ is valid when the keys of its bottom-level nodes are strictly increasing along the list and every node has a tower of positive height.","title":"Valid skip list"},{"id":"ComputerScience/data-structures/search-trees/skip-list/level","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.level","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#level","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For a skip list and a level index $i$ (levels are numbered from $0$ at the bottom), level $i$ is the list of key-value pairs $(k, v)$ of the stored nodes whose tower height $h$ satisfies $i < h$, in the order in which these nodes appear in the bottom-level list of the skip list.","title":"Level of a skip list"},{"id":"ComputerScience/data-structures/search-trees/skip-list/abstraction","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.abstraction","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#abstraction","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"The bottom-level abstraction of a skip list is the association list obtained from its bottom-level nodes, in order, by keeping the key-value pair $(k, v)$ of each node and forgetting its tower height.","title":"Bottom-level abstraction of a skip list"},{"id":"ComputerScience/data-structures/search-trees/skip-list/height-of","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.heightOf","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#height-of","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For a key $k$ and a skip list $S$, the stored tower height of $k$ in $S$ is the tower height of the first bottom-level node of $S$ whose key equals $k$ under the given Boolean equality test on keys; it is undefined when no node has key $k$.","title":"Stored tower height of a key"},{"id":"ComputerScience/data-structures/search-trees/skip-list/advance-at-level","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.advanceAtLevel","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#advance-at-level","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $k$ be a search key, $i$ a zero-based level and $L$ a list of skip-list nodes. Let $P$ be the longest prefix of $L$ whose nodes all have keys less than $k$. If some node of $P$ has tower height greater than $i$ (so it is visible at level $i$), advancing at level $i$ returns the part of $L$ after the last such node; otherwise it returns $L$ unchanged.","title":"One level of skip-list search"},{"id":"ComputerScience/data-structures/search-trees/skip-list/search-window","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.searchWindow","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#search-window","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $k$ be a key and let $H$ be the greatest tower height in a skip list ($H = 0$ if the list is empty or every tower has height $0$). Start with the full list of bottom-level nodes, and for each level $i$ from $H - 1$ down to $0$ in turn (no levels when $H = 0$) replace the current list by the result of advancing at level $i$: within the longest initial segment of the current list whose keys are all less than $k$,\u2026","title":"Skip-list search window"},{"id":"ComputerScience/data-structures/search-trees/skip-list/lookup","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.lookup","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#lookup","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"The lookup of a key $k$ in a skip list computes the search window for $k$, the suffix of the bottom-level nodes reached by descending from the highest level to level $0$, and returns the value of the first node in the window whose key is $k$, or no value if the window has no such node.","title":"Skip-list lookup"},{"id":"ComputerScience/data-structures/search-trees/skip-list/insert","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.insert","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#insert","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"To insert key $k$ with value $v$ and requested height $h$ into a skip list, scan its bottom-level nodes in order, keeping every node whose key is less than $k$. At the first node whose key is not less than $k$: if its key equals $k$, its value is replaced by $v$ and its tower height is kept; if its key is greater than $k$, a new node with key $k$, value $v$ and tower height $\\max(1, h)$ is placed before it. If\u2026","title":"Skip-list insertion"},{"id":"ComputerScience/data-structures/search-trees/skip-list/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.valid_empty","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#valid-empty","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For every linearly ordered type $\\alpha$ and every type $\\beta$, the empty skip list with keys in $\\alpha$ and values in $\\beta$ is valid.","title":"The empty skip list is valid"},{"id":"ComputerScience/data-structures/search-trees/skip-list/level-pairwise","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.level_pairwise","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#level-pairwise","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If a skip list is valid, then for every level index $i$ (levels are numbered from $0$ at the bottom), the keys in the list of key-value pairs visible at level $i$ are strictly increasing: every pair has a strictly smaller key than every pair after it.","title":"Skip-list levels are sorted by key"},{"id":"ComputerScience/data-structures/search-trees/skip-list/level-succ-sublist-level","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.level_succ_sublist_level","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#level-succ-sublist-level","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For every skip list and every level index $i$ (levels are numbered from $0$ at the bottom), the list of key-value pairs visible at level $i + 1$ is a subsequence, not necessarily contiguous, of the list of key-value pairs visible at level $i$.","title":"Each skip-list level is a subsequence of the level below"},{"id":"ComputerScience/data-structures/search-trees/skip-list/abstraction-keys-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.abstraction_keys_nodup","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#abstraction-keys-nodup","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If a skip list is valid, then the keys of the pairs in its bottom-level abstraction, read in order, form a list without duplicates.","title":"Keys of a valid skip list are distinct"},{"id":"ComputerScience/data-structures/search-trees/skip-list/level-keys-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.level_keys_nodup","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#level-keys-nodup","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If a skip list is valid, then for every level index $i$ (levels are numbered from $0$ at the bottom), no key occurs twice in the list of key-value pairs visible at level $i$.","title":"Keys on each skip-list level are distinct"},{"id":"ComputerScience/data-structures/search-trees/skip-list/advance-at-level-suffix","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.advanceAtLevel_suffix","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#advance-at-level-suffix","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For every search key, every level $i$ and every list $L$ of skip-list nodes, advancing the search at level $i$ on $L$ returns a suffix of $L$.","title":"One search step returns a suffix"},{"id":"ComputerScience/data-structures/search-trees/skip-list/search-window-suffix","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.searchWindow_suffix","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#search-window-suffix","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For every skip list and every key $k$, the search window for $k$ is a suffix of the list of the skip list's bottom-level nodes.","title":"The search window is a suffix of the bottom level"},{"id":"ComputerScience/data-structures/search-trees/skip-list/lookup-eq-abstraction","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.lookup_eq_abstraction","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#lookup-eq-abstraction","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For every skip list and every key $k$, the top-down skip-list lookup of $k$ returns the same result as looking up $k$ in the bottom-level abstraction, the list of key-value pairs of all stored nodes in order: the value of the first pair with key $k$, or no value if there is none.","title":"Skip-list search agrees with bottom-level lookup"},{"id":"ComputerScience/data-structures/search-trees/skip-list/valid-insert","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.valid_insert","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#valid-insert","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and $\\beta$ a type. If a skip list with keys in $\\alpha$ and values in $\\beta$ is valid, then for every key $k \\in \\alpha$, value $v \\in \\beta$ and requested height $h \\in \\mathbb{N}$ (including $h = 0$), inserting $k$ with value $v$ and requested height $h$ yields a valid skip list. This covers both a new key and a key already present, whose value is replaced.","title":"Insertion preserves skip-list validity"},{"id":"ComputerScience/data-structures/search-trees/skip-list/lookup-insert","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.lookup_insert","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#lookup-insert","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For every skip list, key $k$, value $v$ and requested height $h$, looking up $k$ after inserting $k$ with value $v$ and requested height $h$ returns $v$.","title":"Lookup after insertion returns the inserted value"},{"id":"ComputerScience/data-structures/search-trees/skip-list/lookup-insert-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.lookup_insert_of_ne","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#lookup-insert-of-ne","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every skip list $S$ with keys in $\\alpha$ and values in $\\beta$, all keys $k$ and $k'$ in $\\alpha$ with $k' \\ne k$, every value $v \\in \\beta$ and every requested height $h \\in \\mathbb{N}$, looking up $k'$ in the skip list obtained by inserting $k$ with value $v$ and requested height $h$ into $S$ gives the same result as looking up $k'$ in $S$.","title":"Skip-list insertion does not change lookup of other keys"},{"id":"ComputerScience/data-structures/search-trees/skip-list/height-of-insert-of-present","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.heightOf_insert_of_present","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#height-of-insert-of-present","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $S$ be a valid skip list in which the key $k$ has stored tower height $h_0$. Then after inserting $k$ into $S$ with any value and any requested height, the stored tower height of $k$ is still $h_0$.","title":"Reinsertion preserves tower height"},{"id":"ComputerScience/data-structures/search-trees/skip-list/height-of-insert-of-absent","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.heightOf_insert_of_absent","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#height-of-insert-of-absent","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If a key $k$ does not occur in a skip list $S$, then after inserting $k$ into $S$ with any value and requested height $h$, the stored tower height of $k$ is $\\max(1, h)$.","title":"Height of a newly inserted key"},{"id":"ComputerScience/data-structures/search-trees/skip-list/erase","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.erase","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#erase","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Given a key $k$ from a linearly ordered type and a skip list $S$, the erasure of $k$ from $S$ is the skip list obtained by scanning the entries of $S$ in order: an entry whose key is less than $k$ is kept and the scan continues; at the first entry whose key is greater than $k$ the scan stops, and that entry and all later ones are kept; at the first entry whose key equals $k$ that entry is removed, the scan stops,\u2026","title":"Skip-list erasure"},{"id":"ComputerScience/data-structures/search-trees/skip-list/erase-empty","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.erase_empty","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#erase-empty","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"For every key $k$, erasing $k$ from the empty skip list gives the empty skip list.","title":"Erasing from the empty skip list"},{"id":"ComputerScience/data-structures/search-trees/skip-list/erase-cons-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.erase_cons_of_lt","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#erase-cons-of-lt","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $e$ be a skip-list entry and $L$ a list of entries. If a key $k$ is less than the key of $e$, then erasing $k$ from the skip list with entry list $e$ followed by $L$ leaves the entry list $e$ followed by $L$ unchanged.","title":"Erasing a key below the first entry"},{"id":"ComputerScience/data-structures/search-trees/skip-list/erase-cons-of-gt","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.erase_cons_of_gt","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#erase-cons-of-gt","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $e$ be a skip-list entry and $L$ a list of entries. If the key of $e$ is less than a key $k$, then erasing $k$ from the skip list with entry list $e$ followed by $L$ gives the entry list consisting of $e$ followed by the entry list obtained by erasing $k$ from the skip list with entry list $L$.","title":"Erasing past a smaller first entry"},{"id":"ComputerScience/data-structures/search-trees/skip-list/erase-cons-of-eq","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.erase_cons_of_eq","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#erase-cons-of-eq","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $e$ be a skip-list entry and $L$ a list of entries, and consider the skip list whose entry list is $e$ followed by $L$. If a key $k$ equals the key of $e$, then erasing $k$ from this skip list leaves the entry list $L$.","title":"Erasing the key of the first entry"},{"id":"ComputerScience/data-structures/search-trees/skip-list/erase-entries-filter","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.erase_entries_filter","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#erase-entries-filter","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If $S$ is a valid skip list and $k$ is a key, then the entry list of the skip list obtained by erasing $k$ from $S$ is the entry list of $S$ with every entry whose key equals $k$ removed, the remaining entries kept in their order.","title":"Erasure on a valid skip list filters out the key"},{"id":"ComputerScience/data-structures/search-trees/skip-list/valid-erase","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.valid_erase","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#valid-erase","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $L$ be a valid skip list, that is, its keys are strictly increasing and every stored tower has positive height. For every key $k$, the skip list obtained from $L$ by erasing $k$ is again valid.","title":"Erasure preserves skip-list validity"},{"id":"ComputerScience/data-structures/search-trees/skip-list/lookup-erase","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.lookup_erase","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#lookup-erase","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"Let $L$ be a valid skip list, that is, its keys are strictly increasing and every stored tower has positive height. For every key $k$, looking up $k$ in the skip list obtained from $L$ by erasing $k$ returns no value.","title":"Erased keys are absent from a skip list"},{"id":"ComputerScience/data-structures/search-trees/skip-list/lookup-erase-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.lookup_erase_of_ne","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#lookup-erase-of-ne","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If $S$ is a valid skip list and $k$, $k'$ are keys with $k' \\ne k$, then looking up $k'$ in the skip list obtained by erasing $k$ from $S$ gives the same result as looking up $k'$ in $S$: the same value, or no value in both.","title":"Erasure preserves lookups of other keys"},{"id":"ComputerScience/data-structures/search-trees/skip-list/height-of-erase","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.heightOf_erase","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#height-of-erase","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If $S$ is a valid skip list and $k$ is a key, then after erasing $k$ from $S$ the key $k$ has no stored tower height.","title":"The erased key has no stored height"},{"id":"ComputerScience/data-structures/search-trees/skip-list/height-of-erase-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.heightOf_erase_of_ne","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#height-of-erase-of-ne","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If $S$ is a valid skip list and $k$, $k'$ are keys with $k' \\ne k$, then the stored tower height of $k'$ in the skip list obtained by erasing $k$ from $S$ is the same as in $S$; in particular $k'$ has no stored height in one exactly when it has none in the other.","title":"Erasure preserves the heights of other keys"},{"id":"ComputerScience/data-structures/search-trees/skip-list/level-erase","kind":"node","lean":"Cslib.Algorithms.Lean.SkipList.level_erase","parent":36791,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/skip-list/index.html#level-erase","scope":"ComputerScience/data-structures/search-trees/skip-list","status":"fully_proved","summary":"If $S$ is a valid skip list, $k$ is a key and $i$ is a natural number, then the level-$i$ list of key-value pairs (levels numbered from $0$) of the skip list obtained by erasing $k$ from $S$ is the level-$i$ list of $S$ with every pair whose key equals $k$ removed, the remaining pairs kept in their order.","title":"Erasure removes the key from every level"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/splay-tree.json","id":"ComputerScience/data-structures/search-trees/splay-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","title":"Splay tree"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/root","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.root?","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#root","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"The optional root key of a binary tree is undefined (none) for the empty tree, and is the key $k$ for a tree whose root node holds $k$.","title":"Root key of a splay tree"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/splay","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.splay","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#splay","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and write $(k; L, R)$ for the binary tree with root key $k$, left subtree $L$ and right subtree $R$. For $x \\in \\alpha$, the splay $S_x(t)$ of a binary tree $t$ with keys in $\\alpha$ is defined by recursion on $t$. The empty tree splays to itself, and $t = (k; L, R)$ is left unchanged if $x = k$, if $x < k$ and $L$ is empty, or if $x > k$ and $R$ is empty. If $x < k$ and\u2026","title":"Splay operation"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/splay-nil","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.splay_nil","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#splay-nil","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For every key $x$ of a linearly ordered type, splaying $x$ in the empty binary tree gives the empty tree.","title":"Splaying the empty tree"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/inorder-splay","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.inorder_splay","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#inorder-splay","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For every key $x$ and every binary tree $t$ over a linearly ordered type, the inorder traversal of the tree obtained by splaying $x$ in $t$ equals the inorder traversal of $t$. No search-tree ordering of $t$ is assumed.","title":"Splaying preserves the inorder traversal"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/mem-splay","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.mem_splay","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#mem-splay","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For keys $x$ and $y$ and a binary tree $t$ over a linearly ordered type, $y$ is a key of the tree obtained by splaying $x$ in $t$ if and only if $y$ is a key of $t$.","title":"Splaying preserves membership"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/num-nodes-splay","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.numNodes_splay","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#num-nodes-splay","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For every binary tree $t$ with keys in a linearly ordered type and every key $x$, the tree obtained by splaying $x$ in $t$ has the same number of nodes as $t$.","title":"Splaying preserves the number of nodes"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/ordered-splay","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.ordered_splay","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#ordered-splay","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys in a linearly ordered type and let $x$ be a key. If $t$ is a binary search tree (at every node, every key of the left subtree is strictly smaller and every key of the right subtree strictly larger than the key of the node), then the tree obtained by splaying $x$ in $t$ is again a binary search tree.","title":"Splaying preserves the search-tree property"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/root-splay-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.root_splay_of_mem","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#root-splay-of-mem","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $t$ be a binary search tree with keys in a linearly ordered type (at every node, every key of the left subtree is strictly smaller and every key of the right subtree strictly larger than the key of the node), and let $x$ be a key that occurs in $t$. Then there are binary trees $l$ and $r$ such that splaying $x$ in $t$ gives the tree whose root holds $x$, with left subtree $l$ and right subtree $r$.","title":"Splaying a present key brings it to the root"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/root-splay-of-mem-2","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.root?_splay_of_mem","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#root-splay-of-mem-2","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $t$ be a binary search tree with keys in a linearly ordered type (at every node, every key of the left subtree is strictly smaller and every key of the right subtree strictly larger than the key of the node), and let $x$ be a key that occurs in $t$. Then the tree obtained by splaying $x$ in $t$ is nonempty and its root key is $x$.","title":"Root key after splaying a present key"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/split","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.split","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#split","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $x \\in \\alpha$, and write $(k; L, R)$ for the binary tree with root key $k$, left subtree $L$ and right subtree $R$. The split of a binary tree with keys in $\\alpha$ at $x$ is the pair of trees defined by recursion: the empty tree splits into two empty trees; if $x < k$ and $L$ splits into $(L_1, L_2)$, then $(k; L, R)$ splits into $(L_1, (k; L_2, R))$; if $k < x$ and\u2026","title":"Search-path split at a key"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/mem-of-mem-split","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.mem_of_mem_split","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#mem-of-mem-split","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For keys $x$ and $y$ and a binary tree $t$ over a linearly ordered type, if $y$ is a key of the left part or of the right part of the split of $t$ at $x$, then $y$ is a key of $t$.","title":"Splay-tree split parts contain only original keys"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/mem-split-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.mem_split_of_mem","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#mem-split-of-mem","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys in a linearly ordered type, and let $x$ and $y$ be keys with $y \\ne x$. If $y$ occurs as a key of $t$, then $y$ occurs as a key of the left part or of the right part of the split of $t$ at $x$. The tree $t$ need not be a binary search tree.","title":"Keys other than the split key survive a split"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/all-split","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.all_split","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#all-split","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $t$ be an ordered binary tree over a linearly ordered type and let $x$ be a key. When $t$ is split at $x$, every key of the left part is strictly less than $x$ and every key of the right part is strictly greater than $x$.","title":"Splitting an ordered tree separates the keys around the split key"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/ordered-split","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.ordered_split","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#ordered-split","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys in a linearly ordered type and let $x$ be a key. If $t$ is a binary search tree (at every node, every key of the left subtree is strictly smaller and every key of the right subtree strictly larger than the key of the node), then the left part and the right part of the split of $t$ at $x$ are both binary search trees.","title":"Both parts of a split search tree are search trees"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.insert","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#insert","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For a key $x$ and a binary tree $t$ over a linearly ordered type, inserting $x$ into $t$ splays $x$ in $t$, splits the resulting tree at $x$ into a left part and a right part, and returns the tree whose root key is $x$, whose left subtree is the left part and whose right subtree is the right part.","title":"Splay-tree insertion"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/access","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.access","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#access","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For a key $x$ and a binary tree $t$ over a linearly ordered type, the access of $x$ in $t$ is the pair $(b, t')$, where $t'$ is the tree obtained by splaying $x$ in $t$ and $b$ is true exactly when $t'$ is nonempty and its root key is $x$.","title":"Splay-tree access"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/contains","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.contains","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#contains","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For a key $x$ and a binary tree $t$ over a linearly ordered type, the membership test of $x$ in $t$ is the Boolean component of the access of $x$ in $t$: it is true exactly when the tree obtained by splaying $x$ in $t$ is nonempty and its root key is $x$.","title":"Splay-tree membership test"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/access-snd","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.access_snd","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#access-snd","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For a key $x$ and a binary tree $t$ over a linearly ordered type, the tree component of the access of $x$ in $t$ is the tree obtained by splaying $x$ in $t$.","title":"The tree returned by a splay-tree access"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/mem-insert","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.mem_insert","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#mem-insert","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For keys $x$ and $y$ and a binary tree $t$ over a linearly ordered type, $y$ is a key of the tree obtained by splay-tree insertion of $x$ into $t$ if and only if $y = x$ or $y$ is a key of $t$.","title":"Keys after splay-tree insertion"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.ordered_insert","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#ordered-insert","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"Let $t$ be a binary tree with keys in a linearly ordered type and let $x$ be a key. If $t$ is a binary search tree (at every node, every key of the left subtree is strictly smaller and every key of the right subtree strictly larger than the key of the node), then the splay-tree insertion of $x$ into $t$ is again a binary search tree.","title":"Splay-tree insertion preserves the search-tree property"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/root-insert","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.root?_insert","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#root-insert","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For every binary tree $t$ with keys in a linearly ordered type and every key $x$, the splay-tree insertion of $x$ into $t$ is a nonempty tree whose root key is $x$.","title":"Splay-tree insertion puts the inserted key at the root"},{"id":"ComputerScience/data-structures/search-trees/splay-tree/contains-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.SplayTree.contains_eq_true_iff","parent":36828,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/splay-tree/index.html#contains-eq-true-iff","scope":"ComputerScience/data-structures/search-trees/splay-tree","status":"fully_proved","summary":"For an ordered binary tree $t$ over a linearly ordered type and a key $x$, the splay-tree membership test of $x$ in $t$ returns true if and only if $x$ is a key of $t$.","title":"Correctness of the splay-tree membership test"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/treap.json","id":"ComputerScience/data-structures/search-trees/treap","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","title":"Treap"},{"id":"ComputerScience/data-structures/search-trees/treap/tree","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#tree","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For a type $\\kappa$ of keys and a type $\\nu$ of values, a treap tree is either empty or a node made of a left subtree, a key in $\\kappa$, a value in $\\nu$, a natural-number priority and a right subtree. Trees are finite, and no ordering of keys or priorities is imposed; the treap invariants are stated separately.","title":"Treap trees"},{"id":"ComputerScience/data-structures/search-trees/treap/all-nodes","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.AllNodes","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#all-nodes","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let $p$ be a predicate on a key, a value and a natural-number priority. A tree satisfies $p$ at every node when $p(k, v, n)$ holds for the key $k$, value $v$ and priority $n$ of each of its nodes. By recursion, this holds for the empty tree, and holds for a node exactly when it holds for the left subtree, $p$ holds at the node's key, value and priority, and it holds for the right subtree.","title":"Treap predicate holding at every node"},{"id":"ComputerScience/data-structures/search-trees/treap/mem-node","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.MemNode","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#mem-node","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"A key $k$, value $v$ and priority $p$ form an entry of a treap tree when some node stores exactly $k$, $v$ and $p$: the empty tree has no entries, and $(k, v, p)$ is an entry of a node with left subtree $L$, root key $r$, root value $w$, root priority $q$ and right subtree $R$ exactly when it is an entry of $L$, or $k = r$, $v = w$ and $p = q$, or it is an entry of $R$.","title":"Entry membership in a treap tree"},{"id":"ComputerScience/data-structures/search-trees/treap/mem-key","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.MemKey","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#mem-key","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"A key $k$ is a key of a treap tree when it is stored at some node: no key belongs to the empty tree, and $k$ belongs to a node with left subtree $L$, root key $r$ and right subtree $R$ exactly when $k$ belongs to $L$, or $k = r$, or $k$ belongs to $R$.","title":"Key membership in a treap tree"},{"id":"ComputerScience/data-structures/search-trees/treap/contents-aux","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.contentsAux","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#contents-aux","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For a tree $t$ and a list $s$ of key-value pairs, the traversal of $t$ onto $s$ is defined by recursion: for the empty tree it is $s$, and for a node with left subtree $l$, key $k$, value $v$ and right subtree $r$ it is the traversal of $l$ onto the list obtained by putting $(k, v)$ in front of the traversal of $r$ onto $s$. The result is the key-value pairs of $t$ in inorder, priorities dropped, followed by $s$.","title":"Treap inorder traversal onto a suffix"},{"id":"ComputerScience/data-structures/search-trees/treap/contents","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.contents","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#contents","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"The contents of a tree $t$ is the list of its key-value pairs in inorder (left subtree, then node, then right subtree), with priorities dropped. It is defined as the inorder traversal of $t$ onto the empty list, which takes linear time.","title":"Treap inorder contents"},{"id":"ComputerScience/data-structures/search-trees/treap/contents-node","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.contents_node","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#contents-node","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For trees $l$ and $r$, a key $k$, a value $v$ and a natural-number priority $n$, the inorder list of key-value pairs of the node with left subtree $l$, key $k$, value $v$, priority $n$ and right subtree $r$ is the inorder list of $l$, followed by $(k, v)$, followed by the inorder list of $r$.","title":"Treap inorder contents of a node"},{"id":"ComputerScience/data-structures/search-trees/treap/size","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.size","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#size","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"The size of a tree is its number of nodes, computed recursively: the empty tree has size $0$, and a node with left subtree $L$ and right subtree $R$ has size equal to the size of $L$ plus one plus the size of $R$.","title":"Size of a treap tree"},{"id":"ComputerScience/data-structures/search-trees/treap/ordered","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.Ordered","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#ordered","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For a key type equipped with a relation $<$, a treap tree is ordered when it is empty, or when it is a node with key $k$ whose left and right subtrees are both ordered, every key $x$ in the left subtree satisfies $x < k$, and every key $y$ in the right subtree satisfies $k < y$.","title":"Ordered treap tree"},{"id":"ComputerScience/data-structures/search-trees/treap/heap-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.HeapOrdered","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#heap-ordered","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"A tree is heap-ordered when the priority of every node is at most the priority of every node below it. Recursively, the empty tree is heap-ordered, and a node with priority $p$ is heap-ordered when its left and right subtrees are heap-ordered and every node in them has priority at least $p$.","title":"Heap order of treap priorities"},{"id":"ComputerScience/data-structures/search-trees/treap/valid","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.Valid","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#valid","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"A binary tree whose nodes each carry a key, a value and a natural-number priority, with keys compared by a relation $<$ (any `LT` instance; no order axioms are assumed), is a valid treap when it satisfies both of the following at every node: every key in the left subtree is less than the node's key and the node's key is less than every key in the right subtree (search-tree order with respect to $<$), and the node's\u2026","title":"Valid treap"},{"id":"ComputerScience/data-structures/search-trees/treap/lookup","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.lookup","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#lookup","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Looking up a key $k$ in a tree with keys in a linearly ordered type follows the binary search tree comparisons. The empty tree gives no value. At a node with key $r$ and value $v$, the lookup continues in the left subtree if $k < r$, continues in the right subtree if $r < k$, and otherwise returns $v$.","title":"Treap lookup"},{"id":"ComputerScience/data-structures/search-trees/treap/lookup-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.lookupWithCost","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#lookup-with-cost","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"The costed lookup of a key $k$ in a tree with keys in a linearly ordered type returns a possibly absent value together with a natural number counting the visited nodes. On the empty tree it returns no value and cost $0$. At a node with key $r$ and value $v$, if $k < r$ it returns the result of the costed lookup of $k$ in the left subtree with cost increased by $1$; if $r < k$ it does the same with the right\u2026","title":"Lookup with cost"},{"id":"ComputerScience/data-structures/search-trees/treap/root-priority","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.rootPriority","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#root-priority","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"The root priority of a treap tree is an optional natural number: it is absent for the empty tree, and for a node it is the priority stored at that node.","title":"Root priority of a treap tree"},{"id":"ComputerScience/data-structures/search-trees/treap/merge","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.merge","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#merge","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"The merge of two treap trees is defined recursively. Merging the empty tree with a tree $T$, or $T$ with the empty tree, gives $T$. If both trees are nodes, with root priorities $p$ for the first and $q$ for the second, then when $p \\le q$ the result keeps the root of the first tree and its left subtree, and its right subtree is the merge of the first tree's right subtree with the second tree; otherwise the result\u2026","title":"Treap merge"},{"id":"ComputerScience/data-structures/search-trees/treap/split-result","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.SplitResult","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#split-result","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For key type $\\kappa$ and value type $\\nu$, a split result consists of a lower tree, an optional entry given by a value in $\\nu$ together with a natural-number priority, and an upper tree, where both trees are treap trees with keys in $\\kappa$ and values in $\\nu$.","title":"Result of a treap split"},{"id":"ComputerScience/data-structures/search-trees/treap/split","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.split","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#split","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linear order, and write $(L, k, v, q, R)$ for the node with left subtree $L$, key $k$, value $v$, priority $q$ and right subtree $R$. Splitting a tree at a pivot $p$ returns a split result, defined by recursion on the tree. The empty tree splits into two empty trees and no entry. For a node $(L, k, v, q, R)$: if $k < p$, split $R$ at $p$ into a lower tree $R_1$, an optional entry $e$ and an\u2026","title":"Treap split"},{"id":"ComputerScience/data-structures/search-trees/treap/insert","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.insert","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#insert","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For a key $k$ of a linearly ordered key type, a value $v$, a priority $p$ and a tree $t$, inserting $k$ with value $v$ and priority $p$ into $t$ splits $t$ at $k$, drops the entry the split finds at $k$ if any, merges the part below $k$ with the single node $(k, v, p)$, and then merges the result with the part above $k$; all merges are priority-guided.","title":"Treap insertion"},{"id":"ComputerScience/data-structures/search-trees/treap/erase","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.erase","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#erase","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For a key $k$ of a linearly ordered key type and a tree $t$, deleting $k$ from $t$ splits $t$ at $k$ and returns the priority-guided merge of the part of the split below $k$ with the part above $k$; the entry the split finds at $k$, if any, is dropped.","title":"Treap deletion"},{"id":"ComputerScience/data-structures/search-trees/treap/all-nodes-iff-forall-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.allNodes_iff_forall_mem","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#all-nodes-iff-forall-mem","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let $p$ be a predicate on a key, a value and a natural-number priority, and let $t$ be a tree. Then $p$ holds at every node of $t$ if and only if $p(k, v, n)$ holds for every key $k$, value $v$ and priority $n$ such that the entry $(k, v, n)$ occurs as a node of $t$.","title":"Treap all-nodes predicate via node membership"},{"id":"ComputerScience/data-structures/search-trees/treap/mem-key-of-mem-node","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.memKey_of_memNode","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#mem-key-of-mem-node","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"If a key $k$, value $v$ and priority $p$ form an entry of a treap tree $t$, that is, some node of $t$ stores exactly $k$, $v$ and $p$, then $k$ is a key of $t$.","title":"Entry membership implies key membership in a treap"},{"id":"ComputerScience/data-structures/search-trees/treap/mem-contents","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.mem_contents","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#mem-contents","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For every tree $t$, key $k$ and value $v$, the pair $(k, v)$ occurs in the inorder key-value list of $t$ if and only if $t$ has a node with key $k$ and value $v$, for some priority.","title":"Inorder contents list the treap's entries"},{"id":"ComputerScience/data-structures/search-trees/treap/key-mem-contents","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.key_mem_contents","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#key-mem-contents","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For a key $k$ and a tree $t$, some pair $(k, v)$ occurs in the inorder list of key-value pairs of $t$ if and only if $k$ is the key of some node of $t$.","title":"Keys occurring in the inorder contents"},{"id":"ComputerScience/data-structures/search-trees/treap/all-nodes-key-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.allNodes_key_iff","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#all-nodes-key-iff","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let $p$ be a predicate on keys and let $t$ be a tree. Then the key of every node of $t$ satisfies $p$ if and only if $p(k)$ holds for every key $k$ that occurs in $t$.","title":"Treap all-nodes predicate on keys via key membership"},{"id":"ComputerScience/data-structures/search-trees/treap/keys","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.keys","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#keys","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"The inorder key list of a tree $t$ is the list of first components of the inorder list of key-value pairs of $t$, that is, the keys of $t$ in inorder.","title":"Inorder key list of a treap"},{"id":"ComputerScience/data-structures/search-trees/treap/keys-node","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.keys_node","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#keys-node","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For trees $l$ and $r$, a key $k$, a value $v$ and a priority $p$, the inorder key list of the node with left subtree $l$, entry $(k, v, p)$ and right subtree $r$ is the inorder key list of $l$, followed by $k$, followed by the inorder key list of $r$.","title":"Inorder keys of a node"},{"id":"ComputerScience/data-structures/search-trees/treap/mem-keys","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.mem_keys","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#mem-keys","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For every treap tree $t$ and key $k$, $k$ occurs in the inorder key list of $t$ if and only if $k$ is a key of $t$.","title":"The inorder key list lists exactly the keys of a treap"},{"id":"ComputerScience/data-structures/search-trees/treap/ordered-keys","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.ordered_keys","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#ordered-keys","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linearly ordered type. If a treap tree $t$ is ordered, then its inorder key list is strictly increasing: each key in the list is smaller than every key after it.","title":"Inorder keys of an ordered treap are strictly increasing"},{"id":"ComputerScience/data-structures/search-trees/treap/keys-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.keys_nodup","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#keys-nodup","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For a linearly ordered key type, if a tree is ordered as a binary search tree (at every node, every key in the left subtree is smaller and every key in the right subtree is larger than the node's key), then its inorder key list has no duplicates.","title":"Treap keys are distinct"},{"id":"ComputerScience/data-structures/search-trees/treap/lookup-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.lookup_eq_some_iff","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#lookup-eq-some-iff","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let $t$ be a tree with keys in a linearly ordered type that is ordered as a binary search tree (every key in a left subtree is smaller and every key in a right subtree is larger than the node's key). For every key $k$ and value $v$, looking up $k$ in $t$ returns $v$ if and only if $t$ has a node with key $k$ and value $v$, for some priority.","title":"Treap lookup finds exactly the stored entries"},{"id":"ComputerScience/data-structures/search-trees/treap/lookup-contents","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.lookup_contents","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#lookup-contents","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let $t$ be a tree with keys in a linearly ordered type that is ordered as a binary search tree (every key in a left subtree is smaller and every key in a right subtree is larger than the node's key). For every key $k$ and value $v$, looking up $k$ in $t$ returns $v$ if and only if the pair $(k, v)$ occurs in the inorder key-value list of $t$.","title":"Treap lookup agrees with the inorder contents"},{"id":"ComputerScience/data-structures/search-trees/treap/lookup-with-cost-result","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.lookupWithCost_result","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#lookup-with-cost-result","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For every tree $t$ with keys in a linearly ordered type and every key $k$, the value component of the costed lookup of $k$ in $t$ (ignoring the visit count) equals the result of looking up $k$ in $t$.","title":"Costed lookup returns the ordinary lookup result"},{"id":"ComputerScience/data-structures/search-trees/treap/lookup-with-cost-le-size","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.lookupWithCost_le_size","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#lookup-with-cost-le-size","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For every tree $t$ with keys in a linearly ordered type and every key $k$, the number of nodes visited by the costed lookup of $k$ in $t$ is at most the number of nodes of $t$.","title":"Lookup cost is at most the tree size"},{"id":"ComputerScience/data-structures/search-trees/treap/heap-root-min","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.heap_root_min","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#heap-root-min","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"If a nonempty tree whose root has priority $p$ is heap-ordered, then every node of the tree, the root included, has priority at least $p$.","title":"The root of a heap-ordered tree has minimal priority"},{"id":"ComputerScience/data-structures/search-trees/treap/mem-key-merge","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.memKey_merge","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#mem-key-merge","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For all trees $l$ and $r$ and every key $k$, $k$ is the key of some node of the merge of $l$ and $r$ if and only if $k$ is the key of some node of $l$ or of some node of $r$.","title":"Keys of a merged treap"},{"id":"ComputerScience/data-structures/search-trees/treap/contents-merge","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.contents_merge","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#contents-merge","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For trees $l$ and $r$, the inorder list of key-value pairs of the priority-guided merge of $l$ and $r$ is the inorder list of $l$ followed by the inorder list of $r$.","title":"Treap merge concatenates inorder contents"},{"id":"ComputerScience/data-structures/search-trees/treap/separated","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.Separated","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#separated","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"For keys equipped with a binary relation $<$, two trees $L$ and $R$ are separated when $a < b$ for every key $a$ occurring at a node of $L$ and every key $b$ occurring at a node of $R$.","title":"Separated treap trees"},{"id":"ComputerScience/data-structures/search-trees/treap/ordered-merge","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.ordered_merge","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#ordered-merge","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linearly ordered type. If treap trees $L$ and $R$ are both ordered and every key of $L$ is smaller than every key of $R$, then the merge of $L$ and $R$ is ordered.","title":"Treap merge preserves search-tree order"},{"id":"ComputerScience/data-structures/search-trees/treap/heap-ordered-merge","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.heapOrdered_merge","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#heap-ordered-merge","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"If two trees $l$ and $r$ are both heap-ordered in their priorities, then the priority-guided merge of $l$ and $r$ is heap-ordered.","title":"Merging preserves heap order"},{"id":"ComputerScience/data-structures/search-trees/treap/ordered-split","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.ordered_split","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#ordered-split","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linearly ordered type and let $t$ be an ordered treap tree. For every pivot key $k$, the lower tree and the upper tree produced by splitting $t$ at $k$ are both ordered.","title":"Treap split preserves search-tree order"},{"id":"ComputerScience/data-structures/search-trees/treap/heap-ordered-split","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.heapOrdered_split","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#heap-ordered-split","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linearly ordered type and let $t$ be a heap-ordered treap tree. For every pivot key $k$, the lower tree and the upper tree produced by splitting $t$ at $k$ are both heap-ordered.","title":"Treap split preserves heap order"},{"id":"ComputerScience/data-structures/search-trees/treap/split-bounds","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.split_bounds","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#split-bounds","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linear order and let $T$ be an ordered tree. For every pivot $p$, every key of the lower tree obtained by splitting $T$ at $p$ is strictly less than $p$, and every key of the upper tree is strictly greater than $p$.","title":"Split bounds for an ordered treap"},{"id":"ComputerScience/data-structures/search-trees/treap/valid-split","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.valid_split","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#valid-split","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linear order. If a tree is a valid treap, then for every pivot $p$, both the lower tree and the upper tree obtained by splitting it at $p$ are valid treaps.","title":"Splitting a treap preserves validity"},{"id":"ComputerScience/data-structures/search-trees/treap/valid-merge","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.valid_merge","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#valid-merge","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linear order. If $L$ and $R$ are valid treaps and every key of $L$ is strictly less than every key of $R$, then the merge of $L$ and $R$ is a valid treap.","title":"Merging separated treaps preserves validity"},{"id":"ComputerScience/data-structures/search-trees/treap/valid-insert","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.valid_insert","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#valid-insert","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linear order. If a tree is a valid treap, that is, its keys are in strict binary-search-tree order and its priorities are in min-heap order, then for every key $k$, value $v$ and natural-number priority $q$, the tree obtained by inserting $k$ with value $v$ and priority $q$ is again a valid treap.","title":"Inserting into a treap preserves validity"},{"id":"ComputerScience/data-structures/search-trees/treap/valid-erase","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.valid_erase","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#valid-erase","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linear order. If a tree is a valid treap, that is, its keys are in strict binary-search-tree order and its priorities are in min-heap order, then for every key $k$ the tree obtained by erasing $k$ from it is again a valid treap.","title":"Erasing from a treap preserves validity"},{"id":"ComputerScience/data-structures/search-trees/treap/contents-split","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.contents_split","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#contents-split","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linearly ordered type, let $t$ be a tree and $k$ a pivot key, and let splitting $t$ at $k$ produce a lower tree $L$, an optional value-priority pair found at $k$, and an upper tree $G$. Then the inorder list of key-value pairs of $t$ is the inorder list of $L$, followed by $(k, v)$ if the split found a pair with value $v$ (and by nothing if it found none), followed by the inorder list of $G$.","title":"Treap inorder contents split at a pivot"},{"id":"ComputerScience/data-structures/search-trees/treap/contents-insert","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.contents_insert","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#contents-insert","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linearly ordered type, let $t$ be a tree, $k$ a key, $v$ a value and $n$ a natural-number priority, and let $L$ and $G$ be the lower and upper trees produced by splitting $t$ at $k$. Then the inorder list of key-value pairs of the tree obtained by inserting $k$ with value $v$ and priority $n$ into $t$ is the inorder list of $L$, followed by $(k, v)$, followed by the inorder list of $G$.","title":"Treap inorder contents after insertion"},{"id":"ComputerScience/data-structures/search-trees/treap/contents-erase","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.contents_erase","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#contents-erase","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let the keys come from a linearly ordered type, let $t$ be a tree and $k$ a key, and let $L$ and $G$ be the lower and upper trees produced by splitting $t$ at $k$. Then the inorder list of key-value pairs of the tree obtained by erasing $k$ from $t$ is the inorder list of $L$ followed by the inorder list of $G$.","title":"Treap inorder contents after erasure"},{"id":"ComputerScience/data-structures/search-trees/treap/lookup-insert-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.lookup_insert_eq","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#lookup-insert-eq","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let $t$ be a valid treap with keys in a linearly ordered type, that is, $t$ is ordered as a binary search tree by key and as a min-heap by priority. For every key $k$, value $v$ and priority $p \\in \\mathbb{N}$, looking up $k$ in the tree obtained by inserting $k$ with value $v$ and priority $p$ into $t$ returns $v$.","title":"Treap lookup after insertion"},{"id":"ComputerScience/data-structures/search-trees/treap/lookup-erase-eq-none","kind":"node","lean":"Cslib.Algorithms.Lean.Treap.Tree.lookup_erase_eq_none","parent":36851,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/treap/index.html#lookup-erase-eq-none","scope":"ComputerScience/data-structures/search-trees/treap","status":"fully_proved","summary":"Let $t$ be a valid treap with keys in a linearly ordered type, that is, $t$ is ordered as a binary search tree by key and as a min-heap by priority. For every key $k$, looking up $k$ in the tree obtained by erasing $k$ from $t$ returns no value.","title":"Erased keys are not found"},{"data_url":"scopes/ComputerScience/data-structures/search-trees/weight-balanced-tree.json","id":"ComputerScience/data-structures/search-trees/weight-balanced-tree","kind":"scope","navigation":"module","parent":36419,"result":false,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","title":"Weight-balanced tree"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/num-nodes-empty","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.numNodes_empty","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#num-nodes-empty","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"The empty binary tree has $0$ nodes.","title":"Number of keys of the empty tree"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/num-nodes-node","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.numNodes_node","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#num-nodes-node","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Write $|t|$ for the number of nodes of a binary tree $t$. For binary trees $l$ and $r$ and a key $k$, the node with key $k$, left subtree $l$ and right subtree $r$ has $|l| + |r| + 1$ nodes.","title":"Number of keys of a node"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/weight","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.weight","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#weight","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"The Nievergelt\u2013Reingold weight $w(t)$ of a binary tree $t$ is its number of keys plus one. Equivalently, it counts the external leaves: the empty tree has weight $1$, and the weight of a node is the sum of the weights of its two children.","title":"Weight of a binary tree"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/balanced","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.Balanced","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#balanced","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Write $|t|$ for the number of keys of a binary tree $t$ and $w(t) = |t| + 1$ for its weight. A binary tree is weight-balanced, or satisfies the invariant $\\mathrm{BB}[1/4]$, when it is empty, or when it is a node with left subtree $L$ and right subtree $R$ such that $L$ and $R$ are weight-balanced, $4\\,w(L) \\le 3(w(L) + w(R))$ and $4\\,w(R) \\le 3(w(L) + w(R))$. Since the weight of a node is $w(L) + w(R)$, this says\u2026","title":"Weight-balanced trees"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/balance","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.balance","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#balance","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"For binary trees $L$, $R$ and a key $k$, write $|t|$ for the number of keys of a tree $t$, $w(t) = |t| + 1$ for its weight, $w = w(L) + w(R)$, and $\\langle X, x, Y \\rangle$ for the node with key $x$, left subtree $X$ and right subtree $Y$. The rebalanced node $\\mathrm{balance}(L, k, R)$ is defined as follows. If $3w < 4\\,w(L)$, then $L = \\langle A, a, B \\rangle$ is a node; if $|B| \\le 2|A|$, the result is the\u2026","title":"Weight-balanced rebalancing"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.insert","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#insert","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"For a key $x$ of a linearly ordered type, the insertion $\\mathrm{insert}(x, t)$ of $x$ into a binary tree $t$ is defined by recursion on $t$. Into the empty tree it gives the single node with key $x$. Into the node $N$ with key $k$, left subtree $L$ and right subtree $R$ it gives $\\mathrm{balance}(\\mathrm{insert}(x, L), k, R)$ if $x < k$, $\\mathrm{balance}(L, k, \\mathrm{insert}(x, R))$ if $k < x$, and $N$ itself if\u2026","title":"Weight-balanced insertion"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/from-list","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.fromList","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#from-list","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"The tree built from a list $x_1, \\dots, x_n$ of elements of a linearly ordered type is obtained from the empty tree by inserting $x_n, x_{n-1}, \\dots, x_1$ in this order with weight-balanced insertion; the empty list gives the empty tree.","title":"Weight-balanced tree from a list"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/balanced-node","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.balanced_node","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#balanced-node","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let $L$ and $R$ be weight-balanced trees and $k$ a key, and write $|t|$ for the number of keys of a tree $t$. If $4(|L| + 1) \\le 3(|L| + |R| + 2)$ and $4(|R| + 1) \\le 3(|L| + |R| + 2)$, then the node with key $k$, left subtree $L$ and right subtree $R$ is weight-balanced.","title":"Building a weight-balanced node"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/balanced-node-iff-weight","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.balanced_node_iff_weight","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#balanced-node-iff-weight","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Write $|t|$ for the number of keys of a tree $t$ and $w(t) = |t| + 1$ for its weight. Let $N$ be the node with key $k$, left subtree $L$ and right subtree $R$. Then $N$ is weight-balanced if and only if $L$ and $R$ are weight-balanced, $4\\,w(L) \\le 3\\,w(N)$ and $4\\,w(R) \\le 3\\,w(N)$. This is the classical $\\mathrm{BB}[1/4]$ invariant: each child has weight at most three quarters of the weight of the node.","title":"Weight form of the balance condition at a node"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/num-nodes-balance","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.numNodes_balance","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#num-nodes-balance","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Write $|t|$ for the number of nodes of a binary tree $t$. For binary trees $l$ and $r$ and a key $k$, the weight-balanced rebalancing of $l$, $k$, $r$ has exactly $|l| + |r| + 1$ nodes.","title":"Weight-balanced rebalancing preserves the number of keys"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/balanced-balance","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.balanced_balance","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#balanced-balance","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let $L$ and $R$ be weight-balanced trees and $k$ a key, and write $|t|$ for the number of keys of a tree $t$. If $4(|L| + 1) \\le 3(|L| + |R| + 2) + 1$ and $4(|R| + 1) \\le 3(|L| + |R| + 2) + 1$, then the rebalanced node $\\mathrm{balance}(L, k, R)$ is weight-balanced.","title":"Weight-balanced rebalancing restores balance"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/mem-balance","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.mem_balance","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#mem-balance","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"For binary trees $l$ and $r$ and a key $k$, an element $x$ occurs in the weight-balanced rebalancing of $l$, $k$, $r$ exactly when $x$ occurs in $l$, $x = k$, or $x$ occurs in $r$. Membership is structural: $x$ is the key of some node.","title":"Membership after weight-balanced rebalancing"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/ordered-balance","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.ordered_balance","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#ordered-balance","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let the keys carry a linear order, let $l$ and $r$ be ordered binary trees (strict binary search trees), and let $k$ be a key such that every key of $l$ is less than $k$ and every key of $r$ is greater than $k$. Then the weight-balanced rebalancing of $l$, $k$, $r$ is ordered.","title":"Weight-balanced rebalancing preserves ordering"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/num-nodes-insert-le","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.numNodes_insert_le","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#num-nodes-insert-le","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let the keys carry a linear order. For every key $x$ and every binary tree $t$, the tree obtained from $t$ by weight-balanced insertion of $x$ has at most one more node than $t$.","title":"Weight-balanced insertion adds at most one key"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/le-num-nodes-insert","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.le_numNodes_insert","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#le-num-nodes-insert","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let the keys carry a linear order. For every key $x$ and every binary tree $t$, the number of nodes of $t$ is at most the number of nodes of the tree obtained from $t$ by weight-balanced insertion of $x$.","title":"Weight-balanced insertion does not decrease the number of keys"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/balanced-insert","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.balanced_insert","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#balanced-insert","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"For every key $x$ of a linearly ordered type and every weight-balanced tree $t$, the tree obtained by inserting $x$ into $t$ with weight-balanced insertion is weight-balanced.","title":"Weight-balanced insertion preserves balance"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/mem-insert","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.mem_insert","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#mem-insert","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let the keys carry a linear order. For all keys $x$, $y$ and every binary tree $t$, $y$ occurs in the tree obtained from $t$ by weight-balanced insertion of $x$ exactly when $y = x$ or $y$ occurs in $t$.","title":"Membership after weight-balanced insertion"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.ordered_insert","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#ordered-insert","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let the keys be linearly ordered. If a binary tree $t$ is ordered as a search tree (at every node, every key of the left subtree is strictly smaller and every key of the right subtree strictly larger than the key at the node), then the tree obtained from $t$ by weight-balanced insertion of any key $x$ is again ordered.","title":"Weight-balanced insertion preserves the search-tree order"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/balanced-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.balanced_fromList","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#balanced-from-list","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"For every list $x_1, \\dots, x_n$ of elements of a linearly ordered type, the tree obtained from the empty tree by inserting $x_n, x_{n-1}, \\dots, x_1$ in this order with weight-balanced insertion is weight-balanced.","title":"Weight-balanced trees built by insertion are balanced"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/ordered-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.ordered_fromList","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#ordered-from-list","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let the keys carry a linear order. For every list $xs$, the weight-balanced tree built from $xs$ by repeated insertion is ordered, that is, a strict binary search tree.","title":"Weight-balanced trees built from a list are ordered"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/mem-from-list","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.mem_fromList","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#mem-from-list","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Let the keys carry a linear order. For every key $x$ and every list $xs$, $x$ occurs in the weight-balanced tree built from $xs$ by repeated insertion exactly when $x$ is an element of $xs$.","title":"Membership in a weight-balanced tree built from a list"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/pow-height-le","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.pow_height_le","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#pow-height-le","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Write $w(t)$ for the weight of a binary tree $t$ (its number of keys plus one), and measure height so that the empty tree has height $0$ and a node has height one more than the taller of its children. If $t$ is weight balanced with parameter $1/4$ (at every node, each child has weight at most three quarters of the weight of the node) and has height $k + 1$ for a natural number $k$, then $w(t) \\ge (4/3)^k$; in\u2026","title":"Exponential weight lower bound for weight-balanced trees"},{"id":"ComputerScience/data-structures/search-trees/weight-balanced-tree/height-le-log2","kind":"node","lean":"Cslib.Algorithms.Lean.WeightBalancedTree.height_le_log2","parent":36903,"result":true,"route":"../roadmap/ComputerScience/data-structures/search-trees/weight-balanced-tree/index.html#height-le-log2","scope":"ComputerScience/data-structures/search-trees/weight-balanced-tree","status":"fully_proved","summary":"Write $|t|$ for the number of keys of a binary tree $t$ and $h(t)$ for its height, where the empty tree has height $0$ and a node has height one more than the larger height of its two subtrees. If $t$ is weight-balanced, then $h(t) \\le \\lfloor \\log_2 ((|t| + 1)^3) \\rfloor + 1$.","title":"Height of a weight-balanced tree is logarithmic"},{"data_url":"scopes/ComputerScience/data-structures/heaps.json","id":"ComputerScience/data-structures/heaps","kind":"scope","parent":36418,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/index.html","scope":"ComputerScience/data-structures/heaps","status":"fully_proved","summary":"Priority queues: binary, d-ary, binomial, Fibonacci and catenable heaps, indexed min-queues, and their sift, decrease-key and merge operations and costs.","title":"Heaps and priority queues"},{"data_url":"scopes/ComputerScience/data-structures/heaps/binary-heap.json","id":"ComputerScience/data-structures/heaps/binary-heap","kind":"scope","navigation":"module","parent":36927,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","title":"Binary heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/heap-parent","kind":"node","lean":"Cslib.Algorithms.Lean.heapParent","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#heap-parent","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"In a binary tree stored in level order in an array indexed from $0$, the parent index of a natural number $i$ is $\\lfloor (i-1)/2 \\rfloor$, where $i - 1$ is truncated subtraction of natural numbers, so the root index $0$ is sent to $0$.","title":"Parent index in a level-order binary tree"},{"id":"ComputerScience/data-structures/heaps/binary-heap/heap-left","kind":"node","lean":"Cslib.Algorithms.Lean.heapLeft","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#heap-left","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"In a binary tree stored in level order in an array indexed from $0$, the left child of the node at index $i$ has index $2i + 1$.","title":"Left child index in a level-order binary tree"},{"id":"ComputerScience/data-structures/heaps/binary-heap/heap-right","kind":"node","lean":"Cslib.Algorithms.Lean.heapRight","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#heap-right","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"In the level-order layout of a binary tree in an array indexed from $0$, the right child of the node at index $i$ is at index $2i + 2$.","title":"Right child index in an array-backed binary tree"},{"id":"ComputerScience/data-structures/heaps/binary-heap/heap-parent-lt-size","kind":"node","lean":"Cslib.Algorithms.Lean.heapParent_lt_size","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#heap-parent-lt-size","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For an array $a$ of length $|a|$ and an index $i$ with $0 < i < |a|$, the parent index $\\lfloor (i-1)/2 \\rfloor$ of $i$ is also less than $|a|$.","title":"The parent of a valid nonroot index is valid"},{"id":"ComputerScience/data-structures/heaps/binary-heap/heap-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.HeapOrdered","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#heap-ordered","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"An array $a$ over a type with a relation $\\le$, indexed from $0$, is heap ordered when every entry other than the root is at least its parent entry: for every index $i$ with $0 < i < |a|$, where $|a|$ is the length of $a$, we have $a_{\\lfloor (i-1)/2 \\rfloor} \\le a_i$.","title":"Heap order on a level-order array"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-up-ready","kind":"node","lean":"Cslib.Algorithms.Lean.SiftUpReady","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-up-ready","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array of length $|a|$ over a type with a relation $\\le$, indexed from $0$, and for $j > 0$ let $p(j) = \\lfloor (j - 1)/2 \\rfloor$ be the parent of $j$. The array $a$ is ready for sift-up at an index $i$ when $i < |a|$, when $a_{p(j)} \\le a_j$ for every $j \\ne i$ with $0 < j < |a|$, and when, if $i > 0$, also $a_{p(i)} \\le a_j$ for every $j$ with $0 < j < |a|$ and $p(j) = i$.","title":"Arrays ready for sift-up"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down-ready","kind":"node","lean":"Cslib.Algorithms.Lean.SiftDownReady","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down-ready","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For an array $a$ over a type ordered by $\\le$, write $p(j) = \\lfloor (j-1)/2 \\rfloor$ for the parent of index $j$. The array $a$ is ready for sift-down at index $i$ when $i$ is a valid index of $a$; $a_{p(j)} \\le a_j$ for every valid index $j > 0$ with $p(j) \\ne i$; and, if $i > 0$, $a_{p(i)} \\le a_j$ for every valid index $j > 0$ with $p(j) = i$.","title":"Sift-down precondition"},{"id":"ComputerScience/data-structures/heaps/binary-heap/smaller-child","kind":"node","lean":"Cslib.Algorithms.Lean.smallerChild","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#smaller-child","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array of length $|a|$ over a linearly ordered type, indexed from $0$, and let $i$ be an index with children $\\ell = 2i + 1$ and $r = 2i + 2$. If $\\ell \\ge |a|$, then $i$ has no smaller child. Otherwise the smaller child of $i$ is $r$ when $r < |a|$ and $a_r < a_\\ell$, and $\\ell$ in every other case, so ties go to the left child.","title":"Smaller child"},{"id":"ComputerScience/data-structures/heaps/binary-heap/smaller-child-gt","kind":"node","lean":"Cslib.Algorithms.Lean.smallerChild_gt","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#smaller-child-gt","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and all indices $i$ and $c$, if $c$ is the smaller child of $i$ in $a$, then $i < c$.","title":"The smaller child has a larger index"},{"id":"ComputerScience/data-structures/heaps/binary-heap/smaller-child-valid","kind":"node","lean":"Cslib.Algorithms.Lean.smallerChild_valid","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#smaller-child-valid","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and all indices $i$ and $c$, if $c$ is the smaller child of $i$ in $a$, then $c$ is less than the length $|a|$ of $a$.","title":"The smaller child is a valid index"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-up","kind":"node","lean":"Cslib.Algorithms.Lean.siftUp","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-up","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Given an array $a$ of length $|a|$ over a linearly ordered type, indexed from $0$, and an index $i$, the sift-up of $a$ from $i$ is defined by recursion on $i$. If $i = 0$ or $i \\ge |a|$, the result is $a$. Otherwise let $p = \\lfloor (i - 1)/2 \\rfloor$ be the parent of $i$; if $a_i < a_p$, the result is the sift-up from $p$ of the array obtained from $a$ by exchanging the entries at positions $i$ and $p$, and\u2026","title":"Sift-up"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down","kind":"node","lean":"Cslib.Algorithms.Lean.siftDown","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Sifting down an array $a$ over a linearly ordered type from index $i$ works as follows. If $i$ has a child in $a$, that is, if $2i+1$ is a valid index, let $c$ be the child of $i$ (among $2i+1$ and $2i+2$, if valid) holding the smaller value, preferring the left child $2i+1$ on ties. If $a_c < a_i$, swap the entries at $i$ and $c$ and continue sifting down from $c$; otherwise, or if $i$ has no child, return the\u2026","title":"Sift-down"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-up-perm","kind":"node","lean":"Cslib.Algorithms.Lean.siftUp_perm","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-up-perm","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every index $i$, the array obtained by sifting up $a$ from $i$ is a permutation of $a$.","title":"Sift-up permutes the array"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down-perm","kind":"node","lean":"Cslib.Algorithms.Lean.siftDown_perm","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down-perm","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every index $i$, the array obtained by sifting down $a$ from $i$ is a permutation of $a$.","title":"Binary-heap sift-down permutes the array"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-up-multiset","kind":"node","lean":"Cslib.Algorithms.Lean.siftUp_multiset","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-up-multiset","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every index $i$, the array obtained by sifting up $a$ from $i$ has the same multiset of entries as $a$.","title":"Binary-heap sift-up preserves the multiset of entries"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down-multiset","kind":"node","lean":"Cslib.Algorithms.Lean.siftDown_multiset","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down-multiset","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every index $i$, the array obtained by sifting down $a$ from $i$ has the same multiset of entries as $a$.","title":"Binary-heap sift-down preserves the multiset of entries"},{"id":"ComputerScience/data-structures/heaps/binary-heap/root-le","kind":"node","lean":"Cslib.Algorithms.Lean.HeapOrdered.root_le","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#root-le","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"If an array $a$ over a linearly ordered type is heap-ordered, then $a_0 \\le a_i$ for every valid index $i$ of $a$.","title":"The root of a heap-ordered binary-heap array is minimal"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-up-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.siftUp_ordered","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-up-ordered","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type and let $i$ be an index such that $a$ is ready for sift-up at $i$. Then the array obtained by sifting up $a$ from $i$ is heap ordered.","title":"Sift-up restores heap order"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.siftDown_ordered","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down-ordered","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type that satisfies the sift-down precondition at a valid index $i$: the only edges that may violate heap order are those leaving $i$, and moving either child of $i$ up to position $i$ keeps the edge entering $i$ ordered. Then the array obtained by sifting down $a$ from $i$ is heap-ordered.","title":"Sift-down restores heap order"},{"id":"ComputerScience/data-structures/heaps/binary-heap/push-sift-up-ready","kind":"node","lean":"Cslib.Algorithms.Lean.push_siftUpReady","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#push-sift-up-ready","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"If an array $a$ of length $n$ over a linearly ordered type is heap-ordered and $x$ is any value, then the array obtained by appending $x$ to $a$ satisfies the sift-up precondition at index $n$, the position of $x$: the only edge that may violate heap order is the one entering index $n$.","title":"Appending to a heap-ordered array prepares it for sift-up"},{"id":"ComputerScience/data-structures/heaps/binary-heap/min-heap","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#min-heap","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"A binary min-heap over a linearly ordered type $\\alpha$ is an array $a$ of elements of $\\alpha$ (its level-order storage), of length $|a|$ and with entries $a_0, \\dots, a_{|a|-1}$, that is heap ordered: for every index $i$ with $0 < i < |a|$, the entry $a_{\\lfloor (i - 1)/2 \\rfloor}$ at the parent index of $i$ is at most $a_i$.","title":"Binary min-heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/elements","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.elements","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#elements","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"The multiset of elements of a binary min-heap is the multiset of the entries of its level-order storage array, counted with multiplicity.","title":"Elements of a binary min-heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/empty","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.empty","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#empty","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"The empty binary min-heap is the min-heap whose level-order storage array has no entries.","title":"Empty binary min-heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/push","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.push","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#push","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"To insert a value $x$ into a binary min-heap $h$, append $x$ at the end of the level-order array of $h$ and sift it up from its index (the old length of the array), repeatedly swapping it with its parent while it is smaller. The result is again a binary min-heap.","title":"Insertion into a binary min-heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/min","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.min?","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#min","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For a binary min-heap $h$, the minimum lookup returns the entry at index $0$ (the root) of the storage array of $h$ when that array is nonempty, and returns no value when it is empty.","title":"Minimum lookup in a binary heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.min?_eq_none_iff","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For a binary min-heap $h$ over a linearly ordered type, looking up the minimum of $h$ returns no value if and only if the storage array of $h$ has size $0$.","title":"Binary heap minimum lookup fails exactly when empty"},{"id":"ComputerScience/data-structures/heaps/binary-heap/min-is-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.min?_isSome_iff","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#min-is-some-iff","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For a binary min-heap $h$ over a linearly ordered type, looking up the minimum of $h$ returns a value if and only if the storage array of $h$ has positive size.","title":"Binary heap minimum lookup succeeds exactly when nonempty"},{"id":"ComputerScience/data-structures/heaps/binary-heap/min-mem","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.min?_mem","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#min-mem","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If looking up the minimum of $h$ returns $x$, then $x$ belongs to the multiset of elements of $h$.","title":"The binary heap minimum lookup returns a stored element"},{"id":"ComputerScience/data-structures/heaps/binary-heap/min-le","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.min?_le","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#min-le","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If looking up the minimum of $h$ returns $x$, then $x \\le y$ for every element $y$ of the multiset of elements of $h$.","title":"The binary heap minimum lookup is a lower bound"},{"id":"ComputerScience/data-structures/heaps/binary-heap/elements-push","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.elements_push","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#elements-push","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type and $x$ a value. The multiset of elements of the heap obtained by inserting $x$ into $h$ is the multiset of elements of $h$ with one more copy of $x$ added.","title":"Elements after binary heap insertion"},{"id":"ComputerScience/data-structures/heaps/binary-heap/count-push-self","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.count_push_self","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#count-push-self","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type and $x$ a value. The multiplicity of $x$ in the multiset of elements of the heap obtained by inserting $x$ into $h$ is one more than its multiplicity in the multiset of elements of $h$.","title":"Binary heap insertion adds one copy of the inserted value"},{"id":"ComputerScience/data-structures/heaps/binary-heap/count-push-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.count_push_of_ne","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#count-push-of-ne","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type, and let $x \\ne y$ be values. Then $y$ occurs in the multiset of elements of the heap obtained by inserting $x$ into $h$ exactly as many times as it occurs in the multiset of elements of $h$.","title":"Binary heap insertion leaves other multiplicities unchanged"},{"id":"ComputerScience/data-structures/heaps/binary-heap/push-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.push_ordered","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#push-ordered","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every binary min-heap $h$ over a linearly ordered type and every value $x$, the level-order array of the heap obtained by inserting $x$ into $h$ is heap-ordered: the entry at every nonroot index $i$ is at least the entry at its parent index $\\lfloor (i-1)/2 \\rfloor$.","title":"Insertion into a binary min-heap preserves heap order"},{"id":"ComputerScience/data-structures/heaps/binary-heap/move-last-to-root","kind":"node","lean":"Cslib.Algorithms.Lean.moveLastToRoot","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#move-last-to-root","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For a nonempty array $a$ of length $n$, this operation swaps the entries at indices $0$ and $n - 1$ and then deletes the final entry. The result has length $n - 1$; when $n \\ge 2$ its entry at index $0$ is $a_{n-1}$ and its entry at each index $0 < i < n - 1$ is $a_i$.","title":"Moving the last entry to the root"},{"id":"ComputerScience/data-structures/heaps/binary-heap/move-last-to-root-sift-down-ready","kind":"node","lean":"Cslib.Algorithms.Lean.moveLastToRoot_siftDownReady","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#move-last-to-root-sift-down-ready","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be a heap-ordered array over a linearly ordered type with at least two entries, and let $b$ be the array obtained from $a$ by moving its last entry to the root. Then $b$ is ready for sifting down at index $0$: index $0$ is valid in $b$, and for every index $j > 0$ of $b$ whose parent index $\\lfloor (j - 1)/2 \\rfloor$ is not $0$, the entry of $b$ at the parent index is at most $b_j$.","title":"Moving the last entry to the root prepares a sift-down"},{"id":"ComputerScience/data-structures/heaps/binary-heap/move-last-to-root-elements","kind":"node","lean":"Cslib.Algorithms.Lean.moveLastToRoot_elements","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#move-last-to-root-elements","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For a nonempty array $a$ over a linearly ordered type, the multiset of entries of the array obtained from $a$ by moving its last entry to the root equals the multiset of entries of $a$ with one occurrence of $a_0$ removed.","title":"Moving the last entry to the root removes the old root"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-tail","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractTail","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-tail","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type whose level-order array is nonempty. If the array has exactly one entry, the result is the empty heap. Otherwise the last entry of the array replaces the root, the array shrinks by one, and the result is the heap obtained by sifting this entry down from the root.","title":"Removing the root of a nonempty binary min-heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For a binary min-heap $h$ over a linearly ordered type, extraction returns nothing when the level-order array of $h$ is empty. Otherwise it returns the root entry (the first entry of the array) together with the heap obtained from $h$ by removing its root and restoring heap order.","title":"Extracting the minimum of a binary min-heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_eq_none_iff","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For a binary min-heap $h$ over a linearly ordered type, extracting the minimum of $h$ returns no result if and only if the level-order storage array of $h$ has size $0$.","title":"Extraction from a binary min-heap fails exactly when it is empty"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-mem","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_mem","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-mem","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds and returns a value $x$ together with a heap $r$, then $x$ belongs to the multiset of elements of $h$.","title":"The value extracted from a binary min-heap is one of its elements"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-le","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_le","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-le","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds and returns a value $x$ together with a heap $r$, then $x \\le y$ for every value $y$ in the multiset of elements of $h$.","title":"The value extracted from a binary min-heap is minimal"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-elements","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_elements","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-elements","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If extracting the minimum of $h$ succeeds and returns the value $x$ and the heap $h'$, then the multiset of elements of $h'$ is obtained from the multiset of elements of $h$ by erasing one occurrence of $x$.","title":"Elements after binary heap extraction"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-cons-elements","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_cons_elements","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-cons-elements","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If extracting the minimum of $h$ succeeds and returns the value $x$ and the heap $h'$, then the multiset of elements of $h$ equals the multiset of elements of $h'$ with one copy of $x$ added.","title":"Binary heap extraction splits off the minimum"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-count","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_count","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-count","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If extracting the minimum of $h$ succeeds and returns the value $x$ and the heap $h'$, then the multiplicity of $x$ in the multiset of elements of $h'$ is its multiplicity in the multiset of elements of $h$ minus one.","title":"Binary heap extraction removes one copy of the minimum"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-count-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_count_of_ne","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-count-of-ne","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If extracting the minimum of $h$ succeeds and returns the value $x$ and the heap $h'$, then every value $y \\ne x$ occurs in the multiset of elements of $h'$ exactly as many times as in the multiset of elements of $h$.","title":"Binary heap extraction leaves other multiplicities unchanged"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-mem-of-ne-iff","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_mem_of_ne_iff","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-mem-of-ne-iff","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type, and suppose extracting the minimum from $h$ succeeds and returns a value $x$ together with a heap $r$. Then every value $y \\ne x$ is an element of $r$ if and only if it is an element of $h$.","title":"Membership after extraction from a binary min-heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/extract-min-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.extractMin_ordered","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#extract-min-ordered","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds and returns a value $x$ together with a heap $r$, then the level-order array of $r$ is heap ordered: every entry other than the root is at least its parent entry.","title":"Extraction from a binary min-heap leaves a heap-ordered array"},{"id":"ComputerScience/data-structures/heaps/binary-heap/log2-heap-parent-add-one","kind":"node","lean":"Cslib.Algorithms.Lean.log2_heapParent_add_one","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#log2-heap-parent-add-one","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every natural number $i > 0$, the parent index $p = \\lfloor (i-1)/2 \\rfloor$ satisfies $\\lfloor \\log_2(p+1) \\rfloor + 1 = \\lfloor \\log_2(i+1) \\rfloor$; that is, in the level-order binary tree the parent of a nonroot index lies one level above it.","title":"The heap parent is one level up"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-up-2","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.siftUp","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-up-2","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Given an array $a$ of length $|a|$ over a linearly ordered type and an index $i$, this computes the sift-up of $a$ at $i$ together with a cost $(c, s) \\in \\mathbb{N} \\times \\mathbb{N}$, where $c$ counts value comparisons and $s$ counts swaps. If $i \\ge |a|$ or $i = 0$, the result is $a$ with cost $(0, 0)$. Otherwise let $p = \\lfloor (i - 1)/2 \\rfloor$ be the parent of $i$. If $a_i < a_p$, the result is this sift-up\u2026","title":"Sift-up with comparison and swap counts"},{"id":"ComputerScience/data-structures/heaps/binary-heap/ret-sift-up","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ret_siftUp","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#ret-sift-up","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every natural number $i$, the array returned by sift-up with comparison and swap counts on $a$ at index $i$ equals the array returned by the plain sift-up of $a$ at $i$.","title":"Counting does not change the result of sift-up"},{"id":"ComputerScience/data-structures/heaps/binary-heap/comparisons-sift-up-le-log2","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.comparisons_siftUp_le_log2","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#comparisons-sift-up-le-log2","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every natural number $i$, the number of value comparisons counted by sift-up with comparison and swap counts on $a$ at index $i$ is at most $\\lfloor \\log_2(i+1) \\rfloor$, the depth of index $i$ in the level-order binary tree.","title":"Comparison bound for sift-up"},{"id":"ComputerScience/data-structures/heaps/binary-heap/swaps-sift-up-le-log2","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.swaps_siftUp_le_log2","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#swaps-sift-up-le-log2","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every natural number $i$, the number of swaps counted by sift-up with comparison and swap counts on $a$ at index $i$ is at most $\\lfloor \\log_2(i+1) \\rfloor$, the depth of index $i$ in the level-order binary tree.","title":"Swap bound for sift-up"},{"id":"ComputerScience/data-structures/heaps/binary-heap/events-sift-up-le-log2","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.events_siftUp_le_log2","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#events-sift-up-le-log2","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every natural number $i$, if sift-up with comparison and swap counts on $a$ at index $i$ counts $c$ comparisons and $s$ swaps, then $c + s \\le 2 \\lfloor \\log_2(i+1) \\rfloor$.","title":"Combined comparison and swap bound for sift-up"},{"id":"ComputerScience/data-structures/heaps/binary-heap/time-sift-up-zero","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.time_siftUp_zero","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#time-sift-up-zero","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type, sift-up with comparison and swap counts on $a$ at index $0$ has cost $(0, 0)$: it counts no comparisons and no swaps.","title":"Sift-up at the root costs nothing"},{"id":"ComputerScience/data-structures/heaps/binary-heap/time-sift-up-of-size-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.time_siftUp_of_size_le","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#time-sift-up-of-size-le","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ of length $|a|$ over a linearly ordered type and every natural number $i$ with $i \\ge |a|$, sift-up with comparison and swap counts on $a$ at index $i$ has cost $(0, 0)$: it counts no comparisons and no swaps.","title":"Sift-up at an out-of-range index costs nothing"},{"id":"ComputerScience/data-structures/heaps/binary-heap/decrease-key","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.decreaseKey","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#decrease-key","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Given a binary min-heap $h$ over a linearly ordered type, with level-order array $a$, an index $i < |a|$ and a value $x \\le a_i$, decrease-key returns the heap whose array is obtained from $a$ by replacing the entry at position $i$ with $x$ and then sifting position $i$ up, repeatedly swapping the entry with its parent while it is strictly smaller than the parent.","title":"Decrease-key in a binary min-heap"},{"id":"ComputerScience/data-structures/heaps/binary-heap/decrease-key-data","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.decreaseKey_data","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#decrease-key-data","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type, with level-order array $a$, let $i < |a|$ and let $x \\le a_i$. The array of the heap obtained by decreasing the key at position $i$ to $x$ equals the result of sifting up position $i$ in the array obtained from $a$ by replacing its entry at position $i$ with $x$.","title":"Array of a binary heap after decrease-key"},{"id":"ComputerScience/data-structures/heaps/binary-heap/decrease-key-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.decreaseKey_ordered","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#decrease-key-ordered","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type, with level-order array $a$, let $i < |a|$ and let $x \\le a_i$. The array of the heap obtained by decreasing the key at position $i$ to $x$ is heap-ordered: the entry at every nonroot position is at least the entry at its parent position.","title":"Decrease-key yields a heap-ordered array"},{"id":"ComputerScience/data-structures/heaps/binary-heap/decrease-key-size","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.decreaseKey_size","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#decrease-key-size","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type, with level-order array $a$, let $i < |a|$ and let $x \\le a_i$. The array of the heap obtained by decreasing the key at position $i$ to $x$ has the same length as $a$.","title":"Decrease-key preserves the heap size"},{"id":"ComputerScience/data-structures/heaps/binary-heap/decrease-key-elements","kind":"node","lean":"Cslib.Algorithms.Lean.MinHeap.decreaseKey_elements","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#decrease-key-elements","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $h$ be a binary min-heap over a linearly ordered type, with level-order array $a$, let $i < |a|$ and let $x \\le a_i$. The multiset of elements of the heap obtained by decreasing the key at position $i$ to $x$ is obtained from the multiset of elements of $h$ by removing one occurrence of $a_i$ and adding one occurrence of $x$.","title":"Multiset of a binary heap after decrease-key"},{"id":"ComputerScience/data-structures/heaps/binary-heap/log2-smaller-child-add-one","kind":"node","lean":"Cslib.Algorithms.Lean.log2_smallerChild_add_one","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#log2-smaller-child-add-one","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type and let $i$ and $c$ be natural numbers. If $c$ is the smaller child of $i$ in $a$, then $\\lfloor \\log_2 (c+1) \\rfloor = \\lfloor \\log_2 (i+1) \\rfloor + 1$; that is, in the level-order binary tree stored in $a$, the node $c$ lies exactly one level below $i$.","title":"The smaller child lies one level below its parent"},{"id":"ComputerScience/data-structures/heaps/binary-heap/log2-smaller-child-le-size","kind":"node","lean":"Cslib.Algorithms.Lean.log2_smallerChild_le_size","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#log2-smaller-child-le-size","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type, of size $|a|$, and let $i$ and $c$ be natural numbers. If $c$ is the smaller child of $i$ in $a$, then $\\lfloor \\log_2 (c+1) \\rfloor \\le \\lfloor \\log_2 |a| \\rfloor$.","title":"The smaller child's level is bounded by the array size"},{"id":"ComputerScience/data-structures/heaps/binary-heap/timed-smaller-child","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.timedSmallerChild","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#timed-smaller-child","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type and $i$ an index, with left child $2i + 1$ and right child $2i + 2$; cost is counted as the pair (value comparisons, array swaps). If both children are valid indices of $a$, the selection costs $(1, 0)$ and returns the right child when its value is strictly less than the value at the left child, and the left child otherwise. If only the left child is a valid index,\u2026","title":"Timed smaller-child selection"},{"id":"ComputerScience/data-structures/heaps/binary-heap/ret-timed-smaller-child","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ret_timedSmallerChild","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#ret-timed-smaller-child","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every index $i$, the value returned by timed child selection at $i$, with its cost discarded, is the smaller child of $i$ in $a$ (an optional index, absent when $i$ has no valid child).","title":"Timed child selection returns the smaller child"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down-2","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.siftDown","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down-2","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For an array $a$ over a linearly ordered type and an index $i$, timed sift-down returns an array together with a cost pair (value comparisons, array swaps), where costs add componentwise. If the left child index $2i+1$ is not a valid index of $a$, it returns $a$ at cost $(0, 0)$. Otherwise let $c$ be the smaller child of $i$: the left child $2i+1$, unless the right child $2i+2$ is a valid index and\u2026","title":"Sift-down with comparison and swap counts"},{"id":"ComputerScience/data-structures/heaps/binary-heap/ret-sift-down","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ret_siftDown","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#ret-sift-down","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"For every array $a$ over a linearly ordered type and every index $i$, the array returned by timed sift-down from $i$, with its cost discarded, equals the array returned by sift-down from $i$.","title":"Timed sift-down returns the sift-down result"},{"id":"ComputerScience/data-structures/heaps/binary-heap/time-timed-smaller-child","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.time_timedSmallerChild","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#time-timed-smaller-child","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type and $i$ an index, and count cost as the pair (value comparisons, array swaps). The timed child selection at $i$ in $a$ has cost $(1, 0)$ if the right child index $2i + 2$ is less than the size of $a$, and cost $(0, 0)$ otherwise.","title":"Cost of child selection"},{"id":"ComputerScience/data-structures/heaps/binary-heap/time-timed-smaller-child-of-none","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.time_timedSmallerChild_of_none","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#time-timed-smaller-child-of-none","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type and $i$ an index. If the smaller-child selection at $i$ in $a$ returns no child, then the timed child selection at $i$, with cost counted as the pair (value comparisons, array swaps), has cost $(0, 0)$.","title":"Child selection at a node without children costs nothing"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down-swaps-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.siftDown_swaps_le","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down-swaps-le","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Write $|a|$ for the size of an array $a$, and let $\\ell(n) = \\lfloor \\log_2 n \\rfloor$ for $n \\ge 1$ and $\\ell(0) = 0$. For every array $a$ over a linearly ordered type and every index $i$, the number of array swaps counted by timed sift-down from $i$ is at most $\\max(0, \\ell(|a|) - \\ell(i+1))$.","title":"Sift-down performs at most one swap per remaining level"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down-comparisons-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.siftDown_comparisons_le","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down-comparisons-le","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Write $|a|$ for the size of an array $a$, and let $\\ell(n) = \\lfloor \\log_2 n \\rfloor$ for $n \\ge 1$ and $\\ell(0) = 0$. For every array $a$ over a linearly ordered type and every index $i$, the number of value comparisons counted by timed sift-down from $i$ is at most $2 \\max(0, \\ell(|a|) - \\ell(i+1))$.","title":"Sift-down makes at most two comparisons per remaining level"},{"id":"ComputerScience/data-structures/heaps/binary-heap/sift-down-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.siftDown_time_le","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#sift-down-time-le","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Write $|a|$ for the size of an array $a$, and let $\\ell(n) = \\lfloor \\log_2 n \\rfloor$ for $n \\ge 1$ and $\\ell(0) = 0$. For every array $a$ over a linearly ordered type and every index $i$, the number of value comparisons plus the number of array swaps counted by timed sift-down from $i$ is at most $3 \\max(0, \\ell(|a|) - \\ell(i+1))$.","title":"Total sift-down cost is at most three per remaining level"},{"id":"ComputerScience/data-structures/heaps/binary-heap/time-sift-down-of-smaller-child-none","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.time_siftDown_of_smallerChild_none","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#time-sift-down-of-smaller-child-none","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type and $i$ an index. If the smaller-child selection at $i$ in $a$ returns no child, then sift-down from $i$, with cost counted as the pair (value comparisons, array swaps), has cost $(0, 0)$: it makes no value comparisons and no array swaps.","title":"Sift-down from a node without children costs nothing"},{"id":"ComputerScience/data-structures/heaps/binary-heap/time-sift-down-of-heap-left-ge","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.time_siftDown_of_heapLeft_ge","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#time-sift-down-of-heap-left-ge","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type, of size $|a|$, and let $i$ be an index with $|a| \\le 2i+1$, so that the left child of $i$ is not a valid index (this includes invalid indices $i$). Then timed sift-down from $i$ has cost pair $(0, 0)$: it counts no comparisons and no swaps.","title":"Sift-down from a leaf costs nothing"},{"id":"ComputerScience/data-structures/heaps/binary-heap/time-sift-down-of-size-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.time_siftDown_of_size_le","parent":36928,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binary-heap/index.html#time-sift-down-of-size-le","scope":"ComputerScience/data-structures/heaps/binary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type, of size $|a|$, and let $i$ be a natural number with $|a| \\le i$. Then timed sift-down from $i$ has cost pair $(0, 0)$: it counts no comparisons and no swaps.","title":"Sift-down from an invalid index costs nothing"},{"data_url":"scopes/ComputerScience/data-structures/heaps/binomial-heap.json","id":"ComputerScience/data-structures/heaps/binomial-heap","kind":"scope","navigation":"module","parent":36927,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","title":"Binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/binomial-tree","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#binomial-tree","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"A binomial tree over a type $\\alpha$ is a node that stores a rank, which is a natural number, a root value in $\\alpha$, and a list of binomial trees over $\\alpha$ as its children. The type imposes no invariant; the binomial-tree shape and heap order are expressed by a separate validity predicate.","title":"Binomial tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/rank","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.rank","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#rank","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"The rank of a binomial tree is the natural number stored at its root node.","title":"Rank of a binomial tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/root","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.root","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#root","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"The root of the binomial tree with rank $r$, root value $a$ and list of children $c$ is $a$.","title":"Root of a binomial tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/children","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.children","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#children","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"The children of a binomial tree are the binomial trees in the list stored at its root node, in their stored order.","title":"Children of a binomial tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/elements","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.elements","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#elements","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"The multiset of elements of a binomial tree with root value $x$ and children $t_1, \\dots, t_k$ is defined recursively as $\\{x\\}$ added to the multiset sum of the multisets of elements of $t_1, \\dots, t_k$; the stored rank plays no role.","title":"Elements of a binomial tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/root-mem-elements","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.root_mem_elements","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#root-mem-elements","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For every binomial tree $t$, the root of $t$ occurs in the multiset represented by $t$.","title":"The root of a binomial tree is one of its elements"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/valid","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.Valid","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#valid","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a type $\\alpha$ with a relation $\\le$, validity of binomial trees over $\\alpha$ is defined inductively: the tree with stored rank $r$, root $a$ and list of children $c_1, \\dots, c_k$ is valid when every child $c_i$ is valid, the ranks of $c_1, \\dots, c_k$ are exactly $r - 1, r - 2, \\dots, 0$ in this order, the multiset of elements of the tree (its root together with the elements of all its children) has exactly\u2026","title":"Valid binomial tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/singleton","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.singleton","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#singleton","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a value $a$ in a linear order, the binomial heap whose list of trees consists of exactly one tree: the tree of rank $0$ with root $a$ and no children.","title":"Singleton binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/link","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.link","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#link","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Over a linear order, the link of binomial trees $s$ and $t$ is defined as follows. If the root of $s$ is at most the root of $t$, it is the tree whose rank is one more than the rank of $s$, whose root is the root of $s$, and whose children are $t$ followed by the children of $s$. Otherwise it is the tree whose rank is one more than the rank of $s$, whose root is the root of $t$, and whose children are $s$ followed\u2026","title":"Linking two binomial trees"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/elements-link","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.elements_link","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#elements-link","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For any two binomial trees $s$ and $t$ over a linearly ordered type, the multiset of elements of the tree obtained by linking $s$ and $t$ is the sum of the multisets of elements of $s$ and $t$.","title":"Linking binomial trees adds their elements"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/valid-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.valid_singleton","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#valid-singleton","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$ and any $a \\in \\alpha$, the singleton binomial tree with rank $0$, root $a$ and no children is valid.","title":"Singleton binomial trees are valid"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/child-valid","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.Valid.child_valid","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#child-valid","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a type $\\alpha$ with a relation $\\le$, every child of a valid binomial tree over $\\alpha$ is itself a valid binomial tree.","title":"Children of a valid binomial tree are valid"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/children-ranks","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.Valid.children_ranks","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#children-ranks","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a type $\\alpha$ with a relation $\\le$, if a valid binomial tree over $\\alpha$ has rank $r$, then the ranks of its children, listed in their stored order, are $r - 1, r - 2, \\dots, 1, 0$; there are no children when $r = 0$.","title":"Child ranks of a valid binomial tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/card-elements","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.Valid.card_elements","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#card-elements","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a type $\\alpha$ with a relation $\\le$, every valid binomial tree of rank $r$ over $\\alpha$ represents a multiset of exactly $2^r$ elements, counted with multiplicity.","title":"Size of a valid binomial tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/rank-le-log2-card","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.Valid.rank_le_log2_card","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#rank-le-log2-card","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"A valid binomial tree of rank $r$ holding $n$ elements, counted with multiplicity, satisfies $r \\le \\lfloor \\log_2 n \\rfloor$.","title":"The rank of a valid binomial tree is logarithmic"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/root-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.Valid.root_le","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#root-le","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a type $\\alpha$ with a relation $\\le$ and a valid binomial tree $t$ over $\\alpha$, the root $a$ of $t$ satisfies $a \\le v$ for every element $v$ of the multiset represented by $t$.","title":"The root of a valid binomial tree is a lower bound"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/link-2","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialTree.Valid.link","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#link-2","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"If $s$ and $t$ are valid binomial trees of equal rank, then the tree obtained by linking $s$ with $t$ is valid.","title":"Linking preserves binomial-tree validity"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/valid-2","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.Valid","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#valid-2","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a type $\\alpha$ with a relation $\\le$, a list of binomial trees over $\\alpha$ is a valid forest when every tree in it is a valid binomial tree and the ranks of its trees are strictly increasing along the list.","title":"Valid binomial forest"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/elements-2","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.elements","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#elements-2","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"The multiset of elements of a list of binomial trees $t_1, \\dots, t_k$ is the multiset sum of the multisets of elements of $t_1, \\dots, t_k$; for the empty list it is the empty multiset.","title":"Elements of a binomial forest"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/ranks-at-least","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.RanksAtLeast","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#ranks-at-least","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a natural number $r$, a list of binomial trees has ranks at least $r$ when every tree in it has rank at least $r$.","title":"Rank lower bound for a forest"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/valid-nil","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.valid_nil","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#valid-nil","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For every type $\\alpha$ with a relation $\\le$, the empty list of binomial trees over $\\alpha$ is a valid binomial forest.","title":"The empty forest is valid"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/sublist","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.Valid.sublist","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#sublist","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a type $\\alpha$ with a relation $\\le$ and lists $L$, $R$ of binomial trees over $\\alpha$, if $L$ is a sublist of $R$ (obtained by deleting entries while keeping the order of the rest) and $R$ is a valid forest, then $L$ is a valid forest.","title":"Validity of binomial forests passes to sublists"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/member-rank-le-log2-card","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.Valid.member_rank_le_log2_card","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#member-rank-le-log2-card","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"If $F$ is a valid binomial forest and $T$ is a tree of $F$ of rank $r$, then $r \\le \\lfloor \\log_2 n \\rfloor$, where $n$ is the number of elements, counted with multiplicity, represented by $F$.","title":"Tree ranks in a valid forest are logarithmic"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/length-le-log2-card-succ","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.Valid.length_le_log2_card_succ","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#length-le-log2-card-succ","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $\\le$. If a valid binomial forest over $\\alpha$ has $n$ elements, counted with multiplicity, then it consists of at most $\\lfloor \\log_2 n \\rfloor + 1$ trees, where $\\lfloor \\log_2 0 \\rfloor$ is taken to be $0$.","title":"A valid binomial forest has logarithmically many trees"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/length-eq-zero-of-card-eq-zero","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.Valid.length_eq_zero_of_card_eq_zero","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#length-eq-zero-of-card-eq-zero","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Let $\\alpha$ be a type with a relation $\\le$. If a list of binomial trees over $\\alpha$ is a valid binomial forest and its multiset of elements has cardinality $0$, then the list has length $0$.","title":"A valid binomial forest with no elements has no trees"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/carry","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.carry","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#carry","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a linearly ordered type, the carry of a binomial tree $t$ into a list $F$ of binomial trees is defined by recursion on $F$. If $F$ is empty, the result is the one-tree list $[t]$. If $F$ starts with a tree $u$ followed by a list $F'$ and the rank of $t$ is less than the rank of $u$, the result is $F$ with $t$ prepended. Otherwise the result is the carry of the link of $t$ and $u$ into $F'$.","title":"Carry insertion into a binomial forest"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/meld-trees","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.meldTrees","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#meld-trees","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Over a linear order, the meld of two lists of binomial trees is defined by recursion on their total length. For every list $L$, the meld of the empty list with $L$ is $L$, and the meld of $L$ with the empty list is $L$. For nonempty lists with first trees $s$ and $t$ and remaining lists $F$ and $G$: if the rank of $s$ is less than the rank of $t$, the meld is $s$ followed by the meld of $F$ with the second list; if\u2026","title":"Melding binomial forests"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/elements-meld-trees","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.elements_meldTrees","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#elements-meld-trees","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For any two lists $L$ and $R$ of binomial trees over a linearly ordered type, the multiset of elements of the meld of $L$ and $R$ is the sum of the multisets of elements of $L$ and $R$.","title":"Melding binomial forests adds their elements"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/meld-trees-correct","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.meldTrees_correct","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#meld-trees-correct","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Let $r$ be a natural number and let $F$ and $G$ be valid binomial forests in which every tree has rank at least $r$. Then the meld of $F$ and $G$ is a valid binomial forest in which every tree has rank at least $r$.","title":"Melding preserves forest validity and rank lower bounds"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/remove-min-tree","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.removeMinTree","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#remove-min-tree","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a list $T$ of binomial trees over a linear order, the result is nothing when $T$ is empty. Otherwise it is the first tree $m$ of $T$ whose root is minimal among the roots of the trees of $T$, together with the list obtained from $T$ by deleting that occurrence of $m$, the remaining trees keeping their order.","title":"Removing a minimum-root tree from a forest"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/remove-min-tree-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.removeMinTree_spec","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#remove-min-tree-spec","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Let $T$ be a list of binomial trees over a linear order, and suppose that removing a minimum tree from $T$ returns the tree $m$ and the list $R$. Then the multiset represented by $m$ plus the multiset represented by the forest $R$ equals the multiset represented by the forest $T$; $m$ is a member of $T$; $R$ is a sublist of $T$; and the root of $m$ is at most the root of every tree in $T$.","title":"Removing a minimum-root tree"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/valid-reverse-children","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialForest.valid_reverse_children","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#valid-reverse-children","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$ and a valid binomial tree $t$ over $\\alpha$, the children of $t$ listed in reverse order form a valid binomial forest: every tree in the list is valid and the ranks strictly increase along the list.","title":"Reversed children of a valid binomial tree form a valid forest"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/binomial-heap","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#binomial-heap","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"A binomial min-heap over a linearly ordered type $\\alpha$ consists of a list of binomial trees over $\\alpha$ together with a proof that this list is a valid binomial forest: every tree in it is a valid binomial tree and the tree ranks are strictly increasing along the list.","title":"Binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/elements-3","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.elements","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#elements-3","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Over a linearly ordered type, the multiset of elements of a binomial heap is the multiset of elements of its list of binomial trees, that is, the multiset sum of the multisets of elements of its trees.","title":"Elements of a binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/rank-le-log2-card-2","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.rank_le_log2_card","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#rank-le-log2-card-2","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Every tree in a binomial heap holding $n$ elements, counted with multiplicity, has rank at most $\\lfloor \\log_2 n \\rfloor$.","title":"Tree ranks in a binomial heap are logarithmic"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/length-le-log2-card-succ-2","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.length_le_log2_card_succ","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#length-le-log2-card-succ-2","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"If a binomial heap over a linearly ordered type has $n$ elements, counted with multiplicity, then its number of trees is at most $\\lfloor \\log_2 n \\rfloor + 1$, where $\\lfloor \\log_2 0 \\rfloor$ is taken to be $0$.","title":"A binomial heap has logarithmically many trees"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/empty","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.empty","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#empty","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a linearly ordered type $\\alpha$, the empty binomial heap over $\\alpha$ is the binomial heap whose list of trees is empty.","title":"Empty binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/meld","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.meld","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#meld","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Over a linear order, the meld of two binomial heaps is the binomial heap whose forest is the meld of the forests of the two heaps.","title":"Melding binomial heaps"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/singleton-2","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.singleton","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#singleton-2","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a value $a$ in a linearly ordered type $\\alpha$, the one-element binomial heap of $a$ is the binomial heap whose list of trees consists of exactly one tree: the binomial tree of rank $0$ with root $a$ and no children.","title":"One-element binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/insert","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.insert","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#insert","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Inserting a value $x$ into a binomial heap $H$ over a linearly ordered type gives the meld of the one-element binomial heap containing $x$ with $H$.","title":"Insertion into a binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/find-min","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.findMin","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#find-min","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"The minimum lookup of a binomial heap over a linearly ordered type returns no value when the heap has no trees, and otherwise returns the least of the root values of its trees, leaving the heap unchanged.","title":"Minimum of a binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/delete-min-trees","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.deleteMinTrees","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#delete-min-trees","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Given a list of binomial trees over a linearly ordered type, if the list is empty there is no result. Otherwise, remove the first tree whose root is minimal among the roots of the list; the result is the root of that tree together with the forest obtained by melding its children, in reversed order, with the remaining trees.","title":"Minimum deletion on a binomial forest"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/delete-min","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.deleteMin","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#delete-min","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"If the binomial heap $H$ has no trees, there is no result. Otherwise, remove from the list of trees of $H$ the first tree whose root is minimal; the result is that root together with the binomial heap whose list of trees is obtained by melding the children of the removed tree, in reversed order, with the remaining trees.","title":"Deleting the minimum of a binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/delete-min-trees-2","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.deleteMin_trees","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#delete-min-trees-2","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For every binomial heap $H$ over a linearly ordered type, deleting the minimum of $H$ and then replacing the remaining heap by its list of trees gives the same result as minimum deletion on the list of trees of $H$: both have no result, or both return the same minimum value and the same list of trees.","title":"Binomial heap deletion agrees with forest deletion"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/elements-meld","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.elements_meld","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#elements-meld","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For any two binomial heaps $H_1$ and $H_2$ over a linearly ordered type, the multiset of elements of the meld of $H_1$ and $H_2$ is the sum of the multisets of elements of $H_1$ and $H_2$.","title":"Melding binomial heaps adds their elements"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/elements-insert","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.elements_insert","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#elements-insert","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For every binomial heap $H$ over a linearly ordered type and every value $v$, the multiset of elements of the heap obtained by inserting $v$ into $H$ is the multiset of elements of $H$ with one more occurrence of $v$.","title":"Insertion into a binomial heap adds one occurrence"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/find-min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.findMin_eq_none_iff","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#find-min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a binomial heap $H$ over a linearly ordered type, the minimum lookup of $H$ returns no value if and only if the list of trees of $H$ is empty.","title":"The minimum lookup fails exactly on the empty binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/find-min-mem","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.findMin_mem","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#find-min-mem","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Let $H$ be a binomial heap over a linearly ordered type. If the minimum lookup of $H$ returns $m$, then $m$ is an element of $H$.","title":"The minimum lookup of a binomial heap returns one of its elements"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/find-min-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.findMin_le","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#find-min-le","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"Let $H$ be a binomial heap over a linearly ordered type. If the minimum lookup of $H$ returns $m$, then $m \\le v$ for every element $v$ of $H$.","title":"The minimum lookup of a binomial heap returns a lower bound"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/delete-min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.deleteMin_eq_none_iff","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#delete-min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a binomial heap $H$ over a linearly ordered type, deleting the minimum of $H$ returns no result if and only if the list of trees of $H$ is empty.","title":"Deleting the minimum fails only on the empty binomial heap"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/delete-min-elements","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.deleteMin_elements","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#delete-min-elements","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a binomial heap $H$ over a linearly ordered type, if deleting the minimum of $H$ succeeds and returns a value $m$ and a heap $H'$, then the multiset of elements of $H'$ is the multiset of elements of $H$ with one occurrence of $m$ removed.","title":"Deleting the minimum of a binomial heap removes one occurrence"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/delete-min-cons-elements","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.deleteMin_cons_elements","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#delete-min-cons-elements","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"For a binomial heap $H$ over a linearly ordered type, if deleting the minimum of $H$ succeeds and returns a value $m$ and a heap $H'$, then the multiset of elements of $H$ equals the multiset of elements of $H'$ with one more occurrence of $m$.","title":"Deleting the minimum of a binomial heap splits its multiset"},{"id":"ComputerScience/data-structures/heaps/binomial-heap/delete-min-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinomialHeap.deleteMin_le","parent":37003,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/binomial-heap/index.html#delete-min-le","scope":"ComputerScience/data-structures/heaps/binomial-heap","status":"fully_proved","summary":"If deleting the minimum of a binomial heap $H$ over a linearly ordered type succeeds and returns a value $m$ together with a remaining heap, then $m \\le v$ for every value $v$ in the multiset of elements of $H$.","title":"The value deleted from a binomial heap is a lower bound for its elements"},{"data_url":"scopes/ComputerScience/data-structures/heaps/d-ary-heap.json","id":"ComputerScience/data-structures/heaps/d-ary-heap","kind":"scope","navigation":"module","parent":36927,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","title":"d-ary heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/normalized-arity","kind":"node","lean":"Cslib.Algorithms.Lean.normalizedArity","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#normalized-arity","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"The normalized arity of a requested arity $r \\in \\mathbb{N}$ is $\\max(2, r)$: the requested arities $0$ and $1$ become $2$ (the binary-heap policy), and every $r \\ge 2$ is left unchanged.","title":"Normalized heap arity"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/normalized-arity-valid","kind":"node","lean":"Cslib.Algorithms.Lean.normalizedArity_valid","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#normalized-arity-valid","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For every natural number $r$, the normalized arity $\\max(2, r)$ of $r$ is at least $2$.","title":"Normalized arities are at least two"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/normalized-arity-eq","kind":"node","lean":"Cslib.Algorithms.Lean.normalizedArity_eq","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#normalized-arity-eq","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For every natural number $d \\ge 2$, the normalized arity $\\max(2, d)$ of $d$ equals $d$.","title":"Normalization leaves a valid arity unchanged"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-parent","kind":"node","lean":"Cslib.Algorithms.Lean.daryParent","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parent index in a "],["math","\\(d\\)"],["text","-ary tree"]],"source":"Parent index in a $d$-ary tree"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-parent","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For natural numbers $d$ and $i$, the $d$-ary parent of $i$ is $\\lfloor (i - 1)/d \\rfloor$, computed in the natural numbers: $i - 1$ is truncated at $0$, so the parent of $0$ is $0$, and division by $0$ gives $0$.","title":"Parent index in a $d$-ary tree"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-child","kind":"node","lean":"Cslib.Algorithms.Lean.daryChild","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Child index in a "],["math","\\(d\\)"],["text","-ary heap array"]],"source":"Child index in a $d$-ary heap array"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-child","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For natural numbers $d$, $i$ and $k$, the child of index $i$ with zero-based offset $k$ in a contiguous level-order $d$-ary tree is the index $d i + k + 1$.","title":"Child index in a $d$-ary heap array"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-parent-child","kind":"node","lean":"Cslib.Algorithms.Lean.daryParent_child","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","The parent of a "],["math","\\(d\\)"],["text","-ary child"]],"source":"The parent of a $d$-ary child"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-parent-child","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d$, $i$ and $k$ be natural numbers with $d \\ge 2$ and $k < d$. Then the $d$-ary parent of the child of $i$ with zero-based offset $k$, that is, of the index $di + k + 1$, is $i$.","title":"The parent of a $d$-ary child"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-parent-lt","kind":"node","lean":"Cslib.Algorithms.Lean.daryParent_lt","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(d\\)"],["text","-ary parent has a smaller index"]],"source":"A $d$-ary parent has a smaller index"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-parent-lt","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d$ and $i$ be natural numbers with $d \\ge 2$ and $i > 0$. Then the $d$-ary parent of $i$ is strictly less than $i$.","title":"A $d$-ary parent has a smaller index"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-parent-lt-size","kind":"node","lean":"Cslib.Algorithms.Lean.daryParent_lt_size","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(d\\)"],["text","-ary parent of a valid index is valid"]],"source":"The $d$-ary parent of a valid index is valid"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-parent-lt-size","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d \\ge 2$ be a natural number, let $a$ be an array over any type, write $|a|$ for its size, and let $i$ be an index with $0 < i < |a|$. Then the $d$-ary parent of $i$ is also less than $|a|$.","title":"The $d$-ary parent of a valid index is valid"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-heap-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.DaryHeapOrdered","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(d\\)"],["text","-ary heap order"]],"source":"$d$-ary heap order"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-heap-ordered","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d$ be a natural number and $a = (a_0, \\dots, a_{n-1})$ an array with entries in a type carrying a relation $\\le$. The array $a$ is $d$-ary heap ordered when $d \\ge 2$ and every nonroot entry is at least the entry at its parent index: $a_{\\lfloor (i-1)/d \\rfloor} \\le a_i$ for every index $i$ with $0 < i < n$.","title":"$d$-ary heap order"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-up-ready","kind":"node","lean":"Cslib.Algorithms.Lean.DarySiftUpReady","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arrays ready for "],["math","\\(d\\)"],["text","-ary sift-up"]],"source":"Arrays ready for $d$-ary sift-up"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-up-ready","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{|a|-1})$ be an array over a type with a relation $\\le$, let $d$ and $i$ be natural numbers, and write $p(j) = \\lfloor (j-1)/d \\rfloor$ for the parent of an index $j > 0$. The array $a$ is ready for sift-up at $i$ when $d \\ge 2$, $i < |a|$, $a_{p(j)} \\le a_j$ for every $0 < j < |a|$ with $j \\ne i$, and, if $i > 0$, $a_{p(i)} \\le a_j$ for every $0 < j < |a|$ with $p(j) = i$.","title":"Arrays ready for $d$-ary sift-up"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-up","kind":"node","lean":"Cslib.Algorithms.Lean.darySiftUp","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(d\\)"],["text","-ary sift-up"]],"source":"$d$-ary sift-up"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-up","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d \\ge 2$, let $a$ be an array over a linearly ordered type, of length $|a|$ with entries $a_0, \\dots, a_{|a|-1}$, let $i$ be a natural number, and write $p(i) = \\lfloor (i-1)/d \\rfloor$ for the parent of $i > 0$. The sift-up of $a$ from $i$ is defined recursively: if $0 < i < |a|$ and $a_i < a_{p(i)}$, it is the sift-up from $p(i)$ of the array obtained from $a$ by swapping the entries at positions $i$ and\u2026","title":"$d$-ary sift-up"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-up-perm","kind":"node","lean":"Cslib.Algorithms.Lean.darySiftUp_perm","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-up-perm","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d \\ge 2$, let $a$ be an array over a linearly ordered type and let $i$ be a natural number. The array obtained from $a$ by $d$-ary sift-up from index $i$ is a permutation of $a$.","title":"Sifting up permutes the array"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-up-multiset","kind":"node","lean":"Cslib.Algorithms.Lean.darySiftUp_multiset","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-up-multiset","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d \\ge 2$, let $a$ be an array over a linearly ordered type and let $i$ be a natural number. The array obtained from $a$ by $d$-ary sift-up from index $i$ has the same multiset of entries as $a$.","title":"Sifting up preserves the multiset of entries"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/root-le","kind":"node","lean":"Cslib.Algorithms.Lean.DaryHeapOrdered.root_le","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","The root of a "],["math","\\(d\\)"],["text","-ary heap-ordered array is a minimum"]],"source":"The root of a $d$-ary heap-ordered array is a minimum"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#root-le","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $d \\ge 2$, and let $a$ be an array of elements of $\\alpha$ that is $d$-ary heap-ordered. Then $a_0 \\le a_i$ for every index $i$ smaller than the size of $a$.","title":"The root of a $d$-ary heap-ordered array is a minimum"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-up-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.darySiftUp_ordered","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sift-up restores "],["math","\\(d\\)"],["text","-ary heap order"]],"source":"Sift-up restores $d$-ary heap order"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-up-ordered","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type, of length $|a|$ with entries $a_0, \\dots, a_{|a|-1}$, let $d$ and $i$ be natural numbers, and write $p(j) = \\lfloor (j-1)/d \\rfloor$ for the parent of an index $j > 0$. Suppose $a$ is ready for sift-up at $i$: $d \\ge 2$, $i < |a|$, $a_{p(j)} \\le a_j$ for every $0 < j < |a|$ with $j \\ne i$, and, if $i > 0$, $a_{p(i)} \\le a_j$ for every $0 < j < |a|$ with $p(j) = i$.\u2026","title":"Sift-up restores $d$-ary heap order"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-push-sift-up-ready","kind":"node","lean":"Cslib.Algorithms.Lean.daryPush_siftUpReady","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pushing onto a "],["math","\\(d\\)"],["text","-ary heap leaves it ready for sifting up"]],"source":"Pushing onto a $d$-ary heap leaves it ready for sifting up"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-push-sift-up-ready","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $d \\ge 2$, and let $a$ be an array over $\\alpha$ that is $d$-ary heap ordered, with size $|a|$. Then for every $x \\in \\alpha$, the array obtained by appending $x$ to $a$ is ready for $d$-ary sifting up at index $|a|$, the position of $x$; in particular every nonroot entry other than $x$ is at least its $d$-ary parent.","title":"Pushing onto a $d$-ary heap leaves it ready for sifting up"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-down-ready","kind":"node","lean":"Cslib.Algorithms.Lean.DarySiftDownReady","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Readiness for "],["math","\\(d\\)"],["text","-ary sift-down"]],"source":"Readiness for $d$-ary sift-down"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-down-ready","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $a$ be an array over a type with a relation $\\le$, and let $d$ and $i$ be natural numbers. Write $|a|$ for the size of $a$ and $p(j)$ for the $d$-ary parent of an index $j$. The array $a$ is ready for $d$-ary sifting down at $i$ when $d \\ge 2$, $i < |a|$, every index $j$ with $0 < j < |a|$ and $p(j) \\ne i$ satisfies $a_{p(j)} \\le a_j$, and, if $i > 0$, every index $j$ with $0 < j < |a|$ and $p(j) = i$ satisfies\u2026","title":"Readiness for $d$-ary sift-down"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-smallest-child-from","kind":"node","lean":"Cslib.Algorithms.Lean.darySmallestChildFrom","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Smallest child among the offsets from "],["math","\\(s\\)"]],"source":"Smallest child among the offsets from $s$"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-smallest-child-from","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $a$ be an array over a linearly ordered type, of length $|a|$, let $d$, $i$ and $s$ be natural numbers, and call $d i + k + 1$ the child of $i$ at offset $k$. Scanning the offsets $s, s + 1, \\dots, d - 1$, this returns the index of a scanned child of $i$ below $|a|$ whose entry in $a$ is least among those, taking the smallest offset among ties, or nothing if no scanned child index is below $|a|$.","title":"Smallest child among the offsets from $s$"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-smallest-child","kind":"node","lean":"Cslib.Algorithms.Lean.darySmallestChild","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Smallest child in a "],["math","\\(d\\)"],["text","-ary heap array"]],"source":"Smallest child in a $d$-ary heap array"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-smallest-child","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $a$ be an array of length $|a|$ over a linearly ordered type and let $d$ and $i$ be natural numbers. The children of $i$ are the indices $d i + k + 1$ for $0 \\le k < d$, and a child is valid when it is below $|a|$. The smallest child of $i$ is the valid child of $i$ with the least entry, taking the smallest offset $k$ among ties, or nothing if $i$ has no valid child.","title":"Smallest child in a $d$-ary heap array"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-smallest-child-valid","kind":"node","lean":"Cslib.Algorithms.Lean.darySmallestChild_valid","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-smallest-child-valid","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $a$ be an array of length $|a|$ over a linearly ordered type and let $d$, $i$ and $c$ be natural numbers. If the smallest valid child of $i$ in the $d$-ary heap array $a$ is $c$, then $c < |a|$.","title":"The smallest child is a valid index"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-smallest-child-gt","kind":"node","lean":"Cslib.Algorithms.Lean.darySmallestChild_gt","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-smallest-child-gt","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d \\ge 2$, let $a$ be an array over a linearly ordered type and let $i$ and $c$ be natural numbers. If the smallest valid child of $i$ in the $d$-ary heap array $a$ is $c$, then $i < c$.","title":"The smallest child lies after its parent"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-down","kind":"node","lean":"Cslib.Algorithms.Lean.darySiftDown","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(d\\)"],["text","-ary sift-down"]],"source":"$d$-ary sift-down"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-down","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $a$ be an array over $\\alpha$, and let $i$ and $d$ be natural numbers with $d \\ge 2$. Sifting down from $i$ in $a$ returns $a$ if $i$ has no valid $d$-ary child in $a$. Otherwise let $c$ be the smallest valid $d$-ary child of $i$, with ties broken toward the lowest child offset: if $a_c < a_i$, swap the entries at $i$ and $c$ and sift down from $c$ in the resulting\u2026","title":"$d$-ary sift-down"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-down-perm","kind":"node","lean":"Cslib.Algorithms.Lean.darySiftDown_perm","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sifting down in a "],["math","\\(d\\)"],["text","-ary heap permutes the array"]],"source":"Sifting down in a $d$-ary heap permutes the array"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-down-perm","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $a$ be an array over $\\alpha$, and let $i$ and $d$ be natural numbers with $d \\ge 2$. The array produced by $d$-ary sifting down from index $i$ in $a$ is a permutation of $a$.","title":"Sifting down in a $d$-ary heap permutes the array"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-down-multiset","kind":"node","lean":"Cslib.Algorithms.Lean.darySiftDown_multiset","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sifting down in a "],["math","\\(d\\)"],["text","-ary heap preserves the multiset of entries"]],"source":"Sifting down in a $d$-ary heap preserves the multiset of entries"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-down-multiset","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $a$ be an array over $\\alpha$, and let $i$ and $d$ be natural numbers with $d \\ge 2$. The array produced by $d$-ary sifting down from index $i$ in $a$ has the same multiset of entries as $a$.","title":"Sifting down in a $d$-ary heap preserves the multiset of entries"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-sift-down-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.darySiftDown_ordered","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sifting down restores "],["math","\\(d\\)"],["text","-ary heap order"]],"source":"Sifting down restores $d$-ary heap order"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-sift-down-ordered","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $d \\ge 2$, and let $a$ be an array over $\\alpha$ that is ready for $d$-ary sifting down at an index $i$. Then the array produced by $d$-ary sifting down from index $i$ in $a$ is $d$-ary heap ordered.","title":"Sifting down restores $d$-ary heap order"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-move-last-to-root","kind":"node","lean":"Cslib.Algorithms.Lean.daryMoveLastToRoot","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-move-last-to-root","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For a nonempty array $a = (a_0, \\dots, a_{n-1})$, the last-to-root move swaps $a_0$ and $a_{n-1}$ and then deletes the last entry, producing an array $b = (b_0, \\dots, b_{n-2})$ of length $n - 1$: when $n \\ge 2$, $b_0 = a_{n-1}$ and $b_i = a_i$ for $0 < i < n - 1$, and $b$ is empty when $n = 1$.","title":"Last-to-root move on an array"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-move-last-to-root-sift-down-ready","kind":"node","lean":"Cslib.Algorithms.Lean.daryMoveLastToRoot_siftDownReady","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Last-to-root move in a "],["math","\\(d\\)"],["text","-ary heap is ready for sift-down"]],"source":"Last-to-root move in a $d$-ary heap is ready for sift-down"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-move-last-to-root-sift-down-ready","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $d$ be a natural number, let $a$ be a $d$-ary heap ordered array with at least two entries over a linearly ordered type, and let $b = (b_0, \\dots, b_{m-1})$ be the array obtained from $a$ by the last-to-root move. Then $b$ is ready for a $d$-ary sift-down at index $0$: $d \\ge 2$, $0 < m$, and $b_{\\lfloor (j-1)/d \\rfloor} \\le b_j$ for every index $j$ with $0 < j < m$ whose parent index $\\lfloor (j-1)/d \\rfloor$\u2026","title":"Last-to-root move in a $d$-ary heap is ready for sift-down"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-move-last-to-root-elements","kind":"node","lean":"Cslib.Algorithms.Lean.daryMoveLastToRoot_elements","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Last-to-root move in a "],["math","\\(d\\)"],["text","-ary heap removes one copy of the root"]],"source":"Last-to-root move in a $d$-ary heap removes one copy of the root"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-move-last-to-root-elements","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For every nonempty array $a = (a_0, \\dots, a_{n-1})$ over a linearly ordered type, the multiset of entries of the array obtained from $a$ by the last-to-root move equals the multiset of entries of $a$ with one occurrence of $a_0$ removed.","title":"Last-to-root move in a $d$-ary heap removes one copy of the root"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/dary-min-heap","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(d\\)"],["text","-ary min-heap"]],"source":"$d$-ary min-heap"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#dary-min-heap","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"A $d$-ary min-heap over a linearly ordered type $\\alpha$ consists of a natural number $d \\ge 2$, its arity, and an array $a = (a_0, \\dots, a_{n-1})$ of elements of $\\alpha$, stored in level order, that is $d$-ary heap ordered: $a_{\\lfloor (i-1)/d \\rfloor} \\le a_i$ for every index $i$ with $0 < i < n$.","title":"$d$-ary min-heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/elements","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.elements","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#elements","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"The multiset of elements of a $d$-ary min-heap $h$ over a linearly ordered type is the multiset of the entries of its level-order storage array, counted with multiplicity.","title":"Elements of a d-ary heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/empty","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.empty","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#empty","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For a natural number $r$ (the requested arity), the empty heap is the $d$-ary min-heap over a linearly ordered type with arity $d = \\max(2, r)$ and an empty storage array; a requested arity below two is thus normalized to two.","title":"Empty d-ary heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/push","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.push","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Insertion into a "],["math","\\(d\\)"],["text","-ary heap"]],"source":"Insertion into a $d$-ary heap"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#push","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Inserting a value $x$ into a $d$-ary min-heap $h$ of arity $d$ gives the heap of the same arity $d$ whose array is obtained by appending $x$ at the end of the level-order array of $h$ and sifting it up from that last index: while the current index $i > 0$ holds an entry strictly smaller than the entry at its parent index $\\lfloor (i-1)/d \\rfloor$, the two entries are swapped and the process continues from the\u2026","title":"Insertion into a $d$-ary heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/min","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.min?","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimum query on a "],["math","\\(d\\)"],["text","-ary min-heap"]],"source":"Minimum query on a $d$-ary min-heap"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#min","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"The minimum query on a $d$-ary min-heap $h$ returns the entry at index $0$ (the root) of the storage array of $h$, or nothing if the array is empty.","title":"Minimum query on a $d$-ary min-heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.min?_eq_none_iff","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","The minimum query on a "],["math","\\(d\\)"],["text","-ary heap fails exactly when it is empty"]],"source":"The minimum query on a $d$-ary heap fails exactly when it is empty"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For a $d$-ary min-heap $h$, the minimum query on $h$ (which returns the entry at index $0$ of the storage array, if any) returns nothing if and only if the storage array of $h$ is empty.","title":"The minimum query on a $d$-ary heap fails exactly when it is empty"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/min-mem","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.min?_mem","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","The minimum query on a "],["math","\\(d\\)"],["text","-ary heap returns an element of the heap"]],"source":"The minimum query on a $d$-ary heap returns an element of the heap"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#min-mem","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap. If the minimum query on $h$ returns $x$, then $x$ belongs to the multiset of elements of $h$.","title":"The minimum query on a $d$-ary heap returns an element of the heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/min-le","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.min?_le","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","The minimum query on a "],["math","\\(d\\)"],["text","-ary heap returns a least element"]],"source":"The minimum query on a $d$-ary heap returns a least element"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#min-le","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap. If the minimum query on $h$ returns $x$, then $x \\le y$ for every element $y$ of the multiset of elements of $h$.","title":"The minimum query on a $d$-ary heap returns a least element"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/elements-push","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.elements_push","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#elements-push","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For a $d$-ary min-heap $h$ over a linearly ordered type and a value $x$, the multiset of elements of the heap obtained by inserting $x$ into $h$ is the multiset of elements of $h$ with one more occurrence of $x$.","title":"Insertion into a d-ary heap adds one occurrence"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/push-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.push_ordered","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Insertion keeps a "],["math","\\(d\\)"],["text","-ary heap ordered"]],"source":"Insertion keeps a $d$-ary heap ordered"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#push-ordered","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap of arity $d$ and let $x$ be a value. Then the level-order array of the heap obtained by inserting $x$ into $h$ is $d$-ary heap-ordered for the same arity $d$.","title":"Insertion keeps a $d$-ary heap ordered"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/count-push-self","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.count_push_self","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Insertion into a "],["math","\\(d\\)"],["text","-ary heap raises the multiplicity by one"]],"source":"Insertion into a $d$-ary heap raises the multiplicity by one"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#count-push-self","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For every $d$-ary min-heap $h$ over a linearly ordered type and every value $x$, the multiplicity of $x$ among the elements of the heap obtained by inserting $x$ into $h$ is one more than its multiplicity among the elements of $h$.","title":"Insertion into a $d$-ary heap raises the multiplicity by one"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/count-push-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.count_push_of_ne","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Insertion into a "],["math","\\(d\\)"],["text","-ary heap preserves other multiplicities"]],"source":"Insertion into a $d$-ary heap preserves other multiplicities"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#count-push-of-ne","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For every $d$-ary min-heap $h$ over a linearly ordered type and all values $x \\ne y$, the multiplicity of $y$ among the elements of the heap obtained by inserting $x$ into $h$ equals the multiplicity of $y$ among the elements of $h$.","title":"Insertion into a $d$-ary heap preserves other multiplicities"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-tail","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractTail","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Removing the root of a "],["math","\\(d\\)"],["text","-ary min-heap"]],"source":"Removing the root of a $d$-ary min-heap"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-tail","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap whose storage array $a$ is nonempty. Removing the root of $h$ gives the heap of arity $d$ whose storage array is empty if $a$ has one entry. Otherwise the last entry of $a$ replaces the root, the last position is dropped, and the new root is repeatedly swapped with its smallest $d$-ary child while that child is smaller.","title":"Removing the root of a $d$-ary min-heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-tail-arity","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractTail_arity","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Removing the root of a "],["math","\\(d\\)"],["text","-ary heap preserves its arity"]],"source":"Removing the root of a $d$-ary heap preserves its arity"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-tail-arity","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For a $d$-ary min-heap $h$ with nonempty storage array, the heap obtained from $h$ by removing its root has the same arity $d$ as $h$.","title":"Removing the root of a $d$-ary heap preserves its arity"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/of-list","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.ofList","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Building a "],["math","\\(d\\)"],["text","-ary min-heap from a list"]],"source":"Building a $d$-ary min-heap from a list"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#of-list","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, let $r \\in \\mathbb{N}$ be a requested arity, and let $d = \\max(2, r)$. The min-heap built from the empty list is the empty min-heap of arity $d$, and the min-heap built from a list with head $x$ and tail $xs$ of elements of $\\alpha$ is obtained by inserting $x$ into the min-heap built from $xs$. Thus the min-heap built from $[x_1, \\dots, x_n]$ starts from the empty min-heap\u2026","title":"Building a $d$-ary min-heap from a list"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/elements-of-list","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.elements_ofList","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#elements-of-list","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For every requested arity $r$ and every list $xs$ over a linearly ordered type, the multiset of elements of the min-heap of arity $\\max(2, r)$ built from $xs$ by successive insertions equals the multiset of entries of $xs$.","title":"Building a heap from a list keeps its elements"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/arity-of-list","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.arity_ofList","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arity of a "],["math","\\(d\\)"],["text","-ary heap built from a list"]],"source":"Arity of a $d$-ary heap built from a list"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#arity-of-list","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For every requested arity $r \\in \\mathbb{N}$ and every list $xs$ over a linearly ordered type, the $d$-ary min-heap built from $xs$ with requested arity $r$ has arity equal to the normalized arity of $r$, namely $\\max(2, r)$.","title":"Arity of a $d$-ary heap built from a list"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extracting the minimum of a "],["math","\\(d\\)"],["text","-ary min-heap"]],"source":"Extracting the minimum of a $d$-ary min-heap"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For a $d$-ary min-heap $h$, extracting the minimum returns nothing if the storage array of $h$ is empty. Otherwise it returns the root entry (the entry at index $0$) together with the heap obtained from $h$ by removing its root: the last entry moves to the root and sifts down along a single $d$-ary child path.","title":"Extracting the minimum of a $d$-ary min-heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin_eq_none_iff","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For a $d$-ary min-heap $h$ over a linearly ordered type, extracting the minimum from $h$ returns no result if and only if the storage array of $h$ is empty.","title":"Extracting the minimum of a d-ary heap fails exactly on the empty heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min-cons-elements","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin_cons_elements","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min-cons-elements","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds, returning a value $x$ and a heap $h'$, then the multiset of elements of $h$ is the multiset of elements of $h'$ with one more occurrence of $x$.","title":"Extracted minimum and remaining d-ary heap recover the original elements"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min-elements","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin_elements","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min-elements","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds, returning a value $x$ and a heap $h'$, then the multiset of elements of $h'$ is obtained from the multiset of elements of $h$ by removing one occurrence of $x$.","title":"Extracting the minimum of a d-ary heap removes one occurrence"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min-count","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin_count","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min-count","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds, returning a value $x$ and a heap $h'$, then the multiplicity of $x$ in the multiset of elements of $h'$ is its multiplicity in the multiset of elements of $h$ minus one.","title":"Extracting the minimum of a d-ary heap lowers its multiplicity by one"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min-mem","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin_mem","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min-mem","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds, returning a value $x$ and a heap $h'$, then $x$ belongs to the multiset of elements of $h$.","title":"The extracted minimum of a d-ary heap is one of its elements"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min-le","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin_le","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min-le","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds, returning a value $x$ and a heap $h'$, then $x \\le y$ for every element $y$ of the multiset of elements of $h$.","title":"The extracted minimum of a d-ary heap is a lower bound"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min-arity","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin_arity","parent":37057,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min-arity","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap over a linearly ordered type. If extracting the minimum from $h$ succeeds, returning a value $x$ and a heap $h'$, then $h'$ has the same arity as $h$.","title":"Extracting the minimum preserves the arity of a d-ary heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/extract-min-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.extractMin_ordered","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Extracting the minimum leaves a heap-ordered "],["math","\\(d\\)"],["text","-ary heap"]],"source":"Extracting the minimum leaves a heap-ordered $d$-ary heap"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#extract-min-ordered","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"Let $h$ be a $d$-ary min-heap. If extracting the minimum of $h$ returns a value $x$ and a heap $h'$ of arity $d'$, then $d' \\ge 2$ and the storage array $a$ of $h'$ is heap-ordered: for every index $i$ with $0 < i < |a|$, the entry $a_{\\lfloor (i-1)/d' \\rfloor}$ is at most $a_i$.","title":"Extracting the minimum leaves a heap-ordered $d$-ary heap"},{"id":"ComputerScience/data-structures/heaps/d-ary-heap/card-elements","kind":"node","lean":"Cslib.Algorithms.Lean.DaryMinHeap.card_elements","parent":37057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Size of a "],["math","\\(d\\)"],["text","-ary min-heap"]],"source":"Size of a $d$-ary min-heap"},"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/d-ary-heap/index.html#card-elements","scope":"ComputerScience/data-structures/heaps/d-ary-heap","status":"fully_proved","summary":"For every $d$-ary min-heap $h$ over a linearly ordered type, the number of elements of $h$, counted with multiplicity, equals the length of its backing array.","title":"Size of a $d$-ary min-heap"},{"data_url":"scopes/ComputerScience/data-structures/heaps/fibonacci-heap.json","id":"ComputerScience/data-structures/heaps/fibonacci-heap","kind":"scope","navigation":"module","parent":36927,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","title":"Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/fibonacci-tree","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#fibonacci-tree","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"A Fibonacci tree over a type $\\alpha$ is a rose tree: a node carrying a stored degree $d \\in \\mathbb{N}$, a root value in $\\alpha$, and a finite list of child Fibonacci trees.","title":"Fibonacci tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/degree","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.degree","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#degree","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The degree of the Fibonacci tree with stored degree $d$, root value $r$ and list of children $c$ is the stored natural number $d$; it is not computed from the children.","title":"Degree of a Fibonacci tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/root","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.root","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#root","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The root of a Fibonacci tree is the value stored at its top node.","title":"Root of a Fibonacci tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/children","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.children","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#children","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The children of the Fibonacci tree with stored degree $d$, root value $r$ and list of subtrees $c$ are the trees of the list $c$, in order.","title":"Children of a Fibonacci tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The multiset $M(t)$ of values stored in a Fibonacci tree $t$, defined recursively: if $t$ has root value $r$ and children $t_1, \\dots, t_k$, then $M(t) = \\{r\\} + M(t_1) + \\cdots + M(t_k)$. The stored degree plays no role.","title":"Multiset of a Fibonacci tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/root-mem-elements","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.root_mem_elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#root-mem-elements","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every Fibonacci tree, its root value belongs to the multiset of elements the tree represents.","title":"The root of a Fibonacci tree is one of its elements"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/card-elements-pos","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.card_elements_pos","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#card-elements-pos","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Every Fibonacci tree represents at least one element: the multiset of elements of any Fibonacci tree has positive size.","title":"Fibonacci trees are nonempty"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/valid","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.Valid","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#valid","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $\\alpha$ carry a relation $\\le$. A Fibonacci tree with stored degree $d$, root $r \\in \\alpha$ and list of children $c$ is valid when every tree in $c$ is valid, $d$ equals the length of $c$, and $r \\le x$ for every element $x$ of the multiset represented by the tree (the root together with the elements of all children).","title":"Valid Fibonacci trees"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/singleton","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.singleton","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#singleton","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a value $x$ of a linearly ordered type, the one-element Fibonacci heap on $x$ has as its only tree the single-node tree with root $x$, and its cached minimum is $x$.","title":"One-element Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/link","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.link","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#link","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $s$ and $t$ be Fibonacci trees over a linearly ordered type. If the root of $s$ is at most the root of $t$, the link of $s$ and $t$ is the tree with the root of $s$, stored degree one more than that of $s$, and children $t$ followed by the children of $s$. Otherwise it is the tree with the root of $t$, stored degree one more than that of $t$, and children $s$ followed by the children of $t$.","title":"Linking two Fibonacci trees"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/degree-link-of-eq","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.degree_link_of_eq","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#degree-link-of-eq","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If two Fibonacci trees $s$ and $t$ over a linearly ordered type have the same degree $d$, then their link has degree $d + 1$.","title":"Degree of the link of two trees of equal degree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-link","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.elements_link","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-link","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For any two Fibonacci trees $S$ and $T$ over a linearly ordered type, the tree obtained by linking $S$ and $T$ represents the sum of the multisets represented by $S$ and by $T$.","title":"Linking adds the elements"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/valid-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.valid_singleton","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#valid-singleton","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. For every $v \\in \\alpha$, the Fibonacci tree of degree $0$ with root $v$ and no children is valid.","title":"Singleton Fibonacci trees are valid"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/child-valid","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.Valid.child_valid","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#child-valid","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Every child of a valid Fibonacci tree is itself a valid Fibonacci tree.","title":"Children of a valid Fibonacci tree are valid"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/degree-eq","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.Valid.degree_eq","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#degree-eq","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every valid Fibonacci tree, its stored degree equals the number of its children.","title":"The degree of a valid Fibonacci tree is its number of children"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/root-le","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.Valid.root_le","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#root-le","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The root of a valid Fibonacci tree is no greater than every element of the multiset the tree represents.","title":"Roots of valid Fibonacci trees are lower bounds"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/degree-lt-card","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.Valid.degree_lt_card","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#degree-lt-card","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every valid Fibonacci tree over a type with a relation $\\le$, the degree of the tree is strictly less than the number of elements, counted with multiplicity, of the multiset the tree represents, since every child contributes at least one element.","title":"The degree of a valid Fibonacci tree is less than its size"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/link-2","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.Valid.link","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#link-2","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If $s$ and $t$ are valid Fibonacci trees over a linearly ordered type, then the link of $s$ and $t$ is a valid Fibonacci tree.","title":"Linking preserves validity of Fibonacci trees"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/valid-2","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.Valid","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#valid-2","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $\\alpha$ carry a relation $\\le$. A Fibonacci forest, a list of Fibonacci trees with values in $\\alpha$, is valid when every tree in it is a valid Fibonacci tree.","title":"Valid Fibonacci forests"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-2","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-2","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Write $M(t)$ for the multiset of values stored in a Fibonacci tree $t$. The multiset represented by a Fibonacci forest, a list of Fibonacci trees $t_1, \\dots, t_k$, is the sum $M(t_1) + \\cdots + M(t_k)$; the empty forest represents the empty multiset.","title":"Multiset of a Fibonacci forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/sorted-degrees","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.SortedDegrees","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#sorted-degrees","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"A Fibonacci forest, a list of Fibonacci trees $t_1, \\dots, t_k$, has sorted degrees when the stored degrees of its trees strictly increase along the list: $\\deg t_i < \\deg t_j$ whenever $i < j$.","title":"Forests with strictly increasing degrees"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/valid-nil","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.valid_nil","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#valid-nil","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $\\alpha$ carry a relation $\\le$. The empty forest, containing no Fibonacci trees with values in $\\alpha$, is valid.","title":"The empty Fibonacci forest is valid"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/valid-append","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.valid_append","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#valid-append","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $\\alpha$ carry a relation $\\le$. If $F$ and $G$ are valid Fibonacci forests with values in $\\alpha$, then the forest consisting of the trees of $F$ followed by the trees of $G$ is valid.","title":"Concatenation preserves validity of Fibonacci forests"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/sublist","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.Valid.sublist","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#sublist","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If a list of Fibonacci trees is a valid forest, then every sublist of it (a subsequence, not necessarily contiguous) is a valid forest.","title":"Sublists of valid Fibonacci forests are valid"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-append","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.elements_append","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-append","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For any two lists $L$ and $R$ of Fibonacci trees, the forest made of the trees of $L$ followed by the trees of $R$ represents the sum of the multisets represented by $L$ and by $R$.","title":"Elements of a concatenated forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/min-root","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.minRoot","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#min-root","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The minimum root of a forest of Fibonacci trees over a linearly ordered type is an optional value: it is absent for the empty forest, and for a nonempty forest it is the least of the roots of its trees.","title":"Minimum root of a forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/min-option","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.minOption","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#min-option","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For optional values $x$ and $y$ of a linearly ordered type, their combined minimum is defined as follows: if $x$ and $y$ are both present, with values $a$ and $b$, it is $\\min(a, b)$; if exactly one of $x$ and $y$ is present, it is that value; and if both are absent, it is absent.","title":"Minimum of two optional values"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/min-root-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.minRoot_eq_none_iff","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#min-root-eq-none-iff","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"A forest of Fibonacci trees over a linearly ordered type has no minimum root exactly when it is empty.","title":"Only the empty forest has no minimum root"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/min-root-spec","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.minRoot_spec","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#min-root-spec","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If a forest $F$ of Fibonacci trees over a linearly ordered type has minimum root $m$, then $m$ is the root of some tree of $F$ and $m$ is at most the root of every tree of $F$.","title":"The minimum root is a least root"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/min-root-append","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.minRoot_append","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#min-root-append","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For forests $F$ and $G$ of Fibonacci trees over a linearly ordered type, the minimum root of $F$ followed by $G$ is the combined minimum of the optional minimum roots of $F$ and of $G$: their minimum if both exist, the one that exists if only one does, and absent if neither does.","title":"Minimum root of a concatenated forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/add-tree","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.addTree","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#add-tree","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Given a Fibonacci tree $t$ and a list of Fibonacci trees, the insertion of $t$ into the list is defined recursively. Inserting $t$ into the empty list gives the one-tree list $[t]$. For a list with first tree $s$ and remaining list $L$: if the stored degree of $t$ is less than the stored degree of $s$, the result is $t$ followed by $s$ and then $L$; if the two stored degrees are equal, the result is the insertion\u2026","title":"Inserting a tree into a forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/consolidate","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.consolidate","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#consolidate","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The consolidation of a list of Fibonacci trees is defined recursively: the consolidation of the empty list is the empty list, and the consolidation of a list with first tree $t$ and remaining list $L$ is the insertion of $t$ into the consolidation of $L$.","title":"Consolidation of a forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-consolidate","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.elements_consolidate","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-consolidate","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every list of Fibonacci trees over a linearly ordered type, its consolidation represents the same multiset as the original list.","title":"Consolidation preserves the elements"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/valid-consolidate","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.valid_consolidate","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#valid-consolidate","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type. If a Fibonacci forest with values in $\\alpha$ is valid, then its consolidation is valid.","title":"Consolidation preserves validity"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/consolidate-sorted","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.consolidate_sorted","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#consolidate-sorted","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every list of Fibonacci trees, the stored degrees of the trees in its consolidation are strictly increasing in list order.","title":"Consolidation sorts root degrees"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/consolidate-degrees-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.consolidate_degrees_nodup","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#consolidate-degrees-nodup","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every list of Fibonacci trees, the trees in its consolidation have pairwise distinct stored degrees.","title":"Consolidation yields distinct root degrees"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/length-le-card-of-sorted","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.length_le_card_of_sorted","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#length-le-card-of-sorted","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $F$ be a valid forest of Fibonacci trees whose degrees are strictly increasing along the list. Then the number of trees in $F$ is at most the number of elements, counted with multiplicity, that $F$ represents.","title":"Degree-sorted forests have at most as many trees as elements"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/remove-min-tree","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.removeMinTree","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#remove-min-tree","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a list of Fibonacci trees over a linearly ordered type, this returns nothing on the empty list. Otherwise it returns the first tree whose root is minimal among all roots in the list, so earlier trees win ties, together with the list of the remaining trees in their original order.","title":"Removing a minimum tree from a Fibonacci forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/remove-min-tree-spec","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.removeMinTree_spec","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#remove-min-tree-spec","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $T$ be a list of Fibonacci trees over a linearly ordered type. If removing a minimum tree from $T$ returns a tree $m$ and a list $R$, then the multiset of elements of $m$ plus the multiset of elements of $R$ equals the multiset of elements of $T$, the tree $m$ belongs to $T$, the list $R$ is a sublist of $T$, and the root of $m$ is at most the root of every tree in $T$.","title":"Minimum-tree removal in a Fibonacci forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/fibonacci-heap","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#fibonacci-heap","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"A Fibonacci min-heap over a linearly ordered type consists of a list of Fibonacci trees, each valid (heap-ordered, and at every node the stored degree equals the number of children), together with a cached minimum equal to the least root value of these trees, absent when the list is empty.","title":"Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-3","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-3","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The multiset represented by a Fibonacci heap over a linearly ordered type is the multiset of its list of trees: the sum, over the trees in the list, of the multisets of values stored in them. The cached minimum plays no role.","title":"Multiset of a Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/empty","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.empty","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#empty","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The Fibonacci heap over a linearly ordered type whose list of trees is empty and whose cached minimum is absent.","title":"Empty Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/singleton-2","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.singleton","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#singleton-2","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a value $x$ of a linearly ordered type, the singleton Fibonacci heap on $x$ has as its only tree the single-node tree with root $x$, stored degree $0$ and no children, and its cached minimum is $x$.","title":"Singleton Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/meld","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.meld","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#meld","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The meld of Fibonacci heaps $H_1$ and $H_2$ is the Fibonacci heap whose root list is the root list of $H_1$ followed by that of $H_2$, and whose cached minimum is the smaller of the two cached minima when both heaps have one, and otherwise the cached minimum of whichever heap has one (none if neither does).","title":"Melding Fibonacci heaps"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/union","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.union","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#union","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The union of Fibonacci heaps $H_1$ and $H_2$ is their meld: the heap whose trees are those of $H_1$ followed by those of $H_2$, and whose cached minimum is the smaller of the two cached minima (the one that exists if a heap is empty, none if both are).","title":"Union of Fibonacci heaps"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/insert","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.insert","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#insert","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Inserting a value $x$ into a Fibonacci heap $H$ gives the meld of the one-element Fibonacci heap containing $x$ with $H$, the one-element heap coming first.","title":"Insertion into a Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/find-min","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.findMin","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#find-min","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Find-min on a Fibonacci heap returns its cached minimum without modifying the heap: the least root value of its trees, or no value when the heap has no trees.","title":"Find-min for Fibonacci heaps"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min-trees","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMinTrees","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min-trees","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Given a list of Fibonacci trees over a linearly ordered type, the result is nothing if the list is empty. Otherwise the first tree whose root is minimal is removed, and the result is its root value together with the consolidation of the forest made of the removed tree's children followed by the remaining trees.","title":"Minimum deletion on a forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMin","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If the root list of the heap is empty, the result is nothing. Otherwise the first tree of the root list whose root is minimal is removed, and the result is its root value together with the heap whose root list is the consolidation of the removed tree's children followed by the remaining trees, and whose cached minimum is the minimum root of this consolidated forest.","title":"Deleting the minimum of a Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min-trees-2","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMin_trees","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min-trees-2","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every Fibonacci heap $H$, deleting the minimum of $H$ and keeping only the returned value and the root list of the resulting heap gives the same result as the root-list minimum deletion applied to the root list of $H$: either both fail, or both return the same value and the same list of trees.","title":"Fibonacci heap minimum deletion agrees with its root-list version"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-empty","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.elements_empty","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-empty","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"The empty Fibonacci heap represents the empty multiset.","title":"The empty Fibonacci heap has no elements"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.elements_singleton","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-singleton","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every value $x$ of a linearly ordered type, the singleton Fibonacci heap on $x$, whose only tree is the single-node tree with root $x$, represents the multiset $\\{x\\}$.","title":"Multiset of a singleton Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-meld","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.elements_meld","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-meld","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For any two Fibonacci heaps $H_1$ and $H_2$ over a linearly ordered type, the meld of $H_1$ and $H_2$ represents the sum of the multisets represented by $H_1$ and by $H_2$.","title":"Melding adds the elements"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-union","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.elements_union","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-union","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For Fibonacci heaps $H_1$ and $H_2$ over a linearly ordered type, the multiset represented by the union of $H_1$ and $H_2$ is the sum of the multisets represented by $H_1$ and by $H_2$.","title":"The union of Fibonacci heaps adds their multisets"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-insert","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.elements_insert","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-insert","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every Fibonacci heap $H$ over a linearly ordered type and every value $x$, the heap obtained by inserting $x$ into $H$ represents the multiset of $H$ with one more occurrence of $x$.","title":"Insertion adds one occurrence"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/card-elements-meld","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.card_elements_meld","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#card-elements-meld","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For Fibonacci heaps $H_1$ and $H_2$, the number of elements of the meld of $H_1$ and $H_2$, counted with multiplicity, is the number of elements of $H_1$ plus the number of elements of $H_2$.","title":"Melding adds the sizes of Fibonacci heaps"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/card-elements-insert","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.card_elements_insert","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#card-elements-insert","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every Fibonacci heap $H$ and every value $x$, the number of elements of the heap obtained by inserting $x$ into $H$, counted with multiplicity, is the number of elements of $H$ plus one.","title":"Insertion increases the size of a Fibonacci heap by one"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/find-min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.findMin_eq_none_iff","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#find-min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a Fibonacci heap $H$ over a linearly ordered type, find-min on $H$ returns no value if and only if $H$ has no trees.","title":"Find-min fails exactly on the empty Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/find-min-mem","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.findMin_mem","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#find-min-mem","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap over a linearly ordered type. If find-min on $H$ returns $m$, then $m$ belongs to the multiset represented by $H$.","title":"Find-min returns an element of the Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/find-min-le","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.findMin_le","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#find-min-le","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap over a linearly ordered type. If find-min on $H$ returns $m$, then $m \\le v$ for every value $v$ in the multiset represented by $H$.","title":"Find-min returns a lower bound of the Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMin_eq_none_iff","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Deleting the minimum of a Fibonacci heap fails if and only if the heap's list of root trees is empty.","title":"Fibonacci heap minimum deletion fails exactly on the empty heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min-elements","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMin_elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min-elements","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If deleting the minimum of a Fibonacci heap $H$ succeeds and returns the value $m$ and the heap $H'$, then the multiset of elements of $H'$ is the multiset of elements of $H$ with one occurrence of $m$ removed.","title":"Fibonacci heap minimum deletion removes one copy of the minimum"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min-cons-elements","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMin_cons_elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min-cons-elements","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If deleting the minimum of a Fibonacci heap $H$ succeeds and returns the value $m$ and the heap $H'$, then the multiset of elements of $H$ is the multiset of elements of $H'$ with one copy of $m$ added.","title":"Fibonacci heap minimum deletion splits off the minimum"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min-le","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMin_le","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min-le","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If deleting the minimum of a Fibonacci heap $H$ succeeds and returns the value $m$, then $m \\le v$ for every element $v$ of $H$.","title":"Fibonacci heap minimum deletion returns a lower bound"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min-sorted-degrees","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMin_sortedDegrees","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min-sorted-degrees","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If deleting the minimum of a Fibonacci heap succeeds and returns the heap $H'$, then the degrees of the root trees of $H'$, in list order, are strictly increasing; consolidation is what makes them so.","title":"Minimum deletion leaves a Fibonacci heap with strictly increasing root degrees"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/delete-min-length-le-card","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.deleteMin_length_le_card","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#delete-min-length-le-card","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If deleting the minimum of a Fibonacci heap succeeds and returns the heap $H'$, then the number of root trees of $H'$ is at most the number of elements of $H'$, counted with multiplicity.","title":"After minimum deletion a Fibonacci heap has at most as many roots as elements"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/subtree-at","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.subtreeAt","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#subtree-at","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a list $p = (i_1, \\dots, i_n)$ of natural numbers and a Fibonacci tree $t$, the subtree of $t$ at $p$ is found by starting at the root and moving to the child with zero-based index $i_1$, then to its child with index $i_2$, and so on. The result is absent when some index is at least the number of children of the current node. The empty path addresses the whole tree $t$.","title":"Subtree of a Fibonacci tree at a path"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-at","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.cutAt","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-at","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a path $p$ of child indices and a Fibonacci tree $t$, the cut of $t$ at $p$ is either undefined or a pair of an extracted subtree and an optional remainder tree. For the empty path, the extracted subtree is $t$ and there is no remainder. For a path consisting of an index $i$ followed by a path $q$, where $t$ has root $r$ and children $c_0, \\dots, c_{k-1}$, the cut is undefined if $i \\ge k$ or if the cut of\u2026","title":"Cutting a subtree out of a Fibonacci tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-root","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.decreaseRoot","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-root","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a Fibonacci tree $t$ and a key $k$, this is the tree whose root value is $k$ and whose stored degree and list of children are those of $t$.","title":"Replacing the root key of a Fibonacci tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/elements-decrease-root","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.elements_decreaseRoot","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#elements-decrease-root","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a Fibonacci tree $t$ and any key $k$, the multiset of elements represented by the tree obtained from $t$ by replacing its root value with $k$ equals the multiset of elements of $t$ with one occurrence of the root of $t$ removed and $k$ added.","title":"Elements after replacing the root key"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-root-2","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.Valid.decreaseRoot","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-root-2","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $T$ be a valid Fibonacci tree over a preordered type and let $k$ be a key that is at most the root of $T$. Then the tree obtained from $T$ by replacing its root by $k$, keeping its degree and children, is valid: the new root is still no greater than every represented element.","title":"Lowering the root keeps a Fibonacci tree valid"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/subtree-at-2","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.Valid.subtreeAt","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#subtree-at-2","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If a Fibonacci tree $t$ is valid and a path $p$ of child indices addresses a subtree $s$ of $t$, then $s$ is valid.","title":"Subtrees of a valid Fibonacci tree are valid"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-at-subtree-at","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.cutAt_subtreeAt","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-at-subtree-at","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $t$ be a Fibonacci tree and $p$ a path of child indices. If cutting $t$ at $p$ extracts a subtree $s$ (with or without a remainder), then $s$ is the subtree of $t$ at $p$.","title":"A cut extracts the addressed subtree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/subtree-at-cut-at","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.subtreeAt_cutAt","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#subtree-at-cut-at","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"If a path $p$ of child indices addresses a subtree $s$ of a Fibonacci tree $t$, then cutting $t$ at $p$ succeeds and extracts exactly $s$, together with some remainder, which may be absent.","title":"Every addressed subtree can be cut out"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-at-eq-self","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.cutAt_eq_self","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-at-eq-self","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $t$ be a Fibonacci tree and $p$ a path of child indices. If cutting $t$ at $p$ extracts a subtree $s$ and leaves no remainder, then $s = t$.","title":"A cut without remainder extracts the whole tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-at-elements","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.cutAt_elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-at-elements","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $t$ be a Fibonacci tree and $p$ a path of child indices. If cutting $t$ at $p$ extracts a subtree $s$ and leaves a remainder tree $t'$, then the multiset of elements of $t$ is the sum of the multisets of elements of $t'$ and of $s$.","title":"Cutting a subtree splits the elements of a Fibonacci tree"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-at-sub-valid","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.cutAt_sub_valid","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-at-sub-valid","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $t$ be a valid Fibonacci tree over a type with a relation $\\le$, and $p$ a path of child indices. If cutting $t$ at $p$ extracts a subtree $s$ (with or without a remainder), then $s$ is valid.","title":"The subtree extracted by a cut from a valid tree is valid"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-at-remainder-valid","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciTree.cutAt_remainder_valid","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-at-remainder-valid","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $t$ be a valid Fibonacci tree over a type with a relation $\\le$, and $p$ a path of child indices. If cutting $t$ at $p$ extracts a subtree and leaves a remainder tree $t'$, then $t'$ is valid: its children are valid, its stored degree equals its number of children, and its root is $\\le$ every element of $t'$.","title":"The remainder of a cut from a valid tree is valid"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-tree-at","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.cutTreeAt","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-tree-at","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For a forest given as a list of Fibonacci trees, an index $i$ and a path $p$ of child indices, the cut is undefined if the list has no tree at position $i$ (counting from $0$) or if cutting that tree $t$ at $p$ is undefined. Otherwise, if cutting $t$ at $p$ extracts the subtree $s$, the result is $s$ together with the list in which the tree at position $i$ is removed (if the cut of $t$ left no remainder) or\u2026","title":"Cutting a subtree out of a Fibonacci forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-tree-at-elements","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.cutTreeAt_elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-tree-at-elements","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $F$ be a forest of Fibonacci trees, $i$ an index and $p$ a path of child indices. If cutting $F$ at the $i$-th tree and path $p$ extracts a subtree $s$ and leaves the forest $F'$, then the multiset of elements of $F$ is the sum of the multisets of elements of $F'$ and of $s$.","title":"Cutting a subtree splits the elements of a Fibonacci forest"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-tree-at-valid","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.cutTreeAt_valid","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-tree-at-valid","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $F$ be a valid forest of Fibonacci trees over a type with a relation $\\le$, $i$ an index and $p$ a path of child indices. If cutting $F$ at index $i$ and path $p$ extracts a subtree $s$ and leaves the forest $F'$, then $s$ is a valid tree and $F'$ is a valid forest.","title":"Cutting a valid forest leaves valid parts"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/cut-tree-at-of-subtree-at","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciForest.cutTreeAt_of_subtreeAt","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#cut-tree-at-of-subtree-at","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $F$ be a forest of Fibonacci trees whose tree at position $i$ is $t$, and let $p$ be a path of child indices such that the subtree of $t$ at $p$ is $s$. Then cutting $F$ at index $i$ and path $p$ is defined and extracts $s$, together with some remaining forest.","title":"Every addressed subtree of a forest can be cut out"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Given a Fibonacci heap $H$ over a linearly ordered type, a root index $i$, a path $p$ of child indices and a key $k$, decrease-key cuts the subtree $S$ at path $p$ out of the $i$-th tree of the root list of $H$, replaces the root of $S$ by $k$, and puts the resulting tree in front of the remaining forest (the root list with the $i$-th tree replaced by what is left of it after the cut, or removed when $S$ is that\u2026","title":"Decrease-key in a Fibonacci heap"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key-trees-of-cut","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey_trees_of_cut","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key-trees-of-cut","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap, $i$ a root index, $p$ a path of child indices and $k$ a key. Suppose that cutting the subtree at path $p$ out of the $i$-th tree of the root list of $H$ succeeds, extracting the subtree $S$ and leaving the forest $R$, and that $k$ is at most the root of $S$. Then the root list of the heap obtained from $H$ by decreasing the key at root index $i$ and path $p$ to $k$ consists of $S$ with\u2026","title":"Root list after decrease-key"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key-eq-self-of-cut-none","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey_eq_self_of_cut_none","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key-eq-self-of-cut-none","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap, $i$ a root index and $p$ a path of child indices. If cutting the subtree at path $p$ out of the $i$-th tree of the root list of $H$ fails, then for every key $k$, decreasing the key at root index $i$ and path $p$ to $k$ returns $H$ unchanged.","title":"Decrease-key at an unresolved address does nothing"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key-eq-self-of-not-le","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey_eq_self_of_not_le","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key-eq-self-of-not-le","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap, $i$ a root index, $p$ a path of child indices and $k$ a key. Suppose that cutting the subtree at path $p$ out of the $i$-th tree of the root list of $H$ succeeds and extracts the subtree $S$, and that $k$ is not at most the root of $S$ (keys are linearly ordered, so $k$ is greater than that root). Then decreasing the key at root index $i$ and path $p$ to $k$ returns $H$ unchanged.","title":"Decrease-key with a larger key does nothing"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key-elements","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey_elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key-elements","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap, $i$ a root index, $p$ a path of child indices and $k$ a key. Suppose that cutting the subtree at path $p$ out of the $i$-th tree of the root list of $H$ succeeds and extracts the subtree $S$, and that $k$ is at most the root of $S$. Then the multiset represented by the heap obtained from $H$ by decreasing the key at root index $i$ and path $p$ to $k$ is the multiset represented by $H$\u2026","title":"Decrease-key replaces one key"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key-elements-of-subtree-at","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey_elements_of_subtreeAt","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key-elements-of-subtree-at","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap, let $T$ be the $i$-th tree of its root list, and let $S$ be the subtree of $T$ at the path $p$ of child indices. If the key $k$ is at most the root of $S$, then the multiset represented by the heap obtained from $H$ by decreasing the key at root index $i$ and path $p$ to $k$ is the multiset represented by $H$ with one occurrence of the root of $S$ removed and one occurrence of $k$ added.","title":"Decrease-key replaces one key, stated by address"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key-card-elements","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey_card_elements","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key-card-elements","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap, $i$ a root index, $p$ a path of child indices and $k$ a key. Suppose that cutting the subtree at path $p$ out of the $i$-th tree of the root list of $H$ succeeds and extracts the subtree $S$, and that $k$ is at most the root of $S$. Then the multiset represented by the heap obtained from $H$ by decreasing the key at root index $i$ and path $p$ to $k$ has the same number of elements,\u2026","title":"Decrease-key preserves the number of elements"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key-find-min","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey_findMin","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key-find-min","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"Let $H$ be a Fibonacci heap whose minimum, as returned by find-min, is $m$, and let $i$ be a root index, $p$ a path of child indices and $k$ a key. Suppose that cutting the subtree at path $p$ out of the $i$-th tree of the root list of $H$ succeeds and extracts the subtree $S$, and that $k$ is at most the root of $S$. Then find-min on the heap obtained from $H$ by decreasing the key at root index $i$ and path $p$\u2026","title":"Minimum after decrease-key"},{"id":"ComputerScience/data-structures/heaps/fibonacci-heap/decrease-key-valid","kind":"node","lean":"Cslib.Algorithms.Lean.FibonacciHeap.decreaseKey_valid","parent":37113,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/fibonacci-heap/index.html#decrease-key-valid","scope":"ComputerScience/data-structures/heaps/fibonacci-heap","status":"fully_proved","summary":"For every Fibonacci heap $H$, root index $i$, path $p$ of child indices and key $k$, the root list of the heap obtained from $H$ by decreasing the key at root index $i$ and path $p$ to $k$ is a valid forest, that is, every tree in it is valid.","title":"Decrease-key yields a valid forest"},{"data_url":"scopes/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue.json","id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","kind":"scope","navigation":"module","parent":36927,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","title":"Indexed minimum priority queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/indexed-min-queue","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.IndexedMinQueue","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#indexed-min-queue","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $W$ be a linearly ordered type, let $W \\cup \\{\\top\\}$ be $W$ with a new greatest element $\\top$ adjoined, and let $n$ be a natural number. An indexed minimum queue over the identities $0, \\dots, n-1$ consists of a binary min-heap whose entries are pairs $(k, v)$ of a key $k \\in W \\cup \\{\\top\\}$ and an identity $v < n$, ordered lexicographically (first by key, then by identity), together with a position vector\u2026","title":"Indexed minimum queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/empty","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.empty","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#empty","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"For every linearly ordered type $W$ and every natural number $n$, the empty indexed minimum queue over the identities $0, \\dots, n-1$, with keys in $W$ with a new greatest element adjoined, has the empty min-heap and a position vector that assigns nothing to each of the $n$ identities, so no identity is present.","title":"Empty indexed minimum queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/member","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.member","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#member","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Costs are counted as five event counts: comparisons, swaps, reads of the position vector, writes to the position vector, and entry appends, pops or replacements. For an indexed minimum queue $q$ over the identities $0, \\dots, n-1$ and an identity $v < n$, the membership test returns true exactly when the position vector of $q$ assigns an index to $v$, and its cost is one position read and no other event.","title":"Membership test in an indexed minimum queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/insert","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.insert","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#insert","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $W$ be a linearly ordered type and let $W \\cup \\{\\top\\}$ be $W$ with a new greatest element $\\top$ adjoined. Costs are counted as five event counts: comparisons, swaps, reads of the position vector, writes to the position vector, and entry appends, pops or replacements. Given an indexed minimum queue $q$ over the identities $0, \\dots, n-1$ with keys in $W \\cup \\{\\top\\}$, whose heap entries (pairs of a key and\u2026","title":"Insertion into an indexed minimum queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/decrease-key","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.decreaseKey","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#decrease-key","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $W$ be a linearly ordered type and let $W \\cup \\{\\top\\}$ be $W$ with a new greatest element $\\top$ adjoined. Costs are counted as five event counts: comparisons, swaps, reads of the position vector, writes to the position vector, and entry appends, pops or replacements. Given an indexed minimum queue $q$ over the identities $0, \\dots, n-1$ with keys in $W \\cup \\{\\top\\}$, whose heap entries (pairs of a key and\u2026","title":"Decrease-key in an indexed minimum queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/extract-min","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.extractMin","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#extract-min","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Costs are counted as five event counts: comparisons, swaps, reads of the position vector, writes to the position vector, and entry appends, pops or replacements. Given an indexed minimum queue $q$, extract-min fails (returns nothing) exactly when the heap of $q$ is empty, at zero cost. Otherwise let $(k, v)$ be the root entry of the heap. If the heap has one entry, extract-min returns $(k, v)$ together with the\u2026","title":"Extract-min in an indexed minimum queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/position-valid","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.position_valid","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#position-valid","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $W$ be a linearly ordered type and let $q$ be an indexed min-queue over $W$ with identities in $\\{0, \\dots, n-1\\}$. If the position table of $q$ records slot $i$ for an identity $v$, then $i$ is less than the number of entries currently stored in the heap array of $q$.","title":"Recorded positions lie inside the heap"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/position-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.position_eq_none_iff","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#position-eq-none-iff","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$ and let $v$ be an identity. The position vector of $q$ records no slot for $v$ if and only if no entry of the heap array of $q$ has identity $v$.","title":"Absent identities carry no heap entry"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/empty-position","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.empty_position","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#empty-position","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"For every linear order $W$, every $n$ and every identity $v \\in \\{0, \\dots, n-1\\}$, the position vector of the empty indexed min queue over $W$ with identities in $\\{0, \\dots, n-1\\}$ records no slot for $v$.","title":"The empty indexed min queue has no present identity"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/insert-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.insert_eq_none_iff","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#insert-eq-none-iff","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$, let $v$ be an identity and let $k \\in W \\cup \\{\\top\\}$. Inserting $v$ with key $k$ into $q$ returns no queue if and only if the position vector of $q$ records a slot for $v$.","title":"Insertion fails exactly on present identities"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/extract-min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.extractMin_eq_none_iff","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#extract-min-eq-none-iff","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$. Extracting the minimum of $q$ returns no result if and only if the heap array of $q$ has size $0$.","title":"Extraction from an indexed min queue fails exactly when it is empty"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/member-ret","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.member_ret","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#member-ret","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$ and let $v$ be an identity. The membership test for $v$ in $q$ returns true if and only if the position vector of $q$ records a slot for $v$.","title":"Membership reads the position vector"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/member-time","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.member_time","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#member-time","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$ and let $v$ be an identity. The cost of the membership test for $v$ in $q$, counted in five event classes (comparisons, swaps, position-vector reads, position-vector writes, and entry appends, pops or replacements), is exactly one position-vector read and no event of any other class.","title":"Membership costs one position read"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/insert-ret","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.insert_ret","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#insert-ret","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$, let $v$ be an identity and let $k \\in W \\cup \\{\\top\\}$, and suppose that inserting $v$ with key $k$ into $q$ returns the queue $q'$. Then the heap of $q'$ is the binary min-heap push of the entry $(k, v)$ onto the heap of $q$; the position vector of $q'$ records a slot $i$ for $v$, with $i$ less than the size of the heap\u2026","title":"Successful insertion into an indexed min queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/decrease-key-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.decreaseKey_eq_none_iff","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#decrease-key-eq-none-iff","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$, let $v$ be an identity and let $k \\in W$. Decreasing the key of $v$ to $k$ in $q$ returns no queue if and only if either the position vector of $q$ records no slot for $v$, or it records a slot $i$ for $v$ with $i$ less than the size of the heap array of $q$ and $k$ is not strictly less, in $W \\cup \\{\\top\\}$, than the\u2026","title":"Decrease-key fails exactly on absent identities and non-decreases"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/decrease-key-ret","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.decreaseKey_ret","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#decrease-key-ret","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$, let $v$ be an identity and let $k \\in W$, and suppose that decreasing the key of $v$ to $k$ in $q$ returns the queue $q'$. Let $a$ be the heap array of $q$. Then the position vector of $q$ records a slot $i$ for $v$ with $i$ less than the size of $a$, $k$ is strictly less, in $W \\cup \\{\\top\\}$, than the key of the entry\u2026","title":"Successful decrease-key"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/extract-min-ret","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.extractMin_ret","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#extract-min-ret","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed min queue over a linear order $W$ with identities in $\\{0, \\dots, n-1\\}$, and suppose that extracting the minimum of $q$ returns the heap entry $e = (k, u)$, with key $k \\in W \\cup \\{\\top\\}$ and identity $u$, together with the queue $q'$. Then binary min-heap extraction on the heap of $q$ returns $e$ together with the heap of $q'$; the position vector of $q'$ records no slot for $u$; and every\u2026","title":"Successful extraction from an indexed min queue"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/insert-time","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.insert_time","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#insert-time","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $W$ be a linearly ordered type and let $W \\cup \\{\\top\\}$ be $W$ with a new greatest element $\\top$ adjoined. Let $q$ be an indexed minimum queue over the identities $0, \\dots, n-1$ with keys in $W \\cup \\{\\top\\}$ whose heap has $s$ entries, let $v < n$ be an identity and let $k \\in W \\cup \\{\\top\\}$ be a key. For each of the five counted events (comparisons, swaps, position reads, position writes, and entry\u2026","title":"Logarithmic event bound for indexed-queue insertion"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/decrease-key-time","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.decreaseKey_time","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#decrease-key-time","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $W$ be a linearly ordered type and let $W \\cup \\{\\top\\}$ be $W$ with a new greatest element $\\top$ adjoined. Let $q$ be an indexed minimum queue over the identities $0, \\dots, n-1$ with keys in $W \\cup \\{\\top\\}$ whose heap has $s$ entries, let $v < n$ be an identity and let $k \\in W$ be a key. For each of the five counted events (comparisons, swaps, position reads, position writes, and entry appends, pops or\u2026","title":"Logarithmic event bound for indexed-queue decrease-key"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/extract-min-time","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.extractMin_time","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#extract-min-time","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $q$ be an indexed minimum queue whose heap has $s$ entries. For each of the five counted events (comparisons, swaps, position reads, position writes, and entry appends, pops or replacements), the number of times that event occurs when extracting the minimum of $q$ is at most $20 (\\lfloor \\log_2 (s + 1) \\rfloor + 1)$.","title":"Logarithmic event bound for indexed-queue extract-min"},{"id":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/repeated-insert-time","kind":"node","lean":"Cslib.Algorithms.Lean.IndexedMinQueue.repeated_insert_time","parent":37205,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/indexed-minimum-priority-queue/index.html#repeated-insert-time","scope":"ComputerScience/data-structures/heaps/indexed-minimum-priority-queue","status":"fully_proved","summary":"Let $W$ be a linearly ordered type and let $W \\cup \\{\\top\\}$ be $W$ with a new greatest element $\\top$ adjoined. Let $q_0, \\dots, q_n$ be indexed minimum queues over the identities $0, \\dots, n-1$ with keys in $W \\cup \\{\\top\\}$, and let $k_0, \\dots, k_{n-1}$ be keys in $W \\cup \\{\\top\\}$. Suppose $q_0$ is the empty queue and, for each $i < n$, inserting identity $i$ with key $k_i$ into $q_i$ succeeds and returns\u2026","title":"Event bound for building an indexed queue by repeated insertion"},{"data_url":"scopes/ComputerScience/data-structures/heaps/leftist-heap.json","id":"ComputerScience/data-structures/heaps/leftist-heap","kind":"scope","parent":36927,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/index.html","scope":"ComputerScience/data-structures/heaps/leftist-heap","status":"fully_proved","title":"Leftist heap"},{"data_url":"scopes/ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs.json","id":"ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs","kind":"scope","navigation":"module","parent":37227,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs/index.html","scope":"ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs","status":"fully_proved","title":"Bottom-up heap construction costs"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs/fdsa-chap15-ex02","kind":"node","lean":"Cslib.Algorithms.Lean.LeftistHeap.ConstructionCost.fdsa_chap15_ex02","parent":37228,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs/index.html#fdsa-chap15-ex02","scope":"ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs","status":"fully_proved","summary":"One adjacent pass costs at most the number of pairs times the merge bound.","title":"Cost bound for one adjacent merging pass"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs/fdsa-chap15-ex01","kind":"node","lean":"Cslib.Algorithms.Lean.LeftistHeap.ConstructionCost.fdsa_chap15_ex01","parent":37228,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs/index.html#fdsa-chap15-ex01","scope":"ComputerScience/data-structures/heaps/leftist-heap/bottom-up-heap-construction-costs","status":"fully_proved","summary":"The multi-pass cost bound (15.2) for equally sized heaps.","title":"Multi-pass cost bound for equally sized heaps"},{"data_url":"scopes/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin.json","id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","kind":"scope","navigation":"module","parent":37227,"result":false,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","title":"Leftist heaps and tail-recursive adjacent merging"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/lheap","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#lheap","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Leftist heap: Empty or Node rank value left right","title":"Leftist heap"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/size","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.size","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#size","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Size (number of nodes)","title":"Number of nodes of a leftist heap"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/rank","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.rank","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#rank","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Rank (right spine length, or stored rank)","title":"Rank of a leftist heap"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/mset-tree","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.mset_tree","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#mset-tree","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Multiset of elements","title":"Multiset of elements of a leftist heap"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/all-le","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.all_le","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#all-le","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"All elements in heap are \u2265 x (for heap ordering)","title":"Lower bound on all elements of a heap"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/heap","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.heap","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#heap","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Min-heap property: root \u2264 all in subtrees","title":"Min-heap property"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/ltree","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.ltree","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#ltree","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Leftist invariant: stored rank is correct and left rank \u2265 right rank For every node `N storedRank x l r`, require:","title":"Leftist invariant with correct stored ranks"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/make","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.make","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#make","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Helper to restore leftist property. Constructs a node whose stored rank is exactly one plus the rank of its resulting right subtree (b.rank + 1 or a.rank + 1 after swap).","title":"Node constructor restoring the leftist property"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/merge-aux","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.mergeAux","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#merge-aux","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Auxiliary fuel-based merge to avoid termination proof complexity. When reconstructing an existing node during recursion, preserves its original stored rank (does not replace it with 0).","title":"Fuel-based auxiliary merge of leftist heaps"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/merge","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.merge","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#merge","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Leftist merge (public API)","title":"Merge of leftist heaps"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/t-merge-aux","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.T_mergeAux","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#t-merge-aux","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Auxiliary fuel-based cost of merge. Complexity model faithful to textbook convention: only calls/steps of `merge` are counted. Reversal of accumulator in `merge_adj2` and other auxiliary operations are not counted.","title":"Fuel-based auxiliary cost of merging"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/t-merge","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.T_merge","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#t-merge","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Cost of merge: number of merge steps (only merge calls counted)","title":"Cost of merging leftist heaps"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/merge-adj","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.merge_adj","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#merge-adj","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Non-tail-recursive merge_adj","title":"Non-tail-recursive adjacent merging of heaps"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/merge-adj2","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.merge_adj2","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#merge-adj2","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Tail-recursive merge_adj2 with accumulator, no use of List.append. Genuinely tail-recursive: uses an accumulator and reverse at base case. Does not use List.append (avoids append operator). Complexity model: only merge steps counted, so `T_merge_adj2 acc ts = T_merge_adj ts` should hold; accumulator reversal overhead is intentionally excluded per textbook convention that only merge calls are counted.","title":"Tail-recursive adjacent merging with an accumulator"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/t-merge-adj","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.T_merge_adj","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#t-merge-adj","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Cost of merge_adj: only merge steps counted per textbook","title":"Cost of non-tail-recursive adjacent merging"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/t-merge-adj2","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.T_merge_adj2","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#t-merge-adj2","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Cost of merge_adj2 (accumulator version): only merge steps counted. Follows textbook convention that only calls/steps of `merge` are counted, so accumulator handling does not contribute to cost. Hence the required comparison `T_merge_adj2 acc ts = T_merge_adj ts` remains valid.","title":"Cost of tail-recursive adjacent merging"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/merge-all2-aux","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.merge_all2Aux","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#merge-all2-aux","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"merge_all2 uses merge_adj2 (tail-recursive variant)","title":"Fuel-based merging of all heaps with tail-recursive passes"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/singleton","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.singleton","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#singleton","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Singleton heap: stored rank 1 = rank E + 1, leftist","title":"Singleton leftist heap"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/lheap-list2","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.lheap_list2","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#lheap-list2","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"lheap_list2: convert list to leftist heap via merge_all2 (tail-recursive adj). Uses merge_all2 which itself uses merge_adj2.","title":"Leftist heap from a list by tail-recursive merging"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/sum-mset","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.sumMSet","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#sum-mset","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Combined multiset of every heap in a list, including duplicates.","title":"Combined multiset of a list of heaps"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/heap-list","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.HeapList","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#heap-list","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Every heap in the list satisfies the min-heap ordering predicate.","title":"Min-heap ordering of every heap in a list"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/ltree-list","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.LTreeList","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#ltree-list","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Every heap in the list has correct stored ranks and is leftist.","title":"Leftist invariant for every heap in a list"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/fdsa-chap15-ex04","kind":"node","lean":"Cslib.Algorithms.Lean.fdsa_chap15_ex04","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#fdsa-chap15-ex04","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"Graded theorem: correctness of lheap_list2 and complexity equivalence. Complexity model faithful to textbook: only merge calls counted.","title":"Correctness and cost equivalence of tail-recursive heap construction"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/mset-lheap-list2","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.mset_lheap_list2","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#mset-lheap-list2","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"List construction preserves the entire input multiset, including duplicates.","title":"List-to-heap construction preserves the multiset"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/heap-lheap-list2","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.heap_lheap_list2","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#heap-lheap-list2","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"List construction satisfies min-heap ordering.","title":"List-to-heap construction gives a min-heap"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/ltree-lheap-list2","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.ltree_lheap_list2","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#ltree-lheap-list2","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"List construction satisfies both stored-rank correctness and leftist balance.","title":"List-to-heap construction gives a leftist tree"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/lheap-list2-correct","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.lheap_list2_correct","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#lheap-list2-correct","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"A reusable per-input form of the three construction-correctness conjuncts.","title":"Per-input correctness of list-to-heap construction"},{"id":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/merge-adj2-eq-reverse-append","kind":"node","lean":"Cslib.Algorithms.Lean.LHeap.merge_adj2_eq_reverse_append","parent":37231,"result":true,"route":"../roadmap/ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin/index.html#merge-adj2-eq-reverse-append","scope":"ComputerScience/data-structures/heaps/leftist-heap/leftist-heaps-and-tail-recursive-adjacent-mergin","status":"fully_proved","summary":"The accumulator is a reversed prefix; newly merged pairs follow it in input order. This additional output characterization is not required by the original exercise. Append appears only in the specification, not the algorithm.","title":"Accumulator characterization of tail-recursive adjacent merging"},{"data_url":"scopes/ComputerScience/data-structures/hashing.json","id":"ComputerScience/data-structures/hashing","kind":"scope","parent":36418,"result":false,"route":"../roadmap/ComputerScience/data-structures/hashing/index.html","scope":"ComputerScience/data-structures/hashing","status":"fully_proved","summary":"Hash tables (chaining, open addressing, cuckoo, perfect), universal hash families and Bloom filters including their false-positive analysis.","title":"Hashing"},{"data_url":"scopes/ComputerScience/data-structures/hashing/bloom-filter.json","id":"ComputerScience/data-structures/hashing/bloom-filter","kind":"scope","navigation":"module","parent":37260,"result":false,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","title":"Bloom filter"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/bloom-filter","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#bloom-filter","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"A Bloom filter over a type $\\alpha$ with $m$ bits consists of a finite list of hash functions $\\alpha \\to \\{0, \\dots, m - 1\\}$ and a bit vector of length $m$.","title":"Bloom filter"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/empty","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.empty","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#empty","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Let $\\alpha$ be a type and $m$ a natural number. Given a list of hash functions $\\alpha \\to \\{0, \\dots, m - 1\\}$, the empty Bloom filter with these hash functions has all of its $m$ bits unset.","title":"Empty Bloom filter"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/insert","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.insert","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#insert","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Inserting an element $x$ into a Bloom filter keeps its hash functions and sets bit $h(x)$ for every hash function $h$ of the filter, leaving all other bits unchanged.","title":"Bloom filter insertion"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/may-contain","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.mayContain","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#may-contain","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"A Bloom filter reports that an element $x$ may be present when bit $h(x)$ is set for every hash function $h$ of the filter, and reports $x$ absent otherwise.","title":"Bloom filter membership query"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/bit-insert-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.bit_insert_iff","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#bit-insert-iff","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Let $F$ be a Bloom filter over $\\alpha$ with $m$ bits, let $x \\in \\alpha$ and let $i \\in \\{0, \\dots, m - 1\\}$. Bit $i$ of the filter obtained by inserting $x$ into $F$ is set if and only if bit $i$ of $F$ is set or some hash function $h$ of $F$ satisfies $h(x) = i$.","title":"Bits set by an insertion"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/may-contain-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.mayContain_iff","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#may-contain-iff","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"For a Bloom filter $F$ and an element $x$, $F$ reports that $x$ may be present if and only if bit $h(x)$ of $F$ is set for every hash function $h$ of $F$.","title":"Membership query in terms of bits"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/may-contain-insert-self","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.mayContain_insert_self","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#may-contain-insert-self","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"For every Bloom filter $F$ and element $x$, the filter obtained by inserting $x$ into $F$ reports that $x$ may be present.","title":"An inserted element is reported present"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/may-contain-insert-mono","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.mayContain_insert_mono","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#may-contain-insert-mono","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Let $F$ be a Bloom filter and let $x$, $y$ be elements. If $F$ reports that $x$ may be present, then so does the filter obtained by inserting $y$ into $F$.","title":"Insertion preserves positive answers"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/may-contain-empty","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.mayContain_empty","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#may-contain-empty","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Let $\\alpha$ be a type, $m$ a natural number and $H$ a nonempty list of hash functions $\\alpha \\to \\{0, \\dots, m - 1\\}$. For every element $x \\in \\alpha$, the empty Bloom filter with hash functions $H$ reports $x$ absent.","title":"The empty Bloom filter reports every element absent"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/insert-all","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.insertAll","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#insert-all","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Insert every element of a list, left to right.","title":"Inserting a list of elements into a Bloom filter"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/hashes-insert-all","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.hashes_insertAll","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#hashes-insert-all","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Insertion keeps the hash functions.","title":"Bloom filter insertion keeps the hash functions"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/bit-insert-all-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.bit_insertAll_iff","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#bit-insert-all-iff","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"After inserting a list, bit `i` is set iff it was set before or some hash function sends some inserted element to `i`.","title":"Bits set after inserting a list into a Bloom filter"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/may-contain-insert-all-empty-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.mayContain_insertAll_empty_iff","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#may-contain-insert-all-empty-iff","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"The false-positive event: after inserting `xs` into an empty filter, `y` is reported present iff every bit selected by `y` is selected by some stored element.","title":"Characterization of the Bloom filter false-positive event"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/set-bits","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.setBits","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#set-bits","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"The bits set by the stored hash values `H a j` (element `a`, hash function `j`).","title":"Bits set by the stored hash values"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/not-mem-set-bits-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.notMem_setBits_iff","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#not-mem-set-bits-iff","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"A bit is unset iff no stored hash value selects it.","title":"A Bloom filter bit is unset iff no stored hash selects it"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/false-positive-prob","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.falsePositiveProb","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#false-positive-prob","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"The exact false-positive rate under uniform hashing: the fraction of outcomes (stored hash values `H`, query hash values `q`) in which every query bit is set.","title":"Exact Bloom filter false-positive rate under uniform hashing"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/card-unset","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.card_unset","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#card-unset","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Exactly `(m - 1) ^ (n * k)` outcomes of the stored hash values leave bit `i` unset.","title":"Number of hash outcomes leaving a given bit unset"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/prob-unset","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.prob_unset","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#prob-unset","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"A given bit is unset with probability `(1 - 1/m) ^ (n * k)`.","title":"Probability that a given Bloom filter bit is unset"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/expected-fraction-set","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.expected_fraction_set","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#expected-fraction-set","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"The expected fraction of set bits is `1 - (1 - 1/m) ^ (n * k)`.","title":"Expected fraction of set bits in a Bloom filter"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/false-positive-prob-eq","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.falsePositiveProb_eq","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#false-positive-prob-eq","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"The exact false-positive rate is the expectation of `(fraction of set bits) ^ k`.","title":"False-positive rate as the expected k-th power of the set fraction"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/classical-le-false-positive-prob","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.classical_le_falsePositiveProb","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#classical-le-false-positive-prob","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Bloom's formula is a lower bound on the exact false-positive rate (Jensen's inequality for `t \u21a6 t ^ k`).","title":"Bloom's formula is a lower bound"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/exp-approx-le-false-positive-prob","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.exp_approx_le_falsePositiveProb","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#exp-approx-le-false-positive-prob","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"The exponential approximation `(1 - exp (-(k n) / m)) ^ k` is also a lower bound.","title":"Exponential approximation is a lower bound on the false-positive rate"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/log-mul-log-one-sub-le","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.log_mul_log_one_sub_le","parent":37261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound "],["math","\\(\\log t \\cdot \\log(1-t) \\le (\\log 2)^2\\)"],["text"," on the unit interval"]],"source":"Bound $\\log t \\cdot \\log(1-t) \\le (\\log 2)^2$ on the unit interval"},"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#log-mul-log-one-sub-le","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"`log t * log (1 - t) \u2264 (log 2) ^ 2` on `(0, 1)`.","title":"Bound $\\log t \\cdot \\log(1-t) \\le (\\log 2)^2$ on the unit interval"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/log-mul-log-one-sub-eq-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.log_mul_log_one_sub_eq_iff","parent":37261,"reading_title":{"format":"inline-parts-v1","parts":[["text","Equality in the log-product bound holds only at "],["math","\\(t = 1/2\\)"]],"source":"Equality in the log-product bound holds only at $t = 1/2$"},"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#log-mul-log-one-sub-eq-iff","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"Equality holds only at `t = 1 / 2`.","title":"Equality in the log-product bound holds only at $t = 1/2$"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/exp-approx","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.expApprox","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#exp-approx","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"The exponential approximation `(1 - exp (-\u03ba n / m)) ^ \u03ba` of the false-positive rate, as a function of a real number `\u03ba` of hash functions.","title":"Exponential approximation of the Bloom filter false-positive rate"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/exp-approx-at-optimum","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.expApprox_at_optimum","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#exp-approx-at-optimum","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"At `\u03ba = (m / n) log 2` half of the bits are expected to be set and the approximation is `(1 / 2) ^ \u03ba`.","title":"Exponential approximation at the optimal number of hash functions"},{"id":"ComputerScience/data-structures/hashing/bloom-filter/exp-approx-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.BloomFilter.expApprox_optimal","parent":37261,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/bloom-filter/index.html#exp-approx-optimal","scope":"ComputerScience/data-structures/hashing/bloom-filter","status":"fully_proved","summary":"The optimal number of hash functions for the exponential approximation: on real `\u03ba > 0`, `(1 - exp (-\u03ba n / m)) ^ \u03ba` is minimal at `\u03ba = (m / n) log 2`.","title":"The optimal number of hash functions"},{"data_url":"scopes/ComputerScience/data-structures/hashing/cuckoo-hashing.json","id":"ComputerScience/data-structures/hashing/cuckoo-hashing","kind":"scope","navigation":"module","parent":37260,"result":false,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","title":"Cuckoo hashing"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/cuckoo-hashing","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#cuckoo-hashing","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Given a type $K$ of keys, a type $V$ of values, a number $m$ of slots, and two hash functions $h_1, h_2 : K \\to \\{0, \\dots, m - 1\\}$, a cuckoo hash table consists of two tables, each a vector of $m$ slots; every slot is either empty or holds a key\u2013value pair $(k, v)$ with $k \\in K$ and $v \\in V$. The structure itself does not require a pair $(k, v)$ to sit at slot $h_1(k)$ of the first table or at slot $h_2(k)$ of\u2026","title":"Cuckoo hash table"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/empty","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.empty","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#empty","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"The empty cuckoo hash table has every slot of both tables vacant.","title":"Empty cuckoo hash table"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/valid","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.Valid","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#valid","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"A cuckoo hash table with hash functions $h_1, h_2$ satisfies the placement invariant when every pair $(k, v)$ in slot $i$ of the first table has $i = h_1(k)$, and every pair $(k, v)$ in slot $i$ of the second table has $i = h_2(k)$.","title":"Placement invariant of a cuckoo hash table"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/stored","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.Stored","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#stored","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"A key $k$ is stored with value $v$ in a cuckoo hash table when the pair $(k, v)$ occupies some slot of the first table or some slot of the second table.","title":"Binding stored in a cuckoo hash table"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/key-det","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.KeyDet","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#key-det","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"A cuckoo hash table is key-deterministic when each key is stored with at most one value: for every key $k$ and values $v_1, v_2$, if the bindings $k \\mapsto v_1$ and $k \\mapsto v_2$ are both stored in the table, then $v_1 = v_2$. Duplicate copies of the same pair are allowed.","title":"Key determinism of a cuckoo hash table"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.valid_empty","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#valid-empty","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"For any hash functions $h_1, h_2$ into $m$ slots, the empty cuckoo hash table, with every slot of both tables vacant, satisfies the placement invariant.","title":"The empty cuckoo hash table satisfies the placement invariant"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/key-det-empty","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.keyDet_empty","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#key-det-empty","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"No key is stored with two different values in the empty cuckoo hash table, vacuously.","title":"The empty cuckoo hash table is key-deterministic"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/insert-aux","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.insertAux","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#insert-aux","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"The eviction loop takes a cuckoo hash table with hash functions $h_1$ and $h_2$, a key\u2013value pair $(k, v)$, a choice of the first or second table, and a natural number as fuel, and either returns a table or fails. With fuel $0$ it fails. With fuel $n + 1$ and the first table chosen, it writes $(k, v)$ into slot $h_1(k)$ of the first table; if that slot was empty it returns the resulting table, and otherwise it\u2026","title":"Cuckoo eviction loop"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"For a cuckoo hash table $T$ with hash functions $h_1$ and $h_2$ and a key $k$, the lookup of $k$ in $T$ returns $v$ if slot $h_1(k)$ of the first table holds a pair $(k, v)$; otherwise it returns $v$ if slot $h_2(k)$ of the second table holds a pair $(k, v)$; otherwise it returns nothing. No other slot is examined.","title":"Cuckoo hash table lookup"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-empty","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_empty","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-empty","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Looking up any key in the empty cuckoo hash table finds no value.","title":"Lookup in the empty cuckoo hash table"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-eq-some-of-table1","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_eq_some_of_table1","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-eq-some-of-table1","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table with hash functions $h_1$ and $h_2$, let $k$ be a key and $v$ a value. If slot $h_1(k)$ of the first table of $T$ holds the pair $(k, v)$, then looking up $k$ in $T$ returns $v$.","title":"Cuckoo lookup finds a key in its first-table slot"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-eq-some-of-table2","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_eq_some_of_table2","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-eq-some-of-table2","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table with hash functions $h_1$ and $h_2$, let $k$ be a key and $v$ a value. If slot $h_2(k)$ of the second table of $T$ holds the pair $(k, v)$ and slot $h_1(k)$ of the first table of $T$ holds no pair whose key is $k$, then looking up $k$ in $T$ returns $v$.","title":"Cuckoo lookup finds a key in its second-table slot"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-sound","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_sound","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-sound","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table with hash functions $h_1$ and $h_2$, let $k$ be a key and $v$ a value. If looking up $k$ in $T$ returns $v$, then the binding $k \\mapsto v$ is stored in some slot of the first or the second table of $T$.","title":"Soundness of cuckoo lookup"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-is-some-of-stored","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_isSome_of_stored","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-is-some-of-stored","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table with hash functions $h_1$ and $h_2$ that satisfies the placement invariant, and let $k$ be a key and $v$ a value. If the binding $k \\mapsto v$ is stored in some slot of either table of $T$, then looking up $k$ in $T$ returns some value.","title":"Cuckoo lookup finds every stored key"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-eq-some-of-stored","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_eq_some_of_stored","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-eq-some-of-stored","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table that satisfies the placement invariant and is key-deterministic. For every key $k$ and value $v$, if the binding $k \\mapsto v$ is stored in $T$, then looking up $k$ in $T$ returns $v$.","title":"Cuckoo lookup finds every stored binding"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-eq-some-iff-stored","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_eq_some_iff_stored","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-eq-some-iff-stored","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table that satisfies the placement invariant and is key-deterministic. For every key $k$ and value $v$, looking up $k$ in $T$ returns $v$ if and only if the binding $k \\mapsto v$ is stored in $T$.","title":"Cuckoo lookup finds exactly the stored bindings"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/slot-has-key","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.slotHasKey","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#slot-has-key","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"A slot of a cuckoo hash table is either empty or holds a pair $(k, v)$ of a key and a value. For a slot $s$ and a key $k'$, this test is true when $s$ holds a pair whose key is $k'$, and false when $s$ is empty or holds a pair with a different key.","title":"Slot holds a key"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/insert","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.insert","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#insert","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Inserting a key $k$ with value $v$ into a cuckoo hash table with hash functions $h_1$ and $h_2$, with fuel $n$, either returns a table or fails. If slot $h_1(k)$ of the first table holds a binding for $k$, that slot is overwritten with $(k, v)$, and slot $h_2(k)$ of the second table is cleared if it also holds a binding for $k$, so no stale competing value remains. Otherwise, if slot $h_2(k)$ of the second table\u2026","title":"Cuckoo hash insertion"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/stored-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.stored_insert","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#stored-insert","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table with hash functions $h_1$ and $h_2$. If inserting (or replacing) the binding $k \\mapsto v$ in $T$ with fuel $f$ succeeds and returns the table $T'$, then the binding $k \\mapsto v$ is stored in some slot of either table of $T'$.","title":"Cuckoo insertion stores the inserted binding"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/valid-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.valid_insert","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#valid-insert","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table that satisfies the placement invariant. If inserting the binding $k \\mapsto v$ into $T$ (which replaces an existing binding for $k$ in place) with some fuel bound succeeds and returns the table $T'$, then $T'$ satisfies the placement invariant.","title":"Cuckoo insertion preserves the placement invariant"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/key-det-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.keyDet_insert","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#key-det-insert","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table that satisfies the placement invariant and is key-deterministic. If inserting a key $k$ with value $v$ into $T$ with some fuel succeeds with result $T'$, then $T'$ is key-deterministic. In particular, replacement leaves no stale copy of $k$ with a different value in the other slot.","title":"Cuckoo insertion preserves key determinism"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_insert","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-insert","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table with hash functions $h_1$ and $h_2$ that satisfies the placement invariant and stores each key with at most one value. If inserting (or replacing) the binding $k \\mapsto v$ in $T$ with fuel $f$ succeeds and returns the table $T'$, then looking up $k$ in $T'$ returns $v$.","title":"Cuckoo lookup returns an inserted value"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/stored-insert-iff-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.stored_insert_iff_of_ne","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#stored-insert-iff-of-ne","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table with hash functions $h_1$ and $h_2$ that satisfies the placement invariant. Suppose inserting (or replacing) the binding $k \\mapsto v$ in $T$ with fuel $f$ succeeds and returns the table $T'$. Then for every key $k' \\ne k$ and every value $v'$, the binding $k' \\mapsto v'$ is stored in $T'$ if and only if it is stored in $T$.","title":"Cuckoo insertion preserves the bindings of other keys"},{"id":"ComputerScience/data-structures/hashing/cuckoo-hashing/lookup-insert-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.CuckooHashing.lookup_insert_of_ne","parent":37289,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/cuckoo-hashing/index.html#lookup-insert-of-ne","scope":"ComputerScience/data-structures/hashing/cuckoo-hashing","status":"fully_proved","summary":"Let $T$ be a cuckoo hash table with hash functions $h_1$ and $h_2$ that satisfies the placement invariant and stores each key with at most one value. If inserting (or replacing) the binding $k \\mapsto v$ in $T$ with fuel $f$ succeeds and returns the table $T'$, then for every key $k' \\ne k$, looking up $k'$ in $T'$ gives the same result as looking up $k'$ in $T$: either both return nothing or both return the same\u2026","title":"Cuckoo insertion leaves lookups of other keys unchanged"},{"data_url":"scopes/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining.json","id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","kind":"scope","navigation":"module","parent":37260,"result":false,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","title":"Hash table with separate chaining"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/hash-table","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#hash-table","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For a key type $\\alpha$, a value type $\\beta$ and a hash function $h : \\alpha \\to \\mathbb{N}$, a hash table with hash function $h$ consists of a positive number $m$ of buckets and, for each bucket index $i < m$, a chain stored at bucket $i$: an association list of key\u2013value pairs whose keys are pairwise distinct. The structure itself does not require keys to lie in their hash-selected bucket.","title":"Hash table with separate chaining"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/empty","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.empty","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#empty","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For a hash function $h$ and a positive integer $m$, the empty hash table with hash function $h$ has $m$ buckets, each holding the empty chain.","title":"Empty hash table"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/lookup","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.lookup","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#lookup","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For a hash table with hash function $h$ and $m$ buckets, looking up a key $k$ searches only the bucket with index $h(k) \\bmod m$: the result is the value bound to $k$ in that bucket's association list, or nothing if $k$ has no binding there.","title":"Hash table lookup"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/insert","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.insert","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#insert","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"Inserting a key $k$ with value $v$ into a hash table with $m$ buckets and hash function $h$ keeps the bucket count and every bucket other than bucket $h(k) \\bmod m$, and replaces the chain at bucket $h(k) \\bmod m$ by that chain with $k$ bound to $v$: any earlier binding of $k$ in that chain is removed and the binding of $k$ to $v$ is added.","title":"Insertion into a hash table"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/valid","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.Valid","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#valid","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"A hash table with hash function $h$ and $m$ buckets is valid when every binding lies in its hash-selected bucket: for every bucket index $i < m$ and every key $k$, if $k$ has a binding in bucket $i$, then $h(k) \\bmod m = i$.","title":"Valid hash table"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/abstraction","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.abstraction","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#abstraction","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"The finite map represented by a hash table with $m$ buckets is built from the list of all bindings of buckets $0, 1, \\dots, m - 1$, concatenated in this order: it binds exactly the keys occurring in this list, and binds each such key to the value of its first binding in the list.","title":"Finite map represented by a hash table"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.valid_empty","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#valid-empty","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For every hash function $h$ and every positive bucket count $m$, the empty hash table with hash function $h$ and $m$ buckets is valid.","title":"The empty hash table is valid"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/lookup-empty","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.lookup_empty","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#lookup-empty","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For every hash function $h$, every positive bucket count $m$ and every key $k$, looking up $k$ in the empty hash table with hash function $h$ and $m$ buckets finds no value.","title":"Lookup in the empty hash table"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/abstraction-empty","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.abstraction_empty","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#abstraction-empty","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For every hash function $h$ and every positive bucket count $m$, the finite map represented by the empty hash table with hash function $h$ and $m$ buckets is the empty finite map.","title":"The empty hash table represents the empty map"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/lookup-insert","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.lookup_insert","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#lookup-insert","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For every hash table $T$, every key $k$ and every value $v$, looking up $k$ in the table obtained by inserting $k$ with value $v$ into $T$ returns $v$.","title":"Hash table lookup after insertion"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/lookup-insert-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.lookup_insert_of_ne","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#lookup-insert-of-ne","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For every hash table $T$, all keys $k \\ne k'$ and every value $v$, looking up $k'$ in the table obtained by inserting $k$ with value $v$ into $T$ gives the same result as looking up $k'$ in $T$, also when $k$ and $k'$ lie in the same bucket.","title":"Insertion does not affect other keys"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/valid-insert","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.valid_insert","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#valid-insert","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"If a hash table is valid, then the table obtained from it by inserting a key $k$ with value $v$ (adding a new binding for $k$, or replacing the existing one) is again valid.","title":"Insertion preserves validity of a hash table"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/lookup-abstraction","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.lookup_abstraction","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#lookup-abstraction","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For every valid hash table $T$ and every key $k$, looking up $k$ in the finite map represented by $T$ gives the same result as looking up $k$ in $T$.","title":"Lookup agrees with the abstraction"},{"id":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/abstraction-insert","kind":"node","lean":"Cslib.Algorithms.Lean.HashTable.abstraction_insert","parent":37314,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/hash-table-with-separate-chaining/index.html#abstraction-insert","scope":"ComputerScience/data-structures/hashing/hash-table-with-separate-chaining","status":"fully_proved","summary":"For every valid hash table $T$, every key $k$ and every value $v$, the finite map represented by the table obtained by inserting $k$ with value $v$ into $T$ equals the finite map represented by $T$ with $k$ bound to $v$, replacing any earlier binding of $k$.","title":"Insertion commutes with the abstraction"},{"data_url":"scopes/ComputerScience/data-structures/hashing/open-addressing.json","id":"ComputerScience/data-structures/hashing/open-addressing","kind":"scope","navigation":"module","parent":37260,"result":false,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","title":"Open addressing"},{"id":"ComputerScience/data-structures/hashing/open-addressing/open-addressing","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#open-addressing","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $K$ be a type of keys, $V$ a type of values, $m$ a natural number, and $p$ a probe sequence assigning to each key $k$ and each $i \\in \\mathbb{N}$ a slot $p(k, i) \\in \\{0, \\dots, m - 1\\}$. An open-addressing hash table with $m$ slots and probe sequence $p$ consists of $m$ slots, each either empty or holding a single pair $(k, v)$ with $k \\in K$ and $v \\in V$.","title":"Open-addressing hash table"},{"id":"ComputerScience/data-structures/hashing/open-addressing/linear-probe","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.linearProbe","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#linear-probe","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Given a hash function $h : K \\to \\mathbb{N}$ and a number of slots $m > 0$, the $i$-th probe of a key $k$ is the slot $(h(k) + i) \\bmod m$ in $\\{0, \\dots, m - 1\\}$.","title":"Linear-probing probe sequence"},{"id":"ComputerScience/data-structures/hashing/open-addressing/double-hash-probe","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.doubleHashProbe","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#double-hash-probe","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Given hash functions $h_1, h_2 : K \\to \\mathbb{N}$ and a number of slots $m > 0$, the $i$-th probe of a key $k$ is the slot $(h_1(k) + i \\cdot h_2(k)) \\bmod m$ in $\\{0, \\dots, m - 1\\}$.","title":"Double-hashing probe sequence"},{"id":"ComputerScience/data-structures/hashing/open-addressing/empty","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.empty","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#empty","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"The open-addressing table with $m$ slots and a given probe sequence in which every slot is empty.","title":"Empty open-addressing table"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup-aux","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookupAux","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup-aux","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Given a table with $m$ slots and probe sequence $p$, a key $k$, a starting index $s$ and a budget $f$ of probes, examine the slots $p(k, s), p(k, s + 1), \\dots$ in order, for at most $f$ probes. The scan returns the value bound to $k$ in the first examined slot that holds a binding for $k$, and returns no value if it first reaches an empty slot or exhausts the budget.","title":"Bounded lookup scan"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookup","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be an open-addressing hash table with $m$ slots and probe sequence $p$, and let $k$ be a key. The lookup of $k$ in $T$ inspects the slots $p(k, 0), p(k, 1), \\dots, p(k, m - 1)$ in this order. It returns the value $v$ of the first inspected slot that holds a pair $(k, v)$, and returns nothing if an empty slot is inspected first or if none of the $m$ inspected slots holds the key $k$.","title":"Lookup in an open-addressing table"},{"id":"ComputerScience/data-structures/hashing/open-addressing/insert-aux","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.insertAux","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#insert-aux","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Given a table with $m$ slots and probe sequence $p$, a key $k$, a value $v$, a starting index $s$ and a budget $f$ of probes, examine the slots $p(k, s), p(k, s + 1), \\dots$ in order, for at most $f$ probes. At the first examined slot that is empty or holds a binding for $k$, the scan returns the table with that slot overwritten by the binding $k \\mapsto v$ and every other slot unchanged; if all $f$ examined slots\u2026","title":"Bounded insertion scan"},{"id":"ComputerScience/data-structures/hashing/open-addressing/insert","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.insert","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#insert","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"To insert the binding $k \\mapsto v$ into a table with $m$ slots and probe sequence $p$, examine the slots $p(k, 0), p(k, 1), \\dots, p(k, m - 1)$ in order. At the first one that is empty or already holds a binding for $k$, write $k \\mapsto v$ there, leaving every other slot unchanged, and return the updated table. If all $m$ examined slots hold bindings for other keys, the insertion fails and returns no table.","title":"Insertion into an open-addressing table"},{"id":"ComputerScience/data-structures/hashing/open-addressing/valid","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.Valid","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#valid","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be an open-addressing table with $m$ slots and probe sequence $p$, where $p(k, i) \\in \\{0, \\dots, m - 1\\}$ is the slot visited by the $i$-th probe of key $k$. Then $T$ is valid when it satisfies two conditions. Probe-closedness: whenever slot $j$ holds a binding $k \\mapsto v$, there is an $i < m$ with $p(k, i) = j$ such that the slots $p(k, i')$ are occupied for all $i' < i$; this is what lets a lookup stop\u2026","title":"Valid open-addressing tables"},{"id":"ComputerScience/data-structures/hashing/open-addressing/stored","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.Stored","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#stored","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"An open-addressing hash table $T$ stores the binding $k \\mapsto v$ when some slot of $T$ holds the pair $(k, v)$.","title":"Stored binding in an open-addressing table"},{"id":"ComputerScience/data-structures/hashing/open-addressing/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.valid_empty","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#valid-empty","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"For all types $K$ and $V$, every number of slots $m$ and every probe sequence $p$, which assigns to each key $k$ and each $i \\in \\mathbb{N}$ a slot $p(k, i) \\in \\{0, \\dots, m - 1\\}$, the empty table with $m$ slots and probe sequence $p$, in which every slot is vacant, is valid: it is probe-closed and no key is stored in two different slots.","title":"The empty open-addressing table is valid"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup-empty","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookup_empty","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup-empty","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"For every key $k$, looking up $k$ in the empty open-addressing table returns no value.","title":"Lookup in the empty open-addressing table"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup-insert","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookup_insert","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup-insert","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"If inserting the binding $k \\mapsto v$ into an open-addressing hash table $T$ succeeds and yields the table $T'$, then looking up $k$ in $T'$ returns $v$. No validity assumption on $T$ is needed.","title":"Lookup after a successful open-addressing insertion"},{"id":"ComputerScience/data-structures/hashing/open-addressing/valid-insert","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.valid_insert","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#valid-insert","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be a valid table with $m$ slots and probe sequence $p$, whose keys have decidable equality. If inserting the binding $k \\mapsto v$ into $T$ (adding it, or replacing the binding already stored for $k$) succeeds and returns the table $T'$, then $T'$ is valid: it is probe-closed and no key is stored in two different slots.","title":"Insertion preserves validity of an open-addressing table"},{"id":"ComputerScience/data-structures/hashing/open-addressing/stored-of-lookup-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.stored_of_lookup_eq_some","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#stored-of-lookup-eq-some","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"If looking up a key $k$ in an open-addressing hash table $T$ returns the value $v$, then $T$ stores the binding $k \\mapsto v$. No validity assumption is needed.","title":"Open-addressing lookup returns only stored bindings"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup-eq-none-of-not-stored","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookup_eq_none_of_not_stored","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup-eq-none-of-not-stored","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be an open-addressing table and $k$ a key such that no slot of $T$ stores a binding $k \\mapsto v$ for any value $v$. Then looking up $k$ in $T$ returns no value. No validity assumption on $T$ is needed.","title":"Lookup of an absent key fails"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup-is-some-of-stored","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookup_isSome_of_stored","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup-is-some-of-stored","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"If a valid open-addressing hash table $T$ stores the binding $k \\mapsto v$, then looking up $k$ in $T$ returns some value.","title":"Valid open-addressing tables find every stored key"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup-is-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookup_isSome_iff","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup-is-some-iff","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be a valid open-addressing hash table and $k$ a key. Looking up $k$ in $T$ returns a value if and only if $T$ stores a binding $k \\mapsto v$ for some value $v$.","title":"Open-addressing lookup succeeds exactly on stored keys"},{"id":"ComputerScience/data-structures/hashing/open-addressing/stored-unique","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.stored_unique","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#stored-unique","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"If a valid open-addressing hash table $T$ stores both $k \\mapsto v_1$ and $k \\mapsto v_2$, then $v_1 = v_2$.","title":"Bindings in a valid open-addressing table are unique"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookup_eq_some_iff","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup-eq-some-iff","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be a valid open-addressing table, $k$ a key and $v$ a value. Then looking up $k$ in $T$ returns $v$ if and only if the binding $k \\mapsto v$ is stored in some slot of $T$.","title":"Lookup in a valid table returns exactly the stored binding"},{"id":"ComputerScience/data-structures/hashing/open-addressing/stored-insert-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.stored_insert_of_ne","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#stored-insert-of-ne","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be an open-addressing hash table, and suppose that inserting the binding $k \\mapsto v$ into $T$ succeeds and yields the table $T'$. Then for every key $k' \\neq k$ and every value $v'$, the table $T'$ stores $k' \\mapsto v'$ if and only if $T$ stores $k' \\mapsto v'$.","title":"Open-addressing insertion leaves other keys' bindings unchanged"},{"id":"ComputerScience/data-structures/hashing/open-addressing/lookup-insert-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.lookup_insert_of_ne","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#lookup-insert-of-ne","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be a valid open-addressing table and suppose inserting the binding $k \\mapsto v$ into $T$ succeeds with result $T'$. Then for every key $k' \\ne k$, looking up $k'$ in $T'$ gives the same result as looking up $k'$ in $T$.","title":"Insertion does not affect lookups of other keys"},{"id":"ComputerScience/data-structures/hashing/open-addressing/occupied-of-insert-eq-none","kind":"node","lean":"Cslib.Algorithms.Lean.OpenAddressing.occupied_of_insert_eq_none","parent":37329,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/open-addressing/index.html#occupied-of-insert-eq-none","scope":"ComputerScience/data-structures/hashing/open-addressing","status":"fully_proved","summary":"Let $T$ be an open-addressing hash table with $m$ slots and probe sequence $p$. If inserting a binding $k \\mapsto v$ into $T$ fails, then for every $i < m$ the slot $p(k, i)$ holds a pair $(k', v')$ with $k' \\neq k$.","title":"Failed open-addressing insertion finds every probed slot taken"},{"data_url":"scopes/ComputerScience/data-structures/hashing/perfect-hashing.json","id":"ComputerScience/data-structures/hashing/perfect-hashing","kind":"scope","navigation":"module","parent":37260,"result":false,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","title":"Perfect hashing"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/perfect-hashing","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#perfect-hashing","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"Fix a type $\\alpha$ of keys, a type $\\beta$ of values and a primary hash function $h : \\alpha \\to \\mathbb{N}$. A two-level (FKS) perfect hash table consists of a number $b > 0$ of buckets and, for each bucket $i < b$, a secondary size $m_i \\in \\mathbb{N}$, a secondary hash function $h_i : \\alpha \\to \\mathbb{N}$ and a secondary table of $m_i$ slots, each of which is empty or holds one binding $(k, v)$ with\u2026","title":"Two-level perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/lookup","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.lookup","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#lookup","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"To look up a key $k$ in a two-level perfect hash table with $b$ buckets, primary hash $h$, secondary sizes $m_i$ and secondary hashes $h_i$, let $i = h(k) \\bmod b$. If $m_i = 0$, the lookup returns nothing. Otherwise it inspects the single slot $h_i(k) \\bmod m_i$ of the secondary table of bucket $i$: if that slot holds a binding $(k, v)$, the lookup returns $v$; if the slot is empty or holds a binding with a\u2026","title":"Lookup in a perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/empty","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.empty","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#empty","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"Given a primary hash function $h$ from keys to natural numbers, a number of buckets $b > 0$, and for each bucket $i < b$ a secondary table size $m_i$ and a secondary hash function $h_i$, the empty table is the two-level perfect hash table with primary hash $h$, $b$ buckets, secondary sizes $m_i$ and secondary hashes $h_i$ in which every slot of every secondary table is vacant.","title":"Empty perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/stored","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.Stored","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#stored","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"A key $k$ and a value $v$ are stored in a two-level perfect hash table $T$ when some secondary slot, in some bucket of $T$, holds the binding $(k, v)$.","title":"Bindings stored in a perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/entries","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.entries","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#entries","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"The entries of a two-level perfect hash table form the list of the bindings $(k, v)$ held in its occupied secondary slots, listed bucket by bucket in increasing bucket order and, within each bucket, in increasing slot order.","title":"Entries of a perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/valid","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.Valid","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#valid","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"Let $T$ be a two-level perfect hash table with primary hash $h$, $b$ buckets, secondary table sizes $m_i$ and secondary hashes $g_i$. Then $T$ is valid when two conditions hold. Placement: whenever slot $j$ of bucket $i$ holds a binding $(k, v)$, we have $i = h(k) \\bmod b$ and $j = g_i(k) \\bmod m_i$. Injectivity: for every bucket $i$, if keys $k_1$ and $k_2$ both occur in bindings stored in bucket $i$ and\u2026","title":"Valid perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/injective-of-placed","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.Valid.injective_of_placed","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#injective-of-placed","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"Let $T$ be a two-level perfect hash table with $b$ buckets and primary hash $h$, and for each bucket $i$ let $m_i$ be its secondary table size and $h_i$ its secondary hash. Suppose every binding $(k, v)$ held in slot $j$ of bucket $i$ satisfies $h(k) \\bmod b = i$ and $j = h_i(k) \\bmod m_i$. Then for every bucket $i$ and all keys $k_1, k_2$ that are each held, with some value, in some slot of bucket $i$, if\u2026","title":"Placement implies secondary injectivity"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/mk-of-placed","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.Valid.mk_of_placed","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#mk-of-placed","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"Let $T$ be a two-level perfect hash table with $b$ buckets, primary hash $h$, secondary sizes $m_i$ and secondary hashes $h_i$. If, whenever slot $j$ of the secondary table of bucket $i$ holds a binding $(k, v)$, we have $h(k) \\bmod b = i$ and $j = h_i(k) \\bmod m_i$, then $T$ is valid.","title":"Placement suffices for validity of a perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/mem-entries-iff","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.mem_entries_iff","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#mem-entries-iff","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For every two-level perfect hash table, every key $k$ and every value $v$, the pair $(k, v)$ belongs to the list of entries of the table (the contents of its occupied secondary slots, bucket by bucket) if and only if some secondary slot of the table holds the binding $(k, v)$.","title":"Entries of a perfect hash table are its stored bindings"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.valid_empty","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#valid-empty","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For every primary hash function, every positive number of buckets, and every choice of secondary table sizes and secondary hash functions, the empty two-level perfect hash table with these parameters is valid: no slot is occupied, so both conditions of validity hold vacuously.","title":"The empty perfect hash table is valid"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/not-stored-empty","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.not_stored_empty","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#not-stored-empty","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For every primary hash function, every number $b > 0$ of buckets, every choice of secondary sizes and secondary hash functions, every key $k$ and every value $v$, no secondary slot of the empty two-level perfect hash table with these data holds the binding $(k, v)$.","title":"The empty perfect hash table stores nothing"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/stored-of-lookup-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.stored_of_lookup_eq_some","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#stored-of-lookup-eq-some","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For a two-level perfect hash table with decidable equality on keys, and with no validity hypothesis, if looking up a key $k$ returns a value $v$, then some secondary slot of the table holds the binding $(k, v)$.","title":"Soundness of perfect-hash lookup"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/lookup-eq-some-of-stored","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.lookup_eq_some_of_stored","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#lookup-eq-some-of-stored","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For a valid two-level perfect hash table, if the binding $(k, v)$ is held in some slot of the table, then looking up $k$ returns $v$.","title":"Lookup finds every stored binding"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/stored-unique","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.stored_unique","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#stored-unique","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"In a valid two-level perfect hash table, if some secondary slots hold bindings $(k, v_1)$ and $(k, v_2)$ with the same key $k$, then $v_1 = v_2$.","title":"Stored values in a valid perfect hash table are unique"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/lookup-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.lookup_eq_none_iff","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#lookup-eq-none-iff","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For a valid two-level perfect hash table and any key $k$, looking up $k$ returns no value if and only if there is no value $v$ such that the binding $(k, v)$ is held in some slot of the table.","title":"Lookup fails exactly on unstored keys"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/lookup-eq-some-of-mem-entries","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.lookup_eq_some_of_mem_entries","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#lookup-eq-some-of-mem-entries","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For a valid two-level perfect hash table, if the pair $(k, v)$ belongs to the list of entries of the table, then looking up $k$ returns $v$.","title":"Lookup finds every entry"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/mem-entries-of-lookup-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.mem_entries_of_lookup_eq_some","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#mem-entries-of-lookup-eq-some","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For a two-level perfect hash table with decidable equality on keys, if looking up a key $k$ returns a value $v$, then $(k, v)$ belongs to the list of entries of the table. No validity hypothesis is needed.","title":"Successful perfect-hash lookups return entries"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/lookup-empty","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.lookup_empty","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#lookup-empty","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For every primary hash function, every number of buckets $b > 0$, all secondary table sizes and secondary hash functions, and every key $k$, looking up $k$ in the empty table returns no value.","title":"Lookup in the empty perfect hash table fails"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/entries-empty","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.entries_empty","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#entries-empty","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For every primary hash function, every number of buckets $b > 0$, and all secondary table sizes and secondary hash functions, the list of entries of the empty table is the empty list.","title":"The empty perfect hash table has no entries"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/bucket-size","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.bucketSize","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#bucket-size","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For a two-level perfect hash table and a bucket $i$, the occupancy $n_i$ of bucket $i$ is the number of slots of the secondary table of bucket $i$ that hold a binding. It is the $n_i$ of the Fredman\u2013Koml\u00f3s\u2013Szemer\u00e9di analysis.","title":"Occupancy of a perfect-hashing bucket"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/total-keys","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.totalKeys","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#total-keys","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For a two-level perfect hash table, write $n_i$ for the number of occupied secondary slots of bucket $i$. The total number of keys of the table is $\\sum_i n_i$, the number of occupied secondary slots over all buckets.","title":"Total number of keys of a perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/total-secondary-space","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.totalSecondarySpace","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#total-secondary-space","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For a two-level perfect hash table, write $m_i$ for the number of slots of the secondary table of bucket $i$. The total secondary space of the table is $\\sum_i m_i$, the number of secondary slots over all buckets.","title":"Total secondary space of a perfect hash table"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/square-sized","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.SquareSized","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#square-sized","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"A two-level perfect hash table with $b$ buckets is square-sized when, for every bucket $i < b$, the size $m_i$ of its secondary table equals $n_i^2$, where $n_i$ is the number of occupied slots of that secondary table.","title":"FKS square sizing rule"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/bucket-size-le-secondary-size","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.bucketSize_le_secondarySize","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#bucket-size-le-secondary-size","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"For every two-level perfect hash table and every bucket $i$, the number $n_i$ of occupied slots of bucket $i$ is at most the number $m_i$ of slots of the secondary table of bucket $i$.","title":"Bucket occupancy is at most the secondary table size"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/secondary-size-le-total-secondary-space","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.secondarySize_le_totalSecondarySpace","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#secondary-size-le-total-secondary-space","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"In a two-level perfect hash table with $b$ buckets and secondary sizes $m_0, \\dots, m_{b-1}$, every bucket $i < b$ satisfies $m_i \\le \\sum_{j < b} m_j$, the total secondary space.","title":"Each secondary table fits in the total secondary space"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/total-secondary-space-eq-sum-sq","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.totalSecondarySpace_eq_sum_sq","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#total-secondary-space-eq-sum-sq","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"Let $T$ be a two-level perfect hash table satisfying the FKS sizing rule, and write $n_i$ for the number of occupied secondary slots of bucket $i$. Then the total secondary space of $T$ equals $\\sum_i n_i^2$, the sum over all buckets $i$.","title":"Total secondary space under square sizing"},{"id":"ComputerScience/data-structures/hashing/perfect-hashing/total-secondary-space-le-total-keys-sq","kind":"node","lean":"Cslib.Algorithms.Lean.PerfectHashing.totalSecondarySpace_le_totalKeys_sq","parent":37353,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/perfect-hashing/index.html#total-secondary-space-le-total-keys-sq","scope":"ComputerScience/data-structures/hashing/perfect-hashing","status":"fully_proved","summary":"If a two-level perfect hash table satisfies the FKS sizing rule, then its total secondary space is at most the square of its total number of keys.","title":"Quadratic space bound under square sizing"},{"data_url":"scopes/ComputerScience/data-structures/hashing/universal-hashing.json","id":"ComputerScience/data-structures/hashing/universal-hashing","kind":"scope","navigation":"module","parent":37260,"result":false,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","title":"Universal hashing"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/collision-count","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.collisionCount","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#collision-count","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For a finite family $H$ of functions from a set $\\alpha$ to $\\{0, 1, \\dots, m - 1\\}$ and keys $x, y \\in \\alpha$, the collision count of $x$ and $y$ in $H$ is the number of functions $h \\in H$ with $h(x) = h(y)$.","title":"Collision count"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/is-universal","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.IsUniversal","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#is-universal","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"A finite family $H$ of functions from a set $\\alpha$ to $\\{0, 1, \\dots, m - 1\\}$ is universal if $m > 0$, $H$ is nonempty, and for all distinct keys $x \\ne y$ in $\\alpha$ the number of functions $h \\in H$ with $h(x) = h(y)$, multiplied by $m$, is at most $|H|$; that is, $x$ and $y$ collide under at most a $1/m$ fraction of the members of $H$.","title":"Universal hash family"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/is-universal-on","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.IsUniversalOn","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#is-universal-on","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $s$ be a finite set of indices from a type $\\iota$, and let $h$ assign to each index $i \\in \\iota$ a function $h_i$ from a set $\\alpha$ to $\\{0, 1, \\dots, m - 1\\}$. The family $h$ is universal on $s$ if $m > 0$, $s$ is nonempty, and for all distinct keys $x \\ne y$ in $\\alpha$ the number of indices $i \\in s$ with $h_i(x) = h_i(y)$, multiplied by $m$, is at most $|s|$.","title":"Indexed universal hash family"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/collision-count-le-card","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.collisionCount_le_card","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#collision-count-le-card","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For every finite family $H$ of functions from a set $\\alpha$ to $\\{0, 1, \\dots, m - 1\\}$ and all keys $x, y \\in \\alpha$, the number of functions $h \\in H$ with $h(x) = h(y)$ is at most $|H|$.","title":"Collision count is at most the family size"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/collision-count-self","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.collisionCount_self","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#collision-count-self","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For every finite family $H$ of functions from a set $\\alpha$ to $\\{0, 1, \\dots, m - 1\\}$ and every key $x \\in \\alpha$, the collision count of $x$ with itself in $H$ equals $|H|$: $x$ collides with itself under every function in $H$.","title":"A key collides with itself under the whole family"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/not-is-universal-empty","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.not_isUniversal_empty","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#not-is-universal-empty","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For every type of keys $\\alpha$ and every natural number $m$, the empty family of functions $\\alpha \\to \\{0, \\dots, m - 1\\}$ is not universal: universality includes nonemptiness, since its meaning is a bound on the collision probability for a uniformly chosen member.","title":"The empty hash family is not universal"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/not-is-universal-on-empty","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.not_isUniversalOn_empty","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#not-is-universal-on-empty","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For every index type $\\iota$, every type of keys $\\alpha$, every natural number $m$ and every family $(h_i)_{i \\in \\iota}$ of functions $h_i \\colon \\alpha \\to \\{0, \\dots, m - 1\\}$, the family is not universal over the empty index set: indexed universality includes nonemptiness of the index set, since there is no uniform random choice from an empty family.","title":"The empty indexed family is not universal"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/is-universal-bound-empty","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.isUniversal_bound_empty","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#is-universal-bound-empty","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For every natural number $m$ and all distinct keys $x \\ne y$ in a set $\\alpha$, the collision count of $x$ and $y$ in the empty family of functions from $\\alpha$ to $\\{0, 1, \\dots, m - 1\\}$, multiplied by $m$, is at most the size of the empty family, which is $0$.","title":"The empty family satisfies the collision bound"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-hash","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.modPrimeHash","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-hash","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For natural numbers $p$ and $m$ with $m > 0$ and elements $a, b$ of $\\mathbb{Z}/p\\mathbb{Z}$, the mod-prime hash function $h_{a,b} \\colon \\mathbb{Z}/p\\mathbb{Z} \\to \\{0, \\dots, m - 1\\}$ is $h_{a,b}(x) = \\operatorname{val}(ax + b) \\bmod m$, where $ax + b$ is computed in $\\mathbb{Z}/p\\mathbb{Z}$ and $\\operatorname{val}(z)$ is the canonical natural-number representative of $z$ (the residue in $\\{0, \\dots, p - 1\\}$\u2026","title":"Mod-prime hash function"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-index","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.ModPrimeIndex","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-index","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For a natural number $p$, the index set of the mod-prime family is the set of pairs $(a, b)$ with $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\neq 0$.","title":"Mod-prime index set"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-hashes","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.modPrimeHashes","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-hashes","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For natural numbers $p$ and $m$ with $m > 0$, the indexed mod-prime family assigns to each index $(a, b)$, with $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\neq 0$, the mod-prime hash function $h_{a,b} \\colon \\mathbb{Z}/p\\mathbb{Z} \\to \\{0, \\dots, m - 1\\}$.","title":"Indexed mod-prime family"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-family","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.modPrimeFamily","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-family","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For $p > 0$ and $m > 0$, the mod-prime family is the finite set of all functions $h_{a,b}$ from $\\mathbb{Z}/p\\mathbb{Z}$ to $\\{0, 1, \\dots, m - 1\\}$ with $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\ne 0$, where $h_{a,b}$ is the mod-prime hash function with index $(a, b)$.","title":"Mod-prime family of distinct functions"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-hash-val","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.modPrimeHash_val","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-hash-val","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For natural numbers $p$ and $m$ with $m > 0$ and elements $a, b, x$ of $\\mathbb{Z}/p\\mathbb{Z}$, the mod-prime hash value $h_{a,b}(x) \\in \\{0, \\dots, m - 1\\}$, read as a natural number, equals $\\operatorname{val}(ax + b) \\bmod m$. Here $ax + b$ is computed in $\\mathbb{Z}/p\\mathbb{Z}$, and $\\operatorname{val}(z)$ is the canonical natural-number representative of $z$: the residue in $\\{0, \\dots, p - 1\\}$ when\u2026","title":"Value of the mod-prime hash"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-hash-eq-iff","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.modPrimeHash_eq_iff","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-hash-eq-iff","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a natural number, $m > 0$ and $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$, and write $\\operatorname{val}(z)$ for the canonical natural-number representative of $z \\in \\mathbb{Z}/p\\mathbb{Z}$ (for $p = 0$, where $\\mathbb{Z}/0\\mathbb{Z} = \\mathbb{Z}$, its absolute value), so that the mod-prime hash function is $h_{a,b}(x) = \\operatorname{val}(ax + b) \\bmod m$. For all keys $x, y \\in \\mathbb{Z}/p\\mathbb{Z}$, we have\u2026","title":"Collisions of a mod-prime hash function"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/pair-map","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.pairMap","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#pair-map","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a natural number and fix two keys $x, y \\in \\mathbb{Z}/p\\mathbb{Z}$. The pair map at $x$ and $y$ sends an index $(a, b)$ of the mod-prime family, that is, $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ with $a \\ne 0$, to the pair $(ax + b, ay + b) \\in (\\mathbb{Z}/p\\mathbb{Z})^2$ of inner hash values at $x$ and $y$.","title":"The pair map of the mod-prime family"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/pair-map-apply","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.pairMap_apply","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#pair-map-apply","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a natural number, let $x, y \\in \\mathbb{Z}/p\\mathbb{Z}$, and let $(a, b)$ be an index of the mod-prime family, so that $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\ne 0$. Then the pair map at $x$ and $y$ sends $(a, b)$ to $(ax + b, ay + b)$.","title":"Evaluating the pair map"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/pair-map-fst-ne-snd","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.pairMap_fst_ne_snd","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#pair-map-fst-ne-snd","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a prime and let $x \\ne y$ be keys in the field $\\mathbb{Z}/p\\mathbb{Z}$. For every index $(a, b)$ of the mod-prime family, that is, $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ with $a \\ne 0$, we have $ax + b \\ne ay + b$.","title":"The two values of the pair map differ"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/pair-equiv","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.pairEquiv","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#pair-equiv","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a prime and let $x \\neq y$ be elements of the field $\\mathbb{Z}/p\\mathbb{Z}$. The pair bijection for $x, y$ is the bijection from the indices $(a, b)$, with $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\neq 0$, to the ordered pairs $(r, s)$ of elements of $\\mathbb{Z}/p\\mathbb{Z}$ with $r \\neq s$ that sends $(a, b)$ to the pair map value $(ax + b, ay + b)$. Its inverse solves the linear system $ax + b = r$,\u2026","title":"Pair bijection of the CLRS counting argument"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/pair-equiv-apply-fst","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.pairEquiv_apply_fst","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#pair-equiv-apply-fst","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a prime and let $x \\neq y$ be elements of $\\mathbb{Z}/p\\mathbb{Z}$. For every index $(a, b)$ with $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\neq 0$, the image of $(a, b)$ under the pair bijection for $x, y$, viewed as an element of $\\mathbb{Z}/p\\mathbb{Z} \\times \\mathbb{Z}/p\\mathbb{Z}$, is the value of the pair map at $(a, b)$, namely $(ax + b, ay + b)$.","title":"The pair bijection is given by the pair map"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/pair-map-injective","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.pairMap_injective","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#pair-map-injective","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a prime and let $x \\ne y$ be keys in $\\mathbb{Z}/p\\mathbb{Z}$. Then the pair map at $x$ and $y$, which sends an index $(a, b)$ of the mod-prime family (so $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\ne 0$) to $(ax + b, ay + b)$, is injective.","title":"The pair map is injective"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-collision-mul-le","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.modPrime_collision_mul_le","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-collision-mul-le","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a prime and $m > 0$, and for $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ with $a \\ne 0$ let $h_{a,b}(x) = ((ax + b) \\bmod p) \\bmod m$ be the mod-prime hash function. For distinct keys $x \\ne y$ in $\\mathbb{Z}/p\\mathbb{Z}$, the number of index pairs $(a, b)$ with $a \\ne 0$ and $h_{a,b}(x) = h_{a,b}(y)$, multiplied by $m$, is at most $p(p - 1)$, the total number of index pairs.","title":"Collision bound for the mod-prime hash functions"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-index-card","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.modPrimeIndex_card","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-index-card","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"For every positive natural number $p$, the set of pairs $(a, b)$ of elements of $\\mathbb{Z}/p\\mathbb{Z}$ with $a \\neq 0$ has exactly $p(p - 1)$ elements.","title":"Number of mod-prime indices"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mod-prime-is-universal-on","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.modPrime_isUniversalOn","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mod-prime-is-universal-on","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p$ be a prime and $m$ a positive natural number. The indexed mod-prime family, whose member at the index $(a, b)$, with $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\neq 0$, is the mod-prime hash function $h_{a,b}$, is universal over the set of all such indices: this set is nonempty, and for all distinct keys $x \\neq y$ in $\\mathbb{Z}/p\\mathbb{Z}$, the number of indices $(a, b)$ with $h_{a,b}(x) = h_{a,b}(y)$,\u2026","title":"Universality of the mod-prime family"},{"id":"ComputerScience/data-structures/hashing/universal-hashing/mem-mod-prime-family","kind":"node","lean":"Cslib.Algorithms.Lean.UniversalHashing.mem_modPrimeFamily","parent":37381,"result":true,"route":"../roadmap/ComputerScience/data-structures/hashing/universal-hashing/index.html#mem-mod-prime-family","scope":"ComputerScience/data-structures/hashing/universal-hashing","status":"fully_proved","summary":"Let $p > 0$ and $m > 0$. A function $h$ from $\\mathbb{Z}/p\\mathbb{Z}$ to $\\{0, 1, \\dots, m - 1\\}$ belongs to the mod-prime family of distinct functions exactly when $h = h_{a,b}$ for some index $(a, b)$ with $a, b \\in \\mathbb{Z}/p\\mathbb{Z}$ and $a \\ne 0$, where $h_{a,b}$ is the mod-prime hash function with index $(a, b)$.","title":"Members of the mod-prime family"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures.json","id":"ComputerScience/data-structures/linear-structures","kind":"scope","parent":36418,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/index.html","scope":"ComputerScience/data-structures/linear-structures","status":"fully_proved","summary":"Sequential containers and their amortized costs: linked lists (singly, doubly, sentinel), stacks with multipop, queues, deques, dynamic tables, binary counters, LRU caches, and generic list operations (ordered insertion, splitting, chunking, inclusive prefixes) that support them.","title":"Lists, stacks, queues and dynamic arrays"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/binary-counter.json","id":"ComputerScience/data-structures/linear-structures/binary-counter","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","title":"Binary counter"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/increment","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.increment","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#increment","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"A $k$-bit counter is a vector $b = (b_0, \\dots, b_{k-1})$ of bits stored lowest bit first. Incrementing $b$ changes the initial run of $1$ bits $b_0, \\dots, b_{t-1}$ to $0$ and then changes the first $0$ bit $b_t$, if there is one, to $1$; if every bit is $1$, all bits become $0$ and no new bit is added. The result is again a $k$-bit counter, and the cost of the increment is the number of bits changed, one unit per\u2026","title":"Binary counter increment"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/increment-value","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.increment_value","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#increment-value","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"Let $b = (b_0, \\dots, b_{k-1})$ be a $k$-bit counter stored lowest bit first, with bits $b_i \\in \\{0, 1\\}$ and unsigned value $v(b) = \\sum_{i < k} b_i 2^i$. If $b'$ is the counter returned by incrementing $b$, then $v(b') = (v(b) + 1) \\bmod 2^k$.","title":"Increment adds one with wraparound"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/increment-time","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.increment_time","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#increment-time","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"Let $b = (b_0, \\dots, b_{k-1})$ be a $k$-bit counter stored lowest bit first, and let $t$ be the length of its initial run of $1$ bits, that is, the largest $t \\le k$ with $b_0 = \\dots = b_{t-1} = 1$. The cost of incrementing $b$ is $t + 1$ if $t < k$, and $t$ if $t = k$.","title":"Exact cost of an increment"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/increment-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.increment_time_le","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#increment-time-le","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every $k$-bit counter $b$ (a vector of $k$ bits, lowest bit first), the cost of incrementing $b$ is at most $k$.","title":"Increment costs at most the width"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For a width $k$, a $k$-bit counter $b$ and $n \\in \\mathbb{N}$, the run of $n$ increments from $b$ is defined recursively. The run of $0$ increments returns $b$ at cost zero. The run of $n + 1$ increments performs the run of $n$ increments, then increments the counter it returned and returns the result; its cost is the cost of the $n$-step run plus the cost of that increment.","title":"Running the binary counter"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run-zero","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run_zero","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run-zero","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every width $k$ and every $k$-bit counter $b$, running zero increments from $b$ returns $b$ unchanged at cost zero.","title":"Running zero increments"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run-ret-succ","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run_ret_succ","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run-ret-succ","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every bit vector $b$ and every natural number $n$, the bit vector returned by running $n + 1$ increments from $b$ is the increment of the bit vector returned by running $n$ increments from $b$.","title":"One more increment"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run-time-succ","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run_time_succ","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run-time-succ","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every width $k$, every $k$-bit counter $b$ and every $n$, the cost of running $n + 1$ successive increments from $b$ equals the cost of running $n$ increments from $b$ plus the cost of incrementing the counter that the $n$-step run returns.","title":"Cost recurrence for the binary counter run"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run-value","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run_value","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run-value","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every width $k$, every $k$-bit counter $b$ and every $n$, let $v$ be the value of $b$ read as an unsigned binary number with the least significant bit first. After $n$ successive increments starting from $b$, the counter read the same way has value $(v + n) \\bmod 2^k$.","title":"Counter value after successive increments"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run-bit-flips","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run_bit_flips","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run-bit-flips","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"Let $k$, $i$ and $n$ be natural numbers with $i < k$, and for each $t$ let $c_t$ be the bit vector returned by running $t$ increments from the all-false bit vector of width $k$. Then the number of $t < n$ such that the bit at position $i$ of $c_t$ (counting from the low-order bit, starting at $0$) differs from the bit at position $i$ of $c_{t+1}$ is $\\lfloor n / 2^i \\rfloor$.","title":"Flip counts of the individual bits"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/increment-time-replicate-true","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.increment_time_replicate_true","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#increment-time-replicate-true","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every $k$, the cost of incrementing the $k$-bit counter whose $k$ bits are all $1$ is exactly $k$.","title":"Incrementing the all-ones counter costs the full width"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run-time-eq","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run_time_eq","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run-time-eq","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For all natural numbers $k$ and $n$, running $n$ increments from the all-false bit vector of width $k$ costs exactly $\\sum_{i=0}^{k-1} \\lfloor n / 2^i \\rfloor$.","title":"Exact cost of increments from zero"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run-time-lt","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run_time_lt","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run-time-lt","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every width $k$ and every $n > 0$, performing $n$ successive increments on the $k$-bit counter that starts with all bits zero costs strictly fewer than $2n$ bit changes in total.","title":"Aggregate cost bound for the binary counter"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/run-time-ge","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.run_time_ge","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#run-time-ge","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For all natural numbers $k$ and $n$ with $k > 0$, running $n$ increments from the all-false bit vector of width $k$ costs at least $n$.","title":"Lower bound on the cost of increments from zero"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"Resetting a bit vector $b$ of width $k$ returns the all-false bit vector of width $k$ and charges cost $1$ for each position of $b$ that holds true, that is, for each bit it changes from true to false.","title":"Resetting a binary counter"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-to-list","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_toList","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-to-list","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every bit vector $b$ of width $k$, the bits returned by resetting $b$ are, in order, $k$ copies of false.","title":"Reset returns the all-false counter"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-value","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_value","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-value","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every bit vector $b$ of width $k$, the bits returned by resetting $b$, read low-order bit first as an unsigned binary numeral, have value $0$.","title":"Reset yields the value zero"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-time","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_time","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-time","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every bit vector $b$ of width $k$, the cost of resetting $b$ equals the number of positions of $b$ that hold true.","title":"Cost of resetting a binary counter"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-time-changes","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_time_changes","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-time-changes","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"Let $b = (b_0, \\dots, b_{k-1})$ be a $k$-bit counter and $b' = (b'_0, \\dots, b'_{k-1})$ the counter returned by resetting $b$. The cost of resetting $b$ equals the number of indices $i < k$ with $b_i \\ne b'_i$.","title":"Reset cost is the Hamming distance"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_time_le","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-time-le","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every $k$-bit counter $b$ (a vector of $k$ bits), the cost of resetting $b$ is at most $k$.","title":"Reset costs at most the width"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-time-zero-width","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_time_zero_width","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-time-zero-width","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"Resetting a bit vector of width $0$ has cost $0$.","title":"Resetting a width-zero counter costs nothing"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-time-replicate-false","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_time_replicate_false","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-time-replicate-false","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every width $k$, resetting the all-false bit vector of width $k$ has cost $0$.","title":"Resetting the all-false counter costs nothing"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-ret-idempotent","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_ret_idempotent","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-ret-idempotent","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every $k$-bit counter $b$, resetting the counter returned by resetting $b$ returns the same counter as resetting $b$ once.","title":"Reset is idempotent"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-time-after-reset","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_time_after_reset","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-time-after-reset","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every $k$-bit counter $b$, resetting the counter returned by resetting $b$ has cost $0$.","title":"A second reset costs nothing"},{"id":"ComputerScience/data-structures/linear-structures/binary-counter/reset-potential","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryCounter.reset_potential","parent":37407,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/binary-counter/index.html#reset-potential","scope":"ComputerScience/data-structures/linear-structures/binary-counter","status":"fully_proved","summary":"For every $k$-bit counter $b$ (a vector of $k$ bits), the cost of resetting $b$ plus the number of $1$ bits in the returned counter equals the number of $1$ bits in $b$.","title":"Potential identity for reset"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/double-ended-queue.json","id":"ComputerScience/data-structures/linear-structures/double-ended-queue","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","title":"Double-ended queue"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/deque","kind":"node","lean":"Cslib.Algorithms.Lean.Deque","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#deque","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"A deque with elements in a type $\\alpha$ consists of two lists of elements of $\\alpha$, the front list and the back list. The front list holds the front part of the sequence in order and the back list holds the back part in reverse order.","title":"Two-list deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/contents","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.contents","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#contents","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"The sequence represented by a deque with front list $f$ and back list $b$ is $f$ followed by the reverse of $b$; it lists the elements from front to back.","title":"Represented sequence of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/empty","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.empty","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#empty","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"The empty deque is the deque whose front list and back list are both empty.","title":"The empty deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/is-empty","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.isEmpty","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#is-empty","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"The emptiness test of a deque returns true exactly when both its front list and its back list are empty.","title":"Emptiness test for a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/is-empty-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.isEmpty_eq_true_iff","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#is-empty-eq-true-iff","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"For every deque $d$, the emptiness test of $d$ returns true if and only if the sequence represented by $d$ is empty.","title":"Correctness of the deque emptiness test"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/push-front","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.pushFront","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#push-front","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"Let $d$ be a deque with front list $f$ and back list $b$, where $b$ is stored in reverse order. Pushing an element $x$ at the front of $d$ gives the deque with front list consisting of $x$ followed by the entries of $f$, and back list $b$.","title":"Push at the front of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/contents-push-front","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.contents_pushFront","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#contents-push-front","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"For every deque $d$ and element $x$, the sequence represented by the deque obtained by pushing $x$ at the front of $d$ is $x$ followed by the sequence represented by $d$.","title":"Pushing at the front of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/push-back","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.pushBack","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#push-back","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"Let $d$ be a deque with front list $f$ and back list $b$, where $b$ is stored in reverse order. Pushing an element $x$ at the back of $d$ gives the deque with front list $f$ and back list consisting of $x$ followed by the entries of $b$.","title":"Push at the back of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/contents-push-back","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.contents_pushBack","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#contents-push-back","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"For every deque $d$ and element $x$, the sequence represented by the deque obtained by pushing $x$ at the back of $d$ is the sequence represented by $d$ followed by $x$.","title":"Pushing at the back of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/peek-front","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.peekFront","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#peek-front","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"Let $d$ be a deque with front list $f$ and back list $b$, where $b$ is stored in reverse order. Peeking at the front of $d$ returns, without removing it, the first element of $f$ when $f$ is nonempty, and otherwise the last element of $b$; when both lists are empty it returns no element.","title":"Peek at the front of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/peek-front-eq-head","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.peekFront_eq_head","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#peek-front-eq-head","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"For every deque $d$, peeking at the front of $d$ returns the first element of the sequence represented by $d$, and returns no element when that sequence is empty.","title":"Peeking at the front gives the first element"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/peek-back","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.peekBack","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#peek-back","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"Let $d$ be a deque with front list $f$ and back list $b$, where $b$ is stored in reverse order. Peeking at the back of $d$ returns, without removing it, the first element of $b$ when $b$ is nonempty, and otherwise the last element of $f$; when both lists are empty it returns no element.","title":"Peek at the back of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/peek-back-eq-get-last","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.peekBack_eq_getLast","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#peek-back-eq-get-last","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"For every deque $d$, peeking at the back of $d$ returns the last element of the sequence represented by $d$ when that sequence is nonempty, and returns nothing when it is empty.","title":"Peeking at the back of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/pop-front","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.popFront","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#pop-front","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"Let $d$ be a deque with front list $f$ and back list $b$, where $b$ is stored in reverse order. If $f$ has first element $v$ and remaining list $f'$, popping the front of $d$ returns $v$ together with the deque with front list $f'$ and back list $b$. If $f$ is empty and the reverse of $b$ has first element $v$ and remaining list $r$, it returns $v$ together with the deque with front list $r$ and empty back list. If\u2026","title":"Pop the front of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/pop-front-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.popFront_eq_none_iff","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#pop-front-eq-none-iff","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"For every deque $d$, popping the front of $d$ returns no result if and only if the sequence represented by $d$ is empty.","title":"Popping the front fails exactly on empty deques"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/contents-eq-cons-of-pop-front-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.contents_eq_cons_of_popFront_eq_some","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#contents-eq-cons-of-pop-front-eq-some","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"If popping the front of a deque $d$ succeeds and returns an element $x$ and a deque $d'$, then the sequence represented by $d$ is $x$ followed by the sequence represented by $d'$.","title":"Popping the front of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/pop-back","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.popBack","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#pop-back","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"Let $d$ be a deque with front list $f$ and back list $b$, where $b$ is stored in reverse order. If $b$ has first element $v$ and remaining list $b'$, popping the back of $d$ returns $v$ together with the deque with front list $f$ and back list $b'$. If $b$ is empty and the reverse of $f$ has first element $v$ and remaining list $r$, it returns $v$ together with the deque with empty front list and back list $r$. If\u2026","title":"Pop the back of a deque"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/pop-back-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.popBack_eq_none_iff","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#pop-back-eq-none-iff","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"For every deque $d$, popping the back of $d$ returns no result if and only if the sequence represented by $d$ is empty.","title":"Popping the back fails exactly on empty deques"},{"id":"ComputerScience/data-structures/linear-structures/double-ended-queue/contents-eq-append-of-pop-back-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.Deque.contents_eq_append_of_popBack_eq_some","parent":37433,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/double-ended-queue/index.html#contents-eq-append-of-pop-back-eq-some","scope":"ComputerScience/data-structures/linear-structures/double-ended-queue","status":"fully_proved","summary":"If popping the back of a deque $d$ succeeds and returns an element $x$ and a deque $d'$, then the sequence represented by $d$ is the sequence represented by $d'$ followed by $x$.","title":"Popping the back of a deque"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/doubly-linked-list.json","id":"ComputerScience/data-structures/linear-structures/doubly-linked-list","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","title":"Doubly linked list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/node","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.Node","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#node","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"A doubly linked list node with payloads in $\\alpha$ and identities in $\\iota$ consists of a payload in $\\alpha$ (the key together with any satellite data), a next pointer and a previous pointer, each of which is either an identity in $\\iota$ or NIL.","title":"Doubly linked list node"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/represents","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.Represents","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#represents","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let a store assign to each identity below $n$ either nothing or a doubly linked list node whose pointers are identities below $n$ or NIL, and let the head be an identity below $n$ or NIL. The store and head represent the list of identities $i_0, \\dots, i_{k-1}$ when these identities are pairwise distinct, the head is $i_0$ (and NIL when $k = 0$), and for each $j < k$ the store entry at $i_j$ is a node whose\u2026","title":"Store representing a doubly linked list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let a store assign to each identity below $n$ either nothing or a doubly linked list node whose pointers are identities below $n$ or NIL, let the head be an identity below $n$ or NIL, and let $x$ be an identity whose store entry is a node, where the store entry of the head is also a node when the head is not NIL. The procedure sets the next pointer of $x$ to the head, then sets the previous pointer of $x$ to NIL,\u2026","title":"Prepending a node to a doubly linked list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-head","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_head","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-head","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Then the head returned by List Prepend on $s$, $\\mathrm{head}$ and $x$ is $x$.","title":"Prepending makes the supplied node the head"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-time","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_time","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-time","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let a store assign to each identity below $n$ either nothing or a doubly linked list node, let the head be an identity below $n$ or NIL, and let $x$ be an identity whose store entry is a node, where the store entry of the head is also a node when the head is not NIL. Then prepending $x$ costs exactly $5$ time units if the head is not NIL and exactly $4$ if it is NIL, counting one unit per pointer assignment and per\u2026","title":"Exact cost of doubly linked list prepend"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-ret","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_ret","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-ret","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Let $u$ be $s$ with the cell at $x$ replaced by the node $s(x)$ with next pointer $\\mathrm{head}$ and previous pointer NIL. Then List Prepend on\u2026","title":"Result of doubly-linked list prepend"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-get","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_get","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-get","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Let $s'$ be the store returned by List Prepend on $s$, $\\mathrm{head}$ and $x$. For every identity $z$, let $c$ be the node $s(x)$ with next\u2026","title":"Cells after doubly-linked list prepend"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-frame","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_frame","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-frame","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Let $s'$ be the store returned by List Prepend on $s$, $\\mathrm{head}$ and $x$. Then $s'(z) = s(z)$ for every identity $z$ with $z \\ne x$ and\u2026","title":"Doubly-linked list prepend leaves other cells unchanged"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-allocated","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_allocated","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-allocated","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Let $s'$ be the store returned by List Prepend on $s$, $\\mathrm{head}$ and $x$. Then for every identity $z$, $s'(z)$ is a node if and only if\u2026","title":"Doubly-linked list prepend preserves allocation"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-payload","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_payload","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-payload","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Let $s'$ be the store returned by List Prepend on $s$, $\\mathrm{head}$ and $x$. Then for every identity $z$, either $s(z)$ and $s'(z)$ are both\u2026","title":"Doubly-linked list prepend preserves payloads"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-node","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_node","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-node","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Suppose moreover that $\\mathrm{head}$ is not $x$ (it may be NIL). Let $s'$ be the store returned by List Prepend on $s$, $\\mathrm{head}$ and $x$.\u2026","title":"Pointers of the node prepended to a doubly-linked list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-old-head","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_oldHead","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-old-head","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Suppose moreover that $\\mathrm{head}$ is an identity $h$ with $h \\ne x$. Let $s'$ be the store returned by List Prepend on $s$, $\\mathrm{head}$\u2026","title":"Old head after doubly-linked list prepend"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-represents","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_represents","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-represents","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Suppose moreover that $s$ with head $\\mathrm{head}$ represents a list $L = (\\ell_1, \\dots, \\ell_k)$ of identities, that is, the $\\ell_i$ are\u2026","title":"Correctness of doubly-linked list prepend"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-disjoint-represents","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_disjoint_represents","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-disjoint-represents","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity $i < n$ either a node or nothing, let $\\mathrm{head}$ be NIL or an identity, and let $x$ be an identity such that $s(x)$ is a node and, when $\\mathrm{head}$ is an identity, $s(\\mathrm{head})$ is a node. Suppose moreover that $s$ with some head $o$ represents a list $L = (\\ell_1, \\dots, \\ell_k)$ of identities, that is, the $\\ell_i$ are distinct,\u2026","title":"Doubly-linked list prepend preserves disjoint lists"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-prepend-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listPrepend_time_le","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-prepend-time-le","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let a store assign to each identity below $n$ either nothing or a doubly linked list node, let the head be an identity below $n$ or NIL, and let $x$ be an identity whose store entry is a node, where the store entry of the head is also a node when the head is not NIL. Then prepending $x$ costs at most $5$ time units, counting one unit per pointer assignment and per NIL test of the old head.","title":"Uniform cost bound for doubly linked list prepend"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let the store be an array of cells, each either empty or holding a node with a payload and optional next and previous pointers to cells, let $h$ be the optional head pointer, and let $x$ be a cell holding a node such that each cell named by its previous or next pointer also holds a node. Following CLRS `LIST-DELETE`, deleting $x$ returns a new store and head as follows. If the node at $x$ has a previous pointer\u2026","title":"Deletion from a doubly linked list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-time","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_time","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-time","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let the store be an array of cells, each either empty or holding a node with optional next and previous pointers, let $h$ be the optional head pointer, and let $x$ be a cell holding a node such that each cell named by its previous or next pointer also holds a node. Counting one unit per NIL test and per executed pointer or head assignment, deleting $x$ from the list with head $h$ costs exactly $4$ units if the node\u2026","title":"Running time of doubly linked list deletion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-head","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_head","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-head","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $h'$ be the header returned by\u2026","title":"Header after doubly linked list deletion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-ret","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_ret","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-ret","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $T$ be $S$ if\u2026","title":"Result of doubly linked list deletion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-get","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_get","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-get","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $S'$ be the store returned by\u2026","title":"Cellwise result of doubly linked list deletion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-frame","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_frame","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-frame","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $S'$ be the store returned by\u2026","title":"Deletion changes only the cells of the neighbors"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-allocated","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_allocated","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-allocated","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $S'$ be the store returned by\u2026","title":"Deletion from a doubly linked list allocates and frees nothing"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-payload","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_payload","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-payload","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $S'$ be the store returned by\u2026","title":"Deletion preserves payloads"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-deleted-record","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_deleted_record","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-deleted-record","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $S'$ be the store returned by\u2026","title":"The deleted node keeps its record"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-disjoint-represents","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_disjoint_represents","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-disjoint-represents","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $S'$ be the store returned by\u2026","title":"Deletion preserves lists avoiding the deleted node's neighbors"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_time_le","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-time-le","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Then List Delete on $S$, $h$ and\u2026","title":"Doubly linked list deletion costs at most four events"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-delete-represents","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listDelete_represents","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-delete-represents","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node (a payload with next and previous pointers, each an index or NIL), let $h$ be a header (an index in $\\{0, \\dots, n-1\\}$ or NIL), and let $x$ be an index whose cell holds a node $X$ such that the cells at the predecessor $X.\\mathrm{prev}$ and the successor $X.\\mathrm{next}$ hold nodes whenever these pointers are not NIL. Let $S'$ and $h'$ be the store and\u2026","title":"Correctness of doubly linked list deletion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"A store is an array of $n$ cells, each either empty or holding a node with a payload and optional next and previous pointers to cells. Given allocated cells $x$ and $y$ such that the successor of $y$, if any, is allocated, inserting $x$ after $y$ performs the five lines of LIST-INSERT in order, each reading the store as updated so far: set the next pointer of $x$ to the next pointer of $y$; set the previous pointer\u2026","title":"Doubly-linked-list insertion (LIST-INSERT)"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert-time","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert_time","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert-time","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"For a store $S$ and allocated cells $x$ and $y$ such that the successor of $y$ in $S$, if any, is allocated, the cost of inserting $x$ after $y$, counting one unit per pointer assignment and per NIL test, is $5$ if $y$ has a successor in $S$ and $4$ if the next pointer of $y$ in $S$ is NIL.","title":"Cost of doubly-linked-list insertion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert-ret","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert_ret","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert-ret","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"For a store $S$ and allocated cells $x$ and $y$ such that the successor of $y$ in $S$, if any, is allocated, let $s$ be the next pointer of $y$ in $S$. Let $S_1$ be $S$ with cell $x$ replaced by the node of $x$ with next pointer $s$ and previous pointer $y$. Let $S_2 = S_1$ if $s$ is NIL; if $s$ is a cell $z$, let $S_2$ be $S_1$ with the previous pointer of the node in cell $z$ of $S_1$ replaced by $x$. Then the\u2026","title":"Result of doubly-linked-list insertion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert-get","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert_get","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert-get","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"For a store $S$, allocated cells $x$ and $y$ such that the successor of $y$ in $S$, if any, is allocated, and any cell $z$, let $s$ be the next pointer of $y$ in $S$. Let $c_1$ be the node of $x$ in $S$ with next pointer replaced by $s$ and previous pointer replaced by $y$ if $z = x$, and the content of cell $z$ in $S$ otherwise. Let $c_2$ be $c_1$ with previous pointer replaced by $x$ if $s = z$, and $c_1$\u2026","title":"Cells after doubly-linked-list insertion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert-frame","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert_frame","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert-frame","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"For a store $S$ and allocated cells $x$ and $y$ such that the successor of $y$ in $S$, if any, is allocated, let $z$ be a cell with $z \\ne x$, $z \\ne y$, and $z$ not the successor of $y$ in $S$. Then cell $z$ of the store returned by inserting $x$ after $y$ equals cell $z$ of $S$.","title":"Frame property of doubly-linked-list insertion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert-allocated","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert_allocated","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert-allocated","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"For a store $S$ and allocated cells $x$ and $y$ such that the successor of $y$ in $S$, if any, is allocated, every cell $z$ of the store returned by inserting $x$ after $y$ is allocated if and only if cell $z$ of $S$ is allocated. No distinctness of $x$, $y$ and the successor of $y$ is assumed.","title":"Doubly-linked-list insertion neither allocates nor frees cells"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert-payload","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert_payload","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert-payload","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"For a store $S$ and allocated cells $x$ and $y$ such that the successor of $y$ in $S$, if any, is allocated, every cell $z$ of the store returned by inserting $x$ after $y$ is empty exactly when cell $z$ of $S$ is empty, and otherwise holds a node with the same payload as the node in cell $z$ of $S$.","title":"Doubly-linked-list insertion preserves payloads"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert-represents","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert_represents","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert-represents","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Suppose the store $S$ with head $h$ represents the list of distinct cells $(i_0, \\dots, i_{m-1})$. Let $k < m$, let $x$ be an allocated cell not among $i_0, \\dots, i_{m-1}$, and assume that cell $i_k$ and its successor in $S$, if any, are allocated. Then the store returned by inserting $x$ after $i_k$, with the same head $h$, represents the list $(i_0, \\dots, i_k, x, i_{k+1}, \\dots, i_{m-1})$.","title":"Insertion into a represented doubly-linked list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-insert-disjoint-represents","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listInsert_disjoint_represents","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-insert-disjoint-represents","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Suppose the store $S$ with head $h$ represents the list of cells $(i_0, \\dots, i_{m-1})$ and with head $h'$ represents a list $L'$ of cells, and that the two lists have no cell in common. Let $k < m$, let $x$ be an allocated cell not in $L'$, and assume that cell $i_k$ and its successor in $S$, if any, are allocated. Then the store returned by inserting $x$ after $i_k$, with head $h'$, still represents $L'$.","title":"Doubly-linked-list insertion preserves disjoint lists"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/circular-represents","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.CircularRepresents","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#circular-represents","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ slots indexed by $\\{0, \\dots, n-1\\}$, each either empty or holding a node with a payload and optional next and previous indices. The store $S$ represents a circular list with sentinel $s$ and data nodes $x_1, \\dots, x_k$ (a list of indices) when the $x_i$ are pairwise distinct and all differ from $s$; slot $s$ holds a node whose next index is $x_1$ and whose previous index is $x_k$ (both\u2026","title":"Circular doubly linked list with a sentinel"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/circular-represents-nil","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.CircularRepresents_nil","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#circular-represents-nil","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ slots indexed by $\\{0, \\dots, n-1\\}$, each either empty or holding a node with a payload and optional next and previous indices, and let $s$ be an index. Then $S$ represents the empty circular list with sentinel $s$ if and only if slot $s$ holds a node whose next index and previous index are both $s$.","title":"Empty circular list with a sentinel"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store with $c$ cells, each either empty or holding a node (a payload, an optional next address and an optional previous address in $\\{0, \\dots, c-1\\}$). Let $x$ be an address whose cell holds a node $\\nu$ with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. Sentinel deletion of $x$ from $S$ is the timed computation that first sets the next address of the\u2026","title":"Deletion from a circular list with a sentinel"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-time","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_time","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-time","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. Then sentinel deletion of $x$ from $S$ has cost exactly $2$, one for each pointer assignment.","title":"Sentinel deletion takes two pointer assignments"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-ret","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_ret","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-ret","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. Let $S_1$ be $S$ with the node at $p$ replaced by a copy whose next address is $n$. Then sentinel deletion of $x$ from $S$ returns $S_1$ with the node of $S_1$ at $n$ replaced by a copy whose previous address is $p$.","title":"The store returned by sentinel deletion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-get","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_get","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-get","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. For every address $z$, let $C$ be the cell of $S$ at $z$ if $z \\ne p$, and that cell with its next address set to $n$ if $z = p$. Then after sentinel deletion of $x$ from $S$ the cell at $z$ is $C$ if $z \\ne n$, and $C$ with its previous address set to $p$\u2026","title":"Cells after sentinel deletion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-frame","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_frame","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-frame","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. For every address $z$ with $z \\ne p$ and $z \\ne n$, the cell at $z$ after sentinel deletion of $x$ from $S$ equals the cell of $S$ at $z$.","title":"Sentinel deletion changes only the two neighbours"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-allocated","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_allocated","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-allocated","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. For every address $z$, the cell at $z$ after sentinel deletion of $x$ from $S$ is nonempty if and only if the cell of $S$ at $z$ is nonempty.","title":"Sentinel deletion preserves allocation"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-payload","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_payload","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-payload","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. For every address $z$, the cell at $z$ after sentinel deletion of $x$ from $S$ and the cell of $S$ at $z$ are either both empty or both hold nodes with the same payload.","title":"Sentinel deletion preserves payloads"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-deleted-record","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_deleted_record","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-deleted-record","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. The cell at $x$ after sentinel deletion of $x$ from $S$ equals the cell of $S$ at $x$, so the deleted node stays allocated and unchanged.","title":"Sentinel deletion keeps the deleted node"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-circular","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_circular","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-circular","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. If $S$ represents a circular doubly linked list with sentinel $s$ and data addresses $\\ell$, and $x$ occurs in $\\ell$, then the store returned by sentinel deletion of $x$ from $S$ represents a circular doubly linked list with sentinel $s$ and the list of\u2026","title":"Sentinel deletion removes a node from its circular list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-delete-disjoint-circular","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelDelete_disjoint_circular","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-delete-disjoint-circular","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a node store and $x$ an address whose cell holds a node with previous address $p$ and next address $n$, where the cells at $p$ and $n$ are nonempty. Suppose $S$ represents a circular doubly linked list with sentinel $s$ and data addresses $\\ell$, with $x$ in $\\ell$, and also one with sentinel $t$ and data addresses $m$, such that no address occurs both among $s$ and $\\ell$ and among $t$ and $m$. Then the\u2026","title":"Sentinel deletion preserves disjoint circular lists"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell $w$ it points to is allocated. Sentinel insertion of $x$ after $y$ performs four pointer assignments in order, each costing one time unit, and returns the resulting store: set the next pointer of $x$ to the next pointer of $y$; set the previous pointer of $x$\u2026","title":"Sentinel list insertion (LIST-INSERT\u2032)"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert-time","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert_time","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert-time","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell it points to is allocated. Then sentinel insertion of $x$ after $y$ in $S$ costs exactly $4$ time units, one per pointer assignment.","title":"Sentinel insertion takes four steps"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert-ret","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert_ret","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert-ret","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell $w$ it points to is allocated; write $S[c]$ for the node at an allocated cell $c$. Then the store returned by sentinel insertion of $x$ after $y$ is obtained from $S$ by three updates, each applied to the result of the previous one: replace the content of $x$\u2026","title":"The store after sentinel insertion"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert-get","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert_get","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert-get","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell it points to is allocated; write $S[c]$ for the node at an allocated cell $c$. Let $S'$ be the store returned by sentinel insertion of $x$ after $y$. Then for every cell $z$, the content of $z$ in $S'$ is obtained from its content in $S$ by three steps in\u2026","title":"Sentinel insertion, cell by cell"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert-frame","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert_frame","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert-frame","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell it points to is allocated. Let $S'$ be the store returned by sentinel insertion of $x$ after $y$. If a cell $z$ differs from $x$ and from $y$, and the next pointer of $y$ in $S$ is not $z$, then $z$ has the same content in $S'$ as in $S$.","title":"Sentinel insertion leaves other cells unchanged"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert-allocated","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert_allocated","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert-allocated","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell it points to is allocated. Let $S'$ be the store returned by sentinel insertion of $x$ after $y$. Then every cell $z$ is allocated in $S'$ if and only if it is allocated in $S$.","title":"Sentinel insertion preserves allocation"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert-payload","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert_payload","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert-payload","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell it points to is allocated. Let $S'$ be the store returned by sentinel insertion of $x$ after $y$. Then for every cell $z$, either $z$ is empty in both $S$ and $S'$, or it holds a node in both and the two nodes have the same payload, including keys and\u2026","title":"Sentinel insertion preserves payloads"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert-circular","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert_circular","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert-circular","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell it points to is allocated. Suppose $S$ circularly represents the list $L$ of cells with sentinel $s$, that $L$ is the concatenation of lists $B$ and $A$, and that $y$ is the last element of $B$, or $y = s$ if $B$ is empty. If $x \\ne s$ and $x$ does not occur\u2026","title":"Sentinel insertion preserves the circular list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/sentinel-insert-disjoint-circular","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.sentinelInsert_disjoint_circular","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#sentinel-insert-disjoint-circular","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $S$ be a store of $n$ cells, each empty or holding a node, in which the cells $x$ and $y$ are allocated, the next pointer of $y$ is not NIL, and the cell it points to is allocated. Suppose $S$ circularly represents the list $L$ with sentinel $s$, that $L$ is the concatenation of lists $B$ and $A$, and that $y$ is the last element of $B$, or $y = s$ if $B$ is empty. Suppose $S$ also circularly represents the\u2026","title":"Sentinel insertion preserves a disjoint circular list"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-search-sentinel","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listSearchSentinel","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-search-sentinel","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let a store have cells indexed by $\\{0, \\dots, m-1\\}$, each empty or holding a node that carries a key, satellite data and optional next and prev links, and let the cell $s$ be the sentinel of some list of data nodes that the store circularly represents. Sentinel search for a key $k$ first overwrites the key of the sentinel with $k$, keeping its satellite data and links, then starts at the successor of $s$ and\u2026","title":"Search in a circular list with a sentinel"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-search-sentinel-ret","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listSearchSentinel_ret","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-search-sentinel-ret","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"If a store circularly represents the list $L$ of data nodes with sentinel $s$, then sentinel search for a key $k$ returns the first node of $L$ (in list order, which starts at the successor of $s$) whose key in the input store is $k$, and returns no node if no node of $L$ has key $k$. In particular, the sentinel is never returned.","title":"Sentinel search returns the first matching node"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-search-sentinel-time","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listSearchSentinel_time","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-search-sentinel-time","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"If a store circularly represents the list $L$ of $n$ data nodes with sentinel $s$, then sentinel search for a key $k$ costs exactly $2i + 4$, where $i$ is the position, counting from $0$, of the first node of $L$ whose key is $k$, and $i = n$ if no node of $L$ has key $k$.","title":"Cost of sentinel search"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-search-sentinel-get","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listSearchSentinel_get","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-search-sentinel-get","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"Let $s$ be the sentinel of some list of data nodes that a store circularly represents, and let $k$ be a key. In the store returned by sentinel search for $k$, the cell $s$ holds the original sentinel node with its key replaced by $k$ and its satellite data and next and prev links unchanged, and every cell $z \\ne s$ holds exactly what it held in the input store.","title":"Sentinel search changes only the sentinel's key"},{"id":"ComputerScience/data-structures/linear-structures/doubly-linked-list/list-search-sentinel-circular","kind":"node","lean":"Cslib.Algorithms.Lean.DoublyLinkedList.listSearchSentinel_circular","parent":37453,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/doubly-linked-list/index.html#list-search-sentinel-circular","scope":"ComputerScience/data-structures/linear-structures/doubly-linked-list","status":"fully_proved","summary":"If a store circularly represents the list $L$ of data nodes with sentinel $s$, then for every key $k$ the store returned by sentinel search for $k$ still circularly represents $L$ with sentinel $s$.","title":"Sentinel search preserves the circular list"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/dynamic-table.json","id":"ComputerScience/data-structures/linear-structures/dynamic-table","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","title":"Dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.Table","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"A dynamic table with items of type $\\alpha$ consists of an array of slots $s_0, \\dots, s_{m-1}$, each either empty or holding an item of type $\\alpha$, together with an item count $n$ satisfying $n \\le m$. The number $m$ of slots is the capacity of the table.","title":"Dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/well-formed","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.WellFormed","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#well-formed","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"A dynamic table with slots $s_0, \\dots, s_{m-1}$ and item count $n$ is well formed when, for every $i < m$, slot $s_i$ is occupied if and only if $i < n$; that is, the occupied slots are exactly the first $n$.","title":"Well-formed dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/well-formed-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.wellFormed_iff","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#well-formed-iff","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"A dynamic table with slots $s_0, \\dots, s_{m-1}$ and item count $n$ is well formed if and only if, for every $i < m$, slot $s_i$ is occupied exactly when $i < n$.","title":"Characterization of well-formed dynamic tables"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/contents","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.contents","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#contents","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with slots $s_0, \\dots, s_{m-1}$, each empty or holding an item, and item count $n \\le m$. The contents of $T$ is the array of the items held by the occupied slots among $s_0, \\dots, s_{n-1}$, in order; empty slots in this prefix are skipped.","title":"Contents of a dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/contents-eq","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.contents_eq","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#contents-eq","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with slots $s_0, \\dots, s_{m-1}$, each empty or holding an item, and item count $n \\le m$. The contents of $T$ is the array obtained from the prefix $s_0, \\dots, s_{n-1}$ by skipping the empty slots and keeping the items of the occupied ones, in order.","title":"Formula for the contents of a dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/empty","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.empty","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#empty","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"The empty dynamic table is the table with no slots (capacity $0$) and item count $0$. It models a table before its first allocation.","title":"Empty dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/empty-eq","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.empty_eq","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#empty-eq","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"The empty dynamic table has an empty slot array, so capacity $0$, and item count $0$.","title":"The empty dynamic table has no slots and no items"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsert","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with slots $s_0, \\dots, s_{m-1}$ and item count $n \\le m$, and let $x$ be an item. Inserting $x$ into $T$ proceeds as follows. If $m = 0$, first replace the slots by a single empty slot. Then, if the table is full, that is, $n$ equals the current capacity $c$, allocate $2c$ empty slots and copy the $c$ old slots into the first $c$ of them, at cost $1$ per copied slot. Finally write $x$\u2026","title":"TABLE-INSERT"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-many","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsertMany","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-many","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Given a dynamic table $T$ and an array $x_0, \\dots, x_{k-1}$ of items, the result of inserting the array into $T$ is obtained by starting from $T$ and inserting $x_0, x_1, \\dots, x_{k-1}$ one at a time with TABLE-INSERT, in array order. Its cost is the sum of the costs of the $k$ insertions; for an empty array the result is $T$ at cost $0$.","title":"Inserting an array of items into a dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-num","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsert_num","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-num","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"For every dynamic table $T$ with item count $n$ and every item $x$, the table returned by inserting $x$ into $T$ with TABLE-INSERT has item count $n + 1$.","title":"Insertion increases the item count by one"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-size","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsert_size","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-size","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with item count $n$ and capacity (number of slots) $m$, and let $x$ be an item. The table returned by inserting $x$ into $T$ with TABLE-INSERT has capacity $1$ if $m = 0$, capacity $2m$ if $m > 0$ and $n = m$, and capacity $m$ otherwise.","title":"Capacity after an insertion"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-time","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsert_time","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-time","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with item count $n$ and capacity $m$, and let $x$ be an item. Inserting $x$ into $T$ with TABLE-INSERT costs $1 + n$ if $m > 0$ and $n = m$, and costs $1$ otherwise.","title":"Cost of an insertion"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-amortized","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsert_amortized","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-amortized","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"For a dynamic table $T$ with item count $n$ and capacity (number of slots) $m$, let $\\Phi(T) = 2n - m$, an integer that may be negative. If inserting an item $x$ into $T$ has cost $c$ and returns the table $T'$, then $c + \\Phi(T') - \\Phi(T) \\le 3$. Here the cost of an insertion is one unit for writing $x$ plus one unit for each old slot copied when a full table is doubled.","title":"Amortized cost of dynamic-table insertion"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-slots","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsert_slots","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-slots","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a well-formed dynamic table with slots $s_0, \\dots, s_{m-1}$ and item count $n$, let $x$ be an item, and let $T'$ be the table returned by inserting $x$ into $T$ with TABLE-INSERT, with capacity $m'$. Then the slots of $T'$ are $s_0, \\dots, s_{n-1}$, followed by a slot holding $x$, followed by $m' - (n + 1)$ empty slots.","title":"Slots after an insertion"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-well-formed","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsert_wellFormed","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-well-formed","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"If $T$ is a well-formed dynamic table and $x$ is an item, then the table returned by inserting $x$ into $T$ with TABLE-INSERT is well formed.","title":"Insertion preserves well-formedness"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-contents","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsert_contents","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-contents","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"If $T$ is a well-formed dynamic table and $x$ is an item, then the contents of the table obtained by inserting $x$ into $T$ are the contents of $T$ followed by $x$.","title":"Insertion appends to the contents"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-many-well-formed","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsertMany_wellFormed","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-many-well-formed","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Call a dynamic table with item count $n$ well formed when, for every slot index $i$, slot $i$ is occupied exactly when $i < n$. If a dynamic table $T$ is well formed, then for every array $xs$, the table obtained from $T$ by inserting the entries of $xs$ one at a time with TABLE-INSERT, in array order, is again well formed.","title":"Repeated insertion preserves well-formedness of a dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-many-contents","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsertMany_contents","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-many-contents","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"If $T$ is a well-formed dynamic table and $x_1, \\dots, x_k$ is an array of items, then inserting $x_1, \\dots, x_k$ into $T$ one at a time, in this order, gives a table whose contents are the contents of $T$ followed by $x_1, \\dots, x_k$.","title":"Inserting an array appends it to the contents"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-many-size-le","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsertMany_size_le","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-many-size-le","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Starting from the empty dynamic table and inserting the items of an array of length $k$ one at a time, the resulting table has capacity (number of slots) at most $2k$.","title":"Capacity after inserting into the empty table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-many-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsertMany_time_le","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-many-time-le","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Starting from the empty dynamic table and inserting the items of an array of length $k$ one at a time, the total cost is at most $3k$, where each insertion costs one unit for writing the new item plus one unit for each old slot copied when a full table is doubled.","title":"Total cost of insertions into the empty table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-insert-many-time-lt","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableInsertMany_time_lt","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-insert-many-time-lt","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Starting from the empty dynamic table and inserting the items of a nonempty array of length $k$ one at a time, the total cost is less than $3k$.","title":"Strict cost bound for nonempty insertion sequences"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/get","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.get?","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#get","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with slots $s_0, \\dots, s_{m-1}$, each empty or holding an item, and item count $n \\le m$. The lookup at logical position $i$ returns the item held by $s_i$ when $i < n$ and $s_i$ is occupied; it returns no value when $i < n$ and $s_i$ is empty, and when $i \\ge n$.","title":"Lookup in a dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/get-eq-none-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.get?_eq_none_of_le","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#get-eq-none-of-le","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with item count $n$. For every natural number $i \\ge n$, the lookup at logical position $i$ in $T$ returns no value.","title":"Lookup beyond the item count returns nothing"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/get-is-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.get?_isSome_iff","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#get-is-some-iff","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a well-formed dynamic table holding $n$ items. For every natural number $i$, the lookup at logical position $i$ in $T$ returns a value if and only if $i < n$.","title":"Lookup succeeds exactly below the item count"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-delete","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableDelete","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-delete","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with slots $s_0, \\dots, s_{m-1}$, each empty or holding an item, and item count $n \\le m$. Deleting the last item of $T$ gives the following table. If $n = 0$, the result is $T$ itself. Otherwise, if $4(n-1) < m$, that is, the load factor after deletion is strictly below $1/4$, the table contracts: the result has slots $s_0, \\dots, s_{n-2}$ followed by $\\lfloor m/2 \\rfloor - (n-1)$ empty\u2026","title":"Deletion from a dynamic table"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-delete-num","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableDelete_num","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-delete-num","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Applying deletion to a dynamic table with item count $n$ gives a table with item count $n - 1$ when $n \\ge 1$, and item count $0$ when $n = 0$.","title":"Item count after deletion"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-delete-size","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableDelete_size","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-delete-size","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with capacity $m$ and item count $n$, and apply deletion to $T$. If $n = 0$, the capacity stays $m$. If $n \\ge 1$ and $4(n-1) < m$, the new capacity is $\\lfloor m/2 \\rfloor$. Otherwise the capacity stays $m$.","title":"Capacity after deletion"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-delete-slots","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableDelete_slots","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-delete-slots","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a well-formed dynamic table with slots $s_0, \\dots, s_{m-1}$ holding $n \\ge 1$ items, and let $m'$ be the capacity of the table obtained by applying deletion to $T$. The slots of that table are the first $n - 1$ slots $s_0, \\dots, s_{n-2}$ of $T$ followed by $m' - (n - 1)$ empty slots.","title":"Slots after deletion"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-delete-well-formed","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableDelete_wellFormed","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-delete-well-formed","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a well-formed dynamic table, that is, its occupied slots are exactly its first $n$ slots, where $n$ is its item count. Then the table obtained by applying deletion to $T$ is again well-formed.","title":"Deletion preserves well-formedness"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/table-delete-contents","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.tableDelete_contents","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#table-delete-contents","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a well-formed dynamic table. The contents of the table obtained by applying deletion to $T$ is the contents of $T$ with its last entry removed; when the contents of $T$ is empty, it stays empty.","title":"Deletion removes the last item from the contents"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/get-delete-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.get?_delete_of_lt","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#get-delete-of-lt","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a well-formed dynamic table holding $n$ items, and let $i$ be a natural number with $i < n - 1$. After applying deletion to $T$, the lookup at logical position $i$ returns the same result as the lookup at position $i$ in $T$.","title":"Deletion preserves lookups below the last position"},{"id":"ComputerScience/data-structures/linear-structures/dynamic-table/get-delete-last","kind":"node","lean":"Cslib.Algorithms.Lean.DynamicTable.get?_delete_last","parent":37516,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/dynamic-table/index.html#get-delete-last","scope":"ComputerScience/data-structures/linear-structures/dynamic-table","status":"fully_proved","summary":"Let $T$ be a dynamic table with item count $n$, where $n - 1$ is read as $0$ when $n = 0$. After applying deletion to $T$, the lookup at logical position $n - 1$ returns no value.","title":"Lookup at the vacated position after deletion"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/join-list-catenable-heap.json","id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","title":"Join list (catenable heap)"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/catenable-heap","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#catenable-heap","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"A catenable heap over a type $\\alpha$ is a persistent unsorted heap, represented as a join list (catenation tree): it is either the empty heap, an element of $\\alpha$ added in front of a catenable heap, or the join of two catenable heaps.","title":"Catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-naive","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toListNaive","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-naive","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The naive flattening of a catenable heap is defined by recursion: the empty heap flattens to the empty list, the heap with an element $x$ added in front of a heap $t$ flattens to $x$ followed by the naive flattening of $t$, and the join of heaps $l$ and $r$ flattens to the naive flattening of $l$ followed by that of $r$.","title":"Naive flattening of a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-aux","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toListAux","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-aux","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"Given a catenable heap $h$ and a list $s$, the accumulating flattening of $h$ onto $s$ is defined by recursion on $h$: for the empty heap it is $s$; for the heap with an element $x$ added in front of a heap $t$ it is $x$ followed by the accumulating flattening of $t$ onto $s$; for the join of heaps $l$ and $r$ it is the accumulating flattening of $l$ onto the accumulating flattening of $r$ onto $s$.","title":"Accumulating flattening of a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toList","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The flattening of a catenable heap $h$ is the list of its elements from left to right, computed as the accumulating flattening of $h$ onto the empty list.","title":"Flattening of a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-multiset","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toMultiset","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-multiset","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The multiset of elements of a catenable heap $h$ is the multiset of the entries of the left-to-right list of elements of $h$.","title":"Multiset of elements of a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/size","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.size","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#size","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The size of a catenable heap is the length of its left-to-right list of elements, that is, the number of its elements counted with multiplicity.","title":"Size of a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-aux-eq","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toListAux_eq","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-aux-eq","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every catenable heap $h$ and list $a$, flattening $h$ from left to right in front of the accumulator $a$ gives the naive left-to-right flattening of $h$ followed by $a$.","title":"Accumulator flattening of a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-eq-to-list-naive","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toList_eq_toListNaive","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-eq-to-list-naive","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every catenable heap $h$, the flattening of $h$ equals the naive flattening of $h$.","title":"Flattening of a catenable heap equals its naive flattening"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-empty","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toList_empty","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-empty","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The flattening of the empty catenable heap is the empty list.","title":"Flattening the empty catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-cons","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toList_cons","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-cons","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For an element $x$ and a catenable heap $t$, the flattening of the heap with $x$ added in front of $t$ is $x$ followed by the flattening of $t$.","title":"Flattening a front node of a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-join","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toList_join","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-join","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For catenable heaps $l$ and $r$, the flattening of the join of $l$ and $r$ is the flattening of $l$ followed by the flattening of $r$.","title":"Flattening a join of catenable heaps"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-multiset-empty","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toMultiset_empty","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-multiset-empty","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The multiset of elements of the empty catenable heap is the empty multiset.","title":"The empty catenable heap has no elements"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-multiset-cons","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toMultiset_cons","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-multiset-cons","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every element $a$ and every catenable heap $t$, the multiset of elements of the heap obtained by adding $a$ in front of $t$ is the multiset of elements of $t$ with one more copy of $a$.","title":"Multiset of a front node"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-multiset-join","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toMultiset_join","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-multiset-join","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For all catenable heaps $L$ and $R$, the multiset of elements of the join node with left child $L$ and right child $R$ is the multiset sum of the multisets of elements of $L$ and $R$ (multiplicities add).","title":"Multiset of a join node"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/size-empty","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.size_empty","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#size-empty","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The empty catenable heap has size $0$.","title":"The empty catenable heap has size zero"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/size-cons","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.size_cons","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#size-cons","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every element $a$ and catenable heap $t$, the size of the heap obtained by adding $a$ in front of $t$ is the size of $t$ plus one.","title":"Size of a heap with an element added in front"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/size-join","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.size_join","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#size-join","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For catenable heaps $\\ell$ and $r$, the size of the join node with left child $\\ell$ and right child $r$ is the size of $\\ell$ plus the size of $r$.","title":"Size of a catenable heap join"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/size-eq-card-to-multiset","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.size_eq_card_toMultiset","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#size-eq-card-to-multiset","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every catenable heap $h$, the size of $h$ equals the cardinality of the multiset of its elements.","title":"Size of a catenable heap as a multiset cardinality"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/make","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.make","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#make","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The operation that creates a catenable heap returns the empty heap and charges cost $0$, since it allocates no node.","title":"Creating an empty catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/insert","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.insert","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#insert","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For an element $x$ and a catenable heap $h$, inserting $x$ into $h$ returns the heap with $x$ added in front of $h$ and charges cost $1$, for the one front node it allocates.","title":"Insertion into a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/union","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.union","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#union","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The union of catenable heaps $h_1$ and $h_2$ allocates one join node, at cost $1$, and returns that join node, whose left child is $h_1$ and whose right child is $h_2$; neither operand is traversed or changed.","title":"Union of catenable heaps"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/make-eq","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.make_eq","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#make-eq","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"Creating a catenable heap returns the empty heap, at cost $0$.","title":"Cost and result of creating a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/insert-eq","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.insert_eq","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#insert-eq","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every element $x$ and catenable heap $h$, inserting $x$ into $h$ returns the heap with $x$ added in front of $h$, at cost exactly $1$.","title":"Cost and result of insertion into a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/union-eq","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.union_eq","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#union-eq","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For all catenable heaps $h_1$ and $h_2$, the union of $h_1$ and $h_2$ returns the join node whose left child is $h_1$ and whose right child is $h_2$, at cost $1$.","title":"Union allocates one join node"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/ret-make","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.ret_make","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#ret-make","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The heap returned by the operation that creates a catenable heap is the empty heap.","title":"Result of creating a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/time-make","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.time_make","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#time-make","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The operation that creates an empty catenable heap has time cost $0$.","title":"Creating an empty catenable heap costs nothing"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/ret-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.ret_insert","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#ret-insert","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every element $x$ and catenable heap $h$, the heap returned by inserting $x$ into $h$ is the heap with $x$ added in front of $h$.","title":"Result of insertion into a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/time-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.time_insert","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#time-insert","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every element $x$ and catenable heap $h$, inserting $x$ into $h$ has time cost exactly $1$.","title":"Insertion into a catenable heap costs one"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/ret-union","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.ret_union","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#ret-union","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For all catenable heaps $h_1$ and $h_2$, the heap returned by their union is the join of $h_1$ and $h_2$, with $h_1$ as left child and $h_2$ as right child, both unchanged.","title":"Result of the union of catenable heaps"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/time-union","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.time_union","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#time-union","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For all catenable heaps $h_1$ and $h_2$, the union of $h_1$ and $h_2$ has time cost exactly $1$, independently of $h_1$ and $h_2$.","title":"Union of catenable heaps costs one"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-make","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toList_make","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-make","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The flattening of the heap returned by the operation that creates a catenable heap is the empty list.","title":"A newly created catenable heap flattens to the empty list"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toList_insert","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-insert","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For an element $x$ and a catenable heap $h$, the flattening of the heap returned by inserting $x$ into $h$ is $x$ followed by the flattening of $h$.","title":"Flattening after insertion into a catenable heap"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-list-union","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toList_union","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-list-union","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For catenable heaps $h_1$ and $h_2$, the flattening of the heap returned by the union of $h_1$ and $h_2$ is the flattening of $h_1$ followed by the flattening of $h_2$.","title":"Flattening a union of catenable heaps"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-multiset-make","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toMultiset_make","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-multiset-make","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The multiset of elements of the heap returned by the make operation is the empty multiset.","title":"A newly made catenable heap has no elements"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-multiset-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toMultiset_insert","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-multiset-insert","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every element $x$ and every catenable heap $h$, the multiset of elements of the heap returned by inserting $x$ into $h$ is the multiset of elements of $h$ with one more copy of $x$.","title":"Insertion adds one copy of the element"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/to-multiset-union","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.toMultiset_union","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#to-multiset-union","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For all catenable heaps $h_1$ and $h_2$, the multiset of elements of the heap returned by the union of $h_1$ and $h_2$ is the multiset sum of the multisets of elements of $h_1$ and $h_2$ (multiplicities add).","title":"Union adds the multisets of elements"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/size-make","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.size_make","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#size-make","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"The heap returned by the operation that creates an empty catenable heap has size $0$.","title":"A newly made catenable heap has size zero"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/size-insert","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.size_insert","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#size-insert","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For every element $x$ and catenable heap $h$, the heap returned by inserting $x$ into $h$ has size one more than the size of $h$.","title":"Insertion into a catenable heap increases its size by one"},{"id":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap/size-union","kind":"node","lean":"Cslib.Algorithms.Lean.CatenableHeap.size_union","parent":37549,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/join-list-catenable-heap/index.html#size-union","scope":"ComputerScience/data-structures/linear-structures/join-list-catenable-heap","status":"fully_proved","summary":"For catenable heaps $h_1$ and $h_2$, the heap returned by the union of $h_1$ and $h_2$ has size equal to the size of $h_1$ plus the size of $h_2$.","title":"Size of a catenable heap union"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/least-recently-used-cache.json","id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","title":"Least-recently-used cache"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/lru-cache","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#lru-cache","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For a type $K$ of keys, a type $V$ of values and a natural number $c$, an LRU cache of capacity $c$ is a list of key-value pairs in $K \\times V$, ordered from most to least recently used, whose keys are pairwise distinct and whose length is at most $c$.","title":"Least-recently-used cache"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/ext","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.ext","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#ext","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Two LRU caches with the same key type, value type and capacity are equal if their lists of entries are equal.","title":"Extensionality of LRU caches"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/erase-key","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.eraseKey","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#erase-key","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Given a key $k$ and a list of key\u2013value pairs, erasing $k$ deletes every pair whose key is $k$ and keeps the other pairs in their original order. When the keys in the list are pairwise distinct, at most one pair is deleted.","title":"Erasing a key from a list of entries"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/empty","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.empty","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#empty","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every capacity $c$, the empty LRU cache of capacity $c$ is the cache with no entries.","title":"Empty LRU cache"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/keys","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.keys","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#keys","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"The key list of an LRU cache is the list of the keys of its entries in the order of the cache, so the most recently used key comes first.","title":"Key list of an LRU cache"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/size","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.size","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#size","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"The size of an LRU cache is the number of entries it stores.","title":"Size of an LRU cache"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Looking up a key $k$ in an LRU cache returns an optional value together with a cache. If the cache stores a binding $(k, v)$, the result is $v$ together with the cache in which that binding is moved to the front and the other entries keep their order; otherwise the result is no value together with the unchanged cache.","title":"LRU cache lookup"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/put","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.put","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#put","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Putting the binding $(k, v)$ into an LRU cache of capacity $c$ removes any entry with key $k$, places $(k, v)$ at the front as the most recently used entry, and then keeps only the first $c$ entries, so the least recently used entries beyond the capacity are evicted.","title":"Insertion into an LRU cache"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/keys-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.keys_nodup","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#keys-nodup","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"The keys stored in any LRU cache are pairwise distinct.","title":"Keys of an LRU cache are distinct"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/size-le-cap","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.size_le_cap","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#size-le-cap","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Every LRU cache of capacity $c$ has size at most $c$.","title":"An LRU cache never exceeds its capacity"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-miss","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_miss","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-miss","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"If the key $k$ is not stored in an LRU cache, then looking up $k$ returns no value and leaves the cache unchanged.","title":"A lookup of an absent key misses"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-hit","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_hit","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-hit","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"If an LRU cache stores the pair $(k, v)$, then looking up $k$ returns $v$.","title":"A lookup of a stored key returns its value"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-promotes","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_promotes","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-promotes","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"If an LRU cache stores the pair $(k, v)$, then after looking up $k$ the most recently used (first) entry of the cache is $(k, v)$.","title":"A successful lookup moves the entry to the front"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-put","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_put","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-put","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache of positive capacity, every key $k$ and every value $v$, looking up $k$ in the cache obtained by putting the binding $(k, v)$ returns $v$.","title":"Lookup after insertion into an LRU cache"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/put-size-le","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.put_size_le","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#put-size-le","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache $C$ of capacity $c$, key $k$ and value $v$, the cache obtained by putting the binding $(k, v)$ into $C$ has size at most $c$.","title":"Insertion respects the capacity of an LRU cache"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/put-keys-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.put_keys_nodup","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#put-keys-nodup","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache $C$, key $k$ and value $v$, the keys of the cache obtained by putting the binding $(k, v)$ into $C$ are pairwise distinct.","title":"Insertion keeps the keys of an LRU cache distinct"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-size-le","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_size_le","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-size-le","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache of capacity $c$ and every key $k$, the cache after looking up $k$ holds at most $c$ entries.","title":"Lookup respects the capacity"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-keys-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_keys_nodup","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-keys-nodup","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache and every key $k$, the keys of the cache obtained by looking up $k$ are pairwise distinct.","title":"Lookups keep keys distinct"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/mem-put-entries","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.mem_put_entries","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#mem-put-entries","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache, key $k$ and value $v$, every key-value pair $p$ stored in the cache after putting the binding $(k, v)$ either equals $(k, v)$, or was stored before and has a key different from $k$.","title":"Entries after insertion"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/mem-get-entries-iff","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.mem_get_entries_iff","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#mem-get-entries-iff","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache and every key $k$, a key-value pair is stored in the cache after looking up $k$ if and only if it was stored in the cache before the lookup.","title":"Lookup preserves the stored bindings"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-perm","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_perm","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-perm","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache and every key $k$, the list of entries of the cache after looking up $k$ is a permutation of the list of entries before.","title":"A lookup permutes the entries"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-size","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_size","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-size","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"For every LRU cache and every key $k$, the cache after looking up $k$ holds exactly as many entries as before.","title":"Lookup preserves the size"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/mem-put-entries-iff-of-not-full","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.mem_put_entries_iff_of_not_full","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#mem-put-entries-iff-of-not-full","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Let an LRU cache of capacity $c$ hold fewer than $c$ entries or already store a binding for the key $k$, and let $v$ be a value. Then a key-value pair $p$ is stored in the cache after putting the binding $(k, v)$ if and only if $p = (k, v)$, or $p$ was stored before and the key of $p$ differs from $k$.","title":"Insertion without eviction"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/put-head","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.put_head?","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#put-head","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Let $C$ be an LRU cache of positive capacity, $k$ a key and $v$ a value. After the binding $(k, v)$ is put into $C$, the first entry of the resulting cache, its most recently used entry, is $(k, v)$.","title":"With positive capacity, the inserted binding becomes the most recent entry"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/get-put-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.get_put_of_ne","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#get-put-of-ne","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Let $C$ be an LRU cache of capacity $c$ that holds fewer than $c$ entries or already stores the key $k$. Then for every key $k' \\ne k$ and every value $v$, looking up $k'$ in the cache obtained by putting the binding $(k, v)$ into $C$ returns the same optional value as looking up $k'$ in $C$.","title":"A put without eviction preserves lookups of other keys"},{"id":"ComputerScience/data-structures/linear-structures/least-recently-used-cache/evicted-is-least-recent","kind":"node","lean":"Cslib.Algorithms.Lean.LruCache.evicted_is_least_recent","parent":37589,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/least-recently-used-cache/index.html#evicted-is-least-recent","scope":"ComputerScience/data-structures/linear-structures/least-recently-used-cache","status":"fully_proved","summary":"Let $C$ be an LRU cache of capacity $c$, let $k'$ be a key not stored in $C$, and let $k$ be a key stored in $C$. If $k$ is no longer stored after putting $k'$ with any value $v$ into $C$, then $C$ held exactly $c$ entries and $k$ is the last key of $C$, that is, the least recently used one.","title":"The evicted key is the least recently used one"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/list-operations.json","id":"ComputerScience/data-structures/linear-structures/list-operations","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","title":"List operations"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/idx-of-lt-iff","kind":"node","lean":"List.Pairwise.idxOf_lt_iff","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#idx-of-lt-iff","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"In a list along which `f` increases, position order agrees with value order.","title":"List positions agree with values of an increasing map"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/product-lex","kind":"node","lean":"List.Pairwise.product_lex","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#product-lex","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"The Cartesian product of two pairwise-related lists is pairwise-related in lexicographic order.","title":"Pairwise relations on list products in lexicographic order"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/get-d-split-lengths","kind":"node","lean":"List.getD_splitLengths","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#get-d-split-lengths","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"A valid indexed chunk begins after the total length of the preceding chunks.","title":"Start position of a chunk in a split list"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/get-d-split-lengths-map","kind":"node","lean":"List.getD_splitLengths_map","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#get-d-split-lengths-map","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Looking up a named chunk uses the first occurrence of its name.","title":"Named chunk lookup uses the first occurrence"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/get-d-split-lengths-replicate","kind":"node","lean":"List.getD_splitLengths_replicate","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#get-d-split-lengths-replicate","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Splitting at a repeated width recovers the corresponding uniform chunk.","title":"Splitting a list at a repeated width"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/eq-of-extract-mul-eq","kind":"node","lean":"List.eq_of_extract_mul_eq","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#eq-of-extract-mul-eq","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Lists of the same uniform-block length are equal when every block extract agrees.","title":"Equality of uniformly split lists from equal block extracts"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/length-flat-map-range","kind":"node","lean":"List.length_flatMap_range","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#length-flat-map-range","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Concatenating `count` lists of length `width` has length `count * width`.","title":"Length of a concatenation of equal-length lists"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/extract-flat-map-range","kind":"node","lean":"List.extract_flatMap_range","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#extract-flat-map-range","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"A uniform-width extract of a concatenated family recovers that family member.","title":"Extracting a member from a uniform-width concatenation"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/exists-length-eq-and-extract-eq","kind":"node","lean":"List.exists_length_eq_and_extract_eq","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#exists-length-eq-and-extract-eq","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Every finite family of equal-length lists is realized by their concatenation.","title":"Every equal-length list family is realized by its concatenation"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through","kind":"node","lean":"List.takeThrough","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"The prefix through the first element satisfying `p`, including that element. If no element satisfies `p`, this is the whole list.","title":"Inclusive first-hit prefix of a list"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-nil","kind":"node","lean":"List.takeThrough_nil","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-nil","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"The inclusive first-hit prefix of an empty list is empty.","title":"Inclusive first-hit prefix of the empty list"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-cons-of-pos","kind":"node","lean":"List.takeThrough_cons_of_pos","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-cons-of-pos","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"If the head satisfies the predicate, it is the whole inclusive first-hit prefix.","title":"Inclusive first-hit prefix when the head is a hit"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-cons-of-neg","kind":"node","lean":"List.takeThrough_cons_of_neg","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-cons-of-neg","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"If the head does not satisfy the predicate, the search continues through the tail.","title":"Inclusive first-hit prefix when the head is not a hit"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-singleton","kind":"node","lean":"List.takeThrough_singleton","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-singleton","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"A singleton is its own inclusive first-hit prefix.","title":"Inclusive first-hit prefix of a singleton"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/mem-of-mem-take-through","kind":"node","lean":"List.mem_of_mem_takeThrough","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#mem-of-mem-take-through","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Every member of an inclusive first-hit prefix belongs to the source list.","title":"Members of an inclusive first-hit prefix lie in the list"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-eq-self-of-any-eq-false","kind":"node","lean":"List.takeThrough_eq_self_of_any_eq_false","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-eq-self-of-any-eq-false","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"If no element satisfies the predicate, the inclusive first-hit prefix is the whole list.","title":"Inclusive first-hit prefix of a list with no hit"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-append-of-any-eq-true","kind":"node","lean":"List.takeThrough_append_of_any_eq_true","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-append-of-any-eq-true","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Once the left list has a hit, appending another list does not change the prefix.","title":"Inclusive first-hit prefix of an append with a hit on the left"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-append-of-any-eq-false","kind":"node","lean":"List.takeThrough_append_of_any_eq_false","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-append-of-any-eq-false","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"If the left list has no hit, the prefix contains all of it and continues into the right list.","title":"Inclusive first-hit prefix of an append with no hit on the left"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/any-take-through","kind":"node","lean":"List.any_takeThrough","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#any-take-through","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"The inclusive first-hit prefix has a hit exactly when the source list has one.","title":"Inclusive first-hit prefix has a hit iff the list does"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-range","kind":"node","lean":"List.takeThrough_range","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-range","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"On `range n`, the prefix ends one past the first hit, or at `n` if there is no hit.","title":"Inclusive first-hit prefix of a range of naturals"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/filter-map-take-through","kind":"node","lean":"List.filterMap_takeThrough","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#filter-map-take-through","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Filtering selected elements from an inclusive prefix commutes with taking the corresponding inclusive prefix of the selected elements.","title":"Inclusive first-hit prefix commutes with filter-map"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-acc","kind":"node","lean":"List.takeThroughAcc","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-acc","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"The prefix through the first element after whose update the accumulator satisfies `stop`, including that element. The initial accumulator is not tested. If no updated accumulator satisfies `stop`, this is the whole list.","title":"Accumulator-stopped inclusive prefix of a list"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-acc-nil","kind":"node","lean":"List.takeThroughAcc_nil","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-acc-nil","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"An empty list has an empty accumulator-stopped prefix.","title":"Accumulator-stopped prefix of the empty list"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-acc-cons-of-stop","kind":"node","lean":"List.takeThroughAcc_cons_of_stop","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-acc-cons-of-stop","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"If the accumulator stops after the head update, the prefix contains only the head.","title":"Accumulator-stopped prefix when the head update stops"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-acc-cons-of-continue","kind":"node","lean":"List.takeThroughAcc_cons_of_continue","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-acc-cons-of-continue","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"If the accumulator continues after the head update, the same updated accumulator processes the tail.","title":"Accumulator-stopped prefix when the head update continues"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-acc-prefix","kind":"node","lean":"List.takeThroughAcc_prefix","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-acc-prefix","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"The accumulator-stopped list is a prefix of its source list.","title":"Accumulator-stopped list is a prefix of its source"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/mem-of-mem-take-through-acc","kind":"node","lean":"List.mem_of_mem_takeThroughAcc","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#mem-of-mem-take-through-acc","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Every member of an accumulator-stopped prefix belongs to the source list.","title":"Members of an accumulator-stopped prefix lie in the list"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-acc-or","kind":"node","lean":"List.takeThroughAcc_or","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-acc-or","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Accumulating disjunction from `false` gives the ordinary inclusive first-hit prefix.","title":"Disjunction accumulator gives the inclusive first-hit prefix"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-acc-first","kind":"node","lean":"List.takeThroughAcc_first","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-acc-first","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Accumulating a projection of the first witness in an `Option` gives the ordinary inclusive first-hit prefix.","title":"First-witness accumulator gives the inclusive first-hit prefix"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/foldl-first-take-through","kind":"node","lean":"List.foldl_first_takeThrough","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#foldl-first-take-through","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Folding the first-witness update over the inclusive tested prefix returns the first witness in the whole source list.","title":"First-witness fold over the inclusive first-hit prefix"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/take-through-acc-eq-map-take-through-sca","kind":"node","lean":"List.takeThroughAcc_eq_map_takeThrough_scanl","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#take-through-acc-eq-map-take-through-sca","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"The recursive ETH operation is the ordinary inclusive first-hit prefix of the elements paired with their updated accumulators.","title":"Accumulator-stopped prefix as a first-hit prefix of a scan"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/length-le-of-mem-to-chunks","kind":"node","lean":"List.length_le_of_mem_toChunks","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#length-le-of-mem-to-chunks","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Let $n > 0$ and let $L$ be a list. Split $L$ from left to right into consecutive chunks of size $n$: elements go into the current chunk until it holds $n$ elements, then a new chunk is started. Every chunk produced has at most $n$ elements.","title":"Chunks of a list have length at most the chunk size"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.orderedInsert","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#ordered-insert","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"For an element $x$ of a linearly ordered type, the ordered insertion of $x$ into the empty list is $[x]$. For a list with head $a$ and tail $\\ell$, it is $x$ followed by the whole list if $x < a$; it is $a$ followed by the ordered insertion of $x$ into $\\ell$ if $a < x$; and it is the list unchanged if $a = x$.","title":"Ordered insertion into a list"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/mem-ordered-insert","kind":"node","lean":"Cslib.Algorithms.Lean.mem_orderedInsert","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#mem-ordered-insert","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Over a linearly ordered type, for all elements $x$, $y$ and every list $\\ell$, the element $y$ belongs to the ordered insertion of $x$ into $\\ell$ if and only if $y = x$ or $y$ belongs to $\\ell$.","title":"Members after ordered insertion"},{"id":"ComputerScience/data-structures/linear-structures/list-operations/ordered-insert-eq-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.orderedInsert_eq_of_mem","parent":37616,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/list-operations/index.html#ordered-insert-eq-of-mem","scope":"ComputerScience/data-structures/linear-structures/list-operations","status":"fully_proved","summary":"Over a linearly ordered type, if every element of a list $\\ell$ is smaller than every later element of $\\ell$ and $x$ belongs to $\\ell$, then the ordered insertion of $x$ into $\\ell$ is $\\ell$ itself.","title":"Inserting a present element into a strictly increasing list"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/singly-linked-list.json","id":"ComputerScience/data-structures/linear-structures/singly-linked-list","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","title":"Singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/node","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.Node","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#node","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"A singly linked list node with payloads in $\\alpha$ and identities in $\\iota$ consists of a payload in $\\alpha$ (the key together with any satellite data) and a next pointer, which is either an identity in $\\iota$ or NIL.","title":"Singly linked list node"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/represents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.Represents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#represents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let a store assign to each identity below $n$ either nothing or a singly linked list node whose next pointer is an identity below $n$ or NIL, and let the head be an identity below $n$ or NIL. The store and head represent the list of identities $i_0, \\dots, i_{k-1}$ when these identities are pairwise distinct, the head is $i_0$ (and NIL when $k = 0$), and for each $j < k$ the store entry at $i_j$\u2026","title":"Store representing a singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/represents-nil","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.represents_nil","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#represents-nil","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be any store assigning to each identity below $n$ either nothing or a singly linked list node. Then $s$ with head NIL represents the empty list of identities.","title":"Representation of the empty singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/represents-allocated","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.represents_allocated","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#represents-allocated","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, and let $\\mathrm{head}$ be an identity below $n$ or NIL. If $s$ and $\\mathrm{head}$ represent the list of identities $i_0, \\dots, i_{k-1}$, then for every identity $i$ in this list the store entry $s(i)$ is a node.","title":"Represented identities hold nodes"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/contents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.contents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#contents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node (a payload together with a next pointer that is an identity below $n$ or NIL). For a list of identities $i_0, \\dots, i_{m-1}$ below $n$, the contents of $s$ at this list are the list $c_0, \\dots, c_{m-1}$, where $c_j$ is the payload of $s(i_j)$ when $s(i_j)$ is a node and $c_j$ is nothing\u2026","title":"Payload list of a singly linked list store"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/contents-length","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.contents_length","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#contents-length","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node. For every list $L$ of identities below $n$, the contents of $s$ at $L$ have the same length as $L$.","title":"The payload list has one entry per identity"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/contents-congr","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.contents_congr","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#contents-congr","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$ and let $s$ and $s'$ be stores assigning to each identity below $n$ either nothing or a singly linked list node. If $L$ is a list of identities below $n$ and $s'(i) = s(i)$ for every $i$ in $L$, then the contents of $s'$ at $L$ equal the contents of $s$ at $L$.","title":"The payload list depends only on the listed cells"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/represents-contents-is-some","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.represents_contents_isSome","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#represents-contents-is-some","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, and let $\\mathrm{head}$ be an identity below $n$ or NIL. If $s$ and $\\mathrm{head}$ represent the list of identities $i_0, \\dots, i_{k-1}$, then every entry of the contents of $s$ at $i_0, \\dots, i_{k-1}$ is a payload rather than nothing.","title":"Represented lists have no missing contents"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/prepend","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.prepend","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#prepend","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let a store assign to each identity below $n$ either nothing or a singly linked list node, let the head be an identity below $n$ or NIL, and let $x$ be an identity whose store entry is a node. Prepending $x$ replaces the store entry at $x$ by the same node with its next pointer set to the head, leaves every other entry unchanged, and returns the updated store together with the new head $x$.","title":"Prepending a node to a singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/prepend-head","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.prepend_head","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#prepend-head","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $\\mathrm{head}$ be an identity below $n$ or NIL, and let $x$ be an identity such that $s(x)$ is a node. Then the head returned by Prepend on $s$, $\\mathrm{head}$ and $x$ is $x$.","title":"Prepending makes the node the head"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/prepend-get","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.prepend_get","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#prepend-get","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $\\mathrm{head}$ be an identity below $n$ or NIL, and let $x$ be an identity such that $s(x)$ is a node. Then for every identity $z$ below $n$, the entry at $z$ of the store returned by Prepend on $s$, $\\mathrm{head}$ and $x$ is the node $s(x)$ with its next pointer replaced by $\\mathrm{head}$\u2026","title":"Store after prepending to a singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/prepend-frame","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.prepend_frame","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#prepend-frame","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Let $S$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $h$ be an identity below $n$ or NIL, and let $x$ be an identity whose store entry is a node. For every identity $z \\ne x$, cell $z$ of the store obtained by prepending $x$ to the list with head $h$ equals cell $z$ of $S$.","title":"Frame property of prepending to a singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/prepend-node","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.prepend_node","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#prepend-node","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $\\mathrm{head}$ be an identity below $n$ or NIL, and let $x$ be an identity such that $s(x)$ is a node. Then the entry at $x$ of the store returned by Prepend on $s$, $\\mathrm{head}$ and $x$ is the node $s(x)$ with its payload kept and its next pointer replaced by $\\mathrm{head}$.","title":"The prepended node points at the old head"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/prepend-represents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.prepend_represents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#prepend-represents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $\\mathrm{head}$ be an identity below $n$ or NIL, and let $x$ be an identity such that $s(x)$ is a node. If $s$ and $\\mathrm{head}$ represent the list of identities $i_0, \\dots, i_{k-1}$ and $x$ is none of $i_0, \\dots, i_{k-1}$, then the store and head returned by Prepend on $s$,\u2026","title":"Prepending conses onto the represented list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/prepend-contents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.prepend_contents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#prepend-contents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Let $S$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $h$ be an identity below $n$ or NIL, let $x$ be an identity whose store entry is a node with payload $a$, and let $(i_0, \\dots, i_{m-1})$ be a list of identities below $n$ not containing $x$. Then the contents, at the list $(x, i_0, \\dots, i_{m-1})$, of the store obtained by prepending $x$ to the list with head\u2026","title":"Contents after prepending to a singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/insert-after","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.insertAfter","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#insert-after","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Let a store $S$ assign to each identity below $n$ either nothing or a singly linked list node whose next pointer is an identity below $n$ or NIL, and let $x$ and $y$ be identities whose store entries are nodes. Inserting $x$ after $y$ first replaces cell $x$ by the node of $x$ with next pointer replaced by the next pointer of $y$ in $S$, then, in the store so obtained, replaces the next pointer of the node in cell\u2026","title":"Inserting a node after another in a singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/insert-after-get","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.insertAfter_get","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#insert-after-get","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Let $S$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $x$ and $y$ be identities whose store entries are nodes, and let $z$ be any identity. Let $c$ be the node of $x$ in $S$ with next pointer replaced by the next pointer of $y$ in $S$ if $z = x$, and the content of cell $z$ in $S$ otherwise. Then cell $z$ of the store obtained by inserting $x$ after $y$ is $c$ with\u2026","title":"Cells after singly-linked-list insertion"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/insert-after-frame","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.insertAfter_frame","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#insert-after-frame","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Let $S$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, and let $x$ and $y$ be identities whose store entries are nodes. For every identity $z$ with $z \\ne y$ and $z \\ne x$, cell $z$ of the store obtained by inserting $x$ after $y$ equals cell $z$ of $S$.","title":"Frame property of singly-linked-list insertion"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/insert-after-payload","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.insertAfter_payload","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#insert-after-payload","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Let $S$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, and let $x$ and $y$ be identities whose store entries are nodes. For every identity $z$, cell $z$ of the store obtained by inserting $x$ after $y$ is empty exactly when cell $z$ of $S$ is empty, and otherwise holds a node with the same payload as the node in cell $z$ of $S$.","title":"Singly-linked-list insertion preserves payloads"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/insert-after-represents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.insertAfter_represents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#insert-after-represents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Suppose the store $S$ with head $h$ represents the list of distinct identities $(i_0, \\dots, i_{m-1})$. Let $k < m$, assume that the store entry of $i_k$ is a node, and let $x$ be an identity whose store entry is a node and which is not among $i_0, \\dots, i_{m-1}$. Then the store obtained by inserting $x$ after $i_k$, with the same head $h$, represents the list $(i_0, \\dots, i_k, x, i_{k+1}, \\dots, i_{m-1})$.","title":"Insertion into a represented singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/insert-after-contents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.insertAfter_contents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#insert-after-contents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Let $S$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $(i_0, \\dots, i_{m-1})$ be a list of identities below $n$, let $k < m$, and let $x$ be an identity such that the store entries of $x$ and $i_k$ are nodes, the node of $x$ having payload $a$. Then the contents of the store obtained by inserting $x$ after $i_k$ at the list\u2026","title":"Contents after singly-linked-list insertion"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/delete-head","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.deleteHead","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#delete-head","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Let a store assign to each identity below $n$ either nothing or a singly linked list node whose next pointer is an identity below $n$ or NIL, and let the head be an identity below $n$ or NIL. Deleting the head returns the store unchanged together with a new head: NIL if the head is NIL or the store entry at the head is empty, and otherwise the next pointer of the node stored at the head.","title":"Deleting the head of a singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/delete-head-represents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.deleteHead_represents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#delete-head-represents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, and suppose that $s$ with head $h$ represents the list of identities $L$, where $h$ is an identity below $n$ or NIL. Let $(s', h')$ be the store and head returned by deleting the head of $s$ with head $h$. Then $s'$ with head $h'$ represents $L$ with its first entry removed, which is the empty\u2026","title":"Deleting the head removes the first identity"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/delete-after","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.deleteAfter","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#delete-after","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node whose next pointer is an identity below $n$ or NIL, and let $y$ be an identity such that $s(y)$ is a node. Deleting after $y$ returns the store that agrees with $s$ except at $y$, where it holds the node $s(y)$ with its next pointer replaced as follows: if the next pointer of $s(y)$ is an identity\u2026","title":"Deleting the successor in a singly linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/delete-after-get","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.deleteAfter_get","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#delete-after-get","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $y$ be an identity such that $s(y)$ is a node, and let $s'$ be the store obtained from $s$ by deleting the successor of $y$. Let $c$ be the node $s(y)$ with its next pointer replaced by the next pointer of $s(m)$ if the next pointer of $s(y)$ is an identity $m$ such that $s(m)$ is a node, and\u2026","title":"Cells after deleting a successor"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/delete-after-frame","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.deleteAfter_frame","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#delete-after-frame","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $y$ be an identity such that $s(y)$ is a node, and let $s'$ be the store obtained from $s$ by deleting the successor of $y$. Then $s'(z) = s(z)$ for every identity $z \\ne y$.","title":"Deleting a successor leaves other cells unchanged"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/delete-after-payload","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.deleteAfter_payload","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#delete-after-payload","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $y$ be an identity such that $s(y)$ is a node, and let $s'$ be the store obtained from $s$ by deleting the successor of $y$. Then for every identity $z$, either $s(z)$ and $s'(z)$ are both nothing, or both are nodes with the same payload.","title":"Deleting a successor preserves payloads"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/delete-after-represents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.deleteAfter_represents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#delete-after-represents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, and suppose that $s$ with head $h$ represents the list of identities $L = (i_0, \\dots, i_{m-1})$. Let $k < m$ be such that $s(i_k)$ is a node, and let $s'$ be the store obtained from $s$ by deleting the successor of $i_k$. Then $s'$ with the same head $h$ represents the list obtained from $L$ by\u2026","title":"Deleting a successor removes it from the represented list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/delete-after-contents","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.deleteAfter_contents","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#delete-after-contents","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, let $L = (i_0, \\dots, i_{m-1})$ be a list of identities below $n$, and let $k < m$ be such that $s(i_k)$ is a node. Let $s'$ be the store obtained from $s$ by deleting the successor of $i_k$. Then the contents of $s'$ at the list obtained from $L$ by removing the entry at position $k + 1$ equal\u2026","title":"Contents after deleting a successor"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/search-aux","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.searchAux","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#search-aux","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Given a node pool with $n$ cells, a target payload $t$, a fuel $f \\in \\mathbb{N}$ and a start $h$ that is either an identity in $\\{0, \\dots, n-1\\}$ or NIL, fuel-bounded search for $t$ from $h$ with fuel $f$ returns nothing if $f = 0$ or $h$ is NIL. Otherwise $h$ is an identity $i$: the search returns nothing if cell $i$ is unallocated, returns $i$ if the node in cell $i$ has payload $t$, and otherwise continues\u2026","title":"Fuel-bounded search in a singly-linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/search","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.search","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#search","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Given a node pool with $n$ cells, a head $h$ that is either an identity in $\\{0, \\dots, n-1\\}$ or NIL, and a target payload $t$, the search for $t$ from $h$ is fuel-bounded search for $t$ from $h$ with fuel $n$. It follows next pointers from $h$ and returns the first visited identity whose node has payload $t$, returning nothing if it reaches NIL or an unallocated cell first, or if $n$ steps run out.","title":"Search in a singly-linked list"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/search-aux-eq-find","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.searchAux_eq_find?","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#search-aux-eq-find","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Suppose a node pool represents, from the head $h$, the list of distinct identities $\\ell$. Then for every target payload $t$ and every fuel $f$ with $f \\ge |\\ell|$, fuel-bounded search for $t$ from $h$ with fuel $f$ returns the first identity in $\\ell$ whose cell holds a node with payload $t$, and returns nothing if no identity in $\\ell$ has this property.","title":"Fuel-bounded search finds the first matching represented identity"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/represents-length-le-capacity","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.represents_length_le_capacity","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#represents-length-le-capacity","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"Fix a capacity $n$, let $s$ be a store assigning to each identity below $n$ either nothing or a singly linked list node, and let $\\mathrm{head}$ be an identity below $n$ or NIL. If $s$ and $\\mathrm{head}$ represent the list of identities $i_0, \\dots, i_{k-1}$, then $k \\le n$.","title":"Represented lists fit in the store"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/search-eq-find","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.search_eq_find?","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#search-eq-find","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"If a node pool represents, from the head $h$, the list of distinct identities $\\ell$, then for every target payload $t$ the search for $t$ from $h$ returns the first identity in $\\ell$ whose cell holds a node with payload $t$, and returns nothing if no identity in $\\ell$ has this property.","title":"Search finds the first matching represented identity"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/search-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.search_eq_some","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#search-eq-some","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"If a node pool represents, from the head $h$, the list of distinct identities $\\ell$, and the search for a target payload $t$ from $h$ returns an identity $x$, then $x$ belongs to $\\ell$ and cell $x$ holds a node with payload $t$.","title":"A successful search returns a matching represented identity"},{"id":"ComputerScience/data-structures/linear-structures/singly-linked-list/search-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.SinglyLinkedList.search_eq_none_iff","parent":37652,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/singly-linked-list/index.html#search-eq-none-iff","scope":"ComputerScience/data-structures/linear-structures/singly-linked-list","status":"fully_proved","summary":"If a node pool represents, from the head $h$, the list of distinct identities $\\ell$, then for every target payload $t$ the search for $t$ from $h$ returns nothing if and only if no identity in $\\ell$ has a cell holding a node with payload $t$.","title":"Search fails exactly when no represented node has the target payload"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/stack-with-multipop.json","id":"ComputerScience/data-structures/linear-structures/stack-with-multipop","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","title":"Stack with multipop"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/push","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.push","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#push","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"Stacks over a type $\\alpha$ are lists of elements of $\\alpha$, with the first element as the top. Pushing $v \\in \\alpha$ onto a stack $s$ returns the stack with top $v$ followed by the elements of $s$, at cost $1$ (one successful push event).","title":"Stack push"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/pop","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.pop","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#pop","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"Stacks over a type $\\alpha$ are lists of elements of $\\alpha$, with the first element as the top. Popping the empty stack returns underflow (no value) at cost $0$. Popping a stack with top $v$ and remaining stack $t$ returns the pair $(v, t)$ at cost $1$ (one successful pop event).","title":"Stack pop"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/top","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.top","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#top","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"Stacks over a type $\\alpha$ are lists of elements of $\\alpha$, with the first element as the top. Reading the top of the empty stack returns underflow (no value) at cost $0$. Reading the top of a stack with top $v$ returns $v$ at cost $1$ (one successful top event); the stack itself is not returned or changed.","title":"Stack top"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/pop-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.pop_eq","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#pop-eq","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"Let $s$ be a stack over a type $\\alpha$ (a list of elements of $\\alpha$ whose first element is the top). If $s$ is empty, popping $s$ returns underflow (no value) at cost $0$. If $s$ has top $h$ and remaining stack $t$, popping $s$ returns the pair $(h, t)$ at cost $1$.","title":"Result and cost of stack pop"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/top-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.top_eq","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#top-eq","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"Let $s$ be a stack over a type $\\alpha$ (a list of elements of $\\alpha$ whose first element is the top). If $s$ is empty, reading the top of $s$ returns underflow (no value) at cost $0$. If $s$ has top $h$, reading the top of $s$ returns $h$ at cost $1$.","title":"Result and cost of reading the stack top"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/operations-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.operations_spec","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#operations-spec","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"Let $\\alpha$ be a type, let $s$ be a stack over $\\alpha$ (a list of elements of $\\alpha$ whose first element is the top) and let $v \\in \\alpha$. Pushing $v$ onto $s$ returns the stack with top $v$ followed by the elements of $s$, at cost $1$. Popping $s$ returns underflow (no value) at cost $0$ if $s$ is empty, and returns the pair $(h, t)$ at cost $1$ if $s$ has top $h$ and remaining stack $t$. Reading the top of\u2026","title":"Result and cost of the stack operations"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/multipop","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.multipop","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#multipop","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For a stack $S$, stored as a list with its top first, and a natural number $k$, MULTIPOP of $S$ with count $k$ repeatedly removes the top element while the stack is nonempty and fewer than $k$ elements have been removed, and returns the remaining stack. Each removed element costs one unit, and nothing else is charged.","title":"MULTIPOP"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/multipop-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.multipop_ret","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#multipop-ret","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For every stack $S$, stored as a list with its top first, and every natural number $k$, the stack returned by MULTIPOP of $S$ with count $k$ is $S$ with its top $k$ elements removed, or the empty stack if $S$ has fewer than $k$ elements.","title":"Result of MULTIPOP"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/multipop-time","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.multipop_time","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#multipop-time","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For every stack $S$ and every natural number $k$, the cost of MULTIPOP of $S$ with count $k$ is $\\min(|S|, k)$, where $|S|$ is the number of elements of $S$.","title":"Cost of MULTIPOP"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/multipop-signed","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.multipopSigned","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#multipop-signed","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For a stack $S$ and an integer $k$, MULTIPOP with integer count $k$ is MULTIPOP of $S$ with the natural count $\\max(k, 0)$, with the same result and cost. In particular, a count $k \\le 0$ removes nothing.","title":"MULTIPOP with an integer count"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/multipop-signed-of-nonpos","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.multipopSigned_of_nonpos","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#multipop-signed-of-nonpos","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For every stack $S$ and every integer $k \\le 0$, MULTIPOP with integer count $k$ applied to $S$ returns $S$ at cost $0$.","title":"MULTIPOP with a nonpositive count does nothing"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/operation","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.Operation","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#operation","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For a type $\\alpha$ of stack entries, a stack operation is one of the following: $\\mathrm{PUSH}(x)$ for an element $x \\in \\alpha$, $\\mathrm{POP}$, or $\\mathrm{MULTIPOP}(k)$ for an integer count $k$.","title":"Stack operations for the MULTIPOP aggregate analysis"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/execute","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.execute","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#execute","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"Given a stack $S$ and a finite sequence of operations, each a PUSH of a value, a POP or a MULTIPOP with an integer count, the operations are applied to $S$ in order. PUSH of $x$ puts $x$ on top at cost $1$; POP removes the top element at cost $1$; MULTIPOP with count $k$ is MULTIPOP with an integer count, costing one unit per element removed. If a POP meets an empty stack, the run stops and reports an error, that\u2026","title":"Running a sequence of stack operations"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/execute-push-eq-bind-push","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.execute_push_eq_bind_push","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#execute-push-eq-bind-push","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For every stack $S$, every value $x$ and every finite sequence $\\sigma$ of stack operations, running PUSH $x$ followed by $\\sigma$ on $S$ has the same result and cost as first performing the standalone PUSH of $x$ on $S$ and then running $\\sigma$ on the resulting stack, the two costs being added.","title":"Running a PUSH agrees with the standalone PUSH"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/execute-pop-eq-bind-pop","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.execute_pop_eq_bind_pop","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#execute-pop-eq-bind-pop","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For every stack $S$ and every finite sequence $\\sigma$ of stack operations, running POP followed by $\\sigma$ on $S$ has the same result and cost as first performing the standalone POP on $S$ and then, if it reports underflow, stopping with an error at no further cost, or, if it removes the top element and leaves the stack $S'$, running $\\sigma$ on $S'$, the two costs being added.","title":"Running a POP agrees with the standalone POP"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/execute-balance","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.execute_balance","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#execute-balance","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For every stack $S$ and every finite sequence $\\sigma$ of stack operations, if running $\\sigma$ on $S$ succeeds with final stack $F$, then the cost of the run plus the number of elements of $F$ equals the number of elements of $S$ plus twice the number of PUSH operations in $\\sigma$.","title":"Cost balance for stack operation sequences"},{"id":"ComputerScience/data-structures/linear-structures/stack-with-multipop/execute-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Stack.execute_time_le","parent":37689,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/stack-with-multipop/index.html#execute-time-le","scope":"ComputerScience/data-structures/linear-structures/stack-with-multipop","status":"fully_proved","summary":"For every stack $S$ and every sequence of $n$ stack operations (PUSH, POP and MULTIPOP), if running the sequence on $S$ succeeds, that is, no POP meets an empty stack, then its cost is at most $|S| + 2n$, where $|S|$ is the number of elements of $S$.","title":"Aggregate cost of stack operation sequences"},{"data_url":"scopes/ComputerScience/data-structures/linear-structures/two-list-queue.json","id":"ComputerScience/data-structures/linear-structures/two-list-queue","kind":"scope","navigation":"module","parent":37406,"result":false,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","title":"Two-list queue"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/potential","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.potential","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#potential","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For a queue stored as an enqueue list and a dequeue list, the potential is the length of the enqueue list: the number of elements that will be reversed when they are moved to the dequeue list.","title":"Potential of a two-list queue"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/dequeue-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.dequeueCost","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#dequeue-cost","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For a queue stored as an enqueue list and a dequeue list, the modeled cost of dequeueing is $1$ when the dequeue list is nonempty, and $m + 1$ when the dequeue list is empty, where $m$ is the length of the enqueue list (the number of elements reversed when they are moved to the dequeue list).","title":"Modeled cost of a dequeue"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/enqueue","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.enqueue","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#enqueue","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"Enqueueing a value $v$ onto a two-list queue $q$ returns the queue obtained from $q$ by adding $v$ at the back, that is, by prepending $v$ to the enqueue list, together with a recorded cost of $1$.","title":"Timed enqueue"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/enqueue-amortized","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.enqueue_amortized","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#enqueue-amortized","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every two-list queue $q$ and every value $v$, the modeled cost of enqueueing $v$ onto $q$ plus the potential of the resulting queue equals the potential of $q$ plus $2$.","title":"Amortized cost of enqueueing"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/dequeue","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.dequeue","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#dequeue","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"Dequeueing from a two-list queue $q$ returns the next element in first-in, first-out order together with the remaining queue, or returns no element together with $q$ itself when $q$ is empty. In both cases the recorded cost is the modeled dequeue cost of $q$.","title":"Timed dequeue"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/dequeue-amortized","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.dequeue_amortized","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#dequeue-amortized","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every two-list queue $q$, the modeled cost of dequeueing from $q$ plus the potential of the resulting queue equals the potential of $q$ plus $1$. This includes the case where $q$ is empty.","title":"Amortized cost of dequeueing"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/operation","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.Operation","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#operation","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"A queue operation over a type $\\alpha$ is either a request to enqueue a value $v \\in \\alpha$ or a request to dequeue. Sequences of such operations are used to state aggregate cost bounds.","title":"Queue operation"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/charge","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.Operation.charge","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#charge","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"The potential-method charge of a queue operation is $2$ for an enqueue and $1$ for a dequeue.","title":"Amortized charge of a queue operation"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/step","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.step","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#step","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For a queue $q$, an enqueue operation with value $v$ performs the costed enqueue of $v$ onto $q$. A dequeue operation performs the costed dequeue of $q$ and keeps the remaining queue and the cost, discarding the removed value; on an empty queue the remaining queue is $q$ itself.","title":"One step of a queue operation"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/step-amortized","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.step_amortized","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#step-amortized","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every queue $q$ and every queue operation $o$, the modeled cost of performing $o$ on $q$ plus the potential of the resulting queue equals the potential of $q$ plus the charge of $o$.","title":"Amortized cost of one queue operation"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/run","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.run","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#run","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"Running a list of queue operations from a queue $q$ performs the operations one step at a time from left to right, starting at $q$. The result is the final queue, and the cost is the sum of the costs of the individual steps; running the empty list returns $q$ at cost $0$.","title":"Running a sequence of queue operations"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/total-charge","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.totalCharge","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#total-charge","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For a list of queue operations, the total charge is the sum of the amortized charges of its operations, where an enqueue is charged $2$ and a dequeue is charged $1$.","title":"Total charge of a sequence of queue operations"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/run-amortized","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.run_amortized","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#run-amortized","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every list of queue operations and every two-list queue $q$, the modeled cost of running the operations from left to right starting at $q$, plus the potential of the final queue, equals the potential of $q$ plus the total charge of the operations.","title":"Amortized cost of a sequence of queue operations"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/run-time-le-potential-add-twice-length","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.Amortized.run_time_le_potential_add_twice_length","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#run-time-le-potential-add-twice-length","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every list of $n$ queue operations and every two-list queue $q$, the modeled cost of running the operations from left to right starting at $q$ is at most the potential of $q$ plus $2n$.","title":"Linear bound on the cost of queue operations"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/contents","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.contents","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#contents","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"A queue is stored as a dequeue list and an enqueue list whose most recently enqueued element comes first. Its contents are its elements in FIFO order: the dequeue list followed by the reverse of the enqueue list.","title":"Contents of a two-list queue"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/to-array-to-list","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.toArray_toList","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#to-array-to-list","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every queue $q$, the array obtained from $q$, read as a list, equals the contents of $q$.","title":"Array conversion lists the contents"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/contents-enqueue","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.contents_enqueue","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#contents-enqueue","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every queue $q$ and value $v$, the contents of the queue obtained by enqueueing $v$ onto $q$ are the contents of $q$ followed by $v$.","title":"Enqueueing appends to the contents"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/is-empty-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.isEmpty_eq_true_iff","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#is-empty-eq-true-iff","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every queue $q$, the emptiness test of $q$ returns true if and only if the contents of $q$ are empty.","title":"The queue emptiness test matches the contents"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/dequeue-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.dequeue?_eq_none_iff","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#dequeue-eq-none-iff","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"For every queue $q$, dequeueing $q$ returns nothing if and only if the contents of $q$ are empty.","title":"Dequeueing fails exactly on empty queues"},{"id":"ComputerScience/data-structures/linear-structures/two-list-queue/contents-eq-cons-of-dequeue-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.Queue.contents_eq_cons_of_dequeue?_eq_some","parent":37707,"result":true,"route":"../roadmap/ComputerScience/data-structures/linear-structures/two-list-queue/index.html#contents-eq-cons-of-dequeue-eq-some","scope":"ComputerScience/data-structures/linear-structures/two-list-queue","status":"fully_proved","summary":"If dequeueing a queue $q$ succeeds and returns an element $a$ and a queue $r$, then the contents of $q$ are $a$ followed by the contents of $r$.","title":"A successful dequeue splits off the first element"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures.json","id":"ComputerScience/data-structures/range-query-structures","kind":"scope","parent":36418,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/index.html","scope":"ComputerScience/data-structures/range-query-structures","status":"fully_proved","summary":"Structures answering range, interval or nearest-point queries: segment trees, Fenwick trees, sparse tables, interval trees, range trees, kd-trees, ball trees and fractional cascading.","title":"Range-query and spatial structures"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/ball-tree.json","id":"ComputerScience/data-structures/range-query-structures/ball-tree","kind":"scope","navigation":"module","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","title":"Ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/ball-tree","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#ball-tree","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For a dimension $d$, a ball tree over the integer points $\\mathbb{Z}^d$ (functions $\\{0, \\dots, d-1\\} \\to \\mathbb{Z}$) is either empty, a leaf holding one point, or a node holding a center point $c \\in \\mathbb{Z}^d$, a natural number $b$ (the squared-radius bound of the node's sphere) and two child ball trees.","title":"Ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/to-list-aux","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.toListAux","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#to-list-aux","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For a ball tree $T$ and a list $A$ of points, the accumulating traversal returns $A$ when $T$ is empty, the point $p$ followed by $A$ when $T$ is a leaf holding $p$, and, when $T$ is a node with children $L$ and $R$, the accumulating traversal of $L$ whose accumulator is the accumulating traversal of $R$ with accumulator $A$.","title":"Accumulating traversal of a ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/to-list","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.toList","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#to-list","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"The point list of a ball tree is the list of the points held at its leaves, from left to right: the empty tree gives the empty list, a leaf holding $p$ gives the list $[p]$, and a node gives the point list of its left subtree followed by the point list of its right subtree. The centers stored at nodes are not included.","title":"Point list of a ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/size","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.size","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#size","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"The size of a ball tree is the number of points held in its leaves: it is $0$ for the empty tree, $1$ for a leaf, and the sum of the sizes of the two children for a node; the center of a node is not counted.","title":"Size of a ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/length-to-list","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.length_toList","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#length-to-list","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For every ball tree $t$, the list of points stored in $t$, read from left to right, has length equal to the size of $t$.","title":"Number of points of a ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/dist-sq","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.distSq","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#dist-sq","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For points $p, q \\in \\mathbb{Z}^d$, the squared distance from $p$ to $q$ is the natural number $\\sum_{i=0}^{d-1} (p_i - q_i)^2$, computed in $\\mathbb{Z}$ and then converted to $\\mathbb{N}$.","title":"Squared Euclidean distance"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/dist-sq-comm","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.distSq_comm","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#dist-sq-comm","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For all points $p, q \\in \\mathbb{Z}^d$, the squared distance from $p$ to $q$ equals the squared distance from $q$ to $p$.","title":"Symmetry of squared distance"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/dist-sq-le-two-mul","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.distSq_le_two_mul","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#dist-sq-le-two-mul","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For all points $a, b, c \\in \\mathbb{Z}^d$, writing $\\delta(x, y)$ for the squared distance from $x$ to $y$, we have $\\delta(a, c) \\le 2\\,(\\delta(a, b) + \\delta(b, c))$.","title":"Relaxed triangle inequality for squared distance"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/radius-sq","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.radiusSq","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#radius-sq","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For a center $c \\in \\mathbb{Z}^d$ and a list $P$ of points of $\\mathbb{Z}^d$, the squared covering radius of $P$ about $c$ is the largest squared distance from $c$ to a point of $P$, or $0$ if $P$ is empty.","title":"Squared covering radius"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/dist-sq-le-radius-sq","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.distSq_le_radiusSq","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#dist-sq-le-radius-sq","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"Let $c \\in \\mathbb{Z}^d$ and let $L$ be a list of points of $\\mathbb{Z}^d$. For every point $p$ of $L$, the squared distance from $c$ to $p$ is at most the squared radius of $L$ around $c$, the largest squared distance from $c$ to a point of $L$.","title":"Points lie within the squared radius"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/search-aux","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.searchAux","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#search-aux","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"Let $d$ be a dimension and write $\\delta(x, y)$ for the squared distance from $x$ to $y$ in $\\mathbb{Z}^d$. For a ball tree $T$ over $\\mathbb{Z}^d$, a query point $q \\in \\mathbb{Z}^d$, a squared radius $r \\in \\mathbb{N}$ and a list $A$ of points, the accumulating search of $T$ for $q$ and $r$ with accumulator $A$ is $A$ when $T$ is empty; when $T$ is a leaf holding $p$, it is $p$ followed by $A$ if\u2026","title":"Accumulating ball-tree search"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/search","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.search","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#search","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"The search of a ball tree $T$ for a query point $q \\in \\mathbb{Z}^d$ and a squared radius $r \\in \\mathbb{N}$ is the accumulating search of $T$ for $q$ and $r$ started with the empty accumulator.","title":"Ball-tree similarity search"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.Valid","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#valid","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"Validity of a ball tree is defined inductively: the empty tree and every leaf are valid, and a node with center $c$ and squared-radius bound $b$ is valid when both of its subtrees are valid and every point in the point list of either subtree lies within squared Euclidean distance $b$ of $c$.","title":"Bounding-sphere invariant of a ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/search-eq-filter","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.search_eq_filter","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#search-eq-filter","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For every ball tree $T$ over $\\mathbb{Z}^d$ satisfying the bounding-sphere invariant, every query point $q \\in \\mathbb{Z}^d$ and every $r \\in \\mathbb{N}$, the search of $T$ for $q$ and $r$ equals the list of points of $T$, from left to right, filtered to the points $p$ whose squared distance from $q$ is at most $r$, in their order. Pruning against the bounding spheres loses no point.","title":"Ball-tree search is exact"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/mem-search-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.mem_search_iff","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#mem-search-iff","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"Let $t$ be a ball tree satisfying the bounding-sphere invariant. For all $q, p \\in \\mathbb{Z}^d$ and every natural number $r$, $p$ is returned by the similarity search of $t$ with query $q$ and squared radius $r$ if and only if $p$ is one of the points stored in $t$ and the squared distance from $q$ to $p$ is at most $r$.","title":"Search of a valid ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/build","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.build","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#build","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"The ball tree built from a list of points of $\\mathbb{Z}^d$ is defined by recursion on the list. The empty list gives the empty tree and a one-point list $[p]$ gives the leaf $p$. A list $L = [x_1, \\dots, x_n]$ with $n \\ge 2$ gives a node whose center is $x_1$, whose bound is the squared radius of $L$ around $x_1$ (the largest squared distance from $x_1$ to a point of $L$), whose left child is the tree built from\u2026","title":"Building a ball tree by halving"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/valid-build","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.valid_build","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#valid-build","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For every list of points in $\\mathbb{Z}^d$, the ball tree built from it by recursive halving satisfies the bounding-sphere invariant.","title":"Ball trees built by halving are valid"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/to-list-build-perm","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.toList_build_perm","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#to-list-build-perm","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For every list $P$ of points of $\\mathbb{Z}^d$, the list of points of the ball tree built from $P$, from left to right, is a permutation of $P$.","title":"A built ball tree stores a permutation of its input"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/mem-to-list-build-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.mem_toList_build_iff","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#mem-to-list-build-iff","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For every list $L$ of points of $\\mathbb{Z}^d$ and every point $p \\in \\mathbb{Z}^d$, $p$ is one of the points stored in the ball tree built from $L$ if and only if $p$ occurs in $L$.","title":"Points of a built ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/size-build","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.size_build","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#size-build","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For every list $P$ of points of $\\mathbb{Z}^d$, the size of the ball tree built from $P$ equals the length of $P$, that is, the number of entries of $P$ counted with repetition.","title":"Size of a built ball tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/search-build","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.search_build","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#search-build","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For every list $P$ of points of $\\mathbb{Z}^d$, every query point $q \\in \\mathbb{Z}^d$ and every $r \\in \\mathbb{N}$, let $T$ be the ball tree built from $P$. The search of $T$ for $q$ and $r$ equals the list of points of $T$, from left to right, filtered to the points $p$ whose squared distance from $q$ is at most $r$, in their order.","title":"Ball-tree search of a built tree"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/search-build-perm","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.search_build_perm","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#search-build-perm","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"For every list $P$ of points of $\\mathbb{Z}^d$, every query point $q \\in \\mathbb{Z}^d$ and every $r \\in \\mathbb{N}$, the search of the ball tree built from $P$ for $q$ and $r$ is a permutation of the sublist of $P$ consisting of the points $p$ whose squared distance from $q$ is at most $r$, kept with their multiplicities.","title":"Ball-tree search of a built tree, up to order"},{"id":"ComputerScience/data-structures/range-query-structures/ball-tree/mem-search-build-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BallTree.mem_search_build_iff","parent":37729,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/ball-tree/index.html#mem-search-build-iff","scope":"ComputerScience/data-structures/range-query-structures/ball-tree","status":"fully_proved","summary":"Let $L$ be a list of points of $\\mathbb{Z}^d$, let $q, p \\in \\mathbb{Z}^d$ and let $r$ be a natural number. Then $p$ is returned by the similarity search with query $q$ and squared radius $r$ on the ball tree built from $L$ if and only if $p$ occurs in $L$ and the squared distance from $q$ to $p$ is at most $r$.","title":"Search of a built ball tree"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/fenwick-tree.json","id":"ComputerScience/data-structures/range-query-structures/fenwick-tree","kind":"scope","navigation":"module","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","title":"Fenwick tree"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/fenwick-tree","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#fenwick-tree","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"A Fenwick tree (binary indexed tree) over a commutative additive monoid $\\alpha$ is an array of elements of $\\alpha$, read as one-indexed interval sums with an unused dummy entry at index $0$. The structure itself imposes no condition on the entries, and the underlying array it represents is not stored; the relation saying that a tree represents a given array is defined separately.","title":"Fenwick tree"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/lowbit","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.lowbit","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#lowbit","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For a natural number $n > 0$, $\\operatorname{lowbit}(n)$ is the largest power of two dividing $n$; for $n = 0$ it is defined as $\\operatorname{lowbit}(0) = 0$.","title":"Lowest set bit"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/lowbit-zero","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.lowbit_zero","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#lowbit-zero","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"By convention, $\\operatorname{lowbit}(0) = 0$.","title":"Lowbit of zero"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/lowbit-pos","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.lowbit_pos","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#lowbit-pos","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every natural number $n > 0$, $\\operatorname{lowbit}(n) > 0$, where $\\operatorname{lowbit}(n)$ is the value of the lowest set bit of $n$ in binary.","title":"The lowest set bit of a positive number is positive"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/lowbit-le","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.lowbit_le","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#lowbit-le","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every natural number $n$, $\\operatorname{lowbit}(n) \\le n$, where $\\operatorname{lowbit}(n)$ is the value of the lowest set bit of $n$ in binary (with $\\operatorname{lowbit}(0) = 0$).","title":"The lowest set bit is at most the number"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/lowbit-dvd","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.lowbit_dvd","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#lowbit-dvd","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every natural number $n > 0$, $\\operatorname{lowbit}(n)$ divides $n$, where $\\operatorname{lowbit}(n)$ is the value of the lowest set bit of $n$ in binary.","title":"The lowest set bit divides the number"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/lowbit-two-pow","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.lowbit_two_pow","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#lowbit-two-pow","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every natural number $n > 0$, $\\operatorname{lowbit}(n)$ is a power of two: $\\operatorname{lowbit}(n) = 2^k$ for some natural number $k$.","title":"The lowest set bit is a power of two"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/lowbit-div-odd","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.lowbit_div_odd","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#lowbit-div-odd","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every natural number $n > 0$, the quotient $n / \\operatorname{lowbit}(n)$ is odd, where $\\operatorname{lowbit}(n)$ is the value of the lowest set bit of $n$ in binary.","title":"Dividing out the lowest set bit leaves an odd number"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/block-sum","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.blockSum","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#block-sum","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For a list $a$ of elements of a commutative additive monoid $\\alpha$ and a natural number $j$, the interval sum at the one-indexed position $j$ is the sum of the $\\operatorname{lowbit}(j)$ entries of $a$ ending at zero-indexed position $j - 1$, where $\\operatorname{lowbit}(j)$ is the largest power of two dividing $j$ (with $\\operatorname{lowbit}(0) = 0$) and entries past the end of $a$ count as $0$.","title":"Fenwick-tree interval sum of a list"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/block-sum-array","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.blockSumArray","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#block-sum-array","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For an array $a$ of elements of a commutative additive monoid $\\alpha$ and a natural number $j$, this is the sum of the $\\operatorname{lowbit}(j)$ entries of $a$ ending at zero-indexed position $j - 1$, where $\\operatorname{lowbit}(j)$ is the largest power of two dividing $j$ (with $\\operatorname{lowbit}(0) = 0$) and entries past the end of $a$ count as $0$.","title":"Interval sum read from an array"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.Valid","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#valid","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"A Fenwick tree $t$ over an additive commutative monoid represents a list $a$ of length $n$ when its stored array has length $n + 1$ (a dummy entry at index $0$ plus one entry per element of $a$) and, for every $j$ with $1 \\le j \\le n$, the stored entry at index $j$ equals the block sum of $a$ at $j$: the sum of the $\\operatorname{lowbit}(j)$ entries of $a$ ending at position $j - 1$ (positions counted from $0$),\u2026","title":"Fenwick tree representing a list"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/size","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.size","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#size","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"The size of a Fenwick tree is the length of its stored array minus one, truncated at $0$; the one accounts for the dummy entry at index $0$. It is read off the stored array in constant time.","title":"Size of a Fenwick tree"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/size-eq-length-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.size_eq_length_of_valid","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#size-eq-length-of-valid","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"If a Fenwick tree $t$ over an additive commutative monoid represents a list $a$, then the size of $t$ equals the length of $a$.","title":"A Fenwick tree has the length of the list it represents"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/build","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.build","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#build","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For a list $a$ of length $n$ over a commutative additive monoid $\\alpha$, the Fenwick tree built from $a$ stores the array of length $n + 1$ whose entry at index $0$ is $0$ and whose entry at each one-indexed position $j$ with $1 \\le j \\le n$ is the sum of the $\\operatorname{lowbit}(j)$ entries of $a$ ending at zero-indexed position $j - 1$, where $\\operatorname{lowbit}(j)$ is the largest power of two dividing $j$.","title":"Building a Fenwick tree from a list"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/size-build","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.size_build","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#size-build","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every list $a$ over an additive commutative monoid, the Fenwick tree built from $a$ has size equal to the length of $a$.","title":"Size of a built Fenwick tree"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/valid-build","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.valid_build","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#valid-build","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every list $a$ over an additive commutative monoid, the Fenwick tree built from $a$ represents $a$.","title":"Building a Fenwick tree yields a valid representation"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/prefix-sum","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.prefixSum","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#prefix-sum","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"Let $t$ be a Fenwick tree over an additive commutative monoid with $n$ underlying entries, let $k$ be a natural number and let $m = \\min(k, n)$. The prefix sum of length $k$ of $t$ is the sum of the stored entries of $t$ at the positions $j_0 = m$, $j_{r+1} = j_r - \\operatorname{lowbit}(j_r)$, taken for as long as $j_r > 0$.","title":"Fenwick tree prefix sum"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/prefix-sum-zero","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.prefixSum_zero","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#prefix-sum-zero","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every Fenwick tree $t$ over an additive commutative monoid, the prefix sum of length $0$ of $t$ is $0$.","title":"The empty prefix sum is zero"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/prefix-sum-eq-take-sum","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.prefixSum_eq_take_sum","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#prefix-sum-eq-take-sum","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"Let $t$ be a Fenwick tree representing a list $a$ of elements of an additive commutative monoid. Then for every natural number $k$, including $k > |a|$, the prefix sum of length $k$ of $t$ equals the sum of the first $\\min(k, |a|)$ entries of $a$.","title":"Correctness of Fenwick tree prefix sums"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/prefix-sum-build","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.prefixSum_build","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#prefix-sum-build","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For every list $a$ of elements of an additive commutative monoid and every natural number $k$, the prefix sum of length $k$ of the Fenwick tree built from $a$ equals the sum of the first $\\min(k, |a|)$ entries of $a$.","title":"Prefix sums of a built Fenwick tree"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/add","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.add","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#add","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"Let $t$ be a Fenwick tree over a commutative additive monoid $\\alpha$, let $n$ be its size (the length of its stored array minus one), let $i$ be a natural number and let $\\delta \\in \\alpha$. Adding $\\delta$ at underlying position $i$ adds $\\delta$ to the stored entries at the one-indexed positions $j_1 = i + 1$ and $j_{k+1} = j_k + \\operatorname{lowbit}(j_k)$, for as long as $j_k \\le n$, where\u2026","title":"Point update of a Fenwick tree"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/add-eq-self-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.add_eq_self_of_le","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#add-eq-self-of-le","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"Let $t$ be a Fenwick tree over a commutative additive monoid $\\alpha$, and let $n$ be its size (the length of its stored array minus one). For every natural number $i \\ge n$ and every $\\delta \\in \\alpha$, the point update of $t$ adding $\\delta$ at position $i$ returns $t$ itself.","title":"Out-of-range point updates leave a Fenwick tree unchanged"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/valid-add","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.valid_add","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#valid-add","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"Let $t$ be a Fenwick tree over an additive commutative monoid that represents a list $a$ of length $n$. For every index $i < n$ (counting from $0$) and every element $\\delta$, the tree obtained from $t$ by adding $\\delta$ at position $i$ represents the list obtained from $a$ by replacing its entry $a_i$ with $a_i + \\delta$.","title":"Point update preserves representation"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/prefix-sum-add","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.prefixSum_add","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#prefix-sum-add","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"Let $t$ be a Fenwick tree representing a list $a$ of elements of an additive commutative monoid, let $i < |a|$ be a position, let $\\delta$ be an element of the monoid, and let $k$ be any natural number, possibly larger than $|a|$. Then the prefix sum of length $k$ of the tree obtained from $t$ by adding $\\delta$ at position $i$ equals the prefix sum of length $k$ of $t$ plus $\\delta$ if $i < k$, and equals the\u2026","title":"Prefix sums after a point update"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/range-sum","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.rangeSum","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#range-sum","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"For a Fenwick tree $t$ over an additive commutative group and natural numbers $\\mathit{lo}$ and $\\mathit{hi}$, the range sum of $t$ over $[\\mathit{lo}, \\mathit{hi})$ is $0$ if $\\mathit{hi} \\le \\mathit{lo}$, and otherwise the prefix sum of length $\\mathit{hi}$ of $t$ minus the prefix sum of length $\\mathit{lo}$ of $t$.","title":"Fenwick tree range sum"},{"id":"ComputerScience/data-structures/range-query-structures/fenwick-tree/range-sum-eq","kind":"node","lean":"Cslib.Algorithms.Lean.FenwickTree.rangeSum_eq","parent":37753,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fenwick-tree/index.html#range-sum-eq","scope":"ComputerScience/data-structures/range-query-structures/fenwick-tree","status":"fully_proved","summary":"Let $t$ be a Fenwick tree representing a list $a = (a_0, \\dots, a_{n-1})$ of elements of an additive commutative group. Then for all natural numbers $\\mathit{lo}$ and $\\mathit{hi}$, the range sum of $t$ over $[\\mathit{lo}, \\mathit{hi})$ equals $\\sum_{\\mathit{lo} \\le i < \\min(\\mathit{hi}, n)} a_i$; in particular it is $0$ when $\\mathit{hi} \\le \\mathit{lo}$.","title":"Correctness of Fenwick tree range sums"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/fractional-cascading.json","id":"ComputerScience/data-structures/range-query-structures/fractional-cascading","kind":"scope","navigation":"module","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","title":"Fractional cascading"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/sample-aux","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.sampleAux","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#sample-aux","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a flag $b$ (true or false) and a list $\\ell$, the alternating selection of $\\ell$ with initial flag $b$ is defined by recursion on $\\ell$: the empty list gives the empty list; for a list with head $a$ and tail $\\ell'$, if $b$ is true it keeps $a$ and continues on $\\ell'$ with flag false, and if $b$ is false it drops $a$ and continues on $\\ell'$ with flag true.","title":"Alternating selection with a flag"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/sample","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.sample","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#sample","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"The sample of a list $[x_0, x_1, x_2, \\dots]$ is the list $[x_0, x_2, x_4, \\dots]$ of every other element, starting with the first, in their original order.","title":"Sample of a catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/sample-aux-sublist","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.sampleAux_sublist","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#sample-aux-sublist","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every flag $b$ (true or false) and every list $\\ell$, the alternating selection of $\\ell$ with initial flag $b$ is a sublist of $\\ell$: it is obtained from $\\ell$ by deleting some elements and keeping the rest in order.","title":"Alternating selection gives a sublist"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/sample-sublist","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.sample_sublist","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#sample-sublist","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every list $\\ell$, the sample of $\\ell$ is a sublist of $\\ell$: it is obtained from $\\ell$ by deleting some elements and keeping the rest in order.","title":"The sample is a sublist"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/mem-of-mem-sample","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.mem_of_mem_sample","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#mem-of-mem-sample","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a catalog (finite list) $L$, every entry of the sample of $L$ (every other entry of $L$, starting with the first) is an entry of $L$.","title":"Entries of a sample are entries of the catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/two-mul-length-sample-le","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.two_mul_length_sample_le","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#two-mul-length-sample-le","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every catalog $L$, twice the length of the sample of $L$ is at most the length of $L$ plus one.","title":"Doubled sample length bound"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-sample-le","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_sample_le","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-sample-le","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every list $xs$ (over any type), the sample of $xs$ has length at most $\\lfloor (|xs| + 1)/2 \\rfloor$, half the length of $xs$ rounded up: for an even length $2m$ the bound is $m$, and for an odd length $2m + 1$ it is $m + 1$.","title":"The sample has at most half the length, rounded up"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-sample-le-length","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_sample_le_length","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-sample-le-length","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every list $xs$ (over any type), the sample of $xs$, every other element starting with the first, has length at most the length of $xs$.","title":"The sample is no longer than the list"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/pairwise-sample","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.pairwise_sample","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#pairwise-sample","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"If a list over a linear order is sorted in nondecreasing order, then so is its sample.","title":"The sample of a sorted list is sorted"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/sorted-merge","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.sortedMerge","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#sorted-merge","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For lists $\\ell_1$ and $\\ell_2$ over a linearly ordered type, their sorted merge is the result of merge sort, with respect to $\\le$, applied to the concatenation of $\\ell_1$ followed by $\\ell_2$.","title":"Sorted merge of two catalogs"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/perm-sorted-merge","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.perm_sortedMerge","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#perm-sorted-merge","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For any lists $\\ell_1$ and $\\ell_2$ over a linear order, the sorted merge of $\\ell_1$ and $\\ell_2$ is a permutation of the concatenation of $\\ell_1$ followed by $\\ell_2$.","title":"A sorted merge rearranges the concatenation"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-sorted-merge","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_sortedMerge","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-sorted-merge","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For catalogs (finite lists) $L$ and $M$ over a linearly ordered type, the sorted merge of $L$ and $M$ has length equal to the length of $L$ plus the length of $M$.","title":"Length of a sorted merge"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/pairwise-sorted-merge","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.pairwise_sortedMerge","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#pairwise-sorted-merge","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For any lists $\\ell_1$ and $\\ell_2$ over a linear order, sorted or not, their sorted merge is sorted in nondecreasing order.","title":"A sorted merge is sorted"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/mem-sorted-merge-iff","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.mem_sortedMerge_iff","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#mem-sorted-merge-iff","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For lists $\\ell_1$ and $\\ell_2$ over a linear order and any element $v$, $v$ belongs to the sorted merge of $\\ell_1$ and $\\ell_2$ if and only if $v$ belongs to $\\ell_1$ or to $\\ell_2$.","title":"Elements of a sorted merge"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/lower-bound","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.lowerBound","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#lower-bound","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ of a linearly ordered type and a catalog (finite list) $L$, the lower bound of $x$ in $L$ is the number of entries of $L$ that are less than $x$, counted with multiplicity. For a sorted catalog this is the first index whose entry is not less than $x$, or the length of $L$ if there is none.","title":"Lower bound of a value in a catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/lower-bound-le-length","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.lowerBound_le_length","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#lower-bound-le-length","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and a catalog $L$ over a linearly ordered type, the lower bound of $x$ in $L$ (the number of entries of $L$ less than $x$) is at most the length of $L$.","title":"Lower bound is at most the catalog length"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/lower-bound-sorted-merge","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.lowerBound_sortedMerge","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#lower-bound-sorted-merge","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and catalogs $L$ and $M$ over a linearly ordered type, the lower bound of $x$ in the sorted merge of $L$ and $M$ equals the lower bound of $x$ in $L$ plus the lower bound of $x$ in $M$, where the lower bound of $x$ in a catalog is the number of its entries less than $x$.","title":"Lower bound in a sorted merge splits"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/lower-bound-sample-le","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.lowerBound_sample_le","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#lower-bound-sample-le","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and a catalog $L$ over a linearly ordered type, the number of entries less than $x$ in the sample of $L$ (every other entry of $L$, starting with the first) is at most the number of entries of $L$ less than $x$.","title":"Lower bound in a sample is at most the lower bound in the catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/count-lt","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.countLT","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#count-lt","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and a catalog $L$, the scan count of $x$ in $L$ is the number of elements of $L$ that are $< x$, computed by a direct scan: it is $0$ for the empty catalog, and for a catalog with head $a$ and tail $L'$ it is the scan count of $x$ in $L'$, plus $1$ if $a < x$.","title":"Scan count of smaller elements"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/count-lt-eq-lower-bound","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.countLT_eq_lowerBound","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#count-lt-eq-lower-bound","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every value $x$ and every catalog $L$, the scan count of elements of $L$ that are $< x$ equals the lower bound of $x$ in $L$.","title":"The scan count equals the lower bound"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/augment-one","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.augmentOne","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#augment-one","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For catalogs $c$ and $a$, the augmentation of $c$ by $a$ is the sorted merge of $c$ with the sample of $a$ (every other element of $a$, starting with the first). In fractional cascading, $a$ is the next augmented catalog.","title":"Augmenting a catalog by the next one"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/pairwise-augment-one","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.pairwise_augmentOne","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#pairwise-augment-one","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For any lists $c$ and $a$ over a linear order, sorted or not, the augmented catalog obtained by merging $c$ with the sample of $a$ is sorted in nondecreasing order.","title":"One augmentation step yields a sorted list"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-augment-one","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_augmentOne","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-augment-one","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For lists $c$ and $a$ over a linearly ordered type, the augmented catalog obtained by merging $c$ with the sample $s(a)$ of $a$ has length $|c| + |s(a)|$.","title":"Length of one augmented catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-augment-one-le","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_augmentOne_le","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-augment-one-le","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For lists $c$ and $a$ over a linearly ordered type, the augmented catalog obtained by merging $c$ with the sample of $a$ has length at most $|c| + \\lfloor (|a| + 1)/2 \\rfloor$, the length of $c$ plus half the length of $a$ rounded up.","title":"Length bound for one augmented catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/mem-augment-one-iff","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.mem_augmentOne_iff","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#mem-augment-one-iff","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For catalogs $c$ and $a$ over a linearly ordered type, a value $v$ is an entry of the augmented catalog of $c$ with $a$ (the sorted merge of $c$ with the sample of $a$) if and only if $v$ is an entry of $c$ or an entry of the sample of $a$.","title":"Membership in an augmented catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/mem-augment-one-imp","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.mem_augmentOne_imp","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#mem-augment-one-imp","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For catalogs $c$ and $a$ over a linearly ordered type, every entry $v$ of the augmented catalog of $c$ with $a$ (the sorted merge of $c$ with the sample of $a$) is an entry of $c$ or an entry of $a$.","title":"Entries of an augmented catalog come from the two catalogs"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/lower-bound-augment-one","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.lowerBound_augmentOne","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#lower-bound-augment-one","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and catalogs $c$ and $a$ over a linearly ordered type, the lower bound of $x$ in the augmented catalog of $c$ with $a$ (the sorted merge of $c$ with the sample of $a$) equals the lower bound of $x$ in $c$ plus the lower bound of $x$ in the sample of $a$, where the lower bound of $x$ in a catalog is the number of its entries less than $x$.","title":"Lower bound in an augmented catalog splits"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/lower-bound-augment-one-le","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.lowerBound_augmentOne_le","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#lower-bound-augment-one-le","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and catalogs $c$ and $a$ over a linearly ordered type, the lower bound of $x$ in the augmented catalog of $c$ with $a$ (the sorted merge of $c$ with the sample of $a$) is at most the lower bound of $x$ in $c$ plus the lower bound of $x$ in $a$, where the lower bound of $x$ in a catalog is the number of its entries less than $x$.","title":"Lower bound in an augmented catalog is at most the sum"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/augment-head","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.augmentHead","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#augment-head","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"Given a catalog $c$ and a list $L$ of catalogs, the result is $c$ itself if $L$ is empty, and otherwise the sorted merge of $c$ with the sample (every other element, starting with the first) of the first catalog of $L$.","title":"Augmenting a catalog by the head of a list"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/augmented","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.augmented","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#augmented","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a list of catalogs $c_1, \\dots, c_k$, the augmented catalogs $A_1, \\dots, A_k$ are built from the last one backwards: $A_k = c_k$, and for $i < k$, $A_i$ is the sorted merge of $c_i$ with the sample (every other element, starting with the first) of $A_{i+1}$. The empty list of catalogs has no augmented catalogs.","title":"Augmented catalogs"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-augmented","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_augmented","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-augmented","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every list $cs$ of catalogs over a linearly ordered type, the list of augmented catalogs built from $cs$ has the same length as $cs$: there is one augmented catalog per catalog.","title":"Number of augmented catalogs"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/augmented-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.augmented_singleton","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#augmented-singleton","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every catalog $c$, the list of augmented catalogs of the one-element list $[c]$ is $[c]$: the only augmented catalog is $c$ itself.","title":"Augmenting a single catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/pairwise-augmented","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.pairwise_augmented","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#pairwise-augmented","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"If every catalog in a list of catalogs over a linear order is sorted in nondecreasing order, then every augmented catalog built from that list is sorted in nondecreasing order.","title":"The augmented catalogs of sorted catalogs are sorted"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-mem-augmented-le","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_mem_augmented_le","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-mem-augmented-le","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every list $cs = [c_1, \\dots, c_k]$ of catalogs over a linearly ordered type, every augmented catalog built from $cs$ has length at most $|c_1| + \\dots + |c_k|$.","title":"Augmented catalogs are bounded by the total input size"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/bridge-ptr","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.bridgePtr","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#bridge-ptr","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a catalog $L$ and a value $v$, the bridge pointer of $v$ into $L$ is the number of elements of $L$ that are $\\le v$.","title":"Bridge pointer"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/bridge-ptr-le-length","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.bridgePtr_le_length","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#bridge-ptr-le-length","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every catalog $L$ and value $v$, the bridge pointer of $v$ into $L$ is at most the length of $L$.","title":"Bridge pointers are at most the catalog length"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/bridges","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.bridges","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#bridges","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For catalogs $C$ and $L$, the bridge pointers of $C$ into $L$ form the list obtained by replacing each entry $v$ of $C$, in order, by the bridge pointer of $v$ into $L$.","title":"Bridge pointer list"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-bridges","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_bridges","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-bridges","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For lists $c$ and $d$ over a linearly ordered type, the list of bridge pointers of $c$ into $d$ has the same length as $c$: there is one bridge pointer per entry of $c$.","title":"Number of bridge pointers"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/le-length-of-mem-bridges","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.le_length_of_mem_bridges","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#le-length-of-mem-bridges","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For lists $c$ and $d$ over a linearly ordered type, every entry $p$ of the list of bridge pointers of $c$ into $d$ satisfies $p \\le |d|$, where $|d|$ is the length of $d$; so every recorded bridge pointer is a boundary index of $d$.","title":"Bridge pointers are at most the next catalog's length"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/search-all","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.searchAll","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#search-all","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and a list of catalogs, the search returns, for each catalog in order, the number of its elements that are $< x$, computed by a direct scan of that catalog. This is the lower-bound position of $x$ in each catalog.","title":"Searching every catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/search-all-nil","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.searchAll_nil","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#search-all-nil","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"Searching the empty list of catalogs for any value $x$ gives the empty list of positions.","title":"Searching no catalogs"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/search-all-cons","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.searchAll_cons","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#search-all-cons","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$, a catalog $c$ and a list of catalogs $cs$, searching $c$ followed by $cs$ for $x$ gives first the direct-scan count of the elements of $c$ that are $< x$, followed by the positions obtained by searching $cs$ for $x$.","title":"Searching a nonempty list of catalogs"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-search-all","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_searchAll","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-search-all","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every value $x$ and every list $cs$ of catalogs over a linearly ordered type, searching each catalog of $cs$ for $x$ gives a list of positions of the same length as $cs$.","title":"One search position per catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/search-all-eq-map-lower-bound","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.searchAll_eq_map_lowerBound","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#search-all-eq-map-lower-bound","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and a list of catalogs $c_1, \\dots, c_k$, searching every catalog for $x$ returns the list whose $i$-th entry is the lower bound of $x$ in $c_i$, that is, the number of elements of $c_i$ that are $< x$.","title":"Correctness of searching every catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/get-elem-search-all","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.getElem?_searchAll","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#get-elem-search-all","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every value $x$, list of catalogs $C_0, \\dots, C_{k-1}$ and index $i$: if $i < k$, the $i$-th position returned by searching all catalogs for $x$ is the lower bound of $x$ in $C_i$; if $i \\ge k$, the search result has no $i$-th entry.","title":"Search positions are lower bounds, index by index"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/search-all-le-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.searchAll_le_of_mem","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#search-all-le-of-mem","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"Let $x$ be a value and let $cs$ be a list of catalogs whose entry at index $i$ is the catalog $c$. Then searching every catalog of $cs$ for $x$ gives, at index $i$, the lower bound of $x$ in $c$ (the number of elements of $c$ that are $< x$), and this lower bound is at most the length of $c$.","title":"Search positions are boundary indices"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/search-augmented","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.searchAugmented","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#search-augmented","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and a list of catalogs, the augmented search builds the augmented catalogs and returns, for each augmented catalog in order, the number of its elements that are $< x$, computed by a direct scan of that catalog.","title":"Searching every augmented catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/length-search-augmented","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.length_searchAugmented","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#length-search-augmented","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For every value $x$ and every list $cs$ of catalogs over a linearly ordered type, searching each augmented catalog built from $cs$ for $x$ gives a list of positions of the same length as $cs$.","title":"One augmented search position per catalog"},{"id":"ComputerScience/data-structures/range-query-structures/fractional-cascading/search-augmented-eq-map-lower-bound","kind":"node","lean":"Cslib.Algorithms.Lean.FractionalCascading.searchAugmented_eq_map_lowerBound","parent":37780,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/fractional-cascading/index.html#search-augmented-eq-map-lower-bound","scope":"ComputerScience/data-structures/range-query-structures/fractional-cascading","status":"fully_proved","summary":"For a value $x$ and a list of catalogs $cs$, searching every augmented catalog of $cs$ for $x$ returns, for each augmented catalog in order, the lower bound of $x$ in it, that is, the number of its elements that are $< x$.","title":"Correctness of augmented search"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/interval-tree.json","id":"ComputerScience/data-structures/range-query-structures/interval-tree","kind":"scope","navigation":"module","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","title":"Interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/interval","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Interval","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#interval","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"An interval is a pair of integers, its low endpoint and its high endpoint, read as the closed interval from the low to the high endpoint. The structure itself imposes no order on the endpoints; the public update operations accept an interval only when its low endpoint is at most its high endpoint.","title":"Interval"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Interval.Valid","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"A closed interval with integer endpoints $a$ (low) and $b$ (high) is valid when its low endpoint is at most its high endpoint, $a \\le b$.","title":"Valid interval"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid-2","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Interval.valid","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid-2","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For an interval $[a, b]$ with integer endpoints, the Boolean validity test returns true exactly when $a \\le b$.","title":"Boolean interval validity test"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/overlaps","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Interval.Overlaps","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#overlaps","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"An interval $I$ with low endpoint $a$ and high endpoint $b$ overlaps an interval $J$ with low endpoint $c$ and high endpoint $d$ when $a \\le d$ and $c \\le b$; in particular, intervals that share only an endpoint overlap.","title":"Overlapping closed intervals"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/overlaps-2","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Interval.overlaps","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#overlaps-2","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For an interval $I$ with low endpoint $a$ and high endpoint $b$ and an interval $J$ with low endpoint $c$ and high endpoint $d$, the overlap test of $I$ against $J$ returns true exactly when $a \\le d$ and $c \\le b$; it compares $a$ with $d$ first and returns false without the second comparison when $a > d$.","title":"Overlap test for closed intervals"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/overlaps-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Interval.overlapsWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#overlaps-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For an interval $I$ with low endpoint $a$ and high endpoint $b$ and an interval $J$ with low endpoint $c$ and high endpoint $d$, the instrumented overlap test of $I$ against $J$ returns the pair (false, $1$) when $a > d$, the pair (false, $2$) when $a \\le d$ and $c > b$, and the pair (true, $2$) when $a \\le d$ and $c \\le b$; the number is the count of endpoint comparisons evaluated.","title":"Overlap test with comparison count"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/overlaps-with-cost-fst","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Interval.overlapsWithCost_fst","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#overlaps-with-cost-fst","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For intervals $I$ and $J$, the Boolean component of the instrumented overlap test of $I$ against $J$ equals the overlap test of $I$ against $J$.","title":"The instrumented overlap test returns the plain result"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/overlaps-with-cost-snd","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Interval.overlapsWithCost_snd","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#overlaps-with-cost-snd","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For intervals $I$ and $J$, the instrumented overlap test of $I$ against $J$ reports $2$ endpoint comparisons when the low endpoint of $I$ is at most the high endpoint of $J$, and $1$ comparison otherwise.","title":"Comparison count of the overlap test"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/entry","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Entry","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#entry","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"An entry of the interval tree consists of an identifier, which is a natural number, and an interval with integer low and high endpoints.","title":"Interval tree entry"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/key","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Entry.key","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#key","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The key of an entry is the pair $(\\ell, i) \\in \\mathbb{Z} \\times \\mathbb{N}$, where $\\ell$ is the low endpoint of the entry's interval and $i$ is the entry's identifier.","title":"Search key of an interval entry"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/compare-key","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.compareKey","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#compare-key","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For keys $(a, i)$ and $(b, j)$ in $\\mathbb{Z} \\times \\mathbb{N}$ (a low endpoint and an occurrence identifier), the comparison returns less-than if $a < b$, or if $a = b$ and $i < j$; greater-than if $a > b$, or if $a = b$ and $i > j$; and equal if $a = b$ and $i = j$.","title":"Lexicographic comparison of keys"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Cost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"A cost record for the interval tree consists of seven natural-number counters, each defaulting to $0$: node visits, key comparisons, endpoint comparisons, cached-maximum reads, cache writes, rotations, and identifier-index operations.","title":"Operation cost counters for interval trees"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/tree","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#tree","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"An interval tree is either empty or a node consisting of a left subtree, an entry, a right subtree and three cached fields: a height and a size (natural numbers) and a maximum high endpoint (an integer). The type imposes no balance or consistency condition on these fields; it is the carrier on which the AVL interval-tree operations act.","title":"Interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cached-max","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cachedMax?","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cached-max","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The cached maximum of an interval tree is no value when the tree is empty, and when the tree is a node it is the integer stored in the node's cached maximum-high-endpoint field.","title":"Cached maximum of an interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/max-with","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.maxWith","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#max-with","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For an integer $a$ and an optional integer $v$, the value is $a$ when $v$ is absent and $\\max(a, v)$ when $v$ is present. It combines a value with an optional subtree maximum without using a sentinel value.","title":"Maximum with an optional value"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/node-max","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.nodeMax","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#node-max","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For subtrees $L$ and $R$ and an entry $e$, the node maximum is the largest of the high endpoint of the interval of $e$, the cached maximum high endpoint stored at the root of $L$, and the one stored at the root of $R$, where an empty subtree stores no cached maximum and contributes nothing.","title":"Maximum high endpoint at a node"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/mk-node","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.mkNode","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#mk-node","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For an interval tree $T$ write $h(T)$ and $|T|$ for its cached height and size, both $0$ for the empty tree. Given interval trees $L$ and $R$ and an entry $e$, this builds the node with left subtree $L$, entry $e$ and right subtree $R$ whose cached height is $\\max(h(L), h(R)) + 1$, whose cached size is $|L| + |R| + 1$, and whose cached maximum high endpoint is the largest of the high endpoint of $e$ and the cached\u2026","title":"Node construction with repaired metadata"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.CacheInvariant","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"An interval tree satisfies the cache invariant when every cached field is exact, recursively. The empty tree satisfies it. A node with stored height $h$, stored size $s$, stored maximum $m$, left subtree $L$, entry $e$ and right subtree $R$ satisfies it when $L$ and $R$ satisfy it, $m$ is at least the high endpoint of every entry of $L$, of $e$ and of every entry of $R$, $m$ equals the high endpoint of one of these\u2026","title":"Cache invariant of an interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/low-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.LowOrdered","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#low-ordered","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The empty tree is low-endpoint ordered, and a node is low-endpoint ordered when both of its subtrees are low-endpoint ordered, every interval in the left subtree has low endpoint at most the low endpoint of the node's interval, and the node's low endpoint is at most the low endpoint of every interval in the right subtree.","title":"Low-endpoint ordered interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid-invariant","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.ValidInvariant","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid-invariant","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"A tree satisfies the validity invariant when every interval stored in it is valid, that is, its low endpoint is at most its high endpoint. Recursively, the empty tree satisfies it, and a node satisfies it when its left subtree does, the interval of its entry is valid, and its right subtree does.","title":"Validity invariant of an interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/rotate-right","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.rotateRight","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#rotate-right","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Write $\\langle L, x, R\\rangle$ for a node with left subtree $L$, entry $x$ and right subtree $R$. The right rotation sends $\\langle\\langle a, x, b\\rangle, y, c\\rangle$ to $\\langle a, x, \\langle b, y, c\\rangle\\rangle$, where both nodes of the result get freshly recomputed height, size and maximum high endpoint. Every tree not of this shape is returned unchanged.","title":"Right rotation"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/rotate-left","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.rotateLeft","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#rotate-left","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Write $\\langle L, x, R\\rangle$ for a node with left subtree $L$, entry $x$ and right subtree $R$. The left rotation sends $\\langle a, x, \\langle b, y, c\\rangle\\rangle$ to $\\langle\\langle a, x, b\\rangle, y, c\\rangle$, where both nodes of the result get freshly recomputed height, size and maximum high endpoint. Every tree not of this shape is returned unchanged.","title":"Left rotation"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/rotate-left-right","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.rotateLeftRight","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#rotate-left-right","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Write $\\langle L, x, R\\rangle$ for a node with left subtree $L$, entry $x$ and right subtree $R$. The left-right rotation sends $\\langle\\langle a, x, \\langle b, y, c\\rangle\\rangle, z, d\\rangle$ to $\\langle\\langle a, x, b\\rangle, y, \\langle c, z, d\\rangle\\rangle$, where all three nodes of the result get freshly recomputed height, size and maximum high endpoint. Every tree not of this shape is returned unchanged.","title":"Left-right double rotation"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/rotate-right-left","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.rotateRightLeft","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#rotate-right-left","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Write $\\langle L, x, R\\rangle$ for a node with left subtree $L$, entry $x$ and right subtree $R$. The right-left rotation sends $\\langle a, x, \\langle\\langle b, y, c\\rangle, z, d\\rangle\\rangle$ to $\\langle\\langle a, x, b\\rangle, y, \\langle c, z, d\\rangle\\rangle$, where all three nodes of the result get freshly recomputed height, size and maximum high endpoint. Every tree not of this shape is returned unchanged.","title":"Right-left double rotation"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/rebalance-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.rebalanceWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#rebalance-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Rebalancing a tree $T$ returns a tree and a cost, where heights are the stored heights. If $T$ is empty, or neither subtree of its root is more than one taller than the other, it returns $T$ unchanged with zero cost. If the left subtree $L$ is more than one taller than the right subtree, it applies a right rotation, costing $2$ cache writes and one rotation, when the left child of $L$ is at least as tall as the\u2026","title":"AVL rebalancing step with cost"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/insert-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.insertKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#insert-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Inserting an entry $x$ into a tree returns a new tree together with an operation cost; keys (low endpoint, then identifier) are compared lexicographically. Into the empty tree, $x$ is inserted as a single node, at a cost of one node visit and one cache write. At a node with entry $y$: if the key of $x$ equals the key of $y$, then $y$ is replaced by $x$, at a cost of one node visit, one key comparison and one cache\u2026","title":"Keyed insertion with cost"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/remove-min-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.removeMinWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#remove-min-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Removing the leftmost entry of an interval tree $T$ returns nothing when $T$ is empty. When the root's left subtree is empty, it returns the root entry, the right subtree, and a cost of one node visit. Otherwise it removes the leftmost entry $m$ of the left subtree recursively (returning nothing if that fails), rebuilds the root node over the new left subtree with recomputed height, size and maximum high endpoint,\u2026","title":"Removing the leftmost entry, with its cost"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/erase-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.eraseKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#erase-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Erasing a key $k \\in \\mathbb{Z} \\times \\mathbb{N}$ from an interval tree returns a new tree together with its cost; the key of an entry is the pair of its low endpoint and identifier, and keys are compared lexicographically. The empty tree is returned unchanged at a cost of one node visit. At a node with entry $x$, the key $k$ is compared with the key of $x$. If $k$ is smaller, $k$ is erased from the left subtree;\u2026","title":"Keyed erasure with cost"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/search-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.searchWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#search-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Given a query interval $q = [c, d]$ and an interval tree $T$, the instrumented overlap search returns an optional entry together with a cost record, defined by recursion on $T$. On the empty tree it returns no entry and records one node visit. At a node with left subtree $L$, entry $e$ with interval $[a, b]$ and right subtree $R$, it first tests overlap by evaluating $a \\le d$ and, only if that holds, $c \\le b$;\u2026","title":"Instrumented overlap search"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/search-with-cost-empty","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.searchWithCost_empty","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#search-with-cost-empty","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For every query interval $q$, the instrumented overlap search of $q$ in the empty interval tree returns no entry, and its cost record counts one node visit and zero for every other counter (key comparisons, endpoint comparisons, cached-maximum reads, cache writes, rotations and index operations).","title":"Instrumented search in the empty tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/search-with-cost-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.searchWithCost_singleton","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#search-with-cost-singleton","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $e$ be an entry whose interval is $[a, b]$, let $q = [c, d]$ be a query interval, and let $k = 2$ if $a \\le d$ and $k = 1$ otherwise. Consider the single-node tree built from $e$ with empty subtrees and exact cached fields. If $[a, b]$ and $[c, d]$ overlap (that is, $a \\le d$ and $c \\le b$), the instrumented overlap search of $q$ in this tree returns $e$ with a cost record of one node visit and $k$ endpoint\u2026","title":"Instrumented search in a one-entry tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/search","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.search","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#search","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The overlap search of a query interval $q$ in an interval tree $T$ returns the optional entry produced by the instrumented overlap search of $q$ in $T$, discarding its cost record.","title":"Interval-tree overlap search"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/oracle-search","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.oracleSearch","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#oracle-search","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For a tree $T$ and a query interval $q$, the oracle search returns the first entry, in the in-order list of entries of $T$, whose interval overlaps $q$ as closed intervals, and returns nothing when no entry's interval overlaps $q$.","title":"Linear-scan overlap search"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/search-complete","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.search_complete","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#search-complete","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree whose cached height, size and maximum high endpoint are exact at every node, and whose low endpoints respect the search order (at every node, low endpoints in the left subtree are at most the node's, and those in the right subtree at least the node's). If some entry stored in $T$ has an interval that overlaps the query interval $Q$ (each interval's low endpoint is at most the other's\u2026","title":"Completeness of interval-tree search"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/search-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.search_eq_none_iff","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#search-eq-none-iff","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree whose cached height, size and maximum high endpoint are exact at every node, and whose low endpoints respect the search order (at every node, low endpoints in the left subtree are at most the node's, and those in the right subtree at least the node's). Then searching $T$ for a query interval $Q$ returns nothing if and only if no entry stored in $T$ has an interval that overlaps $Q$ (each\u2026","title":"Interval-tree search fails exactly when nothing overlaps"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-ok","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheOK","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-ok","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The Boolean cache check returns true on the empty tree. On a node with stored height $h$, stored size $s$, stored maximum $m$, left subtree $L$, entry $e$ and right subtree $R$ it returns true exactly when $h = \\max(\\operatorname{height} L, \\operatorname{height} R) + 1$, $s = \\operatorname{size} L + \\operatorname{size} R + 1$, $m$ is the largest of the high endpoint of $e$ and the cached maxima of those of $L$ and\u2026","title":"Executable cache check for interval trees"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/key-between","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.keyBetween","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#key-between","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Given an optional lower bound, an optional upper bound and a key $k$ (a pair of an integer and a natural number), this test returns true exactly when the lower bound is absent or strictly below $k$, and the upper bound is absent or strictly above $k$. Keys are compared lexicographically: first by the integer, then by the natural number.","title":"Key between optional bounds"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/ordered-valid","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.orderedValid","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#ordered-valid","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Keys of entries are pairs (low endpoint, identifier), ordered lexicographically. Given optional lower and upper key bounds $\\ell$ and $u$, the check accepts the empty tree, and accepts a node with left subtree $L$, entry $e$ and right subtree $R$ exactly when the key $k$ of $e$ is strictly greater than $\\ell$ if $\\ell$ is present and strictly less than $u$ if $u$ is present, the interval of $e$ has low endpoint at\u2026","title":"Ordered-and-valid check for interval trees"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/balanced","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.balanced","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#balanced","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The balance check returns true on the empty tree. On a node whose left and right subtrees have stored heights $h_L$ and $h_R$ (the empty tree has height $0$), it returns true exactly when the node's stored height equals $\\max(h_L, h_R) + 1$, $h_L \\le h_R + 1$, $h_R \\le h_L + 1$, and the check returns true on both subtrees.","title":"Boolean AVL balance check for interval trees"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/balanced-prop","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.BalancedProp","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#balanced-prop","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"An interval tree is AVL balanced with exact stored heights when it is empty, or when it is a node such that, writing $h_L$ and $h_R$ for the stored heights of its left and right subtrees (the empty tree has height $0$), the node's stored height equals $\\max(h_L, h_R) + 1$, $h_L \\le h_R + 1$, $h_R \\le h_L + 1$, and both subtrees are again AVL balanced with exact stored heights.","title":"AVL balance of an interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/key-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.KeyOrdered","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#key-ordered","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The empty tree is key-ordered, and a node is key-ordered when both of its subtrees are key-ordered, the key of every entry in the left subtree is strictly less than the key of the node's entry, and the key of the node's entry is strictly less than the key of every entry in the right subtree. Keys are compared lexicographically: low endpoint first, then identifier.","title":"Key-ordered interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/key-ordered-of-ordered-valid","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.keyOrdered_of_orderedValid","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#key-ordered-of-ordered-valid","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $t$ be an interval tree and let $\\ell$ and $u$ be optional keys in $\\mathbb{Z} \\times \\mathbb{N}$. If the executable ordering check of $t$ with lower bound $\\ell$ and upper bound $u$ succeeds, then $t$ is key-ordered, and every entry $e$ stored in $t$ satisfies $\\ell < \\operatorname{key}(e)$ when $\\ell$ is present and $\\operatorname{key}(e) < u$ when $u$ is present, where $<$ is the lexicographic order on keys\u2026","title":"The executable ordering check implies key ordering"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/low-ordered-of-ordered-valid","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.lowOrdered_of_orderedValid","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#low-ordered-of-ordered-valid","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $t$ be an interval tree and let $\\ell$ and $u$ be optional keys in $\\mathbb{Z} \\times \\mathbb{N}$. If the executable ordering check of $t$ with lower bound $\\ell$ and upper bound $u$ succeeds, then $t$ is low-endpoint ordered, and every entry $e$ stored in $t$ satisfies $\\ell < \\operatorname{key}(e)$ when $\\ell$ is present and $\\operatorname{key}(e) < u$ when $u$ is present, where $<$ is the lexicographic order\u2026","title":"The executable ordering check implies low-endpoint ordering"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid-invariant-of-ordered-valid","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.validInvariant_of_orderedValid","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid-invariant-of-ordered-valid","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree and let $\\ell$ and $u$ be optional lower and upper key bounds. If the executable check of $T$ with bounds $\\ell$ and $u$, which tests that every key lies strictly between its bounds and that every stored interval passes the Boolean validity test, returns true, then every interval stored in $T$ is valid, that is, its low endpoint is at most its high endpoint.","title":"The ordering check implies interval validity"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/remove-min-with-cost-entries-perm","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.removeMinWithCost_entries_perm","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#remove-min-with-cost-entries-perm","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If removing the leftmost entry of a tree $T$ returns an entry $m$, a tree $T'$ and a cost, then the list consisting of $m$ followed by the in-order entries of $T'$ is a permutation of the in-order entries of $T$.","title":"Removing the minimum preserves the other entries"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/insert-key-with-cost-entries-perm","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.insertKeyWithCost_entries_perm","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#insert-key-with-cost-entries-perm","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $t$ be a tree whose entries are strictly ordered by key, and let $x$ be an entry. The in-order list of entries of the tree obtained by inserting $x$ into $t$ by key is a permutation of the list consisting of $x$ followed by those entries of $t$ whose key differs from the key of $x$.","title":"Entries after keyed insertion"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/erase-key-with-cost-entries-perm","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.eraseKeyWithCost_entries_perm","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#erase-key-with-cost-entries-perm","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The key of an entry is the pair $(\\ell, i) \\in \\mathbb{Z} \\times \\mathbb{N}$ of its low endpoint $\\ell$ and identifier $i$; keys are compared lexicographically. Let $T$ be an interval tree whose keys are strictly ordered (every key in a left subtree is lexicographically below its node's key and every key in a right subtree is above it), and let $k \\in \\mathbb{Z} \\times \\mathbb{N}$ be a key. Then the in-order list\u2026","title":"Keyed erasure removes exactly the entries with that key"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/key-nodup-of-key-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.keyNodup_of_keyOrdered","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#key-nodup-of-key-ordered","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If a tree is strictly ordered by key, then the keys of its stored entries are pairwise distinct.","title":"Strictly key-ordered trees have distinct keys"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/remove-min-with-cost-min-le","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.removeMinWithCost_min_le","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#remove-min-with-cost-min-le","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be a key-ordered interval tree: at every node, the keys in the left subtree are strictly smaller and those in the right subtree strictly larger than the node's key, where the key of an entry is the pair (low endpoint, identifier) compared lexicographically. If removing the leftmost entry of $T$ returns the entry $m$ (together with the remaining tree and a cost record), then every entry $e$ stored in $T$ is\u2026","title":"The removed minimum has the least key"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/key-ordered-insert-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.keyOrdered_insertKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#key-ordered-insert-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $t$ be a key-ordered interval tree and let $e$ be an entry. Then the tree returned by the cost-instrumented keyed insertion of $e$ into $t$ (which replaces the entry with the same key, if there is one; the reported cost is discarded) is again key-ordered.","title":"Insertion into an interval tree preserves key ordering"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/key-ordered-remove-min-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.keyOrdered_removeMinWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#key-ordered-remove-min-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $t$ be a key-ordered interval tree. If the cost-instrumented removal of the leftmost entry of $t$ succeeds, returning that entry, a remaining tree $t'$ and a cost, then $t'$ is key-ordered.","title":"Removing the minimum from an interval tree preserves key ordering"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/key-ordered-erase-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.keyOrdered_eraseKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#key-ordered-erase-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $t$ be a key-ordered interval tree and let $k \\in \\mathbb{Z} \\times \\mathbb{N}$ be a key. Then the tree returned by the cost-instrumented keyed deletion of $k$ from $t$ (the reported cost is discarded) is again key-ordered.","title":"Deletion from an interval tree preserves key ordering"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/balanced-prop-rebalance-node","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.balancedProp_rebalance_node","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#balanced-prop-rebalance-node","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $L$ and $R$ be interval trees that are AVL balanced with exact stored heights, with heights $h_L$ and $h_R$ satisfying $h_L \\le h_R + 2$ and $h_R \\le h_L + 2$. Let $N$ be a node with left subtree $L$, entry $e$ and right subtree $R$ whose stored height is $\\max(h_L, h_R) + 1$ (its stored size and stored maximum are arbitrary), and let $B$ be the tree obtained by rebalancing $N$ (the tree component of the\u2026","title":"Rebalancing an interval-tree node with exact stored height restores AVL balance"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/balanced-prop-rebalance-mk-node","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.balancedProp_rebalance_mkNode","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#balanced-prop-rebalance-mk-node","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $L$ and $R$ be interval trees that are AVL balanced with exact stored heights, with heights $h_L$ and $h_R$ satisfying $h_L \\le h_R + 2$ and $h_R \\le h_L + 2$, and let $e$ be an entry. Let $B$ be the tree obtained by rebalancing the node that the smart constructor builds from $L$, $e$ and $R$ (the tree component of the result; the cost is ignored). Then $B$ is AVL balanced with exact stored heights, the height\u2026","title":"Rebalancing a rebuilt interval-tree node restores AVL balance"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/balanced-prop-insert-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.balancedProp_insertKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#balanced-prop-insert-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree that is AVL balanced with exact stored heights, and let $e$ be an entry. The tree $T'$ returned by keyed insertion of $e$ into $T$ (which replaces an entry with the same key, if there is one; we take the tree component of the result and ignore the cost) is again AVL balanced with exact stored heights, and the height of $T'$ is either the height of $T$ or one more.","title":"Keyed insertion preserves AVL balance in an interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/balanced-prop-remove-min-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.balancedProp_removeMinWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#balanced-prop-remove-min-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree that is AVL balanced with exact stored heights. If removing the leftmost entry of $T$ succeeds and returns an entry $m$, a remaining tree $R$ and a cost $c$, then $R$ is AVL balanced with exact stored heights, and the height of $R$ is either the height of $T$ or one less.","title":"Removing the minimum preserves AVL balance in an interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/balanced-prop-erase-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.balancedProp_eraseKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#balanced-prop-erase-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree that is AVL balanced with exact stored heights, and let $k$ be a key (a pair of an integer low endpoint and a natural-number identifier). The tree $T'$ returned by keyed deletion of $k$ from $T$ (the tree component of the result; the cost is ignored) is again AVL balanced with exact stored heights, and the height of $T'$ is either the height of $T$ or one less.","title":"Keyed deletion preserves AVL balance in an interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid-invariant-mk-node","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.validInvariant_mkNode","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid-invariant-mk-node","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If $L$ and $R$ are interval trees satisfying the validity invariant and $e$ is an entry whose interval $[a, b]$ satisfies $a \\le b$, then the node with left subtree $L$, entry $e$ and right subtree $R$, built by the node constructor that recomputes the cached fields, satisfies the validity invariant.","title":"Node construction preserves interval validity"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid-invariant-mem","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.validInvariant_mem","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid-invariant-mem","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If an interval tree $T$ satisfies the validity invariant, then every entry stored in $T$ has an interval $[a, b]$ with $a \\le b$.","title":"Stored intervals of a valid tree are valid"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid-invariant-insert-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.validInvariant_insertKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid-invariant-insert-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree in which every stored interval $[a, b]$ satisfies $a \\le b$, and let $e$ be an entry whose interval $[a, b]$ satisfies $a \\le b$. Then the tree returned by keyed insertion of $e$ into $T$ also has every stored interval $[a, b]$ satisfying $a \\le b$.","title":"Keyed insertion preserves interval validity"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid-invariant-remove-min-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.validInvariant_removeMinWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid-invariant-remove-min-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree in which every stored interval is valid (its low endpoint is at most its high endpoint). If the instrumented removal of the leftmost entry of $T$ succeeds, returning that entry $m$, the remaining tree $T'$ and a cost record $c$, then every interval stored in $T'$ is valid.","title":"Removing the minimum preserves interval validity"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/valid-invariant-erase-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.validInvariant_eraseKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#valid-invariant-erase-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree in which every stored interval $[a, b]$ satisfies $a \\le b$. Then for every key $(\\ell, i)$ with $\\ell$ an integer and $i$ a natural number, the tree returned by keyed erasure of $(\\ell, i)$ from $T$ also has every stored interval $[a, b]$ satisfying $a \\le b$.","title":"Keyed erasure preserves interval validity"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/indexed-interval-tree","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#indexed-interval-tree","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"An indexed interval tree consists of a root interval tree together with an identifier index: a finite map, implemented as a balanced search tree, from natural-number identifiers to integer low endpoints.","title":"Indexed interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/insert-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.insertWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#insert-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an indexed interval tree and let $e$ be an entry with identifier $i$ and interval $[a, b]$. Inserting $e$ into $T$ first tests $a \\le b$. If $a > b$, it returns no tree, at a cost of one endpoint comparison. Otherwise, if the identifier index of $T$ maps $i$ to a low endpoint $\\ell$, the entry with key $(\\ell, i)$ is erased from the root tree of $T$ by keyed erasure; if $i$ is not in the index, the root\u2026","title":"Insertion by identifier with cost"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/erase-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.eraseWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#erase-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"To erase identifier $i$ from an indexed interval tree, look $i$ up in the identifier index. If it is absent, the tree is returned unchanged at a cost of one index operation. If the index maps $i$ to a low endpoint $\\ell$, the entry with key $(\\ell, i)$ is erased from the underlying interval tree, $i$ is removed from the index, and the cost is the cost of the keyed erasure plus two index operations.","title":"Erasure by identifier with cost"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/index-forward-ok","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.indexForwardOK","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#index-forward-ok","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For an indexed interval tree, the forward index check returns true exactly when the identifier index maps the identifier of every stored entry to the low endpoint of that entry's interval.","title":"Forward identifier-index check"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/index-backward-ok","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.indexBackwardOK","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#index-backward-ok","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For an indexed interval tree, the backward index check returns true exactly when, for every identifier $i$ that the identifier index maps to a low endpoint $\\ell$, some stored entry has identifier $i$ and an interval with low endpoint $\\ell$.","title":"Backward identifier-index check"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/check","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.check","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#check","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"For an indexed interval tree, the check returns true exactly when all of the following hold: at every node the cached height and size agree with the values recomputed from its children's cached heights and sizes, and the cached maximum high endpoint equals the largest of the node's own high endpoint and its children's cached maxima; the tree is AVL-balanced (sibling heights differ by at most one) with exact stored\u2026","title":"Representation check for the indexed interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/invariant","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.Invariant","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#invariant","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"An indexed interval tree $T$ satisfies the representation invariant when its root tree has exact cached metadata (heights, sizes, and attained maximum high endpoints), is strictly ordered by the composite key (low endpoint, then identifier), is AVL balanced with exact stored heights, and stores only valid intervals; the identifiers of the stored entries are pairwise distinct; the identifier index maps the\u2026","title":"Representation invariant of an indexed interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-mk-node","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_mkNode","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-mk-node","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If interval trees $L$ and $R$ satisfy the cache invariant (every cached maximum high endpoint, height and size is exact), then for every entry $e$ the node that the smart constructor builds from $L$, $e$ and $R$ also satisfies the cache invariant.","title":"The interval-tree smart constructor preserves the cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-of-cache-ok","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_of_cacheOK","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-of-cache-ok","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If the executable cache check returns true on an interval tree $t$, then $t$ satisfies the cache invariant.","title":"The executable cache check implies the cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-rotate-right","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_rotateRight","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-rotate-right","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If an interval tree $t$ satisfies the cache invariant, then so does the right rotation of $t$, which leaves $t$ unchanged when $t$ is empty or has an empty left subtree.","title":"Right rotation preserves the interval-tree cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-rotate-left","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_rotateLeft","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-rotate-left","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If an interval tree $t$ satisfies the cache invariant, then so does the left rotation of $t$, which leaves $t$ unchanged when $t$ is empty or has an empty right subtree.","title":"Left rotation preserves the interval-tree cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-rotate-left-right","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_rotateLeftRight","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-rotate-left-right","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If an interval tree $t$ satisfies the cache invariant, then so does the left-right double rotation of $t$, which leaves $t$ unchanged unless $t$ has a nonempty left subtree whose right subtree is nonempty.","title":"Left-right rotation preserves the interval-tree cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-rotate-right-left","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_rotateRightLeft","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-rotate-right-left","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If an interval tree $t$ satisfies the cache invariant, then so does the right-left double rotation of $t$, which leaves $t$ unchanged unless $t$ has a nonempty right subtree whose left subtree is nonempty.","title":"Right-left rotation preserves the interval-tree cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-rebalance-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_rebalanceWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-rebalance-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If an interval tree $t$ satisfies the cache invariant, then the tree returned by the rebalancing step at the root of $t$ also satisfies the cache invariant.","title":"Rebalancing preserves the interval-tree cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-insert-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_insertKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-insert-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree satisfying the cache invariant (every cached maximum high endpoint, height and size is exact), and let $e$ be an entry. The tree returned by keyed insertion of $e$ into $T$ (the tree component of the result; the cost is ignored) also satisfies the cache invariant. This includes the case where an entry with the same key is replaced.","title":"Keyed insertion preserves the interval-tree cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-remove-min-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_removeMinWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-remove-min-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $t$ be an interval tree that satisfies the cache invariant. If removing the leftmost entry of $t$ succeeds and returns an entry, a tree $t'$ and an operation cost, then $t'$ satisfies the cache invariant.","title":"Removing the minimum preserves the interval-tree cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/cache-invariant-erase-key-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.Tree.cacheInvariant_eraseKeyWithCost","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#cache-invariant-erase-key-with-cost","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an interval tree satisfying the cache invariant (every cached maximum high endpoint, height and size is exact), and let $k$ be a key (a pair of an integer low endpoint and a natural-number identifier). The tree returned by keyed deletion of $k$ from $T$ (the tree component of the result; the cost is ignored) also satisfies the cache invariant. This includes keys that do not occur in $T$.","title":"Keyed deletion preserves the interval-tree cache invariant"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/invariant-of-check","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.invariant_of_check","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#invariant-of-check","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"If the executable checker returns true on an indexed interval tree $T$, then $T$ satisfies the representation invariant.","title":"Soundness of the invariant checker"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/search-sound","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.search_sound","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#search-sound","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"The search of an indexed interval tree $T$ for a query interval $q = [c, d]$ returns nothing when $c > d$, and otherwise returns the entry, if any, found by the instrumented overlap search of $q$ in the root tree of $T$. For every indexed interval tree $T$, every query interval $q$ and every entry $e$, if this search returns $e$, then $e$ is one of the entries stored in the root tree of $T$ and the interval of $e$\u2026","title":"Soundness of indexed interval-tree search"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/search-complete-of-invariant","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.search_complete_of_invariant","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#search-complete-of-invariant","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an indexed interval tree satisfying its representation invariant, and let $Q$ be a valid query interval (low endpoint at most high endpoint). If some entry stored in $T$ has an interval that overlaps $Q$ (each interval's low endpoint is at most the other's high endpoint), then searching $T$ for $Q$ returns some entry.","title":"Completeness of indexed interval search"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/insert-spec","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.insert_spec","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#insert-spec","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an indexed interval tree satisfying the representation invariant, and let $x$ be an entry whose interval is valid (its low endpoint is at most its high endpoint). If inserting $x$ into $T$ succeeds with result $T'$, then $T'$ satisfies the representation invariant, $x$ is stored in the root tree of $T'$, and an entry $e$ is stored in the root tree of $T'$ if and only if either $e = x$, or $e$ is stored\u2026","title":"Correctness of insertion into an indexed interval tree"},{"id":"ComputerScience/data-structures/range-query-structures/interval-tree/erase-spec","kind":"node","lean":"Cslib.Algorithms.Lean.IntervalTree.IndexedIntervalTree.erase_spec","parent":37830,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/interval-tree/index.html#erase-spec","scope":"ComputerScience/data-structures/range-query-structures/interval-tree","status":"fully_proved","summary":"Let $T$ be an indexed interval tree satisfying the representation invariant, and let $i$ be a natural number. Then the tree obtained by erasing identifier $i$ from $T$ again satisfies the invariant, and an entry is stored in it if and only if it is stored in $T$ and its identifier differs from $i$.","title":"Correctness of erasure from the indexed interval tree"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/k-d-tree.json","id":"ComputerScience/data-structures/range-query-structures/k-d-tree","kind":"scope","navigation":"module","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","title":"k-d tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/kd-tree","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#kd-tree","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"A kd-tree over points of $\\mathbb{Z}^d$ is either empty, a leaf holding one point of $\\mathbb{Z}^d$, or a split node holding a split axis $a \\in \\{0, \\dots, d-1\\}$, a split value $s \\in \\mathbb{Z}$ and two kd-trees, its left and right subtrees.","title":"Kd-trees"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/to-list-aux","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.toListAux","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#to-list-aux","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For a kd-tree $t$ over points of $\\mathbb{Z}^d$ and a list $\\mathit{acc}$ of such points, the accumulator traversal of $t$ onto $\\mathit{acc}$ is $\\mathit{acc}$ when $t$ is empty, the point $p$ followed by $\\mathit{acc}$ when $t$ is a leaf holding $p$, and, when $t$ is a split node with left subtree $L$ and right subtree $R$, the traversal of $L$ onto the traversal of $R$ onto $\\mathit{acc}$.","title":"Kd-tree point traversal with an accumulator"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/to-list","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.toList","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#to-list","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"The point list of a kd-tree is the list of the points stored at its leaves, from left to right.","title":"Point list of a kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/to-list-aux-eq","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.toListAux_eq","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#to-list-aux-eq","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For every kd-tree $t$ over $\\mathbb{Z}^d$ and every list $acc$ of points, the accumulator traversal of $t$ onto $acc$ equals the list of points of $t$ in left-to-right order followed by $acc$.","title":"Accumulator traversal of a kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/to-list-node","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.toList_node","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#to-list-node","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For a split node with axis $a$, split value $s$, left subtree $L$ and right subtree $R$, the point list of the node is the point list of $L$ followed by the point list of $R$.","title":"Point list of a kd-tree split node"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/size","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.size","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#size","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"The size of a kd-tree is defined by recursion: the empty tree has size $0$, a leaf has size $1$, and a split node has the sum of the sizes of its two children. Since points are stored only at leaves, this is the number of points stored in the tree, counted with multiplicity.","title":"Size of a kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/length-to-list","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.length_toList","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#length-to-list","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For every kd-tree $t$, the list of the points stored in $t$, read from left to right, has length equal to the size of $t$, the number of points stored in $t$.","title":"Length of the point list of a kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/next-axis","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.nextAxis","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#next-axis","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For a dimension $d \\ge 1$ and an axis $a \\in \\{0, \\dots, d - 1\\}$, the next split axis is $(a + 1) \\bmod d$, so the axes are visited cyclically.","title":"Next split axis of a kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/in-box","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.inBox","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#in-box","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For $lo, hi, p \\in \\mathbb{Z}^d$, the point $p$ lies in the closed axis-aligned box $[lo, hi]$ when $lo_i \\le p_i \\le hi_i$ for every coordinate $i \\in \\{0, \\dots, d-1\\}$.","title":"Membership in an axis-aligned box"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/report-aux","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.reportAux","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#report-aux","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Points are vectors in $\\mathbb{Z}^d$ with coordinates $p_0, \\dots, p_{d-1}$; the box $[lo, hi]$ consists of the points $p$ with $lo_i \\le p_i \\le hi_i$ for every $i$. For a kd-tree $t$ and a list $acc$ of points, the accumulator report of $t$ for $[lo, hi]$ onto $acc$ is defined by recursion on $t$. For the empty tree it is $acc$. For a leaf $p$ it is $p$ followed by $acc$ if $p$ lies in the box, and $acc$\u2026","title":"Accumulator range report of a kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/report","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.report","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#report","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For a kd-tree $t$ over $\\mathbb{Z}^d$ and $lo, hi \\in \\mathbb{Z}^d$, the range report of $t$ for the box $[lo, hi]$ is the accumulator range report of $t$ for $[lo, hi]$ onto the empty list.","title":"Orthogonal range report of a kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.Valid","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#valid","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"A kd-tree satisfies the kd invariant when at every split node, with axis $a$ and split value $s$, every point of the left subtree has $a$-coordinate at most $s$ and every point of the right subtree has $a$-coordinate at least $s$.","title":"Kd invariant"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/report-aux-eq-filter-append","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.reportAux_eq_filter_append","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#report-aux-eq-filter-append","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $t$ be a valid kd-tree over $\\mathbb{Z}^d$, let $lo, hi \\in \\mathbb{Z}^d$, and let $acc$ be a list of points. The accumulator range report of $t$ for the box $[lo, hi]$ onto $acc$ equals the list of points of $t$, in left-to-right order, that lie in the box (the points $p$ with $lo_i \\le p_i \\le hi_i$ for every coordinate $i$), followed by $acc$.","title":"Accumulator range report of a valid kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/report-eq-filter","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.report_eq_filter","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#report-eq-filter","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For every valid kd-tree $t$ over $\\mathbb{Z}^d$ and all $lo, hi \\in \\mathbb{Z}^d$, the range report of $t$ for the box $[lo, hi]$ equals the list of points of $t$, in left-to-right order, that lie in the box, that is, the points $p$ with $lo_i \\le p_i \\le hi_i$ for every coordinate $i$.","title":"Correctness of the kd-tree range report"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/mem-report-iff","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.mem_report_iff","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#mem-report-iff","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $t$ be a kd-tree over $\\mathbb{Z}^d$ that satisfies the kd invariant, and let $lo, hi, p \\in \\mathbb{Z}^d$. Then $p$ occurs in the range report of $t$ for the box $[lo, hi]$ if and only if $p$ is one of the points stored in $t$ and $lo_i \\le p_i \\le hi_i$ for every coordinate $i$.","title":"Range report of a valid kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/head-coord","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.headCoord","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#head-coord","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For an axis $a \\in \\{0, \\dots, d-1\\}$ and a list of points in $\\mathbb{Z}^d$, this is the $a$-coordinate of the first point of the list, or $0$ if the list is empty.","title":"First-point coordinate"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/build","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.build","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#build","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $d \\ge 1$, let $a \\in \\{0, \\dots, d-1\\}$ be an axis and let $P$ be a list of points in $\\mathbb{Z}^d$. The kd-tree built from $P$ starting at axis $a$ is the empty tree if $P$ is empty and the leaf holding $p$ if $P = [p]$. If $P$ has $n \\ge 2$ points, sort $P$ by $a$-coordinate with merge sort, let $L$ be the first $\\lfloor n/2 \\rfloor$ points of the sorted list and $R$ the remaining ones. The result is the\u2026","title":"Kd-tree construction by median splits"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/valid-build","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.valid_build","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#valid-build","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For $d \\ge 1$, every axis $a$ and every list of points of $\\mathbb{Z}^d$, the kd-tree built from the list by median splits starting at axis $a$ satisfies the kd invariant.","title":"Median-split kd-trees satisfy the kd invariant"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/to-list-build-perm","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.toList_build_perm","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#to-list-build-perm","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For $d \\ge 1$, every axis $a$ and every list of points of $\\mathbb{Z}^d$, the point list of the kd-tree built from the list by median splits starting at axis $a$ is a permutation of the list.","title":"Building a kd-tree preserves its points"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/mem-to-list-build-iff","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.mem_toList_build_iff","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#mem-to-list-build-iff","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $d \\ge 1$, let $a \\in \\{0, \\dots, d-1\\}$ and let $P$ be a list of points in $\\mathbb{Z}^d$. A point $p \\in \\mathbb{Z}^d$ is stored in the kd-tree built from $P$ by median splits starting at axis $a$ if and only if $p$ occurs in $P$.","title":"A built kd-tree holds the same points as its input"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/size-build","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.size_build","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#size-build","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $d \\ge 1$. For every starting axis $a$ and every list $P$ of points of $\\mathbb{Z}^d$, the size of the kd-tree built from $P$ starting at axis $a$ equals the length of $P$ (repeated points counted with multiplicity).","title":"Size of a built kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/build-root","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.buildRoot","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#build-root","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For $d \\ge 1$ and a list of points in $\\mathbb{Z}^d$, this is the kd-tree built from the list by alternating-axis median splits, starting with split axis $0$.","title":"Kd-tree construction from the first axis"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/valid-build-root","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.valid_buildRoot","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#valid-build-root","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For $d \\ge 1$ and every list of points of $\\mathbb{Z}^d$, the kd-tree built from the list by median splits starting at axis $0$ satisfies the kd invariant.","title":"Kd-trees built from axis 0 satisfy the kd invariant"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/to-list-build-root-perm","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.toList_buildRoot_perm","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#to-list-build-root-perm","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"For $d \\ge 1$ and every list of points of $\\mathbb{Z}^d$, the point list of the kd-tree built from the list by median splits starting at axis $0$ is a permutation of the list.","title":"Building a kd-tree from axis 0 preserves its points"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/report-build","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.report_build","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#report-build","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $d \\ge 1$. For every starting axis $a$, every list $P$ of points of $\\mathbb{Z}^d$ and all $lo, hi \\in \\mathbb{Z}^d$, let $T$ be the kd-tree built from $P$ starting at axis $a$. The range report of $T$ for the box $[lo, hi]$ equals the list of points of $T$, in left-to-right order, that lie in the box, that is, the points $p$ with $lo_i \\le p_i \\le hi_i$ for every coordinate $i$.","title":"Range report of a built kd-tree as a filter"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/report-build-perm","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.report_build_perm","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#report-build-perm","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $d \\ge 1$. For every starting axis $a$, every list $P$ of points of $\\mathbb{Z}^d$ and all $lo, hi \\in \\mathbb{Z}^d$, the range report for the box $[lo, hi]$ of the kd-tree built from $P$ starting at axis $a$ is a permutation of the list of entries of $P$ that lie in the box, that is, the points $p$ with $lo_i \\le p_i \\le hi_i$ for every coordinate $i$.","title":"Range report of a built kd-tree up to order"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/mem-report-build-iff","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.mem_report_build_iff","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#mem-report-build-iff","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $d \\ge 1$, let $a \\in \\{0, \\dots, d-1\\}$, let $P$ be a list of points in $\\mathbb{Z}^d$ and let $lo, hi, p \\in \\mathbb{Z}^d$. Then $p$ occurs in the range report, for the box $[lo, hi]$, of the kd-tree built from $P$ by median splits starting at axis $a$ if and only if $p$ occurs in $P$ and $lo_i \\le p_i \\le hi_i$ for every coordinate $i$.","title":"Membership in the range report of a built kd-tree"},{"id":"ComputerScience/data-structures/range-query-structures/k-d-tree/mem-report-build-root-iff","kind":"node","lean":"Cslib.Algorithms.Lean.KdTree.mem_report_buildRoot_iff","parent":37915,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/k-d-tree/index.html#mem-report-build-root-iff","scope":"ComputerScience/data-structures/range-query-structures/k-d-tree","status":"fully_proved","summary":"Let $d \\ge 1$, let $P$ be a list of points in $\\mathbb{Z}^d$ and let $lo, hi, p \\in \\mathbb{Z}^d$. Then $p$ occurs in the range report, for the box $[lo, hi]$, of the kd-tree built from $P$ by median splits starting at axis $0$ if and only if $p$ occurs in $P$ and $lo_i \\le p_i \\le hi_i$ for every coordinate $i$.","title":"Range report of a kd-tree built from the first axis"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/range-tree.json","id":"ComputerScience/data-structures/range-query-structures/range-tree","kind":"scope","navigation":"module","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","title":"Range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/range-tree","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#range-tree","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"A one-dimensional range tree over the integers is either empty, a single leaf holding an integer, or a binary node made of two integers $l$ and $h$ together with a left and a right subtree. The integers $l$ and $h$ are meant to cache the interval $[l, h]$ spanned by the leaf values of the node's subtree; the type does not enforce this, the bounds-correctness predicate does.","title":"Range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/to-list-aux","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.toListAux","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#to-list-aux","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For a range tree $t$ and a list $a$ of integers, $\\operatorname{aux}(t, a)$ is $a$ if $t$ is empty, the list $x$ followed by $a$ if $t$ is a single leaf with value $x$, and $\\operatorname{aux}(L, \\operatorname{aux}(R, a))$ if $t$ is a node with left subtree $L$ and right subtree $R$.","title":"Leaf traversal of a range tree with an accumulator"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/to-list","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.toList","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#to-list","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"The leaf list of a range tree $t$ is the list of the integers stored at its leaves, read from left to right and with repetitions.","title":"Leaf list of a range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/to-list-aux-eq","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.toListAux_eq","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#to-list-aux-eq","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every range tree $t$ and every list $a$ of integers, the accumulator leaf traversal of $t$ started from $a$ equals the leaf list of $t$ followed by $a$.","title":"Range-tree leaf traversal appends to its accumulator"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/to-list-node","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.toList_node","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#to-list-node","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For all integers $lo$, $hi$ and range trees $L$, $R$, the leaf list of the node with stored bounds $lo$, $hi$, left subtree $L$ and right subtree $R$ is the leaf list of $L$ followed by the leaf list of $R$.","title":"Leaf list of a range-tree node"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/size","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.size","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#size","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"The size of a range tree is its number of leaves: $0$ for the empty tree, $1$ for a leaf, and the sum of the sizes of the two children for a node.","title":"Number of leaves of a range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/length-to-list","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.length_toList","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#length-to-list","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every range tree $t$, the list of leaf values of $t$ read from left to right has length equal to the number of leaves of $t$.","title":"Length of the leaf list of a range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/tmin","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.tmin","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#tmin","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For a range tree $t$, $\\operatorname{tmin}(t)$ is $0$ if $t$ is empty, $x$ if $t$ is a single leaf with value $x$, and the stored lower bound $lo$ if $t$ is a node with stored bounds $lo$, $hi$ and two subtrees.","title":"Cached lower bound of a range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/tmax","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.tmax","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#tmax","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For a range tree $t$, $\\operatorname{tmax}(t)$ is $0$ if $t$ is empty, $x$ if $t$ is a single leaf with value $x$, and the stored upper bound $hi$ if $t$ is a node with stored bounds $lo$, $hi$ and two subtrees.","title":"Cached upper bound of a range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/bounds-ok","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.BoundsOk","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#bounds-ok","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For a range tree $s$, write $s_{\\min}$ and $s_{\\max}$ for its cached lower and upper bounds: the stored interval endpoints at a node, the value itself at a leaf, and $0$ for the empty tree. Bounds-validity is defined inductively: the empty tree and every leaf are bounds-valid, and a node with cached interval $[a, b]$, left child $\\ell$ and right child $r$ is bounds-valid when $\\ell$ and $r$ are bounds-valid, each\u2026","title":"Bounds-valid range trees"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.Valid","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#valid","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"A range tree $t$ is valid when it has correct cached bounds (every node has two subtrees with nonempty leaf lists, and its stored lower and upper bounds are the minimum of the lower bounds and the maximum of the upper bounds of its two subtrees, recursively) and its leaf list is sorted in nondecreasing order.","title":"Valid range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/pairwise-to-list","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.Valid.pairwise_toList","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#pairwise-to-list","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"If a range tree $t$ is valid, then its list of leaf values, read from left to right, is sorted in nondecreasing order.","title":"Valid range trees have sorted leaves"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/bounds-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.bounds_of_mem","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#bounds-of-mem","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"Let $t$ be a bounds-valid range tree, and write $t_{\\min}$ and $t_{\\max}$ for its cached lower and upper bounds. Then every leaf value $x$ of $t$ satisfies $t_{\\min} \\le x \\le t_{\\max}$.","title":"Leaf values of a bounds-valid tree lie between the cached bounds"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/bounds-mem-to-list-of-ne-nil","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.bounds_mem_toList_of_ne_nil","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#bounds-mem-to-list-of-ne-nil","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"Let $t$ be a bounds-valid range tree with at least one leaf. Then the cached lower bound of $t$ and the cached upper bound of $t$ are both leaf values of $t$.","title":"The cached bounds of a nonempty bounds-valid range tree are leaf values"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/report-aux","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.reportAux","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#report-aux","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For integers $a$, $b$ and a list $\\mathit{acc}$ of integers, the accumulator range report of a range tree for $[a, b]$ starting from $\\mathit{acc}$ is defined by recursion on the tree. The empty tree returns $\\mathit{acc}$. A leaf with value $x$ returns $x$ followed by $\\mathit{acc}$ if $a \\le x \\le b$, and $\\mathit{acc}$ otherwise. A node with cached interval $[l, h]$ returns $\\mathit{acc}$ if $h < a$ or $b < l$;\u2026","title":"Accumulator range report"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/report","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.report","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#report","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"The range report of a range tree $t$ for integers $a$ and $b$ is the accumulator range report of $t$ for $[a, b]$ starting from the empty list.","title":"Range report"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/report-aux-eq-filter-append","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.reportAux_eq_filter_append","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#report-aux-eq-filter-append","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"Let $t$ be a bounds-valid range tree, let $a$ and $b$ be integers and let $\\mathit{acc}$ be a list of integers. Then the accumulator range report of $t$ for $[a, b]$ starting from $\\mathit{acc}$ is the list of the leaf values $x$ of $t$ with $a \\le x \\le b$, in left-to-right order and with repetitions, followed by $\\mathit{acc}$.","title":"Accumulator range report of a bounds-valid range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/report-eq-filter","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.report_eq_filter","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#report-eq-filter","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"Let $t$ be a range tree satisfying the bounds-correctness predicate: every node has two subtrees with at least one leaf each and caches the least and the greatest leaf value of its subtree. For all integers $a$ and $b$, the range report of $t$ for $[a, b]$ equals the list of leaf values $x$ of $t$ with $a \\le x \\le b$, in left-to-right order.","title":"Correctness of the range report"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/mem-report-iff","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.mem_report_iff","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#mem-report-iff","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"Let $t$ be a range tree satisfying the bounds-correctness predicate: every node has two subtrees with at least one leaf each and caches the least and the greatest leaf value of its subtree. For all integers $a$, $b$ and $x$, the value $x$ occurs in the range report of $t$ for $[a, b]$ if and only if $x$ is a leaf value of $t$ and $a \\le x \\le b$.","title":"Membership in a range report"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/count","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.count","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#count","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For a range tree $t$ and integers $a$ and $b$, the range count of $t$ over $[a, b]$ is the length of the range report of $t$ for $[a, b]$.","title":"Range count"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/count-eq","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.count_eq","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#count-eq","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"Let $t$ be a bounds-valid range tree and let $a$ and $b$ be integers. Then the range count of $t$ over $[a, b]$ equals the number of leaves of $t$ whose value lies in $[a, b]$.","title":"Range count of a bounds-valid tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/build","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.build","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#build","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For a range tree $s$, write $s_{\\min}$ and $s_{\\max}$ for its cached lower and upper bounds. The range tree built from a list $L$ of integers of length $n$ is defined by recursion on $n$: the empty list gives the empty tree, a one-element list $[x]$ gives the leaf $x$, and for $n \\ge 2$ it is the node whose left child $\\ell$ is the tree built from the first $\\lfloor n/2 \\rfloor$ entries of $L$, whose right child\u2026","title":"Building a range tree from a list"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/to-list-build","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.toList_build","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#to-list-build","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every list $xs$ of integers, the leaf list of the range tree built from $xs$ is $xs$.","title":"Building a range tree preserves the list"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/size-build","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.size_build","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#size-build","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every list $\\mathit{xs}$ of integers, the range tree built from $\\mathit{xs}$ has exactly as many leaves as $\\mathit{xs}$ has elements.","title":"Size of a built range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/bounds-ok-build","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.boundsOk_build","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#bounds-ok-build","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every list $L$ of integers, the range tree built from $L$ by recursive halving is bounds-valid.","title":"Built range trees are bounds-valid"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/valid-build-of-sorted","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.valid_build_of_sorted","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#valid-build-of-sorted","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every list $xs$ of integers that is sorted in nondecreasing order, the range tree built from $xs$ is valid.","title":"Building from a sorted list gives a valid range tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/report-build","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.report_build","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#report-build","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every list $\\mathit{xs}$ of integers and all integers $a$ and $b$, the range report for $[a, b]$ on the range tree built from $\\mathit{xs}$ equals the list of elements $x$ of $\\mathit{xs}$ with $a \\le x \\le b$, in their original order.","title":"Range report on a built tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/count-build","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.count_build","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#count-build","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every list $L$ of integers and all integers $a$ and $b$, the range count over $[a, b]$ of the range tree built from $L$ equals the number of entries of $L$, counted with multiplicity, that lie in $[a, b]$.","title":"Range counting on a built tree"},{"id":"ComputerScience/data-structures/range-query-structures/range-tree/mem-report-build-iff","kind":"node","lean":"Cslib.Algorithms.Lean.RangeTree.mem_report_build_iff","parent":37944,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/range-tree/index.html#mem-report-build-iff","scope":"ComputerScience/data-structures/range-query-structures/range-tree","status":"fully_proved","summary":"For every list $L$ of integers and all integers $a$, $b$ and $x$, the value $x$ occurs in the range report for $[a, b]$ on the range tree built from $L$ if and only if $x$ is an entry of $L$ and $a \\le x \\le b$.","title":"Range report membership on a built tree"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/region-quadtree.json","id":"ComputerScience/data-structures/range-query-structures/region-quadtree","kind":"scope","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/index.html","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree","status":"fully_proved","title":"Region quadtree"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants.json","id":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants","kind":"scope","navigation":"module","parent":37974,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/index.html","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants","status":"fully_proved","title":"Quadtree carrier and structural invariants"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/qtree","kind":"node","lean":"Cslib.Algorithms.Lean.QTree","parent":37975,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/index.html#qtree","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants","status":"fully_proved","summary":"A finite region quadtree with a value leaf or four ordered quadrants.","title":"Region quadtree"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/height","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.height","parent":37975,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/index.html#height","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants","status":"fully_proved","summary":"The maximum number of branches from the root to a leaf.","title":"Height of a quadtree"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/compressed","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.compressed","parent":37975,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/index.html#compressed","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants","status":"fully_proved","summary":"Every subtree is compressed and no branch has four identical leaves. Identical nonleaf children are allowed: they need not represent a constant.","title":"Compressed quadtree"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/select-tree","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.selectTree","parent":37975,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/index.html#select-tree","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants","status":"fully_proved","summary":"Selects low/low, low/high, high/low, or high/high, respectively, when `(x, y)` is `(true, true)`, `(true, false)`, `(false, true)`, or `(false, false)`. In lookup, `true` means that the corresponding coordinate is below the split.","title":"Quadrant selection by coordinate flags"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/height-children-le","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.height_children_le","parent":37975,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants/index.html#height-children-le","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/quadtree-carrier-and-structural-invariants","status":"fully_proved","summary":"A bound on branch height bounds all four child heights one level lower.","title":"Height bound for the children of a quadtree branch"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations.json","id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","kind":"scope","navigation":"module","parent":37974,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","title":"Boolean region quadtree operations"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/qc","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.Qc","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#qc","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"A branch constructor that collapses four identical Boolean leaves. This is local compression, not recursive normalization of its arguments.","title":"Compressing branch constructor for Boolean quadtrees"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/get","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.get","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#get","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Boolean lookup with `false` on a nonleaf at scale zero. Leaves ignore coordinates; positive scales use the shared quadrant order.","title":"Boolean quadtree lookup"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/neg","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.neg","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#neg","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Complements every leaf, compressing the rebuilt branches.","title":"Complement of a Boolean quadtree"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/qt-union","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.qtUnion","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#qt-union","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Boolean union, short-circuiting constant regions and compressing branches when both inputs are branches.","title":"Union of Boolean quadtrees"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/qt-diff","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.qtDiff","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#qt-diff","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Ordered Boolean difference: cells in the first region but not the second. A full first region complements the second; an empty second returns the first.","title":"Difference of Boolean quadtrees"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/height-qc-le","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.height_Qc_le","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#height-qc-le","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Local compression never increases a branch's height.","title":"Local compression does not increase quadtree height"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/compressed-qc","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.compressed_Qc","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#compressed-qc","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Four compressed children produce a compressed result.","title":"Local compression of compressed children is compressed"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/get-qc-succ","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.get_Qc_succ","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#get-qc-succ","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Compression preserves positive-scale lookup. Scale zero is excluded: collapsing four `true` leaves changes a branch's zero-scale default.","title":"Local compression preserves lookup at positive scales"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/height-neg-le","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.height_neg_le","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#height-neg-le","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Complementation does not increase height.","title":"Quadtree complementation does not increase height"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/compressed-neg","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.compressed_neg","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#compressed-neg","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Complementation preserves compression of the input.","title":"Quadtree complementation preserves compression"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/get-neg-correct","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.get_neg_correct","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#get-neg-correct","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Complementation negates each lookup whenever the scale bounds height.","title":"Quadtree complementation negates lookup at sufficient scale"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/height-qt-union-le","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.height_qtUnion_le","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#height-qt-union-le","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Union height is bounded by the maximum input height.","title":"Height bound for quadtree union"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/height-qt-diff-le","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.height_qtDiff_le","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#height-qt-diff-le","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Difference height is bounded by the maximum input height.","title":"Height bound for quadtree difference"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/compressed-qt-union","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.compressed_qtUnion","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#compressed-qt-union","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Union preserves compression when both inputs are compressed.","title":"Quadtree union preserves compression"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/compressed-qt-diff","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.compressed_qtDiff","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#compressed-qt-diff","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Difference preserves compression when both inputs are compressed.","title":"Quadtree difference preserves compression"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/get-qt-union-correct","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.get_qtUnion_correct","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#get-qt-union-correct","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Union is pointwise Boolean disjunction at a scale bounding both heights.","title":"Quadtree union is pointwise disjunction at sufficient scale"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/get-qt-diff-correct","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.get_qtDiff_correct","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#get-qt-diff-correct","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"Difference is pointwise conjunction with the second region's complement, provided the scale bounds both heights.","title":"Quadtree difference is pointwise set difference at sufficient scale"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/fdsa-chap13-ex01","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.Boolean.fdsa_chap13_ex01","parent":37981,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations/index.html#fdsa-chap13-ex01","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/boolean-region-quadtree-operations","status":"fully_proved","summary":"The original graded six-part correctness statement, under its original name.","title":"Six-part correctness of Boolean quadtree operations"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees.json","id":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","kind":"scope","navigation":"module","parent":37974,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/index.html","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","status":"fully_proved","title":"Function reconstruction by quadtrees"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/sq-reg","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.sqReg","parent":38000,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/index.html#sq-reg","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","status":"fully_proved","summary":"The represented square: both natural coordinates are below `2 ^ n`.","title":"Square region represented by a quadtree"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/mem-sq-reg","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.mem_sqReg","parent":38000,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/index.html#mem-sq-reg","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","status":"fully_proved","summary":"Membership in the represented square is a pair of strict bounds.","title":"Membership in the represented square"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/get","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.get","parent":38000,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/index.html#get","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","status":"fully_proved","summary":"Polymorphic lookup. A leaf is constant; at zero scale a branch follows its first child at `(0, 0)`. Positive scales select quadrants by the split.","title":"Polymorphic quadtree lookup"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/qt-of-fun","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.qt_of_fun","parent":38000,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/index.html#qt-of-fun","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","status":"fully_proved","summary":"Samples `f` into a full tree. Every branch splits its coordinate square into four equal quadrants, without compressing equal leaves.","title":"Full quadtree sampled from a function"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/height-qt-of-fun","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.height_qt_of_fun","parent":38000,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/index.html#height-qt-of-fun","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","status":"fully_proved","summary":"Reconstruction builds exactly the requested number of branch levels.","title":"Height of the quadtree sampled from a function"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/get-qt-of-fun","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.get_qt_of_fun","parent":38000,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/index.html#get-qt-of-fun","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","status":"fully_proved","summary":"Reading a sampled function recovers its value within the represented square.","title":"Sampled quadtree recovers the function on the represented square"},{"id":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/fdsa-chap13-ex02","kind":"node","lean":"Cslib.Algorithms.Lean.QTree.fdsa_chap13_ex02","parent":38000,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees/index.html#fdsa-chap13-ex02","scope":"ComputerScience/data-structures/range-query-structures/region-quadtree/function-reconstruction-by-quadtrees","status":"fully_proved","summary":"The original graded reconstruction theorem, under its original name.","title":"Correctness of function reconstruction by quadtrees"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/segment-tree.json","id":"ComputerScience/data-structures/range-query-structures/segment-tree","kind":"scope","navigation":"module","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","title":"Segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/segment-tree","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#segment-tree","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"A segment tree with values in $\\alpha$ is either empty, a leaf holding a value in $\\alpha$, or an internal node holding a cached segment aggregate in $\\alpha$, a cached segment length in $\\mathbb{N}$, a left child and a right child, both segment trees.","title":"Segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/agg","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.agg","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#agg","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"The aggregate of a segment tree with values in an additive monoid is $0$ for the empty tree, $x$ for a single leaf with value $x$, and the aggregate cached at the root for an internal node.","title":"Aggregate of a segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/size","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.size","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#size","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"The size of a segment tree is $0$ for the empty tree, $1$ for a leaf, and, for an internal node, the segment length cached at that node.","title":"Size of a segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/to-list-aux","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.toListAux","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#to-list-aux","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"Given a segment tree and a list $acc$, the accumulator traversal returns $acc$ for the empty tree, the list $acc$ with $x$ prepended for a leaf with value $x$, and, for an internal node with left child $L$ and right child $R$, the traversal of $L$ starting from the result of the traversal of $R$ starting from $acc$.","title":"Accumulator traversal of a segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/to-list","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.toList","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#to-list","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"The leaf list of a segment tree is the list of its leaf values from left to right: the empty list for the empty tree, $[x]$ for a leaf with value $x$, and the leaf list of the left child followed by the leaf list of the right child for an internal node.","title":"Leaf list of a segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/to-list-aux-eq","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.toListAux_eq","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#to-list-aux-eq","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every segment tree $t$ and every list $acc$, the accumulator traversal of $t$ starting from $acc$ returns the leaf list of $t$ followed by $acc$.","title":"Accumulator traversal prepends the leaf list"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/to-list-node","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.toList_node","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#to-list-node","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every internal node of a segment tree, with any cached aggregate, any cached length, left child $L$ and right child $R$, its leaf list is the leaf list of $L$ followed by the leaf list of $R$.","title":"Leaf list of an internal node"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.Valid","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#valid","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"Validity of a segment tree over an additive monoid is defined inductively: the empty tree and every leaf are valid, and an internal node with valid left child $L$ and valid right child $R$ is valid when its cached aggregate is the aggregate of $L$ plus the aggregate of $R$ and its cached length is the size of $L$ plus the size of $R$. Here the aggregate and size of the empty tree are $0$ and $0$, of a leaf with\u2026","title":"Valid segment trees"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/length-to-list-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.length_toList_of_valid","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#length-to-list-of-valid","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"If $t$ is a valid segment tree with values in an additive monoid, then the length of the list of leaf values of $t$, read from left to right, equals the size of $t$.","title":"Size of a valid segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/agg-eq-sum-to-list","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.agg_eq_sum_toList","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#agg-eq-sum-to-list","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"If $t$ is a valid segment tree with values in an additive monoid, then the aggregate of $t$ equals the sum of the leaf values of $t$, taken from left to right.","title":"Aggregate of a valid segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"The segment tree built from a list $L$ with entries in an additive monoid is the empty tree if $L$ is empty, and a single leaf with value $x$ if $L = (x)$. If $L$ has length $n \\ge 2$, it is the internal node whose left child is the tree built from the first $\\lfloor n/2 \\rfloor$ entries of $L$, whose right child is the tree built from the remaining entries, and which caches the aggregate of the left child plus the\u2026","title":"Building a segment tree from a list"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/valid-build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.valid_build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#valid-build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every list $xs$ over an additive monoid, the segment tree built from $xs$ is valid.","title":"Built segment trees are valid"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/to-list-build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.toList_build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#to-list-build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every list $xs$ with entries in an additive monoid, the leaf list of the segment tree built from $xs$ is $xs$.","title":"Building a segment tree preserves the list"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/size-build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.size_build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#size-build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every list $xs$ with entries in an additive monoid, the segment tree built from $xs$ has size equal to the length of $xs$.","title":"Size of a built segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/agg-build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.agg_build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#agg-build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every list $L$ with entries in an additive monoid, the aggregate of the segment tree built from $L$ equals the sum of the entries of $L$, taken from left to right.","title":"Aggregate of a built segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/query","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.query","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#query","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For a segment tree $t$ with values in an additive monoid and natural numbers $l, r$, the range query over $[l, r)$, with positions counted from the start of the segment of $t$, is defined recursively. On the empty tree it is $0$. On a leaf with value $x$ it is $x$ if $l = 0 < r$, and $0$ otherwise. On an internal node with cached aggregate $a$, cached size $s$ and children $t_1, t_2$, it is $0$ if $r \\le l$ or\u2026","title":"Range query on a segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/query-eq","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.query_eq","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#query-eq","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"Let $t$ be a valid segment tree with values in an additive monoid, let $(x_0, \\dots, x_{n-1})$ be its leaf values from left to right, and let $l, r$ be natural numbers. The range query over $[l, r)$ on $t$ returns $\\sum_{l \\le k < \\min(r, n)} x_k$, the sum taken in increasing order of $k$; it is $0$ when $\\min(r, n) \\le l$.","title":"Correctness of range queries on a valid segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/query-build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.query_build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#query-build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"Let $L = (x_0, \\dots, x_{n-1})$ be a list with entries in an additive monoid and let $l, r$ be natural numbers. The range query over $[l, r)$ on the segment tree built from $L$ returns $\\sum_{l \\le k < \\min(r, n)} x_k$, the sum taken in increasing order of $k$; it is $0$ when $\\min(r, n) \\le l$.","title":"Range queries on a built segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/update","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.update","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#update","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"The point update of a segment tree over an additive monoid at an index $i \\in \\mathbb{N}$, relative to the tree's segment, with a value $y$ is defined recursively. The empty tree stays empty. A leaf becomes a leaf with value $y$ if $i = 0$ and is unchanged otherwise. At an internal node with left child $L$ and right child $R$: if $i$ is less than the size of $L$, then $L$ is replaced by its point update at $i$ with\u2026","title":"Point update of a segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/size-update","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.size_update","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#size-update","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every segment tree $t$ over an additive monoid, every index $i$ and every value $y$, the point update of $t$ at $i$ with $y$ has the same size as $t$.","title":"Point update preserves the size of a segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/update-2","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.Valid.update","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#update-2","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"If $t$ is a valid segment tree over an additive monoid, then for every natural number $i$ and every value $y$, the point update of $t$ at $i$ with $y$ is valid.","title":"Point update preserves validity"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/to-list-update-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.toList_update_of_lt","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#to-list-update-of-lt","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"Let $t$ be a valid segment tree over an additive monoid and let $i$ be a natural number less than the size of $t$. Then for every value $y$, the leaf list of the point update of $t$ at $i$ with $y$ is the leaf list of $t$ with its entry at position $i$ (counting from $0$) replaced by $y$.","title":"Leaf list after an in-range point update"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/to-list-update-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.toList_update_of_le","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#to-list-update-of-le","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every valid segment tree $t$ over an additive monoid, every index $i$ at least the size of $t$ and every value $y$, the point update of $t$ at $i$ with $y$ has the same leaf list as $t$.","title":"Out-of-range point update leaves the leaf list unchanged"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/to-list-update-build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.toList_update_build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#to-list-update-build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"For every list $xs$ with entries in an additive monoid, every index $i$ smaller than the length of $xs$ and every value $y$, the leaf list of the segment tree built from $xs$, after the point update at $i$ with $y$, is $xs$ with its entry at position $i$ (counting from $0$) replaced by $y$.","title":"Point update of a built segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/agg-update-build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.agg_update_build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#agg-update-build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"Let $L = (x_0, \\dots, x_{n-1})$ be a list with entries in an additive monoid, let $i < n$, and let $y$ be an element of the monoid. Applying the point update at index $i$ with value $y$ to the segment tree built from $L$ gives a tree whose aggregate is the sum, taken from left to right, of the list obtained from $L$ by replacing $x_i$ with $y$.","title":"Aggregate after a point update of a built segment tree"},{"id":"ComputerScience/data-structures/range-query-structures/segment-tree/query-update-build","kind":"node","lean":"Cslib.Algorithms.Lean.SegmentTree.query_update_build","parent":38008,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/segment-tree/index.html#query-update-build","scope":"ComputerScience/data-structures/range-query-structures/segment-tree","status":"fully_proved","summary":"Let $L = (x_0, \\dots, x_{n-1})$ be a list with entries in an additive monoid, let $i < n$, let $y$ be an element of the monoid, and let $L'$ be the list obtained from $L$ by replacing $x_i$ with $y$, with entries $x'_0, \\dots, x'_{n-1}$. Apply the point update at index $i$ with value $y$ to the segment tree built from $L$. For all natural numbers $l, r$, the range query over $[l, r)$ on the result returns\u2026","title":"Range queries after a point update of a built segment tree"},{"data_url":"scopes/ComputerScience/data-structures/range-query-structures/sparse-table.json","id":"ComputerScience/data-structures/range-query-structures/sparse-table","kind":"scope","navigation":"module","parent":37728,"result":false,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","title":"Sparse table"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/sparse-table","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#sparse-table","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"A sparse table over a type $\\alpha$ consists of an array of elements of $\\alpha$, called its data, and an array of arrays of elements of $\\alpha$, called its levels.","title":"Sparse table"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/omin","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.omin","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#omin","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For a type $\\alpha$ with a minimum operation, the option minimum extends $\\min$ to optional values (elements of $\\alpha$ or none): on two elements $a, b$ of $\\alpha$ it returns $\\min(a, b)$, and none is a neutral element on either side.","title":"Option minimum"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/omin-self","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.omin_self","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#omin-self","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For every linearly ordered type $\\alpha$ and every optional value $a$ (an element of $\\alpha$ or none), the option minimum of $a$ with itself is $a$.","title":"Idempotence of the option minimum"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/level","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.level","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#level","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For a list $a$ over a type with a minimum operation $\\min$, the levels of $a$ are defined recursively: level $0$ is $a$, and if level $k$ is $\\ell = (\\ell_0, \\dots, \\ell_{m-1})$, then level $k + 1$ is the list of the values $\\min(\\ell_i, \\ell_{i+2^k})$ for $0 \\le i < m - 2^k$ (empty when $m \\le 2^k$).","title":"Levels of a sparse table"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/level-length","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.level_length","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#level-length","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For a list $a$ of length $n$ over a type with a minimum operation and a natural number $k$, level $k$ of $a$ has length $\\max(n + 1 - 2^k, 0)$: one element for each start index $i$ with $i + 2^k \\le n$.","title":"Length of a level"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/get-elem-level","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.getElem?_level","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#get-elem-level","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be a list over a linearly ordered type, and let $k$ and $i$ be natural numbers with $i + 2^k \\le n$. Then level $k$ of $a$ has an element at index $i$, and it equals $\\min(a_i, a_{i+1}, \\dots, a_{i+2^k-1})$.","title":"Levels hold interval minima"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/next-level","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.nextLevel","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#next-level","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For a type with a minimum operation, an array $(r_0, \\dots, r_{n-1})$ and a natural number $k$, the next level is the array of length $n - 2^k$ (empty when $2^k \\ge n$) whose entry $i$ is $\\min(r_i, r_{i + 2^k})$.","title":"Next level of a sparse table row"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/build-step","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.buildStep","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#build-step","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"Work over a type with a minimum operation $\\min$. Given a pair $(r, L)$ of a current row $r$ (an array of length $|r|$) and an array $L$ of the rows recorded so far, and a natural number $k$, the build step returns the pair $(r', L')$, where $L'$ is $L$ with $r$ appended at the end, and $r'$ is the array whose entry $i$ is $\\min(r_i, r_{i+2^k})$ for each $i$ with $i + 2^k < |r|$.","title":"One step of the sparse-table construction"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/build","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.build","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#build","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For a list $a$ of length $n$ over a type with a minimum operation $\\min$, the sparse table built from $a$ has data $a$ and $\\lfloor \\log_2 n \\rfloor + 1$ stored rows, where $\\lfloor \\log_2 0 \\rfloor$ is taken to be $0$. The rows are produced by iterating the build step for $k = 0, 1, \\dots, \\lfloor \\log_2 n \\rfloor$, starting from the current row $a$ and no recorded rows. So row $0$ is $a$, and entry $i$ of row\u2026","title":"Building a sparse table"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/valid","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.Valid","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#valid","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"A sparse table over a type with a minimum operation, whose data is a list $a$ of length $n$, is valid when it has exactly $m + 1$ levels, where $m = \\lfloor \\log_2 n \\rfloor$ for $n \\ge 1$ and $m = 0$ for $n = 0$, and for each $k \\le m$ its level $k$ is the list $L_k$ defined by $L_0 = a$ and, for $x = L_j$, $L_{j+1}$ is the list of length $|x| - 2^j$ (empty when $2^j \\ge |x|$) whose entry $i$ is\u2026","title":"Valid sparse table"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/build-valid","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.build_valid","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#build-valid","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For every list $a$ over a type with a minimum operation, the sparse table built from $a$ is valid: its stored rows are exactly the levels $0, 1, \\dots, \\lfloor \\log_2 n \\rfloor$ of $a$, in this order, where $n$ is the length of $a$ and $\\lfloor \\log_2 0 \\rfloor$ is taken to be $0$.","title":"Built sparse tables are valid"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/entry","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.entry","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#entry","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For a sparse table $t$ and natural numbers $k$ and $i$, the entry of $t$ at row $k$ and index $i$ is element $i$ of the $k$-th stored row of $t$ (both counted from $0$) when $t$ has a $k$-th row and that row has an element at index $i$; otherwise the entry is undefined. Both lookups are random-access array lookups.","title":"Sparse-table entries"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/entry-eq-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.entry_eq_of_valid","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#entry-eq-of-valid","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"Let $t$ be a valid sparse table over a type with a minimum operation, with data $d$ of length $n$, and let $k \\le \\lfloor \\log_2 n \\rfloor$, where $\\lfloor \\log_2 0 \\rfloor$ is taken to be $0$. Then for every index $i$, the entry of $t$ at row $k$ and index $i$ is defined exactly when level $k$ of $d$ has an element at index $i$, and in that case the two are equal.","title":"Rows of a valid sparse table are its levels"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/entry-build-eq-min","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.entry_build_eq_min?","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#entry-build-eq-min","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be a list over a linearly ordered type, and let $k$ and $i$ be natural numbers with $i + 2^k \\le n$. Then the entry of the sparse table built from $a$ at row $k$ and index $i$ is defined and equals $\\min(a_i, a_{i+1}, \\dots, a_{i+2^k-1})$.","title":"Entries of a built sparse table are interval minima"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/entry-eq-min-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.entry_eq_min?_of_valid","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#entry-eq-min-of-valid","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"Let $t$ be a valid sparse table over a linearly ordered type, with data $d = (d_0, \\dots, d_{n-1})$, and let $k$ and $i$ be natural numbers with $i + 2^k \\le n$. Then the entry of $t$ at row $k$ and index $i$ is defined and equals $\\min(d_i, d_{i+1}, \\dots, d_{i+2^k-1})$.","title":"Entries of a valid sparse table are interval minima"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/query","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.query","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#query","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For a sparse table $t$ over a type with a minimum operation and natural numbers $l, r$, the query over $[l, r)$ returns none when $r \\le l$. Otherwise, with $k = \\lfloor \\log_2 (r - l) \\rfloor$, it returns the option minimum (the minimum, with none as a neutral element on either side) of the entries of $t$ at level $k$ and indices $l$ and $r - 2^k$, in this order, where an entry not stored in $t$ counts as none.","title":"Sparse table range-minimum query"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/query-eq-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.query_eq_of_valid","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#query-eq-of-valid","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"Let $t$ be a valid sparse table over a linearly ordered type whose data is the list $(a_0, \\dots, a_{n-1})$, and let $l, r$ be natural numbers with $r \\le n$. Then the range-minimum query over $[l, r)$ on $t$ returns the minimum of the slice $(a_l, \\dots, a_{r-1})$ (none when $r \\le l$, where the slice is empty).","title":"Correctness of queries on valid sparse tables"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/query-build","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.query_build","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#query-build","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be a list over a linearly ordered type and let $l, r$ be natural numbers with $r \\le n$. Then the range-minimum query over $[l, r)$ on the sparse table built from $a$ returns the minimum of the slice $(a_l, \\dots, a_{r-1})$ (none when $r \\le l$, where the slice is empty).","title":"Correctness of sparse table queries"},{"id":"ComputerScience/data-structures/range-query-structures/sparse-table/query-build-all","kind":"node","lean":"Cslib.Algorithms.Lean.SparseTable.query_build_all","parent":38035,"result":true,"route":"../roadmap/ComputerScience/data-structures/range-query-structures/sparse-table/index.html#query-build-all","scope":"ComputerScience/data-structures/range-query-structures/sparse-table","status":"fully_proved","summary":"For every list $a$ of length $n$ over a linearly ordered type, the range-minimum query over $[0, n)$ on the sparse table built from $a$ returns the minimum of $a$ (none when $a$ is empty).","title":"Whole-range sparse table query"},{"data_url":"scopes/ComputerScience/data-structures/tries-and-integer-dictionaries.json","id":"ComputerScience/data-structures/tries-and-integer-dictionaries","kind":"scope","parent":36418,"result":false,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/index.html","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries","status":"fully_proved","summary":"Prefix- and bit-indexed structures: tries, radix (Patricia) tries, suffix arrays and suffix trees, van Emde Boas trees, x-fast and y-fast tries.","title":"Tries, suffix structures and integer dictionaries"},{"data_url":"scopes/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie.json","id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","kind":"scope","navigation":"module","parent":38055,"result":false,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","title":"Radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/radix-trie","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#radix-trie","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For types $\\alpha$ and $\\beta$, a radix trie with keys that are lists over $\\alpha$ and values in $\\beta$ is a node made of an optional value in $\\beta$ and a finite list of child edges, each a pair of a label, which is a list over $\\alpha$, and a radix trie (the subtree at the end of the edge).","title":"Radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/empty","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.empty","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"The empty radix trie is the single node that stores no value and has no child edges; it contains no key-value entries.","title":"Empty radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/common-prefix","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.commonPrefix","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#common-prefix","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For lists $s$ and $t$ over a type with decidable equality, the longest common prefix of $s$ and $t$ is defined by recursion: it is empty if $s$ or $t$ is empty; if $s$ has first element $a$ and rest $s'$, and $t$ has first element $b$ and rest $t'$, it is $a$ followed by the longest common prefix of $s'$ and $t'$ when $a = b$, and empty when $a \\ne b$.","title":"Longest common prefix"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/common-prefix-self","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.commonPrefix_self","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#common-prefix-self","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For every list $s$ over a type with decidable equality, the longest common prefix of $s$ and $s$ is $s$.","title":"Longest common prefix of a list with itself"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/common-prefix-append-append","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.commonPrefix_append_append","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#common-prefix-append-append","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Write $uv$ for the list $u$ followed by the list $v$. For lists $p$, $s$ and $t$ over a type with decidable equality, the longest common prefix of $ps$ and $pt$ is $p$ followed by the longest common prefix of $s$ and $t$.","title":"A shared prefix factors out of the longest common prefix"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/common-prefix-spec","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.commonPrefix_spec","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#common-prefix-spec","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Let $s$ and $t$ be lists over a type with decidable equality, with longest common prefix $p$. Then there are lists $s'$ and $t'$ such that $s$ is $p$ followed by $s'$, $t$ is $p$ followed by $t'$, and either $s'$ is empty, or $t'$ is empty, or $s'$ and $t'$ are both nonempty with different first elements.","title":"Characterization of the longest common prefix"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/append-drop-common-prefix-left","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.append_drop_commonPrefix_left","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#append-drop-common-prefix-left","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Let $s$ and $t$ be lists over a type with decidable equality, and let $p$ be the longest common prefix of $s$ and $t$. Then $s$ is $p$ followed by the list obtained by dropping the first $|p|$ elements of $s$, where $|p|$ is the length of $p$.","title":"A list is its common prefix with another list followed by the rest"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/common-prefix-drop-spec","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.commonPrefix_drop_spec","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#common-prefix-drop-spec","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Let $s$ and $t$ be lists over a type with decidable equality, let $p$ be their longest common prefix, of length $n$, and let $s'$ and $t'$ be the lists obtained from $s$ and $t$ by dropping their first $n$ elements. Then $s$ is $p$ followed by $s'$, $t$ is $p$ followed by $t'$, and either $s'$ is empty, or $t'$ is empty, or $s'$ and $t'$ are both nonempty with different first elements.","title":"Longest common prefix in terms of dropped remainders"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/append-of-common-prefix-eq","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.append_of_commonPrefix_eq","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#append-of-common-prefix-eq","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Let $s$ and $t$ be lists over a type with decidable equality. If the longest common prefix of $s$ and $t$ is $t$, then $s$ is $t$ followed by the list obtained by dropping the first $|t|$ elements of $s$, where $|t|$ is the length of $t$.","title":"When the common prefix is the second list, the first list extends it"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/common-prefix-eq-of-append","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.commonPrefix_eq_of_append","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#common-prefix-eq-of-append","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Let $s$, $t$ and $u$ be lists over a type with decidable equality. If $s$ is $t$ followed by $u$, then the longest common prefix of $s$ and $t$ is $t$.","title":"The longest common prefix of a list and its prefix"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/find-edge","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.findEdge","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#find-edge","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Given a symbol $c$ and a list of edges, each a pair of a label (a list of symbols) and a subtree, this returns the first edge in the list whose label starts with $c$, or nothing if no label starts with $c$.","title":"Edge search by first symbol"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/replace-edge","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.replaceEdge","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#replace-edge","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Given a symbol $c$, an edge $e$ (a pair of a label and a radix trie) and a list of child edges, this operation replaces the first edge of the list whose label starts with $c$ by $e$ and keeps all other edges in order. If no label in the list starts with $c$, it appends $e$ at the end of the list.","title":"Replacing the child edge for a symbol"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/lookup","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.lookup","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#lookup","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Looking up a key, a list over $\\alpha$, in a radix trie returns an optional value. At a node, the empty key returns the value stored at the node, if any. A nonempty key $k$ with first symbol $c$ goes to the first child edge whose label starts with $c$. If there is no such edge, or its label $\\ell$ is not a prefix of $k$, the lookup returns no value; otherwise it continues in the subtree of that edge with $k$ minus\u2026","title":"Radix trie lookup"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/lookup-node-cons","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.lookup_node_cons","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#lookup-node-cons","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Let $k$ be a nonempty key with first symbol $c$, and consider a node with children $C$ (a list of edges, each a label with a subtree), whatever value the node stores. If no edge of $C$ has a label starting with $c$, looking up $k$ at the node returns no value. Otherwise let $(\\ell, T')$ be the first such edge of $C$: if the longest common prefix of $k$ and $\\ell$ is $\\ell$, the lookup returns the lookup in $T'$ of\u2026","title":"Radix trie lookup of a nonempty key at a node"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/lookup-empty","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.lookup_empty","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#lookup-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For every key $k$, looking up $k$ in the empty radix trie returns no value.","title":"Lookup in the empty radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/insert","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.insert","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Inserting a value $v$ at a key $k$ (a list of symbols) into a radix trie works as follows. If $k$ is empty, the root's value becomes $v$. Otherwise let $c$ be the first symbol of $k$. If no child edge of the root has a label starting with $c$, a new edge labelled $k$ leading to a leaf storing $v$ is added after the existing edges. Otherwise take the first such edge, with label $\\ell$ and subtree $s$, let $q$ be the\u2026","title":"Insertion into a radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/lookup-insert-same","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.lookup_insert_same","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#lookup-insert-same","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For every radix trie $T$, key $k$ and value $v$, looking up $k$ in the trie obtained from $T$ by inserting $v$ at $k$ returns $v$.","title":"Lookup of a key just inserted into a radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/lookup-insert-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.lookup_insert_of_ne","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#lookup-insert-of-ne","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For every radix trie $T$, all keys $k \\ne k'$ and every value $v$, looking up $k$ in the trie obtained from $T$ by inserting $v$ at $k'$ gives the same result as looking up $k$ in $T$.","title":"Insertion into a radix trie leaves lookups of other keys unchanged"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/root-compact","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.RootCompact","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#root-compact","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"A radix trie is path-compressed at its root when its root node stores a value or has at least two child edges.","title":"Path-compressed root"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/valid","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.Valid","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Validity of radix tries is defined inductively: a node is valid when every label of its child edges is nonempty, the first symbols of the labels of its child edges are pairwise distinct, and for each child edge the subtree at its end is valid and path-compressed at its root (its root stores a value or has at least two child edges).","title":"Valid radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/empty-2","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.Valid.empty","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#empty-2","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For all types $\\alpha$ and $\\beta$, the empty radix trie whose keys are lists over $\\alpha$ and whose values lie in $\\beta$ is valid.","title":"The empty radix trie is valid"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/nodup-child-heads","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.Valid.nodup_child_heads","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#nodup-child-heads","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"If a radix-trie node with child edges $(\\ell_1, t_1), \\dots, (\\ell_n, t_n)$ is valid, then the first symbols of the labels $\\ell_1, \\dots, \\ell_n$ are pairwise distinct.","title":"Sibling edge labels start with distinct symbols"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/root-compact-child","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.Valid.rootCompact_child","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#root-compact-child","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"If a radix-trie node is valid, then for each of its child edges, the subtree at the end of that edge is path-compressed at its root: its root stores a value or has at least two child edges.","title":"Children of a valid node are path-compressed"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/insert-2","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.Valid.insert","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#insert-2","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"If $t$ is a valid radix trie, then for every key $k$ and value $v$, the trie obtained by inserting $v$ at $k$ in $t$ is valid.","title":"Insertion preserves validity of radix tries"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/entries","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.entries","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#entries","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"The entries of a radix trie are its key-value pairs, listed recursively. For a node with child edges $(\\ell_1, t_1), \\dots, (\\ell_n, t_n)$ in stored order, the list starts with the pair $(\\varepsilon, v)$, where $\\varepsilon$ is the empty key, if the node stores a value $v$; it continues with the entries of $t_1$ with $\\ell_1$ prepended to every key, then the entries of $t_2$ with $\\ell_2$ prepended, and so on up\u2026","title":"Entries of a radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/keys","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.keys","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#keys","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"The stored keys of a radix trie form the list of the keys of its entries, in the same order.","title":"Stored keys of a radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/lookup-eq-some-iff-mem-entries","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.lookup_eq_some_iff_mem_entries","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#lookup-eq-some-iff-mem-entries","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For a valid radix trie $T$, every key $k$ and every value $v$, looking up $k$ in $T$ returns $v$ if and only if the pair $(k, v)$ occurs in the list of entries of $T$.","title":"Lookup agrees with the entries of a valid radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/mem-keys-iff","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.mem_keys_iff","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#mem-keys-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For a valid radix trie $T$ and every key $k$, $k$ is a stored key of $T$ if and only if looking up $k$ in $T$ returns some value.","title":"Stored keys of a valid radix trie are the keys with a successful lookup"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/keys-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.Valid.keys_nodup","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#keys-nodup","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"If $t$ is a valid radix trie, then its list of stored keys has no duplicates.","title":"Stored keys of a valid radix trie are distinct"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/has-prefix","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.hasPrefix","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#has-prefix","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For a radix trie $t$ and a pattern $p$ (a list of symbols), prefix search returns a boolean computed as follows. If $p$ is empty, it is true exactly when $t$ stores at least one key. Otherwise let $c$ be the first symbol of $p$ and take the first child edge of the root whose label starts with $c$; if there is none, the answer is false. If that edge has label $\\ell$ and subtree $s$, then: when $p$ is a prefix of\u2026","title":"Prefix search in a radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/has-prefix-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.hasPrefix_eq_true_iff","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#has-prefix-eq-true-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For a valid radix trie $t$ and a list $p$ of symbols, prefix search for $p$ in $t$ succeeds if and only if some stored key $k$ of $t$ has $p$ as a prefix.","title":"Prefix search is correct on valid radix tries"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/build-from","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.buildFrom","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#build-from","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"The trie built from the empty list is the empty trie, and the trie built from a list with head $(k, v)$ and tail $L$ is the trie built from $L$ with the value $v$ inserted at the key $k$. Later pairs in the list are thus inserted first, so on duplicate keys the head-most value wins.","title":"Radix trie built from a list of key-value pairs"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/lookup-build-from-exists-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.lookup_buildFrom_exists_of_mem","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#lookup-build-from-exists-of-mem","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For a list $L$ of key-value pairs, if $(k, v)$ belongs to $L$, then looking up $k$ in the radix trie built from $L$ returns some value.","title":"Every listed key is present in the built radix trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/mem-of-lookup-build-from","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.mem_of_lookup_buildFrom","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#mem-of-lookup-build-from","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For every list $L$ of key-value pairs, if looking up a key $k$ in the radix trie built from $L$ returns a value $v$, then the pair $(k, v)$ occurs in $L$.","title":"Lookups in a radix trie built from a list come from the list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/mem-keys-build-from","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.mem_keys_buildFrom","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#mem-keys-build-from","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"For every list $L$ of key-value pairs and every key $k$, $k$ is a stored key of the radix trie built from $L$ if and only if the pair $(k, v)$ occurs in $L$ for some value $v$.","title":"Keys of a radix trie built from a list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/lookup-build-from-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.RadixTrie.lookup_buildFrom_of_mem","parent":38056,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie/index.html#lookup-build-from-of-mem","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/radix-trie","status":"fully_proved","summary":"Let $L$ be a list of key-value pairs, whose keys are lists over $\\gamma$, in which no key occurs twice. If the pair $(k, v)$ occurs in $L$, then looking up $k$ in the radix trie built from $L$ returns $v$.","title":"Lookup in a radix trie built from a list with distinct keys"},{"data_url":"scopes/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array.json","id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","kind":"scope","navigation":"module","parent":38055,"result":false,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","title":"Suffix array"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/suffix","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.suffix","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#suffix","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For a list $s = (s_0, \\dots, s_{n-1})$ over a linearly ordered alphabet and a natural number $i$, the suffix of $s$ at $i$ is the list obtained from $s$ by dropping its first $i$ entries: it is $(s_i, \\dots, s_{n-1})$ when $i < n$ and the empty list when $i \\ge n$.","title":"Suffix of a list at an index"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/suffixes","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.suffixes","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#suffixes","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For a list $s$ of length $n$ over a linearly ordered alphabet, the list of suffixes of $s$ is the list, for $i = 0, 1, \\dots, n$ in this order, of the lists obtained from $s$ by dropping its first $i$ entries. It starts with $s$ itself and ends with the empty list.","title":"Suffixes of a list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/suffix-array","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.suffixArray","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#suffix-array","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For a list $s$ of length $n$ over a linearly ordered alphabet, the suffix array of $s$ is the list of the indices $0, 1, \\dots, n - 1$, sorted so that the suffixes of $s$ starting at them appear in lexicographic order.","title":"Suffix array"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/suffix-array-perm","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.suffixArray_perm","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#suffix-array-perm","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For a list $s$ of length $n$ over a linearly ordered alphabet, the suffix array of $s$ is a permutation of the list $0, 1, \\dots, n - 1$ of all indices below $n$.","title":"The suffix array is a permutation of the indices"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/pairwise-suffix-suffix-array","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.pairwise_suffix_suffixArray","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#pairwise-suffix-suffix-array","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For a list $s$ over a linearly ordered alphabet, the suffixes of $s$ taken in suffix-array order are sorted lexicographically: whenever an index $i$ comes before an index $j$ in the suffix array of $s$, the suffix of $s$ starting at $i$ is lexicographically less than or equal to the suffix of $s$ starting at $j$.","title":"The suffix array sorts the suffixes"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/lcp","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.lcp","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#lcp","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For lists $x$ and $y$, $\\mathrm{lcp}(x, y)$ is the length of the longest common prefix of $x$ and $y$: the number of positions, counted from the start, at which $x$ and $y$ agree before they first differ or one of them ends.","title":"Longest common prefix length"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/take-lcp-eq","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.take_lcp_eq","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#take-lcp-eq","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For lists $x$ and $y$ over a type with decidable equality, let $\\operatorname{lcp}(x, y)$ be the length of their longest common prefix. Then the first $\\operatorname{lcp}(x, y)$ elements of $x$ and the first $\\operatorname{lcp}(x, y)$ elements of $y$ form the same list.","title":"Two lists agree on their longest common prefix"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/take-eq-of-le-lcp","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.take_eq_of_le_lcp","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#take-eq-of-le-lcp","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"Let $x$ and $y$ be lists and let $\\mathrm{lcp}(x, y)$ be the length of their longest common prefix. For every natural number $q \\le \\mathrm{lcp}(x, y)$, the first $q$ elements of $x$ and of $y$ agree.","title":"Common prefixes of every shorter length"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/le-lcp-iff","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.le_lcp_iff","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#le-lcp-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For lists $x$ and $y$, let $\\mathrm{lcp}(x, y)$ be the length of the longest common prefix of $x$ and $y$. For every natural number $q$, we have $q \\le \\mathrm{lcp}(x, y)$ if and only if $q$ is at most the lengths of both $x$ and $y$ and the first $q$ elements of $x$ and of $y$ agree.","title":"Characterization of the longest common prefix length"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/adj-lcp","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.adjLcp","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#adj-lcp","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For a list $s$ and a list of indices $i_0, i_1, \\dots, i_k$, this is the list of length $k$ whose $t$-th entry, for $t < k$, is the length of the longest common prefix of the suffixes of $s$ starting at $i_t$ and at $i_{t+1}$, where the suffix starting at an index at or past the end of $s$ is the empty list. It has one entry per adjacent pair of indices, and it is empty for an empty index list.","title":"Longest-common-prefix lengths of adjacent suffixes"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/lcp-array","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixArray.lcpArray","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#lcp-array","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"For a list $s$ over a linearly ordered alphabet, the LCP array of $s$ is the list of longest-common-prefix lengths of adjacent suffixes in suffix-array order: if the suffix array of $s$ is $i_0, i_1, \\dots, i_k$, then the LCP array has length $k$ and its $t$-th entry, for $t < k$, is the length of the longest common prefix of the suffixes of $s$ starting at $i_t$ and at $i_{t+1}$. It is empty when $s$ is empty.","title":"The LCP array"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/prefix-of-le-of-le","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.prefix_of_le_of_le","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#prefix-of-le-of-le","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"If `x \u2264 y \u2264 z` and `P` is a prefix of both `x` and `z`, then `P` is a prefix of `y`.","title":"Prefix classes are lexicographic intervals"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/take-le-take","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.take_le_take","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#take-le-take","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"Truncation is monotone for the lexicographic order: `x \u2264 y \u2192 x.take m \u2264 y.take m`.","title":"Truncation is monotone for the lexicographic order"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/prefix-suffix-array-of-le-of-le","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.prefix_suffixArray_of_le_of_le","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#prefix-suffix-array-of-le-of-le","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"If rows `r\u2081 \u2264 r\u2082 \u2264 r\u2083` of the suffix array of `s` are in range and the suffixes of rows `r\u2081` and `r\u2083` start with `P`, so does the suffix of row `r\u2082`.","title":"The occurrences of a pattern are contiguous in the suffix array"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/lower-bound","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.lowerBound","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#lower-bound","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"Binary search for the boundary of a predicate on `[lo, hi)`: the first index at which `p` is false, assuming `p` holds on an initial segment of `[lo, hi)` (`lowerBound_spec`). Each call probes the midpoint and recurses on one half.","title":"Binary search for the boundary of a predicate"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/lower-bound-steps","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.lowerBoundSteps","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#lower-bound-steps","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"The number of probes `lowerBound p lo hi` makes.","title":"Number of probes made by the binary search"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/lower-bound-spec","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.lowerBound_spec","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#lower-bound-spec","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"If `p` holds on an initial segment of `[lo, hi)` (whenever it holds at `j`, it holds at every `i` with `lo \u2264 i \u2264 j`), then `lowerBound p lo hi` is the end `k` of that segment: `lo \u2264 k \u2264 hi`, `p` holds on `[lo, k)` and fails on `[k, hi)`.","title":"Binary search is correct"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/lower-bound-steps-le","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.lowerBoundSteps_le","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#lower-bound-steps-le","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"At most `\u230alog\u2082 (hi - lo)\u230b + 1` probes, for any predicate.","title":"Binary search is logarithmic"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/row-key","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.rowKey","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#row-key","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"The suffix of row `r` of the (precomputed) suffix array `sa` of `s`, truncated to `m` letters: the key compared with the pattern during the search. `sa` is computed once by `saSearch` and passed in, so probes do not rebuild the suffix array.","title":"Truncated suffix of a suffix-array row"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/sa-search","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.saSearch","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#sa-search","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"Suffix-array search (Manber and Myers 1993): the row interval `[sp, ep)` of the suffixes of `s` that start with `P`, found by two binary searches over the rows `[0, |s|)`. The suffix array is computed once and shared by both searches.","title":"Suffix-array search for a pattern's row interval"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/sa-search-bounds","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.saSearch_bounds","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#sa-search-bounds","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"The search returns a genuine interval of rows: `sp \u2264 ep \u2264 |s|`.","title":"Suffix-array search returns a valid row interval"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/sa-search-spec","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArraySearch.saSearch_spec","parent":38093,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array/index.html#sa-search-spec","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-array","status":"fully_proved","summary":"A row `r < |s|` lies in `[sp, ep)` exactly when the suffix of row `r` starts with `P`.","title":"Suffix-array search is correct"},{"data_url":"scopes/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree.json","id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","kind":"scope","navigation":"module","parent":38055,"result":false,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","title":"Suffix tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/terminated","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.terminated","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#terminated","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For a list $s = (s_0, \\dots, s_{n-1})$ over $\\alpha$, the terminated string of $s$ is the list $(\\mathrm{some}\\,s_0, \\dots, \\mathrm{some}\\,s_{n-1}, \\mathrm{none})$ over $\\mathrm{Option}\\,\\alpha$: every symbol of $s$ tagged $\\mathrm{some}$, followed by the terminator $\\mathrm{none}$.","title":"Terminated string"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/terminated-drop-eq-suffix-map","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.terminated_drop_eq_suffix_map","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#terminated-drop-eq-suffix-map","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For a list $s$ over $\\alpha$ and a natural number $i \\le |s|$, removing the first $i$ entries of the terminated string of $s$ gives the suffix of $s$ starting at index $i$ with every symbol tagged $\\mathrm{some}$, followed by $\\mathrm{none}$.","title":"Suffixes of the terminated string"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/suffix-entries","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.suffixEntries","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#suffix-entries","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For a list $s = (s_0, \\dots, s_{n-1})$ over $\\alpha$, let $\\hat s = (\\mathrm{some}\\,s_0, \\dots, \\mathrm{some}\\,s_{n-1}, \\mathrm{none})$ be its terminated string. The suffix entries of $s$ are the list of pairs $(\\hat s_{\\ge i}, i)$ for $i = 0, 1, \\dots, n$ in increasing order, where $\\hat s_{\\ge i}$ is $\\hat s$ with its first $i$ entries removed: each nonempty suffix of $\\hat s$ paired with its starting index.","title":"Suffix entries of a string"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/nodup-terminated-drops","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.nodup_terminated_drops","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#nodup-terminated-drops","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"Let $s$ be a list over $\\alpha$. The lists obtained from the terminated string of $s$ by removing its first $i$ symbols, for $i$ ranging over $0, 1, \\ldots, |s|$, are pairwise distinct.","title":"Terminated suffixes are distinct"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/suffix-tree","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.suffixTree","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#suffix-tree","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For a list $s = (s_0, \\dots, s_{n-1})$ over $\\alpha$, let $\\hat s = (\\mathrm{some}\\,s_0, \\dots, \\mathrm{some}\\,s_{n-1}, \\mathrm{none})$ be its terminated string over $\\mathrm{Option}\\,\\alpha$. The suffix tree of $s$ is the radix trie with keys over $\\mathrm{Option}\\,\\alpha$ and natural-number values obtained from the empty trie by inserting, for $i = n, n - 1, \\dots, 0$ in this order, the suffix of $\\hat s$\u2026","title":"Suffix tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/lookup-suffix","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.lookup_suffix","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#lookup-suffix","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"Let $s$ be a list over $\\alpha$ and $i$ a natural number with $i \\le |s|$. Looking up, in the suffix tree of $s$, the terminated string of $s$ with its first $i$ symbols removed returns $i$.","title":"Lookup of a suffix returns its starting index"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/eq-drop-of-lookup-suffix-tree","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.eq_drop_of_lookup_suffixTree","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#eq-drop-of-lookup-suffix-tree","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"Let $s$ be a list over $\\alpha$, let $k$ be a list over $\\alpha$ extended by a fresh terminator symbol, and let $i$ be a natural number. If looking up $k$ in the suffix tree of $s$ returns $i$, then $i \\le |s|$ and $k$ is the terminated string of $s$ with its first $i$ symbols removed.","title":"Lookup in the suffix tree returns starting indices"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/mem-suffix-tree-keys","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.mem_suffixTree_keys","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#mem-suffix-tree-keys","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"Let $s$ be a list over $\\alpha$. A key $k$ is stored in the suffix tree of $s$ if and only if there is a natural number $i \\le |s|$ such that $k$ is the terminated string of $s$ with its first $i$ symbols removed.","title":"The stored keys of a suffix tree are its terminated suffixes"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/stored-keys","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.storedKeys","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#stored-keys","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For a list $s$ over $\\alpha$, the stored keys of $s$ form the list of keys stored in the suffix tree of $s$, in stored sibling order.","title":"Stored keys of the suffix tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/stored-keys-length","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.storedKeys_length","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#stored-keys-length","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For a list $s$ over $\\alpha$, the list of stored keys of the suffix tree of $s$ has length $|s| + 1$.","title":"The suffix tree stores one key per suffix"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/is-prefix-of-map-some-prefix-append-none","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.isPrefix_of_map_some_prefix_append_none","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#is-prefix-of-map-some-prefix-append-none","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For lists $p$ and $q$ over $\\alpha$, view their symbols in $\\alpha$ extended by a fresh terminator symbol. If $p$ is a prefix of $q$ followed by the terminator, then $p$ is a prefix of $q$.","title":"A tagged pattern cannot reach past the terminator"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/is-infix-iff-exists-is-prefix-drop","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.isInfix_iff_exists_isPrefix_drop","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#is-infix-iff-exists-is-prefix-drop","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For lists $s$ and $p$, $p$ is an infix (a contiguous sublist) of $s$ if and only if there is a natural number $i$ such that $p$ is a prefix of the list obtained from $s$ by removing its first $i$ symbols.","title":"Infixes are prefixes of suffixes"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/has-prefix-suffix-tree-iff-exists","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.hasPrefix_suffixTree_iff_exists","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#has-prefix-suffix-tree-iff-exists","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For lists $s$ and $p$ over $\\alpha$, prefix search for $p$ (each symbol viewed in the alphabet extended by the terminator) in the suffix tree of $s$ succeeds if and only if there is a natural number $i$ such that $p$ is a prefix of the list obtained from $s$ by removing its first $i$ symbols.","title":"Suffix-tree search finds exactly the prefixes of suffixes"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/has-prefix-suffix-tree-iff-is-infix","kind":"node","lean":"Cslib.Algorithms.Lean.SuffixTree.hasPrefix_suffixTree_iff_isInfix","parent":38116,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree/index.html#has-prefix-suffix-tree-iff-is-infix","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/suffix-tree","status":"fully_proved","summary":"For lists $s$ and $p$ over $\\alpha$, prefix search for $p$ (each symbol viewed in the alphabet extended by the terminator) in the suffix tree of $s$ succeeds if and only if $p$ occurs in $s$ as a contiguous sublist (an infix).","title":"Suffix-tree search finds exactly the infixes"},{"data_url":"scopes/ComputerScience/data-structures/tries-and-integer-dictionaries/trie.json","id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","kind":"scope","navigation":"module","parent":38055,"result":false,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","title":"Trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/trie","kind":"node","lean":"Cslib.Algorithms.Lean.Trie","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#trie","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"A trie with keys that are lists over $\\alpha$ and values in $\\beta$ is a node consisting of an optional value in $\\beta$ and a finite list of children, each child being a pair of a letter in $\\alpha$ and a trie. The letters labelling the children need not be distinct.","title":"Trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/empty","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.empty","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"The empty trie is the trie consisting of a single root node that stores no value and has no children, so it contains no key-value entries.","title":"Empty trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"The lookup of a key (a list over $\\alpha$) in a trie returns an optional value in $\\beta$, by recursion on the key. For the empty key it is the optional value stored at the root. For a key with first letter $a$ followed by the key $k$, it fails if no child of the root is labelled $a$, and otherwise it is the lookup of $k$ in the first child labelled $a$.","title":"Trie lookup"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/insert","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.insert","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Inserting a value $v$ at a key $k$ (a list of letters) in a trie stores $v$ at the node reached by following $k$ from the root, replacing any value stored there. Along the way, a missing child is created as an empty trie and appended after its siblings, an existing child is updated in place, and all other values and children are unchanged.","title":"Trie insertion"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/valid","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.Valid","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Validity of tries is defined inductively: a trie node with optional value $v$ and child list $c$ is valid when the letters labelling the children in $c$ are pairwise distinct and every child trie in $c$ is valid.","title":"Valid trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/entries","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.entries","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#entries","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"The entries of a trie are the list of its key-value pairs, where a key is a list of letters. If the root stores a value $v$, the list begins with the pair consisting of the empty key and $v$; it continues, for each child in the stored sibling order, with the entries $(k, w)$ of that child, in order, each rewritten as $(a k, w)$, where $a$ is the letter labelling the child and $a k$ is the key $k$ with $a$ prepended.","title":"Entries of a trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-empty","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup_empty","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"For every key $k$, looking up $k$ in the empty trie returns no value.","title":"Lookup in the empty trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-insert-same","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup_insert_same","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-insert-same","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Let $\\alpha$ be a type with decidable equality. For every trie $T$ with keys that are lists over $\\alpha$ and values in $\\beta$, every key $k$ and every value $v \\in \\beta$, looking up $k$ in the trie obtained from $T$ by inserting $v$ at $k$ returns $v$.","title":"Lookup after insertion in a trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-insert-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup_insert_of_ne","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-insert-of-ne","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"For every trie $T$, all keys $k \\ne k'$ and every value $v$, looking up $k$ in the trie obtained by inserting $v$ at $k'$ in $T$ gives the same result as looking up $k$ in $T$.","title":"Trie insertion leaves other keys unchanged"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/nodup-child-keys","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.Valid.nodup_child_keys","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#nodup-child-keys","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"If a trie node with optional value $v$ and child list $c$ is valid, then the letters labelling the children in $c$ are pairwise distinct.","title":"Valid trie nodes have distinct child keys"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/empty-2","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.Valid.empty","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#empty-2","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"The empty trie is valid.","title":"Validity of the empty trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/insert-2","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.Valid.insert","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#insert-2","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"If a trie $T$ is valid, then for every key $k$ (a list of letters) and every value $v$, the trie obtained by inserting $v$ at $k$ in $T$ is valid.","title":"Insertion preserves trie validity"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-eq-some-iff-mem-entries","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup_eq_some_iff_mem_entries","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-eq-some-iff-mem-entries","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"For every valid trie $T$, every key $k$ and every value $v$, looking up $k$ in $T$ returns $v$ if and only if $(k, v)$ is one of the entries of $T$.","title":"Trie lookup agrees with the entries"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-equiv","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.LookupEquiv","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-equiv","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Two tries $T$ and $T'$ are lookup-equivalent when, for every key $k$, looking up $k$ in $T$ gives the same result as looking up $k$ in $T'$.","title":"Lookup equivalence of tries"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/refl","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.LookupEquiv.refl","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#refl","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Every trie is lookup-equivalent to itself.","title":"Reflexivity of trie lookup equivalence"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/symm","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.LookupEquiv.symm","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#symm","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"If a trie $T$ is lookup-equivalent to a trie $T'$, then $T'$ is lookup-equivalent to $T$.","title":"Symmetry of trie lookup equivalence"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/trans","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.LookupEquiv.trans","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#trans","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"If a trie $T_1$ is lookup-equivalent to a trie $T_2$ and $T_2$ is lookup-equivalent to a trie $T_3$, then $T_1$ is lookup-equivalent to $T_3$.","title":"Transitivity of trie lookup equivalence"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/mem-entries-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.LookupEquiv.mem_entries_iff","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#mem-entries-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Let $T$ and $T'$ be valid tries with keys that are lists over $\\alpha$ and values in $\\beta$, and suppose $T$ and $T'$ are lookup-equivalent. Then for every key $k$ and every value $v \\in \\beta$, the pair $(k, v)$ occurs in the entry list of $T$ if and only if it occurs in the entry list of $T'$.","title":"Lookup-equivalent valid tries have the same entries"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/insert-comm","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.LookupEquiv.insert_comm","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#insert-comm","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"For every trie $T$, all keys $k_1 \\ne k_2$ and all values $v_1, v_2$, inserting $v_1$ at $k_1$ and then $v_2$ at $k_2$ gives a trie that is lookup-equivalent to the one obtained by inserting $v_2$ at $k_2$ and then $v_1$ at $k_1$.","title":"Insertions at distinct keys commute in a trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/is-empty-node","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.isEmptyNode","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#is-empty-node","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"The empty-node test is true on a trie node exactly when the node carries no value and has no children; when deletion leaves a child in this state, the child is removed from its parent.","title":"Empty trie nodes"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/is-empty-node-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.isEmptyNode_eq_true_iff","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#is-empty-node-eq-true-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"The empty-node test is true on a trie $t$ exactly when $t$ is the node with no value and no children, that is, the empty trie.","title":"Characterization of empty nodes"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/set-child","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.setChild","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#set-child","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Let $L$ be a list of pairs $(k, c)$ of a key $k$ in $\\alpha$ and a trie $c$ (the children of a trie node), let $a$ be a key and $t$ a trie, and call $t$ empty when it carries no value and has no children. Setting the child at $a$ to $t$ in $L$ gives the following list. If no entry of $L$ has key $a$, the result is $L$ itself when $t$ is empty, and $L$ with $(a, t)$ appended at the end otherwise. If some entry of\u2026","title":"Setting a child in a trie's child list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-set-child-self","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup_setChild_self","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-set-child-self","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Let $L$ be a list of letter\u2013trie pairs, $a$ a letter and $c$ a trie, and let $L'$ be obtained from $L$ by setting the child at $a$ to $c$ (removing the entries at $a$ instead when $c$ has neither value nor children). Looking up $a$ in $L'$ finds nothing when $c$ has neither value nor children, and finds $c$ otherwise.","title":"Looking up a child just set"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-set-child-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup_setChild_of_ne","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-set-child-of-ne","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Let $L$ be a list of letter\u2013trie pairs, $c$ a trie and $a \\ne b$ letters, and let $L'$ be obtained from $L$ by setting the child at $b$ to $c$ (removing the entries at $b$ instead when $c$ has neither value nor children). Then looking up $a$ in $L'$ gives the same result as looking up $a$ in $L$.","title":"Setting a child leaves the other children unchanged"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/set-child-valid","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.setChild_valid","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#set-child-valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Let $L$ be a list of letter\u2013trie pairs whose letters are pairwise distinct and whose tries are all valid, let $c$ be a valid trie and $a$ a letter. Then the list obtained from $L$ by setting the child at $a$ to $c$ (removing the entries at $a$ instead when $c$ has neither value nor children) again has pairwise distinct letters, and every trie in it is valid.","title":"Setting a child preserves the child-list part of validity"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/set-child-2","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.Valid.setChild","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#set-child-2","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Let a trie node with value $v$ and child list $L$ be valid, let $c$ be a valid trie and $a$ a letter. Then the node with value $v$ whose child list is obtained from $L$ by setting the child at $a$ to $c$ (removing the entries at $a$ instead when $c$ has neither value nor children) is valid.","title":"Setting a child preserves validity"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/delete","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.delete","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#delete","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"Deleting a key $k$ (a list of letters) from a trie $t$ works by recursion on $k$. If $k$ is empty, the root of $t$ loses its value and keeps its children. If $k$ consists of a first letter $a$ followed by a key $k'$ and the root of $t$ has no child at $a$, then $t$ is unchanged. Otherwise $k'$ is deleted from the first child at $a$. If the result has neither value nor children, every entry at $a$ is removed from\u2026","title":"Deletion from a trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-delete-self","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup_delete_self","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-delete-self","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"For every trie $t$ and every key $k$, looking up $k$ in the trie obtained from $t$ by deleting $k$ finds no value.","title":"A deleted key is absent"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/lookup-delete-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.lookup_delete_of_ne","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#lookup-delete-of-ne","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"For every trie $t$ and all keys $k \\ne k'$, looking up $k$ in the trie obtained from $t$ by deleting $k'$ gives the same result as looking up $k$ in $t$.","title":"Deletion does not affect other keys"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie/delete-2","kind":"node","lean":"Cslib.Algorithms.Lean.Trie.Valid.delete","parent":38131,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/trie/index.html#delete-2","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/trie","status":"fully_proved","summary":"If a trie $t$ is valid, that is, every node has distinct child keys and valid child tries, then for every key $k$ the trie obtained from $t$ by deleting $k$ is valid.","title":"Deletion preserves trie validity"},{"data_url":"scopes/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree.json","id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","kind":"scope","navigation":"module","parent":38055,"result":false,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","title":"Van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/van-emde-boas","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#van-emde-boas","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For a natural number $m$, a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$ consists of a summary bit vector of length $m$ and, for each $i < m$, a cluster bit vector of length $m$; bit $j$ of cluster $i$ records whether the element $im + j$ is stored. Unlike the fully recursive van Emde Boas tree over universes of size $2^{2^k}$, the summary and the clusters here are plain bit vectors, not\u2026","title":"Two-level van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For a natural number $m$, the empty van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ is the tree in which all $m$ summary bits and all $m^2$ cluster bits ($m$ clusters of $m$ bits each) are unset.","title":"Empty van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/member","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.member","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#member","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"An element $x$ of the universe $\\{0, 1, \\dots, m^2 - 1\\}$ is a member of a two-level van Emde Boas tree $t$ when bit $x \\bmod m$ of cluster $\\lfloor x / m \\rfloor$ of $t$ is set. The summary bits are not consulted.","title":"Membership in a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/insert","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.insert","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $m$ be a natural number, let $t$ be a van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and let $x$ be an element of the universe. Inserting $x$ into $t$ gives the tree obtained from $t$ by setting bit $x \\bmod m$ of cluster $\\lfloor x / m \\rfloor$ and summary bit $\\lfloor x / m \\rfloor$, leaving every other bit unchanged.","title":"Insertion into a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.Valid","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"A two-level van Emde Boas tree $t$ with $m$ clusters is valid when, for every cluster index $i < m$, summary bit $i$ of $t$ is set exactly when cluster $i$ of $t$ is nonempty, that is, when some bit $j < m$ of cluster $i$ is set.","title":"Summary invariant of a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/contents","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.contents","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#contents","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $m$ be a natural number. For a van Emde Boas tree $t$ over the universe $\\{0, 1, \\dots, m^2 - 1\\}$, the set of stored elements of $t$ is the finite set of those elements of the universe that are members of $t$.","title":"Set of stored elements of a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/member-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.member_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#member-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every $m$ and every $x$ in the universe $\\{0, 1, \\dots, m^2 - 1\\}$, $x$ is not a member of the empty two-level van Emde Boas tree over that universe.","title":"The empty van Emde Boas tree has no members"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/member-insert-self","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.member_insert_self","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#member-insert-self","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every two-level van Emde Boas tree $t$ over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and every $x$ in that universe, $x$ is a member of the tree obtained by inserting $x$ into $t$.","title":"An inserted element is a member of a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/member-insert-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.member_insert_of_ne","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#member-insert-of-ne","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every two-level van Emde Boas tree $t$ over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and all $x \\ne y$ in that universe, $y$ is a member of the tree obtained by inserting $x$ into $t$ if and only if $y$ is a member of $t$.","title":"Insertion into a van Emde Boas tree leaves other elements unchanged"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/member-insert","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.member_insert","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#member-insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every two-level van Emde Boas tree $t$ over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and all $x, y$ in that universe, $y$ is a member of the tree obtained by inserting $x$ into $t$ if and only if $y = x$ or $y$ is a member of $t$.","title":"Membership after insertion into a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/member-insert-of-member","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.member_insert_of_member","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#member-insert-of-member","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every two-level van Emde Boas tree $t$ over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and all $x, y$ in that universe, if $y$ is a member of $t$, then $y$ is a member of the tree obtained by inserting $x$ into $t$.","title":"Insertion into a van Emde Boas tree keeps existing members"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.valid_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#valid-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every natural number $m$, the empty two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$ satisfies the summary invariant: each summary bit is set exactly when its cluster is nonempty.","title":"The empty van Emde Boas tree is valid"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/valid-insert","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.valid_insert","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#valid-insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every natural number $m$, every element $x$ of the universe $\\{0, \\dots, m^2 - 1\\}$ and every two-level van Emde Boas tree $t$ over that universe that satisfies the summary invariant (each summary bit is set exactly when its cluster is nonempty), the tree obtained from $t$ by inserting $x$ also satisfies the summary invariant.","title":"Insertion preserves validity of van Emde Boas trees"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/mem-contents","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.mem_contents","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#mem-contents","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every two-level van Emde Boas tree $t$ over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and every $x$ in that universe, $x$ belongs to the finite set of stored elements of $t$ if and only if $x$ is a member of $t$.","title":"The contents of a van Emde Boas tree are its members"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/contents-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.contents_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#contents-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every natural number $m$, the set of stored elements of the empty van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ is empty.","title":"Contents of the empty van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/contents-insert","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.contents_insert","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#contents-insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $m$ be a natural number, let $t$ be a van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and let $x$ be an element of the universe. The set of stored elements of the tree obtained by inserting $x$ into $t$ is the set of stored elements of $t$ together with $x$. No validity assumption on $t$ is needed.","title":"Contents after insertion into a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/min","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.min?","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#min","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $t$ be a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$. If some summary bit of $t$ is set, let $h$ be the least cluster index whose summary bit is set; if cluster $h$ has a set position, let $\\ell$ be the least one. The minimum query on $t$ then returns the element $h \\cdot m + \\ell$. If no summary bit is set, or cluster $h$ has no set position, it returns no element.","title":"Minimum query of a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/min-mem","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.min?_mem","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#min-mem","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $t$ be a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$. If the minimum query on $t$ returns $x$, then $x$ is stored in $t$.","title":"The van Emde Boas minimum is stored"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/min-least","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.min?_least","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#min-least","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $t$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$. If the minimum query on $t$ returns $x$, then $x \\le y$ for every $y$ stored in $t$.","title":"The van Emde Boas minimum is a lower bound"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/min-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.min?_eq_none_iff","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#min-eq-none-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $t$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$. The minimum query on $t$ returns no element if and only if no element of the universe is stored in $t$.","title":"Empty minimum query in a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/min-eq-some-iff-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.min?_eq_some_iff_of_valid","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#min-eq-some-iff-of-valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $t$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$, and let $x$ be in the universe. The minimum query on $t$ returns $x$ if and only if $x$ is stored in $t$ and $x \\le y$ for every $y$ stored in $t$.","title":"Characterization of the van Emde Boas minimum"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/min-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.min?_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#min-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every $m$, the minimum query of the empty two-level van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ returns no element.","title":"The empty van Emde Boas tree has no minimum"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/min-insert-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.min?_insert_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#min-insert-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every $x$ in the universe $\\{0, \\dots, m^2 - 1\\}$, the minimum query on the two-level van Emde Boas tree obtained by inserting $x$ into the empty tree returns $x$.","title":"Minimum of a one-element van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/max","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.max?","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#max","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For a two-level van Emde Boas tree $t$ over the universe $\\{0, 1, \\dots, m^2 - 1\\}$, the maximum query first finds the greatest index $i < m$ whose summary bit is set, and then the greatest index $j < m$ whose bit is set in cluster $i$; it returns the element $i m + j$. It returns no element if no summary bit is set, or if no bit of cluster $i$ is set. When the summary bits record exactly the nonempty clusters,\u2026","title":"Maximum query of a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/max-mem","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.max?_mem","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#max-mem","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every two-level van Emde Boas tree $t$ over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and every $x$ in that universe, if the maximum query of $t$ returns $x$, then $x$ is a member of $t$. No validity hypothesis on $t$ is needed.","title":"The maximum query of a van Emde Boas tree returns a member"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/max-greatest","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.max?_greatest","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#max-greatest","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $m$ be a natural number and let $t$ be a valid van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$. If the maximum query on $t$ returns $x$, then $y \\le x$ for every member $y$ of $t$.","title":"The van Emde Boas maximum query bounds every member"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/max-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.max?_eq_none_iff","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#max-eq-none-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $m$ be a natural number and let $t$ be a valid van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$. The maximum query on $t$ returns no element if and only if no element of the universe is a member of $t$.","title":"When the van Emde Boas maximum query returns nothing"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/max-eq-some-iff-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.max?_eq_some_iff_of_valid","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#max-eq-some-iff-of-valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $m$ be a natural number, let $t$ be a valid van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ and let $x$ be an element of the universe. The maximum query on $t$ returns $x$ if and only if $x$ is a member of $t$ and $y \\le x$ for every member $y$ of $t$.","title":"Correctness of the van Emde Boas maximum query"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/max-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.max?_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#max-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every natural number $m$, the maximum query on the empty van Emde Boas tree over the universe $\\{0, 1, \\dots, m^2 - 1\\}$ returns no element.","title":"Maximum query on the empty van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/max-insert-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.max?_insert_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#max-insert-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $m$ be a natural number and let $x$ be an element of the universe $\\{0, 1, \\dots, m^2 - 1\\}$. The maximum query on the tree obtained by inserting $x$ into the empty van Emde Boas tree over this universe returns $x$.","title":"Maximum query on a one-element van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/successor","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.successor","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#successor","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$ and let $x = hm + \\ell$ with $0 \\le h, \\ell < m$, so that $x$ sits at position $\\ell$ of cluster $h$. If cluster $h$ has a set position after $\\ell$, the successor of $x$ in $T$ is $hm + \\ell'$ for the least such position $\\ell'$. Otherwise, let $h'$ be the least cluster index with $h < h'$ whose summary bit is set; the successor is\u2026","title":"Successor query in a two-level van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/successor-mem","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.successor_mem","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#successor-mem","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$, not necessarily valid, and let $x$ and $y$ be elements of the universe. If the successor of $x$ in $T$ is $y$, then $y$ is a member of $T$.","title":"The van Emde Boas successor is stored"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/successor-gt","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.successor_gt","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#successor-gt","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$, not necessarily valid, and let $x$ and $y$ be elements of the universe. If the successor of $x$ in $T$ is $y$, then $x < y$.","title":"The van Emde Boas successor is strictly larger than its argument"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/successor-least","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.successor_least","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#successor-least","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$ and let $x$, $y$, $z$ be elements of the universe. If the successor of $x$ in $T$ is $y$ and $z$ is a member of $T$ with $x < z$, then $y \\le z$.","title":"No member lies strictly between an element and its van Emde Boas successor"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/successor-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.successor_eq_none_iff","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#successor-eq-none-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$ and let $x$ be an element of the universe. Then $x$ has no successor in $T$ if and only if every member $z$ of $T$ satisfies $z \\le x$.","title":"A van Emde Boas successor is missing exactly when nothing larger is stored"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/successor-eq-some-iff-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.successor_eq_some_iff_of_valid","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#successor-eq-some-iff-of-valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$ and let $x$ and $y$ be elements of the universe. Then the successor of $x$ in $T$ is $y$ if and only if $y$ is a member of $T$, $x < y$, and $y \\le z$ for every member $z$ of $T$ with $x < z$.","title":"The van Emde Boas successor is the least larger member"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/successor-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.successor_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#successor-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every $m$ and every element $x$ of the universe $\\{0, \\dots, m^2 - 1\\}$, $x$ has no successor in the empty two-level van Emde Boas tree over that universe.","title":"The empty van Emde Boas tree has no successors"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/predecessor","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.predecessor","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#predecessor","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$ and let $x = hm + \\ell$ with $0 \\le h, \\ell < m$, so that $x$ sits at position $\\ell$ of cluster $h$. If cluster $h$ has a set position before $\\ell$, the predecessor of $x$ in $T$ is $hm + \\ell'$ for the greatest such position $\\ell'$. Otherwise, let $h'$ be the greatest cluster index with $h' < h$ whose summary bit is set; the\u2026","title":"Predecessor query in a two-level van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/predecessor-mem","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.predecessor_mem","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#predecessor-mem","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$, not necessarily valid, and let $x$ and $y$ be elements of the universe. If the predecessor of $x$ in $T$ is $y$, then $y$ is a member of $T$.","title":"The van Emde Boas predecessor is stored"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/predecessor-lt","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.predecessor_lt","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#predecessor-lt","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $T$ be a two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$, not necessarily valid, and let $x$ and $y$ be elements of the universe. If the predecessor of $x$ in $T$ is $y$, then $y < x$.","title":"The van Emde Boas predecessor is strictly smaller than its argument"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/predecessor-greatest","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.predecessor_greatest","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#predecessor-greatest","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $t$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$, and let $x$, $y$ and $z$ be in the universe. If the predecessor query for $x$ in $t$ returns $y$, and $z$ is stored in $t$ with $z < x$, then $z \\le y$.","title":"Maximality of the van Emde Boas predecessor"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/predecessor-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.predecessor_eq_none_iff","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#predecessor-eq-none-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $t$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$, and let $x$ be in the universe. The predecessor query for $x$ in $t$ returns no element if and only if $x \\le z$ for every $z$ stored in $t$.","title":"Empty predecessor query in a van Emde Boas tree"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/predecessor-eq-some-iff-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.predecessor_eq_some_iff_of_valid","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#predecessor-eq-some-iff-of-valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"Let $t$ be a valid two-level van Emde Boas tree over the universe $\\{0, \\dots, m^2 - 1\\}$, and let $x$ and $y$ be in the universe. The predecessor query for $x$ in $t$ returns $y$ if and only if $y$ is stored in $t$, $y < x$, and $z \\le y$ for every $z$ stored in $t$ with $z < x$.","title":"Characterization of the van Emde Boas predecessor"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/predecessor-empty","kind":"node","lean":"Cslib.Algorithms.Lean.VanEmdeBoas.predecessor_empty","parent":38162,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree/index.html#predecessor-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/van-emde-boas-tree","status":"fully_proved","summary":"For every $x$ in the universe $\\{0, \\dots, m^2 - 1\\}$, the predecessor query for $x$ in the empty two-level van Emde Boas tree returns no element.","title":"Predecessor queries in the empty van Emde Boas tree"},{"data_url":"scopes/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie.json","id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","kind":"scope","navigation":"module","parent":38055,"result":false,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","title":"X-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/xfast-trie","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#xfast-trie","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"An x-fast trie over $w$-bit keys consists of a finite set of stored keys in $\\{0, \\dots, 2^w - 1\\}$ together with, for each level $l \\in \\{0, \\dots, w\\}$, a finite set of natural numbers, intended to hold the $l$-bit prefixes of the stored keys.","title":"x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/prefix-of","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.prefixOf","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#prefix-of","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For natural numbers $l$ and $w$ and a $w$-bit key $x < 2^w$, the length-$l$ prefix of $x$ is the natural number $\\lfloor x / 2^{w-l} \\rfloor$, where $w - l$ is truncated subtraction (it is $0$ when $l > w$). For $l \\le w$ these are the top $l$ bits of $x$, a number below $2^l$.","title":"Bit prefix of a key"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/prefix-of-zero","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.prefixOf_zero","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#prefix-of-zero","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For every $w$-bit key $x < 2^w$, the length-$0$ prefix of $x$ (the trie root) is $0$.","title":"The empty prefix is zero"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/prefix-of-full","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.prefixOf_full","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#prefix-of-full","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For every $w$-bit key $x < 2^w$, the length-$w$ prefix of $x$ equals $x$.","title":"The full-length prefix of a key is the key"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/prefix-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.prefixOf_lt","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#prefix-of-lt","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For natural numbers $l \\le w$ and a $w$-bit key $x < 2^w$, the length-$l$ prefix of $x$ is less than $2^l$.","title":"A prefix fits in its length"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/valid","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.Valid","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"An x-fast trie $t$ over $w$-bit keys is valid when, for every level $l \\in \\{0, \\dots, w\\}$, the level-$l$ set of $t$ equals the set of $l$-bit prefixes (top $l$ bits) of the stored keys of $t$.","title":"Valid x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/empty","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.empty","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"The empty x-fast trie over $w$-bit keys stores no keys, and the prefix set at each of its levels $0, 1, \\dots, w$ is empty.","title":"The empty x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/insert","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.insert","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"Inserting a $w$-bit key $x$ into an x-fast trie $t$ adds $x$ to the keys stored in $t$ and, for each level $0 \\le l \\le w$, adds the $l$-bit prefix of $x$ to the prefix set at level $l$.","title":"Insertion into an x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.valid_empty","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#valid-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For every $w$, the empty x-fast trie over $w$-bit keys, which has no stored keys and every level empty, is valid.","title":"The empty x-fast trie is valid"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/valid-insert","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.valid_insert","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#valid-insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"If $t$ is a valid x-fast trie over $w$-bit keys and $x \\in \\{0, \\dots, 2^w - 1\\}$, then the x-fast trie obtained from $t$ by inserting $x$ is valid.","title":"Insertion preserves validity of an x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/contents-insert","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.contents_insert","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#contents-insert","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For every x-fast trie $t$ over $w$-bit keys and every $w$-bit key $x$, the set of keys stored in the trie obtained by inserting $x$ into $t$ is the set of keys stored in $t$ with $x$ added.","title":"Keys of an x-fast trie after insertion"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/has-prefix","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.hasPrefix","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#has-prefix","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For an x-fast trie $t$ over $w$-bit keys, a level $0 \\le l \\le w$ and a $w$-bit key $x$, the probe of level $l$ of $t$ at $x$ returns true exactly when the $l$-bit prefix of $x$ belongs to the prefix set stored at level $l$ of $t$.","title":"Level probe of an x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/has-prefix-iff","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.hasPrefix_iff","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#has-prefix-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"Let $t$ be a valid x-fast trie over $w$-bit keys, let $0 \\le l \\le w$ and let $x$ be a $w$-bit key. The probe of level $l$ of $t$ at $x$ succeeds if and only if some key $y$ stored in $t$ has the same $l$-bit prefix as $x$.","title":"Level probes of a valid x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/member","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.member","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#member","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For an x-fast trie $t$ over $w$-bit keys and a key $x < 2^w$, the membership query of $x$ in $t$ answers true exactly when the length-$w$ prefix of $x$, which is $x$ itself, lies in the prefix set that $t$ stores for level $w$: a single probe of the deepest level, as in the classical structure.","title":"Membership query in an x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/member-iff","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.member_iff","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#member-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For a valid x-fast trie $t$ over $w$-bit keys and a $w$-bit key $x$, the membership test of $t$ at $x$, which probes the deepest level $w$, returns true if and only if $x$ is stored in $t$.","title":"The membership test of a valid x-fast trie is exact"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/member-insert-self","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.member_insert_self","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#member-insert-self","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For every x-fast trie $t$ over $w$-bit keys and every $w$-bit key $x$, the membership test at $x$ of the trie obtained by inserting $x$ into $t$ returns true.","title":"An inserted key passes the membership test"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/successor","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.successor","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#successor","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For an x-fast trie $t$ over $w$-bit keys and a key $x \\in \\{0, \\dots, 2^w - 1\\}$, the successor of $x$ in $t$ is the least stored key of $t$ that is greater than $x$ when some stored key is greater than $x$; otherwise $x$ has no successor in $t$.","title":"Successor in an x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/predecessor","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.predecessor","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#predecessor","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For an x-fast trie $t$ over $w$-bit keys and a key $x < 2^w$, the predecessor of $x$ in $t$ is the greatest key stored in $t$ that is smaller than $x$, if such a key exists; otherwise the query returns no key.","title":"Predecessor query in an x-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/successor-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.successor_eq_some_iff","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#successor-eq-some-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"Let $t$ be an x-fast trie over $w$-bit keys and let $x$ and $y$ be keys in $\\{0, \\dots, 2^w - 1\\}$. The successor of $x$ in $t$ exists and equals $y$ if and only if $y$ is a stored key of $t$, $x < y$, and $y \\le z$ for every stored key $z$ of $t$ with $x < z$.","title":"Characterization of the x-fast trie successor"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/successor-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.successor_eq_none_iff","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#successor-eq-none-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For an x-fast trie $t$ over $w$-bit keys and a key $x < 2^w$, the successor query of $x$ in $t$ returns no key if and only if no key stored in $t$ is greater than $x$.","title":"When an x-fast trie has no successor"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/predecessor-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.predecessor_eq_some_iff","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#predecessor-eq-some-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For an x-fast trie $t$ over $w$-bit keys and keys $x, y < 2^w$, the predecessor query of $x$ in $t$ returns $y$ if and only if $y$ is stored in $t$, $y < x$, and every key $z$ stored in $t$ with $z < x$ satisfies $z \\le y$.","title":"Characterization of the x-fast trie predecessor"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/predecessor-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.XFastTrie.predecessor_eq_none_iff","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#predecessor-eq-none-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For an x-fast trie $t$ over $w$-bit keys and a key $x < 2^w$, the predecessor query of $x$ in $t$ returns no key if and only if no key stored in $t$ is smaller than $x$.","title":"When an x-fast trie has no predecessor"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/yfast-trie","kind":"node","lean":"Cslib.Algorithms.Lean.YFastTrie","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#yfast-trie","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"A y-fast trie skeleton over $w$-bit keys consists of an x-fast trie over $w$-bit keys, whose stored keys are the representatives, together with a bucket $B_r$, a finite set of keys, for every key $r \\in \\{0, \\dots, 2^w - 1\\}$.","title":"y-fast trie skeleton"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/contents","kind":"node","lean":"Cslib.Algorithms.Lean.YFastTrie.contents","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#contents","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"Let $t$ be a y-fast trie over $w$-bit keys, let $R$ be the set of keys stored in the representative x-fast trie of $t$, and for each $r \\in R$ let $B(r)$ be the bucket owned by $r$. The set of keys stored in $t$ is the union of the buckets of its representatives, $\\bigcup_{r \\in R} B(r)$.","title":"Keys of a y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/mem-contents-iff","kind":"node","lean":"Cslib.Algorithms.Lean.YFastTrie.mem_contents_iff","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#mem-contents-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"For every y-fast trie $t$ over $w$-bit keys and every $w$-bit key $x$, $x$ is stored in $t$ if and only if $x$ lies in the bucket owned by some representative of $t$, that is, by some key stored in the representative x-fast trie of $t$.","title":"Membership in a y-fast trie via representatives' buckets"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/valid-2","kind":"node","lean":"Cslib.Algorithms.Lean.YFastTrie.Valid","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#valid-2","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"A y-fast trie skeleton $t$ over $w$-bit keys, with bucket $B_r$ owned by each key $r$, is valid when its representative x-fast trie is valid, a key $r$ is stored in the representative x-fast trie exactly when its bucket $B_r$ is nonempty, and every key $r$ stored in the representative x-fast trie lies in its own bucket $B_r$.","title":"Valid y-fast trie skeleton"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/rep-mem-contents-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.YFastTrie.rep_mem_contents_of_valid","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#rep-mem-contents-of-valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"If a y-fast trie $t$ over $w$-bit keys is valid, then every key of the representative x-fast trie of $t$ is a stored key of $t$.","title":"Representatives of a valid y-fast trie are stored keys"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/mem-contents-iff-exists-of-valid","kind":"node","lean":"Cslib.Algorithms.Lean.YFastTrie.mem_contents_iff_exists_of_valid","parent":38207,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie/index.html#mem-contents-iff-exists-of-valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/x-fast-trie","status":"fully_proved","summary":"If $t$ is a valid y-fast trie over $w$-bit keys, then a $w$-bit key $x$ is stored in $t$ if and only if $x$ lies in the bucket owned by some $w$-bit key $r$ (such an owner $r$ is then necessarily a representative).","title":"Keys of a valid y-fast trie are the keys in buckets"},{"data_url":"scopes/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie.json","id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","kind":"scope","navigation":"module","parent":38055,"result":false,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","title":"Y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/grouped-yfast-trie","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#grouped-yfast-trie","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For natural numbers $w$ and $B$, a grouped y-fast trie over $w$-bit keys with group bound $B$ consists of an x-fast trie over $w$-bit keys, which holds the representatives, and a list of groups, each a list of $w$-bit keys. The structure itself imposes no condition on these fields; the intended invariants (the representative trie is a valid x-fast trie whose keys are exactly the group maxima, every group is\u2026","title":"Grouped y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/contents","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.contents","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#contents","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"The key set of a grouped y-fast trie $t$ is the finite set of all keys that occur in some group of $t$.","title":"Keys of a grouped y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/mem-contents-iff","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.mem_contents_iff","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#mem-contents-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every grouped y-fast trie $t$ over $w$-bit keys and every key $x$, $x$ lies in the contents of $t$ if and only if $x$ belongs to some group of $t$.","title":"Stored keys are the keys of the groups"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/group-ordered","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.GroupOrdered","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#group-ordered","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"A list of groups, each a list of $w$-bit keys, is group ordered when every key of an earlier group is smaller than every key of a later group.","title":"Ordered list of groups"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/valid","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.Valid","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#valid","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"A grouped y-fast trie $t$ over keys in $\\{0, \\dots, 2^w - 1\\}$ with group bound $B$ is valid when its representative trie is a valid x-fast trie; a key $r$ is stored in the representative trie exactly when $r$ is the last element of some group of $t$; every group of $t$ is nonempty, has at most $B$ keys, and is strictly increasing; and the groups of $t$ are ordered, that is, every key of an earlier group is smaller\u2026","title":"Validity of a grouped y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/find-group","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.findGroup","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#find-group","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For a grouped y-fast trie $t$ and a key $r$, the group of $r$ is the first group of $t$, in the order of the groups, whose last element is $r$, or the empty list if no group of $t$ ends with $r$.","title":"Group of a representative"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/rep-ge","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.repGe","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#rep-ge","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For a grouped y-fast trie $t$ and a key $x$, the result is $x$ itself if the membership test of the representative x-fast trie of $t$ (a single probe of its deepest level) reports $x$ as present. Otherwise it is the successor of $x$ in the representative x-fast trie, that is, the least representative greater than $x$, or nothing if there is none.","title":"Least representative at or above a key"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/member","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.member","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#member","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Given a grouped y-fast trie $t$ over $w$-bit keys and a key $x$, the query first finds a representative $r$: $r = x$ if the membership probe of the representative x-fast trie accepts $x$, and otherwise $r$ is the successor of $x$ in the representative trie. If there is no such $r$, the query returns false. Otherwise it returns whether $x$ belongs to the group owned by $r$, namely the first group of $t$ whose last\u2026","title":"Membership query of a grouped y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/member-iff","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.member_iff","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#member-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $t$ be a valid grouped y-fast trie over $w$-bit keys. Then for every key $x$, the membership query on $t$ (locate the least representative at least $x$, then scan its group) returns true at $x$ if and only if $x$ lies in the contents of $t$.","title":"Correctness of the y-fast trie membership query"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/successor","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.successor","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#successor","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For a grouped y-fast trie $t$ and a key $x$, let $r$ be the least key greater than $x$ stored in the representative x-fast trie of $t$; if there is none, the query returns nothing. Otherwise let $g$ be the first group of $t$ whose last element is $r$, or the empty list if no group has last element $r$. The query then returns the first key of $g$ greater than $x$, or nothing if $g$ has no such key.","title":"Successor query in a grouped y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/successor-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.successor_eq_some_iff","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#successor-eq-some-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $t$ be a valid grouped y-fast trie and let $x$, $y$ be keys. The successor query for $x$ in $t$ returns $y$ if and only if $y$ is stored in $t$, $x < y$, and $y \\le z$ for every key $z$ stored in $t$ with $x < z$.","title":"Successor query returns the least larger stored key"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/successor-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.successor_eq_none_iff","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#successor-eq-none-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $t$ be a valid grouped y-fast trie and $x$ a key. The successor query for $x$ in $t$ returns nothing if and only if no key stored in $t$ is greater than $x$.","title":"Successor query fails exactly when no larger key is stored"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/empty","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.empty","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every key length $w$ and group bound $B$, the empty grouped y-fast trie has the empty x-fast trie as its trie of representatives and has no groups.","title":"The empty grouped y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/valid-empty","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.valid_empty","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#valid-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every key width $w$ and group bound $B$, the empty grouped y-fast trie satisfies the grouped y-fast trie invariant.","title":"The empty grouped y-fast trie is valid"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/member-empty","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.member_empty","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#member-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For all natural numbers $w$ and $B$ and every $w$-bit key $x$, the membership query on the empty grouped y-fast trie over $w$-bit keys with group bound $B$ returns false at $x$.","title":"The empty grouped y-fast trie has no members"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/successor-empty","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.successor_empty","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#successor-empty","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every key $x$, the successor query for $x$ in the empty grouped y-fast trie returns nothing.","title":"The empty grouped y-fast trie has no successors"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/mk-reps","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.mkReps","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#mk-reps","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For a list $rs$ of $w$-bit keys, the representative x-fast trie of $rs$ is obtained from the empty x-fast trie over $w$-bit keys by inserting the elements of $rs$ one at a time, from first to last.","title":"Representative trie of a list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/valid-mk-reps","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.valid_mkReps","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#valid-mk-reps","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every list of $w$-bit keys, the x-fast trie obtained from the empty x-fast trie by inserting the keys of the list one at a time satisfies the x-fast trie invariant: each level holds exactly the prefixes of that length of the stored keys.","title":"Inserting a list of keys into the empty x-fast trie gives a valid trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/contents-mk-reps","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.contents_mkReps","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#contents-mk-reps","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every list $L$ of $w$-bit keys, the x-fast trie obtained by inserting the elements of $L$ one by one into the empty x-fast trie stores exactly the elements of $L$: its key set is the set of elements of $L$.","title":"Keys of the trie of representatives"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/of-groups","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.ofGroups","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#of-groups","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For a list of groups $g_1, \\dots, g_k$, each a list of $w$-bit keys, and any group-size bound $B$, this is the grouped y-fast trie whose groups are $g_1, \\dots, g_k$ in this order and whose representative x-fast trie is obtained from the empty x-fast trie by inserting the last element of each nonempty group, in order. No condition on the groups is checked here; when a group is sorted, its last element is its maximum.","title":"Grouped y-fast trie from a list of groups"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/valid-of-groups","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.valid_ofGroups","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#valid-of-groups","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $w$ and $B$ be natural numbers and let $g_1, \\dots, g_k$ be a list of groups, each a list of keys in $\\{0, \\dots, 2^w - 1\\}$. Suppose every group is nonempty, has at most $B$ keys, and is strictly increasing, and the groups are ordered: every key of an earlier group is smaller than every key of a later group. Then the grouped y-fast trie with group bound $B$ assembled from these groups, whose representatives\u2026","title":"Assembling a valid grouped y-fast trie from groups"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/chunk-groups-aux","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.chunkGroupsAux","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#chunk-groups-aux","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Given a bound $B$, a fuel $n$ and a list $l$ of $w$-bit keys, the fuel-bounded chunking is the empty list of chunks when $n = 0$ or $l$ is empty; otherwise it is the chunk formed by the first $\\min(B, |l|)$ elements of $l$, followed by the fuel-bounded chunking with bound $B$ and fuel $n - 1$ of the remaining elements of $l$.","title":"Fuel-bounded chunking"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/chunk-groups","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.chunkGroups","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#chunk-groups","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Given a bound $B$ and a list $l$ of $w$-bit keys, when $B > 0$ the chunking of $l$ cuts $l$ from the left into consecutive chunks of exactly $B$ elements each, except that the last chunk holds the remaining one to $B$ elements; there are no chunks when $l$ is empty. When $B = 0$, the chunking has no chunks if $l$ is empty and is the single chunk $l$ otherwise.","title":"Chunking a list into groups"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/chunk-groups-zero-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.chunkGroups_zero_of_ne","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#chunk-groups-zero-of-ne","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every nonempty list $l$ of $w$-bit keys, the chunking of $l$ with bound $B = 0$ consists of the single chunk $l$.","title":"Chunking with bound zero"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/flatten-chunk-groups","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.flatten_chunkGroups","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#flatten-chunk-groups","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every bound $B$ (including $B = 0$) and every list $l$ of $w$-bit keys, concatenating the chunks of the chunking of $l$ with bound $B$, in order, gives back $l$.","title":"Chunking preserves the list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/chunk-groups-spec","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.chunkGroups_spec","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#chunk-groups-spec","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $B > 0$ and let $l$ be a strictly increasing list of $w$-bit keys. Then every chunk of the chunking of $l$ into groups of at most $B$ keys is nonempty, has at most $B$ elements, and is strictly increasing.","title":"Chunks of a sorted list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/group-ordered-chunk-groups","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.groupOrdered_chunkGroups","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#group-ordered-chunk-groups","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $B$ be a natural number and let $l$ be a strictly increasing list of $w$-bit keys. Then the chunk groups of $l$ with bound $B$ are group ordered: every key of an earlier chunk is smaller than every key of a later chunk.","title":"Chunks of a sorted list are ordered"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/of-sorted-list","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.ofSortedList","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#of-sorted-list","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For a group bound $B$ and a list $l$ of $w$-bit keys, this is the grouped y-fast trie whose groups, in order, are consecutive pieces of $l$ that together make up $l$, and whose representative x-fast trie is obtained from the empty x-fast trie by inserting the last element of each group, in order. If $B > 0$, the groups are obtained by repeatedly cutting off the first $B$ keys, so every group has exactly $B$ keys\u2026","title":"Grouped y-fast trie from a sorted list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/valid-of-sorted-list","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.valid_ofSortedList","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#valid-of-sorted-list","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $w$ and $B$ be natural numbers with $0 < B$, and let $l$ be a strictly increasing list of keys in $\\{0, \\dots, 2^w - 1\\}$. Then the grouped y-fast trie with group bound $B$ built from $l$ by cutting it into consecutive groups of at most $B$ keys is valid.","title":"Building from a sorted list gives a valid grouped y-fast trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/not-valid-of-sorted-list-zero","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.not_valid_ofSortedList_zero","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#not-valid-of-sorted-list-zero","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every natural number $w$ and every nonempty list $l$ of $w$-bit keys, the grouped y-fast trie with group bound $B = 0$ built from $l$ is not valid.","title":"Building from a nonempty list with group bound zero gives an invalid trie"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/contents-of-sorted-list","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.contents_ofSortedList","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#contents-of-sorted-list","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"For every group bound $B$ (including $B = 0$) and every list $l$ of $w$-bit keys, sorted or not, the key set of the grouped y-fast trie built from $l$ with group bound $B$ is exactly the set of elements of $l$.","title":"Keys of the grouped y-fast trie built from a list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/member-of-sorted-list-iff","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.member_ofSortedList_iff","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#member-of-sorted-list-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $B > 0$ and let $l$ be a strictly increasing list of $w$-bit keys. Then for every key $x$, the membership query on the grouped y-fast trie built from $l$ with group bound $B$ returns true at $x$ if and only if $x$ belongs to $l$.","title":"Membership in a trie built from a sorted list"},{"id":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/successor-of-sorted-list-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.GroupedYFastTrie.successor_ofSortedList_eq_some_iff","parent":38236,"result":true,"route":"../roadmap/ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie/index.html#successor-of-sorted-list-eq-some-iff","scope":"ComputerScience/data-structures/tries-and-integer-dictionaries/y-fast-trie","status":"fully_proved","summary":"Let $B > 0$, let $l$ be a strictly increasing list of $w$-bit keys, and let $x$, $y$ be keys. The successor query for $x$ in the grouped y-fast trie built from $l$ with group bound $B$ returns $y$ if and only if $y$ belongs to $l$, $x < y$, and $y \\le z$ for every $z$ in $l$ with $x < z$.","title":"Successor in the trie built from a sorted list"},{"data_url":"scopes/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees.json","id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees","kind":"scope","parent":36418,"result":false,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/index.html","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees","status":"fully_proved","summary":"Union-find with rank and path compression (including its connected-components application), link-cut trees and Euler-tour trees.","title":"Disjoint sets and dynamic trees"},{"data_url":"scopes/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure.json","id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","kind":"scope","navigation":"module","parent":38270,"result":false,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","title":"Disjoint-set data structure"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/forest","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.Forest","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#forest","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"A parent forest on the vertices $\\{0, \\dots, n-1\\}$ consists of a parent map $p$ from the vertices to themselves (a vertex $x$ with $p(x) = x$ is a root) and a depth certificate $d$ assigning a natural number to every vertex, such that every non-root edge strictly decreases the certificate: if $p(x) \\ne x$, then $d(p(x)) < d(x)$.","title":"Parent forest with a depth certificate"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.find","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a parent forest $F$ with parent map $p$ and a vertex $x$, the root of $x$ is found by following parent pointers from $x$: it is $x$ itself if $p(x) = x$, and otherwise it is the root of $p(x)$.","title":"Root of a vertex in a parent forest"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/path-length","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.pathLength","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#path-length","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a parent forest $F$ with parent map $p$ and a vertex $x$, the path length of $x$ is the number of parent edges followed from $x$ to its root: it is $0$ if $p(x) = x$, and otherwise one more than the path length of $p(x)$.","title":"Path length to the root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.findWithCost","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-with-cost","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a vertex $x$ of a parent forest $F$, find with cost returns the root that find returns for $x$, together with the cost $\\ell(x) + 1$, where $\\ell(x)$ is the path length of $x$ (the number of non-root parent edges that find traverses from $x$). That is, it charges one unit for every visited vertex, including the root.","title":"Find with cost"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/equivalent","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.Equivalent","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#equivalent","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Two vertices $x$ and $y$ of a parent forest $F$ are equivalent when find returns the same root for $x$ as for $y$ in $F$.","title":"Equivalent vertices of a parent forest"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-is-root","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.find_isRoot","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-is-root","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For every vertex $x$ of a parent forest $F$, the vertex $r$ that find returns for $x$ is a root of $F$: the parent of $r$ in $F$ is $r$ itself.","title":"Find returns a root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-parent","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.find_parent","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-parent","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For every vertex $x$ of a parent forest $F$, find returns the same root for the parent of $x$ in $F$ as for $x$.","title":"Find is unchanged along a parent pointer"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-in-class","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.find_in_class","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-in-class","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For every vertex $x$ of a parent forest $F$, the vertex $x$ is equivalent in $F$ to the root that find returns for $x$.","title":"Every vertex is equivalent to its representative"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-with-cost-time","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.findWithCost_time","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-with-cost-time","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For every vertex $x$ of a parent forest $F$, the cost that find with cost charges on $x$ equals $\\ell(x) + 1$, where $\\ell(x)$ is the path length of $x$: the number of non-root parent edges that find traverses from $x$.","title":"The cost of find is the path length plus one"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-with-cost-time-le-depth","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.findWithCost_time_le_depth","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-with-cost-time-le-depth","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For every vertex $x$ of a parent forest $F$ with depth certificate $d$, the cost that find with cost charges on $x$ is at most $d(x) + 1$.","title":"The cost of find is at most the depth certificate plus one"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/link-parent","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.linkParent","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#link-parent","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $F$ be a parent forest with parent map $p$, and let $r$ and $s$ be vertices. The linked parent map sends $r$ to $s$ and agrees with $p$ on every other vertex: it maps $z$ to $s$ if $z = r$, and to $p(z)$ otherwise.","title":"Parent map after linking two roots"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/link-depth","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.linkDepth","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#link-depth","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $F$ be a parent forest with depth certificate $d$, and let $r$ and $s$ be vertices. The linked depth certificate assigns to a vertex $z$ the number $d(z) + d(s) + 1$ if $z$ is equivalent to $r$ in $F$ (that is, $z$ and $r$ have the same root), and $d(z)$ otherwise.","title":"Depth certificate after linking two roots"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/link","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.link","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#link","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $F$ be a parent forest with parent map $p$ and depth certificate $d$, and let $r \\ne s$ be two distinct roots of $F$. Linking $r$ to $s$ gives the parent forest whose parent map is the linked parent map (the parent of $r$ becomes $s$ and all other parents are unchanged) and whose depth certificate is the linked depth certificate (the certificate of every vertex equivalent to $r$ grows by $d(s) + 1$ and all\u2026","title":"Linking two roots of a parent forest"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/union","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.union","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#union","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $F$ be a union-find forest on $n$ vertices: each vertex $z$ has a parent $p(z)$, where roots are their own parents, and a depth certificate $d(z) \\in \\mathbb{N}$ with $d(p(z)) < d(z)$ whenever $p(z) \\ne z$. Let $x$ and $y$ be vertices, and let $r_x$ and $r_y$ be the roots reached from $x$ and $y$ by following parent pointers. If $r_x = r_y$, the union of $x$ and $y$ in $F$ is $F$ itself. Otherwise it is the\u2026","title":"Union in a union-find forest"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-union","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.find_union","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-union","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $F$ be a parent forest, let $x, y, z$ be vertices of $F$, and let $F'$ be the forest obtained from $F$ by the union of $x$ and $y$. If $z$ is equivalent to $x$ in $F$, then find returns for $z$ in $F'$ the root that find returns for $y$ in $F$; otherwise find returns the same root for $z$ in $F'$ as in $F$.","title":"Representatives after a union"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/union-equivalent","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.union_equivalent","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#union-equivalent","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For every parent forest $F$ and vertices $x$ and $y$, the vertices $x$ and $y$ are equivalent in the forest obtained from $F$ by the union of $x$ and $y$.","title":"Union makes its arguments equivalent"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/union-eq-self-of-equivalent","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.union_eq_self_of_equivalent","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#union-eq-self-of-equivalent","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"If $x$ and $y$ are equivalent vertices of a parent forest $F$, then the union of $x$ and $y$ in $F$ is $F$ itself.","title":"Union of equivalent vertices changes nothing"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/equivalent-union-iff","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.equivalent_union_iff","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#equivalent-union-iff","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $F$ be a parent forest and let $x, y, a, b$ be vertices of $F$. Then $a$ and $b$ are equivalent in the forest obtained from $F$ by the union of $x$ and $y$ if and only if $a$ and $b$ are equivalent in $F$, or $a$ is equivalent to $x$ and $b$ is equivalent to $y$ in $F$, or $a$ is equivalent to $y$ and $b$ is equivalent to $x$ in $F$.","title":"Equivalence after a union"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-union-of-unrelated","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.find_union_of_unrelated","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-union-of-unrelated","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $F$ be a parent forest and let $x, y, z$ be vertices of $F$. If $z$ is not equivalent to $x$ in $F$, then find returns the same root for $z$ in the forest obtained from $F$ by the union of $x$ and $y$ as in $F$.","title":"Union preserves the representatives of unrelated classes"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/union-with-cost","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.unionWithCost","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#union-with-cost","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a parent forest $F$ and vertices $x$ and $y$, the costed union returns the union of $x$ and $y$ in $F$ together with a cost equal to the cost of finding the root of $x$ plus the cost of finding the root of $y$ (each one unit per visited vertex, including the root), plus one more unit when the roots of $x$ and $y$ are distinct; when they coincide, union performs no link and none is charged.","title":"Union in a parent forest with cost"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/union-with-cost-time-le-depth","kind":"node","lean":"Cslib.Algorithms.Lean.UnionFind.unionWithCost_time_le_depth","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#union-with-cost-time-le-depth","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For every parent forest $F$ with depth certificate $d$ and vertices $x$ and $y$, the cost charged for the union of $x$ and $y$ in $F$ (two root searches at one unit per visited vertex, plus one unit for the link when the roots differ) is at most $d(x) + d(y) + 3$.","title":"Cost bound for union in a parent forest"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/check-equiv-spec","kind":"node","lean":"Batteries.UnionFind.checkEquiv_spec","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#check-equiv-spec","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Call two natural numbers $a$ and $c$ equivalent in a union-find structure when they have the same root, where a number that is not a node is its own root. Let $U$ be a union-find structure with nodes $0, \\dots, N-1$ and let $x, y < N$. The equivalence query on $x$ and $y$, which runs FIND-SET with path compression on $x$ and then on $y$, returns a union-find structure $U'$ and a boolean $b$ such that $U'$ has $N$\u2026","title":"Correctness of the union-find equivalence query"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/connected-components","kind":"node","lean":"Batteries.UnionFind.connectedComponents","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#connected-components","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Given a natural number $n$ and a list $E$ of pairs of vertices in $\\{0, \\dots, n-1\\}$, CONNECTED-COMPONENTS first builds the union-find structure with $n$ singleton nodes $0, \\dots, n-1$ by $n$ MAKE-SET calls. It then processes the pairs $(u, v)$ of $E$ in list order: it asks whether $u$ and $v$ are equivalent using two FIND-SET calls, keeping the path compression done by the query, and if they are not equivalent\u2026","title":"Connected components via union-find"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/connected-components-size","kind":"node","lean":"Batteries.UnionFind.connectedComponents_size","parent":38271,"reading_title":{"format":"inline-parts-v1","parts":[["text","Connected components allocates "],["math","\\(n\\)"],["text"," nodes"]],"source":"Connected components allocates $n$ nodes"},"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#connected-components-size","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For every natural number $n$ and every list $E$ of pairs of vertices in $\\{0, \\dots, n-1\\}$, the union-find structure returned by CONNECTED-COMPONENTS on $n$ and $E$ has exactly $n$ nodes.","title":"Connected components allocates $n$ nodes"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/connected-components-equiv-iff-reachable","kind":"node","lean":"Batteries.UnionFind.connectedComponents_equiv_iff_reachable","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#connected-components-equiv-iff-reachable","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $n$ be a natural number and let $E$ be a list of pairs of vertices in $\\{0, \\dots, n-1\\}$. For all vertices $a, b < n$, $a$ and $b$ are equivalent in the union-find structure returned by CONNECTED-COMPONENTS on $n$ and $E$ if and only if $a$ and $b$ are joined by a path in the simple graph on $\\{0, \\dots, n-1\\}$ whose edges are the unordered pairs $\\{u, v\\}$ with $(u, v)$ in $E$ and $u \\ne v$.","title":"Connected components computes reachability"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/connected-components-time-spec","kind":"node","lean":"Batteries.UnionFind.connectedComponents_time_spec","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#connected-components-time-spec","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $n$ be a natural number, let $E$ be a list of pairs of vertices in $\\{0, \\dots, n-1\\}$ and let $m$ be the length of $E$. Let $G$ be the simple graph on $\\{0, \\dots, n-1\\}$ whose edges are the unordered pairs $\\{u, v\\}$ with $(u, v)$ in $E$ and $u \\ne v$, and let $k$ be the number of connected components of $G$. Running CONNECTED-COMPONENTS on $n$ and $E$ makes exactly $n$ MAKE-SET calls, $2m$ FIND-SET calls and\u2026","title":"Operation counts of connected components"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-aux","kind":"node","lean":"Batteries.UnionFind.Internal.countedAux","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-aux","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a union-find structure $S$ with node array $A$ and a node $x < |S|$, this is the recursive path-compression routine of FIND, paired with a cost in $\\mathbb{N} \\times \\mathbb{N}$. It charges $(1, 0)$ for testing whether $x$ is a root. If the parent $y$ of $x$ in $A$ equals $x$, it returns $A$ and the root $x$. Otherwise it runs on $y$, obtaining an array $A'$ and a root $r$, charges $(0, 1)$ for one parent\u2026","title":"Cost-counted FIND auxiliary"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-aux-ret","kind":"node","lean":"Batteries.UnionFind.Internal.countedAux_ret","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-aux-ret","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a union-find structure $S$ and a node $x < |S|$, the value returned by the cost-counted FIND auxiliary on $x$ (the compressed node array together with the root of $x$), with its cost discarded, equals the result of the Batteries path-compression auxiliary routine of FIND on $x$.","title":"The cost-counted FIND auxiliary returns the uncounted result"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-aux-time-root","kind":"node","lean":"Batteries.UnionFind.Internal.countedAux_time_root","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-aux-time-root","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $S$ be a union-find structure and $x < |S|$ a node. If the parent of $x$ in $S$ is $x$ itself, then the cost-counted FIND auxiliary on $x$ has cost $(1, 0)$: one root test and no parent writes.","title":"Cost of the FIND auxiliary at a root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-aux-time-step","kind":"node","lean":"Batteries.UnionFind.Internal.countedAux_time_step","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-aux-time-step","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $S$ be a union-find structure and $x < |S|$ a node whose parent $y$ in $S$ differs from $x$. If $c_x$ and $c_y$ are the costs of the cost-counted FIND auxiliary on $x$ and on $y$, then $c_x = (1, 0) + (c_y + (0, 1))$: one root test at $x$, then the run on $y$, then one parent write.","title":"Cost recurrence of the FIND auxiliary at a non-root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-aux-time-balance","kind":"node","lean":"Batteries.UnionFind.Internal.countedAux_time_balance","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-aux-time-balance","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a union-find structure $S$ and a node $x < |S|$, if the cost-counted FIND auxiliary on $x$ has cost $(t, w)$, where $t$ counts root tests and $w$ counts parent writes, then $t = w + 1$.","title":"Root tests exceed parent writes by one"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-find","kind":"node","lean":"Batteries.UnionFind.Internal.countedFind","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-find","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a union-find structure $S$ and a node $x < |S|$, this runs the cost-counted FIND auxiliary on $x$, obtaining a node array $A'$, a root $r$ and a cost $(t, w)$. It returns the union-find structure with node array $A'$, together with $r$ as a node of that structure (whose size equals $|S|$), at cost $(t, w)$.","title":"Cost-counted FIND"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-find-ret","kind":"node","lean":"Batteries.UnionFind.Internal.countedFind_ret","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-find-ret","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a union-find structure $S$ and a node $x < |S|$, the value returned by the cost-counted FIND on $x$ (the compressed union-find structure together with the root of $x$), with its cost discarded, equals the result of the Batteries FIND with path compression on $x$.","title":"The cost-counted FIND returns the FIND result"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-aux-writes-rank","kind":"node","lean":"Batteries.UnionFind.Internal.countedAux_writes_rank","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-aux-writes-rank","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $S$ be a union-find structure, let $R$ be one more than the maximum rank of a node of $S$, and let $x < |S|$ be a node. The number of parent writes performed by the cost-counted FIND auxiliary on $x$ (the second coordinate of its cost) is at most $R - \\operatorname{rank}(x)$.","title":"Parent writes are bounded by the rank gap"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/find-parent-eq-ite-iterate","kind":"node","lean":"Batteries.UnionFind.find_parent_eq_ite_iterate","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#find-parent-eq-ite-iterate","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $S$ be a union-find structure with parent map $p$ (where $p(i) = i$ for $i \\ge |S|$), let $x < |S|$, and let $p'$ be the parent map of the structure produced by FIND with path compression on $x$. Then for every natural number $i$, $p'(i)$ is the root of $x$ if $i = p^k(x)$ for some $k \\ge 0$, and $p'(i) = p(i)$ otherwise.","title":"FIND compresses exactly the original path to the root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/get-elem-find","kind":"node","lean":"Batteries.UnionFind.getElem_find","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#get-elem-find","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $s$ be a union-find structure on the indices $0, \\dots, n-1$ with parent function $p$, let $x < n$, and let $r$ be the root of $x$ in $s$. After find with path compression is run on $x$, for every index $i < n$ the node at $i$ in the resulting array keeps its rank from $s$, and its parent is $r$ if $i = p^k(x)$ for some $k \\ge 0$, and is its original parent $p(i)$ otherwise.","title":"Effect of path-compressing find on each node"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/exists-first-iterate-root-d","kind":"node","lean":"Batteries.UnionFind.exists_first_iterate_rootD","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#exists-first-iterate-root-d","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $S$ be a union-find structure with $n$ nodes and parent map $p$, and let $x < n$. Then there is $d < n$ such that $p^d(x)$ is the root of $x$ and $p^k(x)$ is not the root of $x$ for every $k < d$.","title":"The root is reached in fewer parent steps than there are nodes"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/rank-d-link-aux","kind":"node","lean":"Batteries.UnionFind.LinkInternal.rankD_linkAux","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#rank-d-link-aux","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For an array $A$ of union-find nodes, write $\\rho_A(i)$ for the rank stored at index $i$ of $A$, or $0$ if $i$ is not an index of $A$. Let $x, y$ be indices of $A$ and let $A'$ be the array obtained from $A$ by the Batteries link by rank of $x$ and $y$. Then for every natural number $i$, $\\rho_{A'}(i) = \\rho_A(y) + 1$ if $x \\ne y$, $\\rho_A(x) = \\rho_A(y)$ and $i = y$, and $\\rho_{A'}(i) = \\rho_A(i)$ otherwise.","title":"Ranks after linking two nodes of an array"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/rank-link","kind":"node","lean":"Batteries.UnionFind.rank_link","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#rank-link","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $S$ be a union-find structure, write $\\rho(i)$ for the rank of $i$ in $S$ ($0$ if $i$ is not a node of $S$), and let $x, y$ be nodes of $S$ such that $y$ is a root, that is, the parent of $y$ is $y$. Let $S'$ be the result of linking $x$ and $y$ by rank. Then for every natural number $i$, the rank of $i$ in $S'$ is $\\rho(y) + 1$ if $x \\ne y$, $\\rho(x) = \\rho(y)$ and $i = y$, and is $\\rho(i)$ otherwise.","title":"Ranks after linking in a union-find structure"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/root-d-link-eq-ite","kind":"node","lean":"Batteries.UnionFind.rootD_link_eq_ite","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#root-d-link-eq-ite","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $U$ be a union-find structure on $\\{0, \\dots, n-1\\}$ and let $x, y < n$ be roots of $U$, i.e. each is its own parent. Let $U'$ be obtained by linking $x$ and $y$ by rank: nothing changes if $x = y$; otherwise, if $\\operatorname{rank}(y) < \\operatorname{rank}(x)$ then $y$ gets parent $x$, and if not then $x$ gets parent $y$ and, when the two ranks are equal, the rank of $y$ increases by one. Then for every\u2026","title":"Linking two roots by rank"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/union-carried","kind":"node","lean":"Batteries.UnionFind.LinkInternal.union_carried","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#union-carried","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $U$ be a union-find structure on $\\{0, \\dots, n-1\\}$ and let $x, y < n$. Run find with path compression on $x$ in $U$, giving a structure $U_1$ and a root $r_x$; then run find on $y$ in $U_1$ (not in $U$), giving a structure $U_2$ and a root $r_y$. Then the node array of the union of $x$ and $y$ in $U$ equals the array obtained from the node array of $U_2$ by linking $r_x$ and $r_y$ by rank: nothing changes if\u2026","title":"Union unfolds to two finds and a link"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/rank-find","kind":"node","lean":"Batteries.UnionFind.LinkInternal.rank_find","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#rank-find","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $S$ be a union-find structure and $x$ a node of $S$, and let $S'$ be the structure returned by FIND on $x$ with path compression. For every natural number $i$, the rank of $i$ in $S'$ equals the rank of $i$ in $S$, where the rank of an index that is not a node is $0$.","title":"Path compression preserves ranks"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/rank-union","kind":"node","lean":"Batteries.UnionFind.rank_union","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#rank-union","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $S$ be a union-find structure, write $\\rho(i)$ for the rank of $i$ in $S$ ($0$ if $i$ is not a node of $S$) and $r(i)$ for the root of $i$ in $S$. For nodes $x, y$ of $S$ and every natural number $i$, the rank of $i$ in the union of $x$ and $y$ in $S$ (union by rank with path compression) is $\\rho(r(y)) + 1$ if $r(x) \\ne r(y)$, $\\rho(r(x)) = \\rho(r(y))$ and $r(y) = i$, and is $\\rho(i)$ otherwise.","title":"Ranks after a union"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-link-aux","kind":"node","lean":"Batteries.UnionFind.LinkInternal.countedLinkAux","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-link-aux","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Given an array $A$ of union-find nodes, each storing a parent index and a rank, and indices $x, y$ of $A$, this computation returns an array together with a count $(t, p, w)$ of rank tests $t$, parent writes $p$ and rank writes $w$. If $x = y$ it returns $A$ with count $(0, 0, 0)$. Otherwise it makes one rank test; if the rank of $y$ is less than the rank of $x$, it sets the parent of $y$ to $x$ (one parent write).\u2026","title":"Linking by rank with operation counts"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-link-aux-ret","kind":"node","lean":"Batteries.UnionFind.LinkInternal.countedLinkAux_ret","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-link-aux-ret","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For an array $A$ of union-find nodes and indices $x, y$ of $A$, the array returned by counted linking of $x$ and $y$ in $A$ is exactly the array produced by the uncounted Batteries link by rank of $x$ and $y$ in $A$.","title":"Counted linking returns the linked array"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-link-aux-time","kind":"node","lean":"Batteries.UnionFind.LinkInternal.countedLinkAux_time","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-link-aux-time","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"Let $A$ be an array of union-find nodes, let $x, y$ be indices of $A$, and write $r_x$ and $r_y$ for the ranks stored at $x$ and $y$. Counted linking of $x$ and $y$ in $A$ makes $(t, p, w)$ rank tests, parent writes and rank writes, where $(t, p, w)$ is $(0, 0, 0)$ if $x = y$, is $(1, 1, 0)$ if $x \\ne y$ and $r_y < r_x$, is $(2, 1, 1)$ if $x \\ne y$ and $r_x = r_y$, and is $(2, 1, 0)$ otherwise, that is, if\u2026","title":"Operation counts of linking by rank"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-union","kind":"node","lean":"Batteries.UnionFind.LinkInternal.countedUnion","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-union","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a union-find structure $S$ and indices $x, y$ of $S$, this computation runs counted FIND on $x$ in $S$, giving a structure $S_1$ and a root $r_x$; then counted FIND on $y$ in $S_1$, giving a structure $S_2$ and a root $r_y$; then counted linking of $r_x$ and $r_y$ in the node array of $S_2$. It returns the array produced by the linking, with cost $((a, b), (t, p, w))$, where $a$ and $b$ are the total numbers of\u2026","title":"Union by rank with operation counts"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-union-ret","kind":"node","lean":"Batteries.UnionFind.LinkInternal.countedUnion_ret","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-union-ret","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a union-find structure $S$ and indices $x, y$ of $S$, the array returned by the counted union of $x$ and $y$ in $S$ equals the node array of the Batteries union of $x$ and $y$ in $S$ (union by rank with path compression).","title":"Counted union returns the union's array"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/counted-union-find-balance","kind":"node","lean":"Batteries.UnionFind.LinkInternal.countedUnion_find_balance","parent":38271,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure/index.html#counted-union-find-balance","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/disjoint-set-data-structure","status":"fully_proved","summary":"For a union-find structure $S$ and indices $x, y$ of $S$, in the counted union of $x$ and $y$ in $S$ the total number of root tests made by its two FIND calls equals the total number of parent writes they make plus $2$.","title":"The FIND phases of a union make two more root tests than parent writes"},{"data_url":"scopes/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree.json","id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","kind":"scope","navigation":"module","parent":38270,"result":false,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","title":"Euler tour tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/tree","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#tree","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"A rooted ordered tree with vertices in a type $V$ consists of a root vertex in $V$ together with an ordered list of child subtrees, each a rooted ordered tree with vertices in $V$.","title":"Rooted ordered tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/children-ind","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.children_ind","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#children-ind","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"Let $P$ be a property of trees. Suppose that for every vertex $r$ and all trees $c_1, \\dots, c_k$ (with $k \\ge 0$), if $P$ holds for each $c_i$, then $P$ holds for the tree with root $r$ and child subtrees $c_1, \\dots, c_k$. Then $P$ holds for every tree.","title":"Induction principle for rooted trees"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/root","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.root","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#root","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"The root of the tree with root vertex $r$ and child subtrees $c_1, \\dots, c_k$ is the vertex $r$.","title":"Root of a rooted tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/children","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.children","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#children","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"The children of a tree are the child subtrees of its root, as a list in their given order.","title":"Children of a rooted tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/size","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.size","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#size","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"The size of a tree is its number of nodes, counting repeated vertex labels with multiplicity: the tree with root $r$ and child subtrees $c_1, \\dots, c_k$ has size $1 + \\sum_{i=1}^{k} \\operatorname{size}(c_i)$.","title":"Size of a rooted tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/vertices-aux","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.verticesAux","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#vertices-aux","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For a rooted ordered tree $t$ with root $r$ and child subtrees $c_1, \\dots, c_k$ (in order) and a list $a$, the accumulator traversal $T(t, a)$ is the list whose first entry is $r$ and whose remaining entries are $T(c_1, T(c_2, \\dots, T(c_k, a) \\dots))$; when $k = 0$ the remaining entries are $a$.","title":"Accumulator preorder traversal"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/vertices","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.vertices","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#vertices","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"The preorder vertex list of a rooted ordered tree $t$ is the accumulator preorder traversal of $t$ started from the empty list.","title":"Preorder vertex list of a rooted tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/vertices-aux-eq","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.verticesAux_eq","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#vertices-aux-eq","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For every rooted ordered tree $t$ and every list $a$, the accumulator preorder traversal of $t$ started from $a$ equals the preorder vertex list of $t$ followed by $a$.","title":"Accumulator preorder traversal is append"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/vertices-node","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.vertices_node","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#vertices-node","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For a vertex $r$ and rooted ordered trees $c_1, \\dots, c_k$, the preorder vertex list of the tree with root $r$ and child subtrees $c_1, \\dots, c_k$ (in order) is $r$ followed by the concatenation of the preorder vertex lists of $c_1, \\dots, c_k$, in this order.","title":"Preorder vertex list of a node"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/tour-aux","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.tourAux","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#tour-aux","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"Let $t$ be a tree with root $r$ and child subtrees $c_1, \\dots, c_k$ in order, and let $a$ be a list of vertices. For a list $L$ of vertices, write $r \\cdot L$ for the list $L$ with $r$ put in front. The Euler tour of $t$ with accumulator $a$, written $T(t, a)$, is $r \\cdot L_1$, where $L_{k+1} = a$ and $L_i = T(c_i, r \\cdot L_{i+1})$ for $i = k, \\dots, 1$.","title":"Euler tour with an accumulator"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/tour","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.tour","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#tour","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"The Euler tour of a tree $t$ with root $r$ and child subtrees $c_1, \\dots, c_k$ in order is the list of vertices consisting of $r$, followed, for each $i = 1, \\dots, k$ in turn, by the Euler tour of $c_i$ and then $r$. It is the vertex sequence of the depth-first walk that returns to the parent vertex after each child subtree.","title":"Euler tour"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/tour-aux-eq","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.tourAux_eq","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#tour-aux-eq","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For every tree $t$ and every list $a$ of vertices, the Euler tour of $t$ with accumulator $a$ equals the Euler tour of $t$ followed by $a$.","title":"Accumulator lemma for the Euler tour"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/tour-node","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.tour_node","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#tour-node","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For a vertex $r$ and a list of trees $c_1, \\dots, c_k$, the Euler tour of the tree with root $r$ and child subtrees $c_1, \\dots, c_k$ in order is $r$, followed, for each $i = 1, \\dots, k$ in turn, by the Euler tour of $c_i$ and then $r$.","title":"Recursive formula for the Euler tour"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/arc-tour","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.arcTour","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#arc-tour","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"The arc sequence of a tree $t$ is the list of consecutive pairs of its Euler tour: if the Euler tour of $t$ is $v_0, v_1, \\dots, v_k$, then the arc sequence of $t$ is $(v_0, v_1), (v_1, v_2), \\dots, (v_{k-1}, v_k)$.","title":"Arc sequence of the Euler tour"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/arc-tour-eq-zip","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.arcTour_eq_zip","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#arc-tour-eq-zip","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For every tree $t$, the arc sequence of $t$ is the Euler tour of $t$ zipped with its own tail: if the Euler tour of $t$ is $v_0, v_1, \\dots, v_k$, then the arc sequence of $t$ is $(v_0, v_1), (v_1, v_2), \\dots, (v_{k-1}, v_k)$.","title":"Arc sequence as the zipped Euler tour"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/mem-tour-iff","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.mem_tour_iff","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#mem-tour-iff","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For every tree $t$ and every $v$, $v$ occurs in the Euler tour of $t$ if and only if $v$ is a vertex of $t$.","title":"The Euler tour visits exactly the vertices of the tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.Valid","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#valid","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"A rooted ordered tree is valid when its list of vertices in preorder has no duplicates, that is, every vertex label occurs at most once in the tree.","title":"Valid Euler-tour tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/valid-iff","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.valid_iff","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#valid-iff","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"A tree $t$ is valid if and only if its list of vertices in preorder has no repeated entries.","title":"Valid trees have distinct vertices"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/valid-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.valid_singleton","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#valid-singleton","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For every vertex $r$, the tree with root $r$ and no child subtrees is valid.","title":"A single-vertex tree is valid"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/link","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.link","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#link","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"Let $t_1$ be a tree with root $r$ and child subtrees $c_1, \\dots, c_k$, and let $t_2$ be a tree. The link of $t_1$ and $t_2$ is the tree with root $r$ and child subtrees $c_1, \\dots, c_k, t_2$, so $t_2$ becomes the new last child subtree of the root of $t_1$.","title":"Linking two rooted trees"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/valid-link","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.valid_link","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#valid-link","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"Let $t_1$ and $t_2$ be valid trees such that no vertex of $t_1$ is a vertex of $t_2$. Then the link of $t_1$ and $t_2$, that is, $t_1$ with $t_2$ added as the last child subtree of its root, is valid.","title":"Linking valid trees with disjoint vertices gives a valid tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/not-valid-link-same-root","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.not_valid_link_same_root","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#not-valid-link-same-root","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For every vertex $r$, the link of the single-vertex tree with root $r$ and another single-vertex tree with root $r$ is not valid, since the vertex $r$ occurs twice in it. Hence the link of two valid trees need not be valid when their vertex sets are not disjoint.","title":"Linking two copies of a single vertex is not valid"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/splice-link","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.spliceLink","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#splice-link","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For lists $\\tau_1$ and $\\tau_2$ of vertices and a vertex $r$, the splice of $\\tau_1$ and $\\tau_2$ at $r$ is the list consisting of the entries of $\\tau_1$, followed by the entries of $\\tau_2$, followed by $r$.","title":"Splicing two Euler tours"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/tour-link","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.tour_link","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#tour-link","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"For trees $t_1$ and $t_2$, let $t$ be the link of $t_1$ and $t_2$, that is, $t_1$ with $t_2$ added as the last child subtree of its root. The Euler tour of $t$ is the Euler tour of $t_1$, followed by the Euler tour of $t_2$, followed by the root of $t_1$.","title":"Euler tour of a linked tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/cut-last","kind":"node","lean":"Cslib.Algorithms.Lean.EulerTourTree.Tree.cutLast","parent":38321,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree/index.html#cut-last","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/euler-tour-tree","status":"fully_proved","summary":"Let $t$ be a tree with root $r$ and child subtrees $c_1, \\dots, c_k$. Cutting the last child subtree of $t$ gives no result when $k = 0$; when $k \\ge 1$ it gives the pair consisting of the tree with root $r$ and child subtrees $c_1, \\dots, c_{k-1}$, together with the removed subtree $c_k$.","title":"Cutting the last child subtree"},{"data_url":"scopes/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree.json","id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","kind":"scope","navigation":"module","parent":38270,"result":false,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","title":"Link-cut tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/spine","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.spine","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#spine","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a vertex $r$ and a list of vertices $c_1, \\dots, c_k$, the spine is the rooted tree $r \\to c_1 \\to \\cdots \\to c_k$: its root is $r$, each vertex has the next one in the list as its only child, and the last vertex ($c_k$, or $r$ when $k = 0$) is a leaf.","title":"Path tree on a vertex list"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/of-vertex-list","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.ofVertexList","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#of-vertex-list","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"The forest built from the empty list is empty; the forest built from a list $r, c_1, \\dots, c_k$ consists of the single spine $r \\to c_1 \\to \\cdots \\to c_k$, the rooted tree in which each vertex has the next one as its only child.","title":"Forest from a vertex list"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/find-tree","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.findTree","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#find-tree","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a vertex $x$ and a forest given as a list of trees, the first tree whose vertices contain $x$, or nothing when no tree contains $x$.","title":"Tree of a forest containing a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/remove-tree","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.removeTree","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#remove-tree","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Given a vertex $x$ and a forest, given as a list of rooted ordered trees, removing the trees containing $x$ deletes from the list every tree that has $x$ among its vertices and keeps the remaining trees in their original order.","title":"Removing the trees that contain a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/perm-flat-map-remove-tree","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.perm_flatMap_removeTree","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#perm-flat-map-remove-tree","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $f$ be a forest of rooted ordered trees whose concatenated vertex lists have no duplicates, let $x$ be a vertex, and let $t$ be the first tree of $f$ containing $x$. Then the concatenated vertex list of the forest obtained from $f$ by removing every tree containing $x$, followed by the vertex list of $t$, is a permutation of the concatenated vertex list of $f$.","title":"Removing the tree containing a vertex splits the forest's vertices"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/find-subtree","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.findSubtree","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#find-subtree","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a vertex $x$ and a tree $T$, the subtree of $T$ rooted at the first occurrence of $x$ in preorder, or nothing when $x$ does not occur in $T$.","title":"Subtree rooted at a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/adjacent","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.Adjacent","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#adjacent","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Two vertices $a$ and $b$ are adjacent in a rooted tree $t$ when they are joined by a parent-child edge of $t$ in either direction: some node of $t$ labelled $a$ has a child labelled $b$, or some node of $t$ labelled $b$ has a child labelled $a$.","title":"Adjacency in a rooted tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-left","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.Adjacent.mem_left","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-left","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If $a$ and $b$ are adjacent in a tree $t$, that is, joined by a parent-child edge in either direction, then $a$ is a vertex of $t$.","title":"The first endpoint of a tree edge is a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/symm","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.Adjacent.symm","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#symm","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every rooted ordered tree $T$ and vertices $a$ and $b$, if $a$ and $b$ are adjacent in $T$ (joined by a parent-child edge in either direction), then $b$ and $a$ are adjacent in $T$.","title":"Adjacency in a tree is symmetric"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-right","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.Adjacent.mem_right","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-right","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If $a$ and $b$ are adjacent in a tree $t$, that is, joined by a parent-child edge in either direction, then $b$ is a vertex of $t$.","title":"The second endpoint of a tree edge is a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/adjacent-b","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.adjacentB","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#adjacent-b","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For vertices $a$, $b$ and a rooted tree $t$ with root $r$ and child subtrees $c_1, \\dots, c_k$, the adjacency test of $a$ and $b$ in $t$ returns true exactly when $r = a$ and some $c_i$ has root $b$, or $r = b$ and some $c_i$ has root $a$, or the adjacency test of $a$ and $b$ returns true in some $c_i$.","title":"Adjacency test"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/graft-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.graftAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#graft-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a vertex $y$ and trees $N$ and $T$, grafting $N$ at $y$ in $T$ attaches $N$ as a new last child of the first node of $T$ labelled $y$ in preorder, keeping every existing node and edge; it returns nothing when $y$ does not occur in $T$. This is the tree surgery behind linking two trees: it adds exactly the edge from $y$ to the root of $N$.","title":"Grafting a tree below a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-vertices-graft-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.mem_vertices_graftAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-vertices-graft-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ and $C$ be trees and $y$ a vertex. If grafting $C$ at $y$ in $t$, that is, attaching $C$ as a new last child of a node labelled $y$, produces the tree $t'$, then for every $v$, $v$ is a vertex of $t'$ if and only if $v$ is a vertex of $t$ or a vertex of $C$.","title":"Vertices after grafting"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/root-graft-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.root_graftAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#root-graft-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $T$ and $N$ be rooted ordered trees and $y$ a vertex. If grafting $N$ into $T$ at $y$ is defined and yields the tree $T'$, then the root of $T'$ equals the root of $T$.","title":"Grafting keeps the root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/adjacent-graft-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.adjacent_graftAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#adjacent-graft-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ and $u$ be rooted trees and $y$ a vertex, and suppose that grafting $u$ as a new child under $y$ in $t$ succeeds and yields the tree $t'$. Then for all vertices $a$ and $b$, the vertices $a$ and $b$ are adjacent in $t'$ if and only if they are adjacent in $t$, or adjacent in $u$, or $a = y$ and $b$ is the root of $u$, or $a$ is the root of $u$ and $b = y$.","title":"Edges after grafting"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/perm-vertices-graft-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.perm_vertices_graftAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#perm-vertices-graft-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ and $s$ be rooted ordered trees and $y$ a vertex. If grafting $s$ as a new child at $y$ in $t$ succeeds and yields the tree $t'$, then the vertex list of $t'$ is a permutation of the vertex list of $t$ followed by the vertex list of $s$.","title":"Grafting adds the new subtree's vertices"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/reroot-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.rerootAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#reroot-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a vertex $x$ and a rooted ordered tree $T$ with root $r$, rerooting $T$ at $x$ is defined recursively. If $r = x$, the result is $T$ itself. Otherwise let $C$ be the first child subtree of $r$ that contains $x$, let $p$ be the root of $C$, and let $C'$ be $C$ rerooted at $x$. The result is $C'$ with the tree formed by $r$ and its remaining children (in their original order) attached as a new last child of the\u2026","title":"Rerooting a tree at a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/reroot-at-eq-none-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.rerootAt_eq_none_of_not_mem","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#reroot-at-eq-none-of-not-mem","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every rooted ordered tree $T$ and every vertex $x$ that is not a vertex of $T$, rerooting $T$ at $x$ is undefined (it returns no tree).","title":"Rerooting fails at an absent vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-vertices-reroot-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.mem_vertices_rerootAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-vertices-reroot-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted tree and $x$ a vertex. If rerooting $t$ at $x$ yields the tree $t'$, then a vertex lies in $t'$ if and only if it lies in $t$.","title":"Rerooting preserves the vertex set"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/adjacent-reroot-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.adjacent_rerootAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#adjacent-reroot-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted tree with no repeated vertex, and suppose that rerooting $t$ at a vertex $x$ yields the tree $t'$. Then for all vertices $a$ and $b$, the vertices $a$ and $b$ are adjacent in $t'$ if and only if they are adjacent in $t$.","title":"Rerooting keeps every edge"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/perm-vertices-reroot-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.perm_vertices_rerootAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#perm-vertices-reroot-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted ordered tree and $x$ a vertex. If rerooting $t$ at $x$ yields the tree $t'$, then the vertex list of $t'$ is a permutation of the vertex list of $t$.","title":"Rerooting preserves the vertices"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/prune-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.pruneAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#prune-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a vertex $x$ and a rooted ordered tree $T$, pruning $T$ at $x$ removes every subtree of $T$ whose root is labelled $x$, keeping every other node and edge and the order of children. If the root of $T$ is $x$, nothing remains and pruning is undefined. Otherwise the result is the tree with the same root whose children are, in their original order, the pruned children of the root, with the children labelled $x$\u2026","title":"Pruning the subtrees rooted at a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/prune-at-eq-self-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.pruneAt_eq_self_of_not_mem","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#prune-at-eq-self-of-not-mem","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted ordered tree and $x$ a vertex that does not occur in $t$. Then pruning the subtrees rooted at $x$ from $t$ succeeds and returns $t$ itself.","title":"Pruning at an absent vertex changes nothing"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-vertices-prune-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.mem_vertices_pruneAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-vertices-prune-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted tree and $x$ a vertex. If pruning $t$ at $x$ leaves the tree $t'$, then every vertex of $t'$ is a vertex of $t$ and is different from $x$.","title":"Vertices of a pruned tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-vertices-prune-at-of-mem-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.mem_vertices_pruneAt_of_mem_not_mem","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-vertices-prune-at-of-mem-not-mem","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted tree whose preorder vertex list has no repeated vertex, and let $x$ be a vertex. Suppose the subtree search finds the subtree $s$ of $t$ rooted at $x$, and pruning $t$ at $x$ leaves the tree $t'$. Then every vertex of $t$ that is not a vertex of $s$ is a vertex of $t'$.","title":"Pruning keeps the vertices outside the detached subtree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/perm-vertices-prune-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.perm_vertices_pruneAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#perm-vertices-prune-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted ordered tree whose vertex list has no duplicates and let $x$ be a vertex. If $x$ occurs in $t$, with $s$ the subtree of $t$ rooted at $x$, and pruning at $x$ leaves a tree $t'$ (that is, $x$ is not the root of $t$), then the vertex list of $t$ is a permutation of the vertex list of $t'$ followed by the vertex list of $s$.","title":"Pruning a subtree splits the vertex list"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/root-prune-at","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.root_pruneAt","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#root-prune-at","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $T$ be a rooted ordered tree and $x$ a vertex. If pruning $T$ at $x$ yields a tree $T'$ (that is, the root of $T$ is not $x$), then the root of $T'$ equals the root of $T$.","title":"Pruning keeps the root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/adjacent-prune-at-of-mem-mem","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.adjacent_pruneAt_of_mem_mem","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#adjacent-prune-at-of-mem-mem","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted tree with no repeated vertex and $x$ a vertex, and suppose that removing every subtree rooted at $x$ from $t$ leaves the tree $t'$ (that is, the root of $t$ is not $x$). If $a$ and $b$ are adjacent in $t$ and both are vertices of $t'$, then $a$ and $b$ are adjacent in $t'$.","title":"Pruning keeps surviving edges"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/root-path","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.rootPath","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#root-path","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a vertex $x$ and a rooted ordered tree $T$, the root path of $x$ is defined exactly when $x$ is a vertex of $T$. It is then the list of vertices on the path from the root of $T$ down to $x$, root first and ending with $x$; if $x$ occurs more than once, the path leads to its first occurrence in preorder.","title":"Root-to-vertex path"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/get-last-root-path","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.getLast?_rootPath","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#get-last-root-path","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If $p$ is the path from the root of a tree $T$ down to a vertex $x$ of $T$, then $p$ is nonempty and its last (deepest) vertex is $x$.","title":"A root path ends at its target"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/head-root-path","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.head?_rootPath","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#head-root-path","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a tree and $x$ a vertex. If the root path of $x$ in $t$ (the path from the root of $t$ down to $x$) is defined and equals $p$, then $p$ is nonempty and its first vertex is the root of $t$.","title":"The root path starts at the root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-root-path","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.mem_rootPath","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-root-path","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If $p$ is the path in a tree $t$ from the root down to a vertex $x$, then every vertex of $p$ is a vertex of $t$.","title":"Root paths stay inside the tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/nodup-root-path","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.nodup_rootPath","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#nodup-root-path","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If the preorder vertex list of a rooted tree $t$ has no repeated vertex and $p$ is the path from the root of $t$ down to a vertex $x$, then $p$ has no repeated vertex.","title":"Root paths have distinct vertices"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/is-chain-adjacent-root-path","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.isChain_adjacent_rootPath","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#is-chain-adjacent-root-path","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a tree and $x$ a vertex. If the root path of $x$ in $t$ is defined and equals $p = (p_0, \\dots, p_k)$, then for every $i < k$ the vertices $p_i$ and $p_{i+1}$ are adjacent in $t$, that is, joined by a parent-child edge of $t$ in one direction or the other.","title":"Consecutive vertices of a root path are adjacent"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/parent-child","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.ParentChild","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#parent-child","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For vertices $a$ and $b$ and a rooted ordered tree $t$, $a$ is a parent of $b$ in $t$ when some node of $t$ labelled $a$ has a child subtree whose root is $b$. Preferred edges of a link-cut tree are directed this way, from a node to its preferred child.","title":"Parent-child relation in a rooted tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/adjacent-2","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.ParentChild.adjacent","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#adjacent-2","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every rooted tree $t$ and all vertices $a$ and $b$, if $a$ is the parent of $b$ somewhere in $t$, then $a$ and $b$ are adjacent in $t$.","title":"Parent-child edges are edges"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-left-2","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.ParentChild.mem_left","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-left-2","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If $a$ is the parent of $b$ somewhere in a tree $t$, then $a$ is a vertex of $t$.","title":"The parent endpoint of a tree edge is a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-right-2","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.ParentChild.mem_right","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-right-2","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If $a$ is the parent of $b$ somewhere in a tree $t$, then $b$ is a vertex of $t$.","title":"The child endpoint of a tree edge is a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/is-chain-parent-child-root-path","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.isChain_parentChild_rootPath","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#is-chain-parent-child-root-path","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a tree and $x$ a vertex. If the root path of $x$ in $t$ is defined and equals $p = (p_0, \\dots, p_k)$, then for every $i < k$ the vertex $p_i$ is the parent of $p_{i+1}$ somewhere in $t$.","title":"A root path follows parent-child edges downward"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/path-edges","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.pathEdges","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#path-edges","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a list of vertices $v_0, v_1, \\dots, v_n$, its path edges are the ordered pairs $(v_0, v_1), (v_1, v_2), \\dots, (v_{n-1}, v_n)$ of consecutive vertices, in this order; a list with at most one vertex has no path edges. For a path listed from the root downward each pair is (parent, child), and the preferred edges installed by an access to a vertex $x$ are the path edges of the root-to-$x$ path.","title":"Consecutive edges of a vertex path"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/decomp","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.decomp","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#decomp","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Given a list $E$ of preferred edges, each a pair (parent, preferred child), the decomposition of a tree $T$ is a pair (down, rest): down is the preferred path that starts at the root of $T$ and follows edges of $E$ downward, and rest is the list of all other preferred paths of $T$. When a node has several children joined to it by edges of $E$ (excluded in valid states, where each vertex has at most one preferred\u2026","title":"Preferred-path decomposition of a tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/tree-paths","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.treePaths","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#tree-paths","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a rooted ordered tree and let $E$ be a list of preferred edges, each a pair (parent, child). The preferred paths of $t$ under $E$ are listed as follows: first the downward path of the preferred-path decomposition of $t$ under $E$, which starts at the root of $t$ and follows preferred edges downward, and then all other paths of that decomposition, in the order the decomposition returns them.","title":"Preferred paths of a tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/decomp-fst-head","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.decomp_fst_head?","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#decomp-fst-head","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every list of preferred edges, the downward preferred path in the preferred-path decomposition of a tree is nonempty and its first vertex is the root of the tree.","title":"The downward preferred path starts at the root"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/decomp-perm","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.decomp_perm","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#decomp-perm","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every list of preferred edges, the downward preferred path of a tree followed by the concatenation of the other paths of its preferred-path decomposition is a permutation of the vertex list of the tree.","title":"Preferred paths permute the vertex list"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/perm-tree-paths-flat-map","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.perm_treePaths_flatMap","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#perm-tree-paths-flat-map","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every list $E$ of directed edges and every rooted ordered tree $t$, the concatenation of the preferred paths of $t$ under $E$ is a permutation of the preorder vertex list of $t$.","title":"The preferred paths of a tree rearrange its vertex list"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/decomp-is-chain","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.decomp_isChain","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#decomp-is-chain","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every list of preferred edges, in the preferred-path decomposition of a tree $T$, the downward preferred path and every other path are paths of $T$: any two consecutive vertices are joined by a parent-child edge of $T$, in either direction.","title":"Decomposition paths are paths of the tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/is-chain-of-mem-tree-paths","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.isChain_of_mem_treePaths","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#is-chain-of-mem-tree-paths","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $t$ be a tree and $E$ any list of pairs of vertices, read as preferred (parent, child) edges. For every path $p = (p_0, \\dots, p_k)$ in the preferred-path decomposition of $t$ under $E$ and every $i < k$, the vertices $p_i$ and $p_{i+1}$ are adjacent in $t$, that is, joined by a parent-child edge of $t$ in one direction or the other.","title":"Preferred paths of a tree follow tree edges"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/decomp-fst-eq-root-path","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.decomp_fst_eq_rootPath","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#decomp-fst-eq-root-path","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $T$ be a tree whose vertex list is duplicate-free, let $x$ be a vertex of $T$ with root path $p$ (the path from the root of $T$ down to $x$), and let $E$ be a list of directed edges such that every consecutive pair of $p$, parent first, lies in $E$, and every edge of $E$ whose first vertex lies on $p$ is such a consecutive pair. Then the downward preferred path of the preferred-path decomposition of $T$ under\u2026","title":"Access edges make the root path the downward preferred path"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/state","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#state","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"A state of a link-cut tree over a vertex type $V$ consists of the represented forest, a list of rooted ordered trees with vertices in $V$, and a list of preferred (solid) edges, each a pair $(u, v)$ of vertices meant as the edge from a node $u$ to its preferred child $v$ in the represented forest; the structure itself does not require these pairs to be edges of the forest. The preferred-path decomposition is\u2026","title":"Link-cut tree state"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.Mem","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"A vertex $x$ belongs to a link-cut tree state $s$ when $x$ is a vertex of some tree of the represented forest of $s$.","title":"Vertex of a link-cut tree state"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/connected","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.Connected","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#connected","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Two vertices $x$ and $y$ are connected in a link-cut tree state $s$ when some tree of the represented forest of $s$ has both $x$ and $y$ among its vertices.","title":"Connectivity in a link-cut tree state"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/connected-iff-tour","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.connected_iff_tour","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#connected-iff-tour","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a link-cut tree state $s$ and vertices $x$ and $y$, $x$ and $y$ are connected in $s$ if and only if some tree of the represented forest of $s$ contains both $x$ and $y$ in its Euler tour.","title":"Connectivity via Euler tours"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/valid","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.Valid","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#valid","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"A link-cut tree state $s$ is valid when four conditions hold: the concatenation of the vertex lists of the trees of its represented forest has no duplicates (so the trees are vertex-disjoint); its list of preferred edges has no duplicates; every preferred edge $(a, b)$ is a parent-child edge of some tree of the represented forest, with $a$ the parent of $b$; and any two preferred edges with the same parent are\u2026","title":"Valid link-cut tree states"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/valid-of-preferred-nil","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.valid_of_preferred_nil","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#valid-of-preferred-nil","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state whose list of preferred edges is empty. If the concatenation of the vertex lists of the trees of the represented forest of $s$ has no duplicates, then $s$ is valid.","title":"Validity of states with no preferred edges"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/paths","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.paths","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#paths","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For a link-cut tree state $s$, its preferred paths are obtained by taking, for each represented tree of $s$ in forest order, the preferred paths of that tree under the preferred edges of $s$, and concatenating these lists. Within a tree these are its maximal chains of preferred edges when each vertex has at most one preferred child, each listed from its shallowest to its deepest vertex; a vertex incident to no\u2026","title":"Preferred-path decomposition of a link-cut tree state"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-paths-flat-map","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.mem_paths_flatMap","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-paths-flat-map","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every link-cut tree state $s$ and every vertex $v$, $v$ lies on some path of the preferred-path decomposition of $s$ if and only if $v$ is a vertex of some tree of the represented forest of $s$. No validity hypothesis on $s$ is needed.","title":"Preferred paths cover exactly the vertices of the forest"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/nodup-paths-flat-map","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.nodup_paths_flatMap","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#nodup-paths-flat-map","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"In a valid link-cut tree state, the concatenation of all preferred paths of the decomposition has no repeated vertex: no vertex occurs twice on one path or on two different paths.","title":"Preferred paths cover each vertex at most once"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/pairwise-disjoint-paths","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.pairwise_disjoint_paths","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#pairwise-disjoint-paths","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"In a valid link-cut tree state, any two preferred paths at different positions of the preferred-path decomposition have no vertex in common.","title":"Preferred paths are pairwise vertex-disjoint"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/nodup-of-mem-paths","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.nodup_of_mem_paths","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#nodup-of-mem-paths","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Every preferred path of a valid state has distinct vertices.","title":"Preferred paths have distinct vertices"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/is-chain-of-mem-paths","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.isChain_of_mem_paths","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#is-chain-of-mem-paths","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state. For every path $p = (p_0, \\dots, p_k)$ in the preferred-path decomposition of $s$ and every $i < k$, some represented tree of $s$ has $p_i$ and $p_{i+1}$ adjacent, that is, joined by a parent-child edge in one direction or the other.","title":"Preferred paths of a link-cut tree state follow represented edges"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/access","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.access","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#access","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state and $x$ a vertex. If $x$ is a vertex of some tree of the represented forest of $s$, let $T$ be the first such tree and let $p$ be the path in $T$ from its root down to $x$. Accessing $x$ keeps the represented forest and replaces the preferred edges by those preferred edges of $s$ whose parent endpoint is not on $p$, followed by the edges of $p$, each directed from parent to child.\u2026","title":"Access in a link-cut tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-preferred-access","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.mem_preferred_access","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-preferred-access","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state, $x$ a vertex, and $(u, w)$ a preferred edge of $s$, directed from the parent $u$ to the child $w$. Suppose that, if some tree of the represented forest of $s$ contains $x$, then $u$ does not lie on the path in the first such tree from its root down to the first occurrence of $x$ in preorder. Then $(u, w)$ is still a preferred edge after accessing $x$.","title":"Access keeps off-path preferred edges"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-preferred-access-of-mem-path-edges","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.mem_preferred_access_of_mem_pathEdges","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-preferred-access-of-mem-path-edges","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state and $x$ a vertex, let $t$ be the first tree of the represented forest of $s$ that contains $x$, and let $p$ be the path in $t$ from the root down to the first occurrence of $x$ in preorder. Then every edge of $p$, taken as the pair (parent, child) of consecutive vertices of $p$, is a preferred edge of the state obtained from $s$ by accessing $x$.","title":"Access makes the root path preferred"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/access-forest","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.access_forest","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#access-forest","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every link-cut tree state $s$ and every vertex $x$, accessing $x$ changes only the preferred edges: the represented forest of the resulting state equals the represented forest of $s$.","title":"Access leaves the represented forest unchanged"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/access-last","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.access_last","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#access-last","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a valid link-cut tree state and let $x$ be a vertex of its represented forest. Then, after accessing $x$, the preferred-path decomposition of the new state contains a path $p$ whose last vertex is $x$, and $p$ is the path from the root down to $x$ in some tree of the represented forest of $s$.","title":"After access, the accessed vertex ends its preferred path"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/access-2","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.Valid.access","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#access-2","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If a link-cut tree state $s$ is valid, then for every vertex $x$ the state obtained from $s$ by accessing $x$ is again valid.","title":"Access preserves validity"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/make-root","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.makeRoot","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#make-root","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Given a vertex $x$ and a state $s$, the state $\\operatorname{makeRoot}(x, s)$ has no preferred edges (all edges dashed), and its forest is defined as follows. If $x$ is not a vertex of the forest, the forest is unchanged. Otherwise the first represented tree containing $x$ is rerooted at $x$: the edges on the path from its old root to $x$ are reversed, and when the tree has distinct vertices every undirected edge\u2026","title":"Making a vertex the root in a link-cut tree state"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/make-root-root","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.makeRoot_root","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#make-root-root","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state and $x$ a vertex of its represented forest. After making $x$ the root, some represented tree has root $x$ and contains $x$ as a vertex.","title":"Making a vertex the root installs a tree rooted there"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/connected-make-root","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.connected_makeRoot","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#connected-make-root","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a valid link-cut tree state and let $x$ and $y$ be connected in $s$. Then $x$ and $y$ are still connected in the state obtained from $s$ by making $x$ the root of its represented tree.","title":"Making a vertex the root keeps its connections"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/make-root-2","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.Valid.makeRoot","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#make-root-2","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every valid link-cut tree state $s$ and every vertex $x$, the state obtained from $s$ by making $x$ the root of its represented tree is valid.","title":"Rerooting a link-cut tree state preserves validity"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/link","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.link","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#link","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Given vertices $x$ and $y$ and a state $s$, the state $\\operatorname{link}(x, y, s)$ has no preferred edges (all edges dashed), and its forest is defined as follows. If $x$ or $y$ is not a vertex of the forest, the forest is unchanged. Otherwise let $t_x$ and $t_y$ be the first represented trees containing $x$ and $y$. If $x$ is a vertex of $t_y$, the forest is unchanged. Otherwise $t_x$ is rerooted at $x$ (the\u2026","title":"Linking two trees of a link-cut tree state"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/connected-link","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.connected_link","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#connected-link","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state and let $x$ and $y$ be vertices of its represented forest. Then in the state obtained from $s$ by linking $x$ to $y$, the vertices $x$ and $y$ are connected, that is, some represented tree contains both. No invariant on the state is needed.","title":"Linking connects the endpoints"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/link-2","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.Valid.link","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#link-2","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For every valid link-cut tree state $s$ and all vertices $x$ and $y$, the state obtained from $s$ by linking $x$ and $y$ is valid.","title":"Linking in a link-cut tree preserves validity"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/cut","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.cut","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#cut","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Cutting at a vertex $x$ resets the preferred edges of the state to the empty list and changes its forest as follows. If no tree of the forest contains $x$, the forest is unchanged. Otherwise let $T$ be the first tree containing $x$ and $S$ the first subtree of $T$ rooted at $x$ in preorder. Every tree containing $x$ is removed from the forest, and two trees are appended at the end: first the remainder of $T$ with\u2026","title":"Cutting off the subtree at a vertex"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/not-connected-cut","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.not_connected_cut","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#not-connected-cut","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state, let $t$ be the first tree of its represented forest containing the vertex $x$, and let $u$ be the first subtree of $t$ in preorder whose root is $x$. After cutting $s$ at $x$, the vertex $x$ is not connected to any vertex $y$ that is not a vertex of $u$.","title":"A cut disconnects a vertex from everything outside its subtree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/mem-cut","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.mem_cut","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#mem-cut","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state, let $x$ and $y$ be vertices, and let $t$ be the first tree of the represented forest of $s$ that contains $x$. Suppose no vertex occurs twice in $t$, and let $T_x$ be the subtree of $t$ rooted at $x$. If $y$ is a vertex of $t$ but not of $T_x$, then after cutting $s$ at $x$ both $x$ and $y$ are vertices of the represented forest.","title":"Both sides of a cut remain in the forest"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/cut-2","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.Valid.cut","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#cut-2","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If $s$ is a valid link-cut tree state, then for every vertex $x$ the state obtained from $s$ by cutting off the subtree rooted at $x$ is again valid.","title":"Cutting preserves validity"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/cut-edge","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.cutEdge","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#cut-edge","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"For vertices $x$ and $y$ and a link-cut tree state $s$, cutting the edge between $x$ and $y$ works as follows. If $x$ and $y$ are adjacent in some tree of the represented forest of $s$, the result is obtained from $s$ by first making $x$ the root of its represented tree and then cutting off the subtree rooted at $y$. Otherwise the represented forest is left unchanged. In both cases the list of preferred edges\u2026","title":"Cutting an edge of a link-cut tree"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/cut-edge-eq-cut-make-root-of-adjacent","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.cutEdge_eq_cut_makeRoot_of_adjacent","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#cut-edge-eq-cut-make-root-of-adjacent","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state and let $x$ and $y$ be adjacent in some tree of the represented forest of $s$. Then cutting the edge between $x$ and $y$ in $s$ gives the same state as first making $x$ the root of its represented tree and then cutting off the subtree rooted at $y$.","title":"Edge cut on an adjacent pair"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/cut-edge-eq-of-not-adjacent","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.cutEdge_eq_of_not_adjacent","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#cut-edge-eq-of-not-adjacent","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a link-cut tree state and let $x$ and $y$ be adjacent in no tree of the represented forest of $s$. Then cutting the edge between $x$ and $y$ in $s$ leaves the represented forest unchanged and makes the list of preferred edges empty.","title":"Edge cut on a non-adjacent pair"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/cut-edge-2","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.Valid.cutEdge","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#cut-edge-2","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"If $s$ is a valid link-cut tree state, then for all vertices $x$ and $y$ the state obtained from $s$ by cutting the edge between $x$ and $y$ is again valid.","title":"Edge cutting preserves validity"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/not-connected-cut-edge","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.not_connected_cutEdge","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#not-connected-cut-edge","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a valid link-cut tree state and let $x$ and $y$ be adjacent in some tree of the represented forest of $s$. After the edge cut of $x$ and $y$ in $s$, the vertices $x$ and $y$ are not connected: no represented tree contains both.","title":"Cutting an edge separates its endpoints"},{"id":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/adjacent-cut-edge","kind":"node","lean":"Cslib.Algorithms.Lean.LinkCutTree.State.adjacent_cutEdge","parent":38347,"result":true,"route":"../roadmap/ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree/index.html#adjacent-cut-edge","scope":"ComputerScience/data-structures/disjoint-sets-and-dynamic-trees/link-cut-tree","status":"fully_proved","summary":"Let $s$ be a valid link-cut tree state and let $x$ and $y$ be adjacent in some tree of its represented forest. If $a$ and $b$ are adjacent in some tree of the represented forest of $s$, and neither $a = x$ and $b = y$ nor $a = y$ and $b = x$, then $a$ and $b$ are adjacent in some tree of the represented forest obtained from $s$ by cutting the edge between $x$ and $y$.","title":"An edge cut keeps every other edge"},{"data_url":"scopes/ComputerScience/graph-algorithms.json","id":"ComputerScience/graph-algorithms","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/index.html","scope":"ComputerScience/graph-algorithms","status":"fully_proved","summary":"Algorithms computing properties of explicit graphs (traversal, connectivity, shortest paths, spanning trees, flows, matchings, Euler circuits) with correctness and cost proofs; pure graph theory belongs to GraphTheory, and verifiers or reductions for NP problems belong to TheoryOfComputation.","title":"Graph algorithms"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity","kind":"scope","parent":38431,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity","status":"fully_proved","summary":"Graph representations as adjacency lists and adjacency matrices, breadth-first and depth-first search with predecessor trees, strongly connected components, topological sorting, transitive closure and Hierholzer's Euler circuits.","title":"Graph traversal and connectivity"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","title":"Adjacency list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/directed-adjacency","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#directed-adjacency","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"A directed adjacency graph on a type $V$ consists of a list of vertices that has no duplicates and contains every element of $V$ (so $V$ is finite), together with, for each vertex $v$, an ordered list of the out-neighbors of $v$, in which entries may repeat.","title":"Directed adjacency graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/edge","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.Edge","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For a directed adjacency graph $G$ on $V$ and vertices $s, t \\in V$, there is an edge from $s$ to $t$ in $G$ when $t$ occurs in the adjacency list of $s$.","title":"Edge of a directed adjacency graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/walk","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.Walk","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#walk","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For a directed adjacency graph $G$, a natural number $n$ and vertices $s$ and $t$, there is a walk of length $n$ from $s$ to $t$ in $G$ when there are vertices $v_0, \\dots, v_n$ with $v_0 = s$ and $v_n = t$ such that $(v_i, v_{i+1})$ is an edge of $G$ for every $i < n$.","title":"Walks of a given length"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/to-digraph","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.toDigraph","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#to-digraph","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"The digraph associated with a directed adjacency graph $G$ on a vertex type $V$ is the directed graph on $V$ in which $s$ is adjacent to $t$ exactly when $G$ has an edge from $s$ to $t$. It forgets the order of the adjacency lists and the multiplicity of their entries.","title":"The digraph of a directed adjacency graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/to-digraph-adj","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.toDigraph_adj","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#to-digraph-adj","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For a directed adjacency graph $G$ and vertices $s$ and $t$, $s$ is adjacent to $t$ in the digraph associated with $G$ if and only if $G$ has an edge from $s$ to $t$, that is, $t$ occurs in the adjacency list of $s$.","title":"Adjacency in the associated digraph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/walk-zero-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.walk_zero_iff","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#walk-zero-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"In a directed adjacency graph, there is a walk of length $0$ from a vertex $s$ to a vertex $t$ if and only if $s = t$.","title":"Walks of length zero"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/walk-succ-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.walk_succ_iff","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#walk-succ-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"In a directed adjacency graph $G$, for every natural number $n$ and all vertices $s$ and $t$, there is a walk of length $n + 1$ from $s$ to $t$ if and only if there is a vertex $v$ such that $(s, v)$ is an edge of $G$ and there is a walk of length $n$ from $v$ to $t$.","title":"Splitting off the first edge of a walk"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/append","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.Walk.append","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#append","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph on $V$ and let $s, m, t \\in V$. If $G$ has a walk of length exactly $a$ from $s$ to $m$ and a walk of length exactly $b$ from $m$ to $t$, then $G$ has a walk of length exactly $a + b$ from $s$ to $t$.","title":"Concatenation of walks"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/reachable","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.Reachable","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#reachable","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For a directed adjacency graph $G$ and vertices $s$ and $t$, $t$ is reachable from $s$ when the pair $(s, t)$ lies in the reflexive-transitive closure of the edge relation of $G$: either $s = t$, or a finite sequence of edges of $G$ leads from $s$ to $t$.","title":"Reachability in a directed graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/reachable-iff-exists-walk","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.reachable_iff_exists_walk","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#reachable-iff-exists-walk","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"In a directed adjacency graph $G$, a vertex $t$ is reachable from a vertex $s$ if and only if there is a natural number $n$ such that $G$ has a walk of length exactly $n$ from $s$ to $t$.","title":"Reachability by walks"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/reachable-iff-refl-trans-gen","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.reachable_iff_reflTransGen","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#reachable-iff-refl-trans-gen","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"In a directed adjacency graph $G$, a vertex $t$ is reachable from a vertex $s$ if and only if $(s, t)$ lies in the reflexive-transitive closure of the edge relation of $G$.","title":"Reachability is the reflexive-transitive closure of the edge relation"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/to-lts","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.toLTS","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#to-lts","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"The labelled transition system of a directed adjacency graph $G$ has the vertices of $G$ as states and a single label, and it has a transition from $s$ to $t$ exactly when $G$ has an edge from $s$ to $t$.","title":"The transition system of a directed adjacency graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/reachable-iff-lts","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.reachable_iff_lts","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#reachable-iff-lts","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"In a directed adjacency graph $G$, a vertex $t$ is reachable from a vertex $s$ if and only if, in the unit-labelled labelled transition system of $G$, there is a finite list of labels along which $s$ reaches $t$ by a multistep transition.","title":"Reachability in the associated transition system"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/transpose","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.transpose","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#transpose","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For a directed adjacency graph $G$ on a vertex type with decidable equality, the transpose of $G$ has the same vertex enumeration as $G$, and the adjacency list of a vertex $v$ consists of the vertices $u$ such that $v$ occurs in the adjacency list of $u$ in $G$, each listed once and in the order of the vertex enumeration.","title":"Transpose of a directed adjacency graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/edge-transpose","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.edge_transpose","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#edge-transpose","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"Let $V$ have decidable equality and let $G$ be a directed adjacency graph on $V$. For all $s, t \\in V$, the transpose of $G$ has an edge from $s$ to $t$ if and only if $G$ has an edge from $t$ to $s$.","title":"Edges of the transpose graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/walk-transpose-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.walk_transpose_iff","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#walk-transpose-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For a directed adjacency graph $G$ on a vertex type with decidable equality, every natural number $n$ and all vertices $s$ and $t$, there is a walk of length $n$ from $s$ to $t$ in the transpose of $G$ if and only if there is a walk of length $n$ from $t$ to $s$ in $G$.","title":"Walks in the transpose are reversed walks"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/reachable-transpose-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.reachable_transpose_iff","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#reachable-transpose-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph on a vertex type with decidable equality, and let $G^{\\mathsf{T}}$ be its transpose. For all vertices $s$ and $t$, $t$ is reachable from $s$ in $G^{\\mathsf{T}}$ if and only if $s$ is reachable from $t$ in $G$.","title":"Reachability in the transpose graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/semantically-equivalent","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.SemanticallyEquivalent","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#semantically-equivalent","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"Two directed adjacency graphs $G_1$ and $G_2$ on the same vertex type are semantically equivalent when they have the same edges: for all vertices $s$ and $t$, $G_1$ has an edge from $s$ to $t$ if and only if $G_2$ does.","title":"Semantic equivalence of directed adjacency graphs"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/semantic-equivalent-refl","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.semanticEquivalent_refl","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#semantic-equivalent-refl","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"Every directed adjacency graph $G$ is semantically equivalent to itself.","title":"Semantic equivalence of directed adjacency graphs is reflexive"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/symm","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.SemanticallyEquivalent.symm","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#symm","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"If a directed adjacency graph $G_1$ is semantically equivalent to a directed adjacency graph $G_2$, then $G_2$ is semantically equivalent to $G_1$.","title":"Semantic equivalence of directed adjacency graphs is symmetric"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/trans","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.SemanticallyEquivalent.trans","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#trans","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"If a directed adjacency graph $G_1$ is semantically equivalent to $G_2$, and $G_2$ is semantically equivalent to $G_3$, then $G_1$ is semantically equivalent to $G_3$.","title":"Semantic equivalence of directed adjacency graphs is transitive"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/walk-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.SemanticallyEquivalent.walk_iff","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#walk-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"If two directed adjacency graphs $G_1$ and $G_2$ are semantically equivalent, then for every natural number $n$ and all vertices $s$ and $t$, there is a walk of length $n$ from $s$ to $t$ in $G_1$ if and only if there is a walk of length $n$ from $s$ to $t$ in $G_2$.","title":"Semantically equivalent graphs have the same walks"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/reachable-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.SemanticallyEquivalent.reachable_iff","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#reachable-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"If two directed adjacency graphs $G_1$ and $G_2$ on $V$ are semantically equivalent, then for all $s, t \\in V$, $t$ is reachable from $s$ in $G_1$ if and only if $t$ is reachable from $s$ in $G_2$.","title":"Semantically equivalent graphs have the same reachability"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/transpose-2","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.SemanticallyEquivalent.transpose","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#transpose-2","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For directed adjacency graphs $G_1$ and $G_2$ on a vertex type with decidable equality, if $G_1$ and $G_2$ are semantically equivalent, then their transposes are semantically equivalent.","title":"Transposition preserves semantic equivalence"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/semantic-equivalent-transpose-transpose","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.semanticEquivalent_transpose_transpose","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#semantic-equivalent-transpose-transpose","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph on a vertex type with decidable equality. The transpose of the transpose of $G$ is semantically equivalent to $G$, that is, it has exactly the same edges as $G$.","title":"Transposing twice gives a semantically equivalent graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/has-edge","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.hasEdge","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#has-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For a type $V$ with decidable equality, a directed adjacency graph $G$ on $V$ and vertices $s, t \\in V$, the edge test of $G$ at $(s, t)$ is the Boolean value that is true exactly when $t$ occurs in the adjacency list of $s$.","title":"Boolean edge test"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/has-edge-eq-true","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.hasEdge_eq_true","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#has-edge-eq-true","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For a type $V$ with decidable equality, a directed adjacency graph $G$ on $V$ and vertices $s, t \\in V$, the Boolean edge test of $G$ at $(s, t)$ returns true if and only if $G$ has an edge from $s$ to $t$.","title":"The Boolean edge test is correct"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/erase-edge","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.eraseEdge","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#erase-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"Given a directed-adjacency graph $G$ and vertices $s$ and $t$, erasing the edge $s \\to t$ gives the directed-adjacency graph with the same vertex enumeration as $G$ whose out-neighbour list at $s$ is the out-neighbour list of $s$ in $G$ with every occurrence of $t$ removed (the remaining entries keep their order), and whose out-neighbour list at every other vertex is the same as in $G$.","title":"Erasing a directed edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/vertices-erase-edge","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.vertices_eraseEdge","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#vertices-erase-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For every directed-adjacency graph $G$ and vertices $s$ and $t$, the graph obtained from $G$ by erasing the edge $s \\to t$ has the same vertex enumeration as $G$: the same list of vertices, in the same order.","title":"Erasing an edge keeps the vertex enumeration"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/adjacency-erase-edge-source","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.adjacency_eraseEdge_source","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#adjacency-erase-edge-source","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For every directed-adjacency graph $G$ and vertices $s$ and $t$, the out-neighbour list of $s$ in the graph obtained from $G$ by erasing the edge $s \\to t$ is the out-neighbour list of $s$ in $G$ with every occurrence of $t$ removed, the remaining entries kept in their original order.","title":"Out-neighbours of the source after erasing an edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/adjacency-erase-edge-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.adjacency_eraseEdge_of_ne","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#adjacency-erase-edge-of-ne","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For every directed-adjacency graph $G$ and vertices $s$, $t$ and $u$ with $u \\ne s$, the out-neighbour list of $u$ in the graph obtained from $G$ by erasing the edge $s \\to t$ equals the out-neighbour list of $u$ in $G$.","title":"Erasing an edge leaves the other out-neighbour lists unchanged"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/edge-erase-edge","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.edge_eraseEdge","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#edge-erase-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For every directed-adjacency graph $G$ and vertices $s$, $t$, $u$ and $v$, there is an edge from $u$ to $v$ in the graph obtained from $G$ by erasing the edge $s \\to t$ if and only if there is an edge from $u$ to $v$ in $G$ and it is not the case that both $u = s$ and $v = t$.","title":"Edges after erasing an edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/has-edge-erase-edge-self","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.hasEdge_eraseEdge_self","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#has-edge-erase-edge-self","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For every directed-adjacency graph $G$ and vertices $s$ and $t$, the Boolean edge lookup for $s \\to t$ in the graph obtained from $G$ by erasing the edge $s \\to t$ returns false.","title":"The erased edge is absent"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/erase-edge-idempotent","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.eraseEdge_idempotent","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#erase-edge-idempotent","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For every directed-adjacency graph $G$ and vertices $s$ and $t$, erasing the edge $s \\to t$ from $G$ and then erasing it again gives, at every vertex, the same out-neighbour list as erasing it once.","title":"Erasing an edge twice gives the same out-neighbour lists"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/walk-erase-edge-imp","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.walk_eraseEdge_imp","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#walk-erase-edge-imp","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For every directed-adjacency graph $G$, vertices $s$, $t$, $u$ and $v$, and natural number $n$, if there is a walk of length $n$ from $u$ to $v$ in the graph obtained from $G$ by erasing the edge $s \\to t$, then there is a walk of length $n$ from $u$ to $v$ in $G$.","title":"Walks after erasing an edge are walks in the original graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/reachable-erase-edge-imp","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.reachable_eraseEdge_imp","parent":38433,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list/index.html#reachable-erase-edge-imp","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-list","status":"fully_proved","summary":"For every directed-adjacency graph $G$ and vertices $s$, $t$, $u$ and $v$, if $v$ is reachable from $u$ in the graph obtained from $G$ by erasing the edge $s \\to t$, then $v$ is reachable from $u$ in $G$.","title":"Reachability after erasing an edge implies reachability in the original graph"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","title":"Adjacency matrix"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/adjacency-matrix","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#adjacency-matrix","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For a natural number $n$, an adjacency matrix on $n$ vertices is a directed graph on the vertex set $\\{0, \\dots, n-1\\}$ given by a Boolean-valued function $\\mathrm{adj}$ on ordered pairs of vertices: $\\mathrm{adj}(i, j)$ is true exactly when there is a directed edge from $i$ to $j$.","title":"Adjacency matrix"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/empty","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.empty","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#empty","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For every natural number $n$, the empty adjacency matrix on $n$ vertices is the directed graph on $\\{0, \\dots, n-1\\}$ with no edges: its adjacency function is false on every ordered pair of vertices.","title":"Empty adjacency matrix"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/has-edge","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.hasEdge","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#has-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on the vertex set $\\{0, \\dots, n-1\\}$ and vertices $i, j$, the edge query of $m$ at $(i, j)$ is the Boolean value that is true exactly when $m$ has a directed edge from $i$ to $j$.","title":"Edge query of an adjacency matrix"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/has-edge-empty","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.hasEdge_empty","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#has-edge-empty","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For every natural number $n$ and all vertices $i$ and $j$ among the $n$ vertices, the empty adjacency matrix on $n$ vertices has no edge from $i$ to $j$.","title":"The empty adjacency matrix has no edges"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/add-edge","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.addEdge","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#add-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on the vertex set $\\{0, \\dots, n-1\\}$ and vertices $i$ and $j$, adding the edge from $i$ to $j$ gives the adjacency matrix that has an edge from $i$ to $j$ and agrees with $m$ on every other ordered pair of vertices.","title":"Edge insertion in an adjacency matrix"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/has-edge-add-edge","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.hasEdge_addEdge","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#has-edge-add-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on $n$ vertices and vertices $i$ and $j$, the matrix obtained from $m$ by adding the edge from $i$ to $j$ has an edge from $i$ to $j$.","title":"An inserted edge is present"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/has-edge-add-edge-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.hasEdge_addEdge_of_ne","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#has-edge-add-edge-of-ne","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on $n$ vertices and vertices $i$, $j$, $a$, $b$ with $a \\ne i$ or $b \\ne j$, the matrix obtained from $m$ by adding the edge from $i$ to $j$ has an edge from $a$ to $b$ if and only if $m$ does.","title":"Inserting an edge leaves other pairs unchanged"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/has-edge-add-edge-iff","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.hasEdge_addEdge_iff","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#has-edge-add-edge-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on $n$ vertices and vertices $i$, $j$, $a$, $b$, the matrix obtained from $m$ by adding the edge from $i$ to $j$ has an edge from $a$ to $b$ if and only if $a = i$ and $b = j$, or $m$ has an edge from $a$ to $b$.","title":"Edges after inserting an edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/erase-edge","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.eraseEdge","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#erase-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on the vertex set $\\{0, \\dots, n-1\\}$ and vertices $i$ and $j$, removing the edge from $i$ to $j$ gives the adjacency matrix that has no edge from $i$ to $j$ and agrees with $m$ on every other ordered pair of vertices.","title":"Edge deletion in an adjacency matrix"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/has-edge-erase-edge","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.hasEdge_eraseEdge","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#has-edge-erase-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"Let $m$ be an adjacency matrix on the vertex set $\\{0, \\dots, n-1\\}$ and let $i, j$ be vertices. Then the graph obtained from $m$ by deleting the edge from $i$ to $j$ has no edge from $i$ to $j$.","title":"Deleting an edge removes it"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/has-edge-erase-edge-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.hasEdge_eraseEdge_of_ne","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#has-edge-erase-edge-of-ne","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"Let $m$ be an adjacency matrix on the vertex set $\\{0, \\dots, n-1\\}$ and let $i, j, a, b$ be vertices with $(a, b) \\ne (i, j)$, that is, $a \\ne i$ or $b \\ne j$. Then the graph obtained from $m$ by deleting the edge from $i$ to $j$ has an edge from $a$ to $b$ if and only if $m$ has an edge from $a$ to $b$.","title":"Deleting an edge leaves the other entries unchanged"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/has-edge-erase-edge-iff","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.hasEdge_eraseEdge_iff","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#has-edge-erase-edge-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on $n$ vertices and vertices $i$, $j$, $a$, $b$, the matrix obtained from $m$ by removing the edge from $i$ to $j$ has an edge from $a$ to $b$ if and only if $(a, b) \\ne (i, j)$ and $m$ has an edge from $a$ to $b$.","title":"Edges after deleting an edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/ext","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.ext","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#ext","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"Two adjacency matrices $m$ and $m'$ on the same $n$ vertices are equal if, for all vertices $i$ and $j$, $m$ has an edge from $i$ to $j$ exactly when $m'$ does.","title":"Extensionality of adjacency matrices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/neighbors","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.neighbors","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#neighbors","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on the vertex set $\\{0, \\dots, n-1\\}$ and a vertex $i$, the out-neighbor list of $i$ is the list of all vertices $j$ such that $m$ has a directed edge from $i$ to $j$, in increasing order.","title":"Out-neighbor list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/mem-neighbors","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.mem_neighbors","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#mem-neighbors","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"Let $m$ be an adjacency matrix on the vertex set $\\{0, \\dots, n-1\\}$ and let $i, j$ be vertices. Then $j$ belongs to the out-neighbor list of $i$ if and only if $m$ has a directed edge from $i$ to $j$.","title":"The out-neighbor list matches the edge query"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/neighbors-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.neighbors_nodup","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#neighbors-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on the vertex set $\\{0, \\dots, n-1\\}$ and a vertex $i$, no vertex occurs twice in the out-neighbor list of $i$.","title":"Out-neighbor lists have no duplicates"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/neighbors-pairwise-lt","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.neighbors_pairwise_lt","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#neighbors-pairwise-lt","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For an adjacency matrix $m$ on the vertex set $\\{0, \\dots, n-1\\}$ and a vertex $i$, the out-neighbor list of $i$ is strictly increasing: every vertex in it is smaller than every vertex that comes after it.","title":"Out-neighbor lists are strictly increasing"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/neighbors-empty","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.neighbors_empty","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#neighbors-empty","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"In the graph with no edges on the vertex set $\\{0, \\dots, n-1\\}$, the out-neighbor list of every vertex $i$ is the empty list.","title":"Out-neighbor lists of the empty graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/mem-neighbors-add-edge","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.mem_neighbors_addEdge","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#mem-neighbors-add-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"Let $m$ be an adjacency matrix on the vertex set $\\{0, \\dots, n-1\\}$ and let $i, j$ be vertices. Then $j$ belongs to the out-neighbor list of $i$ in the graph obtained from $m$ by adding the edge from $i$ to $j$.","title":"Adding an edge makes its target an out-neighbor"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/not-mem-neighbors-erase-edge","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.not_mem_neighbors_eraseEdge","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#not-mem-neighbors-erase-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, given by its adjacency matrix, and let $i$ and $j$ be vertices. After the edge from $i$ to $j$ is deleted from $G$ (whether or not $G$ had that edge), $j$ is not an out-neighbor of $i$.","title":"Deleting an edge removes the out-neighbor"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/mem-neighbors-add-edge-iff","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.mem_neighbors_addEdge_iff","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#mem-neighbors-add-edge-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"Let $m$ be an adjacency matrix on the vertex set $\\{0, \\dots, n-1\\}$ and let $i, j, b$ be vertices. Then $b$ belongs to the out-neighbor list of $i$ in the graph obtained from $m$ by adding the edge from $i$ to $j$ if and only if $b = j$ or $b$ belongs to the out-neighbor list of $i$ in $m$.","title":"Out-neighbors after adding an edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/out-degree","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.outDegree","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#out-degree","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"The out-degree of a vertex $i$ of a directed graph given by an adjacency matrix is the number of its out-neighbors, that is, the number of vertices $j$ such that there is an edge from $i$ to $j$.","title":"Out-degree"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/out-degree-empty","kind":"node","lean":"Cslib.Algorithms.Lean.AdjacencyMatrix.outDegree_empty","parent":38470,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix/index.html#out-degree-empty","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/adjacency-matrix","status":"fully_proved","summary":"For every $n$, every vertex of the empty directed graph on the vertices $0, \\dots, n-1$ has out-degree $0$.","title":"Vertices of the empty graph have out-degree zero"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","title":"Breadth-first search"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfsresult","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.BFSResult","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfsresult","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"A breadth-first search result on a vertex type $V$ consists of a distance function, assigning to each vertex either a natural number or none, and a list of vertices. They are meant to record the shortest distance from the source (none when the vertex is unreachable) and the vertices in first-discovery order.","title":"Result of breadth-first search"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfs-model","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.bfsModel","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfs-model","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Given a finite directed graph $G$ with ordered adjacency lists and a source vertex $s$, the list model keeps a list of processed pairs and a queue of pairs $(v, d)$ of a vertex and a distance, starting with no processed pairs and the queue $[(s, 0)]$. It performs one round per vertex of $G$: if the queue is empty, nothing changes; otherwise the front pair $(u, d)$ is removed from the queue and recorded as\u2026","title":"List model of breadth-first search"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfs","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.bfs","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfs","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Given a finite directed graph $G$ with ordered adjacency lists and a source vertex $s$, breadth-first search keeps a FIFO queue of pairs $(v, d)$ of a vertex and a distance and a table of discovered vertices with their distances, starting with the queue $[(s, 0)]$ and $s$ discovered at distance $0$. It performs one round per vertex of $G$: if the queue is empty, nothing changes; otherwise it dequeues the front pair\u2026","title":"Breadth-first search"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfs-eq-bfs-model","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.bfs_eq_bfsModel","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfs-eq-bfs-model","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ with ordered adjacency lists and every source vertex $s$, the result of the executable breadth-first search from $s$ equals the result of the list model of breadth-first search from $s$: both report the same distance (or none) for every vertex and the same first-discovery order.","title":"The executable BFS agrees with the list model"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfs-fin","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.bfsFin","parent":38494,"reading_title":{"format":"inline-parts-v1","parts":[["text","Breadth-first search on the vertices "],["math","\\(0, \\dots, n-1\\)"]],"source":"Breadth-first search on the vertices $0, \\dots, n-1$"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfs-fin","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Given a natural number $n$, an adjacency function assigning to each vertex $i < n$ an ordered list of vertices among $0, \\dots, n-1$, and a source vertex $s < n$, this is the result of breadth-first search from $s$ on the finite directed graph with vertex enumeration $0, 1, \\dots, n-1$ and these adjacency lists.","title":"Breadth-first search on the vertices $0, \\dots, n-1$"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/distance-present-iff-reachable","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.distance_present_iff_reachable","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#distance-present-iff-reachable","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ with ordered adjacency lists and all vertices $s$ and $v$, breadth-first search from $s$ reports some distance for $v$ (rather than none) if and only if $v$ is reachable from $s$ in $G$.","title":"BFS reports a distance exactly for the reachable vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/order-mem-iff-reachable","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.order_mem_iff_reachable","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#order-mem-iff-reachable","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ with ordered adjacency lists and all vertices $s$ and $v$, the vertex $v$ occurs in the first-discovery order of breadth-first search from $s$ if and only if $v$ is reachable from $s$ in $G$.","title":"BFS lists exactly the reachable vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/order-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.order_nodup","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#order-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ with ordered adjacency lists and every vertex $s$, no vertex occurs twice in the first-discovery order of breadth-first search from $s$.","title":"BFS discovers each vertex at most once"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/distance-eq-some-iff-shortest","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.distance_eq_some_iff_shortest","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#distance-eq-some-iff-shortest","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ with ordered adjacency lists, all vertices $s$ and $v$, and every natural number $d$, breadth-first search from $s$ reports distance $d$ for $v$ if and only if there is a walk of exactly $d$ edges from $s$ to $v$ in $G$ and $d \\le k$ for every natural number $k$ such that there is a walk of exactly $k$ edges from $s$ to $v$.","title":"BFS reports exactly the shortest walk length"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/source-distance","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.source_distance","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#source-distance","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ given by ordered adjacency lists and every vertex $s$ of $G$, breadth-first search in $G$ from $s$ reports a distance for $s$ itself, and that distance is $0$ edges.","title":"Breadth-first search places the source at distance zero"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/order-distances-nondecreasing","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.order_distances_nondecreasing","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#order-distances-nondecreasing","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ with ordered adjacency lists and every vertex $s$, the distances reported by breadth-first search from $s$, listed along its first-discovery order, form a nondecreasing sequence, where a vertex without a reported distance counts as $0$.","title":"BFS discovers vertices in nondecreasing distance"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfsforest","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.BFSForest","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfsforest","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"A breadth-first search forest on a vertex type $V$ consists of a breadth-first search result, namely a partial distance function from $V$ to the natural numbers and a list of vertices, together with a partial predecessor function from $V$ to $V$. For a search from a source vertex $s$, these are meant to give the distance from $s$ of each reachable vertex, the reachable vertices in first-discovery order, and the\u2026","title":"Breadth-first search forest"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfs-with-parents","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.bfsWithParents","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfs-with-parents","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Given a finite directed graph $G$ on a vertex type $V$, with ordered adjacency lists, and a source vertex $s$, this runs first-in-first-out breadth-first search from $s$ for $|V|$ rounds. The queue starts with $s$ at distance $0$. Each round tries to dequeue a vertex $u$; if it succeeds, it scans the neighbors of $u$ in adjacency-list order, and each neighbor not yet discovered gets distance one more than $u$,\u2026","title":"Breadth-first search with parent pointers"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/path-to","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.BFSForest.pathTo?","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#path-to","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $p$ be the predecessor function of a breadth-first search forest, let $n$ be the length of its discovery order, and let $s$ and $t$ be vertices. Starting from $v_k = t$, this walks back along $p$, setting $v_{i-1} = p(v_i)$, until it reaches a vertex $v_0$ that tests equal to $s$, and returns the list $v_0, v_1, \\dots, v_k$. It returns no list if the walk meets a vertex that does not test equal to $s$ and has\u2026","title":"Path reconstruction from BFS parents"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfs-with-parents-ret","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.bfsWithParents_ret","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfs-with-parents-ret","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ with ordered adjacency lists and every source vertex $s$, the output of breadth-first search with parents from $s$ in $G$, with its cost and its predecessor table forgotten, equals the result of ordinary FIFO breadth-first search from $s$ in $G$: the same distance table and the same first-discovery order.","title":"Breadth-first search with parents returns the ordinary BFS result"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/predecessor-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.predecessor_eq_none_iff","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#predecessor-eq-none-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph with ordered adjacency lists, $s$ a vertex, and $F$ the predecessor forest computed by breadth-first search with parents from $s$ in $G$. A vertex $v$ has no recorded predecessor in $F$ if and only if $v = s$ or $v$ is not reachable from $s$ along directed edges of $G$.","title":"Exactly the source and the unreachable vertices have no BFS parent"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/predecessor-edge","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.predecessor_edge","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#predecessor-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph with ordered adjacency lists, $s$ a vertex, and $F$ the predecessor forest computed by breadth-first search with parents from $s$ in $G$. If $F$ records $p$ as the predecessor of a vertex $v$, then $G$ has a directed edge from $p$ to $v$, that is, $v$ occurs in the adjacency list of $p$.","title":"BFS parents are joined by graph edges"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/predecessor-distance","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.predecessor_distance","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#predecessor-distance","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph with ordered adjacency lists, $s$ a vertex, and $F$ the predecessor forest computed by breadth-first search with parents from $s$ in $G$. If $F$ records $p$ as the predecessor of a vertex $v$, then there is a natural number $d$ such that $F$ records distance $d$ for $p$ and distance $d + 1$ for $v$.","title":"A BFS parent is one step closer to the source"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/predecessor-eq-find","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.predecessor_eq_find?","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#predecessor-eq-find","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph with ordered adjacency lists, $s$ a vertex, and $F$ the predecessor forest computed by breadth-first search with parents from $s$ in $G$. For every vertex $v \\ne s$, the predecessor that $F$ records for $v$ is the first vertex $u$, in the first-discovery order of ordinary FIFO breadth-first search from $s$ in $G$, whose adjacency list contains $v$; if no vertex in that order has\u2026","title":"The BFS parent is the first discovered in-neighbour"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/predecessor-chain-unique","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.predecessor_chain_unique","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#predecessor-chain-unique","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by adjacency lists, and let $s$ and $v$ be vertices with $v$ reachable from $s$ in $G$. There is exactly one list of vertices that starts at $s$, ends at $v$, and in which each vertex after the first has the preceding vertex as its saved parent in the breadth-first search forest of $G$ from $s$.","title":"Each reachable vertex has a unique saved-parent chain"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/predecessor-chain-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.predecessor_chain_nodup","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#predecessor-chain-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by adjacency lists and let $s$ and $v$ be vertices. If a list of vertices starts at $s$, ends at $v$, and each of its vertices after the first has the preceding vertex as its saved parent in the breadth-first search forest of $G$ from $s$, then no vertex occurs twice in the list.","title":"Saved-parent chains are simple"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/predecessor-relation-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.predecessor_relation_acyclic","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#predecessor-relation-acyclic","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by adjacency lists and let $s$ and $v$ be vertices. In the breadth-first search forest of $G$ from $s$, there is no sequence $v = x_0, x_1, \\dots, x_k = v$ with $k \\ge 1$ in which each $x_i$ is the saved parent of $x_{i+1}$.","title":"The saved-parent relation is acyclic"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/path-to-source","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.BFSForest.pathTo?_source","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#path-to-source","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"For every breadth-first search forest $F$ and every vertex $s$, following the saved parents of $F$ from $s$ back to $s$ returns the one-vertex path $[s]$. This holds for every forest, including one whose discovery order is empty.","title":"Path reconstruction at the source"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/path-to-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.BFSForest.pathTo?_eq_none_iff","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#path-to-eq-none-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by adjacency lists, and let $s$ and $v$ be vertices. Following the saved parents of the breadth-first search forest of $G$ from $s$ back from $v$ towards $s$ returns no path if and only if $v$ is not reachable from $s$ in $G$.","title":"Path reconstruction fails exactly at unreachable vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/path-to-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.BFSForest.pathTo?_spec","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#path-to-spec","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by adjacency lists, let $s$ and $v$ be vertices, and suppose that following the saved parents of the breadth-first search forest of $G$ from $s$ back from $v$ returns a list $P$ of vertices. Then $P$ starts at $s$ and ends at $v$, each vertex of $P$ after the first has the preceding vertex as its saved parent, $G$ has an edge from each vertex of $P$ to the next, no vertex\u2026","title":"Reconstructed BFS paths are shortest paths"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/path-to-time","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.BFSForest.pathTo?_time","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#path-to-time","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by adjacency lists, let $s$ and $v$ be vertices, and suppose breadth-first search from $s$ reports distance $d$ for $v$. Following the saved parents of the breadth-first search forest of $G$ from $s$ back from $v$ costs exactly $3d + 2$, counting one unit for each comparison with $s$, each parent lookup and each vertex prepended to the path.","title":"Cost of path reconstruction for a reachable vertex"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/path-to-time-of-not-reachable","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.BFSForest.pathTo?_time_of_not_reachable","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#path-to-time-of-not-reachable","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by adjacency lists, and let $s$ and $v$ be vertices with $v$ not reachable from $s$ in $G$. Following the saved parents of the breadth-first search forest of $G$ from $s$ back from $v$ costs exactly $2$, counting one unit for each comparison with $s$, each parent lookup and each vertex prepended to the path: one comparison of $v$ with $s$ and one lookup of the missing parent\u2026","title":"Cost of path reconstruction for an unreachable vertex"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfs-with-parents-time","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.bfsWithParents_time","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfs-with-parents-time","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph with vertex list $V$ and adjacency lists $\\mathrm{adj}(u)$, let $s$ be a vertex, and let $L$ be the list of vertices in the order in which breadth-first search from $s$ discovers them. The cost of breadth-first search with parents from $s$, counting dequeue attempts, adjacency checks, enqueues and parent writes, equals $|V| + \\sum_{u \\in L} |\\mathrm{adj}(u)| + |L| + (|L| - 1)$,\u2026","title":"Exact cost of BFS with parents"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/bfs-with-parents-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.BreadthFirstSearch.bfsWithParents_time_le","parent":38494,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search/index.html#bfs-with-parents-time-le","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/breadth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph with vertex list $V$ and adjacency lists $\\mathrm{adj}(u)$, and let $s$ be a vertex. The cost of breadth-first search with parents from $s$, counting dequeue attempts, adjacency checks, enqueues and parent writes, is at most $3|V| + \\sum_{u \\in V} |\\mathrm{adj}(u)|$, where adjacency lists are counted with repetitions.","title":"Linear bound on the cost of BFS with parents"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","title":"Depth-first search"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/traversal","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.Traversal","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#traversal","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"A traversal over a vertex type $V$ consists of three lists: a preorder (discovery order) list of vertices, a postorder (finish order) list of vertices, and a list of parent pairs in $V \\times V$, each stored as $(\\text{child}, \\text{parent})$.","title":"Depth-first traversal record"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/dfs","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.dfs","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#dfs","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Given a directed graph $G$ with ordered adjacency lists and an ordered list of roots, depth-first search processes the roots from left to right, skipping roots already visited. Visiting an unvisited vertex $v$ marks it visited and records its discovery, then processes the out-neighbours of $v$ in adjacency-list order, visiting each one not yet visited at that moment and recording $v$ as its parent, and finally\u2026","title":"Left-first depth-first search"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/steps","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.Internal.Steps","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#steps","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Zero or more independent DFS machine steps.","title":"Sequences of depth-first search machine steps"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/is-left-first-dfs","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.IsLeftFirstDFS","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#is-left-first-dfs","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by ordered adjacency lists and let $r = (r_1, \\dots, r_k)$ be a list of root vertices. A traversal $T$ is a left-first DFS of $G$ from $r$ when some finite run of the following machine ends with an empty work list and with recorded preorder, postorder and (child, parent) list equal to those of $T$.","title":"Left-first depth-first traversals"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/dfs-is-left-first","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.dfs_is_left_first","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#dfs-is-left-first","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For every directed graph $G$ with ordered adjacency lists and every ordered list of roots, the traversal computed by left-first depth-first search of $G$ from these roots conforms to left-first DFS for $G$ and these roots.","title":"Executable DFS meets the left-first specification"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/eq-dfs","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.IsLeftFirstDFS.eq_dfs","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#eq-dfs","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a directed graph with ordered adjacency lists, fix an ordered list of roots, and let $T$ be a traversal. If $T$ conforms to left-first DFS for $G$ and these roots, then $T$ equals the traversal computed by left-first depth-first search of $G$ from these roots.","title":"The left-first DFS specification determines the traversal"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/preorder-sound","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.preorder_sound","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#preorder-sound","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by ordered adjacency lists and let $R$ be a list of roots. Every vertex in the preorder (discovery order) of the depth-first search of $G$ from $R$ is reachable in $G$ from some root in $R$ by a directed path (possibly of length zero).","title":"Depth-first search discovers only reachable vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/preorder-complete","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.preorder_complete","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#preorder-complete","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by ordered adjacency lists and let $R$ be a list of roots. If a vertex $v$ is reachable in $G$ from some root in $R$ by a directed path (possibly of length zero), then $v$ occurs in the preorder (discovery order) of the depth-first search of $G$ from $R$.","title":"Depth-first search discovers every reachable vertex"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/mem-preorder-iff-reachable","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.mem_preorder_iff_reachable","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#mem-preorder-iff-reachable","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices. A vertex $v$ occurs in the discovery order (preorder) of the depth-first search of $G$ from $R$ if and only if there is a root $r$ in $R$ from which $v$ is reachable in $G$ by a directed path of length zero or more.","title":"Depth-first search discovers exactly the reachable vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/preorder-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.preorder_nodup","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#preorder-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ given by ordered adjacency lists and every list $R$ of roots, the preorder (discovery order) of the depth-first search of $G$ from $R$ contains no vertex twice.","title":"Depth-first search discovers each vertex at most once"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/mem-postorder-iff-reachable","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.mem_postorder_iff_reachable","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#mem-postorder-iff-reachable","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices. A vertex $v$ occurs in the finish order (postorder) of the depth-first search of $G$ from $R$ if and only if there is a root $r$ in $R$ from which $v$ is reachable in $G$ by a directed path of length zero or more.","title":"Depth-first search finishes exactly the reachable vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/postorder-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.postorder_nodup","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#postorder-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices. The finish order (postorder) of the depth-first search of $G$ from $R$ contains no vertex twice.","title":"DFS finishes each vertex at most once"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/before","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.Before","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#before","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For elements $x$ and $y$ and a list $xs$, $x$ occurs before $y$ in $xs$ when there are lists $a$, $b$ and $c$ such that $xs$ is the concatenation of $a$, the element $x$, $b$, the element $y$ and $c$; that is, some occurrence of $x$ lies at a strictly earlier position than some occurrence of $y$.","title":"Occurring before in a list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/parent-rel","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.ParentRel","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#parent-rel","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Given a list $P$ of (child, parent) pairs, $p$ is a parent of $c$ in $P$ when the pair $(c, p)$ occurs in $P$.","title":"Parent relation of a list of pairs"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/forest-roots","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.Traversal.forestRoots","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#forest-roots","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"The forest roots of a traversal $T$ are the vertices of the preorder of $T$ that do not occur as the child in any (child, parent) pair recorded by $T$, listed in the order in which they occur in the preorder.","title":"Roots of a DFS forest"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/forest-roots-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.forestRoots_nodup","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#forest-roots-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ given by ordered adjacency lists and every list $r$ of root vertices, the list of forest roots of the left-first depth-first search of $G$ from $r$ contains no vertex twice.","title":"DFS forest roots are distinct"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/forest-root-has-no-parent","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.forestRoot_has_no_parent","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#forest-root-has-no-parent","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by ordered adjacency lists, let $r$ be a list of root vertices, and let $v$ and $p$ be vertices. If $v$ is a forest root of the left-first depth-first search of $G$ from $r$, then $(v, p)$ is not among the (child, parent) pairs recorded by that search.","title":"A DFS forest root has no parent"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/parent-edge","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.parent_edge","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#parent-edge","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices, and run depth-first search on $G$ from $R$. If the pair $(c, p)$ occurs in the list of (child, parent) pairs recorded by the search, then $G$ has a directed edge from $p$ to $c$.","title":"DFS parent pairs are graph edges"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/parent-children-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.parent_children_nodup","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#parent-children-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices, and run depth-first search on $G$ from $R$. In the list of (child, parent) pairs recorded by the search, no vertex occurs as the child of more than one entry; that is, the list of children has no repeated entries.","title":"No vertex is recorded twice as a DFS child"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/mem-forest-roots-or-exists-unique-parent","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.mem_forestRoots_or_existsUnique_parent","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#mem-forest-roots-or-exists-unique-parent","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by ordered adjacency lists, let $r$ be a list of root vertices, and let $v$ be a vertex in the preorder of the left-first depth-first search of $G$ from $r$. Then $v$ is a forest root of that search, or there is exactly one vertex $p$ such that $(v, p)$ is among the (child, parent) pairs recorded by the search.","title":"Every discovered vertex is a forest root or has a unique parent"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/parent-before-child","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.parent_before_child","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#parent-before-child","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices, and run depth-first search on $G$ from $R$. If the pair $(c, p)$ occurs in the list of (child, parent) pairs recorded by the search, then $p$ occurs strictly before $c$ in the discovery order (preorder) of the search.","title":"In preorder, a DFS parent precedes its child"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/child-before-parent-postorder","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.child_before_parent_postorder","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#child-before-parent-postorder","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a directed graph with ordered adjacency lists and fix an ordered list of roots. If $(c, p)$ is one of the (child, parent) pairs recorded by the left-first depth-first search of $G$ from these roots, then $c$ occurs before $p$ in the postorder (finish order) of that search.","title":"A DFS child finishes before its parent"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/decreasing-finish","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.decreasingFinish","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#decreasing-finish","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For a directed graph $G$ with ordered adjacency lists and an ordered list of roots, the decreasing finish order is the reverse of the postorder of the left-first depth-first search of $G$ from those roots: the vertices that search visits, listed from the latest finished to the earliest finished.","title":"Decreasing finish order"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/parent-before-child-decreasing-finish","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.parent_before_child_decreasingFinish","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#parent-before-child-decreasing-finish","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices, and run depth-first search on $G$ from $R$. If the pair $(c, p)$ occurs in the list of (child, parent) pairs recorded by the search, then $p$ occurs strictly before $c$ in the decreasing finish order, the reverse of the finish order.","title":"In decreasing finish order, a DFS parent precedes its child"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/mem-decreasing-finish-iff-reachable","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.mem_decreasingFinish_iff_reachable","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#mem-decreasing-finish-iff-reachable","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by ordered adjacency lists, let $r$ be a list of root vertices, and let $v$ be a vertex. Then $v$ occurs in the decreasing finish order of the left-first depth-first search of $G$ from $r$ if and only if $v$ is reachable in $G$ from some vertex of $r$.","title":"Decreasing finish order lists exactly the vertices reachable from the roots"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/decreasing-finish-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.decreasingFinish_nodup","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#decreasing-finish-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For every directed graph $G$ with ordered adjacency lists and every ordered list of roots, the decreasing finish order of the depth-first search of $G$ from those roots contains no vertex twice.","title":"Decreasing finish order has no repeated vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/parent-chain-reaches-root","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.parent_chain_reaches_root","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#parent-chain-reaches-root","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices, and run depth-first search on $G$ from $R$. Say that $p$ is a recorded parent of $c$ when the pair $(c, p)$ occurs in the list of (child, parent) pairs recorded by the search, and call a discovered vertex with no recorded parent a forest root. Then for every vertex $v$ in the discovery order (preorder) there are $k \\ge 0$ and vertices\u2026","title":"Parent chains in a DFS forest end at a forest root"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/parent-relation-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.parent_relation_acyclic","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#parent-relation-acyclic","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of root vertices, and run depth-first search on $G$ from $R$. Say that $p$ is a recorded parent of $c$ when the pair $(c, p)$ occurs in the list of (child, parent) pairs recorded by the search. Then there are no $k \\ge 1$ and vertices $v_0, v_1, \\dots, v_k$ with $v_k = v_0$ such that each $v_{i+1}$ is a recorded parent of $v_i$.","title":"The DFS parent relation is acyclic"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/forest-root-mem-roots","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.forestRoot_mem_roots","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#forest-root-mem-roots","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph given by ordered adjacency lists, let $r$ be a list of root vertices, and let $v$ be a vertex. If $v$ is a forest root of the left-first depth-first search of $G$ from $r$, then $v$ occurs in $r$.","title":"DFS forest roots are scheduled roots"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/dfs-all","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.dfsAll","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#dfs-all","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For a directed graph $G$ with ordered adjacency lists, the whole-graph traversal is the left-first depth-first search of $G$ whose roots are all vertices of $G$, taken in the graph's fixed vertex enumeration order.","title":"Depth-first search of the whole graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/mem-dfs-all-preorder","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.mem_dfsAll_preorder","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#mem-dfs-all-preorder","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ given by ordered adjacency lists and every vertex $v$ of $G$, $v$ occurs in the preorder of the whole-graph depth-first search of $G$, that is, the left-first depth-first search of $G$ whose roots are all vertices of $G$ in the graph's vertex order.","title":"Whole-graph DFS discovers every vertex"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/mem-dfs-all-postorder","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.mem_dfsAll_postorder","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#mem-dfs-all-postorder","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ given by ordered adjacency lists and every vertex $v$ of $G$, $v$ occurs in the postorder of the whole-graph depth-first search of $G$, that is, the left-first depth-first search of $G$ whose roots are all vertices of $G$ in the graph's vertex order.","title":"Whole-graph DFS finishes every vertex"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/decreasing-finish-all","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.decreasingFinishAll","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#decreasing-finish-all","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For a directed graph $G$ with ordered adjacency lists, the whole-graph decreasing finish order is the decreasing finish order of the left-first depth-first search of $G$ whose roots are all vertices of $G$, taken in the graph's vertex enumeration order; its vertices are listed from the latest finished to the earliest finished.","title":"Whole-graph decreasing finish order"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/mem-decreasing-finish-all","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.mem_decreasingFinishAll","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#mem-decreasing-finish-all","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For every finite directed graph $G$ given by ordered adjacency lists and every vertex $v$ of $G$, $v$ occurs in the whole-graph decreasing finish order of $G$, that is, the decreasing finish order of the left-first depth-first search of $G$ whose roots are all vertices of $G$ in the graph's vertex order.","title":"Whole-graph decreasing finish order contains every vertex"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/decreasing-finish-all-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.decreasingFinishAll_nodup","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#decreasing-finish-all-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"For every directed graph $G$ with ordered adjacency lists, the whole-graph decreasing finish order of $G$ contains no vertex twice.","title":"Whole-graph decreasing finish order has no repeated vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/dfs-exact-steps","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.dfs_exact_steps","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#dfs-exact-steps","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph with ordered adjacency lists $N(v)$, whose entries may repeat, and let $R$ be a list of roots, possibly with repetitions. Run the depth-first search machine from $R$ (each transition drops a pending vertex that is already visited, discovers a new vertex, or finishes a vertex). Let $v_1, \\dots, v_k$ be the preorder of the depth-first search of $G$ from $R$. Then there is a terminal\u2026","title":"Exact transition count of depth-first search"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/dfs-all-exact-steps","kind":"node","lean":"Cslib.Algorithms.Lean.DepthFirstSearch.dfsAll_exact_steps","parent":38524,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search/index.html#dfs-all-exact-steps","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/depth-first-search","status":"fully_proved","summary":"Let $G$ be a finite directed graph whose vertices are listed without repetition as $v_1, \\dots, v_n$, with ordered adjacency lists $N(v)$ whose entries may repeat. Run the depth-first search machine with roots $v_1, \\dots, v_n$ in this order (each transition drops a pending vertex that is already visited, discovers a new vertex, or finishes a vertex). Then there is a terminal machine state, with no work left, whose\u2026","title":"Exact transition count of whole-graph depth-first search"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","title":"Hierholzer's algorithm for Euler circuits"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/edge-list-of","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.edgeListOf","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#edge-list-of","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Given a directed adjacency graph $G$ and a function $R$ assigning to each vertex $u$ a list $R(u)$ of vertices, the edge list of $R$ is the list of pairs $(u, w)$, one for each entry $w$ of $R(u)$, taken over the vertices $u$ in the order of the vertex enumeration of $G$ and, for each $u$, in the order of $R(u)$. A repeated entry gives a repeated pair.","title":"Edge list of an adjacency function"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/edge-list","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.edgeList","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#edge-list","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"The edge list of a directed adjacency graph $G$ is the list of pairs $(u, w)$, one for each entry $w$ of the adjacency list of $u$, taken over the vertices $u$ in the order of the vertex enumeration of $G$ and, for each $u$, in adjacency-list order. A repeated adjacency entry gives a repeated pair, so $G$ is read as a multigraph with one edge occurrence per entry.","title":"Edge list of a directed graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/walk-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.walkEdges","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#walk-edges","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"For a finite sequence of vertices $W = (w_0, w_1, \\dots, w_n)$, the steps of $W$ are the list of consecutive pairs $((w_0, w_1), (w_1, w_2), \\dots, (w_{n-1}, w_n))$, in order. A sequence with at most one vertex has no steps.","title":"Steps of a walk"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/is-balanced","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.IsBalanced","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#is-balanced","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"A finite directed multigraph, read from its adjacency lists with each entry $w$ in the list of $u$ counted as one occurrence of the edge $u \\to w$, is balanced when every vertex is the source of as many edge occurrences as it is the target of.","title":"Balanced directed multigraph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/is-semi-balanced","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.IsSemiBalanced","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#is-semi-balanced","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"A finite directed multigraph is semi-balanced for vertices $s$ and $t$ when, for every vertex $x$, the number of edge occurrences leaving $x$ plus $[x = t]$ equals the number entering $x$ plus $[x = s]$, where $[P]$ is $1$ if $P$ holds and $0$ otherwise. So for $s \\ne t$, one more edge occurrence leaves $s$ than enters it, one more enters $t$ than leaves it, and every other vertex is balanced; for $s = t$ every\u2026","title":"Semi-balanced directed multigraph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/is-euler-path","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.IsEulerPath","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#is-euler-path","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"In a finite directed multigraph, a list of vertices $W = (w_0, \\dots, w_k)$ is an Euler path from $s$ to $t$ when $w_0 = s$, $w_k = t$, and its list of steps $(w_0, w_1), \\dots, (w_{k-1}, w_k)$ is a permutation of the list of edge occurrences of the graph, so that each edge occurrence, counted with multiplicity, is traversed exactly once.","title":"Euler path in a directed multigraph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/is-euler-circuit","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.IsEulerCircuit","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#is-euler-circuit","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"In a finite directed multigraph, a list of vertices $W$ is an Euler circuit from a vertex $s$ when it is an Euler path from $s$ to $s$: it starts and ends at $s$, and its list of consecutive pairs is a permutation of the list of edge occurrences of the graph.","title":"Euler circuit in a directed multigraph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/visit","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.visit","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#visit","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Given a fuel $n$, a vertex $v$, an assignment $R$ of a list of remaining out-neighbours to each vertex, and a list $\\mathit{acc}$ of vertices, the recursion returns a list of vertices and a new assignment, with a cost. With fuel $0$ it returns $(\\mathit{acc}, R)$ at cost $0$. With fuel $n + 1$: if $R(v)$ is empty, it returns $v$ followed by $\\mathit{acc}$ together with $R$, at cost $0$; otherwise $R(v)$ is $w$\u2026","title":"Hierholzer's recursion"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/hierholzer","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.hierholzer","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#hierholzer","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph, read as a multigraph with one edge occurrence per adjacency entry, let $s$ be a vertex, and let $m$ be the number of edge occurrences of $G$, that is, the length of its edge list. Hierholzer's algorithm on $G$ from $s$ runs Hierholzer's recursion from $s$ with fuel $m + 1$, with the adjacency list of each vertex as its list of remaining out-neighbours and the empty list as\u2026","title":"Hierholzer's algorithm"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/walk-edges-append","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.walkEdges_append","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#walk-edges-append","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $A$ and $B$ be nonempty vertex sequences, where $A$ ends at the vertex $a$ and $B$ starts at the vertex $b$. Then the steps of the concatenation $A B$ are the steps of $A$, followed by the step $(a, b)$, followed by the steps of $B$.","title":"Steps of a concatenated walk"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/perm-splice","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.perm_splice","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#perm-splice","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"For an element $e$ and lists $A$, $B$, $C$ of the same type, the list $e \\cdot A \\cdot B \\cdot C$ is a permutation of $B \\cdot e \\cdot A \\cdot C$, where $\\cdot$ denotes concatenation and $e$ stands for the one-element list.","title":"Moving a block of a list in front of an element is a permutation"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/semi-balanced-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.SemiBalancedEdges","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#semi-balanced-edges","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"A list $E$ of ordered pairs of vertices is semi-balanced for vertices $v$ and $t$ when, for every vertex $x$, the number of pairs in $E$ with first component $x$ plus $[x = t]$ equals the number with second component $x$ plus $[x = v]$, where $[P]$ is $1$ if $P$ holds and $0$ otherwise. So for $v \\ne t$, $v$ has one more outgoing than incoming occurrence, $t$ one more incoming than outgoing, and every other vertex\u2026","title":"Semi-balanced edge list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/balanced-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.BalancedEdges","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#balanced-edges","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"A list $E$ of ordered pairs of vertices, read as directed edge occurrences, is balanced when every vertex $x$ occurs as the first component of exactly as many entries of $E$ as it occurs as the second component, counted with multiplicity.","title":"Balanced edge list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/visit-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.visit_spec","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#visit-spec","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $G$ be a finite directed multigraph, let $R$ assign to each vertex $u$ a list $R(u)$ of remaining out-neighbours, and let $E(R)$ be the list of remaining edge occurrences, one pair $(u, w)$ for each entry $w$ of $R(u)$. Let $v$ and $t$ be vertices, $\\mathit{acc}$ a list of vertices and $n$ a natural number with $|E(R)| < n$, and suppose $E(R)$ is semi-balanced for $v$ and $t$: for every vertex $x$, the number\u2026","title":"Contract of Hierholzer's recursion"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/hierholzer-trail","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.hierholzer_trail","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#hierholzer-trail","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph, read as a multigraph with one edge per adjacency entry, and let $s$ and $t$ be vertices such that $G$ is semi-balanced for $s$ and $t$. Then the vertex sequence $W$ returned by Hierholzer's algorithm on $G$ from $s$ starts at $s$ and ends at $t$, and the edge list of $G$ is a permutation of the list of consecutive pairs of $W$ followed by some further list of pairs. So $W$ is\u2026","title":"Hierholzer's algorithm returns a trail in a directed multigraph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/hierholzer-is-euler-path","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.hierholzer_isEulerPath","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#hierholzer-is-euler-path","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph, read as a multigraph with one edge per adjacency entry, and let $s$ and $t$ be vertices such that $G$ is semi-balanced for $s$ and $t$ (at $s$ one more edge occurrence leaves than enters, at $t$ one more enters than leaves, and at every other vertex as many leave as enter; at every vertex when $s = t$) and the source of every edge of $G$ is reachable from $s$. Then the vertex\u2026","title":"Hierholzer's algorithm finds Euler paths in directed multigraphs"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/hierholzer-is-euler-circuit","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.hierholzer_isEulerCircuit","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#hierholzer-is-euler-circuit","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph, read as a multigraph with one edge per adjacency entry, that is balanced (every vertex is the source of exactly as many edge occurrences as it is the target of), and let $s$ be a vertex such that the source of every edge of $G$ is reachable from $s$. Then the vertex sequence returned by Hierholzer's algorithm on $G$ from $s$ is an Euler circuit of $G$ from $s$.","title":"Hierholzer's algorithm finds Euler circuits in directed multigraphs"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/hierholzer-time","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.hierholzer_time","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#hierholzer-time","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph, read as a multigraph with one edge per adjacency entry, and let $s$ and $t$ be vertices such that $G$ is semi-balanced for $s$ and $t$ and the source of every edge of $G$ is reachable from $s$. Then Hierholzer's algorithm on $G$ from $s$ costs exactly $|E|$ units, where $|E|$ is the number of edge occurrences of $G$, that is, the length of its edge list.","title":"Running time of Hierholzer's algorithm"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/is-semi-balanced-of-is-euler-path","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.isSemiBalanced_of_isEulerPath","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#is-semi-balanced-of-is-euler-path","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"If a finite directed multigraph has an Euler path from $s$ to $t$, then it is semi-balanced for $s$ and $t$: counting edge occurrences, $s$ has one more outgoing than incoming edge, $t$ one more incoming than outgoing, and every other vertex is balanced (every vertex, when $s = t$).","title":"A directed multigraph with an Euler path is semi-balanced"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/is-balanced-of-is-euler-circuit","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.isBalanced_of_isEulerCircuit","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#is-balanced-of-is-euler-circuit","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"If a finite directed multigraph has an Euler circuit from some vertex $s$, then it is balanced: every vertex is the source of as many edge occurrences as it is the target of.","title":"A directed multigraph with an Euler circuit is balanced"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/exists-is-euler-path-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.exists_isEulerPath_iff","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#exists-is-euler-path-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph, read as a multigraph with one edge per adjacency entry, and let $s$ and $t$ be vertices such that the source of every edge of $G$ is reachable from $s$. Then $G$ has an Euler path from $s$ to $t$ if and only if $G$ is semi-balanced for $s$ and $t$: at $s$ one more edge occurrence leaves than enters, at $t$ one more enters than leaves, and every other vertex is the source of\u2026","title":"Euler path criterion for directed multigraphs"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/exists-is-euler-circuit-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.exists_isEulerCircuit_iff","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#exists-is-euler-circuit-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph, read as a multigraph with one edge per adjacency entry, and let $s$ be a vertex such that the source of every edge of $G$ is reachable from $s$. Then $G$ has an Euler circuit from $s$ if and only if $G$ is balanced, that is, every vertex is the source of exactly as many edge occurrences as it is the target of.","title":"Euler circuit criterion for directed multigraphs"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/usym","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.usym","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#usym","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"The ordered pair $(a, b)$ of vertices, viewed as the unordered pair $\\{a, b\\}$, so that $(a, b)$ and $(b, a)$ give the same unordered edge.","title":"Unordered edge of an ordered pair"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/udeg","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.udeg","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#udeg","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"For a finite list $R$ of edges of an undirected multigraph, each given as a pair $(a, b)$, the degree of a vertex $x$ is the number of edges in $R$ whose first component is $x$ plus the number whose second component is $x$; parallel edges count separately and a loop $(x, x)$ counts twice.","title":"Degree in an undirected edge list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/even-except","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.EvenExcept","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#even-except","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"For a finite list $R$ of edges of an undirected multigraph and vertices $v$, $t$, the condition holds when for every vertex $x$ the number $\\deg_R(x) + [x = v] + [x = t]$ is even, where $\\deg_R$ is the degree in $R$ (a loop counts twice) and $[P]$ is $1$ if $P$ holds and $0$ otherwise. Thus if $v \\ne t$, exactly $v$ and $t$ have odd degree, and if $v = t$, every degree is even.","title":"Degree parity condition for two endpoints"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/ureach","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.UReach","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#ureach","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"For a list $E$ of edges (ordered pairs of vertices), a vertex $x$ is reachable from a vertex $s$ when there are vertices $s = v_0, v_1, \\dots, v_n = x$ with $n \\ge 0$ such that for each $i < n$, $(v_i, v_{i+1})$ or $(v_{i+1}, v_i)$ belongs to $E$.","title":"Reachability in an undirected multigraph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/is-ueuler-path","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.IsUEulerPath","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#is-ueuler-path","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"For a finite list $E$ of edges of an undirected multigraph and vertices $s$, $t$, a sequence of vertices $W = (w_0, \\dots, w_n)$ is an Euler path of $E$ from $s$ to $t$ when $w_0 = s$, $w_n = t$, and the multiset of unordered pairs $\\{w_{i-1}, w_i\\}$ for $1 \\le i \\le n$ equals the multiset of the edges of $E$ taken as unordered pairs. Thus every edge is traversed exactly once, in either direction.","title":"Euler path in an undirected multigraph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/other-end","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.otherEnd","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#other-end","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"For an edge $e = (a, b)$ and a vertex $v$, the other endpoint of $e$ at $v$ is $b$ if $a = v$, and $a$ otherwise; in particular a loop $(v, v)$ gives $v$.","title":"Other endpoint of an edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/uvisit","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.uvisit","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#uvisit","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"The recursion takes a fuel $f \\in \\mathbb{N}$, a vertex $v$, a list $R$ of remaining edges and an accumulator list $A$ of vertices, and returns a list of vertices and a list of remaining edges. With fuel $0$ it returns $A$ and $R$ unchanged. With fuel $f + 1$: if no edge of $R$ has $v$ as an endpoint, it returns $v$ prepended to $A$, together with $R$; otherwise, with $e$ the first such edge, it removes $e$ from\u2026","title":"Undirected Hierholzer recursion"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/uhierholzer","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.uhierholzer","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#uhierholzer","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Given a list $E$ of undirected edges (ordered pairs of vertices) and a start vertex $s$, run the undirected Hierholzer recursion from $s$ on $E$ with an empty accumulator and fuel $|E| + 1$, and return the list of emitted vertices; its cost is the cost of that recursion.","title":"Undirected Hierholzer's algorithm"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/udeg-erase","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.udeg_erase","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#udeg-erase","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $R$ be a finite list of edges of an undirected multigraph, each an ordered pair of vertices, let $e = (a, b)$ be an edge in $R$, and let $x$ be a vertex. Then the degree of $x$ in $R$ equals the degree of $x$ in the list obtained from $R$ by removing the first occurrence of $e$, plus $[x = a] + [x = b]$, where $[P]$ is $1$ if $P$ holds and $0$ otherwise.","title":"Degree after removing an edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/usym-eq-other-end","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.usym_eq_otherEnd","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#usym-eq-other-end","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"If an ordered pair $e = (a, b)$ of vertices has $a = v$ or $b = v$, then the unordered pair $\\{a, b\\}$ equals $\\{v, w\\}$, where $w$ is the other endpoint of $e$ at $v$.","title":"An edge at a vertex joins it to its other endpoint"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/uvisit-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.uvisit_spec","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#uvisit-spec","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $R$ be a list of edges, $v$, $t$ vertices such that exactly $v$ and $t$ have odd degree in $R$ (no vertex when $v = t$), $A$ a list of vertices, and $f > |R|$ a fuel bound. Run the undirected Hierholzer recursion with fuel $f$ from $v$ on $R$ with accumulator $A$, and let $R'$ be the remaining edge list it returns and $c$ its cost. Then there is a vertex sequence $W$ starting at $v$ and ending at $t$ such that\u2026","title":"Invariant of the undirected Hierholzer recursion"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/uhierholzer-trail","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.uhierholzer_trail","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#uhierholzer-trail","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $E$ be a list of undirected edges and $s$, $t$ vertices such that exactly $s$ and $t$ have odd degree in $E$ (no vertex has odd degree when $s = t$). Then the vertex sequence $W$ returned by the undirected Hierholzer algorithm on $E$ from $s$ starts at $s$ and ends at $t$, and the consecutive pairs of $W$, as unordered pairs, form a sub-multiset of the edges of $E$ as unordered pairs: no edge is traversed more\u2026","title":"Undirected Hierholzer's algorithm returns a trail"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/uhierholzer-is-ueuler-path","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.uhierholzer_isUEulerPath","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#uhierholzer-is-ueuler-path","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $E$ be a list of undirected edges (ordered pairs of vertices) and $s$, $t$ vertices such that exactly $s$ and $t$ have odd degree in $E$ (no vertex has odd degree when $s = t$), and suppose the first endpoint of every edge of $E$ is reachable from $s$ through edges of $E$. Then the vertex sequence returned by the undirected Hierholzer algorithm on $E$ from $s$ is an Euler path of $E$ from $s$ to $t$.","title":"Undirected Hierholzer's algorithm finds Euler paths"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/uhierholzer-is-ueuler-circuit","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.uhierholzer_isUEulerCircuit","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#uhierholzer-is-ueuler-circuit","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $E$ be a list of undirected edges (ordered pairs of vertices) in which every vertex has even degree, and let $s$ be a vertex such that the first endpoint of every edge of $E$ is reachable from $s$ through edges of $E$. Then the vertex sequence returned by the undirected Hierholzer algorithm on $E$ from $s$ is an Euler path of $E$ from $s$ to $s$, that is, an Euler circuit at $s$.","title":"Undirected Hierholzer's algorithm finds Euler circuits"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/uhierholzer-time","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.uhierholzer_time","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#uhierholzer-time","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $E$ be a list of undirected edges and $s$, $t$ vertices such that exactly $s$ and $t$ have odd degree in $E$ (no vertex has odd degree when $s = t$), and suppose the first endpoint of every edge of $E$ is reachable from $s$ through edges of $E$. Then the undirected Hierholzer algorithm on $E$ from $s$ costs exactly $|E|$, the number of edges of $E$, in the cost model that charges one unit per removed edge.","title":"Cost of the undirected Hierholzer algorithm"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/ends-at","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.endsAt","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#ends-at","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"For a vertex $x$, this assigns to an unordered pair $\\{a, b\\}$ the number $[x = a] + [x = b]$ of its endpoints equal to $x$, where $[P]$ is $1$ if $P$ holds and $0$ otherwise; in particular a loop $\\{x, x\\}$ gives $2$.","title":"Multiplicity of a vertex in an unordered edge"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/udeg-eq-of-perm","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.udeg_eq_of_perm","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#udeg-eq-of-perm","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $A$ and $B$ be finite lists of edges of an undirected multigraph such that the multiset of edges of $A$, taken as unordered pairs, equals that of $B$. Then every vertex has the same degree in $A$ as in $B$.","title":"Degrees depend only on the unordered edges"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/even-except-walk-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.evenExcept_walkEdges","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#even-except-walk-edges","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $W = (w_0, \\dots, w_n)$ be a nonempty sequence of vertices with $w_0 = s$ and $w_n = t$, and consider the multigraph whose edges are the consecutive pairs $(w_{i-1}, w_i)$ for $1 \\le i \\le n$. In it every vertex other than $s$ and $t$ has even degree (a loop counts twice); if $s \\ne t$, then $s$ and $t$ have odd degree, and if $s = t$, every vertex has even degree.","title":"Degrees along a walk"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/even-except-of-is-ueuler-path","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.evenExcept_of_isUEulerPath","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#even-except-of-is-ueuler-path","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $E$ be a finite list of edges of an undirected multigraph and let $W$ be an Euler path of $E$ from $s$ to $t$. Then every vertex other than $s$ and $t$ has even degree in $E$ (a loop counts twice); if $s \\ne t$, then $s$ and $t$ have odd degree, and if $s = t$, every vertex has even degree.","title":"Degrees of an undirected Euler path"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/exists-is-ueuler-path-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Hierholzer.exists_isUEulerPath_iff","parent":38562,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits/index.html#exists-is-ueuler-path-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/hierholzer-s-algorithm-for-euler-circuits","status":"fully_proved","summary":"Let $E$ be a finite list of edges of an undirected multigraph (parallel edges and loops allowed), each given as a pair $(a, b)$, and let $s$, $t$ be vertices such that the first endpoint $a$ of every edge is reachable from $s$ along edges of $E$ traversed in either direction. Then $E$ has an Euler path from $s$ to $t$ if and only if, when $s \\ne t$, exactly $s$ and $t$ have odd degree, and, when $s = t$, every\u2026","title":"Euler's theorem for undirected multigraphs"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","title":"Strongly connected components"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/strongly-connected","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.StronglyConnected","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#strongly-connected","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"In a directed adjacency graph $G$, vertices $u$ and $v$ are strongly connected when $v$ is reachable from $u$ and $u$ is reachable from $v$ in $G$.","title":"Strongly connected vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/strongly-connected-refl","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.stronglyConnected_refl","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#strongly-connected-refl","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"In a directed graph $G$, every vertex is strongly connected to itself.","title":"Strong connectivity is reflexive"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/symm","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.StronglyConnected.symm","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#symm","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"In a directed graph $G$, if $u$ is strongly connected to $v$, then $v$ is strongly connected to $u$.","title":"Strong connectivity is symmetric"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/trans","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.StronglyConnected.trans","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#trans","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"In a directed graph $G$, if $u$ is strongly connected to $v$ and $v$ is strongly connected to $w$, then $u$ is strongly connected to $w$.","title":"Strong connectivity is transitive"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/strongly-connected-setoid","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.stronglyConnectedSetoid","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#strongly-connected-setoid","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For a directed graph $G$, the relation on the vertices of $G$ that holds between $u$ and $v$ exactly when $u$ and $v$ are strongly connected in $G$, packaged as an equivalence relation.","title":"Strong connectivity as an equivalence relation"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/same-component-b","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.sameComponentB","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#same-component-b","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For a directed graph $G$ and vertices $u$ and $v$, the test returns true exactly when $v$ occurs in the preorder of the depth-first search of $G$ from the single root $u$, and $u$ occurs in the preorder of the depth-first search of $G$ from the single root $v$.","title":"Executable mutual-reachability test"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/same-component-b-eq-true","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.sameComponentB_eq_true","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#same-component-b-eq-true","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For a directed graph $G$ and vertices $u$ and $v$, the executable mutual-reachability test on $u$ and $v$ returns true if and only if $u$ and $v$ are strongly connected in $G$.","title":"The mutual-reachability test is correct"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/components","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.components","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#components","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"Let $G$ be a directed graph and let $v_1, \\dots, v_n$ be its vertices in decreasing order of finishing time in a depth-first search of the whole graph that takes its roots in the vertex order of $G$. For each $i$, let $C_i$ be the list of those $v_j$, in increasing order of $j$, such that $v_i$ and $v_j$ are strongly connected (tested by two depth-first searches). The output is the list $C_1, \\dots, C_n$ with\u2026","title":"Strongly connected components algorithm"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/components-nonempty","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.components_nonempty","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#components-nonempty","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For every directed graph $G$, each block returned by the strongly connected components algorithm on $G$ contains at least one vertex.","title":"Computed components are nonempty"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/components-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.components_nodup","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#components-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For every directed graph $G$, each block returned by the strongly connected components algorithm on $G$ contains no vertex more than once.","title":"Computed components have no repeated vertices"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/components-pairwise-disjoint","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.components_pairwise_disjoint","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#components-pairwise-disjoint","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For every directed graph $G$, any two blocks at different positions of the list returned by the strongly connected components algorithm on $G$ have no vertex in common.","title":"Computed components are pairwise disjoint"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/mem-flatten-components","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.mem_flatten_components","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#mem-flatten-components","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For a directed graph $G$ and a vertex $v$, $v$ belongs to some block of the computed strongly connected components of $G$ if and only if $v$ occurs in the vertex enumeration of $G$.","title":"The computed components cover the vertex list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/share-component-iff-strongly-connected","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.share_component_iff_stronglyConnected","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#share-component-iff-strongly-connected","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For a directed graph $G$ and vertices $u$ and $v$, some block of the computed strongly connected components of $G$ contains both $u$ and $v$ if and only if $u$ and $v$ are strongly connected in $G$.","title":"Vertices share a computed component exactly when strongly connected"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/component-maximal","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.component_maximal","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#component-maximal","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"Let $G$ be a directed graph, let $B$ be one of the blocks returned by the strongly connected components algorithm on $G$, and let $m$ be a vertex of $B$. Then for every vertex $v$ of $G$, the vertices $m$ and $v$ are strongly connected (each is reachable from the other) if and only if $v$ lies in $B$.","title":"Each computed component is a full mutual-reachability class"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/is-sccpartition","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.IsSCCPartition","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#is-sccpartition","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"A list $\\mathcal{B}$ of vertex lists is an SCC partition of a directed graph $G$ when every block of $\\mathcal{B}$ is nonempty and contains no vertex twice, blocks at different positions of $\\mathcal{B}$ are disjoint, a vertex lies in some block exactly when it is a vertex of $G$, two vertices lie in a common block exactly when they are strongly connected (each is reachable from the other), and for every block $B$,\u2026","title":"SCC partition"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/components-is-sccpartition","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.components_isSCCPartition","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#components-is-sccpartition","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"For every directed graph $G$, the list of blocks returned by the strongly connected components algorithm on $G$ is an SCC partition of $G$: every block is nonempty and contains no vertex twice, blocks at different positions of the list are disjoint, every vertex of $G$ lies in some block, two vertices lie in a common block exactly when they are strongly connected, and for every vertex $m$ of a block $B$ and every\u2026","title":"The computed components form the SCC partition"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/component-reaches","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.ComponentReaches","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#component-reaches","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"In a directed graph $G$, a list $S$ of vertices reaches a list $T$ of vertices when some vertex of $S$ reaches some vertex of $T$ by a directed path in $G$, possibly of length zero.","title":"Reachability between components"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/condensation-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.CondensationAcyclic","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#condensation-acyclic","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"A list $\\mathcal{B}$ of vertex lists of a directed graph $G$ has an acyclic condensation when, for all $S, T \\in \\mathcal{B}$, if some vertex of $S$ reaches some vertex of $T$ in $G$ and some vertex of $T$ reaches some vertex of $S$, then $S = T$. That is, reachability between the blocks is antisymmetric.","title":"Acyclic condensation"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/components-condensation-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.StronglyConnectedComponents.components_condensationAcyclic","parent":38605,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components/index.html#components-condensation-acyclic","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/strongly-connected-components","status":"fully_proved","summary":"Let $G$ be a directed graph and let $S$ and $T$ be blocks returned by the strongly connected components algorithm on $G$. If some vertex of $S$ reaches some vertex of $T$ and some vertex of $T$ reaches some vertex of $S$, then $S = T$.","title":"The condensation of the computed components is acyclic"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","title":"Topological sorting"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/indegree-within","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.indegreeWithin","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#indegree-within","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"For a finite directed graph $G$, a list $R$ of vertices and a vertex $v$, the indegree of $v$ within $R$ is the number of entries $u$ of $R$, counted with multiplicity, such that $G$ has an edge from $u$ to $v$.","title":"Indegree within a vertex list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/indegree","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.indegree","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#indegree","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"The indegree of a vertex $v$ in a finite directed graph $G$ is the number of vertices $u$ such that $G$ has an edge from $u$ to $v$; repeated entries in adjacency lists count once.","title":"Indegree"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/result","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.Result","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#result","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"The result of Kahn's topological sorting algorithm on a graph with vertices of type $\\alpha$ is either a success, carrying a list of vertices, or a failure, carrying the list of vertices processed before the queue emptied and the list of vertices that remain.","title":"Outcome of Kahn's topological sort"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/sort","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.sort","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#sort","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"Run Kahn's algorithm with a first-in first-out queue on a finite directed graph $G$ whose vertices are enumerated as $v_1, \\dots, v_n$. Initially every vertex is remaining, the counter of each vertex is its number of in-neighbours, and the queue holds the vertices with no in-neighbours in enumeration order. At each of $n$ steps, if the queue is nonempty, its front vertex $x$ is removed from the queue and from the\u2026","title":"Kahn's topological sort"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/directed-cycle-in","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.DirectedCycleIn","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#directed-cycle-in","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"Let $G$ be a finite directed graph and $R$ a list of vertices. A list $c = [c_1, \\dots, c_k]$ of vertices is a directed cycle of $G$ in $R$ when $k \\ge 1$, the vertices $c_1, \\dots, c_k$ are pairwise distinct, $G$ has an edge from $c_i$ to $c_{i+1}$ for each $1 \\le i < k$ and an edge from $c_k$ to $c_1$, and every $c_i$ lies in $R$.","title":"Directed cycle within a vertex list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/directed-cycle-in-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.directedCycleIn_iff","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#directed-cycle-in-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"Let $G$ be a finite directed graph and let $R$ and $c = [c_1, \\dots, c_k]$ be lists of vertices. Then $c$ is a directed cycle of $G$ in $R$ if and only if $k \\ge 1$, the vertices $c_1, \\dots, c_k$ are pairwise distinct, $G$ has an edge from $c_i$ to $c_{i+1}$ for every $1 \\le i < k$, $G$ has an edge from $c_k$ to $c_1$, and every $c_i$ lies in $R$.","title":"Characterization of directed cycles within a vertex list"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/is-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.IsAcyclic","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#is-acyclic","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"A finite directed graph $G$ is acyclic when it has no directed cycle, that is, no nonempty list $[c_1, \\dots, c_k]$ of pairwise distinct vertices such that $G$ has an edge from $c_i$ to $c_{i+1}$ for each $1 \\le i < k$ and an edge from $c_k$ to $c_1$.","title":"Acyclic directed graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/is-acyclic-iff-no-directed-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.isAcyclic_iff_no_directedCycle","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#is-acyclic-iff-no-directed-cycle","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"A finite directed graph $G$ is acyclic if and only if there is no directed cycle of $G$ in its vertex list, that is, no nonempty list of pairwise distinct vertices in which each vertex has an edge to the next and the last vertex has an edge to the first.","title":"Acyclicity means having no directed cycle"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/is-topological-order","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.IsTopologicalOrder","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#is-topological-order","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"A list $L$ of vertices of a finite directed graph $G$ is a topological order of $G$ when $L$ has no duplicates, a vertex lies in $L$ exactly when it lies in the vertex enumeration of $G$, and for every edge $u \\to v$ of $G$ the first position of $u$ in $L$ is strictly smaller than the first position of $v$ in $L$.","title":"Topological order"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/success-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.success_nodup","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#success-nodup","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"If Kahn's topological sort on a finite directed graph $G$ returns a success with list $L$, then $L$ has no duplicates.","title":"A successful topological sort has no repeated vertex"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/success-coverage","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.success_coverage","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#success-coverage","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"If Kahn's topological sort on a finite directed graph $G$ returns a success with list $L$, then a vertex lies in $L$ if and only if it lies in the vertex enumeration of $G$.","title":"A successful topological sort lists every vertex"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/success-edge-forward","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.success_edge_forward","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#success-edge-forward","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"If Kahn's topological sort on a finite directed graph $G$ returns a success with list $L$, then for every edge $u \\to v$ of $G$ the first position of $u$ in $L$ is strictly smaller than the first position of $v$ in $L$.","title":"A successful topological sort points every edge forward"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/success-is-topological-order","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.success_isTopologicalOrder","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#success-is-topological-order","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"If Kahn's topological sort on a finite directed graph $G$ returns a success with list $L$, then $L$ is a topological order of $G$.","title":"A successful topological sort is a topological order"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/failure-remainder","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.failure_remainder","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#failure-remainder","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"Let $G$ be a finite directed graph. If deterministic FIFO Kahn topological sorting of $G$ fails after processing the vertex list $P$ and leaving the vertex list $R$ unprocessed, then $R$ is nonempty, neither $P$ nor $R$ has repeated entries, a vertex lies in the vertex list of $G$ if and only if it lies in $P$ or in $R$, and no vertex lies in both $P$ and $R$.","title":"A failed Kahn run splits the vertices into processed and unprocessed ones"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/failure-positive-indegree","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.failure_positive_indegree","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#failure-positive-indegree","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"Let $G$ be a finite directed graph. If deterministic FIFO Kahn topological sorting of $G$ fails after processing the vertex list $P$ and leaving the vertex list $R$ unprocessed, then for every vertex $v$ in $R$ the indegree of $v$ within $R$, the number of entries $u$ of $R$ such that $G$ has an edge from $u$ to $v$, is positive.","title":"Unprocessed vertices of a failed Kahn run have positive indegree among the unprocessed"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/failure-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.failure_cycle","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#failure-cycle","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"Let $G$ be a finite directed graph. If deterministic FIFO Kahn topological sorting of $G$ fails after processing the vertex list $P$ and leaving the vertex list $R$ unprocessed, then $G$ has a directed cycle in $R$: a nonempty list of pairwise distinct vertices of $R$ in which each vertex has an edge to the next and the last vertex has an edge to the first.","title":"A failed Kahn run leaves a directed cycle"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/success-implies-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.success_implies_acyclic","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#success-implies-acyclic","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"If Kahn's topological sort on a finite directed graph $G$ returns a success, then $G$ is acyclic.","title":"Success of Kahn's algorithm implies acyclicity"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/acyclic-implies-success","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.acyclic_implies_success","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#acyclic-implies-success","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"If a finite directed graph $G$, given by a vertex enumeration and adjacency lists, is acyclic, then deterministic FIFO Kahn topological sorting of $G$ returns success with some vertex order.","title":"Kahn's algorithm succeeds on acyclic graphs"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/succeeds-iff-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.TopologicalSort.succeeds_iff_acyclic","parent":38625,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting/index.html#succeeds-iff-acyclic","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/topological-sorting","status":"fully_proved","summary":"For every finite directed graph $G$, Kahn's topological sort on $G$ returns a success if and only if $G$ is acyclic.","title":"Kahn's algorithm succeeds exactly on acyclic graphs"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","title":"Transitive closure"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-stages","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosureStages","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-stages","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"Let $A$ be a Boolean adjacency matrix on the vertices $0, \\dots, n-1$ and let $k \\le n$. The computation first builds the $n \\times n$ Boolean matrix $T^{(0)}$ whose entry $(i, j)$ is true exactly when $i = j$ or $A_{ij}$ is true. Then, for $p = 0, \\dots, k - 1$, it builds a new matrix $T^{(p+1)}$ from the saved matrix $T^{(p)}$ by $T^{(p+1)}_{ij} = T^{(p)}_{ij} \\lor (T^{(p)}_{ip} \\land T^{(p)}_{pj})$, and it\u2026","title":"Staged transitive-closure computation"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosure","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"Given a Boolean adjacency matrix $A$ on the vertices $0, 1, \\dots, n-1$, the algorithm first builds the matrix $t^{(0)}$ with $t^{(0)}_{ij}$ true exactly when $i = j$ or $A_{ij}$ is true. Then, for $k = 0, 1, \\dots, n-1$ in turn, it builds a new matrix $t^{(k+1)}$ with $t^{(k+1)}_{ij} = t^{(k)}_{ij} \\lor (t^{(k)}_{ik} \\land t^{(k)}_{kj})$ for all $i, j$, reading every operand from the saved matrix $t^{(k)}$, and it\u2026","title":"Saved-matrix transitive-closure algorithm"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-stages-zero-cell","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosureStages_zero_cell","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-stages-zero-cell","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"Let $A$ be a Boolean adjacency matrix on the vertices $0, \\dots, n-1$. For all vertices $i$ and $j$, the entry $(i, j)$ of the matrix produced by the staged transitive-closure computation with no pivot stages is true if and only if $i = j$ or $A_{ij}$ is true.","title":"Initial matrix of the transitive-closure computation"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-stages-succ-cell","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosureStages_succ_cell","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-stages-succ-cell","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"Let $A$ be a Boolean adjacency matrix on the vertices $0, \\dots, n-1$, and for $m \\le n$ let $T^{(m)}$ be the matrix produced by the staged transitive-closure computation on $A$ after its first $m$ pivot stages. For every $k < n$ and all vertices $i$ and $j$,","title":"Transition recurrence of the transitive-closure stages"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-initializations","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosure_initializations","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-initializations","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"For every Boolean adjacency matrix on $n$ vertices, the staged transitive-closure computation with no pivot stages (only the initialization of the $n \\times n$ matrix) charges exactly $n^2$ time units, one for each cell assignment.","title":"Cost of initializing the transitive-closure matrix"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-stage-assignments","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosure_stage_assignments","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-stage-assignments","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"For every Boolean adjacency matrix on $n$ vertices and every $k < n$, the time charged by the staged transitive-closure computation with $k + 1$ pivot stages minus the time charged with $k$ pivot stages equals $n^2$: one time unit for each cell assignment of the new stage.","title":"Cost of one transition stage"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-transition-assignment","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosure_transition_assignments","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-transition-assignment","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"Charge one time unit for each cell assignment. For every Boolean adjacency matrix on $n$ vertices, let $a$ be the cost of running the initialization of the saved-matrix transitive-closure algorithm followed by all $n$ of its stages, and let $b$ be the cost of running the initialization alone. Then $a - b = n^3$, where the subtraction is subtraction of natural numbers, truncated at $0$. That is, the $n$ transition\u2026","title":"Cost of the transition stages of transitive closure"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-stages-cell-eq-true-i","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosureStages_cell_eq_true_iff","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-stages-cell-eq-true-i","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"Let $A$ be a Boolean adjacency matrix of a directed graph on the vertices $0, \\dots, n-1$ and let $k \\le n$. For all vertices $i$ and $j$, the entry $(i, j)$ of the matrix produced by the staged transitive-closure computation after its first $k$ pivot stages is true if and only if there are vertices $i = v_0, v_1, \\dots, v_m = j$ with $m \\ge 0$ such that each $A_{v_t v_{t+1}}$ is true and every intermediate vertex\u2026","title":"Invariant of the transitive-closure stages"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-cell-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosure_cell_eq_true_iff","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-cell-eq-true-iff","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"Let $A$ be a Boolean adjacency matrix of a directed graph on the vertices $0, \\dots, n-1$. For all vertices $i$ and $j$, the entry $(i, j)$ of the matrix returned by the transitive-closure algorithm on $A$ is true if and only if $j$ is reachable from $i$ by a sequence of zero or more edges of the graph.","title":"Correctness of the transitive-closure algorithm"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-time","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosure_time","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-time","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"For every Boolean adjacency matrix on $n$ vertices, the transitive-closure algorithm charges exactly $n^2 + n^3$ time units: $n^2$ cell assignments to initialize the matrix and $n^2$ in each of the $n$ transition stages.","title":"Running time of the transitive-closure algorithm"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-time-zero","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosure_time_zero","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-time-zero","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"For the graph with no vertices, the transitive-closure algorithm charges $0$ time units: it performs no cell assignments.","title":"Cost of transitive closure on the empty graph"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/transitive-closure-time-one","kind":"node","lean":"Cslib.Algorithms.Lean.TransitiveClosure.transitiveClosure_time_one","parent":38645,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure/index.html#transitive-closure-time-one","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/transitive-closure","status":"fully_proved","summary":"For every Boolean adjacency matrix on a single vertex, the transitive-closure algorithm charges exactly $2$ time units: one cell assignment for the initialization and one for the single transition stage.","title":"Cost of transitive closure on one vertex"},{"data_url":"scopes/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees.json","id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","kind":"scope","navigation":"module","parent":38432,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","title":"Traversal of implicit binary trees"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/leaves","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.leaves","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#leaves","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"The frontier below `s`, left to right, cut at depth `k`. At depth zero it is `[s]`, even when `step s` branches. Duplicate occurrences are retained.","title":"Depth-cut frontier of a binary branching step"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.round","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"One depth-first round on `(stack, output)`. Terminal states are prepended to the output; a branching state is replaced by its left child followed by its right child.","title":"One depth-first round of binary branching"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/leaves-length-pos","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.leaves_length_pos","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#leaves-length-pos","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"Every depth-cut frontier is nonempty.","title":"Depth-cut frontiers are nonempty"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round-nil-iterate","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.round_nil_iterate","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round-nil-iterate","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"An empty work stack is fixed by any number of rounds.","title":"An empty work stack is fixed by every round"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round-iterate","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.round_iterate","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round-iterate","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"If every frontier occurrence is terminal, exactly `2 * length - 1` rounds process the tree below `s`, preserving the tail `r` and prepending its reversed frontier to `out`. The terminal-frontier hypothesis is essential: a depth cut may contain branching states.","title":"Exact round count for traversing a terminal frontier"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round-iterate-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.round_iterate_of_le","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round-iterate-of-le","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"After at least `2 * length - 1` rounds from `[s]`, the stack is empty and all terminal frontier occurrences have been collected in reverse order.","title":"Enough rounds empty the stack and collect the frontier"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/sum-leaves","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.sum_leaves","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#sum-leaves","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"A natural-valued weight additive across each branch equals the sum of the frontier weights, even when the frontier is cut before termination.","title":"Additive weight of the root is the sum over the frontier"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/leaves-map","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.leaves_map","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#leaves-map","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"A map commuting with both steps transports the ordered frontier, preserving multiplicity.","title":"Commuting maps transport the ordered frontier"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round-map","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.round_map","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round-map","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"A commuting node map transports one stack round and its accumulated output.","title":"Commuting maps transport one depth-first round"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round-iterate-map","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.round_iterate_map","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round-iterate-map","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"A commuting node map transports any number of rounds. No injectivity is required.","title":"Commuting maps transport any number of rounds"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round-count","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.roundCount","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round-count","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"One stack round accumulating natural weights rather than a list of terminal states. The previous count is retained, and each terminal occurrence contributes its weight.","title":"Weight-counting depth-first round"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round-count-iterate","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.roundCount_iterate","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round-count-iterate","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"Counting rounds are the projection of list-output rounds to the sum of output weights.","title":"Counting rounds project list-output rounds"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/round-count-iterate-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.roundCount_iterate_of_le","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#round-count-iterate-of-le","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"With a terminal frontier and enough rounds, the final count is its total weight.","title":"Final count of counting rounds is the frontier weight"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/length-leaves-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.length_leaves_le","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#length-leaves-le","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"With at most `d` further depth steps and `h` further heavy steps, the frontier has at most `\u2211 j \u2208 range (h + 1), d.choose j` occurrences. The source budget hypotheses are retained, including the strict heavy-budget inequality.","title":"Binomial bound on the frontier size under depth and heavy budgets"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/step-eq-none-of-mem-leaves","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.step_eq_none_of_mem_leaves","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#step-eq-none-of-mem-leaves","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"A depth allowance covering the remaining depth budget produces only terminal states.","title":"A large enough depth cut yields only terminal states"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/exists-of-mem-leaves","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.exists_of_mem_leaves","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#exists-of-mem-leaves","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"Every frontier occurrence satisfies the resource invariant at some budgets.","title":"Frontier occurrences satisfy the resource invariant"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/iff-exists-mem-leaves","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.iff_exists_mem_leaves","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#iff-exists-mem-leaves","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"A property split by branches holds at the root iff it holds at some frontier occurrence.","title":"Branch-split properties hold at the root iff at some leaf"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/stack-bound","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.stack_bound","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#stack-bound","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"The depth budget bounds the stack by `D + 1` at every round; each pending state satisfies the resource invariant. This is a list-size bound, not a machine-space model.","title":"Depth budget bounds the depth-first stack size"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/leaves-sim","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.leaves_sim","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#leaves-sim","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"Step transport restricted to a preserved invariant transports the frontier and records that all its occurrences satisfy the invariant.","title":"Invariant-restricted step transport of the frontier"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/total-le-round","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.total_le_round","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#total-le-round","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"One round does not decrease the combined number of pending and collected occurrences.","title":"A round does not decrease pending plus collected occurrences"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/total-le-leaves","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.total_le_leaves","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#total-le-leaves","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"With a terminal frontier, pending and collected occurrences together never exceed its length, including before traversal is complete.","title":"Pending plus collected occurrences are bounded by the frontier length"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/out-of-stack","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.out_of_stack","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#out-of-stack","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"A property of every state ever on the stack holds of each collected leaf.","title":"Stack invariants hold for every collected leaf"},{"id":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/leaves-eq-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.BinaryBranching.leaves_eq_of_le","parent":38658,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees/index.html#leaves-eq-of-le","scope":"ComputerScience/graph-algorithms/traversal-and-connectivity/traversal-of-implicit-binary-trees","status":"fully_proved","summary":"Increasing a depth cut beyond a terminal frontier leaves the list unchanged.","title":"Deepening the cut past a terminal frontier changes nothing"},{"data_url":"scopes/ComputerScience/graph-algorithms/shortest-paths.json","id":"ComputerScience/graph-algorithms/shortest-paths","kind":"scope","parent":38431,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/index.html","scope":"ComputerScience/graph-algorithms/shortest-paths","status":"fully_proved","summary":"Single-source and all-pairs shortest paths: Dijkstra, Bellman-Ford, DAG relaxation, min-plus matrix products and APSP, Floyd-Warshall and Johnson.","title":"Shortest paths"},{"data_url":"scopes/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm.json","id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","kind":"scope","navigation":"module","parent":38682,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","title":"Bellman\u2013Ford algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/edge","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.Edge","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#edge","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"A weighted directed edge on the vertices $0, \\dots, n-1$ consists of a source vertex, a target vertex and an integer weight.","title":"Weighted directed edge"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/to-directed-adjacency","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.toDirectedAdjacency","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#to-directed-adjacency","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For a list of weighted edges on the vertices $\\{0, \\dots, n-1\\}$, the associated ordered directed graph enumerates its vertices as $0, 1, \\dots, n-1$, and the outgoing neighbor list of a vertex $u$ is the list of targets of the edges of the list with source $u$, in the order in which they occur and with repetitions.","title":"Weight-forgetting directed graph"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/path-vertices","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.pathVertices","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#path-vertices","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For a vertex $s$ of $\\{0, \\dots, n-1\\}$ and a list of weighted edges $e_1, \\dots, e_k$ on these vertices, the vertices of the path from $s$ along this list are the list $s, t_1, \\dots, t_k$, where $t_i$ is the target of $e_i$.","title":"Vertices of a path"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/path-weight","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.pathWeight","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#path-weight","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"The weight of a list of integer-weighted edges is the sum of the weights of its entries, an edge occurring several times being counted each time.","title":"Bellman\u2013Ford path weight"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/is-weighted-path","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.IsWeightedPath","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#is-weighted-path","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of weighted directed edges on the vertices $0, \\dots, n-1$. A list $P = (e_1, \\dots, e_k)$ of weighted directed edges is a weighted path in $E$ from $s$ to $t$ when every $e_i$ belongs to $E$, the target of $e_i$ is the source of $e_{i+1}$ for every $i < k$, and either $k = 0$ and $s = t$, or $k \\ge 1$, the source of $e_1$ is $s$ and the target of $e_k$ is $t$.","title":"Weighted path in an edge list"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/edge-to-directed-adjacency-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.edge_toDirectedAdjacency_iff","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#edge-to-directed-adjacency-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For a list $E$ of weighted directed edges on the vertices $0, \\dots, n-1$ and vertices $u$, $v$, the graph obtained from $E$ by forgetting weights has an edge from $u$ to $v$ if and only if some edge in $E$ has source $u$ and target $v$.","title":"Edges of the weight-forgetting graph"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/is-weighted-path-tail","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.isWeightedPath_tail","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#is-weighted-path-tail","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of weighted directed edges on the vertices $0, \\dots, n-1$ and let $s$ and $t$ be vertices. If the list consisting of an edge $e$ followed by a list $R$ of edges is a weighted path in $E$ from $s$ to $t$, then $R$ is a weighted path in $E$ from the target of $e$ to $t$.","title":"Tail of a weighted path"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/is-weighted-path-append-edge","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.isWeightedPath_append_edge","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#is-weighted-path-append-edge","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of weighted directed edges on the vertices $0, \\dots, n-1$ and let $s$, $m$ and $t$ be vertices. If $P$ is a weighted path in $E$ from $s$ to $m$ and $e$ is an edge belonging to $E$ with source $m$ and target $t$, then the list $P$ followed by $e$ is a weighted path in $E$ from $s$ to $t$.","title":"Appending an edge to a weighted path"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/path-weight-append-edge","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.pathWeight_append_edge","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#path-weight-append-edge","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For every list $P$ of weighted directed edges on the vertices $0, \\dots, n-1$ and every weighted directed edge $e$ on these vertices, the weight of the list $P$ followed by $e$ is the weight of $P$ plus the weight of $e$.","title":"Weight of a path with an edge appended"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/is-weighted-path-walk","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.isWeightedPath_walk","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#is-weighted-path-walk","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of weighted directed edges on the vertices $0, \\dots, n-1$ and let $s$ and $t$ be vertices. If a list $P$ of $k$ edges is a weighted path in $E$ from $s$ to $t$, then there is a walk of length $k$ from $s$ to $t$ in the directed graph obtained from $E$ by forgetting the weights.","title":"Weighted paths give walks of the same length"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/reachable-iff-exists-is-weighted-path","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.reachable_iff_exists_isWeightedPath","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#reachable-iff-exists-is-weighted-path","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For a list of weighted edges on the vertices $\\{0, \\dots, n-1\\}$ and vertices $s$ and $t$, the vertex $t$ is reachable from $s$ in the directed graph obtained from the list by forgetting the weights if and only if there is a weighted path from $s$ to $t$ using edges of the list.","title":"Reachability by weighted paths"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/reachable-negative-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.ReachableNegativeCycle","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#reachable-negative-cycle","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For a list of weighted edges on the vertices $\\{0, \\dots, n-1\\}$ and a source vertex $s$, a reachable negative cycle consists of a base vertex $b$, a weighted path $p$ from $s$ to $b$ using edges of the list whose vertices along $p$ from $s$ are pairwise distinct, and a nonempty weighted path $c$ from $b$ to $b$ using edges of the list whose edges have pairwise distinct sources and whose weight is negative.","title":"Reachable negative cycle witness"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/has-reachable-negative-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.HasReachableNegativeCycle","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#has-reachable-negative-cycle","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of weighted directed edges on the vertices $0, \\dots, n-1$ and $s$ a vertex. Then $E$ has a negative cycle reachable from $s$ when there is a reachable negative cycle witness for $E$ and $s$: a base vertex $b$, a weighted path in $E$ from $s$ to $b$ that visits no vertex twice, and a nonempty weighted path in $E$ from $b$ to $b$ whose edge sources are pairwise distinct and whose total weight is\u2026","title":"Existence of a reachable negative cycle"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/distances","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.Distances","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#distances","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For a graph on the vertices $0, \\dots, n-1$, the output of a successful Bellman-Ford run consists of a source vertex $s$, for every vertex a distance that is either an integer or $\\infty$, and for every vertex an optional predecessor edge (a weighted directed edge, or none).","title":"Bellman-Ford distance table"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/path-to","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.Distances.pathTo?","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#path-to","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let the output of a Bellman-Ford run on $n$ vertices have source $s$. The path to a vertex $t$ is built backwards from $t$: at a vertex $v \\ne s$, take the recorded predecessor edge $e$ of $v$, put it in front of the edges collected so far, and continue from the source of $e$. When $s$ is reached within at most $n$ such steps, the result is the collected list of edges (empty if $t = s$). If some vertex on the way\u2026","title":"Bellman-Ford path reconstruction"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/cost","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.Cost","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#cost","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"The cost of a Bellman-Ford run is a pair of natural numbers: the number of edge occurrences checked by relaxation passes, and the number of predecessor edges read or emitted while reconstructing a negative-cycle witness.","title":"Bellman-Ford cost counters"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/total","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.Cost.total","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#total","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"The total of a Bellman\u2013Ford cost record is its number of edge relaxation checks plus its number of negative-cycle witness reconstruction steps.","title":"Total Bellman\u2013Ford cost"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/result","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.Result","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#result","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For a list of weighted edges on the vertices $\\{0, \\dots, n-1\\}$ and a source vertex $s$, a Bellman\u2013Ford result is either a distance record (a source vertex, an optional integer distance for each vertex and an optional predecessor edge for each vertex) or a reachable negative cycle witness for the list and $s$.","title":"Bellman\u2013Ford result"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. Bellman\u2013Ford run on $E$ from $s$ starts with distance $d(s) = 0$ and $d(v) = \\infty$ for every other vertex $v$, and makes $n - 1$ passes over $E$ followed by one final pass. A pass goes through the edges of $E$ in list order, charging one relaxation check per edge occurrence; an edge from $u$ to $v$ of\u2026","title":"Bellman\u2013Ford algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-returns-distances-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_returns_distances_iff","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-returns-distances-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. Bellman-Ford run on $E$ from $s$ returns distances if and only if there is no negative cycle reachable from $s$, that is, no simple path from $s$ to a vertex $b$ together with a nonempty cycle through $b$ along edges of $E$ whose edge sources are pairwise distinct and whose total weight is negative.","title":"Bellman-Ford succeeds exactly when no negative cycle is reachable"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-returns-negative-cycle-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_returns_negativeCycle_iff","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-returns-negative-cycle-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. Bellman-Ford run on $E$ from $s$ returns a reachable negative cycle witness if and only if a negative cycle is reachable from $s$, that is, there are a simple path from $s$ to a vertex $b$ and a nonempty cycle through $b$ along edges of $E$ whose edge sources are pairwise distinct and whose total weight\u2026","title":"Bellman-Ford reports a negative cycle exactly when one is reachable"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-distance-realized","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_distance_realized","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-distance-realized","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. If Bellman-Ford run on $E$ from $s$ returns distances $d$ and the distance $d(t)$ of a vertex $t$ is a finite integer $\\delta$, then there is a walk from $s$ to $t$ along edges of $E$ whose total weight is $\\delta$.","title":"Bellman-Ford distances are realized by walks"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-distance-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_distance_minimal","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-distance-minimal","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. If Bellman-Ford run on $E$ from $s$ returns distances $d$, the distance $d(t)$ of a vertex $t$ is a finite integer $\\delta$, and $P$ is a walk from $s$ to $t$ along edges of $E$ (vertices may repeat), then $\\delta$ is at most the total weight of $P$.","title":"Bellman-Ford distances are lower bounds on walk weights"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-distance-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_distance_eq_none_iff","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-distance-eq-none-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. If Bellman-Ford run on $E$ from $s$ returns distances $d$, with $\\infty$ marking a vertex that has no finite distance, then for every vertex $t$, $d(t) = \\infty$ if and only if $t$ is not reachable from $s$ in the directed graph obtained from $E$ by forgetting the weights.","title":"Bellman-Ford leaves exactly the unreachable vertices at infinite distance"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-predecessor-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_predecessor_spec","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-predecessor-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. Suppose Bellman-Ford run on $E$ from $s$ returns distances $d$ together with predecessor edges, and the predecessor edge of a vertex $t$ is an edge $e$ from $u$ to $v$ of weight $w$. Then $e$ belongs to $E$, $v = t$, and $d(u)$ and $d(t)$ are both finite with $d(t) = d(u) + w$.","title":"Bellman-Ford predecessor edges are tight"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-source-predecessor-eq-none","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_source_predecessor_eq_none","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-source-predecessor-eq-none","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. If Bellman-Ford run on $E$ from $s$ returns distances together with predecessor edges, then $s$ has no predecessor edge.","title":"Bellman-Ford gives the source no predecessor"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-path-to-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_pathTo?_eq_none_iff","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-path-to-eq-none-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. If Bellman-Ford run on $E$ from $s$ returns distances $d$, with $\\infty$ marking a vertex that has no finite distance, together with predecessor edges, then for every vertex $t$, reconstructing a path to $t$ (following predecessor edges back from $t$ for at most $n$ steps until $s$ is reached) fails if\u2026","title":"Bellman-Ford path reconstruction fails exactly at infinite distances"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-path-to-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_pathTo?_spec","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-path-to-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ be a vertex. Suppose Bellman-Ford run on $E$ from $s$ returns distances $d$ together with predecessor edges, and that reconstructing a path to a vertex $t$ (following predecessor edges back from $t$ for at most $n$ steps until $s$ is reached) yields the edge sequence $P$. Then $P$ is a walk from $s$ to $t$ along\u2026","title":"Bellman-Ford reconstructed paths are simple and realize the distances"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/path-to-spec-of-rank","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.Distances.pathTo?_spec_of_rank","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#path-to-spec-of-rank","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"Let $R$ be a distances record on $n$ vertices with source $s$, distance table $d$ (each $d(v)$ is an integer or infinite) and predecessor table $p$; let $E$ be a list of weighted edges; and let $r$ assign to each vertex $v$ a natural number $r(v) < n$. Suppose that $d(s) = 0$, that every vertex $v$ with $d(v)$ infinite has no predecessor, that every vertex $v \\ne s$ with $d(v)$ finite has a predecessor, and that\u2026","title":"Predecessor paths realize the distances under a decreasing rank"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-relaxation-checks","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_relaxationChecks","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-relaxation-checks","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For every list $E$ of weighted edges on $n$ vertices and every source $s$, Bellman-Ford run on $E$ from $s$ charges exactly $(n - 1) \\cdot |E| + |E|$ relaxation checks, where $|E|$ is the length of $E$.","title":"Relaxation checks of Bellman-Ford pass by pass"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-relaxation-checks-eq-mul","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_relaxationChecks_eq_mul","parent":38683,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bellman-Ford makes exactly "],["math","\\(n \\cdot |E|\\)"],["text"," relaxation checks"]],"source":"Bellman-Ford makes exactly $n \\cdot |E|$ relaxation checks"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-relaxation-checks-eq-mul","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For every list $E$ of weighted edges on $n$ vertices and every source $s$, Bellman-Ford run on $E$ from $s$ charges exactly $n \\cdot |E|$ relaxation checks, where $|E|$ is the length of $E$ counting repeated edges.","title":"Bellman-Ford makes exactly $n \\cdot |E|$ relaxation checks"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-cycle-witness-steps-le","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_cycleWitnessSteps_le","parent":38683,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bellman-Ford spends at most "],["math","\\(3n\\)"],["text"," witness steps"]],"source":"Bellman-Ford spends at most $3n$ witness steps"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-cycle-witness-steps-le","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For every list $E$ of weighted edges on $n$ vertices and every source $s$, Bellman-Ford run on $E$ from $s$ charges at most $3n$ negative-cycle witness reconstruction steps.","title":"Bellman-Ford spends at most $3n$ witness steps"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-cycle-witness-steps-eq-zero","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_cycleWitnessSteps_eq_zero","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-cycle-witness-steps-eq-zero","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For every list $E$ of weighted edges on $n$ vertices and every source $s$, if Bellman-Ford run on $E$ from $s$ returns distances rather than a negative cycle, then the run charges zero negative-cycle witness reconstruction steps.","title":"Bellman-Ford spends no witness steps when it returns distances"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-cycle-witness-steps-eq","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_cycleWitnessSteps_eq","parent":38683,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-cycle-witness-steps-eq","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For every list $E$ of weighted edges on $n$ vertices and every source $s$, if Bellman-Ford run on $E$ from $s$ returns a reachable negative cycle whose cycle has $c$ edges and whose path from $s$ to the cycle has $q$ edges, then the run charges exactly $n + c + q$ negative-cycle witness reconstruction steps.","title":"Witness cost of a returned negative cycle"},{"id":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/bellman-ford-total-le","kind":"node","lean":"Cslib.Algorithms.Lean.BellmanFord.bellmanFord_total_le","parent":38683,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bellman-Ford makes at most "],["math","\\(4 n |E|\\)"],["text"," counted steps"]],"source":"Bellman-Ford makes at most $4 n |E|$ counted steps"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm/index.html#bellman-ford-total-le","scope":"ComputerScience/graph-algorithms/shortest-paths/bellman-ford-algorithm","status":"fully_proved","summary":"For every list $E$ of weighted edges on $n$ vertices and every source $s$, the total cost of Bellman-Ford run on $E$ from $s$ (its relaxation checks plus its negative-cycle witness reconstruction steps) is at most $4 \\cdot n \\cdot |E|$, where $|E|$ is the length of $E$.","title":"Bellman-Ford makes at most $4 n |E|$ counted steps"},{"data_url":"scopes/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm.json","id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","kind":"scope","navigation":"module","parent":38682,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","title":"Dijkstra's algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/path-weight","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.pathWeight","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#path-weight","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Given a weight $w(u, v) \\in \\mathbb{N}$ for each ordered pair of vertices, the weight of a list of vertices $v_0, v_1, \\dots, v_k$ is $w(v_0, v_1) + w(v_1, v_2) + \\dots + w(v_{k-1}, v_k)$. In particular the empty list and every one-vertex list have weight $0$.","title":"Endpoint-weight sum along a vertex list"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/has-path-weight","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.HasPathWeight","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#has-path-weight","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph given by adjacency lists, let $w(u, v) \\in \\mathbb{N}$ be a weight for each ordered pair of vertices, let $s$ and $t$ be vertices and let $d$ be a natural number. Then $s$ reaches $t$ with weight $d$ when there is a nonempty list of vertices $v_0, v_1, \\dots, v_k$ with $v_0 = s$, $v_k = t$ and each $v_{i+1}$ in the adjacency list of $v_i$, such that\u2026","title":"Walks of a given weight"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/reachable-iff-exists-has-path-weight","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.reachable_iff_exists_hasPathWeight","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#reachable-iff-exists-has-path-weight","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph given by adjacency lists and let $w(u, v) \\in \\mathbb{N}$ be a weight for each ordered pair of vertices. For vertices $s$ and $t$, $t$ is reachable from $s$ in $G$ if and only if there is a natural number $d$ such that $s$ reaches $t$ with weight $d$ with respect to $w$.","title":"Reachability via weighted walks"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/adjacency-entry-count","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.adjacencyEntryCount","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#adjacency-entry-count","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"For a directed graph $G$ given by an out-neighbour list $\\mathrm{adj}(v)$ for each vertex $v$, the adjacency entry count of $G$ is the total length $\\sum_{v} |\\mathrm{adj}(v)|$ of these lists over all vertices $v$, so repeated entries and self-loops are counted with multiplicity.","title":"Adjacency entry count"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-cost","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.DijkstraCost","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-cost","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"A Dijkstra cost is a pair of natural numbers that counts two kinds of events separately: the number of vertex entries inspected during minimum selection, and the number of adjacency-list entries inspected during relaxation.","title":"Cost of dense Dijkstra"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/total","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.DijkstraCost.total","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#total","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"The total of a Dijkstra cost record is its number of vertex entries inspected by minimum selection plus its number of adjacency entries inspected by relaxation.","title":"Total Dijkstra cost"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-result","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.DijkstraResult","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-result","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"For vertices $0, \\dots, n-1$, a Dijkstra result consists of a source vertex $s$, a distance entry $d(v)$ for each vertex $v$ that is either a natural number or $\\infty$, and a predecessor entry $p(v)$ for each vertex $v$ that is either a vertex or undefined. The entries are stored as arrays of length $n$; $d(v)$ is meant as the computed distance from $s$ to $v$ and $p(v)$ as the vertex before $v$ on a shortest path\u2026","title":"Output of Dijkstra's algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/path-to","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.DijkstraResult.pathTo?","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#path-to","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let a Dijkstra result on $n$ vertices have source $s$ and predecessor entries $p$. The path to a vertex $t$ follows predecessors backwards: starting from $u_0 = t$, as long as $u_i \\ne s$ it sets $u_{i+1} = p(u_i)$. If $u_k = s$ for some $k \\le n$, the result is the list $u_k, u_{k-1}, \\dots, u_0$ from $s$ to $t$. If some $p(u_i)$ is undefined before $s$ is reached, or $s$ is not reached within $n$ steps, there is\u2026","title":"Path recovery from Dijkstra predecessors"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$ given by a fixed enumeration of its vertices and adjacency lists, let $w(u, v) \\in \\mathbb{N}$ be a weight for each ordered pair of vertices, and let $s$ be a source vertex. The algorithm starts with $d(s) = 0$, $d(v) = \\infty$ for $v \\ne s$, no predecessors and no settled vertices, and runs $n$ rounds. In each round it scans all vertices in the order of\u2026","title":"Dense Dijkstra's algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-distance-realized","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_distance_realized","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-distance-realized","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, let $w(u, v)$ be a natural-number weight for every ordered pair of vertices, and let $s$ and $t$ be vertices. If Dijkstra's algorithm on $G$ with weights $w$ from source $s$ outputs a finite distance $d$ for $t$, then $G$ has a walk $s = v_0, v_1, \\dots, v_k = t$, with an edge from each $v_{i-1}$ to $v_i$, of weight $\\sum_{i=1}^{k} w(v_{i-1}, v_i) = d$.","title":"Dijkstra's finite distances are realized by walks"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-distance-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_distance_minimal","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-distance-minimal","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, let $w(u, v)$ be a natural-number weight for every ordered pair of vertices, and let $s$ and $t$ be vertices. If Dijkstra's algorithm on $G$ with weights $w$ from source $s$ outputs a finite distance $d$ for $t$, then $d \\le d'$ for every walk $s = v_0, v_1, \\dots, v_k = t$ in $G$ (with an edge from each $v_{i-1}$ to $v_i$) of weight\u2026","title":"Dijkstra's finite distances are lower bounds on walk weights"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-distance-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_distance_eq_none_iff","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-distance-eq-none-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, let $w(u, v)$ be a natural-number weight for every ordered pair of vertices, and let $s$ and $t$ be vertices. The distance to $t$ output by Dijkstra's algorithm on $G$ with weights $w$ from source $s$ is infinite if and only if $t$ is not reachable from $s$ in $G$.","title":"Dijkstra's distance is infinite exactly at unreachable vertices"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-predecessor-spec","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_predecessor_spec","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-predecessor-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, let $w(u, v)$ be a natural-number weight for every ordered pair of vertices, and let $s$, $t$ and $p$ be vertices. If Dijkstra's algorithm on $G$ with weights $w$ from source $s$ outputs $p$ as the predecessor of $t$, then $G$ has an edge from $p$ to $t$, the distances $d(p)$ and $d(t)$ output for $p$ and $t$ are both finite, and $d(t) = d(p) + w(p, t)$.","title":"Dijkstra's predecessor pointers satisfy the distance equation"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-path-to-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_pathTo?_eq_none_iff","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-path-to-eq-none-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, let $w(u, v)$ be a natural-number weight for every ordered pair of vertices, and let $s$ and $t$ be vertices. Reconstructing a path from $s$ to $t$ by following, for at most $n$ steps, the predecessor pointers output by Dijkstra's algorithm on $G$ with weights $w$ from source $s$ fails if and only if the distance output for $t$ is infinite.","title":"Dijkstra's path reconstruction fails exactly at infinite distance"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-path-to-spec","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_pathTo?_spec","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-path-to-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, let $w(u, v)$ be a natural-number weight for every ordered pair of vertices, and let $s$ and $t$ be vertices. Suppose that following the predecessor pointers output by Dijkstra's algorithm on $G$ with weights $w$ from source $s$ reconstructs a list of vertices $P$ for $t$. Then $P$ is nonempty, say $P = v_0, \\dots, v_k$, with $v_0 = s$ and $v_k = t$;\u2026","title":"Reconstructed Dijkstra paths are simple and have the output weight"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-edge-relaxations","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_edgeRelaxations","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-edge-relaxations","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, let $w(u, v)$ be a natural-number weight for every ordered pair of vertices, and let $s$ be a vertex. The number of adjacency-list entries that Dijkstra's algorithm on $G$ with weights $w$ from source $s$ inspects during relaxation equals the adjacency entry count of $G$, the total length of all out-neighbour lists.","title":"Relaxation count of dense Dijkstra"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-min-scans","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_minScans","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-min-scans","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$, let $w(u, v)$ be a natural-number weight for every ordered pair of vertices, and let $s$ be a vertex. Dijkstra's algorithm on $G$ with weights $w$ from source $s$ inspects exactly $n^2$ vertex entries during minimum selection.","title":"Minimum-selection count of dense Dijkstra"},{"id":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/dijkstra-time","kind":"node","lean":"Cslib.Algorithms.Lean.DirectedAdjacency.dijkstra_time","parent":38719,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm/index.html#dijkstra-time","scope":"ComputerScience/graph-algorithms/shortest-paths/dijkstra-s-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $0, \\dots, n-1$ given by adjacency lists, let $w(u, v) \\in \\mathbb{N}$ be a weight for each ordered pair of vertices, and let $s$ be a vertex. For dense Dijkstra from $s$, the number of vertex entries inspected by minimum selection plus the number of adjacency entries inspected by relaxation is exactly $n^2 + m$, where $m$ is the number of adjacency-list entries of $G$,\u2026","title":"Running time of dense Dijkstra"},{"data_url":"scopes/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm.json","id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","kind":"scope","navigation":"module","parent":38682,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","title":"Floyd\u2013Warshall algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/path-weight","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.pathWeight","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#path-weight","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$ a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$. The weight of a list of vertices $v_0, v_1, \\dots, v_m$ with $m \\ge 1$ is $w(v_0, v_1) + w(v_1, v_2) + \\cdots + w(v_{m-1}, v_m)$, computed in $\\mathbb{Z} \\cup \\{\\infty\\}$ where adding $\\infty$ gives $\\infty$. The empty list and every single-vertex list have weight $0$.","title":"Weight of a vertex path in an extended-integer weight matrix"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/is-weighted-path","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.IsWeightedPath","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#is-weighted-path","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$ a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$, and let $s$ and $t$ be vertices. A list of vertices $p$ is a weighted path from $s$ to $t$ when $p$ is nonempty, its first entry is $s$, its last entry is $t$, and $w(u, v) \\ne \\infty$ for every pair of consecutive entries $u, v$ of $p$.","title":"Finite-weight vertex path in a weight matrix"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/internal-vertices-below","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.InternalVerticesBelow","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#internal-vertices-below","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"For a natural number $k$, a list of vertices in $\\{0, \\dots, n - 1\\}$ has all its intermediate vertices below $k$ when every entry of the list other than its first and its last is less than $k$.","title":"Intermediate vertices below a bound"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/is-stage-shortest-path-weight","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.IsStageShortestPathWeight","parent":38738,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stage-"],["math","\\(k\\)"],["text"," shortest-path weight"]],"source":"Stage-$k$ shortest-path weight"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#is-stage-shortest-path-weight","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$ a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$, let $k$ be a natural number, let $s$ and $t$ be vertices and let $d \\in \\mathbb{Z} \\cup \\{\\infty\\}$. A weighted path from $s$ to $t$ is a nonempty list of vertices that starts at $s$, ends at $t$ and has $w(u, v) \\ne \\infty$ for every pair of consecutive vertices $u, v$; its weight is the sum of these\u2026","title":"Stage-$k$ shortest-path weight"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/is-negative-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.IsNegativeCycle","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#is-negative-cycle","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"For an $n \\times n$ matrix $w$ of weights in $\\mathbb{Z} \\cup \\{+\\infty\\}$, a list of vertices $v_0, \\dots, v_m$ is a negative cycle when $m \\ge 1$, $v_m = v_0$, every weight $w_{v_i v_{i+1}}$ with $i < m$ is finite, the vertices $v_0, \\dots, v_{m-1}$ are pairwise distinct, and the path weight $\\sum_{i < m} w_{v_i v_{i+1}}$ is negative.","title":"Negative cycles"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/has-negative-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.HasNegativeCycle","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#has-negative-cycle","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"An $n \\times n$ matrix $w$ of weights in $\\mathbb{Z} \\cup \\{+\\infty\\}$ has a negative cycle when some list of vertices is a negative cycle for $w$.","title":"Matrices with a negative cycle"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/no-negative-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.NoNegativeCycle","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#no-negative-cycle","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$ a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$. Then $w$ has no negative cycle when there is no list of vertices $v_0, v_1, \\dots, v_m$ with $m \\ge 1$ such that $v_m = v_0$, the vertices $v_0, \\dots, v_{m-1}$ are pairwise distinct, $w(v_{i-1}, v_i) \\ne \\infty$ for $1 \\le i \\le m$, and $w(v_0, v_1) + \\cdots + w(v_{m-1}, v_m) < 0$.","title":"Weight matrices without negative cycles"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/cost","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.Cost","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#cost","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"A cost record for the Floyd\u2013Warshall algorithm is a pair of natural numbers: the number of initialization events (input matrix cells copied into the initial distance table) and the number of min-plus transition events (table entries updated).","title":"Floyd\u2013Warshall cost record"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/total","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.Cost.total","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#total","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"The total of a Floyd\u2013Warshall cost record is its number of initialization events plus its number of min-plus transition events.","title":"Total Floyd\u2013Warshall cost"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Given an $n \\times n$ matrix $w$ of weights in $\\mathbb{Z} \\cup \\{+\\infty\\}$ indexed by the vertices $0, \\dots, n - 1$, the Floyd\u2013Warshall algorithm starts from the table with $d^{(0)}_{ii} = \\min(w_{ii}, 0)$ and $d^{(0)}_{ij} = w_{ij}$ for $i \\ne j$, and for $k = 0, \\dots, n - 1$ computes the table $d^{(k+1)}_{ij} = \\min\\bigl(d^{(k)}_{ij}, d^{(k)}_{ik} + d^{(k)}_{kj}\\bigr)$. It returns the matrix $d^{(n)}$\u2026","title":"Floyd\u2013Warshall algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/has-negative-diagonal","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.hasNegativeDiagonal","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#has-negative-diagonal","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"For an $n \\times n$ matrix $d$ with entries in $\\mathbb{Z} \\cup \\{+\\infty\\}$, the negative-diagonal test on $d$ is the Boolean value that is true exactly when $d_{vv} < 0$ for some vertex $v$.","title":"Negative-diagonal test"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-initializations","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_initializations","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-initializations","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"For every $n$ and every $n \\times n$ matrix of weights in $\\mathbb{Z} \\cup \\{+\\infty\\}$, the Floyd\u2013Warshall algorithm performs exactly $n^2$ initialization events, one for each ordered pair of vertices.","title":"Initialization count of Floyd\u2013Warshall"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-transitions","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_transitions","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-transitions","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"For every $n$ and every $n \\times n$ matrix of weights in $\\mathbb{Z} \\cup \\{+\\infty\\}$, the Floyd\u2013Warshall algorithm performs exactly $n^3$ min-plus transition events, one for each triple $(k, i, j)$ of vertices.","title":"Transition count of Floyd\u2013Warshall"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-time","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_time","parent":38738,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floyd\u2013Warshall performs "],["math","\\(n^2 + n^3\\)"],["text"," steps"]],"source":"Floyd\u2013Warshall performs $n^2 + n^3$ steps"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-time","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"For every $n$ and every $n \\times n$ matrix of weights in $\\mathbb{Z} \\cup \\{+\\infty\\}$, the total number of events counted by the Floyd\u2013Warshall algorithm, initialization events plus min-plus transition events, is exactly $n^2 + n^3$.","title":"Floyd\u2013Warshall performs $n^2 + n^3$ steps"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/has-negative-diagonal-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.hasNegativeDiagonal_eq_true_iff","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#has-negative-diagonal-eq-true-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"For every $n \\times n$ matrix $d$ with entries in $\\mathbb{Z} \\cup \\{+\\infty\\}$, the negative-diagonal test on $d$ returns true if and only if $d_{vv} < 0$ for some vertex $v$.","title":"Characterization of the negative-diagonal test"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-distance-spec","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_distance_spec","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-distance-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$ to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$, suppose that $w$ has no negative cycle, and let $D$ be the distance matrix computed by Floyd\u2013Warshall from $w$. Then for all vertices $s$ and $t$, $D(s, t)$ is at most the weight of every weighted path from $s$ to $t$, and either $D(s, t) = \\infty$ or $D(s, t)$ is the weight of some weighted path from\u2026","title":"Floyd\u2013Warshall computes shortest-path distances"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-distance-realized","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_distance_realized","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-distance-realized","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$ to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$, and let $D$ be the distance matrix computed by Floyd\u2013Warshall from $w$. If $D(s, t) = d$ for vertices $s$, $t$ and an integer $d$, then there is a weighted path from $s$ to $t$ (a nonempty vertex sequence from $s$ to $t$ in which every consecutive pair has finite weight) whose weight, the sum of the\u2026","title":"Finite Floyd\u2013Warshall distances are attained"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-distance-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_distance_minimal","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-distance-minimal","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$ to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$, suppose that $w$ has no negative cycle, and let $D$ be the distance matrix computed by Floyd\u2013Warshall from $w$. Then for all vertices $s$ and $t$ and every weighted path $p$ from $s$ to $t$ (a nonempty vertex sequence from $s$ to $t$ in which every consecutive pair has finite weight), $D(s, t)$ is at\u2026","title":"No path is shorter than the Floyd\u2013Warshall distance"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-distance-eq-top-iff","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_distance_eq_top_iff","parent":38738,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floyd\u2013Warshall reports "],["math","\\(\\infty\\)"],["text"," exactly for unreachable pairs"]],"source":"Floyd\u2013Warshall reports $\\infty$ exactly for unreachable pairs"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-distance-eq-top-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$ to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$, and let $D$ be the distance matrix computed by Floyd\u2013Warshall from $w$. For all vertices $s$ and $t$, $D(s, t) = \\infty$ if and only if there is no weighted path from $s$ to $t$, that is, no nonempty vertex sequence from $s$ to $t$ in which every consecutive pair has finite weight.","title":"Floyd\u2013Warshall reports $\\infty$ exactly for unreachable pairs"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-has-negative-diagonal-sou","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_hasNegativeDiagonal_sound","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-has-negative-diagonal-sou","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$ to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$, and let $D$ be the distance matrix computed by Floyd\u2013Warshall from $w$. If $D(v, v) < 0$ for some vertex $v$, then $w$ has a negative cycle: a vertex sequence $v_0, v_1, \\dots, v_m$ with $m \\ge 1$ and $v_m = v_0$, whose vertices $v_0, \\dots, v_{m-1}$ are pairwise distinct, in which every consecutive pair\u2026","title":"A negative diagonal entry in Floyd\u2013Warshall reveals a negative cycle"},{"id":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/floyd-warshall-has-negative-diagonal-eq","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.floydWarshall_hasNegativeDiagonal_eq_false","parent":38738,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm/index.html#floyd-warshall-has-negative-diagonal-eq","scope":"ComputerScience/graph-algorithms/shortest-paths/floyd-warshall-algorithm","status":"fully_proved","summary":"Let $w$ assign a weight in $\\mathbb{Z} \\cup \\{\\infty\\}$ to each ordered pair of vertices in $\\{0, \\dots, n-1\\}$, and let $D$ be the distance matrix computed by Floyd\u2013Warshall from $w$. If $w$ has no negative cycle, then the negative-diagonal check on $D$ returns false, that is, $D(v, v) \\ge 0$ for every vertex $v$.","title":"Without negative cycles, Floyd\u2013Warshall finds no negative diagonal entry"},{"data_url":"scopes/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products.json","id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","kind":"scope","navigation":"module","parent":38682,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","title":"All-pairs shortest paths by min-plus products"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/is-edge-bounded-shortest-path-weight","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.IsEdgeBoundedShortestPathWeight","parent":38760,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shortest path weight with at most "],["math","\\(r\\)"],["text"," edges"]],"source":"Shortest path weight with at most $r$ edges"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#is-edge-bounded-shortest-path-weight","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $w$ be an $n \\times n$ matrix of weights in $\\mathbb{Z} \\cup \\{\\infty\\}$ on the vertices $\\{0, \\dots, n-1\\}$, let $r$ be a natural number, let $i$ and $j$ be vertices, and let $d \\in \\mathbb{Z} \\cup \\{\\infty\\}$. A weighted path from $i$ to $j$ is a nonempty list $p$ of vertices that starts at $i$, ends at $j$ and has finite weight on each pair of consecutive vertices; its number of edges is its length minus\u2026","title":"Shortest path weight with at most $r$ edges"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/is-edge-bounded-shortest-path-weight-iff","kind":"node","lean":"Cslib.Algorithms.Lean.FloydWarshall.isEdgeBoundedShortestPathWeight_iff","parent":38760,"reading_title":{"format":"inline-parts-v1","parts":[["text","Characterization of the shortest path weight with at most "],["math","\\(r\\)"],["text"," edges"]],"source":"Characterization of the shortest path weight with at most $r$ edges"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#is-edge-bounded-shortest-path-weight-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $w$ be an $n \\times n$ matrix of weights in $\\mathbb{Z} \\cup \\{\\infty\\}$ on the vertices $\\{0, \\dots, n-1\\}$, let $r$ be a natural number, let $i$ and $j$ be vertices, and let $d \\in \\mathbb{Z} \\cup \\{\\infty\\}$. For a weighted path $p$, write $w(p)$ for its weight. Then $d$ is the shortest weight of a path from $i$ to $j$ with at most $r$ edges if and only if $d \\le w(p)$ for every weighted path $p$ from $i$ to\u2026","title":"Characterization of the shortest path weight with at most $r$ edges"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/matrix-accumulate-min-plus-apply","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.matrixAccumulate_minPlus_apply","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#matrix-accumulate-min-plus-apply","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $A$, $B$ and $C$ be $n \\times n$ matrices with entries in $\\mathbb{Z} \\cup \\{\\infty\\}$, viewed in the min-plus semiring. For all $i, j \\in \\{0, \\dots, n-1\\}$, the entry $(i, j)$ of the matrix returned by the matrix accumulator on $A$, $B$ and the accumulator $C$ equals $\\min\\bigl(C_{ij}, \\min_{k} (A_{ik} + B_{kj})\\bigr)$, where $k$ ranges over $\\{0, \\dots, n-1\\}$.","title":"Entries of the min-plus matrix accumulator"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/matrix-accumulate-edge-bounded","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.matrixAccumulate_edgeBounded","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#matrix-accumulate-edge-bounded","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $A$ and $B$ be $n \\times n$ matrices with entries in $\\mathbb{Z} \\cup \\{\\infty\\}$, viewed in the min-plus semiring, and let $r$ be a natural number. Suppose that $B_{jj} = 0$ for every $j$, and that for all $i$ and $j$ the entry $A_{ij}$ is the shortest weight of a path from $i$ to $j$ with at most $r$ edges for the weight matrix $B$. Then for all $i$ and $j$, the entry $(i, j)$ of the matrix returned by the\u2026","title":"One min-plus multiplication extends shortest paths by one edge"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/matrix-pow-edge-bounded","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.matrix_pow_edgeBounded","parent":38760,"reading_title":{"format":"inline-parts-v1","parts":[["text","Min-plus matrix powers give shortest path weights with at most "],["math","\\(r\\)"],["text"," edges"]],"source":"Min-plus matrix powers give shortest path weights with at most $r$ edges"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#matrix-pow-edge-bounded","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $W$ be an $n \\times n$ matrix with entries in $\\mathbb{Z} \\cup \\{\\infty\\}$, viewed in the min-plus semiring, with $W_{ii} = 0$ for every $i$. Then for every natural number $r$ and all $i$ and $j$, the entry $(W^r)_{ij}$ of the $r$-th min-plus power of $W$ is the shortest weight of a path from $i$ to $j$ with at most $r$ edges for the weight matrix $W$.","title":"Min-plus matrix powers give shortest path weights with at most $r$ edges"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/copy-cells","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.copyCells","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#copy-cells","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Given an $n \\times n$ matrix $A$ (a vector of $n$ rows, each a vector of $n$ entries), this computation returns the matrix whose $(i, j)$ entry is $A_{ij}$, traversing the entries row by row and charging a cost of $(1, (0, 0))$ for each entry, that is, one logical assignment in the first cost coordinate and nothing in the other two.","title":"Counted copy of a square matrix"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/copy-cells-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.copyCells_ret","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#copy-cells-ret","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"For every $n \\times n$ matrix $A$ with entries in any type, the value returned by the counted copy of $A$ is $A$ itself.","title":"The counted matrix copy returns its input"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/copy-cells-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.copyCells_time","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#copy-cells-time","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"For every $n \\times n$ matrix $A$ with entries in any type, the counted copy of $A$ has total cost $(n^2, (0, 0))$: exactly $n^2$ logical assignments in the first cost coordinate and zero in the other two.","title":"Cost of the counted matrix copy"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/fill-cells","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fillCells","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#fill-cells","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"For a type $\\alpha$ with a zero element and a natural number $n$, this computation returns the $n \\times n$ matrix over $\\alpha$ all of whose entries are $0$, producing the entries row by row and charging a cost of $(1, (0, 0))$ for each entry, that is, one logical assignment in the first cost coordinate and nothing in the other two.","title":"Counted zero initialization of a square matrix"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/fill-cells-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fillCells_ret","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#fill-cells-ret","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"For every type $\\alpha$ with a zero element and every natural number $n$, the counted zero initialization of size $n$ over $\\alpha$ returns the $n \\times n$ matrix all of whose entries are $0$.","title":"The counted zero initialization returns the zero matrix"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/fill-cells-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fillCells_time","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#fill-cells-time","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"For every type $\\alpha$ with a zero element and every natural number $n$, the counted zero initialization of size $n$ over $\\alpha$ has total cost $(n^2, (0, 0))$: exactly $n^2$ logical assignments in the first cost coordinate and zero in the other two.","title":"Cost of the counted zero initialization"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/faster-apsp","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fasterAPSP","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#faster-apsp","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $W$ be an $n \\times n$ matrix over a type with addition, multiplication and zero. FASTER-APSP copies $W$ into a matrix $L$ and sets $r = 1$. While $r < n - 1$, it fills a new $n \\times n$ matrix $M$ with zeros, sets each entry to $M_{ij} = 0 + L_{i0} L_{0j} + \\dots + L_{i,n-1} L_{n-1,j}$ (adding the terms from left to right), copies $M$ into $L$ and doubles $r$. It returns the final $L$. The run also records\u2026","title":"FASTER-APSP"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/faster-apsp-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fasterAPSP_ret","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#faster-apsp-ret","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $\\alpha$ be a semiring, let $W$ be an $n \\times n$ matrix over $\\alpha$, and let $k = \\lceil \\log_2 (n - 1) \\rceil$ be the least natural number with $n - 1 \\le 2^k$ (so $k = 0$ when $n \\le 2$). The matrix returned by FASTER-APSP on $W$ is the matrix power $W^{2^k}$, with products computed in $\\alpha$.","title":"FASTER-APSP returns a power of the input matrix"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/faster-apsp-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fasterAPSP_time","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#faster-apsp-time","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $\\alpha$ be a type with addition, multiplication and zero, let $W$ be an $n \\times n$ matrix over $\\alpha$, and let $k = \\lceil \\log_2 (n - 1) \\rceil$ be the least natural number with $n - 1 \\le 2^k$. On input $W$, FASTER-APSP performs exactly $n^2 + 2 k n^2$ copy and fill assignments, $k n^3$ scalar additions and $k n^3$ scalar multiplications.","title":"Operation counts of FASTER-APSP"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/faster-apsp-time-is-theta","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fasterAPSP_time_isTheta","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#faster-apsp-time-is-theta","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $\\alpha$ be a type with addition, multiplication and zero, and for each $n$ let $W_n$ be an $n \\times n$ matrix over $\\alpha$. The total number of copy and fill assignments, scalar additions and scalar multiplications that FASTER-APSP performs on $W_n$ is $\\Theta(n^3 \\log n)$ as $n \\to \\infty$.","title":"Asymptotic operation count of FASTER-APSP"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/slow-apsp","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.slowAPSP","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#slow-apsp","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $W$ and $L^{(0)}$ be $n \\times n$ matrices over the min-plus semiring on $\\mathbb{Z} \\cup \\{\\infty\\}$, in which addition is $\\min$ and multiplication is $+$. SLOW-APSP copies $L^{(0)}$ cell by cell and then runs $\\max(n - 1, 0)$ phases. Phase $r$ fills a fresh $n \\times n$ matrix $M$ with $\\infty$, then for each triple $(i, j, k)$ in increasing $i$-$j$-$k$ order sets\u2026","title":"Slow all-pairs shortest paths (SLOW-APSP)"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/slow-apsp-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.slowAPSP_ret","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#slow-apsp-ret","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"For $n \\times n$ matrices $W$ and $L^{(0)}$ over the min-plus semiring on $\\mathbb{Z} \\cup \\{\\infty\\}$, in which addition is $\\min$ and multiplication is $+$, the matrix returned by SLOW-APSP on $W$ and $L^{(0)}$ equals the min-plus matrix product $L^{(0)} W^{n-1}$.","title":"SLOW-APSP returns a min-plus matrix power"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/slow-apsp-edge-bounded","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.slowAPSP_edgeBounded","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#slow-apsp-edge-bounded","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $W$ and $L^{(0)}$ be $n \\times n$ matrices over the min-plus semiring on $\\mathbb{Z} \\cup \\{\\infty\\}$, and let $w_{xy}$ be the underlying value in $\\mathbb{Z} \\cup \\{\\infty\\}$ of $W_{xy}$. Suppose that $w_{ii} = 0$ for every $i$ and that the underlying value of $L^{(0)}_{ij}$ is $0$ if $i = j$ and $\\infty$ otherwise. Then for all $i, j \\in \\{0, \\dots, n-1\\}$, the underlying value $d$ of the $(i, j)$ entry of\u2026","title":"Bounded-edge shortest weights from SLOW-APSP"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/slow-apsp-distance-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.slowAPSP_distance_spec","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#slow-apsp-distance-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $W$ and $L^{(0)}$ be $n \\times n$ matrices over the min-plus semiring on $\\mathbb{Z} \\cup \\{\\infty\\}$, and let $w_{xy}$ be the underlying value in $\\mathbb{Z} \\cup \\{\\infty\\}$ of $W_{xy}$. Suppose that $w_{ii} = 0$ for every $i$, that the underlying value of $L^{(0)}_{ij}$ is $0$ if $i = j$ and $\\infty$ otherwise, and that $w$ has no negative cycle. Then for all $i, j \\in \\{0, \\dots, n-1\\}$, the underlying\u2026","title":"SLOW-APSP computes shortest-path weights"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/slow-apsp-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.slowAPSP_time","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#slow-apsp-time","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"For all $n \\times n$ matrices $W$ and $L^{(0)}$ over the min-plus semiring on $\\mathbb{Z} \\cup \\{\\infty\\}$, the cost of SLOW-APSP on $W$ and $L^{(0)}$ is the triple $(n^2 + 2(n-1)n^2,\\ (n-1)n^3,\\ (n-1)n^3)$ of cell assignments (copies and fills), scalar additions and scalar minimums, where $n - 1$ means $\\max(n - 1, 0)$.","title":"Exact operation counts of SLOW-APSP"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/slow-apsp-time-is-theta","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.slowAPSP_time_isTheta","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#slow-apsp-time-is-theta","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"For any choice, for each $n$, of $n \\times n$ matrices $W_n$ and $L^{(0)}_n$ over the min-plus semiring on $\\mathbb{Z} \\cup \\{\\infty\\}$, the total cost of SLOW-APSP on $W_n$ and $L^{(0)}_n$, that is the number of cell assignments (copies and fills) plus the number of scalar additions plus the number of scalar minimums, is $\\Theta(n^4)$ as $n \\to \\infty$.","title":"SLOW-APSP takes quartic time"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/faster-apsp-edge-bounded","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fasterAPSP_edgeBounded","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#faster-apsp-edge-bounded","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $W$ be an $n \\times n$ matrix with entries in $\\mathbb{Z} \\cup \\{\\infty\\}$, read as edge weights of a directed graph on the vertices $0, \\dots, n - 1$ (weight $\\infty$ meaning no edge), with $W_{ii} = 0$ for every $i$, and let $k = \\lceil \\log_2 (n - 1) \\rceil$ be the least natural number with $n - 1 \\le 2^k$. Run FASTER-APSP on $W$ over the min-plus semiring, whose addition is $\\min$ and whose multiplication\u2026","title":"FASTER-APSP computes edge-bounded shortest-path weights"},{"id":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/faster-apsp-distance-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.fasterAPSP_distance_spec","parent":38760,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products/index.html#faster-apsp-distance-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/all-pairs-shortest-paths-by-min-plus-products","status":"fully_proved","summary":"Let $W$ be an $n \\times n$ matrix with entries in $\\mathbb{Z} \\cup \\{\\infty\\}$, read as edge weights of a directed graph on the vertices $0, \\dots, n - 1$ (weight $\\infty$ meaning no edge), such that $W_{ii} = 0$ for every $i$ and $W$ has no negative cycle. Run FASTER-APSP on $W$ over the min-plus semiring, whose addition is $\\min$ and whose multiplication is $+$. Then for all vertices $i$ and $j$, the returned\u2026","title":"FASTER-APSP computes shortest-path weights"},{"data_url":"scopes/ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm.json","id":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm","kind":"scope","navigation":"module","parent":38682,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/index.html","scope":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm","status":"fully_proved","title":"Johnson's algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/johnson","kind":"node","lean":"Cslib.Algorithms.Lean.Johnson.johnson","parent":38784,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/index.html#johnson","scope":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm","status":"fully_proved","summary":"Let $E$ be a list of directed edges on the vertices $\\{0, \\dots, n-1\\}$ with integer weights. Johnson's algorithm adds a new vertex $s$ with an edge of weight $0$ to every original vertex and runs Bellman\u2013Ford from $s$. If Bellman\u2013Ford reports a negative cycle, the algorithm returns no table. Otherwise let $h(v)$ be the Bellman\u2013Ford distance of $v$ (taken to be $0$ if $v$ is unreached). Each edge from $u$ to $v$ of\u2026","title":"Johnson's algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/johnson-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Johnson.johnson_eq_none_iff","parent":38784,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/index.html#johnson-eq-none-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm","status":"fully_proved","summary":"For every list $E$ of directed edges on the vertices $\\{0, \\dots, n-1\\}$ with integer weights, Johnson's algorithm on $E$ returns no table if and only if there is a vertex $s$ from which a negative cycle of $E$ is reachable.","title":"Johnson's algorithm fails exactly on reachable negative cycles"},{"id":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/johnson-distance-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Johnson.johnson_distance_spec","parent":38784,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/index.html#johnson-distance-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm","status":"fully_proved","summary":"Let $E$ be a list of directed edges on the vertices $\\{0, \\dots, n-1\\}$ with integer weights, and suppose Johnson's algorithm on $E$ returns a table $d$. Then for all vertices $s$ and $t$, $d(s, t) = \\infty$ if and only if $t$ is not reachable from $s$ along edges of $E$. Moreover, if $d(s, t)$ is an integer $x$, then some weighted path in $E$ from $s$ to $t$ has weight $x$, and every weighted path in $E$ from $s$\u2026","title":"Correctness of Johnson's algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/johnson-time-of-success","kind":"node","lean":"Cslib.Algorithms.Lean.Johnson.johnson_time_of_success","parent":38784,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/index.html#johnson-time-of-success","scope":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm","status":"fully_proved","summary":"Let $E$ be a list of $m$ directed edges on the vertices $\\{0, \\dots, n-1\\}$ with integer weights. If Johnson's algorithm on $E$ returns a table, then its counted cost is exactly $n^3 + nm + (n+1)(m+n) + 2n^2 + 4m + 5n + 1$.","title":"Exact cost of a successful run of Johnson's algorithm"},{"id":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/johnson-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Johnson.johnson_time_le","parent":38784,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm/index.html#johnson-time-le","scope":"ComputerScience/graph-algorithms/shortest-paths/johnson-s-algorithm","status":"fully_proved","summary":"For every list $E$ of $m$ directed edges on the vertices $\\{0, \\dots, n-1\\}$ with integer weights, the counted cost of Johnson's algorithm on $E$, including the case where it extracts a negative cycle, is at most $n^3 + nm + (n+1)(m+n) + 2n^2 + 4m + 5n + 1 + 3(n+1)$.","title":"Cost bound for Johnson's algorithm"},{"data_url":"scopes/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs.json","id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","kind":"scope","navigation":"module","parent":38682,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","title":"Shortest paths in directed acyclic graphs"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/cost","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.Cost","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#cost","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"The cost of the one-pass relaxation in topological order consists of three natural-number counters: the number of vertices taken from the order, the number of edge occurrences checked for relaxation, and the number of distance installations (strict improvements or first finite distances).","title":"Cost of the DAG shortest-path pass"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edge occurrences on the vertices $\\{0, \\dots, n-1\\}$, let $s$ be a source vertex, and let $\\sigma$ be a topological order of the directed graph obtained from $E$ by forgetting weights. The relaxation starts from distance estimate $0$ at $s$, infinite estimates at every other vertex and no predecessor edges. It then takes the vertices $u$ of $\\sigma$ in turn and relaxes\u2026","title":"One-pass DAG shortest-path relaxation"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/dag-shortest-paths","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.dagShortestPaths","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#dag-shortest-paths","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Given a list $E$ of integer-weighted edges on the vertices $0, \\dots, n-1$ and a source $s$, run Kahn's topological sort on the directed graph obtained from $E$ by forgetting weights. If it succeeds with order $\\sigma$, start from distance $0$ at $s$ and $\\infty$ elsewhere, relax the outgoing edge occurrences of each vertex in the order $\\sigma$, and return the resulting distances and predecessor edges. If it fails\u2026","title":"Shortest paths in a DAG with a topology check"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-vertex-visits","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_vertexVisits","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-vertex-visits","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edge occurrences on the vertices $\\{0, \\dots, n-1\\}$, let $s$ be a source vertex, and let $\\sigma$ be a topological order of the directed graph obtained from $E$ by forgetting weights. The one-pass DAG shortest-path relaxation of $E$ from $s$ in the order $\\sigma$ is charged exactly $n$ vertex visits.","title":"The DAG shortest-path pass is charged one vertex visit per vertex"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-edge-relaxations","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_edgeRelaxations","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-edge-relaxations","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted edges on $n$ vertices, $s$ a source, and $\\sigma$ a topological order of the directed graph obtained from $E$ by forgetting weights. In the cost of the one-pass relaxation from $s$ in the order $\\sigma$, the number of edge relaxations equals the length of $E$, with identical duplicate edges counted separately.","title":"Edge relaxation count of the DAG pass"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-distance-updates-le","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_distanceUpdates_le","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-distance-updates-le","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted edges on $n$ vertices, $s$ a source, and $\\sigma$ a topological order of the directed graph obtained from $E$ by forgetting weights. In the cost of the one-pass relaxation from $s$ in the order $\\sigma$, the number of distance installations is at most the number of edge relaxations.","title":"Distance updates are bounded by edge relaxations"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-distance-realized","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_distance_realized","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-distance-realized","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted edges on $n$ vertices, $s$ a source, and $\\sigma$ a topological order of the directed graph obtained from $E$ by forgetting weights. If the one-pass relaxation from $s$ in the order $\\sigma$ gives a vertex $t$ the finite distance $d$, then there is a path from $s$ to $t$ made of edge occurrences of $E$ whose edge weights sum to $d$.","title":"Finite DAG distances are path weights"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-source","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_source","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-source","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edge occurrences on the vertices $\\{0, \\dots, n-1\\}$, let $s$ be a source vertex, and let $\\sigma$ be a topological order of the directed graph obtained from $E$ by forgetting weights. The result of the one-pass DAG shortest-path relaxation of $E$ from $s$ in the order $\\sigma$ records $s$ as its source.","title":"The DAG shortest-path pass keeps its source"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-source-distance","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_source_distance","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-source-distance","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edge occurrences on the vertices $\\{0, \\dots, n-1\\}$, let $s$ be a source vertex, and let $\\sigma$ be a topological order of the directed graph obtained from $E$ by forgetting weights. The one-pass DAG shortest-path relaxation of $E$ from $s$ in the order $\\sigma$ ends with distance estimate $0$ at $s$.","title":"The DAG shortest-path pass gives the source distance zero"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-source-predecessor","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_source_predecessor","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-source-predecessor","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edge occurrences on the vertices $\\{0, \\dots, n-1\\}$, let $s$ be a source vertex, and let $\\sigma$ be a topological order of the directed graph obtained from $E$ by forgetting weights. The one-pass DAG shortest-path relaxation of $E$ from $s$ in the order $\\sigma$ records no predecessor edge for $s$.","title":"The source has no DAG shortest-path predecessor"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-distance-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_distance_minimal","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-distance-minimal","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted edges on $n$ vertices, $s$ a source, and $\\sigma$ a topological order of the directed graph obtained from $E$ by forgetting weights. If the one-pass relaxation from $s$ in the order $\\sigma$ gives a vertex $t$ the finite distance $d$, then $d \\le w(p)$ for every path $p$ from $s$ to $t$ made of edge occurrences of $E$, where $w(p)$ is the sum of the weights of the edges of $p$.","title":"Finite DAG distances are at most every path weight"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-distance-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_distance_eq_none_iff","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-distance-eq-none-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted edges on $n$ vertices, $s$ a source, and $\\sigma$ a topological order of the directed graph $G$ obtained from $E$ by forgetting weights. For every vertex $t$, the one-pass relaxation from $s$ in the order $\\sigma$ leaves the distance of $t$ infinite if and only if $t$ is not reachable from $s$ in $G$.","title":"Infinite DAG distances are exactly unreachable vertices"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-predecessor-spec","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_predecessor_spec","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-predecessor-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edge occurrences on the vertices $\\{0, \\dots, n-1\\}$, let $s$ be a source vertex, and let $\\sigma$ be a topological order of the directed graph obtained from $E$ by forgetting weights. Suppose the one-pass DAG shortest-path relaxation of $E$ from $s$ in the order $\\sigma$ records an edge occurrence $e$ from $u$ to $w$ with weight $c$ as the predecessor of a vertex $v$.\u2026","title":"Predecessor edges of the DAG shortest-path pass"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-infinite-predecessor","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_infinite_predecessor","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-infinite-predecessor","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edge occurrences on the vertices $\\{0, \\dots, n-1\\}$, let $s$ be a source vertex, and let $\\sigma$ be a topological order of the directed graph obtained from $E$ by forgetting weights. If the one-pass DAG shortest-path relaxation of $E$ from $s$ in the order $\\sigma$ ends with an infinite distance estimate at a vertex $v$, then it records no predecessor edge for $v$.","title":"Unreached vertices have no DAG shortest-path predecessor"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-finite-predecessor","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_finite_predecessor","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-finite-predecessor","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted edges on $n$ vertices, $s$ a source, and $\\sigma$ a topological order of the directed graph obtained from $E$ by forgetting weights. If the one-pass relaxation from $s$ in the order $\\sigma$ gives a vertex $t \\ne s$ a finite distance, then it records a predecessor edge for $t$.","title":"Finite non-source vertices have predecessors"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/dag-shortest-paths-eq-ok-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.dagShortestPaths_eq_ok_iff","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#dag-shortest-paths-eq-ok-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"For a list $E$ of integer-weighted edges on $n$ vertices, a source $s$, and a distance record $D$, the DAG shortest-path algorithm on $E$ from $s$ returns $D$ if and only if there is a topological order $\\sigma$ of the directed graph obtained from $E$ by forgetting weights such that Kahn's topological sort of that graph succeeds with $\\sigma$ and $D$ is the result of the one-pass relaxation from $s$ in the order\u2026","title":"Successful output of DAG shortest paths"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/dag-shortest-paths-eq-error-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.dagShortestPaths_eq_error_iff","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#dag-shortest-paths-eq-error-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"For a list $E$ of integer-weighted edges on $n$ vertices, a source $s$, and lists $P$ and $R$ of vertices, the DAG shortest-path algorithm on $E$ from $s$ returns the error $(P, R)$ if and only if Kahn's topological sort of the directed graph obtained from $E$ by forgetting weights fails after processing $P$ and leaving $R$.","title":"Failure output of DAG shortest paths"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-path-to-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_pathTo?_eq_none_iff","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-path-to-eq-none-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$, $s$ a source vertex, and $\\sigma$ a topological order of the directed graph with an edge $u \\to v$ for each edge of $E$ from $u$ to $v$. After the one-pass relaxation of $E$ from $s$ in the order $\\sigma$, for every vertex $t$, following the predecessor edges back from $t$ fails to reach $s$ within $n$ steps if and only if the\u2026","title":"Predecessor extraction fails exactly at infinite distances"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/run-path-to-spec","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.run_pathTo?_spec","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#run-path-to-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$, $s$ a source vertex, and $\\sigma$ a topological order of the directed graph with an edge $u \\to v$ for each edge of $E$ from $u$ to $v$. After the one-pass relaxation of $E$ from $s$ in the order $\\sigma$, suppose following the predecessor edges back from a vertex $t$ reaches $s$ within $n$ steps, giving the path $P$. Then $P$\u2026","title":"Paths extracted after topological relaxation are shortest paths"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/dag-shortest-paths-failure-cycle","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.dagShortestPaths_failure_cycle","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#dag-shortest-paths-failure-cycle","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and $s$ a source vertex. If the DAG shortest-path algorithm on $E$ from $s$ fails, reporting the processed vertices and the list $R$ of remaining vertices of its topological sort, then the directed graph with an edge $u \\to v$ for each edge of $E$ from $u$ to $v$ contains a nonempty directed cycle without repeated vertices\u2026","title":"A failed DAG shortest-path run exhibits a cycle"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/dag-shortest-paths-returns-iff-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.dagShortestPaths_returns_iff_acyclic","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#dag-shortest-paths-returns-iff-acyclic","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and $s$ a source vertex. The DAG shortest-path algorithm on $E$ from $s$ succeeds if and only if the directed graph with an edge $u \\to v$ for each edge of $E$ from $u$ to $v$ has no directed cycle.","title":"The DAG shortest-path algorithm succeeds exactly on acyclic graphs"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/dag-shortest-paths-distance-spec","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.dagShortestPaths_distance_spec","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#dag-shortest-paths-distance-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and $s$ a source vertex. Suppose the DAG shortest-path algorithm on $E$ from $s$ succeeds and returns a finite distance $d \\in \\mathbb{Z}$ for a vertex $t$. Then some path from $s$ to $t$ in $E$ has weight $d$, and every path from $s$ to $t$ in $E$ has weight at least $d$. Here a path from $s$ to $t$ in $E$ is a list of edges\u2026","title":"DAG shortest-path distances are shortest-path weights"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/dag-shortest-paths-distance-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.dagShortestPaths_distance_eq_none_iff","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#dag-shortest-paths-distance-eq-none-iff","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and $s$ a source vertex. If the DAG shortest-path algorithm on $E$ from $s$ succeeds, then for every vertex $t$ the returned distance to $t$ is infinite if and only if $t$ is not reachable from $s$ in the directed graph with an edge $u \\to v$ for each edge of $E$ from $u$ to $v$.","title":"Unreachable vertices are exactly those at infinite DAG distance"},{"id":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/dag-shortest-paths-path-to-spec","kind":"node","lean":"Cslib.Algorithms.Lean.DAGShortestPaths.dagShortestPaths_pathTo?_spec","parent":38790,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs/index.html#dag-shortest-paths-path-to-spec","scope":"ComputerScience/graph-algorithms/shortest-paths/shortest-paths-in-directed-acyclic-graphs","status":"fully_proved","summary":"Let $E$ be a list of integer-weighted directed edges on the vertices $\\{0, \\dots, n-1\\}$ and $s$ a source vertex. Suppose the DAG shortest-path algorithm on $E$ from $s$ succeeds, and following its predecessor edges back from a vertex $t$ reaches $s$ within $n$ steps, giving the path $P$. Then $P$ is a path from $s$ to $t$ in $E$, the returned distance to $t$ is finite and equals the weight of $P$, $P$ has fewer\u2026","title":"Extracted DAG shortest paths are shortest paths"},{"data_url":"scopes/ComputerScience/graph-algorithms/spanning-trees-flows-matchings.json","id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings","kind":"scope","parent":38431,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/index.html","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings","status":"fully_proved","summary":"Minimum spanning trees (Kruskal, Prim), maximum flow and minimum cut (Edmonds-Karp), bipartite matching by augmenting paths, and Gale-Shapley stable matching.","title":"Spanning trees, flows and matchings"},{"data_url":"scopes/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm.json","id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","kind":"scope","navigation":"module","parent":38815,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","title":"Edmonds\u2013Karp algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/flow","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.Flow","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#flow","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"A flow on $n$ vertices, labelled $0, \\dots, n-1$, is an integer matrix: it assigns an integer $f(u, v)$ to each ordered pair of vertices $(u, v)$.","title":"Flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/flow-value","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.flowValue","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#flow-value","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For a flow $f$ on $n$ vertices and a vertex $s$, the value of $f$ at $s$ is the net flow leaving $s$, namely the integer $\\sum_v f(s, v)$, where $v$ runs over all vertices.","title":"Flow value"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/is-feasible","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.IsFeasible","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#is-feasible","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function assigning a natural number $c(u, v)$ to each ordered pair of vertices, and let $s$ and $t$ be vertices. A flow $f$ is feasible for $c$, $s$ and $t$ when it is skew symmetric, $f(u, v) = -f(v, u)$ for all $u$ and $v$; it respects the capacities, $f(u, v) \\le c(u, v)$ for all $u$ and $v$; and it is conserved: for every vertex $v$ other than $s$ and $t$, the net flow $\\sum_w f(v, w)$\u2026","title":"Feasible flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/zero-flow","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.zeroFlow","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#zero-flow","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"The zero flow on the vertices $\\{0, \\dots, n - 1\\}$ is the integer-valued matrix $f$ with $f(u, v) = 0$ for all vertices $u$ and $v$.","title":"Zero flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/zero-flow-feasible","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.zeroFlow_feasible","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#zero-flow-feasible","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For every capacity matrix $c$ with natural-number entries on the vertices $\\{0, \\dots, n - 1\\}$ and all vertices $s$ and $t$, the zero flow is feasible for $c$ with source $s$ and sink $t$.","title":"The zero flow is feasible"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/capacity-bounds","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.IsFeasible.capacity_bounds","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#capacity-bounds","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n-1\\}$, let $s$ and $t$ be vertices, and let $f$ be a flow that is feasible for $c$, $s$, $t$. Then for all vertices $u$ and $v$, we have $-c(v, u) \\le f(u, v) \\le c(u, v)$.","title":"Two-sided capacity bounds for a feasible flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/residual-graph","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.residualGraph","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#residual-graph","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix assigning a natural number $c(u, v)$ to each ordered pair of vertices among $0, \\dots, n - 1$, and let $f$ be a flow on these vertices. The residual graph of $c$ and $f$ is the directed adjacency graph whose vertices $0, \\dots, n - 1$ are enumerated in increasing order and in which the adjacency list of $u$ consists of the vertices $v$ with $f(u, v) < c(u, v)$, that is, with positive\u2026","title":"Residual graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/residual-graph-edge-iff","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.residualGraph_edge_iff","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#residual-graph-edge-iff","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries and $f$ an integer matrix (the flow) on the vertices $0, \\dots, n - 1$. For all vertices $u$ and $v$, the residual graph of $c$ and $f$ has an edge from $u$ to $v$ if and only if $f(u, v) < c(u, v)$.","title":"Edges of the residual graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/is-path","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.IsPath","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#is-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"In a directed graph $G$, a list $p$ of vertices is a path from $s$ to $t$ when it is nonempty, its first entry is $s$, its last entry is $t$, every pair of consecutive entries $u, v$ forms a directed edge of $G$ from $u$ to $v$, and no vertex occurs twice in $p$.","title":"Directed path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/path-from-distances","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.pathFromDistances","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#path-from-distances","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph with ordered adjacency lists, $d$ a partial labelling of its vertices by natural numbers, $t$ a vertex, and $\\equiv$ a given Boolean test on pairs of vertices (not required to be equality). The reconstruction from a vertex $u$ with $k$ steps left returns the one-element list $[u]$ when $k = 0$ and $u \\equiv t$ holds, and fails when $k = 0$ and $u \\equiv t$ fails. When $k \\ge 1$, it takes\u2026","title":"Distance-guided path reconstruction"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/path-from-distances-sound","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.pathFromDistances_sound","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#path-from-distances-sound","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $G$ be a directed adjacency graph on a type $V$ whose Boolean test $\\equiv$ on pairs of vertices holds exactly for equal vertices, let $d$ be a partial labelling of the vertices by natural numbers, let $t$ and $u$ be vertices, and let $k$ be a natural number. If the distance-guided reconstruction for $G$, $d$ and $t$ from $u$ with $k$ steps left succeeds and returns a list $p$, then $p$ is a walk from $u$ to\u2026","title":"Soundness of distance-guided path reconstruction"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/shortest-path","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.shortestPath","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#shortest-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $G$ be a finite directed graph on the vertices $\\{0, \\dots, n - 1\\}$ with ordered adjacency lists, and let $s$ be a source and $t$ a sink. Run breadth-first search from $t$ in the reverse graph of $G$; it labels each vertex $v$ from which $t$ is reachable with the distance from $v$ to $t$ in $G$. If $s$ has no label, the search returns no path. Otherwise, starting at $s$ with label $d$, take $d$ steps, each\u2026","title":"Shortest path via breadth-first search"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/shortest-path-sound","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.shortestPath_sound","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#shortest-path-sound","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $G$ be a finite directed graph on the vertices $\\{0, \\dots, n - 1\\}$ and let $s$ and $t$ be vertices. If the breadth-first shortest-path search in $G$ from $s$ to $t$ returns a list $P$ of vertices, then $P$ is a directed path in $G$ from $s$ to $t$ without repeated vertices, and breadth-first search in $G$ from $s$ assigns $t$ a distance $d$ such that $P$ has exactly $d + 1$ vertices.","title":"Soundness of the shortest-path search"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/shortest-path-present-iff-reachable","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.shortestPath_present_iff_reachable","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#shortest-path-present-iff-reachable","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $G$ be a finite directed graph on the vertices $\\{0, \\dots, n - 1\\}$ and let $s$ and $t$ be vertices. The breadth-first shortest-path search in $G$ from $s$ to $t$ returns some path if and only if $t$ is reachable from $s$ in $G$ (every vertex counts as reachable from itself).","title":"The shortest-path search succeeds exactly when the sink is reachable"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/shortest-path-eq-none-iff-not-reachable","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.shortestPath_eq_none_iff_not_reachable","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#shortest-path-eq-none-iff-not-reachable","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $G$ be a finite directed graph on the vertices $\\{0, \\dots, n - 1\\}$ and let $s$ and $t$ be vertices. The breadth-first shortest-path search in $G$ from $s$ to $t$ returns no path if and only if $t$ is not reachable from $s$ in $G$.","title":"The shortest-path search fails exactly when the sink is unreachable"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/path-edges","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.pathEdges","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#path-edges","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"The edges of a list $v_0, v_1, \\dots, v_k$ are the list of its consecutive ordered pairs $(v_0, v_1), (v_1, v_2), \\dots, (v_{k-1}, v_k)$, in this order; the empty list and one-element lists have no edges.","title":"Edges of a path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/residual-amount","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.residualAmount","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#residual-amount","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries and $f$ an integer matrix (the flow) on the vertices $0, \\dots, n - 1$. The residual capacity of an ordered pair of vertices $(u, v)$ is the integer $c(u, v) - f(u, v)$.","title":"Residual capacity"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/path-bottleneck","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.pathBottleneck","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#path-bottleneck","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries and $f$ an integer matrix (the flow) on the same vertices. The bottleneck of a list of vertices $v_0, v_1, \\dots, v_k$ with $k \\ge 1$ is the integer $\\min_{i < k} \\bigl(c(v_i, v_{i+1}) - f(v_i, v_{i+1})\\bigr)$, the least residual capacity of a consecutive pair; a list with at most one vertex has bottleneck $0$.","title":"Bottleneck of a path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/local-delta","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.localDelta","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#local-delta","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For vertices $a$, $b$, $u$, $v$, the unit skew flow of the oriented edge $(a, b)$ at the ordered pair $(u, v)$ is the integer $[u = a \\text{ and } v = b] - [u = b \\text{ and } v = a]$, where $[P]$ is $1$ if $P$ holds and $0$ otherwise.","title":"Unit skew flow along an oriented edge"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/edge-delta","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.edgeDelta","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#edge-delta","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For a list $P$ of elements of a set $V$ and elements $u, v$ of $V$, the edge delta of $P$ at $(u, v)$ is the integer obtained as the number of consecutive pairs of $P$ equal to $(u, v)$ minus the number of consecutive pairs of $P$ equal to $(v, u)$, counted with multiplicity. It is $0$ when $P$ has fewer than two elements.","title":"Edge delta of a path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/augment","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.augment","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#augment","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Given an integer flow matrix $f$ on the vertices $\\{0, \\dots, n-1\\}$, a list of vertices $P$ and an integer $a$, the augmentation of $f$ along $P$ by $a$ is the matrix whose entry at $(u, v)$ is $f(u, v) + a \\cdot d$, where $d$ is the number of consecutive pairs of $P$ equal to $(u, v)$ minus the number of consecutive pairs of $P$ equal to $(v, u)$, counted with multiplicity.","title":"Augmenting a flow along a path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/path-bottleneck-pos","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.pathBottleneck_pos","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#path-bottleneck-pos","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries and $f$ any integer matrix (the flow) on the vertices $0, \\dots, n - 1$, and let $s \\ne t$ be vertices. If $p$ is a duplicate-free path from $s$ to $t$ in the residual graph of $c$ and $f$, then the bottleneck of $p$, the minimum of $c(u, v) - f(u, v)$ over consecutive vertices $u, v$ of $p$, is positive.","title":"Augmenting paths have positive bottleneck"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/augment-feasible","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.augment_feasible","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#augment-feasible","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n-1\\}$, let $s$ and $t$ be vertices, and let $f$ be a flow that is feasible for $c$, $s$, $t$. Let $P = (v_0, \\dots, v_k)$ be a nonempty list of pairwise distinct vertices with $v_0 = s$ and $v_k = t$, and let $a$ be an integer with $0 \\le a$ and $a \\le c(v_i, v_{i+1}) - f(v_i, v_{i+1})$ for every $i < k$. Then the augmentation of\u2026","title":"Augmentation within residual capacities preserves feasibility"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/flow-value-augment","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.flowValue_augment","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#flow-value-augment","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $s \\ne t$ be vertices, $f$ a flow, $a$ an integer, and $p$ a nonempty list of vertices whose first entry is $s$ and whose last entry is $t$. Then the net flow leaving $s$ under the flow obtained by augmenting $f$ along $p$ by $a$ equals the net flow leaving $s$ under $f$, plus $a$.","title":"Augmenting along a path changes the flow value by the amount"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/augment-path-bottleneck-feasible","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.augment_pathBottleneck_feasible","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#augment-path-bottleneck-feasible","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n-1\\}$, let $s$ and $t$ be vertices, and let $f$ be a flow that is feasible for $c$, $s$, $t$. If $P$ is a path from $s$ to $t$ in the residual graph of $f$, that is, a nonempty list of pairwise distinct vertices starting at $s$ and ending at $t$ whose consecutive pairs are residual edges, then the augmentation of $f$ along $P$ by\u2026","title":"Bottleneck augmentation preserves feasibility"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/step","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.step","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#step","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n - 1\\}$, let $s$ be a source and $t$ a sink, and let $f$ be a flow (an integer-valued matrix on pairs of vertices). Run the breadth-first shortest-path search from $s$ to $t$ in the residual graph of $f$ with respect to $c$. If it finds no path, the step returns $f$. If it returns a path $P$, let $b$ be the bottleneck of $P$, the\u2026","title":"One augmenting-path step"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/flow-value-step","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.flowValue_step","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#flow-value-step","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function and $s \\ne t$ vertices. For every flow $f$, the flow obtained from $f$ by one step of bottleneck augmentation either has the same net flow leaving $s$ as $f$, or a strictly larger one.","title":"One augmentation step keeps or strictly raises the flow value"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/run","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.run","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#run","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n - 1\\}$, let $s$ be a source and $t$ a sink, let $k$ be a fuel bound and let $f$ be a flow (an integer-valued matrix on pairs of vertices). The iteration returns $f$ if $k = 0$ or $s = t$. Otherwise it runs the breadth-first shortest-path search from $s$ to $t$ in the residual graph of $f$ with respect to $c$. If the search finds\u2026","title":"Augmenting-path iteration with fuel"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/flow-value-le-run","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.flowValue_le_run","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#flow-value-le-run","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function and $s \\ne t$ vertices. For every flow $f$ and every natural number $k$, the net flow leaving $s$ after running at most $k$ rounds of bottleneck augmentation from $f$ is at least the net flow leaving $s$ under $f$.","title":"The augmentation loop never decreases the flow value"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/source-capacity","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.sourceCapacity","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#source-capacity","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For a capacity matrix $c$ with natural-number entries on the vertices $\\{0, \\dots, n - 1\\}$ and a source $s$, the source capacity is the total capacity leaving $s$, namely $\\sum_{v} c(s, v)$, where $v$ ranges over all vertices, including $s$ itself.","title":"Source capacity"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/edmonds-karp","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.edmondsKarp","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#edmonds-karp","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function assigning a natural number $c(u, v)$ to each ordered pair of vertices, and let $s$ and $t$ be vertices. If $s = t$, the Edmonds\u2013Karp flow is the zero flow. Otherwise it is the result of running at most $C + 1$ rounds of bottleneck augmentation from the zero flow, where $C = \\sum_v c(s, v)$ is the total capacity leaving $s$: each round looks for a shortest path from $s$ to $t$ in the\u2026","title":"Edmonds\u2013Karp maximum flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/edmonds-karp-feasible","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.edmondsKarp_feasible","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#edmonds-karp-feasible","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For every capacity matrix $c$ with natural-number entries on the vertices $\\{0, \\dots, n-1\\}$ and all vertices $s$ and $t$, the flow $f$ returned by the Edmonds\u2013Karp algorithm on $c$, $s$, $t$ is feasible for $c$, $s$, $t$: it is skew-symmetric, satisfies $f(u, v) \\le c(u, v)$ for all vertices $u, v$, and has zero net outflow at every vertex other than $s$ and $t$.","title":"The Edmonds\u2013Karp flow is feasible"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/run-eq-of-no-path","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.run_eq_of_no_path","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#run-eq-of-no-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n - 1\\}$, let $s$ and $t$ be vertices, and let $f$ be a flow (an integer-valued matrix on pairs of vertices). If the breadth-first shortest-path search finds no path from $s$ to $t$ in the residual graph of $f$ with respect to $c$, then for every fuel bound $k$, the augmenting-path iteration started at $f$ with fuel $k$ returns $f$.","title":"The iteration stops when there is no augmenting path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/flow-value-le-source-capacity","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.flowValue_le_sourceCapacity","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#flow-value-le-source-capacity","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function, $s$ and $t$ vertices, and $f$ a flow that is feasible for $c$, $s$ and $t$. Then the net flow leaving $s$ under $f$ is at most $\\sum_v c(s, v)$, the total capacity leaving $s$.","title":"The flow value is at most the source capacity"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/edmonds-karp-no-augmenting-path","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.edmondsKarp_no_augmenting_path","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#edmonds-karp-no-augmenting-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n-1\\}$ and let $s$ and $t$ be distinct vertices. Then $t$ is not reachable from $s$ in the residual graph of the flow returned by the Edmonds\u2013Karp algorithm on $c$, $s$, $t$.","title":"No augmenting path remains after Edmonds\u2013Karp"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/cut-capacity","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.cutCapacity","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#cut-capacity","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For a capacity matrix $c$ with natural-number entries on the vertices $\\{0, \\dots, n-1\\}$ and a set $S$ of vertices, the capacity of the cut $S$ is the integer $\\sum_{u \\in S} \\sum_{v \\notin S} c(u, v)$.","title":"Cut capacity"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/flow-value-le-cut-capacity","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.flowValue_le_cutCapacity","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#flow-value-le-cut-capacity","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function, $s$ and $t$ vertices, and $f$ a flow that is feasible for $c$, $s$ and $t$. For every set $S$ of vertices with $s \\in S$ and $t \\notin S$, the net flow leaving $s$ under $f$ is at most the capacity of the cut $S$, namely $\\sum_{u \\in S} \\sum_{v \\notin S} c(u, v)$.","title":"Weak max-flow min-cut duality"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/residual-cut","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.residualCut","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#residual-cut","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix assigning a natural number $c(u, v)$ to each ordered pair of vertices among $0, \\dots, n - 1$, let $f$ be a flow on these vertices, and let $s$ be a vertex. The residual cut of $c$, $f$ and $s$ is the set of vertices visited by breadth-first search from $s$ in the residual graph of $c$ and $f$, that is, the set of vertices reachable from $s$ in that residual graph.","title":"Residual cut"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/edmonds-karp-min-cut","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.edmondsKarp_minCut","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#edmonds-karp-min-cut","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n-1\\}$, let $s$ and $t$ be distinct vertices, and let $f$ be the flow returned by the Edmonds\u2013Karp algorithm on $c$, $s$, $t$. Then the net flow of $f$ out of $s$ equals the capacity of the cut formed by the vertices reachable from $s$ in the residual graph of $f$.","title":"The Edmonds\u2013Karp flow value equals its residual cut capacity"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/edmonds-karp-maximal","kind":"node","lean":"Cslib.Algorithms.Lean.EdmondsKarp.edmondsKarp_maximal","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#edmonds-karp-maximal","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity matrix with natural-number entries on the vertices $\\{0, \\dots, n-1\\}$, let $s$ and $t$ be distinct vertices, and let $f$ be the flow returned by the Edmonds\u2013Karp algorithm on $c$, $s$, $t$. Then every integer flow $g$ that is feasible for $c$, $s$, $t$ has value at $s$ at most the value of $f$ at $s$, where the value of a flow at $s$ is its net flow out of $s$.","title":"Maximality of the Edmonds\u2013Karp flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/real-cap","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.realCap","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#real-cap","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For a capacity function $c$ on the vertices $0, \\dots, n-1$ with natural-number values, the real capacities are the same values viewed as real numbers, $(u, v) \\mapsto c(u, v) \\in \\mathbb{R}$.","title":"Real capacities"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/arc-flow","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.arcFlow","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#arc-flow","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For an integer flow matrix $f$ on the vertices $0, \\dots, n-1$, its arc flow is the natural-number matrix whose entry at $(u, v)$ is the positive part $\\max(f(u, v), 0)$ of $f(u, v)$.","title":"Arc flow of an integer flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/real-flow","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.realFlow","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#real-flow","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"For an integer flow matrix $f$ on the vertices $0, \\dots, n-1$, its real arc flow is the function $(u, v) \\mapsto \\max(f(u, v), 0)$, the arc flow of $f$, viewed as real numbers.","title":"Real arc flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/flow-eq-arc-flow-sub","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.flow_eq_arcFlow_sub","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#flow-eq-arc-flow-sub","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $f$ be an integer flow matrix on the vertices $0, \\dots, n-1$ that is skew-symmetric, $f(u, v) = -f(v, u)$ for all $u, v$, and let $a(u, v) = \\max(f(u, v), 0)$ be its arc flow. Then $f(u, v) = a(u, v) - a(v, u)$ for all vertices $u$ and $v$.","title":"A skew flow is the difference of its arc flows"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/sum-real-flow-sub","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.sum_realFlow_sub","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#sum-real-flow-sub","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $f$ be a skew-symmetric integer flow matrix on the vertices $0, \\dots, n-1$, so $f(u, v) = -f(v, u)$ for all $u, v$, and let $a(u, v) = \\max(f(u, v), 0)$ be its arc flow, viewed as real numbers. Then for every vertex $v$, $\\sum_w a(v, w) - \\sum_u a(u, v) = \\sum_w f(v, w)$, the net flow leaving $v$.","title":"Net outflow in the arc representation"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/is-feasible-flow-real-flow","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.isFeasibleFlow_realFlow","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#is-feasible-flow-real-flow","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function on the vertices $0, \\dots, n-1$ with natural-number values and let $s, t$ be vertices. If an integer flow matrix $f$ is a feasible skew-symmetric $s$-$t$ flow for $c$, then its arc flow $(u, v) \\mapsto \\max(f(u, v), 0)$, viewed as a real function, is a feasible $s$-$t$ flow for $c$ viewed as real capacities.","title":"A feasible skew flow gives a feasible flow"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/is-stcut-residual-cut","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.isSTCut_residualCut","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#is-stcut-residual-cut","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function on the vertices $0, \\dots, n-1$ with natural-number values and let $s \\ne t$ be vertices. The set of vertices reachable from $s$ in the residual graph of the flow computed by the Edmonds-Karp algorithm on $c$, $s$, $t$ is an $s$-$t$ cut: it contains $s$ and not $t$.","title":"The Edmonds-Karp residual cut separates source and sink"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/edmonds-karp-max-flow-min-cut","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.edmondsKarp_max_flow_min_cut","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#edmonds-karp-max-flow-min-cut","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function on the vertices $0, \\dots, n-1$ with natural-number values, viewed as real capacities, and let $s \\ne t$ be vertices. Let $f$ be the real arc flow of the flow computed by the Edmonds-Karp algorithm on $c$, $s$, $t$ (its positive part, viewed as real numbers), and let $S_0$ be the set of vertices reachable from $s$ in the residual graph of that flow. Then $f$ is a feasible $s$-$t$ flow\u2026","title":"Edmonds-Karp computes a maximum flow and a minimum cut"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/cut-capacity-residual-cut-le","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.cutCapacity_residualCut_le","parent":38816,"reading_title":{"format":"inline-parts-v1","parts":[["text","No "],["math","\\(s\\)"],["text","-"],["math","\\(t\\)"],["text"," cut is smaller than the Edmonds\u2013Karp residual cut"]],"source":"No $s$-$t$ cut is smaller than the Edmonds\u2013Karp residual cut"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#cut-capacity-residual-cut-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function on the vertices $0, \\dots, n-1$ with natural-number values, viewed as real capacities, and let $s \\ne t$ be vertices. Let $S_0$ be the set of vertices reachable from $s$ in the residual graph of the flow computed by the Edmonds\u2013Karp algorithm on $c$, $s$, $t$. Then for every $s$-$t$ cut $S$, the capacity of $S_0$ is at most the capacity of $S$.","title":"No $s$-$t$ cut is smaller than the Edmonds\u2013Karp residual cut"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/exists-nat-max-flow","kind":"node","lean":"Cslib.Algorithms.Lean.Graph.MaxFlowMinCut.exists_nat_max_flow","parent":38816,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm/index.html#exists-nat-max-flow","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/edmonds-karp-algorithm","status":"fully_proved","summary":"Let $c$ be a capacity function on the vertices $0, \\dots, n-1$ with natural-number values, viewed as real capacities, and let $s \\ne t$ be vertices. Then there is a natural-number valued function $f$ on pairs of vertices which, viewed as a real function, is a feasible $s$-$t$ flow for $c$ whose value is at least the value of every feasible real-valued $s$-$t$ flow for $c$.","title":"Integral flow theorem"},{"data_url":"scopes/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm.json","id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","kind":"scope","navigation":"module","parent":38815,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","title":"Gale\u2013Shapley algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/to-order","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.toOrder","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#to-order","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"A permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$, read as a ranking ($\\sigma(k)$ is the $k$-th choice, $\\sigma(0)$ the best), gives the strict total order on $\\{0, \\dots, n-1\\}$ in which $a$ is preferred to $b$ when $\\sigma^{-1}(a) < \\sigma^{-1}(b)$.","title":"Preference order of a ranking"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/mprefs","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.mprefs","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#mprefs","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Given for each proposer $m$ a ranking $\\sigma_m$ of the $n$ receivers ($\\sigma_m(k)$ is the choice of $m$ of rank $k$, rank $0$ the best), the proposers' preference profile assigns to $m$ the strict total order in which $m$ prefers $w$ to $w'$ when $\\sigma_m^{-1}(w) < \\sigma_m^{-1}(w')$.","title":"Proposers' preferences from rankings"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/wprefs","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.wprefs","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#wprefs","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Given for each receiver $w$ a ranking $\\sigma_w$ of the $n$ proposers (a permutation of $\\{0, \\dots, n-1\\}$, with $\\sigma_w(0)$ her first choice), the receivers' preference profile assigns to $w$ the preference order of $\\sigma_w$: $w$ prefers $a$ to $b$ when $\\sigma_w^{-1}(a) < \\sigma_w^{-1}(b)$.","title":"Receivers' preference profile from rankings"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/state","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.State","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#state","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"A state of deferred acceptance with $n$ proposers and $n$ receivers, both numbered $0, \\dots, n-1$, records for each proposer a natural number, the number of proposals he has made (his next proposal goes to his choice of that rank), and for each receiver either the proposer she currently holds or that she holds nobody.","title":"Deferred-acceptance state"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/init","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.init","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#init","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"The initial state with $n$ proposers and $n$ receivers: every proposer has made $0$ proposals and no receiver holds anyone.","title":"Initial state of deferred acceptance"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/free","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.Free","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#free","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"A proposer $m$ is free in a state $s$ of deferred acceptance when no receiver holds $m$ in $s$.","title":"Free proposer"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/new-holder","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.newHolder","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#new-holder","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"In a state $s$ of deferred acceptance, the proposer whom receiver $w$ holds after proposer $m$ proposes to her is $m$ if she holds nobody in $s$ or ranks $m$ strictly better than her current holder $m'$ in her ranking, and $m'$ otherwise.","title":"Receiver's holder after a proposal"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/propose","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.propose","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#propose","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Fix, for each proposer, a ranking of the $n$ receivers and, for each receiver, a ranking of the $n$ proposers (rank $0$ is the first choice). Let $s$ be a state of deferred acceptance in which proposer $m$ has made $k$ proposals. If $k < n$, then $m$ proposes to the receiver $w$ of rank $k$ in his ranking: his number of proposals becomes $k + 1$, and $w$ now holds $m$ if she held nobody or ranks $m$ strictly above\u2026","title":"A proposal"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/chooser","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.chooser","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#chooser","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"In a state $s$ of deferred acceptance with proposers $0, 1, \\dots, n-1$, the chooser returns the least proposer $m$ who is free in $s$ and has made fewer than $n$ proposals in $s$, or no proposer if there is none.","title":"Choosing the next proposer"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/run","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.run","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#run","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Fix rankings for $n$ proposers and $n$ receivers, both numbered $0, \\dots, n-1$. Starting from a state $s$ with a budget of $f$ proposals: if $f = 0$, or if no proposer is both free (held by no receiver) and has made fewer than $n$ proposals, the run returns $s$; otherwise the lowest-numbered such proposer makes a proposal, and the run continues from the resulting state with budget $f - 1$.","title":"Running deferred acceptance"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/gale-shapley","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.galeShapley","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#gale-shapley","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Let each of $n$ proposers rank the $n$ receivers, and each receiver rank the $n$ proposers, with rank $0$ the best. Deferred acceptance is the state reached from the initial state, in which no one has proposed and no receiver holds anyone, by repeating the following step at most $n^2$ times, stopping early once no proposer is both free and has made fewer than $n$ proposals: the least such proposer, having made $p$\u2026","title":"Deferred acceptance"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/inv","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.Inv","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#inv","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Let each proposer $m$ have a ranking $\\sigma_m$ of the $n$ receivers and each receiver $w$ a ranking $\\tau_w$ of the $n$ proposers, where $\\sigma_m(k)$ is the choice of $m$ of rank $k$ (rank $0$ is the best) and $\\tau_w^{-1}(x)$ is the rank of $x$ for $w$. A state $s$ of deferred acceptance satisfies the invariant when: every proposer has made at most $n$ proposals; if $w$ holds $m$, then $m$ has made $k + 1$\u2026","title":"Invariant of deferred acceptance"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/not-blocking","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.not_blocking","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#not-blocking","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Let each proposer $m$ have a ranking $\\sigma_m$ of the $n$ receivers and each receiver $w$ a ranking $\\tau_w$ of the $n$ proposers, where $\\sigma_m^{-1}(x)$ and $\\tau_w^{-1}(x)$ are ranks ($0$ the best). If $\\nu$ is a perfect matching that is stable for the preferences given by these rankings, there are no proposer $m$ and receiver $w$ with $\\sigma_m^{-1}(w) < \\sigma_m^{-1}(\\nu(m))$ and\u2026","title":"A stable matching has no blocking pair"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/le-rank-of-rejected","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.le_rank_of_rejected","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#le-rank-of-rejected","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Let each proposer $m$ have a ranking $\\sigma_m$ of the $n$ receivers ($\\sigma_m(k)$ is the choice of $m$ of rank $k$, rank $0$ the best), and each receiver a ranking of the $n$ proposers. Let $s$ be a state of deferred acceptance satisfying the invariant, let $\\nu$ be a perfect matching that is stable for the preferences given by the rankings, let $m$ be a proposer, and let $b$ be a natural number at most the\u2026","title":"Rejected proposals bound stable partners"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/new-holder-cases","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.newHolder_cases","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#new-holder-cases","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Let $s$ be a state of deferred acceptance, $m$ a proposer and $w$ a receiver with ranking $\\tau_w$ of the proposers ($\\tau_w^{-1}(x)$ is the rank of $x$, rank $0$ the best). Then one of the following holds: $w$ holds nobody in $s$ and holds $m$ after $m$ proposes to her; or $w$ holds some $y$ in $s$ with $\\tau_w^{-1}(m) < \\tau_w^{-1}(y)$ and holds $m$ after $m$ proposes to her; or $w$ holds some $y$ in $s$ with\u2026","title":"Cases of the receiver's new holder"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/remaining","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.remaining","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#remaining","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"For a state $s$ of deferred acceptance with $n$ proposers and $n$ receivers, in which each proposer $m$ has made $k_m$ proposals, the number of proposals still available is $\\sum_{m} \\max(n - k_m, 0)$, the sum running over all $n$ proposers.","title":"Remaining proposals"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/sum-next-propose","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.sum_next_propose","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#sum-next-propose","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Fix rankings for $n$ proposers and $n$ receivers. If proposer $m$ has made fewer than $n$ proposals in a state $s$ of deferred acceptance, then after $m$ proposes, the total number of proposals made by all proposers is one more than in $s$.","title":"A proposal adds one to the proposal count"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/sum-next-gale-shapley-le","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.sum_next_galeShapley_le","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#sum-next-gale-shapley-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"For any rankings of the $n$ receivers by each of the $n$ proposers and of the proposers by each receiver, in the final state of deferred acceptance the numbers of proposals made by the proposers sum to at most $n \\cdot n$.","title":"Deferred acceptance makes at most n\u00b2 proposals"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/held-of-done","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.held_of_done","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#held-of-done","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Fix rankings of the $n$ receivers by each of $n$ proposers and of the proposers by each receiver. Let $s$ be a state of deferred acceptance that satisfies the invariant of deferred acceptance for these rankings, and suppose that no proposer is both free in $s$ and has made fewer than $n$ proposals in $s$. Then every proposer is held by some receiver in $s$.","title":"When deferred acceptance stops, every proposer is held"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/holds-of-held","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.holds_of_held","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#holds-of-held","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Let $s$ be a state of deferred acceptance with $n$ proposers and $n$ receivers. If every proposer is held by some receiver in $s$, then every receiver holds some proposer in $s$.","title":"Every receiver holds someone once every proposer is held"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/partner-w","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.partnerW","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#partner-w","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"In a state $s$ of deferred acceptance with $n$ proposers and $n$ receivers, both numbered $0, \\dots, n-1$, the partner of a receiver $w$ is the proposer she holds in $s$; if she holds nobody, it is the proposer numbered $w$ (a default value).","title":"Receiver's partner"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/partner-m","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.partnerM","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#partner-m","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"In a state $s$ of deferred acceptance with $n$ proposers and $n$ receivers, both numbered $0, \\dots, n-1$, the partner of a proposer $m$ is the lowest-numbered receiver that holds $m$ in $s$; if no receiver holds $m$, it is the receiver numbered $m$ (a default value).","title":"Proposer's partner"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/gs-matching","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.gsMatching","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#gs-matching","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Each proposer with the receiver who holds him at the end.","title":"The Gale-Shapley matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/exists-rank-gs-matching","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.exists_rank_gsMatching","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#exists-rank-gs-matching","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Let each proposer $m$ rank the $n$ receivers by a permutation $\\sigma_m$ of $\\{0, \\dots, n-1\\}$, where $\\sigma_m(k)$ is his choice of rank $k$, let each receiver rank the $n$ proposers, and let $p_m$ be the number of proposals $m$ has made in the final state of deferred acceptance. Then $p_m \\ge 1$, and the Gale-Shapley partner $\\mu(m)$ of $m$ has rank $p_m - 1$ in his ranking, that is,\u2026","title":"A proposer's final partner is his last proposal"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/is-stable-gs-matching","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.isStable_gsMatching","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#is-stable-gs-matching","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"For $n$ proposers and $n$ receivers, each with a ranking of the other side, the perfect matching computed by deferred acceptance is stable for the strict preferences given by these rankings.","title":"The Gale-Shapley matching is stable"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/gs-matching-proposer-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.gsMatching_proposer_optimal","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#gs-matching-proposer-optimal","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"No stable matching gives a proposer a partner he prefers to his Gale-Shapley partner.","title":"Proposer-optimality"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/gs-matching-receiver-pessimal","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.gsMatching_receiver_pessimal","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#gs-matching-receiver-pessimal","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"No stable matching gives a receiver a partner she likes less than her Gale-Shapley partner.","title":"Receiver-pessimality"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/order-rank","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.orderRank","parent":38867,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#order-rank","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"For a strict total order $<$ on $\\{0, \\dots, n-1\\}$, the rank of an element $a$ is the number of elements $b$ with $b < a$ (the elements preferred to $a$), as an element of $\\{0, \\dots, n-1\\}$.","title":"Rank under a strict total order"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/exists-to-order-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.exists_toOrder_eq","parent":38867,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every strict total order on "],["math","\\(\\{0, \\dots, n-1\\}\\)"],["text"," is given by a ranking"]],"source":"Every strict total order on $\\{0, \\dots, n-1\\}$ is given by a ranking"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#exists-to-order-eq","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"For every strict total order $\\prec$ on $\\{0, \\dots, n-1\\}$ there is a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ such that, for all $a$ and $b$, $a \\prec b$ holds exactly when $\\sigma^{-1}(a) < \\sigma^{-1}(b)$.","title":"Every strict total order on $\\{0, \\dots, n-1\\}$ is given by a ranking"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/exists-rankings-gs-matching","kind":"node","lean":"Cslib.Algorithms.Lean.Matching.GaleShapley.exists_rankings_gsMatching","parent":38867,"reading_title":{"format":"inline-parts-v1","parts":[["text","Gale-Shapley serves every preference profile with "],["math","\\(n\\)"],["text"," proposers and "],["math","\\(n\\)"],["text"," receivers"]],"source":"Gale-Shapley serves every preference profile with $n$ proposers and $n$ receivers"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm/index.html#exists-rankings-gs-matching","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/gale-shapley-algorithm","status":"fully_proved","summary":"Let the $n$ proposers and the $n$ receivers both be numbered $0, \\dots, n-1$. Let each proposer $m$ have a strict total order $\\lt _m$ of preference on the receivers and each receiver $w$ a strict total order $\\lt _w$ of preference on the proposers, where $x \\lt _m y$ means that $m$ prefers $x$ to $y$, and likewise for $\\lt _w$. Then there are permutations $\\sigma_m$ (one for each proposer $m$) and $\\tau_w$ (one\u2026","title":"Gale-Shapley serves every preference profile with $n$ proposers and $n$ receivers"},{"data_url":"scopes/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm.json","id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","kind":"scope","navigation":"module","parent":38815,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","title":"Kruskal's algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/raw-edge","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.RawEdge","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#raw-edge","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"A raw edge is an undirected weighted input edge given by three natural numbers: two endpoints, which are not yet checked to be vertices of the graph, and a weight.","title":"Raw input edge"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/graph","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.Graph","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#graph","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For a natural number $n$, an input graph on the vertices $\\{0, \\dots, n-1\\}$ consists of a list of raw edges, kept in input order; the same pair of endpoints may occur on several edges.","title":"Kruskal input graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/weighted-edge","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.WeightedEdge","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#weighted-edge","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"A weighted edge on the vertices $\\{0, \\dots, n-1\\}$ consists of an ordinal, a natural number recording its zero-based position in the original input list, two endpoints in $\\{0, \\dots, n-1\\}$, and a natural-number weight.","title":"Weighted edge"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/key","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.WeightedEdge.key","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#key","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"The sorting key of a weighted edge $e$ is the pair $(w(e), i(e)) \\in \\mathbb{N} \\times \\mathbb{N}$ of its weight $w(e)$ and its input ordinal $i(e)$, the zero-based position of $e$ in the original input edge list.","title":"Sorting key of a weighted edge"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/key-le","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.KeyLE","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#key-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For pairs of natural numbers $(w, i)$ and $(w', i')$, $(w, i) \\preceq (w', i')$ holds when $w < w'$, or $w = w'$ and $i \\le i'$. This is the non-strict lexicographic order on $\\mathbb{N} \\times \\mathbb{N}$; Kruskal's algorithm applies it to keys (weight, input ordinal).","title":"Lexicographic order on edge keys"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/decode-at","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.decodeAt","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#decode-at","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Given natural numbers $n$ and $i$ and a raw edge with source $s$, target $t$ and weight $w$, decoding at ordinal $i$ returns the validated edge on the vertices $\\{0, \\dots, n-1\\}$ with ordinal $i$, source $s$, target $t$ and weight $w$ when $s < n$ and $t < n$, and returns no edge otherwise.","title":"Decoding a raw edge at an ordinal"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/valid-edges-from","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.validEdgesFrom","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#valid-edges-from","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Given $n$, a starting ordinal $o$ and a list of raw edges $e_0, e_1, \\dots, e_{m-1}$, this returns, in input order, the weighted edges on the vertices $\\{0, \\dots, n-1\\}$ obtained from those $e_k$ whose two endpoints are both less than $n$; each keeps the endpoints and weight of $e_k$ and carries the ordinal $o + k$. Edges with an endpoint at least $n$ are omitted, but they still count towards the ordinals of later\u2026","title":"Validating an edge list from a given ordinal"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/valid-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.validEdges","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#valid-edges","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For a graph $G$ on the vertices $\\{0, \\dots, n-1\\}$ given by its list of raw edges $e_0, e_1, \\dots, e_{m-1}$, the valid edges of $G$ are, in input order, the weighted edges obtained from those $e_k$ whose two endpoints are both less than $n$; each keeps the endpoints and weight of $e_k$ and carries its zero-based input ordinal $k$. Edges with an endpoint at least $n$ are omitted.","title":"Valid edges of an input graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/sort-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.sortEdges","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#sort-edges","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Sorting a list of validated edges is merge sort with the comparison $(w, i) \\preceq (w', i')$ if $w < w'$, or $w = w'$ and $i \\le i'$, applied to the pairs (weight, input position) of the edges.","title":"Sorting edges by weight and input position"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/ordered-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.orderedEdges","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#ordered-edges","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For a finite weighted multigraph $G$ on the vertices $\\{0, \\dots, n-1\\}$, the ordered edges of $G$ are its valid input edges (those whose endpoints are both less than $n$), sorted by merge sort in lexicographic order of (weight, input position): by nondecreasing weight, with edges of equal weight in input order.","title":"Valid edges in Kruskal order"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/initial-forest","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.initialForest","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#initial-forest","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"The initial forest on the vertices $\\{0, \\dots, n-1\\}$ is the parent forest in which every vertex is its own parent, so every vertex is a root, and whose depth certificate is $0$ at every vertex.","title":"Discrete parent forest"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/connected-rev","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.ConnectedRev","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#connected-rev","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Connectivity of vertices $u$ and $v$ of $\\{0, \\dots, n-1\\}$ by a list of validated edges, read newest first, is defined by recursion on the list. By the empty list, $u$ and $v$ are connected exactly when $u = v$. By a list whose first edge $e$ has source $a$ and target $b$ and whose remaining edges form the list $L$, the vertices $u$ and $v$ are connected when $u$ and $v$ are connected by $L$, or $u$ is connected\u2026","title":"Connectivity by a newest-first edge list"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/connected","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.Connected","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#connected","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Vertices $u$ and $v$ of $\\{0, \\dots, n-1\\}$ are connected by a list $e_1, \\dots, e_k$ of validated edges when they are connected by the reversed list $e_k, \\dots, e_1$ in the sense of connectivity by a newest-first edge list.","title":"Connectivity by an edge list"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/acyclic-rev","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.AcyclicRev","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#acyclic-rev","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"A list $e_1, \\dots, e_k$ of validated edges on the vertices $\\{0, \\dots, n-1\\}$, read newest first, is newest-first acyclic when, for every $i$, the endpoints of $e_i$ are not connected by the later list $e_{i+1}, \\dots, e_k$ in the sense of connectivity by a newest-first edge list.","title":"Acyclicity of a newest-first edge list"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.Acyclic","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#acyclic","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"A list $e_1, \\dots, e_k$ of validated edges on the vertices $\\{0, \\dots, n-1\\}$ is acyclic when, for every $i$, the endpoints of $e_i$ are not connected by the preceding edges $e_1, \\dots, e_{i-1}$, in the sense of connectivity by an edge list.","title":"Acyclic edge list"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/spans","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.Spans","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#spans","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"A list $L$ of validated edges spans a finite weighted multigraph $G$ on the vertices $\\{0, \\dots, n-1\\}$ when, for every valid input edge of $G$ (one whose endpoints are both less than $n$), the two endpoints of that edge are connected using the edges of $L$.","title":"Edge lists spanning a graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/is-spanning-forest","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.IsSpanningForest","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#is-spanning-forest","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For an input graph $G$ on the vertices $\\{0, \\dots, n-1\\}$, a list $F$ of validated edges is a spanning forest of $G$ when every edge of $F$ is a valid edge of $G$, $F$ is acyclic, and $F$ spans $G$: for every valid edge of $G$, its endpoints are connected by $F$.","title":"Spanning forest of an input graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/is-ordered-spanning-forest","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.IsOrderedSpanningForest","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#is-ordered-spanning-forest","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For an input graph $G$ on the vertices $\\{0, \\dots, n-1\\}$, a list $e_1, \\dots, e_k$ of validated edges is an ordered spanning forest of $G$ when it is a spanning forest of $G$ and its keys are nondecreasing: for all $i < j$, the key of $e_i$ is at most the key of $e_j$, where the key of an edge with weight $w$ and input ordinal $o$ is the pair $(w, o)$ and keys are compared lexicographically.","title":"Ordered spanning forest"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/total-weight","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.totalWeight","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#total-weight","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"The total weight of a list of validated edges is the sum of the weights of its entries, each entry counted once, so parallel edges and repeated entries each contribute their weight.","title":"Total weight of a Kruskal edge list"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/state","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.State","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#state","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"A state of the Kruskal scan on the vertices $\\{0, \\dots, n-1\\}$ consists of a union-find parent forest on these vertices, which records the current partition, and the list of edges accepted so far, newest first.","title":"Kruskal scan state"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/step","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.State.step","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#step","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Let $s$ be a Kruskal scan state and $e$ a validated edge with endpoints $u$ and $v$. If $u$ and $v$ are equivalent in the union-find forest of $s$, processing $e$ leaves $s$ unchanged. Otherwise $e$ is accepted: the new forest is the union of $u$ and $v$ in the forest of $s$, and $e$ is put at the front of the list of accepted edges.","title":"One Kruskal step"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/scan","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.scan","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#scan","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Starting from a Kruskal scan state $s$, scanning a list of validated edges $e_1, \\dots, e_k$ applies one Kruskal step to each edge in list order: the scan of the empty list returns $s$, and the scan of $e_1, \\dots, e_k$ is the scan of $e_2, \\dots, e_k$ from the state obtained by processing $e_1$ in $s$.","title":"Scanning a list of edges"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/has-greedy-cut-property","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.HasGreedyCutProperty","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#has-greedy-cut-property","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Let $G$ be an input graph on the vertices $\\{0, \\dots, n-1\\}$, give each valid edge with weight $w$ and input ordinal $o$ the key $(w, o)$, and compare keys lexicographically. The graph $G$ has the greedy cut property when the following holds for every way of splitting the valid edges of $G$, sorted by key, into an initial segment, an edge $e$ and a final segment. Let $P$ be the union-find partition obtained by\u2026","title":"Greedy cut property of Kruskal's scan"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For a finite weighted multigraph $G$ on the vertices $\\{0, \\dots, n-1\\}$, Kruskal's algorithm starts from the partition of the vertices into singletons and processes the valid input edges of $G$ (those whose endpoints are both less than $n$) in lexicographic order of (weight, input position). An edge whose endpoints lie in different classes of the current partition is accepted and its two classes are merged; an\u2026","title":"Kruskal's algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-edge-mem-valid","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_edge_mem_valid","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-edge-mem-valid","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For every graph $G$ on $n$ vertices, every edge in the list returned by Kruskal's algorithm on $G$ is a valid edge of $G$: an input edge whose two endpoints are both below $n$, carrying its input ordinal.","title":"Kruskal selects only valid input edges"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_acyclic","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-acyclic","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For every graph $G$, the edge list returned by Kruskal's algorithm on $G$ is acyclic: each edge in it joins two vertices that are not connected by the edges before it in the list.","title":"Kruskal's output is acyclic"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-spans","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_spans","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-spans","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For every finite weighted multigraph $G$ on the vertices $\\{0, \\dots, n-1\\}$, given by its list of input edges, and every valid input edge of $G$ (one whose endpoints are both less than $n$), the two endpoints of that edge are connected by the edges that Kruskal's algorithm selects on $G$.","title":"Kruskal's selected edges span the graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-pairwise-key-le","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_pairwise_key_le","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-pairwise-key-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For every graph $G$, if an edge $e$ comes before an edge $f$ in the list returned by Kruskal's algorithm on $G$, then $e$ has smaller weight than $f$, or the same weight as $f$ and input ordinal at most that of $f$.","title":"Kruskal's output is sorted by key"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-greedy-cut-property","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_greedy_cut_property","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-greedy-cut-property","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Let $G$ be a graph and let $e_1, \\dots, e_m$ be its valid edges sorted lexicographically by key (weight, then input ordinal). For a position $k$, let $P_k$ be the partition of the vertices given by the union-find forest after Kruskal's scan, started from the discrete forest, has processed $e_1, \\dots, e_{k-1}$. If the endpoints of $e_k$ lie in different classes of $P_k$, then for every valid edge $c$ of $G$ whose\u2026","title":"Kruskal's greedy cut property"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-light-edge","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_light_edge","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-light-edge","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Let $G$ be a graph and let $e_1, \\dots, e_m$ be its valid edges sorted lexicographically by key (weight, then input ordinal). For a position $k$, let $P_k$ be the partition of the vertices given by the union-find forest after Kruskal's scan, started from the discrete forest, has processed $e_1, \\dots, e_{k-1}$. If the endpoints of $e_k$ lie in different classes of $P_k$, then the weight of $e_k$ is at most the\u2026","title":"Kruskal accepts light edges"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-is-spanning-forest","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_isSpanningForest","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-is-spanning-forest","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For every graph $G$, the edge list returned by Kruskal's algorithm on $G$ is a spanning forest of $G$: every edge in it is a valid edge of $G$, it is acyclic, and it connects the two endpoints of every valid edge of $G$.","title":"Kruskal's output is a spanning forest"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-is-ordered-spanning-forest","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_isOrderedSpanningForest","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-is-ordered-spanning-forest","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"For every graph $G$, the edge list returned by Kruskal's algorithm on $G$ is a spanning forest of $G$ (its edges are valid edges of $G$, it is acyclic, and it connects the endpoints of every valid edge of $G$), and whenever an edge $e$ comes before an edge $f$ in it, the key (weight, input ordinal) of $e$ is lexicographically at most that of $f$.","title":"Kruskal's output is a sorted spanning forest"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/kruskal-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.Kruskal.kruskal_minimal","parent":38898,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm/index.html#kruskal-minimal","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/kruskal-s-algorithm","status":"fully_proved","summary":"Let $G$ be a graph with natural-number edge weights, and let $F$ be any spanning forest of $G$: a list of valid edges of $G$ that is acyclic and connects the endpoints of every valid edge of $G$. Then the total weight of the edge list returned by Kruskal's algorithm on $G$ is at most the total weight of $F$.","title":"Kruskal's algorithm gives a minimum spanning forest"},{"data_url":"scopes/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching.json","id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","kind":"scope","navigation":"module","parent":38815,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","title":"Maximum bipartite matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/is-matching","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.IsMatching","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#is-matching","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let a bipartite graph have left vertices $0, \\dots, a-1$, right vertices $0, \\dots, b-1$ and an adjacency list for each left vertex. A finite set $M$ of pairs $(i, j)$ of a left vertex $i$ and a right vertex $j$ is a matching when every pair in $M$ is an edge ($j$ is in the adjacency list of $i$), no two distinct pairs of $M$ have the same left vertex, and no two distinct pairs of $M$ have the same right vertex.","title":"Matching in a bipartite graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/pairs","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.pairs","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#pairs","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a partner assignment $\\mathrm{mate}$ that sends each right vertex $j < b$ to a left vertex $i < a$ or to nothing, the matched pairs of $\\mathrm{mate}$ form the set of all pairs $(i, j)$ of a left vertex $i < a$ and a right vertex $j < b$ with $\\mathrm{mate}(j) = i$.","title":"Matched pairs of a partner assignment"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/valid","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.Valid","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#valid","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Consider a bipartite graph with left vertices $\\{0, \\dots, a-1\\}$, right vertices $\\{0, \\dots, b-1\\}$ and, for each left vertex $i$, a list of its neighbours. A partner array assigns to each right vertex $j$ either no partner or a left vertex as its partner. It is valid when every partner is a neighbour (if $i$ is the partner of $j$, then $j$ is in the list of neighbours of $i$) and no left vertex is the partner of\u2026","title":"Partner arrays that describe a matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/free","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.Free","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#free","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a partner array $\\mu$, which assigns to each right vertex $j$ either a left vertex $\\mu(j)$ as partner or no partner, a left vertex $i$ is unmatched (free) when no right vertex $j$ has $\\mu(j) = i$.","title":"Unmatched left vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/layers","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.layers","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#layers","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ give each right vertex $j$ of a bipartite graph at most one partner $\\mu(j)$ among the left vertices. The layers are pairs $(Z_k, R_k)$ of a set $Z_k$ of left vertices and a set $R_k$ of right vertices, defined by recursion on $k$: $Z_0$ is the set of unmatched left vertices (those that are the partner of no right vertex) and $R_0 = \\emptyset$; $R_{k+1}$ is the set of right vertices $j$ adjacent to some\u2026","title":"Alternating breadth-first layers"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/exists-mate-of-mem-layers","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.exists_mate_of_mem_layers","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#exists-mate-of-mem-layers","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array and $k$ a natural number. If a left vertex $i$ is reached within $k$ rounds of the layered search and is matched (some right vertex has partner $i$), then some right vertex $j$ reached within $k$ rounds has partner $i$, that is, $\\mu(j) = i$.","title":"Matched reached left vertices are reached through a partner"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/layers-fst-eq-of-eq","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.layers_fst_eq_of_eq","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#layers-fst-eq-of-eq","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For any partner array, if the layered search reaches the same left vertices within $k$ rounds as within $k + 1$ rounds, then for every $m \\ge k$ it reaches exactly those left vertices within $m$ rounds.","title":"The left layers stop growing once two consecutive ones agree"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/layers-fst-stable","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.layers_fst_stable","parent":38932,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rounds "],["math","\\(a\\)"],["text"," and "],["math","\\(a + 1\\)"],["text"," reach the same left vertices"]],"source":"Rounds $a$ and $a + 1$ reach the same left vertices"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#layers-fst-stable","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with $a$ left vertices, for every partner array $\\mu$, the alternating layered search for $\\mu$ reaches the same left vertices within $a + 1$ rounds as within $a$ rounds.","title":"Rounds $a$ and $a + 1$ reach the same left vertices"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/path-to","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.pathTo","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#path-to","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given the adjacency lists $\\mathrm{adj}$ of a bipartite graph, a partner assignment $\\mathrm{mate}$, a level $k$ and a right vertex $j$, the walk back from $j$ at level $k$ is a list of pairs defined by recursion on $k$. At level $0$ it is $[(i, j)]$ for the least unmatched left vertex $i$ with $j \\in \\mathrm{adj}(i)$ and $\\mathrm{mate}(j) \\ne i$, and empty if there is none. At level $k + 1$, if $j$ is reached\u2026","title":"Walking the layers back"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/flip","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.flip","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#flip","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array, assigning to each right vertex either a left partner or none, and let $P$ be a list of pairs $(i, j)$ of a left vertex $i$ and a right vertex $j$. The flip of $\\mu$ along $P$ sends each right vertex $j$ to the left vertex of the first pair in $P$ whose right vertex is $j$, and to $\\mu(j)$ (a partner or none) when no pair of $P$ has right vertex $j$.","title":"Flipping a list of pairs"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/alt-path","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.AltPath","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#alt-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array, giving each right vertex at most one left vertex as its partner, let $k$ be a natural number and $j$ a right vertex. A list $P$ of pairs $(i, j')$ of a left vertex $i$ and a right vertex $j'$ is an alternating path from $j$ within round $k$ when all of the following hold: $P$ is nonempty and its first pair has right vertex $j$; for every pair $(i, j')$ of $P$, $j'$ is a neighbour of\u2026","title":"Alternating path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/is-augmenting-path","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.IsAugmentingPath","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#is-augmenting-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array, $k$ a natural number and $j$ a right vertex. A list $P$ of pairs of a left vertex and a right vertex is an augmenting path from $j$ within round $k$ when $P$ is an alternating path from $j$ within round $k$ and $j$ is unmatched: $\\mu$ gives $j$ no partner.","title":"Augmenting path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/path-to-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.pathTo_spec","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#path-to-spec","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mathrm{mate}$ be a valid partner assignment. For every level $k$ and every right vertex $j$ reached within $k + 1$ rounds of the alternating layered search, the walk back from $j$ at level $k$ is an alternating path from $j$ at level $k$: it starts at $j$, its pairs are non-matching edges, its left vertices are reached within $k$ rounds and its right vertices within $k + 1$ rounds, no vertex occurs twice,\u2026","title":"The walk back is an alternating path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/valid-flip","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.valid_flip","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#valid-flip","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array (partners are neighbours and no left vertex is the partner of two right vertices), and let $P$ be an alternating path for $\\mu$ from a right vertex $j$ with round bound $k$. Then the partner array obtained from $\\mu$ by matching each pair of $P$ and keeping the other partners is valid.","title":"Flipping an alternating path keeps a matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/matched","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.matched","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#matched","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ give each right vertex $j$ of a bipartite graph at most one partner $\\mu(j)$ among the left vertices. The matched right vertices of $\\mu$ are the right vertices $j$ that have a partner $\\mu(j)$.","title":"Matched right vertices"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/matched-flip","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.matched_flip","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#matched-flip","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ give each right vertex of a bipartite graph at most one partner among the left vertices, and let $P$ be an alternating path for $\\mu$ starting at the right vertex $j$. Then the right vertices matched after flipping $P$ are exactly $j$ and the right vertices matched under $\\mu$.","title":"Flipping an alternating path matches its first vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/path-to-is-augmenting-path","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.pathTo_isAugmentingPath","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#path-to-is-augmenting-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array, $k$ a natural number and $j$ a right vertex that is unmatched and is reached within $k + 1$ rounds of the alternating layered search for $\\mu$. Then the walk back from $j$ at level $k$ is an augmenting path from $j$ within round $k$.","title":"The walk back from an unmatched reached vertex is an augmenting path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-matched-flip","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_matched_flip","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-matched-flip","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"If $P$ is an augmenting path for a partner array $\\mu$, then flipping $P$ in $\\mu$ gives a partner array with exactly one more matched right vertex than $\\mu$.","title":"Flipping an augmenting path adds one matched vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/augment","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.augment","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#augment","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array on a bipartite graph with $a$ left vertices and $b$ right vertices. Run the alternating search from the unmatched left vertices for $a + 1$ rounds and take the first right vertex $j$, in the order $0, \\dots, b - 1$, that is unmatched and reached. If there is none, the step fails. Otherwise it returns the partner array obtained from $\\mu$ by flipping the path found by walking back from\u2026","title":"One augmentation step"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/max-matching-aux","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.maxMatchingAux","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#max-matching-aux","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a fuel $f \\in \\mathbb{N}$ and a partner array $\\mu$ (assigning to each right vertex at most one left partner), the loop returns $\\mu$ when $f = 0$. When $f \\ge 1$ it performs one augmentation step on $\\mu$: if the layered search reaches no unmatched right vertex, the loop returns $\\mu$; otherwise it runs with fuel $f - 1$ on the array that this step returns, obtained by flipping the pairs walked back through\u2026","title":"Augmentation loop"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/max-matching","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.maxMatching","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#max-matching","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with right vertices $0, \\dots, b-1$, the partner array obtained by running the augmentation loop with fuel $b + 1$ from the empty matching, in which no right vertex has a partner.","title":"The augmenting-path maximum matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_pairs","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-pairs","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For every partner array $\\mu$, the number of matched pairs $(i, j)$ of $\\mu$, those with $\\mu(j) = i$, equals the number of matched right vertices of $\\mu$.","title":"Matched pairs and matched right vertices are equinumerous"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/is-matching-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.isMatching_pairs","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#is-matching-pairs","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let each right vertex $j$ of a bipartite graph have at most one partner, a left vertex. If the partners are valid for the graph (each partner of $j$ is adjacent to $j$, and no left vertex is the partner of two right vertices), then the set of pairs $(i, j)$ with $i$ the partner of $j$ is a matching of the graph.","title":"A valid partner array gives a matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/augment-some","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.augment_some","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#augment-some","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array that describes a matching. If the augmentation step on $\\mu$ returns a partner array $\\mu'$, then $\\mu'$ also describes a matching, and $\\mu'$ has exactly one more matched right vertex than $\\mu$.","title":"A successful augmentation grows the matching by one edge"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/augment-none","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.augment_none","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#augment-none","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array on a bipartite graph with $a$ left vertices. If the augmentation step on $\\mu$ fails, then no unmatched right vertex is reached within $a + 1$ rounds of the alternating search from the unmatched left vertices.","title":"A failed augmentation reaches no unmatched right vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-le-of-no-free","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_le_of_no_free","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-le-of-no-free","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array on a bipartite graph with $a$ left vertices that describes a matching. If no unmatched right vertex is reached within $a + 1$ rounds of the alternating search from the unmatched left vertices, then no matching has more edges than the matching of $\\mu$: the unreached left vertices and the reached right vertices cover every edge, and each of them is matched, to distinct matching edges.","title":"K\u0151nig's certificate"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/max-matching-aux-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.maxMatchingAux_spec","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#max-matching-aux-spec","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with right vertices $0, \\dots, b-1$, let $\\mu$ be a valid partner array (it assigns to each right vertex at most one left partner, partners are neighbours, and no left vertex is the partner of two right vertices). Let $f \\in \\mathbb{N}$ satisfy $b < |\\mathrm{matched}(\\mu)| + f$, where $\\mathrm{matched}(\\mu)$ is the set of right vertices with a partner, and let $\\mu'$ be the result of the\u2026","title":"Enough augmentations reach a maximum matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/is-matching-max-matching","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.isMatching_maxMatching","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#is-matching-max-matching","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For every bipartite graph with left vertices $0, \\dots, a-1$, right vertices $0, \\dots, b-1$ and an adjacency list for each left vertex, the pairs $(i, j)$ such that the augmenting-path maximum matching algorithm (from the empty matching, repeatedly flip an augmenting path found by the layered search) makes $i$ the partner of $j$ form a matching of the graph: each pair is an edge, and no two distinct pairs share a\u2026","title":"The augmenting-path algorithm returns a matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-le-max-matching","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_le_maxMatching","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-le-max-matching","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with $b$ right vertices, start from the empty matching and repeat the augmentation step until it fails, at most $b + 1$ times. No matching of the graph has more edges than the resulting matching.","title":"Optimality of the augmenting-path matching algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/partner-of","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.partnerOf","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#partner-of","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a partner assignment $\\mathrm{mate}$ that sends each right vertex $j < b$ to a left vertex $i < a$ or to nothing, the partner of a left vertex $i$ is the least right vertex $j$ with $\\mathrm{mate}(j) = i$; it is undefined when there is no such $j$, that is, when $i$ is unmatched.","title":"Partner of a left vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/mark-dead","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.markDead","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#mark-dead","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a set $D$ of right vertices, represented by its indicator function, and a right vertex $j$, the result is the indicator function of $D \\cup \\{j\\}$.","title":"Marking a right vertex dead"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/try-preds","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.tryPreds","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#try-preds","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $j$ be a right vertex, $\\mathrm{ok}$ a test on left vertices, $\\pi$ a map giving each left vertex an optional right vertex (its partner, if any), and $s$ a search taking a right vertex and a dead set (a set of right vertices) to an optional path, a list of pairs of a left and a right vertex, and a new dead set. Given a list of left vertices and a dead set $D$, the procedure goes through the list in order,\u2026","title":"Trying the candidate predecessors of a right vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/search-path","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.searchPath","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#search-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Fix a partner array $\\mu$ assigning each right vertex at most one left partner, and a set $U$ of used left vertices. Write $L_n$ and $R_n$ for the left and right vertices reached within $n$ rounds of the alternating breadth-first layers of $\\mu$ (so $R_0 = \\emptyset$). A left vertex $i$ is unmatched if no right vertex has partner $i$; otherwise its partner is the least right vertex $j'$ with $\\mu(j') = i$. The\u2026","title":"Depth-first search of a Hopcroft\u2013Karp phase"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/try-preds-mono","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.tryPreds_mono","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#try-preds-mono","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a right vertex $j$, a test $\\mathrm{ok}$ on left vertices, a map $\\pi$ giving each left vertex an optional right vertex, and a search $s$ taking a right vertex and a dead set (a set of right vertices) to an optional path and a new dead set, write $T(l, D)$ for the optional path and the dead set returned by trying the left vertices of the list $l$ as predecessors of $j$, starting from the dead set $D$. Suppose\u2026","title":"Trying candidates keeps dead vertices dead"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/try-preds-some","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.tryPreds_some","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#try-preds-some","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a right vertex $j$, a test $\\mathrm{ok}$ on left vertices, a map $\\pi$ giving each left vertex an optional right vertex, and a search $s$ taking a right vertex and a dead set (a set of right vertices) to an optional path and a new dead set, write $T(l, D)$ for the optional path and the dead set returned by trying the left vertices of the list $l$ as predecessors of $j$, starting from the dead set $D$. Suppose\u2026","title":"A path found by trying candidates comes from one candidate"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/try-preds-is-some","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.tryPreds_isSome","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#try-preds-is-some","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a right vertex $j$, a test $\\mathrm{ok}$ on left vertices, a map $\\pi$ giving each left vertex an optional right vertex, and a search $s$ taking a right vertex and a dead set (a set of right vertices) to an optional path and a new dead set, write $T(l, D)$ for the optional path and the dead set returned by trying the left vertices of the list $l$ as predecessors of $j$, starting from the dead set $D$. Suppose\u2026","title":"When trying candidates finds a path"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/search-path-mono","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.searchPath_mono","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#search-path-mono","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For every set of used left vertices, every level $k$, every right vertex $j$ and every set of dead right vertices, each right vertex that is dead before the search at level $k$ from $j$ is still dead after it.","title":"The dead set of a search only grows"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/search-path-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.searchPath_spec","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#search-path-spec","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Suppose the partner array describes a matching, $k$ is a level and $j$ is a right vertex reached within $k + 1$ rounds of the alternating breadth-first search. For every set of used left vertices and every set of dead right vertices, if the search at level $k$ from $j$ finds a path $P$, then $P$ is an alternating path at level $k$ from $j$, no left vertex of $P$ is used, and no right vertex of $P$ was dead before\u2026","title":"Search paths are alternating paths through unused vertices"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/search-path-is-some","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.searchPath_isSome","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#search-path-is-some","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"If the partner array describes a matching, then for every $k$ and every right vertex $j$ reached within $k + 1$ rounds of the alternating breadth-first search, the search at level $k$ from $j$, with no left vertex used and no right vertex dead, finds a path.","title":"Searches from reached vertices succeed when nothing is used or dead"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/mark-l","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.markL","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#mark-l","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a set $U$ of left vertices, represented by its indicator function, and a list $P$ of pairs $(i, j)$ of a left vertex $i$ and a right vertex $j$, the result is the indicator function of $U$ together with the left vertices of the pairs of $P$.","title":"Marking the left vertices of a path used"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/mark-r","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.markR","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#mark-r","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a set $U$ of right vertices, represented by its indicator function, and a list $P$ of pairs $(i, j)$ of a left vertex $i$ and a right vertex $j$, the result is the indicator function of $U$ together with the right vertices of the pairs of $P$.","title":"Marking the right vertices of a path used"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/phase-aux","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.phaseAux","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#phase-aux","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a partner assignment $\\mathrm{mate}$, a level $k$, a list of candidate right vertices, a set $U$ of used left vertices and a set $D$ of used or dead right vertices, the phase search tries the candidates in order. For a candidate $j$ it runs the layered depth-first search from $j$ at level $k$ with used left vertices $U$ and dead set $D$, which returns an optional path and an enlarged dead set $D'$. If it\u2026","title":"Phase search over candidates"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/first-free-round","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.firstFreeRound","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#first-free-round","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a partner array $\\mu$ of a bipartite graph with $a$ left vertices, the first free round of $\\mu$ is the least $d \\le a$ such that some unmatched right vertex is reached within $d + 1$ rounds of the layered search; it is undefined if there is no such $d$.","title":"First free round"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/phase","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.phase","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#phase","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"If no round of the alternating breadth-first search reaches an unmatched right vertex, the phase finds no path. Otherwise let $d$ be the earliest round such that round $d + 1$ reaches one. The phase runs the search at level $d$ from each unmatched right vertex reached within $d + 1$ rounds, in increasing order, starting with no left vertex used and no right vertex dead, and lists the paths found. After each search,\u2026","title":"Hopcroft\u2013Karp phase"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/hk-step","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.hkStep","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#hk-step","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a partner array $\\mu$, consider the phase of $\\mu$, a list of paths, each a list of pairs. If the phase is empty, the step returns nothing; otherwise it returns the flip of $\\mu$ along the concatenation of all the paths of the phase.","title":"Hopcroft\u2013Karp step"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/hopcroft-karp-aux","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.hopcroftKarpAux","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#hopcroft-karp-aux","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Given a fuel $n \\in \\mathbb{N}$ and a partner array $\\mu$, the loop returns $\\mu$ when $n = 0$ or when the Hopcroft\u2013Karp step on $\\mu$ returns nothing; when $n \\ge 1$ and the step returns a partner array $\\mu'$, it returns the loop with fuel $n - 1$ run from $\\mu'$. So it runs Hopcroft\u2013Karp steps while they find paths, at most $n$ times.","title":"Hopcroft\u2013Karp phase loop"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/hopcroft-karp","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.hopcroftKarp","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#hopcroft-karp","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let a bipartite graph have left vertices $0, \\dots, a-1$, right vertices $0, \\dots, b-1$ and an adjacency list for each left vertex. Starting from the empty matching, the Hopcroft\u2013Karp algorithm runs phases, at most $b + 1$ of them, and stops early at the first phase that finds no path: each phase computes the layered search once, collects vertex-disjoint augmenting paths from the unmatched right vertices reached\u2026","title":"Hopcroft\u2013Karp"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/first-free-round-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.firstFreeRound_eq_some","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#first-free-round-eq-some","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a partner array $\\mu$ of a bipartite graph with $a$ left vertices and a natural number $d$, the first free round of $\\mu$ equals $d$ if and only if $d \\le a$, some unmatched right vertex is reached within $d + 1$ rounds of the layered search, and for every $k < d$ no unmatched right vertex is reached within $k + 1$ rounds.","title":"Characterization of the first free round"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/first-free-round-eq-none","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.firstFreeRound_eq_none","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#first-free-round-eq-none","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a partner array $\\mu$ of a bipartite graph with $a$ left vertices, the first free round of $\\mu$ is undefined if and only if no unmatched right vertex is reached within $a + 1$ rounds of the layered search.","title":"When there is no first free round"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/phase-aux-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.phaseAux_spec","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#phase-aux-spec","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mathrm{mate}$ be a valid partner assignment, $k$ a level, $J$ a list of right vertices each unmatched and reached within $k + 1$ rounds of the alternating layered search, and $U_L$, $U_R$ sets of left and right vertices. Then every path that the phase search at level $k$ collects from $J$ with used left vertices $U_L$ and used right vertices $U_R$ is an augmenting path from some right vertex, with left\u2026","title":"Phase paths are disjoint augmenting paths"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/valid-flip-flatten","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.valid_flip_flatten","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#valid-flip-flatten","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array (partners are neighbours and no left vertex is the partner of two right vertices), $k$ a natural number, and $P_1, \\dots, P_r$ paths each of which is an alternating path for $\\mu$, with round bound $k$, from some right vertex. Suppose no left vertex and no right vertex occurs twice in the concatenation $P_1 \\cdots P_r$. Then the partner array obtained from $\\mu$ by matching each\u2026","title":"Flipping disjoint alternating paths at once keeps a matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-matched-flip-flatten","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_matched_flip_flatten","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-matched-flip-flatten","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array and let $P_1, \\dots, P_n$, with $n \\ge 1$, be augmenting paths for $\\mu$, all for the same number $k$ of search rounds, such that no right vertex occurs twice in the concatenation $P_1 P_2 \\cdots P_n$. Then flipping this concatenation in $\\mu$ gives a partner array with at least one more matched right vertex than $\\mu$: the right vertices matched before stay matched, and so does the\u2026","title":"Flipping disjoint augmenting paths matches at least one more vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/phase-ne-nil","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.phase_ne_nil","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#phase-ne-nil","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"If the partner array describes a matching and some unmatched right vertex is reached within $a + 1$ rounds of the alternating breadth-first search, where $a$ is the number of left vertices, then the phase finds at least one path.","title":"A phase finds a path when an unmatched right vertex is reached"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/hk-step-some","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.hkStep_some","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#hk-step-some","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array (a matching of the graph). If the Hopcroft\u2013Karp step on $\\mu$ returns a partner array $\\mu'$, then $\\mu'$ is valid and $|\\mathrm{matched}(\\mu)| + 1 \\le |\\mathrm{matched}(\\mu')|$, where $\\mathrm{matched}(\\nu)$ is the set of right vertices that have a partner in $\\nu$.","title":"A successful Hopcroft\u2013Karp step grows the matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/hk-step-none","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.hkStep_none","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#hk-step-none","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array (a matching of the graph, whose left vertices number $a$). If the Hopcroft\u2013Karp step on $\\mu$ returns nothing, then no right vertex without a partner in $\\mu$ is reached within $a + 1$ rounds of the alternating breadth-first layers of $\\mu$.","title":"A Hopcroft\u2013Karp step that finds nothing"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/hopcroft-karp-aux-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.hopcroftKarpAux_spec","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#hopcroft-karp-aux-spec","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array (a matching of the graph) on $b$ right vertices, and let $n \\in \\mathbb{N}$ be a fuel with $b < |\\mathrm{matched}(\\mu)| + n$, where $\\mathrm{matched}(\\mu)$ is the set of right vertices with a partner in $\\mu$; so $n$ exceeds the number of unmatched right vertices. Let $\\mu'$ be the result of the Hopcroft\u2013Karp phase loop with fuel $n$ from $\\mu$. Then $\\mu'$ is valid, and every\u2026","title":"Correctness of the phase loop"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/is-matching-hopcroft-karp","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.isMatching_hopcroftKarp","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#is-matching-hopcroft-karp","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For every bipartite graph with left vertices $0, \\dots, a-1$, right vertices $0, \\dots, b-1$ and an adjacency list for each left vertex, the pairs $(i, j)$ such that the Hopcroft\u2013Karp algorithm makes $i$ the partner of $j$ form a matching of the graph: each pair is an edge, and no two distinct pairs share a left or a right vertex.","title":"Hopcroft\u2013Karp returns a matching"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-le-hopcroft-karp","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_le_hopcroftKarp","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-le-hopcroft-karp","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"No matching is larger than the result.","title":"Correctness of Hopcroft\u2013Karp"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/mem-layers-length","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.mem_layers_length","parent":38932,"reading_title":{"format":"inline-parts-v1","parts":[["text","The start of an alternating chain of "],["math","\\(L\\)"],["text"," pairs is reached within "],["math","\\(L\\)"],["text"," rounds"]],"source":"The start of an alternating chain of $L$ pairs is reached within $L$ rounds"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#mem-layers-length","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array, let $L \\ge 1$, and let $(i_1, j_1), \\dots, (i_L, j_L)$ be pairs of a left and a right vertex such that each $j_t$ is adjacent to $i_t$ with $\\mu(j_t) \\ne i_t$, each $j_{t+1}$ has partner $\\mu(j_{t+1}) = i_t$ for $t < L$, and $i_L$ is unmatched. Then $j_1$ is among the right vertices reached within $L$ rounds of the alternating layered search for $\\mu$.","title":"The start of an alternating chain of $L$ pairs is reached within $L$ rounds"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/le-length-of-is-augmenting-path","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.le_length_of_isAugmentingPath","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#le-length-of-is-augmenting-path","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ give each right vertex of a bipartite graph at most one partner among the left vertices. If the first free round of $\\mu$ is $d$, then every augmenting path for $\\mu$ has at least $d + 1$ pairs.","title":"No augmenting path is shorter than a phase's"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/search-path-length-le","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.searchPath_length_le","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#search-path-length-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For every set of used left vertices, every level $k$, every right vertex $j$ and every set of dead right vertices, a path found by the search at level $k$ from $j$ has at most $k + 1$ pairs.","title":"Length bound for search paths"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/phase-aux-length-le","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.phaseAux_length_le","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#phase-aux-length-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For every partner assignment $\\mathrm{mate}$, level $k$, list of candidate right vertices and sets of used left and right vertices, every path that the phase search at level $k$ collects from these candidates has at most $k + 1$ pairs.","title":"Length bound for phase paths"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/length-of-mem-phase","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.length_of_mem_phase","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#length-of-mem-phase","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner assignment of a bipartite graph (a matching) whose first free round is $d$. Then every path of the phase of $\\mu$ has exactly $d + 1$ pairs.","title":"Phases use shortest augmenting paths"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/exact-level","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.ExactLevel","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#exact-level","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a partner array $\\mu$, a natural number $m$ and a right vertex $z$, $z$ has exact level $m$ when $z$ is first reached in round $m + 1$ of the layered search for $\\mu$: it is reached within $m + 1$ rounds but not within $m$ rounds.","title":"Exact level of a right vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/is-chain-from","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.IsChainFrom","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#is-chain-from","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph, let each right vertex have at most one partner, a left vertex (the partners need not form a matching), and let $z$ be a right vertex. A list $Q$ of pairs (left vertex, right vertex) is an alternating chain from $z$ when its first pair has right vertex $z$, each of its pairs $(i, j)$ is an edge with $i$ not the partner of $j$, each right vertex after the first is the partner of the preceding\u2026","title":"Alternating chain to an unmatched left vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/no-escape","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.NoEscape","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#no-escape","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array, $U$ a set of left vertices, $D$ a set of right vertices and $z$ a right vertex. Then $z$ has no escape avoiding $U$ and $D$ when there are no $m \\in \\mathbb{N}$ and no list of pairs $(i_1, j_1), \\dots, (i_{m+1}, j_{m+1})$ such that $z$ is first reached in round $m + 1$ of the alternating layered search for $\\mu$, the list is an alternating chain from $z$, no $i_t$ lies in $U$, and no\u2026","title":"No shortest escape"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/tail","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.ExactLevel.tail","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#tail","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array and $z$ a right vertex first reached in round $m + 1$ of the alternating breadth-first layers of $\\mu$, and let $Q$ be an alternating chain for $\\mu$ from $z$ ending at an unmatched left vertex, with exactly $m + 1$ pairs, at least two of them. Let $(i', j')$ be the second pair of $Q$. Then $m \\ge 1$, the right vertex $j'$ is first reached in round $m$, and $Q$ without its first pair is\u2026","title":"The second pair of a shortest alternating chain"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/partner","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.ExactLevel.partner","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#partner","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mathrm{mate}$ be a valid partner assignment for the bipartite graph with adjacency lists $\\mathrm{adj}$. Let $z$ be a right vertex first reached in round $k + 2$ of the alternating layered search, let $i$ be a left vertex reached within $k + 1$ rounds with $z \\in \\mathrm{adj}(i)$ and $\\mathrm{mate}(z) \\ne i$, and let $j'$ be a right vertex with $\\mathrm{mate}(j') = i$. Then $j'$ is first reached in round\u2026","title":"Partners of predecessors lie one level earlier"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/try-preds-dead","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.tryPreds_dead","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#try-preds-dead","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a right vertex $j$, a test $\\mathrm{ok}$ on left vertices, a map $\\pi$ giving each left vertex an optional right vertex, and a search $s$ taking a right vertex and a dead set (a set of right vertices) to an optional path and a new dead set, write $T(l, D)$ for the optional path and the dead set returned by trying the left vertices of the list $l$ as predecessors of $j$, starting from the dead set $D$. Suppose\u2026","title":"Where newly dead vertices come from"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/try-preds-none","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.tryPreds_none","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#try-preds-none","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For a right vertex $j$, a test $\\mathrm{ok}$ on left vertices, a map $\\pi$ giving each left vertex an optional right vertex, and a search $s$ taking a right vertex and a dead set (a set of right vertices) to an optional path and a new dead set, write $T(l, D)$ for the optional path and the dead set returned by trying the left vertices of the list $l$ as predecessors of $j$, starting from the dead set $D$. Suppose\u2026","title":"When trying candidates fails"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/search-path-none-marks","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.searchPath_none_marks","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#search-path-none-marks","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For every set of used left vertices, every level $k$, every right vertex $z$ and every set of dead right vertices, if the search at level $k$ from $z$ finds no path, then $z$ is dead after it.","title":"A failed search marks its start dead"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/search-path-dead","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.searchPath_dead","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#search-path-dead","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Suppose the partner array describes a matching. Run the search at level $k$ from a right vertex $z$ first reached in round $k + 1$, with a set $U$ of used left vertices and an initial set $D$ of dead right vertices, and let $D'$ be the dead set after it. Then every right vertex $y$ in $D'$ but not in $D$ has no escape avoiding $U$ and $D'$: if $y$ is first reached in round $m + 1$, no alternating chain from $y$ to\u2026","title":"Dead vertices have no escape"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/not-dead-of-chain","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.not_dead_of_chain","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#not-dead-of-chain","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array, $U$ a set of left vertices and $D$ a set of right vertices, and let $L$ be a set of left vertices and $R$ a set of right vertices such that every vertex of $U$ lies in $L$ and every vertex of $D$ lies in $R$ or has no escape avoiding $U$ and $D$. Let $m \\in \\mathbb{N}$, let $z$ be a right vertex first reached in round $m + 1$ of the alternating layered search for $\\mu$, and let $S$ be\u2026","title":"Shortest chains avoiding earlier paths contain no dead vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/phase-aux-maximal","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.phaseAux_maximal","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#phase-aux-maximal","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mathrm{mate}$ be a valid partner assignment, let $j$ be a right vertex first reached in round $d + 1$ of the alternating layered search, and let $Q$ be an alternating chain from $j$ with $d + 1$ pairs. Let $J$ be a list of right vertices each first reached in round $d + 1$, $U$ a set of left vertices, $D$ a set of right vertices and $F$ a list of paths, such that every vertex in $U$ is a left vertex of a path\u2026","title":"Invariant of a phase"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/phase-maximal","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.phase_maximal","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#phase-maximal","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Suppose the partner array describes a matching and $d$ is the earliest round such that round $d + 1$ of the alternating breadth-first search reaches an unmatched right vertex. Then for every $k$, every augmenting path $Q$ at level $k$ with exactly $d + 1$ pairs shares a vertex with some path $P$ of the phase: some pair of $Q$ has its left vertex among the left vertices of $P$ or its right vertex among the right\u2026","title":"Phases are maximal"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/mem-layers-snd-of-mate","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.mem_layers_snd_of_mate","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#mem-layers-snd-of-mate","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array (partners are neighbours, and no left vertex is the partner of two right vertices), let $j$ be a right vertex with partner $\\mu(j) = i$, and let $k \\in \\mathbb{N}$. If $i$ is among the left vertices reached within $k$ rounds of the alternating layered search for $\\mu$, then $j$ is among the right vertices reached within $k$ rounds.","title":"A matched right vertex is reached no later than its partner"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/pair-levels-of-chain","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.pair_levels_of_chain","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#pair-levels-of-chain","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a partner array, let $m \\in \\mathbb{N}$, let $z$ be a right vertex first reached in round $m + 1$ of the alternating layered search for $\\mu$, and let $S$ be an alternating chain from $z$ with $m + 1$ pairs. Then for every pair $(i, j)$ of $S$ there is $m' \\in \\mathbb{N}$ such that $j$ is first reached in round $m' + 1$ and $i$ is reached within $m'$ rounds.","title":"Levels along a shortest alternating chain"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/pair-levels-of-mem-phase","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.pair_levels_of_mem_phase","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#pair-levels-of-mem-phase","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with left vertices $0, \\dots, a-1$, let $\\mu$ be a valid partner array (partners are neighbours, and no left vertex is the partner of two right vertices), and let $d \\le a$ be such that round $d + 1$ of the alternating layered search for $\\mu$ reaches an unmatched right vertex and no earlier round does. Then for every pair $(i, j)$ on a path of the Hopcroft\u2013Karp phase for $\\mu$ there is\u2026","title":"Levels of the pairs of a phase"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/free-of-free-flip","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.free_of_free_flip","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#free-of-free-flip","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array (a matching of the graph) whose first free round is defined, equal to some $d$. If a left vertex $i$ is unmatched in the flip of $\\mu$ along the concatenation of all the paths of the phase of $\\mu$, then $i$ is unmatched in $\\mu$.","title":"A phase leaves no new unmatched left vertex"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/layers-flip-phase-subset","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.layers_flip_phase_subset","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#layers-flip-phase-subset","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let the partner array of a bipartite graph with $a$ left vertices be valid, and suppose that some round $r \\le a + 1$ of the layered search reaches an unmatched right vertex, so that a phase runs from the earliest such round. After flipping all the paths that phase finds, for every $k \\in \\mathbb{N}$ the left vertices and the right vertices reached within $k$ rounds of the layered search were already reached within\u2026","title":"Distances do not decrease across a phase"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/le-first-free-round-flip-phase","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.le_firstFreeRound_flip_phase","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#le-first-free-round-flip-phase","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner assignment of a bipartite graph (a matching), and let $\\mu'$ be obtained from $\\mu$ by flipping all the paths of the phase of $\\mu$ at once. If the first free round of $\\mu$ is $d$ and the first free round of $\\mu'$ is $d'$, then $d \\le d'$.","title":"The first free round does not decrease across a phase"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/phase-spec","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.phase_spec","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#phase-spec","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Suppose the partner array describes a matching and $d$ is the earliest round such that round $d + 1$ of the alternating breadth-first search reaches an unmatched right vertex. Then every path of the phase is an augmenting path at level $d$ from some right vertex, and no left vertex and no right vertex occurs twice among the pairs of all the paths of the phase.","title":"The paths of a phase are vertex-disjoint augmenting paths"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/chain-of-chain-flip-phase","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.chain_of_chain_flip_phase","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#chain-of-chain-flip-phase","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array whose first free round is defined, and let $\\mu'$ be obtained from $\\mu$ by flipping all the paths of the phase of $\\mu$. Let $z$ be a right vertex and $m$ a natural number such that $z$ is first reached in round $m + 1$ of the layered search both for $\\mu$ and for $\\mu'$, and let $S$ be an alternating chain for $\\mu'$ from $z$ to an unmatched left vertex with exactly $m + 1$\u2026","title":"Shortest chains from a vertex whose level a phase keeps were chains before it"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/lt-first-free-round-flip-phase","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.lt_firstFreeRound_flip_phase","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#lt-first-free-round-flip-phase","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"The next phase's paths are strictly longer (Hopcroft and Karp). Let $\\mu$ be a valid partner assignment of a bipartite graph (a matching), and let $\\mu'$ be obtained from $\\mu$ by flipping all the paths of the phase of $\\mu$ at once. If the first free round of $\\mu$ is $d$ and the first free round of $\\mu'$ is $d'$, then $d < d'$.","title":"The next phase's paths are strictly longer"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-le-card-pairs-add","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_le_card_pairs_add","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-le-card-pairs-add","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"If no unmatched right vertex is reached by round `k + 1`, every matching has at most as many edges as the current one plus the right vertices first reached in round `k + 1`: the unreached left vertices and the right vertices reached by round `k + 1` cover every edge.","title":"One round of the layered search bounds the deficit"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/mul-card-sub-le","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.mul_card_sub_le","parent":38932,"reading_title":{"format":"inline-parts-v1","parts":[["text","The deficit is at most "],["math","\\(b / d\\)"]],"source":"The deficit is at most $b / d$"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#mul-card-sub-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with right vertices $0, \\dots, b-1$, let $\\mu$ be a valid partner array (partners are neighbours, and no left vertex is the partner of two right vertices), let $M$ be any matching of the graph, and let $d \\in \\mathbb{N}$. Suppose that for every $k < d$, no unmatched right vertex is among the right vertices reached within $k + 1$ rounds of the alternating layered search for $\\mu$. Then\u2026","title":"The deficit is at most $b / d$"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/hopcroft-karp-aux-of-hk-step-none","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.hopcroftKarpAux_of_hkStep_none","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#hopcroft-karp-aux-of-hk-step-none","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"If the Hopcroft\u2013Karp step on a partner array $\\mu$ returns nothing, then for every fuel $f \\in \\mathbb{N}$ the Hopcroft\u2013Karp phase loop with fuel $f$ from $\\mu$ returns $\\mu$.","title":"The phase loop stops when a step finds nothing"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/hopcroft-karp-aux-add","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.hopcroftKarpAux_add","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#hopcroft-karp-aux-add","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"For all $f, g \\in \\mathbb{N}$ and every partner array $\\mu$, the Hopcroft\u2013Karp phase loop with fuel $f + g$ from $\\mu$ returns the same partner array as the loop with fuel $g$ run from the result of the loop with fuel $f$ from $\\mu$.","title":"Fuel adds up in the phase loop"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/first-free-round-hopcroft-karp-aux","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.firstFreeRound_hopcroftKarpAux","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#first-free-round-hopcroft-karp-aux","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array and $c$ a natural number such that the first free round of $\\mu$, if defined, is at least $c$. For every natural number $f$, running at most $f$ Hopcroft\u2013Karp phases from $\\mu$, stopping early when a phase finds no path, gives a valid partner array whose first free round, if defined, is at least $c + f$.","title":"Each phase lengthens the shortest augmenting paths"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-le-hopcroft-karp-aux-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_le_hopcroftKarpAux_of_le","parent":38932,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closing a deficit of "],["math","\\(f\\)"],["text"," in "],["math","\\(f\\)"],["text"," Hopcroft\u2013Karp phases"]],"source":"Closing a deficit of $f$ in $f$ Hopcroft\u2013Karp phases"},"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-le-hopcroft-karp-aux-of-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"Let $\\mu$ be a valid partner array, $M$ a matching of the bipartite graph and $f \\in \\mathbb{N}$. If $M$ has at most $f$ more pairs than the matched pairs of $\\mu$, then $M$ has at most as many pairs as the matched pairs of the partner array returned by the Hopcroft\u2013Karp phase loop with fuel $f$ from $\\mu$.","title":"Closing a deficit of $f$ in $f$ Hopcroft\u2013Karp phases"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/card-le-hopcroft-karp-aux-sqrt","kind":"node","lean":"Cslib.Algorithms.Lean.BipartiteMatching.card_le_hopcroftKarpAux_sqrt","parent":38932,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching/index.html#card-le-hopcroft-karp-aux-sqrt","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/maximum-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with $b$ right vertices, run at most $2 \\lfloor \\sqrt{b} \\rfloor + 2$ Hopcroft\u2013Karp phases from the empty matching, stopping early when a phase finds no augmenting path. Then no matching of the graph has more edges than the resulting one.","title":"Hopcroft and Karp's bound on the number of phases"},{"data_url":"scopes/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm.json","id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","kind":"scope","navigation":"module","parent":38815,"result":false,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","title":"Prim's algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero element, let $W \\cup \\{\\top\\}$ be $W$ with a new greatest element $\\top$ adjoined, let $D$ be a directed adjacency structure on the vertices $0, \\dots, n-1$, with vertex list $L$ and adjacency lists $A(u)$, let $w$ assign a weight $w(\\{u, v\\}) \\in W$ to every unordered pair $\\{u, v\\}$ of vertices, not necessarily distinct, and let $r$ be a vertex. Prim's algorithm\u2026","title":"Prim's algorithm with an indexed binary heap"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim-inspections","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim_inspections","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim-inspections","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero element, let $D$ be a directed adjacency structure on the vertices $0, \\dots, n-1$ with adjacency lists $A(u)$, let $w$ assign a weight in $W$ to every unordered pair of vertices, not necessarily distinct, and let $r$ be a vertex. The number of adjacency entries inspected by Prim's algorithm on $D$, $w$ and $r$ equals the sum of the lengths $|A(u)|$ over all vertices\u2026","title":"Number of adjacency inspections in Prim's algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim-key-tests-le","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim_key_tests_le","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim-key-tests-le","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero element, let $D$ be a directed adjacency structure on the vertices $0, \\dots, n-1$ with adjacency lists $A(u)$, let $w$ assign a weight in $W$ to every unordered pair of vertices, not necessarily distinct, and let $r$ be a vertex. The number of inspected adjacency entries that Prim's algorithm on $D$, $w$ and $r$ finds still in the queue, which is its number of key\u2026","title":"Bound on key comparisons in Prim's algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim-heap-time","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim_heap_time","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim-heap-time","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero element, let $D$ be a directed adjacency structure on the vertices $0, \\dots, n-1$ with adjacency lists $A(u)$, let $w$ assign a weight in $W$ to every unordered pair of vertices, not necessarily distinct, let $r$ be a vertex, and let $a$ be the sum of the lengths $|A(u)|$ over all vertices $u$. Each of the five heap event counts of Prim's algorithm on $D$, $w$ and $r$\u2026","title":"Logarithmic heap cost of Prim's algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim-edge-inspections","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim_edge_inspections","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim-edge-inspections","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero element, let $G$ be a simple graph on the vertices $0, \\dots, n-1$, let $D$ be a directed adjacency structure on the same vertices with adjacency lists $A(u)$, let $w$ assign a weight in $W$ to every unordered pair of vertices, not necessarily distinct, and let $r$ be a vertex. Suppose that for all vertices $u$ and $v$, $v$ occurs in $A(u)$ exactly when $u$ and $v$ are\u2026","title":"Adjacency inspections of Prim's algorithm on a simple graph"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim-root","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim_root","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim-root","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero element, let $D$ be a directed adjacency structure on the vertices $0, \\dots, n-1$, let $w$ assign a weight in $W$ to every unordered pair of vertices, not necessarily distinct, and let $r$ be a vertex. In the output of Prim's algorithm on $D$, $w$ and $r$, the vertex $r$ has no parent.","title":"The root has no parent in Prim's algorithm"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim-parent-edge","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim_parent_edge","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim-parent-edge","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero element, let $D$ be a directed adjacency structure on the vertices $0, \\dots, n-1$ with adjacency lists $A(u)$, let $w$ assign a weight in $W$ to every unordered pair of vertices, not necessarily distinct, and let $r$ be a vertex. For all vertices $u$ and $v$, if Prim's algorithm on $D$, $w$ and $r$ returns $u$ as the parent of $v$, then $v$ occurs in $A(u)$.","title":"Parents returned by Prim's algorithm are adjacency-list neighbors"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim-is-tree","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim_isTree","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim-is-tree","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero element, let $G$ be a connected simple graph on the vertices $0, \\dots, n-1$, let $D$ be a directed adjacency structure on the same vertices with adjacency lists $A(u)$ such that for all vertices $u$ and $v$, $v$ occurs in $A(u)$ exactly when $u$ and $v$ are adjacent in $G$, let $w$ assign a weight in $W$ to every unordered pair of vertices, not necessarily distinct,\u2026","title":"Prim's algorithm returns a tree"},{"id":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/prim-minimum-spanning-tree","kind":"node","lean":"Cslib.Algorithms.Lean.Prim.prim_minimum_spanningTree","parent":39024,"result":true,"route":"../roadmap/ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm/index.html#prim-minimum-spanning-tree","scope":"ComputerScience/graph-algorithms/spanning-trees-flows-matchings/prim-s-algorithm","status":"fully_proved","summary":"Let $G$ be a connected simple graph on the vertices $0, \\dots, n-1$, let $D$ be a directed adjacency structure on the same vertices with adjacency lists $A(u)$ such that for all vertices $u$ and $v$, $v$ occurs in $A(u)$ exactly when $u$ and $v$ are adjacent in $G$, let $w$ assign a real weight $w(e)$ to every unordered pair $e$ of vertices, not necessarily distinct, and let $r$ be a vertex. Let $T$ be the simple\u2026","title":"Prim's algorithm returns a minimum spanning tree"},{"data_url":"scopes/ComputerScience/process-calculi.json","id":"ComputerScience/process-calculi","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/process-calculi/index.html","scope":"ComputerScience/process-calculi","status":"fully_proved","summary":"Algebraic models of concurrent processes: labelled transition systems and their runs, strong and weak bisimulation, observational congruence, CCS (choice, parallel composition, restriction, recursion, finite-state agents) and the pi-calculus.","title":"Process calculi and behavioural equivalence"},{"data_url":"scopes/ComputerScience/process-calculi/session-types.json","id":"ComputerScience/process-calculi/session-types","kind":"scope","parent":39034,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/index.html","scope":"ComputerScience/process-calculi/session-types","status":"fully_proved","summary":"Session types for the \u03c0-calculus: binary protocols and their duality, linear endpoint typing with ownership contexts, subject reduction, progress by priorities, typing up to protocol equivalence, and shared and recursive services.","title":"Session types"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/finite-binary-sessions.json","id":"ComputerScience/process-calculi/session-types/finite-binary-sessions","kind":"scope","parent":39035,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/index.html","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions","status":"fully_proved","title":"Finite binary sessions"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data.json","id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","kind":"scope","navigation":"module","parent":39036,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","title":"Finite sessions with unrestricted name data"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/session-type","kind":"node","lean":"Cslib.SessionTypes.SessionType","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#session-type","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Finite binary protocols carrying unrestricted, passive names.","title":"Finite binary session types"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/dual","kind":"node","lean":"Cslib.SessionTypes.SessionType.dual","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#dual","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Exchange the endpoint's send and receive obligations.","title":"Dual of a session type"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/dual-dual","kind":"node","lean":"Cslib.SessionTypes.SessionType.dual_dual","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#dual-dual","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Duality is an involution, including at completion.","title":"Session type duality is an involution"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/length","kind":"node","lean":"Cslib.SessionTypes.SessionType.length","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#length","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Number of communication actions remaining in a finite protocol.","title":"Length of a session type"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/length-dual","kind":"node","lean":"Cslib.SessionTypes.SessionType.length_dual","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#length-dual","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Opposite endpoint roles have the same number of actions.","title":"Dual session types have equal length"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/context","kind":"node","lean":"Cslib.SessionTypes.Context","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#context","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"One linear endpoint obligation per physical channel in a thread.","title":"Per-channel session context of a thread"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/advance","kind":"node","lean":"Cslib.SessionTypes.advance","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#advance","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Advance just the communicating endpoint; other obligations are unchanged.","title":"Advancing one endpoint of a session context"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/dual-context","kind":"node","lean":"Cslib.SessionTypes.dualContext","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#dual-context","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"The opposite endpoint roles, pointwise.","title":"Pointwise dual of a session context"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/dual-context-advance","kind":"node","lean":"Cslib.SessionTypes.dualContext_advance","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#dual-context-advance","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Advancing dual endpoints preserves pointwise compatibility.","title":"Dualizing a context commutes with advancing an endpoint"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/has-type","kind":"node","lean":"Cslib.SessionTypes.HasType","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#has-type","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Linear typing of finite sequential threads in the existing \u03c0 syntax. `nil` consumes only completed obligations. Input-bound names are unrestricted payloads, never session subjects; there is no parallel duplication rule.","title":"Linear session typing of sequential threads"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/subst-proc","kind":"node","lean":"Cslib.SessionTypes.HasType.substProc","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#subst-proc","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Capture-avoiding payload substitution preserves every linear obligation, including through nested input binders and at nonzero cutoffs.","title":"Payload substitution at any cutoff preserves session typing"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/subst","kind":"node","lean":"Cslib.SessionTypes.HasType.subst","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#subst","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Receiving a closed name preserves the residual thread's typing.","title":"Receiving a closed name preserves session typing"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/finite-support","kind":"node","lean":"Cslib.SessionTypes.HasType.finite_support","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#finite-support","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"A derivation has only finitely many active channel obligations: above some bound every channel is completed. Contexts are total functions.","title":"Session typing derivations have finite support"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/closed-at","kind":"node","lean":"Cslib.SessionTypes.HasType.closedAt","parent":39037,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data/index.html#closed-at","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/finite-sessions-with-unrestricted-name-data","status":"fully_proved","summary":"Every typed thread satisfies the shared \u03c0 name/process scoping predicate. Free session constants remain permitted; this is not \u03bd-closedness.","title":"Session-typed threads are well scoped"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus.json","id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus","kind":"scope","navigation":"module","parent":39036,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/index.html","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus","status":"fully_proved","title":"Binary communication in the existing \u03c0-calculus"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/represent","kind":"node","lean":"Cslib.SessionTypes.represent","parent":39052,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/index.html#represent","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus","status":"fully_proved","summary":"Embed a binary configuration without changing either process.","title":"Binary configuration as a parallel composition"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/represent-injective","kind":"node","lean":"Cslib.SessionTypes.represent_injective","parent":39052,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/index.html#represent-injective","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus","status":"fully_proved","summary":"The representation does not identify distinct configurations.","title":"Injectivity of the binary configuration embedding"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/comm-step","kind":"node","lean":"Cslib.SessionTypes.CommStep","parent":39052,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/index.html#comm-step","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus","status":"fully_proved","summary":"Opposite head guards synchronize, in either endpoint orientation.","title":"Communication step of a binary configuration"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/represent-step","kind":"node","lean":"Cslib.SessionTypes.represent_step","parent":39052,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/index.html#represent-step","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus","status":"fully_proved","summary":"Every session communication is a reduction in the unchanged \u03c0 semantics.","title":"Session communication steps are \u03c0-calculus reductions"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/binary-typed","kind":"node","lean":"Cslib.SessionTypes.BinaryTyped","parent":39052,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/index.html#binary-typed","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus","status":"fully_proved","summary":"Locally linear threads owning opposite roles on every channel. Dual compatibility permits cyclic waits; it does not assert progress.","title":"Dual session typing of a binary configuration"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/preservation","kind":"node","lean":"Cslib.SessionTypes.CommStep.preservation","parent":39052,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus/index.html#preservation","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/binary-communication-in-the-existing-calculus","status":"fully_proved","summary":"Communication consumes dual heads and preserves linear residual typing; payload substitution never changes the session context.","title":"Communication preserves binary session typing"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina.json","id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","kind":"scope","navigation":"module","parent":39036,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","title":"Progress and completion for isolated finite binary sessions"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/single-channel","kind":"node","lean":"Cslib.SessionTypes.SingleChannel","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#single-channel","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"All active obligations of this thread belong to one physical channel.","title":"Single active channel condition for a session context"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/dual","kind":"node","lean":"Cslib.SessionTypes.SingleChannel.dual","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#dual","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"Opposite roles have the same active-channel support.","title":"Dual context keeps the single active channel"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/advance","kind":"node","lean":"Cslib.SessionTypes.SingleChannel.advance","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#advance","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"Communication on the sole session cannot create another dependency.","title":"Advancing the sole channel keeps the single-channel condition"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/eq-nil-of-completed","kind":"node","lean":"Cslib.SessionTypes.HasType.eq_nil_of_completed","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#eq-nil-of-completed","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"A typed thread whose endpoint obligations are completed is inert.","title":"Typed thread with completed obligations is nil"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/active-eq","kind":"node","lean":"Cslib.SessionTypes.SingleChannel.active_eq","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#active-eq","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"Every active endpoint of a single-channel thread is the designated one.","title":"The only active channel is the designated one"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/send-shape","kind":"node","lean":"Cslib.SessionTypes.HasType.send_shape","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#send-shape","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"A send obligation forces a send guard on the sole active channel.","title":"Send obligation on the sole channel forces an output guard"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/recv-shape","kind":"node","lean":"Cslib.SessionTypes.HasType.recv_shape","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#recv-shape","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"A receive obligation forces an input guard on the sole active channel.","title":"Receive obligation on the sole channel forces an input guard"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/progress","kind":"node","lean":"Cslib.SessionTypes.progress","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#progress","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"Dual single-channel threads have either completed or can synchronize. The single-channel hypothesis excludes cross-session dependency cycles.","title":"Progress for dual single-channel session threads"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/completed","kind":"node","lean":"Cslib.SessionTypes.BinaryTyped.completed","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#completed","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"A completed sole protocol means both threads have reached `nil`.","title":"Completed sole protocol means both threads are nil"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/represent-steps","kind":"node","lean":"Cslib.SessionTypes.represent_steps","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#represent-steps","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"Paths of session communication are paths of existing \u03c0 reductions.","title":"Session communication paths are \u03c0-calculus reduction paths"},{"id":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/finite-completion","kind":"node","lean":"Cslib.SessionTypes.finite_completion","parent":39059,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina/index.html#finite-completion","scope":"ComputerScience/process-calculi/session-types/finite-binary-sessions/progress-and-completion-for-isolated-finite-bina","status":"fully_proved","summary":"Every isolated finite dual session admits a finite communication path to completion. This excludes recursion and is not global \u03c0 normalization.","title":"Finite completion of isolated finite binary sessions"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus","kind":"scope","parent":39035,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus","status":"fully_proved","title":"Linear endpoint typing for the \u03c0-calculus"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binary-has-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.binary_hasType","parent":39071,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/index.html#binary-has-type","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus","status":"fully_proved","summary":"Dual original threads embed as disjoint opposite roles on the same physical constants, with exactly their original finite protocol obligations.","title":"Binary session typing embeds into polarized endpoint typing"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","status":"fully_proved","title":"Finite linear \u03c0 fragment and structural closure"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/finite-linear","kind":"node","lean":"Cslib.SessionTypes.Endpoint.FiniteLinear","parent":39073,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/index.html#finite-linear","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","status":"fully_proved","summary":"Existing \u03c0 terms without choice or replication, including their bodies.","title":"Finite linear \u03c0 fragment without choice or replication"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/finite-linear-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.finiteLinear_lift","parent":39073,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/index.html#finite-linear-lift","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","status":"fully_proved","summary":"Lifting cannot add or remove unsupported process constructors.","title":"Lifting preserves and reflects the finite linear fragment"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/finite-linear-swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.finiteLinear_swap","parent":39073,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/index.html#finite-linear-swap","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","status":"fully_proved","summary":"Swapping restrictions cannot add or remove choice or replication.","title":"Restriction swap preserves and reflects the finite linear fragment"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/finite-linear-cong","kind":"node","lean":"Cslib.SessionTypes.Endpoint.finiteLinear_cong","parent":39073,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/index.html#finite-linear-cong","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","status":"fully_proved","summary":"All existing structural congruences preserve the finite linear fragment. In particular, folding replication cannot introduce it.","title":"Structural congruence preserves the finite linear fragment"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/finite-linear-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.finiteLinear_subst","parent":39073,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/index.html#finite-linear-subst","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","status":"fully_proved","summary":"Substitution changes names, not the finite fragment's process shape.","title":"Substitution preserves and reflects the finite linear fragment"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/finite-linear-select-eq","kind":"node","lean":"Cslib.SessionTypes.Endpoint.finiteLinear_select_eq","parent":39073,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/index.html#finite-linear-select-eq","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","status":"fully_proved","summary":"Selection cannot discard a branch of a term in the choice-free fragment.","title":"Selection is trivial on the choice-free fragment"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/finite-linear-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.finiteLinear_step","parent":39073,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure/index.html#finite-linear-step","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-linear-fragment-and-structural-closure","status":"fully_proved","summary":"All existing \u03c0 reduction constructors preserve the finite fragment, including arbitrary pre/post structural congruence and selection.","title":"Reduction preserves the finite linear fragment"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","title":"Polarized endpoint identity"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/role","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Role","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#role","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Opposite capabilities on a single physical \u03c0 name.","title":"Endpoint role on a \u03c0 name"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/key","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#key","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"One endpoint key consists of the existing physical name and a role.","title":"Polarized endpoint key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/opposite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Role.opposite","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#opposite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Reverse the linear role without changing the physical name.","title":"Opposite endpoint role"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/opposite-opposite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Role.opposite_opposite","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#opposite-opposite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Role reversal is an involution.","title":"Role reversal is an involution"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/opposite-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.opposite","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#opposite-2","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"The peer role on the same physical channel.","title":"Peer endpoint key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/opposite-opposite-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.opposite_opposite","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#opposite-opposite-2","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Reversing an endpoint twice recovers its entire key.","title":"Key reversal is an involution"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/opposite-ne","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.opposite_ne","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#opposite-ne","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Opposite roles cannot collide, even on one physical name.","title":"An endpoint key differs from its peer"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/map-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.mapName","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#map-name","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Apply an existing-name transformation, leaving the role untouched.","title":"Renaming the physical name of an endpoint key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/map-name-injective","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.mapName_injective","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#map-name-injective","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Injective physical-name transformations preserve endpoint identity.","title":"Injective renamings act injectively on endpoint keys"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/ext","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.ext","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#ext","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Endpoint identity is exactly its physical name and linear role.","title":"Extensionality for endpoint keys"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/eq-opposite-of-name-eq","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.eq_opposite_of_name_eq","parent":39081,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity/index.html#eq-opposite-of-name-eq","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/polarized-endpoint-identity","status":"fully_proved","summary":"Distinct keys on one physical name must be opposite endpoint roles.","title":"Distinct keys on one name are opposite"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs","status":"fully_proved","title":"Guarded regular protocol graphs"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs/node","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Node","parent":39093,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs/index.html#node","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs","status":"fully_proved","summary":"A graph node: completion or a communication with a bounded continuation. Endpoint payloads include their role, rank, cursor and protocol orientation.","title":"Node of a guarded protocol graph"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs/graph","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Graph","parent":39093,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs/index.html#graph","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/guarded-regular-protocol-graphs","status":"fully_proved","summary":"Finite presentation of guarded equi-recursive protocols. An empty graph has no cursor and therefore cannot denote a protocol.","title":"Finite guarded regular protocol graph"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","status":"fully_proved","title":"Name and capability protocols with guarded regular recursion"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/protocol","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol","parent":39096,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/index.html#protocol","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","status":"fully_proved","summary":"A protocol with explicit payload sort and guarded graph recursion. Payload endpoint roles are transmitted unchanged, not dualized.","title":"Endpoint protocol with guarded regular recursion"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/dual","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.dual","parent":39096,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/index.html#dual","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","status":"fully_proved","summary":"Reverse subject actions and continuations; preserve payload ownership.","title":"Dual of an endpoint protocol"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/dual-dual","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.dual_dual","parent":39096,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/index.html#dual-dual","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","status":"fully_proved","summary":"Duality is involutive for finite and guarded recursive protocols.","title":"Protocol duality is an involution"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/unfold","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.unfold","parent":39096,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/index.html#unfold","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","status":"fully_proved","summary":"Expose one communication guard. Graph successors retain their subject orientation; an endpoint payload retains its independently specified one.","title":"One-step unfolding of a protocol"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/unfold-dual","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.unfold_dual","parent":39096,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/index.html#unfold-dual","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","status":"fully_proved","summary":"Guarded unfolding commutes with duality, including recursive payloads.","title":"Unfolding commutes with protocol duality"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/unfold-ne-regular","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.unfold_ne_regular","parent":39096,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/index.html#unfold-ne-regular","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","status":"fully_proved","summary":"Unfolding always exposes an action or completion, never another silent regular-cursor wrapper. Cycles therefore cannot consist only of unfolding.","title":"Unfolding never yields a regular-cursor protocol"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/unfold-unfold","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.unfold_unfold","parent":39096,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul/index.html#unfold-unfold","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/name-and-capability-protocols-with-guarded-regul","status":"fully_proved","summary":"One root unfolding is enough to expose a guard or completion.","title":"Protocol unfolding is idempotent"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","title":"Finite endpoint ownership maps"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/capability","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#capability","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"One linear protocol obligation with its dependency rank.","title":"Endpoint capability with a dependency rank"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/context","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#context","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Finite ownership map, indexed by polarized keys rather than names.","title":"Finite endpoint ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/dual","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability.dual","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#dual","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"The peer's obligation has the same rank and dual subject protocol.","title":"Dual of an endpoint capability"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/dual-dual","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability.dual_dual","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#dual-dual","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Capability duality does not change rank or lose protocol information.","title":"Capability duality is an involution"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#rekey","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Rekey a finite map when no two owned keys are merged. Global injectivity is unnecessary, which matters for endpoint substitution.","title":"Collision-free rekeying of an ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/mem-rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.mem_rekey","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#mem-rekey","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Rekeying has precisely the image of the original owned keys.","title":"Keys of a rekeyed ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/lookup-rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lookup_rekey","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#lookup-rekey","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Rekeying preserves each capability verbatim at its image key.","title":"Lookups in a rekeyed ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/disjoint-rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.disjoint_rekey","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#disjoint-rekey","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Disjoint ownership remains disjoint when the combined owned domain is rekeyed injectively. Opposite roles are not collapsed to physical names.","title":"Injective rekeying preserves disjoint ownership"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/unique","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.unique","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#unique","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Two lookups of the same linear key cannot give different capabilities.","title":"Uniqueness of capability lookups"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.Completed","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#completed","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"A context has no remaining communication obligation. End capabilities may still be recorded, but an active protocol is never silently discarded.","title":"Completed ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/completed-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.completed_empty","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#completed-empty","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"The empty ownership map is completed.","title":"The empty ownership map is completed"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/rekey-rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_rekey","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#rekey-rekey","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Consecutive collision-free rekeyings compose on the same finite map.","title":"Composition of collision-free rekeyings"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/rekey-id","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_id","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#rekey-id","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Identity rekeying retains the exact finite map, including end slots.","title":"Identity rekeying fixes an ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/rekey-eq","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_eq","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#rekey-eq","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Rekeying that fixes all owned keys retains the exact map.","title":"Rekeying that fixes every owned key fixes the map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/rekey-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.Completed.rekey","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#rekey-2","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Completed protocols remain completed under collision-free name transport.","title":"Rekeying preserves completion"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.Completed.insert","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#insert","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Recording a completed endpoint preserves completion of a finite map.","title":"Inserting a completed capability preserves completion"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/erase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.Completed.erase","parent":39104,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps/index.html#erase","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-endpoint-ownership-maps","status":"fully_proved","summary":"Erasing a key preserves absence of any noncompleted obligation; the empty map is not itself a completion claim about hidden \u03bd processes.","title":"Erasing a key preserves completion"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","title":"Binder transport of finite ownership maps"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lift","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#lift","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"Insert an ambient name binder at any cutoff in an ownership context.","title":"Lifting an ownership map past a binder"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/lookup-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lookup_lift","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#lookup-lift","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"Lifting keeps the entire capability at the corresponding lifted key.","title":"Lookups in a lifted ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.bind","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#bind","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"A restriction binds one physical name and creates its two distinct linear roles. Lifted outer keys cannot be overwritten by these insertions.","title":"Binding a fresh dual pair under restriction"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/fresh-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.fresh_lift","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#fresh-lift","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"Both roles of the newly inserted name are fresh for the lifted context.","title":"Both roles of the new name are fresh for a lifted map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/lookup-bind-positive","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lookup_bind_positive","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#lookup-bind-positive","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"Restriction installs the positive capability without losing it to the opposite role's insertion.","title":"Positive role lookup after restriction binding"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/lookup-bind-negative","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lookup_bind_negative","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#lookup-bind-negative","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"The same restriction installs the negative peer at the SAME name.","title":"Negative role lookup after restriction binding"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/lift-singleton","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lift_singleton","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#lift-singleton","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"Lifting a single capability shifts its name, not its protocol or role.","title":"Lifting a singleton ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/lift-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lift_empty","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#lift-empty","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"Inserting a binder does not manufacture an outer capability.","title":"Lifting the empty ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/bind-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.bind_empty","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#bind-empty","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"In an empty outer context, restriction exposes exactly its two peers.","title":"Restriction binding over the empty map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/entries-bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.entries_bind","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#entries-bind","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"The exact restriction context has two fresh entries plus all shifted outer ownership. No old entry is overwritten by a new endpoint role.","title":"Entries of a restriction-bound ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/fresh-lift-of-fresh","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.fresh_lift_of_fresh","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#fresh-lift-of-fresh","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"Lifting preserves absence of an outer role from its finite context.","title":"Lifting preserves absence of a key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/fresh-bind-of-fresh","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.fresh_bind_of_fresh","parent":39122,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps/index.html#fresh-bind-of-fresh","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/binder-transport-of-finite-ownership-maps","status":"fully_proved","summary":"An absent outer key remains absent after adding the fresh bound dual pair.","title":"Restriction binding preserves absence of an outer key"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","title":"Balanced physical channels with distinct linear roles"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Balanced","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#balanced","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"Every owned role has its opposite role at the same physical name, with the same rank and the dual protocol. This condition may permit deadlocks.","title":"Balanced endpoint ownership"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/balanced-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.balanced_empty","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#balanced-empty","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"An empty external interface is balanced. Internal \u03bd channels still require their own balanced ownership and are not thereby completed.","title":"The empty interface is balanced"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/balanced-bind-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.balanced_bind_empty","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#balanced-bind-empty","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"A real fresh restriction creates a balanced pair on a single physical name.","title":"A fresh restriction pair is balanced"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Balanced.rekey","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#rekey","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"Capture-safe rekeying transports balance when endpoint roles are retained. The injectivity condition is on owned keys, not on arbitrary passive data.","title":"Rekeying preserves balance"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/compatible","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Compatible","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#compatible","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"A partial interface is compatible if opposite roles, WHEN both are present, have dual protocols. A missing peer is not an enabled-edge witness.","title":"Compatibility of a partial endpoint interface"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/compatible-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Balanced.compatible","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#compatible-2","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"A balanced complete interface supplies compatibility for every present pair.","title":"Balanced interfaces are compatible"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/disjoint-subjects-opposite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.disjoint_subjects_opposite","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#disjoint-subjects-opposite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"A dual-compatible pair of physically equal but disjoint owned keys must be opposite roles. The runtime channel name alone does not supply this.","title":"Disjoint owned keys on one name have opposite roles"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/union-left","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Compatible.union_left","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#union-left","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"Any left subcontext of a compatible interface retains compatibility of its present pairs; no global-progress assumption is involved.","title":"Compatibility passes to the left part of a union"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/union-right","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Compatible.union_right","parent":39135,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear/index.html#union-right","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/balanced-physical-channels-with-distinct-linear","status":"fully_proved","summary":"A right child of a disjoint linear split inherits compatibility of its present pairs from the combined interface.","title":"Compatibility passes to the right part of a disjoint split"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","title":"Finite ownership algebra for typing transport"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/rekey-insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_insert","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#rekey-insert","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Rekeying commutes with capability insertion when the combined domain has no key collision. The condition is on ownership, not execution.","title":"Collision-free rekeying commutes with insertion"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/rekey-erase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_erase","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#rekey-erase","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Erasing an owned key commutes with collision-free rekeying.","title":"Collision-free rekeying commutes with erasing an owned key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/rekey-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_union","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#rekey-union","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Collision-free rekeying commutes with a linear parallel split.","title":"Collision-free rekeying commutes with a disjoint union"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/lift-insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lift_insert","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#lift-insert","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Context lifting commutes with endpoint advancement at any cutoff.","title":"Context lifting commutes with insertion"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/fresh-of-disjoint","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.fresh_of_disjoint","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#fresh-of-disjoint","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"A sent endpoint is absent from the other side of a linear split.","title":"Owned keys are absent from the other side of a disjoint split"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/fresh-insert-owned","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.fresh_insert_owned","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#fresh-insert-owned","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Advancing another already-owned carrier does not create a payload alias.","title":"Updating an owned key keeps absent keys absent"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/disjoint-insert-owned","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.disjoint_insert_owned","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#disjoint-insert-owned","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Advancing an already-owned endpoint preserves a disjoint split.","title":"Updating an owned key preserves a disjoint split"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/union-insert-owned","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.union_insert_owned","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#union-insert-owned","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Advancing both peers across a disjoint split advances exactly those two keys in the combined interface, with no other capability change.","title":"Updating owned keys on both sides of a disjoint union"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/disjoint-transfer","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.disjoint_transfer","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#disjoint-transfer","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Capability transfer is a disjoint ownership move, including when the sender and receiver also advance their distinct carrier roles.","title":"Capability transfer with carrier updates preserves disjointness"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/transfer-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.transfer_union","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#transfer-union","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Moving a capability out of one map into the other preserves the exact combined interface, including completed endpoint slots.","title":"Moving a capability across a union preserves the union"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/disjoint-erased-transfer","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.disjoint_erased_transfer","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#disjoint-erased-transfer","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Removing a key from the sender and inserting it at the receiver preserves disjoint endpoint ownership.","title":"Erase-and-insert transfer preserves disjointness"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/transfer-union-insert-owned","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.transfer_union_insert_owned","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#transfer-union-insert-owned","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Delegation plus two carrier advances preserves every other key in the combined context; the sent capability is moved, not duplicated or dropped.","title":"Combined context after delegation and two carrier updates"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/erase-insert-ne","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.erase_insert_ne","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#erase-insert-ne","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Erasing a different endpoint commutes with insertion, which is needed when an unused completed slot passes through a delegation prefix.","title":"Erasing a different key commutes with insertion"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/disjoint-insert-fresh","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.disjoint_insert_fresh","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#disjoint-insert-fresh","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"An insertion disjoint from the other side preserves a linear split, including the unused completed slots transported by structural congruence.","title":"Inserting a key absent from the other side preserves disjointness"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/bind-insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.bind_insert","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#bind-insert","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Inserting an outer endpoint before restriction is the same as inserting its lifted key after the fresh dual pair has been bound.","title":"Restriction binding commutes with inserting an outer key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/insert-union-right-of-fresh","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.insert_union_right_of_fresh","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#insert-union-right-of-fresh","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Inserting a key absent from the left component updates only the right component of a left-biased finite-map union.","title":"Inserting a left-absent key into a union updates the right part"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/lift-erase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lift_erase","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#lift-erase","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Erasing an owned outer key commutes with canonical context lifting.","title":"Context lifting commutes with erasing an owned key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/bind-erase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.bind_erase","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#bind-erase","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Erasing an owned outer role commutes with restriction's fresh dual pair.","title":"Restriction binding commutes with erasing an owned outer key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/erase-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.erase_union","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#erase-union","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Erasure distributes through the exact finite ownership union.","title":"Erasure distributes over union"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/erase-of-not-mem","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.erase_of_notMem","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#erase-of-not-mem","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Erasing an absent key changes no ownership entry.","title":"Erasing an absent key changes nothing"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/lift-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lift_union","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#lift-union","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Canonical lifting preserves a disjoint ownership union exactly.","title":"Context lifting preserves a disjoint union"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/disjoint-bind-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.disjoint_bind_lift","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#disjoint-bind-lift","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"A fresh bound pair belongs only to the left component of a lifted split.","title":"Binding a fresh pair keeps disjointness from a lifted context"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/bind-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.bind_union","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#bind-union","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Scope extrusion keeps the dual pair on the restricted component and lifts exactly the unrelated outer partner ownership.","title":"Restriction binding of a disjoint union"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/insert-of-lookup","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.insert_of_lookup","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#insert-of-lookup","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Reinstalling exactly an already-owned capability changes no entry.","title":"Reinserting an owned capability changes nothing"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/insert-erase-same","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.insert_erase_same","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#insert-erase-same","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Reinstalling a key overrides any previous entry, so prior erasure of that same key is immaterial.","title":"Insertion after erasing the same key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/erase-insert-same","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.erase_insert_same","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#erase-insert-same","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Erasing an inserted key removes its entry, independently of its value.","title":"Erasure after inserting the same key"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/restore-erased-pair","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.restore_erased_pair","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#restore-erased-pair","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Reinstalling the bound pair after removing its old slots restores the same exact map as installing that pair directly.","title":"Reinstalling the bound pair after erasing its slots"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/erase-bind-pair","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.erase_bind_pair","parent":39145,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport/index.html#erase-bind-pair","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-algebra-for-typing-transport","status":"fully_proved","summary":"Removing the two freshly bound roles exposes exactly the lifted external context, including every unrelated completed resource.","title":"Erasing the bound pair leaves the lifted context"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","title":"Pointwise semantic equivalence of actual finite ownership maps"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/context-equivalent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#context-equivalent","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Same lookup domain, identical enclosing capability ranks, and equivalent protocol trees. No matching split or retyping result is stored.","title":"Pointwise equivalence of ownership maps"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/refl","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.refl","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#refl","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Reflexivity of actual matching lookups.","title":"Reflexivity of ownership-map 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equivalence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/symm","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.symm","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#symm","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Symmetry includes both absent and present endpoints.","title":"Symmetry of ownership-map equivalence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/trans","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.trans","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#trans","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Transitivity requires the same intermediate lookup, not a chosen representative.","title":"Transitivity of ownership-map equivalence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/mem-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.mem_iff","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#mem-iff","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Exact key-domain invariance.","title":"Equivalent ownership maps have the same keys"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.completed","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#completed","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Completion invariance uses actual unfolded done, not just empty ownership.","title":"Ownership-map equivalence preserves completion"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/scope","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.scope","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#scope","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Scope of every actual owned key is unchanged with the exact domain.","title":"Ownership-map equivalence preserves key scoping"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.insert","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#insert","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Corresponding insertions preserve literal rank and semantic type equivalence.","title":"Equivalent insertions preserve ownership-map equivalence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/erase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.erase","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#erase","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Both sides erase precisely the same key.","title":"Erasure preserves ownership-map equivalence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/of-lookups","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.of_lookups","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#of-lookups","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Bidirectional actual lookup correspondence suffices for Option.Rel.","title":"Ownership-map equivalence from two-way lookup correspondence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.union","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#union","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Left-biased unions respect equivalent key domains on BOTH sides.","title":"Union preserves ownership-map equivalence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.lift","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#lift","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Canonical binder lift preserves corresponding values and exact domains.","title":"Lifting preserves ownership-map equivalence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.bind","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#bind","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"A chosen exact dual private pair can be preserved or equivalently changed.","title":"Binding equivalent fresh pairs preserves ownership-map equivalence"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/compatible","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.compatible","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#compatible","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Semantic compatibility is invariant, including explicit enclosing ranks.","title":"Ownership-map equivalence preserves semantic compatibility"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/project-keys","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.projectKeys","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#project-keys","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Restrict actual target bindings to the exact key domain of a source map. The implementation is canonical multiset filtering, not a new finite map.","title":"Projection of an ownership map onto another map's keys"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/lookup-project-keys","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lookup_projectKeys","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#lookup-project-keys","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Projection retains precisely the actual target binding at a source key.","title":"Lookups in a key projection"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/mem-project-keys","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.mem_projectKeys","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#mem-project-keys","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"The projected domain is the exact intersection of actual owned domains.","title":"Keys of a projection are the intersection of domains"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/project-keys-none","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.projectKeys_none","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#project-keys-none","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Outside a source domain the projection is literally absent.","title":"Key projection is absent outside the source domain"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/project-keys-lookup","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.projectKeys_lookup","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#project-keys-lookup","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Inside a source domain the actual target value is retained verbatim.","title":"Key projection agrees with the target inside the source domain"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/project-left","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.project_left","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#project-left","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"An equivalent whole map induces an equivalent child with the exact source key domain; no matching target split is supplied by a caller.","title":"Equivalence of a union restricts to its left part"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/split","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.split","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#split","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"GENERAL split derivation from source disjointness and whole-map equivalence. It preserves each child domain and the exact target finite-map union.","title":"Equivalence of a disjoint union splits into equivalent parts"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/singleton","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ContextEquivalent.singleton","parent":39174,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite/index.html#singleton","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/pointwise-semantic-equivalence-of-actual-finite","status":"fully_proved","summary":"Equivalent singleton capabilities preserve their exact polarized key.","title":"Equivalent capabilities give equivalent singleton maps"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge","status":"fully_proved","title":"Lossless finite passive-protocol bridge"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.ofFinite","parent":39198,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/index.html#of-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge","status":"fully_proved","summary":"Embed the original finite passive-name protocol without changing actions.","title":"Embedding of finite passive-name protocols"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/of-finite-dual","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.ofFinite_dual","parent":39198,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/index.html#of-finite-dual","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge","status":"fully_proved","summary":"The finite embedding preserves both protocol orientations.","title":"Finite protocol embedding commutes with duality"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/to-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.toFinite?","parent":39198,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/index.html#to-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge","status":"fully_proved","summary":"Recover a finite passive protocol; richer payloads and recursion are not erased into a falsely finite protocol.","title":"Partial recovery of a finite passive-name protocol"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/to-finite-of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.toFinite?_ofFinite","parent":39198,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/index.html#to-finite-of-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge","status":"fully_proved","summary":"The finite embedding has a literal left inverse.","title":"Recovery is a left inverse of the finite embedding"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/of-finite-injective","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.ofFinite_injective","parent":39198,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge/index.html#of-finite-injective","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/lossless-finite-passive-protocol-bridge","status":"fully_proved","summary":"Different finite protocols remain different in the richer representation.","title":"Injectivity of the finite protocol embedding"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n","status":"fully_proved","title":"Finite ownership maps for the original passive-name typing judgement"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.ofFinite","parent":39204,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/index.html#of-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n","status":"fully_proved","summary":"Turn finitely many original channel slots into polarized finite-map ownership. The supplied rank assignment adds no execution hypothesis.","title":"Polarized ownership map of a finite channel context"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/lookup-of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lookup_ofFinite","parent":39204,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/index.html#lookup-of-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n","status":"fully_proved","summary":"Exact lookup specification for the finite-resource bridge.","title":"Lookup in the finite-context ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/mem-of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.mem_ofFinite","parent":39204,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/index.html#mem-of-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n","status":"fully_proved","summary":"Finite bridge ownership has exactly the bounded constant keys.","title":"Keys of the finite-context ownership map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/of-finite-advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.ofFinite_advance","parent":39204,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/index.html#of-finite-advance","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n","status":"fully_proved","summary":"Advancing an original channel advances precisely its finite-map role.","title":"Finite ownership map commutes with channel advancement"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/disjoint-of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.disjoint_ofFinite","parent":39204,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/index.html#disjoint-of-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n","status":"fully_proved","summary":"Opposite finite roles split without duplicating any endpoint key.","title":"Ownership maps for opposite roles are disjoint"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/completed-of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.completed_ofFinite","parent":39204,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n/index.html#completed-of-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/finite-ownership-maps-for-the-original-passive-n","status":"fully_proved","summary":"Original completed contexts have no active finite-map protocol.","title":"Completed finite contexts give completed ownership maps"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","title":"Linear endpoint typing of the existing \u03c0 syntax"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/has-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#has-type","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Syntax-directed linear typing. Capabilities never contain preservation or progress assumptions. Protocol heads may be guarded regular protocols.","title":"Linear endpoint typing of \u03c0-calculus processes"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/closed-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.closedAt","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#closed-at","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Every endpoint-typed process satisfies the canonical \u03c0 scope predicate.","title":"Endpoint-typed processes are scoped at the ambient depth"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/finite-linear","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.finiteLinear","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#finite-linear","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Typed terms stay in the proved Cong/Step-closed finite fragment.","title":"Endpoint-typed processes lie in the finite linear fragment"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/nil-singleton","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nil_singleton","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#nil-singleton","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"A recorded completed capability is admissible at nil; an active one would fail the completed-context premise.","title":"Nil is typed by a single done capability"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/out-owned","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.out_owned","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#out-owned","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Every output subject is backed by an owned role, never merely by a passive name in the ambient data scope.","title":"Output subjects of endpoint-typed processes are owned"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/nil-insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nil_insert","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#nil-insert","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Completed endpoint slots may be added at nil without creating a communication obligation; active protocols cannot use this rule.","title":"Adding a completed capability at nil preserves typing"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/inp-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.inp_inv","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#inp-inv","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Input inversion exposes its owned role, literal payload sort, and continuation context. Passive and delegated names are not conflated.","title":"Inversion of endpoint typing for input"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/out-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.out_inv","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#out-inv","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Output inversion exposes the actual owned subject, distinct payload sort, and literal continuation resources; delegation consumes the payload role.","title":"Inversion of endpoint typing for output"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/par-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.par_inv","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#par-inv","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Parallel typing exposes an actual disjoint ownership split, not a stored preservation or progress witness.","title":"Inversion of endpoint typing for parallel composition"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/nil-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nil_inv","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#nil-inv","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Nil inversion exposes completion and scope of every recorded endpoint.","title":"Inversion of endpoint typing for nil"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/nil-singleton-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nil_singleton_completed","parent":39211,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax/index.html#nil-singleton-completed","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/linear-endpoint-typing-of-the-existing-syntax","status":"fully_proved","summary":"Any protocol whose guarded head is completed is admissible at a single nil-owned slot, including completed regular graph cursors.","title":"Nil is typed by a single capability with completed head"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/transport-of-the-original-finite-typing-api.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/transport-of-the-original-finite-typing-api","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/transport-of-the-original-finite-typing-api/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/transport-of-the-original-finite-typing-api","status":"fully_proved","title":"Transport of the original finite typing 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map"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/transport-of-the-original-finite-typing-api/of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.ofFinite","parent":39223,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/transport-of-the-original-finite-typing-api/index.html#of-finite","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/transport-of-the-original-finite-typing-api","status":"fully_proved","summary":"Every original finite passive-name derivation embeds with its exact protocol obligations and a bounded finite map of either endpoint role.","title":"Finite passive-name typing embeds into endpoint typing"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport.json","id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport","kind":"scope","navigation":"module","parent":39071,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/index.html","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport","status":"fully_proved","title":"Capture-safe finite ownership 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keys"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/inj-on-under-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.injOn_under_lift","parent":39226,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/index.html#inj-on-under-lift","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport","status":"fully_proved","summary":"Collision freedom of owned keys is stable when entering a binder.","title":"Injectivity of rekeying passes under a binder"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/rekey-under-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_under_lift","parent":39226,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/index.html#rekey-under-lift","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport","status":"fully_proved","summary":"Rekeying under a binder is exactly the lift of the outer rekeying.","title":"Rekeying under a binder is the lift of the outer 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keys"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/inj-on-under-binder","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.injOn_under_binder","parent":39226,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/index.html#inj-on-under-binder","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport","status":"fully_proved","summary":"Rekeying under a binder is injective on both fresh roles and all lifted outer ownership, not just on endpoint keys with different physical names.","title":"Rekeying under a binder is injective on fresh and lifted keys"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/inj-on-insert-owned","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.injOn_insert_owned","parent":39226,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/index.html#inj-on-insert-owned","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport","status":"fully_proved","summary":"Updating an already-owned key introduces no new aliasing obligation.","title":"Injectivity of rekeying survives updating an owned 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name"},{"id":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/inj-on-under-received","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.injOn_under_received","parent":39226,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport/index.html#inj-on-under-received","scope":"ComputerScience/process-calculi/session-types/linear-endpoint-typing-for-the-calculus/capture-safe-finite-ownership-transport","status":"fully_proved","summary":"A linear input's fresh variable role preserves collision freedom of all lifted outer keys, including outer keys of the opposite role.","title":"Injectivity of rekeying after receiving a fresh 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endpoint"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/has-unguarded-input","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasUnguardedInput","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#has-unguarded-input","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"An input is exposed outside every communication guard.","title":"Unguarded input in a process"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/input-cong","kind":"node","lean":"Cslib.SessionTypes.Endpoint.input_cong","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#input-cong","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Every existing structural congruence preserves exposed input presence.","title":"Structural congruence preserves unguarded inputs"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/input-select","kind":"node","lean":"Cslib.SessionTypes.Endpoint.input_select","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#input-select","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Selection cannot hide an input that is present in its selected term.","title":"Selection reflects unguarded inputs"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/source-has-unguarded-input","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Step.source_hasUnguardedInput","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#source-has-unguarded-input","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Every existing \u03c0 step requires an exposed input in its source, including pre-congruence, selection, context closure and replication folding.","title":"Every \u03c0 step source has an unguarded input"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/name-absent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.NameAbsent","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#name-absent","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"The given physical name occurs in no subject, payload or continuation position. Existing binders shift the tracked outer name with canonical lift.","title":"Absence of a physical name from a process"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/out-payload","kind":"node","lean":"Cslib.SessionTypes.Endpoint.NameAbsent.out_payload","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#out-payload","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"A transmitted endpoint payload is an occurrence, even when completed.","title":"An output payload name is not absent"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/name-absent-fresh-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.nameAbsent_fresh_lift","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#name-absent-fresh-lift","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"The inserted name is absent from every position of the actual lifted \u03c0 term, including payloads and all nested input/restriction binders.","title":"The inserted name is absent from a lifted process"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/subst-name-lift-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.substName_liftName","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#subst-name-lift-name","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Substitution removes an inserted binder without changing its shifted outer names. The inserted binder's replacement is arbitrary.","title":"Substitution cancels name lifting"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/subst-proc-lift-proc","kind":"node","lean":"Cslib.SessionTypes.Endpoint.substProc_liftProc","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#subst-proc-lift-proc","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Removing an inserted binder cancels lifting throughout the unchanged \u03c0 syntax, including nested inputs, restrictions, choice and replication.","title":"Substitution cancels process lifting"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/lift-name-injective","kind":"node","lean":"Cslib.SessionTypes.Endpoint.liftName_injective","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#lift-name-injective","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Binder lifting cannot identify two physical names.","title":"Injectivity of name lifting"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/swap-name-at-swap-name-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.swapNameAt_swapNameAt","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#swap-name-at-swap-name-at","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Swapping the same two existing binders twice restores a name.","title":"Name swap is an involution"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/swap-proc-swap-proc","kind":"node","lean":"Cslib.SessionTypes.Endpoint.swapProc_swapProc","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#swap-proc-swap-proc","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Existing process swapping is likewise involutive under every binder.","title":"Process swap is an involution"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/lift-name-zero-ne-var-zero","kind":"node","lean":"Cslib.SessionTypes.Endpoint.liftName_zero_ne_var_zero","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#lift-name-zero-ne-var-zero","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Existing outer names cannot become the fresh innermost binder.","title":"A lifted name is never the innermost binder"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/lift-name-lift-name-zero","kind":"node","lean":"Cslib.SessionTypes.Endpoint.liftName_liftName_zero","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#lift-name-lift-name-zero","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Lifting commutes with entering the innermost name binder.","title":"Commuting a name lift past a lift at cutoff zero"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/lift-name-lift-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.liftName_liftName","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#lift-name-lift-name","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Arbitrary-cutoff lift fusion, including the cutoffs reached under input and restriction binders.","title":"Commutation of name lifts at ordered cutoffs"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/lift-proc-lift-proc","kind":"node","lean":"Cslib.SessionTypes.Endpoint.liftProc_liftProc","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#lift-proc-lift-proc","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Lift fusion throughout existing \u03c0 syntax, at arbitrary ordered cutoffs.","title":"Commutation of process lifts at ordered cutoffs"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/closed-at-lift-name-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.closedAt_liftName_iff","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#closed-at-lift-name-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"At an actual ambient cutoff, lifting reflects as well as preserves canonical name scoping. It is not a second scope definition.","title":"Lifting preserves and reflects name scoping"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/swap-name-at-injective","kind":"node","lean":"Cslib.SessionTypes.Endpoint.swapNameAt_injective","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#swap-name-at-injective","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"Swapping existing binders cannot identify physical names.","title":"Injectivity of name swapping"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/lift-ne-bound","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.lift_ne_bound","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#lift-ne-bound","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"A lifted outer key can never be either freshly bound role at var zero.","title":"A lifted key is never a freshly bound key"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/weaken-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.weaken_completed","parent":39240,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01/index.html#weaken-completed","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-01","status":"fully_proved","summary":"An unused completed endpoint may be added through every current typing constructor. Scope and freshness are explicit; active weakening is not admitted.","title":"Weakening by an unused completed endpoint"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02.json","id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","kind":"scope","navigation":"transparent","parent":39239,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","title":"Weakening by a disjoint completed context to Choice selection preserves process scoping"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/weaken-completed-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.weaken_completed_union","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#weaken-completed-union","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"A disjoint completed context can be reattached to any typed process, with the exact finite-map union required by parallel-nil congruence.","title":"Weakening by a disjoint completed context"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/par-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.par_nil_elim","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#par-nil-elim","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Removing a parallel nil reattaches exactly its completed ownership; no active capability is silently dropped.","title":"Removing a parallel nil preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/under","kind":"node","lean":"Cslib.SessionTypes.Endpoint.under","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#under","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Extend a name transformation under one existing input/\u03bd binder.","title":"Extension of a name map under a binder"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/under-lift-zero","kind":"node","lean":"Cslib.SessionTypes.Endpoint.under_lift_zero","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#under-lift-zero","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"The new binder is fixed and outer names are transformed capture-safely.","title":"Extended name map commutes with lifting outer names"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/map-proc","kind":"node","lean":"Cslib.SessionTypes.Endpoint.mapProc","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#map-proc","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Capture-safe name transformation of existing processes. Binder bodies use `under`; this introduces no new syntax or transition relation.","title":"Capture-safe renaming of processes"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/under-scoped","kind":"node","lean":"Cslib.SessionTypes.Endpoint.under_scoped","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#under-scoped","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Extending a scoped transformation under a binder preserves the canonical scope relation at the incremented source and target depths.","title":"Extending a scoped name map under a binder keeps it scoped"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/under-lift-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.under_liftName","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#under-lift-name","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Under a binder, canonical lifting increases its cutoff by one.","title":"Extending a lift under a binder raises its cutoff"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/map-proc-lift-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.mapProc_liftName","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#map-proc-lift-name","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"The generic syntax transformation specializes to the existing \u03c0 lift.","title":"Process renaming by a lift is process lifting"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/under-subst-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.under_substName","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#under-subst-name","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Under an existing binder, substitution lifts its replacement and increments its cutoff, exactly as PiCalculus.substProc does.","title":"Extending a substitution under a binder raises its cutoff"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/map-proc-subst-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.mapProc_substName","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#map-proc-subst-name","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Subject and payload substitution are exactly the canonical \u03c0 operation, including through all input and restriction binders.","title":"Process renaming by a substitution is process substitution"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/map-proc-closed-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.mapProc_closedAt","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#map-proc-closed-at","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Capture-safe syntax transport preserves the canonical process scope.","title":"Capture-safe renaming preserves process scope"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/under-swap-name-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.under_swapNameAt","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#under-swap-name-at","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Extending a restriction swap under a binder shifts both swapped indices.","title":"Extending a swap under a binder shifts both indices"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/map-proc-swap-name-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.mapProc_swapNameAt","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#map-proc-swap-name-at","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"The generic capture-safe map specializes to the existing \u03c0 restriction swap.","title":"Process renaming by a swap is the restriction swap"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/rename","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.rename","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#rename","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Capture-safe rekeying transports all finite-process endpoint typing, including linear payloads and actual input/\u03bd binders. Collision freedom is required only on owned keys and can be derived from a delegation split.","title":"Collision-free renaming preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.lift","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#lift","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Any ambient binder lift transports full endpoint typing without changing protocols, roles or resource multiplicity.","title":"Binder lifting preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/closed-at-lift-proc-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.closedAt_liftProc_iff","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#closed-at-lift-proc-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Lifting reflects canonical process scoping at an ambient cutoff.","title":"Lifting preserves and reflects process scoping"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/closed-at-swap-name-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.closedAt_swapName_iff","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#closed-at-swap-name-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Swapping two binders inside the ambient scope preserves the canonical name predicate in both directions.","title":"Swapping binders in scope preserves name scoping"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/closed-at-swap-proc-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.closedAt_swapProc_iff","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#closed-at-swap-proc-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Scope preservation by swapping extends through every existing binder.","title":"Swapping binders in scope preserves process scoping"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/closed-at-cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.closedAt_cong_iff","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#closed-at-cong-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Every existing structural congruence transports canonical \u03c0 scoping, including scope extrusion, restriction swapping and replication folding.","title":"Structural congruence preserves process scoping"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/closed-at-select","kind":"node","lean":"Cslib.SessionTypes.Endpoint.closedAt_select","parent":39261,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02/index.html#closed-at-select","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-02","status":"fully_proved","summary":"Selection preserves canonical scope even when actual guarded choices are present in the existing \u03c0 syntax.","title":"Choice selection preserves process scoping"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03.json","id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","kind":"scope","navigation":"transparent","parent":39239,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","title":"Reduction steps preserve process scoping to Capability whose head is a send with a given continuation"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/closed-at-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.closedAt_step","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#closed-at-step","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Every existing \u03c0 step preserves canonical scope, not just direct head communication. Substitution is the existing capture-avoiding one.","title":"Reduction steps preserve process scoping"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/erase-name-absent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.erase_nameAbsent","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#erase-name-absent","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"A role can be erased when its physical name is absent from every subject, payload and binder-adjusted continuation position of the typed term.","title":"Endpoint typing drops a role whose name does not occur"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/absent-lookup-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.absent_lookup_completed","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#absent-lookup-completed","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"An owned capability with no physical-name occurrence must already be completed. This is derived from typing, including endpoint payload positions.","title":"Owned capability with an absent name is completed"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/subst-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.subst_lift","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#subst-lift","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Substitution cancels binder lift on an entire polarized endpoint key.","title":"Substitution cancels lifting on endpoint keys"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/inj-on-received-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.injOn_received_subst","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#inj-on-received-subst","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Linear input substitution is collision-free when the transmitted role was absent from the receiver. Its opposite physical-name role is allowed.","title":"Received-endpoint substitution is injective when the role is fresh"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/rekey-received-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_received_subst","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#rekey-received-subst","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Subject substitution consumes the bound role and inserts exactly the sent key, preserving every other capability and permitting its peer role.","title":"Received-endpoint substitution moves the bound role to the sent key"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/subst-endpoint","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.subst_endpoint","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#subst-endpoint","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Receiving a linear endpoint substitutes its physical name in ALL subject and payload positions, removes the bound role and inserts the sent role exactly once. Capture under input and \u03bd is the canonical \u03c0 substitution.","title":"Endpoint typing under substitution of a transmitted endpoint"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/subst-endpoint-from-split","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.subst_endpoint_from_split","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#subst-endpoint-from-split","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Linear subject substitution with non-aliasing DERIVED from disjoint sender/receiver ownership and the receiver's already-owned carrier.","title":"Linear subject substitution with freshness from disjoint ownership"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/rekey-passive-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.rekey_passive_subst","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#rekey-passive-subst","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Passive input substitution changes no linear context: every owned key is a lifted outer key, so the input binder owns no endpoint role.","title":"Passive substitution leaves the linear context unchanged"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/inj-on-passive-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.injOn_passive_subst","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#inj-on-passive-subst","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Passive substitution is collision-free on all lifted outer endpoint keys.","title":"Passive substitution is injective on lifted outer keys"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/subst-passive","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.subst_passive","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#subst-passive","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Passive payload substitution preserves every linear obligation and cannot introduce an endpoint subject capability.","title":"Endpoint typing under passive name substitution"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/communicating-capabilities-dual","kind":"node","lean":"Cslib.SessionTypes.Endpoint.communicating_capabilities_dual","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#communicating-capabilities-dual","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Disjoint complementary subjects in a compatible interface have exactly dual protocol capabilities, including their ranks and endpoint payload sorts.","title":"Communicating subjects in a compatible split have dual capabilities"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/passive-dual-head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.passive_dual_head","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#passive-dual-head","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Actual endpoint protocols and rank equality force matching passive receive guards at a complementary owned role.","title":"Dual of a passive send head is a passive receive head"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/passive-head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.passive_head_preservation","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#passive-head-preservation","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"At a compatible combined interface with the stated disjoint typed split and scoped passive payload, head communication advances the two complementary roles and no other capability in the exact residual context.","title":"Passive-payload communication preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/endpoint-dual-head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.endpoint_dual_head","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#endpoint-dual-head","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"Duality preserves a delegated payload's role/rank/protocol while reversing only the subject continuation.","title":"Dual of an endpoint send head is an endpoint receive head"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/communicating-ranks-equal","kind":"node","lean":"Cslib.SessionTypes.Endpoint.communicating_ranks_equal","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#communicating-ranks-equal","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"For disjoint owned same-name roles in a compatible combined interface, their capability ranks agree, as required by exact capability duality.","title":"Communicating subjects in a compatible split have equal ranks"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/endpoint-head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.endpoint_head_preservation","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#endpoint-head-preservation","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"At a compatible combined interface with the stated disjoint typed split, owned payload distinct from the carrier and scoped payload name, endpoint communication transfers the payload role once and advances only the carrier.","title":"Endpoint-delegating communication preserves typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/head-comm-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.head_comm_preservation","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#head-comm-preservation","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"For a typed opposite-head pair at an initially compatible interface, direct communication in the existing \u03c0 syntax yields an exact typed residual ownership context, including endpoint delegation.","title":"Head communication preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/matched-output","kind":"node","lean":"Cslib.SessionTypes.Endpoint.MatchedOutput","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#matched-output","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"The actual payload kind and continuation of a matched output head.","title":"Matched output head of a capability"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/sends-to","kind":"node","lean":"Cslib.SessionTypes.Endpoint.SendsTo","parent":39282,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03/index.html#sends-to","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-03","status":"fully_proved","summary":"The old subject's actual head determines its literal continuation, independently of whether the transmitted endpoint is external or \u03bd-bound.","title":"Capability whose head is a send with a given continuation"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04.json","id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","kind":"scope","navigation":"transparent","parent":39239,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","title":"A matched output head is a send with the same continuation to Matching send and receive heads of two capabilities"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/sends-to","kind":"node","lean":"Cslib.SessionTypes.Endpoint.MatchedOutput.sendsTo","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#sends-to","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"The detailed matched output determines the same literal continuation.","title":"A matched output head is a send with the same continuation"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/dual-advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#dual-advance","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Only two actually matched dual subject capabilities advance. Payload ownership transfer is invisible in the combined map; all other entries remain. The payload can be private to a \u03bd binder, so external advancement records its protocol head rather than demanding external ownership of that payload.","title":"Advancement of a matched dual subject pair"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/head-comm-faithful","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.head_comm_faithful","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#head-comm-faithful","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"For a typed opposite-head pair at an initially compatible interface, the packed result retains the literal send head, payload kind and continuation correspondence instead of hiding an unconstrained protocol.","title":"Head communication preserves typing and records the matched output"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/dual-update-lookup-other","kind":"node","lean":"Cslib.SessionTypes.Endpoint.dual_update_lookup_other","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#dual-update-lookup-other","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Outside the two matched subjects, the entire capability is unchanged.","title":"Dual-continuation update leaves other lookups unchanged"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/dual-update-lookup-subject","kind":"node","lean":"Cslib.SessionTypes.Endpoint.dual_update_lookup_subject","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#dual-update-lookup-subject","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"The matched subject's rank is retained at its exact continuation.","title":"Subject lookup after a dual-continuation update"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/mem-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.mem_iff","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#mem-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Both matched old keys remain owned; the update creates no new key.","title":"Dual advancement preserves the owned keys"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/dual-update-compatible","kind":"node","lean":"Cslib.SessionTypes.Endpoint.dual_update_compatible","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#dual-update-compatible","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"From a compatible initial interface, updating both subjects to dual continuations preserves compatibility independently of process typing.","title":"Dual-continuation update preserves compatibility"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/compatible","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.compatible","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#compatible","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"A faithful matched update retains compatibility if the initial interface is compatible.","title":"Dual advancement preserves compatibility"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/dual-update-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.dual_update_balanced","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#dual-update-balanced","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Advancing a matched dual pair preserves full balance when the original interface is balanced, including every untouched opposite-role obligation.","title":"Dual-continuation update preserves balance"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.balanced","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#balanced","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Faithful subject advancement retains balance if the initial external interface is balanced.","title":"Dual advancement preserves balance"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/head-comm-advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.head_comm_advance","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#head-comm-advance","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"A typed opposite-head pair at an initially compatible interface yields a faithful matched update and a compatible typed residual via the packed result.","title":"Head communication yields a dual advancement"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/disjoint-frame","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.disjoint_frame","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#disjoint-frame","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"A matched update leaves a disjoint ownership frame disjoint.","title":"Dual advancement preserves disjointness from a frame"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/frame-left","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.frame_left","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#frame-left","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"A left parallel frame retains all its capabilities and the exact subject head/rank/continuation correspondence of the actual update.","title":"Dual advancement in the left part of a union"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/frame-right","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.frame_right","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#frame-right","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"A right parallel frame requires the actual disjoint ownership split, so its matched entries cannot be shadowed by the left-biased union.","title":"Dual advancement in the right part of a disjoint union"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.swap","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#swap","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Swap ownership of two existing ambient binders without altering any role.","title":"Swapping two ambient binders in an ownership context"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/swap-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.swap","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#swap-2","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Canonical swapping preserves full finite-process endpoint typing.","title":"Binder swapping preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/swap-insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.swap_insert","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#swap-insert","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Swapping maps an inserted capability to the swapped role key verbatim.","title":"Context swapping commutes with insertion"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/swap-double-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.swap_double_lift","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#swap-double-lift","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Two lifts put all outer names above both swapped binder indices.","title":"Swapping the two innermost binders fixes a twice-lifted context"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/swap-bind-bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.swap_bind_bind","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#swap-bind-bind","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Swapping nested restrictions exchanges their complete dual pairs and fixes all lifted outer ownership, rather than forgetting any bound capability.","title":"Swapping nested restrictions exchanges their bound pairs"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/communication-heads","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.CommunicationHeads","parent":39303,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04/index.html#communication-heads","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-04","status":"fully_proved","summary":"Both literal communication heads, with separate payload equivalence.","title":"Matching send and receive heads of two capabilities"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05.json","id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","kind":"scope","navigation":"transparent","parent":39239,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","title":"Semantic advancement of a communicating endpoint pair to Semantic advancement preserves capability ranks"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/semantic-advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#semantic-advance","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Actual old sender/peer values and literal continuations, not raw dual-code identity. Target is EXACTLY the two actual subject insertions.","title":"Semantic advancement of a communicating endpoint pair"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Balanced","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#balanced","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Every actual owned key has a semantic dual peer at equal capability rank.","title":"Semantic balance of a linear context"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/compatible","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Balanced.compatible","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#compatible","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Semantic balance supplies semantic compatibility by actual lookup uniqueness.","title":"Semantic balance implies semantic compatibility"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/balanced-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.balanced_empty","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#balanced-empty","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Empty interface balance is not a completion assertion about private work.","title":"The empty context is semantically balanced"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/balanced-of-raw","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.balanced_of_raw","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#balanced-of-raw","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Literal raw balance is a special case, not the semantic definition.","title":"Raw balance implies semantic balance"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/mem-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.mem_iff","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#mem-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Exact domain preservation includes completed capabilities.","title":"Semantic advancement preserves the context's domain"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/disjoint-frame","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.disjoint_frame","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#disjoint-frame","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"No new key can conflict with a parallel ownership frame.","title":"Semantic advancement preserves disjointness from a frame"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/scope","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.scope","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#scope","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Canonical scope of the whole owned domain is preserved.","title":"Semantic advancement preserves scoping of owned keys"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/compatible-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.compatible","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#compatible-2","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Actual compatible source retains semantic duality after the exact update.","title":"Semantic advancement preserves compatibility"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/balanced-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.balanced","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#balanced-2","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"From a semantically balanced initial interface, actual advancement retains balance with exact old keys and actual new codes.","title":"Semantic advancement preserves balance"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/frame-left","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.frame_left","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#frame-left","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Left frame preserves actual old lookups and all literal head evidence.","title":"Semantic advancement extends from the left part to the whole union"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/frame-right","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.frame_right","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#frame-right","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Right frame uses actual disjointness so neither old subject is shadowed.","title":"Semantic advancement extends from the right part to a disjoint union"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/literal-dual-tail","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.CommunicationHeads.literal_dual_tail","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#literal-dual-tail","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Literal duality of the ORIGINAL codes forces literal tail equality.","title":"Literally dual capabilities have literally dual continuations"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/sends-to","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.CommunicationHeads.sendsTo","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#sends-to","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"The existing raw send-head observation is reused only as a pure head fact.","title":"Communication heads give a send-head observation"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/of-raw","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.of_raw","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#of-raw","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Raw actual dual-code advancement embeds into the semantic relation.","title":"Raw dual advancement is a semantic advancement"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/to-raw","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.to_raw","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#to-raw","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Reverse bridge requires literal ORIGINAL peer compatibility, not merely semantic compatibility or residual retyping.","title":"Semantic advancement is raw under literal compatibility"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.lift","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#lift","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Canonical lifting preserves actual heads, ranks and code-valued payloads.","title":"Lifting preserves semantic advancement"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.swap","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#swap","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Canonical swapping permutes only subject keys, never the protocol codes.","title":"Binder swapping preserves semantic advancement"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/head-advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.head_advance","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#head-advance","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"A typed opposite-head pair at an initially semantically compatible interface yields a faithful semantic advance and a compatible typed residual.","title":"Typed head communication yields a semantic advancement"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/rank-lookup","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.rank_lookup","parent":39324,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05/index.html#rank-lookup","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-05","status":"fully_proved","summary":"Every actual capability retains its enclosing rank at the same key.","title":"Semantic advancement preserves capability ranks"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06.json","id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","kind":"scope","navigation":"transparent","parent":39239,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","title":"Lowering then lifting an outer key returns it to Endpoint typing is preserved by reverse scope extrusion"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/lift-subst-outer-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Key.lift_subst_outer_name","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#lift-subst-outer-name","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Canonical lowering and lifting cancel for a polarized outer key, provided it is not either role of the freshly inserted physical name.","title":"Lowering then lifting an outer key returns it"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/inj-on-lower","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.injOn_lower","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#inj-on-lower","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Canonical lowering is injective on an outer context with no fresh-name role. The inverse is canonical lift, not a guessed fresh constant identity.","title":"Lowering is injective on contexts without fresh-name keys"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/lower","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lower","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#lower","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Remove a fresh ambient binder from a finite outer ownership context.","title":"Lowering of an outer context past a fresh binder"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/lift-lower","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lift_lower","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#lift-lower","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Lifting the recovered outer context restores every key and capability.","title":"Lifting a lowered context recovers it"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/unlift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.unlift","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#unlift","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Typing an actually lifted term under outer-only ownership reflects back to its exact recovered context; no closed-hole premise is required.","title":"Endpoint typing reflects through binder lifting"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/erased-pair-outer","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.erased_pair_outer","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#erased-pair-outer","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Removing both roles of var zero leaves only genuine outer-name keys.","title":"Erasing both roles of the fresh name leaves outer keys"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/lift-zero-injective","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lift_zero_injective","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#lift-zero-injective","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Canonical context lifting is injective, including recorded completed slots.","title":"Context lifting is injective"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/disjoint-lower","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.disjoint_lower","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#disjoint-lower","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Lowering disjoint outer-only contexts preserves their exact key split.","title":"Lowering preserves disjointness of outer contexts"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/bind-insert-outer","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.bind_insert_outer","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#bind-insert-outer","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Installing an outer capability commutes with the entire fresh bound pair.","title":"Outer insertion commutes with binding a fresh pair"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/update-bound-positive","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.update_bound_positive","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#update-bound-positive","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Two updates at the fresh positive/negative roles replace only the bound capability pair and retain the entire lifted external context.","title":"Updating both bound roles, positive first, rebinds the pair"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/update-bound-negative","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.update_bound_negative","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#update-bound-negative","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"A negative sender installs the dual continuation on the positive role; reordering the distinct insertions retains the same complete bound pair.","title":"Updating both bound roles, negative first, rebinds the pair"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/lookup-bind-outer","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.lookup_bind_outer","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#lookup-bind-outer","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"An outer lifted role lookup passes through both fresh bound entries and reflects to the exact old external capability.","title":"Outer lookups pass through a freshly bound pair"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/project-bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.project_bind","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#project-bind","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Exact pure restriction projection. An internal matched pair changes only the bound pair at the same rank; an outer matched pair changes only the external context and retains the entire old bound pair.","title":"Dual advancement under restriction splits into inner and outer cases"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/bind-outer","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.bind_outer","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#bind-outer","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"An actual external matched update lifts through restriction without altering its private pair, even when the payload protocol refers to private data.","title":"Outer dual advancement lifts through restriction"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/project-bind-faithful","kind":"node","lean":"Cslib.SessionTypes.Endpoint.DualAdvance.project_bind_faithful","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#project-bind-faithful","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Projection additionally retains an actual private old-head/continuation transition when the communication is hidden by \u03bd. It does not reduce private advancement to an arbitrary capability with the same rank.","title":"Dual advancement under restriction splits, keeping the private step"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/nu-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nu_inv","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#nu-inv","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Restriction inversion exposes the actual bound dual pair.","title":"Inversion of endpoint typing for restriction"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/scope-forward","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.scope_forward","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#scope-forward","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"The forward canonical scope rule preserves the exact external context and ambient depth, with the dual pair derived from restriction typing.","title":"Endpoint typing is preserved by forward scope extrusion"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/insert-from-absent-partner","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.insert_from_absent_partner","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#insert-from-absent-partner","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"A role missing from one component can be reattached only after actual absence in its typed partner derives completion of the advertised capability.","title":"Endpoint typing absorbs a role absent from its parallel partner"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/restore-scope-pair","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.restore_scope_pair","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#restore-scope-pair","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"In the actual scope split, either freshly bound role can be moved back to the restricted component only when partner absence proves it completed.","title":"Endpoint typing restores both bound roles to the restricted side"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/scope-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.scope_reverse","parent":39345,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06/index.html#scope-reverse","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-06","status":"fully_proved","summary":"Reverse scope transport recovers the exact external ownership split from the actual restriction/parallel typing, with no closed-hole premise.","title":"Endpoint typing is preserved by reverse scope extrusion"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07.json","id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","kind":"scope","navigation":"transparent","parent":39239,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","title":"Endpoint typing is invariant under scope extrusion to Binding a fresh dual pair preserves dual compatibility"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/scope-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.scope_iff","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#scope-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Both directions of the canonical scope congruence preserve exact external polarized ownership and the unchanged ambient binder depth.","title":"Endpoint typing is invariant under scope extrusion"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/par-comm","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.par_comm","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#par-comm","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Parallel commutation swaps a derived disjoint split, not its capabilities.","title":"Parallel commutation preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/par-assoc","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.par_assoc","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#par-assoc","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Parallel reassociation preserves the actual three-way disjoint split.","title":"Parallel reassociation preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/par-assoc-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.par_assoc_reverse","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#par-assoc-reverse","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Reverse reassociation recovers both sub-splits from the supplied typing.","title":"Reverse parallel reassociation preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/par-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.par_nil_intro","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#par-nil-intro","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Adding a parallel nil allocates it no ownership; no active slot is dropped.","title":"Adding a parallel nil preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/nu-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nu_nil_elim","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#nu-nil-elim","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Restriction of nil has exactly the completed external ownership of nil.","title":"Removing a restriction around nil preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/nu-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nu_nil_intro","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#nu-nil-intro","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"A completed fresh pair may be introduced around nil while retaining all original external capabilities and scope.","title":"Adding a restriction around nil preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/nu-swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nu_swap","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#nu-swap","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Nested restriction commutation swaps their derived complete dual pairs.","title":"Swapping nested restrictions preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/nu-swap-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.nu_swap_reverse","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#nu-swap-reverse","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Reverse nested restriction commutation uses the canonical involution.","title":"Reverse restriction swap preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.cong_iff","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#cong-iff","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Every existing canonical structural congruence preserves endpoint typing in both directions with exactly the same depth and ownership. Unsupported choice/replication shapes are rejected by the typing-derived finite invariant.","title":"Structural congruence preserves endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/union-left","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Compatible.union_left","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#union-left","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Left children inherit actual peer compatibility.","title":"Compatibility restricts to the left part of a union"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/union-right","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Compatible.union_right","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#union-right","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"A disjoint right child inherits actual peer compatibility without shadowing.","title":"Compatibility restricts to the right part of a disjoint union"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Compatible.bind","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#bind","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"A chosen exact dual private pair extends a semantically compatible interface.","title":"Binding a dual private pair preserves compatibility"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/bind-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Balanced.bind","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#bind-2","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"At a semantically balanced outer interface, binding an exact dual private pair retains balance for the pair and every actual outer peer.","title":"Binding a dual private pair preserves balance"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/project-bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.project_bind","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#project-bind","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Faithful projection retains an actual private-pair advance OR the actual outer advance and unchanged bound pair. It never chooses an arbitrary Xi.","title":"Semantic advance under a private binding is private or outer"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/bind-outer","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.SemanticAdvance.bind_outer","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#bind-outer","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"An actual outer semantic advance lifts without changing its private pair.","title":"Outer semantic advances lift past a private binding"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/retype","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Balanced.retype","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#retype","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Pointwise semantic recoding retains actual paired domain and cap ranks.","title":"Equivalent retyping preserves semantic balance"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Balanced.union","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#union","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Disjoint balanced components form a balanced ownership frame union.","title":"Union of disjoint balanced contexts is balanced"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Compatible.rekey","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#rekey","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"Injective physical-name rekeying retains partial dual compatibility.","title":"Injective rekeying preserves dual compatibility"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/bind-3","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Compatible.bind","parent":39366,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07/index.html#bind-3","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-07","status":"fully_proved","summary":"A fresh complete bound pair extends any ordinary compatible interface.","title":"Binding a fresh dual pair preserves dual compatibility"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08.json","id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08","kind":"scope","navigation":"transparent","parent":39239,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08/index.html","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08","status":"fully_proved","title":"Subject reduction for endpoint typing at compatible interfaces to Subject reduction for endpoint typing at the empty interface"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08/subject-reduction","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.subject_reduction","parent":39387,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08/index.html#subject-reduction","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08","status":"fully_proved","summary":"From a typed source at an initially compatible interface, all five canonical reduction constructors preserve typing and compatibility, with one matched external subject-pair advancement or an unchanged interface when communication is private. No fragment or enabled-edge premise is required.","title":"Subject reduction for endpoint typing at compatible interfaces"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08/subject-reduction-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.subject_reduction_balanced","parent":39387,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08/index.html#subject-reduction-balanced","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08","status":"fully_proved","summary":"A typed source with a balanced initial interface retains typing and balance after any canonical step. Balance supplies the required compatibility; the exact zero-or-one external advancement evidence is retained.","title":"Subject reduction for endpoint typing at balanced interfaces"},{"id":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08/subject-reduction-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HasType.subject_reduction_empty","parent":39387,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08/index.html#subject-reduction-empty","scope":"ComputerScience/process-calculi/session-types/subject-reduction-for-endpoint-typing/part-08","status":"fully_proved","summary":"An empty-interface typed network retains empty-interface typing through any canonical step. The empty context is balanced, hence compatible, including communication behind restrictions and pre/post congruence.","title":"Subject reduction for endpoint typing at the empty interface"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti.json","id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","kind":"scope","navigation":"module","parent":39035,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","title":"Global progress for session typing with priorities"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/guard-min","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardMin","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#guard-min","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A guard precedes every other active owned endpoint. Transferred payloads are included in ownership until transfer; no enabled edge is assumed.","title":"Guard priority condition on active endpoints"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/guard-min-rejects-peer","kind":"node","lean":"Cslib.SessionTypes.Endpoint.guardMin_rejects_peer","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#guard-min-rejects-peer","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Owning an active peer of the guard at the same rank violates the discipline.","title":"An active same-rank peer violates the guard condition"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardMin.rekey","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#rekey","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Collision-free rekeying preserves guard inequalities and the exact same-endpoint exemption; it cannot silently identify the peer role.","title":"Injective rekeying preserves the guard condition"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/guard-min-singleton","kind":"node","lean":"Cslib.SessionTypes.Endpoint.guardMin_singleton","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#guard-min-singleton","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A guard with only its own endpoint is permitted to keep using that endpoint sequentially at the same rank. Only DIFFERENT keys must rank later.","title":"A guard owning only its own endpoint satisfies the condition"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/insert-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardMin.insert_completed","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#insert-completed","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Adding a genuinely completed role introduces no guard dependency. Completion is checked by the actual unfolded protocol, not a key exemption.","title":"Adding a completed role preserves the guard condition"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/ranked-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#ranked-type","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Linear ownership typing with a strict priority check at every guard. There is no reduction, enabled-edge, progress or preservation field.","title":"Priority-ranked endpoint typing judgement"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/has-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.hasType","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#has-type","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Forgetting priorities recovers the exact existing ownership judgement.","title":"Forgetting priorities recovers endpoint typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/out-guard","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.out_guard","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#out-guard","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Output inversion exposes the priority constraint for the actual owned subject.","title":"Ranked output exposes its subject's guard condition"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/inp-guard","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.inp_guard","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#inp-guard","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Input inversion exposes the priority constraint before either kind of receive.","title":"Ranked input exposes its subject's guard condition"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/nil-of-has-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.nil_of_hasType","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#nil-of-has-type","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Nil has no guard: its refinement is exactly the base completed/scoped context.","title":"Nil typing lifts to ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/input-other-rank","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.input_other_rank","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#input-other-rank","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Every other active owned role must rank strictly after an input subject.","title":"Active roles on other names rank after an input subject"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/par-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.par_inv","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#par-inv","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Refinement preserves the original disjoint parallel ownership split.","title":"Parallel inversion for ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/nu-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.nu_inv","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#nu-inv","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Refinement retains the exact protocol/rank pair installed by restriction.","title":"Restriction inversion for ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/active-payload-rank","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardMin.active_payload_rank","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#active-payload-rank","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Every active delegated payload is included in the pre-transfer guard check.","title":"Active delegated payloads rank after the guard"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/weaken-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.weaken_completed","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#weaken-completed","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Adding a fresh completed capability preserves every guard inequality through all seven rules. No active capability can be added by this theorem.","title":"Weakening ranked typing by a completed capability"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/erase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardMin.erase","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#erase","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Erasing a role can only remove a dependency from a guard check.","title":"Erasing a role preserves the guard condition"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/erase-name-absent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.erase_nameAbsent","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#erase-name-absent","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Actual all-position physical-name absence permits erasure in the refinement too. Subject and payload lookup obligations are never discarded.","title":"Erasing a key whose name is absent preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/weaken-completed-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.weaken_completed_union","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#weaken-completed-union","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Reattaching a disjoint completed finite context preserves all guards.","title":"Weakening ranked typing by a disjoint completed context"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/rename","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.rename","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#rename","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Physical-name rekeying preserves the entire syntax-directed refinement, including all guard priorities, passive/endpoint payloads and nested binders. Injectivity is needed only on owned keys, as in the base ownership judgement.","title":"Collision-free renaming preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/lift-3","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.lift","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#lift-3","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Canonical binder lifting preserves every priority and endpoint rank.","title":"Binder lifting preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/subst-passive","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.subst_passive","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#subst-passive","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Canonical passive substitution transports all priority guards on lifted outer ownership. The bound name owns no endpoint capability in this case.","title":"Passive substitution preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/unlift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.unlift","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#unlift","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Reflection of an actual lifted term recovers every priority and context entry.","title":"Ranked typing descends from a lifted process"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/scope-forward","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.scope_forward","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#scope-forward","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Extrusion lifts the partner's real ownership while retaining every guard.","title":"Forward scope extrusion preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/insert-from-absent-partner","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.insert_from_absent_partner","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#insert-from-absent-partner","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Moving a partner-owned role back requires actual absence to DERIVE completion from plain typing before refined completed weakening is applied.","title":"Adding a role unused by the partner preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/restore-scope-pair","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.restore_scope_pair","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#restore-scope-pair","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Restoring the bound pair adds only absent completed partner roles; all original active ownership and inequalities of the restricted component remain.","title":"Restoring the bound pair preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/scope-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.scope_reverse","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#scope-reverse","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Reverse extrusion derives the entire outer ownership split from typing, removes only genuinely absent partner roles and retains all guard inequalities.","title":"Reverse scope extrusion preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/scope-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.scope_iff","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#scope-iff","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Exact bidirectional canonical scope transport for the same refinement.","title":"Ranked typing is invariant under scope extrusion"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/subst-endpoint","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.subst_endpoint","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#subst-endpoint","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Endpoint substitution transports every ranked subject and payload guard with freshness derived from the actual linear ownership split.","title":"Endpoint substitution preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/subst-endpoint-from-split","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.subst_endpoint_from_split","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#subst-endpoint-from-split","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"The transmitted role's non-aliasing is derived, not a readiness premise.","title":"Disjoint split supplies freshness for endpoint substitution"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/inp-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.inp_inv","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#inp-inv","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Ranked input inversion retains both literal payload sorts and all guards.","title":"Inversion of ranked input typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/out-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.out_inv","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#out-inv","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Ranked output inversion retains the pre-transfer active payload check.","title":"Inversion of ranked output typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/ranked-passive-head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ranked_passive_head","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#ranked-passive-head","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"With the stated disjoint ranked split, scoped passive payload and compatible combined interface, head communication substitutes through the ranked input body and retains both residual guard derivations.","title":"Passive head communication preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/ranked-endpoint-head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.ranked_endpoint_head","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#ranked-endpoint-head","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"With the stated disjoint ranked split, owned non-carrier scoped payload and compatible combined interface, delegation derives substitution freshness from that split and retains the combined payload value.","title":"Endpoint delegation at the head preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/head-comm-faithful","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.head_comm_faithful","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#head-comm-faithful","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A ranked-typed opposite-head pair at an initially compatible interface yields the actual subject/dual-peer heads, payload metadata and exact ranked continuation context after literal communication.","title":"Ranked head communication exposes peers and residual typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/head-comm-advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.head_comm_advance","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#head-comm-advance","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"For a ranked-typed opposite-head pair at a compatible initial interface, the head result yields one faithful update and compatible ranked residual, not a typing witness manufactured by forgetting the refinement.","title":"Ranked head communication yields a dual advance"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/par-comm","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.par_comm","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#par-comm","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Commutation retains the actual disjoint split and both ranked subterms.","title":"Parallel commutation preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/par-assoc","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.par_assoc","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#par-assoc","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Reassociation derives all pairwise disjointness from actual typing.","title":"Parallel reassociation preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/par-assoc-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.par_assoc_reverse","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#par-assoc-reverse","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Reverse reassociation retains exact contexts and guard checks.","title":"Reverse parallel reassociation preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/par-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.par_nil_elim","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#par-nil-elim","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Nil's actual completed ownership can be reattached without new dependencies.","title":"Removing a parallel nil preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/par-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.par_nil_intro","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#par-nil-intro","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Nil introduction allocates an empty ownership context, not an active frame.","title":"Adding a parallel nil preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/nu-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.nu_nil_elim","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#nu-nil-elim","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Restricting nil is equivalent to its original completed/scoped interface.","title":"Removing a restriction around nil preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/nu-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.nu_nil_intro","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#nu-nil-intro","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A completed fresh pair can be introduced around nil without changing guards.","title":"Restricting nil preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/nu-swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.nu_swap","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#nu-swap","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Swapping the two restrictions preserves all guard constraints by the GENERAL capture-safe rename theorem and the exact four-role context equation.","title":"Swapping two restrictions preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/nu-swap-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.nu_swap_reverse","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#nu-swap-reverse","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Reverse restriction swap uses the canonical process involution.","title":"Reverse restriction swap preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.cong_iff","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#cong-iff","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Every canonical Cong constructor preserves the whole priority refinement at exactly the same context and ambient depth. Unsupported sum/replication constructors are impossible by the invariant DERIVED from refinement typing.","title":"Structural congruence preserves ranked typing"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/guard-frontier","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardFrontier","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#guard-frontier","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A parallel frontier consists of nil and actual input/output guards. No restriction is hidden under parallel; guarded continuations are unrestricted.","title":"Parallel guard frontier"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/frontier-normal","kind":"node","lean":"Cslib.SessionTypes.Endpoint.FrontierNormal","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#frontier-normal","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A finite network normal form has all unguarded restrictions outside its parallel frontier. The binder tower can have any finite depth.","title":"Frontier normal form of finite networks"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardFrontier.lift","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#lift","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Lifting maps every frontier subject, payload and guarded continuation.","title":"Lifting preserves guard frontiers"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/lift-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.FrontierNormal.lift","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#lift-2","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Lifting preserves the restriction tower and increments cutoff under binders.","title":"Lifting preserves frontier normal form"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/parallel","kind":"node","lean":"Cslib.SessionTypes.Endpoint.FrontierNormal.parallel","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#parallel","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Two normalized components combine by actual canonical scope extrusion. Every private restriction, on either side, is retained in the resulting tower.","title":"Parallel composition of frontier normal forms normalizes"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/finite-linear-frontier-normal","kind":"node","lean":"Cslib.SessionTypes.Endpoint.finiteLinear_frontier_normal","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#finite-linear-frontier-normal","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"GENERAL normalization of every finite-linear existing \u03c0 term. Prefix bodies are not evaluated or replaced, and no complementary ready pair is assumed.","title":"Every finite linear \u03c0 term has a frontier normal form"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/frontier-normal-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.frontier_normal","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#frontier-normal-2","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Ranked normalization is DERIVED from typing, preserving every guard and all external ownership while exposing all unguarded private restrictions.","title":"Ranked-typed processes have a typed frontier normal form"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/active","kind":"node","lean":"Cslib.SessionTypes.Endpoint.SendsTo.active","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#active","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Every literal communication head is active, for both payload sorts.","title":"Send heads are active for both payload sorts"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/minimum-guard-key","kind":"node","lean":"Cslib.SessionTypes.Endpoint.minimum_guard_key","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#minimum-guard-key","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A minimum active role owned by a prefix must be that prefix's exact subject key, not merely a same-name role. Every other key ranks later.","title":"A minimum active role owned by a guard is its subject key"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/out-active","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.out_active","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#out-active","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Literal output subjects have an active protocol, without any stored reduction or exposed-ready-pair evidence.","title":"Ranked output subjects are active"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/inp-active","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.inp_active","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#inp-active","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Literal input subjects likewise own active receive heads.","title":"Ranked input subjects are active"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/expose-minimum","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardFrontier.expose_minimum","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#expose-minimum","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A minimum active role is exposed as the EXACT subject of an actual leaf. The disjoint ownership split and canonical rearrangement are DERIVED outputs; no ready guard or normalizing typing result is required from the caller.","title":"Minimum active role is the subject of a frontier leaf"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/active-minimum","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Context.active_minimum","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#active-minimum","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A nonempty collection of active finite-map bindings has a least rank. The witness ranges over ALL active roles, not only input guards.","title":"Existence of a least-rank active binding"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/completed-nil","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardFrontier.completed_nil","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#completed-nil","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A frontier with no active ownership cannot contain a live prefix. Completion is the PROCESS congruence to nil, not a completed empty interface.","title":"Completed ranked frontiers are congruent to nil"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/dual-active","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability.dual_active","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#dual-active","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Activity is preserved by protocol duality, including regular unfolding.","title":"Duality preserves activity of a capability"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/minimum-peer-separate","kind":"node","lean":"Cslib.SessionTypes.Endpoint.minimum_peer_separate","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#minimum-peer-separate","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A minimum active role's peer cannot be owned by the same guard leaf. Equal-rank opposite keys violate the actual pre-action GuardMin constraint.","title":"Minimum guard leaf cannot own its dual peer"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/same-name-guards-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.same_name_guards_step","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#same-name-guards-step","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Given disjoint ranked-typed leaves with literal same-name input/output guard shapes and a compatible combined interface, their directions are opposite. The Step is canonical commSelect, possibly after parallel commutation.","title":"Same-name ranked guards are opposite and communicate"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/progress","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardFrontier.progress","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#progress","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"GENERAL balanced ranked frontier progress. A minimum over ALL active owned roles exposes two distinct opposite subject leaves, from which actual canonical communication follows. Completion is process-level Cong to nil.","title":"Progress for balanced ranked guard frontiers"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Balanced.bind","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#bind","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A fresh private pair preserves full dual ownership of an outer balanced interface. Every nonfresh owned role is reflected through canonical lifting.","title":"Binding a fresh dual pair preserves balance"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/subject-reduction","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.subject_reduction","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#subject-reduction","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"From a ranked-typed source with an initially compatible interface, every actual canonical step preserves the refinement and compatibility, with zero or one actual external matched subject advancement. Private head/continuation changes are derived from faithful bound projection.","title":"Subject reduction for ranked typing at compatible interfaces"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/subject-reduction-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.subject_reduction_balanced","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#subject-reduction-balanced","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"For a ranked-typed source, initial balance supplies compatibility and is preserved by the actual canonical step and its exact update.","title":"Ranked subject reduction at balanced interfaces"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/subject-reduction-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.subject_reduction_empty","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#subject-reduction-empty","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Empty-interface ranked typing is preserved by any canonical step at any ambient depth and through pre/post congruence. The empty context supplies balance and therefore the required compatibility.","title":"Ranked subject reduction at the empty interface"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/progress-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.progress_balanced","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#progress-balanced","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"For a ranked-typed process at ANY balanced interface, global progress holds, including disconnected networks. All private channels are tracked through the derived restriction tower. Completion means process congruence to nil, not completion of the interface.","title":"Global progress of ranked processes at balanced interfaces"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/progress-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.progress_empty","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#progress-empty","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"A closed finite ranked network is completed or takes an ACTUAL canonical step. Ambient passive-name depth is arbitrary; no readiness witness is supplied.","title":"Closed ranked networks are completed or can step"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/reachable-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.reachable_empty","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#reachable-empty","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Every state reached by finitely many actual canonical steps retains exact empty-interface ranked typing. This is preservation, not finite termination.","title":"Reachable states of closed ranked networks stay typed"},{"id":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/reachable-progress","kind":"node","lean":"Cslib.SessionTypes.Endpoint.RankedType.reachable_progress","parent":39391,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti/index.html#reachable-progress","scope":"ComputerScience/process-calculi/session-types/global-progress-for-session-typing-with-prioriti","status":"fully_proved","summary":"Global progress holds at every finite reachable state of a closed ranked network. No fairness, eventual liveness or recursive termination is asserted.","title":"Progress at reachable states of closed ranked networks"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence","kind":"scope","navigation":"module","parent":39035,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence","status":"fully_proved","title":"Session typing up to protocol equivalence"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","kind":"scope","navigation":"transparent","parent":39463,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","title":"Payload and continuation branches of an endpoint action to Tree equivalence is a congruence for endpoint sending"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/type-branch","kind":"node","lean":"Cslib.SessionTypes.Endpoint.TypeBranch","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#type-branch","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"The two distinct children of an endpoint action.","title":"Payload and continuation branches of an endpoint action"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/type-label","kind":"node","lean":"Cslib.SessionTypes.Endpoint.TypeLabel","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#type-label","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Observable guarded tree edges. Endpoint metadata is never erased.","title":"Labels of observable protocol-tree edges"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/type-edge","kind":"node","lean":"Cslib.SessionTypes.Endpoint.TypeEdge","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#type-edge","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"One visible tree edge obtained by the existing guarded root unfolding.","title":"Visible edge of a protocol tree after root unfolding"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/protocol-tree","kind":"node","lean":"Cslib.SessionTypes.Endpoint.protocolTree","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#protocol-tree","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Canonical CSLib LTS instance for guarded type-tree observations only.","title":"Labelled transition system of protocol-tree observations"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/tree-equiv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#tree-equiv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Equirecursive regular-tree equality is canonical strong bisimilarity, not equality of graph codes, cursors or finite constructor syntax.","title":"Regular-tree equivalence of protocols as bisimilarity"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/refl","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.refl","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#refl","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Reflexivity comes from the canonical CSLib bisimilarity API.","title":"Protocol tree equivalence is reflexive"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/symm","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.symm","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#symm","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Symmetry comes from the canonical CSLib bisimilarity API.","title":"Protocol tree equivalence is symmetric"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/trans","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.trans","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#trans","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Transitivity comes from composition of canonical CSLib bisimulations.","title":"Protocol tree equivalence is transitive"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/type-edge-unfold","kind":"node","lean":"Cslib.SessionTypes.Endpoint.typeEdge_unfold","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#type-edge-unfold","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Root unfolding exposes exactly the same observation edges.","title":"Root unfolding preserves protocol-tree edges"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/tree-equiv-unfold","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.treeEquiv_unfold","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#tree-equiv-unfold","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"One unfolding is semantically equivalent even though its code changes.","title":"A protocol is tree-equivalent to its unfolding"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/head-match","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HeadMatch","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#head-match","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Exact head and branch correspondence for a relation on child trees. Payload and continuation are related separately; only metadata is literal.","title":"Head and branch matching of protocols up to a child relation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/mono","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HeadMatch.mono","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#mono","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Head matching is monotone in the child relation.","title":"Head matching is monotone in the child relation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/head-match-is-bisimulation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.headMatch_isBisimulation","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#head-match-is-bisimulation","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"A relation matching literal unfolded heads and both child branches is a canonical CSLib bisimulation. This is the coinductive proof rule, not an axiom.","title":"Head-matching relations are bisimulations"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.head","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#head","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Canonical bisimilarity determines literal head metadata and both semantically equivalent child branches. No graph-code equality is required.","title":"Tree-equivalent protocols have matching heads"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/coind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.coind","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#coind","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Coinduction on an explicitly supplied tree relation, checked by CSLib.","title":"Coinduction principle for protocol tree equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/tree-equiv-iff-head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.treeEquiv_iff_head","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#tree-equiv-iff-head","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Exact semantic root/branch characterization, in BOTH directions.","title":"Characterization of tree equivalence by matching heads"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/tree-equiv-unfold-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.treeEquiv_unfold_iff","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#tree-equiv-unfold-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Unfolding on either or both sides leaves the regular tree unchanged.","title":"Unfolding both protocols preserves and reflects tree equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/send-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.sendName","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#send-name","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Constructor congruence matches same-direction name continuations.","title":"Tree equivalence is a congruence for name sending"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/recv-name","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.recvName","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#recv-name","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Constructor congruence also matches input direction separately.","title":"Tree equivalence is a congruence for name receiving"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/send-endpoint","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.sendEndpoint","parent":39464,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01/index.html#send-endpoint","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-01","status":"fully_proved","summary":"Payload and continuation congruence do not conflate the two branches.","title":"Tree equivalence is a congruence for endpoint sending"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","kind":"scope","navigation":"transparent","parent":39463,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","title":"Tree equivalence is a congruence for endpoint receiving to Conversion-typed processes are well scoped"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/recv-endpoint","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.recvEndpoint","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#recv-endpoint","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Endpoint receive congruence preserves literal role/rank metadata.","title":"Tree equivalence is a congruence for endpoint receiving"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/dual","kind":"node","lean":"Cslib.SessionTypes.Endpoint.HeadMatch.dual","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#dual","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Dualizing a head changes subject direction and continuation orientation, but leaves payload orientation and role/rank metadata untouched.","title":"Head matching transfers to dual protocols"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/dual-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.dual","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#dual-2","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Semantic equivalence is preserved by actual protocol duality, including recursive payloads with independently chosen orientation.","title":"Duality preserves protocol tree equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/tree-equiv-dual-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.treeEquiv_dual_iff","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#tree-equiv-dual-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Duality both preserves and reflects equivalence, not only one direction.","title":"Duality preserves and reflects protocol tree equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/tree-equiv-equivalence","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.treeEquiv_equivalence","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#tree-equiv-equivalence","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"The semantic type relation is a genuine equivalence relation.","title":"Protocol tree equivalence is an equivalence relation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/tree-equiv-send-endpoint-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.treeEquiv_sendEndpoint_iff","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#tree-equiv-send-endpoint-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Exact endpoint-send characterization exposes literal metadata equality and separate semantic payload/continuation equivalences for conversion.","title":"Tree equivalence of endpoint sends, componentwise"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/tree-equiv-recv-endpoint-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.treeEquiv_recvEndpoint_iff","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#tree-equiv-recv-endpoint-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Receive characterization preserves the same independent payload metadata.","title":"Tree equivalence of endpoint receives, componentwise"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/done-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Protocol.TreeEquiv.done_iff","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#done-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Activity/completion is invariant under semantic type equivalence.","title":"Tree equivalence preserves completion of a protocol"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/equivalent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability.Equivalent","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#equivalent","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Capability equivalence explicitly retains the enclosing capability rank, which is not an observation of Protocol.TreeEquiv.","title":"Equivalence of endpoint capabilities"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/refl","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability.Equivalent.refl","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#refl","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Reflexivity retains rank and the exact protocol tree.","title":"Reflexivity of capability equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/symm","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability.Equivalent.symm","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#symm","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Symmetry preserves the separately required rank equality.","title":"Symmetry of capability equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/trans","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability.Equivalent.trans","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#trans","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Transitivity composes rank equality and regular-tree equivalence.","title":"Transitivity of capability equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/dual-3","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Capability.Equivalent.dual","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#dual-3","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Duality retains the separate rank and preserves protocol-tree equivalence.","title":"Duality preserves capability equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/compatible","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.Compatible","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#compatible","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Semantic peer compatibility keeps actual capability ranks equal and compares dual protocol trees, not merely their direction or payload metadata.","title":"Semantic peer compatibility of an ownership map"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/compatible-of-raw","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.compatible_of_raw","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#compatible-of-raw","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Old syntactic peer compatibility implies the semantic condition.","title":"Syntactic compatibility implies semantic compatibility"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/has-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#has-type","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Seven syntax-directed rules. No readiness, Step, retyping, preservation or consistent raw representative is a typing premise. Endpoint input installs its actual advertised code; endpoint output transfers the actual owned slot.","title":"Conversion-enabled endpoint typing judgement"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/of-raw","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.of_raw","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#of-raw","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Actual seven-rule induction embeds the raw judgement. This does not transport the raw judgement's substitution or progress theorem to new terms.","title":"Base endpoint typing embeds into conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/one-action-tail","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.one_action_tail","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#one-action-tail","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Local finite-tail conversion: replacing a one-send-then-end protocol by an equivalent code yields a usable output subject, retaining its actual next code and rank. This is not arbitrary context retyping.","title":"Output typing at a code equivalent to one send then end"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/out-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.out_inv","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#out-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Conversion output inversion exposes actual owned payload metadata and semantic comparison, never a supplied retyping result.","title":"Output inversion for conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/closed-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.closedAt","parent":39485,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02/index.html#closed-at","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-02","status":"fully_proved","summary":"Canonical scope follows from the seven conversion rules.","title":"Conversion-typed processes are well scoped"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","kind":"scope","navigation":"transparent","parent":39463,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","title":"Conversion-typed processes have no sum or replication to Peer of a passive send head has an equivalent receive head"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/finite-linear","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.finiteLinear","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#finite-linear","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Typing excludes sum and repl by syntax, not by a caller fragment premise.","title":"Conversion-typed processes have no sum or replication"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/par-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.par_inv","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#par-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Parallel inversion derives its actual disjoint map split.","title":"Parallel inversion for conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/nu-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.nu_inv","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#nu-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"The private pair comes from the actual restriction derivation.","title":"Restriction inversion for conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/nil-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.nil_inv","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#nil-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Nil exposes actual completed ownership and key scope.","title":"Nil inversion for conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/nil-insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.nil_insert","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#nil-insert","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Only a scoped actually completed capability can be added at nil.","title":"Typed nil admits a completed scoped capability"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/weaken-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.weaken_completed","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#weaken-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"An unused completed endpoint may be added through every current typing constructor. Scope and freshness are explicit; active weakening is not admitted.","title":"Weakening conversion typing by a completed capability"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/weaken-completed-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.weaken_completed_union","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#weaken-completed-union","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"A disjoint completed context can be reattached to any typed process, with the exact finite-map union required by parallel-nil congruence.","title":"Weakening conversion typing by a disjoint completed context"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/par-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.par_nil_elim","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#par-nil-elim","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Removing a parallel nil reattaches exactly its completed ownership; no active capability is silently dropped.","title":"Removing a parallel nil preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/erase-name-absent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.erase_nameAbsent","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#erase-name-absent","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"A role can be erased when its physical name is absent from every subject, payload and binder-adjusted continuation position of the typed term.","title":"Erasing a role with an absent name preserves typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/absent-lookup-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.absent_lookup_completed","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#absent-lookup-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"An owned capability with no physical-name occurrence must already be completed. This is derived from typing, including endpoint payload positions.","title":"Owned capabilities with absent names are completed"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/retype","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.retype","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#retype","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"GENERAL arbitrary-depth, arbitrary-context retyping, by all seven typing constructors. Target ownership, ranks and every unfolded branch are derived.","title":"Retyping along ownership-map equivalence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/retype-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.retype_iff","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#retype-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Equivalence transports typing in BOTH directions at the same process/depth.","title":"Ownership-map equivalence preserves typing in both directions"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/rename","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.rename","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#rename","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"GENERAL name transport with collision freedom on actual owned keys and canonical ambient name scope; protocol codes, roles and ranks stay exact.","title":"Collision-free renaming preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.lift","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#lift","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Lift uses the canonical cutoff-aware name and process operations.","title":"Lifting past a binder preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.swap","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#swap","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Swap retains roles, protocols and ranks while permuting two ambient binders.","title":"Swapping adjacent binders preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/subst-endpoint","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.subst_endpoint","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#subst-endpoint","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"GENERAL endpoint substitution: retain the ACTUAL transmitted capability's code and rank at the sent polarized key, allowing its opposite-role alias. Collision freedom is needed only for that exact key and canonical substitution.","title":"Endpoint substitution preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/subst-endpoint-from-split","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.subst_endpoint_from_split","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#subst-endpoint-from-split","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"GENERAL split-derived freshness: actual payload ownership and the receiver's already-owned carrier derive non-aliasing even after advancement.","title":"Endpoint substitution with freshness from a disjoint split"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/subst-passive","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.subst_passive","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#subst-passive","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"GENERAL passive substitution transports all scoped positions but adds NO capability. The input context contains only lifted outer ownership.","title":"Passive name substitution preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/communicating-equivalent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.communicating_equivalent","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#communicating-equivalent","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"At a semantically compatible combined interface, actual disjoint same-name ownership derives opposite roles, literal cap-rank agreement and semantic duality, not raw capability equality.","title":"Communicating subjects have semantically dual capabilities"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/passive-peer-head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.passive_peer_head","parent":39506,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03/index.html#passive-peer-head","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-03","status":"fully_proved","summary":"Semantic duality forces an actual receive-name head and equivalent tail, while keeping the receiver's literal continuation code.","title":"Peer of a passive send head has an equivalent receive head"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","kind":"scope","navigation":"transparent","parent":39463,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","title":"Peer of an endpoint send head has an equivalent receive head to Swapping nested restrictions preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/endpoint-peer-head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.endpoint_peer_head","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#endpoint-peer-head","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Endpoint peer heads preserve literal advertised role/rank, independently equivalent payload and continuation branches, and payload orientation.","title":"Peer of an endpoint send head has an equivalent receive head"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/inp-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.inp_inv","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#inp-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Input inversion retains its actual advertised code and typed continuation.","title":"Input inversion for conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/passive-head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.passive_head_preservation","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#passive-head-preservation","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"GENERAL passive communicating-head preservation at a semantically compatible actual split, with literal receiver-tail and rank evidence.","title":"Passive head communication preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/endpoint-head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.endpoint_head_preservation","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#endpoint-head-preservation","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"At an initially semantically compatible disjoint split, endpoint head communication transfers EXACT v, retaining the receiver's separately equivalent advertised payload/tail codes.","title":"Endpoint delegation preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/head-update-other","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.head_update_other","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#head-update-other","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Exact two-insert head updates leave every other actual lookup unchanged, without requiring the receiver-tail code to equal the sender's dual.","title":"Head update leaves other lookups unchanged"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.head_preservation","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#head-preservation","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"A typed opposite-head pair at an initially semantically compatible interface yields packed literal-head preservation. Both actual heads and payload/tail correspondence are conclusions, not fields in a protocol or admission predicate.","title":"Head communication preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/endpoint-payload-after","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.endpoint_payload_after","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#endpoint-payload-after","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"A transmitted payload slot is literally unchanged in the combined map; its peer role may coexist, but it is neither carrier update key.","title":"Head update leaves the transmitted payload slot unchanged"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/unlift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.unlift","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#unlift","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Canonical lowering removes only actual outer lifted keys, preserving all conversion typing without a closed-hole restriction.","title":"Lowering a lifted process preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/scope-forward","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.scope_forward","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#scope-forward","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"The forward canonical scope rule preserves the exact external context and ambient depth, with the dual pair derived from restriction typing.","title":"Forward scope extrusion preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/insert-from-absent-partner","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.insert_from_absent_partner","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#insert-from-absent-partner","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"A role missing from one component can be reattached only after actual absence in its typed partner derives completion of the advertised capability.","title":"Adding a role unused by the partner preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/restore-scope-pair","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.restore_scope_pair","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#restore-scope-pair","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"In the actual scope split, either freshly bound role can be moved back to the restricted component only when partner absence proves it completed.","title":"Restoring the bound pair to the restricted component"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/scope-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.scope_reverse","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#scope-reverse","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Reverse scope transport recovers the exact external ownership split from the actual restriction/parallel typing, with no closed-hole premise.","title":"Reverse scope extrusion preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/scope-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.scope_iff","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#scope-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Both directions of the canonical scope congruence preserve exact external polarized ownership and the unchanged ambient binder depth.","title":"Scope extrusion preserves conversion typing in both directions"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/par-comm","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.par_comm","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#par-comm","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Parallel commutation swaps a derived disjoint split, not its capabilities.","title":"Parallel commutation preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/par-assoc","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.par_assoc","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#par-assoc","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Parallel reassociation preserves the actual three-way disjoint split.","title":"Parallel reassociation preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/par-assoc-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.par_assoc_reverse","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#par-assoc-reverse","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Reverse reassociation recovers both sub-splits from the supplied typing.","title":"Reverse parallel reassociation preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/par-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.par_nil_intro","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#par-nil-intro","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Adding a parallel nil allocates it no ownership; no active slot is dropped.","title":"Adding a parallel nil preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/nu-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.nu_nil_elim","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#nu-nil-elim","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Restriction of nil has exactly the completed external ownership of nil.","title":"Removing a restriction around nil preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/nu-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.nu_nil_intro","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#nu-nil-intro","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"A completed fresh pair may be introduced around nil while retaining all original external capabilities and scope.","title":"Adding a restriction around nil preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/nu-swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.nu_swap","parent":39527,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04/index.html#nu-swap","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-04","status":"fully_proved","summary":"Nested restriction commutation swaps their derived complete dual pairs.","title":"Swapping nested restrictions preserves conversion typing"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","kind":"scope","navigation":"transparent","parent":39463,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","title":"Reverse restriction swap preserves conversion typing to Erasing a key whose name is absent preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/nu-swap-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.nu_swap_reverse","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#nu-swap-reverse","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Reverse nested restriction commutation uses the canonical involution.","title":"Reverse restriction swap preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.cong_iff","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#cong-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Every existing canonical structural congruence preserves endpoint typing in both directions with exactly the same depth and ownership. Unsupported choice/replication shapes are rejected by the typing-derived finite invariant.","title":"Structural congruence preserves conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/subject-reduction","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.subject_reduction","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#subject-reduction","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"From a typed source at an initially compatible interface, all five canonical reduction constructors preserve typing and compatibility, with one matched external subject-pair advancement or an unchanged interface when communication is private. No fragment or enabled-edge premise is required.","title":"Subject reduction at a compatible conversion interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/subject-reduction-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.subject_reduction_balanced","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#subject-reduction-balanced","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"A typed source with a balanced initial interface retains typing and balance after any canonical step. Balance supplies the required compatibility; the exact zero-or-one external advancement evidence is retained.","title":"Subject reduction at a balanced interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/subject-reduction-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.subject_reduction_empty","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#subject-reduction-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"An empty-interface typed network retains empty-interface typing through any canonical step. The empty context is balanced, hence compatible, including communication behind restrictions and pre/post congruence.","title":"Subject reduction at the empty interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/reachable","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.reachable","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#reachable","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"A typed source at an initially semantically compatible interface retains typing and compatibility through finite actual execution, with a finite trace of faithful advances. This is safety, not progress or termination.","title":"Typing and compatibility persist along finite executions"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/reachable-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.HasType.reachable_empty","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#reachable-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Every finite reachable closed state retains the exact empty interface.","title":"Empty-interface typing persists along finite executions"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/ranked-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#ranked-type","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Linear ownership typing with a strict priority check at every guard. There is no reduction, enabled-edge, progress or preservation field.","title":"Priority-ranked conversion typing judgement"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/has-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.hasType","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#has-type","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Forgetting guard priorities derives the existing conversion judgement.","title":"Forgetting priorities yields conversion typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/of-raw","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.of_raw","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#of-raw","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Explicit seven-case raw-ranked embedding, not theorem-value inheritance.","title":"Ranked typing embeds into conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/par-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.par_inv","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#par-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Ranked parallel typing exposes its actual disjoint ownership split.","title":"Parallel inversion for conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/nu-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.nu_inv","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#nu-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Restriction chooses an actual literal-dual pair.","title":"Restriction inversion for conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/nil-of-has-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.nil_of_hasType","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#nil-of-has-type","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Nil adds no priority premise beyond actual completed and scoped ownership.","title":"Nil typing lifts to conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/out-active","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.out_active","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#out-active","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Literal output is active and checks every other owned active key.","title":"Conversion-ranked output subjects are active"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/inp-active","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.inp_active","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#inp-active","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Literal input is active before either payload kind is received.","title":"Conversion-ranked input subjects are active"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/expose-minimum","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.GuardFrontier.expose_minimum","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#expose-minimum","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"A minimum active role is exposed as the EXACT subject of an actual leaf. The disjoint ownership split and canonical rearrangement are DERIVED outputs; no ready guard or normalizing typing result is required from the caller.","title":"Minimum active role is the subject of a conversion frontier leaf"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/completed-nil","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.GuardFrontier.completed_nil","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#completed-nil","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"A frontier with no active ownership cannot contain a live prefix. Completion is the PROCESS congruence to nil, not a completed empty interface.","title":"Completed conversion-ranked frontiers are congruent to nil"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/minimum-peer-separate","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.minimum_peer_separate","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#minimum-peer-separate","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"A semantic equal-rank active peer cannot inhabit the selected minimum guard.","title":"Minimum guard leaf cannot own an equivalent dual peer"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/weaken-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.weaken_completed","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#weaken-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Adding a fresh completed capability preserves every guard inequality through all seven rules. No active capability can be added by this theorem.","title":"Weakening conversion ranked typing by a completed capability"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/erase-name-absent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.erase_nameAbsent","parent":39548,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05/index.html#erase-name-absent","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-05","status":"fully_proved","summary":"Actual all-position physical-name absence permits erasure in the refinement too. Subject and payload lookup obligations are never discarded.","title":"Erasing a key whose name is absent preserves conversion ranked typing"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","kind":"scope","navigation":"transparent","parent":39463,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","title":"Weakening conversion ranked typing by a disjoint completed context to Adding a role unused by the partner preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/weaken-completed-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.weaken_completed_union","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#weaken-completed-union","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Reattaching a disjoint completed finite context preserves all guards.","title":"Weakening conversion ranked typing by a disjoint completed context"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/retype","kind":"node","lean":"Cslib.SessionTypes.Endpoint.GuardMin.retype","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#retype","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Same domains, literal cap ranks and actual done equivalence preserve GuardMin.","title":"Equivalent contexts preserve the guard condition"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/retype-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.retype","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#retype-2","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"GENERAL arbitrary-depth, arbitrary-context retyping, by all seven typing constructors. Target ownership, ranks and every unfolded branch are derived.","title":"Equivalent contexts preserve conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/retype-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.retype_iff","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#retype-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Equivalence transports typing in BOTH directions at the same process/depth.","title":"Context equivalence transports conversion ranked typing both ways"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/rename","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.rename","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#rename","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"GENERAL name transport with collision freedom on actual owned keys and canonical ambient name scope; protocol codes, roles and ranks stay exact.","title":"Collision-free renaming preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.lift","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#lift","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Lift uses the canonical cutoff-aware name and process operations.","title":"Binder lifting preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.swap","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#swap","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Swap retains roles, protocols and ranks while permuting two ambient binders.","title":"Binder swap preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/subst-endpoint","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.subst_endpoint","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#subst-endpoint","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"GENERAL endpoint substitution: retain the ACTUAL transmitted capability's code and rank at the sent polarized key, allowing its opposite-role alias. Collision freedom is needed only for that exact key and canonical substitution.","title":"Endpoint substitution preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/subst-endpoint-from-split","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.subst_endpoint_from_split","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#subst-endpoint-from-split","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"GENERAL split-derived freshness: actual payload ownership and the receiver's already-owned carrier derive non-aliasing even after advancement.","title":"Split-derived freshness for conversion endpoint substitution"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/subst-passive","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.subst_passive","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#subst-passive","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"GENERAL passive substitution transports all scoped positions but adds NO capability. The input context contains only lifted outer ownership.","title":"Passive substitution preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/inp-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.inp_inv","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#inp-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Input inversion retains its actual advertised code and typed continuation.","title":"Inversion of conversion-ranked input typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/ranked-passive-head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.ranked_passive_head_preservation","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#ranked-passive-head-preservation","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"GENERAL passive communicating-head preservation at a semantically compatible actual split, with literal receiver-tail and rank evidence.","title":"Passive head communication preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/ranked-endpoint-head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.ranked_endpoint_head_preservation","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#ranked-endpoint-head-preservation","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"At an initially semantically compatible disjoint split, endpoint head communication transfers EXACT v, retaining the receiver's separately equivalent advertised payload/tail codes.","title":"Endpoint head communication preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.head_preservation","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#head-preservation","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"A typed opposite-head pair at an initially semantically compatible interface yields packed literal-head preservation. Both actual heads and payload/tail correspondence are conclusions, not fields in a protocol or admission predicate.","title":"Head preservation for a conversion-ranked opposite pair"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/head-advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.head_advance","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#head-advance","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"For a ranked-typed opposite-head pair at an initially semantically compatible interface, head preservation constructs a faithful semantic advance and a compatible ranked residual.","title":"Conversion head communication yields a semantic advance"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/same-name-guards-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.same_name_guards_step","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#same-name-guards-step","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"For disjoint ranked-typed same-name leaves at a semantically compatible combined interface, literal guard shapes have opposite directions and yield an actual canonical Step.","title":"Same-name conversion guards are opposite and communicate"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/progress","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.GuardFrontier.progress","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#progress","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"GENERAL balanced ranked frontier progress. A minimum over ALL active owned roles exposes two distinct opposite subject leaves, from which actual canonical communication follows. Completion is process-level Cong to nil.","title":"Progress for semantically balanced conversion frontiers"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/unlift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.unlift","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#unlift","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Reflection of an actual lifted term recovers every priority and context entry.","title":"Conversion ranked typing descends from a lifted process"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/scope-forward","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.scope_forward","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#scope-forward","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Extrusion lifts the partner's real ownership while retaining every guard.","title":"Forward scope extrusion preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/insert-from-absent-partner","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.insert_from_absent_partner","parent":39569,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06/index.html#insert-from-absent-partner","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-06","status":"fully_proved","summary":"Moving a partner-owned role back requires actual absence to DERIVE completion from conversion typing before refined completed weakening is applied.","title":"Adding a role unused by the partner preserves conversion ranked typing"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","kind":"scope","navigation":"transparent","parent":39463,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","title":"Restoring the bound pair preserves conversion ranked typing to Global progress of conversion-ranked processes at balanced interfaces"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/restore-scope-pair","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.restore_scope_pair","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#restore-scope-pair","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Restoring the bound pair adds only absent completed partner roles; all original active ownership and inequalities of the restricted component remain.","title":"Restoring the bound pair preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/scope-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.scope_reverse","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#scope-reverse","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Reverse extrusion derives the entire outer ownership split from typing, removes only genuinely absent partner roles and retains all guard inequalities.","title":"Reverse scope extrusion preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/scope-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.scope_iff","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#scope-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Exact bidirectional canonical scope transport for the same refinement.","title":"Conversion ranked typing is invariant under scope extrusion"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/par-comm","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.par_comm","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#par-comm","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Commutation retains the actual disjoint split and both ranked subterms.","title":"Parallel commutation preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/par-assoc","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.par_assoc","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#par-assoc","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Reassociation derives all pairwise disjointness from actual typing.","title":"Parallel reassociation preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/par-assoc-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.par_assoc_reverse","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#par-assoc-reverse","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Reverse reassociation retains exact contexts and guard checks.","title":"Reverse parallel reassociation preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/par-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.par_nil_elim","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#par-nil-elim","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Nil's actual completed ownership can be reattached without new dependencies.","title":"Removing a parallel nil preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/par-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.par_nil_intro","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#par-nil-intro","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Nil introduction allocates an empty ownership context, not an active frame.","title":"Adding a parallel nil preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/nu-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.nu_nil_elim","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#nu-nil-elim","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Restricting nil is equivalent to its original completed/scoped interface.","title":"Removing a restriction around nil preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/nu-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.nu_nil_intro","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#nu-nil-intro","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"A completed fresh pair can be introduced around nil without changing guards.","title":"Restricting nil preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/nu-swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.nu_swap","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#nu-swap","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Swapping the two restrictions preserves all guard constraints by the GENERAL capture-safe rename theorem and the exact four-role context equation.","title":"Swapping two restrictions preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/nu-swap-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.nu_swap_reverse","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#nu-swap-reverse","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Reverse restriction swap uses the canonical process involution.","title":"Reverse restriction swap preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.cong_iff","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#cong-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Every canonical Cong constructor preserves the whole priority refinement at exactly the same context and ambient depth. Unsupported sum/replication constructors are impossible by the invariant DERIVED from refinement typing.","title":"Structural congruence preserves conversion ranked typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/subject-reduction","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.subject_reduction","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#subject-reduction","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"From a typed source at an initially compatible interface, all five canonical reduction constructors preserve typing and compatibility, with one matched external subject-pair advancement or an unchanged interface when communication is private. No fragment or enabled-edge premise is required.","title":"Conversion ranked subject reduction at compatible interfaces"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/subject-reduction-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.subject_reduction_balanced","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#subject-reduction-balanced","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"A typed source with a balanced initial interface retains typing and balance after any canonical step. Balance supplies the required compatibility; the exact zero-or-one external advancement evidence is retained.","title":"Conversion ranked subject reduction at balanced interfaces"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/subject-reduction-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.subject_reduction_empty","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#subject-reduction-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"An empty-interface typed network retains empty-interface typing through any canonical step. The empty context is balanced, hence compatible, including communication behind restrictions and pre/post congruence.","title":"Conversion ranked subject reduction at the empty interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/reachable","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.reachable","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#reachable","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"A ranked-typed source at an initially semantically compatible interface retains typing and compatibility through finite actual execution, with a finite trace of faithful advances. This is safety, not progress or termination.","title":"Finite runs keep conversion ranked typing at compatible interfaces"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/reachable-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.reachable_empty","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#reachable-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Every finite reachable closed state retains the exact empty interface.","title":"Reachable closed conversion-ranked states keep the empty interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/frontier-normal","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.frontier_normal","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#frontier-normal","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"Normalization is derived from finite syntax, then NEW ranked Cong keeps ownership.","title":"Conversion-ranked processes have a typed frontier normal form"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/progress-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.progress_balanced","parent":39590,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07/index.html#progress-balanced","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-07","status":"fully_proved","summary":"For a ranked-typed process at ANY balanced interface, global progress holds, including disconnected networks. All private channels are tracked through the derived restriction tower. Completion means process congruence to nil, not completion of the interface.","title":"Global progress of conversion-ranked processes at balanced interfaces"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08.json","id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08","kind":"scope","navigation":"transparent","parent":39463,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08","status":"fully_proved","title":"Closed conversion-ranked networks are completed or can step to Progress at reachable states of closed conversion-ranked networks"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08/progress-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.progress_empty","parent":39611,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08/index.html#progress-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08","status":"fully_proved","summary":"A closed finite ranked network is completed or takes an ACTUAL canonical step. Ambient passive-name depth is arbitrary; no readiness witness is supplied.","title":"Closed conversion-ranked networks are completed or can step"},{"id":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08/reachable-progress","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Conversion.RankedType.reachable_progress","parent":39611,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08/index.html#reachable-progress","scope":"ComputerScience/process-calculi/session-types/session-typing-up-to-protocol-equivalence/part-08","status":"fully_proved","summary":"Global progress holds at every finite reachable state of a closed ranked network. No fairness, eventual liveness or recursive termination is asserted.","title":"Progress at reachable states of closed conversion-ranked networks"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-with-shared-services.json","id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services","kind":"scope","navigation":"module","parent":39035,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services","status":"fully_proved","title":"Session typing with shared services"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01.json","id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","kind":"scope","navigation":"transparent","parent":39614,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","title":"Shared service signature to Idleness is invariant under structural congruence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/service-signature","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceSignature","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#service-signature","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Shared permission describes a transferred endpoint, not owned linear state.","title":"Shared service signature"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/service-env","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceEnv","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#service-env","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Constant-only shared names use the existing Finmap, not a new name foundation.","title":"Shared service environment"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/separated","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#separated","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"An owned linear endpoint is never a declared shared trigger's physical name.","title":"Separation of owned linear names from shared names"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/invocation-priority","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.InvocationPriority","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#invocation-priority","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Shared invocation precedes every actual active owned LINEAR key; none exempt.","title":"Priority of a shared invocation over active owned keys"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/separated-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.separated_empty","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#separated-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Empty captured ownership has no linear/shared alias.","title":"Empty ownership is separated from shared names"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/separated-singleton","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.separated_singleton","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#separated-singleton","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"A singleton received/actual endpoint is separate exactly at its own name.","title":"Singleton context with a nonshared name is separated"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/invocation-priority-singleton","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.invocation_priority_singleton","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#invocation-priority-singleton","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Singleton invocation still compares the ACTIVE payload's actual cap rank.","title":"Invocation priority for a singleton of rank above the priority"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/service-template","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceTemplate","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#service-template","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"A template is a structural tree of declared shared inputs, never active outputs.","title":"Structural shared-service template"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/lift-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceTemplate.lift_iff","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#lift-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Canonical lifting preserves literal shared constants through every structural wrapper.","title":"Service templates are invariant under lifting"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/swap-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceTemplate.swap_iff","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#swap-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Canonical swap preserves template guards, including wrappers under nested binders.","title":"Service templates are invariant under binder swapping"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceTemplate.cong_iff","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#cong-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"ALL16 canonical Cong constructors preserve structural guarded templates in BOTH directions.","title":"Service templates are invariant under structural congruence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/accept-in","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.AcceptIn","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#accept-in","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Actual unguarded shared input presence through structural wrappers; input stops here.","title":"Unguarded shared input on a given name"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/accept-cong","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.accept_cong","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#accept-cong","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"All canonical congruences preserve actual shared-input presence, not execution.","title":"Structural congruence preserves unguarded shared inputs"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/offered-in","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.OfferedIn","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#offered-in","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Persistent services expose input leaves within actual replicated structural templates.","title":"Persistent service offer on a given name"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/offered-accept","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.offered_accept","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#offered-accept","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Every persistent offered service has an actual unguarded input leaf.","title":"Offered services have an unguarded shared input"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/offered-cong","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.offered_cong","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#offered-cong","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"ALL16 actual Cong rules preserve persistent service offering, including wrappers inside replication and actual replica unfolding. No closure constructor is used.","title":"Structural congruence preserves persistent service offers"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/offered","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Offered","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#offered","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Declared signature plus actual persistent guarded service syntax, never execution data.","title":"Declared persistent service offer"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/idle","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Idle","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#idle","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Quiescent structural service templates include waiting declared inputs and wrappers.","title":"Idle process as a structural service template"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/cong-iff-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Offered.cong_iff","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#cong-iff-2","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"All canonical Cong constructors preserve each actual declared persistent offer.","title":"Declared service offers are invariant under congruence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/cong-iff-3","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Idle.cong_iff","parent":39615,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01/index.html#cong-iff-3","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-01","status":"fully_proved","summary":"Idle is precisely structural guarded-template quiescence and is Cong-invariant.","title":"Idleness is invariant under structural congruence"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02.json","id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","kind":"scope","navigation":"transparent","parent":39614,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","title":"Name map fixing every declared shared constant to Inserting a nonshared key preserves separation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/fixes-shared","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.FixesShared","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#fixes-shared","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"A name map leaves every declared shared constant literally fixed.","title":"Name map fixing every declared shared constant"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/non-aliases","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#non-aliases","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Actual owned-key images cannot alias a declared shared physical name.","title":"Nonaliasing of owned-name images with shared constants"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/under","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.FixesShared.under","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#under","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Binder extension still fixes shared constants, which are unchanged by lifting.","title":"Binder extension preserves fixing of shared constants"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/separated-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases.separated","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#separated-2","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"The actual target ownership rekey has static name separation, not a desired type result.","title":"Rekeying along a nonaliasing name map yields separation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/erase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases.erase","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#erase","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Erasure restricts the actual owned domain and cannot add a name-map alias.","title":"Erasing an owned key preserves nonaliasing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/insert-owned","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases.insert_owned","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#insert-owned","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Advancing an already-owned key leaves the name-map domain unchanged.","title":"Updating an already-owned key preserves nonaliasing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/nonaliases-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.nonaliases_empty","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#nonaliases-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Empty BODY ownership imposes no linear-image nonalias obligations.","title":"Every name map is nonaliasing on empty ownership"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/under-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases.under_lift","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#under-lift","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Binder extension maps lifted owned names to lifted images, never new shared constants.","title":"Binder extension preserves nonaliasing of lifted contexts"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/under-received","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases.under_received","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#under-received","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"A newly received parameter remains var0 under the canonical binder map.","title":"Nonaliasing survives adding a received parameter under a binder"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/under-bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases.under_bind","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#under-bind","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Both private-role keys are fresh var0; only actual outer names need nonalias transport.","title":"Nonaliasing survives binding a private session pair"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/union-left","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases.union_left","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#union-left","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Actual left-domain restriction of an ownership union preserves map nonalias.","title":"Nonaliasing restricts to the left part of a union"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/union-right","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.NonAliases.union_right","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#union-right","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Actual right-domain restriction likewise preserves map nonalias.","title":"Nonaliasing restricts to the right part of a union"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/rename","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceTemplate.rename","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#rename","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Structural templates transport through a declared-constant-fixing name map; input BODY typing is handled separately, not hidden in this shape predicate.","title":"Service templates are preserved by renamings fixing shared names"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/rekey","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.InvocationPriority.rekey","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#rekey","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Rekeying retains actual cap values and therefore shared-priority comparisons.","title":"Rekeying preserves shared invocation priority"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/rekey-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.rekey_empty","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#rekey-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"An empty owned context stays empty under the canonical context rekey operation.","title":"Rekeying the empty context gives the empty context"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/rekey-singleton","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.rekey_singleton","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#rekey-singleton","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Rekeying a singleton moves its actual key and retains its capability verbatim.","title":"Rekeying a singleton moves its key and keeps its capability"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/nonaliases-lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.nonaliases_lift","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#nonaliases-lift","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Canonical binder lifting cannot send a nonshared owned name into the shared namespace.","title":"Separation gives nonaliasing for binder lifting"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/nonaliases-swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.nonaliases_swap","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#nonaliases-swap","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Canonical binder swapping likewise preserves actual owned-name nonalias.","title":"Separation gives nonaliasing for binder swapping"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/erase-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.erase","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#erase-2","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Erasing an actual owned key cannot create a shared/linear name alias.","title":"Erasing an owned key preserves separation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/insert","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.insert","parent":39636,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02/index.html#insert","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-02","status":"fully_proved","summary":"Inserting a statically nonshared key preserves physical-name separation.","title":"Inserting a nonshared key preserves separation"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03.json","id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","kind":"scope","navigation":"transparent","parent":39614,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","title":"Binder lifting preserves separation to Inversion of runtime typing for input on a shared name"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.lift","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#lift","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Actual binder lifting preserves nonshared physical names: constants cannot be newly produced by lifting a linear variable.","title":"Binder lifting preserves separation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/bind","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.bind","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#bind","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"A literal private pair has variable names, never declared shared constants.","title":"Binding a private pair preserves separation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/union-left","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.union_left","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#union-left","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"A left child of the actual ownership union retains static shared-name separation.","title":"Separation restricts to the left part of a union"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/union-right","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.union_right","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#union-right","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Disjointness prevents right-child lookup shadowing by the left ownership domain.","title":"Separation restricts to the right part of a disjoint union"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/has-type","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#has-type","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Ten syntax-directed rules over canonical Pi, fixed static ranks and actual ownership. Accept/serve may carry completed ambient annotations, never body capture.","title":"Runtime typing with guarded shared replication"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/of-finite","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.of_finite","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#of-finite","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Real ALL7-rule induction embeds finite ranked typing, including delegation; there is no finite-judgement constructor in the new runtime typing.","title":"Finite ranked typing embeds into runtime typing under separation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/env-extends","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.EnvExtends","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#env-extends","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Old shared declarations retain their EXACT signature in the larger environment.","title":"Extension of a shared service environment"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/env-extend","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceTemplate.env_extend","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#env-extend","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"A declared guarded template remains such under literal signature extension.","title":"Service templates survive environment extension"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/env-extend-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.env_extend","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#env-extend-2","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"General \u0393-extension requires static separation of ALL actual owned names. The proof establishes fresh lifted/private/received BODY names, rather than assuming environment monotonicity or carrying a future/result field.","title":"Environment extension preserves runtime typing under separation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/separated","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.separated","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#separated","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Completed ambient annotations still obey actual shared/linear name separation.","title":"Runtime typing separates owned names from shared services"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/repl-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.repl_completed","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#repl-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Replication has only COMPLETED ambient ownership, not necessarily empty annotation.","title":"Replicated runtime processes have completed ownership"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/repl-body-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.repl_body_empty","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#repl-body-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Replicated BODY ownership is exactly EMPTY, while ambient annotations may be completed.","title":"Typed replication has a body with empty ownership"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/closed-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.closedAt","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#closed-at","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Every rule derives the EXISTING canonical process-scope predicate, including fresh service parameters and empty-capture replicated template bodies.","title":"Runtime-typed processes are scoped at the ambient depth"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/serve-input","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.serve_input","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#serve-input","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"A literal shared-input replica is an admissible structural template, with its received BODY at the fresh singleton and replica BODY at EMPTY ownership.","title":"Typing rule for a replicated input on a shared name"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/nil-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.nil_inv","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#nil-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Nil has only actual completed, scoped, separated ambient ownership.","title":"Inversion of runtime typing for the nil process"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/par-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.par_inv","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#par-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Parallel inversion derives the actual disjoint linear split at unchanged \u0393.","title":"Inversion of runtime typing for parallel composition"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/nu-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.nu_inv","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#nu-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Restriction inversion exposes a real literal-dual bound pair, not a representative.","title":"Inversion of runtime typing for restriction"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/output-subject","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.output_subject","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#output-subject","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"Actual output subjects are linear-owned or declared shared constants.","title":"Output subjects are owned endpoints or declared shared names"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/invoke-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.invoke_inv","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#invoke-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"A declared shared output is necessarily an actual endpoint-transfer invocation.","title":"Output on a declared shared name is a service invocation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/accept-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.accept_inv","parent":39657,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03/index.html#accept-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-03","status":"fully_proved","summary":"A declared shared input has completed ambient annotations and EXACT fresh singleton BODY ownership, not captured ambient capabilities.","title":"Inversion of runtime typing for input on a shared name"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04.json","id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","kind":"scope","navigation":"transparent","parent":39614,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","title":"Inversion of runtime typing for replication to Adding a restriction around nil preserves runtime typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/repl-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.repl_inv","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#repl-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"General replica inversion retains completed/scoped/separated AMBIENT annotations and the structurally guarded BODY typed at EXACT EMPTY ownership.","title":"Inversion of runtime typing for replication"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/service-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.service_inv","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#service-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"A literal input template exposes its declared signature and fresh received BODY; completed ambient annotations are NOT inherited by that BODY.","title":"Inversion of runtime typing for a replicated shared input"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/service-offered","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.service_offered","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#service-offered","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Typing-derived actual literal-input offering survives completed ambient annotations.","title":"Typed replicated shared inputs are offered services"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/insert-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.InvocationPriority.insert_completed","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#insert-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"A completed frame is ignored by shared priority, with NO active-payload exemption.","title":"Inserting a completed entry preserves invocation priority"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/weaken-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.weaken_completed","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#weaken-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"ALL10 rules admit a fresh COMPLETED ambient annotation. Its name is scoped and nonshared; accept/serve BODY ownership is unchanged, never captured.","title":"Weakening runtime typing by one fresh completed entry"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/weaken-completed-union","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.weaken_completed_union","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#weaken-completed-union","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"A disjoint completed/scoped/nonshared finite ambient frame can be reattached. The actual replica/accept BODY is not expanded by those annotations.","title":"Weakening runtime typing by a disjoint completed context"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/erase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.InvocationPriority.erase","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#erase","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Removing an owned key preserves the shared priority inequalities on the remainder.","title":"Erasing an owned key preserves invocation priority"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/absent-lookup-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.absent_lookup_completed","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#absent-lookup-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"An owned name absent from ALL subject/payload/binder-adjusted positions already has an actual done unfolding, including in the shared runtime rules.","title":"Owned names absent from a runtime process are completed"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/erase-name-absent","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.erase_nameAbsent","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#erase-name-absent","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"ALL10 runtime rules support erasure only under real all-position absence. Accept/serve ambient erasure does NOT alter their exact fresh/empty BODY domains.","title":"Erasing a key whose name is absent preserves runtime typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/retype","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.retype","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#retype","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Semantic capability changes do not change owned physical names or \u0393 separation.","title":"Equivalent retyping preserves separation from shared names"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/retype-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.InvocationPriority.retype","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#retype-2","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Actual active-payload ranks preserve the strict shared invocation priority.","title":"Equivalent retyping preserves invocation priority"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/retype-3","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.retype","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#retype-3","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"ALL10 runtime rules retype at arbitrary equivalent linear contexts, with \u0393 UNCHANGED. Actual payload codes, literal cap ranks and empty BODY contexts are retained.","title":"Runtime typing transfers to an equivalent linear context"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/retype-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.retype_iff","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#retype-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Equivalent linear interfaces preserve runtime typing in BOTH directions at exactly the same shared environment, process and ambient depth.","title":"Runtime typing is invariant under equivalent linear contexts"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/lookup-closed-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.lookup_closedAt","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#lookup-closed-at","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Every actual owned key is scoped at the runtime derivation's ambient depth.","title":"Each owned key of a runtime typing is scoped at ambient depth"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/context-closed-at","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.context_closedAt","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#context-closed-at","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Scope applies to the actual complete owned domain, not only active subjects.","title":"Whole owned context of a runtime typing is scoped at ambient depth"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/template-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.template_completed","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#template-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"A typed structurally guarded template owns only completed AMBIENT entries. Its input and replica BODY proofs remain separate and may use their own formal key.","title":"Typed service templates own only completed entries"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/template-reannotate","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.template_reannotate","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#template-reannotate","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Structurally guarded typed templates can replace only their COMPLETED ambient annotations. Fresh input/empty replica BODY proofs are retained unchanged.","title":"Typed service templates retype at any well-formed completed context"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/template-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.template_empty","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#template-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Clearing completed template annotations produces the EXACT empty BODY required by replica admission; this is not unconditional active-resource erasure.","title":"Typed service templates retype at empty ownership"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/nu-nil-elim","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.nu_nil_elim","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#nu-nil-elim","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Removing a nil restriction retains its actual completed/scoped/separated ambient.","title":"Removing a restriction around nil preserves runtime typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/nu-nil-intro","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.nu_nil_intro","parent":39678,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04/index.html#nu-nil-intro","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-04","status":"fully_proved","summary":"Introducing a nil restriction chooses a genuinely completed literal-dual pair.","title":"Adding a restriction around nil preserves runtime typing"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05.json","id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","kind":"scope","navigation":"transparent","parent":39614,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","title":"Runtime typing under renamings that fix and avoid shared names to Request-freeness is invariant under binder swapping"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/rename","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.rename","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#rename","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"ALL10 runtime derivation rules transport actual names with \u0393 unchanged. Shared nonalias is required only on OWNED images, not unrestricted passive names.","title":"Runtime typing under renamings that fix and avoid shared names"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.lift","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#lift","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Canonical binder lift retains the entire runtime derivation and \u0393 literally.","title":"Binder lifting preserves runtime typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/swap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.swap","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#swap","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Canonical swap permutes actual owned binders without changing \u0393 or cap metadata.","title":"Binder swapping preserves runtime typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/nonaliases-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.nonaliases_subst","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#nonaliases-subst","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"An actual nonshared endpoint replacement keeps every owned substituted image nonshared.","title":"Nonshared endpoint substitution keeps owned names nonshared"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/nonaliases-passive","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.nonaliases_passive","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#nonaliases-passive","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Passive var0 is NOT owned: even a shared-constant replacement preserves all actual lifted owned names and creates no linear shared-name alias.","title":"Passive substitution creates no owned shared-name alias"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/subst-endpoint","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.subst_endpoint","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#subst-endpoint","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Actual transmitted endpoint substitution preserves literal role/rank and raw code. Shared nonalias of b is a static consequence of ACTUAL sender ownership/separation.","title":"Runtime typing under substitution of a transmitted endpoint"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/subst-passive","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.subst_passive","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#subst-passive","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Passive substitution has NO owned var0 and adds NO capability. A scoped shared constant can be received as data without becoming a linear/service subject.","title":"Runtime typing under passive name substitution"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/subst-endpoint-from-split","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.subst_endpoint_from_split","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#subst-endpoint-from-split","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Real sender ownership and a disjoint receiver carrier derive exact-role freshness and shared nonalias of the actual transmitted endpoint.","title":"Endpoint substitution with side conditions derived from a split"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/unlift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.unlift","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#unlift","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Reflection of a canonical lifted term retains exact owned capabilities and \u0393.","title":"Runtime typing reflects through binder lifting"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/scope-forward","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.scope_forward","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#scope-forward","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Forward extrusion uses the actual disjoint split and literal bound pair.","title":"Runtime typing is preserved by forward scope extrusion"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/insert-from-absent-partner","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.insert_from_absent_partner","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#insert-from-absent-partner","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Partner ownership may be restored only when actual absence DERIVES completion. Shared nonalias is obtained from the typed owner, not a new unchecked assumption.","title":"Runtime typing absorbs a role absent from its parallel partner"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/restore-scope-pair","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.restore_scope_pair","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#restore-scope-pair","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Restoring a private pair adds only actually absent completed partner roles.","title":"Runtime typing restores both bound roles to the restricted side"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/scope-reverse","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.scope_reverse","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#scope-reverse","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Reverse extrusion derives the exact external split from actual typing and removes only genuinely absent partner roles. No closed-hole or ready premise.","title":"Runtime typing is preserved by reverse scope extrusion"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/scope-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.scope_iff","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#scope-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Exact canonical scope transport preserves SAME \u0393, n and \u0394 in both directions.","title":"Runtime typing is invariant under scope extrusion"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.cong_iff","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#cong-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"ALL16 canonical Cong rules preserve the whole runtime derivation at exactly SAME \u0393/n/\u0394, including guarded replicated templates, completed ambient annotations, and actual runtime delegation continuations. No typing-up-to-Cong constructor.","title":"Structural congruence preserves runtime typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/guarded-frontier","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedFrontier","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#guarded-frontier","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Unguarded frontier atoms are actual guards or existing guarded idle templates.","title":"Guarded frontier of guards and idle service templates"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/guarded-normal","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedNormal","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#guarded-normal","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Actual private restriction towers around a guarded runtime frontier.","title":"Guarded normal form: restrictions over a guarded frontier"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/request-free","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.RequestFree","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#request-free","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"No exposed output is addressed to an actually declared shared constant. Prefix bodies are not inspected. This is syntax only, not a GLOBAL premise.","title":"Process with no exposed output on a declared shared name"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/lift-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.RequestFree.lift_iff","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#lift-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Literal shared constants are unaffected by canonical lifting.","title":"Request-freeness is invariant under lifting"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/swap-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.RequestFree.swap_iff","parent":39699,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05/index.html#swap-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-05","status":"fully_proved","summary":"Canonical binder swap does not turn a nonshared output into a shared one.","title":"Request-freeness is invariant under binder swapping"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06.json","id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","kind":"scope","navigation":"transparent","parent":39614,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","title":"Request-freeness is invariant under structural congruence to Runtime head communication advances or keeps the interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.RequestFree.cong_iff","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#cong-iff","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"All canonical structural laws preserve absence of an exposed shared request.","title":"Request-freeness is invariant under structural congruence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/request-free","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.ServiceTemplate.request_free","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#request-free","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"Existing guarded templates have no exposed output/request.","title":"Service templates are request-free"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedFrontier.lift","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#lift","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"Canonical lifting maps guards and existing idle templates literally.","title":"Lifting preserves guarded frontiers"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/lift-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedNormal.lift","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#lift-2","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"Lifting preserves the actual unguarded private restriction tower.","title":"Lifting preserves guarded normal forms"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/parallel","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedNormal.parallel","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#parallel","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"All private restrictions on either component are extruded canonically; no process execution or ready pair is stored in the normal form.","title":"Parallel composition preserves guarded normal forms up to congruence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/syntax-normal","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.syntax_normal","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#syntax-normal","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"ALL10 runtime rules derive a syntactic normal form internally. Replicas remain actual guarded templates; no recursive process is erased to finite typing.","title":"Runtime-typed processes are congruent to a guarded normal form"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/frontier-normal","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.frontier_normal","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#frontier-normal","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"Actual runtime normalization retains SAME \u0393/n/\u0394 through the proved Cong invariance; this theorem supplies the frontier, never the GLOBAL caller.","title":"Guarded normal form of a typed process keeps its typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/linear-out-active","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.linear_out_active","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#linear-out-active","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"A request-free output has an ACTUAL active linear subject and GuardMin.","title":"Request-free runtime outputs have an active linear subject"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/inp-active-or-completed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.inp_active_or_completed","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#inp-active-or-completed","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"Linear inputs expose active subjects; once-only shared accepts have only completed AMBIENT ownership and may wait quiescently.","title":"Runtime inputs are active linear or have completed ownership"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/completed-idle","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedFrontier.completed_idle","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#completed-idle","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"With no actual requests or active owned keys, typing derives ACTUAL idle service-template syntax, not merely an empty/completed interface.","title":"Request-free completed runtime frontiers are service templates"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/expose-minimum","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedFrontier.expose_minimum","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#expose-minimum","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"A minimum over ALL actual active owned keys exposes the EXACT linear subject of a real runtime leaf. Idle templates/accept ambient are completed; shared outputs are excluded only by the internally established request-free case.","title":"Minimum active key is the subject of a runtime frontier leaf"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/minimum-peer-separate","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.minimum_peer_separate","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#minimum-peer-separate","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"The equal-rank semantic peer of a minimum active role cannot share its actual guard leaf. Opposite roles on the SAME physical name are distinct keys.","title":"Runtime minimum guard leaf cannot own an equivalent dual peer"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/expose-request","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedFrontier.expose_request","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#expose-request","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"A non-request-free frontier yields an ACTUAL declared output leaf through canonical par rearrangement. Neither readiness nor a progress result is supplied.","title":"Non-request-free frontier exposes a shared output up to congruence"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/request-progress","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedFrontier.request_progress","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#request-progress","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"Any actual exposed request fires under the EXISTING all-declared-services availability qualifier, even with ZERO active owned keys or a DONE payload. The Step is constructed from actual offered input syntax, not stored in Available.","title":"Exposed shared requests step when declared services are available"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/shared-head","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.shared_head","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#shared-head","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"A declared shared invocation and ANY typed input worker preserve their EXACT combined linear interface. The actual owned payload is substituted, not the advertised raw protocol; completed receiver annotations remain ambient.","title":"Shared invocation and input worker preserve the combined interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/linear-input-inv","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.linear_input_inv","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#linear-input-inv","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"A nonshared input subject exposes a genuine linear passive or endpoint head. The shared accept alternative is excluded by actual subject nonalias.","title":"Inversion of runtime input on a nonshared subject"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Separated.advance","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#advance","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"Faithful semantic advancement preserves shared/linear separation by its actual owned-domain invariance, including every completed entry.","title":"Semantic advance preserves shared-linear separation"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/passive-head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.passive_head_preservation","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#passive-head-preservation","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"At an INITIALLY compatible disjoint interface, a linear passive head retains both actual carrier codes and every runtime continuation guard at unchanged \u0393.","title":"Passive head communication preserves runtime typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/endpoint-head-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.endpoint_head_preservation","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#endpoint-head-preservation","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"At an INITIALLY compatible split, linear endpoint communication transfers ACTUAL v and keeps the receiver's independent advertised payload/tail codes. The \u0393-preserving runtime substitution retains all continuation priorities.","title":"Endpoint head communication preserves runtime typing"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/head-advance","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.head_advance","parent":39720,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06/index.html#head-advance","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-06","status":"fully_proved","summary":"At an INITIALLY semantically compatible linear interface, actual runtime heads either advance the paired linear subjects faithfully or transfer on a shared service with the EXACT unchanged interface. Mixed shared/linear heads are excluded by source separation, not a finite-fragment erasure or supplied result field.","title":"Runtime head communication advances or keeps the interface"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07.json","id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","kind":"scope","navigation":"transparent","parent":39614,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","title":"Same-name linear guards at a compatible interface communicate to Qualified progress at reachable closed runtime states"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/same-name-linear-guards-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.same_name_linear_guards_step","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#same-name-linear-guards-step","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"Actual disjoint same-name linear leaves at an INITIALLY compatible interface have opposite directions and construct a canonical Step. Shapes/nonalias are internally exposed outputs in the GLOBAL proof, not caller readiness hypotheses.","title":"Same-name linear guards at a compatible interface communicate"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/progress","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.GuardedFrontier.progress","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#progress","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"A typed balanced runtime frontier with actual persistent service availability is syntactically Idle OR takes an ACTUAL canonical Step. Shared requests are handled even if all owned keys are done; minimum/peer leaves are derived internally.","title":"Balanced runtime frontiers are idle or step given availability"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/select-eq","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.select_eq","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#select-eq","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"Canonical choice selection is identity on an actually runtime-typed root: there is no sum rule, although replication is admitted. No finiteLinear erasure.","title":"Choice selection is the identity on runtime-typed processes"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/subject-reduction","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.subject_reduction","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#subject-reduction","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"From a runtime-typed source at an INITIALLY semantically compatible linear interface, ALL5 canonical Step constructors preserve \u0393/n, typing and compatibility. The actual residual interface is unchanged OR has one faithful SemanticAdvance. Shared transfers are exact identity; private projection uses the chosen literal pair.","title":"Subject reduction for runtime typing at compatible interfaces"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/subject-reduction-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.subject_reduction_balanced","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#subject-reduction-balanced","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"An INITIALLY semantically balanced runtime interface derives compatibility and retains balance and the exact zero-or-one external advance after ANY Step.","title":"Subject reduction for runtime typing at balanced interfaces"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/subject-reduction-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.subject_reduction_empty","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#subject-reduction-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"An empty linear interface is initially balanced/compatible and remains EXACTLY empty through any runtime Step, including private sessions and shared services.","title":"Subject reduction for runtime typing at the empty interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/reachable","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.reachable","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#reachable","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"From an INITIALLY compatible runtime interface, finite actual reductions retain typing/compatibility and accumulate a trace of faithful semantic advances. Shared/private identity steps introduce no fake external advancement.","title":"Multi-step subject reduction at compatible interfaces"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/reachable-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.reachable_balanced","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#reachable-balanced","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"Every finite reachable state from an INITIALLY balanced runtime interface retains semantic balance and an ACTUAL faithful advancement trace, not progress.","title":"Multi-step subject reduction at balanced interfaces"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/reachable-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.reachable_empty","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#reachable-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"Finite runtime execution from an empty interface preserves exact empty ownership. This safety result does not identify idle replicas with Cong nil.","title":"Multi-step subject reduction at the empty interface"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/progress-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.progress_balanced","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#progress-balanced","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"GLOBAL progress at arbitrary ambient n/shared \u0393/actual balanced \u0394/process p, qualified ONLY by runtime typing, INITIAL semantic balance and EXISTING persistent availability of ALL declared signatures. Result is actual Idle syntax OR a canonical Step. No frontier/normal form/readiness/Step/fairness/representative is supplied.","title":"Global runtime progress at balanced interfaces given availability"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/progress-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.progress_empty","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#progress-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"Closed empty-interface runtime progress, at arbitrary passive-name depth, requires the SAME actual all-declared-service availability. Idle includes waiting replicas/accepts and nil; no termination or scheduler fairness is asserted.","title":"Closed runtime processes are idle or step given availability"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/reachable-progress","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.reachable_progress","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#reachable-progress","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"Every finite canonical reduction from an initially balanced, available typed runtime retains qualified GLOBAL progress. Safety supplies actual balanced residual ownership/advance trace, and Available.reachable supplies current persistent syntax.","title":"Qualified progress at every reachable runtime state"},{"id":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/reachable-progress-empty","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.HasType.reachable_progress_empty","parent":39741,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07/index.html#reachable-progress-empty","scope":"ComputerScience/process-calculi/session-types/session-typing-with-shared-services/part-07","status":"fully_proved","summary":"Finite reachable empty-interface states retain the same qualified progress; this is one-step safety/progress, not all-schedules eventual responsiveness.","title":"Qualified progress at reachable closed runtime states"},{"data_url":"scopes/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services.json","id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","kind":"scope","navigation":"module","parent":39035,"result":false,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","title":"Recursive sessions through replicated services"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/offered-select","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.offered_select","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#offered-select","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Choice selection cannot discard a persistent SOURCE offer: sums do not expose OfferedIn, and selecting an actual communicating guard exposes no persistent offer.","title":"Selection preserves persistent offers"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/offered-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.offered_step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#offered-step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"ALL5 canonical Step rules retain every actual persistent source service offer. This is valid for arbitrary Pi syntax, WITHOUT a HasType premise.","title":"Persistent service offers survive one reduction step"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Offered.step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Exact declared signature lookup plus an actual persistent offer is retained by every canonical Step, at unchanged shared environment/signature.","title":"Declared signature offers survive one reduction step"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/reachable","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Offered.reachable","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#reachable","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Every finite actual reduction preserves each SOURCE persistent signature offer. No liveness, all-schedules fairness or eventual service invocation is asserted.","title":"Declared signature offers survive finite reduction"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/available","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Available","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#available","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Every declared shared signature has an ACTUAL persistent offer in current syntax. There is NO typing, Step, future trace, ready pair or step-closure field.","title":"Availability of all declared services"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/available-iff-offered","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.available_iff_offered","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#available-iff-offered","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Availability is exactly current declared signature lookup plus persistent syntax.","title":"Availability as offers of all declared signatures"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/cong-iff","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Available.cong_iff","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#cong-iff","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"All canonical structural laws preserve current syntactic availability.","title":"Structural congruence preserves service availability"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/step-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Available.step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#step-2","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Strong actual syntactic availability survives ANY canonical Step.","title":"Service availability survives one reduction step"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/reachable-2","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Available.reachable","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#reachable-2","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Finite actual reductions preserve source syntactic availability; preservation is PROVED here, not encoded into its definition. This is not a progress theorem.","title":"Service availability survives finite reduction"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/no-unguarded-output","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Idle.no_unguarded_output","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#no-unguarded-output","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Idle is syntax-only and excludes a canonical exposed output, including inside parallel, restriction and inert replicated templates.","title":"Idle processes have no unguarded output"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/no-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.Idle.no_step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#no-step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Idle services cannot Step, even through arbitrary canonical congruence.","title":"Idle processes cannot step"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/service-invocation-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.service_invocation_step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#service-invocation-step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Canonical replication unfolds and communicates on a real shared trigger.","title":"Invoking a replicated service is a reduction step"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-graph","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurringGraph","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-graph","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Every edge of this reachable one-state graph performs a session send.","title":"One-state recurring send graph"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-protocol","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurringProtocol","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-protocol","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Initial cursor of the one-state recurring send protocol.","title":"Recurring send protocol"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-cap","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurringCap","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-cap","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The recurring session keeps one literal static capability rank throughout.","title":"Fixed-rank capability of the recurring session"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-unfold","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurring_unfold","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-unfold","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Literal guarded self-loop unfolding, not an unreachable extra graph node.","title":"The recurring protocol unfolds to a send of itself"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-dual-unfold","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurring_dual_unfold","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-dual-unfold","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The peer's literal guarded cursor returns to the same receive protocol.","title":"The dual recurring protocol unfolds to a receive of itself"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-worker","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.senderWorker","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-worker","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Worker code has one session action and a real endpoint-transfer reentry.","title":"Sender worker process"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-worker","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiverWorker","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-worker","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"A received session endpoint survives the worker's passive input binder.","title":"Receiver worker process"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-daemon","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.senderDaemon","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-daemon","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Sender replication captures only shared constants and its per-invocation formal.","title":"Replicated sender daemon"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-daemon","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiverDaemon","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-daemon","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Receiver replication likewise receives, rather than captures, its endpoint.","title":"Replicated receiver daemon"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-request","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.senderRequest","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-request","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Transfer the actual sender-role endpoint to its shared trigger.","title":"Sender-role transfer to its trigger"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-request","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiverRequest","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-request","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Transfer the actual opposite-role endpoint to its shared trigger.","title":"Receiver-role transfer to its trigger"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/pending","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.pending","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#pending","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Persistent services plus TWO pending role transfers, on one physical endpoint.","title":"Pending state with two role transfers"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sent-invoked","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.sentInvoked","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sent-invoked","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"After one trigger invocation, the opposite request and both daemons remain.","title":"State after the sender trigger invocation"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/both-invoked","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.bothInvoked","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#both-invoked","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Both received-role workers can now communicate on their actual session name.","title":"State after both trigger invocations"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.sender_subst","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-subst","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Capture-safe first trigger communication instantiates the actual bound endpoint.","title":"Substitution instantiates the sender worker's endpoint"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiver_subst","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-subst","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Receiver preserves the actual endpoint under its inner passive input binder.","title":"Substitution instantiates the receiver worker's endpoint"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-invocation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.sender_invocation","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-invocation","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The first real trigger Step, leaving both persistent daemons untouched.","title":"Pending state steps by invoking the sender trigger"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-invocation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiver_invocation","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-invocation","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The second actual trigger Step, not a pre-generated finite unrolling.","title":"Sender-invoked state steps by invoking the receiver trigger"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/parallel-interchange","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.parallel_interchange","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#parallel-interchange","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Four canonical parallel components regroup without evaluating any body.","title":"Interchange law for four parallel components"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-return-subst","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiver_return_subst","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-return-subst","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"A passive input removes ONLY its data binder, retaining the actual endpoint.","title":"Data substitution yields the receiver's return request"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/session-and-reentry","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.session_and_reentry","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#session-and-reentry","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Actual session action restores BOTH pending endpoint-role transfers and daemons.","title":"Session action steps back to the pending state"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-signature","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.senderSignature","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-signature","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Static sender and receiver permissions use opposite roles, exact rank7 types.","title":"Sender service signature of the recurring example"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-signature","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiverSignature","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-signature","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Opposite-role receive permission retains the exact literal dual capability.","title":"Receiver service signature of the recurring example"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/services","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.services","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#services","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Two distinct declared constant triggers; declaration alone is not an offer.","title":"Service environment declaring the two triggers"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/endpoint-fresh","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.EndpointFresh","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#endpoint-fresh","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Session endpoints are physically distinct from BOTH declared shared triggers.","title":"Endpoint distinct from both service triggers"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-declared","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.sender_declared","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-declared","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Actual sender signature lookup with literal role, cap rank and raw protocol.","title":"The sender trigger declares the sender signature"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-declared","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiver_declared","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-declared","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The other trigger declares exactly the opposite-role receive permission.","title":"The receiver trigger declares the receiver signature"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/lift","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.EndpointFresh.lift","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#lift","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Actual canonical lifting cannot turn a distinct endpoint into a service constant.","title":"Lifting preserves trigger freshness of an endpoint"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/endpoint-separated","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.endpoint_separated","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#endpoint-separated","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The exact actual-role singleton does not capture a shared trigger as a linear name.","title":"A fresh endpoint singleton is separated from the triggers"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/erase-singleton","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.erase_singleton","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#erase-singleton","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Erasing the actual sole endpoint leaves exactly the empty ownership map. Shared by request and delegation consumers instead of duplicating lookup proofs.","title":"Erasing a singleton's only key leaves the empty map"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/nil-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.nil_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#nil-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Nil embeds the finite ranked fragment with genuinely empty ownership.","title":"Nil is runtime-typed with empty ownership"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-request-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.sender_request_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-request-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Reentry transfers the actual sender capability out, preserving role/rank/code.","title":"The sender reentry request is runtime-typed"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-request-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiver_request_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-request-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Reentry of the opposite role likewise consumes EXACTLY its own linear slot.","title":"The receiver reentry request is runtime-typed"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-worker-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.sender_worker_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-worker-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"One linear session send keeps the same actual guarded protocol cursor and rank.","title":"The sender worker is runtime-typed"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-worker-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiver_worker_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-worker-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"One receive retains the outer received endpoint through its passive input binder.","title":"The receiver worker is runtime-typed"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sender-daemon-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.sender_daemon_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sender-daemon-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The sender daemon starts with NO captured linear capability, only its fresh formal.","title":"The sender daemon is typed with empty ownership"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/receiver-daemon-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.receiver_daemon_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#receiver-daemon-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The receiver daemon likewise captures no linear endpoint, in either orientation.","title":"The receiver daemon is typed with empty ownership"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-interface","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurringInterface","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-interface","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Actual linear interface, one sender role and its opposite receiver role.","title":"Two-role linear interface of the recurring session"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/pending-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.pending_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#pending-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"The real pending state owns two distinct opposite keys, not daemon captures.","title":"The pending state is typed at the two-role interface"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/sent-invoked-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.sentInvoked_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#sent-invoked-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"First invocation transfers the exact sender slot into one worker; frame unchanged.","title":"The state after the sender invocation is typed"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/both-invoked-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.bothInvoked_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#both-invoked-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Both workers own their actual distinct roles after the two trigger communications.","title":"The state after both trigger invocations is typed"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Three phases of actual execution; this does not generate unrolled Pi code.","title":"Phases of the repeated-session execution"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase_step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase-step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Every consecutive phase has a real canonical Step, including session reentry.","title":"Each phase steps to the next"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/repeatable-trace","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.repeatable_trace","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#repeatable-trace","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"For EVERY requested count, a nonzero trace returns to the identical pending code. It has exactly three times (m+1) actual Steps, never a reflexive reachability proof.","title":"Every round count has a trace returning to the pending state"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/session-round-phase","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.session_round_phase","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#session-round-phase","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"EVERY round contains an actual action on the SAME session endpoint b.","title":"Every round has a session action on the same endpoint"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase-typed","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase_typed","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase-typed","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Every actual runtime phase retains the same exact two-role linear interface.","title":"Every phase is typed at the two-role interface"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/typed-repeatable-trace","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.typed_repeatable_trace","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#typed-repeatable-trace","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Arbitrarily long NONZERO actual execution with ownership at EVERY state. The conclusion is a trace; no trace/Step/result is part of service admission.","title":"Arbitrarily long typed repeated-session traces"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase-reachable","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase_reachable","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase-reachable","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Induction on actual Step count reaches each phase of the persistent runtime code.","title":"Every phase is reachable from the pending state"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase-offers","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase_offers","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase-offers","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Actual unguarded daemons offer BOTH declared services at EVERY runtime phase.","title":"Both services are offered at every phase"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-balanced","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurring_balanced","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-balanced","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Both literal opposite-role self-loop slots form an ACTUAL balanced initial interface; balance is derived from lookup/code facts, not a ready-pair hypothesis.","title":"The recurring interface is balanced"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-advance-identity","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurring_advance_identity","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-advance-identity","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"EVERY faithful advance of this actual literal self-loop interface is exact identity. This is a code-level property of the concrete regular cursors, not a claim that arbitrary equivalent recursive codes stay syntactically unchanged.","title":"Semantic advances of the recurring interface are the identity"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/recurring-step-preserved","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.recurring_step_preserved","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#recurring-step-preserved","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"ANY actually runtime-typed process at the concrete recurring interface keeps that EXACT interface after a canonical Step, by GENERAL preservation and the literal self-loop code property. No phase, readiness or schedule hypothesis.","title":"Steps preserve typing at the recurring interface"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase-step-preserved","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase_step_preserved","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase-step-preserved","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"GENERAL runtime subject reduction applies to every recursive phase (hence all THREE phase kinds) in BOTH role orientations, at the exact self-loop interface.","title":"Each next phase is typed by subject reduction"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase-typed-by-preservation","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase_typed_by_preservation","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase-typed-by-preservation","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Every phase's ownership follows from the INITIAL pending typing and GENERAL finite runtime preservation, not repeated application of concrete phase typing.","title":"Every phase is typed by preservation from the pending state"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/initial-available","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.initial_available","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#initial-available","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Initial actual replicated daemons supply EVERY declared signature, not just requests or shared permissions. Availability remains a syntactic predicate.","title":"All declared services are available in the pending state"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase-available-by-reduction","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase_available_by_reduction","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase-available-by-reduction","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"GENERAL persistent-offer reduction closure preserves the real source daemons through every finite actual prefix of the recursive execution.","title":"Services stay available at every phase"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/repeatable-trace-preserved","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.repeatable_trace_preserved","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#repeatable-trace-preserved","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"EVERY m has one nonzero finite SAME-session trace with all neighbor Steps, ALL round markers, ownership derived by GENERAL runtime preservation, and actual persistent availability derived by GENERAL reduction closure at EVERY state.","title":"Repeated-session traces with preserved typing and availability"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/phase-general-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.phase_general_step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#phase-general-step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"GENERAL qualified progress produces an ACTUAL Step at every recursive phase, BOTH roles and arbitrary scoped endpoint. Idle is excluded by exposed output syntax, not by assuming the previously selected execution or readiness.","title":"Every phase of the repeated session can step"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/bound-phase-general-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.bound_phase_general_step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#bound-phase-general-step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"GENERAL empty-interface progress includes the genuine private session in BOTH orientations/all phases. Actual output syntax forces its Step branch.","title":"Every phase of the restricted repeated session can step"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/repeatable-trace-global","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.repeatable_trace_global","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#repeatable-trace-global","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"EVERY m has a SAME-witness counted repeated-session trace, all state types, actual availability and ALL marked session edges, plus GENERAL progress at EVERY state (including the last). This is not all-schedules fairness or termination.","title":"Repeated-session traces with typing, availability and progress"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/accept-invocation-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.accept_invocation_step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#accept-invocation-step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"Actual input syntax supplies an ACTUAL canonical Step with any matching shared-constant output. Payload and continuation are not finite-unrolled, and nested private/parallel/replica wrappers are handled by existing Cong/Step.","title":"Unguarded shared input communicates with a matching output"},{"id":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/offered-invocation-step","kind":"node","lean":"Cslib.SessionTypes.Endpoint.Runtime.offered_invocation_step","parent":39755,"result":true,"route":"../roadmap/ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services/index.html#offered-invocation-step","scope":"ComputerScience/process-calculi/session-types/recursive-sessions-through-replicated-services","status":"fully_proved","summary":"A persistent offered input exposes real canonical communication, including nested templates and DONE endpoint payloads; no typing or active-key premise.","title":"Offered service communicates with a matching shared output"},{"data_url":"scopes/ComputerScience/process-calculi/bisimulation.json","id":"ComputerScience/process-calculi/bisimulation","kind":"scope","navigation":"module","parent":39034,"result":false,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","title":"Bisimulation"},{"id":"ComputerScience/process-calculi/bisimulation/mtr","kind":"node","lean":"Cslib.LTS.HomBisimilarity.mtr","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#mtr","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"Let $s$ and $t$ be bisimilar states of a labelled transition system, and suppose $s$ reaches a state $s'$ by a multistep transition along a finite (possibly empty) list of labels $\\mu_1, \\dots, \\mu_n$. Then there is a state $t'$ such that $t$ reaches $t'$ by a multistep transition along the same list of labels, and $s'$ and $t'$ are bisimilar.","title":"Bisimilarity transfers multistep transitions"},{"id":"ComputerScience/process-calculi/bisimulation/has-out-label-iff","kind":"node","lean":"Cslib.LTS.HomBisimilarity.hasOutLabel_iff","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#has-out-label-iff","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"Let $s$ and $t$ be bisimilar states of a labelled transition system. Then for every label $\\mu$, $s$ has a transition labelled $\\mu$ if and only if $t$ has a transition labelled $\\mu$.","title":"Bisimilar states enable the same labels"},{"id":"ComputerScience/process-calculi/bisimulation/outgoing-labels-eq","kind":"node","lean":"Cslib.LTS.HomBisimilarity.outgoingLabels_eq","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#outgoing-labels-eq","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"Let $s$ and $t$ be bisimilar states of a labelled transition system. Then the set of labels $\\mu$ for which $s$ has a transition labelled $\\mu$ equals the set of labels $\\mu$ for which $t$ has a transition labelled $\\mu$.","title":"Bisimilar states have equal sets of outgoing labels"},{"id":"ComputerScience/process-calculi/bisimulation/of-tr","kind":"node","lean":"Cslib.LTS.\u03c4STr.of_tr","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#of-tr","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the silent closure, the reflexive-transitive closure of the $\\tau$-transition relation. If $s \\xrightarrow{\\tau} t$ is a transition, then $s \\Rightarrow t$.","title":"A silent step lies in the silent closure"},{"id":"ComputerScience/process-calculi/bisimulation/cons","kind":"node","lean":"Cslib.LTS.\u03c4STr.cons","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#cons","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the silent closure, the reflexive-transitive closure of the $\\tau$-transition relation. If $s \\xrightarrow{\\tau} m$ is a transition and $m \\Rightarrow t$, then $s \\Rightarrow t$.","title":"Prepending a silent step to the silent closure"},{"id":"ComputerScience/process-calculi/bisimulation/str-iff-exists-mtr","kind":"node","lean":"Cslib.LTS.\u03c4STr_iff_exists_mtr","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#str-iff-exists-mtr","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the silent closure, the reflexive-transitive closure of the $\\tau$-transition relation. Then $s \\Rightarrow t$ if and only if, for some $n \\in \\mathbb{N}$, there is a multistep transition from $s$ to $t$ whose list of labels consists of $n$ copies of $\\tau$.","title":"The silent closure is made of runs of silent steps"},{"id":"ComputerScience/process-calculi/bisimulation/iff-exists-of-ne","kind":"node","lean":"Cslib.LTS.STr.iff_exists_of_ne","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#iff-exists-of-ne","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the reflexive-transitive closure of the $\\tau$-transition relation. A weak transition $s \\stackrel{\\mu}{\\Longrightarrow} t$ is either the trivial one $s \\stackrel{\\tau}{\\Longrightarrow} s$ or a composite $s \\Rightarrow s_1 \\xrightarrow{\\mu} s_2 \\Rightarrow t$. For a label $\\mu \\ne \\tau$, $s \\stackrel{\\mu}{\\Longrightarrow} t$ holds\u2026","title":"Weak transitions with a visible label"},{"id":"ComputerScience/process-calculi/bisimulation/trans-s-tr","kind":"node","lean":"Cslib.LTS.\u03c4STr.trans_sTr","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#trans-s-tr","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"Let $L$ be a labelled transition system with a silent label $\\tau$. A weak $\\tau$-transition is a sequence of zero or more $\\tau$-steps, and for $\\mu \\ne \\tau$ a weak $\\mu$-transition is zero or more $\\tau$-steps, one $\\mu$-step and zero or more $\\tau$-steps. If a state $s$ reaches a state $m$ by zero or more $\\tau$-steps and there is a weak $\\mu$-transition from $m$ to $t$, then there is a weak $\\mu$-transition\u2026","title":"Silent steps before a weak transition"},{"id":"ComputerScience/process-calculi/bisimulation/trans-str","kind":"node","lean":"Cslib.LTS.STr.trans_\u03c4STr","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#trans-str","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"Let $L$ be a labelled transition system with a silent label $\\tau$. A weak $\\tau$-transition is a sequence of zero or more $\\tau$-steps, and for $\\mu \\ne \\tau$ a weak $\\mu$-transition is zero or more $\\tau$-steps, one $\\mu$-step and zero or more $\\tau$-steps. If there is a weak $\\mu$-transition from a state $s$ to a state $m$ and $m$ reaches a state $t$ by zero or more $\\tau$-steps, then there is a weak\u2026","title":"Silent steps after a weak transition"},{"id":"ComputerScience/process-calculi/bisimulation/to-saturate","kind":"node","lean":"Cslib.LTS.MTr.to_saturate","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#to-saturate","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"Let $L$ be a labelled transition system with a silent label $\\tau$. A weak $\\tau$-transition is a sequence of zero or more $\\tau$-steps, and for $\\mu \\ne \\tau$ a weak $\\mu$-transition is zero or more $\\tau$-steps, one $\\mu$-step and zero or more $\\tau$-steps; the saturation of $L$ is the system on the same states whose $\\mu$-transitions are the weak $\\mu$-transitions of $L$. If $L$ moves from a state $s$ to a state\u2026","title":"Strong traces are weak traces"},{"id":"ComputerScience/process-calculi/bisimulation/iff-hom-bisimilarity-saturate","kind":"node","lean":"Cslib.LTS.HomWeakBisimilarity.iff_homBisimilarity_saturate","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#iff-hom-bisimilarity-saturate","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the reflexive-transitive closure of the $\\tau$-transition relation. The weak transitions $s \\stackrel{\\mu}{\\Longrightarrow} t$ are the trivial ones $s \\stackrel{\\tau}{\\Longrightarrow} s$ and the composites $s \\Rightarrow s_1 \\xrightarrow{\\mu} s_2 \\Rightarrow t$; the saturated system has the same states and has the weak transitions\u2026","title":"Weak bisimilarity is bisimilarity of the saturated system"},{"id":"ComputerScience/process-calculi/bisimulation/tr","kind":"node","lean":"Cslib.LTS.HomWeakBisimilarity.tr","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#tr","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"Let $L$ be a labelled transition system with a silent label $\\tau$. A weak $\\tau$-transition is a sequence of zero or more $\\tau$-steps, and for $\\mu \\ne \\tau$ a weak $\\mu$-transition is zero or more $\\tau$-steps, one $\\mu$-step and zero or more $\\tau$-steps. Two states are weakly bisimilar when some bisimulation of the system whose $\\mu$-transitions are the weak $\\mu$-transitions of $L$ relates them. If $p$ and\u2026","title":"Weak bisimilarity answers a strong step with a weak step"},{"id":"ComputerScience/process-calculi/bisimulation/s-tr","kind":"node","lean":"Cslib.LTS.HomWeakBisimilarity.sTr","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#s-tr","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the reflexive-transitive closure of the $\\tau$-transition relation. The weak transitions $s \\stackrel{\\mu}{\\Longrightarrow} t$ are the trivial ones $s \\stackrel{\\tau}{\\Longrightarrow} s$ and the composites $s \\Rightarrow s_1 \\xrightarrow{\\mu} s_2 \\Rightarrow t$; the saturated system has the same states and has the weak transitions\u2026","title":"Weak bisimilarity transfers weak transitions"},{"id":"ComputerScience/process-calculi/bisimulation/saturate-mtr","kind":"node","lean":"Cslib.LTS.HomWeakBisimilarity.saturate_mtr","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#saturate-mtr","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the reflexive-transitive closure of the $\\tau$-transition relation. The weak transitions $s \\stackrel{\\mu}{\\Longrightarrow} t$ are the trivial ones $s \\stackrel{\\tau}{\\Longrightarrow} s$ and the composites $s \\Rightarrow s_1 \\xrightarrow{\\mu} s_2 \\Rightarrow t$; the saturated system has the same states and has the weak transitions\u2026","title":"Weak bisimilarity transfers weak traces"},{"id":"ComputerScience/process-calculi/bisimulation/to-hom-weak-bisimilarity","kind":"node","lean":"Cslib.LTS.HomBisimilarity.to_homWeakBisimilarity","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#to-hom-weak-bisimilarity","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the reflexive-transitive closure of the $\\tau$-transition relation. The weak transitions $s \\stackrel{\\mu}{\\Longrightarrow} t$ are the trivial ones $s \\stackrel{\\tau}{\\Longrightarrow} s$ and the composites $s \\Rightarrow s_1 \\xrightarrow{\\mu} s_2 \\Rightarrow t$; the saturated system has the same states and has the weak transitions\u2026","title":"Strong bisimilarity implies weak bisimilarity"},{"id":"ComputerScience/process-calculi/bisimulation/has-out-label-saturate-iff","kind":"node","lean":"Cslib.LTS.HomWeakBisimilarity.hasOutLabel_saturate_iff","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#has-out-label-saturate-iff","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the reflexive-transitive closure of the $\\tau$-transition relation. The weak transitions $s \\stackrel{\\mu}{\\Longrightarrow} t$ are the trivial ones $s \\stackrel{\\tau}{\\Longrightarrow} s$ and the composites $s \\Rightarrow s_1 \\xrightarrow{\\mu} s_2 \\Rightarrow t$; the saturated system has the same states and has the weak transitions\u2026","title":"Weakly bisimilar states enable the same weak labels"},{"id":"ComputerScience/process-calculi/bisimulation/outgoing-labels-saturate-eq","kind":"node","lean":"Cslib.LTS.HomWeakBisimilarity.outgoingLabels_saturate_eq","parent":39830,"result":true,"route":"../roadmap/ComputerScience/process-calculi/bisimulation/index.html#outgoing-labels-saturate-eq","scope":"ComputerScience/process-calculi/bisimulation","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, write $\\Rightarrow$ for the reflexive-transitive closure of the $\\tau$-transition relation. The weak transitions $s \\stackrel{\\mu}{\\Longrightarrow} t$ are the trivial ones $s \\stackrel{\\tau}{\\Longrightarrow} s$ and the composites $s \\Rightarrow s_1 \\xrightarrow{\\mu} s_2 \\Rightarrow t$; the saturated system has the same states and has the weak transitions\u2026","title":"Weakly bisimilar states have the same set of weak labels"},{"data_url":"scopes/ComputerScience/process-calculi/calculus-of-communicating-systems.json","id":"ComputerScience/process-calculi/calculus-of-communicating-systems","kind":"scope","navigation":"module","parent":39034,"result":false,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","title":"Calculus of communicating systems"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/process","kind":"node","lean":"Cslib.CCS.Process","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#process","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For a type $A$ of actions, the finite CCS processes over $A$ are built from the inactive process $\\mathbf{0}$ by action prefix $a.P$ (an action $a \\in A$ followed by a process $P$), nondeterministic choice $P + Q$, and parallel composition $P \\mid Q$.","title":"Finite CCS processes"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/raw-step","kind":"node","lean":"Cslib.CCS.Process.RawStep","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#raw-step","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Let $\\sigma$ be a partial function that assigns to some pairs of actions $(a, b)$ an action $\\sigma(a, b)$. The transition relation $P \\xrightarrow{a} P'$ on finite CCS processes (built from prefix $a.P$, choice $P + Q$ and parallel composition $P \\mid Q$) determined by $\\sigma$ is the least relation such that $a.P \\xrightarrow{a} P$; if $P \\xrightarrow{a} P'$ then $P + Q \\xrightarrow{a} P'$,\u2026","title":"Operational semantics of finite CCS with a synchronization function"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/step","kind":"node","lean":"Cslib.CCS.Process.Step","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#step","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The transition relation $P \\xrightarrow{a} P'$ on finite CCS processes is the operational semantics of finite CCS whose synchronization function is undefined on every pair of actions. Thus prefix and choice move as usual, and a parallel composition $P \\mid Q$ moves only by letting one of $P$ and $Q$ move; there are no communication steps.","title":"Transitions of finite CCS without communication"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/lts","kind":"node","lean":"Cslib.CCS.Process.lts","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#lts","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The labelled transition system whose states are the finite CCS processes, whose labels are the actions, and whose transitions $P \\xrightarrow{a} P'$ are those of the communication-free operational semantics of finite CCS.","title":"Labelled transition system of finite CCS"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/choice-lift","kind":"node","lean":"Cslib.CCS.Process.ChoiceLift","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#choice-lift","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given relations $R$ and $S$ on finite CCS processes, write $P_1 + P_2$ for the nondeterministic choice of $P_1$ and $P_2$. The choice lift of $R$ and $S$ relates $P$ to $Q$ exactly when $R(P, Q)$ holds, or $S(P, Q)$ holds, or $P = P_1 + P_2$ and $Q = Q_1 + Q_2$ with $R(P_1, Q_1)$ and $S(P_2, Q_2)$.","title":"Lifting relations through choice"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/prefix-congr","kind":"node","lean":"Cslib.CCS.Process.prefix_congr","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#prefix-congr","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For every action $a$ and all finite CCS processes $P$ and $Q$, if $P$ and $Q$ are strongly bisimilar in the labelled transition system of finite CCS, then the prefixed processes $a.P$ and $a.Q$ are strongly bisimilar there as well.","title":"Prefix preserves strong bisimilarity"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/equation-closure","kind":"node","lean":"Cslib.CCS.Process.EquationClosure","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#equation-closure","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given a relation $E$ on finite CCS processes, its equation closure relates $P$ to $Q$ exactly when $P = Q$ or $E(P, Q)$; that is, it is the reflexive closure of $E$.","title":"Reflexive closure of a relation on processes"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/label","kind":"node","lean":"Cslib.CCS.Label","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#label","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For a set of names, a CCS label is either the silent action $\\tau$, the input on a name $a$, or the output on a name $a$.","title":"CCS labels"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/complementary","kind":"node","lean":"Cslib.CCS.Label.Complementary","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#complementary","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Two CCS labels are complementary when one is the input on a name $a$ and the other is the output on the same name $a$, in either order. No other pair of labels is complementary; in particular the silent label $\\tau$ has no complement.","title":"Complementary labels"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/synchronize","kind":"node","lean":"Cslib.CCS.Label.synchronize","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#synchronize","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Let the names form a type with decidable equality. For labels $a$ and $b$, the synchronization of $a$ with $b$ is the silent action $\\tau$ when $a$ is the input on a name $n$ and $b$ is the output on the same name $n$, or when $a$ is the output on $n$ and $b$ is the input on $n$. For every other pair of labels, including any pair involving $\\tau$, there is no result.","title":"Synchronization of CCS labels"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/communication-step","kind":"node","lean":"Cslib.CCS.Process.CommunicationStep","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#communication-step","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The transition relation on CCS processes with labels over a set of names given by the raw step relation for the synchronization function under which an input and an output on the same name, in either order, synchronize to the silent label $\\tau$, and no other pair of labels synchronizes.","title":"CCS transition relation with communication"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/communication-lts","kind":"node","lean":"Cslib.CCS.Process.communicationLTS","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#communication-lts","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The labelled transition system whose states are CCS processes with labels over a set of names, whose labels are CCS labels over those names, and whose transition relation is the CCS transition relation with communication.","title":"Labelled transition system of CCS with communication"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/communication-parallel-step-iff","kind":"node","lean":"Cslib.CCS.Process.communication_parallel_step_iff","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#communication-parallel-step-iff","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For CCS processes $P$, $Q$, $R$ and a CCS label $a$, the parallel composition $P \\mid Q$ has a transition labelled $a$ to $R$ in the CCS transition relation with communication if and only if one of the following holds: $P$ has a transition labelled $a$ to some $P'$ and $R = P' \\mid Q$; or $Q$ has a transition labelled $a$ to some $Q'$ and $R = P \\mid Q'$; or $a = \\tau$ and there are processes $P'$, $Q'$ and\u2026","title":"Transitions of a parallel composition"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/parallel-lift","kind":"node","lean":"Cslib.CCS.Process.ParallelLift","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#parallel-lift","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given two relations $R$ and $S$ on CCS processes with labels over a set of names, their parallel lift relates a process $U$ to a process $V$ exactly when $R$ relates $U$ to $V$, or $S$ relates $U$ to $V$, or $U = P_1 \\mid Q_1$ and $V = P_2 \\mid Q_2$ where $R$ relates $P_1$ to $P_2$ and $S$ relates $Q_1$ to $Q_2$.","title":"Parallel lift of two relations"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/parallel-congr","kind":"node","lean":"Cslib.CCS.Process.parallel_congr","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#parallel-congr","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For CCS processes $P_1$, $P_2$, $Q_1$, $Q_2$ with labels over a set of names: if $P_1$ and $P_2$ are strongly bisimilar and $Q_1$ and $Q_2$ are strongly bisimilar in the labelled transition system of CCS with communication, then the parallel compositions $P_1 \\mid Q_1$ and $P_2 \\mid Q_2$ are strongly bisimilar there.","title":"Parallel composition preserves strong bisimilarity"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/communication-prefix-congr","kind":"node","lean":"Cslib.CCS.Process.communication_prefix_congr","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#communication-prefix-congr","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For every CCS label $a$ and all CCS processes $P$ and $Q$: if $P$ and $Q$ are strongly bisimilar in the labelled transition system of CCS with communication, then so are the prefixed processes $a.P$ and $a.Q$.","title":"Prefix preserves strong bisimilarity in CCS with communication"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/communication-choice-congr","kind":"node","lean":"Cslib.CCS.Process.communication_choice_congr","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#communication-choice-congr","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For CCS processes $P_1$, $P_2$, $Q_1$, $Q_2$ with labels over a set of names: if $P_1$ and $P_2$ are strongly bisimilar and $Q_1$ and $Q_2$ are strongly bisimilar in the labelled transition system of CCS with communication, then the choices $P_1 + Q_1$ and $P_2 + Q_2$ are strongly bisimilar there.","title":"Choice preserves strong bisimilarity in CCS with communication"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/parallel-comm","kind":"node","lean":"Cslib.CCS.Process.parallel_comm","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#parallel-comm","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For all CCS processes $P$ and $Q$ with labels over a set of names, $P \\mid Q$ and $Q \\mid P$ are strongly bisimilar in the labelled transition system of CCS with communication.","title":"Parallel composition is commutative up to strong bisimilarity"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/parallel-nil-left","kind":"node","lean":"Cslib.CCS.Process.parallel_nil_left","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#parallel-nil-left","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Let the names form a type with decidable equality. For every CCS process $P$ over input, output and silent labels on these names, the parallel composition $0 \\mid P$ of the inactive process $0$ with $P$ is strongly bisimilar to $P$ in the communication labelled transition system.","title":"The inactive process is a left unit for parallel composition"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/parallel-nil-right","kind":"node","lean":"Cslib.CCS.Process.parallel_nil_right","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#parallel-nil-right","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Let the names form a type with decidable equality. For every CCS process $P$ over input, output and silent labels on these names, the parallel composition $P \\mid 0$ of $P$ with the inactive process $0$ is strongly bisimilar to $P$ in the communication labelled transition system.","title":"The inactive process is a right unit for parallel composition"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/visible-label","kind":"node","lean":"Cslib.CCS.VisibleLabel","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#visible-label","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"A visible CCS label, excluding the silent action.","title":"Visible CCS label"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/complement","kind":"node","lean":"Cslib.CCS.VisibleLabel.complement","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#complement","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The complement of the input label on a channel $a$ is the output label on $a$, and the complement of the output label on $a$ is the input label on $a$.","title":"Complement of a visible label"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/to-label","kind":"node","lean":"Cslib.CCS.VisibleLabel.toLabel","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#to-label","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The inclusion of visible labels into CCS labels sends the input on a channel $a$ to the input on $a$, and the output on $a$ to the output on $a$.","title":"Visible labels as CCS labels"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/relabelling","kind":"node","lean":"Cslib.CCS.Relabelling","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#relabelling","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"A finite CCS relabelling over channel names is a finite list of entries $(a, \\ell)$, each pairing a channel name $a$ with a visible label $\\ell$, that is, an input or an output on some channel. The entry $(a, \\ell)$ gives $\\ell$ as the image of the input on $a$; the image of the output on $a$ is then the complement of $\\ell$. The first entry with name $a$ is used, and a name occurring in no entry is mapped to itself.","title":"Relabelling"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/apply","kind":"node","lean":"Cslib.CCS.Relabelling.apply","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#apply","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Let $f$ be a finite relabelling whose table lists the entries $(a_1, \\ell_1), \\dots, (a_k, \\ell_k)$, where each $a_i$ is a channel name and each $\\ell_i$ a visible label. The image of a channel name $a$ under $f$ is $\\ell_i$ for the first index $i$ with $a_i = a$, and the input label on $a$ when no entry has name $a$.","title":"Applying a relabelling to a channel name"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/static-process","kind":"node","lean":"Cslib.CCS.StaticProcess","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#static-process","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Finite CCS processes with the static operators, over a type of channel names, are built from the inactive process $0$ by the prefix $\\alpha.P$ of a CCS action $\\alpha$ (silent, an input or an output) and a process $P$, the choice $P + Q$, the parallel composition $P \\mid Q$, the restriction $P \\setminus L$ of $P$ by a finite list $L$ of names, and the relabelling $P[f]$ of $P$ by a finite relabelling $f$.","title":"Static CCS processes"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/of-process","kind":"node","lean":"Cslib.CCS.StaticProcess.ofProcess","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#of-process","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Each finite CCS process over CCS labels is sent to the process with static operators of the same shape: the inactive process goes to the inactive process, the prefix of an action $\\alpha$ to a process $P$ goes to the prefix of $\\alpha$ to the image of $P$, and a choice or a parallel composition of $P$ and $Q$ goes to the choice or the parallel composition of the images of $P$ and $Q$.","title":"Embedding finite CCS into CCS with static operators"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/relabel-action","kind":"node","lean":"Cslib.CCS.StaticProcess.relabelAction","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#relabel-action","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For a relabelling $f$, the silent action $\\tau$ is relabelled to $\\tau$, an input on a channel $a$ is relabelled to the image of $a$ under $f$, and an output on $a$ is relabelled to the complement of the image of $a$ under $f$, each viewed as a CCS label.","title":"Relabelling an action"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/relabel-action-complementary","kind":"node","lean":"Cslib.CCS.StaticProcess.relabelAction_complementary","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#relabel-action-complementary","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For every relabelling $f$ and all CCS labels $\\alpha$ and $\\beta$, if $\\alpha$ and $\\beta$ are complementary, then the actions obtained by relabelling $\\alpha$ and $\\beta$ with $f$ are complementary.","title":"Relabelling preserves complementary actions"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/allowed","kind":"node","lean":"Cslib.CCS.StaticProcess.Allowed","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#allowed","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given a finite list $L$ of blocked channel names, the silent action $\\tau$ is always permitted, and an input or an output on a channel $a$ is permitted exactly when $a \\notin L$.","title":"Actions permitted by a restriction"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/step-3","kind":"node","lean":"Cslib.CCS.StaticProcess.Step","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#step-3","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The labelled transition relation $P \\xrightarrow{\\alpha} P'$ on static CCS processes, with actions $\\alpha$ the CCS labels (silent $\\tau$, inputs and outputs), is the least relation closed under the following rules, where $\\alpha.P$ is the prefix, $P + Q$ the choice, $P \\mid Q$ the parallel composition, $P \\setminus L$ the restriction of $P$ by a finite list $L$ of channel names, and $P[f]$ the relabelling of $P$\u2026","title":"Transitions of static CCS"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/lts-3","kind":"node","lean":"Cslib.CCS.StaticProcess.lts","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#lts-3","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The labelled transition system of finite CCS with static operators has the finite CCS processes with restriction and relabelling as states and the CCS labels as labels, and its transitions are the steps of the structural operational semantics of these processes.","title":"Transition system of CCS with restriction and relabelling"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/restriction-lift","kind":"node","lean":"Cslib.CCS.StaticProcess.RestrictionLift","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#restriction-lift","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given a finite list $L$ of channel names and a relation $R$ on static CCS processes, the lifted relation relates $P$ to $Q$ exactly when either $R$ relates $P$ to $Q$, or $P = P' \\setminus L$ and $Q = Q' \\setminus L$ for some processes $P'$ and $Q'$ that $R$ relates.","title":"Lifting a relation through a restriction"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/restriction-congr","kind":"node","lean":"Cslib.CCS.StaticProcess.restriction_congr","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#restriction-congr","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For every finite list $L$ of channel names and all static CCS processes $P$ and $Q$, if $P$ and $Q$ are strongly bisimilar in the labelled transition system of finite CCS with static operators, then so are the restricted processes $P \\setminus L$ and $Q \\setminus L$.","title":"Restriction preserves strong bisimilarity"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/relabelling-lift","kind":"node","lean":"Cslib.CCS.StaticProcess.RelabellingLift","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#relabelling-lift","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given a relabelling $f$ and a relation $R$ on finite CCS processes with static operators, the lifted relation is the least relation that relates $P$ to $Q$ and relates the relabelling of $P$ by $f$ to the relabelling of $Q$ by $f$ whenever $R$ relates $P$ to $Q$.","title":"Closure of a relation under one relabelling"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/relabelling-congr","kind":"node","lean":"Cslib.CCS.StaticProcess.relabelling_congr","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#relabelling-congr","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For every relabelling $f$ and all finite CCS processes $P$ and $Q$ with static operators, if $P$ and $Q$ are strongly bisimilar in the labelled transition system of finite CCS with static operators, then the relabellings of $P$ and of $Q$ by $f$ are strongly bisimilar there.","title":"Relabelling preserves strong bisimilarity"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/process-2","kind":"node","lean":"Cslib.CCS.Recursive.Process","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#process-2","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given a set of names and a set of process constants, the recursive CCS processes are generated by the following constructors: the inactive process $\\mathbf{0}$; a process constant $A$; a prefix $\\alpha.P$ for a CCS label $\\alpha$ and a process $P$; a choice $P + Q$; a parallel composition $P \\mid Q$; a restriction $P \\setminus L$ for a finite list $L$ of names; and a relabelling $P[f]$ for a CCS relabelling $f$.","title":"Recursive CCS processes"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/specification","kind":"node","lean":"Cslib.CCS.Recursive.Specification","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#specification","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given a type of channel names and a type of process constants, a recursive specification assigns to each process constant $A$ a defining body: a CCS process expression over those names, which may itself mention process constants (including $A$).","title":"Recursive CCS specification"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/step-2","kind":"node","lean":"Cslib.CCS.Recursive.Process.Step","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#step-2","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Let the channel names have decidable equality and let $\\Delta$ be a recursive specification, assigning to each process constant $A$ its defining body $\\Delta(A)$. The labelled transition relation $P \\xrightarrow{\\alpha} P'$ relative to $\\Delta$ on recursive CCS processes, with actions $\\alpha$ the CCS labels (silent $\\tau$, inputs and outputs), is the least relation closed under the following rules, where\u2026","title":"Transitions of recursive CCS"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/lts-2","kind":"node","lean":"Cslib.CCS.Recursive.Process.lts","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#lts-2","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Assuming equality of names is decidable, for a recursive specification $\\Delta$ the labelled transition system generated by $\\Delta$ has the recursive CCS processes as states and the CCS labels as labels, and its transitions $P \\xrightarrow{\\alpha} P'$ are exactly the steps of the structural operational semantics relative to $\\Delta$.","title":"Labelled transition system of a recursive specification"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/guarded-under","kind":"node","lean":"Cslib.CCS.Recursive.Process.GuardedUnder","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#guarded-under","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"For a Boolean flag $b$, recording whether an action prefix has already been crossed, guardedness of a recursive CCS process under $b$ is defined recursively: the inactive process $\\mathbf{0}$ is guarded under $b$; a constant $A$ is guarded under $b$ exactly when $b$ is true; a prefix $\\alpha.Q$ is guarded under $b$ when $Q$ is guarded under true; a choice $Q + R$ or a parallel composition $Q \\mid R$ is guarded\u2026","title":"Guardedness under a prefix flag"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/guarded","kind":"node","lean":"Cslib.CCS.Recursive.Process.Guarded","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#guarded","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"A recursive CCS process is guarded when all of its constant occurrences lie beneath an action prefix.","title":"Guarded process"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/guarded-under-true","kind":"node","lean":"Cslib.CCS.Recursive.Process.guardedUnder_true","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#guarded-under-true","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Every recursive CCS process $P$ is guarded under the flag $\\mathrm{true}$: once an action prefix has been crossed, every constant occurrence counts as guarded.","title":"Every process is guarded beneath a prefix"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/constant-occurrences","kind":"node","lean":"Cslib.CCS.Recursive.Process.constantOccurrences","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#constant-occurrences","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"The constant occurrences of a recursive CCS process $P$ are the list of the process constants occurring in $P$, from left to right and with repetitions: the inactive process $\\mathbf{0}$ has none; a constant $A$ has the single occurrence $A$; a prefix $\\alpha.Q$, a restriction $Q \\setminus L$ and a relabelling $Q[f]$ have the occurrences of $Q$; and a choice $Q + R$ or a parallel composition $Q \\mid R$ has the\u2026","title":"Constant occurrences of a process"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/constants","kind":"node","lean":"Cslib.CCS.Recursive.Process.constants","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#constants","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Assuming equality of process constants is decidable, the constants of a recursive CCS process $P$ are the list obtained from the constant occurrences of $P$ by removing duplicates, so that every process constant occurring in $P$ appears in it exactly once.","title":"Constants of a process"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/map-constants","kind":"node","lean":"Cslib.CCS.Recursive.Process.mapConstants","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#map-constants","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Given a map $\\rho$ from one set of process constants to another, renaming the constants of a recursive CCS process $P$ by $\\rho$ replaces each occurrence of a constant $A$ by $\\rho(A)$ and leaves $P$ otherwise unchanged: the inactive process, prefixes, choices, parallel compositions, restrictions and relabellings are kept as they are.","title":"Renaming process constants"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/guarded-under-map-constants","kind":"node","lean":"Cslib.CCS.Recursive.Process.guardedUnder_mapConstants","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#guarded-under-map-constants","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"Let $P$ be a recursive CCS process, let $\\rho$ be a map from its process constants to another set of constants, and let $P\\rho$ be the process obtained from $P$ by renaming its constants by $\\rho$. Then for each Boolean flag $b$, $P\\rho$ is guarded under $b$ if and only if $P$ is guarded under $b$.","title":"Renaming constants preserves guardedness"},{"id":"ComputerScience/process-calculi/calculus-of-communicating-systems/guarded-2","kind":"node","lean":"Cslib.CCS.Recursive.Specification.Guarded","parent":39848,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus-of-communicating-systems/index.html#guarded-2","scope":"ComputerScience/process-calculi/calculus-of-communicating-systems","status":"fully_proved","summary":"A recursive specification $\\Delta$, which assigns to each process constant $A$ a body $\\Delta(A)$, is guarded when for every constant $A$ the body $\\Delta(A)$ is a guarded process.","title":"Guarded recursive specification"},{"data_url":"scopes/ComputerScience/process-calculi/labelled-transition-systems.json","id":"ComputerScience/process-calculi/labelled-transition-systems","kind":"scope","navigation":"module","parent":39034,"result":false,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","title":"Labelled transition systems"},{"id":"ComputerScience/process-calculi/labelled-transition-systems/trans","kind":"node","lean":"Cslib.LTS.CanReach.trans","parent":39897,"result":true,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html#trans","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","summary":"In a labelled transition system, say that a state $s$ can reach a state $t$ when there is a finite list of labels along which $s$ makes a multistep transition to $t$. If $s$ can reach $m$ and $m$ can reach $t$, then $s$ can reach $t$.","title":"Reachability in a labelled transition system is transitive"},{"id":"ComputerScience/process-calculi/labelled-transition-systems/shift-seq","kind":"node","lean":"Cslib.Semantics.shiftSeq","parent":39897,"result":true,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html#shift-seq","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","summary":"For a sequence $s_0, s_1, s_2, \\dots$ and a natural number $k$, the shift of the sequence by $k$ is the sequence whose $i$-th term is $s_{i+k}$; that is, the suffix starting at index $k$, re-indexed from $0$.","title":"Shifted sequence"},{"id":"ComputerScience/process-calculi/labelled-transition-systems/finite-run","kind":"node","lean":"Cslib.LTS.FiniteRun","parent":39897,"result":true,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html#finite-run","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","summary":"In a labelled transition system, a sequence of states $s_0, s_1, s_2, \\dots$ is a finite run of length $k$ when for every $i < k$ there is a label $a$ such that there is a transition from $s_i$ to $s_{i+1}$ labelled $a$.","title":"Finite run"},{"id":"ComputerScience/process-calculi/labelled-transition-systems/omega-run","kind":"node","lean":"Cslib.LTS.OmegaRun","parent":39897,"result":true,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html#omega-run","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","summary":"In a labelled transition system, a sequence of states $s_0, s_1, s_2, \\dots$ is an infinite run when for every $i$ there is a label $a$ such that there is a transition from $s_i$ to $s_{i+1}$ labelled $a$.","title":"Infinite run"},{"id":"ComputerScience/process-calculi/labelled-transition-systems/forall-state-of-invariant","kind":"node","lean":"Cslib.LTS.forall_state_of_invariant","parent":39897,"result":true,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html#forall-state-of-invariant","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","summary":"Let $P$ be a predicate on the states of a labelled transition system such that, whenever $P(s)$ holds and there is a transition from $s$ to $t$ with some label, $P(t)$ holds. Then for every infinite run $s_0, s_1, s_2, \\dots$ with $P(s_0)$, the predicate $P(s_k)$ holds for every $k$.","title":"Invariant principle for infinite runs"},{"id":"ComputerScience/process-calculi/labelled-transition-systems/omega-execution-of-omega-run","kind":"node","lean":"Cslib.LTS.omegaExecution_of_omegaRun","parent":39897,"result":true,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html#omega-execution-of-omega-run","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","summary":"Select a witnessing label at each step of an infinite state run.","title":"Infinite state runs lift to labeled executions"},{"id":"ComputerScience/process-calculi/labelled-transition-systems/omega-run-of-omega-execution","kind":"node","lean":"Cslib.LTS.omegaRun_of_omegaExecution","parent":39897,"result":true,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html#omega-run-of-omega-execution","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","summary":"Forget the labels of a canonical infinite execution.","title":"Labeled infinite executions give state runs"},{"id":"ComputerScience/process-calculi/labelled-transition-systems/shift-seq-eq-drop","kind":"node","lean":"Cslib.LTS.shiftSeq_eq_drop","parent":39897,"result":true,"route":"../roadmap/ComputerScience/process-calculi/labelled-transition-systems/index.html#shift-seq-eq-drop","scope":"ComputerScience/process-calculi/labelled-transition-systems","status":"fully_proved","summary":"Function-sequence suffixes agree with canonical `\u03c9Sequence.drop`.","title":"Shifted function sequences agree with \u03c9-sequence drop"},{"data_url":"scopes/ComputerScience/process-calculi/observational-congruence.json","id":"ComputerScience/process-calculi/observational-congruence","kind":"scope","navigation":"module","parent":39034,"result":false,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","title":"Observational congruence"},{"id":"ComputerScience/process-calculi/observational-congruence/rooted-weak-step","kind":"node","lean":"Cslib.LTS.RootedWeakStep","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#rooted-weak-step","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system whose labels include a silent label $\\tau$, there is a rooted weak transition from $s$ to $t$ with label $\\mu$ when there are states $s_1$ and $s_2$ such that $s$ reaches $s_1$ by zero or more $\\tau$-transitions, $s_1 \\xrightarrow{\\mu} s_2$ is a transition, and $s_2$ reaches $t$ by zero or more $\\tau$-transitions. In particular, a rooted weak transition with label $\\tau$ contains at\u2026","title":"Rooted weak transition"},{"id":"ComputerScience/process-calculi/observational-congruence/rooted-weak-step-of-tr","kind":"node","lean":"Cslib.LTS.rootedWeakStep_of_tr","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#rooted-weak-step-of-tr","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, every transition $s \\xrightarrow{\\mu} t$ is a rooted weak transition from $s$ to $t$ with label $\\mu$.","title":"Transitions are rooted weak transitions"},{"id":"ComputerScience/process-calculi/observational-congruence/to-str","kind":"node","lean":"Cslib.LTS.RootedWeakStep.toSTr","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#to-str","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, every rooted weak transition from $s$ to $t$ with label $\\mu$ is a weak transition from $s$ to $t$ with label $\\mu$, where a weak transition with label $\\mu$ is either the trivial step from a state to itself with label $\\tau$, or a transition with label $\\mu$ preceded and followed by zero or more $\\tau$-transitions.","title":"Rooted weak transitions are weak transitions"},{"id":"ComputerScience/process-calculi/observational-congruence/rooted-weak-step-trans","kind":"node","lean":"Cslib.LTS.\u03c4STr.rootedWeakStep_trans","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#rooted-weak-step-trans","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, if $s$ reaches $m$ by zero or more $\\tau$-transitions and there is a rooted weak transition from $m$ to $t$ with label $\\mu$, then there is a rooted weak transition from $s$ to $t$ with label $\\mu$.","title":"Silent steps before a rooted weak transition"},{"id":"ComputerScience/process-calculi/observational-congruence/trans-str","kind":"node","lean":"Cslib.LTS.RootedWeakStep.trans_\u03c4STr","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#trans-str","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with silent action $\\tau$, if there is a rooted weak transition from $s$ to $m$ with label $a$, and $t$ is reachable from $m$ by zero or more $\\tau$-transitions, then there is a rooted weak transition from $s$ to $t$ with label $a$.","title":"Appending silent steps to a rooted weak transition"},{"id":"ComputerScience/process-calculi/observational-congruence/trans-s-tr","kind":"node","lean":"Cslib.LTS.RootedWeakStep.trans_sTr","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#trans-s-tr","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, if there is a rooted weak transition from $s$ to $m$ with label $\\tau$ and a weak transition from $m$ to $t$ with label $\\mu$, then there is a rooted weak transition from $s$ to $t$ with label $\\mu$. Here a weak transition with label $\\mu$ is either the trivial step from a state to itself with label $\\tau$, or a transition with label $\\mu$ preceded and\u2026","title":"A rooted silent step followed by a weak transition is rooted"},{"id":"ComputerScience/process-calculi/observational-congruence/rooted-weakly-bisimilar","kind":"node","lean":"Cslib.LTS.RootedWeaklyBisimilar","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#rooted-weakly-bisimilar","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, states $p$ and $q$ are rooted weakly bisimilar (observation congruent) when every transition $p \\xrightarrow{\\mu} p'$ is matched by a rooted weak transition from $q$ to some $q'$ with label $\\mu$ such that $p'$ and $q'$ are weakly bisimilar, and every transition $q \\xrightarrow{\\mu} q'$ is matched by a rooted weak transition from $p$ to some $p'$ with\u2026","title":"Rooted observation congruence"},{"id":"ComputerScience/process-calculi/observational-congruence/match-rooted-left","kind":"node","lean":"Cslib.LTS.RootedWeaklyBisimilar.match_rootedLeft","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#match-rooted-left","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, if $p$ and $q$ are rooted weakly bisimilar and there is a rooted weak transition from $p$ to $p'$ with label $\\mu$, then there is a state $q'$ with a rooted weak transition from $q$ to $q'$ with label $\\mu$ such that $p'$ and $q'$ are weakly bisimilar.","title":"Observation congruence matches rooted weak transitions"},{"id":"ComputerScience/process-calculi/observational-congruence/trans","kind":"node","lean":"Cslib.LTS.RootedWeaklyBisimilar.trans","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#trans","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with silent action $\\tau$, if $p$ is rooted observationally congruent to $q$ and $q$ is rooted observationally congruent to $r$, then $p$ is rooted observationally congruent to $r$.","title":"Transitivity of rooted observational congruence"},{"id":"ComputerScience/process-calculi/observational-congruence/observation-congruent-equivalence","kind":"node","lean":"Cslib.LTS.observationCongruent_equivalence","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#observation-congruent-equivalence","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In every labelled transition system with a silent label $\\tau$, rooted weak bisimilarity (observation congruence) is reflexive, symmetric and transitive.","title":"Observation congruence is an equivalence relation"},{"id":"ComputerScience/process-calculi/observational-congruence/observation-congruent-of-bisimilar","kind":"node","lean":"Cslib.LTS.observationCongruent_of_bisimilar","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#observation-congruent-of-bisimilar","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, any two strongly bisimilar states are rooted weakly bisimilar.","title":"Strong bisimilarity implies observation congruence"},{"id":"ComputerScience/process-calculi/observational-congruence/to-hom-weak-bisimilarity","kind":"node","lean":"Cslib.LTS.RootedWeaklyBisimilar.to_homWeakBisimilarity","parent":39906,"result":true,"route":"../roadmap/ComputerScience/process-calculi/observational-congruence/index.html#to-hom-weak-bisimilarity","scope":"ComputerScience/process-calculi/observational-congruence","status":"fully_proved","summary":"In a labelled transition system with a silent label $\\tau$, any two rooted weakly bisimilar states are weakly bisimilar.","title":"Observation congruence implies weak bisimilarity"},{"data_url":"scopes/ComputerScience/process-calculi/finite-state-ccs-agents.json","id":"ComputerScience/process-calculi/finite-state-ccs-agents","kind":"scope","navigation":"module","parent":39034,"result":false,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","title":"Finite-state CCS agents"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/sum-of","kind":"node","lean":"Cslib.Algorithms.CCS.sumOf","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#sum-of","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"For a list $l$ of pairs $(a, q)$ of a label and a process, the process that chooses among the prefixes $a.q$ listed in $l$: the empty list gives the inactive process $\\mathbf{0}$, and the list with head $(a, q)$ and tail $r$ gives the choice $a.q + P$ between the prefix $a.q$ and the process $P$ built in the same way from $r$.","title":"Sum of prefixes"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/step-sum-of-iff","kind":"node","lean":"Cslib.Algorithms.CCS.step_sumOf_iff","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#step-sum-of-iff","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"For every recursive specification, every list $l$ of pairs of a label and a process, every label $a$ and every process $q$, the sum of the prefixes listed in $l$ has a transition labelled $a$ to $q$ exactly when $(a, q)$ belongs to $l$.","title":"Transitions of a sum of prefixes"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/realizes","kind":"node","lean":"Cslib.Algorithms.CCS.Realizes","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#realizes","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Fix a recursive specification. A family of processes $P_x$ indexed by codes $x$ realizes a function $s$ that lists, for each code $x$, labelled transitions $(a, y)$ with $a$ a label and $y$ a code, when for every code $x$, every label $a$ and every process $q$, the process $P_x$ has a transition with label $a$ to $q$ exactly when $s(x)$ contains an entry $(a, y)$ with $P_y = q$.","title":"Realization of a code-level transition system"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/constant","kind":"node","lean":"Cslib.Algorithms.CCS.Realizes.constant","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#constant","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Fix a recursive specification, a map $\\iota$ from codes to process constants and a function $s$ listing labelled transitions for each code. Suppose that for every code $x$, writing $(a_1, y_1), \\dots, (a_n, y_n)$ for the entries of $s(x)$ in order, the body of the constant $\\iota(x)$ is $a_1.\\iota(y_1) + (a_2.\\iota(y_2) + \\dots + (a_n.\\iota(y_n) + 0))$, which is $0$ when $n = 0$. Then the processes $\\iota(x)$,\u2026","title":"Constants listing their transitions realize them"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/par-steps","kind":"node","lean":"Cslib.Algorithms.CCS.parSteps","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#par-steps","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Given functions $s$ and $t$ listing the labelled transitions of codes $x$ and $y$, the transitions of the pair $(x, y)$ are listed in three blocks: $(a, (x', y))$ for each entry $(a, x')$ of $s(x)$; then $(a, (x, y'))$ for each entry $(a, y')$ of $t(y)$; then $(\\tau, (x', y'))$ for each entry $(a, x')$ of $s(x)$ and each entry $(b, y')$ of $t(y)$ whose labels are complementary, that is, one is the input and the\u2026","title":"Code-level parallel composition"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/parallel","kind":"node","lean":"Cslib.Algorithms.CCS.Realizes.parallel","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#parallel","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Fix a recursive specification. If the processes $P_x$ realize $s$ and the processes $Q_y$ realize $t$, then the parallel compositions $P_x \\mid Q_y$, indexed by pairs $(x, y)$, realize the code-level parallel composition of $s$ and $t$.","title":"Parallel composition preserves realization"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/allowed-b","kind":"node","lean":"Cslib.Algorithms.CCS.allowedB","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#allowed-b","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"For a list $L$ of blocked names, the test returns true on a label exactly when the label is the silent action $\\tau$, or an input or an output on a name that does not occur in $L$. It is the Boolean form of the condition that restriction by $L$ allows the label.","title":"Boolean restriction test"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/restrict-steps","kind":"node","lean":"Cslib.Algorithms.CCS.restrictSteps","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#restrict-steps","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"For a list $L$ of blocked names and a function $s$ listing the labelled transitions of each code, the restricted transitions of a code $x$ are the entries $(a, y)$ of $s(x)$, in order, whose label $a$ is the silent action $\\tau$ or an input or an output on a name that does not occur in $L$.","title":"Code-level restriction"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/restriction","kind":"node","lean":"Cslib.Algorithms.CCS.Realizes.restriction","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#restriction","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Fix a recursive specification and a list $L$ of blocked names. If the processes $P_x$ realize $s$, then the restricted processes $P_x \\setminus L$ realize the code-level restriction of $s$ by $L$.","title":"Restriction preserves realization"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/forward-check","kind":"node","lean":"Cslib.Algorithms.CCS.forwardCheck","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#forward-check","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Let $s$ list the labelled transitions $(a, x')$ of each implementation code $x$, let $t$ list those of each specification code $y$, and let $R$ be a finite list of code pairs. The forward check succeeds when for every pair $(x, y)$ in $R$ and every entry $(a, x')$ of $s(x)$, either $a = \\tau$ and $(x', y)$ is in $R$, or some entry $(a, y')$ of $t(y)$ with the same label has $(x', y')$ in $R$.","title":"Forward certificate check"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/backward-check","kind":"node","lean":"Cslib.Algorithms.CCS.backwardCheck","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#backward-check","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Let $s$ list the labelled transitions $(a, x')$ of each implementation code $x$, let $t$ list those of each specification code $y$, and let $R$ be a finite list of code pairs. The backward check succeeds when for every pair $(x, y)$ in $R$ and every entry $(a, y')$ of $t(y)$, at least one of the following holds: $a = \\tau$ and $(x, y')$ is in $R$; some entry $(a, x')$ of $s(x)$ with the same label has $(x', y')$ in\u2026","title":"Backward certificate check"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/cert-check","kind":"node","lean":"Cslib.Algorithms.CCS.certCheck","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#cert-check","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"For transition lists $s$ of implementation codes and $t$ of specification codes and a finite list $R$ of code pairs, the certificate check succeeds when both the forward check and the backward check succeed on $s$, $t$ and $R$.","title":"Weak-bisimulation certificate check"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/weakly-bisimilar-of-cert-check","kind":"node","lean":"Cslib.Algorithms.CCS.weaklyBisimilar_of_certCheck","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#weakly-bisimilar-of-cert-check","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Let $f$ realize the code-level transition function $s$ and $g$ realize $t$ for a recursive specification, and let $R$ be a list of code pairs on which the certificate check of $s$, $t$ and $R$ succeeds. Then for every pair $(x, y)$ in $R$, the processes $f(x)$ and $g(y)$ are weakly bisimilar in the labelled transition system of the specification.","title":"Certificate checks give weak bisimilarity"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/root-check","kind":"node","lean":"Cslib.Algorithms.CCS.rootCheck","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#root-check","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Given code-level transition functions $s$ on codes $\\alpha$ and $t$ on codes $\\beta$, each listing labelled steps to codes, a relation $R$ given as a list of code pairs, and codes $x$ and $y$, the root check succeeds exactly when every listed step $(a, x')$ of $s(x)$ is matched by some listed step $(a, y')$ of $t(y)$ with the same label and $(x', y') \\in R$, and every listed step $(a, y')$ of $t(y)$ is matched by\u2026","title":"Direct check of the root condition"},{"id":"ComputerScience/process-calculi/finite-state-ccs-agents/rooted-weakly-bisimilar-of-cert-check","kind":"node","lean":"Cslib.Algorithms.CCS.rootedWeaklyBisimilar_of_certCheck","parent":39919,"result":true,"route":"../roadmap/ComputerScience/process-calculi/finite-state-ccs-agents/index.html#rooted-weakly-bisimilar-of-cert-check","scope":"ComputerScience/process-calculi/finite-state-ccs-agents","status":"fully_proved","summary":"Let $f$ realize the code-level transition function $s$ and $g$ realize $t$ for a recursive specification, and let $R$ be a list of code pairs on which the certificate check of $s$, $t$ and $R$ succeeds. If the root check of $s$, $t$ and $R$ succeeds at codes $x$ and $y$, then the processes $f(x)$ and $g(y)$ are rooted weakly bisimilar (observation congruent) in the labelled transition system of the specification\u2026","title":"Certificate checks give rooted observation congruence"},{"data_url":"scopes/ComputerScience/process-calculi/calculus.json","id":"ComputerScience/process-calculi/calculus","kind":"scope","parent":39034,"result":false,"route":"../roadmap/ComputerScience/process-calculi/calculus/index.html","scope":"ComputerScience/process-calculi/calculus","status":"fully_proved","title":"\u03c0-calculus"},{"data_url":"scopes/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong.json","id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","kind":"scope","navigation":"module","parent":39935,"result":false,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","title":"The \u03c0-calculus: syntax, substitution, structural congruence, reduction"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/name","kind":"node","lean":"Cslib.PiCalculus.Name","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#name","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"A channel name: a free channel `const n`, or the `i`-th enclosing binder `var i`.","title":"Channel name in the \u03c0-calculus"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/proc","kind":"node","lean":"Cslib.PiCalculus.Proc","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#proc","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Processes of the \u03c0-calculus. `inp a p` binds variable `0` in `p`; `nu p` binds variable `0` in `p` as a fresh channel.","title":"Processes of the \u03c0-calculus"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/lift-name","kind":"node","lean":"Cslib.PiCalculus.liftName","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#lift-name","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Shift free variables at or above `cutoff` up by one, e.g. when pushing a process under a binder.","title":"Lifting free de Bruijn indices of a \u03c0-calculus name"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/lift-proc","kind":"node","lean":"Cslib.PiCalculus.liftProc","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#lift-proc","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Shift free variables of a process, entering binders with an increased cutoff.","title":"Lifting free de Bruijn indices of a \u03c0-calculus process"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/subst-name","kind":"node","lean":"Cslib.PiCalculus.substName","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#subst-name","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Substitute the name `n` for the variable at `cutoff`: variables below the cutoff are bound by inner binders and untouched, the variable at the cutoff is replaced by `n`, and variables above shift down by one.","title":"Substitution of a name into a \u03c0-calculus name"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/subst-proc","kind":"node","lean":"Cslib.PiCalculus.substProc","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#subst-proc","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Substitute the name `n` for the variable at `cutoff` in a process. Under a binder, `n` is lifted by one at cutoff `0` so it keeps referring to the same outer names: every free index of `n` refers to a binder above the substitution point, so all of them shift. Lifting at the current cutoff would leave indices below the cutoff captured by the new binder.","title":"Substitution of a name into a \u03c0-calculus process at a cutoff"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/subst","kind":"node","lean":"Cslib.PiCalculus.subst","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#subst","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Substitute the name `n` for the innermost bound variable of `p`.","title":"Substitution for the innermost bound variable of a process"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/swap-name-at","kind":"node","lean":"Cslib.PiCalculus.swapNameAt","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#swap-name-at","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Swap the two de Bruijn indices `cutoff` and `cutoff + 1` in a name: the two binders at that depth exchange places.","title":"Swapping adjacent de Bruijn indices in a name"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/swap-proc","kind":"node","lean":"Cslib.PiCalculus.swapProc","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#swap-proc","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Swap the two binders at depth `cutoff` throughout a process, incrementing the cutoff under binders.","title":"Swapping adjacent binders throughout a process"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/subst-nil","kind":"node","lean":"Cslib.PiCalculus.subst_nil","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#subst-nil","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Substitution leaves a process with no `var 0` untouched: a constant channel stays constant, and the empty process stays empty.","title":"Substitution leaves constant channels and the empty process unchanged"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/cong","kind":"node","lean":"Cslib.PiCalculus.Cong","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#cong","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Structural congruence: `|` is commutative and associative with unit `nil`, `\u03bd` extrudes over `|` (lifting the other side, which cannot mention the fresh name), restriction of `nil` is `nil`, nested restrictions commute (swapping the two bound indices), and replication unfolds.","title":"Structural congruence of \u03c0-calculus processes"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/select","kind":"node","lean":"Cslib.PiCalculus.Select","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#select","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Guarded-choice resolution: discard sum branches to reach a guarded process. Transitivity handles nested sums.","title":"Guarded-choice resolution in the \u03c0-calculus"},{"id":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/step","kind":"node","lean":"Cslib.PiCalculus.Step","parent":39936,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong/index.html#step","scope":"ComputerScience/process-calculi/calculus/the-calculus-syntax-substitution-structural-cong","status":"fully_proved","summary":"Reduction: communication (substituting the sent channel for the input variable) between any selected output and input guards, reduction under `|` and `\u03bd`, and reduction up to structural congruence.","title":"Reduction relation of the \u03c0-calculus"},{"data_url":"scopes/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity.json","id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","kind":"scope","navigation":"module","parent":39935,"result":false,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","title":"Strong free-channel barbed bisimilarity"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/observable","kind":"node","lean":"Cslib.PiCalculus.Observable","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#observable","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"A visible input or output subject. Restricted variables are not subjects of these free-channel observations.","title":"Free-channel input and output observables"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/offer","kind":"node","lean":"Cslib.PiCalculus.Offer","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#offer","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"A selected free-channel guard exposed without reduction. Restrictions hide bound subjects but do not hide free constants.","title":"Free-channel guard offered without reduction"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barb","kind":"node","lean":"Cslib.PiCalculus.Barb","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barb","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Immediate free-channel readiness, up to the existing structural congruence. A witness performs no `Step`.","title":"Immediate free-channel barb"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/behaviour-label","kind":"node","lean":"Cslib.PiCalculus.BehaviourLabel","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#behaviour-label","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Labels for reductions and observation-only self-loops.","title":"Labels for reductions and barb observations"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/observed-lts","kind":"node","lean":"Cslib.PiCalculus.observedLTS","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#observed-lts","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"The existing reduction semantics decorated with barb tests. The observation transitions do not consume a prefix or transmit a name.","title":"Reduction LTS decorated with barb observations"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barbed-bisimilar","kind":"node","lean":"Cslib.PiCalculus.BarbedBisimilar","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barbed-bisimilar","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Strong barbed bisimilarity, using the canonical CSLib bisimulation API. No context closure is built into this relation.","title":"Strong barbed bisimilarity"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/is-bisimulation-observed-lts-iff","kind":"node","lean":"Cslib.PiCalculus.isBisimulation_observedLTS_iff","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#is-bisimulation-observed-lts-iff","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"The decorated LTS has precisely the strong reduction-matching and immediate-barb-preservation clauses of a strong barbed bisimulation.","title":"Decorated-LTS bisimulations are exactly strong barbed bisimulations"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barb-iff","kind":"node","lean":"Cslib.PiCalculus.Cong.barb_iff","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barb-iff","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Structural congruence preserves immediate barbs.","title":"Structural congruence preserves immediate barbs"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/cong-is-barbed-bisimulation","kind":"node","lean":"Cslib.PiCalculus.cong_isBarbedBisimulation","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#cong-is-barbed-bisimulation","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Structural congruence is a strong barbed bisimulation: reductions match one for one, while free-channel observations match immediately.","title":"Structural congruence is a strong barbed bisimulation"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barbed-bisimilar-2","kind":"node","lean":"Cslib.PiCalculus.Cong.barbedBisimilar","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barbed-bisimilar-2","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Every structural law of `Basic` is sound for strong barbed bisimilarity, including scope extrusion and replication unfolding.","title":"Structurally congruent processes are barbed bisimilar"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/has-unguarded-output","kind":"node","lean":"Cslib.PiCalculus.HasUnguardedOutput","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#has-unguarded-output","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"An output not behind a prefix, on any subject (including bound names). This is an overapproximation for arbitrary unguarded sums, not a barb.","title":"Unguarded output in a process"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/has-unguarded-output-iff","kind":"node","lean":"Cslib.PiCalculus.Cong.hasUnguardedOutput_iff","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#has-unguarded-output-iff","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Structural rearrangement cannot expose an output from an input-guarded process. This includes arbitrary uses of replication and scope laws.","title":"Structural congruence preserves unguarded outputs"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/has-unguarded-output-2","kind":"node","lean":"Cslib.PiCalculus.Step.hasUnguardedOutput","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#has-unguarded-output-2","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Every reduction needs an unguarded output in its source, even when communication is private or structural congruence is used arbitrarily.","title":"Every reduction source has an unguarded output"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/not-step-of-no-output","kind":"node","lean":"Cslib.PiCalculus.not_step_of_no_output","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#not-step-of-no-output","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Outputless processes cannot reduce. Input guards may still have observable input barbs; this theorem does not identify them with `nil`.","title":"Processes without unguarded output cannot reduce"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/of-offer","kind":"node","lean":"Cslib.PiCalculus.Barb.of_offer","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#of-offer","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"A directly exposed guard is a barb, with the reflexive structural representative as witness.","title":"Exposed guards are barbs"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barb-nil","kind":"node","lean":"Cslib.PiCalculus.barb_nil","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barb-nil","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"The inactive process has no barb, even after structural rearrangement.","title":"The inactive process has no barbs"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barb-out-iff","kind":"node","lean":"Cslib.PiCalculus.barb_out_iff","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barb-out-iff","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Output readiness sees only its free subject and direction, not its object or guarded continuation.","title":"Barbs of an output prefix"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barb-inp-iff","kind":"node","lean":"Cslib.PiCalculus.barb_inp_iff","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barb-inp-iff","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Waiting inputs are observable, despite being unable to reduce alone.","title":"Barbs of an input prefix"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barb-nu-out-iff","kind":"node","lean":"Cslib.PiCalculus.barb_nu_out_iff","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barb-nu-out-iff","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"A restricted output is visible exactly when its subject is free. Sending the private name on a free subject remains observable.","title":"Barbs of a restricted output prefix"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/barb-iff-2","kind":"node","lean":"Cslib.PiCalculus.BarbedBisimilar.barb_iff","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#barb-iff-2","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Bisimilar processes have the same immediate free-channel barbs.","title":"Barbed bisimilar processes have the same barbs"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/step","kind":"node","lean":"Cslib.PiCalculus.BarbedBisimilar.step","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#step","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"A reduction from the first process is matched by one reduction, with strong barbed bisimilar derivatives.","title":"Barbed bisimilarity matches each reduction by one reduction"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/input-not-barbed-bisimilar-nil","kind":"node","lean":"Cslib.PiCalculus.input_not_barbedBisimilar_nil","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#input-not-barbed-bisimilar-nil","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Waiting for input is not equivalent to doing nothing. Reduction-only bisimilarity would miss this distinction.","title":"Free-channel input is not barbed bisimilar to nil"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/may-barb","kind":"node","lean":"Cslib.PiCalculus.MayBarb","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#may-barb","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"A free-channel barb reachable by finitely many existing reductions, possibly zero. This is a may observation, not a weak equivalence.","title":"May-barb after finitely many reductions"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/reduction-path","kind":"node","lean":"Cslib.PiCalculus.BarbedBisimilar.reductionPath","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#reduction-path","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"A finite reduction path transfers across strong barbed bisimilarity.","title":"Barbed bisimilarity transfers finite reduction paths"},{"id":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/may-barb-iff","kind":"node","lean":"Cslib.PiCalculus.BarbedBisimilar.mayBarb_iff","parent":39950,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity/index.html#may-barb-iff","scope":"ComputerScience/process-calculi/calculus/strong-free-channel-barbed-bisimilarity","status":"fully_proved","summary":"Strong barbed bisimilarity preserves finite may observations.","title":"Barbed bisimilarity preserves may-barbs"},{"data_url":"scopes/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces.json","id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","kind":"scope","navigation":"module","parent":39935,"result":false,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","title":"Scope preservation for \u03c0-calculus names and processes"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/closed-at","kind":"node","lean":"Cslib.PiCalculus.Name.ClosedAt","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#closed-at","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"A name refers only to the `depth` surrounding binders or to a free constant. At depth zero, no variable is permitted.","title":"Name scoped at a binder depth"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/closed-at-2","kind":"node","lean":"Cslib.PiCalculus.Proc.ClosedAt","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#closed-at-2","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"Every variable refers to an enclosing input/restriction binder. `ClosedAt 0` permits free constants, but no dangling de Bruijn variables.","title":"Locally closed processes at a binder depth"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/lift-name-eq-of-closed-at","kind":"node","lean":"Cslib.PiCalculus.liftName_eq_of_closedAt","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#lift-name-eq-of-closed-at","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"Lifting above the ambient binder depth fixes an in-scope name.","title":"Lifting fixes an in-scope name"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/lift-proc-eq-of-closed-at","kind":"node","lean":"Cslib.PiCalculus.liftProc_eq_of_closedAt","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#lift-proc-eq-of-closed-at","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"Lifting above the surrounding binders fixes a locally closed process. At depth zero, insertion under a fresh binder cannot capture its names.","title":"Lifting fixes a locally closed process"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/lift","kind":"node","lean":"Cslib.PiCalculus.Name.ClosedAt.lift","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#lift","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"Inserting a binder at any cutoff preserves name scoping in the larger environment.","title":"Lifting preserves name scoping"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/subst","kind":"node","lean":"Cslib.PiCalculus.Name.ClosedAt.subst","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#subst","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"Removing the binder at `k \u2264 n` preserves name scoping when its replacement is scoped in the remaining environment.","title":"Name substitution preserves scoping"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/lift-2","kind":"node","lean":"Cslib.PiCalculus.Proc.ClosedAt.lift","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#lift-2","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"Shifting a process past a new binder preserves scoping. Inner input and restriction binders increase the cutoff as in `Basic.liftProc`.","title":"Lifting preserves process scoping"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/subst-proc","kind":"node","lean":"Cslib.PiCalculus.Proc.ClosedAt.substProc","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#subst-proc","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"Capture-avoiding substitution removes one ambient binder while preserving every name's scope, including under input and restriction.","title":"Substitution at any cutoff preserves process scoping"},{"id":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/subst-2","kind":"node","lean":"Cslib.PiCalculus.Proc.ClosedAt.subst","parent":39976,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces/index.html#subst-2","scope":"ComputerScience/process-calculi/calculus/scope-preservation-for-calculus-names-and-proces","status":"fully_proved","summary":"Instantiating the innermost binder preserves process scoping.","title":"Instantiating the innermost binder preserves process scoping"},{"data_url":"scopes/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers.json","id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","kind":"scope","navigation":"module","parent":39935,"result":false,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","title":"Structural laws under \u03c0-calculus observers"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/context","kind":"node","lean":"Cslib.PiCalculus.Context","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#context","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"One-hole lexical contexts, including guards, choice and replication.","title":"One-hole lexical process contexts"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/fill","kind":"node","lean":"Cslib.PiCalculus.Context.fill","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#fill","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Fill the lexical hole. Free constants are never captured; locally closed hole processes have no references into surrounding binders.","title":"Lexical filling of a process context"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/fill-2","kind":"node","lean":"Cslib.PiCalculus.Cong.fill","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#fill-2","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Structural congruence survives every one-hole lexical context.","title":"Structural congruence is closed under contexts"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/comp","kind":"node","lean":"Cslib.PiCalculus.Context.comp","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#comp","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Substitute a context into the hole of another context.","title":"Composition of process contexts"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/fill-comp","kind":"node","lean":"Cslib.PiCalculus.Context.fill_comp","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#fill-comp","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Context composition agrees with iterated filling.","title":"Filling a composed context is iterated filling"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/contextual-barbed-equiv","kind":"node","lean":"Cslib.PiCalculus.ContextualBarbedEquiv","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#contextual-barbed-equiv","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Strong barbed equivalence in every lexical process context. For capture-avoiding use, both hole processes must satisfy `Proc.ClosedAt 0`.","title":"Contextual strong barbed equivalence"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/contextual-barbed-equiv-2","kind":"node","lean":"Cslib.PiCalculus.Cong.contextualBarbedEquiv","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#contextual-barbed-equiv-2","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Structural congruence is sound under every observer context, with one-for-one reduction matching and immediate input/output barbs.","title":"Structural congruence implies contextual barbed equivalence"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/fill-3","kind":"node","lean":"Cslib.PiCalculus.ContextualBarbedEquiv.fill","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#fill-3","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Contextual barbed equivalence survives filling a context.","title":"Contextual barbed equivalence is closed under contexts"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/barbed-bisimilar","kind":"node","lean":"Cslib.PiCalculus.ContextualBarbedEquiv.barbedBisimilar","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#barbed-bisimilar","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Contextual barbed equivalence implies standalone barbed bisimilarity.","title":"Contextual barbed equivalence implies barbed bisimilarity"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/fill-avoiding","kind":"node","lean":"Cslib.PiCalculus.Context.fillAvoiding","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#fill-avoiding","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Capture-avoiding filling: each context binder shifts dangling names of the hole, while binders internal to the hole remain untouched.","title":"Capture-avoiding filling of a process context"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/fill-avoiding-eq","kind":"node","lean":"Cslib.PiCalculus.Context.fillAvoiding_eq","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#fill-avoiding-eq","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Lexical and capture-avoiding filling agree for locally closed holes. This is the explicit admissibility condition for the contextual theorem.","title":"Capture-avoiding and lexical filling agree on locally closed holes"},{"id":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/fill-avoiding-barbed-bisimilar","kind":"node","lean":"Cslib.PiCalculus.Cong.fillAvoiding_barbedBisimilar","parent":39986,"result":true,"route":"../roadmap/ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers/index.html#fill-avoiding-barbed-bisimilar","scope":"ComputerScience/process-calculi/calculus/structural-laws-under-calculus-observers","status":"fully_proved","summary":"Every structural law is strongly barbed sound under capture-avoiding contexts, for locally closed hole processes.","title":"Structural laws are barbed sound in capture-avoiding contexts"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi.json","id":"ComputerScience/type-systems-and-lambda-calculi","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi","status":"can_prove","summary":"Typed lambda calculi and type inference: the simply typed lambda calculus (syntax, substitution, normalization, logical relations, normalization by evaluation, elaboration, its models), lambda calculus with references and type safety, and Hindley-Milner let-polymorphism with Algorithm W.","title":"Type systems and lambda calculi"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions.json","id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"can_prove","title":"Calculus of constructions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/srt","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.Srt","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#srt","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"The sorts are $\\mathrm{Prop}$ and $\\mathrm{Type}_i$ for every natural number $i$. In the calculus, $\\mathrm{Prop}$ is impredicative and the $\\mathrm{Type}_i$ form a predicative hierarchy.","title":"Sorts of CC\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/tm","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.Tm","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#tm","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"A term is a variable $\\#i$ with de Bruijn index $i \\in \\mathbb{N}$, a sort $s$ (either $\\mathrm{Prop}$ or $\\mathrm{Type}_i$ for some $i \\in \\mathbb{N}$), an application $f\\,a$ of a term $f$ to a term $a$, an abstraction $\\lambda (x : A).\\, b$ with domain $A$ and body $b$, or a dependent product $\\Pi (x : A).\\, B$ with domain $A$ and codomain $B$.","title":"Terms of the calculus of constructions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/lift","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.lift","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#lift","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"For natural numbers $d$ and $c$, shifting a term $t$ by $d$ above the cutoff $c$ adds $d$ to every variable index $i \\ge c$ and leaves every index $i < c$ unchanged. The cutoff goes up by one under each binder, that is, in the body of a $\\lambda$ and in the codomain of a $\\Pi$ (not in the type annotation or the domain), and sorts are unchanged. Thus the free variables of $t$ with index at least $c$ are increased by\u2026","title":"Shifting de Bruijn indices"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/subst","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.subst","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$ and terms $u$ and $t$, the term $t[k := u]$ is defined by recursion on $t$. For a variable, $\\#i[k := u]$ is $\\#i$ if $i < k$, is $u$ with every free variable raised by $k$ if $i = k$, and is $\\#(i - 1)$ if $i > k$. A sort is unchanged, $(f\\,a)[k := u] = f[k := u]\\,a[k := u]$, $(\\lambda (x : A).\\, b)[k := u] = \\lambda (x : A[k := u]).\\, b[k + 1 := u]$, and\u2026","title":"Substitution for a de Bruijn variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/step","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.Step","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#step","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"One-step beta reduction is the smallest relation on terms that contains every beta step $(\\lambda (x : A).\\, b)\\, a \\to b[a/x]$, where $b[a/x]$ substitutes $a$ for the bound variable $x$ in $b$, and that is closed under every term former: a step inside the function or the argument of an application, inside the type annotation or the body of a $\\lambda$, or inside the domain or the codomain of a $\\Pi$ is a step of\u2026","title":"One-step beta reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/conv","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.Conv","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#conv","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"Beta conversion is the equivalence closure of one-step beta reduction: every term is convertible to itself, and if $t$ is convertible to $u$ and $u$ reduces to $v$ in one step, or $v$ reduces to $u$ in one step, then $t$ is convertible to $v$.","title":"Beta conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/ax","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.Ax","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#ax","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"The axioms say which sort has which sort as its type: $\\mathrm{Prop}$ has type $\\mathrm{Type}_0$, and $\\mathrm{Type}_i$ has type $\\mathrm{Type}_{i+1}$ for every natural number $i$. No other pair of sorts is related.","title":"Sort axioms of CC\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/rl","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.Rl","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#rl","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"The product rules say in which sort $s_3$ a dependent product $\\Pi (x : A).\\, B$ lives when $A$ lives in the sort $s_1$ and $B$ lives in the sort $s_2$. There are exactly three rules: for every sort $s$, a product with domain in $s$ and codomain in $\\mathrm{Prop}$ lives in $\\mathrm{Prop}$ (the impredicativity of $\\mathrm{Prop}$); for all natural numbers $i$ and $j$, a product with domain in $\\mathrm{Type}_i$ and\u2026","title":"Product rules of CC\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/wf","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.Wf","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#wf","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"A context is a list of terms whose head is the most recent binding. Well-formedness of contexts is defined together with the typing judgement $\\Gamma \\vdash t : A$ as the least predicate such that the empty context is well formed, and if $\\Gamma$ is well formed and $\\Gamma \\vdash A : s$ for some sort $s$, then the context obtained from $\\Gamma$ by adding $A$ at the head is well formed.","title":"Well-formed contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/typing","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.Typing","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#typing","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"A context $\\Gamma$ is a list of terms whose head is the type of the most recent variable; write $\\Gamma, A$ for $\\Gamma$ extended by $A$ at the head. The judgement $\\Gamma \\vdash t : A$ ($t$ has type $A$ in $\\Gamma$) is defined together with well-formedness of contexts as the least relation closed under the following rules.","title":"Typing judgement of the calculus of constructions with universes"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/sn","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.SN","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#sn","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"fully_proved","summary":"The strongly normalizing terms are defined inductively: a term $t$ is strongly normalizing when every term to which $t$ reduces in one beta step is strongly normalizing. Equivalently, classically, there is no infinite chain of one-step beta reductions starting from $t$.","title":"Strongly normalizing terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/typing-poly-id","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.typing_polyId","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#typing-poly-id","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"can_prove","summary":"In the empty context, the polymorphic identity $\\lambda (A : \\mathrm{Prop}).\\, \\lambda (x : A).\\, x$ has type $\\Pi (A : \\mathrm{Prop}).\\, \\Pi (x : A).\\, A$.","title":"Typing the polymorphic identity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/typing-poly-id-app","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.typing_polyId_app","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#typing-poly-id-app","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"can_prove","summary":"Write $\\bot$ for $\\Pi (P : \\mathrm{Prop}).\\, P$. In the empty context, the application of the polymorphic identity $\\lambda (A : \\mathrm{Prop}).\\, \\lambda (x : A).\\, x$ to $\\bot$ has type $\\Pi (x : \\bot).\\, \\bot$.","title":"Typing the polymorphic identity applied to falsity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/step-poly-id-app","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.step_polyId_app","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#step-poly-id-app","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"can_prove","summary":"The polymorphic identity $\\lambda (A : \\mathrm{Prop}).\\, \\lambda (x : A).\\, x$ applied to $\\bot = \\Pi (P : \\mathrm{Prop}).\\, P$ reduces in one beta step to $\\lambda (x : \\bot).\\, x$.","title":"Beta step of the applied polymorphic identity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/subject-reduction","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.subject_reduction","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#subject-reduction","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"can_prove","summary":"Let $\\Gamma$ be a context and $t$, $t'$, $A$ terms of the calculus of constructions with universes. If $t$ has type $A$ in $\\Gamma$, that is $\\Gamma \\vdash t : A$, and $t$ reduces to $t'$ in one step of beta reduction (anywhere inside $t$), then $\\Gamma \\vdash t' : A$.","title":"Subject reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/strong-normalization","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.strong_normalization","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#strong-normalization","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"can_prove","summary":"Every well-typed term is strongly normalizing: if $\\Gamma \\vdash t : A$ for some context $\\Gamma$ and some term $A$, then $t$ is strongly normalizing.","title":"Strong normalization of CC\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/consistency","kind":"node","lean":"MathlibExt.TheoryOfComputation.CoCStrongNormalizationWanted.ProgramVerification.CoCStrongNormalization.consistency","parent":40000,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions/index.html#consistency","scope":"ComputerScience/type-systems-and-lambda-calculi/calculus-of-constructions","status":"can_prove","summary":"No closed term has type $\\Pi (P : \\mathrm{Prop}).\\, P$: there is no term $t$ with $\\vdash t : \\Pi (P : \\mathrm{Prop}).\\, P$ in the empty context.","title":"Consistency of CC\u03c9"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference.json","id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","title":"Hindley\u2013Milner type inference"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/expr","kind":"node","lean":"Cslib.HindleyMilner.Expr","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#expr","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"An expression is a variable $x$, an application $e_1\\,e_2$, an abstraction $\\lambda x.\\, e$, or a let-binding $\\mathsf{let}\\ x = e_1\\ \\mathsf{in}\\ e_2$, where $e$, $e_1$, $e_2$ are expressions and variable names $x$ are strings.","title":"Expressions of the \u03bb-calculus with let"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/ty","kind":"node","lean":"Cslib.HindleyMilner.Ty","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#ty","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"A monotype over a set $B$ of base types is a term over the constants $B$: a type variable $\\alpha \\in \\mathbb{N}$, a base type $b \\in B$, or an arrow $\\tau_1 \\to \\tau_2$ of two monotypes (a binary node).","title":"Monotypes"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/scheme","kind":"node","lean":"Cslib.HindleyMilner.Scheme","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#scheme","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"A type scheme over base types $B$ is a term over the constant set $B \\oplus \\mathbb{N}$: its variables $\\alpha \\in \\mathbb{N}$ are the free type variables, a constant $\\mathrm{inl}\\, b$ is the base type $b \\in B$, a constant $\\mathrm{inr}\\, i$ is the bound variable $i$, and a node of two schemes is an arrow.","title":"Type schemes"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/to-scheme","kind":"node","lean":"Cslib.HindleyMilner.toScheme","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#to-scheme","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Every monotype $\\tau$ is read as the type scheme with no bound variables of the same shape: a type variable stays the same type variable, a base type $b$ becomes the constant $\\mathrm{inl}\\, b$, and an arrow $\\tau_1 \\to \\tau_2$ becomes the arrow of the images of $\\tau_1$ and $\\tau_2$.","title":"Monotypes as type schemes"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/eval","kind":"node","lean":"Cslib.HindleyMilner.Scheme.eval","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#eval","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Given maps $f$ and $g$ from natural numbers to monotypes over $B$, a type scheme $\\sigma$ over $B$ evaluates to the monotype obtained by replacing each free type variable $x$ of $\\sigma$ by $f(x)$ and each bound variable $i$ by $g(i)$, keeping every base type $b$ unchanged. An arrow $\\sigma_1 \\to \\sigma_2$ evaluates to the arrow from the evaluation of $\\sigma_1$ to the evaluation of $\\sigma_2$.","title":"Evaluation of a type scheme"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/inst","kind":"node","lean":"Cslib.HindleyMilner.Scheme.inst","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#inst","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Given monotypes $(t_i)_{i \\in \\mathbb{N}}$, the instance of a scheme $\\sigma$ at $(t_i)$ is the monotype obtained from $\\sigma$ by replacing each bound variable $i$ by $t_i$, keeping its free type variables and base types unchanged.","title":"Instantiation of a type scheme"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/subst-ty","kind":"node","lean":"Cslib.HindleyMilner.Scheme.substTy","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#subst-ty","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For a type substitution $s$ and a type scheme $\\sigma$, $s\\sigma$ is the scheme obtained from $\\sigma$ by replacing every free type variable $x$ with the monotype $s(x)$, read as a scheme with no bound variables. The bound variables of $\\sigma$ are left unchanged.","title":"Applying a type substitution to a scheme"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/inst-subst-ty","kind":"node","lean":"Cslib.HindleyMilner.inst_substTy","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#inst-subst-ty","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For every type substitution $s$, every scheme $\\sigma$ and all monotypes $(t_i)_{i \\in \\mathbb{N}}$, instantiating each bound variable $i$ of $s(\\sigma)$ by $s(t_i)$ gives $s$ applied to the instance of $\\sigma$ at $(t_i)$. Here $s$ acts on a scheme through its free type variables only.","title":"Instantiation commutes with substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/subst-ty-congr","kind":"node","lean":"Cslib.HindleyMilner.substTy_congr","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#subst-ty-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"If two type substitutions $s$ and $t$ satisfy $s(x) = t(x)$ for every free type variable $x$ of a type scheme $\\sigma$, then applying $s$ to $\\sigma$ gives the same scheme as applying $t$.","title":"Substitution into a scheme depends only on its free variables"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/ctx","kind":"node","lean":"Cslib.HindleyMilner.Ctx","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#ctx","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"A typing context over base types $B$ is a finite list of bindings $(x, \\sigma)$, each pairing a name $x$ (a string) with a type scheme $\\sigma$ over $B$. A name may be bound more than once; the first binding of a name in the list is the visible one.","title":"Typing context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/lookup","kind":"node","lean":"Cslib.HindleyMilner.Ctx.lookup","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#lookup","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"The lookup of a name $x$ in a typing context $\\Gamma$ is the scheme of the first binding of $x$ in $\\Gamma$ (the visible one), or nothing if $\\Gamma$ has no binding of $x$.","title":"Context lookup"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/visible-vars","kind":"node","lean":"Cslib.HindleyMilner.Ctx.visibleVars","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#visible-vars","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For a typing context $\\Gamma$ and a name $x$, this is the set of free type variables of the scheme that $\\Gamma$ binds to $x$ (the first binding of $x$ in $\\Gamma$), or the empty set if $\\Gamma$ does not bind $x$.","title":"Free type variables of a name's visible scheme"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/ftv","kind":"node","lean":"Cslib.HindleyMilner.Ctx.ftv","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#ftv","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"The free type variables of a typing context $\\Gamma$ are the type variables occurring free in the visible scheme of some name bound in $\\Gamma$.","title":"Free type variables of a context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/ftv-congr","kind":"node","lean":"Cslib.HindleyMilner.Ctx.ftv_congr","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#ftv-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"If two typing contexts $\\Gamma$ and $\\Gamma'$ give every name $x$ the same visible scheme (or both bind no scheme to $x$), then $\\Gamma$ and $\\Gamma'$ have the same free type variables.","title":"Free type variables depend only on visible bindings"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/subst","kind":"node","lean":"Cslib.HindleyMilner.Ctx.subst","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For a type substitution $s$ and a typing context $\\Gamma$, $s\\Gamma$ is the context obtained by applying $s$ to the scheme of every binding of $\\Gamma$, keeping the names and their order.","title":"Applying a type substitution to a context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/lookup-subst-congr","kind":"node","lean":"Cslib.HindleyMilner.Ctx.lookup_subst_congr","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#lookup-subst-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"If two type substitutions $s$ and $t$ agree on every free type variable of a context $\\Gamma$, then for every name $x$ the lookups of $x$ in $s(\\Gamma)$ and in $t(\\Gamma)$ are equal: both give no scheme, or both give the same scheme.","title":"Substitutions agreeing on a context give the same lookups"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/gen","kind":"node","lean":"Cslib.HindleyMilner.gen","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#gen","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Given a typing context $\\Gamma$ and a monotype $\\tau$, the generalization of $\\tau$ with respect to $\\Gamma$ is the type scheme obtained from $\\tau$ by turning each type variable $\\alpha$ that is not free in $\\Gamma$ into the bound variable with index $\\alpha$, while type variables free in $\\Gamma$ stay free.","title":"Generalization of a monotype"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/has-type","kind":"node","lean":"Cslib.HindleyMilner.HasType","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#has-type","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"The syntax-directed Damas\u2013Milner typing relation $\\Gamma \\vdash e : \\tau$ between a context $\\Gamma$, an expression $e$ and a monotype $\\tau$ is the least relation closed under the following rules, where $\\Gamma, x : \\sigma$ is $\\Gamma$ extended by the binding $x : \\sigma$ in front.","title":"Hindley\u2013Milner typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/more-general","kind":"node","lean":"Cslib.HindleyMilner.Scheme.MoreGeneral","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#more-general","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"A scheme $\\sigma'$ is more general than a scheme $\\sigma$ when every instance of $\\sigma$ is also an instance of $\\sigma'$ and every free type variable of $\\sigma'$ is a free type variable of $\\sigma$.","title":"More general type scheme"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/more-general-2","kind":"node","lean":"Cslib.HindleyMilner.Ctx.MoreGeneral","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#more-general-2","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"A context $\\Gamma'$ is more general than a context $\\Gamma$ when the same names are visible in both and, for each such name $x$, the visible scheme of $x$ in $\\Gamma'$ is more general than its visible scheme in $\\Gamma$. Only lookups matter, so shadowed bindings play no role.","title":"More general context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/of-lookup-eq","kind":"node","lean":"Cslib.HindleyMilner.Ctx.MoreGeneral.of_lookup_eq","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#of-lookup-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"If two typing contexts $\\Gamma'$ and $\\Gamma$ give the same lookup result for every name $x$ (the same visible scheme, or none in both), then $\\Gamma'$ is more general than $\\Gamma$.","title":"Contexts with the same lookups are related by generality"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/cons","kind":"node","lean":"Cslib.HindleyMilner.Ctx.MoreGeneral.cons","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#cons","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma'$ and $\\Gamma$ be typing contexts such that $\\Gamma'$ is more general than $\\Gamma$, let $x$ be a name, and let $\\sigma'$ and $\\sigma$ be type schemes such that $\\sigma'$ is more general than $\\sigma$. Then the context obtained by putting the binding $x : \\sigma'$ in front of $\\Gamma'$ is more general than the context obtained by putting the binding $x : \\sigma$ in front of $\\Gamma$.","title":"Extending more general contexts by a binding"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/gen-more-general","kind":"node","lean":"Cslib.HindleyMilner.gen_moreGeneral","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#gen-more-general","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma'$ and $\\Gamma$ be typing contexts such that every free type variable of $\\Gamma'$ is a free type variable of $\\Gamma$, and let $\\tau$ be a monotype. Then the generalization of $\\tau$ with respect to $\\Gamma'$ is more general than the generalization of $\\tau$ with respect to $\\Gamma$.","title":"Generalizing in a context with fewer free variables"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/mono","kind":"node","lean":"Cslib.HindleyMilner.HasType.mono","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#mono","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"If $\\Gamma \\vdash e : \\tau$ and $\\Gamma'$ is a context more general than $\\Gamma$, then $\\Gamma' \\vdash e : \\tau$.","title":"Typing is preserved in more general contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/above","kind":"node","lean":"Cslib.HindleyMilner.above","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#above","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For a finite set $S$ of natural numbers, this is the number $\\max S + 1$, where the maximum of the empty set is taken to be $0$. It is strictly greater than every element of $S$.","title":"Strict upper bound of a finite set of naturals"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/gen-subst-more-general","kind":"node","lean":"Cslib.HindleyMilner.gen_subst_moreGeneral","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#gen-subst-more-general","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma$ be a typing context, $\\tau$ a monotype and $s$ a type substitution. Write $s\\Gamma$ for the context obtained by applying $s$ to every scheme of $\\Gamma$, and let $N$ be one more than the largest free type variable of $s\\Gamma$, with $N = 1$ if $s\\Gamma$ has none. Let $\\rho$ be the substitution sending a type variable $y$ to $s(y)$ if $y$ is free in $\\Gamma$, and to the type variable $y + N$ otherwise.\u2026","title":"Renaming step of the substitution lemma"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/subst-2","kind":"node","lean":"Cslib.HindleyMilner.HasType.subst","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#subst-2","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma$ be a typing context, $e$ an expression and $\\tau$ a monotype with $\\Gamma \\vdash e : \\tau$. Then for every type substitution $s$, $s\\Gamma \\vdash e : s\\tau$, where $s\\Gamma$ applies $s$ to every scheme of $\\Gamma$ and $s\\tau$ applies $s$ to $\\tau$.","title":"Hindley\u2013Milner typing is stable under type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/has-type-var-iff","kind":"node","lean":"Cslib.HindleyMilner.hasType_var_iff","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#has-type-var-iff","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma$ be a typing context, $x$ a name and $\\tau$ a monotype. Then $\\Gamma \\vdash x : \\tau$ holds for the variable expression $x$ if and only if there are a type scheme $\\sigma$ and monotypes $(t_i)_{i \\in \\mathbb{N}}$ such that the lookup of $x$ in $\\Gamma$ gives $\\sigma$ and $\\tau$ is the instance of $\\sigma$ at $(t_i)$.","title":"Typing inversion for variables"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/ne-node-self-left","kind":"node","lean":"Cslib.HindleyMilner.ne_node_self_left","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#ne-node-self-left","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For all monotypes $\\tau_1$ and $\\tau_2$, $\\tau_1 \\ne (\\tau_1 \\to \\tau_2)$.","title":"No monotype is an arrow out of itself"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/not-has-type-lam-self-app","kind":"node","lean":"Cslib.HindleyMilner.not_hasType_lam_self_app","parent":40018,"reading_title":{"format":"inline-parts-v1","parts":[["text","The abstraction "],["math","\\(\\lambda f.\\, f\\, f\\)"],["text"," has no type"]],"source":"The abstraction $\\lambda f.\\, f\\, f$ has no type"},"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#not-has-type-lam-self-app","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For every typing context $\\Gamma$, every variable name $f$ and every monotype $\\tau$, the judgment $\\Gamma \\vdash \\lambda f.\\, f\\, f : \\tau$ does not hold.","title":"The abstraction $\\lambda f.\\, f\\, f$ has no type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/has-type-let-id-app-id","kind":"node","lean":"Cslib.HindleyMilner.hasType_let_id_app_id","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#has-type-let-id-app-id","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For every typing context $\\Gamma$, all variable names $f$ and $x$, and every type variable $\\alpha$, the expression $\\mathtt{let}\\ f = \\lambda x.\\, x\\ \\mathtt{in}\\ f\\, f$ has type $\\alpha \\to \\alpha$ in $\\Gamma$.","title":"`let` generalizes"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/let-subst-ty","kind":"node","lean":"Cslib.HindleyMilner.HasType.let_substTy","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#let-subst-ty","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma$ be a typing context, $s$ a type substitution, $x$ a variable name, $e_1, e_2$ expressions and $\\tau_1, \\tau_2$ monotypes, and write $s\\Gamma$ for the context obtained by applying $s$ to every type scheme of $\\Gamma$. Suppose $e_1$ has type $\\tau_1$ in $\\Gamma$, and $e_2$ has type $\\tau_2$ in the context obtained from $s\\Gamma$ by binding $x$, in front, to the type scheme that results from applying $s$\u2026","title":"The let rule under a type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/bound-sup","kind":"node","lean":"Cslib.HindleyMilner.Scheme.boundSup","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#bound-sup","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"The bound-variable supremum of a type scheme $\\sigma$ is one more than the largest index $i$ of a bound variable occurring in $\\sigma$, and $0$ if no bound variable occurs in $\\sigma$.","title":"Bound-variable supremum of a type scheme"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/w","kind":"node","lean":"Cslib.HindleyMilner.W","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#w","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Given a typing context $\\Gamma$, an expression $e$ and a counter $n$, Algorithm W either fails or returns a substitution, a monotype and a new counter; type variables $\\alpha_m$ with $m \\ge n$ serve as fresh. Write $\\tau_1 \\to \\tau_2$ for the arrow type, $s\\Gamma$ for $\\Gamma$ with the substitution $s$ applied to every type scheme, and $s(\\tau)$ for $s$ applied to a monotype $\\tau$. Algorithm W is defined by\u2026","title":"Algorithm W"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/infer","kind":"node","lean":"Cslib.HindleyMilner.infer","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#infer","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Type inference on an expression $e$ runs Algorithm W on $e$ in the empty context with fresh-variable counter $0$. If W succeeds with a substitution $s$, a monotype $\\tau$ and a counter $n'$, the inferred type of $e$ is $\\tau$; if W fails, $e$ has no inferred type.","title":"Type inference for closed expressions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/has-type-of-w-eq-some","kind":"node","lean":"Cslib.HindleyMilner.hasType_of_W_eq_some","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#has-type-of-w-eq-some","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma$ be a typing context, $e$ an expression and $n$ a natural number. If Algorithm W, run on $\\Gamma$ and $e$ with fresh-variable counter $n$, succeeds and returns a substitution $s$, a monotype $\\tau$ and a counter $n'$, then $e$ has type $\\tau$ in the context $s\\Gamma$ obtained by applying $s$ to every type scheme of $\\Gamma$.","title":"Soundness of Algorithm W"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/has-type-of-infer-eq-some","kind":"node","lean":"Cslib.HindleyMilner.hasType_of_infer_eq_some","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#has-type-of-infer-eq-some","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For every expression $e$ and monotype $\\tau$, if type inference for closed expressions returns $\\tau$ on $e$, then $e$ has type $\\tau$ in the empty context.","title":"Soundness of type inference for closed expressions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/vars-below","kind":"node","lean":"Cslib.HindleyMilner.VarsBelow","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#vars-below","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"A term $t$ has all its variables below $n$ when every variable $z$ occurring in $t$ satisfies $z < n$.","title":"Variables below a bound"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/w-fresh","kind":"node","lean":"Cslib.HindleyMilner.W_fresh","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#w-fresh","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma$ be a typing context and $n$ a natural number such that every free type variable of $\\Gamma$ is less than $n$. If Algorithm W succeeds on $\\Gamma$ and an expression $e$ with counter $n$, returning a substitution $s$, a monotype $\\tau$ and a new counter $n'$, then $n \\le n'$, every type variable of $\\tau$ is less than $n'$, and for every type variable $x < n$ every type variable of $s(x)$ is less than\u2026","title":"Freshness invariants of Algorithm W"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/subst-ty-gen-more-general","kind":"node","lean":"Cslib.HindleyMilner.substTy_gen_moreGeneral","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#subst-ty-gen-more-general","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"For a typing context $\\Gamma$, a monotype $\\tau$ and a type substitution $r$, the scheme obtained by applying $r$ to the free type variables of the generalization of $\\tau$ in $\\Gamma$ is more general than the generalization of $r(\\tau)$ in $r(\\Gamma)$: every instance of the latter is an instance of the former, and every free type variable of the former is a free type variable of the latter.","title":"Substituting into a generalization is more general"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/gen-congr","kind":"node","lean":"Cslib.HindleyMilner.gen_congr","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#gen-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"If two typing contexts $\\Gamma$ and $\\Gamma'$ have the same free type variables, then for every monotype $\\tau$ the generalization of $\\tau$ in $\\Gamma$ equals the generalization of $\\tau$ in $\\Gamma'$.","title":"Generalization depends only on the free variables of the context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/w-complete","kind":"node","lean":"Cslib.HindleyMilner.W_complete","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#w-complete","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Let $\\Gamma$ be a typing context and $n$ a natural number such that every free type variable of $\\Gamma$ is less than $n$. If, for some type substitution $u$, an expression $e$ has a monotype $\\tau'$ in the context $u(\\Gamma)$, then Algorithm W succeeds on $\\Gamma$ and $e$ with counter $n$, returning a substitution $s$, a monotype $\\tau$ and a new counter $n'$, and there is a type substitution $r$ such that\u2026","title":"Completeness of Algorithm W"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/infer-complete","kind":"node","lean":"Cslib.HindleyMilner.infer_complete","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#infer-complete","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"If an expression $e$ has a monotype $\\tau'$ in the empty context, then type inference succeeds on $e$ and returns a monotype $\\tau$ of which $\\tau'$ is a substitution instance: $\\tau' = r(\\tau)$ for some type substitution $r$.","title":"Completeness of type inference for closed expressions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/infer-is-some-iff","kind":"node","lean":"Cslib.HindleyMilner.infer_isSome_iff","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#infer-is-some-iff","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"Type inference succeeds on an expression $e$ if and only if $e$ has some monotype in the empty context.","title":"Type inference succeeds exactly on typable closed expressions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/infer-principal","kind":"node","lean":"Cslib.HindleyMilner.infer_principal","parent":40018,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference/index.html#infer-principal","scope":"ComputerScience/type-systems-and-lambda-calculi/hindley-milner-type-inference","status":"fully_proved","summary":"The inferred type of a closed expression is one of its types, and every type of it is an instance of the inferred one.","title":"Principal types"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus.json","id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","title":"Simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/ty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Ty","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#ty","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Given a type $B$ of base types, the simple types over $B$ are generated by two rules: every base type $b \\in B$ is a simple type, and for simple types $A$ and $C$ the function type $A \\Rightarrow C$ is a simple type.","title":"Simple types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/var","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Var","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#var","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For a context $\\Gamma$ (a list of simple types) and a simple type $A$, the variables of type $A$ in $\\Gamma$ are generated by two rules: the variable $0$ is a variable of type $A$ in the context $A :: \\Gamma$ obtained by prepending $A$ to $\\Gamma$, and if $x$ is a variable of type $A$ in $\\Gamma$, then for every simple type $B$ the successor of $x$ is a variable of type $A$ in $B :: \\Gamma$.","title":"Typed de Bruijn variables of the simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#tm","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $A$ and $B$ range over simple types and $\\Gamma$ over contexts (lists of simple types); write $A \\Rightarrow B$ for the function type from $A$ to $B$ and $A :: \\Gamma$ for the context obtained by prepending $A$ to $\\Gamma$. The terms of type $A$ in context $\\Gamma$ are generated by three rules: every variable of type $A$ in $\\Gamma$ is a term of type $A$ in context $\\Gamma$; if $f$ is a term of type\u2026","title":"Intrinsically typed terms of the simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Ren","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#ren","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For contexts $\\Gamma$ and $\\Delta$ (lists of simple types), a renaming from $\\Gamma$ to $\\Delta$ assigns to every simple type $A$ and every variable of type $A$ in $\\Gamma$ a variable of type $A$ in $\\Delta$.","title":"Renamings of simply typed contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#sub","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For contexts $\\Gamma$ and $\\Delta$ (lists of simple types), a substitution from $\\Gamma$ to $\\Delta$ assigns to every simple type $A$ and every variable of type $A$ in $\\Gamma$ a term of type $A$ in context $\\Delta$.","title":"Simultaneous substitutions of simply typed contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/id-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Var.id","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#id-2","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For every context $\\Gamma$, the identity renaming from $\\Gamma$ to $\\Gamma$ sends each variable to itself.","title":"Identity renaming of a simply typed context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/comp-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Var.comp","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#comp-2","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For contexts $\\Gamma$, $\\Delta$ and $\\Xi$, a renaming $\\rho$ from $\\Gamma$ to $\\Delta$ and a renaming $\\sigma$ from $\\Delta$ to $\\Xi$, the composite of $\\rho$ followed by $\\sigma$ is the renaming from $\\Gamma$ to $\\Xi$ that sends each variable $x$ to $\\sigma(\\rho(x))$.","title":"Composition of renamings of simply typed contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/lift-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Var.lift","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#lift-2","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\rho$ be a renaming from a context $\\Gamma$ to a context $\\Delta$ and let $A$ be a simple type, and write $A :: \\Gamma$ for the context obtained by prepending $A$ to $\\Gamma$. The lift of $\\rho$ is the renaming from $A :: \\Gamma$ to $A :: \\Delta$ that sends the variable $0$ to $0$ and sends the successor of a variable $x$ of $\\Gamma$ to the successor of $\\rho(x)$.","title":"Lifting a renaming under a binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/rename-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.rename","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#rename-2","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For a renaming $\\rho$ from a context $\\Gamma$ to a context $\\Delta$, the renaming $t[\\rho]$ of a term $t$ of type $A$ in context $\\Gamma$ is the term of type $A$ in context $\\Delta$ defined by recursion on $t$: $x[\\rho] = \\rho(x)$ for a variable $x$, $(f\\,a)[\\rho] = f[\\rho]\\,a[\\rho]$ for an application, and $(\\lambda s)[\\rho] = \\lambda (s[\\rho^{\\uparrow}])$ for an abstraction, where $\\rho^{\\uparrow}$ is the lift of\u2026","title":"Renaming a simply typed term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.weaken","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For a term $t$ of type $A$ in a context $\\Gamma$ and a simple type $B$, write $B :: \\Gamma$ for the context obtained by prepending $B$ to $\\Gamma$. The weakening of $t$ is the term of type $A$ in the context $B :: \\Gamma$ obtained by renaming $t$ along the renaming that sends each variable of $\\Gamma$ to its successor in $B :: \\Gamma$.","title":"Weakening a simply typed term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.id","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#id","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For a context $\\Gamma$, the identity substitution from $\\Gamma$ to $\\Gamma$ sends each variable of $\\Gamma$ to the term consisting of that variable alone.","title":"Identity substitution (simply typed lambda calculus)"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/of-ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.ofRen","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#of-ren","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For contexts $\\Gamma$ and $\\Delta$ and a type-preserving renaming $\\rho$ from $\\Gamma$ to $\\Delta$, the induced substitution from $\\Gamma$ to $\\Delta$ sends each variable $x$ of $\\Gamma$ to the term consisting of the variable $\\rho(x)$ alone.","title":"Substitution induced by a renaming (simply typed lambda calculus)"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/lift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.lift","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#lift","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Write $A :: \\Gamma$ for the context $\\Gamma$ extended by a newest variable of type $A$. Given a substitution $\\sigma$ from a context $\\Gamma$ to a context $\\Delta$ and a type $A$, its lift is the substitution from $A :: \\Gamma$ to $A :: \\Delta$ that sends the newest variable to the term consisting of the newest variable alone, and sends the successor of each variable $x$ of $\\Gamma$ to the weakening of the term\u2026","title":"Lifting a substitution under a binder (simply typed lambda calculus)"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/single","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.single","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#single","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Write $A :: \\Gamma$ for the context $\\Gamma$ extended by a newest variable of type $A$. Given a term $a$ of type $A$ in context $\\Gamma$, the single-variable substitution for $a$ is the substitution from $A :: \\Gamma$ to $\\Gamma$ that sends the newest variable to $a$ and sends the successor of each variable $x$ of $\\Gamma$ to the term consisting of the variable $x$ alone.","title":"Single-variable substitution (simply typed lambda calculus)"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.subst","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Given a substitution $\\sigma$ from a context $\\Gamma$ to a context $\\Delta$, the result $t[\\sigma]$ of applying $\\sigma$ to a term $t$ of type $A$ in context $\\Gamma$ is the term of type $A$ in context $\\Delta$ defined by recursion on $t$: $x[\\sigma] = \\sigma(x)$ for a variable $x$, $(f\\,a)[\\sigma] = f[\\sigma]\\,a[\\sigma]$ for an application, and $(\\lambda u)[\\sigma] = \\lambda\\,(u[\\sigma'])$ for an abstraction with\u2026","title":"Simultaneous substitution (simply typed lambda calculus)"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.instantiate","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Write $A :: \\Gamma$ for the context $\\Gamma$ extended by a newest variable of type $A$. Given a term $t$ of type $B$ in context $A :: \\Gamma$ and a term $a$ of type $A$ in context $\\Gamma$, the instantiation of $t$ with $a$ is the term of type $B$ in context $\\Gamma$ obtained by applying to $t$ the single-variable substitution for $a$, which replaces the newest variable by $a$ and leaves the older variables\u2026","title":"Instantiating a binder body (simply typed lambda calculus)"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/cons","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.cons","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#cons","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Given a term $a$ of type $A$ in a context $\\Delta$ and a substitution $\\sigma$ from $\\Gamma$ to $\\Delta$, the extension of $\\sigma$ by $a$ is the substitution from the context $\\Gamma$ extended by a newest variable of type $A$ to $\\Delta$ that sends the newest variable to $a$ and each older variable $x$ of $\\Gamma$ to $\\sigma(x)$.","title":"Extension of a simply typed substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.comp","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#comp","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For substitutions $\\sigma$ from a context $\\Gamma$ to $\\Delta$ and $\\tau$ from $\\Delta$ to $\\Xi$, the composite of $\\sigma$ followed by $\\tau$ is the substitution from $\\Gamma$ to $\\Xi$ that sends each variable $x$ of $\\Gamma$ to the term obtained by applying $\\tau$ to $\\sigma(x)$.","title":"Composition of simply typed substitutions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/rename-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.rename_weaken","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#rename-weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\rho$ be a renaming from the typing context $\\Gamma$ to the typing context $\\Delta$, $t$ a term of type $A$ in context $\\Gamma$, and $B$ a type. Weakening $t$ by a new newest variable of type $B$ and then renaming by the lift of $\\rho$ under that variable gives the same term as renaming $t$ by $\\rho$ and then weakening by a new newest variable of type $B$.","title":"Renaming commutes with weakening for intrinsically typed STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/rename-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.rename_subst","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#rename-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from $\\Gamma$ to $\\Delta$, $\\rho$ a renaming from $\\Delta$ to $\\Xi$ and $t$ a term of type $A$ in $\\Gamma$. Renaming by $\\rho$ the result of applying $\\sigma$ to $t$ gives the result of applying to $t$ the substitution that sends each variable $x$ of $\\Gamma$ to $\\sigma(x)$ renamed by $\\rho$.","title":"Renaming after substitution in the simply typed calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/subst-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.subst_rename","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#subst-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\rho$ be a renaming from the typing context $\\Gamma$ to the typing context $\\Delta$, $\\sigma$ a substitution from $\\Delta$ to the typing context $\\Xi$, and $t$ a term of type $A$ in context $\\Gamma$. Renaming $t$ by $\\rho$ and then applying $\\sigma$ gives the same term as applying to $t$ the substitution that sends each variable $x$ of $\\Gamma$ to $\\sigma(\\rho(x))$.","title":"Renaming then substituting an intrinsically typed STLC term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/subst-lift-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.subst_lift_weaken","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#subst-lift-weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from the typing context $\\Gamma$ to the typing context $\\Delta$, $t$ a term of type $A$ in context $\\Gamma$, and $B$ a type. Weakening $t$ by a new newest variable of type $B$ and then applying the lift of $\\sigma$ under that variable gives the same term as applying $\\sigma$ to $t$ and then weakening by a new newest variable of type $B$.","title":"Lifted substitution commutes with weakening for intrinsically typed STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/subst-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.subst_subst","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#subst-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from the typing context $\\Gamma$ to the typing context $\\Delta$, $\\tau$ a substitution from $\\Delta$ to the typing context $\\Xi$, and $t$ a term of type $A$ in context $\\Gamma$. Applying $\\sigma$ to $t$ and then applying $\\tau$ gives the same term as applying to $t$ the composite substitution, which sends each variable $x$ of $\\Gamma$ to the result of applying $\\tau$ to $\\sigma(x)$.","title":"Composition of substitutions for intrinsically typed STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/comp-id-apply","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.comp_id_apply","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#comp-id-apply","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For every simultaneous substitution $\\sigma$ from a context $\\Gamma$ to a context $\\Delta$ and every variable $x$ of type $A$ in $\\Gamma$, the composite of $\\sigma$ followed by the identity substitution on $\\Delta$ sends $x$ to $\\sigma(x)$.","title":"Right identity law for composition of substitutions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.rename","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Given a substitution $\\sigma$ from the typing context $\\Gamma$ to the typing context $\\Delta$ and a renaming $\\rho$ from $\\Delta$ to the typing context $\\Xi$, the substitution from $\\Gamma$ to $\\Xi$ obtained by renaming images sends each variable $x$ of $\\Gamma$ to the renaming of the term $\\sigma(x)$ by $\\rho$.","title":"Renaming the images of a substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/rename-lift-apply","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.rename_lift_apply","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#rename-lift-apply","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from $\\Gamma$ to $\\Delta$ and $\\rho$ a renaming from $\\Delta$ to $\\Xi$. Lift $\\sigma$ under a binder of type $A$ and rename each of its images by the lift of $\\rho$ under the same binder. At every variable $x$ of the context $\\Gamma$ extended by a newest variable of type $A$, this gives the same term as the lift under that binder of the substitution that renames each image of $\\sigma$\u2026","title":"Lifting commutes with renaming substitution images"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/comp-single-apply","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.comp_single_apply","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#comp-single-apply","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $a$ be a term of type $A$ in a context $\\Gamma$ and $\\sigma$ a substitution from $\\Gamma$ to $\\Delta$, and write $a[\\sigma]$ for the result of applying $\\sigma$ to $a$. The composite of the single substitution of $a$ for the newest variable followed by $\\sigma$ agrees, at every variable $x$ of the context $\\Gamma$ extended by a newest variable of type $A$, with the extension of $\\sigma$ that sends the newest\u2026","title":"Composing a single substitution with a substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/lift-single-apply","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sub.lift_single_apply","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#lift-single-apply","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from $\\Gamma$ to $\\Delta$ and $a$ a term of type $A$ in $\\Delta$. The composite of the lift of $\\sigma$ under a binder of type $A$ followed by the single substitution of $a$ for the newest variable agrees, at every variable $x$ of the context $\\Gamma$ extended by a newest variable of type $A$, with the extension of $\\sigma$ that sends the newest variable to $a$ and each older variable\u2026","title":"Lifting followed by single substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/subst-single-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.subst_single_weaken","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#subst-single-weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $t$ be a term of type $A$ in the typing context $\\Gamma$ and $a$ a term of type $B$ in context $\\Gamma$. Weakening $t$ by a new newest variable of type $B$ and then applying the substitution that replaces the newest variable by $a$ and leaves every older variable unchanged gives back $t$.","title":"Single substitution cancels weakening"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/instantiate-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.instantiate_weaken","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#instantiate-weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"For terms $b$ of type $B$ and $a$ of type $A$ in a context $\\Gamma$, weakening $b$ to the context $\\Gamma$ extended by a newest variable of type $A$ and then instantiating that newest variable by $a$ gives back $b$.","title":"Instantiating a weakened term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/rename-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.rename_instantiate","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#rename-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\rho$ be a renaming from $\\Gamma$ to $\\Delta$, $b$ a term of type $B$ in the context $\\Gamma$ extended by a newest variable of type $A$, and $a$ a term of type $A$ in $\\Gamma$. Renaming by $\\rho$ the instantiation of the newest variable of $b$ by $a$ gives the instantiation of the newest variable of $b'$ by $a'$, where $b'$ is $b$ renamed by the lift of $\\rho$ under the binder and $a'$ is $a$ renamed by $\\rho$.","title":"Renaming commutes with instantiation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/subst-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.subst_instantiate","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#subst-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from the typing context $\\Gamma$ to the typing context $\\Delta$, $s$ a term of type $B$ in the context $\\Gamma$ extended by a newest variable of type $A$, and $a$ a term of type $A$ in context $\\Gamma$. Replacing the newest variable of $s$ by $a$ and then applying $\\sigma$ gives the same term as applying the lift of $\\sigma$ to $s$ and then replacing the newest variable of the result\u2026","title":"Substitution commutes with instantiation for intrinsically typed STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/subst-cons","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.subst_cons","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#subst-cons","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from the typing context $\\Gamma$ to the typing context $\\Delta$, $s$ a term of type $B$ in the context $\\Gamma$ extended by a newest variable of type $A$, and $a$ a term of type $A$ in context $\\Delta$. Applying to $s$ the extension of $\\sigma$ that sends the newest variable to $a$ gives the same term as applying the lift of $\\sigma$ to $s$ and then replacing the newest variable of\u2026","title":"Substituting by an extended substitution, via lifting"},{"id":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/instantiate-eq-subst-cons","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.instantiate_eq_subst_cons","parent":40064,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus/index.html#instantiate-eq-subst-cons","scope":"ComputerScience/type-systems-and-lambda-calculi/simply-typed-lambda-calculus","status":"fully_proved","summary":"Let $b$ be a term of type $B$ in the context $\\Gamma$ extended by a newest variable of type $A$, and $a$ a term of type $A$ in $\\Gamma$. Instantiating the newest variable of $b$ by $a$ gives the same term as applying to $b$ the substitution that sends the newest variable to $a$ and each older variable to itself, that is, the extension of the identity substitution on $\\Gamma$ by $a$.","title":"Instantiation is substitution by an extended identity"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f","kind":"scope","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f","status":"fully_proved","title":"System F"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","title":"The existing untyped beta-conversion term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/model","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Model","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#model","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"Locally closed terms in the pinned CSLib untyped calculus.","title":"Untyped locally closed term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/beta-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.BetaEq","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#beta-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"Equivalence closure of the existing full-beta reduction.","title":"Beta equivalence in the untyped term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/app","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Model.app","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#app","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"Application in the existing untyped term model preserves local closure.","title":"Application in the untyped term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/lc","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Model.lc","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#lc","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"The Boolean carrier invariant implies CSLib's cofinite local closure.","title":"Model terms are locally closed"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/atom","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Model.atom","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#atom","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"A symbolic atom is locally closed; it is not a new System F constant.","title":"Symbolic atoms of the untyped term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/refl","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.BetaEq.refl","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#refl","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"Upstream beta equivalence is reflexive.","title":"Reflexivity of beta equivalence"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/symm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.BetaEq.symm","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#symm","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"Upstream beta equivalence is symmetric.","title":"Symmetry of beta equivalence"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/trans","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.BetaEq.trans","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#trans","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"Upstream beta equivalence is transitive.","title":"Transitivity of beta equivalence"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/app-fun","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.BetaEq.appFun","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#app-fun","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"Application preserves beta equivalence in its function position.","title":"Beta equivalence is a congruence in function position"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/app-arg","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.BetaEq.appArg","parent":40100,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model/index.html#app-arg","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-existing-untyped-beta-conversion-term-model","status":"fully_proved","summary":"Application preserves beta equivalence in its argument position.","title":"Beta equivalence is a congruence in argument position"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","status":"fully_proved","title":"Binary beta-invariant relations"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Rel","parent":40111,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/index.html#rel","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","status":"fully_proved","summary":"A binary relation on the term model, invariant under beta conversion.","title":"Beta-invariant binary relations on the term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/ext","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Rel.ext","parent":40111,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/index.html#ext","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","status":"fully_proved","summary":"Admissible relations are determined by their binary predicates.","title":"Extensionality of beta-invariant relations"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/arrow","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Rel.arrow","parent":40111,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/index.html#arrow","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","status":"fully_proved","summary":"Related functions send every related input pair to related outputs.","title":"Function-space relation of beta-invariant relations"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/universal","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Rel.universal","parent":40111,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/index.html#universal","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","status":"fully_proved","summary":"Universal relatedness, over any index sort, uses impredicative Prop.","title":"Universal quantification over a family of relations"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/empty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Rel.empty","parent":40111,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/index.html#empty","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","status":"fully_proved","summary":"The empty relation is admissible and prevents vacuous universality.","title":"Empty beta-invariant relation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/total","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Rel.total","parent":40111,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/index.html#total","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","status":"fully_proved","summary":"The total binary relation is admissible.","title":"Total beta-invariant relation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/beta-class","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Rel.betaClass","parent":40111,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations/index.html#beta-class","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/binary-beta-invariant-relations","status":"fully_proved","summary":"Both arguments belong to the fixed beta-conversion class of `x`.","title":"Relation of two terms beta-equal to a fixed term"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","title":"Scoped pure System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/ty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#ty","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Pure System F types with `n` available type variables.","title":"Scoped types of pure System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/lift-ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.liftRen","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#lift-ren","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Extend a scoped renaming, preserving the newly bound variable.","title":"Lifting a scoped renaming under a binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/lift-ren-zero","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.liftRen_zero","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#lift-ren-zero","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"The new variable is unaffected by a lifted renaming.","title":"Lifted renaming fixes the newly bound variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/lift-ren-succ","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.liftRen_succ","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#lift-ren-succ","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"A lifted renaming shifts every renamed old variable.","title":"Lifted renaming shifts each renamed old variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.rename","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Capture-avoiding renaming of type variables.","title":"Type-variable renaming in System F types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/lift-sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.liftSub","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#lift-sub","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Lift a type substitution under one type binder.","title":"Lifting a type substitution under a type binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.subst","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Simultaneous, capture-avoiding type substitution.","title":"Simultaneous type substitution in System F types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.instantiate","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Discharge the newest type binder by an arbitrary type.","title":"Instantiating the newest type binder of a type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/lift-sub-zero","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.liftSub_zero","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#lift-sub-zero","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"The fresh type variable is fixed by a lifted substitution.","title":"Lifted type substitution fixes the fresh variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/lift-sub-succ","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.liftSub_succ","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#lift-sub-succ","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Old type-substitution images are weakened under a binder.","title":"Lifted type substitution weakens old images"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#tm","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Pure terms with `n` type variables and `m` term variables.","title":"Scoped terms of pure System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/ctx","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ctx","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#ctx","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"An ordered, unrestricted term context over `n` type variables.","title":"Term variable contexts of System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.tySubst","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Substitute type variables in annotations, arguments, and type binders.","title":"Type substitution in System F terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/rename-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.rename","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#rename-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Rename term variables, lifting only at term binders.","title":"Term-variable renaming in System F terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/ty-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.tyRename","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#ty-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Rename type variables without changing term variables.","title":"Type-variable renaming in System F terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/lift-sub-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.liftSub","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#lift-sub-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Lift a simultaneous term substitution beneath a term lambda.","title":"Lifting a term substitution under a term lambda"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/subst-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.subst","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#subst-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Simultaneous term substitution, avoiding both sorts of capture.","title":"Simultaneous term substitution in System F terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/instantiate-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.instantiate","parent":40119,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f/index.html#instantiate-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/scoped-pure-system-f","status":"fully_proved","summary":"Replace the newest term variable and discharge its binder.","title":"Instantiating the newest term binder of a term"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","title":"System F erasure and closing environments"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/shift-at","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.shiftAt","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#shift-at","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Shift free bound indices in CSLib syntax, preserving indices below `c`.","title":"Shift of untyped bound indices above a cutoff"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/lift-raw","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.liftRaw","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#lift-raw","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Lift a raw erasure environment beneath one scoped term binder.","title":"Lifting a raw erasure environment under a binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/shift-at-closed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.shiftAt_closed","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#shift-at-closed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Introducing a binder does not shift an already locally closed term.","title":"Shifting leaves locally closed terms unchanged"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase-with","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.eraseWith","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase-with","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Erase a scoped term under an arbitrary simultaneous untyped environment.","title":"Type erasure under an untyped environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.erase","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Type erasure into the existing untyped de Bruijn syntax.","title":"Type erasure of System F terms to untyped terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase-with-lc-at","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.eraseWith_lcAt","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase-with-lc-at","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Erasure preserves the closure invariant of its environment images.","title":"Erasure with locally closed images is locally closed"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Close every scoped term variable with a locally closed model term.","title":"Closing System F terms in the term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase-model","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.eraseModel","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase-model","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Erase a term with no free term variables into the existing term model.","title":"Erasure of closed System F terms to the term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase-with-open-rec","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.eraseWith_openRec","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase-with-open-rec","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Opening an erased instance opens exactly its raw environment images.","title":"Opening commutes with erasure under an environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase-with-ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.eraseWith_tySubst","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase-with-ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Type substitution changes no part of the erased untyped term.","title":"Untyped erasure is invariant under type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close-var","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close_var","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close-var","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Closing a variable returns exactly its environment image.","title":"Closing a System F variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close-app","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close_app","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close-app","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Closing commutes with the upstream model's application.","title":"Closing a System F application"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close-tabs","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close_tabs","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close-tabs","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"A type lambda contributes no untyped binder.","title":"Closing erases System F type abstraction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close-tapp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close_tapp","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close-tapp","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"A type argument contributes no untyped application.","title":"Closing erases System F type application"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close-ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close_tySubst","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close-ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Closing is independent of every type substitution.","title":"System F closing is invariant under type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close-beta","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close_beta","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close-beta","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"An erased lambda application beta-converts to its extended body instance.","title":"Beta conversion of a closed System F lambda application"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase-with-shift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.eraseWith_shift","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase-with-shift","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Shifting an erased instance shifts exactly its raw environment images.","title":"Shifting commutes with erasure under an environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase-with-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.eraseWith_rename","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase-with-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Erasing a term-variable renaming precomposes the raw environment.","title":"Erasure of a term-variable renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/erase-with-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.eraseWith_subst","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#erase-with-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Actual term substitution erases by composing raw environments.","title":"Erasure of a term substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close_subst","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Closing an actual substitution composes its model-valued environments.","title":"Closing commutes with System F term substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/close-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Tm.close_instantiate","parent":40138,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments/index.html#close-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/system-f-erasure-and-closing-environments","status":"fully_proved","summary":"Actual binder discharge agrees with extending the closing environment.","title":"Closing an instantiated System F term binder"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f","status":"fully_proved","title":"Impredicative binary interpretation of System F types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/interpret","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.interpret","parent":40160,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/index.html#interpret","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f","status":"fully_proved","summary":"The impredicative binary logical relation denoted by a System F type.","title":"Binary logical relation of a System F type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/interpret-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.interpret_rename","parent":40160,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/index.html#interpret-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f","status":"fully_proved","summary":"Interpreting a renamed type precomposes the relation environment.","title":"Relational interpretation of a renamed System F type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/interpret-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.interpret_subst","parent":40160,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/index.html#interpret-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f","status":"fully_proved","summary":"Type substitution is interpreted by substituting arbitrary relations.","title":"Relational interpretation of a substituted System F type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/interpret-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.interpret_instantiate","parent":40160,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/index.html#interpret-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f","status":"fully_proved","summary":"Universal instantiation inserts the argument's whole logical relation.","title":"Relational interpretation of an instantiated System F type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/interpret-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.interpret_weaken","parent":40160,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f/index.html#interpret-weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/impredicative-binary-interpretation-of-system-f","status":"fully_proved","summary":"Weakening a type does not make it depend on the new arbitrary relation.","title":"Relational interpretation of a weakened System F type"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","title":"Type-substitution algebra for impredicative System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/lift-ren-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.liftRen_id","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#lift-ren-id","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Lifting the identity renaming gives the identity.","title":"Lifted identity renaming is the identity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/lift-ren-comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.liftRen_comp","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#lift-ren-comp","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Lifting preserves composition of renamings.","title":"Lifting preserves composition of renamings"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/rename-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.rename_id","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#rename-id","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Type renaming by the identity does not change a type.","title":"Identity renaming fixes every type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/rename-comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.rename_comp","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#rename-comp","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Two type renamings fuse into their composition.","title":"Successive type renamings fuse into their composition"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/subst-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.subst_rename","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#subst-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Substitution after renaming composes the variable environments.","title":"Substitution after a type renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/rename-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.rename_subst","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#rename-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Renaming after substitution renames each substituted image.","title":"Renaming after a type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/lift-sub-var","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.liftSub_var","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#lift-sub-var","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Lifting the variable substitution gives the variable substitution.","title":"Lifted identity type substitution is the identity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/subst-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.subst_id","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#subst-id","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Substitution of each type variable by itself is the identity.","title":"Identity type substitution fixes every type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/subst-var-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.subst_var_rename","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#subst-var-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"A variable-valued substitution is the corresponding type renaming.","title":"Variable-valued substitution is a type renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/subst-lift-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.subst_lift_rename","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#subst-lift-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Lifted type substitution commutes with weakening an old type.","title":"Lifted type substitution commutes with weakening"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/subst-comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.subst_comp","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#subst-comp","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Two simultaneous type substitutions fuse by substituting each image.","title":"Successive type substitutions fuse into one substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/subst-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.subst_instantiate","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#subst-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Type substitution commutes with discharging a type binder.","title":"Type substitution commutes with instantiation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/instantiate-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Ty.instantiate_rename","parent":40166,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst/index.html#instantiate-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/type-substitution-algebra-for-impredicative-syst","status":"fully_proved","summary":"Instantiating a freshly weakened type restores the original type.","title":"Instantiating a weakened type restores it"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","status":"fully_proved","title":"The five typing rules of pure System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/typing","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing","parent":40180,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/index.html#typing","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","status":"fully_proved","summary":"Extrinsic typing of scoped pure System F terms.","title":"Extrinsic typing of pure System F terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing.tySubst","parent":40180,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/index.html#ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","status":"fully_proved","summary":"Type substitution preserves typing, including every context type.","title":"Type substitution preserves typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/ty-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing.tyRename","parent":40180,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/index.html#ty-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","status":"fully_proved","summary":"Type renaming preserves typing without adding term variables.","title":"Type renaming preserves typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing.rename","parent":40180,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","status":"fully_proved","summary":"A context-respecting term renaming preserves the type.","title":"Term renaming preserves typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing.subst","parent":40180,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","status":"fully_proved","summary":"Correctly typed simultaneous term substitution preserves typing.","title":"Term substitution preserves typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing.instantiate","parent":40180,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/index.html#instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","status":"fully_proved","summary":"Discharging a term binder preserves the body type.","title":"Instantiating a term binder preserves typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/ty-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing.tyInstantiate","parent":40180,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f/index.html#ty-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-five-typing-rules-of-pure-system-f","status":"fully_proved","summary":"Type binder discharge substitutes the type and restores the context.","title":"Instantiating a type binder preserves typing"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","title":"Relational parametricity for pure System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/fundamental","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.fundamental","parent":40188,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html#fundamental","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","summary":"The fundamental binary relational-parametricity theorem of pure System F.","title":"Fundamental theorem of relational parametricity for System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/fundamental-closed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.fundamental_closed","parent":40188,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html#fundamental-closed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","summary":"Every closed typed term is related to itself under arbitrary relations.","title":"Closed typed System F terms are related to themselves"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/poly-id-type","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.polyIdType","parent":40188,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html#poly-id-type","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","summary":"The polymorphic identity type, universally quantified at any type depth.","title":"Polymorphic identity type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/poly-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.polyId","parent":40188,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html#poly-id","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","summary":"The pure polymorphic identity term.","title":"Polymorphic identity term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/poly-id-typed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.polyId_typed","parent":40188,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html#poly-id-typed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","summary":"The polymorphic identity is well typed in any existing term context.","title":"Typing of the polymorphic identity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/polymorphic-identity","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.polymorphic_identity","parent":40188,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html#polymorphic-identity","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","summary":"Every closed inhabitant acts as identity after every type instantiation.","title":"Closed inhabitants of the polymorphic identity type are identities"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/polymorphic-related","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.polymorphic_related","parent":40188,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html#polymorphic-related","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","summary":"Type instantiations preserve every admissible binary input relation.","title":"Polymorphic identity inhabitants preserve admissible relations"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/forall-var-uninhabited","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.forall_var_uninhabited","parent":40188,"reading_title":{"format":"inline-parts-v1","parts":[["text","No closed System F term inhabits "],["math","\\(\\forall X, X\\)"]],"source":"No closed System F term inhabits $\\forall X, X$"},"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f/index.html#forall-var-uninhabited","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/relational-parametricity-for-pure-system-f","status":"fully_proved","summary":"The empty relation rules out every closed inhabitant of `forall X, X`.","title":"No closed System F term inhabits $\\forall X, X$"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","title":"Call-by-value safety for pure System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Value","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#value","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"The two value forms of pure System F.","title":"Value forms of pure System F"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Step","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#step","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"One step of weak, left-to-right call-by-value evaluation.","title":"One-step call-by-value evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Steps","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#steps","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"Zero or more call-by-value evaluation steps.","title":"Multi-step call-by-value evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/preservation","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.preservation","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#preservation","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"A call-by-value step preserves the exact type in arbitrary open contexts.","title":"Type preservation for a call-by-value step"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/canonical-lam","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing.canonical_lam","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#canonical-lam","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"A function-typed value is a term lambda with the declared argument type.","title":"Canonical forms of function-typed values"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/canonical-tabs","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.Typing.canonical_tabs","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#canonical-tabs","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"A universally typed value is a type lambda.","title":"Canonical forms of universally typed values"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/progress","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.progress","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#progress","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"A well-typed term with no free term variables is a value or can step.","title":"Progress for well-typed terms without free term variables"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/preservation-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.preservation_steps","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#preservation-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"Every finite evaluation sequence preserves the exact type.","title":"Type preservation along evaluation sequences"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/safety","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.safety","parent":40197,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f/index.html#safety","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/call-by-value-safety-for-pure-system-f","status":"fully_proved","summary":"No term reachable from a closed well-typed term is stuck.","title":"Type safety for closed well-typed terms"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge","kind":"scope","navigation":"module","parent":40099,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge","status":"fully_proved","title":"The call-by-value erasure bridge"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge/erasure-step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.erasure_step","parent":40207,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge/index.html#erasure-step","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge","status":"fully_proved","summary":"Actual CBV steps preserve beta conversion of closing instances.","title":"Erasure of a System F evaluation step is a beta conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge/erasure-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.SystemF.erasure_steps","parent":40207,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge/index.html#erasure-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f/the-call-by-value-erasure-bridge","status":"fully_proved","summary":"Finite CBV evaluation preserves conversion of erased instances.","title":"Erasure of multistep System F evaluation is a beta conversion"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2","kind":"scope","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2","status":"fully_proved","title":"System F\u03c9"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","title":"Intrinsically kinded type operators for pure F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/kind","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Kind","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#kind","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Kinds of pure F\u03c9. `star` classifies types of terms.","title":"Kinds of pure F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/kvar","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.KVar","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#kvar","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"A type variable with its position and kind certified by the telescope.","title":"Kinded type variables of F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/kren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.KRen","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#kren","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Kind-preserving renamings of type variables.","title":"Kind-preserving renamings of type variables"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/lift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.KRen.lift","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#lift","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Lift a kind-preserving renaming beneath a type binder.","title":"Lifting a renaming under a type binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/ty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#ty","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Type operators, intrinsically indexed by context and result kind.","title":"Intrinsically kinded F\u03c9 type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.rename","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Rename free type variables, lifting at both forms of type binder.","title":"Renaming of F\u03c9 type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Sub","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#sub","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Kind-preserving simultaneous substitutions.","title":"Kind-preserving simultaneous type substitutions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/lift-sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.liftSub","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#lift-sub","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Lift substitution beneath a type binder without capturing old images.","title":"Lifting a type substitution under a type binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.subst","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Capture-avoiding simultaneous substitution of kinded type operators.","title":"Simultaneous substitution of F\u03c9 type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/single","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.single","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#single","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Substitution replacing only the newest type variable.","title":"Substitution for the newest type variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.instantiate","parent":40211,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f/index.html#instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/intrinsically-kinded-type-operators-for-pure-f","status":"fully_proved","summary":"Substitute for the newest variable and discharge its binder.","title":"Instantiation of an F\u03c9 type binder"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","title":"Capture-avoiding binding laws for kinded type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/rename-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.rename_congr","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#rename-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Renaming depends only on its action on kinded variables.","title":"Renaming of type operators depends only on variable images"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/rename-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.rename_id","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#rename-id","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"The identity renaming fixes every type operator.","title":"Identity renaming fixes type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/rename-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.rename_rename","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#rename-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Successive kind-preserving renamings fuse, including beneath binders.","title":"Composition of kind-preserving renamings"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/subst-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.subst_congr","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#subst-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Substitution depends only on the images of kinded variables.","title":"Substitution of type operators depends only on variable images"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/subst-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.subst_id","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#subst-id","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Substituting each variable by itself fixes every type operator.","title":"Identity substitution fixes type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/subst-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.subst_rename","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#subst-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Substitution after renaming is precomposition on the variable images.","title":"Substitution after renaming of type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/rename-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.rename_subst","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#rename-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Renaming after substitution renames each substituted type operator.","title":"Renaming after substitution of type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/subst-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.subst_subst","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#subst-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Capture-avoiding simultaneous substitutions compose.","title":"Composition of simultaneous type-operator substitutions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/instantiate-rename-succ","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.instantiate_rename_succ","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#instantiate-rename-succ","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Discharging a fresh binder leaves a weakened operator unchanged.","title":"Instantiating a weakened type operator leaves it unchanged"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/subst-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.subst_instantiate","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#subst-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Simultaneous substitution commutes with discharging a type binder.","title":"Substitution commutes with type-binder instantiation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/rename-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.rename_instantiate","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#rename-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Renaming commutes with discharging a type binder.","title":"Renaming commutes with type-binder instantiation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/subst-var-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.subst_var_rename","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#subst-var-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Renaming is substitution by the corresponding variable constructors.","title":"Renaming as substitution by variables"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/subst-lift-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.subst_lift_rename","parent":40223,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op/index.html#subst-lift-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/capture-avoiding-binding-laws-for-kinded-type-op","status":"fully_proved","summary":"Lifted type substitution commutes with weakening past the new binder.","title":"Lifted substitution commutes with weakening"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","title":"Semantic kind interpretation and substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/sem","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Kind.Sem","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#sem","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"Interpret kinds over a carrier for term types.","title":"Semantic interpretation of kinds"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/kenv","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.KEnv","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#kenv","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"An environment supplies a value of the declared kind for every variable.","title":"Kinded semantic environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/extend","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.KEnv.extend","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#extend","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"Extend a semantic environment at its newest kinded variable.","title":"Extension of a kinded semantic environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/kind-algebra","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.KindAlgebra","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#kind-algebra","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"The two type constructors at `star`; higher kinds need no extra laws.","title":"Kind algebra of arrow and universal type constructors"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/eval","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.eval","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#eval","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"Interpret a kinded operator by ordinary function abstraction/application.","title":"Semantic evaluation of kinded type operators"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/eval-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.eval_congr","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#eval-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"Interpretation depends only on the pointwise kinded environment.","title":"Evaluation depends only on the environment pointwise"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/eval-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.eval_rename","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#eval-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"Semantic renaming is precomposition of the kinded environment.","title":"Evaluation of a renamed type operator"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/eval-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.eval_subst","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#eval-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"Semantic simultaneous substitution at every kind, including higher kinds.","title":"Evaluation of a substituted type operator"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/eval-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.eval_instantiate","parent":40237,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution/index.html#eval-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/semantic-kind-interpretation-and-substitution","status":"fully_proved","summary":"Semantic instantiation is environment extension by the argument value.","title":"Evaluation of an instantiated type operator"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f","status":"fully_proved","title":"Proved type-beta conversion for pure F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/conv","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv","parent":40247,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/index.html#conv","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f","status":"fully_proved","summary":"Type-beta conversion at a fixed context and kind.","title":"Type-beta conversion in F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/sound","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.sound","parent":40247,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/index.html#sound","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f","status":"fully_proved","summary":"Every type-beta conversion is sound in every semantic kind algebra.","title":"Soundness of type-beta conversion in kind algebras"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.rename","parent":40247,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f","status":"fully_proved","summary":"Kind-preserving renaming respects proved type-beta conversion.","title":"Type-beta conversion is stable under renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.subst","parent":40247,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f","status":"fully_proved","summary":"Kind-preserving substitution respects proved type-beta conversion.","title":"Type-beta conversion is stable under substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/subst-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.subst_congr","parent":40247,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/index.html#subst-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f","status":"fully_proved","summary":"Pointwise convertible substitution images give convertible results.","title":"Pointwise convertible substitutions give convertible results"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/not-conv-arr-all","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.not_conv_arr_all","parent":40247,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f/index.html#not-conv-arr-all","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/proved-type-beta-conversion-for-pure-f","status":"fully_proved","summary":"Term-function arrows cannot convert to quantified types, at any kind.","title":"Function types are not convertible to quantified types"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","title":"Type-beta confluence and conversion inversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/par","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Par","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#par","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Parallel type-beta reduction, retaining the context and kind indices.","title":"Parallel type-beta reduction in F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/refl","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Par.refl","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#refl","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Parallel reduction is reflexive on every kinded operator.","title":"Reflexivity of parallel type-beta reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Par.rename","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Parallel beta reduction is stable under kinded renaming.","title":"Parallel type-beta reduction is stable under renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Par.subst","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Parallel reduction respects simultaneous substitution on both sides.","title":"Parallel type-beta reduction is stable under substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Par.instantiate","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Parallel reduction respects discharging a binder and reducing its image.","title":"Parallel type-beta reduction is stable under instantiation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/develop","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.develop","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#develop","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Contract all originally present type-beta redexes recursively.","title":"Complete development of type-beta redexes"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/to-develop","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Par.to_develop","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#to-develop","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Every parallel reduct reduces to the original complete development.","title":"Parallel reducts reduce to the complete development"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/par-diamond","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.par_diamond","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#par-diamond","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Parallel beta reduction has the diamond property at each kind.","title":"Diamond property of parallel type-beta reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/par-confluent","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.par_confluent","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#par-confluent","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Parallel type-beta reduction is confluent; no termination premise.","title":"Confluence of parallel type-beta reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/red","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Red","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#red","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Finitely many parallel beta reductions, including zero reductions.","title":"Multistep parallel type-beta reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/lam","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Red.lam","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#lam","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Multiple parallel reductions are stable under an operator binder.","title":"Multistep type-beta reduction under an operator abstraction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/all","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Red.all","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#all","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Multiple parallel reductions are stable under a universal binder.","title":"Multistep type-beta reduction under a universal quantifier"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/app","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Red.app","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#app","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Multiple reductions are congruent in both type-application arguments.","title":"Multistep type-beta reduction in type applications"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/arr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Red.arr","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#arr","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Multiple reductions are congruent in both term-function type arguments.","title":"Multistep type-beta reduction in function types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/join","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.join","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#join","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Conversion entails a common parallel-beta reduct (Church-Rosser).","title":"Church\u2013Rosser property of type-beta conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/conv","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Par.conv","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#conv","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Every parallel-beta reduction is a proved conversion.","title":"Parallel type-beta reduction implies conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/conv-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Red.conv","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#conv-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Finitely many parallel-beta reductions are a proved conversion.","title":"Multistep type-beta reduction implies conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/conv-iff-join","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.conv_iff_join","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#conv-iff-join","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Conversion is exactly common-reduct joinability, not semantic equality.","title":"Type-beta conversion is joinability by a common reduct"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/arr-inv","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Red.arr_inv","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#arr-inv","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Reduction of a term-function type keeps its outer constructor.","title":"Reducts of function types are function types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/all-inv","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Red.all_inv","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#all-inv","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Reduction of a quantified type keeps its binder kind and outer shape.","title":"Reducts of quantified types are quantified types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/arr-inj","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.arr_inj","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#arr-inj","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Convertible term-function types have convertible domains and codomains.","title":"Injectivity of function types up to conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/all-kind","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.all_kind","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#all-kind","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Conversion cannot change the kind of a universal binder.","title":"Conversion preserves the kind of a universal binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/all-inj","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.all_inj","parent":40254,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion/index.html#all-inj","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/type-beta-confluence-and-conversion-inversion","status":"fully_proved","summary":"Convertible quantified types at the same kind have convertible bodies.","title":"Injectivity of quantified types up to conversion"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","title":"Higher-kinded relational interpretation of F\u03c9 types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/relational-algebra","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.relationalAlgebra","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#relational-algebra","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"The generic kind algebra instantiated with the banked relation substrate.","title":"Kind algebra of beta-invariant relations"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/interpret","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#interpret","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"A kinded type denotes a relation or an appropriately kinded function.","title":"Relational interpretation of F\u03c9 types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/interpret-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret_subst","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#interpret-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"Relational substitution at every kind, with arbitrary higher-kind values.","title":"Relational interpretation of a substituted F\u03c9 type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/interpret-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.Conv.interpret","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#interpret-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"Proved type-beta conversion is sound for the higher-kinded relations.","title":"Type-beta conversion preserves the relational interpretation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/interpret-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret_rename","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#interpret-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"Type renaming transports an arbitrary higher-kinded relation environment.","title":"Relational interpretation of a renamed F\u03c9 type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/interpret-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret_instantiate","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#interpret-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"A kinded type argument is interpreted before its universal is discharged.","title":"Relational interpretation of an instantiated F\u03c9 type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/interpret-all","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret_all","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#interpret-all","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"A universal quantifies every semantic value of its bound kind.","title":"Relational interpretation of an F\u03c9 universal type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/interpret-arr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret_arr","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#interpret-arr","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"An arrow relates every related pair of inputs at its codomain.","title":"Relational interpretation of an F\u03c9 function type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/interpret-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret_weaken","parent":40278,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ/index.html#interpret-weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-interpretation-of-f-typ","status":"fully_proved","summary":"Weakening ignores the arbitrary new semantic value, at every binder kind.","title":"Relational interpretation of a weakened F\u03c9 type"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","title":"Scoped pure F\u03c9 terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm","parent":40288,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html#tm","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","summary":"Pure terms at a kind telescope and a finite term-variable depth.","title":"Scoped pure F\u03c9 terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/ctx","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ctx","parent":40288,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html#ctx","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","summary":"An ordered unrestricted term context of kind-`star` types.","title":"Term contexts of pure F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.tySubst","parent":40288,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html#ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","summary":"Substitute kinded type variables in all annotations and type arguments.","title":"Type substitution in F\u03c9 terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/ty-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.tyRename","parent":40288,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html#ty-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","summary":"Rename type variables without changing term variables.","title":"Type-variable renaming in F\u03c9 terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.rename","parent":40288,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","summary":"Rename term variables, lifting only beneath term lambdas.","title":"Term-variable renaming in F\u03c9 terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/lift-sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.liftSub","parent":40288,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html#lift-sub","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","summary":"Lift a term substitution beneath one term binder.","title":"Lifting a term substitution under a term binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.subst","parent":40288,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","summary":"Term substitution avoids capture in both namespaces.","title":"Simultaneous term substitution in F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.instantiate","parent":40288,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms/index.html#instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/scoped-pure-f-terms","status":"fully_proved","summary":"Discharge the newest term binder.","title":"Instantiation of an F\u03c9 term binder"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped","status":"fully_proved","title":"Erasure adapter to the existing System F scoped syntax"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/erase-syntax","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.eraseSyntax","parent":40297,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/index.html#erase-syntax","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped","status":"fully_proved","summary":"Erase types using the already banked System F scoped term carrier.","title":"Erasure of F\u03c9 terms to System F syntax"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/erase-syntax-ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.eraseSyntax_tySubst","parent":40297,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/index.html#erase-syntax-ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped","status":"fully_proved","summary":"Erasure ignores all kind-correct type substitutions.","title":"F\u03c9 erasure ignores type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/erase-syntax-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.eraseSyntax_rename","parent":40297,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/index.html#erase-syntax-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped","status":"fully_proved","summary":"Erasure commutes with scoped term-variable renaming.","title":"F\u03c9 erasure commutes with term-variable renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/erase-syntax-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.eraseSyntax_subst","parent":40297,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/index.html#erase-syntax-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped","status":"fully_proved","summary":"Erasure commutes with simultaneous substitution of scoped terms.","title":"F\u03c9 erasure commutes with term substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/erase-syntax-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.eraseSyntax_instantiate","parent":40297,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped/index.html#erase-syntax-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/erasure-adapter-to-the-existing-system-f-scoped","status":"fully_proved","summary":"Erasure commutes with discharging a term binder.","title":"F\u03c9 erasure commutes with term-binder instantiation"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","title":"F\u03c9 closing instances in the single existing term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Close scoped F\u03c9 variables in exactly the existing System F term model.","title":"Closing F\u03c9 terms in the System F term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/erase-model","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.eraseModel","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#erase-model","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Erase a closed-term F\u03c9 expression, with an arbitrary type telescope.","title":"Erasure of closed F\u03c9 terms to the term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close-var","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_var","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close-var","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Closing a variable returns its actual model environment image.","title":"Closing an F\u03c9 variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close-app","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_app","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close-app","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Closing preserves the existing model application constructor.","title":"Closing an F\u03c9 application"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close-tabs","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_tabs","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close-tabs","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Type abstraction adds no binder to an erased closing instance.","title":"Closing erases type abstraction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close-tapp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_tapp","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close-tapp","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Type application adds no application to an erased closing instance.","title":"Closing erases type application"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close-ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_tySubst","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close-ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Any kind-correct type substitution leaves closing instances unchanged.","title":"Closing is invariant under type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close-beta","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_beta","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close-beta","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Applying an erased F\u03c9 lambda beta-converts to its extended body instance.","title":"Beta conversion of a closed F\u03c9 lambda application"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_subst","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Closing composes actual F\u03c9 term substitutions with model environments.","title":"Closing commutes with F\u03c9 term substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/close-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_instantiate","parent":40303,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term/index.html#close-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-closing-instances-in-the-single-existing-term","status":"fully_proved","summary":"Actual term-binder discharge extends the same model closing environment.","title":"Closing an instantiated F\u03c9 term binder"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","status":"fully_proved","title":"Pure F\u03c9 typing with proved type-beta conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/typing","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing","parent":40314,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/index.html#typing","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","status":"fully_proved","summary":"The five pure term rules and the type-beta conversion rule of F\u03c9.","title":"Typing rules of pure F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.tySubst","parent":40314,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/index.html#ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","status":"fully_proved","summary":"Kind-correct type substitution preserves typing in arbitrary contexts.","title":"Type substitution preserves F\u03c9 typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/ty-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.tyRename","parent":40314,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/index.html#ty-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","status":"fully_proved","summary":"Kind-preserving type renaming preserves typing.","title":"Type renaming preserves F\u03c9 typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.rename","parent":40314,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","status":"fully_proved","summary":"A context-respecting term-variable renaming preserves the result type.","title":"Term renaming preserves F\u03c9 typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.subst","parent":40314,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","status":"fully_proved","summary":"Arbitrary correctly typed term substitution preserves typing.","title":"Term substitution preserves F\u03c9 typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.instantiate","parent":40314,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/index.html#instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","status":"fully_proved","summary":"Discharging a typed term binder retains the body result type.","title":"Term-binder instantiation preserves F\u03c9 typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/ty-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.tyInstantiate","parent":40314,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion/index.html#ty-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/pure-f-typing-with-proved-type-beta-conversion","status":"fully_proved","summary":"Discharging a type binder substitutes its result and restores context.","title":"Type-binder instantiation preserves F\u03c9 typing"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","title":"Higher-kinded relational parametricity of pure F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/fundamental","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.fundamental","parent":40322,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#fundamental","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"The full binary higher-kinded fundamental theorem of pure F\u03c9.","title":"Fundamental theorem of higher-kinded parametricity for F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/fundamental-closed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.fundamental_closed","parent":40322,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#fundamental-closed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"Every closed-term typed expression is related to itself at every kind environment.","title":"Closed typed F\u03c9 terms are related to themselves"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/higher-id-type","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.higherIdType","parent":40322,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#higher-id-type","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"A genuine higher-kind identity type: `\u2200 F : * \u2192 *, \u2200 X : *, F X \u2192 F X`.","title":"Higher-kinded identity type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/higher-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.higherId","parent":40322,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#higher-id","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"The pure higher-kinded polymorphic identity term.","title":"Higher-kinded polymorphic identity term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/higher-id-typed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.higherId_typed","parent":40322,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#higher-id-typed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"The higher-kinded identity is typed in every existing term context.","title":"Typing of the higher-kinded identity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/higher-kinded-identity","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.higher_kinded_identity","parent":40322,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#higher-kinded-identity","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"Every closed-term inhabitant is beta-identity at every kind-correct instantiation.","title":"Closed inhabitants of the higher-kinded identity type are identities"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/higher-kinded-related","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.higher_kinded_related","parent":40322,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#higher-kinded-related","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"Different higher-kind type instantiations preserve every admissible binary relation.","title":"Higher-kinded identity inhabitants preserve admissible relations"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/forall-var-uninhabited","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.forall_var_uninhabited","parent":40322,"reading_title":{"format":"inline-parts-v1","parts":[["text","No closed F\u03c9 term inhabits "],["math","\\(\\forall X, X\\)"]],"source":"No closed F\u03c9 term inhabits $\\forall X, X$"},"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#forall-var-uninhabited","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"The empty relation excludes all closed-term inhabitants of `\u2200 X : *, X`.","title":"No closed F\u03c9 term inhabits $\\forall X, X$"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/higher-result-uninhabited","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.higher_result_uninhabited","parent":40322,"reading_title":{"format":"inline-parts-v1","parts":[["text","No closed F\u03c9 term inhabits "],["math","\\(\\forall F, \\forall X, F\\,X\\)"]],"source":"No closed F\u03c9 term inhabits $\\forall F, \\forall X, F\\,X$"},"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f/index.html#higher-result-uninhabited","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/higher-kinded-relational-parametricity-of-pure-f","status":"fully_proved","summary":"A constant empty operator excludes `\u2200 F : * \u2192 *, \u2200 X : *, F X`.","title":"No closed F\u03c9 term inhabits $\\forall F, \\forall X, F\\,X$"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","title":"Conversion-aware weak call-by-value safety for pure F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Value","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#value","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"Weak-CBV values: neither form of lambda evaluates beneath its binder.","title":"Weak call-by-value values of F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Step","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#step","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"Ordinary left-to-right weak call-by-value reduction.","title":"Weak call-by-value reduction of F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/lam-inv","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.lam_inv","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#lam-inv","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"Lambda typing inversion retains the annotation and a proved conversion.","title":"Inversion of F\u03c9 lambda typing up to conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/tabs-inv","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.tabs_inv","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#tabs-inv","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"Type-lambda inversion retains its binder kind and body conversion.","title":"Inversion of F\u03c9 type-lambda typing up to conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/preservation","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.preservation","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#preservation","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"Weak-CBV reduction preserves typing in unrestricted open contexts.","title":"Type preservation for F\u03c9 weak call-by-value reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/arrow","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Value.arrow","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#arrow","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"A value typed at an arrow, despite conversion, is a term lambda.","title":"Canonical forms for F\u03c9 values of function type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/all","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Value.all","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#all","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"A value at a quantified type is a type lambda of the same binder kind.","title":"Canonical forms for F\u03c9 values of quantified type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/progress","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.progress","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#progress","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"Closed-term progress, allowing an arbitrary open kinded-type telescope.","title":"Progress for closed F\u03c9 terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/preservation-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.preservation_steps","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#preservation-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"Every finite weak-CBV reduction sequence preserves typing.","title":"Type preservation along multistep F\u03c9 reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/safety","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Typing.safety","parent":40332,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p/index.html#safety","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/conversion-aware-weak-call-by-value-safety-for-p","status":"fully_proved","summary":"No reachable closed well-typed term is stuck, with type variables allowed.","title":"Type safety of F\u03c9 weak call-by-value reduction"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","title":"The star/forall System F fragment inside pure F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/lift-types","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.liftTypes","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#lift-types","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Extend images of System F type variables under a star-kinded binder.","title":"Lifting System F type-variable images under a star binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.fromSystemF","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Translate System F types with arbitrary kind-`star` variable images.","title":"Translation of System F types into F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.fromSystemF_congr","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"The translation depends only on the images of free type variables.","title":"Type translation depends only on variable images"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-target-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.fromSystemF_target_rename","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-target-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Target kind-preserving renaming commutes with the System F translation.","title":"Type translation commutes with target renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-target-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.fromSystemF_target_subst","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-target-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Target type substitution commutes with the System F translation.","title":"Type translation commutes with target substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-source-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.fromSystemF_source_rename","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-source-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Source type renaming is precomposition of the translated variable images.","title":"Type translation of a source renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-source-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.fromSystemF_source_subst","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-source-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Source type substitution translates to simultaneous operator images.","title":"Type translation of a source substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.fromSystemF_instantiate","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"The System F translation commutes with type-binder instantiation.","title":"Type translation commutes with type-binder instantiation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Translate all five System F term constructors, retaining type binders.","title":"Translation of System F terms into F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-typing","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_typing","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-typing","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Every System F typing derivation is preserved by the star translation.","title":"Term translation preserves System F typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/of-fin","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.KVar.ofFin","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#of-fin","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"The corresponding star-kinded variable in an `n`-star telescope.","title":"Star-kinded variable of an all-star telescope"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/embed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.embed","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#embed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"The canonical variable-for-variable embedding of System F types.","title":"Canonical embedding of System F types into F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/embed-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.embed","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#embed-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"The canonical scoped embedding of System F terms into pure F\u03c9.","title":"Canonical embedding of System F terms into F\u03c9"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/embed-typing","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.embed_typing","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#embed-typing","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Canonical embedding preserves every System F typing derivation.","title":"Canonical embedding preserves System F typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-congr-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_congr","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-congr-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Term translation depends only on the images of its free type variables.","title":"Term translation depends only on type-variable images"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-target-ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_target_tySubst","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-target-ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Operator substitution commutes with translating System F terms.","title":"Term translation commutes with target type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-source-ty-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_source_tySubst","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-source-ty-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Source type substitution commutes with the pure-term translation.","title":"Term translation of a source type substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-ty-rename-succ","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_tyRename_succ","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-ty-rename-succ","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Weakening beneath a type binder agrees in both calculus presentations.","title":"Term translation commutes with weakening under a type binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_rename","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Term-variable renaming commutes with the System F translation.","title":"Term translation commutes with term-variable renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_subst","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Simultaneous term substitution commutes with the System F translation.","title":"Term translation commutes with term substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-instantiate-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_instantiate","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-instantiate-2","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Term-binder instantiation commutes with the System F translation.","title":"Term translation commutes with term-binder instantiation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-ty-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_tyInstantiate","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-ty-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Type-binder discharge agrees for translated System F terms.","title":"Term translation commutes with type-binder instantiation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_value","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-value","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"The System F translation preserves both weak-CBV value forms.","title":"Term translation preserves weak call-by-value values"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/from-system-f-step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.fromSystemF_step","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#from-system-f-step","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"Every System F weak-CBV step is simulated by one F\u03c9 weak-CBV step.","title":"Term translation simulates each weak call-by-value step"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/embed-step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.embed_step","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#embed-step","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"The canonical embedding preserves individual System F evaluation steps.","title":"Canonical embedding preserves single evaluation steps"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/embed-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.embed_steps","parent":40343,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f/index.html#embed-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/the-star-forall-system-f-fragment-inside-pure-f","status":"fully_proved","summary":"The canonical embedding preserves finite System F evaluation sequences.","title":"Canonical embedding preserves multistep evaluation"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter","status":"fully_proved","title":"System F embedding agreement with the single term/relation model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/close-from-system-f","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_fromSystemF","parent":40370,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/index.html#close-from-system-f","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter","status":"fully_proved","summary":"Translated System F terms have exactly the same model closing instances.","title":"Translated System F terms have the same closing instances"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/close-embed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.close_embed","parent":40370,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/index.html#close-embed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter","status":"fully_proved","summary":"Canonical embedding preserves arbitrary model closing instances exactly.","title":"Canonical embedding preserves closing instances"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/erase-model-embed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Tm.eraseModel_embed","parent":40370,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/index.html#erase-model-embed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter","status":"fully_proved","summary":"A System F term and its F\u03c9 embedding have identical model erasures.","title":"Canonical embedding preserves model erasure"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/interpret-from-system-f","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret_fromSystemF","parent":40370,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/index.html#interpret-from-system-f","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter","status":"fully_proved","summary":"The star/forall translation preserves the same binary type interpretation.","title":"Type translation preserves the relational interpretation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/interpret-embed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.Ty.interpret_embed","parent":40370,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter/index.html#interpret-embed","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/system-f-embedding-agreement-with-the-single-ter","status":"fully_proved","summary":"Canonical embedding interprets each source variable by its matching kinded lookup.","title":"Relational interpretation of canonically embedded types"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t.json","id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t","kind":"scope","navigation":"module","parent":40210,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t","status":"fully_proved","title":"F\u03c9 weak-CBV erasure into the existing full-beta term model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t/erasure-step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.erasure_step","parent":40376,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t/index.html#erasure-step","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t","status":"fully_proved","summary":"Every actual F\u03c9 CBV step preserves beta conversion of arbitrary closing instances.","title":"Erasure of an F\u03c9 evaluation step is a beta conversion"},{"id":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t/erasure-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.FOmega.erasure_steps","parent":40376,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t/index.html#erasure-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/system-f-2/f-weak-cbv-erasure-into-the-existing-full-beta-t","status":"fully_proved","summary":"Finite F\u03c9 evaluation preserves beta conversion of the closing instances.","title":"Erasure of multistep F\u03c9 evaluation is a beta conversion"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda.json","id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","title":"Operational semantics of the simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/value-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Value","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#value-2","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"A term of type $A$ in a context $\\Gamma$ is a value exactly when it is a lambda abstraction.","title":"Values of STLC"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Step","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#step","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"One-step reduction on terms of type $A$ in a context $\\Gamma$ is the least relation closed under three left-to-right call-by-value rules: if $f$ reduces to $f'$, then the application $f\\,a$ reduces to $f'\\,a$; if $f$ is a value and $a$ reduces to $a'$, then $f\\,a$ reduces to $f\\,a'$; and if $a$ is a value, then the application of the abstraction of $b$ to $a$ reduces to $b[a]$, the result of substituting $a$ for\u2026","title":"Small-step call-by-value reduction in STLC"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/value-irreducible","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Step.value_irreducible","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#value-irreducible","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"If a term $t$ of type $A$ in a context $\\Gamma$ is a value, then there is no term $t'$ to which $t$ reduces in one call-by-value step.","title":"Values are irreducible"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/closed-function-value-is-lam","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Step.closed_function_value_is_lam","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#closed-function-value-is-lam","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"If a term $f$ of function type $A \\Rightarrow B$ in the empty context is a value, then $f$ is a lambda abstraction: there is a term $b$ of type $B$ in the one-variable context $[A]$ such that $f$ is the abstraction of $b$.","title":"Canonical forms for closed function values"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/progress","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Step.progress","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#progress","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"Every term $t$ of type $A$ in the empty context is either a value or reduces in one call-by-value step to some term $t'$.","title":"Progress for closed STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/deterministic-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Step.deterministic","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#deterministic-2","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"For terms $t$, $t_1$, $t_2$ of type $A$ in a context $\\Gamma$, if $t$ reduces in one call-by-value step both to $t_1$ and to $t_2$, then $t_1 = t_2$.","title":"Small-step call-by-value reduction in STLC is deterministic"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/normal-form","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Step.NormalForm","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#normal-form","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"A term $t$ of type $A$ in a context $\\Gamma$ is in normal form when there is no term $t'$ such that $t$ reduces to $t'$ in one call-by-value step.","title":"Normal forms for call-by-value reduction in STLC"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/closed-normal-form-iff-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Step.closed_normalForm_iff_value","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#closed-normal-form-iff-value","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"A term $t$ of type $A$ in the empty context is in normal form, that is, has no call-by-value reduction step, if and only if it is a value.","title":"Closed normal forms are exactly the values"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Steps","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#steps","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"For terms of type $A$ in a context $\\Gamma$, multi-step reduction is the reflexive transitive closure of one-step call-by-value reduction.","title":"Multi-step call-by-value reduction in STLC"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/eval","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#eval","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"Fix a typing context $\\Gamma$. For terms $t$ and $v$ of the same type in context $\\Gamma$, the big-step evaluation relation $t \\Downarrow v$ is the least relation such that every lambda abstraction $\\lambda.\\,b$ evaluates to itself, and an application $f\\,a$ evaluates to $v$ whenever $f \\Downarrow \\lambda.\\,b$, $a \\Downarrow u$ and $b[u] \\Downarrow v$, where $b[u]$ is the body $b$ with its newest free variable\u2026","title":"Big-step call-by-value evaluation in the simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.value","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#value","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"In the call-by-value simply typed lambda calculus, if a term $t$ of type $A$ in context $\\Gamma$ evaluates by big-step evaluation to a term $v$, written $t \\Downarrow v$, then $v$ is a value, that is, a lambda abstraction.","title":"Big-step evaluation produces values"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/value-self","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.value_self","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#value-self","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"Every value $v$, that is, every lambda abstraction, of any type in any typing context evaluates to itself in the big-step semantics: $v \\Downarrow v$.","title":"Values evaluate to themselves"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/to-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.to_steps","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#to-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"Let $t$ and $v$ be terms of the same type in the same typing context. If $t$ evaluates to $v$ in the big-step semantics, written $t \\Downarrow v$, then $t$ reduces to $v$ in finitely many (possibly zero) left-to-right call-by-value steps.","title":"Big-step evaluation implies small-step evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/anti-step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.anti_step","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#anti-step","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"Let $t$, $t'$ and $v$ be terms of the same type $A$ in the same typing context $\\Gamma$. If $t$ reduces to $t'$ in one left-to-right call-by-value step and $t'$ evaluates to $v$ in the big-step semantics, written $t' \\Downarrow v$, then $t \\Downarrow v$.","title":"Big-step evaluation is closed under backward reduction steps"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/anti-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.anti_steps","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#anti-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"Let $t$, $t'$ and $v$ be terms of the same type $A$ in the same typing context $\\Gamma$. If $t$ reduces to $t'$ in finitely many (possibly zero) left-to-right call-by-value steps and $t'$ evaluates to $v$ in the big-step semantics, written $t' \\Downarrow v$, then $t \\Downarrow v$.","title":"Big-step evaluation is closed under backward multi-step reduction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/of-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.of_steps","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#of-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"Let $t$ and $v$ be terms of the same type in the same typing context. If $t$ reduces to $v$ in finitely many (possibly zero) left-to-right call-by-value steps and $v$ is a value, then $t$ evaluates to $v$ in the big-step semantics, written $t \\Downarrow v$.","title":"Small-step evaluation to a value implies big-step evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/iff-steps-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.iff_steps_value","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#iff-steps-value","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"For terms $t$ and $v$ of the same type in the same typing context, $t$ evaluates to $v$ in the big-step semantics, written $t \\Downarrow v$, if and only if $t$ reduces to $v$ in finitely many (possibly zero) left-to-right call-by-value steps and $v$ is a value, that is, a lambda abstraction.","title":"Big-step evaluation is small-step evaluation to a value"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/deterministic","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.deterministic","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#deterministic","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"For every typing context $\\Gamma$ and type $A$, if a term $t$ of type $A$ in context $\\Gamma$ evaluates in the big-step call-by-value semantics both to $v$ and to $w$, written $t \\Downarrow v$ and $t \\Downarrow w$, then $v = w$.","title":"Determinism of big-step evaluation in the simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/exists-closed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.exists_closed","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#exists-closed","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"For every type $A$ and every closed term $t$ of type $A$, that is, a term in the empty typing context, there is a term $v$ such that $t$ evaluates to $v$ in the big-step call-by-value semantics, written $t \\Downarrow v$.","title":"Closed simply typed terms have a big-step evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/terminates-iff-eval","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.terminates_iff_eval","parent":40379,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda/index.html#terminates-iff-eval","scope":"ComputerScience/type-systems-and-lambda-calculi/operational-semantics-of-the-simply-typed-lambda","status":"fully_proved","summary":"For every closed term $t$ of type $A$, that is, a term in the empty typing context, $t$ terminates (it reduces to a value in finitely many left-to-right call-by-value steps) if and only if there is a term $v$ such that $t$ evaluates to $v$ in the big-step semantics, written $t \\Downarrow v$.","title":"Termination in terms of big-step evaluation"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd.json","id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","title":"Denotational semantics of the simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Ty.denote","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Given an interpretation $I$ assigning a type $I(b)$ to each base type $b$, the denotation $[\\![A]\\!]_I$ of a simple type $A$ is defined by recursion on $A$: a base type $b$ denotes $I(b)$, and a function type $A \\Rightarrow B$ denotes the type of all functions from $[\\![A]\\!]_I$ to $[\\![B]\\!]_I$.","title":"Denotation of simple types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/env","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Env","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#env","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Given an interpretation $I$ assigning a type to each base type and a typing context $\\Gamma$ (a list of simple types), an environment for $\\Gamma$ assigns to every variable $x$ of $\\Gamma$, of any simple type $A$, an element of the denotation $[\\![A]\\!]_I$.","title":"Semantic environments for the denotational semantics of STLC"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/cons","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Env.cons","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#cons","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Given an interpretation $I$ of the base types, a value $a$ in the denotation of a type $A$ and an environment $\\eta$ for a context $\\Gamma$, the extended environment for the context $A :: \\Gamma$ sends the newest variable to $a$ and the successor $\\mathrm{succ}\\,x$ of each variable $x$ of $\\Gamma$ to $\\eta(x)$.","title":"Extending a semantic environment in STLC"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Env.rename","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Given a renaming $\\rho$ from context $\\Gamma$ to context $\\Delta$ (a type-preserving map from the variables of $\\Gamma$ to those of $\\Delta$) and an environment $\\eta$ for $\\Delta$, the renamed environment for $\\Gamma$ sends each variable $x$ of $\\Gamma$ to $\\eta(\\rho(x))$; that is, it is the composite $\\eta \\circ \\rho$.","title":"Renaming an environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.denote","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Given an interpretation $I$ of the base types, the denotation of a well-typed term $t$ of type $A$ in context $\\Gamma$ in an environment $\\eta$ for $\\Gamma$ is the element of the denotation of $A$ defined by recursion on $t$: a variable $x$ denotes $\\eta(x)$, an application $f\\,a$ denotes the denotation of $f$ applied to the denotation of $a$ (both in $\\eta$), and an abstraction $\\lambda.\\,b$ denotes the function\u2026","title":"Denotation of an STLC term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.denote_rename","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every type-preserving renaming $\\rho$ from context $\\Gamma$ to context $\\Delta$, every term $t$ in context $\\Gamma$ and every environment $\\eta$ for $\\Delta$, the denotation in $\\eta$ of the term obtained by renaming the variables of $t$ along $\\rho$ equals the denotation of $t$ in the environment that sends each variable $x$ of $\\Gamma$ to $\\eta(\\rho(x))$.","title":"Renaming lemma for STLC denotation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.denote_weaken","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every term $t$ of type $A$ in context $\\Gamma$, every type $B$, every value $a$ in the denotation of $B$ and every environment $\\eta$ for $\\Gamma$, the weakening of $t$ to the context $B :: \\Gamma$ has, in the environment that extends $\\eta$ by $a$ for the newest variable, the same denotation as $t$ in $\\eta$.","title":"Weakening preserves STLC denotation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.denoteSub","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-sub","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Given an interpretation $I$ of the base types, a type-preserving substitution $\\sigma$ mapping each variable of a context $\\Gamma$ to a term in a context $\\Delta$, and an environment $\\eta$ for $\\Delta$, this is the environment for $\\Gamma$ that sends each variable $x$ to the denotation of $\\sigma(x)$ in $\\eta$.","title":"Denotation of an STLC substitution"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.denote_subst","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every type-preserving substitution $\\sigma$ mapping each variable of a context $\\Gamma$ to a term in a context $\\Delta$, every term $t$ in context $\\Gamma$ and every environment $\\eta$ for $\\Delta$, the denotation in $\\eta$ of $t$ with $\\sigma$ applied simultaneously to its variables equals the denotation of $t$ in the environment that sends each variable $x$ of\u2026","title":"Substitution lemma for STLC denotation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.denote_instantiate","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every term $b$ of type $B$ in context $A :: \\Gamma$, every term $a$ of type $A$ in context $\\Gamma$ and every environment $\\eta$ for $\\Gamma$, the denotation in $\\eta$ of $b$ with its newest variable replaced by $a$ equals the denotation of $b$ in the environment that extends $\\eta$ by the denotation of $a$ in $\\eta$ for the newest variable.","title":"Denotation of an instantiated STLC body"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Step.denote_eq","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, if a term $t$ reduces to $t'$ in one step of call-by-value reduction, then $t$ and $t'$ have the same denotation in every environment $\\eta$ for their context.","title":"One reduction step preserves STLC denotation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-eq-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Steps.denote_eq","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-eq-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, if a term $t$ reduces to $t'$ in zero or more steps of call-by-value reduction, then $t$ and $t'$ have the same denotation in every environment $\\eta$ for their context.","title":"Multi-step reduction preserves STLC denotation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/denote-eq-3","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Eval.denote_eq","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#denote-eq-3","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, if a term $t$ evaluates to $v$ under call-by-value big-step evaluation, then $t$ and $v$ have the same denotation in every environment $\\eta$ for their context.","title":"Big-step evaluation preserves STLC denotation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/sem-eq-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#sem-eq-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types as types. Two terms $t$ and $u$ of the same type $A$ in the same context $\\Gamma$ are semantically equal in $I$ when, for every semantic environment $\\eta$ for $\\Gamma$, the denotation of $t$ in $I$ at $\\eta$ equals the denotation of $u$ in $I$ at $\\eta$.","title":"Semantic equality of terms in a model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/refl","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.refl","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#refl","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every term $t$ is semantically equal to itself in $I$: it has the same denotation as itself in every environment.","title":"Semantic equality is reflexive"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/symm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.symm","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#symm","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types and all terms $t$ and $u$ of the same type in the same context, if $t$ and $u$ are semantically equal in $I$, then $u$ and $t$ are semantically equal in $I$.","title":"Symmetry of semantic equality of terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/trans","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.trans","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#trans","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types and all terms $t$, $u$ and $w$ of the same type in the same context, if $t$ and $u$ are semantically equal in $I$ and $u$ and $w$ are semantically equal in $I$, then $t$ and $w$ are semantically equal in $I$.","title":"Transitivity of semantic equality of terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/app","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.app","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#app","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types. If terms $f$ and $g$ of type $A \\Rightarrow B$ in a context $\\Gamma$ are semantically equal in $I$ (they have the same denotation in every environment), and terms $a$ and $b$ of type $A$ in $\\Gamma$ are semantically equal in $I$, then the applications $f\\,a$ and $g\\,b$ are semantically equal in $I$.","title":"Semantic equality is a congruence for application"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/lam","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.lam","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#lam","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types. If two terms of type $B$ in the context $\\Gamma$ extended by a newest variable of type $A$ are semantically equal in $I$ (they have the same denotation in every environment), then their lambda abstractions, terms of type $A \\Rightarrow B$ in $\\Gamma$, are semantically equal in $I$.","title":"Semantic equality is a congruence for lambda abstraction"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/rename-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.rename","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#rename-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types. If terms $t$ and $u$ of type $A$ in a context $\\Gamma$ are semantically equal in $I$ (they have the same denotation in every environment), then for every type-preserving renaming $\\rho$ of the variables of $\\Gamma$ into a context $\\Delta$, the renamed terms $\\rho(t)$ and $\\rho(u)$ are semantically equal in $I$.","title":"Renaming preserves semantic equality"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.subst","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, if two terms $t$ and $u$ of the same type $A$ in a context $\\Gamma$ are semantically equal in $I$, then for every simultaneous substitution $\\sigma$ from $\\Gamma$ to a context $\\Delta$, the result of applying $\\sigma$ to $t$ and the result of applying $\\sigma$ to $u$ are semantically equal in $I$.","title":"Substitution preserves semantic equality"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/of-step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.of_step","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#of-step","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, if a term $t$ reduces to a term $u$ in one call-by-value evaluation step, then $t$ and $u$ are semantically equal in $I$: they have the same denotation in every environment.","title":"Single-step evaluation is a semantic equation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/of-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.of_steps","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#of-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, if a term $t$ reduces to a term $u$ in zero or more call-by-value evaluation steps, then $t$ and $u$ are semantically equal in $I$: they have the same denotation in every environment.","title":"Multi-step evaluation is a semantic equation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/beta-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.beta","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#beta-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every term $t$ of type $B$ in the context $\\Gamma$ extended by a newest variable of type $A$, and every term $a$ of type $A$ in $\\Gamma$, the redex $(\\lambda.\\,t)\\,a$ is semantically equal in $I$ (has the same denotation in every environment) to the instantiation $t[a]$, which replaces the newest free variable of $t$ by $a$.","title":"Beta is sound for the denotational semantics"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/eta-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.eta","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#eta-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types and every term $f$ of type $A \\Rightarrow B$ in a context $\\Gamma$, let $f'$ be the weakening of $f$ to the context $\\Gamma$ extended by a newest variable $x$ of type $A$. Then the term $\\lambda x.\\, f'\\,x$, which binds $x$, is semantically equal to $f$ in $I$ (has the same denotation in every environment).","title":"Eta is sound for the denotational semantics"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/sub-sem-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SubSemEq","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#sub-sem-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types as types. Two simultaneous substitutions $\\sigma$ and $\\tau$ from a context $\\Gamma$ to a context $\\Delta$ are pointwise semantically equal in $I$ when, for every variable $x$ of $\\Gamma$ (of any type) and every semantic environment $\\eta$ for $\\Delta$, the denotation of the term $\\sigma(x)$ in $I$ at $\\eta$ equals the denotation of the term $\\tau(x)$ in $I$ at $\\eta$.","title":"Pointwise semantic equality of substitutions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/refl-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SubSemEq.refl","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#refl-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types and every simultaneous substitution $\\sigma$ that replaces each variable of a context $\\Gamma$ by a term of the same type in a context $\\Delta$, $\\sigma$ is pointwise semantically equal to itself in $I$: for every variable $x$ of $\\Gamma$, the term $\\sigma(x)$ has the same denotation as itself in every environment for $\\Delta$.","title":"Pointwise semantic equality of substitutions is reflexive"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/symm-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SubSemEq.symm","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#symm-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types and all simultaneous substitutions $\\sigma$ and $\\tau$ from a context $\\Gamma$ to a context $\\Delta$, if $\\sigma$ and $\\tau$ are pointwise semantically equal in $I$, then $\\tau$ and $\\sigma$ are pointwise semantically equal in $I$.","title":"Symmetry of pointwise semantic equality of substitutions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/trans-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SubSemEq.trans","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#trans-2","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types and all simultaneous substitutions $\\sigma$, $\\tau$ and $\\upsilon$ from a context $\\Gamma$ to a context $\\Delta$, if $\\sigma$ and $\\tau$ are pointwise semantically equal in $I$ and $\\tau$ and $\\upsilon$ are pointwise semantically equal in $I$, then $\\sigma$ and $\\upsilon$ are pointwise semantically equal in $I$.","title":"Transitivity of pointwise semantic equality of substitutions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/subst-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.subst_congr","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#subst-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every term $t$ of type $A$ in a context $\\Gamma$, and all simultaneous substitutions $\\sigma$ and $\\tau$ from $\\Gamma$ to a context $\\Delta$ that are pointwise semantically equal in $I$, the result of applying $\\sigma$ to $t$ and the result of applying $\\tau$ to $t$ are semantically equal in $I$.","title":"Semantically equal substitutions give semantically equal results"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/equivalence","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEq.equivalence","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#equivalence","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every context $\\Gamma$ and every type $A$, semantic equality in $I$ of terms of type $A$ in $\\Gamma$ (having the same denotation in every environment) is reflexive, symmetric and transitive.","title":"Semantic equality is an equivalence relation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/tm-ctx","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.TmCtx","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#tm-ctx","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Fix simple types over a type of base types. A one-hole context whose hole has type $A$ in variable context $\\Gamma$, and which produces a term of type $B$ in variable context $\\Delta$, is built by four rules. The hole itself is such a context, with $\\Delta = \\Gamma$ and $B = A$. If $C$ produces type $S \\Rightarrow T$ in $\\Delta$ and $a$ is a term of type $S$ in $\\Delta$, then the application $C\\,a$ produces type\u2026","title":"Typed one-hole contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/plug","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.TmCtx.plug","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#plug","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Let $E$ be a typed one-hole context whose hole has type $A$ in variable context $\\Gamma$ and which produces a term of type $B$ in variable context $\\Delta$. For a term $t$ of type $A$ in $\\Gamma$, the plugged term $E[t]$ of type $B$ in $\\Delta$ is defined by recursion on $E$: $[\\,][t] = t$, $(E\\,a)[t] = E[t]\\,a$, $(f\\,E)[t] = f\\,E[t]$ and $(\\lambda.\\,E)[t] = \\lambda.\\,E[t]$, where $[\\,]$ is the bare hole, $E\\,a$\u2026","title":"Plugging a term into an STLC context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.TmCtx.comp","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#comp","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Let $E_2$ be a typed one-hole context whose hole has type $B$ in variable context $\\Delta$ and which produces a term of type $C$ in context $\\Theta$, and let $E_1$ be one whose hole has type $A$ in context $\\Gamma$ and which produces a term of type $B$ in $\\Delta$. The composite $E_2 \\circ E_1$, with hole of type $A$ in $\\Gamma$ and result of type $C$ in $\\Theta$, replaces the hole of $E_2$ by $E_1$; by recursion\u2026","title":"Composition of STLC contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/plug-comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.TmCtx.plug_comp","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#plug-comp","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Let $E_2$ be a typed one-hole context whose hole has type $B$ in variable context $\\Delta$ and which produces a term of type $C$ in context $\\Theta$, and let $E_1$ be one whose hole has type $A$ in context $\\Gamma$ and which produces a term of type $B$ in $\\Delta$. Then for every term $t$ of type $A$ in $\\Gamma$, $(E_2 \\circ E_1)[t] = E_2[E_1[t]]$.","title":"Plugging into a composite STLC context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/comp-assoc","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.TmCtx.comp_assoc","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#comp-assoc","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Let $E_1$ be a typed one-hole context whose hole has type $A$ in variable context $\\Gamma$ and which produces a term of type $B$ in context $\\Delta$, $E_2$ one with hole of type $B$ in $\\Delta$ producing type $C$ in $\\Theta$, and $E_3$ one with hole of type $C$ in $\\Theta$ producing type $D$ in $\\Xi$. Then $(E_3 \\circ E_2) \\circ E_1 = E_3 \\circ (E_2 \\circ E_1)$.","title":"Associativity of composition of STLC contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/sem-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.TmCtx.semEq","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#sem-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types. Let $t$ and $u$ be terms of type $A$ in variable context $\\Gamma$ that denote the same value in $I$ under every semantic environment. Then for every typed one-hole context $C$ whose hole has type $A$ in context $\\Gamma$ and which produces a term of type $B$ in context $\\Delta$, the terms $C[t]$ and $C[u]$ obtained by plugging $t$ and $u$ into the hole also denote the\u2026","title":"Semantic equality is a congruence for typed contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/model-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ModelEq","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#model-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Two terms $t$ and $u$ of type $A$ in variable context $\\Gamma$ are model-equal when, for every interpretation $I$ of the base types by types in the same universe as the type of base types, they are semantically equal in $I$: with function types interpreted as full function spaces, the denotations of $t$ and $u$ agree at every semantic environment for $\\Gamma$.","title":"Model equality of STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/context","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ModelEq.context","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#context","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Let $E$ be a typed one-hole context whose hole has type $A$ in variable context $\\Gamma$ and which produces a term of type $B$ in context $\\Delta$. If two terms $t$ and $u$ of type $A$ in $\\Gamma$ are model-equal, then $E[t]$ and $E[u]$ are model-equal.","title":"Model equality is a congruence for STLC contexts"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/beta","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ModelEq.beta","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#beta","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every term $b$ of type $B$ in the variable context $\\Gamma$ extended by a newest variable of type $A$, and every term $a$ of type $A$ in $\\Gamma$, the application $(\\lambda.\\,b)\\,a$ is model-equal to $b[a]$, the result of substituting $a$ for the newest variable of $b$.","title":"Model-independent beta equality"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/eta","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ModelEq.eta","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#eta","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"For every term $f$ of type $A \\Rightarrow B$ in variable context $\\Gamma$, the term $\\lambda.\\,(f^{\\uparrow}\\,x_0)$ is model-equal to $f$, where $f^{\\uparrow}$ is $f$ weakened by a newest variable of type $A$ and $x_0$ is that variable.","title":"Model-independent eta equality"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/context-beta","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ModelEq.context_beta","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#context-beta","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Let $E$ be a typed one-hole context whose hole has type $B$ in variable context $\\Gamma$ and which produces a term of type $C$ in context $\\Delta$. For every term $b$ of type $B$ in $\\Gamma$ extended by a newest variable of type $A$, and every term $a$ of type $A$ in $\\Gamma$, the term $E[(\\lambda.\\,b)\\,a]$ is model-equal to $E[b[a]]$, where $b[a]$ is the result of substituting $a$ for the newest variable of $b$.","title":"Beta equality inside an STLC context"},{"id":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/context-eta","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ModelEq.context_eta","parent":40400,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd/index.html#context-eta","scope":"ComputerScience/type-systems-and-lambda-calculi/denotational-semantics-of-the-simply-typed-lambd","status":"fully_proved","summary":"Let $E$ be a typed one-hole context whose hole has type $A \\Rightarrow B$ in variable context $\\Gamma$ and which produces a term of type $C$ in context $\\Delta$. For every term $f$ of type $A \\Rightarrow B$ in $\\Gamma$, the term $E[\\lambda.\\,(f^{\\uparrow}\\,x_0)]$ is model-equal to $E[f]$, where $f^{\\uparrow}$ is $f$ weakened by a newest variable of type $A$ and $x_0$ is that variable.","title":"Eta equality inside an STLC context"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation.json","id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","title":"Normalization by evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/sem","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sem","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#sem","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For a context $\\Gamma$, the semantic values of a simple type in $\\Gamma$ are defined by recursion on the type. At a base type $b$ they are the neutral terms of type $b$ in $\\Gamma$ (variables applied to zero or more normal arguments). At an arrow type $A \\Rightarrow B$ a semantic value is a family that, for every context $\\Delta$ and every renaming from $\\Gamma$ to $\\Delta$, maps semantic values of type $A$ in\u2026","title":"Kripke semantic values"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Sem.rename","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Given a renaming $\\rho$ from a context $\\Gamma$ to a context $\\Delta$, a simple type $A$ and a semantic value $v$ of type $A$ in $\\Gamma$, the renamed value is the semantic value of type $A$ in $\\Delta$ defined by cases on $A$. At a base type it is the neutral term $v$ with its free variables renamed by $\\rho$. At an arrow type $A = B \\Rightarrow C$ it sends a renaming $\\sigma$ from $\\Delta$ to a context $\\Theta$\u2026","title":"Renaming a semantic value"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/sem-env","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEnv","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#sem-env","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For contexts $\\Delta$ and $\\Gamma$, a semantic environment for $\\Gamma$ in $\\Delta$ assigns to every variable of $\\Gamma$, of any type $A$, a Kripke semantic value of type $A$ in $\\Delta$.","title":"Semantic environments for normalization by evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/rename-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEnv.rename","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#rename-2","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Given a renaming $\\rho$ from a context $\\Delta$ to a context $\\Theta$ and a semantic environment $\\eta$ for $\\Gamma$ in $\\Delta$, the renamed environment is the semantic environment for $\\Gamma$ in $\\Theta$ that sends each variable $x$ of type $A$ to the semantic value $\\eta(x)$ renamed along $\\rho$ at type $A$.","title":"Renaming a semantic environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/cons","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEnv.cons","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#cons","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Given a semantic value $a$ of type $A$ in a context $\\Delta$ and a semantic environment $\\eta$ for $\\Gamma$ in $\\Delta$, the extended environment is the semantic environment for the context $A :: \\Gamma$ (the context $\\Gamma$ with a new innermost variable of type $A$) in $\\Delta$ that sends the new innermost variable to $a$ and sends the successor of each variable $x$ of $\\Gamma$ to $\\eta(x)$.","title":"Extending a semantic environment by one variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/identity","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEnv.identity","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#identity","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For a context $\\Gamma$, the identity environment is the semantic environment for $\\Gamma$ in $\\Gamma$ that sends each variable $x$ of type $A$ to the reflection at type $A$ of the neutral term $x$. Reflection turns a neutral term into a semantic value of the same type in the same context.","title":"The identity semantic environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/nbe-eval","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.nbeEval","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#nbe-eval","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Let $t$ be a term of type $T$ in context $\\Gamma$ and let $\\eta$ be a semantic environment interpreting every variable of $\\Gamma$ as a semantic value in context $\\Delta$. The evaluation $\\operatorname{ev}(t, \\eta)$ is the semantic value of type $T$ in context $\\Delta$ defined by recursion on $t$: a variable $x$ evaluates to $\\eta(x)$; an application $f\\,a$ evaluates to the semantic function\u2026","title":"Evaluation into the Kripke semantic domain"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-nf","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeNf","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-nf","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For a term $t$ of type $A$ in context $\\Gamma$, its normal form is the beta-normal, eta-long term of type $A$ in context $\\Gamma$ obtained by reifying, at type $A$, the evaluation of $t$ in the identity semantic environment of $\\Gamma$, which interprets each variable as the reflection of that variable.","title":"Normalization by evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeTm","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-tm","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For a term $t$ of type $A$ in context $\\Gamma$, its normalized term is the term of type $A$ in context $\\Gamma$ underlying the beta-normal, eta-long normal form of $t$ computed by normalization by evaluation.","title":"Normalization of terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/reflect-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.reflect_rename","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#reflect-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every type $A$, every renaming $\\rho$ from a context $\\Gamma$ to a context $\\Delta$, and every neutral term $n$ of type $A$ in context $\\Gamma$, transporting the semantic value obtained by reflecting $n$ at type $A$ along $\\rho$ gives the reflection at type $A$ of the neutral term $n$ renamed by $\\rho$.","title":"Reflection commutes with renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/cons-reflect-rename-identity","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SemEnv.cons_reflect_rename_identity","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#cons-reflect-rename-identity","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Let $\\Gamma$ be a context, $A$ a type, and $\\Gamma, A$ the context $\\Gamma$ extended by a newest variable of type $A$. Transport the identity semantic environment of $\\Gamma$ (which reflects each variable as a neutral value) along the weakening renaming from $\\Gamma$ to $\\Gamma, A$ that sends each variable to its successor, and extend the result by the reflection at type $A$ of the newest variable of $\\Gamma, A$.\u2026","title":"The identity environment under a binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/nbe-eval-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.nbeEval_congr","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#nbe-eval-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Let $t$ be a term of type $A$ in context $\\Gamma$, and let $\\xi$ and $\\zeta$ be semantic environments interpreting the variables of $\\Gamma$ in a context $\\Delta$. If $\\xi(x) = \\zeta(x)$ for every variable $x$ of $\\Gamma$, of any type, then evaluating $t$ in $\\xi$ and evaluating $t$ in $\\zeta$ give the same semantic value.","title":"Evaluation respects pointwise equal environments"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/nbe-eval-ne-to-tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.nbeEval_ne_toTm","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#nbe-eval-ne-to-tm","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every neutral term $n$ of type $A$ in context $\\Gamma$, evaluating the underlying term of $n$ in the identity semantic environment of $\\Gamma$ gives the reflection of $n$ at type $A$.","title":"Evaluating a neutral term reflects it"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-nf-to-tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeNf_toTm","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-nf-to-tm","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every beta-normal, eta-long normal form $n$ of type $A$ in context $\\Gamma$, normalizing the underlying term of $n$ by evaluation returns the normal form $n$ itself.","title":"Normal forms are fixed points of normalization"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-tm-normalize-nf","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeTm_normalizeNf","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-tm-normalize-nf","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every term $t$ of type $A$ in context $\\Gamma$, normalizing the underlying term of the normal form that normalization by evaluation computes from $t$ gives the same term as normalizing $t$.","title":"Renormalizing a computed normal form"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-tm-idem","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeTm_idem","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-tm-idem","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every term $t$ of type $A$ in context $\\Gamma$, normalizing the normalized term of $t$ by evaluation gives the normalized term of $t$ again.","title":"Normalization by evaluation is idempotent"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-tm-to-tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeTm_toTm","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-tm-to-tm","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every eta-long normal form $n$ of type $A$ in context $\\Gamma$, normalizing the underlying term of $n$ returns that term unchanged.","title":"Normalization fixes the terms of normal forms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/to-tm-injective","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Nf.toTm_injective","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#to-tm-injective","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every context $\\Gamma$ and type $A$, two eta-long normal forms of type $A$ in context $\\Gamma$ with the same underlying term are equal.","title":"Erasing normal forms is injective"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/is-canonical","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.IsCanonical","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#is-canonical","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"A term $t$ of type $A$ in context $\\Gamma$ is canonical when it is the underlying term of some beta-normal, eta-long normal form of type $A$ in context $\\Gamma$.","title":"Canonical terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/is-canonical-iff-normalize-tm-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.isCanonical_iff_normalizeTm_eq","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#is-canonical-iff-normalize-tm-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"A term $t$ of type $A$ in context $\\Gamma$ is canonical if and only if normalization by evaluation maps $t$ to itself, that is, the term underlying the normal form computed for $t$ is $t$.","title":"Canonical terms are the fixed points of normalization"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/is-canonical-normalize-tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.isCanonical_normalizeTm","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#is-canonical-normalize-tm","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every term $t$ of type $A$ in context $\\Gamma$, the term obtained by normalizing $t$ by evaluation is canonical.","title":"Normalized terms are canonical"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/rename-3","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.IsCanonical.rename","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#rename-3","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"If a term $t$ of type $A$ in context $\\Gamma$ is canonical, then for every renaming $\\rho$ from $\\Gamma$ to a context $\\Delta$, the term obtained by renaming $t$ along $\\rho$ is canonical.","title":"Renaming preserves canonical terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/canonical-tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CanonicalTm","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#canonical-tm","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For a context $\\Gamma$ and a type $A$, a certified canonical term of type $A$ in context $\\Gamma$ is a term $t$ of type $A$ in context $\\Gamma$ together with a proof that $t$ is a fixed point of normalization by evaluation: the term underlying the normal form computed for $t$ is $t$ itself.","title":"Certified canonical terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/to-canonical-tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Nf.toCanonicalTm","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#to-canonical-tm","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"The canonical term of an eta-long normal form $n$ of type $A$ in context $\\Gamma$ is the underlying term of $n$, certified to be a fixed point of normalization.","title":"Normal forms as canonical terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/to-nf","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CanonicalTm.toNf","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#to-nf","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For a canonical term $t$ of type $A$ in context $\\Gamma$, that is, a term that normalization returns unchanged, its normal form is the eta-long normal form that normalization by evaluation computes from $t$.","title":"The normal form of a canonical term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/is-canonical-exists-unique-nf","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.isCanonical_existsUniqueNf","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#is-canonical-exists-unique-nf","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Let $t$ be a term of type $A$ in context $\\Gamma$. If $t$ is canonical, then there is exactly one beta-normal, eta-long normal form of type $A$ in context $\\Gamma$ whose underlying term is $t$.","title":"Canonical terms have a unique normal form"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/nbe-related","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.NbeRelated","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#nbe-related","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types. For a context $\\Gamma$, an environment $\\eta$ for $\\Gamma$ in $I$, a type $A$, a Kripke semantic value $v$ of type $A$ in $\\Gamma$ and an element $d$ of the denotation of $A$ under $I$, the relation \"$v$ realizes $d$ at $\\eta$\" is defined by recursion on $A$. At a base type, $v$ is a neutral term, and $v$ realizes $d$ when the denotation of $v$, read as a term, in $\\eta$\u2026","title":"Kripke realization relation for NbE"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/rename-4","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.NbeRelated.rename","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#rename-4","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types. Let $v$ be a Kripke semantic value of type $A$ in context $\\Gamma$ that realizes an element $d$ of the denotation of $A$ at an environment $\\eta$ for $\\Gamma$. If $\\rho : \\Gamma \\to \\Delta$ is a renaming and $\\theta$ is an environment for $\\Delta$ with $\\theta \\circ \\rho = \\eta$, then the transport of $v$ along $\\rho$ realizes $d$ at $\\theta$.","title":"Realization is stable under renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/nbe-env-related","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.NbeEnvRelated","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#nbe-env-related","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types, contexts $\\Delta$ and $\\Gamma$, an environment $\\theta$ for $\\Delta$ in $I$, a Kripke semantic environment $\\xi$ assigning to each variable of $\\Gamma$ a semantic value in $\\Delta$, and an environment $\\eta$ for $\\Gamma$ in $I$. Then $\\xi$ realizes $\\eta$ at $\\theta$ when for every type $A$ and every variable $x$ of type $A$ in $\\Gamma$, the semantic value $\\xi(x)$\u2026","title":"Realization of environments"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/nbe-eval-related","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.nbeEval_related","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#nbe-eval-related","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types. Let $t$ be a term of type $A$ in context $\\Gamma$, let $\\xi$ be a Kripke semantic environment assigning to each variable of $\\Gamma$ a semantic value in context $\\Delta$, and let $\\eta$ and $\\theta$ be environments in $I$ for $\\Gamma$ and $\\Delta$. If $\\xi$ realizes $\\eta$ at $\\theta$, then the NbE evaluation of $t$ in $\\xi$ realizes the denotation of $t$ in $\\eta$ at\u2026","title":"Fundamental lemma for NbE evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/identity-related","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.identity_related","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#identity-related","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types and an environment $\\eta$ for a context $\\Gamma$ in $I$. Then the identity Kripke semantic environment on $\\Gamma$, which reflects each variable as a neutral value, realizes $\\eta$ at $\\eta$.","title":"The identity semantic environment realizes every environment"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-tm-denote","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeTm_denote","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-tm-denote","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every interpretation $I$ of the base types, every term $t$ of type $A$ in context $\\Gamma$ and every environment $\\eta$ for $\\Gamma$ in $I$, the NbE normal form of $t$ has the same denotation in $\\eta$ as $t$.","title":"NbE preserves denotation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-tm-sem-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeTm_semEq","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-tm-sem-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"For every interpretation $I$ of the base types and every term $t$ of type $A$ in context $\\Gamma$, $t$ and its NbE normal form are semantically equal in $I$: they have the same denotation in every environment for $\\Gamma$ in $I$.","title":"Every term is semantically equal to its NbE normal form in each model"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/normalize-tm-model-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeTm_modelEq","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#normalize-tm-model-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"Every term $t$ of type $A$ in context $\\Gamma$ is model-equal to its NbE normal form.","title":"Every term is model-equal to its NbE normal form"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/model-eq-of-normalize-tm-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.modelEq_of_normalizeTm_eq","parent":40444,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation/index.html#model-eq-of-normalize-tm-eq","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-by-evaluation","status":"fully_proved","summary":"If two terms $t$ and $u$ of type $A$ in context $\\Gamma$ have the same NbE normal form, then $t$ and $u$ are model-equal.","title":"Equal NbE normal forms imply model equality"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu.json","id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","title":"Normalization of the simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/app-fun","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Steps.appFun","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#app-fun","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $f$ and $f'$ be terms of type $A \\Rightarrow B$ and let $a$ be a term of type $A$, all in the same context. If $f$ evaluates to $f'$ in zero or more call-by-value steps, then the application $f\\,a$ evaluates to $f'\\,a$ in zero or more steps.","title":"Multi-step evaluation is a congruence in function position"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/app-arg","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Steps.appArg","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#app-arg","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $f$ be a term of type $A \\Rightarrow B$ and let $a$, $a'$ be terms of type $A$, all in the same context. If $f$ is a value and $a$ evaluates to $a'$ in zero or more call-by-value steps, then the application $f\\,a$ evaluates to $f\\,a'$ in zero or more steps.","title":"Evaluating the argument of a value function is a multi-step congruence"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/suffix-of-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Steps.suffix_of_value","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#suffix-of-value","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$, $t'$ and $v$ be terms of the same type in the same context, with $v$ a value. If $t$ evaluates to $v$ and $t$ evaluates to $t'$, each in finitely many call-by-value steps, then $t'$ evaluates to $v$ in finitely many steps.","title":"Evaluation to a value passes through every reduct"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/terminates","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Terminates","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#terminates","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"A closed term $t$ terminates when it evaluates, in finitely many call-by-value steps, to a value.","title":"Termination of closed terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/of-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Terminates.of_value","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#of-value","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Every closed term of the simply typed lambda calculus that is a value terminates.","title":"Values terminate"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/anti","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Terminates.anti","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#anti","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$ and $t'$ be closed terms of the same type. If $t$ evaluates to $t'$ in zero or more call-by-value steps and $t'$ terminates, then $t$ terminates.","title":"Termination is closed under backward evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/forward","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Terminates.forward","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#forward","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$ and $t'$ be closed terms of the same type. If $t$ terminates and $t$ evaluates to $t'$ in zero or more call-by-value steps, then $t'$ terminates.","title":"Termination is closed under forward evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/reducible","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Reducible","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#reducible","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Reducibility of a closed term $t$ is defined by recursion on its simple type. At a base type, $t$ is reducible when it terminates. At a function type $A \\Rightarrow B$, $t$ is reducible when it terminates and, for every reducible closed term $a$ of type $A$, the application $t\\,a$ is reducible.","title":"Tait reducibility for closed terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/terminates-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Reducible.terminates","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#terminates-2","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Every reducible closed term of the simply typed lambda calculus terminates.","title":"Reducible terms terminate"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/forward-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Reducible.forward","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#forward-2","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$ and $t'$ be closed terms of the same type. If $t$ is reducible and $t$ evaluates to $t'$ in zero or more call-by-value steps, then $t'$ is reducible.","title":"Reducibility is closed under forward evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/anti-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Reducible.anti","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#anti-2","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$ and $t'$ be closed terms of the same type. If $t$ evaluates to $t'$ in zero or more call-by-value steps and $t'$ is reducible, then $t$ is reducible.","title":"Reducibility is closed under backward evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/iff-of-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Reducible.iff_of_steps","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#iff-of-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$ and $t'$ be closed terms of the same type. If $t$ evaluates to $t'$ in zero or more call-by-value steps, then $t$ is reducible if and only if $t'$ is reducible.","title":"Reducibility is invariant under evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/reducible-sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ReducibleSub","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#reducible-sub","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"A substitution $\\sigma$ that replaces each variable of a context $\\Gamma$ by a closed term of the same type is reducible when $\\sigma(x)$ is reducible for every variable $x$ of $\\Gamma$.","title":"Reducible substitutions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/empty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ReducibleSub.empty","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#empty","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Every substitution from the empty context to closed terms is reducible, since the empty context has no variables.","title":"Substitutions on the empty context are reducible"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/cons","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.ReducibleSub.cons","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#cons","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $\\sigma$ be a substitution that sends each variable of a context $\\Gamma$ to a closed term of the same type, and let $a$ be a closed term of type $A$. If $\\sigma$ and $a$ are reducible, then so is the substitution on the context $A, \\Gamma$ that sends the newest variable to $a$ and each older variable $x$ to $\\sigma(x)$.","title":"Extending a reducible substitution by a reducible term keeps it reducible"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/fundamental","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.fundamental","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#fundamental","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$ be a term of type $A$ in a context $\\Gamma$, and let $\\sigma$ be a reducible substitution sending each variable of $\\Gamma$ to a closed term of the same type. Then the closed term obtained by applying $\\sigma$ to $t$ is reducible.","title":"Fundamental theorem of the reducibility relation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/reducible-closed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.reducible_closed","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#reducible-closed","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Every closed well-typed term of the simply typed lambda calculus is reducible.","title":"Closed terms are reducible"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/terminates-closed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.terminates_closed","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#terminates-closed","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Every closed well-typed term evaluates, in finitely many call-by-value steps, to a value.","title":"Termination of the call-by-value simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/eq-of-value-source","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Steps.eq_of_value_source","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#eq-of-value-source","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $v$ and $t$ be terms of the same type in the same context. If $v$ is a value and $v$ reduces to $t$ in zero or more call-by-value steps, then $v = t$.","title":"Values reduce only to themselves"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/value-unique","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Steps.value_unique","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#value-unique","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$ be a term in any context. If $t$ reduces in zero or more call-by-value steps to a value $v$ and also to a value $w$, then $v = w$.","title":"Uniqueness of the value reached by evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/normalizes","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Normalizes","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#normalizes","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"A closed term $t$ of the simply typed lambda calculus normalizes when it reduces, in zero or more call-by-value steps, to a term in normal form, that is, a term with no further call-by-value reduction step.","title":"Normalizing closed terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/iff-terminates","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Normalizes.iff_terminates","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#iff-terminates","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"A closed well-typed term $t$ normalizes, that is, reduces in zero or more call-by-value steps to a normal form (a term with no further reduction step), if and only if it terminates, that is, reduces in zero or more call-by-value steps to a value.","title":"Normalization and termination coincide"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/normalizes-closed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizes_closed","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#normalizes-closed","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Every closed well-typed term normalizes: it reduces in zero or more call-by-value steps to a normal form.","title":"Normalization of closed terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/normal-form-unique","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalForm_unique","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#normal-form-unique","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"If a closed term $t$ reduces in zero or more call-by-value steps to a normal form $n$ and also to a normal form $m$, then $n = m$.","title":"Uniqueness of normal forms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/closed-function-evaluates-to-lam","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.closed_function_evaluates_to_lam","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#closed-function-evaluates-to-lam","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"For every closed term $f$ of a function type $A \\Rightarrow B$ there is a term $b$ of type $B$ in the context with a single variable of type $A$ such that $f$ reduces, in zero or more call-by-value steps, to the lambda abstraction with body $b$.","title":"Closed functions evaluate to lambda abstractions"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/no-closed-base","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.no_closed_base","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#no-closed-base","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"In the pure, constant-free calculus there is no closed term whose type is a base type.","title":"No closed terms of base type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/same-result","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SameResult","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#same-result","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Two closed terms $t$ and $u$ of the same type have the same result when there is a value $v$ such that both $t$ and $u$ reduce to $v$ in zero or more call-by-value steps.","title":"Closed terms with the same result"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/refl","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SameResult.refl","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#refl","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Every closed term has the same result as itself.","title":"Reflexivity of having the same result"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/symm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SameResult.symm","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#symm","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"If closed terms $t$ and $u$ have the same result, then $u$ and $t$ have the same result.","title":"Symmetry of having the same result"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/trans","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SameResult.trans","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#trans","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"If closed terms $t$ and $u$ have the same result, and $u$ and a closed term $v$ have the same result, then $t$ and $v$ have the same result.","title":"Transitivity of having the same result"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/of-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SameResult.of_steps","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#of-steps","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"If a closed term $t$ reduces to a term $u$ in zero or more call-by-value steps, then $t$ and $u$ have the same result.","title":"Evaluation preserves the result of a closed term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/anti-left","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SameResult.anti_left","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#anti-left","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$, $t'$ and $u$ be closed terms of the same type. If $t$ reduces to $t'$ in zero or more call-by-value steps, and $t'$ and $u$ have the same result, then $t$ and $u$ have the same result.","title":"Left expansion preserves having the same result"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/anti-right","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SameResult.anti_right","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#anti-right","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $t$, $u$ and $u'$ be closed terms of the same type. If $u$ reduces to $u'$ in zero or more call-by-value steps, and $t$ and $u'$ have the same result, then $t$ and $u$ have the same result.","title":"Right expansion preserves having the same result"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/app","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.SameResult.app","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#app","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"Let $f$ and $g$ be closed terms of a function type $A \\Rightarrow B$, and let $a$ and $b$ be closed terms of type $A$. If $f$ and $g$ have the same result and $a$ and $b$ have the same result, then the applications $f\\,a$ and $g\\,b$ have the same result.","title":"Application preserves having the same result"},{"id":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/same-result-equivalence","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.sameResult_equivalence","parent":40480,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu/index.html#same-result-equivalence","scope":"ComputerScience/type-systems-and-lambda-calculi/normalization-of-the-simply-typed-lambda-calculu","status":"fully_proved","summary":"For every simple type $A$, the relation of having the same result is an equivalence relation on the closed terms of type $A$.","title":"Having the same result is an equivalence relation"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul.json","id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","title":"Type checking for the simply typed lambda calculus"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/raw-tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#raw-tm","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Raw terms are extrinsic de Bruijn syntax over a given type of base types: a raw term is either a variable given by a natural-number index $n$, an application $f\\,a$ of two raw terms $f$ and $a$, or a lambda abstraction $\\lambda A.\\,t$ consisting of a simple type $A$ over the base types (the parameter type) and a raw term $t$ (the body). Variables and applications carry no type information.","title":"Raw lambda terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/size","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.size","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#size","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Writing $|t|$ for the size of a raw term $t$, the size counts the syntax nodes of $t$: a variable has size $1$, an application $f\\,a$ has size $|f| + |a| + 1$, and a lambda abstraction $\\lambda A.\\,t$ has size $|t| + 1$, so the type annotation $A$ is not counted.","title":"Size of a raw term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/shift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.shift","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#shift","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For natural numbers $k$ (the amount) and $c$ (the cutoff, $0$ by default), shifting a raw term by $k$ at cutoff $c$ replaces a variable index $n$ by $n + k$ if $c \\le n$ and leaves it unchanged otherwise, shifts both sides of an application by $k$ at cutoff $c$, and sends $\\lambda A.\\,t$ to $\\lambda A.\\,t'$, where $t'$ is $t$ shifted by $k$ at cutoff $c + 1$.","title":"Shifting raw terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/lookup-var","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.lookupVar","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#lookup-var","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Given a context $\\Gamma$, a list of types, and a natural number $n$, looking up the de Bruijn index $n$ in $\\Gamma$ either fails or returns a type $A$ together with an intrinsic variable of type $A$ in $\\Gamma$. It is defined by recursion: lookup fails in the empty context; index $0$ in the context $A :: \\Gamma$, obtained by adding $A$ at the front of $\\Gamma$, returns $A$ with the zero variable; and index $n + 1$\u2026","title":"Variable lookup"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/to-nat","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Var.toNat","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#to-nat","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For a context $\\Gamma$ and a type $A$, a well-typed de Bruijn variable of type $A$ in $\\Gamma$ is sent to its natural-number index, forgetting its type: the variable for the first entry of the context has index $0$, and the successor of a variable $x$ has the index of $x$ plus $1$.","title":"Index of a typed variable"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/lookup-var-to-nat","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.lookupVar_toNat","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#lookup-var-to-nat","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For every context $\\Gamma$, type $A$ and intrinsic variable $x$ of type $A$ in $\\Gamma$, looking up the de Bruijn index of $x$ in $\\Gamma$ succeeds and returns the type $A$ together with $x$.","title":"Lookup recovers an intrinsic variable from its index"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/lookup-var-result-to-nat","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.lookupVar_result_toNat","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#lookup-var-result-to-nat","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For every context $\\Gamma$ and natural number $n$, if looking up the de Bruijn index $n$ in $\\Gamma$ succeeds with a type $A$ and an intrinsic variable $x$ of type $A$ in $\\Gamma$, then the de Bruijn index of $x$ is $n$.","title":"Lookup returns a variable with the queried index"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/erase","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.erase","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#erase","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"The erasure of an intrinsically typed term in a context $\\Gamma$ is the raw term defined by recursion: a variable becomes the raw variable whose index is its position in $\\Gamma$ (counting from $0$ at the front); an application of $f$ to $a$ becomes the raw application of the erasure of $f$ to the erasure of $a$; and a lambda with parameter type $B$ becomes the raw lambda annotated with $B$ whose body is the\u2026","title":"Erasure of intrinsically typed terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/checked","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Checked","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#checked","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For a context $\\Gamma$ of simple types, a checked term in $\\Gamma$ is a pair $(A, t)$ of a simple type $A$ and an intrinsically typed term $t$ of type $A$ in $\\Gamma$.","title":"Checked terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/elaborate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.elaborate","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#elaborate","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. Elaboration in a context $\\Gamma$ of simple types sends a raw term either to failure or to a checked term in $\\Gamma$ (a simple type together with an intrinsically typed term of that type in $\\Gamma$), by recursion on the raw term. A variable with de Bruijn index $n$ elaborates to the variable at position $n$ of $\\Gamma$ (counting from $0$ at the front), with the type\u2026","title":"Elaboration of raw STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/elaborate-erase","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.elaborate_erase","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#elaborate-erase","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. For every intrinsically typed term $t$ of type $A$ in context $\\Gamma$, elaborating the erasure of $t$ in $\\Gamma$ succeeds and returns exactly the pair $(A, t)$. In other words, erasure is a section of elaboration.","title":"Elaboration inverts erasure"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/elaborate-faithful","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.elaborate_faithful","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#elaborate-faithful","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. If elaborating a raw term $r$ in context $\\Gamma$ succeeds with type $A$ and intrinsically typed term $t$, then the erasure of $t$ is exactly $r$.","title":"Elaboration preserves the input syntax"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/erase-injective","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.erase_injective","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#erase-injective","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For a fixed context $\\Gamma$ and type $A$, erasure is injective on intrinsically typed terms of type $A$ in $\\Gamma$: two such terms with the same erasure are equal.","title":"Erasure is injective"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/has-type","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.HasType","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#has-type","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For a context $\\Gamma$ (a list of simple types), a raw term $t$ and a type $A$, the judgment $\\Gamma \\vdash t : A$ is the least relation closed under three rules. A variable with de Bruijn index $n$ has type $A$ in $\\Gamma$ whenever looking up $n$ in $\\Gamma$ succeeds with the type $A$ and some intrinsic variable of type $A$. If $\\Gamma \\vdash f : A \\Rightarrow B$ and $\\Gamma \\vdash a : A$, then the application\u2026","title":"Declarative typing of raw terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/erase-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.HasType.erase","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#erase-2","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For every intrinsically typed term $t$ of type $A$ in context $\\Gamma$, the erasure of $t$ has type $A$ in $\\Gamma$ under the declarative typing judgment for raw terms.","title":"Erasure preserves typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/unique","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.HasType.unique","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#unique","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Let $\\Gamma$ be a context of simple types and $r$ a raw term. If $r$ has type $A$ in $\\Gamma$ and $r$ has type $B$ in $\\Gamma$ under the declarative typing judgment for raw terms, then $A = B$.","title":"Uniqueness of types for raw terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/elaborate-complete","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.HasType.elaborate_complete","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#elaborate-complete","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. If a raw term $r$ has type $A$ in context $\\Gamma$ under the declarative typing judgment, then elaborating $r$ in $\\Gamma$ succeeds and returns the type $A$ together with some intrinsically typed term of type $A$ in $\\Gamma$.","title":"Completeness of elaboration"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/elaborate-sound","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.HasType.elaborate_sound","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#elaborate-sound","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. If elaborating a raw term $r$ in context $\\Gamma$ succeeds with type $A$ (and some intrinsically typed term of type $A$ in $\\Gamma$), then $r$ has type $A$ in $\\Gamma$ under the declarative typing judgment.","title":"Soundness of elaboration"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/iff-elaborates","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.HasType.iff_elaborates","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#iff-elaborates","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. For every context $\\Gamma$, raw term $t$ and type $A$, the declarative typing judgment $\\Gamma \\vdash t : A$ holds if and only if there is an intrinsically typed term $u$ of type $A$ in context $\\Gamma$ such that elaborating $t$ in $\\Gamma$ succeeds and returns the type $A$ together with $u$.","title":"Declarative typing agrees with elaboration"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/infer","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.infer","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#infer","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. Given a context $\\Gamma$ and a raw term $t$, type inference elaborates $t$ in $\\Gamma$; if elaboration succeeds, it returns the type component of the result and forgets the intrinsically typed term, and if elaboration fails, inference fails.","title":"Type inference for raw terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/infer-eq-some-iff","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.infer_eq_some_iff","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#infer-eq-some-iff","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. For every context $\\Gamma$, raw term $t$ and type $A$, type inference for $t$ in $\\Gamma$ returns $A$ if and only if the declarative typing judgment $\\Gamma \\vdash t : A$ holds.","title":"Type inference is sound and complete"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/check","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.check","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#check","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. Given a context $\\Gamma$ (a list of simple types), a raw term $r$ and a simple type $A$, the type checker returns true exactly when type inference for $r$ in $\\Gamma$ returns the type $A$, and false otherwise.","title":"Type checking for raw STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/normalize-raw","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeRaw","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#normalize-raw","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Let the base types have decidable equality. Given a context $\\Gamma$ of simple types and a raw term $r$, raw normalization elaborates $r$ in $\\Gamma$. If elaboration fails, raw normalization fails. If elaboration returns a type $A$ and an intrinsically typed term $t$ of type $A$, raw normalization returns $A$ together with the erasure to a raw term of the normal form of $t$ computed by normalization by evaluation.","title":"Normalization of raw terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/normalize-raw-erase","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeRaw_erase","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#normalize-raw-erase","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. For every intrinsically typed term $t$ of type $A$ in context $\\Gamma$, raw normalization of the erasure of $t$ in $\\Gamma$ succeeds and returns the type $A$ together with the erasure of the normal form of $t$ computed by normalization by evaluation.","title":"Raw normalization of an erased term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/normalize-raw-type","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeRaw_type","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#normalize-raw-type","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Let the base types have decidable equality, let $\\Gamma$ be a context of simple types and let $r$ be a raw term. If normalizing $r$ in $\\Gamma$ succeeds and returns a type $A$ together with a raw term, then type inference for $r$ in $\\Gamma$ returns $A$.","title":"Raw normalization reports the inferred type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/normalize-raw-has-type","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeRaw_hasType","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#normalize-raw-has-type","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. For every context $\\Gamma$ and raw term $t$, if raw normalization of $t$ in $\\Gamma$ succeeds with a type $A$ and a raw term $t'$, then the declarative typing judgment $\\Gamma \\vdash t' : A$ holds.","title":"Raw normalization produces well-typed terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/normalize-raw-idempotent","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.normalizeRaw_idempotent","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#normalize-raw-idempotent","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Assume equality of base types is decidable. For every context $\\Gamma$ and raw term $t$, if raw normalization of $t$ in $\\Gamma$ succeeds with a type $A$ and a raw term $t'$, then raw normalization of $t'$ in $\\Gamma$ succeeds with the same type $A$ and the same term $t'$.","title":"Raw normalization is idempotent"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/lift-ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.liftRen","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#lift-ren","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Given a renaming $\\rho : \\mathbb{N} \\to \\mathbb{N}$ of de Bruijn indices, its lift through one binder is the map $\\mathbb{N} \\to \\mathbb{N}$ sending $0$ to $0$ and $n + 1$ to $\\rho(n) + 1$.","title":"Lifting a numerical renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.rename","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Given a map $\\rho : \\mathbb{N} \\to \\mathbb{N}$ on de Bruijn indices, renaming a raw term by $\\rho$ replaces each variable $n$ by $\\rho(n)$, renames both parts of an application by $\\rho$, and renames the body of a lambda abstraction, keeping its parameter type, by the lifted map $\\rho'$ given by $\\rho'(0) = 0$ and $\\rho'(n + 1) = \\rho(n) + 1$.","title":"Renaming of raw terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/lift-ren-comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.liftRen_comp","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#lift-ren-comp","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For maps $\\rho, \\sigma : \\mathbb{N} \\to \\mathbb{N}$, the lift of $\\sigma \\circ \\rho$ through one binder equals the lift of $\\sigma$ composed after the lift of $\\rho$.","title":"Lifting commutes with composition of numerical renamings"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/rename-comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.rename_comp","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#rename-comp","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For all maps $\\rho, \\sigma : \\mathbb{N} \\to \\mathbb{N}$ and every raw term $t$, renaming $t$ by $\\rho$ and then renaming the result by $\\sigma$ gives the same raw term as renaming $t$ by the composite $\\sigma \\circ \\rho$.","title":"Composition of renamings of raw terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/size-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.size_rename","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#size-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For every map $\\rho : \\mathbb{N} \\to \\mathbb{N}$ and every raw term $t$, the renaming of $t$ by $\\rho$ has the same number of syntax nodes as $t$.","title":"Renaming preserves the size of a raw term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/size-shift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.size_shift","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#size-shift","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For all natural numbers $k$ and $c$ and every raw term $t$, shifting $t$ by $k$ at cutoff $c$ (adding $k$ to each variable index that is at least the cutoff, which grows by one under each binder) gives a raw term with the same number of syntax nodes as $t$.","title":"Shifting preserves the size of a raw term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/indices","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.indices","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#indices","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"The list of de Bruijn indices of a raw term, read from left to right with repetitions: a variable with index $n$ gives $[n]$, an application gives the indices of the function followed by those of the argument, and a lambda abstraction gives the indices of its body, unshifted.","title":"Indices occurring in a raw term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/free-indices","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.freeIndices","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#free-indices","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For a binder depth $d$ and a raw term $t$, the free indices of $t$ at depth $d$ form a list defined by recursion on $t$: a variable with index $n$ contributes $[n - d]$ if $d \\le n$ and the empty list otherwise; an application contributes the free indices of the function followed by those of the argument, both at depth $d$; and a lambda abstraction contributes the free indices of its body at depth $d + 1$.","title":"Free indices of a raw term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/closed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.Closed","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#closed","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"A raw term $t$ is closed when its list of free indices at binder depth $0$ is empty.","title":"Closed raw terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/ren-agrees","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RenAgrees","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#ren-agrees","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"A typed renaming $\\rho$ from context $\\Gamma$ to context $\\Delta$ agrees with a map $r : \\mathbb{N} \\to \\mathbb{N}$ when, for every type $A$ and every variable $x$ of type $A$ in $\\Gamma$, the de Bruijn index of $\\rho(x)$ equals $r$ applied to the de Bruijn index of $x$.","title":"Agreement of a typed and a numerical renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/lift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RenAgrees.lift","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#lift","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Let $\\rho$ be a typed renaming from context $\\Gamma$ to $\\Delta$ and let $r : \\mathbb{N} \\to \\mathbb{N}$ be a numerical renaming that agrees with $\\rho$. Then for every type $A$, the lift of $\\rho$ under a binder of type $A$ agrees with the lift of $r$ through one binder.","title":"Agreement of renamings is preserved by lifting"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RenAgrees.comp","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#comp","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Let $\\rho$ be a typed renaming from context $\\Gamma$ to $\\Delta$, let $\\sigma$ be a typed renaming from $\\Delta$ to $\\Xi$, and let $r, s : \\mathbb{N} \\to \\mathbb{N}$. If $\\rho$ agrees with $r$ and $\\sigma$ agrees with $s$, then the composite renaming that applies $\\rho$ and then $\\sigma$ agrees with $s \\circ r$.","title":"Agreement of renamings is preserved by composition"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/erase-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.erase_rename","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#erase-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Let $\\rho$ be a typed renaming from context $\\Gamma$ to $\\Delta$ and let $r : \\mathbb{N} \\to \\mathbb{N}$ be a numerical renaming that agrees with $\\rho$. Then for every intrinsically typed term $t$ of type $A$ in context $\\Gamma$, the erasure of the renamed term $\\rho(t)$ equals the raw renaming by $r$ (lifted at every binder) of the erasure of $t$.","title":"Erasure commutes with renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/erase-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.Tm.erase_weaken","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#erase-weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"For every intrinsically typed term $t$ in context $\\Gamma$ and every type $B$, the erasure of the weakening of $t$ to the context $\\Gamma$ extended by $B$ equals the raw renaming by the successor map $n \\mapsto n + 1$ (lifted at every binder) of the erasure of $t$. Thus every free index is incremented by one.","title":"Erasure of a weakened term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/rename-eq-on-erase","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.RawTm.rename_eq_on_erase","parent":40516,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul/index.html#rename-eq-on-erase","scope":"ComputerScience/type-systems-and-lambda-calculi/type-checking-for-the-simply-typed-lambda-calcul","status":"fully_proved","summary":"Let $\\rho$ be a type-preserving renaming of variables from context $\\Gamma$ to context $\\Delta$, and let $r, s : \\mathbb{N} \\to \\mathbb{N}$ both agree with $\\rho$: for every variable $x$ of $\\Gamma$, the de Bruijn index of $\\rho(x)$ is the image of the index of $x$ under $r$, and also under $s$. Then for every intrinsically typed term $t$ of type $A$ in context $\\Gamma$, renaming the erasure of $t$ by $r$ and\u2026","title":"Agreeing numerical renamings coincide on erased terms"},{"data_url":"scopes/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references.json","id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","kind":"scope","navigation":"module","parent":39999,"result":false,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","title":"Lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/ty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Ty","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#ty","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Fix a type $B$ of base types. The types over $B$ are generated by the grammar $T ::= b \\mid \\mathsf{Unit} \\mid T_1 \\to T_2 \\mid \\mathsf{Ref}\\,T$ with $b \\in B$: base types, the unit type, function types, and types $\\mathsf{Ref}\\,T$ of reference cells holding values of type $T$.","title":"Types of the simply typed lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/term","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#term","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Fix a type $B$ of base types. The terms are generated by: variables, given as de Bruijn indices $x \\in \\mathbb{N}$; abstractions $\\lambda{:}T.\\,t$, where $T$ is a type over $B$ and $t$ is a term in which index $0$ refers to the abstraction's parameter; applications $t_1\\,t_2$; the unit value $\\mathsf{unit}$; store locations $l \\in \\mathbb{N}$; allocations $\\mathsf{ref}\\,t$; dereferences $!t$; and assignments\u2026","title":"Terms of the simply typed lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.Value","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#value","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"A term is a value if and only if it is an abstraction $\\lambda{:}T.\\,t$, the unit value $\\mathsf{unit}$, or a store location $l$.","title":"Values of the simply typed lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/lift-ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.liftRen","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#lift-ren","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Given a renaming $\\rho : \\mathbb{N} \\to \\mathbb{N}$ of de Bruijn indices, its lift is the renaming that sends $0$ to $0$ and each $x + 1$ to $\\rho(x) + 1$.","title":"Lifting a renaming under a binder in the lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/rename-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.rename","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#rename-2","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Given a renaming $\\rho : \\mathbb{N} \\to \\mathbb{N}$ of de Bruijn indices, renaming a term by $\\rho$ replaces each variable $x$ by $\\rho(x)$; renames the body of an abstraction $\\lambda{:}T.\\,t$ by the lift of $\\rho$ (which sends $0$ to $0$ and each $x + 1$ to $\\rho(x) + 1$), keeping the type annotation $T$; leaves the unit value and store locations unchanged; and acts componentwise on applications, allocations,\u2026","title":"Renaming the variables of a term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/lift-sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.liftSub","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#lift-sub","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Given a substitution $\\sigma$ assigning a term to each de Bruijn index, its lift is the substitution that sends $0$ to the variable $0$ and each $x + 1$ to the term $\\sigma(x)$ renamed by $n \\mapsto n + 1$, that is, with every free variable raised by one.","title":"Lifting a substitution under a binder"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/subst-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.subst","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#subst-2","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Given a substitution $\\sigma$ assigning a term to each de Bruijn index, substituting $\\sigma$ into a term replaces each variable $x$ by $\\sigma(x)$; substitutes into the body of an abstraction $\\lambda{:}T.\\,t$ the lift of $\\sigma$ (which sends $0$ to the variable $0$ and each $x + 1$ to $\\sigma(x)$ with every free variable raised by one), keeping the type annotation $T$; leaves the unit value and store locations\u2026","title":"Parallel substitution in a term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/single-sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.singleSub","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#single-sub","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For a term $s$, the single-variable substitution for $s$ sends the de Bruijn index $0$ to $s$ and each index $x + 1$ to the variable $x$.","title":"Single-variable substitution at index zero"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/instantiate-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.instantiate","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#instantiate-2","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For terms $t$ and $s$, the instantiation of $t$ by $s$ applies to $t$ the parallel substitution that sends the de Bruijn index $0$ to $s$ and each index $x + 1$ to the variable $x$. Thus the free occurrences of variable $0$ in $t$ are replaced by $s$, with the free variables of $s$ raised by the number of binders crossed, and every other free variable of $t$ is lowered by one.","title":"Substitution of a term for variable zero"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/store","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Store","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#store","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"A store $\\mu$ is a finite list of terms; each location $l$ less than the length of $\\mu$ holds the term $\\mu[l]$, the entry of $\\mu$ at index $l$.","title":"Store"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Step","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#step","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"One step of call-by-value evaluation $t \\mid \\mu \\to t' \\mid \\mu'$, from a term $t$ with a store $\\mu$ to a term $t'$ with a store $\\mu'$, is the least relation closed under the following rules, in which $v$, $v_1$ and $v_2$ are values. If $t \\mid \\mu \\to t' \\mid \\mu'$, then $t\\,t_2 \\mid \\mu \\to t'\\,t_2 \\mid \\mu'$, $v_1\\,t \\mid \\mu \\to v_1\\,t' \\mid \\mu'$, $\\mathsf{ref}\\,t \\mid \\mu \\to \\mathsf{ref}\\,t' \\mid \\mu'$,\u2026","title":"One-step evaluation with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Steps","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#steps","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"The relation $t \\mid \\mu \\to^* t' \\mid \\mu'$ holds when the configuration $(t, \\mu)$ reaches $(t', \\mu')$ in zero or more evaluation steps; that is, it is the reflexive-transitive closure of one-step evaluation on pairs of a term and a store.","title":"Multi-step evaluation with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/not-of-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Step.not_of_value","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#not-of-value","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"If $v$ is a value, then there are no stores $\\mu$, $\\mu'$ and term $t'$ such that the configuration $(v, \\mu)$ steps to $(t', \\mu')$: values are normal forms.","title":"Values do not step"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/deterministic","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Step.deterministic","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#deterministic","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"If a configuration $(t, \\mu)$ steps to $(t_1, \\mu_1)$ and also to $(t_2, \\mu_2)$, then $t_1 = t_2$ and $\\mu_1 = \\mu_2$.","title":"Determinism of evaluation with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/rename-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.rename_congr","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#rename-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\rho, \\rho' : \\mathbb{N} \\to \\mathbb{N}$ be renamings of de Bruijn variables with $\\rho(x) = \\rho'(x)$ for every $x \\in \\mathbb{N}$. Then for every term $t$ of the simply typed lambda calculus with references, renaming $t$ by $\\rho$ gives the same term as renaming $t$ by $\\rho'$.","title":"Renaming respects pointwise equality"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/rename-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.rename_id","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#rename-id","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For every term $t$ of the simply typed lambda calculus with references, renaming the de Bruijn variables of $t$ by the identity map of $\\mathbb{N}$ gives back $t$.","title":"Renaming by the identity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/rename-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.rename_rename","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#rename-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For all renamings $\\rho, \\rho' : \\mathbb{N} \\to \\mathbb{N}$ of de Bruijn variables and every term $t$ of the simply typed lambda calculus with references, renaming $t$ by $\\rho$ and then renaming the result by $\\rho'$ gives the same term as renaming $t$ by the composite $\\rho' \\circ \\rho$.","title":"Composition of renamings"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/subst-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.subst_congr","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#subst-congr","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\sigma, \\sigma'$ be parallel substitutions, each sending every de Bruijn index to a term of the simply typed lambda calculus with references, with $\\sigma(x) = \\sigma'(x)$ for every $x \\in \\mathbb{N}$. Then for every term $t$, applying $\\sigma$ to $t$ gives the same term as applying $\\sigma'$ to $t$.","title":"Substitution respects pointwise equality"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/subst-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.subst_id","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#subst-id","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For every term $t$ of the simply typed lambda calculus with references, applying the parallel substitution that sends each de Bruijn index $x$ to the variable term $x$ gives back $t$.","title":"Substitution by variables is the identity"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/subst-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.subst_rename","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#subst-rename","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\rho : \\mathbb{N} \\to \\mathbb{N}$ be a renaming of de Bruijn variables, $\\sigma$ a parallel substitution sending each index to a term of the simply typed lambda calculus with references, and $t$ a term. Renaming $t$ by $\\rho$ and then applying $\\sigma$ gives the same term as applying the substitution $\\sigma \\circ \\rho$ to $t$.","title":"Renaming then substituting"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/rename-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.rename_subst","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#rename-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\rho : \\mathbb{N} \\to \\mathbb{N}$ be a renaming of de Bruijn variables, $\\sigma$ a parallel substitution sending each index to a term of the simply typed lambda calculus with references, and $t$ a term. Applying $\\sigma$ to $t$ and then renaming the result by $\\rho$ gives the same term as applying to $t$ the substitution that sends each index $x$ to the renaming of $\\sigma(x)$ by $\\rho$.","title":"Substituting then renaming"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/subst-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.subst_subst","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#subst-subst","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For all parallel substitutions $\\sigma, \\tau$, each sending every de Bruijn index to a term of the simply typed lambda calculus with references, and every term $t$, applying $\\sigma$ to $t$ and then $\\tau$ to the result gives the same term as applying to $t$ the substitution that sends each index $x$ to the result of applying $\\tau$ to $\\sigma(x)$.","title":"Composition of substitutions in the lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/instantiate-rename-succ","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.instantiate_rename_succ","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#instantiate-rename-succ","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For all terms $t$ and $s$ of the simply typed lambda calculus with references, shift every free de Bruijn variable of $t$ up by one (rename by $x \\mapsto x + 1$), then instantiate the result with $s$, replacing variable $0$ by $s$ and lowering every other variable $x + 1$ to $x$. The outcome is $t$.","title":"Instantiating a shifted term"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/has-type","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#has-type","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"The judgment $\\Gamma \\mid S \\vdash t : T$ (TAPL Figure 13-1), where the context $\\Gamma$ lists the types of the free de Bruijn variables (index $0$ first) and the store typing $S$ lists the types of the locations $0, 1, \\ldots$, is the least relation closed under the following rules, in which $t : T$ abbreviates $\\Gamma \\mid S \\vdash t : T$. A variable $x$ has type $T$ if $\\Gamma$ assigns $T$ to index $x$. If $t$\u2026","title":"Typing judgment for the lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/store-typed","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.StoreTyped","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#store-typed","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ be a variable context and $S$ a store typing. A store $\\mu$, a list of terms whose entry at index $l$ is the contents of location $l$, is well typed under $\\Gamma$ and $S$, written $\\Gamma \\mid S \\vdash \\mu$, when $\\mu$ and $S$ have the same length and, for every location $l$ less than this length, the entry of $\\mu$ at $l$ is a value that has, under $\\Gamma$ and $S$, the type that $S$ assigns to $l$.","title":"Well-typed stores"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/exists-of-get-elem","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.StoreTyped.exists_of_getElem?","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#exists-of-get-elem","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ be a typing context, $S$ a store typing, and $\\mu$ a store (a list of terms, whose entry at index $l$ is the contents of location $l$) that is well typed, $\\Gamma \\mid S \\vdash \\mu$. If $S$ assigns type $T$ to location $l$, then $\\mu$ has an entry $v$ at location $l$, $v$ is a value, and $\\Gamma \\mid S \\vdash v : T$.","title":"Well-typed stores hold a typed value at every location"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/lam-of-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.lam_of_value","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#lam-of-value","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For every context $\\Gamma$ and store typing $S$, if $v$ is a value and $\\Gamma \\mid S \\vdash v : T_1 \\to T_2$, then $v$ is an abstraction $\\lambda{:}T_1.\\, t$ with parameter type $T_1$, for some body $t$.","title":"Canonical forms for function types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/loc-of-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.loc_of_value","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#loc-of-value","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For every context $\\Gamma$ and store typing $S$, if $v$ is a value and $\\Gamma \\mid S \\vdash v : \\mathrm{Ref}\\, T$, then $v$ is a location $l$ to which $S$ assigns type $T$.","title":"Canonical forms for reference types"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/unit-of-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.unit_of_value","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#unit-of-value","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For every context $\\Gamma$ and store typing $S$, if $v$ is a value and $\\Gamma \\mid S \\vdash v : \\mathrm{Unit}$, then $v = \\mathtt{unit}$.","title":"Canonical forms for the unit type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/not-value-base","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.not_value_base","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#not-value-base","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"For every context $\\Gamma$, store typing $S$ and base type $b$, there is no value $v$ with $\\Gamma \\mid S \\vdash v : b$.","title":"No value has a base type"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/id-u","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.idU","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#id-u","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"The identity function $\\lambda x{:}\\mathrm{Unit}.\\ x$ on the unit type.","title":"The identity on Unit"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/body","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.body","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#body","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"The term $\\lambda x{:}\\mathrm{Unit}.\\ (!r)\\,x$, which applies the contents of the reference $r$ to its argument. Here $r$ is a free variable, de Bruijn index $1$ under the binder $\\lambda x$; in Landin's knot it is bound by the outer $\\lambda r$.","title":"The recursive function of Landin's knot, with the cell free"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/knot","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.knot","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#knot","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"The term","title":"Landin's knot"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/body-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.body\u2080","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#body-2","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"The term $b_0 = \\lambda x{:}\\mathrm{Unit}.\\ (!\\ell_0)\\,x$, which applies the contents of store location $\\ell_0 = 0$ to its argument.","title":"The stored recursive function of Landin's knot"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/loop-a","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.loopA","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#loop-a","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"The term $(!\\ell_0)\\ \\mathrm{unit}$: the dereference of the store location $\\ell_0$ (location $0$), applied to the unit value.","title":"Cycle term of Landin's knot"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/c","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.c\u2081","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#c","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"The term $(\\lambda r{:}\\mathrm{Ref}\\,(\\mathrm{Unit}\\to\\mathrm{Unit}).\\ (\\lambda y{:}\\mathrm{Unit}.\\ (!r)\\, \\mathrm{unit})\\ (r := \\lambda x{:}\\mathrm{Unit}.\\ (!r)\\,x))\\ \\ell_0$, where $\\ell_0$ is store location $0$: Landin's knot with its allocation $\\mathrm{ref}\\,(\\lambda x{:}\\mathrm{Unit}.\\ x)$ replaced by $\\ell_0$.","title":"The knot after allocation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/knot-prefix","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.knot_prefix","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#knot-prefix","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Write $\\ell_0$ for store location $0$ and $[t]$ for the store whose only entry, at $\\ell_0$, is $t$, and let $\\mathrm{id} = \\lambda x{:}\\mathrm{Unit}.\\ x$ and $b_0 = \\lambda x{:}\\mathrm{Unit}.\\ (!\\ell_0)\\,x$. Each of the following is one call-by-value step: Landin's knot with the empty store steps to\u2026","title":"The first four steps of Landin's knot"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/knot-cycle","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.knot_cycle","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#knot-cycle","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Write $\\ell_0$ for store location $0$, let $b_0 = \\lambda x{:}\\mathrm{Unit}.\\ (!\\ell_0)\\,x$, and let $[b_0]$ be the store whose only entry, at $\\ell_0$, is $b_0$. In one call-by-value step, $(!\\ell_0)\\,\\mathrm{unit}$ with store $[b_0]$ steps to $b_0\\,\\mathrm{unit}$ with store $[b_0]$, and in one further step $b_0\\,\\mathrm{unit}$ with store $[b_0]$ steps back to $(!\\ell_0)\\,\\mathrm{unit}$ with store $[b_0]$.","title":"The cycle of Landin's knot"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/reachable","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Knot.reachable","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#reachable","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $K$ be Landin's knot, $\\iota = \\lambda x{:}\\mathrm{Unit}.\\,x$, $b_0 = \\lambda x{:}\\mathrm{Unit}.\\,(!\\ell_0)\\,x$ with $\\ell_0$ the store location $0$, $a = (!\\ell_0)\\ \\mathrm{unit}$, $b = b_0\\ \\mathrm{unit}$, $c_1$ the term $K$ with its argument $\\mathrm{ref}\\ \\iota$ replaced by $\\ell_0$, $c_2 = (\\lambda y{:}\\mathrm{Unit}.\\,a)\\,(\\ell_0 := b_0)$ and $c_3 = (\\lambda y{:}\\mathrm{Unit}.\\,a)\\ \\mathrm{unit}$; write\u2026","title":"Configurations on the evaluation path of Landin's knot"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/knot-has-type","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.knot_hasType","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#knot-has-type","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Landin's knot is closed and has type $\\mathrm{Unit}$ in the empty typing context under the empty store typing.","title":"Landin's knot is well typed"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/knot-diverges","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.knot_diverges","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#knot-diverges","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Starting from Landin's knot with the empty store, no sequence of zero or more call-by-value evaluation steps reaches a configuration $(v, \\mu)$ in which the term $v$ is a value, whatever the store $\\mu$.","title":"Landin's knot never evaluates to a value"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/knot-runs-forever","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.knot_runs_forever","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#knot-runs-forever","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"If Landin's knot with the empty store evaluates in zero or more call-by-value steps to a term $t$ with store $\\mu$, then $t$ with store $\\mu$ takes a further step to some term $t'$ with some store $\\mu'$. So the evaluation of the knot runs forever.","title":"Landin's knot never gets stuck"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/of-stlc","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Ty.ofStlc","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#of-stlc","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Fix a type $B$ of base types. The embedding $\\iota$ sends a simple type over $B$ to a type of the calculus with references over $B$, by recursion: $\\iota(b) = b$ for a base type $b \\in B$, and $\\iota(A_1 \\to A_2) = \\iota(A_1) \\to \\iota(A_2)$.","title":"Embedding of simple types into the calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/of-raw","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Term.ofRaw","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#of-raw","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Fix a type $B$ of base types. The embedding $\\iota$ sends a raw (extrinsically typed, binder-annotated) term of the simply typed lambda calculus over $B$ to a term of the calculus with references over $B$, by recursion: a de Bruijn variable $n$ goes to the variable $n$, an application $f\\,a$ goes to $\\iota(f)\\,\\iota(a)$, and an abstraction $\\lambda{:}A.\\,t$ goes to $\\lambda{:}\\iota(A).\\,\\iota(t)$, where $\\iota(A)$\u2026","title":"Embedding of raw STLC terms into the calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/of-raw-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.ofRaw","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#of-raw-2","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Fix a type $B$ of base types, and write $\\iota$ for the embeddings of simple types and of raw terms of the simply typed lambda calculus over $B$ into types and terms of the calculus with references over $B$. Let $\\Gamma$ be a context of simple types, $t$ a raw term and $A$ a simple type such that $t$ has type $A$ in $\\Gamma$. Then for every store typing $S$ we have $\\iota(\\Gamma) \\mid S \\vdash \\iota(t) : \\iota(A)$,\u2026","title":"Typing is preserved by the embedding of raw STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/of-stlc-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.ofStlc","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#of-stlc-2","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Fix a type $B$ of base types, and write $\\iota$ for the embeddings of simple types and of raw terms of the simply typed lambda calculus over $B$ into types and terms of the calculus with references over $B$. Let $\\Gamma$ be a context of simple types, $A$ a simple type and $t$ an intrinsically typed term of type $A$ in $\\Gamma$, and let $\\mathrm{erase}(t)$ be the raw term obtained from $t$ by erasing its typing\u2026","title":"Typing of embedded intrinsically typed STLC terms"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/progress","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.progress","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#progress","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $S$ be a store typing, $t$ a term of some type $T$ in the empty variable context under $S$, and $\\mu$ a store that is well typed under $S$ in the empty context. Then $t$ is a value, or the configuration $(t, \\mu)$ takes an evaluation step to some $(t', \\mu')$.","title":"Progress"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.rename","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#rename","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ and $\\Delta$ be variable contexts, $S$ a store typing, and $\\rho : \\mathbb{N} \\to \\mathbb{N}$ a renaming of variable indices such that whenever $\\Gamma$ assigns type $A$ to variable $x$, $\\Delta$ assigns type $A$ to variable $\\rho(x)$. If a term $t$ has type $T$ under $\\Gamma$ and $S$, then the term obtained from $t$ by renaming its variables by $\\rho$ has type $T$ under $\\Delta$ and $S$.","title":"Renaming preserves typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.weaken","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#weaken","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"If a term $t$ has type $T$ in a variable context $\\Gamma$ under a store typing $S$, then for every type $A$, the term obtained from $t$ by shifting each free de Bruijn variable up by one has type $T$ under $S$ in the context obtained from $\\Gamma$ by adding $A$ in front as variable $0$.","title":"Weakening of typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.subst","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#subst","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ and $\\Delta$ be variable contexts, $S$ a store typing, and $\\sigma$ a substitution sending each variable index to a term, such that whenever $\\Gamma$ assigns type $A$ to variable $x$, the term $\\sigma(x)$ has type $A$ under $\\Delta$ and $S$. If a term $t$ has type $T$ under $\\Gamma$ and $S$, then the term obtained from $t$ by applying $\\sigma$ has type $T$ under $\\Delta$ and $S$.","title":"Parallel substitution preserves typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.instantiate","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#instantiate","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ be a variable context and $S$ a store typing. If a term $t$ has type $T$ under $S$ and the context that extends $\\Gamma$ by a new innermost variable (index $0$) of type $A$, and a term $s$ has type $A$ under $\\Gamma$ and $S$, then substituting $s$ for variable $0$ in $t$ (and lowering every other variable by one) gives a term of type $T$ under $\\Gamma$ and $S$.","title":"Substitution lemma for the lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/extend-store","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.HasType.extendStore","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#extend-store","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ be a variable context and let $S$ and $S'$ be store typings such that $S$ is a prefix of $S'$. If a term $t$ has type $T$ under $\\Gamma$ and $S$, then $t$ has type $T$ under $\\Gamma$ and $S'$.","title":"Store-typing weakening"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/extend-store-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.StoreTyped.extendStore","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#extend-store-2","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ be a variable context, let $S$ and $S'$ be store typings such that $S$ is a prefix of $S'$, and let $\\mu$ be a store that is well typed under $\\Gamma$ and $S$. Then $\\mu$ and $S$ have the same length, and for every location $l$ less than this length, the entry of $\\mu$ at $l$ is a value that has, under $\\Gamma$ and $S'$, the type that $S$ assigns to $l$.","title":"Retyping a well-typed store under an extended store typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/append","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.StoreTyped.append","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#append","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ be a variable context, $S$ a store typing and $\\mu$ a store that is well typed under $\\Gamma$ and $S$. If $v$ is a value of type $T$ under $\\Gamma$ and $S$, then the store obtained by appending $v$ at the end of $\\mu$ is well typed under $\\Gamma$ and the store typing obtained by appending $T$ at the end of $S$.","title":"Allocation preserves store typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/set","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.StoreTyped.set","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#set","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let the store $\\mu$ be well typed under the variable context $\\Gamma$ and the store typing $S$, and let $l$ be a location to which $S$ assigns the type $T$. If $v$ is a value of type $T$ under $\\Gamma$ and $S$, then the store obtained from $\\mu$ by overwriting the entry at $l$ with $v$ is well typed under $\\Gamma$ and $S$.","title":"Assignment preserves store typing"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/preservation","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.preservation","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#preservation","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ be a variable context and $S$ a store typing. If a term $t$ has type $T$ under $\\Gamma$ and $S$, the store $\\mu$ is well typed under $\\Gamma$ and $S$, and $t$ with store $\\mu$ takes one evaluation step to $t'$ with store $\\mu'$, then there is a store typing $S'$ having $S$ as a prefix such that $t'$ has type $T$ under $\\Gamma$ and $S'$ and $\\mu'$ is well typed under $\\Gamma$ and $S'$.","title":"Preservation for the lambda calculus with references"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/preservation-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.Steps.preservation","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#preservation-2","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $\\Gamma$ be a variable context and $S$ a store typing. If a term $t$ has type $T$ under $\\Gamma$ and $S$, the store $\\mu$ is well typed under $\\Gamma$ and $S$, and $t$ with store $\\mu$ evaluates in zero or more steps to $t'$ with store $\\mu'$, then there is a store typing $S'$ having $S$ as a prefix such that $t'$ has type $T$ under $\\Gamma$ and $S'$ and $\\mu'$ is well typed under $\\Gamma$ and $S'$.","title":"Preservation for multi-step evaluation"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/type-safety","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.typeSafety","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#type-safety","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"Let $S$ be a store typing, $T$ a type, $t$ a term of type $T$ in the empty variable context under $S$, and $\\mu$ a store that is well typed under $S$ in the empty context. If the configuration $(t, \\mu)$ evaluates in zero or more steps to $(t', \\mu')$, then $t'$ is a value or $(t', \\mu')$ takes a step to some configuration $(t'', \\mu'')$. So a closed term that is well typed in a well-typed store never reaches a\u2026","title":"Type safety"},{"id":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/type-safety-empty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.References.typeSafety_empty","parent":40559,"result":true,"route":"../roadmap/ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references/index.html#type-safety-empty","scope":"ComputerScience/type-systems-and-lambda-calculi/lambda-calculus-with-references","status":"fully_proved","summary":"A closed term typed under the empty store typing, run from the empty store, never gets stuck: every term it reaches in zero or more steps, together with the store reached alongside it, is a value or can take a further step.","title":"Type safety from the empty store"},{"data_url":"scopes/ComputerScience/programming-language-semantics.json","id":"ComputerScience/programming-language-semantics","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/index.html","scope":"ComputerScience/programming-language-semantics","status":"fully_proved","summary":"Operational and denotational semantics of programming languages and correctness of compilation: the IMP language (syntax, big- and small-step, heap semantics, denotational semantics, stack-machine compilation), PCF with adequacy and contextual equivalence, CPS translation and closure conversion.","title":"Programming language semantics"},{"data_url":"scopes/ComputerScience/programming-language-semantics/closure-conversion.json","id":"ComputerScience/programming-language-semantics/closure-conversion","kind":"scope","navigation":"module","parent":40619,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","title":"Closure conversion"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Source.Value","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#value","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"A source value is a closure of a raw term $b$ (the body of a lambda) over an environment $\\rho$, which is a list of source values.","title":"Source values of closure conversion"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/eval","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Source.Eval","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#eval","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Evaluation $\\rho \\vdash t \\Downarrow v$ of a raw term $t$ to a source value $v$ in an environment $\\rho$ (a list of source values, where variable $n$ refers to the entry of $\\rho$ at index $n$, counting from $0$) is the least relation closed under the following rules, a call-by-value environment semantics that ignores the type annotations of lambdas. A variable $n$ evaluates to the entry of $\\rho$ at index $n$,\u2026","title":"Big-step semantics of the closure-conversion source"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/deterministic","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Source.Eval.deterministic","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#deterministic","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For every environment $\\rho$ of source values and every raw term $t$, if $t$ evaluates in $\\rho$ to source values $v$ and $w$, then $v = w$.","title":"Source evaluation is deterministic"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#tm","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"The terms of the closure-conversion target language, with variables as de Bruijn indices, are: a variable $n$ for each natural number $n$; code $\\mathsf{code}\\,b$ of one parameter, whose body $b$ is evaluated with only its parameter in scope, as variable $0$; application $f\\,a$ of code to an argument; the unit term $\\mathsf{unit}$ (the empty tuple); pairs $(a, b)$; the projections $\\mathsf{fst}\\,e$ and\u2026","title":"Target terms of closure conversion"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/value-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Value","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#value-2","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"A value of the closure-conversion target language is code $\\mathsf{code}\\,b$ for a target term $b$, the unit value $\\mathsf{unit}$, or a pair $(v, w)$ of values.","title":"Target values of closure conversion"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/eval-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm.Eval","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#eval-2","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Evaluation $\\rho \\vdash e \\Downarrow v$ of a target term $e$ to a target value $v$ in an environment $\\rho$ (a list of values, where variable $n$ refers to the entry of $\\rho$ at index $n$, counting from $0$) is the least relation closed under the following rules. A variable $n$ evaluates to the entry of $\\rho$ at index $n$, when that entry exists. Code evaluates to itself:\u2026","title":"Big-step semantics of the closure-conversion target"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/lift-ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm.liftRen","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#lift-ren","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"The lift $r^{\\uparrow}$ of a renaming $r : \\mathbb{N} \\to \\mathbb{N}$ of de Bruijn indices sends $0$ to $0$ and $i + 1$ to $r(i) + 1$.","title":"Lifting a renaming under a binder, for the closure-conversion target"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm.rename","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#rename","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For $r : \\mathbb{N} \\to \\mathbb{N}$, the renaming $t[r]$ of a target term $t$ replaces each variable $i$ by $r(i)$, leaves code $\\mathsf{code}\\,b$ unchanged (its body $b$ is not renamed), fixes $\\mathsf{unit}$, commutes with application, pairing and the two projections, and sends $\\mathsf{let}\\ e\\ \\mathsf{in}\\ b$ to $\\mathsf{let}\\ e[r]\\ \\mathsf{in}\\ b[r^{\\uparrow}]$, where $r^{\\uparrow}$, the lift of $r$ under one\u2026","title":"Renaming target terms"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/shift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm.shift","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#shift","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"The shift of a target term $e$ is its renaming by the successor map $i \\mapsto i + 1$: every free variable of $e$ goes up by one, and code bodies are left unchanged.","title":"Shifting a target term"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/rename-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm.Eval.rename","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#rename-2","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\rho$ and $\\rho'$ be environments and $r : \\mathbb{N} \\to \\mathbb{N}$ a renaming such that, for every index $i$ and value $w$, if $\\rho$ has entry $w$ at index $i$ then $\\rho'$ has entry $w$ at index $r(i)$. If a target term $e$ evaluates to $v$ in $\\rho$, then the renaming of $e$ by $r$ evaluates to $v$ in $\\rho'$.","title":"Evaluation is stable under renaming"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/shift-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm.Eval.shift","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#shift-2","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"If a target term $e$ evaluates to $v$ in an environment $\\rho$, then for every value $x$ the shift of $e$ evaluates to $v$ in the environment $x :: \\rho$ obtained from $\\rho$ by adding $x$ at index $0$.","title":"Evaluation of a shifted term"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/scoped","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm.Scoped","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#scoped","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For a natural number $m$, a target term $e$ is scoped in $m$ when every free variable of $e$ is less than $m$ and every code body in $e$ is closed, that is, scoped in $1$. Inductively: a variable $i$ is scoped in $m$ when $i < m$; $\\mathsf{code}\\,b$ is scoped in every $m$ when $b$ is scoped in $1$; $\\mathsf{unit}$ is scoped in every $m$; an application, a pair or a projection is scoped in $m$ when its immediate\u2026","title":"Well-scoped target terms"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/shift-3","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.Target.Tm.Scoped.shift","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#shift-3","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"If a target term $e$ is scoped in $m$, then its shift is scoped in $m + 1$.","title":"Shifting preserves scoping"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/snds","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.snds","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#snds","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For a natural number $k$ and a target term $e$, the term $\\mathsf{snd}^k\\,e$ applies the second projection $k$ times to $e$: $\\mathsf{snd}^0\\,e = e$ and $\\mathsf{snd}^{k+1}\\,e = \\mathsf{snd}\\,(\\mathsf{snd}^k\\,e)$.","title":"Iterated second projection"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/env-of","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.envOf","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#env-of","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For a map $\\mathit{acc}$ from natural numbers to target terms and a natural number $n$, the environment-packing term of $\\mathit{acc}$ and $n$ is the target term $(\\mathit{acc}(0), (\\mathit{acc}(1), \\dots (\\mathit{acc}(n - 1), \\mathsf{unit})\\dots))$ packing $\\mathit{acc}(0), \\dots, \\mathit{acc}(n - 1)$ as nested pairs ending in $\\mathsf{unit}$. Recursively, it is $\\mathsf{unit}$ for $n = 0$, and for $n + 1$ it is\u2026","title":"Environment-packing term"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/code-acc","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.codeAcc","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#code-acc","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Write $\\#0$ for the target variable $0$ and $\\mathsf{snd}^i$ for $i$ applications of the second projection. The code accessor for index $0$ is $\\mathsf{fst}\\,\\#0$, and the code accessor for index $i + 1$ is $\\mathsf{fst}\\,(\\mathsf{snd}^i\\,(\\mathsf{snd}\\,\\#0))$. When the code parameter $\\#0$ is the pair of the argument and an environment encoded as nested pairs, accessor $0$ fetches the argument and accessor $i + 1$\u2026","title":"Accessors inside converted code"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/convert","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.convert","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#convert","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\mathit{acc}$ be a family of target terms indexed by the natural numbers (accessor $i$ fetches source variable $i$) and $n$ a natural number (the number of source variables in scope). The conversion $\\mathcal{C}(\\mathit{acc}, n, t)$ of a source term $t$ is defined by recursion on $t$. A variable $i$ converts to $\\mathit{acc}(i)$. A lambda with body $b$ (whatever its parameter type) converts to the pair of\u2026","title":"Closure conversion with accessors"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/closure-convert","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.closureConvert","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#closure-convert","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"The closure conversion of a source term $t$ is the conversion of $t$ in a scope of $0$ variables with the accessors that send each index $i$ to the target variable $i$. It is intended for closed terms.","title":"Closure conversion of a program"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/encode-env","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.encodeEnv","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#encode-env","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"The encoding of a list of target values as nested pairs sends the empty list to the unit value and a list with head $w$ and tail $ws$ to the pair of $w$ and the encoding of $ws$. Thus $[w_0, \\dots, w_{k-1}]$ is encoded as $(w_0, (w_1, \\dots (w_{k-1}, \\mathsf{unit})))$.","title":"Encoding a list of target values"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/val-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.ValRel","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#val-rel","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"The relation between source values and target values of closure conversion is the least relation such that, for every source lambda body $b$, every list $\\rho$ of source values and every list $ws$ of target values with $|\\rho| = |ws|$ and $\\rho_i$ related to $ws_i$ for every $i < |\\rho|$, the source closure of $b$ over $\\rho$ is related to the target pair $(\\mathsf{code}\\,c, E)$. Here $c$ is the closure conversion\u2026","title":"Value relation for closure conversion"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/acc-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.AccRel","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#acc-rel","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\rho$ be a list of source values, $\\rho_t$ a list of target values and $\\mathit{acc}$ a family of target terms indexed by the natural numbers. The accessors $\\mathit{acc}$ implement $\\rho$ in $\\rho_t$ when, for every index $i$ less than the length of $\\rho$, there is a target value $w$ such that $\\mathit{acc}(i)$ evaluates to $w$ in the environment $\\rho_t$ and the entry of $\\rho$ at index $i$ is related to\u2026","title":"Accessors implementing a source environment"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/env-of-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.AccRel.envOf","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#env-of-2","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\rho$ be a list of source values, $\\rho_t$ a list of target values and $\\mathit{acc}$ a map from natural numbers to target terms, and suppose the accessors $\\mathit{acc}$ implement $\\rho$ in $\\rho_t$: for every $i < |\\rho|$, the term $\\mathit{acc}(i)$ evaluates in $\\rho_t$ to a target value related to $\\rho_i$ by the value relation. Then there is a list $ws$ of target values with $|ws| = |\\rho|$ and $\\rho_i$\u2026","title":"Packed accessors evaluate to an encoded related environment"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/eval-snds","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.eval_snds","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#eval-snds","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\rho$ be a list of target values, $e$ a target term and $ws = [w_0, \\dots, w_{m-1}]$ a list of target values, and suppose $e$ evaluates in $\\rho$ to the encoding of $ws$, the nested pairs $(w_0, (w_1, \\dots (w_{m-1}, \\mathsf{unit})\\dots))$. Then for every natural number $k \\le m$, the term obtained by applying the second projection $k$ times to $e$ evaluates in $\\rho$ to the encoding of\u2026","title":"Iterated second projections of an encoded environment"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/acc-rel-code-acc","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.accRel_codeAcc","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#acc-rel-code-acc","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $v$ be a source value related to a target value $w$ by the value relation of closure conversion, and let $\\rho$ and $ws$ be lists of source and target values of the same length such that, for every index $i$ below that length, the entry of $\\rho$ at $i$ is related to the entry of $ws$ at $i$. Then the code accessors implement the source environment $v :: \\rho$ ($\\rho$ with $v$ added at index $0$) in the\u2026","title":"Code accessors implement the extended closure environment"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/convert-correct","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.convert_correct","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#convert-correct","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\rho$ be a list of source values, $\\ell$ its length, and $t$ a source term that evaluates to a source value $v$ in the environment $\\rho$. Let $\\rho_t$ be a list of target values and $\\mathit{acc}$ a family of target terms indexed by the natural numbers that implement $\\rho$ in $\\rho_t$. Then there is a target value $w$ such that the conversion of $t$ in a scope of $\\ell$ variables with accessors\u2026","title":"Correctness of closure conversion"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/closure-convert-correct","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.closureConvert_correct","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#closure-convert-correct","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"If a source term $t$ evaluates to a source value $v$ in the empty environment, then there is a target value $w$ such that the closure conversion of $t$ evaluates to $w$ in the empty environment and $v$ is related to $w$ by the value relation of closure conversion.","title":"Correctness of closure conversion in the empty environment"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/convert-scoped","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.convert_scoped","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#convert-scoped","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $m$ and $n$ be natural numbers and $t$ a source term whose list of free indices at binder depth $n$ is empty, that is, $t$ is scoped in $n$ variables. If $\\mathit{acc}$ is a family of target terms indexed by the natural numbers such that $\\mathit{acc}(i)$ is scoped in $m$ for every $i < n$, then the conversion of $t$ in a scope of $n$ variables with accessors $\\mathit{acc}$ is scoped in $m$; in particular,\u2026","title":"Converted terms are well scoped"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/closure-convert-scoped","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.closureConvert_scoped","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#closure-convert-scoped","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"If a source term $t$ is closed (it has no free variables), then its closure conversion is scoped in $0$: it has no free variables, and every code body in it is scoped in $1$, so refers only to its parameter.","title":"Closure conversion of a closed term is well scoped"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/mem-free-indices-succ","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.mem_freeIndices_succ","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#mem-free-indices-succ","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For every raw term $t$ and natural numbers $d$ and $j$, $j$ is a free index of $t$ at binder depth $d + 1$ if and only if $j + 1$ is a free index of $t$ at binder depth $d$.","title":"Free variables one binder deeper"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/captured","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.captured","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#captured","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For a raw term $b$, the captured variables of $b$ form the list of free indices of $b$ at binder depth $1$ with duplicates removed: the de Bruijn indices $j$ such that variable $j + 1$ occurs free in $b$, that is, the free variables of the abstraction $\\lambda.\\,b$, each listed once.","title":"Variables captured by a lambda"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/env-flat","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.envFlat","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#env-flat","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For a map $\\mathrm{acc}$ from natural numbers to target terms and a list $i_1, \\dots, i_n$ of natural numbers, the flat environment is the target term $(\\mathrm{acc}(i_1), (\\mathrm{acc}(i_2), \\dots, (\\mathrm{acc}(i_n), \\mathsf{unit})\\dots))$, which packs $\\mathrm{acc}(i)$ for each $i$ in the list, in order, as nested pairs ending in $\\mathsf{unit}$; the empty list gives $\\mathsf{unit}$.","title":"Flat closure environment"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/flat-acc","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.flatAcc","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#flat-acc","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"For a list $L$ of natural numbers, the flat accessors for $L$ assign a target term to each variable: variable $0$ is sent to $\\mathsf{fst}\\,\\#0$, and variable $j + 1$ is sent to $\\mathsf{fst}\\,(\\mathsf{snd}^k\\,(\\mathsf{snd}\\,\\#0))$. Here $\\#0$ is target variable $0$, $\\mathsf{snd}^k$ applies the second projection $k$ times, and $k$ is the index of the first occurrence of $j$ in $L$, or the length of $L$ if $j$ does\u2026","title":"Accessors inside flat code"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/convert-flat","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.convertFlat","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#convert-flat","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Given accessors $\\mathrm{acc}$, a map from natural numbers to target terms, the flat conversion $C_{\\mathrm{acc}}(t)$ of a raw term $t$ is defined by recursion on $t$. A variable $i$ becomes $\\mathrm{acc}(i)$. A lambda $\\lambda.\\,b$, whose parameter type is ignored, becomes the pair $(\\mathsf{code}\\,C_{\\mathrm{acc}'}(b), e)$. Here $i_1, \\dots, i_n$ are the captured variables of $b$, the environment $e$ is\u2026","title":"Flat closure conversion"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/closure-convert-flat","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.closureConvertFlat","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#closure-convert-flat","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"The flat closure conversion of a raw term $t$ is the flat conversion of $t$ with the accessors $i \\mapsto \\mathsf{var}\\,i$, which send each free variable $i$ of $t$ to the target variable $i$. It is meant for closed terms.","title":"Flat closure conversion of a program"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/flat-val-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.FlatValRel","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#flat-val-rel","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"The flat value relation between source values and target values is the least relation with the following property. Let $b$ be a raw term, $\\rho$ a list of source values of length $|\\rho|$, $c = (c_0, \\dots, c_{m-1})$ the list of variables captured by $\\lambda. b$, and $w_0, \\dots, w_{m-1}$ target values, as many as there are captured variables. If, for every $k < m$, $c_k < |\\rho|$ and the entry of $\\rho$ at index\u2026","title":"Flat value relation"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/flat-acc-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.FlatAccRel","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#flat-acc-rel","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\rho$ be a list of source values of length $|\\rho|$, $\\rho_t$ a list of target values, $\\mathrm{acc}$ a map assigning a target term to each natural number, and $t$ a raw term. The accessors $\\mathrm{acc}$ implement $t$ for $\\rho$ in $\\rho_t$ when, for every free variable $i$ of $t$, $i < |\\rho|$ and $\\mathrm{acc}(i)$ evaluates in the target environment $\\rho_t$ to a target value that is related, by the flat\u2026","title":"Accessors implementing the free variables of a term"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/eval-env-flat","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.eval_envFlat","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#eval-env-flat","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\rho$ be a source environment, $\\rho_t$ a target environment, $\\mathrm{acc}$ a map from natural numbers to target terms, and $i_1, \\dots, i_n$ a list of natural numbers. Write $\\rho_i$ for the entry of $\\rho$ at index $i$, counting from $0$. Suppose every $i$ in the list is less than the length of $\\rho$ and $\\mathrm{acc}(i)$ evaluates in $\\rho_t$ to a target value related to $\\rho_i$ by the flat value\u2026","title":"Evaluating a flat environment"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/env-flat-length","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.envFlat_length","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#env-flat-length","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\mathrm{acc}$ be a map from natural numbers to target terms and $i_1, \\dots, i_n$ a list of natural numbers. If the flat environment $(\\mathrm{acc}(i_1), (\\dots, (\\mathrm{acc}(i_n), \\mathsf{unit})\\dots))$ evaluates in some target environment to the encoding $(w_1, (\\dots, (w_k, \\mathsf{unit})\\dots))$ of a list of target values $w_1, \\dots, w_k$, then $k = n$.","title":"Length of an evaluated flat environment"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/flat-acc-rel-flat-acc","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.flatAccRel_flatAcc","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#flat-acc-rel-flat-acc","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $b$ be a raw term, let $c = (c_0, \\dots, c_{m-1})$ be the list of variables captured by $\\lambda. b$, let $\\rho$ be a list of source values of length $|\\rho|$, write $\\rho_n$ for the entry of $\\rho$ at index $n$ (counting from $0$), and let $w_0, \\dots, w_{m-1}$ be target values such that $c_k < |\\rho|$ and $\\rho_{c_k}$ is related to $w_k$ by the flat value relation for every $k < m$. If a source value $v$ is\u2026","title":"Flat accessors implement the variables of a closure body"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/convert-flat-correct","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.convertFlat_correct","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#convert-flat-correct","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $\\rho$ be a source environment, $\\rho_t$ a target environment, $\\mathrm{acc}$ a map from natural numbers to target terms, and $t$ a raw term that evaluates to a source value $v$ in $\\rho$. Suppose that for every free variable $i$ of $t$, $i$ is less than the length of $\\rho$ and $\\mathrm{acc}(i)$ evaluates in $\\rho_t$ to a target value related by the flat value relation to the entry $\\rho_i$ of $\\rho$ at index\u2026","title":"Correctness of flat closure conversion"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/closure-convert-flat-correct","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.closureConvertFlat_correct","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#closure-convert-flat-correct","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $t$ be a closed raw term, that is, one with no free variables. If $t$ evaluates to a source value $v$ in the empty environment, then the flat closure conversion of $t$ evaluates in the empty target environment to some target value $w$ related to $v$ by the flat value relation.","title":"Correctness of flat closure conversion for closed programs"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/convert-flat-scoped","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.convertFlat_scoped","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#convert-flat-scoped","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"Let $t$ be a raw term, $m$ a natural number and $\\mathrm{acc}$ a map from natural numbers to target terms. If $\\mathrm{acc}(i)$ is scoped in $m$ for every free variable $i$ of $t$, then the flat conversion of $t$ with accessors $\\mathrm{acc}$ is scoped in $m$. In particular every code body it contains is scoped in $1$, so refers only to its parameter.","title":"Flat closure conversion preserves scoping"},{"id":"ComputerScience/programming-language-semantics/closure-conversion/closure-convert-flat-scoped","kind":"node","lean":"Cslib.Languages.LambdaCalculus.ClosureConversion.closureConvertFlat_scoped","parent":40620,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/closure-conversion/index.html#closure-convert-flat-scoped","scope":"ComputerScience/programming-language-semantics/closure-conversion","status":"fully_proved","summary":"If a raw term $t$ is closed, then its flat closure conversion is scoped in $0$: it has no free variables, and every code body in it is scoped in $1$, so refers only to its parameter.","title":"Flat closure conversion of a closed term is well scoped"},{"data_url":"scopes/ComputerScience/programming-language-semantics/continuation-passing-style-translation.json","id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","kind":"scope","navigation":"module","parent":40619,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","title":"Continuation-passing style translation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-ty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbnTy","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-ty","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ of the simply typed lambda calculus. The call-by-name CPS translation $A^\\ast$ of a simple type $A$ is defined by recursion on $A$: a base type is translated to itself, $b^\\ast = b$, and a function type is translated by $(A \\to B)^\\ast = ((A^\\ast \\to R) \\to R) \\to ((B^\\ast \\to R) \\to R)$.","title":"Call-by-name CPS translation of types"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbnComp","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-comp","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"For an answer type $R$ and a simple type $A$, the type of call-by-name computations of type $A$ is $C(A) = (A^\\ast \\to R) \\to R$, where $A^\\ast$ is the call-by-name CPS translation of $A$.","title":"Computation types of the call-by-name CPS translation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-var","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbnVar","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-var","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $C(A) = (A^\\ast \\to R) \\to R$, where $A^\\ast$ is the call-by-name CPS translation of a simple type $A$. A variable of type $A$ in a context $A_1, \\dots, A_n$ is translated to the variable at the same position in the translated context $C(A_1), \\dots, C(A_n)$, where it has type $C(A)$.","title":"Call-by-name CPS translation of variables"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $C(A) = (A^\\ast \\to R) \\to R$, where $A^\\ast$ is the call-by-name CPS translation of a simple type $A$. A term $t$ of type $A$ in a context $A_1, \\dots, A_n$ is translated to a term $\\overline{t}$ of type $C(A)$ in the context $C(A_1), \\dots, C(A_n)$, by recursion on $t$: a variable $x$ is translated to the variable at the same position; an abstraction by\u2026","title":"Call-by-name CPS translation of terms"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.CbnRel","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-rel","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation $I$ of the base types. Write $D(A)$ for the set of values of a simple type $A$ ($D(b) = I(b)$ for a base type $b$, and $D(A \\to B)$ is the set of functions from $D(A)$ to $D(B)$), $A^\\ast$ for the call-by-name CPS translation of $A$, and $C(A) = (A^\\ast \\to R) \\to R$. The relation $a \\sim_A a'$ between $a \\in D(A)$ and $a' \\in D(A^\\ast)$ is defined by recursion on $A$.\u2026","title":"Logical relation for the call-by-name CPS translation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-comp-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.CbnCompRel","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-comp-rel","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation $I$ of the base types; write $D(A)$ for the set of values of a simple type $A$, $A^\\ast$ for its call-by-name CPS translation, $C(A) = (A^\\ast \\to R) \\to R$, and $\\sim_A$ for the call-by-name logical relation between $D(A)$ and $D(A^\\ast)$. A computation $c \\in D(C(A))$ is related to a value $a \\in D(A)$ when there is $v \\in D(A^\\ast)$ with $a \\sim_A v$ such that\u2026","title":"Related computations in the call-by-name CPS translation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-env-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.CbnEnvRel","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-env-rel","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation of the base types, and write $C(A) = (A^\\ast \\to R) \\to R$, where $A^\\ast$ is the call-by-name CPS translation of a simple type $A$. Let $\\eta$ be an environment for a context $\\Gamma = A_1, \\dots, A_n$, assigning to each variable $x$ of type $A$ a value $\\eta(x)$ of type $A$, and let $\\eta'$ be an environment for the translated context $C(A_1), \\dots, C(A_n)$. Then\u2026","title":"Related environments for the call-by-name CPS translation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-correct","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_correct","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-correct","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation of the base types; write $D(A)$ for the set of values of a simple type $A$, $A^\\ast$ for its call-by-name CPS translation, and $\\sim_A$ for the call-by-name logical relation. Let $t$ be a term of type $A$ in a context $\\Gamma$, and let $\\eta$ be an environment for $\\Gamma$ and $\\eta'$ an environment for the translated context such that $\\eta$ and $\\eta'$ are related. Let\u2026","title":"Denotational correctness of the call-by-name CPS translation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-denote-base","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_denote_base","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-denote-base","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation of the base types. Let $t$ be a term of a base type $b$ in a context $\\Gamma$, let $\\eta$ be an environment for $\\Gamma$, and let $\\eta'$ be an environment for the translated context related to $\\eta$. Then for every continuation $k$ from the values of type $b$ to the values of type $R$, the denotation in $\\eta'$ of the call-by-name CPS translation $\\overline{t}$ of $t$,\u2026","title":"The call-by-name CPS translation at base types"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-denote-base-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_denote_base_id","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-denote-base-id","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an interpretation of the base types. Let $t$ be a term of a base type $b$ in a context $\\Gamma$, take the answer type to be $b$, let $\\eta$ be an environment for $\\Gamma$, and let $\\eta'$ be an environment for the translated context related to $\\eta$. Then the denotation in $\\eta'$ of the call-by-name CPS translation $\\overline{t}$ of $t$, applied to the identity function on the values of type $b$, equals the\u2026","title":"Running the call-by-name CPS translation with the identity continuation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-ty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cpsTy","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-ty","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"For a fixed answer type $R$, the translation $\\overline{A}$ of a simple type $A$ is defined by recursion on $A$: $\\overline{b} = b$ for a base type $b$, and $\\overline{A \\to B} = \\overline{A} \\to ((\\overline{B} \\to R) \\to R)$.","title":"Call-by-value CPS translation of types"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/comp-ty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.compTy","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#comp-ty","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"For an answer type $R$ and a simple type $A$, the type of CPS computations returning a value of type $A$ is $(\\overline{A} \\to R) \\to R$, where $\\overline{A}$ is the call-by-value CPS translation of $A$ for answer type $R$.","title":"CPS computation type"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-var","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cpsVar","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-var","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\overline{A}$ for the call-by-value CPS translation of a type $A$. A de Bruijn variable of type $A$ in a context $A_1, \\ldots, A_n$ is sent to the variable at the same position in the translated context $\\overline{A_1}, \\ldots, \\overline{A_n}$, which has type $\\overline{A}$.","title":"CPS translation of variables"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\overline{A}$ for the call-by-value CPS translation of a type $A$ and $\\overline{\\Gamma}$ for the context obtained by translating each type of a context $\\Gamma$. A term $M$ of type $A$ in context $\\Gamma$ is sent to a term $\\overline{M}$ of type $(\\overline{A} \\to R) \\to R$ in context $\\overline{\\Gamma}$, defined by recursion on $M$ by $\\overline{x} = \\lambda k.\\,k\\,x$,\u2026","title":"Call-by-value CPS translation of terms"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.Rel","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#rel","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation $I$ of the base types; write $D(A)$ for the set interpreting a simple type $A$ under $I$ and $\\overline{A}$ for the call-by-value CPS translation of $A$. The relation $a \\sim_A a'$ between $a \\in D(A)$ and $a' \\in D(\\overline{A})$ is defined by recursion on $A$: at a base type $b$, $a \\sim_b a'$ means $a = a'$; at a function type, $f \\sim_{A \\to B} f'$ means that for all\u2026","title":"Logical relation for the CPS translation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/env-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.EnvRel","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#env-rel","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation $I$ of the base types, and write $\\overline{\\Gamma}$ for the context obtained by translating each type of a context $\\Gamma$ by the call-by-value CPS translation. An environment $\\eta$ for $\\Gamma$ and an environment $\\eta'$ for $\\overline{\\Gamma}$ are related when, for every variable $x$ of type $A$ in $\\Gamma$, the value $\\eta(x)$ is related at type $A$, by the CPS\u2026","title":"Related environments"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-correct","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_correct","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-correct","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation $I$ of the base types; write $D(A)$ for the set interpreting a type $A$, $\\overline{A}$, $\\overline{\\Gamma}$ and $\\overline{t}$ for the call-by-value CPS translations of types, contexts and terms, and $[t]_\\eta$ for the denotation of a term $t$ in an environment $\\eta$. Let $t$ be a term of type $A$ in context $\\Gamma$, and let $\\eta$ and $\\eta'$ be related environments\u2026","title":"Denotational correctness of the CPS translation (Meyer\u2013Wand)"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-denote-base","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_denote_base","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-denote-base","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and an interpretation $I$ of the base types; write $D(A)$ for the set interpreting a type $A$, $\\overline{\\Gamma}$ and $\\overline{t}$ for the call-by-value CPS translations of contexts and terms, and $[t]_\\eta$ for the denotation of a term $t$ in an environment $\\eta$. Let $t$ be a term of base type $b$ in context $\\Gamma$, and let $\\eta$ and $\\eta'$ be related environments for $\\Gamma$ and\u2026","title":"CPS translation at a base type"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-denote-base-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_denote_base_id","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-denote-base-id","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an interpretation $I$ of the base types, let $t$ be a term of base type $b$ in context $\\Gamma$, and take the answer type to be $b$; write $\\overline{\\Gamma}$ and $\\overline{t}$ for the call-by-value CPS translations of $\\Gamma$ and $t$, and $[t]_\\eta$ for the denotation of $t$ in an environment $\\eta$. If $\\eta$ and $\\eta'$ are related environments for $\\Gamma$ and $\\overline{\\Gamma}$, then\u2026","title":"CPS translation at a base type with the identity continuation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cpsRen","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-ren","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\llbracket \\Gamma \\rrbracket$ for the context obtained by applying the call-by-value CPS type translation to every entry of $\\Gamma$. A renaming $\\rho$ from $\\Gamma$ to $\\Delta$ induces the renaming from $\\llbracket \\Gamma \\rrbracket$ to $\\llbracket \\Delta \\rrbracket$ that sends the translation of each variable $x$ of $\\Gamma$ (the variable at the same position) to the translation\u2026","title":"Translated renaming"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_rename","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-rename","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\llbracket \\cdot \\rrbracket$ for the call-by-value CPS translation of types, contexts (entrywise) and terms. For every renaming $\\rho$ from a context $\\Gamma$ to a context $\\Delta$ and every term $t$ of type $A$ in $\\Gamma$, the translation of the renamed term $t[\\rho]$ equals the renaming of $\\llbracket t \\rrbracket$ along the translated renaming from\u2026","title":"The call-by-value CPS translation commutes with renaming"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_weaken","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-weaken","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\llbracket \\cdot \\rrbracket$ for the call-by-value CPS translation of types, contexts (entrywise) and terms. For every term $t$ of type $A$ in a context $\\Gamma$ and every type $B$, the translation of the weakening of $t$ to the context $B, \\Gamma$ is the weakening of $\\llbracket t \\rrbracket$ to the context $\\llbracket B \\rrbracket, \\llbracket \\Gamma \\rrbracket$.","title":"The CPS translation commutes with weakening"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/psi-rel","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.PsiRel","parent":40664,"reading_title":{"format":"inline-parts-v1","parts":[["text","Plotkin's value translation "],["math","\\(\\Psi\\)"]],"source":"Plotkin's value translation $\\Psi$"},"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#psi-rel","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$, and write $A^*$ for the call-by-value CPS translation of a type $A$, $\\Gamma^*$ for the context obtained by translating each type of a context $\\Gamma$, and $\\overline{M}$ for the CPS translation of a term $M$. The relation $\\Psi$ between terms of type $A$ in context $\\Gamma$ and terms of type $A^*$ in context $\\Gamma^*$ holds in exactly two cases: a variable $x$ of $\\Gamma$ is related to\u2026","title":"Plotkin's value translation $\\Psi$"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_subst","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-subst","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\llbracket \\cdot \\rrbracket$ for the call-by-value CPS translation of types, contexts (entrywise) and terms; for a variable $x$ of $\\Gamma$, $\\llbracket x \\rrbracket$ is the variable of $\\llbracket \\Gamma \\rrbracket$ at the same position. Let $\\sigma$ be a substitution from $\\Gamma$ to $\\Delta$ and $\\tau$ a substitution from $\\llbracket \\Gamma \\rrbracket$ to\u2026","title":"The CPS translation commutes with substitution of values and variables"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_instantiate","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-instantiate","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\llbracket \\cdot \\rrbracket$ for the call-by-value CPS translation of types, contexts (entrywise) and terms. Let $N$ be a term of type $B$ in a context $A, \\Gamma$, let $v$ be a term of type $A$ in $\\Gamma$, and let $p$ be a term of type $\\llbracket A \\rrbracket$ in $\\llbracket \\Gamma \\rrbracket$ related to $v$ by the value translation $\\Psi$ (a variable is related to the\u2026","title":"The CPS translation commutes with instantiation by a value or variable"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.PsiRel.value","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#value","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$. If a term $v$ of type $A$ in context $\\Gamma$ is related by Plotkin's value translation $\\Psi$ to a term $p$ (of the translated type in the translated context), and $v$ is a value, that is, a $\\lambda$-abstraction, then $p$ is a value.","title":"The value translation preserves values"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/eval-app-lam-iff","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.eval_app_lam_iff","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#eval-app-lam-iff","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Let $N$ be a term of type $D$ in a context $C, \\Xi$ and let $v$ be a value of type $C$ in $\\Xi$. Writing $N[v]$ for $N$ with its newest free variable replaced by $v$, for every term $w$ of type $D$ the application $(\\lambda x. N)\\, v$ evaluates to $w$ under call-by-value big-step evaluation if and only if $N[v]$ evaluates to $w$.","title":"A \u03b2-redex with a value argument evaluates as its contractum"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/eval-app-congr-fun","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.eval_app_congr_fun","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#eval-app-congr-fun","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Let $F$ and $F'$ be terms of type $C \\to D$ in a context $\\Xi$ such that, for every term $u$, $F$ evaluates to $u$ under call-by-value big-step evaluation if and only if $F'$ does. Then for every term $a$ of type $C$ and every term $w$ of type $D$ in $\\Xi$, the application $F\\, a$ evaluates to $w$ if and only if $F'\\, a$ evaluates to $w$.","title":"Evaluation of an application depends on the function only through its results"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/instantiate-ret","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.instantiate_ret","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#instantiate-ret","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Let $V$ be a term of type $X$ and $K$ a term of type $X \\to Z$ in a context $\\Xi$. In the context $X \\to Z, \\Xi$, let $k$ be the newest variable (of type $X \\to Z$) and $V'$ the weakening of $V$. Replacing $k$ by $K$ in the term $k\\, V'$ gives $K\\, V$.","title":"Instantiating a return to the continuation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-simulation","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_simulation","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-simulation","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\llbracket \\cdot \\rrbracket$ for the call-by-value CPS translation of types, contexts (entrywise) and terms. If a term $M$ of type $A$ in a context $\\Gamma$ evaluates to $V$ under call-by-value big-step evaluation, then there is a term $p$ related to $V$ by the value translation $\\Psi$ (a variable is related to the corresponding variable, $\\lambda x. N$ to\u2026","title":"Plotkin's simulation theorem, big-step form"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cps-simulation-lam","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cps_simulation_lam","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cps-simulation-lam","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\llbracket \\cdot \\rrbracket$ for the call-by-value CPS translation of types, contexts (entrywise) and terms. If a term $M$ of type $A \\to B$ in a context $\\Gamma$ evaluates to $\\lambda x. N$ under call-by-value big-step evaluation, then for every value $K$ of type $\\llbracket A \\to B \\rrbracket \\to R$ in $\\llbracket \\Gamma \\rrbracket$ and every term $w$ of type $R$, the application\u2026","title":"Simulation for terms evaluating to an abstraction"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/eval-n","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.EvalN","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#eval-n","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Call-by-name big-step evaluation $M \\Downarrow_n V$ of the simply typed $\\lambda$-calculus relates two terms of the same type in the same context $\\Gamma$. It is the least relation closed under two rules: every abstraction $\\lambda x.\\,N$ evaluates to itself, and if $f \\Downarrow_n \\lambda x.\\,N$ and $N[a/x] \\Downarrow_n v$, then $f\\,a \\Downarrow_n v$. The argument $a$ is substituted for $x$ without being evaluated.","title":"Call-by-name big-step evaluation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbnRen","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-ren","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$. The translation of a context $\\Gamma$ replaces each type $A$ in it by the computation type $(A^* \\to R) \\to R$, where $A^*$ is the call-by-name translation of $A$, and a variable $x$ of $\\Gamma$ becomes the variable $\\overline{x}$ at the same position of the translated context. A renaming $\\rho$ from $\\Gamma$ to $\\Delta$, a type-preserving map from the variables of $\\Gamma$ to those of\u2026","title":"The call-by-name CPS translation of a renaming"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_rename","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-rename","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\overline{t}$ for the call-by-name CPS translation of a term $t$. For a renaming $\\rho$ from a context $\\Gamma$ to a context $\\Delta$, let $\\overline{\\rho}$ be the renaming of the translated contexts that sends the translation of each variable $x$ of $\\Gamma$ to the translation of $\\rho(x)$. Then for every term $t$ in context $\\Gamma$, the translation of the renamed term $t[\\rho]$\u2026","title":"The call-by-name CPS translation commutes with renaming"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_weaken","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-weaken","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\overline{t}$ for the call-by-name CPS translation of a term $t$. For every term $t$ in a context $\\Gamma$ and every type $B$, translating the weakening of $t$ to the context $B, \\Gamma$ (which adds a fresh newest variable of type $B$) gives the weakening of $\\overline{t}$ to the translated context, which adds a fresh newest variable of the translated type $(B^* \\to R) \\to R$,\u2026","title":"The call-by-name CPS translation commutes with weakening"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_subst","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-subst","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\overline{t}$ for the call-by-name CPS translation of a term $t$ and $\\overline{x}$ for the translation of a variable $x$, the variable at the same position of the translated context. Let $\\sigma$ be a substitution from a context $\\Gamma$ to a context $\\Delta$ (assigning to each variable of $\\Gamma$ a term of the same type in context $\\Delta$), and let $\\tau$ be a substitution from\u2026","title":"The call-by-name CPS translation commutes with substitution"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_instantiate","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-instantiate","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$ and write $\\overline{t}$ for the call-by-name CPS translation of a term $t$. Let $N$ be a term of type $B$ in a context $A, \\Gamma$ whose newest variable $x$ has type $A$, and let $a$ be a term of type $A$ in context $\\Gamma$. Then translating the instantiation $N[a/x]$ of $x$ by $a$ gives the instantiation of the newest variable of $\\overline{N}$ by $\\overline{a}$:\u2026","title":"The call-by-name CPS translation commutes with instantiation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_value","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-value","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$. For every closed term $t$, the call-by-name CPS translation of $t$ is a value, that is, a $\\lambda$-abstraction.","title":"The translation of a closed term is a value"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/instantiate-cbn-app","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.instantiate_cbnApp\u2081","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#instantiate-cbn-app","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Let $\\Xi$ be a context and $W$, $V$, $Z$ types, and write $t^{\\uparrow}$ for the weakening of a term $t$ by one new newest variable. Let $F$ be a term of type $((W \\Rightarrow ((V \\Rightarrow Z) \\Rightarrow Z)) \\Rightarrow Z) \\Rightarrow Z$, $N$ a term of type $W$ and $K$ a term of type $V \\Rightarrow Z$, all in context $\\Xi$. In the context $\\Xi$ extended by a newest variable $k$ of type $V \\Rightarrow Z$,\u2026","title":"Administrative step of the call-by-name translation of an application"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/eval-cbn-lam-iff","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.eval_cbn_lam_iff","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#eval-cbn-lam-iff","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$, and write $\\overline{t}$ for the call-by-name CPS translation of a term $t$, $T^*$ for the translation of a type $T$, and $\\Gamma^*$ for the context obtained from $\\Gamma$ by replacing each type $C$ with $(C^* \\to R) \\to R$. Let $N$ be a term of type $B$ in context $A, \\Gamma$, let $K$ be a value of type $(A \\to B)^* \\to R$ in context $\\Gamma^*$, and let $w$ be a term of type $R$ in context\u2026","title":"The translation of an abstraction passes the translated function to its continuation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-eval-n","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_evalN","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-eval-n","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$, and write $\\overline{M}$ for the call-by-name CPS translation of a term $M$ and $T^*$ for the translation of a type $T$. Let $M$ and $V$ be closed terms of type $T$ such that $M$ evaluates to $V$ under call-by-name big-step evaluation. Then for every closed value $K$ of type $T^* \\to R$ and every closed term $w$ of type $R$, the application $\\overline{M}\\,K$ evaluates to $w$ under\u2026","title":"Call-by-name evaluation is preserved by the CPS translation"},{"id":"ComputerScience/programming-language-semantics/continuation-passing-style-translation/cbn-simulation","kind":"node","lean":"Cslib.Languages.LambdaCalculus.Stlc.CPS.cbn_simulation","parent":40664,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/continuation-passing-style-translation/index.html#cbn-simulation","scope":"ComputerScience/programming-language-semantics/continuation-passing-style-translation","status":"fully_proved","summary":"Fix an answer type $R$, and write $\\overline{M}$ for the call-by-name CPS translation of a term $M$ and $T^*$ for the translation of a type $T$. Let $M$ be a closed term of type $A \\to B$ that evaluates to $\\lambda x.\\,N$ under call-by-name big-step evaluation. Then for every closed value $K$ of type $(A \\to B)^* \\to R$ and every closed term $w$ of type $R$, the application $\\overline{M}\\,K$ evaluates to $w$ under\u2026","title":"Plotkin's simulation theorem for the call-by-name CPS translation"},{"data_url":"scopes/ComputerScience/programming-language-semantics/nominal-sets.json","id":"ComputerScience/programming-language-semantics/nominal-sets","kind":"scope","navigation":"module","parent":40619,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","title":"Nominal sets"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/fin-perm-subgroup","kind":"node","lean":"Cslib.Nominal.finPermSubgroup","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#fin-perm-subgroup","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The subgroup of finitely supported permutations of `Atom`: permutations that fix all atoms outside some finite set.","title":"Subgroup of finitely supported permutations of atoms"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/fin-perm","kind":"node","lean":"Cslib.Nominal.FinPerm","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#fin-perm","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The group of finitely supported permutations of `Atom`.","title":"Group of finitely supported permutations"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap","kind":"node","lean":"Cslib.Nominal.swap","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The transposition of two atoms, as a finite permutation.","title":"Transposition of two atoms"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-smul","kind":"node","lean":"Cslib.Nominal.swap_smul","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-smul","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Applying a transposition is applying the underlying `Equiv.swap`.","title":"Action of a transposition"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-mul-self","kind":"node","lean":"Cslib.Nominal.swap_mul_self","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-mul-self","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A transposition is an involution, stated for the `FinPerm` group.","title":"A transposition squares to the identity"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-inv","kind":"node","lean":"Cslib.Nominal.swap_inv","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-inv","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A transposition is its own inverse.","title":"A transposition is its own inverse"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-self","kind":"node","lean":"Cslib.Nominal.swap_self","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-self","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A transposition of an atom with itself is the identity.","title":"Transposition of an atom with itself is the identity"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-comm","kind":"node","lean":"Cslib.Nominal.swap_comm","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-comm","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Transpositions are symmetric.","title":"Symmetry of transpositions"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-conj","kind":"node","lean":"Cslib.Nominal.swap_conj","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-conj","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Conjugation of a transposition: `\u03c3 * swap b c * \u03c3\u207b\u00b9 = swap (\u03c3 b) (\u03c3 c)`.","title":"Conjugation of a transposition"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-smul-of-ne-of-ne","kind":"node","lean":"Cslib.Nominal.swap_smul_of_ne_of_ne","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-smul-of-ne-of-ne","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A transposition fixes every atom distinct from its two swapped atoms.","title":"Transposition fixes atoms other than the swapped pair"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supports","kind":"node","lean":"Cslib.Nominal.Supports","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supports","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"`S` supports `x` when every finite permutation fixing `S` pointwise fixes `x`.","title":"Supporting set of atoms for an element"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/mono","kind":"node","lean":"Cslib.Nominal.Supports.mono","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#mono","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A support can be enlarged: fixing a larger set pointwise is a stronger hypothesis.","title":"Enlarging a support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/nominal-set","kind":"node","lean":"Cslib.Nominal.NominalSet","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#nominal-set","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A type with an action of finite permutations in which every element has a finite support: a *nominal set*.","title":"Nominal set"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/prod","kind":"node","lean":"Cslib.Nominal.Supports.prod","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#prod","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A pair is supported by the union of supports of its components.","title":"A pair is supported by the union of its components' supports"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/fresh","kind":"node","lean":"Cslib.Nominal.Fresh","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#fresh","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"`a` is *fresh* for `x`, written `a # x` in the nominal literature, when some finite support of `x` does not contain `a`.","title":"Freshness of an atom for an element"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/exists-fresh","kind":"node","lean":"Cslib.Nominal.exists_fresh","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#exists-fresh","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Every element has a fresh atom: choose one outside any of its supports.","title":"Existence of fresh atoms"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-smul-of-supports-not-mem","kind":"node","lean":"Cslib.Nominal.swap_smul_of_supports_not_mem","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-smul-of-supports-not-mem","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A transposition of two atoms both outside a support of `x` fixes `x`.","title":"Transposition of atoms outside a support fixes the element"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/swap-smul-of-fresh-fresh","kind":"node","lean":"Cslib.Nominal.swap_smul_of_fresh_fresh","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#swap-smul-of-fresh-fresh","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Swapping two fresh atoms fixes `x`. The proof uses a third atom `c` fresh for both witnessing supports, and the conjugation identity `swap a b = swap a c * swap b c * swap a c`.","title":"Swapping two fresh atoms fixes the element"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/smul-eq-of-fresh","kind":"node","lean":"Cslib.Nominal.smul_eq_of_fresh","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#smul-eq-of-fresh","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A permutation that fixes every atom outside a finite set `F` of fresh atoms fixes `x`, by induction on `F`. If `\u03c0 a = a` we apply the induction hypothesis to `\u03c0`; otherwise we factor `\u03c0 = swap a b * \u03c0'` with `b = \u03c0 a`, where `\u03c0'` fixes `a` and has support inside `F \\ {a}`, and `swap a b` fixes `x` since `a` and `b` are fresh.","title":"Permutations moving only fresh atoms fix the element"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/inter","kind":"node","lean":"Cslib.Nominal.Supports.inter","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#inter","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The intersection of two supports is a support: a permutation fixing the intersection fixes `x`, since every atom it can move (outside some finite support of the permutation, and outside the intersection) is fresh for `x`.","title":"Intersection of two supports is a support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/image","kind":"node","lean":"Cslib.Nominal.Supports.image","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#image","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A permutation moves a support of `x` to a support of `\u03c0 \u2022 x`.","title":"Permutation image of a support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/exists-least-support","kind":"node","lean":"Cslib.Nominal.exists_least_support","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#exists-least-support","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A least finite support exists: among the supports of `x`, one is contained in every support. The proof is by strong induction on a chosen support `S\u2080`: either every element of `S\u2080` lies in every support (so `S\u2080` itself is least), or some `a \u2208 S\u2080` is missing from a support `S\u2090`, and the induction hypothesis applies to the strictly smaller support `S\u2090 \u2229 S\u2080`.","title":"Existence of a least finite support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supp","kind":"node","lean":"Cslib.Nominal.supp","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supp","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The *least* finite support of `x`: a support contained in every support of `x`.","title":"Least finite support of an element"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supp-supports","kind":"node","lean":"Cslib.Nominal.supp_supports","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supp-supports","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The least support is a support.","title":"Least support is a support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supp-subset","kind":"node","lean":"Cslib.Nominal.supp_subset","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supp-subset","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The least support is contained in every support: minimality.","title":"Minimality of the least support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/mem-supp-iff","kind":"node","lean":"Cslib.Nominal.mem_supp_iff","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#mem-supp-iff","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Membership in the least support: `a \u2208 supp (Atom := Atom) x` iff `a` lies in every support of `x`.","title":"Membership in the least support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/fresh-iff-not-mem-supp","kind":"node","lean":"Cslib.Nominal.fresh_iff_not_mem_supp","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#fresh-iff-not-mem-supp","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Freshness is exactly non-membership in the least support.","title":"Freshness is non-membership in the least support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supp-smul","kind":"node","lean":"Cslib.Nominal.supp_smul","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supp-smul","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Support is equivariant: permuting `x` permutes its support.","title":"Equivariance of the least support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supp-atom","kind":"node","lean":"Cslib.Nominal.supp_atom","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supp-atom","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The support of an atom is its singleton.","title":"Support of an atom"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/smul-eq-of-agree-on-supp","kind":"node","lean":"Cslib.Nominal.smul_eq_of_agree_on_supp","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#smul-eq-of-agree-on-supp","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Two permutations that agree on the support of `x` act equally on `x`: their ratio fixes the support pointwise, hence fixes `x`.","title":"Permutations agreeing on the support act equally"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/is-equivariant","kind":"node","lean":"Cslib.Nominal.IsEquivariant","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#is-equivariant","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"An element fixed by every permutation has the empty set as a support.","title":"Equivariant elements of a nominal set"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supports-empty-of-equivariant","kind":"node","lean":"Cslib.Nominal.supports_empty_of_equivariant","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supports-empty-of-equivariant","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"An equivariant element is supported by the empty set.","title":"Equivariant elements are supported by the empty set"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supp-eq-empty-of-equivariant","kind":"node","lean":"Cslib.Nominal.supp_eq_empty_of_equivariant","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supp-eq-empty-of-equivariant","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"An equivariant element has empty support.","title":"Equivariant elements have empty support"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-rel","kind":"node","lean":"Cslib.Nominal.AbsRel","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-rel","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"\u03b1-equivalence relation on atom\u2013element pairs. \u03b1-equivalence: equal pairs, or a renaming of the bound atom by a fresh transposition.","title":"\u03b1-equivalence relation on atom\u2013element pairs"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-rel-refl","kind":"node","lean":"Cslib.Nominal.absRel_refl","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-rel-refl","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Reflexivity of \u03b1-equivalence. \u03b1-equivalence is reflexive.","title":"Reflexivity of \u03b1-equivalence"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-rel-symm","kind":"node","lean":"Cslib.Nominal.absRel_symm","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-rel-symm","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Symmetry of \u03b1-equivalence. \u03b1-equivalence is symmetric. Symmetry uses equivariance of support (`supp_smul`).","title":"Symmetry of \u03b1-equivalence"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-rel-trans","kind":"node","lean":"Cslib.Nominal.absRel_trans","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-rel-trans","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Transitivity of \u03b1-equivalence. \u03b1-equivalence is transitive. Transitivity uses the fact that two permutations agreeing on `supp y` act equally on `y` (`smul_eq_of_agree_on_supp`).","title":"Transitivity of \u03b1-equivalence"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-setoid","kind":"node","lean":"Cslib.Nominal.absSetoid","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-setoid","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The setoid of \u03b1-equivalence.","title":"Setoid of \u03b1-equivalence"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs","kind":"node","lean":"Cslib.Nominal.Abs","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Atom abstraction: `[a]x`, the quotient of `Atom \u00d7 X` by \u03b1-equivalence.","title":"Atom abstraction of a nominal set"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-2","kind":"node","lean":"Cslib.Nominal.abs","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-2","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The abstraction `[a]x`.","title":"Abstraction of an element at a bound atom"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-eq-iff","kind":"node","lean":"Cslib.Nominal.abs_eq_iff","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-eq-iff","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"Equality of abstractions is \u03b1-equivalence.","title":"Equality of abstractions is \u03b1-equivalence"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-eq-of-same-atom","kind":"node","lean":"Cslib.Nominal.abs_eq_of_same_atom","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-eq-of-same-atom","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"An abstraction with the same bound atom determines its body.","title":"Abstractions with the same bound atom have equal bodies"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/abs-smul","kind":"node","lean":"Cslib.Nominal.absSmul","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#abs-smul","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The action of a permutation on abstractions, applied to a representative.","title":"Permutation action on abstractions"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/smul-abs","kind":"node","lean":"Cslib.Nominal.smul_abs","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#smul-abs","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"A permutation applied to an abstraction, on representatives.","title":"Permutation action on abstractions computed on representatives"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supports-erase-abs","kind":"node","lean":"Cslib.Nominal.supports_erase_abs","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supports-erase-abs","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The action of a permutation on an abstraction fixes it when the permutation fixes `supp x \\ {a}` pointwise.","title":"Abstractions are supported by the body's support minus the atom"},{"id":"ComputerScience/programming-language-semantics/nominal-sets/supp-abs","kind":"node","lean":"Cslib.Nominal.supp_abs","parent":40707,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/nominal-sets/index.html#supp-abs","scope":"ComputerScience/programming-language-semantics/nominal-sets","status":"fully_proved","summary":"The support of an abstraction is the body's support minus the bound atom.","title":"Support of an atom abstraction"},{"data_url":"scopes/ComputerScience/programming-language-semantics/operational-semantics-of-imp.json","id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","kind":"scope","navigation":"module","parent":40619,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","title":"Operational semantics of IMP"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/state","kind":"node","lean":"Cslib.IMP.State","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#state","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"A state of an IMP program is a function $\\sigma : \\mathbb{N} \\to \\mathbb{N}$ assigning a natural-number value $\\sigma(i)$ to each variable, where variables are identified by natural numbers $i$.","title":"IMP program states"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/aexp","kind":"node","lean":"Cslib.IMP.AExp","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#aexp","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"The arithmetic expressions of IMP are built inductively from the constants $n$, one for each natural number $n$, and the variables $x_i$, one for each natural number $i$: if $a_1$ and $a_2$ are arithmetic expressions, then so are $a_1 + a_2$, $a_1 - a_2$ and $a_1 \\cdot a_2$.","title":"Arithmetic expressions of IMP"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/bexp","kind":"node","lean":"Cslib.IMP.BExp","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#bexp","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"The Boolean expressions of IMP are built inductively from the constants true and false and the comparisons $a_1 = a_2$ and $a_1 \\le a_2$ of arithmetic expressions $a_1, a_2$: if $b$, $b_1$ and $b_2$ are Boolean expressions, then so are the negation $\\neg b$ and the conjunction $b_1 \\wedge b_2$.","title":"Boolean expressions of IMP"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/cmd","kind":"node","lean":"Cslib.IMP.Cmd","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#cmd","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"The commands of the while-language IMP are built inductively from skip and the assignments $x_i := a$ of an arithmetic expression $a$ to the variable $x_i$ with index $i \\in \\mathbb{N}$: if $c$, $c_1$ and $c_2$ are commands and $b$ is a Boolean expression, then so are the sequential composition $c_1 ; c_2$, the conditional if $b$ then $c_1$ else $c_2$, and the loop while $b$ do $c$.","title":"Commands of IMP"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/eval","kind":"node","lean":"Cslib.IMP.AExp.eval","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#eval","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"The value of an arithmetic expression in a state $\\sigma : \\mathbb{N} \\to \\mathbb{N}$ is the natural number defined by recursion on the expression: a constant $n$ has value $n$, the variable $x_i$ with index $i$ has value $\\sigma(i)$, and for arithmetic expressions $a_1$ and $a_2$, the expressions $a_1 + a_2$, $a_1 - a_2$ and $a_1 \\cdot a_2$ have as value the sum, the truncated difference and the product of the\u2026","title":"Evaluation of IMP arithmetic expressions"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/eval-2","kind":"node","lean":"Cslib.IMP.BExp.eval","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#eval-2","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"The truth value of a Boolean expression in a state $\\sigma : \\mathbb{N} \\to \\mathbb{N}$ is defined by recursion on the expression: the constants true and false have values true and false; for arithmetic expressions $a_1$ and $a_2$, the comparison $a_1 = a_2$ is true exactly when the values of $a_1$ and $a_2$ in $\\sigma$ are equal, and $a_1 \\le a_2$ is true exactly when the value of $a_1$ is at most the value of\u2026","title":"Evaluation of IMP Boolean expressions"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/big-step","kind":"node","lean":"Cslib.IMP.BigStep","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#big-step","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"The big-step relation says when running a command $c$ from a state $s$ terminates in a state $t$; we then say that $c$ takes $s$ to $t$. It is the least relation satisfying the following rules. The command $\\mathtt{skip}$ takes $s$ to $s$. The assignment $x := a$ takes $s$ to the state $s[x \\mapsto a(s)]$, which agrees with $s$ except that $x$ has the value $a(s)$ of the arithmetic expression $a$ in $s$. If $c_1$\u2026","title":"Big-step semantics of IMP"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/deterministic","kind":"node","lean":"Cslib.IMP.BigStep.deterministic","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#deterministic","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"For every command $c$ and every state $s$, if $c$ run from $s$ terminates in a state $t_1$ and also in a state $t_2$ under the big-step semantics, then $t_1 = t_2$.","title":"Determinism of IMP big-step semantics"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/config","kind":"node","lean":"Cslib.IMP.Config","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#config","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"A configuration of IMP is a pair $(c, \\sigma)$ of a command $c$ and a state $\\sigma$.","title":"IMP configuration"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/step","kind":"node","lean":"Cslib.IMP.Step","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#step","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"The single-step relation $\\to$ on IMP configurations is the smallest relation satisfying the following rules, for all variables $x$, arithmetic expressions $a$, Boolean expressions $b$, commands $c, c_1, c_1', c_2$ and states $\\sigma, \\sigma'$. Assignment: $(x := a, \\sigma) \\to (\\mathtt{skip}, \\sigma[x \\mapsto v])$, where $v$ is the value of $a$ in $\\sigma$. Sequencing:\u2026","title":"IMP small-step relation"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/steps","kind":"node","lean":"Cslib.IMP.Steps","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#steps","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"The relation $\\to^*$ on IMP configurations is the reflexive, transitive closure of the small-step relation: $x \\to^* y$ when $y$ is reached from $x$ by a finite sequence of zero or more small steps.","title":"IMP multi-step relation"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/terminates","kind":"node","lean":"Cslib.IMP.Terminates","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#terminates","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"A command $c$ terminates from an initial state $\\sigma$ in a final state $\\tau$ when the configuration $(c, \\sigma)$ reaches $(\\mathtt{skip}, \\tau)$ in finitely many small steps.","title":"Termination of an IMP command"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/deterministic-3","kind":"node","lean":"Cslib.IMP.Step.deterministic","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#deterministic-3","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"For all IMP configurations $x$, $y$, $z$, if $x$ steps to $y$ and $x$ steps to $z$ in one small step, then $y = z$.","title":"Determinism of IMP small-step semantics"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/of-big-step","kind":"node","lean":"Cslib.IMP.Terminates.of_bigStep","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#of-big-step","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"For every IMP command $c$ and states $s$ and $s'$, if $c$ evaluates from $s$ to $s'$ in the big-step semantics, then $c$ started in $s$ terminates by small steps in $s'$: finitely many small steps lead from the configuration $(c, s)$ to $(\\mathrm{skip}, s')$.","title":"Big-step executions terminate in small steps"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/of-step","kind":"node","lean":"Cslib.IMP.BigStep.of_step","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#of-step","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"Let $x = (c, s)$ and $y = (d, t)$ be configurations and $s'$ a state. If $x$ moves to $y$ in one small step and $d$ evaluates from $t$ to $s'$ in the big-step semantics, then $c$ evaluates from $s$ to $s'$ in the big-step semantics.","title":"Prepending one small step to a big-step execution"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/of-steps","kind":"node","lean":"Cslib.IMP.BigStep.of_steps","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#of-steps","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"Let $x = (c, s)$ and $y = (d, t)$ be configurations and $s'$ a state. If $x$ reaches $y$ in finitely many (possibly zero) small steps and $d$ evaluates from $t$ to $s'$ in the big-step semantics, then $c$ evaluates from $s$ to $s'$ in the big-step semantics.","title":"Prepending finitely many small steps to a big-step execution"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/of-terminates","kind":"node","lean":"Cslib.IMP.BigStep.of_terminates","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#of-terminates","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"For every IMP command $c$ and states $s$ and $s'$, if $c$ started in $s$ terminates by small steps in $s'$, then $c$ evaluates from $s$ to $s'$ in the big-step semantics.","title":"Small-step termination implies big-step evaluation"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/terminates-iff-big-step","kind":"node","lean":"Cslib.IMP.terminates_iff_bigStep","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#terminates-iff-big-step","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"For every IMP command $c$ and states $s$ and $s'$, $c$ started in $s$ terminates by small steps in $s'$ if and only if $c$ evaluates from $s$ to $s'$ in the big-step semantics.","title":"Equivalence of small-step and big-step semantics"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/deterministic-2","kind":"node","lean":"Cslib.IMP.Terminates.deterministic","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#deterministic-2","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"For every IMP command $c$ and states $s$, $s_1$, $s_2$, if $c$ started in $s$ terminates by small steps in $s_1$ and also terminates by small steps in $s_2$, then $s_1 = s_2$.","title":"Small-step termination is deterministic"},{"id":"ComputerScience/programming-language-semantics/operational-semantics-of-imp/iff-terminates","kind":"node","lean":"Cslib.IMP.Valid.iff_terminates","parent":40754,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/operational-semantics-of-imp/index.html#iff-terminates","scope":"ComputerScience/programming-language-semantics/operational-semantics-of-imp","status":"fully_proved","summary":"For assertions $P$ and $Q$ on states and an IMP command $c$, the partial-correctness Hoare triple with precondition $P$, command $c$ and postcondition $Q$ is valid if and only if, for all states $s$ and $s'$, whenever $P$ holds in $s$ and $c$ started in $s$ terminates by small steps in $s'$, $Q$ holds in $s'$.","title":"Hoare validity via small-step termination"},{"data_url":"scopes/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine.json","id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","kind":"scope","navigation":"module","parent":40619,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","title":"Compiling IMP to a stack machine"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/instr","kind":"node","lean":"Cslib.IMP.StackMachine.Instr","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#instr","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"An instruction of the stack machine is one of $\\mathsf{push}\\ n$, $\\mathsf{load}\\ x$ and $\\mathsf{store}\\ x$ for natural numbers $n$ and $x$; the arithmetic instructions $\\mathsf{add}$, $\\mathsf{sub}$ and $\\mathsf{mul}$; the comparison and Boolean instructions $\\mathsf{eq}$, $\\mathsf{le}$, $\\mathsf{not}$ and $\\mathsf{and}$; and the jumps $\\mathsf{jump}\\ t$ and $\\mathsf{jumpIfZero}\\ t$ for a natural number $t$. Jump\u2026","title":"Stack machine instructions"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/config","kind":"node","lean":"Cslib.IMP.StackMachine.Config","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#config","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"A configuration of the stack machine is a triple $(pc, \\sigma, s)$ of a program counter $pc \\in \\mathbb{N}$, a stack $\\sigma$, which is a finite list of natural numbers whose head is the top of the stack, and a state $s$ assigning a natural number to each variable.","title":"Stack machine configurations"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/step","kind":"node","lean":"Cslib.IMP.StackMachine.Step","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#step","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"Let $P$ be a program, that is a list of instructions, and write $a :: \\sigma$ for the stack with top $a$ above the stack $\\sigma$. A configuration $(pc, \\sigma, s)$ steps to a configuration $c'$ under $P$ exactly when $pc < |P|$ and, for the instruction at position $pc$ of $P$ (counted from $0$), one of the following holds, where $s[x \\mapsto v]$ is the state $s$ with variable $x$ set to $v$.","title":"Step relation of the stack machine"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/steps","kind":"node","lean":"Cslib.IMP.StackMachine.Steps","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#steps","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"For a program $P$, the multi-step relation on configurations is the reflexive transitive closure of the step relation of the stack machine running $P$: it relates $c$ to $c'$ when $c'$ is obtained from $c$ by zero or more steps.","title":"Multi-step execution of the stack machine"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/deterministic","kind":"node","lean":"Cslib.IMP.StackMachine.Step.deterministic","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#deterministic","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"For every program $P$, the step relation of the stack machine running $P$ is deterministic: if a configuration $x$ steps to $y$ and to $z$, then $y = z$. The fetched instruction and the shape of the stack determine the next configuration.","title":"Determinism of the stack machine"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/no-step-of-length-le","kind":"node","lean":"Cslib.IMP.StackMachine.no_step_of_length_le","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#no-step-of-length-le","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"Let $P$ be a program, that is a list of instructions, and let $(pc, \\sigma, s)$ be a configuration with $|P| \\le pc$. Then the stack machine running $P$ has no step from $(pc, \\sigma, s)$ to any configuration, since there is no instruction at position $pc$ to fetch.","title":"No step past the end of the program"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/alen","kind":"node","lean":"Cslib.IMP.StackMachine.alen","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#alen","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"The instruction count $\\ell(a)$ of an arithmetic expression $a$ is defined by recursion on $a$: a constant or a variable has count $1$, and if $a$ is the sum, difference or product of $a_1$ and $a_2$, then $\\ell(a) = \\ell(a_1) + \\ell(a_2) + 1$.","title":"Instruction count of an arithmetic expression"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/blen","kind":"node","lean":"Cslib.IMP.StackMachine.blen","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#blen","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"The instruction count $\\ell(b)$ of a Boolean expression $b$ is defined by recursion on $b$, using the instruction count $\\ell(a)$ of arithmetic expressions $a$: a truth value has count $1$; the tests $a_1 = a_2$ and $a_1 \\le a_2$ have count $\\ell(a_1) + \\ell(a_2) + 1$; the negation of $b_1$ has count $\\ell(b_1) + 1$; and the conjunction of $b_1$ and $b_2$ has count $\\ell(b_1) + \\ell(b_2) + 1$.","title":"Instruction count of a Boolean expression"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/code-length","kind":"node","lean":"Cslib.IMP.StackMachine.codeLength","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#code-length","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"The instruction count $\\ell(c)$ of an IMP command $c$ is defined by recursion on $c$, using the instruction counts $\\ell(a)$ and $\\ell(b)$ of arithmetic and Boolean expressions: $\\ell(\\mathbf{skip}) = 0$; $\\ell(x := a) = \\ell(a) + 1$; $\\ell(c_1; c_2) = \\ell(c_1) + \\ell(c_2)$; $\\ell(\\mathbf{if}\\ b\\ \\mathbf{then}\\ c_1\\ \\mathbf{else}\\ c_2) = \\ell(b) + 1 + \\ell(c_1) + 1 + \\ell(c_2)$; and\u2026","title":"Instruction count of a command"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/compile-a","kind":"node","lean":"Cslib.IMP.StackMachine.compileA","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#compile-a","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"The compiled code of an arithmetic expression is the list of stack-machine instructions defined by recursion: a constant $n$ compiles to the single instruction push $n$, a variable $x$ to the single instruction load $x$, and the sum, difference or product of $a_1$ and $a_2$ to the code of $a_1$, then the code of $a_2$, then the single instruction add, sub or mul respectively.","title":"Compilation of arithmetic expressions"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/compile-b","kind":"node","lean":"Cslib.IMP.StackMachine.compileB","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#compile-b","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"The compiled code of a Boolean expression is the list of stack-machine instructions defined by recursion, using the compiled code of arithmetic expressions: the truth value true compiles to the single instruction push $1$ and false to push $0$; the test $a_1 = a_2$ (respectively $a_1 \\le a_2$) compiles to the code of $a_1$, then the code of $a_2$, then the equality (respectively order) test instruction; the\u2026","title":"Compilation of Boolean expressions"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/compile","kind":"node","lean":"Cslib.IMP.StackMachine.compile","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#compile","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"The compiled code $C_p(c)$ of an IMP command $c$ at base program counter $p$ is the list of stack-machine instructions defined by recursion on $c$, where jump targets are absolute program counters and $\\ell$ denotes instruction counts: $C_p(\\mathbf{skip})$ is empty; $C_p(x := a)$ is the code of $a$ followed by the instruction store $x$; $C_p(c_1; c_2)$ is $C_p(c_1)$ followed by $C_{p + \\ell(c_1)}(c_2)$. For the\u2026","title":"Compilation of commands"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/fetch-segment","kind":"node","lean":"Cslib.IMP.StackMachine.fetch_segment","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#fetch-segment","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"Let $B$, $S$ and $A$ be lists of instructions and let $i < |S|$ be a natural number. Then the instruction at position $|B| + i$ of the concatenation $B\\,S\\,A$ of the three lists equals the instruction at position $i$ of $S$, positions being counted from $0$.","title":"Fetching from an embedded code segment"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/compile-a-runs","kind":"node","lean":"Cslib.IMP.StackMachine.compileA_runs","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#compile-a-runs","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"Let $a$ be an arithmetic expression with compiled code $C(a)$ and instruction count $\\ell(a)$, let $P$ and $Q$ be lists of instructions, $\\sigma$ a stack and $s$ a state, and write $|P|$ for the length of $P$. The stack machine running the program $P \\, C(a) \\, Q$ (the concatenation of $P$, $C(a)$ and $Q$) goes in finitely many steps from program counter $|P|$, stack $\\sigma$ and state $s$ to program counter\u2026","title":"Compiled arithmetic code pushes the value of the expression"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/compile-b-runs","kind":"node","lean":"Cslib.IMP.StackMachine.compileB_runs","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#compile-b-runs","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"Let $b$ be a Boolean expression with compiled code $C(b)$ and instruction count $\\ell(b)$, let $P$ and $Q$ be lists of instructions, $\\sigma$ a stack and $s$ a state, and write $|P|$ for the length of $P$. The stack machine running the program $P \\, C(b) \\, Q$ (the concatenation of $P$, $C(b)$ and $Q$) goes in finitely many steps from program counter $|P|$, stack $\\sigma$ and state $s$ to program counter\u2026","title":"Compiled Boolean code pushes the truth value of the expression"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/compile-runs","kind":"node","lean":"Cslib.IMP.StackMachine.compile_runs","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#compile-runs","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"Let $c$ be a command and $s$, $s'$ states such that $c$ run from $s$ terminates in $s'$ under the big-step semantics. Let $P$ and $Q$ be lists of instructions, $\\sigma$ a stack, $|P|$ the length of $P$, $C_p(c)$ the compiled code of $c$ at base program counter $p$, and $\\ell(c)$ the instruction count of $c$. Then the stack machine running the program $P \\, C_{|P|}(c) \\, Q$ (the concatenation of $P$, $C_{|P|}(c)$\u2026","title":"Compiled command code simulates big-step execution"},{"id":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/compile-correct","kind":"node","lean":"Cslib.IMP.StackMachine.compile_correct","parent":40775,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine/index.html#compile-correct","scope":"ComputerScience/programming-language-semantics/compiling-imp-to-a-stack-machine","status":"fully_proved","summary":"Let $c$ be a command and $s$, $s'$ states such that $c$ run from $s$ terminates in $s'$ under the big-step semantics. Let $C_0(c)$ be the compiled code of $c$ at base program counter $0$ and $\\ell(c)$ the instruction count of $c$. Then the stack machine running $C_0(c)$ goes in finitely many steps from program counter $0$, the empty stack and state $s$ to program counter $\\ell(c)$, the empty stack and state $s'$.","title":"Correctness of compilation to the stack machine"},{"data_url":"scopes/ComputerScience/programming-language-semantics/denotational-semantics-of-imp.json","id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","kind":"scope","navigation":"module","parent":40619,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","title":"Denotational semantics of IMP"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/while-fun","kind":"node","lean":"Cslib.IMP.whileFun","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#while-fun","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"Given a Boolean expression $b$ (the guard) and a binary relation $B$ on states (the body relation), the functional sends a relation $R$ on states to the relation that relates $s$ to $s'$ exactly when either $b$ evaluates to false in $s$ and $s' = s$, or $b$ evaluates to true in $s$ and there is a state $m$ with $B$ relating $s$ to $m$ and $R$ relating $m$ to $s'$.","title":"One-step unfolding functional of a while loop"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/while-fun-mono","kind":"node","lean":"Cslib.IMP.whileFun_mono","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#while-fun-mono","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"Let $b$ be a Boolean expression, $B$ a binary relation on states, and $\\Phi$ the one-step unfolding functional of a while loop with guard $b$ and body relation $B$. Then $\\Phi$ is monotone for inclusion of relations: for all relations $R_1$ and $R_2$ on states with $R_1 \\subseteq R_2$, we have $\\Phi(R_1) \\subseteq \\Phi(R_2)$.","title":"Monotonicity of the while-loop unfolding functional"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/while-order-hom","kind":"node","lean":"Cslib.IMP.whileOrderHom","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#while-order-hom","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"For a boolean guard $b$ and a relation $B$ between states (the body denotation), this is the map $\\Phi$ on relations between states, ordered by inclusion, where $\\Phi(R)$ relates $s$ to $s'$ exactly when either $b$ is false in $s$ and $s' = s$, or $b$ is true in $s$ and there is a state $m$ with $B(s, m)$ and $R(m, s')$. The map $\\Phi$ is monotone: $R_1 \\subseteq R_2$ implies $\\Phi(R_1) \\subseteq \\Phi(R_2)$.","title":"The while-loop functional as a monotone map"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/denote","kind":"node","lean":"Cslib.IMP.Denote","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#denote","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"The denotation of an IMP command is a binary relation on states, defined by recursion on the command. The command $\\mathtt{skip}$ relates $s$ to $s'$ iff $s' = s$. For a variable identifier $x$ and an arithmetic expression $a$, the assignment $x := a$ relates $s$ to $s'$ iff $s'$ is $s$ with $x$ updated to the value of $a$ in $s$. The sequence $c_1; c_2$ relates $s$ to $s'$ iff there is a state $m$ such that $c_1$\u2026","title":"Relational denotational semantics of IMP"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/while-eq","kind":"node","lean":"Cslib.IMP.Denote.while_eq","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#while-eq","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"Let $b$ be a Boolean expression and $c$ an IMP command, let $W$ be the relational denotation of $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$, and let $\\Phi$ be the one-step unfolding functional of a while loop with guard $b$ and body relation the denotation of $c$. Then $W = \\Phi(W)$.","title":"The while denotation is a fixed point"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/while-iff","kind":"node","lean":"Cslib.IMP.Denote.while_iff","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#while-iff","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"For every Boolean expression $b$, IMP command $c$ and states $s$ and $s'$, the relational denotation of $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$ relates $s$ to $s'$ if and only if either $b$ evaluates to false in $s$ and $s' = s$, or $b$ evaluates to true in $s$ and there is a state $m$ such that the denotation of $c$ relates $s$ to $m$ and the denotation of the loop relates $m$ to $s'$.","title":"Unfolding equation for the while denotation"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/while-ind","kind":"node","lean":"Cslib.IMP.Denote.while_ind","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#while-ind","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"Let $b$ be a Boolean expression, $c$ an IMP command and $P$ a binary relation on states, and let $\\Phi$ be the one-step unfolding functional of a while loop with guard $b$ and body relation the denotation of $c$. If $\\Phi(P) \\subseteq P$, then for all states $s$ and $s'$ related by the relational denotation of $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$, $P$ relates $s$ to $s'$.","title":"Park induction for while loops"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/big-step-of-denote","kind":"node","lean":"Cslib.IMP.Denote.bigStep_of_denote","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#big-step-of-denote","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"For every IMP command $c$ and all states $s$ and $s'$, if the relational denotation of $c$ relates $s$ to $s'$, then executing $c$ from $s$ terminates in $s'$ according to the big-step semantics.","title":"Denotational executions are big-step executions"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/denote-of-big-step","kind":"node","lean":"Cslib.IMP.Denote.denote_of_bigStep","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#denote-of-big-step","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"For every IMP command $c$ and all states $s$ and $s'$, if executing $c$ from $s$ terminates in $s'$ according to the big-step semantics, then the relational denotation of $c$ relates $s$ to $s'$.","title":"Big-step executions are denotational executions"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/iff-big-step","kind":"node","lean":"Cslib.IMP.Denote.iff_bigStep","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#iff-big-step","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"For every IMP command $c$ and all states $s$ and $s'$, the relational denotation of $c$ relates $s$ to $s'$ if and only if executing $c$ from $s$ terminates in $s'$ according to the big-step semantics.","title":"Agreement of the denotational and big-step semantics of IMP"},{"id":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp/deterministic","kind":"node","lean":"Cslib.IMP.Denote.deterministic","parent":40793,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/denotational-semantics-of-imp/index.html#deterministic","scope":"ComputerScience/programming-language-semantics/denotational-semantics-of-imp","status":"fully_proved","summary":"For every IMP command $c$ and all states $s$, $s_1$ and $s_2$, if the relational denotation of $c$ relates $s$ to $s_1$ and $s$ to $s_2$, then $s_1 = s_2$.","title":"The denotational semantics of IMP is deterministic"},{"data_url":"scopes/ComputerScience/programming-language-semantics/heap-semantics-of-imp.json","id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","kind":"scope","navigation":"module","parent":40619,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","title":"Heap semantics of IMP"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/heap","kind":"node","lean":"Cslib.IMP.Heap","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#heap","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"A heap is a finite partial map from addresses to values, where addresses and values are natural numbers: it stores a value $v \\in \\mathbb{N}$ at each address of a finite set of addresses in $\\mathbb{N}$, called the addresses allocated in the heap.","title":"Heap"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/disjoint","kind":"node","lean":"Cslib.IMP.Heap.Disjoint","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#disjoint","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Two heaps $h_1$ and $h_2$ are disjoint when no address is allocated in both: every address allocated in $h_1$ is not allocated in $h_2$.","title":"Disjoint heaps"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/symm","kind":"node","lean":"Cslib.IMP.Heap.Disjoint.symm","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#symm","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"For heaps $h_1$ and $h_2$, if $h_1$ and $h_2$ are disjoint, then $h_2$ and $h_1$ are disjoint.","title":"Symmetry of heap disjointness"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/disjoint-empty-left","kind":"node","lean":"Cslib.IMP.Heap.disjoint_empty_left","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#disjoint-empty-left","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"For every heap $h$, the empty heap is disjoint from $h$ (with the empty heap as the first heap).","title":"The empty heap is disjoint from every heap"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/disjoint-empty-right","kind":"node","lean":"Cslib.IMP.Heap.disjoint_empty_right","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#disjoint-empty-right","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"For every heap $h$, $h$ is disjoint from the empty heap (with the empty heap as the second heap).","title":"Every heap is disjoint from the empty heap"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/erase-union-of-not-mem","kind":"node","lean":"Cslib.IMP.Heap.erase_union_of_notMem","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#erase-union-of-not-mem","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"For heaps $h_1$ and $h_2$, let $h_1 \\cup h_2$ be their left-biased union (at an address allocated in both it keeps the value from $h_1$). If the address $a$ is not allocated in $h_2$, then erasing $a$ from $h_1 \\cup h_2$ gives the union of $h_2$ with the heap obtained from $h_1$ by erasing $a$, taken with that heap on the left.","title":"Erasing an address that the right heap of a union lacks"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/disjoint-insert-left","kind":"node","lean":"Cslib.IMP.Heap.disjoint_insert_left","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#disjoint-insert-left","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"If heaps $h_1$ and $h_2$ are disjoint and the address $a$ is not allocated in $h_2$, then for every value $v$, the heap obtained from $h_1$ by inserting $a$ with value $v$ (overwriting any value $h_1$ stores at $a$) is disjoint from $h_2$.","title":"Inserting an address the other heap lacks preserves disjointness"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/disjoint-erase-left","kind":"node","lean":"Cslib.IMP.Heap.disjoint_erase_left","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#disjoint-erase-left","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"If heaps $h_1$ and $h_2$ are disjoint, then for every address $a$, the heap obtained from $h_1$ by erasing $a$ (removing the address $a$ and its value) is disjoint from $h_2$.","title":"Erasing an address preserves disjointness"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/disjoint-singleton-left","kind":"node","lean":"Cslib.IMP.Heap.disjoint_singleton_left","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#disjoint-singleton-left","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"For every address $a$, value $v$ and heap $h$, the singleton heap that stores $v$ at $a$ and allocates no other address is disjoint from $h$ if and only if $a$ is not allocated in $h$.","title":"Disjointness from a singleton heap"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/insert-erase-eq-self","kind":"node","lean":"Cslib.IMP.Heap.insert_erase_eq_self","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#insert-erase-eq-self","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"If a heap $h$ stores the value $v$ at the address $a$, then erasing $a$ from $h$ and then inserting $a$ with value $v$ gives back $h$.","title":"Reinserting an erased value restores the heap"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/exists-not-mem","kind":"node","lean":"Cslib.IMP.Heap.exists_notMem","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#exists-not-mem","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"For every heap $h$ there is an address that is not allocated in $h$.","title":"Every heap has a free address"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/heap-cmd","kind":"node","lean":"Cslib.IMP.HeapCmd","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#heap-cmd","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Heap commands extend the commands of IMP with heap operations. A heap command is one of: $\\mathbf{skip}$; an assignment $x := a$; a sequence $c_1; c_2$; a conditional $\\mathbf{if}\\ b\\ \\mathbf{then}\\ c_1\\ \\mathbf{else}\\ c_2$; a loop $\\mathbf{while}\\ b\\ \\mathbf{do}\\ c$; a heap read $x := [a]$; a heap write $[a] := e$; an allocation $x := \\mathbf{alloc}(e)$; or a deallocation $\\mathbf{free}(a)$. Here $x$ is a variable\u2026","title":"Heap commands"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/hconfig","kind":"node","lean":"Cslib.IMP.HConfig","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#hconfig","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"A heap configuration is a triple $(c, s, h)$ of a heap command $c$, a store $s$ and a heap $h$.","title":"Heap configurations"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/hstep","kind":"node","lean":"Cslib.IMP.HStep","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#hstep","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"The single-step relation $(c, s, h) \\to (c', s', h')$ on heap configurations is the least relation closed under the rules below. Here $a(s)$ is the value of an expression $a$ in the store $s$, $s[x \\mapsto n]$ is $s$ with variable $x$ set to $n$, $h[\\ell \\mapsto n]$ is $h$ with address $\\ell$ set to $n$, and $h \\setminus \\ell$ is $h$ with address $\\ell$ removed.","title":"Small-step semantics of heap commands"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/hsteps","kind":"node","lean":"Cslib.IMP.HSteps","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#hsteps","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"The relation $(c, s, h) \\to^* (c', s', h')$ on heap configurations is the reflexive-transitive closure of the small-step relation: it holds when the first configuration reaches the second by a finite, possibly empty, sequence of small steps.","title":"Multi-step heap execution"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/safe","kind":"node","lean":"Cslib.IMP.Safe","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#safe","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"A heap configuration, consisting of a heap command, a store and a heap, is safe when every heap configuration reachable from it in zero or more heap steps either has command $\\mathtt{skip}$ or can take a further heap step.","title":"Safe heap configurations"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/of-steps","kind":"node","lean":"Cslib.IMP.Safe.of_steps","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#of-steps","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"If a heap configuration is safe and reaches a second configuration by finitely many (possibly zero) small steps, then the second configuration is safe.","title":"Safety is preserved along executions"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/exists-step-frame","kind":"node","lean":"Cslib.IMP.HStep.exists_step_frame","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#exists-step-frame","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Let $h$ and $h_f$ be disjoint heaps, $c$ a heap command and $s$ a store. If the configuration $(c, s, h)$ can take a small step, then so can $(c, s, h \\cup h_f)$.","title":"Adding a frame heap preserves the ability to step"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/decompose-of-safe-step","kind":"node","lean":"Cslib.IMP.HStep.decompose_of_safe_step","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#decompose-of-safe-step","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Let $h$ and $h_f$ be disjoint heaps, $c$ a heap command and $s$ a store, and suppose that the configuration $(c, s, h)$ can take a small step. If $(c, s, h \\cup h_f)$ takes a small step to $(c', s', h'')$, then there is a heap $h_1$, disjoint from $h_f$, such that $(c, s, h)$ takes a small step to $(c', s', h_1)$ and $h'' = h_1 \\cup h_f$.","title":"Frame decomposition of a single heap step"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/decompose-of-safe","kind":"node","lean":"Cslib.IMP.HSteps.decompose_of_safe","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#decompose-of-safe","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Let $h$ and $h_f$ be disjoint heaps, $c$ a heap command and $s$ a store, and suppose that the configuration $(c, s, h)$ is safe. If $(c, s, h \\cup h_f)$ reaches $(c'', s'', h'')$ by finitely many (possibly zero) small steps, then there is a heap $h_2$, disjoint from $h_f$, such that $(c, s, h)$ reaches $(c'', s'', h_2)$ by finitely many small steps, $h'' = h_2 \\cup h_f$, and $(c'', s'', h_2)$ is safe.","title":"Frame decomposition of heap executions"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/to-heap","kind":"node","lean":"Cslib.IMP.Cmd.toHeap","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#to-heap","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"The embedding sends each plain IMP command to the heap command of the same shape: $\\mathtt{skip}$ goes to $\\mathtt{skip}$, an assignment of an arithmetic expression to a variable goes to the same assignment, a sequence goes to the sequence of the embeddings of its parts, and a conditional or while loop goes to the conditional or while loop with the same guard and with each subcommand replaced by its embedding.","title":"Embedding of IMP into the heap language"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/to-heap-iff","kind":"node","lean":"Cslib.IMP.HStep.toHeap_iff","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#to-heap-iff","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Let $c$ be a plain IMP command, $s$ a store and $h$ a heap, and write $\\overline{d}$ for the embedding of a plain command $d$ into the heap language. Then $(\\overline{c}, s, h)$ takes a heap step to a heap configuration $K$ if and only if there are a plain command $c'$ and a store $s'$ such that $(c, s)$ takes a plain small step to $(c', s')$ and $K = (\\overline{c'}, s', h)$.","title":"Heap steps of embedded commands are plain steps"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/of-to-heap","kind":"node","lean":"Cslib.IMP.HSteps.of_toHeap","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#of-to-heap","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Let $c$ be a plain IMP command, $s$ a store and $h$ a heap, and write $\\overline{d}$ for the embedding of a plain command $d$ into the heap language. If $(\\overline{c}, s, h)$ reaches a heap configuration $(c_2, s_2, h_2)$ in zero or more heap steps, then $h_2 = h$ and there is a plain command $c_2'$ with $c_2 = \\overline{c_2'}$ such that $(c, s)$ reaches $(c_2', s_2)$ in zero or more plain small steps.","title":"Executions of embedded commands are plain executions"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/to-heap-of-steps","kind":"node","lean":"Cslib.IMP.HSteps.toHeap_of_steps","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#to-heap-of-steps","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Let $c, c_2$ be plain IMP commands, $s, s_2$ stores and $h$ a heap, and write $\\overline{d}$ for the embedding of a plain command $d$ into the heap language. If $(c, s)$ reaches $(c_2, s_2)$ in zero or more plain small steps, then $(\\overline{c}, s, h)$ reaches $(\\overline{c_2}, s_2, h)$ in zero or more heap steps.","title":"Plain executions lift to the heap language"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/safe-to-heap","kind":"node","lean":"Cslib.IMP.safe_toHeap","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#safe-to-heap","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"For every plain IMP command $c$ with embedding $\\overline{c}$ into the heap language, every store $s$ and every heap $h$, the heap configuration $(\\overline{c}, s, h)$ is safe.","title":"Embedded plain commands are safe"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/terminates-to-heap","kind":"node","lean":"Cslib.IMP.terminates_toHeap","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#terminates-to-heap","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Let $c$ be a plain IMP command with embedding $\\overline{c}$ into the heap language, let $s, s'$ be stores and let $h$ be a heap. Then $c$ terminates from $s$ with final store $s'$ if and only if $(\\overline{c}, s, h)$ reaches $(\\mathtt{skip}, s', h)$ in zero or more heap steps.","title":"Embedding preserves termination"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/mod-vars","kind":"node","lean":"Cslib.IMP.HeapCmd.modVars","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#mod-vars","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"The set $\\mathrm{mod}(c)$ of variables a heap command $c$ may assign to is defined by recursion on $c$. It is empty for $\\mathbf{skip}$, a heap write $[a] := e$ and a deallocation $\\mathbf{free}(a)$; it is $\\{x\\}$ for an assignment $x := a$, a heap read $x := [a]$ and an allocation $x := \\mathbf{alloc}(e)$; it is $\\mathrm{mod}(c_1) \\cup \\mathrm{mod}(c_2)$ for a sequence $c_1; c_2$ and for a conditional\u2026","title":"Modified variables of a heap command"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/mod-vars-subset","kind":"node","lean":"Cslib.IMP.HStep.modVars_subset","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#mod-vars-subset","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"Write $\\mathrm{mod}(c)$ for the set of variables a heap command $c$ may assign to. If the configuration $(c, s, h)$ takes a small step to $(c', s', h')$, then $\\mathrm{mod}(c') \\subseteq \\mathrm{mod}(c)$.","title":"Steps do not enlarge the modified variables"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/store-agree","kind":"node","lean":"Cslib.IMP.HStep.store_agree","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#store-agree","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"If a heap configuration $(c, s, h)$ takes one heap step to $(c', s', h')$, then $s'(x) = s(x)$ for every variable $x$ outside the set of variables that $c$ may assign to.","title":"A heap step only changes assignable variables"},{"id":"ComputerScience/programming-language-semantics/heap-semantics-of-imp/store-agree-2","kind":"node","lean":"Cslib.IMP.HSteps.store_agree","parent":40805,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/heap-semantics-of-imp/index.html#store-agree-2","scope":"ComputerScience/programming-language-semantics/heap-semantics-of-imp","status":"fully_proved","summary":"If a heap configuration $(c, s, h)$ reaches a heap configuration $(c', s', h')$ in zero or more heap steps, then $s'(x) = s(x)$ for every variable $x$ outside the set of variables that $c$ may assign to.","title":"Heap executions only change assignable variables"},{"data_url":"scopes/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy.json","id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","kind":"scope","navigation":"module","parent":40619,"result":false,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","title":"PCF and computational adequacy"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ty","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ty","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ty","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"PCF types are the simple types over a single ground type, the type of natural numbers.","title":"PCF types"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/nat","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ty.nat","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#nat","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"The ground type $\\mathsf{nat}$ of natural numbers is the unique base type of PCF, whose types are the simple types over a single base type.","title":"Natural number type of PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/var","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Var","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#var","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"PCF uses the typed de Bruijn variables of the simply typed $\\lambda$-calculus over a single base type: a variable of type $A$ in a context $\\Gamma$ is a position in the list $\\Gamma$ at which $A$ occurs.","title":"PCF variables"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/tm","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#tm","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Write $\\mathsf{nat}$ for the ground type of natural numbers and $A \\Rightarrow B$ for the type of functions from $A$ to $B$. The intrinsically typed PCF terms of type $A$ in a context $\\Gamma$ (a list of PCF types) are generated by the following rules: every variable of type $A$ in $\\Gamma$ is a term of type $A$; if $t$ has type $B$ in the context $\\Gamma$ extended by a newest variable of type $A$, then $\\lambda t$\u2026","title":"PCF terms"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ren","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ren","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ren","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For contexts $\\Gamma$ and $\\Delta$ (lists of PCF types), a renaming from $\\Gamma$ to $\\Delta$ is a map sending each variable of $\\Gamma$ of type $A$, for every type $A$, to a variable of $\\Delta$ of the same type $A$; it is the STLC notion of renaming at PCF types.","title":"PCF renamings"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/sub","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Sub","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#sub","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"A substitution from a context $\\Gamma$ to a context $\\Delta$ (lists of PCF types) assigns to every variable of $\\Gamma$, of any type $A$, a PCF term of type $A$ in context $\\Delta$.","title":"Parallel substitutions for PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/numeral-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.numeral","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#numeral-2","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every context $\\Gamma$ and natural number $n$, the numeral of $n$ is the term $\\mathsf{succ}^n(\\mathsf{zero})$ of type $\\mathsf{nat}$ in $\\Gamma$: the numeral of $0$ is $\\mathsf{zero}$, and the numeral of $n + 1$ is $\\mathsf{succ}$ applied to the numeral of $n$.","title":"PCF numerals"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/numeral-injective","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.numeral_injective","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#numeral-injective","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"In every context, the map sending a natural number $n$ to the numeral $\\mathsf{succ}^n(\\mathsf{zero})$ of type $\\mathsf{nat}$ is injective: distinct natural numbers give distinct numerals.","title":"Injectivity of PCF numerals"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.rename","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#rename","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Given a renaming $\\rho$ from a context $\\Gamma$ to a context $\\Delta$, the renaming along $\\rho$ of a PCF term of type $A$ in context $\\Gamma$ is the term of type $A$ in context $\\Delta$ obtained by replacing each variable $x$ by $\\rho(x)$. Under a $\\lambda$-binder, $\\rho$ is replaced by its lift, which sends the bound variable to itself and the successor of each variable $x$ of $\\Gamma$ to the successor of\u2026","title":"Renaming of PCF terms"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.weaken","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#weaken","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $\\Gamma$ be a context and $A$, $B$ types. Weakening sends a PCF term $t$ of type $A$ in context $\\Gamma$ to the term of type $A$ in the context obtained by prepending a new variable of type $B$ to $\\Gamma$, by renaming every free variable of $t$ to its successor, that is, by shifting each de Bruijn index up by one.","title":"Weakening of PCF terms"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/rename-id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.rename_id","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#rename-id","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every context $\\Gamma$, type $A$ and term $t$ of type $A$ in $\\Gamma$, renaming $t$ along the identity renaming of $\\Gamma$ gives $t$.","title":"Renaming a PCF term by the identity"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/id","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Sub.id","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#id","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every context $\\Gamma$, the identity substitution from $\\Gamma$ to $\\Gamma$ sends each variable $x$ of $\\Gamma$ to the term consisting of $x$ alone.","title":"Identity PCF substitution"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/lift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Sub.lift","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#lift","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Given a substitution $\\sigma$ from a context $\\Gamma$ to a context $\\Delta$ and a type $A$, the lift of $\\sigma$ is the substitution from $\\Gamma$ extended by a newest variable of type $A$ to $\\Delta$ extended by a newest variable of type $A$ that sends the newest variable to the newest variable and every older variable $x$ to the weakening of $\\sigma(x)$ by one variable.","title":"Lifting a PCF substitution under a binder"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/single","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Sub.single","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#single","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For a PCF term $a$ of type $A$ in a context $\\Gamma$, this is the substitution from the context $\\Gamma$ extended by a newest variable of type $A$ to $\\Gamma$ that sends the newest variable to $a$ and each older variable $x$ to the variable $x$ of $\\Gamma$.","title":"Single-variable substitution in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/cons","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Sub.cons","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#cons","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Given a substitution $\\sigma$ from a context $\\Gamma$ to a context $\\Delta$ and a term $a$ of type $A$ in $\\Delta$, the extended substitution goes from $\\Gamma$ extended by a newest variable of type $A$ to $\\Delta$: it sends the newest variable to $a$ and every older variable $x$ of $\\Gamma$ to $\\sigma(x)$.","title":"Extending a PCF substitution by a term"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.subst","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#subst","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Given a substitution $\\sigma$ from a context $\\Gamma$ to a context $\\Delta$, applying $\\sigma$ to a PCF term of type $A$ in context $\\Gamma$ gives the term of type $A$ in context $\\Delta$ obtained by replacing each variable $x$ by $\\sigma(x)$. Under a $\\lambda$-binder, $\\sigma$ is replaced by its lift, which sends the bound variable to itself and the successor of each variable $x$ of $\\Gamma$ to the weakening of\u2026","title":"Substitution in PCF terms"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.instantiate","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#instantiate","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Given a term $b$ of type $B$ in a context $\\Gamma$ extended by a newest variable of type $A$, and a term $a$ of type $A$ in $\\Gamma$, the instantiation of $b$ with $a$ is the term of type $B$ in $\\Gamma$ obtained by applying to $b$ the substitution that sends the newest variable to $a$ and every older variable to itself.","title":"Instantiating the newest variable of a PCF term"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/subst-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.subst_rename","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#subst-rename","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $\\rho$ be a renaming from a context $\\Gamma$ to a context $\\Delta$ and $\\sigma$ a substitution from $\\Delta$ to a context $\\Xi$. For every PCF term $t$ of type $A$ in context $\\Gamma$, applying $\\sigma$ to the renaming of $t$ along $\\rho$ gives the result of applying to $t$ the substitution from $\\Gamma$ to $\\Xi$ that sends each variable $x$ of $\\Gamma$ to $\\sigma(\\rho(x))$.","title":"Substitution after renaming in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/rename-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.rename_subst","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#rename-subst","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from a context $\\Gamma$ to a context $\\Delta$ and $\\rho$ a renaming from $\\Delta$ to a context $\\Xi$. For every PCF term $t$ of type $A$ in context $\\Gamma$, renaming along $\\rho$ the result of applying $\\sigma$ to $t$ gives the result of applying to $t$ the substitution from $\\Gamma$ to $\\Xi$ that sends each variable $x$ of $\\Gamma$ to the renaming of $\\sigma(x)$ along $\\rho$.","title":"Renaming after substitution in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/subst-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.subst_subst","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#subst-subst","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Write $t[\\sigma]$ for the result of applying a substitution $\\sigma$ to a PCF term $t$. Let $\\sigma$ be a substitution from a context $\\Gamma$ to a context $\\Delta$ and $\\tau$ a substitution from $\\Delta$ to a context $\\Xi$. For every PCF term $t$ of type $A$ in context $\\Gamma$, $t[\\sigma][\\tau] = t[\\sigma']$, where $\\sigma'$ is the substitution from $\\Gamma$ to $\\Xi$ that sends each variable $x$ of $\\Gamma$ to\u2026","title":"Composition of PCF substitutions"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/instantiate-subst-lift","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.instantiate_subst_lift","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#instantiate-subst-lift","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $\\sigma$ be a substitution from a context $\\Gamma$ to a context $\\Delta$, let $b$ be a term of type $B$ in $\\Gamma$ extended by a newest variable of type $A$, and let $a$ be a term of type $A$ in $\\Delta$. Applying the lift of $\\sigma$ to $b$ and then instantiating the newest variable with $a$ gives the same term as applying to $b$ the extension of $\\sigma$ by $a$, which sends the newest variable to $a$ and\u2026","title":"Instantiation after substitution under a binder"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Value","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#value","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"A closed PCF term is a value when it is a $\\lambda$-abstraction or the numeral $\\mathsf{succ}^n\\,\\mathsf{zero}$ for some natural number $n$.","title":"Values of PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/eq-numeral","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Value.eq_numeral","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#eq-numeral","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"If a closed PCF term $v$ of the ground type of natural numbers is a value, then $v$ is the numeral $\\mathsf{succ}^n\\,\\mathsf{zero}$ for some natural number $n$.","title":"Closed values of ground type are numerals"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/eval","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Eval","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#eval","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Write $\\underline{n}$ for the numeral $\\mathsf{succ}^n\\,\\mathsf{zero}$. For closed PCF terms $t$ and $v$ of the same type, $t \\Downarrow v$ ($t$ evaluates to $v$) is the least relation closed under the following rules, where all terms are closed and $n$ ranges over natural numbers. Every $\\lambda$-abstraction evaluates to itself, and $\\mathsf{zero} \\Downarrow \\mathsf{zero}$. If $t \\Downarrow \\underline{n}$, then\u2026","title":"Big-step call-by-name evaluation of PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/numeral","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Eval.numeral","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#numeral","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every natural number $n$, the closed PCF numeral $\\underline{n} = \\mathsf{succ}^n\\,\\mathsf{zero}$ evaluates to itself under big-step call-by-name evaluation: $\\underline{n} \\Downarrow \\underline{n}$.","title":"Numerals evaluate to themselves"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/of-value","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Eval.of_value","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#of-value","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every type $A$ and every closed PCF term $v$ of type $A$ that is a value, $v \\Downarrow v$ under big-step call-by-name evaluation.","title":"PCF values evaluate to themselves"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/value-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Eval.value","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#value-2","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For closed PCF terms $t$ and $v$ of the same type, if $t \\Downarrow v$ under big-step call-by-name evaluation, then $v$ is a value.","title":"PCF evaluation results are values"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/deterministic","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Eval.deterministic","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#deterministic","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For closed PCF terms $t$, $v$, $w$ of the same type, if $t \\Downarrow v$ and $t \\Downarrow w$ under big-step call-by-name evaluation, then $v = w$.","title":"Determinism of PCF big-step evaluation"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/dom","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ty.dom","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#dom","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"The \u03c9-complete partial order denoted by a PCF type is defined by recursion on the type: the ground type denotes the flat domain of partial natural numbers, and a function type $A \\to B$ denotes the \u03c9-complete partial order of Scott-continuous maps from the domain of $A$ to the domain of $B$.","title":"Domain of a PCF type"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/order-bot","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ty.orderBot","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#order-bot","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every PCF type $A$, the domain denoted by $A$ has a least element, defined by recursion on $A$: at the ground type it is the undefined element of the flat domain of partial natural numbers, and at a function type $B \\to C$ it is the continuous map that sends every argument to the least element of the domain denoted by $C$.","title":"Least elements of PCF domains"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/env","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Env","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#env","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"The \u03c9-complete partial order of environments for a context $\\Gamma$ (a list of PCF types) is defined by recursion on $\\Gamma$: the empty context has the one-point order, and the environments for $\\Gamma$ extended by a newest variable of type $A$ are the pairs $(\\eta, a)$ of an environment $\\eta$ for $\\Gamma$ and an element $a$ of the domain of $A$, ordered componentwise.","title":"Semantic environments for PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/lookup","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Env.lookup","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#lookup","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For a variable $x$ of type $A$ in a context $\\Gamma$, the lookup of $x$ is the Scott-continuous map from environments for $\\Gamma$ to the domain of $A$ defined by recursion on $x$: if $x$ is the newest variable, it sends an environment $(\\eta, a)$ to $a$; if $x$ is the shift of a variable $y$ of the older context, it sends $(\\eta, a)$ to the lookup of $y$ at $\\eta$.","title":"Variable lookup in a PCF environment"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/of-fun","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Env.ofFun","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#of-fun","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $\\Gamma$ be a context, that is, a list of PCF types, and suppose that for every type $A$ and every variable $x$ of type $A$ in $\\Gamma$ we are given a value $f(x)$ in the domain denoted by $A$. The environment built from $f$ is the environment for $\\Gamma$ defined by recursion on $\\Gamma$: for the empty context it is the unique element of the one-point domain, and for the context obtained from $\\Gamma'$ by\u2026","title":"Environment from a family of values"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ifz-hom","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.ifzHom","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ifz-hom","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $\\alpha$ be an \u03c9-complete partial order and let $t$, $s_1$, $s_2$ be Scott-continuous maps from $\\alpha$ to the flat domain of partial natural numbers. Their strict zero test is the Scott-continuous map sending $x \\in \\alpha$ to $s_1(x)$ if $t(x)$ is defined and equal to $0$, to $s_2(x)$ if $t(x)$ is defined and nonzero, and to the undefined value if $t(x)$ is undefined.","title":"Strict zero test as a continuous map"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.denote","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"The denotation of a PCF term $t$ of type $A$ in a context $\\Gamma$ is the Scott-continuous map $[\\![t]\\!]$ from environments for $\\Gamma$ to the domain of $A$, defined by recursion on $t$: a variable denotes its lookup map; $[\\![\\lambda t]\\!]$ is the currying of $[\\![t]\\!]$; $[\\![f\\,a]\\!]\\eta = [\\![f]\\!]\\eta\\,([\\![a]\\!]\\eta)$; $[\\![\\mathsf{zero}]\\!]$ is constantly the defined value $0$;\u2026","title":"Denotation of PCF terms"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-fix-unfold","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.denote_fix_unfold","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-fix-unfold","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every PCF term $t$ of type $A \\to A$ in a context $\\Gamma$ and every environment $\\eta$ for $\\Gamma$, the denotation of $\\mathsf{fix}\\,t$ at $\\eta$ equals the denotation of the application $t\\,(\\mathsf{fix}\\,t)$ at $\\eta$.","title":"Fixed-point equation for the PCF denotation of fix"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-rename","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.denote_rename","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-rename","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $\\rho$ be a type-preserving renaming of the variables of a context $\\Gamma$ into a context $\\Delta$, let $t$ be a PCF term of type $A$ in context $\\Gamma$, and let $\\eta$ be an environment for $\\Delta$. Then the denotation at $\\eta$ of the renaming of $t$ along $\\rho$ equals the denotation of $t$ at the environment for $\\Gamma$ that gives each variable $x$ of $\\Gamma$ the value of the variable $\\rho(x)$ in\u2026","title":"PCF denotation of a renamed term"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-weaken","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.denote_weaken","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-weaken","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ be a PCF term of type $A$ in a context $\\Gamma$, let $\\eta$ be an environment for $\\Gamma$, and let $b$ be an element of the domain of a type $B$. The denotation of the weakening of $t$ (the same term in the context $\\Gamma$ extended by a newest variable of type $B$) at the extended environment $(\\eta, b)$ equals the denotation of $t$ at $\\eta$.","title":"Weakening preserves the PCF denotation"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-subst","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.denote_subst","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-subst","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $\\sigma$ be a type-preserving substitution assigning to each variable of a context $\\Gamma$ a PCF term in a context $\\Delta$, let $t$ be a term of type $A$ in context $\\Gamma$, and let $\\eta$ be an environment for $\\Delta$. Then the denotation at $\\eta$ of the term obtained by applying $\\sigma$ to $t$ equals the denotation of $t$ at the environment for $\\Gamma$ that gives each variable $x$ of $\\Gamma$ the value\u2026","title":"PCF denotation of a substituted term"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-instantiate","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.denote_instantiate","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-instantiate","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ be a PCF term of type $B$ in the context $\\Gamma$ extended by a newest variable of type $A$, let $a$ be a term of type $A$ in context $\\Gamma$, and let $\\eta$ be an environment for $\\Gamma$. Writing $[\\![s]\\!]\\eta$ for the denotation of a term $s$ at $\\eta$, the denotation at $\\eta$ of the term obtained from $t$ by substituting $a$ for the newest variable equals $\\![t]\\!$, the denotation of $t$ at the\u2026","title":"PCF denotation of instantiation"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Step","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#step","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Write $\\underline{n}$ for the numeral $\\mathrm{succ}^n\\,\\mathrm{zero}$, and $b[a]$ for the result of substituting the closed term $a$ for the bound variable of an abstraction body $b$. One-step call-by-name reduction $t \\to t'$ between closed PCF terms of the same type is the least relation closed under these rules: $f\\,a \\to f'\\,a$ whenever $f \\to f'$; $(\\lambda b)\\,a \\to b[a]$, without evaluating $a$;\u2026","title":"Call-by-name reduction step of PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Steps","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#steps","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For closed PCF terms $t$ and $t'$ of the same type, $t \\to^* t'$ holds when $t$ reduces to $t'$ in finitely many, possibly zero, one-step call-by-name reductions; that is, $\\to^*$ is the reflexive transitive closure of one-step reduction.","title":"Multi-step call-by-name reduction of PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/numeral-not-step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.numeral_not_step","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#numeral-not-step","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every natural number $n$, the closed numeral $\\mathrm{succ}^n\\,\\mathrm{zero}$ of type nat does not reduce: there is no closed term $t'$ such that $\\mathrm{succ}^n\\,\\mathrm{zero}$ reduces to $t'$ in one call-by-name step.","title":"PCF numerals are normal forms"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/not-step","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Value.not_step","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#not-step","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"If $v$ is a closed PCF value of type $A$ (an abstraction or a numeral), then there is no closed term $t'$ such that $v$ reduces to $t'$ in one call-by-name step.","title":"PCF values are normal forms"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/deterministic-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Step.deterministic","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#deterministic-2","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"One-step call-by-name reduction of closed PCF terms is deterministic: if a closed term $t$ of type $A$ reduces in one step to $t_1$ and also reduces in one step to $t_2$, then $t_1 = t_2$.","title":"Determinism of PCF call-by-name reduction"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/steps-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Eval.steps","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#steps-2","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ and $v$ be closed PCF terms of type $A$. If $t$ evaluates to $v$ in the big-step call-by-name semantics, then $t$ reduces to $v$ in finitely many (possibly zero) call-by-name steps.","title":"Big-step evaluation implies reduction in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/eval-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Step.eval","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#eval-2","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$, $t'$ and $v$ be closed PCF terms of type $A$. If $t$ reduces to $t'$ in one call-by-name step and $t'$ evaluates to $v$ in the big-step call-by-name semantics, then $t$ evaluates to $v$.","title":"Backward reduction preserves evaluation in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/eval-3","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Steps.eval","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#eval-3","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ and $v$ be closed PCF terms of type $A$. If $t$ reduces to $v$ in finitely many (possibly zero) call-by-name steps and $v$ is a value, then $t$ evaluates to $v$ in the big-step call-by-name semantics.","title":"Reduction to a value implies big-step evaluation in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/eval-iff-steps","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.eval_iff_steps","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#eval-iff-steps","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For closed PCF terms $t$ and $v$ of type $A$, $t$ evaluates to $v$ in the big-step call-by-name semantics if and only if $t$ reduces to $v$ in finitely many (possibly zero) call-by-name steps and $v$ is a value.","title":"Big-step evaluation agrees with small-step reduction in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Eval.denote_eq","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-eq","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $A$ be a PCF type and let $t$ and $v$ be closed terms of type $A$. If $t$ evaluates to $v$ under big-step call-by-name evaluation, then for every environment $\\eta$ of the empty context, the denotations of $t$ and $v$ at $\\eta$ are equal.","title":"Soundness of the PCF denotational semantics"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-numeral","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Eval.denote_numeral","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-numeral","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ be a closed PCF term of the ground type of natural numbers and let $n$ be a natural number. If $t$ evaluates to the numeral for $n$, then for every environment $\\eta$ of the empty context, the denotation of $t$ at $\\eta$ is the defined element $n$ of the flat domain of partial natural numbers.","title":"Closed PCF terms that evaluate to a numeral denote its value"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/approx","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.Approx","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#approx","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For each PCF type $A$, the relation \"$d$ formally approximates $t$\" between an element $d$ of the domain of $A$ and a closed term $t$ of type $A$ is defined by recursion on $A$. At the ground type, whose domain is the flat domain of partial natural numbers, $d$ formally approximates $t$ when, for every natural number $n$ such that $d$ is defined with value $n$, the term $t$ evaluates to the numeral $n$. At a\u2026","title":"Formal approximation relation"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/approx-bot","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.approx_bot","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#approx-bot","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every PCF type $A$ and every closed term $t$ of type $A$, the least element $\\bot$ of the domain of $A$ formally approximates $t$.","title":"The least element approximates every term"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/approx-sup","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.approx_\u03c9Sup","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#approx-sup","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $A$ be a PCF type, $t$ a closed term of type $A$ and $(c_i)_{i \\in \\mathbb{N}}$ a chain in the domain of $A$. If every $c_i$ formally approximates $t$, then the supremum $\\bigsqcup_i c_i$ formally approximates $t$.","title":"Formal approximation is closed under suprema of chains"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/approx-expand","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.approx_expand","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#approx-expand","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $A$ be a PCF type, let $t$ and $t'$ be closed terms of type $A$, and let $d$ be an element of the domain of $A$. If $t$ evaluates to every closed term $v$ to which $t'$ evaluates, and $d$ formally approximates $t'$, then $d$ formally approximates $t$.","title":"Formal approximation is closed under head expansion"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/approx-denote","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.approx_denote","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#approx-denote","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ be a PCF term of type $A$ in a context $\\Gamma$, let $\\eta$ be a semantic environment for $\\Gamma$, and let $\\sigma$ assign to each variable of $\\Gamma$ a closed term of the same type. Suppose that for every variable $x$ of $\\Gamma$, of type $B$ say, the value of $x$ in $\\eta$ formally approximates the closed term $\\sigma(x)$ at type $B$. Then the denotation of $t$ in $\\eta$ formally approximates, at type\u2026","title":"Fundamental lemma of formal approximation"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/adequacy","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.adequacy","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#adequacy","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ be a closed PCF term of ground type $\\mathsf{nat}$ and let $n$ be a natural number. If the denotation of $t$ in the empty environment is defined with value $n$, then $t$ evaluates to the numeral $\\mathsf{succ}^n(\\mathsf{zero})$.","title":"Computational adequacy for PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/eval-numeral-iff","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.eval_numeral_iff","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#eval-numeral-iff","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For every natural number $n$, a closed PCF term $t$ of ground type evaluates to the numeral $n$ if and only if the denotation of $t$ is the defined value $n$.","title":"Evaluation to a numeral agrees with the denotation"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/terminates-iff-dom","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.terminates_iff_dom","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#terminates-iff-dom","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"A closed PCF term $t$ of ground type evaluates to some term if and only if its denotation, an element of the flat domain of partial natural numbers, is defined.","title":"Termination agrees with definedness of the denotation"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ctx","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ctx","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ctx","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Fix a typing context $\\Gamma$ and a type $A$. A one-hole context to $(\\Delta, B)$ has a hole that takes a term of type $A$ in typing context $\\Gamma$ and turns it into a term of type $B$ in typing context $\\Delta$. These contexts are generated by the following rules, for typing contexts $\\Delta$, $\\Delta'$ and types $B$, $C$, with $\\mathbb{N}$ the ground type of natural numbers. The hole itself is a context to\u2026","title":"One-hole contexts of PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/plug","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ctx.plug","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#plug","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $C$ be a one-hole context that turns terms of type $A$ in context $\\Gamma$ into terms of type $B$ in context $\\Delta$, and $t$ a term of type $A$ in context $\\Gamma$. The term $C[t]$ of type $B$ in context $\\Delta$ is defined by recursion on $C$: if $C$ is the hole, $C[t] = t$; if $C$ applies a term former (abstraction, application on either side, successor, predecessor, the zero test at any of its three\u2026","title":"Plugging a term into a PCF context"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ctx.comp","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#comp","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $C$ be a one-hole context whose hole takes terms of type $B$ in typing context $\\Delta$ and which produces terms of type $B'$ in typing context $\\Theta$, and let $D$ be a one-hole context whose hole takes terms of type $A$ in typing context $\\Gamma$ and which produces terms of type $B$ in typing context $\\Delta$. The composite $C \\circ D$ is the context obtained from $C$ by replacing its hole by $D$; its hole\u2026","title":"Composition of PCF contexts"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/plug-comp","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ctx.plug_comp","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#plug-comp","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $D$ be a one-hole context that turns terms of type $B$ in context $\\Delta$ into terms of type $C$ in context $\\Theta$, and let $E$ be one that turns terms of type $A$ in context $\\Gamma$ into terms of type $B$ in context $\\Delta$. Write $K[t]$ for the term obtained by plugging a term $t$ into the hole of a context $K$, and $D \\circ E$ for the context obtained by replacing the hole of $D$ by $E$. Then\u2026","title":"Plugging into a composite PCF context"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/comp-assoc","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Ctx.comp_assoc","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#comp-assoc","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"For one-hole contexts $C$ and $D$ such that the hole of $C$ takes terms of the type and typing context that $D$ produces, write $C \\circ D$ for the context obtained from $C$ by replacing its hole by $D$. For PCF one-hole contexts $C_1$, $C_2$ and $C_3$ such that the hole of $C_1$ takes terms of the type and typing context that $C_2$ produces, and the hole of $C_2$ takes terms of the type and typing context that\u2026","title":"Associativity of PCF context composition"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ctx-equiv","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.CtxEquiv","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ctx-equiv","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Write $C[s]$ for the term obtained by plugging $s$ into the hole of a one-hole context $C$. PCF terms $t$ and $u$ of type $A$ in typing context $\\Gamma$ are contextually equivalent when, for every one-hole context $C$ whose hole takes terms of type $A$ in typing context $\\Gamma$ and which produces closed terms of the ground type of natural numbers, and for every natural number $n$, $C[t]$ evaluates to the numeral\u2026","title":"Contextual equivalence in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-plug-congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.denote_plug_congr","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-plug-congr","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ and $u$ be PCF terms of type $A$ in typing context $\\Gamma$ with the same denotation, and let $C$ be a one-hole context whose hole takes terms of type $A$ in typing context $\\Gamma$. Then the terms $C[t]$ and $C[u]$ obtained by plugging $t$ and $u$ into the hole of $C$ have the same denotation.","title":"Equal denotations are preserved by plugging into PCF contexts"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ctx-equiv-of-denote-eq","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.ctxEquiv_of_denote_eq","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ctx-equiv-of-denote-eq","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"If two PCF terms $t$ and $u$ of the same type in the same typing context have the same denotation, as continuous maps from environments to the domain of their type, then $t$ and $u$ are contextually equivalent.","title":"Equal denotations imply contextual equivalence in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ctx-approx","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.CtxApprox","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ctx-approx","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Write $C[s]$ for the term obtained by plugging $s$ into the hole of a one-hole context $C$. For PCF terms $t$ and $u$ of type $A$ in typing context $\\Gamma$, $t$ contextually approximates $u$ when, for every one-hole context $C$ whose hole takes terms of type $A$ in typing context $\\Gamma$ and which produces closed terms of the ground type of natural numbers, and for every natural number $n$: if $C[t]$ evaluates to\u2026","title":"Contextual approximation in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/denote-plug-mono","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.denote_plug_mono","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#denote-plug-mono","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $C$ be a one-hole context that turns terms of type $A$ in context $\\Gamma$ into terms of type $B$ in context $\\Delta$, and let $t$ and $u$ be terms of type $A$ in context $\\Gamma$. Write $C[s]$ for the term obtained by plugging a term $s$ into the hole of $C$, and $[\\![ s ]\\!]\\,\\eta$ for the denotation of a term $s$ in an environment $\\eta$, ordered in the domain of its type. If\u2026","title":"Plugging into a PCF context is monotone in the denotation"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ctx-approx-of-denote-le","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.ctxApprox_of_denote_le","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ctx-approx-of-denote-le","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ and $u$ be PCF terms of type $A$ in typing context $\\Gamma$, and write $[\\![s]\\!]$ for the denotation of a term $s$. If $[\\![t]\\!]\\eta \\le [\\![u]\\!]\\eta$ in the domain denoted by $A$ for every environment $\\eta$ for $\\Gamma$, then $t$ contextually approximates $u$.","title":"Denotational approximation implies contextual approximation in PCF"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/refl","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.CtxEquiv.refl","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#refl","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Every term $t$ of type $A$ in context $\\Gamma$ is contextually equivalent to itself.","title":"Contextual equivalence in PCF is reflexive"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/congr","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.CtxEquiv.congr","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#congr","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Let $t$ and $u$ be PCF terms of type $A$ in typing context $\\Gamma$. If $t$ and $u$ are contextually equivalent, then for every one-hole context $D$ whose hole takes terms of type $A$ in typing context $\\Gamma$, the terms $D[t]$ and $D[u]$ obtained by plugging $t$ and $u$ into the hole of $D$ are contextually equivalent.","title":"Contextual equivalence in PCF is a congruence"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/refl-2","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.CtxApprox.refl","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#refl-2","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"Every term $t$ of type $A$ in context $\\Gamma$ contextually approximates itself.","title":"Contextual approximation in PCF is reflexive"},{"id":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/ctx-equiv-iff-ctx-approx","kind":"node","lean":"Cslib.Languages.LambdaCalculus.PCF.Tm.ctxEquiv_iff_ctxApprox","parent":40836,"result":true,"route":"../roadmap/ComputerScience/programming-language-semantics/pcf-and-computational-adequacy/index.html#ctx-equiv-iff-ctx-approx","scope":"ComputerScience/programming-language-semantics/pcf-and-computational-adequacy","status":"fully_proved","summary":"PCF terms $t$ and $u$ of the same type in the same typing context are contextually equivalent if and only if $t$ contextually approximates $u$ and $u$ contextually approximates $t$.","title":"Contextual equivalence is mutual contextual approximation in PCF"},{"data_url":"scopes/ComputerScience/machine-learning.json","id":"ComputerScience/machine-learning","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/machine-learning/index.html","scope":"ComputerScience/machine-learning","status":"fully_proved","summary":"Mathematical theory of learning from data, including the probabilistic modelling, inference and sampling methods used in machine learning.","title":"Machine learning"},{"data_url":"scopes/ComputerScience/machine-learning/online-learning-and-boosting.json","id":"ComputerScience/machine-learning/online-learning-and-boosting","kind":"scope","parent":40911,"result":false,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/index.html","scope":"ComputerScience/machine-learning/online-learning-and-boosting","status":"fully_proved","summary":"Mistake and regret bounds for sequential prediction: perceptron, weighted majority, Hedge (multiplicative weights) and AdaBoost.","title":"Online learning and boosting"},{"data_url":"scopes/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm.json","id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","kind":"scope","navigation":"module","parent":40912,"result":false,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","title":"Hedge algorithm"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/update","kind":"node","lean":"Cslib.MachineLearning.Hedge.update","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#update","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"After a round with losses `\u2113`, each expert's weight is multiplied by `exp (-\u03b7 \u2113 i)`.","title":"Hedge weight update"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/weights","kind":"node","lean":"Cslib.MachineLearning.Hedge.weights","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#weights","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $\\eta \\in \\mathbb{R}$ be a learning rate, let $w \\in \\mathbb{R}^n$ be initial weights on $n$ experts, and let $\\ell_1, \\dots, \\ell_T \\in \\mathbb{R}^n$ be the loss vectors of $T$ rounds, given as a list. The weights after these rounds are obtained from $w$ by applying the Hedge update once per round, in the order of the list: round $t$ multiplies the weight of each expert $i$ by $e^{-\\eta \\ell_t(i)}$. For the\u2026","title":"Hedge weights after a sequence of rounds"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/total","kind":"node","lean":"Cslib.MachineLearning.Hedge.total","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#total","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"The total weight of weights $w$ on $n$ experts is $\\sum_i w_i$.","title":"Total weight of the Hedge experts"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/round-loss","kind":"node","lean":"Cslib.MachineLearning.Hedge.roundLoss","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#round-loss","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"For weights $w$ and losses $\\ell$ on $n$ experts, the algorithm's loss in the round is $\\left(\\sum_i w_i \\ell_i\\right) / W$, where $W = \\sum_i w_i$ is the total weight. This is its expected loss when it follows expert $i$ with probability $w_i / W$.","title":"Loss of a Hedge round"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/alg-loss","kind":"node","lean":"Cslib.MachineLearning.Hedge.algLoss","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#alg-loss","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $\\eta$ be a learning rate, $w \\in \\mathbb{R}^n$ initial weights on $n$ experts and $\\ell_1, \\dots, \\ell_T \\in \\mathbb{R}^n$ the losses of $T$ rounds. Set $w_1 = w$ and $w_{t+1}(j) = w_t(j) e^{-\\eta \\ell_t(j)}$. The algorithm's total loss is $\\sum_{t=1}^T \\bigl(\\sum_j w_t(j) \\ell_t(j)\\bigr) / \\sum_j w_t(j)$, the sum over the rounds of the weighted average of that round's losses; a round whose total weight is $0$\u2026","title":"Loss of the Hedge algorithm"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/expert-loss","kind":"node","lean":"Cslib.MachineLearning.Hedge.expertLoss","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#expert-loss","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"For an expert $i$ among $n$ experts and a finite sequence of loss vectors $\\ell_1, \\dots, \\ell_T \\in \\mathbb{R}^n$, the total loss of expert $i$ is $\\sum_{t=1}^T \\ell_t(i)$.","title":"Total loss of an expert"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/exp-neg-mul-le","kind":"node","lean":"Cslib.MachineLearning.Hedge.exp_neg_mul_le","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#exp-neg-mul-le","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"For every real $\\eta$ and every $x \\in [0, 1]$, $e^{-\\eta x} \\le 1 - (1 - e^{-\\eta}) x$.","title":"The exponential lies below its chord on the unit interval"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/total-update-le","kind":"node","lean":"Cslib.MachineLearning.Hedge.total_update_le","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#total-update-le","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $n > 0$, let $w$ be positive weights on $n$ experts, let $\\ell$ be losses with $0 \\le \\ell_i \\le 1$ for every $i$, and let $\\eta$ be a real number. Let $w'$ be the Hedge update of $w$ with learning rate $\\eta$, so that $w'_i = w_i e^{-\\eta \\ell_i}$, and let $W$ and $W'$ be the total weights of $w$ and $w'$. Then $W' \\le W \\exp(-(1 - e^{-\\eta}) L)$, where $L$ is the round's loss under the weights $w$.","title":"A round costs weight in proportion to the algorithm's loss"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/bounded","kind":"node","lean":"Cslib.MachineLearning.Hedge.Bounded","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#bounded","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"A finite sequence of loss vectors $\\ell_1, \\dots, \\ell_T \\in \\mathbb{R}^n$ on $n$ experts has losses in $[0, 1]$ when $0 \\le \\ell_t(i) \\le 1$ for every round $t$ and every expert $i$.","title":"Losses in the unit interval"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/total-weights-le","kind":"node","lean":"Cslib.MachineLearning.Hedge.total_weights_le","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#total-weights-le","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $n > 0$ and let $\\eta$ be a real number. For every sequence of rounds with losses in $[0, 1]$ and all positive initial weights $w$ on $n$ experts, the total weight after running Hedge with learning rate $\\eta$ from $w$ is at most $W \\exp(-(1 - e^{-\\eta}) L)$, where $W$ is the total weight of $w$ and $L$ is the algorithm's total loss over the rounds for the run from $w$ with learning rate $\\eta$.","title":"Weight decay over a Hedge run"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/weights-apply","kind":"node","lean":"Cslib.MachineLearning.Hedge.weights_apply","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#weights-apply","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"For every learning rate $\\eta$, sequence of rounds, initial weights $w$ and expert $i$, the weight of expert $i$ after running Hedge on the rounds from $w$ is $w_i e^{-\\eta L_i}$, where $L_i$ is expert $i$'s total loss over the rounds.","title":"Closed form of the Hedge weights"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/alg-loss-le","kind":"node","lean":"Cslib.MachineLearning.Hedge.algLoss_le","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#alg-loss-le","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $\\eta$ be a real number, and let $\\ell_1, \\dots, \\ell_T$ be loss vectors in $[0, 1]^n$ for $n$ experts. Run Hedge with learning rate $\\eta$ from unit weights, that is $w_1(j) = 1$ and $w_{t+1}(j) = w_t(j) e^{-\\eta \\ell_t(j)}$, and let $x_t(j) = w_t(j) / \\sum_k w_t(k)$ be the distribution played in round $t$. Then for every expert $i$,","title":"The Hedge loss bound"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/round-sq","kind":"node","lean":"Cslib.MachineLearning.Hedge.roundSq","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#round-sq","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"For weights $w$ and losses $\\ell$ on $n$ experts, the round's weighted second moment is $\\left(\\sum_i w_i \\ell_i^2\\right) / W$, where $W = \\sum_i w_i$ is the total weight; that is, $\\sum_i x_i \\ell_i^2$ for the played distribution $x = w / W$.","title":"Weighted second moment of a Hedge round"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/alg-sq","kind":"node","lean":"Cslib.MachineLearning.Hedge.algSq","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#alg-sq","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $\\eta$ be a learning rate, $w \\in \\mathbb{R}^n$ initial weights on $n$ experts and $\\ell_1, \\dots, \\ell_T \\in \\mathbb{R}^n$ the losses of $T$ rounds. Set $w_1 = w$ and $w_{t+1}(j) = w_t(j) e^{-\\eta \\ell_t(j)}$. The summed second moment is $\\sum_{t=1}^T \\bigl(\\sum_j w_t(j) \\ell_t(j)^2\\bigr) / \\sum_j w_t(j)$, the sum over the rounds of the weighted second moment of that round's losses; a round whose total weight\u2026","title":"Summed second moments of the Hedge losses"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/nonneg","kind":"node","lean":"Cslib.MachineLearning.Hedge.Nonneg","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#nonneg","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"A finite sequence of loss vectors $\\ell_1, \\dots, \\ell_T \\in \\mathbb{R}^n$ on $n$ experts has nonnegative losses when $\\ell_t(i) \\ge 0$ for every round $t$ and every expert $i$.","title":"Nonnegative losses"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/total-update-le-sq","kind":"node","lean":"Cslib.MachineLearning.Hedge.total_update_le_sq","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#total-update-le-sq","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $n > 0$, let $w$ be positive weights on $n$ experts, let $\\ell$ be nonnegative losses, and let $\\eta \\ge 0$. Let $w'$ be the Hedge update of $w$ with learning rate $\\eta$, so that $w'_i = w_i e^{-\\eta \\ell_i}$, and let $W$ and $W'$ be the total weights of $w$ and $w'$. Then $W' \\le W \\exp(-\\eta L + \\eta^2 Q)$, where $L$ is the round's loss and $Q$ its weighted second moment under the weights $w$.","title":"A round costs weight in proportion to the algorithm's loss, up to a second-order term"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/total-weights-le-sq","kind":"node","lean":"Cslib.MachineLearning.Hedge.total_weights_le_sq","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#total-weights-le-sq","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $n > 0$ and $\\eta \\ge 0$. For every sequence of rounds with nonnegative losses and all positive initial weights $w$ on $n$ experts, the total weight after running Hedge with learning rate $\\eta$ from $w$ is at most $W \\exp(-\\eta L + \\eta^2 Q)$. Here $W$ is the total weight of $w$, $L$ is the algorithm's total loss over the rounds and $Q$ is the sum of the rounds' weighted second moments, both for the run from\u2026","title":"Weight decay over a Hedge run, second-order form"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/alg-loss-le-sq","kind":"node","lean":"Cslib.MachineLearning.Hedge.algLoss_le_sq","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#alg-loss-le-sq","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Let $\\eta > 0$, and let $\\ell_1, \\dots, \\ell_T$ be loss vectors in $\\mathbb{R}^n$ for $n$ experts with $\\ell_t(j) \\ge 0$ for all $t$ and $j$. Run Hedge with learning rate $\\eta$ from unit weights, that is $w_1(j) = 1$ and $w_{t+1}(j) = w_t(j) e^{-\\eta \\ell_t(j)}$, and let $x_t(j) = w_t(j) / \\sum_k w_t(k)$ be the distribution played in round $t$. Then for every expert $i$,","title":"The second-order Hedge bound"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/weights-append","kind":"node","lean":"Cslib.MachineLearning.Hedge.weights_append","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#weights-append","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"For every learning rate $\\eta$, sequences of rounds $\\lambda$ and $\\lambda'$, and initial weights $w$, the weights after running Hedge on the rounds of $\\lambda$ followed by those of $\\lambda'$, starting from $w$, equal the weights after running it on $\\lambda'$ starting from the weights reached after $\\lambda$ from $w$.","title":"A Hedge run splits at any round"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/alg-loss-append","kind":"node","lean":"Cslib.MachineLearning.Hedge.algLoss_append","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#alg-loss-append","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Fix a learning rate $\\eta$ and $n$ experts. For a finite sequence $\\sigma$ of loss vectors in $\\mathbb{R}^n$ and weights $w \\in \\mathbb{R}^n$, write $L(\\sigma, w)$ for the algorithm's total loss over $\\sigma$ started from $w$, and $W(\\sigma, w)$ for the weights after the rounds of $\\sigma$ started from $w$. Then for all such sequences $\\sigma, \\sigma'$ and all weights $w$, the loss over the concatenation\u2026","title":"The Hedge loss splits at any round"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/alg-sq-append","kind":"node","lean":"Cslib.MachineLearning.Hedge.algSq_append","parent":40913,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm/index.html#alg-sq-append","scope":"ComputerScience/machine-learning/online-learning-and-boosting/hedge-algorithm","status":"fully_proved","summary":"Fix a learning rate $\\eta$ and $n$ experts. For a finite sequence $\\sigma$ of loss vectors in $\\mathbb{R}^n$ and weights $w \\in \\mathbb{R}^n$, write $Q(\\sigma, w)$ for the sum over the rounds of $\\sigma$ of the weighted second moments of the losses, started from $w$, and $W(\\sigma, w)$ for the weights after the rounds of $\\sigma$ started from $w$. Then for all such sequences $\\sigma, \\sigma'$ and all weights $w$,\u2026","title":"The summed second moments split at any round"},{"data_url":"scopes/ComputerScience/machine-learning/online-learning-and-boosting/perceptron.json","id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","kind":"scope","navigation":"module","parent":40912,"result":false,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","title":"Perceptron"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/sign","kind":"node","lean":"Cslib.MachineLearning.Perceptron.sign","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#sign","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"For a commutative ring $K$, the sign of a Boolean label $y$ is the element $\\operatorname{sign}(y) \\in K$ equal to $1$ if $y$ is true and to $-1$ if $y$ is false.","title":"Sign of a Boolean label"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/step","kind":"node","lean":"Cslib.MachineLearning.Perceptron.step","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#step","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a commutative ring with a linear order, $\\iota$ a finite type, and $v \\cdot w = \\sum_{i \\in \\iota} v_i w_i$ the dot product on $K^\\iota$. One perceptron step takes a state $(w, m)$, made of weights $w \\in K^\\iota$ and a mistake count $m \\in \\mathbb{N}$, and a labelled example $(x, y)$ with $x \\in K^\\iota$ and $y$ a Boolean. If $\\operatorname{sign}(y)\\,(w \\cdot x) \\le 0$, that is, $w$ does not classify\u2026","title":"Perceptron step"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/run","kind":"node","lean":"Cslib.MachineLearning.Perceptron.run","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#run","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a commutative ring with a linear order and $\\iota$ a finite type. The run of the perceptron over a finite list $L$ of labelled examples $(x, y)$, with $x \\in K^\\iota$ and $y$ a Boolean, from weights $w \\in K^\\iota$ with $m \\in \\mathbb{N}$ mistakes so far is the pair of final weights and final mistake count obtained by applying the perceptron step to each example of $L$ in order, starting from the state\u2026","title":"Perceptron run"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/train","kind":"node","lean":"Cslib.MachineLearning.Perceptron.train","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#train","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Given a finite list $D$ of labelled examples, a number $n$ of passes and weights $w$, training returns nothing when $n = 0$. When $n = m + 1$, it runs one perceptron pass over $D$ from $w$ with mistake count $0$; if that pass makes no mistake it returns $w$, and otherwise it trains on $D$ for $m$ passes from the weights the pass produced. So training repeats passes until one makes no mistake, for at most $n$\u2026","title":"Perceptron training"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/run-eq","kind":"node","lean":"Cslib.MachineLearning.Perceptron.run_eq","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#run-eq","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a commutative ring with a linear order and $\\iota$ a finite type, and let $L$ be a finite list of labelled examples $(x, y)$ with $x \\in K^\\iota$ and $y$ a Boolean. For all weights $w \\in K^\\iota$ and every natural number $m$, the perceptron run over $L$ from $w$ with $m$ mistakes so far ends with the same weights as the run from $w$ with $0$ mistakes so far, and with mistake count $m + c$, where $c$ is\u2026","title":"The starting mistake count only shifts the final count"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/classifies-of-run-eq-zero","kind":"node","lean":"Cslib.MachineLearning.Perceptron.classifies_of_run_eq_zero","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#classifies-of-run-eq-zero","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a commutative ring with a linear order, $\\iota$ a finite type, and $v \\cdot w = \\sum_{i \\in \\iota} v_i w_i$ the dot product on $K^\\iota$. Let $L$ be a finite list of labelled examples $(x, y)$ with $x \\in K^\\iota$ and $y$ a Boolean, and let $w \\in K^\\iota$. If the perceptron run over $L$ from weights $w$ with $0$ mistakes so far ends with mistake count $0$, then every example $(x, y)$ in $L$ satisfies\u2026","title":"A mistake-free pass classifies every example"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/run-invariant","kind":"node","lean":"Cslib.MachineLearning.Perceptron.run_invariant","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#run-invariant","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a linearly ordered commutative ring, $\\iota$ a finite type, $v \\cdot w = \\sum_{i \\in \\iota} v_i w_i$ the dot product on $K^\\iota$, $u \\in K^\\iota$ and $\\gamma, R \\in K$. Let $L$ be a finite list of labelled examples $(x, y)$ with $x \\in K^\\iota$ and $y$ a Boolean, such that every example in $L$ satisfies $\\gamma \\le \\operatorname{sign}(y)\\,(u \\cdot x)$ and $x \\cdot x \\le R^2$. Let $w \\in K^\\iota$ and\u2026","title":"Invariant of the perceptron run"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/mistakes-mul-sq-le","kind":"node","lean":"Cslib.MachineLearning.Perceptron.mistakes_mul_sq_le","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#mistakes-mul-sq-le","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a linearly ordered commutative ring, $\\iota$ a finite type, $v \\cdot w = \\sum_{i \\in \\iota} v_i w_i$ the dot product on $K^\\iota$, $u \\in K^\\iota$, and $\\gamma, R \\in K$ with $\\gamma > 0$. Let $L$ be a finite list of labelled examples $(x, y)$ with $x \\in K^\\iota$ and $y$ a Boolean, such that every example in $L$ satisfies $\\gamma \\le \\operatorname{sign}(y)\\,(u \\cdot x)$ and $x \\cdot x \\le R^2$. If the\u2026","title":"Novikoff's perceptron mistake bound"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/mistakes-le-div","kind":"node","lean":"Cslib.MachineLearning.Perceptron.mistakes_le_div","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#mistakes-le-div","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $F$ be a linearly ordered field, $\\iota$ a finite type, $v \\cdot w = \\sum_{i \\in \\iota} v_i w_i$ the dot product on $F^\\iota$, $u \\in F^\\iota$, and $\\gamma, R \\in F$ with $\\gamma > 0$. Let $L$ be a finite list of labelled examples $(x, y)$ with $x \\in F^\\iota$ and $y$ a Boolean, such that every example in $L$ satisfies $\\gamma \\le \\operatorname{sign}(y)\\,(u \\cdot x)$ and $x \\cdot x \\le R^2$. If the perceptron\u2026","title":"Novikoff's mistake bound over an ordered field"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/train-classifies","kind":"node","lean":"Cslib.MachineLearning.Perceptron.train_classifies","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#train-classifies","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a commutative ring with a linear order, $\\iota$ a finite type, and $v \\cdot w = \\sum_{i \\in \\iota} v_i w_i$ the dot product on $K^\\iota$. Let $L$ be a finite list of labelled examples $(x, y)$ with $x \\in K^\\iota$ and $y$ a Boolean. For every natural number $n$ and all weights $w, v \\in K^\\iota$, if training on $L$ from $w$ with at most $n$ passes returns $v$, then every example $(x, y)$ in $L$ satisfies\u2026","title":"Trained weights classify every example"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/le-run-of-train-eq-none","kind":"node","lean":"Cslib.MachineLearning.Perceptron.le_run_of_train_eq_none","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#le-run-of-train-eq-none","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a commutative ring with a linear order and $\\iota$ a finite type, and let $L$ be a finite list of labelled examples $(x, y)$ with $x \\in K^\\iota$ and $y$ a Boolean. For every natural number $n$ and all weights $w \\in K^\\iota$, if training on $L$ from $w$ with at most $n$ passes returns no weights, then the perceptron run from $w$ with $0$ mistakes so far over the concatenation of $n$ copies of $L$ ends\u2026","title":"Every failed training pass adds a mistake"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron/train-converges","kind":"node","lean":"Cslib.MachineLearning.Perceptron.train_converges","parent":40935,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/perceptron/index.html#train-converges","scope":"ComputerScience/machine-learning/online-learning-and-boosting/perceptron","status":"fully_proved","summary":"Let $K$ be a linearly ordered commutative ring, let $\\iota$ be a finite index type, write $a \\cdot b = \\sum_{j \\in \\iota} a_j b_j$ for $a, b \\in K^\\iota$, and let the label sign $s(y)$ be $1$ if the Boolean $y$ is true and $-1$ otherwise. Let $D$ be a finite list of examples $(x, y)$ with $x \\in K^\\iota$ and $y$ a Boolean, and let $u \\in K^\\iota$ and $\\gamma, R \\in K$ with $\\gamma > 0$ be such that every example\u2026","title":"The perceptron converges"},{"data_url":"scopes/ComputerScience/machine-learning/online-learning-and-boosting/adaboost.json","id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","kind":"scope","navigation":"module","parent":40912,"result":false,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","title":"AdaBoost"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/err","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.err","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#err","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"For labels $y$, weights $D$ and predictions $h$ on $m$ training points, the weighted training error of $h$ under $D$ is $\\sum_{i : h(i) \\ne y(i)} D(i)$, the total weight of the points that $h$ misclassifies.","title":"Weighted training error"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/alpha","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.alpha","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#alpha","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"A weak hypothesis with weighted error $\\varepsilon \\in \\mathbb{R}$ gets the weight $\\alpha(\\varepsilon) = \\frac{1}{2} \\log \\frac{1 - \\varepsilon}{\\varepsilon}$.","title":"AdaBoost's hypothesis weight"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/normalizer","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.normalizer","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#normalizer","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"For labels $y$, weights $D$ and predictions $h$ on $m$ training points, let $\\varepsilon$ be the weighted training error of $h$ under $D$ and $\\alpha = \\alpha(\\varepsilon)$ the hypothesis weight. The normalizer of the round is","title":"Normalizer of a boosting round"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/reweight","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.reweight","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#reweight","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"For labels $y$, weights $D$ and predictions $h$ on $m$ training points, let $\\varepsilon$ be the weighted training error of $h$ under $D$, $\\alpha = \\alpha(\\varepsilon)$ the hypothesis weight and $Z$ the normalizer of the round. The reweighted weight of point $i$ is","title":"AdaBoost reweighting"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/normalizer-eq","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.normalizer_eq","parent":40948,"reading_title":{"format":"inline-parts-v1","parts":[["text","The normalizer of a round is "],["math","\\(2\\sqrt{\\varepsilon(1 - \\varepsilon)}\\)"]],"source":"The normalizer of a round is $2\\sqrt{\\varepsilon(1 - \\varepsilon)}$"},"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#normalizer-eq","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"Let $y$ be labels on $m$ training points, $D$ real weights on the points with $\\sum_i D(i) = 1$, and $h$ predictions whose weighted training error $\\varepsilon$ under $D$ satisfies $0 < \\varepsilon < 1$. Then the normalizer of the round with weights $D$ and hypothesis $h$ equals $2\\sqrt{\\varepsilon(1 - \\varepsilon)}$.","title":"The normalizer of a round is $2\\sqrt{\\varepsilon(1 - \\varepsilon)}$"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/two-mul-sqrt-le-exp","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.two_mul_sqrt_le_exp","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#two-mul-sqrt-le-exp","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"For real numbers $\\varepsilon$ and $\\gamma$ with $\\gamma \\ge 0$ and $\\varepsilon \\le 1/2 - \\gamma$, we have $2\\sqrt{\\varepsilon (1 - \\varepsilon)} \\le e^{-2\\gamma^2}$, where the square root of a negative number is taken to be $0$.","title":"One round's factor is at most `exp (-2 \u03b3\u00b2)`"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/dist","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.dist","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#dist","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"Given labels $y$ on $m$ training points and a weak learner $\\mathrm{wl}$, which maps weights on the points to a hypothesis's predictions on them, AdaBoost's weights $D_t$ on the points are defined by $D_0(i) = 1/m$ for every point $i$, and $D_{t+1}$ is the reweighting of $D_t$ by the hypothesis $\\mathrm{wl}(D_t)$.","title":"AdaBoost's weights"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/hyp","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.hyp","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#hyp","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"The weak hypothesis $h_t$ of round $t$ is the weak learner's output $\\mathrm{wl}(D_t)$ on the weights $D_t$ of round $t$.","title":"Weak hypothesis of an AdaBoost round"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/eps","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.eps","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#eps","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"The weighted error $\\varepsilon_t$ of round $t$ is the weighted training error of the weak hypothesis $h_t$ of round $t$ under the weights $D_t$ of round $t$.","title":"Weighted error of an AdaBoost round"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/zee","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.zee","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#zee","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"Run AdaBoost on training points with labels $y$, and read labels and predictions as $\\pm 1$. The normalizer of round $t$ is $Z_t = \\sum_i D_t(i) \\exp(-\\alpha_t y_i h_t(i))$, where $D_t$ are AdaBoost's weights in round $t$, $h_t$ is the weak hypothesis of round $t$, and $\\alpha_t$ is its weight.","title":"Normalizer of an AdaBoost round"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/score","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.score","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#score","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"After $T$ rounds, the combined score of training point $i$ is","title":"AdaBoost's combined score"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/train-err","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.trainErr","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#train-err","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"Let AdaBoost run on $m$ training points with labels $y$, read labels and predictions as $\\pm 1$, and let $F_T$ be the combined score after $T$ rounds. The training error after $T$ rounds is the number of training points $i$ with $y_i F_T(i) \\le 0$, divided by $m$; in particular a zero score counts as an error.","title":"Training error of AdaBoost"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/dist-mul-prod","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.dist_mul_prod","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#dist-mul-prod","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"Run AdaBoost with labels $y$ on $m > 0$ training points and a weak learner, and let $D_T$ be the weights on the points after $T$ rounds, $Z_t$ the normalizer of round $t$, and $F_T$ the combined score after $T$ rounds. Then for every $T \\in \\mathbb{N}$ and every training point $i$,","title":"Unrolling the weights"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/train-err-le-prod","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.trainErr_le_prod","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#train-err-le-prod","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"Let AdaBoost run on $m > 0$ training points with labels $y$ and an arbitrary weak learner, and write $Z_t$ for the normalizer of round $t$. Then for every number of rounds $T$, the training error after $T$ rounds is at most $\\prod_{t < T} Z_t$.","title":"The training error is at most the product of the normalizers"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/weak-learner","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.WeakLearner","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#weak-learner","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"A weak learner, which maps weights on the $m$ training points with labels $y$ to a hypothesis on those points, has edge $\\gamma$ if for every weighting $D$ of the training points with $D_i > 0$ for all $i$ and $\\sum_i D_i = 1$, the hypothesis it returns on $D$ has weighted error under $D$ in the interval $(0, 1/2 - \\gamma]$.","title":"The weak-learning assumption"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/train-err-le-prod-sqrt","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.trainErr_le_prod_sqrt","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#train-err-le-prod-sqrt","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"Let AdaBoost run on $m > 0$ training points with labels $y$ and a weak learner with edge $\\gamma \\ge 0$, that is, one that on every weighting $D$ of the training points with $D_i > 0$ for all $i$ and $\\sum_i D_i = 1$ returns a hypothesis whose weighted error under $D$ lies in $(0, 1/2 - \\gamma]$. Write $\\varepsilon_t$ for the weighted error of the weak hypothesis of round $t$ under AdaBoost's weights of that round.\u2026","title":"The Freund\u2013Schapire bound on AdaBoost's training error"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost/train-err-le-exp","kind":"node","lean":"Cslib.MachineLearning.AdaBoost.trainErr_le_exp","parent":40948,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/adaboost/index.html#train-err-le-exp","scope":"ComputerScience/machine-learning/online-learning-and-boosting/adaboost","status":"fully_proved","summary":"Let AdaBoost run on $m > 0$ training points with labels $y$ and a weak learner with edge $\\gamma \\ge 0$, that is, one that on every weighting $D$ of the training points with $D_i > 0$ for all $i$ and $\\sum_i D_i = 1$ returns a hypothesis whose weighted error under $D$ lies in $(0, 1/2 - \\gamma]$. Then for every number of rounds $T$, the training error after $T$ rounds is at most $e^{-2\\gamma^2 T}$.","title":"AdaBoost drives the training error down exponentially"},{"data_url":"scopes/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm.json","id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","kind":"scope","navigation":"module","parent":40912,"result":false,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","title":"Weighted majority algorithm"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/round","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.Round","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#round","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"A round with $n$ experts is a pair $(a, y)$, where $a$ assigns to each expert $i \\in \\{0, \\dots, n - 1\\}$ its binary advice (true or false) and the binary value $y$ is the true outcome.","title":"Round of prediction with expert advice"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/weight-on","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.weightOn","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#weight-on","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Given weights $w_0, \\dots, w_{n-1}$ in a field $K$, advice $a \\in \\{\\mathrm{true}, \\mathrm{false}\\}^n$ and an answer $b \\in \\{\\mathrm{true}, \\mathrm{false}\\}$, the weight on $b$ is $\\sum_{i : a_i = b} w_i$, the total weight of the experts advising $b$.","title":"Weight on an answer"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/predict","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.predict","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#predict","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Let $K$ be a field with a linear order. Given weights in $K$ on $n$ experts and a binary advice (true or false) from each expert, the weighted majority vote is true when the total weight of the experts advising false is at most the total weight of the experts advising true, and false otherwise. In particular ties go to true.","title":"Weighted majority vote"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/update","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.update","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#update","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Given a penalty factor $\\beta$ and weights $w_0, \\dots, w_{n-1}$ in a field $K$, and a round with advice $a \\in \\{\\mathrm{true}, \\mathrm{false}\\}^n$ and outcome $y$, the updated weight of expert $i$ is $w_i$ if $a_i = y$ and $\\beta w_i$ otherwise: after the outcome is revealed, every expert that was wrong has its weight multiplied by $\\beta$.","title":"Weighted majority update"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/weights","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.weights","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#weights","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"For a penalty factor $\\beta$ in a field $K$, the weights after a sequence of rounds $r_1, \\dots, r_k$ starting from weights $w$ in $K$ are obtained by applying the weighted majority update with penalty $\\beta$ for $r_1$, then for $r_2$, and so on up to $r_k$; for the empty sequence they are $w$.","title":"Weights after a run"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/mistakes","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.mistakes","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#mistakes","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Let $K$ be a field with a linear order, let $\\beta \\in K$ be a penalty and let $w$ be initial weights in $K$ on $n$ experts. The weighted majority algorithm processes a list of rounds in order: in each round it predicts the weighted majority vote of the experts' advice under the current weights, and then multiplies the weight of every expert whose advice differed from the true outcome by $\\beta$. Its number of\u2026","title":"Mistakes of the weighted majority algorithm"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/expert-mistakes","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.expertMistakes","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#expert-mistakes","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Given a list of rounds with $n$ experts and an expert $i$, the number of mistakes of $i$ is the number of rounds in the list in which the advice of expert $i$ differs from the true outcome.","title":"Mistakes of an expert"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/total","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.total","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#total","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"The total weight of weights $w_0, \\dots, w_{n-1}$ in a field $K$ is $\\sum_{i < n} w_i$.","title":"Total weight"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/total-update","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.total_update","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#total-update","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Let $\\beta$ be an element of a linearly ordered field $K$, let $w_0, \\dots, w_{n-1} \\in K$ be weights on $n$ experts, and let $r$ be a round with advice $a \\in \\{\\mathrm{true}, \\mathrm{false}\\}^n$ and outcome $y$. If $w'$ are the weights after the update with penalty $\\beta$ on $r$, then $\\sum_{i < n} w'_i = \\sum_{i < n} w_i - (1 - \\beta) \\sum_{i : a_i = \\lnot y} w_i$: the experts that advised wrongly keep a\u2026","title":"Total weight after an update"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/total-update-of-mistake","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.total_update_of_mistake","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#total-update-of-mistake","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, let $\\beta \\in K$ with $\\beta \\le 1$, let $w$ be weights in $K$ on $n$ experts, and let $r$ be a round on which the weighted majority vote under $w$ differs from the true outcome. If $W$ is the total weight of $w$ and $W'$ is the total weight after updating $w$ on $r$ with penalty $\\beta$, then $2 W' \\le (1 + \\beta) W$.","title":"A mistake costs weight"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/total-update-le","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.total_update_le","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#total-update-le","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, let $\\beta \\in K$ with $\\beta \\le 1$, and let $w$ be nonnegative weights in $K$ on $n$ experts. For every round $r$, the total weight after updating $w$ on $r$ with penalty $\\beta$ is at most the total weight of $w$.","title":"An update never increases the total weight"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/total-weights-mul-le","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.total_weights_mul_le","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#total-weights-mul-le","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Let $0 \\le \\beta \\le 1$ in a linearly ordered field $K$, let $w$ be nonnegative weights in $K$ on $n$ experts, and let $\\rho$ be a sequence of rounds. Let $M$ be the number of mistakes the weighted majority algorithm with penalty $\\beta$ makes on $\\rho$ starting from $w$, and let $W$ and $W'$ be the total weights before and after $\\rho$. Then $W' \\cdot 2^M \\le (1 + \\beta)^M \\cdot W$.","title":"Total weight bound in terms of mistakes"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/weights-apply","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.weights_apply","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#weights-apply","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"For a penalty factor $\\beta$ in a field $K$, initial weights $w_0, \\dots, w_{n-1} \\in K$ and a sequence of rounds, the weight of expert $i$ after these rounds is $w_i \\beta^{m_i}$, where $m_i$ is the number of rounds in which the advice of expert $i$ differs from the outcome.","title":"Final weight of an expert"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/mistake-bound-general","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.mistake_bound_general","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#mistake-bound-general","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and let $\\beta \\in K$ with $0 \\le \\beta \\le 1$. Run the weighted majority algorithm with penalty $\\beta$ on a list of rounds with $n$ experts, starting from weight $1$ on every expert, and let $M$ be its number of mistakes. Then every expert $i$, with $m$ mistakes on these rounds, satisfies $\\beta^m 2^M \\le n (1 + \\beta)^M$.","title":"Littlestone\u2013Warmuth mistake bound"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/mistake-bound","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.mistake_bound","parent":40966,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mistake bound for the penalty "],["math","\\(1/2\\)"]],"source":"Mistake bound for the penalty $1/2$"},"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#mistake-bound","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Run the weighted majority algorithm with rational weights and penalty $1/2$ on a list of rounds with $n$ experts, starting from weight $1$ on every expert, and let $M$ be its number of mistakes. Then every expert $i$, with $m$ mistakes on these rounds, satisfies $(4/3)^M \\le n \\cdot 2^m$.","title":"Mistake bound for the penalty $1/2$"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/mistake-bound-nat","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.mistake_bound_nat","parent":40966,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mistake bound for the penalty "],["math","\\(1/2\\)"],["text"," in integers"]],"source":"Mistake bound for the penalty $1/2$ in integers"},"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#mistake-bound-nat","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Run the weighted majority algorithm with rational weights and penalty $1/2$ on a list of rounds with $n$ experts, starting from weight $1$ on every expert, and let $M$ be its number of mistakes. Then every expert $i$, with $m$ mistakes on these rounds, satisfies $4^M \\le n \\cdot 3^M \\cdot 2^m$ in the natural numbers.","title":"Mistake bound for the penalty $1/2$ in integers"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/weights-append","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.weights_append","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#weights-append","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"For a penalty factor $\\beta$ in a field $K$, initial weights $w$ in $K$ and sequences of rounds $\\rho$ and $\\rho'$, the weights after running $\\rho$ followed by $\\rho'$ from $w$ equal the weights after running $\\rho'$ from the weights obtained by running $\\rho$ from $w$.","title":"Weights over a concatenated run"},{"id":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/mistakes-append","kind":"node","lean":"Cslib.MachineLearning.WeightedMajority.mistakes_append","parent":40966,"result":true,"route":"../roadmap/ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm/index.html#mistakes-append","scope":"ComputerScience/machine-learning/online-learning-and-boosting/weighted-majority-algorithm","status":"fully_proved","summary":"Let $K$ be a field with a linear order, let $\\beta \\in K$ be a penalty, let $w$ be weights in $K$ on $n$ experts, and let $R$ and $R'$ be lists of rounds. The number of mistakes of the weighted majority algorithm with penalty $\\beta$ on $R$ followed by $R'$, starting from $w$, equals its number of mistakes on $R$ starting from $w$ plus its number of mistakes on $R'$ starting from the weights reached after $R$.","title":"Mistakes split over concatenated runs"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning.json","id":"ComputerScience/machine-learning/supervised-learning","kind":"scope","parent":40911,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/index.html","scope":"ComputerScience/machine-learning/supervised-learning","status":"fully_proved","summary":"Prediction from labelled data: least squares, ridge, lasso and kernel or Gaussian-process regression (including GBLUP), the bias-variance decomposition, Bayes classifiers and optimal predictors, log loss, decision-tree information gain, backpropagation, and sufficient input subsets.","title":"Supervised learning and regression"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/backpropagation.json","id":"ComputerScience/machine-learning/supervised-learning/backpropagation","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","title":"Backpropagation"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/net","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#net","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Given layer widths $d : \\mathbb{N} \\to \\mathbb{N}$, a feedforward network with widths $d$ is a sequence of arbitrary maps $f_k : \\mathbb{R}^{d(k)} \\to \\mathbb{R}^{d(k+1)}$, one for each $k \\in \\mathbb{N}$, where $f_k$ is the map computed by layer $k$ and $\\mathbb{R}^n$ carries its Euclidean structure.","title":"Feedforward network"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/forward","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net.forward","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#forward","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"For a feedforward network with layers $f_k : \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ between Euclidean spaces, the forward pass through the first $n$ layers is the map $N_n : \\mathbb{R}^{d_0} \\to \\mathbb{R}^{d_n}$ given by $N_0(x) = x$ and $N_{n+1}(x) = f_n(N_n(x))$, that is, $N_n = f_{n-1} \\circ \\cdots \\circ f_0$.","title":"The forward pass"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/backward","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net.backward","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#backward","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $N$ be a feedforward network with layers $f_k : \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ between Euclidean spaces, let $x \\in \\mathbb{R}^{d_0}$ be an input, and write $a_k \\in \\mathbb{R}^{d_k}$ for the forward pass of $x$ through the first $k$ layers. For each $n \\in \\mathbb{N}$, the backward pass at $x$ sends a vector $\\delta \\in \\mathbb{R}^{d_n}$ to $B_n(\\delta) \\in \\mathbb{R}^{d_0}$, defined by\u2026","title":"The backward pass"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/diff-along","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net.DiffAlong","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#diff-along","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $N$ be a feedforward network with layers $f_k : \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ between Euclidean spaces, let $x \\in \\mathbb{R}^{d_0}$ and $n \\in \\mathbb{N}$, and write $a_j \\in \\mathbb{R}^{d_j}$ for the forward pass of $x$ through the first $j$ layers. The network is differentiable along the forward pass of $x$ through its first $n$ layers when $f_j$ is differentiable at $a_j$ for every $j < n$.","title":"Differentiability along the forward pass"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/diff-along-of-differentiable","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net.diffAlong_of_differentiable","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#diff-along-of-differentiable","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $N$ be a feedforward network with layers $f_k : \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ between Euclidean spaces. If every $f_k$ is differentiable on all of $\\mathbb{R}^{d_k}$, then for every input $x \\in \\mathbb{R}^{d_0}$ and every $n \\in \\mathbb{N}$, the layer $f_j$ is differentiable at the forward pass of $x$ through the first $j$ layers for every $j < n$.","title":"Differentiable layers are differentiable along every forward pass"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/differentiable-at-forward","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net.differentiableAt_forward","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#differentiable-at-forward","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $N$ be a feedforward network with layers $f_k : \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ between Euclidean spaces, let $x \\in \\mathbb{R}^{d_0}$ and $n \\in \\mathbb{N}$, and write $N_j : \\mathbb{R}^{d_0} \\to \\mathbb{R}^{d_j}$ for the forward pass through the first $j$ layers. If $f_j$ is differentiable at $N_j(x)$ for every $j < n$, then $N_n$ is differentiable at $x$.","title":"Differentiability of the forward pass"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/backward-eq","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net.backward_eq","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#backward-eq","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $N$ be a feedforward network with layers $f_k : \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ between Euclidean spaces, let $x \\in \\mathbb{R}^{d_0}$ and $n \\in \\mathbb{N}$, and write $N_j : \\mathbb{R}^{d_0} \\to \\mathbb{R}^{d_j}$ for the forward pass through the first $j$ layers. If $f_j$ is differentiable at $N_j(x)$ for every $j < n$, then for every $\\delta \\in \\mathbb{R}^{d_n}$ the backward pass from $\\delta$ at\u2026","title":"The backward pass is the vector-Jacobian product"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/gradient-comp-forward","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.gradient_comp_forward","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#gradient-comp-forward","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $N$ be a feedforward network with layers $f_k : \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ between Euclidean spaces, let $x \\in \\mathbb{R}^{d_0}$ and $n \\in \\mathbb{N}$, and write $N_j$ for the forward pass through the first $j$ layers. Assume that $f_j$ is differentiable at $N_j(x)$ for every $j < n$, and let $\\ell : \\mathbb{R}^{d_n} \\to \\mathbb{R}$ be differentiable at $N_n(x)$. Then\u2026","title":"Backpropagation computes the input gradient"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/shift","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net.shift","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#shift","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"For a feedforward network $N$ with layer widths $d$ and a natural number $j$, the shifted network has layer widths $i \\mapsto d(j + i)$, and its layer $i$ is layer $j + i$ of $N$. It is made of the layers of $N$ from layer $j$ on.","title":"Shifted network"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/forward-add","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.Net.forward_add","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#forward-add","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $N$ be a feedforward network with layers $f_k : \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ between Euclidean spaces, write $N_n$ for its forward pass through the first $n$ layers, and for $j \\in \\mathbb{N}$ write $N^{(j)}_m$ for the forward pass through the first $m$ layers of the shifted network with layers $f_j, f_{j+1}, \\dots$. Then for all $j, m \\in \\mathbb{N}$ and $x \\in \\mathbb{R}^{d_0}$,\u2026","title":"Splitting the forward pass"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/pnet","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.PNet","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#pnet","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Given layer widths $d : \\mathbb{N} \\to \\mathbb{N}$ and parameter dimensions $p : \\mathbb{N} \\to \\mathbb{N}$, a parametrized network is a sequence of arbitrary maps $f_k : \\mathbb{R}^{p(k)} \\times \\mathbb{R}^{d(k)} \\to \\mathbb{R}^{d(k+1)}$, one for each $k \\in \\mathbb{N}$: layer $k$ sends parameters $\\theta \\in \\mathbb{R}^{p(k)}$ and an input $x \\in \\mathbb{R}^{d(k)}$ to $f_k(\\theta, x)$.","title":"Parametrized feedforward network"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/net-2","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.PNet.net","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#net-2","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $P$ be a network with parameters whose layer $k$ is a map $g_k : \\mathbb{R}^{p_k} \\times \\mathbb{R}^{d_k} \\to \\mathbb{R}^{d_{k+1}}$ of the parameters and the input (all spaces Euclidean). For parameters $\\theta = (\\theta_k)_{k \\in \\mathbb{N}}$ with $\\theta_k \\in \\mathbb{R}^{p_k}$, the network at $\\theta$ is the feedforward network whose layer $k$ is\u2026","title":"The network at fixed parameters"},{"id":"ComputerScience/machine-learning/supervised-learning/backpropagation/gradient-param","kind":"node","lean":"Cslib.MachineLearning.Backpropagation.gradient_param","parent":40986,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/backpropagation/index.html#gradient-param","scope":"ComputerScience/machine-learning/supervised-learning/backpropagation","status":"fully_proved","summary":"Let $P$ be a network with parameters whose layer $i$ is a map $g_i : \\mathbb{R}^{p_i} \\times \\mathbb{R}^{d_i} \\to \\mathbb{R}^{d_{i+1}}$ of the parameters and the input (all spaces Euclidean). Fix parameters $\\theta = (\\theta_i)_i$ with $\\theta_i \\in \\mathbb{R}^{p_i}$, let $N$ be the network with layers $f_i = g_i(\\theta_i, \\cdot)$, let $x \\in \\mathbb{R}^{d_0}$ and $k, m \\in \\mathbb{N}$, set $L = k + 1 + m$, and\u2026","title":"Backpropagation computes the parameter gradients"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/bayes-classifier.json","id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","title":"Bayes classifier"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/zero-one-loss","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.zeroOneLoss","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#zero-one-loss","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"For elements $c$ and $y$ of a type $C$ of classes, the 0-1 loss of predicting $y$ when the true class is $c$ is the extended nonnegative real number that equals $0$ if $y = c$ and $1$ if $y \\ne c$.","title":"The 0-1 loss"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/lintegral-zero-one-loss","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.lintegral_zeroOneLoss","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#lintegral-zero-one-loss","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let $C$ be a measurable space in which every singleton is measurable, let $\\nu$ be a measure on $C$, and let $y \\in C$. Writing $\\ell(c, y)$ for the 0-1 loss of predicting $y$ when the true class is $c$, we have $\\int \\ell(c, y) \\, d\\nu(c) = \\nu(C \\setminus \\{y\\})$.","title":"Expected 0-1 loss of a fixed prediction"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/lintegral-zero-one-loss-right","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.lintegral_zeroOneLoss_right","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#lintegral-zero-one-loss-right","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let $C$ be a measurable space with measurable singletons, $\\nu$ a measure on $C$ and $c \\in C$, and let $\\ell(c, y)$ be the 0-1 loss of predicting $y$ when the true class is $c$ ($0$ if $y = c$, $1$ otherwise). Then $\\int_C \\ell(c, y)\\,\\nu(dy) = \\nu(C \\setminus \\{c\\})$.","title":"Integrating the 0-1 loss over the prediction"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/avg-risk-zero-one-loss-deterministic","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.avgRisk_zeroOneLoss_deterministic","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#avg-risk-zero-one-loss-deterministic","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let $C$ be a measurable space of classes with measurable singletons, $P$ a kernel from $C$ to the observation space $\\mathcal X$, $\\pi$ a measure on $C$, $f : \\mathcal X \\to C$ a measurable map, and $\\ell(c, y)$ the 0-1 loss ($0$ if $y = c$, $1$ otherwise). Then the average 0-1 risk of the deterministic kernel $\\delta_f : x \\mapsto \\delta_{f(x)}$, $\\int_C \\int_C \\ell(c, y)\\,(\\delta_f \\circ P)(c)(dy)\\,\\pi(dc)$ with\u2026","title":"The 0-1 risk of a deterministic classifier"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/is-posterior-argmax","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.IsPosteriorArgmax","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#is-posterior-argmax","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let the set $C$ of classes be countable and nonempty with measurable singletons, let $P$ be a finite kernel from $C$ to the observation space $\\mathcal X$ and $\\pi$ a finite measure on $C$, write $P \\circ \\pi$ for the marginal measure on $\\mathcal X$ and $P^\\dagger_\\pi$ for the posterior kernel from $\\mathcal X$ to $C$. A map $f : \\mathcal X \\to C$ is a posterior argmax classifier for $P$ and $\\pi$ if $f$ is\u2026","title":"Posterior argmax classifier"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/is-argmin-estimator","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.IsPosteriorArgmax.isArgminEstimator","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#is-argmin-estimator","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let the set $C$ of classes be countable and nonempty with measurable singletons, let $P$ be a finite kernel from $C$ to the observation space $\\mathcal X$ and $\\pi$ a finite measure on $C$, write $P \\circ \\pi$ for the marginal measure on $\\mathcal X$, $P^\\dagger_\\pi$ for the posterior kernel from $\\mathcal X$ to $C$, and $\\ell(c, y)$ for the 0-1 loss ($0$ if $y = c$, $1$ otherwise). If $f : \\mathcal X \\to C$ is a\u2026","title":"A posterior argmax classifier is an argmin estimator for the 0-1 loss"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/is-posterior-argmax-iff-is-argmin-estima","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.isPosteriorArgmax_iff_isArgminEstimator","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#is-posterior-argmax-iff-is-argmin-estima","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let the set $C$ of classes be countable and nonempty with measurable singletons, let $P$ be a finite kernel from $C$ to the observation space $\\mathcal X$ and $\\pi$ a finite measure on $C$, write $P \\circ \\pi$ for the marginal measure on $\\mathcal X$, $P^\\dagger_\\pi$ for the posterior kernel from $\\mathcal X$ to $C$, and $\\ell(c, y)$ for the 0-1 loss ($0$ if $y = c$, $1$ otherwise). A map $f : \\mathcal X \\to C$ is\u2026","title":"Posterior argmax classifiers are exactly the argmin estimators for the 0-1 loss"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/is-bayes-estimator","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.IsPosteriorArgmax.isBayesEstimator","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#is-bayes-estimator","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"A posterior argmax classifier attains the Bayes risk for the 0-1 loss.","title":"The Bayes classifier is optimal"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/avg-risk-le","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.IsPosteriorArgmax.avgRisk_le","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#avg-risk-le","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let the set $C$ of classes be countable and nonempty with measurable singletons, let $P$ be a finite kernel from $C$ to the observation space $\\mathcal X$ and $\\pi$ a finite measure on $C$, and let $\\ell(c, y)$ be the 0-1 loss ($0$ if $y = c$, $1$ otherwise). For a kernel $\\kappa$ from $\\mathcal X$ to $C$, its average 0-1 risk is $\\int_C \\int_C \\ell(c, y)\\,(\\kappa \\circ P)(c)(dy)\\,\\pi(dc)$, where\u2026","title":"A posterior argmax classifier has minimal 0-1 risk"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/is-posterior-argmax-of-is-map","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.isPosteriorArgmax_of_isMAP","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#is-posterior-argmax-of-is-map","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let the set $C$ of classes be countable and nonempty with measurable singletons, let the observation space $\\mathcal X$ be countable with measurable singletons, let $P$ be a Markov kernel from $C$ to $\\mathcal X$ and $\\pi$ a finite measure on $C$, and write $P \\circ \\pi$ for the marginal measure on $\\mathcal X$. If $f : \\mathcal X \\to C$ assigns to every observation $x$ with $(P \\circ \\pi)(\\{x\\}) \\ne 0$ a maximum a\u2026","title":"MAP classifiers are posterior argmax classifiers"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/exists-is-posterior-argmax","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.exists_isPosteriorArgmax","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#exists-is-posterior-argmax","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let the set $C$ of classes be finite and nonempty with measurable singletons, let $P$ be a finite kernel from $C$ to the observation space $\\mathcal X$ and $\\pi$ a finite measure on $C$. Then there is a posterior argmax classifier $f : \\mathcal X \\to C$ for $P$ and $\\pi$.","title":"Existence of a posterior argmax classifier"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-classifier/bayes-risk-zero-one-loss","kind":"node","lean":"Cslib.MachineLearning.BayesClassifier.bayesRisk_zeroOneLoss","parent":41000,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-classifier/index.html#bayes-risk-zero-one-loss","scope":"ComputerScience/machine-learning/supervised-learning/bayes-classifier","status":"fully_proved","summary":"Let the set $C$ of classes be finite and nonempty with measurable singletons, let $P$ be a finite kernel from $C$ to the observation space $\\mathcal X$ and $\\pi$ a finite measure on $C$, and let $\\ell(c, y)$ be the 0-1 loss ($0$ if $y = c$, $1$ otherwise). Write $P \\circ \\pi$ for the marginal measure on $\\mathcal X$ and $P^\\dagger_\\pi$ for the posterior kernel from $\\mathcal X$ to $C$. For a kernel $\\kappa$ from\u2026","title":"The Bayes risk for the 0-1 loss"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss.json","id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","title":"Bayes estimators for squared and absolute loss"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/sq-loss","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.sqLoss","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#sq-loss","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"The squared loss of predicting $y \\in \\mathbb{R}$ when the true value is $\\theta \\in \\mathbb{R}$ is $(\\theta - y)^2$, regarded as an element of $[0, \\infty]$.","title":"Squared loss"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/lintegral-sq-loss-integral-le","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.lintegral_sqLoss_integral_le","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#lintegral-sq-loss-integral-le","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"Let $\\nu$ be a probability measure on $\\mathbb{R}$ with mean $m = \\int t \\, d\\nu(t)$ (taken to be $0$ when $\\nu$ has no finite first moment). Then for every $y \\in \\mathbb{R}$,","title":"The mean minimizes the expected squared loss"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/posterior-mean","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.posteriorMean","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#posterior-mean","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"Let $P$ be a finite kernel from $\\mathbb{R}$ to a measurable space $\\mathcal{X}$, let $\\pi$ be a finite measure on $\\mathbb{R}$, and write $P^\\dagger\\pi$ for the posterior kernel from $\\mathcal{X}$ to $\\mathbb{R}$. The posterior mean is the function on $\\mathcal{X}$ sending $x$ to $\\int \\theta \\, d(P^\\dagger\\pi)(x)(\\theta)$, the mean of the posterior distribution given the observation $x$ (taken to be $0$ when that\u2026","title":"Posterior mean"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/is-posterior-mean","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.IsPosteriorMean","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#is-posterior-mean","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"Let $P$ be a finite kernel from $\\mathbb{R}$ to a measurable space $\\mathcal{X}$, sending $\\theta$ to the measure $P_\\theta$, let $\\pi$ be a finite measure on $\\mathbb{R}$, and write $P^\\dagger\\pi$ for the posterior kernel from $\\mathcal{X}$ to $\\mathbb{R}$. A function $f : \\mathcal{X} \\to \\mathbb{R}$ is a measurable version of the posterior mean if $f$ is measurable and\u2026","title":"Measurable versions of the posterior mean"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/is-bayes-estimator","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.IsPosteriorMean.isBayesEstimator","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#is-bayes-estimator","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"Let $P$ be a finite kernel from $\\mathbb{R}$ to a measurable space $\\mathcal{X}$, sending a parameter $\\theta$ to the measure $P_\\theta$ on $\\mathcal{X}$, let $\\pi$ be a finite prior measure on $\\mathbb{R}$, and let $f : \\mathcal{X} \\to \\mathbb{R}$ be a measurable version of the posterior mean. Then the deterministic estimator $x \\mapsto f(x)$ is a Bayes estimator for the squared loss\u2026","title":"The posterior mean is a Bayes estimator for squared loss"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/abs-loss","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.absLoss","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#abs-loss","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"The absolute loss of predicting $y \\in \\mathbb{R}$ when the true value is $\\theta \\in \\mathbb{R}$ is $|\\theta - y|$, regarded as an element of $[0, \\infty]$.","title":"Absolute loss"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/lintegral-abs-loss-le-of-is-median","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.lintegral_absLoss_le_of_isMedian","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#lintegral-abs-loss-le-of-is-median","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"Let $\\nu$ be a probability measure on $\\mathbb{R}$ and let $m$ be a median of $\\nu$. Then for every $y \\in \\mathbb{R}$,","title":"The expected absolute loss is minimized at every median"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/is-posterior-median","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.IsPosteriorMedian","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#is-posterior-median","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"Let $P$ be a finite kernel from $\\mathbb{R}$ to a measurable space $\\mathcal{X}$, sending $\\theta$ to the measure $P_\\theta$, let $\\pi$ be a finite measure on $\\mathbb{R}$, and write $P^\\dagger\\pi$ for the posterior kernel from $\\mathcal{X}$ to $\\mathbb{R}$. A function $f : \\mathcal{X} \\to \\mathbb{R}$ is a measurable choice of posterior medians if $f$ is measurable and, for almost every $x$ with respect to the\u2026","title":"Measurable choices of posterior medians"},{"id":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/is-bayes-estimator-2","kind":"node","lean":"Cslib.MachineLearning.OptimalPredictors.IsPosteriorMedian.isBayesEstimator","parent":41013,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss/index.html#is-bayes-estimator-2","scope":"ComputerScience/machine-learning/supervised-learning/bayes-estimators-for-squared-and-absolute-loss","status":"fully_proved","summary":"Let $P$ be a finite kernel from $\\mathbb{R}$ to a measurable space $\\mathcal{X}$, sending a parameter $\\theta$ to the measure $P_\\theta$ on $\\mathcal{X}$, let $\\pi$ be a finite prior measure on $\\mathbb{R}$, and let $f : \\mathcal{X} \\to \\mathbb{R}$ be a measurable choice of posterior medians. Then the deterministic estimator $x \\mapsto f(x)$ is a Bayes estimator for the absolute loss\u2026","title":"A posterior median is a Bayes estimator for absolute loss"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition.json","id":"ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition/index.html","scope":"ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition","status":"fully_proved","title":"Bias\u2013variance decomposition"},{"id":"ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition/integral-sq-sub-of-indep-fun","kind":"node","lean":"Cslib.MachineLearning.BiasVariance.integral_sq_sub_of_indepFun","parent":41023,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition/index.html#integral-sq-sub-of-indep-fun","scope":"ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition","status":"fully_proved","summary":"Let $(\\Omega, \\mu)$ be a probability space, with expectation $\\mathbb{E}$ and variance $\\operatorname{Var}$ taken with respect to $\\mu$. If $Y$ and $\\hat Y$ are independent square-integrable real random variables on $\\Omega$, then $\\mathbb{E}[(Y - \\hat Y)^2] = \\operatorname{Var}(Y) + \\operatorname{Var}(\\hat Y) + (\\mathbb{E}[Y] - \\mathbb{E}[\\hat Y])^2$.","title":"Expected squared error of independent random variables"},{"id":"ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition/integral-sq-noise-sub","kind":"node","lean":"Cslib.MachineLearning.BiasVariance.integral_sq_noise_sub","parent":41023,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition/index.html#integral-sq-noise-sub","scope":"ComputerScience/machine-learning/supervised-learning/bias-variance-decomposition","status":"fully_proved","summary":"Let $(\\Omega, \\mu)$ be a probability space, with expectation $\\mathbb{E}$ and variance $\\operatorname{Var}$ taken with respect to $\\mu$. Let $f$ be a real number, and let $\\varepsilon$ (the noise) and $\\hat Y$ (the predictor) be square-integrable real random variables on $\\Omega$ such that $\\mathbb{E}[\\varepsilon] = 0$ and $\\varepsilon$ is independent of $\\hat Y$. Then the expected squared error of $\\hat Y$ for the\u2026","title":"The bias-variance decomposition"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/cross-entropy-loss.json","id":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/index.html","scope":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss","status":"fully_proved","title":"Cross-entropy loss"},{"id":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/cross-entropy-loss","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.crossEntropyLoss","parent":41026,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/index.html#cross-entropy-loss","scope":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss","status":"fully_proved","summary":"Let $f$ be a probabilistic classifier, assigning to each input $x \\in X$ a probability distribution $f(x)$ on a finite set of labels, and let $D = ((x_1, y_1), \\dots, (x_N, y_N))$ be a labelled sample. The cross-entropy (log) loss of $f$ on $D$, in bits, is the value in $[0, \\infty]$","title":"Cross-entropy loss"},{"id":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/cross-entropy-loss-eq","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.crossEntropyLoss_eq","parent":41026,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/index.html#cross-entropy-loss-eq","scope":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss","status":"fully_proved","summary":"Let $f$ be a probabilistic classifier, assigning to each input $x \\in X$ a probability distribution $f(x)$ on a finite set of labels, and let $D = ((x_1, y_1), \\dots, (x_N, y_N))$ be a labelled sample. The cross-entropy loss of $f$ on $D$ equals the average surprisal of the labels,","title":"The cross-entropy loss is the average negative log-likelihood"},{"id":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/cross-entropy-loss-eq-top-iff","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.crossEntropyLoss_eq_top_iff","parent":41026,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/index.html#cross-entropy-loss-eq-top-iff","scope":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss","status":"fully_proved","summary":"Let $f$ be a probabilistic classifier, assigning to each input $x \\in X$ a probability distribution $f(x)$ on a finite set of labels, and let $D = ((x_1, y_1), \\dots, (x_N, y_N))$ be a labelled sample with $N \\ge 1$. The cross-entropy loss of $f$ on $D$ is $\\infty$ if and only if $f(x_n)(y_n) = 0$ for some $n$.","title":"The cross-entropy loss is infinite exactly when a label is ruled out"},{"id":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/cross-entropy-loss-eq-neg-logb-prod","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.crossEntropyLoss_eq_neg_logb_prod","parent":41026,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/index.html#cross-entropy-loss-eq-neg-logb-prod","scope":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss","status":"fully_proved","summary":"Let $f$ be a probabilistic classifier, assigning to each input $x \\in X$ a probability distribution $f(x)$ on a finite set of labels, and let $D = ((x_1, y_1), \\dots, (x_N, y_N))$ be a labelled sample with $f(x_n)(y_n) \\ne 0$ for every $n$. Then the cross-entropy loss of $f$ on $D$ equals $-\\log_2(L) / N$, where $L = \\prod_{n=1}^{N} f(x_n)(y_n)$ is the likelihood of the sample and the division is computed in\u2026","title":"Cross-entropy loss via the likelihood"},{"id":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/cross-entropy-loss-le-iff-of-ne-zero","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.crossEntropyLoss_le_iff_of_ne_zero","parent":41026,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/index.html#cross-entropy-loss-le-iff-of-ne-zero","scope":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss","status":"fully_proved","summary":"Let $f$ and $g$ be probabilistic classifiers, each assigning to every input $x \\in X$ a probability distribution on a finite set of labels, and let $D = ((x_1, y_1), \\dots, (x_N, y_N))$ be a labelled sample with $N \\ge 1$ such that $f(x_n)(y_n) \\ne 0$ and $g(x_n)(y_n) \\ne 0$ for every $n$. Then the cross-entropy loss of $f$ on $D$ is at most that of $g$ if and only if\u2026","title":"Comparing cross-entropy losses of classifiers that rule out no label"},{"id":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/cross-entropy-loss-le-iff","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.crossEntropyLoss_le_iff","parent":41026,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/cross-entropy-loss/index.html#cross-entropy-loss-le-iff","scope":"ComputerScience/machine-learning/supervised-learning/cross-entropy-loss","status":"fully_proved","summary":"Let $f$ and $g$ be probabilistic classifiers, each assigning to every input $x \\in X$ a probability distribution on a finite set of labels, and let $D = ((x_1, y_1), \\dots, (x_N, y_N))$ be a labelled sample with $N \\ge 1$. Then the cross-entropy loss of $f$ on $D$ is at most that of $g$ if and only if $\\prod_{n=1}^{N} g(x_n)(y_n) \\le \\prod_{n=1}^{N} f(x_n)(y_n)$, that is, $f$ gives the labels at least as large a\u2026","title":"Minimizing cross-entropy is maximum likelihood"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression.json","id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","title":"Gaussian process regression"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/kernel-matrix","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.kernelMatrix","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#kernel-matrix","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"For a kernel $k : X \\times X \\to \\mathbb{R}$ and two families of inputs $a = (a_i)_{i \\in \\alpha}$ and $b = (b_j)_{j \\in \\beta}$ in $X$, the kernel (Gram) matrix $K(a, b)$ is the $\\alpha \\times \\beta$ real matrix with entries $K(a, b)_{ij} = k(a_i, b_j)$.","title":"Kernel matrix"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/joint-cov","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.jointCov","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#joint-cov","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"The joint prior covariance of the test values `f(x*)` and the noisy training targets `y = f(x) + \u03b5`, `\u03b5 ~ N(0, \u03c3\u00b2 I)`: `[[K(x*, x*), K(x*, x)], [K(x, x*), K(x, x) + \u03c3\u00b2 I]]`.","title":"Joint prior covariance in Gaussian-process regression"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/test-part","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.testPart","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#test-part","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"For finite index sets $\\tau$ (test inputs) and $\\nu$ (training inputs), the test-value projection is the $\\tau \\times (\\tau \\sqcup \\nu)$ real matrix with the identity matrix on the columns indexed by $\\tau$ and a zero block on the columns indexed by $\\nu$. It sends a joint vector $(u, y)$ with $u \\in \\mathbb{R}^{\\tau}$ and $y \\in \\mathbb{R}^{\\nu}$ to its block of test values $u$.","title":"Test-value projection"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/target-part","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.targetPart","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#target-part","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"For finite index sets $\\tau$ (test inputs) and $\\nu$ (training inputs), the training-target projection is the $\\nu \\times (\\tau \\sqcup \\nu)$ real matrix with a zero block on the columns indexed by $\\tau$ and the identity matrix on the columns indexed by $\\nu$. It sends a joint vector $(u, y)$ with $u \\in \\mathbb{R}^{\\tau}$ and $y \\in \\mathbb{R}^{\\nu}$ to its block of training targets $y$.","title":"Training-target projection"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/is-unit-det-kernel-matrix-add","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.isUnit_det_kernelMatrix_add","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#is-unit-det-kernel-matrix-add","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"Let $k : X \\times X \\to \\mathbb{R}$ be a kernel and $x = (x_j)_{j \\in \\nu}$ a finite family of inputs whose kernel matrix $K = (k(x_i, x_j))_{i, j \\in \\nu}$ is positive semidefinite. Then for every $\\sigma^2 > 0$ the matrix $K + \\sigma^2 I$ has nonzero determinant, so it is invertible.","title":"Regularized kernel matrices are invertible"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/joint-cov-eq-add","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.jointCov_eq_add","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#joint-cov-eq-add","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"For a kernel $k : X \\times X \\to \\mathbb{R}$, a real number $\\sigma^2$, test inputs $x^* = (x^*_i)_{i \\in \\tau}$ and training inputs $x = (x_j)_{j \\in \\nu}$, the joint covariance of $k$, $\\sigma^2$, $x^*$ and $x$ equals $K + D$, where $K$ is the kernel matrix of the combined family $(x^*, x)$ indexed by $\\tau \\sqcup \\nu$ and $D$ is the diagonal matrix with entry $0$ at each test index and $\\sigma^2$ at each\u2026","title":"The joint covariance as a kernel matrix plus noise"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/joint-cov-pos-semidef","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.jointCov_posSemidef","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#joint-cov-pos-semidef","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"Let $k : X \\times X \\to \\mathbb{R}$ be a kernel, let $x^* = (x^*_i)_{i \\in \\tau}$ and $x = (x_j)_{j \\in \\nu}$ be families of test and training inputs, and let $\\sigma^2 \\ge 0$. If the kernel matrix of the combined family $(x^*, x)$, indexed by $\\tau \\sqcup \\nu$, is positive semidefinite, then the joint covariance of $k$, $\\sigma^2$, $x^*$ and $x$ is positive semidefinite.","title":"The joint covariance is positive semidefinite"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/kernel-matrix-pos-semidef-train","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.kernelMatrix_posSemidef_train","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#kernel-matrix-pos-semidef-train","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"Let $k : X \\times X \\to \\mathbb{R}$ be a kernel and let $x^* = (x^*_i)_{i \\in \\tau}$ and $x = (x_j)_{j \\in \\nu}$ be families of inputs. If the kernel matrix of the combined family $(x^*, x)$, indexed by $\\tau \\sqcup \\nu$, is positive semidefinite, then so is the kernel matrix $(k(x_i, x_j))_{i, j \\in \\nu}$ of $x$.","title":"Training blocks of positive semidefinite kernel matrices"},{"id":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/cond-distrib-test-given-targets","kind":"node","lean":"Cslib.MachineLearning.GaussianProcessRegression.condDistrib_test_given_targets","parent":41033,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/gaussian-process-regression/index.html#cond-distrib-test-given-targets","scope":"ComputerScience/machine-learning/supervised-learning/gaussian-process-regression","status":"fully_proved","summary":"Let $k : X \\times X \\to \\mathbb{R}$ be a kernel, let $x^* = (x^*_i)_{i \\in \\tau}$ (test inputs) and $x = (x_j)_{j \\in \\nu}$ (training inputs) be finite families in $X$, and write $K(a, b)$ for the kernel matrix $(k(a_i, b_j))_{i, j}$. Assume that the kernel matrix of the combined family $(x^*, x)$, indexed by $\\tau \\sqcup \\nu$, is positive semidefinite, and let $\\sigma^2 > 0$. Let $V$ be a measurable random vector\u2026","title":"The Gaussian-process posterior"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees.json","id":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/index.html","scope":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees","status":"fully_proved","title":"Information gain in decision trees"},{"id":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/info-gain","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.infoGain","parent":41043,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/index.html#info-gain","scope":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees","status":"fully_proved","summary":"Let $p$ be a joint probability law on $\\alpha \\times \\beta$ with $\\alpha$ and $\\beta$ finite, the law of an attribute $X$ (first coordinate) and a label $Y$ (second coordinate). The information gain of splitting on $X$ for predicting $Y$ is the real number $H(Y) - H(Y \\mid X)$, where $H(Y)$ is the Shannon entropy in bits of the marginal law of $Y$ and $H(Y \\mid X)$ is the conditional entropy in bits of $Y$ given\u2026","title":"Information gain of a split"},{"id":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/info-gain-eq","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.infoGain_eq","parent":41043,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/index.html#info-gain-eq","scope":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees","status":"fully_proved","summary":"Let $p$ be a joint probability law on $\\alpha \\times \\beta$ with $\\alpha$ and $\\beta$ finite, and let $p_X$ and $p_Y$ be its marginal laws on $\\alpha$ and on $\\beta$. Writing $H$ for Shannon entropy in bits, the information gain of $p$ equals $H(p_X) + H(p_Y) - H(p)$.","title":"Information gain in terms of entropies"},{"id":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/info-gain-nonneg","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.infoGain_nonneg","parent":41043,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/index.html#info-gain-nonneg","scope":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees","status":"fully_proved","summary":"For every joint probability law $p$ on $\\alpha \\times \\beta$ with $\\alpha$ and $\\beta$ finite, the information gain of $p$ is at least $0$.","title":"No split has negative information gain"},{"id":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/of-real-info-gain","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.ofReal_infoGain","parent":41043,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/index.html#of-real-info-gain","scope":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be finite sets carrying \u03c3-algebras in which singletons are measurable, and let $p$ be a joint probability law on $\\alpha \\times \\beta$. The information gain of $p$, regarded as an element of $[0, \\infty]$, equals the mutual information in bits of $p$ viewed as a probability measure on $\\alpha \\times \\beta$.","title":"Information gain is mutual information"},{"id":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/info-gain-eq-zero-iff","kind":"node","lean":"Cslib.MachineLearning.InformationTheory.infoGain_eq_zero_iff","parent":41043,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees/index.html#info-gain-eq-zero-iff","scope":"ComputerScience/machine-learning/supervised-learning/information-gain-in-decision-trees","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be finite sets carrying \u03c3-algebras in which singletons are measurable, and let $p$ be a joint probability law on $\\alpha \\times \\beta$. The information gain of $p$ is zero if and only if $p$, as a measure, equals the product of its marginal law on $\\alpha$ and its marginal law on $\\beta$; that is, exactly when the attribute (first coordinate) and the label (second coordinate) are independent.","title":"Zero information gain means independence"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression.json","id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","title":"Least squares, ridge and lasso regression"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/rss","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.rss","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#rss","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For a real design matrix $X$ with $m$ rows and columns indexed by a finite type $\\iota$, responses $y \\in \\mathbb{R}^m$ and $\\theta \\in \\mathbb{R}^\\iota$, the residual sum of squares is $S(\\theta) = (y - X\\theta) \\cdot (y - X\\theta) = \\sum_{k=1}^{m} (y_k - (X\\theta)_k)^2$.","title":"Residual sum of squares"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/rss-add","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.rss_add","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#rss-add","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real design matrix with $m$ rows and columns indexed by a finite type $\\iota$, let $y \\in \\mathbb{R}^m$, and write $S(\\theta) = (y - X\\theta) \\cdot (y - X\\theta)$ for the residual sum of squares. Then for all $\\theta, v \\in \\mathbb{R}^\\iota$,","title":"Expansion of the residual sum of squares"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/rss-eq-of-normal","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.rss_eq_of_normal","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#rss-eq-of-normal","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real design matrix with $m$ rows and columns indexed by a finite type $\\iota$, let $y \\in \\mathbb{R}^m$, write $S$ for the residual sum of squares, and let $\\theta_0 \\in \\mathbb{R}^\\iota$ solve the normal equations $X^\\top X \\theta_0 = X^\\top y$. Then for every $\\theta \\in \\mathbb{R}^\\iota$,","title":"Pythagoras for least squares"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/rss-le-iff-normal","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.rss_le_iff_normal","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#rss-le-iff-normal","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real design matrix with $m$ rows and columns indexed by a finite type $\\iota$, let $y \\in \\mathbb{R}^m$, and write $S$ for the residual sum of squares. A vector $\\theta_0 \\in \\mathbb{R}^\\iota$ satisfies $S(\\theta_0) \\le S(\\theta)$ for every $\\theta \\in \\mathbb{R}^\\iota$ if and only if $X^\\top X \\theta_0 = X^\\top y$.","title":"Least squares and the normal equations"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/exists-normal","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.exists_normal","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#exists-normal","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, and let $y \\in \\mathbb{R}^m$. Then there is $\\theta \\in \\mathbb{R}^\\iota$ with $X^\\top X \\theta = X^\\top y$: the normal equations always have a solution, whatever the rank of $X$.","title":"Least-squares estimates exist"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/normal-inv","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.normal_inv","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#normal-inv","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$ such that $X^\\top X$ is invertible, and let $y \\in \\mathbb{R}^m$. Then $\\hat\\theta = (X^\\top X)^{-1} X^\\top y$ solves the normal equations $X^\\top X \\hat\\theta = X^\\top y$.","title":"Closed-form solution of the normal equations"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/convex-on-rss","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.convexOn_rss","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#convex-on-rss","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, and let $y \\in \\mathbb{R}^m$. Then the residual sum of squares $\\theta \\mapsto \\sum_k (y_k - (X\\theta)_k)^2$ is a convex function on all of $\\mathbb{R}^\\iota$.","title":"Convexity of the residual sum of squares"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/gauss-likelihood","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.gaussLikelihood","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#gauss-likelihood","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$ and let $v \\ge 0$. The Gaussian likelihood with noise variance $v$ is the function of $\\theta \\in \\mathbb{R}^\\iota$ given by","title":"Gaussian likelihood of linear regression"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/gauss-likelihood-le-iff","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.gaussLikelihood_le_iff","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#gauss-likelihood-le-iff","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$ and $v > 0$, let $L$ be the Gaussian likelihood of linear regression with noise variance $v$, and let $\\mathrm{RSS}(\\theta) = \\sum_k (y_k - (X\\theta)_k)^2$ be the residual sum of squares. Then for all $\\theta, \\theta' \\in \\mathbb{R}^\\iota$, $L(\\theta) \\le L(\\theta')$ if and only if\u2026","title":"The Gaussian likelihood decreases with the residual sum of squares"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/is-max-on-gauss-likelihood-iff","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.isMaxOn_gaussLikelihood_iff","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#is-max-on-gauss-likelihood-iff","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$ and $v > 0$, and let $L$ be the Gaussian likelihood of linear regression with noise variance $v$. Then $\\theta_0 \\in \\mathbb{R}^\\iota$ maximizes $L$ over $\\mathbb{R}^\\iota$ if and only if it solves the normal equations $X^\\top X \\theta_0 = X^\\top y$.","title":"Least squares is maximum likelihood"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/exists-is-max-on-gauss-likelihood","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.exists_isMaxOn_gaussLikelihood","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#exists-is-max-on-gauss-likelihood","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$ and $v > 0$, and let $L$ be the Gaussian likelihood of linear regression with noise variance $v$. Then $L$ attains its maximum on $\\mathbb{R}^\\iota$: there is $\\theta_0 \\in \\mathbb{R}^\\iota$ with $L(\\theta) \\le L(\\theta_0)$ for every $\\theta \\in \\mathbb{R}^\\iota$.","title":"Existence of the Gaussian maximum likelihood estimate"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/ridge","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.ridge","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#ridge","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For a real design matrix $X$ with $m$ rows and columns indexed by a finite type $\\iota$, responses $y \\in \\mathbb{R}^m$, a real number $\\lambda$ and $\\theta \\in \\mathbb{R}^\\iota$, let $S(\\theta) = \\sum_{k=1}^{m} (y_k - (X\\theta)_k)^2$ be the residual sum of squares. The ridge objective is $S(\\theta) + \\lambda \\sum_{i} \\theta_i^2$.","title":"Ridge objective"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/ridge-eq-of-normal","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.ridge_eq_of_normal","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#ridge-eq-of-normal","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$ and $\\lambda \\in \\mathbb{R}$, and write $\\|\\cdot\\|$ for the Euclidean norm and $R(\\theta) = \\|y - X\\theta\\|^2 + \\lambda \\|\\theta\\|^2$ for the ridge objective. If $\\theta_0 \\in \\mathbb{R}^\\iota$ satisfies $(X^\\top X + \\lambda I)\\theta_0 = X^\\top y$, then for every $\\theta \\in \\mathbb{R}^\\iota$,\u2026","title":"Expansion of the ridge objective around a solution of its normal equations"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/ridge-le-of-normal","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.ridge_le_of_normal","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#ridge-le-of-normal","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$ and $\\lambda \\ge 0$, and write $\\|\\cdot\\|$ for the Euclidean norm. If $\\theta_0 \\in \\mathbb{R}^\\iota$ satisfies $(X^\\top X + \\lambda I)\\theta_0 = X^\\top y$, then for every $\\theta \\in \\mathbb{R}^\\iota$, $\\|y - X\\theta_0\\|^2 + \\lambda \\|\\theta_0\\|^2 \\le \\|y - X\\theta\\|^2 + \\lambda \\|\\theta\\|^2$.","title":"Solutions of the ridge normal equations minimize the ridge objective"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/pos-def-ridge","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.posDef_ridge","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#pos-def-ridge","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For every real matrix $X$ with $m$ rows and columns indexed by a finite set $\\iota$ and every real $\\lambda > 0$, the matrix $X^\\top X + \\lambda I$ is positive definite.","title":"The ridge matrix is positive definite"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/ridge-estimate","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.ridgeEstimate","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#ridge-estimate","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For a real matrix $X$ with $m$ rows and columns indexed by a finite set $\\iota$, responses $y \\in \\mathbb{R}^m$ and a real number $\\lambda$, the ridge estimate is $(X^\\top X + \\lambda I)^{-1} X^\\top y \\in \\mathbb{R}^\\iota$.","title":"Ridge estimate"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/ridge-le-ridge-estimate","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.ridge_le_ridgeEstimate","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#ridge-le-ridge-estimate","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$ and $\\lambda > 0$, and write $\\|\\cdot\\|$ for the Euclidean norm. The ridge estimate $\\hat\\theta = (X^\\top X + \\lambda I)^{-1} X^\\top y$ minimizes the ridge objective: for every $\\theta \\in \\mathbb{R}^\\iota$, $\\|y - X\\hat\\theta\\|^2 + \\lambda \\|\\hat\\theta\\|^2 \\le \\|y - X\\theta\\|^2 + \\lambda \\|\\theta\\|^2$.","title":"Closed form of ridge regression"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/prod-gaussian-pdfreal-zero","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.prod_gaussianPDFReal_zero","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#prod-gaussian-pdfreal-zero","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For a finite set $\\iota$, a variance $\\tau \\ge 0$ and $\\theta \\in \\mathbb{R}^\\iota$, the product over $i \\in \\iota$ of the densities of the normal distribution $N(0, \\tau)$ at $\\theta_i$ equals $\\left(\\frac{1}{\\sqrt{2\\pi\\tau}}\\right)^{|\\iota|} \\exp\\left(-\\frac{1}{2\\tau} \\sum_{i \\in \\iota} \\theta_i^2\\right)$.","title":"Density of independent centred Gaussian coordinates"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/ridge-posterior","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.ridgePosterior","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#ridge-posterior","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For $s \\ge 0$ let $\\varphi_s$ be the density of $N(0, s)$, $\\varphi_s(x) = (2 \\pi s)^{-1/2} e^{-x^2 / (2 s)}$, with $\\varphi_0$ the zero function. For a real design matrix $X$ with $m$ rows and columns indexed by a finite type $\\iota$, responses $y \\in \\mathbb{R}^m$, variances $v, \\tau \\ge 0$ and $\\theta \\in \\mathbb{R}^\\iota$, the posterior kernel is the Gaussian likelihood of $y$ times the prior density of $\\theta$:","title":"Posterior kernel of ridge regression"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/ridge-posterior-le-iff","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.ridgePosterior_le_iff","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#ridge-posterior-le-iff","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real design matrix with $m$ rows and columns indexed by a finite type $\\iota$, let $y \\in \\mathbb{R}^m$, and let $v$ and $\\tau$ be positive reals. Let $p$ be the posterior kernel of linear regression with Gaussian noise of variance $v$ and independent $N(0, \\tau)$ priors on the coordinates of the parameter, and let $R$ be the ridge objective with $\\lambda = v / \\tau$. Then for all\u2026","title":"Ridge regression is MAP estimation"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/is-max-on-ridge-posterior","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.isMaxOn_ridgePosterior","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#is-max-on-ridge-posterior","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real design matrix with $m$ rows and columns indexed by a finite type $\\iota$, let $y \\in \\mathbb{R}^m$, and let $v$ and $\\tau$ be positive reals. Let $p$ be the posterior kernel of linear regression with Gaussian noise of variance $v$ and independent $N(0, \\tau)$ priors on the coordinates of the parameter, and let $\\hat\\theta = (X^\\top X + \\lambda I)^{-1} X^\\top y$ be the ridge estimate with\u2026","title":"The ridge estimate maximizes the ridge posterior"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/lasso","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.lasso","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#lasso","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For a real matrix $X$ with $m$ rows and columns indexed by a finite set $\\iota$, responses $y \\in \\mathbb{R}^m$ and a real number $\\lambda$, the lasso objective at $\\theta \\in \\mathbb{R}^\\iota$ is $\\sum_{k=1}^m (y_k - (X\\theta)_k)^2 + \\lambda \\sum_{i \\in \\iota} |\\theta_i|$, the residual sum of squares plus $\\lambda$ times the $\\ell^1$ norm of $\\theta$.","title":"Lasso objective"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/laplace-pdf","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.laplacePDF","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#laplace-pdf","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For a scale $b \\ge 0$, the density of the Laplace distribution with location $0$ and scale $b$ is the function on $\\mathbb{R}$","title":"Laplace density"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/lasso-posterior","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.lassoPosterior","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#lasso-posterior","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$, and let $v \\ge 0$ and $b \\ge 0$. The lasso posterior kernel at $\\theta \\in \\mathbb{R}^\\iota$ is","title":"Lasso posterior kernel"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/lasso-posterior-le-iff","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.lassoPosterior_le_iff","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#lasso-posterior-le-iff","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$, and let $v > 0$ and $b > 0$. Let $P$ be the posterior kernel of linear regression with Gaussian noise of variance $v$ and the Laplace prior with scale $b$ on each coordinate, and let $\\ell$ be the lasso objective with $\\lambda = 2v/b$,\u2026","title":"Lasso is MAP estimation"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/convex-on-lasso","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.convexOn_lasso","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#convex-on-lasso","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$, let $y \\in \\mathbb{R}^m$ and let $\\lambda \\ge 0$. Then the lasso objective $\\theta \\mapsto \\sum_k (y_k - (X\\theta)_k)^2 + \\lambda \\sum_{i \\in \\iota} |\\theta_i|$ is a convex function on all of $\\mathbb{R}^\\iota$.","title":"Convexity of the lasso objective"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/soft-threshold","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.softThreshold","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#soft-threshold","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"For real numbers $t$ and $z$, the soft thresholding of $z$ at $t$ is $S_t(z) = z - t$ if $z > t$; otherwise $S_t(z) = z + t$ if $z < -t$; and $S_t(z) = 0$ in the remaining case.","title":"Soft thresholding"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/soft-threshold-le","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.softThreshold_le","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#soft-threshold-le","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $t \\ge 0$ and $z \\in \\mathbb{R}$, and let $S_t(z)$ be the soft thresholding of $z$ at $t$. Then for every $s \\in \\mathbb{R}$,","title":"Soft thresholding solves the one-dimensional lasso"},{"id":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/lasso-soft-threshold-le","kind":"node","lean":"Cslib.MachineLearning.LeastSquares.lasso_softThreshold_le","parent":41049,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression/index.html#lasso-soft-threshold-le","scope":"ComputerScience/machine-learning/supervised-learning/least-squares-ridge-and-lasso-regression","status":"fully_proved","summary":"Let $X$ be a real matrix with $m$ rows and columns indexed by a finite set $\\iota$ such that $X^\\top X = I$, let $y \\in \\mathbb{R}^m$ and let $\\lambda \\ge 0$. Let $\\hat\\theta \\in \\mathbb{R}^\\iota$ be given by $\\hat\\theta_i = S_{\\lambda/2}((X^\\top y)_i)$, where $S_t$ is soft thresholding at $t$. Then $\\hat\\theta$ minimizes the lasso objective: for every $\\theta \\in \\mathbb{R}^\\iota$,\u2026","title":"Lasso with an orthonormal design"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup.json","id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","title":"Kernel ridge regression and GBLUP"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/pos-def-mul-transpose-add","kind":"node","lean":"Cslib.MachineLearning.GBLUP.posDef_mul_transpose_add","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#pos-def-mul-transpose-add","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"For `\u03bb > 0` the GBLUP matrix `ZZ\u1d40 + \u03bbI` is positive definite.","title":"Positive definiteness of the GBLUP matrix"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/transpose-mul-add-inv-mul-transpose","kind":"node","lean":"Cslib.MachineLearning.GBLUP.transpose_mul_add_inv_mul_transpose","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#transpose-mul-add-inv-mul-transpose","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"The push-through identity `(Z\u1d40Z + \u03bbI)\u207b\u00b9 Z\u1d40 = Z\u1d40 (ZZ\u1d40 + \u03bbI)\u207b\u00b9` for `\u03bb > 0`.","title":"Push-through identity for ridge regression"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/mul-ridge-hat-eq-gblup-hat","kind":"node","lean":"Cslib.MachineLearning.GBLUP.mul_ridgeHat_eq_gblupHat","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#mul-ridge-hat-eq-gblup-hat","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"RR-BLUP = GBLUP (fitted values): `Z (Z\u1d40Z + \u03bbI)\u207b\u00b9 Z\u1d40 = ZZ\u1d40 (ZZ\u1d40 + \u03bbI)\u207b\u00b9` for `\u03bb > 0`.","title":"RR-BLUP = GBLUP"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/ridge-estimate-eq","kind":"node","lean":"Cslib.MachineLearning.GBLUP.ridgeEstimate_eq","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#ridge-estimate-eq","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"The ridge marker effects are back-solved from GBLUP: `(Z\u1d40Z + \u03bbI)\u207b\u00b9 Z\u1d40 y = Z\u1d40 (ZZ\u1d40 + \u03bbI)\u207b\u00b9 y`.","title":"Ridge marker effects are back-solved from GBLUP"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/mul-vec-ridge-estimate","kind":"node","lean":"Cslib.MachineLearning.GBLUP.mulVec_ridgeEstimate","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#mul-vec-ridge-estimate","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"RR-BLUP = GBLUP (predictions): for any genotypes `Z'` (training or new individuals), `Z' \u03b2\u0302 = Z' Z\u1d40 (ZZ\u1d40 + \u03bbI)\u207b\u00b9 y`.","title":"RR-BLUP = GBLUP"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/marker-kernel","kind":"node","lean":"Cslib.MachineLearning.GBLUP.markerKernel","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#marker-kernel","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"The linear marker kernel `k(a, b) = s \u27eaa, b\u27eb = s \u2211\u2c7c a\u2c7c b\u2c7c` on genotype vectors: the covariance of genetic values `a\u1d40\u03b2` and `b\u1d40\u03b2` when the marker effects `\u03b2` are independent with variance `s`.","title":"Linear marker kernel on genotype vectors"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/marker-kernel-apply","kind":"node","lean":"Cslib.MachineLearning.GBLUP.markerKernel_apply","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#marker-kernel-apply","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"Unfolding `markerKernel`.","title":"Formula for the linear marker kernel"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/kernel-matrix-marker-kernel","kind":"node","lean":"Cslib.MachineLearning.GBLUP.kernelMatrix_markerKernel","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#kernel-matrix-marker-kernel","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"The kernel matrix of the marker kernel between genotypes `Z'` and `Z` is `s Z' Z\u1d40`.","title":"Kernel matrix of the marker kernel"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/kernel-matrix-marker-kernel-pos-semidef","kind":"node","lean":"Cslib.MachineLearning.GBLUP.kernelMatrix_markerKernel_posSemidef","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#kernel-matrix-marker-kernel-pos-semidef","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"For `s \u2265 0` the marker kernel is a covariance kernel: its kernel matrix on any inputs is positive semidefinite.","title":"Positive semidefiniteness of the marker kernel matrix"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/gp-mean-marker-kernel","kind":"node","lean":"Cslib.MachineLearning.GBLUP.gpMean_markerKernel","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#gp-mean-marker-kernel","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"For the marker kernel with variance `s > 0` and noise variance `\u03c3\u00b2 > 0`, the predictive-mean matrix `K(Z', Z) (K(Z, Z) + \u03c3\u00b2 I)\u207b\u00b9` of Gaussian-process regression is `Z' Z\u1d40 (ZZ\u1d40 + (\u03c3\u00b2/s) I)\u207b\u00b9`.","title":"GP posterior mean = GBLUP"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/gp-mean-marker-kernel-mul-vec","kind":"node","lean":"Cslib.MachineLearning.GBLUP.gpMean_markerKernel_mulVec","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#gp-mean-marker-kernel-mul-vec","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"With the marker kernel, the GP predictive mean at genotypes `Z'` is `Z' \u03b2\u0302` for the ridge marker effects `\u03b2\u0302` with `\u03bb = \u03c3\u00b2/s`.","title":"GP posterior mean = RR-BLUP"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/cond-distrib-marker-kernel","kind":"node","lean":"Cslib.MachineLearning.GBLUP.condDistrib_markerKernel","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#cond-distrib-marker-kernel","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"If the genetic values `Z' \u03b2` of individuals with genotypes `Z'` and the phenotypes `y = Z \u03b2 + \u03b5` of the training individuals are jointly Gaussian with the covariance of the marker kernel (marker-effect variance `s > 0`, residual variance `\u03c3\u00b2 > 0`), then given `y` the genetic values are Gaussian with mean `Z' Z\u1d40 (ZZ\u1d40 + (\u03c3\u00b2/s) I)\u207b\u00b9 y`, which is the RR-BLUP prediction `Z' \u03b2\u0302` (`gpMean_markerKernel_mulVec`), and\u2026","title":"The GBLUP posterior"},{"id":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/kernel-ridge-fit-eq-mul-vec-ridge-estima","kind":"node","lean":"Cslib.MachineLearning.GBLUP.kernelRidgeFit_eq_mulVec_ridgeEstimate","parent":41079,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup/index.html#kernel-ridge-fit-eq-mul-vec-ridge-estima","scope":"ComputerScience/machine-learning/supervised-learning/kernel-ridge-regression-and-gblup","status":"fully_proved","summary":"In a real RKHS whose kernel has Gram matrix `ZZ\u1d40` at the training inputs, the kernel ridge fit (the minimizer of `\u2211\u1d62 (y\u1d62 - f x\u1d62)\u00b2 + \u03bb \u2016f\u2016\u00b2`, `kernelRidgeRisk_kernelRidgeFit_le`) has fitted values `Z \u03b2\u0302`.","title":"Kernel ridge regression = GBLUP = RR-BLUP"},{"data_url":"scopes/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets.json","id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","kind":"scope","navigation":"module","parent":40985,"result":false,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","title":"Sufficient input subsets"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/is-greedy","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.IsGreedy","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#is-greedy","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"`pick` is a greedy backward-selection rule for `g`: from a nonempty remaining set `R` it masks a feature `i \u2208 R` maximizing `g (R.erase i)`.","title":"Greedy backward-selection rule"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/exists-is-greedy","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.exists_isGreedy","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#exists-is-greedy","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Greedy rules exist for every `g`.","title":"Existence of greedy backward-selection rules"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/chain","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.chain","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#chain","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Backward selection: the list of remaining sets `R\u2080 = R, R\u2081, \u2026, \u2205`, where `R_{t+1}` is `R_t` with the feature chosen by `pick` masked.","title":"Backward-selection chain of remaining feature sets"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/find-sis","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.findSIS","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#find-sis","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"`FindSIS`: un-mask features in reverse order of masking (start from `\u2205`) and return the first set on which the prediction reaches the threshold, or `none`. The paper's procedure does not test `\u2205` itself; the two agree whenever `g \u2205 < \u03c4`, the case the paper considers.","title":"FindSIS procedure for sufficient input subsets"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/chain-of-nonempty","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.chain_of_nonempty","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#chain-of-nonempty","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Unfolding `chain` at a nonempty set.","title":"Backward-selection chain at a nonempty set"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/chain-of-not-nonempty","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.chain_of_not_nonempty","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#chain-of-not-nonempty","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Unfolding `chain` at the empty set (the chain stops).","title":"Backward-selection chain at the empty set"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/subset-of-mem-chain","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.subset_of_mem_chain","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#subset-of-mem-chain","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Every set in the backward-selection chain is a subset of the starting set.","title":"Sets in the backward-selection chain are subsets of the start"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/mem-chain-self","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.mem_chain_self","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#mem-chain-self","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"The starting set is the first set of the chain.","title":"Starting set begins the backward-selection chain"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/find-sis-is-some","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.findSIS_isSome","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#find-sis-is-some","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"If the prediction on the starting set reaches the threshold, `findSIS` returns a set.","title":"FindSIS returns a set when the start reaches the threshold"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/find-sis-sufficient","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.findSIS_sufficient","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#find-sis-sufficient","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Criterion (1): the returned SIS is a subset of the starting features and is sufficient.","title":"FindSIS returns a sufficient subset of the features"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/find-sis-erase-lt","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.findSIS_erase_lt","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#find-sis-erase-lt","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Proposition 1 (first part) of Carter et al.: no feature of the returned SIS can be additionally masked while keeping the prediction at or above the threshold.","title":"Proposition 1 (first part)"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/find-sis-card-le","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.findSIS_card_le","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#find-sis-card-le","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Proposition 1 (second part) of Carter et al.: among the sets met during backward selection, the returned SIS has the smallest cardinality among the sufficient ones.","title":"Proposition 1 (second part)"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/find-sis-minimal-of-monotone","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.findSIS_minimal_of_monotone","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#find-sis-minimal-of-monotone","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"For a monotone set function (masking more features never raises the prediction), the returned SIS is inclusion-minimal: no proper subset is sufficient. (Derived; the paper states only Proposition 1.)","title":"Minimality of FindSIS for monotone set functions"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/sis-collection","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.sisCollection","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#sis-collection","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"`SIScollection`: while the remaining features `R` are sufficient (`\u03c4 \u2264 g R`), find a SIS among them, record it and mask it. Stops if `findSIS` returns an empty set (only possible when `\u03c4 \u2264 g \u2205`).","title":"SIS-collection procedure"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/sis-collection-sufficient","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.sisCollection_sufficient","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#sis-collection-sufficient","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Every set of the SIS-collection is a nonempty sufficient subset of the starting features.","title":"Sets of the SIS-collection are nonempty and sufficient"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/sis-collection-pairwise-disjoint","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.sisCollection_pairwiseDisjoint","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#sis-collection-pairwise-disjoint","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"The sets of the SIS-collection are pairwise disjoint.","title":"Sets of the SIS-collection are pairwise disjoint"},{"id":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/sis-collection-remaining-lt","kind":"node","lean":"Cslib.MachineLearning.SufficientInputSubsets.sisCollection_remaining_lt","parent":41093,"result":true,"route":"../roadmap/ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets/index.html#sis-collection-remaining-lt","scope":"ComputerScience/machine-learning/supervised-learning/sufficient-input-subsets","status":"fully_proved","summary":"Proposition 2 of Carter et al.: if the fully masked input is below the threshold, masking every set of the SIS-collection leaves the prediction below the threshold.","title":"Proposition 2"},{"data_url":"scopes/ComputerScience/machine-learning/unsupervised-learning.json","id":"ComputerScience/machine-learning/unsupervised-learning","kind":"scope","parent":40911,"result":false,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/index.html","scope":"ComputerScience/machine-learning/unsupervised-learning","status":"fully_proved","summary":"Structure in unlabelled data: k-means, archetypal analysis, nonnegative matrix factorization, the successive projection algorithm and probabilistic PCA.","title":"Unsupervised learning and matrix factorization"},{"data_url":"scopes/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis.json","id":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","kind":"scope","navigation":"module","parent":41111,"result":false,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/index.html","scope":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","status":"fully_proved","title":"Archetypal analysis"},{"id":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/rss","kind":"node","lean":"Cslib.MachineLearning.ArchetypalAnalysis.rss","parent":41112,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/index.html#rss","scope":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","status":"fully_proved","summary":"The residual sum of squares of archetypes `z` for the data `x`: each data point is replaced by the closest point of the convex hull of the archetypes. This is only meaningful for `0 < p`: for `p = 0` the hull is empty and `infDist` to the empty set is `0` by convention, so `rss` is `0` vacuously and must not be read as a perfect fit.","title":"Residual sum of squares in archetypal analysis"},{"id":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/feasible","kind":"node","lean":"Cslib.MachineLearning.ArchetypalAnalysis.Feasible","parent":41112,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/index.html#feasible","scope":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","status":"fully_proved","summary":"Archetypes are feasible if each is a mixture of the data points.","title":"Feasible archetypes"},{"id":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/mean","kind":"node","lean":"Cslib.MachineLearning.ArchetypalAnalysis.mean","parent":41112,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/index.html#mean","scope":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","status":"fully_proved","summary":"The sample mean of the data.","title":"Sample mean of the data"},{"id":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/rss-const-mean-le","kind":"node","lean":"Cslib.MachineLearning.ArchetypalAnalysis.rss_const_mean_le","parent":41112,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/index.html#rss-const-mean-le","scope":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","status":"fully_proved","summary":"Cutler\u2013Breiman Proposition 1(i): with a single archetype, the sample mean is feasible and minimizes the RSS.","title":"The sample mean is the optimal single archetype"},{"id":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/exists-frontier-rss-le","kind":"node","lean":"Cslib.MachineLearning.ArchetypalAnalysis.exists_frontier_rss_le","parent":41112,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/index.html#exists-frontier-rss-le","scope":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","status":"fully_proved","summary":"Cutler\u2013Breiman Proposition 1(ii): for at least two archetypes, any feasible archetypes can be replaced by feasible archetypes on the boundary of the convex hull of the data whose convex hull contains the original one; in particular the RSS does not increase.","title":"Feasible archetypes can be moved to the boundary of the hull"},{"id":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/exists-frontier-minimizer","kind":"node","lean":"Cslib.MachineLearning.ArchetypalAnalysis.exists_frontier_minimizer","parent":41112,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/index.html#exists-frontier-minimizer","scope":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","status":"fully_proved","summary":"If the RSS has a feasible minimizer with `p \u2265 2` archetypes, it has one with every archetype on the boundary of the convex hull of the data.","title":"Existence of an archetype minimizer on the hull boundary"},{"id":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/rss-eq-zero-of-subset","kind":"node","lean":"Cslib.MachineLearning.ArchetypalAnalysis.rss_eq_zero_of_subset","parent":41112,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis/index.html#rss-eq-zero-of-subset","scope":"ComputerScience/machine-learning/unsupervised-learning/archetypal-analysis","status":"fully_proved","summary":"Cutler\u2013Breiman Proposition 1(iii): if every data point is a mixture of the archetypes, the RSS vanishes. Requires `0 < p`: for `p = 0` the hull is empty, so `rss` is `0` vacuously and the vanishing would not witness a fit.","title":"The RSS vanishes when the archetypes' convex hull contains the data"},{"data_url":"scopes/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering.json","id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","kind":"scope","navigation":"module","parent":41111,"result":false,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","title":"k-means clustering"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/centroid","kind":"node","lean":"Cslib.MachineLearning.KMeans.centroid","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#centroid","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $x_i$ be points in a real inner product space and $s$ a finite set of indices. The centroid of the points indexed by $s$ is $\\frac{1}{|s|} \\sum_{i \\in s} x_i$; when $s$ is empty it is the zero vector.","title":"Centroid"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/sum-sq-eq","kind":"node","lean":"Cslib.MachineLearning.KMeans.sum_sq_eq","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#sum-sq-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $x_i$ ($i \\in \\iota$) be points in a real inner product space $E$, let $s$ be a finite set of indices, and let $\\mu$ be the centroid of the points $x_i$ with $i \\in s$. Then for every point $c \\in E$,","title":"Sum of squared distances about the centroid"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/nearest","kind":"node","lean":"Cslib.MachineLearning.KMeans.Nearest","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#nearest","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Given points $x_i$ ($i \\in \\iota$) in a normed additive commutative group $E$ and centers $c_0, \\dots, c_{k-1} \\in E$, an assignment $a : \\iota \\to \\{0, \\dots, k-1\\}$ assigns every point to a nearest center when $\\|x_i - c_{a(i)}\\| \\le \\|x_i - c_j\\|$ for every index $i \\in \\iota$ and every $j < k$.","title":"Nearest-center assignment"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/exists-nearest","kind":"node","lean":"Cslib.MachineLearning.KMeans.exists_nearest","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#exists-nearest","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"For any points $x_i$ in a normed additive commutative group and any centers $c_0, \\dots, c_{k-1}$ with $k \\ge 1$, there is an assignment that sends every point to a nearest center.","title":"Nearest-center assignments exist"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/cost","kind":"node","lean":"Cslib.MachineLearning.KMeans.cost","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#cost","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"For points $x_i$ ($i \\in \\iota$, a finite type) in a normed additive commutative group, an assignment $a$ of each point $i$ to a cluster $a(i) \\in \\{0, \\dots, k-1\\}$ and centers $c_0, \\dots, c_{k-1}$, the $k$-means cost is $\\sum_{i \\in \\iota} \\lVert x_i - c_{a(i)} \\rVert^2$, the sum of the squared distances from the points to their assigned centers.","title":"The k-means cost"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/cost-eq-sum-clusters","kind":"node","lean":"Cslib.MachineLearning.KMeans.cost_eq_sum_clusters","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#cost-eq-sum-clusters","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"For points $x_i$ ($i \\in \\iota$, a finite type) in a normed additive commutative group, an assignment $a$ of each point $i$ to a cluster $a(i) \\in \\{0, \\dots, k-1\\}$ and centers $c_0, \\dots, c_{k-1}$, the $k$-means cost equals $\\sum_{j} \\sum_{i : a(i) = j} \\lVert x_i - c_j \\rVert^2$: for each cluster $j$, add up the squared distances from the points assigned to $j$ to its center $c_j$.","title":"The cost as a sum over clusters"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/centers","kind":"node","lean":"Cslib.MachineLearning.KMeans.centers","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#centers","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $x_i$ ($i \\in \\iota$, a finite type) be points in a real inner product space and let $a$ assign each point $i$ to a cluster $a(i) \\in \\{0, \\dots, k-1\\}$. The updated centers send each cluster $j$ to the centroid of the points $x_i$ with $a(i) = j$. A cluster with no points gets the center $0$, which does not affect the $k$-means cost of $a$.","title":"The centroid update step"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/cost-centers-le","kind":"node","lean":"Cslib.MachineLearning.KMeans.cost_centers_le","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#cost-centers-le","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"For a fixed assignment, the centroids are the best centers.","title":"The update step never increases the cost"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/cost-nearest-le","kind":"node","lean":"Cslib.MachineLearning.KMeans.cost_nearest_le","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#cost-nearest-le","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"For fixed centers, nearest-center assignment is optimal.","title":"The assignment step never increases the cost"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/cost-lloyd-le","kind":"node","lean":"Cslib.MachineLearning.KMeans.cost_lloyd_le","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#cost-lloyd-le","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Recompute the centers, then reassign every point to a nearest new center.","title":"One round of Lloyd's algorithm never increases the cost"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/is-run","kind":"node","lean":"Cslib.MachineLearning.KMeans.IsRun","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#is-run","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $x_i$ ($i \\in \\iota$, a finite index set) be points in a real inner product space $E$. Sequences of assignments $a_t : \\iota \\to \\{0, \\dots, k-1\\}$ and of centers $c_t : \\{0, \\dots, k-1\\} \\to E$, indexed by $t \\in \\mathbb{N}$, form a run of Lloyd's algorithm when for every $t$: for each $j < k$ the center $c_{t+1}(j)$ is the centroid of the cluster $\\{i : a_t(i) = j\\}$ (the zero vector if this cluster is\u2026","title":"Run of Lloyd's algorithm"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/antitone","kind":"node","lean":"Cslib.MachineLearning.KMeans.IsRun.antitone","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#antitone","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"The cost never increases along a run of Lloyd's algorithm.","title":"The cost never increases along a run of Lloyd's algorithm"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/exists-run","kind":"node","lean":"Cslib.MachineLearning.KMeans.exists_run","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#exists-run","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $x_i$ ($i \\in \\iota$, a finite type) be points in a real inner product space and let $k \\ge 1$. Every starting assignment $a_0$ of the points to clusters $0, \\dots, k-1$ and every choice of starting centers $c_0$ extend to a run of Lloyd's algorithm: there are assignments $a_t$ and centers $c_t$ for $t \\in \\mathbb{N}$, equal to $a_0$ and $c_0$ at $t = 0$, that form a run.","title":"Runs of Lloyd's algorithm exist"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/soft-assign","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.softAssign","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#soft-assign","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Given weights $w_1, \\dots, w_K \\in \\mathbb{R}$, means $\\mu_1, \\dots, \\mu_K \\in \\mathbb{R}^d$ and a real $v$, the common-variance responsibility of component $k$ at a point $x \\in \\mathbb{R}^d$ is","title":"Responsibilities with a common variance"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/resp-eq-soft-assign","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.resp_eq_softAssign","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#resp-eq-soft-assign","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $\\theta$ be valid parameters of an isotropic Gaussian mixture with $K$ components on $\\mathbb{R}^d$, with weights $w_k$, means $\\mu_k$ and variances all equal to a real number $v$. Then for every point $x \\in \\mathbb{R}^d$ and every component $k$, the responsibility of $k$ at $x$ under $\\theta$ equals the common-variance responsibility $r_v(x, k)$ for the weights $w$ and means $\\mu$.","title":"The Gaussian-mixture responsibilities are the common-variance responsibilities"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/iso-gauss-pdf-div","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.isoGaussPDF_div","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#iso-gauss-pdf-div","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"For all points $\\mu, \\nu, x \\in \\mathbb{R}^d$ and every real $v > 0$, writing $\\mathcal{N}(x \\mid \\mu, vI)$ for the isotropic Gaussian density with mean $\\mu$ and covariance $vI$,","title":"Ratio of isotropic Gaussian densities with a common variance"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/tendsto-exp-div-nhds-gt-zero","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.tendsto_exp_div_nhdsGT_zero","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#tendsto-exp-div-nhds-gt-zero","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"For every real $c < 0$, $\\exp(c / (2v)) \\to 0$ as $v \\to 0^+$.","title":"Exponential decay as the variance tends to zero"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/soft-assign-eq","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.softAssign_eq","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#soft-assign-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"For weights $w_1, \\dots, w_K \\in \\mathbb{R}$, means $\\mu_1, \\dots, \\mu_K \\in \\mathbb{R}^d$, a real $v > 0$, a point $x \\in \\mathbb{R}^d$ and components $j$ and $k$, the common-variance responsibility $r_v(x, k)$ of component $k$ at $x$ for these weights and means satisfies","title":"Common-variance responsibilities relative to a fixed component"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/tendsto-soft-assign","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.tendsto_softAssign","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#tendsto-soft-assign","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $w_1, \\dots, w_K$ be positive weights, $\\mu_1, \\dots, \\mu_K \\in \\mathbb{R}^d$ means and $x \\in \\mathbb{R}^d$ a point, and suppose that $\\mu_j$ is the unique nearest mean to $x$: $\\|x - \\mu_j\\| < \\|x - \\mu_k\\|$ for every $k \\ne j$. Then for every component $k$, the common-variance responsibility $r_v(x, k)$ of $k$ at $x$ for these weights and means tends to $1$ if $k = j$ and to $0$ otherwise as $v \\to 0^+$.","title":"Responsibilities in the zero-variance limit"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/nearest-and-tendsto-soft-assign","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.nearest_and_tendsto_softAssign","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#nearest-and-tendsto-soft-assign","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be points of $\\mathbb{R}^d$, let $\\mu_1, \\dots, \\mu_K \\in \\mathbb{R}^d$ be means with positive weights $w_1, \\dots, w_K$, and let $a$ assign to each $n$ a component $a(n)$ such that $\\|x_n - \\mu_{a(n)}\\| < \\|x_n - \\mu_k\\|$ for every $k \\ne a(n)$. Then $a$ is a nearest-center assignment, that is, $\\|x_n - \\mu_{a(n)}\\| \\le \\|x_n - \\mu_k\\|$ for all $n$ and $k$, and for all $n$ and $k$ the\u2026","title":"The k-means assignment step is the zero-variance limit of the E-step"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/soft-mean","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.softMean","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#soft-mean","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Given weights $w_1, \\dots, w_K \\in \\mathbb{R}$, means $\\mu_1, \\dots, \\mu_K \\in \\mathbb{R}^d$, a real $v$ and data $x_1, \\dots, x_N \\in \\mathbb{R}^d$, the common-variance M-step mean of component $k$ is","title":"Common-variance M-step mean"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/new-mean-eq-soft-mean","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.newMean_eq_softMean","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#new-mean-eq-soft-mean","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $\\theta$ be valid parameters of an isotropic Gaussian mixture with $K$ components on $\\mathbb{R}^d$, with weights $w_k$, means $\\mu_k$ and variances all equal to a real number $v$. Then for data $x_1, \\dots, x_N \\in \\mathbb{R}^d$ and every component $k$, the M-step mean of component $k$ under $\\theta$ equals the common-variance M-step mean of component $k$ for the weights $w$, means $\\mu$, variance $v$ and data\u2026","title":"The Gaussian-mixture M-step mean is the common-variance M-step mean"},{"id":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/tendsto-soft-mean","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.HardEM.tendsto_softMean","parent":41120,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/k-means-clustering/index.html#tendsto-soft-mean","scope":"ComputerScience/machine-learning/unsupervised-learning/k-means-clustering","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data, let $\\mu_1, \\dots, \\mu_K \\in \\mathbb{R}^d$ be means with positive weights $w_1, \\dots, w_K$, and let $a$ assign to each $n$ a component $a(n)$ with $\\|x_n - \\mu_{a(n)}\\| < \\|x_n - \\mu_k\\|$ for every $k \\ne a(n)$. Then for every component $k$ such that $a(n) = k$ for some $n$, the common-variance M-step mean of component $k$ for these weights, means and data tends, as\u2026","title":"The M-step in the zero-variance limit"},{"data_url":"scopes/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization.json","id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","kind":"scope","navigation":"module","parent":41111,"result":false,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","title":"Non-negative matrix factorization"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/objective","kind":"node","lean":"Cslib.MachineLearning.NMF.objective","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#objective","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The NMF objective: the squared Frobenius distance `\u2016X - W H\u2016_F\u00b2 = \u2211 i j, (X i j - (W H) i j)\u00b2` between the data `X` and the product `W * H`.","title":"Objective of nonnegative matrix factorization"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/objective-nonneg","kind":"node","lean":"Cslib.MachineLearning.NMF.objective_nonneg","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#objective-nonneg","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The objective is nonnegative.","title":"Nonnegativity of the NMF objective"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/objective-transpose","kind":"node","lean":"Cslib.MachineLearning.NMF.objective_transpose","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#objective-transpose","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"Transposing exchanges the roles of `W` and `H`: `\u2016X\u1d40 - H\u1d40 W\u1d40\u2016\u00b2 = \u2016X - W H\u2016\u00b2`.","title":"The NMF objective is invariant under transposition"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/exists-minimizer","kind":"node","lean":"Cslib.MachineLearning.NMF.exists_minimizer","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#exists-minimizer","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"For every real matrix `X` and every finite inner index type `k`, the objective attains its infimum over pairs `W, H \u2265 0` (entrywise).","title":"Existence of a minimizer"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/update-h","kind":"node","lean":"Cslib.MachineLearning.NMF.updateH","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#update-h","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The Lee\u2013Seung multiplicative update of `H` (Lee and Seung 2001, Theorem 1): `H a j \u2190 H a j * (W\u1d40 X) a j / (W\u1d40 W H) a j`. An entry with a zero denominator becomes `0` (Lean's `x / 0 = 0`).","title":"Lee\u2013Seung multiplicative update of H"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/update-w","kind":"node","lean":"Cslib.MachineLearning.NMF.updateW","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#update-w","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The Lee\u2013Seung multiplicative update of `W` (Lee and Seung 2001, Theorem 1): `W i a \u2190 W i a * (X H\u1d40) i a / (W H H\u1d40) i a`. An entry with a zero denominator becomes `0`.","title":"Lee\u2013Seung multiplicative update of W"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/sweep","kind":"node","lean":"Cslib.MachineLearning.NMF.sweep","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#sweep","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"One sweep of the Lee\u2013Seung algorithm: update `H`, then update `W` using the new `H`.","title":"One sweep of the Lee\u2013Seung algorithm"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/update-w-eq-transpose","kind":"node","lean":"Cslib.MachineLearning.NMF.updateW_eq_transpose","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#update-w-eq-transpose","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The `W` update is the `H` update of the transposed problem.","title":"The W update is the H update of the transposed problem"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/update-h-nonneg","kind":"node","lean":"Cslib.MachineLearning.NMF.updateH_nonneg","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#update-h-nonneg","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The `H` update keeps `H` nonnegative when `X`, `W`, `H` are nonnegative.","title":"The Lee\u2013Seung H update preserves nonnegativity"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/update-w-nonneg","kind":"node","lean":"Cslib.MachineLearning.NMF.updateW_nonneg","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#update-w-nonneg","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The `W` update keeps `W` nonnegative when `X`, `W`, `H` are nonnegative.","title":"The Lee\u2013Seung W update preserves nonnegativity"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/objective-update-h-le","kind":"node","lean":"Cslib.MachineLearning.NMF.objective_updateH_le","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#objective-update-h-le","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"Lee\u2013Seung, `H` step (Lee and Seung 2001, Theorem 1): for `W, H \u2265 0`, the multiplicative update of `H` does not increase `\u2016X - W H\u2016_F\u00b2`. The proof is the auxiliary-function argument: the quadratic term of the exact expansion is bounded by Lee and Seung's diagonal majorant (here through a weighted Cauchy\u2013Schwarz inequality), and the update minimizes the majorant. `X` may have any sign.","title":"Lee\u2013Seung, `H` step"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/objective-update-w-le","kind":"node","lean":"Cslib.MachineLearning.NMF.objective_updateW_le","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#objective-update-w-le","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"For `W, H \u2265 0`, the multiplicative update of `W` does not increase `\u2016X - W H\u2016_F\u00b2`.","title":"Lee\u2013Seung, `W` step"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/sweep-nonneg","kind":"node","lean":"Cslib.MachineLearning.NMF.sweep_nonneg","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#sweep-nonneg","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"One sweep keeps both factors nonnegative when `X \u2265 0`.","title":"A Lee\u2013Seung sweep preserves nonnegativity"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/objective-sweep-le","kind":"node","lean":"Cslib.MachineLearning.NMF.objective_sweep_le","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#objective-sweep-le","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"One Lee\u2013Seung sweep does not increase the objective, for `X, W, H \u2265 0`.","title":"One Lee\u2013Seung sweep does not increase the objective"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/objective-sweep-iterate-antitone","kind":"node","lean":"Cslib.MachineLearning.NMF.objective_sweep_iterate_antitone","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#objective-sweep-iterate-antitone","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The objective is non-increasing along the Lee\u2013Seung iterates started from `W, H \u2265 0`, for `X \u2265 0`.","title":"The objective is non-increasing along the Lee\u2013Seung iterates"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/grad-h","kind":"node","lean":"Cslib.MachineLearning.NMF.gradH","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#grad-h","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The gradient of the objective with respect to `H`: `2 (W\u1d40 W H - W\u1d40 X)`.","title":"Gradient of the NMF objective with respect to H"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/grad-w","kind":"node","lean":"Cslib.MachineLearning.NMF.gradW","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#grad-w","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The gradient of the objective with respect to `W`: `2 (W H H\u1d40 - X H\u1d40)`.","title":"Gradient of the NMF objective with respect to W"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/has-deriv-at-objective-h","kind":"node","lean":"Cslib.MachineLearning.NMF.hasDerivAt_objective_H","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#has-deriv-at-objective-h","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"`gradH X W H a j` is the partial derivative of the objective in the entry `H a j`.","title":"Partial derivative of the NMF objective in an entry of H"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/has-deriv-at-objective-w","kind":"node","lean":"Cslib.MachineLearning.NMF.hasDerivAt_objective_W","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#has-deriv-at-objective-w","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"`gradW X W H i a` is the partial derivative of the objective in the entry `W i a`.","title":"Partial derivative of the NMF objective in an entry of W"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/mul-grad-h-eq-zero-of-update-h-eq","kind":"node","lean":"Cslib.MachineLearning.NMF.mul_gradH_eq_zero_of_updateH_eq","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#mul-grad-h-eq-zero-of-update-h-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"If `updateH X W H = H`, then `H a j * gradH X W H a j = 0` for every entry.","title":"Complementary slackness at a fixed point of the `H` update"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/mul-grad-w-eq-zero-of-update-w-eq","kind":"node","lean":"Cslib.MachineLearning.NMF.mul_gradW_eq_zero_of_updateW_eq","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#mul-grad-w-eq-zero-of-update-w-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"Complementary slackness at a fixed point of the `W` update.","title":"Complementary slackness at a fixed point of the `W` update"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/is-kktpoint","kind":"node","lean":"Cslib.MachineLearning.NMF.IsKKTPoint","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#is-kktpoint","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The Karush\u2013Kuhn\u2013Tucker conditions of `min \u2016X - W H\u2016_F\u00b2` subject to `W, H \u2265 0`: primal feasibility, dual feasibility (nonnegative gradients) and complementary slackness.","title":"Karush\u2013Kuhn\u2013Tucker point of nonnegative matrix factorization"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/is-kktpoint-of-sweep-eq","kind":"node","lean":"Cslib.MachineLearning.NMF.isKKTPoint_of_sweep_eq","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#is-kktpoint-of-sweep-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"A fixed point of the Lee\u2013Seung sweep with all entries of `W` and `H` positive has both gradients zero, so it satisfies the KKT conditions.","title":"Positive fixed points are KKT points"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/is-separable","kind":"node","lean":"Cslib.MachineLearning.NMF.IsSeparable","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#is-separable","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"`H` is separable: every factor `a` has a pure column, a column `j` of `H` equal to a positive multiple of the unit vector `e_a`. For `X = W * H` this makes column `j` of `X` a positive multiple of column `a` of `W` (a sample containing only factor `a`). Applied to `W\u1d40` it says that every factor has a pure row of `W` (an \"anchor\" feature of factor `a` alone).","title":"Separable factor matrix"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/exists-perm-of-mul-eq-one","kind":"node","lean":"Cslib.MachineLearning.NMF.exists_perm_of_mul_eq_one","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#exists-perm-of-mul-eq-one","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"If `G * K = 1` with `G` and `K` entrywise nonnegative, then there are a permutation `\u03c3` and positive weights `d` with `G a b = d b` if `a = \u03c3 b` and `0` otherwise.","title":"A nonnegative matrix with a nonnegative inverse is monomial"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/eq-of-is-separable","kind":"node","lean":"Cslib.MachineLearning.NMF.eq_of_isSeparable","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#eq-of-is-separable","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"Suppose `X = W * H`, where `W` has a pure row for every factor (`IsSeparable W\u1d40`: for each `a` a row `i` with `W i \u00b7 = e \u2022 e_a`, `e > 0`) and `H` has a pure column for every factor (`IsSeparable H`). Then every factorization `X = W' * H'` with `W', H' \u2265 0` and the same inner dimension agrees with `(W, H)` up to a permutation `\u03c3` and positive scalings `d` of the factors: `W' \u00b7 a = d a \u2022 W \u00b7 (\u03c3 a)` and\u2026","title":"Essential uniqueness of NMF when both factors are separable"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/nonneg-rank","kind":"node","lean":"Cslib.MachineLearning.NMF.nonnegRank","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#nonneg-rank","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The nonnegative rank of `X`: the least `r` such that `X = W * H` with `W : m \u00d7 r` and `H : r \u00d7 n` entrywise nonnegative (Cohen and Rothblum 1993). If `X` has a negative entry no such factorization exists and the value is the junk value `sInf \u2205 = 0`.","title":"Nonnegative rank of a matrix"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/nonneg-rank-le-of-mul-eq","kind":"node","lean":"Cslib.MachineLearning.NMF.nonnegRank_le_of_mul_eq","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#nonneg-rank-le-of-mul-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"A nonnegative factorization with inner dimension `r` bounds the nonnegative rank by `r`.","title":"A nonnegative factorization bounds the nonnegative rank"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/exists-mul-eq-nonneg-rank","kind":"node","lean":"Cslib.MachineLearning.NMF.exists_mul_eq_nonnegRank","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#exists-mul-eq-nonneg-rank","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"A nonnegative matrix has a nonnegative factorization of inner dimension `nonnegRank X`.","title":"Existence of a factorization of nonnegative-rank inner dimension"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/nonneg-rank-le-card-width","kind":"node","lean":"Cslib.MachineLearning.NMF.nonnegRank_le_card_width","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#nonneg-rank-le-card-width","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The nonnegative rank of a nonnegative matrix is at most its number of columns.","title":"Nonnegative rank is at most the number of columns"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/rank-le-nonneg-rank","kind":"node","lean":"Cslib.MachineLearning.NMF.rank_le_nonnegRank","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#rank-le-nonneg-rank","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"The nonnegative rank of a nonnegative matrix is at least its rank.","title":"The nonnegative rank of a nonnegative matrix is at least its rank"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/sum-mul-apply","kind":"node","lean":"Cslib.MachineLearning.NMF.sum_mul_apply","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#sum-mul-apply","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"Column sums of a product: `\u2211 i, (W H) i j = \u2211 a, (\u2211 i, W i a) H a j`.","title":"Column sums of a matrix product"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/sum-mul-apply-of-sum-eq-one","kind":"node","lean":"Cslib.MachineLearning.NMF.sum_mul_apply_of_sum_eq_one","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#sum-mul-apply-of-sum-eq-one","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"Mutational signatures (Alexandrov et al. 2013). Write the mutation-count matrix (mutation types \u00d7 samples) as `W * H`, signatures (columns of `W`) times exposures `H`. If every signature is a probability vector over mutation types, the total mutation count of sample `j` equals the sum of its exposures.","title":"Mutational signatures"},{"id":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/mul-apply-mem-icc","kind":"node","lean":"Cslib.MachineLearning.NMF.mul_apply_mem_Icc","parent":41144,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization/index.html#mul-apply-mem-icc","scope":"ComputerScience/machine-learning/unsupervised-learning/non-negative-matrix-factorization","status":"fully_proved","summary":"Write the bulk matrix (genes \u00d7 samples) as `C * P`, the cell-type profiles `C` times the cell-type proportions `P`. If the proportions of sample `j` are nonnegative and sum to `1` and every cell type expresses gene `g` at a level in `[lo, hi]`, then the bulk level of `g` in sample `j` lies in `[lo, hi]`.","title":"Gene-expression deconvolution"},{"data_url":"scopes/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis.json","id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","kind":"scope","navigation":"module","parent":41111,"result":false,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","title":"Probabilistic principal component analysis"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/inner-to-euclidean-lin","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.inner_toEuclideanLin","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#inner-to-euclidean-lin","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"For every real $d \\times q$ matrix $W$ and all $z \\in \\mathbb{R}^q$ and $t \\in \\mathbb{R}^d$, the Euclidean inner products satisfy $\\langle Wz, t\\rangle = \\langle z, W^\\top t\\rangle$.","title":"The transpose of a real matrix is its adjoint"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/char-fun-map-to-euclidean-lin","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.charFun_map_toEuclideanLin","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#char-fun-map-to-euclidean-lin","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\mathbb{R}^q$ and $W$ a real $d \\times q$ matrix, and write $\\varphi_\\nu(t) = \\int e^{i\\langle x, t\\rangle}\\,d\\nu(x)$ for the characteristic function of a measure $\\nu$. Then the pushforward $W_*\\mu$ of $\\mu$ under $z \\mapsto Wz$ satisfies $\\varphi_{W_*\\mu}(t) = \\varphi_\\mu(W^\\top t)$ for every $t \\in \\mathbb{R}^d$.","title":"Characteristic function of a linear image"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/map-eq-multivariate-gaussian","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.map_eq_multivariateGaussian","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#map-eq-multivariate-gaussian","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $Z$ and $E$ be independent random vectors on a probability space, with values in $\\mathbb{R}^q$ and $\\mathbb{R}^d$, such that $Z$ has the standard Gaussian law $N(0, I_q)$ and $E$ has law $N(0, \\Psi)$ for a positive semidefinite real $d \\times d$ matrix $\\Psi$. Then for every real $d \\times q$ matrix $W$ and every $\\mu \\in \\mathbb{R}^d$, the random vector $WZ + \\mu + E$ has law $N(\\mu, WW^\\top + \\Psi)$.","title":"The marginal of probabilistic PCA and factor analysis is Gaussian"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/ppca-map-eq","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.ppca_map_eq","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#ppca-map-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $Z$ and $E$ be independent random vectors on a probability space, with values in $\\mathbb{R}^q$ and $\\mathbb{R}^d$, such that $Z$ has the standard Gaussian law $N(0, I_q)$ and $E$ has the isotropic law $N(0, \\sigma^2 I_d)$ for a real $\\sigma^2 \\ge 0$. Then for every real $d \\times q$ matrix $W$ and every $\\mu \\in \\mathbb{R}^d$, the random vector $WZ + \\mu + E$ has law $N(\\mu, WW^\\top + \\sigma^2 I_d)$.","title":"The probabilistic PCA marginal"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/factor-analysis-map-eq","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.factorAnalysis_map_eq","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#factor-analysis-map-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $Z$ and $E$ be independent random vectors on a probability space, with values in $\\mathbb{R}^q$ and $\\mathbb{R}^d$, such that $Z$ has the standard Gaussian law $N(0, I_q)$ and $E$ has law $N(0, \\operatorname{diag}\\psi)$ for a vector $\\psi \\in \\mathbb{R}^d$ with $\\psi_i \\ge 0$ for all $i$. Then for every real $d \\times q$ matrix $W$ and every $\\mu \\in \\mathbb{R}^d$, the random vector $WZ + \\mu + E$ has law\u2026","title":"The factor-analysis marginal"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/ppca-cov","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.ppcaCov","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#ppca-cov","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"For a real $d \\times q$ matrix $W$ and a real number $\\sigma^2$, the probabilistic PCA covariance is the $d \\times d$ matrix $W W^\\top + \\sigma^2 I$.","title":"Probabilistic PCA covariance"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/ppca-map-eq-with-density","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.ppca_map_eq_withDensity","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#ppca-map-eq-with-density","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $z$ and $\\varepsilon$ be independent random vectors on a probability space, with $z$ standard Gaussian in $\\mathbb{R}^q$ and $\\varepsilon$ Gaussian in $\\mathbb{R}^d$ with mean $0$ and covariance $\\sigma^2 I$, where $\\sigma^2 > 0$. Then for every real $d \\times q$ matrix $W$ and every $\\mu \\in \\mathbb{R}^d$, the law of $x = W z + \\mu + \\varepsilon$ has density $\\mathcal{N}(x \\mid \\mu, W W^\\top + \\sigma^2 I)$\u2026","title":"The density of the probabilistic PCA observation"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/log-lik","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.logLik","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#log-lik","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"For data $x_1, \\dots, x_N \\in \\mathbb{R}^d$, a mean $\\mu \\in \\mathbb{R}^d$, a real $d \\times q$ loading matrix $W$ and a real number $\\sigma^2$, the probabilistic PCA log-likelihood of the data is","title":"Probabilistic PCA log-likelihood"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/sample-mean","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.sampleMean","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#sample-mean","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"For data points $x_1, \\dots, x_N \\in \\mathbb{R}^d$, the sample mean is $\\bar x = \\frac{1}{N} \\sum_{n=1}^{N} x_n \\in \\mathbb{R}^d$.","title":"Sample mean"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/sample-cov","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.sampleCov","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#sample-cov","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"For data $x_1, \\dots, x_N \\in \\mathbb{R}^d$ with sample mean $\\bar{x}$, the sample covariance is the $d \\times d$ matrix $S = \\frac{1}{N} \\sum_{n=1}^N (x_n - \\bar{x})(x_n - \\bar{x})^\\top$, taken to be $0$ when $N = 0$.","title":"Sample covariance"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/log-lik-of-n-eq-zero","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.logLik_of_N_eq_zero","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#log-lik-of-n-eq-zero","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"For the empty data set ($N = 0$), the probabilistic PCA log-likelihood at every $\\mu \\in \\mathbb{R}^d$, every real $d \\times q$ matrix $W$ and every real number $\\sigma^2$ is $0$.","title":"The log-likelihood of no data is zero"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/log-lik-le-sample-mean","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.logLik_le_sampleMean","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#log-lik-le-sample-mean","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data with sample mean $\\bar{x}$. For every $\\mu \\in \\mathbb{R}^d$, every real $d \\times q$ loading matrix $W$ and every real $\\sigma^2 > 0$, the probabilistic PCA log-likelihood of the data at $(\\mu, W, \\sigma^2)$ is at most its value at $(\\bar{x}, W, \\sigma^2)$.","title":"The mean MLE is the sample mean"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/log-lik-le-gaussian-mle","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.logLik_le_gaussianMLE","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#log-lik-le-gaussian-mle","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data whose sample covariance $S$ is positive definite, and let $\\bar{x}$ be their sample mean. Then for every $\\mu \\in \\mathbb{R}^d$, every real $d \\times q$ matrix $W$ and every real $\\sigma^2 > 0$, the probabilistic PCA log-likelihood of the data at $(\\mu, W, \\sigma^2)$ is at most $\\sum_{n=1}^N \\log \\mathcal{N}(x_n \\mid \\bar{x}, S)$, where $\\mathcal{N}(x \\mid \\bar{x}, S)$\u2026","title":"The probabilistic PCA likelihood is at most the unconstrained Gaussian maximum"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/is-mle","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.IsMLE","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#is-mle","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Given data $x_1, \\dots, x_N \\in \\mathbb{R}^d$, a triple $(\\mu_0, W_0, \\sigma_0^2)$ of a mean $\\mu_0 \\in \\mathbb{R}^d$, a real $d \\times q$ loading matrix $W_0$ and a real number $\\sigma_0^2$ is a probabilistic PCA maximum-likelihood estimate when $\\sigma_0^2 > 0$ and, for every $\\mu \\in \\mathbb{R}^d$, every real $d \\times q$ matrix $W$ and every real $\\sigma^2 > 0$, the probabilistic PCA log-likelihood of the data\u2026","title":"Probabilistic PCA maximum-likelihood estimate"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/is-mle-of-sample-cov-eq","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.isMLE_of_sampleCov_eq","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#is-mle-of-sample-cov-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data with sample mean $\\bar{x}$ and sample covariance $S$. If $W_0$ is a real $d \\times q$ matrix and $\\sigma_0^2 > 0$ is a real number with $S = W_0 W_0^\\top + \\sigma_0^2 I$, then $(\\bar{x}, W_0, \\sigma_0^2)$ is a probabilistic PCA maximum-likelihood estimate for the data.","title":"Probabilistic PCA MLE when the sample covariance has the model form"},{"id":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/log-lik-zero-le","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ProbabilisticPCA.logLik_zero_le","parent":41179,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis/index.html#log-lik-zero-le","scope":"ComputerScience/machine-learning/unsupervised-learning/probabilistic-principal-component-analysis","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data with sample mean $\\bar{x}$ and sample covariance $S$, and suppose $\\operatorname{tr} S > 0$. Then for every $\\mu \\in \\mathbb{R}^d$ and every real $\\sigma^2 > 0$, the probabilistic PCA log-likelihood of the data with the zero $d \\times q$ loading matrix at $(\\mu, 0, \\sigma^2)$ is at most its value at $(\\bar{x}, 0, \\operatorname{tr} S / d)$.","title":"No latent factors: the noise-variance MLE is the average sample variance"},{"data_url":"scopes/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm.json","id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","kind":"scope","navigation":"module","parent":41111,"result":false,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","title":"Successive projection algorithm"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/separable","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.Separable","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#separable","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"A separable data matrix: columns `m j` are sub-convex combinations of `r` linearly independent archetypes `w k`, and every archetype is a column (Gillis\u2013Vavasis Assumption 1).","title":"Separable data matrix"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/residual","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.residual","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#residual","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"The residual of column `j`: its projection onto the orthogonal complement of the span of the columns indexed by `S`.","title":"Residual of a column in the successive projection algorithm"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/is-sparun","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.IsSPARun","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#is-sparun","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"`S` (most recent pick first) is a run of the successive projection algorithm: each pick maximizes the residual norm given the earlier picks (ties broken arbitrarily).","title":"Run of the successive projection algorithm"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/norm-sum-smul-le","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.norm_sum_smul_le","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#norm-sum-smul-le","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"Gillis\u2013Vavasis Lemma 1 (`f = \u2016\u00b7\u2016\u00b2`), bound: a sub-convex combination of vectors of norm at most `M` has norm at most `M`.","title":"Norm bound for a sub-convex combination of vectors"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/eq-single-of-norm-sum-smul-eq","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.eq_single_of_norm_sum_smul_eq","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#eq-single-of-norm-sum-smul-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"Gillis\u2013Vavasis Lemma 1 (`f = \u2016\u00b7\u2016\u00b2`), equality case: for linearly independent vectors of norm at most `M > 0`, a sub-convex combination of norm `M` is a vertex `w k` with `\u2016w k\u2016 = M`.","title":"Equality case of the sub-convex combination norm bound"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/map-nodup","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.IsSPARun.map_nodup","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#map-nodup","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"SPA recovers archetypes (Gillis\u2013Vavasis Theorem 1, noiseless case): in the first `r` picks of any SPA run on a separable matrix with `r` archetypes, the picked columns are pairwise distinct.","title":"SPA recovers archetypes"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/mem-range","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.IsSPARun.mem_range","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#mem-range","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"SPA recovers archetypes (Gillis\u2013Vavasis Theorem 1, noiseless case): each of the first `r` picks of any SPA run is an archetype.","title":"SPA recovers archetypes"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/range-eq","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.IsSPARun.range_eq","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#range-eq","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"After exactly `r` picks, SPA has found every archetype.","title":"Successive projection finds every archetype after r picks"},{"id":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/exists-is-sparun","kind":"node","lean":"Cslib.MachineLearning.MatrixFactorization.SuccessiveProjection.exists_isSPARun","parent":41196,"result":true,"route":"../roadmap/ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm/index.html#exists-is-sparun","scope":"ComputerScience/machine-learning/unsupervised-learning/successive-projection-algorithm","status":"fully_proved","summary":"SPA runs of every length exist (on a nonempty set of columns).","title":"Existence of successive projection runs of every length"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models","kind":"scope","parent":40911,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models","status":"fully_proved","summary":"Bayesian networks and their Markov properties, structure scores and Markov equivalence, causal networks and adjustment, belief propagation and variable elimination, conditional random fields, hidden Markov models (forward-backward, Viterbi, Baum-Welch) and the Kalman filter.","title":"Probabilistic graphical models"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","kind":"scope","navigation":"module","parent":41206,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","title":"Hidden Markov models"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/hmm","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.HMM","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#hmm","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"A hidden Markov model with a set of states $S$, a set of observations $O$ and weights in $R$ consists of an initial weight $\\pi(s) \\in R$ for each state $s$, a transition weight $a(s, s') \\in R$ for each pair of states, the weight of a transition from $s$ to $s'$, and an emission weight $e(s, o) \\in R$ for each state $s$ and observation $o$, the weight of emitting $o$ from $s$.","title":"Hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/path-prob","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.pathProb","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#path-prob","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model $h$ with initial weights $\\pi$, transition weights $a$ and emission weights $b$ in a semiring, observations $x_0, x_1, \\dots$ and $t \\in \\mathbb{N}$, the joint weight of a state path $z = (z_0, \\dots, z_t)$ and $x_0, \\dots, x_t$ is the product, taken from left to right, $\\pi(z_0)\\, b(z_0, x_0)\\, a(z_0, z_1)\\, b(z_1, x_1) \\cdots a(z_{t-1}, z_t)\\, b(z_t, x_t)$.","title":"Joint weight of a state path and the observations"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/likelihood","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.likelihood","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#likelihood","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model $h$ with a finite set of states $S$ and weights in a semiring, observations $x_0, x_1, \\dots$ and $t \\in \\mathbb{N}$, the likelihood of $x_0, \\dots, x_t$ is the sum, over all state paths $z = (z_0, \\dots, z_t) \\in S^{t+1}$, of the joint weight of $z$ and $x_0, \\dots, x_t$.","title":"Likelihood of the observations"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/forward","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.forward","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#forward","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with a finite set of states $S$ and weights in a semiring $R$, with initial weights $\\pi(s)$, transition weights $a(s', s)$ from $s'$ to $s$ and emission weights $e(s, o)$ of the observation $o$ from the state $s$, and let $x$ be an observation sequence. The forward variables are defined by $\\alpha_0(s) = \\pi(s) \\, e(s, x_0)$ and\u2026","title":"Forward variables of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/forward-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.forward_eq","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#forward-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with a finite set of states $S$ and weights in a semiring $R$, and let $x$ be an observation sequence. For every time $t$ and every state $s$, the forward variable $\\alpha_t(s)$ equals $\\sum_z p_t(z)$, the sum running over the state paths $z = (z_0, \\dots, z_t) \\in S^{t+1}$ with $z_t = s$, where $p_t(z)$ is the joint weight of $z$ and the observations $x_0, \\dots, x_t$.","title":"The forward variables are path sums"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/likelihood-eq-sum-forward","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.likelihood_eq_sum_forward","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#likelihood-eq-sum-forward","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with a finite set of states $S$, initial weights $\\pi$, transition weights $a$ and emission weights $b$ in a semiring, and let $x_0, x_1, \\dots$ be observations. Let $\\alpha$ be the forward variables, $\\alpha_0(s) = \\pi(s)\\, b(s, x_0)$ and $\\alpha_{t+1}(s) = \\bigl(\\sum_{s' \\in S} \\alpha_t(s')\\, a(s', s)\\bigr)\\, b(s, x_{t+1})$. Then for every $t \\in \\mathbb{N}$ the likelihood of\u2026","title":"The forward algorithm computes the likelihood"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/viterbi","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.viterbi","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#viterbi","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with a finite nonempty set of states $S$, initial weights $\\pi(s)$, transition weights $a(s', s)$ and emission weights $b(s, o)$ in a linearly ordered commutative semiring $R$, and let $x_0, x_1, \\dots$ be a sequence of observations. The Viterbi variables $V_t(s)$, for $t \\in \\mathbb{N}$ and $s \\in S$, are defined by $V_0(s) = \\pi(s) \\, b(s, x_0)$ and","title":"The Viterbi variables of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/nonneg","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.HMM.Nonneg","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#nonneg","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"A hidden Markov model $h$ with weights in a commutative semiring with a linear order has nonnegative weights when all of its initial weights, transition weights and emission weights are $\\ge 0$: for all states $s, s'$ and every observation $o$, the initial weight of $s$, the weight of the transition from $s$ to $s'$, and the weight of emitting $o$ from $s$ are nonnegative.","title":"Hidden Markov models with nonnegative weights"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/path-prob-le-viterbi","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.pathProb_le_viterbi","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#path-prob-le-viterbi","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with a finite nonempty set of states $S$ and initial weights $\\pi$, transition weights $a$ and emission weights $b$ that are nonnegative elements of a linearly ordered commutative semiring, and let $x_0, x_1, \\dots$ be observations. Let $\\delta$ be the Viterbi variables, $\\delta_0(s) = \\pi(s)\\, b(s, x_0)$ and\u2026","title":"Path weights of a hidden Markov model are bounded by the Viterbi variables"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/exists-path-prob-eq-viterbi","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.exists_pathProb_eq_viterbi","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#exists-path-prob-eq-viterbi","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with a finite nonempty set of states $S$ and weights in a commutative semiring $R$ with a linear order, and let $x$ be an observation sequence. For every time $t$ and every state $s$, there is a state path $z = (z_0, \\dots, z_t)$ with $z_t = s$ whose joint weight with the observations $x_0, \\dots, x_t$ equals the Viterbi variable $V_t(s)$.","title":"The Viterbi variables are attained"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/sup-viterbi-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.sup'_viterbi_eq","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#sup-viterbi-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with a finite nonempty set of states $S$ and initial weights $\\pi$, transition weights $a$ and emission weights $b$ that are nonnegative elements of a linearly ordered commutative semiring, and let $x_0, x_1, \\dots$ be observations. Let $\\delta$ be the Viterbi variables, $\\delta_0(s) = \\pi(s)\\, b(s, x_0)$ and\u2026","title":"Viterbi computes the MAP value"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/sum-vec","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.sumVec","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#sum-vec","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a vector $v = (v_0, \\dots, v_{n-1})$ of length $n$ with entries in an additive commutative monoid, its entry sum is $\\sum_{s < n} v_s$.","title":"Sum of the entries of a vector"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/forward-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.forwardStep","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#forward-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $\\{0, \\dots, n-1\\}$ and weights in a semiring $R$, with transition weight $a(s', s)$ from $s'$ to $s$ and emission weight $e(s, o)$ of the observation $o$ from the state $s$, and let $x$ be an observation sequence. For a table $v \\in R^n$ and a step $t$, the forward step returns the table whose entry at a state $s$ is","title":"One step of the tabulated forward algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/forward-tab","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.forwardTab","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#forward-tab","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $\\{0, \\dots, n-1\\}$ and weights in a semiring $R$, with initial weight $\\pi(s)$ and emission weight $e(s, o)$ of the observation $o$ from the state $s$, and let $x$ be an observation sequence. The forward table is the sequence of vectors in $R^n$ defined by recursion on time: at time $0$ its entry at $s$ is $\\pi(s) \\, e(s, x_0)$, and the table at time $t + 1$ is the\u2026","title":"Tabulated forward algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/likelihood-tab","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.likelihoodTab","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#likelihood-tab","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model $h$ with states $0, \\dots, n - 1$ and weights in a semiring, observations $x_0, x_1, \\dots$ and $t \\in \\mathbb{N}$, the tabulated likelihood at time $t$ is the sum $\\sum_{s < n} v_s$ of the entries of the forward table $v = (v_0, \\dots, v_{n-1})$ at time $t$.","title":"Tabulated likelihood of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/argmax-fin","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.argmaxFin","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#argmax-fin","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $n \\ge 1$ and let $R$ be a linearly ordered set. For a function $f$ from $\\{0, \\dots, n-1\\}$ to $R$, the maximizer of $f$ is the first $i$, in the order $0, \\dots, n-1$, at which $f$ attains its maximum, that is, $f(j) \\le f(i)$ for every $j$.","title":"First maximizer of a function on a finite range"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/viterbi-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.viterbiStep","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#viterbi-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $\\{0, \\dots, n - 1\\}$, where $n \\ge 1$, transition weights $a(s', s)$ and emission weights $b(s, o)$ in a linearly ordered commutative semiring $R$, and let $x_0, x_1, \\dots$ be a sequence of observations. Given a vector $v \\in R^n$ and $t \\in \\mathbb{N}$, the Viterbi step returns the vector in $R^n$ whose entry at state $s$ is","title":"One step of the tabulated Viterbi recursion for a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/viterbi-tab","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.viterbiTab","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#viterbi-tab","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $\\{0, \\dots, n - 1\\}$, where $n \\ge 1$, initial weights $\\pi(s)$, transition weights $a(s', s)$ and emission weights $b(s, o)$ in a linearly ordered commutative semiring $R$, and let $x_0, x_1, \\dots$ be a sequence of observations. The Viterbi table assigns to each $t \\in \\mathbb{N}$ a vector $T_t \\in R^n$: the vector $T_0$ has entry $\\pi(s) \\, b(s, x_0)$ at each state\u2026","title":"The tabulated Viterbi variables of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/back","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.back","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#back","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $\\{0, \\dots, n-1\\}$, where $n \\ge 1$, and weights in a commutative semiring $R$ with a linear order, and let $a(s', s)$ be its transition weight from $s'$ to $s$. For a table $v \\in R^n$ and a state $s$, the backpointer of $s$ is the first state $s'$, in the order $0, \\dots, n-1$, that maximizes $v_{s'} \\, a(s', s)$.","title":"Backpointer of the tabulated Viterbi algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/best-path","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.bestPath","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#best-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $\\{0, \\dots, n-1\\}$, where $n \\ge 1$, and weights in a commutative semiring with a linear order, and fix an observation sequence $x$. For a state $s$, the backtracked path $b_t(s) = (b_t(s)_0, \\dots, b_t(s)_t)$ of length $t + 1$ is defined by recursion on $t$: $b_0(s)$ is the one-state path $(s)$, and $b_{t+1}(s)$ is $b_t(s')$ followed by $s$, where $s'$ is the\u2026","title":"Backtracked path of the tabulated Viterbi algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/path-prob-best-path","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.pathProb_bestPath","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#path-prob-best-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $0, \\dots, n - 1$, where $n \\ge 1$, initial weights $\\pi$, transition weights $a$ and emission weights $b$ in a commutative semiring with a linear order, and let $x_0, x_1, \\dots$ be observations. Let $\\delta$ be the Viterbi variables, $\\delta_0(s) = \\pi(s)\\, b(s, x_0)$ and $\\delta_{t+1}(s) = \\bigl(\\max_{s'} \\delta_t(s')\\, a(s', s)\\bigr)\\, b(s, x_{t+1})$. For every\u2026","title":"The backtracked path of a hidden Markov model attains the Viterbi variable"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/argmax-vec","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.argmaxVec","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#argmax-vec","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $n \\ge 1$ and let $R$ be a linearly ordered set. For a table $v = (v_0, \\dots, v_{n-1}) \\in R^n$, the maximizing index of $v$ is the first index $s$, in the order $0, \\dots, n-1$, such that $v_{s'} \\le v_s$ for every $s'$.","title":"Index of a largest entry of a table"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/viterbi-path","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.viterbiPath","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#viterbi-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $\\{0, \\dots, n - 1\\}$, where $n \\ge 1$, whose weights lie in a linearly ordered commutative semiring $R$, with transition weights $a(s', s)$, and let $x$ be a sequence of observations. For $t \\in \\mathbb{N}$, let $T_0, \\dots, T_t \\in R^n$ be the tabulated Viterbi variables of $h$ on $x$. The Viterbi path of length $t + 1$ is the state path $(z_0, \\dots, z_t)$ in which\u2026","title":"Viterbi's MAP path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/path-prob-le-viterbi-path","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.pathProb_le_viterbiPath","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#path-prob-le-viterbi-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with states $0, \\dots, n - 1$, where $n \\ge 1$, and nonnegative weights in a linearly ordered commutative semiring, and let $x_0, x_1, \\dots$ be observations. For every $t \\in \\mathbb{N}$ and every state path $z = (z_0, \\dots, z_t)$, the joint weight of $z$ and $x_0, \\dots, x_t$ is at most the joint weight of the Viterbi path of length $t + 1$ and $x_0, \\dots, x_t$.","title":"The Viterbi path is a most probable path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/backward","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.backward","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#backward","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with finite state set $S$, transition weights $a$ and emission weights $b$ in a semiring $R$, and let $x_0, x_1, \\dots$ be observations. The backward variables $\\beta_k(t, s) \\in R$, for $k, t \\in \\mathbb{N}$ and $s \\in S$, are defined by $\\beta_0(t, s) = 1$ and","title":"Backward variables of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/bridge","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.bridge","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#bridge","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with finite state set $S$, transition weights $a$ and emission weights $b$ in a semiring $R$, and let $x_0, x_1, \\dots$ be observations. The bridge weights $B_k(t, s, u) \\in R$, for $k, t \\in \\mathbb{N}$ and $s, u \\in S$, are defined from the end by $B_0(t, s, u) = 1$ if $s = u$ and $B_0(t, s, u) = 0$ otherwise, and","title":"Bridge weights of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/state-marg","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.stateMarg","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#state-marg","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model with a finite set of states $S$ and weights in a semiring, observations $x_0, x_1, \\dots$, times $t, k \\in \\mathbb{N}$ and a state $s \\in S$, the joint weight of $x_0, \\dots, x_{t+k}$ and the state $s$ at time $t$ is the sum of the path weights $p_{t+k}(z)$ over all state paths $z = (z_0, \\dots, z_{t+k})$ in $S$ with $z_t = s$, where $p_{t+k}(z)$ is the joint weight of the path $z$ and the\u2026","title":"Joint weight of the observations and a state"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/bridge-succ-front","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.bridge_succ_front","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#bridge-succ-front","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with finite state set $S$, transition weights $a$ and emission weights $b$ in a semiring, let $x_0, x_1, \\dots$ be observations, and write $B_k(t, s, u)$ for the bridge weight of the paths from state $s$ at time $t$ to state $u$ at time $t + k$. Then for all $k, t \\in \\mathbb{N}$ and $s, u \\in S$,","title":"Front decomposition of the bridge weights"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/sum-bridge","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.sum_bridge","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#sum-bridge","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model with a finite set of states $S$ and weights in a semiring, observations $x_0, x_1, \\dots$, all $k, t \\in \\mathbb{N}$ and every state $s \\in S$, the sum over all states $u \\in S$ of the bridge weight from state $s$ at time $t$ to state $u$ at time $t + k$ equals the backward variable $\\beta_{k,t}(s)$, the weight of the observations $x_{t+1}, \\dots, x_{t+k}$ given the state $s$ at time $t$.","title":"Bridge weights sum to the backward variable"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/sum-path-prob-ite-ite","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.sum_pathProb_ite_ite","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#sum-path-prob-ite-ite","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model with a finite set of states $S$ and weights in a semiring, observations $x_0, x_1, \\dots$, all $k, t \\in \\mathbb{N}$ and states $s, u \\in S$, the sum of the path weights $p_{t+k}(z)$ over all state paths $z = (z_0, \\dots, z_{t+k})$ in $S$ with $z_t = s$ and $z_{t+k} = u$ equals $\\alpha_t(s)\\, b_{k,t}(s, u)$. Here $p_{t+k}(z)$ is the joint weight of the path $z$ and the observations\u2026","title":"Paths through two fixed states"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/state-marg-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.stateMarg_eq","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#state-marg-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model with a finite set of states $S$ and weights in a semiring, observations $x_0, x_1, \\dots$, times $t, k \\in \\mathbb{N}$ and a state $s \\in S$, the joint weight of the observations $x_0, \\dots, x_{t+k}$ and the state $s$ at time $t$ equals $\\alpha_t(s)\\,\\beta_{k,t}(s)$, the forward variable $\\alpha_t(s)$ (the weight of $x_0, \\dots, x_t$ with the state at time $t$ equal to $s$) times the\u2026","title":"Forward-backward identity"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/likelihood-eq-sum-forward-mul-backward","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.likelihood_eq_sum_forward_mul_backward","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#likelihood-eq-sum-forward-mul-backward","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with finite state set $S$ and weights in a semiring, and let $x_0, x_1, \\dots$ be observations. Write $\\alpha_t(s)$ for the forward variable (the weight of $x_0, \\dots, x_t$ with the state at time $t$ equal to $s$) and $\\beta_k(t, s)$ for the backward variable. Then for all $t, k \\in \\mathbb{N}$, the likelihood of $x_0, \\dots, x_{t+k}$ equals","title":"The likelihood from the forward and backward variables"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/smoothed","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.smoothed","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#smoothed","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model with a finite set of states $S$ and weights in a semifield $K$, observations $x_0, x_1, \\dots$, times $t, k \\in \\mathbb{N}$ and a state $s \\in S$, the smoothed weight of $s$ at time $t$ is","title":"Smoothed state weights"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/smoothed-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.smoothed_eq","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#smoothed-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $g$ be a hidden Markov model with finite state set $S$ and weights in a semifield, and let $x_0, x_1, \\dots$ be observations. For all $t, k \\in \\mathbb{N}$ and $s \\in S$, the smoothing weight $\\alpha_t(s)\\, \\beta_k(t, s) / L_{t+k}$, where $\\alpha$ and $\\beta$ are the forward and backward variables and $L_{t+k}$ is the likelihood of $x_0, \\dots, x_{t+k}$, equals the joint weight of $x_0, \\dots, x_{t+k}$ and the\u2026","title":"Forward-backward smoothing is a joint-to-likelihood ratio"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/sum-smoothed","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.sum_smoothed","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#sum-smoothed","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model with a finite set of states $S$ and weights in a semifield, observations $x_0, x_1, \\dots$ and times $t, k \\in \\mathbb{N}$, if the likelihood $L_{t+k}$ of $x_0, \\dots, x_{t+k}$ is nonzero, then the smoothed weights $\\gamma_{t,k}(s) = \\alpha_t(s)\\,\\beta_{k,t}(s) / L_{t+k}$, with $\\alpha$ the forward and $\\beta$ the backward variables, sum to $1$ over all states $s \\in S$.","title":"Smoothed weights sum to one"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/smoothed-nonneg","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.smoothed_nonneg","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#smoothed-nonneg","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $g$ be a hidden Markov model with finite state set $S$ whose initial, transition and emission weights are nonnegative elements of a linearly ordered semifield $K$, and let $x_0, x_1, \\dots$ be observations. For all $t, k \\in \\mathbb{N}$ and $s \\in S$, the smoothing weight $\\alpha_t(s)\\, \\beta_k(t, s) / L_{t+k}$ is nonnegative, where $\\alpha$ and $\\beta$ are the forward and backward variables and $L_{t+k}$ is\u2026","title":"Smoothing weights are nonnegative"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/backward-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.backwardStep","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#backward-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model $g$ with states $\\{0, \\dots, n - 1\\}$, transition weights $a$ and emission weights $b$ in a semiring $R$, an observation sequence $x_0, x_1, \\dots$, a vector $v \\in R^n$ and a time $t$, this is the vector in $R^n$ whose entry at state $s$ is $\\sum_{s'} a(s, s')\\, b(s', x_{t+1})\\, v_{s'}$.","title":"One step of the tabulated backward algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/backward-tab","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.backwardTab","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#backward-tab","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a hidden Markov model $g$ with states $\\{0, \\dots, n - 1\\}$, transition weights $a$ and emission weights $b$ in a semiring $R$, and an observation sequence $x_0, x_1, \\dots$, the tables $\\beta^{(k)}_t \\in R^n$ for $k, t \\in \\mathbb{N}$ are defined by recursion on $k$: $\\beta^{(0)}_t$ is the all-ones vector, and $\\beta^{(k+1)}_t$ is one backward step at time $t$ applied to $\\beta^{(k)}_{t+1}$, that is, its entry\u2026","title":"The tabulated backward algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/smooth-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.smoothStep","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#smooth-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For a semifield $K$, vectors $a, b \\in K^n$ and $L \\in K$, this is the vector in $K^n$ whose entry at $s$ is $a_s b_s / L$.","title":"Smoothing step on tables"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/smoothed-tab","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.smoothedTab","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#smoothed-tab","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Let $g$ be a hidden Markov model with states $\\{0, \\dots, n - 1\\}$ and weights in a semifield $K$, and let $x_0, x_1, \\dots$ be observations. For $t, k \\in \\mathbb{N}$, this is the vector in $K^n$ whose entry at state $s$ is $\\alpha_t(s)\\, \\beta_k(t, s) / L_{t+k}$, where $\\alpha_t$ is the tabulated forward vector at time $t$, $\\beta_k(t, \\cdot)$ is the tabulated backward vector for $k$ steps from time $t$, and\u2026","title":"Tabulated forward-backward smoothing"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/posterior-path","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.posteriorPath","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#posterior-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"At each time `t \u2264 T`, a state of largest posterior `smoothed x g t (T - t)` (the first one, via `argmaxFin`).","title":"Posterior decoding"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/smoothed-le-posterior-path","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.smoothed_le_posteriorPath","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#smoothed-le-posterior-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"The posterior path picks a state of largest posterior at each time.","title":"Posterior path maximizes the posterior at each time"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/smoothed-eq-sum-path-prob","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.smoothed_eq_sum_pathProb","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#smoothed-eq-sum-path-prob","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"The posterior of state `s` at time `t \u2264 T`, as a sum over the paths of length `T + 1`: the total weight of the paths through `s` at time `t` over the likelihood.","title":"Smoothed posterior as a sum over state paths"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/expected-correct","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.expectedCorrect","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#expected-correct","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"The posterior expected number of correctly decoded states of a path `z`: the sum over all state paths `w` of `pathProb w / likelihood` times the number of times `t \u2264 T` with `w t = z t`. When the weights are probabilities and the likelihood is positive, this is the expectation under the posterior law of the path; for likelihood `0` it is `0` (`x / 0 = 0`), as is every posterior `smoothed`.","title":"Posterior expected number of correctly decoded states"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/expected-correct-eq-sum-smoothed","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.expectedCorrect_eq_sum_smoothed","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#expected-correct-eq-sum-smoothed","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"For every path `z`, the posterior expected number of times `t \u2264 T` with `w t = z t` is the sum over `t` of the posterior of `z t` at time `t`.","title":"The expected number of correct states is a sum of posteriors"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/expected-correct-le-posterior-path","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.expectedCorrect_le_posteriorPath","parent":41207,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models/index.html#expected-correct-le-posterior-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/hidden-markov-models","status":"fully_proved","summary":"Posterior decoding maximizes the expected number of correctly decoded states when addition preserves the order.","title":"Posterior decoding maximizes the expected number of correctly decoded states"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","kind":"scope","navigation":"module","parent":41206,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","title":"Baum\u2013Welch algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/path-prob-eq-prod","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.pathProb_eq_prod","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#path-prob-eq-prod","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $h$ be an HMM with weights in a commutative semiring, with initial weights $\\pi$, transition weights $a$ and emission weights $b$, and let $x_0, x_1, \\dots$ be observations. For every $T$ and every state path $z = (z_0, \\dots, z_T)$, the path weight of $z$ and $x_0, \\dots, x_T$ is","title":"Path weight as a product"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/valid","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.Valid","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#valid","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"A hidden Markov model with finite state set $S$, finite observation set $O$ and real initial weights $\\pi$, transition weights $a$ and emission weights $b$ is valid when all its weights are strictly positive and form probability distributions: $\\pi(s) > 0$ for every $s$ and $\\sum_{s} \\pi(s) = 1$; $a(s, s') > 0$ for all $s, s'$ and $\\sum_{s'} a(s, s') = 1$ for every $s$; and $b(s, o) > 0$ for all $s, o$ and\u2026","title":"Valid hidden Markov model parameters"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/valid-hmm","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.ValidHMM","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#valid-hmm","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"For finite sets $S$ of states and $O$ of observations, the valid HMMs are the HMMs with states $S$, observations $O$ and real weights whose initial weights are positive and sum to $1$, whose transition weights out of each state are positive and sum to $1$, and whose emission weights from each state are positive and sum to $1$. They form the parameter space of Baum-Welch.","title":"Valid HMMs"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/joint","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.joint","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#joint","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets and let $T$ be a natural number. Let $h$ be a valid hidden Markov model on states $S$ and observations $O$, with positive initial probabilities $\\pi_s$, transition probabilities $a_{ss'}$ and emission probabilities $b_{so}$, each distribution summing to one. The joint of its EM model at an observation sequence $x = (x_0, x_1, \\dots)$ in $O$ and a state path $z = (z_0, \\dots, z_T)$ in\u2026","title":"The EM model of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/marginal-joint","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.marginal_joint","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#marginal-joint","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"For every $T$, every valid HMM $h$ and every observation sequence $x$, the marginal at $h$ and $x$ of the EM model of HMMs with state paths $(z_0, \\dots, z_T)$, that is the sum over all such paths of the path weight of $z$ and $x$, equals the likelihood under $h$ of the observations $x_0, \\dots, x_T$.","title":"The EM marginal of an HMM is its likelihood"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/p-init","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.pInit","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#p-init","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"For a state path $z = (z_0, \\dots, z_T)$ and a state $s$, the initial-state indicator is the real number $1$ if $z_0 = s$ and $0$ otherwise.","title":"Initial-state indicator of a state path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/p-trans","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.pTrans","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#p-trans","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"For a state path $z = (z_0, \\dots, z_T)$ and states $s, s'$, the transition count is the number of $t \\in \\{0, \\dots, T - 1\\}$ with $z_t = s$ and $z_{t+1} = s'$, as a real number.","title":"Transition count of a state path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/p-emit","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.pEmit","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#p-emit","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"For a state path $z = (z_0, \\dots, z_T)$, observations $x_0, x_1, \\dots$, a state $s$ and a symbol $o$, the emission count is the number of times $t \\in \\{0, \\dots, T\\}$ with $z_t = s$ and $x_t = o$, as a real number.","title":"Emission count of a state path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/log-joint","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.log_joint","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#log-joint","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T$ be a natural number, and let $h$ be a valid hidden Markov model on states $S$ and observations $O$, with positive initial probabilities $\\pi_s$, transition probabilities $a_{ss'}$ and emission probabilities $b_{so}$, each distribution summing to one. For every observation sequence $x = (x_0, x_1, \\dots)$ in $O$ and every state path $z = (z_0, \\dots, z_T)$ in $S$, the path\u2026","title":"The log path weight is linear in the path's counts"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/post","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.post","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#post","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"For a valid HMM $h$, observation sequences $x_0, \\dots, x_{N-1}$, an index $n < N$ and a state path $z = (z_0, \\dots, z_T)$, the posterior weight of $z$ is the posterior of $z$ given $x_n$ at $h$ in the EM model of HMMs: the path weight of $z$ and $x_n$ divided by the sum of the path weights of $z'$ and $x_n$ over all state paths $z' = (z'_0, \\dots, z'_T)$.","title":"Posterior weight of a state path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/init-count","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.initCount","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#init-count","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T$ be a natural number, and let $x_1, \\dots, x_N$ be sequences of observations in $O$. For a valid hidden Markov model $h$ on states $S$ and observations $O$ and a state $s$, the expected number of sequences starting in $s$ is","title":"Expected initial-state counts"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/trans-count","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.transCount","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#trans-count","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"For a valid HMM $h$, observation sequences $x_0, \\dots, x_{N-1}$ and states $s, s'$, the expected transition count is $\\sum_{n < N} \\sum_{z} p_h(z \\mid x_n) \\, c_{s s'}(z)$, where $z$ ranges over the state paths $(z_0, \\dots, z_T)$, $p_h(z \\mid x_n)$ is the posterior weight of $z$ for $x_n$, and $c_{s s'}(z)$ is the number of $t < T$ with $z_t = s$ and $z_{t+1} = s'$.","title":"Expected transition count"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/emit-count","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.emitCount","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#emit-count","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T$ be a natural number, and let $x_1, \\dots, x_N$ be sequences $x_n = (x_{n,0}, x_{n,1}, \\dots)$ of observations in $O$. For a valid hidden Markov model $h$ on states $S$ and observations $O$, a state $s$ and an observation $o$, the expected number of emissions of $o$ from $s$ is","title":"Expected emission counts"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/m-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.mStep","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#m-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $h$ be a valid HMM with finite state set $S$ and finite observation set $O$, let $x_0, \\dots, x_{N-1}$ be observation sequences, and consider state paths $(z_0, \\dots, z_T)$. Write $I(s)$, $A(s, s')$ and $E(s, o)$ for the expected numbers, under the posterior of $h$ and summed over the sequences, of sequences starting in $s$, of transitions $s \\to s'$ and of emissions of $o$ from $s$. The M-step is the HMM with\u2026","title":"The Baum-Welch M-step"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/em-q-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.emQ_eq","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#em-q-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T$ be a natural number, and let $x_1, \\dots, x_N$ be sequences $x_n = (x_{n,0}, x_{n,1}, \\dots)$ of observations in $O$. For a valid hidden Markov model $h$ on states $S$ and observations $O$ (positive initial, transition and emission probabilities, each distribution summing to one), write $p_h(x, z)$ for the path weight of the observations $x$ and a state path\u2026","title":"The Baum-Welch objective in terms of expected counts"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/rich","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.Rich","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#rich","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Observation sequences $x_0, \\dots, x_{N-1}$ over a symbol set $O$ are rich for state paths $(z_0, \\dots, z_T)$ when $N > 0$, $T > 0$, and every symbol $o \\in O$ is observed, that is $x_n(t) = o$ for some $n < N$ and some $t \\le T$.","title":"Data rich enough for the Baum-Welch M-step"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/em-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.emStep","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#em-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T \\ge 1$, and let $x_1, \\dots, x_N$ with $N \\ge 1$ be sequences $x_n = (x_{n,0}, x_{n,1}, \\dots)$ of observations in $O$ such that every $o \\in O$ equals $x_{n,t}$ for some $n$ and some $t \\le T$. For a valid hidden Markov model $h$ on states $S$ and observations $O$ (positive initial, transition and emission probabilities, each distribution summing to one), the Baum-Welch step\u2026","title":"One Baum-Welch step"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/em-q-le-em-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.emQ_le_emStep","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#em-q-le-em-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T \\ge 1$, and let $x_1, \\dots, x_N$ with $N \\ge 1$ be sequences $x_n = (x_{n,0}, x_{n,1}, \\dots)$ of observations in $O$ such that every $o \\in O$ equals $x_{n,t}$ for some $n$ and some $t \\le T$. Consider the EM model of a hidden Markov model on states $S$ and observations $O$ whose latent variable is the state path $z = (z_0, \\dots, z_T)$ and whose joint $p_h(x, z)$ is the\u2026","title":"The Baum-Welch update maximizes the EM objective"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/log-lik-le-em-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.logLik_le_emStep","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#log-lik-le-em-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T \\ge 1$, and let $x_1, \\dots, x_N$ with $N \\ge 1$ be sequences $x_n = (x_{n,0}, x_{n,1}, \\dots)$ of observations in $O$ such that every $o \\in O$ equals $x_{n,t}$ for some $n$ and some $t \\le T$. For every valid hidden Markov model $h$ on states $S$ and observations $O$, $\\ell(h) \\le \\ell(\\bar h)$, where $\\bar h$ is the Baum-Welch step applied to $h$ on these data and\u2026","title":"Each Baum-Welch step does not decrease the log-likelihood"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/log-lik-iterate-em-step-monotone","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.logLik_iterate_emStep_monotone","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#log-lik-iterate-em-step-monotone","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T \\ge 1$, and let $x_1, \\dots, x_N$ with $N \\ge 1$ be sequences $x_n = (x_{n,0}, x_{n,1}, \\dots)$ of observations in $O$ such that every $o \\in O$ equals $x_{n,t}$ for some $n$ and some $t \\le T$. Let $h_0$ be a valid hidden Markov model on states $S$ and observations $O$ and let $h_t$ be the result of $t$ Baum-Welch steps from $h_0$ on these data. Then $t \\mapsto \\ell(h_t)$ is\u2026","title":"The log-likelihood is monotone along the Baum-Welch iterates"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/log-lik-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.logLik_eq","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#log-lik-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $S$ and $O$ be finite sets, let $T$ be a natural number, and let $x_1, \\dots, x_N$ be sequences $x_n = (x_{n,0}, x_{n,1}, \\dots)$ of observations in $O$. For every valid hidden Markov model $h$ on states $S$ and observations $O$, the log-likelihood of its EM model (latent state paths $(z_0, \\dots, z_T)$, joint the path weight) on these data equals $\\sum_{n=1}^{N} \\log L_h(x_n)$, where $L_h(x_n)$ is the\u2026","title":"The Baum-Welch log-likelihood is the HMM log-likelihood"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/sum-path-prob-ite-pair","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.sum_pathProb_ite_pair","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#sum-path-prob-ite-pair","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with finite state set $S$, transition weights $a$ and emission weights $b$ in a semiring, and let $x$ be an observation sequence. For all $k, t \\in \\mathbb{N}$ and states $s, s', u$, the sum of the path weights of the state paths $z_0, \\dots, z_{t + 1 + k}$ with $z_t = s$, $z_{t + 1} = s'$ and $z_{t + 1 + k} = u$ equals\u2026","title":"Pairwise path sums with a fixed endpoint"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/pair-marg","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.pairMarg","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#pair-marg","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"For a hidden Markov model with finite state set $S$, an observation sequence $x$, times $t, k \\in \\mathbb{N}$ and states $s, s'$, the pairwise marginal is the joint weight of the observations $x(0), \\dots, x(t + 1 + k)$ and the transition $s \\to s'$ from time $t$ to time $t + 1$: the sum of the path weights of all state paths $z_0, \\dots, z_{t + 1 + k}$ with $z_t = s$ and $z_{t + 1} = s'$.","title":"Pairwise state marginal of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/pair-marg-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.HiddenMarkov.pairMarg_eq","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#pair-marg-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with finite state set $S$, transition weights $a$ and emission weights $b$ in a semiring, and let $x$ be an observation sequence. For all $t, k \\in \\mathbb{N}$ and states $s, s'$, the joint weight of the observations $x(0), \\dots, x(t + 1 + k)$ and the transition $s \\to s'$ from time $t$ to time $t + 1$ equals $\\alpha_t(s)\\, a(s, s')\\, b(s', x(t + 1))\\, \\beta_{k, t + 1}(s')$, where\u2026","title":"Pairwise forward-backward"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/init-count-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.initCount_eq","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#init-count-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $h$ be a valid hidden Markov model with finite state set $S$ and finite observation set $O$, let $T \\in \\mathbb{N}$, and let $x_0, \\dots, x_{N-1}$ be observation sequences. For a sequence $x_n$ write $\\alpha^n_0(s)$ for its forward variable at time $0$ and state $s$, $\\beta^n_{T,0}(s)$ for its backward variable from time $0$ with $T$ remaining steps (the weight of the observations $x_n(1), \\dots, x_n(T)$ given\u2026","title":"The expected starts from forward-backward"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/emit-count-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.emitCount_eq","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#emit-count-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $h$ be a valid hidden Markov model with finite state set $S$ and finite observation set $O$, let $T \\in \\mathbb{N}$, and let $x_0, \\dots, x_{N-1}$ be observation sequences. For a sequence $x_n$ write $\\alpha^n_t(s)$ for its forward variable at time $t$ and state $s$, $\\beta^n_{k,t}(s)$ for its backward variable (the weight of the observations $x_n(t + 1), \\dots, x_n(t + k)$ given the state $s$ at time $t$), and\u2026","title":"The expected emissions from forward-backward"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/trans-count-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.BaumWelch.transCount_eq","parent":41254,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm/index.html#trans-count-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/baum-welch-algorithm","status":"fully_proved","summary":"Let $h$ be a valid hidden Markov model with finite state set $S$, finite observation set $O$, transition weights $a$ and emission weights $b$, let $T \\in \\mathbb{N}$, and let $x_0, \\dots, x_{N-1}$ be observation sequences. For a sequence $x_n$ write $\\alpha^n_t(s)$ for its forward variable at time $t$ and state $s$, $\\beta^n_{k,t}(s)$ for its backward variable (the weight of the observations\u2026","title":"The expected transitions from forward-backward"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","kind":"scope","navigation":"module","parent":41206,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","title":"Conditional random fields"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/chain-crf","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.ChainCRF","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#chain-crf","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $S$ be a type of states, $O$ a type of inputs and $R$ a type of weights with a zero and a relation $\\le$. A linear-chain conditional random field consists of a unary potential $\\phi(x, t, s) \\in R$ and a transition potential $\\psi(x, t, s, s') \\in R$ from $s$ to $s'$, given for every observation sequence $x \\colon \\mathbb{N} \\to O$, every position $t \\in \\mathbb{N}$ and all states $s, s' \\in S$, such that\u2026","title":"Linear-chain conditional random field"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/path-weight","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.pathWeight","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#path-weight","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"For a chain conditional random field with weights in a semiring, unary potentials $u_t(s)$ and transition potentials $\\tau_t(s, s')$ given the observation stream $x$, the unnormalized weight of a label path $z = (z_0, \\dots, z_t)$ is","title":"Weight of a label path in a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/partition","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.partition","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#partition","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"For a chain conditional random field with a finite set of states and weights in a semiring, and an observation stream $x$, the partition function at time $t$ is $Z_t = \\sum_z w_t(z)$, the total weight of all label paths $z$ of length $t + 1$, where $w_t(z)$ is the unnormalized weight of $z$ given $x$.","title":"Partition function of a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/forward","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.forward","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#forward","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"For a chain conditional random field with a finite set of states, weights in a semiring, unary potentials $u_t(s)$ and transition potentials $\\tau_t(s', s)$ given the observation stream $x$, the forward variables are defined by $\\alpha_0(s) = u_0(s)$ and $\\alpha_{t+1}(s) = \\left(\\sum_{s'} \\alpha_t(s')\\, \\tau_t(s', s)\\right) u_{t+1}(s)$, the sum running over all states $s'$.","title":"Forward variables of a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/forward-eq","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.forward_eq","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#forward-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with a finite set of states $S$ and weights in a semiring $R$ carrying a relation $\\le$, and let $x$ be an observation sequence. For every $t \\in \\mathbb{N}$ and every state $s \\in S$, the forward variable $\\alpha_t(s)$ equals the total path weight of the label paths $z = (z_0, \\dots, z_t) \\in S^{t+1}$ that end in $z_t = s$:","title":"The forward variables are path weights"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/partition-eq-sum-forward","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.partition_eq_sum_forward","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#partition-eq-sum-forward","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"For a chain conditional random field with a finite set of states and weights in a semiring, and an observation stream $x$, for every $t$ the partition function at time $t$ equals $\\sum_s \\alpha_t(s)$, the sum over all states $s$ of the forward variables $\\alpha_t(s)$.","title":"The forward algorithm computes the partition function"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/viterbi","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.viterbi","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#viterbi","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain conditional random field with a finite nonempty set of states $S$ and weights in a linearly ordered commutative semiring $R$, and let $x$ be an observation sequence. Write $\\phi_t(s)$ for the unary potential of $c$ at $x$ of state $s$ at position $t$, and $\\psi_t(s', s)$ for its transition potential from $s'$ to $s$ at step $t$. The Viterbi variables $V_t(s)$, for $t \\in \\mathbb{N}$ and\u2026","title":"The Viterbi variables of a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/path-weight-le-viterbi","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.pathWeight_le_viterbi","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#path-weight-le-viterbi","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"For a chain conditional random field with a nonempty finite set of states, weights in a linearly ordered commutative semiring, and an observation stream $x$, for every $t$ and every label path $z = (z_0, \\dots, z_t)$, the weight of $z$ is at most the Viterbi variable at time $t$ of its last state $z_t$.","title":"Path weights of a chain CRF are bounded by the Viterbi variables"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/exists-path-weight-eq-viterbi","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.exists_pathWeight_eq_viterbi","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#exists-path-weight-eq-viterbi","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with a finite nonempty set of states $S$ and weights in a linearly ordered commutative semiring, and let $x$ be an observation sequence. For every $t \\in \\mathbb{N}$ and every state $s \\in S$ there is a label path $z = (z_0, \\dots, z_t) \\in S^{t+1}$ with $z_t = s$ whose path weight equals the Viterbi variable of $s$ at time $t$.","title":"The Viterbi variables of a chain CRF are attained"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/sup-viterbi-eq","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.sup'_viterbi_eq","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#sup-viterbi-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain conditional random field with a finite nonempty set of states $S$ and weights in a linearly ordered commutative semiring $R$, and let $x$ be an observation sequence. For every $t \\in \\mathbb{N}$, the maximum over $s \\in S$ of the Viterbi variable $V_t(s)$ of $c$ at $x$ equals the maximum, over all label paths $z : \\{0, \\dots, t\\} \\to S$, of the weight of $z$.","title":"Viterbi computes the MAP value of a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/cond-prob","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.condProb","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#cond-prob","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with a finite set of states $S$ and weights in a semifield $R$ carrying a partial order $\\le$, and let $x$ be an observation sequence. For $t \\in \\mathbb{N}$, the conditional probability of a label path $z \\in S^{t+1}$ given $x$ is $w_t(z) / Z_t$, where $w_t(z)$ is the path weight of $z$ and $Z_t$ is the partition function, the total path weight of all label paths in $S^{t+1}$; the quotient\u2026","title":"Conditional probability of a label path in a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/sum-cond-prob-eq-one","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.sum_condProb_eq_one","parent":41282,"reading_title":{"format":"inline-parts-v1","parts":[["text","The conditional probabilities sum to "],["math","\\(1\\)"]],"source":"The conditional probabilities sum to $1$"},"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#sum-cond-prob-eq-one","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain conditional random field with finitely many states $S$ and weights in a semifield $R$ equipped with a partial order (not assumed compatible with the operations), let $x$ be an observation sequence, and let $t \\in \\mathbb{N}$. If the partition function of $c$ at $x$ and time $t$ is positive, then the conditional probabilities $p(z \\mid x)$ of all label paths $z : \\{0, \\dots, t\\} \\to S$ of length\u2026","title":"The conditional probabilities sum to $1$"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/cond-prob-nonneg","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.condProb_nonneg","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#cond-prob-nonneg","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with a finite set of states $S$ and weights in a linearly ordered field $R$, and let $x$ be an observation sequence. For every $t \\in \\mathbb{N}$ and every label path $z = (z_0, \\dots, z_t) \\in S^{t+1}$, the conditional probability of $z$ given $x$ is nonnegative.","title":"Conditional probabilities of a chain CRF are nonnegative"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/hmm-to-crf","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.hmmToCRF","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#hmm-to-crf","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with initial weights $\\pi(s)$, transition weights $a(s, s')$ and emission weights $b(s, o)$ in a linearly ordered commutative semiring, all nonnegative. Its chain CRF has, for every observation stream $x$, unary potentials $u_0(s) = \\pi(s)\\, b(s, x_0)$ and $u_t(s) = b(s, x_t)$ for $t \\ge 1$, and transition potentials $\\tau_t(s, s') = a(s, s')$ at every step $t$.","title":"The chain CRF of a hidden Markov model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/hmm-to-crf-path-weight","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.hmmToCRF_pathWeight","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#hmm-to-crf-path-weight","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $h$ be a hidden Markov model with nonnegative weights in a linearly ordered commutative semiring, and let $x$ be an observation stream. For every $t$ and every state path $z = (z_0, \\dots, z_t)$, the weight of $z$ in the chain CRF of $h$ given $x$ equals the joint weight in $h$ of the state path $z$ and the observations $x_0, \\dots, x_t$.","title":"The chain CRF of a hidden Markov model has the same path weights"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/forward-step","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.forwardStep","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#forward-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with states $\\{0, \\dots, n-1\\}$ and weights in a semiring $R$ carrying a relation $\\le$, with unary potential $\\phi_t(s)$ at position $t$ and transition potential $\\psi_t(s', s)$ from $s'$ to $s$ at step $t$, given the observation sequence $x$. For a table $v \\in R^n$ and a step $t$, the forward step returns the table whose entry at a state $s$ is","title":"One step of the tabulated CRF forward algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/forward-tab","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.forwardTab","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#forward-tab","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with states $\\{0, \\dots, n-1\\}$ and weights in a semiring $R$ carrying a relation $\\le$, with unary potential $\\phi_t(s)$ at position $t$ given the observation sequence $x$. The forward table is the sequence of vectors in $R^n$ defined by recursion on time: at time $0$ its entry at $s$ is $\\phi_0(s)$, and the table at time $t + 1$ is the forward step at step $t$ applied to the table at time\u2026","title":"Tabulated CRF forward algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/partition-tab","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.partitionTab","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#partition-tab","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"For a chain conditional random field with states $0, \\dots, n - 1$ and weights in a semiring, and an observation stream $x$, the tabulated partition function at time $t$ is the sum of the entries of the forward table at time $t$, the vector $(\\alpha_t(0), \\dots, \\alpha_t(n - 1))$ of forward variables computed step by step from time $0$.","title":"The tabulated forward algorithm for chain CRFs"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/viterbi-step","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.viterbiStep","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#viterbi-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain conditional random field with states $\\{0, \\dots, n - 1\\}$, where $n \\ge 1$, and weights in a linearly ordered commutative semiring $R$, and let $x$ be an observation sequence. Write $\\phi_t(s)$ for the unary potential of $c$ at $x$ of state $s$ at position $t$, and $\\psi_t(s', s)$ for its transition potential from $s'$ to $s$ at step $t$. Given a vector $v \\in R^n$ and $t \\in \\mathbb{N}$, the\u2026","title":"One step of the tabulated Viterbi recursion for a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/viterbi-tab","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.viterbiTab","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#viterbi-tab","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain conditional random field with states $\\{0, \\dots, n - 1\\}$, where $n \\ge 1$, and weights in a linearly ordered commutative semiring $R$, and let $x$ be an observation sequence. Write $\\phi_0(s)$ for the unary potential of $c$ at $x$ of state $s$ at position $0$. The Viterbi table assigns to each $t \\in \\mathbb{N}$ a vector $T_t \\in R^n$: the vector $T_0$ has entry $\\phi_0(s)$ at each state $s$,\u2026","title":"The tabulated Viterbi variables of a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/back","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.back","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#back","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with states $\\{0, \\dots, n-1\\}$, where $n \\ge 1$, and weights in a linearly ordered commutative semiring $R$, and let $\\psi_t(s', s)$ be its transition potential from $s'$ to $s$ at step $t$ given the observation sequence $x$. For a step $t$, a table $v \\in R^n$ and a state $s$, the backpointer of $s$ is the first state $s'$, in the order $0, \\dots, n-1$, that maximizes\u2026","title":"Backpointer of the tabulated CRF Viterbi algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/best-path","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.bestPath","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#best-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with states $\\{0, \\dots, n-1\\}$, where $n \\ge 1$, and weights in a linearly ordered commutative semiring, and fix an observation sequence $x$. For a state $s$, the backtracked path $b_t(s) = (b_t(s)_0, \\dots, b_t(s)_t)$ of length $t + 1$ is defined by recursion on $t$: $b_0(s)$ is the one-state path $(s)$, and $b_{t+1}(s)$ is $b_t(s')$ followed by $s$, where $s'$ is the backpointer of $s$ at\u2026","title":"Backtracked path of the tabulated CRF Viterbi algorithm"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/path-weight-best-path","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.pathWeight_bestPath","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#path-weight-best-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"For a chain conditional random field with states $0, \\dots, n - 1$ where $n \\ge 1$, weights in a linearly ordered commutative semiring, and an observation stream $x$, for every $t$ and every state $s$ the weight of the backtracked path of length $t + 1$ ending in $s$, obtained by following the backpointers, equals the Viterbi variable at time $t$ of $s$.","title":"The backtracked path of a chain CRF attains the Viterbi variable"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/crf-path","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.crfPath","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#crf-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"Let $c$ be a chain CRF with states $\\{0, \\dots, n-1\\}$, where $n \\ge 1$, and weights in a linearly ordered commutative semiring, and let $x$ be an observation sequence. For $t \\in \\mathbb{N}$, the Viterbi path of length $t + 1$ is the backtracked path ending in the first state, in the order $0, \\dots, n-1$, that maximizes the Viterbi table at time $t$.","title":"The Viterbi path of a chain CRF"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/path-weight-le-crf-path","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.ConditionalRandomField.pathWeight_le_crfPath","parent":41282,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields/index.html#path-weight-le-crf-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/conditional-random-fields","status":"fully_proved","summary":"For a chain conditional random field with states $0, \\dots, n - 1$ where $n \\ge 1$, weights in a linearly ordered commutative semiring, and an observation stream $x$, for every $t$ and every label path $z$ of length $t + 1$, the weight of $z$ is at most the weight of Viterbi's MAP path of length $t + 1$.","title":"The Viterbi path of a chain CRF has maximal weight"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","kind":"scope","navigation":"module","parent":41206,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","title":"Kalman filter"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/belief","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.Belief","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#belief","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"For an index type $n$, a belief about the state consists of a mean $m \\in \\mathbb{R}^n$ and a covariance matrix $P$, an arbitrary real $n \\times n$ matrix. It describes the Gaussian distribution $N(m, P)$ when $P$ is positive semidefinite (the belief is valid); for other matrices Mathlib's multivariate Gaussian is a junk value. Prediction with positive semidefinite process noise, update with positive definite\u2026","title":"Gaussian belief"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/valid","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.Belief.Valid","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#valid","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"A belief is valid when its covariance matrix is positive semidefinite.","title":"Valid Gaussian beliefs"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/law","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.Belief.law","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#law","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"The law of a belief with mean $m$ and covariance $P$ is Mathlib's multivariate Gaussian measure $N(m, P)$ on $\\mathbb{R}^n$. It is the Gaussian distribution with mean $m$ and covariance $P$ when $P$ is positive semidefinite (the belief is valid); otherwise it is the point mass at $m$.","title":"Law of a belief"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/predict","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.predict","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#predict","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Given real $n \\times n$ matrices $F$ and $Q$, the prediction of a belief with mean $m$ and covariance $P$ is the belief with mean $F m$ and covariance $F P F^\\top + Q$. It models the dynamics $x' = F x + w$ with process noise $w \\sim N(0, Q)$.","title":"Kalman prediction"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/gain","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.gain","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#gain","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"For a real $p \\times n$ matrix $H$, a real $p \\times p$ matrix $R$ and a belief with covariance $P$, the Kalman gain is the real $n \\times p$ matrix $K = P H^\\top (H P H^\\top + R)^{-1}$, where the inverse is taken to be $0$ when $H P H^\\top + R$ is singular.","title":"Kalman gain"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/update","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.update","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#update","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Given a real $p \\times n$ observation matrix $H$, a real $p \\times p$ matrix $R$, an observation $y \\in \\mathbb{R}^p$ and a belief with mean $m$ and covariance $P$, the updated belief has mean $m + K (y - H m)$ and covariance $(I - K H) P$, where $K = P H^\\top (H P H^\\top + R)^{-1}$ is the Kalman gain and the matrix inverse is taken to be $0$ when $H P H^\\top + R$ is singular.","title":"Kalman filter update"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/step","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.step","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Given real $n \\times n$ matrices $F$ and $Q$, a real $p \\times n$ matrix $H$, a real $p \\times p$ matrix $R$ and an observation $y \\in \\mathbb{R}^p$, one filter step sends a belief to the update, with $H$, $R$ and $y$, of its prediction with $F$ and $Q$.","title":"Kalman filter step"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/predict-valid","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.predict_valid","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#predict-valid","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Let $F$ and $Q$ be real $n \\times n$ matrices with $Q$ positive semidefinite. If a belief is valid, that is, its covariance $P$ is positive semidefinite, then the predicted belief, with covariance $F P F^\\top + Q$, is valid.","title":"Prediction preserves validity"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/update-cov-eq","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.update_cov_eq","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#update-cov-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Let $H$ be a real $p \\times n$ matrix, $R$ a real $p \\times p$ matrix and $y \\in \\mathbb{R}^p$, and let a belief have covariance $P$. Then the covariance of the belief updated with $H$, $R$ and $y$ equals $P - P H^\\top (H P H^\\top + R)^{-1} H P$. Here $M^{-1}$ is the matrix inverse, taken to be $0$ when $M$ is singular, so no invertibility is assumed.","title":"Schur form of the updated covariance"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/update-valid","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.update_valid","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#update-valid","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Let $H$ be a real $p \\times n$ matrix, $R$ a symmetric positive definite real $p \\times p$ matrix and $y \\in \\mathbb{R}^p$. If a belief is valid, then the belief updated with $H$, $R$ and $y$ is valid.","title":"The Kalman update preserves validity"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/step-valid","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.step_valid","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#step-valid","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Let $F$ and $Q$ be real $n \\times n$ matrices with $Q$ positive semidefinite, $H$ a real $p \\times n$ matrix, $R$ a positive definite real $p \\times p$ matrix and $y \\in \\mathbb{R}^p$. If a belief is valid (its covariance is positive semidefinite), then so is the belief obtained from it by one filter step with $F$, $Q$, $H$, $R$ and the observation $y$.","title":"A filter step preserves validity"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/map-predict","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.map_predict","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#map-predict","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Let $x$ and $w$ be measurable, independent random vectors in $\\mathbb{R}^n$ on a probability space. If $x$ has the law $N(m, P)$ of a belief with mean $m$ and positive semidefinite covariance $P$, and $w$ has law $N(0, Q)$ with $Q$ positive semidefinite, then for every real $n \\times n$ matrix $F$ the random vector $F x + w$ has the law of the predicted belief, $N(F m, F P F^\\top + Q)$.","title":"The predict step is exact"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/cond-distrib-update","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.condDistrib_update","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#cond-distrib-update","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Let $x$ be a measurable random vector in $\\mathbb{R}^n$ and $v$ a measurable random vector in $\\mathbb{R}^p$ on a probability space, with $x$ independent of $v$. Suppose $x$ has the law $N(m, P)$ of a belief with mean $m$ and positive semidefinite covariance $P$, and $v$ has law $N(0, R)$ with $R$ positive semidefinite. Let $H$ be a real $p \\times n$ matrix with $H P H^\\top + R$ invertible, and put $Y = H x + v$.\u2026","title":"The update step is the exact posterior"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/cond-distrib-step","kind":"node","lean":"Cslib.MachineLearning.Sequence.KalmanFilter.condDistrib_step","parent":41308,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter/index.html#cond-distrib-step","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/kalman-filter","status":"fully_proved","summary":"Let $x$ and $w$ be measurable random vectors in $\\mathbb{R}^n$ and $v$ a measurable random vector in $\\mathbb{R}^p$ on a probability space, with $x$ independent of $w$ and the pair $(x, w)$ independent of $v$. Suppose $x$ has the law $N(m, P)$ of a belief with mean $m$ and positive semidefinite covariance $P$, $w$ has law $N(0, Q)$ and $v$ has law $N(0, R)$, with $Q$ and $R$ positive semidefinite. Let $F$ be a real\u2026","title":"One filter step is the exact posterior"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","kind":"scope","navigation":"module","parent":41206,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","title":"Markov equivalence and Bayesian network scores"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/config","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.Config","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#config","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The joint values of the variables in `S`.","title":"Joint configurations of a set of variables"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/rising","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.rising","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#rising","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The rising factorial `a (a + 1) \u22ef (a + n - 1)`, equal to `\u0393(a + n) / \u0393(a)` for `a > 0`.","title":"Rising factorial"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/count","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.count","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#count","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The number of cases in `D` that take the values `c` on the variables `S`.","title":"Count of cases matching a configuration"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/bd-family","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.bdFamily","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#bd-family","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The Bayesian\u2013Dirichlet score of a family `F` with parent set `P \u2286 F` and hyperparameters `\u03b1` on the configurations of `F`.","title":"Bayesian\u2013Dirichlet family score"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/bd-score","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.bdScore","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#bd-score","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The BD score of the structure `Pa` with family hyperparameters `\u03b1`: the product of the family scores.","title":"Bayesian\u2013Dirichlet network score"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/predictive","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.predictive","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#predictive","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The sequential P\u00f3lya\u2013Dirichlet prediction factor for the case `x` after the cases `D`: `\u220f_v (\u03b1_{v,c} + n_{v,c}) / (\u03b1_{v,j} + n_{v,j})`, with `c`, `j` the family and parent values of `x` and `n` the counts in `D`. This is a genuine conditional probability only when `Pa` is a DAG; for arbitrary parent maps (e.g. cycles or self-parents) the factors over all cases need not sum to one.","title":"Sequential P\u00f3lya\u2013Dirichlet prediction factor"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/seq-prob-after","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.seqProbAfter","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#seq-prob-after","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The sequential P\u00f3lya\u2013Dirichlet product for the cases `D`, in order, after the cases `E`: each case contributes its `predictive` factor given all cases before it. A genuine probability of the data only when `Pa` is a DAG (see `predictive`).","title":"Sequential P\u00f3lya\u2013Dirichlet product after prior cases"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/seq-prob","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.seqProb","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#seq-prob","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The sequential P\u00f3lya\u2013Dirichlet product for the cases of `D`, in order. A genuine probability of the data only when `Pa` is a DAG (see `predictive`).","title":"Sequential P\u00f3lya\u2013Dirichlet product of the data"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/sum-count-restrict","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.sum_count_restrict\u2082","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#sum-count-restrict","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The number of cases matching a parent configuration is the sum over the family configurations extending it (`N_j = \u2211_k N_jk`).","title":"Parent configuration counts sum over family configurations"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/seq-prob-eq-bd-score","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.seqProb_eq_bdScore","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#seq-prob-eq-bd-score","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The BD score is the sequential prediction product of the data (hence it does not depend on the order of the cases). This is an identity for all real hyperparameters. Positivity of the hyperparameters only makes every predictive denominator positive; it does not make this product a probability. The product is a genuine probability of the data only when `Pa` is a DAG with no self-parents: for cyclic parent maps the\u2026","title":"The BD score is the sequential prediction product"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/bde-hyper","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.bdeHyper","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#bde-hyper","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"BDe hyperparameters (Heckerman, Geiger and Chickering): `ess` times the weight `\u2211_{x \u21a6 c} p x` of the configuration `c` of `S`. When `\u2211 x, p x = 1`, `p` is a prior joint distribution and this is the prior probability of `c`.","title":"BDe hyperparameters"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/marg-lik","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.margLik","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#marg-lik","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The (unnormalized) marginal likelihood of the data restricted to the variables `S` under the BDe Dirichlet prior on the joint values of `S`. `p` is treated as generalized weights here: the total Dirichlet mass over configurations is `ess * \u2211 x, p x`, so the omitted normalizing factor is `1 / rising (ess * \u2211 x, p x) D.length`, the same for every `S`. When `p` is normalized (`\u2211 x, p x = 1`) this is\u2026","title":"BDe marginal likelihood of a variable set"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/bd-family-bde-hyper","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.bdFamily_bdeHyper","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#bd-family-bde-hyper","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"With BDe hyperparameters, a family score is a ratio of set marginal likelihoods.","title":"BDe family score as a ratio of marginal likelihoods"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/bde-score","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.bdeScore","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#bde-score","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The BDe score of the structure `Pa`.","title":"BDe network score"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/covered-reversal","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.CoveredReversal","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#covered-reversal","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"`Pa'` is obtained from `Pa` by reversing a covered edge `x \u2192 y` (Chickering 1995): `x` is a parent of `y` and `y`'s other parents are exactly `x`'s parents (`Pa y = insert x (Pa x)`); after the reversal `y` has parents `Pa x` and `x` has parents `insert y (Pa x)`. Acyclicity is not required: the score identity holds for any parent-set maps (on a DAG, `x \u2209 Pa x` and `y \u2209 Pa x` hold automatically).","title":"Covered-edge reversal"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/bde-score-covered-reversal","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.bdeScore_coveredReversal","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#bde-score-covered-reversal","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"Covered-edge reversal preserves the BDe score (for a positive weight function `p` \u2014 a prior when `\u2211 x, p x = 1` \u2014 and positive equivalent sample size).","title":"Covered-edge reversal preserves the BDe score"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/bde-score-eq-of-refl-trans-gen","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.bdeScore_eq_of_reflTransGen","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#bde-score-eq-of-refl-trans-gen","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"Structures connected by a sequence of covered-edge reversals have the same BDe score.","title":"Covered-edge reversals preserve the BDe score"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/is-acyclic","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.IsAcyclic","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#is-acyclic","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"The structure has no directed cycle.","title":"Acyclic network structure"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/adjacent","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.Adjacent","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#adjacent","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"`u` and `w` are adjacent (joined by an edge in either direction).","title":"Adjacent vertices in a network structure"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/is-vstructure","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.IsVStructure","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#is-vstructure","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"`u \u2192 w \u2190 z` is a v-structure: two non-adjacent distinct parents of a common child.","title":"V-structure in a directed graph"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/refl-trans-gen-covered-reversal-of-marko","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesianDirichlet.reflTransGen_coveredReversal_of_markovEquivalent","parent":41323,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores/index.html#refl-trans-gen-covered-reversal-of-marko","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/markov-equivalence-and-bayesian-network-scores","status":"fully_proved","summary":"Chickering's transformational characterization (1995): two acyclic structures with the same skeleton and the same v-structures (Markov equivalent by Verma and Pearl) are connected by a sequence of covered-edge reversals.","title":"Chickering's transformational characterization"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","kind":"scope","navigation":"module","parent":41206,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","title":"Bayesian networks"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/sum-snoc","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.sum_snoc","parent":41345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Splitting a sum over "],["math","\\(\\alpha^{n+1}\\)"],["text"," at the last coordinate"]],"source":"Splitting a sum over $\\alpha^{n+1}$ at the last coordinate"},"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#sum-snoc","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $\\alpha$ be a finite type, $M$ an additive commutative monoid, $n$ a natural number and $f : \\alpha^{n+1} \\to M$. Then $\\sum_{x \\in \\alpha^{n+1}} f(x) = \\sum_{y \\in \\alpha^n} \\sum_{a \\in \\alpha} f(y_0, \\dots, y_{n-1}, a)$, where $y = (y_0, \\dots, y_{n-1})$.","title":"Splitting a sum over $\\alpha^{n+1}$ at the last coordinate"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/chain","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.Chain","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#chain","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"A chain of conditional distributions with values in a type $\\alpha$ assigns to every $k \\in \\mathbb{N}$ and every assignment $(x_0, \\dots, x_{k-1})$ of values in $\\alpha$ to the variables $0, \\dots, k - 1$ a finitely supported probability distribution $p_k(\\cdot \\mid x_0, \\dots, x_{k-1})$ on $\\alpha$, the conditional distribution of variable $k$ given the earlier ones.","title":"Chain of conditional distributions"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/joint","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.Chain.joint","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#joint","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"For a chain of conditional distributions with values in a type $\\alpha$, write $p_k(\\cdot \\mid x_0, \\dots, x_{k-1})$ for the conditional distribution of variable $k$ given the earlier ones. The joint distribution $P_n : \\alpha^n \\to \\mathbb{R}$ of the first $n$ variables is defined by the chain rule: $P_0 = 1$ and $P_{n+1}(x_0, \\dots, x_n) = P_n(x_0, \\dots, x_{n-1}) \\, p_n(x_n \\mid x_0, \\dots, x_{n-1})$.","title":"Joint distribution of a chain"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/sum-joint-snoc","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.Chain.sum_joint_snoc","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#sum-joint-snoc","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $c$ be a chain of conditional distributions with values in a finite type $\\alpha$, and write $P_k$ for its joint distribution of the first $k$ variables. For every $n$ and every $y = (y_0, \\dots, y_{n-1}) \\in \\alpha^n$, $\\sum_{a \\in \\alpha} P_{n+1}(y_0, \\dots, y_{n-1}, a) = P_n(y)$.","title":"Marginalizing the last variable of a chain"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/sum-joint","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.Chain.sum_joint","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#sum-joint","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $c$ be a chain of conditional distributions with values in a finite type $\\alpha$, and write $P_n$ for its joint distribution of the first $n$ variables. For every $n$, $\\sum_{x \\in \\alpha^n} P_n(x) = 1$.","title":"The joint distribution of a chain sums to one"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/cond-eq-div","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.Chain.cond_eq_div","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#cond-eq-div","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $c$ be a chain of conditional distributions with values in a type $\\alpha$, let $P_m$ be its joint distribution of the first $m$ variables, and let $p_n(\\cdot \\mid y)$ be its conditional distribution of variable $n$ given values $y \\in \\alpha^n$ of the earlier ones. For every $n$, every $y \\in \\alpha^n$ with $P_n(y) \\ne 0$ and every $a \\in \\alpha$, we have $P_{n+1}(y, a) / P_n(y) = p_n(a \\mid y)$, where\u2026","title":"Conditionals of the joint of a chain"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/joint-eq-prod","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.Chain.joint_eq_prod","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#joint-eq-prod","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"For a chain of conditional distributions with values in a type $\\alpha$, write $p_k(\\cdot \\mid x_0, \\dots, x_{k-1})$ for the conditional distribution of variable $k$ given the earlier ones and $P_n$ for the joint distribution of the first $n$ variables. Then for every $n$ and every $x \\in \\alpha^n$ we have $P_n(x) = \\prod_{k < n} p_k(x_k \\mid x_0, \\dots, x_{k-1})$.","title":"Factorization of the joint of a chain"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/joint-congr","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.Chain.joint_congr","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#joint-congr","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $n \\in \\mathbb{N}$. If two chains of conditional distributions $c$ and $c'$ with values in a type $\\alpha$ have the same conditional distribution of variable $k$ given the earlier variables for every $k < n$, then they have the same joint distribution of the first $n$ variables.","title":"Joint distribution determined by the first conditionals"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/update","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.Chain.update","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#update","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Given a chain $c$ of conditional distributions with values in $\\alpha$, an index $n$ and a family $f$ assigning to each $y \\in \\alpha^n$ a probability distribution $f(y)$ on $\\alpha$, the updated chain has conditional distribution $f$ for variable $n$ and the same conditional distribution as $c$ for every other variable.","title":"Replacing one conditional of a chain"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/exists-chain","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.exists_chain","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#exists-chain","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $\\alpha$ be a finite nonempty type and $n \\in \\mathbb{N}$. For every function $P : \\alpha^n \\to \\mathbb{R}$ with $P(x) \\ge 0$ for all $x$ and $\\sum_x P(x) = 1$, there is a chain of conditional distributions with values in $\\alpha$ whose joint distribution of the first $n$ variables equals $P$.","title":"The chain rule"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/bayes-net","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#bayes-net","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"A Bayesian network with values in a type $\\alpha$ on the nodes $0, \\dots, n - 1$, listed in a topological order, assigns to each node $k$ a finite set of parents among the earlier nodes $0, \\dots, k - 1$, and a conditional probability table: for each assignment of values in $\\alpha$ to the parents of $k$, a finitely supported probability distribution on $\\alpha$ for the value of node $k$.","title":"Bayesian network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/to-chain","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.toChain","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#to-chain","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"A Bayesian network $b$ on nodes $0, \\dots, n - 1$ with values in a nonempty type $\\alpha$ defines a chain of conditional distributions. For $k < n$, the conditional distribution of variable $k$ given values $y_0, \\dots, y_{k-1}$ of the earlier variables is the table of node $k$ applied to the values $y_j$ of its parents $j$. For $k \\ge n$ it is the point mass at a fixed arbitrary element of $\\alpha$.","title":"The chain of a Bayesian network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/joint-2","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.joint","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#joint-2","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $B$ be a Bayesian network on the nodes $0, \\dots, n - 1$ with values in a nonempty type $\\alpha$. The joint distribution of $B$ is the function $P : \\alpha^n \\to \\mathbb{R}$ given by $P(x) = P_n(x)$ for $x \\in \\alpha^n$, where $P_n$ is the joint distribution of the first $n$ variables of the chain of conditional distributions of $B$.","title":"Joint distribution of a Bayesian network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/joint-eq-prod-2","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.joint_eq_prod","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#joint-eq-prod-2","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"The joint is the product over nodes of the probability of the node's value given the values of its parents.","title":"Factorization of a Bayesian network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/sum-joint-2","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.sum_joint","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#sum-joint-2","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $B$ be a Bayesian network on the nodes $0, \\dots, n - 1$ with values in a finite nonempty type $\\alpha$, and let $P$ be its joint distribution. Then $\\sum_{x \\in \\alpha^n} P(x) = 1$.","title":"The joint distribution of a Bayesian network sums to one"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/cond-eq-cpt","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.cond_eq_cpt","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#cond-eq-cpt","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Let $B$ be a Bayesian network on the nodes $0, \\dots, n - 1$ with values in a nonempty type $\\alpha$, and let $P_m$ be the joint distribution of the first $m$ nodes of $B$ viewed as a chain of conditional distributions. For a node $k < n$, write $p_k(\\cdot \\mid z)$ for its conditional probability table at values $z$ of its parents. Let $y = (y_0, \\dots, y_{k-1}) \\in \\alpha^k$ with $P_k(y) \\ne 0$, and write\u2026","title":"The ordered Markov property"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/marginal-coe","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.marginal_coe","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#marginal-coe","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"`FiniteWeight.marginal` on a `Finset` of nodes, as a sum over a decidable filter.","title":"Marginal on a finite set of nodes as a filtered sum"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/cond-indep-coe-iff","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.condIndep_coe_iff","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#cond-indep-coe-iff","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"`FiniteWeight.CondIndep` for `Finset`s of nodes, with the unions taken in `Finset`.","title":"Conditional independence for finite sets of nodes"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/marginal-congr-of-eq-on","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.marginal_congr_of_eqOn","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#marginal-congr-of-eq-on","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"`marginal P S x'` depends only on the values of `x'` on `S`.","title":"A marginal depends only on the fixed variables"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/marginal-le-of-subset","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.marginal_le_of_subset","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#marginal-le-of-subset","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Fixing more variables decreases the probability.","title":"Fixing more variables decreases the marginal"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/marginal-eq-sum-insert","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.marginal_eq_sum_insert","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#marginal-eq-sum-insert","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Summing out one more variable `i \u2209 S`.","title":"Summing out one more variable from a marginal"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/cond-indep-symm","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.condIndep_symm","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#cond-indep-symm","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"Conditional independence is symmetric.","title":"Symmetry of conditional independence"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/cond-indep-mono-left","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.condIndep_mono_left","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#cond-indep-mono-left","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"If `X \u2aeb Y | Z` then `X' \u2aeb Y | Z` for `X' \u2286 X`.","title":"Decomposition"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/cond-indep-weak-union","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.condIndep_weak_union","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#cond-indep-weak-union","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"If `X \u222a W \u2aeb Y | Z` then `X \u2aeb Y | Z \u222a W` (for a nonnegative law).","title":"Weak union"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/is-ancestral","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.IsAncestral","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#is-ancestral","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"A set of nodes is ancestral if it contains the parents of each of its nodes.","title":"Ancestral set of nodes"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/anc-prod","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.ancProd","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#anc-prod","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"The product of the tables of the nodes in `A`, with the uniform distribution on the other nodes. For an ancestral `A` it has the same marginals on `A` as the joint (`marginal_joint_eq_ancProd`).","title":"Product of the tables of an ancestral set"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/anc-prod-univ","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.ancProd_univ","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#anc-prod-univ","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"On all nodes, `ancProd` is the joint.","title":"The ancestral product on all nodes is the joint"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/marginal-joint-eq-anc-prod","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.marginal_joint_eq_ancProd","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#marginal-joint-eq-anc-prod","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"For an ancestral set `A` and `S \u2286 A`, the marginals of the joint on `S` are those of `ancProd A`.","title":"The law of an ancestral set depends only on its tables"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/cond-indep-of-d-separated-finset","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.condIndep_of_dSeparated_finset","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#cond-indep-of-d-separated-finset","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"The global Markov property for `Finset`s of nodes; `condIndep_of_dSeparated` is the statement for arbitrary sets of nodes.","title":"Global Markov property for finite sets of nodes"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/cond-indep-of-d-separated","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.condIndep_of_dSeparated","parent":41345,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks/index.html#cond-indep-of-d-separated","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/bayesian-networks","status":"fully_proved","summary":"The global Markov property (soundness of d-separation: Geiger, Verma & Pearl 1990; Lauritzen, Dawid, Larsen & Leimer 1990): if `Z` d-separates `X` from `Y` in the graph of a Bayesian network, then `X` and `Y` are conditionally independent given `Z` under its joint law. No disjointness of `X`, `Y`, `Z` is needed.","title":"The global Markov property"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","kind":"scope","navigation":"module","parent":41206,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","title":"Causal Bayesian networks"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/pa-val","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.paVal","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#pa-val","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The values of the parents of node `k` in a full assignment `x`.","title":"Parent values of a node"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/intervene","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.intervene","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#intervene","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The mechanism of node `k` is replaced by the point mass at `a`, and all incoming arrows to `k` are deleted (its parent set becomes `\u2205`); every other node keeps its parents and its table. (The new table of `k` is the point mass on the empty parent set; the tables of the other nodes are transported along the equality of their parent sets.)","title":"The intervention `do(X_k = a)`"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/intervene-pa-self","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.intervene_pa_self","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#intervene-pa-self","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The parents of the intervened node are deleted.","title":"Intervention deletes the parents of the intervened node"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/intervene-pa-of-ne","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.intervene_pa_of_ne","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#intervene-pa-of-ne","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The parents of every other node are unchanged by the intervention.","title":"Intervention leaves the parents of other nodes unchanged"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/joint-intervene","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.joint_intervene","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#joint-intervene","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The truncated factorization (Pearl 2009, eq. (3.10)): after `do(X_k = a)` the joint is the product of the tables of all nodes other than `k` on assignments with `x_k = a`, and `0` elsewhere.","title":"The truncated factorization"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/sum-pa-val-intervene","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.sum_paVal_intervene","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#sum-pa-val-intervene","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The intervention `do(X_k = a)` does not change the law of the parents of node `k`.","title":"Intervention preserves the law of the parents"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/sum-intervene-joint-mul","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.sum_intervene_joint_mul","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#sum-intervene-joint-mul","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"Adjustment for the direct causes (Pearl 2009, Theorem 3.2.2; the case `Z = pa(X)` of the back-door adjustment): for every function `f` of the variables, `E[f | do(X_k = a)] = \u2211_z P(pa_k = z) E[f | X_k = a, pa_k = z]`, assuming `P(X_k = a, pa_k = z) > 0` whenever `P(pa_k = z) > 0`.","title":"Adjustment for the direct causes"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/sum-filter-joint-eq-cpt-mul","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.sum_filter_joint_eq_cpt_mul","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#sum-filter-joint-eq-cpt-mul","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The table of node `k` is the conditional law of `X_k` given its parents: `P(X_k = a, pa_k = z) = p_k (a | z) P(pa_k = z)`.","title":"A node's table is its conditional law given its parents"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/sum-intervene-joint-mul-of-cpt-ne-zero","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.sum_intervene_joint_mul_of_cpt_ne_zero","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#sum-intervene-joint-mul-of-cpt-ne-zero","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"Adjustment for the direct causes when the table of node `k` gives the value `a` positive probability at every value of the parents.","title":"Adjustment for the direct causes under table positivity"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/sum-intervene-joint-mul-of-pred","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.sum_intervene_joint_mul_of_pred","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#sum-intervene-joint-mul-of-pred","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"Let `Z` be a set of nodes preceding `k` in the topological order (so `Z` contains no descendant of `k`) that contains the parents of `k`. Then `Z` meets the back-door criterion for every outcome node `y \u2209 Z` (a path from `k` to `y` that starts with an arrow into `k` starts `k \u2190 p` with `p \u2208 pa(k) \u2286 Z`, an interior non-collider, so it is blocked), and for every function `f` of the variables\u2026","title":"Back-door adjustment for a set of predecessors containing the parents"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/sum-filter-restrict-joint-eq-cpt-mul","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.sum_filter_restrict_joint_eq_cpt_mul","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#sum-filter-restrict-joint-eq-cpt-mul","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"For a set `Z` of predecessors of `k` containing its parents, the table of `k` is its conditional law given `Z`: `P(X_k = a, Z = z) = p_k (a | z_{pa k}) P(Z = z)`.","title":"A node's table is its conditional law given a predecessor set"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/sum-intervene-joint-mul-of-pred-of-cpt-n","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.sum_intervene_joint_mul_of_pred_of_cpt_ne_zero","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#sum-intervene-joint-mul-of-pred-of-cpt-n","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"Back-door adjustment for a set of predecessors containing the parents, when the table of node `k` gives the value `a` positive probability at every value of the parents.","title":"Back-door adjustment for a set of predecessors"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/edge","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.Edge","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#edge","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The edge `i \u2192 j` of the network's graph: `i` is a parent of `j`.","title":"Edge of a Bayesian network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/is-descendant","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.IsDescendant","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#is-descendant","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"`j` is a descendant of `i` (every node is its own descendant).","title":"Descendant relation in a Bayesian network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/adj","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.Adj","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#adj","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"`i` and `j` are adjacent in the skeleton of the graph.","title":"Adjacency in the skeleton of a Bayesian network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/is-path","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.IsPath","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#is-path","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"A path of the skeleton from `x` to `y`: a list of distinct nodes, consecutive ones adjacent, starting at `x` and ending at `y`.","title":"Path in the skeleton of a Bayesian network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/blocked-at","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.BlockedAt","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#blocked-at","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The consecutive nodes `u, m, w` of a path are blocked by `Z` (Pearl 2009, Definition 1.2.3): either `m` is a collider `u \u2192 m \u2190 w` none of whose descendants is in `Z`, or `m` is not a collider (a chain or a fork) and `m \u2208 Z`.","title":"Blocking at a node of a path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/path-blocked","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.PathBlocked","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#path-blocked","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"A path is blocked by `Z` if some three consecutive nodes of it are.","title":"Blocked path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/dseparated","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.DSeparated","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#dseparated","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"d-separation (Pearl 1988; Pearl 2009, Definition 1.2.3): `Z` d-separates `X` from `Y` if every path between a node of `X` and a node of `Y` is blocked by `Z`.","title":"d-separation"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/blocked-at-comm","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.blockedAt_comm","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#blocked-at-comm","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"Blocking at a node does not depend on the direction in which the path is read.","title":"Blocking at a node is independent of path direction"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/is-path-reverse","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.isPath_reverse","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#is-path-reverse","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The reverse of a path from `x` to `y` is a path from `y` to `x`.","title":"The reverse of a path is a path"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/path-blocked-reverse","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.pathBlocked_reverse","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#path-blocked-reverse","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"If a path is blocked by `Z`, so is its reverse.","title":"Reversal preserves blocked paths"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/symm","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.DSeparated.symm","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#symm","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"Symmetry of d-separation. d-separation is symmetric.","title":"Symmetry of d-separation"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/mutilate-set","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.mutilateSet","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#mutilate-set","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The network of the intervention `do(X_W = w)` with the arrows into `W` removed: every node `j \u2208 W` has no parents and the point mass at `w j` as its table.","title":"Mutilated network of a set intervention"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/joint-mutilate-set","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.joint_mutilateSet","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#joint-mutilate-set","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"After `do(X_W = w)`, the joint is the product of the tables of the nodes outside `W` on assignments agreeing with `w` on `W`.","title":"The truncated factorization for a set of nodes"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/edge-mutilate-set-iff","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.edge_mutilateSet_iff","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#edge-mutilate-set-iff","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The arrows of `b.mutilateSet W w` are those of `b` not pointing into `W`.","title":"Arrows of the mutilated network"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/of-mutilate-set","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.IsDescendant.of_mutilateSet","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#of-mutilate-set","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"A descendant after removing the arrows into `W` is a descendant in `b`.","title":"Descendants in the mutilated network are descendants in the original"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/eq-of-is-descendant-mutilate-set","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.eq_of_isDescendant_mutilateSet","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#eq-of-is-descendant-mutilate-set","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"After removing the arrows into `W`, a node of `W` is a descendant only of itself.","title":"Intervened nodes descend only from themselves"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/joint-intervene-eq-zero","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.joint_intervene_eq_zero","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#joint-intervene-eq-zero","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"After `do(X_k = a)`, assignments with `x k \u2260 a` have probability `0`.","title":"Assignments contradicting an intervention have probability zero"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/joint-eq-intervene-mul-cpt","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.joint_eq_intervene_mul_cpt","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#joint-eq-intervene-mul-cpt","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"For `x k = a`, `P(x) = P(x | do(X_k = a)) \u00b7 p_k(a | x_{pa k})`.","title":"Joint law as interventional law times the node's table"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/marginal-intervene-of-not-is-descendant","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.marginal_intervene_of_not_isDescendant","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#marginal-intervene-of-not-is-descendant","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The intervention `do(X_k = a)` does not change the law of the non-descendants of `k`.","title":"The intervention `do(X_k = a)` does not change the law of the non-descendants of `k`"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/backdoor-adjustment","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.backdoor_adjustment","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#backdoor-adjustment","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The back-door adjustment (Pearl 1995; Pearl 2009, Theorem 3.3.2): if the set `Z` contains no descendant of the treatment node `k` and blocks every path between `k` and the outcome node `y` that begins with an arrow into `k`, then `P(X_y = v | do(X_k = a)) = \u2211_z P(Z = z) P(X_y = v | X_k = a, Z = z)`, assuming `P(X_k = a, Z = z) > 0` whenever `P(Z = z) > 0`.","title":"The back-door adjustment"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/frontdoor-adjustment","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.BayesNet.frontdoor_adjustment","parent":41376,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks/index.html#frontdoor-adjustment","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/causal-bayesian-networks","status":"fully_proved","summary":"The front-door adjustment (Pearl 1995; Pearl 2009, Theorem 3.3.4): if the set `M` intercepts every directed path from the treatment node `k` to the outcome node `y`, no path from `k` to a node of `M` that begins with an arrow into `k` is open (blocked by `\u2205`), and every path from a node of `M` to `y` that begins with an arrow into that node is blocked by `{k}`, then\u2026","title":"The front-door adjustment"},{"data_url":"scopes/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation.json","id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","kind":"scope","navigation":"module","parent":41206,"result":false,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","title":"Variable elimination and belief propagation"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/tree-model","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#tree-model","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"For a type $\\alpha$ of values and $n \\in \\mathbb{N}$, a pairwise tree model on the vertices $0, \\dots, n$ consists of a parent $p(k + 1) \\in \\{0, \\dots, k\\}$ for each vertex $k + 1$ with $k < n$ (so vertex $0$ is the root and every vertex comes after its parent), a unary factor $\\varphi_i : \\alpha \\to \\mathbb{R}$ for each vertex $i \\le n$, and for each $k < n$ a pairwise factor $\\psi_k(a, b) \\in \\mathbb{R}$ on the\u2026","title":"Pairwise model on a tree"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/parent","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.parent","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#parent","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"In a pairwise tree model on the vertices $0, \\dots, n$, for $k < n$ the parent of vertex $k + 1$ is the parent recorded by the model, an element of $\\{0, \\dots, k\\}$, regarded as one of the vertices $0, \\dots, n$.","title":"Parent of a vertex"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/weight","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.weight","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#weight","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $\\alpha$ be a type and let $m$ be a pairwise model on a tree with vertices $0, \\dots, n$, with unary factors $\\varphi_i : \\alpha \\to \\mathbb{R}$ for each vertex $i$, and with factor $\\psi_k : \\alpha \\times \\alpha \\to \\mathbb{R}$ on the edge between vertex $k + 1$ and its parent $p(k + 1)$ for each $k < n$. The unnormalized weight of an assignment $x : \\{0, \\dots, n\\} \\to \\alpha$ is the real number","title":"Weight of an assignment in a tree model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/last-parent","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.lastParent","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#last-parent","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"In a pairwise tree model on the vertices $0, \\dots, n + 1$, this is the parent of the last vertex $n + 1$, regarded as one of the vertices $0, \\dots, n$ of the tree with the last vertex removed.","title":"Parent of the last vertex"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/message","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.message","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#message","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $\\alpha$ be finite and consider a pairwise tree model on the vertices $0, \\dots, n + 1$ with unary factors $\\varphi_i$ and pairwise factors $\\psi_k$. The message from the last vertex $n + 1$ to its parent is the function $\\mu : \\alpha \\to \\mathbb{R}$ given by $\\mu(a) = \\sum_{b \\in \\alpha} \\psi_n(b, a) \\, \\varphi_{n+1}(b)$, where $\\psi_n$ is the factor of the edge between vertex $n + 1$ and its parent.","title":"Message from the last vertex"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/eliminate","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.eliminate","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#eliminate","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $\\alpha$ be finite, let $m$ be a pairwise tree model on the vertices $0, \\dots, n + 1$ with parents $p$, unary factors $\\varphi_i$ and pairwise factors $\\psi_k$, and let $\\mu$ be the message from the last vertex $n + 1$ to its parent $q = p(n + 1)$. Eliminating the last vertex gives the pairwise tree model on the vertices $0, \\dots, n$ in which vertex $k + 1$ has parent $p(k + 1)$ and edge factor $\\psi_k$ for\u2026","title":"Eliminating the last vertex of a tree model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/sum-weight-mul-eliminate","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.sum_weight_mul_eliminate","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#sum-weight-mul-eliminate","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $\\alpha$ be finite, let $m$ be a pairwise tree model on the vertices $0, \\dots, n + 1$ with weight $w$, and let $m'$ be the model on the vertices $0, \\dots, n$ obtained from $m$ by eliminating the last vertex, with weight $w'$. Then for every function $g : \\alpha \\to \\mathbb{R}$ we have $\\sum_x w(x) \\, g(x_0) = \\sum_y w'(y) \\, g(y_0)$, where $x$ ranges over all assignments $\\{0, \\dots, n + 1\\} \\to \\alpha$ and\u2026","title":"One step of variable elimination"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/root-message","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.rootMessage","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#root-message","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $\\alpha$ be finite. The root message of a pairwise tree model $m$ on the vertices $0, \\dots, n$ is the function $r_m : \\alpha \\to \\mathbb{R}$ defined by recursion on $n$: if $n = 0$, it is the unary factor of the root $0$; if $m$ is a model on the vertices $0, \\dots, n + 1$, it is $r_{m'}$, where $m'$ is the model on the vertices $0, \\dots, n$ obtained from $m$ by eliminating the last vertex.","title":"Root message of a tree model"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/sum-weight-mul-eq","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.sum_weight_mul_eq","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#sum-weight-mul-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $\\alpha$ be finite and let $m$ be a pairwise tree model on the vertices $0, \\dots, n$ with weight $w$ and root message $r : \\alpha \\to \\mathbb{R}$. Then for every function $g : \\alpha \\to \\mathbb{R}$ of the root value, $\\sum_x w(x) \\, g(x_0) = \\sum_{a \\in \\alpha} r(a) \\, g(a)$, where $x$ ranges over all assignments $\\{0, \\dots, n\\} \\to \\alpha$.","title":"Variable elimination is exact"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/partition-eq","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.partition_eq","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#partition-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $\\alpha$ be finite and let $m$ be a pairwise tree model on the vertices $0, \\dots, n$ with weight $w$ and root message $r : \\alpha \\to \\mathbb{R}$. Then the partition function of $m$ is the total root message: $\\sum_x w(x) = \\sum_{a \\in \\alpha} r(a)$, where $x$ ranges over all assignments $\\{0, \\dots, n\\} \\to \\alpha$.","title":"Variable elimination computes the partition function"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/root-marginal-eq","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.rootMarginal_eq","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#root-marginal-eq","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $\\alpha$ be finite and let $m$ be a pairwise tree model on the vertices $0, \\dots, n$ with weight $w$ and root message $r : \\alpha \\to \\mathbb{R}$. Then for every $a \\in \\alpha$ the unnormalized marginal of the root at $a$ equals $r(a)$, that is, $\\sum_x w(x) = r(a)$, where $x$ ranges over the assignments $\\{0, \\dots, n\\} \\to \\alpha$ with $x_0 = a$.","title":"Variable elimination computes the unnormalized root marginal"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/drop-last","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.dropLast","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#drop-last","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"For a tree model with vertices $0, \\dots, n + 1$, the tree model on $0, \\dots, n$ with the same parents, unary factors and edge factors: the last vertex, which is a leaf, is removed together with its unary factor and the factor of its edge.","title":"Removing the last vertex"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/sum-weight-mul-eliminate-cast-succ","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.sum_weight_mul_eliminate_castSucc","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#sum-weight-mul-eliminate-cast-succ","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $m$ be a pairwise tree model on the vertices $0, \\dots, n + 1$ with values in a finite type $\\alpha$, and let $m'$ be the model on the vertices $0, \\dots, n$ obtained from $m$ by summing out its last vertex. Write $w_m$ and $w_{m'}$ for their unnormalized weights. Then for every vertex $v \\le n$ and every function $g : \\alpha \\to \\mathbb{R}$,","title":"Summing out the last vertex preserves weighted sums at the other vertices"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/belief","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.belief","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#belief","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"The belief of each vertex of a tree model over a finite set of values, computed by sum-product message passing: the product of its unary factor, the messages from its subtree and the message from its parent. Recursively, in a model with a single vertex the belief of that vertex is its unary factor; in a model with vertices $0, \\dots, n + 1$ the belief of the last vertex at a value $b$ is its unary factor at $b$\u2026","title":"Sum-product belief"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/down-message","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.downMessage","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#down-message","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"For a tree model with vertices $0, \\dots, n + 1$ over a finite set of values, the message from the parent $p$ of the last vertex to the last vertex at a value $b$ is the sum over all values $a$ of $\\psi(b, a)\\, \\beta(a)$, where $\\psi$ is the factor of the edge between the last vertex and $p$ (last vertex first) and $\\beta(a)$ is the belief of $p$ at $a$ in the model without the last vertex.","title":"Downward message to the last vertex"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/sum-weight-mul-eq-belief","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.sum_weight_mul_eq_belief","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#sum-weight-mul-eq-belief","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $m$ be a pairwise tree model on the vertices $0, \\dots, n$ with values in a finite type $\\alpha$, let $w$ be its unnormalized weight, and let $b_v$ be the belief of vertex $v$ computed by sum-product message passing. Then for every vertex $v$ and every function $g : \\alpha \\to \\mathbb{R}$,","title":"Belief propagation is exact on trees"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/marginal-eq-belief","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.marginal_eq_belief","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#marginal-eq-belief","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"For every tree model over a finite set of values, every vertex $v$ and every value $a$, the belief of $v$ at $a$ is the unnormalized marginal of $v$ at $a$: the sum of the weights of all assignments $x$ with $x_v = a$.","title":"Beliefs are the unnormalized marginals"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/belief-zero","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.belief_zero","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#belief-zero","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"For every tree model over a finite set of values, the belief of the root, vertex $0$, is the root message computed by variable elimination: the two agree at every value.","title":"Belief at the root is the root message"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/nonneg-factors","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.NonnegFactors","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#nonneg-factors","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"A tree model has nonnegative factors when all its unary factors and all its edge factors take only nonnegative values.","title":"Nonnegative factors"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/max-message","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.maxMessage","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#max-message","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"For a tree model with vertices $0, \\dots, n + 1$ over a finite nonempty set of values, the max-product message from the last vertex to its parent at a value $a$ is the maximum over all values $b$ of $\\psi(b, a)\\, \\phi(b)$, where $\\psi$ is the factor of the edge between the last vertex and its parent and $\\phi$ is the unary factor of the last vertex.","title":"Max-product message"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/max-eliminate","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.maxEliminate","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#max-eliminate","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"For a tree model with vertices $0, \\dots, n + 1$ over a finite nonempty set of values, the tree model on $0, \\dots, n$ with the same parents and edge factors whose unary factor at the parent of the last vertex is multiplied by the max-product message of the last vertex; the other unary factors are unchanged.","title":"Maximizing out the last vertex"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/max-weight","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.maxWeight","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#max-weight","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"For a tree model over a finite nonempty set of values, the value computed by maximizing out the vertices from the last one down: for a single vertex it is the maximum of its unary factor over all values, and for more vertices it is the max-product value of the model with the last vertex maximized out.","title":"Max-product value"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/decode","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.decode","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#decode","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"An assignment of a tree model over a finite nonempty set of values, defined by recursion on the number of vertices. For a single vertex, its value is one that maximizes its unary factor. For vertices $0, \\dots, n + 1$, the vertices $0, \\dots, n$ get the decoding of the model with the last vertex maximized out, and the last vertex gets a value $b$ maximizing $\\psi(b, a)\\, \\phi(b)$, where $a$ is the decoded value of\u2026","title":"Max-product decoding"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/weight-le-max-weight","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.weight_le_maxWeight","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#weight-le-max-weight","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $m$ be a pairwise tree model with values in a finite nonempty type $\\alpha$ whose unary and pairwise factors are all nonnegative, and let $M$ be its max-product value, obtained by maximizing out the vertices from the last one down. Then every assignment of values in $\\alpha$ to the vertices has weight at most $M$.","title":"The max-product value bounds every weight"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/weight-decode","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.weight_decode","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#weight-decode","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $m$ be a pairwise tree model with values in a finite nonempty type, and let $M$ be its max-product value, obtained by maximizing out the vertices from the last one down. Then the weight of the assignment produced by max-product decoding equals $M$.","title":"Max-product decoding attains the max-product value"},{"id":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/weight-le-weight-decode","kind":"node","lean":"Cslib.MachineLearning.GraphicalModels.TreeModel.weight_le_weight_decode","parent":41410,"result":true,"route":"../roadmap/ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation/index.html#weight-le-weight-decode","scope":"ComputerScience/machine-learning/probabilistic-graphical-models/variable-elimination-and-belief-propagation","status":"fully_proved","summary":"Let $m$ be a pairwise tree model with values in a finite nonempty type $\\alpha$ whose unary and pairwise factors are all nonnegative. Then every assignment of values in $\\alpha$ to the vertices has weight at most the weight of the assignment produced by max-product decoding.","title":"Max-product decoding finds a MAP assignment"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling","kind":"scope","parent":40911,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling","status":"fully_proved","summary":"Bayes' rule and conjugate models, expectation-maximization for mixtures and latent-variable models, variational mean-field inference, and Monte Carlo, rejection, Gibbs and Metropolis-Hastings sampling.","title":"Bayesian inference and sampling"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","title":"Bayes' rule"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/bayes-posterior","kind":"node","lean":"Cslib.MachineLearning.Bayesian.bayesPosterior","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#bayes-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\kappa$ be a kernel from a measurable space $\\Theta$ of hypotheses to a measurable space $\\mathcal X$ of observations (the likelihood) and $\\mu$ a measure on $\\Theta$ (the prior). The Bayes posterior after observing $x \\in \\mathcal X$ is the measure $\\pi_x$ on $\\Theta$ with","title":"Bayes posterior"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/comp-apply-singleton","kind":"node","lean":"Cslib.MachineLearning.Bayesian.comp_apply_singleton","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#comp-apply-singleton","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\kappa$ be a kernel from a measurable space $\\Theta$ to a measurable space $\\mathcal X$ with measurable singletons, and $\\mu$ a measure on $\\Theta$. For every $x \\in \\mathcal X$,","title":"The evidence is the integral of the likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/comp-mul-bayes-posterior","kind":"node","lean":"Cslib.MachineLearning.Bayesian.comp_mul_bayesPosterior","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#comp-mul-bayes-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\mathcal X$ be a measurable space with measurable singletons, $\\kappa$ a Markov kernel from a measurable space $\\Theta$ to $\\mathcal X$ and $\\mu$ a finite measure on $\\Theta$. Let $x \\in \\mathcal X$ and write $\\pi_x$ for the Bayes posterior of $\\mu$ after observing $x$ under $\\kappa$. Then for every measurable set $s \\subseteq \\Theta$,","title":"Evidence times posterior is the joint mass"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/is-mle","kind":"node","lean":"Cslib.MachineLearning.Bayesian.IsMLE","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#is-mle","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\kappa$ be a kernel from a parameter space $\\Theta$ to a countable observation space $\\mathcal X$ with measurable singletons, assigning to each hypothesis $\\theta$ a measure $\\kappa_\\theta$ on $\\mathcal X$. A hypothesis $\\theta \\in \\Theta$ is a maximum-likelihood hypothesis for the observation $x \\in \\mathcal X$ when no hypothesis gives $x$ higher likelihood: $\\kappa_{\\theta'}(\\{x\\}) \\le \\kappa_\\theta(\\{x\\})$\u2026","title":"Maximum-likelihood hypothesis for a discrete observation"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/is-map","kind":"node","lean":"Cslib.MachineLearning.Bayesian.IsMAP","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#is-map","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\Theta$ and $\\mathcal X$ be countable measurable spaces with measurable singletons, $\\kappa$ a kernel from $\\Theta$ to $\\mathcal X$ and $\\mu$ a measure on $\\Theta$, and write $\\pi_x$ for the Bayes posterior of the prior $\\mu$ after observing $x \\in \\mathcal X$ under the likelihood $\\kappa$. A hypothesis $\\theta \\in \\Theta$ is a maximum a posteriori (MAP) hypothesis for $x$ when\u2026","title":"Maximum a posteriori hypothesis"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/bayes-kernel","kind":"node","lean":"Cslib.MachineLearning.Bayesian.bayesKernel","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#bayes-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\mathcal X$ be a countable measurable space with measurable singletons, $\\kappa$ a kernel from a measurable space $\\Theta$ to $\\mathcal X$, and $\\mu$ a measure on $\\Theta$. The Bayes posterior kernel is the kernel from $\\mathcal X$ to $\\Theta$ that sends each observation $x \\in \\mathcal X$ to the Bayes posterior of the prior $\\mu$ after observing $x$ under the likelihood $\\kappa$.","title":"Bayes posterior kernel"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/posterior-eq-bayes-posterior","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_eq_bayesPosterior","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#posterior-eq-bayes-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let the parameter space $\\Theta$ be a nonempty standard Borel space and the observation space $\\mathcal X$ be countable with measurable singletons. Let $\\kappa$ be a Markov kernel from $\\Theta$ to $\\mathcal X$ (the likelihood), writing $\\kappa_\\theta$ for the probability measure it assigns to $\\theta \\in \\Theta$, let $\\mu$ be a finite measure on $\\Theta$ (the prior), and let $\\kappa^\\dagger\\mu$ be Mathlib's\u2026","title":"Bayes' rule for discrete observations"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/comp-apply-singleton-eq-tsum","kind":"node","lean":"Cslib.MachineLearning.Bayesian.comp_apply_singleton_eq_tsum","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#comp-apply-singleton-eq-tsum","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\Theta$ be a countable measurable space and $\\mathcal X$ a measurable space, both with measurable singletons, let $\\kappa$ be a kernel from $\\Theta$ to $\\mathcal X$ and $\\mu$ a measure on $\\Theta$. For every $x \\in \\mathcal X$,","title":"The evidence as a sum over hypotheses"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/bayes-posterior-apply-singleton","kind":"node","lean":"Cslib.MachineLearning.Bayesian.bayesPosterior_apply_singleton","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#bayes-posterior-apply-singleton","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\Theta$ be a countable measurable space and $\\mathcal X$ a measurable space, both with measurable singletons, let $\\kappa$ be a kernel from $\\Theta$ to $\\mathcal X$ (the likelihood) and $\\mu$ a measure on $\\Theta$ (the prior). For every observation $x \\in \\mathcal X$ and every $\\theta \\in \\Theta$, the Bayes posterior $\\pi_x$ of $\\mu$ after observing $x$ satisfies","title":"Bayes' rule for countably many hypotheses"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/posterior-apply-singleton","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_apply_singleton","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#posterior-apply-singleton","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Bayes' rule for Mathlib's posterior kernel with countably many hypotheses and observations. Let the parameter space $\\Theta$ be a nonempty countable standard Borel space and the observation space $\\mathcal X$ be countable, both with measurable singletons. Let $\\kappa$ be a Markov kernel from $\\Theta$ to $\\mathcal X$ (the likelihood), writing $\\kappa_\\theta$ for the probability measure it assigns to\u2026","title":"Bayes' rule for Mathlib's posterior kernel with countably many hypotheses and observations"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/is-map-iff","kind":"node","lean":"Cslib.MachineLearning.Bayesian.isMAP_iff","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#is-map-iff","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\Theta$ and $\\mathcal X$ be countable measurable spaces with measurable singletons, $\\kappa$ a Markov kernel from $\\Theta$ to $\\mathcal X$, $\\mu$ a finite measure on $\\Theta$, and $x \\in \\mathcal X$ an observation with $(\\kappa \\circ \\mu)(\\{x\\}) \\neq 0$, where $\\kappa \\circ \\mu$ is the composition of $\\mu$ with $\\kappa$. Then $\\theta \\in \\Theta$ is a MAP hypothesis for $x$ under the prior $\\mu$ and the\u2026","title":"MAP hypotheses maximize prior times likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/is-map-iff-is-mle-of-uniform","kind":"node","lean":"Cslib.MachineLearning.Bayesian.isMAP_iff_isMLE_of_uniform","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#is-map-iff-is-mle-of-uniform","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\Theta$ and $\\mathcal X$ be countable measurable spaces with measurable singletons, $\\kappa$ a Markov kernel from $\\Theta$ to $\\mathcal X$, and $\\mu$ a finite measure on $\\Theta$ that is uniform: there is $c \\in [0, \\infty]$, $c \\neq 0$, with $\\mu(\\{\\theta\\}) = c$ for every $\\theta \\in \\Theta$. Let $x \\in \\mathcal X$ be an observation with $(\\kappa \\circ \\mu)(\\{x\\}) \\neq 0$, where $\\kappa \\circ \\mu$ is the\u2026","title":"Under a uniform prior, MAP is maximum likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/is-map-iff-posterior","kind":"node","lean":"Cslib.MachineLearning.Bayesian.isMAP_iff_posterior","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#is-map-iff-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\Theta$ be a nonempty countable standard Borel space, $\\mathcal X$ a countable measurable space with measurable singletons, $\\kappa$ a Markov kernel from $\\Theta$ to $\\mathcal X$, $\\mu$ a finite measure on $\\Theta$, and $x \\in \\mathcal X$ an observation with $(\\kappa \\circ \\mu)(\\{x\\}) \\neq 0$, where $\\kappa \\circ \\mu$ is the composition of $\\mu$ with $\\kappa$. Write $\\kappa^\\dagger_\\mu$ for the posterior\u2026","title":"MAP hypotheses through the posterior kernel"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/comp-eq-with-density-of-mul-eq","kind":"node","lean":"Cslib.MachineLearning.Bayesian.comp_eq_withDensity_of_mul_eq","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#comp-eq-with-density-of-mul-eq","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\nu$ be an s-finite measure on a measurable space $\\Theta$ and $\\rho$ an s-finite measure on a measurable space $\\mathcal{X}$. Let $p : \\Theta \\to [0, \\infty]$, $q : \\Theta \\times \\mathcal{X} \\to [0, \\infty]$, $c : \\mathcal{X} \\to [0, \\infty]$ and $r : \\mathcal{X} \\times \\Theta \\to [0, \\infty]$ be measurable functions such that $\\int r(x, \\theta) \\, d\\nu(\\theta) = 1$ for every $x \\in \\mathcal{X}$ and\u2026","title":"The evidence from a factorization of densities"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/comp-prod-eq-of-mul-eq","kind":"node","lean":"Cslib.MachineLearning.Bayesian.compProd_eq_of_mul_eq","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#comp-prod-eq-of-mul-eq","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\nu$ be an s-finite measure on a measurable space $\\Theta$ and $\\rho$ an s-finite measure on a measurable space $\\mathcal{X}$. Let $p : \\Theta \\to [0, \\infty]$, $q : \\Theta \\times \\mathcal{X} \\to [0, \\infty]$, $c : \\mathcal{X} \\to [0, \\infty]$ and $r : \\mathcal{X} \\times \\Theta \\to [0, \\infty]$ be measurable functions such that $\\int r(x, \\theta) \\, d\\nu(\\theta) = 1$ for every $x \\in \\mathcal{X}$ and\u2026","title":"Disintegration of the joint law from a factorization of densities"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/posterior-ae-eq-of-mul-eq","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_ae_eq_of_mul_eq","parent":41438,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule/index.html#posterior-ae-eq-of-mul-eq","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayes-rule","status":"fully_proved","summary":"Let $\\Theta$ be a nonempty standard Borel space, $\\nu$ an s-finite measure on $\\Theta$ and $\\rho$ an s-finite measure on a measurable space $\\mathcal{X}$. Let $p : \\Theta \\to [0, \\infty]$, $q : \\Theta \\times \\mathcal{X} \\to [0, \\infty]$, $c : \\mathcal{X} \\to [0, \\infty]$ and $r : \\mathcal{X} \\times \\Theta \\to [0, \\infty]$ be measurable functions such that $\\int r(x, \\theta) \\, d\\nu(\\theta) = 1$ for every\u2026","title":"Bayes' rule for densities"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","title":"Conjugate priors"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/binomial-kernel","kind":"node","lean":"Cslib.MachineLearning.Bayesian.binomialKernel","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#binomial-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For a natural number $n$, the binomial kernel sends a real $\\theta$ to the binomial distribution $\\mathrm{Bin}(n, p)$ on $\\mathbb{N}$, the law of the number of successes in $n$ independent coin flips with success probability $p$, where $p$ is $\\theta$ clipped to $[0, 1]$.","title":"Binomial likelihood kernel"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/beta-add-one-left","kind":"node","lean":"Cslib.MachineLearning.Bayesian.beta_add_one_left","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#beta-add-one-left","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For real $a > 0$ and $b > 0$, the beta function $B(a, b) = \\Gamma(a)\\Gamma(b)/\\Gamma(a + b)$ satisfies $B(a + 1, b) = B(a, b) \\cdot \\frac{a}{a + b}$.","title":"Recurrence for the beta function in the first argument"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/beta-pdfreal-mul-pow","kind":"node","lean":"Cslib.MachineLearning.Bayesian.betaPDFReal_mul_pow","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#beta-pdfreal-mul-pow","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $a, b > 0$ be real, $k, m$ natural numbers and $\\theta$ real. Write $f_{a,b}$ for the $\\mathrm{Beta}(a, b)$ density (zero outside $(0, 1)$) and $B(a, b) = \\Gamma(a)\\Gamma(b)/\\Gamma(a + b)$ for the beta function. Then $f_{a,b}(\\theta) \\, \\theta^k (1 - \\theta)^m = \\frac{B(a + k, b + m)}{B(a, b)} f_{a + k, b + m}(\\theta)$.","title":"Conjugacy of the beta density"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/beta-pdf-mul-binomial-kernel","kind":"node","lean":"Cslib.MachineLearning.Bayesian.betaPDF_mul_binomialKernel","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#beta-pdf-mul-binomial-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $a, b > 0$ be real, $n, k$ natural numbers, $m = n - k$ if $k \\le n$ and $m = 0$ otherwise, and $\\theta$ real. Write $f_{a,b}$ for the $\\mathrm{Beta}(a, b)$ density (zero outside $(0, 1)$) and $B$ for the beta function. Then the density $f_{a,b}(\\theta)$ times the probability that the binomial kernel at $\\theta$ gives exactly $k$ successes in $n$ trials equals\u2026","title":"Beta prior times binomial likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/binomial-kernel-comp-beta-measure","kind":"node","lean":"Cslib.MachineLearning.Bayesian.binomialKernel_comp_betaMeasure","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#binomial-kernel-comp-beta-measure","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $a, b > 0$ be real, $n, k$ natural numbers and $B(x, y) = \\Gamma(x)\\Gamma(y)/\\Gamma(x + y)$ the beta function. Under a $\\mathrm{Beta}(a, b)$ prior on the success probability, the marginal probability (the evidence) that the binomial kernel gives exactly $k$ successes in $n$ trials is $\\binom{n}{k} \\frac{B(a + k, b + n - k)}{B(a, b)}$, where $n - k$ is truncated at $0$.","title":"The beta-binomial distribution"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/posterior-binomial-kernel-beta-measure","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_binomialKernel_betaMeasure","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#posterior-binomial-kernel-beta-measure","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $a, b > 0$ be real and $k \\le n$ natural numbers. For the binomial kernel and a $\\mathrm{Beta}(a, b)$ prior, the posterior distribution (Mathlib's posterior kernel) after observing $k$ successes in $n$ trials is $\\mathrm{Beta}(a + k, b + n - k)$.","title":"Beta\u2013binomial conjugacy"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/binomial-kernel-one-comp-beta-measure","kind":"node","lean":"Cslib.MachineLearning.Bayesian.binomialKernel_one_comp_betaMeasure","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#binomial-kernel-one-comp-beta-measure","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $a, b > 0$ be real. Under a $\\mathrm{Beta}(a, b)$ prior on the success probability, the probability that a single trial of the binomial kernel succeeds is the prior mean $a / (a + b)$.","title":"Prior predictive probability of a success"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/posterior-predictive-beta-measure","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_predictive_betaMeasure","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#posterior-predictive-beta-measure","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $a, b > 0$ be real and $k \\le n$ natural numbers. Under a $\\mathrm{Beta}(a, b)$ prior on the success probability, after observing $k$ successes in $n$ trials of the binomial kernel, the posterior predictive probability that one further trial succeeds is $(a + k) / (a + b + n)$.","title":"Posterior predictive of the beta-binomial model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/prob-vectors","kind":"node","lean":"Cslib.MachineLearning.Bayesian.probVectors","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#prob-vectors","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the probability vectors are the $\\theta \\in \\mathbb{R}^{n+1}$ with $\\theta_i \\ge 0$ for every $i$ and $\\sum_i \\theta_i = 1$; they form the parameter space of the categorical model.","title":"Probability vectors"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/categorical-kernel","kind":"node","lean":"Cslib.MachineLearning.Bayesian.categoricalKernel","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#categorical-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the categorical likelihood is the kernel from parameters $\\theta \\in \\mathbb{R}^{n+1}$ to labels $\\{0, \\dots, n\\}$ that sends a probability vector $\\theta$ (all $\\theta_i \\ge 0$ and $\\sum_i \\theta_i = 1$) to the law of one label drawn with probability $\\theta_i$ for label $i$, and sends every other $\\theta$ to the point mass at label $0$.","title":"The categorical likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/seq-likelihood","kind":"node","lean":"Cslib.MachineLearning.Bayesian.seqLikelihood","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#seq-likelihood","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For $\\theta \\in \\mathbb{R}^{n+1}$ and a sequence $s = (s_0, \\dots, s_{N-1})$ of labels in $\\{0, \\dots, n\\}$, the likelihood of $s$ is the extended nonnegative real $\\prod_{m < N} \\theta_{s_m}^+$, where $x^+ = \\max(x, 0)$.","title":"Likelihood of a label sequence"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/categorical-seq-kernel","kind":"node","lean":"Cslib.MachineLearning.Bayesian.categoricalSeqKernel","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#categorical-seq-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For $n, N \\in \\mathbb{N}$, the multinomial sampling model is the kernel from parameters $\\theta \\in \\mathbb{R}^{n+1}$ to sequences $s = (s_0, \\dots, s_{N-1})$ of $N$ labels in $\\{0, \\dots, n\\}$ that sends a probability vector $\\theta$ (all $\\theta_i \\ge 0$ and $\\sum_i \\theta_i = 1$) to the law of $N$ independent draws from $\\theta$, which gives $s$ probability $\\prod_{m < N} \\theta_{s_m}$, and sends every other\u2026","title":"The multinomial sampling model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/counts","kind":"node","lean":"Cslib.MachineLearning.Bayesian.counts","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#counts","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For a sequence $s = (s_0, \\dots, s_{N-1})$ of labels in $\\{0, \\dots, n\\}$ and a label $i$, the count $n_i(s)$ is the number of positions $m < N$ with $s_m = i$.","title":"Label counts"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/seq-likelihood-eq-prod-pow","kind":"node","lean":"Cslib.MachineLearning.Bayesian.seqLikelihood_eq_prod_pow","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#seq-likelihood-eq-prod-pow","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For every $\\theta \\in \\mathbb{R}^{n+1}$ and every sequence $s$ of $N$ labels in $\\{0, \\dots, n\\}$ with label counts $n_i(s)$, the likelihood $\\prod_{m < N} \\theta_{s_m}^+$ of $s$ equals $\\prod_{i = 0}^{n} (\\theta_i^+)^{n_i(s)}$ as extended nonnegative reals, where $x^+ = \\max(x, 0)$.","title":"The sequence likelihood depends only on the label counts"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/with-density-categorical-seq-kernel","kind":"node","lean":"Cslib.MachineLearning.Bayesian.withDensity_categoricalSeqKernel","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#with-density-categorical-seq-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $\\alpha \\in \\mathbb{R}^{n+1}$ with $\\alpha_i > 0$ for every $i$, and let $s$ be a sequence of $N$ labels in $\\{0, \\dots, n\\}$ with label counts $n_i(s)$. The Dirichlet distribution $\\mathrm{Dir}(\\alpha)$ weighted by the density $\\theta \\mapsto P_\\theta(s)$, the probability of $s$ under the multinomial sampling model with parameter $\\theta$, equals $B(\\alpha + n(s)) / B(\\alpha)$ times\u2026","title":"Evidence-weighted Dirichlet prior"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/posterior-categorical-seq-kernel-dirichl","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_categoricalSeqKernel_dirichletMeasure","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#posterior-categorical-seq-kernel-dirichl","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $\\alpha \\in \\mathbb{R}^{n+1}$ with $\\alpha_i > 0$ for every $i$, and let $s$ be a sequence of $N$ labels in $\\{0, \\dots, n\\}$ with label counts $n_i(s)$. Under the multinomial sampling model with the Dirichlet prior $\\mathrm{Dir}(\\alpha)$, the posterior distribution of the parameter after observing $s$ is the Dirichlet distribution with parameters $\\alpha_i + n_i(s)$.","title":"Dirichlet\u2013multinomial conjugacy"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/categorical-kernel-comp-dirichlet-measur","kind":"node","lean":"Cslib.MachineLearning.Bayesian.categoricalKernel_comp_dirichletMeasure","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#categorical-kernel-comp-dirichlet-measur","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $\\alpha \\in \\mathbb{R}^{n+1}$ with $\\alpha_i > 0$ for every $i$. Drawing a parameter $\\theta$ from the Dirichlet distribution $\\mathrm{Dir}(\\alpha)$ and then one label from the categorical likelihood with parameter $\\theta$ gives the label $i \\in \\{0, \\dots, n\\}$ with probability $\\alpha_i / \\sum_j \\alpha_j$, which is the mean of $\\theta_i$ under $\\mathrm{Dir}(\\alpha)$.","title":"Prior predictive of the Dirichlet\u2013categorical model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/posterior-predictive-dirichlet-measure","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_predictive_dirichletMeasure","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#posterior-predictive-dirichlet-measure","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $\\alpha \\in \\mathbb{R}^{n+1}$ with $\\alpha_i > 0$ for every $i$, and let $s$ be a sequence of $N$ labels in $\\{0, \\dots, n\\}$ with label counts $n_i(s)$. Under the multinomial sampling model with the Dirichlet prior $\\mathrm{Dir}(\\alpha)$, drawing a parameter from the posterior after observing $s$ and then one label from the categorical likelihood gives the label $i$ with probability\u2026","title":"Posterior predictive of the Dirichlet\u2013multinomial model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/normal-post-var","kind":"node","lean":"Cslib.MachineLearning.Bayesian.normalPostVar","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#normal-post-var","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For variances $v_0, v \\ge 0$, the posterior variance is $\\frac{v_0 v}{v_0 + v}$, taken to be $0$ when $v_0 = v = 0$. In the normal\u2013normal model with prior variance $v_0$ and known noise variance $v$, this is the variance of the posterior of the mean after one observation.","title":"Posterior variance of the normal\u2013normal model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/normal-post-mean","kind":"node","lean":"Cslib.MachineLearning.Bayesian.normalPostMean","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#normal-post-mean","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For a prior mean $m_0 \\in \\mathbb{R}$, a prior variance $v_0 \\ge 0$, a noise variance $v \\ge 0$ and an observation $x \\in \\mathbb{R}$, the posterior mean is $(v m_0 + v_0 x)/(v_0 + v)$, taken to be $0$ when $v_0 = v = 0$. When $v_0 + v > 0$ it is the precision-weighted average of $m_0$ and $x$.","title":"Posterior mean of the Normal\u2013Normal model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/normal-post-mean-eq","kind":"node","lean":"Cslib.MachineLearning.Bayesian.normalPostMean_eq","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#normal-post-mean-eq","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $m_0, x \\in \\mathbb{R}$, $v_0 > 0$ and $v \\ge 0$. The posterior mean $(v m_0 + v_0 x)/(v_0 + v)$ equals $m_0 + \\frac{v_0}{v_0 + v}(x - m_0)$: it moves from the prior mean $m_0$ toward the observation $x$ by the gain $v_0/(v_0 + v)$.","title":"Posterior mean as a gain update"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/inv-normal-post-var","kind":"node","lean":"Cslib.MachineLearning.Bayesian.inv_normalPostVar","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#inv-normal-post-var","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For variances $v_0, v > 0$, the posterior variance $s = v_0 v/(v_0 + v)$ satisfies $s^{-1} = v_0^{-1} + v^{-1}$.","title":"Precisions add"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/gaussian-pdfreal-mul-gaussian-pdfreal","kind":"node","lean":"Cslib.MachineLearning.Bayesian.gaussianPDFReal_mul_gaussianPDFReal","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#gaussian-pdfreal-mul-gaussian-pdfreal","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $m_0, \\theta, x \\in \\mathbb{R}$ and $v_0, v > 0$, and write $\\varphi_{\\mu, s}(y) = (2 \\pi s)^{-1/2} \\exp(-(y - \\mu)^2 / (2 s))$ for the density of the Gaussian with mean $\\mu$ and variance $s$. With $\\mu(x) = (v m_0 + v_0 x)/(v_0 + v)$ and $s = v_0 v/(v_0 + v)$,","title":"Completing the square"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/measurable-gaussian-pdf-mean","kind":"node","lean":"Cslib.MachineLearning.Bayesian.measurable_gaussianPDF_mean","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#measurable-gaussian-pdf-mean","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For a variance $v \\ge 0$, write $\\varphi_{\\theta, v}(x) = (2 \\pi v)^{-1/2} \\exp(-(x - \\theta)^2 / (2 v))$ for the density at $x$ of the Gaussian with mean $\\theta$ and variance $v$, read as $0$ when $v = 0$, with values in $[0, \\infty]$. Then $(\\theta, x) \\mapsto \\varphi_{\\theta, v}(x)$ is a measurable map on $\\mathbb{R} \\times \\mathbb{R}$.","title":"Measurability of the Gaussian density in its mean and argument"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/gaussian-likelihood","kind":"node","lean":"Cslib.MachineLearning.Bayesian.gaussianLikelihood","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#gaussian-likelihood","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"For a noise variance $v \\ge 0$, the Gaussian likelihood is the kernel from $\\mathbb{R}$ to $\\mathbb{R}$ that sends a parameter $\\theta$ to the law of the observation $x \\sim N(\\theta, v)$, the Gaussian with mean $\\theta$ and variance $v$: for $v > 0$ it is the measure with the $N(\\theta, v)$ density with respect to Lebesgue measure, and for $v = 0$ it is the point mass at $\\theta$.","title":"Gaussian likelihood with known variance"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/gaussian-likelihood-zero-apply","kind":"node","lean":"Cslib.MachineLearning.Bayesian.gaussianLikelihood_zero_apply","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#gaussian-likelihood-zero-apply","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"With noise variance $0$, the Gaussian likelihood sends each $\\theta \\in \\mathbb{R}$ to the Gaussian law with mean $\\theta$ and variance $0$, which is the point mass at $\\theta$: the observation is the parameter.","title":"Noiseless Gaussian likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/gaussian-likelihood-comp-gaussian-real","kind":"node","lean":"Cslib.MachineLearning.Bayesian.gaussianLikelihood_comp_gaussianReal","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#gaussian-likelihood-comp-gaussian-real","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$ and $v_0, v > 0$, and write $N(\\mu, s)$ for the Gaussian law with mean $\\mu$ and variance $s$. If the parameter $\\theta$ has prior $N(m_0, v_0)$ and the observation given $\\theta$ has law $N(\\theta, v)$ (the Gaussian likelihood with noise variance $v$), then the marginal law of the observation is $N(m_0, v_0 + v)$.","title":"The marginal of the data"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/normal-posterior-kernel","kind":"node","lean":"Cslib.MachineLearning.Bayesian.normalPosteriorKernel","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#normal-posterior-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$ and $v_0, v \\ge 0$, let $s = \\frac{v_0 v}{v_0 + v}$ be the posterior variance and, for $x \\in \\mathbb{R}$, let $\\mu(x) = \\frac{v m_0 + v_0 x}{v_0 + v}$ be the posterior mean (with $0 / 0 = 0$). The kernel from $\\mathbb{R}$ to $\\mathbb{R}$ sends $x$ to the measure with density $\\theta \\mapsto \\frac{1}{\\sqrt{2 \\pi s}} e^{-(\\theta - \\mu(x))^2 / (2 s)}$ with respect to Lebesgue measure. For\u2026","title":"Candidate posterior kernel of the normal\u2013normal model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/comp-prod-normal-posterior-kernel","kind":"node","lean":"Cslib.MachineLearning.Bayesian.compProd_normalPosteriorKernel","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#comp-prod-normal-posterior-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$ and $v_0, v > 0$, and write $N(\\mu, s)$ for the Gaussian law with mean $\\mu$ and variance $s$. Let $\\pi = N(m_0, v_0)$ be the prior on the parameter $\\theta$, let $K$ be the Gaussian likelihood with noise variance $v$, so that $K(\\theta) = N(\\theta, v)$, and let $Q$ be the normal posterior kernel, $Q(x) = N(\\mu(x), s)$ with $\\mu(x) = (v m_0 + v_0 x)/(v_0 + v)$ and $s = v_0 v/(v_0 + v)$. For\u2026","title":"Disintegration of the Normal\u2013Normal joint law"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/posterior-gaussian-likelihood-gaussian-r","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_gaussianLikelihood_gaussianReal","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#posterior-gaussian-likelihood-gaussian-r","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$ and $v_0, v > 0$. Take the prior $N(m_0, v_0)$ on the mean $\\theta$ and the likelihood under which an observation $x$ has law $N(\\theta, v)$. Then for Lebesgue-almost every $x \\in \\mathbb{R}$, the posterior distribution of $\\theta$ given $x$ is the normal law with mean $\\frac{v m_0 + v_0 x}{v_0 + v}$ and variance $\\frac{v_0 v}{v_0 + v}$.","title":"Normal\u2013Normal conjugacy with known variance"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/gaussian-likelihood-comp-posterior","kind":"node","lean":"Cslib.MachineLearning.Bayesian.gaussianLikelihood_comp_posterior","parent":41455,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors/index.html#gaussian-likelihood-comp-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/conjugate-priors","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$ and $v_0, v > 0$, and write $N(\\mu, s)$ for the Gaussian law with mean $\\mu$ and variance $s$. Take the prior $N(m_0, v_0)$ on the parameter $\\theta$ and the Gaussian likelihood $\\theta \\mapsto N(\\theta, v)$, and let $\\mu(x) = (v m_0 + v_0 x)/(v_0 + v)$ and $s = v_0 v/(v_0 + v)$. For Lebesgue-almost every observation $x$, a new observation drawn from the likelihood, with $\\theta$\u2026","title":"Normal\u2013Normal posterior predictive"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","title":"Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-post-var","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegPostVar","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-post-var","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"For a prior variance $s_0 \\ge 0$, a noise variance $\\sigma^2 \\ge 0$ and inputs $x \\in \\mathbb{R}^n$, the posterior variance of the weight is the nonnegative real number $s_0 \\sigma^2 / (\\sigma^2 + s_0 \\sum_i x_i^2)$, read as $0$ when the denominator is $0$.","title":"Posterior variance in Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-post-mean","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegPostMean","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-post-mean","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"For a prior mean $m_0 \\in \\mathbb{R}$, a prior variance $s_0 \\ge 0$, a noise variance $\\sigma^2 \\ge 0$, inputs $x \\in \\mathbb{R}^n$ and responses $y \\in \\mathbb{R}^n$, the posterior mean of the weight is the real number $(\\sigma^2 m_0 + s_0 \\sum_i x_i y_i) / (\\sigma^2 + s_0 \\sum_i x_i^2)$, read as $0$ when the denominator is $0$.","title":"Posterior mean in Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/inv-lin-reg-post-var","kind":"node","lean":"Cslib.MachineLearning.Bayesian.inv_linRegPostVar","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#inv-lin-reg-post-var","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"Let $s_0, \\sigma^2 \\ge 0$ be nonzero and $x \\in \\mathbb{R}^n$. Then the posterior variance $v$ of the weight for prior variance $s_0$, noise variance $\\sigma^2$ and inputs $x$ satisfies $1 / v = 1 / s_0 + (\\sum_i x_i^2) / \\sigma^2$.","title":"Precisions add in Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-likelihood-pdfreal","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegLikelihoodPDFReal","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-likelihood-pdfreal","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"For $\\sigma^2 \\ge 0$, $x, y \\in \\mathbb{R}^n$ and $w \\in \\mathbb{R}$, it is the real number $\\prod_i \\varphi(y_i; w x_i, \\sigma^2)$, where $\\varphi(t; \\mu, \\tau) = (2\\pi\\tau)^{-1/2} e^{-(t - \\mu)^2 / (2\\tau)}$ is the Gaussian density with mean $\\mu$ and variance $\\tau$ at $t$, read as $0$ when $\\tau = 0$. For $\\sigma^2 \\ne 0$ it is the Lebesgue density of the responses $y$ given the weight $w$, independent with\u2026","title":"The linear regression likelihood density"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/gaussian-pdfreal-mul-lin-reg-likelihood","kind":"node","lean":"Cslib.MachineLearning.Bayesian.gaussianPDFReal_mul_linRegLikelihoodPDFReal","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#gaussian-pdfreal-mul-lin-reg-likelihood","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$, let $s_0, \\sigma^2 \\ge 0$ be nonzero, and let $x, y \\in \\mathbb{R}^n$. Write $\\varphi(t; \\mu, \\tau)$ for the Gaussian density with mean $\\mu$ and variance $\\tau$ at $t$, $p(w) = \\varphi(w; m_0, s_0)$ for the prior density, $L(y \\mid w) = \\prod_i \\varphi(y_i; w x_i, \\sigma^2)$ for the likelihood density, $m$ and $v$ for the posterior mean and variance of the weight, and\u2026","title":"Completing the square for linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-evidence-pdfreal","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegEvidencePDFReal","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-evidence-pdfreal","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"For $m_0 \\in \\mathbb{R}$, $s_0, \\sigma^2 \\ge 0$ and $x, y \\in \\mathbb{R}^n$, let $m$ and $v$ be the posterior mean and variance of the weight, write $\\varphi(t; \\mu, \\tau) = (2\\pi\\tau)^{-1/2} e^{-(t - \\mu)^2 / (2\\tau)}$ for the Gaussian density with mean $\\mu$ and variance $\\tau$ at $t$ (read as $0$ when $\\tau = 0$), and let $L(y \\mid w) = \\prod_i \\varphi(y_i; w x_i, \\sigma^2)$ be the likelihood density. The\u2026","title":"The evidence density of Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/gaussian-pdfreal-mul-lin-reg-likelihood-2","kind":"node","lean":"Cslib.MachineLearning.Bayesian.gaussianPDFReal_mul_linRegLikelihoodPDFReal'","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#gaussian-pdfreal-mul-lin-reg-likelihood-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$, let $s_0, \\sigma^2 \\ge 0$ be nonzero, let $x, y \\in \\mathbb{R}^n$ and let $w \\in \\mathbb{R}$. Write $\\varphi(t; \\mu, \\tau)$ for the Gaussian density with mean $\\mu$ and variance $\\tau$ at $t$, $L(y \\mid w) = \\prod_i \\varphi(y_i; w x_i, \\sigma^2)$ for the likelihood density, $m$ and $v$ for the posterior mean and variance of the weight, and $Z(y)$ for the evidence density. Then\u2026","title":"Prior times likelihood factors as evidence times posterior"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-likelihood-pdf","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegLikelihoodPDF","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-likelihood-pdf","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"For $\\sigma^2 \\ge 0$, $x, y \\in \\mathbb{R}^n$ and $w \\in \\mathbb{R}$, it is $\\prod_i \\varphi(y_i; w x_i, \\sigma^2)$ as an extended nonnegative real, where $\\varphi(t; \\mu, \\tau)$ is the Gaussian density with mean $\\mu$ and variance $\\tau$ at $t$, read as $0$ when $\\tau = 0$. It is the likelihood density for $\\sigma^2 \\ne 0$ or $n = 0$, and it is $0$ for $\\sigma^2 = 0$ and $n > 0$.","title":"The extended-real linear regression likelihood density"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-likelihood","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegLikelihood","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-likelihood","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"The likelihood of Bayesian linear regression with one feature and known noise variance $\\sigma^2 \\ge 0$, for fixed inputs $x \\in \\mathbb{R}^n$: the kernel from the weight $w \\in \\mathbb{R}$ to the responses in $\\mathbb{R}^n$ under which, given $w$, the responses are independent with $y_i \\sim \\mathcal{N}(w x_i, \\sigma^2)$. For $\\sigma^2 \\ne 0$ it is given by the density $\\prod_i \\varphi(y_i; w x_i, \\sigma^2)$ with\u2026","title":"The likelihood of Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-likelihood-zero-apply","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegLikelihood_zero_apply","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-likelihood-zero-apply","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"For $x \\in \\mathbb{R}^n$ and $w \\in \\mathbb{R}$, the linear regression likelihood with noise variance $0$ and inputs $x$, given the weight $w$, is the Dirac measure at the vector $(w x_i)_i$: the responses are exactly $y_i = w x_i$.","title":"The noiseless linear regression likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-likelihood-apply","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegLikelihood_apply","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-likelihood-apply","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"Let $\\sigma^2 \\ge 0$ be nonzero, $x \\in \\mathbb{R}^n$ and $w \\in \\mathbb{R}$. Given the weight $w$, the linear regression likelihood with noise variance $\\sigma^2$ and inputs $x$ is the product measure $\\bigotimes_i \\mathcal{N}(w x_i, \\sigma^2)$ on $\\mathbb{R}^n$.","title":"The linear regression likelihood is a product of Gaussians"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/measurable-lin-reg-post-mean","kind":"node","lean":"Cslib.MachineLearning.Bayesian.measurable_linRegPostMean","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#measurable-lin-reg-post-mean","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"For a prior mean $m_0 \\in \\mathbb{R}$, a prior variance $s_0 \\ge 0$, a noise variance $\\sigma^2 \\ge 0$ and inputs $x \\in \\mathbb{R}^n$, the posterior mean of the weight, as a function of the responses $y \\in \\mathbb{R}^n$, is a measurable map from $\\mathbb{R}^n$ to $\\mathbb{R}$.","title":"Measurability of the posterior mean in Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-likelihood-comp-gaussian-real","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegLikelihood_comp_gaussianReal","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-likelihood-comp-gaussian-real","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$, let $s_0, \\sigma^2 \\ge 0$ be nonzero, and let $x \\in \\mathbb{R}^n$. If the weight has prior $\\mathcal{N}(m_0, s_0)$ and the responses follow the linear regression likelihood with noise variance $\\sigma^2$ and inputs $x$, then the marginal law of the responses is the measure on $\\mathbb{R}^n$ whose density with respect to Lebesgue measure is the evidence density $Z$ for $m_0$, $s_0$,\u2026","title":"The evidence of Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/posterior-lin-reg-likelihood-gaussian-re","kind":"node","lean":"Cslib.MachineLearning.Bayesian.posterior_linRegLikelihood_gaussianReal","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#posterior-lin-reg-likelihood-gaussian-re","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$, let $s_0 > 0$ and $\\sigma^2 > 0$, and let $x \\in \\mathbb{R}^n$ be fixed inputs. Put the prior $N(m_0, s_0)$ (mean $m_0$, variance $s_0$) on a scalar weight $w$, and let the responses be independent given $w$ with $y_i \\sim N(w x_i, \\sigma^2)$. Then for Lebesgue-almost every response vector $y \\in \\mathbb{R}^n$, the posterior of $w$ given $y$ is Gaussian with mean\u2026","title":"Bayesian linear regression"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-predictive","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegPredictive","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-predictive","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"For a noise variance $\\sigma^2 \\ge 0$ and a new input $x' \\in \\mathbb{R}$, the predictive kernel sends a weight $w \\in \\mathbb{R}$ to the Gaussian law $N(w x', \\sigma^2)$ of the response at $x'$, which is the point mass at $w x'$ when $\\sigma^2 = 0$.","title":"Predictive law of a linear-regression response"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/lin-reg-predictive-comp-posterior","kind":"node","lean":"Cslib.MachineLearning.Bayesian.linRegPredictive_comp_posterior","parent":41487,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression/index.html#lin-reg-predictive-comp-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/bayesian-linear-regression","status":"fully_proved","summary":"Let $m_0 \\in \\mathbb{R}$, let $s_0 > 0$ and $\\sigma^2 > 0$, and let $x \\in \\mathbb{R}^n$ and $x' \\in \\mathbb{R}$. Put the prior $N(m_0, s_0)$ (mean $m_0$, variance $s_0$) on a scalar weight $w$, and let the responses be independent given $w$ with $y_i \\sim N(w x_i, \\sigma^2)$. Then for Lebesgue-almost every response vector $y \\in \\mathbb{R}^n$, integrating the law $N(w x', \\sigma^2)$ of the response at $x'$ against\u2026","title":"Posterior predictive at a new input"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","title":"Expectation\u2013maximization algorithm"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/is-dist","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.IsDist","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#is-dist","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"A function $q : Z \\to \\mathbb{R}$ on a finite set $Z$ is a probability distribution when $q(z) \\ge 0$ for every $z \\in Z$ and $\\sum_{z \\in Z} q(z) = 1$.","title":"Probability distributions on a finite latent space"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/sum-mul-log-sub-le","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.sum_mul_log_sub_le","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#sum-mul-log-sub-le","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a finite set, let $p : Z \\to \\mathbb{R}$ satisfy $p(z) > 0$ for all $z$, and let $q$ be a probability distribution on $Z$. Then $\\sum_{z \\in Z} q(z) (\\log p(z) - \\log q(z)) \\le \\log \\sum_{z \\in Z} p(z)$.","title":"Gibbs' inequality (evidence lower bound form)"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/sum-div-mul-log-sub","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.sum_div_mul_log_sub","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#sum-div-mul-log-sub","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a finite nonempty set, let $p : Z \\to \\mathbb{R}$ satisfy $p(z) > 0$ for all $z$, and write $P = \\sum_{z \\in Z} p(z)$. Then","title":"Equality case of Gibbs' inequality"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/marginal","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.marginal","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#marginal","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"For a joint model $p$ assigning a real number $p_\\theta(x, z)$ to each parameter $\\theta$, observation $x$ and latent value $z$ in a finite latent space $Z$, the marginal is $p_\\theta(x) = \\sum_{z \\in Z} p_\\theta(x, z)$.","title":"Marginal of a latent-variable model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/log-lik","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.logLik","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#log-lik","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"For a joint model $p$ with finite latent space $Z$ and data $x_0, \\dots, x_{N-1}$, the log-likelihood at the parameter $\\theta$ is $\\ell(\\theta) = \\sum_{n < N} \\log p_\\theta(x_n)$, where $p_\\theta(x) = \\sum_{z \\in Z} p_\\theta(x, z)$ is the marginal.","title":"Log-likelihood of a latent-variable model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/posterior","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.posterior","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"For a joint model $p$ with finite latent space $Z$, the posterior of the latent value $z$ given the observation $x$ at the parameter $\\theta$ is $p_\\theta(z \\mid x) = p_\\theta(x, z) / p_\\theta(x)$, where $p_\\theta(x) = \\sum_{z' \\in Z} p_\\theta(x, z')$ is the marginal (the quotient is read as $0$ when the marginal is $0$). Computing it is the E-step of expectation-maximization.","title":"Posterior of a latent-variable model"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/elbo","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.elbo","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#elbo","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, and let $x_1, \\dots, x_N$ be data. For a parameter $\\theta$ and functions $q_1, \\dots, q_N : Z \\to \\mathbb{R}$ (one latent distribution per data point), the evidence lower bound is $\\sum_{n=1}^N \\sum_{z \\in Z} q_n(z) \\bigl(\\log p_\\theta(x_n, z) - \\log q_n(z)\\bigr)$.","title":"Evidence lower bound of latent distributions"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/em-q","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.emQ","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#em-q","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, let $x_1, \\dots, x_N$ be data, and write $p_\\theta(z \\mid x) = p_\\theta(x, z) / \\sum_{z' \\in Z} p_\\theta(x, z')$ for the posterior. For parameters $\\theta'$ and $\\theta$, the EM objective is\u2026","title":"The EM objective"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/post-entropy","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.postEntropy","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#post-entropy","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"For a joint model $p$ with finite latent space $Z$, data $x_0, \\dots, x_{N-1}$ and a parameter $\\theta$, the posterior entropy is $-\\sum_{n < N} \\sum_{z \\in Z} p_\\theta(z \\mid x_n) \\log p_\\theta(z \\mid x_n)$, where $p_\\theta(z \\mid x)$ is the posterior of $z$ given $x$.","title":"Posterior entropy"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/is-dist-posterior","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.isDist_posterior","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#is-dist-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a nonempty finite set and let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, with $p_\\theta(x, z) > 0$ for all $\\theta$, $x$, $z$. For every parameter $\\theta$ and observation $x$, the posterior $z \\mapsto p_\\theta(x, z) / \\sum_{z' \\in Z} p_\\theta(x, z')$ is a probability distribution on $Z$: its values are nonnegative and sum to $1$.","title":"Posteriors are probability distributions"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/elbo-le-log-lik","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.elbo_le_logLik","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#elbo-le-log-lik","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, with $p_\\theta(x, z) > 0$ for all $\\theta$, $x$, $z$, and let $x_1, \\dots, x_N$ be data. For every parameter $\\theta$ and all probability distributions $q_1, \\dots, q_N$ on $Z$,","title":"The evidence lower bound"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/elbo-posterior","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.elbo_posterior","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#elbo-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a nonempty finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, with $p_\\theta(x, z) > 0$ for all $\\theta$, $x$, $z$, and let $x_1, \\dots, x_N$ be data. For every parameter $\\theta$, the posteriors $q_n(z) = p_\\theta(z \\mid x_n) = p_\\theta(x_n, z) / \\sum_{z' \\in Z} p_\\theta(x_n, z')$ satisfy","title":"The bound is tight at the posterior"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/elbo-posterior-eq","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.elbo_posterior_eq","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#elbo-posterior-eq","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, let $x_1, \\dots, x_N$ be data, and write $p_\\theta(z \\mid x) = p_\\theta(x, z) / \\sum_{z' \\in Z} p_\\theta(x, z')$ for the posterior, $Q(\\theta' \\mid \\theta) = \\sum_{n=1}^N \\sum_{z \\in Z} p_\\theta(z \\mid x_n) \\log p_{\\theta'}(x_n, z)$ for the EM objective and\u2026","title":"The ELBO at the posterior is the EM objective plus the entropy"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/em-q-sub-le-log-lik-sub","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.emQ_sub_le_logLik_sub","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#em-q-sub-le-log-lik-sub","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a nonempty finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, with $p_\\theta(x, z) > 0$ for all $\\theta$, $x$, $z$, and let $x_1, \\dots, x_N$ be data. Write $p_\\theta(z \\mid x) = p_\\theta(x, z) / \\sum_{z' \\in Z} p_\\theta(x, z')$ for the posterior,\u2026","title":"The EM inequality"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/log-lik-le-of-em-q-le","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.logLik_le_of_emQ_le","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#log-lik-le-of-em-q-le","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a nonempty finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, with $p_\\theta(x, z) > 0$ for all $\\theta$, $x$, $z$, and let $x_1, \\dots, x_N$ be data. Write $p_\\theta(z \\mid x) = p_\\theta(x, z) / \\sum_{z' \\in Z} p_\\theta(x, z')$ for the posterior,\u2026","title":"The EM theorem"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/log-lik-iterate-monotone","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.ExpectationMaximization.logLik_iterate_monotone","parent":41504,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm/index.html#log-lik-iterate-monotone","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/expectation-maximization-algorithm","status":"fully_proved","summary":"Let $Z$ be a nonempty finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, with $p_\\theta(x, z) > 0$ for all $\\theta$, $x$, $z$, and let $x_1, \\dots, x_N$ be data. Write $p_\\theta(z \\mid x) = p_\\theta(x, z) / \\sum_{z' \\in Z} p_\\theta(x, z')$ for the posterior,\u2026","title":"EM iterates have monotone likelihood"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","title":"Mean-field variational inference"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/evidence","kind":"node","lean":"Cslib.MachineLearning.Variational.evidence","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#evidence","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a finite set and let $p : Z \\to \\mathbb{R}$ give the joint values $p(z) = p(x, z)$ of a fixed observation $x$ with each latent value $z \\in Z$ (an unnormalized posterior). The evidence is $p(x) = \\sum_{z \\in Z} p(z)$.","title":"Evidence of an unnormalized posterior"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/posterior","kind":"node","lean":"Cslib.MachineLearning.Variational.posterior","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a finite set and let $p : Z \\to \\mathbb{R}$ give the joint values $p(z) = p(x, z)$ of a fixed observation $x$ with each latent value $z \\in Z$. The posterior of $z$ is $p(z \\mid x) = p(z) / p(x)$, where $p(x) = \\sum_{z' \\in Z} p(z')$ is the evidence (the quotient is read as $0$ when $p(x) = 0$).","title":"Posterior of an unnormalized joint"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a finite set, let $p : Z \\to \\mathbb{R}$ give the joint values $p(z) = p(x, z)$ of a fixed observation $x$ with each latent value $z \\in Z$, and let $q : Z \\to \\mathbb{R}$. The evidence lower bound of $q$ is $\\mathrm{ELBO}(p, q) = \\sum_{z \\in Z} q(z) \\bigl(\\log p(z) - \\log q(z)\\bigr)$.","title":"Evidence lower bound of a single observation"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/kl","kind":"node","lean":"Cslib.MachineLearning.Variational.kl","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#kl","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a finite set and let $q, r : Z \\to \\mathbb{R}$. The Kullback\u2013Leibler formula of $q$ against $r$ is $\\mathrm{KL}(q \\,\\|\\, r) = \\sum_{z \\in Z} q(z) \\bigl(\\log q(z) - \\log r(z)\\bigr)$; terms with $q(z) = 0$ vanish.","title":"Finite Kullback\u2013Leibler formula"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/kl-nonneg","kind":"node","lean":"Cslib.MachineLearning.Variational.kl_nonneg","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#kl-nonneg","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a finite set, let $q$ be a probability distribution on $Z$, and let $r : Z \\to \\mathbb{R}$ satisfy $r(z) > 0$ for every $z \\in Z$ and $\\sum_{z \\in Z} r(z) = 1$. Then $\\mathrm{KL}(q \\,\\|\\, r) = \\sum_{z \\in Z} q(z) \\bigl(\\log q(z) - \\log r(z)\\bigr) \\ge 0$.","title":"Gibbs' inequality"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo-eq-log-evidence-sub-kl","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo_eq_log_evidence_sub_kl","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo-eq-log-evidence-sub-kl","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a nonempty finite set and let $p : Z \\to \\mathbb{R}$ give the joint values $p(z) = p(x, z)$ of a fixed observation $x$ with each latent value $z \\in Z$, where $p(z) > 0$ for every $z \\in Z$. Let $p(x) = \\sum_{z \\in Z} p(z)$ be the evidence and $p(z \\mid x) = p(z) / p(x)$ the posterior. For $q, r : Z \\to \\mathbb{R}$ write $\\mathrm{ELBO}(p, q) = \\sum_{z \\in Z} q(z) \\bigl(\\log p(z) - \\log q(z)\\bigr)$ and\u2026","title":"The ELBO is the log evidence minus the KL divergence"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo-le-log-evidence","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo_le_log_evidence","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo-le-log-evidence","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a finite set and let $p : Z \\to \\mathbb{R}$ satisfy $p(z) > 0$ for all $z$, with evidence $p(x) = \\sum_{z \\in Z} p(z)$. Then for every probability distribution $q$ on $Z$,","title":"The ELBO is a lower bound"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/kl-posterior-nonneg","kind":"node","lean":"Cslib.MachineLearning.Variational.kl_posterior_nonneg","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#kl-posterior-nonneg","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a nonempty finite set, let $p : Z \\to \\mathbb{R}$ satisfy $p(z) > 0$ for every $z \\in Z$, and let $q$ be a probability distribution on $Z$. Let $\\pi(z) = p(z) / \\sum_{z' \\in Z} p(z')$ be the posterior of $p$. Then $\\mathrm{KL}(q \\,\\|\\, \\pi) = \\sum_{z \\in Z} q(z) \\bigl(\\log q(z) - \\log \\pi(z)\\bigr) \\ge 0$.","title":"The KL divergence to the posterior is nonnegative"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo-posterior","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo_posterior","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo-posterior","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a nonempty finite set and let $p : Z \\to \\mathbb{R}$ satisfy $p(z) > 0$ for all $z$, with evidence $p(x) = \\sum_{z \\in Z} p(z)$ and posterior $p(z \\mid x) = p(z) / p(x)$. Then $\\mathrm{ELBO}(p, p(\\cdot \\mid x)) = \\log p(x)$, where $\\mathrm{ELBO}(p, q) = \\sum_{z} q(z) (\\log p(z) - \\log q(z))$.","title":"The ELBO bound is attained at the posterior"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/em-elbo-eq","kind":"node","lean":"Cslib.MachineLearning.Variational.em_elbo_eq","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#em-elbo-eq","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $Z$ be a finite set, let $p_\\theta(x, z)$ be real numbers indexed by parameters $\\theta$, observations $x$ and latent values $z \\in Z$, let $x_1, \\dots, x_N$ be data, let $\\theta$ be a parameter and let $q_1, \\dots, q_N : Z \\to \\mathbb{R}$. Then the evidence lower bound of expectation-maximization, $\\sum_{n=1}^N \\sum_{z \\in Z} q_n(z) \\bigl(\\log p_\\theta(x_n, z) - \\log q_n(z)\\bigr)$, equals\u2026","title":"The EM evidence lower bound is a sum of per-point ELBOs"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/prod-dist","kind":"node","lean":"Cslib.MachineLearning.Variational.prodDist","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#prod-dist","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets and let $q_1 : A \\to \\mathbb{R}$ and $q_2 : B \\to \\mathbb{R}$. Their mean-field (product) distribution is the function on $A \\times B$ given by $q(a, b) = q_1(a) \\, q_2(b)$.","title":"Mean-field product distribution"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/expected-log","kind":"node","lean":"Cslib.MachineLearning.Variational.expectedLog","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#expected-log","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets and let $p : A \\times B \\to \\mathbb{R}$ and $q_2 : B \\to \\mathbb{R}$. For $a \\in A$, the expected log-joint of $a$ under $q_2$ is $\\mathbb{E}_{q_2}[\\log p(a, \\cdot)] = \\sum_{b \\in B} q_2(b) \\log p(a, b)$.","title":"Expected log-joint"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/cavi","kind":"node","lean":"Cslib.MachineLearning.Variational.cavi","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#cavi","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets, let $p : A \\times B \\to \\mathbb{R}$ be a joint and let $q_2 : B \\to \\mathbb{R}$. The coordinate-ascent update of the first factor is the function $q_1 : A \\to \\mathbb{R}$ given by","title":"Mean-field (CAVI) update of the first factor"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo-prod-dist","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo_prodDist","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo-prod-dist","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets, let $p : A \\times B \\to \\mathbb{R}$ be a joint, and let $q_1$ and $q_2$ be probability distributions on $A$ and $B$, with product $(q_1 q_2)(a, b) = q_1(a) q_2(b)$. Then","title":"ELBO of a product distribution"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo-le-elbo-cavi","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo_le_elbo_cavi","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo-le-elbo-cavi","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets with $A$ nonempty, let $p : A \\times B \\to \\mathbb{R}$ be a joint, and let $q_1$ and $q_2$ be probability distributions on $A$ and $B$. Let $q_1^*$ be the CAVI update of the first factor given $q_2$, $q_1^*(a) = \\exp E(a) / \\sum_{a' \\in A} \\exp E(a')$ with $E(a) = \\sum_{b \\in B} q_2(b) \\log p(a, b)$. Then $\\mathrm{ELBO}(p, q_1 q_2) \\le \\mathrm{ELBO}(p, q_1^* q_2)$, where\u2026","title":"The CAVI update maximizes the ELBO over the first factor"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/swap-joint","kind":"node","lean":"Cslib.MachineLearning.Variational.swapJoint","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#swap-joint","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets and let $p : A \\times B \\to \\mathbb{R}$. The swapped joint of $p$ is the function on $B \\times A$ given by $(b, a) \\mapsto p(a, b)$.","title":"Swapped joint"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/cavi-2","kind":"node","lean":"Cslib.MachineLearning.Variational.cavi\u2082","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#cavi-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets, let $p : A \\times B \\to \\mathbb{R}$ be a joint and let $q_1 : A \\to \\mathbb{R}$. The coordinate-ascent update of the second factor is the function $q_2 : B \\to \\mathbb{R}$ given by","title":"Mean-field (CAVI) update of the second factor"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo-le-elbo-cavi-2","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo_le_elbo_cavi\u2082","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo-le-elbo-cavi-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets with $B$ nonempty, let $p : A \\times B \\to \\mathbb{R}$ be a joint, and let $q_1$ and $q_2$ be probability distributions on $A$ and $B$. Let $q_2^*$ be the CAVI update of the second factor given $q_1$, $q_2^*(b) = \\exp F(b) / \\sum_{b' \\in B} \\exp F(b')$ with $F(b) = \\sum_{a \\in A} q_1(a) \\log p(a, b)$. Then $\\mathrm{ELBO}(p, q_1 q_2) \\le \\mathrm{ELBO}(p, q_1 q_2^*)$, where\u2026","title":"The CAVI update of the second factor does not decrease the ELBO"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/sweep","kind":"node","lean":"Cslib.MachineLearning.Variational.sweep","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#sweep","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets and $p : A \\times B \\to \\mathbb{R}$. One sweep of coordinate ascent (CAVI) sends a pair of factors $(q_1, q_2)$, with $q_1 : A \\to \\mathbb{R}$ and $q_2 : B \\to \\mathbb{R}$, to $(q_1', q_2')$, where $q_1'$ is the CAVI update of the first factor given $q_2$, $q_1'(a) = \\exp(\\sum_b q_2(b) \\log p(a, b)) / \\sum_{a'} \\exp(\\sum_b q_2(b) \\log p(a', b))$, and $q_2'$ is the CAVI update of the\u2026","title":"Mean-field coordinate-ascent sweep"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo-le-elbo-sweep","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo_le_elbo_sweep","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo-le-elbo-sweep","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be nonempty finite sets, let $p : A \\times B \\to \\mathbb{R}$ be a joint, and let $q_1$ and $q_2$ be probability distributions on $A$ and $B$. One sweep replaces $q_1$ by its CAVI update $q_1'$ given $q_2$ and then $q_2$ by its CAVI update $q_2'$ given $q_1'$, where $q_1'(a) \\propto \\exp \\sum_{b} q_2(b) \\log p(a, b)$ and $q_2'(b) \\propto \\exp \\sum_{a} q_1'(a) \\log p(a, b)$, each normalized to sum to\u2026","title":"A sweep of coordinate ascent does not decrease the ELBO"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/elbo-iterate-sweep-monotone","kind":"node","lean":"Cslib.MachineLearning.Variational.elbo_iterate_sweep_monotone","parent":41521,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference/index.html#elbo-iterate-sweep-monotone","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mean-field-variational-inference","status":"fully_proved","summary":"Let $A$ and $B$ be nonempty finite sets, let $p : A \\times B \\to \\mathbb{R}$ be a joint, and let $q_1^{(0)}$ and $q_2^{(0)}$ be probability distributions on $A$ and $B$. A sweep of coordinate ascent replaces $q_1$ by its CAVI update $q_1'(a) \\propto \\exp \\sum_{b} q_2(b) \\log p(a, b)$ and then $q_2$ by its CAVI update $q_2'(b) \\propto \\exp \\sum_{a} q_1'(a) \\log p(a, b)$, each normalized to sum to $1$. Let\u2026","title":"Mean-field coordinate ascent never decreases the ELBO"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","title":"Mixture models"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/sum-mul-log-le-sum-mul-log-div","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.sum_mul_log_le_sum_mul_log_div","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#sum-mul-log-le-sum-mul-log-div","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $Z$ be a finite set, let $c \\colon Z \\to \\mathbb{R}$ be nonnegative counts with positive total $C = \\sum_{z \\in Z} c_z > 0$, and let $w \\colon Z \\to \\mathbb{R}$ be positive weights with $\\sum_{z \\in Z} w_z = 1$. Then $\\sum_{z \\in Z} c_z \\log w_z \\le \\sum_{z \\in Z} c_z \\log(c_z / C)$ (Gibbs' inequality).","title":"The weighted log-likelihood is bounded by its value at the normalized counts"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/bern-pmf","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.bernPMF","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#bern-pmf","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For a vector $p \\in \\mathbb{R}^d$ of success probabilities and a binary vector $x \\in \\{0, 1\\}^d$, the probability of $x$ under the product of $d$ independent Bernoulli distributions with parameters $p_1, \\dots, p_d$ is","title":"Product Bernoulli probability mass function"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/params","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.Params","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#params","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"The parameters of a mixture of $K$ products of $d$ Bernoulli distributions are real mixing weights $\\pi_k$ for $k < K$ and real success probabilities $p_{kj}$ for every component $k < K$ and coordinate $j < d$. No constraint is imposed on these numbers here; the admissible ones are the valid parameters.","title":"Parameters of a Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/valid","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.Params.Valid","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#valid","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Parameters of a mixture of $K$ products of $d$ Bernoulli distributions, with mixing weights $\\pi_k$ and success probabilities $p_{kj}$, are valid when $\\pi_k > 0$ for every $k < K$, $\\sum_{k < K} \\pi_k = 1$, and $0 < p_{kj} < 1$ for every $k < K$ and $j < d$.","title":"Valid parameters of a Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/valid-params","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.ValidParams","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#valid-params","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"The parameter space of EM for a mixture of $K$ products of $d$ Bernoulli distributions is the set of valid parameters: mixing weights $\\pi_k$ and success probabilities $p_{kj}$ ($k < K$, $j < d$) with every $\\pi_k > 0$, $\\sum_{k < K} \\pi_k = 1$ and every $p_{kj}$ strictly between $0$ and $1$.","title":"The parameter space of a Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/joint","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.joint","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#joint","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For valid parameters $\\theta$ of a mixture of $K$ products of $d$ Bernoulli distributions, with mixing weights $\\pi_k$ and success probabilities $p_{kj}$, the joint probability of a binary vector $x \\in \\{0, 1\\}^d$ and a component $k$ is","title":"Joint distribution of the Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/resp","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.resp","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#resp","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $\\theta$ be valid parameters of a mixture of $K$ products of $d$ Bernoulli distributions, and write $p_\\theta(x, k)$ for the joint probability of a binary vector $x \\in \\{0,1\\}^d$ and a component $k < K$ under $\\theta$. The responsibility of $k$ for $x$ is the posterior probability $r_k(x) = p_\\theta(x, k) / \\sum_{k' < K} p_\\theta(x, k')$. Computing it is the E-step of EM.","title":"Responsibilities of a Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/mass","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.mass","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#mass","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For data $x_1, \\dots, x_N \\in \\{0, 1\\}^d$ and valid parameters $\\theta$ of a mixture of $K$ products of $d$ Bernoulli distributions, the effective number of points of component $k$ is $N_k = \\sum_{n=1}^{N} r_{nk}$, where $r_{nk}$ is the posterior probability (responsibility) of component $k$ given $x_n$ under $\\theta$.","title":"Effective number of points of a Bernoulli-mixture component"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/ones","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.ones","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#ones","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For data $x_1, \\dots, x_N \\in \\{0, 1\\}^d$, valid parameters $\\theta$ of a mixture of $K$ products of $d$ Bernoulli distributions, a component $k$ and a coordinate $j$, this is the sum $\\sum_{n \\,:\\, x_{nj} = 1} r_{nk}$, over the points whose $j$-th coordinate is $1$, of the posterior probability $r_{nk}$ of component $k$ given $x_n$ under $\\theta$.","title":"Responsibility-weighted count of ones"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/zeros","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.zeros","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#zeros","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $\\theta$ be valid parameters of a mixture of $K$ products of $d$ Bernoulli distributions, let $x_0, \\dots, x_{N-1} \\in \\{0,1\\}^d$ be data, and let $k < K$ be a component and $j < d$ a coordinate. Write $x_{nj}$ for the $j$-th coordinate of $x_n$ and $r_k(x_n)$ for the responsibility of component $k$ for $x_n$ under $\\theta$. The weighted count is $\\sum_{n < N,\\ x_{nj} = 0} r_k(x_n)$, the sum of the\u2026","title":"Responsibility-weighted count of zeros"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/m-step","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.mStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#m-step","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\{0, 1\\}^d$ be data and $\\theta$ valid parameters of a mixture of $K$ products of $d$ Bernoulli distributions. With $r_{nk}$ the posterior probability of component $k$ given $x_n$ under $\\theta$, $N_k = \\sum_n r_{nk}$ and $A_{kj} = \\sum_n r_{nk} x_{nj}$, the M-step returns the parameters with mixing weights $\\pi'_k = N_k / N$ and success probabilities $p'_{kj} = A_{kj} / N_k$, where a\u2026","title":"The M-step of the Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/rich","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.Rich","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#rich","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Data $x_0, \\dots, x_{N-1} \\in \\{0,1\\}^d$ are rich when $N > 0$ and, for every coordinate $j < d$, some $x_n$ has $j$-th coordinate $1$ and some $x_m$ has $j$-th coordinate $0$. These are the data on which the M-step yields valid parameters.","title":"Rich data for a Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-step","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-step","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\{0, 1\\}^d$ be data with $N \\ge 1$ in which every coordinate takes both values. For valid parameters $\\theta$ of a mixture of $K$ products of $d$ Bernoulli distributions, the EM step maps $\\theta$ to the parameters with mixing weights $\\pi'_k = N_k / N$ and success probabilities $p'_{kj} = A_{kj} / N_k$, where $r_{nk}$ is the posterior probability of component $k$ given $x_n$ under\u2026","title":"One EM step for the Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-q-eq","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.emQ_eq","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-q-eq","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\{0, 1\\}^d$ be data and let $\\theta$ and $\\theta'$ be valid parameters of a mixture of $K$ products of $d$ Bernoulli distributions (positive mixing weights summing to one and success probabilities in $(0, 1)$), where $\\theta'$ has mixing weights $\\pi'_k$ and success probabilities $p'_{kj}$. Write $p_\\theta(x, k)$ for the joint probability of $x$ and component $k$ under $\\theta$ (the mixing\u2026","title":"Decomposition of the EM objective for the Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-q-le-em-step","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.emQ_le_emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-q-le-em-step","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\{0, 1\\}^d$ be rich data: $N > 0$, and for every coordinate $j < d$ some $x_n$ has $j$-th coordinate $1$ and some $x_m$ has $j$-th coordinate $0$. Let $\\theta$ be valid parameters of a mixture of $K$ products of $d$ Bernoulli distributions, and let $\\theta^+$ be the result of the EM step applied to $\\theta$. Then every valid parameter $\\theta'$ satisfies\u2026","title":"The M-step of the Bernoulli mixture maximizes the EM objective"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/log-lik-le-em-step","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.logLik_le_emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#log-lik-le-em-step","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\{0, 1\\}^d$ be rich data: $N > 0$, and for every coordinate $j < d$ some $x_n$ has $j$-th coordinate $1$ and some $x_m$ has $j$-th coordinate $0$. Let $\\theta$ be valid parameters of a mixture of $K$ products of $d$ Bernoulli distributions. Then $\\ell(\\theta) \\le \\ell(\\theta^+)$, where $\\ell$ is the log-likelihood of the data under the mixture's joint distribution and $\\theta^+$ is the\u2026","title":"Each EM step for the Bernoulli mixture does not decrease the log-likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/log-lik-iterate-em-step-monotone","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.BernoulliMixture.logLik_iterate_emStep_monotone","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#log-lik-iterate-em-step-monotone","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\{0, 1\\}^d$ be rich data: $N > 0$, and for every coordinate $j < d$ some $x_n$ has $j$-th coordinate $1$ and some $x_m$ has $j$-th coordinate $0$. Let $\\theta_0$ be valid parameters of a mixture of $K$ products of $d$ Bernoulli distributions, and let $\\theta_t$ be the result of $t$ EM steps from $\\theta_0$. Then $\\ell(\\theta_t)$ is nondecreasing in $t \\in \\mathbb{N}$, where $\\ell$ is the\u2026","title":"The log-likelihood is monotone along the EM iterates of a Bernoulli mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/iso-gauss-pdf","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.isoGaussPDF","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#iso-gauss-pdf","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For a mean $\\mu \\in \\mathbb{R}^d$ and a variance $v \\ge 0$, the isotropic Gaussian density at $x \\in \\mathbb{R}^d$ is $\\mathcal{N}(x \\mid \\mu, v I) = \\prod_{i=1}^{d} \\varphi_{\\mu_i, v}(x_i)$, where $\\varphi_{m, v}$ is the one-dimensional Gaussian density with mean $m$ and variance $v$.","title":"Isotropic Gaussian density"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/log-iso-gauss-pdf","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.log_isoGaussPDF","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#log-iso-gauss-pdf","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For $\\mu, x \\in \\mathbb{R}^d$ and a variance $v > 0$, the isotropic Gaussian density $\\mathcal{N}(x \\mid \\mu, v I)$ with mean $\\mu$ and covariance $v I$ satisfies $\\log \\mathcal{N}(x \\mid \\mu, v I) = -\\tfrac{d}{2} \\log(2 \\pi v) - \\|x - \\mu\\|^2 / (2 v)$.","title":"Log-density of the isotropic Gaussian"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/params-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.Params","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#params-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Parameters of a $K$-component isotropic Gaussian mixture in $d$ dimensions consist, for each component $k < K$, of a mixing weight $\\pi_k \\in \\mathbb{R}$, a mean $\\mu_k \\in \\mathbb{R}^d$ and a variance $\\sigma_k^2 \\ge 0$; component $k$ has covariance $\\sigma_k^2 I$.","title":"Parameters of an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/valid-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.Params.Valid","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#valid-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Parameters of a $K$-component isotropic Gaussian mixture in $d$ dimensions, with mixing weights $\\pi_k$, means $\\mu_k$ and variances $\\sigma_k^2 \\ge 0$, are valid when every $\\pi_k > 0$, $\\sum_k \\pi_k = 1$, and every $\\sigma_k^2 > 0$.","title":"Validity of isotropic Gaussian mixture parameters"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/valid-params-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.ValidParams","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#valid-params-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For a $K$-component isotropic Gaussian mixture in $d$ dimensions, the parameter space of EM is the set of parameters (mixing weights $\\pi_k$, means $\\mu_k \\in \\mathbb{R}^d$ and variances $\\sigma_k^2 \\ge 0$) that are valid: every $\\pi_k > 0$, $\\sum_k \\pi_k = 1$, and every $\\sigma_k^2 > 0$.","title":"The parameter space of isotropic Gaussian mixture EM"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/joint-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.joint","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#joint-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For valid parameters $\\theta$ of a $K$-component isotropic Gaussian mixture in $\\mathbb{R}^d$, with weights $\\pi_k$, means $\\mu_k$ and variances $\\sigma_k^2$, the joint density of a point $x \\in \\mathbb{R}^d$ and a component $k$ is $p_\\theta(x, k) = \\pi_k \\, \\mathcal{N}(x \\mid \\mu_k, \\sigma_k^2 I)$, where $\\mathcal{N}(\\cdot \\mid \\mu, v I)$ is the isotropic Gaussian density.","title":"Joint density of an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/marginal-joint","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.marginal_joint","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#marginal-joint","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For valid parameters $\\theta$ with mixing weights $\\pi_k$, means $\\mu_k$ and variances $\\sigma_k^2$, and every $x \\in \\mathbb{R}^d$, the marginal $p_\\theta(x) = \\sum_k p_\\theta(x, k)$ of the joint density equals $\\sum_k \\pi_k \\, \\mathcal{N}(x \\mid \\mu_k, \\sigma_k^2 I)$, where $\\mathcal{N}(x \\mid \\mu, v I)$ is the isotropic Gaussian density.","title":"Mixture density of an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/resp-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.resp","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#resp-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For valid parameters $\\theta$ with mixing weights $\\pi_k$, means $\\mu_k$ and variances $\\sigma_k^2$, a point $x \\in \\mathbb{R}^d$ and a component $k$, the responsibility of $k$ for $x$ is the posterior $p_\\theta(k \\mid x) = \\pi_k \\mathcal{N}(x \\mid \\mu_k, \\sigma_k^2 I) / \\sum_j \\pi_j \\mathcal{N}(x \\mid \\mu_j, \\sigma_j^2 I)$, where $\\mathcal{N}(x \\mid \\mu, v I)$ is the isotropic Gaussian density. Computing it at\u2026","title":"Responsibilities in an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/nonempty-of-valid","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.nonempty_of_valid","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#nonempty-of-valid","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"If $\\theta$ is a valid parameter of a $K$-component isotropic Gaussian mixture in $d$ dimensions, then $K \\ge 1$.","title":"Valid mixture parameters have at least one component"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/mass-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.mass","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#mass-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Given data points $x_0, \\dots, x_{N-1}$ in $\\mathbb{R}^d$, valid parameters $\\theta$ of a $K$-component isotropic Gaussian mixture and a component $k$, the effective number of points of $k$ is $N_k = \\sum_n r_{nk}$, where $r_{nk}$ is the responsibility of component $k$ for $x_n$ under $\\theta$, the posterior probability of $k$ given $x_n$.","title":"Effective number of points in an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/new-mean","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.newMean","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#new-mean","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Given data points $x_0, \\dots, x_{N-1}$ in $\\mathbb{R}^d$, valid parameters $\\theta$ of a $K$-component isotropic Gaussian mixture and a component $k$, the updated mean of $k$ is the responsibility-weighted mean $\\mu_k' = N_k^{-1} \\sum_n r_{nk} x_n$, where $r_{nk}$ is the responsibility of component $k$ for $x_n$ under $\\theta$ and $N_k = \\sum_n r_{nk}$.","title":"M-step mean update for an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/scatter","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.scatter","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#scatter","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Given data points $x_0, \\dots, x_{N-1}$ in $\\mathbb{R}^d$, valid parameters $\\theta$ of a $K$-component isotropic Gaussian mixture and a component $k$, the scatter of $k$ is $S_k = \\sum_n r_{nk} \\lVert x_n - \\mu_k' \\rVert^2$, where $r_{nk}$ is the responsibility of component $k$ for $x_n$ under $\\theta$, $N_k = \\sum_n r_{nk}$ and $\\mu_k' = N_k^{-1} \\sum_n r_{nk} x_n$ is the responsibility-weighted mean.","title":"Responsibility-weighted scatter in an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/new-var","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.newVar","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#new-var","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Given data points $x_0, \\dots, x_{N-1}$ in $\\mathbb{R}^d$, valid parameters $\\theta$ of a $K$-component isotropic Gaussian mixture and a component $k$, the updated variance of $k$ is $\\sigma_k'^2 = S_k / (d N_k)$, where $r_{nk}$ is the responsibility of component $k$ for $x_n$ under $\\theta$, $N_k = \\sum_n r_{nk}$, $\\mu_k' = N_k^{-1} \\sum_n r_{nk} x_n$ and $S_k = \\sum_n r_{nk} \\lVert x_n - \\mu_k' \\rVert^2$.","title":"M-step variance update for an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/m-step-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.mStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#m-step-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_0, \\dots, x_{N-1}$ be data points in $\\mathbb{R}^d$, let $\\theta$ be valid parameters of a $K$-component isotropic Gaussian mixture, and let $r_{nk}$ be the responsibility of component $k$ for $x_n$ under $\\theta$. With $N_k = \\sum_n r_{nk}$, the M-step returns the parameters with mixing weights $\\pi_k' = N_k / N$, means $\\mu_k' = N_k^{-1} \\sum_n r_{nk} x_n$ and variances $\\sigma_k'^2 = S_k / (d N_k)$, where\u2026","title":"M-step of EM for an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/sum-mul-norm-sub-sq","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.sum_mul_norm_sub_sq","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#sum-mul-norm-sub-sq","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $F$ be a real inner product space, $s$ a finite index set, $w_i \\in \\mathbb{R}$ weights with $W = \\sum_{i \\in s} w_i \\ne 0$, and $x_i \\in F$ points for $i \\in s$, and let $m = W^{-1} \\sum_{i \\in s} w_i x_i$. Then for every $c \\in F$,","title":"The weighted centroid decomposition"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/sum-mul-norm-sub-sq-mean-le","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.sum_mul_norm_sub_sq_mean_le","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#sum-mul-norm-sub-sq-mean-le","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"With positive total weight, the weighted mean minimizes the weighted sum of squared distances. Let $F$ be a real inner product space, $s$ a finite index set, $w_i \\in \\mathbb{R}$ weights with $W = \\sum_{i \\in s} w_i > 0$, and $x_i \\in F$ points for $i \\in s$, and let $m = W^{-1} \\sum_{i \\in s} w_i x_i$ be the weighted mean. Then for every $c \\in F$,","title":"With positive total weight, the weighted mean minimizes the weighted sum of squared distances"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/not-all-equal","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.NotAllEqual","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#not-all-equal","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"A finite family of points $x_0, \\dots, x_{N-1}$ in $\\mathbb{R}^d$ is not all equal when there are indices $n$ and $n'$ with $x_n \\ne x_{n'}$.","title":"Data that are not all equal"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-step-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-step-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be points of $\\mathbb{R}^d$ that are not all equal and let $\\theta$ be valid parameters of a $K$-component isotropic Gaussian mixture. With $r_{nk}$ the responsibility of component $k$ for $x_n$ under $\\theta$ (the E-step), the EM step sends $\\theta$ to the valid parameters with weights $\\pi_k' = N_k / N$, means $\\mu_k' = N_k^{-1} \\sum_n r_{nk} x_n$ and variances $v_k' = S_k / (d N_k)$, where\u2026","title":"The EM step for isotropic Gaussian mixtures"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/comp-term","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.compTerm","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#comp-term","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Given points $x_1, \\dots, x_N$ of $\\mathbb{R}^d$, valid parameters $\\theta$ of a $K$-component isotropic Gaussian mixture, a component $k$, a mean $\\mu \\in \\mathbb{R}^d$ and a real number $v$, the component term is","title":"Component term of the EM objective for isotropic Gaussian mixtures"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-q-eq-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.emQ_eq","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-q-eq-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be points of $\\mathbb{R}^d$, and let $\\theta'$ and $\\theta$ be valid parameters of a $K$-component isotropic Gaussian mixture, where $\\theta'$ has weights $\\pi_k'$, means $\\mu_k'$ and variances $v_k'$. Then the EM objective of the mixture's joint density satisfies","title":"The EM objective of an isotropic Gaussian mixture splits by component"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/sum-mass-mul-log-le","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.sum_mass_mul_log_le","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#sum-mass-mul-log-le","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be $N > 0$ points of $\\mathbb{R}^d$, let $\\theta$ be valid parameters of a $K$-component isotropic Gaussian mixture in $d$ dimensions, and let $N_k = \\sum_n r_{nk}$ be the mass of component $k$, where $r_{nk}$ is the responsibility of component $k$ for $x_n$ under $\\theta$. Then for all real $w_1, \\dots, w_K > 0$ with $\\sum_k w_k = 1$,","title":"Isotropic Gaussian mixtures: the weight update maximizes the weights term"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/comp-term-le","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.compTerm_le","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#comp-term-le","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be points of $\\mathbb{R}^d$ that are not all equal, let $\\theta$ be valid parameters of a $K$-component isotropic Gaussian mixture, let $k$ be a component, and let $r_{nk}$ be the responsibility of component $k$ for $x_n$ under $\\theta$. For every $\\mu \\in \\mathbb{R}^d$ and every real $v > 0$, the component term","title":"The mean and variance updates maximize each component term"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-q-le-em-step-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.emQ_le_emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-q-le-em-step-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be points of $\\mathbb{R}^d$ that are not all equal, let $\\theta$ be valid parameters of a $K$-component isotropic Gaussian mixture, and let $\\theta^+$ be the result of the EM step applied to $\\theta$. Then every valid parameter $\\theta'$ satisfies $Q(\\theta' \\mid \\theta) \\le Q(\\theta^+ \\mid \\theta)$, where $Q$ is the EM objective of the mixture's joint density.","title":"The M-step for isotropic Gaussian mixtures maximizes the EM objective"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/log-lik-le-em-step-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.logLik_le_emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#log-lik-le-em-step-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be points of $\\mathbb{R}^d$ that are not all equal and let $\\theta$ be valid parameters of a $K$-component isotropic Gaussian mixture. Then $\\ell(\\theta) \\le \\ell(\\theta^+)$, where $\\ell$ is the log-likelihood of the data under the mixture and $\\theta^+$ is the result of the EM step applied to $\\theta$.","title":"Each EM step for the Gaussian mixture does not decrease the log-likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/log-lik-iterate-em-step-monotone-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixture.logLik_iterate_emStep_monotone","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#log-lik-iterate-em-step-monotone-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be points of $\\mathbb{R}^d$ that are not all equal, let $\\theta_0$ be valid parameters of a $K$-component isotropic Gaussian mixture, and let $\\theta_t$ be the result of $t$ EM steps from $\\theta_0$. Then $\\ell(\\theta_t)$ is nondecreasing in $t \\in \\mathbb{N}$, where $\\ell$ is the log-likelihood of the data under the mixture.","title":"The log-likelihood is monotone along the EM iterates of an isotropic Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/params-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.Params","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#params-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"A parameter of the $K$-component Gaussian mixture in $d$ dimensions with full covariances consists of, for each component $k < K$, a real mixing weight $\\pi_k$, a mean $\\mu_k \\in \\mathbb{R}^d$ and a real $d \\times d$ covariance matrix $\\Sigma_k$, with no constraints imposed.","title":"Parameters of a full-covariance Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/valid-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.Params.Valid","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#valid-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Parameters $(\\pi_k, \\mu_k, \\Sigma_k)_{k\\lt K}$ of a $K$-component Gaussian mixture in $d$ dimensions are valid when every weight $\\pi_k$ is positive, the weights sum to one, and every covariance matrix $\\Sigma_k$ is positive definite.","title":"Validity of full-covariance Gaussian mixture parameters"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/valid-params-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.ValidParams","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#valid-params-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"The parameter space of EM for the $K$-component Gaussian mixture in $d$ dimensions with full covariances is the set of parameters $(\\pi_k, \\mu_k, \\Sigma_k)_{k\\lt K}$ that are valid: every weight $\\pi_k$ is positive, the weights sum to one, and every covariance matrix $\\Sigma_k$ is positive definite.","title":"Valid parameters of a full-covariance Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/joint-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.joint","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#joint-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For valid parameters $\\theta$ with weights $\\pi_k$, means $\\mu_k$ and covariances $\\Sigma_k$, the joint density of a point $x \\in \\mathbb{R}^d$ and a component $k$ is $p_\\theta(x, k) = \\pi_k \\, \\mathcal{N}(x \\mid \\mu_k, \\Sigma_k)$, where $\\mathcal{N}(x \\mid \\mu, \\Sigma)$ is the multivariate Gaussian density.","title":"Joint density of the full-covariance Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/resp-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.resp","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#resp-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For valid parameters $\\theta = (\\pi_j, \\mu_j, \\Sigma_j)_{j\\lt K}$, a point $x \\in \\mathbb{R}^d$ and a component $k$, write $p_k(x) = \\pi_k \\, \\mathcal{N}(x \\mid \\mu_k, \\Sigma_k)$, where $\\mathcal{N}(\\cdot \\mid \\mu, \\Sigma)$ is the multivariate Gaussian density. The responsibility of $k$ for $x$ is the posterior probability of component $k$ given $x$ (the E-step),","title":"Responsibilities of the full-covariance Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/mass-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.mass","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#mass-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For data $x_1, \\dots, x_N \\in \\mathbb{R}^d$ and valid parameters $\\theta$ with responsibilities $r_{nk}$, the effective number of points of component $k$ is $N_k = \\sum_n r_{nk}$.","title":"Effective number of points of a full-covariance mixture component"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/new-mean-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.newMean","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#new-mean-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Given data $(x_n)_{n\\lt N}$ in $\\mathbb{R}^d$, valid parameters $\\theta$ and a component $k$, let $r_{nk}$ be the responsibility of component $k$ for $x_n$ and let $N_k = \\sum_n r_{nk}$. The mean update is the responsibility-weighted mean","title":"Mean update of the full-covariance Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/new-cov","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.newCov","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#new-cov","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Given data $(x_n)_{n\\lt N}$ in $\\mathbb{R}^d$, valid parameters $\\theta$ and a component $k$, let $r_{nk}$ be the responsibility of component $k$ for $x_n$, let $N_k = \\sum_n r_{nk}$, and let $\\mu_k'$ be the responsibility-weighted mean $N_k^{-1} \\sum_n r_{nk} x_n$. The covariance update is the real $d \\times d$ matrix","title":"Covariance update of the full-covariance Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/m-step-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.mStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#m-step-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For data $x_1, \\dots, x_N \\in \\mathbb{R}^d$ and valid parameters $\\theta$ with responsibilities $r_{nk}$, let $N_k = \\sum_n r_{nk}$. The M-step parameters have weights $\\pi'_k = N_k / N$, means $\\mu'_k = N_k^{-1} \\sum_n r_{nk} x_n$ and covariances $\\Sigma'_k = N_k^{-1} \\sum_n r_{nk} (x_n - \\mu'_k)(x_n - \\mu'_k)^{\\top}$.","title":"M-step of the full-covariance Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/spans-affinely","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.SpansAffinely","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#spans-affinely","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"A family $(x_n)_{n\\lt N}$ of points of $\\mathbb{R}^d$ is in general position when it has at least one point ($N > 0$) and is not contained in an affine hyperplane: for every nonzero $v \\in \\mathbb{R}^d$ there are indices $n, n'$ with $\\langle v, x_n \\rangle \\neq \\langle v, x_{n'} \\rangle$.","title":"Data that span the space affinely"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/new-cov-pos-def","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.newCov_posDef","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#new-cov-pos-def","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $(x_n)_{n\\lt N}$ be data in $\\mathbb{R}^d$ in general position: $N > 0$, and for every nonzero $v \\in \\mathbb{R}^d$ the values $\\langle v, x_n \\rangle$ are not all equal. Then for all valid parameters $\\theta$ of the $K$-component full-covariance Gaussian mixture and every component $k$, the covariance update $\\Sigma_k'$ (the responsibility-weighted sample covariance of the data about the\u2026","title":"The covariance update is positive definite"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-step-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-step-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"For data $x_1, \\dots, x_N \\in \\mathbb{R}^d$ in general position (at least one point, and not all in an affine hyperplane) and valid parameters $\\theta$, the EM step returns the M-step parameters of $\\theta$: with responsibilities $r_{nk}$ under $\\theta$ and $N_k = \\sum_n r_{nk}$, the new weights are $N_k / N$, the new means are the responsibility-weighted means and the new covariances are the\u2026","title":"EM step for the full-covariance Gaussian mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/comp-term-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.compTerm","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#comp-term-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Given points $x_1, \\dots, x_N$ of $\\mathbb{R}^d$, valid parameters $\\theta$ of a $K$-component full-covariance Gaussian mixture, a component $k$, a mean $\\mu \\in \\mathbb{R}^d$ and a real $d \\times d$ matrix $S$, the component term is","title":"Component term of the EM objective for full-covariance Gaussian mixtures"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-q-eq-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.emQ_eq","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-q-eq-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N$ be points of $\\mathbb{R}^d$, and let $\\theta'$ and $\\theta$ be valid parameters of a $K$-component full-covariance Gaussian mixture, where $\\theta'$ has weights $\\pi_k'$, means $\\mu_k'$ and covariance matrices $\\Sigma_k'$. Then the EM objective of the mixture's joint density satisfies","title":"The EM objective of a full-covariance Gaussian mixture splits by component"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/comp-term-le-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.compTerm_le","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#comp-term-le-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data in general position (at least one point, and not all in an affine hyperplane), let $\\theta$ be valid mixture parameters with responsibilities $r_{nk}$, and fix a component $k$. Write $\\mathcal{N}(x \\mid \\mu, S)$ for the multivariate Gaussian density and let $\\mu'_k$ and $\\Sigma'_k$ be the responsibility-weighted mean and the responsibility-weighted sample covariance of\u2026","title":"The mean and covariance updates maximize each component term"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/sum-mass-mul-log-le-2","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.sum_mass_mul_log_le","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#sum-mass-mul-log-le-2","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $(x_n)_{n\\lt N}$ be data in $\\mathbb{R}^d$ with $N > 0$, let $\\theta$ be valid parameters, and let $N_k = \\sum_n r_{nk}$, where $r_{nk}$ is the responsibility of component $k$ for $x_n$. Then for all real weights $w_k > 0$ with $\\sum_k w_k = 1$,","title":"The weight update maximizes the weights term of the full-covariance mixture"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/em-q-le-em-step-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.emQ_le_emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#em-q-le-em-step-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data in general position: $N > 0$, and for every nonzero $v \\in \\mathbb{R}^d$ the values $\\langle v, x_n \\rangle$ are not all equal. Let $\\theta$ be valid parameters of a $K$-component full-covariance Gaussian mixture, and let $\\theta^+$ be the result of the EM step applied to $\\theta$. Then every valid parameter $\\theta'$ satisfies\u2026","title":"The M-step of the full-covariance Gaussian mixture maximizes the EM objective"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/log-lik-le-em-step-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.logLik_le_emStep","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#log-lik-le-em-step-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"Each EM step for the full-covariance Gaussian mixture does not decrease the log-likelihood. Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data in general position (at least one point, and not all in an affine hyperplane) and let $\\theta$ be valid parameters. If $\\theta^{+}$ is the result of one EM step from $\\theta$, then $\\ell(\\theta) \\le \\ell(\\theta^{+})$, where $\\ell$ is the log-likelihood of the data under the\u2026","title":"Each EM step for the full-covariance Gaussian mixture does not decrease the log-likelihood"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/log-lik-iterate-em-step-monotone-3","kind":"node","lean":"Cslib.MachineLearning.LatentVariable.GaussianMixtureFullCov.logLik_iterate_emStep_monotone","parent":41543,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models/index.html#log-lik-iterate-em-step-monotone-3","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/mixture-models","status":"fully_proved","summary":"The log-likelihood is monotone along the EM iterates of a full-covariance Gaussian mixture. Let $x_1, \\dots, x_N \\in \\mathbb{R}^d$ be data in general position: $N > 0$, and for every nonzero $v \\in \\mathbb{R}^d$ the values $\\langle v, x_n \\rangle$ are not all equal. Let $\\theta_0$ be valid parameters of a $K$-component full-covariance Gaussian mixture, and let $\\theta_t$ be the result of $t$ EM steps from\u2026","title":"The log-likelihood is monotone along the EM iterates of a full-covariance Gaussian mixture"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","status":"fully_proved","title":"Monte Carlo and importance sampling"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/mc-estimate","kind":"node","lean":"Cslib.MachineLearning.Sampling.mcEstimate","parent":41605,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/index.html#mc-estimate","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","status":"fully_proved","summary":"Given a function $f : \\alpha \\to \\mathbb{R}$ and samples $X_0, X_1, \\ldots : \\Omega \\to \\alpha$, the Monte Carlo estimate from the first $n$ samples at $\\omega \\in \\Omega$ is $\\frac{1}{n} \\sum_{i < n} f(X_i(\\omega))$, which is $0$ when $n = 0$.","title":"Monte Carlo estimate"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/integral-mc-estimate","kind":"node","lean":"Cslib.MachineLearning.Sampling.integral_mcEstimate","parent":41605,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/index.html#integral-mc-estimate","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$, let $\\nu$ be a measure on $\\alpha$, and let $f : \\alpha \\to \\mathbb{R}$ be measurable and integrable with respect to $\\nu$. If $X_0, X_1, \\ldots : \\Omega \\to \\alpha$ are measurable and the pushforward of $\\mu$ along each $X_i$ is $\\nu$, then for every $n \\ne 0$ the integral with respect to $\\mu$ of the Monte Carlo estimate $\\frac{1}{n} \\sum_{i < n} f(X_i)$ equals $\\int f \\, d\\nu$.","title":"The Monte Carlo estimator is unbiased"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/tendsto-mc-estimate","kind":"node","lean":"Cslib.MachineLearning.Sampling.tendsto_mcEstimate","parent":41605,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/index.html#tendsto-mc-estimate","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","status":"fully_proved","summary":"The Monte Carlo estimator is consistent (strong law of large numbers). Let $\\mu$ be a measure on $\\Omega$, let $\\nu$ be a measure on $\\alpha$, and let $f : \\alpha \\to \\mathbb{R}$ be measurable and integrable with respect to $\\nu$. If $X_0, X_1, \\ldots : \\Omega \\to \\alpha$ are measurable, pairwise independent under $\\mu$, and the pushforward of $\\mu$ along each $X_i$ is $\\nu$, then for $\\mu$-almost every $\\omega$\u2026","title":"The Monte Carlo estimator is consistent"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/is-estimate","kind":"node","lean":"Cslib.MachineLearning.Sampling.isEstimate","parent":41605,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/index.html#is-estimate","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","status":"fully_proved","summary":"Given a weight $w : \\alpha \\to [0, \\infty]$, a function $f : \\alpha \\to \\mathbb{R}$ and samples $X_0, X_1, \\ldots : \\Omega \\to \\alpha$, the importance-sampling estimate from the first $n$ samples at $\\omega \\in \\Omega$ is $\\frac{1}{n} \\sum_{i < n} w(X_i(\\omega)) \\, f(X_i(\\omega))$, where an infinite weight counts as $0$ and the estimate is $0$ when $n = 0$. If the samples are drawn from a proposal measure $\\rho$,\u2026","title":"Importance-sampling estimate"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/integral-weight-mul","kind":"node","lean":"Cslib.MachineLearning.Sampling.integral_weight_mul","parent":41605,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/index.html#integral-weight-mul","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","status":"fully_proved","summary":"Let $\\rho$ be a measure on $\\alpha$, let $w : \\alpha \\to [0, \\infty]$ be measurable with $w < \\infty$ $\\rho$-almost everywhere, and let $\\nu$ be the measure with density $w$ with respect to $\\rho$. Then for every function $f : \\alpha \\to \\mathbb{R}$, $\\int w f \\, d\\rho = \\int f \\, d\\nu$.","title":"Change of measure for importance weights"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/integral-is-estimate","kind":"node","lean":"Cslib.MachineLearning.Sampling.integral_isEstimate","parent":41605,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/index.html#integral-is-estimate","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$, let $\\rho$ be a measure on $\\alpha$, let $w : \\alpha \\to [0, \\infty]$ be measurable with $w < \\infty$ $\\rho$-almost everywhere, and let $\\nu$ be the measure with density $w$ with respect to $\\rho$. Let $f : \\alpha \\to \\mathbb{R}$ be measurable and integrable with respect to $\\nu$. If $X_0, X_1, \\ldots : \\Omega \\to \\alpha$ are measurable and the pushforward of $\\mu$ along each\u2026","title":"Importance sampling is unbiased"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/tendsto-is-estimate","kind":"node","lean":"Cslib.MachineLearning.Sampling.tendsto_isEstimate","parent":41605,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling/index.html#tendsto-is-estimate","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/monte-carlo-and-importance-sampling","status":"fully_proved","summary":"Let $\\mu$ be a measure on $\\Omega$, let $\\rho$ be a measure on $\\alpha$, let $w : \\alpha \\to [0, \\infty]$ be measurable with $w < \\infty$ $\\rho$-almost everywhere, and let $\\nu$ be the measure with density $w$ with respect to $\\rho$. Let $f : \\alpha \\to \\mathbb{R}$ be measurable and integrable with respect to $\\nu$. If $X_0, X_1, \\ldots : \\Omega \\to \\alpha$ are measurable, pairwise independent under $\\mu$, and the\u2026","title":"Importance sampling is consistent"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","title":"Rejection sampling"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/coin","kind":"node","lean":"Cslib.MachineLearning.Sampling.coin","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#coin","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"For $r \\in [0, \\infty]$ with $r \\le 1$, the biased coin with parameter $r$ is the probability mass function on $\\{\\mathrm{true}, \\mathrm{false}\\}$ that gives $\\mathrm{true}$ probability $r$ and $\\mathrm{false}$ probability $1 - r$.","title":"Biased coin"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/accept-prob","kind":"node","lean":"Cslib.MachineLearning.Sampling.acceptProb","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#accept-prob","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"Given probability mass functions $p$ and $q$ on a type $\\alpha$, a constant $M \\in [0, \\infty]$ and a point $x \\in \\alpha$, the acceptance probability of the proposal $x$ is $p(x) / (M\\,q(x))$, computed in $[0, \\infty]$, where $a / 0 = \\infty$ for $a \\ne 0$ and $0 / 0 = 0$.","title":"Acceptance probability"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/trial","kind":"node","lean":"Cslib.MachineLearning.Sampling.trial","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#trial","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"Let $p$ and $q$ be probability mass functions on a type $\\alpha$ (the target and the proposal) and let $M \\in [0, \\infty]$ be an envelope constant with $p(x) \\le M\\,q(x)$ for all $x \\in \\alpha$. One round of rejection sampling for target $p$, proposal $q$ and envelope $M$ is the probability mass function on pairs $(x, b) \\in \\alpha \\times \\{\\mathrm{true}, \\mathrm{false}\\}$ obtained by drawing $x$ from $q$ and then\u2026","title":"One round of rejection sampling"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/trial-apply-true","kind":"node","lean":"Cslib.MachineLearning.Sampling.trial_apply_true","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#trial-apply-true","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"Let $p$ and $q$ be probability mass functions on $\\alpha$ and $M \\in [0, \\infty]$ with $p(y) \\le M\\,q(y)$ for all $y \\in \\alpha$. For every $x \\in \\alpha$, one round of rejection sampling for target $p$, proposal $q$ and envelope $M$ yields the pair $(x, \\mathrm{true})$, that is, proposes $x$ and accepts it, with probability $p(x) / M$.","title":"Probability that a round proposes and accepts a given point"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/tsum-trial-true","kind":"node","lean":"Cslib.MachineLearning.Sampling.tsum_trial_true","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#tsum-trial-true","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"Let $p$ and $q$ be probability mass functions on $\\alpha$ and $M \\in [0, \\infty]$ with $p(x) \\le M\\,q(x)$ for all $x \\in \\alpha$. The sum over $x \\in \\alpha$ of the probabilities that one round of rejection sampling for target $p$, proposal $q$ and envelope $M$ yields $(x, \\mathrm{true})$ equals $M^{-1}$; that is, a round accepts with probability $1 / M$.","title":"Probability that a round accepts"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/tsum-trial-false-add","kind":"node","lean":"Cslib.MachineLearning.Sampling.tsum_trial_false_add","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#tsum-trial-false-add","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"Let $p$ and $q$ be probability mass functions on $\\alpha$ and $M \\in [0, \\infty]$ with $p(x) \\le M\\,q(x)$ for all $x \\in \\alpha$. Let $r$ be the sum over $x \\in \\alpha$ of the probabilities that one round of rejection sampling for target $p$, proposal $q$ and envelope $M$ yields $(x, \\mathrm{false})$, that is, the probability that the round rejects. Then $r + M^{-1} = 1$.","title":"Probability that a round rejects"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/trial-true-div-tsum","kind":"node","lean":"Cslib.MachineLearning.Sampling.trial_true_div_tsum","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#trial-true-div-tsum","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"Let $p$ and $q$ be probability mass functions on $\\alpha$ and $M \\in [0, \\infty]$ with $M \\ne 0$, $M \\ne \\infty$ and $p(y) \\le M\\,q(y)$ for all $y \\in \\alpha$. For one round of rejection sampling for target $p$, proposal $q$ and envelope $M$, write $P(x, \\mathrm{accepted})$ for the probability that the round proposes $x$ and accepts it, and $P(\\mathrm{accepted}) = \\sum_{y \\in \\alpha} P(y, \\mathrm{accepted})$. Then\u2026","title":"Conditional on acceptance, the proposal has the target law"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/tsum-rejections-mul-trial","kind":"node","lean":"Cslib.MachineLearning.Sampling.tsum_rejections_mul_trial","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#tsum-rejections-mul-trial","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"Let $p$ and $q$ be probability mass functions on $\\alpha$ and $M \\in [0, \\infty]$ with $M \\ne 0$, $M \\ne \\infty$ and $p(y) \\le M\\,q(y)$ for all $y \\in \\alpha$. For one round of rejection sampling for target $p$, proposal $q$ and envelope $M$, write $P(x, \\mathrm{accepted})$ for the probability that the round proposes $x$ and accepts it, and $r$ for the sum over $y \\in \\alpha$ of the probabilities that it proposes\u2026","title":"Rejection sampling outputs the target law"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/eq-of-first-step","kind":"node","lean":"Cslib.MachineLearning.Sampling.eq_of_firstStep","parent":41613,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling/index.html#eq-of-first-step","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/rejection-sampling","status":"fully_proved","summary":"Let $p$ and $q$ be probability mass functions on $\\alpha$ and $M \\in [0, \\infty]$ with $M \\ne 0$, $M \\ne \\infty$ and $p(y) \\le M\\,q(y)$ for all $y \\in \\alpha$. For one round of rejection sampling for target $p$, proposal $q$ and envelope $M$, write $P(x, \\mathrm{accepted})$ for the probability that the round proposes $x$ and accepts it, and $r$ for the sum over $y \\in \\alpha$ of the probabilities that it proposes\u2026","title":"Uniqueness for the first-step equation of rejection sampling"},{"data_url":"scopes/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo.json","id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","kind":"scope","navigation":"module","parent":41437,"result":false,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","title":"Markov chain Monte Carlo"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/pmf-kernel","kind":"node","lean":"Cslib.MachineLearning.Sampling.pmfKernel","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#pmf-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\alpha$ be a finite state space with a measurable structure in which singletons are measurable. A transition rule $P$ assigning to each $x \\in \\alpha$ a probability distribution $P(x, \\cdot)$ on $\\alpha$ gives the Markov kernel on $\\alpha$ that sends $x$ to $P(x, \\cdot)$ viewed as a probability measure, so a set $B$ gets mass $\\sum_{y \\in B} P(x, y)$.","title":"Markov kernel of a transition rule"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/is-reversible-pmf-kernel","kind":"node","lean":"Cslib.MachineLearning.Sampling.isReversible_pmfKernel","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#is-reversible-pmf-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\alpha$ be a finite state space with a measurable structure in which singletons are measurable, $\\pi$ a probability distribution on $\\alpha$ and $P$ a transition rule assigning to each $x \\in \\alpha$ a probability distribution $P(x, \\cdot)$ on $\\alpha$. If $\\pi(x) P(x, y) = \\pi(y) P(y, x)$ for all $x, y \\in \\alpha$, then the Markov kernel $\\kappa$ of $P$ is reversible with respect to $\\pi$, viewed as a\u2026","title":"Detailed balance implies reversibility"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/invariant-pmf-kernel","kind":"node","lean":"Cslib.MachineLearning.Sampling.invariant_pmfKernel","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#invariant-pmf-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\alpha$ be a finite state space with a measurable structure in which singletons are measurable, $\\pi$ a probability distribution on $\\alpha$ and $P$ a transition rule assigning to each $x \\in \\alpha$ a probability distribution $P(x, \\cdot)$ on $\\alpha$. If $\\pi(x) P(x, y) = \\pi(y) P(y, x)$ for all $x, y \\in \\alpha$, then $\\pi$, viewed as a measure, is invariant for the Markov kernel $\\kappa$ of $P$: for every\u2026","title":"Detailed balance implies invariance"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/bind-eq-of-detailed-balance","kind":"node","lean":"Cslib.MachineLearning.Sampling.bind_eq_of_detailedBalance","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#bind-eq-of-detailed-balance","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\alpha$ be a finite set, $\\pi$ a probability distribution on $\\alpha$ and $P$ a transition rule assigning to each $x \\in \\alpha$ a probability distribution $P(x, \\cdot)$ on $\\alpha$. If $\\pi(x) P(x, y) = \\pi(y) P(y, x)$ for all $x, y \\in \\alpha$, then one step of $P$ started from $\\pi$ is again distributed as $\\pi$: for every $y \\in \\alpha$, $\\sum_{x \\in \\alpha} \\pi(x) P(x, y) = \\pi(y)$.","title":"Detailed balance gives a stationary distribution"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/mh-accept","kind":"node","lean":"Cslib.MachineLearning.Sampling.mhAccept","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#mh-accept","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"For a probability distribution $\\pi$ on $\\alpha$ and a proposal $Q$ assigning to each $x \\in \\alpha$ a probability distribution $Q(x, \\cdot)$ on $\\alpha$, the acceptance probability of a move from $x$ to $y$ is $a(x, y) = \\min\\bigl(1, \\pi(y) Q(y, x) / (\\pi(x) Q(x, y))\\bigr)$, computed in $[0, \\infty]$ with $c / 0 = \\infty$ for $c \\ne 0$ and $0 / 0 = 0$.","title":"Metropolis\u2013Hastings acceptance probability"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/mh-step","kind":"node","lean":"Cslib.MachineLearning.Sampling.mhStep","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#mh-step","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"For a probability distribution $\\pi$ on $\\alpha$ (the target) and a proposal $Q$ assigning to each $x \\in \\alpha$ a probability distribution $Q(x, \\cdot)$ on $\\alpha$, one Metropolis\u2013Hastings step from $x$ is the distribution of the following move: draw $y$ from $Q(x, \\cdot)$, then move to $y$ with the Metropolis\u2013Hastings acceptance probability $a(x, y)$, and otherwise stay at $x$.","title":"Metropolis\u2013Hastings step"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/mh-detailed-balance","kind":"node","lean":"Cslib.MachineLearning.Sampling.mh_detailedBalance","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#mh-detailed-balance","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on a type $\\alpha$, $Q$ a proposal assigning to each $x \\in \\alpha$ a probability distribution $Q(x, \\cdot)$ on $\\alpha$, and $P(x, y)$ the probability that one Metropolis\u2013Hastings step for target $\\pi$ and proposal $Q$ moves from $x$ to $y$. Then $\\pi(x) P(x, y) = \\pi(y) P(y, x)$ for all $x, y \\in \\alpha$.","title":"Detailed balance for Metropolis\u2013Hastings"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/is-reversible-mh-kernel","kind":"node","lean":"Cslib.MachineLearning.Sampling.isReversible_mhKernel","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#is-reversible-mh-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\alpha$ be a finite state space with a measurable structure in which singletons are measurable, $\\pi$ a probability distribution on $\\alpha$ and $Q$ a proposal assigning to each $x \\in \\alpha$ a probability distribution $Q(x, \\cdot)$ on $\\alpha$. Then the Markov kernel $\\kappa$ of the Metropolis\u2013Hastings step for target $\\pi$ and proposal $Q$ is reversible with respect to $\\pi$, viewed as a measure: for all\u2026","title":"The Metropolis\u2013Hastings kernel is reversible"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/invariant-mh-kernel","kind":"node","lean":"Cslib.MachineLearning.Sampling.invariant_mhKernel","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#invariant-mh-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\alpha$ be a finite state space with a measurable structure in which singletons are measurable, $\\pi$ a probability distribution on $\\alpha$ and $Q$ a proposal assigning to each $x \\in \\alpha$ a probability distribution $Q(x, \\cdot)$ on $\\alpha$. Then $\\pi$, viewed as a measure, is invariant for the Markov kernel $\\kappa$ of the Metropolis\u2013Hastings step for target $\\pi$ and proposal $Q$: for every measurable\u2026","title":"The target is stationary for Metropolis\u2013Hastings"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/cond-norm","kind":"node","lean":"Cslib.MachineLearning.Sampling.condNorm","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#cond-norm","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite. For a coordinate $i \\in \\iota$ and a configuration $x$, the normalizer is $Z_i(x) = \\sum_{s \\in S} \\pi(x[i \\mapsto s]) \\in [0, \\infty]$, where $x[i \\mapsto s]$ is $x$ with coordinate $i$ set to $s$. It normalizes the conditional law of coordinate $i$ given the other coordinates of $x$.","title":"Normalizer of the conditional law"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/cond-law","kind":"node","lean":"Cslib.MachineLearning.Sampling.condLaw","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#cond-law","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite, such that $\\pi(x) \\neq 0$ for every $x$. For a coordinate $i \\in \\iota$ and a configuration $x$, the conditional law of coordinate $i$ given the others is the probability distribution on $S$ that gives $s \\in S$ the mass $\\pi(x[i \\mapsto s]) / Z_i(x)$, where $x[i \\mapsto s]$ is $x$ with coordinate $i$ set\u2026","title":"Conditional law of a coordinate"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/gibbs-step","kind":"node","lean":"Cslib.MachineLearning.Sampling.gibbsStep","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#gibbs-step","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite, such that $\\pi(x) \\neq 0$ for every $x$. The Gibbs update of coordinate $i \\in \\iota$ from $x$ is the probability distribution of $x[i \\mapsto s]$, the configuration $x$ with coordinate $i$ set to $s$, where $s$ is drawn from the conditional law of coordinate $i$ given the other coordinates of $x$.","title":"Gibbs update"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/update-eq-comm","kind":"node","lean":"Cslib.MachineLearning.Sampling.update_eq_comm","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#update-eq-comm","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"For functions $x, y : \\iota \\to S$ and a coordinate $i \\in \\iota$, if setting coordinate $i$ of $x$ to $y(i)$ gives $y$, then setting coordinate $i$ of $y$ to $x(i)$ gives $x$.","title":"Symmetry of differing in one coordinate"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/gibbs-detailed-balance","kind":"node","lean":"Cslib.MachineLearning.Sampling.gibbs_detailedBalance","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#gibbs-detailed-balance","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite, such that $\\pi(x) \\neq 0$ for every $x$, and for a coordinate $i$ let $P_i(x, y)$ be the probability that the Gibbs update of coordinate $i$ moves $x$ to $y$. Then $\\pi(x) P_i(x, y) = \\pi(y) P_i(y, x)$ for all configurations $x$ and $y$.","title":"Detailed balance for a Gibbs update"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/is-reversible-gibbs-kernel","kind":"node","lean":"Cslib.MachineLearning.Sampling.isReversible_gibbsKernel","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#is-reversible-gibbs-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite, such that $\\pi(x) \\neq 0$ for every $x$. For every coordinate $i$, the Markov kernel $K$ of the Gibbs update of coordinate $i$ is reversible with respect to $\\pi$: for all measurable sets $A$ and $B$ of configurations, $\\int_A K(x, B) \\, d\\pi(x) = \\int_B K(x, A) \\, d\\pi(x)$.","title":"Gibbs updates are reversible"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/invariant-gibbs-kernel","kind":"node","lean":"Cslib.MachineLearning.Sampling.invariant_gibbsKernel","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#invariant-gibbs-kernel","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite, such that $\\pi(x) \\neq 0$ for every $x$. For every coordinate $i$, $\\pi$ is an invariant measure of the Markov kernel of the Gibbs update of coordinate $i$: if $x$ is drawn from $\\pi$ and coordinate $i$ is resampled from its conditional law given the other coordinates, the result is again distributed as\u2026","title":"Gibbs sampling leaves the target stationary"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/gibbs-sweep","kind":"node","lean":"Cslib.MachineLearning.Sampling.gibbsSweep","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#gibbs-sweep","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite, such that $\\pi(x) \\neq 0$ for every $x$. For a list $l = (i_1, \\dots, i_n)$ of coordinates, the systematic-scan Gibbs sampler is the Markov kernel on configurations that applies the Gibbs updates of the coordinates $i_1, i_2, \\dots, i_n$ in this order; for the empty list it is the identity kernel.","title":"Systematic-scan Gibbs sampler"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/gibbs-sweep-cons-apply","kind":"node","lean":"Cslib.MachineLearning.Sampling.gibbsSweep_cons_apply","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#gibbs-sweep-cons-apply","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite, such that $\\pi(x) \\neq 0$ for every $x$. For a list $l$ of coordinates let $K_l$ be the systematic-scan Gibbs sampler for $l$, and for a coordinate $i$ let $P_i$ be the Markov kernel of the Gibbs update of coordinate $i$. Then for every configuration $x$, the law $K_{(i, l)}(x)$ of the sampler for the list\u2026","title":"First step of a Gibbs sweep"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/invariant-gibbs-sweep","kind":"node","lean":"Cslib.MachineLearning.Sampling.invariant_gibbsSweep","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#invariant-gibbs-sweep","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Let $\\pi$ be a probability distribution on the configurations $x : \\iota \\to S$, with $\\iota$ and $S$ finite, such that $\\pi(x) \\neq 0$ for every $x$. For every finite list $l$ of coordinates, in any order and possibly with repetitions, $\\pi$ is an invariant measure of the systematic-scan Gibbs sampler that updates the coordinates in the order of $l$.","title":"A full Gibbs sweep leaves the target stationary"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/sum-to-real-pmf","kind":"node","lean":"Cslib.MachineLearning.Sampling.sum_toReal_pmf","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#sum-to-real-pmf","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"The probabilities of a law on a finite type, as reals, sum to `1`.","title":"Probabilities of a finite law sum to one"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/mh-matrix","kind":"node","lean":"Cslib.MachineLearning.Sampling.mhMatrix","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#mh-matrix","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"The transition matrix `P(x, y) = mhStep \u03c0 Q x y` of the Metropolis\u2013Hastings chain, as a real matrix.","title":"Metropolis\u2013Hastings transition matrix"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/mh-matrix-mem-row-stochastic","kind":"node","lean":"Cslib.MachineLearning.Sampling.mhMatrix_mem_rowStochastic","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#mh-matrix-mem-row-stochastic","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"The Metropolis\u2013Hastings matrix is stochastic.","title":"Metropolis\u2013Hastings matrix is stochastic"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/vec-mul-mh-matrix","kind":"node","lean":"Cslib.MachineLearning.Sampling.vecMul_mhMatrix","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#vec-mul-mh-matrix","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"The target is stationary for the Metropolis\u2013Hastings matrix.","title":"Target is stationary for the Metropolis\u2013Hastings matrix"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/is-primitive-mh-matrix","kind":"node","lean":"Cslib.MachineLearning.Sampling.isPrimitive_mhMatrix","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#is-primitive-mh-matrix","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"For a target with full support and a proposal with primitive transition matrix and symmetric support, the Metropolis\u2013Hastings matrix is primitive.","title":"Primitivity of the Metropolis\u2013Hastings matrix"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/exists-tv-dist-mh-matrix-pow-le","kind":"node","lean":"Cslib.MachineLearning.Sampling.exists_tvDist_mhMatrix_pow_le","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#exists-tv-dist-mh-matrix-pow-le","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"Under the hypotheses of `isPrimitive_mhMatrix`, there are `C` and `0 \u2264 \u03c1 < 1` with `d_TV(P\u207f(x, \u00b7), \u03c0) \u2264 C \u03c1\u207f` for every start `x`.","title":"The Metropolis\u2013Hastings chain converges to its target"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/gibbs-sweep-matrix","kind":"node","lean":"Cslib.MachineLearning.Sampling.gibbsSweepMatrix","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#gibbs-sweep-matrix","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"The transition matrix `P(x, y) = gibbsSweep \u03c0 h\u03c0 l x {y}` of the systematic-scan Gibbs sweep along `l`, as a real matrix.","title":"Transition matrix of the systematic-scan Gibbs sweep"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/gibbs-sweep-matrix-mem-row-stochastic","kind":"node","lean":"Cslib.MachineLearning.Sampling.gibbsSweepMatrix_mem_rowStochastic","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#gibbs-sweep-matrix-mem-row-stochastic","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"The Gibbs sweep matrix is stochastic.","title":"Gibbs sweep matrix is stochastic"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/vec-mul-gibbs-sweep-matrix","kind":"node","lean":"Cslib.MachineLearning.Sampling.vecMul_gibbsSweepMatrix","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#vec-mul-gibbs-sweep-matrix","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"The target is stationary for the Gibbs sweep matrix.","title":"Target is stationary for the Gibbs sweep matrix"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/gibbs-sweep-matrix-pos","kind":"node","lean":"Cslib.MachineLearning.Sampling.gibbsSweepMatrix_pos","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#gibbs-sweep-matrix-pos","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"If the sweep visits every coordinate, every transition probability of the sweep is positive.","title":"Positivity of Gibbs sweep transition probabilities"},{"id":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/exists-tv-dist-gibbs-sweep-matrix-pow-le","kind":"node","lean":"Cslib.MachineLearning.Sampling.exists_tvDist_gibbsSweepMatrix_pow_le","parent":41623,"result":true,"route":"../roadmap/ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo/index.html#exists-tv-dist-gibbs-sweep-matrix-pow-le","scope":"ComputerScience/machine-learning/bayesian-inference-and-sampling/markov-chain-monte-carlo","status":"fully_proved","summary":"If the sweep visits every coordinate, there are `C` and `0 \u2264 \u03c1 < 1` with `d_TV(P\u207f(x, \u00b7), \u03c0) \u2264 C \u03c1\u207f` for every start `x`.","title":"The systematic-scan Gibbs sampler converges to its target"},{"data_url":"scopes/ComputerScience/sorting-and-searching.json","id":"ComputerScience/sorting-and-searching","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/index.html","scope":"ComputerScience/sorting-and-searching","status":"can_prove","summary":"Comparison and distribution sorting algorithms (including their expected-cost analyses), and searching and selection in lists and arrays (binary and linear search, quickselect, median of medians, simultaneous min-max). Comparator sorting networks, their depth and the 0-1 principle form the Sorting networks subfield.","title":"Sorting and searching"},{"data_url":"scopes/ComputerScience/sorting-and-searching/sorting-networks.json","id":"ComputerScience/sorting-and-searching/sorting-networks","kind":"scope","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/index.html","scope":"ComputerScience/sorting-and-searching/sorting-networks","status":"fully_proved","summary":"Comparator networks that sort: the bubble network, Batcher's bitonic and odd\u2013even merge networks, depth bounds, and the zero-one and permutation principles.","title":"Sorting networks"},{"data_url":"scopes/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle.json","id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","kind":"scope","navigation":"module","parent":41655,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","title":"Sorting networks and the zero-one principle"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/network","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Network","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#network","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"A comparator network on $n$ wires is a finite list of pairs $(i, j)$ of wires $i, j \\in \\{0, \\dots, n - 1\\}$, each pair standing for the comparator $(i, j)$; the comparators are applied in list order.","title":"Comparator network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/comparator","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.comparator","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#comparator","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"Let $\\alpha$ be a linear order, let $n$ be a natural number, and let $v$ assign a value $v(k) \\in \\alpha$ to each wire $k \\in \\{0, \\dots, n - 1\\}$. For wires $i$ and $j$, the comparator $(i, j)$ sends $v$ to the assignment that has $\\min(v(i), v(j))$ on wire $i$, $\\max(v(i), v(j))$ on wire $j$ when $j \\ne i$, and $v(k)$ on every other wire $k$.","title":"Comparator"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/run","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.run","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#run","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"Let $\\alpha$ be a linear order. The output of a comparator network on an assignment $v$ of a value in $\\alpha$ to each of its $n$ wires is obtained by starting from $v$ and applying the comparators of the network one after another, in list order.","title":"Running a comparator network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/sorts","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Sorts","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#sorts","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"A comparator network on $n$ wires sorts inputs over a linearly ordered type $\\alpha$ if, for every input $v$ assigning a value in $\\alpha$ to each wire, the output of running the network on $v$ is monotone in the wire order: for all wires $i \\le j$, the output value on wire $i$ is at most the output value on wire $j$.","title":"Networks that sort"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/comparator-comp","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.comparator_comp","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#comparator-comp","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be linear orders, let $f : \\alpha \\to \\beta$ be monotone, and let $n$ be a natural number. For all wires $i, j \\in \\{0, \\dots, n - 1\\}$ and every assignment $v$ of a value in $\\alpha$ to each wire, the comparator $(i, j)$ applied to $f \\circ v$ gives $f \\circ w$, where $w$ is the output of the comparator $(i, j)$ on $v$.","title":"Comparators commute with monotone maps"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/run-comp","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.run_comp","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#run-comp","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be linear orders and let $f : \\alpha \\to \\beta$ be monotone. For every comparator network on $n$ wires and every assignment $v$ of a value in $\\alpha$ to each wire, the output of the network on $f \\circ v$ is $f \\circ w$, where $w$ is the output of the network on $v$.","title":"Networks commute with monotone maps"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/comparator-perm","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.comparator_perm","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#comparator-perm","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"Let $\\alpha$ be a linear order and let $n$ be a natural number. For all wires $i, j \\in \\{0, \\dots, n - 1\\}$ and every assignment $v$ of a value in $\\alpha$ to each wire, there is a permutation $\\sigma$ of the wires such that the output of the comparator $(i, j)$ on $v$ is $v \\circ \\sigma$.","title":"A comparator permutes its inputs"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/run-perm","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.run_perm","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#run-perm","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"Let $\\alpha$ be a linear order. For every comparator network on $n$ wires and every assignment $v$ of a value in $\\alpha$ to each wire, there is a permutation $\\sigma$ of the wires such that the output of the network on $v$ is $v \\circ \\sigma$.","title":"The output of a comparator network is a permutation of its input"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/sorts-of-sorts-bool","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.sorts_of_sorts_bool","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#sorts-of-sorts-bool","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"The 0-1 principle (CLRS Theorem 27.2): a network that sorts every Boolean input sorts every input over every linear order.","title":"The 0-1 principle"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/sorts-bool","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.sortsBool","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#sorts-bool","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"The Boolean sorting test of a comparator network on $n$ wires returns true exactly when, for every assignment $v$ of a Boolean value to each wire, the output $w$ of the network on $v$ satisfies $w(i) \\le w(j)$ for all wires $i \\le j$, where $\\mathrm{false} < \\mathrm{true}$.","title":"Boolean sorting test"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/sorts-bool-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.sortsBool_eq_true_iff","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#sorts-bool-eq-true-iff","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"A comparator network passes the Boolean sorting test if and only if it sorts inputs over the Booleans, that is, for every assignment of a Boolean value to each wire, the output of the network is monotone in the wire order.","title":"The Boolean sorting test is correct"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/sorts-of-sorts-bool-2","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.sorts_of_sortsBool","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#sorts-of-sorts-bool-2","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"If a comparator network on $n$ wires passes the Boolean sorting test, then for every linear order $\\alpha$ it sorts inputs over $\\alpha$: for every assignment $v$ of a value in $\\alpha$ to each wire, the output of the network on $v$ is monotone in the wire order.","title":"A network that passes the Boolean sorting test sorts over every linear order"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/spread","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.spread","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#spread","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"For a Boolean input $v$ on $n$ wires, the spread of $v$ is the input with natural-number values whose value on wire $i < n$ is $n + i$ if $v(i)$ is true and $i$ if $v(i)$ is false.","title":"Spread of a Boolean input"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/decide-le-spread","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.decide_le_spread","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#decide-le-spread","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"For every Boolean input $v$ on $n$ wires, applying the threshold map that sends a natural number $k$ to true if $n \\le k$ and to false otherwise to the spread of $v$ gives back $v$: for every wire $i$, the spread of $v$ at $i$ is at least $n$ exactly when $v(i)$ is true.","title":"A Boolean input is a threshold of its spread"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/sorts-of-sorts-injective","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.sorts_of_sorts_injective","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#sorts-of-sorts-injective","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"A network that sorts every input with distinct natural values sorts every input over every linear order.","title":"Sorting distinct inputs suffices"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/rank","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.rank","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#rank","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"For a map $w$ from $\\{0, \\dots, n-1\\}$ to the natural numbers and an index $i < n$, the rank of $w(i)$ among the values of $w$ is the number of indices $j < n$ with $w(j) < w(i)$. It is less than $n$, so it is again an index in $\\{0, \\dots, n-1\\}$.","title":"Rank among the values of an input"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/rank-perm","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.rankPerm","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#rank-perm","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"For an injective map $w$ from $\\{0, \\dots, n-1\\}$ to the natural numbers, the ranking permutation of $w$ is the permutation of $\\{0, \\dots, n-1\\}$ that sends each $i$ to the rank of $w(i)$ among the values of $w$, that is, to the number of indices $j$ with $w(j) < w(i)$. Since $w$ is injective, this map is a bijection.","title":"The ranking permutation"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/sorts-of-sorts-perm","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.sorts_of_sorts_perm","parent":41656,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle/index.html#sorts-of-sorts-perm","scope":"ComputerScience/sorting-and-searching/sorting-networks/sorting-networks-and-the-zero-one-principle","status":"fully_proved","summary":"A comparator network on $n$ wires that sorts each of the $n!$ permutations $\\sigma$ of $\\{0, \\dots, n-1\\}$, viewed as the input whose value on wire $i$ is $\\sigma(i)$, sorts every input over every linear order.","title":"The permutation principle"},{"data_url":"scopes/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter.json","id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","kind":"scope","navigation":"module","parent":41655,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","title":"Bitonic sorter"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/cmp","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.cmp","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#cmp","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $\\alpha$ be a linear order. For wires $i, j \\in \\mathbb{N}$ and an assignment $f$ of values in $\\alpha$ to the wires $x \\in \\mathbb{N}$, the comparator $(i, j)$ applied to $f$ puts $\\min(f(i), f(j))$ on wire $i$, puts $\\max(f(i), f(j))$ on wire $j$ when $j \\ne i$, and leaves every other wire unchanged.","title":"Comparator on natural-number-indexed wires"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/run-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.runN","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#run-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $\\alpha$ be a linear order. Given a list $ps$ of comparators, each a pair $(i, j)$ of natural numbers, and an input $f : \\mathbb{N} \\to \\alpha$ giving the value on each wire, running $ps$ on $f$ applies the comparators of $ps$ to the wire values one after another, from first to last. The comparator $(i, j)$ puts the minimum of the values on wires $i$ and $j$ on wire $i$ and their maximum on wire $j$, and leaves\u2026","title":"Running a comparator network on natural-number-indexed wires"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/within","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.Within","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#within","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"A list $ps$ of comparators, each a pair $(i, j)$ of natural numbers, lies within $[l, h)$ if every comparator $(i, j)$ in $ps$ satisfies $l \\le i < h$ and $l \\le j < h$.","title":"Comparators within an interval"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/run-n-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.runN_of_not_mem","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#run-n-of-not-mem","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $\\alpha$ be a linear order and let $ps$ be a list of comparators $(i, j)$ on $\\mathbb{N}$-indexed wires that all lie within $[l, h)$, that is, $l \\le i < h$ and $l \\le j < h$ for every $(i, j)$ in $ps$. Then for every input $f : \\mathbb{N} \\to \\alpha$ and every wire $x$ with $x < l$ or $h \\le x$, the value of wire $x$ after running $ps$ on $f$ is $f(x)$.","title":"Locality of comparator networks"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/run-n-const","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.runN_const","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#run-n-const","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $\\alpha$ be a linear order, let $l, h \\in \\mathbb{N}$, and let $c$ be a list of comparators on $\\mathbb{N}$-indexed wires each of which joins two wires $x$ with $l \\le x < h$. If an input $f : \\mathbb{N} \\to \\alpha$ takes the same value $a$ on every wire $x$ with $l \\le x < h$, then running $c$ on $f$ gives an output that also takes the value $a$ on every such wire.","title":"Constant blocks stay constant"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/half-cleaner","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.halfCleaner","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#half-cleaner","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For natural numbers $m$ and $b$, the half-cleaner on the block of wires $[b, b + 2m)$ is the list of comparators $(b + i, b + m + i)$ for $i = 0, 1, \\dots, m - 1$, in this order: it compares wire $b + i$ with wire $b + m + i$.","title":"Half-cleaner"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/run-n-half-cleaner","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.runN_halfCleaner","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#run-n-half-cleaner","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $\\alpha$ be a linear order, let $m, b \\in \\mathbb{N}$, and let $f : \\mathbb{N} \\to \\alpha$ be an input on $\\mathbb{N}$-indexed wires. Running the half-cleaner on $[b, b + 2m)$ on $f$ puts $\\min(f(b + i), f(b + m + i))$ on wire $b + i$ and $\\max(f(b + i), f(b + m + i))$ on wire $b + m + i$ for every $i < m$, and leaves the value $f(x)$ on every wire $x$ outside $[b, b + 2m)$ unchanged.","title":"Output of the half-cleaner"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/is-interval","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.IsInterval","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#is-interval","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"A sequence $s : \\mathbb{N} \\to \\{0, 1\\}$ is an interval indicator of length $L$ if there are natural numbers $a \\le e \\le L$ such that for every $i < L$, $s(i) = 1$ exactly when $a \\le i < e$.","title":"Interval indicators"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/is-bitonic","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.IsBitonic","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#is-bitonic","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"A sequence $s : \\mathbb{N} \\to \\{0, 1\\}$ is bitonic of length $L$ if either $s$ or its complement $1 - s$ is an interval indicator of length $L$; that is, on the positions $0, \\dots, L - 1$ it has the form $0^a 1^b 0^c$ or $1^a 0^b 1^c$. Values of $s$ at positions $i \\ge L$ are unconstrained.","title":"Bitonic 0-1 sequences"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/half-clean-bitonic","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.halfClean_bitonic","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#half-clean-bitonic","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $m \\in \\mathbb{N}$ and let $s_0, s_1, \\dots, s_{2m-1}$ be a bitonic sequence of Booleans, ordered by $\\mathrm{false} < \\mathrm{true}$. For $i < m$ put $\\ell_i = \\min(s_i, s_{i+m})$ and $u_i = \\max(s_i, s_{i+m})$. Then either $\\ell_i = \\mathrm{false}$ for every $i < m$ and $u_0, \\dots, u_{m-1}$ is bitonic, or $u_i = \\mathrm{true}$ for every $i < m$ and $\\ell_0, \\dots, \\ell_{m-1}$ is bitonic.","title":"The half-cleaner lemma"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/sorted-on","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.SortedOn","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#sorted-on","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"A Boolean sequence $g : \\mathbb{N} \\to \\mathrm{Bool}$ is sorted on the block $[b, b + L)$ if $g(b + i) \\le g(b + j)$ for all $i \\le j < L$, where Booleans are ordered by $\\mathrm{false} < \\mathrm{true}$.","title":"Sorted on a block"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/bitonic-merge","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.bitonicMerge","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#bitonic-merge","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"The bitonic merger on the block of wires $[b, b + 2^k)$ is the list of comparators defined by recursion on $k$. For $k = 0$ it is empty. On the block $[b, b + 2^{k+1})$ it is the half-cleaner on $[b, b + 2^{k+1})$, followed by the bitonic merger on $[b, b + 2^k)$, followed by the bitonic merger on $[b + 2^k, b + 2^{k+1})$.","title":"Bitonic merger"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/bitonic-merge-sorts","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.bitonicMerge_sorts","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#bitonic-merge-sorts","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $k, b \\in \\mathbb{N}$ and let $f$ assign a Boolean value to each wire $x \\in \\mathbb{N}$, with Booleans ordered by $\\mathrm{false} < \\mathrm{true}$. If the sequence $f(b), f(b + 1), \\dots, f(b + 2^k - 1)$ is bitonic, then running the bitonic merger on the block $[b, b + 2^k)$ on $f$ gives an assignment $g$ that is sorted on that block: $g(b + i) \\le g(b + j)$ whenever $i \\le j < 2^k$.","title":"The bitonic merger sorts every bitonic `0`-`1` input"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/run-n-swap","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.runN_swap","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#run-n-swap","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $ps$ be a list of comparators on $\\mathbb{N}$-indexed wires and let $ps'$ be obtained by replacing each comparator $(i, j)$ in $ps$ by $(j, i)$. For every input $f : \\mathbb{N} \\to \\mathrm{Bool}$ and every wire $x$, the value of wire $x$ after running $ps'$ on $f$ is the negation of the value of wire $x$ after running $ps$ on the complemented input $y \\mapsto \\lnot f(y)$.","title":"Reversing the comparators conjugates the network by complement on Booleans"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/bitonic-sort-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.bitonicSortN","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#bitonic-sort-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"A comparator $(i, j)$ puts the minimum of the values on wires $i$ and $j$ on wire $i$ and the maximum on wire $j$. Batcher's bitonic sorter on the block of wires $[b, b + 2^k)$ is the list of comparators defined by recursion on $k$. For $k = 0$ it is empty. On the block $[b, b + 2^{k+1})$ it is Batcher's bitonic sorter on $[b, b + 2^k)$, followed by Batcher's bitonic sorter on $[b + 2^k, b + 2^{k+1})$ with every\u2026","title":"Batcher's bitonic sorter on a block of wires"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/bitonic-sort-n-sorts","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.bitonicSortN_sorts","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#bitonic-sort-n-sorts","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $k, b \\in \\mathbb{N}$ and let $f$ assign a Boolean value to each wire $x \\in \\mathbb{N}$, with Booleans ordered by $\\mathrm{false} < \\mathrm{true}$. Running Batcher's bitonic sorter on the block $[b, b + 2^k)$ on $f$ gives an assignment $g$ that is sorted on that block: $g(b + i) \\le g(b + j)$ whenever $i \\le j < 2^k$.","title":"The bitonic sorter sorts every `0`-`1` input"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/length-half-cleaner","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.length_halfCleaner","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#length-half-cleaner","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For all natural numbers $m$ and $b$, the half-cleaner on the block of wires $[b, b + 2m)$ has exactly $m$ comparators.","title":"Size of the half-cleaner"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/length-bitonic-merge","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.length_bitonicMerge","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#length-bitonic-merge","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For all natural numbers $k$ and $b$, the bitonic merger on the block of wires $[b, b + 2^k)$ has $k \\cdot 2^{k-1}$ comparators; precisely, twice its number of comparators equals $k \\cdot 2^k$.","title":"Size of the bitonic merger"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/length-bitonic-sort-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.length_bitonicSortN","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#length-bitonic-sort-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For all natural numbers $k$ and $b$, the bitonic sorter on the block of $\\mathbb{N}$-indexed wires $[b, b + 2^k)$ has $k(k+1)2^k/4$ comparators; precisely, four times its number of comparators equals $k(k+1)2^k$.","title":"Size of the bitonic sorter on a block of wires"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/of-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.ofPairs","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#of-pairs","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For a natural number $N$ and a list of comparators on wires indexed by $\\mathbb{N}$, this is the network on the $N$ wires $0, \\dots, N - 1$ that keeps, in their original order, the comparators $(i, j)$ with $i < N$ and $j < N$, and drops all the others.","title":"Restricting comparators to finitely many wires"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/extend-false","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.extendFalse","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#extend-false","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For $N \\in \\mathbb{N}$ and a Boolean vector $v = (v_0, \\dots, v_{N-1})$, its extension by $\\mathrm{false}$ is the function on $\\mathbb{N}$ that takes the value $v_j$ at each $j < N$ and the value $\\mathrm{false}$ at each $j \\ge N$.","title":"Extension of a Boolean vector by false"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/ext-run-of-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.ext_run_ofPairs","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#ext-run-of-pairs","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"Let $N \\in \\mathbb{N}$ and let $P$ be a list of comparators on natural-number-indexed wires, each of which joins two wires below $N$. For every Boolean input $v$ on the wires $0, \\dots, N - 1$, run the network on $N$ wires obtained from $P$ on $v$ and extend its output by $\\mathrm{false}$ to all wires in $\\mathbb{N}$; the result equals the result of running $P$ on the extension of $v$ by $\\mathrm{false}$.","title":"Transported comparator lists agree with the original on Boolean inputs"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/bitonic-sorter","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.bitonicSorter","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#bitonic-sorter","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, Batcher's bitonic sorting network on $2^k$ wires is the comparator network on the wires $0, 1, \\dots, 2^k - 1$ made of the comparators of Batcher's bitonic sorter on the block $[0, 2^k)$ that join two wires below $2^k$ (in fact all of them), applied in the same order.","title":"Batcher's bitonic sorting network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/bitonic-sorter-sorts","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.bitonicSorter_sorts","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#bitonic-sorter-sorts","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For every $k \\in \\mathbb{N}$ and every linearly ordered type $\\alpha$, Batcher's bitonic sorting network on $2^k$ wires sorts inputs over $\\alpha$: for every assignment $v$ of elements of $\\alpha$ to the wires $0, 1, \\dots, 2^k - 1$, the output of the network on $v$ is monotone in the wire order.","title":"Batcher's bitonic sorter sorts every input over every linear order"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/length-bitonic-sorter","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.length_bitonicSorter","parent":41675,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter/index.html#length-bitonic-sorter","scope":"ComputerScience/sorting-and-searching/sorting-networks/bitonic-sorter","status":"fully_proved","summary":"For every natural number $k$, Batcher's bitonic sorting network on $2^k$ wires has $k(k+1)2^k/4$ comparators; precisely, four times its number of comparators equals $k(k+1)2^k$.","title":"Comparator count of Batcher's bitonic sorter"},{"data_url":"scopes/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network.json","id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","kind":"scope","navigation":"module","parent":41655,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","title":"Bubble sorting network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/adjacent","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.adjacent","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#adjacent","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"For $n$ wires and an index $i$, this is the comparator on wires $i$ and $i + 1$ when $i + 1 < n$, and nothing when wire $i + 1$ does not exist.","title":"Adjacent comparator"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/pass","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.pass","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#pass","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"On $n$ wires, the bubble pass ending at wire $m$ is the comparator network $(0, 1), (1, 2), \\ldots, (m - 1, m)$, applied in this order, where a comparator $(i, i + 1)$ is left out when wire $i + 1$ does not exist (that is, when $i + 1 \\ge n$).","title":"Bubble pass"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/passes","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.passes","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#passes","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"On $n$ wires and for a number $m$, this is the comparator network obtained by concatenating the bubble passes ending at wires $m - 1, m - 2, \\ldots, 0$, in this order; for $m = 0$ it is empty.","title":"Successive bubble passes"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/bubble","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.bubble","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#bubble","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"The bubble sorting network on $n$ wires is the comparator network obtained by concatenating the bubble passes ending at wires $n - 1, n - 2, \\ldots, 0$, in this order, where the bubble pass ending at wire $k$ consists of the comparators $(0, 1), (1, 2), \\ldots, (k - 1, k)$.","title":"Bubble sorting network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/pass-spec","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.pass_spec","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#pass-spec","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"Let $w$ be an input on $n$ wires with values in a linearly ordered type, let $m < n$, and let $w'$ be the output of the bubble pass ending at wire $m$ (the comparators $(0, 1), \\ldots, (m - 1, m)$) run on $w$. Then $w'_i = w_i$ for every wire $i > m$; $w'_i \\le w'_m$ for every wire $i \\le m$; and for every wire $i \\le m$ there is a wire $j \\le m$ with $w'_i = w_j$.","title":"Effect of one bubble pass"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/settled","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.Settled","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#settled","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"Let $w$ assign to each of $n$ wires a value in a linearly ordered type, and let $m$ be a natural number. Then $w$ is settled from $m$ when $w_i \\le w_j$ for all wires $m \\le i \\le j$ (the wires $m, \\ldots, n - 1$ are sorted) and $w_i \\le w_j$ for all wires $i < m \\le j$ (they hold the largest values). This is the invariant before the bubble pass ending at wire $m - 1$.","title":"Settled top wires"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/bubble-sorts","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.bubble_sorts","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#bubble-sorts","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"The bubble network sorts every input over every linear order.","title":"The bubble network sorts"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/length-pass","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.length_pass","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#length-pass","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"On $n$ wires, for every $m < n$ the bubble pass ending at wire $m$ has exactly $m$ comparators.","title":"Size of a bubble pass"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/length-bubble","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bubble.length_bubble","parent":41701,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network/index.html#length-bubble","scope":"ComputerScience/sorting-and-searching/sorting-networks/bubble-sorting-network","status":"fully_proved","summary":"For every $n$, the bubble sorting network on $n$ wires has $n(n - 1)/2$ comparators: twice its number of comparators is $n(n - 1)$.","title":"Size of the bubble sorting network"},{"data_url":"scopes/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks.json","id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","kind":"scope","navigation":"module","parent":41655,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","title":"Depth of sorting networks"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/levels","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.levels","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#levels","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Given a list of comparators on $\\mathbb{N}$-indexed wires, each a pair $(i, j)$ of wires, and a starting level $\\ell(x)$ for every wire $x$, process the comparators in order, keeping a current level for each wire, initially $\\ell$: a comparator $(i, j)$ whose wires currently have levels $a$ and $b$ sets the current levels of both wires $i$ and $j$ to $\\max(a, b) + 1$ and leaves every other level unchanged. The\u2026","title":"Comparator levels under earliest scheduling"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/depth-le","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.DepthLE","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#depth-le","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"A list of comparators on wires indexed by $\\mathbb{N}$ has depth at most $d$ when, starting with every wire at level $0$ and scheduling each comparator one level after the later of its two wires, every wire ends at level at most $d$.","title":"Depth bound for a comparator list"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/append","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.DepthLE.append","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#append","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"If a list $p$ of comparators on wires indexed by $\\mathbb{N}$ has depth at most $d$ and a list $q$ of comparators has depth at most $e$, then the concatenation of $p$ followed by $q$ has depth at most $d + e$.","title":"Depth of sequential composition of comparator lists"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/levels-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.levels_of_not_mem","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#levels-of-not-mem","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $ps$ be a list of comparators all of whose wires lie in $[a, b)$. Then for all initial levels $\\ell : \\mathbb{N} \\to \\mathbb{N}$ and every wire $x$ with $x < a$ or $b \\le x$, the level of $x$ after $ps$ starting from $\\ell$ is $\\ell(x)$.","title":"Comparators inside a block leave levels outside it unchanged"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/levels-congr","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.levels_congr","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#levels-congr","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $ps$ be a list of comparators all of whose wires lie in $[a, b)$, and let $\\ell, \\ell' : \\mathbb{N} \\to \\mathbb{N}$ be initial levels that agree on every wire in $[a, b)$. Then for every wire $x$ in $[a, b)$, the level of $x$ after $ps$ starting from $\\ell$ equals the level of $x$ after $ps$ starting from $\\ell'$.","title":"Levels inside a block depend only on initial levels inside it"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/append-disjoint","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.DepthLE.append_disjoint","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#append-disjoint","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $a$, $m$, $b$, $d$ be natural numbers, and let $p$ and $q$ be lists of comparators on wires indexed by $\\mathbb{N}$, where every comparator of $p$ joins two wires in $[a, m)$ and every comparator of $q$ joins two wires in $[m, b)$. If $p$ and $q$ both have depth at most $d$, then the concatenation of $p$ followed by $q$ also has depth at most $d$: the two lists run in parallel.","title":"Parallel composition of comparator lists on adjacent blocks"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/levels-swap","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.levels_swap","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#levels-swap","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"For every list $ps$ of comparators and all initial levels $\\ell : \\mathbb{N} \\to \\mathbb{N}$, the levels after the list obtained from $ps$ by replacing each comparator $(i, j)$ with $(j, i)$, starting from $\\ell$, equal the levels after $ps$ starting from $\\ell$.","title":"Swapping the wires of every comparator does not change levels"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/levels-tab","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.levelsTab","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#levels-tab","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Given a list $ps$ of comparators and a table $t$ of natural numbers $t_0, \\dots, t_{N-1}$, the tabulated levels are computed by processing the comparators in order: a comparator $(i, j)$ sets both entries $t_i$ and $t_j$ to $\\max(t_i, t_j) + 1$, where an entry outside the table reads as $0$ and a write outside the table is ignored.","title":"Tabulated levels"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/get-d-levels-tab","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.getD_levelsTab","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#get-d-levels-tab","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $t$ be a list of natural numbers of length $N$ and let $ps$ be a list of comparators all of whose wires lie in $[0, N)$. Extend $t$ to all wires by setting $t_y = 0$ for $y \\ge N$. Then for every wire $x$, the entry at position $x$ of the tabulated levels of $ps$ starting from the table $t$ (read as $0$ when $x \\ge N$) equals the level of wire $x$ after $ps$, starting from the initial levels $y \\mapsto t_y$.","title":"Tabulated levels agree with levels"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/depth-le-half-cleaner","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.depthLE_halfCleaner","parent":41711,"reading_title":{"format":"inline-parts-v1","parts":[["text","The half-cleaner has depth at most "],["math","\\(1\\)"]],"source":"The half-cleaner has depth at most $1$"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#depth-le-half-cleaner","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"For all natural numbers $m$ and $b$, the half-cleaner on the block of $2m$ wires $b, \\dots, b + 2m - 1$ has depth at most $1$: its comparators join pairwise disjoint pairs of wires.","title":"The half-cleaner has depth at most $1$"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/depth-le-bitonic-merge","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.depthLE_bitonicMerge","parent":41711,"reading_title":{"format":"inline-parts-v1","parts":[["text","The bitonic merger has depth at most "],["math","\\(k\\)"]],"source":"The bitonic merger has depth at most $k$"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#depth-le-bitonic-merge","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"For all natural numbers $k$ and $b$, the bitonic merger on the block of $2^k$ wires $b, b + 1, \\dots, b + 2^k - 1$ has depth at most $k$.","title":"The bitonic merger has depth at most $k$"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/depth-le-bitonic-sort-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.depthLE_bitonicSortN","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#depth-le-bitonic-sort-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"For all natural numbers $k$ and $b$, the comparator list of Batcher's bitonic sorter on the block of $2^k$ wires $b, b + 1, \\dots, b + 2^k - 1$ has depth at most $d$ for some natural number $d$ with $2d \\le k(k+1)$; that is, its depth is at most $k(k+1)/2$.","title":"Depth of the bitonic sorter's comparator list"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/influence","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.influence","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#influence","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Given a list $ps$ of comparators on wires indexed by $\\mathbb{N}$ and an assignment $S$ of a finite set of natural numbers to each wire, the influence sets after $ps$ are computed by processing the comparators in order: a comparator $(i, j)$ replaces the current sets of both wires $i$ and $j$ by their union, and leaves the sets of all other wires unchanged. Starting from $S(x) = \\{x\\}$, the set of a wire tracks the\u2026","title":"Influence sets of wires"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/card-influence-le","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.card_influence_le","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#card-influence-le","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $p$ be a list of comparators on wires indexed by $\\mathbb{N}$. Let $S$ assign to each wire a finite set of natural numbers and let $\\ell$ assign to each wire a natural number, such that $|S(x)| \\le 2^{\\ell(x)}$ for every wire $x$. For each wire $x$, let $L(x)$ be the level of $x$ after $p$ starting from the levels $\\ell$. Then for every wire $x$, the influence set of $x$ after $p$ starting from $S$ has at most\u2026","title":"Influence sets grow at most exponentially with the level"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/run-n-congr-influence","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.runN_congr_influence","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#run-n-congr-influence","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $f_0, g_0 : \\mathbb{N} \\to \\alpha$ be two inputs with values in a linear order $\\alpha$, let $S$ assign to each wire $x$ a finite set $S(x)$ of input positions, and let $F, G : \\mathbb{N} \\to \\alpha$ be wire values such that for every wire $x$, if $f_0$ and $g_0$ agree on $S(x)$, then $F(x) = G(x)$. Then for every list of comparators and every wire $x$, if $f_0$ and $g_0$ agree on the influence set of $x$\u2026","title":"Outputs depend only on the influence set"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/exists-false-run-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.exists_false_runN","parent":41711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Comparators inside "],["math","\\((0, n)\\)"],["text"," keep a false value inside "],["math","\\((0, n)\\)"]],"source":"Comparators inside $(0, n)$ keep a false value inside $(0, n)$"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#exists-false-run-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $n$ be a natural number, let $p$ be a list of comparators, each joining two wires in $[0, n)$, and let $f$ assign a Boolean value to each wire in $\\mathbb{N}$. If $f(y)$ is false for some $y < n$, then after running $p$ on $f$ some wire $y < n$ carries the value false.","title":"Comparators inside $(0, n)$ keep a false value inside $(0, n)$"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/le-two-pow-of-sorts","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.le_two_pow_of_sorts","parent":41711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sorting comparator lists on "],["math","\\(n\\)"],["text"," wires have depth at least "],["math","\\(\\log_2 n\\)"]],"source":"Sorting comparator lists on $n$ wires have depth at least $\\log_2 n$"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#le-two-pow-of-sorts","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $ps$ be a list of comparators on wires indexed by $\\mathbb{N}$, all of whose wires lie in $[0, n)$, and let $d$ be a natural number. Suppose that for every Boolean input $v$ on the wires $0, \\dots, n - 1$, extended by $\\mathrm{false}$ to the wires $\\ge n$, the output $w$ of $ps$ satisfies $w_i \\le w_j$ whenever $i \\le j < n$ (with $\\mathrm{false} < \\mathrm{true}$). If $ps$ has depth at most $d$, then $n \\le 2^d$.","title":"Sorting comparator lists on $n$ wires have depth at least $\\log_2 n$"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/le-of-depth-le-bitonic-sort-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Bitonic.le_of_depthLE_bitonicSortN","parent":41711,"reading_title":{"format":"inline-parts-v1","parts":[["text","The bitonic sorting comparator list on "],["math","\\(2^k\\)"],["text"," wires has depth at least "],["math","\\(k\\)"]],"source":"The bitonic sorting comparator list on $2^k$ wires has depth at least $k$"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#le-of-depth-le-bitonic-sort-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"For all natural numbers $k$ and $d$, if the comparator list of Batcher's bitonic sorter on the wires $0, \\dots, 2^k - 1$ has depth at most $d$, then $k \\le d$.","title":"The bitonic sorting comparator list on $2^k$ wires has depth at least $k$"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/to-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Network.toPairs","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#to-pairs","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"The comparators of a network on $n$ wires, in order, with each comparator $(i, j)$ written as the pair of natural numbers $(i, j)$, forgetting that $i, j < n$.","title":"Comparator list of a network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/depth-le-2","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Network.DepthLE","parent":41711,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#depth-le-2","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"A comparator network on $n$ wires has depth at most $d$ when its list of comparators, read as pairs of wire numbers in $\\mathbb{N}$, has depth at most $d$: starting with every wire at level $0$ and scheduling each comparator one level after the later of its two wires, every wire ends at level at most $d$.","title":"Depth bound for a comparator network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/le-two-pow-of-sorts-2","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.Network.le_two_pow_of_sorts","parent":41711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every sorting network on "],["math","\\(n\\)"],["text"," wires has depth at least "],["math","\\(\\log_2 n\\)"]],"source":"Every sorting network on $n$ wires has depth at least $\\log_2 n$"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#le-two-pow-of-sorts-2","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"Let $C$ be a comparator network on $n$ wires and $d$ a natural number. If $C$ sorts every Boolean input (its output is monotone in the wire order for every input in $\\{\\mathrm{false}, \\mathrm{true}\\}^n$, with $\\mathrm{false} < \\mathrm{true}$) and $C$ has depth at most $d$, then $n \\le 2^d$.","title":"Every sorting network on $n$ wires has depth at least $\\log_2 n$"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/bitonic-sorter-depth-le","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.bitonicSorter_depthLE","parent":41711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Batcher's bitonic sorting network has depth at most "],["math","\\(k(k+1)/2\\)"]],"source":"Batcher's bitonic sorting network has depth at most $k(k+1)/2$"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#bitonic-sorter-depth-le","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"For every natural number $k$, Batcher's bitonic sorting network on $2^k$ wires has depth at most $d$ for some natural number $d$ with $2d \\le k(k+1)$; that is, its depth is at most $k(k+1)/2$.","title":"Batcher's bitonic sorting network has depth at most $k(k+1)/2$"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/le-of-bitonic-sorter-depth-le","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.le_of_bitonicSorter_depthLE","parent":41711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Batcher's bitonic sorter on "],["math","\\(2^k\\)"],["text"," wires has depth at least "],["math","\\(k\\)"]],"source":"Batcher's bitonic sorter on $2^k$ wires has depth at least $k$"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks/index.html#le-of-bitonic-sorter-depth-le","scope":"ComputerScience/sorting-and-searching/sorting-networks/depth-of-sorting-networks","status":"fully_proved","summary":"For all natural numbers $k$ and $d$, if Batcher's bitonic sorting network on $2^k$ wires has depth at most $d$, then $k \\le d$.","title":"Batcher's bitonic sorter on $2^k$ wires has depth at least $k$"},{"data_url":"scopes/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort.json","id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","kind":"scope","navigation":"module","parent":41655,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","title":"Batcher odd\u2013even mergesort"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/relabel","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.relabel","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#relabel","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Given a map $e : \\mathbb{N} \\to \\mathbb{N}$ and a list of comparators, each a pair $(i, j)$ of wires in $\\mathbb{N}$, the relabeled list replaces every comparator $(i, j)$ by $(e(i), e(j))$ and keeps the order of the list.","title":"Relabeling the wires of a comparator list"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/run-n-relabel","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.runN_relabel","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#run-n-relabel","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type and let $e : \\mathbb{N} \\to \\mathbb{N}$ be injective. For every list of comparators, every $f : \\mathbb{N} \\to \\alpha$ and every wire $j$, running the list relabeled by $e$ on $f$ gives at wire $e(j)$ the same value as running the original list on the input $i \\mapsto f(e(i))$ gives at wire $j$.","title":"Running a relabeled network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/run-n-relabel-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.runN_relabel_of_not_mem","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#run-n-relabel-of-not-mem","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $e : \\mathbb{N} \\to \\mathbb{N}$ any map and $f : \\mathbb{N} \\to \\alpha$ an assignment of values to the wires. For every list of comparators and every wire $x$ that is not of the form $e(y)$, running the list relabeled by $e$ on $f$ gives the value $f(x)$ at wire $x$.","title":"Relabeled networks fix the wires outside the image"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/final-layer","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.finalLayer","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#final-layer","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, this is the list of comparators $(2i + 1, 2i + 2)$ for $i = 0, 1, \\dots, 2^{k+1} - 2$, in increasing order of $i$. It compares the neighbouring wires $2i + 1$ and $2i + 2$ and is the last layer of the odd-even merger on $2^{k+2}$ wires.","title":"Last layer of the odd-even merger"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/odd-even-merge","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.oddEvenMerge","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#odd-even-merge","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, the odd-even merger on wires $0, \\dots, 2^{k+1} - 1$ is the comparator list defined by recursion on $k$. For $k = 0$ it is the single comparator $(0, 1)$. For $k + 1$ it is the merger for $k$ with each wire $i$ renamed to $2i$ (merging the even wires), followed by the merger for $k$ with each wire $i$ renamed to $2i + 1$ (merging the odd wires), followed by the comparators $(2i + 1, 2i + 2)$\u2026","title":"Batcher's odd-even merger"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/odd-even-sort-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.oddEvenSortN","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#odd-even-sort-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, odd-even merge sort on wires $0, \\dots, 2^k - 1$ is the comparator list defined by recursion on $k$. For $k = 0$ it is empty. For $k + 1$ it is the sorter for $k$ (sorting wires $0, \\dots, 2^k - 1$), followed by the sorter for $k$ with each wire $i$ renamed to $2^k + i$ (sorting wires $2^k, \\dots, 2^{k+1} - 1$), followed by Batcher's odd-even merger on $2^{k+1}$ wires.","title":"Batcher's odd-even merge sort"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/run-n-final-layer","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.runN_finalLayer","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#run-n-final-layer","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type, $k \\in \\mathbb{N}$ and $f : \\mathbb{N} \\to \\alpha$ an assignment of values to the wires. Running the final layer for $k$ on $f$ gives at a wire $x$ the value $\\min(f(x), f(x + 1))$ if $x$ is odd and $\\lfloor x / 2 \\rfloor < 2^{k+1} - 1$; otherwise the value $\\max(f(x - 1), f(x))$ if $x$ is even, $x > 0$ and $(x - 2) / 2 < 2^{k+1} - 1$; and the value $f(x)$ in all other\u2026","title":"Output of the last layer of the odd-even merger"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/odd-even-merge-merges","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.oddEvenMerge_merges","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#odd-even-merge-merges","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $k \\in \\mathbb{N}$ and let $f$ assign a Boolean value to each wire $0, 1, 2, \\dots$, with false read as $0$ and true as $1$. Let $a, c \\le 2^k$ be such that wires $0, \\dots, 2^k - 1$ hold $a$ zeros followed by ones (wire $i < 2^k$ holds $0$ exactly when $i < a$) and wires $2^k, \\dots, 2^{k+1} - 1$ hold $c$ zeros followed by ones (wire $2^k + i$, for $i < 2^k$, holds $0$ exactly when $i < c$). Then after\u2026","title":"The odd-even merger merges"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/odd-even-sort-n-sorts","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.oddEvenSortN_sorts","parent":41735,"reading_title":{"format":"inline-parts-v1","parts":[["text","Odd-even merge sort sorts every "],["math","\\(0\\)"],["text","-"],["math","\\(1\\)"],["text"," input"]],"source":"Odd-even merge sort sorts every $0$-$1$ input"},"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#odd-even-sort-n-sorts","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $k \\in \\mathbb{N}$ and let $f$ assign a Boolean value to each wire $0, 1, 2, \\dots$, with false read as $0$ and true as $1$. After Batcher's odd-even merge sort on $2^k$ wires runs on $f$, there is $z \\le 2^k$ such that every wire $j < 2^k$ holds $0$ exactly when $j < z$: wires $0, \\dots, 2^k - 1$ hold $z$ zeros followed by ones.","title":"Odd-even merge sort sorts every $0$-$1$ input"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/relabel-within","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.relabel_within","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#relabel-within","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $ps$ be a list of comparators, each a pair of natural numbers, that lies within $[l, h)$, and let $e : \\mathbb{N} \\to \\mathbb{N}$ satisfy $l' \\le e(x) < h'$ for every $x$ with $l \\le x < h$. Then the list obtained from $ps$ by relabeling its wires by $e$ lies within $[l', h')$.","title":"Relabeling keeps comparators within an interval"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/length-odd-even-merge","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.length_oddEvenMerge","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#length-odd-even-merge","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For every $k \\in \\mathbb{N}$, Batcher's odd-even merger on $2^{k+1}$ wires consists of exactly $k \\cdot 2^k + 1$ comparators.","title":"Size of the odd-even merger"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/length-odd-even-sort-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.length_oddEvenSortN","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#length-odd-even-sort-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For every $k \\in \\mathbb{N}$, the number $L$ of comparators of Batcher's odd-even merge sort on $2^k$ wires satisfies $4(L + 1) + k \\cdot 2^k = (k^2 + 4) \\cdot 2^k$, that is, $L = (k^2 - k + 4) \\cdot 2^k / 4 - 1$.","title":"Size of Batcher's odd-even merge sort"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/odd-even-sorter","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.oddEvenSorter","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#odd-even-sorter","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, this is the comparator network on the $2^k$ wires $0, \\dots, 2^k - 1$ given by the comparators of Batcher's odd-even merge sort on $2^k$ wires, in the same order, with any comparator that touches a wire $\\ge 2^k$ dropped.","title":"Batcher's odd-even merge sorting network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/odd-even-sorter-sorts","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.oddEvenSorter_sorts","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#odd-even-sorter-sorts","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For every $k \\in \\mathbb{N}$ and every linearly ordered type $\\alpha$, Batcher's odd-even merge sorting network on $2^k$ wires sorts every input over $\\alpha$: for every assignment $v$ of elements of $\\alpha$ to the wires $0, \\dots, 2^k - 1$, the output of the network on $v$ is monotone in the wire order.","title":"Batcher's odd-even merge sort sorts every input over every linear order"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/levels-relabel","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.levels_relabel","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#levels-relabel","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $e : \\mathbb{N} \\to \\mathbb{N}$ be injective, let $p$ be a comparator list and let $\\ell$ be an initial level assignment. For every wire $j$, the level of wire $e(j)$ after running $p$ with its wires relabeled by $e$, starting from $\\ell$, equals the level of wire $j$ after running $p$ starting from $\\ell \\circ e$.","title":"Levels of a relabeled comparator list"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/levels-relabel-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.levels_relabel_of_not_mem","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#levels-relabel-of-not-mem","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $p$ be a comparator list, let $e : \\mathbb{N} \\to \\mathbb{N}$ be any map and let $\\ell$ be an initial level assignment. For every wire $x$ outside the image of $e$, the level of $x$ after running $p$ with its wires relabeled by $e$, starting from $\\ell$, is $\\ell(x)$.","title":"Relabeled comparator lists leave the other wires alone"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/depth-le-relabel","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.depthLE_relabel","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#depth-le-relabel","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"If a comparator list has depth at most $d$ and $e : \\mathbb{N} \\to \\mathbb{N}$ is injective, then the comparator list obtained by relabeling its wires by $e$ also has depth at most $d$.","title":"Relabeling wires preserves a depth bound"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/levels-relabel-congr","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.levels_relabel_congr","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#levels-relabel-congr","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $p$ be a comparator list and let $e : \\mathbb{N} \\to \\mathbb{N}$ be injective. If two initial level assignments $\\ell$ and $\\ell'$ agree on every wire $e(y)$, then the levels after running $p$ with its wires relabeled by $e$, starting from $\\ell$ and from $\\ell'$, agree on every wire $e(y)$.","title":"Levels of a relabeled comparator list on the image wires"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/depth-le-append-relabel","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.depthLE_append_relabel","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#depth-le-append-relabel","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"Let $e$ and $e'$ be injective maps $\\mathbb{N} \\to \\mathbb{N}$ with disjoint images, that is, $e(y) \\ne e'(y')$ for all $y$ and $y'$. If the comparator lists $p$ and $q$ both have depth at most $d$, then $p$ with its wires relabeled by $e$, followed by $q$ with its wires relabeled by $e'$, has depth at most $d$.","title":"Parallel composition of relabeled comparator lists"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/depth-le-final-layer","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.depthLE_finalLayer","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#depth-le-final-layer","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For every $k$, the final layer of the odd-even merger on $2^{k+2}$ wires, which compares wires $2i + 1$ and $2i + 2$ for each $i < 2^{k+1} - 1$, has depth at most $1$: its comparators are disjoint.","title":"The final layer of the odd-even merger has depth at most one"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/depth-le-odd-even-merge","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.depthLE_oddEvenMerge","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#depth-le-odd-even-merge","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For every $k$, Batcher's odd-even merger on $2^{k+1}$ wires has depth at most $k + 1$.","title":"Depth bound for Batcher's odd-even merger"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/depth-le-odd-even-sort-n","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.OddEven.depthLE_oddEvenSortN","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#depth-le-odd-even-sort-n","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For every $k$, there is a $d$ with $2d \\le k(k+1)$ such that the comparator list of Batcher's odd-even merge sort on wires $0, \\dots, 2^k - 1$ has depth at most $d$; that is, its depth is at most $k(k+1)/2$.","title":"Depth bound for Batcher's odd-even merge sort as a comparator list"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/odd-even-sorter-depth-le","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.oddEvenSorter_depthLE","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#odd-even-sorter-depth-le","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For every $k$, there is a $d$ with $2d \\le k(k+1)$ such that Batcher's odd-even merge sorting network on $2^k$ wires has depth at most $d$; that is, its depth is at most $k(k+1)/2$.","title":"Depth bound for Batcher's odd-even merge sorting network"},{"id":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/le-of-odd-even-sorter-depth-le","kind":"node","lean":"Cslib.Algorithms.Lean.SortingNetworks.le_of_oddEvenSorter_depthLE","parent":41735,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort/index.html#le-of-odd-even-sorter-depth-le","scope":"ComputerScience/sorting-and-searching/sorting-networks/batcher-odd-even-mergesort","status":"fully_proved","summary":"For all $k$ and $d$, if Batcher's odd-even merge sorting network on $2^k$ wires has depth at most $d$, then $k \\le d$.","title":"Lower bound for the depth of Batcher's odd-even merge sorting network"},{"data_url":"scopes/ComputerScience/sorting-and-searching/binary-search.json","id":"ComputerScience/sorting-and-searching/binary-search","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","title":"Binary search"},{"id":"ComputerScience/sorting-and-searching/binary-search/bin-search-contains-eq-is-some","kind":"node","lean":"Cslib.Algorithms.Lean.binSearchContains_eq_isSome","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#bin-search-contains-eq-is-some","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $A$ be an array with entries in a type $\\alpha$, let $k \\in \\alpha$, let $\\mathrm{lt}$ be any Boolean-valued comparison on $\\alpha$, and let $\\ell$ and $h$ be natural numbers. Lean core's Boolean binary search membership test for $k$ in $A$, with comparison $\\mathrm{lt}$ and bounds $\\ell$ and $h$, returns true exactly when Lean core's array binary search with the same arguments returns some element.","title":"Array binary search membership test agrees with binary search"},{"id":"ComputerScience/sorting-and-searching/binary-search/bin-search-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.binSearch_eq_some_iff","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#bin-search-eq-some-iff","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $A = (a_0, \\dots, a_{n-1})$ be an array of length $n$ with entries in a linearly ordered type, with $a_i \\le a_j$ whenever $i < j$. Let $k$ and $a$ be elements of that type and $\\ell$, $h$ natural numbers. Lean core's array binary search for $k$ in $A$, with the strict order $<$ as comparison and inclusive bounds $\\ell$ and $h$, returns $a$ if and only if $a = k$ and there is an index $i < n$ with\u2026","title":"Result of bounded binary search on a sorted array"},{"id":"ComputerScience/sorting-and-searching/binary-search/bin-search-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.binSearch_eq_none_iff","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#bin-search-eq-none-iff","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $A = (a_0, \\dots, a_{n-1})$ be an array of length $n$ with entries in a linearly ordered type, with $a_i \\le a_j$ whenever $i < j$. Let $k$ be an element of that type and $\\ell$, $h$ natural numbers. Lean core's array binary search for $k$ in $A$, with the strict order $<$ as comparison and inclusive bounds $\\ell$ and $h$, returns no element if and only if there is no index $i < n$ with $\\ell \\le i \\le h$ and\u2026","title":"Bounded binary search on a sorted array fails exactly when the key is absent"},{"id":"ComputerScience/sorting-and-searching/binary-search/bin-search-contains-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.binSearchContains_eq_true_iff","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#bin-search-contains-eq-true-iff","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $A = (a_0, \\dots, a_{n-1})$ be an array of length $n$ with entries in a linearly ordered type, with $a_i \\le a_j$ whenever $i < j$. Let $k$ be an element of that type and $\\ell$, $h$ natural numbers. Lean core's Boolean binary search membership test for $k$ in $A$, with the strict order $<$ as comparison and inclusive bounds $\\ell$ and $h$, returns true if and only if there is an index $i < n$ with\u2026","title":"Bounded binary search membership test on a sorted array is correct"},{"id":"ComputerScience/sorting-and-searching/binary-search/bin-search-eq-some-iff-mem","kind":"node","lean":"Cslib.Algorithms.Lean.binSearch_eq_some_iff_mem","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#bin-search-eq-some-iff-mem","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $A$ be an array with entries in a linearly ordered type, sorted so that an earlier entry is never greater than a later one, and let $k$ and $a$ be elements of that type. Lean core's array binary search for $k$ over the whole of $A$, with the strict order $<$ as comparison, returns $a$ if and only if $a = k$ and $k$ is an entry of $A$.","title":"Binary search on a sorted array finds exactly its members"},{"id":"ComputerScience/sorting-and-searching/binary-search/bin-search-eq-none-iff-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.binSearch_eq_none_iff_not_mem","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#bin-search-eq-none-iff-not-mem","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $A$ be an array with entries in a linearly ordered type, sorted so that an earlier entry is never greater than a later one, and let $k$ be an element of that type. Lean core's array binary search for $k$ over the whole of $A$, with the strict order $<$ as comparison, returns no element if and only if $k$ is not an entry of $A$.","title":"Binary search on a sorted array fails exactly on non-members"},{"id":"ComputerScience/sorting-and-searching/binary-search/bin-search-contains-eq-true-iff-mem","kind":"node","lean":"Cslib.Algorithms.Lean.binSearchContains_eq_true_iff_mem","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#bin-search-contains-eq-true-iff-mem","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $A$ be an array with entries in a linearly ordered type, sorted so that an earlier entry is never greater than a later one, and let $k$ be an element of that type. Lean core's Boolean binary search membership test for $k$ over the whole of $A$, with the strict order $<$ as comparison, returns true if and only if $k$ is an entry of $A$.","title":"Binary search membership test on a sorted array is correct"},{"id":"ComputerScience/sorting-and-searching/binary-search/lower-bound","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.lowerBound","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#lower-bound","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Given an array $a = (a_0, \\dots, a_{n-1})$ over a linearly ordered type and a key $k$, lower-bound search maintains a half-open interval $[l, h)$ of indices, starting from $[0, n)$. While $l < h$, it compares $a_m$ with $k$ at the midpoint $m = l + \\lfloor (h - l)/2 \\rfloor$, charging one unit for this comparison, and continues with $[m + 1, h)$ if $a_m < k$ and with $[l, m)$ otherwise. When $l = h$ it returns $l$.","title":"Lower-bound search"},{"id":"ComputerScience/sorting-and-searching/binary-search/lower-bound-spec","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.lowerBound_spec","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#lower-bound-spec","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be an array over a linearly ordered type, sorted in nondecreasing order, let $k$ be a key, and let $\\ell$ be the index returned by lower-bound search for $k$ in $a$. Then $\\ell \\le n$, $a_j < k$ for every $j < \\ell$, and $k \\le a_j$ for every $j$ with $\\ell \\le j < n$.","title":"Correctness of lower-bound search"},{"id":"ComputerScience/sorting-and-searching/binary-search/lower-bound-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.lowerBound_time_le","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#lower-bound-time-le","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"For every array $a$ of length $n$ over a linearly ordered type and every key $k$, lower-bound search for $k$ in $a$ makes at most $\\lceil \\log_2 (n + 1) \\rceil$ element comparisons.","title":"Comparison bound for lower-bound search"},{"id":"ComputerScience/sorting-and-searching/binary-search/binary-search-first","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.binarySearchFirst?","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#binary-search-first","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Given an array $a = (a_0, \\dots, a_{n-1})$ over a linearly ordered type and a key $k$, binary search for the first occurrence of $k$ in $a$ runs lower-bound search for $k$ in $a$, obtaining an index $\\ell$. If $\\ell < n$, it makes one further comparison, charged one unit, and returns $\\ell$ when $a_\\ell = k$ and no index otherwise; if $\\ell \\ge n$, it returns no index without further cost.","title":"Binary search for the first occurrence"},{"id":"ComputerScience/sorting-and-searching/binary-search/binary-search-first-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.binarySearchFirst?_eq_some_iff","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#binary-search-first-eq-some-iff","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be an array over a linearly ordered type, sorted in nondecreasing order, let $k$ be a key and let $i$ be a natural number. Binary search for the first occurrence of $k$ in $a$ returns the index $i$ if and only if $i < n$, $a_i = k$, and $a_j < k$ for every $j < i$.","title":"Binary search returns the first occurrence of the key"},{"id":"ComputerScience/sorting-and-searching/binary-search/binary-search-first-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.binarySearchFirst?_eq_none_iff","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#binary-search-first-eq-none-iff","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be an array over a linearly ordered type, sorted in nondecreasing order, and let $k$ be a key. Binary search for the first occurrence of $k$ in $a$ returns no index if and only if $a_i \\ne k$ for every $i < n$.","title":"Binary search reports absence exactly when the key is absent"},{"id":"ComputerScience/sorting-and-searching/binary-search/binary-search-first-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.binarySearchFirst?_time","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#binary-search-first-time","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"Let $a$ be an array of length $n$ over a linearly ordered type, let $k$ be a key, and let $\\ell$ be the index returned by lower-bound search for $k$ in $a$. The number of element comparisons made by binary search for the first occurrence of $k$ in $a$ equals the number made by lower-bound search for $k$ in $a$, plus $1$ if $\\ell < n$ and plus $0$ otherwise.","title":"Exact comparison count of binary search for the first occurrence"},{"id":"ComputerScience/sorting-and-searching/binary-search/binary-search-first-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.binarySearchFirst?_time_le","parent":41760,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/binary-search/index.html#binary-search-first-time-le","scope":"ComputerScience/sorting-and-searching/binary-search","status":"fully_proved","summary":"For every array $a$ of length $n$ over a linearly ordered type and every key $k$, binary search for the first occurrence of $k$ in $a$ makes at most $\\lceil \\log_2 (n + 1) \\rceil + 1$ element comparisons.","title":"Comparison bound for binary search for the first occurrence"},{"data_url":"scopes/ComputerScience/sorting-and-searching/counting-sort.json","id":"ComputerScience/sorting-and-searching/counting-sort","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/counting-sort/index.html","scope":"ComputerScience/sorting-and-searching/counting-sort","status":"fully_proved","title":"Counting sort"},{"id":"ComputerScience/sorting-and-searching/counting-sort/counting-sort","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.countingSort","parent":41776,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/counting-sort/index.html#counting-sort","scope":"ComputerScience/sorting-and-searching/counting-sort","status":"fully_proved","summary":"Let $k$ be a natural number and let $\\mathrm{key}$ assign to each record a key in $\\{0, \\dots, k-1\\}$. Counting sort of an array $A$ of records returns an array together with a count of charged operations. It starts $k$ counters at zero, counts the records of $A$ with each key, replaces the counters by cumulative counts so that the counter of key $q$ holds the number of records with key at most $q$, and then scans\u2026","title":"Counting sort"},{"id":"ComputerScience/sorting-and-searching/counting-sort/counting-sort-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.countingSort_correct","parent":41776,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/counting-sort/index.html#counting-sort-correct","scope":"ComputerScience/sorting-and-searching/counting-sort","status":"fully_proved","summary":"Let $k$ be a natural number, let $\\mathrm{key}$ assign to each record a key in $\\{0, \\dots, k-1\\}$, and let $A$ be an array of records. The array that counting sort returns on $A$ lists its records in nondecreasing order of key, is a permutation of $A$, and is stable: for every key $q < k$, the records of the output with key $q$, read from left to right, form the same sequence as the records of $A$ with key $q$.","title":"Correctness of counting sort"},{"id":"ComputerScience/sorting-and-searching/counting-sort/counting-sort-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.countingSort_time","parent":41776,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/counting-sort/index.html#counting-sort-time","scope":"ComputerScience/sorting-and-searching/counting-sort","status":"fully_proved","summary":"Let $k$ be a natural number, let $\\mathrm{key}$ assign to each record a key in $\\{0, \\dots, k-1\\}$, and let $A$ be an array of $n$ records. Counting sort on $A$ with this key is charged at most $7n + 3k$ operations, where one operation is charged for each counter initialization, key evaluation, counter read, counter write and output write.","title":"Running time of counting sort"},{"data_url":"scopes/ComputerScience/sorting-and-searching/heapsort.json","id":"ComputerScience/sorting-and-searching/heapsort","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/heapsort/index.html","scope":"ComputerScience/sorting-and-searching/heapsort","status":"fully_proved","title":"Heapsort"},{"id":"ComputerScience/sorting-and-searching/heapsort/heap-sort","kind":"node","lean":"Cslib.Algorithms.Lean.heapSort","parent":41780,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/heapsort/index.html#heap-sort","scope":"ComputerScience/sorting-and-searching/heapsort","status":"fully_proved","summary":"For a list $xs$ of elements of a linearly ordered type, heap sort inserts the entries of $xs$ one at a time, from first to last, into the empty binary min-heap, and then extracts the minimum of the heap as many times as the heap has entries, each time continuing with the remaining heap. The output is the list of the extracted minima in the order in which they were extracted.","title":"Heap sort"},{"id":"ComputerScience/sorting-and-searching/heapsort/heap-sort-multiset","kind":"node","lean":"Cslib.Algorithms.Lean.heapSort_multiset","parent":41780,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/heapsort/index.html#heap-sort-multiset","scope":"ComputerScience/sorting-and-searching/heapsort","status":"fully_proved","summary":"For every list $xs$ of elements of a linearly ordered type, the output of heap sort on $xs$ and $xs$ itself are equal as multisets: every value occurs in both lists the same number of times.","title":"Heap sort preserves multiplicities"},{"id":"ComputerScience/sorting-and-searching/heapsort/heap-sort-perm","kind":"node","lean":"Cslib.Algorithms.Lean.heapSort_perm","parent":41780,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/heapsort/index.html#heap-sort-perm","scope":"ComputerScience/sorting-and-searching/heapsort","status":"fully_proved","summary":"For every list $xs$ of elements of a linearly ordered type, heap sort returns a permutation of $xs$.","title":"Heap sort permutes its input"},{"id":"ComputerScience/sorting-and-searching/heapsort/heap-sort-sorted","kind":"node","lean":"Cslib.Algorithms.Lean.heapSort_sorted","parent":41780,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/heapsort/index.html#heap-sort-sorted","scope":"ComputerScience/sorting-and-searching/heapsort","status":"fully_proved","summary":"For every list $xs$ of elements of a linearly ordered type, heap sort returns a list sorted in nondecreasing order.","title":"Heap sort output is sorted"},{"id":"ComputerScience/sorting-and-searching/heapsort/heap-sort-correct","kind":"node","lean":"Cslib.Algorithms.Lean.heapSort_correct","parent":41780,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/heapsort/index.html#heap-sort-correct","scope":"ComputerScience/sorting-and-searching/heapsort","status":"fully_proved","summary":"For every list $xs$ of elements of a linearly ordered type, heap sort returns a list that is sorted in nondecreasing order and is a permutation of $xs$.","title":"Correctness of heap sort"},{"data_url":"scopes/ComputerScience/sorting-and-searching/insertion-sort.json","id":"ComputerScience/sorting-and-searching/insertion-sort","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","title":"Insertion sort"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/compare-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.compareLE","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#compare-le","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"For elements $x$ and $y$ of a linearly ordered type, this computation returns the truth value of $x \\le y$ and charges one unit of time.","title":"Charged comparison"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/insertion-sort-insert","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.insertionSortInsert","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#insertion-sort-insert","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"Let $a$ be an element of a linearly ordered type. Inserting $a$ into the empty list returns $[a]$ at no cost. Inserting $a$ into a list with head $b$ and tail $l$ first charges one unit for comparing $a \\le b$; if $a \\le b$ it returns $a$ followed by $b$ and $l$, and otherwise it inserts $a$ into $l$, adding that cost, and returns $b$ followed by the result.","title":"Costed insertion into a list"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/insertion-sort","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.insertionSort","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#insertion-sort","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"On a list over a linearly ordered type, insertion sort returns the empty list at no cost when the input is empty. On a list with head $a$ and tail $l$, it sorts $l$ and then inserts $a$ into the result by costed insertion; its cost is the cost of sorting $l$ plus the cost of that insertion. Exactly one unit is charged for each order comparison.","title":"Insertion sort with comparison cost"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/ret-insertion-sort-insert","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ret_insertionSortInsert","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#ret-insertion-sort-insert","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"For an element $a$ and a list $l$ over a linearly ordered type, the list returned by costed insertion of $a$ into $l$ is the ordered insertion of $a$ into $l$ with respect to $\\le$: $a$ is placed immediately before the first element $b$ of $l$ with $a \\le b$, or at the end of $l$ if there is no such element.","title":"Costed insertion returns the ordered insertion"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/ret-insertion-sort","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ret_insertionSort","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#ret-insertion-sort","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"For every list $xs$ over a linearly ordered type, the list returned by insertion sort with comparison costs equals the standard insertion sort of $xs$ with respect to $\\le$, which inserts the head into the sorted tail.","title":"Cost-annotated insertion sort returns the standard insertion sort"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/insertion-sort-eq-insertion-sort-m","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.insertionSort_eq_insertionSortM","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#insertion-sort-eq-insertion-sort-m","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"For a list $l$ over a linearly ordered type in the lowest universe, the costed insertion sort of $l$ equals CSLib's generic monadic insertion sort run in the time-counting monad with the charged comparison: the returned lists agree and the numbers of charged comparisons agree.","title":"Agreement with CSLib's monadic insertion sort"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/insertion-sort-sorted","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.insertionSort_sorted","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#insertion-sort-sorted","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"For every list $l$ over a linearly ordered type, the list returned by insertion sort on $l$ is nondecreasing.","title":"Insertion sort returns a sorted list"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/insertion-sort-perm","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.insertionSort_perm","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#insertion-sort-perm","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"For every list $l$ over a linearly ordered type, the list returned by insertion sort is a permutation of $l$; in particular every element occurs in it exactly as often as in $l$.","title":"Insertion sort permutes its input"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/insertion-sort-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.insertionSort_correct","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#insertion-sort-correct","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"For every list $l$ over a linearly ordered type, the list returned by insertion sort is nondecreasing and is a permutation of $l$.","title":"Correctness of insertion sort"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/insertion-sort-insert-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.insertionSortInsert_time_le","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#insertion-sort-insert-time-le","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"For an element $a$ and a list $l$ over a linearly ordered type, costed insertion of $a$ into $l$ charges at most $n$ comparisons, where $n$ is the length of $l$.","title":"Cost of a single insertion"},{"id":"ComputerScience/sorting-and-searching/insertion-sort/insertion-sort-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.insertionSort_time","parent":41786,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/insertion-sort/index.html#insertion-sort-time","scope":"ComputerScience/sorting-and-searching/insertion-sort","status":"fully_proved","summary":"On a list of length $n$ over a linearly ordered type, insertion sort charges at most $n(n-1)/2$ comparisons.","title":"Comparison bound for insertion sort"},{"data_url":"scopes/ComputerScience/sorting-and-searching/bucket-sort.json","id":"ComputerScience/sorting-and-searching/bucket-sort","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","title":"Bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $R$ be a linearly ordered field with a floor function and let $x_1, \\dots, x_n$ be an array of elements of $R$ that all lie in $[0, 1)$. If $n = 0$, bucket sort returns the empty array at zero cost. Otherwise it creates $n$ empty buckets indexed by $0, \\dots, n - 1$, visits the entries in order and prepends each entry $x$ to bucket $\\lfloor n x \\rfloor$, sorts each bucket with insertion sort, and concatenates\u2026","title":"Bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $R$ be a linearly ordered field with a floor function and let $x_1, \\dots, x_n$ be an array of elements of $R$ that all lie in $[0, 1)$. For $0 \\le i < n$ let $B_i$ be the list of the entries $x_j$ with $\\lfloor n x_j \\rfloor = i$, in decreasing order of $j$, and let $T(B_i)$ be the number of comparisons insertion sort makes on $B_i$. Then the cost of bucket sort on this array equals\u2026","title":"Exact cost of bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-lower","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_lower","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-lower","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $R$ be a linearly ordered field with a floor function. For every array of $n$ elements of $R$ that all lie in $[0, 1)$, the cost of bucket sort on this array is at least $5n$.","title":"Linear lower bound on the cost of bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-le-sum-sq","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_le_sum_sq","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-le-sum-sq","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $R$ be a linearly ordered field with a floor function and let $x_1, \\dots, x_n$ be an array of elements of $R$ that all lie in $[0, 1)$. For $0 \\le i < n$ let $c_i$ be the number of indices $j$ with $\\lfloor n x_j \\rfloor = i$. Then the cost of bucket sort on this array is at most $5n + \\sum_{i=0}^{n-1} c_i^2$.","title":"Bound on the cost of bucket sort via squared bucket sizes"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-empty","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_empty","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-empty","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"For every linearly ordered field $R$ with a floor function, bucket sort applied to the empty array of elements of $R$ returns the empty array at zero cost.","title":"Bucket sort on the empty array"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-perm","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_perm","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-perm","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $R$ be a linearly ordered field with a floor function. For every array of elements of $R$ that all lie in $[0, 1)$, the array returned by bucket sort is a permutation of the input: every element occurs in the output exactly as many times as in the input.","title":"Bucket sort permutes its input"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-sorted","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_sorted","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-sorted","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $R$ be a linearly ordered field with a floor function. For every array of elements of $R$ that all lie in $[0, 1)$, the array returned by bucket sort is nondecreasing: each entry is at most every entry after it.","title":"Bucket sort sorts"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-occupancy-mean","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_occupancy_mean","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-occupancy-mean","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $n \\ge 1$ and let $\\omega_0, \\dots, \\omega_{n-1}$ be independent random variables, each uniformly distributed on $[0, 1)$. For $i < n$, let $N_i$ be the number of indices $j < n$ with $\\lfloor n \\omega_j \\rfloor = i$. Then $\\mathbb{E}[N_i] = 1$.","title":"Expected occupancy of a bucket in bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-occupancy-variance","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_occupancy_variance","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-occupancy-variance","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $n \\ge 1$ and let $\\omega_0, \\dots, \\omega_{n-1}$ be independent random variables, each uniformly distributed on $[0, 1)$. For $i < n$, let $N_i$ be the number of indices $j < n$ with $\\lfloor n \\omega_j \\rfloor = i$. Then $\\operatorname{Var}(N_i) = 1 - 1/n$.","title":"Variance of a bucket's occupancy in bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-occupancy-second-moment","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_occupancy_secondMoment","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-occupancy-second-moment","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $n \\ge 1$ and let $\\omega_0, \\dots, \\omega_{n-1}$ be independent random variables, each uniformly distributed on $[0, 1)$. For $i < n$, let $N_i$ be the number of indices $j < n$ with $\\lfloor n \\omega_j \\rfloor = i$. Then $\\mathbb{E}[N_i^2] = 2 - 1/n$.","title":"Second moment of a bucket's occupancy in bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-measurable","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_measurable","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-measurable","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"For every $n$, the function $T : \\mathbb{R}^n \\to \\mathbb{R}$ that maps $\\omega = (\\omega_0, \\dots, \\omega_{n-1})$ to the number of time steps that bucket sort charges on the array $(\\omega_0, \\dots, \\omega_{n-1})$ if every $\\omega_j$ lies in $[0, 1)$, and to $0$ otherwise, is measurable.","title":"The running time of bucket sort is measurable"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-integrable","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_integrable","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-integrable","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"For every $n$, let $T : \\mathbb{R}^n \\to \\mathbb{R}$ map $\\omega = (\\omega_0, \\dots, \\omega_{n-1})$ to the number of time steps that bucket sort charges on the array $(\\omega_0, \\dots, \\omega_{n-1})$ if every $\\omega_j$ lies in $[0, 1)$, and to $0$ otherwise. Then $T$ is integrable with respect to the product of $n$ copies of the uniform distribution on $[0, 1)$.","title":"The running time of bucket sort is integrable"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-expected-lower","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_expected_lower","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-expected-lower","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $n \\ge 1$ and let $\\omega = (\\omega_0, \\dots, \\omega_{n-1})$ have independent coordinates, each uniformly distributed on $[0, 1)$. Let $T(\\omega)$ be the number of time steps that bucket sort charges on the array $(\\omega_0, \\dots, \\omega_{n-1})$ if every $\\omega_j$ lies in $[0, 1)$, and $T(\\omega) = 0$ otherwise. Then $5n \\le \\mathbb{E}[T]$.","title":"Lower bound on the expected time of bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-expected-upper","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_expected_upper","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-expected-upper","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $n \\ge 1$ and let $\\omega = (\\omega_0, \\dots, \\omega_{n-1})$ have independent coordinates, each uniformly distributed on $[0, 1)$. Let $T(\\omega)$ be the number of time steps that bucket sort charges on the array $(\\omega_0, \\dots, \\omega_{n-1})$ if every $\\omega_j$ lies in $[0, 1)$, and $T(\\omega) = 0$ otherwise. Then $\\mathbb{E}[T] \\le 5n + (n - 1)/2$.","title":"Upper bound on the expected time of bucket sort"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-expected-zero","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_expected_zero","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-expected-zero","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $T$ be the number of time steps that bucket sort charges on the empty array, the only input of length $n = 0$. Its expectation under the product of zero copies of the uniform distribution on $[0, 1)$ is $0$.","title":"Bucket sort on the empty input has expected time zero"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-expected-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_expected_one","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-expected-one","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"Let $\\omega_0$ be uniformly distributed on $[0, 1)$, and let $T(\\omega_0)$ be the number of time steps that bucket sort charges on the one-element array $(\\omega_0)$ if $\\omega_0 \\in [0, 1)$, and $T(\\omega_0) = 0$ otherwise. Then $\\mathbb{E}[T] = 5$.","title":"Expected time of bucket sort on one input"},{"id":"ComputerScience/sorting-and-searching/bucket-sort/bucket-sort-time-expected-is-theta","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bucketSort_time_expected_isTheta","parent":41798,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/bucket-sort/index.html#bucket-sort-time-expected-is-theta","scope":"ComputerScience/sorting-and-searching/bucket-sort","status":"fully_proved","summary":"For each $n$, let $\\omega = (\\omega_0, \\dots, \\omega_{n-1})$ have independent coordinates, each uniformly distributed on $[0, 1)$. Let $T_n(\\omega)$ be the number of time steps that bucket sort charges on the array $(\\omega_0, \\dots, \\omega_{n-1})$ if every $\\omega_j$ lies in $[0, 1)$, and $T_n(\\omega) = 0$ otherwise. Then $\\mathbb{E}[T_n] = \\Theta(n)$ as $n \\to \\infty$.","title":"Bucket sort runs in expected linear time"},{"data_url":"scopes/ComputerScience/sorting-and-searching/linear-search.json","id":"ComputerScience/sorting-and-searching/linear-search","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","title":"Linear search"},{"id":"ComputerScience/sorting-and-searching/linear-search/linear-search","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.linearSearch","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#linear-search","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"Given a predicate $p$ on a type $\\alpha$ and a list $L$ of elements of $\\alpha$, linear search scans $L$ from the front and returns the index of the first element of $L$ that satisfies $p$, or no index if no element does, together with a cost that counts one unit for each evaluation of $p$. On the empty list it returns no index at cost $0$. On a list with head $x$ and tail $T$ it evaluates $p(x)$ at a cost of one\u2026","title":"Linear search"},{"id":"ComputerScience/sorting-and-searching/linear-search/ret-linear-search","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ret_linearSearch","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#ret-linear-search","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"For every predicate $p$ on a type $\\alpha$ and every list $L = (x_0, \\dots, x_{n-1})$ over $\\alpha$, the index returned by linear search for $p$ on $L$ (ignoring its cost) equals the result of the standard library's first-index search: the least $i < n$ with $p(x_i)$ true, or no index if $p(x_i)$ is false for every $i < n$.","title":"Linear search returns the first satisfying index"},{"id":"ComputerScience/sorting-and-searching/linear-search/linear-search-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.linearSearch_eq_some_iff","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#linear-search-eq-some-iff","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"For every predicate $p$ on a type $\\alpha$, every list $L = (x_0, \\dots, x_{n-1})$ over $\\alpha$ and every natural number $i$, linear search for $p$ on $L$ returns the index $i$ if and only if $i < n$, $p(x_i)$ is true, and $p(x_j)$ is false for every $j < i$.","title":"Characterization of a successful linear search"},{"id":"ComputerScience/sorting-and-searching/linear-search/linear-search-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.linearSearch_eq_none_iff","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#linear-search-eq-none-iff","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"For every predicate $p$ on a type $\\alpha$ and every list $L$ over $\\alpha$, linear search for $p$ on $L$ returns no index if and only if $p(x)$ is false for every element $x$ of $L$.","title":"Linear search fails exactly when no element satisfies the predicate"},{"id":"ComputerScience/sorting-and-searching/linear-search/linear-search-time-nil","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.linearSearch_time_nil","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#linear-search-time-nil","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"For every predicate $p$ on a type $\\alpha$, linear search for $p$ on the empty list has cost $0$: it evaluates $p$ zero times.","title":"Cost of linear search on the empty list"},{"id":"ComputerScience/sorting-and-searching/linear-search/linear-search-time-cons","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.linearSearch_time_cons","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#linear-search-time-cons","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"For every predicate $p$ on a type $\\alpha$, every $x \\in \\alpha$ and every list $T$ over $\\alpha$, the cost of linear search for $p$ on the list with head $x$ and tail $T$ is $1$ if $p(x)$ is true, and $1$ plus the cost of linear search for $p$ on $T$ if $p(x)$ is false. The head costs one evaluation of $p$, and the tail is searched only when the head fails.","title":"Cost recurrence for linear search"},{"id":"ComputerScience/sorting-and-searching/linear-search/linear-search-time-eq-of-ret-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.linearSearch_time_eq_of_ret_eq_some","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#linear-search-time-eq-of-ret-eq-some","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"For every predicate $p$ on a type $\\alpha$, every list $L$ over $\\alpha$ and every natural number $i$, if linear search for $p$ on $L$ returns the index $i$, then its cost is $i + 1$: it evaluates $p$ exactly $i + 1$ times.","title":"Cost of a successful linear search"},{"id":"ComputerScience/sorting-and-searching/linear-search/linear-search-time-eq-of-ret-eq-none","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.linearSearch_time_eq_of_ret_eq_none","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#linear-search-time-eq-of-ret-eq-none","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"For every predicate $p$ on a type $\\alpha$ and every list $L$ over $\\alpha$, if linear search for $p$ on $L$ returns no index, then its cost is the length of $L$: it evaluates $p$ once for every element of $L$.","title":"Cost of an unsuccessful linear search"},{"id":"ComputerScience/sorting-and-searching/linear-search/linear-search-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.linearSearch_time_le","parent":41816,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/linear-search/index.html#linear-search-time-le","scope":"ComputerScience/sorting-and-searching/linear-search","status":"fully_proved","summary":"For every predicate $p$ on a type $\\alpha$ and every list $L$ over $\\alpha$, the cost of linear search for $p$ on $L$ is at most the length of $L$: it evaluates $p$ at most once for every element of $L$.","title":"Linear search costs at most the list length"},{"data_url":"scopes/ComputerScience/sorting-and-searching/radix-sort.json","id":"ComputerScience/sorting-and-searching/radix-sort","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/radix-sort/index.html","scope":"ComputerScience/sorting-and-searching/radix-sort","status":"fully_proved","title":"Radix sort"},{"id":"ComputerScience/sorting-and-searching/radix-sort/radix-sort","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.radixSort","parent":41826,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/radix-sort/index.html#radix-sort","scope":"ComputerScience/sorting-and-searching/radix-sort","status":"fully_proved","summary":"Let $b \\ge 2$ and let every record carry a key that is a vector of $d$ digits in $\\{0, \\dots, b-1\\}$, most significant digit first, at positions $0, \\dots, d-1$. Radix sort runs $d$ passes of stable counting sort on the array, the pass for position $i$ sorting by the digit of the key at position $i$, for $i = d-1, d-2, \\dots, 0$, that is, from the least to the most significant digit. Its cost is the total cost of\u2026","title":"Radix sort"},{"id":"ComputerScience/sorting-and-searching/radix-sort/radix-sort-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.radixSort_correct","parent":41826,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/radix-sort/index.html#radix-sort-correct","scope":"ComputerScience/sorting-and-searching/radix-sort","status":"fully_proved","summary":"Let $b \\ge 2$ and let every record carry a key that is a vector $(k_0, \\dots, k_{d-1})$ of $d$ digits in $\\{0, \\dots, b-1\\}$, most significant digit first. For every input array, the output of radix sort is a permutation of the input, is in nondecreasing lexicographic order of keys, and is in nondecreasing order of the numbers $\\sum_{i < d} k_i b^{d-1-i}$ whose base-$b$ digits are the keys. Moreover it is stable:\u2026","title":"Correctness of radix sort"},{"id":"ComputerScience/sorting-and-searching/radix-sort/radix-sort-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.radixSort_time","parent":41826,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/radix-sort/index.html#radix-sort-time","scope":"ComputerScience/sorting-and-searching/radix-sort","status":"fully_proved","summary":"Let $b \\ge 2$ and let every record carry a key that is a vector of $d$ digits in $\\{0, \\dots, b-1\\}$. On an array of $n$ records, radix sort uses at most $d(7n + 3b)$ charged operations, where each counting-sort pass charges one unit for every counter initialization, key evaluation, counter read, counter write and output write.","title":"Cost bound for radix sort"},{"data_url":"scopes/ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order.json","id":"ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order/index.html","scope":"ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order","status":"can_prove","title":"Rearranging a permutation into unimodal order"},{"id":"ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order/unimodal","kind":"node","lean":"MathlibExt.TheoryOfComputation.PermuteToUnimodalWanted.ProgramVerification.Unimodal","parent":41830,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order/index.html#unimodal","scope":"ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order","status":"fully_proved","summary":"An array $a$ of natural numbers is unimodal if it is the concatenation of a strictly increasing array $b$ followed by a strictly decreasing array $b'$, where either part may be empty.","title":"Unimodal arrays"},{"id":"ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order/differences","kind":"node","lean":"MathlibExt.TheoryOfComputation.PermuteToUnimodalWanted.ProgramVerification.differences","parent":41830,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order/index.html#differences","scope":"ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order","status":"fully_proved","summary":"For two vectors $a, b \\in \\mathbb{N}^n$, this is the number of indices $i < n$ with $a_i \\neq b_i$, that is, the Hamming distance between $a$ and $b$.","title":"Number of differing positions"},{"id":"ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order/min-rearrange-correct","kind":"node","lean":"MathlibExt.TheoryOfComputation.PermuteToUnimodalWanted.ProgramVerification.minRearrange_correct","parent":41830,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order/index.html#min-rearrange-correct","scope":"ComputerScience/sorting-and-searching/rearranging-a-permutation-into-unimodal-order","status":"can_prove","summary":"Let $a$ be an array of length $n$ that is a permutation of $1, \\dots, n$, and let $m(a)$ be the value that the program $\\mathrm{minRearrange}$ computes on $a$. Then some unimodal permutation $x$ of $1, \\dots, n$ differs from $a$ in exactly $m(a)$ positions, and every unimodal permutation of $1, \\dots, n$ differs from $a$ in at least $m(a)$ positions. So $m(a)$ is the minimum number of positions in which $a$ differs\u2026","title":"Correctness of the unimodal rearrangement algorithm"},{"data_url":"scopes/ComputerScience/sorting-and-searching/selection-sort.json","id":"ComputerScience/sorting-and-searching/selection-sort","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/selection-sort/index.html","scope":"ComputerScience/sorting-and-searching/selection-sort","status":"fully_proved","title":"Selection sort"},{"id":"ComputerScience/sorting-and-searching/selection-sort/extract-min","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.extractMin","parent":41834,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/selection-sort/index.html#extract-min","scope":"ComputerScience/sorting-and-searching/selection-sort","status":"fully_proved","summary":"Given an initial candidate $c$ in a linearly ordered type and a list $L$, minimum extraction returns a pair $(m, r)$ together with a cost. If $L$ is empty, it returns $c$ with the empty list at cost $0$. If $L$ has head $x$ and tail $L'$, it compares $x$ with $c$ at cost one: if $x < c$, it extracts from $L'$ with candidate $x$, obtaining $(m, r)$, and returns $m$ with the list $c$ followed by $r$; otherwise it\u2026","title":"Minimum extraction"},{"id":"ComputerScience/sorting-and-searching/selection-sort/selection-sort","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.selectionSort","parent":41834,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/selection-sort/index.html#selection-sort","scope":"ComputerScience/sorting-and-searching/selection-sort","status":"fully_proved","summary":"Selection sort of a list $L$ over a linearly ordered type is defined recursively. If $L$ is empty, the result is the empty list at cost $0$. If $L$ has head $x$ and tail $L'$, minimum extraction from $L'$ with initial candidate $x$ yields a minimum $m$ and the list $r$ of remaining elements, and the result is $m$ followed by the selection sort of $r$. The cost is the sum of the costs of all these minimum extractions.","title":"Selection sort"},{"id":"ComputerScience/sorting-and-searching/selection-sort/extract-min-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.extractMin_time","parent":41834,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/selection-sort/index.html#extract-min-time","scope":"ComputerScience/sorting-and-searching/selection-sort","status":"fully_proved","summary":"For every element $c$ of a linearly ordered type and every list $L$ over it, minimum extraction from $L$ with initial candidate $c$ has cost exactly $|L|$, the length of $L$; that is, it performs exactly $|L|$ order comparisons.","title":"Cost of minimum extraction"},{"id":"ComputerScience/sorting-and-searching/selection-sort/extract-min-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.extractMin_correct","parent":41834,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/selection-sort/index.html#extract-min-correct","scope":"ComputerScience/sorting-and-searching/selection-sort","status":"fully_proved","summary":"Let $c$ be an element of a linearly ordered type and $L$ a list over it, and let $(m, r)$ be the pair returned by minimum extraction from $L$ with initial candidate $c$. Then the list $m$ followed by $r$ is a permutation of the list $c$ followed by $L$, $m \\le c$, and $m \\le x$ for every element $x$ of $L$.","title":"Correctness of minimum extraction"},{"id":"ComputerScience/sorting-and-searching/selection-sort/selection-sort-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.selectionSort_correct","parent":41834,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/selection-sort/index.html#selection-sort-correct","scope":"ComputerScience/sorting-and-searching/selection-sort","status":"fully_proved","summary":"For every list $L$ over a linearly ordered type, the list returned by selection sort on $L$ is sorted in nondecreasing order (each entry is $\\le$ every later entry) and is a permutation of $L$, so it preserves duplicate multiplicities.","title":"Correctness of selection sort"},{"id":"ComputerScience/sorting-and-searching/selection-sort/selection-sort-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.selectionSort_time","parent":41834,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/selection-sort/index.html#selection-sort-time","scope":"ComputerScience/sorting-and-searching/selection-sort","status":"fully_proved","summary":"For every list of length $n$ over a linearly ordered type, selection sort has cost exactly $n(n-1)/2$ in the model that charges one unit per order comparison; that is, it performs exactly $n(n-1)/2$ comparisons.","title":"Comparison count of selection sort"},{"data_url":"scopes/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum.json","id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","title":"Simultaneous minimum and maximum"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"Given a list $xs$ over a linearly ordered type, the paired algorithm returns no result when $xs$ is empty and otherwise returns a pair (a candidate minimum and a candidate maximum), counting one unit of time for each comparison of elements. On a one-element list $[a]$ it returns $(a, a)$ without a comparison. On a longer list of odd length both candidates start at the first element without a comparison; on a list\u2026","title":"Simultaneous minimum and maximum"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax_ret","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max-ret","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"For every list $xs$ over a linearly ordered type, the simultaneous minimum and maximum algorithm returns no result on $xs$ when $xs$ is empty, and returns the pair $(\\min xs, \\max xs)$ of the least and greatest elements of $xs$ otherwise.","title":"Correctness of the simultaneous minimum and maximum algorithm"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax_eq_none_iff","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max-eq-none-iff","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"For every list $xs$ over a linearly ordered type, the simultaneous minimum and maximum algorithm returns no result on $xs$ if and only if $xs$ is empty.","title":"The min\u2013max algorithm returns no result exactly on the empty list"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax_eq_some_iff","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max-eq-some-iff","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"For every list $xs$ over a linearly ordered type and all elements $lo$ and $hi$ of that type, the simultaneous minimum and maximum algorithm returns the pair $(lo, hi)$ on $xs$ if and only if $lo$ and $hi$ both belong to $xs$, $lo \\le x$ for every $x$ in $xs$, and $x \\le hi$ for every $x$ in $xs$.","title":"Characterization of the min\u2013max output"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max-nil","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax_nil","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max-nil","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"On the empty list, the simultaneous minimum and maximum algorithm returns no result and makes no comparisons.","title":"The min\u2013max algorithm on the empty list"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax_singleton","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max-singleton","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"For every element $a$ of a linearly ordered type, on the list $[a]$ the simultaneous minimum and maximum algorithm returns the pair $(a, a)$ and makes no comparisons.","title":"The min\u2013max algorithm on a one-element list"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max-time-odd","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax_time_odd","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max-time-odd","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"For every natural number $k$ and every list of length $2k + 1$ over a linearly ordered type, the simultaneous minimum and maximum algorithm makes exactly $3k$ comparisons.","title":"Comparisons of the min\u2013max algorithm on odd-length lists"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max-time-even","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax_time_even","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max-time-even","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"For every natural number $k \\ge 1$ and every list of length $2k$ over a linearly ordered type, the simultaneous minimum and maximum algorithm makes exactly $3k - 2$ comparisons.","title":"Comparisons of the min\u2013max algorithm on even-length lists"},{"id":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/min-max-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.minMax_time","parent":41841,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum/index.html#min-max-time","scope":"ComputerScience/sorting-and-searching/simultaneous-minimum-and-maximum","status":"fully_proved","summary":"For every list of length $n$ over a linearly ordered type, the simultaneous minimum and maximum algorithm makes no comparisons when $n = 0$ and exactly $\\lceil 3n/2 \\rceil - 2$ comparisons when $n \\ge 1$.","title":"Comparison count of the simultaneous minimum and maximum algorithm"},{"data_url":"scopes/ComputerScience/sorting-and-searching/three-way-partitioning.json","id":"ComputerScience/sorting-and-searching/three-way-partitioning","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","title":"Three-way partitioning"},{"id":"ComputerScience/sorting-and-searching/three-way-partitioning/three-way-partition","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ThreeWayPartition","parent":41851,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html#three-way-partition","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","summary":"A three-way partition over a type $\\alpha$ consists of three lists over $\\alpha$, called below, equal and above. They are meant to hold the elements strictly below, equal to, and strictly above a pivot; the record itself imposes no condition on them.","title":"Three-way partition record"},{"id":"ComputerScience/sorting-and-searching/three-way-partitioning/partition3","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3","parent":41851,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html#partition3","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","summary":"Let $p$ be an element of a linearly ordered type $\\alpha$. The three-way partition of a list over $\\alpha$ around $p$, together with the time it is charged, is defined by recursion on the list. The empty list gives three empty lists at no cost. For a list with head $x$ and tail $xs$, one unit of time is charged for comparing $x$ with $p$, the tail $xs$ is partitioned, and $x$ is put at the front of its below list\u2026","title":"Stable three-way partition"},{"id":"ComputerScience/sorting-and-searching/three-way-partitioning/partition3-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_time","parent":41851,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html#partition3-time","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","summary":"For every element $p$ of a linearly ordered type $\\alpha$ and every list $xs$ over $\\alpha$, the time charged by the stable three-way partition of $xs$ around $p$ equals the length of $xs$: one unit for each comparison, one comparison per element.","title":"Cost of the stable three-way partition"},{"id":"ComputerScience/sorting-and-searching/three-way-partitioning/partition3-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_correct","parent":41851,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html#partition3-correct","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","summary":"Let $p$ be an element of a linearly ordered type $\\alpha$, let $xs$ be a list over $\\alpha$, and let $B$, $E$ and $A$ be the below, equal and above lists of the stable three-way partition of $xs$ around $p$. Then the concatenation $B$, $E$, $A$ is a permutation of $xs$, every element $x$ of $B$ satisfies $x < p$, every element of $E$ equals $p$, and every element $x$ of $A$ satisfies $p < x$.","title":"Correctness of the stable three-way partition"},{"id":"ComputerScience/sorting-and-searching/three-way-partitioning/partition3-below-eq-filter","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_below_eq_filter","parent":41851,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html#partition3-below-eq-filter","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","summary":"For every element $p$ of a linearly ordered type $\\alpha$ and every list $xs$ over $\\alpha$, the below list of the stable three-way partition of $xs$ around $p$ is the list of the elements $x$ of $xs$ with $x < p$, in their order in $xs$ and with their multiplicities.","title":"The below list of a three-way partition is a filter"},{"id":"ComputerScience/sorting-and-searching/three-way-partitioning/partition3-equal-eq-filter","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_equal_eq_filter","parent":41851,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html#partition3-equal-eq-filter","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","summary":"For every element $p$ of a linearly ordered type $\\alpha$ and every list $xs$ over $\\alpha$, the equal list of the stable three-way partition of $xs$ around $p$ is the list of the elements $x$ of $xs$ with $x = p$, in their order in $xs$ and with their multiplicities.","title":"The equal list of a three-way partition is a filter"},{"id":"ComputerScience/sorting-and-searching/three-way-partitioning/partition3-above-eq-filter","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_above_eq_filter","parent":41851,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html#partition3-above-eq-filter","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","summary":"For every element $p$ of a linearly ordered type $\\alpha$ and every list $xs$ over $\\alpha$, the above list of the stable three-way partition of $xs$ around $p$ is the list of the elements $x$ of $xs$ with $p < x$, in their order in $xs$ and with their multiplicities.","title":"The above list of a three-way partition is a filter"},{"id":"ComputerScience/sorting-and-searching/three-way-partitioning/partition3-length","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_length","parent":41851,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/three-way-partitioning/index.html#partition3-length","scope":"ComputerScience/sorting-and-searching/three-way-partitioning","status":"fully_proved","summary":"For every element $p$ of a linearly ordered type $\\alpha$ and every list $xs$ over $\\alpha$, the lengths of the below, equal and above lists of the stable three-way partition of $xs$ around $p$ add up to the length of $xs$.","title":"Sizes of the three parts of a three-way partition"},{"data_url":"scopes/ComputerScience/sorting-and-searching/quickselect.json","id":"ComputerScience/sorting-and-searching/quickselect","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","title":"Quickselect"},{"id":"ComputerScience/sorting-and-searching/quickselect/quickselect","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickselect","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#quickselect","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Deterministic quickselect takes a list $L$ over a linear order and a zero-based rank $k$ and returns an element or no element, together with its cost, the number of element comparisons it makes. If $k \\ge |L|$, it returns no element at no cost. Otherwise it runs the following recursion, starting from the list $L$ and the rank $k$. On an empty list it returns no element. On a list with first entry $p$ and remaining\u2026","title":"Quickselect"},{"id":"ComputerScience/sorting-and-searching/quickselect/quickselect-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickselect_eq_none_iff","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#quickselect-eq-none-iff","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Let $L$ be a list over a linear order and let $k$ be a natural number. Quickselect on $L$ with zero-based rank $k$ returns no element if and only if $|L| \\le k$.","title":"Quickselect fails exactly out of range"},{"id":"ComputerScience/sorting-and-searching/quickselect/partition3-count-p-eq-below","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_countP_eq_below","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#partition3-count-p-eq-below","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Let $L$ be a list over a linear order, let $p$ be an element, let $q$ be a Boolean predicate on the elements, and let $B$, $E$, $A$ be the blocks of entries below, equal to and above $p$ in the three-way partition of $L$ around $p$. If $q$ is false on every entry of $E$ and on every entry of $A$, then the number of entries of $L$ satisfying $q$ equals the number of entries of $B$ satisfying $q$, both counted with\u2026","title":"Counting through the below block"},{"id":"ComputerScience/sorting-and-searching/quickselect/partition3-count-p-eq-block-add-above","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_countP_eq_block_add_above","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#partition3-count-p-eq-block-add-above","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Let $L$ be a list over a linear order, let $p$ be an element, let $q$ be a Boolean predicate on the elements, and let $B$, $E$, $A$ be the blocks of entries below, equal to and above $p$ in the three-way partition of $L$ around $p$. If $q$ is true on every entry of $B$ and on every entry of $E$, then the number of entries of $L$ satisfying $q$ equals $|B| + |E|$ plus the number of entries of $A$ satisfying $q$, all\u2026","title":"Counting through the above block"},{"id":"ComputerScience/sorting-and-searching/quickselect/partition3-count-lt","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_count_lt","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#partition3-count-lt","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Let $L$ be a list over a linear order, let $p$ be an element, and let $B$ be the block of entries below $p$ in the three-way partition of $L$ around $p$. Then the number of entries of $L$ that are strictly less than $p$, counted with multiplicity, equals $|B|$.","title":"Entries below the pivot"},{"id":"ComputerScience/sorting-and-searching/quickselect/partition3-count-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.partition3_count_le","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#partition3-count-le","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Let $L$ be a list over a linear order, let $p$ be an element, and let $B$ and $E$ be the blocks of entries below and equal to $p$ in the three-way partition of $L$ around $p$. Then the number of entries of $L$ that are at most $p$, counted with multiplicity, equals $|B| + |E|$.","title":"Entries at most the pivot"},{"id":"ComputerScience/sorting-and-searching/quickselect/quickselect-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickselect_correct","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#quickselect-correct","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Let $L$ be a list over a linear order and let $k$ be a natural number. If quickselect on $L$ with zero-based rank $k$ returns an element $x$, then $x$ is an entry of $L$, and, counting entries of $L$ with multiplicity, the number of entries less than $x$ is at most $k$ and $k$ is less than the number of entries at most $x$.","title":"Correctness of quickselect"},{"id":"ComputerScience/sorting-and-searching/quickselect/quickselect-eq-get-elem-insertion-sort","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickselect_eq_getElem?_insertionSort","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#quickselect-eq-get-elem-insertion-sort","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Let $L$ be a list over a linear order, let $S$ be the insertion sort of $L$ in nondecreasing order, and let $k$ be a natural number. If $k < |L|$, quickselect on $L$ with zero-based rank $k$ returns the entry of $S$ at zero-based position $k$; if $k \\ge |L|$, it returns no element.","title":"Quickselect agrees with sorting"},{"id":"ComputerScience/sorting-and-searching/quickselect/quickselect-eq-some-of-count-rank","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickselect_eq_some_of_count_rank","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#quickselect-eq-some-of-count-rank","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"Let $L$ be a list over a linear order, let $k$ be a natural number and let $x$ be an element. Counting entries of $L$ with multiplicity, if the number of entries less than $x$ is at most $k$ and $k$ is less than the number of entries at most $x$, then quickselect on $L$ with zero-based rank $k$ returns $x$.","title":"Rank inequalities determine the quickselect value"},{"id":"ComputerScience/sorting-and-searching/quickselect/quickselect-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickselect_time","parent":41860,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quickselect/index.html#quickselect-time","scope":"ComputerScience/sorting-and-searching/quickselect","status":"fully_proved","summary":"For every list $L$ over a linear order, of length $n$, and every natural number $k$, quickselect on $L$ with rank $k$ makes at most $n(n-1)/2$ comparisons.","title":"Worst-case comparison bound for quickselect"},{"data_url":"scopes/ComputerScience/sorting-and-searching/median-of-medians.json","id":"ComputerScience/sorting-and-searching/median-of-medians","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","title":"Median of medians"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-groups","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMediansGroups","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-groups","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Given a list $xs$ over a linear order, split $xs$ into consecutive blocks of five elements, the last of which may be shorter. Sort each block $g$ by insertion sort and return, in block order, the entry at position $\\lfloor (|g| - 1)/2 \\rfloor$ (counting from $0$) of each sorted block, that is, its lower median. The cost is the total number of comparisons made by these insertion sorts.","title":"Group medians for median-of-medians selection"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-groups-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMediansGroups_time","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-groups-time","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $xs$ of length $n$ over a linear order, computing the group medians of $xs$ takes at most $2n$ comparisons.","title":"Comparison cost of the group medians"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-groups-mem","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMediansGroups_mem","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-groups-mem","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $xs$ over a linear order, every entry of the list of group medians of $xs$ is an element of $xs$.","title":"Group medians come from the input"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-groups-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMediansGroups_correct","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-groups-correct","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $xs$ be a list over a linear order, and let $g_1, \\dots, g_m$ be its consecutive blocks of five elements, the last of which may be shorter. Then the list of group medians of $xs$ has the same length $m$, say $x_1, \\dots, x_m$, and for each $i$, writing $k_i = \\lfloor (|g_i| - 1)/2 \\rfloor$ and counting elements of $g_i$ with multiplicity: $x_i$ belongs to $g_i$, at most $k_i$ elements of $g_i$ are less than\u2026","title":"Group medians are lower medians of their blocks"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-groups-length","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMediansGroups_length","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-groups-length","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $xs$ of length $n$ over a linear order, the list of group medians of $xs$ has exactly $\\lfloor (n + 4)/5 \\rfloor = \\lceil n/5 \\rceil$ entries.","title":"Number of group medians"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Given a list $L$ over a linear order and a zero-based rank $k$, median-of-medians selection returns either an element of $L$ or nothing, charging one unit of cost per order comparison. If $k \\ge |L|$, where $|L|$ is the length of $L$, it returns nothing at no cost. Otherwise it runs the following recursive procedure on $(L, k)$ with a budget of $|L|$ nested calls, returning nothing if the budget runs out. A list of\u2026","title":"Median-of-medians selection"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-eq-quickselect-of-leng","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_eq_quickselect_of_length_le","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-eq-quickselect-of-leng","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $L$ of length at most $140$ over a linearly ordered type and every natural number $k$, median-of-medians selection of rank $k$ in $L$ coincides with timed quickselect of rank $k$ in $L$: both return the same result and are charged the same number of comparisons.","title":"Median-of-medians selection is quickselect on short lists"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-eq-pure-none-of-length","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_eq_pure_none_of_length_le_rank","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-eq-pure-none-of-length","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $L$ over a linearly ordered type and every natural number $k$ at least the length of $L$, median-of-medians selection of rank $k$ in $L$ returns nothing and is charged zero comparisons.","title":"Out-of-range ranks cost no comparisons"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-base-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_base_time","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-base-time","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $L$ of length $n \\le 140$ over a linearly ordered type and every natural number $k$, median-of-medians selection of rank $k$ in $L$ is charged at most $80n$ comparisons.","title":"Comparison bound for median-of-medians selection on short lists"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-base-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_base_correct","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-base-correct","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $L$ be a list of length at most $140$ over a linearly ordered type and let $k$ be a natural number. If median-of-medians selection of rank $k$ in $L$ returns an element $x$, then $x$ occurs in $L$, the number of entries of $L$ less than $x$, counted with multiplicity, is at most $k$, and $k$ is less than the number of entries of $L$ that are at most $x$.","title":"Correctness of median-of-medians selection on short lists"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_correct","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-correct","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $L$ be a list over a linearly ordered type and let $k$ be a natural number. If median-of-medians selection of rank $k$ in $L$ returns an element $x$, then $x$ occurs in $L$, the number of entries of $L$ less than $x$, counted with multiplicity, is at most $k$, and $k$ is less than the number of entries of $L$ that are at most $x$.","title":"Correctness of median-of-medians selection"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-exists","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_exists","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-exists","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $L$ over a linearly ordered type and every natural number $k$ less than the length of $L$, median-of-medians selection of rank $k$ in $L$ returns some element.","title":"Median-of-medians selection succeeds on valid ranks"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_eq_none_iff","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-eq-none-iff","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $L$ over a linearly ordered type and every natural number $k$, median-of-medians selection of rank $k$ in $L$ returns nothing if and only if $k$ is at least the length of $L$.","title":"Median-of-medians selection fails exactly on out-of-range ranks"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-eq-get-elem-insertion","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_eq_getElem?_insertionSort","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-eq-get-elem-insertion","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $L$ over a linearly ordered type and every natural number $k$, median-of-medians selection of rank $k$ in $L$ returns the entry at zero-based position $k$ of the insertion sort of $L$ by $\\le$ when $k$ is less than the length of $L$, and returns nothing otherwise.","title":"Median-of-medians selection agrees with sorting"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-set-rank","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_set_rank","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-set-rank","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $L$ be a list without repeated entries over a linear order and let $k \\in \\mathbb{N}$. If median-of-medians selection on $L$ at zero-based rank $k$ returns an element $x$, then the set $\\{y \\in L : y \\le x\\}$ has exactly $k + 1$ elements.","title":"Set rank of a median-of-medians result"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-source-rank","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_source_rank","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-source-rank","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $L$ be a list without repeated entries over a linear order and let $i \\ge 1$. If median-of-medians selection on $L$ at zero-based rank $i - 1$ returns an element $x$, then the set $\\{y \\in L : y \\le x\\}$ has exactly $i$ elements.","title":"One-based rank of a median-of-medians result"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-source-exists","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_source_exists","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-source-exists","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $L$ be a list without repeated entries over a linear order, of length $n$, and let $1 \\le i \\le n$. Then median-of-medians selection on $L$ at zero-based rank $i - 1$ returns an element $x$ of $L$, and the set $\\{y \\in L : y \\le x\\}$ has exactly $i$ elements.","title":"Median of medians selects every one-based rank"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-lower-median","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_lower_median","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-lower-median","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $L$ be a nonempty list of length $n$ over a linear order, and let $k = \\lfloor (n - 1)/2 \\rfloor$. Then median-of-medians selection on $L$ at zero-based rank $k$ returns an element $x$ of $L$ such that at most $k$ entries of $L$ are less than $x$ and more than $k$ entries of $L$ are at most $x$, where entries are counted with repetition.","title":"Median of medians finds the lower median"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-upper-median","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_upper_median","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-upper-median","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $L$ be a nonempty list of length $n$ over a linear order, and let $k = \\lfloor n/2 \\rfloor$. Then median-of-medians selection on $L$ at zero-based rank $k$ returns an element $x$ of $L$ such that at most $k$ entries of $L$ are less than $x$ and more than $k$ entries of $L$ are at most $x$, where entries are counted with repetition.","title":"Median of medians finds the upper median"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-child-size-budget","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_child_size_budget","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-child-size-budget","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $n$, $m$ and $p$ be natural numbers with $n \\ge 140$, $m \\le \\lfloor (n + 4)/5 \\rfloor$ and $p \\le \\lfloor 7n/10 \\rfloor + 6$. Then $20m + 20p + n \\le 20n$.","title":"Size budget for the two median-of-medians recursive calls"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-combine-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_combine_time","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-combine-time","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $n$, $m$, $p$, $a$, $c_1$ and $c_2$ be natural numbers with $n \\ge 140$, $m \\le \\lfloor (n + 4)/5 \\rfloor$, $p \\le \\lfloor 7n/10 \\rfloor + 6$, $c_1 \\le 20am$ and $c_2 \\le 20ap$. Then $c_1 + c_2 + an \\le 20an$.","title":"Cost budget for one median-of-medians step"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_time","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-time","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"For every list $L$ of length $n$ over a linear order and every $k \\in \\mathbb{N}$, median-of-medians selection on $L$ at rank $k$ charges at most $80n$ comparisons, where $n$ counts repeated entries.","title":"Median of medians makes linearly many comparisons"},{"id":"ComputerScience/sorting-and-searching/median-of-medians/median-of-medians-linear-recurrence","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.medianOfMedians_linear_recurrence","parent":41871,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/median-of-medians/index.html#median-of-medians-linear-recurrence","scope":"ComputerScience/sorting-and-searching/median-of-medians","status":"fully_proved","summary":"Let $T : \\mathbb{N} \\to \\mathbb{N}$ and $a \\in \\mathbb{N}$. Suppose that $T(n) \\le 20an$ for every $n \\le 140$, and that for every $n > 140$ there are $m, p \\in \\mathbb{N}$ with $m \\le \\lfloor (n + 4)/5 \\rfloor$, $p \\le \\lfloor 7n/10 \\rfloor + 6$ and $T(n) \\le T(m) + T(p) + an$. Then $T(n) \\le 20an$ for every $n \\in \\mathbb{N}$.","title":"Linear solution of the median-of-medians recurrence"},{"data_url":"scopes/ComputerScience/sorting-and-searching/quicksort.json","id":"ComputerScience/sorting-and-searching/quicksort","kind":"scope","navigation":"module","parent":41654,"result":false,"route":"../roadmap/ComputerScience/sorting-and-searching/quicksort/index.html","scope":"ComputerScience/sorting-and-searching/quicksort","status":"fully_proved","title":"Quicksort"},{"id":"ComputerScience/sorting-and-searching/quicksort/quick-sort","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickSort","parent":41895,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quicksort/index.html#quick-sort","scope":"ComputerScience/sorting-and-searching/quicksort","status":"fully_proved","summary":"Deterministic quicksort on lists over a linear order, using the first entry as pivot and counting comparisons. The empty list is returned unchanged at no cost. On a list with first entry $p$ and tail $t$, the algorithm stably splits $t$ into the entries less than $p$, equal to $p$ and greater than $p$, using one comparison per entry of $t$; it then recursively sorts the lesser and greater parts and returns the\u2026","title":"Quicksort"},{"id":"ComputerScience/sorting-and-searching/quicksort/quick-sort-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickSort_correct","parent":41895,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quicksort/index.html#quick-sort-correct","scope":"ComputerScience/sorting-and-searching/quicksort","status":"fully_proved","summary":"For every list $\\ell$ over a linear order, the output of quicksort on $\\ell$ is sorted in nondecreasing order and is a permutation of $\\ell$; in particular every value occurs in the output exactly as often as in $\\ell$.","title":"Correctness of quicksort"},{"id":"ComputerScience/sorting-and-searching/quicksort/quick-sort-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickSort_time","parent":41895,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quicksort/index.html#quick-sort-time","scope":"ComputerScience/sorting-and-searching/quicksort","status":"fully_proved","summary":"For every list of length $n$ over a linear order, quicksort performs at most $n(n-1)/2$ comparisons.","title":"Worst-case comparison bound for quicksort"},{"id":"ComputerScience/sorting-and-searching/quicksort/between-count","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.betweenCount","parent":41895,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quicksort/index.html#between-count","scope":"ComputerScience/sorting-and-searching/quicksort","status":"fully_proved","summary":"For a list $\\ell$ over a linear order and elements $a$ and $b$, this is the number of entries $x$ of $\\ell$, counted with multiplicity, such that $a < x < b$.","title":"Number of list entries strictly between two values"},{"id":"ComputerScience/sorting-and-searching/quicksort/quick-sort-expected-time-uniform-permuta","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickSort_expectedTime_uniformPermutation","parent":41895,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quicksort/index.html#quick-sort-expected-time-uniform-permuta","scope":"ComputerScience/sorting-and-searching/quicksort","status":"fully_proved","summary":"Let $\\ell$ be a list without repeated entries over a linear order. The expected number of comparisons made by quicksort on the list obtained by rearranging the entries of $\\ell$ by a uniformly random permutation of their positions, computed in the extended nonnegative reals $[0, \\infty]$, equals","title":"Expected comparisons of quicksort on a random permutation"},{"id":"ComputerScience/sorting-and-searching/quicksort/quick-sort-expected-time-uniform-permuta-2","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.quickSort_expectedTime_uniformPermutation_toReal","parent":41895,"result":true,"route":"../roadmap/ComputerScience/sorting-and-searching/quicksort/index.html#quick-sort-expected-time-uniform-permuta-2","scope":"ComputerScience/sorting-and-searching/quicksort","status":"fully_proved","summary":"Let $\\ell$ be a list without repeated entries over a linear order. Take the expected number of comparisons made by quicksort on the list obtained by rearranging the entries of $\\ell$ by a uniformly random permutation of their positions, which is an element of $[0, \\infty]$, and convert it to a real number. The result equals the real sum","title":"Expected comparisons of quicksort on a random permutation, as a real number"},{"data_url":"scopes/ComputerScience/dynamic-programming.json","id":"ComputerScience/dynamic-programming","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/index.html","scope":"ComputerScience/dynamic-programming","status":"fully_proved","summary":"Optimization problems solved by tabulated recurrences: knapsack variants, subset sum, rod cutting, matrix-chain ordering, optimal binary search trees, longest common and longest increasing subsequences (including patience sorting) and maximum subarray (Kadane and divide-and-conquer); string edit distance goes to String algorithms.","title":"Dynamic programming"},{"data_url":"scopes/ComputerScience/dynamic-programming/longest-common-subsequence.json","id":"ComputerScience/dynamic-programming/longest-common-subsequence","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/longest-common-subsequence/index.html","scope":"ComputerScience/dynamic-programming/longest-common-subsequence","status":"fully_proved","title":"Longest common subsequence"},{"id":"ComputerScience/dynamic-programming/longest-common-subsequence/longest-common-subsequence","kind":"node","lean":"Cslib.Algorithms.Lean.longestCommonSubsequence","parent":41903,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-common-subsequence/index.html#longest-common-subsequence","scope":"ComputerScience/dynamic-programming/longest-common-subsequence","status":"fully_proved","summary":"Given lists $x = x_1 \\cdots x_m$ and $y = y_1 \\cdots y_n$ over a type with a boolean equality test, fill a table $c_{i,j}$ for $0 \\le i \\le m$ and $0 \\le j \\le n$ by $c_{i,0} = c_{0,j} = 0$, by $c_{i,j} = c_{i-1,j-1} + 1$ when the test reports $x_i = y_j$, and by $c_{i,j} = \\max(c_{i-1,j}, c_{i,j-1})$ otherwise. Then build a witness $w$ by walking back from $(m, n)$ until $i = 0$ or $j = 0$: when the test reports\u2026","title":"Longest common subsequence by dynamic programming"},{"id":"ComputerScience/dynamic-programming/longest-common-subsequence/longest-common-subsequence-correct","kind":"node","lean":"Cslib.Algorithms.Lean.longestCommonSubsequence_correct","parent":41903,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-common-subsequence/index.html#longest-common-subsequence-correct","scope":"ComputerScience/dynamic-programming/longest-common-subsequence","status":"fully_proved","summary":"Let $\\alpha$ be a type whose boolean equality test agrees with equality, let $x$ and $y$ be lists over $\\alpha$, and let $(k, w)$ be the pair returned by the dynamic-programming longest common subsequence algorithm on $x$ and $y$. Then $w$ has length $k$, $w$ is a subsequence (not necessarily contiguous) of both $x$ and $y$, and every common subsequence of $x$ and $y$ has length at most the length of $w$.","title":"Correctness of the longest common subsequence algorithm"},{"id":"ComputerScience/dynamic-programming/longest-common-subsequence/longest-common-subsequence-time","kind":"node","lean":"Cslib.Algorithms.Lean.longestCommonSubsequence_time","parent":41903,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-common-subsequence/index.html#longest-common-subsequence-time","scope":"ComputerScience/dynamic-programming/longest-common-subsequence","status":"fully_proved","summary":"For lists $x$ and $y$ of lengths $m$ and $n$ over a type with a boolean equality test, the abstract time of the dynamic-programming longest common subsequence algorithm on $x$ and $y$ (its number of table and reconstruction transitions) is at most $mn + (m + n)$.","title":"Running time of the longest common subsequence algorithm"},{"data_url":"scopes/ComputerScience/dynamic-programming/longest-increasing-subsequence.json","id":"ComputerScience/dynamic-programming/longest-increasing-subsequence","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","title":"Longest increasing subsequence"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/is-lislength","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsLISLength","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#is-lislength","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For a list $xs$ over a preorder and a natural number $k$, $k$ is the length of a longest strictly increasing subsequence of $xs$ when some strictly increasing subsequence of $xs$ has length $k$ and every strictly increasing subsequence of $xs$ has length at most $k$. Here a subsequence need not be contiguous, and strictly increasing means that each element is strictly less than every later one.","title":"Length of a longest increasing subsequence"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/is-liscell","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsLISCell","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#is-liscell","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For an element $x$, lists $s$ and $zs$, and a natural number $k$, the pair $(k, zs)$ is a table cell for $x$ followed by $s$ when $zs$ is a subsequence (not necessarily contiguous) of $s$, the list $x$ followed by $zs$ is strictly increasing (each element strictly less than every later one) and has length $k$, and for every subsequence $zs'$ of $s$ such that $x$ followed by $zs'$ is strictly increasing, the list\u2026","title":"Table cell of the LIS dynamic program"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/max-above","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxAbove","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#max-above","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Given a value $x$ and a list $t$ of entries $(y, k, zs)$, where $y$ is a value, $k$ a natural number and $zs$ a list of values, this timed computation returns the pair of $k$ and the list $y$ followed by $zs$ for an entry of $t$ with $x < y$ whose $k$ is largest among such entries, provided that $k$ is positive; if no entry with $x < y$ has $k > 0$, it returns $0$ and the empty list. It costs one unit per entry of\u2026","title":"Best table entry above a value"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/lis-table","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.lisTable","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#lis-table","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"The table of a list is a list of triples $(y, k, zs)$, built from the right: the empty list has the empty table, and the table of $x$ followed by $rest$ is the table $t$ of $rest$ with the entry $(x, b + 1, w)$ put in front, where $(b, w)$ is the result of the scan of $t$ for entries above $x$. Building the table costs the total cost of these scans.","title":"Table of the quadratic LIS program"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/max-entry","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxEntry","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#max-entry","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Given a list $t$ of entries $(y, k, zs)$, where $k$ is a natural number and $zs$ is a list with elements of the same type as $y$, this timed computation returns the pair of $k$ and the list $y$ followed by $zs$ for an entry of $t$ whose $k$ is largest, provided that this $k$ is positive, and returns $0$ and the empty list otherwise (in particular when $t$ is empty). It costs one unit per entry of $t$.","title":"Table entry of largest length"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/longest-increasing-subsequence","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.longestIncreasingSubsequence","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#longest-increasing-subsequence","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Given a list $x_0, \\dots, x_{n-1}$ of elements of a preorder with decidable $<$, this timed computation builds, from the right, a table recording for each position $i$ a length and a strictly increasing subsequence starting with $x_i$ (the longest one, by the recurrence $L_i = 1 + \\max \\{ L_j : i < j,\\ x_i < x_j \\}$ with $\\max \\emptyset = 0$), and returns the stored subsequence of an entry of largest length, or the\u2026","title":"Longest increasing subsequence by dynamic programming"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/max-above-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxAbove_time","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#max-above-time","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Let $\\alpha$ be a preorder with decidable $<$. For every value $x \\in \\alpha$ and every list $t$ of entries $(y, k, zs)$, where $y \\in \\alpha$, $k$ is a natural number and $zs$ is a list over $\\alpha$, the scan of $t$ for the best table entry above $x$ costs exactly the length of $t$.","title":"Cost of the scan for the best entry above a value"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/longest-increasing-subsequence-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.longestIncreasingSubsequence_time","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#longest-increasing-subsequence-time","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For every list $xs$ of length $n$ over a preorder, twice the cost of the quadratic longest-increasing-subsequence program on $xs$ equals $n(n + 1)$, so the program costs exactly $n(n + 1)/2$.","title":"Running time of the quadratic LIS program"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/exists-suffix-of-cons-sublist","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.exists_suffix_of_cons_sublist","parent":41907,"reading_title":{"format":"inline-parts-v1","parts":[["text","A subsequence starting with "],["math","\\(y\\)"],["text"," starts at an occurrence of "],["math","\\(y\\)"]],"source":"A subsequence starting with $y$ starts at an occurrence of $y$"},"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#exists-suffix-of-cons-sublist","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For an element $y$ and lists $zs$ and $l$, if the list $y$ followed by $zs$ is a subsequence (not necessarily contiguous) of $l$, then there is a list $s$ such that $y$ followed by $s$ is a suffix of $l$ and $zs$ is a subsequence of $s$.","title":"A subsequence starting with $y$ starts at an occurrence of $y$"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/listable-spec","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.LISTableSpec","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#listable-spec","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"A table $t$ of triples $(y, k, zs)$ satisfies the invariant for a list $l$ when every entry $(y, k, zs)$ of $t$ has a list $s$ such that $y$ followed by $s$ is a suffix of $l$ and $(k, zs)$ is a table cell for $y$ followed by $s$, and conversely, for every element $y$ and list $s$ such that $y$ followed by $s$ is a suffix of $l$, some entry $(y, k, zs)$ of $t$ makes $(k, zs)$ a table cell for $y$ followed by $s$.","title":"Invariant of the LIS table"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/is-liscell-max-above","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.isLISCell_maxAbove","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#is-liscell-max-above","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Let $x$ be an element, $rest$ a list, and $t$ a table of triples $(y, k, zs)$ that satisfies the invariant of the LIS table for $rest$. Let $(b, w)$ be the result of the scan of $t$ for entries above $x$. Then $(b + 1, w)$ is a table cell for $x$ followed by $rest$: $w$ is a subsequence of $rest$, the list $x$ followed by $w$ is strictly increasing and has length $b + 1$, and every strictly increasing list $x$\u2026","title":"The step of the LIS recurrence"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/longest-increasing-subsequence-spec","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.longestIncreasingSubsequence_spec","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#longest-increasing-subsequence-spec","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For every list $xs$ over a preorder, the list returned by the quadratic longest-increasing-subsequence program on $xs$ is a subsequence of $xs$, is strictly increasing (each element strictly less than every later one), and is at least as long as every strictly increasing subsequence of $xs$.","title":"Correctness of the quadratic LIS program"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/is-lislength-longest-increasing-subseque","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.isLISLength_longestIncreasingSubsequence","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#is-lislength-longest-increasing-subseque","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For every list $xs$ over a preorder, the length of the list returned by the quadratic longest-increasing-subsequence program on $xs$ is the length of a longest strictly increasing subsequence of $xs$.","title":"The quadratic LIS program attains the optimal length"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/unique","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsLISLength.unique","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#unique","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For every list $xs$ of elements of a preorder and all natural numbers $k$ and $k'$, if both $k$ and $k'$ are the length of a longest strictly increasing subsequence of $xs$, then $k = k'$.","title":"Uniqueness of the longest increasing subsequence length"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/length-longest-increasing-subsequence-eq","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.length_longestIncreasingSubsequence_eq_iff","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#length-longest-increasing-subsequence-eq","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For every list $xs$ over a preorder and every natural number $k$, the list returned by the quadratic longest-increasing-subsequence program on $xs$ has length $k$ if and only if $k$ is the length of a longest strictly increasing subsequence of $xs$.","title":"Correctness of the quadratic LIS program, by length"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/patience-step","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.patienceStep","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#patience-step","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Given an array $t$ of tails and an element $x$ over a linear order, let $i$ be the index returned by the lower-bound binary search for $x$ in $t$. If $i$ is less than the size of $t$, the step replaces the entry of $t$ at index $i$ by $x$; otherwise it appends $x$ to $t$. Its cost is the number of comparisons made by the binary search.","title":"Patience step"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/patience-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.patienceScan","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#patience-scan","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Given an array of tails and a list over a linear order, the patience scan applies the patience step to the elements of the list in order: on the empty list it returns the tails unchanged, and on a list $x$ followed by $xs$ it applies the patience step for $x$ to the tails and then runs the patience scan of $xs$ from the result. Its cost is the total cost of the steps.","title":"Patience scan"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/patience-length","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.patienceLength","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#patience-length","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Given a list $xs$ over a linear order, patience sorting runs the patience scan of $xs$ starting from the empty array of tails and returns the size of the resulting array. Its cost is the number of comparisons made by the scan.","title":"Patience sorting"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/patience-inv","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.PatienceInv","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#patience-inv","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For lists $P$ and $t = (t_0, \\dots, t_{m-1})$ over a linear order, the invariant holds for the processed prefix $P$ and the tails $t$ when three conditions hold: $t$ is strictly increasing; for every $k < m$ some strictly increasing subsequence of $P$ of length $k + 1$ has last element $t_k$; and every nonempty strictly increasing subsequence of $P$, of length $\\ell$ and with last element $y$, satisfies\u2026","title":"Patience sorting invariant"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.PatienceInv.scan","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#scan","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Let $P$ be a list and $t$ an array of tails over a linear order such that the patience sorting invariant holds for $P$ and the entries of $t$. Then for every list $xs$, the invariant holds for $P$ followed by $xs$ and the entries of the array returned by the patience scan of $xs$ from $t$.","title":"The patience scan preserves the invariant"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/patience-length-spec","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.patienceLength_spec","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#patience-length-spec","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For every list $xs$ over a linear order, the value returned by patience sorting on $xs$ is the length of a longest strictly increasing subsequence of $xs$: some strictly increasing subsequence of $xs$ has this length, and no strictly increasing subsequence of $xs$ is longer.","title":"Correctness of patience sorting"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/patience-length-eq-length-longest-increa","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.patienceLength_eq_length_longestIncreasingSubsequence","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#patience-length-eq-length-longest-increa","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For every list $xs$ over a linear order, the value returned by patience sorting on $xs$ equals the length of the list returned on $xs$ by the quadratic dynamic program for a longest strictly increasing subsequence.","title":"Patience sorting agrees with the quadratic dynamic program"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/patience-step-size-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.patienceStep_size_le","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#patience-step-size-le","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"Let $\\alpha$ be a linear order. For every array $t$ over $\\alpha$ and every element $x \\in \\alpha$, the array returned by the patience step for $x$ on $t$ has size at most the size of $t$ plus $1$.","title":"A patience step grows the tails by at most one"},{"id":"ComputerScience/dynamic-programming/longest-increasing-subsequence/patience-length-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.patienceLength_time_le","parent":41907,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/longest-increasing-subsequence/index.html#patience-length-time-le","scope":"ComputerScience/dynamic-programming/longest-increasing-subsequence","status":"fully_proved","summary":"For every list $xs$ of length $n$ over a linear order, patience sorting on $xs$ uses at most $n \\lceil \\log_2 (n + 1) \\rceil$ comparisons.","title":"Comparison bound for patience sorting"},{"data_url":"scopes/ComputerScience/dynamic-programming/matrix-chain-multiplication.json","id":"ComputerScience/dynamic-programming/matrix-chain-multiplication","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","title":"Matrix chain multiplication"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/parenthesization","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Parenthesization","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#parenthesization","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"A parenthesization of $n$ matrices is a full binary tree whose leaves are labeled by indices in $\\{0, \\dots, n - 1\\}$: either a single leaf naming matrix $i$, or a product node with a left and a right parenthesization as factors. Labels may be repeated, omitted, or out of order.","title":"Parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-indices","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Parenthesization.matrixIndices","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-indices","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"The leaf indices of a parenthesization of $n$ matrices form the list of indices in $\\{0, \\dots, n - 1\\}$ obtained by reading its leaves from left to right: $[i]$ for a single leaf naming matrix $i$, and the concatenation of the lists of the left and right factors for a product node.","title":"Leaf indices of a parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/rows","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Parenthesization.rows","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#rows","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"Let $p = (p_0, \\dots, p_n)$ be a vector of natural numbers, regarding matrix $i$ (for $0 \\le i < n$) as having $p_i$ rows and $p_{i+1}$ columns. The row count of a parenthesization of $n$ matrices with respect to $p$ is $p_i$ for a single leaf naming matrix $i$, and the row count of the left factor for a product node.","title":"Row count of a parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/cols","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Parenthesization.cols","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#cols","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"Let $p = (p_0, \\dots, p_n)$ be a vector of natural numbers, regarding matrix $i$ (for $0 \\le i < n$) as having $p_i$ rows and $p_{i+1}$ columns. The column count of a parenthesization of $n$ matrices with respect to $p$ is $p_{i+1}$ for a single leaf naming matrix $i$, and the column count of the right factor for a product node.","title":"Column count of a parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/dimension-compatible","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Parenthesization.DimensionCompatible","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#dimension-compatible","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"Let $p = (p_0, \\dots, p_n)$ be a vector of natural numbers, regarding matrix $i$ (for $0 \\le i < n$) as having $p_i$ rows and $p_{i+1}$ columns. A parenthesization of $n$ matrices is dimension-compatible with $p$ when, at every product node, the column count of the left factor equals the row count of the right factor, where row and column counts are taken with respect to $p$. A single leaf is always\u2026","title":"Dimension-compatible parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/is-valid","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Parenthesization.IsValid","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#is-valid","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"Let $p = (p_0, \\dots, p_n)$ be a vector of natural numbers, regarding matrix $i$ (for $0 \\le i < n$) as having $p_i$ rows and $p_{i+1}$ columns. A parenthesization of $n$ matrices is valid for $p$ when its leaf indices, read left to right, are exactly $0, 1, \\dots, n - 1$, and it is dimension-compatible with $p$.","title":"Valid parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/scalar-multiplication-cost","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.scalarMultiplicationCost","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#scalar-multiplication-cost","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"Let $p = (p_0, \\dots, p_n)$ be a vector of natural numbers, regarding matrix $i$ (for $0 \\le i < n$) as having $p_i$ rows and $p_{i+1}$ columns, and let $r(T)$ and $c(T)$ be the row and column counts of a parenthesization $T$ with respect to $p$. The scalar multiplication count of a single leaf is $0$, and that of a product node with factors $L$ and $R$ is the sum of the counts of $L$ and $R$ plus $r(L)\\,c(L)\\,c(R)$.","title":"Scalar multiplication count of a parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/cost","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Cost","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#cost","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"A matrix-chain cost is a pair of natural numbers counting the abstract events charged by matrix-chain ordering and reconstruction: the number of split candidates evaluated by the dynamic program, and the number of tree nodes emitted during reconstruction.","title":"Matrix-chain cost"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/total","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Cost.total","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#total","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"The total of a matrix-chain cost is its number of candidate evaluations plus its number of reconstruction nodes.","title":"Total matrix-chain cost"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/result","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.Result","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#result","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"A matrix-chain result for $n$ matrices pairs a natural number, the reported optimal scalar multiplication count, with a parenthesization of $n$ matrices, the reported optimal parenthesization. The structure itself does not require the count to be minimal or to be attained by the parenthesization.","title":"Matrix-chain result"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-chain-order","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.matrixChainOrder","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-chain-order","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"Given $n \\ge 1$ and natural numbers $p_0, \\dots, p_n$, where matrix $i$ of the chain ($0 \\le i < n$) has $p_i$ rows and $p_{i+1}$ columns, the algorithm computes, bottom-up by interval length, values $m_{i,j}$ for $0 \\le i \\le j < n$: $m_{i,i} = 0$ and, for $i < j$, $m_{i,j}$ is the least value of $m_{i,k} + m_{k+1,j} + p_i p_{k+1} p_{j+1}$ over $i \\le k < j$, and the first split point $k$ attaining it is recorded.\u2026","title":"Matrix-chain ordering"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-chain-order-parenthesization-vali","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.matrixChainOrder_parenthesization_valid","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-chain-order-parenthesization-vali","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"For every $n \\ge 1$ and all natural numbers $p_0, \\dots, p_n$, the parenthesization returned by the matrix-chain ordering algorithm on the dimensions $p_0, \\dots, p_n$ is valid: its leaves are the matrices $0, 1, \\dots, n - 1$ in order, each exactly once, and at every multiplication node the column count of the left factor equals the row count of the right factor, where matrix $i$ has $p_i$ rows and $p_{i+1}$\u2026","title":"Matrix-chain ordering returns a valid parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-chain-order-cost-eq","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.matrixChainOrder_cost_eq","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-chain-order-cost-eq","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"For every $n \\ge 1$ and all natural numbers $p_0, \\dots, p_n$, the scalar-multiplication cost of the parenthesization returned by the matrix-chain ordering algorithm on the dimensions $p_0, \\dots, p_n$ equals the optimal cost it reports.","title":"Matrix-chain ordering reports the cost of its parenthesization"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-chain-order-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.matrixChainOrder_minimal","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-chain-order-minimal","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"For every $n \\ge 1$, all natural numbers $p_0, \\dots, p_n$ and every parenthesization $T$ of $n$ matrices that is valid for these dimensions, the optimal cost reported by the matrix-chain ordering algorithm on the dimensions $p_0, \\dots, p_n$ is at most the scalar-multiplication cost of $T$.","title":"Reported matrix-chain cost is a lower bound"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-chain-order-candidate-evaluations","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.matrixChainOrder_candidateEvaluations","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-chain-order-candidate-evaluations","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"For every $n \\ge 1$ and all natural numbers $p_0, \\dots, p_n$, the matrix-chain ordering algorithm run on the dimensions $p_0, \\dots, p_n$ evaluates exactly $\\binom{n+1}{3}$ split candidates.","title":"Split candidates evaluated by matrix-chain ordering"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-chain-order-reconstruction-nodes","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.matrixChainOrder_reconstructionNodes","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-chain-order-reconstruction-nodes","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"For every $n \\ge 1$ and all natural numbers $p_0, \\dots, p_n$, the matrix-chain ordering algorithm run on the dimensions $p_0, \\dots, p_n$ charges exactly $2n - 1$ reconstruction events.","title":"Reconstruction size in matrix-chain ordering"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-chain-order-time","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.matrixChainOrder_time","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-chain-order-time","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"For every $n \\ge 1$ and all natural numbers $p_0, \\dots, p_n$, the total number of events charged by the matrix-chain ordering algorithm run on the dimensions $p_0, \\dots, p_n$ is $\\binom{n+1}{3} + (2n - 1)$.","title":"Exact event count of matrix-chain ordering"},{"id":"ComputerScience/dynamic-programming/matrix-chain-multiplication/matrix-chain-order-time-le-three-mul-cub","kind":"node","lean":"Cslib.Algorithms.Lean.MatrixChain.matrixChainOrder_time_le_three_mul_cube","parent":41932,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/matrix-chain-multiplication/index.html#matrix-chain-order-time-le-three-mul-cub","scope":"ComputerScience/dynamic-programming/matrix-chain-multiplication","status":"fully_proved","summary":"For every $n \\ge 1$ and all natural numbers $p_0, \\dots, p_n$, the total number of events charged by the matrix-chain ordering algorithm run on the dimensions $p_0, \\dots, p_n$ is at most $3n^3$.","title":"Cubic bound for matrix-chain ordering"},{"data_url":"scopes/ComputerScience/dynamic-programming/maximum-subarray-problem.json","id":"ComputerScience/dynamic-programming/maximum-subarray-problem","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","title":"Maximum subarray problem"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/is-max-subarray-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsMaxSubarraySum","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#is-max-subarray-sum","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"For a list $L$ over an additive monoid $\\alpha$ with a linear order, an element $m \\in \\alpha$ is the maximum subarray sum of $L$ when some nonempty contiguous sublist of $L$ has sum $m$ and every nonempty contiguous sublist of $L$ has sum at most $m$.","title":"Being the maximum subarray sum"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/is-max-suffix-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsMaxSuffixSum","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#is-max-suffix-sum","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"For a list $L$ over an additive monoid $\\alpha$ with a linear order, an element $e \\in \\alpha$ is the maximum suffix sum of $L$ when some nonempty suffix of $L$ has sum $e$ and every nonempty suffix of $L$ has sum at most $e$. A nonempty suffix is a contiguous run ending at the last element of $L$.","title":"Being the maximum suffix sum"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/kadane-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kadaneScan","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#kadane-scan","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Given accumulators $e, b$ in an additive monoid $\\alpha$ with a linear order, the scan over the empty list returns $b$ at cost $0$. Over a list with first element $x$ and remaining elements $L$, it sets $e' = \\max(x, e + x)$ and returns the result of the scan of $L$ from the accumulators $e'$ and $\\max(b, e')$, at cost $3$ plus the cost of that scan.","title":"Kadane's scan"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarraySum","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-sum","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type with an additive monoid structure. On the empty list, Kadane's algorithm returns no value. On a list $x_1, x_2, \\dots, x_n$ of elements of $\\alpha$ with $n \\ge 1$, it sets $e = b = x_1$; then for $i = 2, \\dots, n$ in turn it replaces $e$ by $\\max(x_i, e + x_i)$ and then $b$ by $\\max(b, e)$; finally it returns $b$. Each of the steps $i = 2, \\dots, n$ is charged three\u2026","title":"Kadane's algorithm"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/kadane-scan-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kadaneScan_time","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#kadane-scan-time","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order. For all accumulators $e, b \\in \\alpha$ and every list $L$ over $\\alpha$ of length $n$, Kadane's scan of $L$ from the accumulators $e$ and $b$ costs exactly $3n$.","title":"Cost of Kadane's scan"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-sum-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarraySum_time","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-sum-time","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Over an additive monoid $\\alpha$ with a linear order, Kadane's algorithm on a list of length $n \\ge 1$ costs exactly $3(n - 1)$ operations, and on the empty list it costs $0$.","title":"Cost of Kadane's algorithm"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/ne-nil","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsMaxSubarraySum.ne_nil","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#ne-nil","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order. If an element $m \\in \\alpha$ is the maximum subarray sum of a list $L$ over $\\alpha$, then $L$ is nonempty.","title":"A list with a maximum subarray sum is nonempty"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/unique","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsMaxSubarraySum.unique","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#unique","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type with an additive monoid structure, and let $L$ be a list over $\\alpha$. If $m, m' \\in \\alpha$ are both maximum subarray sums of $L$, meaning that each is the sum of some nonempty contiguous run of $L$ and bounds the sum of every nonempty contiguous run of $L$, then $m = m'$.","title":"Uniqueness of the maximum subarray sum"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/concat","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsMaxSuffixSum.concat","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#concat","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order such that $a \\le b$ implies $a + c \\le b + c$. If $e$ is the maximum sum of a nonempty suffix of a list $p$ over $\\alpha$, then for every $x \\in \\alpha$ the maximum sum of a nonempty suffix of the list obtained by appending $x$ to $p$ is $\\max(x, e + x)$.","title":"Maximum suffix sum after appending an element"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/concat-2","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsMaxSubarraySum.concat","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#concat-2","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order such that $a \\le b$ implies $a + c \\le b + c$. Let $p$ be a list over $\\alpha$, let $e$ be the maximum sum of a nonempty suffix of $p$, and let $b$ be the maximum sum of a nonempty contiguous sublist of $p$. Then for every $x \\in \\alpha$, the maximum sum of a nonempty contiguous sublist of the list obtained by appending $x$ to $p$ is $\\max(b, \\max(x, e + x))$.","title":"Maximum subarray sum after appending an element"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/kadane-scan-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kadaneScan_ret","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#kadane-scan-ret","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order such that $a \\le b$ implies $a + c \\le b + c$. Let $p$ be a list over $\\alpha$, let $e$ be the maximum sum of a nonempty suffix of $p$, and let $b$ be the maximum sum of a nonempty contiguous sublist of $p$. Then for every list $r$ over $\\alpha$, the value that Kadane's scan from the accumulators $e$ and $b$ returns on $r$ is the maximum sum of a nonempty\u2026","title":"Invariant of Kadane's scan"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-sum-ret-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarraySum_ret_eq_some_iff","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-sum-ret-eq-some-iff","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order such that $a \\le b$ implies $a + c \\le b + c$. For every list $L$ over $\\alpha$ and every $m \\in \\alpha$, Kadane's algorithm returns $m$ on $L$ if and only if $m$ is the maximum sum of a nonempty contiguous sublist of $L$: some nonempty contiguous sublist of $L$ has sum $m$, and every nonempty contiguous sublist of $L$ has sum at most $m$.","title":"Correctness of Kadane's algorithm"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/sum-le-of-max-subarray-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.sum_le_of_maxSubarraySum","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#sum-le-of-max-subarray-sum","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered type with an additive monoid structure in which $b \\le c$ implies $b + a \\le c + a$. If Kadane's algorithm returns the value $m$ on a list $L$ over $\\alpha$, then every nonempty contiguous run of $L$ has sum at most $m$.","title":"Kadane's algorithm bounds every nonempty run"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/exists-sum-eq-of-max-subarray-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.exists_sum_eq_of_maxSubarraySum","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#exists-sum-eq-of-max-subarray-sum","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order such that $a \\le b$ implies $a + c \\le b + c$. If Kadane's algorithm returns a value $m$ on a list $L$ over $\\alpha$, then some nonempty contiguous sublist of $L$ has sum $m$.","title":"Kadane's algorithm returns an attained sum"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/is-max-prefix-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsMaxPrefixSum","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#is-max-prefix-sum","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order. For a list $xs$ over $\\alpha$ and $p \\in \\alpha$, $p$ is the maximum prefix sum of $xs$ when some nonempty prefix of $xs$ has sum $p$ and every nonempty prefix of $xs$ has sum at most $p$.","title":"Maximum prefix sum"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/suffix-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.suffixScan","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#suffix-scan","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid carrying a linear order. For a nonempty list $x_1, \\dots, x_n$ in $\\alpha$, this timed computation returns the pair $(x_1 + \\dots + x_n, s)$, where $s$ is the maximum of the suffix sums $x_i + \\dots + x_n$ over $1 \\le i \\le n$. It scans from the right: the step for $x_i$ forms $x_i + (x_{i+1} + \\dots + x_n)$ and takes its maximum with the best suffix sum found so far, at a cost of\u2026","title":"Right-to-left suffix scan"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/prefix-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.prefixScan","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#prefix-scan","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"For lists over an additive monoid with a linear order, the prefix scan returns $0$ on the empty list at no cost, and $x$ on a one-element list $[x]$ at cost $2$. On a list with first element $x$ and nonempty tail $t$ it returns $\\max(x, x + b)$, where $b$ is the prefix scan of $t$, at the cost of the scan of $t$ plus $2$.","title":"Prefix scan"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-dc","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarrayDC","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-dc","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"For lists over an additive monoid with a linear order: on the empty list it returns $0$ and on a one-element list $[x]$ it returns $x$, both at no cost. On a list of length $n \\ge 2$, let $m = \\lfloor n/2 \\rfloor$, let $l$ and $r$ be its results on the first $m$ elements and on the remaining $n - m$ elements, let $s$ be the best nonempty suffix sum of the first $m$ elements as computed by the suffix scan, and let\u2026","title":"Divide-and-conquer core of maximum subarray"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-sum-dc","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarraySumDC","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-sum-dc","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"The CLRS algorithm FIND-MAXIMUM-SUBARRAY over an additive monoid with a linear order, returning only the sum. On the empty list it returns no value, at no cost. On a nonempty list it returns the value of the divide-and-conquer core on that list, at the same cost; the core splits a list of length $n \\ge 2$ after its first $\\lfloor n/2 \\rfloor$ elements and returns the largest of its results on the two parts and the\u2026","title":"Divide-and-conquer maximum subarray"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-dccost","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarrayDCCost","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-dccost","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"The function $T : \\mathbb{N} \\to \\mathbb{N}$ given by $T(n) = 0$ for $n \\le 1$ and $T(n) = T(\\lfloor n/2 \\rfloor) + T(n - \\lfloor n/2 \\rfloor) + 2n + 3$ for $n \\ge 2$.","title":"Cost recurrence of divide-and-conquer maximum subarray"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-sum-dc-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarraySumDC_time_le","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-sum-dc-time-le","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order. On a nonempty list over $\\alpha$ of length $n$, the cost $T$ of the divide-and-conquer maximum subarray algorithm satisfies $T + 3 \\le 2n\\lceil \\log_2 n \\rceil + 3n$, that is, $T \\le 2n\\lceil \\log_2 n \\rceil + 3(n - 1)$.","title":"Running time of divide-and-conquer maximum subarray"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/suffix-scan-spec","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.suffixScan_spec","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#suffix-scan-spec","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order. For every nonempty list $xs$ over $\\alpha$, the suffix scan of $xs$ returns a pair whose first component is the sum of $xs$ and whose second component is the maximum sum of a nonempty suffix of $xs$.","title":"Correctness of the suffix scan"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/append","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsMaxSubarraySum.append","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#append","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order such that $b \\le c$ implies $a + b \\le a + c$ and $b + a \\le c + a$, and let $L$ and $R$ be lists over $\\alpha$. Suppose that $l$ is the maximum sum of a nonempty contiguous run of $L$, $r$ is the maximum sum of a nonempty contiguous run of $R$, $s$ is the maximum sum of a nonempty suffix of $L$, and $p$ is the maximum sum of a nonempty prefix of $R$. Then\u2026","title":"Maximum subarray sum of a concatenation"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-sum-dc-ret-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarraySumDC_ret_eq_some_iff","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-sum-dc-ret-eq-some-iff","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order such that $b \\le c$ implies $a + b \\le a + c$ and $b + a \\le c + a$. For every list $xs$ over $\\alpha$ and every $m \\in \\alpha$, the divide-and-conquer algorithm returns $m$ on $xs$ if and only if $m$ is the maximum sum of a nonempty contiguous run of $xs$.","title":"Correctness of divide-and-conquer maximum subarray"},{"id":"ComputerScience/dynamic-programming/maximum-subarray-problem/max-subarray-sum-dc-eq-max-subarray-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.maxSubarraySumDC_eq_maxSubarraySum","parent":41951,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/maximum-subarray-problem/index.html#max-subarray-sum-dc-eq-max-subarray-sum","scope":"ComputerScience/dynamic-programming/maximum-subarray-problem","status":"fully_proved","summary":"Let $\\alpha$ be an additive monoid with a linear order such that $b \\le c$ implies $a + b \\le a + c$ and $b + a \\le c + a$. For every list $xs$ over $\\alpha$, the divide-and-conquer maximum subarray algorithm and Kadane's algorithm return the same result on $xs$.","title":"Divide and conquer agrees with Kadane's algorithm"},{"data_url":"scopes/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem.json","id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","title":"Multiple-choice knapsack problem"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/item","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.Item","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#item","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"A knapsack item with a natural weight and a value in `\u03b2`.","title":"Knapsack item with weight and value"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/is-selection","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.IsSelection","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#is-selection","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"A selection chooses exactly one item from each group, in group order.","title":"Multiple-choice knapsack selection"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/total-weight","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.totalWeight","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#total-weight","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The total weight of a selection.","title":"Total weight of a knapsack selection"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/total-value","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.totalValue","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#total-value","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The total value of a selection.","title":"Total value of a knapsack selection"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/best-packing","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.bestPacking","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#best-packing","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The packing recurrence: the best total value of a selection with total weight at most `c`, `\u22a5` if none exists.","title":"Packing recurrence for multiple-choice knapsack"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/best-covering","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.bestCovering","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#best-covering","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The covering recurrence: the best total value of a selection with total weight at least `L`, `\u22a5` if none exists. The residual target `L - weight` is truncated subtraction.","title":"Covering recurrence for multiple-choice knapsack"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/le-best-packing","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.le_bestPacking","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#le-best-packing","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The packing recurrence is an upper bound: every feasible selection's value is at most `bestPacking`.","title":"Packing recurrence bounds every feasible selection's value"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/exists-eq-of-best-packing-eq-coe","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.exists_eq_of_bestPacking_eq_coe","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#exists-eq-of-best-packing-eq-coe","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The packing recurrence is attained: a finite value is the value of a feasible selection.","title":"Packing recurrence is attained by a feasible selection"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/best-packing-eq-bot-iff","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.bestPacking_eq_bot_iff","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#best-packing-eq-bot-iff","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The packing recurrence is `\u22a5` exactly when no selection fits the capacity.","title":"Packing recurrence is bottom iff no selection fits the capacity"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/le-best-covering","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.le_bestCovering","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#le-best-covering","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The covering recurrence is an upper bound: every selection of total weight at least `L` has value at most `bestCovering`.","title":"Covering recurrence bounds every covering selection's value"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/exists-eq-of-best-covering-eq-coe","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.exists_eq_of_bestCovering_eq_coe","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#exists-eq-of-best-covering-eq-coe","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The covering recurrence is attained: a finite value is the value of a selection of total weight at least `L`.","title":"Covering recurrence is attained by a covering selection"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/best-covering-eq-bot-iff","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.bestCovering_eq_bot_iff","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#best-covering-eq-bot-iff","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The covering recurrence is `\u22a5` exactly when no selection reaches the target.","title":"Covering recurrence is bottom iff no selection reaches the target"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/packing-table","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.packingTable","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#packing-table","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The bottom-up packing table: for each suffix of the groups, the row of best values at capacities `0, \u2026, capacity`, each row computed from the next one by one scan of its group per cell.","title":"Bottom-up packing table for multiple-choice knapsack"},{"id":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/packing-table-eq","kind":"node","lean":"Cslib.Algorithms.Lean.MultipleChoiceKnapsack.packingTable_eq","parent":41977,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/multiple-choice-knapsack-problem/index.html#packing-table-eq","scope":"ComputerScience/dynamic-programming/multiple-choice-knapsack-problem","status":"fully_proved","summary":"The table computes the recurrence at every capacity `0, \u2026, capacity`.","title":"Packing table computes the packing recurrence"},{"data_url":"scopes/ComputerScience/dynamic-programming/optimal-binary-search-tree.json","id":"ComputerScience/dynamic-programming/optimal-binary-search-tree","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/optimal-binary-search-tree/index.html","scope":"ComputerScience/dynamic-programming/optimal-binary-search-tree","status":"fully_proved","title":"Optimal binary search tree"},{"id":"ComputerScience/dynamic-programming/optimal-binary-search-tree/expected-search-cost","kind":"node","lean":"Cslib.Algorithms.Lean.OptimalBST.expectedSearchCost","parent":41992,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/optimal-binary-search-tree/index.html#expected-search-cost","scope":"ComputerScience/dynamic-programming/optimal-binary-search-tree","status":"fully_proved","summary":"Let $R$ be an additive commutative monoid, let $n$ be a natural number, and let $p_0, \\dots, p_{n-1} \\in R$ (key weights) and $q_0, \\dots, q_n \\in R$ (dummy-key weights) be given. Let $T$ be a binary tree whose nodes are labelled by elements of $\\{0, \\dots, n-1\\}$ and whose inorder traversal is $0, 1, \\dots, n-1$, so each key labels exactly one node and $T$ has exactly $n + 1$ empty subtrees. The depth of a node or\u2026","title":"Expected search cost of a binary search tree"},{"id":"ComputerScience/dynamic-programming/optimal-binary-search-tree/optimal-bst","kind":"node","lean":"Cslib.Algorithms.Lean.OptimalBST.optimalBST","parent":41992,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/optimal-binary-search-tree/index.html#optimal-bst","scope":"ComputerScience/dynamic-programming/optimal-binary-search-tree","status":"fully_proved","summary":"Let $R$ be a linearly ordered additive commutative monoid in which addition is monotone, let $n$ be a natural number, and let $p_0, \\dots, p_{n-1}, q_0, \\dots, q_n \\in R$. The procedure (CLRS OPTIMAL-BST, with keys indexed from $0$) fills a cost table $e$ of size $(n+1) \\times (n+1)$, an auxiliary weight table $w$ of the same size, and a root table of size $n \\times n$ with entries in $\\{0, \\dots, n-1\\}$ or empty;\u2026","title":"Optimal binary search tree procedure"},{"id":"ComputerScience/dynamic-programming/optimal-binary-search-tree/optimal-bst-empty","kind":"node","lean":"Cslib.Algorithms.Lean.OptimalBST.optimalBST_empty","parent":41992,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/optimal-binary-search-tree/index.html#optimal-bst-empty","scope":"ComputerScience/dynamic-programming/optimal-binary-search-tree","status":"fully_proved","summary":"Let $R$ be a linearly ordered additive commutative monoid in which addition is monotone. For $n = 0$ keys and a single dummy weight $q_0 \\in R$, the optimal binary search tree procedure returns the $1 \\times 1$ cost table whose only entry is $q_0$, the empty $0 \\times 0$ root table, and running time $0$.","title":"Optimal BST with no keys"},{"id":"ComputerScience/dynamic-programming/optimal-binary-search-tree/optimal-bst-table-spec","kind":"node","lean":"Cslib.Algorithms.Lean.OptimalBST.optimalBST_table_spec","parent":41992,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/optimal-binary-search-tree/index.html#optimal-bst-table-spec","scope":"ComputerScience/dynamic-programming/optimal-binary-search-tree","status":"fully_proved","summary":"Let $R$ be a linearly ordered additive commutative monoid in which addition is monotone, let $n$ be a natural number, let $p_0, \\dots, p_{n-1}, q_0, \\dots, q_n \\in R$, and let $e$ and $\\rho$ be the cost table and the root table returned by the optimal binary search tree procedure on these weights. For $a \\le b \\le n$ write $w(a, b) = \\sum_{i=a}^{b-1} p_i + \\sum_{i=a}^{b} q_i$. Then $e[a][a] = q_a$ for every\u2026","title":"Optimal BST table recurrence"},{"id":"ComputerScience/dynamic-programming/optimal-binary-search-tree/optimal-bst-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.OptimalBST.optimalBST_minimal","parent":41992,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/optimal-binary-search-tree/index.html#optimal-bst-minimal","scope":"ComputerScience/dynamic-programming/optimal-binary-search-tree","status":"fully_proved","summary":"Let $R$ be a linearly ordered additive commutative monoid in which addition is monotone, let $n$ be a natural number, and let $p_0, \\dots, p_{n-1}, q_0, \\dots, q_n \\in R$. Let $e$ be the cost table returned by the optimal binary search tree procedure on these weights. Then $e[0][n]$ is the least expected search cost, with key weights $p$ and dummy-key weights $q$, of a binary tree whose nodes are labelled by\u2026","title":"The optimal BST procedure finds the minimum expected search cost"},{"id":"ComputerScience/dynamic-programming/optimal-binary-search-tree/optimal-bst-time","kind":"node","lean":"Cslib.Algorithms.Lean.OptimalBST.optimalBST_time","parent":41992,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/optimal-binary-search-tree/index.html#optimal-bst-time","scope":"ComputerScience/dynamic-programming/optimal-binary-search-tree","status":"fully_proved","summary":"Let $R$ be a linearly ordered additive commutative monoid in which addition is monotone. For every natural number $n$ and all $p_0, \\dots, p_{n-1}, q_0, \\dots, q_n \\in R$, the running time of the optimal binary search tree procedure on these weights, which is the number of candidate roots it examines, equals $\\binom{n+2}{3}$.","title":"Exact running time of the optimal BST procedure"},{"id":"ComputerScience/dynamic-programming/optimal-binary-search-tree/optimal-bst-time-bounds","kind":"node","lean":"Cslib.Algorithms.Lean.OptimalBST.optimalBST_time_bounds","parent":41992,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/optimal-binary-search-tree/index.html#optimal-bst-time-bounds","scope":"ComputerScience/dynamic-programming/optimal-binary-search-tree","status":"fully_proved","summary":"Let $R$ be a linearly ordered additive commutative monoid in which addition is monotone. For every natural number $n$ and all $p_0, \\dots, p_{n-1}, q_0, \\dots, q_n \\in R$, the running time $T$ of the optimal binary search tree procedure on these weights, which is the number of candidate roots it examines, satisfies $n^3 \\le 6T$ and $T \\le n^3$.","title":"Cubic running time of the optimal BST procedure"},{"data_url":"scopes/ComputerScience/dynamic-programming/rod-cutting.json","id":"ComputerScience/dynamic-programming/rod-cutting","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","title":"Rod cutting"},{"id":"ComputerScience/dynamic-programming/rod-cutting/composition-revenue","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.compositionRevenue","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#composition-revenue","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p = (p_0, \\dots, p_{n-1})$ be a price vector with entries in an additive commutative monoid, where $p_{\\ell - 1}$ is the price of a piece of length $\\ell$. For $k \\le n$ and a composition $(b_1, \\dots, b_m)$ of $k$ into positive parts, its revenue is $p_{b_1 - 1} + \\dots + p_{b_m - 1}$, the total price of its pieces.","title":"Revenue of a rod composition"},{"id":"ComputerScience/dynamic-programming/rod-cutting/composition-revenue-zero","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.compositionRevenue_zero","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#composition-revenue-zero","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p = (p_1, \\dots, p_n)$ be prices in an additive commutative monoid. Every composition of $0$, namely the empty one, has revenue $0$ with respect to $p$.","title":"The empty rod earns nothing"},{"id":"ComputerScience/dynamic-programming/rod-cutting/composition-revenue-single","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.compositionRevenue_single","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#composition-revenue-single","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p = (p_1, \\dots, p_n)$ be prices in an additive commutative monoid, $p_i$ being the price of a piece of length $i$. For $0 < k \\le n$, the revenue of the one-part composition $(k)$ of $k$ is $p_k$.","title":"Revenue of an uncut rod"},{"id":"ComputerScience/dynamic-programming/rod-cutting/composition-revenue-append","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.compositionRevenue_append","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#composition-revenue-append","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p = (p_1, \\dots, p_n)$ be prices in an additive commutative monoid, and define the revenue of a composition $(b_1, \\dots, b_m)$ of some $k \\le n$ to be $p_{b_1} + \\dots + p_{b_m}$. If $c$ is a composition of $k$ and $d$ is a composition of $l$ with $k + l \\le n$, then the revenue of their concatenation, a composition of $k + l$, is the revenue of $c$ plus the revenue of $d$.","title":"Revenue is additive under concatenation"},{"id":"ComputerScience/dynamic-programming/rod-cutting/bottom-up-cut-rod","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.bottomUpCutRod","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#bottom-up-cut-rod","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Given prices $p = (p_1, \\dots, p_n)$ in an additive commutative monoid $\\alpha$ with a linear order, $p_i$ being the price of a piece of length $i$, bottom-up rod cutting fills a table $r_0, \\dots, r_n$ in increasing order by $r_0 = 0$ and $r_j = \\max_{1 \\le i \\le j} (p_i + r_{j-i})$ for $1 \\le j \\le n$, and returns $r_n$. Its cost is the number of candidates $p_i + r_{j-i}$ it evaluates, one for each pair\u2026","title":"Bottom-up rod cutting"},{"id":"ComputerScience/dynamic-programming/rod-cutting/bottom-up-cut-rod-correct","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.bottomUpCutRod_correct","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#bottom-up-cut-rod-correct","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order in which addition is monotone, and let $p = (p_1, \\dots, p_n)$ be prices in $\\alpha$, $p_i$ being the price of a piece of length $i$. The value returned by bottom-up rod cutting on $p$ is the greatest revenue of a composition of $n$: it equals $p_{b_1} + \\dots + p_{b_m}$ for some composition $(b_1, \\dots, b_m)$ of $n$ into positive parts, and it is\u2026","title":"Correctness of bottom-up rod cutting"},{"id":"ComputerScience/dynamic-programming/rod-cutting/bottom-up-cut-rod-zero","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.bottomUpCutRod_zero","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#bottom-up-cut-rod-zero","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"For the empty price vector ($n = 0$) over an additive commutative monoid with a linear order, bottom-up rod cutting returns $0$ at cost $0$.","title":"Bottom-up rod cutting on the empty rod"},{"id":"ComputerScience/dynamic-programming/rod-cutting/bottom-up-cut-rod-time-sum","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.bottomUpCutRod_time_sum","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#bottom-up-cut-rod-time-sum","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"For prices $p = (p_1, \\dots, p_n)$ in an additive commutative monoid with a linear order, the cost of bottom-up rod cutting on $p$ (the number of first-piece candidates it evaluates) equals $\\sum_{j=0}^{n} j$.","title":"Cost of bottom-up rod cutting as a sum over rows"},{"id":"ComputerScience/dynamic-programming/rod-cutting/bottom-up-cut-rod-time-twice","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.bottomUpCutRod_time_twice","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#bottom-up-cut-rod-time-twice","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"For prices $p = (p_1, \\dots, p_n)$ in an additive commutative monoid with a linear order, the cost $T$ of bottom-up rod cutting on $p$ (the number of first-piece candidates it evaluates) satisfies $2T = n(n + 1)$.","title":"Twice the cost of bottom-up rod cutting"},{"id":"ComputerScience/dynamic-programming/rod-cutting/bottom-up-cut-rod-time","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.bottomUpCutRod_time","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#bottom-up-cut-rod-time","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"For prices $p = (p_1, \\dots, p_n)$ in an additive commutative monoid with a linear order, the cost of bottom-up rod cutting on $p$ (the number of first-piece candidates it evaluates) is the triangular number $n(n + 1)/2$, whatever the prices.","title":"Exact cost of bottom-up rod cutting"},{"id":"ComputerScience/dynamic-programming/rod-cutting/bottom-up-cut-rod-time-bounds","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.bottomUpCutRod_time_bounds","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#bottom-up-cut-rod-time-bounds","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"For prices $p = (p_1, \\dots, p_n)$ in an additive commutative monoid with a linear order, the cost $T$ of bottom-up rod cutting on $p$, the number of first-piece candidates it evaluates, satisfies $n^2 \\le 2T$ and $T \\le n^2$.","title":"Bottom-up rod cutting takes quadratic time"},{"id":"ComputerScience/dynamic-programming/rod-cutting/extended-bottom-up-cut-rod","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.extendedBottomUpCutRod","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#extended-bottom-up-cut-rod","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order, and let $p_1, \\dots, p_n \\in \\alpha$ be the prices of pieces of length $1, \\dots, n$ (the price vector's entry $i - 1$ is $p_i$). The algorithm fills a revenue table $r_0, \\dots, r_n$ and a first-cut table $s_1, \\dots, s_n$ for the lengths $j = 0, 1, \\dots, n$ in increasing order: $r_0 = 0$, and for $j \\ge 1$ it scans the candidates $p_i + r_{j-i}$\u2026","title":"Extended bottom-up rod cutting"},{"id":"ComputerScience/dynamic-programming/rod-cutting/extended-bottom-up-cut-rod-correct","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.extendedBottomUpCutRod_correct","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#extended-bottom-up-cut-rod-correct","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p$ be a price vector of length $n$ with entries in a linearly ordered additive commutative monoid (addition is monotone), and let $r = (r_0, \\dots, r_n)$ be the revenue table computed by the extended bottom-up rod-cutting algorithm. For every $j \\le n$, $r_j$ is the greatest revenue of a composition of $j$: some composition of $j$ has revenue $r_j$, and every composition of $j$ has revenue at most $r_j$.","title":"Saved revenues are optimal"},{"id":"ComputerScience/dynamic-programming/rod-cutting/extended-bottom-up-cut-rod-revenue-eq","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.extendedBottomUpCutRod_revenue_eq","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#extended-bottom-up-cut-rod-revenue-eq","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"For every price vector of length $n$ with entries in an additive commutative monoid carrying a linear order, the last entry $r_n$ of the revenue table computed by the extended bottom-up rod-cutting algorithm equals the value returned by the bottom-up rod-cutting algorithm.","title":"The extended algorithm computes the same revenue"},{"id":"ComputerScience/dynamic-programming/rod-cutting/extended-bottom-up-cut-rod-time","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.extendedBottomUpCutRod_time","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#extended-bottom-up-cut-rod-time","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order. For every vector of $n$ prices in $\\alpha$, the extended bottom-up rod-cutting algorithm has cost exactly $n(n+1)/2$, whatever the prices are.","title":"Running time of extended bottom-up rod cutting"},{"id":"ComputerScience/dynamic-programming/rod-cutting/extended-bottom-up-cut-rod-zero","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.extendedBottomUpCutRod_zero","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#extended-bottom-up-cut-rod-zero","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order. On the empty price vector (rod length $0$), the extended bottom-up rod-cutting algorithm returns the revenue table whose single entry is $0$ together with the empty first-cut table, at cost $0$.","title":"Extended bottom-up rod cutting on the empty rod"},{"id":"ComputerScience/dynamic-programming/rod-cutting/extended-bottom-up-cut-rod-first-cut","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.extendedBottomUpCutRod_firstCut","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#extended-bottom-up-cut-rod-first-cut","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p = (p_0, \\dots, p_{n-1})$ be a price vector with entries in an additive commutative monoid carrying a linear order, where $p_{\\ell - 1}$ is the price of a piece of length $\\ell$. Let $r = (r_0, \\dots, r_n)$ be the revenue table and $s = (s_1, \\dots, s_n)$ the saved first cuts computed by the extended bottom-up rod-cutting algorithm, with $s_j$ stored as entry $j - 1$ of the cut vector. For every $j$ with\u2026","title":"The saved first cut attains the saved revenue"},{"id":"ComputerScience/dynamic-programming/rod-cutting/cut-rod-solution","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.cutRodSolution","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#cut-rod-solution","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Given a price vector $p$ of length $n$ with entries in an additive commutative monoid carrying a linear order, where entry $\\ell - 1$ is the price of a piece of length $\\ell$, first run the extended bottom-up rod-cutting algorithm to obtain the saved first cuts $s_1, \\dots, s_n$ (with $s_j$ stored as entry $j - 1$ of its cut vector). Then, starting from the remaining length $L = n$, repeatedly emit a piece of\u2026","title":"Rod cutting with reconstruction"},{"id":"ComputerScience/dynamic-programming/rod-cutting/cut-rod-solution-time","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.cutRodSolution_time","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#cut-rod-solution-time","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p$ be a price vector of length $n$ with entries in an additive commutative monoid carrying a linear order. The rod-cutting reconstruction charges exactly $n(n+1)/2 + m$ events, where $m$ is the number of pieces of the composition of $n$ it returns: one event per evaluated first-cut candidate and one per emitted piece.","title":"Cost of rod-cutting reconstruction"},{"id":"ComputerScience/dynamic-programming/rod-cutting/cut-rod-solution-blocks","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.cutRodSolution_blocks","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#cut-rod-solution-blocks","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p$ be a price vector of length $n$ with entries in an additive commutative monoid carrying a linear order. Let $s$ be the vector of saved first cuts computed by the extended bottom-up rod-cutting algorithm, of length $n$, whose entry $j - 1$ is the cut saved for length $j$, and let $(b_0, \\dots, b_{m-1})$ be the composition of $n$ returned by the rod-cutting reconstruction. For every $i < m$, the index\u2026","title":"Reconstructed pieces are the saved first cuts"},{"id":"ComputerScience/dynamic-programming/rod-cutting/cut-rod-solution-revenue","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.cutRodSolution_revenue","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#cut-rod-solution-revenue","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p$ be a price vector of length $n$ with entries in an additive commutative monoid carrying a linear order, and let $r = (r_0, \\dots, r_n)$ be the revenue table computed by the extended bottom-up rod-cutting algorithm. The revenue of the composition of $n$ returned by the rod-cutting reconstruction equals $r_n$.","title":"Reconstruction earns the final saved revenue"},{"id":"ComputerScience/dynamic-programming/rod-cutting/cut-rod-solution-correct","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.cutRodSolution_correct","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#cut-rod-solution-correct","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $p$ be a price vector of length $n$ with entries in a linearly ordered additive commutative monoid (addition is monotone). The composition of $n$ returned by the rod-cutting reconstruction has the greatest revenue among all compositions of $n$: its revenue is at least the revenue of every composition of $n$.","title":"Rod-cutting reconstruction is optimal"},{"id":"ComputerScience/dynamic-programming/rod-cutting/cut-rod-solution-zero","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.cutRodSolution_zero","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#cut-rod-solution-zero","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"For the empty price vector (rod length $0$) with entries in an additive commutative monoid carrying a linear order, the rod-cutting reconstruction returns the empty composition of $0$ and charges no events.","title":"Rod-cutting reconstruction on the empty rod"},{"id":"ComputerScience/dynamic-programming/rod-cutting/recursive-cut-rod","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.recursiveCutRod","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#recursive-cut-rod","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order, and let $p_1, \\dots, p_n \\in \\alpha$ be prices, $p_i$ being the price of a piece of length $i$. Recursive CUT-ROD returns $r(n)$, where $r(0) = 0$ and, for $1 \\le j \\le n$, $r(j) = \\max(p_1 + r(j - 1), p_2 + r(j - 2), \\dots, p_j + r(0))$. The first cuts $i = 1, \\dots, j$ are scanned in increasing order and every $r(j - i)$ is recomputed by a fresh\u2026","title":"Recursive CUT-ROD"},{"id":"ComputerScience/dynamic-programming/rod-cutting/recursive-cut-rod-correct","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.recursiveCutRod_correct","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#recursive-cut-rod-correct","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order for which addition is monotone, and let $p_1, \\dots, p_n \\in \\alpha$ be prices, $p_i$ being the price of a piece of length $i$. The value returned by recursive CUT-ROD on these prices is the greatest revenue of a composition of $n$: it equals the revenue $p_{i_1} + \\dots + p_{i_k}$ of some composition $n = i_1 + \\dots + i_k$ into positive parts, and\u2026","title":"Correctness of recursive CUT-ROD"},{"id":"ComputerScience/dynamic-programming/rod-cutting/recursive-cut-rod-time","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.recursiveCutRod_time","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#recursive-cut-rod-time","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order, and let $p_1, \\dots, p_n \\in \\alpha$ be prices. Recursive CUT-ROD on these prices, which charges one tick per invocation, makes exactly $2^n$ invocations, counting the root call and every call on an empty rod, whatever the prices are.","title":"Running time of recursive CUT-ROD"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod-aux","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRodAux","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod-aux","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order, let $p = (p_1, \\dots, p_n) \\in \\alpha^n$ be prices, where $p_i$ is the price of a piece of length $i$, and let $c = (c_0, \\dots, c_n)$ be a cache whose entries are each either unknown or a value in $\\alpha$. For $j \\le n$ the procedure returns a value and an updated cache, and counts events, as follows. It first charges one event for probing $c_j$.\u2026","title":"Memoized rod-cutting auxiliary (CLRS MEMOIZED-CUT-ROD-AUX)"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRod","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order and let $p = (p_1, \\dots, p_n) \\in \\alpha^n$ be prices, where $p_i$ is the price of a piece of length $i$. The procedure runs the memoized rod-cutting auxiliary on length $n$ with the cache $(c_0, \\dots, c_n)$ whose entries are all unknown, and returns the value it computes, discarding the final cache. Its event count is that of the auxiliary call.","title":"The memoized cut-rod procedure (CLRS MEMOIZED-CUT-ROD)"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod-aux-hit","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRodAux_hit","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod-aux-hit","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order, let $p \\in \\alpha^n$ be a vector of prices, let $c = (c_0, \\dots, c_n)$ be a cache whose entries are each unknown or a value in $\\alpha$, and let $j \\le n$. If $c_j$ is a known value $v$, then the memoized rod-cutting auxiliary on length $j$ and cache $c$ returns $v$ and the unchanged cache $c$, and charges exactly one event.","title":"A cache hit in memoized rod cutting returns at once"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod-aux-preserves","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRodAux_preserves","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod-aux-preserves","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order, let $p \\in \\alpha^n$ be a vector of prices, let $c = (c_0, \\dots, c_n)$ be a cache whose entries are each unknown or a value in $\\alpha$, let $j \\le n$, and let $c'$ be the cache returned by the memoized rod-cutting auxiliary on length $j$ and cache $c$. Then for every $k \\le n$, if $c_k$ is a known value $v$, then $c'_k = v$; and for every $k$\u2026","title":"The memoized auxiliary preserves known and higher cache entries"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod-zero","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRod_zero","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod-zero","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order. For the empty price vector ($n = 0$), the memoized cut-rod procedure returns $0$ and charges exactly $2$ events: one cache probe and one cache write.","title":"Memoized rod cutting on the empty rod"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod-aux-correct","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRodAux_correct","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod-aux-correct","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered additive commutative monoid (that is, $a \\le b$ implies $a + c \\le b + c$) and let $p = (p_1, \\dots, p_n) \\in \\alpha^n$ be prices, where $p_i$ is the price of a piece of length $i$. Call a cache $c = (c_0, \\dots, c_n)$, whose entries are each unknown or a value in $\\alpha$, valid if for every known entry $c_k = v$ the value $v$ is the greatest composition revenue over compositions\u2026","title":"Correctness of the memoized rod-cutting auxiliary"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod-correct","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRod_correct","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod-correct","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered additive commutative monoid (that is, $a \\le b$ implies $a + c \\le b + c$) and let $p = (p_1, \\dots, p_n) \\in \\alpha^n$ be prices, where $p_i$ is the price of a piece of length $i$. The value returned by the memoized cut-rod procedure on $p$ is the greatest composition revenue over compositions of $n$: it is the revenue of some composition of $n$, and it is at least the revenue of\u2026","title":"Correctness of memoized rod cutting"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod-aux-complete","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRodAux_complete","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod-aux-complete","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be a linearly ordered additive commutative monoid (that is, $a \\le b$ implies $a + c \\le b + c$), let $p = (p_1, \\dots, p_n) \\in \\alpha^n$ be prices, where $p_i$ is the price of a piece of length $i$, and let $j \\le n$. If the memoized rod-cutting auxiliary is run on length $j$ with the cache $(c_0, \\dots, c_n)$ whose entries are all unknown, then for every $k \\le j$ entry $k$ of the returned cache is\u2026","title":"The memoized auxiliary fills the cache up to the requested length"},{"id":"ComputerScience/dynamic-programming/rod-cutting/memoized-cut-rod-time","kind":"node","lean":"Cslib.Algorithms.Lean.RodCutting.memoizedCutRod_time","parent":42000,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/rod-cutting/index.html#memoized-cut-rod-time","scope":"ComputerScience/dynamic-programming/rod-cutting","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a linear order and let $p \\in \\alpha^n$ be a vector of prices. The memoized cut-rod procedure on $p$ charges exactly $n^2 + 2n + 2$ events, where one event is charged per cache probe, per first-cut candidate, and per cache write.","title":"Event count of memoized rod cutting"},{"data_url":"scopes/ComputerScience/dynamic-programming/subset-sum.json","id":"ComputerScience/dynamic-programming/subset-sum","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","title":"Subset sum"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-sums","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSumSums","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-sums","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Given a list $x = (x_1, \\dots, x_n)$ of natural numbers and a target $t \\in \\mathbb{N}$, this timed procedure starts from the list $L_0 = (0)$ and, for $i = 1, \\dots, n$ in order, forms $L_i$ from $L_{i-1}$ in four steps: add $x_i$ to every entry of $L_{i-1}$; merge $L_{i-1}$ with this shifted list (the standard merge of two lists, comparing entries with $\\le$); delete adjacent repeated values; and keep only the\u2026","title":"Attainable sums in exact subset sum"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-sums-mem","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSumSums_mem","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-sums-mem","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Let $x$ be a list of natural numbers, $t$ a natural number, and $L$ the final list of sums computed by the exact subset sum traversal on $x$ with target $t$. A natural number $z$ belongs to $L$ if and only if $z \\le t$ and $z$ is the sum of some sublist of $x$ (a subsequence, not necessarily contiguous).","title":"Characterization of the exact subset sum list"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-sums-zero-mem","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSumSums_zero_mem","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-sums-zero-mem","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every list $x$ of natural numbers and every natural number $t$, the final list of sums computed by the exact subset sum traversal on $x$ with target $t$ contains $0$.","title":"Zero is always a computed sum"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-sums-sorted","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSumSums_sorted","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-sums-sorted","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every list $x$ of natural numbers and every natural number $t$, the final list of sums computed by the exact subset sum traversal on $x$ with target $t$ is strictly increasing; in particular it has no repeated entries.","title":"The exact subset sum list is strictly increasing"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-sums-length-le-pow","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSumSums_length_le_pow","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-sums-length-le-pow","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every list $x$ of natural numbers, of length $n$, and every natural number $t$, the final list of sums computed by the exact subset sum traversal on $x$ with target $t$ has length at most $2^n$.","title":"The number of computed sums is at most two to the power of the input length"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-sums-length-le-target","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSumSums_length_le_target","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-sums-length-le-target","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every list $x$ of natural numbers and every natural number $t$, the final list of sums computed by the exact subset sum traversal on $x$ with target $t$ has length at most $t + 1$.","title":"The number of computed sums is at most the target plus one"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-sums-length-le-sum","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSumSums_length_le_sum","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-sums-length-le-sum","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every list $x$ of natural numbers with sum $S$ and every natural number $t$, the final list of sums computed by the exact subset sum traversal on $x$ with target $t$ has length at most $S + 1$.","title":"The number of computed sums is at most the input total plus one"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-sums-powers-of-two","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSumSums_powers_of_two","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-sums-powers-of-two","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Let $i \\le n$ be natural numbers. On the list $1, 2, 4, \\dots, 2^{i-1}$ (the first $i$ entries of $1, 2, \\dots, 2^{n-1}$) with target $2^n - 1$, the final list of sums computed by the exact subset sum traversal is exactly $0, 1, 2, \\dots, 2^i - 1$ in increasing order.","title":"Prefixes of the powers of two attain every sum"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Given a list $x$ of natural numbers and a target $t \\in \\mathbb{N}$, the procedure first computes the list $L = (\\ell_1, \\dots, \\ell_m)$ of attainable sums of $x$ with target $t$, then scans it from left to right, keeping a running maximum that starts at $0$, and returns $\\max(0, \\ell_1, \\dots, \\ell_m)$. Its counted events are those of computing $L$ plus one per entry of $L$.","title":"Exact subset sum procedure"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-ret","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_ret","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-ret","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every list $x$ of natural numbers and every natural number $t$, the value returned by the exact subset sum algorithm on $x$ with target $t$ equals $\\max(\\cdots\\max(\\max(0, s_1), s_2)\\cdots, s_k)$, where $s_1, \\dots, s_k$ is the final list of sums computed by the exact subset sum traversal on $x$ with target $t$.","title":"Exact subset sum returns the maximum of the computed sums"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-time","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_time","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-time","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Let $x = (x_1, \\dots, x_n)$ be a list of natural numbers and $t \\in \\mathbb{N}$ a target. For $0 \\le i \\le n$ let $L_i$ be the list of attainable sums of the prefix $(x_1, \\dots, x_i)$ with target $t$. For $i < n$ let $M_i$ be the merge of $L_i$ with the list obtained by adding $x_{i+1}$ to every entry of $L_i$, let $c_i$ be the number of comparisons this merge makes, and let $D_i$ be $M_i$ with adjacent repeated\u2026","title":"Event count of exact subset sum"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-time-ge-powers-of-two","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_time_ge_powers_of_two","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-time-ge-powers-of-two","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$, the exact subset-sum procedure on the list $(2^0, 2^1, \\dots, 2^{n-1})$ (empty when $n = 0$) with target $2^n - 1$ makes at least $2^n - 1$ counted events.","title":"Exponential lower bound for exact subset sum on powers of two"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-time-le-exponential","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_time_le_exponential","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-time-le-exponential","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every list $x$ of $n$ natural numbers and every target $t \\in \\mathbb{N}$, the exact subset-sum procedure on $x$ and $t$ makes at most $1 + n + 7(2^n - 1) + 2^n$ counted events.","title":"Exponential upper bound for exact subset sum"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-time-le-numeric","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_time_le_numeric","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-time-le-numeric","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"For every list $x$ of $n$ natural numbers with sum $s$ and every target $t \\in \\mathbb{N}$, the exact subset-sum procedure on $x$ and $t$ makes at most $1 + n + (7n + 1)\\min(t + 1, s + 1)$ counted events.","title":"Pseudo-polynomial bound for exact subset sum"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-time-le-of-target-le-po","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_time_le_of_target_le_pow","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-time-le-of-target-le-po","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Let $x$ be a list of $n$ natural numbers and let $t, d \\in \\mathbb{N}$ with $t \\le n^d$. Then the exact subset-sum procedure on $x$ and $t$ makes at most $1 + n + (7n + 1)(n^d + 1)$ counted events.","title":"Polynomial event bound for exact subset sum with a polynomially bounded target"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-time-le-of-values-le-po","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_time_le_of_values_le_pow","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-time-le-of-values-le-po","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Let $x$ be a list of $n$ natural numbers and let $d \\in \\mathbb{N}$ be such that every entry of $x$ is at most $n^d$. Then for every target $t \\in \\mathbb{N}$ the exact subset-sum procedure on $x$ and $t$ makes at most $1 + n + (7n + 1)(n \\cdot n^d + 1)$ counted events.","title":"Polynomial event bound for exact subset sum with polynomially bounded inputs"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_optimal","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-optimal","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Let $x$ be a list of natural numbers, $t$ a natural number, and $r$ the value returned by the exact subset sum algorithm on $x$ with target $t$. Then $r$ is the sum of some sublist of $x$ (a subsequence, not necessarily contiguous), $r \\le t$, and every sublist of $x$ whose sum is at most $t$ has sum at most $r$.","title":"Optimality of exact subset sum"},{"id":"ComputerScience/dynamic-programming/subset-sum/exact-subset-sum-optimal-of-positive","kind":"node","lean":"Cslib.Algorithms.Lean.ExactSubsetSum.exactSubsetSum_optimal_of_positive","parent":42036,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/subset-sum/index.html#exact-subset-sum-optimal-of-positive","scope":"ComputerScience/dynamic-programming/subset-sum","status":"fully_proved","summary":"Let $x$ be a list of distinct positive natural numbers, $t$ a positive natural number, and $r$ the value returned by the exact subset sum algorithm on $x$ with target $t$. Then $r$ is the sum of some sublist of $x$ (a subsequence, not necessarily contiguous), $r \\le t$, and every sublist of $x$ whose sum is at most $t$ has sum at most $r$.","title":"Optimality of exact subset sum for distinct positive inputs"},{"data_url":"scopes/ComputerScience/dynamic-programming/zero-one-knapsack-problem.json","id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","kind":"scope","navigation":"module","parent":41902,"result":false,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","title":"Zero-one knapsack problem"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/item","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.Item","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#item","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"A zero-one knapsack item consists of a weight $w \\in \\mathbb{N}$ and a value $v \\in \\mathbb{N}$.","title":"Knapsack item"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/result","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.Result","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#result","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"For a list of $n$ knapsack items, a knapsack result consists of a set $S \\subseteq \\{0, \\dots, n - 1\\}$ of item indices, the chosen items, and a natural number, the reported total value.","title":"Knapsack result"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/total-weight","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.totalWeight","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#total-weight","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"For knapsack items $x_0, \\dots, x_{n-1}$ with weights $w_i$ and a set $S \\subseteq \\{0, \\dots, n - 1\\}$ of item indices, the total weight of $S$ is $\\sum_{i \\in S} w_i$.","title":"Total weight of a knapsack selection"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/total-value","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.totalValue","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#total-value","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"For knapsack items $x_0, \\dots, x_{n-1}$ with values $v_i$ and a set $S \\subseteq \\{0, \\dots, n - 1\\}$ of item indices, the total value of $S$ is $\\sum_{i \\in S} v_i$.","title":"Total value of a knapsack selection"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/feasible","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.Feasible","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#feasible","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"For knapsack items $x_0, \\dots, x_{n-1}$ with weights $w_i$ and a capacity $W \\in \\mathbb{N}$, a set $S \\subseteq \\{0, \\dots, n - 1\\}$ of item indices is feasible when its total weight $\\sum_{i \\in S} w_i$ is at most $W$.","title":"Feasible knapsack selection"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/solve","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.solve","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#solve","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"Given knapsack items $x_0, \\dots, x_{n-1}$ with weights $w_i$ and values $v_i$ and a capacity $W \\in \\mathbb{N}$, the algorithm fills a table $V(i, c)$ for $0 \\le i \\le n$ and $0 \\le c \\le W$ by $V(0, c) = 0$ and, for $i < n$, $V(i + 1, c) = \\max(V(i, c), V(i, c - w_i) + v_i)$ if $w_i \\le c$ and $V(i + 1, c) = V(i, c)$ otherwise. Starting from $c = W$, it then inspects the rows $i = n, n - 1, \\dots, 1$: if\u2026","title":"Bottom-up zero-one knapsack algorithm"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/solve-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.solve_correct","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#solve-correct","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"For every list of knapsack items and every capacity $W \\in \\mathbb{N}$, let $S$ and $v$ be the index set and the reported value returned by the bottom-up knapsack algorithm. Then $S$ is feasible for $W$, $v$ is the total value of $S$, and every feasible set $T$ of item indices has total value at most $v$.","title":"Correctness of the zero-one knapsack algorithm"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/solve-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.solve_time","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#solve-time","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"For every list of $n$ knapsack items and every capacity $W \\in \\mathbb{N}$, the bottom-up knapsack algorithm costs exactly $n(W + 1) + n$ ticks: one for each of the $n(W + 1)$ table cells it computes and one for each of the $n$ item rows it inspects during reconstruction.","title":"Exact cost of the knapsack algorithm"},{"id":"ComputerScience/dynamic-programming/zero-one-knapsack-problem/solve-time-le-three-mul","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.Knapsack.solve_time_le_three_mul","parent":42055,"result":true,"route":"../roadmap/ComputerScience/dynamic-programming/zero-one-knapsack-problem/index.html#solve-time-le-three-mul","scope":"ComputerScience/dynamic-programming/zero-one-knapsack-problem","status":"fully_proved","summary":"For every list of $n$ knapsack items and every capacity $W \\in \\mathbb{N}$ with $W \\ge 1$, the bottom-up knapsack algorithm costs at most $3nW$ ticks.","title":"Pseudo-polynomial cost bound for knapsack"},{"data_url":"scopes/ComputerScience/greedy-algorithms-and-matroids.json","id":"ComputerScience/greedy-algorithms-and-matroids","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/index.html","scope":"ComputerScience/greedy-algorithms-and-matroids","status":"fully_proved","summary":"Greedy choice and exchange arguments: the matroid greedy algorithm and its characterization of matroids, graphic matroids, interval scheduling and furthest-in-future offline caching.","title":"Greedy algorithms and matroids"},{"data_url":"scopes/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching.json","id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","kind":"scope","navigation":"module","parent":42065,"result":false,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","title":"Furthest-in-future caching"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/cache-event","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.CacheEvent","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#cache-event","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"A cache event over a type $\\alpha$ of blocks records how one request is served: either a hit on the requested block $r \\in \\alpha$, or a miss on the requested block $r$ together with an optional evicted block $v \\in \\alpha$, absent when nothing is evicted.","title":"Cache events"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/cache-lts","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.cacheLTS","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#cache-lts","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"For a capacity $k \\in \\mathbb{N}$, this is the labelled transition system whose states are lists of blocks (the cache contents, in any order) and whose labels are cache events. Write $|C|$ for the length of a list $C$, $x :: C$ for $C$ with $x$ prepended, and $C \\setminus v$ for $C$ with the first occurrence of $v$ removed. There is a transition from $C$ to $C'$ labelled by a hit on $r$ when $r \\in C$ and $C'$ is a\u2026","title":"Legal cache transitions"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/furthest-in-future","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.furthestInFuture","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#furthest-in-future","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Given a capacity $k$, a sequence of requested pages and an initial cache (a list of pages), the furthest-in-future manager serves the requests in order and returns the list of emitted cache events together with the final cache. A request for a page $r$ already in the cache is a hit and leaves the cache unchanged. Otherwise it is a miss: if $k = 0$, nothing is evicted and the cache becomes empty; if the cache holds\u2026","title":"Furthest-in-future caching"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/furthest-in-future-nil","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.furthestInFuture_nil","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#furthest-in-future-nil","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"For every capacity $k$ and every list $C$ of blocks, the furthest-in-future manager with capacity $k$ on the empty request list from the initial cache $C$ emits no events and ends with the cache $C$, unchanged.","title":"Furthest-in-future on no requests"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/furthest-in-future-cons","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.furthestInFuture_cons","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#furthest-in-future-cons","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Write $x :: L$ for the list $L$ with $x$ prepended, $[\\,]$ for the empty list, $|L|$ for the length of $L$, and $L \\setminus v$ for $L$ with the first occurrence of $v$ removed. Write $\\mathrm{FIF}_k(R, C) = (E, D)$ when the furthest-in-future manager with capacity $k$ on the requests $R$ from the initial cache $C$ emits the events $E$ and ends with the cache $D$. Write $\\mathrm{hit}(r)$ for a hit on $r$,\u2026","title":"One step of the furthest-in-future manager"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/furthest-in-future-legal","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.furthestInFuture_legal","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#furthest-in-future-legal","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k$ be a capacity, $R$ a list of requested blocks and $C$ a duplicate-free list of blocks of length at most $k$. Let $E$ be the events and $D$ the final cache produced by the furthest-in-future manager with capacity $k$ on $R$ from the initial cache $C$. Then $E$ leads from $C$ to $D$ by legal cache transitions for capacity $k$, the blocks requested by the events of $E$ are, in order, exactly $R$, and $D$ is\u2026","title":"The furthest-in-future manager produces legal traces"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/furthest-in-future-zero-capacity","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.furthestInFuture_zero_capacity","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#furthest-in-future-zero-capacity","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Run with capacity $0$ on a request sequence $r_1, \\dots, r_n$ from the empty cache, the furthest-in-future manager emits exactly the events $\\mathrm{miss}(r_1), \\dots, \\mathrm{miss}(r_n)$, one miss per request in order and each evicting no page, and its final cache is empty.","title":"Furthest-in-future caching with zero capacity"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/furthest-in-future-min-misses","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.furthestInFuture_min_misses","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#furthest-in-future-min-misses","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k$ be a capacity, $R$ a list of requested blocks and $C$ a duplicate-free list of blocks of length at most $k$. Let $E'$ be any list of cache events that leads from $C$ to some final cache by legal cache transitions for capacity $k$ and whose events request, in order, exactly the blocks of $R$. Then the furthest-in-future manager with capacity $k$ on $R$ from the initial cache $C$ emits at most as many miss\u2026","title":"Optimality of furthest-in-future caching"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/request","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.CacheEvent.request","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#request","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"The page served by a cache event is the page it requests, whether the event is a hit or a miss.","title":"Page served by a cache event"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/is-miss","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.CacheEvent.isMiss","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#is-miss","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"The miss indicator of a cache event is true if the event is a miss and false if it is a hit.","title":"Miss indicator"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/misses","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.misses","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#misses","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"The number of misses of a finite list of cache events is the number of events in the list that are misses.","title":"Number of misses in a trace"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/cache-lts-valid","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.cacheLTS_valid","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#cache-lts-valid","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k$ be a capacity and let $C \\to C'$ be a legal cache transition for capacity $k$, labelled by any cache event. If $C$ has no duplicate entries and length at most $k$, then $C'$ has no duplicate entries and length at most $k$.","title":"Legal cache transitions preserve valid caches"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/cache-lts-mtr-valid","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.cacheLTS_mtr_valid","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#cache-lts-mtr-valid","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k$ be a capacity and suppose a finite sequence of cache events leads from the cache $C$ to the cache $C'$ by legal cache transitions for capacity $k$. If $C$ has no duplicate entries and length at most $k$, then $C'$ has no duplicate entries and length at most $k$.","title":"Legal cache traces preserve valid caches"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/request-mem-of-tr","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.request_mem_of_tr","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#request-mem-of-tr","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k > 0$ be a capacity. If a legal transition for capacity $k$ takes a cache $C$ to a cache $C'$ by a cache event, then the page served by that event lies in $C'$.","title":"A legal step caches the served page"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/request-mem-of-misses-eq-zero","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.request_mem_of_misses_eq_zero","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#request-mem-of-misses-eq-zero","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k$ be a capacity. If a legal trace of cache events for capacity $k$ starts from a cache $C$ and contains no miss, then every page served by an event of the trace lies in $C$.","title":"A legal trace without misses serves only cached pages"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/one-le-misses","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.one_le_misses","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#one-le-misses","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k$ be a capacity and $C$ a cache listing at most $k$ pages, and consider a legal trace of cache events for capacity $k$ that starts from $C$. Let $p$ be a page in $C$. If at least $k + 1$ distinct pages are each equal to $p$ or served by some event of the trace, then the trace contains at least one miss.","title":"A legal trace misses at least once"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/furthest-in-future-misses-le","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.furthestInFuture_misses_le","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#furthest-in-future-misses-le","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k$ be a capacity, $R$ a list of requested blocks and $C$ a duplicate-free list of blocks of length at most $k$. Let $E'$ be any list of cache events that leads from $C$ to some final cache by legal cache transitions for capacity $k$ and whose events serve, in order, exactly the blocks of $R$. Then the number of misses among the events emitted by the furthest-in-future manager with capacity $k$ on $R$ from the\u2026","title":"Furthest-in-future caching has the fewest misses"},{"id":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/furthest-in-future-requests","kind":"node","lean":"Cslib.Algorithms.Lean.OfflineCaching.furthestInFuture_requests","parent":42066,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching/index.html#furthest-in-future-requests","scope":"ComputerScience/greedy-algorithms-and-matroids/furthest-in-future-caching","status":"fully_proved","summary":"Let $k$ be a capacity, $R$ a list of requested blocks and $C$ a duplicate-free list of blocks of length at most $k$. Then the events emitted by the furthest-in-future manager with capacity $k$ on $R$ from the initial cache $C$ serve, in order, exactly the blocks of $R$.","title":"Furthest-in-future serves exactly the requests"},{"data_url":"scopes/ComputerScience/greedy-algorithms-and-matroids/interval-scheduling.json","id":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling","kind":"scope","navigation":"module","parent":42065,"result":false,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/index.html","scope":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling","status":"fully_proved","title":"Interval scheduling"},{"id":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/interval-compatible","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IntervalCompatible","parent":42085,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/index.html#interval-compatible","scope":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling","status":"fully_proved","summary":"Let each activity $a$ have a start time $s(a)$ and a finish time $f(a)$ in a linearly ordered type, thought of as the closed-open interval $[s(a), f(a))$. Two activities $a$ and $b$ are compatible when $f(a) \\le s(b)$ or $f(b) \\le s(a)$; in particular, activities whose intervals only touch at an endpoint are compatible.","title":"Compatible activities"},{"id":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/symm","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IntervalCompatible.symm","parent":42085,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/index.html#symm","scope":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling","status":"fully_proved","summary":"For any start and finish times and any activities $a$ and $b$, if $a$ is compatible with $b$, then $b$ is compatible with $a$.","title":"Symmetry of interval compatibility"},{"id":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/interval-schedule","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.intervalSchedule","parent":42085,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/index.html#interval-schedule","scope":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling","status":"fully_proved","summary":"Given start and finish times $s$ and $f$ in a linear order and a list $L$ of activities, sort $L$ by finish time with merge sort, then scan the sorted list: the first activity is selected, and each later activity $a$ is selected exactly when $s(a)$ is at least the finish time of the last selected activity. The result is the list of selected activities in sorted order, together with a cost that counts one for each\u2026","title":"Earliest-finish greedy interval scheduling"},{"id":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/interval-schedule-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.intervalSchedule_correct","parent":42085,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/index.html#interval-schedule-correct","scope":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling","status":"fully_proved","summary":"Let $s$ and $f$ assign start and finish times in a linear order to activities, and let $L$ be a list of activities with $s(a) < f(a)$ for every $a$ in $L$. Let $S$ be the list returned by the earliest-finish schedule of $L$. Then $S$ is a subpermutation of $L$ (each activity occurs in $S$ at most as often as in $L$), any two entries of $S$ at distinct positions are compatible, and every subpermutation $M$ of $L$\u2026","title":"Correctness and optimality of earliest-finish interval scheduling"},{"id":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/interval-schedule-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.intervalSchedule_time","parent":42085,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/interval-scheduling/index.html#interval-schedule-time","scope":"ComputerScience/greedy-algorithms-and-matroids/interval-scheduling","status":"fully_proved","summary":"For any start and finish times and any list of $n$ activities, the earliest-finish schedule charges at most $n \\lceil \\log_2 n \\rceil + (n - 1)$ comparisons when $n \\ge 1$, and none when $n = 0$.","title":"Comparison bound for earliest-finish interval scheduling"},{"data_url":"scopes/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm.json","id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","kind":"scope","navigation":"module","parent":42065,"result":false,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","title":"Matroid greedy algorithm"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedyScan","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-scan","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Given an oracle $O$ that accepts or rejects each finite subset of a type $\\alpha$, a finite set $S \\subseteq \\alpha$ and a list $e_1, \\dots, e_n$ of elements of $\\alpha$, the greedy scan processes $e_1, \\dots, e_n$ in order, starting with $S$ as the current set: it replaces the current set $T$ by $T \\cup \\{e_i\\}$ if $O$ accepts $T \\cup \\{e_i\\}$, and leaves it unchanged otherwise. It returns the final current set\u2026","title":"Greedy scan with an independence oracle"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedy","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Given an oracle $O$ that accepts or rejects each finite subset of a type $\\alpha$ and a list $L$ of elements of $\\alpha$, the greedy algorithm starts from the empty set, processes the elements of $L$ in order, adds an element $e$ to the current set $T$ when $O$ accepts $T \\cup \\{e\\}$ and leaves $T$ unchanged otherwise, and returns the final set. It charges one unit of time per element processed.","title":"Greedy algorithm on a matroid"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-scan-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedyScan_time","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-scan-time","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"For every oracle $O$ on finite subsets of a type $\\alpha$, every finite set $S \\subseteq \\alpha$ and every list $L$ of elements of $\\alpha$, the greedy scan with oracle $O$ from $S$ over $L$ takes time exactly the length of $L$.","title":"Running time of the greedy scan"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedy_time","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-time","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"For every oracle $O$ on finite subsets of a type $\\alpha$ and every list $L$ of elements of $\\alpha$, the greedy algorithm with oracle $O$ on $L$ takes time exactly the length of $L$, one oracle query per element.","title":"Running time of the matroid greedy algorithm"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-scan-append","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedyScan_append","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-scan-append","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"For every oracle $O$ on finite subsets of a type $\\alpha$, every finite set $S \\subseteq \\alpha$ and all lists $P$ and $Q$ of elements of $\\alpha$, the set returned by the greedy scan with oracle $O$ from $S$ over $P$ followed by $Q$ equals the set returned by the greedy scan with oracle $O$ over $Q$ starting from the set that the scan with oracle $O$ from $S$ over $P$ returns.","title":"The greedy scan over a concatenation"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-scan-spec","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedyScan_spec","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-scan-spec","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $M$ be a matroid on a type $\\alpha$, let $O$ be an oracle that accepts a finite set $A \\subseteq \\alpha$ exactly when $A$ is independent in $M$, let $S$ be a finite independent set of $M$ and $L$ a list of elements of $\\alpha$, and let $T$ be the set returned by the greedy scan with oracle $O$ from $S$ over $L$. Then $T$ is independent in $M$, and for every element $e$ of $L$ with $e \\notin T$, the set\u2026","title":"Invariant of the greedy scan"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-indep","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedy_indep","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-indep","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $M$ be a matroid on a type $\\alpha$ and let $O$ be an oracle that accepts a finite set $A \\subseteq \\alpha$ exactly when $A$ is independent in $M$. For every list $L$ of elements of $\\alpha$, the set returned by the greedy algorithm with oracle $O$ on $L$ is independent in $M$, and each of its elements occurs in $L$.","title":"The matroid greedy result is independent"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-is-base","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedy_isBase","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-is-base","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $M$ be a matroid on a type $\\alpha$ with ground set $E$, and let $O$ be an oracle that accepts a finite set $A \\subseteq \\alpha$ exactly when $A$ is independent in $M$. For every list $L$ in which an element of $\\alpha$ occurs if and only if it lies in $E$, the set returned by the greedy algorithm with oracle $O$ on $L$ is a base of $M$.","title":"The matroid greedy result is a base"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/card-filter-le-matroid-greedy","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.card_filter_le_matroidGreedy","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#card-filter-le-matroid-greedy","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $M$ be a matroid on a type $\\alpha$, let $O$ be an oracle that accepts a finite set $A \\subseteq \\alpha$ exactly when $A$ is independent in $M$, and let $w : \\alpha \\to \\mathbb{N}$ be a weight function. Let $L$ be a list of elements of $\\alpha$ in nonincreasing order of weight (whenever $a$ occurs before $b$ in $L$, $w(b) \\le w(a)$), and let $G$ be the set returned by the greedy algorithm with oracle $O$ on\u2026","title":"Threshold counts of the matroid greedy result"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedy_optimal","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-optimal","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $M$ be a matroid on a type $\\alpha$, let $O$ be an oracle that accepts a finite set $A \\subseteq \\alpha$ exactly when $A$ is independent in $M$, and let $w : \\alpha \\to \\mathbb{N}$ be a weight function. Let $L$ be a list of elements of $\\alpha$ in nonincreasing order of weight (whenever $a$ occurs before $b$ in $L$, $w(b) \\le w(a)$), and let $G$ be the set returned by the greedy algorithm with oracle $O$ on\u2026","title":"Optimality of the matroid greedy algorithm"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/is-indep-system","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.IsIndepSystem","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#is-indep-system","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"A predicate $\\mathrm{indep}$ on the finite subsets of a type $\\alpha$ is an independence system when the empty set is independent and every subset of an independent set is independent.","title":"Independence system"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-scan-of-subset","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedyScan_of_subset","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-scan-of-subset","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type, let $\\mathrm{indep}$ be an independence system on the finite subsets of $\\alpha$ and let $I$ be an independent set. For every list $L$ of elements of $I$ and every finite set $S \\subseteq I$, the greedy scan of $L$ starting from $S$ returns $S$ together with all elements of $L$.","title":"A scan inside an independent set keeps everything"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/matroid-greedy-scan-of-forall-not","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.matroidGreedyScan_of_forall_not","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#matroid-greedy-scan-of-forall-not","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type, let $\\mathrm{indep}$ be an independence oracle on the finite subsets of $\\alpha$, let $S$ be a finite subset of $\\alpha$ and let $L$ be a list of elements of $\\alpha$ such that $\\mathrm{indep}$ rejects $S \\cup \\{e\\}$ for every $e$ in $L$. Then the greedy scan of $L$ starting from $S$ returns $S$.","title":"A scan that accepts nothing"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/greedy-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.GreedyOptimal","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#greedy-optimal","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite type and let $\\mathrm{indep}$ be an independence oracle on the finite subsets of $\\alpha$. The greedy algorithm is optimal for $\\mathrm{indep}$ when, for every weight function $w : \\alpha \\to \\mathbb{N}$ and every list $L$ that contains every element of $\\alpha$ exactly once, in nonincreasing order of $w$, the set $T$ returned by the matroid greedy algorithm on $L$ with oracle\u2026","title":"Greedy optimality"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/augment-of-greedy-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.augment_of_greedyOptimal","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#augment-of-greedy-optimal","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite type and let $\\mathrm{indep}$ be an independence system on the finite subsets of $\\alpha$ for which the greedy algorithm is optimal. If $I$ and $J$ are independent sets with $|I| < |J|$, then some element $e \\in J$ with $e \\notin I$ makes $I \\cup \\{e\\}$ independent.","title":"Greedy optimality gives the augmentation axiom"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/exists-matroid-of-greedy-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.exists_matroid_of_greedyOptimal","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#exists-matroid-of-greedy-optimal","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite type and let $\\mathrm{indep}$ be an independence system on the finite subsets of $\\alpha$ for which the greedy algorithm is optimal. Then there is a matroid $M$ on $\\alpha$ whose ground set is all of $\\alpha$ such that a finite set $S \\subseteq \\alpha$ is independent in $M$ exactly when $S$ is independent under $\\mathrm{indep}$.","title":"Greedy-optimal independence systems are matroids"},{"id":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/greedy-optimal-iff-exists-matroid","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedyOptimal_iff_exists_matroid","parent":42091,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm/index.html#greedy-optimal-iff-exists-matroid","scope":"ComputerScience/greedy-algorithms-and-matroids/matroid-greedy-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite type and let $\\mathrm{indep}$ be an independence system on the finite subsets of $\\alpha$. The greedy algorithm is optimal for $\\mathrm{indep}$ if and only if there is a matroid $M$ on $\\alpha$ whose ground set is all of $\\alpha$ such that a finite set $S \\subseteq \\alpha$ is independent in $M$ exactly when $S$ is independent under $\\mathrm{indep}$.","title":"Rado\u2013Edmonds theorem"},{"data_url":"scopes/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid.json","id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","kind":"scope","navigation":"module","parent":42065,"result":false,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","title":"Graphic matroid"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/card-connected-component-delete-edges-of","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.card_connectedComponent_deleteEdges_of_not_reachable","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#card-connected-component-delete-edges-of","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"Let $H$ be a simple graph on a finite vertex set and let $ab$ be an edge of $H$. If $a$ and $b$ are not connected in the graph obtained from $H$ by deleting the edge $ab$, then that graph has exactly one more connected component than $H$.","title":"Deleting a bridge adds one component"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/ncard-edge-set-add-card-connected-compon","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.ncard_edgeSet_add_card_connectedComponent","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#ncard-edge-set-add-card-connected-compon","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"Let $V$ be a finite vertex set. For every natural number $n$ and every forest (acyclic simple graph) $F$ on $V$ with exactly $n$ edges, the number of edges of $F$ plus the number of connected components of $F$ equals $|V|$. So a forest on $V$ with $e$ edges has $|V| - e$ connected components.","title":"Edges plus components of a forest"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/exists-insert-is-acyclic","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.exists_insert_isAcyclic","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#exists-insert-is-acyclic","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set, and let $I$ and $J$ be sets of edges of $G$ that each form an acyclic graph. If $|I| < |J|$, then there is an edge $e \\in J$ with $e \\notin I$ such that $I \\cup \\{e\\}$ still forms an acyclic graph.","title":"Augmentation property of forests"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/is-forest-set","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.IsForestSet","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#is-forest-set","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"A set $I$ of unordered pairs of vertices is a forest of a simple graph $G$ when every element of $I$ is an edge of $G$ and the graph with edge set $I$ has no cycles.","title":"Forests of a graph"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/graphic-matroid","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.graphicMatroid","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#graphic-matroid","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set, the graphic matroid of $G$ is the matroid on unordered pairs of vertices whose ground set is the edge set of $G$ and whose independent sets are the forests of $G$, that is, the sets of edges of $G$ that form an acyclic graph.","title":"Graphic matroid"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/chosen","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.chosen","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#chosen","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"The graph of a finite edge set, with decidable adjacency.","title":"Graph of a finite edge set"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/bridge-in","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.bridgeIn","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#bridge-in","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"On a finite vertex set, for a finite set $S$ of unordered pairs of vertices and a pair $e = \\{u, v\\}$, the bridge test returns true exactly when $u$ and $v$ are not connected by a path in the graph whose edges are the non-loop pairs of $S \\setminus \\{e\\}$. When $e \\in S$ is not a loop, this says that $e$ is a bridge of $S$: deleting it disconnects its endpoints.","title":"Bridge test for an edge set"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/bridge-in-iff","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.bridgeIn_iff","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#bridge-in-iff","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"On a finite vertex set, let $S$ be a finite set of unordered pairs of vertices and let $u$, $v$ be vertices. The bridge test of $S$ on the pair $\\{u, v\\}$ returns true if and only if $u$ and $v$ are not connected by a path in the graph whose edges are the non-loop pairs of $S \\setminus \\{\\{u, v\\}\\}$.","title":"Characterization of the bridge test"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/forest-oracle","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.forestOracle","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#forest-oracle","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set, the forest oracle accepts a finite set $S$ of unordered pairs of vertices exactly when every $e \\in S$ is an edge of $G$ and passes the bridge test in $S$: its endpoints are not connected in the graph formed by the other pairs of $S$. It serves as the independence oracle of the graphic matroid of $G$, since a graph is a forest if and only if every edge is a bridge.","title":"Forest oracle"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/forest-oracle-iff","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.forestOracle_iff","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#forest-oracle-iff","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"For a simple graph $G$ on a finite vertex set and a finite set $S$ of unordered pairs of vertices, the forest oracle of $G$ accepts $S$ if and only if $S$ is independent in the graphic matroid of $G$.","title":"The forest oracle decides graphic-matroid independence"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/kruskal-is-forest","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.kruskal_isForest","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#kruskal-is-forest","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex type $V$. For every list $L$ of unordered pairs of vertices, the matroid greedy algorithm run on $L$ with the forest oracle of $G$ returns a forest of $G$: a set of edges of $G$ without cycles.","title":"Kruskal's algorithm returns a forest"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/kruskal-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.kruskal_optimal","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#kruskal-optimal","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex type $V$, let $w$ assign a natural number to each unordered pair of vertices, and let $L$ be a list of unordered pairs of vertices in nonincreasing order of $w$ (each entry weighs at least as much as every later one). Let $T$ be the set returned by the matroid greedy algorithm run on $L$ with the forest oracle of $G$. Then every forest $I$ of $G$ all of whose edges occur\u2026","title":"Optimality of Kruskal's algorithm"},{"id":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/kruskal-is-base","kind":"node","lean":"Cslib.Algorithms.Lean.GraphicMatroid.kruskal_isBase","parent":42109,"result":true,"route":"../roadmap/ComputerScience/greedy-algorithms-and-matroids/graphic-matroid/index.html#kruskal-is-base","scope":"ComputerScience/greedy-algorithms-and-matroids/graphic-matroid","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set and let $L$ be a list of unordered pairs of vertices such that a pair is an edge of $G$ if and only if it occurs in $L$. Then the matroid greedy algorithm run with the forest oracle of $G$ on $L$ returns a base of the graphic matroid of $G$, that is, a maximal forest of $G$.","title":"Kruskal's algorithm returns a maximal forest"},{"data_url":"scopes/ComputerScience/approximation-algorithms.json","id":"ComputerScience/approximation-algorithms","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/approximation-algorithms/index.html","scope":"ComputerScience/approximation-algorithms","status":"fully_proved","summary":"Polynomial-time algorithms with proved approximation ratios for NP-hard optimization: vertex cover, set cover, metric TSP, list scheduling, maximum coverage.","title":"Approximation algorithms"},{"data_url":"scopes/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp.json","id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","kind":"scope","navigation":"module","parent":42123,"result":false,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","title":"Double-tree approximation for metric TSP"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/is-metric-cost","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.IsMetricCost","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#is-metric-cost","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"A cost function $d$ assigning to each pair of vertices a value in a linearly ordered additive commutative monoid is metric when, for all vertices $x$, $y$, $z$, it is symmetric, $d(x, y) = d(y, x)$, zero on the diagonal, $d(x, x) = 0$, nonnegative, $0 \\le d(x, y)$, and satisfies the triangle inequality $d(x, z) \\le d(x, y) + d(y, z)$.","title":"Metric cost function"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/path-from","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.pathFrom","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#path-from","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on the vertices with values in an additive commutative monoid. For a vertex $x$ and a list $L = [y_1, \\dots, y_n]$ of vertices, the cost of the walk from $x$ through $L$ is $d(x, y_1) + d(y_1, y_2) + \\dots + d(y_{n-1}, y_n)$, and $0$ when $L$ is empty.","title":"Cost of a walk"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/tour-cost","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.tourCost","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#tour-cost","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For a cost function $d$ on the vertices with values in an additive commutative monoid, the cost of the closed tour that visits $x_0, x_1, \\dots, x_k$ in order and returns to its first vertex is $\\sum_{i=1}^{k} d(x_{i-1}, x_i) + d(x_k, x_0)$. The empty tour costs $0$.","title":"Cost of a closed tour"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/forest","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.Forest","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#forest","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"A rooted forest with vertices in $\\alpha$ is either empty, or consists of a first tree with root $v$, whose subtrees form a forest (the children of $v$), followed by a forest of the remaining trees (the siblings of $v$).","title":"Rooted forest in left-child right-sibling form"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/preorder","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.Forest.preorder","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#preorder","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"The list of the vertices of a forest in preorder: empty for the empty forest, and for a forest with first root $v$, children forest $C$ and sibling forest $S$, the vertex $v$, followed by the preorder of $C$, followed by the preorder of $S$.","title":"Preorder of a forest"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/weight","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.Forest.weight","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#weight","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For a cost function $d$ and a vertex $p$, the weight of a forest hanging below $p$ is the total cost of its tree edges: each top-level vertex $v$ is joined to $p$ by an edge of cost $d(p, v)$, and each vertex $u$ is joined to each of its children $c$ by an edge of cost $d(u, c)$. The empty forest has weight $0$.","title":"Weight of a forest below a vertex"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/chain","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.Forest.chain","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#chain","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"The forest of a list $x_1, x_2, \\dots, x_n$ of vertices is the path $x_1, x_2, \\dots, x_n$: $x_1$ is its only top-level vertex and each $x_{i+1}$ is the only child of $x_i$. The empty list gives the empty forest.","title":"Path as a forest"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/preorder-walk-aux","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.preorderWalkAux","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#preorder-walk-aux","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For a forest $F$ and a list $a$ of vertices, this is the timed computation defined as follows. On the empty forest it returns $a$ at no cost. On a forest with first root $v$, children forest $C$ and sibling forest $S$, it charges one time unit, walks $S$ with accumulator $a$ to get a list $b$, walks $C$ with accumulator $b$ to get a list $c$, and returns $v$ followed by $c$. It thus returns the vertices of $F$ in\u2026","title":"Preorder walk with an accumulator"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/preorder-walk","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.preorderWalk","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#preorder-walk","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For a forest $F$, the preorder walk of $F$ is the preorder walk of $F$ with the empty list as accumulator: a timed computation that returns the vertices of $F$ in preorder, at one time unit per vertex.","title":"Preorder walk"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/approx-tsptour","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.approxTSPTour","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#approx-tsptour","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"CLRS `APPROX-TSP-TOUR`, steps 3 and 4: given a root $r$ and a forest $F$ (the subtrees of a spanning tree with root $r$), return the tour that lists $r$ followed by the vertices of $F$ in preorder. As a timed computation it charges one unit for $r$ and one unit for each vertex visited by the preorder walk of $F$.","title":"The APPROX-TSP-TOUR algorithm"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/preorder-walk-aux-ret","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.preorderWalkAux_ret","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#preorder-walk-aux-ret","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For every forest $F$ and every list $a$ of vertices, the preorder walk of $F$ with accumulator $a$ returns the vertices of $F$ in preorder followed by $a$.","title":"Result of the preorder walk with an accumulator"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/approx-tsptour-time","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.approxTSPTour_time","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#approx-tsptour-time","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For every root $r$ and forest $F$, the running time of `APPROX-TSP-TOUR` on $r$ and $F$ equals the length of the list of $r$ followed by the vertices of $F$ in preorder: one unit per vertex of the returned tour.","title":"Running time of `APPROX-TSP-TOUR`"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/path-from-le-add","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.pathFrom_le_add","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#path-from-le-add","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on the vertices with values in a linearly ordered additive commutative monoid, which is symmetric, zero on the diagonal, nonnegative, and satisfies the triangle inequality. For vertices $q$, $p$ and a list $L$ of vertices, the cost of the walk from $q$ through $L$ is at most $d(q, p)$ plus the cost of the walk from $p$ through $L$.","title":"Changing the starting vertex of a walk"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/path-from-append-append-le","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.pathFrom_append_append_le","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#path-from-append-append-le","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on the vertices with values in a linearly ordered additive commutative monoid, which is symmetric, zero on the diagonal, nonnegative, and satisfies the triangle inequality. For a vertex $p$ and lists $A$, $B$ of vertices, the cost of the walk that starts at $p$, visits $A$ and then $B$ in order and returns to $p$ is at most the cost of the walk from $p$ through $A$ back to $p$ plus the\u2026","title":"Shortcutting a closed walk through its base point"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/path-from-append-singleton-le","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.pathFrom_append_singleton_le","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#path-from-append-singleton-le","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on the vertices with values in a linearly ordered additive commutative monoid, which is symmetric, zero on the diagonal, nonnegative, and satisfies the triangle inequality. For vertices $v$, $p$ and a list $C$ of vertices, the cost of the walk from $v$ through $C$ ending at $p$ is at most the cost of the walk from $v$ through $C$ ending back at $v$, plus $d(v, p)$.","title":"Rerouting the end of a walk"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/path-from-preorder-le","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.pathFrom_preorder_le","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#path-from-preorder-le","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on the vertices with values in a linearly ordered additive commutative monoid, which is symmetric, zero on the diagonal, nonnegative, and satisfies the triangle inequality. For a forest $F$ and a vertex $p$, let $w$ be the weight of $F$ hanging below $p$: the total cost of the edges joining $p$ to the top-level vertices of $F$ and each vertex of $F$ to its children. Then the cost of the\u2026","title":"The closed preorder walk below a vertex costs at most twice the forest"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/tour-cost-preorder-le","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.tourCost_preorder_le","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#tour-cost-preorder-le","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on the vertices with values in a linearly ordered additive commutative monoid that is symmetric, zero on the diagonal, nonnegative and satisfies the triangle inequality. Let $T$ be a rooted tree with root $r$, given by the forest $F$ of the subtrees of $r$, and let $w(T)$ be its weight, the total cost of its edges. Then the closed tour that visits $r$, then the vertices of $F$ in\u2026","title":"The preorder tour costs at most twice the tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/preorder-chain","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.Forest.preorder_chain","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#preorder-chain","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For every list $L$ of vertices, the preorder of the forest of $L$ (the path through the vertices of $L$ in order) is $L$ itself.","title":"Preorder of a path"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/is-spanning","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.IsSpanning","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#is-spanning","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"The tree with root $r$ and subtree forest $F$ is a spanning tree of a list $V$ of vertices when the vertices of $V$ are distinct and the list of $r$ followed by the vertices of $F$ in preorder is a permutation of $V$: the vertices of the tree are those of $V$, each exactly once.","title":"Rooted spanning tree of a vertex list"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/is-min-spanning","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.IsMinSpanning","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#is-min-spanning","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on the vertices with values in a linearly ordered additive commutative monoid. The tree with root $r$ and subtree forest $F$ is a minimum spanning tree of a list $V$ of vertices for $d$ when it is a spanning tree of $V$ (the vertices of $V$ are distinct and the list of $r$ followed by the vertices of $F$ in preorder is a permutation of $V$) and, for every vertex $r'$ and forest $F'$ such\u2026","title":"Minimum spanning tree of a vertex list"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/is-tour","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.IsTour","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#is-tour","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"A list $H$ is a tour of a list $V$ of vertices when the vertices of $V$ are distinct and $H$ is a permutation of $V$: $H$ lists each vertex of $V$ exactly once and nothing else.","title":"Tour of a vertex list"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/weight-le-tour-cost","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.weight_le_tourCost","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#weight-le-tour-cost","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on the vertices with values in a linearly ordered additive commutative monoid that is symmetric, zero on the diagonal, nonnegative and satisfies the triangle inequality. Let $V$ be a list of distinct vertices. Let $T$ be a minimum spanning tree of $V$ for $d$: a rooted tree whose vertices are the elements of $V$, each once, whose weight (the total cost of its edges) is at most that of\u2026","title":"A minimum spanning tree costs at most any tour"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/approx-tsptour-is-tour","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.approxTSPTour_isTour","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#approx-tsptour-is-tour","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $V$ be a list of distinct vertices, and let $r$ and $F$ be such that $r$ followed by the vertices of the forest $F$ in preorder is a permutation of $V$ (the tree with root $r$ and subtree forest $F$ spans $V$). Then the list returned by `APPROX-TSP-TOUR` on $r$ and $F$ is a tour of $V$: it is a permutation of $V$.","title":"The approximate TSP tour is a tour"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/approx-tsptour-two-approx","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.approxTSPTour_two_approx","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#approx-tsptour-two-approx","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a cost function on vertices with values in a linearly ordered additive commutative monoid that is symmetric, zero on the diagonal, nonnegative and satisfies the triangle inequality. Let $V$ be a list of vertices, let the tree with root $r$ and subtree forest $F$ be a minimum spanning tree of $V$ for $d$, and let $H$ be a tour of $V$. Then the cost under $d$ of the closed tour returned by\u2026","title":"APPROX-TSP-TOUR on a given minimum spanning tree costs at most twice any tour"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/up","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.up","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#up","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ be a parent array on the vertices $\\{0, \\dots, n-1\\}$, assigning to each vertex $v$ either a parent $p(v)$ or no parent. Then $\\operatorname{up}(v)$ is the parent $p(v)$ when $v$ has one, and $v$ itself otherwise, so a root stays put.","title":"One step up a parent array"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/children","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.children","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#children","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ assign to each vertex of $\\{0, \\dots, n-1\\}$ either a parent or nothing. The children of a vertex $v$ are the vertices $c$ whose parent $p(c)$ is $v$, listed in increasing order.","title":"Children in a parent array"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/build-forest","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.buildForest","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#build-forest","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ assign to each vertex of $\\{0, \\dots, n-1\\}$ either a parent or nothing. For $k \\in \\mathbb{N}$ and a vertex $v$, the forest below $v$ of depth $k$ is empty when $k = 0$. For depth $k + 1$ it has one tree for each child $c$ of $v$ (each vertex whose parent is $v$), in increasing order of $c$, namely $c$ with the forest below $c$ of depth $k$ as its subtrees.","title":"Forest of a parent array below a vertex"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/parent-tree","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.ParentTree","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#parent-tree","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ assign to each vertex of $\\{0, \\dots, n-1\\}$ either a parent or nothing, and let a step up move a vertex to its parent, leaving a vertex without parent in place. The parent array $p$ is a parent tree with root $r$ when $r$ has no parent and every vertex reaches $r$ after finitely many steps up.","title":"Parent tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/desc-k","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.DescK","parent":42124,"reading_title":{"format":"inline-parts-v1","parts":[["text","Descendant within "],["math","\\(k\\)"],["text"," levels"]],"source":"Descendant within $k$ levels"},"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#desc-k","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ assign to each vertex of $\\{0, \\dots, n-1\\}$ either a parent or nothing, and let a step up move a vertex to its parent, leaving a vertex without parent in place. For $k \\in \\mathbb{N}$, a vertex $u$ lies strictly below a vertex $v$, at most $k$ levels down, when for some $j < k$ the vertex reached from $u$ after $j$ steps up has parent $v$.","title":"Descendant within $k$ levels"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/weight-build-forest","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.weight_buildForest","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#weight-build-forest","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ be a parent array on the vertices $\\{0, \\dots, n-1\\}$, let $d$ be a cost on ordered pairs of vertices with values in an additive commutative monoid, and let $\\operatorname{up}(u)$ be the parent of $u$, or $u$ itself if $u$ has no parent. For every $k \\in \\mathbb{N}$ and every vertex $v$, let $F$ be the forest of the subtrees of the children of $v$ in $p$, cut off $k$ levels below $v$. Then the weight under\u2026","title":"Weight of the forest built from a parent array"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/desc-k-root","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.descK_root","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#desc-k-root","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ assign to each vertex of $\\{0, \\dots, n-1\\}$ either a parent or nothing, forming a parent tree with root $r$, and let a step up move a vertex to its parent, leaving a vertex without parent in place. Then every vertex $u \\ne r$ lies strictly below $r$, at most $n$ levels down: for some $j < n$, the vertex reached from $u$ after $j$ steps up has parent $r$.","title":"Every non-root vertex lies below the root"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/is-spanning-build-forest","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.isSpanning_buildForest","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#is-spanning-build-forest","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ assign to each vertex of $\\{0, \\dots, n-1\\}$ either a parent or nothing, forming a parent tree with root $r$. Then $r$ together with the forest below $r$ of depth $n$ built from $p$ is a spanning tree of $\\{0, \\dots, n-1\\}$: $r$ followed by the vertices of the forest in preorder lists every vertex exactly once.","title":"The forest of a parent tree spans all vertices"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/complete-adjacency","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.completeAdjacency","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#complete-adjacency","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the complete graph on the vertices $\\{0, \\dots, n-1\\}$ as a directed adjacency: the vertices are listed in increasing order, and the neighbours of each vertex $u$ are all vertices other than $u$, in increasing order.","title":"Complete graph as an ordered adjacency"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/sym-weight","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.symWeight","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#sym-weight","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Given costs $d(u,v)$ on the vertices $\\{0, \\dots, n-1\\}$ with values in a type with commutative addition, the weight of an unordered pair $\\{u, v\\}$ is $d(u,v) + d(v,u)$, which does not depend on the order of $u$ and $v$. When $d$ is symmetric this is twice the cost, which leaves minimum spanning trees unchanged.","title":"Symmetrized edge weights"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/prim-parents","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.primParents","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#prim-parents","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Given costs $d(u,v)$ on the vertices $\\{0, \\dots, n-1\\}$ with values in a linearly ordered type with a zero and commutative addition, and a root $r$, this is the parent array returned by Prim's algorithm run from $r$ on the complete graph on these vertices, with the weight of the edge $\\{u, v\\}$ equal to $d(u,v) + d(v,u)$: it sends each vertex to its parent, or to no parent.","title":"Prim's parent array on the complete graph"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/prim-forest","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.primForest","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#prim-forest","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Given costs $d(u,v)$ on the vertices $\\{0, \\dots, n-1\\}$ with values in a linearly ordered type with a zero and commutative addition, and a root $r$, take Prim's parent array for $d$ and $r$ (Prim's algorithm run from $r$ on the complete graph with edge weights $d(u,v) + d(v,u)$). This forest is the forest of the subtrees below $r$ in that parent array, cut off $n$ levels below $r$, in left-child right-sibling\u2026","title":"Forest of Prim's tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/prim-tsptour","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.primTSPTour","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#prim-tsptour","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Given costs $d(u,v)$ on the vertices $\\{0, \\dots, n-1\\}$ with values in a linearly ordered type with a zero and commutative addition, and a root $r$, the tour is $r$ followed by the vertices of the forest of Prim's tree below $r$ in preorder. Its running time counts one unit per vertex of the tour; running Prim's algorithm and building the forest are not counted.","title":"APPROX-TSP-TOUR with Prim's tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/mem-complete-adjacency","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.mem_completeAdjacency","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#mem-complete-adjacency","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For all vertices $u, v$ of $\\{0, \\dots, n-1\\}$, $v$ is listed among the neighbours of $u$ in the complete graph as an ordered adjacency if and only if $u$ and $v$ are adjacent in the complete simple graph on $\\{0, \\dots, n-1\\}$, that is, $u \\ne v$.","title":"Neighbours in the complete adjacency"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/prim-tree","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.primTree","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#prim-tree","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Given costs $d(u,v)$ on the vertices $\\{0, \\dots, n-1\\}$ with values in a linearly ordered type with a zero and commutative addition, and a root $r$, Prim's tree is the simple graph on these vertices in which distinct vertices $u$ and $v$ are adjacent when one of them is the parent of the other in Prim's parent array for $d$ and $r$.","title":"Prim's tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/prim-tree-is-tree","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.primTree_isTree","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#prim-tree-is-tree","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For costs $d(u,v)$ on the vertices $\\{0, \\dots, n-1\\}$ with values in any linearly ordered type with a zero and commutative addition, and any root $r$, Prim's tree for $d$ and $r$ is a tree on all of $\\{0, \\dots, n-1\\}$, that is, a spanning tree of the complete graph.","title":"Prim's tree is a spanning tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/prim-tree-is-minimum-spanning-tree","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.primTree_isMinimumSpanningTree","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#prim-tree-is-minimum-spanning-tree","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d(u,v)$ be real costs on the vertices $\\{0, \\dots, n-1\\}$, let $r$ be a vertex, and weigh each edge $\\{u, v\\}$ by $d(u,v) + d(v,u)$. Then Prim's tree for $d$ and $r$ is a tree, and for every tree $H$ on these vertices the total weight of the edges of Prim's tree is at most the total weight of the edges of $H$.","title":"Prim's tree is a minimum spanning tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/parent-tree-prim-parents","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.parentTree_primParents","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#parent-tree-prim-parents","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero and a commutative addition, let $d$ assign a weight in $W$ to each ordered pair of vertices of $\\{0, \\dots, n-1\\}$, and let $r$ be a vertex. Then Prim's parent array for $d$ from $r$ is a parent tree with root $r$: $r$ has no parent, and every vertex reaches $r$ after finitely many steps up, where a step up moves a vertex to its parent and leaves a vertex without\u2026","title":"Prim's parent array is a parent tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/is-spanning-prim-forest","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.isSpanning_primForest","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#is-spanning-prim-forest","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $W$ be a linearly ordered type with a zero and a commutative addition, let $d$ assign a weight in $W$ to each ordered pair of vertices of $\\{0, \\dots, n-1\\}$, and let $r$ be a vertex. Then $r$ together with Prim's forest for $d$ and $r$ is a spanning tree of $\\{0, \\dots, n-1\\}$: $r$ followed by the vertices of the forest in preorder lists every vertex exactly once.","title":"Prim's forest spans all vertices"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/prim-tsptour-is-tour","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.primTSPTour_isTour","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#prim-tsptour-is-tour","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"For costs $d(u,v)$ on the vertices $\\{0, \\dots, n-1\\}$ with values in a linearly ordered type with a zero and commutative addition, and any root $r$, the list returned by APPROX-TSP-TOUR with Prim's tree for $d$ and $r$ is a permutation of $0, 1, \\dots, n-1$: it visits every vertex exactly once.","title":"The tour from Prim's tree is a tour"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/sum-edge-finset-eq","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.sum_edgeFinset_eq","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#sum-edge-finset-eq","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $p$ be a parent array on the vertices $\\{0, \\dots, n-1\\}$ in which the root $r$ has no parent and every chain of parents reaches $r$, and let $w$ assign to each unordered pair $\\{u, v\\}$ of vertices (possibly $u = v$) a value in a commutative additive monoid. Write $\\hat p(u)$ for the parent of $u$, or $u$ itself if $u$ has no parent. Then the sum of $w(e)$ over the edges $e$ of the parent graph (distinct $u$\u2026","title":"Edge sums over a parent tree"},{"id":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/prim-tsptour-two-approx","kind":"node","lean":"Cslib.Algorithms.Lean.MetricTSP.primTSPTour_two_approx","parent":42124,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp/index.html#prim-tsptour-two-approx","scope":"ComputerScience/approximation-algorithms/double-tree-approximation-for-metric-tsp","status":"fully_proved","summary":"Let $d$ be a real metric cost on the vertices $\\{0, \\dots, n-1\\}$ (symmetric, zero on the diagonal, nonnegative, and satisfying the triangle inequality), and let $r$ be a vertex. Then the cost of the closed tour through the list returned by APPROX-TSP-TOUR with Prim's tree for $d$ and $r$ is at most twice the cost of the closed tour through $H$, for every list $H$ that visits each of $0, 1, \\dots, n-1$ exactly once.","title":"APPROX-TSP-TOUR with Prim's tree is a 2-approximation"},{"data_url":"scopes/ComputerScience/approximation-algorithms/greedy-set-cover.json","id":"ComputerScience/approximation-algorithms/greedy-set-cover","kind":"scope","navigation":"module","parent":42123,"result":false,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","title":"Greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/best-set","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bestSet","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#best-set","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Given a list $F$ of finite sets and a finite set $U$, the best set of $F$ for $U$ is the first set $S$ in the list $F$ that maximizes $|S \\cap U|$, or $\\emptyset$ when $F$ is empty.","title":"Greedy choice of a set"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F = [S_1, \\dots, S_n]$ be a list of finite sets and $U$ a finite set. The greedy set cover of $U$ from $F$ is computed as follows. Let $T$ be the first $S_i$ in the list that maximizes $|S_i \\cap U|$, or $T = \\emptyset$ when $n = 0$. If $T \\cap U = \\emptyset$, the result is the empty list; otherwise it is $T$ followed by the greedy set cover of $U \\setminus T$ from $F$. Each step, including the last one, costs\u2026","title":"Greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover-mem","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover_mem","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover-mem","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"For every list $F$ of finite sets and every finite set $U$, every set in the list returned by the greedy set cover algorithm on $F$ and $U$ is a member of $F$.","title":"The greedy set cover draws its sets from the family"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover-covers","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover_covers","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover-covers","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets and $U$ a finite set such that every element of $U$ lies in some set of $F$. Then every element of $U$ lies in some set of the list returned by the greedy set cover algorithm on $F$ and $U$.","title":"The greedy set cover is a cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover-length-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover_length_le","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover-length-le","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"For every list $F$ of finite sets and every finite set $U$, the list returned by the greedy set cover algorithm on $F$ and $U$ has length at most $|U|$.","title":"The greedy set cover has at most as many sets as elements"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover_time_le","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover-time-le","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets and $U$ a finite set. The running time of greedy set cover of $U$ from $F$ is at most $(|U| + 1)(|F| + \\sum_{S \\in F} |S|)$, where $|F|$ is the length of the list and the sum runs over the entries of $F$, counted with repetition.","title":"Running time of greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/div-le-harmonic-sub","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.div_le_harmonic_sub","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#div-le-harmonic-sub","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Write $H(n) = 1 + \\frac12 + \\dots + \\frac1n$ for the $n$-th harmonic number, so $H(0) = 0$. For natural numbers $d \\le a \\le k$, we have $\\frac{d}{k} \\le H(a) - H(a - d)$, where $\\frac{d}{k}$ is read as $0$ when $k = 0$.","title":"Lower bound for a difference of harmonic numbers"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/card-le-sum-card-inter","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.card_le_sum_card_inter","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#card-le-sum-card-inter","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $L$ be a list of finite sets and $T$ a finite set such that every element of $T$ lies in some member of $L$. Then $|T| \\le \\sum_{S \\in L} |S \\cap T|$, where the sum runs over the list $L$ with repetitions.","title":"Covering bound by intersections"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover-length-le-sum-harmonic","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover_length_le_sum_harmonic","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover-length-le-sum-harmonic","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets, $U$ a finite set, and $H(n) = 1 + \\frac12 + \\dots + \\frac1n$ the $n$-th harmonic number. For every list $C^*$ of members of $F$ that together cover $U$, the list of sets returned by the greedy set cover algorithm on $F$ and $U$ has length at most $\\sum_{S \\in C^*} H(|S \\cap U|)$, where the sum runs over the list $C^*$ with repetitions.","title":"Potential bound for greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover-approx","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover_approx","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover-approx","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets each with at most $d$ elements, $U$ a finite set, and $H(n) = 1 + \\frac12 + \\dots + \\frac1n$ the $n$-th harmonic number. For every list $C^*$ of members of $F$ that together cover $U$, the list of sets returned by the greedy set cover algorithm on $F$ and $U$ has length at most $|C^*| \\cdot H(d)$, where $|C^*|$ is the length of $C^*$.","title":"Harmonic bound in the largest set size for greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover-approx-card","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover_approx_card","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover-approx-card","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets, $U$ a finite set, and $H(n) = 1 + \\frac12 + \\dots + \\frac1n$ the $n$-th harmonic number. For every list $C^*$ of members of $F$ that together cover $U$, the list of sets returned by the greedy set cover algorithm on $F$ and $U$ has length at most $|C^*| \\cdot H(|U|)$, where $|C^*|$ is the length of $C^*$.","title":"Harmonic bound in the universe size for greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/greedy-set-cover-approx-log","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.greedySetCover_approx_log","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#greedy-set-cover-approx-log","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets and $U$ a finite set. For every list $C^*$ of members of $F$ that together cover $U$, the list of sets returned by the greedy set cover algorithm on $F$ and $U$ has length at most $|C^*| \\cdot (\\ln |U| + 1)$, where $|C^*|$ is the length of $C^*$ and $\\ln 0 = 0$.","title":"Greedy set cover is a logarithmic approximation"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/best-ratio-set","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.bestRatioSet","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#best-ratio-set","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Given a list $F$ of finite sets, a finite set $U$ and a rational cost $w(S)$ for each finite set $S$, this is the first set $S$ in $F$ with $S \\cap U \\ne \\emptyset$ that minimizes the ratio $w(S) / |S \\cap U|$ among the sets of $F$ meeting $U$, or nothing when no set of $F$ meets $U$.","title":"Set with the least cost per new element"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/weighted-greedy-set-cover","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.weightedGreedySetCover","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#weighted-greedy-set-cover","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F = [S_1, \\dots, S_n]$ be a list of finite sets, $U$ a finite set and $w$ a function giving each finite set $S$ a rational cost $w(S)$. The weighted greedy set cover of $U$ from $F$ with costs $w$ is computed as follows. If no $S_i$ meets $U$, the result is the empty list. Otherwise let $T$ be the first $S_i$ in the list with $S_i \\cap U \\ne \\emptyset$ that minimizes $w(S_i) / |S_i \\cap U|$ among the sets of\u2026","title":"Weighted greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/weighted-greedy-set-cover-mem","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.weightedGreedySetCover_mem","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#weighted-greedy-set-cover-mem","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"For every list $F$ of finite sets, finite set $U$ and rational cost function $w$ on finite sets, every set chosen by the weighted greedy set cover of $U$ from $F$ with costs $w$ belongs to $F$.","title":"The weighted greedy set cover uses sets of the family"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/weighted-greedy-set-cover-covers","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.weightedGreedySetCover_covers","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#weighted-greedy-set-cover-covers","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets, $U$ a finite set and $w$ any rational cost function on finite sets. If every element of $U$ lies in some set of $F$, then every element of $U$ lies in some set chosen by the weighted greedy set cover of $U$ from $F$ with costs $w$.","title":"The weighted greedy set cover is a cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/weighted-greedy-set-cover-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.weightedGreedySetCover_time_le","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#weighted-greedy-set-cover-time-le","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F = [S_1, \\dots, S_m]$ be a list of finite sets, $U$ a finite set and $w$ a rational cost function on finite sets. The weighted greedy set cover of $U$ from $F$ with costs $w$ runs in time at most $(|U| + 1)\\,(m + |S_1| + \\dots + |S_m|)$: at most $|U| + 1$ searches for a best set, each costing $m$ plus the total size of the sets of $F$.","title":"Running time of weighted greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/total-cost","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.totalCost","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#total-cost","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"For a function $w$ assigning a rational cost to each finite set, the total cost of a list $C = [S_1, \\dots, S_m]$ of finite sets is $w(S_1) + \\dots + w(S_m)$, a set that occurs several times in $C$ being counted each time.","title":"Total cost of a list of sets"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/weighted-greedy-set-cover-cost-le-sum-ha","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.weightedGreedySetCover_cost_le_sum_harmonic","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#weighted-greedy-set-cover-cost-le-sum-ha","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets, $U$ a finite set and $w$ a rational cost function on finite sets with $w(S) \\ge 0$ for every $S$ in $F$. If $C^*$ is a list of sets from $F$ whose union contains $U$, then the weighted greedy set cover of $U$ from $F$ with costs $w$ has total cost at most $\\sum_{S \\in C^*} w(S)\\, H(|S \\cap U|)$, the sum running over the entries of $C^*$ with repetition, where\u2026","title":"Potential bound for weighted greedy set cover"},{"id":"ComputerScience/approximation-algorithms/greedy-set-cover/weighted-greedy-set-cover-approx","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.weightedGreedySetCover_approx","parent":42171,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/greedy-set-cover/index.html#weighted-greedy-set-cover-approx","scope":"ComputerScience/approximation-algorithms/greedy-set-cover","status":"fully_proved","summary":"Let $F$ be a list of finite sets, $U$ a finite set, $w$ a rational cost function on finite sets with $w(S) \\ge 0$ for every $S$ in $F$, and $d$ a natural number such that every set in $F$ has at most $d$ elements. If $C^*$ is a list of sets from $F$ whose union contains $U$, then the weighted greedy set cover of $U$ from $F$ with costs $w$ has total cost at most $H(d)$ times the total cost of $C^*$, where\u2026","title":"Chv\u00e1tal's bound"},{"data_url":"scopes/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines.json","id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","kind":"scope","navigation":"module","parent":42123,"result":false,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","title":"List scheduling on identical machines"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/is-schedule-of","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.IsScheduleOf","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#is-schedule-of","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $m$ be a natural number. An assignment $\\sigma$ of a list $\\sigma(i)$ of natural numbers (processing times) to each machine $i \\in \\{0, \\dots, m - 1\\}$ is a schedule of a list $J$ of natural numbers when the sum over all machines $i$ of the multisets of entries of $\\sigma(i)$ equals the multiset of entries of $J$, that is, every number occurs in the lists $\\sigma(i)$ together exactly as often as it occurs in $J$.","title":"Schedule of a list of jobs"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/makespan","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.makespan","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#makespan","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"For a schedule $\\sigma$ that gives each of $m$ machines a list of processing times, the load of a machine is the sum of its list, and the makespan of $\\sigma$ is the largest machine load ($0$ when $m = 0$).","title":"Makespan"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/least-loaded","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.leastLoaded","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#least-loaded","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $m \\ge 1$ and let $L$ assign a load $L(i) \\in \\mathbb{N}$ to each machine $i \\in \\{0, \\dots, m - 1\\}$. The least-loaded machine for $L$ is the smallest index $i$ at which $L(i)$ is minimal.","title":"Least-loaded machine"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/list-schedule-scan","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.listScheduleScan","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#list-schedule-scan","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $m \\ge 1$. Given loads $L(i)$ and job lists $\\sigma(i)$ for the machines $i \\in \\{0, \\dots, m - 1\\}$ and a list of remaining jobs, list scheduling returns $\\sigma$ at cost $0$ when no jobs remain. Otherwise it charges cost $m$, takes the first remaining job $p$, appends $p$ to $\\sigma(k)$ and adds $p$ to $L(k)$ for the least-loaded machine $k$ for $L$, and continues in the same way with the other remaining\u2026","title":"List scheduling from an intermediate state"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/list-schedule","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.listSchedule","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#list-schedule","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Given $m \\ge 1$ identical machines and a list of job processing times (natural numbers), start with every machine empty and take the jobs in the given order, appending each job to the list of a currently least-loaded machine (the first one in index order), where the load of a machine is the sum of its jobs. The result assigns to each machine its list of jobs. The cost counted is $m$ per job, one unit for each\u2026","title":"Graham's list scheduling"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/lpt","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.lpt","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#lpt","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Given $m \\ge 1$ identical machines and a list of job processing times (natural numbers), LPT sorts the jobs into nonincreasing order by merge sort and then applies Graham's list scheduling on $m$ machines to the sorted list. The cost counted is the number of comparisons made by merge sort plus the cost of list scheduling, $m$ per job.","title":"Longest processing time first (LPT)"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/is-schedule-of-nil","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.isScheduleOf_nil","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#is-schedule-of-nil","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"For every number $m$ of machines, the assignment of the empty list to each of the $m$ machines is a schedule of the empty list of jobs.","title":"The empty schedule"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/le-makespan","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.IsScheduleOf.le_makespan","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#le-makespan","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $\\sigma$ be a schedule on $m$ machines of a list $J$ of processing times. Then every entry $x$ of $J$ is at most the makespan of $\\sigma$.","title":"Every job is at most the makespan"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/sum-le","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.IsScheduleOf.sum_le","parent":42192,"reading_title":{"format":"inline-parts-v1","parts":[["text","Total work is at most "],["math","\\(m\\)"],["text"," times the makespan"]],"source":"Total work is at most $m$ times the makespan"},"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#sum-le","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"If $\\sigma$ is a schedule of a list $P$ of jobs on $m$ machines, then the total processing time of $P$ is at most $m$ times the makespan of $\\sigma$.","title":"Total work is at most $m$ times the makespan"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/scan-inv","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.ScanInv","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#scan-inv","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"A list $P$ of jobs, machine loads $\\ell_i \\in \\mathbb{N}$ for $i < m$, and a schedule $\\sigma$ on $m$ machines satisfy the invariant when $\\sigma$ is a schedule of $P$ and, for every machine $i$, $\\ell_i$ is the load of machine $i$ in $\\sigma$, the sum of the processing times assigned to it.","title":"List-scheduling state invariant"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/list-schedule-scan-bound","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.listScheduleScan_bound","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#list-schedule-scan-bound","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $m \\ge 1$ and $a, b, M \\in \\mathbb{N}$. Let $\\sigma$ assign job lists to the $m$ machines so that $\\sigma$ is a schedule of a list $P$ of jobs already placed, let $L(i)$ be the sum of $\\sigma(i)$ for each machine $i$, and let $R$ be a list of remaining jobs. Assume that $a$ times the makespan of $\\sigma$ is at most $bM$. Assume also that whenever $P$ followed by $R$ equals $P'$ followed by a job $p$ followed by\u2026","title":"Invariant bound for list scheduling"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/mul-least-loaded-le","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.mul_leastLoaded_le","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#mul-least-loaded-le","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $\\sigma$ be a schedule of a list $P$ of jobs on $m \\ge 1$ machines, with load $\\ell_i$ on machine $i$. Then $m$ times the load of a least-loaded machine is at most the total processing time $T$ of the jobs in $P$, counted with multiplicity: $m \\min_i \\ell_i \\le T$.","title":"A least-loaded machine carries at most the average load"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/list-schedule-approx","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.listSchedule_approx","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#list-schedule-approx","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $m \\ge 1$ and let $J$ be a list of processing times. Let $C$ be the makespan of the schedule that list scheduling on $m$ machines produces from $J$, taking the jobs in the order of $J$, and let $M$ be the makespan of any schedule of $J$ on $m$ machines. Then $mC \\le (2m - 1)M$, that is, $C \\le (2 - 1/m)M$.","title":"Graham's bound for list scheduling"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/two-mul-le-of-loads-gt","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.two_mul_le_of_loads_gt","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#two-mul-le-of-loads-gt","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $\\sigma$ be a schedule of a list $P$ of jobs on $m$ machines, let $p$ and $M$ be natural numbers, and let $d = \\max(M - p, 0)$. If every job in $P$ is at least $p$ and every machine has load (the sum of its jobs) greater than $d$, then $2m \\le n + k$, where $n$ is the number of jobs in $P$ and $k$ is the number of jobs $x$ in $P$ with $p \\le x$ and $x > d$, both counted with multiplicity.","title":"Counting jobs when every machine is heavily loaded"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/count-p-add-count-p-le","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.countP_add_countP_le","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#count-p-add-count-p-le","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $p$ and $M$ be natural numbers with $M < 3p$, and let $\\ell$ be a list of natural numbers (the jobs of one machine) whose sum is at most $M$. Then the number of entries of $\\ell$ that are at least $p$, plus the number of entries that are at least $p$ and greater than $M - p$, is at most $2$. Here $M - p$ is truncated subtraction, equal to $0$ when $M < p$. So $\\ell$ has at most two entries that are at least\u2026","title":"At most two long jobs per machine"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/lpt-step-le","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.lpt_step_le","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#lpt-step-le","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $m \\ge 1$ and let a list of job processing times, in nonincreasing order, be the concatenation of a list $P$, a job $p$ and a list $R$. Let $\\sigma$ be a schedule of the jobs $P$ on $m$ machines, with load $\\ell_i$ on machine $i$, and let $L = \\min_i \\ell_i$ be the load of a least-loaded machine. Then for every schedule $\\tau$ of all the jobs on the $m$ machines, with makespan $M$, we have\u2026","title":"The LPT placement step"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/lpt-approx","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.lpt_approx","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#lpt-approx","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"Let $m \\ge 1$ and let $J$ be a list of natural numbers (job processing times). Let $C$ be the makespan of the schedule of $J$ on $m$ machines produced by the LPT algorithm, and let $M$ be the makespan of any schedule $\\tau$ of $J$ on $m$ machines. Then $3mC \\le (4m - 1)M$, that is, $C \\le (4/3 - 1/(3m))M$.","title":"Graham's bound for LPT scheduling"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/list-schedule-is-schedule-of","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.listSchedule_isScheduleOf","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#list-schedule-is-schedule-of","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"For $m \\ge 1$ and every list $J$ of processing times, the assignment that list scheduling on $m$ machines produces from $J$ is a schedule of $J$.","title":"List scheduling schedules every job"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/list-schedule-time","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.listSchedule_time","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#list-schedule-time","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"For $m \\ge 1$ and every list of $n$ jobs, list scheduling on $m$ machines has cost exactly $nm$, in the cost model that charges $m$ for each job placed (one unit per machine load inspected).","title":"Running time of list scheduling"},{"id":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/lpt-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Scheduling.lpt_time_le","parent":42192,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines/index.html#lpt-time-le","scope":"ComputerScience/approximation-algorithms/list-scheduling-on-identical-machines","status":"fully_proved","summary":"On $m \\ge 1$ machines and a list of $n$ jobs, LPT costs at most $n \\lceil \\log_2 n \\rceil + nm$, counting merge-sort comparisons and machine load inspections.","title":"Running time of LPT"},{"data_url":"scopes/ComputerScience/approximation-algorithms/maximum-n-times-coverage.json","id":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","kind":"scope","navigation":"module","parent":42123,"result":false,"route":"../roadmap/ComputerScience/approximation-algorithms/maximum-n-times-coverage/index.html","scope":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","status":"fully_proved","title":"Maximum n-times coverage"},{"id":"ComputerScience/approximation-algorithms/maximum-n-times-coverage/cover-count","kind":"node","lean":"Cslib.Algorithms.Lean.MaxNTimesCoverage.coverCount","parent":42213,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/maximum-n-times-coverage/index.html#cover-count","scope":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","status":"fully_proved","summary":"The number of times the overlays `O` cover the element `i` (equation (1)).","title":"Cover count of an element under a family of overlays"},{"id":"ComputerScience/approximation-algorithms/maximum-n-times-coverage/n-coverage","kind":"node","lean":"Cslib.Algorithms.Lean.MaxNTimesCoverage.nCoverage","parent":42213,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/maximum-n-times-coverage/index.html#n-coverage","scope":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","status":"fully_proved","summary":"The `n`-times coverage function: the total weight of the elements covered at least `n` times by `O` (equation (2)).","title":"The n-times coverage function"},{"id":"ComputerScience/approximation-algorithms/maximum-n-times-coverage/is-submodular","kind":"node","lean":"Cslib.Algorithms.Lean.MaxNTimesCoverage.IsSubmodular","parent":42213,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/maximum-n-times-coverage/index.html#is-submodular","scope":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","status":"fully_proved","summary":"A set function is submodular if adding an element to a larger set gains at most as much as adding it to a smaller set (the diminishing-returns form, Definition 2 of the paper).","title":"Submodular set function"},{"id":"ComputerScience/approximation-algorithms/maximum-n-times-coverage/cover-count-mono","kind":"node","lean":"Cslib.Algorithms.Lean.MaxNTimesCoverage.coverCount_mono","parent":42213,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/maximum-n-times-coverage/index.html#cover-count-mono","scope":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","status":"fully_proved","summary":"More overlays cover every element at least as often.","title":"Monotonicity of the cover count in the overlays"},{"id":"ComputerScience/approximation-algorithms/maximum-n-times-coverage/n-coverage-mono","kind":"node","lean":"Cslib.Algorithms.Lean.MaxNTimesCoverage.nCoverage_mono","parent":42213,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/maximum-n-times-coverage/index.html#n-coverage-mono","scope":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","status":"fully_proved","summary":"For nonnegative weights the `n`-times coverage function is monotone.","title":"Monotonicity of n-times coverage for nonnegative weights"},{"id":"ComputerScience/approximation-algorithms/maximum-n-times-coverage/not-is-submodular-n-coverage","kind":"node","lean":"Cslib.Algorithms.Lean.MaxNTimesCoverage.not_isSubmodular_nCoverage","parent":42213,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/maximum-n-times-coverage/index.html#not-is-submodular-n-coverage","scope":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","status":"fully_proved","summary":"The `n`-times coverage function is not submodular (Liu et al., Theorem 1): for every `n \u2265 2`, take `n` overlays that each cover a single element of weight `1` once. Adding the last overlay to the other `n - 1` gains `1`; adding it to `\u2205` gains `0`.","title":"The `n`-times coverage function is not submodular"},{"id":"ComputerScience/approximation-algorithms/maximum-n-times-coverage/is-submodular-n-coverage-one","kind":"node","lean":"Cslib.Algorithms.Lean.MaxNTimesCoverage.isSubmodular_nCoverage_one","parent":42213,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/maximum-n-times-coverage/index.html#is-submodular-n-coverage-one","scope":"ComputerScience/approximation-algorithms/maximum-n-times-coverage","status":"fully_proved","summary":"For `n = 1` (weighted maximum coverage) and nonnegative weights, the objective is submodular.","title":"Weighted maximum coverage is submodular"},{"data_url":"scopes/ComputerScience/approximation-algorithms/vertex-cover-2-approximation.json","id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","kind":"scope","navigation":"module","parent":42123,"result":false,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","title":"Vertex cover 2-approximation"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/vertex-cover-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.vertexCoverScan","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#vertex-cover-scan","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Given a finite set $C$ of vertices and a list of pairs of vertices, the scan processes the pairs in order, starting from the cover $C$ and charging one time unit per pair. A pair $(u, v)$ is covered when $u$ or $v$ is already in the current cover; a covered pair is skipped, and an uncovered pair adds both $u$ and $v$ to the cover. The scan returns the cover reached after the last pair.","title":"Vertex cover scan"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/approx-vertex-cover","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.approxVertexCover","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#approx-vertex-cover","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Given a list $L$ of ordered pairs of vertices, APPROX-VERTEX-COVER starts from the empty cover and goes through $L$ in order: a pair $(u, v)$ with $u$ or $v$ already in the cover is skipped, and otherwise both $u$ and $v$ are added to the cover. It returns the final cover, a finite set of vertices, at a cost of one unit per pair of $L$.","title":"APPROX-VERTEX-COVER"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/vertex-cover-scan-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.vertexCoverScan_time","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#vertex-cover-scan-time","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"For every finite set $C$ of vertices and every list $L$ of pairs of vertices, the running time of the vertex cover scan starting from $C$ on $L$ is the length of $L$.","title":"Running time of the vertex cover scan"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/approx-vertex-cover-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.approxVertexCover_time","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#approx-vertex-cover-time","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"For every list $L$ of ordered pairs of vertices, the cost of APPROX-VERTEX-COVER on $L$ is the length of $L$: one unit per listed pair.","title":"Running time of APPROX-VERTEX-COVER"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/subset-vertex-cover-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.subset_vertexCoverScan","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#subset-vertex-cover-scan","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"For every finite set $C$ of vertices and every list of pairs of vertices, the cover returned by the vertex cover scan starting from $C$ on that list contains $C$.","title":"The vertex cover scan only adds vertices"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/approx-vertex-cover-covers","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.approxVertexCover_covers","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#approx-vertex-cover-covers","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"For every list $L$ of ordered pairs of vertices and every pair $(u, v)$ in $L$, the cover that APPROX-VERTEX-COVER returns on $L$ contains $u$ or $v$.","title":"APPROX-VERTEX-COVER covers every listed pair"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/approx-vertex-cover-is-vertex-cover","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.approxVertexCover_isVertexCover","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#approx-vertex-cover-is-vertex-cover","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Let $G$ be a simple graph and $L$ a list of ordered pairs of vertices such that for every edge $uv$ of $G$, $(u, v)$ or $(v, u)$ is in $L$. Then the cover that APPROX-VERTEX-COVER returns on $L$ is a vertex cover of $G$.","title":"APPROX-VERTEX-COVER returns a vertex cover"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/card-vertex-cover-scan-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.card_vertexCoverScan_le","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#card-vertex-cover-scan-le","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Let $G$ be a simple graph, $S$ a vertex cover of $G$, $C_0$ a finite set of vertices with $|C_0| \\le 2|C_0 \\cap S|$, and $L$ a list of ordered pairs of vertices such that $u$ and $v$ are adjacent in $G$ for every $(u, v)$ in $L$. Then the cover $C$ that the vertex cover scan returns on $L$ starting from $C_0$ satisfies $|C| \\le 2|C \\cap S|$.","title":"Counting invariant of the vertex cover scan"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/card-approx-vertex-cover-le-two-mul","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.card_approxVertexCover_le_two_mul","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#card-approx-vertex-cover-le-two-mul","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Let $G$ be a simple graph and $L$ a list of ordered pairs of vertices such that $u$ and $v$ are adjacent in $G$ for every $(u, v)$ in $L$. Then for every vertex cover $S$ of $G$, finite or infinite, the cover $C$ that APPROX-VERTEX-COVER returns on $L$ satisfies $|C| \\le 2|S|$, with sizes in $\\mathbb{N} \\cup \\{\\infty\\}$.","title":"APPROX-VERTEX-COVER is at most twice any vertex cover"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/approx-vertex-cover-two-approximation","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.approxVertexCover_two_approximation","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#approx-vertex-cover-two-approximation","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Let $G$ be a simple graph and $L$ a list of ordered pairs of vertices that lists exactly the edges of $G$, each in at least one orientation: $u$ and $v$ are adjacent in $G$ for every $(u, v)$ in $L$, and for every edge $uv$ of $G$, $(u, v)$ or $(v, u)$ is in $L$. Then the cover $C$ that APPROX-VERTEX-COVER returns on $L$ is a vertex cover of $G$ and $|C| \\le 2\\tau(G)$, where $\\tau(G) \\in \\mathbb{N} \\cup \\{\\infty\\}$\u2026","title":"APPROX-VERTEX-COVER is a 2-approximation"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/vertex-disjoint","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.VertexDisjoint","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#vertex-disjoint","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Two pairs $(a, b)$ and $(c, d)$ of vertices are vertex-disjoint when they share no endpoint: $a \\ne c$, $a \\ne d$, $b \\ne c$ and $b \\ne d$.","title":"Vertex-disjoint pairs"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/exists-matching-vertex-cover-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.exists_matching_vertexCoverScan","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#exists-matching-vertex-cover-scan","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Let $G$ be a simple graph, $C_0$ a finite set of vertices, and $L$ a list of ordered pairs of vertices such that $u$ and $v$ are adjacent in $G$ for every $(u, v)$ in $L$, and let $C$ be the cover that the vertex cover scan returns on $L$ starting from $C_0$. Then there is a sublist $A$ of $L$ such that neither endpoint of any pair in $A$ lies in $C_0$, pairs at different positions of $A$ are vertex-disjoint, a\u2026","title":"Matching invariant of the vertex cover scan"},{"id":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation/approx-vertex-cover-matching","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.approxVertexCover_matching","parent":42221,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/vertex-cover-2-approximation/index.html#approx-vertex-cover-matching","scope":"ComputerScience/approximation-algorithms/vertex-cover-2-approximation","status":"fully_proved","summary":"Let $G$ be a simple graph and $L$ a list of ordered pairs of vertices such that $u$ and $v$ are adjacent in $G$ for every $(u, v)$ in $L$, and let $C$ be the cover that APPROX-VERTEX-COVER returns on $L$. Then there is a sublist $A$ of $L$ such that pairs at different positions of $A$ are vertex-disjoint, every pair in $L$ shares an endpoint with some pair in $A$, a vertex lies in $C$ exactly when it is an endpoint\u2026","title":"The edges APPROX-VERTEX-COVER takes form a maximal matching of the listed pairs"},{"data_url":"scopes/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method.json","id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","kind":"scope","navigation":"module","parent":42123,"result":false,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","title":"Weighted vertex cover by the pricing method"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/pricing-scan","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.pricingScan","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#pricing-scan","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Given rational residual weights $r$ on the vertices, a rational total price $P$ and a list of edges (ordered pairs of vertices), the pricing scan processes the edges in order: an edge $(u, v)$ has price $\\delta = \\min(r(u), r(v))$, which is subtracted from the residual weight of $u$ and of $v$ (once if $u = v$) and added to the total price. The result is the final pair of residual weights and total price, and the\u2026","title":"Pricing scan"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/pricing-vertex-cover","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.pricingVertexCover","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#pricing-vertex-cover","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Given rational weights $w$ on a finite vertex set $V$ and a list of edges (ordered pairs of vertices), run the pricing scan on the list starting from residual weights $w$ and total price $0$, then return the set of vertices whose final residual weight is $0$. On top of the cost of the scan, the selection costs $|V|$ time units.","title":"Pricing method for weighted vertex cover"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/pricing-scan-residual","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.pricingScan_residual","parent":42235,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pricing scan: residual weights stay between "],["math","\\(0\\)"],["text"," and their start"]],"source":"Pricing scan: residual weights stay between $0$ and their start"},"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#pricing-scan-residual","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Let $r$ be rational residual weights with $r(v) \\ge 0$ for every vertex $v$, let $P$ be a rational number and let $L$ be a list of edges. If the pricing scan of $L$ from residual weights $r$ and total price $P$ ends with residual weights $r'$, then $0 \\le r'(v) \\le r(v)$ for every vertex $v$.","title":"Pricing scan: residual weights stay between $0$ and their start"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/pricing-scan-covers","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.pricingScan_covers","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#pricing-scan-covers","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Let $r$ be rational residual weights with $r(v) \\ge 0$ for every vertex $v$, let $P$ be a rational number and let $L$ be a list of edges (ordered pairs of vertices). If the pricing scan of $L$ from residual weights $r$ and total price $P$ ends with residual weights $r'$, then every pair $(u, v)$ in $L$ has $r'(u) = 0$ or $r'(v) = 0$.","title":"Pricing scan: every edge has an exhausted endpoint"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/sum-drop-le-two-mul-price","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.sum_drop_le_two_mul_price","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#sum-drop-le-two-mul-price","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Let $V$ be a finite vertex set, let $r$ be rational residual weights with $r(v) \\ge 0$ for every vertex $v$, let $P$ be a rational number and let $L$ be a list of edges. If the pricing scan of $L$ from residual weights $r$ and total price $P$ ends with residual weights $r'$ and total price $P'$, then $\\sum_{v \\in V} (r(v) - r'(v)) \\le 2 (P' - P)$.","title":"Pricing scan: the vertices pay at most twice the price"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/price-le-sum-drop","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.price_le_sum_drop","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#price-le-sum-drop","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Let $C$ be a finite set of vertices, let $r$ be rational residual weights with $r(v) \\ge 0$ for every vertex $v$, let $P$ be a rational number, and let $L$ be a list of edges (ordered pairs of vertices) such that every pair $(u, v)$ in $L$ has $u \\in C$ or $v \\in C$. If the pricing scan of $L$ from residual weights $r$ and total price $P$ ends with residual weights $r'$ and total price $P'$, then\u2026","title":"Pricing scan: the total price is at most what a cover pays"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/pricing-vertex-cover-covers","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.pricingVertexCover_covers","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#pricing-vertex-cover-covers","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Let $w$ be rational weights on a finite vertex set with $w(v) \\ge 0$ for every vertex $v$, and let $L$ be a list of edges (ordered pairs of vertices). For every pair $(u, v)$ in $L$, $u$ or $v$ belongs to the set returned by the pricing method on $w$ and $L$.","title":"The pricing method covers every listed edge"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/pricing-vertex-cover-is-vertex-cover","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.pricingVertexCover_isVertexCover","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#pricing-vertex-cover-is-vertex-cover","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Let $G$ be a simple graph on a finite vertex set, let $w$ be rational vertex weights with $w(v) \\ge 0$ for every vertex $v$, and let $L$ be a list of ordered pairs of vertices such that for every edge $uv$ of $G$, the pair $(u, v)$ or the pair $(v, u)$ is in $L$. Then the set returned by the pricing method on $w$ and $L$ is a vertex cover of $G$.","title":"The pricing method returns a vertex cover"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/pricing-vertex-cover-weight-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.pricingVertexCover_weight_le","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#pricing-vertex-cover-weight-le","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"Let $w$ be rational weights on a finite vertex set with $w(v) \\ge 0$ for every vertex $v$, let $L$ be a list of edges (ordered pairs of vertices), and let $C$ be a set of vertices such that every pair $(u, v)$ in $L$ has $u \\in C$ or $v \\in C$. Then the total weight of the set returned by the pricing method on $w$ and $L$ is at most $2 \\sum_{v \\in C} w(v)$.","title":"The pricing method weighs at most twice any cover of the listed edges"},{"id":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/pricing-vertex-cover-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.pricingVertexCover_time","parent":42235,"result":true,"route":"../roadmap/ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method/index.html#pricing-vertex-cover-time","scope":"ComputerScience/approximation-algorithms/weighted-vertex-cover-by-the-pricing-method","status":"fully_proved","summary":"For all rational vertex weights $w$ on a finite vertex set $V$ and every list $L$ of edges, the pricing method on $w$ and $L$ takes exactly $|L| + |V|$ time units, where $|L|$ is the length of the list: one unit per listed edge and one per vertex.","title":"Running time of the pricing method"},{"data_url":"scopes/ComputerScience/online-algorithms.json","id":"ComputerScience/online-algorithms","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/online-algorithms/index.html","scope":"ComputerScience/online-algorithms","status":"fully_proved","summary":"Algorithms that act on a request sequence without knowing the future, and competitive-ratio upper and lower bounds: paging, list update, ski rental, online bipartite matching.","title":"Online algorithms"},{"data_url":"scopes/ComputerScience/online-algorithms/competitive-paging.json","id":"ComputerScience/online-algorithms/competitive-paging","kind":"scope","navigation":"module","parent":42246,"result":false,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","title":"Competitive paging"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-step","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruStep","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-step","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For a capacity $k$, a cache $c$ (a list of pages, most recently used first) and a requested page $p$, the LRU step removes the first occurrence of $p$ from $c$ (if any), puts $p$ at the front, and keeps the first $k$ pages of the result. On a miss with a cache of exactly $k \\ge 1$ pages, this evicts the last page, the least recently used one.","title":"One LRU step"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-cache","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruCache","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-cache","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For a capacity $k$, a cache $c$ (a list of pages, most recently used first) and a request sequence $\\sigma$, this is the cache obtained from $c$ by applying one LRU step with capacity $k$ for each request of $\\sigma$, in order.","title":"Cache after LRU serves a request sequence"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-misses","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruMisses","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-misses","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For a capacity $k$, a cache $c$ and a request sequence $\\sigma$, this is the number of requests of $\\sigma$ that are not in the current cache when they are served, where the cache starts at $c$ and after each request is updated by one LRU step with capacity $k$.","title":"Number of LRU misses"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-step-take-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruStep_take_of_mem","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-step-take-of-mem","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k$ be a capacity, $c$ a cache (most recently used first) and $S$ a list of at most $k$ pages equal to the first $|S|$ pages of $c$. If $p \\in S$, then after one LRU step with capacity $k$ serving $p$, the first $|S|$ pages of the cache are $p$ followed by $S$ with the first occurrence of $p$ removed.","title":"The recency invariant for a repeated page"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-step-take-of-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruStep_take_of_not_mem","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-step-take-of-not-mem","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k$ be a capacity, $c$ a cache (most recently used first) and $S$ a list with $|S| + 1 \\le k$ equal to the first $|S|$ pages of $c$. If $p \\notin S$, then after one LRU step with capacity $k$ serving $p$, the first $|S| + 1$ pages of the cache are $p$ followed by $S$.","title":"The recency invariant for a new page"},{"id":"ComputerScience/online-algorithms/competitive-paging/phase-split","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.phaseSplit","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#phase-split","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Given a capacity $k$ and a list $s$ of pages (the pages already in the current phase, most recent first), a request sequence $\\sigma$ is split into a phase and a remainder as follows. The empty sequence splits into two empty parts. For a sequence $p$ followed by $\\sigma'$: if $p$ is in $s$, then $p$ joins the phase and $\\sigma'$ is split with $s$ replaced by $p$ followed by $s$ with its first occurrence of $p$\u2026","title":"Splitting off a phase"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-misses-phase","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruMisses_phase","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","LRU misses at most "],["math","\\(k - |S|\\)"],["text"," times in the rest of a phase"]],"source":"LRU misses at most $k - |S|$ times in the rest of a phase"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-misses-phase","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k$ be a capacity, $c$ a cache (most recently used first) and $S$ a duplicate-free list of at most $k$ pages equal to the first $|S|$ pages of $c$. For every request sequence $\\sigma$, let $\\pi$ be the phase that the phase split cuts from $\\sigma$ with $S$ as the pages already in the phase: the longest prefix of $\\sigma$ whose distinct pages, together with those of $S$, number at most $k$. Then the number of\u2026","title":"LRU misses at most $k - |S|$ times in the rest of a phase"},{"id":"ComputerScience/online-algorithms/competitive-paging/phase-split-snd-cons","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.phaseSplit_snd_cons","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","A phase ends only on a "],["math","\\((k + 1)\\)"],["text","-st distinct page"]],"source":"A phase ends only on a $(k + 1)$-st distinct page"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#phase-split-snd-cons","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k$ be a capacity, let $s$ be a list of pages without repetitions with at most $k$ pages (the pages already in the current phase), and let $\\sigma$ be a request sequence. If the remainder left after splitting off a phase from $\\sigma$ given $s$ starts with a page $q$, then there is a list $S$ of exactly $k$ pairwise distinct pages, none equal to $q$, each of which lies in $s$ or in that phase.","title":"A phase ends only on a $(k + 1)$-st distinct page"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-event","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruEvent","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-event","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For a capacity $k$, a cache $c$ (a list of pages, most recently used first) and a requested page $p$, the LRU event is a hit on $p$ if $p \\in c$. Otherwise it is a miss on $p$ that evicts the last page of $c$ (the least recently used one) if $c$ is nonempty, $k > 0$ and $c$ has at least $k$ pages, and a miss on $p$ without eviction in all other cases.","title":"The event of an LRU step"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-trace","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruTrace","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-trace","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Given a capacity $k$, a cache $c$ and a request sequence $\\sigma$, the events LRU emits on $\\sigma$ from $c$: nothing on the empty sequence, and on a sequence $p$ followed by $\\sigma'$, the event LRU emits when serving $p$ from $c$, followed by the events LRU emits on $\\sigma'$ from the cache after one LRU step with capacity $k$ serving $p$ from $c$.","title":"The LRU trace"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-tr","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lru_tr","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-tr","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k$ be a capacity and $c$ a duplicate-free cache with at most $k$ pages. For every page $p$, the cache system with capacity $k$ has a transition from $c$, labelled by the LRU event for $p$ at $c$, to the cache produced by one LRU step with capacity $k$ serving $p$.","title":"Each LRU step is a legal transition"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-trace-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruTrace_spec","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-trace-spec","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k$ be a capacity, $c$ a cache given as a list of pages without repetitions with at most $k$ pages, and $\\sigma$ a request sequence. Then the sequence of events that LRU emits when serving $\\sigma$ from $c$ is a run of the cache system with capacity $k$ from $c$ to the cache LRU holds after serving $\\sigma$; the pages requested by these events are exactly $\\sigma$, in order; and the number of misses among these\u2026","title":"LRU is a legal cache manager"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-misses-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruMisses_le","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","LRU is "],["math","\\(k\\)"],["text","-competitive"]],"source":"LRU is $k$-competitive"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-misses-le","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$ and let $c$ be a duplicate-free cache with at most $k$ pages. Let $e$ be a legal trace of the cache system with capacity $k$ that starts from a duplicate-free cache with at most $k$ pages, and let $m$ be its number of misses. Then the trace that LRU with capacity $k$ emits when serving, from $c$, the requests of $e$ in order has at most $k m + k$ misses.","title":"LRU is $k$-competitive"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-misses-le-furthest-in-future","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruMisses_le_furthestInFuture","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","LRU is "],["math","\\(k\\)"],["text","-competitive against the offline optimum"]],"source":"LRU is $k$-competitive against the offline optimum"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-misses-le-furthest-in-future","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$, and let $c$ and $c_0$ be duplicate-free caches with at most $k$ pages each. For every request sequence $\\sigma$, the trace that LRU with capacity $k$ emits when serving $\\sigma$ from $c$ has at most $k m + k$ misses, where $m$ is the number of misses of the furthest-in-future manager with capacity $k$ serving $\\sigma$ from $c_0$.","title":"LRU is $k$-competitive against the offline optimum"},{"id":"ComputerScience/online-algorithms/competitive-paging/deterministic-online-paging-algorithm","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.OnlinePager","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#deterministic-online-paging-algorithm","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"A deterministic online paging algorithm with cache size `k`: a state machine whose cache never holds more than `k` pages. Its moves need not be legal transitions; the lower bound holds for any such machine.","title":"Deterministic online paging algorithm"},{"id":"ComputerScience/online-algorithms/competitive-paging/misses","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.OnlinePager.misses","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#misses","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For an online pager $A$, a state $q$ and a request sequence $\\sigma$, the number of misses of $A$ from $q$ on $\\sigma$ counts the requests of $\\sigma$ that are not in the cache of the current state, where the run starts at $q$ and each request moves $A$ to its next state.","title":"Misses of an online pager"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-as-an-online-pager","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruPager","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-as-an-online-pager","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"LRU as an online pager.","title":"LRU as an online pager"},{"id":"ComputerScience/online-algorithms/competitive-paging/lru-pager-misses","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.lruPager_misses","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#lru-pager-misses","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For every cache $c$ of at most $k$ pages and every request sequence $\\sigma$, the number of misses of LRU with capacity $k$, viewed as an online pager and started in the state with cache $c$, on $\\sigma$ equals the number of misses of LRU with capacity $k$ started with cache $c$ on $\\sigma$.","title":"LRU as an online pager has LRU's miss count"},{"id":"ComputerScience/online-algorithms/competitive-paging/outside","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.outside","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#outside","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For a natural number $k$ and a list $c$ of pages from $\\{0, \\dots, k\\}$, the outside page of $c$ is the smallest $x \\in \\{0, \\dots, k\\}$ that does not occur in $c$, or $0$ if every page of $\\{0, \\dots, k\\}$ occurs in $c$.","title":"Outside page of a cache"},{"id":"ComputerScience/online-algorithms/competitive-paging/outside-not-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.outside_not_mem","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#outside-not-mem","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k$ be a natural number and let $c$ be a list of at most $k$ pages from $\\{0, \\dots, k\\}$. Then the outside page of $c$ does not occur in $c$.","title":"The outside page is not cached"},{"id":"ComputerScience/online-algorithms/competitive-paging/adversary","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.adversary","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#adversary","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $A$ be a deterministic online pager with cache size $k$ on the $k + 1$ pages $0, \\dots, k$. From a state $q$ of $A$, the adversary's sequence of $n$ requests is empty for $n = 0$; for $n + 1$ it first requests the smallest page not in the cache of $q$ (page $0$ if there is none) and then continues with the adversary's sequence of $n$ requests from the state that $A$ reaches from $q$ by serving that request.","title":"The paging adversary"},{"id":"ComputerScience/online-algorithms/competitive-paging/misses-adversary","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.misses_adversary","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#misses-adversary","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"The pager misses every request of the adversary's sequence.","title":"The pager misses every request"},{"id":"ComputerScience/online-algorithms/competitive-paging/opt-step","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.optStep","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#opt-step","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Given a cache size $k$, a cache $c$ (a list of pages), a requested page $p$ and the list $f$ of future requests, the offline step returns an event and a new cache. If $p \\in c$, it is a hit on $p$ and the cache is unchanged. Otherwise, if $c$ has fewer than $k$ pages, it is a miss on $p$ with no eviction and the new cache is $p$ followed by $c$. Otherwise, if some page of $c$ does not occur among the first $k - 1$\u2026","title":"Offline paging step"},{"id":"ComputerScience/online-algorithms/competitive-paging/opt-trace","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.optTrace","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#opt-trace","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Given a cache size $k$, the offline trace from a cache $c$ on a list of requests is defined by recursion on the list: on the empty list it is empty, and on a list with first request $p$ and remaining requests $r$ it is the event of the offline step with cache size $k$ on $c$, $p$ and $r$, followed by the offline trace on $r$ from the new cache of that step.","title":"Offline paging trace"},{"id":"ComputerScience/online-algorithms/competitive-paging/find-is-some","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.find?_isSome","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","A full cache has a page outside the next "],["math","\\(k - 1\\)"],["text"," requests"]],"source":"A full cache has a page outside the next $k - 1$ requests"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#find-is-some","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$, let $c$ be a list of $k$ distinct pages and let $f$ be any list of requests. Then the search through $c$ for the first page that does not occur among the first $k - 1$ entries of $f$ succeeds; in particular, some page of $c$ does not occur there.","title":"A full cache has a page outside the next $k - 1$ requests"},{"id":"ComputerScience/online-algorithms/competitive-paging/opt-step-tr","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.optStep_tr","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#opt-step-tr","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$, let $c$ be a cache of at most $k$ distinct pages, let $p$ be a requested page and let $f$ be the list of future requests. Then the event and the new cache produced by the offline step with cache size $k$ on $c$, $p$ and $f$ form a legal transition from $c$ of the caching system with capacity $k$.","title":"Each offline step is a legal transition"},{"id":"ComputerScience/online-algorithms/competitive-paging/opt-trace-legal","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.optTrace_legal","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#opt-trace-legal","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$, let $c$ be a cache of at most $k$ distinct pages and let $\\sigma$ be a list of requests. Then the offline trace with cache size $k$ from $c$ on $\\sigma$ is a legal run of the caching system with capacity $k$ from $c$ to some final cache, and the pages served by its events are, in order, exactly $\\sigma$.","title":"The offline trace is legal"},{"id":"ComputerScience/online-algorithms/competitive-paging/misses-opt-trace-hits","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.misses_optTrace_hits","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#misses-opt-trace-hits","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k$ be a capacity, let $c$ be a cache and let $w$ be a request sequence all of whose requests lie in $c$. Then for every request sequence $\\tau$, the offline strategy with capacity $k$ (on a miss with a full cache, evict the first cached page not among the next $k - 1$ requests, if there is one) started with cache $c$ misses as many times on $w$ followed by $\\tau$ as on $\\tau$ alone.","title":"Hits do not change the offline strategy's misses"},{"id":"ComputerScience/online-algorithms/competitive-paging/mem-of-full","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.mem_of_full","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#mem-of-full","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $c$ be a list of $k$ distinct pages among the $k + 1$ pages $0, \\dots, k$, let $p$ be a page not in $c$, and let $v$ be any page. Then every page $x \\neq v$ lies in the cache obtained from $c$ by removing $v$ and adding $p$.","title":"After a miss on a full cache, every page except the evicted one is present"},{"id":"ComputerScience/online-algorithms/competitive-paging/misses-opt-trace-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.misses_optTrace_le","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","The offline trace misses at most once every "],["math","\\(k\\)"],["text"," requests"]],"source":"The offline trace misses at most once every $k$ requests"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#misses-opt-trace-le","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$ and consider the $k + 1$ pages $0, 1, \\dots, k$. Let $N$ be a natural number, $\\sigma$ a sequence of $n \\le N$ requests to these pages, and $c$ a cache of $|c| \\le k$ distinct pages. Then the number of misses of the offline trace with cache size $k$ from $c$ on $\\sigma$ is at most $(k - |c|) + \\lfloor (n + k - 1) / k \\rfloor = (k - |c|) + \\lceil n / k \\rceil$: after at most $k - |c|$ misses filling\u2026","title":"The offline trace misses at most once every $k$ requests"},{"id":"ComputerScience/online-algorithms/competitive-paging/adversary-vs-optimum","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.adversary_vs_optimum","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","The adversary forces "],["math","\\(n\\)"],["text"," misses while the optimum misses at most "],["math","\\(k + \\lceil n / k \\rceil\\)"],["text"," times"]],"source":"The adversary forces $n$ misses while the optimum misses at most $k + \\lceil n / k \\rceil$ times"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#adversary-vs-optimum","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"The adversary forces $n$ misses while the optimum misses at most $k + \\lceil n / k \\rceil$ times. Let $k \\ge 1$ and let $A$ be a deterministic online pager with cache size $k$ on the $k + 1$ pages $0, \\dots, k$. For every $n$, on the adversary's sequence of $n$ requests started from the initial state of $A$, the pager $A$ started in its initial state misses exactly $n$ times, and the furthest-in-future strategy\u2026","title":"The adversary forces $n$ misses while the optimum misses at most $k + \\lceil n / k \\rceil$ times"},{"id":"ComputerScience/online-algorithms/competitive-paging/not-competitive-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.Online.Paging.not_competitive_of_lt","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","No deterministic online pager is better than "],["math","\\(k\\)"],["text","-competitive"]],"source":"No deterministic online pager is better than $k$-competitive"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#not-competitive-of-lt","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$ and let $A$ be a deterministic online pager with cache size $k$ on the $k + 1$ pages $0, \\dots, k$. For a request sequence $\\sigma$, write $A(\\sigma)$ for the number of misses of $A$ started in its initial state on $\\sigma$, and $\\mathrm{OPT}(\\sigma)$ for the number of misses of the furthest-in-future strategy with capacity $k$ started with an empty cache on $\\sigma$. Then for all natural numbers $c$,\u2026","title":"No deterministic online pager is better than $k$-competitive"},{"id":"ComputerScience/online-algorithms/competitive-paging/policy-misses","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.policyMisses","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#policy-misses","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let a paging policy be given by its step function, which maps a cache (a list of pages) and a requested page to the next cache. Starting from a cache $c$, the policy serves a request sequence $p_1, \\dots, p_n$ by setting $c_0 = c$ and $c_i = \\mathrm{step}(c_{i-1}, p_i)$ for each $i$; its number of misses is the number of indices $i$ with $p_i \\notin c_{i-1}$.","title":"Misses of a paging policy on list caches"},{"id":"ComputerScience/online-algorithms/competitive-paging/policy-misses-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.policyMisses_le","parent":42247,"reading_title":{"format":"inline-parts-v1","parts":[["text","A policy that misses at most "],["math","\\(k\\)"],["text"," times per phase is "],["math","\\(k\\)"],["text","-competitive"]],"source":"A policy that misses at most $k$ times per phase is $k$-competitive"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#policy-misses-le","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$ and consider a paging policy on list caches, given by its step function. Suppose that, started from any cache, the policy misses at most $k$ times on the first phase of every request sequence (the longest prefix of the sequence with at most $k$ distinct pages). Let $T$ be a legal trace of caching with capacity $k$ (a run of the cache transition system) whose initial cache is a list of at most $k$\u2026","title":"A policy that misses at most $k$ times per phase is $k$-competitive"},{"id":"ComputerScience/online-algorithms/competitive-paging/policy-misses-lru-step","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.policyMisses_lruStep","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#policy-misses-lru-step","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For every capacity $k$, the misses of the paging policy whose step is the LRU step with capacity $k$ coincide with the LRU misses with capacity $k$, from every starting cache and on every request sequence.","title":"LRU as a policy on list caches"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-step","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifoStep","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-step","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"The cache is a list of pages, oldest first, and $k$ is the capacity. On a request for a page $p$ already in the cache $c$, the cache stays $c$. Otherwise $p$ is appended at the back of $c$; if $c$ had fewer than $k$ pages, that is the new cache, and if not, the front page, the one loaded earliest, is removed as well.","title":"One step of FIFO paging"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-step-legal","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifoStep_legal","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-step-legal","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"If a list $c$ of pages has no duplicates and at most $k$ entries, then for every page $p$ the cache produced from $c$ by one FIFO step with capacity $k$ on request $p$ also has no duplicates and at most $k$ entries.","title":"One FIFO step keeps the cache legal"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-misses-phase","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifoMisses_phase","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-misses-phase","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $S$ be a list of at most $k$ distinct pages (the pages already in the current phase), let $c$ be any list of pages (a FIFO cache, oldest page first), and let $L$ be a list of distinct pages, each of them in $S$, that is a suffix of $c$. For every request sequence $\\sigma$, let $P$ be the phase split off from $\\sigma$ given $S$: the longest prefix of $\\sigma$ whose distinct pages, together with those of $S$,\u2026","title":"FIFO misses at most `k` times per phase"},{"id":"ComputerScience/online-algorithms/competitive-paging/legal-cache","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.LegalCache","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#legal-cache","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For a type $\\alpha$ of pages and a capacity $k \\in \\mathbb{N}$, a legal cache is a list of pages of $\\alpha$ with no duplicates and at most $k$ entries.","title":"Legal cache"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-cache-legal","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifoCache_legal","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-cache-legal","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"If a list $c$ of pages has no duplicates and at most $k$ entries, then for every finite request sequence $\\sigma$ the cache obtained from $c$ by serving the requests of $\\sigma$ in order, each with one FIFO step of capacity $k$, also has no duplicates and at most $k$ entries.","title":"FIFO keeps the cache legal on every request sequence"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-pager","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifoPager","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-pager","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"The online pager with capacity $k$ whose states are the legal caches of capacity $k$ (lists of at most $k$ distinct pages, oldest first), whose initial state is the empty cache, whose cache in a state is that list itself, and which serves a request $p$ by one FIFO step with capacity $k$.","title":"FIFO paging as an online pager"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-pager-misses","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifoPager_misses","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-pager-misses","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"For every legal cache $q$ of capacity $k$ (at most $k$ distinct pages) and every request sequence $\\sigma$, the number of misses of the FIFO online pager started in $q$ and serving $\\sigma$ equals the number of misses of the policy given by the FIFO step with capacity $k$ serving $\\sigma$ from the cache $q$.","title":"FIFO as an online pager misses as often as the FIFO step policy"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-misses-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifoMisses_le","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-misses-le","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"FIFO is `k`-competitive (Sleator and Tarjan). Let $k \\ge 1$, and let $\\tau$ be a legal trace of the caching system with capacity $k$ that starts from a cache $c_0$ of at most $k$ distinct pages; let $m$ be its number of misses. Then the FIFO online pager with capacity $k$, started in any legal cache (at most $k$ distinct pages) and serving the requests of $\\tau$ in order, misses at most $k m + k$ times.","title":"FIFO is `k`-competitive"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-misses-le-furthest-in-future","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifoMisses_le_furthestInFuture","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-misses-le-furthest-in-future","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$, let $\\sigma$ be a finite request sequence, and let $c_0$ be a list of at most $k$ distinct pages. Let $m$ be the number of misses of the furthest-in-future manager with capacity $k$ serving $\\sigma$ from the cache $c_0$. Then the FIFO online pager with capacity $k$, started in any legal cache (at most $k$ distinct pages), misses at most $k m + k$ of the requests of $\\sigma$.","title":"FIFO is `k`-competitive against the offline optimum"},{"id":"ComputerScience/online-algorithms/competitive-paging/fifo-not-competitive-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.Online.FifoPaging.fifo_not_competitive_of_lt","parent":42247,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/competitive-paging/index.html#fifo-not-competitive-of-lt","scope":"ComputerScience/online-algorithms/competitive-paging","status":"fully_proved","summary":"Let $k \\ge 1$ and let $c$ and $d$ be natural numbers with $c < k d$. For every natural number $b$ there is a finite request sequence $\\sigma$ over $k + 1$ pages with $c \\cdot m + b < d \\cdot f$, where $m$ is the number of misses of the furthest-in-future manager with capacity $k$ serving $\\sigma$ from the empty cache, and $f$ is the number of misses of the FIFO online pager with capacity $k$ serving $\\sigma$ from\u2026","title":"The ratio `k` is tight for FIFO"},{"data_url":"scopes/ComputerScience/online-algorithms/list-update-problem.json","id":"ComputerScience/online-algorithms/list-update-problem","kind":"scope","navigation":"module","parent":42246,"result":false,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","title":"List update problem"},{"id":"ComputerScience/online-algorithms/list-update-problem/access-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.accessCost","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#access-cost","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"The cost of accessing an item $x$ in a list $L$ is its position in $L$, counting from $1$: one more than the index of the first occurrence of $x$ in $L$. If $x$ does not occur in $L$, the cost is the length of $L$ plus one.","title":"Access cost"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtf","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"For a list $L$ and an item $x$, moving $x$ to the front gives the list consisting of $x$ followed by $L$ with its first occurrence of $x$ removed; if $x$ does not occur in $L$, this is $x$ followed by $L$.","title":"Move-to-front"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfCost","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-cost","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Starting from a list $L$, move-to-front serves a request sequence $\\sigma$ one request at a time: a request $x$ costs the position of $x$ in the current list, counting from $1$ (an item not in the list costs its length plus one), and then $x$ is moved to the front (inserted at the front if absent). The cost of move-to-front on $\\sigma$ from $L$ is the sum of these access costs.","title":"Cost of move-to-front"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-final","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfFinal","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-final","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Starting from a list $L$ and serving a request sequence $\\sigma$, move-to-front moves each request $x$ in turn to the front of the current list (inserting it at the front if it is absent). The final list of move-to-front on $\\sigma$ from $L$ is the list it ends with.","title":"Final list of move-to-front"},{"id":"ComputerScience/online-algorithms/list-update-problem/inv-set","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.invSet","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#inv-set","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"For lists $M$ and $L$, the inversion set is the set of pairs $(a, b)$ of items of $M$ with $a$ before $b$ in $M$ and $b$ before $a$ in $L$, that is, the pairs that $M$ and $L$ order differently. Here the position of an item in a list is the index of its first occurrence, and an item that does not occur in the list sits at the position equal to the length of the list.","title":"Inversion set of two lists"},{"id":"ComputerScience/online-algorithms/list-update-problem/inv","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.inv","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#inv","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"For lists $M$ and $L$, the number of inversions of $L$ against $M$ is the size of their inversion set: the number of pairs $(a, b)$ of items of $M$ with $a$ before $b$ in $M$ and $b$ before $a$ in $L$, where the position of an item in a list is the index of its first occurrence, or the length of the list if it does not occur.","title":"Number of inversions between two lists"},{"id":"ComputerScience/online-algorithms/list-update-problem/offline-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.offlineCost","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#offline-cost","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Given lists $L_0, L_1, L_2, \\dots$, the offline cost of a request sequence $\\sigma = x_0 x_1 \\cdots x_{m-1}$ is the sum over $t < m$ of the access cost of $x_t$ in $L_t$ (its position, counting from $1$) plus the number of inversions between $L_t$ and $L_{t+1}$. That is, the algorithm serves request $t$ with $L_t$ and then rearranges it into $L_{t+1}$, paying one for each pair of items whose relative order it\u2026","title":"Offline cost without free exchanges"},{"id":"ComputerScience/online-algorithms/list-update-problem/free-move","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.freeMove","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#free-move","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"A free exchange (Sleator and Tarjan): right after accessing $x$ in a list $L$, move $x$ forward to index $j$ (indices counted from $0$), at no charge. If the index of $x$ in $L$ is at most $j$, the list is unchanged, so $x$ never moves backward for free; otherwise the result is obtained by erasing the first occurrence of $x$ from $L$ and inserting $x$ back so that it has index $j$.","title":"Free move"},{"id":"ComputerScience/online-algorithms/list-update-problem/offline-cost-free","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.offlineCostFree","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#offline-cost-free","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Given lists $L_0, L_1, L_2, \\dots$, target indices $J_0, J_1, J_2, \\dots$ and a request sequence $\\sigma = x_0 x_1 \\cdots x_{m-1}$, let $L_t'$ be the list obtained from $L_t$ by the free exchange that moves $x_t$ forward to index $J_t$ (counting from $0$); it equals $L_t$ if $x_t$ already has index at most $J_t$. The offline cost with free exchanges of $\\sigma$ is the sum over $t < m$ of the access cost of $x_t$ in\u2026","title":"Offline cost with free exchanges"},{"id":"ComputerScience/online-algorithms/list-update-problem/idx-of-erase","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.idxOf_erase","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#idx-of-erase","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list with no repeated entries and let $a \\ne x$ be entries of $M$. After erasing $x$ from $M$, the index of $a$ is unchanged if $a$ comes before $x$ in $M$, and is one less if $a$ comes after $x$ (indices counted from $0$).","title":"Indices after erasing an entry"},{"id":"ComputerScience/online-algorithms/list-update-problem/idx-of-mtf-lt-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.idxOf_mtf_lt_iff","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#idx-of-mtf-lt-iff","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list without duplicates, $x$ an item of $M$, and let $M'$ be the list obtained from $M$ by moving $x$ to the front. For any items $a \\ne x$ and $b \\ne x$ of $M$, $a$ comes before $b$ in $M'$ if and only if $a$ comes before $b$ in $M$.","title":"Move-to-front keeps the relative order of the other items"},{"id":"ComputerScience/online-algorithms/list-update-problem/card-before","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.card_before","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#card-before","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"If $L$ is a list with no repeated entries and $x$ is an entry of $L$, then the number of entries of $L$ whose index is less than the index of $x$ equals the index of $x$ (indices counted from $0$).","title":"Counting the entries before an entry"},{"id":"ComputerScience/online-algorithms/list-update-problem/mem-of-mem-inv-set","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mem_of_mem_invSet","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mem-of-mem-inv-set","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ and $L$ be lists. If a pair $(a, b)$ belongs to the inversion set of $M$ and $L$, then $a$ and $b$ are items of $M$, $a$ comes before $b$ in $M$, and $b$ comes before $a$ in $L$. Here the position of an item in a list is the index of its first occurrence, and an item that does not occur in the list sits at the position equal to the length of the list.","title":"Members of the inversion set"},{"id":"ComputerScience/online-algorithms/list-update-problem/inv-triangle","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.inv_triangle","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#inv-triangle","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$, $L$ and $L'$ be lists with $L$ a rearrangement of $M$. Then the number of inversions of $L'$ against $M$ is at most the number of inversions of $L$ against $M$ plus the number of inversions of $L'$ against $L$.","title":"Triangle inequality for inversions"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-step","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtf_step","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-step","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list of distinct items, $L$ a rearrangement of $M$ (the offline list), and $x$ an item of $M$ at position $k$ in $M$ and position $i$ in $L$, counting from $1$. Let $\\Phi$ be the number of inversions between $M$ and $L$ (the pairs of items that the two lists order differently), and $\\Phi'$ the number of inversions between the list obtained from $M$ by moving $x$ to the front and $L$. Then\u2026","title":"The amortized cost of a move-to-front step"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-perm","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtf_perm","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-perm","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"If $x$ is an item of a list $M$, then the list obtained from $M$ by moving $x$ to the front is a rearrangement (permutation) of $M$.","title":"Move-to-front rearranges the list"},{"id":"ComputerScience/online-algorithms/list-update-problem/free-move-perm","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.freeMove_perm","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#free-move-perm","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"If $x$ is an item of a list $L$, then for every index $j$ (counting from $0$), the list obtained from $L$ by the free move of $x$ to index $j$ is a rearrangement (permutation) of $L$.","title":"A free move rearranges the list"},{"id":"ComputerScience/online-algorithms/list-update-problem/inv-free-move-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.inv_freeMove_le","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#inv-free-move-le","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $L$ be a list without duplicates, $x$ an item of $L$, and $N$ a list that is empty or starts with $x$, all of whose items occur in $L$. For every $j \\in \\mathbb{N}$, let $L'$ be the list obtained from $L$ by moving $x$ forward for free so that exactly $j$ items precede it, where $L' = L$ if at most $j$ items already precede $x$ in $L$. Then the number of inversions of $L'$ against $N$ is at most the number of\u2026","title":"Free moves never add inversions against a list headed by the moved item"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-cost-add-inv-le-free","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfCost_add_inv_le_free","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-cost-add-inv-le-free","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list without duplicates, $\\sigma$ a sequence of $m$ requests, each an item of $M$, $L_0, L_1, \\dots$ lists that are each a rearrangement of $M$, and $J_0, J_1, \\dots$ natural numbers. Accessing the item at position $i$ of a list, counting from $1$, costs $i$. Let $C$ be the total access cost of move-to-front on $\\sigma$ starting from $M$, $F$ the list it ends with, and $D$ the cost on $\\sigma$ of the\u2026","title":"Amortized bound for move-to-front over a request sequence, with free exchanges"},{"id":"ComputerScience/online-algorithms/list-update-problem/offline-cost-free-no-move","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.offlineCostFree_noMove","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#offline-cost-free-no-move","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $K$ be a natural number and let $L_0, L_1, L_2, \\dots$ be lists, each of length at most $K$. Take every free-exchange target to be $K$; this moves nothing, since every item already has index at most $K$. Then for every request sequence $\\sigma$, the offline cost with free exchanges of serving $\\sigma$ with $L_0, L_1, \\dots$ equals the offline cost without free exchanges of serving $\\sigma$ with $L_0, L_1, \\dots$.","title":"Without free moves the two offline costs agree"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-cost-add-inv-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfCost_add_inv_le","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-cost-add-inv-le","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list without duplicates, $\\sigma$ a sequence of $m$ requests, each an item of $M$, and $L_0, L_1, \\dots$ lists that are each a rearrangement of $M$. Accessing the item at position $i$ of a list, counting from $1$, costs $i$. Let $C$ be the total access cost of move-to-front on $\\sigma$ starting from $M$, $F$ the list it ends with, and $D$ the cost on $\\sigma$ of the offline algorithm without free\u2026","title":"Amortized bound for move-to-front over a request sequence"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-cost-le-free","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfCost_le_free","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-cost-le-free","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"In the model of Sleator and Tarjan (1985) with free exchanges, let $M$ be a list of distinct items and $\\sigma$ a sequence of $m$ requests, each an item of $M$. Consider an offline algorithm given by lists $L_0, L_1, \\ldots$, each a rearrangement of $M$ with $L_0 = M$, and natural numbers $J_0, J_1, \\ldots$. It serves request $t$ (counting from $0$) with the list $L_t$ and pays the position of the request in $L_t$\u2026","title":"Move-to-front is 2-competitive"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-cost-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfCost_le","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-cost-le","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list of distinct items and $\\sigma$ a sequence of $m$ requests, each an item of $M$. Consider an offline algorithm that serves request $t$ with a list $L_t$, where every $L_t$ is a rearrangement of $M$ and $L_0 = M$: it pays the position of the request in $L_t$ (counting from $1$), and then one for each pair of items whose relative order differs between $L_t$ and $L_{t+1}$. Then the access cost of\u2026","title":"Move-to-front is 2-competitive without free exchanges"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-cost-le-static","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfCost_le_static","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-cost-le-static","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list of distinct items, $L$ a rearrangement of $M$, and $\\sigma$ a sequence of $m$ requests, each an item of $M$. Then the access cost of move-to-front on $\\sigma$ starting from $M$, plus $m$, is at most twice the cost of serving $\\sigma$ with the fixed list $L$ (the sum over the requests of their positions in $L$, counting from $1$), plus the number of inversions between $M$ and $L$ (the pairs of\u2026","title":"Move-to-front against a static list"},{"id":"ComputerScience/online-algorithms/list-update-problem/deterministic-online-list-update-algorit","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.OnlineListUpdate","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#deterministic-online-list-update-algorit","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"A deterministic online list-update algorithm on the items of `M`: a state machine whose list is always a rearrangement of `M`. Its rearrangements are free, which only strengthens the lower bound.","title":"Deterministic online list-update algorithm"},{"id":"ComputerScience/online-algorithms/list-update-problem/cost","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.OnlineListUpdate.cost","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#cost","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"The cost of a deterministic online list-update algorithm $A$ on a request sequence $x_1, \\dots, x_m$ from a state $q_0$ is $c_1 + \\dots + c_m$, where $q_i$ is the state $A$ moves to from $q_{i-1}$ on request $x_i$ and $c_i$ is the access cost of $x_i$ in the list of state $q_{i-1}$. The empty sequence costs $0$.","title":"Cost of an online list-update algorithm"},{"id":"ComputerScience/online-algorithms/list-update-problem/adversary","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.OnlineListUpdate.adversary","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#adversary","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Given a deterministic online list-update algorithm $A$, a default item $d$, a number $n$ and a state $q$ of $A$, the adversary's sequence is the sequence of $n$ requests that starts in state $q$, always requests the last item of $A$'s current list, and then moves $A$ to its next state on that request. If the current list is empty, the default item $d$ is requested instead.","title":"The list-update adversary"},{"id":"ComputerScience/online-algorithms/list-update-problem/list-ne-nil","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.OnlineListUpdate.list_ne_nil","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#list-ne-nil","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $A$ be a deterministic online list-update algorithm on the items of a nonempty list $M$. Then in every state of $A$, the list of $A$ is nonempty.","title":"The list of an algorithm on a nonempty list is never empty"},{"id":"ComputerScience/online-algorithms/list-update-problem/cost-adversary","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.OnlineListUpdate.cost_adversary","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#cost-adversary","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a nonempty list of $\\ell$ distinct items and let $A$ be a deterministic online list-update algorithm on the items of $M$. For every default item $d$, every natural number $n$ and every state $q$ of $A$, the cost of $A$ from state $q$ on the adversary's sequence of $n$ requests from $q$ with default item $d$ is $n \\ell$.","title":"The adversary makes every access cost the full length of the list"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-online","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfOnline","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-online","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"For a list $M$, online move-to-front is the online list-update algorithm on the items of $M$ whose states are the rearrangements of $M$, starting from $M$ itself: on a request $x$ that is in the current list it moves $x$ to the front, and it ignores requests that are not in the current list.","title":"Online move-to-front"},{"id":"ComputerScience/online-algorithms/list-update-problem/mtf-online-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.mtfOnline_cost","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#mtf-online-cost","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list and $\\sigma$ a request sequence whose requests are all items of $M$. From any state of online move-to-front on $M$, its access cost on $\\sigma$ equals the cost of move-to-front on $\\sigma$ starting from the list of that state.","title":"Online move-to-front pays the move-to-front cost"},{"id":"ComputerScience/online-algorithms/list-update-problem/static-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.staticCost","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#static-cost","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"The static cost of serving a request sequence $\\sigma = x_0 x_1 \\cdots x_{m-1}$ with a fixed list $L$ is the sum over $t < m$ of the access cost of $x_t$ in $L$, its position in $L$ counting from $1$.","title":"Static cost"},{"id":"ComputerScience/online-algorithms/list-update-problem/offline-cost-const","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.offlineCost_const","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#offline-cost-const","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"For every list $L$ and every request sequence $\\sigma$, the offline cost (without free exchanges) of serving $\\sigma$ with the constant sequence of lists $L, L, L, \\dots$ equals the static cost of serving $\\sigma$ with $L$.","title":"The static cost is the offline cost of a constant list"},{"id":"ComputerScience/online-algorithms/list-update-problem/access-cost-add-reverse","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.accessCost_add_reverse","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#access-cost-add-reverse","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"If $L$ is a list of length $\\ell$ with no repeated entries and $x$ is an entry of $L$, then the access cost of $x$ in $L$ plus the access cost of $x$ in the reverse of $L$ equals $\\ell + 1$: the positions of $x$ in $L$ and in its reverse, counting from $1$, add up to $\\ell + 1$.","title":"Positions in a list and in its reverse"},{"id":"ComputerScience/online-algorithms/list-update-problem/static-cost-add-reverse","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.staticCost_add_reverse","parent":42293,"reading_title":{"format":"inline-parts-v1","parts":[["text","A list or its reverse serves any sequence at average cost at most "],["math","\\((\\ell + 1)/2\\)"]],"source":"A list or its reverse serves any sequence at average cost at most $(\\ell + 1)/2$"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#static-cost-add-reverse","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $L$ be a list of $\\ell$ distinct items. For every request sequence $\\sigma$ of $m$ items of $L$, the static cost of serving $\\sigma$ with $L$ plus the static cost of serving $\\sigma$ with the reverse of $L$ equals $m(\\ell + 1)$.","title":"A list or its reverse serves any sequence at average cost at most $(\\ell + 1)/2$"},{"id":"ComputerScience/online-algorithms/list-update-problem/inv-le-sq","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.inv_le_sq","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#inv-le-sq","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"For any lists $M$ and $L$, the number of inversions of $L$ against $M$ is at most $\\ell^2$, where $\\ell$ is the length of $M$.","title":"Inversions are at most the squared length"},{"id":"ComputerScience/online-algorithms/list-update-problem/offline-cost-switch-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.offlineCost_switch_le","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#offline-cost-switch-le","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a list of length $\\ell$ and $R$ a list, and write $\\mathrm{inv}(M, R)$ for the number of inversions between $M$ and $R$. For every request sequence $\\sigma$ of items of $M$, the offline cost (without free exchanges) of serving $\\sigma$ with the lists $M, R, R, R, \\dots$ is at most $\\ell + \\mathrm{inv}(M, R)$ plus the static cost of serving $\\sigma$ with $R$.","title":"Switching lists after the first request"},{"id":"ComputerScience/online-algorithms/list-update-problem/adversary-vs-offline","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.adversary_vs_offline","parent":42293,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#adversary-vs-offline","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a nonempty list of $\\ell$ distinct items, let $A$ be a deterministic online list-update algorithm on the items of $M$, let $d$ be an item and $n$ a natural number, and let $\\sigma$ be the adversary's sequence of $n$ requests against $A$ from its initial state, with default item $d$. Then the cost of $A$ on $\\sigma$ from its initial state is $n \\ell$, and there are lists $L_0, L_1, L_2, \\dots$, each a\u2026","title":"Online and offline costs on the adversary's sequence"},{"id":"ComputerScience/online-algorithms/list-update-problem/not-competitive","kind":"node","lean":"Cslib.Algorithms.Lean.Online.ListUpdate.not_competitive","parent":42293,"reading_title":{"format":"inline-parts-v1","parts":[["text","No deterministic online list-update algorithm beats "],["math","\\(2 - 2/(\\ell + 1)\\)"]],"source":"No deterministic online list-update algorithm beats $2 - 2/(\\ell + 1)$"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/list-update-problem/index.html#not-competitive","scope":"ComputerScience/online-algorithms/list-update-problem","status":"fully_proved","summary":"Let $M$ be a nonempty list of $\\ell$ distinct items and let $A$ be a deterministic online list-update algorithm on the items of $M$. Let $c$, $d$ and $b$ be natural numbers with $c (\\ell + 1) < 2 \\ell d$ (so $d \\ge 1$ and $c / d < 2\\ell / (\\ell + 1) = 2 - 2/(\\ell + 1)$). Then there are a request sequence $\\sigma$, each request an item of $M$, and lists $L_0, L_1, L_2, \\dots$, each a rearrangement of $M$, with\u2026","title":"No deterministic online list-update algorithm beats $2 - 2/(\\ell + 1)$"},{"data_url":"scopes/ComputerScience/online-algorithms/online-bipartite-matching.json","id":"ComputerScience/online-algorithms/online-bipartite-matching","kind":"scope","navigation":"module","parent":42246,"result":false,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","title":"Online bipartite matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/step","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.step","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#step","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Consider a bipartite graph with left vertices $0, \\dots, a-1$ and right vertices $0, \\dots, b-1$, and an assignment $m$ giving each right vertex either a left partner or none. On the arrival of a right vertex $j$, the greedy step returns the assignment that agrees with $m$ at every right vertex other than $j$, and gives $j$ the smallest left neighbour of $j$ that is free in $m$ (the partner of no right vertex), or\u2026","title":"Greedy step of online bipartite matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/greedy","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.greedy","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#greedy","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Consider a bipartite graph with left vertices $0, \\dots, a-1$ and right vertices $0, \\dots, b-1$, an assignment $m$ giving each right vertex either a left partner or none, and a list $j_1, \\dots, j_n$ of right vertices, the arrivals in order. The greedy matching from $m$ on these arrivals is the assignment obtained from $m$ by applying the greedy step of online bipartite matching for the arrival of $j_1$, then of\u2026","title":"Greedy online bipartite matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/step-extends","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.step_extends","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#step-extends","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Let $m$ be an assignment giving each right vertex of a bipartite graph either a left partner or none, and let $j$ be a right vertex with no partner in $m$. If a right vertex $j'$ has partner $i$ in $m$, then $j'$ still has partner $i$ after the greedy step for the arrival of $j$.","title":"Earlier partners survive an arrival"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/done","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.Done","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#done","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Given a partner array, which assigns to each right vertex its partner on the left if it has one, the right vertices of a list $S$ are done when, for every right vertex $j$ of $S$ and every left vertex $i$ adjacent to $j$, $j$ is matched or $i$ is matched.","title":"Done right vertices"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/greedy-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.greedy_spec","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#greedy-spec","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Let $L$ be a list of distinct right vertices, all unmatched in a valid partner array $m$, and let $S$ be a list of right vertices that are done for $m$. Then the partner array obtained from $m$ by the greedy run along $L$ is valid, and every right vertex of $L$ or of $S$ is done for it.","title":"Invariants of the greedy run"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/to-fun","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.toFun","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#to-fun","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"An array $A$ whose entries are each a left vertex or none is read as an assignment of partners to the right vertices $0, \\dots, b-1$: right vertex $j$ gets the entry of $A$ at index $j$ when $j$ is less than the length of $A$, and no partner otherwise.","title":"Partner assignment from an array"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/step-arr","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.stepArr","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#step-arr","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with left vertices $0, \\dots, a - 1$ and right vertices $0, \\dots, b - 1$, take an array whose entry $k$ holds the partner of the right vertex $k$, a missing entry meaning no partner. On the arrival of a right vertex $j$, entry $j$ is set to the first left neighbour of $j$, in the order $0, \\dots, a - 1$, that is free, that is, the partner of none of the right vertices $0, \\dots, b - 1$ in the\u2026","title":"Greedy step on an array"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/greedy-arr","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.greedyArr","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#greedy-arr","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with left vertices $0, \\dots, a - 1$ and right vertices $0, \\dots, b - 1$, start from an array whose entry $k$ holds the partner of the right vertex $k$, a missing entry meaning no partner, and let the right vertices of a list $L$ arrive one after another in the order of $L$. Each arrival performs the greedy step on the array: the entry of the arriving vertex $j$ is set to the first left\u2026","title":"Greedy run on an array"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/greedy-all","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.greedyAll","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#greedy-all","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph with left vertices $0, \\dots, a - 1$ and right vertices $0, \\dots, b - 1$, the greedy online matching is the partner array obtained by starting from the empty matching and letting the right vertices arrive in the order $0, 1, \\dots, b - 1$, each arriving vertex being matched to its first free left neighbour in the order $0, \\dots, a - 1$, or left unmatched if no neighbour is free.","title":"The greedy online matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/greedy-all-maximal","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.greedyAll_maximal","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#greedy-all-maximal","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Every edge has a matched endpoint.","title":"The greedy matching is maximal"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/card-le-two-mul-greedy-all","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.card_le_two_mul_greedyAll","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#card-le-two-mul-greedy-all","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Every matching has at most twice as many edges as the greedy one. An edge of the matching whose right vertex the greedy algorithm matched is charged to that right vertex, any other edge to its left vertex, which the greedy algorithm matched by maximality; a matching charges each greedy edge at most once on each side.","title":"Greedy online matching is `1/2`-competitive"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/is-matching-greedy-all","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.isMatching_greedyAll","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#is-matching-greedy-all","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"The set of pairs $(i, j)$ such that the greedy online matching assigns the left vertex $i$ to the right vertex $j$ is a matching of the bipartite graph.","title":"The greedy online matching is a matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/card-max-matching-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.BipartiteMatching.card_maxMatching_le","parent":42333,"reading_title":{"format":"inline-parts-v1","parts":[["text","Greedy online matching is within a factor "],["math","\\(2\\)"],["text"," of the maximum"]],"source":"Greedy online matching is within a factor $2$ of the maximum"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#card-max-matching-le","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"In a bipartite graph, the maximum matching computed by augmenting paths has at most twice as many edges as the greedy online matching, in which the right vertices arrive in increasing order.","title":"Greedy online matching is within a factor $2$ of the maximum"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/online-matcher","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.OnlineMatcher","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#online-matcher","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"A deterministic online bipartite matching algorithm with $a$ offline (left) vertices $0, \\dots, a - 1$ is given by a proposal rule: on each arrival it receives the neighbour lists (lists of left vertices) of all earlier arrivals, in order, and the neighbour list of the current arrival, and proposes a left vertex to match the arrival to, or none.","title":"Deterministic online bipartite matching algorithm"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/matched","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.matched","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#matched","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Let $A$ be an online matcher with $a$ left vertices, $h$ a history (the neighbour lists of earlier arrivals), $U$ a list of left vertices already matched, and $\\sigma$ a sequence of arrivals, each given by its list of left neighbours. The run of $A$ on $\\sigma$ takes the arrivals in order: on an arrival with neighbour list $N$, $A$ proposes a left vertex, or none, from the current history and $N$; a proposal $i$\u2026","title":"Number of arrivals an online matcher matches"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/is-perfect","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.IsPerfect","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#is-perfect","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"A list $m$ of left vertices is a perfect matching of a sequence $\\sigma$ of arrivals (neighbour lists) when $m$ and $\\sigma$ have the same length, the entries of $m$ are pairwise distinct, and for each $j$ the $j$-th entry of $m$ lies in the $j$-th neighbour list of $\\sigma$.","title":"Perfect matchings of a sequence of arrivals"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/to-adj","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.toAdj","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#to-adj","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"For a sequence $\\sigma = (\\sigma_0, \\dots, \\sigma_{k-1})$ of arrivals, each a list of left vertices in $\\{0, \\dots, a - 1\\}$, the bipartite graph of $\\sigma$ is given by adjacency lists: left vertex $i$ is sent to the list, in increasing order, of the arrivals $j < k$ with $i \\in \\sigma_j$.","title":"Graph of a sequence of arrivals"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/exists-is-matching-of-is-perfect","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.exists_isMatching_of_isPerfect","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#exists-is-matching-of-is-perfect","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Let $\\sigma$ be a sequence of arrivals (neighbour lists) and $m$ a perfect matching of $\\sigma$. Then the bipartite graph of $\\sigma$, in which left vertex $i$ is adjacent to arrival $j$ when $i$ lies in the $j$-th neighbour list, has a matching with as many edges as $\\sigma$ has arrivals.","title":"Perfect matchings of the arrivals are matchings of their graph"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/adversary","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.adversary","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#adversary","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Let $A$ be an online matcher, $h$ a list of earlier neighbour lists and $ps$ a list of gadgets, pairs $(u, v)$ of left vertices. The adversary's sequence of arrivals is empty when $ps$ is empty. Otherwise, for the first gadget $(u, v)$, let $x = v$ if $A$, after the history $h$, proposes $v$ for an arrival with neighbour list $[u, v]$, and $x = u$ otherwise (a proposal of $u$, of no vertex, or of a non-neighbour).\u2026","title":"The adversary for online bipartite matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/adversary-opt","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.adversaryOpt","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#adversary-opt","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Let $A$ be an online matcher, $h$ a list of earlier neighbour lists and $ps$ a list of gadgets, pairs $(u, v)$ of left vertices. The list $\\mathrm{opt}(A, h, ps)$ of left vertices is empty when $ps$ is empty. Otherwise, for the first gadget $(u, v)$, let $x = v$ if $A$, after the history $h$, proposes $v$ for an arrival with neighbour list $[u, v]$, and $x = u$ otherwise; the list starts with $u, v$ in the first\u2026","title":"The adversary's offline matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/gadgets-disjoint","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.GadgetsDisjoint","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#gadgets-disjoint","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"A list of gadgets, pairs $(u, v)$ of left vertices, has distinct vertices when $u \\ne v$ for every gadget $(u, v)$ of the list, and any two gadgets $(u, v)$ and $(u', v')$ at different positions of the list share no vertex: $u \\ne u'$, $u \\ne v'$, $v \\ne u'$ and $v \\ne v'$.","title":"Gadgets with distinct vertices"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/matched-adversary-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.matched_adversary_le","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#matched-adversary-le","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"For every online matcher $A$, every list $h$ of earlier neighbour lists, every list $U$ of left vertices and every list of $k$ gadgets, the run of $A$ on the adversary's arrivals for $A$, $h$ and the gadgets, after the history $h$ and with the left vertices of $U$ already matched, matches at most $k$ arrivals.","title":"The adversary holds every algorithm to one match per gadget"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/is-perfect-adversary-opt","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.isPerfect_adversaryOpt","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#is-perfect-adversary-opt","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"For every online matcher $A$, every list $h$ of earlier neighbour lists and every list of gadgets with distinct vertices, the adversary's offline matching for $A$, $h$ and the gadgets is a perfect matching of the adversary's arrivals for $A$, $h$ and the gadgets. In each gadget, the first arrival takes the vertex the second one does not.","title":"The adversary's arrivals have a perfect matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/gadgets","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.gadgets","parent":42333,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n\\)"],["text"," gadgets on "],["math","\\(2n\\)"],["text"," left vertices"]],"source":"The $n$ gadgets on $2n$ left vertices"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#gadgets","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$, the list of gadgets $(0, 1), (2, 3), \\dots, (2n - 2, 2n - 1)$ of left vertices in $\\{0, \\dots, 2n - 1\\}$, that is, the pairs $(2k, 2k + 1)$ for $k = 0, \\dots, n - 1$ in increasing order of $k$.","title":"The $n$ gadgets on $2n$ left vertices"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/exists-perfect-matched-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.exists_perfect_matched_le","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#exists-perfect-matched-le","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"No deterministic online bipartite matching algorithm beats `1/2` (Karp, Vazirani and Vazirani): against every algorithm with `2 n` left vertices, some sequence of `2 n` arrivals has a perfect matching while the algorithm matches at most `n` of them.","title":"No deterministic online bipartite matching algorithm beats `1/2`"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/exists-is-matching-matched-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.exists_isMatching_matched_le","parent":42333,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(1/2\\)"],["text"," lower bound with a matching of the arrivals' graph"]],"source":"The $1/2$ lower bound with a matching of the arrivals' graph"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#exists-is-matching-matched-le","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"For every $n \\in \\mathbb{N}$ and every online matcher $A$ with $2n$ left vertices, there is a sequence $\\sigma$ of arrivals (neighbour lists) such that the bipartite graph of $\\sigma$, in which left vertex $i$ is adjacent to arrival $j$ when $i$ lies in the $j$-th neighbour list, has a matching with $2n$ edges, while $A$, run on $\\sigma$ from the empty history with no left vertex matched, matches at most $n$\u2026","title":"The $1/2$ lower bound with a matching of the arrivals' graph"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/not-competitive-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.not_competitive_of_lt","parent":42333,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#not-competitive-of-lt","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Let $n$, $b$, $c$, $d$ be natural numbers with $d < 2c$ and $b < n$. For every online matcher $A$ with $2n$ left vertices there are a sequence $\\sigma$ of arrivals and a perfect matching $m$ of $\\sigma$ such that $d \\cdot \\mathrm{ALG} + b < c \\cdot |m|$, where $\\mathrm{ALG}$ is the number of arrivals of $\\sigma$ that $A$ matches when run from the empty history with no left vertex matched, and $|m|$ is the length of\u2026","title":"Finite-size lower bound for deterministic online matching"},{"id":"ComputerScience/online-algorithms/online-bipartite-matching/not-competitive","kind":"node","lean":"Cslib.Algorithms.Lean.Online.MatchingLowerBound.not_competitive","parent":42333,"reading_title":{"format":"inline-parts-v1","parts":[["text","No deterministic online matcher beats ratio "],["math","\\(1/2\\)"],["text",", with any additive constant"]],"source":"No deterministic online matcher beats ratio $1/2$, with any additive constant"},"result":true,"route":"../roadmap/ComputerScience/online-algorithms/online-bipartite-matching/index.html#not-competitive","scope":"ComputerScience/online-algorithms/online-bipartite-matching","status":"fully_proved","summary":"Let $(A_n)_{n \\in \\mathbb{N}}$ be a family of online matchers, $A_n$ having $2n$ left vertices, and let $b$, $c$, $d$ be natural numbers with $d < 2c$. Then for some $n$ there are a sequence $\\sigma$ of arrivals on the left vertices of $A_n$ and a perfect matching $m$ of $\\sigma$ such that $d \\cdot \\mathrm{ALG} + b < c \\cdot |m|$, where $\\mathrm{ALG}$ is the number of arrivals of $\\sigma$ that $A_n$ matches when\u2026","title":"No deterministic online matcher beats ratio $1/2$, with any additive constant"},{"data_url":"scopes/ComputerScience/online-algorithms/ski-rental-problem.json","id":"ComputerScience/online-algorithms/ski-rental-problem","kind":"scope","navigation":"module","parent":42246,"result":false,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","title":"Ski rental problem"},{"id":"ComputerScience/online-algorithms/ski-rental-problem/strategy","kind":"node","lean":"Cslib.Algorithms.Lean.Online.SkiRental.Strategy","parent":42362,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html#strategy","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","summary":"A deterministic strategy is either a day $t \\in \\mathbb{N}$ on which to buy the skis, after renting on the days before it, or the choice to rent forever. Days are numbered from $1$; the value $t = 0$ is also a strategy, and the cost treats it as renting forever.","title":"Deterministic ski-rental strategy"},{"id":"ComputerScience/online-algorithms/ski-rental-problem/cost","kind":"node","lean":"Cslib.Algorithms.Lean.Online.SkiRental.cost","parent":42362,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html#cost","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","summary":"Renting skis costs $1$ per day and buying them costs a natural number $B$. The cost $\\mathrm{cost}_B(s, d) \\in \\mathbb{N}$ of a strategy $s$ for a season of $d \\in \\mathbb{N}$ days is $(t - 1) + B$ if $s$ buys on a day $t$ with $1 \\le t \\le d$ (rent for $t - 1$ days, then buy), and $d$ otherwise, that is, if $s$ rents forever or buys on a day $t$ with $t = 0$ or $t > d$ (rent every day).","title":"Cost of a ski-rental strategy"},{"id":"ComputerScience/online-algorithms/ski-rental-problem/opt","kind":"node","lean":"Cslib.Algorithms.Lean.Online.SkiRental.opt","parent":42362,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html#opt","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","summary":"For a purchase price $B \\in \\mathbb{N}$ and a season of $d \\in \\mathbb{N}$ days, the offline optimum is $\\mathrm{opt}_B(d) = \\min(d, B)$: rent every day or buy on day one, whichever is cheaper.","title":"Offline optimum for ski rental"},{"id":"ComputerScience/online-algorithms/ski-rental-problem/break-even","kind":"node","lean":"Cslib.Algorithms.Lean.Online.SkiRental.breakEven","parent":42362,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html#break-even","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","summary":"For a purchase price $B$, the break-even strategy is the deterministic strategy that buys on day $B$.","title":"Break-even strategy"},{"id":"ComputerScience/online-algorithms/ski-rental-problem/opt-le-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Online.SkiRental.opt_le_cost","parent":42362,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html#opt-le-cost","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","summary":"For every purchase price $B$, every deterministic strategy $s$ and every season length $d$, the offline optimum $\\mathrm{opt}_B(d)$ is at most the cost $\\mathrm{cost}_B(s, d)$ of $s$ for a season of $d$ days.","title":"The offline optimum bounds every strategy's cost"},{"id":"ComputerScience/online-algorithms/ski-rental-problem/cost-break-even-le","kind":"node","lean":"Cslib.Algorithms.Lean.Online.SkiRental.cost_breakEven_le","parent":42362,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html#cost-break-even-le","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","summary":"For every purchase price $B \\ge 1$ and every season length $d$, the break-even strategy $\\mathrm{breakEven}_B$ pays at most $2 - 1/B$ times the offline optimum $\\mathrm{opt}_B(d)$: its cost $\\mathrm{cost}_B(\\mathrm{breakEven}_B, d)$ for a season of $d$ days satisfies $B \\cdot \\mathrm{cost}_B(\\mathrm{breakEven}_B, d) \\le (2B - 1) \\cdot \\mathrm{opt}_B(d)$.","title":"Ski-rental upper bound for the break-even strategy"},{"id":"ComputerScience/online-algorithms/ski-rental-problem/cost-break-even-le-two-mul","kind":"node","lean":"Cslib.Algorithms.Lean.Online.SkiRental.cost_breakEven_le_two_mul","parent":42362,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html#cost-break-even-le-two-mul","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","summary":"For every purchase price $B \\ge 1$ and every season length $d \\in \\mathbb{N}$, the cost $\\mathrm{cost}_B(\\mathrm{breakEven}_B, d)$ of the break-even strategy for a season of $d$ days is at most $2 \\cdot \\mathrm{opt}_B(d)$, twice the offline optimum $\\mathrm{opt}_B(d)$. In particular, the break-even strategy is $2$-competitive.","title":"The break-even strategy pays at most twice the optimum"},{"id":"ComputerScience/online-algorithms/ski-rental-problem/exists-cost-ge","kind":"node","lean":"Cslib.Algorithms.Lean.Online.SkiRental.exists_cost_ge","parent":42362,"result":true,"route":"../roadmap/ComputerScience/online-algorithms/ski-rental-problem/index.html#exists-cost-ge","scope":"ComputerScience/online-algorithms/ski-rental-problem","status":"fully_proved","summary":"Every deterministic strategy pays at least `(2 - 1/B)` times the optimum on some season, `(2 B - 1) * opt \u2264 B * cost`.","title":"Lower bound"},{"data_url":"scopes/ComputerScience/randomized-algorithms.json","id":"ComputerScience/randomized-algorithms","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/randomized-algorithms/index.html","scope":"ComputerScience/randomized-algorithms","status":"fully_proved","summary":"Algorithms whose identity is their use of randomness or whose analysis averages over random inputs: expected running time under input distributions, Fisher-Yates shuffling, the hiring problem, Karger's minimum cut, Freivalds' matrix-product check and MinHash similarity sketches.","title":"Randomized algorithms and probabilistic analysis"},{"data_url":"scopes/ComputerScience/randomized-algorithms/average-case-running-time.json","id":"ComputerScience/randomized-algorithms/average-case-running-time","kind":"scope","navigation":"module","parent":42371,"result":false,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","title":"Average-case running time"},{"id":"ComputerScience/randomized-algorithms/average-case-running-time/expected-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.expectedTime","parent":42372,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html#expected-time","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","summary":"Let $A$ be an algorithm on inputs of a type $\\gamma$ whose running time $T_A(x)$ on each input $x$ is a natural number, and let $D$ be a probability mass function on $\\gamma$. The expected running time of $A$ under $D$ is the extended nonnegative expectation $\\sum_{x \\in \\gamma} D(x)\\, T_A(x) \\in [0, \\infty]$ of the running time, the sum taken over all inputs.","title":"Expected running time"},{"id":"ComputerScience/randomized-algorithms/average-case-running-time/uniform-permutation","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.uniformPermutation","parent":42372,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html#uniform-permutation","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","summary":"For a list $x = (x_0, \\dots, x_{n-1})$, the uniform permutation of $x$ is the probability distribution on lists obtained by choosing a permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$ uniformly at random and outputting $(x_{\\sigma(0)}, \\dots, x_{\\sigma(n-1)})$.","title":"Uniformly random rearrangement of a list"},{"id":"ComputerScience/randomized-algorithms/average-case-running-time/uniform-permutation-length","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.uniformPermutation_length","parent":42372,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html#uniform-permutation-length","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","summary":"For a list $x$, every list in the support of the uniform permutation of $x$ has the same length as $x$.","title":"Rearrangements preserve length"},{"id":"ComputerScience/randomized-algorithms/average-case-running-time/uniform-permutation-perm","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.uniformPermutation_perm","parent":42372,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html#uniform-permutation-perm","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","summary":"For a list $x$, every list in the support of the uniform permutation of $x$ is a permutation of $x$, that is, it has the same entries with the same multiplicities.","title":"Rearrangements are permutations"},{"id":"ComputerScience/randomized-algorithms/average-case-running-time/expected-time-uniform-permutation","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.expectedTime_uniformPermutation","parent":42372,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html#expected-time-uniform-permutation","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","summary":"Let $A$ be an algorithm on lists whose running time $T_A(y)$ on each list $y$ is a natural number, and let $x = (x_0, \\dots, x_{n-1})$ be a list of length $n$. The expected running time of $A$ when its input is $x$ rearranged by a uniformly random permutation equals","title":"Average formula for the expected time over random orderings"},{"id":"ComputerScience/randomized-algorithms/average-case-running-time/expected-time-eq-of-time-eq","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.expectedTime_eq_of_time_eq","parent":42372,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html#expected-time-eq-of-time-eq","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","summary":"Let $A$ be an algorithm on inputs of a type $\\gamma$ whose running time $T_A(x)$ on each input $x$ is a natural number, let $D$ be a probability mass function on $\\gamma$, and let $c$ be a natural number. If $T_A(x) = c$ for every input $x$ in the support of $D$, then the expected running time of $A$ under $D$ equals $c$.","title":"Constant running time on the support gives that expected time"},{"id":"ComputerScience/randomized-algorithms/average-case-running-time/expected-time-uniform-permutation-eq-of","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.expectedTime_uniformPermutation_eq_of_time_eq","parent":42372,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html#expected-time-uniform-permutation-eq-of","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","summary":"Let $A$ be an algorithm on lists whose running time $T_A(y)$ on each list $y$ is a natural number, let $x = (x_0, \\dots, x_{n-1})$ be a list of length $n$, and let $c$ be a natural number. If $T_A(x_{\\sigma(0)}, \\dots, x_{\\sigma(n-1)}) = c$ for every permutation $\\sigma$ of $\\{0, \\dots, n-1\\}$, then the expected running time of $A$ when its input is $x$ rearranged by a uniformly random permutation equals $c$.","title":"Constant running time on all rearrangements gives that expected time"},{"id":"ComputerScience/randomized-algorithms/average-case-running-time/expected-time-uniform-permutation-to-rea","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.expectedTime_uniformPermutation_toReal","parent":42372,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/average-case-running-time/index.html#expected-time-uniform-permutation-to-rea","scope":"ComputerScience/randomized-algorithms/average-case-running-time","status":"fully_proved","summary":"Let $A$ be an algorithm on lists whose running time $T_A(y)$ on each list $y$ is a natural number, and let $x = (x_0, \\dots, x_{n-1})$ be a list of length $n$. The expected running time of $A$ when its input is $x$ rearranged by a uniformly random permutation, converted from $[0, \\infty]$ to a real number, equals","title":"Average formula for the expected time over random orderings, in the reals"},{"data_url":"scopes/ComputerScience/randomized-algorithms/fisher-yates-shuffle.json","id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","kind":"scope","navigation":"module","parent":42371,"result":false,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","title":"Fisher\u2013Yates shuffle"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermute","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"Given a vector $x = (x_0, \\dots, x_{n-1})$ and a draw history $d$, which assigns to each position $i < n$ a number $d_i$ with $0 \\le d_i < n - i$, the Fisher\u2013Yates computation processes the positions $i = 0, 1, \\dots, n - 1$ in increasing order and at position $i$ swaps the entries at positions $i$ and $i + d_i$ of the current vector. It returns the final vector together with a cost pair in\u2026","title":"Fisher\u2013Yates shuffle with supplied draws"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermute_time","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute-time","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"For every vector $x$ of length $n$ and every draw history $d$, which assigns to each position $i < n$ a number $d_i$ with $0 \\le d_i < n - i$, the Fisher\u2013Yates computation on $x$ and $d$ has cost pair $(n, n)$: it retrieves exactly $n$ draws and performs exactly $n$ swaps.","title":"Cost of the Fisher\u2013Yates shuffle"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute-perm","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermute_perm","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute-perm","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"For every vector $x$ of length $n$ and every draw history $d$, which assigns to each position $i < n$ a number $d_i$ with $0 \\le d_i < n - i$, the list of entries of the vector returned by the Fisher\u2013Yates computation on $x$ and $d$ is a permutation of the list of entries of $x$. In particular every value occurs equally often in both, even when values of $x$ repeat.","title":"Fisher\u2013Yates output is a rearrangement of its input"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute-empty","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermute_empty","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute-empty","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"For the empty vector $x$ (of length $0$) and its empty draw history, the Fisher\u2013Yates computation returns $x$ with cost pair $(0, 0)$: it retrieves no draws and performs no swaps.","title":"Fisher\u2013Yates on the empty vector"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermute_singleton","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute-singleton","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"For every vector $x$ of length $1$ and every draw history $d$, the Fisher\u2013Yates computation returns $x$ unchanged with cost pair $(1, 1)$: it still retrieves the single draw $d_0 = 0$ and performs the self-swap at position $0$.","title":"Fisher\u2013Yates on a one-element vector"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute-distribution","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermuteDistribution","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute-distribution","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"For a vector $x$ of length $n$, this is the probability distribution on outcomes (a vector of length $n$ together with a cost pair in $\\mathbb{N} \\times \\mathbb{N}$) obtained by choosing a draw history $d$ uniformly at random among all assignments of a number $d_i$ with $0 \\le d_i < n - i$ to each position $i < n$, and returning the result of the Fisher\u2013Yates computation on $x$ and $d$.","title":"Output distribution of the Fisher\u2013Yates shuffle"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute-distribution-apply","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermuteDistribution_apply","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute-distribution-apply","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"Let $x$ be a vector of length $n$, and consider the draw histories $d$, which assign to each position $i < n$ a number $d_i$ with $0 \\le d_i < n - i$. For every outcome $o$ (a vector of length $n$ together with a cost pair), the probability that the Fisher\u2013Yates output distribution of $x$ assigns to $o$ equals the sum, over all draw histories $d$ for which the Fisher\u2013Yates computation on $x$ and $d$ yields exactly\u2026","title":"Probability of a Fisher\u2013Yates outcome"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute-distribution-eq-map-uni","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermuteDistribution_eq_map_uniform","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute-distribution-eq-map-uni","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"Let $x = (x_0, \\dots, x_{n-1})$ be a vector and let $\\sigma$ be a permutation of $\\{0, \\dots, n - 1\\}$ chosen uniformly at random. Then the Fisher\u2013Yates output distribution of $x$ equals the distribution of the outcome consisting of the vector $(x_{\\sigma(0)}, \\dots, x_{\\sigma(n-1)})$ with cost pair $(n, n)$.","title":"Fisher\u2013Yates produces a uniformly random permutation"},{"id":"ComputerScience/randomized-algorithms/fisher-yates-shuffle/randomly-permute-prefix-probability","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.randomlyPermute_prefix_probability","parent":42381,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/fisher-yates-shuffle/index.html#randomly-permute-prefix-probability","scope":"ComputerScience/randomized-algorithms/fisher-yates-shuffle","status":"fully_proved","summary":"Let $k \\le n$ and let $a_0, \\dots, a_{k-1}$ be distinct elements of $\\{0, \\dots, n - 1\\}$. Run the Fisher\u2013Yates computation on the vector $(0, 1, \\dots, n - 1)$ with every draw history $d$, which assigns to each position $i < n$ a number $d_i$ with $0 \\le d_i < n - i$, and give each history weight $1/n!$. Then the total weight of the histories whose output vector has entry $a_p$ at position $p$ for every $p < k$\u2026","title":"Prefix probabilities of the Fisher\u2013Yates shuffle"},{"data_url":"scopes/ComputerScience/randomized-algorithms/freivalds-algorithm.json","id":"ComputerScience/randomized-algorithms/freivalds-algorithm","kind":"scope","navigation":"module","parent":42371,"result":false,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","title":"Freivalds' algorithm"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Given $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{F}_2 = \\mathbb{Z}/2$, stored as arrays of rows, and a seed $r \\in \\mathbb{F}_2^n$, the trial computes the column vectors $Br$, $A(Br)$ and $Cr$, each by accumulating matrix products into a zero column, and returns true exactly when $A(Br) = Cr$. Its cost is the pair (number of scalar multiplications, number of scalar additions), each multiply-add step counting\u2026","title":"Freivalds trial"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial_time","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-time","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{F}_2$ and every seed $r \\in \\mathbb{F}_2^n$, the Freivalds trial performs exactly $3n^2$ scalar multiplications and exactly $3n^2$ scalar additions.","title":"Cost of the Freivalds trial"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-matrix","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsMatrix","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-matrix","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"An $n \\times n$ array $A$ of rows over $\\mathbb{F}_2 = \\mathbb{Z}/2$ determines the $n \\times n$ matrix over $\\mathbb{F}_2$ whose $(i, j)$ entry is entry $j$ of row $i$ of $A$.","title":"Matrix of a Freivalds input"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-accept-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial_accept_iff","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-accept-iff","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{F}_2$ and every seed $r \\in \\mathbb{F}_2^n$, the Freivalds trial on $A$, $B$, $C$, $r$ returns true if and only if $A(Br) = Cr$.","title":"Acceptance criterion of the Freivalds trial"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-accept-discrepancy-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial_accept_discrepancy_iff","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-accept-discrepancy-iff","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{F}_2$ and every seed $r \\in \\mathbb{F}_2^n$, the Freivalds trial on $A$, $B$, $C$, $r$ returns true if and only if $(AB - C)r = 0$.","title":"Acceptance via the product discrepancy"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-complete","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial_complete","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-complete","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A$, $B$, $C$ be $n \\times n$ matrices over $\\mathbb{F}_2$ with $AB = C$. Then for every seed $r \\in \\mathbb{F}_2^n$ the Freivalds trial on $A$, $B$, $C$, $r$ returns true. This includes the case $n = 0$.","title":"Completeness of the Freivalds trial"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-empty","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial_empty","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-empty","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $0 \\times 0$ matrices $A$, $B$, $C$ over $\\mathbb{F}_2$ and the empty seed $r$, the Freivalds trial returns true at zero cost: it performs no scalar multiplications and no scalar additions.","title":"Freivalds trial on empty matrices"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-acceptance-card-bound","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial_acceptanceCard_bound","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-acceptance-card-bound","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A$, $B$, $C$ be $n \\times n$ matrices over $\\mathbb{F}_2$ with $AB \\ne C$. Then twice the number of seeds $r \\in \\mathbb{F}_2^n$ on which the Freivalds trial returns true is at most the number of all seeds in $\\mathbb{F}_2^n$; that is, at most half of the seeds accept.","title":"Soundness of the Freivalds trial"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-law","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrialLaw","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-law","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$, given as arrays, the Freivalds trial law is the probability distribution on $\\{\\mathrm{true}, \\mathrm{false}\\}$ of the value returned by the Freivalds trial on $A$, $B$, $C$ with seed $r$, where $r$ is drawn uniformly from $\\mathbb{F}_2^n$.","title":"Freivalds trial law"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-seed-point-mass","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsSeed_pointMass","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-seed-point-mass","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$. For every $n$ and every vector $r \\in \\mathbb{F}_2^n$, the uniform probability distribution on $\\mathbb{F}_2^n$ assigns to $r$ the probability $(2^n)^{-1}$.","title":"Uniform mass of a Freivalds seed"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-law-acceptance","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrialLaw_acceptance","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-law-acceptance","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A$, $B$, $C$ be $n \\times n$ matrices over $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$, given as arrays. The probability that the Freivalds trial law of $A$, $B$, $C$ returns true equals the number of seeds $r \\in \\mathbb{F}_2^n$ on which the Freivalds trial on $A$, $B$, $C$ with seed $r$ accepts, divided by the number $|\\mathbb{F}_2^n|$ of all seeds.","title":"Acceptance probability of the Freivalds trial as a seed count"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-law-complete","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrialLaw_complete","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-law-complete","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A$, $B$, $C$ be $n \\times n$ matrices over $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$, given as arrays. If $AB = C$, then the Freivalds trial law of $A$, $B$, $C$ returns true with probability $1$. This includes the case $n = 0$ of empty matrices.","title":"The Freivalds trial accepts a correct product with probability one"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-law-sound","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrialLaw_sound","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-law-sound","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A$, $B$, $C$ be $n \\times n$ matrices over $\\mathbb{F}_2 = \\mathbb{Z}/2\\mathbb{Z}$, given as arrays. If $AB \\ne C$, then the Freivalds trial law of $A$, $B$, $C$ returns true with probability at most $1/2$.","title":"The Freivalds trial accepts a wrong product with probability at most one half"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeat","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Given $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{Z}/2$ and seeds $r_0, \\dots, r_{k-1} \\in (\\mathbb{Z}/2)^n$, run the Freivalds trial on $A$, $B$, $C$ with each seed $r_0, \\dots, r_{k-1}$ in order, including the trials after a rejection, and return the conjunction of their results (true when $k = 0$). The arithmetic cost of the run is the sum of the costs of the $k$ trials.","title":"Repeated Freivalds test on fixed seeds"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeat_time","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-time","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{Z}/2$ and seeds $r_0, \\dots, r_{k-1} \\in (\\mathbb{Z}/2)^n$, the repeated Freivalds test on these seeds performs exactly $3kn^2$ scalar multiplications and $3kn^2$ scalar additions.","title":"Cost of the repeated Freivalds test on fixed seeds"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-accept-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeat_accept_iff","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-accept-iff","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{Z}/2$ and seeds $r_0, \\dots, r_{k-1} \\in (\\mathbb{Z}/2)^n$, the repeated Freivalds test on these seeds accepts if and only if, for every $i < k$, the Freivalds trial on $A$, $B$, $C$ with seed $r_i$ accepts.","title":"Acceptance of the repeated Freivalds test on fixed seeds"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-complete","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeat_complete","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-complete","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A$, $B$, $C$ be $n \\times n$ matrices over $\\mathbb{Z}/2$ with $AB = C$. Then for every $k$, including $k = 0$, and all seeds $r_0, \\dots, r_{k-1} \\in (\\mathbb{Z}/2)^n$, the repeated Freivalds test on these seeds accepts.","title":"Completeness of the repeated Freivalds test on fixed seeds"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-zero","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeat_zero","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-zero","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $n \\times n$ matrices $A$, $B$, $C$ over $\\mathbb{Z}/2$, the repeated Freivalds test on the empty list of seeds ($k = 0$) accepts and performs no scalar multiplications or additions.","title":"Repeated Freivalds test with no trials"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-empty","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeat_empty","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-empty","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For $0 \\times 0$ matrices $A$, $B$, $C$ over $\\mathbb{Z}/2$ and any number $k$ of seeds (each the empty vector), the repeated Freivalds test accepts and performs no scalar multiplications or additions.","title":"Repeated Freivalds test on empty matrices"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-law","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeatLaw","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-law","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For natural numbers $n, k$ and $n \\times n$ matrices $A, B, C$ over $\\mathbb{F}_2$, this is the probability distribution on $\\{\\mathrm{true}, \\mathrm{false}\\}$ of the result returned by the repeated Freivalds executor on $A, B, C$ when its history of $k$ seeds is drawn uniformly from $(\\mathbb{F}_2^n)^k$.","title":"Output distribution of repeated Freivalds trials"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-history-point-mass","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsHistory_pointMass","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-history-point-mass","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"For all natural numbers $n$ and $k$, under the uniform distribution on histories of $k$ seeds, that is, on $(\\mathbb{F}_2^n)^k$, every history has probability exactly $(2^{nk})^{-1}$. This includes the cases $k = 0$ and $n = 0$.","title":"Uniform mass of a Freivalds seed history"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-acceptance-card","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeat_acceptanceCard","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-acceptance-card","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $n, k$ be natural numbers and let $A, B, C$ be $n \\times n$ matrices over $\\mathbb{F}_2$. The number of histories $(r_1, \\dots, r_k)$ of seeds $r_i \\in \\mathbb{F}_2^n$ on which the repeated Freivalds executor accepts equals $a^k$, where $a$ is the number of seeds $r \\in \\mathbb{F}_2^n$ on which a single Freivalds trial accepts.","title":"Counting accepted histories of repeated Freivalds trials"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-law-acceptance-power","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeatLaw_acceptance_power","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-law-acceptance-power","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $n, k$ be natural numbers and let $A, B, C$ be $n \\times n$ matrices over $\\mathbb{F}_2$. If the history of $k$ seeds is drawn uniformly from $(\\mathbb{F}_2^n)^k$, the probability that the repeated Freivalds executor accepts is $p^k$, where $p$ is the probability that a single Freivalds trial accepts on a seed drawn uniformly from $\\mathbb{F}_2^n$.","title":"Acceptance probability of repeated Freivalds trials"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-law-complete","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeatLaw_complete","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-law-complete","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $n, k$ be natural numbers and let $A, B, C$ be $n \\times n$ matrices over $\\mathbb{F}_2$ with $AB = C$. If the history of $k$ seeds is drawn uniformly from $(\\mathbb{F}_2^n)^k$, then the repeated Freivalds executor accepts with probability $1$.","title":"Completeness of repeated Freivalds trials"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-law-sound","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeatLaw_sound","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-law-sound","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $n, k$ be natural numbers and let $A, B, C$ be $n \\times n$ matrices over $\\mathbb{F}_2$ with $AB \\ne C$. If the history of $k$ seeds is drawn uniformly from $(\\mathbb{F}_2^n)^k$, then the repeated Freivalds executor accepts with probability at most $(2^k)^{-1}$.","title":"Soundness of repeated Freivalds trials"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-one-wrong","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivalds_one_wrong","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-one-wrong","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A = B = (1)$ and $C = (0)$ be $1 \\times 1$ matrices over $\\mathbb{Z}/2\\mathbb{Z}$. Then $AB \\neq C$.","title":"The sharp Freivalds example has a wrong product"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-one-accept-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial_one_accept_iff","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-one-accept-iff","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A = B = (1)$ and $C = (0)$ be $1 \\times 1$ matrices over $\\mathbb{Z}/2\\mathbb{Z}$. For every seed $r \\in (\\mathbb{Z}/2\\mathbb{Z})^1$, the Freivalds trial on $A$, $B$, $C$ with seed $r$, which compares $A(Br)$ with $Cr$, accepts if and only if the single entry $r_0$ of $r$ is $0$.","title":"The sharp Freivalds example accepts exactly the zero seed"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-one-acceptance-card","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrial_one_acceptanceCard","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-one-acceptance-card","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A = B = (1)$ and $C = (0)$ be $1 \\times 1$ matrices over $\\mathbb{Z}/2\\mathbb{Z}$. Exactly one seed $r \\in (\\mathbb{Z}/2\\mathbb{Z})^1$ is accepted by the Freivalds trial on $A$, $B$, $C$.","title":"The sharp Freivalds example has exactly one accepted seed"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-trial-law-one-sharp","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsTrialLaw_one_sharp","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-trial-law-one-sharp","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A = B = (1)$ and $C = (0)$ be $1 \\times 1$ matrices over $\\mathbb{Z}/2\\mathbb{Z}$. When the seed is drawn uniformly from $(\\mathbb{Z}/2\\mathbb{Z})^1$, the Freivalds trial on $A$, $B$, $C$ accepts with probability exactly $1/2$.","title":"The sharp Freivalds example is accepted with probability one half"},{"id":"ComputerScience/randomized-algorithms/freivalds-algorithm/freivalds-repeat-law-one-sharp","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.freivaldsRepeatLaw_one_sharp","parent":42391,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/freivalds-algorithm/index.html#freivalds-repeat-law-one-sharp","scope":"ComputerScience/randomized-algorithms/freivalds-algorithm","status":"fully_proved","summary":"Let $A = B = (1)$ and $C = (0)$ be $1 \\times 1$ matrices over $\\mathbb{Z}/2\\mathbb{Z}$, and let $k$ be a natural number. Run $k$ Freivalds trials on $A$, $B$, $C$ with seeds drawn independently and uniformly from $(\\mathbb{Z}/2\\mathbb{Z})^1$, accepting exactly when every trial accepts. Then the run accepts with probability exactly $2^{-k}$.","title":"Exact acceptance probability of repeated trials on the sharp Freivalds example"},{"data_url":"scopes/ComputerScience/randomized-algorithms/hiring-problem.json","id":"ComputerScience/randomized-algorithms/hiring-problem","kind":"scope","navigation":"module","parent":42371,"result":false,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","title":"Hiring problem"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"Given candidates $x_0, \\dots, x_{n-1}$ in a linearly ordered type, the scan interviews them in order while keeping a current manager, initially none, which counts as below every candidate. Candidate $i$ is hired exactly when $x_i$ is strictly greater than the value of the current manager, and then becomes the new manager; ties keep the earlier manager. The scan returns the zero-based index of the final manager\u2026","title":"The hire-assistant scan"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_ret","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-ret","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"For every list $x_0, \\dots, x_{n-1}$ in a linearly ordered type, the manager returned by the hire-assistant scan is none when the list is empty, and otherwise is the smallest index $i$ such that $x_i$ is a maximum of the list.","title":"The hire-assistant scan returns the earliest best candidate"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-hires","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_hires","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-hires","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"For every list $x_0, \\dots, x_{n-1}$ in a linearly ordered type, the list of indices hired by the hire-assistant scan, in chronological order, is the increasing list of all indices $i < n$ such that $x_j < x_i$ for every $j < i$.","title":"The hire-assistant scan hires exactly the strict prefix maxima"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_time","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-time","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"For every list of $n$ candidates in a linearly ordered type, the cost pair of the hire-assistant scan is $(n, h)$, where $h$ is the length of the list of hired indices that the scan returns.","title":"Cost of the hire-assistant scan"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-nil","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_nil","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-nil","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"On the empty list, the hire-assistant scan returns no manager and an empty list of hires, with cost pair $(0, 0)$.","title":"Hire-assistant scan on no candidates"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_singleton","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-singleton","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"For every element $a$ of a linearly ordered type, on the one-element list $[a]$ the hire-assistant scan returns manager index $0$ and hire list $[0]$, with cost pair $(1, 1)$: the single candidate is hired regardless of its quality.","title":"Hire-assistant scan on one candidate"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-hire-bounds","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_hire_bounds","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-hire-bounds","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"For every nonempty list of $n$ candidates in a linearly ordered type, the number of hires $h$ recorded in the cost pair of the hire-assistant scan satisfies $1 \\le h \\le n$.","title":"Hire-count bounds for the hire-assistant scan"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-increasing","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_increasing","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-increasing","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"For every strictly increasing list $x_0 < x_1 < \\dots < x_{n-1}$ in a linearly ordered type, including the empty list, the hire-assistant scan hires the indices $0, 1, \\dots, n - 1$ in that order, and its cost pair is $(n, n)$.","title":"Strictly increasing input hires every candidate"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-expected-hires","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_expectedHires","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-expected-hires","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"Let $x$ be a list of $n$ pairwise distinct elements of a linearly ordered type, and rearrange its entries by a uniformly random permutation of the indices. The expected number of hires, valued in $[0, \\infty]$, that the hire-assistant procedure (interview the entries in order, hiring each one strictly better than every earlier one) makes on the rearranged list equals $\\sum_{i=0}^{n-1} \\frac{1}{i+1}$.","title":"Expected number of hires under a random permutation"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-expected-hires-to-real","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_expectedHires_toReal","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-expected-hires-to-real","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"Let $x$ be a list of $n$ pairwise distinct elements of a linearly ordered type, and rearrange its entries by a uniformly random permutation of the indices. The expected number of hires that the hire-assistant procedure (interview the entries in order, hiring each one strictly better than every earlier one) makes on the rearranged list, an element of $[0, \\infty]$, converted to a real number, equals\u2026","title":"Expected number of hires as a real harmonic sum"},{"id":"ComputerScience/randomized-algorithms/hiring-problem/hire-assistant-expected-hires-to-real-eq","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hireAssistant_expectedHires_toReal_eq_harmonic","parent":42422,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/hiring-problem/index.html#hire-assistant-expected-hires-to-real-eq","scope":"ComputerScience/randomized-algorithms/hiring-problem","status":"fully_proved","summary":"Let $x$ be a list of $n$ pairwise distinct elements of a linearly ordered type, and rearrange its entries by a uniformly random permutation of the indices. The expected number of hires that the hire-assistant procedure (interview the entries in order, hiring each one strictly better than every earlier one) makes on the rearranged list, an element of $[0, \\infty]$, converted to a real number, equals the harmonic\u2026","title":"Expected number of hires is the harmonic number"},{"data_url":"scopes/ComputerScience/randomized-algorithms/karger-s-algorithm.json","id":"ComputerScience/randomized-algorithms/karger-s-algorithm","kind":"scope","navigation":"module","parent":42371,"result":false,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","title":"Karger's algorithm"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/state","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.State","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#state","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"A state over a vertex type $V$ consists of a list of current (super-)vertices in $V$ and a list of edges, each an ordered pair of elements of $V$; repeated pairs are parallel edges.","title":"Multigraph state of Karger's algorithm"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/valid","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.State.Valid","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#valid","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"A multigraph, given by a list of vertices and a list of edges (pairs of vertices, with multiplicity), is valid when its vertex list has no repeated entries and every edge joins two distinct vertices of the list.","title":"Valid multigraph"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/cut-size","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.cutSize","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#cut-size","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For a list of edges (pairs of vertices, repeated pairs counted with multiplicity) and a map $X$ from vertices to booleans, the size of the cut $X$ is the number of edges $(u, v)$ in the list with $X(u) \\ne X(v)$.","title":"Size of a cut"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/is-cut","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.IsCut","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#is-cut","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"A map $X$ from vertices to booleans is a cut of a multigraph state $G$ when some vertex $v$ of $G$ has $X(v) = \\mathrm{true}$ and some vertex $w$ of $G$ has $X(w) = \\mathrm{false}$.","title":"Cuts of a multigraph"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/is-min-cut","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.IsMinCut","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#is-min-cut","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"A natural number $k$ is the minimum cut size of a multigraph state $G$ when some cut of $G$ has size $k$ and every cut of $G$ has size at least $k$.","title":"Minimum cut size"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/rename","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.rename","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#rename","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For vertices $a$ and $b$, renaming $b$ to $a$ is the map on vertices that sends $b$ to $a$ and fixes every other vertex.","title":"Renaming a vertex"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/contract","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.contract","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#contract","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For a multigraph state $G$ and a pair of vertices $e = (a, b)$, the contraction of $G$ along $e$ deletes the first occurrence of $b$ from the vertex list, replaces $b$ by $a$ in both endpoints of every edge, and then deletes every edge whose two endpoints are equal.","title":"Edge contraction"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/success-prob","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.successProb","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#success-prob","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For a natural number $k$, a number $f$ of allowed contractions and a multigraph $G$ with edge list $E_G$, the probability $P_k(f, G)$ that the contraction process on $G$ returns $k$ is defined by recursion on $f$. If $f = 0$, or $G$ has at most two vertices, or $G$ has no edges, then $P_k(f, G)$ is $1$ when $G$ has exactly $k$ edges and $0$ otherwise. Otherwise\u2026","title":"Success probability of the contraction process"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/cut-size-contract","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.cutSize_contract","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#cut-size-contract","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a multigraph state, $e = (a, b)$ a pair of vertices and $Y$ a map from vertices to booleans, and let $\\rho$ send $b$ to $a$ and fix every other vertex. The size of the cut $Y$ in the contraction of $G$ along $e$ equals the size of the cut $Y \\circ \\rho$ in $G$.","title":"Cut sizes under contraction"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/is-min-cut-contract","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.isMinCut_contract","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#is-min-cut-contract","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a valid multigraph state with minimum cut size $k$, and let $X$ be a cut of $G$ of size $k$. If $e = (a, b)$ is an edge of $G$ with $X(a) = X(b)$, then the contraction of $G$ along $e$ also has minimum cut size $k$.","title":"Contraction preserves the minimum cut size"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/valid-contract","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.valid_contract","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#valid-contract","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"If a multigraph $G$ is valid and $e$ is one of its edges, then the contraction of $G$ along $e$ is valid and has exactly one vertex fewer than $G$.","title":"Contraction preserves validity and removes one vertex"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/sum-degree","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.sum_degree","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#sum-degree","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For a valid multigraph $G$, the sum over the vertices $v$ of $G$ of the number of edges crossing the singleton cut $\\{v\\}$ equals twice the number of edges of $G$.","title":"Handshake lemma"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/card-mul-le-two-mul-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.card_mul_le_two_mul_edges","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#card-mul-le-two-mul-edges","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a valid multigraph state with $n \\ge 2$ vertices, $m$ edges (counted with multiplicity) and minimum cut size $k$. Then $n k \\le 2m$: every vertex has degree at least $k$, since its singleton is a cut.","title":"Minimum cut size bounds the number of edges"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/success-prob-nonneg","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.successProb_nonneg","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#success-prob-nonneg","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For every natural number $k$, every number $f$ of allowed contractions and every multigraph state $G$, the probability $P_k(f, G)$ that the contraction process on $G$ returns $k$ satisfies $0 \\le P_k(f, G)$.","title":"The success probability is nonnegative"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/success-prob-le-one","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.successProb_le_one","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#success-prob-le-one","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For every natural number $k$, every number of allowed contractions and every multigraph $G$, the probability that the contraction process on $G$ returns $k$ is at most $1$.","title":"The success probability is at most one"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/success-prob-ge","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.successProb_ge","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#success-prob-ge","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a valid multigraph with $n \\ge 2$ vertices and minimum cut size $k$. Then the contraction process on $G$, with $n$ contractions allowed, returns $k$ with probability at least $2 / (n(n-1))$.","title":"Karger's bound"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/repeat-success-prob","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.repeatSuccessProb","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#repeat-success-prob","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For a multigraph $G$ with $n$ vertices and natural numbers $k$ and $t$, let $p$ be the probability that one run of the contraction process on $G$, with $n$ contractions allowed, returns $k$. The repeated success probability is $1 - (1 - p)^t$, the probability that at least one of $t$ independent runs returns $k$.","title":"Success probability of repeated runs"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/repeat-success-prob-ge-half","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.repeatSuccessProb_ge_half","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#repeat-success-prob-ge-half","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a valid multigraph state with $n \\ge 2$ vertices and minimum cut size $k$, and let $t$ be a natural number with $n(n-1) \\le 2t$. Then the probability that at least one of $t$ independent runs of the contraction process returns $k$ is at least $1/2$.","title":"Repeating Karger's algorithm succeeds with probability at least one half"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/repeat-success-prob-ge","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.repeatSuccessProb_ge","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#repeat-success-prob-ge","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a valid multigraph with $n \\ge 2$ vertices and minimum cut size $k$, and let $c$ and $t$ be natural numbers with $c \\cdot n(n-1) \\le 2t$. Then the probability that at least one of $t$ independent runs of the contraction process on $G$ returns $k$ is at least $1 - (1/2)^c$.","title":"Amplification with high probability"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/repeat-success-prob-ge-one-sub-inv","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.repeatSuccessProb_ge_one_sub_inv","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#repeat-success-prob-ge-one-sub-inv","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a valid multigraph with $n \\ge 2$ vertices and minimum cut size $k$, and let $t$ be a natural number with $\\lceil \\log_2 n \\rceil \\cdot n(n-1) \\le 2t$. Then the probability that at least one of $t$ independent runs of the contraction process on $G$ returns $k$ is at least $1 - 1/n$.","title":"Karger's algorithm succeeds with high probability"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/survive-prob","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.surviveProb","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#survive-prob","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"For a map $X$ from vertices to booleans, a fuel bound $f$ and a multigraph $G$, this is the probability that the contraction process on $G$ (contract an edge chosen uniformly at random, counted with multiplicity, while more than two vertices and at least one edge remain, making at most $f$ contractions) never contracts an edge whose endpoints $X$ separates and ends with exactly two vertices on which $X$ takes both\u2026","title":"Probability that a run outputs a given cut"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/distinct-on","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.DistinctOn","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#distinct-on","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Two maps $X, Y$ from vertices to booleans are distinct on a list of vertices $S$ when some $v \\in S$ has $X(v) \\ne Y(v)$ and some $w \\in S$ has $X(w) = Y(w)$; so on $S$ neither cut is the other or its complement.","title":"Cuts distinct as partitions"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/survive-prob-ge","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.surviveProb_ge","parent":42434,"reading_title":{"format":"inline-parts-v1","parts":[["text","Each minimum cut survives with probability at least "],["math","\\(2/(n(n-1))\\)"]],"source":"Each minimum cut survives with probability at least $2/(n(n-1))$"},"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#survive-prob-ge","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a valid multigraph with $n \\ge 2$ vertices whose minimum cut size $k$ is at least $1$, and let $X$ be a cut of $G$ with exactly $k$ crossing edges. Then the probability that the contraction process on $G$, allowed up to $n$ contractions, outputs the cut $X$ is at least $2/(n(n-1))$.","title":"Each minimum cut survives with probability at least $2/(n(n-1))$"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/sum-survive-prob-le-one","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.sum_surviveProb_le_one","parent":42434,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#sum-survive-prob-le-one","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"Let $G$ be a valid multigraph with vertex list $S$, let $f$ be a number of allowed contractions, and let $X_1, \\dots, X_m$ be Boolean functions on the vertex type that are pairwise distinct as partitions of $S$. Then the probabilities that the contraction process on $G$, with $f$ contractions allowed, outputs $X_1, \\dots, X_m$ add up to at most $1$.","title":"Survival probabilities of distinct cuts sum to at most one"},{"id":"ComputerScience/randomized-algorithms/karger-s-algorithm/two-mul-length-min-cuts-le","kind":"node","lean":"Cslib.Algorithms.Lean.Karger.two_mul_length_minCuts_le","parent":42434,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most "],["math","\\(n(n-1)/2\\)"],["text"," minimum cuts"]],"source":"At most $n(n-1)/2$ minimum cuts"},"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/karger-s-algorithm/index.html#two-mul-length-min-cuts-le","scope":"ComputerScience/randomized-algorithms/karger-s-algorithm","status":"fully_proved","summary":"At most $n(n-1)/2$ minimum cuts (Karger; Kleinberg and Tardos, Section 13.2). Let $G$ be a valid multigraph with $n \\ge 2$ vertices whose minimum cut size $k$ is at least $1$. If $X_1, \\dots, X_m$ are cuts of $G$, each with exactly $k$ crossing edges, that are pairwise distinct as partitions of the vertices of $G$, then $2m \\le n(n-1)$, that is, $m \\le n(n-1)/2$.","title":"At most $n(n-1)/2$ minimum cuts"},{"data_url":"scopes/ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity.json","id":"ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity","kind":"scope","navigation":"module","parent":42371,"result":false,"route":"../roadmap/ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity/index.html","scope":"ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity","status":"fully_proved","title":"MinHash and Jaccard similarity"},{"id":"ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity/min-hash","kind":"node","lean":"Cslib.Bioinformatics.Sketching.MinHash.minHash","parent":42460,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity/index.html#min-hash","scope":"ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity","status":"fully_proved","summary":"The min-hash of `A` under the permutation `\u03c0`: the least element of `\u03c0(A)`, or `\u22a4` if `A` is empty.","title":"Min-hash of a set under a permutation"},{"id":"ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity/card-min-hash-collision","kind":"node","lean":"Cslib.Bioinformatics.Sketching.MinHash.card_minHash_collision","parent":42460,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity/index.html#card-min-hash-collision","scope":"ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity","status":"fully_proved","summary":"The number of permutations of `U` under which `A` and `B` have the same min-hash, times `|A \u222a B|`, equals `|A \u2229 B|` times the number of all permutations.","title":"MinHash, counting form"},{"id":"ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity/uniform-on-min-hash-collision","kind":"node","lean":"Cslib.Bioinformatics.Sketching.MinHash.uniformOn_minHash_collision","parent":42460,"result":true,"route":"../roadmap/ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity/index.html#uniform-on-min-hash-collision","scope":"ComputerScience/randomized-algorithms/minhash-and-jaccard-similarity","status":"fully_proved","summary":"MinHash collision probability (Broder 1997): for a uniformly random permutation `\u03c0` of the finite universe `U` (`uniformOn Set.univ` on `Equiv.Perm U`, for any measurable structure in which singletons are measurable, e.g. `\u22a4`) and sets `A`, `B` with `A \u222a B` nonempty, `P(min \u03c0(A) = min \u03c0(B)) = |A \u2229 B| / |A \u222a B|`, the Jaccard similarity of `A` and `B`.","title":"MinHash collision probability"},{"data_url":"scopes/ComputerScience/parallel-algorithms.json","id":"ComputerScience/parallel-algorithms","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/parallel-algorithms/index.html","scope":"ComputerScience/parallel-algorithms","status":"fully_proved","summary":"Work-span models, Brent's theorem and greedy scheduling, parallel prefix sums (Blelloch, Kogge-Stone) and pointer-jumping list ranking.","title":"Parallel algorithms"},{"data_url":"scopes/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping.json","id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","kind":"scope","navigation":"module","parent":42464,"result":false,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","title":"List ranking by pointer jumping"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/jump","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.jump","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#jump","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"For a successor map $\\mathrm{succ}$ on the nodes $0, \\dots, n-1$, where $\\mathrm{succ}(v)$ is either a node or undefined, the $k$-step jump from $v$ is the node reached by applying $\\mathrm{succ}$ $k$ times starting at $v$, and is undefined if some step is undefined: the $0$-step jump from $v$ is $v$, and the $(k+1)$-step jump is $\\mathrm{succ}$ of the $k$-step jump.","title":"Iterated successor"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/terminates","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.Terminates","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#terminates","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"A successor function $\\mathrm{succ}$ on the nodes $0, \\dots, n - 1$, mapping each node to its successor or to nothing, terminates when from every node $v$ the walk of $n$ steps along $\\mathrm{succ}$ runs off the end, that is, there is no node $n$ steps down the list from $v$.","title":"Terminating successor functions"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rank","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.rank","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rank","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a successor map on the nodes $0, \\dots, n-1$ whose lists terminate. The rank of a node $v$ is the least $k$ such that following $\\mathrm{succ}$ for $k + 1$ steps from $v$ runs off the end of the list, that is, the number of nodes after $v$ in its list.","title":"Rank of a node in a linked list"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/jump-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.jump_eq_none_iff","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#jump-eq-none-iff","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a successor map on the nodes $0, \\dots, n-1$ whose lists terminate. For every node $v$ and every $k \\in \\mathbb{N}$, following $\\mathrm{succ}$ for $k$ steps from $v$ runs off the end of the list if and only if $k > \\operatorname{rank}(v)$.","title":"The characteristic property of the rank"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rank-of-jump","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.rank_of_jump","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rank-of-jump","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a successor map on the nodes $0, \\dots, n-1$ whose lists terminate. If following $\\mathrm{succ}$ for $a$ steps from a node $v$ reaches the node $u$, then $a \\le \\operatorname{rank}(v)$ and $\\operatorname{rank}(u) = \\operatorname{rank}(v) - a$.","title":"Rank decreases along jumps"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/state-2","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.State","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#state-2","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"A pointer-jumping state on the nodes $0, \\dots, n - 1$ assigns to each node a pointer, which is either a node or empty (once it points past the end of the list), and a distance in $\\mathbb{N}$, the distance covered by the pointer.","title":"Pointer-jumping state for list ranking"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/init-2","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.init","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#init-2","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"For a successor map $\\mathrm{succ}$ on the nodes $0, \\dots, n-1$, where $\\mathrm{succ}(v)$ is either a node or undefined (at the end of a list), the initial pointer-jumping state gives each node $v$ the pointer $\\mathrm{succ}(v)$ and the distance $0$ if $\\mathrm{succ}(v)$ is undefined and $1$ otherwise.","title":"Initial state of pointer jumping"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/round-2","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.round","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#round-2","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"In one parallel round, every node $v$ whose pointer is a node $u$ adds the distance of $u$ to its own distance and moves its pointer to the pointer of $u$ (undefined if that of $u$ is); a node whose pointer is undefined keeps its distance and its undefined pointer.","title":"One round of pointer jumping"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rounds-2","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.rounds","parent":42465,"reading_title":{"format":"inline-parts-v1","parts":[["text","The list-ranking state after "],["math","\\(t\\)"],["text"," rounds"]],"source":"The list-ranking state after $t$ rounds"},"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rounds-2","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"For a successor function $\\mathrm{succ}$ on the nodes $0, \\dots, n - 1$, the pointer-jumping state after $t$ rounds is defined by recursion on $t$: after $0$ rounds it is the initial state of $\\mathrm{succ}$, and after $t + 1$ rounds it is one round of pointer jumping applied to the state after $t$ rounds.","title":"The list-ranking state after $t$ rounds"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rounds-spec-2","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.rounds_spec","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rounds-spec-2","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ map each of the nodes $0, \\dots, n - 1$ to its successor or to nothing, and suppose the lists terminate: from every node, $n$ steps run off the end. Write $\\mathrm{rank}(v)$ for the number of nodes after $v$. Then for every $t \\ge 0$ and every node $v$, after $t$ rounds of pointer jumping, the pointer of $v$ is the node $2^t$ steps down the list from $v$ (no node if the list ends before that),\u2026","title":"The invariant of pointer jumping for list ranking"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rounds-dist","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.rounds_dist","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rounds-dist","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a successor map on the nodes $0, \\dots, n-1$ whose lists terminate, and let $t$ be a natural number with $n \\le 2^t$. Then after $t$ rounds of pointer jumping from the initial state, the distance of every node $v$ is the rank of $v$.","title":"After enough rounds, the distances are the ranks"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/tabulate-2","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.tabulate","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#tabulate-2","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"The tabulation of a pointer-jumping state $s$ on the nodes $0, \\dots, n - 1$ is the pair of arrays of length $n$ whose entries at $v$ are the pointer and the distance of $v$ in $s$.","title":"Tabulating a list-ranking state"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/round-tab-2","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.roundTab","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#round-tab-2","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Given an array $P$ of pointers (each a node among $0, \\dots, n-1$ or undefined) and an array $D$ of natural-number distances, one round returns arrays $P'$ and $D'$ of length $n$ where, for each node $v$, $P'[v]$ is $P[u]$ if $P[v] = u$ and is undefined if $P[v]$ is undefined, and $D'[v] = D[v] + D[u]$ if $P[v] = u$ and $D'[v] = D[v]$ if $P[v]$ is undefined. Entries missing from the input arrays are read as\u2026","title":"One pointer-jumping round on arrays"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rounds-tab-2","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.roundsTab","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rounds-tab-2","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"For a successor function $\\mathrm{succ}$ on the nodes $0, \\dots, n - 1$ and $t \\ge 0$, the arrays after $t$ rounds are a pair of arrays (pointers and distances) defined by recursion on $t$: after $0$ rounds they are the tabulation of the initial state of $\\mathrm{succ}$, and after $t + 1$ rounds they are one round on arrays applied to the arrays after $t$ rounds.","title":"List-ranking rounds on arrays"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/list-ranks","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.listRanks","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#list-ranks","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"For a successor map $\\mathrm{succ}$ on the nodes $0, \\dots, n-1$, list ranking by pointer jumping runs $\\lceil \\log_2 n \\rceil$ rounds of pointer jumping on arrays, starting from the initial state, and returns the resulting array of distances, whose entry $v$ is the distance of node $v$.","title":"List ranking by pointer jumping"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/list-ranks-get-d","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListRanking.listRanks_getD","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#list-ranks-get-d","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a successor map on the nodes $0, \\dots, n-1$ whose lists terminate. Then for every node $v$, entry $v$ of the table returned by list ranking by pointer jumping, which runs $\\lceil \\log_2 n \\rceil$ rounds, is the rank of $v$.","title":"Pointer jumping ranks every list in \u2308log\u2082 n\u2309 rounds"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/step-val","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.stepVal","parent":42465,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(k\\)"],["text"," steps ahead"]],"source":"The value $k$ steps ahead"},"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#step-val","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $n$ nodes be linked into lists by a partial successor function, each node $u$ carrying a value $x_u$ in an additive monoid $M$. For $k \\in \\mathbb{N}$ and a node $v$, this is $x_u$ where $u$ is the node $k$ steps after $v$ along its list, and $0$ if the list ends before $k$ steps.","title":"The value $k$ steps ahead"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/seg-sum","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.segSum","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#seg-sum","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $n$ nodes be linked into lists by a partial successor function, and let each node $v$ carry a value $x_v$ in an additive monoid $M$ (not necessarily commutative). For $k \\in \\mathbb{N}$ and a node $v$, the segment sum of length $k$ from $v$ is the sum, in list order, of the values of the first $k$ nodes of the list starting at $v$ (fewer if the list ends sooner); for $k = 0$ it is $0$.","title":"Segment sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/seg-sum-add","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.segSum_add","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#seg-sum-add","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a partial successor function on nodes $0, \\dots, n-1$ and $x_v$ values in an additive monoid $M$. For $k \\in \\mathbb{N}$ write $S_k(v)$ for the sum, in list order, of the values of the nodes reached from $v$ in $0, 1, \\dots, k - 1$ steps along $\\mathrm{succ}$ (a missing node contributes $0$). Then for all $a, b \\in \\mathbb{N}$ and every node $v$, $S_{a+b}(v) = S_a(v) + S_b(u)$ if the node $u$\u2026","title":"Segment sums add up"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/suffix-sum","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.suffixSum","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#suffix-sum","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $n$ nodes be linked into lists by a partial successor function, each node $v$ carrying a value $x_v$ in an additive monoid $M$. The suffix sum of a node $v$ is the segment sum of length $n$ from $v$: the sum, in list order, of the values of the first $n$ nodes of the list starting at $v$ (fewer if it ends sooner).","title":"Suffix sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/suffix-sum-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.suffixSum_eq","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#suffix-sum-eq","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $n$ nodes be linked into lists by a partial successor function such that, starting from any node, $n$ steps run off the end of its list, and let each node $v$ carry a value $x_v$ in an additive monoid $M$. Then for every node $v$, the suffix sum of $v$ is $x_v$ plus the suffix sum of the successor of $v$, or just $x_v$ if $v$ has no successor.","title":"The characteristic property of suffix sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/seg-sum-one","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.segSum_one","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#seg-sum-one","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $n$ nodes be linked into lists by a partial successor function such that, starting from any node, $n$ steps run off the end of its list, and give every node the value $1 \\in \\mathbb{N}$. Then for every $k \\in \\mathbb{N}$ and every node $v$, the segment sum of length $k$ from $v$ (the sum of the values of the first $k$ nodes from $v$, fewer at the end of the list) is $\\min(k, \\operatorname{rank}(v) + 1)$, where\u2026","title":"Segment sums of unit values count nodes"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/suffix-sum-one","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.suffixSum_one","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#suffix-sum-one","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $n$ nodes be linked into lists by a partial successor function such that, starting from any node, $n$ steps run off the end of its list. If every node has value $1 \\in \\mathbb{N}$, then the suffix sum of every node $v$ is $\\operatorname{rank}(v) + 1$, where $\\operatorname{rank}(v)$ is the number of nodes after $v$ in its list.","title":"List ranking is the special case of unit values"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/state","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.State","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#state","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"For $n$ nodes and a type $M$, a state assigns to each node $v$ a pointer, which is either a node or none (past the end of the list), and an accumulated value in $M$. The accumulated value is meant to be the sum of the node values from $v$ up to, but not including, the pointer's target.","title":"State of pointer jumping for suffix sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/init","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.init","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#init","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Given nodes $0, \\dots, n-1$ with a partial successor function $\\mathrm{succ}$ and values $x_v$ in an additive monoid $M$, the initial state gives each node $v$ the pointer $\\mathrm{succ}(v)$ (undefined at the end of a list) and the sum $x_v$.","title":"Initial state of pointer jumping for suffix sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/round","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.round","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#round","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Given a state in which each node $v$ has a pointer $p(v)$, either a node or undefined, and a sum $s(v)$ in an additive monoid $M$, one round produces a new state as follows. If $p(v)$ is undefined, the pointer of $v$ stays undefined and its sum stays $s(v)$. If $p(v) = u$, the new pointer of $v$ is $p(u)$ and its new sum is $s(v) + s(u)$, with $s(u)$ added on the right so that list order is kept.","title":"One round of pointer jumping for suffix sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rounds","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.rounds","parent":42465,"reading_title":{"format":"inline-parts-v1","parts":[["text","State after "],["math","\\(t\\)"],["text"," rounds of pointer jumping for suffix sums"]],"source":"State after $t$ rounds of pointer jumping for suffix sums"},"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rounds","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Given a partial successor function $\\mathrm{succ}$ on nodes $0, \\dots, n-1$, values $x_v$ in an additive monoid $M$ and $t \\in \\mathbb{N}$, the state after $t$ rounds is obtained from the initial state (pointers $\\mathrm{succ}(v)$, sums $x_v$) by applying one round of pointer jumping $t$ times.","title":"State after $t$ rounds of pointer jumping for suffix sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rounds-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.rounds_spec","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rounds-spec","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a partial successor function on nodes $0, \\dots, n-1$ and $x_v$ values in an additive monoid $M$. For every $t \\in \\mathbb{N}$ and every node $v$, after $t$ rounds of pointer jumping the pointer of $v$ is the node reached from $v$ in $2^t$ steps along $\\mathrm{succ}$ (undefined if the list ends sooner), and the sum of $v$ is the sum, in list order, of the values of the nodes reached from $v$\u2026","title":"The invariant of pointer jumping for suffix sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rounds-acc","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.rounds_acc","parent":42465,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pointer-jumping sums are the suffix sums once "],["math","\\(n \\le 2^t\\)"]],"source":"Pointer-jumping sums are the suffix sums once $n \\le 2^t$"},"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rounds-acc","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a partial successor function on nodes $0, \\dots, n-1$ under which every list ends, that is, from every node, the node reached in $n$ steps along $\\mathrm{succ}$ is undefined, and let $x_v$ be values in an additive monoid $M$. If $t \\in \\mathbb{N}$ satisfies $n \\le 2^t$, then for every node $v$ the sum of $v$ after $t$ rounds of pointer jumping is the suffix sum of $v$: the sum, in list order,\u2026","title":"Pointer-jumping sums are the suffix sums once $n \\le 2^t$"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/tabulate","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.tabulate","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#tabulate","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"A suffix-sum pointer-jumping state on $n$ nodes is stored as a pair of arrays of length $n$: the array whose entry $v$ is the pointer of node $v$, and the array whose entry $v$ is the accumulated value of node $v$.","title":"Pointer-jumping state as arrays"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/round-tab","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.roundTab","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#round-tab","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Given an array $P$ of pointers (each a node among $0, \\dots, n-1$ or undefined) and an array $S$ of sums in an additive monoid $M$, one round returns arrays $P'$ and $S'$ of length $n$ where, for each node $v$: if $P[v]$ is undefined, then $P'[v]$ is undefined and $S'[v] = S[v]$; if $P[v] = u$, then $P'[v] = P[u]$ and $S'[v] = S[v] + S[u]$. A read outside an array gives an undefined pointer or the sum $0$.","title":"One round of pointer jumping on arrays"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/rounds-tab","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.roundsTab","parent":42465,"reading_title":{"format":"inline-parts-v1","parts":[["text","Arrays after "],["math","\\(t\\)"],["text"," rounds of pointer jumping for suffix sums"]],"source":"Arrays after $t$ rounds of pointer jumping for suffix sums"},"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#rounds-tab","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Given a partial successor function $\\mathrm{succ}$ on nodes $0, \\dots, n-1$ and values $x_v$ in an additive monoid $M$, the pair of arrays after $0$ rounds stores the pointers $\\mathrm{succ}(v)$ and the sums $x_v$, and the pair after $t + 1$ rounds is one round of pointer jumping on arrays applied to the pair after $t$ rounds.","title":"Arrays after $t$ rounds of pointer jumping for suffix sums"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/list-sums","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.listSums","parent":42465,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#list-sums","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Given a partial successor function $\\mathrm{succ}$ on nodes $0, \\dots, n-1$ and values $x_v$ in an additive monoid $M$, this is the array of sums obtained after $\\lceil \\log_2 n \\rceil$ rounds (the least $k$ with $n \\le 2^k$) of pointer jumping on arrays, starting from the pointers $\\mathrm{succ}(v)$ and the sums $x_v$.","title":"Suffix sums by pointer jumping on arrays"},{"id":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/list-sums-get-d","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.ListPrefix.listSums_getD","parent":42465,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pointer jumping computes all suffix sums in "],["math","\\(\\lceil \\log_2 n \\rceil\\)"],["text"," rounds"]],"source":"Pointer jumping computes all suffix sums in $\\lceil \\log_2 n \\rceil$ rounds"},"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping/index.html#list-sums-get-d","scope":"ComputerScience/parallel-algorithms/list-ranking-by-pointer-jumping","status":"fully_proved","summary":"Let $\\mathrm{succ}$ be a partial successor function on nodes $0, \\dots, n-1$ under which every list ends, that is, from every node, the node reached in $n$ steps along $\\mathrm{succ}$ is undefined, and let $x_v$ be values in an additive monoid $M$. Then for every node $v$, entry $v$ of the array of sums computed by $\\lceil \\log_2 n \\rceil$ rounds of pointer jumping on arrays is the suffix sum of $v$: the sum, in\u2026","title":"Pointer jumping computes all suffix sums in $\\lceil \\log_2 n \\rceil$ rounds"},{"data_url":"scopes/ComputerScience/parallel-algorithms/parallel-prefix-sums.json","id":"ComputerScience/parallel-algorithms/parallel-prefix-sums","kind":"scope","navigation":"module","parent":42464,"result":false,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","title":"Parallel prefix sums"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/seg","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.seg","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#seg","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For a sequence $x : \\mathbb{N} \\to M$ in a monoid $M$ and $a, b \\in \\mathbb{N}$, the window product is the ordered product $x_a x_{a+1} \\cdots x_{b-1}$ of the terms with indices $a \\le j < b$; it is the empty product $1$ when $b \\le a$.","title":"Window product"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/prefix-prod","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.prefixProd","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#prefix-prod","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For a sequence $x_0, x_1, \\dots$ in a monoid $M$ and a natural number $i$, the prefix product at $i$ is $x_0 x_1 \\cdots x_i$, the factors taken in increasing order of index.","title":"Prefix product"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/seg-mul-seg","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.seg_mul_seg","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#seg-mul-seg","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $M$ be a monoid and $x : \\mathbb{N} \\to M$ a sequence, and for $a, b \\in \\mathbb{N}$ let $x_{[a, b)}$ be the ordered product $x_a x_{a+1} \\cdots x_{b-1}$ (equal to $1$ if $b \\le a$). For all $a \\le b \\le c$, $x_{[a, b)} \\, x_{[b, c)} = x_{[a, c)}$.","title":"Products of adjacent windows"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/kogge-stone","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.koggeStone","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#kogge-stone","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Given a sequence $x_0, x_1, \\dots$ in a monoid $M$, the value $s_k(i)$ at position $i$ after $k$ rounds is defined by $s_0(i) = x_i$ and, for each $k$, $s_{k+1}(i) = s_k(i - 2^k) \\, s_k(i)$ if $2^k \\le i$ and $s_{k+1}(i) = s_k(i)$ otherwise.","title":"The Kogge\u2013Stone scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/kogge-stone-eq-seg","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.koggeStone_eq_seg","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#kogge-stone-eq-seg","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $x_0, x_1, \\dots$ be a sequence in a monoid $M$, let $k$ and $i$ be natural numbers, and let $m = \\min(i + 1, 2^k)$. After $k$ rounds of the Kogge\u2013Stone scan, position $i$ holds the product $x_{i+1-m} \\cdots x_i$ of the $m$ consecutive inputs ending at $i$, taken in increasing order of index.","title":"What the scan holds after each round"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/kogge-stone-eq-prefix","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.koggeStone_eq_prefix","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#kogge-stone-eq-prefix","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $x_0, x_1, \\dots$ be a sequence in a monoid $M$ and let $n$ and $K$ be natural numbers with $n \\le 2^K$. After $K$ rounds of the Kogge\u2013Stone scan, every position $i < n$ holds the prefix product $x_0 x_1 \\cdots x_i$.","title":"The scan computes all prefix products"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/round-work","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.roundWork","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#round-work","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"The work of round $k$ on $n$ positions is $\\max(n - 2^k, 0)$, the number of positions $i < n$ with $i \\ge 2^k$, each of which performs one multiplication.","title":"Work of a scan round"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/round-work-eq-card","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.roundWork_eq_card","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#round-work-eq-card","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For natural numbers $n$ and $k$, the work of round $k$ on $n$ positions equals the number of positions $i$ with $2^k \\le i < n$.","title":"The work of a round counts its combining positions"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/work","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.work","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#work","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"The total work of $K$ rounds on $n$ positions is $\\sum_{k=0}^{K-1} \\max(n - 2^k, 0)$, the sum over the rounds $k < K$ of the work $\\max(n - 2^k, 0)$ of round $k$.","title":"Work of the Kogge\u2013Stone scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/work-le","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.work_le","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#work-le","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For all $n, K \\in \\mathbb{N}$, the total work of $K$ rounds of the scan on $n$ positions is at most $nK$.","title":"Work bound for the Kogge\u2013Stone scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/span","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.span","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#span","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"The span of $K$ rounds of the scan is $K$: each round is one parallel step.","title":"Span of the Kogge\u2013Stone scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/kogge-stone-clog-eq-prefix","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.koggeStone_clog_eq_prefix","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#kogge-stone-clog-eq-prefix","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $x_0, x_1, \\dots$ be a sequence in a monoid $M$ and let $n$ be a natural number. After $\\lceil \\log_2 n \\rceil$ rounds of the Kogge\u2013Stone scan (the least $K$ with $n \\le 2^K$), every position $i < n$ holds the prefix product $x_0 x_1 \\cdots x_i$.","title":"The scan computes all prefixes in logarithmically many rounds"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/round-list","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.roundList","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#round-list","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Given a list $y_0, \\dots, y_{n-1}$ in a monoid $M$ and a natural number $k$, one round produces the list of length $n$ whose entry $i$ is $y_{i - 2^k} \\, y_i$ when $2^k \\le i$ and $y_i$ when $i < 2^k$.","title":"One round of the scan on a list"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/scan-list","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.scanList","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#scan-list","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For a list $y = (y_0, \\dots, y_{n-1})$ in a monoid $M$, the list after $K$ rounds is defined by recursion on $K$: after $0$ rounds it is $y$, and after $K + 1$ rounds it is round $K$ applied to the list after $K$ rounds. Here round $k$ sends a list $(z_0, \\dots, z_{n-1})$ to the list of length $n$ whose entry $i$ is $z_i$ for $i < 2^k$ and $z_{i - 2^k} z_i$ for $2^k \\le i < n$.","title":"Tabulated Kogge\u2013Stone scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/get-d-scan-list","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.getD_scanList","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#get-d-scan-list","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $x_0, \\dots, x_{n-1}$ be a list of elements of a monoid $M$, extended to a sequence by $x_j = 1$ for $j \\ge n$. For every number of rounds $K$ and every position $i < n$, entry $i$ of the list produced by $K$ rounds of the tabulated scan equals the value at position $i$ after $K$ rounds of the Kogge\u2013Stone scan applied to the sequence $x$.","title":"The tabulated scan agrees with the round recursion"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/get-d-scan-list-eq-prefix","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.getD_scanList_eq_prefix","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#get-d-scan-list-eq-prefix","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $x_0, \\dots, x_{n-1}$ be a list of elements of a monoid $M$ and let $K$ be a natural number with $n \\le 2^K$. Then for every $i < n$, entry $i$ of the list produced by $K$ rounds of the tabulated scan equals the prefix product $x_0 x_1 \\cdots x_i$.","title":"The tabulated scan computes all prefix products"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/pair-up","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.pairUp","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#pair-up","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For a sequence $x \\colon \\mathbb{N} \\to M$ in a monoid $M$, the sequence of adjacent pair products is the sequence whose term at $i$ is $x(2i) \\cdot x(2i+1)$.","title":"Adjacent pair products of a sequence"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/blelloch","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.blelloch","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#blelloch","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $M$ be a monoid. For a natural number $k$ and a sequence $x = (x_0, x_1, \\dots)$ in $M$, the scan $B_k(x)$ is the sequence in $M$ defined by recursion on $k$: $B_0(x)$ is constantly $1$, and $B_{k+1}(x)$ is obtained from the scan $y = B_k(p)$ of the pair products $p_i = x_{2i} x_{2i+1}$ by setting $B_{k+1}(x)_{2i} = y_i$ and $B_{k+1}(x)_{2i+1} = y_i x_{2i}$ for every $i$.","title":"Blelloch's scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/blelloch-eq-seg","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.blelloch_eq_seg","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#blelloch-eq-seg","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $M$ be a monoid, $k$ a natural number and $x = (x_0, x_1, \\dots)$ a sequence in $M$. For every $j < 2^k$, the value at position $j$ of Blelloch's scan of $x$ on $2^k$ positions is the exclusive prefix product $x_0 x_1 \\cdots x_{j-1}$ (equal to $1$ when $j = 0$).","title":"Blelloch's scan computes the exclusive prefix products"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/blelloch-work","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.blellochWork","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#blelloch-work","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"The number $W(k)$ of monoid multiplications Blelloch's scan performs on $2^k$ positions is defined by $W(0) = 0$ and $W(k+1) = 2^k + W(k) + 2^k$: on $2^{k+1}$ positions, $2^k$ pair products, the recursive scan on the $2^k$ pair products, and $2^k$ products for the odd positions.","title":"Work of Blelloch's scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/blelloch-span","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.blellochSpan","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#blelloch-span","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"The span $S(k)$ of Blelloch's scan on $2^k$ positions is defined by $S(0) = 0$ and $S(k+1) = S(k) + 2$: one parallel step for the pair products, the recursive scan, and one parallel step for the odd positions.","title":"Parallel steps of Blelloch's scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/blelloch-work-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.blellochWork_eq","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#blelloch-work-eq","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For every natural number $k$, the number $W(k)$ of monoid multiplications Blelloch's scan performs on $n = 2^k$ positions satisfies $W(k) + 2 = 2^{k+1}$, so $W(k) = 2^{k+1} - 2 < 2n$.","title":"Blelloch's scan is work-efficient"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/blelloch-span-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.blellochSpan_eq","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#blelloch-span-eq","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For every natural number $k$, the span of Blelloch's scan on $n = 2^k$ positions is $2k = 2 \\log_2 n$.","title":"Blelloch's scan has logarithmic span"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/pair-list","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.pairList","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#pair-list","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"For a list $[a_0, a_1, \\ldots, a_{n-1}]$ of elements of a monoid $M$, the list of adjacent pair products is $[a_0 a_1, a_2 a_3, \\ldots]$: it has length $\\lfloor n/2 \\rfloor$, its $i$-th entry is $a_{2i} a_{2i+1}$, and when $n$ is odd the last element $a_{n-1}$ is dropped.","title":"Adjacent pair products of a list"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/expand","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.expand","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#expand","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $M$ be a monoid. For lists $z = (z_0, z_1, \\dots)$ and $x = (x_0, x_1, \\dots)$ in $M$, the expansion of $z$ along $x$ is the list $(z_0, z_0 x_0, z_1, z_1 x_2, z_2, z_2 x_4, \\dots)$: each $z_i$ is followed by $z_i x_{2i}$, and the list stops at the first $i$ for which $z_i$ or $x_{2i+1}$ does not exist.","title":"Interleaving step of Blelloch's list scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/blelloch-list","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.blellochList","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#blelloch-list","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $M$ be a monoid. For a natural number $k$ and a list $xs = (x_0, x_1, \\dots)$ in $M$, the list scan $L_k(xs)$ is defined by recursion on $k$: $L_0(xs)$ is the one-element list $(1)$, and $L_{k+1}(xs)$ is obtained from the list scan $z = L_k(p)$ of the pair products $p = (x_0 x_1, x_2 x_3, \\dots)$ of $xs$ (a trailing unpaired entry is dropped) by interleaving: $L_{k+1}(xs) = (z_0, z_0 x_0, z_1, z_1 x_2, \\dots)$,\u2026","title":"Blelloch's scan on lists"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/get-d-blelloch-list","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.getD_blellochList","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#get-d-blelloch-list","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $M$ be a monoid, $k$ a natural number and $xs = (x_0, \\dots, x_{2^k - 1})$ a list in $M$ of length $2^k$. For every $j < 2^k$, the entry at position $j$ of Blelloch's list scan of $xs$ in $k$ rounds equals the value at position $j$ of Blelloch's scan on $2^k$ positions of the sequence $x_0, \\dots, x_{2^k - 1}, 1, 1, \\dots$ obtained by extending $xs$ by $1$.","title":"The list implementation computes Blelloch's scan"},{"id":"ComputerScience/parallel-algorithms/parallel-prefix-sums/get-d-blelloch-list-eq-seg","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.PrefixSum.getD_blellochList_eq_seg","parent":42500,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/parallel-prefix-sums/index.html#get-d-blelloch-list-eq-seg","scope":"ComputerScience/parallel-algorithms/parallel-prefix-sums","status":"fully_proved","summary":"Let $M$ be a monoid, $k$ a natural number and $xs = (x_0, \\dots, x_{2^k - 1})$ a list in $M$ of length $2^k$. For every $j < 2^k$, the entry at position $j$ of Blelloch's list scan of $xs$ in $k$ rounds is the exclusive prefix product $x_0 x_1 \\cdots x_{j-1}$ (equal to $1$ when $j = 0$).","title":"The list implementation computes the exclusive prefix products"},{"data_url":"scopes/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem.json","id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","kind":"scope","navigation":"module","parent":42464,"result":false,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","title":"Work\u2013span model and Brent's theorem"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/dag","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Dag","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#dag","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"A computation DAG on $n$ unit-time tasks, numbered $0, \\dots, n - 1$, assigns to each task $v$ a list of predecessors (the tasks $v$ waits for), each numbered below $v$; so the numbering is a topological order.","title":"Computation DAG"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/depth","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.depth","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#depth","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"The depth of a task $v$ is the number of tasks on a longest path ending at $v$: it is $1$ plus the largest depth of a predecessor of $v$, or $1$ if $v$ has no predecessors.","title":"Depth of a task in a computation DAG"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/span","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.span","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#span","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"The span of a computation DAG is the number of tasks on a longest path, that is, the largest depth of a task, and $0$ when there are no tasks.","title":"Span of a computation DAG"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/depth-lt-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.depth_lt_of_mem","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#depth-lt-of-mem","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"In a computation DAG, if $u$ is a predecessor of $v$, then the depth of $u$ is strictly less than the depth of $v$.","title":"Predecessors have smaller depth"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/depths-up-to","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.depthsUpTo","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#depths-up-to","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For $k \\le n$, the list of the depths of tasks $0, \\dots, k - 1$ of a computation DAG on $n$ tasks, computed in topological order: the empty list for $k = 0$, and for $k + 1$ the list for $k$ followed by $1$ plus the largest entry of that list at a predecessor of task $k$ (or $1$ if task $k$ has no predecessors).","title":"Depths of an initial segment of tasks"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/depth-table","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.depthTable","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#depth-table","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"The list of the depths of tasks $0, \\dots, n - 1$ of a computation DAG on $n$ tasks, computed once each in topological order: the entry for task $k$ is $1$ plus the largest entry at a predecessor of $k$, or $1$ if $k$ has no predecessors.","title":"Table of depths"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/get-d-depth-table","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.getD_depthTable","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#get-d-depth-table","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For every task $v$ of a computation DAG, the entry of the table of depths at position $v$ equals the depth of $v$.","title":"The table of depths is correct"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/schedule","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#schedule","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"A schedule of a computation DAG on $n$ tasks on $p$ processors assigns to each task $v$ a step $t(v) \\in \\mathbb{N}$ in which it runs, such that $t(u) < t(v)$ whenever $u$ is a predecessor of $v$, and for every step $s$ at most $p$ tasks $v$ have $t(v) = s$.","title":"Schedule of a computation DAG"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/length","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.length","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#length","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"The length of a schedule is the number of steps it takes: one more than the latest step in which a task runs (steps are numbered from $0$), or $0$ if there are no tasks.","title":"Length of a schedule"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/work-le","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.work_le","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#work-le","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For a computation DAG on $n$ tasks, every schedule of it on $p$ processors that takes $T$ steps satisfies $n \\le p T$. In particular, for $p > 0$ the schedule takes at least $n / p$ steps.","title":"The work law"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/depth-le-time","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.depth_le_time","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#depth-le-time","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For every schedule of a computation DAG on $p$ processors, with steps numbered from $0$, every task $v$ runs in a step $t$ with $\\operatorname{depth}(v) \\le t + 1$.","title":"Depth bounds the step of a task"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/span-le","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.span_le","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#span-le","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"No schedule is shorter than the span.","title":"The span law"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/level-card","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.levelCard","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#level-card","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For a computation DAG and a natural number $d$, the size of level $d$ is the number of tasks of depth $d$.","title":"Level size"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/level-steps","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.levelSteps","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#level-steps","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For a computation DAG and natural numbers $p$ and $d$, let $L_d$ be the number of tasks of depth $d$. The number of steps of level $d$ on $p$ processors is $\\lceil L_d / p \\rceil$ if $p \\ge 1$, and $0$ if $p = 0$.","title":"Steps of a level"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/offset","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.offset","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#offset","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For a computation DAG and natural numbers $p$ and $d$, let $s_e$ be the number of steps of level $e$ on $p$ processors: $\\lceil L_e / p \\rceil$ if $p \\ge 1$ and $0$ if $p = 0$, where $L_e$ is the number of tasks of depth $e$. The first step of level $d$ is $\\sum_{e < d} s_e$.","title":"First step of a level"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/rank","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.rank","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#rank","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"In a computation DAG, the rank of a task $v$ is the number of tasks of the same depth as $v$ with a smaller number than $v$. Taken in numbering order, the tasks of a level have ranks $0, 1, 2, \\dots$","title":"Rank within a level"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/level-time","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.levelTime","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#level-time","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG and $p$ a natural number. If $p \\ge 1$, for each $d$ let $L_d$ be the number of tasks of depth $d$ and $o_d = \\sum_{e < d} \\lceil L_e / p \\rceil$, and for each task $v$ let $r(v)$ be the number of tasks of the same depth as $v$ with a smaller number; the step of a task $v$ is $o_{\\operatorname{depth}(v)} + \\lfloor r(v) / p \\rfloor$. If $p = 0$, the step of every task is $0$.","title":"Step of a task in Brent's schedule"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/depth-eq-of-level-time-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.depth_eq_of_levelTime_eq","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#depth-eq-of-level-time-eq","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG and $p \\ge 1$. If two tasks $u$ and $v$ run in the same step of Brent's level-by-level schedule on $p$ processors, then $u$ and $v$ have the same depth.","title":"No step of Brent's level schedule mixes levels"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/rank-lt-rank","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.rank_lt_rank","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#rank-lt-rank","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"In a computation DAG, if tasks $u$ and $v$ have the same depth and $u$ has a smaller number than $v$, then the rank of $u$ is smaller than the rank of $v$, where the rank of a task is the number of tasks of the same depth with a smaller number. In particular, ranks are distinct within a level.","title":"Ranks increase within a level"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/level-schedule","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.levelSchedule","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#level-schedule","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG and $p \\ge 1$. For each $d$ let $L_d$ be the number of tasks of depth $d$ and $o_d = \\sum_{e < d} \\lceil L_e / p \\rceil$, and for each task $v$ let $r(v)$ be the number of tasks of the same depth as $v$ with a smaller number. Brent's level schedule is the schedule of $G$ on $p$ processors that runs each task $v$ at step $o_{\\operatorname{depth}(v)} + \\lfloor r(v) / p \\rfloor$.","title":"Brent's level schedule"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/level-card-zero","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.levelCard_zero","parent":42529,"reading_title":{"format":"inline-parts-v1","parts":[["text","Level "],["math","\\(0\\)"],["text"," is empty"]],"source":"Level $0$ is empty"},"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#level-card-zero","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"In every computation DAG no task has depth $0$, so the number of tasks of depth $0$ is $0$.","title":"Level $0$ is empty"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/level-schedule-length","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.levelSchedule_length","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#level-schedule-length","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG with $n$ tasks and span $D$, and let $p \\ge 1$. If Brent's level schedule of $G$ on $p$ processors takes $T$ steps, then $pT \\le n + (p - 1) D$, that is, $T \\le (n - D)/p + D$.","title":"Brent's theorem"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/brent","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.brent","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#brent","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG on $n$ unit-time tasks with span $D$, and let $p \\ge 1$. Then some schedule of $G$ on $p$ processors has length $T$ with $pT \\le n + (p - 1)D$, and every schedule of $G$ on $p$ processors has a length $T'$ with $n \\le pT'$ (the work law) and $D \\le T'$ (the span law).","title":"Brent's theorem with the work and span laws"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/load","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.load","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#load","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"The load of step $t$ in a schedule $S$ of a computation DAG is the number of tasks that $S$ runs in step $t$.","title":"Load of a step"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/ready","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.Ready","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#ready","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"In a schedule $S$ of a task graph, a task $v$ is ready at step $t$ if $v$ runs in a step later than $t$ and every predecessor of $v$ runs in a step earlier than $t$.","title":"Ready task"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/greedy","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.Greedy","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#greedy","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"A schedule $S$ of a computation DAG on $p$ processors is greedy if, for every step $t$ and every task that is ready at step $t$, exactly $p$ tasks run in step $t$: a step leaves a ready task waiting only when all $p$ processors are busy.","title":"Greedy schedule"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/greedy-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.greedy_iff","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#greedy-iff","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $S$ be a schedule of a computation DAG on $p$ processors taking $T$ steps. Then $S$ is greedy if and only if, for every step $t < T$ and every task $v$ that is ready at step $t$, exactly $p$ tasks run in step $t$.","title":"Greediness need only be checked at the steps of the schedule"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/incomplete","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.incomplete","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#incomplete","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For a schedule $S$ of a computation DAG on $p$ processors and a natural number $t$, the number of incomplete steps before step $t$ is the number of steps $s < t$ in which $S$ runs fewer than $p$ tasks.","title":"Number of incomplete steps"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/incomplete-lt-depth","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.incomplete_lt_depth","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#incomplete-lt-depth","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"In a greedy schedule, fewer than `depth v` incomplete steps precede the step of task `v`.","title":"Each incomplete step advances the critical path"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/sum-load","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.sum_load","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#sum-load","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"For a schedule $S$ of a task graph with $n$ tasks that takes $T$ steps, the numbers of tasks run in the steps $0, 1, \\dots, T - 1$ add up to $n$.","title":"The step loads add up to the number of tasks"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/mul-complete-le","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.mul_complete_le","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#mul-complete-le","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG with $n$ tasks, and let $S$ be a schedule of $G$ on $p$ processors taking $T$ steps. If $C$ is the number of steps $t < T$ in which $S$ runs at least $p$ tasks (the complete steps, where all $p$ processors are busy), then $pC \\le n$.","title":"Complete steps are bounded by the work"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/incomplete-length-le","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.incomplete_length_le","parent":42529,"reading_title":{"format":"inline-parts-v1","parts":[["text","A greedy schedule has at most "],["math","\\(D\\)"],["text"," incomplete steps"]],"source":"A greedy schedule has at most $D$ incomplete steps"},"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#incomplete-length-le","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG with span $D$, and let $S$ be a greedy schedule of $G$ on $p$ processors taking $T$ steps. Then at most $D$ of the steps $t < T$ are incomplete, that is, run fewer than $p$ tasks.","title":"A greedy schedule has at most $D$ incomplete steps"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/exists-le-time","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.exists_le_time","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#exists-le-time","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $S$ be a schedule of a computation DAG on $p$ processors taking $T$ steps, numbered from $0$. For every step $t < T$ there is a task that $S$ runs at step $t$ or later.","title":"Every step of a schedule has a task running at or after it"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/one-le-load","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.one_le_load","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#one-le-load","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $S$ be a greedy schedule of a task graph on $p$ processors that takes $T$ steps, numbered $0, \\dots, T - 1$. Then every step $t < T$ of $S$ runs at least one task.","title":"A greedy schedule never idles"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/greedy-length-le","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.greedy_length_le","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#greedy-length-le","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG with $n$ tasks and span $D$, and let $S$ be a greedy schedule of $G$ on $p$ processors taking $T$ steps. Then $pT \\le n + (p - 1)D$.","title":"The greedy scheduling bound"},{"id":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/greedy-length","kind":"node","lean":"Cslib.Algorithms.Lean.Parallel.Brent.Schedule.greedy_length","parent":42529,"result":true,"route":"../roadmap/ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem/index.html#greedy-length","scope":"ComputerScience/parallel-algorithms/work-span-model-and-brent-s-theorem","status":"fully_proved","summary":"Let $G$ be a computation DAG with $n$ tasks and span $D$, and let $S$ be a greedy schedule of $G$ on $p$ processors taking $T$ steps. Then $pT \\le n + pD$, that is, $T \\le n/p + D$ when $p \\ge 1$.","title":"The work-span bound for greedy schedules"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms.json","id":"ComputerScience/algebraic-and-numerical-algorithms","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms","status":"fully_proved","summary":"Algorithms on numbers, polynomials and matrices: integer and polynomial multiplication (Karatsuba, FFT and convolution), matrix multiplication (iterative, recursive, Strassen), Horner evaluation, floating-point rounding, modular exponentiation and linear congruences, Miller-Rabin primality testing, and bit-level polynomial and bitvector arithmetic.","title":"Algebraic and numerical algorithms"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists.json","id":"ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists","status":"fully_proved","title":"Bitvectors and Boolean lists"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists/to-bool-list-le","kind":"node","lean":"BitVec.toBoolListLE","parent":42567,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists/index.html#to-bool-list-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists","status":"fully_proved","summary":"The bits of a bitvector in little-endian order, least significant bit first.","title":"Little-endian bits of a bitvector"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists/equiv-bool-list-le","kind":"node","lean":"BitVec.equivBoolListLE","parent":42567,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists/index.html#equiv-bool-list-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists","status":"fully_proved","summary":"Fixed-width little-endian Boolean lists are equivalent to bitvectors of the same width.","title":"Little-endian Boolean lists are equivalent to bitvectors"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists/sum-map-to-nat-eq-count-true","kind":"node","lean":"List.sum_map_toNat_eq_count_true","parent":42567,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists/index.html#sum-map-to-nat-eq-count-true","scope":"ComputerScience/algebraic-and-numerical-algorithms/bitvectors-and-boolean-lists","status":"fully_proved","summary":"The sum of the natural values of a Boolean list is its number of `true` entries.","title":"Sum of a Boolean list counts its true entries"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors.json","id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","title":"Binary polynomial arithmetic on bitvectors"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/degree-of-coeff-list-lt","kind":"node","lean":"Polynomial.degree_ofCoeffList_lt","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#degree-of-coeff-list-lt","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"A polynomial built from a low-coefficient-first list has degree strictly less than the list length, including for an empty list and over a trivial semiring.","title":"Degree bound for a polynomial built from a coefficient list"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/of-coeff-list-of-fn-coeff","kind":"node","lean":"Polynomial.ofCoeffList_ofFn_coeff","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#of-coeff-list-of-fn-coeff","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Reconstruct a polynomial from any fixed-width list containing all its possibly nonzero coefficients.","title":"Reconstructing a polynomial from a fixed-width coefficient list"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/to-bit-vec","kind":"node","lean":"Polynomial.toBitVec","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#to-bit-vec","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Extract the lowest `s` binary coefficients of a polynomial as a fixed-width bitvector. Coefficients at indices at least `s` are deliberately truncated.","title":"Bitvector of the low coefficients of a binary polynomial"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/to-polynomial","kind":"node","lean":"BitVec.toPolynomial","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#to-polynomial","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"The polynomial over `ZMod 2` whose coefficient of `X ^ i` is the `i`-th least-significant bit of `w`.","title":"Binary polynomial of a bitvector"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/coeff-to-polynomial","kind":"node","lean":"BitVec.coeff_toPolynomial","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#coeff-to-polynomial","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Every coefficient of the polynomial encoding is the corresponding least-significant bit; indices outside the fixed width have coefficient zero.","title":"Coefficients of the binary polynomial of a bitvector"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/to-bit-vec-to-polynomial","kind":"node","lean":"BitVec.toBitVec_toPolynomial","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#to-bit-vec-to-polynomial","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Extracting the coefficients of an encoded word recovers the original fixed-width word.","title":"Bitvector encoding recovers the original word"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/to-polynomial-to-bit-vec","kind":"node","lean":"Polynomial.toPolynomial_toBitVec","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#to-polynomial-to-bit-vec","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"A polynomial whose degree is below the selected width is recovered after coefficient extraction and fixed-width encoding.","title":"Polynomial encoding recovers a low-degree polynomial"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/to-polynomial-xor","kind":"node","lean":"BitVec.toPolynomial_xor","parent":42571,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bitwise xor corresponds to polynomial addition over "],["math","\\(\\mathbb{F}_2\\)"]],"source":"Bitwise xor corresponds to polynomial addition over $\\mathbb{F}_2$"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#to-polynomial-xor","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Polynomial encoding turns bitwise xor into addition over `ZMod 2`; this is not the carried integer addition on bitvectors.","title":"Bitwise xor corresponds to polynomial addition over $\\mathbb{F}_2$"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/monic-modulus","kind":"node","lean":"BitVec.monicModulus","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#monic-modulus","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"The monic polynomial obtained by adjoining the width term `X ^ s` to the encoded lower coefficients.","title":"Monic binary modulus of a fixed-width word"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/monic-modulus-monic","kind":"node","lean":"BitVec.monicModulus_monic","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#monic-modulus-monic","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"The binary modulus associated to a fixed-width word is monic, including at width zero.","title":"The binary modulus is monic"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/monic-modulus-nat-degree","kind":"node","lean":"BitVec.monicModulus_natDegree","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#monic-modulus-nat-degree","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"The natural degree of the binary modulus is exactly the width of its lower coefficients.","title":"Degree of the binary modulus"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-equiv","kind":"node","lean":"BitVec.adjoinRootEquiv","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-equiv","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Fixed-width binary words are in bijection with residues modulo their associated monic polynomial. This is an ordinary equivalence because builtin bitvector addition has carries.","title":"Fixed-width binary words as residues modulo a monic polynomial"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-equiv-apply","kind":"node","lean":"BitVec.adjoinRootEquiv_apply","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-equiv-apply","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"The forward equivalence is the canonical quotient constructor applied to the polynomial encoding.","title":"Forward map of the binary word residue equivalence"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-equiv-symm-apply","kind":"node","lean":"BitVec.adjoinRootEquiv_symm_apply","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-equiv-symm-apply","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"The inverse equivalence extracts the canonical remainder's lowest `s` binary coefficients.","title":"Inverse map of the binary word residue equivalence"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-equiv-coeff","kind":"node","lean":"BitVec.adjoinRootEquiv_coeff","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-equiv-coeff","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"The canonical remainder coordinates of an encoded word are exactly its low-first bits.","title":"Remainder coefficients of an encoded binary word are its bits"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-equiv-xor","kind":"node","lean":"BitVec.adjoinRootEquiv_xor","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-equiv-xor","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Bitwise xor corresponds to quotient addition; builtin bitvector addition does not.","title":"Bitwise xor corresponds to residue addition"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-root-pow","kind":"node","lean":"BitVec.adjoinRoot_root_pow","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-root-pow","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"The defining quotient equation reduces `root ^ s` to the encoded lower coefficients.","title":"Reduction of the root power in a binary residue ring"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/mul-xmod","kind":"node","lean":"BitVec.mulXMod","parent":42571,"reading_title":{"format":"inline-parts-v1","parts":[["text","Multiplication by "],["math","\\(X\\)"],["text"," of a binary polynomial residue"]],"source":"Multiplication by $X$ of a binary polynomial residue"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#mul-xmod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Multiply a binary residue by `X`, reducing by `X ^ s + g` when its top bit is set.","title":"Multiplication by $X$ of a binary polynomial residue"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/mul-mod","kind":"node","lean":"BitVec.mulMod","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#mul-mod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Multiply binary residues by low-first Horner evaluation of the second operand's bits.","title":"Modular multiplication of binary polynomial residues"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/pow-mod","kind":"node","lean":"BitVec.powMod","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#pow-mod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Raise a binary residue to a natural power by tail-recursive repeated multiplication.","title":"Modular powering of binary polynomial residues"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/eval-mul-xmod","kind":"node","lean":"BitVec.eval\u2082_mulXMod","parent":42571,"reading_title":{"format":"inline-parts-v1","parts":[["text","Evaluation of modular multiplication by "],["math","\\(X\\)"]],"source":"Evaluation of modular multiplication by $X$"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#eval-mul-xmod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Multiplication by `X` has its expected evaluation in every commutative ring receiving `ZMod 2`.","title":"Evaluation of modular multiplication by $X$"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/eval-mul-mod","kind":"node","lean":"BitVec.eval\u2082_mulMod","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#eval-mul-mod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Horner multiplication has its expected evaluation in every commutative ring receiving `ZMod 2`; the second operand may have any width.","title":"Evaluation of modular Horner multiplication"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/eval-pow-mod","kind":"node","lean":"BitVec.eval\u2082_powMod","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#eval-pow-mod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Repeated multiplication has its expected power evaluation for positive result width.","title":"Evaluation of modular powering"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-equiv-mul-xmod","kind":"node","lean":"BitVec.adjoinRootEquiv_mulXMod","parent":42571,"reading_title":{"format":"inline-parts-v1","parts":[["text","Modular multiplication by "],["math","\\(X\\)"],["text"," matches the quotient root"]],"source":"Modular multiplication by $X$ matches the quotient root"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-equiv-mul-xmod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Modular multiplication by `X` agrees with multiplication by the quotient root.","title":"Modular multiplication by $X$ matches the quotient root"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-equiv-mul-mod","kind":"node","lean":"BitVec.adjoinRootEquiv_mulMod","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-equiv-mul-mod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Modular Horner multiplication agrees with multiplication in the polynomial quotient.","title":"Modular multiplication matches the polynomial quotient"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/adjoin-root-equiv-pow-mod","kind":"node","lean":"BitVec.adjoinRootEquiv_powMod","parent":42571,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors/index.html#adjoin-root-equiv-pow-mod","scope":"ComputerScience/algebraic-and-numerical-algorithms/binary-polynomial-arithmetic-on-bitvectors","status":"fully_proved","summary":"Modular powering agrees with powering in the polynomial quotient, including width zero.","title":"Modular powering matches the polynomial quotient"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform.json","id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","title":"Fast Fourier transform"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/dft-sum","kind":"node","lean":"MathlibExt.Analysis.Fourier.FastFourierTransformWanted.dftSum","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#dft-sum","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a natural number $N$, a complex number $\\omega$, a sequence $f : \\mathbb{N} \\to \\mathbb{C}$ and a natural number $k$, the unnormalized discrete Fourier sum of length $N$ at frequency $k$ is $\\sum_{j=0}^{N-1} \\omega^{jk} f(j)$.","title":"Discrete Fourier sum"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/dft-sum-two-mul","kind":"node","lean":"MathlibExt.Analysis.Fourier.FastFourierTransformWanted.dftSum_two_mul","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#dft-sum-two-mul","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For every natural number $n$, every $\\omega \\in \\mathbb{C}$, every $f : \\mathbb{N} \\to \\mathbb{C}$ and every natural number $K$,","title":"Even\u2013odd splitting of the DFT sum"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/dft-sum-mod","kind":"node","lean":"MathlibExt.Analysis.Fourier.FastFourierTransformWanted.dftSum_mod","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#dft-sum-mod","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $n$ be a natural number and let $\\omega \\in \\mathbb{C}$ satisfy $\\omega^n = 1$. Then for every $f : \\mathbb{N} \\to \\mathbb{C}$ and every natural number $K$, the discrete Fourier sum of length $n$ at frequency $K$ equals the one at frequency $K \\bmod n$:","title":"The DFT sum at a root of unity depends only on the frequency modulo the length"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/cooley-tukey","kind":"node","lean":"MathlibExt.Analysis.Fourier.FastFourierTransformWanted.cooley_tukey","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#cooley-tukey","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For $\\omega \\in \\mathbb{C}$, $f : \\mathbb{N} \\to \\mathbb{C}$ and natural numbers $N$ and $k$, write $D_N(\\omega, f, k) = \\sum_{j < N} \\omega^{jk} f(j)$ for the discrete Fourier sum. Let $n$ be a natural number and let $\\omega \\in \\mathbb{C}$ satisfy $\\omega^{2n} = 1$. Then for every $f : \\mathbb{N} \\to \\mathbb{C}$ and every natural number $K$,","title":"Radix-2 Cooley\u2013Tukey splitting"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-cost","kind":"node","lean":"MathlibExt.Analysis.Fourier.FastFourierTransformWanted.fftCost","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-cost","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a natural number $n$, the closed-form FFT operation count is $n \\lfloor \\log_2 n \\rfloor$, where $\\lfloor \\log_2 n \\rfloor$ is the largest $m$ with $2^m \\le n$ (taken to be $0$ when $n = 0$).","title":"FFT operation count"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-cost-recurrence","kind":"node","lean":"MathlibExt.Analysis.Fourier.FastFourierTransformWanted.fftCost_recurrence","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-cost-recurrence","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $C(n) = n \\lfloor \\log_2 n \\rfloor$ be the closed-form FFT operation count. For every natural number $n > 0$, $C(2n) = 2\\,C(n) + 2n$.","title":"The FFT cost satisfies the radix-2 recurrence"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-split","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftSplit","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-split","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a commutative ring $R$, a natural number $m$ and a vector $x = (x_0, \\dots, x_{2m-1}) \\in R^{2m}$, the split returns the pair of vectors $(x_0, x_2, \\dots, x_{2m-2})$ and $(x_1, x_3, \\dots, x_{2m-1})$ in $R^m$ of the entries of $x$ at even and at odd indices, in order, at a time cost of $4m$ (the charged reads and writes).","title":"Even\u2013odd split for the FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-butterfly","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftButterfly","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-butterfly","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a commutative ring $R$, an element $\\omega \\in R$, vectors $q, r \\in R^m$, a state $(w, y)$ with $w \\in R$ and $y \\in R^{2m}$, and an index $k < m$, the butterfly at $k$ returns the state $(w \\omega, y')$, where $y'$ is obtained from $y$ by setting entry $k$ to $q_k + w r_k$ and entry $k + m$ to $q_k - w r_k$, at a time cost of $8$.","title":"One FFT butterfly"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-butterflies","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftButterflies","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-butterflies","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a commutative ring $R$, an element $\\omega \\in R$ and vectors $q, r \\in R^m$ and $y \\in R^{2m}$, the butterfly pass starts from the state $(1, y)$, applies the FFT butterfly with $\\omega$, $q$ and $r$ at the indices $k = 0, 1, \\dots, m - 1$ in increasing order, and returns the final state; its time cost is the sum of the costs of these $m$ butterflies.","title":"FFT butterfly pass"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a commutative ring $R$, a natural number $d$, an element $\\omega \\in R$ and a vector $x \\in R^{2^d}$, the FFT of $x$ with root $\\omega$ is defined by recursion on $d$. For $d = 0$ it returns $x$ at cost $0$. For $d + 1$ it splits $x$ into its even- and odd-indexed subvectors $e, o \\in R^{2^d}$ (cost $4 \\cdot 2^d$), computes $\\omega^2$ (cost $1$), computes the FFTs $q$ of $e$ and $r$ of $o$ with root $\\omega^2$,\u2026","title":"Radix-two fast Fourier transform"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-array","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftArray","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-array","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $\\omega \\in R$ and $x$ an array over $R$ of length $n$, and let $d = \\lfloor \\log_2 n \\rfloor$ (with $d = 0$ when $n = 0$). If $n = 2^d$, the array FFT runs the radix-two FFT with root $\\omega$ on $x$, viewed as a vector of length $2^d$, and returns its output as an array, with the cost of that run. Otherwise it returns no result, at cost $0$; the length check itself is free.","title":"Radix-two FFT on arrays"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-array-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftArray_eq_none_iff","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-array-eq-none-iff","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $\\omega \\in R$ and $x$ an array over $R$. The array FFT of $x$ with root $\\omega$ returns no result if and only if the length of $x$ is different from $2^d$ for every natural number $d$.","title":"The array FFT rejects exactly the non-powers of two"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-butterfly-ret-low","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftButterfly_ret_low","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-butterfly-ret-low","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $\\omega \\in R$, $m$ a natural number, $q, r$ vectors over $R$ of length $m$, $w \\in R$ a twiddle, $o$ an output vector over $R$ of length $2m$, and $k < m$. Then the butterfly step at index $k$ applied to the state $(w, o)$ produces an output vector whose entry at index $k$ is $q_k + w r_k$.","title":"Lower write of a butterfly step"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-butterfly-ret-high","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftButterfly_ret_high","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-butterfly-ret-high","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $\\omega \\in R$, $m$ a natural number, $q, r$ vectors over $R$ of length $m$, $w \\in R$ a twiddle, $o$ an output vector over $R$ of length $2m$, and $k < m$. Then the butterfly step at index $k$ applied to the state $(w, o)$ produces an output vector whose entry at index $k + m$ is $q_k - w r_k$.","title":"Upper write of a butterfly step"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-butterfly-ret-other","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftButterfly_ret_other","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-butterfly-ret-other","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $\\omega \\in R$, $m$ a natural number, $q, r$ vectors over $R$ of length $m$, $w \\in R$ a twiddle, $o$ an output vector over $R$ of length $2m$, and $k < m$. Then for every index $i < 2m$ with $i \\ne k$ and $i \\ne k + m$, the butterfly step at index $k$ applied to the state $(w, o)$ produces an output vector whose entry at index $i$ is $o_i$.","title":"A butterfly step leaves other entries unchanged"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-butterflies-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftButterflies_ret","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-butterflies-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $\\omega \\in R$, $m$ a natural number, $q = (q_0, \\dots, q_{m-1})$ and $r = (r_0, \\dots, r_{m-1})$ vectors over $R$, and $o$ a vector over $R$ of length $2m$. The butterfly loop with root $\\omega$ on $q$ and $r$, started from twiddle $1$ and output vector $o$, ends with twiddle $\\omega^m$ and an output vector $y$ such that $y_k = q_k + \\omega^k r_k$ and $y_{k+m} = q_k - \\omega^k r_k$\u2026","title":"Result of the butterfly loop"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-split-ret-even","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftSplit_ret_even","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-split-ret-even","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a type, $m$ a natural number and $x = (x_0, \\dots, x_{2m-1})$ a vector of length $2m$ over $R$. For every $k < m$, entry $k$ of the first (even) vector returned by the FFT split of $x$ is $x_{2k}$.","title":"The FFT split extracts the even-indexed entries"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-split-ret-odd","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftSplit_ret_odd","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-split-ret-odd","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a type, $m$ a natural number and $x = (x_0, \\dots, x_{2m-1})$ a vector of length $2m$ over $R$. For every $k < m$, entry $k$ of the second (odd) vector returned by the FFT split of $x$ is $x_{2k+1}$.","title":"The FFT split extracts the odd-indexed entries"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-ret-zero","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_ret_zero","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-ret-zero","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring. For every $\\omega \\in R$ and every vector $x \\in R^{2^0}$, the vector returned by the radix-two FFT of depth $0$ with root $\\omega$ on $x$ is $x$ itself.","title":"The FFT of depth zero returns its input"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-ret-succ-low","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_ret_succ_low","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-ret-succ-low","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $d$ a natural number, $\\omega \\in R$ and $x \\in R^{2^{d+1}}$, with entries indexed from $0$. Let $e, o \\in R^{2^d}$ be the even- and odd-indexed subsequences of $x$ returned by the FFT split, and let $E$ and $O$ be the vectors returned by the radix-two FFT of depth $d$ with root $\\omega \\cdot \\omega$ on $e$ and on $o$. Then for every $k < 2^d$, entry $k$ of the vector returned by the\u2026","title":"Low half of the FFT butterfly recurrence"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-ret-succ-high","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_ret_succ_high","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-ret-succ-high","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $d$ a natural number, $\\omega \\in R$ and $x \\in R^{2^{d+1}}$, with entries indexed from $0$. Let $e, o \\in R^{2^d}$ be the even- and odd-indexed subsequences of $x$ returned by the FFT split, and let $E$ and $O$ be the vectors returned by the radix-two FFT of depth $d$ with root $\\omega \\cdot \\omega$ on $e$ and on $o$. Then for every $k < 2^d$, entry $k + 2^d$ of the vector returned\u2026","title":"High half of the FFT butterfly recurrence"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-split-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftSplit_time","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-split-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For every natural number $m$ and every vector $x$ of length $2m$, splitting $x$ into its even-indexed and its odd-indexed subsequences costs exactly $4m$ charged operations: two reads and two writes for each of the $m$ even/odd pairs.","title":"Cost of the FFT even/odd split"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-butterfly-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftButterfly_time","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-butterfly-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring and $m$ a natural number. For every $\\omega \\in R$, all vectors $q, r \\in R^m$, every current twiddle $w \\in R$ and output vector of length $2m$ over $R$, and every index $k < m$, one butterfly step costs exactly $8$ charged operations: two reads, four scalar operations and two writes.","title":"Cost of one FFT butterfly"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-butterflies-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftButterflies_time","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-butterflies-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring and $m$ a natural number. For every $\\omega \\in R$, all vectors $q, r \\in R^m$ and every output vector of length $2m$ over $R$, the pass that, starting from twiddle $1$, applies one butterfly for each index $k < m$ in increasing order costs exactly $8m$ charged operations.","title":"Cost of the FFT butterfly pass"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-time-add-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_time_add_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-time-add-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring. For every natural number $d$, every $\\omega \\in R$ and every vector $x \\in R^{2^d}$, the radix-two FFT of $x$ with root $\\omega$ costs $T$ charged operations, where $T + 1 = (7d + 1) \\cdot 2^d$; that is, $T = (7d + 1) \\cdot 2^d - 1$.","title":"Exact cost of the radix-two FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-time-add-one-log2","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_time_add_one_log2","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-time-add-one-log2","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring. For every natural number $d$, every $\\omega \\in R$ and every vector $x$ over $R$ of length $n = 2^d$, the radix-two FFT of $x$ with root $\\omega$ costs $T$ charged operations, where $T + 1 = (7 \\log_2 n + 1) \\cdot n$.","title":"Exact cost of the radix-two FFT in terms of the input length"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_time_le","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-time-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring. For every natural number $d$, every $\\omega \\in R$ and every vector $x \\in R^{2^d}$, the radix-two FFT of $x$ with root $\\omega$ costs at most $8 \\cdot 2^d \\cdot d$ charged operations, that is, at most eight per entry per recursion level.","title":"Cost bound for the radix-two FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_correct","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-correct","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For every natural number $d$, every primitive $2^d$-th root of unity $\\omega \\in \\mathbb{C}$, every vector $x = (x_0, \\dots, x_{2^d - 1})$ in $\\mathbb{C}^{2^d}$ and every $k < 2^d$, the $k$-th entry of the radix-two FFT of $x$ with root $\\omega$ equals the discrete Fourier sum $\\sum_{j=0}^{2^d - 1} \\omega^{jk} x_j$.","title":"Correctness of the radix-two FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-array-ret-of-vector","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftArray_ret_of_vector","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-array-ret-of-vector","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $d$ a natural number, $\\omega \\in R$ and $x$ a vector over $R$ of length $2^d$. Then the array FFT of $x$, viewed as an array, with root $\\omega$ returns a result, namely the output of the radix-two FFT of the length-$2^d$ vector $x$ with root $\\omega$, viewed as an array.","title":"The array FFT agrees with the vector FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-array-time-of-vector","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftArray_time_of_vector","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-array-time-of-vector","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $d$ a natural number, $\\omega \\in R$ and $x$ a vector over $R$ of length $2^d$. Then the cost of the array FFT of $x$, viewed as an array, with root $\\omega$ equals the cost of the radix-two FFT of the length-$2^d$ vector $x$ with root $\\omega$.","title":"The array FFT costs the same as the vector FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-array-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftArray_correct","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-array-correct","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $\\omega \\in \\mathbb{C}$ a primitive $2^d$-th root of unity, $x_0, \\dots, x_{2^d - 1}$ complex numbers and $k < 2^d$. Then the array FFT of the array $(x_0, \\dots, x_{2^d - 1})$ with root $\\omega$ returns an array $y$ of length $2^d$ whose entry at index $k$ is the discrete Fourier sum $\\sum_{j=0}^{2^d - 1} \\omega^{jk} x_j$.","title":"The array FFT computes the discrete Fourier transform"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-array-time-add-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftArray_time_add_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-array-time-add-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a commutative ring, $d$ a natural number, $\\omega \\in R$ and $x$ a vector over $R$ of length $2^d$. Then the cost $T$ of the array FFT of $x$, viewed as an array, with root $\\omega$ satisfies $T + 1 = (7d + 1) \\cdot 2^d$.","title":"Exact cost of the array FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-std-add-char-neg-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_stdAddChar_neg_one","parent":42598,"reading_title":{"format":"inline-parts-v1","parts":[["text","The radix-two FFT with root "],["math","\\(e^{-2\\pi i/n}\\)"],["text"," computes the DFT"]],"source":"The radix-two FFT with root $e^{-2\\pi i/n}$ computes the DFT"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-std-add-char-neg-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $n = 2^d$ and let $x_0, \\dots, x_{n-1}$ be complex numbers, indexed by $\\mathbb{Z}/n\\mathbb{Z}$. Run the radix-two FFT on $x$ with sample root $\\omega = e^{-2\\pi i/n}$, the value at $-1$ of the standard additive character of $\\mathbb{Z}/n\\mathbb{Z}$. Then for every $k < n$, entry $k$ of the returned vector equals the discrete Fourier transform coefficient\u2026","title":"The radix-two FFT with root $e^{-2\\pi i/n}$ computes the DFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-std-add-char-one-normalized","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_stdAddChar_one_normalized","parent":42598,"reading_title":{"format":"inline-parts-v1","parts":[["text","The radix-two FFT with root "],["math","\\(e^{2\\pi i/n}\\)"],["text",", divided by "],["math","\\(n\\)"],["text",", computes the inverse DFT"]],"source":"The radix-two FFT with root $e^{2\\pi i/n}$, divided by $n$, computes the inverse DFT"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-std-add-char-one-normalized","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $n = 2^d$ and let $x_0, \\dots, x_{n-1}$ be complex numbers, indexed by $\\mathbb{Z}/n\\mathbb{Z}$. Run the radix-two FFT on $x$ with sample root $\\omega = e^{2\\pi i/n}$, the value at $1$ of the standard additive character of $\\mathbb{Z}/n\\mathbb{Z}$. Then for every $k < n$, entry $k$ of the returned vector multiplied by $1/n$ equals the inverse discrete Fourier transform coefficient\u2026","title":"The radix-two FFT with root $e^{2\\pi i/n}$, divided by $n$, computes the inverse DFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-conjugate","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftConjugate","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-conjugate","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a type with a star operation $z \\mapsto \\bar z$ and a vector $x = (x_0, \\dots, x_{n-1})$ of length $n$ over it, this costed computation returns $(\\bar x_0, \\dots, \\bar x_{n-1})$ and charges $3n$ operations: one read, one conjugation and one write per entry.","title":"Charged entrywise conjugation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-scale","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftScale","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-scale","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a type with a multiplication, a scalar $c$ and a vector $x = (x_0, \\dots, x_{n-1})$ of length $n$ over it, this costed computation returns $(c x_0, \\dots, c x_{n-1})$ and charges $3n$ operations: one read, one multiplication and one write per entry.","title":"Charged entrywise scaling"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field that is a star ring, with star operation $z \\mapsto \\bar z$. For a natural number $d$, an element $\\omega \\in R$ and a vector $x \\in R^{2^d}$, the inverse FFT conjugates every entry of $x$, applies the radix-two FFT with root $\\omega$, conjugates every entry of the result, and multiplies every entry by the inverse $(2^d)^{-1}$ of $2^d$ in $R$ (which is $0$ when $2^d = 0$ in $R$). It charges\u2026","title":"Inverse FFT by conjugation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-pointwise","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPointwise","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-pointwise","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a type $R$ with a multiplication and vectors $x, y$ of length $n$ over $R$, the pointwise product returns the vector of length $n$ whose $i$-th entry is $x_i \\cdot y_i$ for every $i < n$, and charges $4n$ time units: two reads, one multiplication and one write per entry.","title":"Timed pointwise product"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-pad","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPad","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-pad","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For a type $R$ with a zero, a natural number $d$ and a vector $x$ of length $2^d$ over $R$, padding returns the vector of length $2^{d+1}$ whose $i$-th entry is $x_i$ for $i < 2^d$ and $0$ for $2^d \\le i < 2^{d+1}$, and charges $3 \\cdot 2^d$ time units.","title":"Zero padding to twice the length"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircular","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field that is a star ring, $d$ a natural number, $\\omega \\in R$, and $x, y$ vectors of length $2^d$ over $R$. Circular convolution runs the timed radix-two FFT of length $2^d$ with root $\\omega$ on $x$ and then on $y$, multiplies the two transforms pointwise, and applies the timed inverse FFT of length $2^d$ with root $\\omega$ to the product. It returns the result of the inverse transform, and its cost\u2026","title":"FFT circular convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field that is a star ring, $d$ a natural number, $\\omega \\in R$, and $x, y$ vectors of length $2^d$ over $R$. Linear convolution zero-pads $x$ and $y$ to length $2^{d+1}$ and returns the FFT circular convolution of the two padded vectors at length $2^{d+1}$ with root $\\omega$. Its cost is the cost of the two paddings plus the cost of the circular convolution.","title":"FFT linear convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-array","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircularArray","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-array","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field that is a star ring, $\\omega \\in R$, and $x, y$ arrays over $R$, where $x$ has length $n$. If $n = 2^d$ for some natural number $d$ and $y$ also has length $n$, the procedure returns the FFT circular convolution of $x$ and $y$ (read as vectors of length $2^d$) with root $\\omega$, converted to an array, at the cost of that convolution. Otherwise it returns no result, at cost $0$.","title":"FFT circular convolution on arrays"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-array","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinearArray","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-array","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field that is a star ring, $\\omega \\in R$, and $x, y$ arrays over $R$, where $x$ has length $n$. If $n = 2^d$ for some natural number $d$ and $y$ also has length $n$, the procedure returns the FFT linear convolution of $x$ and $y$ (read as vectors of length $2^d$) with root $\\omega$, an array of length $2n$, at the cost of that convolution. Otherwise it returns no result, at cost $0$.","title":"FFT linear convolution on arrays"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-pointwise-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPointwise_ret","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-pointwise-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a type with a multiplication, let $n$ be a natural number and let $x$ and $y$ be vectors of length $n$ over $R$. For every index $i < n$, entry $i$ of the pointwise product of $x$ and $y$ equals $x_i y_i$.","title":"Entries of the pointwise product"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-pad-ret-low","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPad_ret_low","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-pad-ret-low","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a type with a zero, let $d$ be a natural number and let $x$ be a vector of length $2^d$ over $R$. For every index $i < 2^d$, entry $i$ of the zero-padded vector of length $2^{d+1}$ produced from $x$ equals $x_i$.","title":"Zero padding keeps the input prefix"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-pad-ret-high","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPad_ret_high","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-pad-ret-high","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a type with a zero, let $d$ be a natural number and let $x$ be a vector of length $2^d$ over $R$. For every index $i$ with $2^d \\le i < 2^{d+1}$, entry $i$ of the zero-padded vector of length $2^{d+1}$ produced from $x$ is $0$.","title":"Zero padding fills the second half with zeros"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-array-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircularArray_eq_none_iff","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-array-eq-none-iff","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $\\omega \\in R$, and let $x$ and $y$ be arrays over $R$. The array circular-convolution procedure with root $\\omega$ returns no result on $x$ and $y$ if and only if it is not the case that the length of $x$ is a power of two $2^k$ with $k \\ge 0$ and $y$ has the same length as $x$.","title":"Rejection by array circular convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-array-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinearArray_eq_none_iff","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-array-eq-none-iff","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $\\omega \\in R$, and let $x$ and $y$ be arrays over $R$. The array zero-padded linear-convolution procedure with root $\\omega$ returns no result on $x$ and $y$ if and only if it is not the case that the length of $x$ is a power of two $2^k$ with $k \\ge 0$ and $y$ has the same length as $x$.","title":"Rejection by array linear convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-conjugate-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_conjugate_ret","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-conjugate-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $N = 2^d$, let $\\omega \\in \\mathbb{C}$ be a primitive $N$-th root of unity and let $x \\in \\mathbb{C}^N$. Conjugating every entry of $x$, running the FFT procedure with root $\\omega$ and conjugating every entry of the result gives the same vector as running the FFT procedure on $x$ with root $\\overline{\\omega}$.","title":"Conjugating around the FFT conjugates the root"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft_ret","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $N = 2^d$, let $\\omega \\in \\mathbb{C}$ be a primitive $N$-th root of unity and let $x \\in \\mathbb{C}^N$. The output of the inverse FFT procedure on $x$ with root $\\omega$ is the output of the FFT procedure on $x$ with root $\\overline{\\omega}$ with every entry multiplied by $N^{-1}$.","title":"The inverse FFT is a normalized FFT at the conjugate root"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft-correct","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft_correct","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft-correct","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $N = 2^d$, let $\\omega \\in \\mathbb{C}$ be a primitive $N$-th root of unity and let $x \\in \\mathbb{C}^N$. For every $k < N$, entry $k$ of the output of the inverse FFT procedure on $x$ with root $\\omega$ equals $N^{-1}$ times the discrete Fourier sum of length $N$ with root $\\overline{\\omega}$ at frequency $k$ of the sequence $j \\mapsto x_j$ (set to $0$ for $j \\ge N$), that is,\u2026","title":"Correctness of the inverse FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft-std-add-char-neg-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft_stdAddChar_neg_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft-std-add-char-neg-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $N = 2^d$ and $x \\in \\mathbb{C}^N$, and run the inverse FFT procedure on $x$ with root $e^{-2\\pi i/N}$, the value at $-1$ of the standard additive character of $\\mathbb{Z}/N\\mathbb{Z}$. For every $k < N$, entry $k$ of its output equals the canonical inverse discrete Fourier transform of the function on $\\mathbb{Z}/N\\mathbb{Z}$ sending the class of $j < N$ to $x_j$, evaluated at the class of $k$; that is, it\u2026","title":"Inverse FFT at the negative-sign root computes the inverse DFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft-std-add-char-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft_stdAddChar_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft-std-add-char-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $N = 2^d$ and $x \\in \\mathbb{C}^N$, and run the inverse FFT procedure on $x$ with root $e^{2\\pi i/N}$, the value at $1$ of the standard additive character of $\\mathbb{Z}/N\\mathbb{Z}$. For every $k < N$, entry $k$ of its output equals $N^{-1}$ times the canonical discrete Fourier transform of the function on $\\mathbb{Z}/N\\mathbb{Z}$ sending the class of $j < N$ to $x_j$, evaluated at the class of $k$; that is,\u2026","title":"Inverse FFT at the positive-sign root computes the normalized forward DFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft-fft-std-add-char-neg-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft_fft_stdAddChar_neg_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft-fft-std-add-char-neg-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $N = 2^d$ and let $\\omega = e^{-2\\pi i/N}$, the value at $-1$ of the standard additive character of $\\mathbb{Z}/N\\mathbb{Z}$. For every $x \\in \\mathbb{C}^N$, running the FFT procedure on $x$ with root $\\omega$ and then the inverse FFT procedure on its output with root $\\omega$ returns $x$.","title":"Inverse FFT after FFT is the identity at the negative-sign root"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-ifft-std-add-char-neg-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_ifft_stdAddChar_neg_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-ifft-std-add-char-neg-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $N = 2^d$ and let $\\omega = e^{-2\\pi i/N}$, the value at $-1$ of the standard additive character of $\\mathbb{Z}/N\\mathbb{Z}$. For every $x \\in \\mathbb{C}^N$, running the inverse FFT procedure on $x$ with root $\\omega$ and then the FFT procedure on its output with root $\\omega$ returns $x$.","title":"FFT after inverse FFT is the identity at the negative-sign root"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircular_ret","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring conjugation, let $n = 2^d$, $\\omega \\in R$ and $x, y \\in R^n$. The vector returned by FFT circular convolution of $x$ and $y$ with root $\\omega$ equals the vector returned by the inverse FFT with root $\\omega$ applied to the vector returned by pointwise multiplication of $\\hat{x}$ and $\\hat{y}$, where $\\hat{x}$ and $\\hat{y}$ are the vectors returned by the radix-two FFT of $x$\u2026","title":"Output of FFT circular convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_ret","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring conjugation, let $\\omega \\in R$ and $x, y \\in R^{2^d}$. The vector returned by FFT linear convolution of $x$ and $y$ with root $\\omega$ equals the vector returned by FFT circular convolution, at length $2^{d+1}$ with root $\\omega$, of the vectors returned by zero-padding $x$ and $y$ to length $2^{d+1}$.","title":"Output of FFT linear convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-std-add-char-neg-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircular_stdAddChar_neg_one","parent":42598,"reading_title":{"format":"inline-parts-v1","parts":[["text","FFT circular convolution with root "],["math","\\(e^{-2\\pi i/n}\\)"],["text"," computes the cyclic convolution"]],"source":"FFT circular convolution with root $e^{-2\\pi i/n}$ computes the cyclic convolution"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-std-add-char-neg-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $n = 2^d$ and $x, y \\in \\mathbb{C}^n$. Run FFT circular convolution of $x$ and $y$ with root $\\omega = e^{-2\\pi i/n}$, the value at $-1$ of the standard additive character of $\\mathbb{Z}/n\\mathbb{Z}$. Then for every $k < n$, entry $k$ of the returned vector equals $\\sum_{j \\in \\mathbb{Z}/n\\mathbb{Z}} x_j \\, y_{k - j}$, with indices taken modulo $n$.","title":"FFT circular convolution with root $e^{-2\\pi i/n}$ computes the cyclic convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-std-add-char-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircular_stdAddChar_one","parent":42598,"reading_title":{"format":"inline-parts-v1","parts":[["text","FFT circular convolution with root "],["math","\\(e^{2\\pi i/n}\\)"],["text"," computes the cyclic convolution"]],"source":"FFT circular convolution with root $e^{2\\pi i/n}$ computes the cyclic convolution"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-std-add-char-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $n = 2^d$ and $x, y \\in \\mathbb{C}^n$. Run FFT circular convolution of $x$ and $y$ with root $\\omega = e^{2\\pi i/n}$, the value at $1$ of the standard additive character of $\\mathbb{Z}/n\\mathbb{Z}$. Then for every $k < n$, entry $k$ of the returned vector equals $\\sum_{j \\in \\mathbb{Z}/n\\mathbb{Z}} x_j \\, y_{k - j}$, with indices taken modulo $n$.","title":"FFT circular convolution with root $e^{2\\pi i/n}$ computes the cyclic convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft-time-add-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft_time_add_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft-time-add-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star ring structure, let $d$ be a natural number, $N = 2^d$, $\\omega \\in R$ arbitrary and $x \\in R^N$. The number $T$ of operations charged by the inverse FFT procedure on $x$ with root $\\omega$ satisfies $T + 1 = (7d + 10) \\cdot 2^d + 1$, equivalently $T = (7d + 10) \\cdot 2^d$.","title":"Exact operation count of the inverse FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft_time_le","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft-time-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star ring structure, let $d$ be a natural number, $N = 2^d$, $\\omega \\in R$ arbitrary and $x \\in R^N$. The number of operations charged by the inverse FFT procedure on $x$ with root $\\omega$ is at most $8 \\cdot 2^d \\cdot d + 9 \\cdot 2^d + 1$.","title":"Operation bound for the inverse FFT"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-time-add-two","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircular_time_add_two","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-time-add-two","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$, and let $x$ and $y$ be vectors of length $2^d$ over $R$. The cost charged by circular convolution of $x$ and $y$ with root $\\omega$, plus $2$, equals $(21d + 16) \\cdot 2^d$.","title":"Exact cost of FFT circular convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircular_time_le","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-time-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$, and let $x$ and $y$ be vectors of length $2^d$ over $R$. The cost charged by circular convolution of $x$ and $y$ with root $\\omega$ is at most $(21d + 16) \\cdot 2^d$, that is, $21 n \\log_2 n + 16 n$ for the input length $n = 2^d$.","title":"Cost bound for FFT circular convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-time-add-two","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_time_add_two","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-time-add-two","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$, and let $x$ and $y$ be vectors of length $2^d$ over $R$. The cost charged by zero-padded linear convolution of $x$ and $y$ with root $\\omega$, plus $2$, equals $(42d + 80) \\cdot 2^d$.","title":"Exact cost of FFT linear convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_time_le","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-time-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$, and let $x$ and $y$ be vectors of length $2^d$ over $R$. The cost charged by zero-padded linear convolution of $x$ and $y$ with root $\\omega$ is at most $(42d + 80) \\cdot 2^d$, that is, $42 n \\log_2 n + 80 n$ for the input length $n = 2^d$.","title":"Cost bound for FFT linear convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-array-ret-of-vector","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircularArray_ret_of_vector","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-array-ret-of-vector","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$ be arbitrary, and let $x, y$ be vectors over $R$ of length $2^d$. Then array circular FFT convolution with root $\\omega$, run on the arrays underlying $x$ and $y$, succeeds and returns the array underlying the output of vector circular FFT convolution of $x$ and $y$ at size $2^d$ with root $\\omega$.","title":"Array circular FFT convolution agrees with the vector version"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-array-time-of-vector","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircularArray_time_of_vector","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-array-time-of-vector","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$, and let $x, y$ be vectors over $R$ of length $2^d$. The time charged by array circular FFT convolution with root $\\omega$ on the arrays underlying $x$ and $y$ equals the time charged by vector circular FFT convolution of $x$ and $y$ at size $2^d$ with root $\\omega$.","title":"Array circular FFT convolution costs the same as the vector version"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-array-time-add-two","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircularArray_time_add_two","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-array-time-add-two","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$, and let $x, y$ be vectors over $R$ of length $2^d$. The time charged by array circular FFT convolution with root $\\omega$ on the arrays underlying $x$ and $y$, plus $2$, equals $(21d + 16) \\cdot 2^d$.","title":"Exact cost of array circular FFT convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-circular-array-size-of-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftCircularArray_size_of_eq_some","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-circular-array-size-of-eq-some","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $\\omega \\in R$, and let $x, y$ be arrays over $R$. If array circular FFT convolution with root $\\omega$ on $x$ and $y$ returns an array $z$, then $z$ has the same length as $x$.","title":"Output length of array circular FFT convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-array-ret-of-vector","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinearArray_ret_of_vector","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-array-ret-of-vector","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$ be arbitrary, and let $x, y$ be vectors over $R$ of length $2^d$. Then array zero-padded linear FFT convolution with root $\\omega$, run on the arrays underlying $x$ and $y$, succeeds and returns the array underlying the output (of length $2^{d+1}$) of vector zero-padded linear FFT convolution of $x$ and $y$ at size $2^d$\u2026","title":"Array linear FFT convolution agrees with the vector version"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-array-time-of-vector","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinearArray_time_of_vector","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-array-time-of-vector","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$, and let $x, y$ be vectors over $R$ of length $2^d$. The time charged by array zero-padded linear FFT convolution with root $\\omega$ on the arrays underlying $x$ and $y$ equals the time charged by vector zero-padded linear FFT convolution of $x$ and $y$ at size $2^d$ with root $\\omega$.","title":"Array linear FFT convolution costs the same as the vector version"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-array-time-add-two","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinearArray_time_add_two","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-array-time-add-two","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $d$ be a natural number, let $\\omega \\in R$, and let $x, y$ be vectors over $R$ of length $2^d$. The time charged by array zero-padded linear FFT convolution with root $\\omega$ on the arrays underlying $x$ and $y$, plus $2$, equals $(42d + 80) \\cdot 2^d$.","title":"Exact cost of array linear FFT convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-array-size-of-eq-some","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinearArray_size_of_eq_some","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-array-size-of-eq-some","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $R$ be a field with a star-ring structure, let $\\omega \\in R$, and let $x, y$ be arrays over $R$. If array zero-padded linear FFT convolution with root $\\omega$ on $x$ and $y$ returns an array $z$, then the length of $z$ is twice the length of $x$.","title":"Output length of array linear FFT convolution"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-polynomial","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPolynomial","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-polynomial","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"The polynomial of a vector $x = (x_0, \\dots, x_{n-1}) \\in \\mathbb{C}^n$ is $\\sum_{j=0}^{n-1} x_j X^j \\in \\mathbb{C}[X]$, the complex polynomial whose coefficient of $X^j$ is $x_j$ for $j < n$ and $0$ for $j \\ge n$.","title":"Polynomial of a coefficient vector"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-polynomial-coeff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPolynomial_coeff","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-polynomial-coeff","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For every vector $x = (x_0, \\dots, x_{n-1}) \\in \\mathbb{C}^n$ and every natural number $k$, the coefficient of $X^k$ in the polynomial of $x$ is $x_k$ if $k < n$ and $0$ if $k \\ge n$.","title":"Coefficients of the polynomial of a vector"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-polynomial-fft-pad","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPolynomial_fftPad","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-polynomial-fft-pad","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For every natural number $d$ and every vector $x \\in \\mathbb{C}^{2^d}$, the zero padding of $x$ (the vector of length $2^{d+1}$ whose first $2^d$ entries are those of $x$ and whose remaining entries are $0$) has the same polynomial as $x$.","title":"Zero padding preserves the polynomial"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-polynomial-mul-coeff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPolynomial_mul_coeff","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-polynomial-mul-coeff","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $x, y \\in \\mathbb{C}^n$, and set $x_i = y_i = 0$ for $i \\ge n$. For every natural number $k$, the coefficient of $X^k$ in the product of the polynomials of $x$ and $y$ is $\\sum_{j=0}^{k} x_j y_{k-j}$.","title":"Coefficients of a product of vector polynomials"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-polynomial-mul-last","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPolynomial_mul_last","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-polynomial-mul-last","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For every natural number $d$ and all vectors $x, y \\in \\mathbb{C}^{2^d}$, the coefficient of $X^{2 \\cdot 2^d - 1}$ in the product of the polynomials of $x$ and $y$ is $0$.","title":"The last coefficient of a doubled-length product vanishes"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-polynomial-mul-coeff-eq-zero-of-two","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPolynomial_mul_coeff_eq_zero_of_two_mul_le","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-polynomial-mul-coeff-eq-zero-of-two","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"For all vectors $x, y \\in \\mathbb{C}^n$ and every natural number $k \\ge 2n$, the coefficient of $X^k$ in the product of the polynomials of $x$ and $y$ is $0$.","title":"Coefficients of a product beyond twice the length vanish"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-pad-circular-coeff-eq-polynomial","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPad_circular_coeff_eq_polynomial","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-pad-circular-coeff-eq-polynomial","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number and $N = 2^{d+1}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, write $P_x = \\sum_{j < 2^d} x_j X^j$ and $P_y = \\sum_{j < 2^d} y_j X^j$, and let $x', y' \\in \\mathbb{C}^N$ be their zero paddings to length $N$ ($x'_j = x_j$ for $j < 2^d$ and $x'_j = 0$ otherwise, and likewise for $y'$), with indices of $x'$ and $y'$ read modulo $N$. Then for every $k < N$, the coefficient of $X^k$ in $P_x P_y$\u2026","title":"Zero padding turns cyclic convolution into the polynomial product"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-std-add-char-neg-one-coeff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_stdAddChar_neg_one_coeff","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-std-add-char-neg-one-coeff","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^{d+1}$ and $\\omega = e^{-2\\pi i/N}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, write $P_x = \\sum_{j < 2^d} x_j X^j$ and $P_y = \\sum_{j < 2^d} y_j X^j$, and let $z \\in \\mathbb{C}^N$ be the output of the FFT linear convolution of $x$ and $y$ with root $\\omega$, which pads both vectors with zeros to length $N$ and convolves them circularly by FFT. Then for every $k < N$, the entry\u2026","title":"FFT linear convolution gives the product coefficients (Princeton sign)"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-std-add-char-neg-one-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_stdAddChar_neg_one_sum","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-std-add-char-neg-one-sum","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^{d+1}$ and $\\omega = e^{-2\\pi i/N}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, extended by $x_j = y_j = 0$ for $j \\ge 2^d$, let $z \\in \\mathbb{C}^N$ be the output of the FFT linear convolution of $x$ and $y$ with root $\\omega$, which pads both vectors with zeros to length $N$ and convolves them circularly by FFT. Then for every $k < N$,","title":"FFT linear convolution computes the convolution sums (Princeton sign)"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-polynomial-fft-linear-std-add-char-n","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPolynomial_fftLinear_stdAddChar_neg_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-polynomial-fft-linear-std-add-char-n","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^{d+1}$ and $\\omega = e^{-2\\pi i/N}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, write $P_x = \\sum_{j < 2^d} x_j X^j$ and $P_y = \\sum_{j < 2^d} y_j X^j$, and let $z \\in \\mathbb{C}^N$ be the output of the FFT linear convolution of $x$ and $y$ with root $\\omega$, which pads both vectors with zeros to length $N$ and convolves them circularly by FFT. Then\u2026","title":"FFT linear convolution returns the product polynomial (Princeton sign)"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-std-add-char-neg-one-last","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_stdAddChar_neg_one_last","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-std-add-char-neg-one-last","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^{d+1}$ and $\\omega = e^{-2\\pi i/N}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, let $z \\in \\mathbb{C}^N$ be the output of the FFT linear convolution of $x$ and $y$ with root $\\omega$, which pads both vectors with zeros to length $N$ and convolves them circularly by FFT. Then the last entry $z_{N-1}$ is $0$.","title":"The last entry of FFT linear convolution is zero (Princeton sign)"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-std-add-char-one-coeff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_stdAddChar_one_coeff","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-std-add-char-one-coeff","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^{d+1}$ and $\\omega = e^{2\\pi i/N}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, write $P_x = \\sum_{j < 2^d} x_j X^j$ and $P_y = \\sum_{j < 2^d} y_j X^j$, and let $z \\in \\mathbb{C}^N$ be the output of the FFT linear convolution of $x$ and $y$ with root $\\omega$, which pads both vectors with zeros to length $N$ and convolves them circularly by FFT. Then for every $k < N$, the entry\u2026","title":"FFT linear convolution gives the product coefficients (MIT/CLRS sign)"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-std-add-char-one-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_stdAddChar_one_sum","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-std-add-char-one-sum","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^{d+1}$ and $\\omega = e^{2\\pi i/N}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, extended by $x_j = y_j = 0$ for $j \\ge 2^d$, let $z \\in \\mathbb{C}^N$ be the output of the FFT linear convolution of $x$ and $y$ with root $\\omega$, which pads both vectors with zeros to length $N$ and convolves them circularly by FFT. Then for every $k < N$,","title":"FFT linear convolution computes the convolution sums (MIT/CLRS sign)"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-polynomial-fft-linear-std-add-char-o","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftPolynomial_fftLinear_stdAddChar_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-polynomial-fft-linear-std-add-char-o","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^{d+1}$ and $\\omega = e^{2\\pi i/N}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, write $P_x = \\sum_{j < 2^d} x_j X^j$ and $P_y = \\sum_{j < 2^d} y_j X^j$, and let $z \\in \\mathbb{C}^N$ be the output of the FFT linear convolution of $x$ and $y$ with root $\\omega$, which pads both vectors with zeros to length $N$ and convolves them circularly by FFT. Then $\\sum_{k < N} z_k X^k = P_x P_y$\u2026","title":"FFT linear convolution returns the product polynomial (MIT/CLRS sign)"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-linear-std-add-char-one-last","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fftLinear_stdAddChar_one_last","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-linear-std-add-char-one-last","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^{d+1}$ and $\\omega = e^{2\\pi i/N}$. For vectors $x, y \\in \\mathbb{C}^{2^d}$, let $z \\in \\mathbb{C}^N$ be the output of the FFT linear convolution of $x$ and $y$ with root $\\omega$, which pads both vectors with zeros to length $N$ and convolves them circularly by FFT. Then the last entry $z_{N-1}$ is $0$.","title":"The last entry of FFT linear convolution is zero (MIT/CLRS sign)"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/ifft-fft-std-add-char-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.ifft_fft_stdAddChar_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#ifft-fft-std-add-char-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^d$ and $\\omega = e^{2\\pi i/N}$. For every vector $x \\in \\mathbb{C}^N$, running the radix-two FFT with root $\\omega$ on $x$ and then the inverse FFT with root $\\omega$ on its output returns $x$.","title":"Inverse FFT after FFT at the positive root is the identity"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/fft-ifft-std-add-char-one","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.fft_ifft_stdAddChar_one","parent":42598,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform/index.html#fft-ifft-std-add-char-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/fast-fourier-transform","status":"fully_proved","summary":"Let $d$ be a natural number, $N = 2^d$ and $\\omega = e^{2\\pi i/N}$. For every vector $x \\in \\mathbb{C}^N$, running the inverse FFT with root $\\omega$ on $x$ and then the radix-two FFT with root $\\omega$ on its output returns $x$.","title":"FFT after inverse FFT at the positive root is the identity"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding.json","id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","title":"Floating-point rounding"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/rne","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.rne","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#rne","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"For $m$ in a linearly ordered field $K$ with a floor function, $\\operatorname{rne}(m)$ is the integer $\\lfloor m \\rfloor$ if $m - \\lfloor m \\rfloor < 1/2$, the integer $\\lfloor m \\rfloor + 1$ if $m - \\lfloor m \\rfloor > 1/2$, and, when $m - \\lfloor m \\rfloor = 1/2$, whichever of $\\lfloor m \\rfloor$ and $\\lfloor m \\rfloor + 1$ is even.","title":"Nearest-integer rounding with ties to even"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/abs-rne-sub-le-min","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.abs_rne_sub_le_min","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#abs-rne-sub-le-min","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function. For every $m \\in K$, the integer $\\operatorname{rne}(m)$ obtained by rounding $m$ to the nearest integer, ties to even, satisfies $|\\operatorname{rne}(m) - m| \\le m - \\lfloor m \\rfloor$ and $|\\operatorname{rne}(m) - m| \\le \\lfloor m \\rfloor + 1 - m$.","title":"Rounding error bounded by the distances to neighbouring integers"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/abs-rne-sub-le","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.abs_rne_sub_le","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#abs-rne-sub-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function. For every $m \\in K$ and every integer $k$, the integer $\\operatorname{rne}(m)$ obtained by rounding $m$ to the nearest integer, ties to even, satisfies $|\\operatorname{rne}(m) - m| \\le |k - m|$: no integer is closer to $m$ than $\\operatorname{rne}(m)$.","title":"Round to nearest integer"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/abs-rne-sub-le-half","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.abs_rne_sub_le_half","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#abs-rne-sub-le-half","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function. For every $m \\in K$, the integer $\\operatorname{rne}(m)$ obtained by rounding $m$ to the nearest integer, ties to even, satisfies $|\\operatorname{rne}(m) - m| \\le 1/2$.","title":"Rounding to the nearest integer is within one half"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/even-rne-of-abs-sub-eq-half","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.even_rne_of_abs_sub_eq_half","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#even-rne-of-abs-sub-eq-half","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function, let $m \\in K$, and let $\\operatorname{rne}(m)$ be the integer obtained by rounding $m$ to the nearest integer, ties to even. If $|\\operatorname{rne}(m) - m| = 1/2$, that is, $m$ is halfway between two consecutive integers, then $\\operatorname{rne}(m)$ is even.","title":"Ties to even for the nearest integer"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/rne-le-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.rne_le_of_le","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#rne-le-of-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function, let $m \\in K$ and let $k$ be an integer. If $m \\le k$, then $\\operatorname{rne}(m) \\le k$, where $\\operatorname{rne}(m)$ is $m$ rounded to the nearest integer with ties to even.","title":"Rounding to nearest preserves integer upper bounds"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/le-rne-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.le_rne_of_le","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#le-rne-of-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function, let $m \\in K$ and let $k$ be an integer. If $k \\le m$, then $k \\le \\operatorname{rne}(m)$, where $\\operatorname{rne}(m)$ is $m$ rounded to the nearest integer with ties to even.","title":"Rounding to nearest preserves integer lower bounds"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/is-float","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.IsFloat","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#is-float","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"For a natural number $p$, an element $x$ of a linearly ordered field with a floor function is a binary floating-point number with a $p$-bit significand and unbounded exponent when $x = m \\cdot 2^e$ for some integers $m$ and $e$ with $|m| < 2^p$.","title":"Binary floating-point numbers"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/is-float-in","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.IsFloatIn","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#is-float-in","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"For a natural number $p$ and integers $q_{\\min}$ and $q_{\\max}$, an element $x$ of a linearly ordered field with a floor function is a number of the binary format with $p$-bit significands and last-place exponents in $[q_{\\min}, q_{\\max}]$ when $x = m \\cdot 2^e$ for some integers $m$ and $e$ with $|m| < 2^p$ and $q_{\\min} \\le e \\le q_{\\max}$. These are zero, the subnormal numbers and the normal numbers of the format.","title":"Floating-point numbers of a bounded format"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/ulp-exp","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.ulpExp","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#ulp-exp","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"For a natural number $p$ (the significand length) and $x$ in a linearly ordered field $K$ with a floor function, the exponent of the last place of $x$ is the integer $\\lfloor \\log_2 |x| \\rfloor - (p - 1)$. Here $\\lfloor \\log_2 |x| \\rfloor$ is the integer logarithm: the largest integer $n$ with $2^n \\le |x|$ when $x \\ne 0$, and $0$ when $x = 0$.","title":"Exponent of the last place"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/fl","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.fl","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#fl","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"For a natural number $p$ and an element $x$ of a linearly ordered field with a floor function, let $e$ be the exponent of the last place of $x$ at precision $p$, and let $\\operatorname{rne}$ round to the nearest integer with ties to the even one. The rounding of $x$ to nearest, ties to even, is $\\mathrm{fl}_p(x) = \\operatorname{rne}(x / 2^e) \\cdot 2^e$: scale $x$ so that its last place is the units digit, round to\u2026","title":"Round to nearest, ties to even"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/scaled-bounds","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.scaled_bounds","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#scaled-bounds","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function, let $p \\ge 1$ be a natural number and let $x \\in K$ be nonzero. Let $e = \\lfloor \\log_2 |x| \\rfloor - (p - 1)$ be the exponent of the last place of $x$ for $p$-bit significands. Then $2^{p-1} \\le |x / 2^e|$ and $|x / 2^e| < 2^p$.","title":"Bounds on the scaled significand"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/fl-sub-eq","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.fl_sub_eq","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#fl-sub-eq","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $x$ be an element of a linearly ordered field with a floor function, let $p$ be a natural number, and let $e$ be the exponent of the last place of $x$ at precision $p$. Writing $\\operatorname{rne}$ for rounding to the nearest integer with ties to even, the rounding $\\mathrm{fl}_p(x)$ of $x$ to nearest, ties to even, at precision $p$ has the error\u2026","title":"The rounding error at the scale of the last place"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/abs-fl-sub-le","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.abs_fl_sub_le","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#abs-fl-sub-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function and let $p$ be a natural number. For every $x \\in K$, the rounding $\\mathrm{fl}(x)$ of $x$ to nearest, ties to even, with $p$-bit significands satisfies $|\\mathrm{fl}(x) - x| \\le u\\,|x|$, where $u = 2^{-p}$ is the unit roundoff.","title":"The relative error bound"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/abs-fl-sub-le-of-is-float","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.abs_fl_sub_le_of_isFloat","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#abs-fl-sub-le-of-is-float","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function and let $p \\ge 1$. For every $x \\in K$ and every binary floating-point number $f \\in K$ with a $p$-bit significand (and unbounded exponent), the rounding $\\mathrm{fl}(x)$ of $x$ to nearest, ties to even, with $p$-bit significands satisfies $|\\mathrm{fl}(x) - x| \\le |f - x|$: no floating-point number is closer to $x$ than $\\mathrm{fl}(x)$.","title":"Round to nearest for floating-point numbers"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/fl-eq-self","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.fl_eq_self","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#fl-eq-self","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $p \\ge 1$ and let $x$ be an element of a linearly ordered field with a floor function. If $x$ is a binary floating-point number with a $p$-bit significand, that is, $x = m \\cdot 2^e$ for some integers $m$ and $e$ with $|m| < 2^p$, then rounding $x$ to nearest, ties to even, at precision $p$ returns $x$: $\\mathrm{fl}_p(x) = x$.","title":"Floating-point numbers round to themselves"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/even-of-abs-fl-sub-eq","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.even_of_abs_fl_sub_eq","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#even-of-abs-fl-sub-eq","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function, let $p$ be a natural number, let $x \\in K$, and let $e$ be the exponent of the last place of $x$, which is $\\lfloor \\log_2 |x| \\rfloor - (p - 1)$ for $x \\ne 0$. If the rounding $\\mathrm{fl}(x)$ of $x$ to nearest, ties to even, with $p$-bit significands satisfies $|\\mathrm{fl}(x) - x| = 2^e / 2$, then the integer $\\operatorname{rne}(x / 2^e)$, the\u2026","title":"Ties to even for floating-point rounding"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/abs-rne-le","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.abs_rne_le","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#abs-rne-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function, let $p \\ge 1$, let $x \\in K$ be nonzero, and let $e = \\lfloor \\log_2 |x| \\rfloor - (p - 1)$ be the exponent of the last place of $x$. Then $|\\operatorname{rne}(x / 2^e)| \\le 2^p$, where $\\operatorname{rne}$ rounds to the nearest integer, ties to even.","title":"Bound on the rounded significand"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/is-float-fl","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.isFloat_fl","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#is-float-fl","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $p \\ge 1$. For every element $x$ of a linearly ordered field with a floor function, the rounding $\\mathrm{fl}_p(x)$ of $x$ to nearest, ties to even, is a binary floating-point number with a $p$-bit significand: $\\mathrm{fl}_p(x) = m \\cdot 2^e$ for some integers $m$ and $e$ with $|m| < 2^p$.","title":"Rounding gives a floating-point number"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/is-float-in-fl","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.isFloatIn_fl","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#is-float-in-fl","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $p \\ge 1$, let $q_{\\min}$ and $q_{\\max}$ be integers, and let $x$ be an element of a linearly ordered field with a floor function. If $2^{q_{\\min} + p - 1} \\le |x|$ ($x$ is at least the smallest normal number) and $|x| \\le (2^p - 1) \\cdot 2^{q_{\\max}}$ ($x$ is at most the largest finite number), then the rounding $\\mathrm{fl}_p(x)$ of $x$ to nearest, ties to even, is a number of the binary format with $p$-bit\u2026","title":"In the normal range, rounding stays in the format"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/abs-fl-sub-le-of-is-float-in","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.abs_fl_sub_le_of_isFloatIn","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#abs-fl-sub-le-of-is-float-in","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function, let $p \\ge 1$, and let $q_{\\min}$ and $q_{\\max}$ be integers. For every $x \\in K$ and every number $f$ of the binary format with $p$-bit significands and last-place exponents in $[q_{\\min}, q_{\\max}]$, the rounding $\\mathrm{fl}(x)$ of $x$ to nearest, ties to even, with $p$-bit significands satisfies $|\\mathrm{fl}(x) - x| \\le |f - x|$.","title":"Round to nearest in a bounded format"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/exists-sub-eq-of-le","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.exists_sub_eq_of_le","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#exists-sub-eq-of-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $K$ be a linearly ordered field with a floor function and let $p$ be a natural number. Let $x, y \\in K$ and let $m_x, m_y, e_x, e_y$ be integers with $|m_x| < 2^p$, $|m_y| < 2^p$, $x = m_x 2^{e_x}$ and $y = m_y 2^{e_y}$. If $y / 2 \\le x \\le 2y$, then there is an integer $k$ with $|k| < 2^p$ and $x - y = k \\cdot 2^{\\min(e_x, e_y)}$.","title":"The difference of floating-point numbers within a factor of two"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/is-float-sub","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.isFloat_sub","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#is-float-sub","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $p$ be a natural number and let $x$ and $y$ be binary floating-point numbers with $p$-bit significands (and unbounded exponent) in a linearly ordered field with a floor function. If $y / 2 \\le x \\le 2y$, then $x - y$ is also a binary floating-point number with a $p$-bit significand.","title":"Sterbenz's lemma"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/is-float-in-sub","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.isFloatIn_sub","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#is-float-in-sub","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $p$ be a natural number, let $q_{\\min}$ and $q_{\\max}$ be integers, and let $x$ and $y$ be numbers of the binary format with $p$-bit significands whose last place has exponent in $[q_{\\min}, q_{\\max}]$ (subnormal numbers included), in a linearly ordered field with a floor function. If $y / 2 \\le x \\le 2y$, then $x - y$ is also a number of this format.","title":"Sterbenz's lemma in a bounded format"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/fl-sub-eq-sub","kind":"node","lean":"Cslib.Algorithms.Lean.FloatingPoint.fl_sub_eq_sub","parent":42689,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding/index.html#fl-sub-eq-sub","scope":"ComputerScience/algebraic-and-numerical-algorithms/floating-point-rounding","status":"fully_proved","summary":"Let $p \\ge 1$ and let $x$ and $y$ be binary floating-point numbers with $p$-bit significands in a linearly ordered field with a floor function. If $y / 2 \\le x \\le 2y$, then rounding $x - y$ to nearest, ties to even, at precision $p$ returns it exactly: $\\mathrm{fl}_p(x - y) = x - y$.","title":"Floating-point subtraction is exact under Sterbenz's condition"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method.json","id":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","status":"fully_proved","title":"Horner's method"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/horner-eval","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hornerEval","parent":42715,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/index.html#horner-eval","scope":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","status":"fully_proved","summary":"Let $R$ be a commutative semiring, let $x \\in R$, and let $a_0, \\dots, a_{n-1} \\in R$ be coefficients listed in ascending order of degree. Horner evaluation starts with the accumulator $v = 0$ and, for $i = n-1, \\dots, 0$ in turn, replaces $v$ by $a_i + x v$, charging two operations (one multiplication and one addition) for each step. It returns the final accumulator together with the total number of operations\u2026","title":"Horner evaluation with operation count"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/horner-eval-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hornerEval_time","parent":42715,"reading_title":{"format":"inline-parts-v1","parts":[["text","Horner evaluation uses "],["math","\\(2n\\)"],["text"," operations"]],"source":"Horner evaluation uses $2n$ operations"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/index.html#horner-eval-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","status":"fully_proved","summary":"For every commutative semiring $R$, every $x \\in R$ and all coefficients $a_0, \\dots, a_{n-1} \\in R$, Horner evaluation of $a_0, \\dots, a_{n-1}$ at $x$ charges exactly $2n$ operations, one multiplication and one addition per coefficient.","title":"Horner evaluation uses $2n$ operations"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/horner-eval-ret-eq-sum","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hornerEval_ret_eq_sum","parent":42715,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/index.html#horner-eval-ret-eq-sum","scope":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","status":"fully_proved","summary":"For every commutative semiring $R$, every $x \\in R$ and all coefficients $a_0, \\dots, a_{n-1} \\in R$, the value returned by Horner evaluation of $a_0, \\dots, a_{n-1}$ at $x$ is $\\sum_{i=0}^{n-1} a_i x^i$.","title":"Horner evaluation computes the coefficient sum"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/horner-eval-ret-eq-eval","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hornerEval_ret_eq_eval","parent":42715,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/index.html#horner-eval-ret-eq-eval","scope":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","status":"fully_proved","summary":"For every commutative semiring $R$, every $x \\in R$ and all coefficients $a_0, \\dots, a_{n-1} \\in R$, the value returned by Horner evaluation of $a_0, \\dots, a_{n-1}$ at $x$ equals $p(x)$, where $p = a_0 + a_1 X + \\dots + a_{n-1} X^{n-1} \\in R[X]$ is the polynomial with these coefficients.","title":"Horner evaluation is polynomial evaluation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/horner-eval-empty","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hornerEval_empty","parent":42715,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/index.html#horner-eval-empty","scope":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","status":"fully_proved","summary":"For every commutative semiring $R$ and every $x \\in R$, Horner evaluation at $x$ with no coefficients ($n = 0$) returns $0$ and charges $0$ operations.","title":"Horner evaluation of the empty coefficient list"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/horner-eval-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hornerEval_singleton","parent":42715,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/index.html#horner-eval-singleton","scope":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","status":"fully_proved","summary":"For every commutative semiring $R$, every $x \\in R$ and every single coefficient $a_0 \\in R$ ($n = 1$), Horner evaluation of $a_0$ at $x$ returns $a_0$ and charges $2$ operations.","title":"Horner evaluation of a constant polynomial"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/horner-eval-zero","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.hornerEval_zero","parent":42715,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/horner-s-method/index.html#horner-eval-zero","scope":"ComputerScience/algebraic-and-numerical-algorithms/horner-s-method","status":"fully_proved","summary":"For every commutative semiring $R$ and all coefficients $a_0, \\dots, a_n \\in R$ (at least one coefficient), the value returned by Horner evaluation of $a_0, \\dots, a_n$ at $x = 0$ is $a_0$.","title":"Horner evaluation at zero"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm.json","id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","title":"Karatsuba algorithm"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsuba","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"Karatsuba's algorithm on natural numbers $x$ and $y$, viewed as $n$-digit numbers in base $B$, as a computation that returns a natural number together with a cost. If $n \\le 1$, it returns $xy$ at cost $1$. If $n \\ge 2$, let $m = \\lceil n/2 \\rceil$, let $a$ and $b$ be the natural-number quotient and remainder of $x$ by $B^m$, and let $c$ and $d$ be those of $y$ by $B^m$. It runs itself with digit count $m$ on the\u2026","title":"Karatsuba multiplication"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-cost","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsubaCost","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-cost","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"The function $T$ on natural numbers defined by $T(n) = 1$ for $n \\le 1$ and $T(n) = 3\\,T(\\lceil n/2 \\rceil) + n$ for $n \\ge 2$.","title":"Karatsuba cost recurrence"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-middle","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsuba_middle","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-middle","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"For all natural numbers $a, b, c, d$, write $P = |a - b| \\cdot |c - d|$. If $b \\le a$ holds exactly when $d \\le c$, then $ac + bd - P = ad + bc$, where the subtraction is truncated at $0$; otherwise $ac + bd + P = ad + bc$.","title":"Karatsuba's middle coefficient"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsuba_ret","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"For all natural numbers $B$, $n$, $x$ and $y$, Karatsuba multiplication of $x$ and $y$ as $n$-digit numbers in base $B$ returns $xy$.","title":"Correctness of Karatsuba multiplication"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-operands-lt","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsuba_operands_lt","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-operands-lt","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"Let $B \\ge 1$ and $n$ be natural numbers, let $m = \\lceil n/2 \\rceil$, and let $x < B^n$ be a natural number. Then the high part $\\lfloor x / B^m \\rfloor$, the low part $x \\bmod B^m$, and their distance $|\\lfloor x / B^m \\rfloor - x \\bmod B^m|$ are all less than $B^m$.","title":"Size bound for Karatsuba's recursive operands"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-time","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsuba_time","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"For all natural numbers $B$, $n$, $x$ and $y$, the cost of Karatsuba multiplication of $x$ and $y$ as $n$-digit numbers in base $B$ equals $T(n)$, where $T$ is the Karatsuba cost recurrence, $T(n) = 1$ for $n \\le 1$ and $T(n) = 3\\,T(\\lceil n/2 \\rceil) + n$ for $n \\ge 2$.","title":"Cost of Karatsuba multiplication"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-cost-two-pow","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsubaCost_two_pow","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-cost-two-pow","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"Let $T$ be the Karatsuba cost recurrence, $T(n) = 1$ for $n \\le 1$ and $T(n) = 3\\,T(\\lceil n/2 \\rceil) + n$ for $n \\ge 2$. For every natural number $k$, $T(2^k) + 2^{k+1} = 3^{k+1}$.","title":"Karatsuba cost at powers of two"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-cost-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsubaCost_le","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-cost-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"Let $T$ be the Karatsuba cost recurrence, $T(n) = 1$ for $n \\le 1$ and $T(n) = 3\\,T(\\lceil n/2 \\rceil) + n$ for $n \\ge 2$, and let $\\ell(n)$ be the least natural number $k$ with $n \\le 2^k$ (so $\\ell(n) = \\lceil \\log_2 n \\rceil$ for $n \\ge 1$). For every natural number $n$, $T(n) \\le 3^{\\ell(n) + 1}$.","title":"Power-of-three bound on the Karatsuba cost recurrence"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsuba_time_le","parent":42723,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-time-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"Let $\\ell(n)$ be the least natural number $k$ with $n \\le 2^k$ (so $\\ell(n) = \\lceil \\log_2 n \\rceil$ for $n \\ge 1$). For all natural numbers $B$, $n$, $x$ and $y$, the cost of Karatsuba multiplication of $x$ and $y$ as $n$-digit numbers in base $B$ is at most $3^{\\ell(n) + 1}$.","title":"Power-of-three bound on the cost of Karatsuba multiplication"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-cost-le-rpow","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsubaCost_le_rpow","parent":42723,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n^{\\log_2 3}\\)"],["text"," bound for the Karatsuba cost recurrence"]],"source":"The $n^{\\log_2 3}$ bound for the Karatsuba cost recurrence"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-cost-le-rpow","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"Let $T$ be the Karatsuba cost recurrence, $T(n) = 1$ for $n \\le 1$ and $T(n) = 3\\,T(\\lceil n/2 \\rceil) + n$ for $n \\ge 2$. For every natural number $n \\ge 1$, $T(n) \\le 9\\, n^{\\log_2 3}$.","title":"The $n^{\\log_2 3}$ bound for the Karatsuba cost recurrence"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/karatsuba-time-le-rpow","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.karatsuba_time_le_rpow","parent":42723,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(n^{\\log_2 3}\\)"],["text"," bound for Karatsuba multiplication"]],"source":"The $n^{\\log_2 3}$ bound for Karatsuba multiplication"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm/index.html#karatsuba-time-le-rpow","scope":"ComputerScience/algebraic-and-numerical-algorithms/karatsuba-algorithm","status":"fully_proved","summary":"For all natural numbers $B$, $x$, $y$ and $n \\ge 1$, the cost of Karatsuba multiplication of $x$ and $y$ as $n$-digit numbers in base $B$ is at most $9\\, n^{\\log_2 3}$.","title":"The $n^{\\log_2 3}$ bound for Karatsuba multiplication"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms.json","id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","title":"Matrix multiplication algorithms"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/rectangular-matrix-accumulate","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.rectangularMatrixAccumulate","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#rectangular-matrix-accumulate","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a type with an addition and a multiplication, and let $A$, $B$, $C$ be $p \\times q$, $q \\times r$ and $p \\times r$ matrices over $\\alpha$, stored as vectors of rows. Rectangular matrix accumulation adds the product $AB$ to the accumulator $C$: for $i < p$, then $j < r$, then $k < q$, each in increasing order, it replaces $C_{ij}$ by $C_{ij} + A_{ik} B_{kj}$. It returns the final matrix together with\u2026","title":"Rectangular matrix accumulation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/matrix-accumulate","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.matrixAccumulate","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#matrix-accumulate","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a type with an addition and a multiplication, and let $A$, $B$, $C$ be $n \\times n$ matrices over $\\alpha$, stored as vectors of rows. Square matrix accumulation on $A$, $B$, $C$ is rectangular matrix accumulation on $A$, $B$, $C$ with all three dimensions equal to $n$: it adds $AB$ to the accumulator $C$ and returns the result with its cost pair (scalar multiplications, scalar additions).","title":"Square matrix accumulation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/rectangular-matrix-accumulate-apply","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.rectangularMatrixAccumulate_apply","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#rectangular-matrix-accumulate-apply","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a commutative additive monoid with a multiplication, and let $A$, $B$, $C$ be $p \\times q$, $q \\times r$ and $p \\times r$ matrices over $\\alpha$, stored as vectors of rows. For all $i < p$ and $j < r$, the $(i, j)$ entry of the matrix returned by rectangular matrix accumulation on $A$, $B$, $C$ is $C_{ij} + \\sum_{k < q} A_{ik} B_{kj}$.","title":"Entries of rectangular matrix accumulation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/rectangular-matrix-accumulate-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.rectangularMatrixAccumulate_ret","parent":42735,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangular matrix accumulation computes "],["math","\\(C + AB\\)"]],"source":"Rectangular matrix accumulation computes $C + AB$"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#rectangular-matrix-accumulate-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a commutative additive monoid with a multiplication, and let $A$, $B$, $C$ be $p \\times q$, $q \\times r$ and $p \\times r$ matrices over $\\alpha$, stored as vectors of rows. The matrix returned by rectangular matrix accumulation on $A$, $B$, $C$ equals $C + AB$, where $AB$ is the matrix product, $(AB)_{ij} = \\sum_{k < q} A_{ik} B_{kj}$.","title":"Rectangular matrix accumulation computes $C + AB$"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/rectangular-matrix-accumulate-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.rectangularMatrixAccumulate_time","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#rectangular-matrix-accumulate-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a type with an addition and a multiplication, and let $A$, $B$, $C$ be $p \\times q$, $q \\times r$ and $p \\times r$ matrices over $\\alpha$. Rectangular matrix accumulation on $A$, $B$, $C$ has cost $(pqr, pqr)$: it charges exactly $pqr$ scalar multiplications and exactly $pqr$ scalar additions.","title":"Operation count of rectangular matrix accumulation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/rectangular-matrix-accumulate-zero-inner","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.rectangularMatrixAccumulate_zero_inner","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#rectangular-matrix-accumulate-zero-inner","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a type with an addition and a multiplication. If $A$ is a $p \\times 0$ matrix, $B$ a $0 \\times r$ matrix and $C$ a $p \\times r$ matrix over $\\alpha$, then rectangular matrix accumulation on $A$, $B$, $C$ returns $C$ unchanged at cost $(0, 0)$.","title":"Rectangular matrix accumulation with zero inner dimension"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/matrix-accumulate-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.matrixAccumulate_ret","parent":42735,"reading_title":{"format":"inline-parts-v1","parts":[["text","Square matrix accumulation computes "],["math","\\(C + AB\\)"]],"source":"Square matrix accumulation computes $C + AB$"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#matrix-accumulate-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a commutative additive monoid with a multiplication, and let $A$, $B$, $C$ be $n \\times n$ matrices over $\\alpha$, stored as vectors of rows. The matrix returned by square matrix accumulation on $A$, $B$, $C$ equals $C + AB$, where $AB$ is the matrix product, $(AB)_{ij} = \\sum_{k < n} A_{ik} B_{kj}$.","title":"Square matrix accumulation computes $C + AB$"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/matrix-accumulate-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.matrixAccumulate_time","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#matrix-accumulate-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a type with an addition and a multiplication, and let $A$, $B$, $C$ be $n \\times n$ matrices over $\\alpha$. Square matrix accumulation on $A$, $B$, $C$ has cost $(n^3, n^3)$: it charges exactly $n^3$ scalar multiplications and exactly $n^3$ scalar additions.","title":"Operation count of square matrix accumulation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/recursive-matrix-accumulate","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.recursiveMatrixAccumulate","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#recursive-matrix-accumulate","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a type with an addition and a multiplication, let $k$ be a natural number and $n = 2^k$, and let $A$, $B$, $C$ be $n \\times n$ matrices over $\\alpha$, stored as vectors of rows. The procedure adds the product of $A$ and $B$ to the accumulator $C$ by the eight-call recursion of CLRS. When $n = 1$, writing $a$, $b$, $c$ for the single entries of $A$, $B$, $C$, it replaces $c$ by $c + ab$. Otherwise it\u2026","title":"Recursive matrix multiply-accumulate"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/recursive-matrix-accumulate-apply","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.recursiveMatrixAccumulate_apply","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#recursive-matrix-accumulate-apply","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a multiplication, let $k$ be a natural number and $n = 2^k$, and let $A = (a_{ij})$, $B = (b_{ij})$, $C = (c_{ij})$ be $n \\times n$ matrices over $\\alpha$, with indices running from $0$ to $n - 1$. For all indices $i, j < n$, entry $(i, j)$ of the matrix returned by the recursive multiply-accumulate procedure on $A$, $B$, $C$ equals\u2026","title":"Entries of the recursive matrix multiply-accumulate"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/recursive-matrix-accumulate-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.recursiveMatrixAccumulate_ret","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#recursive-matrix-accumulate-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be an additive commutative monoid with a multiplication, let $k$ be a natural number and $n = 2^k$, and let $A = (a_{ij})$, $B = (b_{ij})$, $C$ be $n \\times n$ matrices over $\\alpha$, with indices running from $0$ to $n - 1$. Viewed as a matrix, the result returned by the recursive multiply-accumulate procedure on $A$, $B$, $C$ equals $C + AB$, where $AB$ is the matrix product with entries\u2026","title":"Correctness of the recursive matrix multiply-accumulate"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/recursive-matrix-accumulate-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.recursiveMatrixAccumulate_time","parent":42735,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms/index.html#recursive-matrix-accumulate-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/matrix-multiplication-algorithms","status":"fully_proved","summary":"Let $\\alpha$ be a type with an addition and a multiplication, let $k$ be a natural number and $n = 2^k$, and let $A$, $B$, $C$ be $n \\times n$ matrices over $\\alpha$. The recursive multiply-accumulate procedure on $A$, $B$, $C$ performs exactly $8^k$ scalar multiplications and exactly $8^k$ accumulator additions. Since $8^k = n^3$, each count is the cube of the dimension.","title":"Operation count of the recursive matrix multiply-accumulate"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation.json","id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","title":"Modular exponentiation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For natural numbers $a$, $b$ and $n$, modular exponentiation with base $a$, exponent $b$ and modulus $n$ is the recursive procedure that returns $1$ if $b = 0$. If $b > 0$ is even, it computes the result $d$ for exponent $b/2$ (same base and modulus) and returns $d \\cdot d \\bmod n$. If $b$ is odd, it computes the result $d$ for exponent $b - 1$ and returns $a \\cdot d \\bmod n$. Each even or odd step performs one\u2026","title":"Modular exponentiation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-zero","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_zero","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-zero","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For all natural numbers $a$ and $n$, modular exponentiation with base $a$, exponent $0$ and modulus $n$ returns $1$ and performs no multiplications.","title":"Base case of modular exponentiation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-even","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_even","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-even","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"Let $a$, $b$, $n$ be natural numbers with $b > 0$ and $b$ even. If modular exponentiation on base $a$, exponent $b/2$ and modulus $n$ returns the value $d$ using $t$ multiplications, then on base $a$, exponent $b$ and modulus $n$ it returns the value $(d \\cdot d) \\bmod n$ using $t + 1$ multiplications.","title":"Even step of modular exponentiation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-odd","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_odd","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-odd","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"Let $a$, $b$, $n$ be natural numbers with $b$ odd. If modular exponentiation on base $a$, exponent $b - 1$ and modulus $n$ returns the value $d$ using $t$ multiplications, then on base $a$, exponent $b$ and modulus $n$ it returns the value $(a \\cdot d) \\bmod n$ using $t + 1$ multiplications.","title":"Odd step of modular exponentiation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-ret-zero","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_ret_zero","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-ret-zero","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For all natural numbers $a$ and $n$, modular exponentiation with base $a$, exponent $0$ and modulus $n$ returns $1$.","title":"Modular exponentiation returns 1 at exponent zero"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-time-zero","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_time_zero","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-time-zero","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For all natural numbers $a$ and $n$, modular exponentiation with base $a$, exponent $0$ and modulus $n$ performs no multiplications.","title":"No multiplications at exponent zero"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-ret-even","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_ret_even","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-ret-even","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"Let $a$, $b$, $n$ be natural numbers with $b > 0$ and $b$ even, and let $d$ be the value returned by modular exponentiation on base $a$, exponent $b/2$ and modulus $n$. Then the value returned on base $a$, exponent $b$ and modulus $n$ is $(d \\cdot d) \\bmod n$.","title":"Value of the even step of modular exponentiation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-time-even","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_time_even","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-time-even","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"Let $a$, $b$ and $n$ be natural numbers with $b$ positive and even. Modular exponentiation with base $a$, exponent $b$ and modulus $n$ performs exactly one more multiplication than it does with base $a$, exponent $b/2$ and modulus $n$.","title":"Multiplication count in the even case"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-ret-odd","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_ret_odd","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-ret-odd","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"Let $a$, $b$, $n$ be natural numbers with $b$ odd, and let $d$ be the value returned by modular exponentiation on base $a$, exponent $b - 1$ and modulus $n$. Then the value returned on base $a$, exponent $b$ and modulus $n$ is $(a \\cdot d) \\bmod n$.","title":"Value of the odd step of modular exponentiation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-time-odd","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_time_odd","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-time-odd","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"Let $a$, $b$ and $n$ be natural numbers with $b$ odd. Modular exponentiation with base $a$, exponent $b$ and modulus $n$ performs exactly one more multiplication than it does with base $a$, exponent $b - 1$ and modulus $n$.","title":"Multiplication count in the odd case"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-ret-mod","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_ret_mod","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-ret-mod","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For natural numbers $a$, $b$, $n$ with $n > 0$, including $b = 0$, the value $r$ returned by modular exponentiation on base $a$, exponent $b$ and modulus $n$ satisfies $r \\bmod n = a^b \\bmod n$.","title":"Modular exponentiation is correct up to reduction"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-ret-mod-base","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_ret_mod_base","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-ret-mod-base","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For natural numbers $a$, $b$, $n$ with $n > 0$, the value $r$ returned by modular exponentiation on base $a$, exponent $b$ and modulus $n$ satisfies $r \\bmod n = (a \\bmod n)^b \\bmod n$.","title":"Modular exponentiation with a reduced base"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-ret-is-reduced","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_ret_is_reduced","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-ret-is-reduced","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For natural numbers $a$, $b$, $n$ with $b > 0$, the value $r$ returned by modular exponentiation on base $a$, exponent $b$ and modulus $n$ satisfies $r \\bmod n = r$ (with the convention $r \\bmod 0 = r$).","title":"Modular exponentiation returns a reduced value"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-correct","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_correct","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-correct","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For natural numbers $a$, $b$, $n$ with $b > 0$ and $n > 0$, the value returned by modular exponentiation on base $a$, exponent $b$ and modulus $n$ equals $a^b \\bmod n$.","title":"Correctness of modular exponentiation"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-lt","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_lt","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-lt","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For natural numbers $a$, $b$, $n$ with $b > 0$ and $n > 0$, the value returned by modular exponentiation on base $a$, exponent $b$ and modulus $n$ is less than $n$.","title":"Modular exponentiation returns a value below the modulus"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-is-least-residue","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_is_leastResidue","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-is-least-residue","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For natural numbers $a$, $b$, $n$ with $b > 0$ and $n > 0$, the value $r$ returned by modular exponentiation on base $a$, exponent $b$ and modulus $n$ satisfies $r = a^b \\bmod n$ and $r < n$.","title":"Modular exponentiation returns the least residue"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-zero-is-least-res","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_zero_is_leastResidue","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-zero-is-least-res","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For all natural numbers $a$ and $n$ with $n > 1$, the value returned by modular exponentiation with base $a$, exponent $0$ and modulus $n$ equals $a^0 \\bmod n$ and is less than $n$.","title":"Least residue at exponent zero"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-zero-modulus-one","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_zero_modulus_one","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-zero-modulus-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For every natural number $a$, modular exponentiation with base $a$, exponent $0$ and modulus $1$ returns $1$, whereas $a^0 \\bmod 1 = 0$.","title":"Unreduced result at modulus one"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-time-independent","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_time_independent","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-time-independent","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"For all natural numbers $a$, $a'$, $b$, $n$ and $n'$, modular exponentiation performs the same number of multiplications with base $a$, exponent $b$ and modulus $n$ as with base $a'$, exponent $b$ and modulus $n'$.","title":"The multiplication count depends only on the exponent"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/modular-exponentiation-time-bounds","kind":"node","lean":"Cslib.Algorithms.Lean.ModularExponentiation.modularExponentiation_time_bounds","parent":42748,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation/index.html#modular-exponentiation-time-bounds","scope":"ComputerScience/algebraic-and-numerical-algorithms/modular-exponentiation","status":"fully_proved","summary":"Let $a$, $b$ and $n$ be natural numbers with $b > 0$, and let $\\beta = \\lfloor \\log_2 b \\rfloor + 1$ be the number of binary digits of $b$. The number of multiplications performed by modular exponentiation with base $a$, exponent $b$ and modulus $n$ is at least $\\beta$ and at most $2\\beta - 1$.","title":"Multiplication count of modular exponentiation"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test.json","id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","title":"Miller\u2013Rabin primality test"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/two-adic-split","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.twoAdicSplit","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#two-adic-split","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"For a natural number $m > 0$, the two-adic split of $m$ is the pair $(t, u)$ of natural numbers with $m = 2^t u$ and $u$ odd. For $m = 0$ it is $(0, 0)$.","title":"Splitting off the power of two"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-loop","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witnessLoop","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-loop","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Fix a modulus $n$. With no squarings remaining, the loop at the current value $x$ reports a witness exactly when $x \\ne 1$, at no cost. With $k + 1$ squarings remaining, it costs one modular multiplication and computes $y = x^2 \\bmod n$; it reports a witness if $y = 1$, $x \\ne 1$ and $x \\ne n - 1$, and otherwise continues at $y$ with $k$ squarings remaining.","title":"The Miller\u2013Rabin squaring loop"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witness","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"For natural numbers $a$ and $n$, write $n - 1 = 2^t u$ with $u$ odd (here $n - 1$ is truncated at $0$, and $t = u = 0$ when $n \\le 1$). The test sets $x_0$ to the result of modular exponentiation of $a$ to the power $u$ modulo $n$, which is $a^u \\bmod n$ when $u > 0$ and $1$ when $u = 0$. For $i = 1, \\dots, t$ it computes $x_i = x_{i-1}^2 \\bmod n$ and returns true as soon as $x_i = 1$, $x_{i-1} \\ne 1$ and\u2026","title":"Miller\u2013Rabin witness test"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/miller-rabin","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.millerRabin","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#miller-rabin","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Given a natural number $n$ and a list of bases $a_1, \\dots, a_s$, the test runs the witness test for $n$ on each base in turn. It answers *composite* as soon as some base $a_i$ is a witness to the compositeness of $n$, and *prime* if none is (in particular on the empty list). Its cost is the sum of the costs of the witness tests it runs.","title":"The Miller\u2013Rabin test"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-loop-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witnessLoop_time_le","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-loop-time-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"For all natural numbers $n$, $k$ and $x$, the cost of the squaring loop modulo $n$ started at $x$ with $k$ squarings remaining is at most $k$.","title":"Cost bound for the Miller\u2013Rabin squaring loop"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witness_time_le","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-time-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"For natural numbers $a$ and $n$, write $n - 1 = 2^t u$ with $u$ odd (here $n - 1$ is truncated at $0$, and $t = u = 0$ when $n \\le 1$). The cost of the witness test on base $a$ and modulus $n$ is at most the cost of the modular exponentiation of $a$ to the power $u$ modulo $n$ plus $t$: at most one multiplication for each of the $t$ squarings.","title":"Cost of the Miller\u2013Rabin witness test"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-loop-eq-false","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witnessLoop_eq_false","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-loop-eq-false","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $p$ be a prime and let $k$ and $x$ be natural numbers with $x < p$ and $x^{2^k} \\bmod p = 1$. Then the squaring loop modulo $p$ started at $x$ with $k$ squarings remaining reports no witness.","title":"The squaring loop modulo a prime finds no witness when it ends at 1"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-eq-false-of-prime","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witness_eq_false_of_prime","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-eq-false-of-prime","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n$ be a prime and $a$ a natural number with $0 < a < n$. Then the witness test does not report $a$ as a witness to the compositeness of $n$.","title":"A prime has no Miller\u2013Rabin witnesses in range"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/not-prime-of-witness","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.not_prime_of_witness","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#not-prime-of-witness","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n$ and $a$ be natural numbers with $0 < a < n$. If the witness test reports $a$ as a witness to the compositeness of $n$, then $n$ is not prime.","title":"A Miller\u2013Rabin witness in range proves compositeness"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/miller-rabin-of-prime","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.millerRabin_of_prime","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#miller-rabin-of-prime","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n$ be a prime. For every list of bases $a_1, \\dots, a_s$ (possibly empty) with $0 < a_i < n$ for each $i$, the Miller\u2013Rabin test run on $n$ with these bases answers *prime*.","title":"The Miller\u2013Rabin test never rejects a prime when its bases lie in range"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/not-prime-of-miller-rabin-eq-false","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.not_prime_of_millerRabin_eq_false","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#not-prime-of-miller-rabin-eq-false","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n$ be a natural number and $a_1, \\dots, a_s$ a list of bases with $0 < a_i < n$ for each $i$. If the Miller\u2013Rabin test run on $n$ with these bases answers *composite*, then $n$ is not prime.","title":"With bases in range, a composite answer of the Miller\u2013Rabin test is correct"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-loop-eq-true","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witnessLoop_eq_true","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-loop-eq-true","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n$, $k$ and $x$ be natural numbers with $n > 0$, $x < n$ and $x^{2^k} \\bmod n \\ne 1$. Then the squaring loop modulo $n$ started at $x$ with $k$ squarings remaining reports a witness.","title":"The squaring loop reports a witness when it does not end at 1"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-of-pow-ne-one","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witness_of_pow_ne_one","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-of-pow-ne-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $a$ and $n$ be natural numbers with $n > 1$. If $a^{n-1} \\bmod n \\ne 1$, that is, $a^{n-1} \\not\\equiv 1 \\pmod{n}$, then the Miller\u2013Rabin witness test on base $a$ and modulus $n$ returns true.","title":"Fermat witnesses are Miller\u2013Rabin witnesses"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-loop-eq-false-imp","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witnessLoop_eq_false_imp","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-loop-eq-false-imp","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n$, $k$ and $x$ be natural numbers with $x < n$. If the Miller\u2013Rabin squaring loop modulo $n$, started at $x$ with $k$ squarings to perform, reports no witness, then $x = 1$ or $x^{2^i} \\equiv -1 \\pmod n$ for some $i < k$.","title":"When the squaring loop finds no witness"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/witness-eq-false-imp","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.witness_eq_false_imp","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#witness-eq-false-imp","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n > 1$ and write $n - 1 = 2^t u$ with $u$ odd. If $a$ is a natural number for which the Miller\u2013Rabin witness test $\\mathrm{WITNESS}(a, n)$ returns false, then $a^u \\equiv 1 \\pmod n$, or $a^{2^i u} \\equiv -1 \\pmod n$ for some $i < t$.","title":"What a non-witness satisfies"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/coprime-of-witness-eq-false","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.coprime_of_witness_eq_false","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#coprime-of-witness-eq-false","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n > 1$ and let $a$ be a natural number such that the witness test $\\mathrm{WITNESS}(a, n)$ returns false. Then $a^{n-1} \\equiv 1 \\pmod n$ and $a$ is coprime to $n$.","title":"Non-witnesses pass Fermat's test and are units"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/nonwitnesses","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.nonwitnesses","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#nonwitnesses","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"For a natural number $n$, the non-witnesses of $n$ form the finite set of bases $a$ with $0 < a < n$ for which the Miller\u2013Rabin witness test $\\mathrm{WITNESS}(a, n)$ returns false, that is, the bases that do not witness the compositeness of $n$.","title":"Non-witnesses of a number"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-nonwitnesses-le-of-pow-ne-one","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_nonwitnesses_le_of_pow_ne_one","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-nonwitnesses-le-of-pow-ne-one","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n > 1$, and suppose some $x$ coprime to $n$ has $x^{n-1} \\not\\equiv 1 \\pmod n$. Then at most $(n - 1)/2$ of the bases $1, \\dots, n - 1$ are non-witnesses for $n$.","title":"At most half the bases are non-witnesses when some unit fails Fermat's test"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-nonwitnesses-le-of-carmichael","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_nonwitnesses_le_of_carmichael","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-nonwitnesses-le-of-carmichael","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n$ be a Carmichael number: $n > 2$, $n$ is not prime, and $b^{n-1} \\equiv 1 \\pmod n$ for every $b$ coprime to $n$. Then at most $(n - 1)/2$ of the bases $1, \\dots, n - 1$ are non-witnesses for $n$.","title":"At most half the bases are non-witnesses for a Carmichael number"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-nonwitnesses-le","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_nonwitnesses_le","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-nonwitnesses-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n > 1$ be odd and not prime. Then at most $(n - 1)/2$ of the bases $1, \\dots, n - 1$ are non-witnesses for $n$; equivalently, at least $(n - 1)/2$ of them are witnesses to the compositeness of $n$.","title":"At most half the bases are non-witnesses"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/miller-rabin-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.millerRabin_eq_true_iff","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#miller-rabin-eq-true-iff","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"For every natural number $n$ and every list of bases, the Miller\u2013Rabin test on $n$ with these bases answers *prime* if and only if, for every base $a$ in the list, the witness test $\\mathrm{WITNESS}(a, n)$ returns false, that is, $a$ is not a witness to the compositeness of $n$.","title":"Miller\u2013Rabin answers prime exactly when no base is a witness"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-miller-rabin-errors-le","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_millerRabin_errors_le","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-miller-rabin-errors-le","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n > 1$ be odd and not prime, and let $s$ be a natural number. Among the $(n - 1)^s$ lists $(b_1, \\dots, b_s)$ of bases with $0 < b_i < n$, at most $((n - 1)/2)^s$ make the Miller\u2013Rabin test of $n$ with bases $b_1, \\dots, b_s$ answer *prime*.","title":"Half error bound for Miller\u2013Rabin with several bases"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-nonwitnesses-le-quarter-of-pow-ne-o","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_nonwitnesses_le_quarter_of_pow_ne_one","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-nonwitnesses-le-quarter-of-pow-ne-o","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"At most a quarter of the bases are non-witnesses when some unit fails Fermat's test and the modulus is not a prime power. Let $n > 1$ be odd and not prime, and not of the form $p^e$ with $p$ prime and $e \\ge 2$. Suppose some $z$ coprime to $n$ has $z^{n-1} \\not\\equiv 1 \\pmod n$. Then at most $(n - 1)/4$ of the bases $1, \\dots, n - 1$ are non-witnesses for $n$.","title":"At most a quarter of the bases are non-witnesses when some unit fails Fermat's test and the modulus is not a prime power"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-nonwitnesses-le-quarter-of-carmicha","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_nonwitnesses_le_quarter_of_carmichael","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-nonwitnesses-le-quarter-of-carmicha","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n$ be a Carmichael number: $n > 2$, $n$ is not prime, and $b^{n-1} \\equiv 1 \\pmod n$ for every $b$ coprime to $n$. Then at most $(n - 1)/4$ of the bases $1, \\dots, n - 1$ are non-witnesses for $n$.","title":"At most a quarter of the bases are non-witnesses for a Carmichael number"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-nonwitnesses-le-quarter-of-prime-po","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_nonwitnesses_le_quarter_of_prime_pow","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-nonwitnesses-le-quarter-of-prime-po","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $p$ be a prime and $k \\ge 2$ with $p^k$ odd. Then at most $(p^k - 1)/4$ of the bases $1, \\dots, p^k - 1$ are non-witnesses for $p^k$.","title":"At most a quarter of the bases are non-witnesses for an odd prime power"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-nonwitnesses-le-quarter","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_nonwitnesses_le_quarter","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-nonwitnesses-le-quarter","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n > 1$ be odd and not prime. Then at most $(n - 1)/4$ of the bases $1, \\dots, n - 1$ are non-witnesses for $n$, so one uniformly random base fails to expose $n$ as composite with probability at most $1/4$.","title":"At most a quarter of the bases are non-witnesses"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/card-miller-rabin-errors-le-quarter","kind":"node","lean":"Cslib.Algorithms.Lean.MillerRabin.card_millerRabin_errors_le_quarter","parent":42769,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test/index.html#card-miller-rabin-errors-le-quarter","scope":"ComputerScience/algebraic-and-numerical-algorithms/miller-rabin-primality-test","status":"fully_proved","summary":"Let $n > 1$ be odd and not prime, and let $s$ be a natural number. Among the $(n - 1)^s$ lists $(b_1, \\dots, b_s)$ of bases with $0 < b_i < n$, at most $\\lfloor (n - 1)/4 \\rfloor^s$ make the Miller\u2013Rabin test of $n$ with bases $b_1, \\dots, b_s$ answer *prime*.","title":"Quarter error bound for Miller\u2013Rabin with several bases"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences.json","id":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","status":"fully_proved","title":"Solving linear congruences"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/modular-linear-equation-solver","kind":"node","lean":"Cslib.Algorithms.Lean.ModularLinearEquation.modularLinearEquationSolver","parent":42797,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/index.html#modular-linear-equation-solver","scope":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","status":"fully_proved","summary":"Given natural numbers $a > 0$ and $n > 0$ and an integer $b$, the solver solves $a x \\equiv b \\pmod{n}$ following CLRS. Let $d = \\gcd(a, n)$ and let $y$ be the coefficient of $a$ in the B\u00e9zout identity $d = n u + a y$ computed by the extended Euclidean algorithm on $(n, a)$. If $d \\mid b$, it returns the array $[x_0, x_1, \\dots, x_{d-1}]$, where\u2026","title":"Modular linear equation solver"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/modular-linear-equation-solver-ret","kind":"node","lean":"Cslib.Algorithms.Lean.ModularLinearEquation.modularLinearEquationSolver_ret","parent":42797,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/index.html#modular-linear-equation-solver-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","status":"fully_proved","summary":"Let $a > 0$ and $n > 0$ be natural numbers, $b$ an integer, $d = \\gcd(a, n)$, and $y$ the coefficient of $a$ in the B\u00e9zout identity $d = n u + a y$ computed by the extended Euclidean algorithm on $(n, a)$. If $d \\mid b$, the list returned by the modular linear equation solver on $a$, $b$, $n$ is $[x_0, x_1, \\dots, x_{d-1}]$ with $x_i = \\bigl((y \\cdot (b / d)) \\bmod n + i \\cdot (n / d)\\bigr) \\bmod n$, where\u2026","title":"Output of the modular linear equation solver"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/modular-linear-equation-solver-mem-iff","kind":"node","lean":"Cslib.Algorithms.Lean.ModularLinearEquation.modularLinearEquationSolver_mem_iff","parent":42797,"reading_title":{"format":"inline-parts-v1","parts":[["text","The solver returns exactly the solutions below "],["math","\\(n\\)"]],"source":"The solver returns exactly the solutions below $n$"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/index.html#modular-linear-equation-solver-mem-iff","scope":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","status":"fully_proved","summary":"Let $a > 0$ and $n > 0$ be natural numbers and $b$ an integer. A natural number $x$ occurs in the output of the modular linear equation solver on $a$, $b$, $n$ if and only if $x < n$ and $a x \\equiv b \\pmod{n}$.","title":"The solver returns exactly the solutions below $n$"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/modular-linear-equation-solver-nodup","kind":"node","lean":"Cslib.Algorithms.Lean.ModularLinearEquation.modularLinearEquationSolver_nodup","parent":42797,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/index.html#modular-linear-equation-solver-nodup","scope":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","status":"fully_proved","summary":"For natural numbers $a > 0$ and $n > 0$ and an integer $b$, the entries of the output of the modular linear equation solver on $a$, $b$, $n$ are pairwise distinct.","title":"The solver returns no repeated residues"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/modular-linear-equation-solver-length","kind":"node","lean":"Cslib.Algorithms.Lean.ModularLinearEquation.modularLinearEquationSolver_length","parent":42797,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/index.html#modular-linear-equation-solver-length","scope":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","status":"fully_proved","summary":"For natural numbers $a > 0$ and $n > 0$ and an integer $b$, the output of the modular linear equation solver on $a$, $b$, $n$ has $\\gcd(a, n)$ entries if $\\gcd(a, n)$ divides $b$, and no entries otherwise.","title":"Number of residues returned by the solver"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/modular-linear-equation-solver-empty-iff","kind":"node","lean":"Cslib.Algorithms.Lean.ModularLinearEquation.modularLinearEquationSolver_empty_iff","parent":42797,"reading_title":{"format":"inline-parts-v1","parts":[["text","The solver returns nothing exactly when "],["math","\\(\\gcd(a, n) \\nmid b\\)"]],"source":"The solver returns nothing exactly when $\\gcd(a, n) \\nmid b$"},"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/index.html#modular-linear-equation-solver-empty-iff","scope":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","status":"fully_proved","summary":"For natural numbers $a > 0$ and $n > 0$ and an integer $b$, the modular linear equation solver on $a$, $b$, $n$ returns the empty array if and only if $\\gcd(a, n)$ does not divide $b$.","title":"The solver returns nothing exactly when $\\gcd(a, n) \\nmid b$"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/modular-linear-equation-solver-time-boun","kind":"node","lean":"Cslib.Algorithms.Lean.ModularLinearEquation.modularLinearEquationSolver_time_bounds","parent":42797,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences/index.html#modular-linear-equation-solver-time-boun","scope":"ComputerScience/algebraic-and-numerical-algorithms/solving-linear-congruences","status":"fully_proved","summary":"Let $a > 0$ and $n > 0$ be natural numbers, $b$ an integer and $d = \\gcd(a, n)$, and put $c = 4 + 3d$ if $d \\mid b$ and $c = 0$ otherwise. The cost $T$ of the modular linear equation solver on $a$, $b$, $n$ satisfies $1 + c \\le T \\le 6(2 \\lfloor \\log_2 n \\rfloor + 2) + 1 + c$.","title":"Cost bounds for the modular linear equation solver"},{"data_url":"scopes/ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm.json","id":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm","kind":"scope","navigation":"module","parent":42566,"result":false,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/index.html","scope":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm","status":"fully_proved","title":"Strassen algorithm"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/strassen-matrix-accumulate","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.strassenMatrixAccumulate","parent":42805,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/index.html#strassen-matrix-accumulate","scope":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm","status":"fully_proved","summary":"Let $R$ be a type with a zero, an addition, a subtraction and a multiplication (no ring axioms are assumed). For $k \\ge 0$ and $2^k \\times 2^k$ matrices $A$, $B$, $C$ over $R$, this procedure runs Strassen's seven-product recursion for adding $AB$ to $C$. It returns a $2^k \\times 2^k$ matrix together with a pair of counts: the number of scalar multiplications and the number of scalar additions and subtractions it\u2026","title":"Strassen's matrix accumulator"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/strassen-matrix-accumulate-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.strassenMatrixAccumulate_ret","parent":42805,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/index.html#strassen-matrix-accumulate-ret","scope":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm","status":"fully_proved","summary":"Let $R$ be a non-unital, non-associative ring. For every $k \\ge 0$ and all $2^k \\times 2^k$ matrices $A$, $B$, $C$ over $R$, the matrix returned by Strassen's accumulator on $A$, $B$, $C$ equals $C + AB$, where $AB$ is the usual matrix product, $(AB)_{ij} = \\sum_t A_{it} B_{tj}$.","title":"Correctness of Strassen's matrix accumulator"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/strassen-matrix-accumulate-apply","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.strassenMatrixAccumulate_apply","parent":42805,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/index.html#strassen-matrix-accumulate-apply","scope":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm","status":"fully_proved","summary":"Let $R$ be a non-unital, non-associative ring. For every $k \\ge 0$, all $2^k \\times 2^k$ matrices $A$, $B$, $C$ over $R$ and all indices $i, j$, the $(i, j)$ entry of the matrix returned by Strassen's accumulator on $A$, $B$, $C$ is $C_{ij} + \\sum_{t} A_{it} B_{tj}$, the sum running over all $2^k$ indices $t$.","title":"Entries of Strassen's matrix accumulator"},{"id":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/strassen-matrix-accumulate-time","kind":"node","lean":"Cslib.Algorithms.Lean.Matrix.strassenMatrixAccumulate_time","parent":42805,"result":true,"route":"../roadmap/ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm/index.html#strassen-matrix-accumulate-time","scope":"ComputerScience/algebraic-and-numerical-algorithms/strassen-algorithm","status":"fully_proved","summary":"Let $R$ be a type with a zero, an addition, a subtraction and a multiplication. For every $k \\ge 0$ and all $2^k \\times 2^k$ matrices $A$, $B$, $C$ over $R$, Strassen's accumulator on $A$, $B$, $C$ performs exactly $7^k$ scalar multiplications and exactly $(25 \\cdot 7^k - 22 \\cdot 4^k)/3$ scalar additions and subtractions.","title":"Operation counts of Strassen's matrix accumulator"},{"data_url":"scopes/ComputerScience/computational-geometry.json","id":"ComputerScience/computational-geometry","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/computational-geometry/index.html","scope":"ComputerScience/computational-geometry","status":"fully_proved","summary":"Algorithms on points and segments in the plane: orientation and segment intersection, sweep-line intersection detection, convex hulls and closest pair; geometric search structures stay in Data structures.","title":"Computational geometry"},{"data_url":"scopes/ComputerScience/computational-geometry/closest-pair-of-points-problem.json","id":"ComputerScience/computational-geometry/closest-pair-of-points-problem","kind":"scope","navigation":"module","parent":42810,"result":false,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","title":"Closest pair of points problem"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/dist2","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.dist2","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#dist2","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field. The squared distance between points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of $K^2$ is $(p_1 - q_1)^2 + (p_2 - q_2)^2$.","title":"Squared distance in the plane"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/is-min-pair-dist2","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.IsMinPairDist2","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#is-min-pair-dist2","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, and write $d(p, q) = (p_1 - q_1)^2 + (p_2 - q_2)^2$ for the squared distance between points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of $K^2$. For a list $c_1, \\dots, c_n$ of points of $K^2$, an element $\\delta$ of $K$ is the least pairwise squared distance of the list when $\\delta = d(c_i, c_j)$ for some $i < j$ and $\\delta \\le d(c_i, c_j)$ for all $i < j$.","title":"Least pairwise squared distance"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/min-dist-from","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.minDistFrom","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#min-dist-from","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, and write $d(p, q) = (p_1 - q_1)^2 + (p_2 - q_2)^2$ for the squared distance between points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of $K^2$. For a point $p$ and a list $c_1, \\dots, c_m$ of points of $K^2$, the least squared distance from $p$ to the list is no value if $m = 0$ and $\\min_{1 \\le k \\le m} d(p, c_k)$ otherwise. Computing it is charged one unit of cost per point of the\u2026","title":"Least squared distance from a point to a list"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/brute-min","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.bruteMin","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#brute-min","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a field with a linear order, and write $d(p, q) = (x - x')^2 + (y - y')^2$ for the squared distance between points $p = (x, y)$ and $q = (x', y')$ of $K^2$. On a list $c_1, \\dots, c_n$ of points in $K^2$, the brute-force scan returns no value if $n < 2$, and otherwise returns $\\min_{i < j} d(c_i, c_j)$: for each $i$ it computes the least squared distance from $c_i$ to the later points\u2026","title":"Brute-force closest pair"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/scan-from","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.scanFrom","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#scan-from","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, and write $d(p, q) = (p_1 - q_1)^2 + (p_2 - q_2)^2$ for the squared distance between points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of $K^2$. Let $\\delta \\in K$, let $p = (p_1, p_2)$ be a point of $K^2$ and $c_1, \\dots, c_m$ a list of points $c_l = (x_l, y_l)$ of $K^2$, and let $k \\le m$ be largest such that $(y_l - p_2)^2 < \\delta$ for all $1 \\le l \\le k$. The scan from $p$ with\u2026","title":"Strip scan from a point"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/strip-scan","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.stripScan","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#strip-scan","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, $\\delta \\in K$ and $c_1, \\dots, c_n$ a list of points of $K^2$. For each $i$, the strip scan runs the strip scan from $c_i$ with threshold $\\delta$ on the later points $c_{i+1}, \\dots, c_n$. It returns the least of the values these scans return, or no value if none of them returns a value, and its cost is the sum of their costs.","title":"Closest-pair strip scan"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-rec","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestRec","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-rec","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and $c_1, \\dots, c_n$ a list of points $c_i = (x_i, y_i)$ of $K^2$. If $n \\le 3$, the recursion returns the brute-force scan of the list. Otherwise let $m = \\lfloor n/2 \\rfloor$, and run the recursion on $c_1, \\dots, c_m$ and on $c_{m+1}, \\dots, c_n$. If neither call returns a value, it returns no value. Otherwise let $\\delta$ be the least of the returned values, let $T$ be the\u2026","title":"Closest-pair recursion"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-pair","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestPair","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-pair","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a field with a linear order. Given a list of points in $K^2$, the closest-pair algorithm sorts the points by first coordinate (merge sort, at no cost) and runs the closest-pair recursion on the sorted list. It returns a squared distance or no value, and is charged one unit of cost per squared distance computed.","title":"Divide-and-conquer closest pair"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/is-answer","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.IsAnswer","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#is-answer","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, and write $d(p, q) = (p_1 - q_1)^2 + (p_2 - q_2)^2$ for the squared distance between points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of $K^2$. For a list $c_1, \\dots, c_n$ of points of $K^2$, an optional value $r$ (an element of $K$ or no value) is an answer for the list when every value $\\delta$ that $r$ holds equals $d(c_i, c_j)$ for some $i < j$, and when for all $i < j$, $r$\u2026","title":"Closest-pair answer"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/min-dist-from-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.minDistFrom_spec","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#min-dist-from-spec","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, write $d(u, v) = (x - x')^2 + (y - y')^2$ for the squared distance between points $u = (x, y)$ and $v = (x', y')$ of $K^2$, let $p \\in K^2$, and let $qs$ be a list of points in $K^2$. If the scan for the least squared distance from $p$ to the entries of $qs$ returns a value $\\delta$, then $\\delta = d(p, q)$ for some entry $q$ of $qs$. For every entry $q$ of $qs$, the scan\u2026","title":"The nearest-point scan returns the least squared distance"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/strip-scan-lower","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.stripScan_lower","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#strip-scan-lower","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, write $d(p, q) = (x - x')^2 + (y - y')^2$ for the squared distance between points $p = (x, y)$ and $q = (x', y')$ of $K^2$, and let $\\delta \\in K$. Let $ys$ be a list of points in $K^2$ sorted by second coordinate: whenever an entry precedes another, its second coordinate is at most the other's. If $a = (a_1, a_2)$ and $b = (b_1, b_2)$ are entries of $ys$ at positions $i < j$\u2026","title":"The strip scan bounds every pair with a small vertical gap"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-rec-is-answer","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestRec_isAnswer","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-rec-is-answer","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, write $d(u, v) = (x - x')^2 + (y - y')^2$ for the squared distance between points $u = (x, y)$ and $v = (x', y')$ of $K^2$, and let $xs$ be a list of points in $K^2$ sorted by first coordinate: whenever an entry precedes another, its first coordinate is at most the other's. If the closest-pair recursion returns a value $\\delta$ on $xs$, then $\\delta = d(p, q)$ for some entries\u2026","title":"The closest-pair recursion is correct on lists sorted by first coordinate"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-pair-is-answer","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestPair_isAnswer","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-pair-is-answer","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, write $d(p, q) = (x - x')^2 + (y - y')^2$ for the squared distance between points $p = (x, y)$ and $q = (x', y')$ of $K^2$, and let $ps$ be a list of points in $K^2$. If the divide-and-conquer closest-pair algorithm returns a value $\\delta$ on $ps$, then $\\delta = d(p, q)$ for some entries $p$, $q$ of $ps$ at different positions. For any two entries $p$, $q$ of $ps$ at different\u2026","title":"The closest-pair algorithm returns the answer"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-pair-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestPair_spec","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-pair-spec","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and let $ps$ be a list of at least two points in $K^2$. Then the divide-and-conquer closest-pair algorithm returns a value $\\delta$, and $\\delta$ is the least squared distance between two entries of $ps$ at different positions: some two entries at different positions have squared distance $\\delta$, and any two entries at different positions have squared distance at least $\\delta$.","title":"Correctness of divide-and-conquer closest pair"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-pair-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestPair_eq_none_iff","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-pair-eq-none-iff","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field. For every list of points in $K^2$, the divide-and-conquer closest-pair algorithm returns no value if and only if the list has fewer than two entries.","title":"The closest-pair algorithm returns no value exactly on fewer than two points"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-pair-eq-brute-min","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestPair_eq_bruteMin","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-pair-eq-brute-min","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a linearly ordered field. For every list of points in $K^2$, the divide-and-conquer closest-pair algorithm returns the same value as the brute-force scan (their costs may differ).","title":"Divide and conquer agrees with brute force"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/brute-min-time","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.bruteMin_time","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#brute-min-time","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $K$ be a field with a linear order and let $ps$ be a list of $n$ points in $K^2$. Counting one unit of cost per squared distance computed, twice the cost of the brute-force scan on $ps$ equals $n(n-1)$; that is, the scan computes exactly $n(n-1)/2$ squared distances.","title":"Cost of the brute-force closest-pair scan"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/tagged","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.Tagged","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#tagged","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"A tagged point is a pair $(p, s)$ of a point $p \\in \\mathbb{R}^2$ and a Boolean $s$, read as the side of the dividing line on which $p$ lies: false for left, true for right.","title":"Side-tagged points"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/same-side-far","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.SameSideFar","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#same-side-far","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"For a real $\\delta$, two tagged points $a$ and $b$ are same-side far apart when, if $a$ and $b$ carry the same side, the squared distance between their points is at least $\\delta$.","title":"Same-side points far apart"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/in-strip","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.InStrip","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#in-strip","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Given reals $\\delta$ and $x_{\\mathrm{mid}}$, a tagged point $((x, y), s)$ lies in the strip when $(x - x_{\\mathrm{mid}})^2 < \\delta$, and moreover $x \\le x_{\\mathrm{mid}}$ if its side $s$ is left ($s$ false) and $x_{\\mathrm{mid}} \\le x$ if its side $s$ is right ($s$ true).","title":"Strip points"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/length-le-seven","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.length_le_seven","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#length-le-seven","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $\\delta$ and $x_{\\mathrm{mid}}$ be reals, $p$ a tagged point, $Z$ a list of tagged points, and $L$ the list $p$ followed by $Z$; write $y_a$ for the $y$-coordinate of a tagged point $a$. Suppose every point of $L$ lies in the strip for $\\delta$ and $x_{\\mathrm{mid}}$, any two points of $L$ at different positions with the same side are at squared distance at least $\\delta$, and every point $a$ of $Z$ satisfies\u2026","title":"Packing bound for the strip"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/scan-from-time","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.scanFrom_time","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#scan-from-time","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $\\delta$ be a real, $q = (q_1, q_2)$ a point of $\\mathbb{R}^2$ and $L$ a list of points of $\\mathbb{R}^2$. The cost of the scan from $q$ over $L$ with threshold $\\delta$, one unit per squared distance computed, equals the length of the longest initial segment of $L$ all of whose points $r = (r_1, r_2)$ satisfy $(r_2 - q_2)^2 < \\delta$.","title":"Cost of the scan from one strip point"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/strip-scan-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.stripScan_time_le","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#strip-scan-time-le","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"Let $\\delta$ and $x_{\\mathrm{mid}}$ be reals and $Y$ a list of tagged points that is sorted by nondecreasing $y$-coordinate, whose points all lie in the strip for $\\delta$ and $x_{\\mathrm{mid}}$, and in which any two points at different positions with the same side are at squared distance at least $\\delta$. Then the strip scan with parameter $\\delta$ over the underlying points of $Y$ (tags forgotten) computes at\u2026","title":"Cost of the strip scan"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-rec-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestRec_time_le","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-rec-time-le","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"For every list of $n$ points in $\\mathbb{R}^2$ sorted by nondecreasing $x$-coordinate, the divide-and-conquer recursion of the closest-pair algorithm computes at most $7 n \\lceil \\log_2 n \\rceil + n$ squared distances, where $\\lceil \\log_2 0 \\rceil$ is read as $0$.","title":"Running time of the closest-pair recursion"},{"id":"ComputerScience/computational-geometry/closest-pair-of-points-problem/closest-pair-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.closestPair_time_le","parent":42811,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/closest-pair-of-points-problem/index.html#closest-pair-time-le","scope":"ComputerScience/computational-geometry/closest-pair-of-points-problem","status":"fully_proved","summary":"For every list of $n$ points in $\\mathbb{R}^2$, the closest-pair algorithm computes at most $7 n \\lceil \\log_2 n \\rceil + n$ squared distances, where $\\lceil \\log_2 0 \\rceil$ is read as $0$. In particular it computes $O(n \\log n)$ distances.","title":"Running time of the closest-pair algorithm"},{"data_url":"scopes/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests.json","id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","kind":"scope","navigation":"module","parent":42810,"result":false,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","title":"Orientation and segment intersection tests"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/cross","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.cross","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#cross","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"For $p = (x_p, y_p)$ and $q = (x_q, y_q)$ in $K^2$, where $K$ is a linearly ordered field, the cross product is $p \\times q = x_p y_q - x_q y_p$ (CLRS equation (33.1)).","title":"Cross product of plane vectors"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/direction","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.direction","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#direction","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"For points $p_i, p_j, p_k \\in K^2$, where $K$ is a linearly ordered field, CLRS's DIRECTION$(p_i, p_j, p_k)$ is the cross product $(p_k - p_i) \\times (p_j - p_i)$.","title":"Direction of three points"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/on-segment","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.onSegment","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#on-segment","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"For points $p_i = (x_i, y_i)$, $p_j = (x_j, y_j)$ and $p_k = (x_k, y_k)$ in $K^2$, where $K$ is a linearly ordered field, CLRS's ON-SEGMENT$(p_i, p_j, p_k)$ is the Boolean that is true exactly when $p_k$ lies in the bounding box of $p_i$ and $p_j$, that is, $\\min(x_i, x_j) \\le x_k \\le \\max(x_i, x_j)$ and $\\min(y_i, y_j) \\le y_k \\le \\max(y_i, y_j)$.","title":"The ON-SEGMENT test"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/straddles","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.Straddles","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#straddles","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Two elements $d$ and $e$ of a field with a linear order straddle zero when $d > 0$ and $e < 0$, or $d < 0$ and $e > 0$; that is, $d$ and $e$ are nonzero with opposite signs.","title":"Values of opposite sign"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/collinear-branch","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.collinearBranch","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#collinear-branch","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, $d \\in K$ and $p_i, p_j, p_k \\in K^2$. When $d = 0$, this step costs one unit and returns the value of ON-SEGMENT$(p_i, p_j, p_k)$; when $d \\neq 0$, it costs nothing and returns false.","title":"Collinear-endpoint step of SEGMENTS-INTERSECT"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/segments-intersect","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.segmentsIntersect","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#segments-intersect","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a field with a linear order and let $p_1, p_2, p_3, p_4 \\in K^2$. CLRS's procedure SEGMENTS-INTERSECT$(p_1, p_2, p_3, p_4)$, which tests whether the segments $p_1p_2$ and $p_3p_4$ meet, first computes, at a cost of $4$, the directions $d_1 = \\mathrm{DIRECTION}(p_3, p_4, p_1)$, $d_2 = \\mathrm{DIRECTION}(p_3, p_4, p_2)$, $d_3 = \\mathrm{DIRECTION}(p_1, p_2, p_3)$ and\u2026","title":"The SEGMENTS-INTERSECT procedure"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/segments-intersect-time","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.segmentsIntersect_time","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#segments-intersect-time","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a field with a linear order and let $p_1, p_2, p_3, p_4 \\in K^2$. In the cost model that charges one unit per DIRECTION computation and one per ON-SEGMENT test, SEGMENTS-INTERSECT$(p_1, p_2, p_3, p_4)$ costs at least $4$ and at most $8$.","title":"Running time of SEGMENTS-INTERSECT"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/segments-intersect-ret","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.segmentsIntersect_ret","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#segments-intersect-ret","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a field with a linear order, let $p_1, p_2, p_3, p_4 \\in K^2$, and write $d_1 = \\mathrm{DIRECTION}(p_3, p_4, p_1)$, $d_2 = \\mathrm{DIRECTION}(p_3, p_4, p_2)$, $d_3 = \\mathrm{DIRECTION}(p_1, p_2, p_3)$ and $d_4 = \\mathrm{DIRECTION}(p_1, p_2, p_4)$. The value returned by SEGMENTS-INTERSECT$(p_1, p_2, p_3, p_4)$ is true if and only if at least one of the following holds: $d_1$ and $d_2$ are nonzero with\u2026","title":"SEGMENTS-INTERSECT as a Boolean formula"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/on-segment-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.onSegment_of_mem","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#on-segment-of-mem","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and $p, q, r \\in K^2$. If $r$ lies on the closed segment from $p$ to $q$, then ON-SEGMENT$(p, q, r)$ is true, that is, $r$ lies in the bounding box of $p$ and $q$.","title":"Points of a segment lie in its bounding box"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/direction-eq-zero-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.direction_eq_zero_of_mem","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#direction-eq-zero-of-mem","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and $p, q, r \\in K^2$. If $r$ lies on the closed segment from $p$ to $q$, then the direction $(r - p) \\times (q - p)$ is zero, that is, $r$ is collinear with $p$ and $q$.","title":"Points of a segment have direction zero"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/mem-segment-of-direction-eq-zero","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.mem_segment_of_direction_eq_zero","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#mem-segment-of-direction-eq-zero","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, write $(u_1, u_2) \\times (v_1, v_2) = u_1 v_2 - v_1 u_2$ for the cross product on $K^2$, and let $p, q, r \\in K^2$. If $r$ is collinear with $p$ and $q$, that is, the direction $(r - p) \\times (q - p)$ is zero, and ON-SEGMENT$(p, q, r)$ is true, that is, $r$ lies in the bounding box of $p$ and $q$, then $r$ lies on the closed segment from $p$ to $q$.","title":"A collinear point in the bounding box lies on the segment"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/sub-ne-zero","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.Straddles.sub_ne_zero","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#sub-ne-zero","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"If $d$ and $e$ are elements of a linearly ordered field that straddle zero, that is, they are nonzero with opposite signs, then $d - e \\ne 0$.","title":"Values of opposite sign have nonzero difference"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/param-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.Straddles.param_mem","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#param-mem","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"If $d$ and $e$ are elements of a linearly ordered field that straddle zero, that is, they are nonzero with opposite signs, then $0 \\le d / (d - e) \\le 1$.","title":"Zero-crossing parameter of values of opposite sign"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/straddles-of-combo","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.straddles_of_combo","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#straddles-of-combo","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $d$ and $e$ be nonzero elements of a linearly ordered field. If $a d + b e = 0$ for some $a, b \\ge 0$ in the field with $a + b = 1$, then $d$ and $e$ have opposite signs.","title":"A vanishing convex combination of nonzero values forces opposite signs"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/inter-nonempty-of-straddles","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.inter_nonempty_of_straddles","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#inter-nonempty-of-straddles","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and $p_1, p_2, p_3, p_4 \\in K^2$. Suppose the directions $(p_1 - p_3) \\times (p_4 - p_3)$ and $(p_2 - p_3) \\times (p_4 - p_3)$ are nonzero with opposite signs, and so are $(p_3 - p_1) \\times (p_2 - p_1)$ and $(p_4 - p_1) \\times (p_2 - p_1)$. Then the closed segments $p_1 p_2$ and $p_3 p_4$ have a common point.","title":"Properly crossing segments meet"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/eq-param-of-direction-eq-zero","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.eq_param_of_direction_eq_zero","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#eq-param-of-direction-eq-zero","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and $p, q, r \\in K^2$ with $p \\neq q$. If $(r - p) \\times (q - p) = 0$, then $r = p + \\mu (q - p)$ for some $\\mu \\in K$.","title":"Points with direction zero lie on the line"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/endpoint-mem-of-collinear","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.endpoint_mem_of_collinear","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#endpoint-mem-of-collinear","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and $p_1, p_2, p_3, p_4 \\in K^2$. Suppose the directions $(p_1 - p_3) \\times (p_4 - p_3)$ and $(p_2 - p_3) \\times (p_4 - p_3)$ are both zero and the closed segments $p_1 p_2$ and $p_3 p_4$ have a common point. Then $p_1$ or $p_2$ lies on the segment $p_3 p_4$, or $p_3$ or $p_4$ lies on the segment $p_1 p_2$.","title":"Collinear case of segment intersection"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/straddles-or-endpoint-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.straddles_or_endpoint_mem","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#straddles-or-endpoint-mem","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and $p_1, p_2, p_3, p_4 \\in K^2$, and suppose the closed segments $[p_1, p_2]$ and $[p_3, p_4]$ have a common point. Then either $\\mathrm{DIRECTION}(p_3, p_4, p_1)$ and $\\mathrm{DIRECTION}(p_3, p_4, p_2)$ are nonzero with opposite signs and $\\mathrm{DIRECTION}(p_1, p_2, p_3)$ and $\\mathrm{DIRECTION}(p_1, p_2, p_4)$ are nonzero with opposite signs, or $p_1 \\in [p_3, p_4]$, or\u2026","title":"Meeting segments cross properly or an endpoint lies on the other"},{"id":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/segments-intersect-ret-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.segmentsIntersect_ret_eq_true_iff","parent":42837,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/orientation-and-segment-intersection-tests/index.html#segments-intersect-ret-eq-true-iff","scope":"ComputerScience/computational-geometry/orientation-and-segment-intersection-tests","status":"fully_proved","summary":"Let $K$ be a linearly ordered field and let $p_1, p_2, p_3, p_4 \\in K^2$. The procedure SEGMENTS-INTERSECT$(p_1, p_2, p_3, p_4)$ returns true if and only if the closed segments $[p_1, p_2]$ and $[p_3, p_4]$ have a common point.","title":"Correctness of SEGMENTS-INTERSECT"},{"data_url":"scopes/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm.json","id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","kind":"scope","navigation":"module","parent":42810,"result":false,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","title":"Monotone chain convex hull algorithm"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/pop-non-left","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.popNonLeft","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#pop-non-left","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field, let $p \\in \\Bbbk^2$, and write $u \\times v = u_1 v_2 - v_1 u_2$ for the cross product. Given a stack of points of $\\Bbbk^2$ listed top first, while the stack has at least two points, with top $a$ and next point $b$, and $(a - b) \\times (p - b) \\le 0$ (so $b \\to a \\to p$ is not a strict left turn), remove $a$. The result is the first stack reached that has fewer than two\u2026","title":"Pop step of the monotone-chain scan"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/chain-scan","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.chainScan","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#chain-scan","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order. Given a stack $S$ of points of $\\Bbbk^2$, listed top first, and a list of points $p_1, \\dots, p_n$ of $\\Bbbk^2$, the scan processes $p_1, \\dots, p_n$ in order: for each point $p = p_i$ it removes the top $a$ of the stack as long as the stack has a second element $b$ with $(a_x - b_x)(p_y - b_y) - (p_x - b_x)(a_y - b_y) \\le 0$, and then pushes $p$. Starting from $S$, the\u2026","title":"Monotone chain scan"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHull","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order. The lower chain of a list of points $p_1, \\dots, p_n$ of $\\Bbbk^2$ is computed by Andrew's monotone chain scan: start with an empty stack and process $p_1, \\dots, p_n$ in order; for each point $p = p_i$, remove the top $a$ of the stack as long as the stack has a second element $b$ with $(a_x - b_x)(p_y - b_y) - (p_x - b_x)(a_y - b_y) \\le 0$, then push $p$. The lower chain\u2026","title":"Lower chain"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull-of-points","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHullOfPoints","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull-of-points","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order. For a list $P$ of points of $\\Bbbk^2$, sort $P$ by merge sort into nondecreasing lexicographic $(x, y)$ order and return the lower chain of the sorted list. Only the orientation tests of the lower chain are counted; sorting costs nothing.","title":"Lower chain of unsorted points"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull-sublist","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHull_sublist","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull-sublist","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field and let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$ that are strictly increasing in lexicographic $(x, y)$ order. Then the lower chain of $p_1, \\dots, p_n$ is a sublist of $p_1, \\dots, p_n$: it is $p_{i_1}, \\dots, p_{i_m}$ for some indices $i_1 < \\dots < i_m$.","title":"The lower chain is a sublist of the input"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull-head","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHull_head?","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull-head","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field and let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$ that are strictly increasing in lexicographic $(x, y)$ order. If $n \\ge 1$, the lower chain of $p_1, \\dots, p_n$ starts at $p_1$; if $n = 0$, the lower chain is empty.","title":"The lower chain starts at the first point"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull-get-last","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHull_getLast?","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull-get-last","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field and let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$ that are strictly increasing in lexicographic $(x, y)$ order. If $n \\ge 1$, the lower chain of $p_1, \\dots, p_n$ ends at $p_n$; if $n = 0$, the lower chain is empty.","title":"The lower chain ends at the last point"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull-turn-left","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHull_turn_left","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull-turn-left","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field and let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$ that are strictly increasing in lexicographic $(x, y)$ order, with lower chain $c_0, \\dots, c_{m-1}$. Writing $\\operatorname{cross}(u, v) = u_x v_y - v_x u_y$, for every $i$ with $i + 2 < m$ we have $\\operatorname{cross}(c_{i+1} - c_i, c_{i+2} - c_i) > 0$.","title":"The lower chain turns strictly left"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull-left-of-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHull_left_of_edges","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull-left-of-edges","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field and let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$ that are strictly increasing in lexicographic $(x, y)$ order, and write $\\operatorname{cross}(u, v) = u_x v_y - v_x u_y$. For every $s \\in \\{p_1, \\dots, p_n\\}$ and every pair of consecutive vertices $a, b$ of the lower chain of $p_1, \\dots, p_n$ we have $\\operatorname{cross}(b - a, s - a) \\ge 0$: the point $s$ lies on or to the\u2026","title":"Every input point lies on or left of each lower-chain edge"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHull_time_le","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull-time-le","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order. For every list of $n$ points of $\\Bbbk^2$, computing its lower chain makes at most $2n$ orientation tests (cross-product evaluations).","title":"The lower-chain scan makes linearly many tests"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/lower-hull-of-points-spec","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lowerHullOfPoints_spec","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#lower-hull-of-points-spec","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field, let $P$ be a list of pairwise distinct points of $\\Bbbk^2$ in any order, write $\\preceq$ for the lexicographic $(x, y)$ order on $\\Bbbk^2$, and let $C = c_0, \\dots, c_{m-1}$ be the lower chain of the list obtained by sorting $P$ into $\\preceq$-increasing order. Write $\\operatorname{cross}(u, v) = u_x v_y - v_x u_y$. Then every $c_j$ is a point of $P$; if $P$ is nonempty then\u2026","title":"Correctness of the lower chain for unsorted input"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/upper-hull","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.upperHull","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#upper-hull","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $K$ be a field with a linear order. The upper chain of a list of points $p_1, \\dots, p_n \\in K^2$ is the lower chain of the reversed list $p_n, \\dots, p_1$, together with its cost: the same scan, counting the same orientation tests. For points sorted by $(x, y)$ it runs from the last point back to the first.","title":"Upper chain"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/convex-hull","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.convexHull","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#convex-hull","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order. The convex hull of a list $P$ of points of $\\Bbbk^2$ is the lower chain of $P$ followed by the upper chain of $P$ with its first and last points removed, together with its cost: the cost of computing the lower chain plus the cost of computing the upper chain.","title":"Convex hull by the monotone chain algorithm"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/upper-hull-eq","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.upperHull_eq","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#upper-hull-eq","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order. For every list $p_1, \\dots, p_n$ of points of $\\Bbbk^2$, the upper chain of the list is the image under $p \\mapsto -p$ of the lower chain of $-p_n, \\dots, -p_1$, and the two computations have the same cost.","title":"The upper chain is a reflected lower chain"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/upper-hull-sublist","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.upperHull_sublist","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#upper-hull-sublist","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $K$ be an ordered field and let $p_1, \\dots, p_n \\in K^2$ be strictly increasing in the lexicographic $(x, y)$ order. Then the upper chain of $p_1, \\dots, p_n$ is a sublist of the reversed list $p_n, \\dots, p_1$: it is obtained from $p_n, \\dots, p_1$ by deleting some of the entries.","title":"The upper chain is a subsequence of the reversed input"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/upper-hull-head","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.upperHull_head?","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#upper-hull-head","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$, for an ordered field $\\Bbbk$, strictly increasing in the lexicographic order on $(x, y)$. If $n = 0$ the upper chain is empty; otherwise its first vertex is $p_n$.","title":"The upper chain starts at the last point"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/upper-hull-get-last","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.upperHull_getLast?","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#upper-hull-get-last","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$, for an ordered field $\\Bbbk$, strictly increasing in the lexicographic order on $(x, y)$. If $n = 0$ the upper chain is empty; otherwise its last vertex is $p_1$.","title":"The upper chain ends at the first point"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/upper-hull-turn-left","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.upperHull_turn_left","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#upper-hull-turn-left","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $K$ be an ordered field, let $p_1, \\dots, p_n \\in K^2$ be strictly increasing in the lexicographic $(x, y)$ order, and let $c_0, c_1, \\dots, c_{m-1}$ be the upper chain of $p_1, \\dots, p_n$. Then for every $i$ with $i + 2 < m$, we have $(c_{i+1} - c_i) \\times (c_{i+2} - c_i) > 0$, where $(u_1, u_2) \\times (v_1, v_2) = u_1 v_2 - v_1 u_2$ is the cross product: the vertices $c_i$, $c_{i+1}$, $c_{i+2}$ make a\u2026","title":"The upper chain turns strictly left"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/upper-hull-left-of-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.upperHull_left_of_edges","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#upper-hull-left-of-edges","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$, for an ordered field $\\Bbbk$, strictly increasing in the lexicographic order on $(x, y)$. Then for every input point $s$ and every two consecutive vertices $a, b$ of the upper chain, the cross product $(b - a) \\times (s - a)$ is nonnegative: $s$ lies on or to the left of the edge from $a$ to $b$.","title":"Every input point lies left of the upper chain"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/upper-hull-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.upperHull_time_le","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#upper-hull-time-le","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $K$ be a field with a linear order and let $p_1, \\dots, p_n \\in K^2$ be any $n$ points, sorted or not. Computing the upper chain of $p_1, \\dots, p_n$ costs at most $2n$, where the cost counts one unit per orientation test (one cross-product comparison) and nothing else.","title":"Running time of the upper chain scan"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/convex-hull-subset","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.convexHull_subset","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#convex-hull-subset","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $p_1, \\dots, p_n$ be points of $\\Bbbk^2$, for an ordered field $\\Bbbk$, strictly increasing in the lexicographic order on $(x, y)$. Then every vertex of their convex hull computed by the monotone chain algorithm is one of the $p_i$.","title":"The hull's vertices are input points"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/convex-hull-left-of-edges","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.convexHull_left_of_edges","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#convex-hull-left-of-edges","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $p_1, \\dots, p_n$ be $n \\ge 2$ points of $\\Bbbk^2$, for an ordered field $\\Bbbk$, strictly increasing in the lexicographic order on $(x, y)$, and let $v_1, \\dots, v_m$ be the vertices of their convex hull computed by the monotone chain algorithm. Then for every input point $s$ and every edge $(a, b)$ of the closed polygon $v_1, \\dots, v_m, v_1$ (that is, $a, b$ consecutive in this sequence), the cross product\u2026","title":"The hull polygon contains every input point"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/convex-hull-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.convexHull_time_le","parent":42857,"reading_title":{"format":"inline-parts-v1","parts":[["text","The monotone chain hull makes at most "],["math","\\(4n\\)"],["text"," orientation tests"]],"source":"The monotone chain hull makes at most $4n$ orientation tests"},"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#convex-hull-time-le","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order. For every list of $n$ points of $\\Bbbk^2$, computing their convex hull by the monotone chain algorithm costs at most $4n$, where each orientation test costs one unit.","title":"The monotone chain hull makes at most $4n$ orientation tests"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/turns-left","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.TurnsLeft","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#turns-left","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order and write $(u_1, u_2) \\times (v_1, v_2) = u_1 v_2 - v_1 u_2$ for the cross product on $\\Bbbk^2$. A list $q_1, \\dots, q_k$ of points of $\\Bbbk^2$ turns strictly left when every three consecutive points do: $(q_{i+1} - q_i) \\times (q_{i+2} - q_i) > 0$ for every $i$ with $1 \\le i \\le k - 2$. In particular, every list of fewer than three points turns strictly left.","title":"Lists of points turning strictly left"},{"id":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/convex-hull-turns-left","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.convexHull_turnsLeft","parent":42857,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm/index.html#convex-hull-turns-left","scope":"ComputerScience/computational-geometry/monotone-chain-convex-hull-algorithm","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field and write $(u_1, u_2) \\times (v_1, v_2) = u_1 v_2 - v_1 u_2$ for the cross product on $\\Bbbk^2$. Let $P$ be a list of points of $\\Bbbk^2$ that is strictly increasing in the lexicographic order on $(x, y)$, and suppose the points of $P$ are not all collinear: there are points $a, b, c$ of $P$ with $(b - a) \\times (c - a) \\ne 0$. Let $H$ be the list of points returned by the\u2026","title":"The monotone chain hull of non-collinear points turns strictly left at every vertex"},{"data_url":"scopes/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection.json","id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","kind":"scope","navigation":"module","parent":42810,"result":false,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","title":"Sweep-line segment intersection detection"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/line-y","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.lineY","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#line-y","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ in $K^2$, $K$ a field, and $x \\in K$, this is $p_2 + (x - p_1)(q_2 - p_2)/(q_1 - p_1)$, with division by zero read as $0$. When $p_1 \\ne q_1$ it is the height at $x$ of the line through $p$ and $q$.","title":"Height of a line at an abscissa"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/mem-segment-iff-line-y","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.mem_segment_iff_lineY","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#mem-segment-iff-line-y","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $p$ and $q$ be points of the plane over a linearly ordered field, where a point $z$ has coordinates $(z_1, z_2)$, and suppose $p_1 < q_1$. A point $z$ lies on the segment from $p$ to $q$ if and only if $p_1 \\le z_1 \\le q_1$ and $z_2$ is the height at abscissa $z_1$ of the line through $p$ and $q$.","title":"A non-vertical segment is the graph of its line"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/direction-eq-line-y","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.direction_eq_lineY","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#direction-eq-line-y","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order, let $p, q, r \\in \\Bbbk^2$, writing $p = (p_x, p_y)$ and likewise for $q$ and $r$, with $p_x < q_x$, and let $y_{pq}(x)$ be the height at $x$ of the line through $p$ and $q$. Then $\\mathrm{DIRECTION}(p, q, r) = (q_x - p_x)\\,(y_{pq}(r_x) - r_y)$.","title":"DIRECTION as a scaled vertical offset"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/exists-mem-inter-of-cross","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.exists_mem_inter_of_cross","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#exists-mem-inter-of-cross","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, write $z_1$ for the first coordinate of a point $z \\in K^2$, and let $[p, q]$ and $[p', q']$ be segments in $K^2$ with $p_1 < q_1$ and $p'_1 < q'_1$. For $t \\in K$ let $y(t)$ be the height at $t$ of the line through $p$ and $q$, and $y'(t)$ that of the line through $p'$ and $q'$. Let $a < b$ with $p_1 \\le a$, $p'_1 \\le a$, $b \\le q_1$ and $b \\le q'_1$. If $y(a) < y'(a)$ and\u2026","title":"Crossing of two non-vertical segments"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/exists-leftmost-mem-inter","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.exists_leftmost_mem_inter","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#exists-leftmost-mem-inter","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $K$ be a linearly ordered field, write $x_1$ for the first coordinate of a point $x \\in K^2$, and let $[p, q]$ and $[p', q']$ be segments in $K^2$ with $p_1 < q_1$ and $p'_1 < q'_1$. If $[p, q] \\cap [p', q']$ is nonempty, then it contains a point $m$ with $m_1 \\le w_1$ for every $w \\in [p, q] \\cap [p', q']$.","title":"Leftmost point of the intersection of two segments"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/pairwise-take-while-append-cons-drop-whi","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.pairwise_takeWhile_append_cons_dropWhile_lt","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#pairwise-take-while-append-cons-drop-whi","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $f$ map a type $\\alpha$ to a linearly ordered set, let $T$ be a list over $\\alpha$ with $f(j) < f(k)$ whenever $j$ occurs before $k$, let $i \\in \\alpha$ be such that $f(j) \\ne f(i)$ for every $j$ in $T$, and let $P$ be a predicate that holds for an element $j$ of $T$ exactly when $f(j) < f(i)$. Insert $i$ just before the first element of $T$ not satisfying $P$: the list made of the longest prefix of $T$\u2026","title":"Inserting into a strictly sorted list"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/pairwise-take-while-append-cons-drop-whi-2","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.pairwise_takeWhile_append_cons_dropWhile_le","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#pairwise-take-while-append-cons-drop-whi-2","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $f$ map a type $\\alpha$ to a linearly ordered set, let $T$ be a list over $\\alpha$ with $f(j) \\le f(k)$ whenever $j$ occurs before $k$, let $i \\in \\alpha$, and let $P$ be a predicate that holds for an element $j$ of $T$ exactly when $f(j) < f(i)$. Insert $i$ just before the first element of $T$ not satisfying $P$: the list made of the longest prefix of $T$ satisfying $P$, then $i$, then the rest of $T$. In this\u2026","title":"Inserting into a weakly sorted list"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/head-drop-while-eq-of-unique","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.head?_dropWhile_eq_of_unique","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#head-drop-while-eq-of-unique","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $f$ map a type to a linearly ordered set, let $y$ be a value, and let $T$ be a list sorted non-strictly by $f$, that is, $f(j) \\le f(k)$ whenever $j$ occurs before $k$ in $T$. Suppose $b$ lies in $T$ with $f(b) = y$, and every element $c \\ne b$ of $T$ has $f(c) \\ne y$. Then removing from $T$ its longest prefix of elements $j$ with $f(j) < y$ leaves a list whose first element is $b$.","title":"First element at or above a uniquely attained height"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/exists-append-of-mem-of-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.exists_append_of_mem_of_mem","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#exists-append-of-mem-of-mem","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"If $a$ and $b$ are distinct members of a list $T$, then there are elements $u$, $w$ with either $u = a$ and $w = b$, or $u = b$ and $w = a$, and lists $A$, $M$, $B$ such that $T$ is the concatenation of $A$, the element $u$, $M$, the element $w$ and $B$, in this order.","title":"Splitting a list at two distinct members"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/left","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.left","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#left","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $K$ be a linearly ordered set and $S$ a family of segments in $K^2$, segment $i$ being given by its two endpoints $(a, b)$ in either order. The left endpoint of segment $i$ is $a$ if $a_1 \\le b_1$ and $b$ otherwise, where $a_1$ and $b_1$ are the first coordinates.","title":"Left endpoint of a segment"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/right","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.right","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#right","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Write $z_1$ for the abscissa of a point $z$. If segment $i$ is given by the endpoints $u$ and $v$, in this order, its right endpoint is $v$ when $u_1 \\le v_1$ and $u$ otherwise.","title":"Right endpoint of a segment"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/event","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.Event","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#event","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"For $n$ segments indexed by $0, \\dots, n-1$, an event is a pair $(i, b)$ of a segment index $i$ and a Boolean $b$; it stands for the left endpoint of segment $i$ when $b$ is true and for its right endpoint when $b$ is false.","title":"Sweep event"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/point","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.point","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#point","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"The point of the sweep event $(i, b)$ is the left endpoint of segment $i$ when $b$ is true (a left-endpoint event) and the right endpoint of segment $i$ when $b$ is false.","title":"Endpoint of an event"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/key","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.key","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#key","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $K$ be a linearly ordered set and $S$ a family of segments in $K^2$, each given by its two endpoints. An endpoint event $e$ of $S$ standing for the point $(x, y)$ gets the key $(x, \\varepsilon, y)$, compared lexicographically, where $\\varepsilon = 0$ if $e$ is a left endpoint and $\\varepsilon = 1$ if it is a right endpoint. So events are ordered by $x$; at equal $x$, left endpoints come first; then by $y$. This\u2026","title":"Sweep order on endpoint events"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/events","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.events","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#events","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"For $n$ segments, each given by its two endpoints, the left endpoint of a segment is its endpoint with the smaller $x$-coordinate, or its first given endpoint when both $x$-coordinates agree, and the right endpoint is the other one. The event list consists of the $2n$ events, the left and the right endpoint of each segment, sorted by the $x$-coordinate of the endpoint, with left endpoints before right endpoints at\u2026","title":"Sorted sweep events"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/below","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.below","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#below","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"For a segment $j$, write $l_j$ for its endpoint with the smaller $x$-coordinate, or for its first given endpoint when both $x$-coordinates agree, and $r_j$ for its other endpoint. For a point $q$, segment $j$ is below $q$ when $\\mathrm{DIRECTION}(l_j, r_j, q) < 0$.","title":"Segment strictly below a point"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/test","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.test","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#test","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Given $n$ segments in the plane over a linearly ordered field, each given by its two endpoints, and indices $i, j < n$, this runs CLRS `SEGMENTS-INTERSECT` on the left and right endpoints of segment $i$ and the left and right endpoints of segment $j$. It returns that procedure's Boolean answer together with its cost in `DIRECTION` and `ON-SEGMENT` evaluations.","title":"Intersection test for two input segments"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/test-opt","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.testOpt","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#test-opt","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Given a segment $i$ and possibly a neighbour $j$, return the result of `SEGMENTS-INTERSECT` on segments $i$ and $j$, with its cost, when $j$ is given, and return false at no cost when there is no neighbour.","title":"Testing a segment against an optional neighbour"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/test-pair","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.testPair","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#test-pair","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Given $n$ segments in the plane over a linearly ordered field and two optional segment indices $a$ and $b$ (in the sweep, the segments just below and just above a deleted segment, when they exist), this runs the intersection test of segments $a$ and $b$ when both indices are present, and otherwise returns `false` at no cost.","title":"Testing the neighbours of a deleted segment"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/sweep","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.sweep","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#sweep","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"CLRS `ANY-SEGMENTS-INTERSECT`, lines 3-11: the sweep processes a list of events with a status list $T$ of segments ordered from bottom to top, returns true as soon as an intersection test succeeds, and returns false once the events run out. At the left endpoint $p$ of segment $i$, split $T$ as $A$ followed by $B$, where $A$ is the longest initial part of $T$ whose segments pass strictly below $p$ by the `DIRECTION`\u2026","title":"The any-segments-intersect sweep"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/step","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.step","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#step","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"A step takes an event and a status list $T$ of segments, ordered from bottom to top, and returns a new status list, or no result when an intersection is reported. At the left endpoint $p$ of segment $i$, split $T$ as $A$ followed by $B$, where $A$ is the longest initial part of $T$ whose segments pass strictly below $p$ by the `DIRECTION` test. If `SEGMENTS-INTERSECT` reports that $i$ meets the first segment of $B$\u2026","title":"One step of the sweep"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Starting from a status list $T$, the sweep steps are applied to the events in order, each to the status list left by the previous one. The result is the final status list, or no result as soon as some step reports an intersection; on the empty list of events the result is $T$.","title":"Running the sweep over a list of events"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/key-left-lt-right","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.key_left_lt_right","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#key-left-lt-right","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $K$ be a linearly ordered set and $S$ a family of segments in $K^2$, each given by its two endpoints. For every segment $i$ of $S$, the left endpoint event of $i$ is strictly smaller than its right endpoint event in the sweep order on events (by $x$-coordinate, then left endpoints before right endpoints, then by $y$-coordinate).","title":"Left endpoint events precede right endpoint events"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/left-mem-of-right-mem","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.left_mem_of_right_mem","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#left-mem-of-right-mem","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $K$ be a linearly ordered set and $S$ a family of $n$ segments in $K^2$, each given by its two endpoints. If the sorted sweep event list of $S$ is a list $P$ followed by a list $Q$, and $P$ contains the right endpoint event of a segment $j$, then $P$ also contains the left endpoint event of $j$.","title":"A sweep prefix with a right endpoint has its left endpoint"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/take-while-append-cons-tail-drop-while","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.takeWhile_append_cons_tail_dropWhile","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#take-while-append-cons-tail-drop-while","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"If $i$ occurs in a list $T$ of indices in $\\{0, \\dots, n-1\\}$, then $T$ is the longest initial part of $T$ not containing $i$, followed by $i$, followed by the part of $T$ after the first occurrence of $i$.","title":"Splitting a list at the first occurrence of a member"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run-mem-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run_mem_iff","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run-mem-iff","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $P$ be an initial part of the sorted list of events. If running the sweep over $P$ from the empty status list reports no intersection and ends with status list $T$, then $T$ has no repeated entries, and a segment $j$ belongs to $T$ exactly when the left endpoint event of $j$ is in $P$ and the right endpoint event of $j$ is not.","title":"The sweep status holds the active segments"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/meets","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.Meets","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#meets","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $K$ be a field with a linear order and $S$ a family of segments in $K^2$, each given by its two endpoints. Segments $i$ and $j$ of $S$ meet when the closed segment between the endpoints of $i$ and the closed segment between the endpoints of $j$ have a point in common.","title":"Meeting segments"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run-is-chain","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run_isChain","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run-is-chain","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $P$ be an initial part of the sorted list of events. If running the sweep over $P$ from the empty status list reports no intersection and ends with status list $T$, then no two consecutive segments of $T$ meet.","title":"Neighbours on the sweep status are disjoint"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/exists-meets-of-run-eq-none","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.exists_meets_of_run_eq_none","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#exists-meets-of-run-eq-none","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $S$ be a family of $n$ segments in $K^2$, $K$ a linearly ordered field, and split the sorted list of their $2n$ endpoint events as $P$ followed by $Q$. If running the sweep over $P$ from the empty status list succeeds with status list $T$, and running it over $Q$ from $T$ reports an intersection, then there are two distinct indices $i \\ne j$ such that segments $i$ and $j$ meet.","title":"Soundness of the any-segments-intersect sweep"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/pairwise-lt-of-pairwise-lt","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.pairwise_lt_of_pairwise_lt","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#pairwise-lt-of-pairwise-lt","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let finitely many segments in the plane over a linearly ordered field be given, where a point $z$ has coordinates $(z_1, z_2)$ and segment $j$ has left endpoint $l_j$ and right endpoint $r_j$ with $(l_j)_1 < (r_j)_1$. Write $y_j(t)$ for the height at abscissa $t$ of the line through $l_j$ and $r_j$. Let $p_x$ be such that every point shared by two distinct segments has abscissa at least $p_x$, let $x \\le x' < p_x$,\u2026","title":"The sweep order persists before the leftmost intersection"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/pairwise-le-of-pairwise-lt","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.pairwise_le_of_pairwise_lt","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#pairwise-le-of-pairwise-lt","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let finitely many segments in the plane over a linearly ordered field be given, where a point $z$ has coordinates $(z_1, z_2)$ and segment $j$ has left endpoint $l_j$ and right endpoint $r_j$ with $(l_j)_1 < (r_j)_1$. Write $y_j(t)$ for the height at abscissa $t$ of the line through $l_j$ and $r_j$. Let $p_x$ be such that every point shared by two distinct segments has abscissa at least $p_x$, let $x < p_x$, and\u2026","title":"The sweep order holds weakly at the leftmost intersection"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run-pairwise-lt","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run_pairwise_lt","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run-pairwise-lt","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Fix a number $p_x$ in the field of coordinates. Suppose no segment is vertical (each segment's left endpoint has smaller $x$-coordinate than its right endpoint) and every point common to two distinct segments has $x$-coordinate at least $p_x$. Split the sorted list of events as $P$ followed by $Q$, where the endpoint of every event of $P$ has $x$-coordinate less than $p_x$, and suppose running the sweep over $P$\u2026","title":"Phase one: the status list is strictly sorted between the processed and remaining events"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run-pairwise-le-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run_pairwise_le_of_lt","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run-pairwise-le-of-lt","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Fix a number $p_x$ in the field of coordinates. Suppose no segment is vertical (each segment's left endpoint has smaller $x$-coordinate than its right endpoint) and every point common to two distinct segments has $x$-coordinate at least $p_x$. Split the sorted list of events as $P$ followed by $Q$, where the endpoint of every event of $P$ has $x$-coordinate less than $p_x$ and that of every event of $Q$ has\u2026","title":"The status list is weakly sorted after the events left of a vertical line"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run-pairwise-le","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run_pairwise_le","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run-pairwise-le","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Fix a number $p_x$ in the field of coordinates. Suppose no segment is vertical (each segment's left endpoint has smaller $x$-coordinate than its right endpoint) and every point common to two distinct segments has $x$-coordinate at least $p_x$. Split the sorted list of events as $P_1$, then $P_2$, then $Q$, where the endpoint of every event of $P_1$ has $x$-coordinate less than $p_x$, every event of $P_2$ is a left\u2026","title":"Phase two: inserting left endpoints on a vertical line keeps the status weakly sorted"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run-eq-none-of-lt","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run_eq_none_of_lt","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run-eq-none-of-lt","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $n$ segments in the plane over a linearly ordered field be given, where $z_1$ is the abscissa of a point $z$ and segment $i$ has left endpoint $l_i$ and right endpoint $r_i$ with $(l_i)_1 < (r_i)_1$, and suppose no point lies on three distinct segments. Let $a \\ne b$ be segments that share a point $p$ such that every point shared by two distinct segments has abscissa at least $p_1$, and suppose $(l_a)_1 < p_1$\u2026","title":"Sweep completeness when the leftmost intersection is inside both segments"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run-eq-none-of-left","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run_eq_none_of_left","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run-eq-none-of-left","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $n$ segments in the plane over a linearly ordered field be given, where $z_1$ is the abscissa of a point $z$ and segment $i$ has left endpoint $l_i$ and right endpoint $r_i$ with $(l_i)_1 < (r_i)_1$, and suppose no point lies on three distinct segments. Let $a \\ne b$ be segments such that $l_a$ lies on segment $b$ and every point shared by two distinct segments has abscissa at least $(l_a)_1$. If the left\u2026","title":"Sweep completeness when the leftmost intersection is a left endpoint"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/exists-leftmost-meet","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.exists_leftmost_meet","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#exists-leftmost-meet","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field, let $s_0, \\dots, s_{n-1}$ be closed segments in $\\Bbbk^2$, none of them vertical, and write $z_x$ for the $x$-coordinate of a point $z$. If some $s_i$ and $s_j$ with $i \\ne j$ meet, then there are indices $a \\ne b$ and a point $p \\in s_a \\cap s_b$ such that $p_x \\le z_x$ for all indices $k \\ne l$ and every point $z \\in s_k \\cap s_l$.","title":"A leftmost intersection point"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/run-eq-none-of-meets","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.run_eq_none_of_meets","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#run-eq-none-of-meets","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $n$ segments in the plane over a linearly ordered field be given, none of them vertical (the left endpoint of each has strictly smaller abscissa than its right endpoint), and assume, as CLRS does, that no point lies on three distinct segments. If two distinct segments meet, then the sweep, run on the sorted list of endpoint events from the empty status list, reports an intersection.","title":"Completeness of the segment-intersection sweep"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/event-cost","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.AnySegments.eventCost","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#event-cost","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"For $n$ segments, the charge of an event is $n + 16$ if it is the left endpoint of a segment and $8$ if it is the right endpoint of a segment.","title":"Cost charged to a sweep event"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/any-segments-intersect","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.anySegmentsIntersect","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#any-segments-intersect","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field. Given $n$ segments $s_0, \\dots, s_{n-1}$ in $\\Bbbk^2$, each given by its two endpoints, write $l_t$, the left endpoint of $s_t$, for its endpoint with the smaller $x$-coordinate, or for its first given endpoint when both $x$-coordinates agree, and $r_t$, the right endpoint, for the other one. The procedure sorts the $2n$ endpoints by $x$-coordinate, left endpoints before\u2026","title":"ANY-SEGMENTS-INTERSECT"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/any-segments-intersect-ret-eq-true-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.anySegmentsIntersect_ret_eq_true_iff","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#any-segments-intersect-ret-eq-true-iff","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $\\Bbbk$ be a linearly ordered field and let $s_0, \\dots, s_{n-1}$ be closed segments in $\\Bbbk^2$, each given by its two endpoints in either order. Suppose no segment is vertical (its two endpoints have different $x$-coordinates) and no point lies on three segments $s_i$, $s_j$, $s_k$ with $i$, $j$, $k$ pairwise distinct. Then ANY-SEGMENTS-INTERSECT returns true on these segments if and only if there are\u2026","title":"Correctness of ANY-SEGMENTS-INTERSECT"},{"id":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/any-segments-intersect-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.Geometry.anySegmentsIntersect_time_le","parent":42883,"result":true,"route":"../roadmap/ComputerScience/computational-geometry/sweep-line-segment-intersection-detection/index.html#any-segments-intersect-time-le","scope":"ComputerScience/computational-geometry/sweep-line-segment-intersection-detection","status":"fully_proved","summary":"Let $\\Bbbk$ be a field with a linear order. For every $n$ segments in $\\Bbbk^2$, each given by its two endpoints, ANY-SEGMENTS-INTERSECT costs at most $n^2 + 24n$, so it runs in time $O(n^2)$. Cost counts one unit per evaluation of DIRECTION or ON-SEGMENT.","title":"Running time of ANY-SEGMENTS-INTERSECT"},{"data_url":"scopes/ComputerScience/string-algorithms.json","id":"ComputerScience/string-algorithms","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/index.html","scope":"ComputerScience/string-algorithms","status":"fully_proved","summary":"Exact pattern matching (naive, finite automaton, Knuth-Morris-Pratt, Boyer-Moore, Rabin-Karp, Z algorithm) and edit distance with edit scripts; biological sequence alignment belongs to Biology.","title":"String algorithms"},{"data_url":"scopes/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index.json","id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","title":"Burrows\u2013Wheeler transform and FM-index"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/with-sentinel","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.withSentinel","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#with-sentinel","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The text `t` with the sentinel `\u22a5` appended: `t.map (\u2191) ++ [\u22a5]`. The sentinel is the unique occurrence of `\u22a5` and is smaller than every letter.","title":"Text with an appended sentinel"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/sorted-rotations","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.sortedRotations","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#sorted-rotations","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The starting positions `0, \u2026, |t|` of the rotations of `withSentinel t`, sorted by their rotations in lexicographic order: row `r` of the Burrows\u2013Wheeler matrix is `(withSentinel t).rotate ((sortedRotations t)[r])`.","title":"Sorted rotations of a sentinel-terminated text"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/bwt","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.bwt","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#bwt","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The Burrows\u2013Wheeler transform: the last column of the sorted rotation matrix.","title":"Burrows\u2013Wheeler transform"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/count-less","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.countLess","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#count-less","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"`C[c]`: the number of letters of `L` smaller than `c`.","title":"Count of smaller letters in the FM-index"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/rank","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.rank","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#rank","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"`rank_c(L, r)`: the number of occurrences of `c` among the first `r` letters of `L`.","title":"Rank of a letter in a prefix of the last column"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/lf","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.lf","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#lf","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The LF-mapping computed from the last column alone: `LF(r) = C[L[r]] + rank_{L[r]}(L, r)`.","title":"LF-mapping from the last column"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/invert-aux","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.invertAux","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#invert-aux","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"`k` steps of the inversion walk from row `r`: the letters `L[r], L[LF r], \u2026`.","title":"Steps of the Burrows\u2013Wheeler inversion walk"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/invert","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.invert","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#invert","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"Inversion of the Burrows\u2013Wheeler transform: walk the LF-mapping `|L| - 1` times from row `0` (the rotation that starts with the sentinel) and reverse the letters read.","title":"Inversion of the Burrows\u2013Wheeler transform"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/length-bwt","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.length_bwt","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#length-bwt","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The Burrows\u2013Wheeler transform of `t` has length `|t| + 1` (the sentinel adds one letter).","title":"Length of the Burrows\u2013Wheeler transform"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/map-coe-prefix-append-bot-cons-iff","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.map_coe_prefix_append_bot_cons_iff","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#map-coe-prefix-append-bot-cons-iff","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"A sentinel-free word `P` is a prefix of `u ++ \u22a5 :: v` (letters lifted to `WithBot \u03b1`) exactly when it is a prefix of `u`: the sentinel `\u22a5` stops every match.","title":"The sentinel stops every prefix match"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/pairwise-sorted-rotations","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.pairwise_sortedRotations","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#pairwise-sorted-rotations","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The rows of the Burrows\u2013Wheeler matrix are sorted: the rotations listed by `sortedRotations t` are in lexicographically nondecreasing order.","title":"Rows of the Burrows\u2013Wheeler matrix are sorted"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/sorted-rotations-get-d-lf","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.sortedRotations_getD_lf","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#sorted-rotations-get-d-lf","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"For every row `r`, row `lf (bwt t) r` holds the rotation of row `r` rotated right by one position, i.e. starting one position earlier in the text.","title":"The LF-mapping property"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/invert-bwt","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.invert_bwt","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#invert-bwt","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The Burrows\u2013Wheeler transform is inverted by the LF-mapping walk.","title":"Inversion"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/backward-search","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.backwardSearch","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#backward-search","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"FM-index backward search: the half-open row interval `[sp, ep)` for a pattern, computed from the last column alone, processing the pattern from its last letter to its first. The interval for the suffix and `C[c]` are each computed once per letter, so a pattern of length `m` makes `m` recursive calls.","title":"FM-index backward search"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/backward-search-bounds","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.backwardSearch_bounds","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#backward-search-bounds","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"Backward search always returns a genuine interval of rows, `sp \u2264 ep \u2264 |L|`, on any list `L` (so the difference in `backwardSearch_count` involves no truncated subtraction).","title":"Backward search returns a genuine row interval"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/backward-search-spec","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.backwardSearch_spec","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#backward-search-spec","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"After processing `P`, the interval `[sp, ep)` is exactly the set of rows whose rotation starts with `P`.","title":"Backward search is correct"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/backward-search-count","kind":"node","lean":"Cslib.Bioinformatics.Indexing.BWT.backwardSearch_count","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#backward-search-count","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"`ep - sp` is the number of positions `i \u2264 |t|` at which `P` occurs in `t`.","title":"Counting occurrences"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/rotate-lt-rotate-iff","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.rotate_lt_rotate_iff","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#rotate-lt-rotate-iff","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"For `i, j \u2264 |t|`, rotation `i` of `withSentinel t` is smaller than rotation `j` exactly when suffix `i` is smaller than suffix `j`.","title":"Rotations of `t$` compare as suffixes"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/sorted-rotations-eq-suffix-array","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.sortedRotations_eq_suffixArray","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#sorted-rotations-eq-suffix-array","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"Sorting the rotations of `withSentinel t` orders the starting positions exactly as sorting its suffixes.","title":"The Burrows\u2013Wheeler matrix is the suffix array of `t$`"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/bwt-eq-map-suffix-array","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.bwt_eq_map_suffixArray","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#bwt-eq-map-suffix-array","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"`bwt[r] = t$[SA[r] - 1]`, the letter cyclically before the suffix of row `r` (the sentinel for the row of the whole text, `SA[r] = 0`).","title":"The transform from the suffix array"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/backward-search-iff-prefix","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.backwardSearch_iff_prefix","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#backward-search-iff-prefix","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"Row `r \u2264 |t|` lies in the interval `[sp, ep)` computed by backward search on `bwt t` exactly when `P` is a prefix of the suffix of `t` at position `SA[r]`.","title":"Backward-search intervals are suffix-array intervals"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/occurrences","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.occurrences","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#occurrences","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The occurrences of `P` reported by the FM-index: the suffix-array entries `SA[sp], \u2026, SA[ep - 1]` of the backward-search interval. The BWT and the interval are bound once with `let` and reused, so one query builds the BWT and runs the backward search once, not three times.","title":"Occurrences of a pattern reported by the FM-index"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/mem-occurrences-iff","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.mem_occurrences_iff","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#mem-occurrences-iff","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"`i` is reported for `P` exactly when `i \u2264 |t|` and `P` is a prefix of `t.drop i` (the empty pattern occurs at all `|t| + 1` positions).","title":"The reported positions are exactly the occurrences"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/occurrences-nodup","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.occurrences_nodup","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#occurrences-nodup","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"Each occurrence is reported once.","title":"FM-index reports each occurrence once"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/occurrences-length","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.occurrences_length","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#occurrences-length","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The number of reported positions is the length `ep - sp` of the backward-search interval.","title":"Number of FM-index occurrences equals the interval length"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/suffix-array-get-d-lf","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.suffixArray_getD_lf","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#suffix-array-get-d-lf","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"For a row `r \u2264 |t|`, `LF r` is a row and `SA[LF r] = SA[r] - 1` cyclically (`SA[LF r] = |t|` when `SA[r] = 0`).","title":"LF in suffix-array terms"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/suffix-array-get-d-eq-lf-add-one","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.suffixArray_getD_eq_lf_add_one","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#suffix-array-get-d-eq-lf-add-one","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"`SA[r] = SA[LF r] + 1` when `SA[r] > 0`.","title":"The LF step moves one position left in the text"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/locate-aux","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.locateAux","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#locate-aux","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"`k` steps of sampled-suffix-array locate from row `r`: return the stored value of the first sampled row reached along the LF-mapping, plus the number of LF steps taken. `sample r` is the stored value `SA[r]` of row `r`, if any.","title":"Step-bounded sampled-suffix-array locate"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/locate","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.locate","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#locate","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"Sampled-suffix-array locate (Ferragina and Manzini 2000): the text position of row `r` from the last column `L` and the samples, with at most `|L|` LF steps.","title":"Sampled-suffix-array locate"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/locate-steps","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.locateSteps","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#locate-steps","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"The number of LF steps `locateAux L sample k r` takes.","title":"Number of LF steps taken by sampled locate"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/locate-eq","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.locate_eq","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#locate-eq","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"If every stored value is correct (`sample r = some v \u2192 v = SA[r]` for rows `r \u2264 |t|`) and every row of text position `0` is sampled, then `locate` returns `SA[r]` for every row `r \u2264 |t|`.","title":"Sampled-suffix-array locate is correct"},{"id":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/locate-steps-le","kind":"node","lean":"Cslib.Bioinformatics.Indexing.SuffixArrayBWT.locateSteps_le","parent":42927,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index/index.html#locate-steps-le","scope":"ComputerScience/string-algorithms/burrows-wheeler-transform-and-fm-index","status":"fully_proved","summary":"For `0 < s`, if every row whose text position is a multiple of `s` is sampled, then from row `r \u2264 |t|` locate makes at most `SA[r] mod s` LF steps (for any amount of fuel `k`). At `s = 0` the `s - 1` bound would be `0` while the proof only gives `SA[r] mod 0 = SA[r]` steps.","title":"Locate takes at most `s - 1` steps"},{"data_url":"scopes/ComputerScience/string-algorithms/edit-distance.json","id":"ComputerScience/string-algorithms/edit-distance","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","title":"Edit distance"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-step","kind":"node","lean":"Cslib.Algorithms.Lean.EditStep","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-step","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"An edit step over a type $\\alpha$ is one of four operations, for $a, b \\in \\alpha$: keep $a$ (consume the input value $a$ and reproduce it), delete $a$ (consume $a$), insert $a$ (produce $a$ without consuming input), or substitute $a$ by $b$ (consume $a$ and produce $b$).","title":"Edit step"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-script","kind":"node","lean":"Cslib.Algorithms.Lean.EditScript","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-script","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"An edit script over a type $\\alpha$ is a finite list of edit steps over $\\alpha$, applied from left to right.","title":"Edit script"},{"id":"ComputerScience/string-algorithms/edit-distance/cost","kind":"node","lean":"Cslib.Algorithms.Lean.EditStep.cost","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#cost","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"The cost of an edit step is $0$ for a keep step and $1$ for a deletion, an insertion or a substitution.","title":"Unit cost of an edit step"},{"id":"ComputerScience/string-algorithms/edit-distance/cost-2","kind":"node","lean":"Cslib.Algorithms.Lean.EditScript.cost","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#cost-2","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"The cost of an edit script is the sum of the unit costs of its steps; the empty script has cost $0$.","title":"Cost of an edit script"},{"id":"ComputerScience/string-algorithms/edit-distance/apply","kind":"node","lean":"Cslib.Algorithms.Lean.EditScript.apply?","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#apply","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"Over a type $\\alpha$ with a boolean equality test, running an edit script on an input list processes the steps from left to right and either fails or returns an output list. A keep step with value $a$ consumes the next input value $b$ and outputs $a$; a deletion of $a$ consumes $b$ and outputs nothing; a substitution of $a$ by $c$ consumes $b$ and outputs $c$. Each of these three fails if the input is exhausted or\u2026","title":"Running an edit script"},{"id":"ComputerScience/string-algorithms/edit-distance/transforms","kind":"node","lean":"Cslib.Algorithms.Lean.EditScript.Transforms","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#transforms","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"Over a type with a boolean equality test, an edit script transforms a list $x$ into a list $y$ when running the script on $x$ succeeds and outputs $y$.","title":"An edit script transforming one list into another"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-result","kind":"node","lean":"Cslib.Algorithms.Lean.EditDistanceResult","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-result","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"An edit-distance result over a type $\\alpha$ consists of a natural number, the reported distance, and an edit script over $\\alpha$.","title":"Edit-distance result record"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-cost","kind":"node","lean":"Cslib.Algorithms.Lean.EditDistanceCost","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-cost","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"An edit-distance cost record is a pair of natural numbers: the number of dynamic-programming table cells initialized or transitioned, and the number of operations emitted while reconstructing an edit script.","title":"Edit-distance cost counters"},{"id":"ComputerScience/string-algorithms/edit-distance/total","kind":"node","lean":"Cslib.Algorithms.Lean.EditDistanceCost.total","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#total","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"The total of an edit-distance cost record is the number of table cells initialized or transitioned plus the number of operations emitted during reconstruction.","title":"Total edit-distance cost"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"Given lists $s = s_1 \\cdots s_m$ and $t = t_1 \\cdots t_n$ over a type with a Boolean equality test, the algorithm fills a table $D(i, j)$ for $0 \\le i \\le m$ and $0 \\le j \\le n$ by $D(0, j) = j$, $D(i, 0) = i$ and $D(i, j) = \\min(D(i-1, j-1) + c_{ij}, D(i-1, j) + 1, D(i, j-1) + 1)$ for $i, j \\ge 1$, where $c_{ij}$ is $0$ if the test reports $s_i = t_j$ and $1$ otherwise. It then walks back from $(m, n)$ to\u2026","title":"Edit distance algorithm"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-script-applies","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_script_applies","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-script-applies","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For lists $s$ and $t$ over a type whose Boolean equality test agrees with equality, the edit script returned by the edit distance algorithm on $s$ and $t$ transforms $s$ into $t$.","title":"Correctness of the returned edit script"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-script-cost","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_script_cost","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-script-cost","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For lists $s$ and $t$ over a type with a Boolean equality test, the unit cost of the edit script returned by the edit distance algorithm on $s$ and $t$ equals the distance it returns.","title":"The returned edit script realizes the distance"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_minimal","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-minimal","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"Let $s$ and $t$ be lists over a type whose Boolean equality test agrees with equality. Every edit script that transforms $s$ into $t$ has unit cost at least the distance returned by the edit distance algorithm on $s$ and $t$.","title":"Minimality of the computed edit distance"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-nil-left","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_nil_left","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-nil-left","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For every list $t$ over a type with a Boolean equality test, the distance returned by the edit distance algorithm from the empty list to $t$ is the length of $t$.","title":"Edit distance from the empty list"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-nil-right","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_nil_right","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-nil-right","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For every list $s$ over a type with a Boolean equality test, the distance returned by the edit distance algorithm from $s$ to the empty list is the length of $s$.","title":"Edit distance to the empty list"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-self","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_self","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-self","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For every list $s$ over a type whose Boolean equality test agrees with equality, the distance returned by the edit distance algorithm from $s$ to $s$ is $0$.","title":"Edit distance of a list to itself"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-eq-zero-iff","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_eq_zero_iff","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-eq-zero-iff","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For lists $s$ and $t$ over a type whose Boolean equality test agrees with equality, the distance returned by the dynamic-programming edit distance algorithm on $s$ and $t$ is $0$ if and only if $s = t$.","title":"Edit distance vanishes only on equal lists"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-cell-transitions","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_cellTransitions","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-cell-transitions","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For lists $s$ and $t$ over a type with a Boolean equality test, write $|s|$ and $|t|$ for their lengths. The number of table-cell transitions charged by the dynamic-programming edit distance algorithm on $s$ and $t$ is exactly $(|s| + 1)(|t| + 1)$, one per pair of a prefix of $s$ and a prefix of $t$.","title":"Table cost of the edit distance algorithm"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-reconstruction-steps","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_reconstructionSteps","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-reconstruction-steps","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For lists $s$ and $t$ over a type with a Boolean equality test, the number of reconstruction steps charged by the edit distance algorithm on $s$ and $t$ equals the length of the edit script it returns.","title":"Reconstruction cost of the edit distance algorithm"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-reconstruction-steps-le","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_reconstructionSteps_le","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-reconstruction-steps-le","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For lists $s$ and $t$ over a type with a Boolean equality test, write $|s|$ and $|t|$ for their lengths. The number of reconstruction steps charged by the edit distance algorithm on $s$ and $t$ is at most $|s| + |t|$.","title":"Reconstruction cost bound for edit distance"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-time","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_time","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-time","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"For lists $s$ and $t$ over a type with a Boolean equality test, the total number of events charged by the edit distance algorithm on $s$ and $t$, that is its cell transitions plus its reconstruction steps, equals $(|s| + 1)(|t| + 1) + k$, where $k$ is the length of the edit script it returns.","title":"Running time of the edit distance algorithm"},{"id":"ComputerScience/string-algorithms/edit-distance/exists-inverse","kind":"node","lean":"Cslib.Algorithms.Lean.EditScript.exists_inverse","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#exists-inverse","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"Every script has an inverse of the same cost (swap insertions and deletions, reverse substitutions).","title":"Every edit script has an inverse of the same cost"},{"id":"ComputerScience/string-algorithms/edit-distance/exists-compose","kind":"node","lean":"Cslib.Algorithms.Lean.EditScript.exists_compose","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#exists-compose","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"Two scripts compose into one whose cost is at most the sum of their costs.","title":"Composition of edit scripts is subadditive in cost"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-comm","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_comm","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-comm","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"Symmetry of unit-cost edit distance.","title":"Symmetry"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-triangle","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_triangle","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-triangle","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"The triangle inequality for unit-cost edit distance.","title":"The triangle inequality"},{"id":"ComputerScience/string-algorithms/edit-distance/edit-distance-pos-iff","kind":"node","lean":"Cslib.Algorithms.Lean.editDistance_pos_iff","parent":42960,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/edit-distance/index.html#edit-distance-pos-iff","scope":"ComputerScience/string-algorithms/edit-distance","status":"fully_proved","summary":"Distinct lists are at positive distance.","title":"Edit distance between distinct lists is positive"},{"data_url":"scopes/ComputerScience/string-algorithms/string-matching-problem.json","id":"ComputerScience/string-algorithms/string-matching-problem","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-problem/index.html","scope":"ComputerScience/string-algorithms/string-matching-problem","status":"fully_proved","title":"String matching problem"},{"id":"ComputerScience/string-algorithms/string-matching-problem/match-at","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.MatchAt","parent":42987,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-problem/index.html#match-at","scope":"ComputerScience/string-algorithms/string-matching-problem","status":"fully_proved","summary":"For lists $p$ (the pattern) and $t$ (the text) over a type $\\alpha$ and a natural number $k$, the pattern $p$ matches the text $t$ at the zero-based offset $k$ when $k$ is at most the length of $t$ and $p$ is a prefix of the list obtained from $t$ by dropping its first $k$ entries.","title":"Pattern match at an offset"},{"id":"ComputerScience/string-algorithms/string-matching-problem/match-at-iff","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.matchAt_iff","parent":42987,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-problem/index.html#match-at-iff","scope":"ComputerScience/string-algorithms/string-matching-problem","status":"fully_proved","summary":"For lists $p$ and $t$ and a natural number $k$, the pattern $p$ matches the text $t$ at the zero-based offset $k$ if and only if $k$ is at most the length of $t$ and $p$ is a prefix of the list obtained from $t$ by dropping its first $k$ entries.","title":"Characterization of a match at an offset"},{"id":"ComputerScience/string-algorithms/string-matching-problem/exists-match-at-iff-is-infix","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.exists_matchAt_iff_isInfix","parent":42987,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-problem/index.html#exists-match-at-iff-is-infix","scope":"ComputerScience/string-algorithms/string-matching-problem","status":"fully_proved","summary":"For lists $p$ and $t$, there is an offset $k$ such that the pattern $p$ matches the text $t$ at $k$ if and only if $p$ is an infix of $t$, that is, $t$ is the concatenation of some list $u$, then $p$, then some list $v$.","title":"A pattern occurs in a text iff it matches at some offset"},{"data_url":"scopes/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm.json","id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","title":"Knuth\u2013Morris\u2013Pratt algorithm"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/kmp-search","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kmpSearch","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#kmp-search","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"For lists $p$ (the pattern) and $t$ (the text) over a type with a boolean equality, KMP search for $p$ in $t$ returns either a zero-based offset or no offset, together with a cost: the number of symbol comparisons it makes. If $p$ is empty, it returns the offset $0$ at cost $0$. Otherwise it first builds the prefix table of $p$, a list of entries each pairing a symbol with a failure value. The transition of a table\u2026","title":"Knuth\u2013Morris\u2013Pratt search"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/mk-prefix-table-size","kind":"node","lean":"Array.mkPrefixTable_size","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#mk-prefix-table-size","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"For every pattern $p$, the KMP prefix table of $p$ has exactly as many entries as $p$ has symbols, one for each symbol of $p$.","title":"Size of the KMP prefix table"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/mk-prefix-table-symbols","kind":"node","lean":"Array.mkPrefixTable_symbols","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#mk-prefix-table-symbols","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"Each entry of the Knuth\u2013Morris\u2013Pratt prefix table is a pair consisting of a symbol and a failure length. For every type with a Boolean equality test and every list $p$ over it, the list of first components of the entries of the prefix table built from $p$, read in order, equals $p$.","title":"The KMP prefix table stores the pattern"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/kmp-search-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kmpSearch_eq_some_iff","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#kmp-search-eq-some-iff","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"Let $p$ and $t$ be lists over a type with a lawful boolean equality, and write $|t|$ for the length of $t$. For every natural number $i$, KMP search for $p$ in $t$ returns the offset $i$ if and only if $p$ matches $t$ at $i$ and at no $j < i$, where $p$ matches $t$ at $i$ when $i \\le |t|$ and $p$ is a prefix of the list obtained from $t$ by dropping its first $i$ elements.","title":"KMP returns the first match"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/kmp-search-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kmpSearch_eq_none_iff","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#kmp-search-eq-none-iff","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"For lists $p$ and $t$ over a type with a lawful boolean equality, write $|t|$ for the length of $t$. KMP search for $p$ in $t$ returns no offset if and only if there is no natural number $i$ at which $p$ matches $t$, where $p$ matches $t$ at $i$ when $i \\le |t|$ and $p$ is a prefix of the list obtained from $t$ by dropping its first $i$ elements.","title":"KMP finds no match exactly when no offset matches"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/kmp-search-eq-none-iff-not-is-infix","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kmpSearch_eq_none_iff_not_isInfix","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#kmp-search-eq-none-iff-not-is-infix","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"For lists $p$ (the pattern) and $t$ (the text) over a type with a lawful boolean equality, KMP search for $p$ in $t$ returns no offset if and only if $p$ is not a contiguous sublist of $t$, that is, there are no lists $u$, $v$ with $t = u \\, p \\, v$ (concatenation).","title":"KMP finds no match exactly when the pattern does not occur"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/kmp-search-eq-drop-infix-map-length","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kmpSearch_eq_dropInfix?_map_length","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#kmp-search-eq-drop-infix-map-length","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"Let $p$ and $t$ be lists over a type with a lawful boolean equality, with $p$ nonempty. Let the first-infix split of $t$ by $p$ be the pair $(u, v)$ with $t = u \\, p \\, v$ (concatenation) and $u$ as short as possible, or nothing if $p$ does not occur in $t$. Then the value returned by KMP search for $p$ in $t$ is the length of $u$ if the split exists, and no offset otherwise.","title":"KMP search agrees with the first-infix split"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/kmp-search-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kmpSearch_time_le","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#kmp-search-time-le","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"For any pattern $p$ and text $t$, lists over a type with a boolean equality, KMP search for $p$ in $t$ makes at most $2|p| + 2|t|$ symbol comparisons in total, those made while building the prefix table included, where $|p|$ and $|t|$ are the lengths of $p$ and $t$.","title":"KMP search makes linearly many comparisons"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/mk-prefix-table-failure-spec","kind":"node","lean":"Array.mkPrefixTable_failure_spec","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#mk-prefix-table-failure-spec","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"Let $p$ be a pattern of length $n$ over a type with a lawful boolean equality, and write $p_{\\lt k}$ for its prefix of length $k$. Each entry of the KMP prefix table of $p$ pairs a symbol of $p$ with a failure value. For every $q < n$, the failure value $f$ of entry $q$ (counting from $0$) is the length of the longest proper prefix of $p_{\\lt q+1}$ that is also a suffix of $p_{\\lt q+1}$: $f \\le q$, $p_{\\lt f}$ is a\u2026","title":"The KMP failure function"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/mk-prefix-table-step-spec","kind":"node","lean":"Array.mkPrefixTable_step_spec","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#mk-prefix-table-step-spec","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"Let $p$ be a pattern of length $n$ over a type with a lawful boolean equality, and write $p_{\\lt k}$ for its prefix of length $k$. Let $c$ be a list, $x$ a symbol, and $cx$ the list $c$ followed by $x$. Suppose the state $s \\le n$ is the length of the longest prefix of $p$ that is a suffix of $c$: $p_{\\lt s}$ is a suffix of $c$, and every $k \\le n$ such that $p_{\\lt k}$ is a suffix of $c$ satisfies $k \\le s$. Then\u2026","title":"Correctness of the KMP transition"},{"id":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/kmp-search-nil-pattern","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.kmpSearch_nil_pattern","parent":42991,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/knuth-morris-pratt-algorithm/index.html#kmp-search-nil-pattern","scope":"ComputerScience/string-algorithms/knuth-morris-pratt-algorithm","status":"fully_proved","summary":"For every text $t$, KMP search for the empty pattern in $t$ returns offset $0$ at cost $0$, that is, without any symbol comparisons.","title":"KMP search on the empty pattern"},{"data_url":"scopes/ComputerScience/string-algorithms/naive-string-matching.json","id":"ComputerScience/string-algorithms/naive-string-matching","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","title":"Naive string matching"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"Let $p = p_0 \\cdots p_{m-1}$ be a pattern and $t = t_0 \\cdots t_{n-1}$ a text over an alphabet with a Boolean equality test. If $m \\le n$, the naive matcher handles the shifts $s = 0, 1, \\dots, n - m$ in turn: it compares $p_i$ with $t_{s+i}$ for $i = 0, 1, \\dots$ until a comparison fails or all $m$ symbols are compared, and it reports $s$ if no comparison failed. It returns the reported shifts in the order found,\u2026","title":"Naive string matcher"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches-ret","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches_ret","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches-ret","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"Let $p$ be a pattern of length $m$ and $t$ a text of length $n$. If $m \\le n$, the list of shifts reported by the naive matcher on $p$ and $t$ is the increasing list of all $s \\in \\{0, \\dots, n - m\\}$ such that $p$ is a prefix of the suffix of $t$ starting at the zero-based position $s$, with symbols compared by the Boolean equality of the alphabet. If $m > n$, the list is empty.","title":"Output of naive matching"},{"id":"ComputerScience/string-algorithms/naive-string-matching/mem-naive-matches-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.mem_naiveMatches_iff","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#mem-naive-matches-iff","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"Let $\\alpha$ be a type whose Boolean equality agrees with equality, and write $|t|$ for the length of a list $t$. For every pattern $p$ and text $t$ over $\\alpha$ and every natural number $s$, the naive matcher run on $p$ and $t$ reports the shift $s$ if and only if $p$ matches $t$ at $s$, that is, $s \\le |t|$ and $p$ is a prefix of the suffix of $t$ obtained by dropping its first $s$ symbols.","title":"Correctness of naive string matching"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches-pairwise","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches_pairwise","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches-pairwise","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"For every pattern $p$ and text $t$, the list of shifts reported by the naive matcher on $p$ and $t$ is strictly increasing; in particular, no shift is reported twice.","title":"Naive matching reports shifts in increasing order"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches-time-eq","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches_time_eq","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches-time-eq","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"Let $p = p_0 \\cdots p_{m-1}$ be a pattern and $t = t_0 \\cdots t_{n-1}$ a text. If $m \\le n$, the number of symbol comparisons performed by the naive matcher on $p$ and $t$ is","title":"Exact comparison count of naive matching"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches_time_le","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches-time-le","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"For a pattern of length $m$ and a text of length $n$, the naive matcher performs at most $(n - m + 1) \\cdot m$ symbol comparisons if $m \\le n$, and no comparisons if $m > n$.","title":"Upper bound on the comparisons of naive matching"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches-ret-replicate","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches_ret_replicate","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches-ret-replicate","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"Let $\\alpha$ be a type whose Boolean equality is reflexive, let $x \\in \\alpha$, and let $m \\le n$ be natural numbers. On the pattern consisting of $m$ copies of $x$ and the text consisting of $n$ copies of $x$, the naive matcher reports exactly the shifts $0, 1, \\dots, n - m$ in increasing order, so all overlapping occurrences are reported.","title":"Naive matching on a repeated symbol reports every shift"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches-time-replicate","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches_time_replicate","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches-time-replicate","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"Let $\\alpha$ be a type whose Boolean equality is reflexive, let $x \\in \\alpha$, and let $m \\le n$ be natural numbers. On the pattern consisting of $m$ copies of $x$ and the text consisting of $n$ copies of $x$, the naive matcher performs exactly $(n - m + 1) \\cdot m$ symbol comparisons.","title":"Comparison count of naive matching on a repeated symbol"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches-nil-pattern","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches_nil_pattern","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches-nil-pattern","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"For every text $t$ of length $n$, the naive matcher run on the empty pattern and $t$ reports exactly the shifts $0, 1, \\dots, n$ in increasing order and performs no symbol comparisons.","title":"Naive matching with the empty pattern"},{"id":"ComputerScience/string-algorithms/naive-string-matching/naive-matches-of-length-lt","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.naiveMatches_of_length_lt","parent":43003,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/naive-string-matching/index.html#naive-matches-of-length-lt","scope":"ComputerScience/string-algorithms/naive-string-matching","status":"fully_proved","summary":"If the pattern $p$ is longer than the text $t$, then the naive matcher run on $p$ and $t$ reports no shifts and performs no symbol comparisons.","title":"Naive matching with a pattern longer than the text"},{"data_url":"scopes/ComputerScience/string-algorithms/rabin-karp-algorithm.json","id":"ComputerScience/string-algorithms/rabin-karp-algorithm","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","title":"Rabin\u2013Karp algorithm"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"Given a base $d$, a modulus $q > 0$, a pattern $P = p_0 \\dots p_{m-1}$ and a text $T = t_0 \\dots t_{n-1}$, both lists of natural numbers, this returns the list, in increasing order, of all shifts $s$ with $s + m \\le n$ and $t_{s+k} = p_k$ for every $k < m$. For $m \\ge 1$ it computes by Horner's rule the residues modulo $q$ of $P$ and of the first length-$m$ window of $T$, read as base-$d$ numerals, and updates the\u2026","title":"Rabin\u2013Karp string matching"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-ret-eq-naive-matches","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_ret_eq_naiveMatches","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-ret-eq-naive-matches","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"For every base $d$, modulus $q > 0$ and lists of natural numbers $P$ and $T$, the list of offsets returned by Rabin\u2013Karp matching of $P$ in $T$ with base $d$ and modulus $q$ equals the list returned by naive matching of $P$ in $T$.","title":"Rabin\u2013Karp and naive matching return the same offsets"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-time-eq","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_time_eq","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-time-eq","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"Let $d$ be a base, $q > 0$ a modulus and $P$, $T$ lists of natural numbers with $0 < m \\le |T|$, where $m = |P|$, and put $r = |T| - m$. Index lists from $0$, write $T_{s..}$ for the suffix of $T$ starting at position $s$, and for a list $S$ with at least $m$ entries put $h(S) = \\bigl(\\sum_{i\\lt m} S_i\\, d^{m-1-i}\\bigr) \\bmod q$. Let $B$ be the number of offsets $s < r$ with\u2026","title":"Exact cost of Rabin\u2013Karp matching"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/mem-rabin-karp-matches-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.mem_rabinKarpMatches_iff","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#mem-rabin-karp-matches-iff","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"For every base $d$, every modulus $q > 0$, all lists of natural numbers $P$ (the pattern) and $T$ (the text) and every natural number $s$, Rabin\u2013Karp matching of $P$ in $T$ with base $d$ and modulus $q$ reports the offset $s$ if and only if $P$ matches $T$ at $s$, that is, $s \\le |T|$ and $P$ is a prefix of the suffix of $T$ that starts at position $s$ (counting from $0$).","title":"Correctness of Rabin\u2013Karp matching"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-pairwise","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_pairwise","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-pairwise","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"For every base $d$, modulus $q > 0$ and lists of natural numbers $P$ and $T$, the offsets returned by Rabin\u2013Karp matching of $P$ in $T$ form a strictly increasing list; in particular no offset is reported twice.","title":"Rabin\u2013Karp reports offsets in increasing order"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-nil-pattern","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_nil_pattern","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-nil-pattern","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"For every base $d$, modulus $q > 0$ and list of natural numbers $T$, Rabin\u2013Karp matching of the empty pattern in $T$ returns the list $[0, 1, \\dots, |T|]$ of all offsets from $0$ to $|T|$ in increasing order, at counted cost $0$.","title":"Rabin\u2013Karp matching with the empty pattern"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_time_le","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-time-le","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"Let $d$ be a base, $q > 0$ a modulus and $P$, $T$ lists of natural numbers with $0 < m \\le |T|$, where $m = |P|$, and put $r = |T| - m$. Index lists from $0$, write $T_{s..}$ for the suffix of $T$ starting at position $s$, and for a list $S$ with at least $m$ entries put $h(S) = \\bigl(\\sum_{i\\lt m} S_i\\, d^{m-1-i}\\bigr) \\bmod q$. Let $H$ be the number of hash hits, the offsets $s \\le r$ with $h(P) = h(T_{s..})$.\u2026","title":"Cost bound for Rabin\u2013Karp matching"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-of-length-lt","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_of_length_lt","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-of-length-lt","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"For every base $d$, modulus $q > 0$ and lists of natural numbers $P$ and $T$ with $|T| < |P|$, Rabin\u2013Karp matching of $P$ in $T$ returns no offsets, at counted cost exactly $1 + 2(|P| - 1) + 6|T|$.","title":"Rabin\u2013Karp matching with a pattern longer than the text"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-time-replicate","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_time_replicate","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-time-replicate","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"Let $d$ be a base, $q > 0$ a modulus, $a$ a natural number and $m$, $n$ natural numbers with $0 < m \\le n$. Put $t = \\bigl(a \\sum_{i\\lt m} d^i\\bigr) \\bmod q$, and let $\\beta = 1$ if $t < (a \\cdot (d^{m-1} \\bmod q)) \\bmod q$ and $\\beta = 0$ otherwise. Then Rabin\u2013Karp matching with base $d$ and modulus $q$ of the pattern $a^m$ in the text $a^n$ (the lists of $m$ and of $n$ copies of $a$) has counted cost exactly","title":"Exact cost of Rabin\u2013Karp matching on repeated-symbol words"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-ret-replicate","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_ret_replicate","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-ret-replicate","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"For every base $d$, modulus $q > 0$, symbol $a \\in \\mathbb{N}$ and natural numbers $m \\le n$, Rabin\u2013Karp matching of the pattern $a^m$ in the text $a^n$ (the lists of $m$ and of $n$ copies of $a$) returns $[0, 1, \\dots, n - m]$, that is, every shift in increasing order, including overlapping ones.","title":"Rabin\u2013Karp offsets on repeated-symbol words"},{"id":"ComputerScience/string-algorithms/rabin-karp-algorithm/rabin-karp-matches-matching-time-replica","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.rabinKarpMatches_matching_time_replicate","parent":43014,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/rabin-karp-algorithm/index.html#rabin-karp-matches-matching-time-replica","scope":"ComputerScience/string-algorithms/rabin-karp-algorithm","status":"fully_proved","summary":"Let $d$ be a base, $q > 0$ a modulus, $a$ a natural number and $m$, $n$ natural numbers with $0 < m \\le n$. Let $C$ be the counted cost of Rabin\u2013Karp matching with base $d$ and modulus $q$ of the pattern $a^m$ in the text $a^n$ (the lists of $m$ and of $n$ copies of $a$), minus $8m - 1$, with subtraction truncated at $0$. Then $m(n - m + 1) \\le C \\le 10\\,m(n - m + 1)$.","title":"Matching cost of Rabin\u2013Karp on repeated-symbol words"},{"data_url":"scopes/ComputerScience/string-algorithms/string-matching-with-finite-automata.json","id":"ComputerScience/string-algorithms/string-matching-with-finite-automata","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","title":"String matching with finite automata"},{"id":"ComputerScience/string-algorithms/string-matching-with-finite-automata/compute-transition","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeTransition","parent":43026,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html#compute-transition","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","summary":"Let $P$ be a pattern of length $m$ over the alphabet $\\{0, \\dots, A-1\\}$ and write $P_k$ for its prefix of length $k$. This timed computation returns a table of $(m+1)A$ entries in $\\{0, \\dots, m\\}$, stored row by row: for each state $q \\le m$ and symbol $a < A$, the entry at index $qA + a$ is found by trying $k = \\min(m, q+1)$ and then each smaller value in turn, and is the first $k$ for which $P_k$ is a suffix of\u2026","title":"Transition table of the string-matching automaton"},{"id":"ComputerScience/string-algorithms/string-matching-with-finite-automata/compute-transition-spec","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeTransition_spec","parent":43026,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html#compute-transition-spec","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","summary":"Let $P$ be a pattern of length $m$ over the alphabet $\\{0, \\dots, A-1\\}$ and write $P_k$ for its prefix of length $k$. For every state $q \\le m$ and symbol $a < A$, the entry at index $qA + a$ of the transition table of $P$ is the greatest $k \\le m$ such that $P_k$ is a suffix of the word $P_q a$.","title":"Correctness of the string-matching transition table"},{"id":"ComputerScience/string-algorithms/string-matching-with-finite-automata/compute-transition-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeTransition_time_le","parent":43026,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html#compute-transition-time-le","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","summary":"For every pattern of length $m$ over the alphabet $\\{0, \\dots, A-1\\}$, the cost of computing its transition table is at most $2A(m+1)^3$.","title":"Cost bound for the string-matching transition table"},{"id":"ComputerScience/string-algorithms/string-matching-with-finite-automata/pattern-automaton","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.patternAutomaton","parent":43026,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html#pattern-automaton","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","summary":"For a pattern $P$ of length $m$ over the alphabet $\\{0, \\dots, A-1\\}$, the string-matching automaton of $P$ is the deterministic finite automaton with states $\\{0, \\dots, m\\}$, input symbols $\\{0, \\dots, A-1\\}$, start state $0$, accepting set $\\{m\\}$, and transition from state $q$ on symbol $a$ given by the entry at index $qA + a$ of the transition table of $P$.","title":"String-matching automaton"},{"id":"ComputerScience/string-algorithms/string-matching-with-finite-automata/pattern-automaton-state","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.patternAutomaton_state","parent":43026,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html#pattern-automaton-state","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","summary":"Let $P$ be a pattern of length $m$ over the alphabet $\\{0, \\dots, A-1\\}$ and write $P_k$ for its prefix of length $k$. For every word $w$ over this alphabet, the state that the string-matching automaton of $P$ reaches from its start state after reading $w$ is the greatest $k \\le m$ such that $P_k$ is a suffix of $w$.","title":"The string-matching automaton tracks the longest matched prefix"},{"id":"ComputerScience/string-algorithms/string-matching-with-finite-automata/finite-automaton-matches","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.finiteAutomatonMatches","parent":43026,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html#finite-automaton-matches","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","summary":"Let $P$ be a pattern of length $m$ and $T$ a text over the alphabet $\\{0, \\dots, A-1\\}$. This timed computation first computes the transition table of $P$, then runs through $T$ starting in state $0$: on reading the $i$-th symbol $a$ of $T$ in state $q$ it moves to the state given by the table entry at index $qA + a$, and if that state is $m$ it reports the shift $i - m$. If $m = 0$, the shift $0$ is also reported\u2026","title":"Finite-automaton string matcher"},{"id":"ComputerScience/string-algorithms/string-matching-with-finite-automata/finite-automaton-matches-time","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.finiteAutomatonMatches_time","parent":43026,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html#finite-automaton-matches-time","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","summary":"For every pattern $P$ and text $T$ over the alphabet $\\{0, \\dots, A-1\\}$, where $T$ has length $n$, the cost of the finite-automaton matcher on $P$ and $T$ equals the cost of computing the transition table of $P$, plus $2n$, plus the number of shifts the matcher returns.","title":"Exact cost of the finite-automaton matcher"},{"id":"ComputerScience/string-algorithms/string-matching-with-finite-automata/finite-automaton-matches-ret","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.finiteAutomatonMatches_ret","parent":43026,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/string-matching-with-finite-automata/index.html#finite-automaton-matches-ret","scope":"ComputerScience/string-algorithms/string-matching-with-finite-automata","status":"fully_proved","summary":"For every pattern $P$ and text $T$ over the alphabet $\\{0, \\dots, A-1\\}$, the list of shifts returned by the finite-automaton matcher equals the list returned by the naive string matcher.","title":"Correctness of the finite-automaton matcher"},{"data_url":"scopes/ComputerScience/string-algorithms/z-algorithm.json","id":"ComputerScience/string-algorithms/z-algorithm","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","title":"Z algorithm"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Given a word $s = s_0 \\dots s_{n-1}$ over a type $\\alpha$ with a Boolean equality test, Gusfield's Z-box algorithm returns an array $Z$ of $n$ natural numbers with $Z_0 = n$. It visits $i = 1, \\dots, n - 1$ in turn, keeping a box $[\\ell, r)$, initially empty, that is the rightmost match of a prefix of $s$ found so far. If $\\ell \\le i < r$ and $Z_{i - \\ell} < r - i$, it sets $Z_i = Z_{i - \\ell}$. Otherwise it finds\u2026","title":"Z-algorithm"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z-zero","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ_zero","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z-zero","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type with a Boolean equality test. For every nonempty word of length $n$ over $\\alpha$, the Z-array $Z$ computed by the Z-algorithm has $Z_0 = n$.","title":"The first Z-value is the word length"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z-prefix-iff","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ_prefix_iff","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z-prefix-iff","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type with a lawful Boolean equality test, let $s = s_0 \\dots s_{n-1}$ be a word over $\\alpha$, and let $Z$ be the Z-array computed by the Z-algorithm on $s$. For every position $i < n$ and every natural number $q$, $q \\le Z_i$ holds if and only if $q \\le n - i$ and $s_k = s_{i+k}$ for all $k < q$. Hence $Z_i$ is the length of the longest common prefix of $s$ and its suffix $s_i \\dots s_{n-1}$.","title":"Z-values are longest common prefix lengths"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z-get-elem-le","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ_getElem_le","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z-get-elem-le","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type with a Boolean equality test, not necessarily lawful, and let $s$ be a word of length $n$ over $\\alpha$. For every position $i < n$, the entry at $i$ of the Z-array computed by the Z-algorithm on $s$ is at most $n - i$.","title":"Z-values are at most the suffix length"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ_time_le","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z-time-le","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type with a Boolean equality test, not necessarily lawful. On every word of length $n$ over $\\alpha$, the Z-algorithm has cost at most $3(n - 1)$, read as $0$ when $n = 0$, where the cost counts one unit for each visited position $i \\ge 1$ and one unit for each symbol comparison.","title":"Linear upper bound for the Z-algorithm"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z-time-ge","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ_time_ge","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z-time-ge","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type with a Boolean equality test, not necessarily lawful. On every word of length $n$ over $\\alpha$, the Z-algorithm has cost at least $n - 1$, where the cost counts one unit for each visited position $i \\ge 1$ and one unit for each symbol comparison.","title":"Linear lower bound for the Z-algorithm"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z-replicate","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ_replicate","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z-replicate","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type with a lawful Boolean equality test, let $a \\in \\alpha$ and $n \\in \\mathbb{N}$. The Z-array computed by the Z-algorithm on the word $a^n$ has entry $n - i$ at every position $i < n$.","title":"Z-array of a repeated symbol"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z-replicate-time","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ_replicate_time","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z-replicate-time","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type with a lawful Boolean equality test, let $a \\in \\alpha$ and $n \\in \\mathbb{N}$. The Z-algorithm run on the word $a^n$ has cost exactly $2(n - 1)$, read as $0$ when $n = 0$, where the cost counts one unit for each visited position $i \\ge 1$ and one unit for each symbol comparison. Thus it makes $n - 1$ visits and $n - 1$ comparisons.","title":"Z-algorithm cost on a repeated symbol"},{"id":"ComputerScience/string-algorithms/z-algorithm/compute-z-replicate-push-time","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.computeZ_replicate_push_time","parent":43035,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/z-algorithm/index.html#compute-z-replicate-push-time","scope":"ComputerScience/string-algorithms/z-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a type with a lawful Boolean equality test and let $a \\ne b$ be elements of $\\alpha$. For every $n \\ge 1$, the Z-algorithm run on the word $a^n b$ has cost exactly $3n - 1$, where the cost counts one unit for each visited position $i \\ge 1$ and one unit for each symbol comparison. In terms of the input length $N = n + 1$, this is $3N - 4$.","title":"Z-algorithm cost on a repeated symbol followed by a distinct one"},{"data_url":"scopes/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm.json","id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","kind":"scope","navigation":"module","parent":42926,"result":false,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","title":"Boyer\u2013Moore string-search algorithm"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/preprocess","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.BoyerMoore.preprocess","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#preprocess","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Given a pattern $p = p_0 \\cdots p_{m-1}$ over the alphabet $\\{0, \\dots, \\sigma - 1\\}$, this computes a bad-character table $B$ of length $\\sigma$ and a strong good-suffix table $G$ of length $m$. The table $B$ starts with every entry equal to $m$, and then for $k = 0, 1, \\dots, m - 2$ in turn the entry $B_{p_k}$ is set to $m - 1 - k$. For $G$, let $s_i$ be the length of the longest common suffix of $p_0 \\cdots p_i$\u2026","title":"Boyer\u2013Moore preprocessing"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/preprocess-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.BoyerMoore.preprocess_time_le","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#preprocess-time-le","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"For a pattern of length $m$ over an alphabet of size $\\sigma$, the event cost of the Boyer\u2013Moore preprocessing is at most $\\sigma$ when $m = 0$ and at most $\\sigma + 10m - 6$ when $m \\ge 1$.","title":"Cost of Boyer\u2013Moore preprocessing"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/preprocess-bad-character","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.BoyerMoore.preprocess_badCharacter","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#preprocess-bad-character","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Let $p = p_0 \\cdots p_{m-1}$ be a pattern over the alphabet $\\{0, \\dots, \\sigma - 1\\}$ and let $B$ be the bad-character table computed by the Boyer\u2013Moore preprocessing of $p$. For every symbol $c$, either $B_c = m$ and $p_k \\ne c$ for every $k$ with $k + 1 < m$, or there is $j$ with $j + 1 < m$ and $p_j = c$ such that $B_c = m - 1 - j$ and every $k$ with $k + 1 < m$ and $p_k = c$ satisfies $k \\le j$.","title":"Bad-character table of Boyer\u2013Moore preprocessing"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/preprocess-good-suffix","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.BoyerMoore.preprocess_goodSuffix","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#preprocess-good-suffix","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Let $p = p_0 \\cdots p_{m-1}$ be a pattern over the alphabet $\\{0, \\dots, \\sigma - 1\\}$, let $G$ be the good-suffix table computed by the Boyer\u2013Moore preprocessing of $p$, and let $i < m$. Call a positive integer $e$ an admissible shift at $i$ if $p_{t - e} = p_t$ for every $t < m$ with $i < t$ and $e \\le t$, and $p_{i - e} \\ne p_i$ whenever $e \\le i$. Then $1 \\le G_i \\le m$ and $G_i$ is the least admissible shift\u2026","title":"Good-suffix table of Boyer\u2013Moore preprocessing"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/preprocess-full-match","kind":"node","lean":"Cslib.Algorithms.Lean.StringMatching.BoyerMoore.preprocess_fullMatch","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#preprocess-full-match","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Let $p = p_0 \\cdots p_{m-1}$ be a nonempty pattern over the alphabet $\\{0, \\dots, \\sigma - 1\\}$ and let $G$ be the good-suffix table computed by the Boyer\u2013Moore preprocessing of $p$. Then $1 \\le G_0 \\le m$, $p_{t - G_0} = p_t$ for every $t < m$ with $G_0 \\le t$, and $G_0 \\le e$ for every positive integer $e$ such that $p_{t - e} = p_t$ for every $t < m$ with $e \\le t$. That is, $G_0$ is the least positive period of\u2026","title":"The full-match shift is the least period of the pattern"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/boyer-moore-matches","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.boyerMooreMatches","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#boyer-moore-matches","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Let $p = p_0 \\cdots p_{m-1}$ and $t = t_0 \\cdots t_{n-1}$ be words over the alphabet $\\{0, \\dots, \\sigma - 1\\}$. The matcher returns a list of offsets together with a cost. If $m = 0$ it returns $0, 1, \\dots, n$ at cost $n + 1$, and if $n < m$ it returns the empty list at cost $0$. Otherwise it charges $m + n$ for converting $p$ and $t$ to arrays, computes the bad-character table $B$ and the strong good-suffix\u2026","title":"Boyer\u2013Moore matcher for all occurrences"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/boyer-moore-matches-nil-pattern","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.boyerMooreMatches_nil_pattern","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#boyer-moore-matches-nil-pattern","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"For every text $t$ of length $n$ over the alphabet $\\{0, \\dots, \\sigma - 1\\}$, the Boyer\u2013Moore matcher with the empty pattern and text $t$ returns the list $0, 1, \\dots, n$ of all offsets from $0$ to $n$ in increasing order, at cost $n + 1$.","title":"Boyer\u2013Moore with the empty pattern"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/boyer-moore-matches-of-length-lt","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.boyerMooreMatches_of_length_lt","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#boyer-moore-matches-of-length-lt","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Let $p$ and $t$ be words over the alphabet $\\{0, \\dots, \\sigma - 1\\}$. If $t$ is strictly shorter than $p$, then the Boyer\u2013Moore matcher with pattern $p$ and text $t$ returns the empty list at cost $0$: no preprocessing or search events are charged.","title":"Boyer\u2013Moore with a pattern longer than the text"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/mem-boyer-moore-matches-iff","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.mem_boyerMooreMatches_iff","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#mem-boyer-moore-matches-iff","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"For all words $p$ and $t$ over the alphabet $\\{0, \\dots, \\sigma - 1\\}$ and every natural number $k$, the offset $k$ belongs to the list returned by the Boyer\u2013Moore matcher with pattern $p$ and text $t$ if and only if $p$ occurs in $t$ at offset $k$, that is, $k \\le |t|$ and $p$ is a prefix of the word obtained from $t$ by deleting its first $k$ letters. In particular, overlapping occurrences are all returned.","title":"Boyer\u2013Moore returns exactly the occurrences"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/boyer-moore-matches-pairwise","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.boyerMooreMatches_pairwise","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#boyer-moore-matches-pairwise","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"For all words $p$ and $t$ over the alphabet $\\{0, \\dots, \\sigma - 1\\}$, the list of offsets returned by the Boyer\u2013Moore matcher with pattern $p$ and text $t$ is strictly increasing: each entry is smaller than every entry after it.","title":"Boyer\u2013Moore returns offsets in strictly increasing order"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/boyer-moore-matches-eq-naive-matches","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.boyerMooreMatches_eq_naiveMatches","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#boyer-moore-matches-eq-naive-matches","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"For all words $p$ and $t$ over the alphabet $\\{0, \\dots, \\sigma - 1\\}$, the list of offsets returned by the Boyer\u2013Moore matcher with pattern $p$ and text $t$ equals the list returned by the naive matcher with pattern $p$ and text $t$, entry by entry and in the same order. Only the returned lists are compared, not the costs.","title":"Boyer\u2013Moore agrees with the naive matcher"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/boyer-moore-matches-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.boyerMooreMatches_time_le","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#boyer-moore-matches-time-le","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Let $p$ and $t$ be words over the alphabet $\\{0, \\dots, \\sigma - 1\\}$ of lengths $m$ and $n$. The cost of the Boyer\u2013Moore matcher with pattern $p$ and text $t$ is at most $n + 1$ if $m = 0$, at most $0$ if $n < m$, and at most $\\sigma + n + 11m - 6 + (m + 4)(n - m + 1)$ if $1 \\le m \\le n$.","title":"Worst-case cost of Boyer\u2013Moore matching"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/boyer-moore-matches-time-replicate","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.boyerMooreMatches_time_replicate","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#boyer-moore-matches-time-replicate","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Let $x$ be a letter of the alphabet $\\{0, \\dots, \\sigma - 1\\}$, let $m$ and $n$ be natural numbers with $0 < m \\le n$, and write $x^k$ for the word of $k$ copies of $x$. The cost of the Boyer\u2013Moore matcher with pattern $x^m$ and text $x^n$ equals $m + n + P + (m + 4)(n - m + 1)$, where $P$ is the cost of the Boyer\u2013Moore preprocessing of the pattern $x^m$.","title":"Exact Boyer\u2013Moore cost when pattern and text repeat the same letter"},{"id":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/boyer-moore-matches-time-replicate-appen","kind":"node","lean":"Cslib.Algorithms.Lean.TimeM.boyerMooreMatches_time_replicate_append","parent":43045,"result":true,"route":"../roadmap/ComputerScience/string-algorithms/boyer-moore-string-search-algorithm/index.html#boyer-moore-matches-time-replicate-appen","scope":"ComputerScience/string-algorithms/boyer-moore-string-search-algorithm","status":"fully_proved","summary":"Exact Boyer\u2013Moore cost when the pattern repeats one letter and ends in another, which alone makes up the text. Let $a \\ne b$ be letters of the alphabet $\\{0, \\dots, \\sigma - 1\\}$ and let $m$ and $n$ be natural numbers with $2 \\le m \\le n$. Write $a^{m-1}b$ for the word of $m - 1$ copies of $a$ followed by one $b$, and $b^n$ for the word of $n$ copies of $b$. The cost of the Boyer\u2013Moore matcher with pattern\u2026","title":"Exact Boyer\u2013Moore cost when the pattern repeats one letter and ends in another, which alone makes up the text"},{"data_url":"scopes/ComputerScience/heuristic-search-and-planning.json","id":"ComputerScience/heuristic-search-and-planning","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/index.html","scope":"ComputerScience/heuristic-search-and-planning","status":"fully_proved","summary":"Search and decision procedures from artificial intelligence: A* search, game-tree search with alpha-beta pruning, arc consistency for constraint satisfaction, and value and policy iteration for Markov decision processes.","title":"Heuristic search and planning"},{"data_url":"scopes/ComputerScience/heuristic-search-and-planning/a-search-algorithm.json","id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","kind":"scope","navigation":"module","parent":43060,"result":false,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","title":"A* search algorithm"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/consistent","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.Consistent","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#consistent","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ and let $w$ assign a natural number $w(u, v)$ to each pair of vertices. A heuristic $h$ assigning a natural number to each vertex is consistent (monotone) if $h(u) \\le w(u, v) + h(v)$ for every edge $u \\to v$ of $G$.","title":"Consistent heuristic"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/admissible","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.Admissible","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#admissible","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"For a directed graph $G$ with natural edge weights $w$, a heuristic $h$ with natural values is admissible for a goal vertex $t$ when it never overestimates the weight of a path to $t$: $h(v) \\le d$ for every vertex $v$ and every weight $d$ of a path in $G$ from $v$ to $t$.","title":"Admissible heuristic"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/reduced-weight","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.reducedWeight","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#reduced-weight","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"Let $w$ assign a natural number $w(u, v)$ to each pair of vertices and let $h$ assign a natural number to each vertex. The reduced weight of the pair $(u, v)$ is the natural number $w(u, v) + h(v) - h(u)$, computed with truncated subtraction, so it is $0$ when $h(u) > w(u, v) + h(v)$.","title":"Reduced weight"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/astar-result","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.AStarResult","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#astar-result","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"An A* result on $n$ vertices consists of a Dijkstra result on $n$ vertices (the underlying search, with its source, final keys and predecessor pointers) and a vector that assigns to each vertex $v$ a value $g(v) \\in \\mathbb{N} \\cup \\{\\infty\\}$. In the output of A* from a source $s$ with a consistent heuristic $h$, the underlying search runs on the reduced weights, its keys are $f(v) - h(s)$ where $f = g + h$, and\u2026","title":"Output of A-star search"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/path-to","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.AStarResult.pathTo?","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#path-to","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"Given an A* result $r$ on $n$ vertices and a vertex $v$, follow the predecessor pointers of the underlying Dijkstra search backward from $v$, at most $n$ times, until the source is reached, and return the visited vertices in order from the source to $v$. There is no result if a vertex without a predecessor is met before the source, or if the source is not reached within $n$ steps.","title":"Path recovered from an A-star result"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/a-star","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.aStar","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#a-star","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"A* search. Given a directed graph $G$ on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, a heuristic $h$ with natural values and a source vertex $s$, A* runs Dijkstra's algorithm from $s$ on the reduced weights $w(u, v) + h(v) - h(u)$ at Dijkstra's cost, and returns Dijkstra's result (its distances $g'$ and predecessor pointers) together with the path weights $g(v) = g'(v) + h(s) - h(v)$, where\u2026","title":"A* search"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/path-weight-reduced-weight","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.pathWeight_reducedWeight","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#path-weight-reduced-weight","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, let $h$ be a consistent heuristic with natural values, and let $w_h(x, y) = w(x, y) + h(y) - h(x)$ be the reduced weight, with the subtraction truncated at $0$. For all vertices $u$ and $v$ and every nonempty list of vertices $v_0, v_1, \\dots, v_k$ with $v_0 = u$, $v_k = v$ and an edge of $G$ from each $v_i$ to $v_{i+1}$,\u2026","title":"Reduced weights telescope along paths"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/admissible-of-consistent","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.admissible_of_consistent","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#admissible-of-consistent","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"A consistent heuristic that vanishes at the goal is admissible.","title":"A consistent heuristic that vanishes at the goal is admissible"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/a-star-key","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.aStar_key","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#a-star-key","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"The final keys of A*. Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, let $h$ be a consistent heuristic with natural values and let $s$ be a source vertex. If the path weight $g(v)$ that A* from $s$ returns for a vertex $v$ is finite, then the distance $g'(v)$ that Dijkstra's algorithm from $s$ computes for $v$ on the reduced weights $w(x, y) + h(y) - h(x)$\u2026","title":"The final keys of A*"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/a-star-distance-realized","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.aStar_distance_realized","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#a-star-distance-realized","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"A*'s finite path weights are realized by paths. Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, let $h$ be a consistent heuristic with natural values and let $s$ be a source vertex. If the path weight $g(v)$ that A* from $s$ returns for a vertex $v$ is finite, then some path in $G$ from $s$ to $v$ has weight exactly $g(v)$.","title":"A*'s finite path weights are realized by paths"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/a-star-distance-minimal","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.aStar_distance_minimal","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#a-star-distance-minimal","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"A* with a consistent heuristic is optimal. Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, let $h$ be a consistent heuristic with natural values and let $s$ be a source vertex. If the path weight $g(v)$ that A* from $s$ returns for a vertex $v$ is finite, then $g(v) \\le d$ for every weight $d$ of a path in $G$ from $s$ to $v$.","title":"A* with a consistent heuristic is optimal"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/a-star-distance-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.aStar_distance_eq_none_iff","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#a-star-distance-eq-none-iff","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"A* leaves exactly the unreachable vertices at infinity. Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, let $h$ be any heuristic with natural values and let $s$ be a source vertex. The path weight $g(v)$ that A* from $s$ returns for a vertex $v$ is infinite if and only if $v$ is not reachable from $s$ in $G$.","title":"A* leaves exactly the unreachable vertices at infinity"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/a-star-path-to-spec","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.aStar_pathTo?_spec","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#a-star-path-to-spec","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"A*'s predecessor paths are simple and realize its path weights. Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, let $h$ be a consistent heuristic with natural values and let $s$ be a source vertex. If following the predecessor pointers that A* from $s$ returns, back from a vertex $v$, yields the vertex list $p$ (listed from $s$ to $v$), then the path weight $g(v)$\u2026","title":"A*'s predecessor paths are simple and realize its path weights"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/a-star-predecessor-spec","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.aStar_predecessor_spec","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#a-star-predecessor-spec","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"A*'s predecessor edges are tight. Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, let $h$ be a consistent heuristic with natural values and let $s$ be a source vertex. If A* from $s$ records $u$ as the predecessor of a vertex $v$, then $G$ has an edge from $u$ to $v$, the path weights $g(u)$ and $g(v)$ that A* returns are both finite, and $g(v) = g(u) + w(u, v)$.","title":"A*'s predecessor edges are tight"},{"id":"ComputerScience/heuristic-search-and-planning/a-search-algorithm/a-star-time","kind":"node","lean":"Cslib.Algorithms.Lean.AStar.aStar_time","parent":43061,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/a-search-algorithm/index.html#a-star-time","scope":"ComputerScience/heuristic-search-and-planning/a-search-algorithm","status":"fully_proved","summary":"Cost of A*. Let $G$ be a directed graph on the vertices $\\{0, \\dots, n-1\\}$ with natural edge weights $w$, let $h$ be any heuristic with natural values and let $s$ be a source vertex. In Dijkstra's dense cost model, which counts the vertex entries inspected by minimum selection and the adjacency entries inspected by relaxation, A* from $s$ charges exactly $n^2 + m$ events in total, where $m$ is the number of\u2026","title":"Cost of A*"},{"data_url":"scopes/ComputerScience/heuristic-search-and-planning/ac-3-algorithm.json","id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","kind":"scope","navigation":"module","parent":43060,"result":false,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","title":"AC-3 algorithm"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/binary-csp","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.BinaryCSP","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#binary-csp","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"A binary constraint satisfaction problem with variables in $X$ and values in $D$ consists of a finite list of directed arcs $(x, y)$ with $x, y \\in X$, and, for all variables $x, y$ and values $a, b \\in D$, a Boolean saying whether the arc $(x, y)$ allows $x = a$ together with $y = b$.","title":"Binary constraint satisfaction problem"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/is-solution","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.IsSolution","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#is-solution","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary constraint satisfaction problem with variables $X$ and values $D$, and give each variable $x$ a finite domain $\\mathrm{dom}(x) \\subseteq D$. An assignment $\\sigma : X \\to D$ is a solution of $P$ within $\\mathrm{dom}$ when $\\sigma(x) \\in \\mathrm{dom}(x)$ for every variable $x$ and, for every arc $(x, y)$ of $P$, the arc $(x, y)$ allows the pair $(\\sigma(x), \\sigma(y))$.","title":"Solution of a binary CSP within given domains"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/arc-consistent","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.ArcConsistent","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#arc-consistent","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Given a binary CSP $P$ and domains $\\mathrm{dom}$, an arc $(x, y)$ is consistent when for every value $a \\in \\mathrm{dom}(x)$ there is a value $b \\in \\mathrm{dom}(y)$ such that the arc $(x, y)$ allows $x = a$ together with $y = b$.","title":"Arc consistency"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/symmetric","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.Symmetric","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#symmetric","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"A binary constraint satisfaction problem is symmetric when, for all variables $x, y$ and values $a, b$, the constraint on the ordered pair $(x, y)$ allows $x = a$ together with $y = b$ exactly when the constraint on $(y, x)$ allows $y = b$ together with $x = a$.","title":"Symmetric binary CSP"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/revise","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.revise","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#revise","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary constraint satisfaction problem with values $D$, $\\mathrm{dom}$ finite domains for its variables, and $x$, $y$ variables. The revision of the arc $(x, y)$ is the set of values $a \\in \\mathrm{dom}(x)$ that have a support in the domain of $y$: some $b \\in \\mathrm{dom}(y)$ such that the arc $(x, y)$ allows the pair $(a, b)$.","title":"Revision of an arc"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/indeg","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.indeg","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#indeg","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"For a binary constraint satisfaction problem $P$ and a variable $x$, the in-degree of $x$ is the number of arcs $(z, x)$ of $P$ ending at $x$, counted with multiplicity in the list of arcs of $P$.","title":"In-degree of a variable"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/requeue","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.requeue","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#requeue","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"For a binary constraint satisfaction problem $P$ and variables $x$ and $y$, the arcs requeued after revising $(x, y)$ are the arcs $(z, x)$ of $P$ with $z \\neq y$, in their order in the list of arcs of $P$.","title":"Arcs requeued after a revision"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/potential","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.potential","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#potential","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary constraint satisfaction problem with finitely many variables, $\\mathrm{dom}$ finite domains for its variables and $q$ a queue of arcs. The potential of $\\mathrm{dom}$ and $q$ is","title":"Termination potential of AC-3"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/potential-lt","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.potential_lt","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#potential-lt","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary constraint satisfaction problem with finitely many variables, $\\mathrm{dom}$ finite domains for its variables, $x$ and $y$ variables and $q$ a queue of arcs. Suppose the revision $R$ of the arc $(x, y)$ with respect to $\\mathrm{dom}$ differs from $\\mathrm{dom}(x)$, and let $\\mathrm{dom}'$ agree with $\\mathrm{dom}$ except that $\\mathrm{dom}'(x) = R$. Then the potential of $\\mathrm{dom}'$ with the\u2026","title":"A shrinking revision lowers the potential"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/ac3","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.ac3","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#ac3","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Given a binary CSP $P$ with finitely many variables, domains $\\mathrm{dom}$ and a queue $q$ of pairs of variables, AC-3 returns $\\mathrm{dom}$ when $q$ is empty. Otherwise it removes the first pair $(x, y)$ of $q$ and revises it: the revised domain of $x$ consists of the values $a \\in \\mathrm{dom}(x)$ for which some $b \\in \\mathrm{dom}(y)$ has the arc $(x, y)$ allowing $x = a$ together with $y = b$. If the revised\u2026","title":"The AC-3 algorithm"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/ac3-all","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.ac3All","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#ac3-all","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"For a binary CSP $P$ with finitely many variables and domains $\\mathrm{dom}$, AC-3 on the whole problem is AC-3 run from $\\mathrm{dom}$ with the queue initialized to the arc list of $P$; it returns the final domains together with the number of revisions performed.","title":"AC-3 on the whole problem"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/ac3-subset","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.ac3_subset","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#ac3-subset","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"For a binary CSP $P$ with finitely many variables, domains $\\mathrm{dom}$, a queue $q$ of pairs of variables and a variable $v$, the domain of $v$ returned by AC-3 run from $\\mathrm{dom}$ with queue $q$ is a subset of $\\mathrm{dom}(v)$.","title":"AC-3 only shrinks domains"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/mem-revise-of-is-solution","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.mem_revise_of_isSolution","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#mem-revise-of-is-solution","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary constraint satisfaction problem, $\\mathrm{dom}$ finite domains for its variables and $\\sigma$ a solution of $P$ within $\\mathrm{dom}$. For every arc $(x, y)$ of $P$, the value $\\sigma(x)$ lies in the revision of the arc $(x, y)$ with respect to $\\mathrm{dom}$.","title":"Revision keeps solution values"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/is-solution-ac3","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.isSolution_ac3","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#is-solution-ac3","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary constraint satisfaction problem with finitely many variables, $\\mathrm{dom}$ finite domains for its variables, and $q$ a queue of arcs each of which is an arc of $P$. Then every solution $\\sigma$ of $P$ within $\\mathrm{dom}$ is also a solution of $P$ within the domains returned by AC-3 started from $\\mathrm{dom}$ with queue $q$.","title":"No solution is lost"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/mem-ac3-of-is-solution","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.mem_ac3_of_isSolution","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#mem-ac3-of-is-solution","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary constraint satisfaction problem with finitely many variables and $\\mathrm{dom}$ finite domains for its variables. If $\\sigma$ is a solution of $P$ within $\\mathrm{dom}$, then for every variable $v$ the value $\\sigma(v)$ lies in the domain of $v$ returned by AC-3 run from $\\mathrm{dom}$ with all arcs of $P$ queued.","title":"Solutions survive AC-3"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/not-is-solution-of-ac3-eq-empty","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.not_isSolution_of_ac3_eq_empty","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#not-is-solution-of-ac3-eq-empty","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary constraint satisfaction problem with finitely many variables and $\\mathrm{dom}$ finite domains for its variables. If AC-3, run from $\\mathrm{dom}$ with all arcs of $P$ queued, returns the empty domain for some variable $v$, then no assignment $\\sigma$ is a solution of $P$ within $\\mathrm{dom}$.","title":"An emptied domain rules out solutions"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/arc-consistent-step","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.arcConsistent_step","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#arc-consistent-step","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary CSP whose constraints are symmetric, that is the arc $(x, y)$ allows $(a, b)$ exactly when the arc $(y, x)$ allows $(b, a)$. Let $\\mathrm{dom}$ be domains, let $x \\ne y$ be variables, let $R$ be the set of values $a \\in \\mathrm{dom}(x)$ for which some $b \\in \\mathrm{dom}(y)$ has the arc $(x, y)$ allowing $x = a$ together with $y = b$, and let $\\mathrm{dom}'$ be $\\mathrm{dom}$ with the domain of\u2026","title":"Arc consistency after one AC-3 revision"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/arc-consistent-ac3-of-inv","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.arcConsistent_ac3_of_inv","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#arc-consistent-ac3-of-inv","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"Let $P$ be a binary CSP with finitely many variables whose constraints are symmetric, that is the arc $(x, y)$ allows $(a, b)$ exactly when the arc $(y, x)$ allows $(b, a)$, and whose arcs all join two distinct variables. Let $\\mathrm{dom}$ be domains and $q$ a queue whose entries are arcs of $P$, such that every arc of $P$ not in $q$ is consistent with respect to $\\mathrm{dom}$. Then every arc of $P$ is consistent\u2026","title":"Arc consistency invariant of AC-3"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/arc-consistent-ac3","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.arcConsistent_ac3","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#arc-consistent-ac3","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"AC-3 makes every arc consistent, for constraints given symmetrically on both orientations of an arc and arcs between distinct variables.","title":"AC-3 makes every arc consistent"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/ac3-time-le-potential","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.ac3_time_le_potential","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#ac3-time-le-potential","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"For a binary CSP $P$ with finitely many variables, domains $\\mathrm{dom}$ and a queue $q$ of pairs of variables, the number of revisions AC-3 performs from $\\mathrm{dom}$ with queue $q$ is at most the potential $\\ell + \\sum_x |\\mathrm{dom}(x)| \\cdot \\mathrm{indeg}(x)$, where $\\ell$ is the length of $q$ and $\\mathrm{indeg}(x)$ is the number of entries of the arc list of $P$ ending at $x$.","title":"AC-3 revisions are bounded by the potential"},{"id":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm/ac3-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.ArcConsistency.ac3_time_le","parent":43077,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/ac-3-algorithm/index.html#ac3-time-le","scope":"ComputerScience/heuristic-search-and-planning/ac-3-algorithm","status":"fully_proved","summary":"For a binary CSP $P$ with finitely many variables and initial domains $\\mathrm{dom}$, AC-3 run from $\\mathrm{dom}$ with every arc of $P$ queued performs at most $m + \\sum_x |\\mathrm{dom}(x)| \\cdot \\mathrm{indeg}(x)$ revisions, where $m$ is the length of the arc list of $P$ and $\\mathrm{indeg}(x)$ is the number of entries of that list ending at $x$.","title":"Bound on the number of AC-3 revisions"},{"data_url":"scopes/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning.json","id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","kind":"scope","navigation":"module","parent":43060,"result":false,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","title":"Alpha\u2013beta pruning"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/game-tree","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.GameTree","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#game-tree","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"A game tree with values in a type $\\alpha$ is either a leaf, which carries a value in $\\alpha$ (the value of a terminal position), or an inner node, which carries a finite list of game trees (the positions reachable in one move; the list may be empty). Game trees are built inductively, so every game tree is finite.","title":"Game tree"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/leaves","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.leaves","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#leaves","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"The number of leaves of a game tree is defined recursively: a leaf counts $1$, and an inner node counts the total number of leaves of its children (so an inner node without children counts $0$).","title":"Number of leaves"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/leaves-list","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.leavesList","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#leaves-list","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"The number of leaves of a finite list of game trees is the sum of the numbers of leaves of its members ($0$ for the empty list).","title":"Number of leaves of a list of game trees"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBeta","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"For a player $p$ to move and values $a$ and $b$ in a linear order, alpha-beta pruning with window $[a, b]$ returns the value of a leaf at a cost of one unit (one leaf evaluation), and on an inner node runs the alpha-beta scan of its children for $p$ with window $[a, b]$.","title":"Alpha-beta pruning"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-list","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBetaList","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-list","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Given the player $p$ to move, values $a$ and $b$ in a linear order and a list of game trees (the children of a position), the scan runs alpha-beta pruning for the other player on the children in order, each time with the current bounds $a$ and $b$. The maximizer replaces $a$ by $\\max(a, v)$, where $v$ is the child's returned value, and stops as soon as $b \\le a$, returning the new $a$; if it never stops early it\u2026","title":"Alpha-beta scan of the children"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/le-alpha-beta-list-true","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.le_alphaBetaList_true","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#le-alpha-beta-list-true","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Let values lie in a linear order. For all values $a$ and $b$ and every finite list of game trees, the value returned by the alpha-beta scan of the list with the maximizer to move and window $[a, b]$ is at least $a$.","title":"Lower bound for the maximizer's scan"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-list-false-le","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBetaList_false_le","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-list-false-le","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"For all values $a, b$ in a linear order and every list $ts$ of game trees, the value returned by the alpha-beta scan of $ts$ for the minimizer with lower bound $a$ and upper bound $b$ is at most $b$.","title":"The minimizer's alpha-beta scan stays at or below its upper bound"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBeta_time_le","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-time-le","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"For either player $p$ to move and all values $a, b$ in a linear order, alpha-beta pruning run on a game tree $t$ for $p$ with window $[a, b]$ evaluates at most as many leaves as $t$ has, each leaf evaluation costing one unit.","title":"Alpha-beta never evaluates more leaves than the tree has"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-list-time-le","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBetaList_time_le","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-list-time-le","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"For either player $p$, all values $a, b$ in a linear order and every list $ts$ of game trees, the alpha-beta scan of $ts$ for $p$ with lower bound $a$ and upper bound $b$ evaluates at most as many leaves as the trees in $ts$ have in total, each leaf evaluation costing one unit.","title":"Cost of the alpha-beta scan"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/minimax","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.minimax","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#minimax","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Let values lie in a bounded linear order. The minimax value of a game tree with player $p$ to move (the maximizer or the minimizer) is defined recursively: a leaf is worth the value it carries; an inner node is worth, for the maximizer, the maximum of the minimax values of its children with the minimizer to move, and for the minimizer, the minimum of the minimax values of its children with the maximizer to move. An\u2026","title":"Minimax value"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/minimax-list","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.minimaxList","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#minimax-list","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Let values lie in a bounded linear order. For a player $p$ and a finite list of game trees, the value for $p$ is the best of their minimax values with the other player to move: for the maximizer, the maximum of these values, or $\\bot$ if the list is empty; for the minimizer, the minimum of these values, or $\\top$ if the list is empty.","title":"Minimax value of a list of positions"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-spec","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBeta_spec","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-spec","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Alpha-beta is correct (Knuth and Moore 1975, Theorem 1): for `a \u2264 b`, the result of alpha-beta agrees with the minimax value after clamping both to the window `[a, b]`.","title":"Alpha-beta is correct"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-list-spec","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBetaList_spec","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-list-spec","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Let $p$ be the player to move, $ts$ a list of game trees with values in a bounded linear order, and $a \\le b$. Let $r$ be the value returned by the alpha-beta scan of $ts$ for $p$ with window $[a, b]$, and let $m$ be the best value for $p$ among $ts$: the maximum of the minimax values of the trees in $ts$ for the minimizer ($\\bot$ if $ts$ is empty) when $p$ is the maximizer, and the minimum of their minimax values\u2026","title":"Correctness of the alpha-beta scan"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-le-of-minimax-le","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBeta_le_of_minimax_le","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-le-of-minimax-le","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Let $t$ be a game tree with values in a bounded linear order, $p$ the player to move and $a < b$ values. If the minimax value of $t$ for $p$ is at most $a$, then alpha-beta pruning run on $t$ for $p$ with the window $[a, b]$ returns a value at most $a$.","title":"Alpha-beta fails low when the minimax value is at most the lower bound"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-eq-of-mem-window","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBeta_eq_of_mem_window","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-eq-of-mem-window","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Let $t$ be a game tree with values in a bounded linear order, $p$ the player to move and $a, b$ values. If the minimax value $v$ of $t$ for $p$ satisfies $a < v < b$, then alpha-beta pruning run on $t$ for $p$ with the window $[a, b]$ returns exactly $v$.","title":"Alpha-beta is exact inside the window"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/le-alpha-beta-of-le-minimax","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.le_alphaBeta_of_le_minimax","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#le-alpha-beta-of-le-minimax","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Let values lie in a bounded linear order, let $t$ be a game tree, let $p$ be a player, and let $a < b$. If the minimax value of $t$ with $p$ to move is at least $b$, then the value returned by alpha-beta pruning on $t$ with $p$ to move and window $[a, b]$ is at least $b$.","title":"Alpha-beta at or above the top of the window"},{"id":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/alpha-beta-bot-top","kind":"node","lean":"Cslib.Algorithms.Lean.AlphaBeta.alphaBeta_bot_top","parent":43099,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/alpha-beta-pruning/index.html#alpha-beta-bot-top","scope":"ComputerScience/heuristic-search-and-planning/alpha-beta-pruning","status":"fully_proved","summary":"Let $t$ be a game tree whose values lie in a bounded linear order with least element $\\bot$ and greatest element $\\top$. For either player $p$ to move, alpha-beta pruning run on $t$ for $p$ with the window $[\\bot, \\top]$ returns exactly the minimax value of $t$ for $p$.","title":"Alpha-beta with the full window returns the minimax value"},{"data_url":"scopes/ComputerScience/heuristic-search-and-planning/value-iteration.json","id":"ComputerScience/heuristic-search-and-planning/value-iteration","kind":"scope","navigation":"module","parent":43060,"result":false,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","title":"Value iteration"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/mdp","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.MDP","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#mdp","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"A Markov decision process with states in $S$, actions in $A$ and values in $K$ consists of a reward $r(s, a) \\in K$ for taking action $a$ in state $s$ and, for all states $s, s'$ and actions $a$, a value $p(s, a, s') \\in K$, the probability of moving from $s$ to $s'$ under action $a$.","title":"Markov decision process"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/stochastic","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.MDP.Stochastic","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#stochastic","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"A Markov decision process with finitely many states and transition probabilities $P(s,a,s')$ in a field $K$ with a linear order is stochastic when $0 \\le P(s,a,s')$ for all states $s, s'$ and actions $a$, and $\\sum_{s'} P(s,a,s') = 1$ for every state $s$ and action $a$.","title":"Stochastic Markov decision process"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/q-value","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.qValue","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#q-value","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$ and with rewards $r(s,a)$ and transition probabilities $P(s,a,s')$ in a field $K$, and let $\\gamma \\in K$. The Q-value of action $a$ in state $s$ against a value estimate $V : S \\to K$ is $r(s,a) + \\gamma \\sum_{s'} P(s,a,s') \\, V(s')$.","title":"Q-value of an action"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/bellman","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.bellman","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#bellman","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states $s$, a nonempty finite set of actions $a$, rewards $r(s, a)$ and transition probabilities $p(s, a, s')$ in a field $K$ carrying a linear order, and let $\\gamma \\in K$ be a discount factor. The Bellman optimality operator $T$ sends a function $V$ from states to $K$ to the function","title":"Bellman optimality operator"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/policy-bellman","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyBellman","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#policy-bellman","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a set of actions $A$, and rewards $r(s,a)$ and transition probabilities $P(s,a,s')$ in a field $K$, let $\\gamma \\in K$ be a discount factor, and let $\\pi : S \\to A$ be a deterministic stationary policy. The Bellman operator of $\\pi$ sends a value estimate $V : S \\to K$ to the function\u2026","title":"Bellman operator of a policy"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/value-iteration","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.valueIteration","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#value-iteration","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, a nonempty finite set of actions, and rewards and transition probabilities in a field $K$ carrying a linear order, let $\\gamma \\in K$, and let $T$ be the Bellman optimality operator of $M$ with discount $\\gamma$. For $n \\in \\mathbb{N}$ and a function $V$ from states to $K$, $n$ steps of value iteration from $V$ give the $n$-fold iterate $T^n V$, where\u2026","title":"Value iteration"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/value-iteration-vec","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.valueIterationVec","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#value-iteration-vec","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with states $\\{0, \\dots, m-1\\}$, a nonempty finite set of actions, and rewards and transition probabilities in a field $K$ carrying a linear order, let $\\gamma \\in K$, and let $T$ be the Bellman optimality operator of $M$ with discount $\\gamma$. Tabulated value iteration takes a number of steps $n \\in \\mathbb{N}$ and a vector $v \\in K^m$, read as a function on states: with $0$\u2026","title":"Tabulated value iteration"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/greedy","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.greedy","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#greedy","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, a nonempty finite set of actions, and rewards $r(s, a)$ and transition probabilities $p(s, a, s')$ in a field $K$ carrying a linear order, let $\\gamma \\in K$, and let $V$ be a function from states to $K$. The greedy policy for $V$ chooses in each state $s$ an action $a$ maximizing the action value $r(s, a) + \\gamma \\sum_{s'} p(s, a, s') V(s')$; when\u2026","title":"Greedy policy"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/value-iteration-vec-get","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.valueIterationVec_get","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#value-iteration-vec-get","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with states $\\{0, \\dots, m-1\\}$, a nonempty finite set of actions and values in a field $K$ with a linear order, let $\\gamma \\in K$, and let $T$ be the Bellman optimality operator of $M$ with discount $\\gamma$. For every $n \\in \\mathbb{N}$, every vector $v \\in K^m$ and every state $s$, entry $s$ of the vector computed by $n$ steps of tabulated value iteration from $v$ equals\u2026","title":"Tabulated value iteration is correct"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/policy-bellman-le-bellman","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyBellman_le_bellman","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#policy-bellman-le-bellman","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, and rewards $r(s,a)$ and transition probabilities $P(s,a,s')$ in a field $K$ with a linear order, let $\\gamma \\in K$, and let $\\pi : S \\to A$ be a deterministic stationary policy. For every $V : S \\to K$ and every state $s$, the Bellman operator of $\\pi$ applied to $V$ is at most the Bellman optimality\u2026","title":"A policy's Bellman update is at most the optimal one"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/q-value-greedy","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.qValue_greedy","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#q-value-greedy","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions, and rewards $r(s,a)$ and transition probabilities $P(s,a,s')$ in a field $K$ with a linear order, and let $\\gamma \\in K$. For every $V : S \\to K$ and every state $s$, the action $g$ that the greedy policy for $V$ chooses in $s$ satisfies\u2026","title":"The greedy action attains the Bellman maximum"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/abs-bellman-sub-le","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.abs_bellman_sub_le","parent":43117,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Bellman operator contracts by "],["math","\\(\\gamma\\)"]],"source":"The Bellman operator contracts by $\\gamma$"},"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#abs-bellman-sub-le","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, a nonempty finite set of actions, and rewards and stochastic transition probabilities in a linearly ordered field $K$, let $\\gamma \\ge 0$ in $K$, and let $T$ be the Bellman optimality operator of $M$ with discount $\\gamma$. If two functions $V, W$ from states to $K$ satisfy $|V(s) - W(s)| \\le d$ for every state $s$, where $d \\in K$, then\u2026","title":"The Bellman operator contracts by $\\gamma$"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/bellman-contracting","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.bellman_contracting","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#bellman-contracting","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, a nonempty finite set of actions, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Then the Bellman optimality operator $T$ of $M$ with discount $\\gamma$, acting on real-valued functions on the states with the sup metric, is a contraction with constant $\\gamma$: for all such functions $V$ and $W$,\u2026","title":"Bellman's operator is a contraction"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/optimal-value","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.optimalValue","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#optimal-value","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states $S$, finitely many actions (at least one), real rewards $r(s,a)$, and transition probabilities $P(s,a,s')$ that are nonnegative and sum to $1$ over $s'$, and let $0 \\le \\gamma < 1$. The optimal value of $M$ with discount $\\gamma$ is the unique function $V^* : S \\to \\mathbb{R}$ that is a fixed point of the Bellman optimality operator, that is,\u2026","title":"Optimal value of a discounted Markov decision process"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/policy-value","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#policy-value","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a set of actions $A$, real rewards $r(s,a)$, and transition probabilities $P(s,a,s')$ that are nonnegative and sum to $1$ over $s'$, let $0 \\le \\gamma < 1$, and let $\\pi : S \\to A$ be a deterministic stationary policy. The value of $\\pi$ is the unique function $V^\\pi : S \\to \\mathbb{R}$ that is a fixed point of the Bellman operator of $\\pi$, that\u2026","title":"Value of a policy"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/bellman-optimal-value","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.bellman_optimalValue","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#bellman-optimal-value","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, a nonempty finite set of actions, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Then the optimal value $V^*$ of $M$ with discount $\\gamma$ is a fixed point of the Bellman optimality operator $T$ of $M$ with discount $\\gamma$: $T V^* = V^*$.","title":"The optimal value solves the Bellman optimality equation"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/eq-optimal-value-of-bellman-eq","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.eq_optimalValue_of_bellman_eq","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#eq-optimal-value-of-bellman-eq","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, a nonempty finite set of actions, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. If a real-valued function $V$ on the states satisfies $TV = V$, where $T$ is the Bellman optimality operator of $M$ with discount $\\gamma$, then $V$ is the optimal value of $M$ with discount $\\gamma$.","title":"Uniqueness of the solution of the Bellman optimality equation"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/policy-bellman-policy-value","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyBellman_policyValue","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#policy-bellman-policy-value","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, real rewards $r(s,a)$, and transition probabilities $P(s,a,s')$ that are nonnegative and sum to $1$ over $s'$, let $0 \\le \\gamma < 1$, and let $\\pi$ be a deterministic stationary policy with value $V^\\pi$. Then $V^\\pi$ is a fixed point of the Bellman operator of $\\pi$: for every state $s$,\u2026","title":"A policy's value satisfies its Bellman equation"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/eq-policy-value-of-policy-bellman-eq","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.eq_policyValue_of_policyBellman_eq","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#eq-policy-value-of-policy-bellman-eq","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, real rewards and stochastic transition probabilities, let $0 \\le \\gamma < 1$, and let $\\pi$ be a deterministic stationary policy, a map from states to actions. If a real-valued function $V$ on the states satisfies $T_\\pi V = V$, where $T_\\pi$ is the Bellman operator of $\\pi$ in $M$ with discount $\\gamma$, then $V$ is the value of $\\pi$.","title":"Uniqueness of the solution of a policy's Bellman equation"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/tendsto-value-iteration","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.tendsto_valueIteration","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#tendsto-value-iteration","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states $S$, a nonempty finite set of actions, real rewards and stochastic transition probabilities (nonnegative, each row summing to $1$), and let $0 \\le \\gamma < 1$ be a discount factor. Let $T$ be the Bellman optimality operator of $M$ with discount $\\gamma$. Then for every initial value function $V : S \\to \\mathbb{R}$, the value iterates $T^n V$ converge to\u2026","title":"Value iteration converges"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/dist-value-iteration-le","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.dist_valueIteration_le","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#dist-value-iteration-le","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, a nonempty finite set of actions, real rewards and stochastic transition probabilities, let $0 \\le \\gamma < 1$, and let $T$ be the Bellman optimality operator and $V^*$ the optimal value of $M$ with discount $\\gamma$. For every real-valued function $V$ on the states and every $n \\in \\mathbb{N}$, the result $T^n V$ of $n$ steps of value iteration from\u2026","title":"A priori error bound for value iteration"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/dist-optimal-value-le","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.dist_optimalValue_le","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#dist-optimal-value-le","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, a nonempty finite set of actions, real rewards and stochastic transition probabilities, let $0 \\le \\gamma < 1$, and let $T$ be the Bellman optimality operator and $V^*$ the optimal value of $M$ with discount $\\gamma$. For every real-valued function $V$ on the states, $\\|V - V^*\\|_\\infty \\le \\|V - TV\\|_\\infty / (1 - \\gamma)$, where $\\|\\cdot\\|_\\infty$ is\u2026","title":"Stopping test"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/policy-value-le-optimal-value","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue_le_optimalValue","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#policy-value-le-optimal-value","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, finitely many actions (at least one), real rewards $r(s,a)$, and transition probabilities $P(s,a,s')$ that are nonnegative and sum to $1$ over $s'$, and let $0 \\le \\gamma < 1$. For every deterministic stationary policy $\\pi$, the value $V^\\pi$ of $\\pi$ is at most the optimal value $V^*$ in every state: $V^\\pi(s) \\le V^*(s)$ for all states $s$.","title":"No policy beats the optimal value"},{"id":"ComputerScience/heuristic-search-and-planning/value-iteration/policy-value-greedy","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue_greedy","parent":43117,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/value-iteration/index.html#policy-value-greedy","scope":"ComputerScience/heuristic-search-and-planning/value-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with finitely many states, finitely many actions (at least one), real rewards $r(s,a)$, and transition probabilities $P(s,a,s')$ that are nonnegative and sum to $1$ over $s'$, let $0 \\le \\gamma < 1$, and let $V^*$ be the optimal value. Let $\\pi$ be the greedy policy for $V^*$, which in each state $s$ chooses an action $a$ maximizing\u2026","title":"A greedy policy for the optimal value is optimal"},{"data_url":"scopes/ComputerScience/heuristic-search-and-planning/policy-iteration.json","id":"ComputerScience/heuristic-search-and-planning/policy-iteration","kind":"scope","navigation":"module","parent":43060,"result":false,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","title":"Policy iteration"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/improve","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.improve","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#improve","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, real rewards $r(s, a)$ and stochastic transition probabilities $P(s' \\mid s, a)$, and let $0 \\le \\gamma < 1$. The improvement of a deterministic policy $\\pi : S \\to A$ is a fixed choice of policy greedy for the value $V^\\pi$ of $\\pi$: in each state $s$ it takes a chosen action $a$ maximizing\u2026","title":"Policy improvement step"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/policy-iteration","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyIteration","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#policy-iteration","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Policy iteration run for $n$ steps from a deterministic policy $\\pi : S \\to A$ is the policy obtained from $\\pi$ by applying the policy improvement step $n$ times.","title":"Policy iteration"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/le-policy-value-of-le-policy-bellman","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.le_policyValue_of_le_policyBellman","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#le-policy-value-of-le-policy-bellman","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Let $\\pi$ be a deterministic policy, with Bellman operator $T^\\pi$ and value $V^\\pi$. If $V : S \\to \\mathbb{R}$ satisfies $V(s) \\le (T^\\pi V)(s)$ for every state $s$, then $V(s) \\le V^\\pi(s)$ for every state $s$.","title":"Values below their Bellman image lie below the policy value"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/policy-value-le-policy-value-improve","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue_le_policyValue_improve","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#policy-value-le-policy-value-improve","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Write $V^\\pi$ for the value of a deterministic policy $\\pi : S \\to A$. For every deterministic policy $\\pi$, the improved policy $\\pi'$, which is greedy for $V^\\pi$, satisfies $V^\\pi(s) \\le V^{\\pi'}(s)$ for every state $s$.","title":"Policy improvement theorem"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/policy-value-improve-eq-iff","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue_improve_eq_iff","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#policy-value-improve-eq-iff","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Write $V^\\pi$ for the value of a deterministic policy $\\pi : S \\to A$ and $V^*$ for the optimal value. For every deterministic policy $\\pi$, with improved policy $\\pi'$, we have $V^{\\pi'} = V^\\pi$ if and only if $V^\\pi = V^*$\u2026","title":"Improvement stalls exactly at optimal policies"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/policy-value-improve-ne","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue_improve_ne","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#policy-value-improve-ne","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Write $V^\\pi$ for the value of a deterministic policy $\\pi : S \\to A$ and $V^*$ for the optimal value. If $V^\\pi \\ne V^*$, then the improved policy $\\pi'$ satisfies $V^{\\pi'} \\ne V^\\pi$ (as functions on $S$).","title":"Away from optimality, improvement changes the value"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/policy-value-policy-iteration-mono","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue_policyIteration_mono","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#policy-value-policy-iteration-mono","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Write $V^\\pi$ for the value of a deterministic policy $\\pi : S \\to A$, and $\\pi_k$ for the policy after $k$ steps of policy iteration from $\\pi$. If $i \\le j$, then $V^{\\pi_i}(s) \\le V^{\\pi_j}(s)$ for every state $s$.","title":"Policy iteration never decreases the value"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/policy-value-policy-iteration-of-eq","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue_policyIteration_of_eq","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#policy-value-policy-iteration-of-eq","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Write $V^\\pi$ for the value of a deterministic policy $\\pi : S \\to A$, $V^*$ for the optimal value, and $\\pi_k$ for the policy after $k$ steps of policy iteration from $\\pi$. If $i \\le j$ and $V^{\\pi_i} = V^*$, then\u2026","title":"Once optimal, policy iteration stays optimal"},{"id":"ComputerScience/heuristic-search-and-planning/policy-iteration/policy-value-policy-iteration","kind":"node","lean":"Cslib.Algorithms.Lean.MarkovDecision.policyValue_policyIteration","parent":43142,"result":true,"route":"../roadmap/ComputerScience/heuristic-search-and-planning/policy-iteration/index.html#policy-value-policy-iteration","scope":"ComputerScience/heuristic-search-and-planning/policy-iteration","status":"fully_proved","summary":"Let $M$ be a Markov decision process with a finite set of states $S$, a finite nonempty set of actions $A$, real rewards and stochastic transition probabilities, and let $0 \\le \\gamma < 1$. Write $V^\\pi$ for the value of a deterministic policy $\\pi : S \\to A$, $V^*$ for the optimal value, and $N = |A|^{|S|}$ for the number of deterministic policies. For every deterministic policy $\\pi$, the policy $\\pi_N$ obtained\u2026","title":"Policy iteration terminates"},{"data_url":"scopes/ComputerScience/cryptography.json","id":"ComputerScience/cryptography","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/cryptography/index.html","scope":"ComputerScience/cryptography","status":"fully_proved","summary":"Cryptographic schemes and their correctness and security reductions: the game-based framework of security proofs, symmetric-key encryption and authentication, hash functions and Merkle trees, public-key encryption and key agreement, signatures, identification protocols and commitments.","title":"Cryptography"},{"data_url":"scopes/ComputerScience/cryptography/security-games.json","id":"ComputerScience/cryptography/security-games","kind":"scope","parent":43152,"result":false,"route":"../roadmap/ComputerScience/cryptography/security-games/index.html","scope":"ComputerScience/cryptography/security-games","status":"fully_proved","summary":"The framework of game-based security proofs: adaptive oracle algorithms and their probabilistic semantics, games and advantages, generic security experiments for encryption, MACs and signatures, asymptotic security, the hybrid argument and the difference lemma, and lazy sampling with the PRP/PRF switching lemma.","title":"Security games and proof techniques"},{"data_url":"scopes/ComputerScience/cryptography/security-games/oracle-algorithms.json","id":"ComputerScience/cryptography/security-games/oracle-algorithms","kind":"scope","navigation":"module","parent":43153,"result":false,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","title":"Oracle algorithms"},{"id":"ComputerScience/cryptography/security-games/oracle-algorithms/oracle-sig","kind":"node","lean":"Cslib.Crypto.Framework.OracleSig","parent":43154,"result":true,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html#oracle-sig","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","summary":"For types $X$ and $Y$, this is the polynomial functor whose operations are the queries $x \\in X$, each answered by an element of $Y$.","title":"The signature of one oracle"},{"id":"ComputerScience/cryptography/security-games/oracle-algorithms/oracle-comp","kind":"node","lean":"Cslib.Crypto.Framework.OracleComp","parent":43154,"result":true,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html#oracle-comp","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","summary":"For a type $X$ of queries, a type $Y$ of answers and a type $\\alpha$ of outputs, an adaptive deterministic oracle algorithm is an element of CSLib's free monad over the polynomial functor of one oracle with queries $X$ and answers $Y$. It is built inductively: it either stops with an output in $\\alpha$, or asks a query $x \\in X$ and continues with an algorithm that depends on the answer $y \\in Y$.","title":"Adaptive oracle algorithms"},{"id":"ComputerScience/cryptography/security-games/oracle-algorithms/done","kind":"node","lean":"Cslib.Crypto.Framework.OracleComp.done","parent":43154,"result":true,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html#done","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","summary":"For an output $a \\in \\alpha$, this is the adaptive oracle algorithm with queries in $X$ and answers in $Y$ that asks no query and stops with output $a$.","title":"The oracle algorithm that stops at once"},{"id":"ComputerScience/cryptography/security-games/oracle-algorithms/query","kind":"node","lean":"Cslib.Crypto.Framework.OracleComp.query","parent":43154,"result":true,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html#query","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","summary":"For a point $x \\in X$ and a continuation $k$ assigning an oracle algorithm to each answer $y \\in Y$, the oracle algorithm that first queries $x$ and then continues with $k(y)$, where $y$ is the answer.","title":"Query step"},{"id":"ComputerScience/cryptography/security-games/oracle-algorithms/run","kind":"node","lean":"Cslib.Crypto.Framework.OracleComp.run","parent":43154,"result":true,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html#run","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","summary":"Given a function $O : X \\to Y$, the run of an oracle algorithm with queries in $X$, answers in $Y$ and outputs in $\\alpha$ is its output when every query $x$ is answered by $O(x)$: an algorithm that stops with output $a$ runs to $a$, and one that queries $x$ and continues with $k(y)$ on answer $y$ runs to the run of $k(O(x))$.","title":"Running an oracle algorithm"},{"id":"ComputerScience/cryptography/security-games/oracle-algorithms/queries","kind":"node","lean":"Cslib.Crypto.Framework.OracleComp.queries","parent":43154,"result":true,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html#queries","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","summary":"For an oracle algorithm $c$ and a function $O : X \\to Y$, the queries of $c$ against $O$ are the list of the points of $X$ that $c$ asks, in order, when every query $x$ is answered by $O(x)$: the empty list if $c$ stops, and otherwise the first query $x$ followed by the queries of the continuation on the answer $O(x)$.","title":"Queries of an oracle algorithm"},{"id":"ComputerScience/cryptography/security-games/oracle-algorithms/run-congr","kind":"node","lean":"Cslib.Crypto.Framework.OracleComp.run_congr","parent":43154,"result":true,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html#run-congr","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","summary":"Let $c$ be an oracle algorithm and let $O, O' : X \\to Y$ agree at every query of $c$ against $O$. Then $c$ has the same output against $O$ as against $O'$, and the same list of queries against $O$ as against $O'$.","title":"Locality of oracle runs"},{"id":"ComputerScience/cryptography/security-games/oracle-algorithms/run-bind","kind":"node","lean":"Cslib.Crypto.Framework.OracleComp.run_bind","parent":43154,"result":true,"route":"../roadmap/ComputerScience/cryptography/security-games/oracle-algorithms/index.html#run-bind","scope":"ComputerScience/cryptography/security-games/oracle-algorithms","status":"fully_proved","summary":"Let $c$ be an oracle algorithm with outputs in $\\alpha$, let $g$ map each $a \\in \\alpha$ to an oracle algorithm with outputs in $\\beta$, and let $O : X \\to Y$. Running against $O$ the composition that first runs $c$ and then continues with $g$ applied to its output gives the output of $g(a)$ run against $O$, where $a$ is the output of $c$ run against $O$.","title":"Running a sequential composition"},{"data_url":"scopes/ComputerScience/cryptography/symmetric-key.json","id":"ComputerScience/cryptography/symmetric-key","kind":"scope","parent":43152,"result":false,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/index.html","scope":"ComputerScience/cryptography/symmetric-key","status":"fully_proved","summary":"Perfect secrecy and the one-time pad, pseudorandom generators and stream ciphers, pseudorandom functions and permutations, message authentication, and authenticated encryption by encrypt-then-MAC.","title":"Symmetric-key cryptography"},{"data_url":"scopes/ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group.json","id":"ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group","kind":"scope","navigation":"module","parent":43163,"result":false,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group/index.html","scope":"ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group","status":"fully_proved","title":"One-time pad over a finite group"},{"id":"ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group/group-otp","kind":"node","lean":"Cslib.Crypto.Protocols.PerfectSecrecy.groupOTP","parent":43164,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group/index.html#group-otp","scope":"ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group","status":"fully_proved","summary":"Let $G$ be a finite group. The group one-time pad on $G$ is the private-key encryption scheme with messages, keys and ciphertexts in $G$ whose key $k$ is drawn uniformly from $G$, which encrypts a message $m$ as $k m$, and which decrypts a ciphertext $c$ as $k^{-1} c$.","title":"The group one-time pad"},{"id":"ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group/group-otp-ciphertext-dist-eq-uniform","kind":"node","lean":"Cslib.Crypto.Protocols.PerfectSecrecy.groupOTP_ciphertextDist_eq_uniform","parent":43164,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group/index.html#group-otp-ciphertext-dist-eq-uniform","scope":"ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group","status":"fully_proved","summary":"Let $G$ be a finite group. For every message $m \\in G$, the ciphertext of $m$ under the group one-time pad on $G$, whose key is drawn uniformly from $G$, is uniformly distributed on $G$.","title":"Group one-time pad ciphertexts are uniform"},{"id":"ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group/group-otp-perfectly-secret","kind":"node","lean":"Cslib.Crypto.Protocols.PerfectSecrecy.groupOTP_perfectlySecret","parent":43164,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group/index.html#group-otp-perfectly-secret","scope":"ComputerScience/cryptography/symmetric-key/one-time-pad-over-a-finite-group","status":"fully_proved","summary":"Let $G$ be a finite group. The group one-time pad on $G$ is perfectly secret: for every probability distribution on messages in $G$ and every ciphertext $c$ that occurs with positive probability, the conditional distribution of the message given that the ciphertext is $c$ equals the prior distribution of the message.","title":"The group one-time pad is perfectly secret"},{"data_url":"scopes/ComputerScience/cryptography/symmetric-key/pseudorandom-functions.json","id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","kind":"scope","navigation":"module","parent":43163,"result":false,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","title":"Pseudorandom functions"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/prf-real","kind":"node","lean":"Cslib.Crypto.PRF.prfReal","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#prf-real","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$, $X$, $Y$ and $S$ be finite sets with $K$, $Y$ and $S$ nonempty, let $F : K \\to X \\to Y$ be a keyed function and let $D$ be a distinguisher: an oracle algorithm with coins in $S$ that queries a function $X \\to Y$ and outputs a bit. The real world is the distribution of the output of $D$ with coins $s$ and oracle $F(k, \\cdot)$, where the pair $(k, s)$ is uniform on $K \\times S$.","title":"Real world of the PRF experiment"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/prf-rand","kind":"node","lean":"Cslib.Crypto.PRF.prfRand","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#prf-rand","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $X$, $Y$ and $S$ be finite sets with $Y$ and $S$ nonempty, and let $D$ be a distinguisher: an oracle algorithm with coins in $S$ that queries a function $X \\to Y$ and outputs a bit. The random world is the distribution of the output of $D$ with coins $s$ and oracle $f$, where the pair $(f, s)$ is uniform on $(X \\to Y) \\times S$.","title":"Random world of the PRF experiment"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/prf-advantage","kind":"node","lean":"Cslib.Crypto.PRF.prfAdvantage","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#prf-advantage","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$, $X$, $Y$ and $S$ be finite sets with $K$, $Y$ and $S$ nonempty, let $F : K \\to X \\to Y$ be a keyed function and let $D$ be a distinguisher: an oracle algorithm with coins in $S$ that queries a function $X \\to Y$ and outputs a bit. The PRF advantage of $D$ against $F$ is the real number $|p_{\\mathrm{real}} - p_{\\mathrm{rand}}|$, where $p_{\\mathrm{real}}$ is the probability that $D$ outputs $1$ in the real\u2026","title":"PRF advantage"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/mac-wins","kind":"node","lean":"Cslib.Crypto.PRF.MacWins","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#mac-wins","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"A forger $c$, an adaptive oracle algorithm with queries in $M$, answers in $T$ and output in $M \\times T$, wins against the tagging oracle $O \\colon M \\to T$ when, run with oracle $O$, it outputs a pair $(m, t)$ with $O(m) = t$ and $m$ is not among the queries it asks in that run.","title":"Winning the forgery game"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/mac-game","kind":"node","lean":"Cslib.Crypto.PRF.macGame","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#mac-game","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$ and $R$ be finite nonempty types, let $F \\colon K \\to M \\to T$ be a keyed function, and let $A$ be a forger: for coins $r \\in R$, $A(r)$ is an adaptive oracle algorithm with queries in $M$, answers in $T$ and output in $M \\times T$. The existential-forgery experiment under an adaptive chosen-message attack for the MAC $\\mathrm{Mac}_k(m) = F(k, m)$ draws $(k, r)$ uniformly from $K \\times R$, runs $A(r)$ with\u2026","title":"The MAC forgery experiment"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/mac-reduction","kind":"node","lean":"Cslib.Crypto.PRF.macReduction","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#mac-reduction","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"For a list $L$ of messages in $M$ and a forger $c$, an adaptive oracle algorithm with queries in $M$, answers in $T$ and output in $M \\times T$, the MAC reduction is the oracle algorithm with the same queries and answers and a Boolean output, defined by recursion on $c$. If $c$ stops with output $(m, t)$, it queries $m$ and, on answer $t'$, stops with output true exactly when $t' = t$ and $m \\notin L$. If $c$\u2026","title":"The distinguisher built from a forger"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/mac-game-eq-prf-real","kind":"node","lean":"Cslib.Crypto.PRF.macGame_eq_prfReal","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#mac-game-eq-prf-real","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$ and $R$ be finite nonempty types, let $F \\colon K \\to M \\to T$ be a keyed function, and let $A$ be a forger: for coins $r \\in R$, $A(r)$ is an adaptive oracle algorithm with queries in $M$, answers in $T$ and output in $M \\times T$. Then the output distribution of the existential-forgery experiment for $F$ and $A$ equals the output distribution of the real world of the PRF experiment for $F$ with the\u2026","title":"The MAC reduction in the real PRF world"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/card-mac-wins-mul-le","kind":"node","lean":"Cslib.Crypto.PRF.card_macWins_mul_le","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#card-mac-wins-mul-le","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $M$ and $T$ be finite sets and let $c$ be an adaptive oracle algorithm that queries messages in $M$, receives tags in $T$ and outputs a pair in $M \\times T$. Call a function $f \\colon M \\to T$ a win for $c$ when the run of $c$ with oracle $f$ outputs a pair $(m, t)$ with $f(m) = t$ and $m$ not among the messages queried in that run. If $W$ is the set of functions $M \\to T$ that are wins for $c$, then\u2026","title":"Counting forgeries against a random function"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/prf-rand-mac-reduction-le","kind":"node","lean":"Cslib.Crypto.PRF.prfRand_macReduction_le","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#prf-rand-mac-reduction-le","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $M$ and $T$ be finite sets with $T$ nonempty, and let $A$ be a forger: an oracle algorithm with coins in a finite nonempty set that queries tags in $T$ of messages in $M$ and outputs a message together with a tag. When the tagging oracle is a uniformly random function $M \\to T$ and the coins are uniform, the MAC reduction built from $A$ accepts (that is, $A$ outputs the correct tag of a message it never\u2026","title":"Forging against a random function"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/mac-game-le","kind":"node","lean":"Cslib.Crypto.PRF.macGame_le","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#mac-game-le","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$, $M$, $T$ and $R$ be finite types with $K$, $T$ and $R$ nonempty, let $F \\colon K \\to M \\to T$ be a keyed function, and let $A$ be an adaptive chosen-message forger: for coins $r \\in R$, $A(r)$ is an adaptive oracle algorithm with queries in $M$, answers in $T$ and output in $M \\times T$. Then the probability that $A$ wins the existential-forgery experiment for the MAC $\\mathrm{Mac}_k(m) = F(k, m)$ is at\u2026","title":"The MAC from a PRF, concrete security"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/encrypt","kind":"node","lean":"Cslib.Crypto.PRF.encrypt","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#encrypt","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $Y$ be an additive commutative group of messages and let $f \\colon X \\to Y$ be a function (in the scheme, $f = F_k$ for a key $k$). The encryption of a message $m \\in Y$ with randomness $r \\in X$ is the pair $(r, f(r) + m) \\in X \\times Y$.","title":"Encryption from a keyed function"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/decrypt","kind":"node","lean":"Cslib.Crypto.PRF.decrypt","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#decrypt","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"For a function $F_k \\colon X \\to Y$ into an additive commutative group $Y$, the decryption of a ciphertext $(r, c) \\in X \\times Y$ is $\\mathrm{Dec}(r, c) = c - F_k(r)$.","title":"Decryption with a keyed function"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/decrypt-encrypt","kind":"node","lean":"Cslib.Crypto.PRF.decrypt_encrypt","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#decrypt-encrypt","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $Y$ be an additive commutative group and $F_k \\colon X \\to Y$ a function. For every message $m \\in Y$ and every randomness $r \\in X$, decrypting the ciphertext $\\mathrm{Enc}(m) = (r, F_k(r) + m)$ gives back $m$, that is, $\\mathrm{Dec}(\\mathrm{Enc}(m)) = m$ where $\\mathrm{Dec}(r, c) = c - F_k(r)$.","title":"Correctness of encryption with a keyed function"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/pick","kind":"node","lean":"Cslib.Crypto.PRF.pick","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#pick","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Given a pair of messages $(m_0, m_1) \\in Y \\times Y$ and a challenge bit $b$, the selected message is $m_1$ when $b$ is true and $m_0$ when $b$ is false.","title":"Selecting the challenge message by a bit"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/run-enc","kind":"node","lean":"Cslib.Crypto.PRF.runEnc","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#run-enc","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $X$ be a set, $Y$ an abelian group written additively, $f : X \\to Y$ a function, $q \\in \\mathbb{N}$ and $\\rho : \\{0, \\dots, q - 1\\} \\to X$ the encryption randomness of the queries. Run from query index $i \\in \\mathbb{N}$, an oracle algorithm with messages in $Y$ as queries, answers in $X \\times Y$ and outputs in $\\beta$ either fails or yields its output together with the next query index: if it stops with\u2026","title":"Running an algorithm against the bounded encryption oracle"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/cpaadversary","kind":"node","lean":"Cslib.Crypto.PRF.CPAAdversary","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#cpaadversary","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"A CPA adversary with coins in $R$, against an encryption oracle that takes messages in $Y$ and returns ciphertexts in $X \\times Y$, assigns to each $r \\in R$ an adaptive oracle algorithm with access to the encryption oracle that outputs two challenge messages $(m_0, m_1) \\in Y \\times Y$ together with a second adaptive oracle algorithm, again with access to the encryption oracle, which takes the challenge ciphertext\u2026","title":"CPA adversary"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/cpa-guess","kind":"node","lean":"Cslib.Crypto.PRF.cpaGuess","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#cpa-guess","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $Y$ be an additive commutative group, $f \\colon X \\to Y$ a function, $q$ a natural number, $\\rho \\colon \\{0, \\dots, q - 1\\} \\to X$ the encryption randomness, $r_s \\in X$ the challenge randomness and $b$ a bit. Let $c$ be an oracle algorithm for the encryption oracle that outputs two messages $(m_0, m_1) \\in Y \\times Y$ and a second-phase algorithm. The encryption query with index $i$, counting from $0$ across\u2026","title":"The adversary's guess in the CPA experiment"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/cpa-game","kind":"node","lean":"Cslib.Crypto.PRF.cpaGame","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#cpa-game","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$, $X$ and $R$ be nonempty finite sets, $Y$ an additive commutative group, $q$ a natural number, $F \\colon K \\to X \\to Y$ a keyed function and $A$ a CPA adversary with coins in $R$. The CPA indistinguishability experiment $\\mathrm{PrivK}^{\\mathrm{cpa}}$ for the scheme $\\mathrm{Enc}_k(m) = (r, F_k(r) + m)$ draws, uniformly and independently, a key $k \\in K$, encryption randomness\u2026","title":"The CPA experiment"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/cpa-advantage","kind":"node","lean":"Cslib.Crypto.PRF.cpaAdvantage","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#cpa-advantage","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$, $X$ and $R$ be nonempty finite sets, $Y$ an additive commutative group, $q$ a natural number, $F \\colon K \\to X \\to Y$ a keyed function and $A$ a CPA adversary with coins in $R$. The CPA advantage of $A$ is the real number $|p - 1/2|$, where $p$ is the probability that the CPA experiment for $F$ and $A$ with at most $q$ encryption queries outputs true, that is, that $A$ guesses the challenge bit.","title":"CPA advantage"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/sim-enc","kind":"node","lean":"Cslib.Crypto.PRF.simEnc","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#sim-enc","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $X$ be a set, $Y$ an abelian group written additively, $q \\in \\mathbb{N}$ and $\\rho : \\{0, \\dots, q - 1\\} \\to X$ the encryption randomness of the queries. Started at query index $i \\in \\mathbb{N}$, the simulation turns an oracle algorithm with messages in $Y$ as queries, answers in $X \\times Y$ and outputs in $\\beta$ into an oracle algorithm with queries in $X$ and answers in $Y$ that either fails or outputs\u2026","title":"Simulating the encryption oracle with a function oracle"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/cpa-reduction","kind":"node","lean":"Cslib.Crypto.PRF.cpaReduction","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#cpa-reduction","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $X$ and $R$ be sets, $Y$ an additive commutative group, $q$ a natural number and $A$ a CPA adversary with coins in $R$ against an encryption oracle that takes messages in $Y$ and returns ciphertexts in $X \\times Y$. With its own coins $(\\rho, r_s, b, r)$, where $\\rho \\colon \\{0, \\dots, q - 1\\} \\to X$, $r_s \\in X$, $b$ is a bit and $r \\in R$, and with oracle access to a function $O \\colon X \\to Y$, the\u2026","title":"The PRF distinguisher built from a CPA adversary"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/cpa-game-eq-prf-real","kind":"node","lean":"Cslib.Crypto.PRF.cpaGame_eq_prfReal","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#cpa-game-eq-prf-real","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$, $X$ and $R$ be nonempty finite sets, $Y$ an additive commutative group, $q$ a natural number and $F \\colon K \\to X \\to Y$ a keyed function. For every CPA adversary $A$ with coins in $R$, the CPA experiment for $F$ and $A$ with at most $q$ encryption queries has the same output distribution as the real world of the PRF experiment for $F$ run with the distinguisher built from $A$ for $q$ queries: with the\u2026","title":"The CPA reduction is exact with the real oracle"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/run-enc-congr","kind":"node","lean":"Cslib.Crypto.PRF.runEnc_congr","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#run-enc-congr","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $Y$ be an additive commutative group, let $\\rho_0, \\dots, \\rho_{q-1} \\in X$ be encryption randomness and let $f, f' : X \\to Y$ satisfy $f(\\rho_j) = f'(\\rho_j)$ for every $j < q$. Then for every oracle algorithm $c$ and every natural number $i$, running $c$ against the encryption oracle built from $f$, with query indices starting at $i$, gives the same result as running it against the one built from $f'$. Here\u2026","title":"The encryption oracle depends only on the encryption randomness"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/prf-rand-cpa-reduction","kind":"node","lean":"Cslib.Crypto.PRF.prfRand_cpaReduction","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#prf-rand-cpa-reduction","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $X$ be a finite nonempty set, $Y$ a finite additive commutative group, $q$ a natural number and $A$ a CPA adversary with coins in a finite nonempty set. Let $p$ be the probability that the CPA reduction built from $A$ (whose encryption oracle answers at most $q$ queries) outputs $1$, that is, that $A$ guesses the challenge bit, when its oracle is a uniformly random function $X \\to Y$ and its coins are uniform.\u2026","title":"The CPA reduction against a random function"},{"id":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions/cpa-advantage-le","kind":"node","lean":"Cslib.Crypto.PRF.cpaAdvantage_le","parent":43168,"result":true,"route":"../roadmap/ComputerScience/cryptography/symmetric-key/pseudorandom-functions/index.html#cpa-advantage-le","scope":"ComputerScience/cryptography/symmetric-key/pseudorandom-functions","status":"fully_proved","summary":"Let $K$, $X$ and $R$ be nonempty finite sets, $Y$ a finite additive commutative group, $q$ a natural number and $F \\colon K \\to X \\to Y$ a keyed function, used in the scheme $\\mathrm{Enc}_k(m) = (r, F_k(r) + m)$ with uniform randomness $r \\in X$. For every CPA adversary $A$ with coins in $R$, the CPA advantage of $A$ in the experiment whose encryption oracle answers at most $q$ queries is at most the PRF advantage\u2026","title":"CPA encryption from a PRF, concrete security"},{"data_url":"scopes/ComputerScience/cryptography/hash-functions.json","id":"ComputerScience/cryptography/hash-functions","kind":"scope","parent":43152,"result":false,"route":"../roadmap/ComputerScience/cryptography/hash-functions/index.html","scope":"ComputerScience/cryptography/hash-functions","status":"fully_proved","summary":"Collisions and the birthday bound, the Merkle\u2013Damg\u00e5rd transform and its padding, and Merkle trees with authentication paths.","title":"Hash functions"},{"data_url":"scopes/ComputerScience/cryptography/hash-functions/birthday-bound.json","id":"ComputerScience/cryptography/hash-functions/birthday-bound","kind":"scope","navigation":"module","parent":43194,"result":false,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","title":"Birthday bound"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/collision-game","kind":"node","lean":"Cslib.Crypto.Birthday.collisionGame","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#collision-game","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"For a nonempty finite set $\\alpha$ and a natural number $q$, the collision experiment is the probability distribution on $\\{\\mathrm{true}, \\mathrm{false}\\}$ obtained by drawing a function $s \\colon \\{0, \\dots, q - 1\\} \\to \\alpha$ uniformly at random (equivalently, $q$ independent uniform samples $s(0), \\dots, s(q - 1)$ from $\\alpha$) and outputting true exactly when $s$ is not injective, that is, when $s(i) = s(j)$\u2026","title":"The collision experiment"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/collision-prob","kind":"node","lean":"Cslib.Crypto.Birthday.collisionProb","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#collision-prob","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"For a nonempty finite set $\\alpha$ and a natural number $q$, the collision probability is the probability, as a real number, that the collision experiment with $q$ samples from $\\alpha$ outputs true, that is, that $q$ independent uniform samples from $\\alpha$ contain two equal samples at distinct positions.","title":"Collision probability"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/collision-prob-eq","kind":"node","lean":"Cslib.Crypto.Birthday.collisionProb_eq","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#collision-prob-eq","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set with $N$ elements and $q$ a natural number. The probability that $q$ independent uniform samples from $\\alpha$ contain a collision equals $1 - N (N - 1) \\cdots (N - q + 1) / N^q$, where the product $N (N - 1) \\cdots (N - q + 1)$ has $q$ factors (it is $1$ for $q = 0$ and $0$ for $q > N$).","title":"The exact collision probability"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/collision-prob-le","kind":"node","lean":"Cslib.Crypto.Birthday.collisionProb_le","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#collision-prob-le","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set with $N$ elements and $q$ a natural number. The probability that $q$ independent uniform samples from $\\alpha$ contain a collision is at most $q (q - 1) / (2 N)$.","title":"Birthday upper bound"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/one-sub-exp-le-collision-prob","kind":"node","lean":"Cslib.Crypto.Birthday.one_sub_exp_le_collisionProb","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#one-sub-exp-le-collision-prob","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set with $N$ elements and $q$ a natural number. The probability that $q$ independent uniform samples from $\\alpha$ contain a collision is at least $1 - \\exp(-q (q - 1) / (2 N))$.","title":"Birthday lower bound, exponential form"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/div-one-add-le-collision-prob","kind":"node","lean":"Cslib.Crypto.Birthday.div_one_add_le_collisionProb","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#div-one-add-le-collision-prob","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set with $N$ elements, $q$ a natural number, and $x = q (q - 1) / (2 N)$. The probability that $q$ independent uniform samples from $\\alpha$ contain a collision is at least $x / (1 + x)$.","title":"Birthday lower bound, rational form"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/collision-prob-ge","kind":"node","lean":"Cslib.Crypto.Birthday.collisionProb_ge","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#collision-prob-ge","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set with $N$ elements and $q$ a natural number with $q (q - 1) \\le 2 N$. The probability that $q$ independent uniform samples from $\\alpha$ contain a collision is at least $q (q - 1) / (4 N)$.","title":"Birthday lower bound"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/collision-prob-ge-of-le-sqrt","kind":"node","lean":"Cslib.Crypto.Birthday.collisionProb_ge_of_le_sqrt","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#collision-prob-ge-of-le-sqrt","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set with $N$ elements and $q$ a natural number with $q \\le \\sqrt{2 N}$. The probability that $q$ independent uniform samples from $\\alpha$ contain a collision is at least $q (q - 1) / (4 N)$.","title":"Birthday lower bound as stated in Katz and Lindell"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/collision-prob-ge-of-le-sqrt-card","kind":"node","lean":"Cslib.Crypto.Birthday.collisionProb_ge_of_le_sqrt_card","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#collision-prob-ge-of-le-sqrt-card","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set with $N$ elements and $q$ a natural number with $q \\le \\sqrt{N}$. The probability that $q$ independent uniform samples from $\\alpha$ contain a collision is at least $q (q - 1) / (4 N)$.","title":"Birthday lower bound for at most square-root many samples"},{"id":"ComputerScience/cryptography/hash-functions/birthday-bound/half-le-collision-prob","kind":"node","lean":"Cslib.Crypto.Birthday.half_le_collisionProb","parent":43195,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/birthday-bound/index.html#half-le-collision-prob","scope":"ComputerScience/cryptography/hash-functions/birthday-bound","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set with $N$ elements and $q$ a natural number with $2 N \\le q (q - 1)$. The probability that $q$ independent uniform samples from $\\alpha$ contain a collision is at least $1/2$.","title":"Even odds of a collision"},{"data_url":"scopes/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction.json","id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","kind":"scope","navigation":"module","parent":43194,"result":false,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","title":"Merkle\u2013Damg\u00e5rd construction"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/chain-rev","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.chainRev","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#chain-rev","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function and $iv \\in S$ an initial value. The chaining value of a list of blocks given last block first is defined recursively: the empty list has chaining value $iv$, and a list with first entry $x$ followed by a list $xs$ has chaining value $h(c, x)$, where $c$ is the chaining value of $xs$.","title":"Chaining value of a reversed block list"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/chain","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.chain","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#chain","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function and $iv \\in S$ an initial value. The chaining value of a list of blocks $M = (x_1, \\dots, x_L)$ is $z_L$, where $z_0 = iv$ and $z_i = h(z_{i-1}, x_i)$ for $1 \\le i \\le L$.","title":"Merkle\u2013Damg\u00e5rd chaining value"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/hash","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.hash","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#hash","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function, $iv \\in S$ an initial value and $\\ell : \\mathbb{N} \\to B$ a length encoding. The hash of a list of blocks $M = (x_1, \\dots, x_L)$ is $h(z_L, \\ell(L))$, where $z_0 = iv$ and $z_i = h(z_{i-1}, x_i)$ for $1 \\le i \\le L$.","title":"Merkle\u2013Damg\u00e5rd hash with length strengthening"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/search","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.search","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#search","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function with initial value $iv$, and for a list of blocks written last block first let $z(xs)$ be its chaining value: $z([\\,]) = iv$ and $z(x :: xs) = h(z(xs), x)$. Given two such lists, the search compares their leading compression inputs: on $x :: xs$ and $y :: ys$, if $(z(xs), x) = (z(ys), y)$ it continues with $xs$ and $ys$, and otherwise it returns the pair\u2026","title":"Backward search for a compression collision"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/is-collision","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.IsCollision","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#is-collision","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function. A pair $(p_1, p_2)$ of compression inputs in $S \\times B$ is a collision of $h$ when $p_1 \\ne p_2$ and $h(p_1) = h(p_2)$.","title":"Collision of a compression function"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/find-collision","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.findCollision","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#find-collision","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function, $iv \\in S$ an initial value and $\\ell : \\mathbb{N} \\to B$ a length encoding. Let $M = (x_1, \\dots, x_L)$ and $M' = (x'_1, \\dots, x'_{L'})$ be lists of blocks with chaining values $z_0 = z'_0 = iv$, $z_i = h(z_{i-1}, x_i)$ and $z'_j = h(z'_{j-1}, x'_j)$. The procedure compares the final compression inputs $(z_L, \\ell(L))$ and $(z'_{L'}, \\ell(L'))$ and returns\u2026","title":"Extracting a compression collision from a Merkle\u2013Damg\u00e5rd collision"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/find-collision-spec","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.findCollision_spec","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#find-collision-spec","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function, $iv \\in S$ an initial value and $\\ell : \\mathbb{N} \\to B$ a length encoding, and let $H$ be the Merkle\u2013Damg\u00e5rd hash with length strengthening they define. Write $|M|$ for the number of blocks of a list of blocks $M$. If $M$ and $M'$ are distinct lists of blocks with $H(M) = H(M')$, and $\\ell(|M|) = \\ell(|M'|)$ implies $|M| = |M'|$, then the collision-finding\u2026","title":"A Merkle\u2013Damg\u00e5rd hash collision yields a compression collision"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/hash-collision-game","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.hashCollisionGame","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#hash-collision-game","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $H$ be the Merkle\u2013Damg\u00e5rd hash with length strengthening defined by a compression function $h : S \\times B \\to S$, an initial value $iv \\in S$ and a length encoding $\\ell : \\mathbb{N} \\to B$, and let $N$ be a natural number. An adversary $A$ maps random coins $r$ from a finite nonempty type $R$ to a pair of lists of blocks. The game is the distribution of the Boolean outcome when $r$ is drawn uniformly from\u2026","title":"Hash collision game for Merkle\u2013Damg\u00e5rd"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/compression-collision-game","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.compressionCollisionGame","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#compression-collision-game","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function and let $A$ be an adversary that maps random coins $r$ from a finite nonempty type $R$ either to a pair $(p_1, p_2)$ of compression inputs in $S \\times B$ or to no output. The game is the distribution of the Boolean outcome when $r$ is drawn uniformly from $R$: the adversary wins (outcome true) exactly when $A(r)$ is a pair $(p_1, p_2)$ with $p_1 \\ne p_2$ and\u2026","title":"Compression-function collision game"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/reduction","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.reduction","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#reduction","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function, $iv \\in S$ an initial value, $\\ell : \\mathbb{N} \\to B$ a length encoding and $A$ an adversary that maps random coins $r$ from a finite nonempty type $R$ to a pair $(M, M')$ of lists of blocks. The reduction is the adversary that, on coins $r$, outputs the result of the Merkle\u2013Damg\u00e5rd collision-finding procedure applied to the pair $(M, M') = A(r)$: a pair of\u2026","title":"Merkle\u2013Damg\u00e5rd collision reduction"},{"id":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/hash-collision-game-le","kind":"node","lean":"Cslib.Crypto.MerkleDamgard.hashCollisionGame_le","parent":43206,"result":true,"route":"../roadmap/ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction/index.html#hash-collision-game-le","scope":"ComputerScience/cryptography/hash-functions/merkle-damg-rd-construction","status":"fully_proved","summary":"Let $h : S \\times B \\to S$ be a compression function, $iv \\in S$ an initial value, $\\ell : \\mathbb{N} \\to B$ a length encoding and $N$ a natural number such that $\\ell$ is injective on $\\{0, 1, \\dots, N - 1\\}$. Then for every adversary $A$ that maps random coins from a finite nonempty type $R$ to a pair of lists of blocks, the probability that the reduction built from $A$ wins the compression-function collision\u2026","title":"The Merkle\u2013Damg\u00e5rd transform preserves collision resistance"},{"data_url":"scopes/ComputerScience/cryptography/public-key.json","id":"ComputerScience/cryptography/public-key","kind":"scope","parent":43152,"result":false,"route":"../roadmap/ComputerScience/cryptography/public-key/index.html","scope":"ComputerScience/cryptography/public-key","status":"fully_proved","summary":"Discrete-logarithm, Diffie\u2013Hellman and RSA hardness assumptions, Diffie\u2013Hellman key agreement, ElGamal and RSA encryption with their security and insecurity, deterministic encryption, and key encapsulation with hybrid encryption.","title":"Public-key cryptography"},{"data_url":"scopes/ComputerScience/cryptography/public-key/diffie-hellman-key-exchange.json","id":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange","kind":"scope","navigation":"module","parent":43218,"result":false,"route":"../roadmap/ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/index.html","scope":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange","status":"fully_proved","title":"Diffie\u2013Hellman key exchange"},{"id":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/gpow","kind":"node","lean":"Cslib.Crypto.PublicKey.gpow","parent":43219,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/index.html#gpow","scope":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange","status":"fully_proved","summary":"Let $G$ be a group, $q$ a natural number, $g \\in G$ and $a \\in \\mathbb{Z}/q\\mathbb{Z}$. Then $g^a$ is defined as $g^n$, where $n$ is the natural-number value of $a$: its representative in $\\{0, \\dots, q-1\\}$ when $q > 0$, and $|a|$ when $q = 0$ (in which case $\\mathbb{Z}/q\\mathbb{Z} = \\mathbb{Z}$).","title":"Group exponentiation with modular exponents"},{"id":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/dh-agree","kind":"node","lean":"Cslib.Crypto.PublicKey.dh_agree","parent":43219,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/index.html#dh-agree","scope":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange","status":"fully_proved","summary":"Let $G$ be a group, $q$ a natural number and $g \\in G$. For $x \\in G$ and $c \\in \\mathbb{Z}/q\\mathbb{Z}$ write $x^c$ for $x^n$, where $n$ is the natural-number value of $c$: its representative in $\\{0, \\dots, q-1\\}$ when $q > 0$, and $|c|$ when $q = 0$ (in which case $\\mathbb{Z}/q\\mathbb{Z} = \\mathbb{Z}$). Then $(g^b)^a = (g^a)^b$ for all $a, b \\in \\mathbb{Z}/q\\mathbb{Z}$.","title":"Correctness of Diffie\u2013Hellman key agreement"},{"id":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/dh-key-eq","kind":"node","lean":"Cslib.Crypto.PublicKey.dh_key_eq","parent":43219,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/index.html#dh-key-eq","scope":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange","status":"fully_proved","summary":"Let $G$ be a group, $q$ a natural number and $g \\in G$ with $g^q = 1$. For $x \\in G$ and $c \\in \\mathbb{Z}/q\\mathbb{Z}$ write $x^c$ for $x^n$, where $n$ is the natural-number value of $c$: its representative in $\\{0, \\dots, q-1\\}$ when $q > 0$, and $|c|$ when $q = 0$ (in which case $\\mathbb{Z}/q\\mathbb{Z} = \\mathbb{Z}$). Then $(g^b)^a = g^{ab}$ for all $a, b \\in \\mathbb{Z}/q\\mathbb{Z}$, where $ab$ is the product in\u2026","title":"Value of the Diffie\u2013Hellman shared key"},{"id":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/gpow-equiv","kind":"node","lean":"Cslib.Crypto.PublicKey.gpowEquiv","parent":43219,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/diffie-hellman-key-exchange/index.html#gpow-equiv","scope":"ComputerScience/cryptography/public-key/diffie-hellman-key-exchange","status":"fully_proved","summary":"Let $G$ be a finite group of order $q$, with $q \\neq 0$, and let $g \\in G$ generate $G$, that is, every element of $G$ is an integer power of $g$. Then $a \\mapsto g^a$ is a bijection from $\\mathbb{Z}/q\\mathbb{Z}$ to $G$, where $g^a = g^n$ for the representative $n \\in \\{0, \\dots, q-1\\}$ of $a$. Its inverse is the discrete logarithm to base $g$.","title":"Exponentiation from a generator is a bijection"},{"data_url":"scopes/ComputerScience/cryptography/public-key/elgamal-encryption.json","id":"ComputerScience/cryptography/public-key/elgamal-encryption","kind":"scope","navigation":"module","parent":43218,"result":false,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","title":"ElGamal encryption"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/public-key","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.publicKey","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#public-key","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a commutative group with a fixed element $g$ and let $q$ be a natural number. The public key of a secret key $x \\in \\mathbb{Z}/q$ is $h = g^{v(x)} \\in G$, where $v(x) \\in \\{0, \\dots, q - 1\\}$ is the least nonnegative representative of $x$ if $q \\ge 1$, and $v(x) = |x|$ if $q = 0$ (when $\\mathbb{Z}/0 = \\mathbb{Z}$).","title":"ElGamal public key"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/encrypt","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.encrypt","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#encrypt","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a commutative group with a fixed element $g$ and let $q$ be a natural number. The ElGamal encryption of a message $m \\in G$ under the public key $h \\in G$ with randomness $y \\in \\mathbb{Z}/q$ is the pair $(g^y, h^y m) \\in G \\times G$. Here a power $k^a$ with $k \\in G$ and $a \\in \\mathbb{Z}/q$ means $k^{v(a)}$, where $v(a) \\in \\{0, \\dots, q - 1\\}$ is the least nonnegative representative of $a$ if\u2026","title":"ElGamal encryption"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/decrypt","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.decrypt","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#decrypt","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a commutative group and $q$ a natural number. For $a \\in \\mathbb{Z}/q\\mathbb{Z}$ and $c \\in G$ write $c^a$ for $c^{\\operatorname{val}(a)}$, where $\\operatorname{val}(a) \\in \\mathbb{N}$ is the representative of $a$ in $\\{0, \\dots, q-1\\}$ if $q \\ge 1$ and $|a|$ if $q = 0$. Decrypting a ciphertext $(c_1, c_2) \\in G \\times G$ with the secret key $x \\in \\mathbb{Z}/q\\mathbb{Z}$ gives $c_2 \\cdot (c_1^x)^{-1}$.","title":"ElGamal decryption"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/decrypt-encrypt","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.decrypt_encrypt","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#decrypt-encrypt","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a commutative group, $g \\in G$ and $q$ a natural number. For $a \\in \\mathbb{Z}/q\\mathbb{Z}$ and $c \\in G$ write $c^a$ for $c^{\\operatorname{val}(a)}$, where $\\operatorname{val}(a) \\in \\mathbb{N}$ is the representative of $a$ in $\\{0, \\dots, q-1\\}$ if $q \\ge 1$ and $|a|$ if $q = 0$. For every secret key $x \\in \\mathbb{Z}/q\\mathbb{Z}$, every message $m \\in G$ and every encryption randomness\u2026","title":"Correctness of ElGamal"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/pick","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.pick","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#pick","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group. For a pair of messages $(m_0, m_1) \\in G \\times G$ and a bit $b$, the selected message is $m_1$ if $b$ is true and $m_0$ if $b$ is false.","title":"Message selected by the challenge bit"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/adversary","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.Adversary","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#adversary","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Given a type $G$ of public keys and messages and a type $R$ of random coins, an eavesdropping adversary consists of a choice function, which maps a public key $h \\in G$ and coins $r \\in R$ to a pair of messages in $G \\times G$, and a guessing function, which maps a public key $h \\in G$, a ciphertext in $G \\times G$ and coins $r \\in R$ to a bit.","title":"ElGamal eavesdropping adversary"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/eav-game","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.eavGame","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#eav-game","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a commutative group, $g \\in G$ and $q \\ge 1$. For an eavesdropping adversary $A$ with coins in a finite nonempty type $R$, the experiment draws $x, y \\in \\mathbb{Z}/q$, coins $r \\in R$ and a bit $b$ independently and uniformly. It sets the public key $h = g^x$, lets $A$ choose messages $(m_0, m_1)$ from $h$ and $r$, and gives $A$ the key $h$, the ElGamal ciphertext $(g^y, h^y m_b)$ and the coins $r$,\u2026","title":"ElGamal eavesdropping experiment"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/eav-advantage","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.eavAdvantage","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#eav-advantage","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group, $g \\in G$ and $q \\ge 1$. The eavesdropping advantage of an adversary $A$ with random coins in a finite nonempty type $R$ is the real number $|p - 1/2|$, where $p$ is the probability that $A$ wins the ElGamal eavesdropping experiment for $g$ with exponents in $\\mathbb{Z}/q\\mathbb{Z}$, that is, guesses the challenge bit correctly.","title":"Eavesdropping advantage against ElGamal"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/ddh-real","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.ddhReal","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#ddh-real","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group, $g \\in G$ and $q \\ge 1$ a natural number, and let $D$ be a distinguisher that maps three elements of $G$ and coins in a finite nonempty type $S$ to a bit. The real DDH experiment draws $x, y \\in \\mathbb{Z}/q$ and coins $s \\in S$ independently and uniformly and outputs the distribution of the bit $D(g^x, g^y, g^{xy}, s)$, where $xy$ is the product in $\\mathbb{Z}/q$ and a power\u2026","title":"Real DDH experiment"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/ddh-rand","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.ddhRand","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#ddh-rand","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group, $g \\in G$ and $q \\ge 1$ a natural number, and let $D$ be a distinguisher that maps three elements of $G$ and coins in a finite nonempty type $S$ to a bit. The random DDH experiment draws $x, y, z \\in \\mathbb{Z}/q$ and coins $s \\in S$ independently and uniformly and outputs the distribution of the bit $D(g^x, g^y, g^z, s)$, where a power $g^a$ with $a \\in \\mathbb{Z}/q$ means\u2026","title":"Random DDH experiment"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/ddh-advantage","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.ddhAdvantage","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#ddh-advantage","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group, $g \\in G$ and $q \\ge 1$, and let $D$ be a distinguisher that maps three elements of $G$ and coins in a finite nonempty type $S$ to a bit. The DDH advantage of $D$ is the real number $|p_{\\mathrm{real}} - p_{\\mathrm{rand}}|$, where $p_{\\mathrm{real}}$ and $p_{\\mathrm{rand}}$ are the probabilities that $D$ outputs true in the real and in the random DDH experiment for $g$ with\u2026","title":"Advantage of a DDH distinguisher"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/reduction","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.reduction","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#reduction","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group and let $A$ be an eavesdropping adversary with coins in a finite nonempty type $R$. The distinguisher built from $A$ takes three elements $X, Y, Z \\in G$ and coins consisting of $r \\in R$ and a bit $b$. It lets $A$ choose messages $(m_0, m_1)$ from the public key $X$ and the coins $r$, gives $A$ the public key $X$, the ciphertext $(Y, Z m_b)$ and the coins $r$, where $m_b$ is\u2026","title":"DDH distinguisher from an ElGamal eavesdropper"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/eav-game-eq-ddh-real","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.eavGame_eq_ddhReal","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#eav-game-eq-ddh-real","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a commutative group, $q \\ge 1$ a natural number and $g \\in G$ with $g^q = 1$. For every eavesdropping adversary $A$ with coins in a finite nonempty type $R$, the output distribution of the ElGamal eavesdropping experiment for $A$ (exponents in $\\mathbb{Z}/q$) equals the output distribution of the real DDH experiment run with the distinguisher built from $A$ by the reduction.","title":"ElGamal eavesdropping game equals the real DDH experiment of the reduction"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/ddh-rand-reduction-true","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.ddhRand_reduction_true","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#ddh-rand-reduction-true","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group of order $q$ generated by $g$. For every eavesdropping adversary $A$ with random coins in a finite nonempty type $R$, the DDH distinguisher obtained from $A$ by the reduction outputs true with probability exactly $1/2$ in the random DDH experiment for $g$.","title":"The reduction is a fair coin on random DDH triples"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/eav-advantage-eq-ddh-advantage","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.eavAdvantage_eq_ddhAdvantage","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#eav-advantage-eq-ddh-advantage","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group of order $q$ generated by $g$. For every eavesdropping adversary $A$ with random coins in a finite nonempty type $R$, the eavesdropping advantage of $A$ against ElGamal equals the DDH advantage of the distinguisher obtained from $A$ by the reduction.","title":"ElGamal eavesdropping advantage equals a DDH advantage"},{"id":"ComputerScience/cryptography/public-key/elgamal-encryption/eav-advantage-le","kind":"node","lean":"Cslib.Crypto.PublicKey.ElGamal.eavAdvantage_le","parent":43224,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/elgamal-encryption/index.html#eav-advantage-le","scope":"ComputerScience/cryptography/public-key/elgamal-encryption","status":"fully_proved","summary":"Let $G$ be a finite commutative group of order $q$ generated by $g$, and let $A$ be an eavesdropping adversary with random coins in a finite nonempty type $R$. For every real number $\\varepsilon$, if the DDH advantage of the distinguisher obtained from $A$ by the reduction is at most $\\varepsilon$, then the eavesdropping advantage of $A$ against ElGamal is at most $\\varepsilon$.","title":"DDH bounds transfer to ElGamal"},{"data_url":"scopes/ComputerScience/cryptography/public-key/rsa-cryptosystem.json","id":"ComputerScience/cryptography/public-key/rsa-cryptosystem","kind":"scope","navigation":"module","parent":43218,"result":false,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","title":"RSA cryptosystem"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/pow-eq-self-of-mod-eq","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.pow_eq_self_of_modEq","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#pow-eq-self-of-mod-eq","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Let $p$ be a prime and let $k \\ge 1$ be a natural number with $k \\equiv 1 \\pmod{p - 1}$. Then $x^k = x$ for every $x \\in \\mathbb{Z}/p\\mathbb{Z}$.","title":"Fermat's little theorem with exponents congruent to one"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/pow-eq-pow-of-mod-eq","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.pow_eq_pow_of_modEq","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#pow-eq-pow-of-mod-eq","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Let $p$ be a prime and let $a$ and $b$ be natural numbers with $a, b \\ge 1$ and $a \\equiv b \\pmod{p - 1}$. Then $x^a = x^b$ for every $x \\in \\mathbb{Z}/p\\mathbb{Z}$.","title":"Fermat's little theorem for congruent exponents"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/one-le-mod","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.one_le_mod","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#one-le-mod","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Let $p$, $e$ and $d$ be natural numbers with $p > 2$ and $ed \\equiv 1 \\pmod{p - 1}$. Then $d \\bmod (p - 1) \\ge 1$.","title":"Positivity of the reduced RSA exponent"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/key","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#key","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"A textbook RSA key consists of distinct primes $p$ and $q$ and natural numbers $e$ (the public exponent) and $d$ (the private exponent) such that $ed \\equiv 1 \\pmod{p - 1}$ and $ed \\equiv 1 \\pmod{q - 1}$.","title":"RSA key"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/n","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.n","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#n","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"The modulus of an RSA key with primes $p$ and $q$ is $n = pq$.","title":"RSA modulus"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/encrypt","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.encrypt","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#encrypt","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"For an RSA key with modulus $n = pq$ and public exponent $e$, encryption is the map $\\mathbb{Z}/n\\mathbb{Z} \\to \\mathbb{Z}/n\\mathbb{Z}$ given by $m \\mapsto m^e$.","title":"RSA encryption"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/decrypt","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.decrypt","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#decrypt","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"For an RSA key with modulus $n = pq$ and private exponent $d$, decryption is the map $\\mathbb{Z}/n\\mathbb{Z} \\to \\mathbb{Z}/n\\mathbb{Z}$ given by $c \\mapsto c^d$.","title":"RSA decryption"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/crt","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.crt","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#crt","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"For an RSA key with distinct primes $p$, $q$ and modulus $n = pq$, this is the ring isomorphism $\\mathbb{Z}/n\\mathbb{Z} \\cong \\mathbb{Z}/p\\mathbb{Z} \\times \\mathbb{Z}/q\\mathbb{Z}$ of the Chinese remainder theorem, sending $x$ to $(x \\bmod p, x \\bmod q)$.","title":"Chinese remainder isomorphism for an RSA modulus"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/pow-eq-self-of-crt","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.pow_eq_self_of_crt","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#pow-eq-self-of-crt","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Let an RSA key have primes $p$ and $q$ and modulus $n$, let $k$ be a natural number and $x \\in \\mathbb{Z}/n\\mathbb{Z}$, and let $(x_p, x_q) \\in \\mathbb{Z}/p\\mathbb{Z} \\times \\mathbb{Z}/q\\mathbb{Z}$ be the image of $x$ under the Chinese remainder isomorphism. If $x_p^k = x_p$ and $x_q^k = x_q$, then $x^k = x$.","title":"Checking a power identity through the Chinese remainder theorem"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/pow-ed","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.pow_ed","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#pow-ed","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"For an RSA key with modulus $n$ and exponents $e$ and $d$, every $x \\in \\mathbb{Z}/n\\mathbb{Z}$ satisfies $x^{ed} = x$.","title":"RSA exponent identity"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/decrypt-encrypt","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.decrypt_encrypt","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#decrypt-encrypt","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Decryption inverts encryption on every message, including messages that are not coprime to `n`.","title":"Correctness of textbook RSA"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/encrypt-bijective","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.encrypt_bijective","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#encrypt-bijective","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"For an RSA key with modulus $n = pq$ and public exponent $e$, the encryption map $m \\mapsto m^e$ is a bijection from $\\mathbb{Z}/n\\mathbb{Z}$ to itself.","title":"RSA encryption is a bijection"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/decrypt-encrypt-nat","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.decrypt_encrypt_nat","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#decrypt-encrypt-nat","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"For an RSA key with modulus $n = pq$ and exponents $e$, $d$, every natural number $m < n$ satisfies $(m^e \\bmod n)^d \\bmod n = m$.","title":"Correctness of textbook RSA on natural numbers"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/crt-decrypt","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.crtDecrypt","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#crt-decrypt","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Let $(p, q, e, d)$ be an RSA key with modulus $n = pq$. For natural numbers $d_p$, $d_q$ and $c \\in \\mathbb{Z}/n\\mathbb{Z}$, let $c_p$ and $c_q$ be the images of $c$ in $\\mathbb{Z}/p\\mathbb{Z}$ and $\\mathbb{Z}/q\\mathbb{Z}$. RSA-CRT decryption of $c$ with the exponents $d_p$ and $d_q$ is the element of $\\mathbb{Z}/n\\mathbb{Z}$ that corresponds to the pair $(c_p^{d_p}, c_q^{d_q})$ under the Chinese remainder\u2026","title":"RSA-CRT decryption"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/crt-decrypt-eq","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.crtDecrypt_eq","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#crt-decrypt-eq","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Let $(p, q, e, d)$ be an RSA key with modulus $n = pq$, and let $d_p, d_q \\ge 1$ be natural numbers with $d_p \\equiv d \\pmod{p - 1}$ and $d_q \\equiv d \\pmod{q - 1}$. Then for every $c \\in \\mathbb{Z}/n\\mathbb{Z}$, RSA-CRT decryption of $c$ with the exponents $d_p$ and $d_q$ equals the RSA decryption $c^d$.","title":"RSA-CRT decryption is correct"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/crt-decrypt-mod-eq","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.crtDecrypt_mod_eq","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#crt-decrypt-mod-eq","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Let $(p, q, e, d)$ be an RSA key with modulus $n = pq$ in which both primes are odd, that is $p > 2$ and $q > 2$. Then for every $c \\in \\mathbb{Z}/n\\mathbb{Z}$, RSA-CRT decryption of $c$ with the exponents $d_p = d \\bmod (p - 1)$ and $d_q = d \\bmod (q - 1)$ equals the RSA decryption $c^d$.","title":"RSA-CRT decryption with the standard exponents"},{"id":"ComputerScience/cryptography/public-key/rsa-cryptosystem/of-totient","kind":"node","lean":"Cslib.Crypto.PublicKey.RSA.Key.ofTotient","parent":43241,"result":true,"route":"../roadmap/ComputerScience/cryptography/public-key/rsa-cryptosystem/index.html#of-totient","scope":"ComputerScience/cryptography/public-key/rsa-cryptosystem","status":"fully_proved","summary":"Given distinct primes $p$ and $q$ and natural numbers $e$ and $d$ with $ed \\equiv 1 \\pmod{\\varphi(pq)}$, where $\\varphi$ is Euler's totient function, this is the RSA key with primes $p$ and $q$, public exponent $e$ and private exponent $d$. Its conditions $ed \\equiv 1 \\pmod{p - 1}$ and $ed \\equiv 1 \\pmod{q - 1}$ hold because $\\varphi(pq) = (p - 1)(q - 1)$.","title":"RSA key from the totient condition"},{"data_url":"scopes/ComputerScience/cryptography/signatures-and-protocols.json","id":"ComputerScience/cryptography/signatures-and-protocols","kind":"scope","parent":43152,"result":false,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/index.html","scope":"ComputerScience/cryptography/signatures-and-protocols","status":"fully_proved","summary":"Lamport one-time and Merkle many-time signatures, Schnorr identification and sigma protocols, and hash-based and Pedersen commitments.","title":"Signatures, identification and commitments"},{"data_url":"scopes/ComputerScience/cryptography/signatures-and-protocols/lamport-signature.json","id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","kind":"scope","navigation":"module","parent":43259,"result":false,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","title":"Lamport signature"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/public-key","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.publicKey","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#public-key","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"For a function $f : X \\to Y$ and a secret key $sk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to X$, the public key is the map $pk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to Y$ with $pk(i, b) = f(sk(i, b))$.","title":"Lamport public key"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/sign","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.sign","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#sign","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"The signature of a message $m \\in \\{0, 1\\}^\\ell$ under a secret key $sk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to X$ is $\\sigma \\in X^\\ell$ with $\\sigma_i = sk(i, m_i)$ for every position $i < \\ell$.","title":"Lamport signing"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/verify","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.Verify","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#verify","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"Let $f : X \\to Y$. A signature $\\sigma \\in X^\\ell$ on a message $m \\in \\{0, 1\\}^\\ell$ is valid for a public key $pk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to Y$ when $f(\\sigma_i) = pk(i, m_i)$ for every position $i < \\ell$.","title":"Lamport verification"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/verify-sign","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.verify_sign","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#verify-sign","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"Let $f : X \\to Y$. For every secret key $sk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to X$ and every message $m \\in \\{0, 1\\}^\\ell$, the Lamport signature of $m$ under $sk$ is valid for $m$ under the public key of $sk$.","title":"Correctness of Lamport signatures"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/forger","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.Forger","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#forger","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"A one-time forger for messages of length $\\ell$, with random coins in a set $R$, consists of two maps. The first takes a public key $pk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to Y$ and coins $r \\in R$ and returns the message $m \\in \\{0, 1\\}^\\ell$ whose signature the forger requests. The second takes a public key, a signature $\\sigma \\in X^\\ell$ and coins $r$ and returns a message $m' \\in \\{0, 1\\}^\\ell$ together\u2026","title":"One-time forger"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/wins","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.Forger.Wins","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#wins","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"Let $X$ be a nonempty finite set, $Y$ a set with decidable equality, $f : X \\to Y$ a function, and $\\ell$ a message length; messages are bit strings $m = (m_0, \\dots, m_{\\ell-1})$. A one-time forger $A$ with random coins from a nonempty finite set $R$ wins against a secret key $sk$, which assigns $sk(i, b) \\in X$ to each position $i < \\ell$ and bit $b$, with coins $r \\in R$ when the following holds. Let $pk$ be the\u2026","title":"Winning the Lamport one-time forgery game"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/forge-game","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.forgeGame","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#forge-game","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"Let $f : X \\to Y$ with $X$ finite and nonempty, and let $A$ be a one-time forger for messages of length $\\ell$ with random coins in a finite nonempty set $R$. The one-time forgery game is the distribution on $\\{\\mathrm{true}, \\mathrm{false}\\}$ obtained by drawing a secret key $sk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to X$ and coins $r \\in R$ uniformly and independently, and returning whether $A$ wins against\u2026","title":"One-time forgery game"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/invert-game","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.invertGame","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#invert-game","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"Let $f : X \\to Y$ with $X$ finite and nonempty, and let $I$ be an inverter with random coins in a finite nonempty set $\\Omega$: given $y \\in Y$ and coins $\\omega \\in \\Omega$ it either outputs some $x' \\in X$ or fails. The one-wayness game for $f$ is the distribution on $\\{\\mathrm{true}, \\mathrm{false}\\}$ obtained by drawing $x \\in X$ and $\\omega \\in \\Omega$ uniformly and independently and returning true exactly\u2026","title":"One-wayness game"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/plant","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.plant","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#plant","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"Let $f : X \\to Y$. Given a secret key $sk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to X$, a position $(i, b)$ and a value $y \\in Y$, the planted public key takes the value $y$ at $(i, b)$ and the value $f(sk(j, c))$ at every other position $(j, c)$.","title":"Planted public key"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/invert","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.invert","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#invert","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"Let $f : X \\to Y$, let $A$ be a one-time forger for messages of length $\\ell$ with coins in $R$, and let $y \\in Y$. The inverter's coins are a position $(i, b)$ with $i < \\ell$ and $b \\in \\{0, 1\\}$, a secret key $sk : \\{0, \\dots, \\ell - 1\\} \\times \\{0, 1\\} \\to X$ and forger coins $r \\in R$. It forms the public key $pk$ with $pk(i, b) = y$ and $pk(j, c) = f(sk(j, c))$ for every other $(j, c)$, and runs $A$ on $pk$\u2026","title":"Inverter built from a forger"},{"id":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature/forge-game-le","kind":"node","lean":"Cslib.Crypto.Signatures.Lamport.forgeGame_le","parent":43260,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/lamport-signature/index.html#forge-game-le","scope":"ComputerScience/cryptography/signatures-and-protocols/lamport-signature","status":"fully_proved","summary":"Let $f : X \\to Y$ be a function on a finite nonempty set $X$, let $\\ell \\ge 1$ be the message length, and let $A$ be a one-time forger whose random coins range over a finite nonempty set $R$. Then the probability that $A$ wins the one-time forgery game against the Lamport scheme built from $f$ is at most $2\\ell$ times the probability that the inverter built from $A$ wins the one-wayness game for $f$.","title":"Security reduction for Lamport one-time signatures"},{"data_url":"scopes/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme.json","id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","kind":"scope","navigation":"module","parent":43259,"result":false,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","title":"Schnorr identification scheme"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/accepts","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.Accepts","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#accepts","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $G$ be a commutative group, $q$ a natural number and $g, y \\in G$ (the base and the public key); for $h \\in G$ and $a \\in \\mathbb{Z}/q$ write $h^a$ for $h^{\\bar a}$, where $\\bar a$ is the representative of $a$ in $\\{0, \\dots, q - 1\\}$ (and $\\bar a = |a|$ when $q = 0$, so that $\\mathbb{Z}/q = \\mathbb{Z}$). The verifier accepts a transcript $(I, r, s)$ with $I \\in G$ and $r, s \\in \\mathbb{Z}/q$ when\u2026","title":"Schnorr verifier's acceptance condition"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/respond","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.respond","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#respond","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $q$ be a natural number. For a secret key $x$, prover randomness $k$ and challenge $r$, all in $\\mathbb{Z}/q$, the prover's answer is $s = rx + k \\in \\mathbb{Z}/q$.","title":"Schnorr prover's response"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/accepts-honest","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.accepts_honest","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#accepts-honest","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $G$ be a commutative group, $q \\ge 1$ a natural number and $g \\in G$ with $g^q = 1$; for $h \\in G$ and $a \\in \\mathbb{Z}/q$ write $h^a$ for $h^{\\bar a}$, where $\\bar a \\in \\{0, \\dots, q - 1\\}$ is the representative of $a$. For all $x, k, r \\in \\mathbb{Z}/q$, with public key $y = g^x$, the honest transcript $(g^k, r, rx + k)$ is accepted, that is, $g^{rx + k} (y^r)^{-1} = g^k$.","title":"Completeness of the Schnorr identification scheme"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/extract","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.extract","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#extract","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $q$ be a natural number. For $r, s, r', s' \\in \\mathbb{Z}/q$ (the challenges and answers of two transcripts), the extracted exponent is $(s - s')(r - r')^{-1} \\in \\mathbb{Z}/q$, where $u^{-1}$ is the inverse of $u$ in $\\mathbb{Z}/q$ when $u$ is a unit (and, otherwise, a conventional element of $\\mathbb{Z}/q$ depending on $u$). For $q$ prime and $r \\ne r'$ it is $(s - s')/(r - r')$.","title":"Schnorr knowledge extractor"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/extract-spec","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.extract_spec","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#extract-spec","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $G$ be a commutative group, $q$ a prime and $g, y \\in G$ with $g^q = 1$ and $y^q = 1$; for $h \\in G$ and $a \\in \\mathbb{Z}/q$ write $h^a$ for $h^{\\bar a}$, where $\\bar a \\in \\{0, \\dots, q - 1\\}$ is the representative of $a$. Let $I \\in G$ and $r, s, r', s' \\in \\mathbb{Z}/q$ with $r \\ne r'$, and suppose both transcripts $(I, r, s)$ and $(I, r', s')$ are accepted, that is, $g^s (y^r)^{-1} = I$ and\u2026","title":"Special soundness"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/honest-transcript","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.honestTranscript","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#honest-transcript","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $G$ be a commutative group, $q \\ge 1$ and $g \\in G$; for $a \\in \\mathbb{Z}/q$ write $g^a$ for $g^{\\bar a}$, where $\\bar a \\in \\{0, \\dots, q - 1\\}$ is the representative of $a$. For a secret key $x$, prover randomness $k$ and challenge $r$, all in $\\mathbb{Z}/q$, the honest transcript is the element $(g^k, r, rx + k)$ of $G \\times \\mathbb{Z}/q \\times \\mathbb{Z}/q$.","title":"Honest Schnorr transcript"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/sim-transcript","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.simTranscript","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#sim-transcript","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $G$ be a commutative group, $q \\ge 1$ and $g, y \\in G$; for $h \\in G$ and $a \\in \\mathbb{Z}/q$ write $h^a$ for $h^{\\bar a}$, where $\\bar a \\in \\{0, \\dots, q - 1\\}$ is the representative of $a$. For a challenge $r$ and an answer $s$ in $\\mathbb{Z}/q$, the simulated transcript is the element $(g^s (y^r)^{-1}, r, s)$ of $G \\times \\mathbb{Z}/q \\times \\mathbb{Z}/q$.","title":"Simulated Schnorr transcript"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/transcript","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.transcript","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#transcript","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $G$ be a commutative group, $q \\ge 1$ and $g \\in G$; for $a \\in \\mathbb{Z}/q$ write $g^a$ for $g^{\\bar a}$, where $\\bar a \\in \\{0, \\dots, q - 1\\}$ is the representative of $a$. For a secret key $x \\in \\mathbb{Z}/q$, this is the law of the honest transcript $(g^k, r, rx + k)$ when $(k, r)$ is uniform on $\\mathbb{Z}/q \\times \\mathbb{Z}/q$.","title":"Distribution of honest Schnorr transcripts"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/simulate","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.simulate","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#simulate","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $G$ be a commutative group, $q \\ge 1$ and $g, y \\in G$; for $h \\in G$ and $a \\in \\mathbb{Z}/q$ write $h^a$ for $h^{\\bar a}$, where $\\bar a \\in \\{0, \\dots, q - 1\\}$ is the representative of $a$. The simulator's distribution is the law of the simulated transcript $(g^s (y^r)^{-1}, r, s)$ when $(r, s)$ is uniform on $\\mathbb{Z}/q \\times \\mathbb{Z}/q$; it uses only $g$ and the public key $y$.","title":"Schnorr simulator"},{"id":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/transcript-eq-simulate","kind":"node","lean":"Cslib.Crypto.Identification.Schnorr.transcript_eq_simulate","parent":43272,"result":true,"route":"../roadmap/ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme/index.html#transcript-eq-simulate","scope":"ComputerScience/cryptography/signatures-and-protocols/schnorr-identification-scheme","status":"fully_proved","summary":"Let $G$ be a commutative group, $q \\ge 1$ and $g \\in G$ with $g^q = 1$; for $h \\in G$ and $a \\in \\mathbb{Z}/q$ write $h^a$ for $h^{\\bar a}$, where $\\bar a \\in \\{0, \\dots, q - 1\\}$ is the representative of $a$. For every secret key $x \\in \\mathbb{Z}/q$, with public key $y = g^x$, the honest transcript $(g^k, r, rx + k)$ with $(k, r)$ uniform on $\\mathbb{Z}/q \\times \\mathbb{Z}/q$ has the same distribution as the\u2026","title":"Perfect honest-verifier zero knowledge"},{"data_url":"scopes/ComputerScience/databases.json","id":"ComputerScience/databases","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/databases/index.html","scope":"ComputerScience/databases","status":"fully_proved","summary":"Relational algebra, functional dependencies, normal forms and lossless decomposition, and transaction concurrency control (conflict serializability, two-phase locking).","title":"Databases"},{"data_url":"scopes/ComputerScience/databases/conflict-serializability.json","id":"ComputerScience/databases/conflict-serializability","kind":"scope","navigation":"module","parent":43283,"result":false,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","title":"Conflict serializability"},{"id":"ComputerScience/databases/conflict-serializability/access","kind":"node","lean":"Cslib.Databases.Access","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#access","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"The kind of access an operation makes to its data item is one of two values: a read or a write.","title":"Read or write access"},{"id":"ComputerScience/databases/conflict-serializability/op","kind":"node","lean":"Cslib.Databases.Op","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#op","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"For $n$ transactions, numbered $0, \\dots, n - 1$, and data items of type $X$, an operation consists of a transaction $t \\in \\{0, \\dots, n - 1\\}$, an occurrence identifier $k \\in \\mathbb{N}$, an access, which is a read or a write, and a data item $x \\in X$: transaction $t$ reads or writes $x$, and $k$ makes repeated accesses of $t$ to an item distinct operations.","title":"Operation of a schedule"},{"id":"ComputerScience/databases/conflict-serializability/conflicts","kind":"node","lean":"Cslib.Databases.Op.Conflicts","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#conflicts","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"Two operations of a schedule conflict when they belong to different transactions, access the same data item, and at least one of the two accesses is a write.","title":"Conflicting operations"},{"id":"ComputerScience/databases/conflict-serializability/order-matters","kind":"node","lean":"Cslib.Databases.Op.OrderMatters","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#order-matters","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"Two operations $a$ and $b$ are order-relevant when they belong to the same transaction or they conflict. These are the pairs whose relative order a conflict-equivalent schedule must keep.","title":"Operations whose relative order matters"},{"id":"ComputerScience/databases/conflict-serializability/conflict-equiv","kind":"node","lean":"Cslib.Databases.ConflictEquiv","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#conflict-equiv","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"A schedule $s'$ is conflict-equivalent to a schedule $s$ when $s'$ is a permutation of $s$ and, for any two operations $a$ and $b$ that belong to the same transaction or conflict, an occurrence of $a$ comes before an occurrence of $b$ in $s$ if and only if the same holds in $s'$.","title":"Conflict equivalence"},{"id":"ComputerScience/databases/conflict-serializability/txn-before","kind":"node","lean":"Cslib.Databases.TxnBefore","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#txn-before","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"For a schedule $s$ and transactions $i$ and $j$, transaction $i$ precedes transaction $j$ in $s$ when every operation of $i$ in $s$ occurs in $s$ before every operation of $j$ in $s$.","title":"One transaction before another in a schedule"},{"id":"ComputerScience/databases/conflict-serializability/is-serial","kind":"node","lean":"Cslib.Databases.IsSerial","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#is-serial","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"A schedule $s$ is serial when, for any two distinct transactions $i$ and $j$, either every operation of $i$ in $s$ occurs before every operation of $j$ in $s$, or every operation of $j$ in $s$ occurs before every operation of $i$ in $s$.","title":"Serial schedule"},{"id":"ComputerScience/databases/conflict-serializability/conflict-serializable","kind":"node","lean":"Cslib.Databases.ConflictSerializable","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#conflict-serializable","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"A schedule $s$ is conflict-serializable when some serial schedule is conflict-equivalent to $s$.","title":"Conflict-serializable schedule"},{"id":"ComputerScience/databases/conflict-serializability/precedence-edge","kind":"node","lean":"Cslib.Databases.PrecedenceEdge","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#precedence-edge","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"For a schedule $s$ and transactions $i$ and $j$, there is a precedence edge from $i$ to $j$ in $s$ when some operation of $i$ occurs in $s$ before some operation of $j$ in $s$ that conflicts with it.","title":"Edge of the precedence graph"},{"id":"ComputerScience/databases/conflict-serializability/precedence-graph","kind":"node","lean":"Cslib.Databases.precedenceGraph","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#precedence-graph","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"The precedence (serialization) graph of a schedule $s$ of $n$ transactions is the directed adjacency graph whose vertices are the transactions $0, \\dots, n - 1$, listed in increasing order, and whose adjacency list at a transaction $i$ lists, in increasing order, the transactions $j$ such that there is a precedence edge from $i$ to $j$ in $s$.","title":"Precedence graph"},{"id":"ComputerScience/databases/conflict-serializability/serialize","kind":"node","lean":"Cslib.Databases.serialize","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#serialize","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"For a list $\\sigma$ of transactions and a schedule $s$, the serialization of $s$ along $\\sigma$ sorts the operations of $s$ by the position of the first occurrence of their transaction in $\\sigma$, where a transaction that does not occur in $\\sigma$ gets the length of $\\sigma$ as its position. The sort is stable: operations with the same position keep their relative order in $s$.","title":"Serialization of a schedule along a transaction order"},{"id":"ComputerScience/databases/conflict-serializability/is-acyclic","kind":"node","lean":"Cslib.Databases.ConflictSerializable.isAcyclic","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#is-acyclic","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"If a schedule $s$ in which no operation occurs twice is conflict-serializable, then the precedence graph of $s$ is acyclic.","title":"A conflict-serializable schedule has an acyclic precedence graph"},{"id":"ComputerScience/databases/conflict-serializability/is-serial-serialize","kind":"node","lean":"Cslib.Databases.isSerial_serialize","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#is-serial-serialize","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"Let $s$ be a schedule and $L$ a list in which every transaction occurs. Then the schedule obtained from $s$ by stably sorting its operations by the first position of their transaction in $L$ is serial.","title":"Serializing along a complete transaction order gives a serial schedule"},{"id":"ComputerScience/databases/conflict-serializability/conflict-equiv-serialize","kind":"node","lean":"Cslib.Databases.conflictEquiv_serialize","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#conflict-equiv-serialize","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"Let $s$ be a schedule in which no operation occurs twice, and let $L$ be a topological order of the precedence graph of $s$. Then the schedule obtained from $s$ by stably sorting its operations by the first position of their transaction in $L$ is conflict-equivalent to $s$.","title":"Serializing along a topological order preserves conflict equivalence"},{"id":"ComputerScience/databases/conflict-serializability/conflict-serializable-iff-is-acyclic","kind":"node","lean":"Cslib.Databases.conflictSerializable_iff_isAcyclic","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#conflict-serializable-iff-is-acyclic","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"A schedule without duplicate operations is conflict-serializable exactly when its precedence graph is acyclic.","title":"The serializability theorem"},{"id":"ComputerScience/databases/conflict-serializability/conflict-serializable-of-rank","kind":"node","lean":"Cslib.Databases.conflictSerializable_of_rank","parent":43284,"result":true,"route":"../roadmap/ComputerScience/databases/conflict-serializability/index.html#conflict-serializable-of-rank","scope":"ComputerScience/databases/conflict-serializability","status":"fully_proved","summary":"Let $s$ be a schedule in which no operation occurs twice. If $r$ is a function from the transactions to the natural numbers with $r(i) < r(j)$ whenever the precedence graph of $s$ has an edge from $i$ to $j$, then $s$ is conflict-serializable.","title":"A rank increasing along precedence edges gives conflict serializability"},{"data_url":"scopes/ComputerScience/databases/functional-dependencies.json","id":"ComputerScience/databases/functional-dependencies","kind":"scope","navigation":"module","parent":43283,"result":false,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","title":"Functional dependencies"},{"id":"ComputerScience/databases/functional-dependencies/fd","kind":"node","lean":"Cslib.Databases.FD","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#fd","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"A functional dependency $X \\to Y$ over a type $A$ of attributes consists of two finite sets of attributes $X, Y \\subseteq A$: the determining attributes $X$ (its left side) and the determined attributes $Y$ (its right side).","title":"Functional dependency"},{"id":"ComputerScience/databases/functional-dependencies/holds-in","kind":"node","lean":"Cslib.Databases.FD.HoldsIn","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#holds-in","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $A$ be a type of attributes and $V$ a type of values. A functional dependency $X \\to Y$ on $A$ holds in a relation $r$, a set of tuples $A \\to V$, when any two tuples $t, u \\in r$ that agree on every attribute of $X$ also agree on every attribute of $Y$.","title":"A functional dependency holds in a relation"},{"id":"ComputerScience/databases/functional-dependencies/satisfies","kind":"node","lean":"Cslib.Databases.Satisfies","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#satisfies","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies over attributes $A$, and let $r$ be a relation, given as a set of tuples $A \\to V$ with values in a type $V$. Then $r$ satisfies $F$ when every dependency of $F$ holds in $r$.","title":"A relation satisfies a set of dependencies"},{"id":"ComputerScience/databases/functional-dependencies/entails","kind":"node","lean":"Cslib.Databases.Entails","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#entails","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $A$ be a type of attributes. A finite set $F$ of functional dependencies on $A$ entails a functional dependency $f$ on $A$ when, for every type $V$ and every relation $r$ with values in $V$ (a set of tuples $A \\to V$) that satisfies $F$, the dependency $f$ holds in $r$.","title":"Entailment of functional dependencies"},{"id":"ComputerScience/databases/functional-dependencies/derives","kind":"node","lean":"Cslib.Databases.Derives","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#derives","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies over a set $A$ of attributes. Derivability from $F$ is the least property of functional dependencies over $A$ such that, for all finite sets $X$, $Y$, $Z$ of attributes: every dependency of $F$ is derivable; $X \\to Y$ is derivable whenever $Y \\subseteq X$ (reflexivity); if $X \\to Y$ is derivable, then so is $X \\cup Z \\to Y \\cup Z$ (augmentation); and if $X \\to Y$\u2026","title":"Derivability with Armstrong's axioms"},{"id":"ComputerScience/databases/functional-dependencies/sound","kind":"node","lean":"Cslib.Databases.Derives.sound","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#sound","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"A derivable dependency holds in every relation satisfying `F`.","title":"Armstrong's axioms are sound"},{"id":"ComputerScience/databases/functional-dependencies/decompose","kind":"node","lean":"Cslib.Databases.Derives.decompose","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#decompose","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies and $X$, $Y$, $Z$ finite sets of attributes with $Z \\subseteq Y$. If $X \\to Y$ is derivable from $F$ with Armstrong's axioms, then so is $X \\to Z$.","title":"Decomposition rule"},{"id":"ComputerScience/databases/functional-dependencies/union","kind":"node","lean":"Cslib.Databases.Derives.union","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#union","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies and $X, Y, Z$ finite sets of attributes. If $X \\to Y$ and $X \\to Z$ are derivable from $F$ with Armstrong's axioms, then so is $X \\to Y \\cup Z$.","title":"Union rule"},{"id":"ComputerScience/databases/functional-dependencies/pseudotrans","kind":"node","lean":"Cslib.Databases.Derives.pseudotrans","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#pseudotrans","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies and $X, Y, W, Z$ finite sets of attributes. If $X \\to Y$ and $Y \\cup W \\to Z$ are derivable from $F$ with Armstrong's axioms, then so is $X \\cup W \\to Z$.","title":"Pseudotransitivity rule"},{"id":"ComputerScience/databases/functional-dependencies/union-left","kind":"node","lean":"Cslib.Databases.Derives.union_left","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#union-left","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies and $X, Y$ finite sets of attributes. If $X \\to Y$ is derivable from $F$ with Armstrong's axioms, then so is $X \\to X \\cup Y$.","title":"Adding the left side to the right side"},{"id":"ComputerScience/databases/functional-dependencies/closure-step","kind":"node","lean":"Cslib.Databases.closureStep","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#closure-step","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"For a finite set $F$ of functional dependencies and a finite set $Y$ of attributes, one closure round returns $Y$ together with the right side of every dependency of $F$ whose left side is contained in $Y$, that is, the set $Y \\cup \\bigcup \\{R : (L \\to R) \\in F,\\ L \\subseteq Y\\}$.","title":"One round of the attribute-closure algorithm"},{"id":"ComputerScience/databases/functional-dependencies/closure-loop","kind":"node","lean":"Cslib.Databases.closureLoop","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#closure-loop","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies, $n$ a natural number and $Y$ a finite set of attributes. The loop returns $Y$ if $n = 0$ or if one closure round for $F$ leaves $Y$ unchanged; otherwise it returns the loop with $n - 1$ rounds started from the result of one closure round applied to $Y$. Thus it repeats closure rounds from $Y$ until nothing changes, for at most $n$ rounds.","title":"Iterated closure rounds"},{"id":"ComputerScience/databases/functional-dependencies/attr-closure","kind":"node","lean":"Cslib.Databases.attrClosure","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#attr-closure","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $A$ be a finite set of attributes, $F$ a finite set of functional dependencies over $A$ and $X \\subseteq A$. The attribute closure $X^+$ of $X$ under $F$ is computed by the closure algorithm: starting from $X$, repeat closure rounds (each adds the right side of every dependency of $F$ whose left side is already included) until nothing changes, for at most $|A|$ rounds.","title":"Attribute closure"},{"id":"ComputerScience/databases/functional-dependencies/closure-step-attr-closure","kind":"node","lean":"Cslib.Databases.closureStep_attrClosure","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#closure-step-attr-closure","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $A$ be a finite set of attributes, $F$ a finite set of functional dependencies over $A$ and $X \\subseteq A$. Applying one closure round for $F$ to the attribute closure $X^+$ of $X$ under $F$ returns $X^+$.","title":"The attribute closure is a fixed point of a closure round"},{"id":"ComputerScience/databases/functional-dependencies/rhs-subset-attr-closure","kind":"node","lean":"Cslib.Databases.rhs_subset_attrClosure","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#rhs-subset-attr-closure","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies over a finite set of attributes, $X$ a set of attributes and $X^+$ its attribute closure under $F$. For every dependency $Y \\to Z$ in $F$ with $Y \\subseteq X^+$, also $Z \\subseteq X^+$.","title":"The attribute closure is closed under the dependencies"},{"id":"ComputerScience/databases/functional-dependencies/closure-witness-relation","kind":"node","lean":"Cslib.Databases.closureWitnessRelation","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#closure-witness-relation","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $A$ be a finite set of attributes, $F$ a finite set of functional dependencies over $A$ and $X \\subseteq A$. The closure witness relation for $X$ is the relation with values in $\\{\\mathrm{true}, \\mathrm{false}\\}$ whose tuples are the tuple that is $\\mathrm{true}$ on every attribute and the tuple that is $\\mathrm{true}$ exactly on the attribute closure $X^+$ of $X$ under $F$. These two tuples agree exactly on\u2026","title":"Closure witness relation"},{"id":"ComputerScience/databases/functional-dependencies/holds-in-closure-witness-relation","kind":"node","lean":"Cslib.Databases.holdsIn_closureWitnessRelation","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#holds-in-closure-witness-relation","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies on a finite set of attributes $A$. For every set $X \\subseteq A$ of attributes, the closure witness relation for $X$ under $F$ satisfies $F$.","title":"The closure witness satisfies the dependencies"},{"id":"ComputerScience/databases/functional-dependencies/derives-iff-subset-attr-closure","kind":"node","lean":"Cslib.Databases.derives_iff_subset_attrClosure","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#derives-iff-subset-attr-closure","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $A$ be a finite set of attributes, $F$ a finite set of functional dependencies over $A$ and $X, Y \\subseteq A$. Then $X \\to Y$ is derivable from $F$ with Armstrong's axioms if and only if $Y \\subseteq X^+$, where $X^+$ is the attribute closure of $X$ under $F$ computed by the closure algorithm.","title":"The closure algorithm decides derivability"},{"id":"ComputerScience/databases/functional-dependencies/mem-attr-closure","kind":"node","lean":"Cslib.Databases.mem_attrClosure","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#mem-attr-closure","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies over a finite set of attributes and $X$ a set of attributes. An attribute $a$ lies in the attribute closure $X^+$ of $X$ under $F$, as computed by the closure algorithm, if and only if the dependency $X \\to \\{a\\}$ is derivable from $F$ with Armstrong's axioms.","title":"The closure algorithm computes the attribute closure"},{"id":"ComputerScience/databases/functional-dependencies/derives-of-forall-two-tuple","kind":"node","lean":"Cslib.Databases.derives_of_forall_two_tuple","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#derives-of-forall-two-tuple","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $A$ be a finite set of attributes, $F$ a finite set of functional dependencies over $A$ and $f$ a functional dependency over $A$. If $f$ holds in every relation with values in $\\{\\mathrm{true}, \\mathrm{false}\\}$ that has at most two tuples and satisfies every dependency of $F$, then $f$ is derivable from $F$ with Armstrong's axioms.","title":"Completeness of Armstrong's axioms, two-tuple form"},{"id":"ComputerScience/databases/functional-dependencies/derives-iff-entails","kind":"node","lean":"Cslib.Databases.derives_iff_entails","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#derives-iff-entails","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $A$ be a finite set of attributes, $F$ a finite set of functional dependencies over $A$ and $f$ a functional dependency over $A$. Then $f$ is derivable from $F$ with Armstrong's axioms if and only if $F$ entails $f$, that is, $f$ holds in every relation (with values in any type) that satisfies every dependency of $F$. In other words, Armstrong's axioms are sound and complete for functional dependencies.","title":"Armstrong's theorem"},{"id":"ComputerScience/databases/functional-dependencies/is-superkey","kind":"node","lean":"Cslib.Databases.IsSuperkey","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#is-superkey","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies over a finite set of attributes $A$. A set of attributes $K$ is a superkey for $F$ when the dependency $K \\to A$ is derivable from $F$ with Armstrong's axioms, that is, $K$ determines every attribute.","title":"Superkey"},{"id":"ComputerScience/databases/functional-dependencies/is-candidate-key","kind":"node","lean":"Cslib.Databases.IsCandidateKey","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#is-candidate-key","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies on a finite set of attributes $A$. A set $K \\subseteq A$ of attributes is a candidate key for $F$ when $K$ is a superkey for $F$ and no proper subset of $K$ is a superkey for $F$.","title":"Candidate key"},{"id":"ComputerScience/databases/functional-dependencies/mono","kind":"node","lean":"Cslib.Databases.IsSuperkey.mono","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#mono","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies over a finite set of attributes. If $K$ is a superkey for $F$ and $K \\subseteq K'$, then $K'$ is a superkey for $F$.","title":"Supersets of superkeys are superkeys"},{"id":"ComputerScience/databases/functional-dependencies/eq-of-agree","kind":"node","lean":"Cslib.Databases.IsSuperkey.eq_of_agree","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#eq-of-agree","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies on a finite set of attributes $A$, and let $K$ be a superkey for $F$. Let $V$ be a type of values. If a relation $r$, a set of tuples $A \\to V$, satisfies $F$, then any two tuples $t, u \\in r$ that agree on every attribute of $K$ are equal.","title":"Tuples that agree on a superkey are equal"},{"id":"ComputerScience/databases/functional-dependencies/is-candidate-key-iff","kind":"node","lean":"Cslib.Databases.isCandidateKey_iff","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#is-candidate-key-iff","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies on a finite set of attributes $A$. A set $K \\subseteq A$ of attributes is a candidate key for $F$ if and only if $K$ is a superkey for $F$ and, for every $a \\in K$, the set $K \\setminus \\{a\\}$ is not a superkey for $F$.","title":"Candidate keys are the superkeys minimal under single removals"},{"id":"ComputerScience/databases/functional-dependencies/shrink-key","kind":"node","lean":"Cslib.Databases.shrinkKey","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#shrink-key","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies over a finite set of attributes. Given a list $l$ of attributes and a set of attributes $K$, go through $l$ in order and, for each attribute $a$, remove $a$ from the current set whenever the attribute closure under $F$ of the current set without $a$ is the set of all attributes (that is, the set without $a$ is still a superkey); the result is the set reached at the\u2026","title":"Shrinking a key along a list"},{"id":"ComputerScience/databases/functional-dependencies/find-key","kind":"node","lean":"Cslib.Databases.findKey","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#find-key","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies on a finite set of attributes $A$, and let $l$ be a list of attributes. The key-finding algorithm starts from the set $K = A$ of all attributes and goes through $l$ in order: for each listed attribute $a$, it replaces $K$ by $K \\setminus \\{a\\}$ when the attribute closure of $K \\setminus \\{a\\}$ under $F$ is all of $A$, and keeps $K$ otherwise. It returns the final\u2026","title":"Key-finding algorithm"},{"id":"ComputerScience/databases/functional-dependencies/find-key-is-candidate-key","kind":"node","lean":"Cslib.Databases.findKey_isCandidateKey","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#find-key-is-candidate-key","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies on a finite set of attributes $A$, and let $l$ be a list in which every attribute of $A$ occurs. Then the output of the key-finding algorithm on $F$ and $l$, which starts from the set of all attributes and goes through $l$ in order, removing each listed attribute whenever what remains is still a superkey, is a candidate key for $F$.","title":"The key-finding algorithm returns a candidate key"},{"id":"ComputerScience/databases/functional-dependencies/exists-is-candidate-key-subset","kind":"node","lean":"Cslib.Databases.exists_isCandidateKey_subset","parent":43301,"result":true,"route":"../roadmap/ComputerScience/databases/functional-dependencies/index.html#exists-is-candidate-key-subset","scope":"ComputerScience/databases/functional-dependencies","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies on a finite set of attributes $A$. Every superkey $K$ for $F$ has a subset $K' \\subseteq K$ that is a candidate key for $F$.","title":"Every superkey contains a candidate key"},{"data_url":"scopes/ComputerScience/databases/relational-algebra.json","id":"ComputerScience/databases/relational-algebra","kind":"scope","navigation":"module","parent":43283,"result":false,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","title":"Relational algebra"},{"id":"ComputerScience/databases/relational-algebra/tuple","kind":"node","lean":"Cslib.Databases.Tuple","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#tuple","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For a type $A$ of attributes and a type $D$ of values, a tuple is a partial assignment of values to attributes: a function that assigns to each attribute $a \\in A$ either a value in $D$ or no value, in which case the tuple is undefined at $a$.","title":"Tuple"},{"id":"ComputerScience/databases/relational-algebra/is-over","kind":"node","lean":"Cslib.Databases.Tuple.IsOver","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#is-over","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"A tuple $t$ is over the schema $S$, a finite set of attributes, when $t$ is defined exactly on the attributes of $S$: for every attribute $a$, the value $t(a)$ is defined if and only if $a \\in S$.","title":"Tuple over a schema"},{"id":"ComputerScience/databases/relational-algebra/compatible","kind":"node","lean":"Cslib.Databases.Tuple.Compatible","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#compatible","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Two tuples $t$ and $u$ are compatible (joinable) when they agree on every attribute where both are defined: for every attribute $a$, either $t$ is undefined at $a$, or $u$ is undefined at $a$, or $t(a) = u(a)$.","title":"Compatible tuples"},{"id":"ComputerScience/databases/relational-algebra/merge","kind":"node","lean":"Cslib.Databases.Tuple.merge","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#merge","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"The merge of tuples $t$ and $u$ is the tuple whose value at each attribute $a$ is $t(a)$ if $t$ is defined at $a$, and $u(a)$ otherwise; it is undefined at $a$ when neither $t$ nor $u$ is.","title":"Merge of two tuples"},{"id":"ComputerScience/databases/relational-algebra/compatible-merge-left","kind":"node","lean":"Cslib.Databases.Tuple.compatible_merge_left","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#compatible-merge-left","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For tuples $t$, $u$, $v$: $t$ and $u$ are compatible and the merge of $t$ and $u$ is compatible with $v$ if and only if $t$, $u$, $v$ are pairwise compatible, that is, $t$ and $u$, $t$ and $v$, and $u$ and $v$ are compatible.","title":"Compatibility with a merge, grouped from the left"},{"id":"ComputerScience/databases/relational-algebra/compatible-merge-right","kind":"node","lean":"Cslib.Databases.Tuple.compatible_merge_right","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#compatible-merge-right","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For tuples $t$, $u$, $v$: $u$ and $v$ are compatible and $t$ is compatible with the merge of $u$ and $v$ if and only if $t$, $u$, $v$ are pairwise compatible, that is, $t$ and $u$, $t$ and $v$, and $u$ and $v$ are compatible.","title":"Compatibility with a merge, grouped from the right"},{"id":"ComputerScience/databases/relational-algebra/restrict","kind":"node","lean":"Cslib.Databases.Tuple.restrict","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#restrict","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For a finite set $T$ of attributes and a tuple $t$, the restriction $t|_T$ is the tuple that agrees with $t$ on every attribute in $T$ and is undefined on every attribute outside $T$.","title":"Restriction of a tuple"},{"id":"ComputerScience/databases/relational-algebra/cond","kind":"node","lean":"Cslib.Databases.Cond","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#cond","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"The selection conditions over attributes $A$ and values $D$ are generated by: the condition that always holds; the equality $a = c$ of an attribute $a$ with a constant $c \\in D$; the equality $a = b$ of two attributes $a$ and $b$; the negation of a condition; and the conjunction of two conditions.","title":"Selection conditions"},{"id":"ComputerScience/databases/relational-algebra/eval","kind":"node","lean":"Cslib.Databases.Cond.eval","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#eval","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"The truth value of a selection condition on a tuple $t$ is defined by recursion: the always-true condition holds; $a = c$ holds when $t$ is defined at $a$ with value $c$; $a = b$ holds when $t(a) = t(b)$ as possibly undefined values, so it also holds when $t$ is undefined at both $a$ and $b$; a negation holds when the negated condition fails; and a conjunction holds when both conditions hold.","title":"Truth value of a selection condition"},{"id":"ComputerScience/databases/relational-algebra/attrs","kind":"node","lean":"Cslib.Databases.Cond.attrs","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#attrs","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"The set of attributes mentioned by a selection condition, defined by recursion: it is empty for the always-true condition; for the comparison $a = d$ of an attribute $a$ with a constant $d$ it is $\\{a\\}$; for the comparison $a = b$ of attributes $a$ and $b$ it is $\\{a, b\\}$; for the negation $\\neg \\varphi$ of a condition $\\varphi$ it is the set of attributes of $\\varphi$; and for the conjunction\u2026","title":"Attributes of a selection condition"},{"id":"ComputerScience/databases/relational-algebra/eval-restrict","kind":"node","lean":"Cslib.Databases.Cond.eval_restrict","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#eval-restrict","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Let $c$ be a selection condition, $t$ a tuple and $T$ a finite set of attributes containing every attribute of $c$. Then $c$ has the same truth value on the restriction of $t$ to $T$ as on $t$.","title":"Selection conditions depend only on their attributes"},{"id":"ComputerScience/databases/relational-algebra/is-over-2","kind":"node","lean":"Cslib.Databases.IsOver","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#is-over-2","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"A finite set $r$ of tuples is a relation over the schema $S$, a finite set of attributes, when every tuple $t \\in r$ is over $S$, that is, $t$ is defined exactly on the attributes of $S$.","title":"Relation over a schema"},{"id":"ComputerScience/databases/relational-algebra/select","kind":"node","lean":"Cslib.Databases.select","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For a selection condition $c$ and a finite set $r$ of tuples, the selection $\\sigma_c(r)$ is the set of tuples $t \\in r$ on which $c$ evaluates to true.","title":"Selection"},{"id":"ComputerScience/databases/relational-algebra/select-select","kind":"node","lean":"Cslib.Databases.select_select","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-select","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Write $\\sigma_c$ for selection by a selection condition $c$. For selection conditions $c$ and $c'$ and a finite set $r$ of tuples, $\\sigma_c(\\sigma_{c'}(r)) = \\sigma_{c' \\land c}(r)$: selecting by $c'$ and then by $c$ is the same as selecting once by the conjunction $c' \\land c$.","title":"Cascade of selections"},{"id":"ComputerScience/databases/relational-algebra/select-comm","kind":"node","lean":"Cslib.Databases.select_comm","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-comm","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For all selection conditions $c$ and $c'$ and every finite set $r$ of tuples, selecting by $c'$ and then by $c$ gives the same result as selecting by $c$ and then by $c'$: $\\sigma_c(\\sigma_{c'}(r)) = \\sigma_{c'}(\\sigma_c(r))$.","title":"Selections commute"},{"id":"ComputerScience/databases/relational-algebra/join","kind":"node","lean":"Cslib.Databases.join","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#join","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"The natural join $r \\bowtie s$ of finite sets of tuples $r$ and $s$ is the set of merges of $t$ and $u$ over all pairs of tuples $t \\in r$ and $u \\in s$ that are compatible, that is, agree on every attribute where both are defined. The merge of $t$ and $u$ takes the value of $t$ where $t$ is defined and the value of $u$ otherwise.","title":"Natural join"},{"id":"ComputerScience/databases/relational-algebra/rename","kind":"node","lean":"Cslib.Databases.rename","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#rename","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For a permutation $f$ of the attributes and a finite set $r$ of tuples, the renaming $\\rho_f(r)$ is the finite set of the tuples $t \\circ f^{-1}$ for $t \\in r$: each tuple keeps its values, and its value at an attribute $a$ moves to the attribute $f(a)$.","title":"Renaming of attributes"},{"id":"ComputerScience/databases/relational-algebra/join-comm","kind":"node","lean":"Cslib.Databases.join_comm","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#join-comm","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For all finite sets $r$ and $s$ of tuples, $r \\bowtie s = s \\bowtie r$, where $\\bowtie$ is the natural join.","title":"Commutativity of the natural join"},{"id":"ComputerScience/databases/relational-algebra/join-assoc","kind":"node","lean":"Cslib.Databases.join_assoc","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#join-assoc","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For all finite sets $r$, $s$ and $w$ of tuples, $(r \\bowtie s) \\bowtie w = r \\bowtie (s \\bowtie w)$, where $\\bowtie$ is the natural join.","title":"Associativity of the natural join"},{"id":"ComputerScience/databases/relational-algebra/join-eq-inter","kind":"node","lean":"Cslib.Databases.join_eq_inter","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#join-eq-inter","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"If $r$ and $s$ are relations over the same schema $S$, that is, finite sets of tuples each over $S$, then their natural join is their intersection: $r \\bowtie s = r \\cap s$.","title":"The natural join on a common schema is the intersection"},{"id":"ComputerScience/databases/relational-algebra/select-union","kind":"node","lean":"Cslib.Databases.select_union","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-union","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Assume the set of attributes is finite. Let $c$ be a selection condition and write $\\sigma_c$ for selection by $c$. For finite sets $r$ and $s$ of tuples, $\\sigma_c(r \\cup s) = \\sigma_c(r) \\cup \\sigma_c(s)$.","title":"Selection distributes over union"},{"id":"ComputerScience/databases/relational-algebra/select-sdiff","kind":"node","lean":"Cslib.Databases.select_sdiff","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-sdiff","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Assume the set of attributes is finite. Let $c$ be a selection condition and write $\\sigma_c$ for selection by $c$. For finite sets $r$ and $s$ of tuples, $\\sigma_c(r \\setminus s) = \\sigma_c(r) \\setminus \\sigma_c(s)$.","title":"Selection distributes over difference"},{"id":"ComputerScience/databases/relational-algebra/project","kind":"node","lean":"Cslib.Databases.project","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#project","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For a finite set $T$ of attributes and a finite set $r$ of tuples, the projection $\\pi_T(r)$ is the finite set of the restrictions to $T$ of the tuples of $r$: $\\pi_T(r) = \\{ t|_T : t \\in r \\}$.","title":"Projection of a relation"},{"id":"ComputerScience/databases/relational-algebra/mem-join-iff","kind":"node","lean":"Cslib.Databases.mem_join_iff","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#mem-join-iff","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Let $r$ be a relation over the schema $S$ and $s$ a relation over the schema $S'$, that is, finite sets of tuples each over $S$, respectively each over $S'$. A tuple $t$ belongs to the natural join $r \\bowtie s$ if and only if $t$ is over $S \\cup S'$, the restriction of $t$ to $S$ belongs to $r$, and the restriction of $t$ to $S'$ belongs to $s$.","title":"Membership in a join"},{"id":"ComputerScience/databases/relational-algebra/select-join-left","kind":"node","lean":"Cslib.Databases.select_join_left","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-join-left","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Let $r$ be a relation over a schema $S$, that is, a finite set of tuples each defined exactly on the attributes of $S$, and let $s$ be any finite set of tuples. If every attribute of the selection condition $c$ lies in $S$, then $\\sigma_c(r \\bowtie s) = \\sigma_c(r) \\bowtie s$, where $\\sigma_c$ is selection by $c$ and $\\bowtie$ is the natural join.","title":"Selection pushdown into the left operand of a join"},{"id":"ComputerScience/databases/relational-algebra/select-join-right","kind":"node","lean":"Cslib.Databases.select_join_right","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-join-right","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Let $s$ be a relation over a schema $S'$, that is, a finite set of tuples each defined exactly on the attributes of $S'$, and let $r$ be any finite set of tuples. If every attribute of the selection condition $c$ lies in $S'$, then $\\sigma_c(r \\bowtie s) = r \\bowtie \\sigma_c(s)$, where $\\sigma_c$ is selection by $c$ and $\\bowtie$ is the natural join.","title":"Selection pushdown into the right operand of a join"},{"id":"ComputerScience/databases/relational-algebra/project-project","kind":"node","lean":"Cslib.Databases.project_project","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#project-project","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Assume the set of attributes is finite. For all finite sets of attributes $T$ and $T'$ and every finite set $r$ of tuples, $\\pi_T(\\pi_{T'}(r)) = \\pi_{T \\cap T'}(r)$, where $\\pi_T$ is projection onto $T$.","title":"Cascade of projections"},{"id":"ComputerScience/databases/relational-algebra/project-project-of-subset","kind":"node","lean":"Cslib.Databases.project_project_of_subset","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#project-project-of-subset","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Assume the set of attributes is finite. For finite sets of attributes $T \\subseteq T'$ and every finite set $r$ of tuples, $\\pi_T(\\pi_{T'}(r)) = \\pi_T(r)$, where $\\pi_T$ is projection onto $T$.","title":"Cascade of projections onto a subset"},{"id":"ComputerScience/databases/relational-algebra/select-project","kind":"node","lean":"Cslib.Databases.select_project","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-project","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Assume the set of attributes is finite. Let $c$ be a selection condition, $T$ a finite set of attributes containing every attribute of $c$, and $r$ a finite set of tuples. Then $\\sigma_c(\\pi_T(r)) = \\pi_T(\\sigma_c(r))$, where $\\sigma_c$ is selection by $c$ and $\\pi_T$ is projection onto $T$.","title":"Selection commutes with projection"},{"id":"ComputerScience/databases/relational-algebra/project-union","kind":"node","lean":"Cslib.Databases.project_union","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#project-union","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"For every finite set $T$ of attributes and all finite sets $r$ and $s$ of tuples, the projection onto $T$ of $r \\cup s$ is the union of the projections onto $T$ of $r$ and of $s$: $\\pi_T(r \\cup s) = \\pi_T(r) \\cup \\pi_T(s)$.","title":"Projection distributes over union"},{"id":"ComputerScience/databases/relational-algebra/expr","kind":"node","lean":"Cslib.Databases.Expr","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#expr","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"The relational algebra queries over base relations named by $N$, with attributes $A$ and values $D$, are generated by: the base relation named $n$ for each $n \\in N$; the selection $\\sigma_c(e)$ of a query $e$ by a selection condition $c$; the projection $\\pi_T(e)$ of a query $e$ on a finite set of attributes $T$; the natural join of two queries; the union of two queries; the difference of two queries; and the\u2026","title":"Relational algebra queries"},{"id":"ComputerScience/databases/relational-algebra/eval-2","kind":"node","lean":"Cslib.Databases.Expr.eval","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#eval-2","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Given a database $\\mathrm{db}$, which assigns a finite set of tuples $\\mathrm{db}(n)$ to each relation name $n$, the value of a relational algebra query on $\\mathrm{db}$ is the finite set of tuples defined by recursion on the query: the base relation $n$ has value $\\mathrm{db}(n)$; for a query $e$ with value $v$, the selection $\\sigma_c(e)$ by a selection condition $c$, the projection $\\pi_T(e)$ onto a finite set\u2026","title":"Value of a query"},{"id":"ComputerScience/databases/relational-algebra/schema","kind":"node","lean":"Cslib.Databases.Expr.schema","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#schema","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Given a database schema assigning a finite set $\\mathrm{sch}(n)$ of attributes to each base relation name $n$, the schema of a relational algebra query is either a finite set of attributes or undefined (the query is ill-formed), defined by recursion on the query: the base relation $n$ has schema $\\mathrm{sch}(n)$; if $e$ has schema $S$, then the selection $\\sigma_c(e)$ has schema $S$ when every attribute of the\u2026","title":"Schema of a query"},{"id":"ComputerScience/databases/relational-algebra/conforms","kind":"node","lean":"Cslib.Databases.Expr.Conforms","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#conforms","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Let $N$ be a set of relation names. A database schema assigns to each name $n \\in N$ a finite set of attributes $\\mathrm{sch}(n)$, and a database assigns to each name $n$ a finite set of tuples $\\mathrm{db}(n)$. The database $\\mathrm{db}$ conforms to the database schema $\\mathrm{sch}$ when, for every name $n$, the set $\\mathrm{db}(n)$ is a relation over $\\mathrm{sch}(n)$.","title":"Databases conforming to a schema"},{"id":"ComputerScience/databases/relational-algebra/eval-is-over","kind":"node","lean":"Cslib.Databases.Expr.eval_isOver","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#eval-is-over","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"A well-formed query evaluates, on every conforming database, to a relation over its inferred schema.","title":"Typing is sound"},{"id":"ComputerScience/databases/relational-algebra/equiv","kind":"node","lean":"Cslib.Databases.Expr.Equiv","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#equiv","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Let $\\mathrm{sch}$ be a database schema, assigning a finite set of attributes to each relation name. Two queries $e$ and $e'$ are equivalent with respect to $\\mathrm{sch}$ when there is a finite set of attributes $S$ such that the inferred schemas of $e$ and of $e'$ with respect to $\\mathrm{sch}$ are both $S$ (in particular both queries are well-formed), and $e$ and $e'$ evaluate to the same relation on every\u2026","title":"Query equivalence"},{"id":"ComputerScience/databases/relational-algebra/select-join","kind":"node","lean":"Cslib.Databases.Expr.select_join","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-join","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Fix a database schema $\\mathrm{sch}$ assigning a finite set of attributes to each base relation name. Let $e$ and $e'$ be well-formed queries with schemas $S$ and $S'$, and let $c$ be a selection condition all of whose attributes lie in $S$. Then the query $\\sigma_c(e \\bowtie e')$, the selection by $c$ of the natural join of $e$ and $e'$, is equivalent to the query $\\sigma_c(e) \\bowtie e'$: both are well-formed\u2026","title":"Selection pushdown as a query rewrite"},{"id":"ComputerScience/databases/relational-algebra/select-select-2","kind":"node","lean":"Cslib.Databases.Expr.select_select","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#select-select-2","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Fix a database schema $\\mathrm{sch}$ assigning a finite set of attributes to each base relation name. Let $e$ be a well-formed query with schema $S$, and let $c$ and $c'$ be selection conditions all of whose attributes lie in $S$. Then the query $\\sigma_c(\\sigma_{c'}(e))$, the selection by $c$ of the selection of $e$ by $c'$, is equivalent to the query $\\sigma_{c' \\land c}(e)$, the selection of $e$ by the\u2026","title":"Cascade of selections as a query rewrite"},{"id":"ComputerScience/databases/relational-algebra/project-project-2","kind":"node","lean":"Cslib.Databases.Expr.project_project","parent":43332,"result":true,"route":"../roadmap/ComputerScience/databases/relational-algebra/index.html#project-project-2","scope":"ComputerScience/databases/relational-algebra","status":"fully_proved","summary":"Fix a database schema $\\mathrm{sch}$ assigning a finite set of attributes to each base relation name. Let $e$ be a well-formed query with schema $S$, and let $T \\subseteq T' \\subseteq S$ be finite sets of attributes. Then the query $\\pi_T(\\pi_{T'}(e))$, the projection onto $T$ of the projection of $e$ onto $T'$, is equivalent to the query $\\pi_T(e)$, the projection of $e$ onto $T$: both are well-formed with the\u2026","title":"Cascade of projections as a query rewrite"},{"data_url":"scopes/ComputerScience/databases/lossless-decomposition-and-normal-forms.json","id":"ComputerScience/databases/lossless-decomposition-and-normal-forms","kind":"scope","navigation":"module","parent":43283,"result":false,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","title":"Lossless decomposition and normal forms"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/join-project-eq-of-holds-in","kind":"node","lean":"Cslib.Databases.join_project_eq_of_holdsIn","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#join-project-eq-of-holds-in","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"If the dependency `R\u2081 \u2229 R\u2082 \u2192 R\u2081` holds in a relation `r` over `R\u2081 \u222a R\u2082`, then `r` is the natural join of its projections onto `R\u2081` and `R\u2082`.","title":"Heath's theorem"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/satisfies-on","kind":"node","lean":"Cslib.Databases.SatisfiesOn","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#satisfies-on","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"A relation $r$ satisfies a finite set $F$ of functional dependencies on a set $S$ of attributes when every dependency $X \\to Y$ with $X, Y \\subseteq S$ that is derivable from $F$ by Armstrong's axioms holds in $r$.","title":"Satisfying a set of dependencies on a scheme"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/lossless-join","kind":"node","lean":"Cslib.Databases.LosslessJoin","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#lossless-join","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies, let $R_1$ and $R_2$ be finite sets of attributes, and let $D$ be a type of values. The decomposition of $R_1 \\cup R_2$ into $R_1$ and $R_2$ is lossless for $F$ with values in $D$ when every relation $r$ over $R_1 \\cup R_2$ with values in $D$ that satisfies $F$ on $R_1 \\cup R_2$ (every dependency $X \\to Y$ with $X, Y \\subseteq R_1 \\cup R_2$ that is derivable from\u2026","title":"Lossless decomposition into two schemes"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/lossless-join-of-derives","kind":"node","lean":"Cslib.Databases.losslessJoin_of_derives","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#lossless-join-of-derives","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies and let $R_1$, $R_2$ be sets of attributes. If $R_1 \\cap R_2 \\to R_1$ or $R_1 \\cap R_2 \\to R_2$ is derivable from $F$ by Armstrong's axioms, then for every value type $D$ the decomposition of $R_1 \\cup R_2$ into $R_1$ and $R_2$ is lossless for $F$ with values in $D$.","title":"Lossless binary decompositions from derivable dependencies"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/derives-of-lossless-join","kind":"node","lean":"Cslib.Databases.derives_of_losslessJoin","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#derives-of-lossless-join","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a set of functional dependencies, let $R_1$ and $R_2$ be schemes whose union is the set of all attributes, and let $D$ be a value type containing at least two distinct elements. If the decomposition of $R_1 \\cup R_2$ into $R_1$ and $R_2$ is lossless for $F$ with values in $D$, then $F$ derives $R_1 \\cap R_2 \\to R_1$ or $R_1 \\cap R_2 \\to R_2$.","title":"Converse of the lossless-join test for two schemes"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/lossless-join-iff","kind":"node","lean":"Cslib.Databases.losslessJoin_iff","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#lossless-join-iff","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies, let $R_1$ and $R_2$ be sets of attributes whose union is the set of all attributes, and suppose that the value type $D$ has two distinct elements. Then the decomposition into $R_1$ and $R_2$ is lossless for $F$ with values in $D$ if and only if $R_1 \\cap R_2 \\to R_1$ or $R_1 \\cap R_2 \\to R_2$ is derivable from $F$ by Armstrong's axioms.","title":"Lossless binary decompositions"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/join-unit","kind":"node","lean":"Cslib.Databases.joinUnit","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#join-unit","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"The relation containing only the empty tuple, the unit of the natural join.","title":"Unit relation of the natural join"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/join-projections","kind":"node","lean":"Cslib.Databases.joinProjections","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#join-projections","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"For a relation $r$ (a finite set of tuples) and a list of schemes $S_1, \\dots, S_n$, this is the natural join of the projections $\\pi_{S_1} r, \\dots, \\pi_{S_n} r$, computed as $\\pi_{S_1} r \\bowtie (\\pi_{S_2} r \\bowtie (\\cdots \\bowtie (\\pi_{S_n} r \\bowtie U)))$, where $U$ is the relation containing only the empty tuple; for the empty list it is $U$.","title":"Join of the projections of a relation"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/schema-union","kind":"node","lean":"Cslib.Databases.schemaUnion","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#schema-union","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"For a list $S_1, \\dots, S_n$ of schemes (finite sets of attributes), this is the union $S_1 \\cup \\dots \\cup S_n$; for the empty list it is the empty set.","title":"Union of a list of schemes"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/mem-join-projections","kind":"node","lean":"Cslib.Databases.mem_joinProjections","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#mem-join-projections","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $r$ be a relation over a schema $R$ and let $S_1, \\dots, S_n$ be a list of schemes, each contained in $R$. A tuple $t$ belongs to the natural join of the projections $\\pi_{S_1} r, \\dots, \\pi_{S_n} r$ if and only if $t$ is a tuple over $S_1 \\cup \\dots \\cup S_n$ and, for every $i$, the restriction of $t$ to $S_i$ belongs to $\\pi_{S_i} r$.","title":"Membership in a join of projections"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/is-bcnf","kind":"node","lean":"Cslib.Databases.IsBCNF","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#is-bcnf","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"A scheme $S$ is in Boyce\u2013Codd normal form for a set $F$ of functional dependencies when, for every dependency $X \\to Y$ derivable from $F$ with $X, Y \\subseteq S$, either $Y \\subseteq X$ or $F$ derives $X \\to S$ (that is, $X$ is a superkey of $S$).","title":"Boyce\u2013Codd normal form"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/is-bcnfviolation","kind":"node","lean":"Cslib.Databases.IsBCNFViolation","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#is-bcnfviolation","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a set of functional dependencies and $S$ a scheme, and write $X^+$ for the attribute closure of a set $X$ of attributes under $F$. A set $X \\subseteq S$ violates BCNF in $S$ when $S \\not\\subseteq X^+$ and $X^+ \\cap S \\not\\subseteq X$, that is, when the closure $X^+ \\cap S$ of $X$ within $S$ is neither $X$ nor $S$.","title":"BCNF violations"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/bcnf-candidates","kind":"node","lean":"Cslib.Databases.bcnfCandidates","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#bcnf-candidates","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Given a list $l$ of attributes and a set $S$ of attributes, the candidate violations of BCNF in $S$ are all subsets of the set of attributes that occur in $l$ and lie in $S$, listed once each. Duplicates in $l$ are removed first, so there are $2^k$ candidates when $k$ distinct attributes of $l$ lie in $S$.","title":"Candidate BCNF violations"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/bcnf-decompose","kind":"node","lean":"Cslib.Databases.bcnfDecompose","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#bcnf-decompose","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a set of functional dependencies, $l$ a list of attributes, $n$ a natural number and $S$ a scheme. For $n = 0$ the result is the list $[S]$. For $n > 0$, the candidate violations of BCNF in $S$ built from $l$ are searched in order for the first $X$ that violates BCNF in $S$ for $F$. If there is none, the result is $[S]$. Otherwise, with $X^+$ the attribute closure of $X$ under $F$, the result is the\u2026","title":"BCNF decomposition with bounded depth"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/bcnf-decomposition","kind":"node","lean":"Cslib.Databases.bcnfDecomposition","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#bcnf-decomposition","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a set of functional dependencies, $l$ a list of attributes and $S$ a scheme. The BCNF decomposition of $S$ for $F$ and $l$ is the BCNF decomposition with bounded depth run on $S$ with depth bound $|S|$. At each step, if some candidate violation of BCNF in the current scheme $T$ built from $l$ violates BCNF in $T$ for $F$, the first such $X$ is used: with $X^+$ the attribute closure of $X$ under $F$, $T$\u2026","title":"BCNF decomposition"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/bcnf-decomposition-lossless","kind":"node","lean":"Cslib.Databases.bcnfDecomposition_lossless","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#bcnf-decomposition-lossless","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a set of functional dependencies, $l$ a list of attributes and $S$ a set of attributes, and let $r$ be a relation over $S$ that satisfies every dependency $X \\to Y$ derivable from $F$ with $X, Y \\subseteq S$. Then $r$ is the natural join of its projections onto the schemes of the BCNF decomposition of $S$ for $F$ and $l$.","title":"The BCNF decomposition is lossless"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/is-bcnf-of-mem-bcnf-decomposition","kind":"node","lean":"Cslib.Databases.isBCNF_of_mem_bcnfDecomposition","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#is-bcnf-of-mem-bcnf-decomposition","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a set of functional dependencies and $l$ a list containing every attribute. Then for every scheme $S$, every scheme in the BCNF decomposition of $S$ for $F$ and $l$ is in Boyce\u2013Codd normal form for $F$.","title":"The BCNF decomposition produces BCNF schemes"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/equivalent","kind":"node","lean":"Cslib.Databases.Equivalent","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#equivalent","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Two sets $F$ and $G$ of functional dependencies are equivalent when $F$ derives every dependency of $G$ and $G$ derives every dependency of $F$.","title":"Equivalent sets of functional dependencies"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/project-fds","kind":"node","lean":"Cslib.Databases.projectFDs","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#project-fds","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"For a finite set $F$ of functional dependencies and a set $S$ of attributes, this is the finite set of all dependencies $X \\to Y$ with $X, Y \\subseteq S$ that are derivable from $F$ by Armstrong's axioms, that is, the projection onto $S$ of the closure $F^+$ of $F$.","title":"Projection of the dependency closure onto a scheme"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/dependency-preserving","kind":"node","lean":"Cslib.Databases.DependencyPreserving","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#dependency-preserving","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"A list $l$ of schemes is dependency-preserving for a set $F$ of functional dependencies when every dependency of $F$ is derivable from the union, over the schemes $S$ of $l$, of the sets of dependencies $X \\to Y$ derivable from $F$ with $X, Y \\subseteq S$.","title":"Dependency-preserving decompositions"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/synthesize","kind":"node","lean":"Cslib.Databases.synthesize","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#synthesize","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Given a list $G$ of functional dependencies and a set of attributes $K$, the synthesized decomposition is the list of schemes consisting of $K$ followed by, for each dependency $X \\to Y$ of $G$ in order, the scheme $X \\cup Y$.","title":"Synthesis"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/synthesize-dependency-preserving","kind":"node","lean":"Cslib.Databases.synthesize_dependencyPreserving","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#synthesize-dependency-preserving","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"Let $F$ be a finite set of functional dependencies and $G$ a list of functional dependencies equivalent to $F$ (each derives every dependency of the other). Then for every set of attributes $K$, the decomposition synthesized from $G$ and $K$, with one scheme $X \\cup Y$ for each dependency $X \\to Y$ of $G$ together with the scheme $K$, is dependency-preserving for $F$.","title":"Synthesis is dependency-preserving"},{"id":"ComputerScience/databases/lossless-decomposition-and-normal-forms/synthesize-lossless","kind":"node","lean":"Cslib.Databases.synthesize_lossless","parent":43372,"result":true,"route":"../roadmap/ComputerScience/databases/lossless-decomposition-and-normal-forms/index.html#synthesize-lossless","scope":"ComputerScience/databases/lossless-decomposition-and-normal-forms","status":"fully_proved","summary":"For a cover `G` of `F` and a superkey `K` of all attributes, every relation over all attributes satisfying `F` is the join of its projections onto the synthesized schemes.","title":"Synthesis is lossless"},{"data_url":"scopes/ComputerScience/databases/two-phase-locking.json","id":"ComputerScience/databases/two-phase-locking","kind":"scope","navigation":"module","parent":43283,"result":false,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","title":"Two-phase locking"},{"id":"ComputerScience/databases/two-phase-locking/lock-mode","kind":"node","lean":"Cslib.Databases.LockMode","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#lock-mode","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A lock is held in one of two modes: shared (a read lock) or exclusive (a write lock).","title":"Lock mode"},{"id":"ComputerScience/databases/two-phase-locking/covers","kind":"node","lean":"Cslib.Databases.LockMode.Covers","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#covers","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A lock mode $m'$ covers a lock mode $m$ when a lock in mode $m'$ permits what a lock in mode $m$ permits: an exclusive lock covers both modes, and a shared lock covers only the shared mode.","title":"Lock mode coverage"},{"id":"ComputerScience/databases/two-phase-locking/mode","kind":"node","lean":"Cslib.Databases.Access.mode","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#mode","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"The lock mode an access requires: shared for a read, exclusive for a write.","title":"Lock mode required by an access"},{"id":"ComputerScience/databases/two-phase-locking/action","kind":"node","lean":"Cslib.Databases.Action","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#action","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"For $n$ transactions, numbered $0, \\dots, n-1$, and a type $X$ of data items, an action of a locked schedule is one of three kinds: a read or write operation; a lock request, in which a transaction $t$ acquires a lock on an item $x \\in X$ in a lock mode $m$ (shared or exclusive); or a release, in which a transaction $t$ releases its lock on an item $x \\in X$.","title":"Action of a locked schedule"},{"id":"ComputerScience/databases/two-phase-locking/ops","kind":"node","lean":"Cslib.Databases.ops","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#ops","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"For a locked schedule $s$, the schedule of reads and writes of $s$: its reads and writes in order, with the lock requests and releases erased.","title":"Reads and writes of a locked schedule"},{"id":"ComputerScience/databases/two-phase-locking/holds-before","kind":"node","lean":"Cslib.Databases.HoldsBefore","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#holds-before","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"Let $s$ be a locked schedule, $t$ a transaction, $x$ a data item, $m$ a lock mode and $p$ a natural number. Transaction $t$ holds a lock on $x$ covering mode $m$ just before position $p$ of $s$ when there is a position $q < p$ at which $s$ has a lock request by $t$ on $x$ in some mode $m'$ that covers $m$, and for no position $r$ with $q < r < p$ is the action of $s$ at $r$ a release of $x$ by $t$.","title":"Holding a lock"},{"id":"ComputerScience/databases/two-phase-locking/locks-cover","kind":"node","lean":"Cslib.Databases.LocksCover","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#locks-cover","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A locked schedule $s$ is covered by locks when every read or write is protected by a lock of its transaction: whenever $s$ has at position $p$ a read or write $o$ by transaction $t$ on item $x$, just before position $p$ transaction $t$ holds a lock on $x$ (acquired at an earlier position and not released since) whose mode covers the mode $o$ requires, namely shared for a read and exclusive for a write; an exclusive\u2026","title":"Schedules covered by locks"},{"id":"ComputerScience/databases/two-phase-locking/releases-locks","kind":"node","lean":"Cslib.Databases.ReleasesLocks","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#releases-locks","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A locked schedule $s$ releases its locks when every lock request is followed by a matching release: whenever $s$ has at position $q$ a lock request (in either mode) by transaction $t$ on item $x$, there is a position $r > q$ at which $s$ has a release by $t$ of $x$.","title":"Schedules that release every lock"},{"id":"ComputerScience/databases/two-phase-locking/well-formed","kind":"node","lean":"Cslib.Databases.WellFormed","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#well-formed","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A locked schedule $s$ (a list of reads, writes, lock requests and lock releases of transactions) is well formed when its accesses are covered by locks and it releases its locks: every read or write in $s$ is preceded by a lock request of its own transaction on its item, in a mode covering the access (shared or exclusive for a read, exclusive for a write), with no release of that item by that transaction in between;\u2026","title":"Well-formed locked schedules"},{"id":"ComputerScience/databases/two-phase-locking/legal-at","kind":"node","lean":"Cslib.Databases.LegalAt","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#legal-at","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A locked schedule $s$ is legal just before position $p$ when, for every data item $x$, all distinct transactions $t \\ne t'$ and all lock modes $m$, $m'$: if $t$ holds a lock on $x$ covering $m$ and $t'$ holds a lock on $x$ covering $m'$ just before position $p$ of $s$, then $m$ and $m'$ are both the shared mode. In words, two transactions do not both hold locks on the same item unless both locks are shared.","title":"Lock compatibility at a point"},{"id":"ComputerScience/databases/two-phase-locking/locked-items","kind":"node","lean":"Cslib.Databases.lockedItems","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#locked-items","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"The list of data items locked in a locked schedule $s$: for each lock request of $s$, in order, the item it locks, so an item appears once for each lock request on it.","title":"Locked items"},{"id":"ComputerScience/databases/two-phase-locking/legal","kind":"node","lean":"Cslib.Databases.Legal","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#legal","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A locked schedule $s$ of length $\\ell$ is legal when it satisfies lock compatibility at every point $p \\le \\ell$: just before position $p$, no two distinct transactions both hold locks on the same item unless both locks are shared.","title":"Legal locked schedule"},{"id":"ComputerScience/databases/two-phase-locking/two-phase","kind":"node","lean":"Cslib.Databases.TwoPhase","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#two-phase","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A locked schedule $s$ is two-phase when every lock request of a transaction precedes every release of that transaction: whenever $s$ has at position $p$ a release by transaction $t$ and at position $q$ a lock request by $t$ (on any items, in any mode), then $q < p$.","title":"Two-phase schedules"},{"id":"ComputerScience/databases/two-phase-locking/strict-two-phase","kind":"node","lean":"Cslib.Databases.StrictTwoPhase","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#strict-two-phase","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"A locked schedule $s$ is strict two-phase when it is two-phase and every transaction releases its exclusive locks only after its last read or write: whenever $s$ has at position $p$ a release by transaction $t$ of item $x$ and just before position $p$ transaction $t$ holds an exclusive lock on $x$, every read or write of $t$ in $s$ occurs at a position before $p$.","title":"Strict two-phase schedules"},{"id":"ComputerScience/databases/two-phase-locking/locks-cover-at","kind":"node","lean":"Cslib.Databases.LocksCoverAt","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#locks-cover-at","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"For a locked schedule $s$ and a position $p$: if $s$ has at position $p$ a read or write $o$ by transaction $t$ on item $x$, then just before position $p$ transaction $t$ holds a lock on $x$ whose mode covers the mode $o$ requires (shared for a read, exclusive for a write; an exclusive lock covers both modes). If position $p$ holds a lock request or a release, or lies beyond the end of $s$, the condition holds\u2026","title":"Lock coverage at a position"},{"id":"ComputerScience/databases/two-phase-locking/released-at","kind":"node","lean":"Cslib.Databases.ReleasedAt","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#released-at","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"For a locked schedule $s$ and a position $q$: if $s$ has at position $q$ a lock request (in either mode) by transaction $t$ on item $x$, then there is a position $r$ with $q < r$ and $r$ less than the length of $s$ at which $s$ has a release by $t$ of $x$. If position $q$ holds a read, a write or a release, or lies beyond the end of $s$, the condition holds trivially.","title":"Release of the lock requested at a position"},{"id":"ComputerScience/databases/two-phase-locking/two-phase-at","kind":"node","lean":"Cslib.Databases.TwoPhaseAt","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#two-phase-at","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"For a locked schedule $s$ and positions $p$ and $q$: if $s$ has at position $p$ a release by transaction $t$ and at position $q$ a lock request by the same transaction $t$, then $q < p$. For every other pair of positions, including positions beyond the end of $s$, the condition holds trivially.","title":"Two-phase condition at a pair of positions"},{"id":"ComputerScience/databases/two-phase-locking/strict-at","kind":"node","lean":"Cslib.Databases.StrictAt","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#strict-at","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"For a locked schedule $s$ and positions $p$ and $q$: if $s$ has at position $p$ a release by transaction $t$ of item $x$, just before position $p$ transaction $t$ holds an exclusive lock on $x$, and $s$ has at position $q$ a read or write by $t$, then $q < p$. For every other pair of positions, including positions beyond the end of $s$, the condition holds trivially.","title":"Strictness at a pair of positions"},{"id":"ComputerScience/databases/two-phase-locking/is-unlock-by","kind":"node","lean":"Cslib.Databases.Action.isUnlockBy","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#is-unlock-by","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"For a transaction $t$, this Boolean test on the actions of a locked schedule returns true exactly when the action is a release by $t$ of its lock on some item.","title":"Release test"},{"id":"ComputerScience/databases/two-phase-locking/conflict-serializable","kind":"node","lean":"Cslib.Databases.TwoPhase.conflictSerializable","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#conflict-serializable","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"If the reads and writes of a locked schedule are distinct, are covered by locks, and the schedule is legal and two-phase, then its reads and writes form a conflict-serializable schedule.","title":"The two-phase locking theorem"},{"id":"ComputerScience/databases/two-phase-locking/conflict-serializable-2","kind":"node","lean":"Cslib.Databases.StrictTwoPhase.conflictSerializable","parent":43396,"result":true,"route":"../roadmap/ComputerScience/databases/two-phase-locking/index.html#conflict-serializable-2","scope":"ComputerScience/databases/two-phase-locking","status":"fully_proved","summary":"If the reads and writes of a locked schedule are distinct, are covered by locks, and the schedule is legal and strict two-phase, then its reads and writes form a conflict-serializable schedule.","title":"Strict two-phase locking theorem"},{"data_url":"scopes/ComputerScience/quantum-algorithms.json","id":"ComputerScience/quantum-algorithms","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/quantum-algorithms/index.html","scope":"ComputerScience/quantum-algorithms","status":"fully_proved","summary":"Algorithms for quantum computers and their analysis: Grover search and the classical order-to-factor core of Shor's algorithm.","title":"Quantum algorithms"},{"data_url":"scopes/ComputerScience/quantum-algorithms/grover-s-algorithm.json","id":"ComputerScience/quantum-algorithms/grover-s-algorithm","kind":"scope","navigation":"module","parent":43418,"result":false,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","title":"Grover's algorithm"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/amplitude","kind":"node","lean":"Cslib.Grover.Amplitude","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#amplitude","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For a type $\\alpha$ (the search space), a real amplitude vector indexed by $\\alpha$ is a function $\\psi : \\alpha \\to \\mathbb{R}$, assigning a real amplitude to each point of $\\alpha$.","title":"Real amplitude vectors"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/phase-oracle","kind":"node","lean":"Cslib.Grover.phaseOracle","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#phase-oracle","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For a type $\\alpha$ and a finite set $M \\subseteq \\alpha$ of marked elements, the phase oracle sends a real amplitude vector $\\psi$ on $\\alpha$ to the amplitude vector whose value at $x$ is $-\\psi(x)$ if $x \\in M$ and $\\psi(x)$ if $x \\notin M$; that is, it negates exactly the amplitudes at the marked elements.","title":"Phase oracle"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/amplitude-norm-sq","kind":"node","lean":"Cslib.Grover.amplitudeNormSq","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#amplitude-norm-sq","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For a finite type $\\alpha$ and a real amplitude vector $\\psi : \\alpha \\to \\mathbb{R}$, the squared norm of $\\psi$ is its squared Euclidean norm $\\sum_{x \\in \\alpha} \\psi(x)^2$.","title":"Squared norm of an amplitude vector"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/marked-probability","kind":"node","lean":"Cslib.Grover.markedProbability","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#marked-probability","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For a real amplitude vector $\\psi$ on a type $\\alpha$ and a finite set $M \\subseteq \\alpha$ of marked elements, the marked probability is $\\sum_{x \\in M} \\psi(x)^2$. When $\\alpha$ is finite and $\\psi$ has squared norm $1$, this is the probability of observing a marked outcome.","title":"Probability of the marked outcomes"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/uniform-amplitude","kind":"node","lean":"Cslib.Grover.uniformAmplitude","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#uniform-amplitude","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For a finite type $\\alpha$, the uniform amplitude vector on $\\alpha$ is the real amplitude vector whose value at every point is $1/\\sqrt{|\\alpha|}$, where $|\\alpha|$ is the number of elements of $\\alpha$.","title":"Uniform amplitude vector"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/amplitude-norm-sq-uniform-amplitude","kind":"node","lean":"Cslib.Grover.amplitudeNormSq_uniformAmplitude","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#amplitude-norm-sq-uniform-amplitude","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For every nonempty finite type $\\alpha$, the uniform amplitude vector on $\\alpha$ has squared Euclidean norm $1$.","title":"The uniform amplitude vector is normalized"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/mean-amplitude","kind":"node","lean":"Cslib.Grover.meanAmplitude","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#mean-amplitude","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For a finite type $\\alpha$ and a real amplitude vector $\\psi$ on $\\alpha$, the mean amplitude of $\\psi$ is $\\bigl(\\sum_{x \\in \\alpha} \\psi(x)\\bigr) / |\\alpha|$, the arithmetic mean of its amplitudes. When $\\alpha$ is empty it is $0$.","title":"Mean amplitude"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/diffusion","kind":"node","lean":"Cslib.Grover.diffusion","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#diffusion","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For a finite type $\\alpha$, the diffusion operator sends a real amplitude vector $\\psi$ on $\\alpha$ to the amplitude vector $x \\mapsto 2\\bar\\psi - \\psi(x)$, where $\\bar\\psi$ is the arithmetic mean of the amplitudes of $\\psi$.","title":"Inversion about the mean"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/grover-step","kind":"node","lean":"Cslib.Grover.groverStep","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#grover-step","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite type and $M \\subseteq \\alpha$ a set of marked elements. Write $O_M$ for the phase oracle, which negates the amplitudes at the points of $M$, and $D$ for inversion about the mean, which sends an amplitude vector $\\varphi$ on $\\alpha$ to $x \\mapsto 2\\bar\\varphi - \\varphi(x)$, where $\\bar\\varphi$ is the arithmetic mean of the amplitudes of $\\varphi$. One Grover step sends a real amplitude\u2026","title":"Grover step"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/symmetric-amplitude","kind":"node","lean":"Cslib.Grover.symmetricAmplitude","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#symmetric-amplitude","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $M$ be a set of marked elements of a search space $\\alpha$, and let $a, b \\in \\mathbb{R}$. The symmetric amplitude vector with marked amplitude $a$ and unmarked amplitude $b$ is the real amplitude vector on $\\alpha$ whose value is $a$ at every element of $M$ and $b$ at every element outside $M$.","title":"Symmetric amplitude vector"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/oracle-mean","kind":"node","lean":"Cslib.Grover.oracleMean","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#oracle-mean","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $M$ be a set of marked elements of a finite search space of size $N$, let $M^c$ be the set of unmarked elements, and let $a, b \\in \\mathbb{R}$. The oracle mean is $(|M^c|\\, b - |M|\\, a) / N$.","title":"Mean amplitude after the phase oracle"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/marked-amplitude-next","kind":"node","lean":"Cslib.Grover.markedAmplitudeNext","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#marked-amplitude-next","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $M$ be a set of marked elements of a finite search space of size $N$, let $M^c$ be the set of unmarked elements, and let $a, b \\in \\mathbb{R}$. With $\\mu = (|M^c|\\, b - |M|\\, a) / N$, the next marked amplitude is $2\\mu + a$.","title":"Marked amplitude after a Grover step"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/unmarked-amplitude-next","kind":"node","lean":"Cslib.Grover.unmarkedAmplitudeNext","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#unmarked-amplitude-next","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $M$ be a set of marked elements of a finite search space of size $N$, let $M^c$ be the set of unmarked elements, and let $a, b \\in \\mathbb{R}$. With $\\mu = (|M^c|\\, b - |M|\\, a) / N$, the next unmarked amplitude is $2\\mu - b$.","title":"Unmarked amplitude after a Grover step"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/grover-angle","kind":"node","lean":"Cslib.Grover.groverAngle","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#grover-angle","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"For a finite search space $\\alpha$ and a marked set $M \\subseteq \\alpha$, the Grover angle of $M$ is $\\theta = \\arcsin \\sqrt{|M| / |\\alpha|}$, where the quotient $|M| / |\\alpha|$ is taken to be $0$ when $\\alpha$ is empty.","title":"Grover angle"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/angle-amplitude","kind":"node","lean":"Cslib.Grover.angleAmplitude","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#angle-amplitude","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite search space and $M \\subseteq \\alpha$ a marked set with complement $M^c$. For a real number $\\varphi$, this is the real amplitude vector on $\\alpha$ whose value is $\\sin\\varphi / \\sqrt{|M|}$ at every point of $M$ and $\\cos\\varphi / \\sqrt{|M^c|}$ at every point of $M^c$.","title":"Symmetric Grover state at a given angle"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/grover-iterate","kind":"node","lean":"Cslib.Grover.groverIterate","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#grover-iterate","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite search space and $M \\subseteq \\alpha$ a marked set. For $k \\in \\mathbb{N}$ and a real amplitude vector $\\psi : \\alpha \\to \\mathbb{R}$, the $k$-th Grover iterate of $\\psi$ is the vector obtained by applying the Grover step for $M$ to $\\psi$ $k$ times; the $0$-th iterate is $\\psi$ itself.","title":"Grover iterate"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/amplitude-norm-sq-grover-iterate","kind":"node","lean":"Cslib.Grover.amplitudeNormSq_groverIterate","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#amplitude-norm-sq-grover-iterate","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite search space and $M \\subseteq \\alpha$ a marked set. For every real amplitude vector $\\psi : \\alpha \\to \\mathbb{R}$ and every $k \\in \\mathbb{N}$, let $\\psi_k$ be the vector obtained from $\\psi$ by applying the Grover step for $M$ $k$ times. Then $\\sum_{x \\in \\alpha} \\psi_k(x)^2 = \\sum_{x \\in \\alpha} \\psi(x)^2$.","title":"Grover iteration preserves the squared norm"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/amplitude-norm-sq-grover-iterate-uniform","kind":"node","lean":"Cslib.Grover.amplitudeNormSq_groverIterate_uniform","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#amplitude-norm-sq-grover-iterate-uniform","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite search space, $M \\subseteq \\alpha$ a marked set, and $u$ the uniform amplitude vector, given by $u(x) = 1/\\sqrt{|\\alpha|}$ for every $x \\in \\alpha$. For every $k \\in \\mathbb{N}$, let $\\psi_k$ be the vector obtained from $u$ by applying the Grover step for $M$ $k$ times. Then $\\sum_{x \\in \\alpha} \\psi_k(x)^2 = 1$.","title":"Grover iterates of the uniform state are unit vectors"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/grover-phase","kind":"node","lean":"Cslib.Grover.groverPhase","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#grover-phase","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite search space, $M \\subseteq \\alpha$ a marked set and $\\theta$ the Grover angle of $M$. For $k \\in \\mathbb{N}$, the Grover phase of $M$ after $k$ iterations is the real number $(2k + 1)\\theta$.","title":"Grover phase"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/marked-probability-grover-iterate-unifor","kind":"node","lean":"Cslib.Grover.markedProbability_groverIterate_uniform","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#marked-probability-grover-iterate-unifor","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $\\alpha$ be a finite search space and $M \\subseteq \\alpha$ a marked set such that both $M$ and its complement are nonempty, and let $\\theta = \\arcsin \\sqrt{|M| / |\\alpha|}$ be the Grover angle of $M$. For $k \\in \\mathbb{N}$, let $\\psi_k$ be the vector obtained by applying the Grover step for $M$ $k$ times to the uniform amplitude vector, whose value is $1/\\sqrt{|\\alpha|}$ at every point. Then the marked\u2026","title":"Exact success probability of Grover search"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/floor-iterations","kind":"node","lean":"Cslib.Grover.floorIterations","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#floor-iterations","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $M$ be a set of marked elements of a finite search space of size $N$, and let $\\theta = \\arcsin \\sqrt{|M| / N}$ be its Grover angle. The standard iteration count is $\\lfloor \\pi / (4\\theta) \\rfloor$, the largest natural number at most $\\pi / (4\\theta)$; it is $0$ when $\\theta = 0$, that is, when no element is marked.","title":"Standard Grover iteration count"},{"id":"ComputerScience/quantum-algorithms/grover-s-algorithm/one-step-success-of-four-mul-card-eq","kind":"node","lean":"Cslib.Grover.one_step_success_of_four_mul_card_eq","parent":43419,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/grover-s-algorithm/index.html#one-step-success-of-four-mul-card-eq","scope":"ComputerScience/quantum-algorithms/grover-s-algorithm","status":"fully_proved","summary":"Let $M$ be a nonempty set of marked elements of a finite search space of size $N$ with $4|M| = N$. Starting from the uniform amplitude vector, one Grover iteration produces an amplitude vector whose marked probability is $1$.","title":"One Grover iteration suffices when a quarter is marked"},{"data_url":"scopes/ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding.json","id":"ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding","kind":"scope","navigation":"module","parent":43418,"result":false,"route":"../roadmap/ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding/index.html","scope":"ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding","status":"fully_proved","title":"Shor's reduction of factoring to order finding"},{"id":"ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding/gcd-factor-of-order","kind":"node","lean":"Cslib.Shor.gcd_factor_of_order","parent":43442,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding/index.html#gcd-factor-of-order","scope":"ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding","status":"fully_proved","summary":"Let $N$, $a$ and $r$ be natural numbers with $N > 1$ and $a \\ge 1$. Suppose $a^r \\equiv 1 \\pmod{N}$, $r$ is even, $a^{r/2} \\not\\equiv 1 \\pmod{N}$ and $a^{r/2} \\not\\equiv -1 \\pmod{N}$. Then $d = \\gcd(a^{r/2} - 1, N)$, the value computed by the classical post-processing, is a nontrivial factor of $N$: $d$ divides $N$ and $1 < d < N$.","title":"Shor's order-to-factor reduction"},{"id":"ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding/exists-factor-of-order","kind":"node","lean":"Cslib.Shor.exists_factor_of_order","parent":43442,"result":true,"route":"../roadmap/ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding/index.html#exists-factor-of-order","scope":"ComputerScience/quantum-algorithms/shor-s-reduction-of-factoring-to-order-finding","status":"fully_proved","summary":"Let $N$, $a$ and $r$ be natural numbers with $N > 1$ and $a \\ge 1$. If $a^r \\equiv 1 \\pmod{N}$, $r$ is even, $a^{r/2} \\not\\equiv 1 \\pmod{N}$ and $a^{r/2} \\not\\equiv -1 \\pmod{N}$, then $N$ has a nontrivial factor: there is a natural number $d$ with $d \\mid N$ and $1 < d < N$.","title":"Shor's order-to-factor reduction, existential form"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols.json","id":"ComputerScience/concurrent-systems-and-protocols","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/index.html","scope":"ComputerScience/concurrent-systems-and-protocols","status":"fully_proved","summary":"Specific concurrent or distributed systems and their verified properties: deadlock and its detection, the banker's algorithm, dining philosophers, and communication protocols such as the alternating-bit protocol and go-back-N.","title":"Concurrent systems and protocols"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol.json","id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","title":"Alternating bit protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/name","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.Name","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#name","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The channel names of the alternating-bit protocol are $\\mathit{acc}$ (accept a message from the user), $\\mathit{del}$ (deliver a message to the user), and, for each bit $b$, the internal names $\\mathit{send}_b$ (sender to data channel), $\\mathit{recv}_b$ (data channel to receiver), $\\mathit{reply}_b$ (receiver to acknowledgement channel) and $\\mathit{ack}_b$ (acknowledgement channel to sender).","title":"Alternating-bit channel names"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/sender","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.Sender","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#sender","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"For each bit $b$, the sender has three states: $\\mathrm{accept}(b)$, ready to accept a message to be sent with bit $b$; $\\mathrm{transmit}(b)$, about to transmit a frame tagged with bit $b$; and $\\mathrm{await}(b)$, waiting for the acknowledgement of bit $b$. These six states are all the sender states.","title":"Sender states of the data-free alternating-bit protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/channel","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.Channel","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#channel","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"A one-place channel is either empty or full, holding a single bit.","title":"States of a one-place channel"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/receiver","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.Receiver","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#receiver","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"A state of the alternating-bit receiver is, for a bit $b$, either expecting a frame with bit $b$, about to deliver the message of a frame with bit $b$, or about to acknowledge bit $b$.","title":"Alternating-bit receiver states"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/sender-steps","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.senderSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#sender-steps","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The sender of the alternating-bit protocol has the following transitions. Ready to accept with bit $b$, it can input on $\\mathit{acc}$ and become about to transmit $b$. About to transmit $b$, it can output on $\\mathit{send}_b$ and then await the acknowledgement of $b$. Awaiting the acknowledgement of $b$, it can input on $\\mathit{ack}_b$ and become ready to accept with bit $\\lnot b$, input on\u2026","title":"Alternating-bit sender transitions"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/data-steps","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.dataSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#data-steps","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The one-place data channel of the alternating-bit protocol has the following transitions. When empty, it can input on $\\mathit{send}_b$ and become full with bit $b$, for each bit $b$. When full with bit $b$, it can output on $\\mathit{recv}_b$ and become empty, or lose the bit by a silent step and become empty.","title":"Alternating-bit data channel transitions"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/receiver-steps","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.receiverSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#receiver-steps","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The receiver of the alternating-bit protocol has the following transitions. Expecting bit $b$, it can input on $\\mathit{recv}_b$ and become ready to deliver $b$, or input on $\\mathit{recv}_{\\lnot b}$ and become ready to acknowledge $\\lnot b$. Ready to deliver $b$, it can output on $\\mathit{del}$ and become ready to acknowledge $b$. Ready to acknowledge $b$, it can output on $\\mathit{reply}_b$ and then expect\u2026","title":"Alternating-bit receiver transitions"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/ack-steps","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.ackSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#ack-steps","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"When empty, the lossy one-place acknowledgement channel can, for either bit $b$, input on $\\mathit{reply}_b$ and become full with bit $b$. When full with bit $b$, it can output on $\\mathit{ack}_b$ or make a silent $\\tau$ step (losing the bit), in both cases becoming empty. These are all its transitions.","title":"Transitions of the acknowledgement channel in the data-free alternating-bit protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/buffer-steps","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.bufferSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#buffer-steps","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The one-place buffer has two states, empty (false) and full (true). When empty, its only transition is an input on $\\mathit{acc}$ to the full state; when full, its only transition is an output on $\\mathit{del}$ to the empty state.","title":"Transitions of the data-free one-place buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/agent","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.Agent","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#agent","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"There is one process constant for each sender state, each state of the data channel, each receiver state, each state of the acknowledgement channel, and each state of the one-place buffer (empty or full).","title":"Process constants of the data-free alternating-bit protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/spec","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.spec","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#spec","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The process constants are the states of the five agents: the sender, the data channel, the receiver, the acknowledgement channel and the one-place buffer. For a state $x$ of one of these agents, let $(a_1, x_1), \\dots, (a_n, x_n)$ be the transitions of $x$ as listed by the transition function of its agent, let $X_i$ be the constant of the same agent at state $x_i$, and let $\\mathbf{0}$ be the inactive process. The\u2026","title":"Recursive specification of the data-free alternating-bit protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/lts","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.lts","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#lts","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The labelled transition system whose states are the CCS processes over the alternating-bit channel names and process constants, whose labels are the CCS labels over those names, and whose transitions are the steps of the CCS operational semantics for the recursive specification that defines each process constant (one per state of the sender, the data channel, the receiver, the acknowledgement channel and the\u2026","title":"Alternating-bit transition system"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/internal","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.internal","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#internal","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The internal channel names, all of which the protocol restricts, are $\\mathit{send}_b$, $\\mathit{recv}_b$, $\\mathit{reply}_b$ and $\\mathit{ack}_b$ for both bits $b$, that is, every channel name except $\\mathit{acc}$ and $\\mathit{del}$.","title":"Internal channels of the alternating-bit protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/config","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.Config","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#config","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"A configuration of the alternating-bit protocol consists of a sender state, a state of the data channel, a receiver state and a state of the acknowledgement channel.","title":"Alternating-bit configuration"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/system","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.system","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#system","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"Consider a configuration with sender state $s$, data channel state $d$, receiver state $r$ and acknowledgement channel state $k$. Write $\\mathrm{Sender}(s)$, $\\mathrm{Data}(d)$, $\\mathrm{Receiver}(r)$ and $\\mathrm{Ack}(k)$ for the constants of the four agents at these states, and let $L$ be the set of internal channels $\\mathrm{send}(b)$, $\\mathrm{recv}(b)$, $\\mathrm{reply}(b)$ and $\\mathrm{ack}(b)$ for both bits\u2026","title":"The alternating-bit protocol in a configuration"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/config-steps","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.configSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#config-steps","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The transitions of a configuration of the alternating-bit protocol, listed as pairs of a label and a successor configuration, are obtained by composing the transitions of the sender, the data channel, the receiver and the acknowledgement channel in parallel, in that order (at each stage either side moves alone, or the two sides synchronize on an input and the complementary output on the same channel name, giving a\u2026","title":"Alternating-bit configuration transitions"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/buffer","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.buffer","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#buffer","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"For a Boolean $f$, $\\mathit{buffer}(f)$ is the process constant for the one-place buffer that is full if $f$ is true and empty if $f$ is false.","title":"The data-free one-place buffer in a given state"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/initial","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.initial","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#initial","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The initial configuration of the alternating-bit protocol has the sender ready to accept a message with bit $\\mathrm{false}$, both channels empty, and the receiver expecting bit $\\mathrm{false}$.","title":"Initial alternating-bit configuration"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/abp","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.abp","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#abp","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The process $\\mathit{abp}$ is the protocol in its initial configuration: the parallel composition of the sender, the lossy one-place data channel, the receiver and the lossy one-place acknowledgement channel, with all eight internal channels $\\mathit{send}_b$, $\\mathit{recv}_b$, $\\mathit{reply}_b$ and $\\mathit{ack}_b$ (for both bits $b$) restricted, where the sender is ready to accept a message to be sent with bit\u2026","title":"The data-free alternating-bit protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/buf","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.buf","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#buf","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The process $\\mathit{buf}$ is the one-place buffer in its empty state. By the specification it accepts on $\\mathit{acc}$ and becomes full, and the full buffer delivers on $\\mathit{del}$ and becomes empty again, so $\\mathit{buf} = \\mathit{acc}.\\overline{\\mathit{del}}.\\mathit{buf}$.","title":"The data-free one-place buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/certificate","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.certificate","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#certificate","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The list of every reachable protocol configuration, paired with the buffer state it simulates (full from the accept of a message until its delivery), sorted by silent distance to the next visible action.","title":"Weak-bisimulation certificate for the data-free alternating-bit protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/abp-weakly-bisimilar-buffer","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.abp_weaklyBisimilar_buffer","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#abp-weakly-bisimilar-buffer","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"In the labelled transition system of the protocol's recursive CCS specification, the data-free alternating-bit protocol over lossy one-place channels is weakly bisimilar to the one-place buffer $\\mathit{buf} = \\mathit{acc}.\\overline{\\mathit{del}}.\\mathit{buf}$.","title":"The alternating-bit protocol implements a one-place buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/abp-observation-congruent-buffer","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBit.abp_observationCongruent_buffer","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#abp-observation-congruent-buffer","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"In the labelled transition system of the protocol's recursive CCS specification, the data-free alternating-bit protocol over lossy one-place channels is rooted weakly bisimilar (observation congruent) to the one-place buffer $\\mathit{buf} = \\mathit{acc}.\\overline{\\mathit{del}}.\\mathit{buf}$.","title":"The alternating-bit protocol is observation congruent to a one-place buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/name-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.Name","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#name-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"For bits $b$ and $d$, where $b$ is a sequence bit and $d$ a data bit, the channel names are $\\mathsf{acc}(d)$ (accept the data bit $d$ from the user), $\\mathsf{del}(d)$ (deliver the data bit $d$ to the user), $\\mathsf{send}(b, d)$ (sender to data channel), $\\mathsf{recv}(b, d)$ (data channel to receiver), $\\mathsf{reply}(b)$ (receiver to acknowledgement channel) and $\\mathsf{ack}(b)$ (acknowledgement channel to\u2026","title":"Channel names of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/sender-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.Sender","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#sender-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"For bits $b$ and $d$, a sender state is $\\mathsf{accept}(b)$, ready to accept a message to be sent with sequence bit $b$; $\\mathsf{transmit}(b, d)$, about to transmit the data bit $d$ with sequence bit $b$; or $\\mathsf{await}(b, d)$, waiting for the acknowledgement of $b$ while holding $d$ for retransmission.","title":"Sender states of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/receiver-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.Receiver","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#receiver-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"For bits $b$ and $d$, a receiver state is $\\mathsf{expect}(b)$, expecting a frame with sequence bit $b$; $\\mathsf{deliver}(b, d)$, about to deliver the data bit $d$ of a frame with sequence bit $b$; or $\\mathsf{reply}(b)$, about to acknowledge $b$.","title":"Receiver states of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/sender-steps-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.senderSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#sender-steps-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The transitions of each sender state, as a list of pairs of a label and a successor state, for bits $b$ and $d$: $\\mathsf{accept}(b)$ inputs on $\\mathsf{acc}(d)$ and moves to $\\mathsf{transmit}(b, d)$, for $d = \\mathrm{false}$ and then $d = \\mathrm{true}$; $\\mathsf{transmit}(b, d)$ outputs on $\\mathsf{send}(b, d)$ and moves to $\\mathsf{await}(b, d)$; and $\\mathsf{await}(b, d)$ inputs on $\\mathsf{ack}(b)$ and moves\u2026","title":"Sender transitions of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/data-steps-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.dataSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#data-steps-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The data channel is a lossy one-place channel whose content is either empty or a pair $(b, d)$ of a sequence bit $b$ and a data bit $d$. Its transitions, as a list of pairs of a label and a successor content: the empty channel inputs on $\\mathsf{send}(b, d)$ and then holds $(b, d)$, for the four pairs $(b, d)$ in the order $(\\mathrm{false}, \\mathrm{false})$, $(\\mathrm{false}, \\mathrm{true})$,\u2026","title":"Data-channel transitions of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/receiver-steps-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.receiverSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#receiver-steps-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The transitions of each receiver state, as a list of pairs of a label and a successor state, for bits $b$ and $d$: $\\mathsf{expect}(b)$ inputs on $\\mathsf{recv}(b, d)$ and moves to $\\mathsf{deliver}(b, d)$, or inputs a duplicate on $\\mathsf{recv}(\\lnot b, d)$ and moves to the state $\\mathsf{reply}(\\lnot b)$, these two for $d = \\mathrm{false}$ and then for $d = \\mathrm{true}$; $\\mathsf{deliver}(b, d)$ outputs on\u2026","title":"Receiver transitions of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/ack-steps-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.ackSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#ack-steps-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The transitions of the lossy one-place acknowledgement channel: when empty, it can take an input on $\\mathrm{reply}_b$ for either bit $b$ and then holds $b$; when holding $b$, it can either output on $\\mathrm{ack}_b$ or lose $b$ by a silent step, and in both cases it becomes empty.","title":"Acknowledgement channel of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/buffer-steps-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.bufferSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#buffer-steps-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"When empty, the buffer can take an input on $\\mathrm{acc}_d$ for either bit $d$ and then holds $d$; when holding $d$, its only transition is an output on $\\mathrm{del}_d$, after which it is empty.","title":"Transitions of the one-place buffer of bits"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/agent-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.Agent","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#agent-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"There is one process constant for each sender state, each content of the data channel (empty, or a sequence bit with a data bit), each receiver state, each content of the acknowledgement channel (empty, or a bit), and each content of the one-place buffer of bits (empty, or a bit).","title":"Process constants of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/spec-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.spec","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#spec-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"There is a separate process constant $X_s$ for each state $s$ of each of five components: the sender, the lossy data channel, the receiver, the lossy acknowledgement channel and the one-place buffer of bits. If $(a_1, s_1), \\dots, (a_n, s_n)$ are the listed transitions of $s$ in its component, then the body of $X_s$ is the choice of prefixes $a_1.X_{s_1} + (a_2.X_{s_2} + \\dots + (a_n.X_{s_n} + 0))$, where each\u2026","title":"Recursive specification of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/lts-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.lts","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#lts-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The labelled transition system whose states are the CCS processes over the protocol's channel names and process constants, whose labels are the CCS labels over its channel names, and whose transitions are generated by the protocol's recursive specification.","title":"Transition system of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/internal-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.internal","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#internal-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The list of the channel names that the protocol restricts: $\\mathsf{send}(b, d)$ followed by $\\mathsf{recv}(b, d)$ for the pairs $(b, d)$ in the order $(\\mathrm{false}, \\mathrm{false})$, $(\\mathrm{false}, \\mathrm{true})$, $(\\mathrm{true}, \\mathrm{false})$, $(\\mathrm{true}, \\mathrm{true})$, then $\\mathsf{reply}(\\mathrm{false})$, $\\mathsf{reply}(\\mathrm{true})$, $\\mathsf{ack}(\\mathrm{false})$ and\u2026","title":"Internal channels of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/config-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.Config","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#config-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"A configuration consists of a sender state, the content of the data channel (empty, or a pair of a sequence bit and a data bit), a receiver state, and the content of the acknowledgement channel (empty, or a bit).","title":"Configurations of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/system-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.system","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#system-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"For a configuration $c$ made of a sender state $s$, a data-channel content $d$, a receiver state $r$ and an acknowledgement-channel content $k$, the protocol process at $c$ is $((X_s \\mid X_d) \\mid X_r) \\mid X_k$ with every internal channel restricted, where $X_s$, $X_d$, $X_r$ and $X_k$ are the process constants of these states. The internal channels are all channel names other than accepting and delivering a data\u2026","title":"The alternating-bit protocol with data in a configuration"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/config-steps-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.configSteps","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#config-steps-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"For a configuration, made of a sender state, a data-channel content, a receiver state and an acknowledgement-channel content, the transitions are those of the parallel composition of the sender, the data channel, the receiver and the acknowledgement channel, where one component moves alone or two components synchronize on complementary labels in a silent step. Of these, only the silent steps and the actions on\u2026","title":"Transitions of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/buffer-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.buffer","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#buffer-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"For a content $q$, which is either empty or a bit, the process constant of the one-place buffer of bits holding $q$.","title":"The one-place buffer of bits with given content"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/initial-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.initial","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#initial-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The configuration in which the sender is in state $\\mathsf{accept}(\\mathrm{false})$, ready to accept a message to be sent with sequence bit false, the data channel and the acknowledgement channel are empty, and the receiver is in state $\\mathsf{expect}(\\mathrm{false})$, expecting a frame with sequence bit false.","title":"Initial configuration of the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/abp-data","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.abpData","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#abp-data","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The protocol process in its initial configuration: the sender is ready to accept a data bit, to be sent with sequence bit false, the data channel and the acknowledgement channel are empty, and the receiver expects a frame with sequence bit false. The four agents run in parallel, with all internal channels restricted.","title":"The alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/buf-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.buf","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#buf-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"The process constant of the one-place buffer of bits with empty content.","title":"The empty one-place buffer of bits"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/certificate-2","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.certificate","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#certificate-2","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"A list of 220 pairs, each a protocol configuration together with the buffer content it simulates: every reachable configuration with the buffer content it simulates, listed so that every pair reaches the visible actions of its buffer state through pairs listed earlier.","title":"Weak-bisimulation certificate for the alternating-bit protocol with data"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/abp-data-weakly-bisimilar-buffer","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.abpData_weaklyBisimilar_buffer","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#abp-data-weakly-bisimilar-buffer","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"It is weakly bisimilar to the empty one-place buffer of bits.","title":"The alternating-bit protocol transmits data correctly"},{"id":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/abp-data-observation-congruent-buffer","kind":"node","lean":"Cslib.Algorithms.CCS.AlternatingBitData.abpData_observationCongruent_buffer","parent":43446,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol/index.html#abp-data-observation-congruent-buffer","scope":"ComputerScience/concurrent-systems-and-protocols/alternating-bit-protocol","status":"fully_proved","summary":"In the labelled transition system of the protocol's recursive specification, the alternating-bit protocol with data is observation congruent (rooted weakly bisimilar) to the empty one-place buffer of bits: every transition of either process with some label is matched by the other process through silent steps, then one actual step with the same label, then silent steps, and the two resulting processes are weakly\u2026","title":"The alternating-bit protocol with data is observation congruent to the buffer"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm.json","id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","title":"Banker's algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/system","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#system","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"A system with processes $P$ and resource types $R$ consists of two maps $\\mathrm{claim}, \\mathrm{alloc} : P \\to (R \\to \\mathbb{N})$, where $\\mathrm{claim}(p)$ is the maximum demand of process $p$ for each resource type and $\\mathrm{alloc}(p)$ is the amount of each resource type currently allocated to $p$. No inequality between the two is required.","title":"Claims and allocations of processes"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/need","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.need","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#need","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"The remaining need of a process $p$ in a system $S$ is the resource vector $\\mathrm{claim}(p) - \\mathrm{alloc}(p)$: the maximum claim of $p$ minus its current allocation, computed for each resource type by truncated subtraction of natural numbers, so it is $0$ for every resource type where the allocation exceeds the claim.","title":"Remaining need"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/safe-seq","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.SafeSeq","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#safe-seq","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system with processes $P$ and resource types $R$, where process $p$ has maximum claim $\\mathrm{claim}(p)$ and allocation $\\mathrm{alloc}(p)$ in $R \\to \\mathbb{N}$. A list $p_1, \\dots, p_k$ of processes is a safe sequence from the free resource vector $w : R \\to \\mathbb{N}$ when, for each $i$, the remaining need $\\mathrm{claim}(p_i) - \\mathrm{alloc}(p_i)$ (truncated at zero) is at most\u2026","title":"Safe sequence"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/safe","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.Safe","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#safe","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system of claims and allocations with processes $P$ and resource types $R$. A list of pending processes is safe from the free resource vector $w : R \\to \\mathbb{N}$ when it has no repetitions and some permutation of it is a safe sequence from $w$: the pending processes can all finish, one after another, in some order.","title":"Safe state"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/mono","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.SafeSeq.mono","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#mono","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system of processes with maximum claims and current allocations, and let $\\sigma$ be a list of processes. If $\\sigma$ is a safe sequence for $S$ from a vector $w$ of free resources (a natural number for each resource type), and $w \\le w'$ componentwise, then $\\sigma$ is also a safe sequence for $S$ from $w'$.","title":"Safe sequences are monotone in the free resources"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/erase","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.SafeSeq.erase","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#erase","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system of processes with maximum claims and current allocations, let $w$ be a vector of free resources (a natural number for each resource type), and let $\\sigma$ be a list of processes that is a safe sequence for $S$ from $w$. If $p$ occurs in $\\sigma$, then the list obtained from $\\sigma$ by deleting the first occurrence of $p$ is a safe sequence for $S$ from $w + \\mathrm{alloc}(p)$, where\u2026","title":"Removing a process from a safe sequence"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/exists-runnable","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.Safe.exists_runnable","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#exists-runnable","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system of processes with maximum claims and current allocations, let $w$ be a vector of free resources (a natural number for each resource type), and let $L$ be a nonempty list of pending processes. If $S$ is safe from $w$ for $L$, then some process $p$ in $L$ has remaining need $\\mathrm{need}(p) \\le w$ componentwise, so it can be given its whole remaining need.","title":"A safe state is not deadlocked"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/safe-iff-safe-erase","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.safe_iff_safe_erase","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#safe-iff-safe-erase","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system of processes with maximum claims and current allocations, $w$ a vector of free resources (a natural number for each resource type) and $L$ a list of pending processes without repetitions. If $p$ occurs in $L$ and its remaining need $\\mathrm{need}(p)$ in $S$ satisfies $\\mathrm{need}(p) \\le w$ componentwise, then $S$ is safe from $w$ for $L$ if and only if $S$ is safe from $w + \\mathrm{alloc}(p)$\u2026","title":"Letting a process that fits finish preserves safety"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/safety-check-aux","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.safetyCheckAux","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#safety-check-aux","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system with processes $P$ and a finite set $R$ of resource types, where process $p$ has maximum claim $\\mathrm{claim}(p)$ and allocation $\\mathrm{alloc}(p)$ in $R \\to \\mathbb{N}$. Given a fuel $n \\in \\mathbb{N}$, a free resource vector $w : R \\to \\mathbb{N}$ and a list of pending processes, the algorithm succeeds if the list is empty. Otherwise, if $n > 0$ and some pending process $q$ has remaining\u2026","title":"Banker's safety algorithm with fuel"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/safety-check","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.safetyCheck","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#safety-check","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system of claims and allocations with processes $P$ and a finite set $R$ of resource types. Given a free resource vector $w : R \\to \\mathbb{N}$ and a list of pending processes, the algorithm fails if the list has a repetition. Otherwise it runs the safety algorithm with fuel equal to the length of the list: it repeatedly lets the first pending process whose remaining need (claim minus allocation,\u2026","title":"Banker's safety algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/safety-check-iff","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.safetyCheck_iff","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#safety-check-iff","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"For every system $S$ of claims and allocations over a finite set $R$ of resource types, every free resource vector $w : R \\to \\mathbb{N}$ and every list of pending processes, the banker's safety algorithm succeeds on $w$ and the list if and only if the list is a safe state from $w$.","title":"Correctness of the safety algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/grant","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.grant","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#grant","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"For a system $S$ of processes with maximum claims and current allocations, a process $p$ and a resource vector $r$ (a natural number for each resource type), the system obtained by granting $r$ to $p$ has the same maximum claims as $S$, gives $p$ the allocation $\\mathrm{alloc}(p) + r$ (componentwise sum, where $\\mathrm{alloc}(p)$ is the current allocation of $p$ in $S$), and leaves the allocation of every other\u2026","title":"Granting a request"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/need-grant-self","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.need_grant_self","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#need-grant-self","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"For a system $S$ of processes with maximum claims and current allocations, a process $p$ and a resource vector $r$ (a natural number for each resource type), the remaining need of $p$ in the system obtained by granting $r$ to $p$ equals $\\mathrm{need}(p) - r$, where $\\mathrm{need}(p)$ is the remaining need of $p$ in $S$ and the subtraction is truncated subtraction of natural numbers, taken componentwise.","title":"Remaining need after a grant"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/request","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.request","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#request","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system of claims and allocations with processes $P$ and a finite set $R$ of resource types, let $a : R \\to \\mathbb{N}$ be the vector of free resources, $L$ a list of pending processes, $p$ a process and $r : R \\to \\mathbb{N}$ the resources that $p$ requests. The request is granted when $p$ occurs in $L$, $L$ has no repetitions, $r \\le \\mathrm{need}(p)$ and $r \\le a$ componentwise, where\u2026","title":"Resource-request algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/request-eq-some-iff","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.request_eq_some_iff","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#request-eq-some-iff","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"The request algorithm grants exactly the safe requests within the claim and the free resources. Let $S$ be a system of claims and allocations with processes $P$ and a finite set $R$ of resource types, let $a : R \\to \\mathbb{N}$ be the vector of free resources, $L$ a list of pending processes, $p$ a process and $r : R \\to \\mathbb{N}$ the resources that $p$ requests. Write $\\mathrm{need}(p)$ for the remaining need of\u2026","title":"The request algorithm grants exactly the safe requests within the claim and the free resources"},{"id":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/request-safe","kind":"node","lean":"Cslib.Algorithms.Concurrency.Banker.System.request_safe","parent":43492,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm/index.html#request-safe","scope":"ComputerScience/concurrent-systems-and-protocols/banker-s-algorithm","status":"fully_proved","summary":"Let $S$ be a system of processes with maximum claims and current allocations, $a$ the vector of free resources, $L$ the list of pending processes, $p$ a process and $r$ a resource vector. If the resource-request algorithm grants the request of $r$ by $p$, returning the free vector $a'$ and the system $S'$, then $S'$ is safe from $a'$ for $L$.","title":"Granted requests lead to safe states"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem.json","id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","title":"Byzantine generals problem"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/oral-protocol","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#oral-protocol","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"A synchronous oral-message protocol for a set $G$ of generals, with orders in $V$, local states in $\\sigma$ and messages in $M$, consists of an initial state for each general in $G$, used by the generals other than the commander; an initial state of the commander for each order in $V$; a sending function giving, for each round $r$, each general $j$ in each state $s$ and each general $i$, the message in $M$ that $j$\u2026","title":"Oral-message protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/states-eq-of-inbox-eq","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.states_eq_of_inbox_eq","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#states-eq-of-inbox-eq","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let $P$ be an oral-message protocol for the generals $G$ with states $\\sigma$ and messages $M$. Let $N$ be a type of nodes, node $k$ running the code of the general $\\rho(k) \\in G$, and let $L$ be a property of nodes. For each round $r$ and node $k$, let $s_r(k), s'_r(k) \\in \\sigma$ be two states and $\\mu_r(k), \\mu'_r(k) : G \\to M$ two families of received messages. Suppose that for every $r$ and every node $k$\u2026","title":"Local views determine states"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/init-state","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.initState","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#init-state","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"When the commander $c$ has order $v$, the initial state of general $i$ is the initial state that the protocol assigns to a commander with order $v$ if $i = c$, and the initial state that the protocol assigns to $i$ as a general other than the commander if $i \\ne c$.","title":"Initial state of a general"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/init-state-self","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.initState_self","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#init-state-self","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"When the commander $c$ has order $v$, the initial state of $c$ is the initial state that the protocol assigns to a commander with order $v$.","title":"Initial state of the commander"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/init-state-of-ne","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.initState_of_ne","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#init-state-of-ne","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"If general $i$ is not the commander $c$, then the initial state of $i$ when $c$ has order $v$ is the initial state that the protocol assigns to $i$ as a general other than the commander; in particular, it does not depend on $v$.","title":"Initial state of a lieutenant"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/state","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.state","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#state","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let $P$ be an oral-message protocol for the generals $G$, let $c \\in G$ be the commander, $v$ its order and $S \\subseteq G$ a finite set of traitors, and let $a_r(j, i)$ be the message a traitor $j$ sends general $i$ in round $r$. The state of general $i$ after $r$ rounds is defined by recursion on $r$. After $0$ rounds it is the initial state of $i$ for commander $c$ and order $v$. After $r + 1$ rounds it is the\u2026","title":"Executions of an oral-message protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/inbox","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.inbox","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#inbox","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"In the execution of a protocol $P$ with commander $c$, order $v$, a set of traitors and a behaviour $\\mathrm{adv}$ of the traitors, the messages that general $i$ receives in round $r$ form the function of the sender $j$ whose value is $\\mathrm{adv}(r, j, i)$ if $j$ is a traitor, and otherwise the message that $P$ has $j$ send to $i$ in round $r$ from the state of $j$ after $r$ rounds.","title":"Messages received in a round"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/state-zero","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.state_zero","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#state-zero","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"In the execution of an oral-message protocol $P$ with commander $c$, order $v$, any set of traitors and any traitor behaviour, every general is in its initial state after $0$ rounds: the commander in the initial state given by the order $v$, and every other general in its order-independent initial state.","title":"The start of an execution"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/state-succ","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.state_succ","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#state-succ","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"In the execution of an oral-message protocol $P$ with a given commander, order, set of traitors and traitor behaviour, for every round $r$ and every general $i$, the state of $i$ after $r + 1$ rounds is the state to which $i$ moves by its code for round $r$ from its state after $r$ rounds on the messages it receives in round $r$.","title":"One round of an execution"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/decision","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.decision","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#decision","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"In the execution of a protocol $P$ with commander $c$, order $v$, a set of traitors and a behaviour of the traitors, the decision of general $i$ after $T$ rounds is the value that the decision function of $P$ assigns to $i$ in its state after $T$ rounds.","title":"Decision of a general"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/agreement","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.Agreement","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#agreement","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Consider the execution of a protocol $P$ with commander $c$, order $v$, a set of traitors and a behaviour of the traitors. Condition IC1 holds after $T$ rounds when any two generals $i$ and $j$, neither of which is the commander or a traitor, decide the same value after $T$ rounds of this execution.","title":"Agreement condition (IC1)"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/validity","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.Validity","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#validity","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"In the execution of an oral-message protocol with commander $c$, order $v$, a set of traitors and a traitor behaviour, condition IC2 (validity) holds after $T$ rounds if, provided the commander is loyal, every loyal general other than the commander decides $v$ after $T$ rounds.","title":"Validity in the Byzantine generals problem"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/solves","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.Solves","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#solves","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"An oral-message protocol $P$ for the generals $G$ solves the Byzantine generals problem with commander $c \\in G$ and at most $m$ traitors in $T$ rounds if, for every order $v$, every set of at most $m$ traitors and every behaviour of the traitors (the message each traitor sends to each general in each round), conditions IC1 (agreement) and IC2 (validity) hold after $T$ rounds.","title":"Solving the Byzantine generals problem"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/exists-violation-three","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.exists_violation_three","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#exists-violation-three","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Suppose there are at least two possible orders. For every oral-message protocol for three generals $0$, $1$, $2$ with commander $0$ and every number of rounds $T$, there are an order $v$, a general $t$ and a behaviour of $t$ such that, in the execution with order $v$ in which $t$ is the only traitor, the agreement condition (IC1) or the validity condition (IC2) fails after $T$ rounds.","title":"No protocol for three generals tolerates one traitor"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/not-solves-three","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.not_solves_three","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#not-solves-three","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"No protocol for three generals solves the Byzantine generals problem with one traitor. Suppose there are at least two possible orders. For every oral-message protocol for three generals $0$, $1$, $2$ with commander $0$ and every number of rounds $T$, the protocol does not solve the Byzantine generals problem with commander $0$ and at most one traitor in $T$ rounds.","title":"No protocol for three generals solves the Byzantine generals problem with one traitor"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/simulate","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.simulate","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#simulate","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let $P$ be an oral-message protocol for the generals $G$ with orders in $V$, states $\\sigma$ and messages $M$, let $c \\in G$, let $\\mathrm{grp} : G \\to \\{0, 1, 2\\}$ split the generals into three groups and let $\\mathrm{rep} : \\{0, 1, 2\\} \\to G$. The simulation of $P$ with $c$, $\\mathrm{grp}$ and $\\mathrm{rep}$ is the oral-message protocol for the generals $\\{0, 1, 2\\}$ with orders in $V$, whose states are families\u2026","title":"Three generals simulating a protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/state-simulate","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.state_simulate","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#state-simulate","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let $P$ be an oral-message protocol for a finite set $G$ of generals with messages $M$, let $c \\in G$, and let $\\mathrm{grp} : G \\to \\{0, 1, 2\\}$ and $\\mathrm{rep} : \\{0, 1, 2\\} \\to G$ with $\\mathrm{grp}(c) = 0$. Fix an order $v$, a set $S \\subseteq \\{0, 1, 2\\}$ of traitors and a traitor behaviour $a$ for the simulation of $P$ with $c$, $\\mathrm{grp}$ and $\\mathrm{rep}$: in round $r$ a traitor $h$ sends general $g$\u2026","title":"The simulation is faithful"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/solves-simulate","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.solves_simulate","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#solves-simulate","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let $P$ be an oral-message protocol for a finite set $G$ of generals, let $c \\in G$ and $m, T \\in \\mathbb{N}$, and suppose that $P$ solves the Byzantine generals problem with commander $c$ and at most $m$ traitors in $T$ rounds. Let $\\mathrm{grp} : G \\to \\{0, 1, 2\\}$ and $\\mathrm{rep} : \\{0, 1, 2\\} \\to G$ satisfy $\\mathrm{grp}(c) = 0$, $\\mathrm{grp}(\\mathrm{rep}(g)) = g$ for every $g \\ne 0$, and\u2026","title":"Simulation reduces many generals to three"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/not-solves-of-card-le-three-mul","kind":"node","lean":"Cslib.Distributed.Byzantine.Impossibility.OralProtocol.not_solves_of_card_le_three_mul","parent":43509,"reading_title":{"format":"inline-parts-v1","parts":[["text","No protocol for "],["math","\\(n\\)"],["text"," generals tolerates "],["math","\\(m\\)"],["text"," traitors when "],["math","\\(3 \\le n \\le 3m\\)"]],"source":"No protocol for $n$ generals tolerates $m$ traitors when $3 \\le n \\le 3m$"},"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#not-solves-of-card-le-three-mul","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Suppose there are at least two possible orders and the set of generals is finite with $n$ elements, where $3 \\le n \\le 3m$. Then for every oral-message protocol, every commander $c$ and every number of rounds $T$, the protocol does not solve the Byzantine generals problem with commander $c$ and at most $m$ traitors in $T$ rounds: there are an order, a set of at most $m$ traitors and a behaviour of the traitors for\u2026","title":"No protocol for $n$ generals tolerates $m$ traitors when $3 \\le n \\le 3m$"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/majority","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.majority","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#majority","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"The majority of a finite multiset $s$ of values is the value $x$ such that more than half of the elements of $s$ are equal to $x$ (counted with multiplicity), if there is one (there is at most one), and a fixed default value (retreat) otherwise.","title":"Majority of a multiset"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/majority-eq-of-lt-count","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.majority_eq_of_lt_count","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#majority-eq-of-lt-count","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"If more than half of the elements of a multiset $s$ of values are equal to $x$ (counted with multiplicity), then the majority of $s$ is $x$.","title":"A value filling more than half of a multiset is its majority"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/majority-eq-default","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.majority_eq_default","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#majority-eq-default","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"If, for every value $x$, at most half of the elements of a multiset $s$ of values are equal to $x$ (counted with multiplicity), then the majority of $s$ is the default value (retreat).","title":"Majority falls back to the default value"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/received","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.received","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#received","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Fix a predicate saying which generals are loyal and an adversary $\\mathrm{adv}$ that assigns a value to each pair of a finite sequence of generals and a recipient. The value that lieutenant $p$ receives from commander $c$, who holds the value $v$ and whose superiors form the sequence $\\sigma$, is $v$ if $c$ is loyal, and is $\\mathrm{adv}(\\sigma c, p)$ if $c$ is a traitor, where $\\sigma c$ is $\\sigma$ followed by $c$.","title":"Value received from a commander"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/om","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.om","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#om","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Fix which generals are loyal and a strategy for the traitors, which assigns a value to each list of generals and each recipient. Let $c$ be a commander holding the value $v$, let $\\sigma$ be the list of the commanders above $c$, and let $S$ be a finite set of lieutenants. For a general $q$, let $r_q$ be the value $q$ receives from $c$: $v$ if $c$ is loyal, and otherwise the value the strategy assigns to the list\u2026","title":"The oral messages algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/received-of-loyal","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.received_of_loyal","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#received-of-loyal","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"If the commander $c$ is loyal, then for every value $v$, every list of superiors of $c$ and every general $p$, the value $p$ receives from $c$ is $v$.","title":"A loyal commander sends its value"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/received-of-not-loyal","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.received_of_not_loyal","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#received-of-not-loyal","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Fix which generals are loyal and an adversary $\\mathrm{adv}$ that assigns a value to each pair of a list of generals and a recipient. If the commander $c$ is a traitor, then for every value $v$, every list $\\sigma$ of superiors of $c$ and every general $p$, the value $p$ receives from $c$ is $\\mathrm{adv}(\\sigma c, p)$, where $\\sigma c$ is $\\sigma$ followed by $c$.","title":"A traitorous commander sends the adversary's value"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/om-eq-of-loyal","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.om_eq_of_loyal","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#om-eq-of-loyal","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let $m$ and $k$ be natural numbers, and let a loyal commander $c$ run $\\mathrm{OM}(m)$ with a value $v$ and any list of superiors, sending to a finite set $S$ of lieutenants. If $2k + m < |S| + 1$ (more than $2k + m$ generals when $c \\notin S$) and at most $k$ of the generals in $S \\cup \\{c\\}$ are traitors, then, whatever the traitors send, every loyal lieutenant $p \\in S$ decides $v$.","title":"A loyal commander is obeyed under oral messages"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/om-valid","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.om_valid","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#om-valid","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let a loyal commander $c$ run $\\mathrm{OM}(m)$ with a value $v$ and any list of superiors, sending to a finite set $S$ of lieutenants. If $3m < |S| + 1$ (more than $3m$ generals when $c \\notin S$) and at most $m$ of the generals in $S \\cup \\{c\\}$ are traitors, then, whatever the traitors send, every loyal lieutenant $p \\in S$ decides $v$.","title":"Validity under oral messages (IC2)"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/om-agree","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.om_agree","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#om-agree","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let a commander $c$ run $\\mathrm{OM}(m)$ with a value $v$ and any list of superiors, sending to a finite set $S$ of lieutenants with $c \\notin S$. If there are more than $3m$ generals, that is $3m < |S| + 1$, and at most $m$ of the generals in $S \\cup \\{c\\}$ are traitors, then, whatever the traitors send, any two loyal lieutenants $p, p' \\in S$ decide the same value.","title":"Agreement under oral messages (IC1)"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/om-interactive-consistency","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.om_interactive_consistency","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#om-interactive-consistency","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let a commander $c$ run $\\mathrm{OM}(m)$ with a value $v$ and any list of superiors, sending to a finite set $S$ of lieutenants with $c \\notin S$. If there are more than $3m$ generals, that is $3m < |S| + 1$, and at most $m$ of the generals in $S \\cup \\{c\\}$ are traitors, then, whatever the traitors send, any two loyal lieutenants in $S$ decide the same value (IC1), and if $c$ is loyal, every loyal lieutenant in\u2026","title":"Oral messages achieve interactive consistency"},{"id":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/byzantine-generals","kind":"node","lean":"Cslib.Distributed.Byzantine.OralMessages.byzantine_generals","parent":43509,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem/index.html#byzantine-generals","scope":"ComputerScience/concurrent-systems-and-protocols/byzantine-generals-problem","status":"fully_proved","summary":"Let $G$ be a finite set of more than $3m$ generals, at most $m$ of whom are traitors, and let the commander $c \\in G$ run $\\mathrm{OM}(m)$ with the value $v$, with no superiors and with all the other generals as lieutenants. Then, whatever the traitors send, any two loyal lieutenants decide the same value, and if $c$ is loyal, every loyal lieutenant decides $v$.","title":"The oral messages algorithm solves the Byzantine generals problem"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election.json","id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","title":"Chang\u2013Roberts leader election"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/state","kind":"node","lean":"Cslib.Distributed.ChangRoberts.State","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#state","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"For a ring of $n + 1$ processes $0, \\dots, n$, a global state assigns to each process $k$ the FIFO channel of messages waiting at $k$, oldest first, where each message is the index of the process whose identifier it carries, and records for each process whether it has declared itself leader.","title":"Chang\u2013Roberts global states"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/succ","kind":"node","lean":"Cslib.Distributed.ChangRoberts.succ","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#succ","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"On the ring of $n + 1$ processes $0, \\dots, n$, the successor of process $k$ is $k + 1$ when $k < n$, and $0$ when $k = n$.","title":"Successor on the Chang\u2013Roberts ring"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/pred","kind":"node","lean":"Cslib.Distributed.ChangRoberts.pred","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#pred","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"On a ring of $n + 1$ processes numbered $0, \\dots, n$, the predecessor of $k$ is $k - 1$ if $k \\ne 0$, and $n$ if $k = 0$.","title":"Predecessor on the ring"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/deliver","kind":"node","lean":"Cslib.Distributed.ChangRoberts.deliver","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#deliver","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"On a ring of $n + 1$ processes with identifiers $\\mathrm{id}$, process $k$ receives the oldest message in its channel in state $s$. If the channel of $k$ is empty there is no step. Otherwise let $j$ be the origin of that message, which is removed from the channel: if $\\mathrm{id}(k) < \\mathrm{id}(j)$, process $k$ forwards it by appending $j$ to the end of the channel of its successor on the ring; otherwise, if\u2026","title":"Delivering a message in the Chang\u2013Roberts ring"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/lts","kind":"node","lean":"Cslib.Distributed.ChangRoberts.lts","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#lts","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"For identifiers $\\mathrm{id}$, the algorithm is the labelled transition system whose states are the global states and whose labels are processes: there is a transition from $s$ to $t$ labelled $k$ exactly when delivering the oldest message in the channel of process $k$ in state $s$ gives $t$.","title":"The Chang\u2013Roberts transition system"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/initial","kind":"node","lean":"Cslib.Distributed.ChangRoberts.initial","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#initial","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"In the initial state the channel into each process $k$ holds exactly one message, whose origin is the predecessor of $k$ on the ring (so every process has sent its identifier to its successor), and no process is leader.","title":"Initial state of the Chang\u2013Roberts ring"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/reachable","kind":"node","lean":"Cslib.Distributed.ChangRoberts.Reachable","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#reachable","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"Given a natural-number identifier $\\mathrm{id}(k)$ for each process $k$ of the ring of $n + 1$ processes, a global state $s$ is reachable when the Chang\u2013Roberts transition system with these identifiers leads from the initial state to $s$ by a multistep transition along some finite list of labels, that is, of receiving processes.","title":"Reachable states of Chang\u2013Roberts"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/gap","kind":"node","lean":"Cslib.Distributed.ChangRoberts.gap","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#gap","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"For processes $j$ and $m$ on a ring of $n + 1$ processes numbered $0, \\dots, n$, the gap from $j$ to $m$ is the number of successor steps from $j$ to $m$: it is $m - j$ if $j \\le m$, and $m + (n + 1) - j$ otherwise. It lies in $[0, n]$.","title":"Forward distance on the ring"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/travelled","kind":"node","lean":"Cslib.Distributed.ChangRoberts.travelled","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#travelled","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"For processes $j, k$ of the ring of $n + 1$ processes, the distance travelled by a message from $j$ waiting at $k$ is $n + 1$, the whole ring, when $k = j$, and otherwise the ring distance from $j$ to $k$, the number of links from $j$ forward to $k$.","title":"Distance travelled by a message"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/inv","kind":"node","lean":"Cslib.Distributed.ChangRoberts.Inv","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#inv","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"Given identifiers $\\mathrm{id}$ on a ring of $n + 1$ processes, a state satisfies the invariant when two conditions hold. First, for every message from origin $j$ waiting in the channel of process $k$, every process $m \\ne j$ that the message has already passed has a smaller identifier, $\\mathrm{id}(m) < \\mathrm{id}(j)$; here $m$ has been passed when its forward distance from $j$ is less than the distance the\u2026","title":"The Chang\u2013Roberts invariant"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/travelled-succ","kind":"node","lean":"Cslib.Distributed.ChangRoberts.travelled_succ","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#travelled-succ","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"For processes $j$ and $k$ of the ring with $k \\ne j$, the distance travelled by a message from $j$ waiting at the successor of $k$ is the number of links from $j$ forward to $k$, plus one.","title":"Forwarding adds one link"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/travelled-pred","kind":"node","lean":"Cslib.Distributed.ChangRoberts.travelled_pred","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#travelled-pred","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"For every process $k$ of the ring, the distance travelled by a message from the predecessor of $k$ waiting at $k$ is $1$.","title":"The initial messages have travelled one link"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/inv-mtr","kind":"node","lean":"Cslib.Distributed.ChangRoberts.inv_mtr","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#inv-mtr","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"For every assignment of identifiers, if the invariant holds in a state $s$ and the state $t$ is reached from $s$ by delivering messages along some schedule of receiving processes (a multistep transition of the algorithm's labelled transition system), then the invariant holds in $t$.","title":"The Chang\u2013Roberts invariant is preserved along runs"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/inv-reachable","kind":"node","lean":"Cslib.Distributed.ChangRoberts.inv_reachable","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#inv-reachable","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"For every assignment of identifiers, every state reachable from the initial state satisfies the invariant.","title":"The Chang\u2013Roberts invariant holds in every reachable state"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/leader-max","kind":"node","lean":"Cslib.Distributed.ChangRoberts.leader_max","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#leader-max","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"Let $\\mathrm{id}$ assign a natural-number identifier, not necessarily distinct, to each process $0, \\dots, n$ of the Chang\u2013Roberts ring. In every state reachable with these identifiers, if process $i$ has declared itself leader, then $\\mathrm{id}(m) < \\mathrm{id}(i)$ for every process $m \\ne i$.","title":"Only the maximum is elected"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/leader-unique","kind":"node","lean":"Cslib.Distributed.ChangRoberts.leader_unique","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#leader-unique","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"In every state of the Chang\u2013Roberts algorithm reachable from the initial state, any two processes that have declared themselves leader are equal.","title":"There is at most one leader"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/run","kind":"node","lean":"Cslib.Distributed.ChangRoberts.run","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#run","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"Given a natural-number identifier $\\mathrm{id}(k)$ for each process $k$ of the ring, a global state $s$ and a list $k_1, \\dots, k_r$ of processes, the run applies the Chang\u2013Roberts delivery step at $k_1$, then at $k_2$, and so on. It returns the final state, or no state as soon as some $k_i$ has an empty channel at its turn; on the empty list it returns $s$.","title":"Running a delivery schedule"},{"id":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/run-eq-some-iff","kind":"node","lean":"Cslib.Distributed.ChangRoberts.run_eq_some_iff","parent":43541,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election/index.html#run-eq-some-iff","scope":"ComputerScience/concurrent-systems-and-protocols/chang-roberts-leader-election","status":"fully_proved","summary":"Given a natural-number identifier $\\mathrm{id}(k)$ for each process $k$ of the ring, for all global states $s, t$ and every list $\\kappa$ of processes, running the schedule $\\kappa$ from $s$ returns $t$ if and only if the Chang\u2013Roberts transition system with identifiers $\\mathrm{id}$ has a multistep transition from $s$ to $t$ labelled by $\\kappa$.","title":"Runs are multistep transitions"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait.json","id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","title":"Deadlock and circular wait"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/alloc-state","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#alloc-state","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"A resource-allocation snapshot, for a type $P$ of processes and a type $R$ of resources, records for each resource the process holding it, if any (so each resource has at most one holder), and for each process $p$ and resource $r$ whether $p$ has an outstanding request for $r$. A process never requests a resource it holds.","title":"Resource-allocation snapshot"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/waits-for","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.WaitsFor","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#waits-for","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"In a resource-allocation snapshot $s$, a process $p$ waits for a process $q$ when there is a resource $r$ that $p$ requests in $s$ and that $q$ holds in $s$.","title":"Wait-for relation"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/deadlocked","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.Deadlocked","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#deadlocked","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"In a resource-allocation snapshot $s$, a set $D$ of processes is deadlocked when $D$ is nonempty and every process $p \\in D$ waits in $s$ for some process $q \\in D$, that is, $p$ requests in $s$ a resource that $q$ holds in $s$.","title":"Deadlocked set of processes"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/circular-wait","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.CircularWait","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#circular-wait","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"A resource-allocation snapshot $s$ has circular wait when some process $p$ waits for itself through a chain of one or more wait-for edges in $s$, that is, there are processes $p = p_0, p_1, \\dots, p_k = p$ with $k \\ge 1$ such that $p_{i-1}$ waits for $p_i$ in $s$ for each $1 \\le i \\le k$.","title":"Circular wait"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/not-waits-for-self","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.not_waitsFor_self","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#not-waits-for-self","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"In a resource-allocation snapshot, no process $p$ waits for itself, that is, there is no resource that $p$ both requests and holds.","title":"No process waits for itself directly"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/not-waits-for-of-free","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.not_waitsFor_of_free","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#not-waits-for-of-free","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"In a resource-allocation snapshot, if no resource requested by a process $p$ has a holder, then $p$ waits for no process $q$.","title":"A process whose requests are all free waits for no one"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/exists-deadlocked-of-circular-wait","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.exists_deadlocked_of_circularWait","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#exists-deadlocked-of-circular-wait","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"If a resource-allocation snapshot has circular wait, that is, some process $p$ waits for itself through a chain of one or more wait-for edges, then some set of processes is deadlocked.","title":"Circular wait yields a deadlocked set"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/circular-wait-of-deadlocked","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.circularWait_of_deadlocked","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#circular-wait-of-deadlocked","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"In a resource-allocation snapshot with finitely many processes, if some set of processes is deadlocked, then the snapshot has circular wait: some process waits for itself through a chain of one or more wait-for edges.","title":"A deadlocked set forces circular wait"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/deadlock-iff-circular-wait","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.deadlock_iff_circularWait","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#deadlock-iff-circular-wait","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"In a resource-allocation snapshot with finitely many processes, some set of processes is deadlocked if and only if the snapshot has circular wait, that is, some process waits for itself through a chain of one or more wait-for edges.","title":"Deadlock is circular wait"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/ordered","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.Ordered","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#ordered","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"Let $s$ be a resource-allocation snapshot and let $\\mathrm{rank}$ assign a natural number to each resource. The snapshot $s$ is ordered by $\\mathrm{rank}$ when, for every process $p$, every resource $r$ that $p$ requests in $s$ and every resource $r'$ that $p$ holds in $s$, we have $\\mathrm{rank}(r') < \\mathrm{rank}(r)$: each process requests only resources ranked strictly above every resource it holds.","title":"Havender's resource ordering"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/not-circular-wait-of-ordered","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.not_circularWait_of_ordered","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#not-circular-wait-of-ordered","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"Let $\\mathrm{rank}$ assign a natural number to each resource. If a resource-allocation snapshot is ordered by $\\mathrm{rank}$ (Havender's discipline), that is, $\\mathrm{rank}(r') < \\mathrm{rank}(r)$ whenever some process requests $r$ and holds $r'$, then the snapshot has no circular wait: no process waits for itself through a chain of one or more wait-for edges.","title":"Resource ordering prevents circular wait"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/not-deadlocked-of-ordered","kind":"node","lean":"Cslib.Algorithms.Concurrency.Deadlock.AllocState.not_deadlocked_of_ordered","parent":43560,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait/index.html#not-deadlocked-of-ordered","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-and-circular-wait","status":"fully_proved","summary":"Let $\\mathrm{rank}$ assign a natural number to each resource. If a resource-allocation snapshot with finitely many processes is ordered by $\\mathrm{rank}$ (Havender's discipline), that is, $\\mathrm{rank}(r') < \\mathrm{rank}(r)$ whenever some process requests $r$ and holds $r'$, then no set of processes is deadlocked.","title":"Resource ordering prevents deadlock"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/deadlock-detection.json","id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","title":"Deadlock detection"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/state","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#state","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"For a type $P$ of processes and a type $R$ of resource types, a state consists of two functions $\\mathrm{alloc}, \\mathrm{request} : P \\to R \\to \\mathbb{N}$: $\\mathrm{alloc}(p)(r)$ is the number of instances of resource type $r$ that process $p$ holds, and $\\mathrm{request}(p)(r)$ is the number of instances of $r$ that $p$ is waiting for.","title":"Resource-allocation state"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/reduces","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.Reduces","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#reduces","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Let $P$ be a type of processes and $R$ a type of resource types, and let $S$ be a state that gives each process $p$ an allocation $\\mathrm{alloc}(p)$ and a request $\\mathrm{request}(p)$, both functions $R \\to \\mathbb{N}$ ordered componentwise. A list of processes $p_1, \\dots, p_k$ is a reduction sequence of $S$ from the free vector $w : R \\to \\mathbb{N}$ when, for every $1 \\le i \\le k$, we have\u2026","title":"Reduction sequences"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/erase","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.Reduces.erase","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#erase","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Let $w$ be a resource vector and $p$ a process whose request is at most $w$. If $\\sigma$ is a reduction sequence from $w$, then the list obtained from $\\sigma$ by removing the first occurrence of $p$, if any, is a reduction sequence from $w$ plus the allocation of $p$.","title":"Letting a process that fits finish first preserves reduction sequences"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/eq-nil","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.Reduces.eq_nil","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#eq-nil","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Let $w$ be a resource vector. If $\\sigma$ is a reduction sequence from $w$ and no process of $\\sigma$ has a request at most $w$, then $\\sigma$ is empty.","title":"A reduction sequence in which no process fits is empty"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/reduce-aux","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.reduceAux","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#reduce-aux","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Given the fuel, a number of steps, together with the free resource vector $w$ and a list of pending processes, the algorithm returns two lists: the processes that finished, in order, and the processes left over. With fuel $0$, or when no pending process has a request at most $w$, no process finishes and all the pending processes are left over. Otherwise the fuel is $n + 1$ for some $n$; take the first pending\u2026","title":"The detection algorithm with fuel"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/finished","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.finished","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#finished","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Given the free resource vector $w$ and a list of pending processes, run the detection algorithm: repeatedly take the first pending process whose request is at most what is free, let it finish, remove that entry from the list and add its allocation to what is free, until no pending process has a request at most what is free. This is the list of the processes taken, in the order they were taken.","title":"The processes the detection algorithm lets finish"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/detect","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.detect","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#detect","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Given the free resource vector $w$ and a list of pending processes, repeatedly take the first pending process whose request is at most what is free, let it finish, remove that entry from the list and add its allocation to what is free, until no pending process has a request at most what is free. The processes still pending then are the ones reported as deadlocked.","title":"The detection algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/reduce-aux-disjoint","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.reduceAux_disjoint","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#reduce-aux-disjoint","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Let $w$ be the free resource vector, let the list of pending processes contain no duplicates, and let the fuel be at least the number of pending processes. If $\\sigma$ is a duplicate-free reduction sequence from $w$ all of whose processes are pending, then no process of $\\sigma$ is among the processes that the detection algorithm with that fuel leaves over.","title":"The algorithm with fuel leaves over no process of a reduction sequence"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/detect-stuck","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.detect_stuck","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#detect-stuck","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Let $w$ be the free resource vector. For every process $q$ that the detection algorithm reports, the request of $q$ is not at most $w$ plus the sum of the allocations of the processes that the algorithm lets finish: for at least one resource type, $q$ requests more than is free even after all those processes have released their allocations.","title":"Every reported process is stuck"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/finished-reduces","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.finished_reduces","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#finished-reduces","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"For every free resource vector $w$ and every list of pending processes, the processes that the detection algorithm lets finish, in the order it lets them finish, form a reduction sequence from $w$.","title":"The processes the algorithm lets finish form a reduction sequence"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/mem-detect-iff","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.mem_detect_iff","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#mem-detect-iff","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Let $w$ be the free resource vector and let the list of pending processes contain no duplicates. A process $p$ is reported by the detection algorithm if and only if $p$ is pending and no duplicate-free reduction sequence $\\sigma$ from $w$ made of pending processes contains $p$.","title":"Correctness of deadlock detection"},{"id":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection/detect-eq-nil-iff","kind":"node","lean":"Cslib.Algorithms.Concurrency.DeadlockDetection.State.detect_eq_nil_iff","parent":43573,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/deadlock-detection/index.html#detect-eq-nil-iff","scope":"ComputerScience/concurrent-systems-and-protocols/deadlock-detection","status":"fully_proved","summary":"Let $w$ be the free resource vector and let the list of pending processes contain no duplicates. The detection algorithm, started with $w$ free, reports no process if and only if some reordering of the pending processes is a reduction sequence from $w$.","title":"No deadlock is reported exactly when all the processes can finish"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem.json","id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","title":"Dining philosophers problem"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/state","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.State","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#state","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"A state of the three dining philosophers is a triple $(s_0, s_1, s_2)$ with each $s_i \\in \\{0, 1, 2\\}$ the phase of philosopher $i$: $0$ thinking, $1$ holding the first fork, $2$ eating (holding both forks).","title":"Dining-philosophers state"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/phase","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.phase","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#phase","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"For a state $s = (s_0, s_1, s_2)$ of the three dining philosophers and a philosopher $i \\in \\{0, 1, 2\\}$, the phase of philosopher $i$ in $s$ is the component $s_i$.","title":"Phase of a philosopher"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/set-phase","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.setPhase","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#set-phase","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"For a state $s = (s_0, s_1, s_2)$ of the three dining philosophers, a philosopher $i \\in \\{0, 1, 2\\}$ and a phase $p \\in \\{0, 1, 2\\}$, setting the phase of philosopher $i$ to $p$ gives the state obtained from $s$ by replacing the component $s_i$ by $p$ and keeping the other two components.","title":"Updating the phase of a philosopher"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/protocol","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.Protocol","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#protocol","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"A protocol for three philosophers $i \\in \\{0, 1, 2\\}$ sitting between forks $0, 1, 2$ consists of two maps $\\mathrm{first}, \\mathrm{second}$ from $\\{0, 1, 2\\}$ to itself, the fork philosopher $i$ picks up first and the one it picks up second, such that for every $i$ these are its two neighbouring forks $i$ and $i + 1$ (modulo $3$) in some order: either $\\mathrm{first}(i) = i$ and $\\mathrm{second}(i) = i + 1$, or\u2026","title":"Dining-philosophers protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/held","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.held","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#held","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"Given a protocol and a state, fork $f$ is held exactly when some philosopher $i$ either has $f$ as its first fork and is in phase $1$ or $2$, or has $f$ as its second fork and is in phase $2$ (eating).","title":"Held forks"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/can-move","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.canMove","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#can-move","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"Given a protocol and a state, philosopher $i$ can move exactly when it is thinking (phase $0$) and its first fork is not held by any philosopher, or it holds its first fork (phase $1$) and its second fork is not held by any philosopher, or it is eating (phase $2$), in which case its move puts both forks down.","title":"When a philosopher can move"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/move","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.move","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#move","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The state obtained from $s$ when philosopher $i$ moves: the phase of $i$ advances from thinking ($0$) to holding its first fork ($1$), from $1$ to eating ($2$), and from eating back to thinking; the phases of the other philosophers are unchanged. Whether the move is allowed is not checked here.","title":"Move of a philosopher"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/lts","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.lts","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#lts","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"For a protocol $P$, the labelled transition system on the states whose labels are the philosophers: there is a transition from $s$ to $t$ labelled $i$ exactly when philosopher $i$ can move in $s$ under $P$ and $t$ is the state obtained from $s$ by the move of philosopher $i$.","title":"Transition system of a dining-philosophers protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/eating","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.eating","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#eating","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The valuation of the atoms $i \\in \\{0, 1, 2\\}$ on the states of the dining philosophers: atom $i$ holds at a state exactly when philosopher $i$ is eating there, that is, when its phase is $2$.","title":"The eating valuation"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/initial","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.initial","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#initial","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The state in which all three philosophers are thinking, that is, every phase is $0$.","title":"Initial state of the dining philosophers"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/naive","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.naive","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#naive","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The protocol in which every philosopher $i \\in \\{0, 1, 2\\}$ picks up fork $i$ (its left fork) first and fork $i + 1$ (modulo $3$) second.","title":"Naive dining-philosophers protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/ordered","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.ordered","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#ordered","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The protocol in which every philosopher $i \\in \\{0, 1, 2\\}$ picks up the lower-numbered of its two forks $i$ and $i + 1$ (modulo $3$) first and the higher-numbered one second: philosophers $0$ and $1$ take fork $i$ first, and philosopher $2$ takes fork $0$ first and fork $2$ second.","title":"Resource-ordered dining-philosophers protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/deadlock","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.deadlock","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#deadlock","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The CTL formula $\\neg\\,\\mathsf{EX}\\,\\top$ over the philosophers as atoms. It holds at a state exactly when the state has no successor; in the transition system of a protocol, exactly when no philosopher can move.","title":"The deadlock formula"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/neighbours-apart","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.neighboursApart","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#neighbours-apart","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"For a philosopher $i \\in \\{0, 1, 2\\}$, the CTL formula $\\neg(p_i \\wedge p_{i+1})$ over the atoms $p_0, p_1, p_2$ (one per philosopher), with indices modulo $3$. Under the valuation in which $p_j$ holds when philosopher $j$ eats, it says that the neighbouring philosophers $i$ and $i + 1$ do not eat at the same time.","title":"Neighbours-apart formula"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/naive-deadlock-reachable","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.naive_deadlock_reachable","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#naive-deadlock-reachable","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"Consider the transition system of the naive protocol, in which every philosopher $i$ picks up fork $i$ first and fork $i + 1$ second (indices modulo $3$), with atom $i$ read as \"philosopher $i$ is eating\". Its initial state, where all philosophers are thinking, satisfies the CTL formula $\\mathsf{EF}$ applied to the deadlock formula: from it, a state in which no philosopher can move is reachable.","title":"The naive protocol can deadlock"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/naive-deadlock-state","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.naive_deadlock_state","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#naive-deadlock-state","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the transition system of the naive protocol, with atom $i$ read as \"philosopher $i$ is eating\", the state in which every philosopher holds its first fork (every phase is $1$) satisfies the deadlock formula: no philosopher can move there.","title":"A deadlock state of the naive protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/ordered-deadlock-free","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.ordered_deadlock_free","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#ordered-deadlock-free","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the transition system of the three dining philosophers under the resource-ordered protocol, the initial state, in which everybody thinks, satisfies the CTL formula $\\mathrm{AG}\\,\\mathrm{EX}\\,\\top$: every state reachable from it has a successor, that is, some philosopher can move.","title":"Resource ordering prevents deadlock for the dining philosophers"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/ordered-can-eat","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.ordered_can_eat","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#ordered-can-eat","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the transition system of the three dining philosophers under the resource-ordered protocol, with the atom $p_j$ holding when philosopher $j$ eats, the initial state, in which everybody thinks, satisfies the CTL formula $\\mathrm{AG}\\,\\mathrm{EF}\\,p_i$ for every philosopher $i \\in \\{0, 1, 2\\}$: from every state reachable from the initial state, some state in which philosopher $i$ eats is reachable.","title":"Every philosopher can still eat under resource ordering"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/naive-mutex","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.naive_mutex","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#naive-mutex","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"For every philosopher $i$, in the transition system of the naive protocol, with atom $j$ read as \"philosopher $j$ is eating\", the initial state satisfies $\\mathsf{AG}$ applied to the formula that philosophers $i$ and $i + 1$ (indices modulo $3$) do not both eat: in no state reachable from the initial state, where all philosophers are thinking, are philosophers $i$ and $i + 1$ both eating.","title":"Mutual exclusion under the naive protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/ordered-mutex","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.ordered_mutex","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#ordered-mutex","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the transition system of the three dining philosophers under the resource-ordered protocol, with the atom $p_j$ holding when philosopher $j$ eats, the initial state, in which everybody thinks, satisfies the CTL formula $\\mathrm{AG}\\,\\neg(p_i \\wedge p_{i+1})$ for every philosopher $i \\in \\{0, 1, 2\\}$, with indices modulo $3$: in no state reachable from the initial state do philosophers $i$ and $i + 1$ both eat.","title":"Neighbours never eat together under resource ordering"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/can-move-atomic","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.canMoveAtomic","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#can-move-atomic","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"Philosopher $i$ can move in state $s$ when it thinks and neither of its forks $i$ and $i + 1$ (modulo $3$) is held in $s$, or when it eats (it then puts both forks down); a philosopher holding only its first fork cannot move. A fork is held as in the naive protocol: philosopher $j$ holds fork $j$ when it holds its first fork or eats, and holds fork $j + 1$ when it eats.","title":"Enabled moves under atomic acquisition"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/move-atomic","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.moveAtomic","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#move-atomic","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In a state $s$ (the phases of the three philosophers), the move of philosopher $i$ changes only the phase of $i$: if $i$ is thinking, it starts eating, picking up both forks at once; otherwise it goes back to thinking. The phases of the other two philosophers are unchanged.","title":"Move of a philosopher under atomic acquisition"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/atomic-lts","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.atomicLts","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#atomic-lts","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The labelled transition system on the states of the three philosophers, labelled by the philosopher that moves: there is a step labelled $i$ from $s$ to $t$ exactly when either philosopher $i$ thinks in $s$, neither of its forks $i$ and $i + 1$ (modulo $3$) is held in $s$, and $t$ is $s$ with philosopher $i$ eating; or philosopher $i$ eats in $s$ and $t$ is $s$ with philosopher $i$ thinking. A fork is held as in\u2026","title":"Dining philosophers with atomic fork acquisition"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/seated","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.seated","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#seated","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The number of seated philosophers in a state $s$ is the number of philosophers that hold at least one fork in $s$, that is, whose phase is either holding the first fork or eating.","title":"Number of seated philosophers"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/can-move-butler","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.canMoveButler","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#can-move-butler","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"Philosopher $i$ can move in state $s$ when it can move in $s$ under the naive protocol and, if it is thinking, fewer than two philosophers hold at least one fork in $s$. Under the naive protocol each philosopher $i$ picks up the left fork $i$ first and then the right fork $i + 1$ (modulo $3$), each only when no philosopher holds it, and an eating philosopher can always put both forks down.","title":"Enabled moves under a butler"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/butler-lts","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.butlerLts","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#butler-lts","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"The labelled transition system on the states of the three philosophers, labelled by the philosopher that moves: there is a step labelled $i$ from $s$ to $t$ exactly when philosopher $i$ can move in $s$ under the naive protocol, if philosopher $i$ is thinking then fewer than two philosophers hold a fork in $s$, and $t$ is $s$ with philosopher $i$ advanced one phase (thinking to holding its first fork, to eating, to\u2026","title":"Dining philosophers with a butler"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/atomic-deadlock-free","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.atomic_deadlock_free","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#atomic-deadlock-free","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the dining philosophers transition system with atomic acquisition of both forks, every state reachable from the initial state (all three philosophers thinking) has a successor: some philosopher can move.","title":"Atomic acquisition prevents deadlock"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/atomic-can-eat","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.atomic_can_eat","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#atomic-can-eat","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the dining philosophers transition system with atomic acquisition of both forks, for each philosopher $i \\in \\{0, 1, 2\\}$, from every state reachable from the initial state (all three philosophers thinking) a state in which philosopher $i$ eats is reachable, in zero or more steps.","title":"Under atomic acquisition every philosopher can always still eat"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/atomic-mutex","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.atomic_mutex","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#atomic-mutex","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the dining philosophers transition system with atomic acquisition of both forks, for each philosopher $i \\in \\{0, 1, 2\\}$, in no state reachable from the initial state (all three philosophers thinking) do philosopher $i$ and its neighbour $i + 1$ (modulo $3$) both eat.","title":"Mutual exclusion under atomic acquisition"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/butler-deadlock-free","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.butler_deadlock_free","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#butler-deadlock-free","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the dining philosophers transition system of the naive protocol under a butler, every state reachable from the initial state (all three philosophers thinking) has a successor: some philosopher can move.","title":"A butler prevents deadlock"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/butler-can-eat","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.butler_can_eat","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#butler-can-eat","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the dining philosophers transition system of the naive protocol under a butler, for each philosopher $i \\in \\{0, 1, 2\\}$, from every state reachable from the initial state (all three philosophers thinking) a state in which philosopher $i$ eats is reachable, in zero or more steps.","title":"Under a butler every philosopher can always still eat"},{"id":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/butler-mutex","kind":"node","lean":"Cslib.Algorithms.Concurrency.DiningPhilosophers.butler_mutex","parent":43586,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem/index.html#butler-mutex","scope":"ComputerScience/concurrent-systems-and-protocols/dining-philosophers-problem","status":"fully_proved","summary":"In the dining philosophers transition system of the naive protocol under a butler, for each philosopher $i \\in \\{0, 1, 2\\}$, in no state reachable from the initial state (all three philosophers thinking) do philosopher $i$ and its neighbour $i + 1$ (modulo $3$) both eat.","title":"Mutual exclusion under a butler"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol.json","id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","title":"Go-back-N protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/name","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.Name","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#name","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The channel names of the go-back-N protocol are $\\mathsf{acc}$ (accept a message from the user), $\\mathsf{del}$ (deliver a message to the user) and, for every natural number $k$: $\\mathsf{send}\\,k$ (sender to data channel, a frame with sequence number $k$), $\\mathsf{recv}\\,k$ (data channel to receiver), $\\mathsf{ack}\\,k$ (receiver to sender, the cumulative acknowledgement naming $k$ as the next expected frame) and\u2026","title":"Channel names of go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/sender","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.Sender","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#sender","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"A state of the go-back-N sender is a triple of natural numbers: the base of the window (the sequence number of the oldest unacknowledged frame), the number of outstanding messages (accepted but not yet acknowledged), and the offset from the base of the next frame to transmit.","title":"Go-back-N sender state"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/receiver","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.Receiver","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#receiver","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"A receiver state is one of the following, for a natural number $e$ or $k$: expecting the frame with sequence number $e$; about to deliver $e$, having received the expected frame $e$; or about to acknowledge $k$, the next expected sequence number.","title":"Receiver states of go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/ack-update","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.ackUpdate","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#ack-update","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"For a go-back-N sender with window base $b$, $c$ outstanding messages and next-frame offset $o$, and an acknowledgement $k$, let $d = \\max(k + 3 - b, 0) \\bmod 3$. If $1 \\le d \\le c$, the window slides: the new sender has base $k$, $c - d$ outstanding messages and next-frame offset $\\max(o - d, 0)$. Otherwise the sender is unchanged.","title":"Sender update on an acknowledgement"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/sender-steps","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.senderSteps","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#sender-steps","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The transitions of a sender with window base $b$, $n$ outstanding messages and next offset $o$, as pairs of a label and a successor state, in this order: if $n < 2$, an input on $\\mathsf{acc}$ to $(b, n + 1, o)$; if $o < n$, an output on $\\mathsf{send}\\,((b + o) \\bmod 3)$ to $(b, n, o + 1)$; if $0 < n$, a silent timeout step to $(b, n, 0)$; and, for each $k = 0, 1, 2$, an input on $\\mathsf{ack}\\,k$ to the state\u2026","title":"Sender transitions of go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/data-steps","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.dataSteps","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#data-steps","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The data channel holds at most one frame. When empty, for each sequence number $k \\in \\{0, 1, 2\\}$ it can input on channel $\\mathrm{send}_k$ and then hold frame $k$. When holding frame $k$, it can either output on channel $\\mathrm{recv}_k$ or silently lose the frame by a $\\tau$ step; in both cases it becomes empty.","title":"Transitions of the go-back-N data channel"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/receiver-steps","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.receiverSteps","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#receiver-steps","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The transitions of the receiver, as pairs of a label and a successor state. Expecting $e$, for each $k = 0, 1, 2$ it can input on $\\mathsf{recv}\\,k$ and then is about to deliver $e$ if $k = e$, and about to acknowledge $e$ otherwise. About to deliver $e$, it outputs on $\\mathsf{del}$ and is then about to acknowledge $(e + 1) \\bmod 3$. About to acknowledge $k$, it outputs on $\\mathsf{ack}\\,k$ and then expects $k$.","title":"Receiver transitions of go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/buffer-steps","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.bufferSteps","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#buffer-steps","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The transitions of the go-back-N two-place buffer at occupancy $n \\in \\mathbb{N}$: if $n < 2$ it inputs on $\\mathsf{acc}$ and moves to occupancy $n + 1$, and if $n > 0$ it outputs on $\\mathsf{del}$ and moves to occupancy $n - 1$. It has no other transitions.","title":"Transitions of the two-place buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/ack-channel-steps","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.ackChannelSteps","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#ack-channel-steps","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The lossy one-place acknowledgement channel of the lossy go-back-N variant, given by its transitions in each state. When empty, it inputs on $\\mathsf{reply}_k$ and then holds $k$, for each $k \\in \\{0, 1, 2\\}$. When it holds $k$, it either outputs on $\\mathsf{ack}_k$ or silently loses $k$, and in both cases it becomes empty. It has no other transitions.","title":"Transitions of the lossy acknowledgement channel"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lossy-receiver-steps","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.lossyReceiverSteps","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lossy-receiver-steps","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The transitions of the receiver of the lossy variant, as pairs of a label and a successor state, are those of the receiver of the reliable protocol, except in the state about to acknowledge $k$: there the only transition is an output on $\\mathsf{reply}\\,k$, to the acknowledgement channel, after which the receiver expects $k$.","title":"Receiver transitions of lossy go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/agent","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.Agent","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#agent","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The process constants of the go-back-N specification, one for each state of each component: a sender constant for each sender state, a data-channel constant for each channel content (empty or a natural number), a receiver constant for each receiver state, a buffer constant for each occupancy $n \\in \\mathbb{N}$, an acknowledgement-channel constant for each channel content (empty or a natural number), and a constant\u2026","title":"Process constants of the go-back-N model"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/spec","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.spec","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#spec","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The specification assigns to every process constant the choice, over the transitions listed for its state, of the prefixes $a.K$, where $a$ is the label of the transition and $K$ is the process constant for its target state; an empty list gives the inactive process. The constants are those of the sender, the data channel, the receiver, the two-place buffer, the acknowledgement channel and the receiver of the lossy\u2026","title":"Recursive CCS specification of go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lts","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.lts","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lts","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The labelled transition system whose states are the CCS processes over the go-back-N channel names and process constants, whose labels are the CCS labels over the channel names, and whose transitions are those generated by the go-back-N recursive specification.","title":"Labelled transition system of go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/internal","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.internal","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#internal","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The internal channels of the go-back-N protocol are $\\mathrm{send}_k$ (sender to data channel), $\\mathrm{recv}_k$ (data channel to receiver) and $\\mathrm{ack}_k$ (receiver to sender) for $k \\in \\{0, 1, 2\\}$; all of them are restricted.","title":"Internal channels of go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/config","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.Config","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#config","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"A configuration of the go-back-N protocol consists of a sender state, the content of the data channel (either empty or a single frame with sequence number $k \\in \\mathbb{N}$), and a receiver state.","title":"Go-back-N configurations"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/system","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.system","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#system","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"For a configuration consisting of a sender state $s$, a data channel content $d$ (empty or one frame) and a receiver state $r$, the protocol process is the parallel composition $(S_s \\mid D_d) \\mid R_r$ of the process constants for the sender in state $s$, the data channel holding $d$ and the receiver in state $r$, with the internal channels restricted: the send, receive and acknowledgement channels for the\u2026","title":"Go-back-N protocol process in a configuration"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/config-steps","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.configSteps","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#config-steps","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The transitions of a go-back-N configuration (sender, data channel, receiver) are those of the parallel composition of the three components, in which one component moves alone or two components synchronize on complementary labels in a silent step, restricted to the silent steps and the steps whose label is not on an internal channel ($\\mathsf{send}_k$, $\\mathsf{recv}_k$ or $\\mathsf{ack}_k$ for $k \\in \\{0, 1, 2\\}$).","title":"Transitions of go-back-N configurations"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/buffer","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.buffer","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#buffer","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"For an occupancy $n \\in \\mathbb{N}$, the process constant of the go-back-N specification for the two-place buffer holding $n$ messages.","title":"The two-place buffer process"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/initial","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.initial","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#initial","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The initial configuration of the go-back-N protocol consists of the sender with window base $0$, no outstanding messages and next-frame offset $0$; the empty data channel; and the receiver expecting sequence number $0$.","title":"Initial configuration of go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/go-back-n","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.goBackN","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#go-back-n","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The go-back-N protocol with window size $2$ and sequence numbers modulo $3$ is the recursive CCS process in its initial configuration: the parallel composition of the sender with an empty window based at $0$, the empty lossy one-place data channel, and the receiver expecting $0$, with the internal channels $\\mathrm{send}_k$, $\\mathrm{recv}_k$ and $\\mathrm{ack}_k$ for $k \\in \\{0, 1, 2\\}$ restricted.","title":"The go-back-N protocol"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/buf","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.buf","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#buf","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The two-place buffer process of the go-back-N model with occupancy $0$.","title":"The empty two-place buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/certificate","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.certificate","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#certificate","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"A list of $147$ pairs, each a go-back-N configuration together with the occupancy of the two-place buffer it simulates, namely the number of accepted but undelivered messages. It contains every reachable configuration, and it is ordered so that every pair can reach the visible actions of its buffer state through pairs listed earlier.","title":"Weak-bisimulation certificate for go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/go-back-n-weakly-bisimilar-buffer","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.goBackN_weaklyBisimilar_buffer","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#go-back-n-weakly-bisimilar-buffer","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"Over a lossy one-place data channel, with timeouts and cumulative acknowledgements, the protocol is weakly bisimilar to the empty two-place buffer.","title":"Go-back-N with window 2 implements a two-place buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/go-back-n-observation-congruent-buffer","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.goBackN_observationCongruent_buffer","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#go-back-n-observation-congruent-buffer","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The go-back-N protocol is observation congruent (rooted weakly bisimilar) to the empty two-place buffer: every transition of either process is matched by a rooted weak transition of the other with the same label (silent steps, a step with that label, silent steps), and the resulting processes are weakly bisimilar.","title":"Go-back-N with window 2 is observation congruent to a two-place buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lossy-internal","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.lossyInternal","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lossy-internal","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The list of channels restricted in the lossy variant: the internal channels of the reliable protocol, namely $\\mathsf{send}\\,k$, $\\mathsf{recv}\\,k$ and $\\mathsf{ack}\\,k$ for $k = 0, 1, 2$, followed by the channels $\\mathsf{reply}\\,0$, $\\mathsf{reply}\\,1$ and $\\mathsf{reply}\\,2$ from the receiver to the acknowledgement channel.","title":"Internal channels of lossy go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lossy-config","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.LossyConfig","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lossy-config","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"A configuration of the lossy go-back-N variant is a go-back-N configuration (sender state, data channel content, receiver state) together with the content of the acknowledgement channel, either empty or a single acknowledgement $k \\in \\mathbb{N}$.","title":"Configurations of lossy go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lossy-system","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.lossySystem","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lossy-system","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"Let $c$ be a configuration of the lossy variant, and let $S$, $D$, $R$ and $A$ be the process constants for its sender state, its data-channel content, its receiver state (as a receiver of the lossy variant) and its acknowledgement-channel content. The process of the lossy variant in $c$ is $((S \\mid D) \\mid R) \\mid A$ with the internal channels of the lossy variant restricted.","title":"The lossy go-back-N process in a configuration"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lossy-config-steps","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.lossyConfigSteps","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lossy-config-steps","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The transitions of a configuration of the lossy go-back-N variant come from the parallel composition of the sender, the data channel, the lossy-variant receiver and the acknowledgement channel (in each composition either side moves alone, or the two sides synchronize on complementary input and output labels in a silent step), keeping only the silent transitions and those whose label is not on an internal channel\u2026","title":"Transitions of lossy go-back-N configurations"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lossy-go-back-n","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.lossyGoBackN","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lossy-go-back-n","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The process of the lossy variant of go-back-N in its initial configuration: the sender's window is based at $0$ with no outstanding messages and next offset $0$, the data channel and the acknowledgement channel are empty, and the receiver expects frame $0$.","title":"Go-back-N with a lossy acknowledgement channel"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lossy-step","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.lossy_step","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lossy-step","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"Let $c$ and $c'$ be configurations of the lossy variant and $a$ a label. If $(a, c')$ is one of the configuration-level transitions of $c$, then in the labelled transition system of the go-back-N specification the lossy process in configuration $c$ has a transition labelled $a$ to the lossy process in configuration $c'$.","title":"Configuration transitions are process transitions in lossy go-back-N"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/buffer-silent-closure","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.buffer_silentClosure","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#buffer-silent-closure","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"For every occupancy $n$, every process reachable from the go-back-N buffer with occupancy $n$ by zero or more silent steps is that buffer itself.","title":"The silent closure of the buffer is trivial"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/buffer-weak-step","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.buffer_weakStep","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#buffer-weak-step","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"For every occupancy $n$ and every visible label $a$, if the go-back-N buffer with occupancy $n$ reaches a process $Q$ by a weak transition labelled $a$ (silent steps, one $a$-step, silent steps), then $Q$ is the buffer with some occupancy $m$ such that the buffer at occupancy $n$ has a transition labelled $a$ to occupancy $m$.","title":"Visible weak steps of the buffer"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/stuck-closure","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.stuckClosure","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#stuck-closure","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"The list of the seven configurations of go-back-N with a lossy acknowledgement channel in which the sender's window has base $0$ with two outstanding messages, the receiver holds frame $1$ about to deliver it, and the acknowledgement channel is empty, while the sender's offset of the next frame to transmit and the data channel are, in order: offset $2$ and empty, offset $0$ and empty, offset $1$ holding frame $0$,\u2026","title":"Silent closure of the stuck lossy go-back-N configuration"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/stuck-not-has-out-label-acc","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.stuck_not_hasOutLabel_acc","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#stuck-not-has-out-label-acc","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"Consider go-back-N with a lossy acknowledgement channel in the configuration where the sender's window has base $0$ with two outstanding messages, both transmitted (offset of the next frame $2$), the data channel and the acknowledgement channel are empty, and the receiver holds frame $1$ about to deliver it. This process has no weak transition labelled by accepting a message: no sequence of silent steps followed by\u2026","title":"The stuck lossy go-back-N configuration refuses new messages"},{"id":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/lossy-go-back-n-not-weakly-bisimilar-buf","kind":"node","lean":"Cslib.Algorithms.CCS.GoBackN.lossyGoBackN_not_weaklyBisimilar_buffer","parent":43619,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol/index.html#lossy-go-back-n-not-weakly-bisimilar-buf","scope":"ComputerScience/concurrent-systems-and-protocols/go-back-n-protocol","status":"fully_proved","summary":"With a lossy acknowledgement channel and the deliver-then-acknowledge receiver, go-back-N is not weakly bisimilar to the two-place buffer. Accept two messages, deliver the first and lose its acknowledgement, and let the receiver take the second frame: the buffer now holds one message and accepts another, but the protocol cannot accept before it delivers.","title":"Lossy acknowledgements break the buffer specification"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/lamport-clocks.json","id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","title":"Lamport clocks"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/event","kind":"node","lean":"Cslib.Distributed.LamportClock.Event","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#event","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"An event of an execution, for a type $P$ of processes, consists of the process in $P$ it belongs to and, if it is a receipt, a natural number: the position in the execution of the event whose message it receives. The structure does not require that position to be earlier than the event itself.","title":"Event"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/well-formed","kind":"node","lean":"Cslib.Distributed.LamportClock.WellFormed","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#well-formed","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"An execution $e_0, \\dots, e_{n-1}$ is well formed when every receipt names an earlier event: whenever $e_j$ receives the message sent by the event at position $i$, then $i < j$.","title":"Well-formed executions"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/step","kind":"node","lean":"Cslib.Distributed.LamportClock.Step","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#step","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"In an execution, position $i$ precedes position $j$ by one step when $i < j$, both positions hold events, and either the two events belong to the same process or the event at $j$ receives the message of the event at $i$.","title":"One step of happened-before"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/happens-before","kind":"node","lean":"Cslib.Distributed.LamportClock.HappensBefore","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#happens-before","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"Lamport's happened-before relation on the positions of an execution is the transitive closure of one step of happened-before: $a$ happened before $b$ when there is a chain $a = i_0, i_1, \\dots, i_k = b$ with $k \\ge 1$ in which each $i_m$ precedes $i_{m+1}$ by one step.","title":"Happened-before"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/lt","kind":"node","lean":"Cslib.Distributed.LamportClock.HappensBefore.lt","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#lt","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"If position $a$ happened before position $b$ in an execution, then $a < b$.","title":"Happened-before goes forward in the execution"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/clock-step","kind":"node","lean":"Cslib.Distributed.LamportClock.clockStep","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#clock-step","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"The state is a counter $c(p) \\in \\mathbb{N}$ for each process $p$ together with the timestamps assigned so far. An event $e$ of process $p$ gets the timestamp $t = \\max(c(p), s) + 1$, where $s$ is the timestamp already assigned at the position that $e$ receives from; $s = 0$ if $e$ is not a receipt or if no timestamp has been assigned at that position. The step sets $c(p)$ to $t$ and appends $t$ to the timestamps.","title":"One step of the Lamport clock algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/run","kind":"node","lean":"Cslib.Distributed.LamportClock.run","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#run","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"The clock algorithm over an execution applies one step of the algorithm to each event in order, starting from all process counters $0$ and no timestamps. The result is the final counters together with the timestamps assigned to the events, in order.","title":"Running the Lamport clock algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/timestamps","kind":"node","lean":"Cslib.Distributed.LamportClock.timestamps","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#timestamps","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"The Lamport timestamps $t_0, \\dots, t_{n-1}$ of an execution $e_0, \\dots, e_{n-1}$ are computed by Lamport's clock algorithm. Every process keeps a counter, initially $0$. For $k = 0, \\dots, n-1$ in turn, let $c$ be the current counter of the process of $e_k$, and let $m = t_i$ if $e_k$ receives the message sent by the event at position $i$ with $i < k$, and $m = 0$ otherwise. Then $t_k = \\max(c, m) + 1$, and the\u2026","title":"Lamport timestamps"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/timestamps-append","kind":"node","lean":"Cslib.Distributed.LamportClock.timestamps_append","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#timestamps-append","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"Let $e_0, \\dots, e_{n-1}$ be an execution, and extend it by appending further events. For every $i < n$, the Lamport timestamp of the event at position $i$ is the same in the extended execution as in the original one.","title":"Appending events does not change earlier timestamps"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/timestamp-le-clock","kind":"node","lean":"Cslib.Distributed.LamportClock.timestamp_le_clock","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#timestamp-le-clock","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"After the Lamport clock algorithm has run over an execution, the counter of each process is at least the timestamp of every event of that process in the execution.","title":"Process counters dominate their timestamps"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/timestamps-get-d-take","kind":"node","lean":"Cslib.Distributed.LamportClock.timestamps_getD_take","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#timestamps-get-d-take","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"Let $e_0, \\dots, e_{n-1}$ be an execution and let $i < k \\le n$. The Lamport timestamp of $e_i$ in the whole execution equals its Lamport timestamp in the prefix $e_0, \\dots, e_{k-1}$.","title":"A timestamp depends only on a prefix containing its event"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/step-lt","kind":"node","lean":"Cslib.Distributed.LamportClock.step_lt","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#step-lt","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"If position $i$ precedes position $j$ by one step of happened-before in an execution, then the Lamport timestamp of $i$ is smaller than the Lamport timestamp of $j$.","title":"One step of happened-before raises the timestamp"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/timestamp-lt-of-recv","kind":"node","lean":"Cslib.Distributed.LamportClock.timestamp_lt_of_recv","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#timestamp-lt-of-recv","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"Let $e_0, \\dots, e_{n-1}$ be a well-formed execution and let $t_k$ be the Lamport timestamp of $e_k$. If $j < n$ and $e_j$ receives the message sent by the event at position $i$, then $t_i < t_j$.","title":"A receipt is stamped later than its send"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/timestamp-lt-of-proc","kind":"node","lean":"Cslib.Distributed.LamportClock.timestamp_lt_of_proc","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#timestamp-lt-of-proc","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"Let $e_0, \\dots, e_{n-1}$ be an execution and let $t_k$ be the Lamport timestamp of $e_k$. If $i < j < n$ and $e_i$ and $e_j$ belong to the same process, then $t_i < t_j$.","title":"Lamport timestamps increase along each process"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/clock-condition","kind":"node","lean":"Cslib.Distributed.LamportClock.clock_condition","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#clock-condition","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"In every execution, if the event at position $a$ happened before the event at position $b$, then the Lamport timestamp of $a$ is smaller than the Lamport timestamp of $b$ (Lamport 1978).","title":"The clock condition"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/exists-events","kind":"node","lean":"Cslib.Distributed.LamportClock.HappensBefore.exists_events","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#exists-events","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"If position $a$ happened before position $b$ in an execution, then the execution has an event at position $a$ and an event at position $b$.","title":"Happened-before relates positions of the execution"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/total-before","kind":"node","lean":"Cslib.Distributed.LamportClock.TotalBefore","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#total-before","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"Fix a linear order on the processes. In an execution $e_0, \\dots, e_{n-1}$ with Lamport timestamps $t_0, \\dots, t_{n-1}$, position $a$ precedes position $b$ when $a < n$, $b < n$, and either $t_a < t_b$, or $t_a = t_b$ and the process of $e_a$ is smaller than the process of $e_b$.","title":"Lamport's total order"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/total-before-of-happens-before","kind":"node","lean":"Cslib.Distributed.LamportClock.totalBefore_of_happensBefore","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#total-before-of-happens-before","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"In every execution, if the event at position $a$ happened before the event at position $b$, then position $a$ precedes position $b$ in Lamport's total order.","title":"Lamport's total order extends happened-before"},{"id":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks/total-before-total","kind":"node","lean":"Cslib.Distributed.LamportClock.totalBefore_total","parent":43655,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/lamport-clocks/index.html#total-before-total","scope":"ComputerScience/concurrent-systems-and-protocols/lamport-clocks","status":"fully_proved","summary":"In every execution, for any two distinct positions $a \\ne b$ that both hold events, $a$ precedes $b$ or $b$ precedes $a$ in Lamport's total order.","title":"Lamport's order is total"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/two-phase-commit.json","id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","title":"Two-phase commit"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/coord-state","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.CoordState","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#coord-state","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"In two-phase commit the coordinator is in one of three states: waiting (collecting votes), committed (it has decided commit) or aborted (it has decided abort).","title":"Coordinator states"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/part-state","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.PartState","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#part-state","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"A participant is in one of four states: working (it has not voted), prepared (it voted yes and waits for the decision), committed, or aborted (it voted no, or learned of an abort).","title":"Participant states in two-phase commit"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/state","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.State","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#state","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"For a set $P$ of participants, a global state consists of a coordinator state together with a participant state for each participant $p \\in P$.","title":"Global states of two-phase commit"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/action","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.Action","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#action","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"The actions of the two-phase commit protocol over a set $P$ of participants are: a participant $p \\in P$ votes yes; a participant $p$ votes no; the coordinator decides commit; the coordinator decides abort; a participant $p$ learns the commit decision; and a participant $p$ learns the abort decision.","title":"Two-phase commit actions"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/set-part","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.State.setPart","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#set-part","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"For a global state $s$, a participant $p$ and a participant state $x$, the updated state has the same coordinator state as $s$, gives participant $p$ the state $x$, and leaves the state of every other participant unchanged.","title":"Updating one participant's state"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/step","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.Step","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#step","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"The protocol steps from a global state $s$ to a global state $t$ by an action exactly in the following cases. A working participant $p$ may vote yes, after which $p$ is prepared, or vote no, after which $p$ is aborted. If the coordinator is waiting and every participant is prepared, the coordinator may decide commit, after which it is committed. If the coordinator is waiting, it may decide abort, after which it is\u2026","title":"Transitions of two-phase commit"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/lts","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.lts","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#lts","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"For a set $P$ of participants, the labelled transition system whose states are the global states of the protocol, whose labels are the protocol actions, and whose transitions are the protocol steps.","title":"Two-phase commit as a transition system"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/initial","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.initial","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#initial","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"The initial state of the two-phase commit protocol: the coordinator is waiting and every participant is working.","title":"Initial state of two-phase commit"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/reachable","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.Reachable","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#reachable","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"A global state $s$ is reachable when some finite sequence of protocol actions leads by protocol steps from the initial state, in which the coordinator is waiting and every participant is working, to $s$.","title":"Reachable states of two-phase commit"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/step-2","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.Reachable.step","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#step-2","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"If a global state $s$ of the two-phase commit protocol is reachable and some action takes $s$ to $t$ in one protocol step, then $t$ is reachable.","title":"Reachable states are closed under steps"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/inv","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.Inv","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#inv","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"A global state $s$ of the two-phase commit protocol satisfies the invariant when two conditions hold: if the coordinator has committed, then every participant is prepared or committed; and every committed participant has a committed coordinator.","title":"Two-phase commit invariant"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/inv-mtr","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.inv_mtr","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#inv-mtr","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"If a state $s$ of the two-phase commit protocol satisfies the inductive invariant and a state $t$ is reached from $s$ by a finite sequence of transitions, then $t$ satisfies the invariant.","title":"Multistep transitions preserve the two-phase commit invariant"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/inv-reachable","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.inv_reachable","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#inv-reachable","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"For every set $P$ of participants, every global state of the two-phase commit protocol that is reachable from the initial state satisfies the inductive invariant, that is, both of the following hold: if the coordinator has committed, then every participant is prepared or committed; and every committed participant has a committed coordinator.","title":"The two-phase commit invariant holds in reachable states"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/atomicity","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.atomicity","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#atomicity","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"For every set $P$ of participants, in every reachable state of the two-phase commit protocol, if some participant $p$ has committed, then no participant $q$ has aborted.","title":"Atomicity between participants"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/coord-committed-imp-all-voted-yes","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.coord_committed_imp_all_voted_yes","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#coord-committed-imp-all-voted-yes","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"In every reachable state of the two-phase commit protocol, if the coordinator has decided commit, then every participant voted yes, that is, every participant is prepared or committed.","title":"Validity for the coordinator"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/committed-imp-all-voted-yes","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.committed_imp_all_voted_yes","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#committed-imp-all-voted-yes","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"In every reachable state of the two-phase commit protocol, if some participant has committed, then every participant voted yes, that is, every participant is prepared or committed.","title":"Validity for participants"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/aborted-imp-not-committed","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.aborted_imp_not_committed","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#aborted-imp-not-committed","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"In every reachable state of the two-phase commit protocol, if some participant has aborted, then the coordinator has not decided commit.","title":"An aborted participant rules out a coordinator commit"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/part-committed-coord-committed","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.part_committed_coord_committed","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#part-committed-coord-committed","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"In every reachable state of the two-phase commit protocol, if some participant has committed, then the coordinator has committed.","title":"A committed participant has a committed coordinator"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/coord-committed-part-not-aborted","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.coord_committed_part_not_aborted","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#coord-committed-part-not-aborted","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"In every reachable state of the two-phase commit protocol, if the coordinator has decided commit, then no participant has aborted.","title":"A committed coordinator has no aborted participant"},{"id":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit/coord-aborted-part-not-committed","kind":"node","lean":"Cslib.Distributed.TwoPhaseCommit.coord_aborted_part_not_committed","parent":43675,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/two-phase-commit/index.html#coord-aborted-part-not-committed","scope":"ComputerScience/concurrent-systems-and-protocols/two-phase-commit","status":"fully_proved","summary":"In every reachable state of the two-phase commit protocol, if the coordinator has decided abort, then no participant has committed.","title":"An aborted coordinator has no committed participant"},{"data_url":"scopes/ComputerScience/concurrent-systems-and-protocols/vector-clocks.json","id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","kind":"scope","navigation":"module","parent":43445,"result":false,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","title":"Vector clocks"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/stamp","kind":"node","lean":"Cslib.Distributed.VectorClock.stamp","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#stamp","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"Let $P$ be the set of processes. Given the current vectors $V_q \\in \\mathbb{N}^P$ of the processes $q$ and the timestamps $T_0, T_1, \\dots$ assigned so far, the vector clock algorithm stamps an event $e$ of process $p$ as follows. Let $m = T_i$ if $e$ receives the message sent by the event at position $i$ and $T_i$ has been assigned, and $m = 0$ otherwise. The stamp of $e$ is the pointwise maximum of $V_p$ and $m$,\u2026","title":"Vector clock stamp of an event"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/clock-step","kind":"node","lean":"Cslib.Distributed.VectorClock.clockStep","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#clock-step","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"The state of the algorithm holds a vector $V_q \\in \\mathbb{N}^P$ for each process $q$, where $P$ is the set of processes, and the timestamps assigned so far, kept in an array for constant-time append and lookup. On an event $e$ of process $p$, the step replaces $V_p$ by the stamp $s$ of $e$ computed from the current state, leaves the other vectors unchanged, and appends $s$ to the timestamps.","title":"One step of the vector clock algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/run","kind":"node","lean":"Cslib.Distributed.VectorClock.run","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#run","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"The state of the vector clock algorithm after an execution is obtained by starting from the state in which every process's vector is $0$ and no timestamps have been assigned, and applying one step of the algorithm to each event of the execution in order.","title":"Running the vector clock algorithm"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/timestamps","kind":"node","lean":"Cslib.Distributed.VectorClock.timestamps","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#timestamps","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"The vector timestamps of an execution are the list of vectors that the vector clock algorithm, run over the execution, assigns to its events, in order.","title":"Vector timestamps of an execution"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/run-get-d","kind":"node","lean":"Cslib.Distributed.VectorClock.run_getD","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#run-get-d","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"For every execution and every natural number $i$, the entry at index $i$ of the array of timestamps built by the vector clock algorithm on the execution equals the $i$-th vector timestamp of the execution, where an index out of range gives the zero vector in both lookups.","title":"Array lookup agrees with the timestamp list"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/timestamps-get-d-eq-stamp","kind":"node","lean":"Cslib.Distributed.VectorClock.timestamps_getD_eq_stamp","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#timestamps-get-d-eq-stamp","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"For every execution and every position $j$ of an event $e$ of it, the vector timestamp of $j$ equals the stamp of $e$ computed from the state of the vector clock algorithm after the first $j$ events of the execution.","title":"Timestamps are stamps from the preceding state"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/timestamp-le-vector","kind":"node","lean":"Cslib.Distributed.VectorClock.timestamp_le_vector","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#timestamp-le-vector","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"For every execution and every position $i$ of an event of it, the vector timestamp of $i$ is pointwise at most the vector of that event's process in the state of the vector clock algorithm after the whole execution.","title":"Each process's vector dominates its events' timestamps"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/vector-eq","kind":"node","lean":"Cslib.Distributed.VectorClock.vector_eq","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#vector-eq","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"Run the vector clock algorithm over an execution, starting from all vectors $0$. For every process $p$, the vector held by $p$ at the end is either the zero vector or the vector timestamp of some event of $p$ in the execution.","title":"A process's final vector is zero or one of its timestamps"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/one-le-own","kind":"node","lean":"Cslib.Distributed.VectorClock.one_le_own","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#one-le-own","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"For every position $j$ of an event of an execution, if $p$ is the process of that event, then the $p$-entry of the vector timestamp of $j$ is at least $1$.","title":"An event's own entry is positive"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/vector-clock-le-of-step","kind":"node","lean":"Cslib.Distributed.VectorClock.vectorClock_le_of_step","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#vector-clock-le-of-step","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"In an execution, write $V_i$ for the vector timestamp that the vector clock algorithm gives to the event at position $i$. If the event at position $j$ follows the event at position $i$ by one step of happened-before, then $V_i(q) \\le V_j(q)$ for every process $q$.","title":"Vector timestamps are monotone along one step of happened-before"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/vector-clock-le-of-happens-before","kind":"node","lean":"Cslib.Distributed.VectorClock.vectorClock_le_of_happensBefore","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#vector-clock-le-of-happens-before","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"In an execution, write $V_i$ for the vector timestamp that the vector clock algorithm gives to the event at position $i$. If the event at position $a$ happened before the event at position $b$, then $V_a(q) \\le V_b(q)$ for every process $q$.","title":"Vector timestamps are monotone along happened-before"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/eq-or-happens-before-of-le","kind":"node","lean":"Cslib.Distributed.VectorClock.eq_or_happensBefore_of_le","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#eq-or-happens-before-of-le","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"Let $a$ and $b$ be positions of events of an execution, and let $p$ be the process of the event at $a$. If the $p$-entry of the vector timestamp of $a$ is at most the $p$-entry of the vector timestamp of $b$, then $a = b$ or $a$ happened before $b$.","title":"Own-entry comparison detects happened-before"},{"id":"ComputerScience/concurrent-systems-and-protocols/vector-clocks/happens-before-iff-lt","kind":"node","lean":"Cslib.Distributed.VectorClock.happensBefore_iff_lt","parent":43696,"result":true,"route":"../roadmap/ComputerScience/concurrent-systems-and-protocols/vector-clocks/index.html#happens-before-iff-lt","scope":"ComputerScience/concurrent-systems-and-protocols/vector-clocks","status":"fully_proved","summary":"For two events of an execution, `a` happened before `b` exactly when `a`'s timestamp is smaller than `b`'s in the pointwise order.","title":"Vector clocks characterize happened-before"},{"data_url":"scopes/ComputerScience/program-logics-and-verification.json","id":"ComputerScience/program-logics-and-verification","kind":"scope","parent":36417,"result":false,"route":"../roadmap/ComputerScience/program-logics-and-verification/index.html","scope":"ComputerScience/program-logics-and-verification","status":"fully_proved","summary":"Logics for proving programs correct: Hoare logic for partial and total correctness with soundness and relative completeness, weakest preconditions, verification-condition generation, and separation logic.","title":"Program logics and verification"},{"data_url":"scopes/ComputerScience/program-logics-and-verification/hoare-logic.json","id":"ComputerScience/program-logics-and-verification/hoare-logic","kind":"scope","navigation":"module","parent":43710,"result":false,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","title":"Hoare logic"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/assertion","kind":"node","lean":"Cslib.IMP.Assertion","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#assertion","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"An assertion is a predicate on program states: it assigns a proposition to each state. Assertions serve as the preconditions and postconditions of Hoare triples.","title":"Assertion"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/entails","kind":"node","lean":"Cslib.IMP.Assertion.Entails","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#entails","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"An assertion $P$ entails an assertion $Q$ when every state that satisfies $P$ also satisfies $Q$.","title":"Entailment of assertions"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/subst","kind":"node","lean":"Cslib.IMP.Assertion.subst","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#subst","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $P$ be an assertion, $x$ a variable and $e$ an arithmetic expression. The assertion $P[e/x]$ holds in a state $s$ exactly when $P$ holds in the state obtained from $s$ by setting the value of $x$ to the value of $e$ in $s$.","title":"Substitution in an assertion"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/valid","kind":"node","lean":"Cslib.IMP.Valid","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#valid","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For assertions $P$ and $Q$ and a command $c$, the partial-correctness Hoare triple $\\lbrace P \\rbrace\\ c\\ \\lbrace Q \\rbrace$ is valid when, for all states $s$ and $t$, if $s$ satisfies $P$ and executing $c$ from $s$ terminates in $t$ under the big-step semantics, then $t$ satisfies $Q$.","title":"Validity of a Hoare triple"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/while-loop","kind":"node","lean":"Cslib.IMP.Valid.whileLoop","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#while-loop","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $I$ be an assertion, $b$ a Boolean expression and $c$ a command. Write $I \\wedge b$ for the assertion that $I$ holds and $b$ evaluates to true, and $I \\wedge \\neg b$ for the assertion that $I$ holds and $b$ evaluates to false. If the partial-correctness Hoare triple $\\lbrace I \\wedge b \\rbrace\\ c\\ \\lbrace I \\rbrace$ is valid, then so is the triple\u2026","title":"Invariant rule for while loops"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/wlp","kind":"node","lean":"Cslib.IMP.wlp","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#wlp","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For a command $c$ and an assertion $Q$, the weakest liberal precondition of $c$ with respect to $Q$ is the assertion that holds in a state $s$ exactly when every state $t$ in which an execution of $c$ from $s$ terminates, under the big-step semantics, satisfies $Q$.","title":"Weakest liberal precondition"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/while-loop-2","kind":"node","lean":"Cslib.IMP.wlp.whileLoop","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#while-loop-2","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $b$ be a Boolean expression, $c$ a command and $Q$ an assertion. Write $\\mathrm{wlp}(d, R)$ for the weakest liberal precondition of a command $d$ with respect to an assertion $R$, and let $W$ be the loop $\\mathbf{while}\\ b\\ \\mathbf{do}\\ c$. Then $\\mathrm{wlp}(W, Q)$ is the assertion that holds in a state $s$ exactly when both of the following hold: if $b$ evaluates to false in $s$, then $Q$ holds in $s$; and if\u2026","title":"Weakest liberal precondition unfolding of a while loop"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/invariant-le-while","kind":"node","lean":"Cslib.IMP.wlp.invariant_le_while","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#invariant-le-while","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $I$ and $Q$ be assertions, $b$ a Boolean expression and $c$ a command. Suppose that every state satisfying $I$ in which $b$ evaluates to true satisfies the weakest liberal precondition of $c$ with respect to $I$, and that every state satisfying $I$ in which $b$ evaluates to false satisfies $Q$. Then $I$ entails the weakest liberal precondition of the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$ with respect to $Q$.","title":"Loop invariants entail the weakest liberal precondition of the loop"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/tvalid","kind":"node","lean":"Cslib.IMP.TValid","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#tvalid","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For assertions $P$ and $Q$ on states and a command $c$, the total-correctness Hoare triple with precondition $P$, command $c$ and postcondition $Q$ is valid when every state $s$ satisfying $P$ has a terminating big-step execution of $c$ from $s$ ending in a state $t$ that satisfies $Q$.","title":"Validity of total-correctness Hoare triples"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/while-variant","kind":"node","lean":"Cslib.IMP.TValid.whileVariant","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#while-variant","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $I$ be an assertion (the invariant), $V$ a function from states to natural numbers (the variant), $b$ a Boolean guard and $c$ a command. Suppose that for every state $s$ satisfying $I$ in which $b$ evaluates to true, there is a terminating big-step execution of $c$ from $s$ ending in a state $s'$ that satisfies $I$ with $V(s') < V(s)$. Then the total-correctness Hoare triple with precondition $I$, command\u2026","title":"Variant rule for while loops"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/valid-2","kind":"node","lean":"Cslib.IMP.TValid.valid","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#valid-2","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For assertions $P$, $Q$ and a command $c$, if the total-correctness Hoare triple with precondition $P$, command $c$ and postcondition $Q$ is valid, then the partial-correctness Hoare triple with the same precondition, command and postcondition is valid.","title":"Total correctness implies partial correctness"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/terminates","kind":"node","lean":"Cslib.IMP.TValid.terminates","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#terminates","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"If the total-correctness Hoare triple with precondition $P$, command $c$ and postcondition $Q$ is valid and a state $s$ satisfies $P$, then there is a state $t$ such that $c$, started in $s$, reaches $\\mathtt{skip}$ with state $t$ in finitely many small steps, and $t$ satisfies $Q$.","title":"Total correctness gives a terminating small-step run"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/wp","kind":"node","lean":"Cslib.IMP.wp","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#wp","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For a command $c$ and a postcondition $Q$, the weakest precondition $\\mathrm{wp}(c, Q)$ is the assertion that holds at a state $s$ exactly when there is a terminating big-step execution of $c$ from $s$ ending in a state that satisfies $Q$.","title":"Weakest precondition for total correctness"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/while-iff","kind":"node","lean":"Cslib.IMP.wp.while_iff","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#while-iff","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For a Boolean guard $b$, a loop body $c$, a postcondition $Q$ and a state $s$, the state $s$ satisfies the total-correctness weakest precondition $\\mathrm{wp}(\\mathtt{while}\\ b\\ \\mathtt{do}\\ c, Q)$ if and only if either $b$ evaluates to false in $s$ and $s$ satisfies $Q$, or $b$ evaluates to true in $s$ and $s$ satisfies $\\mathrm{wp}(c, \\mathrm{wp}(\\mathtt{while}\\ b\\ \\mathtt{do}\\ c, Q))$.","title":"Weakest-precondition unfolding of a while loop"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/le-wlp","kind":"node","lean":"Cslib.IMP.wp.le_wlp","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#le-wlp","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For every command $c$, postcondition $Q$ and state $s$, if $s$ satisfies the total-correctness weakest precondition $\\mathrm{wp}(c, Q)$, then $s$ satisfies the weakest liberal precondition $\\mathrm{wlp}(c, Q)$.","title":"Total-correctness weakest precondition implies the weakest liberal precondition"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/derivable","kind":"node","lean":"Cslib.IMP.Derivable","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#derivable","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Here an assertion is a predicate on states, and an assertion $P$ entails an assertion $P'$ when every state satisfying $P$ satisfies $P'$. For an assertion $P$ and a Boolean expression $b$, the assertion $P \\land b$ holds in a state when $P$ holds there and $b$ evaluates to true in it, and the assertion $P \\land \\lnot b$ holds in a state when $P$ holds there and $b$ evaluates to false in it. Derivability of\u2026","title":"Derivability in Hoare logic for partial correctness"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/sound","kind":"node","lean":"Cslib.IMP.Derivable.sound","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#sound","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Every partial-correctness Hoare triple $\\{P\\}\\,c\\,\\{Q\\}$ that is derivable in the partial-correctness proof system is valid: for all states $s$ and $s'$, if $s$ satisfies $P$ and $c$ has a terminating big-step execution from $s$ to $s'$, then $s'$ satisfies $Q$.","title":"Soundness of Hoare logic for partial correctness"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/strengthen","kind":"node","lean":"Cslib.IMP.Derivable.strengthen","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#strengthen","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $P$, $P'$ and $Q$ be assertions and $c$ a command. If $P$ entails $P'$ (every state satisfying $P$ satisfies $P'$) and the partial-correctness triple $\\{P'\\}\\,c\\,\\{Q\\}$ is derivable in the partial-correctness proof system, then so is $\\{P\\}\\,c\\,\\{Q\\}$.","title":"Precondition strengthening for partial correctness"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/derivable-wlp","kind":"node","lean":"Cslib.IMP.derivable_wlp","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#derivable-wlp","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For every command $c$ and every assertion $Q$, the triple whose precondition is the weakest liberal precondition of $c$ with respect to $Q$, whose command is $c$ and whose postcondition is $Q$ is derivable in partial-correctness Hoare logic.","title":"The weakest liberal precondition gives a derivable triple"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/complete","kind":"node","lean":"Cslib.IMP.Derivable.complete","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#complete","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For all assertions $P$ and $Q$ and every command $c$, if the total-correctness triple with precondition $P$, command $c$ and postcondition $Q$ is valid, then it is derivable in total-correctness Hoare logic.","title":"Relative completeness of total-correctness Hoare logic"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/derivable-iff-valid","kind":"node","lean":"Cslib.IMP.derivable_iff_valid","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#derivable-iff-valid","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"A Hoare triple, given by a precondition $P$, a command $c$ and a postcondition $Q$, is derivable in the default inference system on Hoare triples, which is partial-correctness Hoare logic, if and only if it is valid for partial correctness.","title":"Partial-correctness Hoare logic as an inference system"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/derivable-iff-denote","kind":"node","lean":"Cslib.IMP.derivable_iff_denote","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#derivable-iff-denote","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For all assertions $P$ and $Q$ and every command $c$, the partial-correctness triple with precondition $P$, command $c$ and postcondition $Q$ is derivable if and only if, for all states $s$ and $t$ such that $P$ holds at $s$ and the pair $(s, t)$ lies in the relational denotation of $c$, $Q$ holds at $t$.","title":"Partial-correctness Hoare logic matches the denotational semantics"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/iterates","kind":"node","lean":"Cslib.IMP.Iterates","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#iterates","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $b$ be a Boolean expression and $c$ a command. For a natural number $n$ and states $s$, $s'$, the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$ runs from $s$ to $s'$ in $n$ iterations when, started in $s$, it exits in $s'$ after exactly $n$ executions of the body $c$. This is the least relation such that the loop runs from $s$ to $s$ in $0$ iterations whenever $b$ evaluates to false in $s$, and the loop runs from\u2026","title":"Loop runs with a given number of iterations"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/big-step","kind":"node","lean":"Cslib.IMP.Iterates.bigStep","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#big-step","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $b$ be a boolean expression, $c$ a command, $n$ a natural number and $s$, $t$ states. If the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$, started in $s$, exits in $t$ after exactly $n$ executions of the body $c$, then the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$ runs from $s$ to $t$ in the big-step semantics.","title":"Counted loop runs are big-step executions"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/exists-of-big-step","kind":"node","lean":"Cslib.IMP.Iterates.exists_of_bigStep","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#exists-of-big-step","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $b$ be a Boolean expression, $c$ a command and $s$, $s'$ states. If the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$ has a terminating big-step execution from $s$ to $s'$, then there is a natural number $n$ such that the loop, started in $s$, exits in $s'$ after exactly $n$ executions of the body $c$.","title":"Every terminating loop run has an iteration count"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/count-unique","kind":"node","lean":"Cslib.IMP.Iterates.count_unique","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#count-unique","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $b$ be a boolean expression, $c$ a command and $s$ a state. If the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$, started in $s$, exits in a state $t$ after exactly $n$ executions of the body and also exits in a state $t'$ after exactly $m$ executions of the body, then $n = m$.","title":"Uniqueness of the loop iteration count"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/iteration-count","kind":"node","lean":"Cslib.IMP.iterationCount","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#iteration-count","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $b$ be a Boolean expression, $c$ a command and $s$ a state. If there are a natural number $n$ and a state $s'$ such that the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$, started in $s$, exits in $s'$ after exactly $n$ executions of the body $c$, then the iteration count of the loop from $s$ is such an $n$; otherwise it is $0$.","title":"Loop iteration count"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/iteration-count-eq","kind":"node","lean":"Cslib.IMP.Iterates.iterationCount_eq","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#iteration-count-eq","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $b$ be a Boolean expression, $c$ a command, $n$ a natural number and $s$, $s'$ states. If the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$, started in $s$, exits in $s'$ after exactly $n$ executions of the body $c$, then the iteration count of the loop from $s$ is $n$.","title":"The iteration count of a terminating loop run"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/iteration-count-eq-zero","kind":"node","lean":"Cslib.IMP.iterationCount_eq_zero","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#iteration-count-eq-zero","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $b$ be a Boolean expression, $c$ a command and $s$ a state. If the loop $\\mathtt{while}\\ b\\ \\mathtt{do}\\ c$ has no terminating big-step execution from $s$ to any state, then the iteration count of the loop from $s$ is $0$.","title":"The iteration count of a nonterminating loop is zero"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/tderivable","kind":"node","lean":"Cslib.IMP.TDerivable","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#tderivable","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Derivability of total-correctness Hoare triples $\\{P\\}\\,c\\,\\{Q\\}$, for assertions $P$, $Q$ (predicates on states) and IMP commands $c$, is the least relation closed under the following rules. In an assertion, $b$ for a boolean expression means that $b$ evaluates to true and $\\neg b$ that it evaluates to false. Skip: $\\{Q\\}\\,\\mathtt{skip}\\,\\{Q\\}$. Assignment: for a variable $x$ and an arithmetic expression $a$,\u2026","title":"Total-correctness Hoare logic"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/sound-2","kind":"node","lean":"Cslib.IMP.TDerivable.sound","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#sound-2","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For all assertions $P$ and $Q$ and every command $c$, if the triple $\\{P\\}\\,c\\,\\{Q\\}$ is derivable in total-correctness Hoare logic, then it is valid for total correctness: every state satisfying $P$ has a terminating big-step execution of $c$ ending in a state that satisfies $Q$.","title":"Soundness of total-correctness Hoare logic"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/strengthen-2","kind":"node","lean":"Cslib.IMP.TDerivable.strengthen","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#strengthen-2","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"Let $P$, $P'$ and $Q$ be assertions and $c$ a command. If $P$ entails $P'$ (every state satisfying $P$ satisfies $P'$) and the triple $\\{P'\\}\\,c\\,\\{Q\\}$ is derivable in total-correctness Hoare logic, then so is $\\{P\\}\\,c\\,\\{Q\\}$.","title":"Precondition strengthening for total correctness"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/tderivable-wp","kind":"node","lean":"Cslib.IMP.tderivable_wp","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#tderivable-wp","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For every IMP command $c$ and every assertion $Q$ (a predicate on states), the triple $\\{\\mathrm{wp}(c, Q)\\}\\,c\\,\\{Q\\}$ is derivable in the total-correctness Hoare logic. Here $\\mathrm{wp}(c, Q)$ is the semantic weakest precondition: it holds in a state $s$ exactly when there is a terminating execution of $c$ from $s$ ending in a state satisfying $Q$.","title":"The weakest-precondition triple is derivable in total-correctness Hoare logic"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/complete-2","kind":"node","lean":"Cslib.IMP.TDerivable.complete","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#complete-2","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For all assertions $P$ and $Q$ and every command $c$, if the total-correctness Hoare triple $\\{P\\}\\,c\\,\\{Q\\}$ is valid, then it is derivable in total-correctness Hoare logic.","title":"Total-correctness Hoare logic is relatively complete"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/derivable-2","kind":"node","lean":"Cslib.IMP.TDerivable.derivable","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#derivable-2","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For all assertions $P$ and $Q$ and every command $c$, if the triple with precondition $P$, command $c$ and postcondition $Q$ is derivable in total-correctness Hoare logic, then it is derivable in partial-correctness Hoare logic.","title":"Total-correctness derivations give partial-correctness derivations"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/tderivable-iff-tvalid","kind":"node","lean":"Cslib.IMP.tderivable_iff_tvalid","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#tderivable-iff-tvalid","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For all assertions $P$, $Q$ (predicates on states) and every IMP command $c$, the triple $\\{P\\}\\,c\\,\\{Q\\}$ is derivable in the total-correctness Hoare logic if and only if it is totally valid.","title":"Soundness and relative completeness of total-correctness Hoare logic"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/tderivable-iff-denote","kind":"node","lean":"Cslib.IMP.tderivable_iff_denote","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#tderivable-iff-denote","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For all assertions $P$, $Q$ (predicates on states) and every IMP command $c$, the triple $\\{P\\}\\,c\\,\\{Q\\}$ is derivable in the total-correctness Hoare logic if and only if for every state $s$ satisfying $P$ there is a state $s'$ such that $(s, s')$ lies in the relational denotation of $c$ and $s'$ satisfies $Q$.","title":"Total-correctness Hoare logic is sound and complete for the denotational semantics"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/triple","kind":"node","lean":"Cslib.IMP.Triple","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#triple","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"A Hoare triple $\\{P\\}\\,c\\,\\{Q\\}$ consists of a precondition $P$ and a postcondition $Q$, both assertions (predicates on states), and an IMP command $c$. It is the judgement form of both the partial-correctness and the total-correctness Hoare logic.","title":"Hoare triples"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/total-correctness","kind":"node","lean":"Cslib.IMP.TotalCorrectness","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#total-correctness","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"An empty type used only as a label: derivability of Hoare triples in the total-correctness Hoare logic is registered as an inference system under this tag, while partial-correctness derivability is the default inference system on Hoare triples.","title":"Total-correctness tag for Hoare triples"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/derivable-iff-valid-2","kind":"node","lean":"Cslib.IMP.Triple.derivable_iff_valid","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#derivable-iff-valid-2","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"For every Hoare triple $\\{P\\}\\,c\\,\\{Q\\}$, the triple is derivable in the default inference system on Hoare triples, which is partial-correctness Hoare logic, if and only if it is valid for partial correctness.","title":"Derivability in the default inference system on Hoare triples is validity"},{"id":"ComputerScience/program-logics-and-verification/hoare-logic/derivable-in-total-correctness-iff-tvali","kind":"node","lean":"Cslib.IMP.Triple.derivableIn_totalCorrectness_iff_tvalid","parent":43711,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/hoare-logic/index.html#derivable-in-total-correctness-iff-tvali","scope":"ComputerScience/program-logics-and-verification/hoare-logic","status":"fully_proved","summary":"A Hoare triple, given by a precondition $P$, a command $c$ and a postcondition $Q$, is derivable in the inference system on Hoare triples tagged total correctness, which is total-correctness Hoare logic, if and only if it is valid for total correctness.","title":"Total-correctness Hoare logic as an inference system"},{"data_url":"scopes/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests.json","id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","kind":"scope","navigation":"module","parent":43710,"result":false,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","title":"Kleene algebra with tests"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/star-simulation","kind":"node","lean":"KleeneAlgebra.star_simulation","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#star-simulation","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"A one-step simulation lifts to arbitrary finite iteration. The simulator `p` need not be invertible or a test.","title":"A one-step simulation lifts to finite iteration"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-star-absorption","kind":"node","lean":"KleeneAlgebra.test_star_absorption","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-star-absorption","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"An idempotent subidentity preserved by one step is preserved by iteration. This is a corollary of `star_simulation`; the test assumptions are explicit, with no additional algebraic structure.","title":"Tests preserved by one step are preserved by iteration"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-set","kind":"node","lean":"Cslib.RelationAlgebra.testSet","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-set","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"The diagonal relation restricted to an assertion.","title":"Diagonal test relation of an assertion"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/mem-test-set","kind":"node","lean":"Cslib.RelationAlgebra.mem_testSet","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#mem-test-set","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"Membership in a diagonal test.","title":"Membership in a diagonal test relation"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-set-le-one","kind":"node","lean":"Cslib.RelationAlgebra.testSet_le_one","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-set-le-one","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"A test is a subidentity, not an arbitrary relation.","title":"Diagonal tests are subidentities"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-set-mul-test-set","kind":"node","lean":"Cslib.RelationAlgebra.testSet_mul_testSet","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-set-mul-test-set","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"Sequential tests conjoin their assertions without changing state.","title":"Composing diagonal tests conjoins their assertions"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-mul-le-mul-test-iff","kind":"node","lean":"Cslib.RelationAlgebra.test_mul_le_mul_test_iff","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-mul-le-mul-test-iff","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"The verification inequality is precisely universal partial correctness: every execution from `P` ends in `Q`.","title":"Verification inequality characterizes partial correctness"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-mul-eq-test-mul-test-iff","kind":"node","lean":"Cslib.RelationAlgebra.test_mul_eq_test_mul_test_iff","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-mul-eq-test-mul-test-iff","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"A postcondition can be absorbed on the right exactly when it holds on every execution starting in the precondition.","title":"Postcondition absorption characterizes partial correctness"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-mul-test-compl-eq-bot-iff","kind":"node","lean":"Cslib.RelationAlgebra.test_mul_test_compl_eq_bot_iff","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-mul-test-compl-eq-bot-iff","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"Classically, verification means that no execution from `P` can reach the complement of `Q`.","title":"Partial correctness as unreachability of the negated postcondition"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-set-empty","kind":"node","lean":"Cslib.RelationAlgebra.testSet_empty","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-set-empty","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"The false assertion is the zero relation.","title":"Test of the false assertion is the zero relation"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-set-univ","kind":"node","lean":"Cslib.RelationAlgebra.testSet_univ","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-set-univ","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"The true assertion is the identity relation.","title":"Test of the true assertion is the identity relation"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-set-compl","kind":"node","lean":"Cslib.RelationAlgebra.testSet_compl","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-set-compl","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"Boolean test negation is complement relative to identity, not the full relation-algebra complement (which includes off-diagonal pairs).","title":"Test of a negated assertion is the complement within identity"},{"id":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/test-set-union","kind":"node","lean":"Cslib.RelationAlgebra.testSet_union","parent":43753,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/kleene-algebra-with-tests/index.html#test-set-union","scope":"ComputerScience/program-logics-and-verification/kleene-algebra-with-tests","status":"fully_proved","summary":"Union of assertions is union of their diagonal tests.","title":"Test of a union of assertions is the union of tests"},{"data_url":"scopes/ComputerScience/program-logics-and-verification/separation-logic.json","id":"ComputerScience/program-logics-and-verification/separation-logic","kind":"scope","navigation":"module","parent":43710,"result":false,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","title":"Separation logic"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/hassertion","kind":"node","lean":"Cslib.IMP.HAssertion","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#hassertion","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"A heap assertion is a predicate on a store and a heap: it assigns a proposition to every store $s$ and every heap $h$. Heap assertions serve as the preconditions and postconditions of separation logic.","title":"Heap assertion"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/entails","kind":"node","lean":"Cslib.IMP.HAssertion.Entails","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#entails","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"A heap assertion $P$ entails a heap assertion $Q$ when, for every store $s$ and every heap $h$, if $P$ holds of $s$ and $h$, then $Q$ holds of $s$ and $h$.","title":"Entailment of heap assertions"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/emp","kind":"node","lean":"Cslib.IMP.HAssertion.emp","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#emp","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"The heap assertion $\\mathrm{emp}$ holds of a store $s$ and a heap $h$ exactly when $h$ is the empty heap, whatever the store $s$ is.","title":"Empty-heap assertion"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/pure","kind":"node","lean":"Cslib.IMP.HAssertion.pure","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#pure","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"For a proposition $p$, the pure assertion of $p$ holds of a store $s$ and a heap $h$ exactly when $h$ is the empty heap and $p$ is true.","title":"Pure assertion"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/points-to","kind":"node","lean":"Cslib.IMP.HAssertion.pointsTo","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#points-to","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"For natural numbers $a$ and $v$, the assertion $a \\mapsto v$ holds of a store $s$ and a heap $h$ exactly when $h$ is the singleton heap that maps the address $a$ to the value $v$ and allocates no other address.","title":"Points-to assertion"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/star","kind":"node","lean":"Cslib.IMP.HAssertion.star","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#star","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"For heap assertions $P$ and $Q$, the assertion $P \\star Q$ holds of a store $s$ and a heap $h$ when there are disjoint heaps $h_1$ and $h_2$ with $h = h_1 \\cup h_2$ such that $P(s, h_1)$ and $Q(s, h_2)$.","title":"Separating conjunction"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/hexists","kind":"node","lean":"Cslib.IMP.HAssertion.hexists","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#hexists","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"For a function $p$ assigning a heap assertion $p(n)$ to each natural number $n$ (an address or a value), the assertion $\\exists n,\\ p(n)$ holds of a store $s$ and a heap $h$ exactly when $p(n)$ holds of $s$ and $h$ for some natural number $n$.","title":"Existential heap assertion"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/hand","kind":"node","lean":"Cslib.IMP.HAssertion.hand","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#hand","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"For heap assertions $P$ and $Q$, the assertion $P \\wedge Q$ holds of a store $s$ and a heap $h$ exactly when $P$ holds of $s$ and $h$ and $Q$ holds of $s$ and the same heap $h$.","title":"Conjunction of heap assertions"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/sep-valid","kind":"node","lean":"Cslib.IMP.SepValid","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#sep-valid","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"For heap assertions $P$, $Q$ and a heap command $c$, the triple $\\lbrace P \\rbrace\\ c\\ \\lbrace Q \\rbrace$ is valid (in the fault-avoiding sense) when, for every store $s$ and heap $h$ with $P(s, h)$, the configuration $(c, s, h)$ is safe (no configuration reachable from it is stuck at a command other than $\\mathtt{skip}$), and whenever $(c, s, h)$ reaches $(\\mathtt{skip}, s', h')$ in finitely many heap steps,\u2026","title":"Separation-logic triple validity"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/while-loop","kind":"node","lean":"Cslib.IMP.SepValid.whileLoop","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#while-loop","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $I$ be a heap assertion, $b$ a Boolean expression and $c$ a heap command. Write $I \\wedge b$ for the assertion that holds of a store $s$ and a heap $h$ when $I(s, h)$ holds and $b$ evaluates to true in $s$, and $I \\wedge \\neg b$ for the one that holds when $I(s, h)$ holds and $b$ evaluates to false in $s$. If the triple $\\lbrace I \\wedge b \\rbrace\\ c\\ \\lbrace I \\rbrace$ is valid, then so is the triple\u2026","title":"Separation-logic while rule"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/frame","kind":"node","lean":"Cslib.IMP.SepValid.frame","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#frame","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $P$, $Q$ and $R$ be heap assertions and $c$ a heap command. Suppose the separation-logic triple with precondition $P$, command $c$ and postcondition $Q$ is valid, and suppose that whenever $R$ holds of a store $s$ and a heap $h$, it also holds of $s'$ and $h$ for every store $s'$ that agrees with $s$ on every variable outside the set of variables that $c$ may assign to. Then the triple with precondition\u2026","title":"Frame rule"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/of-valid","kind":"node","lean":"Cslib.IMP.SepValid.of_valid","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#of-valid","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $P$ and $Q$ be store assertions and $c$ a command of plain IMP. If the partial-correctness Hoare triple with precondition $P$, command $c$ and postcondition $Q$ is valid, then the separation-logic triple for the embedding of $c$ into heap IMP is valid with the precondition that holds of a store $s$ and any heap exactly when $P(s)$ holds and the postcondition that holds of a store $s$ and any heap exactly when\u2026","title":"Hoare validity embeds into separation validity"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/of-valid-frame","kind":"node","lean":"Cslib.IMP.SepValid.of_valid_frame","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#of-valid-frame","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $P$ and $Q$ be store assertions, $R$ a heap assertion and $c$ a command of plain IMP, and write $\\hat c$ for the embedding of $c$ into heap IMP. Suppose the partial-correctness Hoare triple with precondition $P$, command $c$ and postcondition $Q$ is valid, and suppose that whenever $R$ holds of a store $s$ and a heap $h$, it also holds of $s'$ and $h$ for every store $s'$ that agrees with $s$ on every variable\u2026","title":"Framed embedding of Hoare validity"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/store-irrel","kind":"node","lean":"Cslib.IMP.StoreIrrel","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#store-irrel","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"A heap assertion $P$ is store-irrelevant when for all stores $s, s'$ and every heap $h$, $P(s, h)$ implies $P(s', h)$; that is, whether $P$ holds does not depend on the store.","title":"Store-irrelevant heap assertions"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/frame-of-store-irrel","kind":"node","lean":"Cslib.IMP.frame_of_storeIrrel","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#frame-of-store-irrel","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $R$ be a store-irrelevant heap assertion and $c$ a heap command. For all stores $s, s'$ and every heap $h$, if $R(s, h)$ holds and $s'(x) = s(x)$ for every variable $x$ outside the set of variables that $c$ may assign to, then $R(s', h)$ holds.","title":"Store-irrelevant frames satisfy the frame-rule side condition"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/star-and-left","kind":"node","lean":"Cslib.IMP.star_and_left","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#star-and-left","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $p$ be a predicate on stores and $P$, $R$ heap assertions. Write $A \\star B$ for the separating conjunction of heap assertions $A$ and $B$ (the heap splits into two disjoint parts satisfying $A$ and $B$, respectively, at the same store), and $p \\land P$ for the heap assertion that holds at store $s$ and heap $h$ when $p(s)$ and $P(s, h)$ both hold. Then $(p \\land P) \\star R = p \\land (P \\star R)$.","title":"Pulling a store fact out of a separating conjunction"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/star-swap-right","kind":"node","lean":"Cslib.IMP.star_swap_right","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#star-swap-right","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"For heap assertions $P$, $Q$ and $R$, write $A \\star B$ for the separating conjunction of heap assertions $A$ and $B$ (the heap splits into two disjoint parts satisfying $A$ and $B$, respectively, at the same store). Then $P \\star (Q \\star R)$ entails $(P \\star R) \\star Q$: every store and heap satisfying the first assertion satisfy the second.","title":"Rearranging a separating conjunction"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/read-local","kind":"node","lean":"Cslib.IMP.read_local","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#read-local","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $x$ and $y$ be variables, let $a$ and $v$ be natural numbers, and let $p$ and $q$ be predicates on stores. Write $s[x := n]$ for the store that agrees with $s$ except that it maps $x$ to $n$, and $a \\mapsto v$ for the heap assertion that holds when the heap is the single cell at address $a$ holding $v$. If every store $s$ satisfying $p$ has $s(y) = a$ and $q(s[x := v])$, then the separation-logic triple for the\u2026","title":"Local rule for a heap read"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/assign-step","kind":"node","lean":"Cslib.IMP.assign_step","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#assign-step","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $x$ and $y$ be variables, let $p$ and $q$ be predicates on stores, and let $H$ be a store-irrelevant heap assertion. Write $s[x := n]$ for the store that agrees with $s$ except that it maps $x$ to $n$. If every store $s$ satisfying $p$ has $q(s[x := s(y)])$, then the separation-logic triple for the assignment $x := y$ is valid with precondition \"$p(s)$ and $H(s, h)$\" and postcondition \"$q(s)$ and $H(s, h)$\",\u2026","title":"Assignment rule with a store-irrelevant heap assertion"},{"id":"ComputerScience/program-logics-and-verification/separation-logic/exists","kind":"node","lean":"Cslib.IMP.SepValid.exists","parent":43767,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/separation-logic/index.html#exists","scope":"ComputerScience/program-logics-and-verification/separation-logic","status":"fully_proved","summary":"Let $c$ be a heap command, let $\\iota$ be an arbitrary index (any type or proposition), and let $P_i$ and $Q_i$ be heap assertions for each $i$ in $\\iota$. If for every $i$ the triple with precondition $P_i$, command $c$ and postcondition $Q_i$ is valid, then the triple with precondition \"there is $i$ with $P_i$\", command $c$ and postcondition \"there is $i$ with $Q_i$\" is valid.","title":"Existential rule for separation-logic triples"},{"data_url":"scopes/ComputerScience/program-logics-and-verification/verification-condition-generation.json","id":"ComputerScience/program-logics-and-verification/verification-condition-generation","kind":"scope","navigation":"module","parent":43710,"result":false,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","title":"Verification-condition generation"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/acmd","kind":"node","lean":"Cslib.IMP.ACmd","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#acmd","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"An annotated command is built like an IMP command, from skip, assignments $x := a$ of an arithmetic expression $a$ to a variable $x$ (a natural number), sequential composition $C_1;\\,C_2$ and conditionals $\\mathrm{if}\\ b\\ \\mathrm{then}\\ C_1\\ \\mathrm{else}\\ C_2$ on a boolean expression $b$, except that every while loop $\\{I\\}\\ \\mathrm{while}\\ b\\ \\mathrm{do}\\ C$ also carries an assertion $I$, its invariant.","title":"Annotated commands"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/strip","kind":"node","lean":"Cslib.IMP.ACmd.strip","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#strip","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"The IMP command $\\mathrm{strip}(C)$ underlying an annotated command $C$ is obtained by deleting the invariant of every while loop: skip, assignments, sequential composition and conditionals are kept with their parts stripped, and $\\{I\\}\\ \\mathrm{while}\\ b\\ \\mathrm{do}\\ C'$ becomes $\\mathrm{while}\\ b\\ \\mathrm{do}\\ \\mathrm{strip}(C')$.","title":"Erasing annotations"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/pre","kind":"node","lean":"Cslib.IMP.ACmd.pre","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#pre","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"For an annotated command $C$ and an assertion $Q$ (the postcondition), the precondition $\\mathrm{pre}(C, Q)$ is the assertion computed backwards as follows: $\\mathrm{pre}(\\mathrm{skip}, Q) = Q$; $\\mathrm{pre}(x := a, Q)$ holds in a state $s$ iff $Q$ holds in the state obtained from $s$ by giving $x$ the value of $a$ in $s$; $\\mathrm{pre}(C_1;\\,C_2, Q) = \\mathrm{pre}(C_1, \\mathrm{pre}(C_2, Q))$; the precondition of\u2026","title":"Precondition of an annotated command"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/vc","kind":"node","lean":"Cslib.IMP.ACmd.vc","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#vc","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"For an annotated command $C$ and an assertion $Q$ (the postcondition), the verification condition $\\mathrm{vc}(C, Q)$ is the proposition defined as follows, where $\\mathrm{pre}$ is the precondition of an annotated command: $\\mathrm{vc}(\\mathrm{skip}, Q)$ and $\\mathrm{vc}(x := a, Q)$ are true; $\\mathrm{vc}(C_1;\\,C_2, Q)$ holds iff $\\mathrm{vc}(C_1, \\mathrm{pre}(C_2, Q))$ and $\\mathrm{vc}(C_2, Q)$ hold; the\u2026","title":"Verification condition of an annotated command"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/vc-sound","kind":"node","lean":"Cslib.IMP.ACmd.vc_sound","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#vc-sound","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"For an annotated command $C$ and an assertion $Q$, write $\\mathrm{pre}(C, Q)$ and $\\mathrm{vc}(C, Q)$ for the precondition and the verification condition of $C$ for the postcondition $Q$, and $\\mathrm{strip}(C)$ for the IMP command obtained from $C$ by erasing its loop invariants. For every such $C$ and $Q$, if $\\mathrm{vc}(C, Q)$ holds, then the Hoare triple $\\{\\mathrm{pre}(C, Q)\\}\\ \\mathrm{strip}(C)\\ \\{Q\\}$ is\u2026","title":"Soundness of verification conditions"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/vc-valid","kind":"node","lean":"Cslib.IMP.ACmd.vc_valid","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#vc-valid","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"Let $C$ be an annotated command, let $\\mathrm{strip}(C)$ be the IMP command obtained from $C$ by erasing its loop invariants, and let $P$, $Q$ be assertions. If the verification condition $\\mathrm{vc}(C, Q)$ holds and $P$ entails the precondition $\\mathrm{pre}(C, Q)$, then the partial-correctness Hoare triple $\\{P\\}\\ \\mathrm{strip}(C)\\ \\{Q\\}$ is valid.","title":"Verification conditions yield valid Hoare triples"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/pre-entails-vc","kind":"node","lean":"Cslib.IMP.ACmd.pre_entails_vc","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#pre-entails-vc","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"Let $C$ be an annotated command and $Q$, $Q'$ assertions such that $Q$ entails $Q'$, and write $\\mathrm{pre}(C, Q)$ and $\\mathrm{vc}(C, Q)$ for the precondition and the verification condition of $C$ for the postcondition $Q$. If $\\mathrm{vc}(C, Q)$ holds, then $\\mathrm{pre}(C, Q)$ entails $\\mathrm{pre}(C, Q')$ and $\\mathrm{vc}(C, Q')$ holds.","title":"Weakening the postcondition of an annotated command"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/vc-complete","kind":"node","lean":"Cslib.IMP.ACmd.vc_complete","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#vc-complete","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"Let $c$ be an IMP command and $P$, $Q$ assertions such that the Hoare triple $\\{P\\}\\ c\\ \\{Q\\}$ is derivable in the partial-correctness Hoare logic. Then there is an annotated command $C$ such that erasing its loop invariants gives $c$, that is $\\mathrm{strip}(C) = c$, the verification condition $\\mathrm{vc}(C, Q)$ holds, and $P$ entails the precondition $\\mathrm{pre}(C, Q)$.","title":"Completeness of verification conditions for derivable triples"},{"id":"ComputerScience/program-logics-and-verification/verification-condition-generation/vc-complete-of-valid","kind":"node","lean":"Cslib.IMP.ACmd.vc_complete_of_valid","parent":43788,"result":true,"route":"../roadmap/ComputerScience/program-logics-and-verification/verification-condition-generation/index.html#vc-complete-of-valid","scope":"ComputerScience/program-logics-and-verification/verification-condition-generation","status":"fully_proved","summary":"Let $c$ be an IMP command and $P$, $Q$ assertions such that the partial-correctness Hoare triple $\\{P\\}\\ c\\ \\{Q\\}$ is valid. Then there is an annotated command $C$ such that erasing its loop invariants gives $c$, that is $\\mathrm{strip}(C) = c$, the verification condition $\\mathrm{vc}(C, Q)$ holds, and $P$ entails the precondition $\\mathrm{pre}(C, Q)$.","title":"Completeness of verification conditions for valid triples"},{"area":"Applied Mathematics","data_url":"chapters/InformationTheory.json","id":"InformationTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/InformationTheory/index.html","scope":"InformationTheory","status":"can_prove","summary":"Information theory quantifies information, uncertainty, compression, and transmission through entropy and coding.","title":"Information Theory"},{"data_url":"scopes/InformationTheory/source-coding.json","id":"InformationTheory/source-coding","kind":"scope","parent":43798,"result":false,"route":"../roadmap/InformationTheory/source-coding/index.html","scope":"InformationTheory/source-coding","status":"fully_proved","summary":"Lossless compression of information sources by variable-length codes, including prefix codes, the Kraft\u2013McMillan inequality and Shannon's source coding bounds.","title":"Source coding"},{"id":"InformationTheory/source-coding/kraft-mcmillan-lengths","kind":"node","lean":"MathlibExt.InformationTheory.KraftMcMillan.kraft_mcmillan_lengths","parent":43799,"result":true,"route":"../roadmap/InformationTheory/source-coding/index.html#kraft-mcmillan-lengths","scope":"InformationTheory/source-coding","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of source symbols, let $\\beta$ be a finite alphabet with $D \\ge 2$ letters, and let $l : \\alpha \\to \\mathbb{N}$. There is a prefix-free code assigning to each $a \\in \\alpha$ a word over $\\beta$ of length $l(a)$ if and only if $\\sum_{a \\in \\alpha} D^{-l(a)} \\le 1$.","title":"Kraft's inequality for prefix codes"},{"data_url":"scopes/InformationTheory/source-coding/prefix-codes.json","id":"InformationTheory/source-coding/prefix-codes","kind":"scope","navigation":"module","parent":43799,"result":false,"route":"../roadmap/InformationTheory/source-coding/prefix-codes/index.html","scope":"InformationTheory/source-coding/prefix-codes","status":"fully_proved","title":"Prefix codes"},{"id":"InformationTheory/source-coding/prefix-codes/is-prefix-free","kind":"node","lean":"MathlibExt.InformationTheory.IsPrefixFree","parent":43801,"result":true,"route":"../roadmap/InformationTheory/source-coding/prefix-codes/index.html#is-prefix-free","scope":"InformationTheory/source-coding/prefix-codes","status":"fully_proved","summary":"A code $c$ assigning to each source symbol a finite word over an alphabet is prefix-free if for all distinct symbols $a_1 \\ne a_2$ the word $c(a_1)$ is not a prefix of $c(a_2)$. Since the condition covers all ordered pairs of distinct symbols, both prefix directions are forbidden.","title":"Prefix-free code"},{"id":"InformationTheory/source-coding/prefix-codes/eq-of-is-prefix","kind":"node","lean":"MathlibExt.InformationTheory.IsPrefixFree.eq_of_isPrefix","parent":43801,"result":true,"route":"../roadmap/InformationTheory/source-coding/prefix-codes/index.html#eq-of-is-prefix","scope":"InformationTheory/source-coding/prefix-codes","status":"fully_proved","summary":"If $c$ is a prefix-free code and the codeword $c(a_1)$ is a prefix of $c(a_2)$, then $a_1 = a_2$.","title":"Prefix-related codewords come from the same symbol"},{"id":"InformationTheory/source-coding/prefix-codes/injective","kind":"node","lean":"MathlibExt.InformationTheory.IsPrefixFree.injective","parent":43801,"result":true,"route":"../roadmap/InformationTheory/source-coding/prefix-codes/index.html#injective","scope":"InformationTheory/source-coding/prefix-codes","status":"fully_proved","summary":"If $c$ is a prefix-free code and $c(a_1) = c(a_2)$, then $a_1 = a_2$.","title":"Prefix-free codes are injective"},{"id":"InformationTheory/source-coding/prefix-codes/eq-of-mem-nil","kind":"node","lean":"MathlibExt.InformationTheory.IsPrefixFree.eq_of_mem_nil","parent":43801,"result":true,"route":"../roadmap/InformationTheory/source-coding/prefix-codes/index.html#eq-of-mem-nil","scope":"InformationTheory/source-coding/prefix-codes","status":"fully_proved","summary":"If a prefix-free code $c$ assigns the empty word to a symbol $a_0$, then every symbol $a$ equals $a_0$.","title":"A prefix-free code with an empty codeword has one symbol"},{"data_url":"scopes/InformationTheory/source-coding/huffman-coding.json","id":"InformationTheory/source-coding/huffman-coding","kind":"scope","navigation":"module","parent":43799,"result":false,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","title":"Huffman coding"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-tree","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-tree","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"A Huffman tree over an alphabet $\\alpha$ is either a leaf, which carries a symbol $a \\in \\alpha$ and a natural-number weight $w$, or a fork, which joins a left and a right Huffman tree over $\\alpha$.","title":"Huffman tree"},{"id":"InformationTheory/source-coding/huffman-coding/weighted-leaves","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.weightedLeaves","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#weighted-leaves","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The list of pairs $(s, w)$, one for each leaf of a Huffman tree, where $s$ is the symbol and $w$ the weight at that leaf, read from left to right.","title":"Weighted leaves of a Huffman tree"},{"id":"InformationTheory/source-coding/huffman-coding/symbols","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.symbols","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#symbols","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The list of the symbols at the leaves of a Huffman tree, read from left to right, with one entry per leaf.","title":"List of leaf symbols of a Huffman tree"},{"id":"InformationTheory/source-coding/huffman-coding/total-weight","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.totalWeight","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#total-weight","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The sum of the weights at all leaves of a Huffman tree, counted once per leaf.","title":"Total weight of a Huffman tree"},{"id":"InformationTheory/source-coding/huffman-coding/weighted-path-length","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.weightedPathLength","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#weighted-path-length","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The weighted path length of a Huffman tree is defined recursively: a single leaf has weighted path length $0$, and the tree with left subtree $L$ and right subtree $R$ has weighted path length equal to the weighted path length of $L$ plus that of $R$ plus the total weight of $L$ plus the total weight of $R$.","title":"Weighted path length"},{"id":"InformationTheory/source-coding/huffman-coding/weighted-path-length-fork","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.weightedPathLength_fork","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#weighted-path-length-fork","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For Huffman trees $L$ and $R$, the weighted path length of the tree with left subtree $L$ and right subtree $R$ equals the weighted path length of $L$ plus the weighted path length of $R$ plus the total weight of $L$ plus the total weight of $R$.","title":"Weighted path length of a merged tree"},{"id":"InformationTheory/source-coding/huffman-coding/encoded-length","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.encodedLength","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#encoded-length","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The encoded length of a leaf of weight $w$ is $w$ (its lone symbol receives a one-bit codeword). The encoded length of the tree with left subtree $L$ and right subtree $R$ is $P(L) + P(R) + W(L) + W(R)$, where $P$ is the weighted path length and $W$ the total weight; this is the weighted path length of that tree.","title":"Encoded length of a coding tree"},{"id":"InformationTheory/source-coding/huffman-coding/encoded-length-fork","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.encodedLength_fork","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#encoded-length-fork","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For any two coding trees $L$ and $R$, the encoded length of the tree with left subtree $L$ and right subtree $R$ equals its weighted path length.","title":"Encoded length of a fork"},{"id":"InformationTheory/source-coding/huffman-coding/path-code","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.pathCode?","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#path-code","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a Huffman tree $T$ and a symbol $s$, the path code of $s$ in $T$ records the edges on the path from the root of $T$ to the leftmost leaf labelled $s$, writing false for a left edge and true for a right edge. If $T$ is a single leaf labelled $s$, the path code is empty; if no leaf of $T$ is labelled $s$, there is no path code.","title":"Root-to-leaf path code"},{"id":"InformationTheory/source-coding/huffman-coding/code","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.code?","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#code","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree and $a$ a symbol, and write $0$ for the bit false and $1$ for the bit true. If $T$ is a single leaf labelled $s$, the codeword of $a$ is the one-bit word $0$ when $a = s$ and is undefined otherwise. Otherwise the codeword of $a$ records the edges from the root of $T$ to the leftmost leaf labelled $a$, with $0$ for a left edge and $1$ for a right edge; it is undefined when no leaf of $T$ is\u2026","title":"Codeword lookup in a Huffman tree"},{"id":"InformationTheory/source-coding/huffman-coding/decode-path","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.decodePath?","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#decode-path","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree and $w$ a bit string, and write $0$ for the bit false and $1$ for the bit true. Following $w$ in $T$ starts at the root and reads the bits of $w$ one at a time, moving to the left subtree on $0$ and to the right subtree on $1$, until it reaches a leaf; it then returns the label of that leaf together with the unread bits of $w$. It fails if $w$ runs out before a leaf is reached. In\u2026","title":"Following a path in a Huffman tree"},{"id":"InformationTheory/source-coding/huffman-coding/decode-one","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.decodeOne?","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#decode-one","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree and $w$ a bit string, and write $0$ for the bit false and $1$ for the bit true. If $T$ is a single leaf labelled $s$, decoding one codeword from $w$ returns $s$ together with the rest of $w$ when $w$ starts with $0$, and fails otherwise. If $T$ is a fork with left subtree $L$ and right subtree $R$, it fails when $w$ is empty; when $w$ is a bit $b$ followed by $w'$, it follows $w'$ from the\u2026","title":"Decoding one codeword"},{"id":"InformationTheory/source-coding/huffman-coding/decode","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.decode?","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#decode","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree. Decoding the empty bit string with $T$ returns the empty sequence of symbols. To decode a nonempty bit string $w$, decode one codeword from $w$ with $T$, obtaining a symbol $s$ and the unread bits $w'$, and return $s$ followed by the decoding of $w'$; decoding fails if decoding one codeword from $w$ fails or if decoding $w'$ fails.","title":"Decoding a bit string"},{"id":"InformationTheory/source-coding/huffman-coding/encode","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.encode?","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#encode","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Given a coding tree $T$, the empty message encodes to the empty bit string, and a message $s_1 s_2 \\cdots s_n$ encodes to the concatenation, in order, of the codewords of $s_1, \\dots, s_n$ in $T$. If some $s_i$ has no codeword in $T$, the message has no encoding.","title":"Huffman encoding of a message"},{"id":"InformationTheory/source-coding/huffman-coding/leaf-symbol","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.LeafSymbol","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#leaf-symbol","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a Huffman tree $T$ over an alphabet $\\alpha$, the type of leaf symbols of $T$ is the subtype of those $a \\in \\alpha$ that occur in the list of leaf symbols of $T$.","title":"Type of leaf symbols of a Huffman tree"},{"id":"InformationTheory/source-coding/huffman-coding/code-of-leaf","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.codeOfLeaf","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#code-of-leaf","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a Huffman tree $T$ and a symbol $s$ occurring at a leaf of $T$, the codeword of $s$ is the word that the codeword lookup of $T$ returns for $s$, or the empty word if the lookup fails.","title":"Codeword of a leaf symbol"},{"id":"InformationTheory/source-coding/huffman-coding/code-is-some-of-mem-symbols","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.code?_isSome_of_mem_symbols","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#code-is-some-of-mem-symbols","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree and $s$ a symbol occurring at a leaf of $T$. Then the codeword lookup of $s$ in $T$ succeeds.","title":"Every leaf symbol has a codeword"},{"id":"InformationTheory/source-coding/huffman-coding/code-code-of-leaf","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.code?_codeOfLeaf","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#code-code-of-leaf","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree and $s$ a symbol occurring at a leaf of $T$. Then the codeword lookup of $s$ in $T$ succeeds and returns the codeword of $s$ as a leaf symbol of $T$.","title":"Codeword lookup agrees with the leaf code"},{"id":"InformationTheory/source-coding/huffman-coding/prefix-free","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.prefixFree","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#prefix-free","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For every Huffman tree $T$, the codeword map on the symbols occurring as leaves of $T$ is prefix-free: if $a$ and $b$ are distinct leaf symbols of $T$, then the codeword of $a$ is not a prefix of the codeword of $b$.","title":"Huffman tree codes are prefix-free"},{"id":"InformationTheory/source-coding/huffman-coding/decode-code-of-leaf","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.decode_codeOfLeaf","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#decode-code-of-leaf","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree and let $s_1, \\dots, s_n$ ($n \\ge 0$) be symbols occurring at leaves of $T$, with codewords $c(s_1), \\dots, c(s_n)$ as leaf symbols of $T$. Decoding the concatenation $c(s_1) c(s_2) \\cdots c(s_n)$ with $T$ succeeds and returns the sequence $s_1, \\dots, s_n$.","title":"Decoding inverts the leaf code"},{"id":"InformationTheory/source-coding/huffman-coding/encode-code-of-leaf","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.encode_codeOfLeaf","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#encode-code-of-leaf","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree and let $s_1, \\dots, s_n$ ($n \\ge 0$) be symbols occurring at leaves of $T$, with codewords $c(s_1), \\dots, c(s_n)$ as leaf symbols of $T$. Encoding $s_1, \\dots, s_n$ with $T$ succeeds and returns the concatenation $c(s_1) c(s_2) \\cdots c(s_n)$.","title":"Encoding concatenates leaf codewords"},{"id":"InformationTheory/source-coding/huffman-coding/decode-encode-leaf-symbols","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.decode_encode_leafSymbols","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#decode-encode-leaf-symbols","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree and let $s_1, \\dots, s_n$ ($n \\ge 0$) be symbols occurring at leaves of $T$. Encoding $s_1, \\dots, s_n$ with $T$ and then decoding the resulting bit string with $T$, failing if the encoding fails, returns $s_1, \\dots, s_n$. In particular, the encoding succeeds.","title":"Decoding inverts encoding"},{"id":"InformationTheory/source-coding/huffman-coding/path-code-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.pathCode?_eq_none_iff","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#path-code-eq-none-iff","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a Huffman tree $T$ over an alphabet $\\alpha$ with decidable equality and a symbol $a \\in \\alpha$, the root-to-leaf path code of $a$ in $T$ is undefined exactly when $a$ is not among the leaf symbols of $T$.","title":"A symbol has a path code exactly when it labels a leaf"},{"id":"InformationTheory/source-coding/huffman-coding/weighted-path-length-eq-sum-path-lengths","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.weightedPathLength_eq_sum_pathLengths","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#weighted-path-length-eq-sum-path-lengths","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman tree whose leaf symbols are pairwise distinct. Then the weighted path length of $T$ equals the sum, over the leaves of $T$ with symbol $s$ and weight $w$, of $w$ times the length of the root-to-leaf path of $s$ in $T$ (the depth of that leaf, with the root at depth zero).","title":"Weighted path length as a sum over leaves"},{"id":"InformationTheory/source-coding/huffman-coding/encoded-length-eq-sum-code-lengths","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.encodedLength_eq_sum_codeLengths","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#encoded-length-eq-sum-code-lengths","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a coding tree whose leaf symbols are pairwise distinct. For a symbol $s$, let $\\ell(s)$ be the length of the codeword of $s$ in $T$, or $0$ if $s$ has no codeword. Then the encoded length of $T$ equals $\\sum_{(s, w)} w \\cdot \\ell(s)$, the sum over the weighted leaves $(s, w)$ of $T$.","title":"Encoded length is the total weighted codeword length"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-priority","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanPriority","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-priority","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"A Huffman priority is a pair $(w, s)$ of natural numbers: a weight $w$ and a serial number $s$.","title":"Huffman priority"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-work-item","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWorkItem","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-work-item","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"A work item over an alphabet $\\alpha$ pairs a Huffman priority, that is, a natural-number weight and a natural-number serial, with a Huffman tree over $\\alpha$.","title":"Huffman work item"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-two-min","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTwoMin","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-two-min","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"A record consisting of two Huffman priorities, the first and the second, together with a binary min-heap of Huffman priorities, the rest.","title":"Result of a two-minimum extraction"},{"id":"InformationTheory/source-coding/huffman-coding/extract-two-min","kind":"node","lean":"Cslib.Algorithms.Lean.extractTwoMin?","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#extract-two-min","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Given a binary min-heap $h$ of Huffman priorities, extract the minimum of $h$, obtaining a priority $p_1$ and a heap $h_1$, then extract the minimum of $h_1$, obtaining a priority $p_2$ and a heap $h_2$. The result is the triple $(p_1, p_2, h_2)$; if either extraction meets an empty heap, there is no result.","title":"Extracting two minima from a Huffman heap"},{"id":"InformationTheory/source-coding/huffman-coding/extract-two-min-cons-elements","kind":"node","lean":"Cslib.Algorithms.Lean.extractTwoMin_cons_elements","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#extract-two-min-cons-elements","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"If extracting two minima from a binary min-heap $h$ of Huffman priorities succeeds with first priority $p_1$, second priority $p_2$ and remaining heap $h'$, then the multiset of elements of $h$ equals the multiset of elements of $h'$ with $p_1$ and $p_2$ added, counted with multiplicity.","title":"Extracting two minima preserves the heap's elements"},{"id":"InformationTheory/source-coding/huffman-coding/extract-two-min-greedy","kind":"node","lean":"Cslib.Algorithms.Lean.extractTwoMin_greedy","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#extract-two-min-greedy","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Order Huffman priorities lexicographically by weight and then serial number. If extracting two minima from a binary min-heap $h$ of Huffman priorities succeeds with first priority $p_1$, second priority $p_2$ and remaining heap $h'$, then $p_1 \\le q$ for every element $q$ of $h$, and $p_2 \\le q$ for every $q$ that is $p_2$ or an element of $h'$.","title":"Two-minimum extraction is greedy"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-table-forest","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanTableForest","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-table-forest","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a table $T$ whose cells each hold at most one work item, the work forest of $T$ is the list of the work items held in the cells of $T$, in increasing cell order; empty cells are skipped.","title":"Work forest of a table"},{"id":"InformationTheory/source-coding/huffman-coding/take-serial","kind":"node","lean":"Cslib.Algorithms.Lean.takeSerial?","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#take-serial","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Given a serial number $s$ and a table $t$ of optional Huffman work items, look up the entry of $t$ at index $s$. If that entry is a work item whose serial number is $s$, the result is its tree together with the table obtained from $t$ by emptying the entry at $s$; otherwise (the index is out of bounds, the entry is empty, or it holds an item with a different serial number) the removal fails.","title":"Removing a work item by serial"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-forest-leaves","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanForestLeaves","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-forest-leaves","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The weighted leaves of a list of Huffman work items are the concatenation, in list order, of the weighted leaves of the items' trees.","title":"Weighted leaves of a Huffman work forest"},{"id":"InformationTheory/source-coding/huffman-coding/take-serial-item","kind":"node","lean":"Cslib.Algorithms.Lean.takeSerial?_item","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#take-serial-item","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $t$ be a table of optional Huffman work items, and call the list of work items present in a table its work forest. If removing serial number $s$ from $t$ succeeds and returns a tree $T$ and an updated table $t'$, then there is a work item $w$ with tree $T$ and serial number $s$ such that $w$ followed by the work forest of $t'$ is a permutation of the work forest of $t$.","title":"Removing a serial removes one work item from the forest"},{"id":"InformationTheory/source-coding/huffman-coding/take-serial-size","kind":"node","lean":"Cslib.Algorithms.Lean.takeSerial?_size","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#take-serial-size","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $t$ be a table of optional Huffman work items. If removing serial number $s$ from $t$ succeeds and returns a tree together with an updated table $t'$, then $t'$ has the same size as $t$.","title":"Removing a serial preserves the table size"},{"id":"InformationTheory/source-coding/huffman-coding/take-serial-get-elem-of-ne","kind":"node","lean":"Cslib.Algorithms.Lean.takeSerial?_getElem?_of_ne","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#take-serial-get-elem-of-ne","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $t$ be a table of optional Huffman work items. If removing serial number $s$ from $t$ succeeds and returns a tree together with an updated table $t'$, then for every index $j \\ne s$ the entry of $t'$ at $j$ equals the entry of $t$ at $j$ (both are absent when $j$ is out of bounds).","title":"Removing a serial leaves the other table entries unchanged"},{"id":"InformationTheory/source-coding/huffman-coding/take-serial-is-some-of-stored","kind":"node","lean":"Cslib.Algorithms.Lean.takeSerial?_isSome_of_stored","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#take-serial-is-some-of-stored","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $t$ be a table of optional Huffman work items and let $w$ be a work item with serial number $s$. If the entry of $t$ at index $s$ is $w$, then removing serial $s$ from $t$ succeeds and returns the tree of $w$ together with an updated table.","title":"A stored work item can be removed by its serial"},{"id":"InformationTheory/source-coding/huffman-coding/initial-huffman-forest-from","kind":"node","lean":"Cslib.Algorithms.Lean.initialHuffmanForestFrom","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#initial-huffman-forest-from","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Given a starting serial number $s$ and a list of pairs $(a_0, w_0), \\dots, (a_{k-1}, w_{k-1})$ of symbols and natural-number weights, this is the list of $k$ Huffman work items whose $i$-th entry has priority (weight $w_i$, serial $s + i$) and whose tree is the leaf with symbol $a_i$ and weight $w_i$.","title":"Numbered leaf work items"},{"id":"InformationTheory/source-coding/huffman-coding/initial-huffman-forest","kind":"node","lean":"Cslib.Algorithms.Lean.initialHuffmanForest","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#initial-huffman-forest","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a list of pairs $(a_0, w_0), \\dots, (a_{k-1}, w_{k-1})$ of symbols and natural-number weights, the initial forest is the list of $k$ Huffman work items whose $i$-th entry has priority (weight $w_i$, serial $i$) and whose tree is the leaf with symbol $a_i$ and weight $w_i$.","title":"Initial Huffman forest"},{"id":"InformationTheory/source-coding/huffman-coding/initial-huffman-table","kind":"node","lean":"Cslib.Algorithms.Lean.initialHuffmanTable","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#initial-huffman-table","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a list of $k$ pairs $(a_0, w_0), \\dots, (a_{k-1}, w_{k-1})$ of symbols and natural-number weights, the initial table is the array of length $k$ of optional Huffman work items whose cell $i$ is present and holds the work item with priority (weight $w_i$, serial $i$) and tree the leaf with symbol $a_i$ and weight $w_i$; that is, it is the initial Huffman forest with every entry marked present.","title":"Initial Huffman table"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-table-forest-initial","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanTableForest_initial","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-table-forest-initial","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For every list of weighted symbols, the work forest read off the initial serial-indexed table of the list is exactly the initial leaf forest of the list, in the same order.","title":"The initial table holds the initial forest"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-table-forest-push-some","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanTableForest_push_some","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-table-forest-push-some","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For every table $T$ whose cells each hold at most one work item and every work item $x$, the work forest of the table obtained from $T$ by appending a new last cell holding $x$ is the work forest of $T$ followed by $x$.","title":"Appending to the table appends to the forest"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-heap-of-forest","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanHeapOfForest","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-heap-of-forest","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The heap of a list of Huffman work items is obtained from the empty binary min-heap by inserting the priorities of the items one at a time, in list order.","title":"Heap of a Huffman work forest"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-loop","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanLoop","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-loop","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The loop takes a fuel $n$, a next serial number $s$, a binary min-heap $H$ of Huffman priorities, ordered lexicographically by weight and then serial number, and a table $T$ whose cells each hold at most one work item. Taking serial $i$ from a table succeeds when cell $i$ holds a work item whose priority has serial number $i$, and returns that item's tree together with the table with cell $i$ emptied. With fuel\u2026","title":"Bounded Huffman merge loop"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-loop-with-lookup-count-aux","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanLoopWithLookupCountAux","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-loop-with-lookup-count-aux","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The loop takes a fuel $n$, a next serial number $s$, a binary min-heap $H$ of Huffman priorities, ordered lexicographically by weight and then serial number, a table $T$ whose cells each hold at most one work item, and a counter $c$. Taking serial $i$ from a table succeeds when cell $i$ holds a work item whose priority has serial number $i$, and returns that item's tree together with the table with cell $i$\u2026","title":"Huffman loop with an accumulated lookup count"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-loop-with-lookup-count","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanLoopWithLookupCount","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-loop-with-lookup-count","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a fuel $n$, a next serial number $s$, a binary min-heap $H$ of Huffman priorities and a table $T$ whose cells each hold at most one work item, the Huffman loop with lookup count on $(n, s, H, T)$ is the Huffman loop with an accumulated lookup count on $(n, s, H, T)$ started with counter $0$. It runs the bounded sequence of Huffman merges and counts indexed table lookups, one for each work item taken from the\u2026","title":"Huffman loop with lookup count"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-loop-with-lookup-count-eq","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanLoopWithLookupCount_eq","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-loop-with-lookup-count-eq","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For every fuel $n$, next serial number $s$, binary min-heap $H$ of Huffman priorities and table $T$ whose cells each hold at most one work item, the Huffman loop with lookup count on $(n, s, H, T)$ fails exactly when the bounded Huffman merge loop on $(n, s, H, T)$ fails, and returns $(t, 2n + 1)$ when the merge loop returns the tree $t$: two indexed table reads for each of the $n$ merges and one for the final step.","title":"Lookup count of the instrumented Huffman loop"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-loop-with-lookup-count-map-fst","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanLoopWithLookupCount_map_fst","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-loop-with-lookup-count-map-fst","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For every fuel $n$, next serial number $s$, binary min-heap $H$ of Huffman priorities and table $T$ whose cells each hold at most one work item, discarding the count from the result of the Huffman loop with lookup count on $(n, s, H, T)$ gives the result of the bounded Huffman merge loop on $(n, s, H, T)$: the two fail together, and otherwise they return the same tree.","title":"Discarding the lookup count recovers the Huffman loop"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-loop-lookup-count","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanLoop_lookupCount","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-loop-lookup-count","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $k$ be a merge budget, $n$ a next serial number, $h$ a binary min-heap of Huffman priorities and $t$ a serial-indexed table of optional Huffman work items. If the Huffman loop with budget $k$ on $n$, $h$ and $t$ returns a tree $T$, then the Huffman loop with lookup counting on the same arguments returns $T$ together with the count $2k + 1$ of indexed payload-table reads.","title":"Indexed reads of a successful Huffman loop"},{"id":"InformationTheory/source-coding/huffman-coding/huffman","kind":"node","lean":"Cslib.Algorithms.Lean.huffman","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $L$ be a list of $n$ pairs $(a_0, w_0), \\dots, (a_{n-1}, w_{n-1})$ of symbols and natural-number weights, with decidable equality on symbols. The construction fails when $L$ is empty or two pairs share a symbol. Otherwise it starts from $n$ leaves, the leaf for $(a_i, w_i)$ having priority (weight $w_i$, serial $i$), and performs $n - 1$ merges: each merge removes the two trees whose priorities are least in the\u2026","title":"Huffman's algorithm"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-weighted-leaves-perm","kind":"node","lean":"Cslib.Algorithms.Lean.huffman_weightedLeaves_perm","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-weighted-leaves-perm","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $L$ be a list of pairs $(a, w)$ of a symbol $a$ and a natural-number weight $w$, with decidable equality on symbols. If the Huffman construction on $L$ succeeds with a tree $T$, then the list of weighted leaves of $T$, read left to right, is a permutation of $L$.","title":"Huffman's construction keeps the weighted leaves"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-prefix-free","kind":"node","lean":"Cslib.Algorithms.Lean.huffman_prefixFree","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-prefix-free","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"If the Huffman construction succeeds on a list of weighted symbols and returns a tree $T$, then the code of $T$ on its leaf symbols is prefix-free: for any two distinct leaf symbols of $T$, neither codeword is a prefix of the other.","title":"Huffman codes are prefix-free"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-weight-tree","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-weight-tree","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"A Huffman weight tree is a full binary tree whose leaves carry only real weights: it is either a leaf with a weight $w \\in \\mathbb{R}$, or a fork joining a left and a right Huffman weight tree.","title":"Huffman weight tree"},{"id":"InformationTheory/source-coding/huffman-coding/weights","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.weights","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#weights","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The leaf weights of a Huffman weight tree form a multiset of reals that forgets where the leaves sit: a leaf of weight $w$ gives $\\{w\\}$, and a tree with subtrees $L$ and $R$ gives the multiset sum of the leaf weights of $L$ and those of $R$.","title":"Multiset of leaf weights"},{"id":"InformationTheory/source-coding/huffman-coding/total-weight-2","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.totalWeight","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#total-weight-2","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The total weight of a Huffman weight tree is the sum of its leaf weights: a leaf of weight $w$ has total weight $w$, and a tree with subtrees $L$ and $R$ has total weight equal to the total weight of $L$ plus that of $R$.","title":"Total leaf weight of a Huffman weight tree"},{"id":"InformationTheory/source-coding/huffman-coding/path-cost","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.pathCost","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#path-cost","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a Huffman weight tree $T$, write $w(T)$ for its total leaf weight. The weighted external path length $P(T)$ of $T$ is defined by the merge recurrence: a leaf $T$ has $P(T) = 0$, and a tree $T$ with subtrees $L$ and $R$ has $P(T) = P(L) + P(R) + w(L) + w(R)$.","title":"Weighted external path length of a Huffman weight tree"},{"id":"InformationTheory/source-coding/huffman-coding/height","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.height","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#height","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The height of a weight tree is the number of edges on a longest path from the root to a leaf: a leaf has height $0$, and a fork with subtrees $L$ and $R$ has height $1 + \\max(\\mathrm{height}(L), \\mathrm{height}(R))$.","title":"Height of a weight tree"},{"id":"InformationTheory/source-coding/huffman-coding/contains-at","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.ContainsAt","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#contains-at","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For a real number $v$, a weight tree $T$ and a natural number $d$, the relation \"$T$ contains $v$ at depth $d$\" is defined inductively: the leaf of weight $v$ contains $v$ at depth $0$, and if $L$ contains $v$ at depth $d$ or $R$ contains $v$ at depth $d$, then the fork with left subtree $L$ and right subtree $R$ contains $v$ at depth $d + 1$. Thus it holds exactly when some leaf of $T$ of weight $v$ lies at depth\u2026","title":"Weight at a given depth"},{"id":"InformationTheory/source-coding/huffman-coding/reweight","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.ContainsAt.reweight","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#reweight","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Suppose a leaf of weight $v$ occurs at depth $d$ in a Huffman weight tree $T$, and let $r$ be a real number. Then there are a multiset $R$ of reals and a Huffman weight tree $T'$ such that the multiset of leaf weights of $T$ is $\\{v\\} + R$, that of $T'$ is $\\{r\\} + R$, the total leaf weight of $T'$ is that of $T$ plus $r - v$, the weighted external path length of $T'$ is that of $T$ plus $d(r - v)$, and $T'$ has\u2026","title":"Reweighting one leaf occurrence"},{"id":"InformationTheory/source-coding/huffman-coding/raise-bounded","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.raiseBounded","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#raise-bounded","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman weight tree whose height is at most a natural number $B$, and write $W(T)$ for its multiset of leaf weights. Let $v \\le r$ be real numbers and let $S$ be a multiset of reals with $\\{r\\} + W(T) = \\{v\\} + S$. Then there is a Huffman weight tree $T'$ with multiset of leaf weights $S$ and height at most $B$ whose weighted external path length is at most that of $T$ plus $B(r - v)$.","title":"Raising one leaf weight in a bounded-height tree"},{"id":"InformationTheory/source-coding/huffman-coding/raise-pair","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.raisePair","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#raise-pair","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a Huffman weight tree whose height is at most a natural number $B$, and write $W(T)$ for its multiset of leaf weights. Let $f, s, \\ell, n$ be real numbers with $\\ell \\le n$, $\\ell \\le f$ and $n \\le s$, and let $S$ be a multiset of reals with $\\{f, s\\} + W(T) = \\{\\ell, n\\} + S$. Then there is a Huffman weight tree $T'$ with multiset of leaf weights $S$ whose weighted external path length is at most that\u2026","title":"Raising two leaf weights in a bounded-height tree"},{"id":"InformationTheory/source-coding/huffman-coding/exists-cherry-contraction","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.exists_cherry_contraction","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#exists-cherry-contraction","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a weight tree of height $h \\ge 1$. Then there are real numbers $x$ and $y$, a multiset $M$ of real numbers and a real number $c$ such that the multiset of leaf weights of $T$ is $\\{x, y\\} + M$, the path cost of $T$ is $c + hx + hy$, and for every real number $z$ there is a weight tree $T'$ of height at most $h$ whose multiset of leaf weights is $\\{z\\} + M$ and whose path cost is $c + (h - 1)z$.","title":"Contracting a deepest sibling pair"},{"id":"InformationTheory/source-coding/huffman-coding/contract-least-pair","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.contractLeastPair","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#contract-least-pair","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $T$ be a weight tree, let $a$ and $b$ be real numbers and let $M$ be a multiset of real numbers such that the multiset of leaf weights of $T$ is $\\{a, b\\} + M$, $a \\le b$, and $b \\le w$ for every $w \\in M$. Then there is a weight tree $T'$ whose multiset of leaf weights is $\\{a + b\\} + M$ and whose path cost satisfies $\\mathrm{cost}(T') + a + b \\le \\mathrm{cost}(T)$, where $\\mathrm{cost}$ denotes path cost.","title":"Contracting the two least weights"},{"id":"InformationTheory/source-coding/huffman-coding/merge-trace","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.MergeTrace","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#merge-trace","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The relation \"$c$ is a two-minimum merge cost of the multiset $W$ of real weights\" is defined inductively by two rules. A singleton multiset $\\{w\\}$ has merge cost $0$. If $a \\le b$, every element of the multiset $R$ is at least $b$, and $c$ is a merge cost of $\\{a + b\\} + R$, then $a + b + c$ is a merge cost of $\\{a, b\\} + R$. Thus a merge cost is the total cost of repeatedly replacing two smallest weights by\u2026","title":"Two-minimum merge trace"},{"id":"InformationTheory/source-coding/huffman-coding/le-path-cost","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanWeightTree.MergeTrace.le_pathCost","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#le-path-cost","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $W$ be a multiset of real weights and let $c$ be a merge cost of $W$ certified by a two-minimum merge trace. Then every Huffman weight tree whose multiset of leaf weights is $W$ has weighted external path length at least $c$.","title":"Two-minimum merge costs lower-bound weighted path length"},{"id":"InformationTheory/source-coding/huffman-coding/erase-symbols","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.eraseSymbols","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#erase-symbols","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Erasing the symbols of a Huffman tree gives the weight tree of the same shape: a leaf with symbol $s$ and natural-number weight $w$ becomes the leaf of real weight $w$, and a fork with subtrees $L$ and $R$ becomes the fork whose subtrees are the erasures of $L$ and $R$.","title":"Erasing symbols from a Huffman tree"},{"id":"InformationTheory/source-coding/huffman-coding/erase-symbols-weights","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.eraseSymbols_weights","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#erase-symbols-weights","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For every Huffman tree $T$, the multiset of leaf weights of the weight tree obtained from $T$ by erasing its symbols is the multiset of the weights $w$, viewed as real numbers, over the weighted leaves $(s, w)$ of $T$.","title":"Erasing symbols preserves leaf weights"},{"id":"InformationTheory/source-coding/huffman-coding/erase-symbols-total-weight","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.eraseSymbols_totalWeight","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#erase-symbols-total-weight","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For every Huffman tree $T$, the total weight of the weight tree obtained from $T$ by erasing its symbols equals the total weight of $T$, viewed as a real number.","title":"Erasing symbols preserves total weight"},{"id":"InformationTheory/source-coding/huffman-coding/erase-symbols-path-cost","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.eraseSymbols_pathCost","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#erase-symbols-path-cost","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"For every Huffman tree $T$, the path cost of the weight tree obtained from $T$ by erasing its symbols equals the weighted path length of $T$, viewed as a real number.","title":"Erasing symbols preserves weighted path length"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-forest-priorities","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanForestPriorities","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-forest-priorities","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The priorities of a list of Huffman work items form the multiset consisting of the priority of each item in the list, counted with multiplicity.","title":"Priorities of a Huffman work forest"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-forest-weights","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanForestWeights","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-forest-weights","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The weights of a list of Huffman work items form the multiset consisting of the total leaf weight of each item's tree, taken as a real number, counted with multiplicity.","title":"Weights of a Huffman work forest"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-forest-cost","kind":"node","lean":"Cslib.Algorithms.Lean.huffmanForestCost","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-forest-cost","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"The cost of a list of Huffman work items is the real number obtained by summing, over the items in the list, the weighted path length of each item's tree.","title":"Cost of a Huffman work forest"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-loop-invariant","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanLoopInvariant","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-loop-invariant","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $n$ be a natural number, $H$ a binary min-heap of Huffman priorities and $T$ an array of optional Huffman work items, and let $F$ be the list of work items stored in $T$, in array order. The invariant holds when the multiset of elements of $H$ equals the multiset of priorities of the items of $F$; the serial numbers of the items of $F$ are pairwise distinct; every item of $F$ has serial number less than $n$;\u2026","title":"Huffman loop invariant"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-merge-trace","kind":"node","lean":"Cslib.Algorithms.Lean.huffman_mergeTrace","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-merge-trace","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $L$ be a list of symbol\u2013weight pairs with natural-number weights, over a symbol type with decidable equality, and suppose the Huffman construction returns the tree $t$ on $L$. Then the weighted path length of $t$, as a real number, is the cost of a two-minimum merge trace for the multiset of the weights of $L$, taken as reals: starting from that multiset, repeatedly replace two weights $x \\le y$ such that every\u2026","title":"Huffman's construction follows a two-minimum merge trace"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-is-some","kind":"node","lean":"Cslib.Algorithms.Lean.huffman_isSome","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-is-some","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $L$ be a nonempty list of symbol\u2013weight pairs with natural-number weights, over a symbol type with decidable equality. If no symbol occurs in $L$ more than once, then the Huffman construction returns a tree on $L$.","title":"Huffman's construction succeeds on nonempty inputs with distinct symbols"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-eq-none-iff","kind":"node","lean":"Cslib.Algorithms.Lean.huffman_eq_none_iff","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-eq-none-iff","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $L$ be a list of symbol\u2013weight pairs with natural-number weights, over a symbol type with decidable equality. The Huffman construction returns no tree on $L$ if and only if $L$ is empty or some symbol occurs in $L$ more than once.","title":"Huffman's construction fails exactly on empty or repeated-symbol inputs"},{"id":"InformationTheory/source-coding/huffman-coding/admissible","kind":"node","lean":"Cslib.Algorithms.Lean.HuffmanTree.Admissible","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#admissible","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Given an input list of symbols with natural-number weights, a Huffman tree is admissible for the input when its list of weighted leaves, read from left to right, is a permutation of the input list: the leaves of the tree carry exactly the input's (symbol, weight) pairs, with the same multiplicities.","title":"Admissible Huffman trees"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-optimal-of-merge-trace","kind":"node","lean":"Cslib.Algorithms.Lean.huffman_optimal_of_mergeTrace","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-optimal-of-merge-trace","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $L$ be a list of symbol\u2013weight pairs with natural-number weights, and let $t$ and $t'$ be coding trees with symbols of the same type. Suppose the weighted path length of $t$, as a real number, is the cost of a two-minimum merge trace for the multiset of the weights of $L$, taken as reals (repeatedly replace two weights $x \\le y$, with every other weight at least $y$, by $x + y$ until one weight remains, the\u2026","title":"A two-minimum merge trace gives a minimum weighted path length"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.huffman_optimal","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-optimal","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $L$ be a list of symbol\u2013weight pairs with natural-number weights, over a symbol type with decidable equality, and suppose the Huffman construction returns the tree $t$ on $L$. Then every coding tree $t'$ (a full binary tree whose leaves carry symbols and weights) whose weighted leaves, listed left to right, form a permutation of $L$ has weighted path length at least that of $t$.","title":"Optimality of Huffman coding"},{"id":"InformationTheory/source-coding/huffman-coding/huffman-encoded-length-optimal","kind":"node","lean":"Cslib.Algorithms.Lean.huffman_encodedLength_optimal","parent":43806,"result":true,"route":"../roadmap/InformationTheory/source-coding/huffman-coding/index.html#huffman-encoded-length-optimal","scope":"InformationTheory/source-coding/huffman-coding","status":"fully_proved","summary":"Let $L$ be a list of symbol\u2013weight pairs with natural-number weights, over a symbol type with decidable equality, and suppose Huffman's algorithm returns the tree $t$ on $L$. Then every Huffman tree $t'$ that is admissible for $L$, that is, whose weighted leaves listed from left to right form a permutation of $L$, has encoded length at least that of $t$.","title":"Huffman's algorithm minimizes the encoded length"},{"data_url":"scopes/InformationTheory/source-coding/run-length-encoding.json","id":"InformationTheory/source-coding/run-length-encoding","kind":"scope","navigation":"module","parent":43799,"result":false,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","title":"Run-length encoding"},{"id":"InformationTheory/source-coding/run-length-encoding/compress-from","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.compressFrom","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#compress-from","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"For a bit $c$, a natural number $r$ and a finite bit list $w$, the compression $C(c, r, w)$ of $w$ from the pending run $(c, r)$ is the list of natural numbers defined by recursion on $w$. If $w$ is empty, $C(c, r, w)$ is the one-entry list $(r)$. If $w$ is a bit $b$ followed by a list $w'$, then $C(c, r, w)$ is $r$ followed by $C(\\lnot c, 1, w')$ when $b \\ne c$; it is $255$, then $0$, followed by $C(c, 1, w')$\u2026","title":"Run-length compression from a pending run"},{"id":"InformationTheory/source-coding/run-length-encoding/compress","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.compress","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#compress","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"The run-length compression of a finite bit list $w$ is $C(0, 0, w)$, the compression of $w$ from an empty pending run of zero bits. It lists the lengths of the maximal runs of $w$, alternating between zeros and ones and starting with a possibly empty run of zeros; a run of length $n > 255$ is written as $255, 0, 255, 0, \\dots, 255, 0, m$ with $1 \\le m \\le 255$.","title":"Run-length compression"},{"id":"InformationTheory/source-coding/run-length-encoding/expand-from","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.expandFrom","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#expand-from","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"For a bit $c$ and a list $(n_1, \\dots, n_k)$ of natural numbers, the expansion $E(c, (n_1, \\dots, n_k))$ is the bit list obtained by concatenating, for $i = 1, \\dots, k$ in order, $n_i$ copies of $c$ when $i$ is odd and $n_i$ copies of $\\lnot c$ when $i$ is even.","title":"Run-length expansion from a starting bit"},{"id":"InformationTheory/source-coding/run-length-encoding/expand","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.expand","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#expand","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"The run-length expansion of a list $(n_1, \\dots, n_k)$ of natural numbers is $E(0, (n_1, \\dots, n_k))$, the expansion from the zero bit: the bit list obtained by concatenating, for $i = 1, \\dots, k$ in order, $n_i$ zeros when $i$ is odd and $n_i$ ones when $i$ is even.","title":"Run-length expansion"},{"id":"InformationTheory/source-coding/run-length-encoding/compress-from-le","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.compressFrom_le","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#compress-from-le","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"For every bit $c$, every natural number $r \\le 255$ and every finite bit list $w$, every entry of the compression $C(c, r, w)$ of $w$ from the pending run $(c, r)$ is at most $255$.","title":"Compression from a short pending run emits counts at most 255"},{"id":"InformationTheory/source-coding/run-length-encoding/compress-le","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.compress_le","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#compress-le","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"For every finite bit list $w$, every entry of the run-length compression of $w$ is at most $255$.","title":"Run lengths fit in 8 bits"},{"id":"InformationTheory/source-coding/run-length-encoding/expand-from-length","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.expandFrom_length","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#expand-from-length","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"For every bit $c$ and every list $(n_1, \\dots, n_k)$ of natural numbers, the expansion $E(c, (n_1, \\dots, n_k))$ has length $n_1 + \\dots + n_k$.","title":"Length of an expansion from a starting bit"},{"id":"InformationTheory/source-coding/run-length-encoding/expand-length","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.expand_length","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#expand-length","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"For every list $(n_1, \\dots, n_k)$ of natural numbers, the run-length expansion of $(n_1, \\dots, n_k)$ has length $n_1 + \\dots + n_k$.","title":"Length of a run-length expansion"},{"id":"InformationTheory/source-coding/run-length-encoding/expand-from-compress-from","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.expandFrom_compressFrom","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#expand-from-compress-from","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"For every bit $c$, every natural number $r$ and every finite bit list $w$, the expansion from the starting bit $c$ of the compression $C(c, r, w)$ is the list of $r$ copies of $c$ followed by $w$.","title":"Expansion recovers the pending run and the input"},{"id":"InformationTheory/source-coding/run-length-encoding/expand-compress","kind":"node","lean":"Cslib.Algorithms.Lean.RunLength.expand_compress","parent":43888,"result":true,"route":"../roadmap/InformationTheory/source-coding/run-length-encoding/index.html#expand-compress","scope":"InformationTheory/source-coding/run-length-encoding","status":"fully_proved","summary":"For every finite bit list $w$, the run-length expansion of the run-length compression of $w$ is $w$.","title":"Run-length expansion inverts compression"},{"data_url":"scopes/InformationTheory/source-coding/shannon-s-source-coding-theorem.json","id":"InformationTheory/source-coding/shannon-s-source-coding-theorem","kind":"scope","navigation":"module","parent":43799,"result":false,"route":"../roadmap/InformationTheory/source-coding/shannon-s-source-coding-theorem/index.html","scope":"InformationTheory/source-coding/shannon-s-source-coding-theorem","status":"fully_proved","title":"Shannon's source coding theorem"},{"id":"InformationTheory/source-coding/shannon-s-source-coding-theorem/is-prefix-free","kind":"node","lean":"MathlibExt.InformationTheory.ShannonSourceCodingWanted.IsPrefixFree","parent":43899,"result":true,"route":"../roadmap/InformationTheory/source-coding/shannon-s-source-coding-theorem/index.html#is-prefix-free","scope":"InformationTheory/source-coding/shannon-s-source-coding-theorem","status":"fully_proved","summary":"A code $c$ assigning to each element of $\\alpha$ a finite word over $\\beta$ is prefix-free when, for all distinct $a_1, a_2 \\in \\alpha$, the word $c(a_1)$ is not a prefix of the word $c(a_2)$. This is the general notion of a prefix-free code.","title":"Prefix-free code (Shannon source coding)"},{"id":"InformationTheory/source-coding/shannon-s-source-coding-theorem/shannon-source-coding-lower-bound","kind":"node","lean":"MathlibExt.InformationTheory.ShannonSourceCodingWanted.shannon_source_coding_lower_bound","parent":43899,"result":true,"route":"../roadmap/InformationTheory/source-coding/shannon-s-source-coding-theorem/index.html#shannon-source-coding-lower-bound","scope":"InformationTheory/source-coding/shannon-s-source-coding-theorem","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be finite types with $q = |\\beta| \\ge 2$, and let $p : \\alpha \\to \\mathbb{R}$ satisfy $p(a) > 0$ for every $a \\in \\alpha$ and $\\sum_{a \\in \\alpha} p(a) = 1$. Let $c$ assign to each $a \\in \\alpha$ a finite word $c(a)$ over $\\beta$, of length $|c(a)|$, such that $c$ is injective and its set of codewords is uniquely decodable: any two finite lists of codewords with the same concatenation are\u2026","title":"Shannon source coding lower bound"},{"id":"InformationTheory/source-coding/shannon-s-source-coding-theorem/shannon-source-coding-upper-bound","kind":"node","lean":"MathlibExt.InformationTheory.ShannonSourceCodingWanted.shannon_source_coding_upper_bound","parent":43899,"result":true,"route":"../roadmap/InformationTheory/source-coding/shannon-s-source-coding-theorem/index.html#shannon-source-coding-upper-bound","scope":"InformationTheory/source-coding/shannon-s-source-coding-theorem","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be finite types with $q = |\\beta| \\ge 2$, and let $p : \\alpha \\to \\mathbb{R}$ satisfy $p(a) > 0$ for every $a \\in \\alpha$ and $\\sum_{a \\in \\alpha} p(a) = 1$. Then there is a prefix-free code $c$ assigning to each $a \\in \\alpha$ a finite word $c(a)$ over $\\beta$, of length $|c(a)|$, whose expected length $\\sum_{a \\in \\alpha} p(a)\\,|c(a)|$ is strictly less than\u2026","title":"Shannon source coding upper bound"},{"data_url":"scopes/InformationTheory/error-correcting-codes.json","id":"InformationTheory/error-correcting-codes","kind":"scope","parent":43798,"result":false,"route":"../roadmap/InformationTheory/error-correcting-codes/index.html","scope":"InformationTheory/error-correcting-codes","status":"can_prove","summary":"Codes that detect or correct transmission errors, including linear codes, Hamming and Reed\u2013Solomon codes, cyclic redundancy checks, and the Hamming-space geometry used to bound them.","title":"Error-correcting codes"},{"data_url":"scopes/InformationTheory/error-correcting-codes/bounds-on-code-size.json","id":"InformationTheory/error-correcting-codes/bounds-on-code-size","kind":"scope","navigation":"module","parent":43903,"result":false,"route":"../roadmap/InformationTheory/error-correcting-codes/bounds-on-code-size/index.html","scope":"InformationTheory/error-correcting-codes/bounds-on-code-size","status":"fully_proved","title":"Bounds on code size"},{"id":"InformationTheory/error-correcting-codes/bounds-on-code-size/singleton-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.SingletonBound.singleton_bound","parent":43904,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/bounds-on-code-size/index.html#singleton-bound","scope":"InformationTheory/error-correcting-codes/bounds-on-code-size","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite alphabet with $q$ letters, and let $n$ and $d$ be natural numbers with $1 \\le d \\le n$. If $C$ is a set of words of length $n$ over $\\alpha$ in which any two distinct words are at Hamming distance at least $d$, then $C$ has at most $q^{n - d + 1}$ elements.","title":"Singleton bound"},{"id":"InformationTheory/error-correcting-codes/bounds-on-code-size/ball","kind":"node","lean":"Hamming.ball","parent":43904,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/bounds-on-code-size/index.html#ball","scope":"InformationTheory/error-correcting-codes/bounds-on-code-size","status":"fully_proved","summary":"For finite types $\\iota$ and $F$, a word $x : \\iota \\to F$ and a natural number $e$, the closed Hamming ball of radius $e$ about $x$ is the finite set of all words $y : \\iota \\to F$ whose Hamming distance from $x$ is at most $e$.","title":"Closed Hamming ball"},{"id":"InformationTheory/error-correcting-codes/bounds-on-code-size/card-ball","kind":"node","lean":"Hamming.card_ball","parent":43904,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/bounds-on-code-size/index.html#card-ball","scope":"InformationTheory/error-correcting-codes/bounds-on-code-size","status":"fully_proved","summary":"For finite types $\\iota$ and $F$, every word $x : \\iota \\to F$ and every natural number $e$, the closed Hamming ball of radius $e$ about $x$ has $\\sum_{k=0}^{e} \\binom{|\\iota|}{k} (|F| - 1)^k$ elements.","title":"Size of a Hamming ball"},{"id":"InformationTheory/error-correcting-codes/bounds-on-code-size/card-ball-fin","kind":"node","lean":"Hamming.card_ball_fin","parent":43904,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/bounds-on-code-size/index.html#card-ball-fin","scope":"InformationTheory/error-correcting-codes/bounds-on-code-size","status":"fully_proved","summary":"Let $n \\ge 1$ and let $F$ be a finite alphabet with $q \\ge 2$ letters. For every word $x \\in F^n$ and every natural number $e$, the closed Hamming ball of radius $e$ about $x$ has $\\sum_{k=0}^{e} \\binom{n}{k} (q - 1)^k$ elements.","title":"Size of a Hamming ball of fixed-length words"},{"id":"InformationTheory/error-correcting-codes/bounds-on-code-size/hamming-sphere-packing-bound","kind":"node","lean":"MathlibExt.Combinatorics.Coding.HammingSpherePacking.hamming_sphere_packing_bound","parent":43904,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/bounds-on-code-size/index.html#hamming-sphere-packing-bound","scope":"InformationTheory/error-correcting-codes/bounds-on-code-size","status":"fully_proved","summary":"Let $\\alpha$ be a finite alphabet with $q \\ge 2$ letters, and let $n$ and $t$ be natural numbers. If $C$ is a set of words of length $n$ over $\\alpha$ in which any two distinct words are at Hamming distance at least $2t + 1$, then","title":"Hamming bound"},{"data_url":"scopes/InformationTheory/error-correcting-codes/cyclic-redundancy-checks.json","id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","kind":"scope","navigation":"module","parent":43903,"result":false,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","title":"Cyclic redundancy checks"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/encode","kind":"node","lean":"Cslib.CodingTheory.CRC.encode","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#encode","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field, let $g$ be a polynomial over $F$ (the generator), and let $r$ be the degree of $g$ (with $r = 0$ when $g = 0$). The CRC encoding of a polynomial $m$ over $F$ (the message) is $m X^r - (m X^r \\bmod g)$: the message shifted up by $r$ positions, minus the remainder of the shifted message modulo $g$.","title":"CRC encoder"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/encode-eq-add","kind":"node","lean":"Cslib.CodingTheory.CRC.encode_eq_add","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#encode-eq-add","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field of characteristic 2 (for example $\\mathbb{F}_2$), let $g$ and $m$ be polynomials over $F$, and let $r$ be the degree of $g$ (with $r = 0$ when $g = 0$). Then the CRC encoding of $m$ with generator $g$ equals $m X^r + (m X^r \\bmod g)$: the encoder appends the remainder to the shifted message.","title":"CRC encoding in characteristic 2"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/dvd-encode","kind":"node","lean":"Cslib.CodingTheory.CRC.dvd_encode","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#dvd-encode","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field. For all polynomials $g$ and $m$ over $F$, the polynomial $g$ divides the CRC encoding of $m$ with generator $g$.","title":"CRC codewords are multiples of the generator"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/coeff-encode-add","kind":"node","lean":"Cslib.CodingTheory.CRC.coeff_encode_add","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#coeff-encode-add","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field and $g \\ne 0$ a polynomial over $F$ of degree $r$. For every message polynomial $m$ over $F$ and every natural number $i$, the coefficient of $X^{i+r}$ in the CRC encoding of $m$ with generator $g$ equals the coefficient of $X^i$ in $m$.","title":"CRC encoding is systematic"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/dvd-encode-add-iff","kind":"node","lean":"Cslib.CodingTheory.CRC.dvd_encode_add_iff","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#dvd-encode-add-iff","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field, let $g$, $m$ and $e$ be polynomials over $F$, and let $c$ be the CRC encoding of $m$ with generator $g$. Then $g$ divides $c + e$ if and only if $g$ divides $e$: a codeword plus an error pattern passes the check exactly when $g$ divides the error.","title":"CRC detection criterion"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/not-dvd-c-mul-x-pow","kind":"node","lean":"Cslib.CodingTheory.CRC.not_dvd_C_mul_X_pow","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#not-dvd-c-mul-x-pow","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field and $g$ a polynomial over $F$ with at least two nonzero coefficients. Then $g$ does not divide $a X^i$ for any nonzero $a \\in F$ and any natural number $i$.","title":"Single-symbol errors are detected"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/is-burst-le","kind":"node","lean":"Cslib.CodingTheory.CRC.IsBurstLE","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#is-burst-le","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field and $L$ a natural number. A polynomial $e$ over $F$ is a burst of length at most $L$ when $e = X^i b$ for some natural number $i$ and some polynomial $b$ over $F$ with $b(0) \\ne 0$ and $\\deg b < L$. Then the nonzero coefficients of $e$ lie in positions $i, \\dots, i + L - 1$, the first one nonzero.","title":"Burst errors"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/is-burst-le-of-support","kind":"node","lean":"Cslib.CodingTheory.CRC.isBurstLE_of_support","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#is-burst-le-of-support","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field, let $e \\ne 0$ be a polynomial over $F$, and let $i$ and $L$ be natural numbers. If every $j$ for which the coefficient of $X^j$ in $e$ is nonzero satisfies $i \\le j < i + L$, then $e$ is a burst of length at most $L$.","title":"Errors confined to a window are bursts"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/not-dvd-of-is-burst-le","kind":"node","lean":"Cslib.CodingTheory.CRC.not_dvd_of_isBurstLE","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#not-dvd-of-is-burst-le","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field, let $g$ be a polynomial over $F$ that is not divisible by $X$, and let $L$ be a natural number with $L \\le \\deg g$. If a polynomial $e$ over $F$ is a burst of length at most $L$, then $g$ does not divide $e$.","title":"Short bursts are detected"},{"id":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks/detects","kind":"node","lean":"Cslib.CodingTheory.CRC.detects","parent":43910,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/cyclic-redundancy-checks/index.html#detects","scope":"InformationTheory/error-correcting-codes/cyclic-redundancy-checks","status":"fully_proved","summary":"Let $F$ be a field, let $g$ be a polynomial over $F$ with at least two nonzero coefficients that is not divisible by $X$, let $m$ be a polynomial over $F$, and let $c$ be the CRC encoding of $m$ with generator $g$. Then $g$ does not divide $c + aX^i$ for any nonzero $a \\in F$ and any natural number $i$, and $g$ does not divide $c + e$ for any polynomial $e$ over $F$ that is a burst of length at most $\\deg g$. That\u2026","title":"CRC error-detection guarantee"},{"data_url":"scopes/InformationTheory/error-correcting-codes/linear-codes.json","id":"InformationTheory/error-correcting-codes/linear-codes","kind":"scope","navigation":"module","parent":43903,"result":false,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","title":"Linear codes"},{"id":"InformationTheory/error-correcting-codes/linear-codes/linear-code","kind":"node","lean":"Cslib.CodingTheory.LinearCode","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#linear-code","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field and $\\iota$ an index set, not necessarily finite. A linear code over $F$ with coordinates indexed by $\\iota$ is an $F$-linear subspace of the space of words, the functions $\\iota \\to F$.","title":"Linear code"},{"id":"InformationTheory/error-correcting-codes/linear-codes/encode","kind":"node","lean":"Cslib.CodingTheory.LinearCode.encode","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#encode","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $G$ be a $\\kappa \\times \\iota$ matrix over a field $F$, with $\\kappa$ finite. The encoder of $G$ sends a message $m \\in F^\\kappa$ to the word $mG \\in F^\\iota$ given by $(mG)_j = \\sum_{k \\in \\kappa} m_k G_{kj}$, the combination of the rows of $G$ with coefficients $m$.","title":"Encoder of a generator matrix"},{"id":"InformationTheory/error-correcting-codes/linear-codes/of-generator","kind":"node","lean":"Cslib.CodingTheory.LinearCode.ofGenerator","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#of-generator","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field, $\\kappa$ a finite index set, $\\iota$ any index set and $G$ a $\\kappa \\times \\iota$ matrix over $F$. The code generated by $G$ is the linear code of all words $m G$ with $m : \\kappa \\to F$, the image of the encoder $m \\mapsto m G$; equivalently, the span of the rows of $G$.","title":"Code generated by a matrix"},{"id":"InformationTheory/error-correcting-codes/linear-codes/syndrome","kind":"node","lean":"Cslib.CodingTheory.LinearCode.syndrome","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#syndrome","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field, $\\iota$ a finite set and $\\rho$ a set. For a matrix $H \\in F^{\\rho \\times \\iota}$ (a parity-check matrix) and a word $y \\in F^\\iota$, the syndrome of $y$ is the vector $Hy \\in F^\\rho$.","title":"Syndrome"},{"id":"InformationTheory/error-correcting-codes/linear-codes/of-parity-check","kind":"node","lean":"Cslib.CodingTheory.LinearCode.ofParityCheck","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#of-parity-check","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field, $\\iota$ a finite index set, $\\rho$ any index set and $H$ a $\\rho \\times \\iota$ matrix over $F$. The code with parity-check matrix $H$ is the linear code of all words $c : \\iota \\to F$ whose syndrome $H c$ is zero, that is, the kernel of $c \\mapsto H c$.","title":"Code with a given parity-check matrix"},{"id":"InformationTheory/error-correcting-codes/linear-codes/syndrome-add-of-mem","kind":"node","lean":"Cslib.CodingTheory.LinearCode.syndrome_add_of_mem","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#syndrome-add-of-mem","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field, $\\iota$ a finite set and $H$ a parity-check matrix over $F$ with columns indexed by $\\iota$. If $c \\in F^\\iota$ is a codeword of the code with parity-check matrix $H$ and $e \\in F^\\iota$ is any word, then the received word $c + e$ has the same syndrome as $e$, that is, $H(c + e) = He$.","title":"The syndrome depends only on the error"},{"id":"InformationTheory/error-correcting-codes/linear-codes/of-generator-eq-of-parity-check-iff","kind":"node","lean":"Cslib.CodingTheory.LinearCode.ofGenerator_eq_ofParityCheck_iff","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#of-generator-eq-of-parity-check-iff","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field, $\\kappa$ and $\\iota$ finite index sets and $\\rho$ any index set, and let $G$ be a $\\kappa \\times \\iota$ matrix and $H$ a $\\rho \\times \\iota$ matrix over $F$. The code generated by $G$ equals the code with parity-check matrix $H$ if and only if $G H^{\\mathsf{T}} = 0$ and $\\operatorname{rank} G + \\operatorname{rank} H$ equals $|\\iota|$, the length of the code.","title":"Generator/parity-check duality"},{"id":"InformationTheory/error-correcting-codes/linear-codes/encode-from-cols-one-inl","kind":"node","lean":"Cslib.CodingTheory.LinearCode.encode_fromCols_one_inl","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#encode-from-cols-one-inl","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field, $\\kappa$ a finite set and $A$ a $\\kappa \\times \\rho$ matrix over $F$. For the systematic generator matrix $G = [I \\mid A]$, whose columns are indexed by the disjoint union $\\kappa \\sqcup \\rho$, the encoding $mG$ of every message $m \\in F^\\kappa$ agrees with $m$ on the coordinates in $\\kappa$: $(mG)_i = m_i$ for every $i \\in \\kappa$.","title":"The systematic encoder copies the message"},{"id":"InformationTheory/error-correcting-codes/linear-codes/systematic-eq","kind":"node","lean":"Cslib.CodingTheory.LinearCode.systematic_eq","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#systematic-eq","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field, $\\kappa$ and $\\rho$ finite sets, and $A$ a $\\kappa \\times \\rho$ matrix over $F$. Then the code generated by $G = [I_\\kappa \\mid A]$ equals the code with parity-check matrix $H = [-A^{\\mathsf{T}} \\mid I_\\rho]$, both being codes whose coordinates are indexed by the disjoint union of $\\kappa$ and $\\rho$.","title":"Systematic generator and parity-check matrices"},{"id":"InformationTheory/error-correcting-codes/linear-codes/dot-form","kind":"node","lean":"Cslib.CodingTheory.dotForm","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#dot-form","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field and $\\iota$ a finite set. The dot product form is the bilinear form on $F^\\iota$ that sends a pair of words $(x, y)$ to $x \\cdot y = \\sum_{i \\in \\iota} x_i y_i$.","title":"Dot product bilinear form"},{"id":"InformationTheory/error-correcting-codes/linear-codes/dual","kind":"node","lean":"Cslib.CodingTheory.LinearCode.dual","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#dual","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field and $\\iota$ a finite set. The dual $C^\\perp$ of a linear code $C \\subseteq F^\\iota$ is the linear code of all words $x \\in F^\\iota$ with $x \\cdot c = \\sum_{i \\in \\iota} x_i c_i = 0$ for every codeword $c \\in C$.","title":"Dual code"},{"id":"InformationTheory/error-correcting-codes/linear-codes/finrank-add-finrank-dual","kind":"node","lean":"Cslib.CodingTheory.LinearCode.finrank_add_finrank_dual","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#finrank-add-finrank-dual","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field and $\\iota$ a finite set. For every linear code $C \\subseteq F^\\iota$, $\\dim C + \\dim C^\\perp = |\\iota|$, where $C^\\perp$ is the dual code.","title":"Dimension of the dual code"},{"id":"InformationTheory/error-correcting-codes/linear-codes/dual-of-generator","kind":"node","lean":"Cslib.CodingTheory.LinearCode.dual_ofGenerator","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#dual-of-generator","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $G$ be a matrix over a field $F$ with finitely many rows and finitely many columns. Then the dual of the code generated by $G$ (the span of its rows) is the code with parity-check matrix $G$ (the words $x$ with $Gx = 0$).","title":"Dual of the code generated by a matrix"},{"id":"InformationTheory/error-correcting-codes/linear-codes/dual-of-parity-check","kind":"node","lean":"Cslib.CodingTheory.LinearCode.dual_ofParityCheck","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#dual-of-parity-check","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $H$ be a matrix over a field $F$ with finitely many rows and finitely many columns. Then the dual of the code with parity-check matrix $H$ (the words $x$ with $Hx = 0$) is the code generated by $H$ (the span of its rows).","title":"Dual of the code with a given parity-check matrix"},{"id":"InformationTheory/error-correcting-codes/linear-codes/min-dist","kind":"node","lean":"Cslib.CodingTheory.LinearCode.minDist","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#min-dist","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"The minimum distance of a linear code $C$ of finite length is the least Hamming distance (number of coordinates where they differ) between two distinct codewords of $C$, taken in $\\mathbb{N} \\cup \\{\\infty\\}$; it is $\\infty$ if $C$ has at most one codeword.","title":"Minimum distance"},{"id":"InformationTheory/error-correcting-codes/linear-codes/min-weight","kind":"node","lean":"Cslib.CodingTheory.LinearCode.minWeight","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#min-weight","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"The minimum weight of a linear code $C$ of finite length is the least weight of a nonzero codeword of $C$, where the weight of a word is its number of nonzero coordinates; it is taken in $\\mathbb{N} \\cup \\{\\infty\\}$ and is $\\infty$ if $C$ has no nonzero codeword.","title":"Minimum weight"},{"id":"InformationTheory/error-correcting-codes/linear-codes/hamming-dist-eq-hamming-norm-sub","kind":"node","lean":"Cslib.CodingTheory.hammingDist_eq_hammingNorm_sub","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#hamming-dist-eq-hamming-norm-sub","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field and $\\iota$ a finite set. For all words $x, y \\in F^\\iota$, the Hamming distance between $x$ and $y$ (the number of coordinates where they differ) equals the Hamming weight of $x - y$ (its number of nonzero coordinates).","title":"Hamming distance is the weight of the difference"},{"id":"InformationTheory/error-correcting-codes/linear-codes/min-dist-eq-min-weight","kind":"node","lean":"Cslib.CodingTheory.LinearCode.minDist_eq_minWeight","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#min-dist-eq-min-weight","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"For a linear code of finite length over a field, the minimum distance equals the minimum weight.","title":"Minimum distance equals minimum weight"},{"id":"InformationTheory/error-correcting-codes/linear-codes/syndrome-decode","kind":"node","lean":"Cslib.CodingTheory.LinearCode.syndromeDecode","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#syndrome-decode","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Syndrome decoding with a lookup `table` from syndromes to error patterns: subtract the error pattern the table assigns to the syndrome of the received word.","title":"Syndrome decoding"},{"id":"InformationTheory/error-correcting-codes/linear-codes/is-syndrome-table","kind":"node","lean":"Cslib.CodingTheory.LinearCode.IsSyndromeTable","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#is-syndrome-table","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $H$ be a $\\rho \\times \\iota$ matrix over a field $F$, with $\\iota$ finite, and let $t$ be a natural number. A syndrome table for $H$, that is, a map $T$ from syndromes $s \\in F^\\rho$ to error patterns $T(s) \\in F^\\iota$, is correct up to radius $t$ when, for every word $e \\in F^\\iota$ of Hamming weight at most $t$, the word $T(He)$ has the same syndrome as $e$, that is $H \\, T(He) = He$, and has Hamming weight\u2026","title":"Syndrome table correct up to a radius"},{"id":"InformationTheory/error-correcting-codes/linear-codes/syndrome-decode-add-of-hamming-norm-le","kind":"node","lean":"Cslib.CodingTheory.LinearCode.syndromeDecode_add_of_hammingNorm_le","parent":43921,"reading_title":{"format":"inline-parts-v1","parts":[["text","Syndrome decoding corrects every error of weight at most "],["math","\\(t\\)"]],"source":"Syndrome decoding corrects every error of weight at most $t$"},"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#syndrome-decode-add-of-hamming-norm-le","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $H$ be a parity-check matrix over a field $F$, let $C$ be the code with parity-check matrix $H$, let $d \\in \\mathbb{N} \\cup \\{\\infty\\}$ be its minimum distance, and let $t \\in \\mathbb{N}$ with $2t < d$. If a syndrome table is correct up to radius $t$ for $H$, then for every codeword $c \\in C$ and every word $e$ of Hamming weight at most $t$, syndrome decoding with this table sends the received word $c + e$ to\u2026","title":"Syndrome decoding corrects every error of weight at most $t$"},{"id":"InformationTheory/error-correcting-codes/linear-codes/syndrome-decode-add-of-min-dist-eq","kind":"node","lean":"Cslib.CodingTheory.LinearCode.syndromeDecode_add_of_minDist_eq","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#syndrome-decode-add-of-min-dist-eq","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $H$ be a parity-check matrix over a field $F$ such that the code $C$ with parity-check matrix $H$ has finite minimum distance $d$, and let $t = \\lfloor (d - 1)/2 \\rfloor$. If a syndrome table is correct up to radius $t$ for $H$, then for every codeword $c \\in C$ and every word $e$ of Hamming weight at most $t$, syndrome decoding with this table sends the received word $c + e$ to $c$.","title":"Syndrome decoding up to half the minimum distance"},{"id":"InformationTheory/error-correcting-codes/linear-codes/exists-is-syndrome-table","kind":"node","lean":"Cslib.CodingTheory.LinearCode.exists_isSyndromeTable","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#exists-is-syndrome-table","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $H$ be a matrix over a field $F$ whose columns are indexed by a finite set. For every natural number $t$ there is a syndrome table for $H$ that is correct up to radius $t$.","title":"Existence of correct syndrome tables"},{"id":"InformationTheory/error-correcting-codes/linear-codes/search-table","kind":"node","lean":"Cslib.CodingTheory.LinearCode.searchTable","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#search-table","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a field, $\\iota$ and $\\rho$ finite sets, $H$ a $\\rho \\times \\iota$ parity-check matrix over $F$, and $t \\in \\mathbb{N}$, where the words $F^\\iota$ come with a fixed enumeration. The search table sends a syndrome $s \\in F^\\rho$ to the first word $e \\in F^\\iota$ in the enumeration with Hamming weight at most $t$ and syndrome $He = s$, and to $0$ if there is no such word.","title":"Syndrome table by exhaustive search"},{"id":"InformationTheory/error-correcting-codes/linear-codes/card-filter-mem","kind":"node","lean":"Cslib.CodingTheory.LinearCode.card_filter_mem","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#card-filter-mem","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"If $C$ is a linear code of length $n$ over a finite field $F$, then exactly $|F|^{\\dim C}$ of the $|F|^n$ words of length $n$ lie in $C$, where $\\dim C$ is the dimension of $C$ over $F$.","title":"Number of codewords of a linear code"},{"id":"InformationTheory/error-correcting-codes/linear-codes/finrank-add-le-of-le-min-dist","kind":"node","lean":"Cslib.CodingTheory.LinearCode.finrank_add_le_of_le_minDist","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#finrank-add-le-of-le-min-dist","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $C$ be a linear code of length $n$ over a finite field $F$, and let $d \\le n$ be a natural number that is at most the minimum distance of $C$. Then $\\dim C + d \\le n + 1$.","title":"Singleton bound for linear codes, dimension form"},{"id":"InformationTheory/error-correcting-codes/linear-codes/min-dist-le-length","kind":"node","lean":"Cslib.CodingTheory.LinearCode.minDist_le_length","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#min-dist-le-length","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"If $C$ is a linear code of length $n$ over a field and $C$ is not the zero subspace, then the minimum distance of $C$ is finite and at most $n$.","title":"Minimum distance is at most the length"},{"id":"InformationTheory/error-correcting-codes/linear-codes/min-dist-le-of-ne-bot","kind":"node","lean":"Cslib.CodingTheory.LinearCode.minDist_le_of_ne_bot","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#min-dist-le-of-ne-bot","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a finite field and $C$ a linear code of length $n$ and dimension $k$ over $F$ that is not the zero subspace. Then the minimum distance of $C$ is at most $n - k + 1$.","title":"Singleton bound for linear codes"},{"id":"InformationTheory/error-correcting-codes/linear-codes/pow-finrank-mul-sum-le-of-lt-min-dist","kind":"node","lean":"Cslib.CodingTheory.LinearCode.pow_finrank_mul_sum_le_of_lt_minDist","parent":43921,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/linear-codes/index.html#pow-finrank-mul-sum-le-of-lt-min-dist","scope":"InformationTheory/error-correcting-codes/linear-codes","status":"fully_proved","summary":"Let $F$ be a finite field with $q$ elements and $C$ a linear code of length $n$ and dimension $k$ over $F$. If $t$ is a natural number such that $2t$ is less than the minimum distance of $C$, then","title":"Hamming bound for linear codes"},{"data_url":"scopes/InformationTheory/error-correcting-codes/hamming-7-4-code.json","id":"InformationTheory/error-correcting-codes/hamming-7-4-code","kind":"scope","navigation":"module","parent":43903,"result":false,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","title":"Hamming (7,4) code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/g","kind":"node","lean":"Cslib.CodingTheory.Hamming74.G","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#g","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"$G$ is the $4 \\times 7$ matrix over $\\mathbb{F}_2$, with columns indexed by the positions $1, \\dots, 7$, whose $k$-th row encodes the $k$-th message bit: if $p$ is the $k$-th of the message positions $3, 5, 6, 7$, the row has entry $1$ at position $p$ and at each of the parity positions $1, 2, 4$ that occurs in the binary expansion of $p$, and entry $0$ elsewhere.","title":"Generator matrix of the Hamming (7,4) code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/h","kind":"node","lean":"Cslib.CodingTheory.Hamming74.H","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#h","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"$H$ is the $3 \\times 7$ matrix over $\\mathbb{F}_2$, with columns indexed by the positions $1, \\dots, 7$, whose column at position $p$ is the binary expansion of $p$, least significant bit first: its entry in row $i$, for $i = 0, 1, 2$, is the $i$-th binary digit of $p$.","title":"Parity-check matrix of the Hamming (7,4) code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/code","kind":"node","lean":"Cslib.CodingTheory.Hamming74.code","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#code","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $H$ be the $3 \\times 7$ matrix over $\\mathbb{F}_2$ whose column $j$, for $j = 1, \\dots, 7$, is the binary expansion of $j$, least significant bit first. The Hamming code is the binary linear code of length $7$ with parity-check matrix $H$: the words $c \\in \\mathbb{F}_2^7$ with $Hc = 0$.","title":"The Hamming (7,4) code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/encode","kind":"node","lean":"Cslib.CodingTheory.Hamming74.encode","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#encode","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $G$ be the $4 \\times 7$ generator matrix over $\\mathbb{F}_2$ with rows $(1,1,1,0,0,0,0)$, $(1,0,0,1,1,0,0)$, $(0,1,0,1,0,1,0)$ and $(1,1,0,1,0,0,1)$. A message $m \\in \\mathbb{F}_2^4$ is encoded as the word $mG \\in \\mathbb{F}_2^7$, the combination of the rows of $G$ with coefficients $m$.","title":"Encoder of the Hamming code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/message","kind":"node","lean":"Cslib.CodingTheory.Hamming74.message","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#message","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"The message of a word $y = (y_1, \\dots, y_7) \\in \\mathbb{F}_2^7$ is $(y_3, y_5, y_6, y_7) \\in \\mathbb{F}_2^4$, its bits at the positions that are not powers of two.","title":"Message bits of a word"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/error-pos","kind":"node","lean":"Cslib.CodingTheory.Hamming74.errorPos","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#error-pos","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"For a syndrome $s = (s_0, s_1, s_2) \\in \\mathbb{F}_2^3$, read its entries as bits of the number $p = s_0 + 2s_1 + 4s_2 \\in \\{0, \\dots, 7\\}$. If $p = 0$ there is no error position; otherwise the error position is $p - 1 \\in \\{0, \\dots, 6\\}$.","title":"Error position of a syndrome"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/table","kind":"node","lean":"Cslib.CodingTheory.Hamming74.table","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#table","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Number the positions of a word in $\\mathbb{F}_2^7$ as $1, \\dots, 7$. For a syndrome $s = (s_0, s_1, s_2) \\in \\mathbb{F}_2^3$, read each $s_i$ as $0$ or $1$ and let $p = s_0 + 2 s_1 + 4 s_2$. The table sends $s$ to the zero word if $p = 0$, and otherwise to the word whose only nonzero entry is a $1$ at position $p$.","title":"Syndrome table of the Hamming (7,4) code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/decode","kind":"node","lean":"Cslib.CodingTheory.Hamming74.decode","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#decode","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $H$ be the $3 \\times 7$ parity-check matrix of the Hamming code over $\\mathbb{F}_2$, whose column $j$, for $j = 1, \\dots, 7$, is the binary expansion of $j$, least significant bit first. Given a word $y \\in \\mathbb{F}_2^7$, compute its syndrome $s = (s_0, s_1, s_2) = Hy \\in \\mathbb{F}_2^3$. If $s = 0$, return $y$. Otherwise read the entries of $s$ as bits of the number\u2026","title":"Syndrome decoder of the Hamming code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/error-pos-syndrome-single","kind":"node","lean":"Cslib.CodingTheory.Hamming74.errorPos_syndrome_single","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#error-pos-syndrome-single","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $H$ be the parity-check matrix of the Hamming (7,4) code, and index the coordinates of words in $\\mathbb{F}_2^7$ by $0, \\dots, 6$, so that index $i$ is position $i + 1$. For each index $i$, let $e_i \\in \\mathbb{F}_2^7$ be the word with entry $1$ at index $i$ and $0$ elsewhere. Then the error position of the syndrome $He_i$ is defined and equals $i$.","title":"The syndrome of a single-bit error locates it"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/is-syndrome-table-table","kind":"node","lean":"Cslib.CodingTheory.Hamming74.isSyndromeTable_table","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#is-syndrome-table-table","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $H$ be the parity-check matrix of the Hamming (7,4) code and $T$ the syndrome table of its decoder. Then $T$ is a correct syndrome table up to radius $1$ for $H$: for every word $e \\in \\mathbb{F}_2^7$ of Hamming weight at most $1$, the word $T(He)$ has Hamming weight at most $1$ and the same syndrome as $e$, that is, $H \\, T(He) = He$.","title":"The Hamming decoder's syndrome table is correct"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/g-mul-h-transpose","kind":"node","lean":"Cslib.CodingTheory.Hamming74.G_mul_H_transpose","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#g-mul-h-transpose","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $G$ be the generator matrix and $H$ the parity-check matrix of the Hamming (7,4) code over $\\mathbb{F}_2$. Then $G H^{\\mathsf{T}} = 0$, the $4 \\times 3$ zero matrix: every row of $G$ is orthogonal to every row of $H$, so every row of $G$ satisfies the parity checks given by $H$.","title":"The generator matrix is orthogonal to the parity-check matrix"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/of-generator-g-eq-code","kind":"node","lean":"Cslib.CodingTheory.Hamming74.ofGenerator_G_eq_code","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#of-generator-g-eq-code","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"The code generated by the generator matrix $G$, that is, the set of words $mG$ for $m \\in \\mathbb{F}_2^4$, equals the Hamming (7,4) code, the set of words $c \\in \\mathbb{F}_2^7$ with $Hc = 0$ for the parity-check matrix $H$.","title":"The Hamming generator matrix generates the Hamming code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/three-le-min-weight-code","kind":"node","lean":"Cslib.CodingTheory.Hamming74.three_le_minWeight_code","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#three-le-min-weight-code","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Every nonzero codeword of the Hamming (7,4) code over $\\mathbb{F}_2$ has at least $3$ nonzero coordinates; that is, the minimum weight of the code is at least $3$.","title":"The Hamming (7,4) code has minimum weight at least 3"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/min-dist-code","kind":"node","lean":"Cslib.CodingTheory.Hamming74.minDist_code","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#min-dist-code","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"The minimum distance of the Hamming (7,4) code is $3$.","title":"Minimum distance of the Hamming (7,4) code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/decode-add","kind":"node","lean":"Cslib.CodingTheory.Hamming74.decode_add","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#decode-add","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"For every codeword $c$ of the Hamming code and every word $e \\in \\mathbb{F}_2^7$ of Hamming weight at most $1$, decoding $c + e$ returns $c$.","title":"The decoder corrects every error of weight at most 1"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/decode-encode-add-single","kind":"node","lean":"Cslib.CodingTheory.Hamming74.decode_encode_add_single","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#decode-encode-add-single","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"For every message $m \\in \\mathbb{F}_2^4$ and every position $i$ among the $7$, flipping bit $i$ of the encoding of $m$ and then decoding returns the encoding of $m$.","title":"Every single-bit error is corrected"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/decode-mem-code","kind":"node","lean":"Cslib.CodingTheory.Hamming74.decode_mem_code","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#decode-mem-code","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"For every word $y \\in \\mathbb{F}_2^7$, the decoding of $y$ belongs to the Hamming code.","title":"The decoder always outputs a codeword"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/hamming-dist-decode-le-one","kind":"node","lean":"Cslib.CodingTheory.Hamming74.hammingDist_decode_le_one","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#hamming-dist-decode-le-one","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $D$ be the Hamming (7,4) decoder. For every word $y \\in \\mathbb{F}_2^7$, the Hamming distance between $y$ and $D(y)$ is at most $1$.","title":"The Hamming decoder changes at most one bit"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/exists-unique-hamming-dist-le-one","kind":"node","lean":"Cslib.CodingTheory.Hamming74.existsUnique_hammingDist_le_one","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#exists-unique-hamming-dist-le-one","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Every word lies within distance 1 of exactly one codeword.","title":"The Hamming code is perfect"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/hamming-dist-decode-le","kind":"node","lean":"Cslib.CodingTheory.Hamming74.hammingDist_decode_le","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#hamming-dist-decode-le","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $D$ be the Hamming (7,4) decoder and $C \\subseteq \\mathbb{F}_2^7$ the Hamming code. For every word $y \\in \\mathbb{F}_2^7$ and every codeword $c \\in C$, the Hamming distance between $y$ and $D(y)$ is at most the Hamming distance between $y$ and $c$.","title":"No codeword is closer than the Hamming decoder's output"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/pow-finrank-mul-sum-le","kind":"node","lean":"Cslib.CodingTheory.Hamming74.pow_finrank_mul_sum_le","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#pow-finrank-mul-sum-le","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $k$ be the dimension of the Hamming (7,4) code over $\\mathbb{F}_2$. Then $2^k \\cdot \\sum_{i=0}^{1} \\binom{7}{i} (2 - 1)^i \\le 2^7$.","title":"Hamming bound for the Hamming (7,4) code"},{"id":"InformationTheory/error-correcting-codes/hamming-7-4-code/pow-finrank-mul-sum-eq","kind":"node","lean":"Cslib.CodingTheory.Hamming74.pow_finrank_mul_sum_eq","parent":43951,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/hamming-7-4-code/index.html#pow-finrank-mul-sum-eq","scope":"InformationTheory/error-correcting-codes/hamming-7-4-code","status":"fully_proved","summary":"Let $k$ be the dimension of the Hamming (7,4) code over $\\mathbb{F}_2$. Then $2^k \\cdot \\sum_{i=0}^{1} \\binom{7}{i} (2 - 1)^i = 2^7$: the Hamming bound for one error holds with equality.","title":"The Hamming bound is attained"},{"data_url":"scopes/InformationTheory/error-correcting-codes/locally-testable-codes.json","id":"InformationTheory/error-correcting-codes/locally-testable-codes","kind":"scope","navigation":"module","parent":43903,"result":false,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"can_prove","title":"Locally testable codes"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/binary-code","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.BinaryCode","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#binary-code","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"A binary code of length $n$ is a linear subspace of $\\mathbb{F}_2^n$, the vector space of words of length $n$ over the two-element field $\\mathbb{F}_2$.","title":"Binary code"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/dim","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.dim","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#dim","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"The dimension of a binary code $C \\subseteq \\mathbb{F}_2^n$ is its dimension as a vector space over $\\mathbb{F}_2$.","title":"Dimension of a binary code"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/rate","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.rate","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#rate","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"The rate of a binary code $C \\subseteq \\mathbb{F}_2^n$ is its dimension over $\\mathbb{F}_2$ divided by its length $n$, as a real number; it is $0$ when $n = 0$.","title":"Rate of a binary code"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/distance","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.distance","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#distance","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"The distance of a binary code $C \\subseteq \\mathbb{F}_2^n$ is the smallest Hamming weight (number of nonzero coordinates) of a nonzero codeword of $C$, and is $0$ if $C = \\{0\\}$.","title":"Minimum distance of a binary code"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/normalised-distance","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.normalisedDistance","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#normalised-distance","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"The normalised distance of a binary code $C \\subseteq \\mathbb{F}_2^n$ is its distance divided by its length $n$, as a real number; it is $0$ when $n = 0$.","title":"Normalised distance of a binary code"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/distance-from","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.distanceFrom","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#distance-from","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"For a binary code $C \\subseteq \\mathbb{F}_2^n$ and a word $f \\in \\mathbb{F}_2^n$, the distance from $f$ to $C$ is the minimum, over all codewords $c \\in C$, of the Hamming distance between $f$ and $c$, that is, the number of coordinates in which $f$ and $c$ differ.","title":"Distance from a word to a binary code"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/normalised-distance-from","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.normalisedDistanceFrom","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#normalised-distance-from","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"For a binary code $C \\subseteq \\mathbb{F}_2^n$ and a word $f \\in \\mathbb{F}_2^n$, the normalised distance from $f$ to $C$ is the distance from $f$ to $C$ (the Hamming distance from $f$ to a closest codeword) divided by the length $n$, as a nonnegative real number; it is $0$ when $n = 0$.","title":"Normalised distance from a word to a binary code"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/is-good","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.IsGood","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#is-good","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"Let $(n_i)_{i \\in \\mathbb{N}}$ be natural numbers and, for each $i$, let $C_i \\subseteq \\mathbb{F}_2^{n_i}$ be a binary code of length $n_i$. The family $(C_i)$ is good if $n_i \\to \\infty$ as $i \\to \\infty$ and there is a real number $\\varepsilon > 0$ such that for every $i$ both the rate and the normalised distance of $C_i$ are at least $\\varepsilon$.","title":"Good family of codes"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/card-le","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.cardLE","parent":43974,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subsets with at most "],["math","\\(q\\)"],["text"," elements"]],"source":"Subsets with at most $q$ elements"},"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#card-le","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"For natural numbers $n$ and $q$, this is the collection of all subsets $S \\subseteq \\{0, 1, \\dots, n - 1\\}$ with at most $q$ elements.","title":"Subsets with at most $q$ elements"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/ltc","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.LTC","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#ltc","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"fully_proved","summary":"Let $C \\subseteq \\mathbb{F}_2^n$ be a binary code, let $q$ be a natural number and let $\\kappa \\ge 0$ be a real number. A $\\kappa$-local tester for $C$ with $q$ queries consists of a probability distribution on the subsets $S \\subseteq \\{0, 1, \\dots, n - 1\\}$ with at most $q$ elements, together with, for each such $S$, a set $V_S \\subseteq \\mathbb{F}_2^S$ of allowed local views, such that: at least $\\kappa$ times\u2026","title":"Locally testable code"},{"id":"InformationTheory/error-correcting-codes/locally-testable-codes/theorem-1-2","kind":"node","lean":"MathlibExt.Combinatorics.GoodLTCodesWanted.GoodLTC.theorem_1_2","parent":43974,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/locally-testable-codes/index.html#theorem-1-2","scope":"InformationTheory/error-correcting-codes/locally-testable-codes","status":"can_prove","summary":"For every real number $\\rho$ with $0 \\le \\rho < 1$ there exist a natural number $q$, a real number $\\kappa > 0$, lengths $n_i \\in \\mathbb{N}$ and binary codes $C_i$ of length $n_i$ for $i \\in \\mathbb{N}$ such that the family $(C_i)_{i \\in \\mathbb{N}}$ is good and, for every $i$, the code $C_i$ is $\\kappa$-locally testable with $q$ queries and has rate at least $\\rho$.","title":"Good locally testable codes of every rate below one"},{"data_url":"scopes/InformationTheory/error-correcting-codes/reed-solomon-codes.json","id":"InformationTheory/error-correcting-codes/reed-solomon-codes","kind":"scope","navigation":"module","parent":43903,"result":false,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","title":"Reed\u2013Solomon codes"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/encode","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.encode","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#encode","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"For a field $F$ and points $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$, the encoding of a polynomial $p \\in F[X]$ is the word $(p(\\alpha_0), \\dots, p(\\alpha_{n-1}))$ in $F^n$.","title":"Reed-Solomon encoder"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/encode-lin","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.encodeLin","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#encode-lin","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"For a field $F$ and points $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$, the $F$-linear map $F[X] \\to F^n$ sending a polynomial $p$ to $(p(\\alpha_0), \\dots, p(\\alpha_{n-1}))$.","title":"Reed-Solomon encoder as a linear map"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/code","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.code","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#code","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ and let $k$ be a natural number. The Reed-Solomon code $\\mathrm{RS}(\\alpha, k)$ is the linear code of length $n$ over $F$ consisting of the words $(p(\\alpha_0), \\dots, p(\\alpha_{n-1}))$ for the polynomials $p \\in F[X]$ of degree less than $k$.","title":"Reed-Solomon code"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/card-eval-eq-zero-le","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.card_eval_eq_zero_le","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#card-eval-eq-zero-le","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1}$ be pairwise distinct elements of $F$, and let $p \\in F[X]$ be a nonzero polynomial. Then the number of indices $i < n$ with $p(\\alpha_i) = 0$ is at most the degree of $p$.","title":"Root count at distinct points"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/le-hamming-norm-encode","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.le_hammingNorm_encode","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#le-hamming-norm-encode","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct and let $k \\le n$. For every nonzero polynomial $p \\in F[X]$ of degree less than $k$, the encoding $(p(\\alpha_0), \\dots, p(\\alpha_{n-1}))$ has at least $n - k + 1$ nonzero coordinates.","title":"Nonzero Reed-Solomon codewords are heavy"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/inj-on-encode","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.injOn_encode","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#inj-on-encode","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct and let $k \\le n$. If two polynomials $p, q \\in F[X]$ of degree less than $k$ satisfy $p(\\alpha_i) = q(\\alpha_i)$ for every $i < n$, then $p = q$.","title":"The Reed-Solomon encoder is injective on low-degree polynomials"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/finrank-code","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.finrank_code","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#finrank-code","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct and let $k \\le n$. The Reed-Solomon code of the points $\\alpha$ with parameter $k$ has dimension $k$ over $F$.","title":"Dimension of a Reed-Solomon code"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/le-min-dist-code","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.le_minDist_code","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#le-min-dist-code","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct and let $k \\in \\mathbb{N}$. The Reed-Solomon code of the points $\\alpha$ with parameter $k$ has minimum distance at least $\\max(n - k, 0) + 1$.","title":"Lower bound on the minimum distance of a Reed-Solomon code"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/exists-codeword-hamming-norm-le","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.exists_codeword_hammingNorm_le","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#exists-codeword-hamming-norm-le","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ (not necessarily distinct) and let $1 \\le k \\le n$. There is a nonzero polynomial $p \\in F[X]$ of degree less than $k$ whose encoding $(p(\\alpha_0), \\dots, p(\\alpha_{n-1}))$ has at most $n - k + 1$ nonzero coordinates.","title":"A low-weight Reed-Solomon codeword"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/min-dist-code","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.minDist_code","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#min-dist-code","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field (not necessarily finite), let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct and let $1 \\le k \\le n$. The Reed-Solomon code of the points $\\alpha$ with parameter $k$ has minimum distance exactly $n - k + 1$.","title":"Reed-Solomon codes are MDS"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/min-dist-code-eq-singleton","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.minDist_code_eq_singleton","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#min-dist-code-eq-singleton","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a finite field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct and let $1 \\le k \\le n$. The Reed-Solomon code $C$ of the points $\\alpha$ with parameter $k$ has minimum distance $d = n - \\dim C + 1$, so the Singleton bound $d \\le n - \\dim C + 1$ for linear codes holds with equality.","title":"Reed-Solomon codes meet the Singleton bound"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/is-key-solution","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.IsKeySolution","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#is-key-solution","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$, let $k, e \\in \\mathbb{N}$ and let $y \\in F^n$ be a received word. A pair of polynomials $(Q, E)$ over $F$ solves the key equations for $y$ when $E$ is monic of degree $e$, $Q$ has degree less than $e + k$, and $Q(\\alpha_i) = y_i E(\\alpha_i)$ for every $i < n$.","title":"Berlekamp-Welch key equations"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/exists-is-key-solution","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.exists_isKeySolution","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#exists-is-key-solution","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ and let $p \\in F[X]$ have degree less than $k$. If a received word $y \\in F^n$ differs from the encoding $(p(\\alpha_0), \\dots, p(\\alpha_{n-1}))$ in at most $e$ coordinates, then the Berlekamp-Welch key equations for $y$ with parameters $k$ and $e$ have a solution $(Q, E)$.","title":"The key equations are solvable"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/eq-mul-of-is-key-solution","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.eq_mul_of_isKeySolution","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#eq-mul-of-is-key-solution","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct and let $k, e$ be natural numbers with $2e + k \\le n$. Let $p \\in F[X]$ have degree less than $k$, and let $y \\in F^n$ be within Hamming distance $e$ of $(p(\\alpha_0), \\dots, p(\\alpha_{n-1}))$. If $(Q, E)$ solves the key equations for $y$ ($E$ is monic of degree $e$, $\\deg Q < e + k$, and $Q(\\alpha_i) = y_i E(\\alpha_i)$ for all\u2026","title":"Every solution of the key equations reveals the message"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/div-by-monic-eq-of-is-key-solution","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.divByMonic_eq_of_isKeySolution","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#div-by-monic-eq-of-is-key-solution","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct and let $k, e$ be natural numbers with $2e + k \\le n$. Let $p \\in F[X]$ have degree less than $k$, and let $y \\in F^n$ be within Hamming distance $e$ of $(p(\\alpha_0), \\dots, p(\\alpha_{n-1}))$. If $(Q, E)$ solves the key equations for $y$ ($E$ is monic of degree $e$, $\\deg Q < e + k$, and $Q(\\alpha_i) = y_i E(\\alpha_i)$ for all\u2026","title":"The quotient of a key-equation solution is the message"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/key-solver","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.KeySolver","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#key-solver","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"For a field $F$, points $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ and parameters $k, e \\in \\mathbb{N}$, a solver for the Berlekamp-Welch key equations is a procedure that takes a received word $y \\in F^n$ and either returns a pair of polynomials $(Q, E)$ over $F$ or reports failure, such that every returned pair solves the key equations for $y$, and a pair is returned whenever the key equations for $y$ have a solution.","title":"Solver for the key equations"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/of-choice","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.KeySolver.ofChoice","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#of-choice","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"For points $\\alpha_0, \\dots, \\alpha_{n-1}$ in a field $F$ and natural numbers $k$ and $e$, this is the solver for the key equations that, on a received word $y \\in F^n$, returns a pair $(Q, E)$ of polynomials solving the key equations for $y$ (that is, $E$ is monic of degree $e$, $\\deg Q < e + k$ and $Q(\\alpha_i) = y_i E(\\alpha_i)$ for all $i$), picked by the axiom of choice, whenever such a pair exists, and\u2026","title":"A key-equation solver by choice"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/decode","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.decode","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#decode","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$, let $k, e$ be natural numbers and let $S$ be a solver for the key equations with error budget $e$. On a received word $y \\in F^n$ the decoder runs $S$ on $y$ and reports failure if $S$ returns nothing. If $S$ returns a pair $(Q, E)$ (the solver guarantees that $E$ is monic), let $q$ be the quotient of $Q$ by $E$; the decoder returns $q$ if $q$ has\u2026","title":"Berlekamp-Welch decoder"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/decode-eq-some-iff","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.decode_eq_some_iff","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#decode-eq-some-iff","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct, let $k, e$ be natural numbers with $2e + k \\le n$ (the unique-decoding regime), and let $S$ be a solver for the key equations with error budget $e$. For every polynomial $p \\in F[X]$ and every received word $y \\in F^n$, the Berlekamp-Welch decoder with solver $S$ returns $p$ on $y$ if and only if $p$ has degree less than $k$ and the\u2026","title":"What the decoder returns"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/decode-eq-none-iff","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.decode_eq_none_iff","parent":43986,"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#decode-eq-none-iff","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct, let $k, e$ be natural numbers with $2e + k \\le n$, and let $S$ be a solver for the key equations with error budget $e$. For every received word $y \\in F^n$, the Berlekamp-Welch decoder with solver $S$ reports failure on $y$ if and only if, for every polynomial $p \\in F[X]$ of degree less than $k$, the Hamming distance between $y$\u2026","title":"The decoder fails exactly when no codeword is close"},{"id":"InformationTheory/error-correcting-codes/reed-solomon-codes/decode-encode-add","kind":"node","lean":"Cslib.CodingTheory.ReedSolomon.decode_encode_add","parent":43986,"reading_title":{"format":"inline-parts-v1","parts":[["text","Berlekamp-Welch corrects every error of weight at most "],["math","\\(\\lfloor (n-k)/2 \\rfloor\\)"]],"source":"Berlekamp-Welch corrects every error of weight at most $\\lfloor (n-k)/2 \\rfloor$"},"result":true,"route":"../roadmap/InformationTheory/error-correcting-codes/reed-solomon-codes/index.html#decode-encode-add","scope":"InformationTheory/error-correcting-codes/reed-solomon-codes","status":"fully_proved","summary":"Let $F$ be a field, let $\\alpha_0, \\dots, \\alpha_{n-1} \\in F$ be pairwise distinct, let $k \\le n$, put $t = \\lfloor (n-k)/2 \\rfloor$, and let $S$ be a solver for the key equations with error budget $t$. For every polynomial $p \\in F[X]$ of degree less than $k$ and every error word $\\varepsilon \\in F^n$ with at most $t$ nonzero entries, the Berlekamp-Welch decoder with solver $S$, run on the received word\u2026","title":"Berlekamp-Welch corrects every error of weight at most $\\lfloor (n-k)/2 \\rfloor$"},{"data_url":"scopes/InformationTheory/entropy-and-mutual-information.json","id":"InformationTheory/entropy-and-mutual-information","kind":"scope","parent":43798,"result":false,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/index.html","scope":"InformationTheory/entropy-and-mutual-information","status":"fully_proved","summary":"Shannon entropy, conditional entropy, mutual information and Kullback\u2013Leibler divergence, the basic measures of uncertainty and dependence between random variables.","title":"Entropy and mutual information"},{"data_url":"scopes/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence.json","id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","kind":"scope","navigation":"module","parent":44008,"result":false,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","title":"Shannon entropy and Kullback\u2013Leibler divergence"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/shannon-entropy","kind":"node","lean":"InformationTheory.shannonEntropy","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#shannon-entropy","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"The Shannon entropy in bits of a probability mass function $p$ on a finite type $\\alpha$ is $H(p) = -\\sum_{a \\in \\alpha} p(a) \\log_2 p(a)$, where a term with $p(a) = 0$ contributes $0$.","title":"Shannon entropy"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/shannon-entropy-pure","kind":"node","lean":"InformationTheory.shannonEntropy_pure","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#shannon-entropy-pure","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For every outcome $a$ of a finite alphabet, the point mass at $a$ has Shannon entropy $0$.","title":"A point mass has zero entropy"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/shannon-entropy-map-const","kind":"node","lean":"InformationTheory.shannonEntropy_map_const","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#shannon-entropy-map-const","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For any probability mass function $p$ on a type $\\alpha$ and any point $b$ of a finite alphabet $\\beta$, the image of $p$ under the constant map $\\alpha \\to \\beta$ with value $b$ has Shannon entropy $0$.","title":"A constant observation has zero entropy"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/shannon-entropy-map-equiv","kind":"node","lean":"InformationTheory.shannonEntropy_map_equiv","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#shannon-entropy-map-equiv","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For a probability mass function $p$ on a finite alphabet $\\alpha$ and a bijection $e : \\alpha \\to \\beta$ onto a finite alphabet $\\beta$, the image of $p$ under $e$ has the same Shannon entropy as $p$.","title":"Entropy is invariant under relabeling"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/shannon-entropy-map-prod-singleton","kind":"node","lean":"InformationTheory.shannonEntropy_map_prod_singleton","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#shannon-entropy-map-prod-singleton","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For a probability mass function $p$ on a finite alphabet $\\alpha$, the image of $p$ under $a \\mapsto (a, 0)$ in $\\alpha \\times \\{0\\}$ has the same Shannon entropy as $p$.","title":"Appending a singleton coordinate preserves entropy"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/to-real-kl-div-eq-sum","kind":"node","lean":"InformationTheory.toReal_klDiv_eq_sum","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#to-real-kl-div-eq-sum","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"Let $\\mu$ and $\\nu$ be finite measures on a finite type $\\alpha$ in which every singleton is measurable, with $\\mu \\ll \\nu$ and $\\mu(\\alpha) = \\nu(\\alpha)$. Then the Kullback\u2013Leibler divergence $\\mathrm{KL}(\\mu \\parallel \\nu)$, taken as a real number, equals $\\sum_{a \\in \\alpha} \\mu(\\{a\\}) \\log (\\mu(\\{a\\}) / \\nu(\\{a\\}))$, where $\\log$ is the natural logarithm and a term with $\\mu(\\{a\\}) = 0$ is $0$.","title":"Kullback\u2013Leibler divergence on a finite space"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/to-real-kl-div-to-measure","kind":"node","lean":"InformationTheory.toReal_klDiv_toMeasure","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#to-real-kl-div-to-measure","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"Let $p$ and $q$ be probability mass functions on a finite type $\\alpha$ in which every singleton is measurable, such that $p(a) = 0$ whenever $q(a) = 0$, and let $P$ and $Q$ be the measures they define. Then the Kullback\u2013Leibler divergence $\\mathrm{KL}(P \\parallel Q)$, taken as a real number, equals $\\sum_{a \\in \\alpha} p(a) \\log (p(a) / q(a))$, where $\\log$ is the natural logarithm and a term with $p(a) = 0$ is $0$.","title":"Kullback\u2013Leibler divergence of finite probability laws"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/surprisal","kind":"node","lean":"InformationTheory.surprisal","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#surprisal","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"The surprisal (information content) in bits of an outcome of probability $q$ is $-\\log_2 q$, with value $\\infty$ when $q = 0$. Formally $q$ ranges over the extended nonnegative reals $[0, \\infty]$: the surprisal of $0$ is $\\infty$, the surprisal of $q \\in (0, \\infty)$ is $\\max(0, -\\log_2 q)$, and the surprisal of $\\infty$ is $0$.","title":"Surprisal"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/cross-entropy","kind":"node","lean":"InformationTheory.crossEntropy","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#cross-entropy","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For probability mass functions $p$ and $q$ on a finite alphabet, the cross-entropy of $q$ relative to $p$ is $H(p, q) = \\sum_a p(a) \\cdot (-\\log_2 q(a))$ in bits, a value in $[0, \\infty]$. Here $-\\log_2 q(a)$ is the surprisal of $a$ under $q$, which is $\\infty$ when $q(a) = 0$, and a term with $p(a) = 0$ contributes $0$. So $H(p, q)$ is $\\infty$ exactly when $p$ charges an outcome that $q$ rules out.","title":"Cross-entropy"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/cross-entropy-pure","kind":"node","lean":"InformationTheory.crossEntropy_pure","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#cross-entropy-pure","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For an outcome $y$ of a finite alphabet and a probability mass function $q$ on it, the cross-entropy $H(\\delta_y, q)$ of $q$ relative to the point mass $\\delta_y$ at $y$ equals the surprisal $-\\log_2 q(y)$ of $y$ under $q$, which is $\\infty$ when $q(y) = 0$.","title":"Cross-entropy of a point mass"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/lintegral-surprisal","kind":"node","lean":"InformationTheory.lintegral_surprisal","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#lintegral-surprisal","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For probability mass functions $p$ and $q$ on a finite alphabet, the expected surprisal $\\mathbb{E}_{a \\sim p}[-\\log_2 q(a)]$ (with $-\\log_2 0 = \\infty$), taken as the integral of a $[0, \\infty]$-valued function with respect to the law $p$, equals the cross-entropy $H(p, q)$.","title":"Cross-entropy is expected surprisal"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/cross-entropy-eq-shannon-entropy-add-kl","kind":"node","lean":"InformationTheory.crossEntropy_eq_shannonEntropy_add_klDiv","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#cross-entropy-eq-shannon-entropy-add-kl","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For probability mass functions $p$ and $q$ on a finite alphabet, the cross-entropy $H(p, q)$ in bits equals $H(p) + D(p \\parallel q) / \\ln 2$, where $H(p)$ is the Shannon entropy of $p$ in bits and $D(p \\parallel q)$ is the Kullback\u2013Leibler divergence of $p$ from $q$ with the natural logarithm. The identity holds in $[0, \\infty]$ with no support hypothesis: when $p$ charges an outcome that $q$ rules out, both sides\u2026","title":"Cross-entropy is entropy plus divergence"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/shannon-entropy-le-cross-entropy","kind":"node","lean":"InformationTheory.shannonEntropy_le_crossEntropy","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#shannon-entropy-le-cross-entropy","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"The cross-entropy of any prediction `q` is at least the entropy of the truth `p`.","title":"Gibbs' inequality"},{"id":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/cross-entropy-eq-shannon-entropy-iff","kind":"node","lean":"InformationTheory.crossEntropy_eq_shannonEntropy_iff","parent":44009,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence/index.html#cross-entropy-eq-shannon-entropy-iff","scope":"InformationTheory/entropy-and-mutual-information/shannon-entropy-and-kullback-leibler-divergence","status":"fully_proved","summary":"For probability mass functions $p$ and $q$ on a finite alphabet, the cross-entropy $H(p, q)$ in bits equals the Shannon entropy $H(p)$ in bits if and only if $q = p$. Thus log loss is a strictly proper scoring rule.","title":"Equality in Gibbs' inequality"},{"data_url":"scopes/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information.json","id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","kind":"scope","navigation":"module","parent":44008,"result":false,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","title":"Conditional entropy and mutual information"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/quantized-conditional-entropy","kind":"node","lean":"InformationTheory.quantizedConditionalEntropy","parent":44024,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional entropy of a finite observation given a sub-"],["math","\\(\\sigma\\)"],["text","-algebra"]],"source":"Conditional entropy of a finite observation given a sub-$\\sigma$-algebra"},"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#quantized-conditional-entropy","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $m$ be a sub-$\\sigma$-algebra of the $\\sigma$-algebra of $\\Omega$, let $n \\in \\mathbb{N}$, and let $f : \\Omega \\to \\{0, \\dots, n-1\\}$ be measurable. For each $i < n$ let $P_i = \\mathbb{E}_\\mu[\\mathbf{1}_{\\{f = i\\}} \\mid m]$ be the conditional probability of $\\{f = i\\}$ given $m$. The quantized conditional entropy of $f$ given $m$, in bits, is","title":"Conditional entropy of a finite observation given a sub-$\\sigma$-algebra"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/quantized-conditional-entropy-singleton","kind":"node","lean":"InformationTheory.quantizedConditionalEntropy_singleton","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#quantized-conditional-entropy-singleton","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$ and let $m$ be a sub-$\\sigma$-algebra of the $\\sigma$-algebra of $\\Omega$. The quantized conditional entropy given $m$ of the observation that is constantly $0$, with values in the one-element set $\\{0\\}$, is $0$.","title":"One-valued observations have zero conditional entropy"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/quantized-conditional-entropy-const","kind":"node","lean":"InformationTheory.quantizedConditionalEntropy_const","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#quantized-conditional-entropy-const","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $m$ be a sub-$\\sigma$-algebra of the $\\sigma$-algebra of $\\Omega$, and let $c \\in \\{0, \\dots, n-1\\}$ for some $n \\in \\mathbb{N}$. The quantized conditional entropy given $m$ of the constant observation $\\omega \\mapsto c$ is $0$.","title":"Constant quantized observations have zero conditional entropy"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/conditional-shannon-entropy","kind":"node","lean":"InformationTheory.conditionalShannonEntropy","parent":44024,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional Shannon entropy given a sub-"],["math","\\(\\sigma\\)"],["text","-algebra"]],"source":"Conditional Shannon entropy given a sub-$\\sigma$-algebra"},"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#conditional-shannon-entropy","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $m$ be a sub-$\\sigma$-algebra of the $\\sigma$-algebra of $\\Omega$, and let $X : \\Omega \\to Y$ be a measurable map into a measurable space $Y$. The conditional Shannon entropy of $X$ given $m$ is the supremum in $[0, \\infty]$, over all $n \\in \\mathbb{N}$ and all measurable maps $q : Y \\to \\{0, \\dots, n-1\\}$, of the quantized conditional entropy\u2026","title":"Conditional Shannon entropy given a sub-$\\sigma$-algebra"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/conditional-shannon-entropy-const","kind":"node","lean":"InformationTheory.conditionalShannonEntropy_const","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#conditional-shannon-entropy-const","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $m$ be a sub-$\\sigma$-algebra of the $\\sigma$-algebra of $\\Omega$, and let $y$ be a point of a measurable space $Y$. The conditional Shannon entropy given $m$ of the constant output $\\omega \\mapsto y$ is $0$.","title":"Conditional entropy of a constant output vanishes"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/measure-mutual-information","kind":"node","lean":"InformationTheory.measureMutualInformation","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#measure-mutual-information","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on the product $X \\times Y$ of two measurable spaces, with marginals $\\mu_X$ and $\\mu_Y$. The mutual information of $\\mu$, in bits and possibly infinite, is","title":"Mutual information of a joint probability measure"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/to-real-kl-div-prod","kind":"node","lean":"InformationTheory.toReal_klDiv_prod","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#to-real-kl-div-prod","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be finite sets with measurable structures in which singletons are measurable, let $p$ be a probability law on $\\alpha \\times \\beta$ with marginals $p_X$ and $p_Y$, and write $H$ for Shannon entropy in bits. Then the Kullback\u2013Leibler divergence of $p$ from the product measure $p_X \\otimes p_Y$, computed with the natural logarithm and taken as a real number, equals\u2026","title":"Divergence of a finite joint law from the product of its marginals"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/shannon-entropy-le-add","kind":"node","lean":"InformationTheory.shannonEntropy_le_add","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#shannon-entropy-le-add","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $p$ be a probability law on $\\alpha \\times \\beta$, where $\\alpha$ and $\\beta$ are finite, with marginals $p_X$ and $p_Y$. Writing $H$ for Shannon entropy in bits, $H(p) \\le H(p_X) + H(p_Y)$.","title":"Subadditivity of entropy"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/measure-mutual-information-to-measure","kind":"node","lean":"InformationTheory.measureMutualInformation_toMeasure","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#measure-mutual-information-to-measure","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $p$ be a probability distribution on $\\alpha \\times \\beta$, where $\\alpha$ and $\\beta$ are finite measurable spaces in which every singleton is measurable, viewed as the joint law of a pair $(X, Y)$. Write $H(X, Y)$, $H(X)$ and $H(Y)$ for the Shannon entropies of $p$ and of its first and second marginals. Then the mutual information of the probability measure defined by $p$, in bits, is the real number\u2026","title":"Mutual information of a finite joint law"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/measure-mutual-information-to-measure-eq","kind":"node","lean":"InformationTheory.measureMutualInformation_toMeasure_eq_zero_iff","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#measure-mutual-information-to-measure-eq","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $p$ be a discrete probability distribution on $\\alpha \\times \\beta$, where $\\alpha$ and $\\beta$ are measurable spaces. The mutual information of the probability measure defined by $p$ is $0$ if and only if this measure equals the product of the probability measures defined by the two marginals of $p$.","title":"Mutual information vanishes exactly under independence"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/finite-conditional-entropy","kind":"node","lean":"InformationTheory.finiteConditionalEntropy","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#finite-conditional-entropy","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $p$ be a probability distribution on $\\alpha \\times \\beta$, where $\\alpha$ and $\\beta$ are finite, viewed as the joint law of a pair $(X, Y)$, and let $p_X(a) = \\sum_{b \\in \\beta} p(a, b)$ be its first marginal. The conditional entropy of $Y$ given $X$, in bits, is","title":"Conditional entropy of a finite joint law"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/shannon-entropy-eq-add-finite-conditiona","kind":"node","lean":"InformationTheory.shannonEntropy_eq_add_finiteConditionalEntropy","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#shannon-entropy-eq-add-finite-conditiona","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $p$ be a probability law on $\\alpha \\times \\beta$, where $\\alpha$ and $\\beta$ are finite, and let $p_X$ be its first marginal. Write $H(X, Y)$ for the Shannon entropy of $p$, $H(X)$ for the Shannon entropy of $p_X$, and $H(Y \\mid X) = -\\sum_{a, b} p(a, b) \\log_2 (p(a, b) / p_X(a))$ for the finite conditional entropy of $p$, all in bits. Then $H(X, Y) = H(X) + H(Y \\mid X)$.","title":"Chain rule for entropy"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/finite-conditional-entropy-le","kind":"node","lean":"InformationTheory.finiteConditionalEntropy_le","parent":44024,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information/index.html#finite-conditional-entropy-le","scope":"InformationTheory/entropy-and-mutual-information/conditional-entropy-and-mutual-information","status":"fully_proved","summary":"Let $p$ be a probability distribution on $\\alpha \\times \\beta$, where $\\alpha$ and $\\beta$ are finite, viewed as the joint law of a pair $(X, Y)$. Then the conditional entropy $H(Y \\mid X)$ of $p$ is at most the Shannon entropy $H(Y)$ of the second marginal of $p$, both in bits.","title":"Conditioning does not increase entropy"},{"data_url":"scopes/InformationTheory/entropy-and-mutual-information/conditional-mutual-information.json","id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","kind":"scope","navigation":"module","parent":44008,"result":false,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","title":"Conditional mutual information"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/sum-to-real-mul-comp","kind":"node","lean":"InformationTheory.sum_toReal_mul_comp","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#sum-to-real-mul-comp","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"Summing a function of `f t` against `p` is summing against the pushforward `p.map f`.","title":"Sums against a pushforward of a finite law"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/le-map-apply","kind":"node","lean":"InformationTheory.le_map_apply","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#le-map-apply","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"An atom's mass is at most the mass of its image.","title":"Atom mass is at most the mass of its image"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/to-real-le-map-apply","kind":"node","lean":"InformationTheory.toReal_le_map_apply","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#to-real-le-map-apply","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"An atom's mass is at most the mass of its image, as real numbers.","title":"Atom mass is at most the mass of its image, in reals"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/shannon-entropy-mul-log-two","kind":"node","lean":"InformationTheory.shannonEntropy_mul_log_two","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#shannon-entropy-mul-log-two","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"Entropy times `log 2` as a `p log p` sum.","title":"Shannon entropy as a p log p sum"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/sum-mul-log-map","kind":"node","lean":"InformationTheory.sum_mul_log_map","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#sum-mul-log-map","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"`\u2211 p(t) log (p.map f)(f t) = -H(p.map f) log 2`.","title":"Log-sum against a pushforward equals its negative entropy"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/shannon-entropy-map-equiv-comp","kind":"node","lean":"InformationTheory.shannonEntropy_map_equiv_comp","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#shannon-entropy-map-equiv-comp","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"Relabeling the image of a map by a bijection preserves entropy.","title":"Entropy of a pushforward is invariant under relabeling"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-mutual-information","kind":"node","lean":"InformationTheory.finiteMutualInformation","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-mutual-information","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"Mutual information `I(X; Y)` of a finite joint law `p` of `(X, Y)`, in bits: the real value of `measureMutualInformation` on `p`, which is finite here (`finiteMutualInformation_eq`). Each finite alphabet carries the discrete \u03c3-algebra.","title":"Mutual information of a finite joint law"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-mutual-information-eq","kind":"node","lean":"InformationTheory.finiteMutualInformation_eq","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-mutual-information-eq","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"`I(X; Y) = H(X) + H(Y) - H(X, Y)` for a finite joint law.","title":"Mutual information in terms of entropies"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-conditional-mutual-information","kind":"node","lean":"InformationTheory.finiteConditionalMutualInformation","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-conditional-mutual-information","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"The conditional mutual information `I(X; Y | Z)` of a finite joint law `p` of `(X, Y, Z)`, in bits: `\u2211 p(x, y, z) log\u2082 (p(x, y, z) p(z) / (p(x, z) p(y, z)))`.","title":"Conditional mutual information of a finite joint law"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-conditional-mutual-information-eq","kind":"node","lean":"InformationTheory.finiteConditionalMutualInformation_eq","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-conditional-mutual-information-eq","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"`I(X; Y | Z) = H(X, Z) + H(Y, Z) - H(X, Y, Z) - H(Z)`.","title":"Conditional mutual information in terms of entropies"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-conditional-mutual-information-no","kind":"node","lean":"InformationTheory.finiteConditionalMutualInformation_nonneg","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-conditional-mutual-information-no","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"Conditional mutual information is nonnegative.","title":"Nonnegativity of conditional mutual information"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-conditional-mutual-information-eq-2","kind":"node","lean":"InformationTheory.finiteConditionalMutualInformation_eq_zero_iff","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-conditional-mutual-information-eq-2","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"`I(X; Y | Z) = 0` exactly when `X` and `Y` are conditionally independent given `Z`: `p(x, y, z) p(z) = p(x, z) p(y, z)` for every atom.","title":"Vanishing conditional mutual information iff conditional independence"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-mutual-information-chain-rule","kind":"node","lean":"InformationTheory.finiteMutualInformation_chain_rule","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-mutual-information-chain-rule","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"`I(X; (Y, Z)) = I(X; Z) + I(X; Y | Z)`. Here `p` is read as a joint law of `X` and the pair `(Y, Z)`.","title":"Chain rule for mutual information"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-mutual-information-le-left-of-mar","kind":"node","lean":"InformationTheory.finiteMutualInformation_le_left_of_markov","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-mutual-information-le-left-of-mar","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"Data-processing inequality, first half: if `X` and `Z` are conditionally independent given `Y` (a Markov chain `X - Y - Z`, stated as `I(X; Z | Y) = 0` for the joint law `p` of `(X, Y, Z)`), then `I(X; Z) \u2264 I(X; Y)`.","title":"Data-processing inequality"},{"id":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information/finite-mutual-information-le-right-of-ma","kind":"node","lean":"InformationTheory.finiteMutualInformation_le_right_of_markov","parent":44038,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/conditional-mutual-information/index.html#finite-mutual-information-le-right-of-ma","scope":"InformationTheory/entropy-and-mutual-information/conditional-mutual-information","status":"fully_proved","summary":"Data-processing inequality, second half: under the same Markov condition, `I(X; Z) \u2264 I(Y; Z)`.","title":"Data-processing inequality"},{"data_url":"scopes/InformationTheory/entropy-and-mutual-information/chow-liu-trees.json","id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","kind":"scope","navigation":"module","parent":44008,"result":false,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","title":"Chow\u2013Liu trees"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/dependence-forest","kind":"node","lean":"InformationTheory.DependenceForest","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#dependence-forest","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"A dependence forest on the variables `\u03b9` (Chow\u2013Liu 1968): each variable has at most one parent, and `rank` strictly increases from parent to child, so the parent relation has no cycles.","title":"Dependence forest"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/approx","kind":"node","lean":"InformationTheory.DependenceForest.approx","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#approx","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"Chow\u2013Liu's product approximation `P_\u03c4(x) = \u220f_i P(x_i | x_{j(i)})` of a joint law `p`, with the marginal `P(x_i)` for a parentless `i`. The conditional `P(x_i | x_j)` is `P(x_i, x_j) / P(x_j)`; where `P(x_j) = 0` the quotient is `0`, and such `x` have `P(x) = 0`.","title":"Chow\u2013Liu product approximation of a joint law"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/approx-nonneg","kind":"node","lean":"InformationTheory.DependenceForest.approx_nonneg","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#approx-nonneg","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"The approximation is nonnegative.","title":"Nonnegativity of the Chow\u2013Liu approximation"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/approx-pos","kind":"node","lean":"InformationTheory.DependenceForest.approx_pos","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#approx-pos","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"`P \u226a P_\u03c4`: the approximation is positive wherever `p` is.","title":"Joint law is absolutely continuous w.r.t. its Chow\u2013Liu approximation"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/sum-approx","kind":"node","lean":"InformationTheory.DependenceForest.sum_approx","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#sum-approx","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"The approximation is a probability law: `\u2211_x P_\u03c4(x) = 1`.","title":"Chow\u2013Liu approximation sums to one"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/approx-pmf","kind":"node","lean":"InformationTheory.DependenceForest.approxPMF","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#approx-pmf","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"Chow\u2013Liu's approximation `P_\u03c4` as a probability mass function.","title":"Chow\u2013Liu approximation as a probability mass function"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/weight","kind":"node","lean":"InformationTheory.DependenceForest.weight","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#weight","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"The mutual-information weight `\u2211_i I(X_i; X_{j(i)})` of a dependence forest, in bits.","title":"Mutual-information weight of a dependence forest"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/divergence","kind":"node","lean":"InformationTheory.DependenceForest.divergence","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#divergence","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"The divergence `D(P \u2016 P_\u03c4) = \u2211_x P(x) log\u2082 (P(x) / P_\u03c4(x))` of `P` from Chow\u2013Liu's approximation, in bits (`divergence_eq_klDiv`).","title":"Divergence from the Chow\u2013Liu approximation"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/divergence-eq-kl-div","kind":"node","lean":"InformationTheory.DependenceForest.divergence_eq_klDiv","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#divergence-eq-kl-div","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"`divergence` is Mathlib's Kullback\u2013Leibler divergence `klDiv` of `P` relative to `P_\u03c4`, converted to bits; the finite alphabet carries the discrete \u03c3-algebra.","title":"Chow\u2013Liu divergence as a Kullback\u2013Leibler divergence"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/divergence-eq","kind":"node","lean":"InformationTheory.DependenceForest.divergence_eq","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#divergence-eq","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"`D(P \u2016 P_\u03c4) = \u2211_i H(X_i) - H(X) - \u2211_i I(X_i; X_{j(i)})`.","title":"Chow\u2013Liu identity"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/divergence-nonneg","kind":"node","lean":"InformationTheory.DependenceForest.divergence_nonneg","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#divergence-nonneg","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"The divergence of `P` from Chow\u2013Liu's approximation is nonnegative, so `\u2211_i I(X_i; X_{j(i)}) \u2264 \u2211_i H(X_i) - H(X)` for every dependence forest.","title":"Nonnegativity of the Chow\u2013Liu divergence"},{"id":"InformationTheory/entropy-and-mutual-information/chow-liu-trees/divergence-le-divergence-iff","kind":"node","lean":"InformationTheory.DependenceForest.divergence_le_divergence_iff","parent":44054,"result":true,"route":"../roadmap/InformationTheory/entropy-and-mutual-information/chow-liu-trees/index.html#divergence-le-divergence-iff","scope":"InformationTheory/entropy-and-mutual-information/chow-liu-trees","status":"fully_proved","summary":"Among dependence forests, a smaller divergence is the same as a larger mutual-information weight, so minimizing `D(P \u2016 P_\u03c4)` over dependence trees is maximizing the total mutual information `\u2211_i I(X_i; X_{j(i)})` of their edges.","title":"Chow\u2013Liu optimality"},{"data_url":"scopes/InformationTheory/information-inequalities.json","id":"InformationTheory/information-inequalities","kind":"scope","parent":43798,"result":false,"route":"../roadmap/InformationTheory/information-inequalities/index.html","scope":"InformationTheory/information-inequalities","status":"fully_proved","summary":"Inequalities relating entropies, divergences and mutual information, such as the log-sum, Pinsker, Fano, data-processing and Shearer inequalities.","title":"Information inequalities"},{"id":"InformationTheory/information-inequalities/sum-mul-log-sum-div-sum-le-sum-mul-log-d","kind":"node","lean":"Real.sum_mul_log_sum_div_sum_le_sum_mul_log_div","parent":44067,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/index.html#sum-mul-log-sum-div-sum-le-sum-mul-log-d","scope":"InformationTheory/information-inequalities","status":"fully_proved","summary":"Let $s$ be a finite subset of a type $\\iota$ and let $p, q : \\iota \\to \\mathbb{R}$ satisfy $p_i \\ge 0$ and $q_i > 0$ for all $i \\in s$. Then","title":"Log-sum inequality"},{"id":"InformationTheory/information-inequalities/csiszar-kullback-pinsker","kind":"node","lean":"MathlibExt.InformationTheory.Pinsker.csiszar_kullback_pinsker","parent":44067,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/index.html#csiszar-kullback-pinsker","scope":"InformationTheory/information-inequalities","status":"fully_proved","summary":"Let $\\alpha$ be a finite nonempty set and let $p, q : \\alpha \\to \\mathbb{R}$ satisfy $p(a) > 0$ and $q(a) > 0$ for every $a \\in \\alpha$ and $\\sum_a p(a) = \\sum_a q(a) = 1$. Then, with the natural logarithm,","title":"Csisz\u00e1r\u2013Kullback\u2013Pinsker inequality for strictly positive distributions"},{"data_url":"scopes/InformationTheory/information-inequalities/data-processing-inequality.json","id":"InformationTheory/information-inequalities/data-processing-inequality","kind":"scope","navigation":"module","parent":44067,"result":false,"route":"../roadmap/InformationTheory/information-inequalities/data-processing-inequality/index.html","scope":"InformationTheory/information-inequalities/data-processing-inequality","status":"fully_proved","title":"Data-processing inequality"},{"id":"InformationTheory/information-inequalities/data-processing-inequality/measure-mutual-information-map-swap","kind":"node","lean":"InformationTheory.measureMutualInformation_map_swap","parent":44070,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/data-processing-inequality/index.html#measure-mutual-information-map-swap","scope":"InformationTheory/information-inequalities/data-processing-inequality","status":"fully_proved","summary":"`I(Y; X) = I(X; Y)`.","title":"Symmetry of mutual information"},{"id":"InformationTheory/information-inequalities/data-processing-inequality/measure-mutual-information-comp-le","kind":"node","lean":"InformationTheory.measureMutualInformation_comp_le","parent":44070,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/data-processing-inequality/index.html#measure-mutual-information-comp-le","scope":"InformationTheory/information-inequalities/data-processing-inequality","status":"fully_proved","summary":"If `Z` is produced from `Y` by a channel `\u03ba` that does not see `X` (a Markov chain `X \u2192 Y \u2192 Z`), then `I(X; Z) \u2264 I(X; Y)`. The joint law of `(X, Z)` is `(Kernel.id \u2225\u2096 \u03ba) \u2218\u2098 \u03bc`.","title":"Data-processing inequality"},{"id":"InformationTheory/information-inequalities/data-processing-inequality/measure-mutual-information-map-le","kind":"node","lean":"InformationTheory.measureMutualInformation_map_le","parent":44070,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/data-processing-inequality/index.html#measure-mutual-information-map-le","scope":"InformationTheory/information-inequalities/data-processing-inequality","status":"fully_proved","summary":"`I(X; g(Y)) \u2264 I(X; Y)` for a measurable `g`.","title":"Processing cannot increase information"},{"data_url":"scopes/InformationTheory/information-inequalities/fano-s-inequality.json","id":"InformationTheory/information-inequalities/fano-s-inequality","kind":"scope","navigation":"module","parent":44067,"result":false,"route":"../roadmap/InformationTheory/information-inequalities/fano-s-inequality/index.html","scope":"InformationTheory/information-inequalities/fano-s-inequality","status":"fully_proved","title":"Fano's inequality"},{"id":"InformationTheory/information-inequalities/fano-s-inequality/fano-inequality-general","kind":"node","lean":"MathlibExt.InformationTheory.Fano.fano_inequality_general","parent":44074,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/fano-s-inequality/index.html#fano-inequality-general","scope":"InformationTheory/information-inequalities/fano-s-inequality","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be finite nonempty sets with $|\\alpha| \\ge 2$, let $p : \\alpha \\times \\beta \\to \\mathbb{R}$ satisfy $p(a, b) > 0$ for all $a, b$ and $\\sum_{a, b} p(a, b) = 1$, and let $g : \\beta \\to \\alpha$. Put $p_Y(b) = \\sum_a p(a, b)$, let $P_e = \\sum_b \\sum_{a \\ne g(b)} p(a, b)$ be the probability that $g$ guesses wrong, and let $H(X \\mid Y) = -\\sum_{a, b} p(a, b) \\log (p(a, b) / p_Y(b))$, where $\\log$\u2026","title":"Fano's inequality, general form"},{"id":"InformationTheory/information-inequalities/fano-s-inequality/fano-inequality","kind":"node","lean":"MathlibExt.InformationTheory.Fano.fano_inequality","parent":44074,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/fano-s-inequality/index.html#fano-inequality","scope":"InformationTheory/information-inequalities/fano-s-inequality","status":"fully_proved","summary":"Let $\\alpha$ and $\\beta$ be finite nonempty sets with $|\\alpha| \\ge 2$, let $p : \\alpha \\times \\beta \\to \\mathbb{R}$ satisfy $p(a, b) > 0$ for all $a, b$ and $\\sum_{a, b} p(a, b) = 1$, and let $g : \\beta \\to \\alpha$ be an estimator. Put $p_Y(b) = \\sum_a p(a, b)$, let $P_e = \\sum_b \\sum_{a \\ne g(b)} p(a, b)$ be the probability that $g$ guesses wrong, and let\u2026","title":"Fano's inequality"},{"data_url":"scopes/InformationTheory/information-inequalities/shearer-s-entropy-inequality.json","id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","kind":"scope","navigation":"module","parent":44067,"result":false,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","title":"Shearer's entropy inequality"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-marginal","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerMarginal","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-marginal","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a finite type, $n$ a natural number and $\\mu$ a real-valued function on the strings $x = (x_0, \\dots, x_{n-1}) \\in \\alpha^n$. For a set $S \\subseteq \\{0, \\dots, n-1\\}$ of coordinates, write $x|_S : S \\to \\alpha$ for the restriction of a string $x$ to $S$. For a map $y : S \\to \\alpha$, the $S$-marginal of $\\mu$ at $y$ is the sum $\\sum_{x \\in \\alpha^n,\\ x|_S = y} \\mu(x)$ of $\\mu(x)$ over all strings\u2026","title":"Marginal on a set of coordinates"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-full-entropy","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerFullEntropy","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-full-entropy","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"For a finite set $\\alpha$, a natural number $n$ and a real-valued function $\\mu$ on $\\alpha^n$, the full (joint) entropy of $\\mu$ is $H(\\mu) = -\\sum_{x \\in \\alpha^n} \\mu(x) \\log \\mu(x)$, where $\\log t$ is the natural logarithm of $|t|$ for $t \\neq 0$ and $\\log 0 = 0$.","title":"Joint entropy"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-marginal-entropy","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerMarginalEntropy","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-marginal-entropy","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"For a finite set $\\alpha$, a natural number $n$, a real-valued function $\\mu$ on $\\alpha^n$ and a set $S \\subseteq \\{0, \\dots, n-1\\}$ of coordinates, let $P_S$ be the marginal of $\\mu$ on $S$: for $y \\in \\alpha^S$, $P_S(y)$ is the sum of $\\mu(x)$ over the $x \\in \\alpha^n$ whose restriction to $S$ is $y$. The marginal entropy of $\\mu$ on $S$ is $H_S(\\mu) = -\\sum_{y \\in \\alpha^S} P_S(y) \\log P_S(y)$, where $\\log t$\u2026","title":"Marginal entropy"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-marginal-nonneg","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerMarginal_nonneg","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-marginal-nonneg","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a finite set, $n$ a natural number, $\\mu$ a real-valued function on $\\alpha^n$ with $\\mu(x) \\ge 0$ for all $x$, and $S$ a set of coordinates in $\\{0, \\dots, n-1\\}$. Then the marginal $P_S$ of $\\mu$ on $S$ satisfies $P_S(y) \\ge 0$ for every $y \\in \\alpha^S$.","title":"Marginals of a nonnegative function are nonnegative"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-marginal-sum","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerMarginal_sum","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-marginal-sum","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a finite set, $n$ a natural number, $\\mu$ a real-valued function on $\\alpha^n$ with $\\sum_{x \\in \\alpha^n} \\mu(x) = 1$, and $S$ a set of coordinates in $\\{0, \\dots, n-1\\}$. Then the marginal $P_S$ of $\\mu$ on $S$ satisfies $\\sum_{y \\in \\alpha^S} P_S(y) = 1$.","title":"Marginals preserve total mass"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-marginal-entropy-mono","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerMarginalEntropy_mono","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-marginal-entropy-mono","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a finite set, $n$ a natural number, and $\\mu$ a real-valued function on $\\alpha^n$ with $\\mu(x) \\ge 0$ for all $x$. If $T \\subseteq S$ are sets of coordinates in $\\{0, \\dots, n-1\\}$, then the marginal entropy of $\\mu$ on $T$ is at most the marginal entropy of $\\mu$ on $S$.","title":"Monotonicity of marginal entropy"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-marginal-entropy-submodular","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerMarginalEntropy_submodular","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-marginal-entropy-submodular","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a finite set, $n$ a natural number, and $\\mu$ a probability mass function on $\\alpha^n$, that is, $\\mu(x) \\ge 0$ for all $x$ and $\\sum_{x \\in \\alpha^n} \\mu(x) = 1$. Writing $H_S$ for the marginal entropy of $\\mu$ on a set $S \\subseteq \\{0, \\dots, n-1\\}$ of coordinates, any two such sets $A$ and $B$ satisfy $H_{A \\cup B} + H_{A \\cap B} \\le H_A + H_B$.","title":"Submodularity of marginal entropy"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-marginal-entropy-empty","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerMarginalEntropy_empty","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-marginal-entropy-empty","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a finite set, $n$ a natural number, and $\\mu$ a real-valued function on $\\alpha^n$ with $\\sum_{x \\in \\alpha^n} \\mu(x) = 1$. Then the marginal entropy of $\\mu$ on the empty set of coordinates is $0$.","title":"Marginal entropy on the empty set vanishes"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-marginal-entropy-univ","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearerMarginalEntropy_univ","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-marginal-entropy-univ","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"For a finite set $\\alpha$, a natural number $n$ and a real-valued function $\\mu$ on $\\alpha^n$, the marginal entropy of $\\mu$ on the set $\\{0, \\dots, n-1\\}$ of all coordinates equals the full entropy of $\\mu$.","title":"Marginal entropy on all coordinates is the full entropy"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-entropy-inequality-general","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearer_entropy_inequality_general","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-entropy-inequality-general","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set, $n$ a natural number, and $\\mu$ a probability mass function on $\\alpha^n$, that is, $\\mu(x) \\ge 0$ for all $x \\in \\alpha^n$ and $\\sum_{x \\in \\alpha^n} \\mu(x) = 1$. Write $H(\\mu)$ for the full entropy of $\\mu$ and, for a set $S$ of coordinates in $\\{0, \\dots, n-1\\}$, write $H_S(\\mu)$ for the marginal entropy of $\\mu$ on $S$. Let $C$ be a set of subsets of $\\{0, \\dots, n-1\\}$\u2026","title":"Shearer's entropy inequality for every cover multiplicity"},{"id":"InformationTheory/information-inequalities/shearer-s-entropy-inequality/shearer-entropy-inequality","kind":"node","lean":"MathlibExt.InformationTheory.Shearer.shearer_entropy_inequality","parent":44077,"result":true,"route":"../roadmap/InformationTheory/information-inequalities/shearer-s-entropy-inequality/index.html#shearer-entropy-inequality","scope":"InformationTheory/information-inequalities/shearer-s-entropy-inequality","status":"fully_proved","summary":"Let $\\alpha$ be a nonempty finite set, $n$ a natural number, and $\\mu$ a probability mass function on $\\alpha^n$, that is, $\\mu(x) \\ge 0$ for all $x \\in \\alpha^n$ and $\\sum_{x \\in \\alpha^n} \\mu(x) = 1$. Write $H(\\mu)$ for the full entropy of $\\mu$ and, for a set $S$ of coordinates in $\\{0, \\dots, n-1\\}$, write $H_S(\\mu)$ for the marginal entropy of $\\mu$ on $S$. Let $C$ be a set of subsets of $\\{0, \\dots, n-1\\}$\u2026","title":"Shearer's entropy inequality"},{"data_url":"scopes/InformationTheory/channel-simulation.json","id":"InformationTheory/channel-simulation","kind":"scope","parent":43798,"result":false,"route":"../roadmap/InformationTheory/channel-simulation/index.html","scope":"InformationTheory/channel-simulation","status":"fully_proved","summary":"Simulating a noisy channel with shared randomness and auxiliary random variables, including one-shot channel simulation and the functional representation lemma.","title":"Channel simulation"},{"data_url":"scopes/InformationTheory/channel-simulation/functional-representation-lemma.json","id":"InformationTheory/channel-simulation/functional-representation-lemma","kind":"scope","navigation":"module","parent":44089,"result":false,"route":"../roadmap/InformationTheory/channel-simulation/functional-representation-lemma/index.html","scope":"InformationTheory/channel-simulation/functional-representation-lemma","status":"fully_proved","title":"Functional representation lemma"},{"id":"InformationTheory/channel-simulation/functional-representation-lemma/auxiliary-conditioning","kind":"node","lean":"InformationTheory.auxiliaryConditioning","parent":44090,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/functional-representation-lemma/index.html#auxiliary-conditioning","scope":"InformationTheory/channel-simulation/functional-representation-lemma","status":"fully_proved","summary":"For measurable spaces $X$, $Y$ and $S$, this is the sigma-algebra on $(X \\times Y) \\times S$ generated by the projection $((x, y), s) \\mapsto s$ onto $S$, regarded as a sub-sigma-algebra of the product sigma-algebra on $(X \\times Y) \\times S$.","title":"Sigma-algebra of the auxiliary coordinate"},{"id":"InformationTheory/channel-simulation/functional-representation-lemma/input-auxiliary-conditioning","kind":"node","lean":"InformationTheory.inputAuxiliaryConditioning","parent":44090,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/functional-representation-lemma/index.html#input-auxiliary-conditioning","scope":"InformationTheory/channel-simulation/functional-representation-lemma","status":"fully_proved","summary":"For measurable spaces $X$, $Y$ and $S$, this is the sigma-algebra on $(X \\times Y) \\times S$ generated by the map $((x, y), s) \\mapsto (x, s)$ into $X \\times S$ with its product sigma-algebra, regarded as a sub-sigma-algebra of the product sigma-algebra on $(X \\times Y) \\times S$.","title":"Sigma-algebra of the input and auxiliary coordinates"},{"id":"InformationTheory/channel-simulation/functional-representation-lemma/general-conditional-entropy","kind":"node","lean":"InformationTheory.generalConditionalEntropy","parent":44090,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/functional-representation-lemma/index.html#general-conditional-entropy","scope":"InformationTheory/channel-simulation/functional-representation-lemma","status":"fully_proved","summary":"For a probability measure $\\nu$ on $(X \\times Y) \\times S$, $H(Y \\mid S)$ is the conditional Shannon entropy under $\\nu$, in bits and with values in $[0, \\infty]$, of the coordinate $((x, y), s) \\mapsto y$ given the sigma-algebra generated by the coordinate $((x, y), s) \\mapsto s$.","title":"Conditional entropy of the output given the auxiliary variable"},{"id":"InformationTheory/channel-simulation/functional-representation-lemma/general-recovery-entropy","kind":"node","lean":"InformationTheory.generalRecoveryEntropy","parent":44090,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/functional-representation-lemma/index.html#general-recovery-entropy","scope":"InformationTheory/channel-simulation/functional-representation-lemma","status":"fully_proved","summary":"For a probability measure $\\nu$ on $(X \\times Y) \\times S$, $H(Y \\mid X, S)$ is the conditional Shannon entropy under $\\nu$, in bits and with values in $[0, \\infty]$, of the coordinate $((x, y), s) \\mapsto y$ given the sigma-algebra generated by the map $((x, y), s) \\mapsto (x, s)$ into $X \\times S$.","title":"Conditional entropy of the output given input and auxiliary variable"},{"id":"InformationTheory/channel-simulation/functional-representation-lemma/is-general-representation","kind":"node","lean":"InformationTheory.IsGeneralRepresentation","parent":44090,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/functional-representation-lemma/index.html#is-general-representation","scope":"InformationTheory/channel-simulation/functional-representation-lemma","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $X \\times Y$ and $\\nu$ a probability measure on $(X \\times Y) \\times S$. Then $\\nu$ is a functional representation of $\\mu$ when all of the following hold: the image of $\\nu$ under $((x, y), s) \\mapsto (x, y)$ is $\\mu$; the mutual information of the image of $\\nu$ under $((x, y), s) \\mapsto (x, s)$ is zero; the conditional entropy $H(Y \\mid X, S)$ of $\\nu$ is zero; and there is\u2026","title":"Functional representation of a joint law"},{"id":"InformationTheory/channel-simulation/functional-representation-lemma/marginal-eq","kind":"node","lean":"InformationTheory.IsGeneralRepresentation.marginal_eq","parent":44090,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/functional-representation-lemma/index.html#marginal-eq","scope":"InformationTheory/channel-simulation/functional-representation-lemma","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $X \\times Y$ and let $\\nu$ be a probability measure on $(X \\times Y) \\times S$ that is a functional representation of $\\mu$. Then the image of $\\nu$ under the projection $((x, y), s) \\mapsto (x, y)$ is $\\mu$.","title":"A functional representation has the prescribed joint law"},{"id":"InformationTheory/channel-simulation/functional-representation-lemma/exists-recovery","kind":"node","lean":"InformationTheory.IsGeneralRepresentation.exists_recovery","parent":44090,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/functional-representation-lemma/index.html#exists-recovery","scope":"InformationTheory/channel-simulation/functional-representation-lemma","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on $X \\times Y$ and let $\\nu$ be a probability measure on $(X \\times Y) \\times S$ that is a functional representation of $\\mu$. Then there is a measurable map $g : X \\times S \\to Y$ such that $g(x, s) = y$ for $\\nu$-almost every $((x, y), s)$.","title":"A functional representation has a measurable recovery map"},{"data_url":"scopes/InformationTheory/channel-simulation/one-shot-channel-simulation.json","id":"InformationTheory/channel-simulation/one-shot-channel-simulation","kind":"scope","navigation":"module","parent":44089,"result":false,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","title":"One-shot channel simulation"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/finite-joint-law","kind":"node","lean":"InformationTheory.FiniteJointLaw","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#finite-joint-law","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"A finite joint law consists of positive integers $n_X$ and $n_Y$ and a probability distribution on $\\{0, \\dots, n_X - 1\\} \\times \\{0, \\dots, n_Y - 1\\}$.","title":"Finite joint law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/input-law","kind":"node","lean":"InformationTheory.FiniteJointLaw.inputLaw","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#input-law","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"The input law of a finite joint law $p$ on $\\{0, \\dots, n_X - 1\\} \\times \\{0, \\dots, n_Y - 1\\}$ is its first marginal: the law on $\\{0, \\dots, n_X - 1\\}$ that is the image of $p$ under $(x, y) \\mapsto x$.","title":"Input law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/output-law","kind":"node","lean":"InformationTheory.FiniteJointLaw.outputLaw","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#output-law","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"The output law of a finite joint law $p$ of an input $X$ and an output $Y$ is the law of $Y$: the marginal of $p$ on the output alphabet, obtained by pushing $p$ forward along the projection onto the second coordinate.","title":"Output law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/mutual-information","kind":"node","lean":"InformationTheory.FiniteJointLaw.mutualInformation","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#mutual-information","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"The mutual information of a finite joint law $p$ of $(X, Y)$ is $I(X; Y) = H(X) + H(Y) - H(X, Y)$, where $H(X)$ and $H(Y)$ are the Shannon entropies in bits of the input and output laws of $p$, and $H(X, Y)$ is the Shannon entropy in bits of $p$.","title":"Mutual information"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/finite-extension","kind":"node","lean":"InformationTheory.FiniteExtension","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#finite-extension","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"A finite extension of a finite joint law $p$ on $\\{0, \\dots, n_X - 1\\} \\times \\{0, \\dots, n_Y - 1\\}$ consists of a positive integer $n_S$ and a probability law on triples $(x, y, s)$ with $s \\in \\{0, \\dots, n_S - 1\\}$ whose marginal on the pairs $(x, y)$ is exactly $p$.","title":"Finite extension of a joint law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/trivial","kind":"node","lean":"InformationTheory.FiniteExtension.trivial","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#trivial","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"The trivial extension of a finite joint law $p$ of an input $X$ and an output $Y$ has the one-letter auxiliary alphabet $\\{0\\}$ and, as its law, the law of $((X, Y), 0)$ for $(X, Y)$ distributed according to $p$; its $(X, Y)$ marginal is $p$.","title":"Trivial extension"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/auxiliary-law","kind":"node","lean":"InformationTheory.FiniteExtension.auxiliaryLaw","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#auxiliary-law","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For a finite extension $q$ of a finite joint law $p$, viewed as the law of a triple $(X, Y, S)$ with $S$ taking values in $\\{0, \\dots, n_S - 1\\}$, the auxiliary law is the marginal law of $S$ under $q$.","title":"Auxiliary marginal of an extension"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/input-auxiliary-law","kind":"node","lean":"InformationTheory.FiniteExtension.inputAuxiliaryLaw","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#input-auxiliary-law","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Let $q$ be a finite extension of a finite joint law of $(X, Y)$, that is, a law of $((X, Y), S)$ with an auxiliary variable $S$. Its input-auxiliary law is the joint law of $(X, S)$: the image of the law of $q$ under $((x, y), s) \\mapsto (x, s)$.","title":"Input-auxiliary law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/output-auxiliary-law","kind":"node","lean":"InformationTheory.FiniteExtension.outputAuxiliaryLaw","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#output-auxiliary-law","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Let $q$ be a finite extension of a finite joint law of $(X, Y)$, that is, a law of $((X, Y), S)$ with an auxiliary variable $S$. Its output-auxiliary law is the joint law of $(Y, S)$: the image of the law of $q$ under $((x, y), s) \\mapsto (y, s)$.","title":"Output-auxiliary law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/auxiliary-information","kind":"node","lean":"InformationTheory.FiniteExtension.auxiliaryInformation","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#auxiliary-information","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For a finite extension $q$ of a finite joint law $p$, viewed as the law of a triple $(X, Y, S)$ whose $(X, Y)$ marginal is $p$, this is $I(X; S) = H(X) + H(S) - H(X, S)$, where $H(X)$ is the Shannon entropy in bits of the $X$ marginal of $p$, and $H(S)$ and $H(X, S)$ are the Shannon entropies in bits of the $S$ and $(X, S)$ marginals of $q$.","title":"Mutual information between the input and the auxiliary"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/conditional-entropy","kind":"node","lean":"InformationTheory.FiniteExtension.conditionalEntropy","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#conditional-entropy","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For a finite extension $q$ of a finite joint law $p$, viewed as the law of a triple $(X, Y, S)$ whose $(X, Y)$ marginal is $p$, this is $H(Y \\mid S) = H(Y, S) - H(S)$, where $H(Y, S)$ and $H(S)$ are the Shannon entropies in bits of the $(Y, S)$ and $S$ marginals of $q$.","title":"Conditional entropy of the output given the auxiliary"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/output-given-input-auxiliary-entropy","kind":"node","lean":"InformationTheory.FiniteExtension.outputGivenInputAuxiliaryEntropy","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#output-given-input-auxiliary-entropy","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For a finite extension of a finite joint law, with input $X$, output $Y$ and auxiliary $S$, the conditional entropy of $Y$ given $(X, S)$ is $H(Y \\mid X, S) = H(X, Y, S) - H(X, S)$, where $H$ denotes Shannon entropy in bits, $H(X, Y, S)$ is the entropy of the law of the extension and $H(X, S)$ is the entropy of its $(X, S)$ marginal.","title":"Conditional entropy of the output given the input and the auxiliary"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/output-given-input-auxiliary-entropy-eq","kind":"node","lean":"InformationTheory.FiniteExtension.outputGivenInputAuxiliaryEntropy_eq_zero_of_output_subsingleton","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#output-given-input-auxiliary-entropy-eq","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Let $p$ be a finite joint law of an input $X$ and an output $Y$ whose output alphabet has exactly one letter. Then for every finite extension of $p$ by an auxiliary $S$, the conditional entropy $H(Y \\mid X, S)$ of the output given the input and the auxiliary is $0$.","title":"Zero output entropy given input and auxiliary for a one-letter output"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/trivial-auxiliary-law","kind":"node","lean":"InformationTheory.FiniteExtension.trivial_auxiliaryLaw","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#trivial-auxiliary-law","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For every finite joint law $p$, the auxiliary law of the trivial extension of $p$ is the point mass at $0$, the only letter of its one-letter auxiliary alphabet $\\{0\\}$.","title":"Auxiliary law of the trivial extension"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/trivial-auxiliary-information","kind":"node","lean":"InformationTheory.FiniteExtension.trivial_auxiliaryInformation","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#trivial-auxiliary-information","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For every finite joint law $p$, the mutual information $I(X; S)$ between the input $X$ and the auxiliary $S$ in the trivial extension of $p$, whose auxiliary alphabet has a single letter, is $0$.","title":"The trivial extension has zero auxiliary information"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/trivial-conditional-entropy","kind":"node","lean":"InformationTheory.FiniteExtension.trivial_conditionalEntropy","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#trivial-conditional-entropy","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For every finite joint law $p$ of an input $X$ and an output $Y$, the conditional entropy $H(Y \\mid S)$ of the output given the auxiliary $S$ in the trivial extension of $p$ equals the Shannon entropy $H(Y)$ of the output law of $p$.","title":"Conditional entropy in the trivial extension"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/input-auxiliary-law-map-fst","kind":"node","lean":"InformationTheory.FiniteExtension.inputAuxiliaryLaw_map_fst","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#input-auxiliary-law-map-fst","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Let $q$ be a finite extension of a finite joint law $p$ of $(X, Y)$ by an auxiliary variable $S$. Then the marginal of $X$ under the joint law of $(X, S)$ in $q$ equals the law of $X$ under $p$.","title":"Input marginal of the input-auxiliary law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/output-auxiliary-law-map-fst","kind":"node","lean":"InformationTheory.FiniteExtension.outputAuxiliaryLaw_map_fst","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#output-auxiliary-law-map-fst","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Let $q$ be a finite extension of a finite joint law $p$ of $(X, Y)$ by an auxiliary variable $S$. Then the marginal of $Y$ under the joint law of $(Y, S)$ in $q$ equals the law of $Y$ under $p$.","title":"Output marginal of the output-auxiliary law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/input-auxiliary-law-map-snd","kind":"node","lean":"InformationTheory.FiniteExtension.inputAuxiliaryLaw_map_snd","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#input-auxiliary-law-map-snd","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Let $q$ be a finite extension of a finite joint law of $(X, Y)$ by an auxiliary variable $S$. Then the marginal of $S$ under the joint law of $(X, S)$ in $q$ equals the law of $S$ in $q$.","title":"Auxiliary marginal of the input-auxiliary law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/output-auxiliary-law-map-snd","kind":"node","lean":"InformationTheory.FiniteExtension.outputAuxiliaryLaw_map_snd","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#output-auxiliary-law-map-snd","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Let $q$ be a finite extension of a finite joint law of $(X, Y)$ by an auxiliary variable $S$. Then the marginal of $S$ under the joint law of $(Y, S)$ in $q$ equals the law of $S$ in $q$.","title":"Auxiliary marginal of the output-auxiliary law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/conditional-entropy-eq-zero-of-output-su","kind":"node","lean":"InformationTheory.FiniteExtension.conditionalEntropy_eq_zero_of_output_subsingleton","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#conditional-entropy-eq-zero-of-output-su","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"If the output alphabet of a finite joint law $p$ of $(X, Y)$ has exactly one element, then for every finite extension of $p$ by an auxiliary variable $S$, the conditional entropy $H(Y \\mid S)$ is $0$.","title":"Conditional entropy vanishes for a one-letter output"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/conditional-prefix-optimum","kind":"node","lean":"InformationTheory.conditionalPrefixOptimum","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#conditional-prefix-optimum","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For a finite joint law $p$ of $(X, Y)$, this is the infimum in the extended reals of the conditional entropy $H(Y \\mid S)$ in bits over all finite extensions of $p$ by an auxiliary variable $S$ such that $I(X; S) = 0$ and $H(Y \\mid X, S) = 0$. The infimum of the empty set is $+\\infty$.","title":"Conditional-prefix optimum of a joint law"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/worst-case-optimum","kind":"node","lean":"InformationTheory.worstCaseOptimum","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#worst-case-optimum","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For a real number $t$, the worst-case optimum at $t$ is the supremum, taken in the extended reals, of the conditional-prefix optimum of $p$ over all finite joint laws $p$ of an input $X$ and an output $Y$ whose mutual information $I(X; Y)$, in bits, equals $t$; it is $-\\infty$ when no such law exists.","title":"Worst-case conditional-prefix optimum"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/channel-simulation-constant","kind":"node","lean":"InformationTheory.channelSimulationConstant","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#channel-simulation-constant","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"The constant $c_c^*$ is the limit superior in the extended reals, as the real number $t$ tends to $+\\infty$, of $W(t) - t - \\log_2 t$, where $W(t)$ is the worst-case optimum at $t$: the supremum of the conditional-prefix optimum over all finite joint laws whose mutual information in bits is exactly $t$.","title":"Channel simulation constant"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/has-uniform-outer-constant","kind":"node","lean":"InformationTheory.HasUniformOuterConstant","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#has-uniform-outer-constant","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"A real number $\\alpha$ is a uniform outer constant when there is a real $\\beta > 0$ such that for every finite joint law $p$ of $(X, Y)$ there is a finite extension of $p$ by an auxiliary variable $S$ with $I(X; S) = 0$ and $H(Y \\mid X, S) = 0$ and","title":"Uniform outer constant for channel simulation"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/uniform-outer-constant-infimum","kind":"node","lean":"InformationTheory.uniformOuterConstantInfimum","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#uniform-outer-constant-infimum","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Call a real number $\\alpha$ a uniform outer constant when there is a real $\\beta > 0$ such that every finite joint law $p$ of an input $X$ and an output $Y$ has a finite extension by an auxiliary $S$ with $I(X; S) = 0$, $H(Y \\mid X, S) = 0$ and $H(Y \\mid S) \\le I(X; Y) + \\log_2(I(X; Y) + \\beta) + \\alpha$, where entropies and mutual informations are in bits. This is the infimum of the set of uniform outer constants,\u2026","title":"Infimum of the uniform outer constants"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/conditional-prefix-optimum-le","kind":"node","lean":"InformationTheory.conditionalPrefixOptimum_le","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#conditional-prefix-optimum-le","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"Let $p$ be a finite joint law of $(X, Y)$ and let $q$ be a finite extension of $p$ by an auxiliary variable $S$ with $I(X; S) = 0$ and $H(Y \\mid X, S) = 0$. Then the conditional-prefix optimum of $p$ is at most the conditional entropy $H(Y \\mid S)$ of $q$.","title":"Each admissible extension bounds the conditional-prefix optimum"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/conditional-prefix-optimum-eq-zero-of-ou","kind":"node","lean":"InformationTheory.conditionalPrefixOptimum_eq_zero_of_output_subsingleton","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#conditional-prefix-optimum-eq-zero-of-ou","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"If the output alphabet of a finite joint law $p$ has exactly one element, then the conditional-prefix optimum of $p$ equals $0$.","title":"The conditional-prefix optimum vanishes for a one-letter output"},{"id":"InformationTheory/channel-simulation/one-shot-channel-simulation/conditional-prefix-optimum-le-worst-case","kind":"node","lean":"InformationTheory.conditionalPrefixOptimum_le_worstCaseOptimum","parent":44098,"result":true,"route":"../roadmap/InformationTheory/channel-simulation/one-shot-channel-simulation/index.html#conditional-prefix-optimum-le-worst-case","scope":"InformationTheory/channel-simulation/one-shot-channel-simulation","status":"fully_proved","summary":"For every finite joint law $p$ of $(X, Y)$, the conditional-prefix optimum of $p$ is at most the worst-case optimum at $t = I(X; Y)$, the mutual information of $p$ in bits.","title":"The worst-case optimum dominates each conditional-prefix optimum"},{"area":"Applied Mathematics","data_url":"chapters/QuantumInformation.json","id":"QuantumInformation","kind":"scope","parent":0,"result":false,"route":"../roadmap/QuantumInformation/index.html","scope":"QuantumInformation","status":"can_prove","summary":"Quantum information studies states, channels, measurement, entanglement, communication, and quantum computation.","title":"Quantum Information"},{"data_url":"scopes/QuantumInformation/quantum-foundations.json","id":"QuantumInformation/quantum-foundations","kind":"scope","parent":44128,"result":false,"route":"../roadmap/QuantumInformation/quantum-foundations/index.html","scope":"QuantumInformation/quantum-foundations","status":"can_prove","summary":"Quantum foundations studies what quantum mechanics rules out and allows at the level of measurement outcomes, including contextuality, Bell nonlocality and bounds on quantum correlations.","title":"Quantum foundations"},{"id":"QuantumInformation/quantum-foundations/kochen-specker-finite-real-three","kind":"node","lean":"MathlibExt.QuantumInformation.Finite.KochenSpeckerWanted.kochen_specker_finite_real_three","parent":44129,"reading_title":{"format":"inline-parts-v1","parts":[["text","Kochen\u2013Specker theorem in "],["math","\\(\\mathbb{R}^3\\)"]],"source":"Kochen\u2013Specker theorem in $\\mathbb{R}^3$"},"result":true,"route":"../roadmap/QuantumInformation/quantum-foundations/index.html#kochen-specker-finite-real-three","scope":"QuantumInformation/quantum-foundations","status":"fully_proved","summary":"There exist a natural number $N$ and unit vectors $v_1, \\dots, v_N \\in \\mathbb{R}^3$ such that no function $c : \\{1, \\dots, N\\} \\to \\{0, 1\\}$ satisfies both of the following: $c(i) + c(j) \\le 1$ whenever $v_i$ and $v_j$ are orthogonal, and $c(i) + c(j) + c(k) = 1$ whenever $(v_i, v_j, v_k)$ is an orthonormal triple.","title":"Kochen\u2013Specker theorem in $\\mathbb{R}^3$"},{"id":"QuantumInformation/quantum-foundations/tsirelson-bound-tight","kind":"node","lean":"MathlibExt.QuantumInformation.Finite.TsirelsonTight.tsirelson_bound_tight","parent":44129,"result":true,"route":"../roadmap/QuantumInformation/quantum-foundations/index.html#tsirelson-bound-tight","scope":"QuantumInformation/quantum-foundations","status":"fully_proved","summary":"There exist $4 \\times 4$ complex matrices $A_0, A_1, B_0, B_1$ such that each of them is Hermitian and squares to the identity matrix, each $A_i$ commutes with each $B_j$, and the CHSH operator $A_0 B_0 + A_0 B_1 + A_1 B_0 - A_1 B_1$ has $2\\sqrt{2}$ as an eigenvalue.","title":"Tsirelson's bound is attained"},{"data_url":"scopes/QuantumInformation/quantum-foundations/gleason-s-theorem.json","id":"QuantumInformation/quantum-foundations/gleason-s-theorem","kind":"scope","navigation":"module","parent":44129,"result":false,"route":"../roadmap/QuantumInformation/quantum-foundations/gleason-s-theorem/index.html","scope":"QuantumInformation/quantum-foundations/gleason-s-theorem","status":"can_prove","title":"Gleason's theorem"},{"id":"QuantumInformation/quantum-foundations/gleason-s-theorem/is-orth-proj","kind":"node","lean":"MathlibExt.Analysis.GleasonWanted.Analysis.IsOrthProj","parent":44132,"result":true,"route":"../roadmap/QuantumInformation/quantum-foundations/gleason-s-theorem/index.html#is-orth-proj","scope":"QuantumInformation/quantum-foundations/gleason-s-theorem","status":"fully_proved","summary":"A continuous linear operator $P$ on a complex Hilbert space $H$ is an orthogonal projection if it is idempotent, $P^2 = P$, and self-adjoint, $P^* = P$.","title":"Orthogonal projection"},{"id":"QuantumInformation/quantum-foundations/gleason-s-theorem/frame-function","kind":"node","lean":"MathlibExt.Analysis.GleasonWanted.Analysis.FrameFunction","parent":44132,"result":true,"route":"../roadmap/QuantumInformation/quantum-foundations/gleason-s-theorem/index.html#frame-function","scope":"QuantumInformation/quantum-foundations/gleason-s-theorem","status":"fully_proved","summary":"Let $H$ be a complex Hilbert space. A frame function on the projection lattice of $H$ is a function $\\mu$ from the continuous linear operators on $H$ to $\\mathbb{R}$ such that $\\mu(P) \\ge 0$ for every orthogonal projection $P$ on $H$, $\\mu(P + Q) = \\mu(P) + \\mu(Q)$ for all orthogonal projections $P$ and $Q$ on $H$ with $PQ = 0$, and $\\mu(I) = 1$ for the identity operator $I$ on $H$.","title":"Frame function on projections"},{"id":"QuantumInformation/quantum-foundations/gleason-s-theorem/re-tr","kind":"node","lean":"MathlibExt.Analysis.GleasonWanted.Analysis.reTr","parent":44132,"result":true,"route":"../roadmap/QuantumInformation/quantum-foundations/gleason-s-theorem/index.html#re-tr","scope":"QuantumInformation/quantum-foundations/gleason-s-theorem","status":"fully_proved","summary":"For a continuous linear operator $A$ on a finite-dimensional complex Hilbert space $H$, let $\\operatorname{tr}(A) \\in \\mathbb{C}$ be the trace of $A$ as a $\\mathbb{C}$-linear map on $H$. The real part of the trace of $A$ is the real number $\\operatorname{Re} \\operatorname{tr}(A)$.","title":"Real part of the trace"},{"id":"QuantumInformation/quantum-foundations/gleason-s-theorem/gleason-theorem-finite","kind":"node","lean":"MathlibExt.Analysis.GleasonWanted.Analysis.gleason_theorem_finite","parent":44132,"result":true,"route":"../roadmap/QuantumInformation/quantum-foundations/gleason-s-theorem/index.html#gleason-theorem-finite","scope":"QuantumInformation/quantum-foundations/gleason-s-theorem","status":"can_prove","summary":"Let $H$ be a finite-dimensional complex Hilbert space of dimension at least $3$, write $\\operatorname{tr}$ for the trace of a $\\mathbb{C}$-linear map on $H$, and let $\\mu$ be a frame function on the projection lattice of $H$. Then there is a unique continuous linear operator $\\rho$ on $H$ such that $\\rho$ is positive, $\\operatorname{Re} \\operatorname{tr}(\\rho) = 1$, and\u2026","title":"Gleason's theorem in finite dimensions"},{"id":"QuantumInformation/quantum-foundations/gleason-s-theorem/sphere-frame-function","kind":"node","lean":"MathlibExt.Analysis.GleasonWanted.Analysis.SphereFrameFunction","parent":44132,"result":true,"route":"../roadmap/QuantumInformation/quantum-foundations/gleason-s-theorem/index.html#sphere-frame-function","scope":"QuantumInformation/quantum-foundations/gleason-s-theorem","status":"fully_proved","summary":"Let $H$ be a complex Hilbert space. A frame function on the unit sphere of $H$ is a function $f$ from the unit vectors of $H$ to $\\mathbb{R}$ such that $f(x) \\ge 0$ for every unit vector $x$ in $H$ and, for every Hilbert basis $(b_i)_{i \\in \\iota}$ of $H$, the family $(f(b_i))_{i \\in \\iota}$ is unconditionally summable with sum $1$.","title":"Frame function on the unit sphere"},{"id":"QuantumInformation/quantum-foundations/gleason-s-theorem/gleason-theorem-separable","kind":"node","lean":"MathlibExt.Analysis.GleasonWanted.Analysis.gleason_theorem_separable","parent":44132,"result":true,"route":"../roadmap/QuantumInformation/quantum-foundations/gleason-s-theorem/index.html#gleason-theorem-separable","scope":"QuantumInformation/quantum-foundations/gleason-s-theorem","status":"can_prove","summary":"Let $H$ be a separable complex Hilbert space whose dimension over $\\mathbb{C}$ is at least $3$, and let $f$ be a frame function on the unit sphere of $H$. Then there is a positive continuous linear operator $\\rho$ on $H$ such that $f(x) = \\operatorname{Re} \\langle x, \\rho x \\rangle$ for every unit vector $x$ in $H$.","title":"Gleason's theorem for separable Hilbert spaces"},{"data_url":"scopes/QuantumInformation/quantum-operations.json","id":"QuantumInformation/quantum-operations","kind":"scope","parent":44128,"result":false,"route":"../roadmap/QuantumInformation/quantum-operations/index.html","scope":"QuantumInformation/quantum-operations","status":"fully_proved","summary":"Quantum operations and channels covers the physically allowed transformations of quantum states, such as isometries and completely positive maps, and impossibility results like the no-cloning theorem.","title":"Quantum operations and channels"},{"id":"QuantumInformation/quantum-operations/no-cloning-two-state","kind":"node","lean":"MathlibExt.QuantumInformation.Finite.NoCloning.no_cloning_two_state","parent":44139,"result":true,"route":"../roadmap/QuantumInformation/quantum-operations/index.html#no-cloning-two-state","scope":"QuantumInformation/quantum-operations","status":"fully_proved","summary":"Let $H$ be a complex inner product space and let $\\psi, \\varphi, b \\in H$ be unit vectors with $0 < |\\langle \\psi, \\varphi \\rangle| < 1$. Then there is no $\\mathbb{C}$-linear isometric isomorphism $U$ of $H \\otimes_{\\mathbb{C}} H$ onto itself with $U(\\psi \\otimes b) = \\psi \\otimes \\psi$ and $U(\\varphi \\otimes b) = \\varphi \\otimes \\varphi$.","title":"No-cloning theorem for two states"},{"area":"Applied Mathematics","data_url":"chapters/GameTheory.json","id":"GameTheory","kind":"scope","parent":0,"result":false,"route":"../roadmap/GameTheory/index.html","scope":"GameTheory","status":"can_prove","summary":"Game theory studies strategic and combinatorial interaction, equilibria, winning conditions, and optimal play.","title":"Game Theory"},{"data_url":"scopes/GameTheory/combinatorial-games.json","id":"GameTheory/combinatorial-games","kind":"scope","parent":44141,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/index.html","scope":"GameTheory/combinatorial-games","status":"can_prove","summary":"Combinatorial game theory studies two-player games of perfect information without chance: impartial games such as Nim and its variants (P-positions, Sprague\u2013Grundy values, remoteness), partizan games such as Snort, and Maker\u2013Breaker positional games.","title":"Combinatorial game theory"},{"data_url":"scopes/GameTheory/combinatorial-games/accelerated-zeckendorf-game.json","id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","title":"Accelerated Zeckendorf game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-move","kind":"node","lean":"MetaMathlibExt.AccelZeckMove","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-move","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$ and $F_{i+1} = F_i + F_{i-1}$, and call a finitely supported function $s : \\mathbb{N} \\to \\mathbb{N}$ a state, $s(i)$ being the number of copies of $F_i$. A move takes a state $s$ to a state $t$ if for some multiplicity $m > 0$ one of the following holds: the move $m \\cdot C_i$ with $i \\ge 2$, where $s(i-1) \\ge m$, $s(i) \\ge m$ and $t$ is obtained from $s$ by\u2026","title":"Moves of the Accelerated Zeckendorf Game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-move-single-one-two","kind":"node","lean":"MetaMathlibExt.accelZeckMove_single_one_two","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-move-single-one-two","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$, $F_{i+1} = F_i + F_{i-1}$. Passing from the state consisting of two copies of $F_1$ to the state consisting of one copy of $F_2$ is a move of the Accelerated Zeckendorf Game.","title":"Two ones combine into a two in the Accelerated Zeckendorf Game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/not-accel-zeck-move-zero","kind":"node","lean":"MetaMathlibExt.not_accelZeckMove_zero","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#not-accel-zeck-move-zero","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"No move takes the empty state (no copies of any $F_i$) to any state.","title":"No move from the empty state"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/is-accel-zeck-game","kind":"node","lean":"MetaMathlibExt.IsAccelZeckGame","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#is-accel-zeck-game","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"A sequence of states $g_0, g_1, \\dots$ is a complete Accelerated Zeckendorf Game on $n$ with $L$ moves if $g_0$ consists of $n$ copies of $F_1$, for every $j < L$ some move takes $g_j$ to $g_{j+1}$, and no move is possible from $g_L$. The states after $g_L$ play no role.","title":"Complete Accelerated Zeckendorf Games"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/is-accel-zeck-game-zero","kind":"node","lean":"MetaMathlibExt.isAccelZeckGame_zero","parent":44143,"reading_title":{"format":"inline-parts-v1","parts":[["text","The complete game on "],["math","\\(0\\)"]],"source":"The complete game on $0$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#is-accel-zeck-game-zero","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"The sequence whose states are all the empty state is a complete Accelerated Zeckendorf Game on $0$ with $0$ moves.","title":"The complete game on $0$"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-losing","kind":"node","lean":"MetaMathlibExt.AccelZeckLosing","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-losing","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"The losing and the winning states of the Accelerated Zeckendorf Game are defined by simultaneous induction: a state is losing when every move from it leads to a winning state, and a state is winning when some move from it leads to a losing state. In particular every state from which no move is possible is losing.","title":"Losing states of the Accelerated Zeckendorf Game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-winning","kind":"node","lean":"MetaMathlibExt.AccelZeckWinning","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-winning","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"A state $s$ is winning (the player to move has a winning strategy) if some move takes $s$ to a losing state. Winning and losing states are defined by simultaneous induction: a state is losing if every move from it reaches a winning state.","title":"Winning states of the Accelerated Zeckendorf Game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-winning-iff","kind":"node","lean":"MetaMathlibExt.accelZeckWinning_iff","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-winning-iff","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"A state $s$ is winning if and only if some move takes $s$ to a losing state.","title":"Characterization of winning states"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-losing-of-terminal","kind":"node","lean":"MetaMathlibExt.accelZeckLosing_of_terminal","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-losing-of-terminal","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"Every state of the Accelerated Zeckendorf Game from which no move is possible is a losing state.","title":"Terminal states are losing in the Accelerated Zeckendorf Game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-winning-single-one-two","kind":"node","lean":"MetaMathlibExt.accelZeckWinning_single_one_two","parent":44143,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first player wins the game on "],["math","\\(2\\)"]],"source":"The first player wins the game on $2$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-winning-single-one-two","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"The state consisting of two copies of $F_1$ is winning.","title":"The first player wins the game on $2$"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/zeckendorf-state","kind":"node","lean":"MetaMathlibExt.zeckendorfState","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#zeckendorf-state","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$ and $F_{i+1} = F_i + F_{i-1}$, and record a game state as a finitely supported function $s : \\mathbb{N} \\to \\mathbb{N}$, where $s(i)$ is the number of copies of $F_i$. Every $n \\in \\mathbb{N}$ has a unique Zeckendorf decomposition $n = F_{i_1} + \\cdots + F_{i_r}$ with $r \\ge 0$, indices $i_1, \\dots, i_r \\ge 1$ and $i_t \\ge i_{t+1} + 2$ for $1 \\le t < r$. The\u2026","title":"Zeckendorf decomposition as a game state"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/zeckendorf-state-zero","kind":"node","lean":"MetaMathlibExt.zeckendorfState_zero","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#zeckendorf-state-zero","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"The Zeckendorf state of $0$ is the empty game state, with no copies of any Fibonacci number.","title":"The Zeckendorf state of zero is empty"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/zeckendorf-state-four","kind":"node","lean":"MetaMathlibExt.zeckendorfState_four","parent":44143,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Zeckendorf state of "],["math","\\(4\\)"]],"source":"The Zeckendorf state of $4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#zeckendorf-state-four","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"fully_proved","summary":"The Zeckendorf state of $4$ consists of one copy of $F_3$ and one copy of $F_1$, reflecting the Zeckendorf decomposition $4 = F_3 + F_1 = 3 + 1$.","title":"The Zeckendorf state of $4$"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-game-terminates-at-zeckendorf","kind":"node","lean":"MetaMathlibExt.accelZeckGame_terminates_at_zeckendorf","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-game-terminates-at-zeckendorf","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$, $F_{i+1} = F_i + F_{i-1}$. Let $n$ be a natural number, let $Z(n)$ be the number of terms of the Zeckendorf decomposition of $n$, and let $IZ(n)$ be the sum of the indices $k$ of its terms $F_k$. Then there is no infinite sequence of moves starting from $n$ copies of $F_1$. Moreover, every complete Accelerated Zeckendorf Game on $n$ (a sequence of moves from $n$\u2026","title":"The Accelerated Zeckendorf Game terminates at the Zeckendorf decomposition"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-game-sharp-lower-bound","kind":"node","lean":"MetaMathlibExt.accelZeckGame_sharp_lower_bound","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-game-sharp-lower-bound","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$, $F_{i+1} = F_i + F_{i-1}$. Let $n \\ge 1$ and let $F_k$ be the largest Fibonacci number with $F_k \\le n$, the largest term of the Zeckendorf decomposition of $n$. Then every complete Accelerated Zeckendorf Game on $n$ (a sequence of moves from $n$ copies of $F_1$ to a state from which no move is possible) has at least $k - 1$ moves, and some complete game on $n$\u2026","title":"Sharp lower bound on the length of an Accelerated Zeckendorf Game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-game-lower-bound","kind":"node","lean":"MetaMathlibExt.accelZeckGame_lower_bound","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-game-lower-bound","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$, $F_{i+1} = F_i + F_{i-1}$. Let $n \\ge 1$ and let $F_k$ be the largest Fibonacci number with $F_k \\le n$, the largest term of the Zeckendorf decomposition of $n$. Then every complete Accelerated Zeckendorf Game on $n$ (a sequence of moves from $n$ copies of $F_1$ to a state from which no move is possible) has at least $k - 1$ moves.","title":"Lower bound on the length of an Accelerated Zeckendorf Game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-move-fib-combine","kind":"node","lean":"MetaMathlibExt.accelZeckMove_fib_combine","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-move-fib-combine","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$, $F_{j+1} = F_j + F_{j-1}$, extended by $F_0 = 1$ and $F_{-1} = 0$. Let $p \\ge 2$ and $i \\ge 1$, and let $s$ be a state with $F_{i-1}$ copies of $F_{p-1}$, $F_i$ copies of $F_p$ and no copy of $F_{p+1}$. Let $t$ be the state obtained from $s$ by replacing $F_{i-1}$ copies of each of $F_{p-1}$ and $F_p$ by $F_{i-1}$ copies of $F_{p+1}$. Then passing from $s$ to $t$\u2026","title":"Moving a Fibonacci number of pairs in the Accelerated Zeckendorf Game"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-game-sharp-lower-bound-fib","kind":"node","lean":"MetaMathlibExt.accelZeckGame_sharp_lower_bound_fib","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-game-sharp-lower-bound-fib","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$, $F_{i+1} = F_i + F_{i-1}$, and let $k \\ge 1$. Every complete Accelerated Zeckendorf Game on $F_k$ (a sequence of moves from $F_k$ copies of $F_1$ to a state from which no move is possible) has at least $k - 1$ moves, and some complete game on $F_k$ has exactly $k - 1$ moves.","title":"Sharp lower bound for the Accelerated Zeckendorf Game on a Fibonacci number"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-unique-winning-first-move","kind":"node","lean":"MetaMathlibExt.accelZeck_unique_winning_first_move","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-unique-winning-first-move","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"For every $n$, if the state consisting of $n$ copies of $F_1$ is winning, then exactly one state reachable from it in one move is losing: when the first player has a winning strategy in the game on $n$, exactly one first move preserves it.","title":"Unique winning first move"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-conjecture17-implies-conjectu","kind":"node","lean":"MetaMathlibExt.accelZeck_conjecture17_implies_conjecture16","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-conjecture17-implies-conjectu","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$, $F_{i+1} = F_i + F_{i-1}$. Suppose that for every $n$ such that the state consisting of $n$ copies of $F_1$ is losing, and every $k$ with $3k \\le n$, the state consisting of $k$ copies of $F_3$ and $n - 3k$ copies of $F_1$ is losing. Then for every $n > 9$ the state consisting of $n$ copies of $F_1$ (the starting position of the Accelerated Zeckendorf Game on\u2026","title":"Accelerated Zeckendorf Game: Conjecture 1.7 implies Conjecture 1.6"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-conjecture17-le-five","kind":"node","lean":"MetaMathlibExt.accelZeck_conjecture17_le_five","parent":44143,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-conjecture17-le-five","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"Index the Fibonacci numbers as $F_1 = 1$, $F_2 = 2$, $F_{i+1} = F_i + F_{i-1}$. Let $n$ be such that the state consisting of $n$ copies of $F_1$ (the starting position of the Accelerated Zeckendorf Game on $n$) is losing. Then for every $k$ with $1 \\le k \\le 5$ and $3k \\le n$, the state consisting of $k$ copies of $F_3$ and $n - 3k$ copies of $F_1$ is losing.","title":"Accelerated Zeckendorf Game: Conjecture 1.7 in its first five cases"},{"id":"GameTheory/combinatorial-games/accelerated-zeckendorf-game/accel-zeck-odd-unique-losing-move","kind":"node","lean":"MetaMathlibExt.accelZeck_odd_unique_losing_move","parent":44143,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unique losing move from "],["math","\\(k\\)"],["text"," copies of "],["math","\\(F_2\\)"],["text"," for odd "],["math","\\(k\\)"]],"source":"Unique losing move from $k$ copies of $F_2$ for odd $k$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/accelerated-zeckendorf-game/index.html#accel-zeck-odd-unique-losing-move","scope":"GameTheory/combinatorial-games/accelerated-zeckendorf-game","status":"can_prove","summary":"Let $k$ be odd and $n \\ge 2k$, and suppose that for every $i < k$ with $3i \\le n$ the state with $i$ copies of $F_3$, $n - 3i$ copies of $F_1$ and nothing else is losing. Then $n \\ge 3k$; the state $T$ with $k$ copies of $F_3$, $n - 3k$ copies of $F_1$ and nothing else is losing and is reached in one move from the state $S$ with $k$ copies of $F_2$, $n - 2k$ copies of $F_1$ and nothing else; and every losing state\u2026","title":"Unique losing move from $k$ copies of $F_2$ for odd $k$"},{"data_url":"scopes/GameTheory/combinatorial-games/chocolate-bar-games.json","id":"GameTheory/combinatorial-games/chocolate-bar-games","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","title":"Chocolate bar games"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/has-nsproperty","kind":"node","lean":"MetaMathlibExt.HasNSProperty","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#has-nsproperty","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"A function $h : \\mathbb{N} \\to \\mathbb{N}$ has the *NS property* if it is monotone (weakly increasing) and, for all $z, z' \\in \\mathbb{N}$ and every natural number $i \\ge 1$, $\\lfloor z / 2^i \\rfloor = \\lfloor z' / 2^i \\rfloor$ implies $\\lfloor h(z) / 2^{i-1} \\rfloor = \\lfloor h(z') / 2^{i-1} \\rfloor$.","title":"NS property"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/has-nsproperty-const","kind":"node","lean":"MetaMathlibExt.hasNSProperty_const","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#has-nsproperty-const","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"For every natural number $c$, the constant function $\\mathbb{N} \\to \\mathbb{N}$ with value $c$ has the NS property.","title":"Constant functions have the NS property"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/not-has-nsproperty-id","kind":"node","lean":"MetaMathlibExt.not_hasNSProperty_id","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#not-has-nsproperty-id","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"The identity function on the natural numbers does not have the NS property: $\\lfloor 0/2 \\rfloor = \\lfloor 1/2 \\rfloor$, but $0 \\ne 1$.","title":"The identity function lacks the NS property"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/has-nsproperty-extend-const","kind":"node","lean":"MetaMathlibExt.hasNSProperty_extend_const","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#has-nsproperty-extend-const","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $z \\in \\mathbb{N}$ and let $h : \\mathbb{N} \\to \\mathbb{N}$ satisfy $h(a) \\le h(b)$ whenever $a \\le b \\le z$, and $\\lfloor h(a) / 2^{i-1} \\rfloor = \\lfloor h(b) / 2^{i-1} \\rfloor$ whenever $a, b \\le z$, $i \\ge 1$ and $\\lfloor a / 2^i \\rfloor = \\lfloor b / 2^i \\rfloor$. Then the function sending $x$ to $h(x)$ for $x \\le z$ and to $h(z)$ for $x > z$ has the NS property.","title":"Constant extension preserves the NS property"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/min-div-two-pow-eq-of-div-two-pow-eq","kind":"node","lean":"MetaMathlibExt.min_div_two_pow_eq_of_div_two_pow_eq","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#min-div-two-pow-eq-of-div-two-pow-eq","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $a, b, c, k \\in \\mathbb{N}$. If $\\lfloor a / 2^k \\rfloor = \\lfloor b / 2^k \\rfloor$, then $\\lfloor \\min(a, c) / 2^k \\rfloor = \\lfloor \\min(b, c) / 2^k \\rfloor$.","title":"Taking a minimum preserves quotients by powers of two"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/min-div-two-pow-eq","kind":"node","lean":"MetaMathlibExt.HasNSProperty.min_div_two_pow_eq","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#min-div-two-pow-eq","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $h : \\mathbb{N} \\to \\mathbb{N}$ have the NS property and let $z, z', k \\in \\mathbb{N}$ with $k \\ge 1$ and $\\lfloor z' / 2^k \\rfloor = \\lfloor z / 2^k \\rfloor$. Then for every $y \\in \\mathbb{N}$, $\\lfloor \\min(h(z'), y) / 2^{k-1} \\rfloor = \\lfloor \\min(h(z), y) / 2^{k-1} \\rfloor$.","title":"Quotients of a truncated NS function by powers of two"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/exists-of-lt-xor","kind":"node","lean":"MetaMathlibExt.HasNSProperty.exists_of_lt_xor","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#exists-of-lt-xor","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $g : \\mathbb{N} \\to \\mathbb{N}$ have the NS property, and let $x, z, c$ be natural numbers with $x \\le g(z)$ and $c < x \\oplus z$, where $\\oplus$ is nim-sum (bitwise exclusive or). Then $c = u \\oplus z$ for some $u < x$, or $c = \\min(g(w), x) \\oplus w$ for some $w < z$.","title":"Mex step for chocolate bars with one function"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/exists-of-lt-xor-xor","kind":"node","lean":"MetaMathlibExt.HasNSProperty.exists_of_lt_xor_xor","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#exists-of-lt-xor-xor","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $g, h : \\mathbb{N} \\to \\mathbb{N}$ have the NS property, and let $x, y, z, c$ be natural numbers with $x \\le g(z)$, $y \\le h(z)$ and $c < x \\oplus y \\oplus z$, where $\\oplus$ is nim-sum (bitwise exclusive or). Then $c = u \\oplus y \\oplus z$ for some $u < x$, or $c = x \\oplus v \\oplus z$ for some $v < y$, or $c = \\min(g(w), x) \\oplus \\min(h(w), y) \\oplus w$ for some $w < z$.","title":"Mex step for chocolate bars with two functions"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/exists-of-lt-foldl-xor","kind":"node","lean":"MetaMathlibExt.exists_of_lt_foldl_xor","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#exists-of-lt-foldl-xor","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $m$ be a natural number, let $f_1, \\dots, f_m : \\mathbb{N} \\to \\mathbb{N}$ have the NS property, and let $x_1, \\dots, x_m, z, c$ be natural numbers with $x_t \\le f_t(z)$ for all $t$ and $c < x_1 \\oplus \\cdots \\oplus x_m \\oplus z$, where $\\oplus$ is nim-sum (bitwise exclusive or). Then either there are an index $i$ and some $u < x_i$ such that $c$ is the nim-sum $x_1 \\oplus \\cdots \\oplus x_m \\oplus z$ with $x_i$\u2026","title":"Mex step for chocolate bars with many functions"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/image-xor-xor-eq","kind":"node","lean":"MetaMathlibExt.HasNSProperty.image_xor_xor_eq","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#image-xor-xor-eq","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $g, h : \\mathbb{N} \\to \\mathbb{N}$ have the NS property and let $x, y, z \\in \\mathbb{N}$ with $x \\le g(z)$ and $y \\le h(z)$. Writing $\\oplus$ for the nim-sum (bitwise exclusive or) of natural numbers, the sets $\\{x \\oplus y \\oplus w : w < z\\}$ and $\\{\\min(g(w), x) \\oplus \\min(h(w), y) \\oplus w : w < z\\}$ are equal, where $w$ ranges over the natural numbers.","title":"Nim-sum image identity for two NS functions"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/image-xor-eq","kind":"node","lean":"MetaMathlibExt.HasNSProperty.image_xor_eq","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#image-xor-eq","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $g : \\mathbb{N} \\to \\mathbb{N}$ have the NS property and let $x, z \\in \\mathbb{N}$ with $x \\le g(z)$. Writing $\\oplus$ for the nim-sum (bitwise exclusive or) of natural numbers, the sets $\\{x \\oplus w : w < z\\}$ and $\\{\\min(g(w), x) \\oplus w : w < z\\}$ are equal, where $w$ ranges over the natural numbers.","title":"Nim-sum image identity for one NS function"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/has-nsproperty-div-two-pow","kind":"node","lean":"MetaMathlibExt.hasNSProperty_div_two_pow","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#has-nsproperty-div-two-pow","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"For every natural number $m \\ge 1$, the function $x \\mapsto \\lfloor x / 2^m \\rfloor$ on $\\mathbb{N}$ has the NS property.","title":"Halving functions have the NS property"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/div-two-pow-eq-iff-exists-block","kind":"node","lean":"MetaMathlibExt.div_two_pow_eq_iff_exists_block","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#div-two-pow-eq-iff-exists-block","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $i$ be a natural number and let $z < z'$ be natural numbers. Then $\\lfloor z / 2^i \\rfloor = \\lfloor z' / 2^i \\rfloor$ if and only if there is a natural number $d$ with $d \\cdot 2^i \\le z$ and $z' < (d + 1) \\cdot 2^i$.","title":"Equal quotients by a power of two lie in one dyadic block"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/exists-split-of-div-two-pow-lt","kind":"node","lean":"MetaMathlibExt.exists_split_of_div_two_pow_lt","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#exists-split-of-div-two-pow-lt","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $z, z', i$ be natural numbers with $\\lfloor z / 2^i \\rfloor < \\lfloor z' / 2^i \\rfloor$. Then there are natural numbers $c$, $s$ and $t$ with $s \\ge i$, $t < 2^s$, $z = c \\cdot 2^{s+1} + t$ and $c \\cdot 2^{s+1} + 2^s \\le z'$.","title":"Dyadic splitting of two numbers with different quotients"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/indicator-ici-even-iff-has-nsproperty","kind":"node","lean":"MetaMathlibExt.indicator_Ici_even_iff_hasNSProperty","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#indicator-ici-even-iff-has-nsproperty","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ be monotone, with $f(x) \\le 1$ for all $x$ and $f(x) = 1$ for some $x$. Then $f$ has the NS property if and only if there is an even $m \\in \\mathbb{N}$ such that $f$ is the indicator function of $\\{n \\in \\mathbb{N} : n \\ge m\\}$, that is, $f(n) = 1$ for $n \\ge m$ and $f(n) = 0$ for $n < m$.","title":"Indicator functions with the NS property"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/chocolate-bar-grundy","kind":"node","lean":"MetaMathlibExt.chocolateBarGrundy","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#chocolate-bar-grundy","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"Let $f_1, \\dots, f_s : \\mathbb{N} \\to \\mathbb{N}$. For coordinates $x = (x_1, \\dots, x_s) \\in \\mathbb{N}^s$ and a width $z \\in \\mathbb{N}$, the position $(x, z)$ stands for the chocolate bar $\\mathrm{CB}(f_1, \\dots, f_s, x_1, \\dots, x_s, z)$, whose column at width $w \\le z$ extends $\\min(f_t(w), x_t) + 1$ cells along axis $t$. A move either lowers one coordinate $x_i$ to some $u < x_i$, keeping the width $z$, or\u2026","title":"Grundy number of a chocolate bar"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/chocolate-bar-grundy-eq","kind":"node","lean":"MetaMathlibExt.chocolateBarGrundy_eq","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#chocolate-bar-grundy-eq","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"Let $f_1, \\dots, f_s$ be functions $\\mathbb{N} \\to \\mathbb{N}$ and write $G(x, z)$ for the Grundy number of the chocolate bar with coordinates $x = (x_1, \\dots, x_s) \\in \\mathbb{N}^s$ and width $z \\in \\mathbb{N}$. For all $x$ and $z$, $G(x, z)$ is the least natural number $m$ such that $m \\ne G(x', z)$ for every $x'$ obtained from $x$ by replacing one coordinate $x_i$ with some $u < x_i$, and $m \\ne G(x^{(w)}, w)$\u2026","title":"Mex recursion for the Grundy number of a chocolate bar"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/chocolate-bar-grundy-zero","kind":"node","lean":"MetaMathlibExt.chocolateBarGrundy_zero","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#chocolate-bar-grundy-zero","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"For all functions $f_1, \\dots, f_s : \\mathbb{N} \\to \\mathbb{N}$, the chocolate bar $\\mathrm{CB}(f_1, \\dots, f_s, 0, \\dots, 0, 0)$, whose coordinates and width are all $0$, has Grundy number $0$: it is a single bitter square and has no move.","title":"The single bitter square has Grundy number zero"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/height-chocolate-bar-grundy","kind":"node","lean":"MetaMathlibExt.heightChocolateBarGrundy","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#height-chocolate-bar-grundy","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"Let $s \\in \\mathbb{N}$ and $F : \\mathbb{N}^s \\to \\mathbb{N}$. The $(s + 1)$-dimensional chocolate bar $\\mathrm{CB}(F, x_1, \\dots, x_s, y)$ has, at each $u = (u_1, \\dots, u_s)$ with $u_t \\le x_t$ for all $t$, a column of height $\\min(F(u), y) + 1$. A move either lowers one coordinate $x_i$ to some $a < x_i$, which clamps the height to $\\min(F(x'), y)$ for the new position $x'$, or lowers the height $y$ to some\u2026","title":"Grundy number of a chocolate bar with a height function"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/height-chocolate-bar-grundy-eq","kind":"node","lean":"MetaMathlibExt.heightChocolateBarGrundy_eq","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#height-chocolate-bar-grundy-eq","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"Let $s \\in \\mathbb{N}$ and $F : \\mathbb{N}^s \\to \\mathbb{N}$, and write $G(x, y)$ for the Grundy number of the chocolate bar $\\mathrm{CB}(F, x_1, \\dots, x_s, y)$. For all $x \\in \\mathbb{N}^s$ and $y \\in \\mathbb{N}$, $G(x, y)$ is the least natural number $m$ such that $m \\ne G(x', \\min(F(x'), y))$ for every $x'$ obtained from $x$ by lowering one coordinate $x_i$ to some $u < x_i$, and $m \\ne G(x, v)$ for every\u2026","title":"Mex recursion for chocolate bars with a height function"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/height-chocolate-bar-grundy-zero","kind":"node","lean":"MetaMathlibExt.heightChocolateBarGrundy_zero","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#height-chocolate-bar-grundy-zero","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"For every $s \\in \\mathbb{N}$ and every $F : \\mathbb{N}^s \\to \\mathbb{N}$, the chocolate bar $\\mathrm{CB}(F, 0, \\dots, 0, 0)$ has no move, so its Grundy number is $0$.","title":"The single bitter cube has Grundy number zero"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/nim-pass-indicator","kind":"node","lean":"MetaMathlibExt.nimPassIndicator","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#nim-pass-indicator","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"For $s$ piles with thresholds $m_0, \\dots, m_{s-1}$ and pile sizes $x_0, \\dots, x_{s-1}$ (natural numbers), the pass indicator is $1$ if $m_j \\le x_j$ for some pile $j$ (the pass may be used), and $0$ if $x_j < m_j$ for every pile $j$.","title":"Pass indicator of Nim with a restricted pass"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/nim-pass-indicator-le-one","kind":"node","lean":"MetaMathlibExt.nimPassIndicator_le_one","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#nim-pass-indicator-le-one","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"For every $s \\in \\mathbb{N}$ and all $m, x \\in \\mathbb{N}^s$, the pass indicator of $m$ at $x$ is at most $1$; that is, it takes values in $\\{0, 1\\}$.","title":"The Nim pass indicator is at most one"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/nim-pass-indicator-zero","kind":"node","lean":"MetaMathlibExt.nimPassIndicator_zero","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#nim-pass-indicator-zero","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"fully_proved","summary":"Let $s \\in \\mathbb{N}$ and $m \\in \\mathbb{N}^s$ with $m_i > 0$ for every $i < s$. Then the pass indicator of $m$ takes the value $0$ at the zero tuple $(0, \\dots, 0)$.","title":"The pass is unavailable from the empty position"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/image-foldl-xor-eq-image-min","kind":"node","lean":"MetaMathlibExt.image_foldl_xor_eq_image_min","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#image-foldl-xor-eq-image-min","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $m \\in \\mathbb{N}$, let $f_0, \\dots, f_{m-1} : \\mathbb{N} \\to \\mathbb{N}$ have the NS property, and let $x_0, \\dots, x_{m-1}, z \\in \\mathbb{N}$ satisfy $x_t \\le f_t(z)$ for every $t < m$. Then","title":"Nim-sum sets under clamping by functions with the NS property"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/has-nsproperty-iff-chocolate-bar-grundy","kind":"node","lean":"MetaMathlibExt.hasNSProperty_iff_chocolateBarGrundy_depth_zero","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#has-nsproperty-iff-chocolate-bar-grundy","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ be monotone. Then $f$ has the NS property if and only if, for all $x, z \\in \\mathbb{N}$ with $x \\le f(z)$, the chocolate bar $\\mathrm{CB}(f, 0, x, 0, z)$ of width $z$ (first function $f$, first coordinate $x$, second function constantly $0$, second coordinate $0$) has Grundy number $x \\oplus z$, where $\\oplus$ denotes nim-sum (bitwise exclusive or).","title":"The NS property characterises nim-sum Grundy numbers of two-dimensional chocolate bars"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/sections-has-nsproperty-iff-height-choco","kind":"node","lean":"MetaMathlibExt.sections_hasNSProperty_iff_heightChocolateBarGrundy_eq_xor","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#sections-has-nsproperty-iff-height-choco","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $F : \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$ be monotone for the componentwise order. Then the sections $x \\mapsto F(n, x)$ and $w \\mapsto F(w, n)$ have the NS property for every $n \\in \\mathbb{N}$ if and only if, for all natural numbers $w$, $x$, $y$ with $y \\le F(w, x)$, the Grundy number of the chocolate bar $\\mathrm{CB}(F, w, x, y)$, whose column at $(u, v)$ (with $u \\le w$, $v \\le x$) has height\u2026","title":"NS property of sections and the three-dimensional chocolate bar"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/has-nsproperty-and-iff-chocolate-bar-gru","kind":"node","lean":"MetaMathlibExt.hasNSProperty_and_iff_chocolateBarGrundy_eq_xor","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#has-nsproperty-and-iff-chocolate-bar-gru","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $g, h : \\mathbb{N} \\to \\mathbb{N}$ be monotone. Then $g$ and $h$ both have the NS property if and only if, for all natural numbers $x, y, z$ with $x \\le g(z)$ and $y \\le h(z)$, the Grundy number of the chocolate bar $\\mathrm{CB}(g, h, x, y, z)$ equals $x \\oplus y \\oplus z$, where $\\oplus$ is nim-sum (bitwise exclusive or).","title":"Nim-sum Grundy numbers of chocolate bars with two functions"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/forall-has-nsproperty-iff-chocolate-bar","kind":"node","lean":"MetaMathlibExt.forall_hasNSProperty_iff_chocolateBarGrundy_eq_xor_four","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#forall-has-nsproperty-iff-chocolate-bar","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $f_1, f_2, f_3, f_4 : \\mathbb{N} \\to \\mathbb{N}$ be monotone. Then all four have the NS property if and only if, for all natural numbers $x_1, x_2, x_3, x_4, z$ with $x_t \\le f_t(z)$ for every $t$, the Grundy number of the chocolate bar $\\mathrm{CB}(f_1, f_2, f_3, f_4, x_1, x_2, x_3, x_4, z)$ equals $x_1 \\oplus x_2 \\oplus x_3 \\oplus x_4 \\oplus z$, where $\\oplus$ is nim-sum (bitwise exclusive or).","title":"Nim-sum Grundy numbers of chocolate bars with four functions"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/height-chocolate-bar-grundy-add-div-eq-z","kind":"node","lean":"MetaMathlibExt.heightChocolateBarGrundy_add_div_eq_zero_iff","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#height-chocolate-bar-grundy-add-div-eq-z","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $k$ be a natural number with $k \\equiv 3 \\pmod 4$, and let $F(w, x) = \\lfloor (w + x) / k \\rfloor$. For all $w, x, y \\in \\mathbb{N}$ with $y \\le F(w, x)$, the three-dimensional chocolate bar $\\mathrm{CB}(F, w, x, y)$ has Grundy number $0$ (is a P-position) if and only if $w \\oplus x \\oplus y = 0$, where $\\oplus$ denotes nim-sum (bitwise exclusive or).","title":"P-positions of a three-dimensional chocolate bar with floor-quotient height"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/indicator-sections-iff-height-chocolate","kind":"node","lean":"MetaMathlibExt.indicator_sections_iff_heightChocolateBarGrundy_eq_xor","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#indicator-sections-iff-height-chocolate","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $s \\in \\mathbb{N}$ and let $F : \\mathbb{N}^s \\to \\mathbb{N}$ be monotone for the coordinatewise order, with $F(x) \\le 1$ for all $x$. For $x \\in \\mathbb{N}^s$, $i < s$ and $t \\in \\mathbb{N}$ write $x[i \\mapsto t]$ for $x$ with its $i$-th coordinate replaced by $t$, and suppose that for all $x$ and $i$ there is a $t$ with $F(x[i \\mapsto t]) = 1$. Write $\\oplus$ for nim-sum. Then the following are equivalent: (a)\u2026","title":"Height-one chocolate bars with nim-sum Grundy numbers"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/nim-pass-indicator-sections-has-nsproper","kind":"node","lean":"MetaMathlibExt.nimPassIndicator_sections_hasNSProperty_iff","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#nim-pass-indicator-sections-has-nsproper","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $s \\in \\mathbb{N}$ and $m \\in \\mathbb{N}^s$ with $m_i > 0$ for every $i < s$, and let $F : \\mathbb{N}^s \\to \\mathbb{N}$ be the pass indicator of $m$: $F(x) = 1$ if $m_j \\le x_j$ for some $j$, and $F(x) = 0$ otherwise. For $x \\in \\mathbb{N}^s$, $i < s$ and $t \\in \\mathbb{N}$ write $x[i \\mapsto t]$ for $x$ with its $i$-th coordinate replaced by $t$. Then the following are equivalent: (a) for all\u2026","title":"Sections of the Nim pass indicator with the NS property"},{"id":"GameTheory/combinatorial-games/chocolate-bar-games/nim-pass-grundy-eq-xor-iff","kind":"node","lean":"MetaMathlibExt.nimPassGrundy_eq_xor_iff","parent":44166,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/chocolate-bar-games/index.html#nim-pass-grundy-eq-xor-iff","scope":"GameTheory/combinatorial-games/chocolate-bar-games","status":"can_prove","summary":"Let $s \\in \\mathbb{N}$ and $m \\in \\mathbb{N}^s$ with $m_i > 0$ for every $i < s$, and let $F : \\mathbb{N}^s \\to \\mathbb{N}$ be the pass indicator of $m$: $F(x) = 1$ if $m_j \\le x_j$ for some $j$, and $F(x) = 0$ otherwise. Write $\\oplus$ for nim-sum. Then the following are equivalent: (a) for all $x \\in \\mathbb{N}^s$ and $p \\in \\mathbb{N}$ with $p \\le F(x)$, the Grundy number of the chocolate bar\u2026","title":"Nim with a restricted pass"},{"data_url":"scopes/GameTheory/combinatorial-games/fast-van-der-waerden-game.json","id":"GameTheory/combinatorial-games/fast-van-der-waerden-game","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","title":"Fast van der Waerden game"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/contains-copy","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.ContainsCopy","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#contains-copy","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"Let $S$ and $M$ be finite sets of natural numbers. $M$ contains a homothetic copy of $S$ if there are a natural number $a > 0$ and an integer $b$ such that $as + b \\in M$ for every $s \\in S$, that is, $aS + b \\subseteq M$. This is Maker's winning condition.","title":"Containing a homothetic copy"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/maker-wins-within","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.MakerWinsWithin","parent":44200,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maker wins within "],["math","\\(p\\)"],["text"," moves"]],"source":"Maker wins within $p$ moves"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#maker-wins-within","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"Let $S$ be a finite set of natural numbers. In the $S$-game, Maker and Breaker alternately claim unclaimed natural numbers, and Maker wins once the numbers Maker has claimed contain a copy $aS + b = \\{as + b : s \\in S\\}$ of $S$, where $a$ is a positive integer and $b$ is an integer. Let $M$ and $B$ be finite sets of natural numbers (claimed by Maker and by Breaker) and $p$ a natural number, with Maker to move.\u2026","title":"Maker wins within $p$ moves"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/is-rat-copy","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.IsRatCopy","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#is-rat-copy","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets of rational numbers. $B$ is a copy of $A$ if there are rationals $a > 0$ and $b$ such that $B = aA + b = \\{ax + b : x \\in A\\}$.","title":"Copies of finite sets of rationals"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/is-rat-reflection","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.IsRatReflection","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#is-rat-reflection","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"Let $A$ and $B$ be finite sets of rational numbers. $B$ is a reflection of $A$ if there are rationals $a < 0$ and $b$ such that $B = aA + b = \\{ax + b : x \\in A\\}$.","title":"Reflections of finite sets of rationals"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/is-symmetric-of-type","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.IsSymmetricOfType","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#is-symmetric-of-type","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"Let $S = \\{s_0 < s_1 < \\dots < s_{n-1}\\}$ be a finite set of $n$ rational numbers and let $i, j < n$. Then $S$ is called symmetric of type $(i, j)$ if $i < j$ and $S \\setminus \\{s_i\\}$ is a copy of $S \\setminus \\{s_j\\}$.","title":"Symmetric sets of a given type"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/is-symmetric","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.IsSymmetric","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#is-symmetric","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"A finite set $S$ of rational numbers is symmetric if it is symmetric of some type $(i, j)$ with $i, j < |S|$.","title":"Symmetric sets of rationals"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/zero-geometric-set","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.zeroGeometricSet","parent":44200,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zero-geometric set "],["math","\\(S_2(n, k)\\)"]],"source":"The zero-geometric set $S_2(n, k)$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#zero-geometric-set","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"For a natural number $n$ and a rational number $k$, $S_2(n, k)$ is the finite set of rationals $\\{0\\} \\cup \\{k^i : 0 \\le i < n - 1\\}$, that is, $\\{0, 1, k, \\dots, k^{n-2}\\}$ when $n \\ge 2$ and $\\{0\\}$ when $n \\le 1$.","title":"The zero-geometric set $S_2(n, k)$"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/contains-copy-empty","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.containsCopy_empty","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#contains-copy-empty","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"Every finite set $M$ of natural numbers contains a homothetic copy of the empty set.","title":"Every finite set contains a copy of the empty set"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/maker-wins-within-zero","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.makerWinsWithin_zero","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#maker-wins-within-zero","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"Let $S$ be a finite set of natural numbers, and let $M$ and $B$ be finite sets of natural numbers. Maker wins the $S$-game within $0$ moves from the position where Maker has claimed $M$ and Breaker has claimed $B$ if and only if $M$ contains a copy $aS + b = \\{as + b : s \\in S\\}$ of $S$, where $a$ is a positive integer and $b$ is an integer.","title":"Winning within zero moves"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/is-rat-copy-refl","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.isRatCopy_refl","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#is-rat-copy-refl","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"For every finite set $A$ of rational numbers, $A$ is a copy of $A$.","title":"Every finite set of rationals is a copy of itself"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/zero-geometric-set-one","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.zeroGeometricSet_one","parent":44200,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zero-geometric set "],["math","\\(S_2(1, k)\\)"],["text"," is "],["math","\\(\\{0\\}\\)"]],"source":"The zero-geometric set $S_2(1, k)$ is $\\{0\\}$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#zero-geometric-set-one","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"For every rational number $k$, the zero-geometric set $S_2(1, k)$ equals $\\{0\\}$.","title":"The zero-geometric set $S_2(1, k)$ is $\\{0\\}$"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/zero-geometric-set-add-two","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.zeroGeometricSet_add_two","parent":44200,"reading_title":{"format":"inline-parts-v1","parts":[["text","The zero-geometric set "],["math","\\(S_2(n + 2, k)\\)"]],"source":"The zero-geometric set $S_2(n + 2, k)$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#zero-geometric-set-add-two","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"fully_proved","summary":"For every natural number $n$ and every rational $k$, $S_2(n + 2, k) = \\{0\\} \\cup \\{k^i : 0 \\le i \\le n\\} = \\{0, 1, k, \\dots, k^n\\}$.","title":"The zero-geometric set $S_2(n + 2, k)$"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/maker-wins-within-card-of-card-le-three","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.makerWinsWithin_card_of_card_le_three","parent":44200,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maker wins the "],["math","\\(S\\)"],["text","-game in "],["math","\\(|S|\\)"],["text"," moves when "],["math","\\(|S| \\le 3\\)"]],"source":"Maker wins the $S$-game in $|S|$ moves when $|S| \\le 3$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#maker-wins-within-card-of-card-le-three","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"If $S$ is a finite set of natural numbers with $|S| \\le 3$, then, starting from the position where no number has been claimed, Maker wins the $S$-game within $|S|$ moves.","title":"Maker wins the $S$-game in $|S|$ moves when $|S| \\le 3$"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/is-symmetric-of-maker-wins-within-card","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.isSymmetric_of_makerWinsWithin_card","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#is-symmetric-of-maker-wins-within-card","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"Let $S$ be a finite set of natural numbers with at least two elements. If Maker, moving first with no numbers claimed, can force a win in the game on $S$ within $|S|$ selections, then $S$, viewed as a set of rational numbers, is symmetric.","title":"A fast Maker win forces symmetry"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/is-symmetric-of-type-cases","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.isSymmetricOfType_cases","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#is-symmetric-of-type-cases","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"Let $S$ be a finite set of rational numbers with $n \\ge 4$ elements. If $S$ is symmetric of type $(i, j)$, with positions counted from $0$, then $(i, j)$ is $(0, n - 1)$, $(1, n - 1)$ or $(0, n - 2)$. In the paper's one-based numbering these are the types $(1, n)$, $(2, n)$ and $(1, n - 1)$.","title":"Possible types of a symmetric set"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/zero-geometric-set-of-is-symmetric-of-ty","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.zeroGeometricSet_of_isSymmetricOfType","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#zero-geometric-set-of-is-symmetric-of-ty","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"Let $S = \\{s_0 < s_1 < \\dots < s_{n-1}\\}$ be a finite set of $n$ rational numbers, and let $i, j < n$ be such that $S$ is symmetric of type $(i, j)$, with positions counted from $0$. If $i = 1$ and $j = n - 1$, then there is a rational number $k > 1$ such that the zero-geometric set $S_2(n, k)$ is a copy of $S$. If $i = 0$ and $j = n - 2$, then there is a rational number $k > 1$ such that $S_2(n, k)$ is a\u2026","title":"Zero-geometric copies of symmetric sets"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/is-symmetric-of-type-unique","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.isSymmetricOfType_unique","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#is-symmetric-of-type-unique","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"A finite set $S$ of rational numbers with at least four elements is symmetric of at most one type: if $S$ is symmetric of types $(i, j)$ and $(i', j')$, then $i = i'$ and $j = j'$.","title":"Uniqueness of the symmetry type"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/maker-wins-within-of-hales-jewett","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.makerWinsWithin_of_halesJewett","parent":44200,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hales\u2013Jewett bound for the "],["math","\\(S\\)"],["text","-game"]],"source":"Hales\u2013Jewett bound for the $S$-game"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#maker-wins-within-of-hales-jewett","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"Let $S$ be a finite set of natural numbers with $|S| = n$, and let $N$ be a natural number such that every 2-coloring of $S^N$ has a monochromatic combinatorial line. Then, starting from the position where no number has been claimed, Maker wins the $S$-game within $\\lceil n^N / 2 \\rceil$ moves.","title":"Hales\u2013Jewett bound for the $S$-game"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/not-maker-wins-within-card-of-five-le","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.not_makerWinsWithin_card_of_five_le","parent":44200,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maker cannot win the "],["math","\\(S\\)"],["text","-game in "],["math","\\(|S|\\)"],["text"," moves when "],["math","\\(|S| \\ge 5\\)"]],"source":"Maker cannot win the $S$-game in $|S|$ moves when $|S| \\ge 5$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#not-maker-wins-within-card-of-five-le","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"If $S$ is a finite set of natural numbers with $|S| \\ge 5$, then, starting from the position where no number has been claimed, Maker does not win the $S$-game within $|S|$ moves.","title":"Maker cannot win the $S$-game in $|S|$ moves when $|S| \\ge 5$"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/maker-wins-within-four-iff-is-symmetric","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.makerWinsWithin_four_iff_isSymmetric","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#maker-wins-within-four-iff-is-symmetric","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"If $S$ is a set of four natural numbers, then, starting from the position where no number has been claimed, Maker wins the $S$-game within $4$ moves if and only if $S$, viewed as a set of rationals, is symmetric.","title":"Four-element sets with a four-move win are the symmetric ones"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/maker-wins-within-five-of-card-eq-four","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.makerWinsWithin_five_of_card_eq_four","parent":44200,"reading_title":{"format":"inline-parts-v1","parts":[["text","Maker wins the "],["math","\\(S\\)"],["text","-game in five moves when "],["math","\\(|S| = 4\\)"]],"source":"Maker wins the $S$-game in five moves when $|S| = 4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#maker-wins-within-five-of-card-eq-four","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"If $S$ is a set of four natural numbers, then, starting from the position where no number has been claimed, Maker wins the $S$-game within $5$ moves.","title":"Maker wins the $S$-game in five moves when $|S| = 4$"},{"id":"GameTheory/combinatorial-games/fast-van-der-waerden-game/maker-wins-within-of-tree","kind":"node","lean":"MetaMathlibExt.FastVanDerWaerdenGame.makerWinsWithin_of_tree","parent":44200,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/fast-van-der-waerden-game/index.html#maker-wins-within-of-tree","scope":"GameTheory/combinatorial-games/fast-van-der-waerden-game","status":"can_prove","summary":"Let $S$ be a finite set of natural numbers and $p$ a natural number. Let $T$ be a finite rooted tree with root $r$ whose vertices are pairwise distinct rationals, such that every path from $r$ down to a vertex has at most $p$ vertices and some such path has exactly $p$ vertices (so $T$ has height $p - 1$). Suppose that for every leaf $v$ the set of vertices on the path from $r$ to $v$ contains a copy $aS + b$ of\u2026","title":"Tree criterion for fast wins"},{"data_url":"scopes/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game.json","id":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/index.html","scope":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","status":"can_prove","title":"Hamiltonicity Maker\u2013Breaker game"},{"id":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/complete-board","kind":"node","lean":"MathlibExt.Combinatorics.HamiltonicityGameCriticalBiasOWR1782Problem013Wanted.completeBoard","parent":44223,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edges of the complete graph "],["math","\\(K_n\\)"]],"source":"Edges of the complete graph $K_n$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/index.html#complete-board","scope":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","status":"fully_proved","summary":"For a natural number $n$, the complete board is the finite set of edges of the complete graph $K_n$ on the vertex set $\\{0, 1, \\dots, n - 1\\}$, that is, the set of all unordered pairs $\\{i, j\\}$ of distinct vertices $i \\ne j$.","title":"Edges of the complete graph $K_n$"},{"id":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/free-elems","kind":"node","lean":"MathlibExt.Combinatorics.HamiltonicityGameCriticalBiasOWR1782Problem013Wanted.freeElems","parent":44223,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/index.html#free-elems","scope":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","status":"fully_proved","summary":"For finite sets $X$ (the board), $M$ (the elements claimed by Maker) and $R$ (the elements claimed by Breaker), the free elements are the elements of $X$ claimed by neither player, that is, the set $X \\setminus (M \\cup R)$.","title":"Free elements of a position"},{"id":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/maker-can-force","kind":"node","lean":"MathlibExt.Combinatorics.HamiltonicityGameCriticalBiasOWR1782Problem013Wanted.MakerCanForce","parent":44223,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/index.html#maker-can-force","scope":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","status":"fully_proved","summary":"Fix a finite board $X$, a bias $b \\in \\mathbb{N}$ and a property of finite sets, Maker's target. At a position $(M, R)$, where $M$ and $R$ are the sets claimed so far by Maker and by Breaker, an element of $X$ is free if it lies in neither $M$ nor $R$. The relation \"Maker, to move at $(M, R)$, can force a win\" is the least one satisfying two rules: it holds when no element of $X$ is free and $M$ satisfies the\u2026","title":"Maker can force a win in the biased Maker-Breaker game"},{"id":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/maker-wins-breaker-first","kind":"node","lean":"MathlibExt.Combinatorics.HamiltonicityGameCriticalBiasOWR1782Problem013Wanted.MakerWinsBreakerFirst","parent":44223,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/index.html#maker-wins-breaker-first","scope":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","status":"fully_proved","summary":"For a finite board $X$, a bias $b \\in \\mathbb{N}$ and a target property of Maker's claimed set, Maker wins the $(1 : b)$ Maker-Breaker game on $X$ in which Breaker moves first when, for every set $F \\subseteq X$ of exactly $\\min(b, |X|)$ elements that Breaker may claim in his first move, Maker can force a win when it is his turn at the position where he has claimed nothing and Breaker has claimed $F$.","title":"Maker wins the biased game with Breaker moving first"},{"id":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/hamiltonian-target","kind":"node","lean":"MathlibExt.Combinatorics.HamiltonicityGameCriticalBiasOWR1782Problem013Wanted.HamiltonianTarget","parent":44223,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/index.html#hamiltonian-target","scope":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","status":"fully_proved","summary":"For a natural number $n$, a set $C$ of unordered pairs of vertices from $\\{0, 1, \\dots, n - 1\\}$ meets the Hamiltonicity target when the simple graph on these $n$ vertices whose edges are the pairs of distinct vertices in $C$ is Hamiltonian: it has a cycle passing through every vertex exactly once, with the convention that the one-vertex graph is Hamiltonian.","title":"Maker's Hamiltonicity target"},{"id":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/hamiltonicity-critical-bias","kind":"node","lean":"MathlibExt.Combinatorics.HamiltonicityGameCriticalBiasOWR1782Problem013Wanted.hamiltonicityCriticalBias","parent":44223,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/index.html#hamiltonicity-critical-bias","scope":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","status":"fully_proved","summary":"This is the statement that the critical bias of the Hamiltonicity game is $(1 + o(1))\\, n / \\ln n$. In the $(1 : b)$ Maker-Breaker game on the edges of the complete graph $K_n$ with Breaker moving first, Breaker claims $b$ free edges per turn (all remaining free edges if fewer than $b$ are left), Maker claims one free edge per turn, and Maker wins if, once every edge is claimed, his edges contain a Hamilton cycle.\u2026","title":"The Hamiltonicity critical-bias property"},{"id":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/hamiltonicity-critical-bias-holds","kind":"node","lean":"MathlibExt.Combinatorics.HamiltonicityGameCriticalBiasOWR1782Problem013Wanted.hamiltonicityCriticalBias_holds","parent":44223,"reading_title":{"format":"inline-parts-v1","parts":[["text","The critical bias of the Hamiltonicity game is "],["math","\\((1 + o(1))\\, n / \\ln n\\)"]],"source":"The critical bias of the Hamiltonicity game is $(1 + o(1))\\, n / \\ln n$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game/index.html#hamiltonicity-critical-bias-holds","scope":"GameTheory/combinatorial-games/hamiltonicity-maker-breaker-game","status":"can_prove","summary":"Consider the $(1 : b)$ Maker-Breaker game on the edges of the complete graph $K_n$ in which Breaker moves first: Breaker claims $b$ free edges per turn (all remaining free edges if fewer than $b$ are left), Maker claims one free edge per turn, and Maker wins if, once every edge is claimed, his edges contain a Hamilton cycle. For every real $\\varepsilon > 0$ there is $n_0$ such that for every $n \\ge n_0$, Maker has\u2026","title":"The critical bias of the Hamiltonicity game is $(1 + o(1))\\, n / \\ln n$"},{"data_url":"scopes/GameTheory/combinatorial-games/nim.json","id":"GameTheory/combinatorial-games/nim","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/nim/index.html","scope":"GameTheory/combinatorial-games/nim","status":"fully_proved","title":"Nim"},{"id":"GameTheory/combinatorial-games/nim/bouton-nim-p-positions","kind":"node","lean":"MetaMathlibExt.BoutonNim.bouton_nim_p_positions","parent":44231,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim/index.html#bouton-nim-p-positions","scope":"GameTheory/combinatorial-games/nim","status":"fully_proved","summary":"Let a Nim position be a finite list of heap sizes $a_1, \\dots, a_n \\in \\mathbb{N}$, with nim-sum $a_1 \\oplus \\cdots \\oplus a_n$, where $\\oplus$ is bitwise exclusive or and the empty list has nim-sum $0$. A move chooses an index $i$ and replaces $a_i$ by a natural number $y < a_i$. Then: (i) if the position has nim-sum $0$, every move leads to a position with nonzero nim-sum; (ii) if the position has nonzero\u2026","title":"Bouton's move properties in Nim"},{"id":"GameTheory/combinatorial-games/nim/nim-sum-eq-zero-iff-last-eq-xor-fold","kind":"node","lean":"MetaMathlibExt.nim_sum_eq_zero_iff_last_eq_xor_fold","parent":44231,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim/index.html#nim-sum-eq-zero-iff-last-eq-xor-fold","scope":"GameTheory/combinatorial-games/nim","status":"fully_proved","summary":"For natural numbers $a_1, \\dots, a_n$ (the preceding piles) and $b$ (the last pile), the nim-sum $a_1 \\oplus \\cdots \\oplus a_n \\oplus b$ is $0$ if and only if $b = a_1 \\oplus \\cdots \\oplus a_n$, where $\\oplus$ is bitwise exclusive or and the nim-sum of no piles is $0$.","title":"Nim-sum zero via the last pile"},{"data_url":"scopes/GameTheory/combinatorial-games/nim-on-a-cube.json","id":"GameTheory/combinatorial-games/nim-on-a-cube","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","title":"Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-rotation-move","kind":"node","lean":"MetaMathlibExt.CubeRotationMove","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-rotation-move","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"fully_proved","summary":"Label the faces of a cube $f_a, f_b, f_c, f_A, f_B, f_C$, where $f_A$, $f_B$, $f_C$ are opposite $f_a$, $f_b$, $f_c$, and record a position as the tuple $p = (a, b, c, A, B, C)$ of numbers of tokens on these faces. A rotation is one of the three belts of four faces $\\{f_a, f_b, f_A, f_B\\}$, $\\{f_a, f_c, f_A, f_C\\}$ and $\\{f_b, f_c, f_B, f_C\\}$. In the game $C_R$ there is a move from a position $p$ to a position $q$\u2026","title":"Rotation move on the cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-rotation-move-examples","kind":"node","lean":"MetaMathlibExt.cubeRotationMove_examples","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-rotation-move-examples","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"fully_proved","summary":"Write positions as tuples $(a, b, c, A, B, C)$ of numbers of tokens on the faces $f_a, f_b, f_c, f_A, f_B, f_C$ of a cube. In the game $C_R$ there is a rotation move from $(0, 0, 0, 0, 0, 1)$ to the empty position $(0, 0, 0, 0, 0, 0)$, and there is no rotation move from the empty position.","title":"Rotation moves at the empty cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-vertex-move","kind":"node","lean":"MetaMathlibExt.CubeVertexMove","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-vertex-move","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"fully_proved","summary":"Label the faces of a cube $f_a, f_b, f_c, f_A, f_B, f_C$, where $f_A$, $f_B$, $f_C$ are opposite $f_a$, $f_b$, $f_c$, and record a position as the tuple $p = (a, b, c, A, B, C)$ of numbers of tokens on these faces. In the game $C_V$ there is a move from a position $p$ to a position $q$ when, for some choice of one face from each opposite pair $\\{f_a, f_A\\}$, $\\{f_b, f_B\\}$, $\\{f_c, f_C\\}$ (three faces that meet at\u2026","title":"Vertex move on the cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-edge-move","kind":"node","lean":"MetaMathlibExt.CubeEdgeMove","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-edge-move","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"fully_proved","summary":"Label the faces of a cube $f_a, f_b, f_c, f_A, f_B, f_C$, where $f_A$, $f_B$, $f_C$ are opposite $f_a$, $f_b$, $f_c$, and record a position as the tuple $p = (a, b, c, A, B, C)$ of numbers of tokens on these faces. In the game $C_E$ there is a move from a position $p$ to a position $q$ when, for some two faces from different opposite pairs (two faces that share an edge), $q$ agrees with $p$ on the other four faces,\u2026","title":"Edge move on the cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-face-move","kind":"node","lean":"MetaMathlibExt.CubeFaceMove","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-face-move","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"fully_proved","summary":"Label the faces of a cube $f_a, f_b, f_c, f_A, f_B, f_C$, where $f_A$, $f_B$, $f_C$ are opposite $f_a$, $f_b$, $f_c$, and record a position as the tuple $p = (a, b, c, A, B, C)$ of numbers of tokens on these faces. In the game $C_F$ there is a move from a position $p$ to a position $q$ when, for some face, $q$ agrees with $p$ on the other five faces, $q \\le p$ on every face, and $q \\ne p$; that is, at least one\u2026","title":"Face move on the cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-vertex-edge-face-move-examples","kind":"node","lean":"MetaMathlibExt.cubeVertexEdgeFaceMove_examples","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-vertex-edge-face-move-examples","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"fully_proved","summary":"Write positions as tuples $(a, b, c, A, B, C)$ of numbers of tokens on the faces $f_a, f_b, f_c, f_A, f_B, f_C$ of a cube. There is a vertex move (game $C_V$) from $(1, 1, 1, 0, 0, 0)$ to the empty position $(0, 0, 0, 0, 0, 0)$, an edge move (game $C_E$) from $(1, 1, 0, 0, 0, 0)$ to the empty position, and a face move (game $C_F$) from $(1, 1, 1, 0, 0, 0)$ to $(0, 1, 1, 0, 0, 0)$. No vertex move, edge move or face\u2026","title":"Examples of vertex, edge and face moves"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-rotation-nim-p-positions","kind":"node","lean":"MetaMathlibExt.cubeRotationNim_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-rotation-nim-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write positions of the game $C_R$ as tuples $(a, b, c, A, B, C)$ of numbers of tokens on the faces $f_a, f_b, f_c, f_A, f_B, f_C$, where $f_A$, $f_B$, $f_C$ are opposite $f_a$, $f_b$, $f_c$. Let $\\mathcal P$ be the set of positions with $a + A = b + B = c + C$. Then no rotation move leads from a position in $\\mathcal P$ to a position in $\\mathcal P$, and from every position outside $\\mathcal P$ there is a rotation\u2026","title":"P-positions of rotation Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-vertex-nim-p-positions","kind":"node","lean":"MetaMathlibExt.cubeVertexNim_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-vertex-nim-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write a position of the cube game as $p = (a, b, c, A, B, C)$, the numbers of tokens on the six faces of a cube, where the faces carrying $A$, $B$, $C$ are opposite those carrying $a$, $b$, $c$; a vertex move removes at least one token in total from the three faces at one vertex (one face from each opposite pair). Let $P$ be the set of positions with $a = A$, $b = B$ and $c = C$. Then no vertex move leads from a\u2026","title":"P-positions of vertex Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-edge-nim-p-positions","kind":"node","lean":"MetaMathlibExt.cubeEdgeNim_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-edge-nim-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write positions of the game $C_E$ as tuples $(a, b, c, A, B, C)$ of numbers of tokens on the faces $f_a, f_b, f_c, f_A, f_B, f_C$, where $f_A$, $f_B$, $f_C$ are opposite $f_a$, $f_b$, $f_c$, and let $\\oplus$ be the Nim-sum (bitwise exclusive or) of natural numbers. Let $\\mathcal P$ be the set of positions with $a \\oplus A = b \\oplus B = c \\oplus C$. Then no edge move leads from a position in $\\mathcal P$ to a\u2026","title":"P-positions of edge Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-vertex-nim-misere-p-positions","kind":"node","lean":"MetaMathlibExt.cubeVertexNim_misere_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-vertex-nim-misere-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write a position of the cube game as $p = (a, b, c, A, B, C)$, the numbers of tokens on the six faces of a cube, where the faces carrying $A$, $B$, $C$ are opposite those carrying $a$, $b$, $c$; a vertex move removes at least one token in total from the three faces at one vertex (one face from each opposite pair). Let $P$ be any set of positions such that $p \\in P$ if and only if $p$ has a vertex move and no vertex\u2026","title":"Mis\u00e8re P-positions of vertex Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-rotation-nim-misere-p-positions","kind":"node","lean":"MetaMathlibExt.cubeRotationNim_misere_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-rotation-nim-misere-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write positions of the game $C_R$ as tuples $p = (a, b, c, A, B, C)$ of numbers of tokens on the faces $f_a, f_b, f_c, f_A, f_B, f_C$, where $f_A$, $f_B$, $f_C$ are opposite $f_a$, $f_b$, $f_c$. Let $\\mathcal P$ be any set of positions such that a position lies in $\\mathcal P$ if and only if it has a rotation move and every rotation move from it leads to a position outside $\\mathcal P$; this is the set of\u2026","title":"Mis\u00e8re P-positions of rotation Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/cube-face-nim-misere-p-positions","kind":"node","lean":"MetaMathlibExt.cubeFaceNim_misere_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#cube-face-nim-misere-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write positions of the game $C_F$ (Nim with six heaps) as tuples $p = (a, b, c, A, B, C)$ of numbers of tokens on the faces $f_a, f_b, f_c, f_A, f_B, f_C$, where $f_A$, $f_B$, $f_C$ are opposite $f_a$, $f_b$, $f_c$, and let $\\oplus$ be the Nim-sum (bitwise exclusive or) of natural numbers. Let $\\mathcal P$ be any set of positions such that a position lies in $\\mathcal P$ if and only if it has a face move and every\u2026","title":"Mis\u00e8re P-positions of face Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/slow-cube-vertex-nim-p-positions","kind":"node","lean":"MetaMathlibExt.slowCubeVertexNim_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#slow-cube-vertex-nim-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write a position of the cube game as $p = (a, b, c, A, B, C)$, the numbers of tokens on the six faces of a cube, where the faces carrying $A$, $B$, $C$ are opposite those carrying $a$, $b$, $c$. A slow vertex move is a vertex move (removing at least one token in total from the three faces at one vertex, one face from each opposite pair) in which each face loses at most one token. Let $P$ be the set of positions for\u2026","title":"P-positions of slow vertex Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/slow-cube-edge-nim-p-positions","kind":"node","lean":"MetaMathlibExt.slowCubeEdgeNim_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#slow-cube-edge-nim-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write a position of the cube game as $p = (a, b, c, A, B, C)$, the numbers of tokens on the six faces of a cube, where the faces carrying $A$, $B$, $C$ are opposite those carrying $a$, $b$, $c$. A slow edge move is an edge move (removing at least one token in total from two faces in different opposite pairs) in which each face loses at most one token. Let $P$ be the set of positions for which $a + A$, $b + B$ and\u2026","title":"P-positions of slow edge Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/slow-cube-face-nim-p-positions","kind":"node","lean":"MetaMathlibExt.slowCubeFaceNim_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#slow-cube-face-nim-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write a position of the cube game as $p = (a, b, c, A, B, C)$, the numbers of tokens on the six faces of a cube. A slow face move is a face move that removes exactly one token from one face. Let $P$ be the set of positions with $a + b + c + A + B + C$ even. Then no slow face move leads from a position in $P$ to a position in $P$, and from every position not in $P$ some slow face move leads to a position in $P$.","title":"P-positions of slow face Nim on a cube"},{"id":"GameTheory/combinatorial-games/nim-on-a-cube/slow-cube-rotation-nim-p-positions","kind":"node","lean":"MetaMathlibExt.slowCubeRotationNim_pPositions","parent":44234,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/nim-on-a-cube/index.html#slow-cube-rotation-nim-p-positions","scope":"GameTheory/combinatorial-games/nim-on-a-cube","status":"can_prove","summary":"Write a position of the cube game as $p = (a, b, c, A, B, C)$, the numbers of tokens on the six faces of a cube, where the faces carrying $A$, $B$, $C$ are opposite those carrying $a$, $b$, $c$. A slow rotation move is a rotation move (removing at least one token in total from the four faces of one rotation, that is from the faces of two opposite pairs) in which each face loses at most one token. Let $P$ be any set\u2026","title":"P-positions of slow rotation Nim on a cube"},{"data_url":"scopes/GameTheory/combinatorial-games/snort.json","id":"GameTheory/combinatorial-games/snort","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","title":"Snort"},{"id":"GameTheory/combinatorial-games/snort/snort-mover-wins","kind":"node","lean":"MetaMathlibExt.snortMoverWins","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-mover-wins","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"Let $G$ be a finite simple graph, let $k$ be a natural number, and let $A$ and $B$ be finite sets of vertices of $G$, the vertices coloured so far by the player to move and by the opponent. A legal move colours a vertex $v$ that lies in neither $A$ nor $B$ and has no neighbour in $B$. With move bound $0$ the player to move loses. With move bound $k + 1$ the player to move wins at $(A, B)$ exactly when some legal\u2026","title":"Snort positions with a move bound"},{"id":"GameTheory/combinatorial-games/snort/snort-first-player-wins","kind":"node","lean":"MetaMathlibExt.SnortFirstPlayerWins","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-first-player-wins","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"Let $G$ be a finite simple graph. In Snort on $G$ the two players alternately colour a vertex that is not yet coloured and is not adjacent to any vertex coloured by the opponent, and a player with no legal move loses. The first player wins Snort on $G$ when the player who moves first has a winning strategy from the position in which no vertex is coloured.","title":"First player wins Snort"},{"id":"GameTheory/combinatorial-games/snort/snort-second-player-wins","kind":"node","lean":"MetaMathlibExt.SnortSecondPlayerWins","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-second-player-wins","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"Let $G$ be a finite simple graph. In Snort on $G$, two players alternately colour a vertex of $G$ that is not yet coloured, each with their own colour, and a player may not colour a vertex adjacent to a vertex of the opponent's colour; a player with no legal move loses. The second player wins Snort on $G$ if, starting from the position in which no vertex is coloured, the player who moves first has no winning\u2026","title":"Second-player win in Snort"},{"id":"GameTheory/combinatorial-games/snort/is-snort-response-pair","kind":"node","lean":"MetaMathlibExt.IsSnortResponsePair","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-snort-response-pair","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"In Snort on a finite graph $G$, a pair $(u, v)$ of vertices is a response pair if $u \\neq v$, $u$ and $v$ are not adjacent (so the first player colouring $u$ and the second answering $v$ is legal), and the first player, now to move, loses from the position in which the first player has coloured $u$ and the second player has coloured $v$.","title":"Snort response pairs"},{"id":"GameTheory/combinatorial-games/snort/is-opposition","kind":"node","lean":"MetaMathlibExt.IsOpposition","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-opposition","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"An opposition of a graph $G$ is an automorphism $f$ of $G$ such that for every vertex $v$, $f(f(v)) = v$, $f(v) \\neq v$, and $f(v)$ is not adjacent to $v$; that is, an involutive automorphism moving every vertex outside its closed neighbourhood.","title":"Oppositions"},{"id":"GameTheory/combinatorial-games/snort/is-opposable","kind":"node","lean":"MetaMathlibExt.IsOpposable","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-opposable","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"A graph $G$ is opposable if it has an opposition: an automorphism $f$ of $G$ such that for every vertex $v$, $f(f(v)) = v$, $f(v) \\neq v$, and $f(v)$ is not adjacent to $v$.","title":"Opposable graphs"},{"id":"GameTheory/combinatorial-games/snort/is-almost-opposable","kind":"node","lean":"MetaMathlibExt.IsAlmostOpposable","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-almost-opposable","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"A graph $G$ is almost opposable if there are a vertex $u$ and a set $S$ of vertices with $u \\in S \\subseteq N[u]$, where $N[u]$ is the closed neighbourhood of $u$ ($u$ together with its neighbours), such that the subgraph $G - S$ induced on the vertices outside $S$ is opposable.","title":"Almost opposable graph"},{"id":"GameTheory/combinatorial-games/snort/is-compatibly-almost-opposable","kind":"node","lean":"MetaMathlibExt.IsCompatiblyAlmostOpposable","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-compatibly-almost-opposable","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"A graph $G$ is compatibly almost opposable if there are a vertex $u$, a set $S$ of vertices with $u \\in S \\subseteq N[u]$ (the closed neighbourhood of $u$), an automorphism $\\alpha$ of $G$ with $\\alpha(\\alpha(v)) = v$ for every vertex $v$, and an opposition $f$ of the subgraph $G - S$ induced on the vertices outside $S$, such that $\\alpha(x) = f(x)$ for every vertex $x \\notin S$.","title":"Compatibly almost opposable graphs"},{"id":"GameTheory/combinatorial-games/snort/is-opposition-matching","kind":"node","lean":"MetaMathlibExt.IsOppositionMatching","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-opposition-matching","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"Let $K$ be a graph. A subgraph $M$ of $K$ is an opposition matching of $K$ if it is a perfect matching of $K$ and for any two distinct edges $ab$ and $cd$ of $M$, $ac \\in E(K) \\iff bd \\in E(K)$ and $ad \\in E(K) \\iff bc \\in E(K)$.","title":"Opposition matchings"},{"id":"GameTheory/combinatorial-games/snort/formula-product","kind":"node","lean":"MetaMathlibExt.formulaProduct","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#formula-product","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"Let $G$ and $H$ be graphs with vertex sets $V$ and $W$, and let $\\varphi$ be a Boolean function of four Boolean arguments. The product $G \\varphi H$ is the graph on $V \\times W$ in which two distinct vertices $(g_1, h_1)$ and $(g_2, h_2)$ are adjacent exactly when $\\varphi(g_1 g_2 \\in E(G), h_1 h_2 \\in E(H), g_1 = g_2, h_1 = h_2)$ is true.","title":"Graph product given by a Boolean formula"},{"id":"GameTheory/combinatorial-games/snort/strong-prod","kind":"node","lean":"MetaMathlibExt.strongProd","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#strong-prod","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"For simple graphs $G$ on $V$ and $H$ on $W$, the strong product $G \\boxtimes H$ is the simple graph on $V \\times W$ in which $(g_1, h_1)$ and $(g_2, h_2)$ are adjacent if and only if $g_1 = g_2$ and $h_1 h_2$ is an edge of $H$, or $g_1 g_2$ is an edge of $G$ and $h_1 = h_2$, or $g_1 g_2$ is an edge of $G$ and $h_1 h_2$ is an edge of $H$.","title":"Strong product of graphs"},{"id":"GameTheory/combinatorial-games/snort/firework","kind":"node","lean":"MetaMathlibExt.firework","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#firework","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"For a graph $G$, vertices $u$ and $v$ of $G$ and a natural number $n$, the $n$-firework of $G$ at $(u, v)$ is the graph obtained from $G$ by adding $2n$ new vertices, $n$ of which are adjacent only to $u$ and the other $n$ adjacent only to $v$.","title":"Firework of a graph"},{"id":"GameTheory/combinatorial-games/snort/strong-grid","kind":"node","lean":"MetaMathlibExt.strongGrid","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#strong-grid","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$ and natural numbers $n_0, \\dots, n_{k-1}$, the $k$-dimensional strong grid $P_{n_0} \\boxtimes \\cdots \\boxtimes P_{n_{k-1}}$ is the simple graph whose vertices are the tuples $(x_0, \\dots, x_{k-1})$ of integers with $0 \\le x_i < n_i$ for each $i$, in which two vertices $x$ and $y$ are adjacent if and only if $x \\ne y$ and $|x_i - y_i| \\le 1$ for every $i$.","title":"Strong grid"},{"id":"GameTheory/combinatorial-games/snort/cartesian-grid","kind":"node","lean":"MetaMathlibExt.cartesianGrid","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#cartesian-grid","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$ and side lengths $n_1, \\dots, n_k \\in \\mathbb{N}$, the $k$-dimensional Cartesian grid has as vertices the tuples $(x_1, \\dots, x_k)$ of integers with $0 \\le x_i < n_i$ for each $i$, and two tuples are adjacent when they differ in exactly one coordinate, and there by exactly one. It is the Cartesian product $P_{n_1} \\square \\cdots \\square P_{n_k}$ of the paths $P_{n_i}$ on $n_i$ vertices.","title":"Cartesian grid"},{"id":"GameTheory/combinatorial-games/snort/king-graph","kind":"node","lean":"MetaMathlibExt.kingGraph","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#king-graph","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the king's graph of the $n \\times m$ board is the simple graph whose vertices are the pairs $(i, j)$ with $0 \\le i < n$ and $0 \\le j < m$, in which two distinct pairs $(i, j)$ and $(i', j')$ are adjacent when $|i - i'| \\le 1$ and $|j - j'| \\le 1$.","title":"King's graph"},{"id":"GameTheory/combinatorial-games/snort/knight-graph","kind":"node","lean":"MetaMathlibExt.knightGraph","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#knight-graph","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the knight's graph of the $n \\times m$ board is the simple graph whose vertices are the pairs $(i, j)$ with $0 \\le i < n$ and $0 \\le j < m$, in which two pairs $(i, j)$ and $(i', j')$ are adjacent when either $|i - i'| = 1$ and $|j - j'| = 2$, or $|i - i'| = 2$ and $|j - j'| = 1$.","title":"Knight's graph"},{"id":"GameTheory/combinatorial-games/snort/bishop-graph","kind":"node","lean":"MetaMathlibExt.bishopGraph","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#bishop-graph","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the bishop's graph on the $n \\times m$ board has as vertices the cells $(i, j)$ with $0 \\le i < n$ and $0 \\le j < m$, and two distinct cells $(i, j)$ and $(i', j')$ are adjacent when they lie on a common diagonal, that is, when $|i - i'| = |j - j'|$.","title":"Bishop's graph"},{"id":"GameTheory/combinatorial-games/snort/queen-graph","kind":"node","lean":"MetaMathlibExt.queenGraph","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#queen-graph","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"For natural numbers $n$ and $m$, the queen's graph of the $n \\times m$ board is the simple graph whose vertices are the pairs $(i, j)$ with $0 \\le i < n$ and $0 \\le j < m$, in which two distinct pairs $(i, j)$ and $(i', j')$ are adjacent when $i = i'$, or $j = j'$, or $|i - i'| = |j - j'|$; that is, when they share a row, a column or a diagonal.","title":"Queen's graph"},{"id":"GameTheory/combinatorial-games/snort/snort-two-vertices","kind":"node","lean":"MetaMathlibExt.snort_two_vertices","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-two-vertices","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"The first player wins Snort on the complete graph $K_2$, and the second player wins Snort on the graph with two vertices and no edges.","title":"Snort on two vertices"},{"id":"GameTheory/combinatorial-games/snort/snort-two-by-two","kind":"node","lean":"MetaMathlibExt.snort_two_by_two","parent":44251,"reading_title":{"format":"inline-parts-v1","parts":[["text","Snort on the "],["math","\\(2 \\times 2\\)"],["text"," queen's and rook's graphs"]],"source":"Snort on the $2 \\times 2$ queen's and rook's graphs"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-two-by-two","scope":"GameTheory/combinatorial-games/snort","status":"fully_proved","summary":"The first player wins Snort on the queen's graph of the $2 \\times 2$ board (the Peaceable Queens game), and the second player wins Snort on the rook's graph of the $2 \\times 2$ board (the Peaceable Rooks game).","title":"Snort on the $2 \\times 2$ queen's and rook's graphs"},{"id":"GameTheory/combinatorial-games/snort/snort-second-player-wins-of-is-opposable","kind":"node","lean":"MetaMathlibExt.snortSecondPlayerWins_of_isOpposable","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-second-player-wins-of-is-opposable","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"If a finite simple graph $G$ is opposable, then the second player wins Snort on $G$.","title":"The second player wins Snort on opposable graphs"},{"id":"GameTheory/combinatorial-games/snort/is-opposable-iff-exists-opposition-match","kind":"node","lean":"MetaMathlibExt.isOpposable_iff_exists_oppositionMatching","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-opposable-iff-exists-opposition-match","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"A graph $G$ is opposable if and only if its complement $\\overline{G}$ has an opposition matching.","title":"Oppositions and opposition matchings"},{"id":"GameTheory/combinatorial-games/snort/is-opposable-of-blocks","kind":"node","lean":"MetaMathlibExt.isOpposable_of_blocks","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-opposable-of-blocks","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"Let $G$ be a graph, let $p$ be a map from its vertices to an index set, with blocks $B_i = \\{v : p(v) = i\\}$, and let $f$ be a map from the vertices of $G$ to themselves. Suppose that for every index $i$ there is an opposition of the subgraph of $G$ induced on $B_i$ that agrees with $f$ on $B_i$, and that for all vertices $u, w$ with $p(u) \\neq p(w)$ we have $uw \\in E(G) \\iff f(u)f(w) \\in E(G)$ and\u2026","title":"Gluing oppositions of blocks"},{"id":"GameTheory/combinatorial-games/snort/is-opposable-formula-product","kind":"node","lean":"MetaMathlibExt.isOpposable_formulaProduct","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-opposable-formula-product","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"Let $G$ and $H$ be graphs, let $\\varphi$ be a Boolean function of four truth values, and let $G \\varphi H$ be the formula product, in which distinct pairs $(g_1, h_1)$ and $(g_2, h_2)$ are adjacent when $\\varphi$ is true at the atoms $g_1g_2 \\in E(G)$, $h_1h_2 \\in E(H)$, $g_1 = g_2$, $h_1 = h_2$. Suppose that for all distinct non-adjacent vertices $g_1, g_2$ of $G$ and every vertex $h$ of $H$, $\\varphi$ is false at\u2026","title":"Formula products with an opposable first factor"},{"id":"GameTheory/combinatorial-games/snort/is-opposable-products","kind":"node","lean":"MetaMathlibExt.isOpposable_products","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-opposable-products","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"If $G$ is an opposable graph, then for every graph $H$ the Cartesian product $G \\mathbin{\\square} H$, the strong product $G \\boxtimes H$, and the tensor, lexicographic, co-normal and homomorphic products of $G$ with $H$ are opposable. Here distinct pairs $(g_1, h_1)$ and $(g_2, h_2)$ are adjacent in the tensor product when $g_1g_2 \\in E(G)$ and $h_1h_2 \\in E(H)$; in the lexicographic product when $g_1g_2 \\in E(G)$,\u2026","title":"Standard products with an opposable first factor"},{"id":"GameTheory/combinatorial-games/snort/snort-second-player-wins-firework","kind":"node","lean":"MetaMathlibExt.snortSecondPlayerWins_firework","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-second-player-wins-firework","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"Let $G$ be a finite simple graph and let $(u, v)$ be a response pair of $G$ whose distance in $G$ is at least $3$ (infinite distance, when $u$ and $v$ lie in different components, is allowed). If the second player wins Snort on $G$, then for every natural number $n$ at least the number $|V(G)|$ of vertices of $G$, the second player wins Snort on the $n$-firework of $G$ at $(u, v)$.","title":"Fireworks preserve second-player wins in Snort"},{"id":"GameTheory/combinatorial-games/snort/exists-large-tree-snort-second-player-wi","kind":"node","lean":"MetaMathlibExt.exists_large_tree_snortSecondPlayerWins","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#exists-large-tree-snort-second-player-wi","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"For every natural number $N$ there is a tree $T$ on $n \\ge N$ vertices such that the second player wins Snort on $T$.","title":"Arbitrarily large trees on which the second player wins Snort"},{"id":"GameTheory/combinatorial-games/snort/snort-first-player-wins-of-is-almost-opp","kind":"node","lean":"MetaMathlibExt.snortFirstPlayerWins_of_isAlmostOpposable","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-first-player-wins-of-is-almost-opp","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"For every finite simple graph $G$ that is almost opposable, the first player wins Snort on $G$.","title":"Almost opposable graphs are first-player wins in Snort"},{"id":"GameTheory/combinatorial-games/snort/is-compatibly-almost-opposable-strong-pr","kind":"node","lean":"MetaMathlibExt.isCompatiblyAlmostOpposable_strongProd","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-compatibly-almost-opposable-strong-pr","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"If $G$ and $H$ are compatibly almost opposable graphs, then their strong product $G \\boxtimes H$ is compatibly almost opposable.","title":"Strong products of compatibly almost opposable graphs"},{"id":"GameTheory/combinatorial-games/snort/is-almost-opposable-strong-prod","kind":"node","lean":"MetaMathlibExt.isAlmostOpposable_strongProd","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-almost-opposable-strong-prod","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"Let $G$ and $H$ be graphs. If $G$ is almost opposable and $H$ has a universal vertex, that is, a vertex adjacent to every other vertex of $H$, then the strong product $G \\boxtimes H$ is almost opposable.","title":"Strong products with a graph that has a universal vertex"},{"id":"GameTheory/combinatorial-games/snort/is-compatibly-almost-opposable-box-prod","kind":"node","lean":"MetaMathlibExt.isCompatiblyAlmostOpposable_boxProd","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-compatibly-almost-opposable-box-prod","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"Let $G$ and $H$ be compatibly almost opposable graphs. Suppose that for some vertex $u$ of $G$ there are an involutive automorphism $\\alpha$ of $G$ and an opposition $f$ of the induced subgraph $G - u$ on the vertices other than $u$ such that $\\alpha(x) = f(x)$ for every vertex $x \\neq u$. Then the Cartesian product $G \\square H$ is compatibly almost opposable.","title":"Cartesian products of compatibly almost opposable graphs"},{"id":"GameTheory/combinatorial-games/snort/is-almost-opposable-strong-grid","kind":"node","lean":"MetaMathlibExt.isAlmostOpposable_strongGrid","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#is-almost-opposable-strong-grid","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"Let $k \\ge 1$ and $n_1, \\dots, n_k \\ge 1$. Then the $k$-dimensional strong grid $P_{n_1} \\boxtimes \\cdots \\boxtimes P_{n_k}$, the strong product of the paths $P_{n_i}$ on $n_i$ vertices, is almost opposable.","title":"Almost opposability of strong grids"},{"id":"GameTheory/combinatorial-games/snort/cartesian-grid-opposability","kind":"node","lean":"MetaMathlibExt.cartesianGrid_opposability","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#cartesian-grid-opposability","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"Let $k \\ge 1$ and $n_1, \\dots, n_k \\ge 1$, and let $G = P_{n_1} \\square \\cdots \\square P_{n_k}$ be the $k$-dimensional Cartesian grid, the Cartesian product of the paths $P_{n_i}$ on $n_i$ vertices. Then $G$ is almost opposable if and only if at most one of $n_1, \\dots, n_k$ is even, and $G$ is opposable if more than one of them is even.","title":"Opposability of Cartesian grids"},{"id":"GameTheory/combinatorial-games/snort/snort-first-player-wins-king-graph","kind":"node","lean":"MetaMathlibExt.snortFirstPlayerWins_kingGraph","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-first-player-wins-king-graph","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $m \\ge 1$, the first player wins Snort on the king's graph of the $n \\times m$ board (the Peaceable Kings game).","title":"First player wins Peaceable Kings"},{"id":"GameTheory/combinatorial-games/snort/snort-second-player-wins-knight-graph-if","kind":"node","lean":"MetaMathlibExt.snortSecondPlayerWins_knightGraph_iff","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-second-player-wins-knight-graph-if","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"For natural numbers $n \\ge 1$ and $m \\ge 1$, the second player wins Snort on the knight's graph of the $n \\times m$ board (the Peaceable Knights game) if and only if $n$ is even or $m$ is even.","title":"Snort on the knight's graph"},{"id":"GameTheory/combinatorial-games/snort/snort-second-player-wins-bishop-graph-if","kind":"node","lean":"MetaMathlibExt.snortSecondPlayerWins_bishopGraph_iff","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-second-player-wins-bishop-graph-if","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"For all natural numbers $n \\ge 1$ and $m \\ge 1$, the second player wins Snort on the bishop's graph of the $n \\times m$ board (the Peaceable Bishops game) if and only if $n$ or $m$ is even.","title":"Second player wins Peaceable Bishops exactly when a side is even"},{"id":"GameTheory/combinatorial-games/snort/snort-second-player-wins-rook-graph-iff","kind":"node","lean":"MetaMathlibExt.snortSecondPlayerWins_rookGraph_iff","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-second-player-wins-rook-graph-iff","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"For natural numbers $n \\ge 1$ and $m \\ge 1$, the second player wins Snort on the rook's graph of the $n \\times m$ board (the Peaceable Rooks game) if and only if $n$ and $m$ are both even.","title":"Snort on the rook's graph"},{"id":"GameTheory/combinatorial-games/snort/snort-first-player-wins-queen-graph","kind":"node","lean":"MetaMathlibExt.snortFirstPlayerWins_queenGraph","parent":44251,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/snort/index.html#snort-first-player-wins-queen-graph","scope":"GameTheory/combinatorial-games/snort","status":"can_prove","summary":"Let $n \\ge 1$ and $m \\ge 1$ be natural numbers such that $n$ or $m$ is odd. Then the first player wins Snort on the queen's graph of the $n \\times m$ board (the Peaceable Queens game).","title":"First player wins Peaceable Queens on boards with an odd side"},{"data_url":"scopes/GameTheory/combinatorial-games/sprague-grundy-theorem.json","id":"GameTheory/combinatorial-games/sprague-grundy-theorem","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","title":"Sprague\u2013Grundy theorem"},{"id":"GameTheory/combinatorial-games/sprague-grundy-theorem/p-positions-unique","kind":"node","lean":"MetaMathlibExt.pPositions_unique","parent":44290,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html#p-positions-unique","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","summary":"Write $x \\to y$ when there is a move from position $x$ to position $y$, and assume there is no infinite sequence of moves $x_0 \\to x_1 \\to x_2 \\to \\cdots$. Call a set $P$ of positions stable if no move leads from a position in $P$ to a position in $P$, and absorbing if from every position outside $P$ some move leads to a position in $P$. Then any two stable and absorbing sets of positions are equal.","title":"Uniqueness of the P-positions of a terminating impartial game"},{"id":"GameTheory/combinatorial-games/sprague-grundy-theorem/impartial-game","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.ImpartialGame","parent":44290,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html#impartial-game","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","summary":"An impartial game consists of a type of positions, a starting position, for each position $p$ a finite set of positions reachable from $p$ in one move, and a rank function assigning a natural number to each position such that every move strictly decreases the rank: if $q$ is reachable from $p$ in one move, then $\\mathrm{rank}(q) < \\mathrm{rank}(p)$.","title":"Impartial game"},{"id":"GameTheory/combinatorial-games/sprague-grundy-theorem/is-p","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.IsP","parent":44290,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html#is-p","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","summary":"In an impartial game $G$, the P-positions are defined by well-founded recursion along moves: a position $p$ is a P-position if and only if no position reachable from $p$ in one move is a P-position. Positions that are not P-positions are called N-positions.","title":"P-positions"},{"id":"GameTheory/combinatorial-games/sprague-grundy-theorem/game-sum","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.gameSum","parent":44290,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html#game-sum","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","summary":"The sum $G + H$ of two impartial games $G$ and $H$ is the impartial game whose positions are the pairs $(a, b)$ of a position $a$ of $G$ and a position $b$ of $H$, and in which a move changes exactly one component: from $(a, b)$ one may move to $(a', b)$ for any move from $a$ to $a'$ in $G$, or to $(a, b')$ for any move from $b$ to $b'$ in $H$. Its starting position is the pair of the starting positions of $G$ and\u2026","title":"Sum of impartial games"},{"id":"GameTheory/combinatorial-games/sprague-grundy-theorem/nim-heap","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.nimHeap","parent":44290,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html#nim-heap","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","summary":"For a natural number $n$, the Nim heap of size $n$ is the impartial game whose positions are $0, 1, \\dots, n$, whose starting position is $n$, and in which a move from $k$ goes to any $j < k$; the rank of a position is its value.","title":"Nim heap"},{"id":"GameTheory/combinatorial-games/sprague-grundy-theorem/equiv","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.Equiv","parent":44290,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html#equiv","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","summary":"Two impartial games $G$ and $H$ are equivalent when no third game tells them apart as summands: for every impartial game $K$, the starting position of the sum $G + K$ is a P-position if and only if the starting position of the sum $H + K$ is a P-position. The starting position of a sum is the pair of the starting positions of its summands.","title":"Equivalence of impartial games"},{"id":"GameTheory/combinatorial-games/sprague-grundy-theorem/is-p-iff","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.isP_iff","parent":44290,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html#is-p-iff","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","summary":"In an impartial game $G$, a position $p$ is a P-position if and only if every position reachable from $p$ in one move is not a P-position.","title":"Recursive characterization of P-positions"},{"id":"GameTheory/combinatorial-games/sprague-grundy-theorem/sprague-grundy","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.sprague_grundy","parent":44290,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sprague-grundy-theorem/index.html#sprague-grundy","scope":"GameTheory/combinatorial-games/sprague-grundy-theorem","status":"fully_proved","summary":"Every impartial game $G$ is equivalent to a Nim heap: there is a natural number $n$ such that $G$ is equivalent to the Nim heap of size $n$.","title":"Sprague\u2013Grundy theorem"},{"data_url":"scopes/GameTheory/combinatorial-games/impartial-pebbling-games.json","id":"GameTheory/combinatorial-games/impartial-pebbling-games","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","title":"Impartial pebbling games"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-step","kind":"node","lean":"MetaMathlibExt.pebblingStep","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\((k+1:k)\\)"],["text","-pebbling move"]],"source":"A $(k+1:k)$-pebbling move"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-step","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Let $a = (a_0, \\dots, a_{n-1})$ be a distribution of pebbles on the vertices $0, \\dots, n-1$, let $k$ be a natural number and let $i$ and $j$ be vertices. The position after the $(k+1:k)$-pebbling move from $i$ to $j$ is the distribution $b$ with $b_j = a_j + k$, $b_i = a_i - (k + 1)$ if $i \\ne j$, and $b_x = a_x$ for every other vertex $x$. When $i \\ne j$ and $a_i \\ge k + 1$, this removes $k + 1$ pebbles from $i$\u2026","title":"A $(k+1:k)$-pebbling move"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/sum-pebbling-step-add-one","kind":"node","lean":"MetaMathlibExt.sum_pebblingStep_add_one","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#sum-pebbling-step-add-one","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Let $n, k \\in \\mathbb{N}$, let $a$ assign $a(x) \\in \\mathbb{N}$ pebbles to each vertex $x \\in \\{0, \\dots, n-1\\}$, and let $i \\ne j$ be vertices with $a(i) \\ge k + 1$. Then the pebbling step from $i$ to $j$, which removes $k + 1$ pebbles from $i$ and adds $k$ pebbles to $j$, gives a position $b$ with $\\sum_x b(x) + 1 = \\sum_x a(x)$.","title":"A pebbling move removes exactly one pebble"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-moves","kind":"node","lean":"MetaMathlibExt.pebblingMoves","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moves of the "],["math","\\((k+1:k)\\)"],["text","-pebbling game"]],"source":"Moves of the $(k+1:k)$-pebbling game"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-moves","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertices $0, \\dots, n-1$, let $k$ be a natural number and let $a = (a_0, \\dots, a_{n-1})$ be a distribution of pebbles on the vertices. This is the finite set of distributions reachable from $a$ by one $(k+1:k)$-pebbling move on $G$: for adjacent vertices $i$ and $j$ with $a_i \\ge k + 1$, remove $k + 1$ pebbles from $i$ and add $k$ pebbles to $j$.","title":"Moves of the $(k+1:k)$-pebbling game"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/mem-pebbling-moves","kind":"node","lean":"MetaMathlibExt.mem_pebblingMoves","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#mem-pebbling-moves","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertices $0, \\dots, n-1$, let $k$ be a natural number, and let $a$ and $b$ be distributions of pebbles on the vertices of $G$. Then $b$ is reachable from $a$ by one $(k+1:k)$-pebbling move on $G$ if and only if there are adjacent vertices $i$ and $j$ of $G$ such that $i$ holds at least $k + 1$ pebbles in $a$ and $b$ is obtained from $a$ by removing $k + 1$ pebbles from $i$ and\u2026","title":"Successors of a pebbling position"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-game","kind":"node","lean":"MetaMathlibExt.pebblingGame","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\((k+1:k)\\)"],["text","-pebbling game"]],"source":"The $(k+1:k)$-pebbling game"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-game","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertices $0, \\dots, n-1$ and let $k$ be a natural number. The impartial two-player $(k+1:k)$-pebbling game $\\Gamma^{(k+1:k)}_G$ has as positions all distributions $a = (a_0, \\dots, a_{n-1})$ of pebbles on the vertices, with $a_i \\in \\mathbb{N}$. A move from $a$ chooses adjacent vertices $i$ and $j$ with $a_i \\ge k + 1$, removes $k + 1$ pebbles from $i$ and adds $k$ pebbles to $j$.\u2026","title":"The $(k+1:k)$-pebbling game"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-game-is-p-of-forall-le","kind":"node","lean":"MetaMathlibExt.pebblingGame_isP_of_forall_le","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positions with at most "],["math","\\(k\\)"],["text"," pebbles on every vertex are P-positions"]],"source":"Positions with at most $k$ pebbles on every vertex are P-positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-game-is-p-of-forall-le","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertices $0, \\dots, n-1$ and let $k$ be a natural number. In the $(k+1:k)$-pebbling game on $G$, every position with at most $k$ pebbles on each vertex is a P-position.","title":"Positions with at most $k$ pebbles on every vertex are P-positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-all-ntotals","kind":"node","lean":"MetaMathlibExt.pebblingAllNTotals","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-all-ntotals","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Let $G$ be a simple graph on the vertices $0, \\dots, n-1$ and let $k$ be a natural number. In the $(k+1:k)$-pebbling game on $G$, a move removes $k+1$ pebbles from a vertex and adds $k$ pebbles to an adjacent vertex. This is the set of natural numbers $m$ such that every position with $m$ pebbles in total (a number of pebbles on each vertex, summing to $m$) is an N-position, that is, not a P-position.","title":"Totals at which every pebbling game is an N-game"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/zero-not-mem-pebbling-all-ntotals","kind":"node","lean":"MetaMathlibExt.zero_not_mem_pebblingAllNTotals","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#zero-not-mem-pebbling-all-ntotals","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"For every simple graph $G$ on the vertex set $\\{0, \\dots, n-1\\}$ and every $k \\in \\mathbb{N}$, the number $0$ does not belong to the set of totals $m$ such that every $(k+1:k)$-pebbling game on $G$ with $m$ pebbles is an N-game. Indeed, the empty position admits no move, so it is a P-game.","title":"The total zero is never an all-N total"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-two-is-n-iff","kind":"node","lean":"MetaMathlibExt.pebbling_completeTwo_isN_iff","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Winning positions of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(K_2\\)"]],"source":"Winning positions of $(2:1)$-pebbling on $K_2$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-two-is-n-iff","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $a = (a_0, a_1)$ be a position of the $(2:1)$-pebbling game on the complete graph $K_2$, with $a_0$ and $a_1$ pebbles on its two vertices. If $a_0 + a_1$ is even, then $a$ is an N-position (that is, not a P-position) if and only if $a_0 \\not\\equiv a_1 \\pmod 3$. If $a_0 + a_1$ is odd, then $a$ is an N-position if and only if $a_0 \\equiv a_1 \\pmod 3$.","title":"Winning positions of $(2:1)$-pebbling on $K_2$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-all-ntotals-complete-two-eq-emp","kind":"node","lean":"MetaMathlibExt.pebblingAllNTotals_completeTwo_eq_empty","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","No all-N threshold for "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(K_2\\)"]],"source":"No all-N threshold for $(2:1)$-pebbling on $K_2$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-all-ntotals-complete-two-eq-emp","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"For every natural number $m$, some $(2:1)$-pebbling game on the complete graph $K_2$ with $m$ pebbles in total is a P-game; that is, $m^{(2:1)}(K_2)$, the least total at which every game is an N-game, does not exist.","title":"No all-N threshold for $(2:1)$-pebbling on $K_2$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/is-least-pebbling-all-ntotals-complete-t","kind":"node","lean":"MetaMathlibExt.isLeast_pebblingAllNTotals_completeThree_one","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","All-N threshold of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(K_3\\)"]],"source":"All-N threshold of $(2:1)$-pebbling on $K_3$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#is-least-pebbling-all-ntotals-complete-t","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"The least number $m$ such that every $(2:1)$-pebbling game on the complete graph $K_3$ with $m$ pebbles in total is an N-game, written $m^{(2:1)}(K_3)$, exists and equals $7$.","title":"All-N threshold of $(2:1)$-pebbling on $K_3$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/is-least-pebbling-all-ntotals-complete-f","kind":"node","lean":"MetaMathlibExt.isLeast_pebblingAllNTotals_completeFour_one","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","All-N threshold of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(K_4\\)"]],"source":"All-N threshold of $(2:1)$-pebbling on $K_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#is-least-pebbling-all-ntotals-complete-f","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"The least number $m$ such that every $(2:1)$-pebbling game on the complete graph $K_4$ with $m$ pebbles in total is an N-game, written $m^{(2:1)}(K_4)$, exists and equals $23$.","title":"All-N threshold of $(2:1)$-pebbling on $K_4$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/is-least-pebbling-all-ntotals-complete-o","kind":"node","lean":"MetaMathlibExt.isLeast_pebblingAllNTotals_complete_odd","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","All-N threshold of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(K_n\\)"],["text"," for odd "],["math","\\(n\\)"]],"source":"All-N threshold of $(2:1)$-pebbling on $K_n$ for odd $n$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#is-least-pebbling-all-ntotals-complete-o","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"For every odd $n \\ge 5$, the least number $m$ such that every $(2:1)$-pebbling game on the complete graph $K_n$ with $m$ pebbles in total is an N-game, written $m^{(2:1)}(K_n)$, exists and equals $n + 2$.","title":"All-N threshold of $(2:1)$-pebbling on $K_n$ for odd $n$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/is-least-pebbling-all-ntotals-complete-e","kind":"node","lean":"MetaMathlibExt.isLeast_pebblingAllNTotals_complete_even","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","All-N threshold of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(K_n\\)"],["text"," for even "],["math","\\(n\\)"]],"source":"All-N threshold of $(2:1)$-pebbling on $K_n$ for even $n$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#is-least-pebbling-all-ntotals-complete-e","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"For every even $n \\ge 6$, the least number $m$ such that every $(2:1)$-pebbling game on the complete graph $K_n$ with $m$ pebbles in total is an N-game, written $m^{(2:1)}(K_n)$, exists and equals $n + 7$.","title":"All-N threshold of $(2:1)$-pebbling on $K_n$ for even $n$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/is-least-pebbling-all-ntotals-complete-t-2","kind":"node","lean":"MetaMathlibExt.isLeast_pebblingAllNTotals_completeThree","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","All-N threshold of "],["math","\\((k+1:k)\\)"],["text","-pebbling on "],["math","\\(K_3\\)"]],"source":"All-N threshold of $(k+1:k)$-pebbling on $K_3$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#is-least-pebbling-all-ntotals-complete-t-2","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"For every positive integer $k$, the least number $m$ such that every $(k+1:k)$-pebbling game on the complete graph $K_3$ with $m$ pebbles in total is an N-game, written $m^{(k+1:k)}(K_3)$, exists and equals $6k + 1$.","title":"All-N threshold of $(k+1:k)$-pebbling on $K_3$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-is-p-two-kadd-th","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_isP_twoKAddThree","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-is-p-two-kadd-th","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a positive integer and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Every position with $2k + 3$ pebbles on one vertex and none on the other two is a P-position.","title":"Pebbling on the triangle: a single pile of 2k+3 pebbles is a P-position"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-is-p-two-ksub-on","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_isP_twoKSubOne","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-is-p-two-ksub-on","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a positive integer and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Let $\\ell_1 \\le \\ell_2$ be natural numbers with $\\ell_1 + \\ell_2 \\le k - 1$. Every position whose three piles are $2k - 1 - \\ell_1 - \\ell_2$, $k - \\ell_1$ and $k - \\ell_2$ in some order is a P-position.","title":"Pebbling on the triangle: a two-parameter family of P-positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-is-p-four-kadd-o","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_isP_fourKAddOne","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-is-p-four-kadd-o","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a positive integer and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Every position whose three piles are $4k + 1$, $k - 1$ and $k - 1$ in some order is a P-position.","title":"Pebbling on the triangle: piles 4k+1, k-1, k-1 form a P-position"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-response-lowers","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_response_lowersMax","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-response-lowers","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a positive integer and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Let $\\ell$ be a natural number with $\\ell \\le 2k - 1$, and let $a$ be a position with $6k - 2\\ell$ pebbles in total in which some pile has at least $k + 1$ pebbles and every pile has at most $3k - \\ell$ pebbles. Then for every\u2026","title":"Pebbling on the triangle: a reply that brings every pile below the bound"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-is-p-of-le","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_isP_of_le","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-is-p-of-le","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a positive integer and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Let $\\ell$ be a natural number with $\\ell \\le 2k - 1$. Every position with $6k - 2\\ell$ pebbles in total in which every pile is at most $3k - \\ell$ is a P-position.","title":"Pebbling on the triangle: total 6k-2\u2113 with no pile above half is a P-position"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-is-n-of-le","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_isN_of_le","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-is-n-of-le","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a positive integer and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Let $\\ell$ be an integer with $1 \\le \\ell \\le 2k - 1$. Every position with $6k - 2\\ell + 1$ pebbles in total in which every pile is at most $4k - \\ell + 1$ and some pile is at most $2k - \\ell$ is an N-position, that is, not a\u2026","title":"Pebbling on the triangle: N-positions with total 6k-2\u2113+1 and a small pile"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-is-n-of-le-four","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_isN_of_le_fourK","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-is-n-of-le-four","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a positive integer and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Every position with $6k - 1$ pebbles in total in which every pile is at most $4k$ is an N-position, that is, not a P-position.","title":"Pebbling on the triangle: total 6k-1 with every pile at most 4k"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-is-n-of-zero","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_isN_of_zero","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-is-n-of-zero","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a natural number and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Let $\\ell$ be an integer with $1 \\le \\ell \\le k - 1$, and let $a_1, a_2$ be natural numbers with $a_1 + a_2 = 6k - 2\\ell + 1$ and $a_2 \\le k$. Every position whose three piles are $a_1$, $a_2$ and $0$ in some order is an N-position,\u2026","title":"Pebbling on the triangle: N-positions with an empty vertex"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-complete-three-is-n-of-ge-four","kind":"node","lean":"MetaMathlibExt.pebbling_completeThree_isN_of_ge_fourKAddOne","parent":44299,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-complete-three-is-n-of-ge-four","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $k$ be a positive integer and consider the $(k+1:k)$-pebbling game on the complete graph $K_3$, where a move removes $k+1$ pebbles from one vertex and adds $k$ pebbles to another. Every position with $6k - 1$ pebbles in total in which some pile is at least $4k + 1$, other than the positions whose three piles are $4k + 1$, $k - 1$ and $k - 1$ in some order, is an N-position, that is, not a P-position.","title":"Pebbling on the triangle: total 6k-1 with a pile of at least 4k+1"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-pattern-aaab","kind":"node","lean":"MetaMathlibExt.C4PatternAAAB","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(X_{aaab}\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The set $X_{aaab}$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-pattern-aaab","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$, with indices read modulo $4$, lies in $X_{aaab}$ when there is an index $i$ such that $a_i \\equiv a_{i+1} \\equiv a_{i+2} \\pmod 3$ and $a_{i-1} \\not\\equiv a_i \\pmod 3$.","title":"The set $X_{aaab}$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-pattern-abab","kind":"node","lean":"MetaMathlibExt.C4PatternABAB","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(X_{abab}\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The set $X_{abab}$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-pattern-abab","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$ lies in $X_{abab}$ when $a_0 \\equiv a_2 \\pmod 3$, $a_1 \\equiv a_3 \\pmod 3$ and $a_0 \\not\\equiv a_1 \\pmod 3$.","title":"The set $X_{abab}$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-pattern-aabc","kind":"node","lean":"MetaMathlibExt.C4PatternAABC","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(X_{aabc}\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The set $X_{aabc}$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-pattern-aabc","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$, with indices read modulo $4$, lies in $X_{aabc}$ when there is an index $i$ with $a_i \\equiv a_{i+1} \\pmod 3$ such that $a_i$, $a_{i+2}$ and $a_{i-1}$ are pairwise incongruent modulo $3$.","title":"The set $X_{aabc}$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-pattern-abac","kind":"node","lean":"MetaMathlibExt.C4PatternABAC","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(X_{abac}\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The set $X_{abac}$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-pattern-abac","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$, with indices read modulo $4$, lies in $X_{abac}$ when there is $i \\in \\{0, 1\\}$ with $a_i \\equiv a_{i+2} \\pmod 3$ such that $a_i$, $a_{i+1}$ and $a_{i-1}$ are pairwise incongruent modulo $3$.","title":"The set $X_{abac}$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-pattern-aabbhat","kind":"node","lean":"MetaMathlibExt.C4PatternAABBHat","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\hat X_{aabb}\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The set $\\hat X_{aabb}$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-pattern-aabbhat","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$, with indices read modulo $4$, lies in $\\hat X_{aabb}$ when there is $i \\in \\{0, 1\\}$ with $a_i \\equiv a_{i+1} \\pmod 3$ and $a_{i+2} \\equiv a_{i-1} \\pmod 3$.","title":"The set $\\hat X_{aabb}$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-mod-four-aabc","kind":"node","lean":"MetaMathlibExt.C4ModFourAABC","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(Y_{aabc}\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The set $Y_{aabc}$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-mod-four-aabc","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$, with indices read modulo $4$, lies in $Y_{aabc}$ when there is an index $i$ with $a_{i+2} \\equiv a_i + 2 \\pmod 4$ and $a_{i+1} \\equiv a_{i-1} + 1 \\pmod 4$.","title":"The set $Y_{aabc}$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-mod-four-abachat","kind":"node","lean":"MetaMathlibExt.C4ModFourABACHat","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\hat Y_{abac}\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The set $\\hat Y_{abac}$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-mod-four-abachat","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$, with indices read modulo $4$, lies in $\\hat Y_{abac}$ when there is an index $i$ with $a_i \\equiv a_{i+2} \\pmod 4$ and $a_{i+1} \\equiv a_{i-1} + 1 \\pmod 4$.","title":"The set $\\hat Y_{abac}$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-mod-four-ababhat","kind":"node","lean":"MetaMathlibExt.C4ModFourABABHat","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(\\hat Y_{abab}\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The set $\\hat Y_{abab}$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-mod-four-ababhat","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$ lies in $\\hat Y_{abab}$ when $a_0 \\equiv a_2 \\pmod 4$ and $a_1 \\equiv a_3 \\pmod 4$.","title":"The set $\\hat Y_{abab}$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-mod-four-ababhat-compl","kind":"node","lean":"MetaMathlibExt.C4ModFourABABHatCompl","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","The complement set "],["math","\\((\\hat Y_{abab})^c\\)"],["text"," of "],["math","\\(C_4\\)"],["text"," positions"]],"source":"The complement set $(\\hat Y_{abab})^c$ of $C_4$ positions"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-mod-four-ababhat-compl","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"A position $a = (a_0, a_1, a_2, a_3)$ of natural-number pebble counts on the vertices $v_0, v_1, v_2, v_3$ of $C_4$ lies in $(\\hat Y_{abab})^c$ when its total $a_0 + a_1 + a_2 + a_3$ is even and $a$ does not lie in $\\hat Y_{abab}$.","title":"The complement set $(\\hat Y_{abab})^c$ of $C_4$ positions"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-pattern-examples","kind":"node","lean":"MetaMathlibExt.c4Pattern_examples","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Examples of the mod-"],["math","\\(3\\)"],["text"," pattern classes on "],["math","\\(C_4\\)"]],"source":"Examples of the mod-$3$ pattern classes on $C_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-pattern-examples","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Writing a distribution of pebbles on the vertices $v_0, v_1, v_2, v_3$ of $C_4$ as $(a_0, a_1, a_2, a_3)$, we have $(0,0,0,0) \\in \\hat X_{aabb}$, $(1,0,0,0) \\in X_{aaab}$, $(0,1,0,1) \\in X_{abab}$, $(0,0,1,2) \\in X_{aabc}$, $(0,1,0,2) \\in X_{abac}$, and $(0,0,0,0) \\notin X_{aaab}$.","title":"Examples of the mod-$3$ pattern classes on $C_4$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-mod-four-examples","kind":"node","lean":"MetaMathlibExt.c4ModFour_examples","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Examples of positions in the sets "],["math","\\(Y\\)"],["text"," for "],["math","\\(C_4\\)"]],"source":"Examples of positions in the sets $Y$ for $C_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-mod-four-examples","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"fully_proved","summary":"Writing $(a_0, a_1, a_2, a_3)$ for the position with $a_j$ pebbles on vertex $v_j$ of $C_4$: the position $(2, 0, 0, 1)$ lies in $Y_{aabc}$, the position $(0, 1, 0, 0)$ lies in $\\hat Y_{abac}$, the position $(0, 0, 0, 0)$ lies in $\\hat Y_{abab}$, and the position $(1, 1, 0, 0)$ lies in $(\\hat Y_{abab})^c$.","title":"Examples of positions in the sets $Y$ for $C_4$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-pattern-invariant-rotational-symmetri","kind":"node","lean":"MetaMathlibExt.c4Pattern_invariant_rotationalSymmetric_reflectional","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Invariance of the mod-"],["math","\\(3\\)"],["text"," pattern classes under symmetric replies on "],["math","\\(C_4\\)"]],"source":"Invariance of the mod-$3$ pattern classes under symmetric replies on $C_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-pattern-invariant-rotational-symmetri","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Consider the $(2:1)$-pebbling game on the cycle $C_4$ with vertices $v_0, v_1, v_2, v_3$, indices read modulo $4$, and let $a = (a_0, a_1, a_2, a_3)$ be a position with $a_i$ pebbles on $v_i$. Let $j$ be an index with $a_j \\ge 2$, let $\\varepsilon \\in \\{1, -1\\}$, and let $b$ be the position obtained from $a$ by the move from $v_j$ to $v_{j+\\varepsilon}$, which removes two pebbles from $v_j$ and adds one pebble to\u2026","title":"Invariance of the mod-$3$ pattern classes under symmetric replies on $C_4$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/c4-pattern-closed-under-moves","kind":"node","lean":"MetaMathlibExt.c4Pattern_closed_under_moves","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closure of the mod-"],["math","\\(3\\)"],["text"," pattern classes under moves on "],["math","\\(C_4\\)"]],"source":"Closure of the mod-$3$ pattern classes under moves on $C_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#c4-pattern-closed-under-moves","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Let $a$ and $b$ be distributions of pebbles on the vertices of the $4$-cycle $C_4$ such that $b$ is obtained from $a$ by one $(2:1)$-pebbling move, that is, by removing two pebbles from a vertex holding at least two pebbles and adding one pebble to an adjacent vertex. If $a \\in \\hat X_{aabb} \\cup X_{abac}$, then $b \\in \\hat X_{aabb} \\cup X_{abac}$; and if $a \\in X_{aaab} \\cup X_{abab} \\cup X_{aabc}$, then\u2026","title":"Closure of the mod-$3$ pattern classes under moves on $C_4$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-cycle-four-is-p-iff-even","kind":"node","lean":"MetaMathlibExt.pebbling_cycleFour_isP_iff_even","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Parity decides "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(C_4\\)"],["text"," in "],["math","\\(\\hat X_{aabb} \\cup X_{abac}\\)"]],"source":"Parity decides $(2:1)$-pebbling on $C_4$ in $\\hat X_{aabb} \\cup X_{abac}$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-cycle-four-is-p-iff-even","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Consider the $(2:1)$-pebbling game on the cycle $C_4$ with vertices $v_0, v_1, v_2, v_3$, where $v_i$ is adjacent to $v_{i \\pm 1}$ (indices mod $4$), and let $a = (a_0, a_1, a_2, a_3)$ be a position with $a_i$ pebbles on $v_i$. Suppose that $a$ lies in $\\hat X_{aabb} \\cup X_{abac}$, where $\\hat X_{aabb}$ means that for some $i \\in \\{0, 1\\}$, $a_i \\equiv a_{i+1}$ and $a_{i+2} \\equiv a_{i-1} \\pmod 3$, and $X_{abac}$\u2026","title":"Parity decides $(2:1)$-pebbling on $C_4$ in $\\hat X_{aabb} \\cup X_{abac}$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-cycle-four-is-p-of-mod-four-aba","kind":"node","lean":"MetaMathlibExt.pebbling_cycleFour_isP_of_modFourABABHat","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positions in "],["math","\\(\\hat Y_{abab}\\)"],["text"," are P-positions of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(C_4\\)"]],"source":"Positions in $\\hat Y_{abab}$ are P-positions of $(2:1)$-pebbling on $C_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-cycle-four-is-p-of-mod-four-aba","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Consider the $(2:1)$-pebbling game on the cycle $C_4$ with vertices $v_0, v_1, v_2, v_3$, where $v_i$ is adjacent to $v_{i \\pm 1}$ (indices mod $4$). Every position $a = (a_0, a_1, a_2, a_3)$, with $a_i$ pebbles on $v_i$, that lies in $\\hat Y_{abab}$, that is, with $a_0 \\equiv a_2$ and $a_1 \\equiv a_3 \\pmod 4$, is a P-position.","title":"Positions in $\\hat Y_{abab}$ are P-positions of $(2:1)$-pebbling on $C_4$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-cycle-four-is-p-of-mod-four-aba-2","kind":"node","lean":"MetaMathlibExt.pebbling_cycleFour_isP_of_modFourABACHat","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positions in "],["math","\\((X_{abab} \\cup X_{aabc} \\cup X_{aaab}) \\cap \\hat Y_{abac}\\)"],["text"," are P-positions of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(C_4\\)"]],"source":"Positions in $(X_{abab} \\cup X_{aabc} \\cup X_{aaab}) \\cap \\hat Y_{abac}$ are P-positions of $(2:1)$-pebbling on $C_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-cycle-four-is-p-of-mod-four-aba-2","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Positions in $(X_{abab} \\cup X_{aabc} \\cup X_{aaab}) \\cap \\hat Y_{abac}$ are P-positions of $(2:1)$-pebbling on $C_4$. Consider the $(2:1)$-pebbling game on the cycle $C_4$ with vertices $v_0, v_1, v_2, v_3$, where $v_i$ is adjacent to $v_{i \\pm 1}$ (indices mod $4$), and let $a = (a_0, a_1, a_2, a_3)$ be a position with $a_i$ pebbles on $v_i$. Suppose that $a$ lies in $X_{abab} \\cup X_{aabc} \\cup X_{aaab}$, where\u2026","title":"Positions in $(X_{abab} \\cup X_{aabc} \\cup X_{aaab}) \\cap \\hat Y_{abac}$ are P-positions of $(2:1)$-pebbling on $C_4$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-cycle-four-is-n-of-mod-four-aab","kind":"node","lean":"MetaMathlibExt.pebbling_cycleFour_isN_of_modFourAABC","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positions in "],["math","\\(Y_{aabc}\\)"],["text"," are N-positions of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(C_4\\)"]],"source":"Positions in $Y_{aabc}$ are N-positions of $(2:1)$-pebbling on $C_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-cycle-four-is-n-of-mod-four-aab","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Consider the $(2:1)$-pebbling game on the cycle $C_4$ with vertices $v_0, v_1, v_2, v_3$, where $v_i$ is adjacent to $v_{i \\pm 1}$ (indices mod $4$). Let $a = (a_0, a_1, a_2, a_3)$ be a position, with $a_i$ pebbles on $v_i$, that lies in $Y_{aabc}$: for some $i$, $a_{i+2} \\equiv a_i + 2$ and $a_{i+1} \\equiv a_{i-1} + 1 \\pmod 4$. Then $a$ is an N-position, that is, not a P-position.","title":"Positions in $Y_{aabc}$ are N-positions of $(2:1)$-pebbling on $C_4$"},{"id":"GameTheory/combinatorial-games/impartial-pebbling-games/pebbling-cycle-four-is-n-of-mod-four-aba","kind":"node","lean":"MetaMathlibExt.pebbling_cycleFour_isN_of_modFourABABHatCompl","parent":44299,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positions in "],["math","\\((X_{abab} \\cup X_{aabc} \\cup X_{aaab}) \\cap (\\hat Y_{abab})^c\\)"],["text"," are N-positions of "],["math","\\((2:1)\\)"],["text","-pebbling on "],["math","\\(C_4\\)"]],"source":"Positions in $(X_{abab} \\cup X_{aabc} \\cup X_{aaab}) \\cap (\\hat Y_{abab})^c$ are N-positions of $(2:1)$-pebbling on $C_4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-pebbling-games/index.html#pebbling-cycle-four-is-n-of-mod-four-aba","scope":"GameTheory/combinatorial-games/impartial-pebbling-games","status":"can_prove","summary":"Positions in $(X_{abab} \\cup X_{aabc} \\cup X_{aaab}) \\cap (\\hat Y_{abab})^c$ are N-positions of $(2:1)$-pebbling on $C_4$. Consider the $(2:1)$-pebbling game on the cycle $C_4$ with vertices $v_0, v_1, v_2, v_3$, where $v_i$ is adjacent to $v_{i \\pm 1}$ (indices mod $4$), and let $a = (a_0, a_1, a_2, a_3)$ be a position with $a_i$ pebbles on $v_i$. Suppose that $a$ lies in $X_{abab} \\cup X_{aabc} \\cup X_{aaab}$,\u2026","title":"Positions in $(X_{abab} \\cup X_{aabc} \\cup X_{aaab}) \\cap (\\hat Y_{abab})^c$ are N-positions of $(2:1)$-pebbling on $C_4$"},{"data_url":"scopes/GameTheory/combinatorial-games/impartial-placement-games.json","id":"GameTheory/combinatorial-games/impartial-placement-games","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","title":"Impartial placement games"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/placement-terminal","kind":"node","lean":"MetaMathlibExt.placementTerminal","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#placement-terminal","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of cells and let $E$ be a predicate on subsets of $\\alpha$, the condition under which play has ended. A position $S \\subseteq \\alpha$ (the set of occupied cells) is terminal when $E(S)$ holds or $S = \\alpha$: play has ended, or every cell is occupied. These are the positions with no options.","title":"Terminal positions of a placement game"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/placement-ult","kind":"node","lean":"MetaMathlibExt.placementUlt","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#placement-ult","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of cells and let $E$ be a predicate on subsets of $\\alpha$, the condition under which play has ended. A position $S \\subseteq \\alpha$ is an ult when it is not terminal but has a terminal option: $S$ is not terminal, and there is a cell $c \\notin S$ such that $S \\cup \\{c\\}$ is terminal.","title":"Ults of a placement game"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/placement-penult","kind":"node","lean":"MetaMathlibExt.placementPenult","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#placement-penult","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of cells and let $E$ be a predicate on subsets of $\\alpha$, the condition under which play has ended. A position $S \\subseteq \\alpha$ is a penult when it is not terminal and all of its options are ults: $S$ is not terminal, and $S \\cup \\{c\\}$ is an ult for every cell $c \\notin S$.","title":"Penults of a placement game"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/placement-game","kind":"node","lean":"MetaMathlibExt.placementGame","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#placement-game","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of cells and let $E$ be a predicate on subsets of $\\alpha$, the condition under which play has ended. The placement game is the impartial game whose positions are the subsets $S \\subseteq \\alpha$ (the occupied cells), starting from the empty board $\\emptyset$. If $E(S)$ holds there are no moves from $S$; otherwise the moves from $S$ lead to the positions $S \\cup \\{c\\}$ with\u2026","title":"The placement game"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/mem-placement-game-moves","kind":"node","lean":"MetaMathlibExt.mem_placementGame_moves","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#mem-placement-game-moves","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of cells and $E$ a predicate on subsets of $\\alpha$ saying when play has ended. In the placement game on $\\alpha$ with end predicate $E$, a position $T$ is a move from a position $S$ if and only if $E(S)$ fails and $T = S \\cup \\{c\\}$ for some cell $c \\notin S$.","title":"Moves of a placement game"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/placement-first-player-wins","kind":"node","lean":"MetaMathlibExt.placementFirstPlayerWins","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#placement-first-player-wins","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of cells and $E$ a predicate on subsets of $\\alpha$ saying when play has ended. The player to move from a position $S \\subseteq \\alpha$ wins the placement game on $\\alpha$ with end predicate $E$ when $S$ is not a P-position of that game under normal play, in which the player unable to move loses.","title":"First-player win in a placement game"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/placement-first-player-wins-iff","kind":"node","lean":"MetaMathlibExt.placementFirstPlayerWins_iff","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#placement-first-player-wins-iff","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of cells and $E$ a predicate on subsets of $\\alpha$ saying when play has ended. In the placement game on $\\alpha$ with end predicate $E$, the player to move from a position $S \\subseteq \\alpha$ wins if and only if $E(S)$ fails and there is a cell $c \\notin S$ such that the player to move from $S \\cup \\{c\\}$ does not win.","title":"Normal-play recursion for placement games"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/placement-terminal-univ","kind":"node","lean":"MetaMathlibExt.placementTerminal_univ","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#placement-terminal-univ","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Let $\\alpha$ be a finite set of cells. For every predicate $E$ on subsets of $\\alpha$, the full board $\\alpha$ is a terminal position of the placement game with ending condition $E$.","title":"The full board is terminal"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/tic-ended","kind":"node","lean":"MetaMathlibExt.ticEnded","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#tic-ended","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Write the cells of the $n \\times n$ board as pairs $(r, c)$ (row, column) in $\\{0, \\dots, n-1\\}^2$. Impartial Tic has ended at a set $S$ of occupied cells when some row or some column is full: there is a row $r$ with $(r, c) \\in S$ for every column $c$, or there is a column $c$ with $(r, c) \\in S$ for every row $r$.","title":"End of Impartial Tic"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/not-tic-ended-empty","kind":"node","lean":"MetaMathlibExt.not_ticEnded_empty","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#not-tic-ended-empty","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"For $n \\ge 1$, play of Impartial Tic has not ended on the empty $n \\times n$ board: no row and no column is full.","title":"Impartial Tic has not ended on the empty board"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/tak-adjacent","kind":"node","lean":"MetaMathlibExt.takAdjacent","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#tak-adjacent","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Two cells $p = (p_1, p_2)$ and $q = (q_1, q_2)$ of the $n \\times n$ board $\\{0, \\dots, n-1\\}^2$, written as (row, column), are orthogonally adjacent when they lie in the same row and their columns differ by $1$ ($p_1 = q_1$ and $|p_2 - q_2| = 1$), or they lie in the same column and their rows differ by $1$ ($p_2 = q_2$ and $|p_1 - q_1| = 1$).","title":"Orthogonal adjacency"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/tak-has-road","kind":"node","lean":"MetaMathlibExt.takHasRoad","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#tak-has-road","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"Write the cells of the $n \\times n$ board as pairs (row, column) in $\\{0, \\dots, n-1\\}^2$. Impartial Tak has ended at a set $S$ of occupied cells when the tokens contain an orthogonal path from an edge of the board to the opposite edge: there are a token $a \\in S$ and a cell $b$ joined by a sequence $a = p_0, p_1, \\dots, p_k = b$ with $k \\ge 0$, in which each $p_{i+1}$ is a token in $S$ orthogonally adjacent to\u2026","title":"Roads in Impartial Tak"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/not-tak-has-road-empty","kind":"node","lean":"MetaMathlibExt.not_takHasRoad_empty","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#not-tak-has-road-empty","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"For every $n$, the empty $n \\times n$ board contains no road of Impartial Tak.","title":"The empty board has no Tak road"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/weighted-cross-weight","kind":"node","lean":"MetaMathlibExt.weightedCrossWeight","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#weighted-cross-weight","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"For natural numbers $a$ and $b$, the weight $w(a, b)$ is $0$ if $a, b \\in \\{0, 4\\}$; otherwise it is $2$ if $a = 2$ and $1 \\le b \\le 3$, or $b = 2$ and $1 \\le a \\le 3$; and it is $1$ in all other cases. On the $5 \\times 5$ window $0 \\le a, b \\le 4$ this puts weight $0$ at the four corners, $2$ on the central plus (the centre $(2, 2)$ and its four orthogonal neighbours) and $1$ elsewhere.","title":"Weights of the Weighted Cross"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/sum-weighted-cross-weight","kind":"node","lean":"MetaMathlibExt.sum_weightedCrossWeight","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#sum-weighted-cross-weight","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"fully_proved","summary":"The weights $w(a, b)$ of the Weighted Cross over the $5 \\times 5$ window sum to $26$: $\\sum_{a=0}^{4} \\sum_{b=0}^{4} w(a, b) = 26$.","title":"Total weight of the Weighted Cross"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tic-first-player-wins-iff-odd","kind":"node","lean":"MetaMathlibExt.impartialTic_firstPlayerWins_iff_odd","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tic-first-player-wins-iff-odd","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"For every natural number $n$, under normal play the first player wins Impartial Tic on the $n \\times n$ board, starting from the empty board, if and only if $n$ is odd; for even $n$ the second player wins.","title":"The first player wins Impartial Tic exactly on odd boards"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-not-first-player-wins-of-e","kind":"node","lean":"MetaMathlibExt.impartialTak_not_firstPlayerWins_of_even","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-not-first-player-wins-of-e","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the impartial placement game on the $n \\times n$ board in which the players alternately place a token on an empty cell, play ends once the tokens contain a road, and the player unable to move loses. If $n$ is even, the first player does not win from the empty $n \\times n$ board, that is, the second player wins.","title":"The second player wins Impartial Tak on even boards"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-five-first-player-wins","kind":"node","lean":"MetaMathlibExt.impartialTak_five_firstPlayerWins","parent":44342,"reading_title":{"format":"inline-parts-v1","parts":[["text","The first player wins Impartial Tak on the "],["math","\\(5 \\times 5\\)"],["text"," board"]],"source":"The first player wins Impartial Tak on the $5 \\times 5$ board"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-five-first-player-wins","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the impartial placement game on the $n \\times n$ board in which the players alternately place a token on an empty cell, play ends once the tokens contain a road, and the player unable to move loses. On the $5 \\times 5$ board the first player wins from the empty board.","title":"The first player wins Impartial Tak on the $5 \\times 5$ board"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tic-dual-penult-card-le","kind":"node","lean":"MetaMathlibExt.impartialTic_dual_penult_card_le","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tic-dual-penult-card-le","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Let $n \\ge 5$. If $S$ is a penult of Impartial Tic on the $n \\times n$ board, then at most $4(n - 2)$ cells of the board lie outside $S$.","title":"Token bound for dual Impartial Tic penults"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tic-dual-penult-card-le-of-fou","kind":"node","lean":"MetaMathlibExt.impartialTic_dual_penult_card_le_of_four","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tic-dual-penult-card-le-of-fou","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"If $S$ is a penult of Impartial Tic on the $4 \\times 4$ board, then at most $9$ cells of the board lie outside $S$.","title":"Dual Impartial Tic penults on the 4 \u00d7 4 board have at most 9 tokens"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tic-exists-dual-penult-card","kind":"node","lean":"MetaMathlibExt.impartialTic_exists_dual_penult_card","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tic-exists-dual-penult-card","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Let $n \\ge 5$ and let $x$ be a natural number with $2n \\le x \\le 4(n - 2)$. Then there is a penult $S$ of Impartial Tic on the $n \\times n$ board such that exactly $x$ cells of the board lie outside $S$.","title":"Token counts of dual Impartial Tic penults"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-exists-penult-card","kind":"node","lean":"MetaMathlibExt.impartialTak_exists_penult_card","parent":44342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Impartial Tak penults of every size from "],["math","\\(n^2 - 4(n - 2) - 2\\)"],["text"," to "],["math","\\(n^2 - 2n\\)"]],"source":"Impartial Tak penults of every size from $n^2 - 4(n - 2) - 2$ to $n^2 - 2n$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-exists-penult-card","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the placement game on the $n \\times n$ board whose play ends once the tokens contain a road. Let $n \\ge 4$ and let $x$ be a natural number with $n^2 - 4(n - 2) - 2 \\le x \\le n^2 - 2n$. Then there is an Impartial Tak penult on the $n \\times n$ board with exactly $x$ tokens.","title":"Impartial Tak penults of every size from $n^2 - 4(n - 2) - 2$ to $n^2 - 2n$"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-is-greatest-penult-card","kind":"node","lean":"MetaMathlibExt.impartialTak_isGreatest_penult_card","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-is-greatest-penult-card","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the placement game on the $n \\times n$ board whose play ends once the tokens contain a road. For every $n \\ge 2$, the largest number of tokens in an Impartial Tak penult on the $n \\times n$ board is $U(n) = n^2 - 2n$: some penult has exactly $n^2 - 2n$ tokens and no penult has more.","title":"The largest Impartial Tak penult"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-exists-snake-penult","kind":"node","lean":"MetaMathlibExt.impartialTak_exists_snake_penult","parent":44342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Impartial Tak penults with about "],["math","\\(n^2/3\\)"],["text"," tokens"]],"source":"Impartial Tak penults with about $n^2/3$ tokens"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-exists-snake-penult","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the placement game on the $n \\times n$ board whose play ends once the tokens contain a road. For every $n \\ge 6$ there is an Impartial Tak penult $S$ on the $n \\times n$ board whose number of tokens $|S|$ satisfies $3|S| + 8 = n^2 + 4n$ if $n \\equiv 1$ or $4 \\pmod 6$, $3|S| + 12 = n^2 + 4n$ if $n \\equiv 2$ or $3 \\pmod 6$, $3|S| + 16 = n^2 + 6n$ if $n \\equiv 5 \\pmod 6$, and $3|S| + 18 = n^2 + 6n$ if\u2026","title":"Impartial Tak penults with about $n^2/3$ tokens"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-penult-cross-nonempty","kind":"node","lean":"MetaMathlibExt.impartialTak_penult_cross_nonempty","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-penult-cross-nonempty","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the placement game on the $n \\times n$ board whose play ends once the tokens contain a road. Let $S$ be an Impartial Tak penult on the $n \\times n$ board and let $c$ be an interior cell, that is, a cell in neither the first nor the last row and in neither the first nor the last column. Then $S$ contains $c$ or a cell orthogonally adjacent to $c$: every Cross (an interior cell together with its four\u2026","title":"Every Cross meets an Impartial Tak penult"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-penult-card-ge-fifth","kind":"node","lean":"MetaMathlibExt.impartialTak_penult_card_ge_fifth","parent":44342,"reading_title":{"format":"inline-parts-v1","parts":[["text","An "],["math","\\(n^2/5\\)"],["text"," lower bound for Impartial Tak penults"]],"source":"An $n^2/5$ lower bound for Impartial Tak penults"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-penult-card-ge-fifth","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the placement game on the $n \\times n$ board whose play ends once the tokens contain a road. There is a real constant $C$ such that for every natural number $n$, every Impartial Tak penult on the $n \\times n$ board has at least $n^2/5 - Cn$ tokens; that is, every penult has at least $n^2/5 + O(n)$ tokens.","title":"An $n^2/5$ lower bound for Impartial Tak penults"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-penult-thick-cross","kind":"node","lean":"MetaMathlibExt.impartialTak_penult_thickCross","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-penult-thick-cross","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the placement game on the $n \\times n$ board whose play ends once the tokens contain a road. Index the rows and columns by $0, \\dots, n - 1$, let $S$ be an Impartial Tak penult on the $n \\times n$ board, and let $r, c$ be natural numbers with $r + 4 \\le n$ and $c + 4 \\le n$. The Thick Cross at $(r, c)$ is the $4 \\times 4$ window of cells in rows $r, \\dots, r + 3$ and columns $c, \\dots, c + 3$ with\u2026","title":"Every Thick Cross holds three tokens of an Impartial Tak penult"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-penult-card-ge-quarter","kind":"node","lean":"MetaMathlibExt.impartialTak_penult_card_ge_quarter","parent":44342,"reading_title":{"format":"inline-parts-v1","parts":[["text","An "],["math","\\(n^2/4\\)"],["text"," lower bound for Impartial Tak penults"]],"source":"An $n^2/4$ lower bound for Impartial Tak penults"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-penult-card-ge-quarter","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the placement game on the $n \\times n$ board whose play ends once the tokens contain a road. There is a real constant $C$ such that for every natural number $n$, every Impartial Tak penult on the $n \\times n$ board has at least $n^2/4 - Cn$ tokens; that is, every penult has at least $n^2/4 + O(n)$ tokens.","title":"An $n^2/4$ lower bound for Impartial Tak penults"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-penult-weighted-cross","kind":"node","lean":"MetaMathlibExt.impartialTak_penult_weightedCross","parent":44342,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-penult-weighted-cross","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Let $S$ be a penult of Impartial Tak on the $n \\times n$ board, with rows and columns numbered from $0$. For all natural numbers $r, c$ with $r + 5 \\le n$ and $c + 5 \\le n$, consider the $5 \\times 5$ window of rows $r, \\dots, r + 4$ and columns $c, \\dots, c + 4$, and give each token of $S$ in the window, lying in row $i$ and column $j$ of the board, the Weighted Cross weight of position $(i - r, j - c)$. Then the\u2026","title":"Weighted Cross bound for Impartial Tak penults"},{"id":"GameTheory/combinatorial-games/impartial-placement-games/impartial-tak-penult-card-ge-weighted-cr","kind":"node","lean":"MetaMathlibExt.impartialTak_penult_card_ge_weightedCross","parent":44342,"reading_title":{"format":"inline-parts-v1","parts":[["text","A "],["math","\\(\\frac{7}{26}(n - 4)^2\\)"],["text"," lower bound for Impartial Tak penults"]],"source":"A $\\frac{7}{26}(n - 4)^2$ lower bound for Impartial Tak penults"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/impartial-placement-games/index.html#impartial-tak-penult-card-ge-weighted-cr","scope":"GameTheory/combinatorial-games/impartial-placement-games","status":"can_prove","summary":"Impartial Tak is the placement game on the $n \\times n$ board whose play ends once the tokens contain a road. For every $n \\ge 5$, every Impartial Tak penult $S$ on the $n \\times n$ board satisfies $7(n - 4)^2 \\le 26|S|$, that is, it has at least $\\frac{7}{26}(n - 4)^2$ tokens.","title":"A $\\frac{7}{26}(n - 4)^2$ lower bound for Impartial Tak penults"},{"data_url":"scopes/GameTheory/combinatorial-games/maximum-nim.json","id":"GameTheory/combinatorial-games/maximum-nim","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","title":"Maximum Nim"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-mex","kind":"node","lean":"MetaMathlibExt.maximumNimMex","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-mex","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For a finite set $s$ of natural numbers, $\\operatorname{mex}(s)$ is the least natural number that does not belong to $s$.","title":"Minimum excludant of a finite set"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-mex-eq-iff","kind":"node","lean":"MetaMathlibExt.maximumNimMex_eq_iff","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-mex-eq-iff","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"Let $s$ be a finite set of natural numbers and $m$ a natural number. Then $\\operatorname{mex}(s)$, the least natural number not in $s$, equals $m$ if and only if $m \\notin s$ and every natural number $n < m$ lies in $s$.","title":"Characterization of the minimum excludant"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For a rule function $f : \\mathbb{N} \\to \\mathbb{N}$, the Grundy number $G_f(x)$ of a pile of $x$ stones in Maximum Nim is defined by recursion on $x$ as the mex (least natural number not in the set) of $\\{G_f(x - u) : 1 \\le u \\le \\min(f(x), x)\\}$, the Grundy numbers of the positions reachable from $x$ by removing $u$ stones.","title":"Grundy numbers of Maximum Nim"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-eq","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_eq","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-eq","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For every rule function $f : \\mathbb{N} \\to \\mathbb{N}$ and every pile size $x \\in \\mathbb{N}$, the Grundy number $G_f(x)$ of a pile of $x$ stones in Maximum Nim with rule function $f$ equals the mex (least natural number not in the set) of $\\{G_f(x - u) : 1 \\le u \\le \\min(f(x), x)\\}$.","title":"Recursive formula for Maximum Nim Grundy numbers"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-div-two-five","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_div_two_five","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-div-two-five","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"In Maximum Nim with rule function $f(x) = \\lfloor x/2 \\rfloor$, a pile of $5$ stones has Grundy number $1$.","title":"Grundy number of a pile of five in halving Maximum Nim"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-div-two-four","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_div_two_four","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-div-two-four","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"In Maximum Nim with rule function $f(x) = \\lfloor x/2 \\rfloor$, a pile of $4$ stones has Grundy number $2$.","title":"Grundy number of a pile of four in halving Maximum Nim"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-div-two-three","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_div_two_three","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-div-two-three","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"In Maximum Nim with rule function $f(x) = \\lfloor x/2 \\rfloor$, a pile of $3$ stones has Grundy number $0$.","title":"Grundy number of a pile of three in halving Maximum Nim"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-game","kind":"node","lean":"MetaMathlibExt.maximumNimGame","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-game","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For a rule function $f : \\mathbb{N} \\to \\mathbb{N}$, the Maximum Nim game with rule $f$ is the impartial game whose positions are the pile sizes $x \\in \\mathbb{N}$, in which the moves from $x$ go to $x - u$ for each $u$ with $1 \\le u \\le \\min(f(x), x)$. The rank of a position is the pile size itself, and the starting position is the empty pile $0$.","title":"The Maximum Nim game"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-game-is-p-iff","kind":"node","lean":"MetaMathlibExt.maximumNimGame_isP_iff","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-game-is-p-iff","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ be a rule function and let $x$ be a pile size. The position $x$ of the Maximum Nim game with rule $f$ is a P-position if and only if, for every $u$ with $1 \\le u \\le \\min(f(x), x)$, the position $x - u$ is not a P-position.","title":"P-positions of Maximum Nim"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-h","kind":"node","lean":"MetaMathlibExt.maximumNimH","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Maximum Nim map "],["math","\\(h_k\\)"]],"source":"The Maximum Nim map $h_k$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-h","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For natural numbers $k$ and $x$, let $h_k(x) = x + \\lfloor x/(k-1) \\rfloor + 1$. The paper takes $k \\ge 2$; for $k \\le 1$ the definition gives $h_k(x) = x + 1$, because $k - 1$ is truncated to $0$ and $\\lfloor x/0 \\rfloor$ is taken to be $0$.","title":"The Maximum Nim map $h_k$"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-h-three-zero","kind":"node","lean":"MetaMathlibExt.maximumNimH_three_zero","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(h_3(0) = 1\\)"],["text"," of the Maximum Nim map"]],"source":"The value $h_3(0) = 1$ of the Maximum Nim map"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-h-three-zero","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For the map $h_k(x) = x + \\lfloor x/(k-1) \\rfloor + 1$ on the natural numbers, we have $h_3(0) = 1$.","title":"The value $h_3(0) = 1$ of the Maximum Nim map"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-h-three-two","kind":"node","lean":"MetaMathlibExt.maximumNimH_three_two","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","The value "],["math","\\(h_3(2) = 4\\)"],["text"," of the Maximum Nim map"]],"source":"The value $h_3(2) = 4$ of the Maximum Nim map"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-h-three-two","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For the map $h_k(x) = x + \\lfloor x/(k-1) \\rfloor + 1$ on the natural numbers, we have $h_3(2) = 4$.","title":"The value $h_3(2) = 4$ of the Maximum Nim map"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-elimination-order-aux","kind":"node","lean":"MetaMathlibExt.josephusEliminationOrderAux","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-elimination-order-aux","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For natural numbers $k$ and $s$ (the starting position), a fuel $f$ and a list $c$ of natural numbers (the current circle), the simulation returns the empty list when $f = 0$ or $c$ is empty. Otherwise, with $L$ the length of $c$ and $i = (s + k - 1) \\bmod L$, where the subtraction is truncated at $0$, it returns the entry of $c$ at zero-based position $i$ followed by the result of the simulation with the same $k$,\u2026","title":"Fuel-bounded Josephus simulation"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-elimination-order","kind":"node","lean":"MetaMathlibExt.josephusEliminationOrder","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-elimination-order","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For natural numbers $k$ and $n$, place $1, \\dots, n$ in order on a circle and repeatedly remove one entry. When $L$ entries remain and the current position is $s$ (zero-based, initially $s = 0$), the entry at zero-based position $i = (s + k - 1) \\bmod L$ is removed, where the subtraction is truncated at $0$, and $i$ becomes the current position, so the next count starts at the entry after the removed one. The\u2026","title":"Josephus elimination order"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-elimination-order-three-ten","kind":"node","lean":"MetaMathlibExt.josephusEliminationOrder_three_ten","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-elimination-order-three-ten","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"The Josephus elimination order with step $3$ on the circle $1, \\dots, 10$ is $3, 6, 9, 2, 7, 1, 8, 5, 10, 4$: the entries are removed in the order $3, 6, 9, 2, 7, 1, 8, 5, 10$, and the survivor $4$ comes last.","title":"Josephus elimination order for ten people and step three"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-elimination-order-two-three","kind":"node","lean":"MetaMathlibExt.josephusEliminationOrder_two_three","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-elimination-order-two-three","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"The Josephus elimination order with step $2$ on the circle $1, 2, 3$ is $2, 1, 3$: the entry $2$ is removed first, then the entry $1$, and the survivor $3$ comes last.","title":"Josephus elimination order for three people and step two"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-elimination-order-length","kind":"node","lean":"MetaMathlibExt.josephusEliminationOrder_length","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-elimination-order-length","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For all natural numbers $k$ and $n$, the Josephus elimination order with step $k$ on the circle $1, \\dots, n$ has exactly $n$ entries.","title":"Length of the Josephus elimination order"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-elimination-order-perm","kind":"node","lean":"MetaMathlibExt.josephusEliminationOrder_perm","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-elimination-order-perm","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For all natural numbers $k$ and $n$, the Josephus elimination order with step $k$ on the circle $1, \\dots, n$ is a permutation of the list $1, 2, \\dots, n$; in particular, each of $1, \\dots, n$ occurs in it exactly once.","title":"The Josephus elimination order is a permutation of the circle"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-elimination-order-get-last","kind":"node","lean":"MetaMathlibExt.josephusEliminationOrder_getLast?","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-elimination-order-get-last","scope":"GameTheory/combinatorial-games/maximum-nim","status":"fully_proved","summary":"For natural numbers $k \\ge 1$ and $n \\ge 1$, the Josephus elimination order with step $k$ on the circle $1, \\dots, n$ is nonempty and its last entry is $J_k(n)$, the value at $n$ of the Josephus function for elimination step $k$.","title":"The survivor is the last entry of the Josephus elimination order"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-eq-zero-iff-is-p","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_eq_zero_iff_isP","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-eq-zero-iff-is-p","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(0) = 0$ and $f(m-1) \\le f(m) \\le f(m-1) + 1$ for every $m \\ge 1$. Then for every pile size $x \\in \\mathbb{N}$, the Grundy number of $x$ in Maximum Nim with rule function $f$ is $0$ if and only if $x$ is a P-position (a previous-player win under normal play) of that game.","title":"Grundy value zero characterizes P-positions in Maximum Nim"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-eq-rule-of-lt","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_eq_rule_of_lt","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-eq-rule-of-lt","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(0) = 0$ and $f(m-1) \\le f(m) \\le f(m-1) + 1$ for every $m \\ge 1$, and let $G$ be the Grundy number of Maximum Nim with rule function $f$. If $x \\ge 1$ and $f(x-1) < f(x)$, then $G(x) = f(x)$.","title":"Grundy number where the rule function increases"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-eq-sub-rule-succ","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_eq_sub_rule_succ","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-eq-sub-rule-succ","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $f : \\mathbb{N} \\to \\mathbb{N}$ satisfy $f(0) = 0$ and $f(m-1) \\le f(m) \\le f(m-1) + 1$ for every $m \\ge 1$, and let $G$ be the Grundy number of Maximum Nim with rule function $f$. If $x \\ge 1$ and $f(x) = f(x-1)$, then $G(x) = G(x - f(x) - 1)$.","title":"Grundy number where the rule function is flat"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-sub-div-sub-one-eq","kind":"node","lean":"MetaMathlibExt.maximumNim_sub_div_sub_one_eq","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-sub-div-sub-one-eq","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ and $x$ be natural numbers such that $x$ is not a multiple of $k$. Then $x - \\lfloor x/k \\rfloor - 1 = \\lfloor (k-1)x/k \\rfloor$.","title":"Floor identity for non-multiples in Maximum Nim"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-div-eq-of-dvd","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_div_eq_of_dvd","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-div-eq-of-dvd","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ and let $G$ be the Grundy number of Maximum Nim with rule function $f(y) = \\lfloor y/k \\rfloor$. Every positive multiple $x$ of $k$ satisfies $G(x) = x/k$.","title":"Maximum Nim Grundy numbers at multiples of the divisor"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-eq-f-of-not-dvd","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_eq_F_of_not_dvd","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-eq-f-of-not-dvd","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ and let $G$ be the Grundy number of Maximum Nim with rule function $f(y) = \\lfloor y/k \\rfloor$. If $x$ is a natural number not divisible by $k$, then $G(x) = G(\\lfloor (k-1)x/k \\rfloor)$.","title":"Maximum Nim Grundy numbers at non-multiples of the divisor"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-sub-eq-zero","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_sub_eq_zero","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-sub-eq-zero","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"For every natural number $k \\ge 2$ and every natural number $j$ with $1 \\le j \\le k$, a pile of $k - j$ stones has Grundy number $0$ in Maximum Nim with rule function $f(y) = \\lfloor y/k \\rfloor$.","title":"Floor-rule Maximum Nim: positions below k have Grundy number zero"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-not-dvd-mul-sub-of-lt","kind":"node","lean":"MetaMathlibExt.maximumNim_not_dvd_mul_sub_of_lt","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","A Maximum Nim non-divisibility lemma for "],["math","\\(s < t\\)"]],"source":"A Maximum Nim non-divisibility lemma for $s < t$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-not-dvd-mul-sub-of-lt","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k, n, t, h, s, i$ be natural numbers with $h \\ge 1$, $k = nt + h$, $h \\le n$ and $s < t$. If $1 \\le i \\le n$, then $k$ does not divide $n(k - s) - i$.","title":"A Maximum Nim non-divisibility lemma for $s < t$"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-not-dvd-mul-sub-of-ne","kind":"node","lean":"MetaMathlibExt.maximumNim_not_dvd_mul_sub_of_ne","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","A Maximum Nim non-divisibility lemma for "],["math","\\(i \\ne h\\)"]],"source":"A Maximum Nim non-divisibility lemma for $i \\ne h$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-not-dvd-mul-sub-of-ne","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k, n, t, h, i$ be natural numbers with $t \\ge 1$, $h \\ge 1$, $k = nt + h$ and $h \\le n$. If $1 \\le i \\le n$ and $i \\ne h$, then $k$ does not divide $n(k - t) - i$.","title":"A Maximum Nim non-divisibility lemma for $i \\ne h$"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-mul-sub-eq-decomposit","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_mul_sub_eq_decomposition","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","Floor-rule Maximum Nim: Grundy reduction when "],["math","\\(n = kt + u\\)"]],"source":"Floor-rule Maximum Nim: Grundy reduction when $n = kt + u$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-mul-sub-eq-decomposit","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k$, $t$, $u$ and $n$ be natural numbers with $k \\ge 2$, $t \\ge 1$, $u \\le k - 1$ and $n = kt + u$. Write $G(x)$ for the Grundy number of a pile of $x$ stones in Maximum Nim with rule function $f(y) = \\lfloor y/k \\rfloor$. Then for every natural number $i$ with $1 \\le i \\le n$ that is not divisible by $k$,","title":"Floor-rule Maximum Nim: Grundy reduction when $n = kt + u$"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-mul-sub-eq-f-eq-pred","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_mul_sub_eq_F_eq_pred","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-mul-sub-eq-f-eq-pred","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $n$, $t$, $h$ be natural numbers with $t \\ge 1$ and $1 \\le h \\le n$, and let $k = nt + h$. Write $G(x)$ for the Grundy number of a pile of $x$ stones in Maximum Nim with rule function $f(y) = \\lfloor y/k \\rfloor$, and let $F(x) = \\lfloor (k-1)x/k \\rfloor$. Then for every natural number $i$ with $1 \\le i \\le n$,","title":"Floor-rule Maximum Nim: one-step Grundy reduction when k = nt + h"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-mul-sub-eq-iterate-f","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_mul_sub_eq_iterate_F","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-mul-sub-eq-iterate-f","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $n$, $t$, $h$ be natural numbers with $t \\ge 1$ and $1 \\le h \\le n$, and let $k = nt + h$. Write $G(x)$ for the Grundy number of a pile of $x$ stones in Maximum Nim with rule function $f(y) = \\lfloor y/k \\rfloor$, let $F(x) = \\lfloor (k-1)x/k \\rfloor$, and let $F^t$ be the $t$-fold iterate of $F$. Then for every natural number $i$ with $1 \\le i \\le n$,","title":"Floor-rule Maximum Nim: t-step Grundy reduction when k = nt + h"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-grundy-mul-sub-eq-succ-itera","kind":"node","lean":"MetaMathlibExt.maximumNimGrundy_mul_sub_eq_succ_iterate_F","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-grundy-mul-sub-eq-succ-itera","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $n$, $t$, $h$ be natural numbers with $t \\ge 1$ and $1 \\le h \\le n$, and let $k = nt + h$. Write $G(x)$ for the Grundy number of a pile of $x$ stones in Maximum Nim with rule function $f(y) = \\lfloor y/k \\rfloor$, let $F(x) = \\lfloor (k-1)x/k \\rfloor$, and let $F^{t+1}$ be the $(t+1)$-fold iterate of $F$. Then for every natural number $i$ with $1 \\le i \\le h - 1$,","title":"Floor-rule Maximum Nim: (t+1)-step Grundy reduction when k = nt + h"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-h-strict-mono","kind":"node","lean":"MetaMathlibExt.maximumNimH_strictMono","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Maximum Nim map "],["math","\\(h_k\\)"],["text"," is strictly increasing"]],"source":"The Maximum Nim map $h_k$ is strictly increasing"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-h-strict-mono","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and let $h_k$ be the map $x \\mapsto x + \\lfloor x/(k-1) \\rfloor + 1$ on the natural numbers. For all natural numbers $x$ and $x'$ with $x < x'$, we have $h_k(x) < h_k(x')$.","title":"The Maximum Nim map $h_k$ is strictly increasing"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-h-succ-le","kind":"node","lean":"MetaMathlibExt.maximumNimH_succ_le","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Maximum Nim map "],["math","\\(h_k\\)"],["text"," exceeds its argument"]],"source":"The Maximum Nim map $h_k$ exceeds its argument"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-h-succ-le","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and let $h_k$ be the map $x \\mapsto x + \\lfloor x/(k-1) \\rfloor + 1$ on the natural numbers. Every natural number $x$ satisfies $x < x + 1 \\le h_k(x)$.","title":"The Maximum Nim map $h_k$ exceeds its argument"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-h-iterate-tendsto-at-top","kind":"node","lean":"MetaMathlibExt.maximumNimH_iterate_tendsto_atTop","parent":44373,"reading_title":{"format":"inline-parts-v1","parts":[["text","Iterates of the Maximum Nim map "],["math","\\(h_k\\)"],["text"," tend to infinity"]],"source":"Iterates of the Maximum Nim map $h_k$ tend to infinity"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-h-iterate-tendsto-at-top","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number and let $h_k$ be the map $x \\mapsto x + \\lfloor x/(k-1) \\rfloor + 1$ on the natural numbers. For every natural number $x_0$, the iterates $h_k^p(x_0)$ tend to infinity as $p \\to \\infty$.","title":"Iterates of the Maximum Nim map $h_k$ tend to infinity"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-h-iterate-bounds","kind":"node","lean":"MetaMathlibExt.maximumNimH_iterate_bounds","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-h-iterate-bounds","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ be a natural number, let $h_k(x) = x + \\lfloor x/(k-1) \\rfloor + 1$ on natural numbers, and define the real maps $h_k^-(x) = x + \\frac{x}{k-1}$ and $h_k^+(x) = x + \\frac{x}{k-1} + 1$. Writing $g^p$ for the $p$-fold iterate of a map $g$, for all natural numbers $p$ and $x$: (i) $(h_k^-)^p(x) = \\left(\\frac{k}{k-1}\\right)^p x$, (ii) $(h_k^+)^p(x) = \\left(\\frac{k}{k-1}\\right)^p (x + k - 1) - (k - 1)$, and\u2026","title":"Linear bounds on h_k and their iterates"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-position-complement-eq-maximum","kind":"node","lean":"MetaMathlibExt.josephus_position_complement_eq_maximumNimGrundy","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-position-complement-eq-maximum","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ and $1 \\le m \\le n$. Write $o_n(m)$ for the position, counting from $1$, at which $m$ appears in the Josephus elimination order of the circle $1, \\dots, n$ in which every $k$-th person is removed and the survivor is listed last, and write $G$ for the Grundy number of Maximum Nim with rule function $f(y) = \\lfloor y/k \\rfloor$. Then $n - o_n(m) = G(nk - m)$.","title":"Josephus elimination positions as Maximum Nim Grundy numbers"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-eq-mul-sub-iterate-maximum-nim","kind":"node","lean":"MetaMathlibExt.josephus_eq_mul_sub_iterate_maximumNimH_zero","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-eq-mul-sub-iterate-maximum-nim","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ and $n \\ge 1$ be natural numbers, let $h_k(x) = x + \\lfloor x/(k-1) \\rfloor + 1$ for natural numbers $x$, and write $h_k^p$ for the $p$-th iterate of $h_k$. Then there is a natural number $p \\ge 1$ with $h_k^{p-1}(0) < n(k-1) \\le h_k^p(0)$ and $J_k(n) = nk - h_k^p(0)$, where $J_k$ is the Josephus function for elimination step $k$.","title":"Josephus survivor via iteration from zero"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-eq-mul-sub-iterate-maximum-nim-2","kind":"node","lean":"MetaMathlibExt.josephus_eq_mul_sub_iterate_maximumNimH_pred","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-eq-mul-sub-iterate-maximum-nim-2","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ and $n \\ge 2$ be natural numbers, let $h_k(x) = x + \\lfloor x/(k-1) \\rfloor + 1$ for natural numbers $x$, and write $h_k^p$ for the $p$-th iterate of $h_k$. Then there is a natural number $p \\ge 1$ with $h_k^{p-1}(k-1) < n(k-1) \\le h_k^p(k-1)$ and $J_k(n) = nk - h_k^p(k-1)$, where $J_k$ is the Josephus function for elimination step $k$.","title":"Josephus survivor via the shifted iteration"},{"id":"GameTheory/combinatorial-games/maximum-nim/maximum-nim-h-iterate-floor-le-mul-le-it","kind":"node","lean":"MetaMathlibExt.maximumNimH_iterate_floor_le_mul_le_iterate_ceil","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#maximum-nim-h-iterate-floor-le-mul-le-it","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ and $n \\ge 1$ be natural numbers, let $h_k(x) = x + \\lfloor x/(k-1) \\rfloor + 1$ on natural numbers, write $h_k^p$ for its $p$-fold iterate, and let","title":"Logarithmic bounds on the iterates of h_k"},{"id":"GameTheory/combinatorial-games/maximum-nim/josephus-eq-mul-sub-iterate-maximum-nim-3","kind":"node","lean":"MetaMathlibExt.josephus_eq_mul_sub_iterate_maximumNimH_between_log_bounds","parent":44373,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/maximum-nim/index.html#josephus-eq-mul-sub-iterate-maximum-nim-3","scope":"GameTheory/combinatorial-games/maximum-nim","status":"can_prove","summary":"Let $k \\ge 2$ and $n \\ge 2$ be natural numbers, let $h_k(x) = x + \\lfloor x/(k-1) \\rfloor + 1$ for natural numbers $x$, write $h_k^p$ for the $p$-th iterate of $h_k$, and define the real numbers $v = \\log n / (\\log k - \\log(k-1))$ and $w = (\\log(n+1) - \\log 2) / (\\log k - \\log(k-1))$, where $\\log$ is the natural logarithm. Then there is a natural number $p \\ge 1$ with $\\lfloor w \\rfloor \\le p \\le \\lceil v \\rceil$\u2026","title":"Logarithmic bounds for the Josephus iteration count"},{"data_url":"scopes/GameTheory/combinatorial-games/smith-s-remoteness-function.json","id":"GameTheory/combinatorial-games/smith-s-remoteness-function","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/smith-s-remoteness-function/index.html","scope":"GameTheory/combinatorial-games/smith-s-remoteness-function","status":"fully_proved","title":"Smith's remoteness function"},{"id":"GameTheory/combinatorial-games/smith-s-remoteness-function/smith-remoteness","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.smithRemoteness","parent":44415,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/smith-s-remoteness-function/index.html#smith-remoteness","scope":"GameTheory/combinatorial-games/smith-s-remoteness-function","status":"fully_proved","summary":"For an impartial game $G$, the remoteness $R(p)$ of a position $p$ of $G$ is defined by well-founded recursion along moves; the options of $p$ are the positions reachable from $p$ in one move. If $p$ has no options, then $R(p) = 0$. Otherwise, if some option $q$ of $p$ has $R(q)$ even, then $R(p)$ is $1$ plus the least even value of $R(q)$ over the options $q$ of $p$; if every option $q$ of $p$ has $R(q)$ odd, then\u2026","title":"Smith's remoteness function"},{"id":"GameTheory/combinatorial-games/smith-s-remoteness-function/smith-remoteness-eq","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.smithRemoteness_eq","parent":44415,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/smith-s-remoteness-function/index.html#smith-remoteness-eq","scope":"GameTheory/combinatorial-games/smith-s-remoteness-function","status":"fully_proved","summary":"Let $G$ be an impartial game, $p$ a position of $G$, and $R$ the set of remotenesses of the options of $p$ (the positions reachable from $p$ in one move). If $R$ contains an even number, the remoteness of $p$ is $1$ plus the least even element of $R$; otherwise, if $R$ is nonempty, it is $1$ plus the largest element of $R$; and if $R$ is empty, it is $0$.","title":"Recursion for Smith's remoteness"},{"id":"GameTheory/combinatorial-games/smith-s-remoteness-function/smith-remoteness-of-moves-eq-empty","kind":"node","lean":"MetaMathlibExt.SpragueGrundy.smithRemoteness_of_moves_eq_empty","parent":44415,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/smith-s-remoteness-function/index.html#smith-remoteness-of-moves-eq-empty","scope":"GameTheory/combinatorial-games/smith-s-remoteness-function","status":"fully_proved","summary":"In an impartial game, the remoteness of a terminal position (a position with no moves) is $0$.","title":"Smith's remoteness vanishes at terminal positions"},{"data_url":"scopes/GameTheory/combinatorial-games/game-of-euclid.json","id":"GameTheory/combinatorial-games/game-of-euclid","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","title":"Game of Euclid"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition-aux","kind":"node","lean":"MetaMathlibExt.euclidPPositionAux","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition-aux","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"The fuel-bounded evaluation of the P-positions of Euclid is the function $E(f, a, b)$ of natural numbers $f, a, b$ with Boolean values defined by recursion on the fuel $f$. With no fuel, $E(0, a, b)$ is true. Otherwise $E(f + 1, a, b)$ is true when $a = b$, $a = 0$ or $b = 0$; when $0 < a < b$ it is true iff $E(f, a, b - ja)$ is false for every $j \\ge 1$ with $ja < b$; and when $0 < b < a$ it is true iff\u2026","title":"Fuel-bounded Euclid classifier"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition","kind":"node","lean":"MetaMathlibExt.euclidPPosition","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"For natural numbers $a$ and $b$, the Euclid P-position test at $(a, b)$ is the value at $(a, b)$ of the fuel-bounded evaluation of the P-positions of Euclid with fuel $a + b$.","title":"Euclid P-position test"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-moves","kind":"node","lean":"MetaMathlibExt.euclidMoves","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-moves","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"The moves of Euclid from a pair $(a, b)$ of natural numbers are as follows. If $a = b$, $a = 0$ or $b = 0$, there are none. If $0 < a < b$, the moves are to the pairs $(a, b - ja)$ for every integer $j \\ge 1$ with $ja < b$; if $0 < b < a$, the moves are to the pairs $(a - jb, b)$ for every integer $j \\ge 1$ with $jb < a$.","title":"Euclid moves"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-moves-rank-lt","kind":"node","lean":"MetaMathlibExt.euclidMoves_rank_lt","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-moves-rank-lt","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"If $(a', b')$ is a Euclid move from $(a, b)$, then $a' + b' < a + b$.","title":"Euclid moves lower the sum"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-game","kind":"node","lean":"MetaMathlibExt.euclidGame","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-game","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"Euclid as an impartial game: its positions are the pairs $(a, b)$ of natural numbers, its moves from $(a, b)$ are the Euclid moves, the rank of $(a, b)$ is $a + b$, and its starting position is $(1, 1)$.","title":"The game Euclid"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition-aux-eq-true-iff","kind":"node","lean":"MetaMathlibExt.euclidPPositionAux_eq_true_iff","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition-aux-eq-true-iff","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"For all natural numbers $f$, $a$ and $b$ with $a + b \\le f$, the fuel-bounded evaluation of the P-positions of Euclid with fuel $f$ returns true at $(a, b)$ if and only if $(a, b)$ is a normal-play P-position of the Euclid game.","title":"Correctness of the fuel-bounded Euclid classifier"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition-eq-true-iff","kind":"node","lean":"MetaMathlibExt.euclidPPosition_eq_true_iff","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition-eq-true-iff","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"For all natural numbers $a$ and $b$, the Euclid P-position test returns true at $(a, b)$ if and only if $(a, b)$ is a normal-play P-position of the Euclid game.","title":"The Euclid P-position test is correct"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition-small","kind":"node","lean":"MetaMathlibExt.euclidPPosition_small","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition-small","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"The Euclid P-position test returns true at the pairs $(1, 1)$ and $(3, 2)$ and false at the pairs $(2, 1)$ and $(1, 2)$.","title":"Small P-positions of Euclid"},{"id":"GameTheory/combinatorial-games/game-of-euclid/calkin-wilf-row","kind":"node","lean":"MetaMathlibExt.calkinWilfRow","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#calkin-wilf-row","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"Row $k$ of the Calkin\u2013Wilf tree is a list of numerator\u2013denominator pairs, the pair $(a, b)$ standing for $a/b$. Row $0$ is the single pair $(1, 1)$, and row $k + 1$ is obtained from row $k$ by replacing each pair $(a, b)$, in order, by its two children $(a, a + b)$ and $(a + b, b)$.","title":"Rows of the Calkin\u2013Wilf tree"},{"id":"GameTheory/combinatorial-games/game-of-euclid/calkin-wilf-row-one-two","kind":"node","lean":"MetaMathlibExt.calkinWilfRow_one_two","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#calkin-wilf-row-one-two","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"fully_proved","summary":"As lists of numerator\u2013denominator pairs, row $1$ of the Calkin\u2013Wilf tree is $(1, 2), (2, 1)$ and row $2$ is $(1, 3), (3, 2), (2, 3), (3, 1)$, in this order.","title":"The first rows of the Calkin\u2013Wilf tree"},{"id":"GameTheory/combinatorial-games/game-of-euclid/count-euclid-pposition-calkin-wilf-row","kind":"node","lean":"MetaMathlibExt.count_euclidPPosition_calkinWilfRow","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#count-euclid-pposition-calkin-wilf-row","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Let $L_k$ be the number of P-positions of Euclid in row $k$ of the Calkin\u2013Wilf tree. Then $L_0 = 1$, $L_1 = 0$, $L_{k+2} = L_{k+1} + 2L_k$ for every $k$, and $3L_k = 2^k + 2(-1)^k$ for every $k$, that is, $L_k = 2^k/3 + 2(-1)^k/3$.","title":"Number of P-positions of Euclid in a row of the Calkin\u2013Wilf tree"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition-tops-of-arch","kind":"node","lean":"MetaMathlibExt.euclidPPosition_tops_of_arch","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition-tops-of-arch","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Call a pair $(a, b)$ of natural numbers a P-position when the Euclid P-position test returns true at $(a, b)$, and read it as the vertex $a/b$ of the Calkin\u2013Wilf tree, with son $a/(a+b)$ and daughter $(a+b)/b$. Let $m, n$ be positive integers. If $m/n$ is a P-position, then so are its son's daughter $(2m+n)/(m+n)$ and its daughter's son $(m+n)/(m+2n)$.","title":"Distinguishing the tops of arches"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition-branch-step","kind":"node","lean":"MetaMathlibExt.euclidPPosition_branch_step","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition-branch-step","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Call a pair $(a, b)$ of natural numbers a P-position when the Euclid P-position test returns true at $(a, b)$, and read it as the vertex $a/b$ of the Calkin\u2013Wilf tree, with son $a/(a+b)$ and daughter $(a+b)/b$. Let $x, y$ be positive integers. If the daughter $(x+y)/y$ of $x/y$ is a P-position, then so is its niece $(2x+y)/(x+y)$, the daughter of its sibling $x/(x+y)$. If the son $x/(x+y)$ of $x/y$ is a P-position,\u2026","title":"Establishing branches: nieces and nephews of P-positions"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition-aunt-xor-cousin","kind":"node","lean":"MetaMathlibExt.euclidPPosition_aunt_xor_cousin","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition-aunt-xor-cousin","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Call a pair $(a, b)$ of natural numbers a P-position when the Euclid P-position test returns true at $(a, b)$, and let its son be $s(a, b) = (a, a + b)$ and its daughter $d(a, b) = (a + b, b)$, as in the Calkin\u2013Wilf tree. Let $x, y$ be positive integers, let $r = (x, y)$, let $q$ be one of the two children $s(r)$ and $d(r)$ of $r$, and let $q'$ be the other one. If the daughter $d(q)$ is a P-position, then exactly\u2026","title":"Aunt or cousin of a P-position"},{"id":"GameTheory/combinatorial-games/game-of-euclid/euclid-pposition-self-similar","kind":"node","lean":"MetaMathlibExt.euclidPPosition_self_similar","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#euclid-pposition-self-similar","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Call a pair $(a, b)$ of natural numbers a P-position when the Euclid P-position test returns true at $(a, b)$, and read it as the vertex $a/b$ of the Calkin\u2013Wilf tree, with son $a/(a+b)$ and daughter $(a+b)/b$. Let $V$ be a vertex in some row of the Calkin\u2013Wilf tree that is a P-position. Then for every finite sequence $w$ of steps, each step going to the son or to the daughter, the vertex reached from $V$ by\u2026","title":"Self-similarity of the P-positions of Euclid"},{"id":"GameTheory/combinatorial-games/game-of-euclid/count-arch-tops-calkin-wilf-row","kind":"node","lean":"MetaMathlibExt.count_arch_tops_calkinWilfRow","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#count-arch-tops-calkin-wilf-row","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"For $k \\ge 2$ let $T_k$ be the number of vertices of row $k$ of the Calkin\u2013Wilf tree of the form $(2m+n)/(m+n)$ with $m/n$ a P-position of Euclid in row $k - 2$. Then $3T_k = 2^{k-2} + 2(-1)^k$, that is, $T_k = 2^{k-2}/3 + 2(-1)^k/3$.","title":"Number of arch tops in a row of the Calkin\u2013Wilf tree"},{"id":"GameTheory/combinatorial-games/game-of-euclid/arch-branch-levels","kind":"node","lean":"MetaMathlibExt.arch_branch_levels","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#arch-branch-levels","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"On numerator\u2013denominator pairs write $s(x, y) = (x, x + y)$ for the son and $t(x, y) = (x + y, y)$ for the daughter of a vertex of the Calkin\u2013Wilf tree. Let $a, b, c, d, k$ be natural numbers with $0 < b < a$ and $0 < c < d$. Then $t(s^k(a - b, b))$, the daughter of the $k$-fold son of $(a - b, b)$, has first entry $(k+1)a - kb$ and second entry $ka - (k-1)b$, and $s(t^k(c, d - c))$, the son of the $k$-fold\u2026","title":"Levels of the branches of an arch"},{"id":"GameTheory/combinatorial-games/game-of-euclid/fibonacci-arch-positions","kind":"node","lean":"MetaMathlibExt.fibonacci_arch_positions","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#fibonacci-arch-positions","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Let $F_j$ be the Fibonacci numbers with $F_0 = F_1 = 1$ and $F_{j+2} = F_{j+1} + F_j$. Call a pair $(a, b)$ of natural numbers a P-position when the Euclid P-position test returns true at $(a, b)$, read it as the vertex $a/b$ of the Calkin\u2013Wilf tree, and let $s(a/b) = a/(a+b)$ be its son and $d(a/b) = (a+b)/b$ its daughter; write $s^k$ and $d^k$ for the $k$-fold iterates. Let $n \\ge 1$ and $k \\ge 0$, and let\u2026","title":"The Fibonacci arches"},{"id":"GameTheory/combinatorial-games/game-of-euclid/fibonacci-arch-tops","kind":"node","lean":"MetaMathlibExt.fibonacci_arch_tops","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#fibonacci-arch-tops","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Index the Fibonacci numbers as in the source, $F_0 = F_1 = 1$ and $F_{j+2} = F_{j+1} + F_j$. Write $a/b$ for the position $(a, b)$ of Euclid, and call $a/(a+b)$ its son and $(a+b)/b$ its daughter. Let $n \\ge 1$ and $k \\ge 0$ be integers, and let $X = (F_{2n+1} + kF_{2n-1})/(F_{2n+1} + (k-1)F_{2n-1})$ and $Y = (F_{2n} + kF_{2n-2})/(F_{2n} + (k+1)F_{2n-2})$. Then $X$ is a P-position of Euclid, and so are its son's\u2026","title":"Fibonacci arch tops"},{"id":"GameTheory/combinatorial-games/game-of-euclid/count-max-remoteness-calkin-wilf-row","kind":"node","lean":"MetaMathlibExt.count_max_remoteness_calkinWilfRow","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#count-max-remoteness-calkin-wilf-row","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"For every $k \\ge 0$, the number of positions in row $k$ of the Calkin\u2013Wilf tree whose Smith remoteness in the Euclid game is $k$ equals $2^{k/2}$ if $k$ is even and $2^{(k+1)/2}$ if $k$ is odd.","title":"Positions of a Calkin\u2013Wilf row with maximal optimal play in Euclid"},{"id":"GameTheory/combinatorial-games/game-of-euclid/max-remoteness-path-alternates","kind":"node","lean":"MetaMathlibExt.max_remoteness_path_alternates","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#max-remoteness-path-alternates","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"In the Calkin\u2013Wilf tree the vertex $a/b$ has son $a/(a+b)$ and daughter $(a+b)/b$. Let $w = (w_0, \\dots, w_{\\ell-1})$ be a sequence of steps, each \"son\" or \"daughter\", and let $a/b$ be the vertex reached from the root $1/1$ by following $w$. If the Smith remoteness of the position $(a, b)$ of Euclid equals $\\ell$, then $w_{2i} \\ne w_{2i+1}$ for every $i$ with $2i + 1 < \\ell$; that is, steps $2i+1$ and $2i+2$ are\u2026","title":"Maximal-remoteness paths alternate in pairs"},{"id":"GameTheory/combinatorial-games/game-of-euclid/golden-bounds-of-max-remoteness-n","kind":"node","lean":"MetaMathlibExt.golden_bounds_of_max_remoteness_N","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#golden-bounds-of-max-remoteness-n","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. If the pair $(m, n)$, read as the fraction $m/n$, lies in row $k$ of the Calkin\u2013Wilf tree, $n < m$, the position $(m, n)$ of Euclid is not a P-position, and its Smith remoteness in Euclid equals $k$, then $\\varphi < m/n < \\varphi + 1$.","title":"Golden-ratio bounds for N-positions of maximal remoteness above 1"},{"id":"GameTheory/combinatorial-games/game-of-euclid/golden-bounds-of-max-remoteness-n-symm","kind":"node","lean":"MetaMathlibExt.golden_bounds_of_max_remoteness_N_symm","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#golden-bounds-of-max-remoteness-n-symm","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. If the pair $(m, n)$, read as the fraction $m/n$, lies in row $k$ of the Calkin\u2013Wilf tree, $m < n$, the position $(m, n)$ of Euclid is not a P-position, and its Smith remoteness in Euclid equals $k$, then $1/(\\varphi + 1) < m/n < 1/\\varphi$.","title":"Golden-ratio bounds for N-positions of maximal remoteness below 1"},{"id":"GameTheory/combinatorial-games/game-of-euclid/golden-bounds-of-max-remoteness-p","kind":"node","lean":"MetaMathlibExt.golden_bounds_of_max_remoteness_P","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#golden-bounds-of-max-remoteness-p","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. If the pair $(m, n)$, read as the fraction $m/n$, lies in row $k$ of the Calkin\u2013Wilf tree, $m \\ne n$, the position $(m, n)$ of Euclid is a P-position, and its Smith remoteness in Euclid equals $k$, then $(\\varphi + 2)/(\\varphi + 1) < m/n < \\varphi$ or $1/\\varphi < m/n < (\\varphi + 1)/(\\varphi + 2)$.","title":"Golden-ratio bounds for P-positions of maximal remoteness"},{"id":"GameTheory/combinatorial-games/game-of-euclid/continued-fraction-max-remoteness","kind":"node","lean":"MetaMathlibExt.continued_fraction_max_remoteness","parent":44419,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/game-of-euclid/index.html#continued-fraction-max-remoteness","scope":"GameTheory/combinatorial-games/game-of-euclid","status":"can_prove","summary":"For natural numbers $a_0, \\dots, a_r$ write $[a_0, \\dots, a_r]$ for the continued fraction $a_0 + 1/(a_1 + 1/(\\cdots + 1/a_r))$, and $1^j$ for a block of $j$ ones; read a positive rational $u/v$ in lowest terms as the position $(u, v)$ of the Euclid game. Let $n \\ge 1$, $i$ and $k$ be natural numbers. If the position of $[1^{n-1}, 2]$ lies in row $k$ of the Calkin\u2013Wilf tree, its Smith remoteness is $k$. If the\u2026","title":"Continued fractions with maximal optimal play in Euclid"},{"data_url":"scopes/GameTheory/combinatorial-games/slow-nim.json","id":"GameTheory/combinatorial-games/slow-nim","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","title":"Slow Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-after","kind":"node","lean":"MetaMathlibExt.slowANimAfter","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-after","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $p$ be a position of Slow$A$-Nim on $n$ stacks, with $p_i$ tokens on stack $i$, and let $S$ be a set of stacks. The position after playing on $S$ is obtained by removing one token from each stack of $S$: its stack $i$ holds $p_i - 1$ if $i \\in S$ and $p_i$ otherwise, where an empty stack of $S$ stays empty.","title":"Playing on a set of stacks"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-moves","kind":"node","lean":"MetaMathlibExt.slowANimMoves","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-moves","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $A$ be a finite set of natural numbers and $p$ a position on $n$ stacks. The options of $p$ in Slow$A$-Nim are the positions obtained from $p$ by choosing a nonempty set $S$ of stacks with $|S| \\in A$, each holding at least one token, and removing one token from each stack of $S$.","title":"Options in Slow A-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/sum-lt-of-mem-slow-anim-moves","kind":"node","lean":"MetaMathlibExt.sum_lt_of_mem_slowANimMoves","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moves in Slow"],["math","\\(A\\)"],["text","-Nim decrease the token count"]],"source":"Moves in Slow$A$-Nim decrease the token count"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#sum-lt-of-mem-slow-anim-moves","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For every finite set $A$ of natural numbers and all positions $p$, $q$ of Slow$A$-Nim on $n$ stacks, if $q$ is reached from $p$ by one move, then the total number of tokens strictly decreases: $\\sum_i q_i < \\sum_i p_i$.","title":"Moves in Slow$A$-Nim decrease the token count"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-game","kind":"node","lean":"MetaMathlibExt.slowANimGame","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Slow"],["math","\\(A\\)"],["text","-Nim"]],"source":"Slow$A$-Nim"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-game","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For $n \\in \\mathbb{N}$ and a finite set $A$ of natural numbers, Slow$A$-Nim $\\mathrm{SN}(n, A)$ is the impartial game whose positions assign a number $p_i$ of tokens to each of $n$ labelled stacks $i$, and whose options from $p$ are obtained by choosing a nonempty set $S$ of stacks with $|S| \\in A$, each holding at least one token, and removing one token from each stack of $S$. Its rank is the total number of\u2026","title":"Slow$A$-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-moves-zero","kind":"node","lean":"MetaMathlibExt.slowANimMoves_zero","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-moves-zero","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For every finite set $A$ of natural numbers, the position of Slow$A$-Nim on $n$ stacks in which every stack is empty has no options.","title":"The empty position has no options"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-is-p-zero","kind":"node","lean":"MetaMathlibExt.slowANim_isP_zero","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-is-p-zero","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For every $n$ and every finite set $A$ of natural numbers, the position of Slow$A$-Nim $\\mathrm{SN}(n, A)$ with all $n$ stacks empty is a P-position, since it has no options.","title":"The empty position is a P-position"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduced","kind":"node","lean":"MetaMathlibExt.SlowANimReduced","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reduced positions of Slow "],["math","\\(A\\)"],["text","-Nim"]],"source":"Reduced positions of Slow $A$-Nim"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduced","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"A position $p$ of Slow $A$-Nim on $n$ stacks is reduced when for every stack $i$ there is a terminal position $t$ (one with no options) that is reachable from $p$ by a sequence of zero or more moves and has $t_i = 0$.","title":"Reduced positions of Slow $A$-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduction","kind":"node","lean":"MetaMathlibExt.slowANimReduction","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reduction of a Slow"],["math","\\(A\\)"],["text","-Nim position"]],"source":"Reduction of a Slow$A$-Nim position"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduction","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $A$ be a finite set of natural numbers and let $p$ be a position of Slow$A$-Nim on $n$ stacks. For each stack $i$, let $u_i(p)$ be the least height of stack $i$ over all terminal positions (positions with no moves) reachable from $p$ by a sequence of zero or more moves. The reduction of $p$ is the position $r(p)$ with $r(p)_i = p_i - u_i(p)$ for every stack $i$.","title":"Reduction of a Slow$A$-Nim position"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduced-zero","kind":"node","lean":"MetaMathlibExt.slowANimReduced_zero","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduced-zero","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For every finite set $A$ of natural numbers and every $n$, the position of Slow $A$-Nim on $n$ stacks in which every stack is empty is reduced.","title":"The empty position is reduced"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduction-le","kind":"node","lean":"MetaMathlibExt.slowANimReduction_le","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduction-le","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For every finite set $A$ of natural numbers, every position $p$ of Slow$A$-Nim on $n$ stacks and every stack $i$, the reduction $r(p)$ of $p$ satisfies $r(p)_i \\le p_i$.","title":"The reduction never adds tokens"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-legal-seq","kind":"node","lean":"MetaMathlibExt.SlowANimLegalSeq","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-legal-seq","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $A$ be a finite set of natural numbers and $p$ a position of Slow$A$-Nim on $n$ stacks. A finite sequence $S_1, \\dots, S_m$ of sets of stacks is legal from $p$ when it is empty, or when $S_1$ is nonempty, $|S_1| \\in A$, every stack in $S_1$ holds at least one token in $p$, and $S_2, \\dots, S_m$ is legal from the position obtained from $p$ by removing one token from each stack of $S_1$.","title":"Legal move sequences in Slow A-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-legal-seq-nil","kind":"node","lean":"MetaMathlibExt.slowANimLegalSeq_nil","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-legal-seq-nil","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For every finite set $A$ of natural numbers and every position $p$ of Slow$A$-Nim on $n$ stacks, the empty sequence of moves is legal from $p$.","title":"The empty move sequence is legal"},{"id":"GameTheory/combinatorial-games/slow-nim/is-pposition-set","kind":"node","lean":"MetaMathlibExt.IsPPositionSet","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#is-pposition-set","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $R$ be a move relation on a type of positions, where $R\\,x\\,y$ means that $y$ is an option of $x$. A set $P$ of positions is a set of P-positions for $R$ when no position of $P$ has an option in $P$, and every position outside $P$ has an option in $P$.","title":"Set of P-positions"},{"id":"GameTheory/combinatorial-games/slow-nim/eq","kind":"node","lean":"MetaMathlibExt.IsPPositionSet.eq","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#eq","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $R$ be a move relation on a type of positions, where $R\\,x\\,y$ means that $y$ is an option of $x$, and suppose that there is no infinite sequence of moves (the relation \"$y$ is an option of $x$\" is well-founded). If $P$ and $Q$ are both sets of P-positions for $R$, then $P = Q$.","title":"Uniqueness of the set of P-positions"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-partial-structure","kind":"node","lean":"MetaMathlibExt.slowANim_partialStructure","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Partial structure of the P-positions of Slow "],["math","\\(A\\)"],["text","-Nim"]],"source":"Partial structure of the P-positions of Slow $A$-Nim"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-partial-structure","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $A \\subseteq \\{1, \\dots, n\\}$ and let $p$ be a position of Slow $A$-Nim on $n$ stacks. If every stack height of $p$ is even, then $p$ is a P-position. If the number of stacks of $p$ with odd height lies in $A$, then $p$ is not a P-position.","title":"Partial structure of the P-positions of Slow $A$-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-one-odd-is-p-iff","kind":"node","lean":"MetaMathlibExt.slowANim_one_odd_isP_iff","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-one-odd-is-p-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $k$ be odd with $1 \\le k \\le n$ and let $A = \\{1, k\\}$. A position $p = (p_1, \\dots, p_n)$ of Slow$A$-Nim on $n$ stacks is a P-position if and only if its total number of tokens $p_1 + \\dots + p_n$ is even.","title":"P-positions of Slow A-Nim with A = {1, k}, k odd"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-one-n-even-is-p-iff","kind":"node","lean":"MetaMathlibExt.slowANim_one_n_even_isP_iff","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-one-n-even-is-p-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $n \\ge 1$ be even and let $A = \\{1, n\\}$. A position $p = (p_1, \\dots, p_n)$ of Slow$A$-Nim on $n$ stacks is a P-position if and only if its total number of tokens $p_1 + \\dots + p_n$ and its smallest stack $\\min_i p_i$ are both even.","title":"P-positions of Slow A-Nim with A = {1, n}, n even"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-one-two-three-is-p-iff","kind":"node","lean":"MetaMathlibExt.slowANim_one_two_three_isP_iff","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","P-positions of Slow "],["math","\\(\\{1,2\\}\\)"],["text","-Nim on three stacks"]],"source":"P-positions of Slow $\\{1,2\\}$-Nim on three stacks"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-one-two-three-is-p-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"In Slow $A$-Nim with $A = \\{1, 2\\}$ on three stacks, let $p = (p_0, p_1, p_2)$ be a position with $p_0 \\le p_1 \\le p_2$. Then $p$ is a P-position if and only if its three stack heights are all even or all odd.","title":"P-positions of Slow $\\{1,2\\}$-Nim on three stacks"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-one-two-four-is-p-iff","kind":"node","lean":"MetaMathlibExt.slowANim_one_two_four_isP_iff","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","P-positions of Slow "],["math","\\(\\{1,2\\}\\)"],["text","-Nim on four stacks"]],"source":"P-positions of Slow $\\{1,2\\}$-Nim on four stacks"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-one-two-four-is-p-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"In Slow $A$-Nim with $A = \\{1, 2\\}$ on four stacks, let $p = (p_0, p_1, p_2, p_3)$ be a position with $p_0 \\le p_1 \\le p_2 \\le p_3$. Then $p$ is a P-position if and only if either all four stack heights are even, or $p_0$ is even and $p_1$, $p_2$, $p_3$ are odd.","title":"P-positions of Slow $\\{1,2\\}$-Nim on four stacks"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-one-two-five-is-p-iff","kind":"node","lean":"MetaMathlibExt.slowANim_one_two_five_isP_iff","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-one-two-five-is-p-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $A = \\{1, 2\\}$ and let $p = (p_1, \\dots, p_5)$ be a position of Slow$A$-Nim on five stacks in canonical form, that is, $p_1 \\le p_2 \\le \\dots \\le p_5$. Then $p$ is a P-position if and only if all five stacks are even, or $p_1, p_2$ are even and $p_3, p_4, p_5$ are odd, or $p_1, p_2, p_5$ are odd and $p_3, p_4$ are even, or $p_1, p_2, p_3, p_4$ are odd and $p_5$ is even.","title":"P-positions of Slow A-Nim with A = {1, 2} on five stacks"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduced-iff-nirb","kind":"node","lean":"MetaMathlibExt.slowANimReduced_iff_nirb","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduced-iff-nirb","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $A \\subseteq \\{1, \\dots, n\\}$ be nonempty with least element $a$, and let $p$ be a position of Slow $A$-Nim on $n$ stacks. Then $p$ is reduced if and only if $a \\cdot M \\le \\sum_i p_i$, where $M$ is the largest stack height of $p$ (taken to be $0$ when $n = 0$).","title":"NIRB criterion for reduced positions"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-legal-seq-reduction-iff","kind":"node","lean":"MetaMathlibExt.slowANimLegalSeq_reduction_iff","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-legal-seq-reduction-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $A \\subseteq \\{1, \\dots, n\\}$ and let $p$ be a position of Slow$A$-Nim on $n$ stacks, with reduction $r(p)$. A finite sequence of moves, each given by its set of stacks, is legal from $r(p)$ if and only if it is legal from $p$.","title":"Legal move sequences depend only on the reduction"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-is-p-iff-of-reduction-eq","kind":"node","lean":"MetaMathlibExt.slowANim_isP_iff_of_reduction_eq","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-is-p-iff-of-reduction-eq","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $A \\subseteq \\{1, \\dots, n\\}$. If two positions $p$ and $q$ of Slow$A$-Nim $\\mathrm{SN}(n, A)$ have the same reduction, $r(p) = r(q)$, then $p$ is a P-position if and only if $q$ is a P-position.","title":"Positions with the same reduction have the same outcome"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduced-min","kind":"node","lean":"MetaMathlibExt.slowANimReduced_min","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduced-min","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $A \\subseteq \\{1, \\dots, n\\}$ and let $p$ be a reduced position of Slow $A$-Nim on $n$ stacks. Then for every $m \\in \\mathbb{N}$ the position $\\mathrm{ctt}(p, m)$ with stack heights $\\mathrm{ctt}(p, m)_i = \\min(p_i, m)$ is reduced.","title":"Chopping a reduced position keeps it reduced"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduction-eq-min","kind":"node","lean":"MetaMathlibExt.slowANimReduction_eq_min","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduction-eq-min","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $A \\subseteq \\{1, \\dots, n\\}$ and let $p$ be a position of Slow$A$-Nim on $n$ stacks, with largest stack height $M = \\max_i p_i$ ($M = 0$ when $n = 0$). For $m \\in \\mathbb{N}$ write $\\mathrm{ctt}(p, m)$ for the position whose stack $i$ has height $\\min(p_i, m)$. If $m \\le M$, the position $\\mathrm{ctt}(p, m)$ is reduced, and $\\mathrm{ctt}(p, m')$ is not reduced for any $m'$ with $m < m' \\le M$, then the\u2026","title":"The reduction chops at the largest reduced height"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-general-reduction","kind":"node","lean":"MetaMathlibExt.slowANim_generalReduction","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-general-reduction","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $A \\subseteq \\{1, \\dots, n\\}$ be nonempty with least element $a$, and let $p$ be a reduced position of Slow$A$-Nim $\\mathrm{SN}(n, A)$ with largest stack height $M$ and total $\\Sigma(p) = \\sum_i p_i$. Let $S$ be a set of $\\ell \\in A$ stacks, each holding at least one token, and suppose that the position obtained from $p$ by removing one token from each stack of $S$ is not reduced. Then $a M \\le \\Sigma(p)$ and,\u2026","title":"General reduction lemma"},{"id":"GameTheory/combinatorial-games/slow-nim/is-pposition-set-extend","kind":"node","lean":"MetaMathlibExt.isPPositionSet_extend","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#is-pposition-set-extend","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $M$ and $E$ be move relations on a type of positions, let $P$ be a set of P-positions for $M$, and let $Q \\subseteq P$. Suppose that (1) no element of $P \\setminus Q$ has an $E$-move to an element of $P \\setminus Q$, (2) every position outside $P$ that has an $M$-move into $Q$ has an $M$-move or an $E$-move into $P \\setminus Q$, and (3) every position of $Q$ has an $E$-move into $P \\setminus Q$. Then\u2026","title":"P-positions after adding moves"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduced-graph-move","kind":"node","lean":"MetaMathlibExt.SlowANimReducedGraphMove","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moves of the playable game graph of Slow "],["math","\\(A\\)"],["text","-Nim"]],"source":"Moves of the playable game graph of Slow $A$-Nim"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduced-graph-move","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"The playable (reduced) game graph of Slow $A$-Nim on $n$ stacks has the reduced positions as its positions. For reduced positions $x$ and $y$, there is a move from $x$ to $y$ when $y$ is the reduction $r(q)$ of some option $q$ of $x$ in Slow $A$-Nim, that is, of some position reached from $x$ by one legal move.","title":"Moves of the playable game graph of Slow $A$-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/not-slow-anim-reduced-graph-move-zero","kind":"node","lean":"MetaMathlibExt.not_slowANimReducedGraphMove_zero","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#not-slow-anim-reduced-graph-move-zero","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For every $n$ and every finite set $A$ of natural numbers, the position of Slow$A$-Nim $\\mathrm{SN}(n, A)$ with all $n$ stacks empty has no move to any position in the playable game graph of $\\mathrm{SN}(n, A)$.","title":"The empty position is terminal in the playable game graph"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-is-p-iff-reduction-mem","kind":"node","lean":"MetaMathlibExt.slowANim_isP_iff_reduction_mem","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","P-positions of Slow "],["math","\\(A\\)"],["text","-Nim via the playable game graph"]],"source":"P-positions of Slow $A$-Nim via the playable game graph"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-is-p-iff-reduction-mem","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $n \\in \\mathbb{N}$ and $A \\subseteq \\{1, \\dots, n\\}$. Let $\\tilde P$ be a set of P-positions for the moves of the playable game graph of Slow$A$-Nim $\\mathrm{SN}(n, A)$, whose positions are the reduced positions: no position in $\\tilde P$ has a move to a position in $\\tilde P$, and every reduced position outside $\\tilde P$ has a move to a position in $\\tilde P$. Then a position $p$ of $\\mathrm{SN}(n, A)$ is a\u2026","title":"P-positions of Slow $A$-Nim via the playable game graph"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-n-sub-one-option-of-reduced","kind":"node","lean":"MetaMathlibExt.slowANim_n_sub_one_option_of_reduced","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-n-sub-one-option-of-reduced","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $n \\ge 2$, $k = n - 1$ and $A = \\{k\\}$, so that a move removes one token from each of exactly $k$ nonempty stacks. Let $p$ be a reduced position of Slow$A$-Nim on $n$ stacks, with $s = (p_1 + \\dots + p_n) \\bmod 2k$ and $o$ odd stacks. Let $p'$ be obtained from $p$ by playing on a set $S$ of $k$ stacks, each holding at least one token in $p$, and let $j$ be the stack not in $S$. If $p'$ is reduced, then\u2026","title":"Options without reduction in Slow {n - 1}-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-n-sub-one-option-of-not-reduce","kind":"node","lean":"MetaMathlibExt.slowANim_n_sub_one_option_of_not_reduced","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-n-sub-one-option-of-not-reduce","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $n \\ge 2$, $k = n - 1$ and $A = \\{k\\}$. Let $p$ be a reduced position of Slow$A$-Nim on $n$ stacks, with $s = (p_1 + \\dots + p_n) \\bmod 2k$, $o$ odd stacks, and maximum $M = \\max_i p_i$ attained by $\\alpha$ stacks. Let $p'$ be obtained from $p$ by playing on a set $S$ of $k$ stacks, each holding at least one token in $p$, and suppose that $p'$ is not reduced. Write $r = r(p')$ for its reduction,\u2026","title":"Options with reduction in Slow {n - 1}-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduced-mem-p-positions-n-sub","kind":"node","lean":"MetaMathlibExt.slowANimReduced_mem_pPositions_n_sub_one_iff","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","P-positions of the reduced game of Slow "],["math","\\(\\{n-1\\}\\)"],["text","-Nim"]],"source":"P-positions of the reduced game of Slow $\\{n-1\\}$-Nim"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduced-mem-p-positions-n-sub","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $n \\ge 2$ and $A = \\{n - 1\\}$, and let $\\tilde P$ be a set of P-positions of the playable game graph of Slow $A$-Nim on $n$ stacks: no move of that graph joins two elements of $\\tilde P$, and every reduced position outside $\\tilde P$ has a move into $\\tilde P$. For a reduced position $q$, let $s$ be the remainder of $\\sum_i q_i$ modulo $2(n - 1)$ and let $o$ be the number of stacks of $q$ of odd height. Then\u2026","title":"P-positions of the reduced game of Slow $\\{n-1\\}$-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-reduced-mem-p-positions-n-sub-2","kind":"node","lean":"MetaMathlibExt.slowANimReduced_mem_pPositions_n_sub_one_n_iff","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","P-positions of the reduced game of Slow "],["math","\\(\\{n-1, n\\}\\)"],["text","-Nim"]],"source":"P-positions of the reduced game of Slow $\\{n-1, n\\}$-Nim"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-reduced-mem-p-positions-n-sub-2","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $n \\ge 2$ and $A = \\{n - 1, n\\}$, and let $\\tilde P$ be a set of P-positions of the playable game graph of Slow $A$-Nim on $n$ stacks: no move of that graph joins two elements of $\\tilde P$, and every reduced position outside $\\tilde P$ has a move into $\\tilde P$. For a reduced position $q$, let $s$ be the remainder of $\\sum_i q_i$ modulo $2n - 2$ and let $o$ be the number of stacks of $q$ of odd height. Then\u2026","title":"P-positions of the reduced game of Slow $\\{n-1, n\\}$-Nim"},{"id":"GameTheory/combinatorial-games/slow-nim/is-exact-slow-nim-critical","kind":"node","lean":"MetaMathlibExt.IsExactSlowNimCritical","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(m\\)"],["text","-critical positions of exact slow "],["math","\\(k\\)"],["text","-NIM"]],"source":"$m$-critical positions of exact slow $k$-NIM"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#is-exact-slow-nim-critical","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k$ and $m$ be natural numbers and let $\\mathrm{NIM}(k + 1, k)$ be Slow$A$-Nim on $k + 1$ piles with $A = \\{k\\}$: a move chooses a nonempty set of exactly $k$ nonempty piles and removes one stone from each. Let $R$ be Smith's remoteness function of this game. A position $x = (x_0, \\dots, x_k)$ is $m$-critical when $R(x) = m$ and $R(y) \\ne m$ for every position $y < x$, where $y < x$ means $y_i \\le x_i$ for\u2026","title":"$m$-critical positions of exact slow $k$-NIM"},{"id":"GameTheory/combinatorial-games/slow-nim/is-exact-slow-nim-critical-zero","kind":"node","lean":"MetaMathlibExt.isExactSlowNimCritical_zero","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","The empty position is "],["math","\\(0\\)"],["text","-critical"]],"source":"The empty position is $0$-critical"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#is-exact-slow-nim-critical-zero","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k$ be a natural number and let $\\mathrm{NIM}(k + 1, k)$ be Slow$A$-Nim on $k + 1$ piles with $A = \\{k\\}$, in which a move chooses a nonempty set of exactly $k$ nonempty piles and removes one stone from each. The position $x$ of $\\mathrm{NIM}(k + 1, k)$ in which all $k + 1$ piles are empty is $0$-critical: its Smith remoteness is $0$, and no position $y$ with $y_i \\le x_i$ for every $i$ and $y \\ne x$ has\u2026","title":"The empty position is $0$-critical"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-mrule-kept","kind":"node","lean":"MetaMathlibExt.exactSlowNimMRuleKept","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-mrule-kept","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For a position $x \\in \\mathbb{N}^{n+1}$ with piles indexed by $0, \\dots, n$: if some entry of $x$ is even, the kept pile is the least index $i$ such that $x_i$ is even and $x_i \\le x_j$ for every $j$ with $x_j$ even; otherwise it is the least index $i$ such that $x_j \\le x_i$ for every $j$.","title":"Pile kept by the M-rule"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-mrule-move","kind":"node","lean":"MetaMathlibExt.exactSlowNimMRuleMove","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-mrule-move","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For a position $x$ on $n + 1$ piles, the M-move keeps one pile, chosen by the M-rule as the first pile holding the smallest even entry if some entry of $x$ is even, and otherwise the first pile holding a largest entry. It removes one stone from every other pile, where an empty pile stays empty.","title":"The M-move"},{"id":"GameTheory/combinatorial-games/slow-nim/slow-anim-moves-singleton-eq-empty-iff","kind":"node","lean":"MetaMathlibExt.slowANimMoves_singleton_eq_empty_iff","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["text","Terminal positions of exact slow "],["math","\\(k\\)"],["text","-NIM"]],"source":"Terminal positions of exact slow $k$-NIM"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#slow-anim-moves-singleton-eq-empty-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $n$ and $k$ be natural numbers and let $x = (x_0, \\dots, x_{n-1})$ be a position of Slow$A$-Nim on $n$ piles with $A = \\{k\\}$. Then $x$ has no options if and only if $k = 0$ or fewer than $k$ of the piles are nonempty.","title":"Terminal positions of exact slow $k$-NIM"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-mrule-length-aux","kind":"node","lean":"MetaMathlibExt.exactSlowNimMRuleLengthAux","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-mrule-length-aux","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k$ and $f$ be natural numbers and let $x$ be a position on $k + 1$ piles. Call $x$ terminal when $k = 0$ or fewer than $k$ of its piles are nonempty. The value at $f$ and $x$ is $0$ when $f = 0$ or $x$ is terminal; otherwise it is $1$ plus the value at $f - 1$ and the position reached from $x$ by the M-move. Thus it counts the M-moves played from $x$, stopping at a terminal position or after $f$ moves.","title":"Fuel-bounded length of M-rule play"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-mrule-length-aux-succ","kind":"node","lean":"MetaMathlibExt.exactSlowNimMRuleLengthAux_succ","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-mrule-length-aux-succ","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k, f \\in \\mathbb{N}$ and $x \\in \\mathbb{N}^{k+1}$, and write $M_f(x)$ for the number of M-rule steps from $x$ in Slow$\\{k\\}$-Nim on $k + 1$ stacks, counted for at most $f$ steps. Then $M_{f+1}(x) = 0$ if $x$ has no options, and otherwise $M_{f+1}(x) = 1 + M_f(x')$, where $x'$ is the position obtained from $x$ by the M-move.","title":"Successor step of the truncated M-rule length"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-mrule-length","kind":"node","lean":"MetaMathlibExt.exactSlowNimMRuleLength","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-mrule-length","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k$ be a natural number and let $x = (x_0, \\dots, x_k)$ be a position on $k + 1$ piles. Then $M(x)$ is the number of M-moves played from $x$ until a terminal position (one with $k = 0$ or fewer than $k$ nonempty piles) is reached, computed as the fuel-bounded length of M-rule play with fuel $x_0 + \\dots + x_k$.","title":"M-rule length"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-mrule-length-three-three","kind":"node","lean":"MetaMathlibExt.exactSlowNimMRuleLength_three_three_three","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-mrule-length-three-three","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"In exact slow $2$-NIM on three piles, $\\mathrm{NIM}(3, 2)$, where a move removes one stone from each of exactly two nonempty piles, the M-rule length of the position $(3, 3, 3)$ is $M(3, 3, 3) = 4$.","title":"M-rule length of the position (3, 3, 3)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-mrule-length-three-five-f","kind":"node","lean":"MetaMathlibExt.exactSlowNimMRuleLength_three_five_five","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-mrule-length-three-five-f","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"In exact slow $2$-NIM on three piles, $\\mathrm{NIM}(3, 2)$, where a move removes one stone from each of exactly two nonempty piles, the M-rule length of the position $(3, 5, 5)$ is $M(3, 5, 5) = 6$.","title":"M-rule length of the position (3, 5, 5)"},{"id":"GameTheory/combinatorial-games/slow-nim/is-exact-slow-nim-basic-a0","kind":"node","lean":"MetaMathlibExt.IsExactSlowNimBasicA0","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#is-exact-slow-nim-basic-a0","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k$ and $b$ be natural numbers. A vector $z = (z_0, \\dots, z_k)$ of $k + 1$ natural numbers is basic of type (A0) for $b$ when $z_0 + \\dots + z_k = kb$, $z_i \\le b$ for every $i$, $b$ is even, and every $z_i$ is even.","title":"Basic vectors of type (A0)"},{"id":"GameTheory/combinatorial-games/slow-nim/is-exact-slow-nim-basic-a1","kind":"node","lean":"MetaMathlibExt.IsExactSlowNimBasicA1","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#is-exact-slow-nim-basic-a1","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k$ and $b$ be natural numbers. A vector $z = (z_0, \\dots, z_k)$ of $k + 1$ natural numbers is basic of type (A1) for $b$ when $z_0 + \\dots + z_k = kb$, $z_i \\le b$ for every $i$, $b$ is odd, and exactly one $z_i$ is even.","title":"Basic vectors of type (A1)"},{"id":"GameTheory/combinatorial-games/slow-nim/is-exact-slow-nim-basic-b","kind":"node","lean":"MetaMathlibExt.IsExactSlowNimBasicB","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#is-exact-slow-nim-basic-b","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k$ and $b$ be natural numbers. A vector $z = (z_0, \\dots, z_k)$ of $k + 1$ natural numbers is basic of type (B) for $b$ when $z_0 + \\dots + z_k = kb + k - 1$, $z_i < b$ for every $i$, $b$ is even, and every $z_i$ is odd. Here $kb + k - 1$ is computed in the natural numbers, so it is $0$ when $k = 0$.","title":"Basic vectors of type (B)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-basic-value","kind":"node","lean":"MetaMathlibExt.exactSlowNimBasicValue","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-basic-value","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$ and $x \\in \\mathbb{N}^{k+1}$, the basic-vector value $B(x)$ is the largest $b \\le \\sum_i x_i$ for which some $z \\le x$ (componentwise) is a basic vector of type (A0), (A1) or (B) with parameter $b$; such a $b$ exists, since $z = 0$ is basic of type (A0) with parameter $0$. For $k \\ge 1$ the bound is automatic, because every basic $z \\le x$ with parameter $b$ has\u2026","title":"Basic-vector value"},{"id":"GameTheory/combinatorial-games/slow-nim/exists-supports-exact-slow-nim-basic-val","kind":"node","lean":"MetaMathlibExt.exists_supports_exactSlowNimBasicValue","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exists-supports-exact-slow-nim-basic-val","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"Let $k$ be a natural number, let $x$ be a position on $k + 1$ piles, and let $b = B(x)$ be its basic value. Then some $z \\le x$ (componentwise) is a basic vector of type (A0), (A1) or (B) with parameter $b$, that is, $z$ supports $x$. Here type (A0) means $\\sum_i z_i = kb$, $z_i \\le b$ for all $i$, and $b$ and every $z_i$ are even; type (A1) means $\\sum_i z_i = kb$, $z_i \\le b$ for all $i$, $b$ is odd and exactly\u2026","title":"The basic value is attained"},{"id":"GameTheory/combinatorial-games/slow-nim/is-exact-slow-nim-critical-iff","kind":"node","lean":"MetaMathlibExt.isExactSlowNimCritical_iff","parent":44445,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(m\\)"],["text","-critical positions of exact slow "],["math","\\(k\\)"],["text","-NIM with "],["math","\\(k + 1\\)"],["text"," piles (Theorem 1)"]],"source":"$m$-critical positions of exact slow $k$-NIM with $k + 1$ piles (Theorem 1)"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#is-exact-slow-nim-critical-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $k \\ge 1$ and $m$ be natural numbers, and let $x = (x_0, \\dots, x_k)$ be a position of $\\mathrm{NIM}(k + 1, k)$, the game on $k + 1$ piles in which a move removes one stone from each of exactly $k$ nonempty piles. Call $x$ $m$-critical when its Smith remoteness is $m$ and no position $y < x$ (that is, $y_i \\le x_i$ for every $i$ and $y \\ne x$) has remoteness $m$. Then $x$ is $m$-critical if and only if either\u2026","title":"$m$-critical positions of exact slow $k$-NIM with $k + 1$ piles (Theorem 1)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-critical-of-even","kind":"node","lean":"MetaMathlibExt.exactSlowNim_critical_of_even","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-critical-of-even","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $k \\ge 1$, let $m > 0$ be even, and let $x \\in \\mathbb{N}^{k+1}$ be an $m$-critical position of Slow$\\{k\\}$-Nim on $k + 1$ stacks. Then the entries of $x$ are either all odd or all even. If they are all even, then some entry satisfies $x_i < m$; every option $y$ of $x$ with $y_i < x_i$ for every $i$ such that $x_i = m$ is $(m - 1)$-critical; and the position $x'$ obtained from $x$ by the M-move satisfies\u2026","title":"Moves from critical positions with even parameter (Corollary 1 (i))"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-critical-of-odd","kind":"node","lean":"MetaMathlibExt.exactSlowNim_critical_of_odd","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-critical-of-odd","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $k \\ge 2$, let $m$ be odd, and let $x \\in \\mathbb{N}^{k+1}$ be an $m$-critical position of Slow$\\{k\\}$-Nim on $k + 1$ stacks. Then $x$ has exactly one even entry $x_i$. The position $x'$ obtained from $x$ by removing one token from every pile other than $i$ is an option of $x$, it is the only option of $x$ that is $(m - 1)$-critical, and it is the position reached from $x$ by the M-move.","title":"Unique critical-lowering move for odd parameter (Corollary 1 (ii))"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-odd-is-p-iff","kind":"node","lean":"MetaMathlibExt.exactSlowNim_odd_isP_iff","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-odd-is-p-iff","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $0 < k$ and let $\\mathrm{NIM}(k + 1, k)$ be Slow$A$-Nim on $k + 1$ stacks with $A = \\{k\\}$. A position $x = (x_0, \\dots, x_k)$ of $\\mathrm{NIM}(k + 1, k)$ all of whose entries are odd is a P-position if and only if there is a natural number $m$ such that $x_0 + \\dots + x_k = km + k - 1$, $x_i < m$ for every $i$, and $m$ is even.","title":"Proposition 1 (odd P-positions of exact slow k-NIM)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-mrule-length-eq-smith-rem","kind":"node","lean":"MetaMathlibExt.exactSlowNimMRuleLength_eq_smithRemoteness","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-mrule-length-eq-smith-rem","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $0 < k$ and let $\\mathrm{NIM}(k + 1, k)$ be exact slow $k$-NIM on $k + 1$ piles, that is, Slow$A$-Nim on $k + 1$ stacks with $A = \\{k\\}$, where a move removes one stone from each of exactly $k$ nonempty piles. For every position $x$ of $\\mathrm{NIM}(k + 1, k)$, the M-rule length $M(x)$ equals Smith's remoteness $R(x)$ of $x$ in $\\mathrm{NIM}(k + 1, k)$.","title":"Theorem 2 (the M-rule realizes Smith's remoteness)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-basic-value-of-supports-b","kind":"node","lean":"MetaMathlibExt.exactSlowNimBasicValue_of_supports_basicB","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-basic-value-of-supports-b","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $k \\ge 1$, let $B$ be the basic-vector value on $\\mathbb{N}^{k+1}$, and let $x, z \\in \\mathbb{N}^{k+1}$ with $z \\le x$ componentwise, where $z$ is a basic vector of type (B) with parameter $B(x)$. Then $x = z$, and every option $x'$ of $x$ in Slow$\\{k\\}$-Nim on $k + 1$ stacks (a move removes one token from each of exactly $k$ nonempty stacks) satisfies $B(x') = B(x) - 1$, where subtraction is truncated at $0$.","title":"Positions supported by a (B) basic vector (Lemma 1)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-basic-value-of-supports-b-2","kind":"node","lean":"MetaMathlibExt.exactSlowNimBasicValue_of_supports_basicA0","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-basic-value-of-supports-b-2","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $k \\ge 1$, let $B$ be the basic-vector value on $\\mathbb{N}^{k+1}$, and let $x, z \\in \\mathbb{N}^{k+1}$ with $z \\le x$ componentwise, where $z$ is a basic vector of type (A0) with parameter $B(x)$. Let $j$ be a pile with $x_i > 0$ for every $i \\ne j$, and let $x'$ be obtained from $x$ by removing one token from every pile other than $j$. Then $B(x') \\le B(x) - 1$ and $B(x')$ is odd. Moreover, if $z_j < B(x)$,\u2026","title":"Moves at a position supported by an (A0) basic vector (Lemma 2)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-basic-value-of-supports-b-3","kind":"node","lean":"MetaMathlibExt.exactSlowNimBasicValue_of_supports_basicA1","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-basic-value-of-supports-b-3","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $k \\ge 1$, let $B$ be the basic-vector value on $\\mathbb{N}^{k+1}$, and let $x, z \\in \\mathbb{N}^{k+1}$ with $z \\le x$ componentwise, where $z$ is a basic vector of type (A1) with parameter $B(x)$. Then every option $x'$ of $x$ in Slow$\\{k\\}$-Nim on $k + 1$ stacks (a move removes one token from each of exactly $k$ nonempty stacks) satisfies $B(x') \\ge B(x) - 2$, and the position $x''$ obtained from $x$ by the\u2026","title":"Moves at a position supported by an (A1) basic vector (Lemma 3)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-basic-value-eq-smith-remo","kind":"node","lean":"MetaMathlibExt.exactSlowNimBasicValue_eq_smithRemoteness","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-basic-value-eq-smith-remo","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"Let $k \\ge 1$ and consider Slow$\\{k\\}$-Nim on $k + 1$ stacks, in which a move removes one token from each of exactly $k$ nonempty stacks. For every position $x \\in \\mathbb{N}^{k+1}$, the basic-vector value $B(x)$ equals Smith's remoteness of $x$ in this game.","title":"Basic-vector value equals Smith's remoteness (Lemma 4)"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-binary-nat-encoding","kind":"node","lean":"MetaMathlibExt.exactSlowNimBinaryNatEncoding","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-binary-nat-encoding","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"The bitstring encoding of $\\mathbb{N}$ that writes $0$ as the empty string and a positive integer $n$ as its binary digits, least significant bit first, together with Mathlib's decoding map, which recovers $n$ from its encoding. It is a definition, not an instance; Theorem 3 installs it locally so that lists of piles are encoded as self-delimiting binary blocks.","title":"Binary encoding of natural numbers"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-remoteness-of-list","kind":"node","lean":"MetaMathlibExt.exactSlowNimRemotenessOfList","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-remoteness-of-list","scope":"GameTheory/combinatorial-games/slow-nim","status":"fully_proved","summary":"For a list $(x_0, \\dots, x_{n-1})$ of natural numbers of length $n$, the value is Smith's remoteness of the position $(x_0, \\dots, x_{n-1})$ of Slow$A$-Nim on $n$ stacks with $A = \\{n - 1\\}$, where $0 - 1 = 0$. For $n = k + 1$ this game is exact slow $k$-NIM $\\mathrm{NIM}(k + 1, k)$, so $k$ is part of the input; the empty list gives the game on no stacks.","title":"Remoteness of exact slow k-NIM from a list of piles"},{"id":"GameTheory/combinatorial-games/slow-nim/exact-slow-nim-remoteness-is-poly-time","kind":"node","lean":"MetaMathlibExt.exactSlowNimRemoteness_isPolyTime","parent":44445,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/slow-nim/index.html#exact-slow-nim-remoteness-is-poly-time","scope":"GameTheory/combinatorial-games/slow-nim","status":"can_prove","summary":"The map sending a list $(x_0, \\dots, x_{n-1})$ of natural numbers of length $n$ to Smith's remoteness of the position $(x_0, \\dots, x_{n-1})$ of Slow$A$-Nim on $n$ stacks with $A = \\{n - 1\\}$ (where $0 - 1 = 0$) is polynomial-time computable. Here natural numbers, in the input and the output, are encoded in binary, and a list is encoded as the concatenation of the self-delimiting forms of the encodings of its\u2026","title":"Theorem 3 (Smith's remoteness of exact slow k-NIM is polynomial-time computable)"},{"data_url":"scopes/GameTheory/combinatorial-games/princess-and-the-roses-game.json","id":"GameTheory/combinatorial-games/princess-and-the-roses-game","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","title":"Princess and the Roses game"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/impartial-game-equiv-iff-is-p-game-sum","kind":"node","lean":"MetaMathlibExt.impartialGame_equiv_iff_isP_gameSum","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#impartial-game-equiv-iff-is-p-game-sum","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"Two finite impartial games $G$ and $H$ are equivalent if and only if their sum $G + H$ is a P-position, that is, the position of $G + H$ in which both $G$ and $H$ are at their starting positions is a P-position.","title":"Equivalence of impartial games via their sum"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/is-p-start-iff-of-equiv-nim-heap","kind":"node","lean":"MetaMathlibExt.isP_start_iff_of_equiv_nimHeap","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#is-p-start-iff-of-equiv-nim-heap","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"Let $G$ be a finite impartial game and $n$ a natural number such that $G$ is equivalent to the Nim heap $*n$, that is, $G$ has Sprague\u2013Grundy value $n$. Then the starting position of $G$ is a P-position if and only if $n = 0$.","title":"P-positions are the games of Sprague\u2013Grundy value zero"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/sprague-grundy-value-eq-mex-options","kind":"node","lean":"MetaMathlibExt.spragueGrundyValue_eq_mex_options","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#sprague-grundy-value-eq-mex-options","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"Let $G$ be a finite impartial game and $n$ a natural number such that $G$ is equivalent to the Nim heap $*n$, that is, $G$ has Sprague\u2013Grundy value $n$. For a move from the start position of $G$ to a position $q$, write $G_q$ for the game $G$ started at $q$. Then no $G_q$ is equivalent to $*n$, and for every natural number $m < n$ some $G_q$ is equivalent to $*m$. That is, the Sprague\u2013Grundy value of $G$ is the mex\u2026","title":"Mex rule for Sprague\u2013Grundy values"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses","kind":"node","lean":"MetaMathlibExt.princessRoses","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$ and $p \\in \\mathbb{N}^k$, the Princess and the Roses game (P&R) with $k$ labelled piles started at $p$ is the impartial game whose positions are the $q \\in \\mathbb{N}^k$, with $q_i$ objects on pile $i$, and whose moves remove one object from one nonempty pile, or one object from each of two distinct nonempty piles. It is Slow$A$-Nim $\\mathrm{SN}(k, \\{1, 2\\})$ started at $p$, and its rank is\u2026","title":"The Princess and the Roses game"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/mem-slow-anim-moves-one-two","kind":"node","lean":"MetaMathlibExt.mem_slowANimMoves_one_two","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#mem-slow-anim-moves-one-two","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"Let $p$ and $q$ be positions of Slow$A$-Nim on $k$ stacks with $A = \\{1, 2\\}$, which is the Princess and the Roses game. Then $q$ is an option of $p$ if and only if there are stacks $i \\le j$ with $p_i \\ge 1$ and $p_j \\ge 1$ such that $q_l = p_l - 1$ for $l \\in \\{i, j\\}$ and $q_l = p_l$ for every other stack $l$; that is, $q$ removes one token from stack $i$ when $i = j$, and one token from each of stacks $i$ and\u2026","title":"Options in Princess and the Roses"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/zero-zero-mem-princess-roses-moves-one-o","kind":"node","lean":"MetaMathlibExt.zero_zero_mem_princessRoses_moves_one_one","parent":44507,"reading_title":{"format":"inline-parts-v1","parts":[["text","In two-pile Princess and the Roses, "],["math","\\((1, 1)\\)"],["text"," moves to "],["math","\\((0, 0)\\)"]],"source":"In two-pile Princess and the Roses, $(1, 1)$ moves to $(0, 0)$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#zero-zero-mem-princess-roses-moves-one-o","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"In the Princess and the Roses game on two piles, $(0, 0)$ is an option of the position $(1, 1)$: one may take one object from each pile.","title":"In two-pile Princess and the Roses, $(1, 1)$ moves to $(0, 0)$"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-five-piles-is-p-iff","kind":"node","lean":"MetaMathlibExt.princessRoses_five_piles_isP_iff","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-five-piles-is-p-iff","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"Let $a \\ge b \\ge c \\ge d \\ge e$ be natural numbers. The Princess and the Roses position $(a, b, c, d, e)$ is a P-position if and only if the parities of $a, b, c, d, e$ are, in this order, (even, even, even, even, even), (odd, odd, odd, even, even), (even, odd, odd, odd, odd) or (odd, even, even, odd, odd).","title":"Lim's theorem on five-pile P&R"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-two-piles-equiv-nim-heap","kind":"node","lean":"MetaMathlibExt.princessRoses_two_piles_equiv_nimHeap","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-two-piles-equiv-nim-heap","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $a \\ge b$, the Princess and the Roses position $(a, b)$ is equivalent to the Nim heap $*n$, that is, it has Sprague\u2013Grundy value $n$, where $n = 0$ if $a$ and $b$ are both even, $n = 1$ if $a$ is odd and $b$ is even, $n = 2$ if $a$ and $b$ are both odd, and $n = 3$ if $a$ is even and $b$ is odd.","title":"Sprague\u2013Grundy values of two-pile Princess and the Roses"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-three-piles-equiv-nim-hea","kind":"node","lean":"MetaMathlibExt.princessRoses_three_piles_equiv_nimHeap","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-three-piles-equiv-nim-hea","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $a \\ge b \\ge c$, the Princess and the Roses position $(a, b, c)$ is equivalent to the Nim heap $*n$, that is, it has Sprague\u2013Grundy value $n$, where $n = 0$ if $a$ and $b$ both have the parity of $c$, $n = 1$ if $a$ differs from $c$ in parity and $b$ does not, $n = 2$ if $a$ and $b$ both differ from $c$ in parity, and $n = 3$ if $b$ differs from $c$ in parity and $a$ does not.","title":"Sprague\u2013Grundy values of three-pile Princess and the Roses"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-four-piles-equiv-add-thre","kind":"node","lean":"MetaMathlibExt.princessRoses_four_piles_equiv_add_three_largest","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-four-piles-equiv-add-thre","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $a \\ge b \\ge c \\ge d$ and every natural number $n$, the Princess and the Roses positions $(a, b, c, d)$ and $(a + n, b + n, c + n, d)$ are equivalent; that is, they have the same Sprague\u2013Grundy value.","title":"Adding to the three largest piles of four-pile P&R"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/simplified-princess-roses-move-list","kind":"node","lean":"MetaMathlibExt.simplifiedPrincessRosesMoveList","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#simplified-princess-roses-move-list","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"The simplified Princess and the Roses game on four sorted piles $(a, b, c, d)$ has seven moves, given as the amounts removed from $(a, b, c, d)$: $(1, 0, 0, 0)$, $(0, 1, 0, 0)$, $(0, 0, 0, 1)$, $(1, 1, 0, 0)$, $(1, 0, 0, 1)$, $(0, 1, 0, 1)$ and $(0, 0, 1, 1)$.","title":"Moves of simplified P&R"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/simplified-princess-roses-moves","kind":"node","lean":"MetaMathlibExt.simplifiedPrincessRosesMoves","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#simplified-princess-roses-moves","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"The seven simplified Princess and the Roses moves are the removal vectors $(1, 0, 0, 0)$, $(0, 1, 0, 0)$, $(0, 0, 0, 1)$, $(1, 1, 0, 0)$, $(1, 0, 0, 1)$, $(0, 1, 0, 1)$ and $(0, 0, 1, 1)$. For a quadruple $(a, b, c, d)$ of natural numbers, its options are the quadruples $(a - r_1, b - r_2, c - r_3, d - r_4)$ for those moves $(r_1, r_2, r_3, r_4)$ with $r_1 \\le a$, $r_2 \\le b$, $r_3 \\le c$ and $r_4 \\le d$ whose\u2026","title":"Options in simplified Princess and the Roses"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/mem-simplified-princess-roses-moves","kind":"node","lean":"MetaMathlibExt.mem_simplifiedPrincessRosesMoves","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#mem-simplified-princess-roses-moves","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"For natural numbers $a, b, c, d$ and $a', b', c', d'$, the position $[a', b', c', d']$ is an option of the simplified P&R position $[a, b, c, d]$ if and only if $a' \\le a$, $b' \\le b$, $c' \\le c$, $d' \\le d$, $a' \\ge b' \\ge c' \\ge d'$, and the amounts removed $(a - a', b - b', c - c', d - d')$ form one of the seven simplified P&R moves $(1, 0, 0, 0)$, $(0, 1, 0, 0)$, $(0, 0, 0, 1)$, $(1, 1, 0, 0)$, $(1, 0, 0, 1)$,\u2026","title":"Characterization of simplified P&R options"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/simplified-princess-roses-moves-rank-lt","kind":"node","lean":"MetaMathlibExt.simplifiedPrincessRosesMoves_rank_lt","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#simplified-princess-roses-moves-rank-lt","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"For all quadruples $(a, b, c, d)$ and $(a', b', c', d')$ of natural numbers, if $(a', b', c', d')$ is an option of $(a, b, c, d)$ in simplified Princess and the Roses, then $a' + b' + c' + d' < a + b + c + d$.","title":"Simplified P&R moves decrease the number of objects"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/simplified-princess-roses-moves-three-th","kind":"node","lean":"MetaMathlibExt.simplifiedPrincessRosesMoves_three_three_three_zero","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#simplified-princess-roses-moves-three-th","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"In simplified Princess and the Roses, the position $(3, 3, 3, 0)$ has no options.","title":"A terminal position of simplified P&R"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/simplified-princess-roses","kind":"node","lean":"MetaMathlibExt.simplifiedPrincessRoses","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#simplified-princess-roses","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"For a quadruple $p$ of natural numbers, simplified Princess and the Roses started at $p$ is the impartial game whose positions are all quadruples $(a, b, c, d)$ of natural numbers, whose options from a position are its simplified Princess and the Roses options (the results of the seven simplified moves that take no pile below zero and leave the piles in non-increasing order), whose rank is the total number of\u2026","title":"Simplified Princess and the Roses"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-equiv-simplified-princess","kind":"node","lean":"MetaMathlibExt.princessRoses_equiv_simplifiedPrincessRoses","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-equiv-simplified-princess","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $a \\ge b \\ge c \\ge d$ and every natural number $n$, the Princess and the Roses position $(a, b, c, d)$ is equivalent to the simplified P&R position $[a + n, b + n, c + n, d]$; that is, the two have the same Sprague\u2013Grundy value.","title":"Four-pile P&R is equivalent to simplified P&R"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-move-list","kind":"node","lean":"MetaMathlibExt.princessRosesHelperMoveList","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-move-list","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"The list of the seven moves of P&R Helper on a position $[x, y, z]$, each given as the triple of amounts removed from $(x, y, z)$: in order, $(1, 1, 1)$, $(1, 0, 0)$, $(1, 2, 1)$, $(1, 1, 2)$, $(0, 1, 0)$, $(0, 1, 1)$ and $(0, 0, 1)$. As changes to the position these are $(-1, -1, -1)$, $(-1, 0, 0)$, $(-1, -2, -1)$, $(-1, -1, -2)$, $(0, -1, 0)$, $(0, -1, -1)$ and $(0, 0, -1)$.","title":"Moves of P&R Helper"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-moves","kind":"node","lean":"MetaMathlibExt.princessRosesHelperMoves","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-moves","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"The P&R Helper game is played on triples of natural numbers with the seven moves that subtract $(1, 1, 1)$, $(1, 0, 0)$, $(1, 2, 1)$, $(1, 1, 2)$, $(0, 1, 0)$, $(0, 1, 1)$ or $(0, 0, 1)$. The options of a position $(x, y, z)$ are the triples $(x - u, y - v, z - w)$ for those moves $(u, v, w)$ with $u \\le x$, $v \\le y$, $w \\le z$ and $x - u \\le y - v \\le z - w$.","title":"Options in P&R Helper"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/mem-princess-roses-helper-moves","kind":"node","lean":"MetaMathlibExt.mem_princessRosesHelperMoves","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#mem-princess-roses-helper-moves","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"For natural numbers $x, y, z, x', y', z'$, the position $[x', y', z']$ is an option of the P&R Helper position $[x, y, z]$ if and only if $x' \\le x$, $y' \\le y$, $z' \\le z$, $x' \\le y' \\le z'$, and the amounts removed $(x - x', y - y', z - z')$ form one of the seven P&R Helper moves $(1, 1, 1)$, $(1, 0, 0)$, $(1, 2, 1)$, $(1, 1, 2)$, $(0, 1, 0)$, $(0, 1, 1)$ and $(0, 0, 1)$.","title":"Characterization of P&R Helper options"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-moves-rank-lt","kind":"node","lean":"MetaMathlibExt.princessRosesHelperMoves_rank_lt","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-moves-rank-lt","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"For all triples $(x, y, z)$ and $(x', y', z')$ of natural numbers, if $(x', y', z')$ is an option of $(x, y, z)$ in the P&R Helper game, then $x' + y' + z' < x + y + z$.","title":"P&R Helper moves decrease the coordinate sum"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-moves-small","kind":"node","lean":"MetaMathlibExt.princessRosesHelperMoves_small","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-moves-small","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"In the P&R Helper game, the position $(0, 0, 0)$ has no options, and the options of $(1, 1, 1)$ are exactly $(0, 0, 0)$ and $(0, 1, 1)$.","title":"Small P&R Helper positions"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper","kind":"node","lean":"MetaMathlibExt.princessRosesHelper","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"fully_proved","summary":"For a triple $p$ of natural numbers, P&R Helper started at $p$ is the impartial game whose positions are the triples $(x, y, z)$ of natural numbers, whose options from $(x, y, z)$ are the triples $(x - u, y - v, z - w)$ for $(u, v, w)$ among $(1, 1, 1)$, $(1, 0, 0)$, $(1, 2, 1)$, $(1, 1, 2)$, $(0, 1, 0)$, $(0, 1, 1)$ and $(0, 0, 1)$ with $u \\le x$, $v \\le y$, $w \\le z$ and $x - u \\le y - v \\le z - w$, whose rank is\u2026","title":"The P&R Helper game"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-equiv-princess-roses-help","kind":"node","lean":"MetaMathlibExt.princessRoses_equiv_princessRosesHelper","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-equiv-princess-roses-help","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $a \\ge b \\ge c \\ge d$, the Princess and the Roses position $(a, b, c, d)$ is equivalent to the P&R Helper position $[d, b - c + d, a - c + d]$; that is, the two have the same Sprague\u2013Grundy value.","title":"Four-pile P&R reduces to P&R Helper"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-zer","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_zero_iff","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-zer","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $x \\le y \\le z$, the P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $0$ (that is, has Sprague\u2013Grundy value $0$) if and only if $x$, $y$ and $z$ are all even.","title":"P&R Helper positions of Sprague\u2013Grundy value 0"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-one","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_one_iff_of_lt_add","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-one","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $x \\le y \\le z$ with $z < x + y$ (the union of the regions $U$ and $R$), the P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $1$ (that is, has Sprague\u2013Grundy value $1$) if and only if one of the following holds: $(x, y, z) = (1, 1, 1)$; or $y = x$, $z = x$ and $x \\equiv 3 \\pmod 4$; or $y = x$, $z = x + 1$, $x \\equiv 1 \\pmod 4$ and $x \\ne 1$; or $y = x + 1$,\u2026","title":"P&R Helper positions of Sprague\u2013Grundy value 1 on regions U and R"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-two","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_two_iff_of_mem_U","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-two","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $x \\le y \\le z$ with $z < x + y$ and $y \\le 2x + 1$ (the region $U$), write $s = y + z - x$. The P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $2$ (that is, has Sprague\u2013Grundy value $2$) if and only if one of the following holds: (a) $y = x$, $z = x$, $x \\equiv 1 \\pmod 4$ and $x \\ne 1$; (b) $y = x$, $z = x + 1$ and $x \\equiv 3 \\pmod 4$; (c) $y = x + 1$,\u2026","title":"P&R Helper positions of Sprague\u2013Grundy value 2 on region U"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-thr","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_three_iff_of_mem_U","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-thr","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $x \\le y \\le z$ with $z < x + y$ and $y \\le 2x + 1$ (the region $U$), write $s = y + z - x$. The P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $3$ (that is, has Sprague\u2013Grundy value $3$) if and only if one of the following holds: (a) $(x, y, z)$ is one of $(1, 2, 2)$, $(2, 2, 3)$, $(5, 6, 6)$, $(6, 6, 7)$, $(3, 4, 5)$, $(5, 5, 8)$; (b) $y = x$, $z = x + 1$,\u2026","title":"P&R Helper positions of Sprague\u2013Grundy value 3 on region U"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-fou","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_four_iff_of_mem_U","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-fou","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $x \\le y \\le z$ with $z < x + y$ and $y \\le 2x + 1$ (the region $U$), write $s = y + z - x$. The P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $4$ (that is, has Sprague\u2013Grundy value $4$) if and only if one of the following holds: (a) $(x, y, z)$ is one of $(2, 3, 3)$, $(5, 6, 7)$, $(4, 4, 5)$, $(7, 8, 8)$; (b) $y = x$, $z = x + 1$, $x \\equiv 2 \\pmod 4$ and\u2026","title":"P&R Helper positions of Sprague\u2013Grundy value 4 on region U"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-fiv","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_five_iff_of_mem_U","parent":44507,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sprague\u2013Grundy value "],["math","\\(5\\)"],["text"," of P&R Helper on region "],["math","\\(U\\)"]],"source":"Sprague\u2013Grundy value $5$ of P&R Helper on region $U$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-fiv","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"Let $x \\le y \\le z$ be natural numbers with $z < x + y$ and $y \\le 2x + 1$ (the region $U$). The P&R Helper position $[x, y, z]$ is equivalent to the Nim heap $*5$, that is, has Sprague\u2013Grundy value $5$, if and only if one of the following holds:","title":"Sprague\u2013Grundy value $5$ of P&R Helper on region $U$"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-six","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_six_iff_of_mem_U","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-six","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"For natural numbers $x \\le y \\le z$ with $z < x + y$ and $y \\le 2x + 1$ (the region $U$), the P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $6$ (that is, has Sprague\u2013Grundy value $6$) if and only if $y = x + 2$, $z = x + 5$ and $x \\equiv 1 \\pmod 8$; and it is never equivalent to the Nim heap of size $7$.","title":"P&R Helper positions of Sprague\u2013Grundy value 6 or 7 on region U"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-iff","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_iff_of_mem_R","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-iff","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"Let $x \\le y \\le z$ be natural numbers with $z < x + y$ and $y > 2x + 1$ (the region $R$), and write $s = y + z - x$. For every natural number $n$, the P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $n$ (that is, has Sprague\u2013Grundy value $n$) if and only if one of the following holds: $n = 0$ and $x$, $y$, $z$ are all even; $n = 1$ and $s \\equiv 1 \\pmod 4$; $n = 2$, $x$ is even and\u2026","title":"Sprague\u2013Grundy values of P&R Helper on region R"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-iff-2","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_iff_of_mem_L","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-iff-2","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"Let $x \\le y \\le z$ be natural numbers with $z \\ge x + y$ and $y \\le 2x$ (the region $L$). For every natural number $n$, the P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $n$ (that is, has Sprague\u2013Grundy value $n$) if and only if one of the following holds: $n = 0$ and $x$, $y$, $z$ are all even; $n = 1$, $x$ and $y$ are even and $z$ is odd; $n = 2$ and $y + 2z - 2x \\equiv 3 \\pmod 4$;\u2026","title":"Sprague\u2013Grundy values of P&R Helper on region L"},{"id":"GameTheory/combinatorial-games/princess-and-the-roses-game/princess-roses-helper-equiv-nim-heap-iff-3","kind":"node","lean":"MetaMathlibExt.princessRosesHelper_equiv_nimHeap_iff_of_mem_D","parent":44507,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/princess-and-the-roses-game/index.html#princess-roses-helper-equiv-nim-heap-iff-3","scope":"GameTheory/combinatorial-games/princess-and-the-roses-game","status":"can_prove","summary":"Let $x \\le y \\le z$ be natural numbers with $z \\ge x + y$ and $y > 2x$ (the region $D$). For every natural number $n$, the P&R Helper game started at $[x, y, z]$ is equivalent to the Nim heap of size $n$ (that is, has Sprague\u2013Grundy value $n$) if and only if $n$ is the number assigned to the parities of $(x, y, z)$ as follows: $0$ to (even, even, even), $1$ to (even, even, odd), $2$ to (even, odd, odd), $3$ to\u2026","title":"Sprague\u2013Grundy values of P&R Helper on region D"},{"data_url":"scopes/GameTheory/combinatorial-games/sylver-coinage.json","id":"GameTheory/combinatorial-games/sylver-coinage","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","title":"Sylver coinage"},{"id":"GameTheory/combinatorial-games/sylver-coinage/sylver-legal-moves","kind":"node","lean":"MetaMathlibExt.sylverLegalMoves","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#sylver-legal-moves","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"fully_proved","summary":"For a set $M$ of natural numbers (a position of Sylver Coinage), the set $L(M)$ of legal moves consists of the natural numbers that are not finite sums of elements of $M$, repetitions allowed, i.e. the natural numbers outside the additive submonoid of $\\mathbb{N}$ generated by $M$. Since $0$ is the empty sum, every legal move is positive.","title":"Legal moves in Sylver Coinage"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-pposition","kind":"node","lean":"MetaMathlibExt.IsSylverPPosition","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-pposition","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"fully_proved","summary":"A set $M$ of natural numbers is a P-position of Sylver Coinage when $1 \\notin M$ and, for every legal move $q$ of $M$, the position $M \\cup \\{q\\}$ is an N-position. P-positions and N-positions are defined together, as a mutual inductive definition. In particular a position whose only legal move is $1$ is a P-position (the player to move must name $1$ and loses).","title":"P-positions of Sylver Coinage"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-nposition","kind":"node","lean":"MetaMathlibExt.IsSylverNPosition","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-nposition","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"fully_proved","summary":"A position $M$ of Sylver Coinage is an N-position (the player to move wins) when either $1 \\in M$, so that the previous player named $1$ and lost, or some legal move $q$ of $M$ leads to a P-position $M \\cup \\{q\\}$. N-positions and P-positions are defined together inductively by these rules.","title":"N-positions of Sylver Coinage"},{"id":"GameTheory/combinatorial-games/sylver-coinage/zero-not-mem-sylver-legal-moves","kind":"node","lean":"MetaMathlibExt.zero_notMem_sylverLegalMoves","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#zero-not-mem-sylver-legal-moves","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"fully_proved","summary":"For every set $M$ of natural numbers, $0$ is not a legal move of the Sylver Coinage position $M$.","title":"Zero is never a legal move"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-pposition-two-three","kind":"node","lean":"MetaMathlibExt.isSylverPPosition_two_three","parent":44542,"reading_title":{"format":"inline-parts-v1","parts":[["text","The position "],["math","\\(\\{2, 3\\}\\)"],["text"," is a P-position"]],"source":"The position $\\{2, 3\\}$ is a P-position"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-pposition-two-three","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"fully_proved","summary":"In Sylver Coinage, the position $\\{2, 3\\}$ is a P-position: its only legal move is $1$.","title":"The position $\\{2, 3\\}$ is a P-position"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-nposition-two","kind":"node","lean":"MetaMathlibExt.isSylverNPosition_two","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-nposition-two","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"fully_proved","summary":"The position $\\{2\\}$ of Sylver Coinage is an N-position: the player to move wins by naming $3$, which leaves the P-position $\\{2, 3\\}$.","title":"Opening with 2 loses"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-pposition-prime-opening","kind":"node","lean":"MetaMathlibExt.isSylverPPosition_prime_opening","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-pposition-prime-opening","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"For every prime $p > 3$, the position $\\{p\\}$ of Sylver Coinage is a P-position: a first player who opens with $p$ wins.","title":"Hutchings' theorem"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-end","kind":"node","lean":"MetaMathlibExt.IsSylverEnd","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-end","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"fully_proved","summary":"Let $M$ be a position of Sylver Coinage with set of legal moves $L(M)$. A natural number $x$ is an end of $M$ if $x \\in L(M)$ and every $y \\in L(M)$ with $y \\ne x$ is still a legal move of the position $M \\cup \\{x\\}$; that is, naming $x$ eliminates no other legal move.","title":"Ends of a Sylver Coinage position"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-end-of-is-greatest","kind":"node","lean":"MetaMathlibExt.isSylverEnd_of_isGreatest","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-end-of-is-greatest","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"fully_proved","summary":"Let $M$ be a position of Sylver Coinage whose set of legal moves $L(M)$ has a greatest element $F$. Then $F$ is an end of $M$.","title":"The largest legal move is an end"},{"id":"GameTheory/combinatorial-games/sylver-coinage/sylver-ender-eliminates-largest-move","kind":"node","lean":"MetaMathlibExt.sylverEnder_eliminates_largest_move","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#sylver-ender-eliminates-largest-move","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Let $M$ be a set of natural numbers whose set of legal moves $L(M)$ has a greatest element $F > 1$, and suppose that $F$ is the only end of $M$ (so $M$ is an ender). Then for every legal move $x$ of $M$ with $x < F$, the number $F$ is not a legal move of $M \\cup \\{x\\}$.","title":"In an ender, every smaller move eliminates the largest legal move"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-nposition-of-ender","kind":"node","lean":"MetaMathlibExt.isSylverNPosition_of_ender","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-nposition-of-ender","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Let $M$ be a position of Sylver Coinage whose set of legal moves $L(M)$ has a greatest element $F$ with $F > 1$. If every end of $M$ equals $F$ (that is, $M$ is an ender), then $M$ is an N-position.","title":"Enders with largest legal move above 1 are N-positions"},{"id":"GameTheory/combinatorial-games/sylver-coinage/numerical-semigroup-two-generated-symmet","kind":"node","lean":"MetaMathlibExt.numericalSemigroup_two_generated_symmetric","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#numerical-semigroup-two-generated-symmet","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Let $a$, $b$ and $F$ be natural numbers and let $S = \\langle a, b \\rangle$ be the additive submonoid of $\\mathbb{N}$ generated by $a$ and $b$. If $F \\notin S$ and every natural number $n > F$ lies in $S$ (so $F$ is the Frobenius number of $S$), then $x \\in S \\iff F - x \\notin S$ for every natural number $x \\le F$.","title":"Two-generated numerical semigroups are symmetric"},{"id":"GameTheory/combinatorial-games/sylver-coinage/numerical-semigroup-four-pseudo-symmetri","kind":"node","lean":"MetaMathlibExt.numericalSemigroup_four_pseudoSymmetric_iff","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#numerical-semigroup-four-pseudo-symmetri","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Write $\\langle A \\rangle$ for the additive submonoid of $\\mathbb{N}$ generated by a set $A$. Let $a$, $b$ and $F$ be natural numbers with $4 < a < b$, $a \\notin \\langle 4 \\rangle$ and $b \\notin \\langle 4, a \\rangle$ (so $\\{4, a, b\\}$ is the minimal generating set of $S = \\langle 4, a, b \\rangle$), and suppose that $F \\notin S$ and that every natural number $n > F$ lies in $S$. Then $S$ is pseudo-symmetric, meaning\u2026","title":"Pseudo-symmetric semigroups of multiplicity 4 and embedding dimension 3"},{"id":"GameTheory/combinatorial-games/sylver-coinage/sylver-position-canonical-eq-minimal-gen","kind":"node","lean":"MetaMathlibExt.sylverPosition_canonical_eq_minimalGenerators","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#sylver-position-canonical-eq-minimal-gen","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Let $M$ be a finite set of natural numbers whose set of legal moves $L(M)$ is finite, and suppose that removing any element changes the legal moves: $L(M \\setminus \\{a\\}) \\ne L(M)$ for every $a \\in M$. Let $S$ be the additive submonoid of $\\mathbb{N}$ generated by $M$. Then $M$ is the set of nonzero elements $s \\in S$ such that whenever $s = a + b$ with $a, b \\in S$, one of $a$ and $b$ is $0$; that is, $M$ is the\u2026","title":"A canonical position is the minimal generating set of its semigroup"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-nposition-four-odd-even","kind":"node","lean":"MetaMathlibExt.isSylverNPosition_four_odd_even","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-nposition-four-odd-even","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Let $x \\ge 5$ be an odd natural number. For every even legal move $b$ of the position $\\{4, x\\}$, the position $\\{4, x, b\\}$ is an N-position.","title":"Even replies to 4 and an odd number at least 5 lose"},{"id":"GameTheory/combinatorial-games/sylver-coinage/exists-unique-sylver-response-four-odd","kind":"node","lean":"MetaMathlibExt.existsUnique_sylver_response_four_odd","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#exists-unique-sylver-response-four-odd","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"For every odd natural number $x \\ge 5$ there is exactly one legal move $q$ of the position $\\{4, x\\}$ such that $\\{4, x, q\\}$ is a P-position, and every legal move $q$ of $\\{4, x\\}$ with $\\{4, x, q\\}$ a P-position satisfies $q \\equiv x - 2 \\pmod 4$.","title":"Sicherman's claim on replies to 4 and an odd number"},{"id":"GameTheory/combinatorial-games/sylver-coinage/exists-unique-sylver-four-odd-partner","kind":"node","lean":"MetaMathlibExt.existsUnique_sylver_four_odd_partner","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#exists-unique-sylver-four-odd-partner","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"For every odd natural number $q > 3$ there is exactly one odd natural number $x$ such that $\\{4, x\\}$ is an N-position and $\\{4, x, q\\}$ is a P-position.","title":"Odd numbers above 3 have a unique odd partner after 4"},{"id":"GameTheory/combinatorial-games/sylver-coinage/sylver-four-pair-exists-unique","kind":"node","lean":"MetaMathlibExt.sylver_four_pair_existsUnique","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#sylver-four-pair-exists-unique","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Let $q > 3$ be an odd natural number. Then $\\{4, q\\}$ is an N-position of Sylver Coinage, and there is a unique odd natural number $x_q$ such that $\\{4, x_q\\}$ is an N-position and $\\{4, q, x_q\\}$ is a P-position.","title":"Unique odd partner after 4"},{"id":"GameTheory/combinatorial-games/sylver-coinage/sylver-four-pair-bounds","kind":"node","lean":"MetaMathlibExt.sylver_four_pair_bounds","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#sylver-four-pair-bounds","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Let $q$ and $x$ be odd natural numbers with $7 < q < x$ such that $\\{4, x\\}$ is an N-position and $\\{4, x, q\\}$ is a P-position of Sylver Coinage. Then $2q < x < 3q$.","title":"Conway's bounds for 4-pairs"},{"id":"GameTheory/combinatorial-games/sylver-coinage/is-sylver-pposition-four-odd-pair-even","kind":"node","lean":"MetaMathlibExt.isSylverPPosition_four_odd_pair_even","parent":44542,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/sylver-coinage/index.html#is-sylver-pposition-four-odd-pair-even","scope":"GameTheory/combinatorial-games/sylver-coinage","status":"can_prove","summary":"Let $x > 1$ be a natural number. If $x \\equiv 1 \\pmod 4$, then $\\{4, x, x + 2, 6\\}$ is a P-position; if $x \\equiv 7 \\pmod 8$, then $\\{4, x, x + 2, 10\\}$ is a P-position; if $x \\equiv 5 \\pmod 8$, then $\\{4, x, x + 6, 10\\}$ is a P-position; and if $x \\equiv 1 \\pmod 8$, then $\\{4, x, x + 6, 14\\}$ is a P-position.","title":"Four-element P-positions containing 4"},{"data_url":"scopes/GameTheory/combinatorial-games/wythoff-s-game.json","id":"GameTheory/combinatorial-games/wythoff-s-game","kind":"scope","navigation":"module","parent":44142,"result":false,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","title":"Wythoff's game"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-extra-move-grundy","kind":"node","lean":"MetaMathlibExt.wythoffExtraMoveGrundy","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-extra-move-grundy","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Positions are triples $(x, y, p)$ with $x, y \\in \\mathbb{N}$ and $p \\in \\{0, 1\\}$. A move from $(x, y, p)$ goes to $(u, y, p)$ with $u < x$, to $(x, v, p)$ with $v < y$, or to $(x - t, y - t, p)$ with $1 \\le t \\le \\min(x, y)$ (the moves of a Queen in Wythoff's game), or, when $p = 1$, it is an extra move to $(x, y, 0)$. There are two variants. In the strict variant the extra move is the one-time pass of Wythoff's\u2026","title":"Grundy numbers of Wythoff's game with an extra move"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-grundy","kind":"node","lean":"MetaMathlibExt.wythoffPassGrundy","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-grundy","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Wythoff's game with a pass has positions $(x, y, p)$ with $x, y \\in \\mathbb{N}$ and $p \\in \\{0, 1\\}$, where $p = 1$ means the one-time pass is still available. A move goes from $(x, y, p)$ to $(u, y, p)$ with $u < x$, to $(x, v, p)$ with $v < y$, or to $(x - t, y - t, p)$ with $1 \\le t \\le \\min(x, y)$, or, when $p = 1$ and $x + y > 0$, it is the pass to $(x, y, 0)$. $\\mathcal{G}_1(x, y, p)$ is the Grundy number of\u2026","title":"Grundy numbers of Wythoff's game with a pass"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-plus-stone-grundy","kind":"node","lean":"MetaMathlibExt.wythoffPlusStoneGrundy","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-plus-stone-grundy","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"For natural numbers $x, y$ and $p \\in \\{0, 1\\}$, $\\mathcal{G}_2(x, y, p)$ is the Grundy number of the position $(x, y, p)$ in the sum of Wythoff's game on $(x, y)$ and a pile of $p$ stones. That is, it is the Grundy function of Wythoff's game with the extra move from $(x, y, 1)$ to $(x, y, 0)$ always available, including at $(0, 0, 1)$.","title":"Grundy numbers of Wythoff's game plus one stone"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-set-a","kind":"node","lean":"MetaMathlibExt.wythoffPassSetA","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(A\\)"]],"source":"The set $A$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-set-a","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"$A$ is the finite set of pairs of natural numbers $A = \\{(0, 0), (1, 2), (2, 1), (3, 5), (4, 7), (5, 3), (7, 4)\\}$. These are the positions $(x, y, 0)$ of the set $A$ in the source, with the third coordinate dropped.","title":"The set $A$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-set-b","kind":"node","lean":"MetaMathlibExt.wythoffPassSetB","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(B\\)"]],"source":"The set $B$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-set-b","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"$B$ is the finite set of pairs of natural numbers $B = \\{(0, 0), (1, 3), (3, 1), (2, 5), (5, 2), (4, 8), (8, 4), (6, 7), (7, 6)\\}$. These are the positions $(x, y, 1)$ of the set $B$ in the source, with the third coordinate dropped.","title":"The set $B$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-set-c","kind":"node","lean":"MetaMathlibExt.wythoffPassSetC","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(C\\)"]],"source":"The set $C$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-set-c","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"$C$ is the finite set of pairs of natural numbers $C = \\{(0, 1), (1, 0), (2, 2), (3, 6), (6, 3), (4, 8), (8, 4), (5, 7), (7, 5)\\}$. These are the positions $(x, y, 1)$ of the set $C$ in the source, with the third coordinate dropped.","title":"The set $C$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-extra-move-grundy-eq","kind":"node","lean":"MetaMathlibExt.wythoffExtraMoveGrundy_eq","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-extra-move-grundy-eq","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"In both variants (strict: the pass of Wythoff's game with a pass; non-strict: a pile of one stone), the Grundy number of a position $(x, y, p)$ with $x, y \\in \\mathbb{N}$ and $p \\in \\{0, 1\\}$ is the least natural number $m$ that differs from the Grundy number of $(u, y, p)$ for every $u < x$, from that of $(x, v, p)$ for every $v < y$, and from that of $(x - t, y - t, p)$ for every $t$ with $1 \\le t \\le x$ and\u2026","title":"Mex recursion for the Grundy numbers of Wythoff's game with an extra move"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-grundy-zero-zero","kind":"node","lean":"MetaMathlibExt.wythoffPassGrundy_zero_zero","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-grundy-zero-zero","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $\\mathcal{G}_1(x, y, p)$ be the Grundy numbers of Wythoff's game with a pass, where $p = 1$ means the pass is still available. Then $\\mathcal{G}_1(0, 0, p) = 0$ for both $p = 0$ and $p = 1$: the terminal position $(0, 0)$ has Grundy number $0$ with or without the pass available, since the pass is not allowed from a terminal position.","title":"The terminal position of Wythoff's game with a pass"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-plus-stone-grundy-zero-zero-true","kind":"node","lean":"MetaMathlibExt.wythoffPlusStoneGrundy_zero_zero_true","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wythoff's game plus one stone at "],["math","\\((0, 0, 1)\\)"]],"source":"Wythoff's game plus one stone at $(0, 0, 1)$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-plus-stone-grundy-zero-zero-true","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"In the sum of Wythoff's game and a pile of one stone, the position $(0, 0, 1)$ (no Wythoff moves left, the stone present) has Grundy number $\\mathcal{G}_2(0, 0, 1) = 1$: its only move is to the terminal position $(0, 0, 0)$.","title":"Wythoff's game plus one stone at $(0, 0, 1)$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-set-b-card-wythoff-pass-set","kind":"node","lean":"MetaMathlibExt.wythoffPassSetB_card_wythoffPassSetC_card","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sets "],["math","\\(B\\)"],["text"," and "],["math","\\(C\\)"],["text"," have nine elements each"]],"source":"The sets $B$ and $C$ have nine elements each"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-set-b-card-wythoff-pass-set","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"The sets $B = \\{(0, 0), (1, 3), (3, 1), (2, 5), (5, 2), (4, 8), (8, 4), (6, 7), (7, 6)\\}$ and $C = \\{(0, 1), (1, 0), (2, 2), (3, 6), (6, 3), (4, 8), (8, 4), (5, 7), (7, 5)\\}$ of pairs of natural numbers each have exactly $9$ elements.","title":"The sets $B$ and $C$ have nine elements each"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-set-a-card","kind":"node","lean":"MetaMathlibExt.wythoffPassSetA_card","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The set "],["math","\\(A\\)"],["text"," has seven elements"]],"source":"The set $A$ has seven elements"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-set-a-card","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"The set $A = \\{(0, 0), (1, 2), (2, 1), (3, 5), (4, 7), (5, 3), (7, 4)\\}$ of pairs of natural numbers has exactly $7$ elements.","title":"The set $A$ has seven elements"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-grundy-eq-zero-iff","kind":"node","lean":"MetaMathlibExt.wythoffGrundy_eq_zero_iff","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-grundy-eq-zero-iff","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\mathcal{G}_1(x, y, 0)$ be the Grundy number of the position $(x, y, 0)$ of Wythoff's game with a pass, in which the pass has been used; this is the Grundy number of $(x, y)$ in classical Wythoff's game. Then $\\mathcal{G}_1(x, y, 0) = 0$ (that is, $(x, y)$ is a P-position) if and only if there is $n \\in \\mathbb{N}$ with $(x, y) = (\\lfloor n\\varphi \\rfloor, \\lfloor n\\varphi \\rfloor + n)$ or\u2026","title":"Wythoff's theorem"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-grundy-one-near-zero","kind":"node","lean":"MetaMathlibExt.wythoffGrundy_one_near_zero","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-grundy-one-near-zero","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\mathcal{G}_1(x, y, 0)$ be the Grundy number of the position $(x, y, 0)$ of Wythoff's game with a pass, in which the pass has been used; this is the Grundy number of $(x, y)$ in classical Wythoff's game. If $\\mathcal{G}_1(x, y, 0) = 1$, then there are $v, w \\in \\mathbb{N}$ with $\\mathcal{G}_1(v, w, 0) = 0$ and $(x - v)^2 + (y - w)^2 \\le 20$: every position with Grundy number $1$ lies within Euclidean distance\u2026","title":"Grundy-one positions of Wythoff's game lie near P-positions"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-p-positions-le-eight","kind":"node","lean":"MetaMathlibExt.wythoffPass_pPositions_le_eight","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","P-positions of Wythoff's game with a pass for "],["math","\\(x, y \\le 8\\)"]],"source":"P-positions of Wythoff's game with a pass for $x, y \\le 8$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-p-positions-le-eight","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\mathcal{G}_1(x, y, p)$ be the Grundy number of Wythoff's game with a one-time pass ($p = 1$: the pass is still available; $p = 0$: classical Wythoff's game), and $\\mathcal{G}_2(x, y, p)$ the Grundy number of the sum of Wythoff's game and a pile of $p$ stones. Let $A = \\{(0, 0), (1, 2), (2, 1), (3, 5), (5, 3), (4, 7), (7, 4)\\}$, $B = \\{(0, 0), (1, 3), (3, 1), (2, 5), (5, 2), (4, 8), (8, 4), (6, 7), (7, 6)\\}$\u2026","title":"P-positions of Wythoff's game with a pass for $x, y \\le 8$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-moves-meet-set-b-set-c","kind":"node","lean":"MetaMathlibExt.wythoffPass_moves_meet_setB_setC","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moves into the sets "],["math","\\(B\\)"],["text"," and "],["math","\\(C\\)"],["text"," away from the corner"]],"source":"Moves into the sets $B$ and $C$ away from the corner"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-moves-meet-set-b-set-c","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $B = \\{(0, 0), (1, 3), (3, 1), (2, 5), (5, 2), (4, 8), (8, 4), (6, 7), (7, 6)\\}$ and $C = \\{(0, 1), (1, 0), (2, 2), (3, 6), (6, 3), (4, 8), (8, 4), (5, 7), (7, 5)\\}$. For natural numbers $x, y$, call the pairs $(u, y)$ with $u < x$ the horizontal moves from $(x, y)$, the pairs $(x, v)$ with $v < y$ its vertical moves, and the pairs $(x - t, y - t)$ with $1 \\le t \\le \\min(x, y)$ its diagonal moves. Suppose that\u2026","title":"Moves into the sets $B$ and $C$ away from the corner"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-moves-meet-set-b-iff-set-c","kind":"node","lean":"MetaMathlibExt.wythoffPass_moves_meet_setB_iff_setC","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Moves from a large position meet "],["math","\\(B\\)"],["text"," exactly when they meet "],["math","\\(C\\)"]],"source":"Moves from a large position meet $B$ exactly when they meet $C$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-moves-meet-set-b-iff-set-c","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $B = \\{(0, 0), (1, 3), (3, 1), (2, 5), (5, 2), (4, 8), (8, 4), (6, 7), (7, 6)\\}$ and $C = \\{(0, 1), (1, 0), (2, 2), (3, 6), (6, 3), (4, 8), (8, 4), (5, 7), (7, 5)\\}$. Let $x, y$ be natural numbers with $x \\ge 9$ or $y \\ge 9$. Then some pair $(u, y)$ with $u < x$ lies in $B$ if and only if some such pair lies in $C$; some pair $(x, v)$ with $v < y$ lies in $B$ if and only if some such pair lies in $C$; and some\u2026","title":"Moves from a large position meet $B$ exactly when they meet $C$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-grundy-eq-zero-iff-plus-sto","kind":"node","lean":"MetaMathlibExt.wythoffPassGrundy_eq_zero_iff_plusStone","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-grundy-eq-zero-iff-plus-sto","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\mathcal{G}_1(x, y, p)$ be the Grundy numbers of Wythoff's game with a pass, where $p = 1$ means the pass is still available, and let $\\mathcal{G}_2(x, y, p)$ be the Grundy numbers of the sum of Wythoff's game and a pile of $p \\in \\{0, 1\\}$ stones. If $x \\ge 9$ or $y \\ge 9$, then $\\mathcal{G}_1(x, y, 1) = 0$ if and only if $\\mathcal{G}_2(x, y, 1) = 0$: away from the $9 \\times 9$ corner, Wythoff's game with an\u2026","title":"Pass versus a pile of one stone in Wythoff's game"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-grundy-true-eq-zero-iff","kind":"node","lean":"MetaMathlibExt.wythoffPassGrundy_true_eq_zero_iff","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-grundy-true-eq-zero-iff","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\mathcal{G}_1(x, y, p)$ be the Grundy numbers of Wythoff's game with a pass, where $p = 1$ means the pass is still available and $p = 0$ means it has been used, so that $\\mathcal{G}_1(x, y, 0)$ is the Grundy number of $(x, y)$ in classical Wythoff's game. If $x \\ge 9$ or $y \\ge 9$, then $\\mathcal{G}_1(x, y, 1) = 0$ if and only if $\\mathcal{G}_1(x, y, 0) = 1$.","title":"Away from the corner, P-positions with a pass are Wythoff's Grundy-one positions"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-pass-p-position-near-wythoff-p-p","kind":"node","lean":"MetaMathlibExt.wythoffPass_pPosition_near_wythoff_pPosition","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-pass-p-position-near-wythoff-p-p","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\mathcal{G}_1(x, y, p)$ be the Grundy number of Wythoff's game with a one-time pass, where $p = 1$ means the pass is still available and $p = 0$ gives classical Wythoff's game. For all natural numbers $x, y$ with $\\mathcal{G}_1(x, y, 1) = 0$ there exist natural numbers $v, w$ with $\\mathcal{G}_1(v, w, 0) = 0$ and $(x - v)^2 + (y - w)^2 \\le 20$, that is, $(v, w)$ lies within Euclidean distance $\\sqrt{20}$ of\u2026","title":"P-positions with the pass available lie near Wythoff P-positions"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-move","kind":"node","lean":"MetaMathlibExt.wythoffTerminalMove","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-move","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $\\ell$ be a natural number. In the game $K^\\ell$, whose positions are pairs $(x, y)$ of natural numbers (the sizes of two piles of tokens), there is a move from $(x, y)$ to $(x', y')$ when $x + y > \\ell$ and either $y' = y$ and $x' < x$, or $x' = x$ and $y' < y$, or $x = x' + k$ and $y = y' + k$ for some $k > 0$. Thus the positions with $x + y \\le \\ell$ are terminal, and from any other position a move removes a\u2026","title":"Moves of Wythoff's game with terminal positions"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/not-wythoff-terminal-move-of-terminal","kind":"node","lean":"MetaMathlibExt.not_wythoffTerminalMove_of_terminal","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#not-wythoff-terminal-move-of-terminal","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $\\ell$ be a natural number. In the game $K^\\ell$ (Wythoff's game in which the positions $(x, y)$ with $x + y \\le \\ell$ are terminal), there is no move from a position $(x, y)$ with $x + y \\le \\ell$ to any position.","title":"No moves from terminal positions"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-move-zero-one-zero","kind":"node","lean":"MetaMathlibExt.wythoffTerminalMove_zero_one_zero","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","A move to the origin in "],["math","\\(K^0\\)"]],"source":"A move to the origin in $K^0$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-move-zero-one-zero","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"In the game $K^0$ (Wythoff's game in which only $(0, 0)$ is terminal), there is a move from the position $(1, 0)$ to the position $(0, 0)$.","title":"A move to the origin in $K^0$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-mex-a","kind":"node","lean":"MetaMathlibExt.wythoffMexA","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-mex-a","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Fix a natural number $\\ell$. The sequence $(a_n)_{n \\ge 0}$ is defined recursively: $a_n$ is the least natural number greater than $\\ell$ that differs from $a_i$ and from $a_i + i + \\ell + 1$ for every $i < n$. Writing $b_i = a_i + i + \\ell + 1$, this says that $a_n$ is the least natural number outside $\\{0, 1, \\dots, \\ell\\} \\cup \\{a_i, b_i : i < n\\}$; in particular $a_0 = \\ell + 1$ and $b_0 = 2\\ell + 2$.","title":"The mex sequence of Wythoff's game with terminal positions"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-mex-b","kind":"node","lean":"MetaMathlibExt.wythoffMexB","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-mex-b","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Fix a natural number $\\ell$ and let $(a_n)$ be the mex sequence of $K^\\ell$ (Wythoff's game in which the positions $(x, y)$ with $x + y \\le \\ell$ are terminal). Its companion sequence is $b_n = a_n + n + \\ell + 1$ for every natural number $n$.","title":"The companion of the mex sequence"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-mex-a-zero","kind":"node","lean":"MetaMathlibExt.wythoffMexA_zero","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The term "],["math","\\(a_0\\)"],["text"," of the mex sequence"]],"source":"The term $a_0$ of the mex sequence"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-mex-a-zero","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"For every natural number $\\ell$, the mex sequence $(a_n)$ of $K^\\ell$ (Wythoff's game in which the positions $(x, y)$ with $x + y \\le \\ell$ are terminal) satisfies $a_0 = \\ell + 1$.","title":"The term $a_0$ of the mex sequence"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-mex-a-one","kind":"node","lean":"MetaMathlibExt.wythoffMexA_one","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The term "],["math","\\(a_1\\)"],["text"," of the mex sequence"]],"source":"The term $a_1$ of the mex sequence"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-mex-a-one","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $\\ell \\ge 1$, let $K^\\ell$ be Wythoff's game in which the positions $(x, y)$ with $x + y \\le \\ell$ are terminal, and let $(a_n)$ be the mex sequence of $K^\\ell$. Then $a_1 = \\ell + 2$.","title":"The term $a_1$ of the mex sequence"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-mex-b-zero","kind":"node","lean":"MetaMathlibExt.wythoffMexB_zero","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The term "],["math","\\(b_0\\)"],["text"," of the companion sequence"]],"source":"The term $b_0$ of the companion sequence"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-mex-b-zero","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"For every natural number $\\ell$, the companion sequence $(b_n)$ of the mex sequence of $K^\\ell$ (Wythoff's game in which the positions $(x, y)$ with $x + y \\le \\ell$ are terminal) satisfies $b_0 = 2\\ell + 2$.","title":"The term $b_0$ of the companion sequence"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-pset","kind":"node","lean":"MetaMathlibExt.wythoffTerminalPSet","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Predicted P-positions of "],["math","\\(K^\\ell\\)"]],"source":"Predicted P-positions of $K^\\ell$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-pset","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $\\ell$ be a natural number, let $a_n$ be the least natural number $m > \\ell$ with $m \\ne a_i$ and $m \\ne b_i$ for all $i < n$, where $b_i = a_i + i + \\ell + 1$ (the mex sequences of $K^\\ell$). This is the set of positions $(x, y)$ of natural numbers with $x + y \\le \\ell$ (the terminal positions of $K^\\ell$), together with the pairs $(a_n, b_n)$ and $(b_n, a_n)$ for $n \\ge 0$.","title":"Predicted P-positions of $K^\\ell$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/mem-wythoff-terminal-pset-of-terminal","kind":"node","lean":"MetaMathlibExt.mem_wythoffTerminalPSet_of_terminal","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#mem-wythoff-terminal-pset-of-terminal","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $\\ell$ be a natural number. Every position $(x, y) \\in \\mathbb{N} \\times \\mathbb{N}$ with $x + y \\le \\ell$ belongs to the predicted P-position set of the game $K^\\ell$ (Wythoff's game in which the positions with $x + y \\le \\ell$ are terminal).","title":"Terminal positions lie in the predicted P-position set"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/komak-g","kind":"node","lean":"MetaMathlibExt.komakG","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Komak sequence "],["math","\\(g\\)"]],"source":"The Komak sequence $g$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#komak-g","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $\\varphi$ be the golden ratio. The sequence $g : \\mathbb{N} \\to \\mathbb{N}$ of Komak et al. is defined by recursion on $n$: if some $m < n$ satisfies $\\lfloor n\\varphi \\rfloor = \\lfloor m\\varphi^2 \\rfloor + 1$, then $g(n) = 1 - g(m)$ for the least such $m$; otherwise $g(n) = 1$.","title":"The Komak sequence $g$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/komak-g-zero","kind":"node","lean":"MetaMathlibExt.komakG_zero","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#komak-g-zero","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"The sequence $g$ of Komak et al. satisfies $g(0) = 1$.","title":"Value of the Komak sequence at 0"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/komak-g-one","kind":"node","lean":"MetaMathlibExt.komakG_one","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#komak-g-one","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"The sequence $g$ of Komak et al. satisfies $g(1) = 0$: with $\\varphi$ the golden ratio, $\\lfloor \\varphi \\rfloor = 1 = \\lfloor 0 \\cdot \\varphi^2 \\rfloor + 1$, so $g(1) = 1 - g(0) = 0$.","title":"Value of the Komak sequence at 1"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/fib-greedy-digits","kind":"node","lean":"MetaMathlibExt.fibGreedyDigits","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#fib-greedy-digits","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $F_0 = 1$, $F_1 = 2$ and $F_{i+2} = F_{i+1} + F_i$. For $k, n \\in \\mathbb{N}$, the greedy digits of $n$ at positions $k - 1, \\dots, 0$ are the list of $k$ digits, most significant first, computed with a remainder $r$ that starts at $n$: for $j = k - 1, \\dots, 0$ in turn, the digit at position $j$ is $1$ if $F_j \\le r$, in which case $F_j$ is subtracted from $r$, and $0$ otherwise. The digits $1$ and $0$ are\u2026","title":"Greedy Fibonacci digits"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/fib-digit-count","kind":"node","lean":"MetaMathlibExt.fibDigitCount","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#fib-digit-count","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $F_0 = 1$, $F_1 = 2$ and $F_{i+2} = F_{i+1} + F_i$. For $n, s, k_0 \\in \\mathbb{N}$, the digit count of $n$ searched from $k_0$ in at most $s$ steps is the least $k$ with $k_0 \\le k < k_0 + s$ and $n < F_k$ if there is one, and $k_0 + s$ otherwise.","title":"Fibonacci digit count"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/fib-rep","kind":"node","lean":"MetaMathlibExt.fibRep","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#fib-rep","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $F_0 = 1$, $F_1 = 2$ and $F_{i+2} = F_{i+1} + F_i$. The greedy Fibonacci representation $\\mathrm{rep}_F(n)$ of $n \\in \\mathbb{N}$ is the word of digits $d_{k-1} \\cdots d_0$, most significant first, where $k$ is the least integer with $n < F_k$ and the digits are chosen greedily: starting from the remainder $n$, for $j = k - 1, \\dots, 0$ in turn, $d_j = 1$ and $F_j$ is subtracted from the remainder if $F_j$ is\u2026","title":"Greedy Fibonacci representation"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/fib-rep-examples","kind":"node","lean":"MetaMathlibExt.fibRep_examples","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#fib-rep-examples","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"With $F_0 = 1$, $F_1 = 2$ and $F_{i+2} = F_{i+1} + F_i$, the greedy Fibonacci representations are $\\mathrm{rep}_F(0) = \\varepsilon$ (the empty word), $\\mathrm{rep}_F(2) = 10$, $\\mathrm{rep}_F(4) = 101$, $\\mathrm{rep}_F(7) = 1010$ and $\\mathrm{rep}_F(12) = 10101$.","title":"Greedy Fibonacci representations of 0, 2, 4, 7 and 12"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/is-fibonacci-automatic","kind":"node","lean":"MetaMathlibExt.IsFibonacciAutomatic","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#is-fibonacci-automatic","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"A sequence $u : \\mathbb{N} \\to \\mathbb{N}$ is Fibonacci-automatic if there are a finite set $Q$ of states, a transition function $\\delta : Q \\times \\{0, 1\\} \\to Q$, an initial state $q_0 \\in Q$ and an output map $\\tau : Q \\to \\mathbb{N}$ such that for every $n$, $u(n) = \\tau(q)$, where $q$ is the state reached from $q_0$ by reading the digits of the greedy Fibonacci representation $\\mathrm{rep}_F(n)$ with $\\delta$,\u2026","title":"Fibonacci-automatic sequences"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/is-fibonacci-automatic-const","kind":"node","lean":"MetaMathlibExt.isFibonacciAutomatic_const","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#is-fibonacci-automatic-const","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"For every $c \\in \\mathbb{N}$, the constant sequence $n \\mapsto c$ is Fibonacci-automatic.","title":"Constant sequences are Fibonacci-automatic"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/morph-iterate-wythoff-terminal-one-examp","kind":"node","lean":"MetaMathlibExt.morphIterate_wythoffTerminal_one_example","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","A prefix of the fourth iterate of "],["math","\\(f_1\\)"]],"source":"A prefix of the fourth iterate of $f_1$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#morph-iterate-wythoff-terminal-one-examp","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $f_1$ be the map on the alphabet $\\{0, 1, 2, 3, 4\\}$ given by $0 \\mapsto 01$, $1 \\mapsto 2$, $2 \\mapsto 34$, $3 \\mapsto 31$ and $4 \\mapsto 2$, extended to finite words by concatenation. The first six letters of the word $f_1^4(0)$ are $0, 1, 2, 3, 4, 3$.","title":"A prefix of the fourth iterate of $f_1$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/larsson-set-two","kind":"node","lean":"MetaMathlibExt.larssonSetTwo","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Larsson's set "],["math","\\(R_2\\)"]],"source":"Larsson's set $R_2$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#larsson-set-two","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Let $\\varphi = (1 + \\sqrt{5})/2$ be the golden ratio. The set $R_2 \\subseteq \\mathbb{N} \\times \\mathbb{N}$ consists of $(0, 0)$ together with, for every natural number $n$, the pairs $(n, 2n + 1)$ and $(2n + 1, n)$ and the pairs $(2\\lfloor n\\varphi \\rfloor + 2, 2\\lfloor n\\varphi^2 \\rfloor + 2)$ and $(2\\lfloor n\\varphi^2 \\rfloor + 2, 2\\lfloor n\\varphi \\rfloor + 2)$.","title":"Larsson's set $R_2$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/larsson-set-two-examples","kind":"node","lean":"MetaMathlibExt.larssonSetTwo_examples","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sample positions in Larsson's set "],["math","\\(R_2\\)"]],"source":"Sample positions in Larsson's set $R_2$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#larsson-set-two-examples","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"The positions $(0, 0)$, $(0, 1)$ and $(2, 2)$ belong to Larsson's set $R_2$.","title":"Sample positions in Larsson's set $R_2$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/larsson-set-three","kind":"node","lean":"MetaMathlibExt.larssonSetThree","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Larsson's set "],["math","\\(R_3\\)"]],"source":"Larsson's set $R_3$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#larsson-set-three","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"Larsson's set $R_3 \\subseteq \\mathbb{N} \\times \\mathbb{N}$ consists of $(0, 0)$ together with the pairs $(n, 2n + 1)$, $(2n + 1, n)$, $(n, 2n + 2)$ and $(2n + 2, n)$ for all $n \\ge 0$.","title":"Larsson's set $R_3$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/larsson-set-three-examples","kind":"node","lean":"MetaMathlibExt.larssonSetThree_examples","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sample positions in Larsson's set "],["math","\\(R_3\\)"]],"source":"Sample positions in Larsson's set $R_3$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#larsson-set-three-examples","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"fully_proved","summary":"The positions $(0, 0)$, $(0, 2)$ and $(3, 1)$ belong to Larsson's set $R_3$.","title":"Sample positions in Larsson's set $R_3$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-p-positions","kind":"node","lean":"MetaMathlibExt.wythoffTerminal_pPositions","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-p-positions","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\ell \\ge 1$. Let $a_n$ be the least natural number $m > \\ell$ with $m \\ne a_i$ and $m \\ne b_i$ for all $i < n$, where $b_i = a_i + i + \\ell + 1$. Let $P$ be the set of positions $(x, y)$ with $x + y \\le \\ell$, together with the pairs $(a_n, b_n)$ and $(b_n, a_n)$ for $n \\ge 0$. Then $P$ is stable, meaning no move of $K^\\ell$ leads from a position of $P$ to a position of $P$, and absorbing, meaning from every\u2026","title":"P-positions of Wythoff's game with terminal positions"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-one-p-positions","kind":"node","lean":"MetaMathlibExt.wythoffTerminal_one_pPositions","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","P-positions of "],["math","\\(K^1\\)"]],"source":"P-positions of $K^1$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-one-p-positions","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $a_n$ be the least natural number $m > 1$ with $m \\ne a_i$ and $m \\ne b_i$ for all $i < n$, where $b_i = a_i + i + 2$; so $(a_0, b_0) = (2, 4)$. Let $P$ be the set of positions $(x, y)$ with $x + y \\le 1$, together with the pairs $(a_n, b_n)$ and $(b_n, a_n)$ for $n \\ge 0$. Then $P$ is stable, meaning no move of $K^1$ leads from a position of $P$ to a position of $P$, and absorbing, meaning from every position\u2026","title":"P-positions of $K^1$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-two-p-positions","kind":"node","lean":"MetaMathlibExt.wythoffTerminal_two_pPositions","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-two-p-positions","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"In the game $K^2$ on two piles, a move from a position $(x, y)$ with $x + y > 2$ removes a positive number of tokens from one pile or the same positive number from both piles, and no move is allowed from a position with $x + y \\le 2$. Let $a_n$ be the least integer $m > 2$ that differs from $a_i$ and $b_i$ for every $i < n$, and let $b_n = a_n + n + 3$, so that $(a_0, b_0) = (3, 6)$. Let $P$ consist of the\u2026","title":"P-positions of Wythoff's game when positions of total at most 2 are terminal"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-mex-properties","kind":"node","lean":"MetaMathlibExt.wythoffMex_properties","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Structure of the mex sequences of "],["math","\\(K^\\ell\\)"]],"source":"Structure of the mex sequences of $K^\\ell$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-mex-properties","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\ell \\ge 1$. Let $a_n$ be the least natural number $m > \\ell$ with $m \\ne a_i$ and $m \\ne b_i$ for all $i < n$, where $b_i = a_i + i + \\ell + 1$ (the mex sequences of the game $K^\\ell$). Then $(a_n)$ and $(b_n)$ are strictly increasing; a natural number $j$ satisfies $j > \\ell$ if and only if $j = a_n$ or $j = b_n$ for some $n$; $a_n \\ne b_m$ for all $n$ and $m$; every $j > \\ell$ equals $b_n - a_n$ for exactly\u2026","title":"Structure of the mex sequences of $K^\\ell$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-mex-floor-golden-ratio","kind":"node","lean":"MetaMathlibExt.wythoffMex_floor_goldenRatio","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-mex-floor-golden-ratio","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\ell \\ge 1$, let $\\varphi$ be the golden ratio, let $K^\\ell$ be Wythoff's game in which the positions $(x, y)$ with $x + y \\le \\ell$ are terminal, and let $(a_n)$ be the mex sequence of $K^\\ell$ and $(b_n)$ its companion sequence. Then there are integers $\\lambda_n$ for $n \\ge 0$ and an integer $C$ with $|\\lambda_n| \\le C$ for all $n$ such that, for every natural number $n$,\u2026","title":"Golden-ratio formula for the mex sequences"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-mex-tendsto-golden-ratio","kind":"node","lean":"MetaMathlibExt.wythoffMex_tendsto_goldenRatio","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Growth of the mex sequences of "],["math","\\(K^\\ell\\)"]],"source":"Growth of the mex sequences of $K^\\ell$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-mex-tendsto-golden-ratio","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\ell \\ge 1$. Let $a_n$ be the least natural number $m > \\ell$ with $m \\ne a_i$ and $m \\ne b_i$ for all $i < n$, where $b_i = a_i + i + \\ell + 1$ (the mex sequences of the game $K^\\ell$), and let $\\varphi = (1 + \\sqrt{5})/2$ be the golden ratio. Then $a_n / n \\to \\varphi$ and $b_n / n \\to \\varphi^2$ as $n \\to \\infty$.","title":"Growth of the mex sequences of $K^\\ell$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/komak-p-positions","kind":"node","lean":"MetaMathlibExt.komak_pPositions","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Komak's description of the P-positions of "],["math","\\(K^2\\)"]],"source":"Komak's description of the P-positions of $K^2$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#komak-p-positions","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio and $g$ the sequence of Komak et al. Let $S$ be the set of pairs $(x, y)$ of natural numbers with $x + y \\le 2$, together with the pairs $(\\lfloor n\\varphi \\rfloor + g(n) - 1, \\lfloor n\\varphi^2 \\rfloor + g(n))$ and $(\\lfloor n\\varphi^2 \\rfloor + g(n), \\lfloor n\\varphi \\rfloor + g(n) - 1)$ for all $n \\ge 0$. In the game $K^2$ (Wythoff's game in which the positions with\u2026","title":"Komak's description of the P-positions of $K^2$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/komak-g-hofstadter","kind":"node","lean":"MetaMathlibExt.komakG_hofstadter","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#komak-g-hofstadter","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $h$ be Hofstadter's $G$-sequence (OEIS A005206), given by $h(0) = 0$ and $h(n) = n - h(h(n - 1))$ for $n \\ge 1$. The sequence $g$ of Komak et al. satisfies $g(0) = 1$, $g(1) = 0$, and for every $n \\ge 2$, $g(n) = 1 - g(h(n - 1))$ if $h(n - 2) < h(n - 1)$, and $g(n) = 1$ otherwise.","title":"Hofstadter recursion for the Komak sequence"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/is-fibonacci-automatic-komak-g","kind":"node","lean":"MetaMathlibExt.isFibonacciAutomatic_komakG","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","The Komak sequence "],["math","\\(g\\)"],["text"," is Fibonacci-automatic"]],"source":"The Komak sequence $g$ is Fibonacci-automatic"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#is-fibonacci-automatic-komak-g","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"The sequence $g$ of Komak et al. is Fibonacci-automatic.","title":"The Komak sequence $g$ is Fibonacci-automatic"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-one-fib-rep","kind":"node","lean":"MetaMathlibExt.wythoffTerminal_one_fibRep","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibonacci characterization of the P-positions of "],["math","\\(K^1\\)"]],"source":"Fibonacci characterization of the P-positions of $K^1$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-one-fib-rep","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"For a natural number $n$, let $\\operatorname{rep}_F(n)$ be its greedy Fibonacci representation for $F_0 = 1$, $F_1 = 2$, $F_{i+2} = F_{i+1} + F_i$: a word over $\\{0, 1\\}$, most significant digit first, without leading zeros, empty for $n = 0$. Let $S$ be the set of positions $(x, y)$ with $x + y \\le 1$, together with the pairs $(a, b)$ and $(b, a)$ for natural numbers $a \\le b$ such that $\\operatorname{rep}_F(a)$\u2026","title":"Fibonacci characterization of the P-positions of $K^1$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/exists-fibonacci-automatic-wythoff-termi","kind":"node","lean":"MetaMathlibExt.exists_fibonacciAutomatic_wythoffTerminal_three","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#exists-fibonacci-automatic-wythoff-termi","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. Let $a_n$ be the least integer greater than $3$ that differs from $a_i$ and $b_i$ for all $i < n$, and let $b_n = a_n + n + 4$. Then there is a Fibonacci-automatic sequence $g_3 : \\mathbb{N} \\to \\mathbb{N}$ with $g_3(n) \\le 2$ for all $n$ such that, for every $n \\ge 0$, $a_n = \\lfloor (n + 2)\\varphi \\rfloor + g_3(n + 1) - 1$ and\u2026","title":"Golden-ratio formula for Wythoff's game with terminal positions up to total 3"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/exists-fibonacci-automatic-wythoff-termi-2","kind":"node","lean":"MetaMathlibExt.exists_fibonacciAutomatic_wythoffTerminal_four","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#exists-fibonacci-automatic-wythoff-termi-2","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. Let $a_n$ be the least natural number greater than $4$ that differs from $a_i$ and $b_i$ for all $i < n$, and let $b_n = a_n + n + 5$. Then there is a Fibonacci-automatic sequence $g_4 : \\mathbb{N} \\to \\mathbb{N}$ with $g_4(n) \\le 2$ for all $n$ such that, for every $n \\ge 0$, $a_n = \\lfloor (n + 2)\\varphi \\rfloor + g_4(n + 1)$ and\u2026","title":"Golden-ratio formula for Wythoff's game with terminal positions up to total 4"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-one-morphic","kind":"node","lean":"MetaMathlibExt.wythoffTerminal_one_morphic","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-one-morphic","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $a_n$ be the least natural number $m > 1$ that differs from $a_i$ and $b_i$ for every $i < n$, and let $b_n = a_n + n + 2$, so that $(a_0, b_0) = (2, 4)$. Let $f_1$ be the morphism on the alphabet $\\{0, 1, 2, 3, 4\\}$ given by $0 \\mapsto 0\\,1$, $1 \\mapsto 2$, $2 \\mapsto 3\\,4$, $3 \\mapsto 3\\,1$, $4 \\mapsto 2$ and extended to finite words by concatenation, and let $g_1$ send the letters $0, 1, 3$ to $a$ and the\u2026","title":"Morphic coding of the Wythoff mex pairs at level one"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-two-morphic","kind":"node","lean":"MetaMathlibExt.wythoffTerminal_two_morphic","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-two-morphic","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $a_n$ be the least integer $m > 2$ that differs from $a_i$ and $b_i$ for every $i < n$, and let $b_n = a_n + n + 3$, so that $(a_0, b_0) = (3, 6)$. Let $f_2$ be the morphism on the alphabet $\\{0, 1, \\dots, 15\\}$ given by $0 \\mapsto 0\\,1$, $1 \\mapsto 2$, $2 \\mapsto 3\\,4$, $3 \\mapsto 5\\,6$, $4 \\mapsto 7$, $5 \\mapsto 8\\,9$, $6 \\mapsto 10$, $7 \\mapsto 11\\,12$, $8 \\mapsto 10\\,13$, $9 \\mapsto 14$,\u2026","title":"Morphic coding of the Wythoff mex pairs at level two"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-three-morphic","kind":"node","lean":"MetaMathlibExt.wythoffTerminal_three_morphic","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-three-morphic","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $a_n$ be the least integer $m > 3$ that differs from $a_i$ and $b_i$ for every $i < n$, and let $b_n = a_n + n + 4$, so that $(a_0, b_0) = (4, 8)$. Let $f_3$ be the morphism on the alphabet $\\{0, 1, \\dots, 21\\}$ given by $0 \\mapsto 0\\,1$, $1 \\mapsto 2$, $2 \\mapsto 3\\,4$, $3 \\mapsto 5\\,6$, $4 \\mapsto 7$, $5 \\mapsto 8\\,9$, $6 \\mapsto 10$, $7 \\mapsto 11\\,12$, $8 \\mapsto 13\\,12$, $9 \\mapsto 14$,\u2026","title":"Morphic coding of the Wythoff mex pairs at level three"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/blocking-wythoff-two-p-positions","kind":"node","lean":"MetaMathlibExt.blockingWythoff_two_pPositions","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#blocking-wythoff-two-p-positions","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio, and let $R_2$ consist of $(0, 0)$ and, for every $n \\ge 0$, the positions $(n, 2n + 1)$, $(2n + 1, n)$, $(2\\lfloor n\\varphi \\rfloor + 2, 2\\lfloor n\\varphi^2 \\rfloor + 2)$ and $(2\\lfloor n\\varphi^2 \\rfloor + 2, 2\\lfloor n\\varphi \\rfloor + 2)$. A Wythoff move from a position of two piles removes a positive number of tokens from one pile, or the same positive number from both. Then\u2026","title":"Larsson's P-positions of Blocking Wythoff Nim with one blocked option"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/blocking-wythoff-three-p-positions","kind":"node","lean":"MetaMathlibExt.blockingWythoff_three_pPositions","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#blocking-wythoff-three-p-positions","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $R_3$ consist of $(0, 0)$ and, for every $n \\ge 0$, the positions $(n, 2n + 1)$, $(2n + 1, n)$, $(n, 2n + 2)$ and $(2n + 2, n)$. A Wythoff move from a position of two piles removes a positive number of tokens from one pile, or the same positive number from both. Then every position in $R_3$ has at most two Wythoff moves to positions in $R_3$, and every position outside $R_3$ has at least three Wythoff moves to\u2026","title":"Larsson's P-positions of Blocking Wythoff Nim with two blocked options"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/wythoff-terminal-no-redundant-move","kind":"node","lean":"MetaMathlibExt.wythoffTerminal_no_redundant_move","parent":44563,"reading_title":{"format":"inline-parts-v1","parts":[["text","No redundant move in "],["math","\\(K^\\ell\\)"],["text"," for "],["math","\\(1 \\le \\ell \\le 4\\)"]],"source":"No redundant move in $K^\\ell$ for $1 \\le \\ell \\le 4$"},"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#wythoff-terminal-no-redundant-move","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $1 \\le \\ell \\le 4$. Let $a_n$ be the least natural number $m > \\ell$ with $m \\ne a_i$ and $m \\ne b_i$ for all $i < n$, where $b_i = a_i + i + \\ell + 1$, and let $P$ be the set of positions $(x, y)$ with $x + y \\le \\ell$ together with the pairs $(a_n, b_n)$ and $(b_n, a_n)$ for $n \\ge 0$. Let $v$ be one of the move vectors $(i, 0)$, $(0, i)$ or $(i, i)$ with $i > 0$. Then there is a position $p \\notin P$ such\u2026","title":"No redundant move in $K^\\ell$ for $1 \\le \\ell \\le 4$"},{"id":"GameTheory/combinatorial-games/wythoff-s-game/blocking-wythoff-no-redundant-move","kind":"node","lean":"MetaMathlibExt.blockingWythoff_no_redundant_move","parent":44563,"result":true,"route":"../roadmap/GameTheory/combinatorial-games/wythoff-s-game/index.html#blocking-wythoff-no-redundant-move","scope":"GameTheory/combinatorial-games/wythoff-s-game","status":"can_prove","summary":"Let $\\varphi$ be the golden ratio. Let $R_2$ consist of $(0, 0)$ and, for every $n \\ge 0$, the positions $(n, 2n + 1)$, $(2n + 1, n)$, $(2\\lfloor n\\varphi \\rfloor + 2, 2\\lfloor n\\varphi^2 \\rfloor + 2)$ and $(2\\lfloor n\\varphi^2 \\rfloor + 2, 2\\lfloor n\\varphi \\rfloor + 2)$, and let $R_3$ consist of $(0, 0)$ and, for every $n \\ge 0$, the positions $(n, 2n + 1)$, $(2n + 1, n)$, $(n, 2n + 2)$ and $(2n + 2, n)$. A\u2026","title":"Blocking Wythoff Nim with one or two blocked options has no redundant move"},{"data_url":"scopes/GameTheory/social-choice.json","id":"GameTheory/social-choice","kind":"scope","parent":44141,"result":false,"route":"../roadmap/GameTheory/social-choice/index.html","scope":"GameTheory/social-choice","status":"fully_proved","summary":"Social choice theory studies rules that aggregate individual preferences into a collective ranking or choice, and the impossibility theorems that limit them.","title":"Social choice theory"},{"data_url":"scopes/GameTheory/social-choice/arrow-s-impossibility-theorem.json","id":"GameTheory/social-choice/arrow-s-impossibility-theorem","kind":"scope","navigation":"module","parent":44626,"result":false,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","title":"Arrow's impossibility theorem"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/strict-total-order","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.StrictTotalOrder","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#strict-total-order","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"A strict total order on a type $\\alpha$ consists of a relation $<$ on $\\alpha$ that is irreflexive ($a < a$ holds for no $a$), transitive ($a < b$ and $b < c$ imply $a < c$) and total on distinct elements ($a \\ne b$ implies $a < b$ or $b < a$). Here it models a strict preference over alternatives.","title":"Strict preference orders for Arrow's theorem"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/mk","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.StrictTotalOrder.mk","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#mk","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"A relation $<$ on a type $\\alpha$ that is irreflexive ($a < a$ holds for no $a$), transitive ($a < b$ and $b < c$ imply $a < c$) and total on distinct elements ($a \\ne b$ implies $a < b$ or $b < a$) defines a strict total order on $\\alpha$.","title":"Strict total order from a relation (Arrow's theorem)"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/lt","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.StrictTotalOrder.lt","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#lt","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"For a strict total order $T$ on a type $\\alpha$, this is its underlying binary relation $\\lt _T$ on $\\alpha$.","title":"Preference relation of a strict total order (Arrow's theorem)"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/irrefl","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.StrictTotalOrder.irrefl","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#irrefl","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"Let $T$ be a strict total order on a type $\\alpha$, and write $a \\lt _T b$ for its relation. Then for every $a \\in \\alpha$, $a \\lt _T a$ does not hold.","title":"Irreflexivity of strict preference (Arrow's theorem)"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/trans","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.StrictTotalOrder.trans","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#trans","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"If $<$ is a strict total order on a type $\\alpha$ and $a, b, c \\in \\alpha$ satisfy $a < b$ and $b < c$, then $a < c$.","title":"Strict preference orders for Arrow's theorem are transitive"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/total","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.StrictTotalOrder.total","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#total","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"If $<$ is a strict total order on a type $\\alpha$ and $a, b \\in \\alpha$ are distinct, then $a < b$ or $b < a$.","title":"Strict preference orders for Arrow's theorem are total"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/profile","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.Profile","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#profile","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"A profile assigns to each voter a strict total order on the alternatives.","title":"Preference profile (Arrow's theorem)"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/social-welfare-function","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.SocialWelfareFunction","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#social-welfare-function","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"A social welfare function maps each unrestricted profile to a social strict total order.","title":"Social welfare function"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/is-pareto","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.IsPareto","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#is-pareto","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"A social welfare function $F$ satisfies the Pareto condition if for every profile $p$ and all alternatives $a, b$, whenever every voter strictly prefers $a$ to $b$ in $p$, the social order $F(p)$ also strictly prefers $a$ to $b$.","title":"Pareto condition"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/is-iia","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.IsIIA","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#is-iia","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"A social welfare function $F$ satisfies independence of irrelevant alternatives if for all profiles $p_1, p_2$ and all alternatives $a, b$ the following holds: if every voter strictly prefers $a$ to $b$ in $p_1$ exactly when that voter strictly prefers $a$ to $b$ in $p_2$, then $F(p_1)$ strictly prefers $a$ to $b$ exactly when $F(p_2)$ does.","title":"Independence of irrelevant alternatives"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/is-dictator","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.IsDictator","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#is-dictator","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"A voter $d$ is a dictator for a social welfare function $F$ if for every profile $p$ and all alternatives $a, b$, whenever $d$ strictly prefers $a$ to $b$ in $p$, the social order $F(p)$ also strictly prefers $a$ to $b$.","title":"Dictator of a social welfare function"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/is-dictatorial","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.IsDictatorial","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#is-dictatorial","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"A social welfare function $F$ is dictatorial if some voter $d$ is a dictator for $F$: for every profile $p$ and all alternatives $a, b$, whenever $d$ strictly prefers $a$ to $b$ in $p$, the social order $F(p)$ also strictly prefers $a$ to $b$.","title":"Dictatorial social welfare function"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/arrow-impossibility-general","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.arrow_impossibility_general","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#arrow-impossibility-general","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"Suppose there are finitely many alternatives, at least three of them, and finitely many voters, at least one. Then every social welfare function, which assigns a strict total order of the alternatives to each profile of strict total orders (one per voter), that satisfies the Pareto condition and independence of irrelevant alternatives is dictatorial.","title":"Arrow's impossibility theorem for strict preferences, general form"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/arrow-impossibility","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.ArrowWanted.arrow_impossibility","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#arrow-impossibility","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"Suppose there are finitely many alternatives, at least three of them, and finitely many voters, at least one. Then every social welfare function, which assigns a strict total order of the alternatives to each profile of strict total orders (one per voter), that satisfies the Pareto condition and independence of irrelevant alternatives is dictatorial.","title":"Arrow's impossibility theorem for strict preferences"},{"id":"GameTheory/social-choice/arrow-s-impossibility-theorem/arrow-impossibility-2","kind":"node","lean":"MetaMathlibExt.arrowImpossibility","parent":44627,"result":true,"route":"../roadmap/GameTheory/social-choice/arrow-s-impossibility-theorem/index.html#arrow-impossibility-2","scope":"GameTheory/social-choice/arrow-s-impossibility-theorem","status":"fully_proved","summary":"Let $A$ be a type of alternatives containing three distinct elements, and let $E$ be a finite nonempty set of voters. A profile $P$ assigns to every voter $v$ a relation $P_v$ on $A$; it is admissible if, for every $v \\in E$, $P_v$ is a total preorder (reflexive, transitive, and $P_v(x, y)$ or $P_v(y, x)$ for all $x, y \\in A$). For a relation $R$ on $A$, say $R$ strictly ranks $a$ above $b$ if $R(a, b)$ holds and\u2026","title":"Arrow's impossibility theorem for weak preferences"},{"data_url":"scopes/GameTheory/social-choice/condorcet-s-jury-theorem.json","id":"GameTheory/social-choice/condorcet-s-jury-theorem","kind":"scope","navigation":"module","parent":44626,"result":false,"route":"../roadmap/GameTheory/social-choice/condorcet-s-jury-theorem/index.html","scope":"GameTheory/social-choice/condorcet-s-jury-theorem","status":"fully_proved","title":"Condorcet's jury theorem"},{"id":"GameTheory/social-choice/condorcet-s-jury-theorem/condorcet-jury-theorem","kind":"node","lean":"MetaMathlibExt.condorcet_jury_theorem","parent":44643,"result":true,"route":"../roadmap/GameTheory/social-choice/condorcet-s-jury-theorem/index.html#condorcet-jury-theorem","scope":"GameTheory/social-choice/condorcet-s-jury-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $X_0, X_1, \\dots$ be measurable maps from $\\Omega$ to $\\{\\text{true}, \\text{false}\\}$ (the votes, true meaning a correct vote), and let $p$ be a number with $1/2 < p < 1$. Suppose that each vote is correct with probability $p$, that is $\\mu(X_i = \\text{true}) = p$ for every $i$, and that the events $\\{X_i = \\text{true}\\}$ are independent: for\u2026","title":"Condorcet's jury theorem"},{"id":"GameTheory/social-choice/condorcet-s-jury-theorem/condorcet-jury-theorem-of-i-indep-fun","kind":"node","lean":"MetaMathlibExt.condorcet_jury_theorem_of_iIndepFun","parent":44643,"result":true,"route":"../roadmap/GameTheory/social-choice/condorcet-s-jury-theorem/index.html#condorcet-jury-theorem-of-i-indep-fun","scope":"GameTheory/social-choice/condorcet-s-jury-theorem","status":"fully_proved","summary":"Let $\\mu$ be a probability measure on a measurable space $\\Omega$, let $X_0, X_1, \\dots$ be measurable maps from $\\Omega$ to $\\{\\text{true}, \\text{false}\\}$ (the votes, true meaning a correct vote) that are mutually independent random variables under $\\mu$, and let $p$ be a number with $1/2 < p < 1$. Suppose that each vote is correct with probability $p$, that is $\\mu(X_i = \\text{true}) = p$ for every $i$. For\u2026","title":"Condorcet's jury theorem for mutually independent votes"},{"data_url":"scopes/GameTheory/social-choice/gibbard-satterthwaite-theorem.json","id":"GameTheory/social-choice/gibbard-satterthwaite-theorem","kind":"scope","navigation":"module","parent":44626,"result":false,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","title":"Gibbard\u2013Satterthwaite theorem"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/strict-total-order","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.StrictTotalOrder","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#strict-total-order","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"A strict total order on a type $\\alpha$ is a binary relation $<$ on $\\alpha$ that is irreflexive ($a < a$ holds for no $a$), transitive ($a < b$ and $b < c$ imply $a < c$) and total on distinct elements ($a \\ne b$ implies $a < b$ or $b < a$). In the Gibbard\u2013Satterthwaite development it models a voter's strict preference over alternatives.","title":"Strict preference orders for the Gibbard\u2013Satterthwaite theorem"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/mk","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.StrictTotalOrder.mk","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#mk","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"A binary relation $<$ on a type $\\alpha$ that is irreflexive ($a < a$ holds for no $a$), transitive ($a < b$ and $b < c$ imply $a < c$) and total on distinct elements ($a \\ne b$ implies $a < b$ or $b < a$) defines a strict total order on $\\alpha$ whose relation is $<$.","title":"Building a strict preference order in the Gibbard\u2013Satterthwaite setting"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/lt","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.StrictTotalOrder.lt","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#lt","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"The relation of a strict total order $T$ on a type $\\alpha$ is its underlying binary relation $\\lt _T$ on $\\alpha$, which is irreflexive, transitive, and relates any two distinct elements in one direction or the other.","title":"Relation of a strict preference order"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/irrefl","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.StrictTotalOrder.irrefl","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#irrefl","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"Let $T$ be a strict total order on a type $\\alpha$, with relation $\\lt _T$. Then $a \\lt _T a$ fails for every $a \\in \\alpha$.","title":"Irreflexivity of strict preference"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/trans","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.StrictTotalOrder.trans","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#trans","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"Let $<$ be the relation of a strict total order on a type $\\alpha$. If $a, b, c \\in \\alpha$ satisfy $a < b$ and $b < c$, then $a < c$.","title":"Transitivity of strict preferences in the Gibbard\u2013Satterthwaite setting"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/total","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.StrictTotalOrder.total","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#total","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"Let $<$ be the relation of a strict total order on a type $\\alpha$. If $a, b \\in \\alpha$ and $a \\ne b$, then $a < b$ or $b < a$.","title":"Strict preferences in the Gibbard\u2013Satterthwaite setting are total"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/profile","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.Profile","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#profile","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"For a type $A$ of alternatives and a type $V$ of voters, a profile is a function assigning to each voter $i \\in V$ a strict total order $\\lt _i$ on $A$, the strict preference of voter $i$.","title":"Preference profiles for the Gibbard\u2013Satterthwaite theorem"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/social-choice-function","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.SocialChoiceFunction","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#social-choice-function","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"For a type $A$ of alternatives and a type $V$ of voters, a resolute social choice function is a function $F$ sending each profile $p$ (an assignment of a strict total order on $A$ to each voter in $V$) to a single chosen alternative $F(p) \\in A$.","title":"Social choice functions"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/is-top-ranked","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.IsTopRanked","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#is-top-ranked","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"For a strict total order $o$ on the alternatives, write $x \\succ_o y$ when $x$ is related to $y$ by $o$ ($o$ ranks $x$ above $y$). An alternative $t$ is top-ranked by $o$ when $t \\succ_o b$ for every alternative $b \\ne t$.","title":"Top-ranked alternative"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/is-onto","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.IsOnto","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#is-onto","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"A resolute social choice function $F$ is onto when every alternative $a$ is chosen at some profile, that is, $F(p) = a$ for some profile $p$.","title":"Onto social choice function"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/diff-at-most-at","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.DiffAtMostAt","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#diff-at-most-at","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"Let $p_1$ and $p_2$ be profiles, each assigning a strict total order on the alternatives to every voter, and let $i$ be a voter. Then $p_2$ differs from $p_1$ at most at voter $i$ when $p_1(j) = p_2(j)$ for every voter $j \\ne i$.","title":"Profiles differing only at one voter"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/is-strategy-proof","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.IsStrategyProof","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#is-strategy-proof","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"For a strict total order $P$, write $x \\succ_P y$ when $x$ is related to $y$ by $P$ ($P$ ranks $x$ above $y$). A resolute social choice function $F$ is strategy-proof when, for all profiles $p_1$, $p_2$ and every voter $i$ such that $p_1(j) = p_2(j)$ for all voters $j \\ne i$, it is not the case that $F(p_2) \\succ_{p_1(i)} F(p_1)$: no voter gains by misreporting alone, judged by the true order $p_1(i)$.","title":"Strategy-proof social choice function"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/is-dictator","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.IsDictator","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#is-dictator","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"For a resolute social choice function $F$, a voter $d$ is a dictator when, for every profile $p$, the chosen alternative $F(p)$ is top-ranked in the preference order $p(d)$ of voter $d$.","title":"Dictator of a social choice function"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/is-dictatorial","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.IsDictatorial","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#is-dictatorial","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"A resolute social choice function $F$ is dictatorial when some voter is a dictator for $F$: there is a voter $d$ such that, for every profile $p$, the chosen alternative $F(p)$ is top-ranked in the preference order $p(d)$ of voter $d$.","title":"Dictatorial social choice function"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/gibbard-satterthwaite-general","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.gibbard_satterthwaite_general","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#gibbard-satterthwaite-general","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"Let the set of alternatives be finite and the set of voters be finite and nonempty, and suppose there are at least three distinct alternatives. Then every resolute social choice function on profiles of strict preference orders that is onto and strategy-proof is dictatorial.","title":"Gibbard\u2013Satterthwaite theorem, finiteness form"},{"id":"GameTheory/social-choice/gibbard-satterthwaite-theorem/gibbard-satterthwaite","kind":"node","lean":"MathlibExt.GameTheory.SocialChoice.GibbardSatterthwaiteWanted.gibbard_satterthwaite","parent":44646,"result":true,"route":"../roadmap/GameTheory/social-choice/gibbard-satterthwaite-theorem/index.html#gibbard-satterthwaite","scope":"GameTheory/social-choice/gibbard-satterthwaite-theorem","status":"fully_proved","summary":"Let the set of alternatives be finite and the set of voters be finite and nonempty, and suppose there are at least three distinct alternatives. Then every resolute social choice function on profiles of strict preference orders that is onto and strategy-proof is dictatorial.","title":"Gibbard\u2013Satterthwaite theorem"},{"data_url":"scopes/GameTheory/cooperative-games.json","id":"GameTheory/cooperative-games","kind":"scope","parent":44141,"result":false,"route":"../roadmap/GameTheory/cooperative-games/index.html","scope":"GameTheory/cooperative-games","status":"fully_proved","summary":"Cooperative game theory studies games in which coalitions of players can make binding agreements, and solution concepts such as the core and the Shapley value for dividing a coalition's worth.","title":"Cooperative game theory"},{"data_url":"scopes/GameTheory/cooperative-games/shapley-value.json","id":"GameTheory/cooperative-games/shapley-value","kind":"scope","navigation":"module","parent":44663,"result":false,"route":"../roadmap/GameTheory/cooperative-games/shapley-value/index.html","scope":"GameTheory/cooperative-games/shapley-value","status":"fully_proved","title":"Shapley value"},{"id":"GameTheory/cooperative-games/shapley-value/coop-game","kind":"node","lean":"MathlibExt.GameTheory.ShapleyValue.CoopGame","parent":44664,"result":true,"route":"../roadmap/GameTheory/cooperative-games/shapley-value/index.html#coop-game","scope":"GameTheory/cooperative-games/shapley-value","status":"fully_proved","summary":"Given a set $N$ of players, a cooperative game on $N$ is a function $v$ that assigns to every coalition $S \\subseteq N$ a real number $v(S)$, the worth of $S$, such that $v(\\emptyset) = 0$.","title":"Cooperative game with transferable utility"},{"id":"GameTheory/cooperative-games/shapley-value/is-null-player","kind":"node","lean":"MathlibExt.GameTheory.ShapleyValue.IsNullPlayer","parent":44664,"result":true,"route":"../roadmap/GameTheory/cooperative-games/shapley-value/index.html#is-null-player","scope":"GameTheory/cooperative-games/shapley-value","status":"fully_proved","summary":"Let $v$ be a cooperative game on a finite set $N$ of players. A player $i \\in N$ is a null player of $v$ if $v(S \\cup \\{i\\}) = v(S)$ for every coalition $S \\subseteq N$ with $i \\notin S$.","title":"Null player"},{"id":"GameTheory/cooperative-games/shapley-value/is-efficient","kind":"node","lean":"MathlibExt.GameTheory.ShapleyValue.IsEfficient","parent":44664,"result":true,"route":"../roadmap/GameTheory/cooperative-games/shapley-value/index.html#is-efficient","scope":"GameTheory/cooperative-games/shapley-value","status":"fully_proved","summary":"Let $N$ be a finite set of players and $\\varphi$ an $\\mathbb{R}$-linear map from cooperative games on $N$ (added and scaled pointwise) to payoff vectors in $\\mathbb{R}^N$. Then $\\varphi$ is efficient if for every game $v$ the payoffs add up to the worth of the grand coalition: $\\sum_{i \\in N} \\varphi(v)_i = v(N)$.","title":"Efficient value"},{"id":"GameTheory/cooperative-games/shapley-value/is-symmetric","kind":"node","lean":"MathlibExt.GameTheory.ShapleyValue.IsSymmetric","parent":44664,"result":true,"route":"../roadmap/GameTheory/cooperative-games/shapley-value/index.html#is-symmetric","scope":"GameTheory/cooperative-games/shapley-value","status":"fully_proved","summary":"Let $N$ be a finite set of players and $\\varphi$ an $\\mathbb{R}$-linear map from cooperative games on $N$ (added and scaled pointwise) to payoff vectors in $\\mathbb{R}^N$. Then $\\varphi$ is symmetric if for every game $v$ and every permutation $\\sigma$ of $N$ with $v(\\sigma(S)) = v(S)$ for all coalitions $S \\subseteq N$, every player $i$ satisfies $\\varphi(v)_{\\sigma(i)} = \\varphi(v)_i$.","title":"Symmetric value"},{"id":"GameTheory/cooperative-games/shapley-value/satisfies-null-player-axiom","kind":"node","lean":"MathlibExt.GameTheory.ShapleyValue.SatisfiesNullPlayerAxiom","parent":44664,"result":true,"route":"../roadmap/GameTheory/cooperative-games/shapley-value/index.html#satisfies-null-player-axiom","scope":"GameTheory/cooperative-games/shapley-value","status":"fully_proved","summary":"Let $N$ be a finite set of players and $\\varphi$ an $\\mathbb{R}$-linear map from cooperative games on $N$ (added and scaled pointwise) to payoff vectors in $\\mathbb{R}^N$. Then $\\varphi$ satisfies the null player axiom if $\\varphi(v)_i = 0$ for every game $v$ and every null player $i$ of $v$.","title":"Null player axiom"},{"data_url":"scopes/GameTheory/cooperative-games/bondareva-shapley-theorem.json","id":"GameTheory/cooperative-games/bondareva-shapley-theorem","kind":"scope","navigation":"module","parent":44663,"result":false,"route":"../roadmap/GameTheory/cooperative-games/bondareva-shapley-theorem/index.html","scope":"GameTheory/cooperative-games/bondareva-shapley-theorem","status":"fully_proved","title":"Bondareva\u2013Shapley theorem"},{"id":"GameTheory/cooperative-games/bondareva-shapley-theorem/is-core-allocation","kind":"node","lean":"MathlibExt.GameTheory.BondarevaShapleyWanted.IsCoreAllocation","parent":44670,"result":true,"route":"../roadmap/GameTheory/cooperative-games/bondareva-shapley-theorem/index.html#is-core-allocation","scope":"GameTheory/cooperative-games/bondareva-shapley-theorem","status":"fully_proved","summary":"A payoff vector $x \\colon N \\to \\mathbb{R}$ is a core allocation of a cooperative game $v$ on a finite set $N$ of players if it is efficient, $\\sum_{i \\in N} x_i = v(N)$, and coalitionally rational, $v(S) \\le \\sum_{i \\in S} x_i$ for every coalition $S \\subseteq N$.","title":"Core allocation"},{"id":"GameTheory/cooperative-games/bondareva-shapley-theorem/is-balanced","kind":"node","lean":"MathlibExt.GameTheory.BondarevaShapleyWanted.IsBalanced","parent":44670,"result":true,"route":"../roadmap/GameTheory/cooperative-games/bondareva-shapley-theorem/index.html#is-balanced","scope":"GameTheory/cooperative-games/bondareva-shapley-theorem","status":"fully_proved","summary":"A cooperative game $v$ on a finite set $N$ of players is balanced if for every family of real weights $(\\lambda_S)_{S \\subseteq N}$ with $\\lambda_S \\ge 0$ for every coalition $S \\subseteq N$ and $\\sum_{S \\subseteq N,\\, i \\in S} \\lambda_S = 1$ for every player $i \\in N$, one has $\\sum_{S \\subseteq N} \\lambda_S \\, v(S) \\le v(N)$.","title":"Balanced cooperative game"},{"id":"GameTheory/cooperative-games/bondareva-shapley-theorem/bondareva-shapley","kind":"node","lean":"MathlibExt.GameTheory.BondarevaShapleyWanted.bondareva_shapley","parent":44670,"result":true,"route":"../roadmap/GameTheory/cooperative-games/bondareva-shapley-theorem/index.html#bondareva-shapley","scope":"GameTheory/cooperative-games/bondareva-shapley-theorem","status":"fully_proved","summary":"Let $v$ be a cooperative game with transferable utility on a finite set $N$ of players. Then $v$ has a core allocation $x \\colon N \\to \\mathbb{R}$, that is, the core of $v$ is nonempty, if and only if $v$ is balanced.","title":"Bondareva\u2013Shapley theorem"},{"area":"Applied Mathematics","data_url":"chapters/Biology.json","id":"Biology","kind":"scope","parent":0,"result":false,"route":"../roadmap/Biology/index.html","scope":"Biology","status":"can_state","summary":"Mathematical biology models populations, genomes, and biochemical networks, from coalescents and reaction dynamics to the algorithms that read biological sequences.","title":"Mathematical Biology"},{"data_url":"scopes/Biology/population-genetics.json","id":"Biology/population-genetics","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/population-genetics/index.html","scope":"Biology/population-genetics","status":"can_state","summary":"Mathematical models of allele frequencies in populations under random mating, selection, mutation, migration and genetic drift, including the Wright\u2013Fisher and Moran models and their diffusion limits.","title":"Population genetics"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory.json","id":"Biology/population-genetics/diffusion-theory","kind":"scope","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/index.html","scope":"Biology/population-genetics/diffusion-theory","status":"can_state","summary":"The Wright\u2013Fisher diffusion as the continuum limit of allele-frequency dynamics: generator and semigroup, fixation probabilities, absorption and sojourn times, mutation and stationary laws, allele age and the Poisson random field.","title":"Diffusion theory"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion.json","id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","title":"Wright\u2013Fisher diffusion"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#generator","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The backward generator of the Wright\u2013Fisher diffusion with genic selection, time in units of `2N` generations and scaled selection coefficient `\u03c3 = 2Ns`: `(L_\u03c3 u)(x) = x (1 - x) / 2 \u00b7 u''(x) + \u03c3 x (1 - x) \u00b7 u'(x)`.","title":"Backward generator of the Wright\u2013Fisher diffusion with selection"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Kimura's fixation probability of an allele at initial frequency `x` with scaled selection coefficient `\u03c3 = 2Ns`: `(1 - e^{-2\u03c3x}) / (1 - e^{-2\u03c3})`, and `x` in the neutral case.","title":"Kimura's fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/mean-absorption-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAbsorptionTime","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#mean-absorption-time","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Kimura\u2013Ohta mean time to absorption (loss or fixation) of a neutral allele at frequency `x`, in units of `2N` generations: `-2 (x log x + (1 - x) log (1 - x))`. In generations this is `-4N (x ln x + (1 - x) ln (1 - x))`.","title":"Kimura\u2013Ohta mean absorption time of a neutral allele"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/wright-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.wrightDensity","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#wright-density","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Wright's (unnormalized) stationary density of the Wright\u2013Fisher diffusion with scaled mutation rates `\u03b1` (towards the allele) and `\u03b2` (away from it): `x ^ (2\u03b1 - 1) (1 - x) ^ (2\u03b2 - 1)` (real powers).","title":"Wright's stationary density of the Wright\u2013Fisher diffusion"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-slope","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraSlope","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-slope","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The constant `K \u03c3` such that `(kimuraFixation \u03c3)' x = K \u03c3 \u00b7 e^{-2\u03c3x}`: `1` if `\u03c3 = 0`, and `2\u03c3 / (1 - e^{-2\u03c3})` otherwise.","title":"Slope constant of Kimura's fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/one-sub-exp-ne-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.one_sub_exp_ne_zero","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#one-sub-exp-ne-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"For `\u03c3 \u2260 0` the denominator `1 - e^{-2\u03c3}` of Kimura's formula does not vanish.","title":"Denominator of Kimura's formula is nonzero"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-fixation-zero-right","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation_zero_right","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-fixation-zero-right","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"An allele that is absent is never fixed.","title":"Kimura's fixation probability vanishes at frequency zero"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-fixation-one-right","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation_one_right","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-fixation-one-right","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"An allele that is already fixed stays fixed.","title":"Kimura's fixation probability equals one at frequency one"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-fixation-neutral","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation_neutral","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-fixation-neutral","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Neutral case: the fixation probability equals the initial frequency.","title":"Neutral fixation probability equals the initial frequency"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-slope-ne-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraSlope_ne_zero","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-slope-ne-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The slope constant of Kimura's formula never vanishes.","title":"Slope constant of Kimura's formula is nonzero"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/has-deriv-at-kimura-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasDerivAt_kimuraFixation","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#has-deriv-at-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The derivative of Kimura's fixation probability: `u'(x) = K \u03c3 \u00b7 e^{-2\u03c3x}`.","title":"Derivative of Kimura's fixation probability at a point"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/continuous-kimura-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.continuous_kimuraFixation","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#continuous-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Kimura's fixation probability is continuous in the initial frequency.","title":"Continuity of Kimura's fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/deriv-kimura-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.deriv_kimuraFixation","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#deriv-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The derivative of Kimura's fixation probability, as a function.","title":"Derivative of Kimura's fixation probability as a function"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/generator-kimura-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_kimuraFixation","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#generator-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The fixation probability solves the backward equation `x (1 - x) / 2 \u00b7 u'' + \u03c3 x (1 - x) \u00b7 u' = 0` (everywhere).","title":"Kimura (1962)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/eq-kimura-fixation-of-generator-eq-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.eq_kimuraFixation_of_generator_eq_zero","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#eq-kimura-fixation-of-generator-eq-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"If `u` is continuous on `[0, 1]`, differentiable on `(0, 1)` with derivative `u'` differentiable on `(0, 1)`, solves the backward equation `L_\u03c3 u = 0` on `(0, 1)`, and satisfies `u 0 = 0`, `u 1 = 1`, then `u` is Kimura's fixation probability on `[0, 1]`.","title":"Uniqueness of the fixation probability (Kimura 1962)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/eq-kimura-fixation-of-cont-diff-on","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.eq_kimuraFixation_of_contDiffOn","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#eq-kimura-fixation-of-cont-diff-on","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Uniqueness of the fixation probability, `C\u00b2` form: a function continuous on `[0, 1]`, `C\u00b2` on `(0, 1)`, solving `L_\u03c3 u = 0` on `(0, 1)` with `u 0 = 0`, `u 1 = 1` equals Kimura's fixation probability on `[0, 1]`.","title":"Uniqueness of the fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-fixation-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation_nonneg","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-fixation-nonneg","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Kimura's fixation probability is nonnegative at nonnegative frequencies, for every selection coefficient `\u03c3`.","title":"Nonnegativity of Kimura's fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-fixation-le-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation_le_one","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-fixation-le-one","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Kimura's fixation probability is at most `1` at frequencies `y \u2264 1`, for every selection coefficient `\u03c3`.","title":"Kimura's fixation probability is at most one"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/kimura-fixation-mem-icc","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation_mem_Icc","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#kimura-fixation-mem-icc","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Kimura's fixation probability is a probability on `[0, 1]`.","title":"Kimura's fixation probability lies in the unit interval"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/lt-kimura-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.lt_kimuraFixation","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#lt-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Advantageous alleles (`\u03c3 > 0`) are fixed with probability strictly larger than their initial frequency `x \u2208 (0, 1)`.","title":"Advantageous alleles fix more often than their initial frequency"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/mean-absorption-time-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAbsorptionTime_zero","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#mean-absorption-time-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The neutral mean absorption time vanishes at `x = 0`.","title":"Neutral mean absorption time vanishes at frequency zero"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/mean-absorption-time-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAbsorptionTime_one","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#mean-absorption-time-one","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The neutral mean absorption time vanishes at `x = 1`.","title":"Neutral mean absorption time vanishes at frequency one"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/has-deriv-at-mean-absorption-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasDerivAt_meanAbsorptionTime","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#has-deriv-at-mean-absorption-time","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"On `(0, 1)`, `T'(x) = -2 (log x - log (1 - x))`.","title":"Derivative of the neutral mean absorption time"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/generator-mean-absorption-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_meanAbsorptionTime","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#generator-mean-absorption-time","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"The neutral mean absorption time solves `L_0 T = -1` on `(0, 1)`.","title":"Kimura\u2013Ohta (1969)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/generator-heterozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_heterozygosity","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#generator-heterozygosity","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"`H(x) = 2x(1 - x)` is an eigenfunction of the neutral generator with eigenvalue `-1`, so `E[H_t] = H_0 e^{-t}` (time in units of `2N` generations).","title":"Decay of heterozygosity"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/generator-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_pow","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#generator-pow","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"Moment duality with Kingman's coalescent at generator level: the neutral generator maps `y \u21a6 y ^ n` to `C(n, 2) (x ^ (n - 1) - x ^ n)`, which is the generator of the block-counting process (jumps `n \u2192 n - 1` at rate `C(n, 2)`) applied to `n \u21a6 x ^ n`.","title":"Moment duality with Kingman's coalescent"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/deriv-flux-wright-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.deriv_flux_wrightDensity","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#deriv-flux-wright-density","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"On `(0, 1)`, with infinitesimal variance `a(x) = x (1 - x)` and mutation drift `b(x) = \u03b1 (1 - x) - \u03b2 x`, Wright's density `\u03c0(x) = x ^ (2\u03b1 - 1) (1 - x) ^ (2\u03b2 - 1)` satisfies `(a \u03c0)' = 2 b \u03c0`, i.e. the stationary forward (Fokker\u2013Planck) equation `\u00bd (a \u03c0)' - b \u03c0 = 0` with zero probability flux.","title":"Wright's stationary density (zero-flux form)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/integral-wright-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_wrightDensity","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#integral-wright-density","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0` the stationary density integrates to the Beta function `B(2\u03b1, 2\u03b2) = \u0393(2\u03b1) \u0393(2\u03b2) / \u0393(2\u03b1 + 2\u03b2)`, so `wrightDensity \u03b1 \u03b2 x / B(2\u03b1, 2\u03b2)` is the Beta`(2\u03b1, 2\u03b2)` probability density on `(0, 1)`.","title":"Normalization of Wright's density"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/integral-wright-density-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_wrightDensity_pos","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#integral-wright-density-pos","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0` Wright's density has a finite, positive total mass on `(0, 1)`.","title":"Wright's density has finite positive total mass"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/interval-integrable-wright-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.intervalIntegrable_wrightDensity","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#interval-integrable-wright-density","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"For `\u03b1, \u03b2 > 0` Wright's density is integrable on `[0, 1]`.","title":"Integrability of Wright's density on the unit interval"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/integral-binomial-heterozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_binomial_heterozygosity","parent":44677,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion/index.html#integral-binomial-heterozygosity","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion","status":"fully_proved","summary":"If the number `X` of copies of an allele among `M` gene copies (`M = 2N` for diploids) is `Binomial(M, x)`, then the new frequency `p' = X / M` satisfies `E[2 p' (1 - p')] = (1 - 1/M) \u00b7 2x(1 - x)`.","title":"Heterozygosity decay in one Wright\u2013Fisher generation"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model.json","id":"Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model/index.html","scope":"Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model","status":"fully_proved","title":"Diffusion limit of the Wright\u2013Fisher model"},{"id":"Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model/integral-binomial-eq-bernstein-approxima","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_binomial_eq_bernsteinApproximation","parent":44709,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model/index.html#integral-binomial-eq-bernstein-approxima","scope":"Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model","status":"fully_proved","summary":"For `M` gene copies and current frequency `x`, the expected value of `f` at the next frequency, `\u222b k, f (k / M) \u2202Bin(M, x)`, is the `M`-th Bernstein approximation of `f` on `[0, 1]` evaluated at `x`.","title":"One Wright\u2013Fisher generation as a Bernstein approximation"},{"id":"Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model/tendsto-wf-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.tendsto_wfGenerator","parent":44709,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model/index.html#tendsto-wf-generator","scope":"Biology/population-genetics/diffusion-theory/diffusion-limit-of-the-wright-fisher-model","status":"fully_proved","summary":"For `f` of class `C\u00b2` and `x \u2208 [0, 1]`, the one-generation generator of the neutral Wright\u2013Fisher chain with `M` gene copies, sped up by the factor `M` (time in units of `M` generations), converges to the neutral Wright\u2013Fisher diffusion generator `(L_0 f)(x) = x (1 - x) / 2 \u00b7 f''(x)`.","title":"Wright\u2013Fisher diffusion limit, generator form (Feller 1951; Voronovskaya 1932)"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif.json","id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"can_state","title":"Fixation and absorption in the Wright\u2013Fisher diffusion"},{"id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/is-wright-fisher-solution","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.KimuraFixationProcess.IsWrightFisherSolution","parent":44712,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html#is-wright-fisher-solution","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"fully_proved","summary":"`X` solves the martingale problem for the Wright\u2013Fisher diffusion with scaled genic selection coefficient `\u03c3 = 2Ns` (time in units of `2N` generations) with respect to the filtration `\u2131` and the measure `\u03bc`: `X` takes values in `[0, 1]`, has continuous paths, is adapted to `\u2131`, and for every `C\u00b2` function `f` the process `f (X t) - f (X 0) - \u222b\u2080\u1d57 generator \u03c3 f (X s) ds` is an `\u2131`-martingale.","title":"Martingale problem for the Wright\u2013Fisher diffusion"},{"id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/absorption-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.KimuraFixationProcess.absorptionTime","parent":44712,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html#absorption-time","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"fully_proved","summary":"The absorption time of a `[0, 1]`-valued path: the first time it is at `0` or `1` (the infimum of the empty set is `0` in `\u211d\u22650`; the path is absorbed almost surely, see `ae_exists_absorbed`).","title":"Absorption time of a frequency path"},{"id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/cont-diff-kimura-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.contDiff_kimuraFixation","parent":44712,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html#cont-diff-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"fully_proved","summary":"Kimura's fixation probability is a `C\u00b2` function of the initial frequency on all of `\u211d`.","title":"Kimura's fixation probability is twice continuously differentiable"},{"id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/kimura-slope-pos","kind":"node","parent":44712,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html#kimura-slope-pos","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"can_state","summary":"The slope constant of Kimura's formula is positive.","title":"Positivity of Kimura's slope constant"},{"id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/strict-mono-kimura-fixation","kind":"node","parent":44712,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html#strict-mono-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"can_state","summary":"Kimura's fixation probability is strictly increasing in the initial frequency.","title":"Strict monotonicity of Kimura's fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/ae-exists-absorbed","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.KimuraFixationProcess.ae_exists_absorbed","parent":44712,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html#ae-exists-absorbed","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"fully_proved","summary":"A solution of the Wright\u2013Fisher martingale problem is almost surely absorbed: from some time on it is constantly `0` or constantly `1`.","title":"Absorption in finite time"},{"id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/measure-real-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.KimuraFixationProcess.measureReal_fixation","parent":44712,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html#measure-real-fixation","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"fully_proved","summary":"For a solution of the Wright\u2013Fisher martingale problem with scaled selection coefficient `\u03c3 = 2Ns` started at frequency `x`, the probability that the allele is eventually fixed is `(1 - e^{-2\u03c3x}) / (1 - e^{-2\u03c3})` (`x` when `\u03c3 = 0`).","title":"Kimura's fixation probability, process form"},{"id":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/integral-absorption-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.KimuraFixationProcess.integral_absorptionTime","parent":44712,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif/index.html#integral-absorption-time","scope":"Biology/population-genetics/diffusion-theory/fixation-and-absorption-in-the-wright-fisher-dif","status":"fully_proved","summary":"For a solution of the neutral Wright\u2013Fisher martingale problem started at frequency `x`, the expected time to loss or fixation, in units of `2N` generations, is `-2 (x log x + (1 - x) log (1 - x))`.","title":"Kimura\u2013Ohta mean absorption time, process form"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion.json","id":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/index.html","scope":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion","status":"fully_proved","title":"Sojourn times of the Wright\u2013Fisher diffusion"},{"id":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/neutral-green","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.neutralGreen","parent":44721,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/index.html#neutral-green","scope":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion","status":"fully_proved","summary":"The neutral Wright\u2013Fisher Green's function (sojourn density), time in units of `2N` generations (Ewens 2004, Chapter 4): `G(p, x) = 2 (1 - p) / (1 - x)` for `x \u2264 p` and `G(p, x) = 2 p / x` for `p < x`. In the diffusion reading, for `0 < x < 1`, `G(p, x) dx` is the expected time a neutral allele started at frequency `p` spends at frequencies in `(x, x + dx)` before loss or fixation.","title":"Neutral Wright\u2013Fisher Green's function"},{"id":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/generator-integral-neutral-green","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_integral_neutralGreen","parent":44721,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/index.html#generator-integral-neutral-green","scope":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion","status":"fully_proved","summary":"For continuous `g` and `p \u2208 (0, 1)`, the function `u(q) = \u222b x in 0..1, G(q, x) g(x)` solves the neutral backward equation `L_0 u = -g` at `p`. In the diffusion reading, `u(p)` is the expected value of `\u222b\u2080^T g(X_t) dt` up to the absorption time `T`.","title":"Green's representation (Ewens 2004, Chapter 4)"},{"id":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/integral-neutral-green-left","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_neutralGreen_left","parent":44721,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/index.html#integral-neutral-green-left","scope":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion","status":"fully_proved","summary":"Boundary value at `0`: an allele that is absent spends no time in `(0, 1)`, so `\u222b x in 0..1, G(0, x) g(x) = 0`.","title":"Sojourn integral vanishes for an absent allele"},{"id":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/integral-neutral-green-right","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_neutralGreen_right","parent":44721,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/index.html#integral-neutral-green-right","scope":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion","status":"fully_proved","summary":"Boundary value at `1`: a fixed allele spends no time in `(0, 1)`, so `\u222b x in 0..1, G(1, x) g(x) = 0`.","title":"Sojourn integral vanishes for a fixed allele"},{"id":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/integral-neutral-green","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_neutralGreen","parent":44721,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion/index.html#integral-neutral-green","scope":"Biology/population-genetics/diffusion-theory/sojourn-times-of-the-wright-fisher-diffusion","status":"fully_proved","summary":"For `p \u2208 [0, 1]`, the total expected sojourn time `\u222b x in 0..1, G(p, x)` is the Kimura\u2013Ohta mean absorption time `-2 (p log p + (1 - p) log (1 - p))` (units of `2N` generations).","title":"Mean absorption time as a sojourn integral"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/poisson-random-field-model.json","id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","title":"Poisson random field model"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/prf-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.prfDensity","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#prf-density","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"The Sawyer\u2013Hartl (1992) Poisson-random-field density: the expected number of sites per unit frequency at which the derived allele has frequency `x`, for scaled mutation rate `\u03b8 = 4Nu` and scaled genic selection `\u03c3 = 2Ns`: `\u03b8 \u00b7 u_\u03c3(1 - x) / (x (1 - x))` with `u_\u03c3 = kimuraFixation \u03c3`, i.e. `\u03b8 (1 - e^{-2\u03c3(1 - x)}) / ((1 - e^{-2\u03c3}) x (1 - x))`, and `\u03b8 / x` when `\u03c3 = 0`. The model domain is `\u03b8 > 0` and `x \u2208 (0, 1)`; at\u2026","title":"Sawyer\u2013Hartl Poisson random field density"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/prf-density-of-ne-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.prfDensity_of_ne_zero","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#prf-density-of-ne-zero","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"The Poisson-random-field density with selection, explicitly: `\u03b8 (1 - e^{-2\u03c3(1 - x)}) / ((1 - e^{-2\u03c3}) x (1 - x))` for `\u03c3 \u2260 0`.","title":"Closed form of the Poisson random field density with selection"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/prf-density-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.prfDensity_zero","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#prf-density-zero","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"The neutral Poisson-random-field density is `\u03b8 / x` (for `x \u2260 1`).","title":"Neutral Poisson random field density"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/prf-density-zero-eq-neutral-green","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.prfDensity_zero_eq_neutralGreen","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#prf-density-zero-eq-neutral-green","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"A new neutral mutation enters at a small frequency `p`; with mutations arriving at rate `\u03b8 / (2p)` per unit time (so that the mutational input of frequency is `\u03b8 / 2`), the expected sojourn density at `x \u2265 p` is `\u03b8 / (2p) \u00b7 G(p, x) = \u03b8 / x`, the neutral Poisson-random-field density.","title":"The neutral density from the Green's function"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/forward-prf-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.forward_prfDensity","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#forward-prf-density","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"On `(0, 1)` the weighted density `F(x) = x (1 - x) \u03c6(x)` of the Poisson-random-field density solves the stationary Fokker\u2013Planck equation `\u00bd F'' - \u03c3 F' = 0` of the Wright\u2013Fisher diffusion with genic selection `\u03c3` (the forward equation `\u00bd (x (1 - x) \u03c6)'' - (\u03c3 x (1 - x) \u03c6)' = 0`).","title":"Stationary forward equation (Sawyer\u2013Hartl 1992)"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/current-prf-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.current_prfDensity","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#current-prf-density","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"On `(0, 1)` the current `J = \u03c3 F - F' / 2` of the stationary forward equation, for the weighted density `F(x) = x (1 - x) \u03c6(x)`, is the constant `\u03b8 \u00b7 u_\u03c3'(0) / 2` (`u_\u03c3'(0) = kimuraSlope \u03c3 = 2\u03c3 / (1 - e^{-2\u03c3})`, and `1` for `\u03c3 = 0`), so `J = \u03b8 \u03c3 / (1 - e^{-2\u03c3})`, and `J = \u03b8 / 2` in the neutral case: the mutational input of frequency at `0`, carried through `(0, 1)` to fixation at `1`.","title":"Probability current"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/tendsto-weighted-prf-density-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.tendsto_weighted_prfDensity_zero","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#tendsto-weighted-prf-density-zero","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"As `x \u2192 0\u207a` the weighted density `x (1 - x) \u03c6(x)` tends to `\u03b8`. (This is not the probability current, which is `current_prfDensity`.)","title":"Boundary normalization at `0`"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/tendsto-weighted-prf-density-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.tendsto_weighted_prfDensity_one","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#tendsto-weighted-prf-density-one","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"As `x \u2192 1\u207b` the weighted density `x (1 - x) \u03c6(x)` tends to `0`.","title":"Boundary normalization at `1`"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/eq-prf-density-of-forward","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.eq_prfDensity_of_forward","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#eq-prf-density-of-forward","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"Let `\u03b8 \u2260 0` and let `\u03c6` be a density on `(0, 1)` whose weighted density `x (1 - x) \u03c6(x)` extends to a function `F` continuous on `[0, 1]`, twice differentiable on `(0, 1)`, solving the stationary forward equation `\u00bd F'' - \u03c3 F' = 0` there, with boundary normalizations `F(0) = \u03b8` and `F(1) = 0`. Then `\u03c6` is the Sawyer\u2013Hartl density on `(0, 1)`.","title":"ODE characterization of the Poisson-random-field density"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/expected-sfs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.expectedSFS","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#expected-sfs","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"The expected unfolded site-frequency spectrum of a sample of `n` gene copies under the Poisson random field: the expected number of sites where exactly `i` of the `n` sampled copies carry the derived allele, `E \u03be_i = \u222b x in 0..1, C(n, i) x ^ i (1 - x) ^ (n - i) \u03c6(x)`. The site-frequency spectrum consists of the segregating classes `1 \u2264 i < n`. The value at `i = n` is the expected number of sites at which every\u2026","title":"Expected site frequency spectrum under the Poisson random field"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/expected-sfs-neutral","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.expectedSFS_neutral","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#expected-sfs-neutral","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"With `\u03c3 = 0`, for the segregating classes `1 \u2264 i < n` the expected number of sites with `i` derived copies in a sample of `n` is `\u03b8 / i`.","title":"Neutral site-frequency spectrum (Fu 1995)"},{"id":"Biology/population-genetics/diffusion-theory/poisson-random-field-model/expected-sfs-neutral-self","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.expectedSFS_neutral_self","parent":44727,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/poisson-random-field-model/index.html#expected-sfs-neutral-self","scope":"Biology/population-genetics/diffusion-theory/poisson-random-field-model","status":"fully_proved","summary":"With `\u03c3 = 0` the expected number of sites at which all `n \u2265 1` sampled copies carry the derived allele is `\u03b8 / n`. These sites are monomorphic in the sample, so this is not a class of the site-frequency spectrum.","title":"Expected number of sites derived in the whole neutral sample"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di.json","id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","title":"Selection duality and general one-dimensional diffusions"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/asg-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.asgGenerator","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#asg-generator","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The generator of the line-counting process of the ancestral selection graph with scaled selection coefficient `\u03c3` (time in units of `2N` generations, the scaling of `generator`): from `n` lines, branching `n \u2192 n + 1` at rate `n \u03c3` and coalescence `n \u2192 n - 1` at rate `C(n, 2)`.","title":"Generator of the ancestral selection graph line-counting process"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/generator-one-sub-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_one_sub_pow","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#generator-one-sub-pow","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Selection duality (Mano 2009) at generator level, with the line-counting process of the ancestral selection graph (Krone\u2013Neuhauser 1997) as the dual: the Wright\u2013Fisher generator with genic selection maps `y \u21a6 (1 - y) ^ n` to `C(n, 2) ((1 - x)^(n-1) - (1 - x)^n) + n \u03c3 ((1 - x)^(n+1) - (1 - x)^n)`. This identity holds for every real `\u03c3`; the dual is a Markov chain only for `\u03c3 \u2265 0`.","title":"Selection duality (Mano 2009)"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/generator-one-sub-pow-eq-asg-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_one_sub_pow_eq_asgGenerator","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#generator-one-sub-pow-eq-asg-generator","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The selection duality in operator form: `L_\u03c3` applied to `y \u21a6 (1 - y) ^ n` equals the ancestral-selection-graph generator applied to `k \u21a6 (1 - x) ^ k` at `n`.","title":"Selection duality in operator form"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/asg-stationary","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.asgStationary","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#asg-stationary","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The stationary law of the ancestral-selection-graph line-counting process with `\u03c3 > 0`: `\u03c0_k = (2\u03c3)^k / k! / (e^{2\u03c3} - 1)` for `k \u2265 1` and `\u03c0_0 = 0`, a Poisson`(2\u03c3)` law conditioned to be positive. For `\u03c3 = 0` the stationary law is `\u03b4\u2081`: branching vanishes, coalescence reaches state `1`, and state `1` is absorbing, so `\u03c0_1 = 1` and `\u03c0_k = 0` otherwise (the limit of the `\u03c3 > 0` formula as `\u03c3 \u2192 0`).","title":"Stationary law of the ancestral selection graph line counts"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/asg-stationary-detailed-balance","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.asgStationary_detailed_balance","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#asg-stationary-detailed-balance","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Detailed balance of the ancestral-selection-graph chain: `\u03c0_k \u00b7 (k \u03c3) = \u03c0_{k+1} \u00b7 C(k + 1, 2)` for every `k` (the flux `k \u2192 k + 1` equals the flux `k + 1 \u2192 k`). This is an identity for every real `\u03c3`.","title":"Detailed balance"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/asg-stationary-global-balance","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.asgStationary_global_balance","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#asg-stationary-global-balance","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Global balance (stationarity `\u03c0 Q = 0`) of the ancestral-selection-graph chain at state `k + 1`: the outflow `\u03c0_{k+1} ((k + 1) \u03c3 + C(k + 1, 2))` equals the inflow `\u03c0_k (k \u03c3) + \u03c0_{k+2} C(k + 2, 2)` (for `k = 0` the inflow from the unreachable state `0` vanishes).","title":"Global balance"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/asg-stationary-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.asgStationary_nonneg","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#asg-stationary-nonneg","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"For `\u03c3 > 0` the stationary weights are nonnegative.","title":"Stationary weights under selection are nonnegative"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/has-sum-asg-stationary-mul-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasSum_asgStationary_mul_pow","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#has-sum-asg-stationary-mul-pow","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The probability generating function of the stationary law: for `\u03c3 > 0` and every real `y`, `\u2211\u2096 \u03c0_k y^k = (e^{2\u03c3y} - 1) / (e^{2\u03c3} - 1)`.","title":"Generating function of the stationary law under selection"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/has-sum-asg-stationary","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasSum_asgStationary","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#has-sum-asg-stationary","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"For `\u03c3 > 0` the stationary weights sum to `1`.","title":"Stationary weights under selection sum to one"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/has-sum-asg-stationary-mul-pow-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasSum_asgStationary_mul_pow_zero","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#has-sum-asg-stationary-mul-pow-zero","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The neutral stationary law is `\u03b4\u2081`: its generating function is `y`.","title":"Neutral stationary law is the point mass at one"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/has-sum-asg-stationary-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasSum_asgStationary_zero","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#has-sum-asg-stationary-zero","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The neutral stationary weights sum to `1`.","title":"Neutral stationary weights sum to one"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/one-sub-kimura-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.one_sub_kimuraFixation","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#one-sub-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The loss probability in Kimura's formula, written with positive exponents: for `\u03c3 \u2260 0`, `1 - kimuraFixation \u03c3 x = (e^{2\u03c3(1 - x)} - 1) / (e^{2\u03c3} - 1)`.","title":"Loss probability in Kimura's fixation formula"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/has-sum-asg-stationary-mul-one-sub-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasSum_asgStationary_mul_one_sub_pow","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#has-sum-asg-stationary-mul-one-sub-pow","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Loss probability from the stationary ancestral selection graph (Mano 2009): for `\u03c3 > 0`, `\u2211\u2096 \u03c0_k (1 - x)^k = 1 - kimuraFixation \u03c3 x`, i.e. the loss probability of the favoured allele at frequency `x` is the probability that all lines of the stationary ancestral selection graph carry the unfavoured allele. Only the series identity is formalized; the sampling interpretation is not.","title":"Loss probability from the stationary ancestral selection graph"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/diffusion-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.diffusionGenerator","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#diffusion-generator","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The backward generator of a one-dimensional diffusion with drift `b` and infinitesimal variance `a`: `(L u)(x) = a(x) / 2 \u00b7 u''(x) + b(x) \u00b7 u'(x)`.","title":"Backward generator of a one-dimensional diffusion"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/scale-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.scaleDensity","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#scale-density","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale density `\u03c8(y) = exp (-\u222b\u2080^y 2 b(z) / a(z) dz)` of the diffusion with drift `b` and variance `a` (Karlin\u2013Taylor 1981, Chapter 15).","title":"Scale density of a one-dimensional diffusion"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/scale-function","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.scaleFunction","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#scale-function","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale function `S(p) = \u222b\u2080^p \u03c8(y) dy`, normalized by `S(0) = 0`.","title":"Scale function of a one-dimensional diffusion"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/fixation-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.fixationProb","parent":44740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dirichlet ratio "],["math","\\(S(p)/S(1)\\)"],["text"," of the scale function"]],"source":"Dirichlet ratio $S(p)/S(1)$ of the scale function"},"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#fixation-prob","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The analytic Dirichlet ratio `S(p) / S(1)` for drift `b` and variance `a` (Kimura 1962; Ewens 2004, Chapter 4). This is the solution of the backward boundary-value problem `L u = 0`, `u(0) = 0`, `u(1) = 1` (see `eq_fixationProb_of_diffusionGenerator_eq_zero`). It is the hitting probability `P_p(hit 1 before 0)` only when both boundaries are Feller-accessible; that is not implied by `IsRegularDiffusion` (see the\u2026","title":"Dirichlet ratio $S(p)/S(1)$ of the scale function"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/green-function","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.greenFunction","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#green-function","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The Green's function (analytic Green kernel) of `-L` with Dirichlet conditions at `0` and `1` (Ewens 2004, Chapter 4; Karlin\u2013Taylor 1981, Chapter 15): `G(p, x) = 2 S(min p x) (S(1) - S(max p x)) / (S(1) a(x) \u03c8(x))`. In the diffusion reading (which requires Feller-accessible boundaries), `G(p, x) dx` is the expected time spent in `(x, x + dx)` before absorption.","title":"Dirichlet Green's function of a one-dimensional diffusion"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/is-regular-diffusion","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion","parent":44740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Standing regularity hypotheses for a diffusion on "],["math","\\((0, 1)\\)"]],"source":"Standing regularity hypotheses for a diffusion on $(0, 1)$"},"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#is-regular-diffusion","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Regularity hypotheses for a diffusion on `(0, 1)` with drift `b` and variance `a`: `a` is positive and continuous on `(0, 1)`, `b / a` is continuous on `(0, 1)` and integrable on `[0, 1]` (so the scale function is finite up to both boundaries). The values of `b` and `a` at `0` and `1` are irrelevant. These are standing hypotheses for this file, not Feller's classification of regular boundaries: they do not imply\u2026","title":"Standing regularity hypotheses for a diffusion on $(0, 1)$"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/scale-density-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.scaleDensity_pos","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#scale-density-pos","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale density is positive.","title":"Positivity of the scale density"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/scale-function-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.scaleFunction_zero","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#scale-function-zero","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale function vanishes at `0`.","title":"Scale function vanishes at zero"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/continuous-on-scale-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion.continuousOn_scaleDensity","parent":44740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuity of the scale density on "],["math","\\((0, 1)\\)"]],"source":"Continuity of the scale density on $(0, 1)$"},"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#continuous-on-scale-density","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale density is continuous on `[0, 1]`.","title":"Continuity of the scale density on $(0, 1)$"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/has-deriv-at-scale-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion.hasDerivAt_scaleDensity","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#has-deriv-at-scale-density","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"`\u03c8' = -(2 b / a) \u03c8` inside the interval.","title":"Derivative of the scale density"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/has-deriv-at-scale-function","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion.hasDerivAt_scaleFunction","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#has-deriv-at-scale-function","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"`S' = \u03c8` inside the interval.","title":"Derivative of the scale function is the scale density"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/continuous-on-scale-function","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion.continuousOn_scaleFunction","parent":44740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuity of the scale function on "],["math","\\((0, 1)\\)"]],"source":"Continuity of the scale function on $(0, 1)$"},"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#continuous-on-scale-function","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale function is continuous on `[0, 1]`.","title":"Continuity of the scale function on $(0, 1)$"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/strict-mono-on-scale-function","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion.strictMonoOn_scaleFunction","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#strict-mono-on-scale-function","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale function is strictly increasing on `[0, 1]`.","title":"Scale function is strictly increasing"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/scale-function-one-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion.scaleFunction_one_pos","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#scale-function-one-pos","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"`S(1) > 0`, so the scale-function ratio and the Green's function have a positive denominator.","title":"Scale function is positive at one"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/fixation-prob-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion.fixationProb_one","parent":44740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dirichlet ratio equals one at "],["math","\\(p = 1\\)"]],"source":"Dirichlet ratio equals one at $p = 1$"},"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#fixation-prob-one","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"A fixed allele stays fixed.","title":"Dirichlet ratio equals one at $p = 1$"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/fixation-prob-mem-icc","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.IsRegularDiffusion.fixationProb_mem_Icc","parent":44740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dirichlet ratio lies in "],["math","\\((0, 1)\\)"]],"source":"Dirichlet ratio lies in $(0, 1)$"},"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#fixation-prob-mem-icc","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale-function ratio lies in `[0, 1]`.","title":"Dirichlet ratio lies in $(0, 1)$"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/fixation-prob-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.fixationProb_zero","parent":44740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Dirichlet ratio vanishes at "],["math","\\(p = 0\\)"]],"source":"Dirichlet ratio vanishes at $p = 0$"},"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#fixation-prob-zero","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"An absent allele is never fixed (for every `b`, `a`).","title":"Dirichlet ratio vanishes at $p = 0$"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/diffusion-generator-fixation-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.diffusionGenerator_fixationProb","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#diffusion-generator-fixation-prob","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale-function ratio `S / S(1)` solves the backward equation `a / 2 \u00b7 u'' + b \u00b7 u' = 0` on `(0, 1)`.","title":"Kimura (1962)"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/eq-of-diffusion-generator-eq-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.eq_of_diffusionGenerator_eq_zero","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#eq-of-diffusion-generator-eq-zero","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"If `u` is continuous on `[0, 1]`, differentiable on `(0, 1)` with derivative differentiable on `(0, 1)`, and `L u = 0` on `(0, 1)`, then `u = u(0) + (u(1) - u(0)) \u00b7 S / S(1)` on `[0, 1]`.","title":"Solutions of `L u = 0`"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/eq-fixation-prob-of-diffusion-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.eq_fixationProb_of_diffusionGenerator_eq_zero","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#eq-fixation-prob-of-diffusion-generator","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"A function continuous on `[0, 1]`, differentiable on `(0, 1)` with derivative differentiable on `(0, 1)`, solving `L u = 0` on `(0, 1)` with `u(0) = 0` and `u(1) = 1` equals `S / S(1)` on `[0, 1]`.","title":"Uniqueness of the scale-function ratio (Kimura 1962)"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/eq-of-diffusion-generator-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.eq_of_diffusionGenerator_eq","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#eq-of-diffusion-generator-eq","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Two functions continuous on `[0, 1]`, differentiable on `(0, 1)` with derivatives differentiable on `(0, 1)`, with `L u = L v` on `(0, 1)` and the same values at `0` and `1`, agree on `[0, 1]`. Together with `diffusionGenerator_integral_greenFunction` this identifies the Green integral as the solution of `L v = -g`, `v(0) = v(1) = 0` whenever it has this regularity.","title":"Uniqueness for the Dirichlet problem"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/diffusion-generator-integral-green-funct","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.diffusionGenerator_integral_greenFunction","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#diffusion-generator-integral-green-funct","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Let `g` be continuous on `(0, 1)` and assume the two halves of the Green integral are integrable away from the opposite boundary: `S g / (a \u03c8)` on `[0, q]` and `(S(1) - S) g / (a \u03c8)` on `[q, 1]` for every `q \u2208 (0, 1)` (true for every `g` continuous on `[0, 1]` under genic selection, where `S / (a \u03c8)` is bounded near `0` and `(S(1) - S) / (a \u03c8)` near `1`). Then `v(q) = \u222b\u2080\u00b9 G(q, x) g(x) dx` solves `L v = -g` on\u2026","title":"Green's representation (Ewens 2004, Chapter 4; Karlin\u2013Taylor 1981, Chapter 15)"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/integral-green-function-zero-left","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_greenFunction_zero_left","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#integral-green-function-zero-left","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Dirichlet condition at `0`: `\u222b\u2080\u00b9 G(0, x) g(x) dx = 0` (for every `b`, `a`, `g`).","title":"Dirichlet condition of the Green's function at zero"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/integral-green-function-one-left","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_greenFunction_one_left","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#integral-green-function-one-left","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Dirichlet condition at `1`: `\u222b\u2080\u00b9 G(1, x) g(x) dx = 0` (for every `b`, `a`, `g`).","title":"Dirichlet condition of the Green's function at one"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/diffusion-generator-wright-fisher","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.diffusionGenerator_wrightFisher","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#diffusion-generator-wright-fisher","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The general generator with drift `\u03c3 x (1 - x)` and variance `x (1 - x)` is the Wright\u2013Fisher generator with genic selection.","title":"Wright\u2013Fisher generator with genic selection as a general diffusion"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/is-regular-diffusion-wright-fisher","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.isRegularDiffusion_wrightFisher","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#is-regular-diffusion-wright-fisher","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The Wright\u2013Fisher diffusion with genic selection is regular.","title":"Wright\u2013Fisher diffusion with genic selection is regular"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/scale-density-wright-fisher","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.scaleDensity_wrightFisher","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#scale-density-wright-fisher","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The scale density of the Wright\u2013Fisher diffusion with genic selection is `e^{-2\u03c3y}` on `[0, 1]`.","title":"Scale density of the Wright\u2013Fisher diffusion with genic selection"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/fixation-prob-wright-fisher","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.fixationProb_wrightFisher","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#fixation-prob-wright-fisher","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"For genic selection the general scale-function ratio is `kimuraFixation \u03c3 p` on `[0, 1]`.","title":"Kimura's formula as a corollary"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/green-function-neutral","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.greenFunction_neutral","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#green-function-neutral","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The general Green's function of the neutral Wright\u2013Fisher diffusion (`b = 0`, `a = x (1 - x)`) is the neutral Green's function of `Sojourn.lean` for `x \u2208 (0, 1)`.","title":"General Green's function agrees with the neutral Wright\u2013Fisher one"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/dominance-drift","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.dominanceDrift","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#dominance-drift","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The drift `2 c x (1 - x) (h + (1 - 2h) x)` of an allele at frequency `x` with genotype fitnesses `1`, `1 + h s`, `1 + s` (for zero, one and two copies), `c = N s`, time in units of `2N` generations (Kimura 1962). For `h = 1/2` it is genic selection with `\u03c3 = c`.","title":"Drift of an allele under selection with dominance"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/is-regular-diffusion-dominance","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.isRegularDiffusion_dominance","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#is-regular-diffusion-dominance","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The diffusion with dominance is regular.","title":"Diffusion with dominance is regular"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/fixation-prob-dominance","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.fixationProb_dominance","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#fixation-prob-dominance","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"With `D = 2h - 1`, for `p \u2208 [0, 1]`, `u(p) = \u222b\u2080^p e^{-2cDx(1-x) - 2cx} dx / \u222b\u2080\u00b9 e^{-2cDx(1-x) - 2cx} dx`.","title":"Kimura's scale-function ratio with dominance (Kimura 1962)"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/conditional-green","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.conditionalGreen","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#conditional-green","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The sojourn density conditioned on fixation, the Doob `h`-transform of the Green's function by the fixation probability `u`: `G(p, x) u(x) / u(p)` (Kimura\u2013Ohta 1969; Ewens 2004, Chapter 4). At `p = 0` it is the junk value `0`.","title":"Sojourn density conditioned on fixation"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/diffusion-generator-mul-integral-conditi","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.diffusionGenerator_mul_integral_conditionalGreen","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#diffusion-generator-mul-integral-conditi","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Under the hypotheses of `diffusionGenerator_integral_greenFunction` for `g = u` (the fixation probability), the function `w(q) = u(q) \u00b7 \u222b\u2080\u00b9 G*(q, x) dx`, with `G*` the conditional sojourn density, solves `L w = -u` on `(0, 1)` (Ewens 2004, Chapter 4).","title":"Conditional mean time to fixation, backward equation"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/neutral-fixation-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.neutralFixationTime","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#neutral-fixation-time","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The neutral conditional mean time to fixation `t\u0304\u2081(p) = -2 (1 - p) log (1 - p) / p`, in units of `2N` generations (Kimura\u2013Ohta 1969). It is meaningful for `p \u2208 (0, 1)`; at `p = 0` the division gives the junk value `0` (the limit is `2`, see `tendsto_neutralFixationTime_zero`).","title":"Neutral conditional mean time to fixation"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/neutral-loss-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.neutralLossTime","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#neutral-loss-time","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The neutral conditional mean time to loss `t\u0304\u2080(p) = -2 p log p / (1 - p)`, in units of `2N` generations (Kimura\u2013Ohta 1969). It is meaningful for `p \u2208 (0, 1)`.","title":"Neutral conditional mean time to loss"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/neutral-loss-time-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.neutralLossTime_eq","parent":44740,"reading_title":{"format":"inline-parts-v1","parts":[["text","Neutral loss time equals fixation time at "],["math","\\(1 - p\\)"]],"source":"Neutral loss time equals fixation time at $1 - p$"},"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#neutral-loss-time-eq","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"Loss of an allele at frequency `p` is fixation of the other allele at `1 - p`.","title":"Neutral loss time equals fixation time at $1 - p$"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/integral-neutral-green-mul-div","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_neutralGreen_mul_div","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#integral-neutral-green-mul-div","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"For `p \u2208 (0, 1)`, `\u222b\u2080\u00b9 G(p, x) \u00b7 x / p dx = -2 (1 - p) log (1 - p) / p`, with `G` the neutral Green's function and `u(x) = x` the neutral fixation probability.","title":"Neutral conditional fixation time as an `h`-transformed sojourn integral"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/integral-neutral-green-mul-one-sub-div","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_neutralGreen_mul_one_sub_div","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#integral-neutral-green-mul-one-sub-div","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"For `p \u2208 (0, 1)`, `\u222b\u2080\u00b9 G(p, x) \u00b7 (1 - x) / (1 - p) dx = -2 p log p / (1 - p)`.","title":"Neutral conditional loss time"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/integral-conditional-green-neutral","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_conditionalGreen_neutral","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#integral-conditional-green-neutral","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"For the neutral Wright\u2013Fisher diffusion (`b = 0`, `a = x (1 - x)`) and `p \u2208 (0, 1)`, the total conditional sojourn time `\u222b\u2080\u00b9 G(p, x) u(x) / u(p) dx` is `-2 (1 - p) log (1 - p) / p`.","title":"The general `h`-transform specializes to Kimura\u2013Ohta"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/generator-mul-neutral-fixation-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_mul_neutralFixationTime","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#generator-mul-neutral-fixation-time","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"The neutral conditional fixation time satisfies the conditioned backward equation `L_0 (p \u00b7 t\u0304\u2081(p)) = -p` on `(0, 1)`.","title":"Backward equation for the neutral conditional fixation time"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/tendsto-neutral-fixation-time-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.tendsto_neutralFixationTime_zero","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#tendsto-neutral-fixation-time-zero","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"A new neutral mutation that is destined to fix takes on average `2` units (`4N` generations) to do so (Kimura\u2013Ohta 1969).","title":"`t\u0304\u2081(0\u207a) = 2`"},{"id":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/mul-neutral-fixation-time-add-mul-neutra","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.mul_neutralFixationTime_add_mul_neutralLossTime","parent":44740,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di/index.html#mul-neutral-fixation-time-add-mul-neutra","scope":"Biology/population-genetics/diffusion-theory/selection-duality-and-general-one-dimensional-di","status":"fully_proved","summary":"For `p \u2208 (0, 1)`, `p \u00b7 t\u0304\u2081(p) + (1 - p) \u00b7 t\u0304\u2080(p) = -2 (p log p + (1 - p) log (1 - p))`, the unconditional mean absorption time.","title":"Total expectation"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect.json","id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"can_state","title":"Allele age and the Maruyama effect"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/kimura-slope-pos","kind":"node","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#kimura-slope-pos","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"can_state","summary":"The slope constant of Kimura's formula is positive.","title":"Positivity of the slope constant in Kimura's formula"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/strict-mono-kimura-fixation","kind":"node","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#strict-mono-kimura-fixation","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"can_state","summary":"Kimura's fixation probability is strictly increasing in the initial frequency.","title":"Kimura's fixation probability is strictly increasing"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/kimura-fixation-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation_pos","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#kimura-fixation-pos","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"An allele present at positive frequency has positive fixation probability.","title":"Positivity of Kimura's fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/kimura-fixation-lt-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.kimuraFixation_lt_one","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#kimura-fixation-lt-one","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"An allele at frequency below `1` is lost with positive probability.","title":"Kimura's fixation probability is less than one"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/selection-green","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.selectionGreen","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#selection-green","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The sojourn density (Green's function) of the Wright\u2013Fisher diffusion with genic selection, time in units of `2N` generations, `\u03c3 = 2Ns`. With `u = kimuraFixation \u03c3` and `u'(x) = kimuraSlope \u03c3 \u00b7 e^{-2\u03c3x}`,","title":"Sojourn density of the Wright\u2013Fisher diffusion with selection"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/selection-green-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.selectionGreen_zero","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#selection-green-zero","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For `\u03c3 = 0` and `0 < x < 1` the sojourn density is the neutral Green's function.","title":"Neutral case of the sojourn density"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/green-function-eq-selection-green","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.greenFunction_eq_selectionGreen","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#green-function-eq-selection-green","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For genic selection `b x = \u03c3 x (1 - x)`, `a x = x (1 - x)`, the general Green's function `greenFunction` is `selectionGreen \u03c3` on `[0, 1] \u00d7 (0, 1)`. The scale function is `S(y) = u(y) / u'(0)` with `u = kimuraFixation \u03c3` and `u'(0) = kimuraSlope \u03c3`, since both have derivative `\u03c8(y) = e^{-2\u03c3y}` and vanish at `0`; substituting that and `S(1) = 1 / u'(0)` into `greenFunction` gives `selectionGreen`.","title":"The general Green kernel specializes to Kimura's sojourn density"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/generator-integral-selection-green","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_integral_selectionGreen","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#generator-integral-selection-green","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Green's representation with selection (Ewens 2004, Chapter 4; the integrated form of the paper's `-\u03b4(x - p) = s p (1 - p) \u2202\u03a8/\u2202p + p (1 - p) / (4N) \u2202\u00b2\u03a8/\u2202p\u00b2`). For continuous `g` and `p \u2208 (0, 1)`, `v(q) = \u222b\u2080\u00b9 \u03a8(q, x) g(x) dx` solves `L_\u03c3 v = -g` at `p`; with `integral_selectionGreen_left` and `integral_selectionGreen_right` (`v(0) = v(1) = 0`) this generalizes `generator_integral_neutralGreen` to `\u03c3 \u2260 0`. In the\u2026","title":"Green's representation with selection"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/integral-selection-green-left","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_selectionGreen_left","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#integral-selection-green-left","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Boundary value at `0`: an absent allele spends no time in `(0, 1)`, so `\u222b\u2080\u00b9 \u03a8(0, x) g(x) dx = 0` for every `g`.","title":"Sojourn density from frequency zero vanishes"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/integral-selection-green-right","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_selectionGreen_right","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#integral-selection-green-right","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Boundary value at `1`: a fixed allele spends no time in `(0, 1)`, so `\u222b\u2080\u00b9 \u03a8(1, x) g(x) dx = 0` for every `g`.","title":"Sojourn density from frequency one vanishes"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/generator-selection-green-of-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_selectionGreen_of_lt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#generator-selection-green-of-lt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Below the target frequency the sojourn density solves the homogeneous backward equation in its first argument: `L_\u03c3 \u03a8(\u00b7, x) = 0` at every `p < x` (pointwise form of the paper's `-\u03b4(x - p) = L_\u03c3 \u03a8(\u00b7, x)`).","title":"Sojourn density solves the backward equation below the target"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/generator-selection-green-of-gt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.generator_selectionGreen_of_gt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#generator-selection-green-of-gt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Above the target frequency: `L_\u03c3 \u03a8(\u00b7, x) = 0` at every `p > x`.","title":"Sojourn density solves the backward equation above the target"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/continuous-selection-green-left","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.continuous_selectionGreen_left","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#continuous-selection-green-left","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The sojourn density is continuous in the starting frequency (in particular at `p = x`).","title":"Continuity of the sojourn density in the starting frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/has-deriv-within-at-selection-green-iic","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasDerivWithinAt_selectionGreen_Iic","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#has-deriv-within-at-selection-green-iic","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Left derivative at the diagonal: `\u2202_p \u03a8(x\u207b, x) = 2 (1 - u(x)) / (x (1 - x))`. Together with `hasDerivWithinAt_selectionGreen_Ici` this gives the jump `-2 / (x (1 - x))` of `\u2202_p \u03a8` at `p = x`, i.e. `L_\u03c3 \u03a8(\u00b7, x) = -\u03b4_x` in the distributional sense.","title":"Left derivative of the sojourn density at the diagonal"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/has-deriv-within-at-selection-green-ici","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.hasDerivWithinAt_selectionGreen_Ici","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#has-deriv-within-at-selection-green-ici","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Right derivative at the diagonal: `\u2202_p \u03a8(x\u207a, x) = -2 u(x) / (x (1 - x))`.","title":"Right derivative of the sojourn density at the diagonal"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/age-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.ageDensity","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#age-density","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The Maruyama\u2013Kimura conditional sojourn density (the paper's `\u03a8_z(p, z)`): given the initial frequency `p` and the current frequency `x`, the density of mean time spent at frequency `z` along the path, `\u03a8(p, z) \u03a8(z, x) / \u03a8(p, x)` (Maruyama and Kimura 1975; Kiezun et al. 2013, Methods). Time in units of `2N` generations.","title":"Maruyama\u2013Kimura conditional sojourn density"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-below","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaBelow","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-below","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The paper's closed form `\u03a8\u2081(z)` for `0 < z < x` (Kiezun et al. 2013, Methods), in units of `2N` generations with `\u03c3 = 2Ns`: `(e^{-2\u03c3(1-z)} - 1)(e^{-2\u03c3z} - 1) / (\u03c3 z (1 - z) (1 - e^{-2\u03c3}))`, and its continuous extension `2` for `\u03c3 = 0`. It does not depend on the current frequency `x`.","title":"Closed-form sojourn density below the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The paper's closed form `\u03a8\u2082(z)` for `x < z < 1` (Kiezun et al. 2013, Methods), in units of `2N` generations with `\u03c3 = 2Ns`: `(e^{-2\u03c3(1-z)} - 1)(e^{2\u03c3(1-z)} - 1)(1 - e^{2\u03c3x}) / (\u03c3 z (1 - z)(e^{-2\u03c3(1-x)} - 1)(1 - e^{2\u03c3}))`, and its continuous extension `2 x (1 - z) / ((1 - x) z)` for `\u03c3 = 0`.","title":"Closed-form sojourn density above the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/mean-age","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAge","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#mean-age","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The mean age of an allele at current frequency `x` (the paper's `a(x)`), in units of `2N` generations: `\u222b\u2080^x \u03a8\u2081(z) dz + \u222b_x^1 \u03a8\u2082(z) dz`.","title":"Mean age of an allele at a given frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-below-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaBelow_eq","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-below-eq","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2081` through Kimura's fixation probability: `\u03a8\u2081(z) = 2 u(z)(1 - u(z)) / (z (1 - z) u'(z))` on `(0, 1)`.","title":"Sojourn density below the current frequency via fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove_eq","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above-eq","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2082` through Kimura's fixation probability: `\u03a8\u2082(z) = 2 u(x)(1 - u(z))\u00b2 / ((1 - u(x)) z (1 - z) u'(z))` on `(0, 1)`.","title":"Sojourn density above the current frequency via fixation probability"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/age-density-of-le-of-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.ageDensity_of_le_of_le","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#age-density-of-le-of-le","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For `0 < p \u2264 z \u2264 x < 1` the conditional sojourn density is the paper's `\u03a8\u2081(z)`; it depends neither on `p` nor on `x`.","title":"Closed form below the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/age-density-of-le-of-ge","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.ageDensity_of_le_of_ge","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#age-density-of-le-of-ge","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For `0 < p \u2264 x \u2264 z < 1` the conditional sojourn density is the paper's `\u03a8\u2082(z)`; it does not depend on `p`.","title":"Closed form above the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-below-generations","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaBelow_generations","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-below-generations","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"With `N` diploid individuals, selection coefficient `s \u2260 0` and time in generations, the paper prints `\u03a8\u2081(z) = (e^{-\u03b3(1-z)} - 1)(e^{-\u03b3z} - 1) / (s z (1 - z)(1 - e^{-\u03b3}))` with `\u03b3 = 4Ns`; it equals `2N \u00b7 maruyamaBelow (2Ns) z`.","title":"Paper units"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above-generations","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove_generations","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above-generations","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The paper's `\u03a8\u2082(z) = (e^{-\u03b3(1-z)} - 1)(e^{\u03b3(1-z)} - 1)(1 - e^{\u03b3x}) / (s z (1 - z)(e^{-\u03b3(1-x)} - 1)(1 - e^{\u03b3}))`, `\u03b3 = 4Ns`, equals `2N \u00b7 maruyamaAbove (2Ns) x z`.","title":"Paper units"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/mean-age-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAge_zero","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#mean-age-zero","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The mean age of a neutral allele at frequency `x \u2208 (0, 1)` is `-2 x log x / (1 - x)` in units of `2N` generations (`-4N x ln x / (1 - x)` generations).","title":"Kimura and Ohta (1973)"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-below-eq-sinh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaBelow_eq_sinh","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-below-eq-sinh","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2081` in hyperbolic form: `2 sinh(\u03c3z) sinh(\u03c3(1 - z)) / (\u03c3 z (1 - z) sinh \u03c3)` for `\u03c3 \u2260 0`.","title":"Hyperbolic form of the sojourn density below the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above-eq-sinh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove_eq_sinh","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above-eq-sinh","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2082` in hyperbolic form: `2 sinh(\u03c3(1 - z))\u00b2 sinh(\u03c3x) / (\u03c3 sinh \u03c3 sinh(\u03c3(1 - x)) z (1 - z))` for `\u03c3 \u2260 0`.","title":"Hyperbolic form of the sojourn density above the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-below-neg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaBelow_neg","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-below-neg","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2081` is the same for `\u03c3` and `-\u03c3`.","title":"Symmetry"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above-neg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove_neg","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above-neg","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2082` is the same for `\u03c3` and `-\u03c3`.","title":"Symmetry"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-below-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaBelow_pos","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-below-pos","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2081` is positive on `(0, 1)`.","title":"Positivity of the sojourn density below the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-below-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaBelow_lt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-below-lt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For `\u03c3 \u2260 0`, `\u03a8\u2081(z) < 2`, its neutral value, for every `z \u2208 (0, 1)`.","title":"Maruyama effect, below the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove_lt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above-lt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For `\u03c3 \u2260 0` and `0 < x < z < 1`, `\u03a8\u2082(z)` is strictly below its neutral value `2 x (1 - z) / ((1 - x) z)`.","title":"Maruyama effect, above the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove_pos","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above-pos","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2082` is positive for `0 < x < z < 1`.","title":"Positivity of the sojourn density above the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above-one-right","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove_one_right","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above-one-right","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"At `z = 1` the closed form `\u03a8\u2082` vanishes (a null set for the integral).","title":"Sojourn density above the current frequency vanishes at one"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/interval-integrable-maruyama-below","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.intervalIntegrable_maruyamaBelow","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#interval-integrable-maruyama-below","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2081` is integrable on `[0, x]` (it is bounded by `2`).","title":"Integrability of the sojourn density below the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/interval-integrable-maruyama-above","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.intervalIntegrable_maruyamaAbove","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#interval-integrable-maruyama-above","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"`\u03a8\u2082` is integrable on `[x, 1]` for `x \u2208 (0, 1)` (it is bounded by `2 / (1 - x)`).","title":"Integrability of the sojourn density above the current frequency"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/mean-age-neg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAge_neg","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#mean-age-neg","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"The mean age is the same for `\u03c3` and `-\u03c3`.","title":"Symmetry"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/mean-age-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAge_lt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#mean-age-lt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For `\u03c3 \u2260 0` and `x \u2208 (0, 1)`, an allele under selection, advantageous or deleterious, is on average strictly younger than a neutral allele at the same frequency.","title":"The Maruyama effect (Kiezun et al. 2013)"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/tendsto-integral-age-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.tendsto_integral_ageDensity","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#tendsto-integral-age-density","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Mean age as the limit of conditional sojourn times (Maruyama 1974; Kiezun et al. 2013, Methods): for `x \u2208 (0, 1)`, the total conditional sojourn time `\u222b\u2080\u00b9 \u03a8_z(p, z) dz` of a path from initial frequency `p` to current frequency `x` tends to the mean age `a(x)` as `p \u2192 0\u207a` (an allele arises at frequency close to `0`).","title":"Mean age as the limit of conditional sojourn times"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-below-lt-of-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaBelow_lt_of_lt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-below-lt-of-lt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For `0 \u2264 \u03c3 < \u03c4` and `z \u2208 (0, 1)`, `\u03a8\u2081` at `\u03c4` is strictly below `\u03a8\u2081` at `\u03c3`.","title":"Stronger selection, smaller `\u03a8\u2081`"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/maruyama-above-lt-of-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.maruyamaAbove_lt_of_lt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#maruyama-above-lt-of-lt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"For `0 \u2264 \u03c3 < \u03c4` and `0 < x < z < 1`, `\u03a8\u2082` at `\u03c4` is strictly below `\u03a8\u2082` at `\u03c3`.","title":"Stronger selection, smaller `\u03a8\u2082`"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/mean-age-lt-mean-age-of-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAge_lt_meanAge_of_lt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#mean-age-lt-mean-age-of-lt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"At a fixed frequency `x \u2208 (0, 1)`, the mean age is strictly decreasing in `\u03c3 \u2265 0`. Kiezun et al. (2013, Figure 1b) show this numerically from the closed forms; here it is proved.","title":"Stronger selection, younger alleles"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/mean-age-lt-mean-age-of-abs-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.meanAge_lt_meanAge_of_abs_lt","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#mean-age-lt-mean-age-of-abs-lt","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Stronger selection, younger alleles, for either sign of selection: if `|\u03c3| < |\u03c4|` then an allele at frequency `x \u2208 (0, 1)` under selection `\u03c4` is on average strictly younger than one under selection `\u03c3`.","title":"Stronger selection, younger alleles"},{"id":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/lintegral-mul-eq-lintegral-sojourn-mul","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.lintegral_mul_eq_lintegral_sojourn_mul","parent":44797,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect/index.html#lintegral-mul-eq-lintegral-sojourn-mul","scope":"Biology/population-genetics/diffusion-theory/allele-age-and-the-maruyama-effect","status":"fully_proved","summary":"Mean time as a sum of sojourn times (the identity behind Maruyama and Kimura 1975). For a jointly measurable family of transition densities `w p x t \u2265 0` on `(0, 1)` satisfying the Chapman\u2013Kolmogorov equation for positive times, `\u222b\u2080^\u221e t w(p, x, t) dt = \u222b\u2080\u00b9 \u03a8(p, y) \u03a8(y, x) dy` with `\u03a8(p, x) = \u222b\u2080^\u221e w(p, x, t) dt`, in `\u211d\u22650\u221e`. Dividing by `\u03a8(p, x)` reads `\u222b\u2080\u00b9 \u03a8(p, y) \u03a8(y, x) / \u03a8(p, x) dy` (the integral of `ageDensity`\u2026","title":"Mean time as a sum of sojourn times"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation.json","id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","title":"Wright\u2013Fisher diffusion with mutation"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/mutation-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.mutationGenerator","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#mutation-generator","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"The backward generator of the Wright\u2013Fisher diffusion with two-way mutation, time in units of `2N` generations and scaled mutation rates `\u03b8\u2081 = 4N u\u2081` (towards the allele) and `\u03b8\u2082 = 4N u\u2082` (away from it): `(L f)(x) = x (1 - x) / 2 \u00b7 f''(x) + (\u03b8\u2081 (1 - x) - \u03b8\u2082 x) / 2 \u00b7 f'(x)`.","title":"Wright\u2013Fisher generator with two-way mutation"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/mutation-generator-zero-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.mutationGenerator_zero_zero","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#mutation-generator-zero-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"Without mutation the generator is the neutral Wright\u2013Fisher generator `generator 0`.","title":"Mutation generator without mutation is the neutral generator"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/beta-pdfreal-eq-wright-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.betaPDFReal_eq_wrightDensity","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#beta-pdfreal-eq-wright-density","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"On `(0, 1)` the Beta`(\u03b8\u2081, \u03b8\u2082)` density is Wright's unnormalized density in the scaled rates `\u03b1 = \u03b8\u2081 / 2`, `\u03b2 = \u03b8\u2082 / 2`, divided by the Beta function `B(\u03b8\u2081, \u03b8\u2082)`.","title":"Beta density as Wright's stationary density"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/cont-diff-on-beta-pdfreal","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.contDiffOn_betaPDFReal","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#cont-diff-on-beta-pdfreal","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"The Beta`(\u03b8\u2081, \u03b8\u2082)` density is analytic (`C^\u03c9`), in particular smooth, on `(0, 1)`, so the derivatives in `flux_betaPDFReal` and `forwardEquation_betaPDFReal` exist.","title":"Smoothness of the Beta density on the open unit interval"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/flux-beta-pdfreal","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.flux_betaPDFReal","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#flux-beta-pdfreal","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"On `(0, 1)` the Beta`(\u03b8\u2081, \u03b8\u2082)` density `\u03c0` satisfies `\u00bd (x (1 - x) \u03c0)' - (\u03b8\u2081 (1 - x) - \u03b8\u2082 x) / 2 \u00b7 \u03c0 = 0`. The identity holds for all real `\u03b8\u2081, \u03b8\u2082` (it is linear in `\u03c0`; Mathlib's density is identically `0` when `B(\u03b8\u2081, \u03b8\u2082) = 0`); `\u03c0` is a probability density when `\u03b8\u2081, \u03b8\u2082 > 0` (`isProbabilityMeasureBeta`).","title":"Zero probability flux (Wright 1931)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/forward-equation-beta-pdfreal","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.forwardEquation_betaPDFReal","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#forward-equation-beta-pdfreal","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"On `(0, 1)` the Beta`(\u03b8\u2081, \u03b8\u2082)` density `\u03c0` solves the stationary Fokker\u2013Planck equation `\u00bd (a \u03c0)'' - (b \u03c0)' = 0` with `a(x) = x (1 - x)` and `b(x) = (\u03b8\u2081 (1 - x) - \u03b8\u2082 x) / 2`. Holds for all real `\u03b8\u2081, \u03b8\u2082` (see `flux_betaPDFReal`).","title":"Stationary forward equation (Wright 1931)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/interval-integrable-mutation-generator-m","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.intervalIntegrable_mutationGenerator_mul_betaPDFReal","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#interval-integrable-mutation-generator-m","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"For `\u03b8\u2081, \u03b8\u2082 > 0` and `f` twice continuously differentiable on `[0, 1]`, `L f \u00b7 \u03c0` is integrable on `[0, 1]`, so the integral in `integral_mutationGenerator_mul_betaPDFReal` is a genuine (not junk) value.","title":"Integrability of the mutation generator against the Beta density"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/integral-mutation-generator-mul-beta-pdf","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_mutationGenerator_mul_betaPDFReal","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#integral-mutation-generator-mul-beta-pdf","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"For `\u03b8\u2081, \u03b8\u2082 > 0` and every `f` twice continuously differentiable on `[0, 1]`, `\u222b\u2080\u00b9 (L f)(x) \u03c0(x) dx = 0` for the Beta`(\u03b8\u2081, \u03b8\u2082)` density `\u03c0`. (`L f` uses `deriv`, which agrees with the one-sided derivatives inside `(0, 1)`; its values at the endpoints do not affect the integral.) The proof is integration by parts: `L f \u00b7 \u03c0 = (x ^ \u03b8\u2081 (1 - x) ^ \u03b8\u2082 f' / (2 B(\u03b8\u2081, \u03b8\u2082)))'` on `(0, 1)` because the flux vanishes, and this\u2026","title":"Weak stationarity of Wright's law (density form)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/integrable-mutation-generator-beta-measu","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integrable_mutationGenerator_betaMeasure","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#integrable-mutation-generator-beta-measu","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"For `\u03b8\u2081, \u03b8\u2082 > 0` and `f` twice continuously differentiable on `[0, 1]`, `L f` is integrable against Wright's stationary law `Beta(\u03b8\u2081, \u03b8\u2082)`.","title":"Integrability of the mutation generator under Wright's stationary law"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/integral-mutation-generator-beta-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_mutationGenerator_betaMeasure","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#integral-mutation-generator-beta-measure","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"For `\u03b8\u2081, \u03b8\u2082 > 0` and every `f` twice continuously differentiable on `[0, 1]`, `\u222b L f d(Beta(\u03b8\u2081, \u03b8\u2082)) = 0`.","title":"Weak stationarity of Wright's law (Wright 1931; Ewens 2004, ch. 4)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/mutation-generator-id","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.mutationGenerator_id","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#mutation-generator-id","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"The generator applied to the frequency itself is the mutation drift: `L x = (\u03b8\u2081 (1 - x) - \u03b8\u2082 x) / 2`.","title":"Mutation generator applied to the allele frequency"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/mutation-generator-heterozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.mutationGenerator_heterozygosity","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#mutation-generator-heterozygosity","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"The generator applied to the heterozygosity `H(x) = 2x (1 - x)`: `L H = -2x (1 - x) + (\u03b8\u2081 (1 - x) - \u03b8\u2082 x) (1 - 2x)` (drift decreases `H` at rate `1`, mutation restores it).","title":"Mutation generator applied to heterozygosity"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/mutated-freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.mutatedFreq","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#mutated-freq","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"Allele frequency after one generation of two-way mutation, before random sampling: an `a` copy mutates to `A` with probability `u\u2081` and an `A` copy to `a` with probability `u\u2082`, so the frequency `x` of `A` becomes `(1 - u\u2082) x + u\u2081 (1 - x) \u2208 [0, 1]`.","title":"Allele frequency after one generation of two-way mutation"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/integral-binomial-mutated-freq-drift","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_binomial_mutatedFreq_drift","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#integral-binomial-mutated-freq-drift","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"One generation of the Wright\u2013Fisher chain on `M` gene copies (`M = 2N`): mutation moves the frequency `x` to `p' = mutatedFreq u\u2081 u\u2082 x`, then the new count is `Binomial(M, p')`. With `u\u2081 = \u03b8\u2081 / (2M)`, `u\u2082 = \u03b8\u2082 / (2M)` (`\u03b8\u1d62 = 4N u\u1d62`), the expected change of frequency per `2N = M` generations is exactly the drift coefficient of `mutationGenerator`: `M \u00b7 (E[k / M] - x) = (\u03b8\u2081 (1 - x) - \u03b8\u2082 x) / 2`, for every `M`.","title":"Drift of the Wright\u2013Fisher chain with mutation (Ewens 2004, ch. 4)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/integral-binomial-mutated-freq-sq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.integral_binomial_mutatedFreq_sq","parent":44843,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation/index.html#integral-binomial-mutated-freq-sq","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-diffusion-with-mutation","status":"fully_proved","summary":"In the setting of `integral_binomial_mutatedFreq_drift`, the scaled second moment of the frequency change is `M \u00b7 E[(k / M - x) ^ 2] = p' (1 - p') + (\u03b8\u2081 (1 - x) - \u03b8\u2082 x) ^ 2 / (4M)` with `p' = mutatedFreq u\u2081 u\u2082 x = x + (\u03b8\u2081 (1 - x) - \u03b8\u2082 x) / (2M)`. As `M \u2192 \u221e` with `\u03b8\u2081, \u03b8\u2082, x` fixed this tends to `x (1 - x)`, the diffusion coefficient of `mutationGenerator` (the limit itself is not stated here).","title":"Infinitesimal variance of the Wright\u2013Fisher chain with mutation"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality.json","id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","title":"Wright\u2013Fisher semigroup and moment duality"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/cauchy-seq-apply-of-dense","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.ChainLimit.cauchySeq_apply_of_dense","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#cauchy-seq-apply-of-dense","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Uniformly contractive operators that are Cauchy on a dense set are Cauchy everywhere.","title":"Cauchy property of uniform contractions extends from a dense set"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/tendsto-apply-of-dense","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.ChainLimit.tendsto_apply_of_dense","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#tendsto-apply-of-dense","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Limits of uniformly contractive operators extend from a dense set to the whole space.","title":"Limits of uniform contractions extend from a dense set"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/dense-polynomial-functions","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.ChainLimit.dense_polynomialFunctions","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#dense-polynomial-functions","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Weierstrass approximation: polynomial functions are dense in `C([0, 1], \u211d)`.","title":"Weierstrass approximation theorem on the unit interval"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/floor-mul-eq-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.ChainLimit.floor_mul_eq_zero","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#floor-mul-eq-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"If `t \u2264 0`, then the natural floor of `t * M` is zero.","title":"Natural floor of a nonpositive multiple is zero"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/pow-apply-one","kind":"node","lean":"ContinuousLinearMap.pow_apply_one","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#pow-apply-one","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Every power of a continuous linear endomorphism fixing `1` also fixes `1`.","title":"Powers of a unital endomorphism fix the constant function"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/pow-nonneg","kind":"node","lean":"ContinuousLinearMap.pow_nonneg","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#pow-nonneg","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Every power of a positive continuous linear endomorphism is positive.","title":"Powers of a positive endomorphism are positive"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfStep","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-step","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"One generation of the neutral Wright\u2013Fisher chain with `M` gene copies, acting on continuous functions of the allele frequency: `(wfStep M f)(x) = \u2211_k f(k/M) C(M, k) x^k (1 - x)^(M - k)`, which for `M \u2265 1` is the expected value of `f` at the next frequency `Bin(M, x) / M` (the `M`-th Bernstein operator, Mathlib's `bernsteinApproximation`; the expectation is `ProbabilityTheory.integral_binomial`). It is a linear\u2026","title":"One generation of the neutral Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-step-apply","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfStep_apply","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-step-apply","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"`wfStep M` is the Bernstein approximation operator.","title":"Wright\u2013Fisher step is the Bernstein operator"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/norm-wf-step-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.norm_wfStep_le","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#norm-wf-step-le","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"One Wright\u2013Fisher generation is a contraction for the sup norm.","title":"One Wright\u2013Fisher generation is a sup-norm contraction"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-step-apply-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfStep_apply_one","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-step-apply-one","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"One Wright\u2013Fisher generation fixes the constant function `1`.","title":"One Wright\u2013Fisher generation fixes constants"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-step-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfStep_nonneg","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-step-nonneg","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"One Wright\u2013Fisher generation maps nonnegative functions to nonnegative functions.","title":"One Wright\u2013Fisher generation preserves nonnegativity"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/norm-wf-step-pow-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.norm_wfStep_pow_le","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#norm-wf-step-pow-le","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Any number of Wright\u2013Fisher generations is a contraction for the sup norm.","title":"Iterated Wright\u2013Fisher generations are sup-norm contractions"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/sum-binomial-mul-div-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.sum_binomial_mul_div_pow","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#sum-binomial-mul-div-pow","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"For `M \u2265 1` gene copies, `\u2211_i C(M, i) x^i (1 - x)^(M - i) (i/M)^n = \u2211_k S(n, k) M_(k) / M^n \u00b7 x^k`: the `n`-th moment of the frequency after one generation equals the generating function of the number of distinct parents of `n` sampled genes. Holds for every real `x`.","title":"Moment duality, one generation (polynomial identity)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/integral-binomial-div-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.integral_binomial_div_pow","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#integral-binomial-div-pow","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"If the allele count after one neutral Wright\u2013Fisher generation with `M \u2265 1` gene copies is `Bin(M, x)`, then `E_x[(X\u2081)^n] = \u2211_k wfLineageProb M n k \u00b7 x^k`, where `X\u2081` is the new frequency and `wfLineageProb M n k` is the probability that `n` genes have `k` distinct parents.","title":"Moment duality, one generation"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-step-x-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfStep_X_pow","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-step-x-pow","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"One generation maps `x ^ n` to `\u2211_k wfLineageProb M n k x ^ k`.","title":"One Wright\u2013Fisher generation acting on monomials"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-step-pow-x-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfStep_pow_X_pow","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-step-pow-x-pow","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"For `M \u2265 1` and `n \u2264 N`, `m` generations of the Wright\u2013Fisher chain map `x ^ n` to `\u2211_k ((wfLineageMatrix N M)^m) n k \u00b7 x^k`: the `n`-th moment of the frequency after `m` generations is the generating function of the number of ancestors, `m` generations back, of `n` sampled genes.","title":"Moment duality, `m` generations"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/dual-moment-poly","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.dualMomentPoly","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#dual-moment-poly","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The dual moment polynomial `\u2211_k P_t(n \u2192 k) X^k`, where for `t \u2265 0`, `P_t(n \u2192 k) = exp (t \u2022 G) n k` is the probability that `n` lineages of Kingman's coalescent have `k` ancestors time `t` earlier (time in units of `M = 2N` generations); evaluated at `x` it is `E_n[x^{A_t}]` for the block-counting process `A`.","title":"Dual moment polynomial of Kingman's coalescent"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/poly-semigroup","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.polySemigroup","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#poly-semigroup","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The neutral Wright\u2013Fisher diffusion semigroup on polynomials, defined by moment duality on the monomial basis: `polySemigroup t (X ^ n) = dualMomentPoly t n`. It is meaningful for `t \u2265 0`; the formula is defined, and the group law holds, for every real `t`.","title":"Wright\u2013Fisher semigroup on polynomials by moment duality"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/poly-semigroup-x-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.polySemigroup_X_pow","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#poly-semigroup-x-pow","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The dual semigroup on a monomial is the dual moment polynomial.","title":"Dual semigroup on a monomial"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/poly-semigroup-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.polySemigroup_zero","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#poly-semigroup-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"At time `0` the dual semigroup is the identity.","title":"Dual semigroup at time zero is the identity"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/poly-semigroup-add","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.polySemigroup_add","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#poly-semigroup-add","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Semigroup law on polynomials, from `exp ((s + t) \u2022 G) = exp (t \u2022 G) * exp (s \u2022 G)` and truncation consistency; holds for all real `s`, `t`.","title":"Semigroup law"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/poly-semigroup-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.polySemigroup_one","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#poly-semigroup-one","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The dual semigroup fixes the constant polynomial `1` (zero lineages stay zero).","title":"Dual semigroup fixes the constant polynomial one"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/tendsto-wf-step-pow-floor-to-continuous","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.tendsto_wfStep_pow_floor_toContinuousMapOn","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#tendsto-wf-step-pow-floor-to-continuous","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"For `t \u2265 0` and every polynomial `p`, `\u230atM\u230b` generations of the Wright\u2013Fisher chain with `M` gene copies applied to `p` converge uniformly on `[0, 1]` to `polySemigroup t p` as `M \u2192 \u221e` (moment duality plus Kingman's theorem for the lineage count).","title":"Convergence on polynomials"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/cauchy-seq-wf-step-pow-floor","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.cauchySeq_wfStep_pow_floor","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#cauchy-seq-wf-step-pow-floor","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"For every continuous `f`, the sequence `(wfStep M)^\u230atM\u230b f` is Cauchy in `C([0, 1], \u211d)`: it converges on the dense set of polynomials and the operators are contractions.","title":"Rescaled Wright\u2013Fisher iterates form a Cauchy sequence"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The neutral Wright\u2013Fisher diffusion semigroup on `C([0, 1], \u211d)`, time in units of `M = 2N` generations: `wfSemigroup t f` is the uniform limit of `(wfStep M)^\u230atM\u230b f` as `M \u2192 \u221e` (`tendsto_wfStep_pow_floor`). For `t < 0` it is the identity (a junk value: `\u230atM\u230b\u208a = 0`).","title":"Neutral Wright\u2013Fisher diffusion semigroup"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/tendsto-wf-step-pow-floor","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.tendsto_wfStep_pow_floor","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#tendsto-wf-step-pow-floor","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"For every continuous `f`, `\u230atM\u230b` generations of the neutral Wright\u2013Fisher chain with `M` gene copies applied to `f` converge uniformly on `[0, 1]` to `wfSemigroup t f`.","title":"Diffusion approximation of the transition operators (Trotter\u2013Kato, concrete form)"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/norm-wf-semigroup-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.norm_wfSemigroup_le","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#norm-wf-semigroup-le","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"`sup_{[0,1]} |wfSemigroup t f| \u2264 sup_{[0,1]} |f|`.","title":"Contraction"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-to-continuous-map-on","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_toContinuousMapOn","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-to-continuous-map-on","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"On polynomials the diffusion semigroup is the dual semigroup, for `t \u2265 0`.","title":"Diffusion semigroup agrees with the dual semigroup on polynomials"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-of-nonpos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_of_nonpos","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-of-nonpos","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"For `t \u2264 0`, `wfSemigroup t` is the identity (at `t = 0` by definition of a semigroup; for `t < 0` this is a junk value of the definition).","title":"Wright\u2013Fisher semigroup at nonpositive times is the identity"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_zero","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"At time `0` the diffusion semigroup is the identity.","title":"Wright\u2013Fisher diffusion semigroup at time zero is the identity"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-x-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_X_pow","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-x-pow","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Moment duality for the diffusion (Ethier & Griffiths 1993). For `t \u2265 0`, the semigroup applied to `x \u21a6 x ^ n` is `\u2211_k P_t(n \u2192 k) x^k`, with `P_t = exp (t \u2022 G)` the transition matrix of Kingman's block-counting process `A`; in process language `E_x[(X_t)^n] = E_n[x^{A_t}]` (the processes themselves are not constructed here).","title":"Moment duality for the diffusion"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-add","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_add","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-add","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Semigroup law for `s, t \u2265 0`.","title":"Semigroup law"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-apply-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_apply_one","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-apply-one","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The diffusion semigroup fixes constants (it is Markov: total mass `1`).","title":"Wright\u2013Fisher diffusion semigroup fixes constants"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_nonneg","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-nonneg","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The diffusion semigroup maps nonnegative functions to nonnegative functions.","title":"Positivity"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/tendsto-wf-semigroup-nhds-ge-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.tendsto_wfSemigroup_nhdsGE_zero","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#tendsto-wf-semigroup-nhds-ge-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"`wfSemigroup t f \u2192 f` uniformly as `t \u2192 0\u207a`, for every continuous `f`.","title":"Strong continuity"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/poly-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.polyGenerator","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#poly-generator","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The neutral Wright\u2013Fisher generator on polynomials, `x (1 - x) / 2 \u00b7 p''`.","title":"Neutral Wright\u2013Fisher generator on polynomials"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/eval-poly-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.eval_polyGenerator","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#eval-poly-generator","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"`polyGenerator` is the neutral Wright\u2013Fisher generator `generator 0` evaluated on `p`.","title":"Polynomial generator agrees with the neutral Wright\u2013Fisher generator"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/tendsto-wf-semigroup-sub-div","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.tendsto_wfSemigroup_sub_div","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#tendsto-wf-semigroup-sub-div","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"For every polynomial `p`, `(wfSemigroup t p - p) / t \u2192 x (1 - x) / 2 \u00b7 p''` uniformly on `[0, 1]` as `t \u2192 0\u207a`.","title":"Generator on polynomials"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/has-deriv-at-wf-semigroup-to-continuous","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.hasDerivAt_wfSemigroup_toContinuousMapOn","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#has-deriv-at-wf-semigroup-to-continuous","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"At every positive time, differentiating the existing Wright\u2013Fisher semigroup on a polynomial equals applying the semigroup to its generator.","title":"Polynomial semigroup dynamics"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-second-increment-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_second_increment_le","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-second-increment-le","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The centered second moment grows at most linearly: `T_t[(y-x)\u00b2](x) \u2264 t/4`.","title":"Second-moment bound for the Wright\u2013Fisher semigroup"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-fourth-increment-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroup_fourth_increment_le","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-fourth-increment-le","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The fourth centered increment of the neutral Wright\u2013Fisher semigroup satisfies the uniform Kolmogorov bound `T_t[(y-x)\u2074](x) \u2264 (3/16)t\u00b2`.","title":"Fourth-moment Kolmogorov bound for the Wright\u2013Fisher semigroup"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-positive","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroupPositive","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-positive","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The Wright\u2013Fisher semigroup bundled as a positive continuous linear operator.","title":"Wright\u2013Fisher semigroup as a positive operator"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-positive-apply","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroupPositive_apply","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-positive-apply","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Bundling the operator as positive leaves its action on continuous functions unchanged.","title":"Action of the positive Wright\u2013Fisher operator"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-positive-apply-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroupPositive_apply_one","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-positive-apply-one","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The positive Wright\u2013Fisher operator fixes the constant function `1`.","title":"Positive Wright\u2013Fisher operator fixes constants"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-semigroup-positive-add","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfSemigroupPositive_add","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-semigroup-positive-add","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The positive-operator form of the Wright\u2013Fisher semigroup law.","title":"Semigroup law for the positive Wright\u2013Fisher operator"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-transition-kernel","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfTransitionKernel","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-transition-kernel","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"The transition kernel of the neutral Wright\u2013Fisher diffusion at time `t`. For nonnegative `t` it is the probabilistic transition corresponding to `wfSemigroup t`; for negative `t`, both the operator and kernel are the identity junk value inherited from `wfSemigroup`.","title":"Transition kernel of the neutral Wright\u2013Fisher diffusion"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/integral-wf-transition-kernel","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.integral_wfTransitionKernel","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#integral-wf-transition-kernel","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Integration against the transition kernel agrees with the Wright\u2013Fisher semigroup on every continuous test function and every initial allele frequency.","title":"Wright\u2013Fisher transition kernel represents the semigroup"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-transition-kernel-of-nonpos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfTransitionKernel_of_nonpos","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-transition-kernel-of-nonpos","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"For nonpositive times, the Wright\u2013Fisher transition is the identity kernel.","title":"Wright\u2013Fisher transition kernel is the identity at nonpositive times"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-transition-kernel-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfTransitionKernel_zero","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-transition-kernel-zero","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"At time zero the Wright\u2013Fisher transition is the identity kernel.","title":"Wright\u2013Fisher transition kernel is the identity at time zero"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-transition-kernel-zero-apply","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfTransitionKernel_zero_apply","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-transition-kernel-zero-apply","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Pointwise identity law for the transition kernel at time zero.","title":"Wright\u2013Fisher transitions at time zero are Dirac masses"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/wf-transition-kernel-add","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.wfTransitionKernel_add","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#wf-transition-kernel-add","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Chapman\u2013Kolmogorov law for the Wright\u2013Fisher transition kernels. With Mathlib's convention, `K_t \u2218\u2096 K_s` first evolves for time `s` and then for time `t`.","title":"Chapman\u2013Kolmogorov equation for the Wright\u2013Fisher diffusion"},{"id":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/integral-wf-transition-kernel-x-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.WrightFisherSemigroup.integral_wfTransitionKernel_X_pow","parent":44859,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality/index.html#integral-wf-transition-kernel-x-pow","scope":"Biology/population-genetics/diffusion-theory/wright-fisher-semigroup-and-moment-duality","status":"fully_proved","summary":"Kernel form of Wright\u2013Fisher moment duality: transition-kernel moments are the transition probabilities of Kingman's block-counting process.","title":"Kernel form of Wright\u2013Fisher moment duality"},{"data_url":"scopes/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus.json","id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","kind":"scope","navigation":"module","parent":44676,"result":false,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","title":"Construction of the neutral Wright\u2013Fisher diffusion"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/sample-path","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.SamplePath","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#sample-path","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"Canonical sample paths before continuous modification.","title":"Canonical sample paths of the raw Wright\u2013Fisher process"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/neutral-kernel","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.neutralKernel","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#neutral-kernel","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The neutral Wright--Fisher transition kernel at nonnegative time.","title":"Neutral Wright\u2013Fisher transition kernel at nonnegative time"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/neutral-kernel-add","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.neutralKernel_add","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#neutral-kernel-add","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"Chapman--Kolmogorov for the nonnegative-time specialization.","title":"Chapman\u2013Kolmogorov for the nonnegative-time Wright\u2013Fisher kernel"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/raw-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.rawMeasure","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#raw-measure","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The canonical probability measure on `NNReal \u2192 I`.","title":"Canonical path measure of the raw Wright\u2013Fisher process"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/raw-process","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.rawProcess","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#raw-process","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The raw real-valued coordinate process, normalized at time zero on every sample rather than merely almost surely.","title":"Raw Wright\u2013Fisher coordinate process"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/map-eval-raw-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.map_eval_rawMeasure","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#map-eval-raw-measure","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"One-time marginals of the canonical raw path measure.","title":"One-time marginals of the raw Wright\u2013Fisher path measure"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/map-eval-zero-raw-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.map_eval_zero_rawMeasure","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#map-eval-zero-raw-measure","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The time-zero coordinate of the canonical path law is the prescribed initial state.","title":"Time-zero marginal of the raw path measure is the initial state"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/map-pair-raw-measure-of-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.map_pair_rawMeasure_of_lt","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#map-pair-raw-measure-of-lt","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The ordered two-time law is the first marginal followed by the transition over the time gap.","title":"Two-time law of the raw Wright\u2013Fisher path measure"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/raw-process-ae-eq-coordinate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.rawProcess_ae_eq_coordinate","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#raw-process-ae-eq-coordinate","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"Exact normalization changes no fixed-time law.","title":"Normalized raw process agrees with the coordinates almost surely"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/kolmogorov-constant","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.kolmogorovConstant","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#kolmogorov-constant","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The explicit Kolmogorov constant `3/16`.","title":"Kolmogorov constant for the Wright\u2013Fisher continuity bound"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/is-kolmogorov-process-raw-process","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.isKolmogorovProcess_rawProcess","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#is-kolmogorov-process-raw-process","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The normalized raw coordinate process satisfies the explicit fourth-moment Kolmogorov condition with `p=4`, `q=2`, and `M=3/16`.","title":"Kolmogorov continuity condition for the raw Wright\u2013Fisher process"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/continuous-realization","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.ContinuousRealization","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#continuous-realization","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"A selected continuous modification of the raw Wright--Fisher process, with all pathwise properties stated on every sample.","title":"Continuous realization of the raw Wright\u2013Fisher process"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/continuous-realization-2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.continuousRealization","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#continuous-realization-2","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The continuous modification selected by Kolmogorov's theorem.","title":"Continuous modification selected by Kolmogorov's theorem"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/modified-process","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.modifiedProcess","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#modified-process","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The concrete continuous process used by the final realization.","title":"Continuous modified Wright\u2013Fisher process"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/map-restrict-modified-eq-raw","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.map_restrict_modified_eq_raw","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#map-restrict-modified-eq-raw","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The modified process has the raw finite-dimensional laws, proved on each finite set by intersecting its fixed-time modification events.","title":"Modified process has the raw finite-dimensional laws"},{"id":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/map-restrict-modified-eq-coordinate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Diffusion.NeutralWrightFisherRaw.map_restrict_modified_eq_coordinate","parent":44912,"result":true,"route":"../roadmap/Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus/index.html#map-restrict-modified-eq-coordinate","scope":"Biology/population-genetics/diffusion-theory/construction-of-the-neutral-wright-fisher-diffus","status":"fully_proved","summary":"The modified finite-dimensional laws also equal the original canonical coordinate laws after coercion from `I` to `\u211d`.","title":"Modified process has the canonical coordinate laws"},{"data_url":"scopes/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran.json","id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","title":"Ewens sampling formula and the Chinese restaurant process"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/ewens-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.ewensProb","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#ewens-prob","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The Ewens sampling formula: the probability, under the neutral infinite-alleles model with scaled mutation rate `\u03b8 = 4Nu`, that a sample of `n` genes has allelic partition `\u03c0` (`\u03c0.parts.count j` alleles carried by exactly `j` genes each): `n! * (\u220f j, (\u03b8 / j) ^ (a j) / (a j)!) / \u03b8^{(n)}` (Ewens 1972), for `\u03b8 > 0`.","title":"Ewens sampling formula"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/sum-ewens-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.sum_ewensProb","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#sum-ewens-prob","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The Ewens sampling formula is a probability distribution on the partitions of `n` (Ewens 1972; equivalently, Cauchy's cycle-type formula `\u2211_{\u03c3 \u2208 S_n} \u03b8^{#cycles \u03c3} = \u03b8^{(n)}`).","title":"The Ewens sampling formula is a probability distribution"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/ewens-prob-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.ewensProb_nonneg","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#ewens-prob-nonneg","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The Ewens sampling formula is nonnegative for `\u03b8 > 0`; with `sum_ewensProb` it is a probability distribution on the partitions of `n`.","title":"Nonnegativity of the Ewens sampling formula"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/sum-ewens-prob-mul-card-parts","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.sum_ewensProb_mul_card_parts","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#sum-ewens-prob-mul-card-parts","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Expected number of alleles in a sample of `n` genes under the Ewens sampling formula: `E[K_n] = \u2211_{i < n} \u03b8 / (\u03b8 + i)` (Ewens 1972).","title":"Expected number of alleles"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/hoppe-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.hoppeStep","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#hoppe-step","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"One step of Hoppe's urn on allelic partitions (Hoppe 1984): from a partition `\u03c0` of `k`, the next gene carries a new allele with probability `\u03b8 / (\u03b8 + k)` (target parts `1 ::\u2098 \u03c0.parts`), and copies an allele carried by `j` genes with probability `j * a j / (\u03b8 + k)`, where `a j = \u03c0.parts.count j` (target parts `(j + 1) ::\u2098 \u03c0.parts.erase j`). The value is the transition probability from `\u03c0` to `\u03c0'`.","title":"One step of Hoppe's urn"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/hoppe-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.hoppeLaw","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#hoppe-law","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The law of the allelic partition after `n` draws from Hoppe's urn, started from the empty sample: `hoppeLaw \u03b8 (n + 1) \u03c0' = \u2211 \u03c0, hoppeLaw \u03b8 n \u03c0 * hoppeStep \u03b8 \u03c0 \u03c0'`.","title":"Law of the allelic partition from Hoppe's urn"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/hoppe-step-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.hoppeStep_nonneg","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#hoppe-step-nonneg","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Hoppe-urn transition probabilities are nonnegative.","title":"Hoppe urn transition probabilities are nonnegative"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/sum-hoppe-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.sum_hoppeStep","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#sum-hoppe-step","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The Hoppe-urn transition is stochastic: from every partition of `k` the probabilities of the partitions of `k + 1` sum to one.","title":"Hoppe urn transition is stochastic"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/hoppe-law-eq-ewens-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.hoppeLaw_eq_ewensProb","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#hoppe-law-eq-ewens-prob","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Hoppe's urn generates the Ewens sampling formula (Hoppe 1984): after `n` draws, the allelic partition has law `ewensProb \u03b8`.","title":"Hoppe's urn generates the Ewens sampling formula"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/hoppe-law-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.hoppeLaw_nonneg","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#hoppe-law-nonneg","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The law of Hoppe's urn after `n` draws is nonnegative for `\u03b8 > 0`.","title":"Nonnegativity of the Hoppe urn law"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/new-table","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.newTable","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#new-table","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Seat customer `n` at a new table: the partition of `Fin (n + 1)` whose blocks are the blocks of `\u03c0` (relabeled into `Fin (n + 1)`) together with the singleton `{n}`.","title":"Seating a customer at a new table"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/join-table","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.joinTable","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#join-table","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Seat customer `n` at the occupied table `B` of `\u03c0`: the block `B` becomes `B \u222a {n}` and the other blocks are unchanged.","title":"Seating a customer at an occupied table"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/parts-new-table","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.parts_newTable","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#parts-new-table","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The blocks of `newTable \u03c0`.","title":"Blocks of the partition after opening a new table"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/parts-join-table","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.parts_joinTable","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#parts-join-table","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The blocks of `joinTable \u03c0 B hB`.","title":"Blocks of the partition after joining a table"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/seat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.seat","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#seat","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Seat customer `n` according to the choice `c`: `none` opens a new table, `some B` joins the occupied table `B`.","title":"Seating a customer by a seating choice"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/seat-weight","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.seatWeight","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#seat-weight","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The probability of each seating choice for customer `n` given the partition `\u03c0` of the first `n` customers: `\u03b8 / (n + \u03b8)` for a new table, `|B| / (n + \u03b8)` for the table `B`.","title":"Seating probabilities in the Chinese restaurant process"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/seat-weight-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.seatWeight_nonneg","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#seat-weight-nonneg","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The seating probabilities are nonnegative for `\u03b8 > 0`.","title":"Nonnegativity of seating probabilities"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/sum-seat-weight","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.sum_seatWeight","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#sum-seat-weight","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The seating probabilities sum to one for `\u03b8 > 0`.","title":"Seating probabilities sum to one"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/seat-choice","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.seatChoice","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#seat-choice","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The seating choice of customer `n`, as a probability distribution.","title":"Seating choice distribution of a customer"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crpStep","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-step","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"One step of the Chinese restaurant process: the law of the partition of `Fin (n + 1)` given the partition `\u03c0` of the first `n` customers.","title":"One step of the Chinese restaurant process"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The Chinese restaurant process with parameter `\u03b8 > 0`: the law of the random partition of the customers `Fin n` into tables, built one customer at a time by `crpStep` (Aldous 1985, \u00a711; Pitman 2006, ch. 3).","title":"Chinese restaurant process"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crpProb","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-prob","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The closed-form law of the Chinese restaurant process, `\u03b8^{|\u03c0|} \u220f_{B \u2208 \u03c0} (|B| - 1)! / \u03b8^{(n)}` (Pitman 2006, \u00a73.1). Every block is nonempty, so `|B| - 1` is never a truncated natural subtraction.","title":"Closed-form law of the Chinese restaurant process"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/comap-cast-succ-seat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.comap_castSucc_seat","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#comap-cast-succ-seat","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Removing the last customer undoes every seating: restricting `seat \u03c0 c` to the first `n` customers gives back `\u03c0`.","title":"Removing the last customer undoes a seating"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/seat-injective","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.seat_injective","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#seat-injective","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Different seating choices give different partitions.","title":"Injectivity of seating"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/exists-seat-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.exists_seat_eq","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#exists-seat-eq","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Every partition of `Fin (n + 1)` arises by seating customer `n` into its restriction to the first `n` customers.","title":"Every partition arises from seating the last customer"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-prob-seat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crpProb_seat","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-prob-seat","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"One seating multiplies the closed-form law by the seating probability.","title":"Seating multiplies the closed-form law by the seating probability"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-apply","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp_apply","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-apply","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Law of the Chinese restaurant process (Pitman 2006, \u00a73.1): `P(\u03c0) = \u03b8^{|\u03c0|} \u220f_{B \u2208 \u03c0} (|B| - 1)! / \u03b8^{(n)}`.","title":"Law of the Chinese restaurant process"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-to-real","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp_toReal","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-to-real","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The law of the Chinese restaurant process as a real number.","title":"Chinese restaurant process law as a real number"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-comap-perm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp_comap_perm","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-comap-perm","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Exchangeability, pointwise: relabeling the customers by a permutation `\u03c3` does not change the probability of a partition.","title":"Exchangeability"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-map-comap-perm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp_map_comap_perm","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-map-comap-perm","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Exchangeability of the Chinese restaurant process (Aldous 1985, \u00a711): the law of the partition relabeled by any permutation of the customers is the law of the partition.","title":"Exchangeability of the Chinese restaurant process"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-map-restrict","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp_map_restrict","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-map-restrict","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Restricting the random partition of `Fin (n + 1)` to the first `n` customers gives the process with `n` customers.","title":"Consistency of the Chinese restaurant process"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-map-restrict-cast-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp_map_restrict_castLE","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-map-restrict-cast-le","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Restricting the random partition of `Fin n` to the first `m \u2264 n` customers gives the process with `m` customers.","title":"Consistency for every sample size"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/block-sizes","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.blockSizes","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#block-sizes","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The allelic partition of a set partition of `Fin n`: the multiset of its block sizes, a partition of the integer `n`.","title":"Allelic partition of a set partition"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-map-block-sizes-eq-hoppe-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp_map_blockSizes_eq_hoppeLaw","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-map-block-sizes-eq-hoppe-law","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The law of the multiset of block sizes after `n` customers is the law `Ewens.hoppeLaw` of Hoppe's urn on allelic partitions after `n` draws.","title":"The block sizes follow Hoppe's urn"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/crp-map-block-sizes","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.crp_map_blockSizes","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#crp-map-block-sizes","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"The allelic partition of the Chinese restaurant process follows the Ewens sampling formula (Ewens 1972; Hoppe 1984): the multiset of block sizes after `n` customers has law `Ewens.ewensProb \u03b8`.","title":"The allelic partition of the Chinese restaurant process follows the Ewens sampling formula"},{"id":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/sum-crp-to-real-mul-card-parts","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ChineseRestaurant.sum_crp_toReal_mul_card_parts","parent":44929,"result":true,"route":"../roadmap/Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran/index.html#sum-crp-to-real-mul-card-parts","scope":"Biology/population-genetics/ewens-sampling-formula-and-the-chinese-restauran","status":"fully_proved","summary":"Expected number of tables (expected number of alleles in a sample of `n` genes, Ewens 1972): `E[|\u03c0|] = \u2211_{i < n} \u03b8 / (\u03b8 + i)`.","title":"Expected number of tables"},{"data_url":"scopes/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m.json","id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","title":"Haldane's survival probability of a beneficial mutation"},{"id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/pgf-poisson-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Haldane.pgf_poissonMeasure","parent":44966,"result":true,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html#pgf-poisson-measure","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","summary":"The Poisson(`r`) generating function: `\u2211 e^{-r} r^k / k! x^k = exp (r (x - 1))`.","title":"Generating function of the Poisson distribution"},{"id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/mean-poisson-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Haldane.mean_poissonMeasure","parent":44966,"result":true,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html#mean-poisson-measure","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","summary":"The Poisson(`r`) law has mean `r`.","title":"Mean of the Poisson distribution"},{"id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/survival-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Haldane.survivalProb","parent":44966,"result":true,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html#survival-prob","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","summary":"Haldane's survival probability `\u03c0(s) = 1 - q(s)` of a mutant with selective advantage `s` per generation, whose number of copies in the next generation is Poisson with mean `1 + s`. The model needs `s \u2265 -1`; for `s < -1` the mean is truncated to `0` (`Real.toNNReal`), the mutant leaves no copies, and `\u03c0(s) = 0`.","title":"Haldane's survival probability of a mutant"},{"id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/one-sub-survival-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Haldane.one_sub_survivalProb","parent":44966,"result":true,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html#one-sub-survival-prob","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","summary":"`\u03c0(s)` solves Haldane's equation `1 - \u03c0 = exp (-(1 + s) \u03c0)`.","title":"Haldane's equation for the survival probability"},{"id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/survival-prob-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Haldane.survivalProb_pos","parent":44966,"result":true,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html#survival-prob-pos","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","summary":"A beneficial mutant survives with positive probability.","title":"Beneficial mutant survives with positive probability"},{"id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/survival-prob-lt-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Haldane.survivalProb_lt_one","parent":44966,"result":true,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html#survival-prob-lt-one","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","summary":"The survival probability is less than `1`: the mutant leaves no copies with positive probability `exp (-(1 + s))`. (For `s < -1`, outside the model, `\u03c0(s) = 0`.)","title":"Survival probability is less than one"},{"id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/survival-prob-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Haldane.survivalProb_nonneg","parent":44966,"result":true,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html#survival-prob-nonneg","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","summary":"The survival probability is non-negative.","title":"Nonnegativity of the survival probability"},{"id":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/tendsto-survival-prob-div","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Haldane.tendsto_survivalProb_div","parent":44966,"reading_title":{"format":"inline-parts-v1","parts":[["text","Haldane's approximation "],["math","\\(\\pi(s) \\approx 2s\\)"]],"source":"Haldane's approximation $\\pi(s) \\approx 2s$"},"result":true,"route":"../roadmap/Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m/index.html#tendsto-survival-prob-div","scope":"Biology/population-genetics/haldane-s-survival-probability-of-a-beneficial-m","status":"fully_proved","summary":"Haldane's approximation (Haldane 1927): `\u03c0(s) / s \u2192 2` as `s \u2192 0\u207a`.","title":"Haldane's approximation $\\pi(s) \\approx 2s$"},{"data_url":"scopes/Biology/population-genetics/hardy-weinberg-law.json","id":"Biology/population-genetics/hardy-weinberg-law","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","title":"Hardy\u2013Weinberg law"},{"id":"Biology/population-genetics/hardy-weinberg-law/transmit","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.transmit","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#transmit","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"Mendel's law of segregation: a parent with genotype `g` transmits its maternal allele `g.1` or its paternal allele `g.2`, each with probability `1/2`.","title":"Mendel's law of segregation"},{"id":"Biology/population-genetics/hardy-weinberg-law/gamete","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.gamete","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#gamete","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"The law of a random gamete of a generation with genotype law `G`: pick a parent from `G`, then one of its two alleles by `transmit`. Its mass at `a` is the frequency of allele `a` (`gamete_apply`).","title":"Law of a random gamete"},{"id":"Biology/population-genetics/hardy-weinberg-law/offspring","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.offspring","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#offspring","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"The offspring genotype law under random union of gametes: the maternal and paternal alleles are independent gametes of the parental generation.","title":"Offspring genotype law under random union of gametes"},{"id":"Biology/population-genetics/hardy-weinberg-law/transmit-apply","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.transmit_apply","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#transmit-apply","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"`transmit g` puts mass `1/2` on each of the two alleles of `g` (mass `1` on `a` for a homozygote `g = (a, a)`).","title":"Allele transmission law of a parental genotype"},{"id":"Biology/population-genetics/hardy-weinberg-law/gamete-apply","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.gamete_apply","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#gamete-apply","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"The gamete law is the allele frequency: `gamete G a = \u00bd \u2211 b, G (a, b) + \u00bd \u2211 b, G (b, a)`, half the expected number of copies of `a` in a parental genotype.","title":"Gamete law equals the allele frequency"},{"id":"Biology/population-genetics/hardy-weinberg-law/offspring-apply","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.offspring_apply","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#offspring-apply","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"Hardy\u2013Weinberg proportions (Hardy 1908, Weinberg 1908): after one generation of random mating the genotype law is the product `p \u2297 p` of the parental allele frequencies `p = gamete G`.","title":"Hardy\u2013Weinberg proportions"},{"id":"Biology/population-genetics/hardy-weinberg-law/gamete-offspring","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.gamete_offspring","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#gamete-offspring","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"Allele frequencies are conserved under random mating: the gamete law of the offspring equals that of the parents.","title":"Allele frequencies are conserved"},{"id":"Biology/population-genetics/hardy-weinberg-law/offspring-offspring","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.offspring_offspring","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#offspring-offspring","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"The Hardy\u2013Weinberg proportions reached after one generation of random mating are a fixed point of random mating.","title":"Hardy\u2013Weinberg equilibrium"},{"id":"Biology/population-genetics/hardy-weinberg-law/freq-true-add-freq-false","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.freq_true_add_freq_false","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#freq-true-add-freq-false","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"The two allele frequencies sum to one: `p + q = 1`.","title":"The two allele frequencies sum to one"},{"id":"Biology/population-genetics/hardy-weinberg-law/offspring-true-true","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.offspring_true_true","parent":44975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hardy\u2013Weinberg frequency "],["math","\\(p^2\\)"],["text"," of the AA homozygote"]],"source":"Hardy\u2013Weinberg frequency $p^2$ of the AA homozygote"},"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#offspring-true-true","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"After one generation of random mating the homozygote `AA` has frequency `p\u00b2`.","title":"Hardy\u2013Weinberg frequency $p^2$ of the AA homozygote"},{"id":"Biology/population-genetics/hardy-weinberg-law/offspring-false-false","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.offspring_false_false","parent":44975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hardy\u2013Weinberg frequency "],["math","\\(q^2\\)"],["text"," of the aa homozygote"]],"source":"Hardy\u2013Weinberg frequency $q^2$ of the aa homozygote"},"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#offspring-false-false","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"After one generation of random mating the homozygote `aa` has frequency `q\u00b2`.","title":"Hardy\u2013Weinberg frequency $q^2$ of the aa homozygote"},{"id":"Biology/population-genetics/hardy-weinberg-law/offspring-heterozygote","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.offspring_heterozygote","parent":44975,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hardy\u2013Weinberg frequency "],["math","\\(2pq\\)"],["text"," of the heterozygote"]],"source":"Hardy\u2013Weinberg frequency $2pq$ of the heterozygote"},"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#offspring-heterozygote","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"After one generation of random mating the (unordered) heterozygote `Aa`, i.e. the two ordered genotypes `(A, a)` and `(a, A)`, has frequency `2pq`.","title":"Hardy\u2013Weinberg frequency $2pq$ of the heterozygote"},{"id":"Biology/population-genetics/hardy-weinberg-law/hw-deviation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.hwDeviation","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#hw-deviation","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"The deviation `D = P(AA) - p\u00b2` of a biallelic genotype law from Hardy\u2013Weinberg proportions, where `p` is the frequency of allele `A` (`true`).","title":"Deviation from Hardy\u2013Weinberg proportions"},{"id":"Biology/population-genetics/hardy-weinberg-law/hw-deviation-offspring","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.HardyWeinberg.hwDeviation_offspring","parent":44975,"result":true,"route":"../roadmap/Biology/population-genetics/hardy-weinberg-law/index.html#hw-deviation-offspring","scope":"Biology/population-genetics/hardy-weinberg-law","status":"fully_proved","summary":"The deviation from Hardy\u2013Weinberg proportions vanishes after one generation of random mating, whatever the parental genotype law.","title":"Hardy\u2013Weinberg proportions after one generation of random mating"},{"data_url":"scopes/Biology/population-genetics/infinite-alleles-model-and-homozygosity.json","id":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/infinite-alleles-model-and-homozygosity/index.html","scope":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","status":"fully_proved","title":"Infinite-alleles model and homozygosity"},{"id":"Biology/population-genetics/infinite-alleles-model-and-homozygosity/homozygosity-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.InfiniteAlleles.homozygosityStep","parent":44990,"result":true,"route":"../roadmap/Biology/population-genetics/infinite-alleles-model-and-homozygosity/index.html#homozygosity-step","scope":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","status":"fully_proved","summary":"One generation of drift and mutation in the infinite-alleles model with `M` gene copies and mutation probability `u` per copy: `F' = (1 - u)\u00b2 (1/M + (1 - 1/M) F)`. The model domain is `2 \u2264 M` (two distinct copies must exist) and `0 \u2264 u \u2264 1`; the theorems below assume only `1 \u2264 M`, and `M = 1` is an algebraic extension with no biological meaning. For `M = 0` the definition uses the junk value `1/0 = 0`.","title":"Homozygosity recursion in the infinite-alleles model"},{"id":"Biology/population-genetics/infinite-alleles-model-and-homozygosity/eq-homozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.InfiniteAlleles.eqHomozygosity","parent":44990,"result":true,"route":"../roadmap/Biology/population-genetics/infinite-alleles-model-and-homozygosity/index.html#eq-homozygosity","scope":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","status":"fully_proved","summary":"The equilibrium homozygosity `F* = (1 - u)\u00b2 / (M - (M - 1)(1 - u)\u00b2)`. Biologically meaningful for `2 \u2264 M`; its denominator is positive for `1 \u2264 M` and `0 < u \u2264 1`, the hypotheses of the theorems below (`M = 1` being an algebraic extension).","title":"Equilibrium homozygosity in the infinite-alleles model"},{"id":"Biology/population-genetics/infinite-alleles-model-and-homozygosity/homozygosity-step-eq-homozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.InfiniteAlleles.homozygosityStep_eqHomozygosity","parent":44990,"result":true,"route":"../roadmap/Biology/population-genetics/infinite-alleles-model-and-homozygosity/index.html#homozygosity-step-eq-homozygosity","scope":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","status":"fully_proved","summary":"`F*` is a fixed point of the homozygosity recursion.","title":"Equilibrium homozygosity is a fixed point of the recursion"},{"id":"Biology/population-genetics/infinite-alleles-model-and-homozygosity/homozygosity-step-eq-self-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.InfiniteAlleles.homozygosityStep_eq_self_iff","parent":44990,"result":true,"route":"../roadmap/Biology/population-genetics/infinite-alleles-model-and-homozygosity/index.html#homozygosity-step-eq-self-iff","scope":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","status":"fully_proved","summary":"For `0 < u \u2264 1`, `F*` is the unique fixed point of the homozygosity recursion.","title":"Uniqueness of the equilibrium homozygosity"},{"id":"Biology/population-genetics/infinite-alleles-model-and-homozygosity/tendsto-iterate-homozygosity-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.InfiniteAlleles.tendsto_iterate_homozygosityStep","parent":44990,"result":true,"route":"../roadmap/Biology/population-genetics/infinite-alleles-model-and-homozygosity/index.html#tendsto-iterate-homozygosity-step","scope":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","status":"fully_proved","summary":"For `0 < u \u2264 1` the homozygosity converges to `F*` from every initial value `F\u2080`.","title":"Convergence of homozygosity to equilibrium"},{"id":"Biology/population-genetics/infinite-alleles-model-and-homozygosity/tendsto-eq-homozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.InfiniteAlleles.tendsto_eqHomozygosity","parent":44990,"result":true,"route":"../roadmap/Biology/population-genetics/infinite-alleles-model-and-homozygosity/index.html#tendsto-eq-homozygosity","scope":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","status":"fully_proved","summary":"With `M = 2N` gene copies and mutation probability `u = \u03b8/(4N)` (so that `\u03b8 = 4Nu`), the equilibrium homozygosity tends to `1/(1 + \u03b8)` as `N \u2192 \u221e`.","title":"Kimura\u2013Crow limit"},{"id":"Biology/population-genetics/infinite-alleles-model-and-homozygosity/tendsto-one-sub-eq-homozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.InfiniteAlleles.tendsto_one_sub_eqHomozygosity","parent":44990,"reading_title":{"format":"inline-parts-v1","parts":[["text","Limit "],["math","\\(\\theta/(1+\\theta)\\)"],["text"," of the equilibrium heterozygosity"]],"source":"Limit $\\theta/(1+\\theta)$ of the equilibrium heterozygosity"},"result":true,"route":"../roadmap/Biology/population-genetics/infinite-alleles-model-and-homozygosity/index.html#tendsto-one-sub-eq-homozygosity","scope":"Biology/population-genetics/infinite-alleles-model-and-homozygosity","status":"fully_proved","summary":"The equilibrium heterozygosity `H* = 1 - F*` tends to `\u03b8/(1 + \u03b8)` as `N \u2192 \u221e` with `M = 2N` and `u = \u03b8/(4N)`.","title":"Limit $\\theta/(1+\\theta)$ of the equilibrium heterozygosity"},{"data_url":"scopes/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio.json","id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","title":"Linkage disequilibrium and Haldane's map function"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/allele-freq-fst","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.alleleFreqFst","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#allele-freq-fst","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"The frequency `\u2211 b, x (a, b)` of allele `a` at the first locus.","title":"Allele frequency at the first locus"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/allele-freq-snd","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.alleleFreqSnd","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#allele-freq-snd","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"The frequency `\u2211 a, x (a, b)` of allele `b` at the second locus.","title":"Allele frequency at the second locus"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/linkage-diseq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.linkageDiseq","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#linkage-diseq","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"The linkage disequilibrium `D = x_AB x_ab - x_Ab x_aB`.","title":"Linkage disequilibrium of a two-locus gamete law"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/recombine","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.recombine","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#recombine","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"One generation of random mating with recombination fraction `r`: `recombine r x g = (1 - r) x_g + r p_{g.1} q_{g.2}`.","title":"Random mating with recombination"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/recombine-eq-sub","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.recombine_eq_sub","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#recombine-eq-sub","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"The classical form of the recursion: coupling gametes (`AB`, `ab`) lose `r D` and repulsion gametes (`Ab`, `aB`) gain `r D`. This identity holds for every real `r`; the biological domain is `0 \u2264 r \u2264 1/2`.","title":"Classical form of the recombination recursion"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/sum-recombine","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.sum_recombine","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#sum-recombine","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Recombination keeps gamete frequencies summing to `1`.","title":"Recombination preserves total gamete frequency"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/recombine-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.recombine_nonneg","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#recombine-nonneg","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Recombination keeps gamete frequencies nonnegative for `0 \u2264 r \u2264 1`.","title":"Recombination preserves nonnegativity of gamete frequencies"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/allele-freq-fst-recombine","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.alleleFreqFst_recombine","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#allele-freq-fst-recombine","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Allele frequencies at the first locus are unchanged by random mating (for every real `r`).","title":"Recombination preserves first-locus allele frequencies"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/allele-freq-snd-recombine","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.alleleFreqSnd_recombine","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#allele-freq-snd-recombine","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Allele frequencies at the second locus are unchanged by random mating (for every real `r`).","title":"Recombination preserves second-locus allele frequencies"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/linkage-diseq-recombine","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.linkageDiseq_recombine","parent":44998,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linkage disequilibrium decays by the factor "],["math","\\(1 - r\\)"]],"source":"Linkage disequilibrium decays by the factor $1 - r$"},"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#linkage-diseq-recombine","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Linkage disequilibrium decays by the factor `1 - r` per generation: `D' = (1 - r) D` (for every real `r`).","title":"Linkage disequilibrium decays by the factor $1 - r$"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/linkage-diseq-iterate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.linkageDiseq_iterate","parent":44998,"reading_title":{"format":"inline-parts-v1","parts":[["text","Linkage disequilibrium after "],["math","\\(t\\)"],["text"," generations"]],"source":"Linkage disequilibrium after $t$ generations"},"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#linkage-diseq-iterate","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"After `t` generations, `D_t = (1 - r)^t D_0`.","title":"Linkage disequilibrium after $t$ generations"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/tendsto-linkage-diseq-iterate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.tendsto_linkageDiseq_iterate","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#tendsto-linkage-diseq-iterate","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Linkage disequilibrium tends to `0` whenever `0 < r \u2264 1`; this contains the biological range `0 < r \u2264 1/2`. For `r = 0` (complete linkage) `D` is constant.","title":"Linkage disequilibrium tends to zero"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/haldane","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.haldane","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#haldane","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Haldane's map function `r(d) = (1 - e^{-2d}) / 2`: the recombination fraction between two loci at map distance `d` Morgans when crossovers form a Poisson process (no interference).","title":"Haldane's map function"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/has-sum-poisson-odd","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.hasSum_poisson_odd","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#has-sum-poisson-odd","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"The probability that a Poisson count of mean `d` is odd, as a series: `\u2211_{k odd} e^{-d} d^k / k! = (1 - e^{-2d}) / 2` (Haldane 1919). The series identity holds for every real `d`; it is a probability for `0 \u2264 d`.","title":"Series for the probability of an odd Poisson count"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/poisson-measure-real-odd","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.poissonMeasure_real_odd","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#poisson-measure-real-odd","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Haldane's map function (Haldane 1919): if the number of crossovers in an interval of map length `d` Morgans is Poisson with mean `d`, the probability of an odd number of crossovers, i.e. of recombination between the ends of the interval, is `(1 - e^{-2d}) / 2`.","title":"Haldane's map function from Poisson crossovers"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/haldane-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.haldane_nonneg","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#haldane-nonneg","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Haldane's map function is nonnegative at nonnegative distances.","title":"Nonnegativity of Haldane's map function"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/haldane-lt-half","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.haldane_lt_half","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#haldane-lt-half","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"The recombination fraction is always below `1/2` (for every real `d`).","title":"Haldane's recombination fraction is below one half"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/tendsto-haldane-at-top","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.tendsto_haldane_atTop","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#tendsto-haldane-at-top","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Loci far apart are unlinked in the limit: `r \u2192 1/2` as `d \u2192 \u221e`.","title":"Distant loci are unlinked in the limit"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/neg-half-log-haldane","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.neg_half_log_haldane","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#neg-half-log-haldane","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"The inverse map function `d = -\u00bd log(1 - 2r)` recovers the map distance.","title":"Inverse Haldane map function recovers map distance"},{"id":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/haldane-neg-half-log","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Linkage.haldane_neg_half_log","parent":44998,"result":true,"route":"../roadmap/Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio/index.html#haldane-neg-half-log","scope":"Biology/population-genetics/linkage-disequilibrium-and-haldane-s-map-functio","status":"fully_proved","summary":"Every `r < 1/2` is the recombination fraction at the distance `-\u00bd log(1 - 2r)`. The hypothesis keeps `log` on positive arguments: at `r = 1/2` the right side would use the junk value `log 0 = 0`. The distance is nonnegative exactly when `0 \u2264 r`.","title":"Every recombination fraction below one half is attained"},{"data_url":"scopes/Biology/population-genetics/luria-delbr-ck-distribution.json","id":"Biology/population-genetics/luria-delbr-ck-distribution","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","title":"Luria\u2013Delbr\u00fcck distribution"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/clone-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.cloneProb","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#clone-prob","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The Lea\u2013Coulson clone-size law `P(size = j) = 1 / (j (j + 1))` for `j \u2265 1` (`P(size = 0) = 0`).","title":"Lea\u2013Coulson clone-size law"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/conv-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.convPow","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#conv-pow","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The `k`-fold convolution power of a sequence on `\u2115`: `p^{*0} = \u03b4\u2080` and `p^{*(k+1)}(n) = \u2211_{i \u2264 n} p(i) p^{*k}(n - i)`.","title":"Convolution power of a sequence on the naturals"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/ld-pmf","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.ldPmf","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#ld-pmf","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The Luria\u2013Delbr\u00fcck probability of `n` mutants with expected mutation number `m`: the compound-Poisson law `\u2211_k e^{-m} m^k / k! \u00b7 cloneProb^{*k}(n)`. Since clones have size at least `1`, only `k \u2264 n` contribute.","title":"Luria\u2013Delbr\u00fcck distribution"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/clone-pgf","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.clonePgf","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#clone-pgf","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The clone-size generating function `h(z) = 1 + (1 / z - 1) log (1 - z)`.","title":"Closed form of the clone-size generating function"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/integral-clone-size","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.integral_cloneSize","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#integral-clone-size","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"A clone founded `\u03c4` before the end has geometric size `e^{-\u03c4} (1 - e^{-\u03c4})^{j-1}`; mixing over the founding-time density `e^{-\u03c4}` on `(0, \u221e)` gives `1 / (j (j + 1))` for `j \u2265 1`.","title":"Clone-size law from the founding-time density"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/has-sum-clone-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.hasSum_cloneProb","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#has-sum-clone-prob","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The clone-size law sums to `1`.","title":"The clone-size law sums to one"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/has-sum-clone-prob-mul-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.hasSum_cloneProb_mul_pow","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#has-sum-clone-prob-mul-pow","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"For `0 < z < 1`, `\u2211\u2c7c z\u02b2 / (j (j + 1)) = 1 + (1 / z - 1) log (1 - z)`.","title":"Clone-size generating function"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/ld-pmf-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.ldPmf_zero","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#ld-pmf-zero","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The probability of no mutants is `e^{-m}`.","title":"`p\u2080`"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/p0-method","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.p0_method","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#p0-method","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The `p\u2080` method (Luria\u2013Delbr\u00fcck 1943): the expected number of mutations is recovered from the proportion of cultures with no mutants, `m = -log p\u2080`.","title":"The `p\u2080` method"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/ld-pmf-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.ldPmf_nonneg","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#ld-pmf-nonneg","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The Luria\u2013Delbr\u00fcck probabilities are nonnegative for `m \u2265 0`.","title":"Nonnegativity of Luria\u2013Delbr\u00fcck probabilities"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/has-sum-ld-pmf","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.hasSum_ldPmf","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#has-sum-ld-pmf","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The Luria\u2013Delbr\u00fcck probabilities sum to `1` (for `m \u2265 0`).","title":"The Luria\u2013Delbr\u00fcck probabilities sum to one"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/has-sum-ld-pmf-mul-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.hasSum_ldPmf_mul_pow","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#has-sum-ld-pmf-mul-pow","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"For `m \u2265 0` and `0 < z < 1`, `\u2211\u2099 p\u2099 z\u207f = exp (m (h(z) - 1)) = exp (m (1 / z - 1) log (1 - z))`.","title":"Lea\u2013Coulson generating function"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/exp-lea-coulson-eq-rpow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.exp_leaCoulson_eq_rpow","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#exp-lea-coulson-eq-rpow","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"The Lea\u2013Coulson generating function in power form: `exp (m (1 / z - 1) log (1 - z)) = (1 - z)^{m (1 - z) / z}` for `z < 1`, `z \u2260 0`.","title":"Power form of the Lea\u2013Coulson generating function"},{"id":"Biology/population-genetics/luria-delbr-ck-distribution/ld-pmf-recursion","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.LuriaDelbruck.ldPmf_recursion","parent":45019,"result":true,"route":"../roadmap/Biology/population-genetics/luria-delbr-ck-distribution/index.html#ld-pmf-recursion","scope":"Biology/population-genetics/luria-delbr-ck-distribution","status":"fully_proved","summary":"For `n \u2265 1`, `p\u2099 = (m / n) \u2211_{i < n} p\u1d62 / (n - i + 1)` (all real `m`).","title":"Ma\u2013Sandri\u2013Sarkar recursion"},{"data_url":"scopes/Biology/population-genetics/moran-model.json","id":"Biology/population-genetics/moran-model","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/moran-model/index.html","scope":"Biology/population-genetics/moran-model","status":"fully_proved","title":"Moran model"},{"id":"Biology/population-genetics/moran-model/moran-up","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranUp","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-up","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"Probability that one step of the Moran process with `N` individuals and mutant relative fitness `r` takes the mutant count from `i` to `i + 1`: a mutant is chosen to reproduce (probability `r i / (r i + N - i)`) and a wild type is chosen to die (probability `(N - i) / N`). Meaningful for `0 < r` and `i \u2264 N` (the real subtraction `N - i` is negative for `i > N`).","title":"Up-step probability of the Moran process"},{"id":"Biology/population-genetics/moran-model/moran-down","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranDown","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-down","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"Probability that one step of the Moran process takes the mutant count from `i` to `i - 1`: a wild type is chosen to reproduce (probability `(N - i) / (r i + N - i)`) and a mutant is chosen to die (probability `i / N`). Meaningful for `0 < r` and `i \u2264 N`.","title":"Down-step probability of the Moran process"},{"id":"Biology/population-genetics/moran-model/moran-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranFixation","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-fixation","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"The fixation probability of the Moran process started from `i` mutants out of `N`: `(1 - r\u207b\u00b9 ^ i) / (1 - r\u207b\u00b9 ^ N)` for `r \u2260 1`, and `i / N` in the neutral case `r = 1`. The model's domain is `0 < r`, `N \u2265 1`, `i \u2264 N`; outside it the value is the closed formula evaluated with Lean's total division (e.g. `x / 0 = 0`) and carries no meaning.","title":"Fixation probability of the Moran process"},{"id":"Biology/population-genetics/moran-model/moran-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranProb","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-prob","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"One-step transition probabilities of the Moran process on the states `Fin (N + 1)`: `a_i` to `i + 1`, `b_i` to `i - 1`, `1 - a_i - b_i` to `i`, and `0` elsewhere.","title":"One-step transition probabilities of the Moran process"},{"id":"Biology/population-genetics/moran-model/moran-up-last","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranUp_last","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-up-last","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"No mutant can be added when the mutant type is fixed.","title":"No up-step from the fixed state of the Moran process"},{"id":"Biology/population-genetics/moran-model/moran-down-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranDown_zero","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-down-zero","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"No mutant can be removed when the mutant type is absent.","title":"No down-step from the empty state of the Moran process"},{"id":"Biology/population-genetics/moran-model/mul-moran-down","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.mul_moranDown","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#mul-moran-down","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"The ratio of the down- and up-step probabilities is `1 / r`: `r \u00b7 b_i = a_i`.","title":"Ratio of Moran down- and up-step probabilities"},{"id":"Biology/population-genetics/moran-model/moran-up-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranUp_nonneg","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-up-nonneg","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"The up-step probability is nonnegative on the states `0 \u2264 i \u2264 N`.","title":"Nonnegativity of the Moran up-step probability"},{"id":"Biology/population-genetics/moran-model/moran-down-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranDown_nonneg","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-down-nonneg","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"The down-step probability is nonnegative on the states `0 \u2264 i \u2264 N`.","title":"Nonnegativity of the Moran down-step probability"},{"id":"Biology/population-genetics/moran-model/moran-up-add-moran-down-le-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranUp_add_moranDown_le_one","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-up-add-moran-down-le-one","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"On the states `0 \u2264 i \u2264 N` the up- and down-step probabilities sum to at most `1`, so the probability `1 - a_i - b_i` of staying is nonnegative.","title":"Moran up- and down-step probabilities sum to at most one"},{"id":"Biology/population-genetics/moran-model/sum-moran-prob-mul","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.sum_moranProb_mul","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#sum-moran-prob-mul","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"One-step expectation of a function `f` of the mutant count from state `i`: `\u2211 j, P i j f j = a_i f (i + 1) + b_i f (i - 1) + (1 - a_i - b_i) f i` (the boundary terms vanish at `i = 0` and `i = N`).","title":"One-step expectation under the Moran process"},{"id":"Biology/population-genetics/moran-model/moran-matrix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranMatrix","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-matrix","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"The Moran process with `N` individuals and mutant relative fitness `r > 0`, as a stochastic matrix on the mutant counts `Fin (N + 1)`.","title":"Moran process as a stochastic matrix"},{"id":"Biology/population-genetics/moran-model/moran-fixation-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranFixation_zero","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-fixation-zero","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"A mutant that is absent is never fixed: `\u03c6_0 = 0`.","title":"An absent mutant is never fixed"},{"id":"Biology/population-genetics/moran-model/moran-fixation-self","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranFixation_self","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-fixation-self","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"A mutant that is already fixed stays fixed: `\u03c6_N = 1` (for `N \u2260 0`).","title":"A fixed mutant stays fixed"},{"id":"Biology/population-genetics/moran-model/moran-fixation-eq-first-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.moranFixation_eq_firstStep","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#moran-fixation-eq-first-step","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"The fixation probability satisfies `\u03c6_i = a_i \u03c6_{i+1} + b_i \u03c6_{i-1} + (1 - a_i - b_i) \u03c6_i`. This is the first-step equation of the chain for the interior states `0 < i < N`; the identity is algebraic (it follows from `r \u00b7 b_i = a_i`), so it is stated for every `i \u2265 1`.","title":"First-step equation (Moran 1958)"},{"id":"Biology/population-genetics/moran-model/eq-moran-fixation","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.eq_moranFixation","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#eq-moran-fixation","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"Any `\u03c8` with `\u03c8 0 = 0`, `\u03c8 N = 1` that satisfies the first-step equation `\u03c8 i = a_i \u03c8 (i+1) + b_i \u03c8 (i-1) + (1 - a_i - b_i) \u03c8 i` at every `0 < i < N` equals `moranFixation r N` on `0 \u2264 i \u2264 N`.","title":"Uniqueness of the fixation probability"},{"id":"Biology/population-genetics/moran-model/tendsto-mat-pow-moran-matrix-last","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.tendsto_matPow_moranMatrix_last","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#tendsto-mat-pow-moran-matrix-last","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"For `N \u2260 0`, the probability that the Moran chain started at `i` is in state `N` after `t` steps tends to `\u03c6_i` as `t \u2192 \u221e`. Since `N` absorbs, this `t`-step probability is the probability of fixation by time `t`, so its limit is the probability of eventual fixation.","title":"Fixation probability of the chain"},{"id":"Biology/population-genetics/moran-model/tendsto-mat-pow-moran-matrix-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.tendsto_matPow_moranMatrix_zero","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#tendsto-mat-pow-moran-matrix-zero","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"For `N \u2260 0`, the probability that the Moran chain started at `i` is in state `0` after `t` steps tends to `1 - \u03c6_i`; together with `tendsto_matPow_moranMatrix_last`, absorption at `0` or `N` is certain.","title":"Loss probability of the chain"},{"id":"Biology/population-genetics/moran-model/kimura-fixation-half","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.kimuraFixation_half","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#kimura-fixation-half","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"Kimura's diffusion fixation probability at scaled selection `\u03c3 / 2` is `(1 - e^{-\u03c3 x}) / (1 - e^{-\u03c3})` (for `\u03c3 \u2260 0`).","title":"Closed form of Kimura's fixation probability"},{"id":"Biology/population-genetics/moran-model/tendsto-moran-fixation-diffusion","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Moran.tendsto_moranFixation_diffusion","parent":45034,"result":true,"route":"../roadmap/Biology/population-genetics/moran-model/index.html#tendsto-moran-fixation-diffusion","scope":"Biology/population-genetics/moran-model","status":"fully_proved","summary":"With relative fitness `r_N = 1 + \u03c3 / N` and initial count `i_N = \u230ax N\u230b`, the fixation probability `\u03c6^{(N)}_{i_N}` tends to Kimura's `kimuraFixation (\u03c3 / 2) x`, which is `(1 - e^{-\u03c3 x}) / (1 - e^{-\u03c3})` for `\u03c3 \u2260 0` (`kimuraFixation_half`) and `x` for `\u03c3 = 0`. The initial frequency `x` is meaningful in `[0, 1]`; the limit of the closed formula holds for every `x \u2265 0`, so only `0 \u2264 x` is assumed (for negative `x`,\u2026","title":"Diffusion limit of the Moran fixation probability"},{"data_url":"scopes/Biology/population-genetics/mutation-selection-balance.json","id":"Biology/population-genetics/mutation-selection-balance","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","title":"Mutation\u2013selection balance"},{"id":"Biology/population-genetics/mutation-selection-balance/lin-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.linStep","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#lin-step","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"One generation of the linear model: `x \u21a6 (1 - s) x + U`.","title":"Linear mutation\u2013selection recursion"},{"id":"Biology/population-genetics/mutation-selection-balance/lin-step-iterate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.linStep_iterate","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#lin-step-iterate","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"Closed-form orbit: `x_t = U / s + (1 - s)^t (x_0 - U / s)`.","title":"Closed-form orbit of the linear mutation\u2013selection recursion"},{"id":"Biology/population-genetics/mutation-selection-balance/lin-step-eq-self-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.linStep_eq_self_iff","parent":45055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unique fixed point "],["math","\\(U/s\\)"],["text"," of the linear recursion"]],"source":"Unique fixed point $U/s$ of the linear recursion"},"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#lin-step-eq-self-iff","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"`U / s` is the unique fixed point.","title":"Unique fixed point $U/s$ of the linear recursion"},{"id":"Biology/population-genetics/mutation-selection-balance/tendsto-lin-step-iterate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.tendsto_linStep_iterate","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#tendsto-lin-step-iterate","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"For `0 < s < 2` every orbit converges to `U / s`.","title":"Global attraction"},{"id":"Biology/population-genetics/mutation-selection-balance/de-novo-fraction","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.deNovoFraction","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#de-novo-fraction","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"The fraction of LoF alleles in the next generation that are new mutations, from frequency `x`: `U / ((1 - s) x + U)`.","title":"Fraction of de novo loss-of-function alleles"},{"id":"Biology/population-genetics/mutation-selection-balance/de-novo-fraction-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.deNovoFraction_eq","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#de-novo-fraction-eq","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"At equilibrium the de novo fraction is `s_het`.","title":"At equilibrium the de novo fraction is `s_het`"},{"id":"Biology/population-genetics/mutation-selection-balance/kernel","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.kernel","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#kernel","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"The transition probability from `i` LoF copies among `N` to `j` copies among `N'`: binomial sampling of `N'` copies with success probability `(1 - s)(i / N) + U`.","title":"Binomial transition kernel for loss-of-function counts"},{"id":"Biology/population-genetics/mutation-selection-balance/mean-freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.meanFreq","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#mean-freq","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"The mean LoF frequency `\u2211 \u03bc_i (i / N)` of a distribution `\u03bc` on `Fin (N + 1)`.","title":"Mean loss-of-function frequency of a distribution"},{"id":"Biology/population-genetics/mutation-selection-balance/push-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.pushStep","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#push-step","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"One generation of the chain acting on distributions.","title":"One generation of the chain on distributions"},{"id":"Biology/population-genetics/mutation-selection-balance/sum-kernel","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.sum_kernel","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#sum-kernel","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"The kernel rows have total mass one (for every real sampling probability).","title":"Rows of the transition kernel have total mass one"},{"id":"Biology/population-genetics/mutation-selection-balance/kernel-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.kernel_nonneg","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#kernel-nonneg","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"The kernel entries are probabilities when the sampling probability is in `[0, 1]`.","title":"Transition kernel entries are probabilities"},{"id":"Biology/population-genetics/mutation-selection-balance/sum-kernel-mul","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.sum_kernel_mul","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#sum-kernel-mul","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"The binomial mean: `\u2211_j kernel(i, j) (j / N') = (1 - s)(i / N) + U` for `N' > 0`.","title":"Binomial mean of the transition kernel"},{"id":"Biology/population-genetics/mutation-selection-balance/mean-freq-push-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.meanFreq_pushStep","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#mean-freq-push-step","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"For any sizes `N`, `N' > 0` and any distribution `\u03bc` of total mass one, the mean frequency after one generation is `(1 - s) m + U`.","title":"Demography-free mean, one step"},{"id":"Biology/population-genetics/mutation-selection-balance/sum-push-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.sum_pushStep","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#sum-push-step","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"Pushing a distribution forward preserves total mass.","title":"Pushing a distribution forward preserves total mass"},{"id":"Biology/population-genetics/mutation-selection-balance/law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.law","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#law","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"The law of the LoF count in generation `t` for population sizes `N t` (any demography), started from `\u03bc\u2080`.","title":"Law of the loss-of-function count under any demography"},{"id":"Biology/population-genetics/mutation-selection-balance/mean-freq-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.meanFreq_law","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#mean-freq-law","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"For every sequence of positive population sizes, `E[X_t]` follows the deterministic linear recursion (derived from the binomial mean). The identity holds for all real `s, U`; `law` is a probability law when `0 \u2264 U \u2264 s \u2264 1` and `\u03bc\u2080` is one.","title":"Demography-free mean"},{"id":"Biology/population-genetics/mutation-selection-balance/tendsto-mean-freq-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.tendsto_meanFreq_law","parent":45055,"reading_title":{"format":"inline-parts-v1","parts":[["text","Expected loss-of-function frequency converges to "],["math","\\(U/s\\)"]],"source":"Expected loss-of-function frequency converges to $U/s$"},"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#tendsto-mean-freq-law","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"The expected LoF frequency converges to `U / s` under any demography.","title":"Expected loss-of-function frequency converges to $U/s$"},{"id":"Biology/population-genetics/mutation-selection-balance/mean-fitness","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.meanFitness","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#mean-fitness","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"Mean fitness under Hardy\u2013Weinberg with LoF frequency `q` and fitnesses `1, 1 - s, 1 - t`: `1 - 2 q (1 - q) s - q\u00b2 t`.","title":"Mean fitness under Hardy\u2013Weinberg proportions"},{"id":"Biology/population-genetics/mutation-selection-balance/selected","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.selected","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#selected","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"LoF frequency after selection: `q (1 - s (1 - q) - t q) / w\u0304`.","title":"Loss-of-function frequency after selection"},{"id":"Biology/population-genetics/mutation-selection-balance/diploid-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.diploidStep","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#diploid-step","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"One generation: selection, then mutation of wild-type alleles at rate `U`.","title":"One diploid generation of selection and mutation"},{"id":"Biology/population-genetics/mutation-selection-balance/mean-fitness-mult","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.meanFitness_mult","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#mean-fitness-mult","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"With multiplicative fitness, `w\u0304 = (1 - s q)\u00b2` for every `q`.","title":"Mean fitness under multiplicative fitness"},{"id":"Biology/population-genetics/mutation-selection-balance/diploid-step-mult-fixed","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.diploidStep_mult_fixed","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#diploid-step-mult-fixed","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"`U / s` is an equilibrium of the multiplicative diploid model (an algebraic identity for `s \u2260 0`, `U \u2260 1`; a frequency in `[0, 1]` when `0 \u2264 U \u2264 s`).","title":"`U / s` is an equilibrium"},{"id":"Biology/population-genetics/mutation-selection-balance/mean-fitness-mult-at-equilibrium","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.meanFitness_mult_at_equilibrium","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#mean-fitness-mult-at-equilibrium","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"At the multiplicative equilibrium the mean fitness is `(1 - U)\u00b2`, independent of `s`. Requires `U \u2260 1`, so that `U / s` is genuinely a fixed point (`diploidStep_mult_fixed`): for `U = 1` the point `U / s` is not an equilibrium.","title":"Mean fitness at the multiplicative mutation\u2013selection equilibrium"},{"id":"Biology/population-genetics/mutation-selection-balance/load-mult-at-equilibrium","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.load_mult_at_equilibrium","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#load-mult-at-equilibrium","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"`1 - w\u0304 = 2U - U\u00b2` at the multiplicative equilibrium (`U \u2260 1`, so `U / s` is a fixed point).","title":"Haldane\u2013Muller load"},{"id":"Biology/population-genetics/mutation-selection-balance/mean-fitness-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.meanFitness_pos","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#mean-fitness-pos","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"Mean fitness is positive for `0 \u2264 q < 1` and `s, t \u2264 1` (nonnegative fitnesses).","title":"Positivity of mean fitness for allele frequencies below one"},{"id":"Biology/population-genetics/mutation-selection-balance/le-div-of-diploid-step-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MutationSelectionBalance.le_div_of_diploidStep_eq","parent":45055,"result":true,"route":"../roadmap/Biology/population-genetics/mutation-selection-balance/index.html#le-div-of-diploid-step-eq","scope":"Biology/population-genetics/mutation-selection-balance","status":"fully_proved","summary":"Equilibria lie at or below `U / s` when homozygotes are at least multiplicatively unfit (`2s - s\u00b2 \u2264 t`, which includes recessive lethality `t = 1`): every fixed point `q \u2208 [0, 1)` of the diploid map satisfies `q \u2264 U / s`.","title":"Equilibria lie at or below `U / s`"},{"data_url":"scopes/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price.json","id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","title":"Natural selection, Fisher's theorem and the Price equation"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/mean-fitness","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.meanFitness","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#mean-fitness","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"The mean fitness `w\u0304 = \u2211 i, p i * w i` of a population with type frequencies `p` and fitnesses `w`.","title":"Mean fitness of a population"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/fitness-variance","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.fitnessVariance","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#fitness-variance","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"The variance of fitness `Var_p(w) = \u2211 i, p i * (w i - w\u0304) ^ 2` under the type frequencies `p`.","title":"Variance of fitness"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/select","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.select","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#select","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"One generation of haploid selection: `select p w i = p i * w i / w\u0304`.","title":"One generation of haploid selection"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/mean-fitness-select-sub","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.meanFitness_select_sub","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#mean-fitness-select-sub","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"Fisher's fundamental theorem of natural selection in its exact haploid, discrete-generation form: the change in mean fitness over one generation of selection is the variance of fitness divided by the mean fitness, `w\u0304(p') - w\u0304(p) = Var_p(w) / w\u0304(p)` (cf. Fisher 1930, who states the theorem for the additive genetic variance). The identity needs only `\u2211 p = 1` and `w\u0304 \u2260 0`; the model domain is the simplex with\u2026","title":"Fisher's fundamental theorem of natural selection"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/mean-fitness-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.meanFitness_pos","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#mean-fitness-pos","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"Mean fitness is positive on the simplex when every fitness is positive.","title":"Positivity of mean fitness on the simplex"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/mean-fitness-le-mean-fitness-select","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.meanFitness_le_meanFitness_select","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#mean-fitness-le-mean-fitness-select","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"Mean fitness never decreases under haploid selection (corollary of Fisher's theorem).","title":"Mean fitness never decreases under haploid selection"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/select-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.select_nonneg","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#select-nonneg","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"Haploid selection keeps frequencies nonnegative.","title":"Haploid selection preserves nonnegative frequencies"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/sum-select","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.sum_select","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#sum-select","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"Haploid selection keeps the frequencies summing to `1` whenever `w\u0304 \u2260 0` (for `w\u0304 = 0` the division by zero would give `0`).","title":"Haploid selection preserves total frequency one"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/marginal-fitness","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.marginalFitness","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#marginal-fitness","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"The marginal fitness `\u2211 j, (w i j + w j i) / 2 * p j` of allele `i`: the mean fitness of a genotype carrying `i` whose other allele is drawn from the frequencies `p`, averaging the two positions at which `i` can sit in the ordered genotype.","title":"Marginal fitness of an allele"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/marginal-fitness-of-symm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.marginalFitness_of_symm","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#marginal-fitness-of-symm","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"For symmetric genotype fitnesses the marginal fitness is the row marginal `\u2211 j, w i j * p j`.","title":"Marginal fitness for symmetric genotype fitnesses"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/diploid-mean-fitness","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidMeanFitness","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#diploid-mean-fitness","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"The mean fitness `w\u0304 = \u2211 i, \u2211 j, p i * p j * w i j` of a population in Hardy\u2013Weinberg proportions with allele frequencies `p` and genotype fitnesses `w`.","title":"Diploid mean fitness under Hardy\u2013Weinberg proportions"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/diploid-select","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidSelect","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#diploid-select","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"One generation of diploid viability selection under random mating, followed by Mendelian transmission of either allele of each survivor: `diploidSelect p w i = p i * (\u2211 j, (w i j + w j i) / 2 * p j) / w\u0304`.","title":"One generation of diploid viability selection"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/diploid-mean-fitness-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidMeanFitness_pos","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#diploid-mean-fitness-pos","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"Diploid mean fitness is positive on the simplex when every genotype fitness is positive.","title":"Positivity of diploid mean fitness"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/diploid-select-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidSelect_nonneg","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#diploid-select-nonneg","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"Diploid viability selection keeps allele frequencies nonnegative.","title":"Diploid selection preserves nonnegative allele frequencies"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/sum-diploid-select","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.sum_diploidSelect","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#sum-diploid-select","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"Diploid viability selection keeps allele frequencies summing to `1`.","title":"Diploid selection preserves total allele frequency one"},{"id":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/price","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.price","parent":45082,"result":true,"route":"../roadmap/Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price/index.html#price","scope":"Biology/population-genetics/natural-selection-fisher-s-theorem-and-the-price","status":"fully_proved","summary":"The Price equation (exact, finite population). Individual `i` has fitness `w i` (number of offspring), trait value `z i`, and mean offspring trait value `z' i`. With population means `w\u0304 = \ud835\udd3c i, w i`, `z\u0304 = \ud835\udd3c i, z i` and offspring mean trait `z\u0304' = \u2211 i, w i * z' i / \u2211 i, w i`, `w\u0304 (z\u0304' - z\u0304) = Cov(w, z) + E[w (z' - z)]`, where `Cov` is the population covariance (Price 1970). Nonnegativity of `w` is not needed; only\u2026","title":"Price equation"},{"data_url":"scopes/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection.json","id":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/index.html","scope":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","status":"fully_proved","title":"Baum\u2013Eagon inequality for diploid selection"},{"id":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/diploid-mean-fitness-diploid-select-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidMeanFitness_diploidSelect_pos","parent":45099,"result":true,"route":"../roadmap/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/index.html#diploid-mean-fitness-diploid-select-pos","scope":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","status":"fully_proved","summary":"Mean fitness stays positive after one generation of diploid viability selection (for nonnegative frequencies and fitnesses; normalization is not needed).","title":"Mean fitness stays positive under diploid selection"},{"id":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/two-mul-rel-entropy-le-log-diploid-mean","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.two_mul_relEntropy_le_log_diploidMeanFitness_diploidSelect","parent":45099,"result":true,"route":"../roadmap/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/index.html#two-mul-rel-entropy-le-log-diploid-mean","scope":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","status":"fully_proved","summary":"With nonnegative allele frequencies and genotype fitnesses and positive mean fitness, one generation of selection raises the log mean fitness by at least twice the relative entropy `\u2211 i, p'_i log (p'_i / p_i)` of the new allele frequencies `p'` from the old ones `p` (Baum\u2013Eagon's argument for the quadratic form `w\u0304`). Alleles absent before selection stay absent (`p_i = 0 \u2192 p'_i = 0`), so the summands with `p_i = 0`\u2026","title":"Diploid Baum\u2013Eagon mean-fitness inequality, sharp form"},{"id":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/diploid-mean-fitness-le-diploid-mean-fit","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidMeanFitness_le_diploidMeanFitness_diploidSelect","parent":45099,"result":true,"route":"../roadmap/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/index.html#diploid-mean-fitness-le-diploid-mean-fit","scope":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","status":"fully_proved","summary":"Mean fitness never decreases under diploid multi-allele viability selection with random mating (Mulholland\u2013Smith 1959; Scheuer\u2013Mandel 1959; Kingman 1961).","title":"Mean fitness never decreases"},{"id":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/diploid-mean-fitness-diploid-select-eq-i","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidMeanFitness_diploidSelect_eq_iff","parent":45099,"result":true,"route":"../roadmap/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/index.html#diploid-mean-fitness-diploid-select-eq-i","scope":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","status":"fully_proved","summary":"Mean fitness is unchanged by one generation of selection exactly when the allele frequencies are at an equilibrium of the selection map.","title":"Mean fitness is unchanged exactly at selection equilibria"},{"id":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/diploid-mean-fitness-lt-diploid-mean-fit","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidMeanFitness_lt_diploidMeanFitness_diploidSelect","parent":45099,"result":true,"route":"../roadmap/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/index.html#diploid-mean-fitness-lt-diploid-mean-fit","scope":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","status":"fully_proved","summary":"Away from equilibria, mean fitness strictly increases under diploid viability selection.","title":"Diploid selection strictly increases mean fitness off equilibria"},{"id":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/has-deriv-at-diploid-mean-fitness-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.hasDerivAt_diploidMeanFitness_two","parent":45099,"result":true,"route":"../roadmap/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/index.html#has-deriv-at-diploid-mean-fitness-two","scope":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","status":"fully_proved","summary":"For two alleles with frequencies `(x, 1 - x)`, the mean fitness `w\u0304(x) = x\u00b2 w\u2080\u2080 + x (1 - x) (w\u2080\u2081 + w\u2081\u2080) + (1 - x)\u00b2 w\u2081\u2081` has derivative `2 (m\u2080 - m\u2081)`, twice the difference of the marginal fitnesses. This holds for every real `x`.","title":"Derivative of two-allele mean fitness"},{"id":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/diploid-select-two-sub","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Selection.diploidSelect_two_sub","parent":45099,"result":true,"route":"../roadmap/Biology/population-genetics/baum-eagon-inequality-for-diploid-selection/index.html#diploid-select-two-sub","scope":"Biology/population-genetics/baum-eagon-inequality-for-diploid-selection","status":"fully_proved","summary":"Wright's formula (Wright 1937): for two alleles with frequencies `(x, 1 - x)`, one generation of viability selection changes the frequency of allele `0` by `\u0394x = x (1 - x) (dw\u0304/dx) / (2 w\u0304)`. The identity is algebraic and holds for every real `x` with `w\u0304(x) \u2260 0`; the model domain is `x \u2208 [0, 1]` with nonnegative fitnesses.","title":"Wright's formula"},{"data_url":"scopes/Biology/population-genetics/wright-s-island-model.json","id":"Biology/population-genetics/wright-s-island-model","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","title":"Wright's island model"},{"id":"Biology/population-genetics/wright-s-island-model/island-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandStep","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-step","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"One generation of Wright's island-model identity-by-descent recursion for a deme of `M` gene copies (`M = 2N` for `N` diploids) receiving a fraction `m` of migrants from an infinite pool: `F' = (1 - m) ^ 2 * (1 / M + (1 - 1 / M) * F)`. Biologically meaningful for `M \u2265 2` and `0 \u2264 m \u2264 1`; `M = 1` is an algebraic extension (no pair of distinct copies exists), and at `M = 0` Lean's `1 / 0 = 0` gives a junk value.","title":"Identity-by-descent recursion in Wright's island model"},{"id":"Biology/population-genetics/wright-s-island-model/island-equilibrium","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandEquilibrium","parent":45107,"reading_title":{"format":"inline-parts-v1","parts":[["text","Equilibrium "],["math","\\(F_{ST}\\)"],["text"," of Wright's island model"]],"source":"Equilibrium $F_{ST}$ of Wright's island model"},"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-equilibrium","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"The equilibrium identity-by-descent probability (Wright's `F_ST`) of the island model: `F* = (1 - m) ^ 2 / (M - (M - 1) (1 - m) ^ 2)`. Biologically meaningful for `M \u2265 2` and `0 \u2264 m \u2264 1`; the denominator is positive already for `M \u2265 1` (`islandEquilibrium_denom_pos`), where `M = 1` is an algebraic extension.","title":"Equilibrium $F_{ST}$ of Wright's island model"},{"id":"Biology/population-genetics/wright-s-island-model/island-factor-mem-ico","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandFactor_mem_Ico","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-factor-mem-ico","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"For `M \u2265 1` and `0 \u2264 m \u2264 1`, the per-generation contraction factor `(1 - m) ^ 2 (1 - 1 / M)` lies in `[0, 1)`.","title":"Island-model contraction factor lies in (0, 1)"},{"id":"Biology/population-genetics/wright-s-island-model/island-equilibrium-denom-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandEquilibrium_denom_pos","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-equilibrium-denom-pos","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"The denominator `M - (M - 1) (1 - m) ^ 2` of the equilibrium is positive (at least `1`) for `M \u2265 1` and `0 \u2264 m \u2264 1`.","title":"Positivity of the island-model equilibrium denominator"},{"id":"Biology/population-genetics/wright-s-island-model/island-step-island-equilibrium","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandStep_islandEquilibrium","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-step-island-equilibrium","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"`F*` is a fixed point of the island-model recursion.","title":"Island-model equilibrium is a fixed point of the recursion"},{"id":"Biology/population-genetics/wright-s-island-model/eq-island-equilibrium-of-island-step-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.eq_islandEquilibrium_of_islandStep_eq","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#eq-island-equilibrium-of-island-step-eq","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"`F*` is the only fixed point of the island-model recursion.","title":"Uniqueness of the island-model fixed point"},{"id":"Biology/population-genetics/wright-s-island-model/island-step-iterate-sub","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandStep_iterate_sub","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-step-iterate-sub","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"After `t` generations, `F_t - F* = ((1 - m) ^ 2 (1 - 1 / M)) ^ t (F_0 - F*)`.","title":"Geometric approach to equilibrium"},{"id":"Biology/population-genetics/wright-s-island-model/tendsto-island-step-iterate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.tendsto_islandStep_iterate","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#tendsto-island-step-iterate","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"For `M \u2265 1` and `0 \u2264 m \u2264 1`, the island-model recursion converges to `F*` from every initial value `F\u2080`.","title":"Convergence to equilibrium"},{"id":"Biology/population-genetics/wright-s-island-model/island-equilibrium-mem-icc","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandEquilibrium_mem_Icc","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-equilibrium-mem-icc","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"The equilibrium identity-by-descent probability lies in `[0, 1]`.","title":"Island-model equilibrium lies in (0, 1)"},{"id":"Biology/population-genetics/wright-s-island-model/island-equilibrium-mem-ico","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandEquilibrium_mem_Ico","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-equilibrium-mem-ico","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"With migration (`0 < m \u2264 1`) the equilibrium identity-by-descent probability lies in `[0, 1)`.","title":"Island-model equilibrium is below one under migration"},{"id":"Biology/population-genetics/wright-s-island-model/island-equilibrium-zero-right","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.islandEquilibrium_zero_right","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#island-equilibrium-zero-right","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"Isolated demes (`m = 0`): the equilibrium is complete identity by descent, `F* = 1`.","title":"Complete identity by descent in isolated demes"},{"id":"Biology/population-genetics/wright-s-island-model/tendsto-island-equilibrium","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.IslandModel.tendsto_islandEquilibrium","parent":45107,"result":true,"route":"../roadmap/Biology/population-genetics/wright-s-island-model/index.html#tendsto-island-equilibrium","scope":"Biology/population-genetics/wright-s-island-model","status":"fully_proved","summary":"For demes of `N` diploids (`M = 2N` gene copies) and migration fraction `m = \u03ba / (4N)`, so that `4 N m = \u03ba` is held fixed, the equilibrium `F*` tends to `1 / (1 + \u03ba)` as `N \u2192 \u221e`.","title":"Wright's `F_ST = 1 / (1 + 4 N m)` (Wright 1931)"},{"data_url":"scopes/Biology/population-genetics/wright-fisher-markov-chain.json","id":"Biology/population-genetics/wright-fisher-markov-chain","kind":"scope","navigation":"module","parent":44675,"result":false,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","title":"Wright\u2013Fisher Markov chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/wright-fisher","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.wrightFisher","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#wright-fisher","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"The neutral Wright\u2013Fisher transition matrix on `M` gene copies (Ewens 2004, ch. 3): `P(i, j) = (M choose j) (i/M)^j (1 - i/M)^(M - j)`, the binomial law `Bin(M, i/M)` of the number of `A` copies among `M` draws with replacement. The model needs `M \u2265 1`; for `M = 0` (where `i / M` is Lean's junk value `0`) the matrix is the one-state chain, which is `Bin(0, p) = \u03b4\u2080` for every `p`.","title":"Neutral Wright\u2013Fisher transition matrix"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/coe-z","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.coe_z","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#coe-z","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"The frequency of allele `A` in state `i` is `bernstein.z i = i / M` (for `M = 0` it is Lean's junk value `i / 0 = 0`).","title":"Allele frequency of a Wright\u2013Fisher state"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/wright-fisher-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.wrightFisher_prob","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#wright-fisher-prob","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Entries of the Wright\u2013Fisher matrix.","title":"Entries of the Wright\u2013Fisher matrix"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/heterozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.heterozygosity","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#heterozygosity","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"The heterozygosity `H(j) = 2 (j/M) (1 - j/M)` of state `j`: the probability that two gene copies drawn with replacement carry different alleles.","title":"Heterozygosity of a Wright\u2013Fisher state"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/wright-fisher-prob-eq-binomial","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.wrightFisher_prob_eq_binomial","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#wright-fisher-prob-eq-binomial","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"The rows of the Wright\u2013Fisher matrix are Mathlib's binomial laws `Bin(M, i/M)`.","title":"Wright\u2013Fisher rows are binomial laws"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/sum-prob-mul-wright-fisher","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.sum_prob_mul_wrightFisher","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#sum-prob-mul-wright-fisher","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Martingale property of the allele count (one step): `\u2211 j, P(i, j) j = i`.","title":"One-step martingale property of the Wright\u2013Fisher allele count"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/sum-mat-pow-mul-wright-fisher","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.sum_matPow_mul_wrightFisher","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#sum-mat-pow-mul-wright-fisher","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Martingale property of the allele count (`t` steps): `\u2211 j, P^t(i, j) j = i`.","title":"Multi-step martingale property of the Wright\u2013Fisher allele count"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/wright-fisher-prob-zero-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.wrightFisher_prob_zero_zero","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#wright-fisher-prob-zero-zero","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"The state `0` (allele `A` lost) is absorbing.","title":"Loss of an allele is absorbing in the Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/wright-fisher-prob-last-last","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.wrightFisher_prob_last_last","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#wright-fisher-prob-last-last","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"The state `M` (allele `A` fixed) is absorbing.","title":"Fixation of an allele is absorbing in the Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/sum-prob-mul-heterozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.sum_prob_mul_heterozygosity","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#sum-prob-mul-heterozygosity","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Heterozygosity decays by the factor `1 - 1/M` per generation (one step). For `M = 0` both sides are `0` because `H \u2261 0`.","title":"One-step decay of heterozygosity under Wright\u2013Fisher drift"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/sum-mat-pow-mul-heterozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.sum_matPow_mul_heterozygosity","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#sum-mat-pow-mul-heterozygosity","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Heterozygosity decays geometrically (Ewens 2004, ch. 3): `\u2211 j, P^t(i, j) H(j) = (1 - 1/M)^t H(i)`. For `M = 0` both sides are `0` because `H \u2261 0` (the factor `1 - 1/0 = 1` is then irrelevant).","title":"Geometric decay of heterozygosity under Wright\u2013Fisher drift"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/tendsto-mat-pow-interior","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.tendsto_matPow_interior","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#tendsto-mat-pow-interior","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"The probability of sitting in an interior state `0 < j < M` at time `t` tends to `0`.","title":"Interior Wright\u2013Fisher states vanish in the limit"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/tendsto-mat-pow-last","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.tendsto_matPow_last","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#tendsto-mat-pow-last","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Fixation probability (Ewens 2004, ch. 3): started from `i` copies, allele `A` is fixed with limiting probability `i / M`, i.e. `P^t(i, M) \u2192 i / M`.","title":"Fixation probability in the Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/tendsto-mat-pow-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.tendsto_matPow_zero","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#tendsto-mat-pow-zero","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Loss probability: started from `i` copies, allele `A` is lost with limiting probability `1 - i / M`, i.e. `P^t(i, 0) \u2192 1 - i / M`.","title":"Limiting loss probability of the Wright\u2013Fisher transition matrix"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/tendsto-measure-real-eq-last","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.tendsto_measureReal_eq_last","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#tendsto-measure-real-eq-last","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Path-level fixation probability: if `X` is a Wright\u2013Fisher chain (in the sense of `IsFiniteMarkovChain`) started from `i` copies, then `P(X_t = M) \u2192 i / M`.","title":"Limiting fixation probability of a Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/measure-exists-absorbed","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.measure_exists_absorbed","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#measure-exists-absorbed","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Almost-sure absorption: a Wright\u2013Fisher chain is almost surely eventually absorbed in `0` or `M` (Ewens 2004, ch. 3).","title":"Almost-sure absorption of the Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/measure-real-exists-eq-last","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.measureReal_exists_eq_last","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#measure-real-exists-eq-last","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Path-level fixation probability (Ewens 2004, ch. 3): a Wright\u2013Fisher chain started from `i` copies ever reaches `M` (fixation of `A`) with probability `i / M`.","title":"Fixation probability of a Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/measure-real-exists-eq-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.measureReal_exists_eq_zero","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#measure-real-exists-eq-zero","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Path-level loss probability: a Wright\u2013Fisher chain started from `i` copies ever reaches `0` (loss of `A`) with probability `1 - i / M`.","title":"Loss probability of a Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/measure-markov-chain-measure-exists-abso","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.measure_markovChainMeasure_exists_absorbed","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#measure-markov-chain-measure-exists-abso","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"The finite Wright\u2013Fisher chain under the canonical Markov law almost surely reaches an endpoint. The law and chain are not premises supplied by a caller.","title":"Finite Wright\u2013Fisher chain is absorbed almost surely"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/measure-real-markov-chain-measure-exists","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.measureReal_markovChainMeasure_exists_eq_last","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#measure-real-markov-chain-measure-exists","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Under the canonical finite Wright\u2013Fisher law, the probability of ever fixing at `M` equals the initial allele frequency `i / M`.","title":"Fixation probability of the finite Wright\u2013Fisher chain"},{"id":"Biology/population-genetics/wright-fisher-markov-chain/measure-real-markov-chain-measure-exists-2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.WrightFisherChain.measureReal_markovChainMeasure_exists_eq_zero","parent":45120,"result":true,"route":"../roadmap/Biology/population-genetics/wright-fisher-markov-chain/index.html#measure-real-markov-chain-measure-exists-2","scope":"Biology/population-genetics/wright-fisher-markov-chain","status":"fully_proved","summary":"Under the canonical finite Wright\u2013Fisher law, the probability of ever losing the allele at `0` equals `1 - i / M`.","title":"Loss probability of an allele in the finite Wright\u2013Fisher chain"},{"data_url":"scopes/Biology/coalescent-theory.json","id":"Biology/coalescent-theory","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/coalescent-theory/index.html","scope":"Biology/coalescent-theory","status":"can_prove","summary":"Backward-in-time models of the genealogy of a sample of genes, from Kingman's coalescent and its generalizations to the site frequency spectrum, recombination and population structure.","title":"Coalescent theory"},{"data_url":"scopes/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events.json","id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","title":"Hudson\u2013Kaplan bound on recombination events"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/four-gametes","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.FourGametes","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#four-gametes","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"Sites `i` and `j` show all four gametes: for every pair of alleles `(a, b)` some haplotype carries `a` at `i` and `b` at `j`. This reuses `Cslib.Bioinformatics.Phylogenetics.Splits.FourGametes` applied to the site columns `(M \u00b7 i)` and `(M \u00b7 j)`; it is not a second definition of the four-gamete predicate.","title":"Four-gamete condition for two sites"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/carriers","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.carriers","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#carriers","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"The haplotypes carrying allele `a` at site `i`; when `a` is the derived allele this is the clade defined by the mutation at `i`.","title":"Haplotypes carrying an allele at a site"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/separates","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.Separates","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#separates","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"A breakpoint `g` (a cut immediately to the left of site `g`) separates sites `i` and `j` when `i < g \u2264 j`.","title":"Breakpoint separating two sites"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/is-breakpoint-set","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.IsBreakpointSet","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#is-breakpoint-set","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"A set `B` of breakpoints is valid for the sample `M` when it separates every pair of sites `i < j` showing all four gametes.","title":"Valid set of recombination breakpoints"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/is-disjoint-family","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.IsDisjointFamily","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#is-disjoint-family","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"A family `F` of site pairs `(i, j)` consists of four-gamete intervals (`i < j`, all four gametes at `i` and `j`) that are pairwise disjoint: of two distinct intervals, one ends at or before the start of the other, so no breakpoint separates both.","title":"Disjoint family of four-gamete intervals"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/sweep-step","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.sweepStep","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#sweep-step","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"One step of the sweep at right endpoint `j`. The list `acc` holds the intervals kept so far, most recent first; the next interval must start at or after the right end of the most recent one (at or after site `0` if none was kept). Among the admissible four-gamete intervals ending at `j`, the shortest one is kept.","title":"One step of the Hudson\u2013Kaplan sweep"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/sweep","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.sweep","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#sweep","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"The intervals kept by Hudson and Kaplan's left-to-right sweep, most recent first.","title":"Hudson\u2013Kaplan left-to-right sweep"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/rm","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.rm","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#rm","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"The number of intervals kept by the sweep.","title":"Hudson and Kaplan's `R_m`"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/nodup-sweep","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.nodup_sweep","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#nodup-sweep","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"The intervals kept by the sweep are distinct.","title":"Intervals kept by the Hudson\u2013Kaplan sweep are distinct"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/is-disjoint-family-sweep","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.isDisjointFamily_sweep","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#is-disjoint-family-sweep","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"The intervals kept by the sweep are pairwise disjoint four-gamete intervals.","title":"Hudson\u2013Kaplan sweep yields disjoint four-gamete intervals"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/is-breakpoint-set-sweep","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.isBreakpointSet_sweep","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#is-breakpoint-set-sweep","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"Cutting immediately to the left of the right endpoint of every kept interval gives a valid breakpoint set.","title":"Sweep cuts form a valid breakpoint set"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/card-sweep-snd","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.card_sweep_snd","parent":45143,"reading_title":{"format":"inline-parts-v1","parts":[["text","The sweep yields exactly "],["math","\\(R_m\\)"],["text"," distinct breakpoints"]],"source":"The sweep yields exactly $R_m$ distinct breakpoints"},"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#card-sweep-snd","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"The right endpoints of the kept intervals are distinct, so there are `R_m` of them.","title":"The sweep yields exactly $R_m$ distinct breakpoints"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/card-le-card-of-is-breakpoint-set","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.card_le_card_of_isBreakpointSet","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#card-le-card-of-is-breakpoint-set","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"Validity of the bound (Hudson\u2013Kaplan 1985): a valid breakpoint set has at least as many points as any family of pairwise disjoint four-gamete intervals, since each interval needs its own breakpoint.","title":"Validity of the bound"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/rm-le-card-of-is-breakpoint-set","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.rm_le_card_of_isBreakpointSet","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#rm-le-card-of-is-breakpoint-set","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"`R_m` is a lower bound (Hudson\u2013Kaplan 1985): every valid breakpoint set has at least `R_m` points.","title":"`R_m` is a lower bound"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/exists-is-breakpoint-set-card-eq-rm","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.exists_isBreakpointSet_card_eq_rm","parent":45143,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(R_m\\)"],["text"," is the minimum size of a valid breakpoint set"]],"source":"$R_m$ is the minimum size of a valid breakpoint set"},"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#exists-is-breakpoint-set-card-eq-rm","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"The bound is attained: some valid breakpoint set has exactly `R_m` points, so `R_m` is the minimum size of a valid breakpoint set.","title":"$R_m$ is the minimum size of a valid breakpoint set"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/card-le-rm","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.card_le_rm","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#card-le-rm","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"No family of pairwise disjoint four-gamete intervals has more than `R_m` members.","title":"Greedy optimality"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/exists-is-disjoint-family-card-eq-rm","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.exists_isDisjointFamily_card_eq_rm","parent":45143,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(R_m\\)"],["text"," is the maximum number of disjoint four-gamete intervals"]],"source":"$R_m$ is the maximum number of disjoint four-gamete intervals"},"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#exists-is-disjoint-family-card-eq-rm","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"The sweep attains the maximum: `R_m` is the largest size of a family of pairwise disjoint four-gamete intervals.","title":"$R_m$ is the maximum number of disjoint four-gamete intervals"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/not-four-gametes-iff","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.not_fourGametes_iff","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#not-four-gametes-iff","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"Two sites miss one of the four gametes exactly when, for some choice of the alleles `a` at `i` and `b` at `j` (the derived alleles), the carrier sets are nested or disjoint, as two clades of one rooted tree are.","title":"Characterization of site pairs failing the four-gamete test"},{"id":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/rm-le-card-of-nested-or-disjoint","kind":"node","lean":"Cslib.Bioinformatics.PopulationGenetics.HudsonKaplan.rm_le_card_of_nested_or_disjoint","parent":45143,"result":true,"route":"../roadmap/Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events/index.html#rm-le-card-of-nested-or-disjoint","scope":"Biology/coalescent-theory/hudson-kaplan-bound-on-recombination-events","status":"fully_proved","summary":"Suppose the breakpoints `B` cut the sequence into blocks such that, within a block, the derived-allele clades of every two sites are nested or disjoint for some choice of the derived alleles (as for sites whose mutations sit on one tree under the infinite-sites model). Then there are at least `R_m` breakpoints.","title":"Hudson and Kaplan's bound for tree-like blocks"},{"data_url":"scopes/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator.json","id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","title":"Kingman's coalescent and Watterson's estimator"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/sum-inv-choose-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.sum_inv_choose_two","parent":45163,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of reciprocals of binomial coefficients "],["math","\\(\\binom{k}{2}\\)"]],"source":"Sum of reciprocals of binomial coefficients $\\binom{k}{2}$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#sum-inv-choose-two","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"`\u2211_{k=2}^n (k.choose 2)\u207b\u00b9 = 2 (1 - 1/n)` for `n \u2265 1` (telescoping `2/(k(k-1))`).","title":"Sum of reciprocals of binomial coefficients $\\binom{k}{2}$"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/sum-mul-inv-choose-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.sum_mul_inv_choose_two","parent":45163,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted sum of reciprocals of "],["math","\\(\\binom{k}{2}\\)"],["text"," is twice a harmonic number"]],"source":"Weighted sum of reciprocals of $\\binom{k}{2}$ is twice a harmonic number"},"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#sum-mul-inv-choose-two","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"`\u2211_{k=2}^n k (k.choose 2)\u207b\u00b9 = 2 H_{n-1}`.","title":"Weighted sum of reciprocals of $\\binom{k}{2}$ is twice a harmonic number"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/kingman-times","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#kingman-times","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"Kingman's coalescent for a sample of size `n`, through its holding times: `T k` is the time (in units of `2N` generations) during which the genealogy has exactly `k` lineages, for `2 \u2264 k \u2264 n`. These are independent, and `T k` is exponential with rate `k.choose 2`. The values of `T k` for `k \u2209 [2, n]` are irrelevant.","title":"Holding times of Kingman's coalescent"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/tmrca","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.tmrca","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#tmrca","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"Time to the most recent common ancestor of the sample.","title":"Time to the most recent common ancestor"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/total-length","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.totalLength","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#total-length","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"Total branch length of the sample genealogy: `k` branches during `T k`.","title":"Total branch length of a coalescent genealogy"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/choose-two-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.choose_two_pos","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#choose-two-pos","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"The number of pairs `k.choose 2` is positive for `k \u2208 [2, n]`.","title":"Positivity of the number of pairs"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/integrable-t","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integrable_T","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#integrable-t","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"Each holding time is integrable.","title":"Integrability of coalescent holding times"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/integral-t","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integral_T","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#integral-t","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"`E[T k] = (k.choose 2)\u207b\u00b9`.","title":"Expected coalescent holding time"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/integral-tmrca","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integral_tmrca","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#integral-tmrca","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"The expected time to the most recent common ancestor is `2 (1 - 1/n)`.","title":"Expected time to the most recent common ancestor"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/integrable-total-length","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integrable_totalLength","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#integrable-total-length","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"The total branch length is integrable.","title":"Integrability of the total branch length"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/integral-total-length","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integral_totalLength","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#integral-total-length","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"The expected total branch length is `2 H_{n-1}`.","title":"Expected total branch length of the coalescent"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/variance-tmrca","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.variance_tmrca","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#variance-tmrca","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"The variance of the time to the most recent common ancestor is `\u2211_{k=2}^n ((k.choose 2)\u207b\u00b9)\u00b2` (this uses independence).","title":"Variance of the time to the most recent common ancestor"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/integral-seg-sites","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integral_segSites","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#integral-seg-sites","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"Infinite-sites mutation at rate `\u03b8/2` per lineage per unit time: if the number of segregating sites `S` is integrable and has conditional mean `(\u03b8/2) L` given the genealogy \u03c3-algebra `m`, then `E S = \u03b8 H_{n-1}`. (Integrability matters: Mathlib's conditional expectation of a non-integrable function is `0`.)","title":"Expected number of segregating sites"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/integrable-watterson","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integrable_watterson","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#integrable-watterson","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"Watterson's estimator `S / H_{n-1}` is integrable when `S` is.","title":"Integrability of Watterson's estimator"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/watterson-unbiased","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.watterson_unbiased","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#watterson-unbiased","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"`E[S / H_{n-1}] = \u03b8` for a sample of size `n \u2265 2`.","title":"Watterson's estimator is unbiased"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/pi-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.piMeasure","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#pi-measure","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"The canonical probability space for the Kingman holding times of a sample of size `n`: the product of the exponential laws `Exp (k.choose 2)`, `2 \u2264 k \u2264 n`.","title":"Canonical probability space for Kingman holding times"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/of-pi","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.ofPi","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#of-pi","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"The coordinate projections of `piMeasure n` form a Kingman coalescent; in particular `KingmanTimes` is inhabited for every sample size.","title":"Kingman coalescent from coordinate projections"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/no-coal-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.noCoalProb","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#no-coal-prob","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"In a Wright\u2013Fisher population of `M` gene copies, the probability that `k` lineages, choosing their parents independently and uniformly, have `k` distinct parents.","title":"Probability of no coalescence in one Wright\u2013Fisher generation"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/no-coal-prob-eq-desc-factorial-div","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.noCoalProb_eq_descFactorial_div","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#no-coal-prob-eq-desc-factorial-div","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"`noCoalProb M k` is the falling factorial `M (M-1) \u22ef (M-k+1)` divided by `M ^ k`.","title":"No-coalescence probability as a falling factorial ratio"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/tendsto-no-coal-prob-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tendsto_noCoalProb_one","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#tendsto-no-coal-prob-one","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"`noCoalProb M k \u2192 1` as `M \u2192 \u221e`.","title":"No-coalescence probability tends to one"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/tendsto-coalescence-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tendsto_coalescence_rate","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#tendsto-coalescence-rate","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"For `M` gene copies (`M = 2N`), the per-generation probability that some of `k` lineages coalesce is `(k.choose 2) / M + o(1/M)`; i.e. in time units of `M` generations, coalescences occur at rate `k.choose 2`.","title":"Kingman's coalescence rate"},{"id":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/tendsto-no-coal-prob-pow-floor","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tendsto_noCoalProb_pow_floor","parent":45163,"result":true,"route":"../roadmap/Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator/index.html#tendsto-no-coal-prob-pow-floor","scope":"Biology/coalescent-theory/kingman-s-coalescent-and-watterson-s-estimator","status":"fully_proved","summary":"After `\u230at M\u230b` Wright\u2013Fisher generations (`t \u2265 0` coalescent time units), the probability that `k` lineages have not coalesced tends to `exp (-(k.choose 2) t)`, the survival function of the exponential law with rate `k.choose 2`.","title":"Wright\u2013Fisher coalescence times converge to exponential laws"},{"data_url":"scopes/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula.json","id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","title":"Ancestral lineage counting and Tavar\u00e9's formula"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/wf-lineage-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.wfLineageProb","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#wf-lineage-prob","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"One-generation transition probability of the ancestral lineage count in a neutral Wright\u2013Fisher population of `M` gene copies: `k` lineages pick their parents uniformly and independently among `M` copies, and `j` distinct parents occur with probability `stirlingSecond k j * M.descFactorial j / M ^ k`.","title":"Wright\u2013Fisher transition probability of the lineage count"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/wf-lineage-matrix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.wfLineageMatrix","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#wf-lineage-matrix","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The one-generation transition matrix of the ancestral lineage count of a sample of at most `n` gene copies in a Wright\u2013Fisher population of `M` copies; row `k`, column `j` is `wfLineageProb M k j`.","title":"Wright\u2013Fisher transition matrix of the lineage count"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/block-counting-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.blockCountingGenerator","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#block-counting-generator","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The rate matrix of Kingman's block-counting (pure-death) process on `{0, \u2026, n}`: `k` lineages become `k - 1` at rate `k.choose 2`, time in units of `M = 2N` generations.","title":"Generator of Kingman's block-counting process"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/tavare-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.tavareProb","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#tavare-prob","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"Tavar\u00e9's closed form for the probability that `i` lineages of Kingman's coalescent have `j` ancestors `t` time units (`2N` generations each) earlier, for `1 \u2264 j \u2264 i`.","title":"Tavar\u00e9's formula for the number of ancestral lineages"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/tendsto-smul-wf-lineage-matrix-sub-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.tendsto_smul_wfLineageMatrix_sub_one","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#tendsto-smul-wf-lineage-matrix-sub-one","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The scaled one-generation increment `M \u2022 (wfLineageMatrix n M - 1)` converges entrywise to the block-counting generator: `k` lineages coalesce to `k - 1` with probability `(k.choose 2) / M + o(1/M)`, and to fewer with probability `O(1/M\u00b2)`.","title":"Scaled Wright\u2013Fisher lineage matrix converges to Kingman's generator"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/tendsto-wf-lineage-matrix-pow-floor","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.tendsto_wfLineageMatrix_pow_floor","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#tendsto-wf-lineage-matrix-pow-floor","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"For `t \u2265 0`, the `\u230at M\u230b`-generation transition matrix of the Wright\u2013Fisher ancestral lineage count converges, as the population size `M` tends to infinity, to the transition matrix `exp (t \u2022 G)` of the pure-death process with rates `k.choose 2`.","title":"Kingman's theorem for the lineage count"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/block-counting-right-eigen","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.blockCountingRightEigen","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#block-counting-right-eigen","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"Right eigenvectors of the block-counting generator, as columns: column `k` has entries `i_{(k)} / i^{(k)}` (falling over rising factorial) and eigenvalue `-(k.choose 2)`.","title":"Right eigenvectors of Kingman's block-counting generator"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/block-counting-left-eigen","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.blockCountingLeftEigen","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#block-counting-left-eigen","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"Left eigenvectors of the block-counting generator, as rows: row `0` is the indicator of state `0`, and for `1 \u2264 k` row `k` has entries `(2k-1) (-1)^{k-j} j^{(k-1)} / (j! (k-j)!)` for `j \u2264 k` and `0` for `j > k`; the eigenvalue is `-(k.choose 2)`.","title":"Left eigenvectors of Kingman's block-counting generator"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/block-counting-eigenvalue","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.blockCountingEigenvalue","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#block-counting-eigenvalue","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The eigenvalues of the block-counting generator, `-(k.choose 2)`.","title":"Eigenvalues of Kingman's block-counting generator"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/block-counting-generator-mul-right-eigen","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.blockCountingGenerator_mul_rightEigen","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#block-counting-generator-mul-right-eigen","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The columns of `blockCountingRightEigen n` are right eigenvectors: `G V = V D`.","title":"Right eigenvector equation for the block-counting generator"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/block-counting-left-eigen-mul-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.blockCountingLeftEigen_mul_generator","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#block-counting-left-eigen-mul-generator","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The rows of `blockCountingLeftEigen n` are left eigenvectors: `W G = D W`.","title":"Left eigenvector equation for the block-counting generator"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/block-counting-left-eigen-mul-right-eige","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.blockCountingLeftEigen_mul_rightEigen","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#block-counting-left-eigen-mul-right-eige","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The left and right eigenvector matrices are mutually inverse: `W V = 1`.","title":"Left and right eigenvector matrices are mutually inverse"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/exp-smul-block-counting-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.exp_smul_blockCountingGenerator","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#exp-smul-block-counting-generator","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The spectral decomposition of the transition matrix of the block-counting process: `exp (t \u2022 G) = V diag(exp (-(k.choose 2) t)) W`.","title":"Spectral decomposition of the block-counting transition matrix"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/exp-block-counting-generator-apply","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.exp_blockCountingGenerator_apply","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#exp-block-counting-generator-apply","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The transition probabilities of Kingman's block-counting process from `i` to `j` lineages in time `t`, for `1 \u2264 j \u2264 i \u2264 n`, are given by `tavareProb`.","title":"Tavar\u00e9's formula"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/exp-smul-block-counting-generator-cast-l","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.exp_smul_blockCountingGenerator_castLE","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#exp-smul-block-counting-generator-cast-l","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"For `n \u2264 N`, the top-left `(n + 1) \u00d7 (n + 1)` block of the transition matrix `exp (t \u2022 G)` of the block-counting process on `{0, \u2026, N}` is the transition matrix of the process on `{0, \u2026, n}` (the process only moves down). Holds for every real `t`.","title":"Truncation consistency"},{"id":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/exp-smul-block-counting-generator-of-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanBlockCounting.exp_smul_blockCountingGenerator_of_lt","parent":45186,"result":true,"route":"../roadmap/Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula/index.html#exp-smul-block-counting-generator-of-lt","scope":"Biology/coalescent-theory/ancestral-lineage-counting-and-tavar-s-formula","status":"fully_proved","summary":"The block-counting process never gains lineages: `exp (t \u2022 G) i j = 0` for `i < j`. Holds for every real `t`.","title":"The block-counting process never gains lineages"},{"data_url":"scopes/Biology/coalescent-theory/expected-site-frequency-spectrum.json","id":"Biology/coalescent-theory/expected-site-frequency-spectrum","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","title":"Expected site frequency spectrum"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/merge-blocks","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.mergeBlocks","parent":45203,"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#merge-blocks","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"Merge the blocks at positions `i < j` of the block-size list `b`: remove both and append a block of size `b[i] + b[j]`.","title":"Merging two blocks of a block-size list"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/block-pairs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.blockPairs","parent":45203,"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#block-pairs","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"The unordered pairs of positions `i < j` among `k` blocks.","title":"Unordered pairs of blocks"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/random-joining","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.randomJoining","parent":45203,"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#random-joining","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"The random-joining law: `randomJoining m b ps` is the probability that `m` steps of random joining started from the block-size list `b` visit the list of block-size lists `ps` (starting with `b` itself). Each step merges a pair of blocks chosen uniformly among all pairs; the process stops once fewer than two blocks remain.","title":"Random-joining law on block-size lists"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/join-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.joinStep","parent":45203,"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#join-step","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"The transition operator of random joining acting on a statistic `g` of block-size lists: the average of `g` over the lists obtained from `b` by merging a uniformly chosen pair of blocks (`g []` once fewer than two blocks remain). By `exists_finset_randomJoining`, its `j`-th iterate at `b` is the expectation of `g` at step `j` of random joining started from `b`.","title":"Transition operator of random joining"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/sfs-length","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.sfsLength","parent":45203,"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#sfs-length","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"The total length of the branches subtending exactly `i` of the `n` sample copies, for the block-size history `ps` (entry `n - k` lists the block sizes while there are `k` lineages) and holding times `T`: `\u2211_{k=2}^{n} #{blocks of size i among k} \u00b7 T k`.","title":"Total branch length subtending i sample copies"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/exists-finset-random-joining","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.exists_finset_randomJoining","parent":45203,"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#exists-finset-random-joining","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"The random-joining law has finite support, and the expectation of a statistic of the `j`-th visited list (`j \u2264 m`) is the `j`-th iterate of `joinStep`.","title":"Random joining has finite support and iterates the join step"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/join-step-iterate-count","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.joinStep_iterate_count","parent":45203,"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#join-step-iterate-count","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"Expected number of blocks of size `i` when `k` blocks remain, for random joining started from `n` singletons: `k \u00b7 C(n-i-1, k-2) / C(n-1, k-1)` (Fu 1995; Griffiths\u2013Tavar\u00e9 1998). Equivalently, a given one of the `k` blocks has size `i` with probability `C(n-i-1, k-2) / C(n-1, k-1)`.","title":"Expected number of blocks of size `i`"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/sum-block-count-mul-inv-choose-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.sum_blockCount_mul_inv_choose_two","parent":45203,"reading_title":{"format":"inline-parts-v1","parts":[["text","Expected block counts weighted by coalescence times sum to "],["math","\\(2/i\\)"]],"source":"Expected block counts weighted by coalescence times sum to $2/i$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#sum-block-count-mul-inv-choose-two","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"Summing the expected block counts against `E T_k = (k.choose 2)\u207b\u00b9` gives `2 / i`: `\u2211_{k=2}^{n} k C(n-i-1, k-2) / C(n-1, k-1) \u00b7 (k.choose 2)\u207b\u00b9 = 2 / i` for `1 \u2264 i < n`.","title":"Expected block counts weighted by coalescence times sum to $2/i$"},{"id":"Biology/coalescent-theory/expected-site-frequency-spectrum/integral-sfs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SiteFrequencySpectrum.integral_sfs","parent":45203,"result":true,"route":"../roadmap/Biology/coalescent-theory/expected-site-frequency-spectrum/index.html#integral-sfs","scope":"Biology/coalescent-theory/expected-site-frequency-spectrum","status":"fully_proved","summary":"Expected site frequency spectrum (Fu 1995). Let the genealogy of a sample of `n` gene copies consist of Kingman holding times `K.T` and an independent block-size history `\u03c4` with the random-joining law. If, given the genealogy (the \u03c3-algebra generated by `\u03c4` and the holding times), the number `\u03be i` of sites whose derived allele is carried by exactly `i` copies is integrable and has conditional mean `(\u03b8 / 2) L_i`\u2026","title":"Expected site frequency spectrum"},{"data_url":"scopes/Biology/coalescent-theory/partition-valued-kingman-coalescent.json","id":"Biology/coalescent-theory/partition-valued-kingman-coalescent","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"can_prove","title":"Partition-valued Kingman coalescent"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/sample-partition","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.SamplePartition","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#sample-partition","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The set partitions of a sample of `n` labeled genes `{0, \u2026, n - 1}`.","title":"Set partitions of a labeled sample"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/is-merge","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.IsMerge","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#is-merge","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"`\u03c0'` arises from `\u03c0` by merging two blocks: `\u03c0` refines `\u03c0'` and has exactly one more block (so exactly two blocks of `\u03c0` lie in one block of `\u03c0'`, and the others are blocks of `\u03c0'`).","title":"Partition obtained by merging two blocks"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/jump-matrix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.jumpMatrix","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#jump-matrix","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The transition matrix of the jump chain of Kingman's coalescent: from a partition with `k \u2265 2` blocks, move to each of the `k.choose 2` partitions obtained by merging two blocks with probability `1 / k.choose 2`; a partition with at most one block is absorbing.","title":"Jump chain of Kingman's coalescent"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/jump-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.jumpLaw","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#jump-law","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The law of the jump chain after `m` jumps, started from the partition into singletons: `jumpLaw n m \u03c0` is the probability that the partition after `m` coalescences is `\u03c0`.","title":"Law of Kingman's jump chain after m jumps"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/kingman-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.kingmanProb","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#kingman-prob","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The probability `(n - k)! k! (k - 1)! \u220f_{B \u2208 \u03c0} |B|! / (n! (n - 1)!)` of a partition `\u03c0` with `k` blocks of a sample of size `n` (meaningful for `1 \u2264 k \u2264 n`; for the empty sample it gives `1` to the empty partition).","title":"Kingman's formula"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/partition-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.partitionRate","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#partition-rate","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The rate matrix of Kingman's partition-valued coalescent: each pair of blocks merges at rate `1`, so a partition with `k` blocks is left at total rate `k.choose 2` (time in units of `2N` generations).","title":"Rate matrix of Kingman's partition-valued coalescent"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/time-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.timeLaw","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#time-law","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The time-`t` marginal of Kingman's `n`-coalescent started from singletons: `timeLaw n t \u03c0 = exp (t \u2022 partitionRate n) \u22a5 \u03c0` is the probability that the ancestral partition of the sample at time `t \u2265 0` (in units of `2N` generations) is `\u03c0`. The matrix exponential is defined for every real `t`; the identities below hold for every real `t`.","title":"Time-t marginal of Kingman's coalescent"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/restrict-matrix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.restrictMatrix","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#restrict-matrix","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The restriction map as a `0/1` matrix: entry `(\u03c0, \u03c1)` is `1` iff `\u03c0.comap Fin.castSucc = \u03c1`.","title":"Restriction matrix on sample partitions"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/card-is-merge","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.card_isMerge","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#card-is-merge","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"A partition with `k` blocks has exactly `k.choose 2` partitions that arise from it by merging two blocks.","title":"Mergers"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/two-mul-sum-prod-factorial-is-merge","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.two_mul_sum_prod_factorial_isMerge","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#two-mul-sum-prod-factorial-is-merge","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Summing `\u220f_{B \u2208 \u03c0} |B|!` over the partitions `\u03c0` that merge to `\u03c0'` gives `(n - #\u03c0'.parts) / 2 \u00b7 \u220f_{B \u2208 \u03c0'} |B|!`: a block of size `b` has `(2^b - 2) / 2` splits into two blocks, and `\u2211_{\u2205 \u2260 A \u228a B} |A|! |B \\ A|! = (b - 1) b!`.","title":"Splitting identity"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/kingman-prob-bot","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.kingmanProb_bot","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#kingman-prob-bot","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Kingman's formula gives probability `1` to the partition into singletons.","title":"Kingman's formula at the partition into singletons"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/kingman-prob-of-card","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.kingmanProb_of_card","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#kingman-prob-of-card","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Kingman's formula for a partition with `k` blocks.","title":"Kingman's formula for a partition with k blocks"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/sum-kingman-prob-is-merge","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.sum_kingmanProb_isMerge","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#sum-kingman-prob-is-merge","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"For a partition `\u03c0'` with `k` blocks, `1 \u2264 k < n`, the Kingman probabilities of the partitions merging to `\u03c0'` sum to `(k + 1).choose 2` times the Kingman probability of `\u03c0'`.","title":"Backward equation of Kingman's formula"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/jump-matrix-mem-row-stochastic","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.jumpMatrix_mem_rowStochastic","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#jump-matrix-mem-row-stochastic","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The jump-chain transition matrix is row-stochastic.","title":"Kingman's jump-chain matrix is row-stochastic"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/jump-law-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.jumpLaw_eq","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#jump-law-eq","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Kingman's formula (Kingman 1982; Durrett 2008, ch. 1). For `m < n`, after `m` coalescences the jump chain started from singletons is at a partition `\u03c0` with probability `kingmanProb n \u03c0 = (n - k)! k! (k - 1)! \u220f_{B \u2208 \u03c0} |B|! / (n! (n - 1)!)` if `\u03c0` has `k = n - m` blocks, and `0` otherwise. (For `m \u2265 n - 1` the chain has been absorbed at the one-block partition.)","title":"Kingman's formula"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/sum-jump-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.sum_jumpLaw","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#sum-jump-law","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The jump chain's law after `m` jumps sums to one (its entries are nonnegative by `jumpMatrix_mem_rowStochastic`).","title":"Jump-chain law sums to one"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/sum-kingman-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.sum_kingmanProb","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#sum-kingman-prob","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Kingman's formula is a probability distribution on the partitions with `k` blocks, for `1 \u2264 k \u2264 n`. Equivalently, `\u2211_{\u03c0, k blocks} \u220f_{B \u2208 \u03c0} |B|! = n! (n - 1)! / ((n - k)! k! (k - 1)!)`, the Lah number `L(n, k)`.","title":"Kingman's formula is a probability distribution"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/kingman-prob-comap-perm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.kingmanProb_comap_perm","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#kingman-prob-comap-perm","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Relabeling the sample by a permutation (`Finpartition.comap`) preserves Kingman's formula.","title":"Kingman's formula is invariant under relabeling"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/jump-law-comap-perm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.jumpLaw_comap_perm","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#jump-law-comap-perm","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Exchangeability of the jump chain: for `m < n`, relabeling the sample does not change the probability of a partition after `m` coalescences.","title":"Exchangeability"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/num-blocks","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.numBlocks","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#num-blocks","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The number of blocks of a partition of a sample of `n`, as an element of `Fin (n + 1)` (a state of the block-counting process).","title":"Number of blocks of a sample partition"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/kingman-matrix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.kingmanMatrix","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#kingman-matrix","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Kingman's formula arranged by number of blocks: entry `(k, \u03c0)` is `kingmanProb n \u03c0` if `\u03c0` has `k` blocks and `0` otherwise. For `1 \u2264 k \u2264 n`, row `k` is the law of the jump chain after `n - k` jumps (`jumpLaw_eq`).","title":"Kingman's formula arranged by number of blocks"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/kingman-matrix-last","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.kingmanMatrix_last","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#kingman-matrix-last","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Row `n` of `kingmanMatrix n` is the point mass at the partition into singletons.","title":"Last row of Kingman's matrix is the singleton point mass"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/block-counting-generator-mul-kingman-mat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.blockCountingGenerator_mul_kingmanMatrix","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#block-counting-generator-mul-kingman-mat","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Kingman's formula intertwines the block-counting generator with the partition rate matrix: `G * \u039b = \u039b * Q`. This is the backward equation `sum_kingmanProb_isMerge` in matrix form.","title":"Kingman's formula intertwines the block-counting generator with the partition rate matrix"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/time-law-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.timeLaw_eq","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#time-law-eq","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The time-`t` marginal of the partition-valued coalescent (Kingman 1982): for every real `t`, the probability of a partition `\u03c0` with `k` blocks is the probability that the block-counting process goes from `n` to `k` lineages in time `t`, times Kingman's formula.","title":"The time-`t` marginal of the partition-valued coalescent"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/time-law-eq-tavare-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.timeLaw_eq_tavareProb","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#time-law-eq-tavare-prob","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The time-`t` marginal with Tavar\u00e9's closed form for the block-counting factor: for a partition `\u03c0` of a sample of `n` (so `\u03c0` has at least one block when `n \u2265 1`), `P(\u03a0_t = \u03c0) = tavareProb n k t \u00b7 kingmanProb n \u03c0` with `k = #\u03c0.parts`.","title":"Time-t marginal via Tavar\u00e9's formula"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/time-law-comap-perm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.timeLaw_comap_perm","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#time-law-comap-perm","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Exchangeability of the time-`t` marginal.","title":"Exchangeability"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/sum-partition-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.sum_partitionRate","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#sum-partition-rate","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Each row of the partition rate matrix sums to `0`: a partition with `k` blocks has `k.choose 2` mergers (`card_isMerge`), each at rate `1`.","title":"Rows of the partition rate matrix sum to zero"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/is-rate-matrix-partition-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.isRateMatrix_partitionRate","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#is-rate-matrix-partition-rate","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The partition rate matrix is a rate matrix: off-diagonal entries are `0` or `1`, and each row sums to `0` (`sum_partitionRate`).","title":"Partition rate matrix is a rate matrix"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/time-law-eq-ctmc-transition","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.timeLaw_eq_ctmcTransition","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#time-law-eq-ctmc-transition","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"`timeLaw` is a row of the transition function `P t = exp (t \u2022 partitionRate n)` of the continuous-time chain with rate matrix `partitionRate n`.","title":"Time-t marginal as a continuous-time chain transition"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/ctmc-transition-partition-rate-mem-row-s","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.ctmcTransition_partitionRate_mem_rowStochastic","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#ctmc-transition-partition-rate-mem-row-s","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"For `t \u2265 0` the transition matrix `exp (t \u2022 partitionRate n)` is row-stochastic.","title":"Partition coalescent transition matrix is row-stochastic"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/time-law-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.timeLaw_nonneg","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#time-law-nonneg","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"For `t \u2265 0` the time-`t` marginal is nonnegative; with `sum_timeLaw` it is a probability distribution on the partitions of the sample.","title":"Nonnegativity of the time-t marginal"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/sum-time-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.sum_timeLaw","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#sum-time-law","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The time-`t` marginal sums to one over all partitions: `\u2211_\u03c0 timeLaw n t \u03c0 = 1` (for every real `t`, since the rows of the rate matrix sum to `0`).","title":"Time-t marginal sums to one"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/partition-rate-mul-restrict-matrix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.partitionRate_mul_restrictMatrix","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#partition-rate-mul-restrict-matrix","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Lumpability of restriction (Kingman 1982): restricting to the first `n` genes intertwines the rate matrices of the `(n + 1)`- and `n`-coalescents.","title":"Lumpability of restriction"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/sum-time-law-restrict","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescent.sum_timeLaw_restrict","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#sum-time-law-restrict","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"Consistency (Kingman 1982): the time-`t` law of the restriction of the `(n + 1)`-coalescent to the first `n` genes is the time-`t` law of the `n`-coalescent.","title":"Consistency"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/path-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescentWanted.pathProb","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#path-prob","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"fully_proved","summary":"The probability that the jump chain of Kingman's `n`-coalescent, started from the partition into singletons, follows the list of partitions `l`: `0` unless `l` starts at the partition into singletons, and otherwise the product of the one-step transition probabilities `jumpMatrix n` along `l`. The empty list has probability `0`.","title":"Path probability of the Kingman coalescent jump chain"},{"id":"Biology/coalescent-theory/partition-valued-kingman-coalescent/sum-path-prob-restrict","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.PartitionCoalescentWanted.sum_pathProb_restrict","parent":45213,"result":true,"route":"../roadmap/Biology/coalescent-theory/partition-valued-kingman-coalescent/index.html#sum-path-prob-restrict","scope":"Biology/coalescent-theory/partition-valued-kingman-coalescent","status":"can_prove","summary":"Consistency of the jump chains (Kingman 1982): for `n \u2265 1`, restricting each partition of a path of the `(n + 1)`-jump chain (a list of `n + 1` partitions) to the first `n` genes and deleting consecutive repetitions gives a list of `n` partitions of `Fin n` distributed as the path of the `n`-jump chain.","title":"Consistency of the jump chains"},{"data_url":"scopes/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees.json","id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","title":"Ranked labeled histories of coalescent trees"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/lineage-pair","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LineagePair","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#lineage-pair","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"The unordered pairs of lineages when there are `k` lineages labeled `0, \u2026, k - 1`.","title":"Unordered pairs of lineages"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/ranked-history","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.RankedHistory","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#ranked-history","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"A ranked labeled history of `n` labeled genes, encoded by its coalescence events: at step `i < n - 1` there are `n - i` lineages and the step records the pair that merges (the merged lineage takes the smaller label; the others keep their relative order).","title":"Ranked labeled history"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/card-lineage-pair","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.card_lineagePair","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#card-lineage-pair","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"The number of pairs of `k` lineages is `k.choose 2`.","title":"Number of pairs of lineages"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/merge-blocks","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.mergeBlocks","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#merge-blocks","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"Merge the lineages (blocks of sampled genes) at positions `a < b` of a list of blocks: the merged block replaces position `a`, and position `b` is removed.","title":"Merging two lineage blocks"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/blocks-after","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.blocksAfter","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#blocks-after","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"The lineages (as blocks of sampled genes, indexed by lineage label) after the first `t` coalescence events of a history, starting from the singletons `{0}, \u2026, {n - 1}`.","title":"Lineage blocks after the first coalescence events"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/blocks-of","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.blocksOf","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#blocks-of","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"The ancestral partitions of the sample along a history, as lists of blocks indexed by lineage label; entry `t` (for `t < n`) is the partition after `t` coalescence events.","title":"Ancestral partitions along a history, indexed by lineage label"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/partitions-of","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.partitionsOf","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#partitions-of","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"The ancestral partitions of the sample along a history, as sets of blocks (forgetting the lineage labels): the ranked labeled history in the sense of Edwards (1970).","title":"Ancestral partitions along a history as sets of blocks"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/prod-choose-two-mul-two-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.prod_choose_two_mul_two_pow","parent":45250,"reading_title":{"format":"inline-parts-v1","parts":[["text","Product of pair counts times a power of two is "],["math","\\(n!(n-1)!\\)"]],"source":"Product of pair counts times a power of two is $n!(n-1)!$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#prod-choose-two-mul-two-pow","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"`\u220f_{k=2}^n C(k, 2) * 2^{n-1} = n! (n - 1)!`.","title":"Product of pair counts times a power of two is $n!(n-1)!$"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/prod-fin-choose-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.prod_fin_choose_two","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#prod-fin-choose-two","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"Reindexing: `\u220f_{i < n - 1} C(n - i, 2) = \u220f_{k=2}^n C(k, 2)`.","title":"Reindexing the product of pair counts"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/card-ranked-history","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.card_rankedHistory","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#card-ranked-history","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"`\u220f_{k=2}^n C(k, 2)` (Edwards 1970).","title":"Number of ranked labeled histories"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/card-ranked-history-mul-two-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.card_rankedHistory_mul_two_pow","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#card-ranked-history-mul-two-pow","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"`#histories * 2^{n-1} = n! (n - 1)!`, i.e. there are `n! (n - 1)! / 2^{n-1}` ranked labeled histories of `n` labeled genes (Edwards 1970).","title":"Number of ranked labeled histories"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/random-joining","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.randomJoining","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#random-joining","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"Random joining (Edwards 1970; Kingman 1982): the pair merging at each step is uniform among the current `(n - i).choose 2` pairs and independent of the earlier steps. Kingman (1982) shows that this is the law of the tree topology of the coalescent.","title":"Random joining"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/random-joining-singleton-eq-inv","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.randomJoining_singleton_eq_inv","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#random-joining-singleton-eq-inv","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"Under random joining each history has probability `\u220f_{k=2}^n C(k, 2)\u207b\u00b9`.","title":"Every ranked history is equally likely under random joining"},{"id":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/random-joining-singleton","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.randomJoining_singleton","parent":45250,"result":true,"route":"../roadmap/Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees/index.html#random-joining-singleton","scope":"Biology/coalescent-theory/ranked-labeled-histories-of-coalescent-trees","status":"fully_proved","summary":"Each ranked labeled history has probability `2^{n-1} / (n! (n - 1)!)` under random joining (Edwards 1970; Tajima 1983).","title":"Each ranked labeled history has probability `2^{n-1} / (n! (n - 1)!)`"},{"data_url":"scopes/Biology/coalescent-theory/multispecies-coalescent.json","id":"Biology/coalescent-theory/multispecies-coalescent","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","title":"Multispecies coalescent"},{"id":"Biology/coalescent-theory/multispecies-coalescent/species-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.SpeciesTree","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#species-tree","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"A rooted binary species tree on the species `Fin n` with branch lengths: `leaf i` is species `i`, and `node l r t` is the ancestral species of the species below `l` and `r`, where `t` is the length, in coalescent units (`2N` generations), of the branch above the node, i.e. the time during which the ancestral population of `l` and `r` exists before it merges with a sister population. The length stored at the root\u2026","title":"Rooted binary species tree with branch lengths"},{"id":"Biology/coalescent-theory/multispecies-coalescent/nonneg-lengths","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.SpeciesTree.NonnegLengths","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#nonneg-lengths","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Every branch length of the species tree, including the unused one stored at the root, is nonnegative. On such trees `geneTreeProb` is a probability law; for a negative length it is only a signed weight.","title":"Species tree with nonnegative branch lengths"},{"id":"Biology/coalescent-theory/multispecies-coalescent/taxa","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.SpeciesTree.taxa","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#taxa","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The species below a node.","title":"Species below a node of a species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/clusters","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.SpeciesTree.clusters","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#clusters","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The clusters (clades) of a species tree: the sets of species below its vertices, including the singletons and the full set of species.","title":"Clusters of a species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/shape","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.SpeciesTree.shape","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#shape","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The topology of a species tree, forgetting branch lengths, as a planar rooted binary tree.","title":"Topology of a species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/sum-tavare-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.sum_tavareProb","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#sum-tavare-prob","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"From `m \u2265 1` lineages, the number of ancestors at time `t` is one of `1, \u2026, m`. (The rows of `exp (t \u2022 G)` restricted to the states `\u2265 1` sum to one because `G` kills the indicator of `{1, \u2026, n}`; an identity for every real `t`.)","title":"Tavar\u00e9's probabilities sum to one"},{"id":"Biology/coalescent-theory/multispecies-coalescent/tavare-prob-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tavareProb_nonneg","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#tavare-prob-nonneg","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Tavar\u00e9's probabilities are nonnegative for `t \u2265 0`: `exp (t \u2022 G)` is the entrywise limit of powers of the Wright\u2013Fisher transition matrix (`tendsto_wfLineageMatrix_pow_floor`), whose entries are nonnegative. (For `t < 0` they can be negative or exceed one.)","title":"Tavar\u00e9's probabilities are nonnegative"},{"id":"Biology/coalescent-theory/multispecies-coalescent/merge-along","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.mergeAlong","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#merge-along","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The lineages after the first `t` coalescence events of the ranked history `h` (of `m` lineages), starting from the lineages `bs`, each given as the set of sampled genes it is ancestral to. Events follow the encoding of `RankedHistory`: the pair `{a, b}` with `a < b` merges into position `a`, and position `b` is removed (`mergeBlocks`). With the singletons as starting lineages this is `blocksAfter`\u2026","title":"Lineages after the first coalescence events of a ranked history"},{"id":"Biology/coalescent-theory/multispecies-coalescent/merge-along-singletons","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.mergeAlong_singletons","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#merge-along-singletons","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Starting from the singletons, `mergeAlong` is `blocksAfter`.","title":"Merging lineages from singletons gives the ancestral blocks"},{"id":"Biology/coalescent-theory/multispecies-coalescent/clades-along","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.cladesAlong","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#clades-along","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The clusters formed during the first `t` coalescence events of `h` from the lineages `bs`: every lineage present at some step `u \u2264 t`, as the set of sampled genes it is ancestral to.","title":"Clusters formed along a ranked history"},{"id":"Biology/coalescent-theory/multispecies-coalescent/branch-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.branchStep","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#branch-step","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"One population of the multispecies coalescent along a branch of length `t`. It receives `m` lineages `bs` (and the clusters `acc` formed below it). Under Kingman's coalescent the number `j` of lineages left at the top of the branch has law `tavareProb m j t` (Tavar\u00e9's formula, `exp_blockCountingGenerator_apply`), and, independently of the coalescence times, the order of the `m - j` coalescences is the beginning of\u2026","title":"Coalescence along one branch of the species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/exit-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.exitLaw","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#exit-law","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The law of the lineages leaving the top of the branch above a vertex of the species tree, in continuation-passing form: `exitLaw T k` is the expectation of `k m bs acc`, where `m` is the number of exiting lineages, `bs` the lineages and `acc` the clusters formed below. The two daughter populations evolve independently, and their lineages enter the ancestral population together.","title":"Law of the lineages leaving a species-tree branch"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The multispecies coalescent (Pamilo & Nei 1988; Degnan & Salter 2005), one lineage sampled per species: the probability that the gene tree, given by its set of clusters, lies in the event `E`. Below the root, each branch population runs Kingman's coalescent for the branch length (`branchStep`); in the root population all remaining lineages coalesce by random joining. The definition accepts any real branch lengths;\u2026","title":"The multispecies coalescent"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-node-length","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_node_length","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-node-length","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The branch length stored at the root of a species tree does not affect the gene tree.","title":"Root branch length does not affect gene-tree probabilities"},{"id":"Biology/coalescent-theory/multispecies-coalescent/exit-law-eq-const","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.exitLaw_eq_const","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#exit-law-eq-const","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"If a continuation takes the constant value `c` whenever at least one lineage exits, then so does `exitLaw` (Tavar\u00e9's probabilities sum to one and random joining is a probability).","title":"Exit law of a constant continuation"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-true","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_true","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-true","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"For every species tree, the event \"any gene tree\" has weight one (an identity for every real branch length). With `geneTreeProb_nonneg` and `geneTreeProb_or` this makes `geneTreeProb T` a probability law when `T.NonnegLengths`.","title":"Normalization"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-eq-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_eq_one","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-eq-one","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"An event containing every gene tree has weight one, for every species tree.","title":"Total gene-tree probability is one"},{"id":"Biology/coalescent-theory/multispecies-coalescent/exit-law-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.exitLaw_nonneg","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#exit-law-nonneg","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"On a species tree with nonnegative branch lengths, `exitLaw` of a continuation that is nonnegative whenever at least one lineage exits is nonnegative (Tavar\u00e9's probabilities are nonnegative, `tavareProb_nonneg`).","title":"Nonnegativity of the exit law"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_nonneg","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-nonneg","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Gene-tree probabilities are nonnegative on species trees with nonnegative branch lengths. (For a negative branch length they need not be.)","title":"Gene-tree probabilities are nonnegative"},{"id":"Biology/coalescent-theory/multispecies-coalescent/integral-random-joining","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.integral_randomJoining","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#integral-random-joining","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"An expectation under random joining is the average over the `\u220f_{k=2}^m C(k, 2)` ranked histories.","title":"Expectation under random joining as an average over histories"},{"id":"Biology/coalescent-theory/multispecies-coalescent/integral-random-joining-rat-cast","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.integral_randomJoining_ratCast","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#integral-random-joining-rat-cast","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"`integral_randomJoining` for a rational-valued function, with a rational result (used to evaluate random-joining probabilities by computation).","title":"Rational-valued expectation under random joining"},{"id":"Biology/coalescent-theory/multispecies-coalescent/random-joining-real","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.randomJoining_real","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#random-joining-real","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The probability of an event under random joining is the fraction of ranked histories in it.","title":"Random-joining probability as a fraction of ranked histories"},{"id":"Biology/coalescent-theory/multispecies-coalescent/exit-law-add","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.exitLaw_add","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#exit-law-add","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"`exitLaw` is additive in the continuation.","title":"Additivity of the exit law"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-or","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_or","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-or","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The probabilities of two disjoint events of gene trees add up. With `geneTreeProb_nonneg` and `geneTreeProb_true`, `geneTreeProb T` is a probability law when `T.NonnegLengths`.","title":"Additivity"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-le-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_le_one","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-le-one","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"On a species tree with nonnegative branch lengths, every gene-tree probability is at most one.","title":"Gene-tree probabilities are at most one"},{"id":"Biology/coalescent-theory/multispecies-coalescent/tavare-prob-two-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tavareProb_two_two","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two lineages remain distinct with probability "],["math","\\(e^{-t}\\)"]],"source":"Two lineages remain distinct with probability $e^{-t}$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#tavare-prob-two-two","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Two lineages are still distinct after time `t` with probability `e^{-t}`.","title":"Two lineages remain distinct with probability $e^{-t}$"},{"id":"Biology/coalescent-theory/multispecies-coalescent/tavare-prob-two-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tavareProb_two_one","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two lineages coalesce with probability "],["math","\\(1 - e^{-t}\\)"]],"source":"Two lineages coalesce with probability $1 - e^{-t}$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#tavare-prob-two-one","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Two lineages have coalesced within time `t` with probability `1 - e^{-t}`.","title":"Two lineages coalesce with probability $1 - e^{-t}$"},{"id":"Biology/coalescent-theory/multispecies-coalescent/tavare-prob-three-three","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tavareProb_three_three","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three lineages remain distinct with probability "],["math","\\(e^{-3t}\\)"]],"source":"Three lineages remain distinct with probability $e^{-3t}$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#tavare-prob-three-three","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Three lineages are still distinct after time `t` with probability `e^{-3t}`.","title":"Three lineages remain distinct with probability $e^{-3t}$"},{"id":"Biology/coalescent-theory/multispecies-coalescent/tavare-prob-three-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tavareProb_three_two","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three lineages have two ancestors at time "],["math","\\(t\\)"]],"source":"Three lineages have two ancestors at time $t$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#tavare-prob-three-two","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Three lineages have two ancestors at time `t` with probability `(3/2)(e^{-t} - e^{-3t})`.","title":"Three lineages have two ancestors at time $t$"},{"id":"Biology/coalescent-theory/multispecies-coalescent/tavare-prob-three-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tavareProb_three_one","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three lineages have one ancestor at time "],["math","\\(t\\)"]],"source":"Three lineages have one ancestor at time $t$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#tavare-prob-three-one","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Three lineages have one ancestor at time `t` with probability `1 - (3/2) e^{-t} + (1/2) e^{-3t}`.","title":"Three lineages have one ancestor at time $t$"},{"id":"Biology/coalescent-theory/multispecies-coalescent/triple","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.triple","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#triple","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The three-species tree `((0, 1), 2)` with internal branch length `t`.","title":"Three-species tree ((0, 1), 2)"},{"id":"Biology/coalescent-theory/multispecies-coalescent/triple-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tripleTree","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#triple-tree","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The gene tree on three genes with cherry `c` (a pair of genes), as a set of clusters.","title":"Three-gene tree with a given cherry"},{"id":"Biology/coalescent-theory/multispecies-coalescent/clusters-triple","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.clusters_triple","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#clusters-triple","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The species tree `((0, 1), 2)` has the clusters of the gene tree with cherry `{0, 1}`.","title":"Clusters of the three-species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-triple-matching","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_triple_matching","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-triple-matching","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Three-species concordance (Hudson 1983; Pamilo & Nei 1988): with one lineage per species and internal branch length `t`, the gene tree matches the species tree with probability `1 - (2/3) e^{-t}`. (An identity for every real `t`; branch lengths are `t \u2265 0`.)","title":"Three-species concordance"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-triple-cherry-02","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_triple_cherry_02","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Probability of the discordant three-species gene tree with cherry "],["math","\\(\\{0, 2\\}\\)"]],"source":"Probability of the discordant three-species gene tree with cherry $\\{0, 2\\}$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-triple-cherry-02","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"With one lineage per species and internal branch length $t$, the gene tree of the species tree $((0, 1), 2)$ whose cherry is $\\{0, 2\\}$ has probability $\\frac{1}{3} e^{-t}$. (An identity for every real $t$; branch lengths are $t \\ge 0$.)","title":"Probability of the discordant three-species gene tree with cherry $\\{0, 2\\}$"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-triple-cherry-12","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_triple_cherry_12","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Probability of the discordant three-species gene tree with cherry "],["math","\\(\\{1, 2\\}\\)"]],"source":"Probability of the discordant three-species gene tree with cherry $\\{1, 2\\}$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-triple-cherry-12","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"With one lineage per species and internal branch length $t$, the gene tree of the species tree $((0, 1), 2)$ whose cherry is $\\{1, 2\\}$ has probability $\\frac{1}{3} e^{-t}$. (An identity for every real $t$; branch lengths are $t \\ge 0$.)","title":"Probability of the discordant three-species gene tree with cherry $\\{1, 2\\}$"},{"id":"Biology/coalescent-theory/multispecies-coalescent/triple-no-anomaly","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.triple_no_anomaly","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#triple-no-anomaly","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"No anomalous gene trees for three species (Degnan & Rosenberg 2006): for a positive internal branch, each discordant gene tree is strictly less probable than the matching one.","title":"No anomalous gene trees for three species"},{"id":"Biology/coalescent-theory/multispecies-coalescent/tendsto-no-coal-prob-pow-floor-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tendsto_noCoalProb_pow_floor_two","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#tendsto-no-coal-prob-pow-floor-two","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The three-species probabilities come from the Wright\u2013Fisher model: in a population of `M` gene copies, the lineages of species `0` and `1` have distinct parents in each of the `\u230at M\u230b` generations of the internal branch (`t \u2265 0` coalescent units) with probability `noCoalProb M 2 ^ \u230at M\u230b`, which tends to `tavareProb 2 2 t = e^{-t}`, the weight of the case in which three lineages reach the root population.","title":"Wright\u2013Fisher limit of the no-coalescence probability"},{"id":"Biology/coalescent-theory/multispecies-coalescent/caterpillar","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.caterpillar","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#caterpillar","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The asymmetric (caterpillar) species tree `(((0, 1), 2), 3)`: `x` is the length of the deeper internal branch (above the ancestor of `0, 1, 2`) and `y` that of the shallower one (above the ancestor of `0, 1`), as in Degnan & Rosenberg (2006).","title":"Caterpillar four-species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/balanced","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.balanced","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#balanced","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The symmetric (balanced) species tree `((0, 1), (2, 3))` with internal branch lengths `y\u2081` (above the ancestor of `0, 1`) and `y\u2082` (above the ancestor of `2, 3`).","title":"Balanced four-species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/caterpillar-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.caterpillarTree","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#caterpillar-tree","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The asymmetric gene tree `(((a, b), c), d)` on four genes, as a set of clusters.","title":"Caterpillar four-gene tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/balanced-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.balancedTree","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#balanced-tree","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The symmetric gene tree `((a, b), (c, d))` on four genes, as a set of clusters.","title":"Balanced four-gene tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/clusters-caterpillar","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.clusters_caterpillar","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#clusters-caterpillar","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The species tree `(((0, 1), 2), 3)` has the clusters of the gene tree `(((0, 1), 2), 3)`.","title":"Clusters of the caterpillar species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/dr-f","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.drF","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#dr-f","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Degnan & Rosenberg (2006), eq. (1): `f(x, y)`, the probability of the matching gene tree `(((A, B), C), D)` on the caterpillar species tree.","title":"Degnan\u2013Rosenberg probability of the matching caterpillar gene tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/dr-g","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.drG","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#dr-g","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Degnan & Rosenberg (2006), eq. (2): `g(x, y)`, the probability of the gene tree `((A, C), (B, D))` (and of `((A, D), (B, C))`) on the caterpillar species tree.","title":"Degnan\u2013Rosenberg formula g for the gene tree ((A, C), (B, D))"},{"id":"Biology/coalescent-theory/multispecies-coalescent/dr-h","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.drH","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#dr-h","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Degnan & Rosenberg (2006), eq. (3): `h(x, y)`, the probability of the gene tree `((A, B), (C, D))` on the caterpillar species tree.","title":"Degnan\u2013Rosenberg formula h for the gene tree ((A, B), (C, D))"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-caterpillar-matching","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_caterpillar_matching","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-caterpillar-matching","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The matching gene tree on the caterpillar species tree has probability `f(x, y)` (Degnan & Rosenberg 2006, eq. (1)). An identity for all real `x, y`.","title":"The matching gene tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-caterpillar-ac-bd","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_caterpillar_AC_BD","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-caterpillar-ac-bd","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The symmetric gene tree `((A, C), (B, D))` on the caterpillar species tree has probability `g(x, y)` (Degnan & Rosenberg 2006, eq. (2)).","title":"Probability of the gene tree ((A, C), (B, D)) on the caterpillar"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-caterpillar-ad-bc","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_caterpillar_AD_BC","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-caterpillar-ad-bc","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The symmetric gene tree `((A, D), (B, C))` on the caterpillar species tree has probability `g(x, y)`, the same as `((A, C), (B, D))` (Degnan & Rosenberg 2006).","title":"Probability of the gene tree ((A, D), (B, C)) on the caterpillar"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-caterpillar-ab-cd","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_caterpillar_AB_CD","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-caterpillar-ab-cd","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The symmetric gene tree `((A, B), (C, D))` on the caterpillar species tree has probability `h(x, y)` (Degnan & Rosenberg 2006, eq. (3)).","title":"Probability of the gene tree ((A, B), (C, D)) on the caterpillar"},{"id":"Biology/coalescent-theory/multispecies-coalescent/anomaly-a","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.anomalyA","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#anomaly-a","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Degnan & Rosenberg (2006), eq. (4): the boundary `a(x)` of the four-species anomaly zone, for a deeper internal branch of length `x > 0` (at `x = 0` the denominator vanishes and Lean's `a / 0 = 0` gives the junk value `log (2/3 - 0)`; `a(x) \u2192 \u221e` as `x \u2192 0\u207a`).","title":"Boundary of the four-species anomaly zone"},{"id":"Biology/coalescent-theory/multispecies-coalescent/anomaly-b","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.anomalyB","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#anomaly-b","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Degnan & Rosenberg (2006), eq. (5): the boundary `b(x)` below which the caterpillar species tree has three anomalous gene trees (defined for `x \u2265 0`).","title":"Boundary for three anomalous gene trees"},{"id":"Biology/coalescent-theory/multispecies-coalescent/dr-g-lt-dr-h","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.drG_lt_drH","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(g(x, y) < h(x, y)\\)"],["text"," for a positive shallower branch"]],"source":"$g(x, y) < h(x, y)$ for a positive shallower branch"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#dr-g-lt-dr-h","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"For a positive shallower branch, the symmetric gene tree `((A, B), (C, D))` is strictly more probable than `((A, C), (B, D))`: `g(x, y) < h(x, y)` (Degnan & Rosenberg 2006).","title":"$g(x, y) < h(x, y)$ for a positive shallower branch"},{"id":"Biology/coalescent-theory/multispecies-coalescent/dr-f-lt-dr-h-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.drF_lt_drH_iff","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#dr-f-lt-dr-h-iff","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The boundary `a(x)` of the anomaly zone (Degnan & Rosenberg 2006): for a deeper internal branch `x > 0`, the symmetric gene tree `((A, B), (C, D))` is more probable than the matching gene tree, `f(x, y) < h(x, y)`, if and only if `y < a(x)`.","title":"The boundary `a(x)` of the anomaly zone"},{"id":"Biology/coalescent-theory/multispecies-coalescent/dr-f-lt-dr-g-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.drF_lt_drG_iff","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Degnan\u2013Rosenberg criterion: "],["math","\\(f(x, y) < g(x, y)\\)"],["text"," iff "],["math","\\(y < b(x)\\)"]],"source":"Degnan\u2013Rosenberg criterion: $f(x, y) < g(x, y)$ iff $y < b(x)$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#dr-f-lt-dr-g-iff","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Degnan & Rosenberg (2006): for `x \u2265 0`, the gene tree `((A, C), (B, D))` (equivalently `((A, D), (B, C))`) is more probable than the matching gene tree, `f(x, y) < g(x, y)`, if and only if `y < b(x)`.","title":"Degnan\u2013Rosenberg criterion: $f(x, y) < g(x, y)$ iff $y < b(x)$"},{"id":"Biology/coalescent-theory/multispecies-coalescent/anomaly-b-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.anomalyB_zero","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["text","Value "],["math","\\(\\log(7/6)\\)"],["text"," of the three-anomaly boundary at zero"]],"source":"Value $\\log(7/6)$ of the three-anomaly boundary at zero"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#anomaly-b-zero","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"`b(0) = log (7/6)` (Degnan & Rosenberg 2006): if `y \u2265 log (7/6)`, then `y \u2265 b(x)` for every `x \u2265 0` (`anomalyB_le_anomalyB_zero`).","title":"Value $\\log(7/6)$ of the three-anomaly boundary at zero"},{"id":"Biology/coalescent-theory/multispecies-coalescent/anomaly-b-le-anomaly-b-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.anomalyB_le_anomalyB_zero","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#anomaly-b-le-anomaly-b-zero","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"`b` is maximal at `0`: `b(x) \u2264 log (7/6)` for `x \u2265 0`, so for `y \u2265 log (7/6)` at most one gene tree is anomalous.","title":"Three-anomaly boundary is maximal at zero"},{"id":"Biology/coalescent-theory/multispecies-coalescent/tendsto-anomaly-a","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.tendsto_anomalyA","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#tendsto-anomaly-a","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"`a(x) \u2192 \u221e` as `x \u2192 0\u207a` (Degnan & Rosenberg 2006): very short deep branches produce an anomalous gene tree however long the shallower branch is.","title":"Anomaly-zone boundary blows up for short deep branches"},{"id":"Biology/coalescent-theory/multispecies-coalescent/caterpillar-anomalous-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.caterpillar_anomalous_iff","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#caterpillar-anomalous-iff","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Anomalous gene tree on four species (Degnan & Rosenberg 2006): on the caterpillar species tree with deeper branch `x > 0` and shallower branch `y`, the symmetric gene tree `((A, B), (C, D))` is more probable than the gene tree matching the species tree if and only if `y < a(x)`. With `caterpillar_le_matching` this gives `caterpillar_exists_anomalous_iff`.","title":"Anomalous gene tree on four species"},{"id":"Biology/coalescent-theory/multispecies-coalescent/caterpillar-anomalous-ac-bd-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.caterpillar_anomalous_AC_BD_iff","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#caterpillar-anomalous-ac-bd-iff","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"On the caterpillar species tree with `x \u2265 0`, the gene trees `((A, C), (B, D))` and `((A, D), (B, C))` are more probable than the matching gene tree if and only if `y < b(x)`.","title":"Anomaly of ((A, C), (B, D)) and ((A, D), (B, C)) on the caterpillar"},{"id":"Biology/coalescent-theory/multispecies-coalescent/is-balanced-gene-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.IsBalancedGeneTree","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#is-balanced-gene-tree","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"`G` is one of the three balanced (symmetric) gene trees on four genes.","title":"Balanced gene tree on four genes"},{"id":"Biology/coalescent-theory/multispecies-coalescent/caterpillar-le-matching","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.caterpillar_le_matching","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#caterpillar-le-matching","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"On the caterpillar species tree with nonnegative branch lengths, no asymmetric gene tree is more probable than the matching one (Degnan & Rosenberg 2006): only the three balanced gene trees can be anomalous.","title":"Only balanced gene trees can be anomalous on the caterpillar"},{"id":"Biology/coalescent-theory/multispecies-coalescent/dr-g-le-dr-h","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.drG_le_drH","parent":45265,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(g(x, y) \\le h(x, y)\\)"],["text"," for a nonnegative shallower branch"]],"source":"$g(x, y) \\le h(x, y)$ for a nonnegative shallower branch"},"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#dr-g-le-dr-h","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"`g(x, y) \u2264 h(x, y)` for a nonnegative shallower branch (equality at `y = 0`).","title":"$g(x, y) \\le h(x, y)$ for a nonnegative shallower branch"},{"id":"Biology/coalescent-theory/multispecies-coalescent/caterpillar-exists-anomalous-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.caterpillar_exists_anomalous_iff","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#caterpillar-exists-anomalous-iff","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The four-species anomaly zone (Degnan & Rosenberg 2006, Definition 1 and Figure 2): the caterpillar species tree with deeper branch `x > 0` and shallower branch `y \u2265 0` has an anomalous gene tree, one strictly more probable than the gene tree matching the species tree, if and only if `y < a(x)`.","title":"The four-species anomaly zone"},{"id":"Biology/coalescent-theory/multispecies-coalescent/balanced-no-anomaly","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.balanced_no_anomaly","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#balanced-no-anomaly","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"No anomalous gene trees on the balanced four-species tree (Degnan & Rosenberg 2006): for nonnegative branch lengths, no gene tree is more probable than the one matching the species tree `((0, 1), (2, 3))`. Given the numbers of lineages entering the root population, the matching tree is already the most probable outcome of random joining.","title":"No anomalous gene trees on the balanced four-species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/displays-split","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.DisplaysSplit","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#displays-split","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"A rooted gene tree on four genes `G` (a set of clusters) displays the quartet split `ab | cd`, where `{c, d}` is the complement of `{a, b}`: forgetting the root, the internal edge separates `{a, b}` from `{c, d}`; equivalently `{a, b}` or `{c, d}` is a cluster.","title":"Gene tree displaying a quartet split"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-caterpillar-split-01","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_caterpillar_split_01","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-caterpillar-split-01","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"No anomalous unrooted quartets, caterpillar species tree (Allman, Degnan & Rhodes 2011): the unrooted gene quartet matches the species quartet `01 | 23` with probability `1 - (2/3) e^{-y}`, where `y` is the length of the internal edge of the unrooted species quartet (the shallower internal branch).","title":"No anomalous unrooted quartets, caterpillar species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-caterpillar-split-02","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_caterpillar_split_02","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-caterpillar-split-02","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"On the caterpillar species tree the discordant unrooted gene quartet `02 | 13` has probability `(1/3) e^{-y}`.","title":"Probability of quartet split 02|13 on the caterpillar tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-caterpillar-split-03","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_caterpillar_split_03","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-caterpillar-split-03","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"On the caterpillar species tree the discordant unrooted gene quartet `03 | 12` has probability `(1/3) e^{-y}`.","title":"Probability of quartet split 03|12 on the caterpillar tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-balanced-split-01","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_balanced_split_01","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-balanced-split-01","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"No anomalous unrooted quartets, balanced species tree (Allman, Degnan & Rhodes 2011): on `((0, 1), (2, 3))` the unrooted gene quartet matches `01 | 23` with probability `1 - (2/3) e^{-(y\u2081 + y\u2082)}`, where `y\u2081 + y\u2082` is the internal edge of the unrooted species quartet.","title":"No anomalous unrooted quartets, balanced species tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-balanced-split-02","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_balanced_split_02","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-balanced-split-02","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"On the balanced species tree the discordant unrooted gene quartet `02 | 13` has probability `(1/3) e^{-(y\u2081 + y\u2082)}`.","title":"Probability of quartet split 02|13 on the balanced tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/gene-tree-prob-balanced-split-03","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.geneTreeProb_balanced_split_03","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#gene-tree-prob-balanced-split-03","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"On the balanced species tree the discordant unrooted gene quartet `03 | 12` has probability `(1/3) e^{-(y\u2081 + y\u2082)}`.","title":"Probability of quartet split 03|12 on the balanced tree"},{"id":"Biology/coalescent-theory/multispecies-coalescent/split-01-most-probable","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.split_01_most_probable","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#split-01-most-probable","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The species quartet maximizes quartet support (Allman, Degnan & Rhodes 2011; Mirarab et al. 2014): for a positive internal edge, the unrooted gene quartet matching the species quartet is strictly the most probable of the three, on both four-species shapes. Since the expected quartet support of a candidate four-species tree (one quartet) is the probability of its quartet, the species tree maximizes it.","title":"The species quartet maximizes quartet support"},{"id":"Biology/coalescent-theory/multispecies-coalescent/is-clade","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.IsClade","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#is-clade","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The set `S` of sampled genes is a clade of the ranked history `h`: it is a block of one of the ancestral partitions of `h`, i.e. the set of descendants of some lineage.","title":"Clade of a ranked history"},{"id":"Biology/coalescent-theory/multispecies-coalescent/random-joining-is-clade","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.randomJoining_isClade","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#random-joining-is-clade","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"Clade probability (Rosenberg 2003): under random joining, a fixed set of `k` among `m` genes, `1 \u2264 k < m`, is a clade with probability `2m / (k (k + 1) C(m, k))`.","title":"Clade probability"},{"id":"Biology/coalescent-theory/multispecies-coalescent/random-joining-is-clade-of-le-five","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.MultispeciesCoalescent.randomJoining_isClade_of_le_five","parent":45265,"result":true,"route":"../roadmap/Biology/coalescent-theory/multispecies-coalescent/index.html#random-joining-is-clade-of-le-five","scope":"Biology/coalescent-theory/multispecies-coalescent","status":"fully_proved","summary":"The case `m \u2264 5` of `randomJoining_isClade`.","title":"Clade probability under random joining for at most five genes"},{"data_url":"scopes/Biology/coalescent-theory/segregating-sites-and-pairwise-differences.json","id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","title":"Segregating sites and pairwise differences"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-a1","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaA1","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's constant "],["math","\\(a_1\\)"]],"source":"Tajima's constant $a_1$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-a1","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`a\u2081 = \u2211_{i=1}^{n-1} 1/i` (Tajima 1989).","title":"Tajima's constant $a_1$"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-a2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaA2","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's constant "],["math","\\(a_2\\)"]],"source":"Tajima's constant $a_2$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-a2","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`a\u2082 = \u2211_{i=1}^{n-1} 1/i\u00b2` (Tajima 1989).","title":"Tajima's constant $a_2$"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-b1","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaB1","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's constant "],["math","\\(b_1\\)"]],"source":"Tajima's constant $b_1$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-b1","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`b\u2081 = (n + 1) / (3 (n - 1))` (Tajima 1989), for `n \u2265 2`.","title":"Tajima's constant $b_1$"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-b2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaB2","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's constant "],["math","\\(b_2\\)"]],"source":"Tajima's constant $b_2$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-b2","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`b\u2082 = 2 (n\u00b2 + n + 3) / (9 n (n - 1))` (Tajima 1989), for `n \u2265 2`.","title":"Tajima's constant $b_2$"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-c1","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaC1","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's constant "],["math","\\(c_1\\)"]],"source":"Tajima's constant $c_1$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-c1","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`c\u2081 = b\u2081 - 1/a\u2081` (Tajima 1989), for `n \u2265 2`.","title":"Tajima's constant $c_1$"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-c2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaC2","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's constant "],["math","\\(c_2\\)"]],"source":"Tajima's constant $c_2$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-c2","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`c\u2082 = b\u2082 - (n + 2)/(a\u2081 n) + a\u2082/a\u2081\u00b2` (Tajima 1989), for `n \u2265 2`.","title":"Tajima's constant $c_2$"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-e1","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaE1","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's constant "],["math","\\(e_1\\)"]],"source":"Tajima's constant $e_1$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-e1","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`e\u2081 = c\u2081 / a\u2081` (Tajima 1989), for `n \u2265 2`.","title":"Tajima's constant $e_1$"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-e2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaE2","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's constant "],["math","\\(e_2\\)"]],"source":"Tajima's constant $e_2$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-e2","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`e\u2082 = c\u2082 / (a\u2081\u00b2 + a\u2082)` (Tajima 1989), for `n \u2265 2`.","title":"Tajima's constant $e_2$"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/tajima-a1-eq-harmonic","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaA1_eq_harmonic","parent":45342,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's "],["math","\\(a_1\\)"],["text"," is a harmonic number"]],"source":"Tajima's $a_1$ is a harmonic number"},"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#tajima-a1-eq-harmonic","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`a\u2081` is the harmonic number `H_{n-1}`.","title":"Tajima's $a_1$ is a harmonic number"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/variance-total-length","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.variance_totalLength","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#variance-total-length","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"Variance of the total branch length (Watterson 1975): `Var L = \u2211_{k=2}^n k\u00b2 Var T_k = 4 \u2211_{j=1}^{n-1} 1/j\u00b2` (uses independence of the holding times).","title":"Variance of the total branch length"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/variance-seg-sites","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.variance_segSites","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#variance-seg-sites","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"Variance of the number of segregating sites (Watterson 1975; Tajima 1989): if, given the genealogy \u03c3-algebra `m`, the number `S` of segregating sites has conditional mean and conditional variance `(\u03b8/2) L` (as for a Poisson count with mean `\u03b8 L / 2`), then `Var S = \u03b8 a\u2081 + \u03b8\u00b2 a\u2082`.","title":"Variance of the number of segregating sites"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/sample-pairs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.samplePairs","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#sample-pairs","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"The pairs of genes in a sample of size `n`.","title":"Pairs of genes in a sample"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/mean-pair-diff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.meanPairDiff","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#mean-pair-diff","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"The mean number of pairwise differences `\u03c0 = C(n, 2)\u207b\u00b9 \u2211_{i<j} D_{ij}` (Tajima 1983), where `D s` is the number of sites at which the two sequences of the pair `s` differ.","title":"Mean number of pairwise differences"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/integral-mean-pair-diff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.integral_meanPairDiff","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#integral-mean-pair-diff","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"Expected pairwise differences (Tajima 1983): if, for each pair `s` of sampled genes, the coalescence time `\u03c4 s` of the pair is exponential with rate `1` and the number `D s` of differences has conditional mean `\u03b8 \u03c4 s` given the genealogy \u03c3-algebra `m` (mutations at rate `\u03b8 / 2` on each of the two branches of length `\u03c4 s`), then `E \u03c0 = \u03b8`.","title":"Expected pairwise differences"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/integral-tajima-numerator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integral_tajimaNumerator","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#integral-tajima-numerator","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"Tajima's numerator has mean zero (Tajima 1989): under the hypotheses of `integral_meanPairDiff` and Watterson's (`E[S | m] = (\u03b8/2) L`), `E[\u03c0 - S / a\u2081] = 0` for a sample of size `n \u2265 2`.","title":"Tajima's numerator has mean zero"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/integral-tajima-var-estimator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integral_tajimaVarEstimator","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#integral-tajima-var-estimator","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"Unbiased estimator of Tajima's variance term (Tajima 1989): with `S` as in `variance_segSites`, `E[e\u2081 S + e\u2082 S (S - 1)] = c\u2081 \u03b8 + c\u2082 \u03b8\u00b2` for `n \u2265 2`. (By Tajima's analysis `c\u2081 \u03b8 + c\u2082 \u03b8\u00b2` is `Var(\u03c0 - S / a\u2081)`.)","title":"Unbiased estimator of Tajima's variance term"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/sum-sample-pairs-card-inter","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.sum_samplePairs_card_inter","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#sum-sample-pairs-card-inter","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"Among the pairs of a sample of size `n`, one equals a given pair `s`, `2 (n - 2)` share exactly one gene with it and `C(n - 2, 2)` are disjoint from it.","title":"Counting sample pairs by overlap with a given pair"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/variance-mean-pair-diff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.variance_meanPairDiff","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#variance-mean-pair-diff","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"Variance of the mean number of pairwise differences (Tajima 1983): `Var \u03c0 = b\u2081 \u03b8 + b\u2082 \u03b8\u00b2` for a sample of size `n \u2265 2`, under the conditional-moment model of the module docstring: `\u03c4 s ~ Exp(1)`, `E[\u03c4 s \u03c4 t] = 4/3` (one shared gene), `11/9` (disjoint pairs); the shared path lengths `\u2113`; `E[D s | m] = \u03b8 \u03c4 s` and `E[D s D t | m] = \u03b8\u00b2 \u03c4 s \u03c4 t + (\u03b8 / 2) \u2113 s t`.","title":"Variance of the mean number of pairwise differences"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/pair-separate-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.pairSeparateProb","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#pair-separate-prob","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"The probability `(k - 1) (n + 1) / ((n - 1) (k + 1))` that a given pair of a sample of size `n` is still in two distinct lineages while the genealogy has `k` lineages (`2 \u2264 k \u2264 n`, `n \u2265 2`; at `n = 1` the definition divides by zero and returns `0`).","title":"Probability that a pair lies in separate lineages"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/sum-pair-separate-prob-div-choose","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.sum_pairSeparateProb_div_choose","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#sum-pair-separate-prob-div-choose","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`\u2211_{k=2}^n p_k / C(k, 2) = 1`: the mean coalescence time of a pair is `1`.","title":"Mean coalescence time of a pair is one"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/sum-mul-pair-separate-prob-div-choose-sq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.sum_mul_pairSeparateProb_div_choose_sq","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#sum-mul-pair-separate-prob-div-choose-sq","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`\u2211_{k=2}^n k p_k / C(k, 2)\u00b2 = (n + 2) / n`.","title":"Sum of pair-separation probabilities over squared pair counts"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/covariance-mean-pair-diff-seg-sites","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.covariance_meanPairDiff_segSites","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#covariance-mean-pair-diff-seg-sites","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"`Cov(\u03c0, S) = \u03b8 + (n + 2) \u03b8\u00b2 / (2 n)` for `n \u2265 2`, if `\u03c4 s ~ Exp(1)`, `E[\u03c4 s L] = 2 a\u2081 + (n + 2) / n` (`L` the total branch length), `E[D s | m] = \u03b8 \u03c4 s`, `E[S | m] = (\u03b8 / 2) L` and `E[D s S | m] = (\u03b8\u00b2 / 2) \u03c4 s L + \u03b8 \u03c4 s` (the path of each pair lies in the tree). This is the covariance behind Tajima's (1989) `c\u2081, c\u2082`.","title":"Covariance of `\u03c0` and `S`"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/variance-tajima-numerator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.variance_tajimaNumerator","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#variance-tajima-numerator","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"Tajima's variance identity (Tajima 1989): `Var(\u03c0 - S / a\u2081) = c\u2081 \u03b8 + c\u2082 \u03b8\u00b2` for a sample of size `n \u2265 2`, under the hypotheses of `variance_meanPairDiff`, `covariance_meanPairDiff_segSites` and `variance_segSites`.","title":"Tajima's variance identity"},{"id":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences/integral-pair-time-mul-total-length","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.KingmanTimes.integral_pairTime_mul_totalLength","parent":45342,"result":true,"route":"../roadmap/Biology/coalescent-theory/segregating-sites-and-pairwise-differences/index.html#integral-pair-time-mul-total-length","scope":"Biology/coalescent-theory/segregating-sites-and-pairwise-differences","status":"fully_proved","summary":"If the coalescence time of a pair is `\u03c4 = \u2211_{k=2}^n \u03c7 k T k`, where `\u03c7 k` indicates that the pair is still in two distinct lineages while there are `k` lineages, and the topology is independent of the holding times in the sense `E[\u03c7 k T k T k'] = p k E[T k T k']` with `p k = pairSeparateProb n k`, then `E[\u03c4 L] = 2 a\u2081 + (n + 2) / n` for `n \u2265 2`. Only this factorization is used, so `\u03c7 k` need not be an indicator in\u2026","title":"Pair coalescence time against the total branch length"},{"data_url":"scopes/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio.json","id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"can_prove","title":"Site frequency spectrum under variable population size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/sfs-kernel","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.sfsKernel","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#sfs-kernel","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The Griffiths\u2013Tavar\u00e9 kernel `k C(n-i-1, k-2) / C(n-1, k-1)`: the expected number of branches subtending exactly `i` of the `n` sample copies while `k` lineages remain (`SiteFrequencySpectrum.joinStep_iterate_count`). Meaningful for `1 \u2264 i < n`, `2 \u2264 k \u2264 n`; outside that range the natural-number subtractions give junk values.","title":"Griffiths\u2013Tavar\u00e9 kernel"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/variable-sfs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.variableSFS","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#variable-sfs","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The expected SFS entry `(\u03b8 / 2) \u2211_{k=2}^{n} sfsKernel n i k \u00b7 t k` of a sample of size `n` whose mean holding times are `t k = E T_k` (Griffiths & Tavar\u00e9 1998, Polanski & Kimmel 2003).","title":"Expected site frequency spectrum under variable population size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/integral-sfs-of-indep","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.integral_sfs_of_indep","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#integral-sfs-of-indep","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Expected SFS under a variable population size (Griffiths & Tavar\u00e9 1998; Polanski & Kimmel 2003). Let the genealogy of a sample of `n` copies consist of holding times `T k` (integrable for `2 \u2264 k \u2264 n`, otherwise arbitrary) and an independent block-size history `\u03c4` with the random-joining law. If, given the genealogy, the count `\u03be i` is integrable with conditional mean `(\u03b8 / 2) L_i`, then\u2026","title":"Expected SFS under a variable population size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/variable-sfs-inv-choose-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.variableSFS_inv_choose_two","parent":45367,"reading_title":{"format":"inline-parts-v1","parts":[["text","Variable-size spectrum reduces to Fu's "],["math","\\(\\theta/i\\)"],["text"," at constant size"]],"source":"Variable-size spectrum reduces to Fu's $\\theta/i$ at constant size"},"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#variable-sfs-inv-choose-two","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"With the constant-size means `E T_k = 1 / C(k,2)`, the variable-size SFS is Fu's `\u03b8 / i`.","title":"Variable-size spectrum reduces to Fu's $\\theta/i$ at constant size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/sfs-kernel-eq-zero-of-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.sfsKernel_eq_zero_of_lt","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#sfs-kernel-eq-zero-of-lt","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The kernel row `i` vanishes beyond `k = n - i + 1`: a branch subtending `i` copies cannot exist while more than `n - i + 1` lineages remain.","title":"Griffiths\u2013Tavar\u00e9 kernel vanishes beyond the diagonal"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/sfs-kernel-diag-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.sfsKernel_diag_pos","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#sfs-kernel-diag-pos","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The kernel entry at `k = n - i + 1` is `(n - i + 1) / C(n-1, i-1) > 0`.","title":"Positivity of the Griffiths\u2013Tavar\u00e9 kernel on the diagonal"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/eq-of-variable-sfs-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.eq_of_variableSFS_eq","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#eq-of-variable-sfs-eq","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"If two vectors of mean holding times give the same expected SFS for some `\u03b8 \u2260 0`, they agree at every `k = 2, \u2026, n`. This is a direct consequence of `sfsKernel_eq_zero_of_lt` and `sfsKernel_diag_pos` (the kernel is triangular with nonzero diagonal); it is recorded here as an observation, not as a cited theorem.","title":"The expected SFS determines the mean holding times"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/theta-weighted","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.thetaWeighted","parent":45367,"reading_title":{"format":"inline-parts-v1","parts":[["text","Achaz's weighted estimator of "],["math","\\(\\theta\\)"]],"source":"Achaz's weighted estimator of $\\theta$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#theta-weighted","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Achaz's weighted estimator `\u03b8_\u03c9 = \u2211_{i=1}^{n-1} \u03c9_i i \u03be_i` (Achaz 2009, with the weights already normalized).","title":"Achaz's weighted estimator of $\\theta$"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/theta-pi","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.thetaPi","parent":45367,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's "],["math","\\(\\theta_\\pi\\)"],["text"," in site frequency spectrum form"]],"source":"Tajima's $\\theta_\\pi$ in site frequency spectrum form"},"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#theta-pi","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Tajima's `\u03b8_\u03c0` in its SFS form `\u2211_i i (n - i) \u03be_i / C(n,2)` (weights `(n - i) / C(n,2)`).","title":"Tajima's $\\theta_\\pi$ in site frequency spectrum form"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/theta-h","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.thetaH","parent":45367,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fay and Wu's "],["math","\\(\\theta_H\\)"]],"source":"Fay and Wu's $\\theta_H$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#theta-h","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Fay & Wu's `\u03b8_H = \u2211_i i\u00b2 \u03be_i / C(n,2)` (weights `i / C(n,2)`).","title":"Fay and Wu's $\\theta_H$"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/fay-wu-h","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.fayWuH","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#fay-wu-h","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Fay & Wu's `H = \u03b8_\u03c0 - \u03b8_H` (Fay & Wu 2000).","title":"Fay and Wu's H statistic"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/fu-li-dnumerator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.fuLiDNumerator","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#fu-li-dnumerator","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The numerator `S - a\u2081 \u03be\u2081` of Fu & Li's `D` with an outgroup, where `S = \u2211_i \u03be_i` and the external-branch mutations are the derived singletons `\u03be\u2081` (Fu & Li 1993).","title":"Numerator of Fu and Li's D"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/fu-li-fnumerator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.fuLiFNumerator","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#fu-li-fnumerator","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The numerator `\u03b8_\u03c0 - \u03be\u2081` of Fu & Li's `F` with an outgroup (Fu & Li 1993), with `\u03b8_\u03c0` in its SFS form.","title":"Numerator of Fu and Li's F"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/integral-theta-weighted","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.integral_thetaWeighted","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#integral-theta-weighted","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Achaz's unbiasedness (Achaz 2009): if `E \u03be_i = \u03b8 / i` for `1 \u2264 i \u2264 n - 1` (the constant-size spectrum, `SiteFrequencySpectrum.integral_sfs`), every weighted estimator with `\u2211_i \u03c9_i = 1` has mean `\u03b8`.","title":"Achaz's unbiasedness"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/integral-fay-wu-h","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.integral_fayWuH","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#integral-fay-wu-h","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Fay & Wu's `H` has mean zero under the constant-size spectrum (`n \u2265 2`).","title":"Fay and Wu's H has mean zero at constant size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/integral-fu-li-dnumerator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.integral_fuLiDNumerator","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#integral-fu-li-dnumerator","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The numerator of Fu & Li's `D` has mean zero under the constant-size spectrum (`n \u2265 2`).","title":"Numerator of Fu and Li's D has mean zero at constant size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/integral-fu-li-fnumerator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.integral_fuLiFNumerator","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#integral-fu-li-fnumerator","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The numerator of Fu & Li's `F` has mean zero under the constant-size spectrum (`n \u2265 2`).","title":"Numerator of Fu and Li's F has mean zero at constant size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/integral-theta-weighted-kingman","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.integral_thetaWeighted_kingman","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#integral-theta-weighted-kingman","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Achaz's unbiasedness under Kingman's coalescent, with the hypotheses of `SiteFrequencySpectrum.integral_sfs`.","title":"Achaz's estimator is unbiased under Kingman's coalescent"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/cum-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.cumRate","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#cum-rate","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The cumulative coalescence intensity `R(s) = \u222b_0^s 1/\u03b7` of a relative population-size function `\u03b7` (pairs coalesce at rate `1/\u03b7(s)` at time `s` before the present). Meaningful for positive `\u03b7` with `1/\u03b7` locally integrable.","title":"Cumulative coalescence intensity"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/first-coal-mean","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.firstCoalMean","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#first-coal-mean","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The expected time to the first coalescence among `m` lineages, `c_m = \u222b_0^\u221e exp(-C(m,2) R(s)) ds`, in a population of relative size `\u03b7` (Bhaskar & Song's `E[T_{m,m}]`). When the integrand is not integrable (for example `\u03b7 \u2261 0`), this is Lean's junk value `0`; for piecewise-constant histories it is integrable (`choose_two_mul_firstCoalMean`).","title":"Expected time to the first coalescence"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/piecewise-history","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#piecewise-history","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"A piecewise-constant population-size history with `K` pieces: piece `j < K` starts at time `start j` (backward from the present, `start 0 = 0`, strictly increasing) and has relative size `size j > 0`; the last piece extends to infinity. Adjacent pieces may have equal sizes, so this also covers histories with fewer than `K` distinct pieces. Values at indices `j \u2265 K` are unused. The case `K = 0` is degenerate (its\u2026","title":"Piecewise-constant population-size history"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/size-fn","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.sizeFn","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#size-fn","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The population-size function of a piecewise-constant history: at time `s \u2265 0` it is the size of the piece `j` with `start j \u2264 s < start (j + 1)` (the last piece has no end). Equal to `0` for `s < 0` (junk; only `s \u2265 0` is meaningful).","title":"Size function of a piecewise-constant history"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/start-mono","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.start_mono","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#start-mono","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The start times are monotone on the pieces.","title":"Monotonicity of piece start times"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/start-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.start_nonneg","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#start-nonneg","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Every piece starts at a nonnegative time.","title":"Nonnegativity of piece start times"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/size-fn-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.sizeFn_eq","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#size-fn-eq","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"On piece `j` the size function equals `size j`.","title":"Size function on a piece equals the piece size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/cum-rate-piece","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.cumRate_piece","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#cum-rate-piece","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"On piece `j` (including its right end), `1 / sizeFn` is integrable on `[0, s]` and the cumulative rate is affine: `R(s) = R(start j) + (s - start j) / size j`.","title":"Cumulative rate is affine on each piece"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/choose-two-mul-first-coal-mean","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.choose_two_mul_firstCoalMean","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#choose-two-mul-first-coal-mean","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"For a history with `K \u2265 1` pieces and `m \u2265 2`, writing `a = C(m,2)` and `R = cumRate h.sizeFn`, `a c_m = size 0 + \u2211_{j=1}^{K-1} (size j - size (j-1)) exp(-a R(start j))`. In particular the integral defining `c_m` converges.","title":"Closed form of the first-coalescence times"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/exists-piece","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.exists_piece","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#exists-piece","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Every time `s \u2265 0` lies in some piece.","title":"Every nonnegative time lies in some piece"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/size-fn-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.sizeFn_pos","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#size-fn-pos","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The size function is positive on `[0, \u221e)`.","title":"Positivity of the piecewise size function"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/cum-rate-start-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.cumRate_start_lt","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#cum-rate-start-lt","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The coalescent-time breakpoints `R(start j)` are strictly increasing.","title":"Coalescent-time breakpoints are strictly increasing"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/dual","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.dual","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#dual","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The dual history in coalescent time: piece `j` starts at `R(start j)` and has size `1 / size j`.","title":"Dual history in coalescent time"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/size-fn-eq-inv-dual","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.sizeFn_eq_inv_dual","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#size-fn-eq-inv-dual","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The size at time `s` is the inverse of the dual history's size at coalescent time `R(s)`.","title":"Size function is the inverse of the dual size in coalescent time"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/cum-rate-dual-cum-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.cumRate_dual_cumRate","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#cum-rate-dual-cum-rate","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The dual cumulative rate inverts the cumulative rate.","title":"Dual cumulative rate inverts the cumulative rate"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/cum-rate-strict-mono-on","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.cumRate_strictMonoOn","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#cum-rate-strict-mono-on","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The cumulative rate is strictly increasing on `[0, \u221e)`.","title":"Cumulative rate is strictly increasing"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/size-fn-eq-of-first-coal-mean-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.sizeFn_eq_of_firstCoalMean_eq","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#size-fn-eq-of-first-coal-mean-eq","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Bhaskar\u2013Song identifiability (Bhaskar & Song 2014, Corollary 7, \"determined\" half, stated for the first-coalescence times): two piecewise-constant histories with `K` pieces whose expected first-coalescence times `c_m` agree for `2 \u2264 m \u2264 n`, with `n \u2265 2K`, have the same population size at every time `s \u2265 0`. The proof writes `C(m,2) c_m` as an exponential sum in `C(m,2)` over at most `2K - 1` rates\u2026","title":"Bhaskar\u2013Song identifiability"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/exists-ne-first-coal-mean-eq-of-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.exists_ne_firstCoalMean_eq_of_lt","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#exists-ne-first-coal-mean-eq-of-lt","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Tightness of `n \u2265 2K` (Bhaskar & Song 2014, Corollary 7): for every `K \u2265 2` there are two piecewise-constant histories with `K` pieces whose sizes differ at some time `s \u2265 0` but whose first-coalescence times `c_m` agree for `2 \u2264 m \u2264 2K - 1`; so `sizeFn_eq_of_firstCoalMean_eq` fails for `n = 2K - 1`. The construction is explicit: with `z = 2^{-C(m,2)}`, the two histories have coalescent-time breakpoints\u2026","title":"Tightness of `n \u2265 2K`"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/measurable-size-fn","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.measurable_sizeFn","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#measurable-size-fn","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The size function is measurable.","title":"Measurability of the piecewise size function"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/exists-le-size-fn-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.exists_le_sizeFn_le","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#exists-le-size-fn-le","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"On `[0, \u221e)` the size function lies between the smallest and the largest piece size.","title":"Size function lies between the extreme piece sizes"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/exists-size-fn-eq-ico","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.exists_sizeFn_eq_Ico","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#exists-size-fn-eq-ico","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The size function is constant on an interval to the right of every `s \u2265 0`.","title":"Size function is locally constant to the right"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/exists-ne-first-coal-mean-eq-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.PiecewiseHistory.exists_ne_firstCoalMean_eq_two","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#exists-ne-first-coal-mean-eq-two","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"The two-piece case of `exists_ne_firstCoalMean_eq_of_lt`.","title":"Distinct two-piece histories with equal first-coalescence times"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/first-coal-mean-comp-max","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.firstCoalMean_comp_max","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#first-coal-mean-comp-max","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"Only the values on `[0, \u221e)` enter the first-coalescence times.","title":"First-coalescence times depend only on sizes at nonnegative times"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/exists-ne-ae-first-coal-mean-eq-icc","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFS.exists_ne_ae_firstCoalMean_eq_Icc","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#exists-ne-ae-first-coal-mean-eq-icc","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"fully_proved","summary":"For every `n` there are two measurable population-size functions, bounded between positive constants and different on a set of positive measure in `[0, \u221e)`, with the same `c_m` for `2 \u2264 m \u2264 n` (two `(n + 2)`-piece histories from `PiecewiseHistory.exists_ne_firstCoalMean_eq_of_lt`, extended to `s < 0` by their size at `0`). This is the finite-sample shape of the statement of Myers, Fefferman & Patterson (2008);\u2026","title":"Non-identifiability from a bounded sample size"},{"id":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/exists-ne-ae-first-coal-mean-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.VariableSFSWanted.exists_ne_ae_firstCoalMean_eq","parent":45367,"result":true,"route":"../roadmap/Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio/index.html#exists-ne-ae-first-coal-mean-eq","scope":"Biology/coalescent-theory/site-frequency-spectrum-under-variable-populatio","status":"can_prove","summary":"Non-identifiability of general histories (Myers, Fefferman & Patterson 2008): there are two measurable population-size functions, bounded between positive constants, that differ on a set of positive Lebesgue measure in `[0, \u221e)` and have the same expected first-coalescence time `c_m` for every `m \u2265 2` (hence the same expected SFS for every sample size).","title":"Non-identifiability of general histories"},{"data_url":"scopes/Biology/coalescent-theory/skyline-plot-estimates-of-population-size.json","id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","title":"Skyline plot estimates of population size"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/skyline-times","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SkylineTimes","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#skyline-times","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"The coalescent intervals of a sample of `n` genes under the classic skyline model with piecewise-constant population-size parameter `\u03b8 k` while there are `k` lineages: `T k` is exponential with rate `C(k, 2) / \u03b8 k`, independently over `2 \u2264 k \u2264 n`.","title":"Coalescent intervals under the classic skyline model"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/skyline-log-lik","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.skylineLogLik","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#skyline-log-lik","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"The log-likelihood of `\u03b8` for one coalescent interval of length `w` with `c = C(k, 2)` pairs of lineages: the log-density `log (c / \u03b8) - c w / \u03b8` of `Exp(c / \u03b8)` at `w`.","title":"Log-likelihood of one coalescent interval in the skyline model"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/skyline-log-lik-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.skylineLogLik_le","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#skyline-log-lik-le","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"The skyline estimate maximizes the interval likelihood (Pybus, Rambaut & Harvey 2000): for `c, w, \u03b8 > 0`, `\u2113(\u03b8) \u2264 \u2113(c w)`.","title":"The skyline estimate maximizes the interval likelihood"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/skyline-log-lik-eq-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.skylineLogLik_eq_iff","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#skyline-log-lik-eq-iff","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"For `c, w, \u03b8 > 0`, `\u2113(\u03b8) = \u2113(c w)` iff `\u03b8 = c w`.","title":"Uniqueness of the skyline maximizer"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/sum-skyline-log-lik-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.sum_skylineLogLik_le","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#sum-skyline-log-lik-le","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"For positive `c i`, `w i`, `\u03b8 i` over a finite set of intervals, `\u2211 \u2113_i(\u03b8_i) \u2264 \u2211 \u2113_i(c_i w_i)`.","title":"The skyline estimates jointly maximize the likelihood"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/sum-skyline-log-lik-eq-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.sum_skylineLogLik_eq_iff","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#sum-skyline-log-lik-eq-iff","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"Equality in `sum_skylineLogLik_le` holds iff `\u03b8 i = c i w i` for every interval.","title":"Uniqueness of the joint maximizer"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/skyline-estimate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.skylineEstimate","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#skyline-estimate","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"The classic skyline estimate `\u03b8\u0302_k = C(k, 2) w_k` for the interval with `k` lineages.","title":"Classic skyline estimate of population size"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/integral-skyline-estimate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SkylineTimes.integral_skylineEstimate","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#integral-skyline-estimate","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"`E[C(k, 2) T k] = \u03b8 k`.","title":"The skyline estimate is unbiased"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/variance-skyline-estimate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SkylineTimes.variance_skylineEstimate","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#variance-skyline-estimate","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"`Var[C(k, 2) T k] = \u03b8 k\u00b2`.","title":"Variance of the skyline estimate"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/i-indep-fun-skyline-estimate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SkylineTimes.iIndepFun_skylineEstimate","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#i-indep-fun-skyline-estimate","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"The skyline estimates are independent across intervals.","title":"The skyline estimates are independent"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/skyline-pi-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.skylinePiMeasure","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#skyline-pi-measure","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"The canonical probability space for the skyline model: the product of the exponential laws `Exp(C(k, 2) / \u03b8 k)`, `2 \u2264 k \u2264 n`.","title":"Canonical probability space for the skyline model"},{"id":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size/of-pi","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.SkylineTimes.ofPi","parent":45411,"result":true,"route":"../roadmap/Biology/coalescent-theory/skyline-plot-estimates-of-population-size/index.html#of-pi","scope":"Biology/coalescent-theory/skyline-plot-estimates-of-population-size","status":"fully_proved","summary":"The coordinate projections of `skylinePiMeasure n \u03b8` form a skyline model whenever the `\u03b8 k` are positive; in particular `SkylineTimes` is inhabited.","title":"Coordinate projections form a skyline model"},{"data_url":"scopes/Biology/coalescent-theory/structured-coalescent.json","id":"Biology/coalescent-theory/structured-coalescent","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","title":"Structured coalescent"},{"id":"Biology/coalescent-theory/structured-coalescent/pair-state","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.PairState","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#pair-state","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"The state of two ancestral lineages in the island model: in the same deme, in different demes, or coalesced (absorbing).","title":"State of two lineages in the island model"},{"id":"Biology/coalescent-theory/structured-coalescent/sum-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.PairState.sum_eq","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#sum-eq","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"A sum over the three states.","title":"Sum over the island-model pair states"},{"id":"Biology/coalescent-theory/structured-coalescent/rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.rate","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#rate","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"The transition rates `q(s, t)` (`s \u2260 t`) of the two-lineage structured coalescent in the finite island model with `d` demes and `M = 4Nm`, time in units of `2N` generations: same deme \u2192 coalesced at rate `1`, same deme \u2192 different demes at rate `M`, different demes \u2192 same deme at rate `M / (d - 1)`; all other rates (and the diagonal) are `0`. Meaningful for `d \u2265 2` and `M \u2265 0`.","title":"Transition rates of the two-lineage structured coalescent"},{"id":"Biology/coalescent-theory/structured-coalescent/exit-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.exitRate","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#exit-rate","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"The total rate `q\u209b = \u2211\u209c q(s, t)` of leaving the state `s`.","title":"Exit rate of a structured-coalescent state"},{"id":"Biology/coalescent-theory/structured-coalescent/first-step-equations","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.FirstStepEquations","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#first-step-equations","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"The first-step (backward Kolmogorov) equations for the mean time to coalescence: `h coalesced = 0`, and at each transient state `s`, `\u2211\u209c q(s, t) (h t - h s) = -1`. (The vector of mean absorption times of a finite absorbing continuous-time Markov chain satisfies these equations; this file does not construct the chain.)","title":"First-step equations for mean coalescence time"},{"id":"Biology/coalescent-theory/structured-coalescent/within-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.withinTime","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#within-time","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"The mean coalescence time `T\u2080 = d` of two lineages in the same deme (units of `2N` generations).","title":"Mean coalescence time within a deme"},{"id":"Biology/coalescent-theory/structured-coalescent/between-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.betweenTime","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#between-time","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"The mean coalescence time `T\u2081 = d + (d - 1) / M` of two lineages in different demes (units of `2N` generations). Meaningful for `M > 0`; `betweenTime d 0 = d` is a junk value (the true mean is infinite).","title":"Mean coalescence time between demes"},{"id":"Biology/coalescent-theory/structured-coalescent/mean-coalescence-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.meanCoalescenceTime","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#mean-coalescence-time","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"The closed-form mean time to coalescence from each state: `T\u2080`, `T\u2081`, and `0` once coalesced. `firstStepEquations_iff` shows it is the unique solution of the first-step equations.","title":"Closed-form mean coalescence times in the island model"},{"id":"Biology/coalescent-theory/structured-coalescent/first-step-equations-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.firstStepEquations_iff","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#first-step-equations-iff","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"For `d \u2265 2` demes and `M = 4Nm > 0`, the first-step equations have the unique solution `T\u2080 = d`, `T\u2081 = d + (d - 1) / M`. In particular `T\u2080` does not depend on the migration rate.","title":"Mean coalescence times in the finite island model (Strobeck 1987; Slatkin 1991)"},{"id":"Biology/coalescent-theory/structured-coalescent/mean-coalescence-time-eq-first-step","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.meanCoalescenceTime_eq_firstStep","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#mean-coalescence-time-eq-first-step","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"At each transient state `s` the exit rate is positive and the mean coalescence time is the mean holding time `1 / q\u209b` plus the mean time from the next state, `h s = 1 / q\u209b + \u2211\u209c (q(s, t) / q\u209b) h t`.","title":"First-step analysis in jump-chain form"},{"id":"Biology/coalescent-theory/structured-coalescent/total-time","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.totalTime","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#total-time","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"Slatkin's mean coalescence time `T\u0304 = T\u2080 / d + (d - 1) T\u2081 / d` of two genes drawn from two demes chosen independently and uniformly at random.","title":"Slatkin's mean total coalescence time"},{"id":"Biology/coalescent-theory/structured-coalescent/slatkin-fst","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.slatkinFST","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#slatkin-fst","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"Slatkin's (1991) `F_ST = (T\u0304 - T\u2080) / T\u0304` for the finite island model.","title":"Slatkin's F_ST for the finite island model"},{"id":"Biology/coalescent-theory/structured-coalescent/total-time-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.totalTime_eq","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#total-time-eq","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"`T\u0304 = d + (d - 1)\u00b2 / (d M)` for `d \u2265 1`.","title":"Closed form of Slatkin's mean total coalescence time"},{"id":"Biology/coalescent-theory/structured-coalescent/total-time-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.totalTime_pos","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#total-time-pos","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"The denominator of Slatkin's `F_ST` is positive: `0 < T\u0304` for `d \u2265 1` and `M > 0`.","title":"Positivity of Slatkin's mean total coalescence time"},{"id":"Biology/coalescent-theory/structured-coalescent/slatkin-fst-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.slatkinFST_eq","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#slatkin-fst-eq","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"For `d \u2265 2` demes and `M = 4Nm > 0`, `F_ST = 1 / (1 + M d\u00b2 / (d - 1)\u00b2)`.","title":"Slatkin's `F_ST` for the finite island model (Slatkin 1991)"},{"id":"Biology/coalescent-theory/structured-coalescent/slatkin-fst-mem-ioo","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.slatkinFST_mem_Ioo","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#slatkin-fst-mem-ioo","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"Slatkin's `F_ST` lies in `(0, 1)` for `d \u2265 2` and `M > 0`.","title":"Slatkin's F_ST lies strictly between 0 and 1"},{"id":"Biology/coalescent-theory/structured-coalescent/tendsto-slatkin-fst","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.StructuredCoalescent.tendsto_slatkinFST","parent":45424,"result":true,"route":"../roadmap/Biology/coalescent-theory/structured-coalescent/index.html#tendsto-slatkin-fst","scope":"Biology/coalescent-theory/structured-coalescent","status":"fully_proved","summary":"As the number of demes grows, Slatkin's `F_ST` tends to Wright's `1 / (1 + M) = 1 / (1 + 4Nm)` for every `M > 0`.","title":"Many demes (Slatkin 1991)"},{"data_url":"scopes/Biology/coalescent-theory/two-locus-coalescent.json","id":"Biology/coalescent-theory/two-locus-coalescent","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","title":"Two-locus coalescent"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-state","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.TwoLocusState","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-state","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The states of the two-locus ancestral process of a sample of two sequences.","title":"States of the two-locus ancestral process"},{"id":"Biology/coalescent-theory/two-locus-coalescent/sum-univ","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.TwoLocusState.sum_univ","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#sum-univ","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"A sum over the six states.","title":"Sum over the two-locus ancestral states"},{"id":"Biology/coalescent-theory/two-locus-coalescent/coalesced-a","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.TwoLocusState.coalescedA","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#coalesced-a","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Whether locus `A` has found its most recent common ancestor.","title":"Coalescence indicator at locus A"},{"id":"Biology/coalescent-theory/two-locus-coalescent/coalesced-b","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.TwoLocusState.coalescedB","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#coalesced-b","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Whether locus `B` has found its most recent common ancestor.","title":"Coalescence indicator at locus B"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusRate","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-rate","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The rate matrix of the two-locus ancestral process of a sample of two, with time in units of `2 N` generations and `\u03c1 = 4 N r`: coalescence at rate `1` for every pair of ancestral lineages, recombination at rate `\u03c1 / 2` for every lineage ancestral at both loci.","title":"Rate matrix of the two-locus coalescent"},{"id":"Biology/coalescent-theory/two-locus-coalescent/is-rate-matrix-two-locus-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.isRateMatrix_twoLocusRate","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#is-rate-matrix-two-locus-rate","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"For `\u03c1 \u2265 0`, `twoLocusRate \u03c1` is a rate matrix.","title":"The two-locus coalescent rate matrix is a rate matrix"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-denom","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusDenom","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-denom","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The common denominator `\u03c1\u00b2 + 13 \u03c1 + 18` of the two-locus moments.","title":"Common denominator of the two-locus moments"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-denom-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusDenom_pos","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-denom-pos","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The denominator `\u03c1\u00b2 + 13 \u03c1 + 18` is positive for `\u03c1 \u2265 0`.","title":"Positivity of the two-locus moment denominator"},{"id":"Biology/coalescent-theory/two-locus-coalescent/eq-one-of-mean-time-a","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.eq_one_of_meanTimeA","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#eq-one-of-mean-time-a","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"For `\u03c1 \u2265 0`, the backward equations of `E_i T_A` (`m i = 0` where `A` has coalesced, `(Q m) i = -1` elsewhere) force `m i = 1` in every state where `A` has not coalesced: locus `A` alone is Kingman's coalescent of two genes.","title":"Mean coalescence time at locus `A`"},{"id":"Biology/coalescent-theory/two-locus-coalescent/eq-one-of-mean-time-b","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.eq_one_of_meanTimeB","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#eq-one-of-mean-time-b","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Mean coalescence time at locus `B` (the mirror image of `eq_one_of_meanTimeA`).","title":"Mean coalescence time at locus `B`"},{"id":"Biology/coalescent-theory/two-locus-coalescent/eq-two-of-second-moment-a","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.eq_two_of_secondMomentA","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#eq-two-of-second-moment-a","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"If `m` solves the mean equations of `eq_one_of_meanTimeA` and `g` the second-moment equations (`g i = 0` where `A` has coalesced, `(Q g) i = -2 m i` elsewhere), then `g i = 2` where `A` has not coalesced, so the variance read off from these equations is `2 - 1\u00b2 = 1`.","title":"Second moment of `T_A`"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-cov","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusCov","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-cov","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The covariance `Cov(T_A, T_B) = (\u03c1 + 18) / (\u03c1\u00b2 + 13 \u03c1 + 18)` of the coalescence times at two loci of two sequences sampled linked (Hudson 1983; Kaplan & Hudson 1985).","title":"Covariance of coalescence times at two linked loci"},{"id":"Biology/coalescent-theory/two-locus-coalescent/covariance-of-product-moment","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.covariance_of_productMoment","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#covariance-of-product-moment","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Covariance of coalescence times (Hudson 1983; Kaplan & Hudson 1985). Let `mA`, `mB` solve the mean equations of the two loci and `f` the product-moment equations (`f i = 0` where either locus has coalesced, `(Q f) i = -(mA i + mB i)` where neither has). Then `f - mA mB` is `(\u03c1 + 18) / (\u03c1\u00b2 + 13\u03c1 + 18)` from `linked`, `6 / (\u03c1\u00b2 + 13\u03c1 + 18)` from `partlySplit` and `4 / (\u03c1\u00b2 + 13\u03c1 + 18)` from `split`. This is a statement\u2026","title":"Covariance of coalescence times"},{"id":"Biology/coalescent-theory/two-locus-coalescent/product-moment","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.productMoment","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#product-moment","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The explicit solution of the product-moment equations: `1 + Cov` in the three states where neither locus has coalesced, `0` elsewhere.","title":"Solution of the two-locus product-moment equations"},{"id":"Biology/coalescent-theory/two-locus-coalescent/mean-time-a","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.meanTimeA","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#mean-time-a","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The mean coalescence time at locus `A`: `1` before `A` coalesces, `0` after.","title":"Mean coalescence time at locus A"},{"id":"Biology/coalescent-theory/two-locus-coalescent/mean-time-b","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.meanTimeB","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#mean-time-b","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The mean coalescence time at locus `B`: `1` before `B` coalesces, `0` after.","title":"Mean coalescence time at locus B"},{"id":"Biology/coalescent-theory/two-locus-coalescent/product-moment-solution","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.productMoment_solution","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#product-moment-solution","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The systems of `covariance_of_productMoment` are solvable: `meanTimeA`, `meanTimeB` and `productMoment \u03c1` satisfy them.","title":"The explicit two-locus moments solve the moment equations"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-cov-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusCov_zero","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-cov-zero","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Complete linkage: `Cov(T_A, T_B) = 1 = Var T_A`.","title":"Two-locus covariance under complete linkage"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-cov-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusCov_pos","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-cov-pos","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The covariance is positive for `\u03c1 \u2265 0`.","title":"Positivity of the two-locus covariance"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-cov-le-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusCov_le_one","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-cov-le-one","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The covariance is at most `Var T_A = 1` for `\u03c1 \u2265 0` (outside this domain the formula can exceed `1`; e.g. `twoLocusCov (-1) = 17 / 6`).","title":"Two-locus covariance is at most one"},{"id":"Biology/coalescent-theory/two-locus-coalescent/antitone-on-two-locus-cov","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.antitoneOn_twoLocusCov","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#antitone-on-two-locus-cov","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The covariance decreases with the recombination rate.","title":"Two-locus covariance decreases with recombination rate"},{"id":"Biology/coalescent-theory/two-locus-coalescent/tendsto-two-locus-cov-at-top","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.tendsto_twoLocusCov_atTop","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#tendsto-two-locus-cov-at-top","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Free recombination: `Cov(T_A, T_B) \u2192 0` as `\u03c1 \u2192 \u221e`.","title":"Two-locus covariance vanishes under free recombination"},{"id":"Biology/coalescent-theory/two-locus-coalescent/diseq-poly","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.diseqPoly","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#diseq-poly","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The linkage disequilibrium `D = x_AB x_ab - x_Ab x_aB` as a polynomial.","title":"Linkage disequilibrium polynomial"},{"id":"Biology/coalescent-theory/two-locus-coalescent/freq-fst-poly","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.freqFstPoly","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#freq-fst-poly","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The frequency `p = x_AB + x_Ab` of allele `A` as a polynomial.","title":"Frequency of allele A as a polynomial"},{"id":"Biology/coalescent-theory/two-locus-coalescent/freq-snd-poly","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.freqSndPoly","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#freq-snd-poly","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The frequency `q = x_AB + x_aB` of allele `B` as a polynomial.","title":"Frequency of allele B as a polynomial"},{"id":"Biology/coalescent-theory/two-locus-coalescent/diseq-zpoly","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.diseqZPoly","parent":45442,"reading_title":{"format":"inline-parts-v1","parts":[["text","Weighted disequilibrium polynomial "],["math","\\(D(1-2p)(1-2q)\\)"]],"source":"Weighted disequilibrium polynomial $D(1-2p)(1-2q)$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#diseq-zpoly","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"`D (1 - 2p)(1 - 2q)` as a polynomial.","title":"Weighted disequilibrium polynomial $D(1-2p)(1-2q)$"},{"id":"Biology/coalescent-theory/two-locus-coalescent/het-prod-poly","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.hetProdPoly","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#het-prod-poly","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"`\u03a0 = p (1 - p) q (1 - q)` as a polynomial.","title":"Product of heterozygosities polynomial"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusGenerator","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-generator","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The generator of the two-locus Wright\u2013Fisher diffusion on polynomials in the gamete frequencies, time in units of `2 N` generations, `\u03c1 = 4 N r`, symmetric mutation at rate `\u03bd` per locus in each direction (`\u03bd = \u03b8 / 2`, `\u03b8 = 4 N u`): `L f = \u00bd \u2211 x_g (\u03b4_{gh} - x_h) \u2202_g \u2202_h f + \u2211 (-(\u03c1/2) \u03b5_g D + \u03bd (x_{g'} + x_{g''} - 2 x_g)) \u2202_g f`.","title":"Generator of the two-locus Wright\u2013Fisher diffusion"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-generator-x","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusGenerator_X","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-generator-x","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Without mutation the drift of a gamete frequency is `2 N` times its per-generation change under `Linkage.recombine` when `\u03c1 = 4 N r`.","title":"Drift of gamete frequencies matches recombination"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-generator-diseq-sq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusGenerator_diseqSq","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-generator-diseq-sq","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"On the simplex, `L D\u00b2 = -(3 + \u03c1 + 8\u03bd) D\u00b2 + D (1 - 2p)(1 - 2q) + p (1 - p) q (1 - q)` (for all real `\u03c1`, `\u03bd`).","title":"Hill\u2013Robertson / Ohta\u2013Kimura, first row"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-generator-diseq-z","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusGenerator_diseqZ","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-generator-diseq-z","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"On the simplex, `L (D z) = 4 D\u00b2 - (5 + \u03c1/2 + 8\u03bd) D z` with `z = (1 - 2p)(1 - 2q)`.","title":"Hill\u2013Robertson / Ohta\u2013Kimura, second row"},{"id":"Biology/coalescent-theory/two-locus-coalescent/two-locus-generator-het-prod","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.twoLocusGenerator_hetProd","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#two-locus-generator-het-prod","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"On the simplex, `L \u03a0 = D z - (2 + 8\u03bd) \u03a0 + \u03bd (p (1 - p) + q (1 - q))`; for `\u03bd = 0` the three rows close.","title":"Hill\u2013Robertson / Ohta\u2013Kimura, third row"},{"id":"Biology/coalescent-theory/two-locus-coalescent/is-stationary-law","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.IsStationaryLaw","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#is-stationary-law","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"A stationary law of the two-locus diffusion in the weak (basic adjoint) sense: a probability measure on the gamete-frequency simplex with `\u222b L f d\u03bc = 0` for every polynomial `f`.","title":"Stationary law of the two-locus diffusion"},{"id":"Biology/coalescent-theory/two-locus-coalescent/sigma-dsq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.sigmaDSq","parent":45442,"reading_title":{"format":"inline-parts-v1","parts":[["text","Stationary standardized linkage disequilibrium "],["math","\\(\\sigma_d^2\\)"]],"source":"Stationary standardized linkage disequilibrium $\\sigma_d^2$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#sigma-dsq","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The stationary standardized disequilibrium `\u03c3_d\u00b2 = (10 + \u03c1 + 16\u03bd) / (22 + 13\u03c1 + \u03c1\u00b2 + 128\u03bd + 24\u03c1\u03bd + 128\u03bd\u00b2)`.","title":"Stationary standardized linkage disequilibrium $\\sigma_d^2$"},{"id":"Biology/coalescent-theory/two-locus-coalescent/sigma-dsq-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.IsStationaryLaw.sigmaDSq_eq","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#sigma-dsq-eq","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"For a stationary law with `\u03c1, \u03bd \u2265 0` and `E \u03a0 > 0`, `E D\u00b2 / E[p (1 - p) q (1 - q)] = sigmaDSq \u03c1 \u03bd`.","title":"Ohta\u2013Kimura `\u03c3_d\u00b2`"},{"id":"Biology/coalescent-theory/two-locus-coalescent/sigma-dsq-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.sigmaDSq_zero","parent":45442,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ohta\u2013Kimura formula for "],["math","\\(\\sigma_d^2\\)"],["text"," without mutation"]],"source":"Ohta\u2013Kimura formula for $\\sigma_d^2$ without mutation"},"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#sigma-dsq-zero","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"The low-mutation value `\u03c3_d\u00b2 = (10 + \u03c1) / (22 + 13\u03c1 + \u03c1\u00b2)` (Ohta & Kimura 1971).","title":"Ohta\u2013Kimura formula for $\\sigma_d^2$ without mutation"},{"id":"Biology/coalescent-theory/two-locus-coalescent/tendsto-sigma-dsq-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescent.tendsto_sigmaDSq_zero","parent":45442,"reading_title":{"format":"inline-parts-v1","parts":[["text","Low-mutation limit of "],["math","\\(\\sigma_d^2\\)"]],"source":"Low-mutation limit of $\\sigma_d^2$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#tendsto-sigma-dsq-zero","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"`\u03c3_d\u00b2 \u2192 (10 + \u03c1) / (22 + 13\u03c1 + \u03c1\u00b2)` as the mutation rate tends to `0`, for `\u03c1 \u2265 0`.","title":"Low-mutation limit of $\\sigma_d^2$"},{"id":"Biology/coalescent-theory/two-locus-coalescent/lintegral-coalescence-time-a","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescentPath.lintegral_coalescenceTimeA","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#lintegral-coalescence-time-a","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"For the two-locus ancestral process started from two linked sequences, the hitting time `T_A` of the states where locus `A` has coalesced has mean `1`.","title":"Mean coalescence time at locus `A`, path form"},{"id":"Biology/coalescent-theory/two-locus-coalescent/lintegral-coalescence-time-b","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescentPath.lintegral_coalescenceTimeB","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#lintegral-coalescence-time-b","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"From two linked sequences, the hitting time `T_B` of the states where locus `B` has coalesced has mean `1`.","title":"Mean coalescence time at locus `B`, path form"},{"id":"Biology/coalescent-theory/two-locus-coalescent/lintegral-coalescence-time-a-mul-coalesc","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescentPath.lintegral_coalescenceTimeA_mul_coalescenceTimeB","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#lintegral-coalescence-time-a-mul-coalesc","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Product moment of the two coalescence times, path form (Hudson 1983; Kaplan & Hudson 1985): from two linked sequences, `E (T_A T_B) = 1 + (\u03c1 + 18) / (\u03c1\u00b2 + 13\u03c1 + 18)`.","title":"Product moment of the two coalescence times, path form"},{"id":"Biology/coalescent-theory/two-locus-coalescent/covariance-coalescence-times","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TwoLocusCoalescentPath.covariance_coalescenceTimes","parent":45442,"result":true,"route":"../roadmap/Biology/coalescent-theory/two-locus-coalescent/index.html#covariance-coalescence-times","scope":"Biology/coalescent-theory/two-locus-coalescent","status":"fully_proved","summary":"Covariance of the coalescence times, path form (Hudson 1983; Kaplan & Hudson 1985): for the two-locus ancestral process started from two linked sequences, `E (T_A T_B) - E T_A E T_B = (\u03c1 + 18) / (\u03c1\u00b2 + 13\u03c1 + 18)`, with all three expectations finite lower Lebesgue integrals of the hitting times.","title":"Covariance of the coalescence times, path form"},{"data_url":"scopes/Biology/coalescent-theory/coalescents.json","id":"Biology/coalescent-theory/coalescents","kind":"scope","navigation":"module","parent":45142,"result":false,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html","scope":"Biology/coalescent-theory/coalescents","status":"can_prove","title":"\u039b-coalescents"},{"id":"Biology/coalescent-theory/coalescents/merger-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.mergerRate","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#merger-rate","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"Pitman's merger rate `\u03bb_{b,k} = \u222b x ^ (k - 2) (1 - x) ^ (b - k) \u039b(dx)`: while there are `b` blocks, the rate at which a given set of `k` of them merges into one block, for `2 \u2264 k \u2264 b` (Pitman 1999, Theorem 1; Sagitov 1999). Powers are natural-number powers, so `0 ^ 0 = 1`. Outside `2 \u2264 k \u2264 b` no such merger exists, and the rate is defined to be `0`.","title":"Pitman's merger rate"},{"id":"Biology/coalescent-theory/coalescents/total-merger-rate","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.totalMergerRate","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#total-merger-rate","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The total merger rate `\u03bb_b = \u2211_{k=2}^b (b.choose k) \u03bb_{b,k}` while there are `b` blocks (Pitman 1999; Berestycki 2009, \u00a73.1).","title":"Total merger rate of a \u039b-coalescent"},{"id":"Biology/coalescent-theory/coalescents/block-counting-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.blockCountingGenerator","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#block-counting-generator","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The generator of the block-counting process of the \u039b-coalescent on the states `0, \u2026, n` (state `b` means `b` blocks): from `b` blocks the process jumps to `j < b` blocks at rate `(b.choose (b + 1 - j)) \u03bb_{b, b + 1 - j}` (a `(b + 1 - j)`-merger), and the diagonal is `-\u03bb_b`. States `0` and `1` are absorbing (state `0` is never reached from `b \u2265 1`).","title":"Generator of the \u039b-coalescent block-counting process"},{"id":"Biology/coalescent-theory/coalescents/merger-rate-of-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.mergerRate_of_le","parent":45484,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pitman's integral formula for merger rates when "],["math","\\(2 \\le k \\le b\\)"]],"source":"Pitman's integral formula for merger rates when $2 \\le k \\le b$"},"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#merger-rate-of-le","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The merger rate inside its domain.","title":"Pitman's integral formula for merger rates when $2 \\le k \\le b$"},{"id":"Biology/coalescent-theory/coalescents/merger-rate-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.mergerRate_nonneg","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#merger-rate-nonneg","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"Merger rates are nonnegative.","title":"Nonnegativity of \u039b-coalescent merger rates"},{"id":"Biology/coalescent-theory/coalescents/merger-rate-two-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.mergerRate_two_two","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#merger-rate-two-two","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"Two blocks merge at rate `\u039b [0, 1]`; `\u039b` is often normalized to a probability measure, so that this pair rate is `1` as in Kingman's coalescent.","title":"Pair merger rate of a \u039b-coalescent"},{"id":"Biology/coalescent-theory/coalescents/total-merger-rate-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.totalMergerRate_nonneg","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#total-merger-rate-nonneg","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The total merger rate is nonnegative.","title":"Nonnegativity of the total merger rate"},{"id":"Biology/coalescent-theory/coalescents/is-rate-matrix-block-counting-generator","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.isRateMatrix_blockCountingGenerator","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#is-rate-matrix-block-counting-generator","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The block-counting generator is a rate matrix: its jump rates are nonnegative and each row sums to `0`, since the jump rates out of `b` blocks add up to `\u03bb_b`.","title":"\u039b-coalescent block-counting generator is a rate matrix"},{"id":"Biology/coalescent-theory/coalescents/merger-rate-le-measure-univ","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.mergerRate_le_measure_univ","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#merger-rate-le-measure-univ","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"A merger rate is at most the total mass of `\u039b`.","title":"Merger rates are bounded by the total mass of \u039b"},{"id":"Biology/coalescent-theory/coalescents/merger-rate-consistent","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.mergerRate_consistent","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#merger-rate-consistent","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"Pitman's consistency relation `\u03bb_{b,k} = \u03bb_{b+1,k} + \u03bb_{b+1,k+1}` for `2 \u2264 k \u2264 b` (Pitman 1999; Berestycki 2009, \u00a73.1): a given `k`-set of `b` blocks merges, with an extra block present, either without it or together with it.","title":"Pitman's consistency relation"},{"id":"Biology/coalescent-theory/coalescents/sq-mul-sum-choose-mul-pow","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.sq_mul_sum_choose_mul_pow","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#sq-mul-sum-choose-mul-pow","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The polynomial identity behind the closed form of the total merger rate: `x ^ 2 \u2211_{k=2}^b (b.choose k) x ^ (k - 2) (1 - x) ^ (b - k) = 1 - (1 - x) ^ b - b x (1 - x) ^ (b - 1)` for every real `x` and every `b` (binomial theorem).","title":"Binomial identity behind the total merger rate"},{"id":"Biology/coalescent-theory/coalescents/total-merger-rate-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.totalMergerRate_eq","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#total-merger-rate-eq","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The total merger rate in closed form (Berestycki 2009, \u00a73.1): `\u03bb_b = (b.choose 2) \u039b {0} + \u222b_{(0,1]} (1 - (1 - x) ^ b - b x (1 - x) ^ (b - 1)) / x ^ 2 \u039b(dx)`. The atom at `0` (the Kingman part) is split off, so no division by `0` occurs.","title":"The total merger rate in closed form"},{"id":"Biology/coalescent-theory/coalescents/total-merger-rate-succ","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.totalMergerRate_succ","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#total-merger-rate-succ","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"Adding a block raises the total merger rate by `b \u03bb_{b+1,2}`: `\u03bb_{b+1} = \u03bb_b + b \u03bb_{b+1,2}`. The new events are exactly the binary mergers of the new block with one of the `b` old ones (a consequence of the consistency relation).","title":"Recursion for the total merger rate"},{"id":"Biology/coalescent-theory/coalescents/total-merger-rate-mono","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.totalMergerRate_mono","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#total-merger-rate-mono","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The total merger rate `\u03bb_b` is nondecreasing in the number of blocks `b`.","title":"Monotonicity of the total merger rate"},{"id":"Biology/coalescent-theory/coalescents/eq-of-merger-rate-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.eq_of_mergerRate_eq","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#eq-of-merger-rate-eq","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"A finite measure on `[0, 1]` is determined by its merger rates; indeed by the rates `\u03bb_{k,k} = \u222b x ^ (k - 2) \u039b(dx)` of complete mergers, which are its moments (Pitman 1999, Theorem 1).","title":"Uniqueness in Pitman's representation"},{"id":"Biology/coalescent-theory/coalescents/merger-rate-dirac-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.mergerRate_dirac_zero","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#merger-rate-dirac-zero","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"Kingman's coalescent is the \u039b-coalescent with `\u039b = \u03b4\u2080`: only binary mergers occur, each pair merging at rate `1` (Pitman 1999; Berestycki 2009, \u00a73.1).","title":"Kingman's coalescent"},{"id":"Biology/coalescent-theory/coalescents/total-merger-rate-dirac-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.totalMergerRate_dirac_zero","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#total-merger-rate-dirac-zero","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"For Kingman's coalescent the total merger rate is the number of pairs, `\u03bb_b = b.choose 2`.","title":"Total merger rate of Kingman's coalescent"},{"id":"Biology/coalescent-theory/coalescents/merger-rate-volume","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.mergerRate_volume","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#merger-rate-volume","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"The Bolthausen\u2013Sznitman coalescent (`\u039b` = Lebesgue measure on `[0, 1]`) has merger rates given by the Beta integral `\u03bb_{b,k} = (k - 2)! (b - k)! / (b - 1)!` for `2 \u2264 k \u2264 b` (Bolthausen\u2013Sznitman 1998; Pitman 1999; Berestycki 2009, \u00a73.1).","title":"Merger rates of the Bolthausen\u2013Sznitman coalescent"},{"id":"Biology/coalescent-theory/coalescents/total-merger-rate-volume","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescent.totalMergerRate_volume","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#total-merger-rate-volume","scope":"Biology/coalescent-theory/coalescents","status":"fully_proved","summary":"For the Bolthausen\u2013Sznitman coalescent the total merger rate is `\u03bb_b = b - 1` (`b \u2265 1`).","title":"Total merger rate of the Bolthausen\u2013Sznitman coalescent"},{"id":"Biology/coalescent-theory/coalescents/exists-measure-of-consistent","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.LambdaCoalescentWanted.exists_measure_of_consistent","parent":45484,"result":true,"route":"../roadmap/Biology/coalescent-theory/coalescents/index.html#exists-measure-of-consistent","scope":"Biology/coalescent-theory/coalescents","status":"can_prove","summary":"Pitman's representation theorem, existence half (Pitman 1999, Theorem 1): every consistent family of nonnegative merger rates `r b k`, `2 \u2264 k \u2264 b`, is the family of merger rates `\u03bb_{b,k}` of some finite measure `\u039b` on `[0, 1]`. (Uniqueness of `\u039b` is `LambdaCoalescent.eq_of_mergerRate_eq`.)","title":"Pitman's representation theorem, existence half"},{"data_url":"scopes/Biology/bioinformatics.json","id":"Biology/bioinformatics","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/bioinformatics/index.html","scope":"Biology/bioinformatics","status":"fully_proved","summary":"Algorithms and statistical models on biological sequences and data: sequence alignment, genome assembly, motifs, RNA secondary structure, clustering and network reconstruction.","title":"Bioinformatics"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment.json","id":"Biology/bioinformatics/sequence-alignment","kind":"scope","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/index.html","scope":"Biology/bioinformatics/sequence-alignment","status":"fully_proved","summary":"Pairwise and multiple alignment of biological sequences: Needleman\u2013Wunsch and Smith\u2013Waterman, affine gaps, banded, linear-space, semi-global and wavefront algorithms.","title":"Sequence alignment"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment.json","id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","title":"Pairwise sequence alignment"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/alignment","kind":"node","lean":"Cslib.Algorithms.Lean.Alignment","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#alignment","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"An alignment over an alphabet $\\alpha$ is a finite list of columns, where each column is a pair $(u, v)$ and each of $u$ and $v$ is either a letter of $\\alpha$ or a gap.","title":"Alignment"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/is-alignment","kind":"node","lean":"Cslib.Algorithms.Lean.IsAlignment","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#is-alignment","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"A list $A$ of columns, each a pair of optional letters, is an alignment of sequences $x$ and $y$ when no column has both entries empty, the letters present in the first entries of the columns, read in order, form $x$, and the letters present in the second entries, read in order, form $y$.","title":"Valid alignment of two sequences"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/column-score","kind":"node","lean":"Cslib.Algorithms.Lean.columnScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#column-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Given a substitution score $\\sigma$, which assigns an integer $\\sigma(a, b)$ to all letters $a, b \\in \\alpha$, and a gap score $g \\in \\mathbb{Z}$, a column pairing letters $a$ and $b$ scores $\\sigma(a, b)$, a column pairing a letter with a gap (on either side) scores $g$, and a column with a gap on both sides scores $0$.","title":"Score of an alignment column"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/alignment-score","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#alignment-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Given a substitution score $\\sigma$, which assigns an integer $\\sigma(a, b)$ to all letters $a, b \\in \\alpha$, and a gap score $g \\in \\mathbb{Z}$, the score of an alignment $A$ over $\\alpha$ is the sum, over all columns of $A$, of the score of the column with respect to $\\sigma$ and $g$.","title":"Score of an alignment"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/is-alignment-append","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_append","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#is-alignment-append","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"If $A_1$ is an alignment of sequences $x_1$ and $y_1$, and $A_2$ is an alignment of sequences $x_2$ and $y_2$, then the concatenation of $A_1$ and $A_2$ is an alignment of the concatenation of $x_1$ and $x_2$ and the concatenation of $y_1$ and $y_2$.","title":"Concatenation of alignments"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-aux","kind":"node","lean":"Cslib.Algorithms.Lean.nwScoreAux","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-aux","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$, and write $|x|$ for the length of a sequence $x$ and $ax$ for the sequence with first element $a$ followed by $x$. For a fuel $n \\in \\mathbb{N}$ and sequences $x, y$ over $\\alpha$, the value $N_n(x, y) \\in \\mathbb{Z}$ is defined by $N_0(x, y) = 0$, by $N_{n+1}(x, y) = g\\,|y|$ if $x$ is empty and\u2026","title":"Fuel-bounded Needleman\u2013Wunsch recursion"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Given a type $\\alpha$, a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$, the Needleman\u2013Wunsch score $\\mathrm{NW}(xs, ys)$ of two lists over $\\alpha$ is defined by recursion on their total length: if one of the lists is empty, it is $g$ times the length of the other; and for lists $a\\,xs$ and $b\\,ys$ with heads $a$, $b$ and tails $xs$, $ys$,","title":"Needleman\u2013Wunsch score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-nil-left","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_nil_left","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-nil-left","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$, gap score $g \\in \\mathbb{Z}$ and sequence $y$ over $\\alpha$ of length $|y|$, the Needleman\u2013Wunsch score of the empty sequence and $y$ with respect to $\\sigma$ and $g$ is $g\\,|y|$: one gap per letter.","title":"Aligning the empty sequence with a sequence"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-nil-right","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_nil_right","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-nil-right","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$, gap score $g \\in \\mathbb{Z}$ and sequence $x$ over $\\alpha$ of length $|x|$, the Needleman\u2013Wunsch score of $x$ and the empty sequence with respect to $\\sigma$ and $g$ is $g\\,|x|$: one gap per letter.","title":"Aligning a sequence with the empty sequence"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-eq","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_eq","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-eq","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score and $g \\in \\mathbb{Z}$ a gap score, and write $N(x, y)$ for the Needleman\u2013Wunsch score of sequences $x, y$ over $\\alpha$ with respect to $\\sigma$ and $g$. For all letters $a, b \\in \\alpha$ and sequences $x, y$, writing $ax$ for the sequence with first element $a$ followed by $x$,","title":"The Needleman\u2013Wunsch recurrence"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/alignments-aux","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentsAux","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#alignments-aux","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Given a fuel $n \\in \\mathbb{N}$ and sequences $x$ and $y$ over $\\alpha$, this lists alignments column by column from the front. With fuel $0$ the list is empty. With fuel $n + 1$: two empty sequences give the single empty alignment; if only $y$ is nonempty, with first letter $b$ and rest $y'$, each alignment of $x$ and $y'$ with fuel $n$ is preceded by the column pairing a gap with $b$; if only $x$ is nonempty,\u2026","title":"Fuel-bounded enumeration of alignments"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/alignments","kind":"node","lean":"Cslib.Algorithms.Lean.alignments","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#alignments","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For sequences $x$ and $y$ over $\\alpha$, this is the list produced by the fuel-bounded enumeration of alignments of $x$ and $y$ with fuel $|x| + |y| + 1$.","title":"The list of all alignments"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/mem-alignments","kind":"node","lean":"Cslib.Algorithms.Lean.mem_alignments","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#mem-alignments","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For sequences $x$ and $y$ and a list $A$ of columns, $A$ occurs in the enumerated list of all alignments of $x$ and $y$ if and only if $A$ is an alignment of $x$ and $y$.","title":"The enumerated alignments are exactly the valid ones"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/alignment-score-le-nw-score","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentScore_le_nwScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#alignment-score-le-nw-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score and $g \\in \\mathbb{Z}$ a gap score. For all sequences $x$ and $y$ over $\\alpha$ and every valid alignment $A$ of $x$ and $y$, the score of $A$ with respect to $\\sigma$ and $g$ is at most the Needleman\u2013Wunsch score of $x$ and $y$.","title":"Needleman\u2013Wunsch is optimal"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/exists-alignment-score-eq-nw-score","kind":"node","lean":"Cslib.Algorithms.Lean.exists_alignmentScore_eq_nwScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#exists-alignment-score-eq-nw-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score and $g \\in \\mathbb{Z}$ a gap score. For all sequences $x$ and $y$ over $\\alpha$ there is a valid alignment $A$ of $x$ and $y$ whose score with respect to $\\sigma$ and $g$ equals the Needleman\u2013Wunsch score of $x$ and $y$.","title":"Needleman\u2013Wunsch is attained"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-front","kind":"node","lean":"Cslib.Algorithms.Lean.swCellFront","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-front","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$, and write $ax$ for the sequence with first element $a$ followed by $x$. For a fuel $n \\in \\mathbb{N}$ and sequences $x, y$ over $\\alpha$, the value $F_n(x, y) \\in \\mathbb{Z}$ is defined by $F_0(x, y) = 0$, by $F_{n+1}(x, y) = 0$ if $x$ or $y$ is empty, and by","title":"Fuel-bounded Smith\u2013Waterman recursion"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-score","kind":"node","lean":"Cslib.Algorithms.Lean.swCellScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Given a substitution score $\\sigma(a, b) \\in \\mathbb{Z}$ for letters $a, b$ and a gap score $g \\in \\mathbb{Z}$, and writing $sc$ for a sequence $s$ followed by a letter $c$, the terminal Smith\u2013Waterman cell $H(x, y)$ of two sequences $x$ and $y$ is the integer defined by $H(x, y) = 0$ when $x$ or $y$ is empty, and","title":"The terminal Smith\u2013Waterman cell"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-score-nonneg","kind":"node","lean":"Cslib.Algorithms.Lean.swCellScore_nonneg","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-score-nonneg","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma$, every gap score $g$ and all sequences $x$ and $y$, the terminal Smith\u2013Waterman cell $H(x, y)$ satisfies $H(x, y) \\ge 0$.","title":"The terminal Smith\u2013Waterman cell is nonnegative"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-score-nil-left","kind":"node","lean":"Cslib.Algorithms.Lean.swCellScore_nil_left","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-score-nil-left","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For a substitution score $\\sigma$, a gap score $g$ and every sequence $y$, the terminal Smith\u2013Waterman cell of the empty sequence and $y$ is $0$.","title":"The terminal cell vanishes when the first sequence is empty"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-score-nil-right","kind":"node","lean":"Cslib.Algorithms.Lean.swCellScore_nil_right","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-score-nil-right","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For a substitution score $\\sigma$, a gap score $g$ and every sequence $x$, the terminal Smith\u2013Waterman cell of $x$ and the empty sequence is $0$.","title":"The terminal cell vanishes when the second sequence is empty"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-score-eq","kind":"node","lean":"Cslib.Algorithms.Lean.swCellScore_eq","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-score-eq","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ a gap score, and for a sequence $s$ and a letter $c$ write $sc$ for $s$ followed by $c$. For all sequences $x$ and $y$ and all letters $a$ and $b$, the terminal Smith\u2013Waterman cell $H$ satisfies","title":"The Smith\u2013Waterman recurrence"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/suffix-of-snoc","kind":"node","lean":"Cslib.Algorithms.Lean.suffix_of_snoc","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#suffix-of-snoc","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"If $s$ is a suffix of the list obtained by appending an element $a$ at the end of a list $xs$, then either $s$ is empty, or there is a suffix $s'$ of $xs$ such that $s$ is $s'$ with $a$ appended at the end.","title":"Suffixes of a list with an element appended"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/suffix-append-snoc","kind":"node","lean":"Cslib.Algorithms.Lean.suffix_append_snoc","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#suffix-append-snoc","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"If a list $s$ is a suffix of a list $xs$, then for every element $a$, the list $s$ with $a$ appended at the end is a suffix of $xs$ with $a$ appended at the end.","title":"Appending an element preserves suffixes"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/is-alignment-reverse","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_reverse","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#is-alignment-reverse","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For a list of columns $A$ and sequences $x$ and $y$, $A$ is an alignment of the reversal of $x$ and the reversal of $y$ if and only if the reversal of $A$ (its columns in reverse order) is an alignment of $x$ and $y$.","title":"Reversing an alignment"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/alignment-score-reverse","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentScore_reverse","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#alignment-score-reverse","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$, gap score $g \\in \\mathbb{Z}$ and alignment $A$, the alignment obtained by listing the columns of $A$ in reverse order has the same score as $A$.","title":"Reversal preserves alignment score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-reverse","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_reverse","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-reverse","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Reversing both sequences preserves the optimal global score.","title":"Needleman\u2013Wunsch is reversal-invariant"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-snoc","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_snoc","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-snoc","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score and $g \\in \\mathbb{Z}$ a gap score, and write $N(x, y)$ for the Needleman\u2013Wunsch score of sequences $x, y$ over $\\alpha$ with respect to $\\sigma$ and $g$. For all sequences $x, y$ and letters $a, b \\in \\alpha$, writing $xa$ for the sequence $x$ followed by the letter $a$,","title":"The Needleman\u2013Wunsch recurrence, end-peeling"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/list-max","kind":"node","lean":"Cslib.Algorithms.Lean.listMax","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#list-max","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For a list $l$ of integers, this is the largest of $0$ and the entries of $l$; in particular the empty list gives $0$.","title":"Maximum of an integer list with floor zero"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/suffix-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.suffixPairs","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#suffix-pairs","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For lists $xs$ and $ys$, the list of all pairs $(u, v)$ where $u$ is a suffix of $xs$ and $v$ is a suffix of $ys$, the empty lists and $xs$, $ys$ themselves included.","title":"Pairs of suffixes"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/suffix-max","kind":"node","lean":"Cslib.Algorithms.Lean.suffixMax","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#suffix-max","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Given a substitution score $\\sigma$ and an integer gap score $g$, the suffix maximum of lists $xs$ and $ys$ is the maximum of the Needleman\u2013Wunsch scores of all pairs $(u, v)$ with $u$ a suffix of $xs$ and $v$ a suffix of $ys$.","title":"Suffix maximum of Needleman\u2013Wunsch scores"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-le-suffix-max","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_le_suffixMax","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-le-suffix-max","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score and $g \\in \\mathbb{Z}$ a gap score. If $(x', y')$ is a suffix pair of sequences $x$ and $y$ over $\\alpha$, that is, $x'$ is a suffix of $x$ and $y'$ is a suffix of $y$, then the Needleman\u2013Wunsch score of $x'$ and $y'$ is at most the suffix maximum of $x$ and $y$, both with respect to $\\sigma$ and $g$.","title":"Suffix pairs score at most the suffix maximum"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/suffix-max-le","kind":"node","lean":"Cslib.Algorithms.Lean.suffixMax_le","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#suffix-max-le","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score, $g$ an integer gap score, $xs$ and $ys$ lists and $y$ an integer. If every pair $(u, v)$ with $u$ a suffix of $xs$ and $v$ a suffix of $ys$ has Needleman\u2013Wunsch score at most $y$, then the maximum Needleman\u2013Wunsch score over all such suffix pairs is at most $y$.","title":"Upper bounds on the suffix maximum"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/exists-nw-score-eq-suffix-max","kind":"node","lean":"Cslib.Algorithms.Lean.exists_nwScore_eq_suffixMax","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#exists-nw-score-eq-suffix-max","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$, a gap score $g \\in \\mathbb{Z}$ and sequences $x$ and $y$ over $\\alpha$, there is a suffix pair $(x', y')$ of $x$ and $y$, that is, a suffix $x'$ of $x$ and a suffix $y'$ of $y$, whose Needleman\u2013Wunsch score equals the suffix maximum of $x$ and $y$.","title":"The suffix maximum is attained"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/zero-le-suffix-max","kind":"node","lean":"Cslib.Algorithms.Lean.zero_le_suffixMax","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#zero-le-suffix-max","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For every scoring function $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$, every gap score $g \\in \\mathbb{Z}$ and all lists $x$ and $y$ over $\\alpha$, the suffix maximum of $x$ and $y$ (the largest Needleman\u2013Wunsch score, under $\\sigma$ and $g$, of a suffix of $x$ against a suffix of $y$) is at least $0$, since the pair of empty suffixes scores $0$.","title":"The suffix maximum is nonnegative"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/is-alignment-append-some-none","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_append_some_none","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#is-alignment-append-some-none","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"If $A$ is an alignment of sequences $x$ and $y$ and $a$ is a letter, then $A$ followed by the column with $a$ in the first row and a gap in the second row is an alignment of $x$ followed by $a$ and of $y$.","title":"Extending an alignment by a letter of the first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/is-alignment-append-none-some","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_append_none_some","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#is-alignment-append-none-some","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"If $A$ is an alignment of sequences $x$ and $y$ and $b$ is a letter, then $A$ followed by the column with a gap in the first row and $b$ in the second row is an alignment of $x$ and of $y$ followed by $b$.","title":"Extending an alignment by a letter of the second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-le-nw-score-snoc-left","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_le_nwScore_snoc_left","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-le-nw-score-snoc-left","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score, $g \\in \\mathbb{Z}$ a gap score, and write $N(x, y)$ for the Needleman\u2013Wunsch score of sequences $x, y$ over $\\alpha$ with respect to $\\sigma$ and $g$. For all sequences $x, y$ and every letter $a \\in \\alpha$, writing $xa$ for $x$ with $a$ appended at the end, $N(x, y) + g \\le N(xa, y)$.","title":"Appending to the first sequence loses at most one gap"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-le-nw-score-snoc-right","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_le_nwScore_snoc_right","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-le-nw-score-snoc-right","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score, $g \\in \\mathbb{Z}$ a gap score, and write $N(x, y)$ for the Needleman\u2013Wunsch score of sequences $x, y$ over $\\alpha$ with respect to $\\sigma$ and $g$. For all sequences $x, y$ and every letter $b \\in \\alpha$, writing $yb$ for $y$ with $b$ appended at the end, $N(x, y) + g \\le N(x, yb)$.","title":"Appending to the second sequence loses at most one gap"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-le-sw-cell-score-of-suffix","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_le_swCellScore_of_suffix","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-le-sw-cell-score-of-suffix","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score and $g \\le 0$ a gap score. If $x'$ is a suffix of a sequence $x$ and $y'$ is a suffix of a sequence $y$ over $\\alpha$, then the Needleman\u2013Wunsch score of $x'$ and $y'$ is at most the terminal Smith\u2013Waterman cell score of $x$ and $y$, both with respect to $\\sigma$ and $g$.","title":"Suffix pairs score at most the terminal Smith\u2013Waterman cell"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-score-le-suffix-max","kind":"node","lean":"Cslib.Algorithms.Lean.swCellScore_le_suffixMax","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-score-le-suffix-max","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma$, every gap score $g$ of either sign and all sequences $x$ and $y$, the terminal Smith\u2013Waterman cell $H(x, y)$ is at most the maximum Needleman\u2013Wunsch score over all pairs of a suffix of $x$ and a suffix of $y$.","title":"The terminal cell is at most the suffix maximum"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/suffix-max-le-sw-cell-score","kind":"node","lean":"Cslib.Algorithms.Lean.suffixMax_le_swCellScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#suffix-max-le-sw-cell-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g \\le 0$ an integer gap score. For all lists $xs$ and $ys$, the maximum Needleman\u2013Wunsch score over all pairs of a suffix of $xs$ and a suffix of $ys$ is at most the terminal Smith\u2013Waterman cell score of $xs$ and $ys$.","title":"The suffix maximum is at most the terminal Smith\u2013Waterman cell"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-score-eq-suffix-max","kind":"node","lean":"Cslib.Algorithms.Lean.swCellScore_eq_suffixMax","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-score-eq-suffix-max","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For a substitution score $\\sigma$, a gap score $g \\le 0$ and all sequences $x$ and $y$, the terminal Smith\u2013Waterman cell $H(x, y)$ equals the maximum Needleman\u2013Wunsch score over all pairs of a suffix of $x$ and a suffix of $y$ (the empty suffixes included).","title":"For nonpositive gaps, the terminal Smith\u2013Waterman cell is the best suffix-pair score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/prefix-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.prefixPairs","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#prefix-pairs","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For lists $xs$ and $ys$, the list of all pairs $(p, q)$ where $p$ is a prefix of $xs$ and $q$ is a prefix of $ys$, the empty lists and $xs$, $ys$ themselves included.","title":"Pairs of prefixes"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/infix-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.infixPairs","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#infix-pairs","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For sequences $x$ and $y$ over $\\alpha$, the list of infix pairs of $x$ and $y$ is obtained by going through each prefix $p$ of $x$ and each prefix $q$ of $y$ and, for each such $(p, q)$, listing all pairs $(x', y')$ of a suffix $x'$ of $p$ and a suffix $y'$ of $q$.","title":"Infix pairs"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/local-score","kind":"node","lean":"Cslib.Algorithms.Lean.localScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#local-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For an integer substitution score $\\sigma$, an integer gap score $g$ and sequences $x$ and $y$, the local score of $x$ and $y$ is the maximum of the Needleman\u2013Wunsch score of $(x', y')$ over all pairs of an infix $x'$ of $x$ and an infix $y'$ of $y$ (contiguous and possibly empty).","title":"Local alignment score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-score-le-local-score","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_le_localScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-score-le-local-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ be a substitution score and $g \\in \\mathbb{Z}$ a gap score. If $(x', y')$ is an infix pair of sequences $x$ and $y$ over $\\alpha$, that is, $x'$ is a suffix of a prefix of $x$ and $y'$ is a suffix of a prefix of $y$, then the Needleman\u2013Wunsch score of $x'$ and $y'$ is at most the local score of $x$ and $y$, both with respect to $\\sigma$ and $g$.","title":"Infix pairs score at most the local score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/local-score-le","kind":"node","lean":"Cslib.Algorithms.Lean.localScore_le","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#local-score-le","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer substitution score, $g$ an integer gap score, $x$ and $y$ sequences and $c$ an integer. If the Needleman\u2013Wunsch score of every pair $(x', y')$ with $x'$ an infix of $x$ and $y'$ an infix of $y$ is at most $c$, then the local score of $x$ and $y$ is at most $c$.","title":"Upper bounds on the local score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-score","kind":"node","lean":"Cslib.Algorithms.Lean.swScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Given a substitution score $\\sigma$ and a gap score $g$, the Smith\u2013Waterman score of two sequences $x$ and $y$ is the maximum of the terminal Smith\u2013Waterman cell $H(p, q)$ over all pairs of a prefix $p$ of $x$ and a prefix $q$ of $y$ (the empty prefixes included).","title":"The Smith\u2013Waterman score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-cell-score-le-sw-score","kind":"node","lean":"Cslib.Algorithms.Lean.swCellScore_le_swScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-cell-score-le-sw-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For a substitution score $\\sigma$, a gap score $g$, sequences $x$ and $y$, a prefix $p$ of $x$ and a prefix $q$ of $y$, the terminal Smith\u2013Waterman cell $H(p, q)$ is at most the Smith\u2013Waterman score of $x$ and $y$.","title":"Each prefix-pair cell is at most the Smith\u2013Waterman score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/sw-score-le","kind":"node","lean":"Cslib.Algorithms.Lean.swScore_le","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#sw-score-le","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"For a substitution score $\\sigma$, a gap score $g$, sequences $x$ and $y$ and an integer $c$, if the terminal Smith\u2013Waterman cell $H(p, q)$ is at most $c$ for every prefix $p$ of $x$ and every prefix $q$ of $y$, then the Smith\u2013Waterman score of $x$ and $y$ is at most $c$.","title":"Bounding the Smith\u2013Waterman score by its cells"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/local-score-eq-sw-score","kind":"node","lean":"Cslib.Algorithms.Lean.localScore_eq_swScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#local-score-eq-sw-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer-valued substitution score on letters and let $g$ be an integer gap score with $g \\le 0$. Then for all sequences $x$ and $y$, the local score of $x$ and $y$ (the maximum Needleman\u2013Wunsch score over pairs of infixes) equals their Smith\u2013Waterman score (the maximum terminal-cell score over pairs of prefixes).","title":"The local score equals the Smith\u2013Waterman score"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/next-row-aux","kind":"node","lean":"Cslib.Bioinformatics.Alignment.NeedlemanWunsch.nextRowAux","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#next-row-aux","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"The cells `F[i+1][j+1], F[i+1][j+2], \u2026` of a row from the remaining letters `b :: ys` of the second sequence, the remaining cells `up :: prev` of the previous row, the previous row's cell `diag` to the upper left and the cell `left` just computed. A previous row that is too short ends the row early (this does not happen for rows of the table, `nextRow_eq`).","title":"Remaining cells of a Needleman\u2013Wunsch table row"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/next-row","kind":"node","lean":"Cslib.Bioinformatics.Alignment.NeedlemanWunsch.nextRow","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#next-row","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"The row of the table for the prefix extended by `a`, from the previous row: the first cell is one more gap than the previous row's first cell, then the recurrence left to right.","title":"Next row of the Needleman\u2013Wunsch table"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-rows-aux","kind":"node","lean":"Cslib.Bioinformatics.Alignment.NeedlemanWunsch.nwRowsAux","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-rows-aux","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"The rows of the table, filled top to bottom over the letters of the first sequence; the result is the last row. Row `0` is `F[0][j] = j * gap`.","title":"Row-by-row filling of the Needleman\u2013Wunsch table"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-rows","kind":"node","lean":"Cslib.Bioinformatics.Alignment.NeedlemanWunsch.nwRows","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-rows","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"The last row `F[|xs|][\u00b7]` of the Needleman\u2013Wunsch table.","title":"Last row of the Needleman\u2013Wunsch table"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-tab","kind":"node","lean":"Cslib.Bioinformatics.Alignment.NeedlemanWunsch.nwTab","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-tab","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"Its last cell `F[|xs|][|ys|]`.","title":"The Needleman\u2013Wunsch table"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/next-row-eq","kind":"node","lean":"Cslib.Bioinformatics.Alignment.NeedlemanWunsch.nextRow_eq","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#next-row-eq","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"From the row of the prefix `px`, `nextRow` computes the row of `px ++ [a]`: the cells `nwScore (px ++ [a]) py` for the prefixes `py` of `ys`, shortest first.","title":"One row of the table"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-rows-eq","kind":"node","lean":"Cslib.Bioinformatics.Alignment.NeedlemanWunsch.nwRows_eq","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-rows-eq","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"The last row of the table is `F[|xs|][j] = nwScore xs (ys.take j)`: the cells `nwScore xs py` over the prefixes `py` of `ys`, shortest first.","title":"The last row of the table"},{"id":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/nw-tab-eq-nw-score","kind":"node","lean":"Cslib.Bioinformatics.Alignment.NeedlemanWunsch.nwTab_eq_nwScore","parent":45507,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment/index.html#nw-tab-eq-nw-score","scope":"Biology/bioinformatics/sequence-alignment/pairwise-sequence-alignment","status":"fully_proved","summary":"The table computes the optimal global alignment score (Needleman\u2013Wunsch 1970), for every substitution score `\u03c3` and gap score `gap`.","title":"The table computes the optimal global alignment score"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment.json","id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","title":"Banded sequence alignment"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-from","kind":"node","lean":"Cslib.Algorithms.Lean.bandedFrom","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-from","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $w \\in \\mathbb{N}$ be a band width. For natural numbers $i$, $j$ and an alignment $A$, the band check of $A$ with width $w$ from offset $(i, j)$ is true exactly when every prefix $P$ of $A$ (including the empty prefix and $A$ itself) satisfies $|(i + p) - (j + q)| \\le w$, where $p$ and $q$ are the numbers of columns of $P$ whose first, respectively second, entry is present.","title":"Band check from a starting offset"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/is-banded","kind":"node","lean":"Cslib.Algorithms.Lean.IsBanded","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#is-banded","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $w \\in \\mathbb{N}$. An alignment $A$ is banded with width $w$ when, for every prefix of $A$ (including the empty prefix and $A$ itself), the number of columns of that prefix with a letter in the first row and the number with a letter in the second row differ by at most $w$.","title":"Banded alignments"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/abs-le-of-between","kind":"node","lean":"Cslib.Algorithms.Lean.abs_le_of_between","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#abs-le-of-between","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"For integers $a$, $d$, $t$ and $w$, if $|a| \\le w$, $|a - d| \\le w$ and $0 \\le t \\le d$, then $|a - t| \\le w$.","title":"Bound at an intermediate point, difference form"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/abs-le-of-between-add","kind":"node","lean":"Cslib.Algorithms.Lean.abs_le_of_between_add","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#abs-le-of-between-add","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"For integers $a$, $d$, $t$ and $w$, if $|a| \\le w$, $|a + d| \\le w$ and $0 \\le t \\le d$, then $|a + t| \\le w$.","title":"Bound at an intermediate point, sum form"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-from-final","kind":"node","lean":"Cslib.Algorithms.Lean.bandedFrom_final","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-from-final","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $w, i, j \\in \\mathbb{N}$ and let $A$ be an alignment. If the band check of $A$ with width $w$ from offset $(i, j)$ is true, then $|(i + p) - (j + q)| \\le w$, where $p$ and $q$ are the numbers of columns of $A$ whose first, respectively second, entry is present.","title":"A passing band check bounds the final offset"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/is-banded-length-diff","kind":"node","lean":"Cslib.Algorithms.Lean.isBanded_length_diff","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#is-banded-length-diff","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $w \\in \\mathbb{N}$. If $A$ is a valid alignment of sequences $x$ and $y$ and $A$ is banded with width $w$, then the lengths of $x$ and $y$ differ by at most $w$, that is $\\bigl| |x| - |y| \\bigr| \\le w$.","title":"Banded alignments force close lengths"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/diag-align","kind":"node","lean":"Cslib.Algorithms.Lean.diagAlign","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#diag-align","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"For sequences $x = x_1 \\dots x_m$ and $y = y_1 \\dots y_n$, let $k = \\min(m, n)$. The diagonal alignment of $x$ and $y$ consists of the columns $(x_1, y_1), \\dots, (x_k, y_k)$, which pair letters by position whether or not they are equal, followed by one column for each remaining letter of the longer sequence, in order, pairing that letter with a gap.","title":"Diagonal alignment"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/is-alignment-diag-align","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_diagAlign","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#is-alignment-diag-align","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"For all sequences $x$ and $y$, the diagonal alignment of $x$ and $y$ is a valid alignment of $x$ and $y$: none of its columns consists of two gaps, and deleting the gaps from its first and second rows gives $x$ and $y$ respectively.","title":"The diagonal alignment is a valid alignment"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-from-map-gap-y","kind":"node","lean":"Cslib.Algorithms.Lean.bandedFrom_map_gapY","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-from-map-gap-y","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $w, i, j \\in \\mathbb{N}$, let $y$ be a list of length $n$, and let $A$ be the alignment with one column for each element $b$ of $y$, in order, with empty first entry and second entry $b$. If $|i - j| \\le w$ and $|i - (j + n)| \\le w$, then the band check of $A$ with width $w$ from offset $(i, j)$ is true.","title":"Columns using only the second sequence stay in the band"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-from-map-gap-x","kind":"node","lean":"Cslib.Algorithms.Lean.bandedFrom_map_gapX","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-from-map-gap-x","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $w, i, j \\in \\mathbb{N}$, let $x$ be a list of length $n$, and let $A$ be the alignment with one column for each element $a$ of $x$, in order, with first entry $a$ and empty second entry. If $|i - j| \\le w$ and $|(i + n) - j| \\le w$, then the band check of $A$ with width $w$ from offset $(i, j)$ is true.","title":"Columns using only the first sequence stay in the band"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-from-diag-align","kind":"node","lean":"Cslib.Algorithms.Lean.bandedFrom_diagAlign","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-from-diag-align","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $w, i, j \\in \\mathbb{N}$ and let $x$, $y$ be lists of lengths $m$ and $n$. If $|i - j| \\le w$ and $|(i + m) - (j + n)| \\le w$, then the band check of the diagonal alignment of $x$ and $y$ with width $w$ from offset $(i, j)$ is true.","title":"The diagonal alignment stays in the band"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/is-banded-diag-align","kind":"node","lean":"Cslib.Algorithms.Lean.isBanded_diagAlign","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#is-banded-diag-align","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $w \\in \\mathbb{N}$. If two sequences $x$ and $y$ have lengths differing by at most $w$, that is $\\bigl| |x| - |y| \\bigr| \\le w$, then the diagonal alignment of $x$ and $y$ is banded with width $w$.","title":"The diagonal alignment is banded"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-score","kind":"node","lean":"Cslib.Algorithms.Lean.bandedScore","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-score","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$, assigning an integer $\\sigma(a, b)$ to each pair of letters $a$, $b$, an integer gap score $g$ and a band width $w \\in \\mathbb{N}$. The banded alignment score of two sequences $x$ and $y$ is the maximum, over all valid alignments of $x$ and $y$ that are banded with width $w$, of the (integer) alignment score with respect to $\\sigma$ and $g$. It has no value (it is the empty\u2026","title":"Banded alignment score"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-score-le-nw-score","kind":"node","lean":"Cslib.Algorithms.Lean.bandedScore_le_nwScore","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-score-le-nw-score","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer substitution score on an alphabet, $g$ an integer gap score and $w \\in \\mathbb{N}$ a band width. For sequences $x$ and $y$ over the alphabet and an integer $s$, if the banded score of $x$ and $y$ (under $\\sigma$, $g$ and width $w$) is defined and equals $s$, then $s$ is at most the Needleman\u2013Wunsch score of $x$ and $y$.","title":"The banded score is at most the Needleman\u2013Wunsch score"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-score-eq-nw-score-of-banded-optim","kind":"node","lean":"Cslib.Algorithms.Lean.bandedScore_eq_nwScore_of_banded_optimal","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-score-eq-nw-score-of-banded-optim","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer substitution score on an alphabet, $g$ an integer gap score and $w \\in \\mathbb{N}$ a band width. For sequences $x$ and $y$ over the alphabet, if some alignment of $x$ and $y$ is banded with width $w$ and has score (under $\\sigma$ and $g$) equal to the Needleman\u2013Wunsch score of $x$ and $y$, then the banded score of $x$ and $y$ (under $\\sigma$, $g$ and width $w$) is defined and equals their\u2026","title":"Banded alignment is exact when some optimal alignment is banded"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-score-none-of-length-diff-gt","kind":"node","lean":"Cslib.Algorithms.Lean.bandedScore_none_of_length_diff_gt","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-score-none-of-length-diff-gt","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer substitution score on an alphabet, $g$ an integer gap score and $w \\in \\mathbb{N}$ a band width. For sequences $x$ and $y$ over the alphabet, of lengths $m$ and $n$, with $w < |m - n|$, the banded score of $x$ and $y$ (under $\\sigma$, $g$ and width $w$) is undefined: no alignment of $x$ and $y$ is banded with width $w$.","title":"No banded score when the lengths differ by more than the band width"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-score-some-of-length-diff-le","kind":"node","lean":"Cslib.Algorithms.Lean.bandedScore_some_of_length_diff_le","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-score-some-of-length-diff-le","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$, assigning an integer $\\sigma(a, b)$ to each pair of letters $a$, $b$, an integer gap score $g$ and a band width $w \\in \\mathbb{N}$. If two sequences $x$ and $y$ have lengths differing by at most $w$, that is $\\bigl| |x| - |y| \\bigr| \\le w$, then the banded alignment score of $x$ and $y$ (for $\\sigma$, $g$ and $w$) is defined: it equals some integer $s$.","title":"The banded score is defined when the lengths are close"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/gap-columns","kind":"node","lean":"Cslib.Algorithms.Lean.gapColumns","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#gap-columns","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"The number of gap columns of an alignment: columns with exactly one letter.","title":"Number of gap columns of an alignment"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-from-of-abs-add-gap-columns-le","kind":"node","lean":"Cslib.Algorithms.Lean.bandedFrom_of_abs_add_gapColumns_le","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-from-of-abs-add-gap-columns-le","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"Along every prefix, the offset difference is at most the starting difference plus the number of gap columns: an alignment starting at `(i, j)` stays in the band of width `w` as soon as `|i - j| + gapColumns A \u2264 w`.","title":"An alignment with few gap columns stays in the band from its start"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/is-banded-of-gap-columns-le","kind":"node","lean":"Cslib.Algorithms.Lean.isBanded_of_gapColumns_le","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#is-banded-of-gap-columns-le","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"An alignment with at most `w` gap columns is `w`-banded.","title":"An alignment with at most w gap columns is w-banded"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/alignment-score-le-neg-gap-columns","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentScore_le_neg_gapColumns","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#alignment-score-le-neg-gap-columns","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"With nonpositive substitution scores and gap score at most `-1`, an alignment scores at most minus its number of gap columns.","title":"Alignment score is at most minus the number of gap columns"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/is-banded-of-neg-le-alignment-score","kind":"node","lean":"Cslib.Algorithms.Lean.isBanded_of_neg_le_alignmentScore","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#is-banded-of-neg-le-alignment-score","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"The band lemma (Ukkonen 1985): with nonpositive substitution scores and gap score at most `-1`, every alignment scoring at least `-w` (cost at most `w`) is `w`-banded.","title":"The band lemma"},{"id":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/banded-score-eq-nw-score-of-neg-le","kind":"node","lean":"Cslib.Algorithms.Lean.bandedScore_eq_nwScore_of_neg_le","parent":45567,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/banded-sequence-alignment/index.html#banded-score-eq-nw-score-of-neg-le","scope":"Biology/bioinformatics/sequence-alignment/banded-sequence-alignment","status":"fully_proved","summary":"The banded optimum is exact when the optimal score is at least `-w` (optimal cost at most `w`), for nonpositive substitution scores and gap score at most `-1`: some optimal alignment exists and, by the band lemma, it is `w`-banded.","title":"The banded optimum is exact"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment.json","id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","title":"Gotoh affine-gap alignment"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gap-state","kind":"node","lean":"Cslib.Algorithms.Lean.GapState","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gap-state","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"A gap state is one of three values: $\\mathrm{none}$, $\\mathrm{gapX}$ and $\\mathrm{gapY}$. The affine-gap score and the Gotoh recursion carry one as the kind of the previous column: $\\mathrm{gapX}$ after a column pairing an element of the first sequence with a gap, $\\mathrm{gapY}$ after a column pairing a gap with an element of the second sequence, and $\\mathrm{none}$ otherwise, including before the first column.","title":"Gap states"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gap-cost","kind":"node","lean":"Cslib.Algorithms.Lean.gapCost","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gap-cost","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Given an integer gap-open cost $o$, an integer gap-extend cost $e$ and gap states $p$ (of the previous column) and $c$ (of the current gap column), the cost of the current column is $e$ if $p = c$ (the column continues a run) and $o$ otherwise (it opens a run).","title":"Gap cost"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/affine-score-aux","kind":"node","lean":"Cslib.Algorithms.Lean.affineScoreAux","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#affine-score-aux","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix an integer substitution score $\\sigma$, an integer gap-open cost $o$ and an integer gap-extend cost $e$. The affine-gap score of an alignment started from a gap state $p$ is the sum of its column scores, computed left to right while carrying a gap state that starts at $p$ and is updated after each column. The empty alignment scores $0$. A column pairing an element $a$ of the first sequence with an element $b$\u2026","title":"Affine-gap score from a gap state"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/affine-score","kind":"node","lean":"Cslib.Algorithms.Lean.affineScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#affine-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Given an integer substitution score $\\sigma$, an integer gap-open cost $o$ and an integer gap-extend cost $e$, the affine-gap score of an alignment is its affine-gap score from a gap state, started from the state $\\mathrm{none}$ (no open gap run).","title":"Affine-gap alignment score"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-aux","kind":"node","lean":"Cslib.Algorithms.Lean.gotohAux","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-aux","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix an integer-valued scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$. A gap state records what the previous alignment column was: a match ($\\mathrm{none}$), a gap in the second sequence ($\\mathrm{gapX}$) or a gap in the first sequence ($\\mathrm{gapY}$). For gap states $p$, $q$ let $c(p, q)$ be the gap cost, $g_e$ if $p = q$ and $g_o$ otherwise. Write\u2026","title":"Fuel-bounded Gotoh recursion"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-none","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContNone","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-none","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Given a substitution score $\\sigma$, a gap-open score $g_o$ and a gap-extend score $g_e$, this is the value of the Gotoh dynamic program on sequences $x$ and $y$ when no gap run is open (the previous column was a match, or there is no previous column): the best affine-gap score of an alignment of $x$ and $y$, where a match column pairing $a$ with $b$ scores $\\sigma(a, b)$ and a gap column scores $g_e$ if it\u2026","title":"Gotoh value with no open gap run"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-gap-x","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContGapX","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-gap-x","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix an integer-valued scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$. For sequences $u$ and $v$, the value $X(u, v)$ is the fuel-bounded Gotoh recursion on $u$ and $v$, run with fuel equal to the sum of their lengths and started in the gap state where the previous column was a gap in $v$.","title":"Gotoh continuation value after a gap in the second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-gap-y","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContGapY","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-gap-y","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix an integer-valued scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$. For sequences $u$ and $v$, the value $Y(u, v)$ is the fuel-bounded Gotoh recursion on $u$ and $v$, run with fuel equal to the sum of their lengths and started in the gap state where the previous column was a gap in $u$.","title":"Gotoh continuation value after a gap in the first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-score","kind":"node","lean":"Cslib.Algorithms.Lean.gotohScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"For a substitution score $\\sigma$ and gap-open and gap-extend scores $g_o$ and $g_e$, the Gotoh score of sequences $x$ and $y$ is the Gotoh value of $x$ against $y$ with no open gap run, since no gap run is open before the first column.","title":"Gotoh score"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gap-run-cost","kind":"node","lean":"Cslib.Algorithms.Lean.gapRunCost","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gap-run-cost","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Given an integer gap-open cost $o$, an integer gap-extend cost $e$, gap states $p$ (of the column before the run) and $c$ (of the run's columns) and a length $k \\in \\mathbb{N}$, the cost of a run of $k$ gap columns is $k e$ if $p = c$ (the run continues an open run); otherwise it is $0$ for $k = 0$ and $o + (k - 1) e$ for $k \\ge 1$.","title":"Cost of a gap run"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gap-run-cost-succ","kind":"node","lean":"Cslib.Algorithms.Lean.gapRunCost_succ","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gap-run-cost-succ","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"For an integer gap-open cost $o$, an integer gap-extend cost $e$, gap states $p$ and $c$ and $k \\in \\mathbb{N}$, the gap cost of a column in state $c$ after state $p$, plus the cost of a run of $k$ columns in state $c$ that follows state $c$, equals the cost of a run of $k + 1$ columns in state $c$ that follows state $p$.","title":"Adding a column to a gap run"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/is-alignment-of-cons-match","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_of_cons_match","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#is-alignment-of-cons-match","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $a$ and $b$ be elements, let $xs$ and $ys$ be sequences, and let $A$ be an alignment. If the alignment whose first column is the match column $(a, b)$, followed by the columns of $A$, is a valid alignment of the sequence $a$ followed by $xs$ and the sequence $b$ followed by $ys$, then $A$ is a valid alignment of $xs$ and $ys$.","title":"Removing a leading match column from a valid alignment"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/is-alignment-of-cons-gap-x","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_of_cons_gapX","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#is-alignment-of-cons-gap-x","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $A$ be an alignment, $a$ an element and $x$, $y$ sequences. If prepending to $A$ the column that pairs $a$ with a gap gives a valid alignment of $a x$ (the sequence $x$ with $a$ prepended) and $y$, then $A$ is a valid alignment of $x$ and $y$.","title":"Removing a leading column with a gap in the second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/is-alignment-of-cons-gap-y","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_of_cons_gapY","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#is-alignment-of-cons-gap-y","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $A$ be an alignment, $b$ an element and $x$, $y$ sequences. If prepending to $A$ the column that pairs a gap with $b$ gives a valid alignment of $x$ and $b y$ (the sequence $y$ with $b$ prepended), then $A$ is a valid alignment of $x$ and $y$.","title":"Removing a leading column with a gap in the first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/is-alignment-cons-match","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_cons_match","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#is-alignment-cons-match","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"If $A$ is a valid alignment of sequences $x$ and $y$, then for all elements $a$ and $b$, the alignment obtained by prepending to $A$ the column that pairs $a$ with $b$ is a valid alignment of $a x$ and $b y$ (the sequences $x$ and $y$ with $a$ and $b$ prepended).","title":"Prepending a match column"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/is-alignment-cons-gap-x","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_cons_gapX","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#is-alignment-cons-gap-x","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"If $A$ is a valid alignment of sequences $x$ and $y$, then for every element $a$, the alignment obtained by prepending to $A$ the column that pairs $a$ with a gap is a valid alignment of $a x$ (the sequence $x$ with $a$ prepended) and $y$.","title":"Prepending a column with a gap in the second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/is-alignment-cons-gap-y","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_cons_gapY","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#is-alignment-cons-gap-y","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"If $A$ is a valid alignment of sequences $x$ and $y$, then for every element $b$, the alignment obtained by prepending to $A$ the column that pairs a gap with $b$ is a valid alignment of $x$ and $b y$ (the sequence $y$ with $b$ prepended).","title":"Prepending a column with a gap in the first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/affine-score-le-gotoh-score","kind":"node","lean":"Cslib.Algorithms.Lean.affineScore_le_gotohScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#affine-score-le-gotoh-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer substitution score, $o$ an integer gap-open cost and $e$ an integer gap-extend cost. For all sequences $x$ and $y$ and every valid alignment $A$ of $x$ and $y$, the affine-gap score of $A$ is at most the Gotoh score of $x$ and $y$, both taken with $\\sigma$, $o$ and $e$.","title":"Gotoh is optimal"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/exists-affine-score-eq-gotoh-score","kind":"node","lean":"Cslib.Algorithms.Lean.exists_affineScore_eq_gotohScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#exists-affine-score-eq-gotoh-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer substitution score, $o$ an integer gap-open cost and $e$ an integer gap-extend cost. For all sequences $x$ and $y$ there is a valid alignment $A$ of $x$ and $y$ whose affine-gap score equals the Gotoh score of $x$ and $y$, both taken with $\\sigma$, $o$ and $e$.","title":"Gotoh is attained"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-none-eq","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContNone_eq","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-none-eq","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix an integer-valued scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$. For sequences $u$, $v$ let $N(u, v)$, $X(u, v)$ and $Y(u, v)$ be the Gotoh continuation values when the previous column was a match, a gap in $v$ and a gap in $u$ respectively, and write $a \\cdot u$ for the sequence with first letter $a$ followed by $u$. For all letters $a$, $b$ and\u2026","title":"First-column recurrence for the continuation after a match"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-gap-x-eq","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContGapX_eq","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-gap-x-eq","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix an integer-valued scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$. For sequences $u$, $v$ let $N(u, v)$, $X(u, v)$ and $Y(u, v)$ be the Gotoh continuation values when the previous column was a match, a gap in $v$ and a gap in $u$ respectively, and write $a \\cdot u$ for the sequence with first letter $a$ followed by $u$. For all letters $a$, $b$ and\u2026","title":"First-column recurrence for the continuation after a gap in the second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-gap-y-eq","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContGapY_eq","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-gap-y-eq","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix an integer-valued scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$. For sequences $u$, $v$ let $N(u, v)$, $X(u, v)$ and $Y(u, v)$ be the Gotoh continuation values when the previous column was a match, a gap in $v$ and a gap in $u$ respectively, and write $a \\cdot u$ for the sequence with first letter $a$ followed by $u$. For all letters $a$, $b$ and\u2026","title":"First-column recurrence for the continuation after a gap in the first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-none-nil-left","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContNone_nil_left","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-none-nil-left","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $g_o$ and $g_e$ be the gap-open and gap-extend scores. For every sequence $y$ of length $n$, the Gotoh value with no open gap run of the empty sequence against $y$ equals the score of a freshly opened run of $n$ gaps in the first sequence: $0$ if $n = 0$, and $g_o + (n - 1) g_e$ if $n \\ge 1$.","title":"Gotoh value against an empty first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-none-nil-right","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContNone_nil_right","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-none-nil-right","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $g_o$ and $g_e$ be the gap-open and gap-extend scores. For every sequence $x$ of length $n$, the Gotoh value with no open gap run of $x$ against the empty sequence equals the score of a freshly opened run of $n$ gaps in the second sequence: $0$ if $n = 0$, and $g_o + (n - 1) g_e$ if $n \\ge 1$.","title":"Gotoh value against an empty second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-gap-x-nil-left","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContGapX_nil_left","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-gap-x-nil-left","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix a scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$, and let $X(u, v)$ be the Gotoh continuation value of sequences $u$, $v$ when the previous column was a gap in $v$. For every sequence $v$ of length $k$, the value $X(\\varepsilon, v)$, where $\\varepsilon$ is the empty sequence, is the cost of one freshly opened gap run of length $k$: it is $0$ if\u2026","title":"Continuation after a gap in the second sequence, empty first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-gap-x-nil-right","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContGapX_nil_right","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-gap-x-nil-right","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix a scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$, and let $X(u, v)$ be the Gotoh continuation value of sequences $u$, $v$ when the previous column was a gap in $v$. For every sequence $u$ of length $k$, the value $X(u, \\varepsilon)$, where $\\varepsilon$ is the empty sequence, equals $k \\, g_e$: the open gap run is extended by $k$ columns.","title":"Continuation after a gap in the second sequence, empty second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-gap-y-nil-left","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContGapY_nil_left","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-gap-y-nil-left","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix a scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$, and let $Y(u, v)$ be the Gotoh continuation value of sequences $u$, $v$ when the previous column was a gap in $u$. For every sequence $v$ of length $k$, the value $Y(\\varepsilon, v)$, where $\\varepsilon$ is the empty sequence, equals $k \\, g_e$: the open gap run is extended by $k$ columns.","title":"Continuation after a gap in the first sequence, empty first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-cont-gap-y-nil-right","kind":"node","lean":"Cslib.Algorithms.Lean.gotohContGapY_nil_right","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-cont-gap-y-nil-right","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Fix a scoring function $\\sigma$ on pairs of letters and integer gap-open and gap-extend penalties $g_o$ and $g_e$, and let $Y(u, v)$ be the Gotoh continuation value of sequences $u$, $v$ when the previous column was a gap in $u$. For every sequence $u$ of length $k$, the value $Y(u, \\varepsilon)$, where $\\varepsilon$ is the empty sequence, is the cost of one freshly opened gap run of length $k$: it is $0$ if\u2026","title":"Continuation after a gap in the first sequence, empty second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/affine-score-aux-eq-alignment-score","kind":"node","lean":"Cslib.Algorithms.Lean.affineScoreAux_eq_alignmentScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#affine-score-aux-eq-alignment-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer substitution score and $g$ an integer. If the gap-open cost and the gap-extend cost both equal $g$, then for every alignment $A$ and every gap state $p$, the affine-gap score of $A$ started from state $p$, with substitution score $\\sigma$, equals the alignment score of $A$ with substitution score $\\sigma$ and gap penalty $g$.","title":"Linear gaps as affine gaps, from any gap state"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/affine-score-eq-alignment-score","kind":"node","lean":"Cslib.Algorithms.Lean.affineScore_eq_alignmentScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#affine-score-eq-alignment-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer substitution score and $g$ an integer. If the gap-open cost and the gap-extend cost both equal $g$, then for every alignment $A$, the affine-gap score of $A$ with substitution score $\\sigma$ equals the alignment score of $A$ with substitution score $\\sigma$ and gap penalty $g$.","title":"Linear gaps are affine gaps with equal costs"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-score-eq-nw-score","kind":"node","lean":"Cslib.Algorithms.Lean.gotohScore_eq_nwScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-score-eq-nw-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ an integer. If the gap-open and gap-extend scores both equal $g$, then for all sequences $x$ and $y$ the Gotoh score (optimal affine-gap alignment score) of $x$ and $y$ equals the Needleman\u2013Wunsch score of $x$ and $y$ with substitution score $\\sigma$ and gap score $g$.","title":"Gotoh with equal gap scores is Needleman\u2013Wunsch"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/cell-spec","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.cellSpec","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#cell-spec","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The specification of a cell: the three continuation values `(M, X, Y)` of the suffix pair `(xs, ys)`, given that the previous column was a match, a gap in `ys`, or a gap in `xs`.","title":"Specification of a Gotoh table cell"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/step","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.step","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#step","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The cell of `(a :: xs, b :: ys)` from the cells `diag` of `(xs, ys)`, `up` of `(xs, b :: ys)` and `left` of `(a :: xs, ys)`: a match continues from `M`, a gap in `ys` from `X` (extending if the previous column was one), a gap in `xs` from `Y`.","title":"Gotoh cell recurrence from diagonal, upper and left cells"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/step-x","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.stepX","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#step-x","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The cell of `(a :: xs, [])` from the cell `o` of `(xs, [])`: one more gap column in `ys`.","title":"Gotoh cell recurrence against an empty second suffix"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/first-row","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.firstRow","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#first-row","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The row of the suffix `[]` of `xs`: the cells of `([], t)` for the suffixes `t` of `ys`, longest first, each one gap column in `xs` more than the next.","title":"Gotoh table row of the empty first suffix"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/next-row","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.nextRow","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#next-row","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The row of `a :: xs` from the row `prev` of `xs`, over the remaining suffixes of `ys`, longest first; each cell uses the cell to its right in the same row. A row `prev` that is too short gives junk cells (this does not happen for rows of the table, `nextRow_eq`).","title":"Gotoh table row from the previous row"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-rows","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.gotohRows","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-rows","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The row of the suffix `xs` (of the first sequence) of the table, computed from the rows of its shorter suffixes.","title":"Gotoh table row of a suffix of the first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-tab","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.gotohTab","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-tab","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The `M` value of the cell `(xs, ys)`.","title":"The Gotoh table"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/first-row-eq","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.firstRow_eq","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#first-row-eq","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"`firstRow ys` holds the cells of `([], t)` for the suffixes `t` of `ys`.","title":"The first row"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/next-row-eq","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.nextRow_eq","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#next-row-eq","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"From the row of the suffix `xs`, `nextRow` computes the row of `a :: xs`: the cells `cellSpec (a :: xs) t` for the suffixes `t` of `ys`, longest first.","title":"One row of the table"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-rows-eq","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.gotohRows_eq","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-rows-eq","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The row of `xs` holds `cellSpec xs t` for every suffix `t` of `ys`, longest first.","title":"The rows of the table"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-tab-eq-gotoh-score","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.gotohTab_eq_gotohScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-tab-eq-gotoh-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"The table computes the optimal affine-gap score (Gotoh 1982): `gotohTab = gotohScore`.","title":"The table computes the optimal affine-gap score"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/affine-score-le-gotoh-tab","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.affineScore_le_gotohTab","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#affine-score-le-gotoh-tab","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Every alignment of `xs` and `ys` has affine-gap score at most `gotohTab`.","title":"The table is an upper bound"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/exists-affine-score-eq-gotoh-tab","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.exists_affineScore_eq_gotohTab","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#exists-affine-score-eq-gotoh-tab","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"Some alignment of `xs` and `ys` has affine-gap score `gotohTab`.","title":"The table is attained"},{"id":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/gotoh-tab-eq-nw-score","kind":"node","lean":"Cslib.Bioinformatics.Alignment.Gotoh.gotohTab_eq_nwScore","parent":45591,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment/index.html#gotoh-tab-eq-nw-score","scope":"Biology/bioinformatics/sequence-alignment/gotoh-affine-gap-alignment","status":"fully_proved","summary":"With `gapOpen = gapExtend = gap`, the Gotoh table computes the linear-gap Needleman\u2013Wunsch score.","title":"Gotoh table with equal gap costs gives the Needleman\u2013Wunsch score"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment.json","id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","title":"Hirschberg's linear-space alignment"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/best-argmax","kind":"node","lean":"Cslib.Algorithms.Lean.bestArgmax","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#best-argmax","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"For a function $f : \\mathbb{N} \\to \\mathbb{Z}$ and $n \\in \\mathbb{N}$, the argmax of $f$ over $\\{0, \\dots, n\\}$ is the least $k \\le n$ such that $f(j) \\le f(k)$ for all $j \\le n$.","title":"Argmax over an initial segment"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/best-argmax-le","kind":"node","lean":"Cslib.Algorithms.Lean.bestArgmax_le","parent":45637,"reading_title":{"format":"inline-parts-v1","parts":[["text","The argmax over "],["math","\\(\\{0, \\dots, n\\}\\)"],["text"," is at most "],["math","\\(n\\)"]],"source":"The argmax over $\\{0, \\dots, n\\}$ is at most $n$"},"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#best-argmax-le","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"For every function $f : \\mathbb{N} \\to \\mathbb{Z}$ and every $n \\in \\mathbb{N}$, the argmax of $f$ over $\\{0, \\dots, n\\}$ is at most $n$.","title":"The argmax over $\\{0, \\dots, n\\}$ is at most $n$"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/best-argmax-spec","kind":"node","lean":"Cslib.Algorithms.Lean.bestArgmax_spec","parent":45637,"reading_title":{"format":"inline-parts-v1","parts":[["text","The argmax over "],["math","\\(\\{0, \\dots, n\\}\\)"],["text"," maximizes the function there"]],"source":"The argmax over $\\{0, \\dots, n\\}$ maximizes the function there"},"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#best-argmax-spec","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Let $f : \\mathbb{N} \\to \\mathbb{Z}$, let $n \\in \\mathbb{N}$ and let $m$ be the argmax of $f$ over $\\{0, \\dots, n\\}$. Then $f(k) \\le f(m)$ for every $k \\le n$.","title":"The argmax over $\\{0, \\dots, n\\}$ maximizes the function there"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/split-score-le-nw-score-ys","kind":"node","lean":"Cslib.Algorithms.Lean.splitScore_le_nwScore_ys","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#split-score-le-nw-score-ys","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer-valued substitution score on an alphabet, let $g$ be an integer gap score, and write $\\mathrm{NW}(x, y)$ for the optimal Needleman\u2013Wunsch global alignment score of sequences $x$ and $y$ under $\\sigma$ and $g$. For sequences $x$, $y$ and natural numbers $k$, $m$, let $x_{\\lt k}$ be the first $k$ letters of $x$ (all of $x$ if $k$ is at least its length) and $x_{\\ge k}$ the remaining\u2026","title":"A split of the second sequence scores at most the Needleman\u2013Wunsch optimum"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/split-score-le-nw-score-xs","kind":"node","lean":"Cslib.Algorithms.Lean.splitScore_le_nwScore_xs","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#split-score-le-nw-score-xs","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Let $\\sigma$ be an integer-valued substitution score on an alphabet, let $g$ be an integer gap score, and write $\\mathrm{NW}(x, y)$ for the optimal Needleman\u2013Wunsch global alignment score of sequences $x$ and $y$ under $\\sigma$ and $g$. For sequences $x$, $y$ and natural numbers $m$, $k$, let $x_{\\lt m}$ be the first $m$ letters of $x$ (all of $x$ if $m$ is at least its length) and $x_{\\ge m}$ the remaining\u2026","title":"A split of the first sequence scores at most the Needleman\u2013Wunsch optimum"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/exists-split-eq-nw-score-ys","kind":"node","lean":"Cslib.Algorithms.Lean.exists_split_eq_nwScore_ys","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#exists-split-eq-nw-score-ys","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a type $\\alpha$, a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$. Write $S(u, v)$ for the Needleman\u2013Wunsch score of lists $u$, $v$, and $u_{\\lt m}$ for the first $m$ elements of $u$ and $u_{\\ge m}$ for the rest. For all lists $xs$, $ys$ over $\\alpha$ and every $m \\le |ys|$, there is $k \\le |xs|$ with\u2026","title":"Every cut of the second list extends to an optimal split"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/exists-split-eq-nw-score-xs","kind":"node","lean":"Cslib.Algorithms.Lean.exists_split_eq_nwScore_xs","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#exists-split-eq-nw-score-xs","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a type $\\alpha$, a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$. Write $S(u, v)$ for the Needleman\u2013Wunsch score of lists $u$, $v$, and $u_{\\lt m}$ for the first $m$ elements of $u$ and $u_{\\ge m}$ for the rest. For all lists $xs$, $ys$ over $\\alpha$ and every $m \\le |xs|$, there is $k \\le |ys|$ with\u2026","title":"Every cut of the first list extends to an optimal split"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/nw-score-eq-best-split-ys","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_eq_bestSplit_ys","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#nw-score-eq-best-split-ys","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$ assigning an integer to each pair of letters and an integer gap score $g$, and write $\\mathrm{NW}$ for the Needleman\u2013Wunsch score. For a sequence $s$, write $s_{\\lt j}$ for its first $j$ letters and $s_{\\ge j}$ for the remaining letters. Let $x$, $y$ be sequences and $m \\le |y|$, and let $k$ be the first index $j \\in \\{0, \\dots, |x|\\}$ that maximizes\u2026","title":"Hirschberg split identity, splitting the second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/nw-score-eq-best-split-xs","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_eq_bestSplit_xs","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#nw-score-eq-best-split-xs","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$ assigning an integer to each pair of letters and an integer gap score $g$, and write $\\mathrm{NW}$ for the Needleman\u2013Wunsch score. For a sequence $s$, write $s_{\\lt j}$ for its first $j$ letters and $s_{\\ge j}$ for the remaining letters. Let $x$, $y$ be sequences and $m \\le |x|$, and let $k$ be the first index $j \\in \\{0, \\dots, |y|\\}$ that maximizes\u2026","title":"Hirschberg split identity, splitting the first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/nw-score-singleton-singleton","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_singleton_singleton","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#nw-score-singleton-singleton","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma$ assigning an integer to each pair of letters, integer gap score $g$ and letters $a$, $b$, the Needleman\u2013Wunsch score of the one-letter sequences $[a]$ and $[b]$ is $\\max(\\sigma(a, b), g + g)$: the better of matching $a$ with $b$ and gapping both.","title":"Needleman\u2013Wunsch score of two single letters"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/small-align","kind":"node","lean":"Cslib.Algorithms.Lean.smallAlign","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#small-align","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$ assigning an integer to each pair of letters and an integer gap score $g$, and write a column as a pair whose missing entry, a gap, is marked $\\bot$. The base-case alignment of sequences $x$ and $y$ is the empty alignment if both are empty; the single column $(\\bot, b)$ if $x$ is empty and $y = [b]$; the single column $(a, \\bot)$ if $x = [a]$ and $y$ is empty; if $x = [a]$ and\u2026","title":"Base-case alignment for Hirschberg"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/is-alignment-small-align","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_smallAlign","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#is-alignment-small-align","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma$ assigning an integer to each pair of letters, integer gap score $g$ and sequences $x$, $y$ with $|x| \\le 1$ and $|y| \\le 1$, the base-case alignment of $x$ and $y$ is a valid alignment of $x$ and $y$.","title":"The base-case alignment is valid"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/alignment-score-small-align-eq-nw-score","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentScore_smallAlign_eq_nwScore","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#alignment-score-small-align-eq-nw-score","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a type $\\alpha$, a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$. For all lists $xs$, $ys$ over $\\alpha$ with $|xs| \\le 1$ and $|ys| \\le 1$, the alignment score of the base-case alignment of $xs$ and $ys$ equals the Needleman\u2013Wunsch score of $xs$ and $ys$.","title":"The base-case alignment is optimal"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/hirschberg-align","kind":"node","lean":"Cslib.Algorithms.Lean.hirschbergAlign","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#hirschberg-align","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$ assigning an integer to each pair of letters and an integer gap score $g$, and write $\\mathrm{NW}$ for the Needleman\u2013Wunsch score. For a sequence $s$, write $s_{\\lt j}$ for its first $j$ letters and $s_{\\ge j}$ for the remaining letters. The Hirschberg reconstruction of sequences $x$ and $y$ with fuel $n$ is the alignment defined by recursion on $n$. With fuel $0$ it is the empty\u2026","title":"Hirschberg reconstruction with fuel"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/hirschberg-align-small","kind":"node","lean":"Cslib.Algorithms.Lean.hirschbergAlign_small","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#hirschberg-align-small","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a type $\\alpha$, a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$. For every $n \\in \\mathbb{N}$ and all lists $xs$, $ys$ over $\\alpha$ with $|xs| \\le 1$ and $|ys| \\le 1$, the Hirschberg reconstruction of $xs$ and $ys$ with fuel $n + 1$ is the base-case alignment of $xs$ and $ys$.","title":"Hirschberg reconstruction on lists of length at most one"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/hirschberg-align-split-ys","kind":"node","lean":"Cslib.Algorithms.Lean.hirschbergAlign_split_ys","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#hirschberg-align-split-ys","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$ assigning an integer to each pair of letters and an integer gap score $g$, and write $\\mathrm{NW}$ for the Needleman\u2013Wunsch score. For a sequence $s$, write $s_{\\lt j}$ for its first $j$ letters and $s_{\\ge j}$ for the remaining letters. Let $x$, $y$ be sequences with $|y| \\ge |x|$ that do not both have length at most $1$, let $m = \\lfloor |y|/2 \\rfloor$, and let $k$ be the first\u2026","title":"Hirschberg recursion step splitting the second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/hirschberg-align-split-xs","kind":"node","lean":"Cslib.Algorithms.Lean.hirschbergAlign_split_xs","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#hirschberg-align-split-xs","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a type $\\alpha$, a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$. Write $H_n(u, v)$ for the Hirschberg reconstruction of lists $u$, $v$ with fuel $n$, $S(u, v)$ for their Needleman\u2013Wunsch score, and $u_{\\lt m}$, $u_{\\ge m}$ for the first $m$ elements of $u$ and the rest. Let $xs$, $ys$ be lists over $\\alpha$ that are not both of length at most $1$, with\u2026","title":"Hirschberg reconstruction splits the longer first list"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/is-alignment-hirschberg-align","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_hirschbergAlign","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#is-alignment-hirschberg-align","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma$ assigning an integer to each pair of letters, integer gap score $g$, natural number $n$ and sequences $x$, $y$ with $|x| + |y| \\le n$, the Hirschberg reconstruction of $x$ and $y$ with fuel $n$ is a valid alignment of $x$ and $y$.","title":"Hirschberg reconstruction with enough fuel is valid"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/alignment-score-hirschberg-align","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentScore_hirschbergAlign","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#alignment-score-hirschberg-align","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a type $\\alpha$, a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$. For every fuel $n \\in \\mathbb{N}$ and all lists $xs$, $ys$ over $\\alpha$ with $|xs| + |ys| \\le n$, the alignment score of the Hirschberg reconstruction of $xs$ and $ys$ with fuel $n$ equals the Needleman\u2013Wunsch score of $xs$ and $ys$.","title":"Hirschberg reconstruction with enough fuel attains the optimum"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/hirschberg","kind":"node","lean":"Cslib.Algorithms.Lean.hirschberg","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#hirschberg","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$ assigning an integer to each pair of letters and an integer gap score $g$. The Hirschberg alignment of sequences $x$ and $y$ is the Hirschberg reconstruction of $x$ and $y$ with fuel $|x| + |y|$.","title":"Hirschberg's alignment algorithm"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/is-alignment-hirschberg","kind":"node","lean":"Cslib.Algorithms.Lean.isAlignment_hirschberg","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#is-alignment-hirschberg","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma$ assigning an integer to each pair of letters, integer gap score $g$ and sequences $x$, $y$, the Hirschberg alignment of $x$ and $y$ is a valid alignment of $x$ and $y$.","title":"Hirschberg reconstruction is valid"},{"id":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/alignment-score-hirschberg","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentScore_hirschberg","parent":45637,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment/index.html#alignment-score-hirschberg","scope":"Biology/bioinformatics/sequence-alignment/hirschberg-s-linear-space-alignment","status":"fully_proved","summary":"Fix a type $\\alpha$, a substitution score $\\sigma : \\alpha \\to \\alpha \\to \\mathbb{Z}$ and a gap score $g \\in \\mathbb{Z}$. For all lists $xs$, $ys$ over $\\alpha$, the alignment score of the Hirschberg alignment of $xs$ and $ys$ equals the Needleman\u2013Wunsch score of $xs$ and $ys$.","title":"Hirschberg reconstruction attains the Needleman\u2013Wunsch optimum"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment.json","id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","title":"Multiple sequence alignment"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/msalignment","kind":"node","lean":"Cslib.Algorithms.Lean.MSAlignment","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#msalignment","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"A multiple alignment with $k$ rows over an alphabet $\\alpha$ is a finite list of columns, each of which assigns to every row $i \\in \\{0, \\dots, k-1\\}$ either a letter of $\\alpha$ or a gap. No condition ties the rows to given sequences.","title":"Multiple alignments"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/is-msalignment","kind":"node","lean":"Cslib.Algorithms.Lean.IsMSAlignment","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#is-msalignment","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"A multiple alignment $A$ of $k$ rows is a valid multiple alignment of sequences $s_0, \\dots, s_{k-1}$ when no column of $A$ consists only of gaps and, for every $i < k$, deleting the gaps from row $i$ of $A$ gives $s_i$.","title":"Valid multiple alignment"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/induced-pair","kind":"node","lean":"Cslib.Algorithms.Lean.inducedPair","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#induced-pair","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"Given a multiple alignment $A$ of $k$ sequences and rows $i, j < k$, the pairwise alignment induced by $A$ on rows $i$ and $j$ is obtained by keeping, for each column $c$ of $A$ in order, the pair $(c_i, c_j)$, and dropping the columns in which both $c_i$ and $c_j$ are gaps.","title":"Induced pairwise alignment"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/induced-pair-ne-none-none","kind":"node","lean":"Cslib.Algorithms.Lean.inducedPair_ne_none_none","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#induced-pair-ne-none-none","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"For a multiple alignment $A$ of $k$ sequences and rows $i, j < k$, no column of the pairwise alignment induced by $A$ on rows $i$ and $j$ is a gap in both rows.","title":"Induced alignments have no all-gap column"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/induced-pair-filter-map-fst","kind":"node","lean":"Cslib.Algorithms.Lean.inducedPair_filterMap_fst","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#induced-pair-filter-map-fst","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"For a multiple alignment $A$ of $k$ sequences and rows $i, j < k$, deleting the gaps from the first row of the pairwise alignment induced by $A$ on rows $i$ and $j$ gives the same sequence as deleting the gaps from row $i$ of $A$.","title":"Letters of the first row of an induced alignment"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/induced-pair-filter-map-snd","kind":"node","lean":"Cslib.Algorithms.Lean.inducedPair_filterMap_snd","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#induced-pair-filter-map-snd","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"For a multiple alignment $A$ of $k$ sequences and rows $i, j < k$, deleting the gaps from the second row of the pairwise alignment induced by $A$ on rows $i$ and $j$ gives the same sequence as deleting the gaps from row $j$ of $A$.","title":"Letters of the second row of an induced alignment"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/induced-pair-is-alignment","kind":"node","lean":"Cslib.Algorithms.Lean.inducedPair_isAlignment","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#induced-pair-is-alignment","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"Let $A$ be a valid multiple alignment of sequences $s_0, \\dots, s_{k-1}$. Then for all rows $i, j < k$, the pairwise alignment induced by $A$ on rows $i$ and $j$ is a valid pairwise alignment of $s_i$ and $s_j$.","title":"Induced pairwise alignments are valid"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/pair-finset","kind":"node","lean":"Cslib.Algorithms.Lean.pairFinset","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#pair-finset","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"For $k \\in \\mathbb{N}$, the finite set of all pairs $(i, j)$ with $i, j \\in \\{0, \\dots, k-1\\}$ and $i < j$, so that each unordered pair of distinct rows appears exactly once.","title":"Pairs of rows"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/column-sp","kind":"node","lean":"Cslib.Algorithms.Lean.columnSP","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#column-sp","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ assign an integer score to each pair of letters and let $g$ be an integer gap score. For a column $c$ of a multiple alignment of $k$ sequences, which gives each row $r < k$ either a letter or a gap $c_r$, the sum-of-pairs score of $c$ is $\\sum_{i < j < k} s(c_i, c_j)$, where $s$ is the pairwise column score: $\\sigma(a, b)$ for letters $a$ and $b$, $g$ for a letter against a gap, and $0$ for two gaps.","title":"Sum-of-pairs score of a column"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/msa-score","kind":"node","lean":"Cslib.Algorithms.Lean.msaScore","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#msa-score","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"Fix a score $\\sigma(a, b) \\in \\mathbb{Z}$ for each pair of letters $a, b$ and a gap score $g \\in \\mathbb{Z}$. For two entries $x, y$, each a letter or a gap, let $s(x, y) = \\sigma(x, y)$ if both are letters, $s(x, y) = g$ if exactly one is a gap, and $s(x, y) = 0$ if both are gaps. The sum-of-pairs score of a multiple alignment $A$ of $k$ sequences is the sum, over the columns $c$ of $A$, of\u2026","title":"Sum-of-pairs score"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/alignment-score-induced-pair","kind":"node","lean":"Cslib.Algorithms.Lean.alignmentScore_inducedPair","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#alignment-score-induced-pair","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ assign an integer score to each pair of letters and let $g$ be an integer gap score. For a multiple alignment $A$ of $k$ sequences and rows $i, j < k$, the score (with respect to $\\sigma$ and $g$) of the pairwise alignment induced by $A$ on rows $i$ and $j$ equals the sum, over all columns $c$ of $A$, of the column score of the pair $(c_i, c_j)$.","title":"Score of an induced pairwise alignment"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/msa-score-eq-sum-induced","kind":"node","lean":"Cslib.Algorithms.Lean.msaScore_eq_sum_induced","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#msa-score-eq-sum-induced","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"Let $\\sigma$ assign an integer score to each pair of letters and let $g$ be an integer gap score. For every multiple alignment $A$ of $k$ sequences, the sum-of-pairs score of $A$ equals the sum, over all pairs of rows $i < j < k$, of the score of the pairwise alignment induced by $A$ on rows $i$ and $j$.","title":"The SP score is the sum of the induced pairwise scores"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/msa-score-le-sum-nw-score","kind":"node","lean":"Cslib.Algorithms.Lean.msaScore_le_sum_nwScore","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#msa-score-le-sum-nw-score","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"Fix a score $\\sigma(a, b) \\in \\mathbb{Z}$ for each pair of letters $a, b$ of an alphabet $\\alpha$ and a gap score $g \\in \\mathbb{Z}$. Let $s_0, \\dots, s_{k-1}$ be sequences over $\\alpha$ and let $A$ be a valid multiple alignment of them: no column of $A$ consists only of gaps, and deleting the gaps from row $i$ of $A$ leaves $s_i$. Then the sum-of-pairs score of $A$ under $\\sigma$ and $g$ is at most\u2026","title":"The sum-of-pairs score is bounded by the pairwise optima"},{"id":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/exists-is-msalignment","kind":"node","lean":"Cslib.Algorithms.Lean.exists_isMSAlignment","parent":45660,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment/index.html#exists-is-msalignment","scope":"Biology/bioinformatics/sequence-alignment/multiple-sequence-alignment","status":"fully_proved","summary":"For every natural number $n$ and every family of $k$ sequences $s_0, \\dots, s_{k-1}$ whose lengths sum to at most $n$, there is a valid multiple alignment of $s_0, \\dots, s_{k-1}$.","title":"Every family of sequences has a multiple alignment"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models.json","id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","title":"Pair hidden Markov models"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/alignments-of-aux","kind":"node","lean":"Cslib.Bioinformatics.HMM.alignmentsOfAux","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#alignments-of-aux","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"All alignments of `xs : List \u03b1` with `ys : List \u03b2`, by fuel recursion on the first column (the same recursion as `Cslib.Algorithms.Lean.alignmentsAux`, for two alphabets). Implementation detail: use `alignmentsOf`, whose fuel `xs.length + ys.length + 1` is enough.","title":"Fuel-bounded enumeration of two-alphabet alignments"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/alignments-of","kind":"node","lean":"Cslib.Bioinformatics.HMM.alignmentsOf","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#alignments-of","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"All alignments of a sequence over `\u03b1` with a sequence over `\u03b2`: lists of columns `Option \u03b1 \u00d7 Option \u03b2` with no `(none, none)` column whose rows filter to `xs` and `ys` (`mem_alignmentsOf`), each listed once (`nodup_alignmentsOf`). For `\u03b2 = \u03b1` this is `Cslib.Algorithms.Lean.alignments` (`alignmentsOf_eq_alignments`).","title":"All alignments of sequences over two alphabets"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/alignments-of-cons-cons","kind":"node","lean":"Cslib.Bioinformatics.HMM.alignmentsOf_cons_cons","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#alignments-of-cons-cons","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The alignments of two nonempty sequences, by their first column.","title":"Alignments of two nonempty sequences by first column"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/alignments-of-nil-cons","kind":"node","lean":"Cslib.Bioinformatics.HMM.alignmentsOf_nil_cons","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#alignments-of-nil-cons","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The alignments of the empty sequence with a nonempty one.","title":"Alignments of the empty sequence with a nonempty one"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/alignments-of-cons-nil","kind":"node","lean":"Cslib.Bioinformatics.HMM.alignmentsOf_cons_nil","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#alignments-of-cons-nil","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The alignments of a nonempty sequence with the empty one.","title":"Alignments of a nonempty sequence with the empty one"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/mem-alignments-of","kind":"node","lean":"Cslib.Bioinformatics.HMM.mem_alignmentsOf","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#mem-alignments-of","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"No `(none, none)` column, and the rows filter to the two sequences.","title":"Membership in `alignmentsOf` is validity"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/nodup-alignments-of","kind":"node","lean":"Cslib.Bioinformatics.HMM.nodup_alignmentsOf","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#nodup-alignments-of","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"Each alignment is listed once.","title":"Each alignment is listed once"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/alignments-of-eq-alignments","kind":"node","lean":"Cslib.Bioinformatics.HMM.alignmentsOf_eq_alignments","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#alignments-of-eq-alignments","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"On one alphabet, `alignmentsOf` is `Cslib.Algorithms.Lean.alignments`.","title":"Two-alphabet alignments agree with single-alphabet alignments"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/alignment-automaton","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#alignment-automaton","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"A weighted automaton reading an alignment of a sequence over `\u03b1` with a sequence over `\u03b2` column by column with a control state in `Q`: the weight of reading a column in a control state, the next control state, and the final weight of a control state. Weights are in `R`.","title":"Weighted alignment automaton"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/align-weight","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.alignWeight","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#align-weight","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The weight of an alignment read from control state `q`: the product of the column weights along the alignment, then the final weight.","title":"Weight of an alignment read by an automaton"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/sum-dpaux","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.sumDPAux","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#sum-dpaux","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The sum recursion with fuel (the fuel bounds the recursion depth). Implementation detail: use `sumDP`, whose fuel `xs.length + ys.length + 1` is enough.","title":"Fuel-bounded sum recursion over alignments"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/sum-dp","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.sumDP","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#sum-dp","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The total weight of all alignments of `xs` and `ys` read from control state `q`, by the first-column recursion (`sumDP_eq_sum`).","title":"Total alignment weight by first-column recursion"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/sum-dp-eq-sum","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.sumDP_eq_sum","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#sum-dp-eq-sum","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The sum recursion is the sum over alignments.","title":"The sum recursion is the sum over alignments"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/sum-dp-cons-cons","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.sumDP_cons_cons","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#sum-dp-cons-cons","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The sum recursion on two nonempty sequences.","title":"Sum recursion on two nonempty sequences"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/sum-dp-nil-cons","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.sumDP_nil_cons","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#sum-dp-nil-cons","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The sum recursion when the first sequence is empty.","title":"Sum recursion with an empty first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/sum-dp-cons-nil","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.sumDP_cons_nil","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#sum-dp-cons-nil","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The sum recursion when the second sequence is empty.","title":"Sum recursion with an empty second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/sum-dp-nil-nil","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.sumDP_nil_nil","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#sum-dp-nil-nil","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The sum recursion on two empty sequences is the final weight.","title":"Sum recursion on two empty sequences is the final weight"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/max-dpaux","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.maxDPAux","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#max-dpaux","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The max recursion with fuel. Implementation detail: use `maxDP`.","title":"Fuel-bounded max recursion over alignments"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/max-dp","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.maxDP","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#max-dp","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The max-recursion value of an alignment of `xs` and `ys` read from control state `q`, by the first-column recursion (`alignWeight_le_maxDP`, `exists_alignWeight_eq_maxDP`). It is the largest alignment weight only when the weights are nonnegative; in general it is the value of the recursively chosen alignment.","title":"Max-recursion alignment value from a control state"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/nonneg","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.Nonneg","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#nonneg","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"Nonnegative column weights. The final weights may have any sign: the max recursion bounds every alignment weight without that (`alignWeight_le_maxDP`).","title":"Nonnegative column weights of an alignment automaton"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/align-weight-le-max-dp","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.alignWeight_le_maxDP","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#align-weight-le-max-dp","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"Every alignment weighs at most the max recursion (nonnegative weights).","title":"Every alignment weighs at most the max recursion"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/exists-align-weight-eq-max-dp","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.exists_alignWeight_eq_maxDP","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#exists-align-weight-eq-max-dp","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The max recursion is attained by an alignment.","title":"The max recursion is attained"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/max-dp-cons-cons","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.maxDP_cons_cons","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#max-dp-cons-cons","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The max recursion on two nonempty sequences.","title":"Max recursion on two nonempty sequences"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/max-dp-nil-cons","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.maxDP_nil_cons","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#max-dp-nil-cons","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The max recursion when the first sequence is empty.","title":"Max recursion with an empty first sequence"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/max-dp-cons-nil","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.maxDP_cons_nil","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#max-dp-cons-nil","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The max recursion when the second sequence is empty.","title":"Max recursion with an empty second sequence"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/max-dp-nil-nil","kind":"node","lean":"Cslib.Bioinformatics.HMM.AlignmentAutomaton.maxDP_nil_nil","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#max-dp-nil-nil","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The max recursion on two empty sequences is the final weight.","title":"Max recursion on two empty sequences is the final weight"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/pair-state","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairState","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#pair-state","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The emitting states of a pair HMM: `M` emits an aligned pair, `X` a letter of the first sequence only, `Y` a letter of the second sequence only.","title":"Emitting states of a pair HMM"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/col-state","kind":"node","lean":"Cslib.Bioinformatics.HMM.colState","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#col-state","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The state that emits an alignment column: `M` for `(some a, some b)`, `X` for `(some a, none)`, `Y` for `(none, some b)` (and `M`, unused, for the invalid `(none, none)`).","title":"Pair HMM state emitting an alignment column"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/pair-hmm","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#pair-hmm","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"A pair HMM over the alphabet `\u03b1` with weights in `R` (Durbin et al., chapter 4): the transition weights from the silent begin state (`start`) and between emitting states (`trans`), to the silent end state (`stop` from an emitting state, `startStop` for the empty alignment), and the emission weights of the three emitting states.","title":"Pair hidden Markov model"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/emit-col","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.emitCol","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#emit-col","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The emission weight of an alignment column (`0` for the invalid `(none, none)`).","title":"Emission weight of an alignment column"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/to-automaton","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.toAutomaton","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#to-automaton","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The pair HMM as an alignment automaton: the control state is the previous HMM state (`none` for the begin state).","title":"Pair HMM as an alignment automaton"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/align-prob","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.alignProb","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#align-prob","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The joint weight of the sequences and the state path of an alignment `A` (the path visits `colState c` for each column `c`): the product of the transition and emission weights along the path, from the begin state to the end state.","title":"Joint weight of sequences and an alignment's state path"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/likelihood","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.likelihood","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#likelihood","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The total weight `P(xs, ys)` of a pair HMM: the total weight of all state paths that emit `xs` and `ys` (`likelihood_eq_sum`), by the first-column recursion.","title":"Total weight of a sequence pair under a pair HMM"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/likelihood-eq-sum","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.likelihood_eq_sum","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#likelihood-eq-sum","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The sum over all alignments of `xs` and `ys` of the joint weight of the sequences and the alignment.","title":"The pair-HMM recursion computes the total weight"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/viterbi","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.viterbi","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#viterbi","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The Viterbi value of a pair HMM: the max-recursion value for `xs`, `ys`. It is the largest joint weight over alignments when the HMM weights are nonnegative (see `alignProb_le_viterbi`).","title":"Viterbi value of a pair HMM"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/nonneg-2","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.Nonneg","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#nonneg-2","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"Nonnegative transition weights into emitting states and nonnegative emission weights (the transitions into the end state may have any sign: the Viterbi bound does not need them).","title":"Nonnegative weights of a pair HMM"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/to-automaton-nonneg","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.toAutomaton_nonneg","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#to-automaton-nonneg","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"A pair HMM with nonnegative weights is a nonnegative alignment automaton.","title":"Nonnegative pair HMM gives a nonnegative alignment automaton"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/align-prob-le-viterbi","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.alignProb_le_viterbi","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#align-prob-le-viterbi","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"Every alignment weighs at most the Viterbi value.","title":"Every alignment weighs at most the Viterbi value"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/exists-align-prob-eq-viterbi","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.exists_alignProb_eq_viterbi","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#exists-align-prob-eq-viterbi","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The Viterbi value is attained by an alignment: the Viterbi alignment.","title":"The Viterbi value is attained"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/is-memoryless","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.IsMemoryless","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#is-memoryless","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The pair HMM's transition weights do not depend on the current state: every transition into an emitting state `s` (from the begin state or from any emitting state) has weight `\u03c0 s`, and every transition into the end state has weight `\u03c4`.","title":"Memoryless pair HMM"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/background","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.background","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#background","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The independence-model weight for `xs` and `ys` (Durbin et al., chapter 4): `\u03b7\u00b2 (1 - \u03b7)^(|xs| + |ys|) \u220f q xs \u220f q ys`. This is a probability when `q` is a normalized nonnegative distribution and `0 < \u03b7 < 1`, as required by `LinearGapLogOdds`.","title":"Independence-model weight of a sequence pair"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/linear-gap-log-odds","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.LinearGapLogOdds","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#linear-gap-log-odds","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"The hypotheses of the log-odds correspondence: a memoryless pair HMM with normalized positive transition and emission probabilities, normalized positive background frequencies and `0 < \u03b7 < 1`, whose per-column log-odds scores against the independence model are the integer scores `\u03c3 a b` (aligned pair) and `gap` (gap column).","title":"Hypotheses of the linear-gap log-odds correspondence"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/log-align-prob-div-background","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.log_alignProb_div_background","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#log-align-prob-div-background","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"For a memoryless pair HMM, the log-odds of an alignment against the independence model is its linear-gap alignment score plus `log (\u03c4 / \u03b7\u00b2)`.","title":"Log-odds of one alignment"},{"id":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/log-viterbi-div-background","kind":"node","lean":"Cslib.Bioinformatics.HMM.PairHMM.log_viterbi_div_background","parent":45675,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models/index.html#log-viterbi-div-background","scope":"Biology/bioinformatics/sequence-alignment/pair-hidden-markov-models","status":"fully_proved","summary":"For a memoryless pair HMM whose per-column log-odds scores are `\u03c3 a b` and `gap`, the log-odds Viterbi score against the independence model is the Needleman\u2013Wunsch score with substitution scores `\u03c3` and linear gap score `gap`, plus the length term `log (\u03c4 / \u03b7\u00b2)`.","title":"Log-odds Viterbi is linear-gap Needleman\u2013Wunsch"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models.json","id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","title":"Profile hidden Markov models"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/profile-state","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileState","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#profile-state","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"The state types of a profile HMM node: match, insert, delete.","title":"State types of a profile HMM node"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/profile-hmm","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#profile-hmm","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"A profile HMM with `L` consensus columns over the alphabet `\u03b1`, with weights in `R`. `trans k s t` is the weight of a transition from the state of type `s` at node `k` (`0 \u2264 k \u2264 L`; node `0` is the begin state, read as type `M`) to a state of type `t`: to `M` or `D` at node `k + 1`, or to `I` at node `k`. `stop s` is the weight of the transition from the state of type `s` at node `L` to the end state.","title":"Profile hidden Markov model"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/columns","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.columns","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#columns","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"The consensus columns `[1, \u2026, L]`.","title":"Consensus columns of a profile HMM"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/is-plan7","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.IsPlan7","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#is-plan7","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"HMMER's Plan 7 restrictions (Eddy 1998): no delete-to-insert and no insert-to-delete transitions, and no insert states at node `0` (the begin state) or node `L`.","title":"HMMER Plan 7 restrictions"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/to-automaton","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.toAutomaton","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#to-automaton","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"The profile HMM as an alignment automaton reading alignments of a sequence to the columns `[1, \u2026, L]`. The control state is the type of the previous state and the current node (the last column consumed); it starts at `(M, 0)`, the begin state. Reading `(some a, some j)` visits `M j`, `(some a, none)` visits the insert state of the current node, `(none, some j)` visits `D j`; the invalid column `(none, none)` has\u2026","title":"Profile HMM as an alignment automaton"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/path-prob","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.pathProb","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#path-prob","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"The weight of a path through the model, given as an alignment `A` of the sequence to the columns: the product of its transition and emission weights from the begin state to the end state.","title":"Weight of a path through a profile HMM"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/forward","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.forward","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#forward","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"The forward value of a sequence `x`: the total weight of all paths through the model that emit `x` (`forward_eq_sum`), by the recursion on one residue and one column.","title":"Forward value of a sequence under a profile HMM"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/forward-eq-sum","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.forward_eq_sum","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#forward-eq-sum","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"The sum, over all alignments of `x` to the columns `[1, \u2026, L]` (all paths through the model emitting `x`, each once), of the path weights.","title":"Forward computes the total path weight"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/viterbi","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.viterbi","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#viterbi","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"The Viterbi value of a sequence `x`: the value selected by the max recursion (`pathProb_le_viterbi`, `exists_pathProb_eq_viterbi`). Under nonnegative weights it is the largest weight of a path through the model emitting `x`.","title":"Viterbi value of a sequence under a profile HMM"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/nonneg","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.Nonneg","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#nonneg","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"Nonnegative transition and emission weights (the transitions to the end state may have any sign: the Viterbi bound does not need them).","title":"Nonnegative weights of a profile HMM"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/to-automaton-nonneg","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.toAutomaton_nonneg","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#to-automaton-nonneg","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"A profile HMM with nonnegative weights is a nonnegative alignment automaton.","title":"Nonnegative profile HMM gives a nonnegative alignment automaton"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/path-prob-le-viterbi","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.pathProb_le_viterbi","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#path-prob-le-viterbi","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"Every path weighs at most the Viterbi value.","title":"Every path weighs at most the Viterbi value"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/exists-path-prob-eq-viterbi","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.exists_pathProb_eq_viterbi","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#exists-path-prob-eq-viterbi","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"Some path (alignment of `x` to the profile) has weight equal to the Viterbi value. Under nonnegative weights this attained value is the largest path weight (`pathProb_le_viterbi`); without that hypothesis attainment alone does not imply maximality.","title":"The Viterbi value is attained"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/pseudocount-emission","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.pseudocountEmission","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#pseudocount-emission","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"The emission estimate of a match state from the residues `col` that a multiple alignment places in its column (gaps omitted), with pseudocounts `pc` (Durbin et al., chapter 5; Laplace's rule is `pc = 1`): `(count a + pc a) / (|col| + \u2211 b, pc b)`.","title":"Pseudocount emission estimate of a match state"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/sum-pseudocount-emission","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.sum_pseudocountEmission","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#sum-pseudocount-emission","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"With positive pseudocounts, the estimated emissions sum to one.","title":"Pseudocount emission estimates sum to one"},{"id":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/pseudocount-emission-pos","kind":"node","lean":"Cslib.Bioinformatics.HMM.ProfileHMM.pseudocountEmission_pos","parent":45720,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models/index.html#pseudocount-emission-pos","scope":"Biology/bioinformatics/sequence-alignment/profile-hidden-markov-models","status":"fully_proved","summary":"With positive pseudocounts, every residue gets a positive emission estimate, including residues absent from the column.","title":"Pseudocount emission estimates are positive"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment.json","id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","title":"Semi-global and overlap alignment"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/zero-le-list-max","kind":"node","lean":"Cslib.Algorithms.Lean.zero_le_listMax","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#zero-le-list-max","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"For every list $L$ of integers, the maximum of $L$ with floor $0$ (which is $0$ for the empty list) is at least $0$.","title":"The floored list maximum is nonnegative"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/overlap-pairs","kind":"node","lean":"Cslib.Algorithms.Lean.overlapPairs","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#overlap-pairs","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"The overlap pairs of sequences $x$ and $y$ are the pairs $(x', y')$ in which $x'$ is a suffix of $x$ and $y'$ is a prefix of $y$, or $x'$ is a prefix of $x$ and $y'$ is a suffix of $y$. Prefixes and suffixes include the empty sequence and the whole sequence; the pairs are listed with the suffix\u2013prefix pairs first, and the list can contain repetitions.","title":"Overlap pairs"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/overlap-score","kind":"node","lean":"Cslib.Algorithms.Lean.overlapScore","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#overlap-score","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Given a substitution score $\\sigma$ and a gap score $g$, the overlap score of sequences $s$ and $t$ is the maximum, with floor $0$, of the Needleman\u2013Wunsch scores of the overlap pairs of $s$ and $t$: the pairs $(s', t')$ where $s'$ is a suffix of $s$ and $t'$ a prefix of $t$, or $s'$ is a prefix of $s$ and $t'$ a suffix of $t$.","title":"Overlap alignment score"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/nw-score-le-overlap-score","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_le_overlapScore","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#nw-score-le-overlap-score","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ a gap score. If $(x', y')$ is an overlap pair of sequences $x$ and $y$, then the Needleman\u2013Wunsch score of $x'$ against $y'$ is at most the overlap score of $x$ and $y$, both taken with $\\sigma$ and $g$.","title":"Overlap-pair scores are at most the overlap score"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/overlap-score-le","kind":"node","lean":"Cslib.Algorithms.Lean.overlapScore_le","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#overlap-score-le","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Fix a substitution score $\\sigma$ and a gap score $g$, and sequences $s$ and $t$. If an integer $b$ is at least the Needleman\u2013Wunsch score of every overlap pair of $s$ and $t$ (a suffix of $s$ with a prefix of $t$, or a prefix of $s$ with a suffix of $t$), then the overlap score of $s$ and $t$ is at most $b$.","title":"Upper bounds for the overlap score"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/exists-nw-score-eq-overlap-score","kind":"node","lean":"Cslib.Algorithms.Lean.exists_nwScore_eq_overlapScore","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#exists-nw-score-eq-overlap-score","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ a gap score. For all sequences $x$ and $y$ there is an overlap pair $(x', y')$ of $x$ and $y$ such that the Needleman\u2013Wunsch score of $x'$ against $y'$ (with $\\sigma$ and $g$) equals the overlap score of $x$ and $y$.","title":"The overlap score is attained"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/overlap-score-nonneg","kind":"node","lean":"Cslib.Algorithms.Lean.overlapScore_nonneg","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#overlap-score-nonneg","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"For every substitution score $\\sigma$, gap score $g$ and sequences $s$ and $t$, the overlap score of $s$ and $t$ is at least $0$.","title":"The overlap score is nonnegative"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/infixes","kind":"node","lean":"Cslib.Algorithms.Lean.infixes","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#infixes","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"The infixes of a sequence $y$ are its contiguous substrings: the sequences $l$ such that $y$ is the concatenation of $s$, $l$ and $t$ for some sequences $s$ and $t$, including the empty sequence and $y$ itself. They are listed as the suffixes of the prefixes of $y$, so the list can contain repetitions.","title":"Infixes"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/fitting-score","kind":"node","lean":"Cslib.Algorithms.Lean.fittingScore","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#fitting-score","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ a gap score. The fitting score of a sequence $x$ against a sequence $y$ is the maximum, over all infixes $l$ of $y$ (including the empty one), of the Needleman\u2013Wunsch score of $x$ against $l$ with $\\sigma$ and $g$: the best global alignment score of $x$ against a substring of $y$.","title":"Fitting score"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/nw-score-le-fitting-score","kind":"node","lean":"Cslib.Algorithms.Lean.nwScore_le_fittingScore","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#nw-score-le-fitting-score","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ a gap score. If $l$ is an infix of a sequence $y$, then for every sequence $x$ the Needleman\u2013Wunsch score of $x$ against $l$ is at most the fitting score of $x$ against $y$, both taken with $\\sigma$ and $g$.","title":"Infix scores are at most the fitting score"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/fitting-score-le","kind":"node","lean":"Cslib.Algorithms.Lean.fittingScore_le","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#fitting-score-le","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score, $g$ a gap score, $x$ and $y$ sequences and $b$ an integer. If the Needleman\u2013Wunsch score of $x$ against every infix of $y$ (with $\\sigma$ and $g$) is at most $b$, then the fitting score of $x$ against $y$ is at most $b$.","title":"Upper bounds on the fitting score"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/exists-nw-score-eq-fitting-score","kind":"node","lean":"Cslib.Algorithms.Lean.exists_nwScore_eq_fittingScore","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#exists-nw-score-eq-fitting-score","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ a gap score. For all sequences $x$ and $y$ there is an infix $l$ of $y$ such that the Needleman\u2013Wunsch score of $x$ against $l$ (with $\\sigma$ and $g$) equals the fitting score of $x$ against $y$.","title":"The fitting score is attained"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/overlap-score-le-local-score","kind":"node","lean":"Cslib.Algorithms.Lean.overlapScore_le_localScore","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#overlap-score-le-local-score","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ a gap score. For all sequences $x$ and $y$, the overlap score of $x$ and $y$ is at most the local score of $x$ and $y$, both taken with $\\sigma$ and $g$.","title":"The overlap score is at most the local score"},{"id":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/fitting-score-le-local-score","kind":"node","lean":"Cslib.Algorithms.Lean.fittingScore_le_localScore","parent":45737,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment/index.html#fitting-score-le-local-score","scope":"Biology/bioinformatics/sequence-alignment/semi-global-and-overlap-alignment","status":"fully_proved","summary":"Let $\\sigma$ be a substitution score and $g$ a gap score. For all sequences $x$ and $y$, the fitting score of $x$ against $y$ is at most the local score of $x$ and $y$, both taken with $\\sigma$ and $g$.","title":"The fitting score is at most the local score"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment.json","id":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/index.html","scope":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","status":"fully_proved","title":"Smith\u2013Waterman local alignment"},{"id":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/next-row-aux","kind":"node","lean":"Cslib.Bioinformatics.Alignment.SmithWaterman.nextRowAux","parent":45752,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/index.html#next-row-aux","scope":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","status":"fully_proved","summary":"The cells `H[i+1][j+1], H[i+1][j+2], \u2026` of a row from the remaining letters `b :: ys` of the second sequence, the remaining cells `up :: prev` of the previous row, the previous row's cell `diag` to the upper left and the cell `left` just computed. A previous row that is too short ends the row early (this does not happen for rows of the table, `nextRow_eq`).","title":"Auxiliary row computation for the Smith\u2013Waterman table"},{"id":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/next-row","kind":"node","lean":"Cslib.Bioinformatics.Alignment.SmithWaterman.nextRow","parent":45752,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/index.html#next-row","scope":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","status":"fully_proved","summary":"The row of the table for the prefix extended by `a`, from the previous row: `H[i+1][0] = 0`, then the recurrence left to right.","title":"Next row of the Smith\u2013Waterman table"},{"id":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/sw-tab-aux","kind":"node","lean":"Cslib.Bioinformatics.Alignment.SmithWaterman.swTabAux","parent":45752,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/index.html#sw-tab-aux","scope":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","status":"fully_proved","summary":"The Smith\u2013Waterman table, filled row by row over the remaining letters of the first sequence: `row` is the current row and `best` the largest cell of the earlier rows.","title":"Row-by-row filling of the Smith\u2013Waterman table"},{"id":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/sw-tab","kind":"node","lean":"Cslib.Bioinformatics.Alignment.SmithWaterman.swTab","parent":45752,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/index.html#sw-tab","scope":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","status":"fully_proved","summary":"The largest cell of the `(|xs| + 1) \u00d7 (|ys| + 1)` table, with the zero row `H[0][j] = 0` to start.","title":"The Smith\u2013Waterman table"},{"id":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/next-row-eq","kind":"node","lean":"Cslib.Bioinformatics.Alignment.SmithWaterman.nextRow_eq","parent":45752,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/index.html#next-row-eq","scope":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","status":"fully_proved","summary":"From the row of the prefix `px`, `nextRow` computes the row of `px ++ [a]`: the cells `H[i+1][j] = swCellScore \u03c3 gap (px ++ [a]) py` for the prefixes `py` of `ys`, shortest first.","title":"One row of the table"},{"id":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/sw-tab-eq-sw-score","kind":"node","lean":"Cslib.Bioinformatics.Alignment.SmithWaterman.swTab_eq_swScore","parent":45752,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/index.html#sw-tab-eq-sw-score","scope":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","status":"fully_proved","summary":"`swTab` is the largest cell `swCellScore \u03c3 gap px py` over all pairs of prefixes.","title":"The table computes the Smith\u2013Waterman score"},{"id":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/sw-tab-eq-local-score","kind":"node","lean":"Cslib.Bioinformatics.Alignment.SmithWaterman.swTab_eq_localScore","parent":45752,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment/index.html#sw-tab-eq-local-score","scope":"Biology/bioinformatics/sequence-alignment/smith-waterman-local-alignment","status":"fully_proved","summary":"The table computes the optimal local alignment score (Smith\u2013Waterman 1981, `gap \u2264 0`): `swTab` is the maximum, over all pairs of infixes of `xs` and `ys` (including the empty ones, which give the floor `0`), of their optimal global alignment score `nwScore`.","title":"The table computes the optimal local alignment score"},{"data_url":"scopes/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm.json","id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","kind":"scope","navigation":"module","parent":45506,"result":false,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","title":"Wavefront alignment algorithm"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/linear-penalties","kind":"node","lean":"Cslib.Algorithms.Lean.LinearPenalties","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#linear-penalties","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"Linear gap penalties: match `0`, mismatch `mismatch \u2265 1`, each gap column `gap \u2265 1`.","title":"Linear gap penalties"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/column-cost","kind":"node","lean":"Cslib.Algorithms.Lean.LinearPenalties.columnCost","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#column-cost","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The cost of one alignment column.","title":"Cost of an alignment column under linear gap penalties"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/cost","kind":"node","lean":"Cslib.Algorithms.Lean.LinearPenalties.cost","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#cost","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The cost of an alignment: the sum of its column costs.","title":"Cost of an alignment under linear gap penalties"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/max-offset","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.maxOffset","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#max-offset","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The largest valid offset on diagonal `k`: `h \u2264 |ys|` and `v = h - k \u2264 |xs|`.","title":"Largest valid offset on a diagonal"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/extend","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.extend","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#extend","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"Follow matching letters along diagonal `k` from offset `h`, with fuel.","title":"Fuelled extension of matches along a diagonal"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/slide","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.slide","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#slide","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The match extension (\"slide\") along diagonal `k` from offset `h`.","title":"Match extension (slide) along a diagonal"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/wavefront-aux","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.wavefrontAux","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#wavefront-aux","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The wavefront recurrence with fuel (fuel `s + 1` suffices; see `wavefront`). `none` stands for an unreachable diagonal. Offsets proposed by an operation are clamped to the largest valid offset of the target diagonal before sliding.","title":"Fuelled wavefront recurrence"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/wavefront","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.wavefront","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#wavefront","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The furthest-reaching offset of score `s` on diagonal `k = h - v`.","title":"Furthest-reaching wavefront offset"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/wavefront-sound","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.wavefront_sound","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#wavefront-sound","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"Soundness: a wavefront offset is a valid cell of its diagonal, reachable by an alignment of the corresponding prefixes with cost at most `s`.","title":"Soundness of the wavefront algorithm"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/wavefront-complete","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.wavefront_complete","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#wavefront-complete","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"Completeness: every alignment of a pair of prefixes is dominated on its diagonal by the wavefront of its exact cost.","title":"Completeness of the wavefront algorithm"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/exists-wavefront-end-iff","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.exists_wavefront_end_iff","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#exists-wavefront-end-iff","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The wavefront algorithm is exact: the end cell is reached by some score `s' \u2264 s` iff some alignment of `xs` and `ys` costs at most `s`. Hence the least such `s` is the optimal cost.","title":"Exactness of the wavefront alignment algorithm"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/score","kind":"node","lean":"Cslib.Algorithms.Lean.LinearPenalties.score","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#score","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The substitution score in the maximization convention of `nwScore`: `0` for a match and `-mismatch` for a mismatch (the gap score is `-gap`).","title":"Substitution score from linear gap penalties"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/alignment-score-eq-neg-cost","kind":"node","lean":"Cslib.Algorithms.Lean.LinearPenalties.alignmentScore_eq_neg_cost","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#alignment-score-eq-neg-cost","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"Scores are negated costs.","title":"Alignment score is the negated linear-penalty cost"},{"id":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/exists-wavefront-end-iff-nw-score","kind":"node","lean":"Cslib.Algorithms.Lean.Wavefront.exists_wavefront_end_iff_nwScore","parent":45760,"result":true,"route":"../roadmap/Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm/index.html#exists-wavefront-end-iff-nw-score","scope":"Biology/bioinformatics/sequence-alignment/wavefront-alignment-algorithm","status":"fully_proved","summary":"The end cell is reached by some score `s' \u2264 s` iff `s` is at least the optimal Needleman\u2013Wunsch cost.","title":"The wavefront algorithm computes the optimal alignment cost"},{"data_url":"scopes/Biology/bioinformatics/aracne-gene-network-reconstruction.json","id":"Biology/bioinformatics/aracne-gene-network-reconstruction","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/aracne-gene-network-reconstruction/index.html","scope":"Biology/bioinformatics/aracne-gene-network-reconstruction","status":"fully_proved","title":"ARACNE gene network reconstruction"},{"id":"Biology/bioinformatics/aracne-gene-network-reconstruction/aracne","kind":"node","lean":"Cslib.Bioinformatics.GeneNetworks.Aracne.aracne","parent":45775,"result":true,"route":"../roadmap/Biology/bioinformatics/aracne-gene-network-reconstruction/index.html#aracne","scope":"Biology/bioinformatics/aracne-gene-network-reconstruction","status":"fully_proved","summary":"The network ARACNE reconstructs from symmetric pairwise mutual informations `I` with independence threshold `I\u2080` and DPI tolerance zero (Margolin et al. 2006): `{i, k}` is an edge iff `i \u2260 k`, `I\u2080 < I i k`, and no third gene `j` with `I\u2080 < I i j` and `I\u2080 < I j k` has `I i k < min (I i j) (I j k)`.","title":"ARACNE gene network reconstruction"},{"id":"Biology/bioinformatics/aracne-gene-network-reconstruction/le-aracne-of-is-acyclic","kind":"node","lean":"Cslib.Bioinformatics.GeneNetworks.Aracne.le_aracne_of_isAcyclic","parent":45775,"result":true,"route":"../roadmap/Biology/bioinformatics/aracne-gene-network-reconstruction/index.html#le-aracne-of-is-acyclic","scope":"Biology/bioinformatics/aracne-gene-network-reconstruction","status":"fully_proved","summary":"If the true network `T` is a forest whose edges all carry mutual information above `I\u2080`, and `I a b \u2264 min (I a c) (I c b)` whenever `c` separates `a` from `b` in `T` (every walk from `a` to `b` visits `c`), then ARACNE keeps every edge of `T`.","title":"No false negatives"},{"id":"Biology/bioinformatics/aracne-gene-network-reconstruction/aracne-le-of-is-tree","kind":"node","lean":"Cslib.Bioinformatics.GeneNetworks.Aracne.aracne_le_of_isTree","parent":45775,"result":true,"route":"../roadmap/Biology/bioinformatics/aracne-gene-network-reconstruction/index.html#aracne-le-of-is-tree","scope":"Biology/bioinformatics/aracne-gene-network-reconstruction","status":"fully_proved","summary":"If the true network `T` is a tree and `I a b < min (I a c) (I c b)` whenever `c` separates `a` from `b` in `T`, then every edge ARACNE keeps is an edge of `T`.","title":"No false positives"},{"id":"Biology/bioinformatics/aracne-gene-network-reconstruction/aracne-eq-of-is-tree","kind":"node","lean":"Cslib.Bioinformatics.GeneNetworks.Aracne.aracne_eq_of_isTree","parent":45775,"result":true,"route":"../roadmap/Biology/bioinformatics/aracne-gene-network-reconstruction/index.html#aracne-eq-of-is-tree","scope":"Biology/bioinformatics/aracne-gene-network-reconstruction","status":"fully_proved","summary":"ARACNE recovers tree networks exactly (Margolin et al. 2006, Theorem 1): if the true network `T` is a tree whose edges all carry mutual information above `I\u2080`, and the strict data-processing inequality holds on every separated triple, then ARACNE returns `T`. Strictness is the explicit form of the paper's exclusion of exact cancellations; with equality ARACNE can keep an indirect edge (see the tests).","title":"ARACNE recovers tree networks exactly"},{"id":"Biology/bioinformatics/aracne-gene-network-reconstruction/le-aracne-of-is-max-weight-spanning-tree","kind":"node","lean":"Cslib.Bioinformatics.GeneNetworks.Aracne.le_aracne_of_isMaxWeightSpanningTree","parent":45775,"result":true,"route":"../roadmap/Biology/bioinformatics/aracne-gene-network-reconstruction/index.html#le-aracne-of-is-max-weight-spanning-tree","scope":"Biology/bioinformatics/aracne-gene-network-reconstruction","status":"fully_proved","summary":"Chow\u2013Liu trees survive ARACNE (Margolin et al. 2006, Theorem 2): a spanning tree `T` of maximum total weight `\u2211_{{i, k} \u2208 T} I i k` among all spanning trees, whose edges all carry mutual information above `I\u2080`, is a subnetwork of the ARACNE network.","title":"Chow\u2013Liu trees survive ARACNE"},{"data_url":"scopes/Biology/bioinformatics/chromhmm-chromatin-state-model.json","id":"Biology/bioinformatics/chromhmm-chromatin-state-model","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","title":"ChromHMM chromatin-state model"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/ones-count","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.onesCount","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#ones-count","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The weight `\u2211 o with o j = true, c o` of the binary vectors with mark `j` present.","title":"Weight of binary vectors with a mark present"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/zeros-count","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.zerosCount","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#zeros-count","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The weight `\u2211 o with o j = false, c o` of the binary vectors with mark `j` absent.","title":"Weight of binary vectors with a mark absent"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/factored-update","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.factoredUpdate","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#factored-update","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The factored (per-mark) update `ones j / (ones j + zeros j)`.","title":"Factored per-mark update"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/sum-bern-pmf","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.sum_bernPMF","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#sum-bern-pmf","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"A product of Bernoulli distributions sums to one over the binary vectors, for every parameter vector `p` (the identity is algebraic: `\u220f j, (p j + (1 - p j)) = 1`).","title":"Product of Bernoulli distributions sums to one"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/sum-mul-log-bern-pmf","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.sum_mul_log_bernPMF","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#sum-mul-log-bern-pmf","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The expected log-likelihood of weighted binary vectors under a product of Bernoulli distributions with parameters in `(0, 1)` splits over the marks.","title":"Expected product-Bernoulli log-likelihood splits over marks"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/sum-mul-log-bern-pmf-le-factored-update","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.sum_mul_log_bernPMF_le_factoredUpdate","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#sum-mul-log-bern-pmf-le-factored-update","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The factored update maximizes the expected log-likelihood over all products of Bernoulli distributions with parameters in `(0, 1)`, for nonnegative weights `c` in which every mark is both present and absent with positive weight.","title":"The factored update maximizes the expected log-likelihood"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/factored-update-pos","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.factoredUpdate_pos","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#factored-update-pos","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The factored update is positive when both counts are.","title":"Positivity of the factored update"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/factored-update-lt-one","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.factoredUpdate_lt_one","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#factored-update-lt-one","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The factored update is below one when both counts are positive.","title":"Factored update is below one"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/params","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.Params","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#params","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"ChromHMM parameters: initial and transition weights, and the presence probability of each mark in each state.","title":"ChromHMM parameters"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/to-hmm","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.Params.toHMM","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#to-hmm","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The HMM of ChromHMM parameters: emissions are products of Bernoulli distributions.","title":"HMM of ChromHMM parameters"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/valid","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.Params.Valid","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#valid","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"Valid parameters: positive initial and transition distributions, mark probabilities in `(0, 1)`.","title":"Valid ChromHMM parameters"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/valid-params","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.ValidParams","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#valid-params","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The parameter space of ChromHMM.","title":"Parameter space of ChromHMM"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/to-hmm-2","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.Params.Valid.toHMM","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#to-hmm-2","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"A valid ChromHMM parameter is a valid HMM.","title":"Valid ChromHMM parameters give a valid HMM"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/to-valid-hmm","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.toValidHMM","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#to-valid-hmm","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The valid HMM of a valid ChromHMM parameter.","title":"Valid HMM built from a valid ChromHMM parameter"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/joint","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.joint","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#joint","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The EM model of ChromHMM: the joint weight of a state path and a sequence of mark vectors.","title":"ChromHMM joint weight of a state path and mark vectors"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/mark-ones","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.markOnes","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#mark-ones","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The expected number of bins in state `s` with mark `j` present.","title":"Expected number of bins in a state with a mark present"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/mark-zeros","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.markZeros","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#mark-zeros","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The expected number of bins in state `s` with mark `j` absent.","title":"Expected number of bins in a state with a mark absent"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/m-step","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.mStep","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#m-step","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The Baum-Welch initial and transition updates, and the factored emission update `prob s j = markOnes / (markOnes + markZeros)`.","title":"The ChromHMM M-step"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/mark-ones-eq","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.markOnes_eq","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#mark-ones-eq","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"`markOnes s j` is the posterior expected number of bins in state `s` at which mark `j` is present.","title":"The factored update in posterior form"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/rich","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.Rich","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#rich","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"Data on which the ChromHMM M-step is valid: a sequence, a transition, and every mark both present and absent somewhere.","title":"Rich data for the ChromHMM M-step"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/m-step-valid","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.mStep_valid","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#m-step-valid","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The ChromHMM update is a valid parameter on rich data.","title":"The ChromHMM update is a valid parameter"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/em-step","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.emStep","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#em-step","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"One ChromHMM step, on rich data.","title":"One ChromHMM step on rich data"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/joint-pos","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.joint_pos","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#joint-pos","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The ChromHMM joint weight is positive.","title":"Positivity of the ChromHMM joint weight"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/em-q-le-em-step","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.emQ_le_emStep","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#em-q-le-em-step","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The ChromHMM update maximizes `Q(\u00b7 | \u03b8)` over all valid product-Bernoulli HMMs.","title":"The ChromHMM update maximizes `Q(\u00b7 | \u03b8)`"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/log-lik-le-em-step","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.logLik_le_emStep","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#log-lik-le-em-step","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"Each ChromHMM step does not decrease the log-likelihood.","title":"Each ChromHMM step does not decrease the log-likelihood"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/log-lik-iterate-em-step-monotone","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.logLik_iterate_emStep_monotone","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#log-lik-iterate-em-step-monotone","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The log-likelihood is monotone along the ChromHMM iterates.","title":"The log-likelihood is monotone along the ChromHMM iterates"},{"id":"Biology/bioinformatics/chromhmm-chromatin-state-model/log-lik-eq","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ChromHMM.logLik_eq","parent":45781,"result":true,"route":"../roadmap/Biology/bioinformatics/chromhmm-chromatin-state-model/index.html#log-lik-eq","scope":"Biology/bioinformatics/chromhmm-chromatin-state-model","status":"fully_proved","summary":"The EM log-likelihood is the sum of the HMM log-likelihoods of the sequences.","title":"EM log-likelihood as a sum of HMM log-likelihoods"},{"data_url":"scopes/Biology/bioinformatics/de-bruijn-graph-genome-assembly.json","id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","title":"De Bruijn graph genome assembly"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/kmers","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.kmers","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#kmers","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"The `k`-mers of `s` in order of position: the length-`k` windows `(s.drop i).take k` for `i + k \u2264 |s|`.","title":"The k-mers of a string"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/kmers-impl","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.kmersImpl","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#kmers-impl","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"Executable form of `kmers`: the suffixes of `s` are generated once by `List.tails`, so the cost is `O(|s| \u00b7 k)` rather than the `\u0398(|s|\u00b2)` of recomputing `s.drop i` for every `i`. The compiler uses it in place of `kmers` (`kmers_eq_kmersImpl`).","title":"Efficient executable computation of k-mers"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/kmer-edge","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.kmerEdge","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#kmer-edge","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"The edge of the de Bruijn multigraph contributed by a `k`-mer `w`: from its prefix `(k-1)`-mer `w.dropLast` to its suffix `(k-1)`-mer `w.tail`.","title":"De Bruijn edge of a k-mer"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/de-bruijn-edges","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.deBruijnEdges","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#de-bruijn-edges","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"The de Bruijn multigraph of `s` (Pevzner\u2013Tang\u2013Waterman 2001) as its list of edge occurrences: one edge per `k`-mer occurrence, in order of position.","title":"De Bruijn multigraph of a string"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/spell","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.spell","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#spell","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"The string spelled by a walk of `(k-1)`-mers: the first vertex followed by the last letter of each later vertex.","title":"String spelled by a walk of (k-1)-mers"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/zip-tail-kmers","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.zip_tail_kmers","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#zip-tail-kmers","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"Its steps are exactly the edges contributed by the `k`-mers of `s`, one per occurrence, in order of position.","title":"The `(k-1)`-mer walk of `s` traverses the de Bruijn multigraph in order"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/zip-tail-kmers-perm","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.zip_tail_kmers_perm","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#zip-tail-kmers-perm","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"The `k`-mer walk of `s` is an Eulerian trail of its de Bruijn multigraph: its steps are a permutation of the edge occurrences, each traversed once.","title":"The `k`-mer walk of `s` is an Eulerian trail"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/spell-kmers","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.spell_kmers","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#spell-kmers","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"The `(k-1)`-mer walk of `s` spells `s` back (`2 \u2264 k`, `k - 1 \u2264 |s|`).","title":"The (k-1)-mer walk of a string spells the string"},{"id":"Biology/bioinformatics/de-bruijn-graph-genome-assembly/kmers-spell-perm","kind":"node","lean":"Cslib.Bioinformatics.Assembly.DeBruijn.kmers_spell_perm","parent":45809,"result":true,"route":"../roadmap/Biology/bioinformatics/de-bruijn-graph-genome-assembly/index.html#kmers-spell-perm","scope":"Biology/bioinformatics/de-bruijn-graph-genome-assembly","status":"fully_proved","summary":"Every Eulerian trail spells a string with the same `k`-mer multiset (`2 \u2264 k`): if `W` is a walk of `(k-1)`-mers whose steps are a permutation of the edge occurrences of the de Bruijn multigraph of `s`, then the `k`-mers of the spelled string are a permutation of those of `s`. (The empty walk is allowed: it is a trail only when `s` has no `k`-mers, and spells `[]`.)","title":"Every Eulerian trail spells a string with the same `k`-mer multiset"},{"data_url":"scopes/Biology/bioinformatics/gps-model-of-chip-seq-binding-events.json","id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","title":"GPS model of ChIP-seq binding events"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/survivor-weight","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.survivorWeight","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#survivor-weight","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"The surviving weight `max (N j - \u03b1) 0` of event `j`, for expected read counts `N` and sparseness parameter `\u03b1`.","title":"Surviving weight of a GPS event"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/m-step","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.mStep","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#m-step","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"`\u03c0 j = max (N j - \u03b1) 0 / \u2211 k, max (N k - \u03b1) 0`.","title":"The GPS M-step with component elimination"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/penalized-q","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.penalizedQ","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#penalized-q","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"The expected complete-data log penalized likelihood of GPS, up to an additive constant: `\u2211 j, (N j - \u03b1) log (\u03c0 j)` (the expected log-likelihood `\u2211 j, N j log (\u03c0 j)` plus the log of the negative Dirichlet prior). Meaningful for positive `\u03c0`.","title":"Expected penalized complete-data log-likelihood of GPS"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/truncated-q","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.truncatedQ","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#truncated-q","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"The truncated objective `\u2211 j, max (N j - \u03b1) 0 * log (\u03c0 j)`.","title":"Truncated GPS objective"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/survivor-weight-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.survivorWeight_nonneg","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#survivor-weight-nonneg","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"Surviving weights are nonnegative.","title":"Nonnegativity of surviving weights"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/sum-survivor-weight-pos","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.sum_survivorWeight_pos","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#sum-survivor-weight-pos","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"The total surviving weight is positive iff some event has `\u03b1 < N j`.","title":"Total surviving weight is positive iff some event survives"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/m-step-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.mStep_nonneg","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#m-step-nonneg","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"The update is nonnegative.","title":"Nonnegativity of the GPS M-step update"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/sum-m-step","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.sum_mStep","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#sum-m-step","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"The update sums to one when some event survives.","title":"GPS M-step update sums to one when some event survives"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/m-step-eq-zero-iff","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.mStep_eq_zero_iff","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#m-step-eq-zero-iff","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"When some event survives, event `j` gets probability zero iff `N j \u2264 \u03b1`.","title":"Elimination"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/m-step-pos-iff","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.mStep_pos_iff","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#m-step-pos-iff","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"When some event survives, event `j` keeps positive probability iff `\u03b1 < N j`.","title":"GPS M-step keeps positive probability exactly on surviving events"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/m-step-zero","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.mStep_zero","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#m-step-zero","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"With `\u03b1 = 0` and nonnegative counts the update is the ordinary mixture M-step `N j / \u2211 N`.","title":"GPS M-step at zero sparseness is the ordinary mixture M-step"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/truncated-q-le-m-step","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.truncatedQ_le_mStep","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#truncated-q-le-m-step","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"The update is an upper bound for the truncated objective over the positive probability vectors (derived characterization of the elimination rule). Upper-bound statement only: this does not prove the value at `mStep` is the supremum (no approximation or `sSup` theorem is given). When elimination occurs `mStep` itself is not a positive vector (eliminated coordinates are zero), so it is the boundary point on the face\u2026","title":"The update is an upper bound for the truncated objective"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/truncated-q-eq-penalized-q","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.truncatedQ_eq_penalizedQ","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#truncated-q-eq-penalized-q","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"When every event survives, the penalized and truncated objectives agree.","title":"Truncated and penalized objectives agree when every event survives"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/m-step-of-forall-lt","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.mStep_of_forall_lt","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#m-step-of-forall-lt","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"When every event survives, the update is `(N j - \u03b1) / \u2211 k, (N k - \u03b1)`.","title":"GPS M-step formula when every event survives"},{"id":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events/penalized-q-le-m-step","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.GPS.penalizedQ_le_mStep","parent":45819,"result":true,"route":"../roadmap/Biology/bioinformatics/gps-model-of-chip-seq-binding-events/index.html#penalized-q-le-m-step","scope":"Biology/bioinformatics/gps-model-of-chip-seq-binding-events","status":"fully_proved","summary":"When every remaining event survives (`\u03b1 < N j` for all `j`), the update maximizes the GPS expected complete-data log penalized likelihood over the positive probability vectors.","title":"The M-step of the restructured model"},{"data_url":"scopes/Biology/bioinformatics/lander-waterman-coverage-model.json","id":"Biology/bioinformatics/lander-waterman-coverage-model","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","title":"Lander\u2013Waterman coverage model"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/covers","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.Covers","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#covers","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"A read of length `L` starting at position `s` of a circular genome of length `G` covers base `x` when `x` is one of `s, s + 1, \u2026, s + L - 1 (mod G)`, i.e. `(x - s) mod G < L`.","title":"Read covering a base of a circular genome"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/depth","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.depth","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#depth","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"The depth of coverage of base `x`: the number of reads covering it.","title":"Depth of coverage of a base"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/normalized-count","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.normalizedCount","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#normalized-count","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"The totalized normalized count `|E| / G ^ N` of a set of start tuples. For `0 < G`, this is the probability when the `N` start positions are independent and uniform on `Fin G`. If `G = 0` and `N > 0`, the sample space is empty and this value is zero, so it is not a probability law. For `N = 0`, there is one empty start tuple, even when `G = 0`.","title":"Normalized count of read start tuples"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/normalized-count-eq-uniform-of-finset","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.normalizedCount_eq_uniformOfFinset","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#normalized-count-eq-uniform-of-finset","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"`normalizedCount` agrees with Mathlib's totalized uniform measure on all start tuples. This measure is a probability law only when the sample space is nonempty.","title":"Normalized count agrees with the totalized uniform measure"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/card-uncovered","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.card_uncovered","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#card-uncovered","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"For `L \u2264 G`, base `x` is uncovered by all `N` reads for exactly `(G - L) ^ N` of the `G ^ N` start tuples.","title":"Exact coverage count"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/card-depth-eq","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.card_depth_eq","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#card-depth-eq","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"For `L \u2264 G`, base `x` is covered by exactly `k` reads for `N.choose k * L ^ k * (G - L) ^ (N - k)` of the start tuples.","title":"Exact depth law (count form)"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/normalized-count-uncovered","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.normalizedCount_uncovered","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#normalized-count-uncovered","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"Exact uncovered probability (Clarke\u2013Carbon, Lander\u2013Waterman, circular genome): `(1 - L / G) ^ N` for `L \u2264 G`.","title":"Exact uncovered probability"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/normalized-count-depth-eq","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.normalizedCount_depth_eq","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#normalized-count-depth-eq","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"The depth at a fixed base is binomial with `N` trials and success probability `L / G`, for `L \u2264 G`.","title":"Exact depth law"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/tendsto-normalized-count-uncovered","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.tendsto_normalizedCount_uncovered","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#tendsto-normalized-count-uncovered","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"With read length `L` fixed, genome lengths `G N \u2265 L`, and coverage `N L / G_N \u2192 c` as the number of reads `N \u2192 \u221e`, the probability that a given base is uncovered tends to `e^{-c}`.","title":"Lander\u2013Waterman limit for the uncovered fraction"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/tendsto-normalized-count-depth","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.tendsto_normalizedCount_depth","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#tendsto-normalized-count-depth","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"Under the hypotheses of `tendsto_normalizedCount_uncovered`, the probability that a given base is covered by exactly `k` reads tends to `e^{-c} c^k / k.factorial`.","title":"The coverage depth tends to Poisson(`c`)"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/no-successor","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.NoSuccessor","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#no-successor","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"Read `j` has no successor when no other read starts in the `L - 1` positions after it, `starts j + 1, \u2026, starts j + L - 1 (mod G)` (reads starting at the same position are not successors of each other). In Lander and Waterman's terms such a read is counted as ending an island, but several no-successor reads can lie in one island, so this is not the island count.","title":"Read with no successor"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/sum-card-no-successor","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.sum_card_noSuccessor","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#sum-card-no-successor","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"For `L \u2264 G`, summed over the reads, the number of start tuples in which read `j` has no successor is `N * G * (G - (L - 1)) ^ (N - 1)`. This counts no-successor reads, not island ends: a start tuple can have several no-successor reads in one island.","title":"Count of no-successor reads (exact)"},{"id":"Biology/bioinformatics/lander-waterman-coverage-model/expected-no-successor","kind":"node","lean":"Cslib.Bioinformatics.Assembly.LanderWaterman.expected_noSuccessor","parent":45835,"result":true,"route":"../roadmap/Biology/bioinformatics/lander-waterman-coverage-model/index.html#expected-no-successor","scope":"Biology/bioinformatics/lander-waterman-coverage-model","status":"fully_proved","summary":"For `0 < G` and `L \u2264 G`, the expected number of reads with no successor is `N (1 - (L - 1) / G) ^ (N - 1)`. The `0 < G` hypothesis is essential: at `G = 0` and `N > 0`, the right-hand expression is `N` while the sum of totalized normalized counts is zero.","title":"Expected number of no-successor reads (exact)"},{"data_url":"scopes/Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure.json","id":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/index.html","scope":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure","status":"fully_proved","title":"Nussinov algorithm for RNA secondary structure"},{"id":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/is-structure","kind":"node","lean":"Cslib.Bioinformatics.RNA.Nussinov.IsStructure","parent":45849,"result":true,"route":"../roadmap/Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/index.html#is-structure","scope":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure","status":"fully_proved","summary":"A secondary structure on `s` (Nussinov et al. 1978): a finite set of base pairs `(i, j)`, each closing a hairpin with at least `m` unpaired positions strictly inside (`i + m < j`), joining complementary bases (`pair s[i] s[j]`, which also puts both positions inside `s`), with every position in at most one pair and no two pairs crossing (`i < k < j < l`).","title":"RNA secondary structure"},{"id":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/list-max","kind":"node","lean":"Cslib.Bioinformatics.RNA.Nussinov.listMax","parent":45849,"result":true,"route":"../roadmap/Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/index.html#list-max","scope":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure","status":"fully_proved","summary":"The largest of a list of naturals (`0` for the empty list).","title":"Maximum of a list of natural numbers"},{"id":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/nussinov","kind":"node","lean":"Cslib.Bioinformatics.RNA.Nussinov.nussinov","parent":45849,"result":true,"route":"../roadmap/Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/index.html#nussinov","scope":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure","status":"fully_proved","summary":"The first base `a` of `a :: rest` is either unpaired, or paired with `rest[k]` for some `k \u2265 m` with `pair a rest[k]`, splitting the rest into the enclosed part `rest.take k` and the outer part `rest.drop (k + 1)`. Exponential-time specification; the usual `O(n\u00b3)` table memoizes it on the intervals of `s`.","title":"The Nussinov recursion"},{"id":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/card-le-nussinov","kind":"node","lean":"Cslib.Bioinformatics.RNA.Nussinov.card_le_nussinov","parent":45849,"result":true,"route":"../roadmap/Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/index.html#card-le-nussinov","scope":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure","status":"fully_proved","summary":"Every secondary structure has at most `nussinov pair m s` pairs.","title":"Soundness bound"},{"id":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/exists-is-structure-card-eq","kind":"node","lean":"Cslib.Bioinformatics.RNA.Nussinov.exists_isStructure_card_eq","parent":45849,"result":true,"route":"../roadmap/Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure/index.html#exists-is-structure-card-eq","scope":"Biology/bioinformatics/nussinov-algorithm-for-rna-secondary-structure","status":"fully_proved","summary":"Some secondary structure has exactly `nussinov pair m s` pairs, so the recursion computes the maximum number of pairs over all non-crossing structures.","title":"Optimality"},{"data_url":"scopes/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding.json","id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","title":"McCaskill partition function for RNA folding"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/can-pair","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.CanPair","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#can-pair","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"Positions `i` and `j` of `s` both lie in `s` and carry complementary bases.","title":"Complementary base pairing of two positions"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/inside","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.inside","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#inside","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"Inside recursion over a commutative semiring `R`: `inside pair m s w i len` is the weighted sum over the secondary structures supported in the interval `[i, i + len)` (proved in `inside_eq_sum`). Base `i` is either unpaired, or paired with `i + 1 + k` for some `k \u2265 m` with `CanPair pair s i (i + 1 + k)`, which contributes `w i (i + 1 + k)` times the inside values of the enclosed interval `[i + 1, i + 1 + k)` and of\u2026","title":"Inside recursion"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/outside","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.outside","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#outside","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"Outside recursion (McCaskill 1990) over a commutative semiring `R`, for a sequence of length `n = s.length`: `outside pair m s w i j` is the weighted sum of the parts outside the pair `(i, j)` of all structures containing it (proved in `outside_mul_eq_sum`). Either no pair encloses `(i, j)`, leaving independent structures on `[0, i)` and `[j + 1, n)`, or there is a unique innermost enclosing pair `(a, b)` (`a < i`,\u2026","title":"Outside recursion"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/structures-in","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.structuresIn","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#structures-in","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"The secondary structures on `s` (`IsStructure pair m s`) all of whose pairs lie in the interval `[i, i + len)`.","title":"Secondary structures within an interval"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/structures","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.structures","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#structures","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"All secondary structures on `s` with minimum hairpin `m` (see `mem_structures`).","title":"Set of all secondary structures of a sequence"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/weight","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.weight","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#weight","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"The weight of a structure: the product of the weights of its base pairs.","title":"Weight of a secondary structure"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/mem-structures","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.mem_structures","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#mem-structures","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"`structures` is exactly the set of secondary structures on `s`.","title":"Membership in the set of secondary structures"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/inside-eq-sum","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.inside_eq_sum","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#inside-eq-sum","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"The inside recursion sums the weights of all secondary structures supported in the interval `[i, i + len)`, over any commutative semiring.","title":"Inside algorithm"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/inside-eq-sum-structures","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.inside_eq_sum_structures","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#inside-eq-sum-structures","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"The inside value of the whole sequence sums the weights of all its secondary structures.","title":"Inside value sums the weights of all secondary structures"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/outside-mul-eq-sum","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.outside_mul_eq_sum","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#outside-mul-eq-sum","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"Outside algorithm (McCaskill 1990): for a candidate pair `(i, j)` (`i + m < j` and `CanPair pair s i j`), the outside value times the pair's weight times the inside value of the enclosed interval `[i + 1, j)` sums the weights of all secondary structures containing `(i, j)`.","title":"Outside algorithm"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/inside-min-tropical","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.inside_minTropical","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#inside-min-tropical","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"With every base pair weighted `-1` in the min-plus semiring `MinTropical (WithTop \u2124)` (addition `min`, multiplication `+`), a structure with `k` pairs has weight `-k`, so the inside value of the whole sequence is minus the Nussinov maximum number of base pairs.","title":"Tropical semiring"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/waterman-number","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.watermanNumber","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#waterman-number","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"Waterman's recursion for the number `S_m(n)` of secondary structures on `n` points with minimum hairpin `m` when every pair is allowed: `S(0) = 1` and `S(n + 1) = S(n) + \u2211_{m \u2264 k < n} S(k) S(n - 1 - k)` (Waterman 1978; Stein\u2013Waterman 1979). For `m = 0` these are the Motzkin numbers (`watermanNumber_zero_eq_motzkinNumber`); for `m = 1` the Stein\u2013Waterman numbers `1, 1, 1, 2, 4, 8, 17, 37, \u2026` (OEIS A004148).","title":"Waterman's recursion"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/card-structures-eq-waterman-number","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.card_structures_eq_watermanNumber","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#card-structures-eq-waterman-number","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"The number of secondary structures on `n` points with minimum hairpin `m`, when every pair of positions may pair, is Waterman's number `S_m(n)`.","title":"Counting semiring"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/waterman-number-zero-eq-motzkin-number","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.watermanNumber_zero_eq_motzkinNumber","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#waterman-number-zero-eq-motzkin-number","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"For `m = 0` (no hairpin constraint) Waterman's numbers are the Motzkin numbers `M(n) = \u2211_k C(n, 2k) C(k)`: the number of non-crossing partial matchings on `n` points.","title":"Waterman numbers with no hairpin constraint are Motzkin numbers"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/energy","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.energy","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#energy","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"An additive base-pair energy model: the free energy of a structure is the sum of the energies `e i j` of its base pairs (in the same units as `RT`).","title":"Additive base-pair energy model"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/partition-function","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.partitionFunction","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#partition-function","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"The partition function `Z = \u2211_S exp (-E(S) / RT)` over all secondary structures `S` on `s` with minimum hairpin `m`, for the additive energy `e` at thermal energy `RT` (physically `0 < RT`; the identities below hold for every real `RT`).","title":"McCaskill partition function of RNA secondary structures"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/inside-boltzmann-eq-partition-function","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.inside_boltzmann_eq_partitionFunction","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#inside-boltzmann-eq-partition-function","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"McCaskill's algorithm (inside part): over the sum-product semiring `\u211d` with Boltzmann weights `exp (-e i j / RT)`, the inside value of the whole sequence is the partition function. This is an algebraic identity valid for every real `RT`.","title":"McCaskill's algorithm"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/pair-prob","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.pairProb","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#pair-prob","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"The base-pair probability `P(i, j)`: the total weight of the structures containing the pair `(i, j)`, divided by the total weight `Z` of all structures. For nonnegative weights `Z \u2265 1` (the empty structure has weight `1`), so the denominator is positive; for Boltzmann weights this is the probability of `(i, j)` under the Boltzmann distribution on structures.","title":"Base-pair probability"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/pair-prob-eq-outside","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.pairProb_eq_outside","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#pair-prob-eq-outside","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"For a candidate pair `(i, j)`, `P(i, j) = outside(i, j) \u00b7 w(i, j) \u00b7 inside(i + 1, j) / inside(0, n)`.","title":"McCaskill's base-pair probabilities"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/one-le-sum-weight","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.one_le_sum_weight","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#one-le-sum-weight","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"For nonnegative weights the total weight of all structures is at least `1`, the weight of the empty structure; so the denominator of `pairProb` is positive.","title":"Total structure weight is at least one"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/pair-prob-nonneg","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.pairProb_nonneg","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#pair-prob-nonneg","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"Base-pair probabilities are nonnegative for nonnegative weights.","title":"Nonnegativity of base-pair probabilities"},{"id":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/sum-pair-prob-le-one","kind":"node","lean":"Cslib.Bioinformatics.RNA.PartitionFunction.sum_pairProb_le_one","parent":45855,"result":true,"route":"../roadmap/Biology/bioinformatics/mccaskill-partition-function-for-rna-folding/index.html#sum-pair-prob-le-one","scope":"Biology/bioinformatics/mccaskill-partition-function-for-rna-folding","status":"fully_proved","summary":"For nonnegative weights, the probabilities that base `i` pairs with some later base `j` or some earlier base `j` sum to at most `1` (`\u2211_j (P(i, j) + P(j, i)) \u2264 1`).","title":"Each base pairs at most once"},{"data_url":"scopes/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly.json","id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","title":"Omnitigs and safe walks in genome assembly"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/multigraph","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.Multigraph","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#multigraph","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"A directed multigraph given by the source and target of each edge.","title":"Directed multigraph"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/is-walk","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.IsWalk","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#is-walk","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"A list of edges is a walk when consecutive edges are incident (head to tail).","title":"Walk in a directed multigraph"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/is-path","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.IsPath","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#is-path","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"A path: a nonempty walk whose nodes `src e\u2080, tgt e\u2080, \u2026, tgt e_\u2113` are distinct, except that the first and last node may coincide.","title":"Path in a directed multigraph"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/is-circular-covering","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.IsCircularCovering","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#is-circular-covering","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"A circular edge-covering walk: a nonempty closed walk through every edge.","title":"Circular edge-covering walk"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/is-circular-subwalk","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.IsCircularSubwalk","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#is-circular-subwalk","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"`w` is a subwalk of the circular walk `A`: it is read off `A` from some start position, wrapping around (possibly several times).","title":"Subwalk of a circular walk"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/is-safe","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.IsSafe","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#is-safe","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"Safe walk: a subwalk of every circular edge-covering walk.","title":"Safe walk in a multigraph"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/is-omnitig","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.IsOmnitig","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#is-omnitig","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"Omnitig (Tomescu\u2013Medvedev, Definition 5): for all `1 \u2264 i \u2264 j \u2264 t` (`t + 1 = w.length`), no path from `v_j = src e_j` to `v_i = src e_i` has first edge `\u2260 e_j` and last edge `\u2260 e_{i-1}`.","title":"Omnitig"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/is-safe-2","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.IsOmnitig.isSafe","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#is-safe-2","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"Omnitigs are safe (Theorem 1, (2) \u21d2 (1)).","title":"Omnitigs are safe walks"},{"id":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/is-omnitig-of-is-safe","kind":"node","lean":"Cslib.Bioinformatics.Assembly.Omnitig.isOmnitig_of_isSafe","parent":45878,"result":true,"route":"../roadmap/Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly/index.html#is-omnitig-of-is-safe","scope":"Biology/bioinformatics/omnitigs-and-safe-walks-in-genome-assembly","status":"fully_proved","summary":"Safe walks are omnitigs (Theorem 1, (1) \u21d2 (2)), when a circular edge-covering walk exists (otherwise every walk is vacuously safe).","title":"Safe walks are omnitigs"},{"data_url":"scopes/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms.json","id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","title":"Optimal leaf ordering of dendrograms"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/dendrogram","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.Dendrogram","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#dendrogram","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"A rooted binary tree whose leaves are labeled by `\u03b1`: the shape of a dendrogram (merge heights are irrelevant for leaf ordering).","title":"Dendrogram as a leaf-labeled binary tree"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/leaves","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.Dendrogram.leaves","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#leaves","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"The leaves of a dendrogram, left to right.","title":"Leaves of a dendrogram"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/num-internal","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.Dendrogram.numInternal","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#num-internal","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"The number of internal (merge) nodes.","title":"Number of internal nodes of a dendrogram"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/orderings","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.Dendrogram.orderings","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#orderings","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"The leaf orders obtained by choosing, at each internal node, which child is drawn first. Each choice pattern contributes one entry, so the list has `2 ^ numInternal t` entries.","title":"Leaf orderings of a dendrogram"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/adj-score","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.adjScore","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#adj-score","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"The score of a leaf order: the sum of the similarities of adjacent leaves.","title":"Adjacent-similarity score of a leaf order"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/opt-score","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.optScore","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#opt-score","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"The best score among the leaf orders of `t` (never `\u22a5`: `optScore_ne_bot`).","title":"Optimal leaf-ordering score"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/opt-ends","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.optEnds","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#opt-ends","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"The best score among the leaf orders of `t` that start with `i` and end with `j` (`\u22a5` if there is none).","title":"Optimal leaf-ordering score with fixed end leaves"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/join-ends","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.joinEnds","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#join-ends","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"One side of the recursion: the left part ranges over orders of a subtree with leaves `Ll` and best-score table `Ml`, the right part over a subtree with leaves `Lr` and table `Mr`; the two are joined at the adjacent pair `(k, l)`.","title":"Joining two subtree orders at an adjacent leaf pair"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/best-ends","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.bestEnds","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#best-ends","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"The dynamic program of Bar-Joseph, Gifford and Jaakkola: `bestEnds S t i j` is computed from the tables of the two children, trying both child orders.","title":"Bar-Joseph\u2013Gifford\u2013Jaakkola dynamic program for leaf ordering"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/length-orderings","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.length_orderings","parent":45888,"reading_title":{"format":"inline-parts-v1","parts":[["text","A dendrogram with m internal nodes has "],["math","\\(2^m\\)"],["text"," leaf orders"]],"source":"A dendrogram with m internal nodes has $2^m$ leaf orders"},"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#length-orderings","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"A dendrogram with `m` internal nodes has `2 ^ m` leaf orders (one per flip pattern).","title":"A dendrogram with m internal nodes has $2^m$ leaf orders"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/perm-of-mem-orderings","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.perm_of_mem_orderings","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#perm-of-mem-orderings","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"Every leaf order of `t` is a permutation of its leaves.","title":"Leaf orders are permutations of the leaves"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/best-ends-eq-opt-ends","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.bestEnds_eq_optEnds","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#best-ends-eq-opt-ends","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"Correctness of the optimal leaf ordering recursion (Bar-Joseph, Gifford and Jaakkola 2001): the dynamic program computes, for every end pair `(i, j)`, the best score of a leaf order of `t` starting at `i` and ending at `j`.","title":"Correctness of the optimal leaf ordering recursion"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/opt-score-eq-sup-best-ends","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.optScore_eq_sup_bestEnds","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#opt-score-eq-sup-best-ends","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"The optimal leaf-ordering score is the best `bestEnds` value over end pairs of leaves.","title":"Optimal leaf-ordering score via the dynamic program"},{"id":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/opt-score-ne-bot","kind":"node","lean":"Cslib.Bioinformatics.Clustering.OptimalLeafOrdering.optScore_ne_bot","parent":45888,"result":true,"route":"../roadmap/Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms/index.html#opt-score-ne-bot","scope":"Biology/bioinformatics/optimal-leaf-ordering-of-dendrograms","status":"fully_proved","summary":"Every dendrogram has a leaf order, so the optimal score is a real number.","title":"Optimal leaf-ordering score is finite"},{"data_url":"scopes/Biology/bioinformatics/position-weight-matrices.json","id":"Biology/bioinformatics/position-weight-matrices","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","title":"Position weight matrices"},{"id":"Biology/bioinformatics/position-weight-matrices/score","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.score","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#score","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The score `\u2211\u1d62 W i (s i)` of a site `s` of width `w` under a position weight matrix `W`.","title":"Score of a site under a position weight matrix"},{"id":"Biology/bioinformatics/position-weight-matrices/site-lik","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.siteLik","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#site-lik","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The likelihood `\u220f\u1d62 p i (s i)` of a site under the product (column-independent) site model `p`.","title":"Site likelihood under a column-independent model"},{"id":"Biology/bioinformatics/position-weight-matrices/bg-lik","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.bgLik","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#bg-lik","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The likelihood `\u220f\u2096 q (x k)` of a sequence of length `m` under the background of independent letters with law `q`.","title":"Background likelihood of a sequence"},{"id":"Biology/bioinformatics/position-weight-matrices/log-odds","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.logOdds","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#log-odds","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The log-odds matrix `log (p i b / q b)` of a site model against a background (natural logarithm).","title":"Log-odds matrix of a site model"},{"id":"Biology/bioinformatics/position-weight-matrices/score-log-odds","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.score_logOdds","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#score-log-odds","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The log-odds score is a log-likelihood ratio (Staden 1989; Stormo 2000): for positive site model and background, `score (logOdds p q) s = log (siteLik p s / bgLik q s)`.","title":"The log-odds score is a log-likelihood ratio"},{"id":"Biology/bioinformatics/position-weight-matrices/sum-site-lik","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.sum_siteLik","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#sum-site-lik","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"A product of column distributions is a distribution on sites.","title":"Product site likelihoods form a distribution"},{"id":"Biology/bioinformatics/position-weight-matrices/sum-bg-lik","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.sum_bgLik","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#sum-bg-lik","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The background law of independent letters is a distribution on sequences.","title":"Background likelihoods form a distribution"},{"id":"Biology/bioinformatics/position-weight-matrices/sum-mul-site-lik-le-of-sum-mul-bg-lik-le","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.sum_mul_siteLik_le_of_sum_mul_bgLik_le","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#sum-mul-site-lik-le-of-sum-mul-bg-lik-le","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Thresholding the log-odds score is most powerful (Neyman\u2013Pearson): a test `\u03c6` of the background (null) against the site model whose background size is at most that of the threshold test `score > t` has at most its power.","title":"Thresholding the log-odds score is most powerful"},{"id":"Biology/bioinformatics/position-weight-matrices/info-content","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.infoContent","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#info-content","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The information content `\u2211_b p b log\u2082 (p b / q b)` (in bits) of a PWM column `p` against the background `q` (Schneider et al. 1986). Letters with `p b = 0` contribute `0`.","title":"Information content of a position weight matrix column"},{"id":"Biology/bioinformatics/position-weight-matrices/info-content-eq-kl-div","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.infoContent_eq_klDiv","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#info-content-eq-kl-div","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"For laws `P \u226a Q`, the information content of `P` against `Q` is `KL(P \u2016 Q) / log 2`, Mathlib's `klDiv` in bits.","title":"Information content is a Kullback\u2013Leibler divergence"},{"id":"Biology/bioinformatics/position-weight-matrices/info-content-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.infoContent_nonneg","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#info-content-nonneg","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Information content is nonnegative (Gibbs' inequality) for a column distribution and a positive background distribution.","title":"Information content is nonnegative"},{"id":"Biology/bioinformatics/position-weight-matrices/info-content-uniform","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.infoContent_uniform","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#info-content-uniform","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Information content against the uniform background (Schneider et al. 1986): `log\u2082 |\u03b1| - H(p)`, the maximal entropy minus the column entropy.","title":"Information content against the uniform background"},{"id":"Biology/bioinformatics/position-weight-matrices/info-content-uniform-le","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.infoContent_uniform_le","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#info-content-uniform-le","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Against the uniform background, a column carries at most `log\u2082 |\u03b1|` bits.","title":"Information content is at most the log of the alphabet size"},{"id":"Biology/bioinformatics/position-weight-matrices/info-content-uniform-le-two","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.infoContent_uniform_le_two","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#info-content-uniform-le-two","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"For a four-letter alphabet (DNA) and the uniform background, a column carries between `0` and `2` bits.","title":"A DNA column carries at most two bits"},{"id":"Biology/bioinformatics/position-weight-matrices/sum-site-lik-mul-score-log-odds","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.sum_siteLik_mul_score_logOdds","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#sum-site-lik-mul-score-log-odds","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Under the site model, `E[score] = log 2 \u00b7 \u2211\u1d62 IC(p\u1d62 \u2016 q)`. This is an identity for every `q` (both sides use the same logarithms); it is an expected log-likelihood ratio for a positive background.","title":"The expected log-odds score is the total information content"},{"id":"Biology/bioinformatics/position-weight-matrices/score-step","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.scoreStep","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#score-step","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"One column of the score-distribution recursion: from the distribution `d` of the score of the first columns, the distribution after adding a column `c` with background `q`, `\u2211_b q b \u00b7 (d shifted by c b)`.","title":"One column step of the score-distribution recursion"},{"id":"Biology/bioinformatics/position-weight-matrices/score-dist","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.scoreDist","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#score-dist","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The background distribution of the score of an integer matrix `M`, computed column by column from the point mass at `0` (Staden 1989; Touzet and Varr\u00e9 2007).","title":"Background score distribution of an integer matrix"},{"id":"Biology/bioinformatics/position-weight-matrices/score-dist-apply","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.scoreDist_apply","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#score-dist-apply","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"`scoreDist q M t` is `\u2211_{s : score M s = t} \u220f\u1d62 q (s i)`, for every `q`. When `q` is a distribution this is the background probability of the score `t` (the weights sum to `1` by `sum_bgLik`).","title":"The column recursion computes the exact score distribution"},{"id":"Biology/bioinformatics/position-weight-matrices/p-value","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.pValue","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#p-value","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The exact p-value of the score threshold `t`, summed over the computed distribution.","title":"Exact p-value of a score threshold"},{"id":"Biology/bioinformatics/position-weight-matrices/p-value-eq","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.pValue_eq","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#p-value-eq","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"`pValue q M t = \u2211_{s : score M s \u2265 t} \u220f\u1d62 q (s i)`, the background probability `P_q(score M \u2265 t)` when `q` is a distribution.","title":"Exact p-values"},{"id":"Biology/bioinformatics/position-weight-matrices/abs-score-sub-score-round-le","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.abs_score_sub_score_round_le","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#abs-score-sub-score-round-le","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Rounding `c \u00b7 W` to integers changes the rescaled score of every site by at most `w / (2c)`.","title":"Scaling and rounding error"},{"id":"Biology/bioinformatics/position-weight-matrices/partition-fn","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.partitionFn","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#partition-fn","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The column partition function `Z = \u2211_b q b e^{-\u03bb \u03b5 b}` of energies `\u03b5` at inverse temperature `\u03bb`.","title":"Column partition function of an energy matrix"},{"id":"Biology/bioinformatics/position-weight-matrices/boltzmann-pwm","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.boltzmannPWM","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#boltzmann-pwm","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The Boltzmann PWM `q b e^{-\u03bb \u03b5 i b} / Z\u1d62` of an energy matrix `\u03b5`.","title":"Boltzmann position weight matrix"},{"id":"Biology/bioinformatics/position-weight-matrices/selected-law","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.selectedLaw","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#selected-law","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The law of background sites selected with weight `e^{-\u03bb E(s)}` for the additive energy `E s = score \u03b5 s = \u2211\u1d62 \u03b5 i (s i)`: `q(s) e^{-\u03bb E(s)} / \u2211_{s'} q(s') e^{-\u03bb E(s')}`.","title":"Law of energy-selected background sites"},{"id":"Biology/bioinformatics/position-weight-matrices/partition-fn-pos","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.partitionFn_pos","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#partition-fn-pos","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The partition function of a positive background is positive.","title":"Positivity of the column partition function"},{"id":"Biology/bioinformatics/position-weight-matrices/is-dist-boltzmann-pwm","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.isDist_boltzmannPWM","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#is-dist-boltzmann-pwm","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Each column of the Boltzmann PWM is a distribution.","title":"Columns of the Boltzmann matrix are distributions"},{"id":"Biology/bioinformatics/position-weight-matrices/selected-law-eq-site-lik","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.selectedLaw_eq_siteLik","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#selected-law-eq-site-lik","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Boltzmann selection of background sites with an additive energy yields the product site model with columns `boltzmannPWM`. This is an identity for every `q` (both sides vanish when some `Z\u1d62 = 0`); it is a statement about laws for positive `q`.","title":"Berg\u2013von Hippel"},{"id":"Biology/bioinformatics/position-weight-matrices/log-odds-boltzmann-pwm","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.logOdds_boltzmannPWM","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#log-odds-boltzmann-pwm","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"`log (p i b / q b) = -\u03bb \u03b5 i b - log Z\u1d62` for the Boltzmann PWM.","title":"Log-odds are affine in the energies"},{"id":"Biology/bioinformatics/position-weight-matrices/score-log-odds-boltzmann-pwm","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.score_logOdds_boltzmannPWM","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#score-log-odds-boltzmann-pwm","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The log-odds score of a selected site is `-\u03bb E(s)` up to the constant `\u2211\u1d62 log Z\u1d62`.","title":"Log-odds score of a selected site is its scaled energy up to a constant"},{"id":"Biology/bioinformatics/position-weight-matrices/site-pos","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.sitePos","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#site-pos","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The position `j + i` of motif column `i` for a motif starting at `j`, in a sequence of length `n + w` (so `n + 1` possible starts).","title":"Sequence position of a motif column"},{"id":"Biology/bioinformatics/position-weight-matrices/window","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.window","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#window","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The window of width `w` of `x` starting at `j`.","title":"Window of a sequence at a given start"},{"id":"Biology/bioinformatics/position-weight-matrices/window-ratio","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.windowRatio","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#window-ratio","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The likelihood ratio of the window at `j` under the site model `\u03b8` against the background `q`, and `1` when there is no motif occurrence.","title":"Likelihood ratio of a motif window"},{"id":"Biology/bioinformatics/position-weight-matrices/motif-joint","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.motifJoint","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#motif-joint","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The joint likelihood of a sequence `x` and a latent occurrence `z` with prior `prior z` and start `start z` (`none`: no occurrence): background letters outside the window, site-model letters inside. One occurrence per sequence (OOPS) is `Z = Fin (n + 1)`, `start = some`, `prior` uniform; zero or one (ZOOPS) is `Z = Option (Fin (n + 1))`, `start = id`, `prior none = 1 - \u03b3` and `prior (some j) = \u03b3 / (n + 1)` for a\u2026","title":"Joint likelihood of a sequence and a motif occurrence"},{"id":"Biology/bioinformatics/position-weight-matrices/motif-joint-of-start-eq-some","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.motifJoint_of_start_eq_some","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#motif-joint-of-start-eq-some","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"For a positive background and an occurrence at `j`, the joint is the prior times the product of `\u03b8 (k - j)` inside the window and `q` outside.","title":"The joint is the window model"},{"id":"Biology/bioinformatics/position-weight-matrices/motif-joint-of-start-eq-none","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.motifJoint_of_start_eq_none","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#motif-joint-of-start-eq-none","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"With no occurrence, the joint is the prior times the background likelihood.","title":"Joint likelihood with no motif occurrence"},{"id":"Biology/bioinformatics/position-weight-matrices/pos-pwm","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.PosPWM","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#pos-pwm","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"PWMs with positive entries and columns summing to `1`: the EM parameter space.","title":"Positive position weight matrices"},{"id":"Biology/bioinformatics/position-weight-matrices/motif-model","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.motifModel","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#motif-model","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The motif model as an EM model on `PosPWM`.","title":"Motif model as an EM model"},{"id":"Biology/bioinformatics/position-weight-matrices/motif-model-pos","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.motifModel_pos","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#motif-model-pos","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The motif model is positive for a positive background and a positive prior.","title":"Positivity of the motif model"},{"id":"Biology/bioinformatics/position-weight-matrices/posterior-motif-model","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.posterior_motifModel","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#posterior-motif-model","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The posterior of the occurrence `z` is proportional to its prior times the likelihood ratio of its window (Lawrence and Reilly 1990; Bailey and Elkan 1995).","title":"The E-step in closed form"},{"id":"Biology/bioinformatics/position-weight-matrices/expected-counts","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.expectedCounts","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#expected-counts","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The expected count of letter `b` in motif column `i` under the posteriors at `\u03b8`: `\u2211\u2098 \u2211_z P(z | x\u2098, \u03b8) [start z = some j \u2227 x\u2098 (j + i) = b]`.","title":"Expected letter counts in motif columns"},{"id":"Biology/bioinformatics/position-weight-matrices/covers","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.Covers","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#covers","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Every letter can occur in every motif column: for each column `i` and letter `b` some sequence has `b` at offset `i` of some possible occurrence. This keeps the M-step positive.","title":"Coverage condition for motif columns"},{"id":"Biology/bioinformatics/position-weight-matrices/expected-counts-pos","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.expectedCounts_pos","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#expected-counts-pos","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Under `Covers`, every expected count is positive.","title":"Positivity of expected motif counts"},{"id":"Biology/bioinformatics/position-weight-matrices/m-step","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.mStep","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#m-step","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The M-step: normalized expected letter counts, `\u03b8' i b = c i b / \u2211_{b'} c i b'`.","title":"M-step for position weight matrices"},{"id":"Biology/bioinformatics/position-weight-matrices/em-q-le-em-q-m-step","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.emQ_le_emQ_mStep","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#em-q-le-em-q-m-step","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The M-step maximizes the EM objective over PWMs with positive entries and normalized columns: for every `\u03b8'`, `Q(\u03b8' | \u03b8) \u2264 Q(mStep \u03b8 | \u03b8)` (Gibbs' inequality per column).","title":"The M-step maximizes the EM objective"},{"id":"Biology/bioinformatics/position-weight-matrices/log-lik-iterate-m-step-monotone","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.logLik_iterate_mStep_monotone","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#log-lik-iterate-m-step-monotone","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The motif likelihood is nondecreasing along the EM iterates.","title":"The motif likelihood is nondecreasing along the EM iterates"},{"id":"Biology/bioinformatics/position-weight-matrices/pair-mgf","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.pairMGF","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#pair-mgf","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The moment generating function `\u2211\u2090 \u2211_b p\u2090 p_b e^{\u03bb s_{ab}}` of a pair score under independent letters with law `p`.","title":"Moment generating function of a pair score"},{"id":"Biology/bioinformatics/position-weight-matrices/exists-unique-pair-mgf-eq-one","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.existsUnique_pairMGF_eq_one","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#exists-unique-pair-mgf-eq-one","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Karlin\u2013Altschul `\u03bb` (Karlin and Altschul 1990): if the expected score under a positive background distribution is negative and some score is positive, there is a unique `\u03bb > 0` with `\u2211 p\u2090 p_b e^{\u03bb s_{ab}} = 1`.","title":"Karlin\u2013Altschul `\u03bb`"},{"id":"Biology/bioinformatics/position-weight-matrices/target-freq","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.targetFreq","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#target-freq","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The implicit target frequencies `q_{ab} = p\u2090 p_b e^{\u03bb s_{ab}}` of a score matrix.","title":"Implicit target frequencies of a score matrix"},{"id":"Biology/bioinformatics/position-weight-matrices/is-dist-target-freq","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.isDist_targetFreq","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#is-dist-target-freq","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"At a root of `pairMGF p s \u03bb = 1` the target frequencies form a distribution on pairs.","title":"Target frequencies form a distribution on pairs"},{"id":"Biology/bioinformatics/position-weight-matrices/score-eq-log-target-freq","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.score_eq_log_targetFreq","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#score-eq-log-target-freq","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Every such score matrix is a log-odds matrix (Karlin and Altschul 1990): `s_{ab} = log (q_{ab} / (p\u2090 p_b)) / \u03bb` for every `\u03bb > 0`; at the Karlin\u2013Altschul root the `q_{ab}` form a distribution (`isDist_targetFreq`), the implicit target frequencies.","title":"Every such score matrix is a log-odds matrix"},{"id":"Biology/bioinformatics/position-weight-matrices/rel-entropy-eq-and-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.relEntropy_eq_and_nonneg","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#rel-entropy-eq-and-nonneg","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The relative entropy of the target frequencies (Altschul 1991): `\u03bb \u2211 q_{ab} s_{ab} = \u2211 q_{ab} log (q_{ab} / (p\u2090 p_b))`, which is nonnegative.","title":"The relative entropy of the target frequencies"},{"id":"Biology/bioinformatics/position-weight-matrices/sum-target-freq-mul-pos","kind":"node","lean":"Cslib.Bioinformatics.Motifs.PWM.sum_targetFreq_mul_pos","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#sum-target-freq-mul-pos","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"At the Karlin\u2013Altschul root `\u03bb > 0`, the expected score `\u2211 q_{ab} s_{ab}` under the target frequencies is positive.","title":"The target score is positive"},{"id":"Biology/bioinformatics/position-weight-matrices/one-hot","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.oneHot","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#one-hot","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The one-hot encoding of a letter: `oneHot a b = 1` if `b = a` and `0` otherwise.","title":"One-hot encoding of a letter"},{"id":"Biology/bioinformatics/position-weight-matrices/conv","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.conv","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#conv","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The response of a convolution (cross-correlation) filter `W` of width `w` on the one-hot encoded sequence `x` at the window starting at `i`: `\u2211 k, \u2211 b, W k b * oneHot (x (i + k)) b`.","title":"Convolution filter response at a window"},{"id":"Biology/bioinformatics/position-weight-matrices/window-2","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.window","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#window-2","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The window of length `w` of `x` starting at `i`.","title":"Window of a sequence starting at a position"},{"id":"Biology/bioinformatics/position-weight-matrices/conv-eq-pwm-score","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.conv_eq_pwmScore","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#conv-eq-pwm-score","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The convolution response is the PWM score of the window.","title":"The convolution response is the PWM score of the window"},{"id":"Biology/bioinformatics/position-weight-matrices/max-pool","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.maxPool","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#max-pool","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Global max pooling over the `m + 1` windows of a sequence of length `m + w`.","title":"Global max pooling"},{"id":"Biology/bioinformatics/position-weight-matrices/le-max-pool","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.le_maxPool","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#le-max-pool","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Every window's PWM score is at most the max-pooled output.","title":"Every window's PWM score is at most the max-pooled output"},{"id":"Biology/bioinformatics/position-weight-matrices/exists-eq-max-pool","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.exists_eq_maxPool","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#exists-eq-max-pool","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Some window attains the max-pooled output.","title":"Max pooling returns the best PWM hit"},{"id":"Biology/bioinformatics/position-weight-matrices/max-pool-relu","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.maxPool_relu","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#max-pool-relu","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Pooling `max (conv + \u03b2) 0` over the windows gives `max (maxPool + \u03b2) 0`.","title":"Bias and ReLU commute with max pooling"},{"id":"Biology/bioinformatics/position-weight-matrices/conv-shift","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.conv_shift","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#conv-shift","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Dropping the first letter shifts the responses by one window.","title":"Translation equivariance"},{"id":"Biology/bioinformatics/position-weight-matrices/rc-seq","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.rcSeq","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#rc-seq","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The reverse complement of the length-`L` prefix of `x`: position `j < L` holds the complement of `x (L - 1 - j)`.","title":"Reverse complement of a sequence prefix"},{"id":"Biology/bioinformatics/position-weight-matrices/rc-filter","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.rcFilter","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#rc-filter","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The reverse-complement filter: `rcFilter W k b = W (Fin.rev k) (comp b)`.","title":"Reverse-complement filter"},{"id":"Biology/bioinformatics/position-weight-matrices/conv-rc-filter","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.conv_rcFilter","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#conv-rc-filter","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"On a sequence of length `m + w`, its response at window `i \u2264 m` is the response of `W` on the reverse complement at window `m - i`.","title":"The reverse-complement filter scans the other strand"},{"id":"Biology/bioinformatics/position-weight-matrices/max-pool-rc-filter","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.maxPool_rcFilter","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#max-pool-rc-filter","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"Max pooling the reverse-complement filter equals max pooling `W` on the reverse complement.","title":"Max pooling the reverse-complement filter equals max pooling `W` on the reverse complement"},{"id":"Biology/bioinformatics/position-weight-matrices/max-pool-congr","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.maxPool_congr","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#max-pool-congr","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The responses only read the positions `< m + w` of a sequence of length `m + w`.","title":"Filter responses read only positions within the sequence length"},{"id":"Biology/bioinformatics/position-weight-matrices/both-strands","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.bothStrands","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#both-strands","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The both-strand score: the larger of the max-pooled responses of `W` and of its reverse complement.","title":"Both-strand convolution score"},{"id":"Biology/bioinformatics/position-weight-matrices/both-strands-rc-seq","kind":"node","lean":"Cslib.Bioinformatics.Regulatory.ConvolutionPWM.bothStrands_rcSeq","parent":45903,"result":true,"route":"../roadmap/Biology/bioinformatics/position-weight-matrices/index.html#both-strands-rc-seq","scope":"Biology/bioinformatics/position-weight-matrices","status":"fully_proved","summary":"The both-strand score is invariant under reverse complementing the sequence, for an involutive complement.","title":"The both-strand score is invariant under reverse complementing the sequence"},{"data_url":"scopes/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions.json","id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","title":"Profile scores for amino acid substitutions"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/profile-score","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.profileScore","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#profile-score","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The profile log-odds score `log (p a / q a)` of residue `a`: `p` is the position-specific profile probability (estimated from aligned homologs), `q` the background probability. Meaningful for `p a, q a > 0`; otherwise Lean's `log 0 = 0` and `x / 0 = 0` give junk.","title":"Profile log-odds score of a residue"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/sum-profile-score-eq-log-likelihood-rati","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.sum_profileScore_eq_log_likelihood_ratio","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#sum-profile-score-eq-log-likelihood-rati","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"For a sequence `s` over independent positions with profiles `p j` and background `q` (all positive), the summed profile score is the log of the likelihood ratio `\u220f p_j(s_j) / \u220f q(s_j)` of the profile model against the background.","title":"The score is a log-likelihood ratio"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/sum-mul-profile-score-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.sum_mul_profileScore_nonneg","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#sum-mul-profile-score-nonneg","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"If `p` and `q` are probability vectors and `q` is positive, the expected score under the profile, `\u2211 p a log (p a / q a)` (the Kullback\u2013Leibler divergence), is nonnegative.","title":"Gibbs' inequality for the profile score"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/sum-mul-profile-score-nonpos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.sum_mul_profileScore_nonpos","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#sum-mul-profile-score-nonpos","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"If `p` and `q` are probability vectors and `p` is positive, the expected score under the background, `\u2211 q a log (p a / q a)`, is nonpositive.","title":"Expected profile score under the background is nonpositive"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/psic-difference","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.psicDifference","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#psic-difference","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The PSIC score difference of a substitution from the wild-type residue `wt` to the mutant residue `mt`: `profileScore p q wt - profileScore p q mt`.","title":"PSIC score difference of a substitution"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/psic-difference-eq-log","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.psicDifference_eq_log","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#psic-difference-eq-log","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The PSIC difference is the log of the ratio of the two residues' likelihood ratios.","title":"PSIC difference as a log ratio of likelihood ratios"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/psic-difference-pos-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.psicDifference_pos_iff","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#psic-difference-pos-iff","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The PSIC difference is positive iff the wild type is more enriched in the profile, relative to background, than the mutant.","title":"Positivity criterion for the PSIC difference"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/naive-bayes","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#naive-bayes","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"A two-class naive Bayes model: class *damaging* has prior `prior \u2208 (0, 1)`, class *benign* the complement; each feature `j` takes values in the finite type `X j` with class-conditional probabilities `damaging j` and `benign j`, assumed positive; features are conditionally independent given the class. The tables are estimated from training data.","title":"Two-class naive Bayes model"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/joint","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes.joint","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#joint","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The joint probability of class `c` (`true` = damaging) and feature values `x`: `\u03c0 \u220f P_D(x_j)` or `(1 - \u03c0) \u220f P_B(x_j)`.","title":"Joint probability of class and features in naive Bayes"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/sum-joint","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes.sum_joint","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#sum-joint","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The joint law of class and features is a probability law.","title":"Naive Bayes joint law is a probability law"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/posterior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes.posterior","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#posterior","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The posterior probability that the variant is damaging given feature values `x` (Bayes' rule on the joint law).","title":"Naive Bayes posterior probability of damage"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/log-likelihood-ratio","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes.logLikelihoodRatio","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#log-likelihood-ratio","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The log-likelihood ratio `log (P_D(v) / P_B(v))` of value `v` of feature `j`.","title":"Log-likelihood ratio of a feature value"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/score","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes.score","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#score","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The naive-Bayes score: prior log-odds plus the features' log-likelihood ratios.","title":"Naive Bayes score"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/posterior-eq-sigmoid","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes.posterior_eq_sigmoid","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#posterior-eq-sigmoid","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The posterior probability of *damaging* is the logistic function of the score.","title":"Naive-Bayes posterior"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/posterior-lt-posterior-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes.posterior_lt_posterior_iff","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#posterior-lt-posterior-iff","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"The posterior is strictly monotone in the score.","title":"Naive Bayes posterior is strictly monotone in the score"},{"id":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/posterior-update-le-of-monotone","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.PolyPhen.NaiveBayes.posterior_update_le_of_monotone","parent":45972,"result":true,"route":"../roadmap/Biology/bioinformatics/profile-scores-for-amino-acid-substitutions/index.html#posterior-update-le-of-monotone","scope":"Biology/bioinformatics/profile-scores-for-amino-acid-substitutions","status":"fully_proved","summary":"Let feature `j` carry a numerical value `g v` (for example a PSIC-difference bin) such that its likelihood ratio `P_D(v) / P_B(v)` is monotone in `g v`. With the other features fixed, the posterior is monotone in `g`.","title":"Monotonicity in a feature value"},{"data_url":"scopes/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm.json","id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","kind":"scope","navigation":"module","parent":45505,"result":false,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","title":"Zuker's minimum free energy algorithm"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/loop-energy","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.LoopEnergy","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#loop-energy","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"A loop (nearest-neighbor) energy model: energies of hairpin loops closed by `(i, j)`, of stacks, bulges and interior loops closed by `(i, j)` with inner pair `(k, l)`, and the affine multiloop parameters `a` (per multiloop), `b` (per branch) and `c` (per unpaired base). The exterior loop has energy `0`. Units: free energy, e.g. kcal/mol.","title":"Nearest-neighbor loop energy model"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/encloses","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.Encloses","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#encloses","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"`q` lies strictly inside the pair `p`: `p.1 < q.1` and `q.2 < p.2`.","title":"Strict enclosure of one base pair by another"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/branches","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.branches","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#branches","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"The branches of the loop closed by `p` in `P`: the pairs of `P` strictly inside `p` that are not strictly inside another pair of `P` strictly inside `p`.","title":"Branches of a loop closed by a base pair"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/exterior-branches","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.exteriorBranches","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#exterior-branches","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"The exterior branches of `P`: the pairs not strictly inside any pair of `P`.","title":"Exterior branches of a secondary structure"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/exists-unique-closing-pair","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.existsUnique_closing_pair","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#exists-unique-closing-pair","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"In a secondary structure every pair is either an exterior branch or a branch of exactly one pair of the structure (and not both): each pair is shared by the loop it closes and exactly one enclosing loop, possibly the exterior loop.","title":"Unique loop decomposition"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/loop-energy-2","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.loopEnergy","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#loop-energy-2","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"The energy of the loop closed by `p = (i, j)` in `P`: a hairpin with no branch, a stack, bulge or interior loop with one branch `(k, l)`, and the affine multiloop energy `a + b \u00b7 r + c \u00b7 u` with `r \u2265 2` branches and `u = (j - i - 1) - \u2211 (l - k + 1)` unpaired bases.","title":"Energy of the loop closed by a base pair"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/free-energy","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.freeEnergy","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#free-energy","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"The free energy of a structure in the loop model: the sum of the energies of the loops closed by its pairs (the exterior loop contributes `0`).","title":"Free energy of a structure in the loop model"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/loop-energy-eq-hairpin","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.loopEnergy_eq_hairpin","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#loop-energy-eq-hairpin","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"A pair with no branch closes a hairpin loop.","title":"A pair with no branch closes a hairpin loop"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/loop-energy-eq-interior","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.loopEnergy_eq_interior","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#loop-energy-eq-interior","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"A pair with exactly one branch `q` closes a stack, bulge or interior loop with inner pair `q`.","title":"A pair with one branch closes a stack, bulge or interior loop"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/loop-energy-eq-multiloop","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.loopEnergy_eq_multiloop","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#loop-energy-eq-multiloop","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"A pair with at least two branches closes a multiloop with the affine energy.","title":"A pair with two or more branches closes a multiloop"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/exists-unique-loop-of-pos","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.existsUnique_loop_of_pos","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#exists-unique-loop-of-pos","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"Every position has a unique innermost strict container (not loop membership): a position `x` either lies strictly inside no pair, or there is a unique pair `p` with `p.1 < x < p.2` such that no branch of `p` has `x` strictly inside it. This is an innermost-container assignment, not the conventional loop decomposition of positions: endpoints are excluded from the loop they close (a position `x = q.1` or `x = q.2` is\u2026","title":"Every position has a unique innermost strict container"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/zuker-vw","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.zukerVW","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#zuker-vw","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"Zuker's recursion on the interval `[i, i + len)`, with `j = i + len - 1`, returning `(V, WM)`: the least of the hairpin energy `hairpin i j`, the interior-loop energies `interior i j k l + V(k, l)` over `i < k < l < j`, and the multiloop energies `a + WM(i + 1, u) + WM(u + 1, j - 1)` over `i < u < j - 1`; unpaired base) is the least of `V + b`, `WM(i + 1, j) + c` (base `i` unpaired), `WM(i, j - 1) + c` (base `j`\u2026","title":"Zuker's recursion"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/zuker-exterior","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.zukerExterior","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#zuker-exterior","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"The exterior-loop recursion on the prefix `[0, len)`: `W(0) = 0` and `W(len + 1) = min (W(len), min_{i \u2264 len} W(i) + V(i, len + 1 - i))` (the last base is unpaired, or closes an exterior branch `(i, len)`).","title":"Zuker exterior-loop recursion"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/zuker-mfe","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.zukerMFE","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#zuker-mfe","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"Zuker's minimum free energy of `s`: the exterior recursion on the whole sequence.","title":"Zuker minimum free energy"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/zuker-mfe-eq-inf","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.zukerMFE_eq_inf","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#zuker-mfe-eq-inf","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"Zuker's algorithm (Zuker and Stiegler 1981; Zuker and Sankoff 1984), here with the affine multiloop energy of Jaeger, Turner and Zuker (1989): the Zuker recursion computes the minimum free energy over all secondary structures on `s` with minimum hairpin `m`, for any real loop energies (of either sign). The minimum ranges over a finite set that contains the empty structure, so it is never `\u22a4`.","title":"Zuker's algorithm"},{"id":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/zuker-mfe-le-zero","kind":"node","lean":"Cslib.Bioinformatics.RNA.Zuker.zukerMFE_le_zero","parent":45989,"result":true,"route":"../roadmap/Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm/index.html#zuker-mfe-le-zero","scope":"Biology/bioinformatics/zuker-s-minimum-free-energy-algorithm","status":"fully_proved","summary":"The minimum free energy is at most `0`, the energy of the open chain (the empty structure).","title":"Minimum free energy is at most zero"},{"data_url":"scopes/Biology/phylogenetics.json","id":"Biology/phylogenetics","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/phylogenetics/index.html","scope":"Biology/phylogenetics","status":"fully_proved","summary":"Reconstruction of evolutionary trees from sequence data: substitution models, likelihood and parsimony, distance methods, tree metrics, splits and the enumeration of phylogenetic trees.","title":"Phylogenetics"},{"data_url":"scopes/Biology/phylogenetics/counting-binary-phylogenetic-trees.json","id":"Biology/phylogenetics/counting-binary-phylogenetic-trees","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","title":"Counting binary phylogenetic trees"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/plane-tree","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.TreeCount.PlaneTree","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#plane-tree","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"A planar rooted binary tree with leaves labeled by `\u03b1`: a leaf `leaf a`, or an internal vertex `node l r` with left subtree `l` and right subtree `r` (the children are ordered).","title":"Planar rooted binary tree with labeled leaves"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/leaves","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.TreeCount.PlaneTree.leaves","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#leaves","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"The multiset of leaf labels.","title":"Leaf labels of a planar binary tree"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/leaves-ne-zero","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.TreeCount.PlaneTree.leaves_ne_zero","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#leaves-ne-zero","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"Every planar tree has at least one leaf.","title":"Every planar binary tree has a leaf"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/swap","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.TreeCount.PlaneTree.Swap","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#swap","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"One swap: exchange the two children of one internal vertex (anywhere in the tree).","title":"Swap of the children of an internal vertex"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/on","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.TreeCount.PlaneTree.On","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#on","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"The planar drawings of rooted binary trees on the taxa `X`: planar trees whose leaves are labeled by `X`, each taxon labeling exactly one leaf.","title":"Planar drawings of rooted binary trees on a taxon set"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/rooted-binary-tree","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.TreeCount.RootedBinaryTree","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#rooted-binary-tree","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"The rooted binary phylogenetic trees on the taxa `X`: planar drawings with leaf set `X` modulo exchanging the two children of internal vertices, i.e. rooted binary trees with unordered children and leaves labeled bijectively by `X`, up to label-preserving isomorphism.","title":"Rooted binary phylogenetic tree"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/nat-card-rooted-binary-tree","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.TreeCount.nat_card_rootedBinaryTree","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#nat-card-rooted-binary-tree","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"On `n \u2265 1` labeled taxa there are `(2 n - 3)\u203c = 1 \u00b7 3 \u00b7 5 \u22ef (2 n - 3)` rooted binary phylogenetic trees (`1` tree for `n = 1` and `n = 2`, `3` for `n = 3`, `15` for `n = 4`). For `n = 1` the natural-number subtraction gives `2 - 3 = 0` and `0\u203c = 1`, the conventional value `(-1)\u203c = 1` of the formula; the empty set of taxa (excluded here) has no tree (`isEmpty_rootedBinaryTree_empty`).","title":"Schr\u00f6der's count of rooted binary phylogenetic trees"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/unrooted-binary-tree","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.UnrootedBinaryTree","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#unrooted-binary-tree","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"An unrooted binary phylogenetic tree on the taxa `X`: a finite tree `G` on a vertex type `V` whose degree-one vertices are labeled bijectively by `X` (`label`) and whose other vertices all have degree three.","title":"Unrooted binary phylogenetic tree"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/iso","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.UnrootedBinaryTree.Iso","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#iso","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"Two unrooted binary phylogenetic trees are isomorphic when some graph isomorphism maps the leaf with each label to the leaf with the same label.","title":"Isomorphism of unrooted binary phylogenetic trees"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/is-binary-hierarchy","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.IsBinaryHierarchy","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#is-binary-hierarchy","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"A binary hierarchy on a finite set `S`: a family of nonempty subsets of `S` containing `S` and every singleton, any two members nested or disjoint, and every member with at least two elements the disjoint union of two members. These are the cluster systems of the rooted binary phylogenetic trees on `S` (Semple & Steel 2003, Section 3.1).","title":"Binary hierarchy on a finite set"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/is-binary-hierarchy-filter","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.isBinaryHierarchy_filter","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#is-binary-hierarchy-filter","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"The members of a binary hierarchy below a member form a binary hierarchy.","title":"Binary hierarchy restricted below a member is a binary hierarchy"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/nat-card-binary-hierarchy","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.nat_card_binaryHierarchy","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#nat-card-binary-hierarchy","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"On a nonempty finite set `S` there are `(2 |S| - 3)\u203c` binary hierarchies, as many as rooted binary phylogenetic trees on `S`: a tree corresponds to the leaf sets of its subtrees.","title":"The number of binary hierarchies"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/cluster","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.cluster","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#cluster","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"The cluster of a vertex `v` seen from the leaf labeled `a`: the taxa `x \u2260 a` whose leaf lies beyond `v`, i.e. `v` is on the path from the leaf `a` to the leaf `x`.","title":"Cluster of a vertex seen from a leaf"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/cluster-set","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.clusterSet","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#cluster-set","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"The clusters of all vertices seen from the leaf labeled `a`. (The cluster of the leaf `a` itself is the set of all other taxa, which is also the cluster of its neighbor.)","title":"Cluster set of a tree seen from a leaf"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/cluster-set-eq-of-iso","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.clusterSet_eq_of_iso","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#cluster-set-eq-of-iso","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"Isomorphic trees have the same clusters.","title":"Isomorphic trees have the same clusters"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/is-binary-hierarchy-cluster-set","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.isBinaryHierarchy_clusterSet","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#is-binary-hierarchy-cluster-set","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"The clusters of a tree seen from a leaf form a binary hierarchy on the other taxa (Semple & Steel 2003, Section 3.1).","title":"Clusters of an unrooted tree form a binary hierarchy"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/iso-of-cluster-set-eq","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.iso_of_clusterSet_eq","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#iso-of-cluster-set-eq","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"Two trees with the same clusters seen from the leaf `a` are isomorphic: the clusters determine the tree.","title":"Clusters determine an unrooted binary tree"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/exists-cluster-set-eq","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.exists_clusterSet_eq","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#exists-cluster-set-eq","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"Every binary hierarchy on the taxa other than `a` is the cluster set, seen from the leaf `a`, of some unrooted binary phylogenetic tree (vertices: the leaf `a` and the members of the hierarchy; each member is joined to the smallest member strictly containing it).","title":"Every binary hierarchy is a cluster set of a tree"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/quot-equiv-hierarchy","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.quotEquivHierarchy","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#quot-equiv-hierarchy","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"Unrooted binary phylogenetic trees up to label-preserving isomorphism correspond to the binary hierarchies on the taxa other than `a`, by their clusters seen from the leaf `a`.","title":"Unrooted binary trees correspond to binary hierarchies"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/nat-card-unrooted-binary-tree-of-two-le","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.nat_card_unrootedBinaryTree_of_two_le","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#nat-card-unrooted-binary-tree-of-two-le","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"The count `(2 n - 5)\u203c` of unrooted binary phylogenetic trees holds from `n = 2` taxa on: for `n = 2` the single tree is one edge and `2 \u00b7 2 - 5 = 0` in `\u2115`, `0\u203c = 1` (the conventional value `(-1)\u203c = 1`). For `n \u2264 1` there is no tree while the formula gives `1`.","title":"Number of unrooted binary phylogenetic trees"},{"id":"Biology/phylogenetics/counting-binary-phylogenetic-trees/nat-card-unrooted-binary-tree","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.UnrootedTreeCount.nat_card_unrootedBinaryTree","parent":46007,"result":true,"route":"../roadmap/Biology/phylogenetics/counting-binary-phylogenetic-trees/index.html#nat-card-unrooted-binary-tree","scope":"Biology/phylogenetics/counting-binary-phylogenetic-trees","status":"fully_proved","summary":"On `n \u2265 3` labeled taxa there are `(2 n - 5)\u203c = 1 \u00b7 3 \u22ef (2 n - 5)` unrooted binary phylogenetic trees up to label-preserving isomorphism.","title":"The number of unrooted binary phylogenetic trees"},{"data_url":"scopes/Biology/phylogenetics/felsenstein-s-pruning-algorithm.json","id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","title":"Felsenstein's pruning algorithm"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/tree","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.Tree","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#tree","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"A rooted binary tree with leaf labels `L`; an internal node `node P l Q r` carries the transition weights `P` and `Q` (from the node's state to the child's state) on the edges to its left child `l` and right child `r`.","title":"Rooted binary tree with transition weights"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/num-internal","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.Tree.numInternal","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#num-internal","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"The number of internal nodes.","title":"Number of internal nodes of a tree"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/num-leaves","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.Tree.numLeaves","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#num-leaves","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"The number of leaves.","title":"Number of leaves of a tree"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/labeling","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.Labeling","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#labeling","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"A labeling of the non-root nodes of a tree by states: at every internal node, the states of its two children and labelings of the two subtrees.","title":"Labeling of the non-root nodes of a tree"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/card-labeling","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.card_labeling","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#card-labeling","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"There are `|S| ^ (2 \u00b7 #internal nodes)` labelings (one state for each non-root node), so the brute-force sum `bruteLik` has exponentially many terms.","title":"Number of labelings of a tree"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/weight","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.weight","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#weight","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"The weight of a labeling given root state `s`: the product over all edges of the transition weight from the parent's state to the child's state, times the leaf weight `obs l` at the state of every leaf `l`.","title":"Weight of a tree labeling"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/brute-lik","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.bruteLik","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#brute-lik","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"The likelihood of the leaf data conditioned on root state `s`, by brute force: the sum of the weights of all labelings of the non-root nodes.","title":"Brute-force likelihood given the root state"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/likelihood","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.likelihood","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#likelihood","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"The site likelihood with root distribution `\u03c0`: `\u2211 s, \u03c0 s * bruteLik obs t s`.","title":"Site likelihood of a phylogenetic tree"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/prune","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.prune","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#prune","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"Felsenstein's pruning recursion: the partial likelihood of the subtree `t` given state `s` at its root.","title":"Felsenstein's pruning recursion"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/brute-lik-leaf","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.bruteLik_leaf","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#brute-lik-leaf","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"At a leaf, the brute-force likelihood is the leaf weight.","title":"Brute-force likelihood at a leaf"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/brute-lik-node","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.bruteLik_node","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#brute-lik-node","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"The brute-force likelihood factors over the two subtrees of the root (the distributive law behind pruning).","title":"Brute-force likelihood factors over subtrees"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/prune-eq-brute-lik","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.prune_eq_bruteLik","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#prune-eq-brute-lik","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"Felsenstein's pruning algorithm computes the likelihood conditioned on the root state.","title":"Felsenstein's pruning algorithm computes the likelihood conditioned on the root state"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/likelihood-eq-sum-prune","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.likelihood_eq_sum_prune","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#likelihood-eq-sum-prune","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"`likelihood obs t \u03c0 = \u2211 s, \u03c0 s * prune obs t s`.","title":"The site likelihood by pruning"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/prune-tab","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.pruneTab","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#prune-tab","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"Felsenstein's pruning algorithm, tabulated on the states `Fin n`: the vector of partial likelihoods of each subtree is computed once, with `O(n\u00b2)` operations per internal node.","title":"Tabulated Felsenstein pruning algorithm"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/prune-tab-get","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.pruneTab_get","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#prune-tab-get","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"The tabulated algorithm computes the pruning recursion.","title":"Tabulated pruning computes the pruning recursion"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/likelihood-eq-sum-prune-tab","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.likelihood_eq_sum_pruneTab","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#likelihood-eq-sum-prune-tab","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"The site likelihood from the tabulated partial likelihoods at the root.","title":"Site likelihood from tabulated partial likelihoods"},{"id":"Biology/phylogenetics/felsenstein-s-pruning-algorithm/sum-prod-site-weight-eq-prod-likelihood","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinPruning.sum_prod_siteWeight_eq_prod_likelihood","parent":46029,"result":true,"route":"../roadmap/Biology/phylogenetics/felsenstein-s-pruning-algorithm/index.html#sum-prod-site-weight-eq-prod-likelihood","scope":"Biology/phylogenetics/felsenstein-s-pruning-algorithm","status":"fully_proved","summary":"For an alignment whose sites `k : \u03b9` evolve independently along the same tree with leaf data `obs k`, the joint likelihood (the sum over all choices of a root state and a labeling at every site of the product of the per-site weights) is the product of the per-site likelihoods.","title":"Independent sites"},{"data_url":"scopes/Biology/phylogenetics/jukes-cantor-model.json","id":"Biology/phylogenetics/jukes-cantor-model","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","title":"Jukes\u2013Cantor model"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-prob","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcProb","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-prob","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The Jukes\u2013Cantor transition probability from base `i` to base `j` over a branch of length `t`, with substitution rate `\u03bc`.","title":"Jukes\u2013Cantor transition probability"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcRate","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-rate","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The Jukes\u2013Cantor rate matrix: rate `\u03bc / 3` to each other base, `-\u03bc` on the diagonal.","title":"Jukes\u2013Cantor rate matrix"},{"id":"Biology/phylogenetics/jukes-cantor-model/p-diff","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.pDiff","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#p-diff","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The probability that the two ends of a branch of length `t` carry different bases, given base `i` at the start.","title":"Probability that the ends of a branch differ"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-distance","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcDistance","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-distance","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The Jukes\u2013Cantor distance: the expected number of substitutions per site corresponding to a proportion `p` of differing sites. Meaningful only for `0 \u2264 p < 3/4`; outside that range the value is Lean's totalized logarithm, not a distance (at `p = 3/4` it is `0`).","title":"Jukes\u2013Cantor distance"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-rate-nonneg","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcRate_nonneg","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-rate-nonneg","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"Off-diagonal Jukes\u2013Cantor rates are nonnegative when `0 \u2264 \u03bc`.","title":"Nonnegativity of off-diagonal Jukes\u2013Cantor rates"},{"id":"Biology/phylogenetics/jukes-cantor-model/sum-jc-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.sum_jcRate","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#sum-jc-rate","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"Each row of the Jukes\u2013Cantor rate matrix sums to `0`.","title":"Rows of the Jukes\u2013Cantor rate matrix sum to zero"},{"id":"Biology/phylogenetics/jukes-cantor-model/sum-jc-prob","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.sum_jcProb","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#sum-jc-prob","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"Each row of the Jukes\u2013Cantor transition matrix sums to `1`.","title":"Rows of the Jukes\u2013Cantor transition matrix sum to one"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-prob-nonneg","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcProb_nonneg","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-prob-nonneg","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The Jukes\u2013Cantor transition probabilities are nonnegative when `0 \u2264 \u03bc * t`.","title":"Nonnegativity of Jukes\u2013Cantor transition probabilities"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-prob-zero","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcProb_zero","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-prob-zero","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"At time `0` the Jukes\u2013Cantor transition matrix is the identity.","title":"Jukes\u2013Cantor transition matrix at time zero is the identity"},{"id":"Biology/phylogenetics/jukes-cantor-model/sum-jc-prob-mul-jc-prob","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.sum_jcProb_mul_jcProb","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#sum-jc-prob-mul-jc-prob","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"Chapman\u2013Kolmogorov equation for the Jukes\u2013Cantor model.","title":"Chapman\u2013Kolmogorov equation"},{"id":"Biology/phylogenetics/jukes-cantor-model/sum-jc-prob-mul-jc-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.sum_jcProb_mul_jcRate","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#sum-jc-prob-mul-jc-rate","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The right-hand side of the forward equation, in closed form.","title":"Right-hand side of the Jukes\u2013Cantor forward equation in closed form"},{"id":"Biology/phylogenetics/jukes-cantor-model/has-deriv-at-jc-prob","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.hasDerivAt_jcProb","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#has-deriv-at-jc-prob","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"Kolmogorov forward equation for the Jukes\u2013Cantor model: `d/dt P(t) = P(t) Q` with `Q = jcRate \u03bc`.","title":"Kolmogorov forward equation"},{"id":"Biology/phylogenetics/jukes-cantor-model/p-diff-eq","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.pDiff_eq","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#p-diff-eq","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The probability that the two ends of a branch differ is `3/4 \u00b7 (1 - exp (-4 \u03bc t / 3))`, whatever the starting base.","title":"Closed form of the Jukes\u2013Cantor difference probability"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-stochastic-matrix","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcStochasticMatrix","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-stochastic-matrix","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The Jukes\u2013Cantor transition matrix over a branch with `0 \u2264 \u03bc * t`, packaged as a `StochasticMatrix` on the four states `Fin (3 + 1)` so that the finite Markov-chain API (matrix powers, stationarity, detailed balance) applies to it.","title":"Jukes\u2013Cantor transition matrix as a stochastic matrix"},{"id":"Biology/phylogenetics/jukes-cantor-model/mat-pow-jc-stochastic-matrix","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.matPow_jcStochasticMatrix","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#mat-pow-jc-stochastic-matrix","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The `n`-step matrix of the Jukes\u2013Cantor chain observed every `t` time units is the transition matrix over the branch length `n * t` (Chapman\u2013Kolmogorov).","title":"Chapman\u2013Kolmogorov equation for the Jukes\u2013Cantor chain"},{"id":"Biology/phylogenetics/jukes-cantor-model/detailed-balance-jc-stochastic-matrix","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.detailedBalance_jcStochasticMatrix","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#detailed-balance-jc-stochastic-matrix","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The uniform distribution satisfies detailed balance with the Jukes\u2013Cantor matrix (the matrix is symmetric), hence is stationary.","title":"Detailed balance of the Jukes\u2013Cantor chain"},{"id":"Biology/phylogenetics/jukes-cantor-model/p-diff-nonneg","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.pDiff_nonneg","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#p-diff-nonneg","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The probability that the ends of a branch differ is nonnegative when `0 \u2264 \u03bc * t`.","title":"Nonnegativity of the Jukes\u2013Cantor difference probability"},{"id":"Biology/phylogenetics/jukes-cantor-model/p-diff-lt-three-quarters","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.pDiff_lt_three_quarters","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#p-diff-lt-three-quarters","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The probability that the ends of a branch differ is below the saturation value `3/4`, for all real `\u03bc` and `t` (it is negative when `\u03bc * t < 0`, see `pDiff_mem_Ico`).","title":"Jukes\u2013Cantor difference probability is below 3/4"},{"id":"Biology/phylogenetics/jukes-cantor-model/p-diff-mem-ico","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.pDiff_mem_Ico","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#p-diff-mem-ico","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"For `0 \u2264 \u03bc * t` the probability that the ends of a branch differ lies in `[0, 3/4)`, the domain of `jcDistance`.","title":"Jukes\u2013Cantor difference probability lies in (0, 3/4)"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-distance-p-diff","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcDistance_pDiff","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-distance-p-diff","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"`-(3/4) \u00b7 log (1 - 4/3 \u00b7 pDiff \u03bc t i) = \u03bc * t`, the expected number of substitutions per site along the branch.","title":"The Jukes\u2013Cantor distance correction"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-rate-matrix","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcRateMatrix","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-rate-matrix","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The Jukes\u2013Cantor rate matrix as a `Matrix`.","title":"Jukes\u2013Cantor rate matrix as a matrix"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-eigenbasis","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcEigenbasis","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-eigenbasis","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"An eigenbasis of the Jukes\u2013Cantor rate matrix, as columns: the constant vector (eigenvalue `0`) and three vectors with zero sum (eigenvalue `-4 \u03bc / 3`).","title":"Eigenbasis of the Jukes\u2013Cantor rate matrix"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-eigenbasis-unit","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcEigenbasisUnit","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-eigenbasis-unit","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"The eigenbasis as a unit, with its explicit inverse.","title":"Invertible Jukes\u2013Cantor eigenbasis"},{"id":"Biology/phylogenetics/jukes-cantor-model/smul-jc-rate-matrix-eq","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.smul_jcRateMatrix_eq","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#smul-jc-rate-matrix-eq","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"Diagonalization of `t \u2022 jcRateMatrix \u03bc`.","title":"Diagonalization of the Jukes\u2013Cantor rate matrix"},{"id":"Biology/phylogenetics/jukes-cantor-model/jc-prob-eq-exp","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.JukesCantor.jcProb_eq_exp","parent":46047,"result":true,"route":"../roadmap/Biology/phylogenetics/jukes-cantor-model/index.html#jc-prob-eq-exp","scope":"Biology/phylogenetics/jukes-cantor-model","status":"fully_proved","summary":"`jcProb \u03bc t = exp (t \u2022 jcRateMatrix \u03bc)`.","title":"The Jukes\u2013Cantor transition matrix is the matrix exponential of the rate matrix"},{"data_url":"scopes/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction.json","id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","title":"Maximum-likelihood ancestral reconstruction"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/nonneg-weights","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.NonnegWeights","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#nonneg-weights","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"All transition weights in the tree are nonnegative.","title":"Nonnegative transition weights on a tree"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/argmax-state","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.argmaxState","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#argmax-state","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"A state maximizing `f` (ties broken by choice).","title":"State maximizing a function"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/le-argmax-state","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.le_argmaxState","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#le-argmax-state","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"`argmaxState f` maximizes `f`.","title":"The argmax state maximizes the function"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/sup-eq-argmax-state","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.sup'_eq_argmaxState","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#sup-eq-argmax-state","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"The value at `argmaxState f` is the maximum `univ.sup'` of `f`.","title":"Value at the argmax state is the maximum"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/max-prune","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.maxPrune","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#max-prune","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"The max-product (Viterbi) form of Felsenstein's pruning recursion: the largest weight of a labeling of the subtree `t` given state `s` at its root.","title":"Max-product pruning recursion"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/traceback","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.traceback","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#traceback","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"Pupko's traceback: given the state `s` of the root of `t`, label each child with a state maximizing the transition weight times the child's `maxPrune`, and recurse.","title":"Pupko's traceback"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/max-prune-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.maxPrune_nonneg","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#max-prune-nonneg","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"With nonnegative weights, `maxPrune` is nonnegative.","title":"Nonnegativity of max-product pruning"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/weight-traceback","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.weight_traceback","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#weight-traceback","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"No sign condition is needed.","title":"The traced labeling attains `maxPrune`"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/weight-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.weight_nonneg","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#weight-nonneg","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"With nonnegative weights, every labeling has nonnegative weight.","title":"Nonnegativity of labeling weights"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/weight-le-max-prune","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.weight_le_maxPrune","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#weight-le-max-prune","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"`maxPrune` bounds the weight of every labeling (nonnegative weights).","title":"`maxPrune` bounds the weight of every labeling"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/max-prune-eq-sup","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.maxPrune_eq_sup'","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#max-prune-eq-sup","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"`maxPrune obs t s` is the maximum over all labelings `z` of the non-root nodes of `weight obs t s z`.","title":"The max-product recursion computes the maximal labeling weight"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/joint-root","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.jointRoot","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#joint-root","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"The root state of Pupko's reconstruction: it maximizes `\u03c0 s * maxPrune obs t s`.","title":"Root state of Pupko's reconstruction"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/pupko","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.pupko","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#pupko","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"The root state `jointRoot` and the traced labeling.","title":"Pupko's joint reconstruction"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/joint-lik-pupko","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.jointLik_pupko","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#joint-lik-pupko","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"The likelihood of Pupko's reconstruction is `max_s \u03c0 s * maxPrune obs t s`.","title":"Likelihood of Pupko's joint reconstruction"},{"id":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/pupko-is-joint-ml","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.AncestralReconstruction.pupko_isJointML","parent":46073,"result":true,"route":"../roadmap/Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction/index.html#pupko-is-joint-ml","scope":"Biology/phylogenetics/maximum-likelihood-ancestral-reconstruction","status":"fully_proved","summary":"With nonnegative root, transition and leaf weights, no root state `s` and labeling `z` of the non-root nodes has larger joint likelihood `\u03c0 s * weight obs t s z` than Pupko's reconstruction.","title":"Pupko et al. (2000): joint maximum likelihood"},{"data_url":"scopes/Biology/phylogenetics/neighbor-joining.json","id":"Biology/phylogenetics/neighbor-joining","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","title":"Neighbor joining"},{"id":"Biology/phylogenetics/neighbor-joining/initial-dist","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.initialDist","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#initial-dist","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"The dissimilarity between clusters at the start of neighbor joining: on singletons, `initialDist d {x} {y} = d x y` (only singleton values are used).","title":"Initial cluster dissimilarity in neighbor joining"},{"id":"Biology/phylogenetics/neighbor-joining/q-criterion","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.qCriterion","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#q-criterion","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"The Studier\u2013Keppler form of the neighbor-joining criterion on the cluster set `S`: `Q(a, b) = (|S| - 2) D a b - \u2211_{c \u2208 S} D a c - \u2211_{c \u2208 S} D b c`.","title":"Neighbor-joining Q-criterion"},{"id":"Biology/phylogenetics/neighbor-joining/join-dist","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.joinDist","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#join-dist","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"The cluster dissimilarity after joining `a` and `b`: the new cluster `a \u222a b` is at `(D a c + D b c - D a b) / 2` from every other cluster `c`, and at `0` from itself; other values are unchanged.","title":"Cluster dissimilarity after a neighbor-joining step"},{"id":"Biology/phylogenetics/neighbor-joining/njrun","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.NJRun","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#njrun","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"`NJRun d S D C`: some run of neighbor joining on the dissimilarity `d` reaches the cluster set `S` with cluster dissimilarity `D`, having formed the clusters in `C`. A join is allowed while at least four clusters remain, for any pair of distinct clusters minimizing `qCriterion`.","title":"Runs of neighbor joining"},{"id":"Biology/phylogenetics/neighbor-joining/is-binary-phylogenetic-tree","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.IsBinaryPhylogeneticTree","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#is-binary-phylogenetic-tree","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"`G` with the taxon placement `\u03b9` is a binary phylogenetic `X`-tree: `G` is a tree, `\u03b9` is injective, the leaves (degree-one vertices) are exactly the taxa, and every other vertex has degree three.","title":"Binary phylogenetic tree"},{"id":"Biology/phylogenetics/neighbor-joining/side","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.side","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#side","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"The taxa on the `a`-side of the edge `{a, b}`: those whose vertex is still joined to `a` once the edge `{a, b}` is deleted.","title":"Taxa on one side of a tree edge"},{"id":"Biology/phylogenetics/neighbor-joining/split-gain","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.splitGain","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#split-gain","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"The contribution of one split `{A, A\u1d9c}` to the neighbor-joining score of the pair `x, y`: the number of taxa on the far side of the split when `x, y` lie on the same side, and `1` when the split separates them.","title":"Contribution of a split to the neighbor-joining score"},{"id":"Biology/phylogenetics/neighbor-joining/side-swap","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.side_swap","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#side-swap","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"In a tree, the two sides of an edge partition the taxa.","title":"Two sides of a tree edge partition the taxa"},{"id":"Biology/phylogenetics/neighbor-joining/side-compat","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.side_compat","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#side-compat","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"In a tree, the splits of any two edges are compatible: one side of the first edge lies inside one side of the second.","title":"Splits of two tree edges are compatible"},{"id":"Biology/phylogenetics/neighbor-joining/side-leaf","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.side_leaf","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#side-leaf","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"The leaf side of a pendant edge holds only the leaf's taxon.","title":"Leaf side of a pendant edge"},{"id":"Biology/phylogenetics/neighbor-joining/side-nonempty","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.side_nonempty","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#side-nonempty","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"Every side of every edge of a binary phylogenetic tree holds a taxon.","title":"Every edge side of a binary phylogenetic tree is nonempty"},{"id":"Biology/phylogenetics/neighbor-joining/side-eq-union","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.side_eq_union","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#side-eq-union","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"At an internal vertex `a` with neighbor `b`, the `a`-side of `{a, b}` is the disjoint union of the far sides of the other two edges at `a`.","title":"Edge side at an internal vertex as a disjoint union"},{"id":"Biology/phylogenetics/neighbor-joining/dist-eq-sum-darts","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.dist_eq_sum_darts","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#dist-eq-sum-darts","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"The path length between two taxa is half the sum, over the darts (oriented edges) of the tree, of the lengths of the edges whose split separates them.","title":"Split decomposition of a tree metric"},{"id":"Biology/phylogenetics/neighbor-joining/exists-eq-pair-of-is-max","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.exists_eq_pair_of_isMax","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#exists-eq-pair-of-is-max","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"For a compatible system of splits with positive weights, containing every trivial split and binary (every side with at least two taxa is the disjoint union of two sides), on at least four taxa, a pair maximizing the total `splitGain` is a side of some split.","title":"Cherry lemma for split systems"},{"id":"Biology/phylogenetics/neighbor-joining/q-criterion-singleton","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.qCriterion_singleton","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#q-criterion-singleton","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"The Q-criterion of two singleton clusters at the start of neighbor joining: `Q({x}, {y}) = (|X| - 2) d x y - \u2211_z d x z - \u2211_z d y z`.","title":"Q-criterion of two singleton clusters"},{"id":"Biology/phylogenetics/neighbor-joining/sub-sum-sub-sum-eq-neg-sum-split-gain","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.sub_sum_sub_sum_eq_neg_sum_splitGain","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#sub-sum-sub-sum-eq-neg-sum-split-gain","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"If `d x y` is half the total weight of the splits `A e` separating `x` and `y`, then `(|X| - 2) d x y - \u2211_z d x z - \u2211_z d y z = -\u2211 e, w e * splitGain (A e) x y`.","title":"The Q-criterion of a split-decomposed dissimilarity"},{"id":"Biology/phylogenetics/neighbor-joining/eq-of-side-eq-singleton","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.eq_of_side_eq_singleton","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#eq-of-side-eq-singleton","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"A side holding a single taxon `z` is the leaf side of `z`'s pendant edge.","title":"A singleton side is a pendant leaf side"},{"id":"Biology/phylogenetics/neighbor-joining/adj-adj-of-side-eq-pair","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.adj_adj_of_side_eq_pair","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#adj-adj-of-side-eq-pair","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"If a side of the edge `{a, b}` is the pair `{x, y}`, then `a` is a common neighbor of the leaves `\u03b9 x` and `\u03b9 y`: they form a cherry.","title":"A two-taxon side forms a cherry"},{"id":"Biology/phylogenetics/neighbor-joining/exists-adj-adj-of-q-criterion-min","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.exists_adj_adj_of_qCriterion_min","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#exists-adj-adj-of-q-criterion-min","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"(Saitou & Nei 1987; Studier & Keppler 1988; Bryant 2005). Let `d` be the path-length metric of a binary phylogenetic `X`-tree `(G, \u03b9)` with positive edge lengths `w` and at least four taxa. If the distinct taxa `x, y` minimize the Q-criterion over all pairs of distinct singleton clusters at the start of neighbor joining, then `\u03b9 x` and `\u03b9 y` have a common neighbor.","title":"On a tree metric, a pair of taxa minimizing the Q-criterion is a cherry"},{"id":"Biology/phylogenetics/neighbor-joining/exists-adj-adj-of-join-init","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.NJRun.exists_adj_adj_of_join_init","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#exists-adj-adj-of-join-init","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"On the path-length metric of a binary phylogenetic `X`-tree with positive edge lengths and at least four taxa, any pair of clusters that neighbor joining may join from the initial singletons consists of two taxa with a common neighbor.","title":"The first join of neighbor joining on a tree metric joins a cherry"},{"id":"Biology/phylogenetics/neighbor-joining/exists-split-system-of-partition","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.exists_splitSystem_of_partition","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#exists-split-system-of-partition","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"Let `S` partition the taxa of a binary phylogenetic `X`-tree `(G, \u03b9)` with positive edge lengths `w` and path-length metric `d` into sides of edges. Then there is a system of splits `A e` (`e : E`) of the set of clusters with positive weights `w' e`, closed under complement, pairwise compatible, binary, and containing every trivial split, such that edge with both endpoints internal; splits separating `s` from `t`,\u2026","title":"The path metric on a partition into sides"},{"id":"Biology/phylogenetics/neighbor-joining/nj-run-splits","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.njRun_splits","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#nj-run-splits","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"Let `d` be the path-length metric of a binary phylogenetic `X`-tree `(G, \u03b9)` with positive edge lengths `w`. Then every complete run of neighbor joining (stopped with at most three clusters) forms exactly the splits of the internal edges of the tree: a set of taxa `A` or its complement is a formed cluster if and only if `A` is the set of taxa on the `a`-side of an edge `{a, b}` with both endpoints internal.","title":"Neighbor joining is exact on tree metrics"},{"id":"Biology/phylogenetics/neighbor-joining/nj-run-splits-of-join","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.njRun_splits_of_join","parent":46089,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining/index.html#nj-run-splits-of-join","scope":"Biology/phylogenetics/neighbor-joining","status":"fully_proved","summary":"Let `(G, \u03b9)` be a binary phylogenetic `X`-tree and `D` any dissimilarity. Suppose that whenever a run of neighbor joining on `D` has reached a partition `S` of the taxa into sides of edges of the tree (with at least four clusters) and joins two of its clusters, their union is again a side of an edge. Then every complete run of neighbor joining on `D` (stopped with at most three clusters) forms exactly the splits of\u2026","title":"Neighbor joining forms the splits of a tree when every join forms a side"},{"data_url":"scopes/Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining.json","id":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/index.html","scope":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining","status":"fully_proved","title":"Atteson's theorem on neighbor joining"},{"id":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/exists-eq-pair-of-q-criterion-min-of-abs","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.exists_eq_pair_of_qCriterion_min_of_abs_sub_lt","parent":46113,"result":true,"route":"../roadmap/Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/index.html#exists-eq-pair-of-q-criterion-min-of-abs","scope":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining","status":"fully_proved","summary":"The selection step of Atteson's theorem (Atteson 1999), with a proof after Mihaescu, Levy & Pachter (2009, Theorem 16). Let `A e` (`e : E`) be a compatible binary system of splits with positive weights `w`, containing every trivial split and closed under complement, on at least four taxa, and let `d x y` be half the total weight of the indices whose side separates `x` and `y`. Let `\u03bc` be a lower bound on the weight\u2026","title":"Robust cherry lemma for split systems"},{"id":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/exists-adj-adj-of-q-criterion-min-of-abs","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.exists_adj_adj_of_qCriterion_min_of_abs_sub_lt","parent":46113,"result":true,"route":"../roadmap/Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/index.html#exists-adj-adj-of-q-criterion-min-of-abs","scope":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining","status":"fully_proved","summary":"The selection step of Atteson's theorem (Atteson 1999), with a proof after Mihaescu, Levy & Pachter (2009, Theorem 16). Let `d` be the path-length metric of a binary phylogenetic `X`-tree `(G, \u03b9)` with positive edge lengths `w` and at least four taxa, and let `\u03bc` be a lower bound on the lengths of the internal edges (edges with no taxon at either end). Let `D` be a symmetric dissimilarity with zero diagonal such\u2026","title":"Robust cherry lemma for neighbor joining"},{"id":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/exists-adj-adj-of-join-init-of-abs-sub-l","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.NJRun.exists_adj_adj_of_join_init_of_abs_sub_lt","parent":46113,"result":true,"route":"../roadmap/Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/index.html#exists-adj-adj-of-join-init-of-abs-sub-l","scope":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining","status":"fully_proved","summary":"Under the hypotheses of `exists_adj_adj_of_qCriterion_min_of_abs_sub_lt`, any pair of clusters that neighbor joining run on `D` may join from the initial singletons consists of two taxa with a common neighbor.","title":"The first join of neighbor joining on a perturbed tree metric joins a cherry"},{"id":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/exists-avg","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.NJRun.exists_avg","parent":46113,"result":true,"route":"../roadmap/Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/index.html#exists-avg","scope":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining","status":"fully_proved","summary":"Along every run of neighbor joining on a symmetric dissimilarity `d` with zero diagonal there are weights `\u03b1 s` (nonnegative, supported on `s`, summing to `1`) and row terms `\u03ba s` such that the cluster dissimilarity vanishes on the diagonal and `D s t = \u2211 x, \u2211 y, \u03b1 s x * \u03b1 t y * d x y - \u03ba s - \u03ba t` for distinct current clusters. Joining `a, b` takes `\u03b1 (a \u222a b) = (\u03b1 a + \u03b1 b) / 2` and\u2026","title":"Neighbor joining averages up to row shifts"},{"id":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/q-criterion-eq-sub-of-shift","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.qCriterion_eq_sub_of_shift","parent":46113,"result":true,"route":"../roadmap/Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/index.html#q-criterion-eq-sub-of-shift","scope":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining","status":"fully_proved","summary":"The shifting lemma (Mihaescu, Levy & Pachter 2009, Lemma 4): if `D'` agrees with `D` on the diagonal of `S` and `D' s t = D s t + f s + f t` for distinct `s, t \u2208 S`, then the Q-criterion of every pair of distinct clusters drops by the same amount `2 \u2211_{c \u2208 S} f c`, so both have the same minimizing pairs.","title":"The shifting lemma"},{"id":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/nj-run-splits-of-abs-sub-lt","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoining.njRun_splits_of_abs_sub_lt","parent":46113,"result":true,"route":"../roadmap/Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining/index.html#nj-run-splits-of-abs-sub-lt","scope":"Biology/phylogenetics/atteson-s-theorem-on-neighbor-joining","status":"fully_proved","summary":"Atteson's theorem: neighbor joining has `\u2113\u221e` radius `1/2` (Atteson 1999). Let `d` be the path-length metric of a binary phylogenetic `X`-tree `(G, \u03b9)` with positive edge lengths `w`, let `\u03bc` be a lower bound on the lengths of its internal edges (edges with no taxon at either end), and let `D` be a symmetric dissimilarity with zero diagonal such that `|D x y - d x y| < \u03bc / 2` for all distinct taxa. Then every\u2026","title":"Atteson's theorem: neighbor joining has `\u2113\u221e` radius `1/2`"},{"data_url":"scopes/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution.json","id":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/index.html","scope":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","status":"fully_proved","title":"Neighbor joining and balanced minimum evolution"},{"id":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/cherry-coeff","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoiningBME.cherryCoeff","parent":46120,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/index.html#cherry-coeff","scope":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","status":"fully_proved","summary":"The generalized Pauplin coefficient of the pair of clusters `u, v` in the tree `T_ab` obtained from the star on `S` by joining `a` and `b`: `1/2` for `{a, b}`; `1/(2(|S| - 2))` when one of `u, v` is `a` or `b` and the other is not; `1/(|S| - 2)` when neither is. Values on the diagonal `u = v` are never used.","title":"Generalized Pauplin coefficient of a cherry tree"},{"id":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/cherry-length","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoiningBME.cherryLength","parent":46120,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/index.html#cherry-length","scope":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","status":"fully_proved","summary":"The balanced minimum-evolution length of `T_ab` for the dissimilarity `D`: half the sum over ordered pairs of distinct clusters of `cherryCoeff S a b u v * D u v`, which is `\u2211_{u<v} cherryCoeff S a b u v * D u v` when `D` is symmetric.","title":"Balanced minimum-evolution length of a cherry tree"},{"id":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/cherry-length-eq","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoiningBME.cherryLength_eq","parent":46120,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/index.html#cherry-length-eq","scope":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","status":"fully_proved","summary":"For `D` symmetric and zero on the diagonal of `S`, `|S| \u2265 3` and distinct `a, b \u2208 S`: `cherryLength S D a b = (\u2211_{u \u2260 v \u2208 S} D u v + Q(a, b)) / (2 (|S| - 2))`.","title":"The BME length of a join in terms of the Q-criterion"},{"id":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/q-criterion-le-iff-cherry-length-le","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoiningBME.qCriterion_le_iff_cherryLength_le","parent":46120,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/index.html#q-criterion-le-iff-cherry-length-le","scope":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","status":"fully_proved","summary":"The BME length of `T_ab` on the current clusters for the current cluster dissimilarity `D`. This is the local star identity; it is not the full Gascuel\u2013Steel comparison on all taxa.","title":"Minimizing `Q` is minimizing the local star BME length of the join"},{"id":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/symm-diag","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoiningBME.NJRun.symm_diag","parent":46120,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/index.html#symm-diag","scope":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","status":"fully_proved","summary":"Along a run of neighbor joining on a symmetric `d` with zero diagonal, the cluster dissimilarity stays symmetric and vanishes on every current cluster.","title":"Neighbor joining preserves symmetry and zero diagonal"},{"id":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/is-min-q-criterion-iff-cherry-length","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoiningBME.NJRun.isMin_qCriterion_iff_cherryLength","parent":46120,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/index.html#is-min-q-criterion-iff-cherry-length","scope":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","status":"fully_proved","summary":"At every state of a run on a symmetric dissimilarity with zero diagonal and at least three clusters, a pair of distinct clusters minimizes the Q-criterion (the pairs NJ may join) if and only if joining it gives the smallest `cherryLength`, the BME length of the join tree `T_ab` on the current clusters for the current cluster dissimilarity `D`. This is the local star identity only: it does not compare partially\u2026","title":"Q-minimality is cherry-length minimality at a run state"},{"id":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/cherry-length-eq-length","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.NeighborJoiningBME.cherryLength_eq_length","parent":46120,"result":true,"route":"../roadmap/Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution/index.html#cherry-length-eq-length","scope":"Biology/phylogenetics/neighbor-joining-and-balanced-minimum-evolution","status":"fully_proved","summary":"Let `T_ab` carry real edge lengths: `\u2113 c` on the pendant edge of each cluster `c \u2208 S` and `m` on the edge between `u` and the center. If `D` is its path-length dissimilarity (`D a b = \u2113 a + \u2113 b`, `D x c = \u2113 x + m + \u2113 c` for `x \u2208 {a, b}` and `c` another cluster, `D c c' = \u2113 c + \u2113 c'` for two other distinct clusters), then `cherryLength S D a b = \u2211_{c \u2208 S} \u2113 c + m`.","title":"The coefficients give the tree length"},{"data_url":"scopes/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution.json","id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","title":"Pauplin's formula and balanced minimum evolution"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/rtree","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.RTree","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#rtree","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"A rooted binary tree with leaf labels in `X`; `node a l b r` has the child `l` at the end of an edge of length `a` and the child `r` at the end of an edge of length `b`.","title":"Rooted binary tree with edge lengths"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/utree","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.UTree","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#utree","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"An unrooted binary tree, rooted at its edge of length `len` that joins the roots of the rooted binary trees `left` and `right`.","title":"Unrooted binary tree with edge lengths"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/pauplin-coeff","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.pauplinCoeff","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#pauplin-coeff","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"The Pauplin coefficient `2 ^ (1 - \u03c4)` of a pair of leaves at topological distance `\u03c4`.","title":"Pauplin coefficient of a pair of leaves"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/length","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.RTree.length","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#length","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"The total length of the edges.","title":"Total edge length of a rooted binary tree"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/leaves","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.RTree.leaves","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#leaves","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"The leaves, each with its label, its distance to the root and its number of edges to the root.","title":"Leaves of a rooted binary tree with root distances"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/pairs","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.RTree.pairs","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#pairs","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"The unordered pairs of leaves, each with the two labels, the length of the path between the two leaves and its number of edges.","title":"Leaf pairs of a rooted binary tree with path lengths"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/kraft","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.RTree.kraft","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#kraft","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"Kraft's equality for a rooted binary tree: `\u2211_{leaves} 2^{-depth} = 1`.","title":"Kraft's equality"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/length-2","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.UTree.length","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#length-2","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"The total length of the edges.","title":"Total edge length of an unrooted binary tree"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/labels","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.UTree.labels","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#labels","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"The leaf labels (with multiplicity).","title":"Leaf labels of an unrooted binary tree"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/pairs-2","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.UTree.pairs","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#pairs-2","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"The unordered pairs of leaves, each with the two labels, the length `d_T(i, j)` of the path between the two leaves and its number of edges `\u03c4_ij`.","title":"Leaf pairs of an unrooted binary tree with path lengths"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/bme-length","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.UTree.bmeLength","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#bme-length","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"The balanced minimum-evolution length of the topology of `T` for a dissimilarity `d` (Desper and Gascuel 2002): `\u2211_{i<j} 2^{1-\u03c4_ij} (d(i, j) + d(j, i)) / 2`. It depends on `T` only through the topological distances `\u03c4_ij`. Each term is symmetrized, so the value does not depend on the orientation `T.pairs` happens to assign to a pair: swapping the children of a node, or the two sides of the root edge, leaves\u2026","title":"Balanced minimum-evolution length"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/bme-length-path-length","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.UTree.bmeLength_pathLength","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#bme-length-path-length","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"Pauplin's formula (Pauplin 2000): the total edge length of a binary tree with real edge lengths is `\u2211_{i<j} 2^{1-\u03c4_ij} d_T(i, j)`, the sum over pairs of leaves of the path length weighted by `2` to the power one minus the number of edges on the path.","title":"Pauplin's formula"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/bme-length-eq-length","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.UTree.bmeLength_eq_length","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#bme-length-eq-length","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"If `d` is the path-length dissimilarity of `T` on its leaves and `d` is symmetric, `bmeLength T d = length T`. Symmetry is needed because `bmeLength` averages the two orientations of each pair.","title":"The balanced minimum-evolution length of a tree metric is the tree length"},{"id":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/kraft-leaf","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.BalancedMinimumEvolution.UTree.kraft_leaf","parent":46128,"result":true,"route":"../roadmap/Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution/index.html#kraft-leaf","scope":"Biology/phylogenetics/pauplin-s-formula-and-balanced-minimum-evolution","status":"fully_proved","summary":"Kraft equality for unrooted binary trees: for every leaf `i` of a tree with distinct leaf labels, `\u2211_{j \u2260 i} 2^{1-\u03c4_ij} = 1`.","title":"Kraft equality"},{"data_url":"scopes/Biology/phylogenetics/phylocsf-coding-classifier.json","id":"Biology/phylogenetics/phylocsf-coding-classifier","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","title":"PhyloCSF coding classifier"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/leaf-data","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.leafData","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#leaf-data","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"The leaf data of an alignment column `col : L \u2192 S`: the indicator of the observed codon at each leaf.","title":"Leaf data of an alignment column"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/site-lik","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.siteLik","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#site-lik","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"The likelihood of one alignment column under the model `(t, \u03c0)`.","title":"Likelihood of one alignment column"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/align-lik","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.alignLik","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#align-lik","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"The likelihood of an alignment of `m` independent columns: the product of the site likelihoods.","title":"Likelihood of an alignment of independent columns"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/score","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.score","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#score","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"The PhyloCSF score (natural-log scale): the sum over the columns of the log-likelihood ratio of the coding model `(tC, \u03c0C)` against the non-coding model `(tN, \u03c0N)`. The published PhyloCSF score is reported in decibans; see `score_decibans` for the conversion. Thresholds for this `score` are in natural-log units.","title":"The PhyloCSF score"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/score-decibans","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.score_decibans","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#score-decibans","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"`10 * log\u2081\u2080(L_C / L_N)`, i.e. `(10 / log 10) * score`, since `log\u2081\u2080 x = log x / log 10`.","title":"The published PhyloCSF score in decibans"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/score-decibans-eq","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.score_decibans_eq","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#score-decibans-eq","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"Decibans are the natural-log score rescaled by `10 / log 10`.","title":"Deciban score as a rescaled natural-log score"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/site-lik-eq-sum-prune","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.siteLik_eq_sum_prune","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#site-lik-eq-sum-prune","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"Each site likelihood is computed by Felsenstein's pruning algorithm.","title":"Site likelihood by Felsenstein's pruning algorithm"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/score-eq-log-div","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.score_eq_log_div","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#score-eq-log-div","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"The score is the log-likelihood ratio of the alignment when all site likelihoods are positive.","title":"The score is the log-likelihood ratio of the alignment"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/score-test","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.scoreTest","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#score-test","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"The threshold test on the score: call the alignment coding (`1`) iff the score exceeds the cutoff `c`.","title":"Score threshold test for calling an alignment coding"},{"id":"Biology/phylogenetics/phylocsf-coding-classifier/weighted-power-le-score-test","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.PhyloCSF.weighted_power_le_scoreTest","parent":46143,"result":true,"route":"../roadmap/Biology/phylogenetics/phylocsf-coding-classifier/index.html#weighted-power-le-score-test","scope":"Biology/phylogenetics/phylocsf-coding-classifier","status":"fully_proved","summary":"With positive site likelihoods under both models, a randomized test `\u03c6` whose weighted size under the non-coding weights is at most that of `scoreTest c` has at most its weighted power under the coding weights. This is the dominated Neyman\u2013Pearson inequality for the unnormalized alignment weights `alignLik`; it becomes a statement about statistical size and power only under additional normalization hypotheses (\u2026","title":"PhyloCSF's score threshold test is most powerful for unnormalized weights"},{"data_url":"scopes/Biology/phylogenetics/small-parsimony-problem.json","id":"Biology/phylogenetics/small-parsimony-problem","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","title":"Small parsimony problem"},{"id":"Biology/phylogenetics/small-parsimony-problem/cost","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.Cost","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#cost","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"The cost semiring: `MinTropical (WithTop \u2115)`, with `min` as addition and `+` as multiplication.","title":"Min-plus cost semiring"},{"id":"Biology/phylogenetics/small-parsimony-problem/edge-cost","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.edgeCost","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#edge-cost","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"The total substitution cost of a labeling `x` of the non-root vertices of `t` with root state `s`: the sum over all edges of the cost `untrop (P a b)` from the parent's state `a` to the child's state `b`.","title":"Total substitution cost of a labeling"},{"id":"Biology/phylogenetics/small-parsimony-problem/agrees","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.Agrees","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#agrees","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"The labeling `x` (with root state `s`) agrees with the observed states `\u03c3` at every leaf.","title":"Labeling agrees with the observed leaf states"},{"id":"Biology/phylogenetics/small-parsimony-problem/leaf-obs","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.leafObs","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#leaf-obs","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"The leaf weights of the observed states `\u03c3`: the tropical one (cost `0`) at the observed state and the tropical zero (cost `\u22a4`) at every other state.","title":"Tropical leaf weights of observed states"},{"id":"Biology/phylogenetics/small-parsimony-problem/untrop-weight","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.untrop_weight","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#untrop-weight","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"The weight of a labeling in the tropical semiring is its edge cost if it agrees with the leaf states and `\u22a4` otherwise.","title":"Tropical weight of a labeling equals its edge cost"},{"id":"Biology/phylogenetics/small-parsimony-problem/sankoff","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.sankoff","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#sankoff","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"The minimum cost of the subtree `t` given the state `s` at its root, computed by Felsenstein's pruning recursion over the tropical semiring.","title":"Sankoff's algorithm"},{"id":"Biology/phylogenetics/small-parsimony-problem/sankoff-eq-inf","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.sankoff_eq_inf","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#sankoff-eq-inf","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"`sankoff \u03c3 t s` is the minimum total edge cost over all labelings of the non-root vertices that agree with the observed states at the leaves (with root state `s`).","title":"Sankoff's algorithm computes the small-parsimony cost"},{"id":"Biology/phylogenetics/small-parsimony-problem/sankoff-le-edge-cost","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.sankoff_le_edgeCost","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#sankoff-le-edge-cost","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"Sankoff's cost is a lower bound for the cost of every labeling that agrees with the leaf states.","title":"Sankoff's cost is a lower bound"},{"id":"Biology/phylogenetics/small-parsimony-problem/exists-edge-cost-eq-sankoff","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.exists_edgeCost_eq_sankoff","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#exists-edge-cost-eq-sankoff","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"Sankoff's cost is attained: if it is finite, some labeling that agrees with the leaf states has exactly this cost. (It is `\u22a4` exactly when every agreeing labeling uses a forbidden change, or when `t` is a single leaf whose observed state is not `s`.)","title":"Sankoff's cost is attained"},{"id":"Biology/phylogenetics/small-parsimony-problem/sankoff-leaf","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.sankoff_leaf","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#sankoff-leaf","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"Sankoff's recursion at a leaf: cost `0` at the observed state, `\u22a4` elsewhere.","title":"Sankoff's recursion at a leaf"},{"id":"Biology/phylogenetics/small-parsimony-problem/sankoff-node","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.sankoff_node","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#sankoff-node","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"Sankoff's recursion at an internal vertex: the cost of state `s` is the sum, over the two children, of the minimum over the child's state `a` of the edge cost from `s` to `a` plus the child's cost at `a`.","title":"Sankoff's recursion at an internal vertex"},{"id":"Biology/phylogenetics/small-parsimony-problem/unit-cost","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.unitCost","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#unit-cost","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"The unit cost: `0` for no change and `1` for a change of state.","title":"Unit substitution cost"},{"id":"Biology/phylogenetics/small-parsimony-problem/is-unit-cost","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.IsUnitCost","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#is-unit-cost","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"Every edge of the tree carries the unit cost.","title":"Tree with unit costs on every edge"},{"id":"Biology/phylogenetics/small-parsimony-problem/fitch","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.fitch","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#fitch","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"The set of candidate states at the root of the subtree and the number of changes counted so far. The edge weights of the tree are ignored. Each child is evaluated once per node, so the recursion makes one call per vertex.","title":"Fitch's algorithm"},{"id":"Biology/phylogenetics/small-parsimony-problem/fitch-score-eq","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.fitch_score_eq","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#fitch-score-eq","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"On a binary tree with unit costs, Fitch's count equals the minimum, over root states, of Sankoff's cost.","title":"Fitch's algorithm computes the small-parsimony score"},{"id":"Biology/phylogenetics/small-parsimony-problem/fitch-set-eq","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Parsimony.fitch_set_eq","parent":46154,"result":true,"route":"../roadmap/Biology/phylogenetics/small-parsimony-problem/index.html#fitch-set-eq","scope":"Biology/phylogenetics/small-parsimony-problem","status":"fully_proved","summary":"On a binary tree with unit costs, a root state is in Fitch's set if and only if Sankoff's cost at that state is the minimum.","title":"Fitch's root set is the set of optimal root states"},{"data_url":"scopes/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei.json","id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","title":"Pulley principle, LogDet distances and Felsenstein zone"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/is-reversible-semigroup","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.IsReversibleSemigroup","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#is-reversible-semigroup","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"A family of transition matrices `P t`, indexed by branch length, is a reversible semigroup for the distribution `\u03c0`: it satisfies the Chapman\u2013Kolmogorov equation and detailed balance `\u03c0 a P t a b = \u03c0 b P t b a`. With rows summing to `1`, detailed balance makes `\u03c0` stationary; the pulley principle uses only the two equations below.","title":"Reversible semigroup of transition matrices"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/prune-node-comm","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.prune_node_comm","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#prune-node-comm","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Swapping the two subtrees of a node does not change its partial likelihoods; with `likelihood_node_comm` this lets the rerooting move `likelihood_reroot` reach either child.","title":"Partial likelihoods are invariant under swapping subtrees"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/likelihood-node-comm","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.likelihood_node_comm","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#likelihood-node-comm","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Swapping the two subtrees of the root does not change the likelihood.","title":"Likelihood is invariant under swapping root subtrees"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/likelihood-node-eq","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.likelihood_node_eq","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#likelihood-node-eq","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"With the root on the edge between the subtrees `l` and `r`, at distances `s` and `t`, the likelihood is that of the root placed at the root of `l`, with `r` at distance `s + t`.","title":"The root can be pushed to the end of its edge"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/likelihood-slide","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.likelihood_slide","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#likelihood-slide","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Moving the root along its edge does not change the likelihood (Felsenstein 1981).","title":"Pulley principle, sliding"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/likelihood-reroot","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.likelihood_reroot","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#likelihood-reroot","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Let the root sit on the edge between an internal vertex `x` (at distance `s\u2081`) and the subtree `C` (at distance `s\u2082`), where `x` has the subtrees `A` and `B` at distances `a` and `b`. Moving the root to the edge between `x` and `A`, at distance `u\u2081` from `A` and `u\u2082` from `x` with `u\u2081 + u\u2082 = a`, gives the same likelihood: the unrooted tree and its edge lengths are unchanged (Felsenstein 1981).","title":"Pulley principle, rerooting"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/is-reversible-semigroup-ctmc-transition","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.isReversibleSemigroup_ctmcTransition","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#is-reversible-semigroup-ctmc-transition","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"A reversible continuous-time model gives a reversible semigroup: if `\u03c0` is in detailed balance with the generator `Q`, the transition functions `exp (t \u2022 Q)` form a reversible semigroup for `\u03c0`.","title":"Reversible continuous-time models give reversible semigroups"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/paralinear","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.paralinear","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#paralinear","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The LogDet (paralinear) distance of a joint distribution matrix `F` (`F i j = \u2119(X = i, Y = j)`): `-log |det F| + (log det \u03a0_X + log det \u03a0_Y) / 2`, where the diagonal marginal matrices have determinants `\u220f i, \u2211 j, F i j` and `\u220f j, \u2211 i, F i j`. It is a distance only when `det F \u2260 0` and the marginals are positive; otherwise the value is Lean's totalized logarithm (`log 0 = 0`).","title":"LogDet (paralinear) distance"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/paralinear-diagonal","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.paralinear_diagonal","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#paralinear-diagonal","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The distance of a vertex to itself is `0`: the joint distribution of `X` with itself is the diagonal matrix of its (positive) marginal.","title":"LogDet distance of a vertex to itself is zero"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/vec-mul-pos","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.vecMul_pos","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#vec-mul-pos","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Under a nonnegative, nonsingular transition matrix, a positive distribution stays positive: every column of `M` has a positive entry. This puts the LogDet logarithms in their domain.","title":"Nonsingular transition matrices preserve positive distributions"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/paralinear-three-point","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.paralinear_threePoint","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#paralinear-three-point","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Additivity of the LogDet distance at a separating vertex (Steel 1994; Lockhart et al. 1994; Lake 1994). Let `Y` have distribution `p` (positive) and let `X` and `Z` be conditionally independent given `Y`, with transition matrices `M` (from `Y` to `X`) and `N` (from `Y` to `Z`) whose rows sum to `1` and whose determinants are nonzero. The joint distribution matrices are `F_XY = M\u1d40 \u03a0_Y`, `F_YZ = \u03a0_Y N` and\u2026","title":"Additivity of the LogDet distance at a separating vertex"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/paralinear-chain-step","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.paralinear_chain_step","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#paralinear-chain-step","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"For a Markov chain `X \u2192 Y \u2192 Z` with `X` distributed as `p` (positive), transition matrices `A` (nonnegative) and `B` with unit row sums and nonzero determinants, `d(X, Z) = d(X, Y) + d(Y, Z)`, where `F_XZ = \u03a0_X A B`, `F_XY = \u03a0_X A` and `F_YZ = \u03a0_Y B` with `Y` distributed as `p A`.","title":"Additivity of the LogDet distance along a chain, one step"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/path-prod","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.pathProd","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#path-prod","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The product `M 0 * M 1 * \u22ef * M (n - 1)` of the transition matrices along the first `n` edges of a path.","title":"Product of transition matrices along a path"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/paralinear-path-prod","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.paralinear_pathProd","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#paralinear-path-prod","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Additivity of the LogDet distance along a path (Steel 1994; Lockhart et al. 1994): for a Markov chain `v\u2080 \u2192 v\u2081 \u2192 \u22ef \u2192 v_n` along a path, with positive initial distribution `p` and nonnegative nonsingular transition matrices `M i` with unit row sums, the LogDet distance between the ends is the sum of the distances of consecutive vertices. The joint distribution of `v\u2080, v_n` is `\u03a0_{v\u2080} M 0 \u22ef M (n - 1)`, and that of\u2026","title":"Additivity of the LogDet distance along a path"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/two-state","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.twoState","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#two-state","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The symmetric two-state model on one branch: the state changes with probability `p`. As a model, `0 \u2264 p \u2264 1/2`.","title":"Symmetric two-state model on a branch"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/quartet","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.quartet","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#quartet","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The quartet `((A, B), (C, D))` on the taxa `0 = A`, `1 = B`, `2 = C`, `3 = D`, with change probabilities `pA, pB, pC, pD` on the pendant edges and `pI` on the internal edge. The root sits at the `AB` end of the internal edge (the edge to the `AB` cherry vertex has change probability `0`); by the pulley principle its position does not matter.","title":"Four-taxon quartet tree model"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/site-obs","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.siteObs","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#site-obs","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The leaf weights of a site pattern `\u03c3`: the indicator of the observed state.","title":"Leaf weights of a site pattern"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/pattern-prob","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.patternProb","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#pattern-prob","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The probability of the site pattern `\u03c3` on the quartet, with the uniform root distribution (the stationary distribution of the symmetric model).","title":"Probability of a site pattern on a quartet"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/sum-pattern-prob","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.sum_patternProb","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#sum-pattern-prob","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The pattern probabilities sum to `1`, for all parameters.","title":"Site pattern probabilities sum to one"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/pattern-prob-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.patternProb_nonneg","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#pattern-prob-nonneg","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The pattern probabilities are nonnegative for change probabilities in `[0, 1]`.","title":"Nonnegativity of site pattern probabilities"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/supports","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.Supports","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#supports","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The site pattern `\u03c3` supports the split `{a, b} | {c, d}`: `a, b` agree, `c, d` agree, and the two pairs differ (a parsimony-informative pattern for that split).","title":"Site pattern supporting a split"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/split-freq","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.splitFreq","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#split-freq","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The expected frequency of the patterns supporting the split `{a, b} | {c, d}`.","title":"Expected frequency of patterns supporting a split"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/split-freq-wrong-sub-true","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.splitFreq_wrong_sub_true","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#split-freq-wrong-sub-true","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"With change probability `P` on the long non-sister branches to `A` and `C` and `Q` on the other three branches, the expected frequency of patterns supporting the wrong split `AC | BD` exceeds that of the true split `AB | CD` by `(1 - 2 Q) (P ^ 2 - Q (1 - Q))`, for all real `P, Q`. This is Felsenstein's equations (16)\u2013(17) (his four-species model has equal probabilities of change `0 \u2192 1` and reversion `1 \u2192 0`).","title":"Felsenstein (1978), the exact comparison"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/split-freq-wrong-sub-third","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.splitFreq_wrong_sub_third","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#split-freq-wrong-sub-third","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The wrong split `AC | BD` against the third split `AD | BC`: the difference of expected frequencies is `(1 - 2 Q) (P - Q) ^ 2`.","title":"Wrong split against third split frequency difference"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/split-freq-true-sub-third","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.splitFreq_true_sub_third","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#split-freq-true-sub-third","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The true split `AB | CD` against the third split `AD | BC`: the difference of expected frequencies is `Q (1 - 2 Q) (1 - 2 P)`.","title":"True split against third split frequency difference"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/quartet-shape","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.quartetShape","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#quartet-shape","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The rooted quartet `((a, b), (c, d))` with unit costs, for Fitch's algorithm.","title":"Rooted quartet with unit costs"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/parsimony-score","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.parsimonyScore","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#parsimony-score","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The Fitch parsimony score of the site pattern `\u03c3` on the quartet `((a, b), (c, d))`; it is the small-parsimony score by `Parsimony.fitch_score_eq`.","title":"Fitch parsimony score on a quartet"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/parsimony-score-sub","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.parsimonyScore_sub","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#parsimony-score-sub","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"On four taxa, the parsimony scores of two quartets differ only on informative patterns: the score on `AB | CD` minus the score on `AC | BD` is `1` on patterns supporting `AC | BD`, `-1` on patterns supporting `AB | CD`, and `0` otherwise; likewise for `AD | BC`.","title":"Difference of parsimony scores of two quartets"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/expected-score","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.expectedScore","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#expected-score","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The expected Fitch parsimony score of one site on the quartet `((a, b), (c, d))`.","title":"Expected Fitch parsimony score on a quartet"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/expected-score-wrong-lt","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.expectedScore_wrong_lt","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#expected-score-wrong-lt","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The Felsenstein zone (Felsenstein 1978): if `0 \u2264 Q < 1/2` and `P ^ 2 > Q (1 - Q)`, the wrong quartet `AC | BD` has strictly smaller expected parsimony score per site than both the true quartet `AB | CD` and the third quartet `AD | BC`. The statement is algebraic in `P`; the model has `0 \u2264 P \u2264 1/2`.","title":"The Felsenstein zone"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/ae-eventually-parsimony-wrong","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.ae_eventually_parsimony_wrong","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#ae-eventually-parsimony-wrong","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Parsimony is inconsistent in the Felsenstein zone (Felsenstein 1978): if the sites `\u03c3 i` are pairwise independent and identically distributed with the pattern distribution of the quartet, and `0 \u2264 Q < 1/2`, `P ^ 2 > Q (1 - Q)`, then almost surely, for all long enough alignments, the wrong quartet `AC | BD` has a strictly smaller total Fitch score than the true quartet `AB | CD` and the third quartet `AD | BC`.","title":"Parsimony is inconsistent in the Felsenstein zone"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/jc-path","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.jcPath","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#jc-path","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The JC transition matrix along a path whose branch lengths are listed in `ts`: the product of the branch transition matrices `jcProb \u03bc t`.","title":"Jukes\u2013Cantor transition matrix along a path"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/jc-path-eq","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.jcPath_eq","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#jc-path-eq","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The JC transition matrix along a path is the transition matrix of the total length.","title":"Chapman\u2013Kolmogorov along a path"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/jc-distance-jc-path","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.jcDistance_jcPath","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#jc-distance-jc-path","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"JC-corrected distances are path lengths (Jukes & Cantor 1969): the JC distance of the probability that the two ends of a path differ is `\u03bc` times the total branch length.","title":"JC-corrected distances are path lengths"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/p-hat","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.pHat","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#p-hat","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The empirical p-distance of the taxa `x, y` over the first `n` sites of the alignment `seq` (`seq i x` is the base of taxon `x` at site `i`): the proportion of sites where they differ.","title":"Empirical p-distance"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/jc-hat","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.jcHat","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#jc-hat","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"The JC-corrected empirical distance `jcDistance (pHat seq n x y)`.","title":"Jukes\u2013Cantor corrected empirical distance"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/njrecovers-within","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.NJRecoversWithin","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#njrecovers-within","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Neighbor joining recovers the tree within radius `r` of the dissimilarity `d`: every complete run of neighbor joining on a symmetric dissimilarity `D` with zero diagonal and `|D x y - d x y| < r` for distinct taxa forms exactly the splits of the internal edges of `(G, \u03b9)`. Atteson's theorem gives this for the path-length metric of a binary phylogenetic tree with `r` half the shortest internal edge length\u2026","title":"Neighbor joining recovers a tree within a radius"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/ae-tendsto-p-hat","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.ae_tendsto_pHat","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#ae-tendsto-p-hat","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"For pairwise independent, identically distributed sites, the empirical p-distance of every pair of taxa converges almost surely to the probability that the two taxa differ at a site.","title":"Strong law for the p-distances"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/ae-eventually-nj-run-splits-jc-hat","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.ae_eventually_njRun_splits_jcHat","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#ae-eventually-nj-run-splits-jc-hat","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Statistical consistency of neighbor joining under Jukes\u2013Cantor (Saitou & Nei 1987; Atteson 1999). Let `(G, \u03b9)` be a binary phylogenetic `X`-tree with path-length metric `d` for edge lengths `w`, and let the sites be pairwise independent and identically distributed, with the two-leaf marginals of the JC model with rate `\u03bc` and uniform root: for every path `p` from `x` to `y`,\u2026","title":"Statistical consistency of neighbor joining under Jukes\u2013Cantor"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/nj-recovers-within-of-le","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.njRecoversWithin_of_le","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#nj-recovers-within-of-le","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Let `d` be the path-length metric of a binary phylogenetic `X`-tree `(G, \u03b9)` with positive edge lengths `w` whose internal edges (no taxon at either end) have length at least `m`. For `\u03bc > 0`, neighbor joining recovers the tree within radius `\u03bc m / 2` of the scaled metric `\u03bc d`. This is `njRun_splits_of_abs_sub_lt` (Atteson 1999) for the edge lengths `\u03bc w`.","title":"Atteson's radius for the scaled tree metric"},{"id":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/ae-eventually-nj-run-splits-jc-hat-of-le","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.FelsensteinZone.ae_eventually_njRun_splits_jcHat_of_le","parent":46171,"result":true,"route":"../roadmap/Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei/index.html#ae-eventually-nj-run-splits-jc-hat-of-le","scope":"Biology/phylogenetics/pulley-principle-logdet-distances-and-felsenstei","status":"fully_proved","summary":"Statistical consistency of neighbor joining under Jukes\u2013Cantor (Saitou & Nei 1987; Atteson 1999), without a robustness hypothesis. Let `(G, \u03b9)` be a binary phylogenetic `X`-tree with path-length metric `d` for positive edge lengths `w`, whose internal edges have length at least `m > 0`, and let the sites be pairwise independent and identically distributed, with the two-leaf marginals of the JC model with rate\u2026","title":"Statistical consistency of neighbor joining under Jukes\u2013Cantor"},{"data_url":"scopes/Biology/phylogenetics/time-reversible-nucleotide-substitution-models.json","id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","title":"Time-reversible nucleotide substitution models"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/gtr-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.gtrRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#gtr-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The GTR rate matrix (Tavar\u00e9 1986) with base frequencies `\u03c0` and exchangeabilities `s`: `s i j * \u03c0 j` off the diagonal and minus the off-diagonal row sum on it. The diagonal of `s` is not used.","title":"General time-reversible (GTR) rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/sum-gtr-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.sum_gtrRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#sum-gtr-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The rows of the GTR matrix sum to `0`, for all parameters.","title":"Rows of the GTR rate matrix sum to zero"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-rate-matrix-gtr-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isRateMatrix_gtrRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-rate-matrix-gtr-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"With nonnegative base frequencies and nonnegative off-diagonal exchangeabilities, the GTR matrix is a rate matrix (the diagonal `s i i` is not used by `gtrRate`).","title":"GTR matrix is a rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-detailed-balance-gtr-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isDetailedBalance_gtrRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-detailed-balance-gtr-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"With symmetric exchangeabilities, the GTR matrix is in detailed balance with its base frequencies, `\u03c0 i Q i j = \u03c0 j Q j i` (Tavar\u00e9 1986).","title":"GTR is reversible"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/vec-mul-gtr-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.vecMul_gtrRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#vec-mul-gtr-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The base frequencies are invariant for the GTR generator: `\u03c0 Q = 0`.","title":"Base frequencies are invariant for the GTR generator"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/vec-mul-ctmc-transition-gtr-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.vecMul_ctmcTransition_gtrRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#vec-mul-ctmc-transition-gtr-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The base frequencies are stationary for the GTR transition function at every time.","title":"Base frequencies are stationary for the GTR chain"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-detailed-balance-ctmc-transition-gtr","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isDetailedBalance_ctmcTransition_gtrRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-detailed-balance-ctmc-transition-gtr","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"Detailed balance of the GTR transition function with the base frequencies, `\u03c0 i P t i j = \u03c0 j P t j i` for every `t` (time-reversibility of the GTR chain).","title":"Time-reversibility of the GTR chain"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/hky-exchangeability","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.hkyExchangeability","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#hky-exchangeability","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The HKY85 exchangeabilities: `\u03ba` for a transition (indices of equal parity: `A \u2194 G`, `C \u2194 T` in the order `A, C, G, T`), `1` for a transversion. The diagonal (also `\u03ba`) is not used by `gtrRate`.","title":"HKY85 exchangeabilities"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/hky-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.hkyRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#hky-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The HKY85 rate matrix (Hasegawa\u2013Kishino\u2013Yano 1985) with base frequencies `\u03c0` and transition/transversion ratio `\u03ba`: rate `\u03ba \u03c0 j` for a transition to `j`, `\u03c0 j` for a transversion.","title":"HKY85 rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/f81-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.f81Rate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#f81-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The F81 rate matrix (Felsenstein 1981): rate `\u03c0 j` to every other base `j`.","title":"F81 rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/k80-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.k80Rate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#k80-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The K80 rate matrix (Kimura 1980): equal base frequencies `1/4`, rate `\u03ba / 4` for a transition and `1 / 4` for each transversion.","title":"K80 rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-rate-matrix-hky-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isRateMatrix_hkyRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-rate-matrix-hky-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"With nonnegative base frequencies and `\u03ba \u2265 0`, the HKY85 matrix is a rate matrix.","title":"HKY85 matrix is a rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-rate-matrix-f81-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isRateMatrix_f81Rate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-rate-matrix-f81-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"With nonnegative base frequencies, the F81 matrix is a rate matrix.","title":"F81 matrix is a rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-rate-matrix-k80-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isRateMatrix_k80Rate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-rate-matrix-k80-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"With `\u03ba \u2265 0`, the K80 matrix is a rate matrix.","title":"K80 matrix is a rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-detailed-balance-hky-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isDetailedBalance_hkyRate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-detailed-balance-hky-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"HKY85 is in detailed balance with its base frequencies.","title":"Detailed balance of the HKY85 model"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-detailed-balance-f81-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isDetailedBalance_f81Rate","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-detailed-balance-f81-rate","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"F81 is in detailed balance with its base frequencies.","title":"Detailed balance of the F81 model"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/hky-rate-one","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.hkyRate_one","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#hky-rate-one","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"HKY85 with `\u03ba = 1` is F81.","title":"HKY85 with \u03ba = 1 is F81"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/gtr-rate-jukes-cantor","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.gtrRate_jukesCantor","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#gtr-rate-jukes-cantor","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"Equal base frequencies `1/4` and equal exchangeabilities `4 \u03bc / 3` give the Jukes\u2013Cantor rate matrix `jcRateMatrix \u03bc`.","title":"JC69 is the equal-rates GTR model"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/ctmc-transition-gtr-rate-jukes-cantor","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.ctmcTransition_gtrRate_jukesCantor","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#ctmc-transition-gtr-rate-jukes-cantor","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"The equal-rates GTR transition function is the Jukes\u2013Cantor transition probability `jcProb \u03bc t` (by `JukesCantor.jcProb_eq_exp`).","title":"Equal-rates GTR chain is the Jukes\u2013Cantor model"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/is-rate-matrix-jc-rate-matrix","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.isRateMatrix_jcRateMatrix","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#is-rate-matrix-jc-rate-matrix","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"For `\u03bc \u2265 0` the Jukes\u2013Cantor matrix is a rate matrix.","title":"Jukes\u2013Cantor matrix is a rate matrix"},{"id":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models/k80-rate-one","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SubstitutionModels.k80Rate_one","parent":46213,"result":true,"route":"../roadmap/Biology/phylogenetics/time-reversible-nucleotide-substitution-models/index.html#k80-rate-one","scope":"Biology/phylogenetics/time-reversible-nucleotide-substitution-models","status":"fully_proved","summary":"K80 with `\u03ba = 1` is JC69 with substitution rate `\u03bc = 3/4`.","title":"K80 with \u03ba = 1 is the Jukes\u2013Cantor model"},{"data_url":"scopes/Biology/phylogenetics/codon-substitution-models-and-dn-ds.json","id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","title":"Codon substitution models and dN/dS"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/codon","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.Codon","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#codon","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"A codon: three nucleotides, each in `Fin 4` (order `A, C, G, T`).","title":"Codon as a triple of nucleotides"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/diff-positions","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.diffPositions","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#diff-positions","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The positions at which two codons differ.","title":"Positions at which two codons differ"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/is-single-change","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.IsSingleChange","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#is-single-change","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"Two codons differ at exactly one position.","title":"Codons differing at exactly one position"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/is-transition-change","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.IsTransitionChange","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#is-transition-change","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"Every differing position is a transition (same parity: `A \u2194 G` or `C \u2194 T`).","title":"Codon change consisting of transitions"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/is-single-change-comm","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.isSingleChange_comm","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#is-single-change-comm","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"Differing at exactly one position is symmetric.","title":"Symmetry of single-position codon changes"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/is-transition-change-comm","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.isTransitionChange_comm","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#is-transition-change-comm","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"Being a transition change is symmetric.","title":"Symmetry of transition codon changes"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/gy-exchange","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.gyExchange","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#gy-exchange","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The Goldman\u2013Yang exchangeability between codons `i` and `j`: `0` unless they differ at one position; otherwise `\u03ba` for a transition (`1` for a transversion), times `\u03c9` if the amino acid changes.","title":"Goldman\u2013Yang codon exchangeability"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/gy-exchange-symm","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.gyExchange_symm","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#gy-exchange-symm","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The Goldman\u2013Yang exchangeabilities are symmetric.","title":"Symmetry of Goldman\u2013Yang exchangeabilities"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/gy-exchange-eq","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.gyExchange_eq","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#gy-exchange-eq","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The nonsynonymous exchangeabilities are linear in `\u03c9`, the synonymous ones do not involve it.","title":"Dependence of Goldman\u2013Yang exchangeabilities on \u03c9"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/gy-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.gyRate","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#gy-rate","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The Goldman\u2013Yang rate matrix (Goldman & Yang 1994): the GTR matrix with codon frequencies `\u03c0` and exchangeabilities `gyExchange`.","title":"Goldman\u2013Yang codon rate matrix"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/syn-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.synRate","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#syn-rate","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The stationary rate of synonymous substitutions per codon, `\u03c1_S = \u2211_{i \u2260 j, aa i = aa j} \u03c0_i Q_ij`. This is the stationary rate when `\u2211 i, \u03c0 i = 1`; for unnormalized weights it is scaled by `(\u2211 i, \u03c0 i) ^ 2`.","title":"Stationary rate of synonymous substitutions"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/nonsyn-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.nonsynRate","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#nonsyn-rate","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The stationary rate of nonsynonymous substitutions per codon, `\u03c1_N = \u2211_{aa i \u2260 aa j} \u03c0_i Q_ij`. This is the stationary rate when `\u2211 i, \u03c0 i = 1`; for unnormalized weights it is scaled by `(\u2211 i, \u03c0 i) ^ 2`.","title":"Stationary rate of nonsynonymous substitutions"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/syn-sites","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.synSites","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#syn-sites","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The number of synonymous sites per codon, counted under `\u03c9 = 1`: `S = 3 \u03c1_S(1) / (\u03c1_S(1) + \u03c1_N(1))`.","title":"Number of synonymous sites per codon"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/nonsyn-sites","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.nonsynSites","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#nonsyn-sites","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The number of nonsynonymous sites per codon, counted under `\u03c9 = 1`: `N = 3 \u03c1_N(1) / (\u03c1_S(1) + \u03c1_N(1))`.","title":"Number of nonsynonymous sites per codon"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/d-s","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.dS","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#d-s","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The synonymous rate per synonymous site, `dS = \u03c1_S(\u03c9) / S`.","title":"Synonymous substitution rate per synonymous site (dS)"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/d-n","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.dN","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#d-n","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The nonsynonymous rate per nonsynonymous site, `dN = \u03c1_N(\u03c9) / N`.","title":"Nonsynonymous substitution rate per nonsynonymous site (dN)"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/gy-rate-of-ne","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.gyRate_of_ne","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#gy-rate-of-ne","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"Off the diagonal the Goldman\u2013Yang rate is `gyExchange i j * \u03c0 j`.","title":"Off-diagonal entries of the Goldman\u2013Yang rate matrix"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/is-rate-matrix-gy-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.isRateMatrix_gyRate","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#is-rate-matrix-gy-rate","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"With nonnegative frequencies, `\u03ba \u2265 0` and `\u03c9 \u2265 0`, the Goldman\u2013Yang matrix is a rate matrix.","title":"Goldman\u2013Yang matrix is a rate matrix"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/is-detailed-balance-gy-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.isDetailedBalance_gyRate","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#is-detailed-balance-gy-rate","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"It is in detailed balance with its codon frequencies.","title":"The Goldman\u2013Yang chain is reversible"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/syn-rate-eq","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.synRate_eq","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#syn-rate-eq","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The synonymous rate does not depend on `\u03c9`: `\u03c1_S(\u03c9) = \u03c1_S(1)`.","title":"Synonymous rate does not depend on \u03c9"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/nonsyn-rate-eq","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.nonsynRate_eq","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#nonsyn-rate-eq","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"The nonsynonymous rate is linear in `\u03c9`: `\u03c1_N(\u03c9) = \u03c9 \u03c1_N(1)`.","title":"Nonsynonymous rate is linear in \u03c9"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/syn-sites-add-nonsyn-sites","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.synSites_add_nonsynSites","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#syn-sites-add-nonsyn-sites","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"There are three sites per codon: `S + N = 3` when `\u03c1_S(1) + \u03c1_N(1) \u2260 0`.","title":"Synonymous and nonsynonymous sites sum to three"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/d-n-div-d-s","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.dN_div_dS","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#d-n-div-d-s","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"`dN / dS = \u03c9` in the Goldman\u2013Yang model with sites counted under `\u03c9 = 1` (derived; see the module docstring for the site definition), provided `\u03c1_S(1) > 0` and `\u03c1_N(1) > 0`.","title":"`dN / dS = \u03c9`"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/syn-rate-pos-of-mem","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.synRate_pos_of_mem","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#syn-rate-pos-of-mem","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"`\u03c1_S(1) > 0` if the frequencies are nonnegative, `\u03ba > 0`, and some synonymous single change joins two codons of positive frequency.","title":"Positivity of the synonymous rate"},{"id":"Biology/phylogenetics/codon-substitution-models-and-dn-ds/nonsyn-rate-pos-of-mem","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.CodonModel.nonsynRate_pos_of_mem","parent":46235,"result":true,"route":"../roadmap/Biology/phylogenetics/codon-substitution-models-and-dn-ds/index.html#nonsyn-rate-pos-of-mem","scope":"Biology/phylogenetics/codon-substitution-models-and-dn-ds","status":"fully_proved","summary":"`\u03c1_N(1) > 0` if the frequencies are nonnegative, `\u03ba > 0`, and some nonsynonymous single change joins two codons of positive frequency.","title":"Positivity of the nonsynonymous rate"},{"data_url":"scopes/Biology/phylogenetics/tree-metrics-and-the-four-point-condition.json","id":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/tree-metrics-and-the-four-point-condition/index.html","scope":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition","status":"fully_proved","title":"Tree metrics and the four-point condition"},{"id":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition/is-tree-metric","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.Buneman.IsTreeMetric","parent":46261,"result":true,"route":"../roadmap/Biology/phylogenetics/tree-metrics-and-the-four-point-condition/index.html#is-tree-metric","scope":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition","status":"fully_proved","summary":"`d` is a tree metric: there are a finite tree `G` on a vertex type `V`, positive edge lengths `w`, and a placement `\u03b9` of the taxa on vertices such that `d x y` is the total length of the (unique) path from `\u03b9 x` to `\u03b9 y`.","title":"Tree metric"},{"id":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition/four-point-condition","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.Buneman.FourPointCondition","parent":46261,"result":true,"route":"../roadmap/Biology/phylogenetics/tree-metrics-and-the-four-point-condition/index.html#four-point-condition","scope":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition","status":"fully_proved","summary":"The four-point condition: `d x y + d z t \u2264 max (d x z + d y t) (d x t + d y z)`.","title":"Four-point condition"},{"id":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition/four-point-condition-2","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.Buneman.IsTreeMetric.fourPointCondition","parent":46261,"result":true,"route":"../roadmap/Biology/phylogenetics/tree-metrics-and-the-four-point-condition/index.html#four-point-condition-2","scope":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition","status":"fully_proved","summary":"Tree metrics satisfy the four-point condition (the easy direction of Buneman's theorem).","title":"Tree metrics satisfy the four-point condition"},{"id":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition/is-tree-metric-of-four-point-condition","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.Buneman.isTreeMetric_of_fourPointCondition","parent":46261,"result":true,"route":"../roadmap/Biology/phylogenetics/tree-metrics-and-the-four-point-condition/index.html#is-tree-metric-of-four-point-condition","scope":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition","status":"fully_proved","summary":"A symmetric dissimilarity on a finite set that vanishes on the diagonal and satisfies the four-point condition is a tree metric.","title":"Buneman's theorem, converse direction"},{"id":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition/is-tree-metric-iff-four-point-condition","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.Buneman.isTreeMetric_iff_fourPointCondition","parent":46261,"result":true,"route":"../roadmap/Biology/phylogenetics/tree-metrics-and-the-four-point-condition/index.html#is-tree-metric-iff-four-point-condition","scope":"Biology/phylogenetics/tree-metrics-and-the-four-point-condition","status":"fully_proved","summary":"A symmetric dissimilarity on a finite set that vanishes on the diagonal is a tree metric if and only if it satisfies the four-point condition.","title":"Buneman's four-point theorem"},{"data_url":"scopes/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma.json","id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","title":"Splits, compatibility, ultrametrics and UPGMA"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/is-hierarchy","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.IsHierarchy","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#is-hierarchy","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"A hierarchy on a finite set `S`: a family of nonempty subsets of `S` containing `S` and every singleton, any two members nested or disjoint. These are the cluster systems of the rooted phylogenetic trees on `S` (Semple & Steel 2003, Section 3.5).","title":"Hierarchy on a finite set"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/is-hierarchy-2","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.IsBinaryHierarchy.isHierarchy","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#is-hierarchy-2","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"A binary hierarchy is a hierarchy.","title":"A binary hierarchy is a hierarchy"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/is-hierarchy-completion","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.isHierarchy_completion","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#is-hierarchy-completion","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Adding `S` and the singletons to a family of pairwise nested-or-disjoint nonempty subsets of a nonempty `S` gives a hierarchy.","title":"Pairwise nested-or-disjoint clusters form a hierarchy"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/exists-is-binary-hierarchy-superset","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.exists_isBinaryHierarchy_superset","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#exists-is-binary-hierarchy-superset","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Every family of pairwise nested-or-disjoint nonempty subsets of a nonempty `S` is contained in a binary hierarchy on `S`, the cluster set of a rooted binary phylogenetic tree on `S` (Semple & Steel 2003, Proposition 3.5.2 and the refinement of a rooted tree to a binary one).","title":"Hierarchies are refined by binary trees"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/card-add-one","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.IsBinaryHierarchy.card_add_one","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#card-add-one","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"A binary hierarchy on `S` has `2 |S| - 1` members: the `|S|` singletons and `|S| - 1` internal clusters (stated without subtraction).","title":"Number of members of a binary hierarchy"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/compatible","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.Compatible","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#compatible","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The splits `A | A\u1d9c` and `B | B\u1d9c` are compatible: one of `A \u2229 B`, `A \u2229 B\u1d9c`, `A\u1d9c \u2229 B`, `A\u1d9c \u2229 B\u1d9c` is empty (Semple & Steel 2003, Section 3.1).","title":"Compatible splits"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/symm","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.Compatible.symm","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#symm","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Compatibility is symmetric.","title":"Symmetry of split compatibility"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/compatible-compl-left","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.compatible_compl_left","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#compatible-compl-left","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Compatibility depends only on the split, not on the side used to store it.","title":"Compatibility is unchanged by complementing the first split"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/compatible-compl-right","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.compatible_compl_right","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#compatible-compl-right","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Compatibility depends only on the split, not on the side used to store it.","title":"Compatibility is unchanged by complementing the second split"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/compatible-iff-nested-or-disjoint","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.compatible_iff_nested_or_disjoint","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#compatible-iff-nested-or-disjoint","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"For the sides not containing a fixed taxon `a`, compatibility is being nested or disjoint: splits seen from a root are clusters.","title":"Compatible clusters are nested or disjoint"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/four-gametes","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.FourGametes","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#four-gametes","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The binary characters `\u03c7` and `\u03c8` show all four gametes `(0,0), (0,1), (1,0), (1,1)` on the taxa.","title":"Four-gamete condition for binary characters"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/char-split","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.charSplit","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#char-split","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The split of a binary character: the taxa in state `true`.","title":"Split of a binary character"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/compatible-iff-not-four-gametes","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.compatible_iff_not_fourGametes","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#compatible-iff-not-four-gametes","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Two binary characters give compatible splits iff they do not show all four gametes (Buneman 1971; Estabrook, Johnson & McMorris 1976).","title":"The four-gamete test"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/split-set","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.splitSet","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#split-set","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The split set of `T` computed from the leaf `a`: both sides of every cluster of `T` seen from `a`. It equals `edgeSplits T` for every `a` (`splitSet_eq_edgeSplits`).","title":"Split set of a tree computed from a leaf"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/not-mem-of-mem-cluster-set","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.notMem_of_mem_clusterSet","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#not-mem-of-mem-cluster-set","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The clusters seen from `a` do not contain `a`.","title":"Clusters seen from a leaf omit that leaf"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/edge-side","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.edgeSide","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#edge-side","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The side of the edge `p \u2013 v` containing `v`: the taxa whose leaf is closer to `v` than to `p`.","title":"Side of a tree edge containing a vertex"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/edge-splits","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.edgeSplits","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#edge-splits","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The edge splits of `T`: the sides of all its edges (both sides of every edge).","title":"Edge splits of a tree"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/compl-edge-side","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.compl_edgeSide","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#compl-edge-side","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The two sides of an edge are complementary.","title":"The two sides of a tree edge are complementary"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/cluster-eq-edge-side","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.cluster_eq_edgeSide","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#cluster-eq-edge-side","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The cluster of a vertex `v` seen from `a` is the side of `v` of the edge from `v` to its neighbor `p` toward `a`.","title":"Clusters seen from a leaf are edge sides"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/split-set-eq-edge-splits","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.splitSet_eq_edgeSplits","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#split-set-eq-edge-splits","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The split set computed from any leaf is the set of edge splits.","title":"Split set from any leaf equals the edge splits"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/split-set-eq-split-set","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.splitSet_eq_splitSet","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#split-set-eq-split-set","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The split set does not depend on the leaf used to compute it.","title":"The split set does not depend on the leaf used to compute it"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/compatible-of-mem-split-set","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.compatible_of_mem_splitSet","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#compatible-of-mem-split-set","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Any two splits of one tree are compatible.","title":"Splits computed from a leaf are pairwise compatible"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/compatible-of-mem-edge-splits","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.compatible_of_mem_edgeSplits","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#compatible-of-mem-edge-splits","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Any two splits of one tree are compatible.","title":"Edge splits of a tree are pairwise compatible"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/splits-equivalence","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.splits_equivalence","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#splits-equivalence","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Splits-equivalence theorem, binary form (Buneman 1971; Semple & Steel 2003, Theorem 3.1.4): on at least two taxa, a family `F` of proper splits (both sides nonempty) is contained in the split set of some unrooted binary phylogenetic tree iff its members are pairwise compatible. The tree is a binary refinement of the tree whose splits are exactly `F` and the trivial ones.","title":"Splits-equivalence theorem"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/is-perfect-phylogeny","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.IsPerfectPhylogeny","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#is-perfect-phylogeny","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"`T` is a perfect phylogeny for the binary character matrix `M` (taxa by characters): the split of every non-constant column is a split of `T`, i.e. every character changes state on exactly one edge.","title":"Perfect phylogeny for a binary character matrix"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/exists-perfect-phylogeny-iff","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.exists_perfectPhylogeny_iff","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#exists-perfect-phylogeny-iff","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Perfect phylogeny and the four-gamete condition (Estabrook, Johnson & McMorris 1976; Gusfield 1991): a binary character matrix on at least two taxa has a perfect phylogeny iff no two of its columns show all four gametes.","title":"Perfect phylogeny and the four-gamete condition"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/rf-dist","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.rfDist","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#rf-dist","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The Robinson\u2013Foulds distance: the number of splits in exactly one of the two trees. Each split is counted once, by its side away from the leaf `a` (its cluster seen from `a`); trivial splits are common to all trees and cancel. The value does not depend on `a` (`rfDist_eq_rfDist`).","title":"Robinson\u2013Foulds distance"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/two-mul-rf-dist","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.two_mul_rfDist","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#two-mul-rf-dist","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Each split is counted twice in the symmetric difference of the split sets (once per side), so the Robinson\u2013Foulds distance is half of it.","title":"Robinson\u2013Foulds distance as half the split symmetric difference"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/rf-dist-eq-rf-dist","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.rfDist_eq_rfDist","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#rf-dist-eq-rf-dist","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The Robinson\u2013Foulds distance does not depend on the leaf used to compute it.","title":"Robinson\u2013Foulds distance is independent of the base leaf"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/rf-dist-self","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.rfDist_self","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#rf-dist-self","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The Robinson\u2013Foulds distance from a tree to itself is zero.","title":"Robinson\u2013Foulds distance from a tree to itself is zero"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/rf-dist-comm","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.rfDist_comm","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#rf-dist-comm","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The Robinson\u2013Foulds distance is symmetric.","title":"Symmetry of the Robinson\u2013Foulds distance"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/rf-dist-triangle","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.rfDist_triangle","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#rf-dist-triangle","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The Robinson\u2013Foulds distance satisfies the triangle inequality.","title":"Triangle inequality for the Robinson\u2013Foulds distance"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/rf-dist-eq-zero-iff","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.rfDist_eq_zero_iff","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#rf-dist-eq-zero-iff","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The Robinson\u2013Foulds distance vanishes exactly on isomorphic trees (Robinson & Foulds 1981).","title":"Robinson\u2013Foulds distance vanishes exactly on isomorphic trees"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/rf-dist-congr","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.rfDist_congr","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#rf-dist-congr","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The Robinson\u2013Foulds distance is invariant under isomorphism, so it is a metric on the trees up to isomorphism.","title":"Isomorphism invariance of the Robinson\u2013Foulds distance"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/rf-dist-le","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.rfDist_le","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#rf-dist-le","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"On `n \u2265 3` taxa it is at most `2 (n - 3)`, the number of nontrivial splits of two binary trees (Robinson & Foulds 1981).","title":"Maximum of the Robinson\u2013Foulds distance"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/majority-splits","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.majoritySplits","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#majority-splits","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The majority splits of a finite family of trees `T i`: the clusters, seen from `a`, of more than half of the trees (repetitions in the family count).","title":"Majority splits of a family of trees"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/nested-or-disjoint-of-mem-majority-split","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.nested_or_disjoint_of_mem_majoritySplits","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#nested-or-disjoint-of-mem-majority-split","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Majority splits are compatible (Margush & McMorris 1981): two clusters each in more than half of the trees are both clusters of a common tree (pigeonhole), hence nested or disjoint.","title":"Majority splits are compatible"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/exists-tree-majority-splits-subset","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.exists_tree_majoritySplits_subset","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#exists-tree-majority-splits-subset","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The majority splits are clusters of one unrooted binary phylogenetic tree (a binary refinement of the majority-rule consensus tree).","title":"The majority-rule consensus tree"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/bootstrap-support","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.bootstrapSupport","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#bootstrap-support","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Bootstrap support (Felsenstein 1985) of the split with side `A` away from `a`, for a tree estimator `est` and an alignment `M` of `m` sites: the fraction of the `m ^ m` site resamplings with replacement `w : Fin m \u2192 Fin m` (uniform draws, whose site counts are multinomial) for which the tree estimated from the resampled alignment `M \u2218 w` has the cluster `A`. This is the exact bootstrap proportion; Felsenstein's\u2026","title":"Bootstrap support"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/bootstrap-support-mem-icc","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.bootstrapSupport_mem_Icc","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#bootstrap-support-mem-icc","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Bootstrap support is a frequency: it lies in `[0, 1]`.","title":"Bootstrap support lies in the unit interval"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/one-half-lt-bootstrap-support-iff","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.one_half_lt_bootstrapSupport_iff","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#one-half-lt-bootstrap-support-iff","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"A split has bootstrap support above one half iff it is a majority split of the bootstrap replicate trees; so the splits with support above one half are pairwise compatible and form one tree (`exists_tree_majoritySplits_subset`).","title":"Bootstrap support above one half means a majority split"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/is-ultrametric","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.IsUltrametric","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#is-ultrametric","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"`d` is an ultrametric on the taxa: symmetric, zero exactly on the diagonal, and satisfying the three-point condition `d x z \u2264 max (d x y) (d y z)` (equivalently, of the three distances among any three taxa the two largest are equal).","title":"Ultrametric on taxa"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/is-dated-tree-metric","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.IsDatedTreeMetric","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#is-dated-tree-metric","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"`d` is the metric of a dated rooted tree, encoded by its clusters: a hierarchy `H` on the taxa with dates `h` (`0` at the leaves, strictly increasing from a cluster to any cluster strictly containing it) such that `d x y` is the date of the smallest cluster containing `x` and `y` (their most recent common ancestor). With edge lengths `(h D - h C) / 2` from each cluster `C` to its parent `D`, `d x y` is the path\u2026","title":"Metric of a dated rooted tree"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/four-point-condition","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.IsUltrametric.fourPointCondition","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#four-point-condition","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Ultrametrics satisfy the four-point condition (Semple & Steel 2003, Section 7.2), so every ultrametric on a finite set is a tree metric (`IsUltrametric.isTreeMetric`).","title":"Ultrametrics satisfy the four-point condition"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/is-tree-metric","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.IsUltrametric.isTreeMetric","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#is-tree-metric","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Every ultrametric on a finite set is a tree metric (via Buneman's four-point theorem).","title":"Every ultrametric is a tree metric"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/is-ultrametric-iff-is-dated-tree-metric","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.isUltrametric_iff_isDatedTreeMetric","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#is-ultrametric-iff-is-dated-tree-metric","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Ultrametrics are dated tree metrics (Semple & Steel 2003, Theorem 7.2.5): on a nonempty finite set of taxa, a dissimilarity satisfies the three-point condition (and is a metric) iff it is the metric of a dated rooted tree.","title":"Ultrametrics are dated tree metrics"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/avg-dist","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.avgDist","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#avg-dist","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The average linkage between two clusters: the mean of `d x y` over `x \u2208 C`, `y \u2208 D`.","title":"Average linkage between clusters"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/upgma-reaches","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.UpgmaReaches","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#upgma-reaches","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The states reachable by UPGMA (Sokal & Michener 1958) on `d`: a partition `P` of the taxa into blocks and the heights `c x y` recorded so far. It starts from the singletons; each step merges two distinct blocks `C`, `D` of least average linkage (ties broken arbitrarily) and records `avgDist d C D` as the height at which every `x \u2208 C` and `y \u2208 D` are joined (the distance between them in the output tree; their common\u2026","title":"States reachable by UPGMA"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/upgma-exact","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.upgma_exact","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#upgma-exact","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"In every UPGMA run on an ultrametric, the recorded height of two taxa in the same block is their distance. In particular a completed run (one block) reconstructs `d`, i.e. the dated tree, exactly (Semple & Steel 2003, Section 7.2).","title":"UPGMA is exact on ultrametrics"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/exists-upgma-reaches-univ","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.exists_upgmaReaches_univ","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#exists-upgma-reaches-univ","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"UPGMA always completes: from the singletons it reaches the single block `univ`.","title":"UPGMA always completes"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/triplet","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.Triplet","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#triplet","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"A rooted triplet `xy|z`: `x` and `y` are closer to each other than to `z`.","title":"Rooted triplet"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/displays-triplet","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.DisplaysTriplet","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#displays-triplet","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The cluster system `H` displays the triplet `xy|z`: some cluster contains `x` and `y` but not `z`.","title":"Cluster system displaying a triplet"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/has-aho-cut","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.HasAhoCut","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#has-aho-cut","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"The Aho graph of the triplets `R` on `S'` (vertices `S'`, an edge `x y` for each `xy|z \u2208 R` with `x, y, z \u2208 S'`) has a cut: a partition of `S'` into two nonempty parts with no edge across, i.e. the graph is disconnected.","title":"Cut in the Aho graph"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/eq-of-displays-triplet-iff","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.eq_of_displaysTriplet_iff","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#eq-of-displays-triplet-iff","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"A rooted binary tree is determined by its triplets (Semple & Steel 2003, Theorem 6.4.1): two binary hierarchies on `S` displaying the same triplets are equal.","title":"A rooted binary tree is determined by its triplets"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/build-spec","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.build_spec","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#build-spec","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"Correctness of BUILD (Aho, Sagiv, Szymanski & Ullman 1981; Semple & Steel 2003, Section 6.4): a set `R` of triplets on a nonempty `S`, none of the degenerate form `xx|x` (which no tree displays and no cut detects), is displayed by a rooted binary phylogenetic tree on `S` iff for every subset `S'` of at least two taxa the Aho graph of `R` on `S'` is disconnected. BUILD splits along the components and recurses; the\u2026","title":"Correctness of BUILD"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/displays-quartet","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.DisplaysQuartet","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#displays-quartet","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"`T` displays the quartet `xy|zw`: some edge split of `T` has `x, y` on one side and `z, w` on the other.","title":"Tree displaying a quartet"},{"id":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/iso-of-displays-quartet-iff","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.Splits.iso_of_displaysQuartet_iff","parent":46267,"result":true,"route":"../roadmap/Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma/index.html#iso-of-displays-quartet-iff","scope":"Biology/phylogenetics/splits-compatibility-ultrametrics-and-upgma","status":"fully_proved","summary":"An unrooted binary tree is determined by its quartets (Semple & Steel 2003, Corollary 6.3.8): two trees displaying the same quartets are isomorphic. Already the quartets containing the taxon `a` suffice; they are the rooted triplets seen from `a`.","title":"An unrooted binary tree is determined by its quartets"},{"data_url":"scopes/Biology/phylogenetics/evolutionary-conservation-scores.json","id":"Biology/phylogenetics/evolutionary-conservation-scores","kind":"scope","navigation":"module","parent":46006,"result":false,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","title":"Evolutionary conservation scores"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/mcs-approx","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.MotifConservation.mcsApprox","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#mcs-approx","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The conserved count `k` of a motif with `N` occurrences, standardized under `Bin(N, p)`: `(k - N p) / \u221a(N p (1 - p))`. This is an approximation to Xie et al.'s MCS, not their exact score.","title":"Binomial z-score approximation to the motif conservation score"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/mcs-approx-strict-mono","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.MotifConservation.mcsApprox_strictMono","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#mcs-approx-strict-mono","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The MCS approximation is strictly increasing in the conserved count, for `N \u2265 1` and `p \u2208 (0, 1)`.","title":"Motif conservation score approximation is strictly increasing"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/mcs-approx-sum-eq","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.MotifConservation.mcsApprox_sum_eq","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#mcs-approx-sum-eq","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The MCS approximation of the number of conserved occurrences among the first `n` is the de Moivre\u2013Laplace standardized sum.","title":"Motif conservation score approximation as a de Moivre\u2013Laplace sum"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/tendsto-prob-le-mcs-approx","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.MotifConservation.tendsto_prob_le_mcsApprox","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#tendsto-prob-le-mcs-approx","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"If the conservation indicators of the occurrences are i.i.d. Bernoulli(`p`) with `0 < p < 1`, then for every `z`, `P(z \u2264 MCS approximation of the first n occurrences)` tends to the standard normal mass of `[z, \u221e)`.","title":"De Moivre\u2013Laplace for the MCS approximation"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/siphy-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.siphyRate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#siphy-rate","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The SiPhy rate matrix `Q a b = \u03c9 R a b \u03c0 b` for `a \u2260 b` (Garber et al. 2009, Eq. 10), with rows summing to zero: `\u03c9` times the GTR matrix with exchangeabilities `R` and base frequencies `\u03c0`.","title":"The SiPhy rate matrix"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/is-rate-matrix-siphy-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.isRateMatrix_siphyRate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#is-rate-matrix-siphy-rate","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"With `\u03c9 \u2265 0`, nonnegative `R` and nonnegative `\u03c0`, the SiPhy matrix is a rate matrix.","title":"SiPhy matrix is a rate matrix"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/is-detailed-balance-siphy-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.isDetailedBalance_siphyRate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#is-detailed-balance-siphy-rate","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"For symmetric `R` it is in detailed balance with `\u03c0`, for every `\u03c9`.","title":"`\u03c0` is the equilibrium of the SiPhy matrix"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/expected-log-lik","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.expectedLogLik","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#expected-log-lik","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The expected complete-data log-likelihood of a rate matrix `Q` and root distribution `\u03c0` (Garber et al. 2009, Eq. 6): `\u2211_{a \u2260 b} n a b log Q a b + \u2211 a, \u03c4 a Q a a + \u2211 b, r b log \u03c0 b`, for expected substitution counts `n`, expected state durations `\u03c4` and the root posterior `r`.","title":"The expected complete-data log-likelihood"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/total-count","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.totalCount","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#total-count","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The expected total number of substitutions `N = \u2211_{a \u2260 b} n a b`.","title":"Expected total number of substitutions"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/in-count","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.inCount","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#in-count","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The weight `c b = \u2211_{a \u2260 b} n a b + r b` of state `b`: substitutions into `b` plus the root posterior of `b`.","title":"Weight of a state: substitutions into it plus root posterior"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/exit-weight","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.exitWeight","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#exit-weight","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The time-weighted exchange rate into `b`, `e b = \u2211_{a \u2260 b} \u03c4 a R a b`.","title":"Time-weighted exchange rate into a state"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/pi-objective","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.piObjective","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#pi-objective","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The part of Eq. 6 that depends on `\u03c0` (for fixed `\u03c9`): `\u2211 b, (c b * log (\u03c0 b) - d b * \u03c0 b)`.","title":"Part of the SiPhy objective depending on the constraint vector"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/exchange-term","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.exchangeTerm","parent":46325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Constant "],["math","\\(n \\log R\\)"],["text"," term of the SiPhy objective"]],"source":"Constant $n \\log R$ term of the SiPhy objective"},"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#exchange-term","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The `n log R` constant of Eq. 6.","title":"Constant $n \\log R$ term of the SiPhy objective"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/expected-log-lik-siphy-rate","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.expectedLogLik_siphyRate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#expected-log-lik-siphy-rate","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"For `\u03c9 > 0`, positive off-diagonal `R` and positive `\u03c0`, `expectedLogLik = N log \u03c9 + \u2211_{a \u2260 b} n a b log R a b + piObjective c (\u03c9 e) \u03c0`.","title":"Eq. 6 for the SiPhy matrix"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/omega-update","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.omegaUpdate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#omega-update","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The SiPhy rate update (Eq. 7, for fixed `\u03c0`): `\u03c9 = N / \u2211 b, e b \u03c0 b`.","title":"The SiPhy rate update"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/expected-log-lik-le-omega-update","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.expectedLogLik_le_omegaUpdate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#expected-log-lik-le-omega-update","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The rate update maximizes Eq. 6 over `\u03c9 > 0` for a fixed positive `\u03c0`, when the expected number of substitutions `N` and the time-weighted rate `\u2211 b, e b \u03c0 b` are positive.","title":"The rate update maximizes Eq. 6 over `\u03c9 > 0`"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/expected-log-lik-lt-omega-update","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.expectedLogLik_lt_omegaUpdate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#expected-log-lik-lt-omega-update","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"Any other `\u03c9 > 0` gives a strictly smaller value of Eq. 6.","title":"The rate update is the unique maximizer"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/exists-unique-lagrange","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.existsUnique_lagrange","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#exists-unique-lagrange","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"For positive `c` and nonnegative `d` on a nonempty finite alphabet there is exactly one `\u03b3` with `d b + \u03b3 > 0` for all `b` and `\u2211 b, c b / (d b + \u03b3) = 1`.","title":"The Lagrange multiplier of the `\u03c0` update exists and is unique"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/pi-objective-lt","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.piObjective_lt","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#pi-objective-lt","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The `\u03c0` update is the unique maximizer of `piObjective c d` over the positive probability vectors: with `\u03b3` as in `existsUnique_lagrange`, every other positive `\u03c0` summing to one has a strictly smaller objective than `b \u21a6 c b / (d b + \u03b3)`.","title":"The `\u03c0` update is the unique maximizer"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/pi-update","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.piUpdate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#pi-update","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The SiPhy constraint-vector update (Eq. 8, for fixed `\u03c9`): `\u03c0 b = c b / (\u03c9 e b + \u03b3)`, with `\u03b3` the Lagrange multiplier of `existsUnique_lagrange`.","title":"The SiPhy constraint-vector update"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/expected-log-lik-lt-pi-update","kind":"node","lean":"MathlibExt.Biology.Phylogenetics.SiPhy.expectedLogLik_lt_piUpdate","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#expected-log-lik-lt-pi-update","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The constraint-vector update is the unique maximizer of Eq. 6 over the positive probability vectors `\u03c0` (the rate matrix `\u03c9 R \u03c0` and the root distribution both use `\u03c0`), for fixed `\u03c9 > 0`, positive off-diagonal `R` and positive weights `c`.","title":"The constraint-vector update is the unique maximizer of Eq. 6"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/separates","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.Separates","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#separates","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"Whether the leaf set `S` meets both sides of an edge with leaf set `B` below it, inside a tree with leaf set `U`: `S \u2229 B` and `S \u2229 (U \\ B)` are nonempty.","title":"Leaf set meeting both sides of a tree edge"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/nonneg-weights","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.NonnegWeights","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#nonneg-weights","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"Edge lengths are nonnegative: every dart's undirected edge has nonnegative length.","title":"Nonnegative edge lengths of a tree"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/total-length","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.totalLength","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#total-length","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The total branch length: half the sum of the edge lengths over the darts, since each undirected edge is traversed once in each direction.","title":"Total branch length of a tree"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.bls","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The Branch Length Score (unnormalized): the total branch length of the smallest subtree of `T` connecting the leaves `S`, as half the sum over the darts of the edge length times the indicator that `S` meets both sides (`edgeSide`) of the dart's edge.","title":"The Branch Length Score"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-frac","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.blsFrac","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-frac","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The normalized Branch Length Score, `bls S T w / totalLength T w`, as reported by Kheradpour et al. (a fraction of the total branch length).","title":"Normalized branch length score"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.bls_nonneg","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-nonneg","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The BLS is nonnegative for nonnegative edge lengths.","title":"Branch length score is nonnegative for nonnegative edge lengths"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-le-total-length","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.bls_le_totalLength","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-le-total-length","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The BLS is at most the total branch length.","title":"Branch length score is at most the tree length for nonnegative lengths"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-mono","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.bls_mono","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-mono","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"With nonnegative edge lengths, more conserving species never lower the BLS.","title":"Monotonicity"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-empty","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.bls_empty","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-empty","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The BLS of the empty set is `0`.","title":"Branch length score of the empty set is zero"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-singleton","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.bls_singleton","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-singleton","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"A single species has BLS `0` (a motif present in one genome is not conserved).","title":"A single species has BLS `0`"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-leaves","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.bls_leaves","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-leaves","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The BLS of the set of all leaves is the total branch length.","title":"All species give the whole tree"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-union-le","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.bls_union_le","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-union-le","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"For sets of species `S`, `S'` that share a species, the subtree connecting `S \u222a S'` is at most as long as the two subtrees together.","title":"Subadditivity"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-frac-nonneg","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.blsFrac_nonneg","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-frac-nonneg","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The normalized BLS is nonnegative.","title":"Normalized branch length score is nonnegative for nonnegative lengths"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/bls-frac-le-one","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.blsFrac_le_one","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#bls-frac-le-one","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The normalized BLS is at most `1` when the tree has positive total length.","title":"Normalized branch length score is at most one for nonnegative lengths"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/edge-side-of-leaf","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.edgeSide_of_leaf","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#edge-side-of-leaf","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"The taxa whose leaf is closer to the leaf vertex than to its neighbor are exactly that leaf's taxon.","title":"The side of a pendant edge is the singleton of its leaf"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/dart-sum-edge","kind":"node","lean":"Cslib.Bioinformatics.Phylogenetics.BranchLengthScore.dart_sum_edge","parent":46325,"result":true,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/index.html#dart-sum-edge","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"fully_proved","summary":"A sum over darts of a function of the undirected edge is twice the sum over the edges: the two darts of each edge contribute equally.","title":"Sum over darts is twice the sum over edges"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/binom-tail","kind":"node","parent":46325,"result":false,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/binom-tail.html","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"can_state","summary":"The upper tail `P(X \u2265 k) = \u2211_{i = k}^{n} C(n, i) p^i (1 - p)^(n - i)` of the binomial law `Bin(n, p)`.","title":"Upper tail of the binomial distribution"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/binom-tail-zero","kind":"node","parent":46325,"result":false,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/binom-tail-zero.html","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"can_state","summary":"`P(X \u2265 0) = 1`, for every real `p` (the binomial theorem).","title":"Binomial upper tail at zero equals one"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/binom-tail-of-lt","kind":"node","parent":46325,"reading_title":{"format":"inline-parts-v1","parts":[["text","Binomial upper tail vanishes beyond "],["math","\\(n\\)"]],"source":"Binomial upper tail vanishes beyond $n$"},"result":false,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/binom-tail-of-lt.html","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"can_state","summary":"`P(X \u2265 k) = 0` for `k > n`.","title":"Binomial upper tail vanishes beyond $n$"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/binom-tail-succ-le","kind":"node","parent":46325,"result":false,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/binom-tail-succ-le.html","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"can_state","summary":"The tail is nonincreasing in `k` for `p \u2208 [0, 1]`.","title":"The tail is nonincreasing in `k`"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/has-deriv-at-binom-tail","kind":"node","parent":46325,"result":false,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/has-deriv-at-binom-tail.html","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"can_state","summary":"Derivative of the binomial upper tail in the success probability: for `1 \u2264 k \u2264 n`, `d/dp P(Bin(n, p) \u2265 k) = n C(n - 1, k - 1) p^(k-1) (1-p)^(n-k)` at every real `p` (the sum telescopes).","title":"Derivative of the binomial upper tail"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/monotone-on-binom-tail","kind":"node","parent":46325,"result":false,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/monotone-on-binom-tail.html","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"can_state","summary":"The binomial upper tail is nondecreasing in `p` on `[0, 1]`, for every `n` and `k`.","title":"The binomial upper tail is nondecreasing in `p` on `(0, 1)`"},{"id":"Biology/phylogenetics/evolutionary-conservation-scores/strict-mono-on-binom-tail","kind":"node","parent":46325,"result":false,"route":"../roadmap/Biology/phylogenetics/evolutionary-conservation-scores/strict-mono-on-binom-tail.html","scope":"Biology/phylogenetics/evolutionary-conservation-scores","status":"can_state","summary":"The binomial upper tail is strictly increasing in `p` on `[0, 1]` when `1 \u2264 k \u2264 n`.","title":"The binomial upper tail is strictly increasing in `p` on `(0, 1)`"},{"data_url":"scopes/Biology/statistical-genetics.json","id":"Biology/statistical-genetics","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/statistical-genetics/index.html","scope":"Biology/statistical-genetics","status":"fully_proved","summary":"Statistical methods for genome-wide association studies, heritability estimation from summary statistics, fine-mapping, Mendelian randomization and gene constraint scores.","title":"Statistical genetics"},{"data_url":"scopes/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization.json","id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","title":"Bayesian fine-mapping and colocalization"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/wakefield-abf","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.wakefieldABF","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#wakefield-abf","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Wakefield's approximate Bayes factor: the `N(0, V + W)` density of the estimate `b` (its marginal law under the alternative `\u03b2 ~ N(0, W)`) divided by its `N(0, V)` density (the null `\u03b2 = 0`). Wakefield (2009) reports the reciprocal, null over alternative; coloc and SuSiE use this orientation, in which large values favor association. For `V = 0` the null density is Mathlib's junk value `0` and so is the ratio; the\u2026","title":"Wakefield's approximate Bayes factor"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/wakefield-abf-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.wakefieldABF_eq","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#wakefield-abf-eq","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"With `z = b / \u221aV`, `ABF = \u221a(V / (V + W)) \u00b7 exp (z\u00b2 W / (2 (V + W)))`.","title":"Wakefield's closed form"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/wakefield-abf-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.wakefieldABF_pos","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#wakefield-abf-pos","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Wakefield's ABF is strictly positive on its sampling-variance domain.","title":"Positivity of Wakefield's approximate Bayes factor"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/gaussian-real-eq-with-density-wakefield","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.gaussianReal_eq_withDensity_wakefieldABF","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#gaussian-real-eq-with-density-wakefield","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The alternative marginal `N(0, V + W)` has density `wakefieldABF V W` with respect to the null law `N(0, V)`.","title":"Wakefield's Bayes factor as a density ratio of Gaussians"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/rn-deriv-map-add-ae-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.rnDeriv_map_add_ae_eq","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#rn-deriv-map-add-ae-eq","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"If `\u03b2 ~ N(0, W)` and the sampling error `\u03b5 ~ N(0, V)` are independent, the law of the estimate `\u03b2 + \u03b5` (alternative) has density `wakefieldABF V W` with respect to the law of `\u03b5` (null).","title":"The ABF is the Bayes factor"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/ols-effect","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.olsEffect","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#ols-effect","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The marginal least-squares effect estimate `x\u1d40 y / x\u1d40 x` of a variant with genotype vector `x`.","title":"Marginal least-squares effect estimate"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/ols-var","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.olsVar","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#ols-var","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The sampling variance `\u03c3\u00b2 / x\u1d40 x` of `olsEffect x y` for residual variance `\u03c3\u00b2`.","title":"Sampling variance of the least-squares effect estimate"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/ser-cov","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.serCov","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#ser-cov","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The covariance `\u03c3\u2080\u00b2 x x\u1d40 + \u03c3\u00b2 I` of `y = x \u03b2 + e` for `\u03b2 ~ N(0, \u03c3\u2080\u00b2)` and `e ~ N(0, \u03c3\u00b2 I)`.","title":"Covariance of the single-effect regression model"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/pos-def-ser-cov","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.posDef_serCov","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#pos-def-ser-cov","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"`serCov \u03c32 \u03c302 x` is positive definite for `\u03c3\u00b2 > 0`.","title":"Single-effect regression covariance is positive definite"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/map-ser-eq-with-density","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.map_ser_eq_withDensity","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#map-ser-eq-with-density","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"If `y = x \u03b2 + e` with `\u03b2 ~ N(0, \u03c3\u2080\u00b2)` independent of `e ~ N(0, \u03c3\u00b2 I)`, then `y` has density `multivariateGaussianPDFReal 0 (serCov \u03c32 \u03c302 x)`.","title":"The marginal law of the phenotype given the causal variant"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/multivariate-gaussian-pdfreal-ser-cov-di","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.multivariateGaussianPDFReal_serCov_div","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#multivariate-gaussian-pdfreal-ser-cov-di","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The regression Bayes factor is the Wakefield ABF (Wang et al. 2020): the density of `y` when variant `x` carries an `N(0, \u03c3\u2080\u00b2)` effect, divided by its density with no effect, is the ABF of the summary statistics `olsEffect x y` and `olsVar \u03c32 x`.","title":"The regression Bayes factor is the Wakefield ABF"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/prior-weights","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.PriorWeights","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#prior-weights","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"A finite probability distribution used as the prior over the single causal variant.","title":"Prior weights over the causal variant"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/exists-weight-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.PriorWeights.exists_weight_pos","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#exists-weight-pos","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"A normalized non-negative finite prior assigns positive weight to some variant.","title":"A normalized prior gives some variant positive weight"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/evidence-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.PriorWeights.evidence_pos","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#evidence-pos","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Strictly positive evidence has a strictly positive prior-weighted marginal likelihood.","title":"Positivity of the prior-weighted marginal likelihood"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/normalized-weighted-evidence","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.normalizedWeightedEvidence","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#normalized-weighted-evidence","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Algebraic normalization of arbitrary weights times evidence. This is useful even outside the probability domain; it takes Lean's value `0` when the denominator vanishes.","title":"Normalized weighted evidence"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/pip","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.pip","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#pip","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Posterior inclusion probability from a normalized non-negative prior and variant evidence. When every evidence value is positive, `PriorWeights.evidence_pos` makes the denominator strictly positive, and `pip_nonneg` and `sum_pip` prove that these values form a probability distribution.","title":"Posterior inclusion probability"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/pip-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.pip_nonneg","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#pip-nonneg","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Posterior inclusion probabilities are non-negative when every evidence value is positive.","title":"Nonnegativity of posterior inclusion probabilities"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/sum-pip","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.sum_pip","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#sum-pip","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Posterior inclusion probabilities normalize to one when every evidence value is positive.","title":"Posterior inclusion probabilities sum to one"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/ser-likelihood-ratio-eq-normalized-weigh","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.serLikelihoodRatio_eq_normalizedWeightedEvidence","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#ser-likelihood-ratio-eq-normalized-weigh","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The unconditional algebraic single-effect likelihood-ratio identity for arbitrary real weights. Its right side is accurately named algebraic normalization, not a posterior probability.","title":"Single-effect likelihood-ratio identity"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/ser-posterior-evidence-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.serPosterior_evidence_pos","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#ser-posterior-evidence-pos","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The prior-weighted single-effect evidence is strictly positive on the regression domain.","title":"Positivity of the single-effect regression evidence"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/ser-posterior-eq-pip","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.serPosterior_eq_pip","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#ser-posterior-eq-pip","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Posterior inclusion probabilities of the single-effect regression (Wang et al. 2020): Bayes' rule for the causal variant with a normalized non-negative prior equals `pip` applied to the strictly positive Wakefield ABFs.","title":"Posterior inclusion probabilities of the single-effect regression"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/cond-distrib-ser-effect","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.condDistrib_ser_effect","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#cond-distrib-ser-effect","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The effect posterior of the single-effect regression (Wang et al. 2020): given `y = x \u03b2 + e` with `\u03b2 ~ N(0, \u03c3\u2080\u00b2)` independent of `e ~ N(0, \u03c3\u00b2 I)`, for almost every `y` the effect is normal with mean `\u03c3\u2080\u00b2 / (\u03c3\u2080\u00b2 + s\u00b2) \u00b7 \u03b2\u0302` and variance `\u03c3\u2080\u00b2 s\u00b2 / (\u03c3\u2080\u00b2 + s\u00b2)`, where `\u03b2\u0302 = olsEffect x y` and `s\u00b2 = olsVar \u03c32 x`.","title":"The effect posterior of the single-effect regression"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-hypothesis","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.ColocHypothesis","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-hypothesis","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The five coloc hypotheses: no association (`h0`), trait 1 only (`h1`), trait 2 only (`h2`), two distinct causal variants (`h3`), one shared causal variant (`h4`).","title":"The five colocalization hypotheses"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-hypothesis-2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocHypothesis","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-hypothesis-2","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The hypothesis of a configuration `(causal variant of trait 1, causal variant of trait 2)`, `none` meaning no causal variant.","title":"Colocalization hypothesis of a causal configuration"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-prior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.ColocPrior","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-prior","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Non-negative coloc configuration-weight parameters (Giambartolomei et al. 2014): `p1`, `p2`, `p12` for a variant causal for trait 1 only, trait 2 only, and both. The cited hypothesis weights use null weight `1`. The bound also supports a separate normalized categorical-prior model.","title":"Colocalization prior parameters"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/p0","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.ColocPrior.p0","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#p0","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The remaining null mass in the optional normalized categorical-prior model.","title":"Null mass of the categorical colocalization prior"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/categorical-prior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.ColocPrior.categoricalPrior","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#categorical-prior","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"A normalized categorical prior, distinct from the cited coloc relative weights.","title":"Normalized categorical colocalization prior"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/weight","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.ColocPrior.weight","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#weight","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The cited coloc unnormalized configuration weight, whose H0 weight is `1`.","title":"Unnormalized colocalization configuration weight"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/categorical-prior-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.ColocPrior.categoricalPrior_nonneg","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#categorical-prior-nonneg","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Categorical configuration priors are non-negative.","title":"Nonnegativity of categorical configuration priors"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/weight-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.ColocPrior.weight_nonneg","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#weight-nonneg","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Cited coloc configuration weights are non-negative.","title":"Nonnegativity of colocalization configuration weights"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-bf","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocBF","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-bf","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The Bayes factor of a configuration against the null: the product of the two traits' Bayes factors, `1` for a trait without a causal variant (the two studies are independent).","title":"Bayes factor of a colocalization configuration"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-bf-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocBF_nonneg","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-bf-nonneg","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"A configuration Bayes factor is non-negative when both traits' Bayes factors are.","title":"Nonnegativity of configuration Bayes factors"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-closed-form","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocClosedForm","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-closed-form","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The cited coloc closed-form algebraic scores. H0 has score `1`; H1--H4 retain the per-configuration coefficients of Giambartolomei et al. (2014). They are unnormalized hypothesis weights, and hence have a probability interpretation, when the Bayes factors are non-negative.","title":"Closed-form colocalization hypothesis scores"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-categorical-closed-form","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocCategoricalClosedForm","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-categorical-closed-form","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Closed forms for the optional normalized categorical-prior model. This differs from the cited coloc formula only in using its remaining null mass rather than the cited H0 weight `1`.","title":"Closed-form scores for the categorical colocalization prior"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/sum-coloc-weight-mul-coloc-bf","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.sum_colocWeight_mul_colocBF","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#sum-coloc-weight-mul-coloc-bf","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The coloc closed forms (Giambartolomei et al. 2014): the sum of unnormalized configuration weight times Bayes factor over each hypothesis equals its closed form.","title":"The coloc closed forms"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-closed-form-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocClosedForm_nonneg","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-closed-form-nonneg","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Cited coloc closed-form scores are non-negative on the Bayes-factor domain.","title":"Nonnegativity of closed-form colocalization scores"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-evidence-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocEvidence_pos","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-evidence-pos","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Cited coloc evidence is strictly positive for non-negative Bayes factors because H0 has unnormalized weight `1`.","title":"Positivity of colocalization evidence"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/sum-categorical-prior-mul-coloc-bf","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.sum_categoricalPrior_mul_colocBF","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#sum-categorical-prior-mul-coloc-bf","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The corresponding closed forms for the optional normalized categorical-prior model.","title":"Categorical-prior hypothesis scores equal their closed forms"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/sum-categorical-prior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.ColocPrior.sum_categoricalPrior","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#sum-categorical-prior","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The optional categorical configuration priors sum to one.","title":"Categorical colocalization priors sum to one"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-posterior-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocPosterior_eq","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-posterior-eq","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The unconditional algebraic coloc normalization identity: each hypothesis's configuration score divided by the total score equals its closed-form score divided by the closed-form total. For arbitrary real `bf1` and `bf2` these ratios need not be probabilities. See `colocPosterior_nonneg` and `sum_colocPosterior_eq_one` for the probability-domain results.","title":"Colocalization normalization identity"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-posterior-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocPosterior_nonneg","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-posterior-nonneg","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Normalized cited coloc hypothesis weights are non-negative for non-negative Bayes factors.","title":"Nonnegativity of colocalization posteriors"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/sum-coloc-posterior-eq-one","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.sum_colocPosterior_eq_one","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#sum-coloc-posterior-eq-one","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"Normalized cited coloc hypothesis weights sum to one for non-negative Bayes factors.","title":"Colocalization posteriors sum to one"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/coloc-categorical-posterior-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.colocCategoricalPosterior_eq","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#coloc-categorical-posterior-eq","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The normalized posterior identity for the separate categorical-prior model.","title":"Posterior identity for the categorical colocalization prior"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/is-credible-set","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.IsCredibleSet","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#is-credible-set","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"A `\u03c1`-credible set: its total posterior inclusion probability is at least `\u03c1`.","title":"Credible set of variants"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/sorted-prefix","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.sortedPrefix","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#sorted-prefix","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The first `k` variants in the order `\u03c3`.","title":"Prefix of variants in a given order"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/card-sorted-prefix","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.card_sortedPrefix","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#card-sorted-prefix","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"A prefix of length `k \u2264 card \u03b9` has `k` elements.","title":"Cardinality of a sorted prefix"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/sum-le-sum-sorted-prefix","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.sum_le_sum_sortedPrefix","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#sum-le-sum-sorted-prefix","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The `k` largest values carry the most mass: if `\u03c3` sorts `\u03b1` in decreasing order, no set of `S.card` variants has more mass than the prefix of that length.","title":"Sorted prefix carries the most posterior mass"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/le-card-of-is-credible-set","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.le_card_of_isCredibleSet","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#le-card-of-is-credible-set","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"The shortest credible prefix has minimum size (Maller et al. 2012; Wang et al. 2020): if `\u03c3` sorts `\u03b1` in decreasing order and no prefix shorter than `k` is `\u03c1`-credible, every `\u03c1`-credible set has at least `k` elements.","title":"The shortest credible prefix has minimum size"},{"id":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/exists-minimal-credible-prefix","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FineMapping.exists_minimal_credible_prefix","parent":46371,"result":true,"route":"../roadmap/Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization/index.html#exists-minimal-credible-prefix","scope":"Biology/statistical-genetics/bayesian-fine-mapping-and-colocalization","status":"fully_proved","summary":"If the total mass is at least `\u03c1`, some prefix of length `k` is `\u03c1`-credible, has `k` elements, and every `\u03c1`-credible set has at least `k` elements.","title":"A minimum-size credible set exists and is a sorted prefix"},{"data_url":"scopes/Biology/statistical-genetics/bolt-lmm-mixed-model-association.json","id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","title":"BOLT-LMM mixed-model association"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#model","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The variance components of the linear mixed model `y = X \u03b2 + g + e` with `g ~ N(0, \u03c3_g\u00b2 K)` and `e ~ N(0, \u03c3_e\u00b2 I)` on `n` individuals: a positive semidefinite genetic relationship matrix `K`, a genetic variance `\u03c3_g\u00b2 \u2265 0` and an environmental variance `\u03c3_e\u00b2 > 0`.","title":"Variance components of the linear mixed model"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/v","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.V","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#v","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The marginal phenotypic covariance `V = \u03c3_g\u00b2 K + \u03c3_e\u00b2 I` of `y = X \u03b2 + g + e`.","title":"Marginal phenotypic covariance matrix"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/pos-def-v","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.posDef_V","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#pos-def-v","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"`V = \u03c3_g\u00b2 K + \u03c3_e\u00b2 I` is positive definite.","title":"Positive definiteness of the phenotypic covariance"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/log-lik","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.logLik","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#log-lik","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The marginal log-likelihood of the mean `\u03bc`: the log of the `N(\u03bc, V)` density at `y`, `-(n log 2\u03c0 + log det V + (y - \u03bc)\u1d40 V\u207b\u00b9 (y - \u03bc)) / 2` (`det V > 0` by `posDef_V`).","title":"Marginal log-likelihood of the mean"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/gls","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.gls","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#gls","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The generalized least-squares estimator `\u03b2\u0302 = (X\u1d40 V\u207b\u00b9 X)\u207b\u00b9 X\u1d40 V\u207b\u00b9 y`.","title":"Generalized least-squares estimator"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/proj","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.proj","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#proj","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The residual-forming matrix `P = V\u207b\u00b9 - V\u207b\u00b9 X (X\u1d40 V\u207b\u00b9 X)\u207b\u00b9 X\u1d40 V\u207b\u00b9`.","title":"Residual-forming matrix"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/log-lik-le-log-lik-gls","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.logLik_le_logLik_gls","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#log-lik-le-log-lik-gls","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"For `X` of full column rank, every mean `X \u03b2` has marginal log-likelihood at most that of the GLS fit `X \u03b2\u0302`.","title":"GLS maximizes the marginal likelihood"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/log-lik-eq-log-lik-gls-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.logLik_eq_logLik_gls_iff","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#log-lik-eq-log-lik-gls-iff","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"For `X` of full column rank, `X \u03b2` attains the maximal marginal log-likelihood only at `\u03b2 = \u03b2\u0302`.","title":"The GLS maximizer is unique"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/log-lik-gls","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.logLik_gls","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#log-lik-gls","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The maximal marginal log-likelihood is `-(n log 2\u03c0 + log det V + y\u1d40 P y) / 2`.","title":"Maximal marginal log-likelihood"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/proj-mul-x","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.proj_mul_X","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#proj-mul-x","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"`P X = 0`: the residual-forming matrix annihilates the fixed effects.","title":"Residual-forming matrix annihilates the fixed effects"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/proj-mul-v-mul-proj","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.proj_mul_V_mul_proj","parent":46421,"reading_title":{"format":"inline-parts-v1","parts":[["text","Identity "],["math","\\(PVP = P\\)"],["text"," for the residual-forming matrix"]],"source":"Identity $PVP = P$ for the residual-forming matrix"},"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#proj-mul-v-mul-proj","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"`P V P = P`. With `P X = 0`, this says that under the null model `y = X \u03b2 + \u03b5` with `Cov \u03b5 = V` the score `x\u1d40 P y = x\u1d40 P \u03b5` has variance `x\u1d40 P V P x = x\u1d40 P x`.","title":"Identity $PVP = P$ for the residual-forming matrix"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/profile-log-lik","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.profileLogLik","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#profile-log-lik","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The profile marginal log-likelihood of a SNP effect `\u03b3` for the SNP genotype vector `x`: `sup_\u03b2 logLik y (X \u03b2 + \u03b3 x)`.","title":"Profile log-likelihood of a SNP effect"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/score-stat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.scoreStat","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#score-stat","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The score statistic `(x\u1d40 P y)\u00b2 / (x\u1d40 P x)` for one SNP `x`. Its denominator is positive when `x` is not in the column space of `X` (`info_pos`); otherwise the statistic is `0` by the convention `a / 0 = 0` and carries no information.","title":"Score statistic for one SNP"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/profile-log-lik-eq-quadratic","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.profileLogLik_eq_quadratic","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#profile-log-lik-eq-quadratic","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"`\u2113(\u03b3) = \u2113(0) + \u03b3 x\u1d40 P y - \u03b3\u00b2 x\u1d40 P x / 2`, so its score at `\u03b3 = 0` is `x\u1d40 P y` and its information is `x\u1d40 P x`.","title":"The profile log-likelihood of a SNP effect is an exact quadratic"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/has-deriv-at-profile-log-lik","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.hasDerivAt_profileLogLik","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#has-deriv-at-profile-log-lik","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The score of a SNP effect at `\u03b3 = 0` is `x\u1d40 P y`.","title":"The score of a SNP effect"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/score-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.score_eq","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#score-eq","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The score is evaluated at the null GLS fit: `x\u1d40 P y = x\u1d40 V\u207b\u00b9 (y - X \u03b2\u0302)`.","title":"Score at the null generalized least-squares fit"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/info-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.info_pos","parent":46421,"reading_title":{"format":"inline-parts-v1","parts":[["text","Information of a SNP outside the column space of "],["math","\\(X\\)"],["text"," is positive"]],"source":"Information of a SNP outside the column space of $X$ is positive"},"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#info-pos","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The information `x\u1d40 P x` is positive when the SNP `x` is not in the column space of `X`.","title":"Information of a SNP outside the column space of $X$ is positive"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/blup","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.blup","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#blup","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The BLUP `\u011d = G V\u207b\u00b9 (y - X \u03b2\u0302)` of the genetic effects, with `G = \u03c3_g\u00b2 K` (Henderson 1975).","title":"Best linear unbiased predictor of genetic effects"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/is-mmesolution","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.IsMMESolution","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#is-mmesolution","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"`(\u03b2, g)` solves Henderson's mixed-model equations with `R = \u03c3_e\u00b2 I` and `G = \u03c3_g\u00b2 K`, i.e. the specialization of `MixedModel.IsMMESolution` to `Z = 1` (every individual has its own genetic effect), `G = \u03c3_g\u00b2 K`, `R = \u03c3_e\u00b2 I`. In unfolded form this is `X\u1d40 R\u207b\u00b9 X \u03b2 + X\u1d40 R\u207b\u00b9 g = X\u1d40 R\u207b\u00b9 y` and `R\u207b\u00b9 X \u03b2 + (R\u207b\u00b9 + G\u207b\u00b9) g = R\u207b\u00b9 y`. The equations are meaningful when `G` is invertible (`isMMESolution_iff`); otherwise `G\u207b\u00b9` is\u2026","title":"Solution of Henderson's mixed-model equations"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/is-mmesolution-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMM.Model.isMMESolution_iff","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#is-mmesolution-iff","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"For positive definite `K`, `\u03c3_g\u00b2 > 0` and `X` of full column rank, `(\u03b2, g)` solves the mixed-model equations iff `\u03b2 = \u03b2\u0302` and `g = \u03c3_g\u00b2 K V\u207b\u00b9 (y - X \u03b2\u0302)`. This is the specialization of `MixedModel.isMMESolution_iff_gls` to `Z = 1`, `G = \u03c3_g\u00b2 K`, `R = \u03c3_e\u00b2 I`; the generic theorem gives `\u03b2 = glsMatrix X V *\u1d65 y` and `g = G V\u207b\u00b9 (y - X \u03b2\u0302)`, which are `Model.gls` and `Model.blup` by the two conversion lemmas below.","title":"Henderson's mixed-model equations give the GLS estimator and the BLUP"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/mixture-prior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.mixturePrior","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#mixture-prior","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The BOLT-LMM prior on a SNP effect: the mixture `p N(0, v\u2081) + (1 - p) N(0, v\u2082)` of two centered Gaussians with variances `v\u2081, v\u2082` (`v\u2082 = 0` gives the spike-and-slab prior with a point mass at `0`).","title":"Two-Gaussian mixture prior on a SNP effect"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/is-probability-measure-mixture-prior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.isProbabilityMeasure_mixturePrior","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#is-probability-measure-mixture-prior","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The mixture prior is a probability measure for a mixing weight `p \u2264 1`.","title":"Mixture prior is a probability measure"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/integrable-sq-mixture-prior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.integrable_sq_mixturePrior","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#integrable-sq-mixture-prior","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The mixture prior has a finite second moment.","title":"Mixture prior has finite second moment"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/is-admissible","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.IsAdmissible","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#is-admissible","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"An admissible variational factor for the prior `\u03c0`: a probability measure with a finite second moment and finite KL divergence from `\u03c0` (`q \u226a \u03c0` with integrable log-likelihood ratio).","title":"Admissible variational factor"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/mean","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.mean","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#mean","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The mean `\u222b b dq` of a factor.","title":"Mean of a variational factor"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/col-norm-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.colNormSq","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#col-norm-sq","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The squared norm `\u2016x\u2c7c\u2016\u00b2 = \u2211\u1d62 X i j \u00b2` of the genotype column of SNP `j`.","title":"Squared norm of a genotype column"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/expected-rss","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.expectedRSS","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#expected-rss","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The expected residual sum of squares `\u2016y - X \u03bc\u2016\u00b2 + \u2211\u2c7c \u2016x\u2c7c\u2016\u00b2 v\u2c7c` of the product of the factors `q`, with means `\u03bc\u2c7c` and variances `v\u2c7c` (`integral_rss_pi`).","title":"Expected residual sum of squares under the variational factors"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/integral-rss-pi","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.integral_rss_pi","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#integral-rss-pi","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"For probability factors with finite second moments, `\ud835\udd3c_{\u220f q\u2c7c} \u2016y - X \u03b2\u2016\u00b2 = \u2016y - X \u03bc\u2016\u00b2 + \u2211\u2c7c \u2016x\u2c7c\u2016\u00b2 v\u2c7c`.","title":"The expected residual sum of squares of a product of factors"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/elbo","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.elbo","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#elbo","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The mean-field evidence lower bound `-(n/2) log (2\u03c0 \u03c3_e\u00b2) - \ud835\udd3c_q \u2016y - X \u03b2\u2016\u00b2 / (2 \u03c3_e\u00b2) - \u2211\u2c7c KL(q\u2c7c \u2016 \u03c0)`, meaningful for `\u03c3_e\u00b2 > 0`. It is the ELBO for admissible factors (finite KL); for an inadmissible factor `toReal` would turn an infinite divergence into `0`, so all results assume admissibility.","title":"Mean-field evidence lower bound"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/partial-residual","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.partialResidual","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#partial-residual","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The partial residual `r\u2c7c = y - \u2211_{k \u2260 j} x\u2096 \u03bc\u2096` of the other SNPs' posterior means.","title":"Partial residual of a SNP"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/potential","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.potential","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#potential","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The CAVI potential `f\u2c7c(b) = (b x\u2c7c\u1d40 r\u2c7c - \u2016x\u2c7c\u2016\u00b2 b\u00b2 / 2) / \u03c3_e\u00b2`: the part of the expected log-likelihood `\ud835\udd3c_{q_{-j}}[log p(y | \u03b2)]` that depends on `\u03b2\u2c7c = b` (`exists_elbo_update_eq`).","title":"Coordinate-ascent potential for one SNP effect"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/cavi-update","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.caviUpdate","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#cavi-update","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The CAVI update of factor `j`: the prior tilted by the potential, `q\u2c7c \u221d \u03c0 \u00b7 exp f\u2c7c = exp \ud835\udd3c_{q_{-j}}[log p(y, \u03b2)]`.","title":"Coordinate-ascent update of one variational factor"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/cavi-step","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.caviStep","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#cavi-step","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"One coordinate-ascent step: replace factor `j` by its CAVI update.","title":"One coordinate-ascent step"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/cavi-run","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.caviRun","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#cavi-run","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"Coordinate ascent along the update schedule `s`: at time `t` factor `s t` is updated.","title":"Coordinate ascent along an update schedule"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/exists-elbo-update-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.exists_elbo_update_eq","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#exists-elbo-update-eq","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"With the other factors fixed, the ELBO of `q` with factor `j` replaced by an admissible `q'` is `\u222b f\u2c7c dq' - KL(q' \u2016 \u03c0) + C`.","title":"The ELBO as a function of one factor"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/is-admissible-cavi-update","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.isAdmissible_caviUpdate","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#is-admissible-cavi-update","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The CAVI update is an admissible factor, for a probability prior with a finite second moment.","title":"Coordinate-ascent update is an admissible factor"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/elbo-update-eq-sub","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.elbo_update_eq_sub","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#elbo-update-eq-sub","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"Replacing factor `j` by an admissible `q'` gives `ELBO(caviStep q j) - KL(q' \u2016 caviUpdate q j)`.","title":"The CAVI update maximizes the ELBO over its factor, with exact gap"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/elbo-le-elbo-cavi-step","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.elbo_le_elbo_caviStep","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#elbo-le-elbo-cavi-step","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"Coordinate ascent does not decrease the ELBO.","title":"Coordinate ascent does not decrease the ELBO"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/monotone-elbo-cavi-run","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.monotone_elbo_caviRun","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#monotone-elbo-cavi-run","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"The ELBO is nondecreasing along coordinate ascent, for any update schedule.","title":"The ELBO is nondecreasing along coordinate ascent"},{"id":"Biology/statistical-genetics/bolt-lmm-mixed-model-association/is-coord-max-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.BOLTLMMVariational.isCoordMax_iff","parent":46421,"result":true,"route":"../roadmap/Biology/statistical-genetics/bolt-lmm-mixed-model-association/index.html#is-coord-max-iff","scope":"Biology/statistical-genetics/bolt-lmm-mixed-model-association","status":"fully_proved","summary":"No admissible change of factor `j` increases the ELBO iff `q\u2c7c` is the expected-log-joint update `\u03c0 \u00b7 exp f\u2c7c / Z`.","title":"Stationary points of coordinate ascent"},{"data_url":"scopes/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf.json","id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","title":"Gene constraint scores (pLI and LOEUF)"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/poisson-lik","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.poissonLik","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#poisson-lik","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The Poisson likelihood `e^{-\u03bc} \u03bc^o / (o!)` of an observed count `o` at mean `\u03bc`.","title":"Poisson likelihood of an observed count"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/poisson-lik-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.poissonLik_pos","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#poisson-lik-pos","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The likelihood is positive at a positive mean.","title":"Positivity of the Poisson likelihood"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/flat-posterior-density","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.flatPosterior_density","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#flat-posterior-density","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The likelihood `\u03b3 \u21a6 P(o | Poisson(\u03b3 E))` times `E` is the Gamma density of shape `o + 1` and rate `E`. For `E > 0` that density integrates to one, so `E` is the normalizing constant of the flat-prior posterior (the identity itself is algebraic and holds for every real `E`).","title":"Flat-prior posterior of `\u03b3`"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/oe-upper","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.oeUpper","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#oe-upper","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The upper confidence bound for the observed/expected ratio at level `\u03b1`: `u_\u03b1(o) / E`, with `u_\u03b1` Garwood's bound for the Poisson mean.","title":"Upper confidence bound for the observed/expected ratio"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/loeuf","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.loeuf","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#loeuf","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The upper end of the 90% interval, `oeUpper (1/20)`.","title":"LOEUF"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/gamma-measure-real-iic-oe-upper","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.gammaMeasure_real_Iic_oeUpper","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#gamma-measure-real-iic-oe-upper","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The Gamma(`o + 1`, `E`) posterior puts mass exactly `1 - \u03b1` below `oeUpper \u03b1 E o`.","title":"The flat-prior posterior quantile is Garwood's bound"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/oe-upper-coverage","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.oeUpper_coverage","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#oe-upper-coverage","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"For every true ratio `\u03b3 \u2265 0` and `E > 0`, when `o ~ Poisson(\u03b3 E)`, `P(\u03b3 \u2264 oeUpper \u03b1 E o) \u2265 1 - \u03b1`.","title":"Exact coverage"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/oe-upper-zero","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.oeUpper_zero","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#oe-upper-zero","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"With no observed variants the bound is `log (1 / \u03b1) / E`.","title":"Observed/expected upper bound with no observed variants"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/loeuf-zero","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.loeuf_zero","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#loeuf-zero","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"LOEUF of a gene with no observed LoF variants is `log 20 / E`.","title":"LOEUF of a gene with no observed LoF variants"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/oe-upper-strict-mono","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.oeUpper_strictMono","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#oe-upper-strict-mono","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The bound is strictly increasing in the observed count.","title":"Observed/expected upper bound increases with the observed count"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/oe-upper-strict-anti-on","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.oeUpper_strictAntiOn","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#oe-upper-strict-anti-on","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The bound is strictly decreasing in the expected count `E > 0`.","title":"Observed/expected upper bound decreases with the expected count"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/class-posterior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.classPosterior","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#class-posterior","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The posterior probability of class `c` for a gene with `o` observed and `E` expected LoF variants, in the Poisson mixture with prior weights `w` and class rates `rate`: `w_c P(o | rate_c E) / \u2211_{c'} w_{c'} P(o | rate_{c'} E)`. pLI is this at the class with the smallest rate.","title":"Posterior class probability in the Poisson mixture"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/sum-class-posterior","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.sum_classPosterior","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#sum-class-posterior","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The class posteriors sum to one.","title":"Class posteriors sum to one"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/poisson-lik-succ","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.poissonLik_succ","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#poisson-lik-succ","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The likelihood recursion `P(o + 1 | \u03bc) = P(o | \u03bc) \u03bc / (o + 1)`.","title":"Recursion for the Poisson likelihood"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/class-posterior-antitone","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.classPosterior_antitone","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#class-posterior-antitone","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The posterior of a class `c\u2080` whose rate is the smallest is antitone in `o`, for every prior.","title":"pLI is antitone in the observed count"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/class-posterior-strict-anti","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GeneConstraint.classPosterior_strictAnti","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#class-posterior-strict-anti","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"Strictly antitone when some class has a strictly larger rate.","title":"Class posterior is strictly antitone in the observed count"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/pos-weights","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.PosWeights","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#pos-weights","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"Strictly positive class weights summing to one.","title":"Strictly positive class weights summing to one"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/gene-data","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.GeneData","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#gene-data","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"A gene's data: observed LoF count `o` and positive expected count `E`.","title":"Loss-of-function count data for a gene"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/joint","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.joint","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#joint","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The joint likelihood of a gene's data and class `z`: `w_z P(o | rate_z E)`.","title":"Joint likelihood of gene data and constraint class"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/joint-pos","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.joint_pos","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#joint-pos","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The joint likelihood is positive for positive rates.","title":"Positivity of the Poisson mixture joint likelihood"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/mass","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.mass","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#mass","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The expected number of genes in class `z` under the posteriors at weights `w`: `\u2211_n P_w(z | o_n)`.","title":"Expected number of genes in a constraint class"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/mass-pos","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.mass_pos","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#mass-pos","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The class masses are positive when there is at least one gene.","title":"Positivity of the class masses"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/sum-mass","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.sum_mass","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#sum-mass","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The class masses add up to the number of genes.","title":"Class masses sum to the number of genes"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/em-step","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.emStep","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#em-step","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The EM update `w'_z = (1/N) \u2211_n P_w(z | o_n)`, for `N > 0` genes and positive rates.","title":"EM update of the Poisson mixture class weights"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/em-q-le-em-step","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.emQ_le_emStep","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#em-q-le-em-step","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The M-step maximizes `Q(\u00b7 | w)` over admissible weights.","title":"The M-step maximizes `Q(\u00b7 | w)`"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/log-lik-le-log-lik-em-step","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.logLik_le_logLik_emStep","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#log-lik-le-log-lik-em-step","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"EM does not decrease the likelihood of the class weights.","title":"EM does not decrease the likelihood"},{"id":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/log-lik-iterate-em-step-monotone","kind":"node","lean":"Cslib.Bioinformatics.StatisticalGenetics.PoissonClassEM.logLik_iterate_emStep_monotone","parent":46462,"result":true,"route":"../roadmap/Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf/index.html#log-lik-iterate-em-step-monotone","scope":"Biology/statistical-genetics/gene-constraint-scores-pli-and-loeuf","status":"fully_proved","summary":"The log-likelihood is monotone along the EM iterates.","title":"Monotonicity of the log-likelihood along the class-weight EM iterates"},{"data_url":"scopes/Biology/statistical-genetics/imputation-of-association-summary-statistics.json","id":"Biology/statistical-genetics/imputation-of-association-summary-statistics","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","title":"Imputation of association summary statistics"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#model","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"Square-integrable statistics `z_t` (untyped marker) and `z_o` (typed markers) with raw second moments `\ud835\udd3c z_o z_o\u1d40 = \u03a3_oo`, `\ud835\udd3c z_t z_o = \u03a3_ot`, `\ud835\udd3c z_t\u00b2 = \u03a3_tt`. These are the covariances of the ImpG null model when the statistics are centred.","title":"Second-moment model for summary-statistic imputation"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/imputed","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.imputed","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#imputed","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"The imputed statistic `\u1e91 = w\u1d40 z_o` with weights `w`.","title":"Imputed summary statistic"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/integral-imputed-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.integral_imputed_sq","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#integral-imputed-sq","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"`\ud835\udd3c \u1e91\u00b2 = w\u1d40 \u03a3_oo w`.","title":"Second moment of the imputed statistic"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/integral-imputed-mul","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.integral_imputed_mul","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#integral-imputed-mul","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"`\ud835\udd3c \u1e91 z_t = w\u1d40 \u03a3_ot`.","title":"Cross moment of the imputed and untyped statistics"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/integral-error-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.integral_error_sq","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#integral-error-sq","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"The mean squared imputation error: `\ud835\udd3c (z_t - \u1e91)\u00b2 = \u03a3_tt - 2 w\u1d40 \u03a3_ot + w\u1d40 \u03a3_oo w`.","title":"Mean squared imputation error"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/integral-imputed-sq-of-normal","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.integral_imputed_sq_of_normal","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#integral-imputed-sq-of-normal","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"For the ImpG weights (`\u03a3_oo w = \u03a3_ot`, i.e. `w = \u03a3_oo\u207b\u00b9 \u03a3_ot` when `\u03a3_oo` is invertible) the imputed statistic has second moment `w\u1d40 \u03a3_ot = \u03a3_to \u03a3_oo\u207b\u00b9 \u03a3_ot`; for `\u03a3_tt = 1` this is `1 -` the linear-prediction residual variance, ImpG's `r\u00b2_pred`.","title":"Second moment of the statistic imputed with ImpG weights"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/integral-error-sq-of-normal","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.integral_error_sq_of_normal","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#integral-error-sq-of-normal","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"For the ImpG weights (`\u03a3_oo w = \u03a3_ot`) the mean squared imputation error is `\u03a3_tt - w\u1d40 \u03a3_ot = \u03a3_tt - \u03a3_to \u03a3_oo\u207b\u00b9 \u03a3_ot`, the linear-prediction residual variance (the conditional variance only under the Gaussian ImpG model).","title":"Mean squared error of imputation with ImpG weights"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/sigma-oo-transpose","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.sigmaOO_transpose","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#sigma-oo-transpose","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"The moment matrix `\u03a3_oo` is symmetric.","title":"Symmetry of the typed-marker moment matrix"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/integral-error-sq-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.integral_error_sq_eq","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#integral-error-sq-eq","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"If `\u03a3_oo w = \u03a3_ot`, then for any weights `w'`, `\ud835\udd3c (z_t - w'\u1d40 z_o)\u00b2 = \ud835\udd3c (z_t - w\u1d40 z_o)\u00b2 + (w' - w)\u1d40 \u03a3_oo (w' - w)`.","title":"Excess error of a linear imputation"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/integral-error-sq-le","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.integral_error_sq_le","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#integral-error-sq-le","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"The weights solving `\u03a3_oo w = \u03a3_ot` minimize the mean squared error `\ud835\udd3c (z_t - w'\u1d40 z_o)\u00b2` over all weights `w'`.","title":"ImpG is the best linear imputation"},{"id":"Biology/statistical-genetics/imputation-of-association-summary-statistics/integral-normalized-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SummaryImputation.Model.integral_normalized_sq","parent":46490,"result":true,"route":"../roadmap/Biology/statistical-genetics/imputation-of-association-summary-statistics/index.html#integral-normalized-sq","scope":"Biology/statistical-genetics/imputation-of-association-summary-statistics","status":"fully_proved","summary":"For any weights `w` with `w\u1d40 \u03a3_oo w > 0` (for example the ridge weights solving `(\u03a3_oo + \u03bb I) w = \u03a3_ot`), the statistic `\u1e91 / \u221a(w\u1d40 \u03a3_oo w)` has second moment `1`, so it has unit variance under the centred null.","title":"Normalized imputed statistic"},{"data_url":"scopes/Biology/statistical-genetics/latent-causal-variable-model.json","id":"Biology/statistical-genetics/latent-causal-variable-model","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","title":"Latent causal variable model"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#model","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The latent causal variable model of O'Connor and Price (2018, Equation 3) on a probability space: `\u03b1_k = q_k \u03c0 + \u03b3_k` with `\u03c0` independent of `(\u03b3\u2081, \u03b3\u2082)`, the law of `(\u03b3\u2081, \u03b3\u2082)` invariant under each sign flip, `\u03c0` of mean zero and unit variance, and both `\u03b1_k` of unit second moment. Fourth moments are assumed finite.","title":"Latent causal variable model"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/alpha1","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.alpha1","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#alpha1","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The marginal effect `\u03b1\u2081 = q\u2081 \u03c0 + \u03b3\u2081` on trait 1.","title":"Marginal effect on the first trait in the latent causal variable model"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/alpha2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.alpha2","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#alpha2","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The marginal effect `\u03b1\u2082 = q\u2082 \u03c0 + \u03b3\u2082` on trait 2.","title":"Marginal effect on the second trait in the latent causal variable model"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/kurtosis","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.kurtosis","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#kurtosis","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The excess kurtosis `\u03ba_\u03c0 = E(\u03c0\u2074) - 3` of the effect on the latent variable.","title":"Excess kurtosis of the latent effect"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/integral-alpha1-mul-alpha2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.integral_alpha1_mul_alpha2","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#integral-alpha1-mul-alpha2","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The genetic correlation is `E(\u03b1\u2081 \u03b1\u2082) = q\u2081 q\u2082` (O'Connor and Price 2018, after Equation 1).","title":"Genetic correlation in the latent causal variable model"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/integral-alpha1-pow-three-mul-alpha2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.integral_alpha1_pow_three_mul_alpha2","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#integral-alpha1-pow-three-mul-alpha2","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"Mixed fourth moment (O'Connor and Price 2018, Equation 2): `E(\u03b1\u2081\u00b3 \u03b1\u2082) = \u03ba_\u03c0 q\u2081\u00b3 q\u2082 + 3 q\u2081 q\u2082`.","title":"Mixed fourth moment"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/integral-alpha1-mul-alpha2-pow-three","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.integral_alpha1_mul_alpha2_pow_three","parent":46502,"reading_title":{"format":"inline-parts-v1","parts":[["text","Mixed fourth moment "],["math","\\(E(\\alpha_1 \\alpha_2^3)\\)"]],"source":"Mixed fourth moment $E(\\alpha_1 \\alpha_2^3)$"},"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#integral-alpha1-mul-alpha2-pow-three","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The mirror image of Equation 2: `E(\u03b1\u2081 \u03b1\u2082\u00b3) = \u03ba_\u03c0 q\u2081 q\u2082\u00b3 + 3 q\u2081 q\u2082`.","title":"Mixed fourth moment $E(\\alpha_1 \\alpha_2^3)$"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/q1-sq-le-one","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.q1_sq_le_one","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#q1-sq-le-one","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"Unit variance of `\u03b1\u2081` forces `q\u2081\u00b2 \u2264 1` (the non-mediated part has variance `1 - q\u2081\u00b2`).","title":"Mediated coefficient of the first trait is at most one in size"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/q2-sq-le-one","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.q2_sq_le_one","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#q2-sq-le-one","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"Unit variance of `\u03b1\u2082` forces `q\u2082\u00b2 \u2264 1`.","title":"Mediated coefficient of the second trait is at most one in size"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/one-le-integral-pi-pow-four","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.one_le_integral_pi_pow_four","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#one-le-integral-pi-pow-four","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"`E(\u03c0\u2074) \u2265 (E \u03c0\u00b2)\u00b2 = 1`, so the excess kurtosis is at least `-2`.","title":"Fourth moment of the latent effect is at least one"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/centred-moment-left","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.centred_moment_left","parent":46502,"reading_title":{"format":"inline-parts-v1","parts":[["text","Centred mixed fourth moment "],["math","\\(E(\\alpha_1^3 \\alpha_2)\\)"]],"source":"Centred mixed fourth moment $E(\\alpha_1^3 \\alpha_2)$"},"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#centred-moment-left","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The centred mixed fourth moment: `E(\u03b1\u2081\u00b3 \u03b1\u2082) - 3 \u03c1_g = \u03ba_\u03c0 q\u2081\u00b3 q\u2082`.","title":"Centred mixed fourth moment $E(\\alpha_1^3 \\alpha_2)$"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/centred-moment-right","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.centred_moment_right","parent":46502,"reading_title":{"format":"inline-parts-v1","parts":[["text","Centred mixed fourth moment "],["math","\\(E(\\alpha_1 \\alpha_2^3)\\)"]],"source":"Centred mixed fourth moment $E(\\alpha_1 \\alpha_2^3)$"},"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#centred-moment-right","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The centred mixed fourth moment: `E(\u03b1\u2081 \u03b1\u2082\u00b3) - 3 \u03c1_g = \u03ba_\u03c0 q\u2081 q\u2082\u00b3`.","title":"Centred mixed fourth moment $E(\\alpha_1 \\alpha_2^3)$"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/abs-centred-moment-le-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.abs_centredMoment_le_iff","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#abs-centred-moment-le-iff","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"If `\u03ba_\u03c0 \u2260 0` and `\u03c1_g = q\u2081 q\u2082 \u2260 0`, then `|E(\u03b1\u2081 \u03b1\u2082\u00b3) - 3 \u03c1_g| \u2264 |E(\u03b1\u2081\u00b3 \u03b1\u2082) - 3 \u03c1_g|` iff `q\u2082\u00b2 \u2264 q\u2081\u00b2`.","title":"The centred moments order `q\u2081\u00b2` and `q\u2082\u00b2`"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/neg-two-le-kurtosis","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.neg_two_le_kurtosis","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#neg-two-le-kurtosis","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"`\u03ba_\u03c0 \u2265 -2`.","title":"Excess kurtosis of the latent effect is at least \u22122"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/abs-moment-le-iff-of-kurtosis-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.abs_moment_le_iff_of_kurtosis_pos","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#abs-moment-le-iff-of-kurtosis-pos","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"If `\u03ba_\u03c0 > 0` and `\u03c1_g \u2260 0`, then `|E(\u03b1\u2081 \u03b1\u2082\u00b3)| \u2264 |E(\u03b1\u2081\u00b3 \u03b1\u2082)|` iff `q\u2082\u00b2 \u2264 q\u2081\u00b2` (O'Connor and Price 2018, after Equation 2, which states the forward implication).","title":"The paper's uncentred comparison, under its sign condition"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/abs-moment-le-iff-of-kurtosis-neg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.abs_moment_le_iff_of_kurtosis_neg","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#abs-moment-le-iff-of-kurtosis-neg","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"If `\u03ba_\u03c0 < 0` and `\u03c1_g \u2260 0`, then `|E(\u03b1\u2081 \u03b1\u2082\u00b3)| \u2264 |E(\u03b1\u2081\u00b3 \u03b1\u2082)|` iff `q\u2081\u00b2 \u2264 q\u2082\u00b2`. So the paper's uncentred criterion needs `\u03ba_\u03c0 > 0`; the centred criterion `abs_centredMoment_le_iff` does not.","title":"The uncentred comparison is reversed when `\u03ba_\u03c0 < 0`"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/gcp","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.gcp","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#gcp","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The genetic causality proportion `gcp = (log|q\u2082| - log|q\u2081|) / (log|q\u2082| + log|q\u2081|)` (O'Connor and Price 2018, Equation 4). Meaningful for `0 < |q\u2081|, |q\u2082| \u2264 1` and `\u03c1_g\u00b2 < 1`; for `|q\u2081| = |q\u2082| = 1` the denominator vanishes and the value is `0` by the convention `x / 0 = 0`.","title":"Genetic causality proportion"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/gcp-nonneg-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.gcp_nonneg_iff","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#gcp-nonneg-iff","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"`gcp \u2265 0` iff `q\u2082\u00b2 \u2264 q\u2081\u00b2`, for nonzero `q\u2081, q\u2082` with `\u03c1_g\u00b2 = (q\u2081 q\u2082)\u00b2 < 1`.","title":"Sign of the genetic causality proportion"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/rpow-gcp","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.rpow_gcp","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#rpow-gcp","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The genetic causality proportion solves the paper's Equation 1: `q\u2082\u00b2 / q\u2081\u00b2 = (\u03c1_g\u00b2) ^ gcp` with `\u03c1_g = q\u2081 q\u2082`, for nonzero `q\u2081, q\u2082` with `\u03c1_g\u00b2 < 1`.","title":"Genetic causality proportion solves the defining equation"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/gcp-le-one","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.gcp_le_one","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#gcp-le-one","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"`gcp \u2264 1` when `q\u2081\u00b2 \u2264 1` (for nonzero `q\u2081, q\u2082` with `\u03c1_g\u00b2 < 1`).","title":"Genetic causality proportion is at most 1"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/neg-one-le-gcp","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.neg_one_le_gcp","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#neg-one-le-gcp","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"`-1 \u2264 gcp` when `q\u2082\u00b2 \u2264 1` (for nonzero `q\u2081, q\u2082` with `\u03c1_g\u00b2 < 1`).","title":"Genetic causality proportion is at least \u22121"},{"id":"Biology/statistical-genetics/latent-causal-variable-model/gcp-nonneg-of-abs-moment-le","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LatentCausalVariable.Model.gcp_nonneg_of_abs_moment_le","parent":46502,"result":true,"route":"../roadmap/Biology/statistical-genetics/latent-causal-variable-model/index.html#gcp-nonneg-of-abs-moment-le","scope":"Biology/statistical-genetics/latent-causal-variable-model","status":"fully_proved","summary":"The paper's criterion for `gcp \u2265 0` (O'Connor and Price 2018, after Equation 2), under the sign condition `\u03ba_\u03c0 > 0` that it needs: if `\u03c1_g \u2260 0`, `\u03c1_g\u00b2 < 1` and `|E(\u03b1\u2081 \u03b1\u2082\u00b3)| \u2264 |E(\u03b1\u2081\u00b3 \u03b1\u2082)|`, then `gcp \u2265 0`.","title":"The paper's criterion for `gcp \u2265 0`"},{"data_url":"scopes/Biology/statistical-genetics/ld-score-regression.json","id":"Biology/statistical-genetics/ld-score-regression","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","title":"LD-score regression"},{"id":"Biology/statistical-genetics/ld-score-regression/ld-corr","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.ldCorr","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#ld-corr","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The normalized cross product `r j k = X_j\u1d40 X_k / N` of markers `j` and `k`. For a standardized genotype matrix (`IsStandardized`) it is the sample correlation of the two markers; for other matrices it is only a scale-dependent dot product.","title":"Normalized cross product of genotype markers"},{"id":"Biology/statistical-genetics/ld-score-regression/ld-score","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.ldScore","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#ld-score","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The LD score `\u2113 j = \u2211 k, r j k ^ 2` of marker `j` (an LD score in the sense of the cited paper for a standardized genotype matrix).","title":"LD score of a marker"},{"id":"Biology/statistical-genetics/ld-score-regression/is-standardized","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.IsStandardized","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#is-standardized","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The genotype matrix is standardized: every marker column has mean `0` and sum of squares `N` (unit sample variance).","title":"Standardized genotype matrix"},{"id":"Biology/statistical-genetics/ld-score-regression/ld-corr-self","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.IsStandardized.ldCorr_self","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#ld-corr-self","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"For a standardized matrix each marker has correlation `1` with itself.","title":"A standardized marker has correlation one with itself"},{"id":"Biology/statistical-genetics/ld-score-regression/ld-corr-sq-le-one","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.IsStandardized.ldCorr_sq_le_one","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#ld-corr-sq-le-one","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"For a standardized matrix the correlations lie in `[-1, 1]` (Cauchy\u2013Schwarz).","title":"Correlations of standardized markers lie in (-1, 1)"},{"id":"Biology/statistical-genetics/ld-score-regression/model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.Model","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#model","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The random-effects model of LD-score regression (Bulik-Sullivan et al. 2015), stated at the level of raw second moments, for `N` individuals and `M` markers with heritability `h2`.","title":"Random-effects model of LD-score regression"},{"id":"Biology/statistical-genetics/ld-score-regression/h2-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.Model.h2_nonneg","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#h2-nonneg","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The heritability is nonnegative when there is at least one marker.","title":"Nonnegativity of heritability in the LD-score model"},{"id":"Biology/statistical-genetics/ld-score-regression/h2-le-one","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.Model.h2_le_one","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#h2-le-one","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The heritability is at most `1` when there is at least one individual.","title":"Heritability is at most one in the LD-score model"},{"id":"Biology/statistical-genetics/ld-score-regression/pheno","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.Model.pheno","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#pheno","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The phenotype `y = X \u03b2 + \u03b5`.","title":"Phenotype in the LD-score model"},{"id":"Biology/statistical-genetics/ld-score-regression/beta-hat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.Model.betaHat","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#beta-hat","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The `N`-normalized score `\u03b2\u0302 j = X_j\u1d40 y / N` of marker `j`. For a standardized column (`\u2211 i, X i j ^ 2 = N`) it is the marginal least-squares estimate of the effect of marker `j`; in general that estimate has denominator `\u2211 i, X i j ^ 2` instead of `N`.","title":"Normalized marginal score in the LD-score model"},{"id":"Biology/statistical-genetics/ld-score-regression/chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.Model.chiSq","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#chi-sq","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The association statistic `\u03c7\u00b2 j = N * \u03b2\u0302 j ^ 2` of marker `j`.","title":"Association statistic in the LD-score model"},{"id":"Biology/statistical-genetics/ld-score-regression/integral-chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.Model.integral_chiSq","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#integral-chi-sq","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The exact expectation of the association statistic built from the `N`-normalized score, for an arbitrary genotype matrix: `\ud835\udd3c[\u03c7\u00b2 j] = N h\u00b2 / M * \u2113 j + (1 - h\u00b2) * (\u2211 i, X i j ^ 2) / N`. (For `N = 0` both sides are `0` by the convention `x / 0 = 0`.)","title":"Expected association statistic for an arbitrary genotype matrix"},{"id":"Biology/statistical-genetics/ld-score-regression/integral-chi-sq-of-standardized","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScore.Model.integral_chiSq_of_standardized","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#integral-chi-sq-of-standardized","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"LD-score regression (Bulik-Sullivan et al. 2015): for a standardized genotype matrix (`IsStandardized`, so that `ldCorr` is the sample correlation and `ldScore` the LD score) and `N \u2260 0`, `\ud835\udd3c[\u03c7\u00b2 j] = N h\u00b2 / M * \u2113 j + (1 - h\u00b2)`. The expected association statistic is affine in the LD score, with slope `N h\u00b2 / M` and intercept `1 - h\u00b2`. The proof uses only that column `j` has sum of squares `N`; the rest of the\u2026","title":"LD-score regression"},{"id":"Biology/statistical-genetics/ld-score-regression/mean-ldscore","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.meanLDScore","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#mean-ldscore","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The mean LD score `\u2113\u0304 = (1 / M) \u2211 j, \u2113 j`.","title":"Mean LD score"},{"id":"Biology/statistical-genetics/ld-score-regression/mean-integral-chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.mean_integral_chiSq","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#mean-integral-chi-sq","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"Polygenic inflation: for a standardized genotype matrix with `N, M \u2260 0`, the mean expected association statistic is `1 - h\u00b2 + N h\u00b2 \u2113\u0304 / M`.","title":"Polygenic inflation of the mean association statistic"},{"id":"Biology/statistical-genetics/ld-score-regression/pheno-strat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.phenoStrat","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#pheno-strat","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The phenotype with a stratification shift: `y i = (X \u03b2) i + \u03b5 i + s * c i`.","title":"Phenotype with population stratification"},{"id":"Biology/statistical-genetics/ld-score-regression/beta-hat-strat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.betaHatStrat","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#beta-hat-strat","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The `N`-normalized score of marker `j` under stratification, `X_j\u1d40 y / N`.","title":"Normalized score under stratification"},{"id":"Biology/statistical-genetics/ld-score-regression/chi-sq-strat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.chiSqStrat","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#chi-sq-strat","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The association statistic `N * \u03b2\u0302 j ^ 2` under stratification.","title":"Association statistic under stratification"},{"id":"Biology/statistical-genetics/ld-score-regression/integral-chi-sq-strat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.integral_chiSqStrat","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#integral-chi-sq-strat","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"Stratification inflation: for a probability measure and mean-zero effects and noise, `\ud835\udd3c[\u03c7\u00b2 j] = N h\u00b2 / M * \u2113 j + (1 - h\u00b2) (\u2211 i, X i j ^ 2) / N + N s\u00b2 (X_j\u1d40 c / N)\u00b2`.","title":"Expected association statistic under stratification"},{"id":"Biology/statistical-genetics/ld-score-regression/integral-chi-sq-strat-of-standardized","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.integral_chiSqStrat_of_standardized","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#integral-chi-sq-strat-of-standardized","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"Stratification inflation for a standardized genotype matrix and `N \u2260 0`: `\ud835\udd3c[\u03c7\u00b2 j] = N h\u00b2 / M * \u2113 j + (1 - h\u00b2) + N s\u00b2 (X_j\u1d40 c / N)\u00b2`.","title":"Expected statistic under stratification for standardized genotypes"},{"id":"Biology/statistical-genetics/ld-score-regression/integral-chi-sq-strat-of-const","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.integral_chiSqStrat_of_const","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#integral-chi-sq-strat-of-const","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"With a constant stratification term (`s\u00b2 (X_j\u1d40 c / N)\u00b2 = a` for every marker), the expected statistic is exactly affine in the LD score, with slope `N h\u00b2 / M` and intercept `1 - h\u00b2 + N a`.","title":"LD-score regression with constant stratification"},{"id":"Biology/statistical-genetics/ld-score-regression/mean-integral-chi-sq-strat-of-const","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.mean_integral_chiSqStrat_of_const","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#mean-integral-chi-sq-strat-of-const","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"With a constant stratification term `N a`, the mean expected statistic is `N h\u00b2 \u2113\u0304 / M + (1 - h\u00b2 + N a)`.","title":"Mean association statistic with constant stratification"},{"id":"Biology/statistical-genetics/ld-score-regression/attenuation-ratio","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.attenuationRatio","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#attenuation-ratio","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The attenuation ratio `(intercept - 1) / (mean \u03c7\u00b2 - 1)` of Loh et al. (2018).","title":"Attenuation ratio"},{"id":"Biology/statistical-genetics/ld-score-regression/attenuation-ratio-of-const","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.attenuationRatio_of_const","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#attenuation-ratio-of-const","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"With a constant stratification term `N a`, the attenuation ratio of the exact intercept `1 - h\u00b2 + N a` and the mean expected statistic is `(N a - h\u00b2) / (N h\u00b2 \u2113\u0304 / M + N a - h\u00b2)`.","title":"Attenuation ratio under constant stratification"},{"id":"Biology/statistical-genetics/ld-score-regression/confounding-share-of-const","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.confoundingShare_of_const","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#confounding-share-of-const","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"With a constant stratification term `N a`, the share of the mean inflation above the exact null level `1 - h\u00b2` that is due to stratification is `N a / (N a + N h\u00b2 \u2113\u0304 / M)`.","title":"Confounding share of inflation under constant stratification"},{"id":"Biology/statistical-genetics/ld-score-regression/confounding-share-mem-icc","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDScoreInflation.confoundingShare_mem_Icc","parent":46525,"result":true,"route":"../roadmap/Biology/statistical-genetics/ld-score-regression/index.html#confounding-share-mem-icc","scope":"Biology/statistical-genetics/ld-score-regression","status":"fully_proved","summary":"The confounding share `N a / (N a + N h\u00b2 \u2113\u0304 / M)` lies in `[0, 1]` when `a`, `h\u00b2` and `\u2113\u0304` are nonnegative (as they are in the model: `a` is a square, `h\u00b2 \u2265 0` by `Model.h2_nonneg`, LD scores are sums of squares).","title":"Confounding share lies between 0 and 1"},{"data_url":"scopes/Biology/statistical-genetics/genomic-prediction-from-summary-statistics.json","id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","title":"Genomic prediction from summary statistics"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-score-mul-score","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.integral_score_mul_score","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-score-mul-score","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"In a target population whose marker genotypes `x j` have raw second moments `\u222b x j * x k = [j = k]` (standardized, uncorrelated markers), the scores `x\u1d40 a` and `x\u1d40 b` of two effect vectors have raw cross moment `a\u1d40 b`. With mean-zero genotypes and a probability measure this is the covariance of the genetic value `x\u1d40 \u03b2` and the polygenic score `x\u1d40 \u03b2\u0302`.","title":"Cross moment of two polygenic scores"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/prsmodel","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#prsmodel","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The effect-estimate model behind polygenic-score accuracy (Daetwyler et al. 2008; Dudbridge 2013), at the level of raw second moments, for `M` independent markers: true effects `beta k` with `\u222b \u03b2 k \u03b2 l = (h\u00b2 / M) [k = l]`, estimation errors `err k` with `\u222b e k e l = s\u00b2 [k = l]`, orthogonal to the effects. The estimates are `\u03b2\u0302 = \u03b2 + e` (`betaHat`). For a probability measure and mean-zero effects and errors these\u2026","title":"Effect-estimate model for polygenic score accuracy"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/h2-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.h2_nonneg","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#h2-nonneg","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The model forces `h\u00b2 \u2265 0` when there is at least one marker.","title":"Nonnegativity of heritability in the effect-estimate model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/s2-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.s2_nonneg","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#s2-nonneg","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The model forces `s\u00b2 \u2265 0` when there is at least one marker.","title":"Nonnegativity of the sampling error variance of effect estimates"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/beta-hat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.betaHat","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#beta-hat","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The estimated effects `\u03b2\u0302 = \u03b2 + e`.","title":"Estimated marker effects"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/accuracy-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.accuracySq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#accuracy-sq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The squared correlation of the genetic value and the polygenic score as a ratio of expectations, `\ud835\udd3c[\u03b2\u1d40\u03b2\u0302]\u00b2 / (\ud835\udd3c[\u03b2\u1d40\u03b2] \ud835\udd3c[\u03b2\u0302\u1d40\u03b2\u0302])`: by `integral_score_mul_score`, `\u03b2\u1d40\u03b2\u0302`, `\u03b2\u1d40\u03b2` and `\u03b2\u0302\u1d40\u03b2\u0302` are the covariance and variances of `x\u1d40\u03b2` and `x\u1d40\u03b2\u0302` over a target population of standardized uncorrelated markers. It is not the expectation of a squared correlation.","title":"Squared accuracy of a polygenic score"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-beta-mul-beta-hat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.integral_beta_mul_betaHat","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-beta-mul-beta-hat","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"`\ud835\udd3c[\u03b2\u1d40\u03b2\u0302] = h\u00b2` (for at least one marker).","title":"Expected cross product of true and estimated effects"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-beta-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.integral_beta_sq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-beta-sq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"`\ud835\udd3c[\u03b2\u1d40\u03b2] = h\u00b2` (for at least one marker).","title":"Expected squared norm of the true effects"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-beta-hat-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.integral_betaHat_sq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-beta-hat-sq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"`\ud835\udd3c[\u03b2\u0302\u1d40\u03b2\u0302] = h\u00b2 + M s\u00b2` (for at least one marker).","title":"Expected squared norm of the estimated effects"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/accuracy-sq-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.accuracySq_eq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#accuracy-sq-eq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"Polygenic-score accuracy as a ratio of expectations: `r\u00b2_g = h\u00b2 / (h\u00b2 + M s\u00b2)`. (If `h\u00b2 = 0` both sides are `0`.)","title":"Polygenic-score accuracy"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/accuracy-sq-eq-of-s2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.accuracySq_eq_of_s2","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#accuracy-sq-eq-of-s2","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"With sampling variance `s\u00b2 = 1 / N`, `r\u00b2_g = N h\u00b2 / (N h\u00b2 + M)`.","title":"Daetwyler\u2013Dudbridge formula"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/has-orthogonal-columns","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.HasOrthogonalColumns","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#has-orthogonal-columns","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The training genotype matrix has orthogonal columns with sum of squares `N`, `\u2211 i, X i k * X i l = N [k = l]`; for centered columns this says that the markers are uncorrelated, with unit sample variance, in the training sample.","title":"Training genotype matrix with orthogonal standardized columns"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/of-ldscore","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.ofLDScore","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#of-ldscore","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The effect-estimate model obtained from the LD-score model when the effects are estimated by the normalized scores `\u03b2\u0302 = X\u1d40 y / N` on a training matrix with orthogonal standardized columns: the error is `e k = \u2211 i, X i k \u03b5 i / N` and its sampling variance is `(1 - h\u00b2) / N`.","title":"Effect-estimate model derived from the LD-score model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/of-ldscore-beta-hat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.ofLDScore_betaHat","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#of-ldscore-beta-hat","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The estimates of `ofLDScore` are the normalized scores `X\u1d40 y / N` of the LD-score model.","title":"Estimates of the derived model are normalized scores"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/accuracy-sq-of-ldscore","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.PRSModel.accuracySq_ofLDScore","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#accuracy-sq-of-ldscore","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"With orthogonal standardized training columns, `\u03b2\u0302 = X\u1d40 y / N` and residual variance `1 - h\u00b2`, `r\u00b2_g = N h\u00b2 / (N h\u00b2 + M (1 - h\u00b2))`. The cited `N h\u00b2 / (N h\u00b2 + M)` takes the residual variance to be `1`.","title":"Polygenic-score accuracy in the LD-score model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/marginal-estimate","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.marginalEstimate","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#marginal-estimate","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The marginal least-squares estimate `x_j\u1d40 y / x_j\u1d40 x_j` of marker `j` (single-marker regression without intercept, on centered genotypes and phenotype as in Yang et al. 2012).","title":"Marginal least-squares estimate of a marker effect"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/gram-diag","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.gramDiag","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#gram-diag","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The diagonal `D = diag(X\u1d40X)` of the Gram matrix.","title":"Diagonal of the genotype Gram matrix"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/transpose-mul-vec-eq-gram-diag-mul-vec","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.transpose_mulVec_eq_gramDiag_mulVec","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#transpose-mul-vec-eq-gram-diag-mul-vec","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"`X\u1d40 y = D \u03b2\u0302` for the marginal estimates `\u03b2\u0302` (Yang et al. 2012). This holds for every genotype matrix: a column with `x_j\u1d40 x_j = 0` is zero, so both sides vanish there.","title":"Genotype-phenotype cross products from marginal estimates"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/cojo-joint","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.cojo_joint","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#cojo-joint","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"COJO joint estimate (Yang et al. 2012, eq. (5)): the joint least-squares estimate `(X\u1d40X)\u207b\u00b9 X\u1d40 y` equals `(X\u1d40X)\u207b\u00b9 D \u03b2\u0302` in terms of the marginal estimates `\u03b2\u0302`, so it can be computed from summary statistics and `X\u1d40X`. (Stated for every `X`; it is the least-squares estimate when `X\u1d40X` is invertible.)","title":"COJO joint estimate"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/cojo-joint-of-diagonal","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.cojo_joint_of_diagonal","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#cojo-joint-of-diagonal","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"For markers uncorrelated in the sample (`X\u1d40X` diagonal) without zero columns, the joint estimate is the vector of marginal estimates.","title":"Joint estimate equals marginal estimates for uncorrelated markers"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/conditional-estimate","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.conditionalEstimate","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#conditional-estimate","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The unconditional algebraic formula from Yang et al. 2012, eq. (15): `(X\u2082\u1d40X\u2082)\u207b\u00b9 X\u2082\u1d40 y - (X\u2082\u1d40X\u2082)\u207b\u00b9 X\u2082\u1d40X\u2081 (X\u2081\u1d40X\u2081)\u207b\u00b9 X\u2081\u1d40 y`. When both Gram determinants are units, this has the usual interpretation as the block-2 fit conditional on the block-1 fit. For a singular Gram matrix, Mathlib's nonsingular inverse is zero and the value can be a junk value.","title":"Conditional estimate of second-block effects (COJO)"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/conditional-estimate-eq-summary","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.conditionalEstimate_eq_summary","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#conditional-estimate-eq-summary","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"COJO conditional algebraic formula from summary statistics (Yang et al. 2012, eq. (17)): `b\u0302\u2082|b\u0302\u2081 = (X\u2082\u1d40X\u2082)\u207b\u00b9 D\u2082 \u03b2\u0302\u2082 - (X\u2082\u1d40X\u2082)\u207b\u00b9 X\u2082\u1d40X\u2081 (X\u2081\u1d40X\u2081)\u207b\u00b9 D\u2081 \u03b2\u0302\u2081`.","title":"COJO conditional algebraic formula from summary statistics"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/conditional-estimate-eq-residual","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.conditionalEstimate_eq_residual","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#conditional-estimate-eq-residual","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"Unconditionally, the conditional formula equals the nonsingular-inverse formula applied to the algebraic residual `y - X\u2081 ((X\u2081\u1d40X\u2081)\u207b\u00b9 X\u2081\u1d40 y)`. If both Gram determinants are units, these are the successive least-squares fits on blocks 1 and 2. With a singular Gram matrix they need not minimize residual error. This is in general not the block-2 part of the joint formula on both blocks (the formulas agree when\u2026","title":"Conditional estimate as a fit to the first-block residual"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/strat-ldscore","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.stratLDScore","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#strat-ldscore","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The stratified normalized-cross-product score `\u2113(j, C) = \u2211_{k \u2208 C} (X_j\u1d40 X_k / N)\u00b2`. For a standardized genotype matrix this is the stratified LD score of marker `j` with respect to annotation category `C` (Finucane et al. 2015).","title":"Stratified LD score"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/stratified-model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.StratifiedModel","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#stratified-model","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The stratified LD-score model (Finucane et al. 2015), at the level of raw second moments: annotation categories `cat c \u2286 {markers}` (possibly overlapping), per-category contributions `tau c`, effects with `\u222b \u03b2 k \u03b2 l = [k = l] \u2211_c \u03c4_c [k \u2208 cat c]`, noise with `\u222b \u03b5 i \u03b5 i' = \u03c3\u00b2_e [i = i']`, effects orthogonal to noise. The cited model normalizes the phenotypic variance so that `\u03c3\u00b2_e \u2248 1`; the confounding term `N a` is\u2026","title":"Stratified LD-score model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/pheno","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.StratifiedModel.pheno","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#pheno","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The phenotype `y = X \u03b2 + \u03b5`.","title":"Phenotype in the stratified LD-score model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/beta-hat-2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.StratifiedModel.betaHat","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#beta-hat-2","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The `N`-normalized score `\u03b2\u0302 j = X_j\u1d40 y / N`.","title":"Normalized score in the stratified LD-score model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.StratifiedModel.chiSq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#chi-sq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The association statistic `\u03c7\u00b2 j = N \u03b2\u0302 j \u00b2`.","title":"Association statistic in the stratified LD-score model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.StratifiedModel.integral_chiSq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-chi-sq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The exact expectation of the association statistic in the stratified model, for any genotype matrix: `\ud835\udd3c[\u03c7\u00b2 j] = N \u2211_c \u03c4_c \u2113(j, cat c) + \u03c3\u00b2_e (\u2211 i, X i j \u00b2) / N`.","title":"Expected association statistic in the stratified LD-score model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-chi-sq-of-standardized","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.StratifiedModel.integral_chiSq_of_standardized","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-chi-sq-of-standardized","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"Stratified LD-score regression (Finucane et al. 2015): for a standardized genotype matrix and `N \u2260 0`, `\ud835\udd3c[\u03c7\u00b2 j] = N \u2211_c \u03c4_c \u2113(j, cat c) + \u03c3\u00b2_e`. Full-matrix standardization makes every normalized cross product in `stratLDScore` a sample correlation.","title":"Stratified LD-score regression"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-chi-sq-single-causal","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.StratifiedModel.integral_chiSq_single_causal","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-chi-sq-single-causal","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"One causal variant (the LD-proxy noncentrality; Pritchard & Przeworski 2001): if the genotype matrix is standardized and the only category is the single causal marker `c` with variance `\u03c4`, then for a marker `m` and `N \u2260 0`, `\ud835\udd3c[\u03c7\u00b2 m] = N \u03c4 r m c \u00b2 + \u03c3\u00b2_e`.","title":"One causal variant"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/sample-corr","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.sampleCorr","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#sample-corr","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The in-sample normalized cross product `\u2211_{i \u2208 S} X i j X i k / |S|` of markers `j`, `k` in the sample `S` of a pool of individuals (the sample LD correlation for a matrix standardized within `S`).","title":"In-sample LD correlation"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/cross-trait-model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.CrossTraitModel","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#cross-trait-model","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The bivariate LD-score model (Bulik-Sullivan et al. 2015b), at the level of the cross second moments that enter `\ud835\udd3c[z\u2081 z\u2082]`: a pool of `n` genotyped individuals with genotype matrix `X`, two GWAS samples `S\u2081`, `S\u2082` of the pool (possibly overlapping), effects `\u03b2\u2081`, `\u03b2\u2082` with `\u222b \u03b2\u2081 k \u03b2\u2082 l = (\u03c1_g / M) [k = l]` (genetic covariance `\u03c1_g`), noises `\u03b5\u2081`, `\u03b5\u2082` with `\u222b \u03b5\u2081 i \u03b5\u2082 i' = \u03c1_e [i = i']` (residual cross-moment `\u03c1_e`\u2026","title":"Bivariate LD-score model"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/z1","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.CrossTraitModel.z1","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#z1","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The scaled marginal score `z\u2081 j = \u221aN\u2081 X_j\u1d40 y\u2081 / N\u2081` of marker `j` for trait 1, computed on `S\u2081` with `y\u2081 = X \u03b2\u2081 + \u03b5\u2081`. The model does not assert the normalization needed for a z-score.","title":"Scaled marginal score for the first trait"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/z2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.CrossTraitModel.z2","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#z2","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The analogous scaled marginal score `z\u2082 j` of marker `j` for trait 2, computed on `S\u2082`.","title":"Scaled marginal score for the second trait"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-z-mul-z","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.CrossTraitModel.integral_z_mul_z","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-z-mul-z","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"With `N\u2081 = |S\u2081|`, `N\u2082 = |S\u2082|`, `\ud835\udd3c[z\u2081 j z\u2082 j] = \u221a(N\u2081 N\u2082) (\u03c1_g / M) \u2211_k r\u00b9 j k r\u00b2 j k + \u03c1_e (\u2211_{i \u2208 S\u2081 \u2229 S\u2082} X i j \u00b2) / \u221a(N\u2081 N\u2082)`, with the in-sample normalized cross products `r\u00b9`, `r\u00b2` (`sampleCorr`).","title":"Cross-trait scaled-score identity, exact form"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/integral-z-mul-z-of-same-ld","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.CrossTraitModel.integral_z_mul_z_of_sameLD","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#integral-z-mul-z-of-same-ld","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"If the two samples have the same normalized cross products with marker `j` and the overlap marker is standardized at `j` (`\u2211_{i \u2208 S\u2081 \u2229 S\u2082} X i j \u00b2 = N_s`), then `\ud835\udd3c[z\u2081 j z\u2082 j] = \u221a(N\u2081 N\u2082) \u03c1_g \u2113_j / M + \u03c1_e N_s / \u221a(N\u2081 N\u2082)`. Here `\u03c1_e` is only the residual cross-moment specified by `CrossTraitModel`; this theorem does not state the Bulik-Sullivan phenotypic-correlation overlap formula or establish z-score normalization.","title":"Equal normalized-cross-product specialization of the residual cross-moment identity"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/total","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.total","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#total","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The number of individuals `N = \u2211 (r i + s i)`.","title":"Number of individuals in a case-control genotype table"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/numer","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.numer","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#numer","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The trend numerator `U = \u2211 i, i (r i S - s i R)` with `R = \u2211 r`, `S = \u2211 s`.","title":"Numerator of the Cochran\u2013Armitage trend statistic"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/score-var","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.scoreVar","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#score-var","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The pooled genotype-score variance `\u2211 i\u00b2 n_i / N - (\u2211 i n_i / N)\u00b2`, `n_i = r i + s i`.","title":"Pooled genotype-score variance"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/trend-stat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.trendStat","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#trend-stat","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The Cochran\u2013Armitage trend statistic with scores `0, 1, 2` (Armitage 1955): `N U\u00b2 / (R S (N \u2211 i\u00b2 n_i - (\u2211 i n_i)\u00b2))`.","title":"Cochran\u2013Armitage trend statistic"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/corr-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.corrSq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#corr-sq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The squared sample correlation of case status (`1` for cases, `0` for controls) and the genotype score over the `N` individuals of the table.","title":"Squared correlation of case status and genotype score"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/allele-freq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.alleleFreq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#allele-freq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The pooled frequency `p = \u2211 i n_i / (2 N)` of the counted allele.","title":"Pooled allele frequency"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/pearson2x2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.pearson2x2","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#pearson2x2","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"Pearson's statistic `n (a d - b c)\u00b2 / ((a + b)(c + d)(a + c)(b + d))` of a `2 \u00d7 2` table `[[a, b], [c, d]]` with `n = a + b + c + d`.","title":"Pearson's chi-squared statistic of a 2 \u00d7 2 table"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/allelic-stat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.allelicStat","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#allelic-stat","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The allelic test: Pearson's statistic of the `2 \u00d7 2` table of case and control allele counts (`2N` alleles).","title":"Allelic test statistic"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/trend-stat-eq-total-mul-corr-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.trendStat_eq_total_mul_corrSq","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#trend-stat-eq-total-mul-corr-sq","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The trend test is `N r\u00b2` (Armitage 1955): the Cochran\u2013Armitage statistic is `N` times the squared sample correlation of case status and genotype score. (In degenerate tables, where a denominator vanishes, both sides are `0` by the convention `x / 0 = 0`.)","title":"The trend test is `N r\u00b2`"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/score-var-sub-hwe-var","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.scoreVar_sub_hweVar","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#score-var-sub-hwe-var","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The observed score variance minus the Hardy\u2013Weinberg score variance `2 p q`: `Var - 2 p (1 - p) = (4 n\u2080 n\u2082 - n\u2081\u00b2) / (2 N\u00b2)`.","title":"Departure of the genotype-score variance from Hardy\u2013Weinberg"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/allelic-stat-mul-hwe-var","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.allelicStat_mul_hweVar","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#allelic-stat-mul-hwe-var","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"Allelic versus trend test (Sasieni 1997): with cases and controls present, a pooled allele frequency strictly between `0` and `1` and a positive score variance, `allelic \u00b7 2 p q = trend \u00b7 Var`.","title":"Allelic versus trend test"},{"id":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics/allelic-stat-eq-trend-stat-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.GenomicPrediction.ArmitageTrend.allelicStat_eq_trendStat_iff","parent":46552,"result":true,"route":"../roadmap/Biology/statistical-genetics/genomic-prediction-from-summary-statistics/index.html#allelic-stat-eq-trend-stat-iff","scope":"Biology/statistical-genetics/genomic-prediction-from-summary-statistics","status":"fully_proved","summary":"The allelic test equals the trend test iff the pooled sample is in Hardy\u2013Weinberg proportions (`n\u2081\u00b2 = 4 n\u2080 n\u2082`; Sasieni 1997), whenever the trend numerator is nonzero.","title":"The allelic test equals the trend test iff the pooled sample is in Hardy\u2013Weinberg proportions"},{"data_url":"scopes/Biology/statistical-genetics/mendelian-randomization.json","id":"Biology/statistical-genetics/mendelian-randomization","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","title":"Mendelian randomization"},{"id":"Biology/statistical-genetics/mendelian-randomization/linear-moment-model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.LinearMomentModel","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#linear-moment-model","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"An observational linear moment model: variables `Z`, `X`, `Y` and `U` on a probability space, all square integrable, with the algebraic decomposition `Y = \u03b1 + \u03b2 X + U`. The residual `U` is an arbitrary observed-variable function. Consequently `outcome_eq` does not exclude direct effects of `Z` under interventions, and `\u03b2` is only the coefficient in this decomposition. Orthogonality (`Cov(Z, U) = 0`) and nonzero\u2026","title":"Linear moment model for instrumental variables"},{"id":"Biology/statistical-genetics/mendelian-randomization/covariance-outcome","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.LinearMomentModel.covariance_outcome","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#covariance-outcome","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The covariance of any square-integrable variable with the outcome splits along the observational decomposition: `Cov(V, Y) = \u03b2 Cov(V, X) + Cov(V, U)`.","title":"Covariance with the outcome in a linear moment model"},{"id":"Biology/statistical-genetics/mendelian-randomization/covariance-div-covariance","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.LinearMomentModel.covariance_div_covariance","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#covariance-div-covariance","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The Wald covariance identity: if `Cov(Z, U) = 0` and `Cov(Z, X) \u2260 0`, then `Cov(Z, Y) / Cov(Z, X) = \u03b2`. For a two-valued `Z` the ratio is the grouping estimator of Wald (1940), the difference of the group means of `Y` over that of `X`. This is a moment-condition identity, not a formal causal-identification result.","title":"Wald ratio identity"},{"id":"Biology/statistical-genetics/mendelian-randomization/covariance-div-covariance-of-indep-fun","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.LinearMomentModel.covariance_div_covariance_of_indepFun","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#covariance-div-covariance-of-indep-fun","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The Wald covariance identity when the instrument is independent of the residual.","title":"Wald ratio identity for an instrument independent of the residual"},{"id":"Biology/statistical-genetics/mendelian-randomization/covariance-div-variance-div","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.LinearMomentModel.covariance_div_variance_div","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#covariance-div-variance-div","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The Wald ratio as a ratio of regression slopes: the slope of `Y` on `Z` divided by the slope of `X` on `Z` (the form used with SNP\u2013outcome and SNP\u2013exposure associations) is `\u03b2`. This is again an observational moment identity.","title":"Wald ratio as a ratio of regression slopes"},{"id":"Biology/statistical-genetics/mendelian-randomization/covariance-div-variance","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.LinearMomentModel.covariance_div_variance","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#covariance-div-variance","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The ordinary least-squares coefficient identity for `Y` on `X` is `\u03b2 + Cov(X, U) / Var(X)`, which differs from `\u03b2` exactly when `Cov(X, U) \u2260 0`.","title":"Bias of the ordinary least-squares coefficient under confounding"},{"id":"Biology/statistical-genetics/mendelian-randomization/ratio-estimate","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.ratioEstimate","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#ratio-estimate","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The ratio estimate `\u03b2\u0302_{Yj} / \u03b2\u0302_{Xj}` of SNP `j` (Lean's `x / 0 = 0` gives `0` when `\u03b2\u0302_{Xj} = 0`; such SNPs carry weight `0` in `ivwWeight`).","title":"Ratio estimate of a single SNP"},{"id":"Biology/statistical-genetics/mendelian-randomization/ivw-weight","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.ivwWeight","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#ivw-weight","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The first-order inverse-variance weight `w_j = \u03b2\u0302_{Xj}\u00b2 / \u03c3_j\u00b2` of the ratio estimate of SNP `j`, where `\u03c3_j` is the standard error of `\u03b2\u0302_{Yj}` (Burgess, Butterworth & Thompson 2013).","title":"Inverse-variance weight of a ratio estimate"},{"id":"Biology/statistical-genetics/mendelian-randomization/ivw","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.ivw","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#ivw","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The inverse-variance-weighted (IVW) estimator `\u2211 \u03b2\u0302_{Xj} \u03b2\u0302_{Yj} \u03c3_j\u207b\u00b2 / \u2211 \u03b2\u0302_{Xj}\u00b2 \u03c3_j\u207b\u00b2` (Burgess, Butterworth & Thompson 2013). The standard errors are positive in the model; it is the junk value `0` if every `\u03b2\u0302_{Xj} = 0`.","title":"Inverse-variance-weighted estimator"},{"id":"Biology/statistical-genetics/mendelian-randomization/weighted-ssr","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.weightedSSR","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#weighted-ssr","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The weighted residual sum of squares `\u2211 (\u03b2\u0302_{Yj} - b \u03b2\u0302_{Xj})\u00b2 / \u03c3_j\u00b2` of the regression through the origin with slope `b`.","title":"Weighted residual sum of squares through the origin"},{"id":"Biology/statistical-genetics/mendelian-randomization/ivw-eq-weighted-mean","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.ivw_eq_weightedMean","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#ivw-eq-weighted-mean","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The IVW estimator is the weighted mean of the ratio estimates with weights `w_j = \u03b2\u0302_{Xj}\u00b2 / \u03c3_j\u00b2`. This holds for all data: a SNP with `\u03b2\u0302_{Xj} = 0` has weight `0` and contributes nothing to either side.","title":"IVW estimator as a weighted mean of ratio estimates"},{"id":"Biology/statistical-genetics/mendelian-randomization/ivw-le-weighted-ssr","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.ivw_le_weightedSSR","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#ivw-le-weighted-ssr","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The IVW estimator is the weighted least-squares slope through the origin: it minimizes the weighted residual sum of squares. This holds for all data: if every `\u03b2\u0302_{Xj} = 0` the sum does not depend on the slope (and `ivw` is the junk value `0`), and a SNP with `\u03c3_j = 0` has weight `0` on both sides by Lean's `x / 0 = 0`.","title":"IVW estimator minimizes the weighted residual sum of squares"},{"id":"Biology/statistical-genetics/mendelian-randomization/sum-ivw-weight-pos","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.sum_ivwWeight_pos","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#sum-ivw-weight-pos","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The total weight is positive as soon as some SNP has `\u03b2\u0302_{Xj} \u2260 0`.","title":"Positivity of the total IVW weight"},{"id":"Biology/statistical-genetics/mendelian-randomization/ivw-of-proportional","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.ivw_of_proportional","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#ivw-of-proportional","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"Without error, IVW recovers the proportionality coefficient: if `\u03b2\u0302_{Yj} = \u03b2 \u03b2\u0302_{Xj}` for all `j` and some `\u03b2\u0302_{Xj} \u2260 0`, then the IVW estimate is `\u03b2`. This algebraic statement does not itself give `\u03b2` a causal interpretation.","title":"IVW recovers an exact proportionality coefficient"},{"id":"Biology/statistical-genetics/mendelian-randomization/integral-ivw","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.integral_ivw","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#integral-ivw","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"Unbiasedness of IVW (Burgess, Butterworth & Thompson 2013), with the SNP\u2013exposure associations `\u03b2\u0302_{Xj}` treated as fixed: if `\u03b2\u0302_{Yj} = \u03b2 \u03b2\u0302_{Xj} + \u03b5_j` with integrable mean-zero errors, the IVW estimator has mean `\u03b2`.","title":"Unbiasedness of the IVW estimator"},{"id":"Biology/statistical-genetics/mendelian-randomization/variance-ivw","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.variance_ivw","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#variance-ivw","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The first-order variance of IVW (Burgess, Butterworth & Thompson 2013): with the `\u03b2\u0302_{Xj}` fixed and `\u03b2\u0302_{Yj} = \u03b2 \u03b2\u0302_{Xj} + \u03b5_j`, where the errors are square integrable, pairwise uncorrelated (for instance independent) and `Var(\u03b5_j) = \u03c3_j\u00b2`, the IVW estimator has variance `(\u2211 w_j)\u207b\u00b9 = (\u2211 \u03b2\u0302_{Xj}\u00b2 / \u03c3_j\u00b2)\u207b\u00b9`.","title":"Variance of the IVW estimator"},{"id":"Biology/statistical-genetics/mendelian-randomization/weighted-mean","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.weightedMean","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#weighted-mean","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The weighted mean `\u2211 v_j a_j / \u2211 v_j`.","title":"Weighted mean"},{"id":"Biology/statistical-genetics/mendelian-randomization/weighted-cov","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.weightedCov","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#weighted-cov","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The weighted covariance `\u2211 v_j (a_j - \u0101_v)(b_j - b\u0304_v) / \u2211 v_j`, where `\u0101_v` and `b\u0304_v` are weighted means.","title":"Weighted covariance"},{"id":"Biology/statistical-genetics/mendelian-randomization/egger-slope","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.eggerSlope","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#egger-slope","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The MR-Egger slope (Bowden, Davey Smith & Burgess 2015): the slope of the weighted linear regression of `\u03b2\u0302_{Yj}` on `\u03b2\u0302_{Xj}` with an intercept, `Cov_v(\u03b2\u0302_X, \u03b2\u0302_Y) / Var_v(\u03b2\u0302_X)`. The paper's weights are `v_j = \u03c3_j\u207b\u00b2`.","title":"MR-Egger slope"},{"id":"Biology/statistical-genetics/mendelian-randomization/egger-intercept","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.eggerIntercept","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#egger-intercept","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The MR-Egger intercept `\u03b2\u0302\u0304_Y - slope \u00b7 \u03b2\u0302\u0304_X` (weighted means).","title":"MR-Egger intercept"},{"id":"Biology/statistical-genetics/mendelian-randomization/egger-ssr","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.eggerSSR","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#egger-ssr","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The weighted residual sum of squares `\u2211 v_j (\u03b2\u0302_{Yj} - a - b \u03b2\u0302_{Xj})\u00b2` of the line with intercept `a` and slope `b`.","title":"Weighted residual sum of squares of a line"},{"id":"Biology/statistical-genetics/mendelian-randomization/egger-ssr-le","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.eggerSSR_le","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#egger-ssr-le","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The MR-Egger line is the weighted least-squares line: for nonnegative weights and `\u03b2\u0302_X` of nonzero weighted variance (so the total weight is positive and `\u03b2\u0302_X` is not constant on the support of the weights), it minimizes the weighted residual sum of squares.","title":"MR-Egger line is the weighted least-squares line"},{"id":"Biology/statistical-genetics/mendelian-randomization/egger-slope-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.eggerSlope_eq","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#egger-slope-eq","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The MR-Egger slope under pleiotropy (Bowden, Davey Smith & Burgess 2015): if the SNP\u2013outcome associations are `\u0393_j = \u03b1_j + \u03b2 \u03b3_j`, with pleiotropic terms `\u03b1_j` and SNP\u2013exposure associations `\u03b3_j`, the Egger slope of `\u0393` on `\u03b3` is `\u03b2 + Cov_v(\u03b3, \u03b1) / Var_v(\u03b3)`.","title":"MR-Egger slope under pleiotropy"},{"id":"Biology/statistical-genetics/mendelian-randomization/egger-intercept-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.eggerIntercept_eq","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#egger-intercept-eq","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The MR-Egger intercept under pleiotropy: it is the weighted mean pleiotropic effect `\u1fb1_v` minus `Cov_v(\u03b3, \u03b1) / Var_v(\u03b3) \u00b7 \u03b3\u0304_v`.","title":"MR-Egger intercept under pleiotropy"},{"id":"Biology/statistical-genetics/mendelian-randomization/egger-slope-eq-of-inside","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.eggerSlope_eq_of_inside","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#egger-slope-eq-of-inside","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The finite-sample InSIDE identity (Bowden, Davey Smith & Burgess 2015): if the pleiotropic terms are uncorrelated with the instrument strengths in the weighted sense, `Cov_v(\u03b1, \u03b3) = 0`, the MR-Egger slope is the coefficient `\u03b2` and the intercept is the weighted mean pleiotropic term `\u1fb1_v`. This identity alone does not encode a causal interpretation.","title":"InSIDE identity for MR-Egger"},{"id":"Biology/statistical-genetics/mendelian-randomization/ivw-of-pleiotropy","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.ivw_of_pleiotropy","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#ivw-of-pleiotropy","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"Directional pleiotropy biases IVW: if `\u0393_j = \u03b1_j + \u03b2 \u03b3_j`, the IVW estimate is `\u03b2 + \u2211 \u03b3_j \u03b1_j \u03c3_j\u207b\u00b2 / \u2211 \u03b3_j\u00b2 \u03c3_j\u207b\u00b2`. For example a common pleiotropic term `\u03b1_j = \u03b1 \u2260 0` biases IVW whenever `\u2211 \u03b3_j \u03c3_j\u207b\u00b2 \u2260 0`, whereas a constant `\u03b1` satisfies InSIDE, so `eggerSlope_eq_of_inside` gives MR-Egger slope `\u03b2` and intercept `\u03b1`.","title":"Directional pleiotropy biases the IVW estimator"},{"id":"Biology/statistical-genetics/mendelian-randomization/lintegral-enorm-div-eq-top","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.lintegral_enorm_div_eq_top","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#lintegral-enorm-div-eq-top","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"A ratio with a Gaussian denominator has no first moment (a classical fact, recalled in the weak-instrument analysis of Burgess & Thompson 2011): if `D ~ N(m, v)` with `v > 0` and `N` is independent of `D` and not almost surely `0`, then `\ud835\udd3c|N / D| = \u221e`. Applied to the ratio estimator `\u03b2\u0302_Y / \u03b2\u0302_X` with a normally distributed SNP\u2013exposure estimate `\u03b2\u0302_X`.","title":"Ratio with a Gaussian denominator has infinite first moment"},{"id":"Biology/statistical-genetics/mendelian-randomization/not-integrable-div-gaussian","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.not_integrable_div_gaussian","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#not-integrable-div-gaussian","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The ratio of two independent normal variables, the denominator with positive variance and the numerator not almost surely zero (`(mN, vN) \u2260 (0, 0)`), is not integrable.","title":"Ratio of independent normal variables is not integrable"},{"id":"Biology/statistical-genetics/mendelian-randomization/compliance-type","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.ComplianceType","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#compliance-type","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The four compliance types of Angrist, Imbens & Rubin (1996), determined by the pair of potential treatments `(D(0), D(1))`.","title":"Compliance types of Angrist, Imbens and Rubin"},{"id":"Biology/statistical-genetics/mendelian-randomization/compliance-type-2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.complianceType","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#compliance-type-2","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The compliance type of the potential treatments `(D(0), D(1)) = (d\u2080, d\u2081)`.","title":"Compliance type of a pair of potential treatments"},{"id":"Biology/statistical-genetics/mendelian-randomization/potential-outcomes","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#potential-outcomes","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"A potential-outcomes model with a binary instrument and a binary treatment (Imbens & Angrist 1994; Angrist, Imbens & Rubin 1996): the assigned instrument `Z`, the potential treatments `D(z)` and the potential outcomes `Y(z, d)`, all measurable. The observed treatment is `D(Z)` and the observed outcome `Y(Z, D(Z))`. The identifying assumptions (independence, exclusion, monotonicity, relevance) are separate predicates.","title":"Potential-outcomes model with binary instrument and treatment"},{"id":"Biology/statistical-genetics/mendelian-randomization/observed-treatment","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.observedTreatment","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#observed-treatment","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The observed treatment `D = D(Z)`.","title":"Observed treatment in a potential-outcomes model"},{"id":"Biology/statistical-genetics/mendelian-randomization/observed-outcome","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.observedOutcome","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#observed-outcome","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The observed outcome `Y = Y(Z, D(Z))`.","title":"Observed outcome in a potential-outcomes model"},{"id":"Biology/statistical-genetics/mendelian-randomization/type","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.type","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#type","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The compliance type of a unit.","title":"Compliance type of a unit"},{"id":"Biology/statistical-genetics/mendelian-randomization/compliers","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.compliers","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#compliers","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The compliers, `D(0) = 0` and `D(1) = 1`.","title":"Compliers in a potential-outcomes model"},{"id":"Biology/statistical-genetics/mendelian-randomization/independence","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.Independence","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#independence","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"Independence (random assignment): the instrument is independent of the vector of all potential treatments and potential outcomes.","title":"Independence assumption for an instrument"},{"id":"Biology/statistical-genetics/mendelian-randomization/exclusion","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.Exclusion","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#exclusion","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"Exclusion: the instrument affects the outcome only through the treatment, `Y(0, d) = Y(1, d)`.","title":"Exclusion restriction for an instrument"},{"id":"Biology/statistical-genetics/mendelian-randomization/monotonicity","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.Monotonicity","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#monotonicity","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"Monotonicity: assignment never decreases the treatment, `D(0) \u2264 D(1)`; there are no defiers.","title":"Monotonicity assumption (no defiers)"},{"id":"Biology/statistical-genetics/mendelian-randomization/arm-mean","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.armMean","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#arm-mean","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The mean of `f` in the arm `Z = z`, `\ud835\udd3c[f | Z = z]`, the expectation under the conditional measure. It is the junk value `0` when `P(Z = z) = 0`.","title":"Mean of a variable within an instrument arm"},{"id":"Biology/statistical-genetics/mendelian-randomization/treated","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.treated","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#treated","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The observed treatment as a `0`/`1` real variable.","title":"Observed treatment as a 0-1 real variable"},{"id":"Biology/statistical-genetics/mendelian-randomization/wald-estimand","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.waldEstimand","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#wald-estimand","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The Wald (instrumental-variable) estimand `(\ud835\udd3c[Y | Z = 1] - \ud835\udd3c[Y | Z = 0]) / (\ud835\udd3c[D | Z = 1] - \ud835\udd3c[D | Z = 0])`.","title":"Wald instrumental-variable estimand"},{"id":"Biology/statistical-genetics/mendelian-randomization/arm-mean-treated-sub","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.armMean_treated_sub","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#arm-mean-treated-sub","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The first stage (Imbens & Angrist 1994): under independence and monotonicity, with both arms of positive probability, the difference in treatment rates between the arms is the proportion of compliers.","title":"First stage: treatment-rate difference equals complier share"},{"id":"Biology/statistical-genetics/mendelian-randomization/arm-mean-outcome-sub","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.armMean_outcome_sub","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#arm-mean-outcome-sub","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The reduced form (Imbens & Angrist 1994): under independence, exclusion and monotonicity, with integrable potential outcomes and both arms of positive probability, the difference in mean outcomes between the arms is `\ud835\udd3c[(Y(z, 1) - Y(z, 0)) 1_complier]`, for either `z`.","title":"Reduced form: outcome difference is the complier-weighted effect"},{"id":"Biology/statistical-genetics/mendelian-randomization/wald-estimand-eq-late-totalized","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.waldEstimand_eq_late_totalized","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#wald-estimand-eq-late-totalized","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The unrestricted algebraic identity behind LATE. If the complier set has measure zero, both sides totalize to `0`; therefore this theorem is not an identification or complier-average claim at that boundary. Use `waldEstimand_eq_late` for the source-faithful result with relevance.","title":"Unrestricted algebraic identity behind the LATE theorem"},{"id":"Biology/statistical-genetics/mendelian-randomization/wald-estimand-eq-late","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MendelianRandomization.PotentialOutcomes.waldEstimand_eq_late","parent":46602,"result":true,"route":"../roadmap/Biology/statistical-genetics/mendelian-randomization/index.html#wald-estimand-eq-late","scope":"Biology/statistical-genetics/mendelian-randomization","status":"fully_proved","summary":"The local average treatment effect theorem (Imbens & Angrist 1994, Theorem 1; Angrist, Imbens & Rubin 1996): under independence, exclusion, monotonicity and relevance, with integrable potential outcomes and both arms of positive probability, the Wald estimand is the average treatment effect among compliers, `\ud835\udd3c[Y(1) - Y(0) | complier]` (read at either instrument value `z`, which exclusion makes irrelevant).\u2026","title":"Local average treatment effect theorem"},{"data_url":"scopes/Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici.json","id":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/index.html","scope":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici","status":"fully_proved","title":"Posterior of the heterozygous selection coefficient"},{"id":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/balance-mean","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.ShetPosterior.balanceMean","parent":46648,"result":true,"route":"../roadmap/Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/index.html#balance-mean","scope":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici","status":"fully_proved","summary":"Mean PTV allele count `n U / s` at deterministic mutation\u2013selection balance, for `n` sampled chromosomes, PTV mutation rate `U` and heterozygous selection coefficient `s > 0`.","title":"Mean PTV count at mutation-selection balance"},{"id":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/balance-mean-antitone-on","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.ShetPosterior.balanceMean_antitoneOn","parent":46648,"result":true,"route":"../roadmap/Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/index.html#balance-mean-antitone-on","scope":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici","status":"fully_proved","summary":"The balance mean is antitone in `s > 0` for `n U \u2265 0`.","title":"Balance mean is decreasing in the selection coefficient"},{"id":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/posterior-tail-antitone-of-antitone-on","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.ShetPosterior.posteriorTail_antitone_of_antitoneOn","parent":46648,"result":true,"route":"../roadmap/Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/index.html#posterior-tail-antitone-of-antitone-on","scope":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici","status":"fully_proved","summary":"Posterior of `s_het` is stochastically nonincreasing in the PTV count, any antitone mean: for a prior `\u03bc` on `s_het` concentrated on `(0, \u221e)`, a Poisson count with mean `\u03bb` nonnegative and antitone on `(0, \u221e)`, integrable likelihoods and positive marginal likelihoods, `P(s_het > t \u2223 k') \u2264 P(s_het > t \u2223 k)` whenever `k \u2264 k'`.","title":"Posterior of `s_het` is stochastically nonincreasing in the PTV count, any antitone mean"},{"id":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/posterior-tail-balance-antitone","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.ShetPosterior.posteriorTail_balance_antitone","parent":46648,"result":true,"route":"../roadmap/Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici/index.html#posterior-tail-balance-antitone","scope":"Biology/statistical-genetics/posterior-of-the-heterozygous-selection-coeffici","status":"fully_proved","summary":"With the balance mean `n U / s_het` (`n, U > 0`) and any probability prior on `s_het \u2208 (0, \u221e)`, the posterior probability `P(s_het > t \u2223 k)` is nonincreasing in the observed PTV count `k`.","title":"Cassa et al. (2017) model"},{"data_url":"scopes/Biology/statistical-genetics/principal-component-correction-for-stratificatio.json","id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","title":"Principal-component correction for stratification"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The adjusted (residualized) `y`: the residual of the least-squares regression of `y` on the subspace `V` (`y` minus its orthogonal projection onto `V`).","title":"Residualized vector after adjustment for a subspace"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted-mem-orthogonal","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted_mem_orthogonal","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted-mem-orthogonal","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Residuals are orthogonal to the adjustment subspace.","title":"Residuals lie in the orthogonal complement"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/inner-adjusted-eq-zero","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.inner_adjusted_eq_zero","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#inner-adjusted-eq-zero","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Residuals are orthogonal to every vector of the subspace, in particular to every axis.","title":"Residuals are orthogonal to every axis"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted-eq-star-projection-orthogonal","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted_eq_starProjection_orthogonal","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted-eq-star-projection-orthogonal","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The residual is the orthogonal projection onto `V\u15ee`.","title":"Residual as projection onto the orthogonal complement"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted-add","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted_add","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted-add","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Residualization is additive.","title":"Additivity of residualization"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted-smul","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted_smul","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted-smul","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Residualization is homogeneous.","title":"Homogeneity of residualization"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted-eq-zero-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted_eq_zero_iff","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted-eq-zero-iff","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The residual vanishes exactly on the subspace.","title":"Residual vanishes exactly on the subspace"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted-add-of-mem","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted_add_of_mem","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted-add-of-mem","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Adding a vector of the subspace (stratification lying in the axis span) does not change the residual.","title":"Stratification within the axis span does not change the residual"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted-of-mem-orthogonal","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted_of_mem_orthogonal","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted-of-mem-orthogonal","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Vectors orthogonal to the subspace are their own residuals: stratification outside the axis span is not removed.","title":"Vectors orthogonal to the subspace are their own residuals"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/adjusted-span-orthonormal","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.adjusted_span_orthonormal","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#adjusted-span-orthonormal","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Regressing on an orthonormal family of axes one axis at a time gives the joint residual: `y - \u2211\u2096 \u27eaa\u2096, y\u27eb a\u2096`.","title":"Residual with respect to an orthonormal family of axes"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/frisch-waugh-lovell","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.frischWaughLovell","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#frisch-waugh-lovell","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Frisch\u2013Waugh\u2013Lovell, uniqueness: in any least-squares decomposition `y = c + \u03b2 g + e` with `c \u2208 V` and `e \u22a5 V \u2294 span {g}`, the coefficient `\u03b2` is the coefficient of the regression of the residualized `y` on the residualized `g`, and the residual `e` is the residual of that regression.","title":"Frisch\u2013Waugh\u2013Lovell theorem, uniqueness"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/exists-frisch-waugh-lovell","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.exists_frischWaughLovell","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#exists-frisch-waugh-lovell","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Frisch\u2013Waugh\u2013Lovell, existence: the coefficient of the residual-on-residual regression gives a least-squares decomposition of `y` on `V \u2294 span {g}`.","title":"Frisch\u2013Waugh\u2013Lovell theorem, existence"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.chiSq","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#chi-sq","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The EIGENSTRAT association statistic `(N - K - 1) \u27ear_g, r_y\u27eb\u00b2 / (\u2016r_g\u2016\u00b2 \u2016r_y\u2016\u00b2)` of the adjusted genotype `r_g` and adjusted phenotype `r_y`: `(N - K - 1)` times their squared cosine, which is their squared sample correlation when both are centered (Price et al. 2006). When either adjusted vector is zero the ratio is the junk value `0`.","title":"EIGENSTRAT association statistic"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/chi-sq-add-of-mem","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.chiSq_add_of_mem","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#chi-sq-add-of-mem","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Adding vectors of `V` to the phenotype and the genotype does not change the EIGENSTRAT statistic.","title":"Stratification in the axis span is removed"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/chi-sq-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.chiSq_nonneg","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#chi-sq-nonneg","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The EIGENSTRAT statistic is nonnegative when `K + 1 \u2264 N`.","title":"Nonnegativity of the EIGENSTRAT statistic"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/chi-sq-of-mem","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.chiSq_of_mem","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#chi-sq-of-mem","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"A genotype lying in the axis span carries no signal after adjustment (its residual is zero, and the statistic is the junk value `0` of the zero denominator).","title":"Genotypes in the axis span carry no signal after adjustment"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/allele-freq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.alleleFreq","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#allele-freq","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The allele-frequency estimate `p\u0302\u2c7c = \u03bc\u2c7c / 2` of marker `j`, where `\u03bc\u2c7c` is the mean variant-allele count of column `j` (autosomal diploid data, counts in `{0, 1, 2}`).","title":"Allele-frequency estimate of a marker"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/normalized-genotype","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.normalizedGenotype","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#normalized-genotype","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Patterson's normalized genotype matrix `(C i j - \u03bc\u2c7c) / \u221a(p\u0302\u2c7c (1 - p\u0302\u2c7c))`. For a monomorphic marker (`p\u0302\u2c7c \u2208 {0, 1}`) the denominator is `0` and the column is the junk value `0`; Patterson et al. use polymorphic markers only.","title":"Patterson's normalized genotype matrix"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/relatedness-matrix","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.relatednessMatrix","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#relatedness-matrix","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Patterson's `N \u00d7 N` matrix `\u03a8 = M M\u1d40 / (number of markers)` of the normalized genotypes, whose top eigenvectors are the axes of variation.","title":"Patterson's relatedness matrix"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/relatedness-matrix-pos-semidef","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.relatednessMatrix_posSemidef","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#relatedness-matrix-pos-semidef","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Patterson's matrix `\u03a8` is positive semidefinite.","title":"Patterson's relatedness matrix is positive semidefinite"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/relatedness-matrix-mul-vec-one","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.relatednessMatrix_mulVec_one","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#relatedness-matrix-mul-vec-one","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The all-ones vector is in the kernel of `\u03a8` (columns of the normalized matrix are centered), so at most `N - 1` eigenvalues are nonzero.","title":"All-ones vector lies in the kernel of the relatedness matrix"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/axes","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.axes","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#axes","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The axes of variation: an orthonormal eigenbasis of `\u03a8`, in decreasing eigenvalue order.","title":"Axes of variation"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/axis-eigenvalues","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.axisEigenvalues","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#axis-eigenvalues","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The eigenvalues of `\u03a8` in decreasing order, matching `axes`.","title":"Eigenvalues of the axes of variation"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/top-axes-span","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.topAxesSpan","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#top-axes-span","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The span of the top `K` axes of variation, the subspace EIGENSTRAT adjusts for.","title":"Span of the top axes of variation"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/axes-mem-top-axes-span","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.axes_mem_topAxesSpan","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#axes-mem-top-axes-span","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Each of the top `K` axes lies in their span.","title":"Each top axis lies in the span of the top axes"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/finrank-top-axes-span","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.finrank_topAxesSpan","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#finrank-top-axes-span","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The top `K` axes span a `K`-dimensional subspace.","title":"Dimension of the span of the top axes"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/num-significant","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.numSignificant","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#num-significant","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Patterson's sequential count of significant axes: among the first `K` statistics, the length of the leading run of statistics exceeding the threshold `t` (testing stops at the first non-significant one).","title":"Sequential count of significant axes"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/num-significant-le","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.numSignificant_le","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#num-significant-le","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"At most `K` axes are declared significant.","title":"At most K axes are declared significant"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/lt-num-significant-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.lt_numSignificant_iff","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#lt-num-significant-iff","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"The first `numSignificant stat t K` indices are exactly those `k < K` whose statistics and all earlier ones exceed the threshold.","title":"Characterization of the axes counted as significant"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/lt-stat-of-lt-num-significant","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.lt_stat_of_lt_numSignificant","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#lt-stat-of-lt-num-significant","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"Every axis counted as significant has a statistic above the threshold.","title":"Significant axes have statistics above the threshold"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/num-significant-pos-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.numSignificant_pos_iff","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#num-significant-pos-iff","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"At least one axis is significant iff the leading statistic exceeds the threshold.","title":"Criterion for at least one significant axis"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/num-significant-antitone","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.numSignificant_antitone","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#num-significant-antitone","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"A higher threshold declares no more axes significant.","title":"Count of significant axes decreases with the threshold"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/num-significant-eq-card-of-antitone","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.numSignificant_eq_card_of_antitone","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#num-significant-eq-card-of-antitone","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"For decreasing statistics (such as ordered eigenvalues compared with a fixed threshold) the sequential count is the number of statistics above the threshold.","title":"Significant-axis count for decreasing statistics"},{"id":"Biology/statistical-genetics/principal-component-correction-for-stratificatio/measure-lt-num-significant-le","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Eigenstrat.measure_lt_numSignificant_le","parent":46653,"result":true,"route":"../roadmap/Biology/statistical-genetics/principal-component-correction-for-stratificatio/index.html#measure-lt-num-significant-le","scope":"Biology/statistical-genetics/principal-component-correction-for-stratificatio","status":"fully_proved","summary":"If, under the null hypothesis, the `k`-th statistic exceeds `t` with probability at most `1 - F t` (the Tracy\u2013Widom approximation with `F` the TW\u2081 distribution function, taken here as a hypothesis), then the rule declares more than `k` significant axes with probability at most `1 - F t`.","title":"Conditional error bound"},{"data_url":"scopes/Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio.json","id":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/index.html","scope":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio","status":"fully_proved","title":"Proximal contamination in mixed-model association"},{"id":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/mm-chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.ProximalContamination.mmChiSq","parent":46688,"result":true,"route":"../roadmap/Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/index.html#mm-chi-sq","scope":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio","status":"fully_proved","summary":"The mixed-model (generalized-least-squares score) statistic `\u03c7\u00b2(V) = (x\u1d40 V\u207b\u00b9 y)\u00b2 / (x\u1d40 V\u207b\u00b9 x)` of the marker `x` for the phenotype `y` under the phenotypic covariance `V`. For positive definite `V` and `x \u2260 0` the denominator is positive.","title":"Mixed-model association chi-squared statistic"},{"id":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/inv-add-smul-vec-mul-vec-mul-vec","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.ProximalContamination.inv_add_smul_vecMulVec_mulVec","parent":46688,"result":true,"route":"../roadmap/Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/index.html#inv-add-smul-vec-mul-vec-mul-vec","scope":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio","status":"fully_proved","summary":"For positive definite `V\u2080` and `c \u2265 0`, `(V\u2080 + c x x\u1d40)\u207b\u00b9 x = (1 + c x\u1d40 V\u2080\u207b\u00b9 x)\u207b\u00b9 \u2022 V\u2080\u207b\u00b9 x`.","title":"Sherman\u2013Morrison applied to the candidate marker"},{"id":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/mm-chi-sq-add-smul-vec-mul-vec","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.ProximalContamination.mmChiSq_add_smul_vecMulVec","parent":46688,"result":true,"route":"../roadmap/Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/index.html#mm-chi-sq-add-smul-vec-mul-vec","scope":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio","status":"fully_proved","summary":"Including the candidate marker `x` in the covariance, `V = V\u2080 + c x x\u1d40` with `V\u2080` positive definite and `c \u2265 0`, divides the mixed-model statistic by `1 + c x\u1d40 V\u2080\u207b\u00b9 x`, for every phenotype vector `y`: `\u03c7\u00b2(V\u2080 + c x x\u1d40) = \u03c7\u00b2(V\u2080) / (1 + c x\u1d40 V\u2080\u207b\u00b9 x)`.","title":"Proximal contamination"},{"id":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/mm-chi-sq-add-smul-vec-mul-vec-le","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.ProximalContamination.mmChiSq_add_smul_vecMulVec_le","parent":46688,"result":true,"route":"../roadmap/Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio/index.html#mm-chi-sq-add-smul-vec-mul-vec-le","scope":"Biology/statistical-genetics/proximal-contamination-in-mixed-model-associatio","status":"fully_proved","summary":"Including the candidate marker in the covariance never increases the mixed-model statistic: `\u03c7\u00b2(V\u2080 + c x x\u1d40) \u2264 \u03c7\u00b2(V\u2080)` for positive definite `V\u2080` and `c \u2265 0`.","title":"Including the marker in the covariance never increases the statistic"},{"data_url":"scopes/Biology/statistical-genetics/signed-ld-profile-regression.json","id":"Biology/statistical-genetics/signed-ld-profile-regression","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","title":"Signed LD profile regression"},{"id":"Biology/statistical-genetics/signed-ld-profile-regression/ld-matrix","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SignedLDProfile.ldMatrix","parent":46693,"result":true,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html#ld-matrix","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","summary":"The in-sample LD matrix `R\u0302 = X\u1d40 X / N`, with entries `ldCorr X j k`.","title":"In-sample LD matrix"},{"id":"Biology/statistical-genetics/signed-ld-profile-regression/model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SignedLDProfile.Model","parent":46693,"result":true,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html#model","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","summary":"The linear phenotype model of SLDP regression at the level of first moments: integrable random effects `\u03b2` with means `meanBeta` and integrable noise `\u03b5` of mean zero. (SLDP's model takes `meanBeta = \u03bc v` for a signed annotation `v`; second moments are not needed for the first-moment identity.)","title":"First-moment phenotype model of SLDP regression"},{"id":"Biology/statistical-genetics/signed-ld-profile-regression/pheno","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SignedLDProfile.Model.pheno","parent":46693,"result":true,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html#pheno","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","summary":"The phenotype `y = X \u03b2 + \u03b5`.","title":"Phenotype in the SLDP regression model"},{"id":"Biology/statistical-genetics/signed-ld-profile-regression/alpha-hat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SignedLDProfile.Model.alphaHat","parent":46693,"result":true,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html#alpha-hat","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","summary":"The vector `\u03b1\u0302 = X\u1d40 y / N` of marginal correlations between the (standardized) markers and the phenotype.","title":"Marginal marker-phenotype correlations in SLDP regression"},{"id":"Biology/statistical-genetics/signed-ld-profile-regression/integral-alpha-hat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SignedLDProfile.Model.integral_alphaHat","parent":46693,"result":true,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html#integral-alpha-hat","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","summary":"Conditional on the genotype matrix, `\ud835\udd3c \u03b1\u0302 = R\u0302 \ud835\udd3c \u03b2` with `R\u0302 = X\u1d40 X / N`.","title":"First-moment identity, exact in-sample form"},{"id":"Biology/statistical-genetics/signed-ld-profile-regression/integral-alpha-hat-of-mean","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SignedLDProfile.Model.integral_alphaHat_of_mean","parent":46693,"result":true,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html#integral-alpha-hat-of-mean","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","summary":"SLDP first-moment identity (Reshef et al. 2018, Methods, Equation 5): if the true effects have mean `\u03bc v`, then `\ud835\udd3c \u03b1\u0302 = \u03bc R\u0302 v`, a multiple of the signed LD profile `R\u0302 v` of `v`.","title":"SLDP first-moment identity"},{"id":"Biology/statistical-genetics/signed-ld-profile-regression/integral-gls-estimator","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SignedLDProfile.Model.integral_gls_estimator","parent":46693,"result":true,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html#integral-gls-estimator","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","summary":"For any weight matrix `W` and signed LD profile `s = R\u0302 v` with `\u27eas, s\u27eb_W = s\u1d40 W s \u2260 0`, if `\ud835\udd3c \u03b2 = \u03bc v` then `\ud835\udd3c [\u27eas, \u03b1\u0302\u27eb_W / \u27eas, s\u27eb_W] = \u03bc`, where `\u27eaa, b\u27eb_W = a\u1d40 W b`. SLDP uses the inverse of `\u03a9 = \u03c3\u00b2 R\u00b2 + R / N` as `W`.","title":"Unbiasedness of the weighted least-squares estimator"},{"id":"Biology/statistical-genetics/signed-ld-profile-regression/integral-gls-estimator-of-background","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SignedLDProfile.Model.integral_gls_estimator_of_background","parent":46693,"result":true,"route":"../roadmap/Biology/statistical-genetics/signed-ld-profile-regression/index.html#integral-gls-estimator-of-background","scope":"Biology/statistical-genetics/signed-ld-profile-regression","status":"fully_proved","summary":"Unbiasedness with covariates (the signed background model): suppose the true effects have mean `\u03bc v + U \u03b3` for background annotations `U` (columns `u_b`) with arbitrary coefficients `\u03b3`, let `A = R\u0302 U` be their signed LD profiles, and let `z\u22a5 = R\u0302 v - A w` be the weighted residual of the signed LD profile of `v` on `A` (`A\u1d40 W z\u22a5 = 0`) for a symmetric weight matrix `W`. If `\u27eaz\u22a5, z\u22a5\u27eb_W \u2260 0`, then\u2026","title":"Unbiasedness with covariates"},{"data_url":"scopes/Biology/statistical-genetics/stratified-ld-score-regression.json","id":"Biology/statistical-genetics/stratified-ld-score-regression","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","title":"Stratified LD-score regression"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#model","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The random-effects model of stratified LD-score regression at the level of raw second moments: marker `k` has effect variance `sigma2 k`, the noise has variance `sigmaE2`, and effects and noise are orthogonal in `L\u00b2`. As in `LDScore.Model`, the measure is arbitrary, no means are assumed, and all statements are conditional on the genotype matrix.","title":"Random-effects model of stratified LD-score regression"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/pheno","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.pheno","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#pheno","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The phenotype `y = X \u03b2 + \u03b5`.","title":"Phenotype in the stratified LD-score model"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/beta-hat","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.betaHat","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#beta-hat","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The `N`-normalized score `\u03b2\u0302 j = X_j\u1d40 y / N` of marker `j` (as in `LDScore.Model.betaHat`).","title":"Normalized marker score in stratified LD-score regression"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.chiSq","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#chi-sq","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The association statistic `\u03c7\u00b2 j = N * \u03b2\u0302 j ^ 2` of marker `j`.","title":"Association chi-squared statistic in stratified LD-score regression"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/of-ldscore","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.ofLDScore","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#of-ldscore","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The LD-score regression model of `LDScore` is the stratified model with equal per-marker variances `h\u00b2 / M` and noise variance `1 - h\u00b2`.","title":"LD-score regression model as a stratified model"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/chi-sq-of-ldscore","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.chiSq_ofLDScore","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#chi-sq-of-ldscore","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The stratified statistic of `ofLDScore D` is the statistic of `LDScore`.","title":"Stratified statistic agrees with the LD-score statistic"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/integral-chi-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.integral_chiSq","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#integral-chi-sq","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"Expected statistic with per-marker variances, for an arbitrary genotype matrix: `\ud835\udd3c[\u03c7\u00b2 j] = N \u2211_k r_jk\u00b2 \u03c3\u00b2_k + \u03c3\u00b2_e (\u2211_i X_ij\u00b2) / N`. (For `N = 0` both sides are `0`.)","title":"Expected statistic with per-marker variances"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/integral-chi-sq-of-standardized","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.integral_chiSq_of_standardized","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#integral-chi-sq-of-standardized","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"For a column with sum of squares `N` (a standardized marker) and `N \u2260 0`: `\ud835\udd3c[\u03c7\u00b2 j] = N \u2211_k r_jk\u00b2 \u03c3\u00b2_k + \u03c3\u00b2_e`.","title":"Expected chi-squared statistic of a standardized marker"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/annot-ldscore","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.annotLDScore","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#annot-ldscore","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The LD score of marker `j` with respect to annotation `C`: `\u2113(j, C) = \u2211_k a_{kC} r_jk\u00b2` (Finucane et al. 2015 for binary annotations; Gazal et al. 2017 for continuous ones).","title":"Annotation LD score of a marker"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/integral-chi-sq-of-annot","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.integral_chiSq_of_annot","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#integral-chi-sq-of-annot","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"Stratified LD-score regression (Finucane et al. 2015; continuous annotations, Gazal et al. 2017): if the per-marker variances are linear in the annotations, `\u03c3\u00b2_k = \u2211_C \u03c4_C a_{kC}`, then for a standardized genotype matrix and `N \u2260 0`, `\ud835\udd3c[\u03c7\u00b2 j] = N \u2211_C \u03c4_C \u2113(j, C) + \u03c3\u00b2_e`.","title":"Stratified LD-score regression"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/integral-chi-sq-of-alpha","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.integral_chiSq_of_alpha","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#integral-chi-sq-of-alpha","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"Frequency-dependent architecture (the \u03b1 model of Schoech et al. 2019): if the per-allele effect variance of marker `k` with allele frequency `p_k` is proportional to `[p_k (1 - p_k)]^\u03b1`, the variance of the effect of the standardized genotype is `\u03c3\u00b2_k = c [p_k (1 - p_k)]^(1 + \u03b1)`, and then, for a standardized genotype matrix and `N \u2260 0`, `\ud835\udd3c[\u03c7\u00b2 j] = N c \u2211_k [p_k (1 - p_k)]^(1 + \u03b1) r_jk\u00b2 + \u03c3\u00b2_e`. (The real power is\u2026","title":"Frequency-dependent architecture"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/cat-h2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.catH2","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#cat-h2","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The heritability `h\u00b2(C) = \u2211_{k \u2208 C} \u03c3\u00b2_k` of category `C`, for a labeling `cat` of the markers by categories.","title":"Heritability of a marker category"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/cat-size","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.catSize","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#cat-size","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The number `M_C` of markers in category `C`.","title":"Number of markers in a category"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/enrichment","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.enrichment","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#enrichment","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The enrichment of category `C`: its proportion of heritability divided by its proportion of markers, `(h\u00b2(C) / h\u00b2) / (M_C / M)` with `h\u00b2 = \u2211_k \u03c3\u00b2_k` (Finucane et al. 2015).","title":"Heritability enrichment of a category"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/sum-size-mul-enrichment","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.sum_size_mul_enrichment","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#sum-size-mul-enrichment","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"For a partition of the markers into categories (a labeling `cat`) and nonzero total heritability, the marker-proportion-weighted enrichments sum to one: `\u2211_C (M_C / M) * Enrichment(C) = 1`. Empty categories contribute `0` on both sides of each term.","title":"Marker-proportion-weighted enrichments sum to one"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/design","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.design","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#design","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"The design matrix `[1 | N \u2113(\u00b7, C)]` of stratified LD-score regression with a free intercept; column `none` is the intercept and column `some C` the scaled annotation LD score.","title":"Design matrix of stratified LD-score regression"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/injective-mul-vec-iff-rank-eq-card","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.injective_mulVec_iff_rank_eq_card","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#injective-mul-vec-iff-rank-eq-card","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"A real matrix has injective `mulVec` iff it has full column rank.","title":"Injective matrix action iff full column rank"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/integral-chi-sq-eq-design-mul-vec","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.Model.integral_chiSq_eq_design_mulVec","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#integral-chi-sq-eq-design-mul-vec","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"Under the annotation model, the vector of expected statistics is the design matrix applied to the parameter vector `(\u03c3\u00b2_e, \u03c4)` (intercept first). Hence the parameters are determined by the expected statistics of all markers, whatever their values, iff `(design X a).rank = Fintype.card (Option \u03b9)` (`injective_mulVec_iff_rank_eq_card`).","title":"Expected statistics are the design matrix times the parameters"},{"id":"Biology/statistical-genetics/stratified-ld-score-regression/sigma2-sub-sigma2-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.StratifiedLDScore.sigma2_sub_sigma2_eq","parent":46702,"result":true,"route":"../roadmap/Biology/statistical-genetics/stratified-ld-score-regression/index.html#sigma2-sub-sigma2-eq","scope":"Biology/statistical-genetics/stratified-ld-score-regression","status":"fully_proved","summary":"LDSC-SEG contrast (Finucane et al. 2018): for per-SNP variances `\u03c3\u00b2_k = \u2211_C \u03c4_C a_{kC}`, two SNPs `k`, `k'` whose annotations agree except on annotation `cS` differ in per-SNP variance by `\u03c4_{cS} (a_{k cS} - a_{k' cS})`. With `cS` the indicator of a gene set `S` contained in the control gene set `G` (whose indicator is one of the other annotations), `k \u2208 S` and `k' \u2208 G \u2216 S` with equal values of all other\u2026","title":"LDSC-SEG contrast"},{"data_url":"scopes/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics.json","id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","kind":"scope","navigation":"module","parent":46370,"result":false,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","title":"Testing LD consistency of association statistics"},{"id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/dentist-s","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDOutlierTest.dentistS","parent":46722,"result":true,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html#dentist-s","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","summary":"The DENTIST-S statistic `(z_i - r z_c) ^ 2 / (1 - r ^ 2)` of a variant with `z`-score `zi`, given the `z`-score `zc` of the lead (putative causal) variant and the reference correlation `r` between them (Kanai et al. 2022, Equation 1). Meaningful for `r ^ 2 < 1`; for `r ^ 2 = 1` it is `0` by the convention `x / 0 = 0`.","title":"DENTIST-S statistic"},{"id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/dentist-mean","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDOutlierTest.dentistMean","parent":46722,"result":true,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html#dentist-mean","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","summary":"The mean `(\u03c1 - r) \u03bb / \u221a(1 - r\u00b2)` of the standardized DENTIST difference when the true correlation is `\u03c1`, the reference correlation is `r` and the lead variant has mean `\u03bb`.","title":"Mean of the standardized DENTIST difference"},{"id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/dentist-variance","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDOutlierTest.dentistVariance","parent":46722,"result":true,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html#dentist-variance","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","summary":"The variance `(1 - 2 r \u03c1 + r\u00b2) / (1 - r\u00b2)` of the standardized DENTIST difference when the true correlation is `\u03c1` and the reference correlation is `r`.","title":"Variance of the standardized DENTIST difference"},{"id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/map-standardized-eq-gaussian-real","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDOutlierTest.map_standardized_eq_gaussianReal","parent":46722,"result":true,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html#map-standardized-eq-gaussian-real","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","summary":"Let `(z_i, z_c)` be jointly Gaussian with unit variances, correlation `\u03c1` (the covariance, as the variances are `1`), and means `\u03c1 \u03bb` and `\u03bb`. For a reference correlation `r` with `r ^ 2 < 1`, the standardized difference `(z_i - r z_c) / \u221a(1 - r\u00b2)` is Gaussian with mean `dentistMean \u03c1 r \u03bb` and variance `dentistVariance \u03c1 r`.","title":"Exact law of the standardized DENTIST difference"},{"id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/map-dentist-s-eq-map-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDOutlierTest.map_dentistS_eq_map_sq","parent":46722,"result":true,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html#map-dentist-s-eq-map-sq","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","summary":"When the reference correlation equals the true correlation `\u03c1` (`\u03c1 ^ 2 < 1`), the DENTIST-S statistic has the law of the square of a standard normal variable.","title":"DENTIST-S is chi-squared with one degree of freedom under correct LD"},{"id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/map-dentist-s-eq-chi-square","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDOutlierTest.map_dentistS_eq_chiSquare","parent":46722,"result":true,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html#map-dentist-s-eq-chi-square","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","summary":"Under the model of `map_standardized_eq_gaussianReal` with reference correlation equal to the true correlation `\u03c1` (`\u03c1 ^ 2 < 1`), the DENTIST-S statistic has the `chiSquareMeasure 1` law, for every value `\u03bb` of the lead variant's mean.","title":"DENTIST-S is exactly `\u03c7\u00b2\u2081` when the reference LD is right"},{"id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/dentist-variance-eq-one-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDOutlierTest.dentistVariance_eq_one_iff","parent":46722,"result":true,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html#dentist-variance-eq-one-iff","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","summary":"The standardized DENTIST difference has unit variance iff the reference correlation is `0` or equals the true correlation (for `r ^ 2 < 1`).","title":"Unit-variance criterion for the DENTIST difference"},{"id":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/dentist-mean-eq-zero-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.LDOutlierTest.dentistMean_eq_zero_iff","parent":46722,"result":true,"route":"../roadmap/Biology/statistical-genetics/testing-ld-consistency-of-association-statistics/index.html#dentist-mean-eq-zero-iff","scope":"Biology/statistical-genetics/testing-ld-consistency-of-association-statistics","status":"fully_proved","summary":"The standardized DENTIST difference has mean zero iff the reference correlation equals the true correlation or the lead variant has mean zero (for `r ^ 2 < 1`).","title":"Zero-mean criterion for the DENTIST difference"},{"data_url":"scopes/Biology/population-structure.json","id":"Biology/population-structure","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/population-structure/index.html","scope":"Biology/population-structure","status":"fully_proved","summary":"f-statistics, admixture graphs, ancestry models and linkage-disequilibrium methods for inferring the relationships and mixture histories of populations.","title":"Population structure and admixture"},{"data_url":"scopes/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm.json","id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","title":"Admixture model and the FRAPPE EM algorithm"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/params","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.Params","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#params","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"Parameters `\u03b8 = (Q, F)` of the admixture model: positive ancestry proportions `Q i k` summing to one over `k`, and reference-allele frequencies `F k j` strictly between `0` and `1`.","title":"Parameters of the admixture model"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/qf","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.qf","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#qf","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The reference-allele probability `qf \u03b8 i j = \u2211 k, Q i k * F k j` of a copy of SNP `j` in individual `i`.","title":"Reference-allele probability in the admixture model"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/joint","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.joint","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#joint","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The joint weight of an allele copy `(i, j, a)` and its ancestral population `k`: `Q i k * (if a then F k j else 1 - F k j)`.","title":"Joint weight of an allele copy and its ancestral population"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/joint-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.joint_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#joint-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The joint weight is positive (the standing hypothesis of the generic EM).","title":"Positivity of the admixture joint weight"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/marginal-eq","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.marginal_eq","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#marginal-eq","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"Reference with probability `qf \u03b8 i j`, alternative with probability `1 - qf \u03b8 i j`.","title":"The marginal of an allele copy"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/qf-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.qf_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#qf-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The reference-allele probability is positive.","title":"Positivity of the reference-allele probability"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/qf-lt-one","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.qf_lt_one","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#qf-lt-one","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The reference-allele probability is below one.","title":"Reference-allele probability is less than one"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/is-ref","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.isRef","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#is-ref","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"Whether copy `c` of the genotype `g i j` (with `g i j` reference copies) carries the reference allele: copies `c < g i j` do.","title":"Reference-allele indicator for a genotype copy"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/copy-allele","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.copyAllele","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#copy-allele","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The observation `(i, j, a)` of copy `c` of genotype `g i j`.","title":"Observation of one allele copy of a genotype"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/polymorphic","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.Polymorphic","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#polymorphic","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"Every SNP is polymorphic in the sample: some individual carries a reference allele and some individual carries an alternative allele. Without it the M-step frequency is `0` or `1`.","title":"Polymorphic genotype data"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/data","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.data","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#data","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The `2 |I| |J|` allele-copy observations of the genotype matrix `g`, indexed by `Fin`.","title":"Allele-copy observations of a genotype matrix"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/sum-data","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.sum_data","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#sum-data","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"A sum over the indexed data is a sum over individuals, SNPs and copies.","title":"Sum over allele-copy data as a triple sum"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/admixture-log-lik","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.admixtureLogLik","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#admixture-log-lik","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The admixture log-likelihood `\u2113(Q, F) = \u2211 i, \u2211 j, (g i j * log (qf \u03b8 i j) + (2 - g i j) * log (1 - qf \u03b8 i j))`.","title":"Admixture log-likelihood"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/log-lik-eq-admixture-log-lik","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.logLik_eq_admixtureLogLik","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#log-lik-eq-admixture-log-lik","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The EM log-likelihood of the allele copies is the admixture log-likelihood.","title":"The EM log-likelihood of the allele copies is the admixture log-likelihood"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/admixture-log-lik-le-of-em-q-le","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.admixtureLogLik_le_of_emQ_le","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#admixture-log-lik-le-of-em-q-le","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"A step that does not decrease `Q(\u00b7 | \u03b8)` does not decrease the admixture log-likelihood.","title":"The EM theorem for the admixture model"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/admixture-log-lik-iterate-monotone","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.admixtureLogLik_iterate_monotone","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#admixture-log-lik-iterate-monotone","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The admixture log-likelihood is monotone along the iterates of any M-step `M` with `Q(M \u03b8 | \u03b8) \u2265 Q(\u03b8 | \u03b8)`.","title":"The admixture log-likelihood is monotone along the iterates"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/resp","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.resp","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#resp","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The responsibility `r = P(z = k | copy c of (i, j))` (the E-step).","title":"Ancestry responsibility of an allele copy (E-step)"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/resp-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.resp_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#resp-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The responsibilities are positive.","title":"Positivity of the admixture responsibilities"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/sum-resp","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.sum_resp","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#sum-resp","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The responsibilities of a copy sum to one over the populations.","title":"Admixture responsibilities sum to one"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/anc-count","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.ancCount","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#anc-count","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The expected number of allele copies of individual `i` from population `k`.","title":"Expected ancestry count of an individual"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/m-step-q","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.mStepQ","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#m-step-q","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The FRAPPE update of the ancestry proportions, `Q' i k = (\u2211 j, \u2211 c, r) / (2 J)`.","title":"FRAPPE update of the ancestry proportions"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/sum-anc-count","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.sum_ancCount","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#sum-anc-count","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The expected ancestry counts of an individual sum to its `2 J` allele copies.","title":"Expected ancestry counts sum to the number of allele copies"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/anc-count-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.ancCount_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#anc-count-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The expected ancestry counts are positive when there is at least one SNP.","title":"Positivity of the expected ancestry counts"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/m-step-q-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.mStep_Q_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#m-step-q-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The updated ancestry proportions are positive.","title":"Positivity of the updated ancestry proportions"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/m-step-q-sum","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.mStep_Q_sum","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#m-step-q-sum","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The updated ancestry proportions of each individual sum to one.","title":"Updated ancestry proportions sum to one"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/pop-count","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.popCount","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#pop-count","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The expected number of allele copies of SNP `j` from population `k`.","title":"Expected population count of a SNP"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/ref-count","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.refCount","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#ref-count","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The expected number of reference copies of SNP `j` from population `k`.","title":"Expected reference-allele count of a SNP per population"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/alt-count","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.altCount","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#alt-count","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The expected number of alternative copies of SNP `j` from population `k`.","title":"Expected alternative-allele count of a SNP per population"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/m-step-f","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.mStepF","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#m-step-f","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The FRAPPE update of the allele frequencies, `F' k j = (\u2211 i, \u2211 c reference, r) / (\u2211 i, \u2211 c, r)`.","title":"FRAPPE update of the allele frequencies"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/ref-count-add-alt-count","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.refCount_add_altCount","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#ref-count-add-alt-count","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"Reference plus alternative expected counts give the population's expected count.","title":"Reference and alternative counts add to the population count"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/ref-count-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.refCount_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#ref-count-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"On polymorphic data the expected reference counts are positive.","title":"Positivity of expected reference counts on polymorphic data"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/alt-count-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.altCount_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#alt-count-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"On polymorphic data the expected alternative counts are positive.","title":"Positivity of expected alternative counts on polymorphic data"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/pop-count-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.popCount_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#pop-count-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"On polymorphic data the expected population counts are positive.","title":"Positivity of expected population counts on polymorphic data"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/m-step-f-pos","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.mStep_F_pos","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#m-step-f-pos","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"On polymorphic data the updated allele frequencies are positive.","title":"Positivity of the updated allele frequencies"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/m-step-f-lt-one","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.mStep_F_lt_one","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#m-step-f-lt-one","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"On polymorphic data the updated allele frequencies are below one.","title":"Updated allele frequencies are less than one"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/em-step","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.emStep","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#em-step","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The FRAPPE EM step `\u03b8 \u21a6 (Q', F')`, with `Q' = mStepQ` and `F' = mStepF`; it lies in the strictly positive parameter set when `J` is nonempty and every SNP is polymorphic.","title":"The FRAPPE EM step"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/em-q-eq","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.emQ_eq","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#em-q-eq","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"`Q(\u03b8' | \u03b8)` splits into an ancestry term per individual and a two-point term per population and SNP.","title":"Decomposition of the admixture EM objective"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/em-q-le-em-step","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.emQ_le_emStep","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#em-q-le-em-step","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"No parameter `\u03b8'` in the strictly positive parameter set has a larger `Q(\u03b8' | \u03b8)` than the closed-form update `emStep g hg \u03b8`.","title":"The FRAPPE update maximizes `Q(\u00b7 | \u03b8)`"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/admixture-log-lik-le-em-step","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.admixtureLogLik_le_emStep","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#admixture-log-lik-le-em-step","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"Each FRAPPE EM step does not decrease the admixture log-likelihood.","title":"Each FRAPPE EM step does not decrease the admixture log-likelihood"},{"id":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/admixture-log-lik-iterate-em-step-monoto","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.AdmixtureEM.admixtureLogLik_iterate_emStep_monotone","parent":46732,"result":true,"route":"../roadmap/Biology/population-structure/admixture-model-and-the-frappe-em-algorithm/index.html#admixture-log-lik-iterate-em-step-monoto","scope":"Biology/population-structure/admixture-model-and-the-frappe-em-algorithm","status":"fully_proved","summary":"The admixture log-likelihood is monotone along the FRAPPE EM iterates.","title":"The admixture log-likelihood is monotone along the FRAPPE EM iterates"},{"data_url":"scopes/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil.json","id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","title":"Fourier computation of weighted linkage disequilibrium"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/cell-sum","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.cellSum","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#cell-sum","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The total `U k = \u2211_{x : c x = k} u x` of a per-SNP quantity `u` over grid cell `k`.","title":"Total of a per-SNP quantity over a grid cell"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/pair-sum","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.pairSum","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#pair-sum","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The brute-force sum `\u2211_{x, y : c y = c x + j} u x v y` over ordered SNP pairs whose grid cells differ by `j`.","title":"Brute-force pair sum at a cell offset"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/pair-sum-eq-sum-cell-sum","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.pairSum_eq_sum_cellSum","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#pair-sum-eq-sum-cell-sum","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"Grouping pairs by cell offset: if every SNP lies in a cell `c x < K`, the pair sum at offset `j` is the lag-`j` cross-correlation of the cell totals.","title":"Pair sum as cross-correlation of cell totals"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/pad","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.pad","parent":46773,"reading_title":{"format":"inline-parts-v1","parts":[["text","Zero padding of a sequence to "],["math","\\(\\mathbb{Z}/N\\mathbb{Z}\\)"]],"source":"Zero padding of a sequence to $\\mathbb{Z}/N\\mathbb{Z}$"},"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#pad","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"Zero padding: the sequence `U` read on `ZMod N` through `k \u21a6 U k.val`. When `U` vanishes from `K` on and `K \u2264 N`, this is `U` followed by `N - K` zeros.","title":"Zero padding of a sequence to $\\mathbb{Z}/N\\mathbb{Z}$"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/autocorr-dft","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.autocorrDFT","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#autocorr-dft","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The transform computation of the lag-`j` autocorrelation: the inverse discrete Fourier transform of the power spectrum `|\ud835\udcd5 \u0168|^2` of the zero-padded sequence, read at `j`.","title":"Autocorrelation via the discrete Fourier transform"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/autocorr-dft-eq","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.autocorrDFT_eq","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#autocorr-dft-eq","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"Zero-padded circular autocorrelation equals the linear one: if `U k = 0` for `k \u2265 K`, `2 K \u2264 N + 1` and `j < K`, then `\ud835\udcd5\u207b(|\ud835\udcd5 \u0168|^2)(j) = \u2211_{k < K} U k U (k + j)`.","title":"Zero-padded circular autocorrelation equals linear autocorrelation"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/mean-geno","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.meanGeno","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#mean-geno","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The sample mean genotype `\u1e21(x) = (1/m) \u2211 i, g i x` at SNP `x` over `m` individuals.","title":"Sample mean genotype at a SNP"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/sample-cov","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.sampleCov","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#sample-cov","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The unbiased sample covariance `\u1e91(x, y) = (1/(m - 1)) \u2211 i, (g i x - \u1e21 x) (g i y - \u1e21 y)` of the genotypes at SNPs `x` and `y` (Loh et al. 2013, Eq. 5; meaningful for `m \u2265 2`).","title":"Unbiased sample covariance of genotypes"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/brute-weighted-ld","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.bruteWeightedLD","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#brute-weighted-ld","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The brute-force weighted-LD sum at cell offset `j`: `\u2211_{x, y : c y = c x + j} \u1e91(x, y) w(x) w(y)`, costing a pass over all pairs.","title":"Brute-force weighted linkage disequilibrium sum"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/centred-weighted","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.centredWeighted","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#centred-weighted","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The centred weighted genotype `(g i x - \u1e21 x) w x` of individual `i` at SNP `x`.","title":"Centred weighted genotype"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/dft-weighted-ld","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.dftWeightedLD","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#dft-weighted-ld","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"ALDER's transform algorithm: for each individual, the zero-padded autocorrelation of the cell totals of its centred weighted genotypes, computed through `\ud835\udcd5`; summed over individuals and divided by `m - 1`.","title":"ALDER's Fourier-transform weighted LD computation"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/dft-weighted-ld-eq-brute-weighted-ld","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.dftWeightedLD_eq_bruteWeightedLD","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#dft-weighted-ld-eq-brute-weighted-ld","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The transform algorithm computes the brute-force weighted-LD sum exactly, for every lag `j < K`, when all SNPs lie in cells `c x < K` and the transform length satisfies `2 K \u2264 N + 1`.","title":"ALDER's transform algorithm equals the brute-force weighted LD"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/pair-count","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.pairCount","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#pair-count","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The number of ordered SNP pairs at cell offset `j`, by which ALDER divides the weighted sum.","title":"Number of SNP pairs at a cell offset"},{"id":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/autocorr-dft-cell-count","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.WeightedLDFFT.autocorrDFT_cellCount","parent":46773,"result":true,"route":"../roadmap/Biology/population-structure/fourier-computation-of-weighted-linkage-disequil/index.html#autocorr-dft-cell-count","scope":"Biology/population-structure/fourier-computation-of-weighted-linkage-disequil","status":"fully_proved","summary":"The pair count is the zero-padded autocorrelation of the cell occupancy counts.","title":"Pair count as autocorrelation of cell occupancy counts"},{"data_url":"scopes/Biology/population-structure/li-stephens-haplotype-copying-model.json","id":"Biology/population-structure/li-stephens-haplotype-copying-model","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","title":"Li\u2013Stephens haplotype copying model"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/obs","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.Obs","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#obs","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The observation at a site: the column of reference alleles and the target allele.","title":"Site observation in the Li\u2013Stephens model"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/obs-2","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.obs","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#obs-2","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The observation sequence for the reference panel `H` (haplotype `k` carries allele `H k t` at site `t`) and the target haplotype `x`.","title":"Observation sequence of a reference panel and target haplotype"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/trans","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.trans","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#trans","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The copying transition probability: switch with probability `\u03c1` to a uniformly random reference haplotype (possibly the same one), otherwise keep copying `k`: `(1 - \u03c1) [k = k'] + \u03c1 / K`.","title":"Li\u2013Stephens copying transition probability"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/emit","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.emit","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#emit","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The emission probability: `1 - \u03b5` if the target allele equals the allele of the copied haplotype `k`, and the mismatch probability `\u03b5` otherwise.","title":"Li\u2013Stephens emission probability"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/hmm","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.hmm","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#hmm","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The Li\u2013Stephens copying model with `K` reference haplotypes, switch probability `\u03c1` and mismatch probability `\u03b5`, as a hidden Markov model with a uniform initial copied haplotype.","title":"The Li\u2013Stephens copying model"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/trans-row-sum","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.trans_row_sum","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#trans-row-sum","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"Each transition row sums to `1` (when `K` is invertible in `R`, e.g. `K \u2260 0` over `\u211d`).","title":"Li\u2013Stephens transition rows sum to one"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/init-sum","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.init_sum","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#init-sum","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The initial law (uniform over the `K` haplotypes) sums to `1`.","title":"Uniform initial law over haplotypes sums to one"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/emit-sum-bool","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.emit_sum_bool","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#emit-sum-bool","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"With biallelic sites, the two emission probabilities at a site sum to `1` over the target allele, for each fixed panel column `col` (the emissions are conditional on the panel).","title":"Biallelic emission probabilities sum to one"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/fast-step","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.fastStep","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#fast-step","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"One step of the collapsed forward recursion on the previous site's forward variables `v`, at output state `k`: `e_k ((1 - \u03c1) v k + \u03c1 (\u2211 j, v j) / K)`. This is the pointwise specification; each call evaluates the length-`K` sum, so use `fastStepTab` for the `O(K)` per-site step.","title":"One step of the collapsed forward recursion"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/fast-forward","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.fastForward","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#fast-forward","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The forward variables of the collapsed recursion, as a function of site and state. This is the untabulated specification (it re-evaluates earlier sites inside each sum); `fastForwardTab` is the `O(t K)` implementation.","title":"Forward variables of the collapsed recursion"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/sum-mul-trans","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.sum_mul_trans","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#sum-mul-trans","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The sum over predecessors collapses: `\u2211 j, v j * trans \u03c1 j k = (1 - \u03c1) v k + \u03c1 (\u2211 v) / K`.","title":"Collapse of the sum over predecessor haplotypes"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/fast-forward-eq-forward","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.fastForward_eq_forward","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#fast-forward-eq-forward","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The collapsed forward recursion computes the forward variables of the Li\u2013Stephens HMM.","title":"The collapsed forward recursion computes the forward variables"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/likelihood-eq-sum-fast-forward","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.likelihood_eq_sum_fastForward","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#likelihood-eq-sum-fast-forward","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The likelihood of the target haplotype (given the panel) is the sum of the collapsed forward variables.","title":"The likelihood of the target haplotype"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/fast-step-tab","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.fastStepTab","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#fast-step-tab","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"One tabulated `O(K)` step: the total of the previous table is computed once.","title":"Tabulated O(K) forward step"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/fast-forward-tab","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.fastForwardTab","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#fast-forward-tab","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The `O(K)` forward variables, tabulated site by site: `O(t K)` operations in total.","title":"Tabulated O(K) forward variables"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/fast-forward-tab-get","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.fastForwardTab_get","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#fast-forward-tab-get","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The tabulated `O(K)` recursion computes the forward variables of the Li\u2013Stephens HMM.","title":"Tabulated recursion computes the Li\u2013Stephens forward variables"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/likelihood-tab","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.likelihoodTab","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#likelihood-tab","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The tabulated Li\u2013Stephens likelihood in `O(t K)` operations.","title":"The tabulated Li\u2013Stephens likelihood"},{"id":"Biology/population-structure/li-stephens-haplotype-copying-model/likelihood-tab-eq","kind":"node","lean":"Cslib.Bioinformatics.PopulationStructure.LiStephens.likelihoodTab_eq","parent":46788,"result":true,"route":"../roadmap/Biology/population-structure/li-stephens-haplotype-copying-model/index.html#likelihood-tab-eq","scope":"Biology/population-structure/li-stephens-haplotype-copying-model","status":"fully_proved","summary":"The tabulated `O(K)` recursion computes the likelihood of the target haplotype.","title":"Tabulated recursion computes the haplotype likelihood"},{"data_url":"scopes/Biology/population-structure/patterson-s-f-statistics.json","id":"Biology/population-structure/patterson-s-f-statistics","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","title":"Patterson's f-statistics"},{"id":"Biology/population-structure/patterson-s-f-statistics/f2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f2","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f2","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"The statistic `f2 X Y = \ud835\udd3c[(p X - p Y) ^ 2]`, the expected squared allele-frequency difference between populations `X` and `Y`.","title":"f2 statistic"},{"id":"Biology/population-structure/patterson-s-f-statistics/f3","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f3","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f3","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"The statistic `f3 C A B = \ud835\udd3c[(p C - p A) * (p C - p B)]` of a target population `C` against two source populations `A` and `B`.","title":"f3 statistic"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f4","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"The statistic `f4 A B C D = \ud835\udd3c[(p A - p B) * (p C - p D)]`.","title":"f4 statistic"},{"id":"Biology/population-structure/patterson-s-f-statistics/f2-self","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f2_self","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f2-self","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f2 X X = 0`.","title":"f2 of a population with itself is zero"},{"id":"Biology/population-structure/patterson-s-f-statistics/f2-comm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f2_comm","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f2-comm","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f2` is symmetric.","title":"Symmetry of the f2 statistic"},{"id":"Biology/population-structure/patterson-s-f-statistics/f2-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f2_nonneg","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f2-nonneg","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f2` is nonnegative.","title":"Nonnegativity of the f2 statistic"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4-comm-pairs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f4_comm_pairs","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4-comm-pairs","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f4` is symmetric under exchanging the two pairs.","title":"f4 is symmetric under exchanging the two pairs"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4-swap-left","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f4_swap_left","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4-swap-left","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f4` changes sign when the first pair is swapped.","title":"f4 changes sign when the first pair is swapped"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4-swap-right","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f4_swap_right","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4-swap-right","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f4` changes sign when the second pair is swapped.","title":"f4 changes sign when the second pair is swapped"},{"id":"Biology/population-structure/patterson-s-f-statistics/f2-eq-f4","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f2_eq_f4","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f2-eq-f4","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f2` is the special case `f4 A B A B`.","title":"f2 as a special case of f4"},{"id":"Biology/population-structure/patterson-s-f-statistics/f3-eq-f4","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f3_eq_f4","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f3-eq-f4","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f3` is the special case `f4 C A C B`.","title":"f3 as a special case of f4"},{"id":"Biology/population-structure/patterson-s-f-statistics/integrable-sub-sq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.integrable_sub_sq","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#integrable-sub-sq","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"A difference of square-integrable frequencies has integrable square.","title":"Square of a difference of square-integrable frequencies is integrable"},{"id":"Biology/population-structure/patterson-s-f-statistics/f3-eq-f2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f3_eq_f2","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f3-eq-f2","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f3` in terms of `f2`: `f3 C A B = (f2 C A + f2 C B - f2 A B) / 2`.","title":"f3 in terms of f2"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4-eq-f2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.f4_eq_f2","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4-eq-f2","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f4` in terms of `f2`: `f4 A B C D = (f2 A D + f2 B C - f2 A C - f2 B D) / 2`.","title":"f4 in terms of f2"},{"id":"Biology/population-structure/patterson-s-f-statistics/drift-model","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#drift-model","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"The admixture-graph drift model, stated at the level of second moments.","title":"Admixture-graph drift model"},{"id":"Biology/population-structure/patterson-s-f-statistics/freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.freq","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#freq","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"The allele frequency of population `X`: the root frequency plus the weighted sum of the drift increments of the edges traversed by lineages from `X`.","title":"Allele frequency of a population in the drift model"},{"id":"Biology/population-structure/patterson-s-f-statistics/len-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.len_nonneg","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#len-nonneg","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Drift lengths are nonnegative.","title":"Drift lengths are nonnegative"},{"id":"Biology/population-structure/patterson-s-f-statistics/mem-lp-sum-incr","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.memLp_sum_incr","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#mem-lp-sum-incr","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"A weighted sum of increments is square integrable.","title":"Weighted sum of drift increments is square integrable"},{"id":"Biology/population-structure/patterson-s-f-statistics/mem-lp-freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.memLp_freq","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#mem-lp-freq","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Population frequencies are square integrable.","title":"Population frequencies are square integrable"},{"id":"Biology/population-structure/patterson-s-f-statistics/integral-sum-mul-sum","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.integral_sum_mul_sum","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#integral-sum-mul-sum","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Bilinearity of the second moment of increments: `\ud835\udd3c[(\u2211 e, a e * incr e) * (\u2211 f, b f * incr f)] = \u2211 e, a e * b e * len e`.","title":"Bilinearity of the second moment of drift increments"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4-freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f4_freq","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4-freq","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f4` in the drift model: the root cancels, and `f4 A B C D` is the drift length shared by the paths `A\u2013B` and `C\u2013D`, weighted by `(w A e - w B e) * (w C e - w D e)`.","title":"f4 in the drift model"},{"id":"Biology/population-structure/patterson-s-f-statistics/f2-freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f2_freq","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f2-freq","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f2` in the drift model: `f2 X Y = \u2211 e, (w X e - w Y e) ^ 2 * len e`.","title":"f2 in the drift model"},{"id":"Biology/population-structure/patterson-s-f-statistics/f3-freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f3_freq","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f3-freq","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f3` in the drift model: `f3 C A B = \u2211 e, (w C e - w A e) * (w C e - w B e) * len e`.","title":"f3 in the drift model"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4-eq-zero-of-disjoint","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f4_eq_zero_of_disjoint","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4-eq-zero-of-disjoint","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`f4 A B C D = 0` when no edge contributes to both the `A\u2013B` and the `C\u2013D` path. On a tree with split `AB | CD` the paths `A\u2013B` and `C\u2013D` share no edge, so this hypothesis holds.","title":"f4 vanishes for edge-disjoint paths"},{"id":"Biology/population-structure/patterson-s-f-statistics/has-binary-loadings","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.HasBinaryLoadings","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#has-binary-loadings","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"The model has binary loadings if every lineage weight is `0` or `1`.","title":"Binary loadings of a drift model"},{"id":"Biology/population-structure/patterson-s-f-statistics/lineage","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.lineage","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#lineage","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"The edges of weight `1` for population `X`; on a tree, the root-to-`X` lineage.","title":"Lineage of a population in the drift model"},{"id":"Biology/population-structure/patterson-s-f-statistics/f3-nonneg-of-has-binary-loadings","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f3_nonneg_of_hasBinaryLoadings","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f3-nonneg-of-has-binary-loadings","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"With binary loadings `f3` is nonnegative; on a tree it is the drift length of the path from `C` to the `A\u2013B` path.","title":"f3 is nonnegative under binary loadings"},{"id":"Biology/population-structure/patterson-s-f-statistics/f2-eq-sum-symm-diff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f2_eq_sum_symmDiff","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f2-eq-sum-symm-diff","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"With binary loadings `f2 X Y` is the total drift length of the edges in exactly one of the weight-`1` edge sets of `X` and `Y`. On a tree, where these sets are the root-to-`X` and root-to-`Y` paths, this is the additive tree distance between `X` and `Y`.","title":"f2 as a symmetric-difference drift length under binary loadings"},{"id":"Biology/population-structure/patterson-s-f-statistics/not-has-binary-loadings-of-f3-neg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.not_hasBinaryLoadings_of_f3_neg","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#not-has-binary-loadings-of-f3-neg","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Three-population test: a negative `f3 C A B` is incompatible with binary loadings, hence with every tree model.","title":"Three-population test: negative f3 rules out a tree"},{"id":"Biology/population-structure/patterson-s-f-statistics/f3-admixture","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f3_admixture","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f3-admixture","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"If `C` is a mixture of `A` and `B` in proportions `\u03b1 : 1 - \u03b1` (with no subsequent drift), then `f3 C A B = -\u03b1 (1 - \u03b1) f2 A B`.","title":"f3 of a pure admixture"},{"id":"Biology/population-structure/patterson-s-f-statistics/f3-admixture-drift","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f3_admixture_drift","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f3-admixture-drift","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Admixture followed by drift: if `C` is an `\u03b1 : 1 - \u03b1` mixture of `A` and `B` that then drifts along a private edge `c` (not traversed by `A` or `B`), then `f3 C A B = len c - \u03b1 (1 - \u03b1) f2 A B`; post-admixture drift can mask the admixture signal.","title":"f3 of an admixture followed by drift"},{"id":"Biology/population-structure/patterson-s-f-statistics/f3-neg-of-admixture","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f3_neg_of_admixture","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f3-neg-of-admixture","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Three-population test, converse direction: a nondegenerate pure admixture (`0 < \u03b1 < 1`, `f2 A B > 0`) yields a negative `f3 C A B`.","title":"Nondegenerate pure admixture gives negative f3"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4-admixture-right","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f4_admixture_right","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4-admixture-right","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"If `X` is an `\u03b1 : 1 - \u03b1` mixture of `Y` and `C`, then `f4 A O X C = \u03b1 * f4 A O Y C` (bilinearity of `f4`, together with `f4 A O C C = 0`).","title":"f4 of an admixed population scales with the mixture proportion"},{"id":"Biology/population-structure/patterson-s-f-statistics/f4-ratio","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f4_ratio","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#f4-ratio","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"f4-ratio estimation of the admixture proportion. f4-ratio estimation (Patterson et al. 2012). Let `X` be an `\u03b1 : 1 - \u03b1` mixture of a source lineage `Y` and of `C`, where `Y` is unsampled but sister to the sampled population `B` relative to the `A\u2013O` path, so that `f4 A O Y C = f4 A O B C`. If this value is nonzero, the ratio `f4 A O X C / f4 A O B C` of two estimable statistics recovers the admixture proportion `\u03b1`.","title":"f4-ratio estimation of the admixture proportion"},{"id":"Biology/population-structure/patterson-s-f-statistics/mean-freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.meanFreq","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#mean-freq","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Mean allele frequency `q\u0304 = \u2211 i, c i * q i` of a population made of subpopulations with weights `c i` and allele frequencies `q i`.","title":"Mean allele frequency of a subdivided population"},{"id":"Biology/population-structure/patterson-s-f-statistics/het-sub","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.hetSub","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#het-sub","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Mean within-subpopulation (Hardy\u2013Weinberg) heterozygosity `H_S = \u2211 i, c i * 2 q i (1 - q i)`.","title":"Mean within-subpopulation heterozygosity"},{"id":"Biology/population-structure/patterson-s-f-statistics/het-total","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.hetTotal","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#het-total","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Total heterozygosity `H_T = 2 q\u0304 (1 - q\u0304)` expected under random mating of the pooled population.","title":"Total heterozygosity of the pooled population"},{"id":"Biology/population-structure/patterson-s-f-statistics/wahlund","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.wahlund","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#wahlund","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"Wahlund identity: for weights summing to `1`, `H_T - H_S = 2 \u2211 i, c i * (q i - q\u0304) ^ 2`, twice the weighted variance of subpopulation frequencies.","title":"Wahlund identity"},{"id":"Biology/population-structure/patterson-s-f-statistics/het-sub-le-het-total","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.hetSub_le_hetTotal","parent":46807,"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#het-sub-le-het-total","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"The Wahlund effect: with nonnegative weights summing to `1`, subdivision lowers heterozygosity, `H_S \u2264 H_T`.","title":"Wahlund effect"},{"id":"Biology/population-structure/patterson-s-f-statistics/fst-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.fst_nonneg","parent":46807,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonnegativity of the fixation index "],["math","\\(F_{ST}\\)"]],"source":"Nonnegativity of the fixation index $F_{ST}$"},"result":true,"route":"../roadmap/Biology/population-structure/patterson-s-f-statistics/index.html#fst-nonneg","scope":"Biology/population-structure/patterson-s-f-statistics","status":"fully_proved","summary":"`F_ST = (H_T - H_S) / H_T` is nonnegative for nonnegative weights summing to `1`. The ratio is defined only when `H_T > 0` (at a fixed allele `H_T = H_S = 0`), so that is assumed.","title":"Nonnegativity of the fixation index $F_{ST}$"},{"data_url":"scopes/Biology/population-structure/abba-baba-test-and-f-statistic-estimators.json","id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","title":"ABBA\u2013BABA test and f-statistic estimators"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/config-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.configProb","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#config-prob","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Probability of the derived/ancestral configuration `x` when haplotype `i` is sampled independently and carries the derived allele with probability `q i`.","title":"Probability of a derived-allele configuration"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/sample-expect","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.sampleExpect","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#sample-expect","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Expectation of a function `g` of the configuration under independent sampling of one haplotype per population at derived-allele frequencies `q`.","title":"Expectation under independent haplotype sampling"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/abba","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.abba","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#abba","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Unpolarized `ABBA` indicator for populations `(P1, P2, P3, O)`: `P1` agrees with `O`, `P2` with `P3`, and `P1` differs from `P2` (the patterns `ABBA` and `BAAB`).","title":"Unpolarized ABBA site-pattern indicator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/baba","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.baba","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#baba","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Unpolarized `BABA` indicator for populations `(P1, P2, P3, O)`: `P1` agrees with `P3`, `P2` with `O`, and `P1` differs from `P2` (the patterns `BABA` and `ABAB`).","title":"Unpolarized BABA site-pattern indicator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/abba-pol","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.abbaPol","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#abba-pol","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Polarized `ABBA` indicator (Durand et al. 2011): `P1` and `O` ancestral, `P2` and `P3` derived.","title":"Polarized ABBA site-pattern indicator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/baba-pol","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.babaPol","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#baba-pol","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Polarized `BABA` indicator (Durand et al. 2011): `P1` and `P3` derived, `P2` and `O` ancestral.","title":"Polarized BABA site-pattern indicator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/sum-config-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.sum_configProb","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#sum-config-prob","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"The configuration probabilities sum to `1`.","title":"Configuration probabilities sum to one"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/sample-expect-prod","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.sampleExpect_prod","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#sample-expect-prod","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"`sampleExpect` of a product of functions of single coordinates factorizes: `E[\u220f i, g i (x i)] = \u220f i, (g i true * q i + g i false * (1 - q i))`.","title":"Sampling expectation of a product factorizes"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/sample-expect-add","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.sampleExpect_add","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#sample-expect-add","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"`sampleExpect` is additive in the function.","title":"Additivity of the sampling expectation"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/sample-expect-sub","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.sampleExpect_sub","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#sample-expect-sub","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"`sampleExpect` commutes with subtraction of functions.","title":"Sampling expectation commutes with subtraction"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-eq-sample-expect","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_eq_sampleExpect","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-eq-sample-expect","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"`sampleExpect` is the expectation under independent sampling: if the derived-allele indicators `X i` are independent random variables with `P(X i = true) = q i`, then `E[g (X 0, \u2026, X (n - 1))] = sampleExpect g q`.","title":"Sampling expectation is an expectation of independent indicators"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/baba-sub-abba","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.baba_sub_abba","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#baba-sub-abba","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"The site-pattern identity: `1[BABA] - 1[ABBA] = (x\u2081 - x\u2082) (x\u2083 - x\u2084)` for every configuration of derived-allele indicators.","title":"ABBA\u2013BABA site-pattern identity"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/sample-expect-baba-sub-abba","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.sampleExpect_baba_sub_abba","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#sample-expect-baba-sub-abba","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"ABBA\u2013BABA identity (Patterson et al. 2012; Durand et al. 2011): under independent sampling of one haplotype per population at derived-allele frequencies `q` (any reals; frequencies in `[0, 1]` in the biological model), `E[BABA - ABBA] = (q\u2081 - q\u2082) (q\u2083 - q\u2084)`.","title":"ABBA\u2013BABA identity"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/sample-expect-baba-pol-sub-abba-pol","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.sampleExpect_babaPol_sub_abbaPol","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#sample-expect-baba-pol-sub-abba-pol","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Polarized form (Durand et al. 2011, with `q 3` the derived-allele frequency in the outgroup): `E[BABA_pol - ABBA_pol] = (q\u2081 - q\u2082) q\u2083 (1 - q\u2084)`. When the outgroup is fixed for the ancestral allele (`q 3 = 0`) this is `(q\u2081 - q\u2082) (q\u2083 - q\u2084)`, as in the unpolarized identity.","title":"Expected polarized ABBA\u2013BABA difference"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-sample-expect-baba-sub-abba","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_sampleExpect_baba_sub_abba","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-sample-expect-baba-sub-abba","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"When the frequencies `p X \u03c9` are themselves random (drift, and the choice of locus) and one haplotype is sampled from each population independently given the frequencies, the expected `BABA - ABBA` difference is `f4 P1 P2 P3 O`.","title":"ABBA\u2013BABA equals `f4`"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/f4-eq-sum-of-sub-eq-sum","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.f4_eq_sum_of_sub_eq_sum","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#f4-eq-sum-of-sub-eq-sum","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Linearity of `f4` in its first pair: if `p A - p B = \u2211 l, c l * (p (C l) - p (D l))` pointwise, then `f4 A B Y Z = \u2211 l, c l * f4 (C l) (D l) Y Z`.","title":"Linearity of the f4-statistic in its first pair"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/f4-affine-combination","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.f4_affine_combination","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#f4-affine-combination","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"qpAdm model equation (Haak et al. 2015, Supplementary Information): if the target `T` is an admixture of sources `S k` with weights `w k` summing to `1` (`p T = \u2211 k, w k * p (S k)` pointwise; the weights may be negative, as in fitted qpAdm models), then for all `X`, `Y`, `Z`, `f4 T X Y Z = \u2211 k, w k * f4 (S k) X Y Z`. The merged `DriftModel.f4_admixture_right` is a two-source drift-model instance (mixture in the\u2026","title":"qpAdm model equation"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/f4-matrix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.f4Matrix","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#f4-matrix","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"The qpWave matrix of left populations `L 0, \u2026, L n` and right populations `R 0, \u2026, R r`: entry `(i, j)` is `f4 (L (i + 1)) (L 0) (R (j + 1)) (R 0)`.","title":"qpWave matrix of f4-statistics"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/rank-f4-matrix-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.rank_f4Matrix_le","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#rank-f4-matrix-le","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"qpWave rank bound (Reich et al. 2012, Supplementary Information; Patterson et al. 2012): if the frequency of every left population `L i` is an affine combination `\u2211 l, a i l * p (S l)` (`\u2211 l, a i l = 1`, coefficients of any sign) of the frequencies of `k + 1` source populations, then the matrix `f4 (L i) (L 0) (R j) (R 0)` (`i, j \u2265 1`) has rank at most `k`; with `k` sources, at most `k - 1`. The right populations\u2026","title":"qpWave rank bound"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/het-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.hetHat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#het-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Unbiased estimator of `p (1 - p)` from `x` derived copies among `m` sampled copies: `x (m - x) / (m (m - 1))` (meaningful for `m \u2265 2`).","title":"Unbiased heterozygosity estimator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/f2-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.f2Hat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#f2-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"The unbiased `f2` estimator of Patterson et al. (2012), Appendix: `(\u00e2 - b\u0302)\u00b2 - \u00e2 (1 - \u00e2) / (m_A - 1) - b\u0302 (1 - b\u0302) / (m_B - 1)` with `\u00e2 = x_A / m_A`, written as `(x_A / m_A - x_B / m_B)\u00b2 - x_A (m_A - x_A) / (m_A\u00b2 (m_A - 1)) - \u2026`.","title":"Unbiased f2 estimator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/f3-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.f3Hat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#f3-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"The unbiased `f3` estimator of Patterson et al. (2012), Appendix: `(\u0109 - \u00e2) (\u0109 - b\u0302) - \u0109 (1 - \u0109) / (m_C - 1)`.","title":"Unbiased f3 estimator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/f4-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.f4Hat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#f4-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"The `f4` estimator `(\u00e2 - b\u0302) (\u0109 - d\u0302)`; it needs no correction for four distinct populations.","title":"f4 estimator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/hudson-den-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.hudsonDenHat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#hudson-den-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Hudson's `F_ST` denominator estimator `\u00e2 (1 - b\u0302) + b\u0302 (1 - \u00e2)` (Bhatia et al. 2013); the numerator estimator is `f2Hat`.","title":"Estimator of Hudson's F_ST denominator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/hudson-den","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.hudsonDen","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#hudson-den","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Hudson's `F_ST` denominator at one site: `p_A (1 - p_B) + p_B (1 - p_A)`, the probability that two copies, one from each population, differ. The numerator is `(p_A - p_B) ^ 2`.","title":"Hudson's F_ST denominator at one site"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/hudson-fst","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.hudsonFst","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#hudson-fst","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Hudson's `F_ST` over sites `s` in the ratio-of-averages form of Bhatia et al. (2013): `\u2211 s, (p_A s - p_B s) ^ 2 / \u2211 s, hudsonDen (p_A s) (p_B s)`. It is defined when the denominator is positive (it vanishes only if every site is fixed for the same allele in both populations); otherwise Lean's value `0` is a junk value.","title":"Hudson's F_ST (ratio of averages)"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/sq-sub-le-hudson-den","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.sq_sub_le_hudsonDen","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#sq-sub-le-hudson-den","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"At each site Hudson's numerator is at most the denominator for frequencies in `[0, 1]`: the difference is `p_A (1 - p_A) + p_B (1 - p_B) \u2265 0`.","title":"Hudson's numerator is at most the denominator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/hudson-fst-mem-icc","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.hudsonFst_mem_Icc","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#hudson-fst-mem-icc","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Hudson's `F_ST` lies in `[0, 1]` for frequencies in `[0, 1]`. (When the denominator vanishes the junk value `0` also lies in `[0, 1]`, so no positivity hypothesis is needed.)","title":"Hudson's F_ST lies between 0 and 1"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-div-of-has-law-binomial","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_div_of_hasLaw_binomial","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-div-of-has-law-binomial","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"The sample frequency is unbiased: for `X ~ Bin(m, p)` with `m \u2265 1`, `E[X / m] = p`. (For `m = 0`, `X / 0 = 0` is a junk value, hence `m \u2265 1`.)","title":"Sample allele frequency is unbiased"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-het-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_hetHat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-het-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"For `X ~ Bin(m, p)` with `m \u2265 2`, `E[X (m - X) / (m (m - 1))] = p (1 - p)`.","title":"Unbiased heterozygosity"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-f2-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_f2Hat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-f2-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Unbiased `f2` (Patterson et al. 2012, Appendix): with independent binomial samples `X_A ~ Bin(m_A, p_A)`, `X_B ~ Bin(m_B, p_B)`, `m_A, m_B \u2265 2`, `E[f2Hat] = (p_A - p_B) ^ 2`, the population `f2` at the site.","title":"Unbiased `f2`"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-f3-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_f3Hat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-f3-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Unbiased `f3` (Patterson et al. 2012, Appendix): with pairwise independent binomial samples, `m_C \u2265 2` and `m_A, m_B \u2265 1`, `E[f3Hat] = (p_C - p_A) (p_C - p_B)`.","title":"Unbiased `f3`"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-f4-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_f4Hat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-f4-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"For four populations sampled independently across the two pairs (`m \u2265 1` each), `E[f4Hat] = (p_A - p_B) (p_C - p_D)`; no correction is needed.","title":"Unbiased `f4`"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-hudson-den-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_hudsonDenHat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-hudson-den-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Hudson's denominator is unbiased (Bhatia et al. 2013): with independent binomial samples (`m \u2265 1` each), `E[hudsonDenHat] = hudsonDen p_A p_B`.","title":"Hudson's denominator is unbiased"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-sum-f2-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_sum_f2Hat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-sum-f2-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Summed over sites, Hudson's numerator estimator `\u2211 s, f2Hat` is unbiased for `\u2211 s, (p_A s - p_B s) ^ 2` (independence is needed only within each site).","title":"Unbiasedness of the summed f2 estimator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-sum-hudson-den-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_sum_hudsonDenHat","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-sum-hudson-den-hat","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Summed over sites, Hudson's denominator estimator is unbiased for `\u2211 s, hudsonDen (p_A s) (p_B s)`.","title":"Unbiasedness of the summed Hudson denominator estimator"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-sum-div-integral-sum","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.ADMIXTOOLS.integral_sum_div_integral_sum","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-sum-div-integral-sum","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"Hudson's `F_ST`, ratio of averages (Bhatia et al. 2013): the ratio of the expected summed numerator and denominator estimators is the population `hudsonFst`. This is a statement about the ratio of expectations; the expectation of the ratio of the estimated sums is in general not `hudsonFst`.","title":"Hudson's `F_ST`, ratio of averages"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/integral-sample-expect-abba-eq-baba","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.integral_sampleExpect_abba_eq_baba","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#integral-sample-expect-abba-eq-baba","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"In the drift model, if no edge lies on both the `P1\u2013P2` and the `P3\u2013O` path (as on the tree `((P1, P2), P3), O`), then the expected unpolarized `ABBA` and `BABA` counts are equal. Thus `BABA - ABBA` and its sign reversal have expectation `0`. The integrability hypotheses make both expectations meaningful.","title":"The unpolarized D-statistic numerator vanishes on a tree"},{"id":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators/f4-affine-combination-2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatistics.DriftModel.f4_affine_combination","parent":46848,"result":true,"route":"../roadmap/Biology/population-structure/abba-baba-test-and-f-statistic-estimators/index.html#f4-affine-combination-2","scope":"Biology/population-structure/abba-baba-test-and-f-statistic-estimators","status":"fully_proved","summary":"If the lineage weights of `T` are an affine combination `\u2211 k, w k \u2022 weight (S k)` of those of the sources (`\u2211 k, w k = 1`, weights of any sign), then `f4 T X Y Z = \u2211 k, w k * f4 (S k) X Y Z`.","title":"qpAdm model equation in the drift model"},{"data_url":"scopes/Biology/population-structure/admixture-graphs.json","id":"Biology/population-structure/admixture-graphs","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","title":"Admixture graphs"},{"id":"Biology/population-structure/admixture-graphs/admixture-graph","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#admixture-graph","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"An admixture graph on the nodes `V` (Patterson et al. 2012): a rooted directed acyclic graph in which every non-root node has one or two parents, with admixture proportions.","title":"Admixture graph"},{"id":"Biology/population-structure/admixture-graphs/edge","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.Edge","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#edge","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The edges `(child, parent)` of the graph.","title":"Edges of an admixture graph"},{"id":"Biology/population-structure/admixture-graphs/visit","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.visit","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#visit","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"`visit X c` is the probability that the lineage of a gene sampled in `X`, traced toward the root with parents chosen according to the admixture proportions, passes through `c`.","title":"Probability that a lineage passes through a node"},{"id":"Biology/population-structure/admixture-graphs/weight","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.weight","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#weight","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"`weight X e` is the probability that the lineage of a gene sampled in `X` traverses the edge `e = (c, p)`: it reaches `c` and then descends from `p`.","title":"Lineage weight of an edge in an admixture graph"},{"id":"Biology/population-structure/admixture-graphs/path-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.pathProb","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#path-prob","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The probability of a root-ward path `[v\u2080, v\u2081, \u2026, v_k]`: the product of the proportions `prop v\u1d62 v\u1d62\u208a\u2081`.","title":"Probability of a root-ward path in an admixture graph"},{"id":"Biology/population-structure/admixture-graphs/paths","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.paths","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#paths","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The root-ward paths from `X`: the lists `[X = v\u2080, v\u2081, \u2026, v_k = root]` with each `v\u1d62\u208a\u2081` a parent of `v\u1d62`.","title":"Root-ward paths from a node of an admixture graph"},{"id":"Biology/population-structure/admixture-graphs/visit-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.visit_eq","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#visit-eq","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"First-step equation for `visit`.","title":"First-step equation for visit probabilities"},{"id":"Biology/population-structure/admixture-graphs/visit-self","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.visit_self","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#visit-self","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"A lineage passes through its own node with probability `1`.","title":"A lineage visits its own node with probability one"},{"id":"Biology/population-structure/admixture-graphs/visit-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.visit_nonneg","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#visit-nonneg","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"Visit probabilities are nonnegative.","title":"Nonnegativity of visit probabilities"},{"id":"Biology/population-structure/admixture-graphs/visit-le-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.visit_le_one","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#visit-le-one","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"Visit probabilities are at most `1`.","title":"Visit probabilities are at most one"},{"id":"Biology/population-structure/admixture-graphs/visit-root","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.visit_root","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#visit-root","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"Every lineage reaches the root.","title":"Every lineage reaches the root"},{"id":"Biology/population-structure/admixture-graphs/paths-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.paths_eq","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#paths-eq","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"First-step equation for `paths`.","title":"First-step equation for root-ward paths"},{"id":"Biology/population-structure/admixture-graphs/prop-le-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.prop_le_one","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#prop-le-one","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"Admixture proportions are at most `1`.","title":"Admixture proportions are at most one"},{"id":"Biology/population-structure/admixture-graphs/visit-eq-zero-of-mem-parents","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.visit_eq_zero_of_mem_parents","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#visit-eq-zero-of-mem-parents","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"Acyclicity: a lineage from a parent `Y` of `X` never passes through `X`.","title":"A lineage from a parent never passes through its child"},{"id":"Biology/population-structure/admixture-graphs/weight-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.weight_nonneg","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#weight-nonneg","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"Lineage weights are nonnegative.","title":"Nonnegativity of lineage weights"},{"id":"Biology/population-structure/admixture-graphs/weight-le-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.weight_le_one","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#weight-le-one","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"Lineage weights are at most `1`.","title":"Lineage weights are at most one"},{"id":"Biology/population-structure/admixture-graphs/weight-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.weight_eq","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#weight-eq","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"First-step recursion for lineage weights: a lineage from `X` traverses `e` either as its first step (if `e` starts at `X`) or after moving to a parent `Y`, chosen with probability `prop X Y`.","title":"First-step recursion for lineage weights"},{"id":"Biology/population-structure/admixture-graphs/weight-admixture","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.weight_admixture","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#weight-admixture","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The recursion at an admixture node `X` with parents `A \u2260 B`: with `\u03b1 = prop X A`, `w X = \u03b1 w A + (1 - \u03b1) w B + w\u2080`, where `w\u2080` puts weight `\u03b1` on the edge `(X, A)`, `1 - \u03b1` on `(X, B)` and `0` elsewhere.","title":"Lineage weight recursion at an admixture node"},{"id":"Biology/population-structure/admixture-graphs/weight-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.weight_tree","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#weight-tree","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The recursion at a tree node `X` with the single parent `Y`: `w X = w Y + 1_{(X, Y)}`.","title":"Lineage weight recursion at a tree node"},{"id":"Biology/population-structure/admixture-graphs/weight-eq-sum-paths","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.weight_eq_sum_paths","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#weight-eq-sum-paths","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"Path-sum form of the lineage weights (Patterson et al. 2012): `weight X e` is the sum, over the root-ward paths from `X` that traverse `e`, of the products of the admixture proportions along the path.","title":"Path-sum formula for lineage weights"},{"id":"Biology/population-structure/admixture-graphs/is-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.IsTree","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#is-tree","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The graph is a tree: no node has two parents.","title":"Tree-shaped admixture graph"},{"id":"Biology/population-structure/admixture-graphs/visit-eq-zero-or-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.visit_eq_zero_or_one","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#visit-eq-zero-or-one","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"On a tree every visit probability is `0` or `1`.","title":"Visit probabilities on a tree are zero or one"},{"id":"Biology/population-structure/admixture-graphs/weight-eq-zero-or-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.weight_eq_zero_or_one","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#weight-eq-zero-or-one","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"On a tree every lineage weight is `0` or `1`.","title":"Lineage weights on a tree are zero or one"},{"id":"Biology/population-structure/admixture-graphs/drift-model","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.driftModel","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#drift-model","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The drift model of an admixture graph: the populations are the nodes, the root frequency is `root`, the edge increments are `incr` (square integrable and pairwise uncorrelated), the drift length of an edge is the second moment of its increment, and the weights are the lineage weights of the graph.","title":"Drift model of an admixture graph"},{"id":"Biology/population-structure/admixture-graphs/has-binary-loadings-of-is-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.hasBinaryLoadings_of_isTree","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#has-binary-loadings-of-is-tree","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"A tree-shaped admixture graph induces a drift model with binary loadings (all weights `0` or `1`).","title":"A tree-shaped admixture graph has binary loadings"},{"id":"Biology/population-structure/admixture-graphs/f3-nonneg-of-is-tree","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.f3_nonneg_of_isTree","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#f3-nonneg-of-is-tree","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The three-population test on a tree-shaped admixture graph: `f3 C A B \u2265 0`.","title":"Three-population test is nonnegative on a tree"},{"id":"Biology/population-structure/admixture-graphs/f3-admixture","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.f3_admixture","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#f3-admixture","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"The three-population statistic at an admixture node `X` with parents `A \u2260 B`, proportions `\u03b1 = prop X A` and `\u03b2 = prop X B` (`\u03b1 + \u03b2 = 1`), in a drift model with the graph's weights: `f3 X A B = \u03b1\u00b2 \u2113_{XA} + \u03b2\u00b2 \u2113_{XB} - \u03b1 \u03b2 f2 A B`, where `\u2113_{XA}`, `\u2113_{XB}` are the drift lengths of the two admixture edges.","title":"Three-population statistic at an admixture node"},{"id":"Biology/population-structure/admixture-graphs/f3-admixture-of-len-eq-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureGraph.f3_admixture_of_len_eq_zero","parent":46888,"result":true,"route":"../roadmap/Biology/population-structure/admixture-graphs/index.html#f3-admixture-of-len-eq-zero","scope":"Biology/population-structure/admixture-graphs","status":"fully_proved","summary":"With no drift on the admixture edges, an admixture node `X` with parents `A \u2260 B` and proportion `\u03b1 = prop X A` has `f3 X A B = -\u03b1 (1 - \u03b1) f2 A B` (the hypothesis of the seed's `DriftModel.f3_admixture`, here derived from the graph).","title":"Three-population statistic at an admixture node without edge drift"},{"data_url":"scopes/Biology/population-structure/admixture-linkage-disequilibrium.json","id":"Biology/population-structure/admixture-linkage-disequilibrium","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","title":"Admixture linkage disequilibrium"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/allele-count","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.alleleCount","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#allele-count","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The `0/1` count of the `'1'` allele carried by a gamete at one locus.","title":"Allele count of a gamete at one locus"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/allele-cov","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.alleleCov","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#allele-cov","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The covariance of the allele counts at the two loci of one gamete drawn from the gamete frequencies `x`: `\u2211 g, x g a(g.1) a(g.2) - (\u2211 g, x g a(g.1)) (\u2211 g, x g a(g.2))`.","title":"Covariance of allele counts within a gamete"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/genotype-cov","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.genotypeCov","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#genotype-cov","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The covariance of the unphased diploid genotypes `X = X\u2081 + X\u2082 \u2208 {0, 1, 2}` at the first locus and `Y = Y\u2081 + Y\u2082` at the second, for an individual made of two gametes `g`, `h` drawn independently from `x` (random union of gametes). This is `z(x, y)` of Loh et al. (2013), Eq. 1.","title":"Covariance of unphased diploid genotypes"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/allele-cov-eq-linkage-diseq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.alleleCov_eq_linkageDiseq","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#allele-cov-eq-linkage-diseq","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The covariance of the allele counts of one gamete is the linkage disequilibrium `D = x_AB x_ab - x_Ab x_aB` (frequencies summing to `1`).","title":"Gamete allele covariance equals linkage disequilibrium"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/genotype-cov-eq-two-mul","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.genotypeCov_eq_two_mul","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#genotype-cov-eq-two-mul","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The diploid genotype covariance is twice the haplotype one, `z = 2 D`: the two same-gamete terms each contribute `D` and the two cross-gamete terms vanish by independence (Loh et al. 2013, Eqs. 2\u20133).","title":"Diploid genotype covariance is twice the linkage disequilibrium"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/bern-freq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.bernFreq","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#bern-freq","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The frequency `p` of the `'1'` allele (`b = true`) or `1 - p` of the `'0'` allele.","title":"Bernoulli allele frequency"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/source-mix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.sourceMix","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#source-mix","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Gamete frequencies right after the pulse: with probability `\u03b1` the gamete comes from `A`, where the loci are in linkage equilibrium with `'1'` frequencies `pA` and `qA`, and otherwise from `B` (frequencies `pB`, `qB`). The biological domain is `\u03b1, pA, qA, pB, qB \u2208 [0, 1]`; the identities below hold for all reals.","title":"Gamete frequencies after an admixture pulse"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/sum-source-mix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.sum_sourceMix","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#sum-source-mix","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The pulse gamete frequencies sum to `1`.","title":"Pulse gamete frequencies sum to one"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/linkage-diseq-source-mix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.linkageDiseq_sourceMix","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#linkage-diseq-source-mix","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Admixture LD right after the pulse: `D\u2080 = \u03b1 (1 - \u03b1) \u03b4(x) \u03b4(y)`, with `\u03b4(x) = pA - pB`, `\u03b4(y) = qA - qB` (Chakraborty and Weiss 1988).","title":"Admixture linkage disequilibrium right after the pulse"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/linkage-diseq-admixed","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.linkageDiseq_admixed","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#linkage-diseq-admixed","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Admixture LD after `n` generations of random mating with recombination fraction `r`: `D_n = (1 - r)^n \u03b1 (1 - \u03b1) \u03b4(x) \u03b4(y)`.","title":"Decay of admixture linkage disequilibrium over generations"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/sum-iterate-recombine","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.sum_iterate_recombine","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#sum-iterate-recombine","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Recombination keeps gamete frequencies summing to `1` over any number of generations.","title":"Recombination preserves total gamete frequency"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/genotype-cov-admixed","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.genotypeCov_admixed","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#genotype-cov-admixed","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The diploid genotype covariance in the admixed population `n` generations after the pulse: `z(x, y) = 2 \u03b1 (1 - \u03b1) (1 - r)^n \u03b4(x) \u03b4(y)` (Loh et al. 2013, Eq. 3, with the exact discrete-generation factor `(1 - r)^n` in place of `e^{-n d}`).","title":"Diploid genotype covariance in an admixed population"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/ancestry-pulse","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.ancestryPulse","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#ancestry-pulse","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Joint law of the local-ancestry labels at the two loci right after the pulse (`true` = descends from `A`): every gamete is entirely from `A` (probability `\u03b1`) or entirely from `B`. This is `sourceMix` with perfectly differentiated markers (`pA = qA = 1`, `pB = qB = 0`).","title":"Joint law of local ancestry after an admixture pulse"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/allele-cov-ancestry-pulse","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.alleleCov_ancestryPulse","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#allele-cov-ancestry-pulse","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Pulse admixture: after `n` generations of random mating the local-ancestry indicators at two loci with recombination fraction `r` have covariance `Cov(I_x, I_y) = \u03b1 (1 - \u03b1) (1 - r)^n` (for every real `\u03b1`, `r`; the biological domain is `0 \u2264 \u03b1 \u2264 1`, `0 \u2264 r \u2264 1/2`). The ancestry labels obey the two-locus random-mating recursion of `Linkage.recombine` (Chakraborty and Weiss 1988).","title":"Covariance of local-ancestry indicators after pulse admixture"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/one-sub-haldane","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.one_sub_haldane","parent":46917,"reading_title":{"format":"inline-parts-v1","parts":[["text","Haldane's map function: "],["math","\\(1 - r = e^{-d}\\cosh d\\)"]],"source":"Haldane's map function: $1 - r = e^{-d}\\cosh d$"},"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#one-sub-haldane","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"With Haldane's map function, `1 - r = e^{-d} cosh d` for map length `d`.","title":"Haldane's map function: $1 - r = e^{-d}\\cosh d$"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/pow-one-sub-haldane","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.pow_one_sub_haldane","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#pow-one-sub-haldane","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Exact decay in Morgans: `(1 - r)^n = e^{-n d} (cosh d)^n` when `r = haldane d`.","title":"Exact decay factor under Haldane's map function"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/exp-neg-mul-le-pow-one-sub-haldane","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.exp_neg_mul_le_pow_one_sub_haldane","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#exp-neg-mul-le-pow-one-sub-haldane","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The continuous decay `e^{-n d}` used by Chakraborty and Weiss (1988) and Loh et al. (2013) is a lower bound for the exact factor `(1 - r)^n` under Haldane's map (all real `d`; the biological domain is `0 \u2264 d`).","title":"Continuous decay is a lower bound for the exact factor under Haldane's map"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/allele-cov-ancestry-pulse-haldane","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.alleleCov_ancestryPulse_haldane","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#allele-cov-ancestry-pulse-haldane","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Local-ancestry covariance in Morgans: `Cov(I_x, I_y) = \u03b1 (1 - \u03b1) e^{-n d} (cosh d)^n` for loci `d` Morgans apart.","title":"Local-ancestry covariance in Morgans"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/weighted-ld","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.weightedLD","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#weighted-ld","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The expected ALDER two-site weighted-LD statistic `\ud835\udd3c[z(x, y) w(x) w(y)]` (Loh et al. 2013, Appendix A, without distance binning). The loci `x` and `y` drift independently: `\u03c9.1` and `\u03c9.2` are independent draws from `\u03bc`, and `p X \u03c9` is the frequency of the `'1'` allele in population `X`. The covariance `z(x, y)` is that of the population formed `n` generations ago as an `\u03b1 : 1 - \u03b1` mixture of `A` and `B`\u2026","title":"Expected ALDER weighted linkage disequilibrium"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/weighted-ld-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.weightedLD_eq","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#weighted-ld-eq","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"The expected weighted LD factors as `2 \u03b1 (1 - \u03b1) (1 - r)^n (f4 A B R S)^2`: by independence of the loci, `\ud835\udd3c[\u03b4(x) w(x) \u03b4(y) w(y)] = \ud835\udd3c[\u03b4 w]^2` and `\ud835\udd3c[\u03b4 w] = f4 A B R S`. The identity holds with the Bochner-integral conventions as stated; it is meaningful when the frequencies are square integrable, as in every `DriftModel`.","title":"Weighted linkage disequilibrium in terms of the f4 statistic"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/weighted-ld-sources","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.DriftModel.weightedLD_sources","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#weighted-ld-sources","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Weights from the true mixing populations (`R = A`, `S = B`): the amplitude is `2 \u03b1 \u03b2 F2(A, B)^2` (Loh et al. 2013, Eq. 4).","title":"Weighted LD amplitude with the true source populations"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/f4-eq-f2-of-branch","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.DriftModel.f4_eq_f2_of_branch","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#f4-eq-f2-of-branch","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Branch points on the `A\u2013B` path. Suppose that `A = A\u2033 + uA`, `A\u2032 = A\u2033 + uA'`, `B = B\u2033 + uB`, `B\u2032 = B\u2033 + uB'` at the level of lineage weights, where `w A\u2033 - w B\u2033` is disjoint in support from each of `uA`, `uB`, `uA'`, `uB'`, and each of `uA`, `uB` is disjoint from each of `uA'`, `uB'`. On a tree this says that the paths `A\u2033\u2013B\u2033`, `A\u2033\u2013A`, `B\u2033\u2013B`, `A\u2033\u2013A\u2032`, `B\u2033\u2013B\u2032` share no edge (the vectors may be signed, so every\u2026","title":"f4 equals f2 for references branching off the path"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/weighted-ld-branch","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.DriftModel.weightedLD_branch","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#weighted-ld-branch","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"Two diverged references `A\u2032`, `B\u2032` branching off the `A\u2013B` path at `A\u2033` and `B\u2033`: the amplitude is `2 \u03b1 \u03b2 F2(A\u2033, B\u2033)^2` (Loh et al. 2013, Eq. 6).","title":"Weighted LD amplitude with two diverged references"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/weighted-ld-admixed-ref","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.DriftModel.weightedLD_admixedRef","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#weighted-ld-admixed-ref","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"One reference `A` and the admixed population `C` itself, with no drift in `C` after the pulse (`w C = \u03b1 w A + (1 - \u03b1) w B`): the weights are `p A - p C = \u03b2 \u03b4` and the amplitude is `2 \u03b1 \u03b2^3 F2(A, B)^2` (Loh et al. 2013, Eq. 7).","title":"Weighted LD amplitude with the admixed population as reference"},{"id":"Biology/population-structure/admixture-linkage-disequilibrium/weighted-ld-one-ref","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.AdmixtureLD.DriftModel.weightedLD_oneRef","parent":46917,"result":true,"route":"../roadmap/Biology/population-structure/admixture-linkage-disequilibrium/index.html#weighted-ld-one-ref","scope":"Biology/population-structure/admixture-linkage-disequilibrium","status":"fully_proved","summary":"One diverged reference `R\u2032` branching off the `A\u2013B` path at `R\u2033` (so `A = R\u2033 + uA`, `B = R\u2033 + uB`, `R\u2032 = R\u2033 + uR` with pairwise disjoint supports) and the admixed population `C` with no post-pulse drift: the weights `p R\u2032 - p C` give the amplitude `2 \u03b1 \u03b2 (\u03b1 F2(A, R\u2033) - \u03b2 F2(B, R\u2033))^2` (Loh et al. 2013, Eq. 8).","title":"Weighted LD amplitude with one diverged reference"},{"data_url":"scopes/Biology/population-structure/basis-theorem-for-f-statistics.json","id":"Biology/population-structure/basis-theorem-for-f-statistics","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","title":"Basis theorem for f-statistics"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f4-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f4Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f4-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"The per-locus `f4` form `x \u21a6 (x A - x B) * (x C - x D)` on frequency vectors `x : P \u2192 \u211d`.","title":"Per-locus f4 form"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f2-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f2Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f2-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"The per-locus `f2` form `x \u21a6 (x A - x B) ^ 2`.","title":"Per-locus f2 form"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f3-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f3Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f3-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"The per-locus `f3` form `x \u21a6 (x C - x A) * (x C - x B)`.","title":"Per-locus f3 form"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f2-form-eq-f4-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f2Form_eq_f4Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f2-form-eq-f4-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"`f2Form A B = f4Form A B A B`.","title":"Per-locus f2 form as a special f4 form"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f-space","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.fSpace","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f-space","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"The space `V_F` spanned by all per-locus f-statistics (all `f4Form`; `f2Form` and `f3Form` are special cases).","title":"Space spanned by the per-locus f-statistics"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f4-form-mem-f-space","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f4Form_mem_fSpace","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f4-form-mem-f-space","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"Every `f4Form` lies in `V_F`.","title":"Every f4 form lies in the f-statistic space"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f2-form-mem-f-space","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f2Form_mem_fSpace","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f2-form-mem-f-space","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"Every `f2Form` lies in `V_F`.","title":"Every f2 form lies in the f-statistic space"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f3-form-mem-f-space","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f3Form_mem_fSpace","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f3-form-mem-f-space","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"Every `f3Form` lies in `V_F`.","title":"Every f3 form lies in the f-statistic space"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f4-eq-integral-f4-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f4_eq_integral_f4Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f4-eq-integral-f4-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"The `f4` statistic is the integral of the `f4` form of the frequency vector.","title":"f4 statistic as the integral of the f4 form"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/integral-sum-smul-f2-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.integral_sum_smul_f2Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#integral-sum-smul-f2-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"A linear combination of `f2` forms integrates to the same linear combination of `f2` statistics, for square-integrable frequencies.","title":"Integral of a linear combination of f2 forms"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f2-form-comm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f2Form_comm","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f2-form-comm","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"`f2Form` is symmetric.","title":"Symmetry of the per-locus f2 form"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f2-form-self","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f2Form_self","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f2-form-self","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"`f2Form A A = 0`.","title":"Per-locus f2 form of a population with itself is zero"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f3-form-comm","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f3Form_comm","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f3-form-comm","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"`f3Form` is symmetric in its last two arguments.","title":"Per-locus f3 form is symmetric in its sources"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f3-form-self-left","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f3Form_self_left","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f3-form-self-left","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"`f3Form C C B = 0`.","title":"Per-locus f3 form vanishes when target equals first source"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f4-form-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f4Form_eq","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f4-form-eq","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"`f4Form` in terms of `f2Form`: the polarization identity behind `f4_eq_f2`.","title":"Polarization identity for the per-locus f4 form"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/f2-form-eq-f3-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.f2Form_eq_f3Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#f2-form-eq-f3-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"`f2Form` in terms of the `f3` forms with base `o`.","title":"Per-locus f2 form in terms of f3 forms with a base population"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/linear-independent-f2-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.linearIndependent_f2Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#linear-independent-f2-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"Patterson's `f2` basis: the forms `f2Form i j`, `i < j`, are linearly independent.","title":"Linear independence of Patterson's f2 basis"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/span-f2-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.span_f2Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#span-f2-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"Patterson's `f2` basis: the forms `f2Form i j`, `i < j`, span `V_F`, i.e. every f-statistic is a linear combination of the `f2 P\u1d62 P\u2c7c`, `i < j`.","title":"Patterson's f2 basis spans the f-statistic space"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/linear-independent-f3-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.linearIndependent_f3Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#linear-independent-f3-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"Patterson's `f3` basis with base population `o`: the forms `f3Form o i j` with `i \u2264 j`, `i \u2260 o`, `j \u2260 o` are linearly independent.","title":"Linear independence of Patterson's f3 basis"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/span-f3-form","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.span_f3Form","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#span-f3-form","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"Patterson's `f3` basis with base population `o`: the forms `f3Form o i j` with `i \u2264 j`, `i \u2260 o`, `j \u2260 o` span `V_F`.","title":"Patterson's f3 basis spans the f-statistic space"},{"id":"Biology/population-structure/basis-theorem-for-f-statistics/finrank-f-space","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsBasis.finrank_fSpace","parent":46943,"result":true,"route":"../roadmap/Biology/population-structure/basis-theorem-for-f-statistics/index.html#finrank-f-space","scope":"Biology/population-structure/basis-theorem-for-f-statistics","status":"fully_proved","summary":"The f-statistics of `n` populations span a space of dimension `n choose 2`. (A linear order on the populations, used to index the `f2` basis, is chosen in the proof.)","title":"Dimension of the f-statistic space is n choose 2"},{"data_url":"scopes/Biology/population-structure/coalescence-times-f-statistics-and-pca.json","id":"Biology/population-structure/coalescence-times-f-statistics-and-pca","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","title":"Coalescence times, f-statistics and PCA"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/het-pair","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.hetPair","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#het-pair","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"The probability `a (1 - b) + b (1 - a)` that a gene from a population with allele frequency `a` and a gene from a population with allele frequency `b` differ at a biallelic site.","title":"Probability that two genes differ at a biallelic site"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/sq-sub-eq-het-pair","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.sq_sub_eq_hetPair","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#sq-sub-eq-het-pair","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"`(a - b) ^ 2 = hetPair a b - (hetPair a a + hetPair b b) / 2`.","title":"Squared frequency difference via pairwise heterozygosity"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/mean-het","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.meanHet","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#mean-het","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"The expected heterozygosity `\u03c0 X Y = \ud835\udd3c[hetPair (p X) (p Y)]` between populations `X` and `Y` (within population `X` when `Y = X`).","title":"Expected heterozygosity between populations"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/integrable-het-pair","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.integrable_hetPair","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#integrable-het-pair","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"`hetPair` of square-integrable frequencies is integrable on a finite measure space.","title":"Integrability of pairwise heterozygosity"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/f2-eq-mean-het","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.f2_eq_meanHet","parent":46965,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peter's formula for "],["math","\\(f_2\\)"],["text"," via expected heterozygosity"]],"source":"Peter's formula for $f_2$ via expected heterozygosity"},"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#f2-eq-mean-het","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"Peter (2016): `f2 A B = \u03c0 A B - (\u03c0 A A + \u03c0 B B) / 2`.","title":"Peter's formula for $f_2$ via expected heterozygosity"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/coalescent-model","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.CoalescentModel","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#coalescent-model","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"The relation between allele frequencies and coalescence times under neutral infinite-sites mutation, taken as the model assumption: the expected heterozygosity between populations `X` and `Y` is `\u03b8` times the expected coalescence time `coalTime X Y` of a gene from `X` and a gene from `Y` (two genes drawn independently, so with replacement, when `X = Y`). Time is in units of `2N` generations and `\u03b8 = 4Nu`. The\u2026","title":"Coalescent model linking heterozygosity and coalescence times"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/f2-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.CoalescentModel.f2_eq","parent":46965,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(f_2\\)"],["text"," in terms of expected coalescence times"]],"source":"$f_2$ in terms of expected coalescence times"},"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#f2-eq","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"`f2 A B = \u03b8 (\ud835\udd3c T_AB - (\ud835\udd3c T_AA + \ud835\udd3c T_BB) / 2)`.","title":"$f_2$ in terms of expected coalescence times"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/f3-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.CoalescentModel.f3_eq","parent":46965,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(f_3\\)"],["text"," in terms of expected coalescence times"]],"source":"$f_3$ in terms of expected coalescence times"},"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#f3-eq","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"`f3 C A B = \u03b8 / 2 * (\ud835\udd3c T_CA + \ud835\udd3c T_CB - \ud835\udd3c T_AB - \ud835\udd3c T_CC)`.","title":"$f_3$ in terms of expected coalescence times"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/f4-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.CoalescentModel.f4_eq","parent":46965,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(f_4\\)"],["text"," in terms of expected coalescence times"]],"source":"$f_4$ in terms of expected coalescence times"},"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#f4-eq","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"`f4 A B C D = \u03b8 / 2 * (\ud835\udd3c T_AD + \ud835\udd3c T_BC - \ud835\udd3c T_AC - \ud835\udd3c T_BD)`.","title":"$f_4$ in terms of expected coalescence times"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/col-mean","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.colMean","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#col-mean","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"The column mean `(1 / n) \u2211 i, Z i s` of locus `s` over the `n` samples.","title":"Column mean of a genotype matrix"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/centered","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.centered","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#centered","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"The genotype matrix with columns centred over the samples.","title":"Column-centred genotype matrix"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/gen-cov","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.genCov","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#gen-cov","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"The sample covariance `C i j = (1 / L) \u2211 s, X i s * X j s` of the centred genotype matrix `X` over the `L` loci.","title":"Sample covariance of the centred genotype matrix"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/pair-diff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.pairDiff","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#pair-diff","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"The number of pairwise differences `d i j = \u2211 s, (Z i s - Z j s) ^ 2` (for `0`/`1` haplotypes, the number of loci at which samples `i` and `j` differ).","title":"Number of pairwise differences between samples"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/double-center","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.doubleCenter","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#double-center","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"The double centring `t\u0304 i\u00b7 + t\u0304 \u00b7j - t i j - t\u0304 \u00b7\u00b7` of a square matrix `t` (row, column and grand means over the `n` indices).","title":"Double centring of a square matrix"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/gen-cov-eq-double-center","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.genCov_eq_doubleCenter","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#gen-cov-eq-double-center","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"McVean (2009), deterministic form: the covariance of the centred genotype matrix is the double centring of the pairwise-difference matrix, divided by `2 L`.","title":"Genotype covariance as double-centred pairwise differences"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/mc-vean-model","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.McVeanModel","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#mc-vean-model","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"McVean's genealogical setting for PCA, taken as the model assumption: random genotypes `Z i s` (sample `i`, locus `s`), square integrable, whose expected numbers of pairwise differences are `\u03b8` times the expected pairwise coalescence times `coalTime i j` (neutral infinite sites, time in units of `2N` generations, `\u03b8` the total scaled mutation rate of the `L` loci). For `\u03b8 = 0` the coalescence times are not\u2026","title":"McVean's genealogical model for PCA"},{"id":"Biology/population-structure/coalescence-times-f-statistics-and-pca/integral-gen-cov","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.CoalescentFStatistics.McVeanModel.integral_genCov","parent":46965,"result":true,"route":"../roadmap/Biology/population-structure/coalescence-times-f-statistics-and-pca/index.html#integral-gen-cov","scope":"Biology/population-structure/coalescence-times-f-statistics-and-pca","status":"fully_proved","summary":"McVean (2009): the expected genotype covariance is `\u03b8 / (2 L)` times the double centring of the expected pairwise coalescence times, `\ud835\udd3c[C i j] = \u03b8 / (2 L) * (t\u0304 i\u00b7 + t\u0304 \u00b7j - t i j - t\u0304 \u00b7\u00b7)`.","title":"McVean's theorem on expected genotype covariance"},{"data_url":"scopes/Biology/population-structure/centred-population-covariance-and-f-statistics.json","id":"Biology/population-structure/centred-population-covariance-and-f-statistics","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","title":"Centred population covariance and f-statistics"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/pop-mean","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.popMean","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#pop-mean","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"The mean frequency `p\u0304 \u03c9 = (1 / n) \u2211 X, p X \u03c9` across the `n` populations.","title":"Mean frequency across populations"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/centred-cov","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.centredCov","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#centred-cov","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"The centred leaf covariance `K X Y = \ud835\udd3c[(p X - p\u0304) * (p Y - p\u0304)]`: the second moments of the frequencies centred across populations (not over `\u03a9`).","title":"Centred leaf covariance"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/f2-eq-centred-cov","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.f2_eq_centredCov","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#f2-eq-centred-cov","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"`f2` from the centred leaf covariance: `f2 X Y = K X X + K Y Y - 2 K X Y`.","title":"f2 from the centred leaf covariance"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/centred-cov-eq-double-center-f2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.centredCov_eq_doubleCenter_f2","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#centred-cov-eq-double-center-f2","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"The centred leaf covariance from `f2`: `K` is half the double centring of the `f2` matrix, `K X Y = (f\u03042 X\u00b7 + f\u03042 \u00b7Y - f2 X Y - f\u03042 \u00b7\u00b7) / 2`.","title":"Centred leaf covariance as the double centring of f2"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/f2-eq-iff-centred-cov-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.f2_eq_iff_centredCov_eq","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#f2-eq-iff-centred-cov-eq","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"f-statistics determine exactly the centred leaf covariance. f-statistics see a model only through its centred leaf covariance: two square-integrable frequency models have the same `f2` at every pair (hence the same `f3` and `f4`, by `f3_eq_f2` and `f4_eq_f2`) if and only if they have the same centred leaf covariance.","title":"f-statistics determine exactly the centred leaf covariance"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/star-weight","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.starWeight","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#star-weight","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"Lineage weights of the star tree on three populations `0 = A`, `1 = B`, `2 = C`: edge `e` is the leaf edge of population `e`.","title":"Lineage weights of the three-population star tree"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/admix-weight","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.admixWeight","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#admix-weight","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"Lineage weights of the admixture model: `A` and `B` sit on edges `0` and `1`, and `C` is an `\u03b1 : 1 - \u03b1` mixture of `A` and `B` followed by drift on its private edge `2`.","title":"Lineage weights of the three-population admixture model"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/star-len","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.starLen","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#star-len","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"Edge lengths of the star tree matching the admixture model with edge lengths `a`, `b`, `c`: `(1 - \u03b1)(a + b)`, `\u03b1 (a + b)` and `c - \u03b1 (1 - \u03b1)(a + b)`.","title":"Edge lengths of the star tree matching an admixture model"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/admix-weight-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.admixWeight_two","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#admix-weight-two","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"In the admixture model, `C` is the `\u03b1 : 1 - \u03b1` mixture of `A` and `B` plus its private edge.","title":"Admixed population's weights in the admixture model"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/has-binary-loadings-star","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.hasBinaryLoadings_star","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#has-binary-loadings-star","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"The star-tree model has binary loadings.","title":"Star-tree model has binary loadings"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/not-has-binary-loadings-admix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.not_hasBinaryLoadings_admix","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#not-has-binary-loadings-admix","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"For `\u03b1 \u2209 {0, 1}` the admixture model does not have binary loadings.","title":"Admixture model lacks binary loadings"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/f2-star-eq-f2-admix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.f2_star_eq_f2_admix","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#f2-star-eq-f2-admix","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"The star tree and the admixture model have the same `f2` at every pair of populations.","title":"Star tree and admixture model have equal f2"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/f4-star-eq-f4-admix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.f4_star_eq_f4_admix","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#f4-star-eq-f4-admix","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"The star tree and the admixture model have the same `f4` (hence the same `f3`) at every quadruple of populations.","title":"Star tree and admixture model have equal f4"},{"id":"Biology/population-structure/centred-population-covariance-and-f-statistics/star-len-nonneg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.FStatisticsCovariance.starLen_nonneg","parent":46983,"result":true,"route":"../roadmap/Biology/population-structure/centred-population-covariance-and-f-statistics/index.html#star-len-nonneg","scope":"Biology/population-structure/centred-population-covariance-and-f-statistics","status":"fully_proved","summary":"The matching star tree has nonnegative edge lengths when `0 \u2264 \u03b1 \u2264 1`, `a, b \u2265 0`, and the private drift of `C` is at least `\u03b1 (1 - \u03b1)(a + b)`, i.e. when `f3 C A B \u2265 0` in the admixture model.","title":"Nonnegativity of the matching star-tree edge lengths"},{"data_url":"scopes/Biology/population-structure/fitting-admixture-graphs-by-least-squares.json","id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","title":"Fitting admixture graphs by least squares"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/score-s1","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.scoreS1","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#score-s1","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"The qpGraph score `S1(g) = -\u00bd (g - f)\u1d40 Q\u207b\u00b9 (g - f)` (Patterson et al. 2012, eq. 6) of predicted statistics `g` against estimated statistics `f` with error covariance `Q`.","title":"Generalized least-squares qpGraph score S1"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/score-s2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.scoreS2","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#score-s2","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"The least-squares qpGraph score `S2(g) = -\u2211\u1d62 (g\u1d62 - f\u1d62)\u00b2 / (Q\u1d62\u1d62 + \u03bb)` (Patterson et al. 2012, eq. 7). A term with `Q\u1d62\u1d62 + \u03bb = 0` is `0` by the convention `x / 0 = 0`.","title":"Least-squares qpGraph score S2"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/score-s1-basis-change","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.scoreS1_basis_change","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#score-s1-basis-change","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"`S1` is basis independent: under an invertible change of basis `A` of the statistics, with the covariance transformed to `A Q A\u1d40`, the score is unchanged. (The identity holds for every `Q`; for singular `Q` both sides are `0` by Mathlib's convention `Q\u207b\u00b9 = 0`.)","title":"Basis independence of the qpGraph score S1"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/exists-score-s2-basis-change-ne","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.exists_scoreS2_basis_change_ne","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#exists-score-s2-basis-change-ne","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"`S2` is not basis independent (Patterson et al. 2012, after eq. 7): there are a positive definite covariance `Q`, an invertible change of basis `A` and statistics `f`, `g` such that, for every `\u03bb`, the score changes.","title":"The qpGraph score S2 depends on the basis"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/score-s2-eq-score-s1-diagonal","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.scoreS2_eq_scoreS1_diagonal","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#score-s2-eq-score-s1-diagonal","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"`S2` is twice `S1` with the covariance replaced by the diagonal matrix of the `Q\u1d62\u1d62 + \u03bb`, when these are nonzero.","title":"Score S2 as score S1 with a diagonal covariance"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/design-matrix","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.designMatrix","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#design-matrix","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"The qpGraph design matrix of basis statistics `f4 (A i) (B i) (C i) (D i)` for lineage weights `w`: entry `(i, e)` is `(w A\u1d62 e - w B\u1d62 e) * (w C\u1d62 e - w D\u1d62 e)`.","title":"qpGraph design matrix of basis f4 statistics"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/f4-eq-design-matrix-mul-vec","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.f4_eq_designMatrix_mulVec","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#f4-eq-design-matrix-mul-vec","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"In the drift model, with the admixture weights fixed, the predicted statistics are linear in the edge lengths: `g = B \u2113` for the design matrix `B`.","title":"Predicted f4 statistics are linear in the edge lengths"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/score-s1-le-of-normal-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.scoreS1_le_of_normal_eq","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#score-s1-le-of-normal-eq","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"GLS maximizes `S1`: for positive definite `Q`, a solution `\u2113\u0302` of the normal equations `B\u1d40 Q\u207b\u00b9 B \u2113\u0302 = B\u1d40 Q\u207b\u00b9 f` maximizes `\u2113 \u21a6 S1(B \u2113)` over all `\u2113`.","title":"Solutions of the normal equations maximize the score S1"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/eq-of-score-s1-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.eq_of_scoreS1_eq","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#eq-of-score-s1-eq","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"Uniqueness of the GLS maximizer: for positive definite `Q` and a design matrix of full column rank (`B.mulVec` injective), a vector `\u2113` with the same score as a solution `\u2113\u0302` of the normal equations equals `\u2113\u0302`.","title":"Uniqueness of the generalized least-squares maximizer"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/is-unit-det-gram","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.isUnit_det_gram","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#is-unit-det-gram","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"For positive definite `Q` and a design matrix of full column rank, the GLS Gram matrix `B\u1d40 Q\u207b\u00b9 B` is invertible.","title":"Invertibility of the generalized least-squares Gram matrix"},{"id":"Biology/population-structure/fitting-admixture-graphs-by-least-squares/normal-eq-gls","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.QpGraph.normal_eq_gls","parent":46998,"result":true,"route":"../roadmap/Biology/population-structure/fitting-admixture-graphs-by-least-squares/index.html#normal-eq-gls","scope":"Biology/population-structure/fitting-admixture-graphs-by-least-squares","status":"fully_proved","summary":"The closed-form GLS estimate `\u2113\u0302 = (B\u1d40 Q\u207b\u00b9 B)\u207b\u00b9 B\u1d40 Q\u207b\u00b9 f` solves the normal equations when the Gram matrix is invertible.","title":"Closed-form generalized least-squares estimate of edge lengths"},{"data_url":"scopes/Biology/population-structure/outgroup-f3-and-admixed-branches.json","id":"Biology/population-structure/outgroup-f3-and-admixed-branches","kind":"scope","navigation":"module","parent":46731,"result":false,"route":"../roadmap/Biology/population-structure/outgroup-f3-and-admixed-branches/index.html","scope":"Biology/population-structure/outgroup-f3-and-admixed-branches","status":"fully_proved","title":"Outgroup f3 and admixed branches"},{"id":"Biology/population-structure/outgroup-f3-and-admixed-branches/f3-eq-sum-inter-symm-diff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TreeFStatistics.f3_eq_sum_inter_symmDiff","parent":47010,"result":true,"route":"../roadmap/Biology/population-structure/outgroup-f3-and-admixed-branches/index.html#f3-eq-sum-inter-symm-diff","scope":"Biology/population-structure/outgroup-f3-and-admixed-branches","status":"fully_proved","summary":"With binary loadings, `f3 O A B` is the total drift length of the edges shared by the paths `O\u2013A` and `O\u2013B` (on a tree, the path from `O` to the `A\u2013B` path).","title":"f3 statistic as shared drift on a tree"},{"id":"Biology/population-structure/outgroup-f3-and-admixed-branches/f3-outgroup","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TreeFStatistics.f3_outgroup","parent":47010,"result":true,"route":"../roadmap/Biology/population-structure/outgroup-f3-and-admixed-branches/index.html#f3-outgroup","scope":"Biology/population-structure/outgroup-f3-and-admixed-branches","status":"fully_proved","summary":"Outgroup `f3` (Raghavan et al. 2014): with binary loadings, if the lineage of `O` shares no edge with the lineages of `A` and `B`, then `f3 O A B` is the drift from the outgroup to the root plus the drift shared by `A` and `B`.","title":"Outgroup f3 statistic"},{"id":"Biology/population-structure/outgroup-f3-and-admixed-branches/f3-outgroup-le-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TreeFStatistics.f3_outgroup_le_iff","parent":47010,"result":true,"route":"../roadmap/Biology/population-structure/outgroup-f3-and-admixed-branches/index.html#f3-outgroup-le-iff","scope":"Biology/population-structure/outgroup-f3-and-admixed-branches","status":"fully_proved","summary":"For a fixed outgroup `O` and population `A`, the outgroup `f3` ranks candidates `B` by the drift they share with `A`.","title":"Outgroup f3 ranks populations by shared drift"},{"id":"Biology/population-structure/outgroup-f3-and-admixed-branches/mixed-drift","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TreeFStatistics.mixedDrift","parent":47010,"result":true,"route":"../roadmap/Biology/population-structure/outgroup-f3-and-admixed-branches/index.html#mixed-drift","scope":"Biology/population-structure/outgroup-f3-and-admixed-branches","status":"fully_proved","summary":"MixMapper's mixed drift `\u03b1\u00b2 \u2113_a + (1 - \u03b1)\u00b2 \u2113_b + \u2113_c`: the only combination of the terminal drift lengths that enters the `f2` statistics of an admixed population.","title":"MixMapper mixed drift"},{"id":"Biology/population-structure/outgroup-f3-and-admixed-branches/f2-admixed-branch","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TreeFStatistics.f2_admixedBranch","parent":47010,"result":true,"route":"../roadmap/Biology/population-structure/outgroup-f3-and-admixed-branches/index.html#f2-admixed-branch","scope":"Biology/population-structure/outgroup-f3-and-admixed-branches","status":"fully_proved","summary":"The admixed-branch identity (Lipson et al. 2013): if `Mx` is an `\u03b1 : 1 - \u03b1` mixture of the endpoints of the edges `a` (below `A`) and `b` (below `B`), followed by drift on `c`, and `A`, `B`, `X` do not use `a`, `b`, `c`, then `f2 Mx X = \u03b1 f2 A X + (1 - \u03b1) f2 B X - \u03b1 (1 - \u03b1) f2 A B + mixedDrift \u03b1 \u2113_a \u2113_b \u2113_c`.","title":"Admixed-branch identity for f2 statistics"},{"id":"Biology/population-structure/outgroup-f3-and-admixed-branches/alpha-eq-of-f2","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TreeFStatistics.alpha_eq_of_f2","parent":47010,"result":true,"route":"../roadmap/Biology/population-structure/outgroup-f3-and-admixed-branches/index.html#alpha-eq-of-f2","scope":"Biology/population-structure/outgroup-f3-and-admixed-branches","status":"fully_proved","summary":"Identifiability of the mixture proportion (our formulation): with the split points `A`, `B` given, two references `X\u2081`, `X\u2082` satisfying the hypotheses of `f2_admixedBranch` and with different contrasts `f2 A X\u1d62 - f2 B X\u1d62` determine `\u03b1` from `f2` statistics.","title":"Identifiability of the admixture proportion from f2 statistics"},{"id":"Biology/population-structure/outgroup-f3-and-admixed-branches/mixed-drift-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.TreeFStatistics.mixedDrift_eq","parent":47010,"result":true,"route":"../roadmap/Biology/population-structure/outgroup-f3-and-admixed-branches/index.html#mixed-drift-eq","scope":"Biology/population-structure/outgroup-f3-and-admixed-branches","status":"fully_proved","summary":"Given `\u03b1`, the mixed drift is determined by the `f2` statistics of one reference `X`.","title":"Mixed drift determined by f2 statistics"},{"data_url":"scopes/Biology/quantitative-genetics.json","id":"Biology/quantitative-genetics","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/quantitative-genetics/index.html","scope":"Biology/quantitative-genetics","status":"fully_proved","summary":"The genetics of continuously varying traits: decomposition of genetic variance, resemblance between relatives, kinship, mixed models and the response to selection.","title":"Quantitative genetics"},{"data_url":"scopes/Biology/quantitative-genetics/breeder-s-equation.json","id":"Biology/quantitative-genetics/breeder-s-equation","kind":"scope","navigation":"module","parent":47018,"result":false,"route":"../roadmap/Biology/quantitative-genetics/breeder-s-equation/index.html","scope":"Biology/quantitative-genetics/breeder-s-equation","status":"fully_proved","title":"Breeder's equation"},{"id":"Biology/quantitative-genetics/breeder-s-equation/model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Breeder.Model","parent":47019,"result":true,"route":"../roadmap/Biology/quantitative-genetics/breeder-s-equation/index.html#model","scope":"Biology/quantitative-genetics/breeder-s-equation","status":"fully_proved","summary":"The linear parent\u2013offspring model on a population described by a probability measure `\u03bc`: the parental (mid-parent) value `parent` and the offspring value `offspring` are integrable, `mean` is the population mean of the parental value, and the offspring value regresses linearly on the parental value, `\ud835\udd3c[O | P] = mean + h2 * (P - mean)` (equivalently `O = mean + h2 (P - mean) + e` with `\ud835\udd3c[e | P] = 0`). Here `h2` is\u2026","title":"Linear parent\u2013offspring regression model"},{"id":"Biology/quantitative-genetics/breeder-s-equation/selection-differential","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Breeder.Model.selectionDifferential","parent":47019,"result":true,"route":"../roadmap/Biology/quantitative-genetics/breeder-s-equation/index.html#selection-differential","scope":"Biology/quantitative-genetics/breeder-s-equation","status":"fully_proved","summary":"The selection differential `S = \ud835\udd3c[P | selected] - \u03bc` of the selected group `s`.","title":"Selection differential"},{"id":"Biology/quantitative-genetics/breeder-s-equation/response","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Breeder.Model.response","parent":47019,"result":true,"route":"../roadmap/Biology/quantitative-genetics/breeder-s-equation/index.html#response","scope":"Biology/quantitative-genetics/breeder-s-equation","status":"fully_proved","summary":"The response to selection `R = \ud835\udd3c[O | selected] - \u03bc`: the mean offspring value of the selected parents, relative to the population mean.","title":"Response to selection"},{"id":"Biology/quantitative-genetics/breeder-s-equation/response-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Breeder.Model.response_eq","parent":47019,"result":true,"route":"../roadmap/Biology/quantitative-genetics/breeder-s-equation/index.html#response-eq","scope":"Biology/quantitative-genetics/breeder-s-equation","status":"fully_proved","summary":"The breeder's equation (Lush 1937): if the selected group `s` is determined by the parental value (an event of `\u03c3(P)`) and has positive mass, the response to selection is the heritability times the selection differential, `R = h\u00b2 S`.","title":"Breeder's equation"},{"id":"Biology/quantitative-genetics/breeder-s-equation/selection-differential-univ","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Breeder.Model.selectionDifferential_univ","parent":47019,"result":true,"route":"../roadmap/Biology/quantitative-genetics/breeder-s-equation/index.html#selection-differential-univ","scope":"Biology/quantitative-genetics/breeder-s-equation","status":"fully_proved","summary":"Selecting the whole population gives no selection differential, `S = 0`.","title":"Selecting the whole population gives zero selection differential"},{"id":"Biology/quantitative-genetics/breeder-s-equation/response-univ","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Breeder.Model.response_univ","parent":47019,"result":true,"route":"../roadmap/Biology/quantitative-genetics/breeder-s-equation/index.html#response-univ","scope":"Biology/quantitative-genetics/breeder-s-equation","status":"fully_proved","summary":"Selecting the whole population gives no response, `R = 0`.","title":"Selecting the whole population gives zero response"},{"data_url":"scopes/Biology/quantitative-genetics/dominance-linkage-disequilibrium.json","id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","kind":"scope","navigation":"module","parent":47018,"result":false,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","title":"Dominance linkage disequilibrium"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/random-union","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#random-union","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"A diploid two-locus genotype under random union of gametes: the haplotypes `(X\u2081, Y\u2081)` and `(X\u2082, Y\u2082)` are independent and identically distributed, with square-integrable allele codings.","title":"Two-locus diploid genotype under random union of gametes"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/freq-a","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.freqA","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#freq-a","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The allele frequency (mean coding) `p = E X\u2081` at locus `A`.","title":"Allele frequency at the first locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/freq-b","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.freqB","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#freq-b","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The allele frequency (mean coding) `q = E Y\u2081` at locus `B`.","title":"Allele frequency at the second locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/var-a","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.varA","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#var-a","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The haplotype variance `V_A = E (X\u2081 - p)\u00b2` at locus `A`.","title":"Haplotype variance at the first locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/var-b","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.varB","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#var-b","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The haplotype variance `V_B = E (Y\u2081 - q)\u00b2` at locus `B`.","title":"Haplotype variance at the second locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/ld-d","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.ldD","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#ld-d","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The gametic linkage disequilibrium `D = E (X\u2081 - p) (Y\u2081 - q)`.","title":"Gametic linkage disequilibrium"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/add-a","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.addA","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#add-a","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The additive coding `a_A = (X\u2081 - p) + (X\u2082 - p)` (centred allele count) at locus `A`.","title":"Additive coding at the first locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/add-b","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.addB","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#add-b","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The additive coding `a_B = (Y\u2081 - q) + (Y\u2082 - q)` at locus `B`.","title":"Additive coding at the second locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/dom-a","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.domA","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#dom-a","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The dominance coding `d_A = (X\u2081 - p) (X\u2082 - p)` at locus `A`.","title":"Dominance coding at the first locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/dom-b","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.domB","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#dom-b","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The dominance coding `d_B = (Y\u2081 - q) (Y\u2082 - q)` at locus `B`.","title":"Dominance coding at the second locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/integral-add-a-mul-add-b","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.integral_addA_mul_addB","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#integral-add-a-mul-add-b","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"Additive LD: `E(a_A a_B) = 2 D`.","title":"Additive linkage disequilibrium is twice the gametic one"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/integral-add-a-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.integral_addA_sq","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#integral-add-a-sq","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The additive coding has second moment `E(a_A\u00b2) = 2 V_A`.","title":"Second moment of the additive coding at the first locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/integral-add-b-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.integral_addB_sq","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#integral-add-b-sq","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The additive coding has second moment `E(a_B\u00b2) = 2 V_B`.","title":"Second moment of the additive coding at the second locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/integral-dom-a-mul-dom-b","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.integral_domA_mul_domB","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#integral-dom-a-mul-dom-b","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"Dominance LD: `E(d_A d_B) = D\u00b2`.","title":"Dominance linkage disequilibrium is the square of the gametic one"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/integral-dom-a-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.integral_domA_sq","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#integral-dom-a-sq","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The dominance coding has second moment `E(d_A\u00b2) = V_A\u00b2`.","title":"Second moment of the dominance coding at the first locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/integral-dom-b-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.integral_domB_sq","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#integral-dom-b-sq","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The dominance coding has second moment `E(d_B\u00b2) = V_B\u00b2`.","title":"Second moment of the dominance coding at the second locus"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/integral-add-a-mul-dom-b","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.integral_addA_mul_domB","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#integral-add-a-mul-dom-b","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The additive coding at `A` and the dominance coding at `B` are uncorrelated: `E(a_A d_B) = 0`.","title":"Additive and dominance codings at different loci are uncorrelated"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/add-corr-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.addCorr_eq","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#add-corr-eq","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The correlation of the additive codings is the gametic correlation `r = D / \u221a(V_A V_B)`.","title":"Correlation of additive codings equals the gametic correlation"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/dom-corr-eq-add-corr-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.RandomUnion.domCorr_eq_addCorr_sq","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#dom-corr-eq-add-corr-sq","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"Dominance LD is the square of additive LD (Palmer et al. 2023, graphical abstract): the correlation of the dominance codings at two loci is the square of the correlation of their additive codings.","title":"Dominance LD is the square of additive LD"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/dominance-encoding","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.dominanceEncoding","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#dominance-encoding","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"The dominance encoding of a biallelic genotype `g \u2208 {0, 1, 2}` (count of the allele with frequency `p`): `p\u00b2`, `-p (1 - p)`, `(1 - p)\u00b2`. Dividing by `-p (1 - p)` gives the encoding `-p / (1 - p)`, `1`, `-(1 - p) / p` of Palmer et al. (2023).","title":"Dominance encoding of a biallelic genotype"},{"id":"Biology/quantitative-genetics/dominance-linkage-disequilibrium/dom-code-eq-dominance-encoding","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.DominanceLD.domCode_eq_dominanceEncoding","parent":47026,"result":true,"route":"../roadmap/Biology/quantitative-genetics/dominance-linkage-disequilibrium/index.html#dom-code-eq-dominance-encoding","scope":"Biology/quantitative-genetics/dominance-linkage-disequilibrium","status":"fully_proved","summary":"For `0`/`1` allele indicators the dominance coding `(X\u2081 - p) (X\u2082 - p)` depends only on the genotype `X\u2081 + X\u2082` and equals its dominance encoding.","title":"Dominance coding of allele indicators equals the dominance encoding"},{"data_url":"scopes/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance.json","id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","kind":"scope","navigation":"module","parent":47018,"result":false,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","title":"Fisher's decomposition of genetic variance"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/mean-value","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.meanValue","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#mean-value","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The mean genotypic value `\u03bc = \u2211 i \u2211 j p i p j G i j` under Hardy\u2013Weinberg proportions.","title":"Mean genotypic value under Hardy\u2013Weinberg proportions"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/avg-effect","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.avgEffect","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#avg-effect","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The average effect of allele `i`, `\u03b1 i = \u2211 j p j G i j - \u03bc`: the mean value of the genotypes carrying `i` (the other gene drawn from the population) minus the population mean. This is the average excess; under Hardy\u2013Weinberg proportions it is Fisher's average effect, the coefficient of the least-squares additive fit (`domVar_le_sum_sq`).","title":"Average effect of an allele"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/dom-dev","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.domDev","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#dom-dev","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The dominance deviation `\u03b4 i j = G i j - \u03bc - \u03b1 i - \u03b1 j`.","title":"Dominance deviation of a genotype"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/gen-var","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.genVar","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#gen-var","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The genetic variance `V_G = \u2211 i \u2211 j p i p j (G i j - \u03bc) \u00b2`.","title":"Genetic variance under Hardy\u2013Weinberg proportions"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/add-var","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.addVar","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#add-var","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The additive genetic variance `V_A = 2 \u2211 i p i \u03b1 i \u00b2`, the variance of the breeding value `\u03b1 i + \u03b1 j`.","title":"Additive genetic variance"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/dom-var","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.domVar","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#dom-var","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The dominance variance `V_D = \u2211 i \u2211 j p i p j \u03b4 i j \u00b2`.","title":"Dominance variance"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/genotypic-value-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.genotypicValue_eq","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#genotypic-value-eq","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The decomposition `G i j = \u03bc + \u03b1 i + \u03b1 j + \u03b4 i j` (an identity by definition of `\u03b4`).","title":"Decomposition of the genotypic value"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/sum-mul-avg-effect","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.sum_mul_avgEffect","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#sum-mul-avg-effect","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The average effects have mean zero: `\u2211 i p i \u03b1 i = 0`.","title":"Average effects have mean zero"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/sum-mul-dom-dev-right","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.sum_mul_domDev_right","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#sum-mul-dom-dev-right","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"Orthogonality of the dominance deviations: `\u2211 j p j \u03b4 i j = 0` for every `i`.","title":"Orthogonality of dominance deviations in the second allele"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/sum-mul-dom-dev-left","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.sum_mul_domDev_left","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#sum-mul-dom-dev-left","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"Orthogonality of the dominance deviations for a symmetric genotypic value: `\u2211 i p i \u03b4 i j = 0` for every `j`.","title":"Orthogonality of dominance deviations in the first allele"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/dom-var-le-sum-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.domVar_le_sum_sq","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#dom-var-le-sum-sq","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"Among all fits `c + a i + a j` of the genotypic value by a function additive over the two genes, the additive fitted genotypic value `\u03bc + \u03b1 i + \u03b1 j` minimizes the weighted squared error `\u2211 p i p j (G i j - c - a i - a j) \u00b2`; its error is `V_D`.","title":"Least squares (Fisher 1918)"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/gen-var-eq-add-var-add-dom-var","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.genVar_eq_addVar_add_domVar","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#gen-var-eq-add-var-add-dom-var","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"`V_G = V_A + V_D`.","title":"Fisher's variance decomposition"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/cov-parent-offspring","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.covParentOffspring","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#cov-parent-offspring","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The covariance of the genotypic values of a parent and its offspring: the parent has genotype `(i, j)` (frequency `p i p j`) and transmits `cond b i j` (each with probability `1/2`), and the offspring's other gene `k` comes from an unrelated mate (frequency `p k`).","title":"Parent\u2013offspring covariance of genotypic values"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/cov-half-sibs","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.covHalfSibs","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#cov-half-sibs","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The covariance of the genotypic values of half sibs: the common parent has genotype `(i, j)` and transmits `cond b i j` to one offspring and, independently, `cond b' i j` to the other; the second genes `k`, `l` come from two unrelated mates.","title":"Half-sib covariance of genotypic values"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/cov-full-sibs","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.covFullSibs","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#cov-full-sibs","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"The covariance of the genotypic values of full sibs: the sire `(i, j)` and the dam `(k, l)` are unrelated; the first offspring receives `cond b i j` and `cond e k l`, the second, independently, `cond b' i j` and `cond e' k l`.","title":"Full-sib covariance of genotypic values"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/cov-parent-offspring-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.covParentOffspring_eq","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#cov-parent-offspring-eq","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"`Cov(P, O) = V_A / 2`.","title":"Parent and offspring (Fisher 1918)"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/cov-half-sibs-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.covHalfSibs_eq","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#cov-half-sibs-eq","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"`Cov(HS) = V_A / 4`.","title":"Half sibs (Fisher 1918)"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/cov-full-sibs-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.covFullSibs_eq","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#cov-full-sibs-eq","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"`Cov(FS) = V_A / 2 + V_D / 4`.","title":"Full sibs (Fisher 1918)"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/biallelic-freq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.biallelicFreq","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#biallelic-freq","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"Allele frequencies at a biallelic locus: `A\u2081` (`true`) has frequency `q\u2081` and `A\u2082` (`false`) has frequency `1 - q\u2081`.","title":"Allele frequencies at a biallelic locus"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/biallelic-value","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.biallelicValue","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#biallelic-value","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"Genotypic values at a biallelic locus on Falconer's scale: `A\u2081A\u2081 \u21a6 a`, `A\u2081A\u2082 \u21a6 d`, `A\u2082A\u2082 \u21a6 -a` (the origin is the midpoint of the homozygotes).","title":"Genotypic values at a biallelic locus on Falconer's scale"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/add-var-biallelic","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.addVar_biallelic","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#add-var-biallelic","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"With `p = q\u2081` and `q = 1 - q\u2081`, `V_A = 2pq (a + d (q - p)) \u00b2`. An algebraic identity, valid for every real `q\u2081`; it is a variance for `0 \u2264 q\u2081 \u2264 1`.","title":"Additive variance at a biallelic locus (Falconer & Mackay 1996, ch. 8)"},{"id":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/dom-var-biallelic","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.FisherDecomposition.domVar_biallelic","parent":47048,"result":true,"route":"../roadmap/Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance/index.html#dom-var-biallelic","scope":"Biology/quantitative-genetics/fisher-s-decomposition-of-genetic-variance","status":"fully_proved","summary":"`V_D = (2pqd) \u00b2`, for every real `q\u2081`.","title":"Dominance variance at a biallelic locus (Falconer & Mackay 1996, ch. 8)"},{"data_url":"scopes/Biology/quantitative-genetics/covariance-between-relatives.json","id":"Biology/quantitative-genetics/covariance-between-relatives","kind":"scope","navigation":"module","parent":47018,"result":false,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","title":"Covariance between relatives"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/locus-model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#locus-model","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"A single locus under random union of gametes: allele frequencies `p` on a finite allele set and symmetric genotypic values `g a b` (the mean phenotype of genotype `{a, b}`).","title":"Single-locus model under random union of gametes"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/mean","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.mean","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#mean","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The population mean genotypic value `\u03bc = \u2211 p_a p_b g(a, b)`.","title":"Population mean genotypic value at a single locus"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/alpha","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.alpha","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#alpha","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Fisher's average effect of allele `a`: `\u03b1 a = \u2211_b p_b g(a, b) - \u03bc`.","title":"Fisher's average effect of an allele at a single locus"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/delta","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.delta","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#delta","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The dominance deviation `\u03b4 a b = g(a, b) - \u03bc - \u03b1 a - \u03b1 b`.","title":"Dominance deviation at a single locus"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-g","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.varG","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-g","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The genetic variance `V_G = \u2211 p_a p_b (g(a, b) - \u03bc)\u00b2`.","title":"Genetic variance at a single locus"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-a","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.varA","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-a","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The additive variance `V_A = 2 \u2211 p_a \u03b1_a\u00b2`.","title":"Additive variance at a single locus"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-d","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.varD","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-d","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The dominance variance `V_D = \u2211 p_a p_b \u03b4_{ab}\u00b2`.","title":"Dominance variance at a single locus"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/sum-p-alpha","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.sum_p_alpha","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#sum-p-alpha","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Average effects are centred: `\u2211 p_a \u03b1_a = 0`.","title":"Average effects are centred"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/sum-p-delta","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.sum_p_delta","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#sum-p-delta","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Dominance deviations are orthogonal to every allele: `\u2211_b p_b \u03b4_{ab} = 0`.","title":"Dominance deviations are orthogonal to every allele"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-g-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.varG_eq","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-g-eq","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Fisher decomposition of the genetic variance: `V_G = V_A + V_D`.","title":"Fisher decomposition"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-a-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.varA_nonneg","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-a-nonneg","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The additive variance is nonnegative.","title":"Nonnegativity of the single-locus additive variance"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-d-nonneg","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.varD_nonneg","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-d-nonneg","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The dominance variance is nonnegative.","title":"Nonnegativity of the single-locus dominance variance"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-d-le","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.varD_le","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-d-le","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The additive fit is least squares (Fisher definition of average effects): every additive fit `c + h a + h b` has mean squared error at least `V_D`, the error of `\u03bc + \u03b1 a + \u03b1 b`.","title":"The additive fit is least squares"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/rel-prod","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.relProd","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#rel-prod","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The mean product of genotypic values of two non-inbred relatives that share `0`, `1`, `2` alleles identical by descent with probabilities `k\u2080, k\u2081, k\u2082`: given the IBD state, the non-shared alleles are independent draws from `p`.","title":"Mean product of genotypic values of two relatives"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/rel-cov-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.relCov_eq","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#rel-cov-eq","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Covariance between relatives at one locus (Fisher 1918; in the IBD form of Falconer and Mackay ch. 9): for IBD probabilities summing to one, `E[G_X G_Y] - \u03bc\u00b2 = (k\u2081 / 2 + k\u2082) V_A + k\u2082 V_D`, i.e. `2\u03c6 V_A + \u0394 V_D` with kinship `\u03c6 = k\u2081 / 4 + k\u2082 / 2` and `\u0394 = k\u2082`.","title":"Covariance between relatives"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/cockerham-components","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.CockerhamComponents","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#cockerham-components","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Cockerham\u2013Kempthorne variance components up to order `n`: `V i j = V_{A^i D^j}` is the component of `i` additive and `j` dominance factors (`V 1 0 = V_A`, `V 0 1 = V_D`, `V 2 0 = V_AA`, \u2026). They are nonnegative, there is no `(0, 0)` component, and the genetic covariance of relatives with `2\u03c6` and `\u0394` is `\u2211_{i, j \u2264 n} (2\u03c6)^i \u0394^j V i j`. This structure is the stated representation; it is proved for one locus by\u2026","title":"Cockerham\u2013Kempthorne variance components"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/cov","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.CockerhamComponents.cov","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#cov","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The genetic covariance of relatives with coefficients `2\u03c6 = twoPhi` and `\u0394 = \u0394`.","title":"Genetic covariance of relatives in Cockerham form"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-a-2","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.CockerhamComponents.varA","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-a-2","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The additive variance `V_A = V 1 0` (zero if `n = 0`).","title":"Additive variance of the Cockerham components"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/falconer","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.CockerhamComponents.falconer","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#falconer","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Falconer's estimator under the ACE model, in covariance form: monozygotic twins have `2\u03c6 = \u0394 = 1`, dizygotic twins (full sibs) `2\u03c6 = 1/2`, `\u0394 = 1/4`, and both share the same common-environment covariance `C`; the estimator is `2 (Cov_MZ - Cov_DZ)`.","title":"Falconer's estimator"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/falconer-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.CockerhamComponents.falconer_eq","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#falconer-eq","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Falconer's estimator is `V_A` plus the non-additive components weighted by `2 (1 - 2^{-i} 4^{-j})` (derived from Cockerham's coefficients).","title":"Falconer's estimator in terms of Cockerham components"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/var-a-le-falconer","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.CockerhamComponents.varA_le_falconer","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#var-a-le-falconer","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The phantom-heritability inequality in covariance form: `V_A \u2264 2 (Cov_MZ - Cov_DZ)`.","title":"Phantom heritability inequality"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/falconer-eq-var-a-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.CockerhamComponents.falconer_eq_varA_iff","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#falconer-eq-var-a-iff","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Equality holds iff every non-additive component of order `\u2264 n` vanishes.","title":"Equality case of the phantom heritability inequality"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/narrow-le-falconer-heritability","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.CockerhamComponents.narrow_le_falconerHeritability","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#narrow-le-falconer-heritability","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"Phantom heritability on the correlation scale (Zuk et al. 2012): with phenotypic variance `V_P > 0`, the narrow-sense heritability `V_A / V_P` is at most the ACE estimate `2 (r_MZ - r_DZ)`, where `r = Cov / V_P`.","title":"Phantom heritability"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/to-components","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.toComponents","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#to-components","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The one-locus model as Cockerham components of order `1`: `V 1 0 = V_A`, `V 0 1 = V_D`, `V 1 1 = 0`.","title":"Single-locus model as Cockerham components"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/rel-cov-eq-cov","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.relCov_eq_cov","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#rel-cov-eq-cov","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"The one-locus covariance between relatives is the Cockerham form of `toComponents`.","title":"Single-locus covariance between relatives in Cockerham form"},{"id":"Biology/quantitative-genetics/covariance-between-relatives/falconer-eq-of-locus","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.CovarianceRelatives.LocusModel.falconer_eq_of_locus","parent":47071,"result":true,"route":"../roadmap/Biology/quantitative-genetics/covariance-between-relatives/index.html#falconer-eq-of-locus","scope":"Biology/quantitative-genetics/covariance-between-relatives","status":"fully_proved","summary":"With monozygotic twins (`k\u2082 = 1`), dizygotic twins (`k = (1/4, 1/2, 1/4)`) and a common environment `C`, `2 (Cov_MZ - Cov_DZ) = V_A + (3/2) V_D`.","title":"One-locus phantom heritability, unconditional"},{"data_url":"scopes/Biology/quantitative-genetics/genomic-relationship-matrices.json","id":"Biology/quantitative-genetics/genomic-relationship-matrices","kind":"scope","navigation":"module","parent":47018,"result":false,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","title":"Genomic relationship matrices"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/ibdmodel","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#ibdmodel","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"Mal\u00e9cot's identity-by-descent model at one biallelic locus, at the level of second moments. The individuals are `\u03b9`; individual `i` carries the gene copies `(i, 0)` (paternal) and `(i, 1)` (maternal), and `allele c \u2208 {0, 1}` indicates the reference allele on copy `c`. The reference allele has frequency `p \u2208 (0, 1)` (a polymorphic locus), and `ibd c d` is the probability that the copies `c` and `d` are identical by\u2026","title":"Mal\u00e9cot's identity-by-descent model"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/genotype","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.genotype","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#genotype","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The genotype (reference-allele count) `x_i \u2208 {0, 1, 2}` of individual `i`.","title":"Genotype of an individual at a biallelic locus"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/kinship","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.kinship","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#kinship","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The kinship coefficient `\u03c6_ij`: the probability that a copy drawn at random from `i` and a copy drawn at random from `j` are identical by descent.","title":"Kinship coefficient"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/inbreeding","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.inbreeding","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#inbreeding","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The inbreeding coefficient `F_i`: the probability that the two copies of `i` are identical by descent.","title":"Inbreeding coefficient"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/grm","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.grm","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#grm","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The genetic relationship matrix entry of Yang et al. (2010) at one locus, `A_ij = (x_i - 2p)(x_j - 2p) / (2p(1 - p))`.","title":"Genetic relationship matrix entry"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/grm-diag","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.grmDiag","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#grm-diag","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The diagonal estimator of GCTA (Yang et al. 2011) at one locus, `A_ii = 1 + (x_i\u00b2 - (1 + 2p) x_i + 2p\u00b2) / (2p(1 - p))`.","title":"GCTA diagonal relationship estimator"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integrable-allele","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integrable_allele","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integrable-allele","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The allele indicators are integrable.","title":"Integrability of allele indicators"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integrable-allele-mul","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integrable_allele_mul","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integrable-allele-mul","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"Products of allele indicators are integrable.","title":"Integrability of products of allele indicators"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/ibd-self","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.ibd_self","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#ibd-self","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"A gene copy is identical by descent with itself: `ibd c c = 1`.","title":"A gene copy is identical by descent with itself"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/ibd-comm","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.ibd_comm","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#ibd-comm","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"Identity by descent is symmetric.","title":"Symmetry of identity by descent"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/kinship-self","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.kinship_self","parent":47098,"reading_title":{"format":"inline-parts-v1","parts":[["text","Self-kinship equals "],["math","\\((1 + F_i)/2\\)"]],"source":"Self-kinship equals $(1 + F_i)/2$"},"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#kinship-self","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"`\u03c6_ii = (1 + F_i) / 2`.","title":"Self-kinship equals $(1 + F_i)/2$"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/kinship-comm","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.kinship_comm","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#kinship-comm","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"Kinship is symmetric.","title":"Symmetry of kinship"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integrable-genotype","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integrable_genotype","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integrable-genotype","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"Genotypes are integrable.","title":"Integrability of genotypes"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integrable-genotype-mul","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integrable_genotype_mul","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integrable-genotype-mul","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"Products of genotypes are integrable.","title":"Integrability of products of genotypes"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integral-genotype","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integral_genotype","parent":47098,"reading_title":{"format":"inline-parts-v1","parts":[["text","Expected genotype equals "],["math","\\(2p\\)"]],"source":"Expected genotype equals $2p$"},"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integral-genotype","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"`\ud835\udd3c[x_i] = 2p`.","title":"Expected genotype equals $2p$"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integral-genotype-mul","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integral_genotype_mul","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integral-genotype-mul","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"`\ud835\udd3c[x_i x_j] = 4p\u00b2 + 4p(1 - p) \u03c6_ij`.","title":"Second moment of genotypes in terms of kinship"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integral-centered-mul","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integral_centered_mul","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integral-centered-mul","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The genetic relationship covariance (Yang et al. 2010): `\ud835\udd3c[(x_i - 2p)(x_j - 2p)] = 2p(1 - p) \u00b7 2 \u03c6_ij`.","title":"The genetic relationship covariance"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integral-grm","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integral_grm","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integral-grm","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The GRM is unbiased for twice the kinship (Yang et al. 2010): `\ud835\udd3c[A_ij] = 2 \u03c6_ij`, for every pair `i, j` (including `i = j`, where `2 \u03c6_ii = 1 + F_i`).","title":"The GRM is unbiased for twice the kinship"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integral-grm-self","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integral_grm_self","parent":47098,"reading_title":{"format":"inline-parts-v1","parts":[["text","Diagonal GRM entry is unbiased for "],["math","\\(1 + F_i\\)"]],"source":"Diagonal GRM entry is unbiased for $1 + F_i$"},"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integral-grm-self","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The diagonal GRM entry is unbiased for `1 + F_i`: `\ud835\udd3c[A_ii] = 1 + F_i`.","title":"Diagonal GRM entry is unbiased for $1 + F_i$"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integral-sub-sq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integral_sub_sq","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integral-sub-sq","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The KING moment (Manichaikul et al. 2010), general form: `\ud835\udd3c[(x_i - x_j)\u00b2] = 2p(1 - p)(1 + F_i) + 2p(1 - p)(1 + F_j) - 8p(1 - p) \u03c6_ij`.","title":"The KING moment"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integral-sub-sq-of-inbreeding-eq-zero","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integral_sub_sq_of_inbreeding_eq_zero","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integral-sub-sq-of-inbreeding-eq-zero","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The KING moment (Manichaikul et al. 2010) for two non-inbred individuals: `\ud835\udd3c[(x_i - x_j)\u00b2] = 4p(1 - p)(1 - 2\u03c6_ij)`.","title":"The KING moment"},{"id":"Biology/quantitative-genetics/genomic-relationship-matrices/integral-grm-diag","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Relatedness.IBDModel.integral_grmDiag","parent":47098,"result":true,"route":"../roadmap/Biology/quantitative-genetics/genomic-relationship-matrices/index.html#integral-grm-diag","scope":"Biology/quantitative-genetics/genomic-relationship-matrices","status":"fully_proved","summary":"The GCTA diagonal estimator is unbiased for `1 + F_i` (Yang et al. 2011): `\ud835\udd3c[1 + (x_i\u00b2 - (1 + 2p) x_i + 2p\u00b2) / (2p(1 - p))] = 1 + F_i`.","title":"The GCTA diagonal estimator is unbiased for `1 + F_i`"},{"data_url":"scopes/Biology/quantitative-genetics/lande-s-selection-gradient.json","id":"Biology/quantitative-genetics/lande-s-selection-gradient","kind":"scope","navigation":"module","parent":47018,"result":false,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","title":"Lande's selection gradient"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/integral-sub-mul-gaussian-real","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.integral_sub_mul_gaussianReal","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#integral-sub-mul-gaussian-real","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"Stein's lemma for the normal law `N(\u03bc, v)`, `v > 0`: if `g` is differentiable with derivative `g'` and `g`, `(z - \u03bc) g z` and `g'` are integrable, then `\ud835\udd3c[(z - \u03bc) g z] = v \ud835\udd3c[g' z]` (Stein 1981, Lemma 1).","title":"Stein's lemma"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/mean-fitness","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.meanFitness","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#mean-fitness","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"The mean fitness `w\u0304(\u03bc) = \ud835\udd3c[W z]` of a population whose trait is distributed as `N(\u03bc, v)`.","title":"Mean fitness of a normally distributed trait"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/selection-differential","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.selectionDifferential","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#selection-differential","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"The directional selection differential `S = \ud835\udd3c[(z - \u03bc) W z] / w\u0304`: the mean trait of the selected parents, weighting each individual by its fitness, minus the mean trait `\u03bc` before selection. If `w\u0304 = 0` the division returns the junk value `0`.","title":"Directional selection differential"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/has-deriv-at-mean-fitness","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.hasDerivAt_meanFitness","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#has-deriv-at-mean-fitness","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"The derivative of mean fitness with respect to the trait mean is `d/d\u03bc w\u0304(\u03bc) = \ud835\udd3c[(z - \u03bc) W z] / v`, for bounded measurable fitness and `v > 0`.","title":"Derivative of mean fitness with respect to the trait mean"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/selection-differential-eq-mul-deriv-log","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.selectionDifferential_eq_mul_deriv_log","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#selection-differential-eq-mul-deriv-log","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"Lande's equation for one normally distributed trait (Lande 1979): the selection differential is the phenotypic variance times the derivative of log mean fitness with respect to the trait mean, `S = v \u00b7 d/d\u03bc log w\u0304(\u03bc)`.","title":"Lande's equation"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/selection-differential-div-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.selectionDifferential_div_eq","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#selection-differential-div-eq","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"The Lande\u2013Arnold selection gradient for one normally distributed trait: for a differentiable bounded fitness function with integrable derivative, `\u03b2 = S / v = \ud835\udd3c[W' z] / w\u0304`, the average slope of relative fitness `W / w\u0304` (Lande\u2013Arnold 1983).","title":"The Lande\u2013Arnold selection gradient"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/mean-fitness-mv","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.meanFitnessMV","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#mean-fitness-mv","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"The mean fitness `w\u0304(\u03bc) = \ud835\udd3c[W z]` of a population whose trait vector is distributed as `N(\u03bc, P)`.","title":"Mean fitness of a multivariate normal trait"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/selection-differential-mv","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.selectionDifferentialMV","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#selection-differential-mv","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"The directional selection differential `S = \ud835\udd3c[W z \u00b7 (z - \u03bc)] / w\u0304` of a trait vector distributed as `N(\u03bc, P)`: the fitness-weighted mean trait vector of the selected parents minus the mean `\u03bc` before selection. If `w\u0304 = 0` it is the junk value `0`.","title":"Multivariate directional selection differential"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/has-gradient-at-mean-fitness-mv","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.hasGradientAt_meanFitnessMV","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#has-gradient-at-mean-fitness-mv","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"The gradient of mean fitness with respect to the vector of trait means is `\u2207_\u03bc w\u0304(\u03bc) = P\u207b\u00b9 \ud835\udd3c[W z \u00b7 (z - \u03bc)]`, for positive definite `P` and bounded measurable `W`.","title":"Lande's gradient of mean fitness for multivariate traits"},{"id":"Biology/quantitative-genetics/lande-s-selection-gradient/selection-differential-mv-eq","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.SelectionGradient.selectionDifferentialMV_eq","parent":47121,"result":true,"route":"../roadmap/Biology/quantitative-genetics/lande-s-selection-gradient/index.html#selection-differential-mv-eq","scope":"Biology/quantitative-genetics/lande-s-selection-gradient","status":"fully_proved","summary":"Lande's equation (Lande 1979): for a multivariate normal trait vector, the selection differential is the phenotypic covariance matrix applied to the gradient of log mean fitness with respect to the trait means, `S = P \u2207_\u03bc log w\u0304(\u03bc)`.","title":"Lande's equation"},{"data_url":"scopes/Biology/quantitative-genetics/linear-mixed-models-and-blup.json","id":"Biology/quantitative-genetics/linear-mixed-models-and-blup","kind":"scope","navigation":"module","parent":47018,"result":false,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","title":"Linear mixed models and BLUP"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/phen-cov","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.phenCov","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#phen-cov","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"The phenotypic covariance `V = Z G Z\u1d40 + R` of the linear mixed model `y = X b + Z u + e` with `Cov u = G`, `Cov e = R`.","title":"Phenotypic covariance of a linear mixed model"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/is-mmesolution","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.IsMMESolution","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#is-mmesolution","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"Henderson's mixed-model equations for `y = X b + Z u + e`, `Cov u = G`, `Cov e = R`: `[X\u1d40 R\u207b\u00b9 X, X\u1d40 R\u207b\u00b9 Z; Z\u1d40 R\u207b\u00b9 X, Z\u1d40 R\u207b\u00b9 Z + G\u207b\u00b9] [b; u] = [X\u1d40 R\u207b\u00b9 y; Z\u1d40 R\u207b\u00b9 y]`.","title":"Solution of Henderson's mixed-model equations"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/pos-def-phen-cov","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.posDef_phenCov","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#pos-def-phen-cov","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"`V = Z G Z\u1d40 + R` is positive definite.","title":"Positive definiteness of the phenotypic covariance"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/is-mmesolution-iff","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.isMMESolution_iff","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#is-mmesolution-iff","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"Henderson's mixed-model equations (Henderson 1975; Robinson 1991): for positive definite `G` and `R` and `V = Z G Z\u1d40 + R`, `(b, u)` solves the mixed-model equations if and only if `b` solves the generalized least-squares normal equations `X\u1d40 V\u207b\u00b9 X b = X\u1d40 V\u207b\u00b9 y` and `u = G Z\u1d40 V\u207b\u00b9 (y - X b)`.","title":"Henderson's mixed-model equations"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/is-mmesolution-iff-gls","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.isMMESolution_iff_gls","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#is-mmesolution-iff-gls","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"The MME solution is the GLS estimate and the BLUP (Henderson 1975): if moreover `X\u1d40 V\u207b\u00b9 X` is invertible, the mixed-model equations have the unique solution `b\u0302 = (X\u1d40 V\u207b\u00b9 X)\u207b\u00b9 X\u1d40 V\u207b\u00b9 y` (the GLS estimate) and `\u00fb = G Z\u1d40 V\u207b\u00b9 (y - X b\u0302)`.","title":"The MME solution is the GLS estimate and the BLUP"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/cond-distrib-mixed-model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.condDistrib_mixedModel","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#cond-distrib-mixed-model","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"BLUP as a conditional law (Henderson 1975; Robinson 1991): in the Gaussian mixed model `y = X b + Z u + e` with fixed effects `b`, random effects `u ~ N(0, G)` and residuals `e ~ N(0, R)` independent, `G` and `R` positive definite, the conditional law of `u` given `y` is `N(G Z\u1d40 V\u207b\u00b9 (y - X b), G - G Z\u1d40 V\u207b\u00b9 Z G)` with `V = Z G Z\u1d40 + R` (for almost every `y`).","title":"BLUP as a conditional law"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/cond-exp-mixed-model","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.condExp_mixedModel","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#cond-exp-mixed-model","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"BLUP is the conditional expectation in the Gaussian mixed model: `\ud835\udd3c[u | y] = G Z\u1d40 V\u207b\u00b9 (y - X b)` almost surely.","title":"BLUP is the conditional expectation"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/cond-distrib-ldpred-inf","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.condDistrib_ldpredInf","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#cond-distrib-ldpred-inf","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"The LDpred-inf posterior (Vilhj\u00e1lmsson et al. 2015; the likelihood `\u03b2\u0302 | \u03b2 ~ N(R \u03b2, R / N)` of Zhu and Stephens 2017): with `M` markers, `N` individuals, heritability `h\u00b2 > 0`, effects `\u03b2 ~ N(0, h\u00b2 / M \u00b7 I)`, an LD matrix `R` that is positive definite, and marginal estimates `\u03b2\u0302 = R \u03b2 + \u03b5` with `\u03b5 ~ N(0, R / N)` independent of `\u03b2`, the posterior law of `\u03b2` given `\u03b2\u0302` is\u2026","title":"The LDpred-inf posterior"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/marginal-effect","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.marginalEffect","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#marginal-effect","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"The `N`-normalized marginal association estimate `\u03b2\u0302_j = X_j\u1d40 y / N` of marker `j` for the phenotype `y = X \u03b2 + \u03b5` (fixed effects `\u03b2`, noise `\u03b5`).","title":"Marginal association estimate of a marker"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/integral-marginal-effect","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.integral_marginalEffect","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#integral-marginal-effect","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"The mean of the marginal estimates (Zhu and Stephens 2017): for fixed effects `\u03b2` and noise with mean zero, `\ud835\udd3c[X_j\u1d40 y / N] = (R \u03b2)_j` with `R = X\u1d40 X / N` (`ldCorr`).","title":"The mean of the marginal estimates"},{"id":"Biology/quantitative-genetics/linear-mixed-models-and-blup/integral-marginal-effect-sub-mul","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.MixedModel.integral_marginalEffect_sub_mul","parent":47132,"result":true,"route":"../roadmap/Biology/quantitative-genetics/linear-mixed-models-and-blup/index.html#integral-marginal-effect-sub-mul","scope":"Biology/quantitative-genetics/linear-mixed-models-and-blup","status":"fully_proved","summary":"The raw residual-product second moment of the marginal estimates (Zhu and Stephens 2017): if `\ud835\udd3c[\u03b5_i \u03b5_i'] = \u03c3\u00b2 [i = i']`, then `\ud835\udd3c[(\u03b2\u0302_j - (R \u03b2)_j)(\u03b2\u0302_k - (R \u03b2)_k)] = \u03c3\u00b2 / N \u00b7 R_jk`.","title":"The raw residual-product second moment of the marginal estimates"},{"data_url":"scopes/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding.json","id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","kind":"scope","navigation":"module","parent":47018,"result":false,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","title":"Pedigree kinship and inbreeding"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/pedigree","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#pedigree","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"A pedigree on the individuals `Fin n`, numbered so that parents precede their children (every finite pedigree, a finite DAG, has such a numbering). `parents i = some g` means that `g 0` is the father and `g 1` the mother of `i`; `parents i = none` means that `i` is a founder. Founders are unrelated and not inbred (their `2 n_F` gene copies are distinct).","title":"Pedigree"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/is-parent","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.IsParent","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#is-parent","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"`a` is a parent of `b`.","title":"Parent relation in a pedigree"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/is-ancestor","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.IsAncestor","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#is-ancestor","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"`a` is an ancestor of `b` (a parent, a parent of a parent, \u2026).","title":"Ancestor relation in a pedigree"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/no-selfing","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.NoSelfing","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#no-selfing","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"No individual is the offspring of a selfing (father \u2260 mother), as in Wright's setting of a bisexual population.","title":"Pedigree without selfing"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/coins","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.Coins","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#coins","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"A gene-dropping outcome: `\u03c9 i k` is the Mendelian coin of the copy `(i, k)` (`k = 0` the paternal copy, `k = 1` the maternal copy), saying which copy (`0` or `1`) of the parent `g k` of `i` it inherits. Under gene dropping all `4 ^ n` outcomes are equally likely.","title":"Gene-dropping outcome"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/origin","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.origin","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#origin","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The founder copy from which the gene copy `c = (i, k)` descends under the gene-dropping outcome `\u03c9`: a founder copy is its own origin; a copy of a non-founder descends from the copy `\u03c9 i k` of its parent `g k`.","title":"Founder origin of a gene copy"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/ibd","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.ibd","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#ibd","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The probability, under gene dropping (independent fair Mendelian coins), that the gene copies `c` and `d` are identical by descent, i.e. descend from the same founder copy.","title":"Probability of identity by descent under gene dropping"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/gene-drop-kinship","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.geneDropKinship","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#gene-drop-kinship","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The kinship (coancestry) coefficient under gene dropping: the probability that a copy drawn at random from `i` and a copy drawn at random from `j` are identical by descent.","title":"Kinship coefficient under gene dropping"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/inbreeding","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.inbreeding","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#inbreeding","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The inbreeding coefficient under gene dropping: the probability that the two copies of `i` are identical by descent.","title":"Inbreeding coefficient under gene dropping"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The kinship coefficient by Mal\u00e9cot's recursion, computed in the order of the numbering (the tabular method): for `i < j`, `\u03c6 i j = (\u03c6 i f + \u03c6 i m) / 2` with `f, m` the parents of `j` and `\u03c6 i j = 0` if `j` is a founder; `\u03c6 i i = (1 + \u03c6 f m) / 2` with `f, m` the parents of `i` and `\u03c6 i i = 1 / 2` for a founder; symmetric for `j < i`.","title":"Kinship coefficient by Mal\u00e9cot's recursion"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-inbreeding","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinshipInbreeding","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-inbreeding","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The inbreeding coefficient by Mal\u00e9cot's recursion: `F i = \u03c6 f m` for the parents `f, m` of `i`, and `F i = 0` for a founder.","title":"Inbreeding coefficient by Mal\u00e9cot's recursion"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-of-lt-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship_of_lt_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-of-lt-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The tabular recursion for `i < j` with `j` a non-founder.","title":"Tabular kinship when the second individual is a later non-founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-of-lt-of-parents-eq-none","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship_of_lt_of_parents_eq_none","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-of-lt-of-parents-eq-none","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The tabular recursion for `i < j` with `j` a founder.","title":"Tabular kinship when the second individual is a later founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-self-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship_self_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-self-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The tabular recursion on the diagonal for a non-founder: `\u03c6 i i = (1 + \u03c6 f m) / 2`.","title":"Tabular self-kinship of a non-founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-self-of-parents-eq-none","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship_self_of_parents_eq_none","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-self-of-parents-eq-none","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The tabular recursion on the diagonal for a founder: `\u03c6 i i = 1 / 2`.","title":"Tabular self-kinship of a founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-of-gt-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship_of_gt_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-of-gt-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The tabular recursion for `j < i` with `i` a non-founder.","title":"Tabular kinship when the first individual is a later non-founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-of-gt-of-parents-eq-none","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship_of_gt_of_parents_eq_none","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-of-gt-of-parents-eq-none","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The tabular recursion for `j < i` with `i` a founder.","title":"Tabular kinship when the first individual is a later founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-comm","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship_comm","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-comm","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The tabular kinship is symmetric.","title":"Symmetry of tabular kinship"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/anc-paths","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.ancPaths","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#anc-paths","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The ancestral paths from `i`: the lists `[i = v\u2080, v\u2081, \u2026, v_k]` with each `v_{t+1}` a parent of `v_t` (`k \u2265 0`; the path `[i]` has no edge).","title":"Ancestral paths from an individual"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/paths-to","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.pathsTo","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#paths-to","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The ancestral paths from `i` that end at `A`.","title":"Ancestral paths ending at a given ancestor"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/head-of-mem-anc-paths","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.head?_of_mem_ancPaths","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#head-of-mem-anc-paths","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Every path in `ancPaths i` starts at `i` and is nonempty.","title":"Ancestral paths are nonempty with the starting individual as head"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/origin-of-parents-eq-none","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.origin_of_parents_eq_none","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#origin-of-parents-eq-none","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"A founder copy is its own origin.","title":"A founder gene copy is its own origin"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/origin-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.origin_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#origin-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"A copy of a non-founder descends from the chosen copy of the corresponding parent.","title":"Origin of a gene copy of a non-founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/lt","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.IsAncestor.lt","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#lt","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Ancestors precede their descendants.","title":"Ancestors precede their descendants"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/is-ancestor-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.isAncestor_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#is-ancestor-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"A parent is an ancestor.","title":"Parents are ancestors"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/ibd-self","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.ibd_self","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#ibd-self","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Two copies are identical by descent of themselves with probability `1`.","title":"A gene copy is identical by descent with itself"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/ibd-comm","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.ibd_comm","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#ibd-comm","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Identity by descent is symmetric.","title":"Symmetry of identity by descent"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/ibd-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.ibd_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#ibd-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"One step of Mal\u00e9cot's recursion for identity by descent.","title":"Mal\u00e9cot's recursion step for identity by descent"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/gene-drop-kinship-self","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.geneDropKinship_self","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#gene-drop-kinship-self","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"`\u03c6 i i = (1 + F i) / 2` under gene dropping.","title":"Self-kinship under gene dropping"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/inbreeding-of-parents-eq-none","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.inbreeding_of_parents_eq_none","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#inbreeding-of-parents-eq-none","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The inbreeding coefficient of a founder is `0`.","title":"Founders are not inbred"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/inbreeding-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.inbreeding_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#inbreeding-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The inbreeding coefficient of `i` is the kinship of its parents, `F i = \u03c6 f m`, under gene dropping.","title":"Mal\u00e9cot"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/gene-drop-kinship-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.geneDropKinship_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#gene-drop-kinship-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Mal\u00e9cot's recursion under gene dropping: if `j \u2260 i` is not an ancestor of `i` and has parents `f, m`, then `\u03c6 i j = (\u03c6 i f + \u03c6 i m) / 2`.","title":"Mal\u00e9cot's recursion"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/gene-drop-kinship-of-parents-eq-none","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.geneDropKinship_of_parents_eq_none","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#gene-drop-kinship-of-parents-eq-none","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"A founder `j \u2260 i` that is not an ancestor of `i` has kinship `0` with `i`.","title":"Zero kinship with an unrelated founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/gene-drop-kinship-comm","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.geneDropKinship_comm","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#gene-drop-kinship-comm","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Gene-dropping kinship is symmetric.","title":"Symmetry of gene-dropping kinship"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/gene-drop-kinship-eq-kinship","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.geneDropKinship_eq_kinship","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#gene-drop-kinship-eq-kinship","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The kinship coefficient computed by Mal\u00e9cot's recursion is the probability of identity by descent under gene dropping.","title":"Mal\u00e9cot's recursion computes gene-dropping kinship"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/inbreeding-eq-kinship-inbreeding","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.inbreeding_eq_kinshipInbreeding","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#inbreeding-eq-kinship-inbreeding","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The inbreeding coefficient computed by Mal\u00e9cot's recursion is the probability of identity by descent of the two copies of `i` under gene dropping.","title":"Mal\u00e9cot's recursion computes the inbreeding coefficient"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/mem-anc-paths-of-parents-eq-none","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.mem_ancPaths_of_parents_eq_none","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#mem-anc-paths-of-parents-eq-none","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The only ancestral path from a founder is `[i]`.","title":"Ancestral paths from a founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/mem-anc-paths-of-parents-eq-some","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.mem_ancPaths_of_parents_eq_some","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#mem-anc-paths-of-parents-eq-some","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"The ancestral paths from a non-founder: `[i]` and `i :: q` for a path `q` from a parent.","title":"Ancestral paths from a non-founder"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/le-of-mem-anc-paths","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.le_of_mem_ancPaths","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#le-of-mem-anc-paths","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Every individual on an ancestral path from `i` is `i` or precedes it.","title":"Individuals on an ancestral path are its start or precede it"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/exists-eq-cons-of-mem-anc-paths","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.exists_eq_cons_of_mem_ancPaths","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#exists-eq-cons-of-mem-anc-paths","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Every ancestral path from `i` starts with `i`.","title":"Ancestral path as its start followed by a tail"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/eq-singleton-of-mem-anc-paths","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.eq_singleton_of_mem_ancPaths","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#eq-singleton-of-mem-anc-paths","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"A path from `i` ending at `i` is `[i]`.","title":"An ancestral path returning to its start is trivial"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/singleton-mem-anc-paths","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.singleton_mem_ancPaths","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#singleton-mem-anc-paths","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"`[i]` is an ancestral path from `i`.","title":"Trivial ancestral path"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/kinship-eq-sum-paths","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.kinship_eq_sum_paths","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#kinship-eq-sum-paths","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Wright's path formula in kinship form: `\u03c6 i j` is the sum, over common ancestors `A` (including `i` or `j` themselves) and pairs of ancestral paths `p` from `i` to `A` and `p'` from `j` to `A` that share no individual other than `A`, of `(1/2) ^ (n\u2081 + n\u2082 + 1) * (1 + F A)`, where `n\u2081 + n\u2082 + 1 = p.length + p'.length - 1` is the number of individuals on the loop.","title":"Wright's path formula"},{"id":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/inbreeding-eq-sum-paths","kind":"node","lean":"MathlibExt.Biology.StatisticalGenetics.Kinship.Pedigree.inbreeding_eq_sum_paths","parent":47144,"result":true,"route":"../roadmap/Biology/quantitative-genetics/pedigree-kinship-and-inbreeding/index.html#inbreeding-eq-sum-paths","scope":"Biology/quantitative-genetics/pedigree-kinship-and-inbreeding","status":"fully_proved","summary":"Wright's path formula for the inbreeding coefficient (Wright 1922): if `i` has parents `f = g 0` and `m = g 1`, then `F_i = \u03a3 (1/2) ^ (n\u2081 + n\u2082 + 1) (1 + F_A)` over common ancestors `A` of `f` and `m` (including `f` or `m` themselves) and pairs of ancestral paths from `f` and `m` to `A` (with `n\u2081` and `n\u2082` edges) that share only `A`. Here `F` is the gene-dropping inbreeding coefficient.","title":"Wright's path formula"},{"data_url":"scopes/Biology/systems-biology.json","id":"Biology/systems-biology","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/systems-biology/index.html","scope":"Biology/systems-biology","status":"can_prove","summary":"Dynamics of biochemical reaction networks and gene regulatory circuits: mass-action kinetics, enzyme kinetics, ultrasensitivity, feedback, metabolic flux analysis and gene expression noise.","title":"Systems biology"},{"data_url":"scopes/Biology/systems-biology/coherent-feed-forward-loop.json","id":"Biology/systems-biology/coherent-feed-forward-loop","kind":"scope","navigation":"module","parent":47189,"result":false,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","title":"Coherent feed-forward loop"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/relaxation","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.relaxation","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#relaxation","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The variation-of-constants solution of `z' = P(t) - \u03b1 z`, `z(0) = z\u2080`: `z(t) = z\u2080 e^{-\u03b1 t} + \u222b\u2080\u1d57 e^{-\u03b1 (t - s)} P(s) ds`. Used for `t \u2265 0`.","title":"Variation-of-constants solution of a linear relaxation equation"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/has-deriv-at-relaxation","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.hasDerivAt_relaxation","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#has-deriv-at-relaxation","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The relaxation solves `z' = P - \u03b1 z` at every time where the (locally integrable) production rate `P` is continuous.","title":"Relaxation solution satisfies its differential equation"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/and-gate","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.andGate","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#and-gate","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"AND-gate production in the logic approximation: rate `\u03b2` when `K_x < x` and `K_y < y`, else `0`.","title":"AND-gate production rate in the logic approximation"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/simple-gate","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.simpleGate","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#simple-gate","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"Single-input production in the logic approximation: rate `\u03b2` when `K_x < x`, else `0`.","title":"Single-input production rate in the logic approximation"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/y-on","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.yOn","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#y-on","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"`Y` after an ON step of `Sx` from `Y(0) = 0`: `Y(t) = (\u03b2_y/\u03b1_y)(1 - e^{-\u03b1_y t})`.","title":"Intermediate factor after an ON step"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/on-delay","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.onDelay","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#on-delay","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The ON-step delay `T_on = (1/\u03b1_y) log(1 / (1 - K_yz / Y_st))`, `Y_st = \u03b2_y/\u03b1_y`: the time at which `Y` crosses the activation threshold `K_yz` of `Z`. Meaningful for `0 < K_yz < Y_st`.","title":"ON-step delay of the feed-forward loop"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/has-deriv-at-y-on","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.hasDerivAt_yOn","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#has-deriv-at-y-on","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"`yOn` solves `Y' = \u03b2_y - \u03b1_y Y` with `Y(0) = 0` (for `\u03b1_y \u2260 0`; at `\u03b1_y = 0` Lean's `\u03b2_y / 0 = 0` makes `yOn` identically `0`, which is not a solution).","title":"Intermediate factor solves its production\u2013decay equation"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/on-delay-pos","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.onDelay_pos","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#on-delay-pos","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The ON delay is positive.","title":"Positivity of the ON delay"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/y-on-gt-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.yOn_gt_iff","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#y-on-gt-iff","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"`Y` exceeds `K_yz` exactly after the ON delay: `K_yz < Y(t) \u2194 T_on < t`.","title":"Intermediate factor exceeds the threshold exactly after the ON delay"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/ffl-on","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.fflOn","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#ffl-on","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The FFL target after an ON step of `Sx` at `t = 0` from `Y(0) = Z(0) = 0`, with `X* = 1` and `K_xz < 1`.","title":"Feed-forward loop target after an ON step"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/simple-on","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.simpleOn","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#simple-on","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The simple-regulation target after an ON step of `Sx` at `t = 0` from `Z(0) = 0`.","title":"Simple-regulation target after an ON step"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/ffl-on-eq-zero","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.ffl_on_eq_zero","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#ffl-on-eq-zero","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The FFL target stays at `0` until `T_on`.","title":"ON delay"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/ffl-on-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.ffl_on_eq","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#ffl-on-eq","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"After `T_on` the FFL target relaxes from `0` towards `\u03b2_z/\u03b1_z`: `Z(t) = (\u03b2_z/\u03b1_z)(1 - e^{-\u03b1_z (t - T_on)})` (for `\u03b1_z > 0`, `K_xz < 1`).","title":"Delayed rise"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/simple-on-pos","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.simple_on_pos","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#simple-on-pos","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"Simple regulation responds without delay: `Z(t) = (\u03b2_z/\u03b1_z)(1 - e^{-\u03b1_z t}) > 0` for all `t > 0`.","title":"Simple regulation responds without delay"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/ffl-off","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.fflOff","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#ffl-off","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The FFL target after `Sx` is removed at `t = 0` (`X* = 0`), starting from `Z(0) = z\u2080`, for an arbitrary trajectory `Y` of the intermediate factor.","title":"Feed-forward loop target after an OFF step"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/simple-off","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.simpleOff","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#simple-off","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"The simple-regulation target after `Sx` is removed at `t = 0`, from `Z(0) = z\u2080`.","title":"Simple-regulation target after an OFF step"},{"id":"Biology/systems-biology/coherent-feed-forward-loop/ffl-off-eq-simple-off","kind":"node","lean":"MathlibExt.Biology.Dynamics.FeedForwardLoop.ffl_off_eq_simple_off","parent":47190,"result":true,"route":"../roadmap/Biology/systems-biology/coherent-feed-forward-loop/index.html#ffl-off-eq-simple-off","scope":"Biology/systems-biology/coherent-feed-forward-loop","status":"fully_proved","summary":"For a nonnegative threshold `K_xz`, whatever `Y` does, the FFL and simple regulation decay identically after an OFF step: `Z(t) = z\u2080 e^{-\u03b1_z t}`.","title":"No OFF delay"},{"data_url":"scopes/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore.json","id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","kind":"scope","navigation":"module","parent":47189,"result":false,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"can_prove","title":"Complex balancing and the deficiency-zero theorem"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/reaction-network","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#reaction-network","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"A chemical reaction network on the species `S` with reactions indexed by `\u03c1`: reaction `r` turns the source complex `src r` into the product complex `prod r` (stoichiometric coefficients).","title":"Chemical reaction network"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/stoichiometric-subspace","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.stoichiometricSubspace","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#stoichiometric-subspace","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The stoichiometric subspace: the span of the reaction vectors `prod r - src r`.","title":"Stoichiometric subspace"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/mass-action-field","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.massActionField","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#mass-action-field","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The mass-action vector field with rate constants `k`: `x \u21a6 \u2211_r k r (\u220f_s x_s ^ (src r s)) (prod r - src r)`.","title":"Mass-action vector field"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/rate","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.rate","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#rate","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The mass-action rate `k r * \u220f_s c_s ^ (src r s)` of reaction `r` at concentrations `c`.","title":"Mass-action reaction rate"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/is-complex-balanced","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.IsComplexBalanced","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#is-complex-balanced","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"Concentrations `c` are complex balanced for the rate constants `k`: at every complex `z`, the total rate of the reactions with source `z` equals the total rate of the reactions with product `z`. (Complexes that occur in no reaction give `0 = 0`.)","title":"Complex-balanced concentrations"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/pseudo-helmholtz","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.pseudoHelmholtz","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#pseudo-helmholtz","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The pseudo-Helmholtz function `G(c) = \u2211_s c_s (log c_s - log c*_s - 1)` relative to the reference concentrations `cstar` (Horn & Jackson 1972). It is meaningful for positive `c` and `cstar`; elsewhere `Real.log` takes junk values.","title":"Pseudo-Helmholtz function"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/helmholtz-rate","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.helmholtzRate","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#helmholtz-rate","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The rate of change of the pseudo-Helmholtz function along the mass-action vector field, `\u2211_s (log c_s - log c*_s) f_s(c)`: the derivative of `pseudoHelmholtz cstar` at `c > 0` applied to `massActionField k c` (`hasDerivAt_pseudoHelmholtz`).","title":"Rate of change of the pseudo-Helmholtz function"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/sum-rate-mul-prod-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.sum_rate_mul_prod_eq","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#sum-rate-mul-prod-eq","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"Complex balance at `c` rewrites the rate-weighted sum of any function of the products as the same sum over the sources (for all real `c` and `k`).","title":"Complex balance exchanges source and product sums"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/mass-action-field-eq-zero-of-is-complex","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.massActionField_eq_zero_of_isComplexBalanced","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#mass-action-field-eq-zero-of-is-complex","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"A complex-balanced state is an equilibrium of the mass-action system (for all real `c` and `k`). Source: Horn & Jackson (1972).","title":"Complex-balanced states are equilibria"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/helmholtz-rate-nonpos","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.helmholtzRate_nonpos","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#helmholtz-rate-nonpos","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"If `c*` is a positive complex-balanced equilibrium for positive rate constants, the pseudo-Helmholtz function does not increase along the mass-action vector field at any positive `c`. Source: Horn & Jackson (1972); Feinberg (2019), ch. 15.","title":"Horn\u2013Jackson inequality"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/helmholtz-rate-eq-zero-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.helmholtzRate_eq_zero_iff","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#helmholtz-rate-eq-zero-iff","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"Under the hypotheses of `helmholtzRate_nonpos`, the rate of change of the pseudo-Helmholtz function vanishes at a positive `c` exactly when `c` is an equilibrium. Source: Horn & Jackson (1972).","title":"Equality in the Horn\u2013Jackson inequality"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/is-complex-balanced-of-mass-action-field","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.isComplexBalanced_of_massActionField_eq_zero","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#is-complex-balanced-of-mass-action-field","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"If a mass-action system with positive rate constants has a positive complex-balanced equilibrium, then every positive equilibrium is complex balanced.","title":"Horn\u2013Jackson"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/positive-equilibria-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.positiveEquilibria_eq","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#positive-equilibria-eq","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"If `c*` is a positive complex-balanced equilibrium for positive rate constants, the positive equilibria are exactly the positive `c` whose `log c - log c*` is orthogonal (for the dot product) to the stoichiometric subspace. Source: Horn & Jackson (1972); Feinberg (2019), ch. 15.","title":"Horn\u2013Jackson: the positive equilibria"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/exists-pos-sub-mem-log-orthogonal","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.exists_pos_sub_mem_log_orthogonal","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#exists-pos-sub-mem-log-orthogonal","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"For any subspace `W` of `S \u2192 \u211d` and positive `a`, `b` there is a positive `c` with `c - b \u2208 W` and `log c - log a` orthogonal to `W` (for the dot product). Source: Birch (1963); Horn & Jackson (1972).","title":"Birch's theorem, existence"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/eq-of-sub-mem-log-orthogonal","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.eq_of_sub_mem_log_orthogonal","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#eq-of-sub-mem-log-orthogonal","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"Two positive vectors whose difference lies in a subspace `W` and whose logarithms differ by a vector orthogonal to `W` are equal.","title":"Birch's theorem, uniqueness"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/exists-unique-equilibrium-of-is-complex","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.existsUnique_equilibrium_of_isComplexBalanced","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#exists-unique-equilibrium-of-is-complex","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"If a mass-action system with positive rate constants has a positive complex-balanced equilibrium, then each positive stoichiometric compatibility class `(x\u2080 + \ud835\udcae) \u2229 \u211d_{>0}^S`, `x\u2080 > 0`, contains exactly one positive equilibrium. Source: Horn & Jackson (1972); Feinberg (2019), ch. 15.","title":"Horn\u2013Jackson: one equilibrium per compatibility class"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/has-deriv-at-pseudo-helmholtz","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.hasDerivAt_pseudoHelmholtz","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#has-deriv-at-pseudo-helmholtz","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The pseudo-Helmholtz function has time derivative `helmholtzRate` along a solution of the mass-action system, at any time where the solution is positive.","title":"Pseudo-Helmholtz function along positive mass-action solutions"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/antitone-on-pseudo-helmholtz","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.antitoneOn_pseudoHelmholtz","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#antitone-on-pseudo-helmholtz","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"Along a solution of a mass-action system with positive rate constants and a positive complex-balanced equilibrium `c*`, positive and differentiable (two-sided) at every `t \u2265 0`, `G(x t)` is nonincreasing on `[0, \u221e)`.","title":"The pseudo-Helmholtz function does not increase along solutions"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/complexes","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.complexes","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#complexes","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The complexes of the network: the sources and products of its reactions.","title":"Complexes of a chemical reaction network"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/reacts","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.Reacts","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#reacts","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The directed reaction relation on complexes: `y \u27f6 y'` if some reaction has source `y` and product `y'`.","title":"Directed reaction relation on complexes"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/linkage-graph","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.linkageGraph","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#linkage-graph","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The undirected reaction graph on the complexes: distinct complexes are adjacent when one reacts to the other.","title":"Linkage graph of a reaction network"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/num-linkage-classes","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.numLinkageClasses","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#num-linkage-classes","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The number of linkage classes: connected components of the undirected reaction graph.","title":"Number of linkage classes of a reaction network"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/deficiency","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.deficiency","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#deficiency","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The deficiency `\u03b4 = n - \u2113 - s` (number of complexes, linkage classes, dimension of the stoichiometric subspace).","title":"Deficiency of a chemical reaction network"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/is-weakly-reversible","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.IsWeaklyReversible","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#is-weakly-reversible","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"fully_proved","summary":"The network is weakly reversible: for every reaction `y \u2192 y'` there is a directed path of reactions from `y'` back to `y`.","title":"Weakly reversible reaction network"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/exists-pos-is-complex-balanced","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.exists_pos_isComplexBalanced","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#exists-pos-is-complex-balanced","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"can_prove","summary":"A weakly reversible mass-action network of deficiency zero has, for any positive rate constants, a positive complex-balanced equilibrium. Together with `existsUnique_equilibrium_of_isComplexBalanced` this gives `existsUnique_equilibrium`.","title":"Horn\u2013Feinberg: complex balancing for every choice of rate constants"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/exists-unique-equilibrium","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.existsUnique_equilibrium","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#exists-unique-equilibrium","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"can_prove","summary":"A weakly reversible mass-action network of deficiency zero has, for any positive rate constants, exactly one equilibrium in each positive stoichiometric compatibility class.","title":"Deficiency-zero theorem, weakly reversible case"},{"id":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/not-exists-positive-equilibrium","kind":"node","lean":"MathlibExt.Biology.Dynamics.ComplexBalanced.ReactionNetwork.not_exists_positive_equilibrium","parent":47208,"result":true,"route":"../roadmap/Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore/index.html#not-exists-positive-equilibrium","scope":"Biology/systems-biology/complex-balancing-and-the-deficiency-zero-theore","status":"can_prove","summary":"A mass-action network of deficiency zero that is not weakly reversible has no positive equilibrium for any positive rate constants.","title":"Deficiency-zero theorem, non-weakly-reversible case"},{"data_url":"scopes/Biology/systems-biology/goldbeter-koshland-ultrasensitivity.json","id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","kind":"scope","navigation":"module","parent":47189,"result":false,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","title":"Goldbeter\u2013Koshland ultrasensitivity"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/rate-ratio","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.rateRatio","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#rate-ratio","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"The value of `V\u2081 / V\u2082` for which the mole fraction `W` of modified protein is at steady state (Goldbeter\u2013Koshland 1981, Eq. 8): `W (1 - W + K\u2081) / ((1 - W) (W + K\u2082))`. Meaningful for `0 < W < 1` and `K\u2081, K\u2082 > 0`; at `W = 1` Lean's `a / 0 = 0` gives the junk value `0`.","title":"Rate ratio at a given steady-state modified fraction"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/goldbeter-koshland","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.goldbeterKoshland","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#goldbeter-koshland","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"The Goldbeter\u2013Koshland function: the steady-state mole fraction of modified protein when `V\u2081 / V\u2082 = v` (Goldbeter\u2013Koshland 1981, Eq. 7, rationalized so that it also covers `v = 1`, Eq. 9). Meaningful for `v, K\u2081, K\u2082 > 0`.","title":"Goldbeter\u2013Koshland function"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/response-coefficient","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.responseCoefficient","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#response-coefficient","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"The response coefficient `R_v`: the ratio of the values of `V\u2081 / V\u2082` giving 90% and 10% modified protein at steady state (Goldbeter\u2013Koshland 1981).","title":"Response coefficient of a covalent modification cycle"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/hill","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.hill","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#hill","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"The fraction of a Hill response `S\u207f / (K\u207f + S\u207f)` with half-saturation constant `K` and Hill coefficient `n` (meaningful for `S, K > 0`, `n > 0`).","title":"Hill response function"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/steady-state-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.steadyState_iff","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#steady-state-iff","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"Steady state of the modification cycle (Goldbeter\u2013Koshland 1981, Eq. 8). For `V\u2082 > 0`, `K\u2081, K\u2082 > 0` and `0 < W < 1`, modification and demodification balance, `V\u2081 (1 - W) / (K\u2081 + 1 - W) = V\u2082 W / (K\u2082 + W)`, iff `V\u2081 / V\u2082 = rateRatio K\u2081 K\u2082 W`.","title":"Steady state of the modification cycle"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/strict-mono-on-rate-ratio","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.strictMonoOn_rateRatio","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#strict-mono-on-rate-ratio","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"`rateRatio K\u2081 K\u2082` is strictly increasing on `(0, 1)` for `K\u2081, K\u2082 > 0`: it is the product of the positive increasing functions `W / (W + K\u2082)` and `1 + K\u2081 / (1 - W)`.","title":"Rate ratio is strictly increasing in the modified fraction"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/goldbeter-koshland-mem-ioo","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.goldbeterKoshland_mem_Ioo","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#goldbeter-koshland-mem-ioo","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"For `v, K\u2081, K\u2082 > 0` the Goldbeter\u2013Koshland function lies in `(0, 1)`.","title":"Goldbeter\u2013Koshland function lies strictly between 0 and 1"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/rate-ratio-goldbeter-koshland","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.rateRatio_goldbeterKoshland","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#rate-ratio-goldbeter-koshland","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"The Goldbeter\u2013Koshland function is a steady state (Goldbeter\u2013Koshland 1981, Eq. 7): for `v, K\u2081, K\u2082 > 0`, `rateRatio K\u2081 K\u2082 (goldbeterKoshland v K\u2081 K\u2082) = v`.","title":"The Goldbeter\u2013Koshland function is a steady state"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/eq-goldbeter-koshland","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.eq_goldbeterKoshland","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#eq-goldbeter-koshland","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"Uniqueness of the steady state (Goldbeter\u2013Koshland 1981, Eq. 7). For `K\u2081, K\u2082 > 0`, if `0 < W < 1` and `rateRatio K\u2081 K\u2082 W = v` with `v > 0`, then `W` is the Goldbeter\u2013Koshland function.","title":"Uniqueness of the steady state"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/goldbeter-koshland-one","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.goldbeterKoshland_one","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#goldbeter-koshland-one","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"Equal maximal rates (Goldbeter\u2013Koshland 1981, Eq. 9): for `V\u2081 = V\u2082` and `K\u2081, K\u2082 > 0` the steady-state modified fraction is `K\u2082 / (K\u2081 + K\u2082)`.","title":"Equal maximal rates"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/response-coefficient-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.responseCoefficient_eq","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#response-coefficient-eq","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"The response coefficient (Goldbeter\u2013Koshland 1981, Eq. 10): for `K\u2081, K\u2082 > -0.1`, `R_v = 81 (K\u2081 + 0.1)(K\u2082 + 0.1) / ((K\u2081 + 0.9)(K\u2082 + 0.9))`.","title":"The response coefficient"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/one-lt-response-coefficient","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.one_lt_responseCoefficient","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#one-lt-response-coefficient","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"For `K\u2081, K\u2082 > 0` the response coefficient exceeds `1`: the switch is never infinitely sharp.","title":"Response coefficient exceeds 1"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/response-coefficient-lt-81","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.responseCoefficient_lt_81","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#response-coefficient-lt-81","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"For `K\u2081, K\u2082 > 0` the response coefficient is below the Michaelis\u2013Menten value `81`.","title":"Response coefficient is below the Michaelis\u2013Menten value 81"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/tendsto-response-coefficient-at-top","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.tendsto_responseCoefficient_atTop","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#tendsto-response-coefficient-at-top","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"As both `K\u2081, K\u2082 \u2192 \u221e` the response coefficient tends to `81`, the value of a Michaelis\u2013Menten response.","title":"First-order limit"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/tendsto-response-coefficient-zero","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.tendsto_responseCoefficient_zero","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#tendsto-response-coefficient-zero","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"As both `K\u2081, K\u2082 \u2192 0` the response coefficient tends to `1`.","title":"Zero-order ultrasensitivity"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/hill-ratio-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.hill_ratio_eq","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#hill-ratio-eq","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"Hill response coefficient (Goldbeter\u2013Koshland 1981, Eq. 11). For a Hill response with `n > 0` and `K > 0`, if `S\u2089, S\u2081 > 0` give 90% and 10% response, then `S\u2089 / S\u2081 = 81 ^ (1 / n)`.","title":"Hill response coefficient"},{"id":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity/michaelis-menten-ratio-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.GoldbeterKoshland.michaelisMenten_ratio_eq","parent":47236,"result":true,"route":"../roadmap/Biology/systems-biology/goldbeter-koshland-ultrasensitivity/index.html#michaelis-menten-ratio-eq","scope":"Biology/systems-biology/goldbeter-koshland-ultrasensitivity","status":"fully_proved","summary":"Michaelis\u2013Menten response (`n = 1`): going from 10% to 90% of the response takes an 81-fold increase of the stimulus.","title":"Michaelis\u2013Menten response"},{"data_url":"scopes/Biology/systems-biology/integral-feedback-and-perfect-adaptation.json","id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","kind":"scope","navigation":"module","parent":47189,"result":false,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","title":"Integral feedback and perfect adaptation"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/linear-system","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#linear-system","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"A single-input single-output linear time-invariant system `\u1e8b = A x + u b`, `y = c \u2b1d x + d u` with state space `n \u2192 \u211d`.","title":"Single-input single-output linear time-invariant system"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/field","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.field","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#field","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"The vector field `A x + u b` at state `x` and input `u`.","title":"Vector field of a linear time-invariant system"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/output","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.output","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#output","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"The output `c \u2b1d x + d u`, measured from its adapted value.","title":"Output of a linear time-invariant system"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/has-perfect-adaptation","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.HasPerfectAdaptation","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#has-perfect-adaptation","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"For every constant input `u`, the output vanishes at every equilibrium.","title":"Perfect adaptation"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/has-integral-feedback","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.HasIntegralFeedback","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#has-integral-feedback","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"A linear functional `z = k \u2b1d x` of the state integrates the output, `\u017c = k \u2b1d (A x + u b) = y`, for every state and input.","title":"Integral feedback"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/dc-gain","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.dcGain","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#dc-gain","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"The steady-state (DC) gain `d - c \u2b1d A\u207b\u00b9 b`; for nonsingular `A` the output at the equilibrium for input `u` is `dcGain * u`.","title":"Steady-state (DC) gain of a linear system"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/has-perfect-adaptation-of-has-integral-f","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.hasPerfectAdaptation_of_hasIntegralFeedback","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#has-perfect-adaptation-of-has-integral-f","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"Integral feedback implies perfect adaptation.","title":"Integral feedback implies perfect adaptation"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/field-eq-zero-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.field_eq_zero_iff","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#field-eq-zero-iff","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"For nonsingular `A`, the unique equilibrium for input `u` is `x = -u \u2022 A\u207b\u00b9 b`.","title":"Unique equilibrium of a nonsingular linear system"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/output-equilibrium","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.output_equilibrium","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#output-equilibrium","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"For nonsingular `A`, the output at the equilibrium for input `u` is `dcGain * u`.","title":"Output at equilibrium equals DC gain times input"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/has-perfect-adaptation-iff-dc-gain-eq-ze","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.hasPerfectAdaptation_iff_dcGain_eq_zero","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#has-perfect-adaptation-iff-dc-gain-eq-ze","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"For nonsingular `A`, perfect adaptation holds iff the steady-state gain vanishes.","title":"Perfect adaptation iff the DC gain vanishes"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/has-integral-feedback-iff-has-perfect-ad","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.hasIntegralFeedback_iff_hasPerfectAdaptation","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#has-integral-feedback-iff-has-perfect-ad","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"Integral feedback is necessary (linear systems with nonsingular `A`): perfect adaptation holds iff some linear functional of the state integrates the output; one such functional is `k = c A\u207b\u00b9`.","title":"Integral feedback is necessary"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/has-deriv-at-integral","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.LinearSystem.hasDerivAt_integral","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#has-deriv-at-integral","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"Along a trajectory `\u1e8b(t) = A x(t) + u(t) b`, the functional `k \u2b1d x(t)` of an integral-feedback system has derivative `y(t) = c \u2b1d x(t) + d u(t)`: it is the time integral of the output.","title":"Integral-feedback functional is the time integral of the output"},{"id":"Biology/systems-biology/integral-feedback-and-perfect-adaptation/output-eq-zero-of-integral-feedback","kind":"node","lean":"MathlibExt.Biology.Dynamics.IntegralFeedback.output_eq_zero_of_integral_feedback","parent":47254,"result":true,"route":"../roadmap/Biology/systems-biology/integral-feedback-and-perfect-adaptation/index.html#output-eq-zero-of-integral-feedback","scope":"Biology/systems-biology/integral-feedback-and-perfect-adaptation","status":"fully_proved","summary":"Integral feedback gives robust perfect adaptation (nonlinear form). Let `F p` be the vector field and `y p` the output of a system with parameters `p`, and let `z` be a differentiable state function with derivative `z' x` satisfying `z' x (F p x) = y p x` for all `p`, `x` (so `\u017c = y` along every trajectory, for every parameter). Then the output vanishes at every equilibrium of every `F p`.","title":"Integral feedback gives robust perfect adaptation"},{"data_url":"scopes/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise.json","id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","kind":"scope","navigation":"module","parent":47189,"result":false,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","title":"Intrinsic and extrinsic gene expression noise"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/intrinsic-noise-sq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.intrinsicNoiseSq","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#intrinsic-noise-sq","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"Two-reporter intrinsic noise strength `\u03b7_int\u00b2 = \u27e8(X - Y)\u00b2\u27e9 / (2 \u27e8X\u27e9\u27e8Y\u27e9)`.","title":"Intrinsic noise strength"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/extrinsic-noise-sq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.extrinsicNoiseSq","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#extrinsic-noise-sq","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"Two-reporter extrinsic noise strength `\u03b7_ext\u00b2 = (\u27e8X Y\u27e9 - \u27e8X\u27e9\u27e8Y\u27e9) / (\u27e8X\u27e9\u27e8Y\u27e9)`.","title":"Extrinsic noise strength"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/total-noise-sq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.totalNoiseSq","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#total-noise-sq","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"Two-reporter total noise strength `\u03b7_tot\u00b2 = (\u27e8X\u00b2 + Y\u00b2\u27e9 / 2 - \u27e8X\u27e9\u27e8Y\u27e9) / (\u27e8X\u27e9\u27e8Y\u27e9)`.","title":"Total noise strength"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/total-noise-sq-eq-add","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.totalNoiseSq_eq_add","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#total-noise-sq-eq-add","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"Two-reporter decomposition (Swain et al. 2002; Elowitz et al. 2002): for any square-integrable reporters, `\u03b7_tot\u00b2 = \u03b7_int\u00b2 + \u03b7_ext\u00b2`.","title":"Two-reporter decomposition"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/is-identical-reporter-pair","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.IsIdenticalReporterPair","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#is-identical-reporter-pair","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"Two reporters of the same gene given the extrinsic \u03c3-algebra `m`: equal conditional means, equal conditional second moments, and conditionally uncorrelated. Conditional independence with identical conditional laws given `m` implies these three conditions.","title":"Identical reporter pair"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/integral-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.IsIdenticalReporterPair.integral_eq","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#integral-eq","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"Identical reporters have the same mean.","title":"Identical reporters have equal means"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/covariance-eq-variance-cond-exp","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.IsIdenticalReporterPair.covariance_eq_variance_condExp","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#covariance-eq-variance-cond-exp","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"For identical reporters the covariance is the extrinsic variance: `Cov(X, Y) = Var(E[X | m])`.","title":"Covariance of identical reporters equals extrinsic variance"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/integral-sub-sq-div-two-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.IsIdenticalReporterPair.integral_sub_sq_div_two_eq","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#integral-sub-sq-div-two-eq","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"For identical reporters half the mean squared difference is the intrinsic variance: `\u27e8(X - Y)\u00b2\u27e9 / 2 = \u27e8Var(X | m)\u27e9`.","title":"Half mean squared reporter difference equals intrinsic variance"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/extrinsic-noise-sq-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.IsIdenticalReporterPair.extrinsicNoiseSq_eq","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#extrinsic-noise-sq-eq","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"The two-reporter extrinsic noise is `Var(E[X | m]) / \u27e8X\u27e9\u00b2` (Swain et al. Eq. 8).","title":"Two-reporter formula for extrinsic noise"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/intrinsic-noise-sq-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.IsIdenticalReporterPair.intrinsicNoiseSq_eq","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#intrinsic-noise-sq-eq","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"The two-reporter intrinsic noise is `\u27e8Var(X | m)\u27e9 / \u27e8X\u27e9\u00b2` (Swain et al. Eq. 8).","title":"Two-reporter formula for intrinsic noise"},{"id":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/total-noise-sq-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.ExpressionNoise.IsIdenticalReporterPair.totalNoiseSq_eq","parent":47268,"result":true,"route":"../roadmap/Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise/index.html#total-noise-sq-eq","scope":"Biology/systems-biology/intrinsic-and-extrinsic-gene-expression-noise","status":"fully_proved","summary":"The two-reporter total noise is `Var(X) / \u27e8X\u27e9\u00b2`, and it splits into the intrinsic and extrinsic parts by the law of total variance.","title":"Total noise splits into intrinsic and extrinsic parts"},{"data_url":"scopes/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s.json","id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","kind":"scope","navigation":"module","parent":47189,"result":false,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","title":"Michaelis\u2013Menten kinetics and the quasi-steady state"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/michaelis-constant","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.michaelisConstant","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#michaelis-constant","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The Michaelis constant `K_M = (kb + k\u2082)/k\u2081` (Briggs\u2013Haldane). Meaningful for `k\u2081 > 0`.","title":"Michaelis constant"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/substrate-rate","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.substrateRate","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#substrate-rate","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The mass-action rate of the substrate, `s' = -k\u2081 e s + kb c`.","title":"Mass-action rate of the substrate"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/complex-rate","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.complexRate","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#complex-rate","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The mass-action rate of the complex, `c' = k\u2081 e s - (kb + k\u2082) c`.","title":"Mass-action rate of the enzyme\u2013substrate complex"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/is-solution","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.IsSolution","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#is-solution","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"A solution of the Michaelis\u2013Menten mass-action system on `[0, \u221e)`: `s' = -k\u2081 e s + kb c`, `c' = k\u2081 e s - (kb + k\u2082) c`, `e' = -c'`, `p' = k\u2082 c`.","title":"Solution of the Michaelis\u2013Menten mass-action system"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/enzyme-conservation","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.IsSolution.enzyme_conservation","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#enzyme-conservation","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"`e(t) + c(t) = e(0) + c(0)`.","title":"Enzyme conservation"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/substrate-conservation","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.IsSolution.substrate_conservation","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#substrate-conservation","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"`s(t) + c(t) + p(t) = s(0) + c(0) + p(0)`.","title":"Mass conservation"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/qss-complex","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.qssComplex","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#qss-complex","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The quasi-steady-state value of the complex, `c = e\u2080 s / (K_M + s)`.","title":"Quasi-steady-state value of the complex"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/mm-rate","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.mmRate","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#mm-rate","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The Michaelis\u2013Menten rate law `v(s) = V_max s / (K_M + s)`.","title":"Michaelis\u2013Menten rate law"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/complex-rate-eq-zero-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.complexRate_eq_zero_iff","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#complex-rate-eq-zero-iff","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"With total enzyme `e\u2080` (so `e = e\u2080 - c`), `k\u2081 > 0` and `K_M + s \u2260 0`, the complex is stationary iff `c = e\u2080 s / (K_M + s)`.","title":"Quasi-steady-state manifold"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/product-rate-qss","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.productRate_qss","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#product-rate-qss","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"On the quasi-steady-state manifold the product forms at rate `k\u2082 c = V_max s / (K_M + s)`, `V_max = k\u2082 e\u2080`.","title":"Briggs\u2013Haldane rate law"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/substrate-rate-qss","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.substrateRate_qss","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#substrate-rate-qss","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"On the quasi-steady-state manifold the substrate is consumed at the Michaelis\u2013Menten rate: `s' = -V_max s / (K_M + s)`.","title":"Substrate consumption at the Michaelis\u2013Menten rate under QSSA"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/mm-rate-strict-mono","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.mmRate_strictMono","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#mm-rate-strict-mono","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The rate law is strictly increasing on `s \u2265 0` for `V_max, K_M > 0`.","title":"Michaelis\u2013Menten rate is strictly increasing"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/mm-rate-michaelis-constant","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.mmRate_michaelisConstant","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#mm-rate-michaelis-constant","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The Michaelis constant is the half-saturation substrate level: `v(K_M) = V_max / 2`.","title":"Michaelis constant is the half-saturation substrate level"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/mm-rate-lt","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.mmRate_lt","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#mm-rate-lt","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The rate law is below its maximum: `v(s) < V_max` for `s \u2265 0`, `V_max, K_M > 0`.","title":"Michaelis\u2013Menten rate stays below its maximum"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/tendsto-mm-rate","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.tendsto_mmRate","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#tendsto-mm-rate","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"Saturation: `v(s) \u2192 V_max` as `s \u2192 \u221e`.","title":"Saturation of the Michaelis\u2013Menten rate"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/fast-transient","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.fastTransient","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#fast-transient","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The fast transient of the complex with the substrate frozen at `s\u0304`: `c(t) = c* + (c\u2080 - c*) e^{-k\u2081 (s\u0304 + K_M) t}`, where `c* = e\u2080 s\u0304 / (K_M + s\u0304)`.","title":"Fast transient of the complex with frozen substrate"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/has-deriv-at-fast-transient","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.hasDerivAt_fastTransient","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#has-deriv-at-fast-transient","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The fast transient solves the boundary-layer equation `c' = k\u2081 (e\u2080 - c) s\u0304 - (kb + k\u2082) c` for `k\u2081 > 0` and `K_M + s\u0304 \u2260 0`.","title":"Fast transient solves the boundary-layer equation"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/tendsto-fast-transient","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.tendsto_fastTransient","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#tendsto-fast-transient","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"The fast transient converges to the quasi-steady-state value for `k\u2081 > 0` and `s\u0304 + K_M > 0`.","title":"Fast transient converges to the quasi-steady state"},{"id":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/abs-sub-reduced-le","kind":"node","lean":"MathlibExt.Biology.Dynamics.MichaelisMenten.abs_sub_reduced_le","parent":47280,"result":true,"route":"../roadmap/Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s/index.html#abs-sub-reduced-le","scope":"Biology/systems-biology/michaelis-menten-kinetics-and-the-quasi-steady-s","status":"fully_proved","summary":"Rigorous QSSA (Tikhonov 1952; Heineken\u2013Tsuchiya\u2013Aris 1967). On the slow time scale `\u03c4 = e\u2080 t`, the substrate of the full Michaelis\u2013Menten system with `s(0) = s\u2080`, `c(0) = 0`, `e(0) = e\u2080` stays within `C e\u2080` of the solution `S\u0304` of the reduced equation `dS\u0304/d\u03c4 = -k\u2082 S\u0304 / (K_M + S\u0304)`, `S\u0304(0) = s\u2080`, uniformly on `[0, T]`.","title":"Rigorous QSSA"},{"data_url":"scopes/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu.json","id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","kind":"scope","navigation":"module","parent":47189,"result":false,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","title":"Stochastic mass action, metabolic control and flux balance"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/propensity","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.propensity","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#propensity","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The stochastic mass-action propensity of reaction `r` in state `x` (molecule counts): `k r * \u220f_s x_s (x_s - 1) \u22ef (x_s - src_s + 1)`, the falling factorial `Nat.descFactorial x_s src_s` (it vanishes when `x_s < src_s`). Anderson, Craciun & Kurtz (2010), \u00a72.","title":"Stochastic mass-action propensity"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/fire","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.fire","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#fire","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The state reached from `x` by one firing of reaction `r`, `x - src r + prod r` (meaningful when `src r \u2264 x`, the only case in which the propensity is nonzero).","title":"State after one reaction firing"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/unfire","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.unfire","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#unfire","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The state from which one firing of reaction `r` leads to `x`, `x - prod r + src r` (meaningful when `prod r \u2264 x`).","title":"State before one reaction firing"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/poisson-weight","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.poissonWeight","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#poisson-weight","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The product-form Poisson weight `\u220f_s c_s ^ x_s e^{-c_s} / Nat.factorial x_s` of a state `x`.","title":"Product-form Poisson weight of a state"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/class-generator","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.classGenerator","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#class-generator","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The generator of the stochastic mass-action chain restricted to a finite set `\u03a9` of states: `Q x y = \u2211_r \u03bb_r(x) ([fire r x = y] - [x = y])`. It is a rate matrix when `\u03a9` is closed under firing (`isRateMatrix_classGenerator`).","title":"Generator of the mass-action chain on a finite state class"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/poisson-weight-mul-sum-propensity","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.poissonWeight_mul_sum_propensity","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#poisson-weight-mul-sum-propensity","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Global balance of the product-form Poisson weights (Anderson, Craciun & Kurtz 2010, Thm 4.1): if `c` is complex balanced for the deterministic mass-action system with rate constants `k`, then at every state `x` the total outflow `\u03c0(x) \u2211_r \u03bb_r(x)` equals the total inflow `\u2211_{r : prod r \u2264 x} \u03c0(x - prod r + src r) \u03bb_r(x - prod r + src r)` for `\u03c0 = poissonWeight c`. This is an algebraic identity valid for all real `k`\u2026","title":"Global balance of the product-form Poisson weights"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/is-rate-matrix-class-generator","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.isRateMatrix_classGenerator","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#is-rate-matrix-class-generator","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The restricted generator is a rate matrix when the rate constants are nonnegative and `\u03a9` is closed under the reactions.","title":"Restricted mass-action generator is a rate matrix"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/poisson-weight-vec-mul-class-generator","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.poissonWeight_vecMul_classGenerator","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#poisson-weight-vec-mul-class-generator","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Anderson\u2013Craciun\u2013Kurtz on a finite class (Anderson, Craciun & Kurtz 2010, Thm 4.1\u20134.2): if `c` is complex balanced and every state of `\u03a9` from which a reaction leads into `\u03a9` lies in `\u03a9` (`hbwd`), the Poisson weights restricted to `\u03a9` are invariant for the restricted generator: `\u03c0 Q = 0`, for all real `k` and `c`. For a closed communicating class of a weakly reversible network with positive rate constants `hbwd`\u2026","title":"Anderson\u2013Craciun\u2013Kurtz on a finite class"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/class-stationary-law","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.classStationaryLaw","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#class-stationary-law","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The Poisson weights normalized on a finite set `\u03a9` of states.","title":"Normalized Poisson stationary law on a finite state class"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/class-stationary-law-spec","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.classStationaryLaw_spec","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#class-stationary-law-spec","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Stationary law of the stochastic mass-action chain on a finite class (Anderson, Craciun & Kurtz 2010, Thm 4.1\u20134.2). Let the rate constants be nonnegative, `c > 0` complex balanced, and `\u03a9` a nonempty finite set of states closed under the reactions in both directions. Then the restricted generator is a rate matrix, and the normalized product-form Poisson law `\u03c0(x) \u221d \u220f_s c_s^{x_s} e^{-c_s} / Nat.factorial x_s` on `\u03a9`\u2026","title":"Stationary law of the stochastic mass-action chain on a finite class"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/flux","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.flux","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#flux","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Reaction rates `v_i(e, s) = e_i f_i(s)` proportional to enzyme levels `e`.","title":"Reaction fluxes proportional to enzyme levels"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/steady-flux","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.steadyFlux","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#steady-flux","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The steady-state fluxes `J(e) = v(e, \u03c3(e))` along a steady-state map `\u03c3`.","title":"Steady-state fluxes along a steady-state map"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/flux-control","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.fluxControl","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#flux-control","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The flux control coefficient `C^{J_j}_{e_i} = (e_i / J_j) \u2202J_j/\u2202e_i`. It is meaningful when `J_j \u2260 0` and the steady-state flux is differentiable at `e`; otherwise Lean's `x / 0 = 0` and `fderiv` of a non-differentiable map, `0`, give junk values.","title":"Flux control coefficient"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/conc-control","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.concControl","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#conc-control","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The concentration control coefficient `C^{s_m}_{e_i} = (e_i / s_m) \u2202s_m/\u2202e_i`, meaningful for a positive concentration `s_m` and a steady state differentiable at `e`.","title":"Concentration control coefficient"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/elasticity","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.elasticity","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#elasticity","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The elasticity `\u03b5^{v_i}_{s_m} = (s_m / v_i) \u2202v_i/\u2202s_m`, meaningful for a nonzero rate `v_i` and a rate law differentiable at `s`.","title":"Elasticity of a reaction rate"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/is-regular-steady-state","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.IsRegularSteadyState","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#is-regular-steady-state","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"A regular steady state at enzyme levels `e\u2080`: `\u03c3` is differentiable at `e\u2080`, the rate laws are differentiable at `\u03c3 e\u2080`, `N v(e, \u03c3 e) = 0` for `e` near `e\u2080`, and the Jacobian `\u03b4 \u21a6 N \u2202_s v(e\u2080, \u03c3 e\u2080) \u03b4` is injective (it is square, so this is its invertibility).","title":"Regular steady state of a metabolic network"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/exists-is-regular-steady-state","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.exists_isRegularSteadyState","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#exists-is-regular-steady-state","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"If `s\u2080` is a steady state at enzyme levels `e\u2080`, the rate laws are continuously differentiable at `s\u2080`, and the Jacobian `\u03b4 \u21a6 N (e\u2080 \u2299 f'(s\u2080) \u03b4)` is injective, then the steady state continues to a map `\u03c3` with `\u03c3 e\u2080 = s\u2080` satisfying `IsRegularSteadyState`.","title":"Regular steady states exist by the implicit function theorem"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/sum-conc-control","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.sum_concControl","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#sum-conc-control","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Concentration summation theorem (Heinrich & Rapoport 1974): for a positive steady-state concentration `s_m`, `\u2211_i C^{s_m}_{e_i} = 0`. The positivity hypothesis is the domain on which the scaled concentration control coefficients have their biological meaning.","title":"Concentration summation theorem"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/sum-flux-control","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.sum_fluxControl","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#sum-flux-control","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Flux summation theorem (Kacser & Burns 1973; Heinrich & Rapoport 1974): for a nonzero steady-state flux `J_j`, `\u2211_i C^{J_j}_{e_i} = 1`.","title":"Flux summation theorem"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/sum-flux-control-mul-elasticity","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.sum_fluxControl_mul_elasticity","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#sum-flux-control-mul-elasticity","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Flux connectivity theorem (Kacser & Burns 1973): when all steady-state fluxes are nonzero, `\u2211_i C^{J_j}_{e_i} \u03b5^{v_i}_{s_m} = 0` for every flux `J_j` and metabolite `s_m`.","title":"Flux connectivity theorem"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/fba-feasible","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.fbaFeasible","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#fba-feasible","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The feasible fluxes of flux balance analysis: `S v = b`, `l \u2264 v \u2264 u`.","title":"Feasible fluxes of flux balance analysis"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/fba-value","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.fbaValue","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#fba-value","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The optimal value `max {c \u2b1d\u1d65 v | S v = b, l \u2264 v \u2264 u}` of flux balance analysis (the supremum; it is `0` by convention when the problem is infeasible).","title":"Optimal value of flux balance analysis"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/fba-dual-feasible","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.fbaDualFeasible","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#fba-dual-feasible","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Dual feasibility for flux balance analysis: `y` prices the metabolites, `\u03b1, \u03b2 \u2265 0` the upper and lower flux bounds, and `S\u1d40 y + \u03b1 - \u03b2 = c`.","title":"Dual feasibility for flux balance analysis"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/fba-dual-value","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.fbaDualValue","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#fba-dual-value","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The dual objective `b \u2b1d\u1d65 y + u \u2b1d\u1d65 \u03b1 - l \u2b1d\u1d65 \u03b2`.","title":"Dual objective of flux balance analysis"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/exists-fba-value-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.exists_fbaValue_eq","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#exists-fba-value-eq","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"A feasible flux balance problem attains its optimum.","title":"Feasible flux balance problem attains its optimum"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/dot-product-le-fba-dual-value","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.dotProduct_le_fbaDualValue","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#dot-product-le-fba-dual-value","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Weak duality for flux balance analysis.","title":"Weak duality for flux balance analysis"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/exists-fba-dual-value-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.exists_fbaDualValue_eq","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#exists-fba-dual-value-eq","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Strong duality for flux balance analysis: a feasible problem has a dual feasible point whose value is the optimal value.","title":"Strong duality for flux balance analysis"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/fba-value-concave","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.fbaValue_concave","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#fba-value-concave","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The optimal value is concave in the right-hand side `b` on the set of feasible `b`.","title":"Concavity of the flux balance optimum in the right-hand side"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/exists-optimal-dual-iff-supergradient","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.exists_optimal_dual_iff_supergradient","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#exists-optimal-dual-iff-supergradient","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Shadow prices (Orth, Thiele & Palsson 2010): at a feasible `b`, the metabolite prices `y` of the optimal dual solutions are exactly the supergradients of the optimal value: `fbaValue b' \u2264 fbaValue b + y \u2b1d\u1d65 (b' - b)` for every feasible `b'`.","title":"Shadow prices"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/sinkhorn-scaling-unique","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.sinkhorn_scaling_unique","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#sinkhorn-scaling-unique","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Two diagonal scalings of a positive matrix to doubly stochastic form differ by a positive scalar.","title":"Uniqueness of Sinkhorn scaling"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/exists-unique-symmetric-balancing","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.existsUnique_symmetric_balancing","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#exists-unique-symmetric-balancing","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Symmetric matrix balancing (Sinkhorn 1964; the \"equal visibility\" normalization of Imakaev et al. 2012): a symmetric matrix with positive entries has a unique positive vector `d` with `diag(d) A diag(d)` doubly stochastic.","title":"Symmetric matrix balancing"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/has-total-support","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.HasTotalSupport","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#has-total-support","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"A square matrix has total support if it is nonzero and every positive entry lies on a positive diagonal: for `A i j > 0` there is a permutation `\u03c3` with `\u03c3 i = j` and `A k (\u03c3 k) > 0` for all `k` (Sinkhorn & Knopp 1967; Csima & Datta 1972).","title":"Total support of a square matrix"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/has-total-support-of-balancing","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.hasTotalSupport_of_balancing","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#has-total-support-of-balancing","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Necessity of total support (Sinkhorn & Knopp 1967; Csima & Datta 1972): if a positive diagonal scaling `diag(d) A diag(d)` is doubly stochastic, then `A` has total support. The proof uses Birkhoff's theorem. The converse for symmetric nonnegative `A` is `exists_symmetric_balancing_of_hasTotalSupport`.","title":"Necessity of total support"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/influences","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.Influences","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#influences","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"In the interaction graph of `f : {0,1}\u207f \u2192 {0,1}\u207f` there is an edge `j \u2192 i` when the `i`-th coordinate of `f` depends on the `j`-th variable.","title":"Interaction graph of a Boolean network"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/is-acyclic","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.IsAcyclic","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#is-acyclic","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The interaction graph of `f` has no directed cycle (loops included).","title":"Acyclic interaction graph"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/robert-fixed-point","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.robert_fixed_point","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#robert-fixed-point","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Robert's theorem (Robert 1980): if the interaction graph of `f : {0,1}\u207f \u2192 {0,1}\u207f` is acyclic, `f` has a unique fixed point `p`, and every synchronous trajectory reaches it within `n` steps.","title":"Robert's theorem"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/is-in-tree","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.IsInTree","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#is-in-tree","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"`t : \u03b9 \u2192 \u03b9` is a spanning in-tree rooted at `r` (each `i \u2260 r` points to its parent `t i`): `t r = r` and iterating `t` from any state reaches `r`.","title":"Spanning in-tree rooted at a state"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/tree-weight","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.treeWeight","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#tree-weight","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"The total weight `\u2211_t \u220f_{i \u2260 r} Q i (t i)` of the spanning in-trees rooted at `r`.","title":"Total weight of spanning in-trees"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/is-irreducible","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.IsIrreducible","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#is-irreducible","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"A rate matrix is irreducible if every state reaches every other state through positive jump rates.","title":"Irreducible rate matrix"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/tree-weight-vec-mul-eq-zero","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.treeWeight_vecMul_eq_zero","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#tree-weight-vec-mul-eq-zero","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"Markov chain tree theorem (King & Altman 1956; Hill 1966): the in-tree weights are invariant for every matrix with zero row sums, `w Q = 0`. The proof matches both sides with the maps `g : \u03b9 \u2192 \u03b9` whose functional graph has its cycle through `j` (Anantharam & Tsoucas 1989).","title":"Markov chain tree theorem"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/tree-weight-pos","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.treeWeight_pos","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#tree-weight-pos","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"For an irreducible rate matrix every in-tree weight is positive.","title":"In-tree weights of an irreducible rate matrix are positive"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/eq-smul-tree-weight-of-vec-mul-eq-zero","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.eq_smul_treeWeight_of_vecMul_eq_zero","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#eq-smul-tree-weight-of-vec-mul-eq-zero","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"For an irreducible rate matrix every invariant vector is a multiple of the in-tree weights: the stationary law is proportional, at each state, to the weight of the spanning in-trees rooted there.","title":"Uniqueness in the Markov chain tree theorem"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/exists-finset-optimal-fba-dual","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.exists_finset_optimal_fbaDual","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#exists-finset-optimal-fba-dual","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"There is a finite set `T` of pairs `(y, u \u2b1d\u1d65 \u03b1 - l \u2b1d\u1d65 \u03b2)` with `(y, \u03b1, \u03b2)` dual feasible, such that for every feasible `b` some pair in `T` is optimal: `y \u2b1d\u1d65 b + \u03ba = fbaValue S l u c b`.","title":"Finitely many dual solutions suffice"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/exists-finset-is-least-fba-value","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.exists_finset_isLeast_fbaValue","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#exists-finset-is-least-fba-value","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"There are finitely many affine functions `b \u21a6 y \u2b1d\u1d65 b + \u03ba` whose minimum is the optimal value at every feasible right-hand side `b`.","title":"The FBA value is piecewise linear"},{"id":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/has-fderiv-at-fba-value-of-unique","kind":"node","lean":"MathlibExt.Biology.Dynamics.SystemsBiology.hasFDerivAt_fbaValue_of_unique","parent":47300,"result":true,"route":"../roadmap/Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu/index.html#has-fderiv-at-fba-value-of-unique","scope":"Biology/systems-biology/stochastic-mass-action-metabolic-control-and-flu","status":"fully_proved","summary":"If `b` is an interior point of the set of feasible right-hand sides and `y` is the only metabolite price vector of an optimal dual solution, then the optimal value is differentiable at `b` with derivative `b' \u21a6 y \u2b1d\u1d65 b'`.","title":"Shadow prices are derivatives when unique"},{"data_url":"scopes/Biology/detecting-selection.json","id":"Biology/detecting-selection","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/detecting-selection/index.html","scope":"Biology/detecting-selection","status":"fully_proved","summary":"Statistical tests of neutrality and genome scans for selection based on polymorphism, divergence and haplotype structure, and models of the distribution of fitness effects.","title":"Detecting natural selection"},{"data_url":"scopes/Biology/detecting-selection/composite-selection-scores.json","id":"Biology/detecting-selection/composite-selection-scores","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","title":"Composite selection scores"},{"id":"Biology/detecting-selection/composite-selection-scores/composite-model","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#composite-model","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"A composite-score model (Grossman et al. 2010): a prior probability `prior` that the site is under selection, and for each of the `k` component statistics a positive score function `sel` under selection and `neut` under neutrality (unnormalized densities: only positivity is assumed). Conditionally on the hypothesis the components are independent, which is what makes the joint scores the products below. The\u2026","title":"Composite selection score model"},{"id":"Biology/detecting-selection/composite-selection-scores/bayes-factor","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.bayesFactor","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#bayes-factor","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"The Bayes factor of component `i` at the observed value: the ratio of its score function under selection to its score function under neutrality.","title":"Bayes factor of a component statistic"},{"id":"Biology/detecting-selection/composite-selection-scores/cms","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.cms","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#cms","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"The CMS score: the product of the component Bayes factors.","title":"Composite of multiple signals (CMS) score"},{"id":"Biology/detecting-selection/composite-selection-scores/posterior","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.posterior","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#posterior","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"The normalized composite score: the normalized ratio of the products of the component score functions. This is the posterior probability of selection given the observed component values when the score functions are normalized densities (see `sum_joint_eq_one` for the normalization hypothesis).","title":"Normalized composite score"},{"id":"Biology/detecting-selection/composite-selection-scores/bayes-factor-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.bayesFactor_pos","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#bayes-factor-pos","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"Every component Bayes factor is positive.","title":"Positivity of component Bayes factors"},{"id":"Biology/detecting-selection/composite-selection-scores/cms-eq-div","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.cms_eq_div","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#cms-eq-div","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"The CMS score is the ratio of the two joint densities.","title":"CMS score as a ratio of joint densities"},{"id":"Biology/detecting-selection/composite-selection-scores/posterior-odds","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.posterior_odds","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#posterior-odds","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"Bayes-rule identity for conditionally independent components: the posterior odds of selection are the prior odds times the CMS score. This is an algebraic identity for the normalized score; it is a genuine Bayes rule only under the normalization hypothesis of `sum_joint_eq_one`.","title":"Posterior odds equal prior odds times the CMS score"},{"id":"Biology/detecting-selection/composite-selection-scores/log-posterior-odds","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.log_posterior_odds","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#log-posterior-odds","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"On the log scale the evidence is additive: the log posterior odds are the log prior odds plus the sum of the log Bayes factors of the component statistics.","title":"Additivity of evidence on the log-odds scale"},{"id":"Biology/detecting-selection/composite-selection-scores/posterior-lt-posterior-of-cms-lt","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.posterior_lt_posterior_of_cms_lt","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#posterior-lt-posterior-of-cms-lt","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"The posterior probability is a strictly increasing function of the CMS score, so the two rankings of sites agree.","title":"Posterior probability is strictly increasing in the CMS score"},{"id":"Biology/detecting-selection/composite-selection-scores/sum-joint-eq-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.CompositeModel.sum_joint_eq_one","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#sum-joint-eq-one","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"The model is a probability model: with normalised component score functions (each a density with respect to a common base measure), the joint distribution of the observations sums to one. This is the validity hypothesis that makes `posterior` a genuine Bayes posterior.","title":"Normalization of the composite score model"},{"id":"Biology/detecting-selection/composite-selection-scores/escape-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.escapeProb","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#escape-prob","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"The probability that a lineage at position `y` escapes a sweep located at `x`, with compound strength parameter `\u03b1 > 0`: `1 - exp (-\u03b1 |y - x|)` (Kim and Stephan 2002; Nielsen et al. 2005). The parameter `\u03b1` is proportional to the recombination rate per unit distance divided by the selection coefficient.","title":"Probability of escaping a selective sweep"},{"id":"Biology/detecting-selection/composite-selection-scores/escape-prob-eq-zero-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.escapeProb_eq_zero_iff","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#escape-prob-eq-zero-iff","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"No lineage escapes at the swept site itself, and nowhere else.","title":"Escape probability vanishes only at the swept site"},{"id":"Biology/detecting-selection/composite-selection-scores/escape-prob-inj","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.escapeProb_inj","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#escape-prob-inj","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"The escape-probability profile determines the sweep location and its strength. Positivity of one of the two strength parameters is enough: the location is the unique zero of that profile, and the strengths then agree by injectivity of `exp`.","title":"Escape profile determines sweep location and strength"},{"id":"Biology/detecting-selection/composite-selection-scores/mixture-weight-unique","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.mixture_weight_unique","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#mixture-weight-unique","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"A mixture of two distinct laws determines its mixing weight.","title":"Uniqueness of the mixing weight of a two-component mixture"},{"id":"Biology/detecting-selection/composite-selection-scores/sweep-param-unique","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweep_param_unique","parent":47347,"result":true,"route":"../roadmap/Biology/detecting-selection/composite-selection-scores/index.html#sweep-param-unique","scope":"Biology/detecting-selection/composite-selection-scores","status":"fully_proved","summary":"Identifiability of the sweep parameters from the per-site frequency spectra. If the neutral spectrum `f` and the post-sweep spectrum `g` differ, and two sweep parameter pairs give the same mixture at every position, then the parameters coincide. This is what the composite likelihood of SweepFinder maximises over.","title":"Identifiability of sweep parameters from frequency spectra"},{"data_url":"scopes/Biology/detecting-selection/distribution-of-fitness-effects.json","id":"Biology/detecting-selection/distribution-of-fitness-effects","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","title":"Distribution of fitness effects"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/abs-expected-sfs-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.abs_expectedSFS_le","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#abs-expected-sfs-le","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The expected number of sites with `i` derived copies in a sample of `n`, `1 \u2264 i < n`, is bounded uniformly in the selection coefficient: `|E \u03be_i(\u03c3)| \u2264 |\u03b8| C(n, i)`.","title":"Uniform bound on the expected site-frequency spectrum"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/dfe-sfs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.dfeSFS","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#dfe-sfs","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The expected unfolded site-frequency spectrum under the distribution of fitness effects `\u03bc` (a law on the scaled selection coefficient `\u03c3 = 2Ns`): `\u222b E \u03be_i(\u03c3) d\u03bc(\u03c3)`, where each new mutation draws its own `\u03c3` from `\u03bc`.","title":"Site-frequency spectrum under a distribution of fitness effects"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/dfe-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.dfeDensity","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#dfe-density","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The Poisson-random-field frequency density averaged over the distribution of fitness effects `\u03bc`: `\u222b \u03c6_\u03c3(x) d\u03bc(\u03c3)`.","title":"Frequency density averaged over a distribution of fitness effects"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/integrable-expected-sfs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.integrable_expectedSFS","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#integrable-expected-sfs","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"For `1 \u2264 i < n`, the expected spectrum `E \u03be_i(\u03c3)` is integrable in `\u03c3` against every finite measure.","title":"Integrability of the expected spectrum in the selection coefficient"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/dfe-sfs-eq-integral-dfe-density","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.dfeSFS_eq_integral_dfeDensity","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#dfe-sfs-eq-integral-dfe-density","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"For a finite measure `\u03bc` of selection coefficients and a segregating class `1 \u2264 i < n`, the DFE-averaged spectrum is the binomial sampling integral of the DFE-averaged frequency density: `\u222b E \u03be_i(\u03c3) d\u03bc(\u03c3) = \u222b\u2080\u00b9 C(n, i) x ^ i (1 - x) ^ (n - i) (\u222b \u03c6_\u03c3(x) d\u03bc(\u03c3)) dx`.","title":"Mixture identity"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/dfe-sfs-dirac","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.dfeSFS_dirac","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#dfe-sfs-dirac","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"A DFE concentrated on one selection coefficient gives the single-`\u03c3` spectrum.","title":"Spectrum under a point-mass DFE is the single-selection spectrum"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/dfe-sfs-add-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.dfeSFS_add_measure","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#dfe-sfs-add-measure","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The spectrum is additive in the (finite) DFE.","title":"DFE spectrum is additive in the measure"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/dfe-sfs-smul-measure","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.dfeSFS_smul_measure","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#dfe-sfs-smul-measure","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The spectrum is homogeneous in the DFE.","title":"DFE spectrum is homogeneous in the measure"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/gamma-dfe","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.gammaDFE","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#gamma-dfe","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The gamma distribution of fitness effects (Boyko et al. 2008): the scaled selection coefficient is `\u03c3 = -T` with `T ~ Gamma(a, r)`, shape `a > 0` and rate `r > 0` (scale `1 / r`), so new mutations are deleterious with mean strength `a / r`.","title":"Gamma distribution of fitness effects"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/is-probability-measure-gamma-dfe","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.isProbabilityMeasure_gammaDFE","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#is-probability-measure-gamma-dfe","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The gamma DFE is a probability law.","title":"Gamma DFE is a probability measure"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/gamma-dfe-ioi-zero","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.gammaDFE_Ioi_zero","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#gamma-dfe-ioi-zero","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"Under the gamma DFE no mutation is beneficial.","title":"No mutation is beneficial under the gamma DFE"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/integral-id-gamma-dfe","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.integral_id_gammaDFE","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#integral-id-gamma-dfe","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The mean scaled selection coefficient under the gamma DFE is `-a / r`.","title":"Mean selection coefficient under the gamma DFE"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/variance-id-gamma-dfe","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.variance_id_gammaDFE","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#variance-id-gamma-dfe","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The variance of the scaled selection coefficient under the gamma DFE is `a / r ^ 2`.","title":"Variance of the selection coefficient under the gamma DFE"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/gamma-dfe-map-const-mul","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.gammaDFE_map_const_mul","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#gamma-dfe-map-const-mul","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"Rescaling a gamma DFE by `c > 0` (for example from `s` to `\u03c3 = 2Ns`, `c = 2N`) keeps the shape and divides the rate by `c`.","title":"Rescaling a gamma DFE divides its rate"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/gamma-dfe-real-iio","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.gammaDFE_real_Iio","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#gamma-dfe-real-iio","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"Under the gamma DFE the fraction of new mutations with `\u03c3 < -c`, `c \u2265 0`, is the gamma tail `\u0393(a, r c) / \u0393(a)`.","title":"Fraction of strongly deleterious mutations"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/gamma-dfe-real-ico","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.gammaDFE_real_Ico","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#gamma-dfe-real-ico","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The fraction of new mutations whose strength `-\u03c3` lies in `(c\u2081, c\u2082]`, `0 \u2264 c\u2081 \u2264 c\u2082`, is `(\u0393(a, r c\u2081) - \u0393(a, r c\u2082)) / \u0393(a)`.","title":"Fraction of mutations in a strength interval under the gamma DFE"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/dfe-sfs-gamma-dfe","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.dfeSFS_gammaDFE","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#dfe-sfs-gamma-dfe","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The spectrum under the gamma DFE as an integral against the gamma law of the strength.","title":"Site-frequency spectrum under the gamma DFE"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/poisson-log-lik","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.poissonLogLik","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#poisson-log-lik","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The Poisson log-likelihood `\u2211 (k_i log E_i - E_i - log k_i!)` of counts `k_i` with means `E_i`. Meaningful for `E_i > 0`; at `E_i = 0` Lean's `log 0 = 0` gives a junk value.","title":"Poisson log-likelihood of counts"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/pi-poisson-measure-real-singleton","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.pi_poissonMeasure_real_singleton","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#pi-poisson-measure-real-singleton","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The joint law of independent Poisson counts with means `E_i` gives the count vector `k` the probability `\u220f e^{-E_i} E_i ^ {k_i} / (k_i!)`.","title":"Probability of a count vector under independent Poisson laws"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/log-pi-poisson-measure-real-singleton","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.log_pi_poissonMeasure_real_singleton","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#log-pi-poisson-measure-real-singleton","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"For independent Poisson counts with positive means `E_i`, the log-probability of the observed counts `k` is `poissonLogLik E k`.","title":"The Poisson-random-field log-likelihood"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/has-fderiv-at-poisson-log-lik","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.hasFDerivAt_poissonLogLik","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#has-fderiv-at-poisson-log-lik","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"The derivative of the log-likelihood in the parameter `\u03c8`, when the means `E_i(\u03c8)` are positive and differentiable at `\u03c8`: `\u2211 (k_i / E_i(\u03c8) - 1) \u00b7 DE_i(\u03c8)`.","title":"Derivative of the Poisson log-likelihood"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/score-eq-zero-of-is-max-on","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.score_eq_zero_of_isMaxOn","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#score-eq-zero-of-is-max-on","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"If `\u03c8` maximizes the Poisson log-likelihood over a parameter set `S` that is a neighborhood of `\u03c8`, and the means are positive and differentiable at `\u03c8`, then `\u2211 (k_i / E_i(\u03c8) - 1) \u00b7 DE_i(\u03c8) = 0`.","title":"Score equations"},{"id":"Biology/detecting-selection/distribution-of-fitness-effects/poisson-log-lik-le-profile","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.DistributionOfFitnessEffects.poissonLogLik_le_profile","parent":47363,"result":true,"route":"../roadmap/Biology/detecting-selection/distribution-of-fitness-effects/index.html#poisson-log-lik-le-profile","scope":"Biology/detecting-selection/distribution-of-fitness-effects","status":"fully_proved","summary":"With means `E_i = \u03b8 f_i` linear in the mutation parameter, positive `f_i` and at least one count, every `\u03b8 > 0` has likelihood at most that of `\u03b8\u0302 = \u2211 k_i / \u2211 f_i`.","title":"Profile maximum in `\u03b8`"},{"data_url":"scopes/Biology/detecting-selection/ewens-watterson-test.json","id":"Biology/detecting-selection/ewens-watterson-test","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","title":"Ewens\u2013Watterson test"},{"id":"Biology/detecting-selection/ewens-watterson-test/cycle-type-weight","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.cycleTypeWeight","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#cycle-type-weight","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"The number of permutations of `n` letters with cycle type `\u03c0`, `w(\u03c0) = n! / \u220f j, j ^ (a j) * (a j)!` with `a j = \u03c0.parts.count j` (Cauchy's formula).","title":"Number of permutations with a given cycle type"},{"id":"Biology/detecting-selection/ewens-watterson-test/num-alleles-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.numAllelesProb","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#num-alleles-prob","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"The probability `P_\u03b8(K = k)` that a sample of `n` genes carries exactly `k` distinct alleles under the Ewens sampling formula.","title":"Probability of a given number of alleles under the Ewens formula"},{"id":"Biology/detecting-selection/ewens-watterson-test/ewens-cond-prob","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.ewensCondProb","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#ewens-cond-prob","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"The conditional probability of the allelic partition `\u03c0` given that the number of alleles is `K = K(\u03c0) = \u03c0.parts.card`, under the Ewens sampling formula with parameter `\u03b8`.","title":"Ewens conditional probability given the number of alleles"},{"id":"Biology/detecting-selection/ewens-watterson-test/cond-upper-pvalue","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.condUpperPValue","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#cond-upper-pvalue","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"The conditional upper-tail p-value of a statistic `T` of the allelic partition: the conditional probability, given the observed number of alleles `K(\u03c0)`, that `T` is at least its observed value `T \u03c0`. It is written with the cycle-type weights and so does not involve `\u03b8` (by `ewensCondProb_eq` it is the conditional tail probability for every `\u03b8 > 0`).","title":"Conditional upper-tail p-value of the Ewens\u2013Watterson test"},{"id":"Biology/detecting-selection/ewens-watterson-test/homozygosity","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.homozygosity","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#homozygosity","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"The sample homozygosity `F(\u03c0) = \u2211_i (n_i / n)\u00b2`, the sum over the alleles of the squared sample frequencies (Watterson 1978); `n_i` ranges over the parts of `\u03c0`.","title":"Sample homozygosity of an allelic partition"},{"id":"Biology/detecting-selection/ewens-watterson-test/cycle-type-weight-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.cycleTypeWeight_pos","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#cycle-type-weight-pos","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"The cycle-type weight is positive.","title":"Positivity of the cycle-type weight"},{"id":"Biology/detecting-selection/ewens-watterson-test/ewens-prob-eq-mul-cycle-type-weight","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.ewensProb_eq_mul_cycleTypeWeight","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#ewens-prob-eq-mul-cycle-type-weight","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"Factorization of the Ewens sampling formula (Ewens 1972): `P_\u03b8(\u03c0) = \u03b8 ^ K(\u03c0) / \u03b8^{(n)} * w(\u03c0)`, with `K(\u03c0) = \u03c0.parts.card` the number of alleles.","title":"Factorization of the Ewens sampling formula"},{"id":"Biology/detecting-selection/ewens-watterson-test/ewens-cond-prob-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.ewensCondProb_eq","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#ewens-cond-prob-eq","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"`K` is sufficient for `\u03b8` (Ewens 1972): for `\u03b8 > 0`, the conditional law of the allelic partition given the number of alleles is `w(\u03c0) / \u2211_{K(\u03c0') = K(\u03c0)} w(\u03c0')`, which does not depend on `\u03b8`.","title":"`K` is sufficient for `\u03b8`"},{"id":"Biology/detecting-selection/ewens-watterson-test/sum-cycle-type-weight-eq-stirling-first","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.sum_cycleTypeWeight_eq_stirlingFirst","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#sum-cycle-type-weight-eq-stirling-first","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"The cycle-type weights of the partitions of `n` into `k` parts sum to the unsigned Stirling number of the first kind `|s(n, k)|`.","title":"Cauchy's count by number of cycles"},{"id":"Biology/detecting-selection/ewens-watterson-test/num-alleles-prob-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.numAllelesProb_eq","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#num-alleles-prob-eq","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"Law of the number of alleles (Ewens 1972): `P_\u03b8(K = k) = |s(n, k)| \u03b8 ^ k / \u03b8^{(n)}`. As an identity between the defining sums it holds for every real `\u03b8`; it is a probability for `\u03b8 > 0`.","title":"Law of the number of alleles"},{"id":"Biology/detecting-selection/ewens-watterson-test/ewens-cond-prob-eq-stirling-first","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.ewensCondProb_eq_stirlingFirst","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#ewens-cond-prob-eq-stirling-first","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"The Ewens\u2013Watterson conditional law (Ewens 1972; Watterson 1978): for `\u03b8 > 0`, `P(\u03c0 | K = K(\u03c0)) = w(\u03c0) / |s(n, K(\u03c0))| = n! / (|s(n, k)| \u220f j, j ^ (a j) (a j)!)`.","title":"The Ewens\u2013Watterson conditional law"},{"id":"Biology/detecting-selection/ewens-watterson-test/sum-upper-tail-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.sum_upperTail_le","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#sum-upper-tail-le","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"For nonnegative masses `q` on a finite set `s` and a statistic `T`, the upper-tail p-value `p x = \u2211_{y \u2208 s, T x \u2264 T y} q y` satisfies `\u2211_{x \u2208 s, p x \u2264 t} q x \u2264 t` for every `t \u2265 0`.","title":"Validity of a discrete upper-tail p-value"},{"id":"Biology/detecting-selection/ewens-watterson-test/sum-cond-upper-pvalue-le-cond","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.sum_condUpperPValue_le_cond","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#sum-cond-upper-pvalue-le-cond","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"For `\u03b8 > 0`, every number of alleles `k` and every level `t \u2265 0`, the conditional probability given `K = k` that the conditional upper-tail p-value of `T` is at most `t` is at most `t`.","title":"Conditional validity of the Ewens\u2013Watterson test"},{"id":"Biology/detecting-selection/ewens-watterson-test/sum-cond-upper-pvalue-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Ewens.sum_condUpperPValue_le","parent":47387,"result":true,"route":"../roadmap/Biology/detecting-selection/ewens-watterson-test/index.html#sum-cond-upper-pvalue-le","scope":"Biology/detecting-selection/ewens-watterson-test","status":"fully_proved","summary":"Validity of the Ewens\u2013Watterson test (Watterson 1978): for every `\u03b8 > 0` and every level `t \u2265 0`, the Ewens probability that the conditional upper-tail p-value of `T` is at most `t` is at most `t`. With `T = homozygosity` this is the test against excess homozygosity, with `T = -homozygosity` the test against a deficit of homozygosity.","title":"Validity of the Ewens\u2013Watterson test"},{"data_url":"scopes/Biology/detecting-selection/extended-haplotype-homozygosity.json","id":"Biology/detecting-selection/extended-haplotype-homozygosity","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","title":"Extended haplotype homozygosity"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/window","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.window","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#window","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"The window of radius `d` of haplotype `i`: its alleles at the `d` sites closest to the core, in the direction indexed by `\u2115`.","title":"Haplotype window around the core"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/window-eq-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.window_eq_iff","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#window-eq-iff","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"Two haplotypes have the same window of radius `d` exactly when they agree at every site `k < d`.","title":"Equality of haplotype windows"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/concordant-pairs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.concordantPairs","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#concordant-pairs","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"The number of ordered pairs of distinct haplotypes of `C` whose windows of radius `d` agree.","title":"Concordant haplotype pairs at a given radius"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/ehh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ehh","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#ehh","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"Extended haplotype homozygosity at radius `d` of the haplotype set `C`: the probability that two distinct haplotypes drawn from `C` without replacement agree at all `d` sites nearest the core (Sabeti et al. 2002). For `#C \u2264 1` the denominator vanishes and the junk value `0` is returned; every statement below that needs the probabilistic reading assumes `2 \u2264 #C`.","title":"Extended haplotype homozygosity (EHH)"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/ehh-eq-zero-of-card-le-one","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ehh_eq_zero_of_card_le_one","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#ehh-eq-zero-of-card-le-one","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"Outside the domain `2 \u2264 #C` the definition returns the junk value `0`.","title":"EHH is zero for a set of at most one haplotype"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/ehh-zero-window","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ehh_zero_window","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#ehh-zero-window","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"At radius `0` every pair of haplotypes agrees, so `EHH(0) = 1`.","title":"EHH at radius zero equals one"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/ehh-antitone","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ehh_antitone","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#ehh-antitone","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"EHH is nonincreasing in the radius: widening the window can only break ties.","title":"EHH is nonincreasing in the radius"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/ehh-eq-sum-counts","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ehh_eq_sum_counts","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#ehh-eq-sum-counts","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"EHH as the sampling-without-replacement homozygosity of the partition of `C` into window types: `EHH(d) = \u2211_t c_t (c_t - 1) / (#C (#C - 1))`, where `c_t` is the number of haplotypes of `C` with window `t`. This is the form in which EHH is usually defined.","title":"EHH as homozygosity of window-type counts"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/concordant-pairs-core","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.concordantPairsCore","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#concordant-pairs-core","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"The number of ordered pairs of distinct haplotypes of `C` whose core alleles (under `core`) agree and whose windows of radius `d` agree. The extended haplotype includes the core, so pairs carrying different core alleles are never concordant, even when their outward windows match.","title":"Concordant haplotype pairs sharing a core allele"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/ehh-core","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ehhCore","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#ehh-core","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"Extended haplotype homozygosity with core-allele labels: the probability that two distinct haplotypes drawn from `C` without replacement share both their core allele and their window of radius `d`. For a set `C` of carriers of a single core allele this is the same count as `ehh`; for a pooled set `D` carrying several core alleles it excludes the cross-core pairs that plain `ehh` would wrongly count as concordant.","title":"Extended haplotype homozygosity with core-allele labels"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/rehh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.rehh","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#rehh","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"Relative extended haplotype homozygosity: the core-labelled EHH of the core-allele carriers `C` divided by the core-labelled EHH of the pooled set `D` of haplotypes carrying the other core alleles. The denominator counts only pairs sharing a core allele, so cross-core pairs never inflate it. This is the statistic of the long-range haplotype test (Sabeti et al. 2002). It is a junk value `0` when the denominator\u2026","title":"Relative extended haplotype homozygosity (REHH)"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/one-lt-rehh-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.one_lt_rehh_iff","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#one-lt-rehh-iff","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"With a positive reference EHH, the relative EHH exceeds `1` exactly when the core allele's haplotypes are the more homozygous ones.","title":"Criterion for relative EHH to exceed one"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/pair-no-recomb","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.pairNoRecomb","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#pair-no-recomb","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"The probability that no recombination falls on the genealogy of a pair of lineages within genetic distance `d` of the core, given that the pair coalesces at time `t` (time in units of `2N` generations, `\u03c1 = 4Nr` the scaled recombination rate per site): the two lineages carry recombination events at total rate `\u03c1 * d` until they coalesce.","title":"Probability of no recombination on a pair's genealogy"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/pair-no-recomb-antitone","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.pairNoRecomb_antitone","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#pair-no-recomb-antitone","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"The no-recombination probability decays in the distance from the core.","title":"No-recombination probability decays with distance"},{"id":"Biology/detecting-selection/extended-haplotype-homozygosity/integral-pair-no-recomb","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.integral_pairNoRecomb","parent":47402,"result":true,"route":"../roadmap/Biology/detecting-selection/extended-haplotype-homozygosity/index.html#integral-pair-no-recomb","scope":"Biology/detecting-selection/extended-haplotype-homozygosity","status":"fully_proved","summary":"Averaging over the pair coalescence time `T\u2082`, which is exponential of rate `1` in units of `2N` generations under the neutral coalescent, the probability that no recombination falls on the pair's genealogy within distance `d` is `1 / (1 + \u03c1 d)`. In a mutation-free model, since a recombined lineage can still carry an identical haplotype, this is a lower bound for `E[EHH(d)]`; without that assumption the bound is\u2026","title":"Mean no-recombination probability under the neutral coalescent"},{"data_url":"scopes/Biology/detecting-selection/hudson-kreitman-aguad-test.json","id":"Biology/detecting-selection/hudson-kreitman-aguad-test","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","title":"Hudson\u2013Kreitman\u2013Aguad\u00e9 test"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-poly-mean","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaPolyMean","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-poly-mean","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"HKA's expected number of segregating sites `\u03b8 C(n)` in a sample of `n` genes with scaled mutation rate `\u03b8` (use `f \u03b8` for species `B`).","title":"HKA expected number of segregating sites"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-poly-var","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaPolyVar","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-poly-var","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"HKA's variance of the number of segregating sites, `\u03b8 C(n) + \u03b8\u00b2 C\u2082(n)`.","title":"HKA variance of the number of segregating sites"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-div-mean","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaDivMean","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-div-mean","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"HKA's expected divergence `\u03b8 (T + (1 + f) / 2)`.","title":"HKA expected divergence"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-div-var","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaDivVar","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-div-var","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"HKA's variance of the divergence, `\u03b8 (T + (1 + f) / 2) + (\u03b8 (1 + f) / 2)\u00b2`.","title":"HKA variance of the divergence"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/integral-hka-divergence","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.integral_hkaDivergence","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#integral-hka-divergence","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"Expected divergence (Hudson, Kreitman & Aguad\u00e9 1987): if the ancestral coalescence time `\u03c4` is exponential with rate `2 / (1 + f)` and, given the genealogy \u03c3-algebra `m`, the divergence `D` has conditional mean `\u03b8 (T + \u03c4)`, then `E D = \u03b8 (T + (1 + f) / 2)` (for `f > -1`). No integrability hypothesis on `D` is needed: if `D` were not integrable its conditional expectation would be `0` (Mathlib's convention), and the\u2026","title":"Expected divergence"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/variance-hka-divergence","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.variance_hkaDivergence","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#variance-hka-divergence","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"Variance of the divergence (Hudson, Kreitman & Aguad\u00e9 1987): under the hypotheses of `integral_hkaDivergence`, if moreover `D` is square integrable with conditional variance `\u03b8 (T + \u03c4)` given `m` (as for a Poisson count), then `Var D = \u03b8 (T + (1 + f) / 2) + (\u03b8 (1 + f) / 2)\u00b2`.","title":"Variance of the divergence"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-statistic","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaStatistic","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-statistic","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"The HKA statistic for `L` loci with polymorphism counts `SA i`, `SB i`, divergences `D i`, parameters `\u03b8 i`, `T`, `f` and sample sizes `nA`, `nB`: the sum over loci of the squared standardized deviations of the three counts from their HKA expectations.","title":"Hudson\u2013Kreitman\u2013Aguad\u00e9 statistic"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-poly-var-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaPolyVar_pos","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-poly-var-pos","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"HKA's variances are positive in the biological domain: `\u03b8 > 0`, `n \u2265 2`.","title":"Positivity of the HKA polymorphism variance"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-div-var-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaDivVar_pos","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-div-var-pos","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"HKA's divergence variance is positive for `\u03b8 > 0`, `T \u2265 0`, `f > -1`.","title":"Positivity of the HKA divergence variance"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/integral-hka-statistic","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.integral_hkaStatistic","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#integral-hka-statistic","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"If every count is square integrable with the HKA mean and variance (as derived above under the neutral model), with `\u03b8 i > 0`, `T \u2265 0`, `f > 0` and `nA, nB \u2265 2`, then `E X\u00b2 = 3 L`.","title":"The HKA statistic has mean `3 L` at the true parameters"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-equations","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaEquations","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-equations","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"HKA's estimating equations (Hudson, Kreitman & Aguad\u00e9 1987) for parameters `\u03b8 i`, `T`, `f` given the counts: the totals over loci of polymorphism in `A`, in `B` and of divergence match their expectations, and at each locus the total of the three counts matches its expectation. (Only `L - 1` of the per-locus equations are independent of the totals.)","title":"HKA's estimating equations"},{"id":"Biology/detecting-selection/hudson-kreitman-aguad-test/hka-equations-expected","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.hkaEquations_expected","parent":47418,"result":true,"route":"../roadmap/Biology/detecting-selection/hudson-kreitman-aguad-test/index.html#hka-equations-expected","scope":"Biology/detecting-selection/hudson-kreitman-aguad-test","status":"fully_proved","summary":"The true parameters solve HKA's estimating equations at the expected counts.","title":"True parameters solve the HKA estimating equations"},{"data_url":"scopes/Biology/detecting-selection/mcdonald-kreitman-test.json","id":"Biology/detecting-selection/mcdonald-kreitman-test","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","title":"McDonald\u2013Kreitman test"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/alpha-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.alphaHat","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#alpha-hat","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"The Smith\u2013Eyre-Walker estimator of the proportion of adaptive nonsynonymous substitutions, `\u03b1\u0302 = 1 - (D_s P_n) / (D_n P_s)`.","title":"Smith\u2013Eyre-Walker estimator of adaptive substitutions"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/neutrality-index","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.neutralityIndex","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#neutrality-index","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"The neutrality index `NI = (P_n / P_s) / (D_n / D_s)`.","title":"Neutrality index"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/alpha-hat-eq-one-sub-neutrality-index","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.alphaHat_eq_one_sub_neutralityIndex","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#alpha-hat-eq-one-sub-neutrality-index","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"`\u03b1\u0302 = 1 - NI`, for all real counts (the junk values of division agree).","title":"Adaptive proportion estimator equals one minus the neutrality index"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/mkmodel","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#mkmodel","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"Expected counts of the MK table without a slightly deleterious class (see the module docstring): synonymous and nonsynonymous mutational targets `Ls`, `Ln`, neutral fraction `f\u2080` of nonsynonymous mutations, polymorphisms and neutral fixed differences per unit of neutral target `polyRate`, `divRate` (`\u03b8_P`, `\u03bb_D` in the module docstring), and `A` expected adaptive substitutions.","title":"Expected-count model of the McDonald\u2013Kreitman table"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/eps","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel.EPs","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#eps","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"Expected synonymous polymorphisms `E P_s = \u03b8_P L_s`.","title":"Expected synonymous polymorphisms"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/epn","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel.EPn","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#epn","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"Expected nonsynonymous polymorphisms `E P_n = \u03b8_P f\u2080 L_n`.","title":"Expected nonsynonymous polymorphisms"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/eds","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel.EDs","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#eds","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"Expected synonymous fixed differences `E D_s = \u03bb_D L_s`.","title":"Expected synonymous fixed differences"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/edn","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel.EDn","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#edn","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"Expected nonsynonymous fixed differences `E D_n = \u03bb_D f\u2080 L_n + A`.","title":"Expected nonsynonymous fixed differences"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/alpha","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel.alpha","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#alpha","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"The proportion of nonsynonymous substitutions that are adaptive, `\u03b1 = A / E D_n`.","title":"Proportion of adaptive nonsynonymous substitutions"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/alpha-hat-expected","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel.alphaHat_expected","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#alpha-hat-expected","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"(Smith & Eyre-Walker 2002): `1 - (E D_s E P_n) / (E D_n E P_s) = A / E D_n`.","title":"`\u03b1\u0302` at the expected counts is the proportion of adaptive substitutions"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/prop-nonsyn-eq-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel.propNonsyn_eq_iff","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#prop-nonsyn-eq-iff","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"The proportions of nonsynonymous changes among expected fixed differences and expected polymorphisms agree iff there are no adaptive substitutions.","title":"The MK null is neutrality"},{"id":"Biology/detecting-selection/mcdonald-kreitman-test/tendsto-alpha-hat","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.McDonaldKreitman.MKModel.tendsto_alphaHat","parent":47431,"result":true,"route":"../roadmap/Biology/detecting-selection/mcdonald-kreitman-test/index.html#tendsto-alpha-hat","scope":"Biology/detecting-selection/mcdonald-kreitman-test","status":"fully_proved","summary":"Let `Dn i, Ds i, Pn i, Ps i` be the counts at locus `i`; suppose each of the four sequences consists of pairwise independent, identically distributed, integrable variables whose means are the expected counts of `M` (the loci are exchangeable copies of the model). Then almost surely `\u03b1\u0302` computed from the counts summed over the first `m` loci converges to `\u03b1 = A / E D_n` as `m \u2192 \u221e`.","title":"Consistency of `\u03b1\u0302` over loci"},{"data_url":"scopes/Biology/detecting-selection/selective-sweeps.json","id":"Biology/detecting-selection/selective-sweeps","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","title":"Selective sweeps"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-traj","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepTraj","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-traj","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The logistic trajectory of a favourable allele with genic selection coefficient `s`, starting from frequency `x\u2080`: the solution of `x' = s x (1 - x)` with `x 0 = x\u2080` (Maynard Smith and Haigh 1974).","title":"Logistic trajectory of a selective sweep"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-traj-mem-ioo","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepTraj_mem_Ioo","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-traj-mem-ioo","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The trajectory stays strictly between `0` and `1`.","title":"Sweep trajectory stays strictly between 0 and 1"},{"id":"Biology/detecting-selection/selective-sweeps/has-deriv-at-sweep-traj","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.hasDerivAt_sweepTraj","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#has-deriv-at-sweep-traj","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The logistic equation: `x'(t) = s x(t) (1 - x(t))`.","title":"Sweep trajectory solves the logistic equation"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-duration","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepDuration","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-duration","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The time the logistic trajectory with selection coefficient `s` takes to go from frequency `a` to frequency `b`: `(1/s) log (b (1 - a) / (a (1 - b)))`, the difference of the logits divided by `s`.","title":"Duration of a logistic sweep between two frequencies"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-traj-sweep-duration","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepTraj_sweepDuration","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-traj-sweep-duration","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"`sweepDuration s a b` is the exact time at which the trajectory started at `a` reaches `b`.","title":"Sweep trajectory reaches its target at the sweep duration"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-duration-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepDuration_pos","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-duration-pos","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The time to reach a higher frequency is positive.","title":"Positivity of the sweep duration"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-duration-full","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepDuration_full","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-duration-full","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"A complete sweep in a population of `N` diploids, from a single copy `1 / (2N)` to `1 - 1 / (2N)`, takes exactly `(2 / s) log (2N - 1)` generations in the deterministic logistic model. (The familiar `(2 / s) log (2Ns)` is the conditional expected duration of the corresponding stochastic sweep, a different quantity.)","title":"Duration of a complete deterministic sweep"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-ehh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepEhh","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-ehh","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The probability that two carriers of a swept allele are identical over a window of genetic length `d`, in the star-genealogy model of Voight et al. (2006): all carriers coalesce at the start of the sweep, a time `T` ago, and the pair is identical exactly when neither of its two lineages recombined, each at rate `r * d` per unit time.","title":"EHH in the star-genealogy sweep model"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-ehh-eq-poisson-sq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepEhh_eq_poisson_sq","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-ehh-eq-poisson-sq","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The star-genealogy EHH is the square of the probability that a Poisson number of recombination events with mean `r d T` on one lineage is zero.","title":"Star-genealogy EHH as a squared Poisson zero probability"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-ehh-antitone","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepEhh_antitone","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-ehh-antitone","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"Haplotype homozygosity decays with the window length.","title":"Star-genealogy EHH decays with window length"},{"id":"Biology/detecting-selection/selective-sweeps/ihh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ihh","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#ihh","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The one-sided integrated haplotype homozygosity: the area under the EHH decay curve in one direction from the focal site. Voight et al.'s iHH integrates in both directions; see `ihh_twoSided`.","title":"One-sided integrated haplotype homozygosity"},{"id":"Biology/detecting-selection/selective-sweeps/ihh-two-sided","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ihh_twoSided","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#ihh-two-sided","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The two-sided integrated haplotype homozygosity of Voight et al.: the sum of the one-sided integrals in the two chromosome directions.","title":"Two-sided integrated haplotype homozygosity"},{"id":"Biology/detecting-selection/selective-sweeps/ihh-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ihh_eq","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#ihh-eq","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"In the star-genealogy model the one-sided integrated haplotype homozygosity is `1 / (2 r T)`.","title":"One-sided iHH in the star-genealogy model"},{"id":"Biology/detecting-selection/selective-sweeps/ihh-two-sided-eq","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ihh_twoSided_eq","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#ihh-two-sided-eq","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"In the star-genealogy model the two-sided integrated haplotype homozygosity is `1 / (r T)`.","title":"Two-sided iHH in the star-genealogy model"},{"id":"Biology/detecting-selection/selective-sweeps/ihs","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ihs","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#ihs","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The integrated haplotype score of Voight et al. (2006): the log ratio of the integrated haplotype homozygosities of the ancestral and the derived allele, whose star genealogies have ages `T_A` and `T_D`. The factor of two between the one-sided and two-sided integrals cancels in this ratio.","title":"Integrated haplotype score (iHS)"},{"id":"Biology/detecting-selection/selective-sweeps/ihs-eq-log-div","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ihs_eq_log_div","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#ihs-eq-log-div","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"In the star-genealogy model the integrated haplotype score is the log ratio of the two ages, so the recombination rate cancels.","title":"iHS as the log ratio of genealogy ages"},{"id":"Biology/detecting-selection/selective-sweeps/ihs-neg-iff","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.ihs_neg_iff","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#ihs-neg-iff","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"The integrated haplotype score is negative exactly when the derived allele's star genealogy is the younger one, i.e. when the derived allele swept faster.","title":"iHS is negative exactly when the derived allele is younger"},{"id":"Biology/detecting-selection/selective-sweeps/sweep-ehh-sweep-duration-full","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.sweepEhh_sweepDuration_full","parent":47444,"result":true,"route":"../roadmap/Biology/detecting-selection/selective-sweeps/index.html#sweep-ehh-sweep-duration-full","scope":"Biology/detecting-selection/selective-sweeps","status":"fully_proved","summary":"Combining the exact sweep duration with the star-genealogy decay: after a complete sweep of a favourable allele with selection coefficient `s` in a population of `N` diploids, the extended haplotype homozygosity at genetic distance `d` is the power law `(2N - 1) ^ (-(4 r d / s))`.","title":"Power-law EHH after a complete sweep"},{"data_url":"scopes/Biology/detecting-selection/cross-population-haplotype-homozygosity.json","id":"Biology/detecting-selection/cross-population-haplotype-homozygosity","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","title":"Cross-population haplotype homozygosity"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/is-exchangeable-pair","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.IsExchangeablePair","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#is-exchangeable-pair","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"The pair of statistics `(X, Y)` is exchangeable under `\u03bc`: the joint law is unchanged by swapping the two coordinates. This is an additional calibration assumption for a two-group haplotype test, stronger than the ordinary neutral null. Defined as `IdentDistrib` of the two orderings, so the measurability of the joint laws is part of the hypothesis rather than restored separately in each theorem.","title":"Exchangeable pair of statistics"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/map-comp","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.IsExchangeablePair.map_comp","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#map-comp","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"Applying the same measurable function to both statistics preserves exchangeability.","title":"Exchangeability is preserved by a common measurable map"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/map-sub-neg","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.IsExchangeablePair.map_sub_neg","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#map-sub-neg","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"The difference of an exchangeable pair has a symmetric law.","title":"Difference of an exchangeable pair has a symmetric law"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/measure-le-sub","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.IsExchangeablePair.measure_le_sub","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#measure-le-sub","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"Equal tails: the upper tail of `X - Y` at `t` has the same probability as the lower tail at `-t`.","title":"Equal tails of the difference of an exchangeable pair"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/measure-abs-sub","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.IsExchangeablePair.measure_abs_sub","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#measure-abs-sub","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"A two-sided rejection region splits into two tails of equal probability.","title":"Two-sided rejection region splits into equal tails"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/measure-le-log-ratio","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.IsExchangeablePair.measure_le_logRatio","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#measure-le-log-ratio","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"For positive statistics, the log ratio \u2014 the XP-EHH form \u2014 has equal tails.","title":"Equal tails of the log ratio of an exchangeable pair"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/measure-le-ratio","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.IsExchangeablePair.measure_le_ratio","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#measure-le-ratio","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"The same statement for the ratio itself, the form used for relative EHH: the probability that the ratio exceeds `c` equals the probability that it falls below `1 / c`.","title":"Equal tails of the ratio of an exchangeable pair"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/xpehh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.xpehh","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#xpehh","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"The cross-population extended haplotype homozygosity score (Sabeti et al. 2007): the log ratio of the integrated haplotype homozygosities of the same core region in two populations. Both arguments must be positive for the score to be meaningful. Outside that domain `Real.log` is totalized by Mathlib, not by the stated junk value: `Real.log 0 = 0` and `Real.log (-x) = Real.log x`, so for example\u2026","title":"Cross-population extended haplotype homozygosity (XP-EHH)"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/xpehh-ihh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.xpehh_ihh","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#xpehh-ihh","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"XP-EHH computed from two star-genealogy sweep models with the same recombination rate is the log ratio of the two ages, with the opposite sign to `ihs`.","title":"XP-EHH in star-genealogy sweep models"},{"id":"Biology/detecting-selection/cross-population-haplotype-homozygosity/measure-le-xpehh","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.SelectionScan.measure_le_xpehh","parent":47463,"result":true,"route":"../roadmap/Biology/detecting-selection/cross-population-haplotype-homozygosity/index.html#measure-le-xpehh","scope":"Biology/detecting-selection/cross-population-haplotype-homozygosity","status":"fully_proved","summary":"Under exchangeability, the XP-EHH score has a symmetric law: the two tails of the two-sided test carry the same probability.","title":"Symmetry of the XP-EHH law under exchangeability"},{"data_url":"scopes/Biology/detecting-selection/tajima-s-d.json","id":"Biology/detecting-selection/tajima-s-d","kind":"scope","navigation":"module","parent":47346,"result":false,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","title":"Tajima's D"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-d","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaD","parent":47474,"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-d","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"Tajima's `D` (Tajima 1989) for a sample of `n` genes with mean number of pairwise differences `\u03c0` and `S` segregating sites: `D = (\u03c0 - S / a\u2081) / \u221a(e\u2081 S + e\u2082 S (S - 1))`. It is meaningful for `n \u2265 4` and `S \u2265 1` (`tajimaVarEstimator_pos`); otherwise Lean's junk values for `x / 0` and `\u221a` of a nonpositive number apply.","title":"Tajima's `D`"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-c1-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaC1_two","parent":47474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's "],["math","\\(c_1\\)"],["text"," vanishes for two genes"]],"source":"Tajima's $c_1$ vanishes for two genes"},"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-c1-two","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"For two genes `c\u2081 = 0`.","title":"Tajima's $c_1$ vanishes for two genes"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-c2-two","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaC2_two","parent":47474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's "],["math","\\(c_2\\)"],["text"," vanishes for two genes"]],"source":"Tajima's $c_2$ vanishes for two genes"},"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-c2-two","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"For two genes `c\u2082 = 0`.","title":"Tajima's $c_2$ vanishes for two genes"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-c1-three","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaC1_three","parent":47474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's "],["math","\\(c_1\\)"],["text"," vanishes for three genes"]],"source":"Tajima's $c_1$ vanishes for three genes"},"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-c1-three","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"For three genes `c\u2081 = 0`.","title":"Tajima's $c_1$ vanishes for three genes"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-c2-three","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaC2_three","parent":47474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tajima's "],["math","\\(c_2\\)"],["text"," vanishes for three genes"]],"source":"Tajima's $c_2$ vanishes for three genes"},"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-c2-three","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"For three genes `c\u2082 = 0`.","title":"Tajima's $c_2$ vanishes for three genes"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-c1-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaC1_pos","parent":47474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positivity of Tajima's "],["math","\\(c_1\\)"],["text"," for at least four genes"]],"source":"Positivity of Tajima's $c_1$ for at least four genes"},"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-c1-pos","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"`c\u2081 = (n + 1) / (3 (n - 1)) - 1 / a\u2081 > 0` for `n \u2265 4`.","title":"Positivity of Tajima's $c_1$ for at least four genes"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-c2-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaC2_pos","parent":47474,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positivity of Tajima's "],["math","\\(c_2\\)"],["text"," for at least four genes"]],"source":"Positivity of Tajima's $c_2$ for at least four genes"},"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-c2-pos","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"`c\u2082 = b\u2082 - (n + 2) / (a\u2081 n) + a\u2082 / a\u2081\u00b2 > 0` for `n \u2265 4`.","title":"Positivity of Tajima's $c_2$ for at least four genes"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-var-estimator-pos","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaVarEstimator_pos","parent":47474,"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-var-estimator-pos","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"The denominator of Tajima's `D` is well defined: `e\u2081 S + e\u2082 S (S - 1) > 0` for `n \u2265 4` and `S \u2265 1`.","title":"Positivity of the denominator of Tajima's D"},{"id":"Biology/detecting-selection/tajima-s-d/tajima-chebyshev-test","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.tajimaChebyshevTest","parent":47474,"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#tajima-chebyshev-test","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"The rejection region of the Chebyshev test of neutrality at level `\u03b1` and hypothesized `\u03b8\u2080`: `|\u03c0 - S / a\u2081| \u2265 \u221a((c\u2081 \u03b8\u2080 + c\u2082 \u03b8\u2080\u00b2) / \u03b1)`.","title":"Chebyshev test of neutrality"},{"id":"Biology/detecting-selection/tajima-s-d/measure-tajima-chebyshev-test-le","kind":"node","lean":"MathlibExt.Biology.PopulationGenetics.Coalescent.measure_tajimaChebyshevTest_le","parent":47474,"result":true,"route":"../roadmap/Biology/detecting-selection/tajima-s-d/index.html#measure-tajima-chebyshev-test-le","scope":"Biology/detecting-selection/tajima-s-d","status":"fully_proved","summary":"If Tajima's numerator `\u03c0 - S / a\u2081` is square integrable with mean `0` and variance `c\u2081 \u03b8\u2080 + c\u2082 \u03b8\u2080\u00b2 > 0` (as under the standard neutral model, by `KingmanTimes.integral_tajimaNumerator` and `KingmanTimes.variance_tajimaNumerator`), then for every `\u03b1 > 0` the test rejects with probability at most `\u03b1`.","title":"Level of the Chebyshev test"},{"data_url":"scopes/Biology/cancer-genomics.json","id":"Biology/cancer-genomics","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/cancer-genomics/index.html","scope":"Biology/cancer-genomics","status":"fully_proved","summary":"Mathematical models for tumour sequencing data: purity and copy number, cancer cell fractions, mutation detection power, mutational signatures and the timing of genomic events.","title":"Cancer genomics"},{"data_url":"scopes/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores.json","id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","kind":"scope","navigation":"module","parent":47485,"result":false,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","title":"GISTIC permutation null for copy-number scores"},{"id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/permuted-score","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.GisticNull.permutedScore","parent":47486,"result":true,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html#permuted-score","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","summary":"The permuted score at marker `m`: `\u2211 j, x j (\u03c3 j m)`.","title":"Permuted GISTIC score at a marker"},{"id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/permutation-null","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.GisticNull.permutationNull","parent":47486,"result":true,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html#permutation-null","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","summary":"The GISTIC permutation null at marker `m`: the probability that `\u2211 j, x j (\u03c3 j m) = g` when the permutations `\u03c3 j` of the markers are independent and uniform.","title":"GISTIC permutation null distribution"},{"id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/score-count","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.GisticNull.scoreCount","parent":47486,"result":true,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html#score-count","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","summary":"The number of marker choices `u : Fin S \u2192 Fin M` (one marker per sample) with `\u2211 j, x j (u j) = g`.","title":"Count of marker choices with a given score"},{"id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/permutation-null-eq-count","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.GisticNull.permutationNull_eq_count","parent":47486,"result":true,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html#permutation-null-eq-count","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","summary":"(Mermel et al. 2011): the probability of the score `g` is the fraction of the `M ^ S` marker choices with that score, whatever the marker `m`.","title":"The GISTIC permutation null is the law of a sum of independent per-sample draws"},{"id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/permutation-null-zero","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.GisticNull.permutationNull_zero","parent":47486,"result":true,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html#permutation-null-zero","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","summary":"With no samples the null is the point mass at `0`.","title":"GISTIC null with no samples is a point mass at zero"},{"id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/permutation-null-succ","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.GisticNull.permutationNull_succ","parent":47486,"result":true,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html#permutation-null-succ","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","summary":"Adding sample `S` convolves the null of the first `S` samples with the empirical law of the new sample, `P_{S+1}(g) = \u2211_v (1 / M) P_S(g - x_S(v))`.","title":"Convolution step"},{"id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/window-min","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.GisticNull.windowMin","parent":47486,"result":true,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html#window-min","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","summary":"The minimum of `y` over the window of `n + 1` consecutive markers starting at `i` (GeneGISTIC's per-sample gene-level value for a gene covering those markers).","title":"Window minimum over consecutive markers"},{"id":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/gene-null-eq-count","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.GisticNull.geneNull_eq_count","parent":47486,"result":true,"route":"../roadmap/Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores/index.html#gene-null-eq-count","scope":"Biology/cancer-genomics/gistic-permutation-null-for-copy-number-scores","status":"fully_proved","summary":"GeneGISTIC null (Mermel et al. 2011): permuting the per-sample minima of all running windows of `n + 1` markers gives, at every window `i`, the law of a sum of independent draws of one window minimum per sample.","title":"GeneGISTIC null"},{"data_url":"scopes/Biology/cancer-genomics/molecular-timing-of-copy-number-gains.json","id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","kind":"scope","navigation":"module","parent":47485,"result":false,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","title":"Molecular timing of copy-number gains"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/gain-history","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#gain-history","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"A single copy-number event on a region: `copies` original copies, of which copy `i` has `desc i` copies after the event (`0` if lost, `2` if duplicated).","title":"Copy-number gain history"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/final-copies","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.finalCopies","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#final-copies","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"The copy number `c\u2081 = \u2211 i, desc i` after the event.","title":"Copy number after a copy-number event"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/amplified","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.amplified","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#amplified","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"The number of original copies amplified by the event (`desc i \u2265 2`).","title":"Number of amplified copies"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/expected-count","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.expectedCount","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#expected-count","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"Expected number of mutations of multiplicity `m`: `\u03bc t #{i | desc i = m} + [m = 1] \u03bc c\u2081 (1 - t)` (per-copy rate `\u03bc`, event at time `t` of the normalized history `[0, 1]`).","title":"Expected number of mutations of a given multiplicity"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/sum-mul-expected-count","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.sum_mul_expectedCount","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#sum-mul-expected-count","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"`\u2211_m m E_m = \u03bc c\u2081`, summing over all multiplicities up to a bound `B \u2265 max(1, max_i desc i)`.","title":"The multiplicity-weighted total is time-invariant"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/sum-expected-count-amplified","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.sum_expectedCount_amplified","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#sum-expected-count-amplified","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"The mutations of multiplicity at least two number `\u03bc t #{i | desc i \u2265 2}` in expectation: they all arose before the event, on an amplified copy.","title":"Expected count of mutations on amplified copies"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/final-copies-pos","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.finalCopies_pos","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#final-copies-pos","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"An event that amplifies some copy leaves a positive copy number.","title":"Positive copy number after an amplifying event"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/rate-eq","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.rate_eq","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#rate-eq","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"`\u03bc = \u2211_m m E_m / c\u2081` (`c\u2081 > 0`).","title":"The mutation rate from the spectrum"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/timing-eq","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.timing_eq","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#timing-eq","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"When some copy is amplified and `\u03bc \u2260 0`, `t = c\u2081 \u2211_{m \u2265 2} E_m / (#{i | desc i \u2265 2} \u2211_m m E_m)`.","title":"The timing closed form"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/expected-count-injective","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.expectedCount_injective","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#expected-count-injective","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"If some copy is amplified, two parameter pairs `(\u03bc, t)`, `(\u03bc', t')` with `\u03bc, \u03bc' \u2260 0` and the same expected multiplicity spectrum are equal.","title":"Identifiability"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/single-gain","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.singleGain","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#single-gain","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"A single-copy gain `1 + 1 \u2192 2 + 1`: one copy duplicated, the other kept.","title":"Single-copy gain"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/cn-loh","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.cnLOH","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#cn-loh","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"Copy-neutral loss of heterozygosity `1 + 1 \u2192 2 + 0`: one copy duplicated, the other lost.","title":"Copy-neutral loss of heterozygosity"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/wgd","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.wgd","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#wgd","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"Whole-genome doubling of a diploid region `1 + 1 \u2192 2 + 2`.","title":"Whole-genome doubling event"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/single-gain-timing","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.singleGain_timing","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#single-gain-timing","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"Single gain (`2 + 1`): `t = 3 E\u2082 / (E\u2081 + 2 E\u2082)`, for `\u03bc \u2260 0`.","title":"Single gain"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/cn-loh-timing","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.cnLOH_timing","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#cn-loh-timing","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"Copy-neutral LOH (`2 + 0`): `t = 2 E\u2082 / (E\u2081 + 2 E\u2082)`, for `\u03bc \u2260 0`.","title":"Copy-neutral LOH"},{"id":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains/wgd-timing","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.MutationTiming.GainHistory.wgd_timing","parent":47495,"result":true,"route":"../roadmap/Biology/cancer-genomics/molecular-timing-of-copy-number-gains/index.html#wgd-timing","scope":"Biology/cancer-genomics/molecular-timing-of-copy-number-gains","status":"fully_proved","summary":"Whole-genome doubling (`2 + 2`): `t = 2 E\u2082 / (E\u2081 + 2 E\u2082)`, for `\u03bc \u2260 0`.","title":"Whole-genome doubling"},{"data_url":"scopes/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf.json","id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","kind":"scope","navigation":"module","parent":47485,"result":false,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","title":"Mutational signatures and Bayesian NMF"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/kl-div","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.klDiv","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#kl-div","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The Kullback\u2013Leibler `\u03b2 = 1` divergence `d(x | y) = x log (x / y) - x + y` (Tan\u2013F\u00e9votte, Eq. 4). With Mathlib's `0 * log _ = 0` it gives `d(0 | y) = y`, the correct value; `y > 0` is needed when `x > 0`.","title":"Generalized Kullback\u2013Leibler divergence"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/recon","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.recon","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#recon","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The reconstruction `(W H)_fn = \u2211_k w_fk h_kn`.","title":"Reconstruction matrix in nonnegative matrix factorization"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/data-fit","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.dataFit","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#data-fit","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The data-fit term `D_1(V | W H) = \u2211_{f,n} d(v_fn | (W H)_fn)`.","title":"Data-fit term of the ARD-NMF objective"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/objective-h","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.objectiveH","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#objective-h","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The `H`-dependent part `D_1(V | W H) / \u03c6 + \u2211_{k,n} h_kn / \u03bb_k` of the `\u2113\u2081`-ARD objective (Tan\u2013F\u00e9votte, Eq. 20, `\u03b2 = 1`).","title":"H-dependent part of the ARD-NMF objective"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/update-h","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.updateH","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#update-h","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The `\u2113\u2081`-ARD multiplicative update of `H` for `\u03b2 = 1` (Tan\u2013F\u00e9votte, Eq. 32, Algorithm 2): `h_kn \u2190 h_kn (\u2211_f w_fk v_fn / (W H)_fn) / (\u2211_f w_fk + \u03c6 / \u03bb_k)`.","title":"ARD multiplicative update of H"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/objective-h-update-h-le","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.objectiveH_updateH_le","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#objective-h-update-h-le","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The `H` update is monotone (Tan\u2013F\u00e9votte 2013, \u00a7IV-B, `\u03b2 = 1`): it does not increase `D_1(V | W H) / \u03c6 + \u2211 h_kn / \u03bb_k`, for `\u03c6 > 0`, `\u03bb > 0`, `V \u2265 0`, `W \u2265 0` with no zero row, and `H > 0` entrywise.","title":"The `H` update is monotone"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/objective-w","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.objectiveW","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#objective-w","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The `W`-dependent part `D_1(V | W H) / \u03c6 + \u2211_{f,k} w_fk / \u03bb_k` of the `\u2113\u2081`-ARD objective (Tan\u2013F\u00e9votte, Eq. 20, `\u03b2 = 1`).","title":"W-dependent part of the ARD-NMF objective"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/update-w","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.updateW","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#update-w","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The `\u2113\u2081`-ARD multiplicative update of `W` for `\u03b2 = 1` (Tan\u2013F\u00e9votte, Algorithm 2): `w_fk \u2190 w_fk (\u2211_n v_fn h_kn / (W H)_fn) / (\u2211_n h_kn + \u03c6 / \u03bb_k)`.","title":"ARD multiplicative update of W"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/objective-w-update-w-le","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.objectiveW_updateW_le","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#objective-w-update-w-le","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"It does not increase `D_1(V | W H) / \u03c6 + \u2211 w_fk / \u03bb_k`, for `\u03c6 > 0`, `\u03bb > 0`, `V \u2265 0`, `H \u2265 0` with no zero column, and `W > 0` entrywise (the transpose of `objectiveH_updateH_le`).","title":"The `W` update is monotone"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/relevance","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.relevance","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#relevance","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The relevance update `\u03bb_k = (\u2016w_k\u2016\u2081 + \u2016h_k\u2016\u2081 + b) / c` (Tan\u2013F\u00e9votte, Eq. 33, `\u2113\u2081` prior; `c = F + N + a + 1`).","title":"Relevance update in ARD-NMF"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/relevance-is-min-on","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.relevance_isMinOn","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#relevance-is-min-on","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The relevance update is the exact minimizer (Tan\u2013F\u00e9votte, \u00a7IV-C): for `A > 0` and `c > 0`, `\u03bb \u21a6 A / \u03bb + c log \u03bb` attains its minimum over `\u03bb > 0` at `\u03bb = A / c`.","title":"The relevance update is the exact minimizer"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/relevance-unique","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.relevance_unique","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#relevance-unique","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"If `\u03bb > 0` does as well as `A / c`, then `\u03bb = A / c`.","title":"The minimizer is unique"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/div-le-relevance","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.div_le_relevance","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#div-le-relevance","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The relevance never falls below its floor `b / c` (`W, H \u2265 0`, `c > 0`).","title":"Lower bound on the relevance update"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/relevance-eq-floor-iff","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.relevance_eq_floor_iff","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#relevance-eq-floor-iff","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"Pruning (Tan\u2013F\u00e9votte, \u00a7IV-D): the relevance equals its floor `b / c` exactly when the `k`-th column of `W` and the `k`-th row of `H` are zero (`W, H \u2265 0`, `c > 0`).","title":"Pruning"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/update-h-eq-zero","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.updateH_eq_zero","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#update-h-eq-zero","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The multiplicative update maps a zero entry of `H` to zero.","title":"A pruned entry stays pruned"},{"id":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/update-w-eq-zero","kind":"node","lean":"Cslib.Bioinformatics.MutationalSignatures.ARDNMF.updateW_eq_zero","parent":47512,"result":true,"route":"../roadmap/Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf/index.html#update-w-eq-zero","scope":"Biology/cancer-genomics/mutational-signatures-and-bayesian-nmf","status":"fully_proved","summary":"The multiplicative update maps a zero entry of `W` to zero.","title":"A pruned entry stays pruned"},{"data_url":"scopes/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra.json","id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","kind":"scope","navigation":"module","parent":47485,"result":false,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","title":"Tumour purity, ploidy and copy-ratio algebra"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/mix-copy","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.mixCopy","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#mix-copy","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Average absolute copy number `\u03b1 q + 2 (1 - \u03b1)` of a locus with copy number `q` in the cancer cells, in a sample of purity `\u03b1` whose normal cells are diploid (Carter et al. 2012). With `q = \u03c4` this is the sample ploidy `D` of Eq. 1; with the local copy number it is `D_s` of Eq. 2.","title":"Average absolute copy number of a tumor sample"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/copy-ratio","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.copyRatio","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#copy-ratio","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Relative copy ratio `R = (\u03b1 q + 2 (1 - \u03b1)) / (\u03b1 \u03c4 + 2 (1 - \u03b1))` of a locus with copy number `q`, purity `\u03b1` and cancer-cell ploidy `\u03c4` (Carter et al. 2012, Eq. 1).","title":"Relative copy ratio"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/slope","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.slope","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#slope","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The copy-ratio slope `\u03b1 / D` of Eq. 1 (Carter et al. 2012), the copy ratio of one extra copy per cancer cell.","title":"Copy-ratio slope"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/intercept","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.intercept","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#intercept","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The copy-ratio intercept `2 (1 - \u03b1) / D` of Eq. 1 (Carter et al. 2012), the copy ratio of a homozygously deleted locus.","title":"Copy-ratio intercept"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/somatic-af","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.somaticAF","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#somatic-af","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Expected allelic fraction `\u03b1 s / (\u03b1 q + 2 (1 - \u03b1))` of a clonal somatic point mutation with multiplicity `s` (copies per cancer cell) at a locus of copy number `q` (Carter et al. 2012, Eq. 2; Eq. 10 is `s = 1`).","title":"Expected allelic fraction of a somatic mutation"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/germline-af","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.germlineAF","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#germline-af","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Expected allelic fraction `((1 - \u03b1) + \u03b1 g) / (\u03b1 q + 2 (1 - \u03b1))` of a heterozygous germline variant present in `g` copies per cancer cell at a locus of copy number `q` (Carter et al. 2012, Eq. 11).","title":"Expected allelic fraction of a germline variant"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/scaled-purity","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.scaledPurity","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#scaled-purity","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The purity `\u03b1 / (m - (m - 1) \u03b1)` that, together with all copy numbers multiplied by `m`, reproduces the copy ratios and allelic fractions of purity `\u03b1` (derived from Carter et al. 2012, Eqs. 1, 2, 11).","title":"Rescaled purity"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/mix-copy-pos","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.mixCopy_pos","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#mix-copy-pos","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The mixture copy number is positive in the biological domain `0 \u2264 \u03b1 \u2264 1`, `0 < q`.","title":"Positivity of the mixture copy number"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/copy-ratio-eq-slope-mul-add-intercept","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.copyRatio_eq_slope_mul_add_intercept","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#copy-ratio-eq-slope-mul-add-intercept","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"`R = (\u03b1 / D) q + 2 (1 - \u03b1) / D`, for all real arguments.","title":"Eq. 1 is affine in `q`"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/sum-mul-copy-ratio-eq-one","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.sum_mul_copyRatio_eq_one","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#sum-mul-copy-ratio-eq-one","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"If the segment weights `w` sum to one and the ploidy is the weighted mean copy number `\u03c4 = \u2211 w q`, the weighted mean copy ratio is `1` (Carter et al. 2012, Methods: \"correctly calibrated copy-ratio data always has mean = 1\").","title":"Calibrated copy ratios have mean `1`"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/somatic-af-eq-slope-mul-div-copy-ratio","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.somaticAF_eq_slope_mul_div_copyRatio","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#somatic-af-eq-slope-mul-div-copy-ratio","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Allelic fractions are determined by the copy-ratio slope (Carter et al. 2012, Eq. 2): `\u03b1 s / D_s = (\u03b1 / D) s / R`, whenever `D \u2260 0`.","title":"Allelic fractions are determined by the copy-ratio slope"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/purity-eq","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.purity_eq","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#purity-eq","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Inverse of Eq. 1, purity: `\u03b1 = 2 b / (2 b + \u03b4)` for the slope `b` and intercept `\u03b4`, when `0 < \u03b1 \u2264 1` and `0 < \u03c4`.","title":"Purity from copy-ratio slope and intercept"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/ploidy-eq","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.ploidy_eq","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#ploidy-eq","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Inverse of Eq. 1, ploidy: `\u03c4 = (1 - \u03b4) / b` for the slope `b` and intercept `\u03b4`, when `0 < \u03b1 \u2264 1` and `0 < \u03c4`.","title":"Ploidy from copy-ratio slope and intercept"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/bij-on-slope-intercept","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.bijOn_slope_intercept","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#bij-on-slope-intercept","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The map `(\u03b1, \u03c4) \u21a6 (\u03b1 / D, 2 (1 - \u03b1) / D)` is a bijection from `{0 < \u03b1 \u2264 1, 0 < \u03c4}` onto `{0 < b, 0 \u2264 \u03b4 < 1}`.","title":"Purity and ploidy are identifiable from the slope and intercept of Eq. 1"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/scaled-purity-denom-pos","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.scaledPurity_denom_pos","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#scaled-purity-denom-pos","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The denominator `m - (m - 1) \u03b1 = m (1 - \u03b1) + \u03b1` of the rescaled purity is positive for `m > 0` and `0 \u2264 \u03b1 \u2264 1`.","title":"Positivity of the rescaled purity denominator"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/scaled-purity-mem","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.scaledPurity_mem","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#scaled-purity-mem","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The rescaled purity is again a purity: `0 \u2264 \u03b1 / (m - (m - 1) \u03b1) \u2264 1` for `m > 0` and `0 \u2264 \u03b1 \u2264 1`.","title":"Rescaled purity lies in (0, 1)"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/copy-ratio-scale","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.copyRatio_scale","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#copy-ratio-scale","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Scaling every copy number and the ploidy by `m > 0` and the purity to `\u03b1 / (m - (m - 1) \u03b1)` leaves every copy ratio of Eq. 1 unchanged.","title":"Ploidy-scaling non-identifiability, copy ratios"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/somatic-af-scale","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.somaticAF_scale","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#somatic-af-scale","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Ploidy-scaling non-identifiability, somatic allelic fractions (Eq. 2): with copy number and multiplicity both scaled by `m > 0` and the purity rescaled, the allelic fraction is unchanged.","title":"Ploidy-scaling non-identifiability, somatic allelic fractions"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/germline-af-scale","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.germlineAF_scale","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#germline-af-scale","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Ploidy-scaling non-identifiability, germline allelic fractions (Eq. 11): with copy number and germline multiplicity both scaled by `m > 0` and the purity rescaled, the allelic fraction is unchanged.","title":"Ploidy-scaling non-identifiability, germline allelic fractions"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/scaled-purity-ne","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.scaledPurity_ne","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#scaled-purity-ne","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The rescaled solution is a genuinely different purity: for `0 < \u03b1 < 1` and `m \u2260 1`, `\u03b1 / (m - (m - 1) \u03b1) \u2260 \u03b1`.","title":"Rescaled purity differs from the original purity"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/copy-ratio-doubling","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.copyRatio_doubling","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#copy-ratio-doubling","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Doubling every copy number and the ploidy and replacing the purity by `\u03b1 / (2 - \u03b1)` reproduces every copy ratio (the `m = 2` case of `copyRatio_scale`).","title":"Genome doubling"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/somatic-af-nonneg","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.somaticAF_nonneg","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#somatic-af-nonneg","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"Somatic allelic fractions are nonnegative in the biological domain.","title":"Nonnegativity of somatic allelic fractions"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/somatic-af-le-one","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.somaticAF_le_one","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#somatic-af-le-one","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"A somatic allelic fraction is at most one when the multiplicity does not exceed the copy number (`s \u2264 q`).","title":"Somatic allelic fraction is at most one"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/somatic-af-mono-purity","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.somaticAF_mono_purity","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#somatic-af-mono-purity","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The somatic allelic fraction increases with the purity (`0 \u2264 s`, `0 < q`).","title":"Somatic allelic fraction increases with purity"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/somatic-af-anti-copy","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.somaticAF_anti_copy","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#somatic-af-anti-copy","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The somatic allelic fraction decreases with the local copy number (`0 \u2264 \u03b1 \u2264 1`, `0 \u2264 s`, `0 < q \u2264 q'`).","title":"Somatic allelic fraction decreases with copy number"},{"id":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/somatic-af-mono-mult","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.Absolute.somaticAF_mono_mult","parent":47529,"result":true,"route":"../roadmap/Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra/index.html#somatic-af-mono-mult","scope":"Biology/cancer-genomics/tumour-purity-ploidy-and-copy-ratio-algebra","status":"fully_proved","summary":"The somatic allelic fraction increases with the multiplicity (`0 \u2264 \u03b1 \u2264 1`, `0 < q`).","title":"Somatic allelic fraction increases with multiplicity"},{"data_url":"scopes/Biology/cancer-genomics/detection-power-for-somatic-mutations.json","id":"Biology/cancer-genomics/detection-power-for-somatic-mutations","kind":"scope","navigation":"module","parent":47485,"result":false,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","title":"Detection power for somatic mutations"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/binom-pmf","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.binomPmf","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#binom-pmf","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The binomial probability `C(n, i) p^i (1 - p)^(n - i)` of `i` successes in `n` trials, as a function of `i : \u2115`. It vanishes for `i > n` (the binomial coefficient is zero there, so the truncated subtraction `n - i` is harmless).","title":"Binomial probability mass function"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The binomial upper tail `\u2119_p(X \u2265 m) = \u2211_{m \u2264 i \u2264 n} C(n, i) p^i (1 - p)^(n - i)`, Carter's `P(m)` with `p = \u03b5 / 3`.","title":"Binomial upper tail probability"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/binom-pmf-nonneg","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.binomPmf_nonneg","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#binom-pmf-nonneg","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Binomial probabilities are nonnegative for `0 \u2264 p \u2264 1`.","title":"Nonnegativity of binomial probabilities"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/binom-pmf-pos","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.binomPmf_pos","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#binom-pmf-pos","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Binomial probabilities are positive on `{0, \u2026, n}` for `0 < p < 1`.","title":"Positivity of binomial probabilities"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/binom-pmf-eq-zero","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.binomPmf_eq_zero","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#binom-pmf-eq-zero","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Binomial probabilities vanish above `n`.","title":"Binomial probabilities vanish above n"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/sum-binom-pmf","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.sum_binomPmf","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#sum-binom-pmf","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The binomial law sums to one (the binomial theorem), for every real `p`.","title":"Binomial probabilities sum to one"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-nonneg","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_nonneg","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-nonneg","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Tail probabilities are nonnegative for `0 \u2264 p \u2264 1`.","title":"Nonnegativity of binomial tail probabilities"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-zero","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_zero","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-zero","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"`P(0) = 1`, for every real `p`.","title":"Binomial upper tail from zero is one"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-eq-zero","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_eq_zero","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-eq-zero","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"`P(m) = 0` for `m > n`.","title":"Binomial upper tail beyond n is zero"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-eq-one-sub-sum","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_eq_one_sub_sum","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-eq-one-sub-sum","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Carter's form of the tail: `P(m) = 1 - \u2211_{i < m} Binom(i | n, p)`, for every `m`.","title":"Binomial upper tail as one minus a lower sum"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-eq-add","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_eq_add","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-eq-add","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"One step of the tail recursion: `P(m) = Binom(m | n, p) + P(m + 1)`.","title":"Recursion for the binomial upper tail"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-antitone","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_antitone","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-antitone","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"`m \u21a6 P(m)` is antitone for `0 \u2264 p \u2264 1`.","title":"Binomial upper tail is decreasing"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-mono","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_mono","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-mono","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"For `0 \u2264 p \u2264 p' \u2264 1`, `\u2119_p(X \u2265 m) \u2264 \u2119_{p'}(X \u2265 m)`.","title":"Binomial tails increase with the success probability"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/binom-pmf-le-binom-pmf-of-le-div","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.binomPmf_le_binomPmf_of_le_div","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#binom-pmf-le-binom-pmf-of-le-div","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"For `a \u2264 n` and `0 \u2264 p \u2264 p' \u2264 a / n`, `Binom(a | n, p) \u2264 Binom(a | n, p')`.","title":"The binomial likelihood is unimodal in `p`, increasing part"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/binom-pmf-le-binom-pmf-of-div-le","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.binomPmf_le_binomPmf_of_div_le","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#binom-pmf-le-binom-pmf-of-div-le","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"For `a \u2264 n` and `a / n \u2264 p \u2264 p' \u2264 1`, `Binom(a | n, p') \u2264 Binom(a | n, p)`.","title":"The binomial likelihood is unimodal in `p`, decreasing part"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/critical-count","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.criticalCount","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#critical-count","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Carter's critical count `k = min {m | P(m) \u2264 FPR}` for the null success probability `p\u2080 = \u03b5 / 3` (Online Methods, before Eq. 9).","title":"Critical count of the mutation detection test"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/randomization","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.randomization","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#randomization","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The randomization probability `d = (FPR - P(k)) / (P(k - 1) - P(k))` of Eq. 9.","title":"Randomization probability of the detection test"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/power","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.power","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#power","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The power `Pow(n, f) = P_f(X \u2265 k) + d Binom(k - 1 | n, f)` of the exact-size randomized test (Carter et al. 2012, Eq. 9, with the randomization at `k - 1`; see the module doc).","title":"Power of the randomized mutation detection test"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-critical-count-le","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_criticalCount_le","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-critical-count-le","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The critical count meets the false-positive bound: `P(k) \u2264 FPR` (for `FPR \u2265 0`).","title":"Critical count meets the false-positive bound"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/critical-count-le","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.criticalCount_le","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#critical-count-le","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The critical count is at most `n + 1` (for `FPR \u2265 0`).","title":"Upper bound on the critical count"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/critical-count-pos","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.criticalCount_pos","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#critical-count-pos","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The critical count is positive when `FPR < 1`.","title":"Positivity of the critical count"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/lt-upper-tail-critical-count-sub-one","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.lt_upperTail_criticalCount_sub_one","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#lt-upper-tail-critical-count-sub-one","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Minimality of the critical count: `FPR < P(k - 1)` (for `0 \u2264 FPR < 1`).","title":"Minimality of the critical count"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/upper-tail-sub-one-sub","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.upperTail_sub_one_sub","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#upper-tail-sub-one-sub","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The gap `P(k - 1) - P(k)` is the null probability of the boundary point `k - 1`.","title":"Tail gap at the critical count"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/randomization-mem","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.randomization_mem","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#randomization-mem","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"`0 \u2264 d < 1` for `0 \u2264 FPR < 1`.","title":"The randomization is a probability"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/power-self","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.power_self","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#power-self","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Exact size (Carter et al. 2012, Eq. 9 with the randomization at `k - 1`): under the null `f = p\u2080` the rejection probability is exactly `FPR`, for `0 \u2264 FPR < 1`.","title":"Exact size"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/power-eq-mix","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.power_eq_mix","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#power-eq-mix","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"The power is the mixture `(1 - d) P_f(X \u2265 k) + d P_f(X \u2265 k - 1)` of two tail probabilities.","title":"Detection power as a mixture of tail probabilities"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/power-mono","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.power_mono","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#power-mono","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"`Pow(n, f) \u2264 Pow(n, f')` for `0 \u2264 f \u2264 f' \u2264 1`.","title":"Power increases with the allelic fraction"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/power-somatic-af-mono-purity","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.power_somaticAF_mono_purity","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#power-somatic-af-mono-purity","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Power increases with purity at the ABSOLUTE allelic fraction `\u03b1 s / (\u03b1 q + 2 (1 - \u03b1))` of a clonal mutation of multiplicity `0 \u2264 s \u2264 q` (Carter et al. 2012, Eqs. 2, 9, 10).","title":"Power increases with purity"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/power-somatic-af-anti-copy","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.power_somaticAF_anti_copy","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#power-somatic-af-anti-copy","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Power decreases with local copy number at the ABSOLUTE allelic fraction, for a clonal mutation of multiplicity `0 \u2264 s \u2264 q \u2264 q'` (Carter et al. 2012, Eqs. 2, 9, 10).","title":"Power decreases with local copy number"},{"id":"Biology/cancer-genomics/detection-power-for-somatic-mutations/power-somatic-af-mono-mult","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.DetectionPower.power_somaticAF_mono_mult","parent":47556,"result":true,"route":"../roadmap/Biology/cancer-genomics/detection-power-for-somatic-mutations/index.html#power-somatic-af-mono-mult","scope":"Biology/cancer-genomics/detection-power-for-somatic-mutations","status":"fully_proved","summary":"Power increases with multiplicity at the ABSOLUTE allelic fraction, for `0 \u2264 s \u2264 s' \u2264 q` (Carter et al. 2012, Eqs. 2, 9).","title":"Power increases with multiplicity"},{"data_url":"scopes/Biology/cancer-genomics/cancer-cell-fraction-estimation.json","id":"Biology/cancer-genomics/cancer-cell-fraction-estimation","kind":"scope","navigation":"module","parent":47485,"result":false,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","title":"Cancer cell fraction estimation"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/tail-ratio","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.tailRatio","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#tail-ratio","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"Posterior probability that the parameter `c i` exceeds `t`, for prior weights `prior` and likelihoods `lik i` of the observed data: `\u2211_{c i > t} prior i lik i / \u2211_i prior i lik i`.","title":"Posterior tail probability on a grid"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/posterior-tail-mono-of-tp2","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.posteriorTail_mono_of_tp2","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#posterior-tail-mono-of-tp2","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"If the kernels `L` (data `x`) and `L'` (data `x'`) satisfy the TP2 inequality `L' i L j \u2265 L i L' j` whenever `c j \u2264 c i`, then for a nonnegative prior every posterior tail `P(c > t)` is at least as large under `L'` as under `L` (both normalizers positive).","title":"Monotone likelihood ratio implies stochastically larger posteriors"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/binom-pmf-mul-le-mul","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.binomPmf_mul_le_mul","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#binom-pmf-mul-le-mul","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"The binomial family is TP2 (monotone likelihood ratio in the success probability): for `a \u2264 a'` and `0 \u2264 p \u2264 p' \u2264 1`, `Binom(a | n, p') Binom(a' | n, p) \u2264 Binom(a' | n, p') Binom(a | n, p)`.","title":"The binomial family is TP2"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/posterior-tail","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.posteriorTail","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#posterior-tail","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"Landau et al.'s posterior probability that the CCF exceeds `t`, given `a` of `n` reads, purity `\u03b1`, copy number `q`, a prior `prior` on the grid points `grid k`: `\u2211_{grid k > t} \u03c0_k Binom(a | n, f(grid k)) / \u2211_k \u03c0_k Binom(a | n, f(grid k))`, with `f(c) = \u03b1 c / (\u03b1 q + 2 (1 - \u03b1))`.","title":"Posterior probability that the cancer cell fraction exceeds a threshold"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/is-clonal","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.IsClonal","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#is-clonal","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"Landau et al.'s clonal call: `P(CCF > t | a) > \u03b3` (they take `t = 0.95`, `\u03b3 = 0.5`).","title":"Clonal call for a mutation"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/grid-model","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.GridModel","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#grid-model","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"Hypotheses on the CCF grid model: purity `0 < \u03b1 \u2264 1`, copy number `q > 0`, grid in `[0, q]` and within the defining CCF bound `\u2264 1`, a nonnegative prior, and one grid point of positive prior weight whose allelic fraction lies strictly inside `(0, 1)` (this makes every posterior well defined).","title":"Grid model for the cancer cell fraction"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/normalizer-pos","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.GridModel.normalizer_pos","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#normalizer-pos","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"In a grid model every posterior normalizer `\u2211_k \u03c0_k Binom(a | n, f(grid k))`, `a \u2264 n`, is positive.","title":"Positivity of the grid posterior normalizer"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/posterior-tail-mono","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.posteriorTail_mono","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#posterior-tail-mono","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"The CCF posterior is stochastically increasing in the read count (Landau et al. 2013 with the monotone-likelihood-ratio argument): `P(CCF > t | a) \u2264 P(CCF > t | a')` for `a \u2264 a' \u2264 n`.","title":"The CCF posterior is stochastically increasing in the read count"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/mono","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.IsClonal.mono","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#mono","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"The clonal call is upward closed in the read count.","title":"The clonal call is upward closed in the read count"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/exists-threshold","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.exists_threshold","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#exists-threshold","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"For every CCF threshold `t` and posterior level `\u03b3` there is `a\u2080` such that, for `a \u2264 n`, the mutation is called clonal iff `a\u2080 \u2264 a` (`a\u2080 = n + 1` when no count is called clonal).","title":"The clonal call is a read-count threshold"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/ccf-mle","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.ccfMLE","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#ccf-mle","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"The continuous maximum-likelihood CCF `\u0109 = min 1 (a / (n \u03ba))` for the allelic-fraction slope `\u03ba = \u03b1 / (\u03b1 q + 2 (1 - \u03b1))`. Landau et al. (2013) use the grid posterior; this continuous maximizer over `[0, 1]` is derived here.","title":"Maximum-likelihood cancer cell fraction"},{"id":"Biology/cancer-genomics/cancer-cell-fraction-estimation/is-max-on-ccf-mle","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.CancerCellFraction.isMaxOn_ccfMLE","parent":47587,"result":true,"route":"../roadmap/Biology/cancer-genomics/cancer-cell-fraction-estimation/index.html#is-max-on-ccf-mle","scope":"Biology/cancer-genomics/cancer-cell-fraction-estimation","status":"fully_proved","summary":"For `0 < \u03b1 \u2264 1`, `q \u2265 1`, `a \u2264 n` and `0 < n`, the likelihood `c \u21a6 Binom(a | n, f(c))` on `[0, 1]` is maximized at `\u0109 = min 1 (a / (n \u03ba))`.","title":"The continuous CCF MLE"},{"data_url":"scopes/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery.json","id":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","kind":"scope","navigation":"module","parent":47485,"result":false,"route":"../roadmap/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/index.html","scope":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","status":"fully_proved","title":"Saturation power for cancer gene discovery"},{"id":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/saturation-power","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.SaturationPower.saturationPower","parent":47600,"result":true,"route":"../roadmap/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/index.html#saturation-power","scope":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","status":"fully_proved","summary":"Power of the one-sided binomial test of Lawrence et al. (2014): the probability, when each of `N` patients is mutated with probability `p\u2081`, of observing a count `x` whose background p-value `\u2119_{p\u2080}(X \u2265 x)` is at most `\u03b3`.","title":"Saturation power of the binomial gene discovery test"},{"id":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/lawrence-p0","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.SaturationPower.lawrenceP0","parent":47600,"result":true,"route":"../roadmap/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/index.html#lawrence-p0","scope":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","status":"fully_proved","summary":"Background probability `p\u2080 = 1 - (1 - \u03bc f_g)^(3L/4)` that a patient carries at least one non-silent background mutation in a gene of `L` coding bases (Lawrence et al. 2014).","title":"Background mutation probability per patient"},{"id":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/lawrence-p1","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.SaturationPower.lawrenceP1","parent":47600,"result":true,"route":"../roadmap/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/index.html#lawrence-p1","scope":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","status":"fully_proved","summary":"Signal probability `p\u2081 = p\u2080 + r (1 - m)` for a gene mutated in a fraction `r` of patients above background, with miss rate `m` (Lawrence et al. 2014).","title":"Signal mutation probability per patient"},{"id":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/tendsto-saturation-power","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.SaturationPower.tendsto_saturationPower","parent":47600,"result":true,"route":"../roadmap/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/index.html#tendsto-saturation-power","scope":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","status":"fully_proved","summary":"For `0 \u2264 p\u2080 < p\u2081 \u2264 1` and any level `\u03b3 > 0`, the power of the binomial test tends to `1` as the number of patients grows.","title":"Power tends to one"},{"id":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/eventually-le-saturation-power","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.SaturationPower.eventually_le_saturationPower","parent":47600,"result":true,"route":"../roadmap/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/index.html#eventually-le-saturation-power","scope":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","status":"fully_proved","summary":"For every target power `\u03b2 < 1` there is a cohort size beyond which the power is at least `\u03b2` (Lawrence et al. 2014 take `\u03b2 = 0.9`).","title":"`N_\u03b2` exists"},{"id":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/lawrence-p0-mem","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.SaturationPower.lawrenceP0_mem","parent":47600,"result":true,"route":"../roadmap/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/index.html#lawrence-p0-mem","scope":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","status":"fully_proved","summary":"The background probability is a probability below one: `0 \u2264 p\u2080 < 1` for `0 \u2264 \u03bc f_g < 1` and `L \u2265 0`.","title":"Background mutation probability lies in (0, 1)"},{"id":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/lawrence-p0-lt-lawrence-p1","kind":"node","lean":"MathlibExt.Biology.CancerGenomics.SaturationPower.lawrenceP0_lt_lawrenceP1","parent":47600,"result":true,"route":"../roadmap/Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery/index.html#lawrence-p0-lt-lawrence-p1","scope":"Biology/cancer-genomics/saturation-power-for-cancer-gene-discovery","status":"fully_proved","summary":"A driver mutated above background raises the per-patient probability: `p\u2080 < p\u2080 + r (1 - m)` for `r > 0` and `m < 1`.","title":"Driver genes raise the per-patient mutation probability"},{"data_url":"scopes/Biology/population-dynamics.json","id":"Biology/population-dynamics","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/population-dynamics/index.html","scope":"Biology/population-dynamics","status":"fully_proved","summary":"Differential-equation models of population growth, species interaction and evolutionary change, such as logistic growth, predator\u2013prey systems, replicator dynamics and quasispecies.","title":"Population and evolutionary dynamics"},{"data_url":"scopes/Biology/population-dynamics/eigen-s-quasispecies-error-threshold.json","id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","kind":"scope","navigation":"module","parent":47608,"result":false,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","title":"Eigen's quasispecies error threshold"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/master-field","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.masterField","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#master-field","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"The master-sequence vector field of the single-peak quasispecies without back mutation: `x' = \u03c3 Q x - (\u03c3 x + (1 - x)) x`, where `\u03c3` is the master fitness relative to the mutants and `Q` the probability of an error-free copy. Meaningful for `\u03c3 > 1`, `0 \u2264 Q \u2264 1`, `0 \u2264 x \u2264 1`; it is a polynomial and is defined for all reals.","title":"Master-sequence vector field of the single-peak quasispecies"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/master-equilibrium","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.masterEquilibrium","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#master-equilibrium","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"The interior equilibrium `x* = (\u03c3 Q - 1) / (\u03c3 - 1)` of the master frequency. For `\u03c3 = 1` (no selective advantage) Lean's `x / 0 = 0` gives the junk value `0`; every result below assumes `1 < \u03c3`.","title":"Interior equilibrium of the master-sequence frequency"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/master-field-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.masterField_eq","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#master-field-eq","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"The master-sequence equation is logistic: `x' = x ((\u03c3 Q - 1) - (\u03c3 - 1) x)`.","title":"Master-sequence equation is logistic"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/master-field-eq-zero-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.masterField_eq_zero_iff","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#master-field-eq-zero-iff","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"The equilibria of the master frequency are `0` and `x* = (\u03c3 Q - 1)/(\u03c3 - 1)`.","title":"Equilibria of the master-sequence frequency"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/one-sub-master-equilibrium","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.one_sub_masterEquilibrium","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#one-sub-master-equilibrium","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"The mutant frequency at the interior equilibrium is `1 - x* = \u03c3 (1 - Q)/(\u03c3 - 1)`.","title":"Mutant frequency at the interior equilibrium"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/master-equilibrium-pos-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.masterEquilibrium_pos_iff","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#master-equilibrium-pos-iff","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"For `\u03c3 > 1`, the interior equilibrium is positive iff `\u03c3 Q > 1`.","title":"Error threshold"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/master-equilibrium-le-one","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.masterEquilibrium_le_one","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#master-equilibrium-le-one","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"For `\u03c3 > 1` and `Q \u2264 1` the interior equilibrium is at most `1`.","title":"Interior quasispecies equilibrium is at most 1"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/one-lt-mul-pow-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.one_lt_mul_pow_iff","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#one-lt-mul-pow-iff","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"For per-site fidelity `0 < q < 1` and `\u03c3 > 0`, `\u03c3 q ^ L > 1` iff `L < log \u03c3 / (-log q)`.","title":"Maximal genome length"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/tendsto-master-equilibrium","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.tendsto_masterEquilibrium","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#tendsto-master-equilibrium","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"The interior equilibrium depends continuously on `Q`, so any sequence of fidelities with `Q_L \u2192 Qlim` has `x*(Q_L) \u2192 x*(Qlim)`.","title":"Continuity of the interior equilibrium in the copying fidelity"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/tendsto-master-equilibrium-fidelity","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.tendsto_masterEquilibrium_fidelity","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#tendsto-master-equilibrium-fidelity","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"With per-site fidelity `q_L = 1 - \u03bc / L`, so that the expected number of mutations per genome copy is `\u03bc`, the error-free probability is `q_L ^ L \u2192 e^{-\u03bc}` and the master equilibrium converges to `(\u03c3 e^{-\u03bc} - 1)/(\u03c3 - 1)`.","title":"Fixed-`q_L` limit"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/master-equilibrium-exp-pos-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.masterEquilibrium_exp_pos_iff","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#master-equilibrium-exp-pos-iff","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"In the fixed-`q_L` limit the master sequence persists iff `\u03bc < log \u03c3`.","title":"Eigen's error threshold"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/tendsto-master-of-one-lt","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.tendsto_master_of_one_lt","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#tendsto-master-of-one-lt","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"If `\u03c3 > 1`, `\u03c3 Q > 1` and `x` solves the master equation on `[0, \u221e)` from a positive master frequency, then `x t \u2192 x*`.","title":"Convergence above the threshold"},{"id":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold/tendsto-master-zero-of-le-one","kind":"node","lean":"MathlibExt.Biology.Dynamics.Quasispecies.tendsto_master_zero_of_le_one","parent":47609,"result":true,"route":"../roadmap/Biology/population-dynamics/eigen-s-quasispecies-error-threshold/index.html#tendsto-master-zero-of-le-one","scope":"Biology/population-dynamics/eigen-s-quasispecies-error-threshold","status":"fully_proved","summary":"If `\u03c3 > 1`, `\u03c3 Q \u2264 1` and `x` solves the master equation on `[0, \u221e)` from a positive master frequency, then `x t \u2192 0`.","title":"Loss of the master at or below the threshold"},{"data_url":"scopes/Biology/population-dynamics/logistic-growth-equation.json","id":"Biology/population-dynamics/logistic-growth-equation","kind":"scope","navigation":"module","parent":47608,"result":false,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","title":"Logistic growth equation"},{"id":"Biology/population-dynamics/logistic-growth-equation/logistic","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.logistic","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#logistic","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"Explicit solution of the logistic equation `x' = r x (1 - x / K)` with `x 0 = x\u2080`: `K x\u2080 e^{r t} / (K + x\u2080 (e^{r t} - 1))`.","title":"Explicit solution of the logistic equation"},{"id":"Biology/population-dynamics/logistic-growth-equation/logistic-zero","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.logistic_zero","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#logistic-zero","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"The logistic solution starts at `x\u2080` (for `K \u2260 0`).","title":"Initial value of the logistic solution"},{"id":"Biology/population-dynamics/logistic-growth-equation/logistic-denom-pos","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.logistic_denom_pos","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#logistic-denom-pos","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"In the biological domain (`K > 0`, `x\u2080 > 0`, `r t \u2265 0`, or `x\u2080 \u2264 K`) the denominator of the logistic solution is positive.","title":"Positivity of the logistic solution's denominator"},{"id":"Biology/population-dynamics/logistic-growth-equation/has-deriv-at-logistic","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.hasDerivAt_logistic","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#has-deriv-at-logistic","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"Wherever the denominator `K + x\u2080 (e^{r t} - 1)` is nonzero, `x' = r x (1 - x / K)` (all real parameters, `K \u2260 0`).","title":"The explicit solution solves the logistic equation"},{"id":"Biology/population-dynamics/logistic-growth-equation/eq-on-logistic-of-has-deriv-within-at","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.eqOn_logistic_of_hasDerivWithinAt","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#eq-on-logistic-of-has-deriv-within-at","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"Uniqueness of the logistic solution on `[0, T]`: in the biological domain (`K > 0`, `x\u2080 > 0`, `r \u2265 0`), any function continuous on `[0, T]`, with right derivative `r x (1 - x / K)` on `[0, T)` and `x 0 = x\u2080`, equals the explicit solution on `[0, T]`.","title":"Uniqueness of the logistic solution"},{"id":"Biology/population-dynamics/logistic-growth-equation/tendsto-logistic-at-top","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.tendsto_logistic_atTop","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#tendsto-logistic-at-top","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"For `r > 0` and `x\u2080 > 0`, `x(t) \u2192 K` as `t \u2192 \u221e` (the identity needs no sign condition on `K`; biologically `K > 0`).","title":"Convergence to the carrying capacity"},{"id":"Biology/population-dynamics/logistic-growth-equation/tendsto-logistic-carrying-capacity-at-to","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.tendsto_logistic_carryingCapacity_atTop","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#tendsto-logistic-carrying-capacity-at-to","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"For fixed `r`, `x\u2080`, `t`, the logistic solution tends to the exponential-growth solution `x\u2080 e^{r t}` as the carrying capacity `K \u2192 \u221e`.","title":"Exponential limit"},{"id":"Biology/population-dynamics/logistic-growth-equation/logit-logistic","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.logit_logistic","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#logit-logistic","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"Log-odds grow linearly for a frequency (`K = 1`): for `0 < x\u2080 < 1` and all `r`, `t`, `log (x(t) / (1 - x(t))) = log (x\u2080 / (1 - x\u2080)) + r t`.","title":"Log-odds grow linearly"},{"id":"Biology/population-dynamics/logistic-growth-equation/logistic-transit-time","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.logistic_transitTime","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#logistic-transit-time","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"Starting from frequency `a \u2208 (0, 1)` with `r \u2260 0`, the logistic frequency reaches `1 - a` at time `(2 / r) log ((1 - a) / a)`.","title":"Exact deterministic transit time"},{"id":"Biology/population-dynamics/logistic-growth-equation/logistic-transit-time-two-n","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.logistic_transitTime_twoN","parent":47623,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logistic transit time from "],["math","\\(1/(2N)\\)"],["text"," to "],["math","\\(1 - 1/(2N)\\)"]],"source":"Logistic transit time from $1/(2N)$ to $1 - 1/(2N)$"},"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#logistic-transit-time-two-n","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"The transit time from `1 / (2N)` to `1 - 1 / (2N)` is `(2 / r) log (2N - 1)`, for `N \u2265 1` and `r \u2260 0`.","title":"Logistic transit time from $1/(2N)$ to $1 - 1/(2N)$"},{"id":"Biology/population-dynamics/logistic-growth-equation/strict-mono-logistic","kind":"node","lean":"MathlibExt.Biology.Dynamics.Logistic.strictMono_logistic","parent":47623,"result":true,"route":"../roadmap/Biology/population-dynamics/logistic-growth-equation/index.html#strict-mono-logistic","scope":"Biology/population-dynamics/logistic-growth-equation","status":"fully_proved","summary":"For `r > 0` and `0 < x\u2080 < K` the logistic solution is strictly increasing, so the transit time of `logistic_transitTime` is the unique time at which `1 - a` is reached.","title":"Logistic solution is strictly increasing"},{"data_url":"scopes/Biology/population-dynamics/lotka-volterra-predator-prey-model.json","id":"Biology/population-dynamics/lotka-volterra-predator-prey-model","kind":"scope","navigation":"module","parent":47608,"result":false,"route":"../roadmap/Biology/population-dynamics/lotka-volterra-predator-prey-model/index.html","scope":"Biology/population-dynamics/lotka-volterra-predator-prey-model","status":"fully_proved","title":"Lotka\u2013Volterra predator\u2013prey model"},{"id":"Biology/population-dynamics/lotka-volterra-predator-prey-model/first-integral","kind":"node","lean":"MathlibExt.Biology.Dynamics.LotkaVolterra.firstIntegral","parent":47635,"result":true,"route":"../roadmap/Biology/population-dynamics/lotka-volterra-predator-prey-model/index.html#first-integral","scope":"Biology/population-dynamics/lotka-volterra-predator-prey-model","status":"fully_proved","summary":"The Lotka\u2013Volterra first integral `V(x, y) = \u03b4 x - \u03b3 log x + \u03b2 y - \u03b1 log y` for the system `x' = \u03b1 x - \u03b2 x y`, `y' = \u03b4 x y - \u03b3 y`. It is meaningful on the open positive quadrant `x, y > 0`; elsewhere `Real.log` takes junk values (`log 0 = 0`, `log (-x) = log x`).","title":"First integral of the Lotka\u2013Volterra system"},{"id":"Biology/population-dynamics/lotka-volterra-predator-prey-model/has-deriv-at-first-integral","kind":"node","lean":"MathlibExt.Biology.Dynamics.LotkaVolterra.hasDerivAt_firstIntegral","parent":47635,"result":true,"route":"../roadmap/Biology/population-dynamics/lotka-volterra-predator-prey-model/index.html#has-deriv-at-first-integral","scope":"Biology/population-dynamics/lotka-volterra-predator-prey-model","status":"fully_proved","summary":"If `x' = \u03b1 x - \u03b2 x y` and `y' = \u03b4 x y - \u03b3 y` at time `t` and `x t, y t > 0`, then `t \u21a6 V(x t, y t)` has derivative `0` at `t`.","title":"Lotka\u2013Volterra first integral"},{"id":"Biology/population-dynamics/lotka-volterra-predator-prey-model/first-integral-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.LotkaVolterra.firstIntegral_eq","parent":47635,"result":true,"route":"../roadmap/Biology/population-dynamics/lotka-volterra-predator-prey-model/index.html#first-integral-eq","scope":"Biology/population-dynamics/lotka-volterra-predator-prey-model","status":"fully_proved","summary":"If `(x, y)` solves the Lotka\u2013Volterra system on `[0, \u221e)` and both populations stay positive there, then `V(x t, y t) = V(x 0, y 0)` for all `t \u2265 0`.","title":"Positive solutions lie on level curves"},{"id":"Biology/population-dynamics/lotka-volterra-predator-prey-model/first-integral-equilibrium-le","kind":"node","lean":"MathlibExt.Biology.Dynamics.LotkaVolterra.firstIntegral_equilibrium_le","parent":47635,"result":true,"route":"../roadmap/Biology/population-dynamics/lotka-volterra-predator-prey-model/index.html#first-integral-equilibrium-le","scope":"Biology/population-dynamics/lotka-volterra-predator-prey-model","status":"fully_proved","summary":"For positive parameters and `x, y > 0`, `V(\u03b3 / \u03b4, \u03b1 / \u03b2) \u2264 V(x, y)`.","title":"The coexistence equilibrium minimizes the first integral"},{"data_url":"scopes/Biology/population-dynamics/replicator-dynamics.json","id":"Biology/population-dynamics/replicator-dynamics","kind":"scope","navigation":"module","parent":47608,"result":false,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","title":"Replicator dynamics"},{"id":"Biology/population-dynamics/replicator-dynamics/mean-fitness","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.meanFitness","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#mean-fitness","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"The mean fitness `f\u0304(x) = \u2211 i, x i * f x i` of the population state `x` for the fitness function `f`.","title":"Mean fitness of a population state"},{"id":"Biology/population-dynamics/replicator-dynamics/is-solution","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.IsSolution","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#is-solution","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"`x` solves the replicator equation `x_i' = x_i (f_i(x) - f\u0304(x))` for all times `t \u2265 0`.","title":"Solution of the replicator equation"},{"id":"Biology/population-dynamics/replicator-dynamics/is-rest-point","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.IsRestPoint","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#is-rest-point","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"`x` is a rest point of the replicator equation: `x_i (f_i(x) - f\u0304(x)) = 0` for all `i`.","title":"Rest point of the replicator equation"},{"id":"Biology/population-dynamics/replicator-dynamics/is-rest-point-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.isRestPoint_iff","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#is-rest-point-iff","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"If every strategy is present (`x i > 0`), then `x` is a rest point iff every strategy has fitness equal to the mean fitness.","title":"Rest points in the interior are equalizers"},{"id":"Biology/population-dynamics/replicator-dynamics/has-deriv-at-sum","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.IsSolution.hasDerivAt_sum","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#has-deriv-at-sum","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"The total frequency `\u2211 x_i` evolves by `(\u2211 x_i)' = f\u0304(x) (1 - \u2211 x_i)`.","title":"Evolution of the total frequency under replicator dynamics"},{"id":"Biology/population-dynamics/replicator-dynamics/sum-eq-one","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.IsSolution.sum_eq_one","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#sum-eq-one","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"For continuous fitness `f`, a solution with `\u2211 x_i(0) = 1` satisfies `\u2211 x_i(t) = 1` for all `t \u2265 0`.","title":"Invariance of the simplex hyperplane"},{"id":"Biology/population-dynamics/replicator-dynamics/eq-zero-of-eq-zero","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.IsSolution.eq_zero_of_eq_zero","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#eq-zero-of-eq-zero","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"For continuous fitness `f`, a strategy absent at time `0` (`x_i(0) = 0`) stays absent for all `t \u2265 0`.","title":"Invariance of the faces of the simplex"},{"id":"Biology/population-dynamics/replicator-dynamics/pos-of-pos","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.IsSolution.pos_of_pos","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#pos-of-pos","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"For continuous fitness `f`, a strategy with positive frequency at time `0` has positive frequency for all `t \u2265 0`.","title":"Present strategies stay present"},{"id":"Biology/population-dynamics/replicator-dynamics/mem-simplex","kind":"node","lean":"MathlibExt.Biology.Dynamics.Replicator.IsSolution.mem_simplex","parent":47640,"result":true,"route":"../roadmap/Biology/population-dynamics/replicator-dynamics/index.html#mem-simplex","scope":"Biology/population-dynamics/replicator-dynamics","status":"fully_proved","summary":"For continuous fitness `f`, a solution starting in the standard simplex (`x i \u2265 0` for all `i` and `\u2211 i, x i = 1`) stays in it for all `t \u2265 0`.","title":"Invariance of the simplex (Hofbauer\u2013Sigmund 1998, \u00a77.1)"},{"data_url":"scopes/Biology/pattern-formation.json","id":"Biology/pattern-formation","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/pattern-formation/index.html","scope":"Biology/pattern-formation","status":"can_prove","summary":"Reaction\u2013diffusion models of biological spread and spatial patterning, including travelling waves, morphogen gradients and Turing instabilities.","title":"Reaction\u2013diffusion and pattern formation"},{"data_url":"scopes/Biology/pattern-formation/fisher-kpp-travelling-waves.json","id":"Biology/pattern-formation/fisher-kpp-travelling-waves","kind":"scope","navigation":"module","parent":47650,"result":false,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"can_prove","title":"Fisher\u2013KPP travelling waves"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/is-wave-profile","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.IsWaveProfile","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#is-wave-profile","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"`U` is a traveling-wave profile of speed `c` for `\u2202u/\u2202t = D \u2202\u00b2u/\u2202x\u00b2 + r u (1 - u)`: it is twice differentiable with first derivative `U'` and `U'' = -(c U' + r U (1 - U)) / D`, which for `D \u2260 0` is the profile equation `D U'' + c U' + r U (1 - U) = 0`.","title":"Traveling-wave profile of the Fisher\u2013KPP equation"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/is-front","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.IsFront","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#is-front","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"A traveling front of speed `c`: a wave profile with `0 < U < 1`, `U \u2192 1` at `-\u221e` and `U \u2192 0` at `+\u221e`.","title":"Traveling front of the Fisher\u2013KPP equation"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/has-deriv-at-wave-t","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.hasDerivAt_wave_t","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#has-deriv-at-wave-t","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"Time derivative of the wave `u(x, t) = U(x - c t)`: `\u2202u/\u2202t = -c U'(x - c t)`.","title":"Time derivative of a traveling wave"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/has-deriv-at-wave-x","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.hasDerivAt_wave_x","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#has-deriv-at-wave-x","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"Space derivative of the wave: `\u2202u/\u2202x = U'(x - c t)`.","title":"Space derivative of a traveling wave"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/has-deriv-at-wave-xx","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.hasDerivAt_wave_xx","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#has-deriv-at-wave-xx","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"Second space derivative of the wave; with `hD : D \u2260 0` it gives the PDE `-c U' = D U'' + r U (1 - U)`, i.e. `\u2202u/\u2202t = D \u2202\u00b2u/\u2202x\u00b2 + r u (1 - u)`.","title":"Traveling wave solves the Fisher\u2013KPP equation"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/not-eventually-pos-of-sq-lt","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.not_eventually_pos_of_sq_lt","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#not-eventually-pos-of-sq-lt","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"If `D, r > 0` and `c\u00b2 < 4 r D`, a wave profile tending to `0` at `+\u221e` is not eventually positive. (This excludes `|c| < 2 \u221a(r D)`, not `c < 2 \u221a(r D)` for negative `c`.)","title":"No traveling wave with `c\u00b2 < 4 r D`"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/not-is-front-of-sq-lt","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.not_isFront_of_sq_lt","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#not-is-front-of-sq-lt","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"A front of speed `c` needs `c\u00b2 \u2265 4 r D`; for `c \u2265 0` this is `c \u2265 2 \u221a(r D)`.","title":"Minimal speed of Fisher\u2013KPP traveling fronts"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/az-rate","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.azRate","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#az-rate","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"The rate `\u03ba = \u221a(r / (6 D))` of the Ablowitz\u2013Zeppetella wave.","title":"Rate of the Ablowitz\u2013Zeppetella wave"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/az-speed","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.azSpeed","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#az-speed","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"The Ablowitz\u2013Zeppetella speed `c = 5 \u221a(r D / 6)` (`5 / \u221a6` when `D = r = 1`).","title":"Speed of the Ablowitz\u2013Zeppetella wave"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/az-wave","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.azWave","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#az-wave","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"The Ablowitz\u2013Zeppetella profile `U(z) = (1 + C exp (\u03ba z))\u207b\u00b2`.","title":"Ablowitz\u2013Zeppetella wave profile"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/az-wave-deriv","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.azWaveDeriv","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#az-wave-deriv","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"The derivative of the Ablowitz\u2013Zeppetella profile, `-2 \u03ba C e^{\u03ba z} (1 + C e^{\u03ba z})\u207b\u00b3`.","title":"Derivative of the Ablowitz\u2013Zeppetella profile"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/is-front-az-wave","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.isFront_azWave","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#is-front-az-wave","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"For `D, r, C > 0`, `(1 + C exp (z \u221a(r / (6 D))))\u207b\u00b2` is a front of speed `5 \u221a(r D / 6)`.","title":"The Ablowitz\u2013Zeppetella front"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/two-mul-sqrt-lt-az-speed","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.two_mul_sqrt_lt_azSpeed","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#two-mul-sqrt-lt-az-speed","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"fully_proved","summary":"The Ablowitz\u2013Zeppetella speed exceeds the minimal speed: `2 \u221a(r D) < 5 \u221a(r D / 6)`.","title":"Ablowitz\u2013Zeppetella speed exceeds the minimal front speed"},{"id":"Biology/pattern-formation/fisher-kpp-travelling-waves/exists-is-front-of-le","kind":"node","lean":"MathlibExt.Biology.Dynamics.FisherKPP.exists_isFront_of_le","parent":47651,"result":true,"route":"../roadmap/Biology/pattern-formation/fisher-kpp-travelling-waves/index.html#exists-is-front-of-le","scope":"Biology/pattern-formation/fisher-kpp-travelling-waves","status":"can_prove","summary":"Kolmogorov\u2013Petrovsky\u2013Piskunov: fronts exist at every speed `c \u2265 2 \u221a(r D)`, and they are strictly decreasing.","title":"Kolmogorov\u2013Petrovsky\u2013Piskunov: fronts exist at every speed `c \u2265 2 \u221a(r D)`"},{"data_url":"scopes/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model.json","id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","kind":"scope","navigation":"module","parent":47650,"result":false,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","title":"Morphogen gradients and the French flag model"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/decay-length","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.decayLength","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#decay-length","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"The decay length `\u03bb = \u221a(D / k)` of a morphogen with diffusion coefficient `D` and linear degradation rate `k` (both positive in the model).","title":"Decay length of a morphogen gradient"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/profile","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.profile","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#profile","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"The steady-state gradient `c(x) = (j / \u221a(D k)) e^{-x/\u03bb}`, `\u03bb = \u221a(D / k)`, for source flux `j` at `x = 0`. Meaningful for `D, k > 0`.","title":"Steady-state exponential morphogen profile"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/threshold-position","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.thresholdPosition","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#threshold-position","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"The French-flag boundary for threshold `\u03b8`: `x_\u03b8 = \u03bb log(c(0) / \u03b8)`, where `c(0) = j / \u221a(D k)`. Meaningful for `D, k, j, \u03b8 > 0`; it is negative when `\u03b8 > c(0)` (no cell is above threshold).","title":"French-flag boundary position for a threshold"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/is-steady-state","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.IsSteadyState","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#is-steady-state","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"A bounded steady state of the synthesis\u2013diffusion\u2013degradation model on `[0, \u221e)`, with `c' = dc/dx`: `D c'' = k c` for `x > 0`, `c` and `c'` continuous up to the source, source flux `-D c'(0) = j`, and `c` bounded on `[0, \u221e)`.","title":"Bounded steady state of the synthesis\u2013diffusion\u2013degradation model"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/decay-length-pos","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.decayLength_pos","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#decay-length-pos","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"The decay length is positive for positive `D` and `k`.","title":"Positivity of the decay length"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/div-decay-length-sq","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.div_decayLength_sq","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#div-decay-length-sq","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"`D / \u03bb\u00b2 = k`.","title":"Diffusion coefficient over squared decay length equals degradation rate"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/sqrt-mul-eq","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.sqrt_mul_eq","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#sqrt-mul-eq","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"`\u221a(D k) = D / \u03bb`, so the source amplitude is `c(0) = j \u03bb / D`.","title":"Source amplitude in terms of the decay length"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/has-deriv-at-profile","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.hasDerivAt_profile","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#has-deriv-at-profile","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"The profile has slope `c'(x) = -c(x)/\u03bb`.","title":"Slope of the morphogen profile"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/is-steady-state-profile","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.isSteadyState_profile","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#is-steady-state-profile","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"The exponential profile is a bounded steady state with source flux `j`.","title":"Exponential profile is a bounded steady state"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/eq-profile-of-is-steady-state","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.eq_profile_of_isSteadyState","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#eq-profile-of-is-steady-state","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"Every bounded steady state with source flux `j` is the exponential profile on `[0, \u221e)`.","title":"Uniqueness"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/profile-strict-anti","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.profile_strictAnti","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#profile-strict-anti","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"The profile is strictly decreasing in `x` for a positive source.","title":"Morphogen profile is strictly decreasing"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/profile-threshold-position","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.profile_thresholdPosition","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#profile-threshold-position","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"The profile equals `\u03b8` at the boundary `x_\u03b8`.","title":"Profile equals the threshold at the French-flag boundary"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/profile-eq-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.profile_eq_iff","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#profile-eq-iff","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"For positive parameters and threshold `\u03b8`, `c(x) = \u03b8` exactly at `x = x_\u03b8 = \u03bb log(c(0)/\u03b8)`.","title":"French-flag boundary"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/le-profile-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.le_profile_iff","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#le-profile-iff","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"A cell at `x` is at or above threshold `\u03b8` iff it lies within the boundary `x_\u03b8`: `\u03b8 \u2264 c(x) \u2194 x \u2264 x_\u03b8`.","title":"French-flag regions"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/threshold-position-mul","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.thresholdPosition_mul","parent":47666,"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#threshold-position-mul","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"Scaling the source flux by `a > 0` moves every French-flag boundary by `\u03bb log a`.","title":"Boundary shift"},{"id":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/threshold-position-two-mul","kind":"node","lean":"MathlibExt.Biology.Dynamics.MorphogenGradient.thresholdPosition_two_mul","parent":47666,"reading_title":{"format":"inline-parts-v1","parts":[["text","Doubling the source flux shifts boundaries by "],["math","\\(\\lambda \\log 2\\)"]],"source":"Doubling the source flux shifts boundaries by $\\lambda \\log 2$"},"result":true,"route":"../roadmap/Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model/index.html#threshold-position-two-mul","scope":"Biology/pattern-formation/morphogen-gradients-and-the-french-flag-model","status":"fully_proved","summary":"Doubling the source flux shifts every French-flag boundary away from the source by exactly `\u03bb log 2`.","title":"Doubling the source flux shifts boundaries by $\\lambda \\log 2$"},{"data_url":"scopes/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability.json","id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","kind":"scope","navigation":"module","parent":47650,"result":false,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","title":"Turing patterns and diffusion-driven instability"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/diffusion-matrix","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.diffusionMatrix","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#diffusion-matrix","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The diffusion matrix `D = diag(d_u, d_v)` of a two-species reaction-diffusion system.","title":"Diffusion matrix of a two-species system"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/mode-matrix","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.modeMatrix","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#mode-matrix","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The mode matrix `J - k\u00b2 D` governing the Fourier mode of wavenumber `k` of the linearized system `w_t = D w_xx + J w`, where `J` is the Jacobian of the kinetics at the steady state (convention: `J i j` is the partial derivative of the `i`-th reaction term with respect to the `j`-th species, so `J = [[f_u, f_v], [g_u, g_v]]`).","title":"Mode matrix of a linearized reaction-diffusion system"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/is-unstable-mode","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.IsUnstableMode","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#is-unstable-mode","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Fourier mode of wavenumber `k` is *linearly unstable*: the mode matrix `J - k\u00b2 D` has a complex eigenvalue `\u03bc` with `0 < Re \u03bc`.","title":"Linearly unstable Fourier mode"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/fourier-mode","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.fourierMode","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#fourier-mode","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The complex Fourier mode `w(t, x) = e^{\u03bc t} cos(k x) v` with growth rate `\u03bc`, wavenumber `k` and amplitude vector `v`. (The real part of a complex solution of the real linear system is a real solution.)","title":"Complex Fourier mode"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/is-linearized-solution","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.IsLinearizedSolution","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#is-linearized-solution","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"`w` is a classical solution of the linearized reaction-diffusion system `w_t = D w_xx + J w` on `\u211d \u00d7 \u211d`: `w(t, \u00b7)` is twice differentiable with first and second derivatives `wx`, `wxx`, and `w(\u00b7, x)` is differentiable with the prescribed time derivative.","title":"Classical solution of the linearized reaction-diffusion system"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/dispersion-det","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.dispersionDet","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#dispersion-det","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The dispersion polynomial `h(q) = d_u d_v q\u00b2 - (d_v f_u + d_u g_v) q + det J`, so that `det (J - k\u00b2 D) = h(k\u00b2)` (`det_modeMatrix`).","title":"Dispersion polynomial"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/critical-wavenumber-sq","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.criticalWavenumberSq","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#critical-wavenumber-sq","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The squared critical wavenumber `k_c\u00b2 = (d_v f_u + d_u g_v) / (2 d_u d_v)`, the minimizer of the dispersion polynomial (meaningful for `d_u, d_v > 0`; division by zero gives `0`).","title":"Squared critical wavenumber"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/band-discriminant","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.bandDiscriminant","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#band-discriminant","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The discriminant `(d_v f_u + d_u g_v)\u00b2 - 4 d_u d_v det J` of the dispersion polynomial.","title":"Discriminant of the dispersion polynomial"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/band-lower","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.bandLower","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#band-lower","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Lower end `q\u208b = (S - \u221a\u0394) / (2 d_u d_v)` of the unstable band of squared wavenumbers, where `S = d_v f_u + d_u g_v` and `\u0394 = bandDiscriminant` (meaningful when `0 < \u0394`, `d_u, d_v > 0`).","title":"Lower end of the unstable wavenumber band"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/band-upper","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.bandUpper","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#band-upper","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Upper end `q\u208a = (S + \u221a\u0394) / (2 d_u d_v)` of the unstable band of squared wavenumbers (meaningful when `0 < \u0394`, `d_u, d_v > 0`).","title":"Upper end of the unstable wavenumber band"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/neumann-wavenumber","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.neumannWavenumber","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#neumann-wavenumber","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Neumann wavenumber `k_n = n \u03c0 / L` of the mode `cos(n \u03c0 x / L)` on `[0, L]`. Meaningful only for `0 < L`; at `L = 0` the totalized division gives `0`, so every labeled Neumann mode collapses to the homogeneous mode.","title":"Neumann wavenumber on an interval"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/trace-mode-matrix","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.trace_modeMatrix","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#trace-mode-matrix","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"`tr (J - k\u00b2 D) = tr J - k\u00b2 (d_u + d_v)`.","title":"Trace of the mode matrix"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/det-mode-matrix","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.det_modeMatrix","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#det-mode-matrix","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"`det (J - k\u00b2 D) = h(k\u00b2)`.","title":"Determinant of the mode matrix is the dispersion polynomial"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/exists-eigenvalue-re-pos-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.exists_eigenvalue_re_pos_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#exists-eigenvalue-re-pos-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"A real `2 \u00d7 2` matrix has a complex eigenvalue with positive real part iff its trace is positive or its determinant is negative.","title":"Routh\u2013Hurwitz for `2 \u00d7 2` matrices"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/is-unstable-mode-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.isUnstableMode_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#is-unstable-mode-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Mode `k` is unstable iff `0 < tr (J - k\u00b2 D)` or `det (J - k\u00b2 D) < 0`.","title":"Instability criterion for a Fourier mode via trace and determinant"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/fourier-mode-is-linearized-solution-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.fourierMode_isLinearizedSolution_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#fourier-mode-is-linearized-solution-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Fourier mode `e^{\u03bc t} cos(k x) v` is a classical solution of the linearized system `w_t = D w_xx + J w` iff `(J - k\u00b2 D) v = \u03bc v`.","title":"Dispersion relation"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/norm-fourier-mode-zero","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.norm_fourierMode_zero","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#norm-fourier-mode-zero","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The amplitude of a Fourier mode: `\u2016w(t, 0)\u2016 = e^{Re \u03bc \u00b7 t} \u2016v\u2016`.","title":"Amplitude of a Fourier mode"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/norm-fourier-mode-le","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.norm_fourierMode_le","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#norm-fourier-mode-le","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"`\u2016w(t, 0)\u2016` is the amplitude of the mode: `\u2016w(t, x)\u2016 \u2264 \u2016w(t, 0)\u2016` for every `x`.","title":"Fourier mode is largest at the origin"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/tendsto-norm-fourier-mode-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.tendsto_norm_fourierMode_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#tendsto-norm-fourier-mode-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"A nonzero Fourier mode grows without bound iff its growth rate has positive real part.","title":"Fourier mode grows unboundedly iff its growth rate is positive"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/is-unstable-mode-iff-exists-growing","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.isUnstableMode_iff_exists_growing","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#is-unstable-mode-iff-exists-growing","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"A Fourier mode grows iff `J - k\u00b2 D` has an eigenvalue with positive real part.","title":"A Fourier mode grows iff `J - k\u00b2 D` has an eigenvalue with positive real part"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/not-is-unstable-mode-zero","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.not_isUnstableMode_zero","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#not-is-unstable-mode-zero","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The homogeneous mode `k = 0` is stable when the kinetics are stable (`tr J < 0`, `0 < det J`).","title":"Homogeneous mode is stable for stable kinetics"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/is-unstable-mode-iff-dispersion-det-neg","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.isUnstableMode_iff_dispersionDet_neg","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#is-unstable-mode-iff-dispersion-det-neg","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"If `tr J < 0` and `d_u, d_v \u2265 0`, mode `k` is unstable iff `h(k\u00b2) < 0`.","title":"Mode instability iff the dispersion polynomial is negative"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/turing-instability","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.turing_instability","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#turing-instability","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"If the steady state is stable without diffusion (`tr J < 0`, `0 < det J`), `d_u, d_v \u2265 0`, and `det (J - k\u00b2 D) < 0`, then the homogeneous mode is stable while mode `k` is unstable.","title":"Diffusion-driven instability (Turing 1952)"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/dispersion-det-critical-wavenumber-sq","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.dispersionDet_criticalWavenumberSq","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#dispersion-det-critical-wavenumber-sq","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The value of the dispersion polynomial at its minimizer: `h(k_c\u00b2) = det J - (d_v f_u + d_u g_v)\u00b2 / (4 d_u d_v)`.","title":"Minimum value of the dispersion polynomial"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/dispersion-det-critical-wavenumber-sq-le","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.dispersionDet_criticalWavenumberSq_le","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#dispersion-det-critical-wavenumber-sq-le","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The critical squared wavenumber minimizes the dispersion polynomial over all real `q`.","title":"Critical squared wavenumber minimizes the dispersion polynomial"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/two-mul-sqrt-lt-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.two_mul_sqrt_lt_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#two-mul-sqrt-lt-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Turing condition in squared form: `2 \u221a(d_u d_v det J) < S` iff `0 < S` and `4 d_u d_v det J < S\u00b2`, where `S = d_v f_u + d_u g_v`. This holds for all real inputs: when `d_u d_v det J < 0`, `Real.sqrt` returns `0` and both sides reduce to `0 < S`.","title":"Turing condition in squared form"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/exists-dispersion-det-neg-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.exists_dispersionDet_neg_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#exists-dispersion-det-neg-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"For `d_u, d_v > 0` and `0 < det J`, the dispersion polynomial is negative at some `q \u2265 0` iff `2 \u221a(d_u d_v det J) < d_v f_u + d_u g_v`.","title":"Turing's criterion for diffusion-driven instability"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/exists-is-unstable-mode-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.exists_isUnstableMode_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#exists-is-unstable-mode-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"If the steady state is stable without diffusion (`tr J < 0`, `0 < det J`) and `d_u, d_v > 0`, some Fourier mode is unstable iff `d_v f_u + d_u g_v > 2 \u221a(d_u d_v det J)`.","title":"Turing's criterion"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/is-unstable-mode-sqrt-critical-wavenumbe","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.isUnstableMode_sqrt_criticalWavenumberSq","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#is-unstable-mode-sqrt-critical-wavenumbe","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Under Turing's criterion the critical wavenumber `k_c = \u221a(k_c\u00b2)` is itself unstable (and `0 < k_c\u00b2`).","title":"Critical wavenumber is unstable under Turing's criterion"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/critical-wavenumber-sq-of-threshold","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.criticalWavenumberSq_of_threshold","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#critical-wavenumber-sq-of-threshold","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"At the bifurcation threshold `d_v f_u + d_u g_v = 2 \u221a(d_u d_v det J)` the critical squared wavenumber is `k_c\u00b2 = \u221a(det J / (d_u d_v))` and `h(k_c\u00b2) = 0`.","title":"Critical wavenumber at the Turing bifurcation threshold"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/mul-neg-of-exists-is-unstable-mode","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.mul_neg_of_exists_isUnstableMode","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#mul-neg-of-exists-is-unstable-mode","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"If the steady state is stable without diffusion and `d_u, d_v \u2265 0`, diffusion-driven instability forces `f_u g_v < 0`.","title":"`f_u` and `g_v` have opposite signs"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/lt-of-exists-is-unstable-mode","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.lt_of_exists_isUnstableMode","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#lt-of-exists-is-unstable-mode","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"In the activator-inhibitor sign pattern (`u` is the activator, `0 < f_u`; then `g_v < 0` follows from `tr J < 0`), diffusion-driven instability of a kinetically stable state forces `d_u < d_v`.","title":"The inhibitor diffuses faster"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/dispersion-det-neg-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.dispersionDet_neg_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#dispersion-det-neg-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"For `d_u, d_v > 0`, the dispersion polynomial is negative exactly on the open band `(q\u208b, q\u208a)`, which is nonempty iff the discriminant is positive.","title":"Dispersion polynomial is negative exactly on the unstable band"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/has-deriv-at-fourier-mode-neumann-wavenu","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.hasDerivAt_fourierMode_neumannWavenumber","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#has-deriv-at-fourier-mode-neumann-wavenu","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Neumann mode `cos(k_n x)` has zero flux at both ends of `[0, L]`: the `x`-derivative of the Fourier mode with wavenumber `k_n = n \u03c0 / L` vanishes at `x = 0` and `x = L`.","title":"Neumann modes have zero flux at the boundary"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/is-unstable-mode-neumann-wavenumber-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.isUnstableMode_neumannWavenumber_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#is-unstable-mode-neumann-wavenumber-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"On `[0, L]` with Neumann boundary conditions, if `tr J < 0` and `d_u, d_v > 0`, mode `n` is unstable iff `k_n\u00b2 = (n \u03c0 / L)\u00b2` lies in the unstable band `(q\u208b, q\u208a)` (which requires a positive discriminant).","title":"Finite domain"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/finite-set-of-is-unstable-mode-neumann-w","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.finite_setOf_isUnstableMode_neumannWavenumber","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#finite-set-of-is-unstable-mode-neumann-w","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Only finitely many Neumann modes are unstable on `[0, L]` (for `d_u, d_v, L > 0`, any `J`).","title":"Only finitely many Neumann modes are unstable"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/not-is-unstable-mode-neumann-wavenumber","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.not_isUnstableMode_neumannWavenumber_of_le","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#not-is-unstable-mode-neumann-wavenumber","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"If the first nonhomogeneous mode already lies beyond the band, `q\u208a \u2264 (\u03c0 / L)\u00b2`, then no Neumann mode is unstable on `[0, L]`.","title":"Small domains admit no pattern"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/exists-neumann-mem-band","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.exists_neumann_mem_band","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#exists-neumann-mem-band","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Every long enough interval has a squared Neumann wavenumber inside a given nonempty open band `(qlo, qhi)` with `0 < qhi`: there is `L\u2080 > 0` such that for all `L \u2265 L\u2080` some `(n \u03c0 / L)\u00b2` lies in `(qlo, qhi)`.","title":"Long intervals have a squared Neumann wavenumber in any given band"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/exists-is-unstable-mode-neumann-wavenumb","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.exists_isUnstableMode_neumannWavenumber","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#exists-is-unstable-mode-neumann-wavenumb","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Under Turing's criterion, every long enough interval `[0, L]` has an unstable Neumann mode.","title":"Large domains pattern"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/schnakenberg","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.schnakenberg","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#schnakenberg","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Schnakenberg's reaction terms `(f, g) = (a - u + u\u00b2 v, b - u\u00b2 v)` as a map on `x = (u, v) : Fin 2 \u2192 \u211d`, with source rates `a, b` (biologically `a, b > 0`).","title":"Schnakenberg reaction kinetics"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/schnakenberg-steady-state","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.schnakenbergSteadyState","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#schnakenberg-steady-state","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The homogeneous steady state `(u\u2080, v\u2080) = (a + b, b / (a + b)\u00b2)` of the Schnakenberg kinetics (meaningful for `a + b \u2260 0`; at `a + b = 0` the division gives the junk `v\u2080 = 0`).","title":"Steady state of the Schnakenberg kinetics"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/schnakenberg-jacobian","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.schnakenbergJacobian","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#schnakenberg-jacobian","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Jacobian of the Schnakenberg kinetics at its steady state, `[[(b - a) / (a + b), (a + b)\u00b2], [-2 b / (a + b), -(a + b)\u00b2]]`.","title":"Jacobian of the Schnakenberg kinetics at its steady state"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/schnakenberg-steady-state-2","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.schnakenberg_steadyState","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#schnakenberg-steady-state-2","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"`(a + b, b / (a + b)\u00b2)` is a steady state of the Schnakenberg kinetics when `a + b \u2260 0`.","title":"Schnakenberg steady state is an equilibrium"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/jacobian-matrix-schnakenberg","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.jacobianMatrix_schnakenberg","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#jacobian-matrix-schnakenberg","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Jacobian matrix of the Schnakenberg kinetics at the steady state is `schnakenbergJacobian a b`.","title":"Computation of the Schnakenberg Jacobian at the steady state"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/det-schnakenberg-jacobian","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.det_schnakenbergJacobian","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#det-schnakenberg-jacobian","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"`det J = (a + b)\u00b2` for the Schnakenberg Jacobian (`a + b \u2260 0`).","title":"Determinant of the Schnakenberg Jacobian"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/trace-schnakenberg-jacobian-neg-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.trace_schnakenbergJacobian_neg_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#trace-schnakenberg-jacobian-neg-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Schnakenberg steady state is stable without diffusion iff `b - a < (a + b)\u00b3` (for `0 < a + b`; then `0 < det J` automatically).","title":"Stability of the Schnakenberg steady state without diffusion"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/schnakenberg-turing-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.schnakenberg_turing_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#schnakenberg-turing-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Turing instability of the Schnakenberg kinetics (Murray 2003, ch. 2). With `d_u = 1`, `d_v = d > 0`, `0 < a + b` and `b - a < (a + b)\u00b3` (stable without diffusion), some Fourier mode is unstable iff `2 \u221ad (a + b)\u00b2 < d (b - a) - (a + b)\u00b3`.","title":"Turing instability of the Schnakenberg kinetics"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/gierer-meinhardt","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.giererMeinhardt","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#gierer-meinhardt","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"Gierer\u2013Meinhardt activator-inhibitor kinetics in Murray's nondimensional form `(f, g) = (a - b u + u\u00b2 / v, u\u00b2 - v)` on `x = (u, v)` (biologically `a, b > 0` and `u, v > 0`; at `v = 0` the division gives the junk `u\u00b2 / 0 = 0`).","title":"Gierer\u2013Meinhardt activator-inhibitor kinetics"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/gierer-meinhardt-steady-state","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.giererMeinhardtSteadyState","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#gierer-meinhardt-steady-state","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The homogeneous steady state `(u\u2080, v\u2080) = ((a + 1) / b, ((a + 1) / b)\u00b2)` of the Gierer\u2013Meinhardt kinetics (meaningful for `b \u2260 0`, `a + 1 \u2260 0`).","title":"Steady state of the Gierer\u2013Meinhardt kinetics"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/gierer-meinhardt-jacobian","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.giererMeinhardtJacobian","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#gierer-meinhardt-jacobian","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Jacobian of the Gierer\u2013Meinhardt kinetics at its steady state, `[[b (1 - a) / (1 + a), -(b / (a + 1))\u00b2], [2 (a + 1) / b, -1]]`.","title":"Jacobian of the Gierer\u2013Meinhardt kinetics at its steady state"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/gierer-meinhardt-steady-state-2","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.giererMeinhardt_steadyState","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#gierer-meinhardt-steady-state-2","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"`((a + 1) / b, ((a + 1) / b)\u00b2)` is a steady state of the Gierer\u2013Meinhardt kinetics when `a + 1 \u2260 0` and `b \u2260 0`.","title":"Gierer\u2013Meinhardt steady state is an equilibrium"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/jacobian-matrix-gierer-meinhardt","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.jacobianMatrix_giererMeinhardt","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#jacobian-matrix-gierer-meinhardt","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Jacobian matrix of the Gierer\u2013Meinhardt kinetics at the steady state is `giererMeinhardtJacobian a b`.","title":"Computation of the Gierer\u2013Meinhardt Jacobian at the steady state"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/det-gierer-meinhardt-jacobian","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.det_giererMeinhardtJacobian","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#det-gierer-meinhardt-jacobian","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"`det J = b` for the Gierer\u2013Meinhardt Jacobian (`a + 1 \u2260 0`, `b \u2260 0`).","title":"Determinant of the Gierer\u2013Meinhardt Jacobian"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/trace-gierer-meinhardt-jacobian-neg-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.trace_giererMeinhardtJacobian_neg_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#trace-gierer-meinhardt-jacobian-neg-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"The Gierer\u2013Meinhardt steady state is stable without diffusion iff `b (1 - a) < 1 + a` (for `0 < a + 1`; then `0 < det J = b` for `0 < b`).","title":"Stability of the Gierer\u2013Meinhardt steady state without diffusion"},{"id":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/gierer-meinhardt-turing-iff","kind":"node","lean":"MathlibExt.Biology.Dynamics.TuringPatterns.giererMeinhardt_turing_iff","parent":47683,"result":true,"route":"../roadmap/Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability/index.html#gierer-meinhardt-turing-iff","scope":"Biology/pattern-formation/turing-patterns-and-diffusion-driven-instability","status":"fully_proved","summary":"With `d_u = 1`, `d_v = d > 0`, `0 < a + 1`, `0 < b` and `b (1 - a) < 1 + a` (stable without diffusion), some Fourier mode is unstable iff `2 \u221a(d b) < d b (1 - a) / (1 + a) - 1`.","title":"Turing instability of the Gierer\u2013Meinhardt kinetics"},{"data_url":"scopes/Biology/epidemiology.json","id":"Biology/epidemiology","kind":"scope","parent":44674,"result":false,"route":"../roadmap/Biology/epidemiology/index.html","scope":"Biology/epidemiology","status":"fully_proved","summary":"Compartmental, renewal and branching-process models of infectious disease transmission and their reproduction numbers.","title":"Mathematical epidemiology"},{"data_url":"scopes/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number.json","id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","kind":"scope","navigation":"module","parent":47739,"result":false,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","title":"Euler\u2013Lotka equation and the reproduction number"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/reproduction-number","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.reproductionNumber","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#reproduction-number","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"The reproductive number implied by growth rate `r` and generation-interval law `g`: `R = 1 / M_g(-r)` (Wallinga\u2013Lipsitch 2007, eq. 2.9).","title":"Reproductive number from growth rate and generation interval"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/convex-order-le","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.ConvexOrderLE","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#convex-order-le","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"`g\u2081 \u2264cx g\u2082` if `\u222b \u03c6 dg\u2081 \u2264 \u222b \u03c6 dg\u2082` for every convex `\u03c6 : \u211d \u2192 \u211d` that is integrable under both laws (Shaked\u2013Shanthikumar 2007, \u00a73.A).","title":"Convex order"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/renewal-exp-iff","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.renewal_exp_iff","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#renewal-exp-iff","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"For `c \u2260 0`, the exponential `b(t) = c e^{r t}` solves `b(t) = R \u222b b(t - a) g(da)` for every `t` iff `R M_g(-r) = 1`. This is meaningful when `a \u21a6 e^{-r a}` is `g`-integrable; otherwise Lean's integral and `M_g(-r)` are both `0` and both sides of the equivalence are false.","title":"Renewal equation for exponential growth"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/renewal-exp-iff-reproduction-number","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.renewal_exp_iff_reproductionNumber","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#renewal-exp-iff-reproduction-number","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"Under the hypotheses of `renewal_exp_iff`, exponential growth at rate `r` solves the renewal equation iff `R = 1 / M_g(-r)`.","title":"Euler\u2013Lotka"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/reproduction-number-pos","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.reproductionNumber_pos","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#reproduction-number-pos","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"The reproductive number is positive when `M_g(-r)` is finite.","title":"Positivity of the reproductive number"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/reproduction-number-dirac","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.reproductionNumber_dirac","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#reproduction-number-dirac","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"Fixed generation interval `T`: `R = e^{r T}` (for all real `r`, `T`).","title":"Fixed generation interval"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/reproduction-number-gamma-measure","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.reproductionNumber_gammaMeasure","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#reproduction-number-gamma-measure","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"Gamma generation interval with shape `\u03b1 > 0` and rate `\u03b2 > 0`: for `r > -\u03b2`, `R = (1 + r / \u03b2)^\u03b1`.","title":"Gamma generation interval"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/reproduction-number-exp-measure","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.reproductionNumber_expMeasure","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#reproduction-number-exp-measure","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"Exponential generation interval with mean `T > 0` (rate `1 / T`): for `r T > -1`, `R = 1 + r T` (Wallinga\u2013Lipsitch 2007, eq. 3.1).","title":"Exponential generation interval"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/reproduction-number-le-of-convex-order-l","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.reproductionNumber_le_of_convexOrderLE","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#reproduction-number-le-of-convex-order-l","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"If `g\u2081 \u2264cx g\u2082` and both have finite `M(-r)`, then `R(g\u2082) \u2264 R(g\u2081)` at growth rate `r` (any real `r`). In particular, among generation intervals with a given mean, the fixed interval maximizes `R`.","title":"More spread lowers `R`"},{"id":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/reproduction-number-le-exp-mean","kind":"node","lean":"MathlibExt.Biology.Epidemiology.EulerLotka.reproductionNumber_le_exp_mean","parent":47740,"result":true,"route":"../roadmap/Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number/index.html#reproduction-number-le-exp-mean","scope":"Biology/epidemiology/euler-lotka-equation-and-the-reproduction-number","status":"fully_proved","summary":"If the generation interval has finite mean `T\u0304 = \u222b a g(da)` and finite `M_g(-r)`, then `R \u2264 e^{r T\u0304}` (any real `r`); Wallinga and Lipsitch (2007, \u00a73) note that the fixed interval gives this upper bound.","title":"Jensen bound"},{"data_url":"scopes/Biology/epidemiology/kermack-mckendrick-sir-model.json","id":"Biology/epidemiology/kermack-mckendrick-sir-model","kind":"scope","navigation":"module","parent":47739,"result":false,"route":"../roadmap/Biology/epidemiology/kermack-mckendrick-sir-model/index.html","scope":"Biology/epidemiology/kermack-mckendrick-sir-model","status":"fully_proved","title":"Kermack\u2013McKendrick SIR model"},{"id":"Biology/epidemiology/kermack-mckendrick-sir-model/is-solution","kind":"node","lean":"MathlibExt.Biology.Epidemiology.SIR.IsSolution","parent":47751,"result":true,"route":"../roadmap/Biology/epidemiology/kermack-mckendrick-sir-model/index.html#is-solution","scope":"Biology/epidemiology/kermack-mckendrick-sir-model","status":"fully_proved","summary":"`(S, I, R)` solves the Kermack\u2013McKendrick SIR system with transmission rate `\u03b2`, removal rate `\u03b3` and population size `N` for all times `t \u2265 0`: `S' = -\u03b2 S I / N`, `I' = \u03b2 S I / N - \u03b3 I`, `R' = \u03b3 I`.","title":"Solution of the Kermack\u2013McKendrick SIR system"},{"id":"Biology/epidemiology/kermack-mckendrick-sir-model/total-eq","kind":"node","lean":"MathlibExt.Biology.Epidemiology.SIR.IsSolution.total_eq","parent":47751,"result":true,"route":"../roadmap/Biology/epidemiology/kermack-mckendrick-sir-model/index.html#total-eq","scope":"Biology/epidemiology/kermack-mckendrick-sir-model","status":"fully_proved","summary":"Along a solution, `S + I + R` is constant on `[0, \u221e)`.","title":"Conservation of the population"},{"id":"Biology/epidemiology/kermack-mckendrick-sir-model/deriv-infected-pos-iff","kind":"node","lean":"MathlibExt.Biology.Epidemiology.SIR.IsSolution.deriv_infected_pos_iff","parent":47751,"result":true,"route":"../roadmap/Biology/epidemiology/kermack-mckendrick-sir-model/index.html#deriv-infected-pos-iff","scope":"Biology/epidemiology/kermack-mckendrick-sir-model","status":"fully_proved","summary":"If initially `I(0) > 0`, with `\u03b3 > 0` and `N > 0`, the number of infectives initially increases, `I'(0) > 0`, iff `R\u2080 S(0) / N > 1` with `R\u2080 = \u03b2 / \u03b3`.","title":"Threshold theorem (Kermack\u2013McKendrick 1927)"},{"id":"Biology/epidemiology/kermack-mckendrick-sir-model/susceptible-mul-exp-eq","kind":"node","lean":"MathlibExt.Biology.Epidemiology.SIR.IsSolution.susceptible_mul_exp_eq","parent":47751,"result":true,"route":"../roadmap/Biology/epidemiology/kermack-mckendrick-sir-model/index.html#susceptible-mul-exp-eq","scope":"Biology/epidemiology/kermack-mckendrick-sir-model","status":"fully_proved","summary":"For `\u03b3 \u2260 0`, the quantity `S(t) e^{R\u2080 R(t) / N}` (`R\u2080 = \u03b2 / \u03b3`) is constant along a solution on `[0, \u221e)`.","title":"Conserved quantity behind the final-size relation"},{"id":"Biology/epidemiology/kermack-mckendrick-sir-model/final-size","kind":"node","lean":"MathlibExt.Biology.Epidemiology.SIR.IsSolution.final_size","parent":47751,"result":true,"route":"../roadmap/Biology/epidemiology/kermack-mckendrick-sir-model/index.html#final-size","scope":"Biology/epidemiology/kermack-mckendrick-sir-model","status":"fully_proved","summary":"If along a solution (`\u03b3 \u2260 0`) `S(t) \u2192 S\u221e` and `R(t) \u2192 R\u221e` as `t \u2192 \u221e`, then `S\u221e = S(0) e^{-R\u2080 (R\u221e - R(0)) / N}` with `R\u2080 = \u03b2 / \u03b3`. See `final_size_eq` for the classical form.","title":"Final-size relation (Kermack\u2013McKendrick 1927)"},{"id":"Biology/epidemiology/kermack-mckendrick-sir-model/final-size-eq","kind":"node","lean":"MathlibExt.Biology.Epidemiology.SIR.IsSolution.final_size_eq","parent":47751,"result":true,"route":"../roadmap/Biology/epidemiology/kermack-mckendrick-sir-model/index.html#final-size-eq","scope":"Biology/epidemiology/kermack-mckendrick-sir-model","status":"fully_proved","summary":"If the population is `N = S + I + R`, nobody is removed initially (`R(0) = 0`), `\u03b3 \u2260 0`, `S(t) \u2192 S\u221e` and the epidemic dies out (`I(t) \u2192 0`), then `S\u221e = S(0) e^{-R\u2080 (N - S\u221e) / N}` with `R\u2080 = \u03b2 / \u03b3`.","title":"Final-size equation (Kermack\u2013McKendrick 1927)"},{"data_url":"scopes/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio.json","id":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","kind":"scope","navigation":"module","parent":47739,"result":false,"route":"../roadmap/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/index.html","scope":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","status":"fully_proved","title":"Next-generation matrix and the basic reproduction number"},{"id":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/r0","kind":"node","lean":"MathlibExt.Biology.Epidemiology.NextGenerationMatrix.R0","parent":47758,"result":true,"route":"../roadmap/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/index.html#r0","scope":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","status":"fully_proved","summary":"The basic reproduction number `R\u2080 = \u03c1(F V\u207b\u00b9)`: the spectral radius (largest modulus of a complex eigenvalue) of the next-generation matrix `F V\u207b\u00b9` (van den Driessche\u2013Watmough 2002, Eq. (4)). It is meaningful when `V` is invertible (for a nonsingular M-matrix it is).","title":"Basic reproduction number of the next-generation matrix"},{"id":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/is-hurwitz-sub-iff-r0-lt-one","kind":"node","lean":"MathlibExt.Biology.Epidemiology.NextGenerationMatrix.isHurwitz_sub_iff_R0_lt_one","parent":47758,"result":true,"route":"../roadmap/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/index.html#is-hurwitz-sub-iff-r0-lt-one","scope":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","status":"fully_proved","summary":"`R\u2080` as a threshold for the linearized infected subsystem (van den Driessche\u2013Watmough 2002, proof of Theorem 2). If `F` is entrywise nonnegative and `V` is a nonsingular M-matrix, then every eigenvalue of `F - V` has negative real part iff `R\u2080 = \u03c1(F V\u207b\u00b9) < 1`.","title":"`R\u2080` as a threshold for the linearized infected subsystem"},{"id":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/exists-re-pos-iff-one-lt-r0","kind":"node","lean":"MathlibExt.Biology.Epidemiology.NextGenerationMatrix.exists_re_pos_iff_one_lt_R0","parent":47758,"result":true,"route":"../roadmap/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/index.html#exists-re-pos-iff-one-lt-r0","scope":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","status":"fully_proved","summary":"`R\u2080 > 1` gives an unstable mode (van den Driessche\u2013Watmough 2002, proof of Theorem 2). If `F` is entrywise nonnegative and `V` is a nonsingular M-matrix, then `F - V` has an eigenvalue with positive real part iff `R\u2080 = \u03c1(F V\u207b\u00b9) > 1`.","title":"`R\u2080 > 1` gives an unstable mode"},{"id":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/mem-spectrum-from-blocks-zero","kind":"node","lean":"MathlibExt.Biology.Epidemiology.NextGenerationMatrix.mem_spectrum_fromBlocks_zero\u2081\u2082","parent":47758,"result":true,"route":"../roadmap/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/index.html#mem-spectrum-from-blocks-zero","scope":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","status":"fully_proved","summary":"The complex eigenvalues of a block lower-triangular real matrix `[[A, 0], [C, D]]` are those of `A` together with those of `D`.","title":"Eigenvalues of a block lower-triangular matrix"},{"id":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/is-hurwitz-from-blocks-zero-iff","kind":"node","lean":"MathlibExt.Biology.Epidemiology.NextGenerationMatrix.isHurwitz_fromBlocks_zero\u2081\u2082_iff","parent":47758,"result":true,"route":"../roadmap/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/index.html#is-hurwitz-from-blocks-zero-iff","scope":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","status":"fully_proved","summary":"A block lower-triangular real matrix `[[A, 0], [C, D]]` is Hurwitz iff both diagonal blocks are.","title":"Hurwitz criterion for block lower-triangular matrices"},{"id":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/global-solution-decay-of-r0-lt-one","kind":"node","lean":"MathlibExt.Biology.Epidemiology.NextGenerationMatrix.globalSolution_decay_of_R0_lt_one","parent":47758,"result":true,"route":"../roadmap/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/index.html#global-solution-decay-of-r0-lt-one","scope":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","status":"fully_proved","summary":"Threshold theorem, stability half, conditional decay form (van den Driessche\u2013Watmough 2002, Theorem 2). Let `x\u2080` be an equilibrium of `x' = f(x)` on `\u211d^{\u03b9 \u2295 \u03ba}` (infected compartments `\u03b9`, the others `\u03ba`), with `f` differentiable at `x\u2080` and Jacobian `[[F - V, 0], [C, D]]` as in their Lemma 1, where `F \u2265 0`, `V` is a nonsingular M-matrix and `D` is Hurwitz. If `R\u2080 = \u03c1(F V\u207b\u00b9) < 1`, then every solution defined on all\u2026","title":"Threshold theorem, stability half, conditional decay form"},{"id":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/exists-re-pos-jacobian-of-one-lt-r0","kind":"node","lean":"MathlibExt.Biology.Epidemiology.NextGenerationMatrix.exists_re_pos_jacobian_of_one_lt_R0","parent":47758,"result":true,"route":"../roadmap/Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio/index.html#exists-re-pos-jacobian-of-one-lt-r0","scope":"Biology/epidemiology/next-generation-matrix-and-the-basic-reproductio","status":"fully_proved","summary":"Threshold theorem, instability half, spectral form (van den Driessche\u2013Watmough 2002, Theorem 2). With `F \u2265 0` and `V` a nonsingular M-matrix, if `R\u2080 = \u03c1(F V\u207b\u00b9) > 1` then the Jacobian `[[F - V, 0], [C, D]]` at the disease-free equilibrium has an eigenvalue with positive real part (whatever `C` and `D`).","title":"Threshold theorem, instability half, spectral form"},{"data_url":"scopes/Biology/epidemiology/superspreading-and-negative-binomial-offspring.json","id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","kind":"scope","navigation":"module","parent":47739,"result":false,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","title":"Superspreading and negative-binomial offspring"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/nb-pmf","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.nbPmf","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#nb-pmf","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"The negative-binomial offspring probability with mean `R` and dispersion `k`: `P(Z = n) = C(k + n - 1, n) p^k (1 - p)^n` with `p = k / (k + R)` (Lloyd-Smith et al. 2005). Meaningful for `k > 0`, `R \u2265 0`.","title":"Negative-binomial offspring probabilities"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/nb-pgf","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.nbPgf","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#nb-pgf","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"The negative-binomial generating function `G_k(s) = (1 + (R / k) (1 - s))^{-k}`.","title":"Negative-binomial probability generating function"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/nb-offspring","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.nbOffspring","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#nb-offspring","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"The negative-binomial offspring law as a measure on `\u2115`.","title":"Negative-binomial offspring distribution"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/nb-pmf-nonneg","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.nbPmf_nonneg","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#nb-pmf-nonneg","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"The negative-binomial probabilities are nonnegative for `k > 0`, `R \u2265 0`.","title":"Nonnegativity of the negative-binomial probabilities"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/has-sum-nb-pmf-mul-pow","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.hasSum_nbPmf_mul_pow","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#has-sum-nb-pmf-mul-pow","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"For `k > 0`, `R \u2265 0` and `s \u2208 [0, 1]`, `\u2211\u2099 P(Z = n) s\u207f = (1 + (R / k) (1 - s))^{-k}`.","title":"Generating-function identity"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/is-probability-measure-nb-offspring","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.isProbabilityMeasure_nbOffspring","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#is-probability-measure-nb-offspring","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"The negative-binomial offspring law is a probability measure (`k > 0`, `R \u2265 0`).","title":"Negative-binomial offspring law is a probability measure"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/pgf-nb-offspring","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.pgf_nbOffspring","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#pgf-nb-offspring","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"The generating function of the negative-binomial offspring law is `nbPgf` on `[0, 1]`.","title":"Generating function of the negative-binomial law in closed form"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/nb-pgf-antitone","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.nbPgf_antitone","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#nb-pgf-antitone","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"For `R \u2265 0`, `s \u2208 [0, 1]` and `0 < k\u2081 \u2264 k\u2082`, `G_{k\u2082}(s) \u2264 G_{k\u2081}(s)`; equivalently `k \u21a6 k log (1 + x / k)` is nondecreasing for `x \u2265 0`.","title":"Monotonicity in the dispersion"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/nb-pgf-one-dispersion","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.nbPgf_one_dispersion","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#nb-pgf-one-dispersion","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"`k = 1` is the geometric offspring law: `G_1(s) = 1 / (1 + R (1 - s))`.","title":"Negative-binomial law with dispersion one is geometric"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/tendsto-nb-pgf-at-top","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.tendsto_nbPgf_atTop","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#tendsto-nb-pgf-at-top","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"As `k \u2192 \u221e`, `G_k(s) \u2192 e^{-R (1 - s)}` (for all real `R`, `s`).","title":"Poisson limit"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/extinction-prob-le-of-pgf-le","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.extinctionProb_le_of_pgf_le","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#extinction-prob-le-of-pgf-le","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"If the offspring generating functions satisfy `f\u2082 \u2264 f\u2081` on `[0, 1]`, then the extinction probabilities satisfy `q\u2082 \u2264 q\u2081`.","title":"Comparison principle for Galton\u2013Watson extinction"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/is-least-extinction-prob-nb-offspring","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.isLeast_extinctionProb_nbOffspring","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#is-least-extinction-prob-nb-offspring","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"The extinction probability of the negative-binomial epidemic is the least fixed point of `G_k` in `[0, 1]`.","title":"Negative-binomial extinction probability is the least fixed point"},{"id":"Biology/epidemiology/superspreading-and-negative-binomial-offspring/extinction-prob-nb-offspring-antitone","kind":"node","lean":"MathlibExt.Biology.Epidemiology.Superspreading.extinctionProb_nbOffspring_antitone","parent":47766,"result":true,"route":"../roadmap/Biology/epidemiology/superspreading-and-negative-binomial-offspring/index.html#extinction-prob-nb-offspring-antitone","scope":"Biology/epidemiology/superspreading-and-negative-binomial-offspring","status":"fully_proved","summary":"Superspreading raises the extinction probability (Lloyd-Smith et al. 2005): at a fixed mean `R \u2265 0`, the probability `q_k` that the outbreak started by one case goes extinct is nonincreasing in the dispersion `k > 0`.","title":"Superspreading raises the extinction probability"},{"area":"Applied Mathematics","data_url":"chapters/StatisticalMechanics.json","id":"StatisticalMechanics","kind":"scope","parent":0,"result":false,"route":"../roadmap/StatisticalMechanics/index.html","scope":"StatisticalMechanics","status":"can_prove","summary":"Statistical mechanics studies lattice models of interacting spins and bonds through Gibbs measures, partition functions, correlation inequalities, and phase transitions.","title":"Statistical Mechanics"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model.json","id":"StatisticalMechanics/random-cluster-model","kind":"scope","parent":47780,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/index.html","scope":"StatisticalMechanics/random-cluster-model","status":"fully_proved","summary":"The random-cluster (Fortuin\u2013Kasteleyn) model and its Edwards\u2013Sokal coupling with the Potts and Ising models: finite-volume measures, the FKG inequality and comparison with Bernoulli percolation, infinite-volume limits, translation invariance and ergodicity, the free energy, agreement of free and wired measures, and the critical point.","title":"Random-cluster model"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation","kind":"scope","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation","status":"fully_proved","title":"Edwards\u2013Sokal coupling and the FK representation"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/fv-magnetization","kind":"node","lean":"StatMech.IsingFK.fvMagnetization","parent":47782,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/index.html#fv-magnetization","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation","status":"fully_proved","summary":"The magnetization `\u27e8\u03c3_0\u27e9` at the origin under the plus-boundary Ising measure on the box `\u039b_n` at inverse temperature `\u03b2` and zero field.","title":"Finite-volume plus-boundary magnetization at the origin"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model","status":"fully_proved","title":"The Ising model as the two-state Potts model"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model/to-ising","kind":"node","lean":"StatMech.IsingFK.toIsing","parent":47784,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model/index.html#to-ising","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model","status":"fully_proved","summary":"The Ising configuration of a two-state Potts coloring, via `Fin 2 \u2243 Bool`.","title":"Ising configuration of a two-state Potts coloring"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model/beta-of-p","kind":"node","lean":"StatMech.IsingFK.betaOfP","parent":47784,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model/index.html#beta-of-p","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model","status":"fully_proved","summary":"The Ising inverse temperature `-(1/2) log (1 - p)` corresponding to the FK edge parameter `p`. See also `pToBeta` for general `q` (`pToBeta_two_eq_betaOfP`).","title":"Ising inverse temperature of an FK edge parameter"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model/p-of-beta","kind":"node","lean":"StatMech.IsingFK.pOfBeta","parent":47784,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model/index.html#p-of-beta","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-as-the-two-state-potts-model","status":"fully_proved","summary":"The FK edge parameter `p = 1 - e^{-2\u03b2}` of the Ising model at inverse temperature `\u03b2`. See also `betaToP` for general `q` (`betaToP_two_eq_pOfBeta`).","title":"FK edge parameter of an Ising inverse temperature"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic","status":"fully_proved","title":"The Potts critical temperature and the FK critical parameter"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/p-to-beta","kind":"node","lean":"StatMech.IsingFK.pToBeta","parent":47788,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/index.html#p-to-beta","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic","status":"fully_proved","summary":"The Potts inverse temperature `-((q - 1)/q) log (1 - p)` corresponding to the FK edge parameter `p`.","title":"Potts inverse temperature of an FK edge parameter"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/beta-to-p","kind":"node","lean":"StatMech.IsingFK.betaToP","parent":47788,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/index.html#beta-to-p","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic","status":"fully_proved","summary":"The FK edge parameter `1 - e^{-q\u03b2/(q-1)}` of the `q`-state Potts model at inverse temperature `\u03b2`.","title":"FK edge parameter of a Potts inverse temperature"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/p-to-beta-two-eq-beta-of-p","kind":"node","lean":"StatMech.IsingFK.pToBeta_two_eq_betaOfP","parent":47788,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inverse-temperature map of the Potts model at "],["math","\\(q = 2\\)"],["text"," is the Ising map"]],"source":"Inverse-temperature map of the Potts model at $q = 2$ is the Ising map"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/index.html#p-to-beta-two-eq-beta-of-p","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic","status":"fully_proved","summary":"At `q = 2` the general parametrization `pToBeta` is `betaOfP`.","title":"Inverse-temperature map of the Potts model at $q = 2$ is the Ising map"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/beta-to-p-two-eq-p-of-beta","kind":"node","lean":"StatMech.IsingFK.betaToP_two_eq_pOfBeta","parent":47788,"reading_title":{"format":"inline-parts-v1","parts":[["text","FK edge-parameter map of the Potts model at "],["math","\\(q = 2\\)"],["text"," is the Ising map"]],"source":"FK edge-parameter map of the Potts model at $q = 2$ is the Ising map"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/index.html#beta-to-p-two-eq-p-of-beta","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic","status":"fully_proved","summary":"At `q = 2` the general parametrization `betaToP` is `pOfBeta`.","title":"FK edge-parameter map of the Potts model at $q = 2$ is the Ising map"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/beta-c","kind":"node","lean":"StatMech.IsingFK.betaC","parent":47788,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/index.html#beta-c","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic","status":"fully_proved","summary":"The critical inverse temperature `sup {\u03b2 > 0 : m*(\u03b2) = 0}` of a spontaneous-magnetization curve `m*`. See also `Ising.betaC`, the infimum of the inverse temperatures with positive magnetization.","title":"Critical inverse temperature of a magnetization curve"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/ising-beta-c","kind":"node","lean":"StatMech.IsingFK.isingBetaC","parent":47788,"reading_title":{"format":"inline-parts-v1","parts":[["text","Critical inverse temperature of the "],["math","\\(d\\)"],["text","-dimensional Ising model"]],"source":"Critical inverse temperature of the $d$-dimensional Ising model"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic/index.html#ising-beta-c","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-potts-critical-temperature-and-the-fk-critic","status":"fully_proved","summary":"The critical inverse temperature of the `d`-dimensional Ising model, `sup {\u03b2 > 0 : m*(\u03b2) = 0}`.","title":"Critical inverse temperature of the $d$-dimensional Ising model"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph","status":"fully_proved","title":"The q-state Potts model on a finite graph"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/mono-indicator","kind":"node","lean":"StatMech.Potts.monoIndicator","parent":47795,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/index.html#mono-indicator","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph","status":"fully_proved","summary":"The indicator of a monochromatic edge: `\ud835\udfd9[\u03c3 x = \u03c3 y]` on the unordered pair `s(x, y)`.","title":"Monochromatic-edge indicator"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/agreement","kind":"node","lean":"StatMech.Potts.agreement","parent":47795,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/index.html#agreement","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph","status":"fully_proved","summary":"The agreement count: the number of edges of `G` whose endpoints carry the same spin.","title":"Agreement count of a spin configuration"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/potts-weight","kind":"node","lean":"StatMech.Potts.pottsWeight","parent":47795,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/index.html#potts-weight","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph","status":"fully_proved","summary":"The Potts Boltzmann weight `exp (\u03b2 J \u00b7 agreement G \u03c3)`.","title":"Potts Boltzmann weight"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/potts-z","kind":"node","lean":"StatMech.Potts.pottsZ","parent":47795,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/index.html#potts-z","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph","status":"fully_proved","summary":"The Potts partition function `Z = \u2211_\u03c3 exp (\u03b2 J \u00b7 agreement G \u03c3)`.","title":"Potts partition function"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/potts-prob","kind":"node","lean":"StatMech.Potts.pottsProb","parent":47795,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/index.html#potts-prob","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph","status":"fully_proved","summary":"The Potts probability of the spin configuration `\u03c3`.","title":"Potts probability of a spin configuration"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/potts-pmf","kind":"node","lean":"StatMech.Potts.pottsPMF","parent":47795,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph/index.html#potts-pmf","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-q-state-potts-model-on-a-finite-graph","status":"fully_proved","summary":"The Potts Gibbs measure on `V \u2192 Fin q` as a probability mass function.","title":"Potts Gibbs measure as a probability mass function"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","title":"The Edwards\u2013Sokal coupling"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-edge-factor","kind":"node","lean":"StatMech.FK.esEdgeFactor","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-edge-factor","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The Edwards\u2013Sokal factor of an edge `e`: `1 - p` if `e` is closed in `\u03c9`, and `p` times the indicator that `\u03c3` agrees at the endpoints if `e` is open.","title":"Edwards\u2013Sokal edge factor"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-weight","kind":"node","lean":"StatMech.FK.esWeight","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-weight","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The Edwards\u2013Sokal joint weight of a spin configuration `\u03c3` and a bond configuration `\u03c9`: the product of `esEdgeFactor` over the edges of `G`.","title":"Edwards\u2013Sokal joint weight"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/const-on-open","kind":"node","lean":"StatMech.FK.ConstOnOpen","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#const-on-open","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"`\u03c3` is constant on every open edge of `\u03c9`, i.e. on every open cluster.","title":"Spin configuration constant on open clusters"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/const-on-open-equiv","kind":"node","lean":"StatMech.FK.constOnOpenEquiv","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#const-on-open-equiv","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"Colorings constant on open clusters correspond to colorings of the connected components of the open subgraph.","title":"Cluster-constant colorings as colorings of clusters"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/card-const-on-open","kind":"node","lean":"StatMech.FK.card_constOnOpen","parent":47802,"reading_title":{"format":"inline-parts-v1","parts":[["text","There are "],["math","\\(q^{k(\\omega)}\\)"],["text"," colorings constant on the open clusters"]],"source":"There are $q^{k(\\omega)}$ colorings constant on the open clusters"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#card-const-on-open","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"There are `q ^ k(\u03c9)` colorings constant on the open clusters, where `k(\u03c9)` is the number of clusters.","title":"There are $q^{k(\\omega)}$ colorings constant on the open clusters"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-weight-sum","kind":"node","lean":"StatMech.FK.esWeight_sum","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-weight-sum","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"Spin marginal of the Edwards\u2013Sokal weight: summing over spins gives the random-cluster weight `p^{o(\u03c9)} (1 - p)^{c(\u03c9)} q^{k(\u03c9)}`.","title":"Edwards\u2013Sokal weight summed over spins is the random-cluster weight"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-coeff","kind":"node","lean":"StatMech.FK.esCoeff","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-coeff","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The Edwards\u2013Sokal factor of an edge `e` written for an arbitrary pair `(e, b)` with `b` the state of `e`; edges outside `G` get factor `1`.","title":"Edwards\u2013Sokal factor of an arbitrary coordinate and state"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-first-const","kind":"node","lean":"StatMech.FK.esFirstConst","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-first-const","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The normalizing constant `e^{-\u03b2J |E(G)|}` relating the bond marginal to the Potts weight.","title":"Normalizing constant relating the bond sum to the Potts weight"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-weight-sum-eq-potts-weight","kind":"node","lean":"StatMech.FK.esWeight_sum_eq_pottsWeight","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-weight-sum-eq-potts-weight","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"Bond marginal of the Edwards\u2013Sokal weight: with `p = 1 - e^{-\u03b2J}`, summing over bond configurations gives `e^{-\u03b2J |E|}` times the Potts weight of `\u03c3`.","title":"Edwards\u2013Sokal bond sum is a multiple of the Potts weight"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-z","kind":"node","lean":"StatMech.FK.esZ","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-z","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The Edwards\u2013Sokal partition function: the total joint weight.","title":"Edwards\u2013Sokal partition function"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-second-marginal","kind":"node","lean":"StatMech.FK.esSecondMarginal","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-second-marginal","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The bond marginal of the normalized Edwards\u2013Sokal measure.","title":"Bond marginal of the Edwards\u2013Sokal measure"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/es-first-marginal","kind":"node","lean":"StatMech.FK.esFirstMarginal","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#es-first-marginal","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The spin marginal of the normalized Edwards\u2013Sokal measure.","title":"Spin marginal of the Edwards\u2013Sokal measure"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/edwards-sokal-second-marginal","kind":"node","lean":"StatMech.FK.edwardsSokal_secondMarginal","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#edwards-sokal-second-marginal","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The bond marginal of the Edwards\u2013Sokal measure is the random-cluster measure with parameters `(1 - e^{-\u03b2J}, q)`.","title":"Edwards\u2013Sokal bond marginal is the random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/edwards-sokal-first-marginal","kind":"node","lean":"StatMech.FK.edwardsSokal_firstMarginal","parent":47802,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling/index.html#edwards-sokal-first-marginal","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling","status":"fully_proved","summary":"The spin marginal of the Edwards\u2013Sokal measure is the `q`-state Potts measure at inverse temperature `\u03b2` and coupling `J`.","title":"Edwards\u2013Sokal spin marginal is the Potts measure"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli","status":"fully_proved","title":"Compatible colorings in the Edwards\u2013Sokal coupling"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/compatible","kind":"node","lean":"StatMech.IsingFK.Compatible","parent":47817,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/index.html#compatible","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli","status":"fully_proved","summary":"A coloring `\u03c3` is compatible with the bond configuration `\u03c9` if every open edge of `G` joins vertices of the same color. See also `FK.ConstOnOpen` (`compatible_iff_constOnOpen`).","title":"Compatibility of a coloring with a bond configuration"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/mono-bool","kind":"node","lean":"StatMech.IsingFK.monoBool","parent":47817,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/index.html#mono-bool","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli","status":"fully_proved","summary":"Whether an edge is monochromatic under the coloring `\u03c3`.","title":"Monochromatic edge indicator"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/disagree-edges","kind":"node","lean":"StatMech.IsingFK.disagreeEdges","parent":47817,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/index.html#disagree-edges","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli","status":"fully_proved","summary":"The edges of `G` whose endpoints have different colors.","title":"Edges with differently colored endpoints"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/coloring","kind":"node","lean":"StatMech.IsingFK.coloring","parent":47817,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/index.html#coloring","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli","status":"fully_proved","summary":"The coloring of the vertices obtained from a coloring `\u03c4` of the open clusters of `\u03c9`.","title":"Vertex coloring induced by a coloring of open clusters"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/es-joint-prob","kind":"node","lean":"StatMech.IsingFK.esJointProb","parent":47817,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli/index.html#es-joint-prob","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/compatible-colorings-in-the-edwards-sokal-coupli","status":"fully_proved","summary":"The joint Edwards\u2013Sokal probability of the coloring `\u03c3` and the bond configuration `\u03c9`.","title":"Edwards\u2013Sokal joint probability"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","title":"The Edwards\u2013Sokal coupling with wired boundary conditions"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/boundary-fixed","kind":"node","lean":"StatMech.IsingFK.BoundaryFixed","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#boundary-fixed","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The coloring `\u03c3` equals `b` on every boundary vertex.","title":"Coloring fixed to a color on the boundary"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/potts-weight-wired","kind":"node","lean":"StatMech.IsingFK.pottsWeightWired","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#potts-weight-wired","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The Potts weight restricted to colorings with boundary color `b`.","title":"Potts weight with fixed boundary color"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/potts-zwired","kind":"node","lean":"StatMech.IsingFK.pottsZWired","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#potts-zwired","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The partition function of the Potts model with boundary color `b`.","title":"Potts partition function with fixed boundary color"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/potts-prob-wired","kind":"node","lean":"StatMech.IsingFK.pottsProbWired","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#potts-prob-wired","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The Gibbs probability of the Potts model with boundary color `b`.","title":"Potts Gibbs probability with fixed boundary color"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/es-weight-wired","kind":"node","lean":"StatMech.IsingFK.esWeightWired","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#es-weight-wired","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The Edwards\u2013Sokal weight restricted to colorings with boundary color `b`.","title":"Edwards\u2013Sokal weight with fixed boundary color"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/es-zwired","kind":"node","lean":"StatMech.IsingFK.esZWired","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#es-zwired","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The total wired Edwards\u2013Sokal weight.","title":"Total wired Edwards\u2013Sokal weight"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/es-wired-first-marginal","kind":"node","lean":"StatMech.IsingFK.esWiredFirstMarginal","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#es-wired-first-marginal","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The spin marginal of the wired Edwards\u2013Sokal measure.","title":"Spin marginal of the wired Edwards\u2013Sokal measure"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/wired-sub","kind":"node","lean":"StatMech.IsingFK.wiredSub","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#wired-sub","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The graph of open edges of `\u03c9` together with a clique on the boundary vertices.","title":"Open-edge graph with a boundary clique"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/const-on-wired","kind":"node","lean":"StatMech.IsingFK.ConstOnWired","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#const-on-wired","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The coloring `\u03c3` is constant along the edges of `wiredSub`.","title":"Coloring constant on the wired graph"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/const-on-wired-equiv","kind":"node","lean":"StatMech.IsingFK.constOnWiredEquiv","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#const-on-wired-equiv","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"Colorings constant on the clusters of `wiredSub` correspond to colorings of those clusters.","title":"Colorings constant on wired clusters as cluster colorings"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/pinned-fun-equiv","kind":"node","lean":"StatMech.IsingFK.pinnedFunEquiv","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#pinned-fun-equiv","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"Functions to `Fin q` taking the value `b` at `c\u2080` correspond to functions on the complement of `c\u2080`. See also `Equiv.piSplitAt`.","title":"Functions pinned at a point as functions on its complement"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/es-wired-second-marginal","kind":"node","lean":"StatMech.IsingFK.esWiredSecondMarginal","parent":47823,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c/index.html#es-wired-second-marginal","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-edwards-sokal-coupling-with-wired-boundary-c","status":"fully_proved","summary":"The bond marginal of the wired Edwards\u2013Sokal measure.","title":"Bond marginal of the wired Edwards\u2013Sokal measure"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","status":"fully_proved","title":"Ising two-point function via the Edwards\u2013Sokal coupling"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/ising-spin","kind":"node","lean":"StatMech.FK.isingSpin","parent":47836,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/index.html#ising-spin","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","status":"fully_proved","summary":"The Ising spin value of a two-state color: `0 \u21a6 -1`, `1 \u21a6 1`.","title":"Ising spin value of a two-state color"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/connected","kind":"node","lean":"StatMech.FK.Connected","parent":47836,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/index.html#connected","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","status":"fully_proved","summary":"`x` and `y` are joined by a path of open edges of `\u03c9` in `G` (`x \u2194 y`).","title":"Open connection between two vertices"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/conn-prob","kind":"node","lean":"StatMech.FK.connProb","parent":47836,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/index.html#conn-prob","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","status":"fully_proved","summary":"The random-cluster connection probability `\u03c6_{G,p,q}(x \u2194 y)`.","title":"Random-cluster connection probability"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/eq-fun-equiv","kind":"node","lean":"StatMech.FK.eqFunEquiv","parent":47836,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/index.html#eq-fun-equiv","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","status":"fully_proved","summary":"For `a \u2260 b`, functions `C \u2192 Fin q` with `f a = f b` correspond to functions on `C \\ {b}`.","title":"Functions equal at two points as functions omitting one point"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/const-eq-equiv","kind":"node","lean":"StatMech.FK.constEqEquiv","parent":47836,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/index.html#const-eq-equiv","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","status":"fully_proved","summary":"Colorings constant on open clusters with `\u03c3 x = \u03c3 y` correspond to colorings of the clusters that agree on the clusters of `x` and `y`.","title":"Correspondence for cluster-constant colorings equal at two vertices"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/es-two-point","kind":"node","lean":"StatMech.FK.esTwoPoint","parent":47836,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/index.html#es-two-point","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","status":"fully_proved","summary":"The Ising two-point function `\u27e8\u03c3_x \u03c3_y\u27e9` at bond parameter `p`, computed as the Edwards\u2013Sokal spin marginal.","title":"Ising two-point function from the Edwards\u2013Sokal coupling"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/es-two-point-eq-conn-prob","kind":"node","lean":"StatMech.FK.esTwoPoint_eq_connProb","parent":47836,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ising two-point function equals the "],["math","\\(q = 2\\)"],["text"," connection probability"]],"source":"Ising two-point function equals the $q = 2$ connection probability"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c/index.html#es-two-point-eq-conn-prob","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/ising-two-point-function-via-the-edwards-sokal-c","status":"fully_proved","summary":"The Ising two-point function equals the random-cluster connection probability with `q = 2`: `\u27e8\u03c3_x \u03c3_y\u27e9 = \u03c6_{p,2}(x \u2194 y)` (Edwards\u2013Sokal 1988; Grimmett, *The Random-Cluster Model*, 2006, Theorem 1.16).","title":"Ising two-point function equals the $q = 2$ connection probability"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function","status":"fully_proved","title":"The wired Edwards\u2013Sokal one-point function"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/wired-colour-equiv","kind":"node","lean":"StatMech.IsingFK.wiredColourEquiv","parent":47844,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/index.html#wired-colour-equiv","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function","status":"fully_proved","summary":"Colorings constant on wired clusters with boundary color `b` correspond to colorings of the clusters of `wiredSub` with the boundary cluster colored `b`.","title":"Wired colorings as cluster colorings with a pinned boundary cluster"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/conn-to-bdry","kind":"node","lean":"StatMech.IsingFK.ConnToBdry","parent":47844,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/index.html#conn-to-bdry","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function","status":"fully_proved","summary":"The vertex `x` is joined by an open path of `\u03c9` to a boundary vertex.","title":"Open connection of a vertex to the boundary"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/es-wired-one-point","kind":"node","lean":"StatMech.IsingFK.esWiredOnePoint","parent":47844,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/index.html#es-wired-one-point","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function","status":"fully_proved","summary":"The one-point function `\u27e8\u03c3_x\u27e9` of the wired Edwards\u2013Sokal measure with boundary color `b` and `q = 2`.","title":"Wired Edwards\u2013Sokal one-point function"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/wired-conn-to-bdry-prob","kind":"node","lean":"StatMech.IsingFK.wiredConnToBdryProb","parent":47844,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/index.html#wired-conn-to-bdry-prob","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function","status":"fully_proved","summary":"The probability `\u03c6^w(x \u2194 \u2202)` that `x` is connected to the boundary under the wired random-cluster measure with `q = 2`.","title":"Wired random-cluster probability of connecting to the boundary"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/es-wired-one-point-eq-wired-conn-to-bdry","kind":"node","lean":"StatMech.IsingFK.esWiredOnePoint_eq_wiredConnToBdryProb","parent":47844,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function/index.html#es-wired-one-point-eq-wired-conn-to-bdry","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-wired-edwards-sokal-one-point-function","status":"fully_proved","summary":"With `+` boundary condition, the Ising magnetization at `x` equals the probability that `x` is connected to the boundary in the wired FK measure with `q = 2` (Edwards and Sokal, Phys. Rev. D 38 (1988); Grimmett, *The Random-Cluster Model*, Theorem 1.16).","title":"Edwards\u2013Sokal one-point identity"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po","status":"fully_proved","title":"The Ising model in a box as a wired two-state Potts model"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/ising-bond","kind":"node","lean":"StatMech.IsingFK.isingBond","parent":47850,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/index.html#ising-bond","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po","status":"fully_proved","summary":"The Ising bond `\u03c3_x \u03c3_y` of a two-state coloring, with color `0` as spin `+1`. See also `Ising.bond` for Boolean configurations.","title":"Ising bond variable of a two-state coloring"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/ising-wired-weight","kind":"node","lean":"StatMech.IsingFK.isingWiredWeight","parent":47850,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/index.html#ising-wired-weight","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po","status":"fully_proved","summary":"The Ising weight `exp (\u03b2 \u2211_e \u03c3_x \u03c3_y)` of a two-state coloring with boundary color `0`, and `0` otherwise.","title":"Ising weight of a coloring with plus boundary"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/ising-wired-z","kind":"node","lean":"StatMech.IsingFK.isingWiredZ","parent":47850,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/index.html#ising-wired-z","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po","status":"fully_proved","summary":"The partition function of the Ising model with `+` boundary condition on a graph.","title":"Plus-boundary Ising partition function on a graph"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/ising-wired-one-point","kind":"node","lean":"StatMech.IsingFK.isingWiredOnePoint","parent":47850,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/index.html#ising-wired-one-point","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po","status":"fully_proved","summary":"The one-point function `\u27e8\u03c3_x\u27e9\u207a` of the Ising model with `+` boundary condition on a graph.","title":"Plus-boundary Ising one-point function on a graph"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/b-to-f","kind":"node","lean":"StatMech.IsingFK.bToF","parent":47850,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/index.html#b-to-f","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po","status":"fully_proved","summary":"The two-state color of a Boolean spin: `true \u21a6 0`, `false \u21a6 1`.","title":"Two-state color of a Boolean spin"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/lat-to-box","kind":"node","lean":"StatMech.IsingFK.latToBox","parent":47850,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po/index.html#lat-to-box","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-ising-model-in-a-box-as-a-wired-two-state-po","status":"fully_proved","summary":"A configuration in the box `\u039b_n` as a two-state coloring of `\u039b_{n+1}`, with color `0` (spin `+`) outside `\u039b_n`.","title":"Box configuration as a coloring of the next larger box"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f","status":"fully_proved","title":"The lattice magnetization as a wired one-point function on the box graph"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f/f-to-b","kind":"node","lean":"StatMech.IsingFK.fToB","parent":47857,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f/index.html#f-to-b","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f","status":"fully_proved","summary":"The Boolean spin of a two-state color: `0 \u21a6 true`.","title":"Boolean spin of a two-state color"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f/box-to-lat","kind":"node","lean":"StatMech.IsingFK.boxToLat","parent":47857,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f/index.html#box-to-lat","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f","status":"fully_proved","summary":"The configuration in `\u039b_n` obtained by restricting a two-state coloring of `\u039b_{n+1}`.","title":"Restriction of a box coloring to the inner box"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f/hbx-lat-box-equiv","kind":"node","lean":"StatMech.IsingFK.hbx_latBoxEquiv","parent":47857,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box configurations as box-graph colorings with boundary color "],["math","\\(0\\)"]],"source":"Box configurations as box-graph colorings with boundary color $0$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f/index.html#hbx-lat-box-equiv","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/the-lattice-magnetization-as-a-wired-one-point-f","status":"fully_proved","summary":"Configurations in `\u039b_n` correspond to two-state colorings of `\u039b_{n+1}` with boundary color `0`.","title":"Box configurations as box-graph colorings with boundary color $0$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability","status":"fully_proved","title":"Magnetization equals FK percolation probability, through box graphs"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability/origin-mem-box-box-verts","kind":"node","lean":"StatMech.IsingFK.origin_mem_box_boxVerts","parent":47861,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability/index.html#origin-mem-box-box-verts","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability","status":"fully_proved","summary":"The origin lies in every box. Same statement as `Ising.origin_mem_box`, which this module does not import.","title":"Origin lies in every box"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability/box-origin","kind":"node","lean":"StatMech.IsingFK.boxOrigin","parent":47861,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability/index.html#box-origin","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability","status":"fully_proved","summary":"The origin as a vertex of the box graph.","title":"Origin as a vertex of the box graph"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability/box-boundary-conn-profile","kind":"node","lean":"StatMech.IsingFK.boxBoundaryConnProfile","parent":47861,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability/index.html#box-boundary-conn-profile","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/magnetization-equals-fk-percolation-probability","status":"fully_proved","summary":"The probability `\u03c6^w_n(0 \u2194 \u2202\u039b_n)` that the origin is connected to the boundary of the box graph of `\u039b_n` under the wired FK measure with `q = 2`.","title":"Wired probability that the origin reaches the box boundary"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi.json","id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi","kind":"scope","navigation":"module","parent":47782,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/index.html","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi","status":"fully_proved","title":"Potts correlations and random-cluster connectivity"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/potts-agree-prob","kind":"node","lean":"StatMech.FK.pottsAgreeProb","parent":47865,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/index.html#potts-agree-prob","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi","status":"fully_proved","summary":"The Potts two-point agreement probability `\u03bc_{G,\u03b2,J,q}(\u03c3_x = \u03c3_y)`.","title":"Potts two-point agreement probability"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/edwards-sokal-correlation","kind":"node","lean":"StatMech.FK.edwards_sokal_correlation","parent":47865,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/index.html#edwards-sokal-correlation","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi","status":"fully_proved","summary":"For the `q`-state Potts model on a finite graph with `p = 1 - e^{-\u03b2J} \u2208 (0, 1)`, `\u03bc_{\u03b2,J,q}(\u03c3_x = \u03c3_y) - 1/q = (1 - 1/q) \u03c6_{p,q}(x \u2194 y)`. The hypothesis `hp1` holds for every real `\u03b2J` and `hq` follows from `[NeZero q]`; the only restriction is `hp`, i.e. `\u03b2J > 0` (ferromagnetic). (Edwards\u2013Sokal, Phys. Rev. D 38, 1988; Grimmett, *The Random-Cluster Model*, 2006, Theorem 1.16.)","title":"Edwards\u2013Sokal correlation identity"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/edwards-sokal-correlation-2","kind":"node","lean":"StatMech.FK.edwards_sokal_correlation'","parent":47865,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edwards\u2013Sokal correlation identity in "],["math","\\((q-1)/q\\)"],["text"," form"]],"source":"Edwards\u2013Sokal correlation identity in $(q-1)/q$ form"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/index.html#edwards-sokal-correlation-2","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi","status":"fully_proved","summary":"The Edwards\u2013Sokal correlation identity with the prefactor written as `(q - 1)/q`.","title":"Edwards\u2013Sokal correlation identity in $(q-1)/q$ form"},{"id":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/potts-agree-prob-ge","kind":"node","lean":"StatMech.FK.pottsAgreeProb_ge","parent":47865,"reading_title":{"format":"inline-parts-v1","parts":[["text","Potts two-point agreement at least "],["math","\\(1/q\\)"],["text"," for "],["math","\\(q \\ge 2\\)"],["text",", "],["math","\\(\\beta J > 0\\)"]],"source":"Potts two-point agreement at least $1/q$ for $q \\ge 2$, $\\beta J > 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi/index.html#potts-agree-prob-ge","scope":"StatisticalMechanics/random-cluster-model/edwards-sokal-coupling-and-the-fk-representation/potts-correlations-and-random-cluster-connectivi","status":"fully_proved","summary":"For `q \u2265 2` and `\u03b2J > 0`, two Potts spins agree with probability at least `1/q`.","title":"Potts two-point agreement at least $1/q$ for $q \\ge 2$, $\\beta J > 0$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs.json","id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","kind":"scope","navigation":"module","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","title":"Random-cluster measures on finite graphs"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/open-sub","kind":"node","lean":"StatMech.FK.openSub","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#open-sub","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The open subgraph of `G` under the edge configuration `\u03c9`: the edges of `G` that `\u03c9` declares open.","title":"Open subgraph of an edge configuration"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/open-sub-eq-inf-from-edge-set","kind":"node","lean":"StatMech.FK.openSub_eq_inf_fromEdgeSet","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Open subgraph as the meet of "],["math","\\(G\\)"],["text"," with the open-edge graph"]],"source":"Open subgraph as the meet of $G$ with the open-edge graph"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#open-sub-eq-inf-from-edge-set","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The open subgraph is `G \u2293 fromEdgeSet {e | \u03c9 e = true}` in Mathlib's terms.","title":"Open subgraph as the meet of $G$ with the open-edge graph"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/num-clusters","kind":"node","lean":"StatMech.FK.numClusters","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#num-clusters","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The number `k(\u03c9)` of open clusters: connected components of the open subgraph, isolated vertices included.","title":"Number of open clusters"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/edge-product","kind":"node","lean":"StatMech.FK.edgeProduct","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#edge-product","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The Bernoulli edge product `\u220f_{e \u2208 E(G)} (p if e open, 1 - p if e closed)`.","title":"Bernoulli edge product"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-weight","kind":"node","lean":"StatMech.FK.fkWeight","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-weight","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The unnormalized random-cluster weight `p ^ o(\u03c9) (1 - p) ^ c(\u03c9) q ^ k(\u03c9)`.","title":"Unnormalized random-cluster weight"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-z","kind":"node","lean":"StatMech.FK.fkZ","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-z","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The ambient FK normalizer, summing over every bit assignment on `Sym2 V`. Each loop or nonedge contributes a factor of two relative to the edge partition function `ecz_fkZEdge` defined in `EdgePartitionFunction`.","title":"Ambient FK normalizer"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-prob","kind":"node","lean":"StatMech.FK.fkProb","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-prob","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The probability of an ambient configuration: `fkWeight G p q \u03c9 / fkZ G p q`. For `0 < p < 1` and `q > 0`, its graph-edge marginal is the random-cluster law and unused coordinates are uniform.","title":"Probability of an ambient FK configuration"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-pmf","kind":"node","lean":"StatMech.FK.fkPMF","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-pmf","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The ambient extension of the random-cluster edge law as a `PMF` on all `Sym2 V` configurations, with independent uniform bits on loops and nonedges.","title":"Ambient random-cluster law as a probability mass function"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-weight-one","kind":"node","lean":"StatMech.FK.fkWeight_one","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","At "],["math","\\(q = 1\\)"],["text"," the random-cluster weight is the Bernoulli product"]],"source":"At $q = 1$ the random-cluster weight is the Bernoulli product"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-weight-one","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"At `q = 1` the random-cluster weight is the Bernoulli product weight.","title":"At $q = 1$ the random-cluster weight is the Bernoulli product"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/agrees-off","kind":"node","lean":"StatMech.FK.AgreesOff","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#agrees-off","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"`\u03c9` agrees with `\u03c8` on every edge outside `F`.","title":"Agreement of configurations off an edge set"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/cond-fibre","kind":"node","lean":"StatMech.FK.condFibre","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#cond-fibre","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The fibre of configurations agreeing with `\u03c8` off `F`.","title":"Fibre of configurations agreeing off an edge set"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/induced-fk-z","kind":"node","lean":"StatMech.FK.inducedFkZ","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#induced-fk-z","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The total FK weight of the fibre `condFibre F \u03c8`.","title":"Total FK weight of a fibre"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/induced-fk-prob","kind":"node","lean":"StatMech.FK.inducedFkProb","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#induced-fk-prob","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The FK measure induced on the fibre `condFibre F \u03c8`: FK weight divided by the fibre's total weight.","title":"FK measure induced on a fibre"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/cond-fk-prob","kind":"node","lean":"StatMech.FK.condFkProb","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#cond-fk-prob","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The random-cluster measure conditioned on agreeing with `\u03c8` off `F`.","title":"Random-cluster measure conditioned on the edges off a set"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/cond-fk-prob-eq-induced-fk-prob","kind":"node","lean":"StatMech.FK.condFkProb_eq_inducedFkProb","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#cond-fk-prob-eq-induced-fk-prob","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Conditioned on the states of the edges outside `F`, the random-cluster measure is the FK weight renormalized on the fibre (Grimmett, *The Random-Cluster Model*, 2006, Lemma 4.13).","title":"Domain Markov property (finite volume)"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/proj-off","kind":"node","lean":"StatMech.FK.projOff","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#proj-off","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The configuration that closes every edge of `F` and agrees with `\u03c8` elsewhere.","title":"Closing the edges of a set in a configuration"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fibre-reps","kind":"node","lean":"StatMech.FK.fibreReps","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fibre-reps","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Representatives of the fibres of `F`: configurations with every edge of `F` closed.","title":"Fibre representatives with every edge of a set closed"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/depends-on-outside","kind":"node","lean":"StatMech.FK.DependsOnOutside","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#depends-on-outside","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Membership in `B` depends only on the states of edges outside `F`.","title":"Event depending only on edges outside a set"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/open-sub-set-open-eq-set-closed-of-not-m","kind":"node","lean":"StatMech.FK.openSub_setOpen_eq_setClosed_of_notMem","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#open-sub-set-open-eq-set-closed-of-not-m","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Off the edges of `G`, flipping `e` does not change the open subgraph.","title":"Toggling a non-edge leaves the open subgraph unchanged"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/num-clusters-set-open-le","kind":"node","lean":"StatMech.FK.numClusters_setOpen_le","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#num-clusters-set-open-le","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Opening an edge does not increase the number of clusters. This is a special case of Mathlib's `SimpleGraph.ConnectedComponent.card_le_card_of_le`, which proves it.","title":"Opening an edge does not increase the cluster count"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/num-clusters-set-closed-le-set-open-succ","kind":"node","lean":"StatMech.FK.numClusters_setClosed_le_setOpen_succ","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#num-clusters-set-closed-le-set-open-succ","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Opening an edge lowers the number of clusters by at most one.","title":"Opening an edge lowers the cluster count by at most one"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/edge-rest","kind":"node","lean":"StatMech.FK.edgeRest","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#edge-rest","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The Bernoulli product over the edges of `G` other than `e`.","title":"Bernoulli product over the edges other than one edge"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/cond-open","kind":"node","lean":"StatMech.FK.condOpen","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#cond-open","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The conditional probability under `\u03c6_{G,p,q}` that `e` is open given the states of all other edges: `\u03c6(setOpen e \u03c9) / (\u03c6(setOpen e \u03c9) + \u03c6(setClosed e \u03c9))`.","title":"Conditional probability that an edge is open given the rest"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/c-fe","kind":"node","lean":"StatMech.FK.cFE","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#c-fe","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The finite-energy constant `min (p / (p + q (1 - p))) (1 - p)`.","title":"Finite-energy constant of the random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/finite-energy","kind":"node","lean":"StatMech.FK.finite_energy","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#finite-energy","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Finite-energy property of the random-cluster measure. For `p \u2208 (0, 1)` and `q \u2265 1`, the conditional probability that an edge is open given all other edges lies in `[cFE p q, 1 - cFE p q]`. For edges of `G` the sharper bounds `p / (p + q (1 - p))` and `p` are proved along the way. See G. Grimmett, *The Random-Cluster Model* (2006), Chapter 3.","title":"Finite-energy property"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/edge-product-w","kind":"node","lean":"StatMech.FK.edgeProductW","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#edge-product-w","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The inhomogeneous Bernoulli product `\u220f_{e \u2208 E(G)} (pf e if e open, 1 - pf e if closed)`.","title":"Inhomogeneous Bernoulli edge product"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-weight-w","kind":"node","lean":"StatMech.FK.fkWeightW","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-weight-w","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The inhomogeneous random-cluster weight `edgeProductW G pf \u03c9 \u00b7 q ^ k(\u03c9)`.","title":"Inhomogeneous random-cluster weight"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-zw","kind":"node","lean":"StatMech.FK.fkZW","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-zw","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The partition function of the inhomogeneous random-cluster measure.","title":"Partition function of the inhomogeneous random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-prob-w","kind":"node","lean":"StatMech.FK.fkProbW","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-prob-w","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The inhomogeneous random-cluster probability `fkWeightW / fkZW`.","title":"Inhomogeneous random-cluster probability"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-mean-w","kind":"node","lean":"StatMech.FK.fkMeanW","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-mean-w","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The expectation of `g` under the inhomogeneous random-cluster measure.","title":"Expectation under the inhomogeneous random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-cov-w","kind":"node","lean":"StatMech.FK.fkCovW","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-cov-w","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The covariance `E[f g] - E[f] E[g]` under the inhomogeneous random-cluster measure (a finite form of Mathlib's `ProbabilityTheory.covariance`).","title":"Covariance under the inhomogeneous random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/coord-r","kind":"node","lean":"StatMech.FK.coordR","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#coord-r","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The real-valued state `\ud835\udfd9[e open]` of the edge `e`.","title":"Real-valued indicator that an edge is open"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/edge-product-woff","kind":"node","lean":"StatMech.FK.edgeProductWOff","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#edge-product-woff","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The inhomogeneous Bernoulli product over the edges of `G` other than `e\u2080`.","title":"Inhomogeneous Bernoulli product over all edges but one"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-numer-w","kind":"node","lean":"StatMech.FK.fkNumerW","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-numer-w","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The unnormalized expectation `\u2211_\u03c9 g(\u03c9) fkWeightW G pf q \u03c9`.","title":"Unnormalized inhomogeneous random-cluster expectation"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/has-deriv-at-fk-mean-w","kind":"node","lean":"StatMech.FK.hasDerivAt_fkMeanW","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#has-deriv-at-fk-mean-w","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Differential formula for the random-cluster measure: the derivative of `\u03c6[g]` in the parameter of the edge `e\u2080` is `Cov(g, \ud835\udfd9[e\u2080 open]) / (p (1 - p))` with `p = pf e\u2080`. See G. Grimmett, *The Random-Cluster Model* (2006), Chapter 3.","title":"Differential formula"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-prob-wof","kind":"node","lean":"StatMech.FK.fkProbWOf","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-prob-wof","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The probability of the event `A` under the inhomogeneous random-cluster measure.","title":"Probability of an event under the inhomogeneous random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/has-deriv-at-param-of-beta","kind":"node","lean":"StatMech.FK.hasDerivAt_paramOfBeta","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Derivative in "],["math","\\(\\beta\\)"],["text"," of the edge parameter "],["math","\\(1 - e^{-\\beta J}\\)"]],"source":"Derivative in $\\beta$ of the edge parameter $1 - e^{-\\beta J}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#has-deriv-at-param-of-beta","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The derivative of the edge parameter `p(\u03b2) = 1 - exp (-\u03b2 J)` is `J exp (-\u03b2 J)`.","title":"Derivative in $\\beta$ of the edge parameter $1 - e^{-\\beta J}$"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/beta-params","kind":"node","lean":"StatMech.FK.betaParams","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edge parameters "],["math","\\(p_e(\\beta) = 1 - e^{-\\beta J_e}\\)"]],"source":"Edge parameters $p_e(\\beta) = 1 - e^{-\\beta J_e}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#beta-params","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The edge parameters `p_e(\u03b2) = 1 - exp (-\u03b2 J_e)`.","title":"Edge parameters $p_e(\\beta) = 1 - e^{-\\beta J_e}$"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/beta-score","kind":"node","lean":"StatMech.FK.betaScore","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logarithmic "],["math","\\(\\beta\\)"],["text","-derivative of the random-cluster weight"]],"source":"Logarithmic $\\beta$-derivative of the random-cluster weight"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#beta-score","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The logarithmic `\u03b2`-derivative of the random-cluster weight under `betaParams`: `\u2211_{e \u2208 E(G)} (J_e / p_e) (\ud835\udfd9[e open] - p_e)`.","title":"Logarithmic $\\beta$-derivative of the random-cluster weight"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/has-deriv-at-fk-mean-w-beta-sum","kind":"node","lean":"StatMech.FK.hasDerivAt_fkMeanW_beta_sum","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#has-deriv-at-fk-mean-w-beta-sum","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Differential formula in `\u03b2` for the random-cluster measure with edge parameters `1 - exp (-\u03b2 J_e)`: `d/d\u03b2 \u03c6_\u03b2[g] = \u2211_e (J_e / p_e) Cov(g, \ud835\udfd9[e open])`. See G. Grimmett, *The Random-Cluster Model* (2006), Chapter 3.","title":"Differential formula in `\u03b2`"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/open-count","kind":"node","lean":"StatMech.FK.openCount","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#open-count","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The number `o(\u03c9)` of open edges of `G` in `\u03c9`.","title":"Number of open edges"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/closed-count","kind":"node","lean":"StatMech.FK.closedCount","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#closed-count","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The number `c(\u03c9)` of closed edges of `G` in `\u03c9`.","title":"Number of closed edges"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/tilt-factor","kind":"node","lean":"StatMech.FK.tiltFactor","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#tilt-factor","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The tilt factor `\u03bb = p\u2082 (1 - p\u2081) / ((1 - p\u2082) p\u2081)`, the ratio of the odds of an open edge at `p\u2082` and at `p\u2081`.","title":"Tilt factor between two edge parameters"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-expect","kind":"node","lean":"StatMech.FK.fkExpect","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-expect","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The expectation `\u03c6_{G,p,q}[X] = \u2211_\u03c9 \u03c6_{G,p,q}(\u03c9) X(\u03c9)` of an observable `X`.","title":"Random-cluster expectation of an observable"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/re-cfg","kind":"node","lean":"StatMech.FK.reCfg","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#re-cfg","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The configuration `\u03c9 \u2218 \u03c3` on edges: the edge `e` is open in `reCfg \u03c3 \u03c9` iff `\u03c3 e` is open in `\u03c9`. See also `reCfgIso`, the same relabelling along an equivalence `V \u2243 W`.","title":"Relabelling of an edge configuration by a vertex permutation"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/re-cfg-equiv","kind":"node","lean":"StatMech.FK.reCfgEquiv","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#re-cfg-equiv","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Relabelling by `\u03c3` as a permutation of the configuration space, with inverse relabelling by `\u03c3\u207b\u00b9`.","title":"Relabelling as a permutation of the configuration space"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-open-sub-iso","kind":"node","lean":"StatMech.FK.fkTI_openSubIso","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-open-sub-iso","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"An automorphism `\u03c3` of `G` is a graph isomorphism from the open subgraph of the relabelled configuration `reCfg \u03c3 \u03c9` to the open subgraph of `\u03c9`.","title":"Automorphism as an isomorphism of open subgraphs"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-fk-prob-re-cfg","kind":"node","lean":"StatMech.FK.fkTI_fkProb_reCfg","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-fk-prob-re-cfg","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Graph automorphisms preserve the free FK probability. See also `fvs_fkProb_reCfgIso` for isomorphisms between two graphs.","title":"Graph automorphisms preserve the free FK probability"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-wired-graph-iso","kind":"node","lean":"StatMech.FK.fkTI_wiredGraphIso","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-wired-graph-iso","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"An automorphism `\u03c3` of `G` preserving the boundary set is a graph isomorphism between the wired graphs (open edges plus all boundary pairs) of `reCfg \u03c3 \u03c9` and `\u03c9`.","title":"Boundary-preserving automorphism as an isomorphism of wired graphs"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-wired-fk-prob-re-cfg","kind":"node","lean":"StatMech.FK.fkTI_wiredFkProb_reCfg","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-wired-fk-prob-re-cfg","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Graph automorphisms preserving the boundary preserve the wired FK probability.","title":"Boundary-preserving automorphisms preserve the wired FK probability"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-edge-marg-prob-fk-prob-re-cfg","kind":"node","lean":"StatMech.FK.fkTI_edgeMargProb_fkProb_reCfg","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-edge-marg-prob-fk-prob-re-cfg","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Under the free FK measure, the edges `e` and `\u03c3 e` are open with the same probability.","title":"Graph automorphisms preserve free FK edge marginals"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-edge-marg-prob-wired-fk-prob-re-cf","kind":"node","lean":"StatMech.FK.fkTI_edgeMargProb_wiredFkProb_reCfg","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-edge-marg-prob-wired-fk-prob-re-cf","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Under the wired FK measure, the edges `e` and `\u03c3 e` are open with the same probability.","title":"Boundary-preserving automorphisms preserve wired FK edge marginals"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/box-sym","kind":"node","lean":"StatMech.FK.BoxSym","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box symmetry of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Box symmetry of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#box-sym","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"A symmetry of `\u2124^d` that preserves nearest-neighbour adjacency and every centered box `\u039b_n = [-n, n]^d`, such as a coordinate reflection or a coordinate permutation.","title":"Box symmetry of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/lift","kind":"node","lean":"StatMech.FK.BoxSym.lift","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#lift","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The restriction of a box symmetry to a permutation of the vertices of the box `\u039b_n`.","title":"Restriction of a box symmetry to a box"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-wired-density-shift-inv","kind":"node","lean":"StatMech.FK.fkTI_wired_density_shift_inv","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box symmetries preserve the wired "],["math","\\(q = 2\\)"],["text"," density of box edges"]],"source":"Box symmetries preserve the wired $q = 2$ density of box edges"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-wired-density-shift-inv","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Box symmetries preserve the wired `q = 2` infinite-volume edge density. See also `rot_wired_density_boxSym` for the statement on edges of `\u2124^d`.","title":"Box symmetries preserve the wired $q = 2$ density of box edges"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-free-density-shift-inv","kind":"node","lean":"StatMech.FK.fkTI_free_density_shift_inv","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box symmetries preserve the free "],["math","\\(q = 2\\)"],["text"," density of box edges"]],"source":"Box symmetries preserve the free $q = 2$ density of box edges"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-free-density-shift-inv","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Box symmetries preserve the free `q = 2` infinite-volume edge density. See also `rot_free_density_boxSym` for the statement on edges of `\u2124^d`.","title":"Box symmetries preserve the free $q = 2$ density of box edges"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-sign-flip-equiv","kind":"node","lean":"StatMech.FK.fkTI_signFlipEquiv","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reflection of "],["math","\\(\\mathbb{Z}^d\\)"],["text"," in one coordinate"]],"source":"Reflection of $\\mathbb{Z}^d$ in one coordinate"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-sign-flip-equiv","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The reflection `x \u21a6 x` with the `j`-th coordinate negated.","title":"Reflection of $\\mathbb{Z}^d$ in one coordinate"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fk-ti-sign-flip","kind":"node","lean":"StatMech.FK.fkTI_signFlip","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fk-ti-sign-flip","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The reflection in the `j`-th coordinate as a box symmetry.","title":"Coordinate reflection as a box symmetry"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/re-cfg-iso","kind":"node","lean":"StatMech.FK.reCfgIso","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#re-cfg-iso","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The pull-back of a configuration `\u03c9` on the edges of `W` along `\u03c3 : V \u2243 W`: the edge `e` is open in `reCfgIso \u03c3 \u03c9` iff `\u03c3 e` is open in `\u03c9`. For `V = W` it is `reCfg` (`fvs_reCfgIso_eq_reCfg`).","title":"Pull-back of an edge configuration along a vertex bijection"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/re-cfg-iso-equiv","kind":"node","lean":"StatMech.FK.reCfgIsoEquiv","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#re-cfg-iso-equiv","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Pull-back along `\u03c3` as an equivalence of configuration spaces, with inverse the pull-back along `\u03c3\u207b\u00b9`.","title":"Pull-back of configurations as an equivalence"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-re-cfg-iso-eq-re-cfg","kind":"node","lean":"StatMech.FK.fvs_reCfgIso_eq_reCfg","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-re-cfg-iso-eq-re-cfg","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"For a permutation `\u03c3` of `V`, pulling back along `\u03c3` is the relabelling `reCfg \u03c3`.","title":"Pull-back along a permutation is relabelling"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-open-sub-iso","kind":"node","lean":"StatMech.FK.fvs_openSubIso","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-open-sub-iso","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"An isomorphism `\u03c3 : G \u2243 H` is a graph isomorphism from the open subgraph of the pulled-back configuration `reCfgIso \u03c3 \u03c9` to the open subgraph of `\u03c9`.","title":"Open-subgraph isomorphism for a pulled-back configuration"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-fk-prob-re-cfg-iso","kind":"node","lean":"StatMech.FK.fvs_fkProb_reCfgIso","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-fk-prob-re-cfg-iso","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Isomorphic graphs have the same free FK probability, up to relabelling by the isomorphism. See also `fkTI_fkProb_reCfg`, the special case of an automorphism.","title":"Isomorphic graphs have equal free FK probabilities after relabelling"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-wired-graph-iso","kind":"node","lean":"StatMech.FK.fvs_wiredGraphIso","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-wired-graph-iso","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"An isomorphism `\u03c3 : G \u2243 H` matching the boundary sets is a graph isomorphism between the wired graphs of `reCfgIso \u03c3 \u03c9` and `\u03c9`.","title":"Wired-graph isomorphism for a pulled-back configuration"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-wired-fk-prob-re-cfg-iso","kind":"node","lean":"StatMech.FK.fvs_wiredFkProb_reCfgIso","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-wired-fk-prob-re-cfg-iso","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Isomorphic graphs with matching boundary sets have the same wired FK probability, up to relabelling. See also `fkTI_wiredFkProb_reCfg`.","title":"Boundary-matching isomorphisms preserve wired FK probabilities"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-finite-volume-shift-invariant","kind":"node","lean":"StatMech.FK.fvs_finiteVolume_shift_invariant","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-finite-volume-shift-invariant","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"Graph isomorphisms transport the wired and the free FK edge marginals: `\u03c3 e` is open in `H` with the same probability as `e` in `G`.","title":"Boundary-matching isomorphisms transport FK edge marginals"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-trans-box","kind":"node","lean":"StatMech.FK.fvs_transBox","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translated box in "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Translated box in $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-trans-box","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The translated box `v + \u039b_m = {y \u2208 \u2124^d : \u2016y - v\u2016_\u221e \u2264 m}`.","title":"Translated box in $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-trans-box-verts","kind":"node","lean":"StatMech.FK.fvs_transBoxVerts","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-trans-box-verts","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The vertices of the translated box `v + \u039b_m`, as a subtype of `\u2124^d`.","title":"Vertex type of a translated box"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-trans-equiv","kind":"node","lean":"StatMech.FK.fvs_transEquiv","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-trans-equiv","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The translation `x \u21a6 x + v` from the centered box `\u039b_m` onto `v + \u039b_m`.","title":"Translation from the centered box onto a translated box"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-trans-box-graph","kind":"node","lean":"StatMech.FK.fvs_transBoxGraph","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-trans-box-graph","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The nearest-neighbour graph of `\u2124^d` restricted to `v + \u039b_m`.","title":"Nearest-neighbour graph on a translated box"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-trans-box-boundary","kind":"node","lean":"StatMech.FK.fvs_transBoxBoundary","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-trans-box-boundary","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The vertex boundary of `v + \u039b_m`: the translate of the vertex boundary of `\u039b_m`.","title":"Vertex boundary of a translated box"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-box-translation-edge-marg","kind":"node","lean":"StatMech.FK.fvs_box_translation_edgeMarg","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-box-translation-edge-marg","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The wired and free FK edge marginals on the translated box `v + \u039b_m` are the marginals on the centered box `\u039b_m`, transported by `x \u21a6 x + v`.","title":"Translation covariance of FK edge marginals on boxes"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/fvs-trans-box-ne-box","kind":"node","lean":"StatMech.FK.fvs_transBox_ne_box","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","Some translated box leaves the centered box for "],["math","\\(d \\ge 1\\)"]],"source":"Some translated box leaves the centered box for $d \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#fvs-trans-box-ne-box","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"For `d \u2265 1`, some translated box `v + \u039b_m` is not contained in `\u039b_m`, so translation is not a box symmetry in the sense of `BoxSym`.","title":"Some translated box leaves the centered box for $d \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/open-sub-mono","kind":"node","lean":"StatMech.FK.openSub_mono","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#open-sub-mono","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The open subgraph is monotone in the configuration. The same statement is `StatMech.Sharpness.openSub_mono` in `BkCriterion`.","title":"Monotonicity of the open subgraph in the configuration"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/conn-event","kind":"node","lean":"StatMech.FK.connEvent","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#conn-event","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The event that `x` and `y` are joined by a path of open edges of `G`.","title":"Event that two vertices are joined by open edges"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/two-point-fun","kind":"node","lean":"StatMech.FK.twoPointFun","parent":47870,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#two-point-fun","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"The two-point function of the free random-cluster measure on a finite graph: the probability `\u03c6(x \u2194 y)` that `x` and `y` are connected by open edges.","title":"Two-point function of the free random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/two-point-fun-fkg","kind":"node","lean":"StatMech.FK.twoPointFun_fkg","parent":47870,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG inequality for connection events for "],["math","\\(q \\ge 1\\)"]],"source":"FKG inequality for connection events for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs/index.html#two-point-fun-fkg","scope":"StatisticalMechanics/random-cluster-model/random-cluster-measures-on-finite-graphs","status":"fully_proved","summary":"FKG for connection events: for `q \u2265 1`, `\u03c6(x \u2194 y) \u03c6(u \u2194 v) \u2264 \u03c6(x \u2194 y, u \u2194 v)`.","title":"FKG inequality for connection events for $q \\ge 1$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste.json","id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","kind":"scope","navigation":"module","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","title":"FKG inequality and monotonicity of random-cluster measures"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/edge-marginal-open","kind":"node","lean":"StatMech.FK.edgeMarginalOpen","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#edge-marginal-open","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The probability `\u03c6_{G,p,q}(e open)` that the edge `e` is open.","title":"Probability that an edge is open under the random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/edge-density-bounds","kind":"node","lean":"StatMech.FK.edge_density_bounds","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edge density lies between "],["math","\\(p/(p+q(1-p))\\)"],["text"," and "],["math","\\(p\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Edge density lies between $p/(p+q(1-p))$ and $p$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#edge-density-bounds","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1`, every edge of `G` is open with probability between `p / (p + q(1 - p))` and `p`.","title":"Edge density lies between $p/(p+q(1-p))$ and $p$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/card-connected-component-submodular","kind":"node","lean":"StatMech.FK.card_connectedComponent_submodular","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#card-connected-component-submodular","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The number of connected components of simple graphs on a finite vertex set is supermodular: `#comp(G\u2081) + #comp(G\u2082) \u2264 #comp(G\u2081 \u2294 G\u2082) + #comp(G\u2081 \u2293 G\u2082)`.","title":"Supermodularity of the number of connected components"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/num-clusters-supermodular","kind":"node","lean":"StatMech.FK.numClusters_supermodular","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#num-clusters-supermodular","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The cluster count of the random-cluster model is supermodular in the configuration.","title":"Supermodularity of the random-cluster cluster count"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-prob-fkglattice-condition","kind":"node","lean":"StatMech.FK.fkProb_FKGLatticeCondition","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-prob-fkglattice-condition","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` and `p \u2208 (0, 1)`, the random-cluster measure satisfies the FKG lattice condition `\u03c6(a) \u03c6(b) \u2264 \u03c6(a \u2294 b) \u03c6(a \u2293 b)`.","title":"FKG lattice condition for the random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-prob-positively-associated","kind":"node","lean":"StatMech.FK.fkProb_positively_associated","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-prob-positively-associated","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1`, increasing functions are positively correlated under `\u03c6_{G,p,q}`: `\u03c6(f) \u03c6(g) \u2264 \u03c6(f g)`. (Fortuin\u2013Kasteleyn\u2013Ginibre, Comm. Math. Phys. 22, 1971; Grimmett, *The Random-Cluster Model*, 2006, Theorem 3.8.)","title":"FKG inequality for the random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-prob-positively-associated-events","kind":"node","lean":"StatMech.FK.fkProb_positively_associated_events","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-prob-positively-associated-events","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The FKG inequality for increasing events: `\u03c6(A) \u03c6(B) \u2264 \u03c6(A \u2229 B)` for `q \u2265 1`.","title":"FKG inequality for increasing random-cluster events"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/ker-lap","kind":"node","lean":"StatMech.FK.kerLap","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#ker-lap","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The kernel of the Laplacian of the open subgraph of `\u03c9`, a subspace of `V \u2192 \u211d`.","title":"Kernel of the Laplacian of the open subgraph"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/num-clusters-eq-finrank","kind":"node","lean":"StatMech.FK.numClusters_eq_finrank","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#num-clusters-eq-finrank","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The number of open clusters equals the dimension of the kernel of the Laplacian of the open subgraph.","title":"Number of open clusters equals the dimension of the Laplacian kernel"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-prob-cross","kind":"node","lean":"StatMech.FK.fkProb_cross","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holley's cross inequality for FK measures, "],["math","\\(p_1 \\le p_2\\)"],["text",", "],["math","\\(q \\ge 1\\)"]],"source":"Holley's cross inequality for FK measures, $p_1 \\le p_2$, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-prob-cross","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Holley's cross inequality for two random-cluster measures with `p\u2081 \u2264 p\u2082` and the same `q \u2265 1`.","title":"Holley's cross inequality for FK measures, $p_1 \\le p_2$, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-prob-stochastically-increasing","kind":"node","lean":"StatMech.FK.fkProb_stochastically_increasing","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-prob-stochastically-increasing","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` and `p\u2081 \u2264 p\u2082`, `\u03c6_{p\u2081,q}(A) \u2264 \u03c6_{p\u2082,q}(A)` for every increasing event `A` (Grimmett, *The Random-Cluster Model*, 2006, Theorem 3.21).","title":"Monotonicity of the random-cluster measure in `p`"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/ch-open-count","kind":"node","lean":"StatMech.FK.chOpenCount","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#ch-open-count","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The number of open edges of `G` in `\u03c9`. Same as `openCount` in `Tilt`.","title":"Number of open edges of a configuration"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/diff-set","kind":"node","lean":"StatMech.FK.diffSet","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#diff-set","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The edges of `G` closed in `b` and open in `c`.","title":"Edges closed in one configuration and open in another"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/num-clusters-add-open-count-mono","kind":"node","lean":"StatMech.FK.numClusters_add_openCount_mono","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#num-clusters-add-open-count-mono","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `b \u2264 c`, `k(b) + o(b) \u2264 k(c) + o(c)`: opening an edge lowers the cluster count by at most one.","title":"Cluster count plus open-edge count is monotone"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/reduced-density","kind":"node","lean":"StatMech.FK.reducedDensity","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reduced density "],["math","\\(p/(p + q(1-p))\\)"]],"source":"Reduced density $p/(p + q(1-p))$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#reduced-density","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The reduced density `p / (p + q(1 - p))`: the Bernoulli parameter dominated by `\u03c6_{p,q}`.","title":"Reduced density $p/(p + q(1-p))$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-monotone-in-p","kind":"node","lean":"StatMech.FK.fk_monotone_in_p","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Random-cluster measure increases stochastically in "],["math","\\(p\\)"],["text"," when "],["math","\\(q \\ge 1\\)"]],"source":"Random-cluster measure increases stochastically in $p$ when $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-monotone-in-p","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` the random-cluster measure is stochastically increasing in `p`.","title":"Random-cluster measure increases stochastically in $p$ when $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-bernoulli-sandwich","kind":"node","lean":"StatMech.FK.fk_bernoulli_sandwich","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-bernoulli-sandwich","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1`, `P_{p/(p + q(1 - p))} \u227c \u03c6_{p,q} \u227c P_p`, with every law extended uniformly to loops and nonedges. On graph-edge marginals, this gives the usual FK/Bernoulli comparison (Fortuin 1972; Grimmett, *The Random-Cluster Model*, 2006, Theorem 3.21).","title":"Comparison inequalities"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/mixed-supermodular","kind":"node","lean":"StatMech.FK.mixed_supermodular","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#mixed-supermodular","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Mixed supermodularity of cluster counts: `k(a) + k\u00b9(b) \u2264 k(a \u2293 b) + k\u00b9(a \u2294 b)`, where `k\u00b9` counts clusters with the boundary wired.","title":"Mixed supermodularity of free and wired cluster counts"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/free-wired-weight-cross","kind":"node","lean":"StatMech.FK.freeWiredWeight_cross","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holley's condition for free and wired weights when "],["math","\\(q \\ge 1\\)"]],"source":"Holley's condition for free and wired weights when $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#free-wired-weight-cross","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Holley's condition for the free and wired weights when `q \u2265 1`. See also `bcWeight_cross`, which contains this as the case of boundary wirings `\u22a5 \u2264 clique`.","title":"Holley's condition for free and wired weights when $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/free-wired-prob-cross","kind":"node","lean":"StatMech.FK.freeWiredProb_cross","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holley's condition for free and wired probabilities when "],["math","\\(q \\ge 1\\)"]],"source":"Holley's condition for free and wired probabilities when $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#free-wired-prob-cross","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Holley's condition for the free and wired probabilities when `q \u2265 1`.","title":"Holley's condition for free and wired probabilities when $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-prob-le-wired-fk-prob-increasing","kind":"node","lean":"StatMech.FK.fkProb_le_wiredFkProb_increasing","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free\u2013wired domination on a finite graph for "],["math","\\(q \\ge 1\\)"]],"source":"Free\u2013wired domination on a finite graph for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-prob-le-wired-fk-prob-increasing","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` the free random-cluster measure gives every increasing event at most the probability the wired measure gives it.","title":"Free\u2013wired domination on a finite graph for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/free-finite-measure-dominated","kind":"node","lean":"StatMech.FK.freeFiniteMeasure_dominated","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free box measure dominated by the wired one for "],["math","\\(q \\ge 1\\)"]],"source":"Free box measure dominated by the wired one for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#free-finite-measure-dominated","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` the free box measure is stochastically dominated by the wired box measure on `\u2124^d`: `\u03c6\u2070_n \u227c \u03c6\u00b9_n`.","title":"Free box measure dominated by the wired one for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/num-clusters-bc","kind":"node","lean":"StatMech.FK.numClustersBC","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#num-clusters-bc","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The number of clusters of `\u03c9` under the boundary wiring `C`: connected components of the open subgraph of `\u03c9` joined with `C`.","title":"Number of clusters under a boundary wiring"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/mixed-supermodular-bc","kind":"node","lean":"StatMech.FK.mixed_supermodular_bc","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#mixed-supermodular-bc","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Mixed supermodularity of cluster counts for nested wirings `C \u2264 C'`: `k_C(a) + k_C'(b) \u2264 k_C(a \u2293 b) + k_C'(a \u2294 b)`.","title":"Mixed supermodularity of cluster counts for nested wirings"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-weight","kind":"node","lean":"StatMech.FK.bcWeight","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-weight","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The unnormalized random-cluster weight with boundary wiring `C`: `edgeProduct G p \u03c9 \u00b7 q ^ numClustersBC G C \u03c9`.","title":"Random-cluster weight with a boundary wiring"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-z","kind":"node","lean":"StatMech.FK.bcZ","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-z","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The partition function with boundary wiring `C`.","title":"Partition function with a boundary wiring"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob","kind":"node","lean":"StatMech.FK.bcProb","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The random-cluster probability with boundary wiring `C`, `bcWeight / bcZ`.","title":"Random-cluster probability with a boundary wiring"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-weight-cross","kind":"node","lean":"StatMech.FK.bcWeight_cross","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holley's condition for weights with nested wirings, "],["math","\\(q \\ge 1\\)"]],"source":"Holley's condition for weights with nested wirings, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-weight-cross","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Holley's condition for the weights with nested wirings `C \u2264 C'` and `q \u2265 1`.","title":"Holley's condition for weights with nested wirings, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-cross","kind":"node","lean":"StatMech.FK.bcProb_cross","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holley's condition for probabilities with nested wirings, "],["math","\\(q \\ge 1\\)"]],"source":"Holley's condition for probabilities with nested wirings, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-cross","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Holley's condition for the probabilities with nested wirings `C \u2264 C'` and `q \u2265 1`.","title":"Holley's condition for probabilities with nested wirings, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-positively-associated","kind":"node","lean":"StatMech.FK.bcProb_positively_associated","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG inequality for the random-cluster measure, "],["math","\\(q \\ge 1\\)"]],"source":"FKG inequality for the random-cluster measure, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-positively-associated","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"FKG inequality for the random-cluster measure (Fortuin, Kasteleyn and Ginibre, 1971): for `q \u2265 1` and any boundary wiring `C`, increasing functions are positively correlated under `\u03c6_C`. See Grimmett, *The Random-Cluster Model* (2006), Chapter 3.","title":"FKG inequality for the random-cluster measure, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-mono-bc","kind":"node","lean":"StatMech.FK.bcProb_mono_bc","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Random-cluster comparison of boundary conditions for "],["math","\\(q \\ge 1\\)"]],"source":"Random-cluster comparison of boundary conditions for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-mono-bc","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Comparison of boundary conditions for the random-cluster measure: for `q \u2265 1` and wirings `C \u2264 C'`, `\u03c6_C(A) \u2264 \u03c6_C'(A)` for every increasing event `A`. This follows from Holley's inequality (Holley, 1974); see Grimmett, *The Random-Cluster Model* (2006), Chapter 4.","title":"Random-cluster comparison of boundary conditions for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-dlr-sandwich","kind":"node","lean":"StatMech.FK.bcProb_dlr_sandwich","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-dlr-sandwich","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` and `C` inside the boundary clique, `\u03c6^0(A) \u2264 \u03c6^C(A) \u2264 \u03c6^1(A)` for every increasing event `A`.","title":"Free\u2013wired sandwich for boundary graphs inside the boundary clique"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-dlr-sandwich-top","kind":"node","lean":"StatMech.FK.bcProb_dlr_sandwich_top","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-dlr-sandwich-top","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The free lower bound together with the bound by the boundary condition `\u22a4` (all vertices wired).","title":"Sandwich between the free and all-wired boundary conditions"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/edge-marg-prob","kind":"node","lean":"StatMech.FK.edgeMargProb","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#edge-marg-prob","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The probability `\u03bc(\u03c9(e) = 1)` that the coordinate `e` is open under the mass function `\u03bc`.","title":"Edge marginal of a mass function"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/event-mass-prob","kind":"node","lean":"StatMech.FK.eventMassProb","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#event-mass-prob","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The mass `\u03bc(A)` of an event under the mass function `\u03bc`.","title":"Mass of an event under a mass function"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/is-monotone-coupling-fun","kind":"node","lean":"StatMech.FK.IsMonotoneCouplingFun","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#is-monotone-coupling-fun","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"`P` is a monotone coupling of the mass functions `\u03bc` and `\u03bd`: a nonnegative mass function on pairs with marginals `\u03bc` and `\u03bd`, supported on pairs `(\u03c9, \u03c9')` with `\u03c9 \u2264 \u03c9'`.","title":"Monotone coupling of two mass functions"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/event-mass-eq-of-edge-marg-eq","kind":"node","lean":"StatMech.FK.eventMass_eq_of_edgeMarg_eq","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#event-mass-eq-of-edge-marg-eq","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Two monotonically coupled measures with the same edge marginals give the same mass to every increasing event.","title":"Monotone couplings with equal edge marginals agree on up-sets"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-event-mass","kind":"node","lean":"StatMech.FK.bcEventMass","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-event-mass","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The mass `\u03c6^C_{G,p,q}(A)` of an event under the random-cluster measure with boundary graph `C`.","title":"Event mass under the random-cluster measure with a boundary graph"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-event-mass-le-qpow","kind":"node","lean":"StatMech.FK.bcEventMass_le_qpow","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary-graph comparison "],["math","\\(\\varphi^{C'}(A) \\le q^N \\varphi^C(A)\\)"]],"source":"Boundary-graph comparison $\\varphi^{C'}(A) \\le q^N \\varphi^C(A)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-event-mass-le-qpow","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"If the cluster counts with boundary graphs `C \u2264 C'` differ by at most `N`, then `\u03c6^C(A) \u2264 q^N \u03c6^{C'}(A)` for nonnegative event indicators.","title":"Boundary-graph comparison $\\varphi^{C'}(A) \\le q^N \\varphi^C(A)$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-le-boundary-clique","kind":"node","lean":"StatMech.FK.bc_le_boundaryClique","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-le-boundary-clique","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"A wiring whose edges join boundary vertices lies below the boundary clique.","title":"Boundary-supported wirings lie below the boundary clique"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-bot-eq-fk-prob","kind":"node","lean":"StatMech.FK.bcProb_bot_eq_fkProb","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-bot-eq-fk-prob","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The empty wiring gives the free random-cluster measure.","title":"Empty wiring gives the free random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-clique-eq-wired-fk-prob","kind":"node","lean":"StatMech.FK.bcProb_clique_eq_wiredFkProb","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-clique-eq-wired-fk-prob","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The boundary clique gives the wired random-cluster measure.","title":"Boundary clique gives the wired random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/fk-prob-le-bc-prob-le-wired-fk-prob","kind":"node","lean":"StatMech.FK.fkProb_le_bcProb_le_wiredFkProb","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary wirings lie between free and wired when "],["math","\\(q \\ge 1\\)"]],"source":"Boundary wirings lie between free and wired when $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#fk-prob-le-bc-prob-le-wired-fk-prob","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` and a wiring `C` supported on the boundary, free \u2264 `\u03c6_C` \u2264 wired on every increasing event.","title":"Boundary wirings lie between free and wired when $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-monotone-in-p","kind":"node","lean":"StatMech.FK.bcProb_monotone_in_p","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary-graph FK measure increases in "],["math","\\(p\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Boundary-graph FK measure increases in $p$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-monotone-in-p","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1`, the random-cluster measure with boundary graph `C` is increasing in `p` on increasing events.","title":"Boundary-graph FK measure increases in $p$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/induced-bc-z","kind":"node","lean":"StatMech.FK.inducedBcZ","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#induced-bc-z","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The partition function of the fibre of `\u03c8` off `F`: the sum of `bcWeight` over configurations agreeing with `\u03c8` off `F`.","title":"Partition function of a conditioning fibre"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/cond-bc-prob","kind":"node","lean":"StatMech.FK.condBcProb","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#cond-bc-prob","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"The random-cluster measure with wiring `C` conditioned on agreeing with `\u03c8` off the edge set `F`.","title":"Conditional random-cluster measure with a boundary wiring"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/cond-bc-prob-cross","kind":"node","lean":"StatMech.FK.condBcProb_cross","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holley's condition for nested-wiring conditional measures, "],["math","\\(q \\ge 1\\)"]],"source":"Holley's condition for nested-wiring conditional measures, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#cond-bc-prob-cross","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Holley's condition for the conditional measures with nested wirings `C \u2264 C'` and `q \u2265 1`.","title":"Holley's condition for nested-wiring conditional measures, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/cond-bc-prob-mono-bc","kind":"node","lean":"StatMech.FK.condBcProb_mono_bc","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional measures increase with the wiring for "],["math","\\(q \\ge 1\\)"]],"source":"Conditional measures increase with the wiring for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#cond-bc-prob-mono-bc","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` and `C \u2264 C'`, the conditional measure with wiring `C` is dominated by the one with wiring `C'` on increasing events.","title":"Conditional measures increase with the wiring for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/bc-prob-eq-fibre-mass-mul-cond-bc-prob","kind":"node","lean":"StatMech.FK.bcProb_eq_fibreMass_mul_condBcProb","parent":47948,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#bc-prob-eq-fibre-mass-mul-cond-bc-prob","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Disintegration: `\u03c6_C(\u03c9)` is the mass of the fibre of `\u03c9` times the conditional probability of `\u03c9` in that fibre.","title":"Disintegration of a boundary-wired measure over conditioning fibres"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/wired-fk-prob-inner-dominated","kind":"node","lean":"StatMech.FK.wiredFkProb_inner_dominated","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wirings with the boundary clique dominate the wired measure, "],["math","\\(q \\ge 1\\)"]],"source":"Wirings with the boundary clique dominate the wired measure, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#wired-fk-prob-inner-dominated","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"For `q \u2265 1` the wired measure is dominated by `\u03c6_C'` for any wiring `C'` containing the boundary clique.","title":"Wirings with the boundary clique dominate the wired measure, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/wired-fk-prob-inner-dominated-cond","kind":"node","lean":"StatMech.FK.wiredFkProb_inner_dominated_cond","parent":47948,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired measure conditionally dominated by larger wirings, "],["math","\\(q \\ge 1\\)"]],"source":"Wired measure conditionally dominated by larger wirings, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste/index.html#wired-fk-prob-inner-dominated-cond","scope":"StatisticalMechanics/random-cluster-model/fkg-inequality-and-monotonicity-of-random-cluste","status":"fully_proved","summary":"Conditional version of `wiredFkProb_inner_dominated`.","title":"Wired measure conditionally dominated by larger wirings, $q \\ge 1$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures","kind":"scope","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures","status":"fully_proved","title":"Infinite-volume random-cluster measures"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/dfi-open-edge-count","kind":"node","lean":"StatMech.FK.dfi_openEdgeCount","parent":48000,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/index.html#dfi-open-edge-count","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures","status":"fully_proved","summary":"The number of open edges of `\u03c9` in the finite set `E`, as a real number.","title":"Number of open edges in a finite edge set"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-edge-open-event","kind":"node","lean":"StatMech.FK.boxEdgeOpenEvent","parent":48000,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/index.html#box-edge-open-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures","status":"fully_proved","summary":"The event that the box edge `eb` of `\u039b_N` is open.","title":"Event that a box edge is open"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","title":"Finite- and infinite-volume FK measures on \u2124^d"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/wired-fk-weight","kind":"node","lean":"StatMech.FK.wiredFkWeight","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#wired-fk-weight","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The unnormalized wired random-cluster weight `edgeProduct G p \u03c9 \u00b7 q ^ k\u00b9(\u03c9)`, where `k\u00b9` counts clusters after wiring all boundary vertices together.","title":"Wired random-cluster weight"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/wired-fk-z","kind":"node","lean":"StatMech.FK.wiredFkZ","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#wired-fk-z","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The wired partition function: the sum of `wiredFkWeight` over all configurations.","title":"Wired random-cluster partition function"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/wired-fk-prob","kind":"node","lean":"StatMech.FK.wiredFkProb","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#wired-fk-prob","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The wired random-cluster probability of a configuration, `wiredFkWeight / wiredFkZ`.","title":"Wired random-cluster probability of a configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/wired-fk-pmf","kind":"node","lean":"StatMech.FK.wiredFkPMF","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#wired-fk-pmf","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The wired random-cluster measure on a finite graph, as a probability mass function.","title":"Wired random-cluster measure as a probability mass function"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/box-verts","kind":"node","lean":"StatMech.FK.boxVerts","parent":48003,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vertices of the box "],["math","\\((-n, n)^d\\)"]],"source":"Vertices of the box $(-n, n)^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#box-verts","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The vertices of the box `\u039b_n = [-n, n]^d \u2286 \u2124^d`, as a subtype.","title":"Vertices of the box $(-n, n)^d$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/box-graph","kind":"node","lean":"StatMech.FK.boxGraph","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#box-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The nearest-neighbour graph on the box `\u039b_n`, induced from the hypercubic lattice.","title":"Nearest-neighbour graph on a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/box-boundary","kind":"node","lean":"StatMech.FK.boxBoundary","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#box-boundary","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The vertex boundary of the box: sites of `\u039b_n` that are not in `\u039b_(n-1)`.","title":"Vertex boundary of a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/edge-incl","kind":"node","lean":"StatMech.FK.edgeIncl","parent":48003,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inclusion of box edges into "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Inclusion of box edges into $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#edge-incl","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The inclusion of edges of the box into edges of `\u2124^d`.","title":"Inclusion of box edges into $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/extend-edge","kind":"node","lean":"StatMech.FK.extendEdge","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#extend-edge","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"Extend a configuration on the box to `\u2124^d` by declaring every edge outside the box closed.","title":"Extension of a box configuration by closed edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/free-finite-measure","kind":"node","lean":"StatMech.FK.freeFiniteMeasure","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#free-finite-measure","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The free random-cluster measure `\u03c6\u2070_n` on the box `\u039b_n`, pushed to the configuration space of `\u2124^d` by `extendEdge`.","title":"Free random-cluster measure on a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/wired-finite-measure","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#wired-finite-measure","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"The wired random-cluster measure `\u03c6\u00b9_n` on the box `\u039b_n` (boundary vertices wired), pushed to the configuration space of `\u2124^d` by `extendEdge`.","title":"Wired random-cluster measure on a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/free-infinite-volume","kind":"node","lean":"StatMech.FK.freeInfiniteVolume","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#free-infinite-volume","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"A free infinite-volume random-cluster measure on `\u2124^d`: a weak limit of a subsequence of `freeFiniteMeasure d n`, chosen by Prokhorov compactness.","title":"Free infinite-volume random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/free-infinite-volume-is-limit","kind":"node","lean":"StatMech.FK.freeInfiniteVolume_isLimit","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#free-infinite-volume-is-limit","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"`freeInfiniteVolume` is the weak limit of `freeFiniteMeasure` along some strictly increasing sequence of box sizes.","title":"Free infinite-volume measure is a subsequential weak limit"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/wired-infinite-volume","kind":"node","lean":"StatMech.FK.wiredInfiniteVolume","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#wired-infinite-volume","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"A wired infinite-volume random-cluster measure on `\u2124^d`: a weak limit of a subsequence of `wiredFiniteMeasure d n`, chosen by Prokhorov compactness.","title":"Wired infinite-volume random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/wired-infinite-volume-is-limit","kind":"node","lean":"StatMech.FK.wiredInfiniteVolume_isLimit","parent":48003,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d/index.html#wired-infinite-volume-is-limit","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/finite-and-infinite-volume-fk-measures-on-d","status":"fully_proved","summary":"`wiredInfiniteVolume` is the weak limit of `wiredFiniteMeasure` along some strictly increasing sequence of box sizes.","title":"Wired infinite-volume measure is a subsequential weak limit"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","title":"The wired box measure as a conditional measure on the next box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/box-vert-incl","kind":"node","lean":"StatMech.FK.boxVertIncl","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#box-vert-incl","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The inclusion of the vertices of `\u039b_n` into `\u039b_{n+1}`. The case `m = n + 1` of `boxVertInclLE`.","title":"Inclusion of box vertices into the next box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/box-graph-emb","kind":"node","lean":"StatMech.FK.boxGraphEmb","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#box-graph-emb","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"`\u039b_n` as an induced subgraph of `\u039b_{n+1}`.","title":"Box as an induced subgraph of the next box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/inner-edge","kind":"node","lean":"StatMech.FK.innerEdge","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#inner-edge","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The inclusion of edges (unordered pairs) of `\u039b_n` into `\u039b_{n+1}`.","title":"Inclusion of box edges into the next box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/is-inner-vert","kind":"node","lean":"StatMech.FK.IsInnerVert","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#is-inner-vert","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"A vertex of `\u039b_{n+1}` lies in `\u039b_n`.","title":"Vertex of the next box lying in the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-ext","kind":"node","lean":"StatMech.FK.crossExt","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The extension of a configuration of `\u039b_n` to `\u039b_{n+1}` that closes every edge not coming from `\u039b_n`.","title":"Extension of a configuration closing the annulus edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/inner-bdry","kind":"node","lean":"StatMech.FK.innerBdry","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#inner-bdry","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"A vertex of `\u039b_{n+1}` on the vertex boundary of `\u039b_n`.","title":"Vertex on the boundary of the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-wired-graph","kind":"node","lean":"StatMech.FK.crossWiredGraph","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-wired-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The open subgraph of `crossExt \u03c9` in `\u039b_{n+1}` with the boundary of `\u039b_n` wired.","title":"Extended open graph with the inner box boundary wired"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/inner-wired-graph","kind":"node","lean":"StatMech.FK.innerWiredGraph","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#inner-wired-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The open subgraph of `\u03c9` in `\u039b_n` with the boundary of `\u039b_n` wired.","title":"Open graph of a box configuration with the box boundary wired"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/inner-wired-hom","kind":"node","lean":"StatMech.FK.innerWiredHom","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#inner-wired-hom","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The inclusion of the wired open graph of `\u039b_n` into `crossWiredGraph`.","title":"Inclusion of the wired inner graph into the extended wired graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-component-equiv-fun","kind":"node","lean":"StatMech.FK.crossComponentEquivFun","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-component-equiv-fun","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"Components of `crossWiredGraph` from wired components of `\u03c9` in `\u039b_n` and annulus vertices.","title":"Map from inner wired components and annulus vertices to components"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-component-equiv","kind":"node","lean":"StatMech.FK.crossComponentEquiv","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-component-equiv","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"Components of `crossWiredGraph` are the wired components of `\u03c9` in `\u039b_n` plus one singleton per annulus vertex.","title":"Extended-graph components as inner components plus annulus vertices"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/annulus-card","kind":"node","lean":"StatMech.FK.annulusCard","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#annulus-card","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The number of vertices of the annulus `\u039b_{n+1} \\ \u039b_n`.","title":"Number of annulus vertices between consecutive boxes"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/non-inner-edge-card","kind":"node","lean":"StatMech.FK.nonInnerEdgeCard","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#non-inner-edge-card","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The number of edges of `\u039b_{n+1}` not coming from `\u039b_n`.","title":"Number of next-box edges not coming from the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-shift","kind":"node","lean":"StatMech.FK.crossShift","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-shift","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The constant `(1 - p)^{#annulus edges} q^{#annulus vertices}` relating the extended weight to the wired weight on `\u039b_n`.","title":"Annulus constant linking the extended and wired inner weights"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/inner-edge-finset","kind":"node","lean":"StatMech.FK.innerEdgeFinset","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#inner-edge-finset","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The edges of `\u039b_{n+1}` coming from `\u039b_n`.","title":"Edges of the next box coming from the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/inner-restrict","kind":"node","lean":"StatMech.FK.innerRestrict","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#inner-restrict","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The restriction of a configuration of `\u039b_{n+1}` to the edges of `\u039b_n`.","title":"Restriction of a next-box configuration to the inner edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/inner-fibre-equiv","kind":"node","lean":"StatMech.FK.innerFibreEquiv","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#inner-fibre-equiv","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"Configurations of `\u039b_{n+1}` closed off the inner edges correspond to configurations of `\u039b_n`.","title":"Configurations closed off the inner edges as inner-box configurations"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cond-bc-prob-cross-ext-eq-wired-fk-prob","kind":"node","lean":"StatMech.FK.condBcProb_crossExt_eq_wiredFkProb","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cond-bc-prob-cross-ext-eq-wired-fk-prob","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"Conditioned on every edge outside `\u039b_n` being closed, the random-cluster measure on `\u039b_{n+1}` with the boundary of `\u039b_n` wired is the wired random-cluster measure on `\u039b_n` (Grimmett, *The Random-Cluster Model*, 2006, Lemma 4.13).","title":"Domain Markov property for the wired box measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/is-annulus-wiring","kind":"node","lean":"StatMech.FK.IsAnnulusWiring","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#is-annulus-wiring","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"Every edge of `D` joins vertices of the annulus `\u039b_{n+1} \\ \u039b_n`.","title":"Wiring with every edge inside the annulus"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-wired-graph-add","kind":"node","lean":"StatMech.FK.crossWiredGraphAdd","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-wired-graph-add","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"`crossWiredGraph` with the extra wiring `D` added.","title":"Extended wired graph with an extra wiring added"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/annulus-vert","kind":"node","lean":"StatMech.FK.AnnulusVert","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#annulus-vert","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The vertices of the annulus `\u039b_{n+1} \\ \u039b_n`.","title":"Vertices of the annulus between consecutive boxes"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/annulus-subgraph","kind":"node","lean":"StatMech.FK.annulusSubgraph","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#annulus-subgraph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The restriction of `D` to the annulus vertices.","title":"Restriction of a wiring to the annulus vertices"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/inner-wired-hom-add","kind":"node","lean":"StatMech.FK.innerWiredHomAdd","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#inner-wired-hom-add","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The inclusion of the wired open graph of `\u039b_n` into `crossWiredGraphAdd`.","title":"Inclusion of the wired inner graph into the extra-wired graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/annulus-hom-add","kind":"node","lean":"StatMech.FK.annulusHomAdd","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#annulus-hom-add","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The inclusion of the annulus wiring into `crossWiredGraphAdd`.","title":"Inclusion of the annulus wiring into the extra-wired graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-component-equiv-add-fun","kind":"node","lean":"StatMech.FK.crossComponentEquivAddFun","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-component-equiv-add-fun","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"Components of `crossWiredGraphAdd` from wired components of `\u03c9` in `\u039b_n` and components of the annulus wiring.","title":"Map from inner and annulus-wiring components to extra-wired ones"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-component-equiv-add","kind":"node","lean":"StatMech.FK.crossComponentEquivAdd","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-component-equiv-add","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"Components of `crossWiredGraphAdd` are the wired components of `\u03c9` in `\u039b_n` plus the components of the annulus wiring.","title":"Extra-wired components as inner plus annulus-wiring components"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/cross-shift-add","kind":"node","lean":"StatMech.FK.crossShiftAdd","parent":48019,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o/index.html#cross-shift-add","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-box-measure-as-a-conditional-measure-o","status":"fully_proved","summary":"The constant `(1 - p)^{#annulus edges} q^{#components of the annulus wiring}`.","title":"Annulus constant counting the components of an extra wiring"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","title":"Boundary conditions induced on a box by the outer annulus"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/psi-ext","kind":"node","lean":"StatMech.FK.psiExt","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"Glue a configuration `\u03c9` on `\u039b_n` into an outer configuration `\u03c8` on `\u039b_(n+1)`: edges of `\u039b_n` follow `\u03c9`, the others follow `\u03c8`.","title":"Gluing a box configuration into an outer configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/outside-open-graph","kind":"node","lean":"StatMech.FK.outsideOpenGraph","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#outside-open-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The open edges of `\u03c8` on `\u039b_(n+1)` that are not edges of `\u039b_n`.","title":"Open edges of an outer configuration outside the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/outside-graph","kind":"node","lean":"StatMech.FK.outsideGraph","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#outside-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The outer graph: open outer edges of `\u03c8` together with the clique on `\u2202\u039b_(n+1)`.","title":"Outer open graph plus the outer boundary clique"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/induced-wiring","kind":"node","lean":"StatMech.FK.inducedWiring","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#induced-wiring","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The wiring induced on `\u039b_n` by `\u03c8`: distinct vertices of `\u039b_n` connected in the outer graph.","title":"Wiring induced on a box through the outer graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/inner-image-graph","kind":"node","lean":"StatMech.FK.innerImageGraph","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#inner-image-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The open edges of `\u03c9`, viewed as edges of `\u039b_(n+1)`.","title":"Inner open edges viewed in the larger box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/psi-wired-graph-eq-sup","kind":"node","lean":"StatMech.FK.psiWiredGraph_eq_sup","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#psi-wired-graph-eq-sup","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The wired open graph of the glued configuration is the union of the inner open edges and the outer graph.","title":"Wired glued open graph is the union of inner and outer graphs"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/full-graph","kind":"node","lean":"StatMech.FK.fullGraph","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#full-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The wired open graph of the glued configuration on `\u039b_(n+1)`.","title":"Wired open graph of a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/inner-graph","kind":"node","lean":"StatMech.FK.innerGraph","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#inner-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The open graph of `\u03c9` on `\u039b_n` with the induced wiring added.","title":"Inner open graph with the induced wiring"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/full-graph-reachable-box-vert-incl","kind":"node","lean":"StatMech.FK.fullGraph_reachable_boxVertIncl","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#full-graph-reachable-box-vert-incl","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"Two vertices of `\u039b_n` are connected in the glued graph iff they are connected in `innerGraph`.","title":"Glued and inner graphs have the same connections on the box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/outside-only","kind":"node","lean":"StatMech.FK.OutsideOnly","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#outside-only","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"A vertex of `\u039b_(n+1)` whose outer component contains no vertex of `\u039b_n`.","title":"Outer vertex whose outer component avoids the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/is-outside-only","kind":"node","lean":"StatMech.FK.IsOutsideOnly","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#is-outside-only","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"`OutsideOnly` as a predicate on components of the outer graph.","title":"Outer component avoiding the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/outside-only-comp","kind":"node","lean":"StatMech.FK.OutsideOnlyComp","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#outside-only-comp","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The components of the outer graph that avoid `\u039b_n`.","title":"Outer components avoiding the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/full-component-map","kind":"node","lean":"StatMech.FK.fullComponentMap","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#full-component-map","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The map from components of `innerGraph` and outer components avoiding `\u039b_n` to components of the glued graph.","title":"Component map into the wired glued graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/full-component-equiv","kind":"node","lean":"StatMech.FK.fullComponentEquiv","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#full-component-equiv","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"Components of the glued graph are components of `innerGraph` plus the outer components avoiding `\u039b_n`.","title":"Component decomposition of the wired glued graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/outside-shift","kind":"node","lean":"StatMech.FK.outsideShift","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#outside-shift","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The number of outer components avoiding `\u039b_n`.","title":"Number of outer components avoiding the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/num-clusters-bc-psi-ext","kind":"node","lean":"StatMech.FK.numClustersBC_psiExt","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#num-clusters-bc-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The wired cluster count of a glued configuration is the induced cluster count of `\u03c9` plus `outsideShift d n \u03c8`.","title":"Wired cluster count of a glued configuration splits additively"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/psi-edge-factor","kind":"node","lean":"StatMech.FK.psiEdgeFactor","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#psi-edge-factor","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The Bernoulli edge product over the edges of `\u039b_(n+1)` outside `\u039b_n`, evaluated at `\u03c8`.","title":"Outer Bernoulli edge factor of an outer configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/edge-product-psi-ext","kind":"node","lean":"StatMech.FK.edgeProduct_psiExt","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#edge-product-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The edge product of a glued configuration factorizes into the inner edge product and `psiEdgeFactor`.","title":"Edge product of a glued configuration factorizes"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/psi-shift","kind":"node","lean":"StatMech.FK.psiShift","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#psi-shift","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The factor depending only on `\u03c8`: `psiEdgeFactor \u00b7 q ^ outsideShift`.","title":"Outer weight factor of an outer configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/cond-bc-prob-psi-ext-eq-bc-prob","kind":"node","lean":"StatMech.FK.condBcProb_psiExt_eq_bcProb","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#cond-bc-prob-psi-ext-eq-bc-prob","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"Domain Markov property: the wired measure on `\u039b_(n+1)`, conditioned on the outer configuration `\u03c8`, is the random-cluster measure on `\u039b_n` with the induced wiring.","title":"Domain Markov property for the wired measure on nested boxes"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/induced-wiring-le-wired","kind":"node","lean":"StatMech.FK.inducedWiring_le_wired","parent":48047,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#induced-wiring-le-wired","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"The induced wiring lies below the clique on `\u2202\u039b_n`.","title":"Induced wiring lies below the inner boundary clique"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/cond-bc-prob-psi-ext-dominated-wired","kind":"node","lean":"StatMech.FK.condBcProb_psiExt_dominated_wired","parent":48047,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired box measure dominates the induced-wiring measure for "],["math","\\(q \\ge 1\\)"]],"source":"Wired box measure dominates the induced-wiring measure for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute/index.html#cond-bc-prob-psi-ext-dominated-wired","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/boundary-conditions-induced-on-a-box-by-the-oute","status":"fully_proved","summary":"For `q \u2265 1`, the measure on `\u039b_n` with the induced wiring is dominated by the wired measure on `\u039b_n`.","title":"Wired box measure dominates the induced-wiring measure for $q \\ge 1$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits","status":"fully_proved","title":"Ordering of free and wired subsequential limits"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits/weak-limit-extremality-of-common-subseq","kind":"node","lean":"StatMech.FK.weakLimit_extremality_of_common_subseq","parent":48070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free limit at most wired limit along a common subsequence, "],["math","\\(q \\ge 1\\)"]],"source":"Free limit at most wired limit along a common subsequence, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits/index.html#weak-limit-extremality-of-common-subseq","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits","status":"fully_proved","summary":"If free and wired box measures converge weakly to `\u03bd\u2080` and `\u03bd\u2081` along a common subsequence and `q \u2265 1`, then `\u03bd\u2080(A) \u2264 \u03bd\u2081(A)` for every clopen increasing event `A`.","title":"Free limit at most wired limit along a common subsequence, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits/exists-common-subseq-weak-limits","kind":"node","lean":"StatMech.FK.exists_common_subseq_weak_limits","parent":48070,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits/index.html#exists-common-subseq-weak-limits","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits","status":"fully_proved","summary":"Free and wired box measures have weak limits along a common strictly increasing subsequence.","title":"Common subsequence with free and wired weak limits"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits/exists-infinite-volume-extremal","kind":"node","lean":"StatMech.FK.exists_infiniteVolume_extremal","parent":48070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ordered free and wired subsequential limits exist for "],["math","\\(q \\ge 1\\)"]],"source":"Ordered free and wired subsequential limits exist for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits/index.html#exists-infinite-volume-extremal","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/ordering-of-free-and-wired-subsequential-limits","status":"fully_proved","summary":"For `q \u2265 1` there are a subsequence and weak limits `\u03c60` (free) and `\u03c61` (wired) along it with `\u03c60(A) \u2264 \u03c61(A)` for all clopen increasing `A`.","title":"Ordered free and wired subsequential limits exist for $q \\ge 1$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro","status":"fully_proved","title":"The FK percolation probability as a limit of crossing probabilities"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro/i-inter-crossing-event-eq-percolation-ev","kind":"node","lean":"StatMech.FK.iInter_crossingEvent_eq_percolationEvent","parent":48074,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro/index.html#i-inter-crossing-event-eq-percolation-ev","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro","status":"fully_proved","summary":"The crossing events intersect to the percolation event. See also `Percolation.percolationEvent_eq_iInter` (`Sharpness`), the same fact indexed from `n + 1`.","title":"Crossing events intersect to the percolation event"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro/fk-boundary-conn-real","kind":"node","lean":"StatMech.FK.fkBoundaryConnReal","parent":48074,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro/index.html#fk-boundary-conn-real","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro","status":"fully_proved","summary":"The probability, under the chosen wired infinite-volume measure, that the origin is connected to the complement of `\u039b_(n-1)`.","title":"Wired probability that the origin connects to the outside of a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro/fk-boundary-conn-real-tendsto-fk-theta","kind":"node","lean":"StatMech.FK.fkBoundaryConnReal_tendsto_fkTheta","parent":48074,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro/index.html#fk-boundary-conn-real-tendsto-fk-theta","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-fk-percolation-probability-as-a-limit-of-cro","status":"fully_proved","summary":"The crossing probabilities of the wired infinite-volume measure converge to the percolation probability `fkTheta`.","title":"Wired crossing probabilities converge to the percolation probability"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","title":"Two-point functions of infinite-volume FK measures"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/iv-two-point-sup","kind":"node","lean":"StatMech.FK.ivTwoPointSup","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#iv-two-point-sup","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"The supremum over `n` of the masses `\u03bc n A` of an event `A`.","title":"Supremum of the masses of an event along a measure sequence"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/iv-two-point-inf","kind":"node","lean":"StatMech.FK.ivTwoPointInf","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#iv-two-point-inf","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"The infimum over `n` of the masses `\u03bc n A` of an event `A`.","title":"Infimum of the masses of an event along a measure sequence"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/iv-two-point-inf-is-limit","kind":"node","lean":"StatMech.FK.ivTwoPointInf_isLimit","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#iv-two-point-inf-is-limit","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"Along a stochastically decreasing sequence of measures, the mass of a clopen increasing event converges to its infimum.","title":"Masses of clopen increasing events converge along decreasing measures"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/box-restrict","kind":"node","lean":"StatMech.FK.boxRestrict","parent":48078,"reading_title":{"format":"inline-parts-v1","parts":[["text","Restriction of a configuration of "],["math","\\(\\mathbb{Z}^d\\)"],["text"," to a box"]],"source":"Restriction of a configuration of $\\mathbb{Z}^d$ to a box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#box-restrict","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"Restriction of a configuration of `\u2124^d` to the edges of the box `\u039b_n`.","title":"Restriction of a configuration of $\\mathbb{Z}^d$ to a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/is-clopen-box-restrict-preimage","kind":"node","lean":"StatMech.FK.isClopen_boxRestrict_preimage","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#is-clopen-box-restrict-preimage","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"Cylinder events of the box `\u039b_N` are clopen.","title":"Cylinder events of a box are clopen"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/measurable-set-box-restrict-preimage","kind":"node","lean":"StatMech.FK.measurableSet_boxRestrict_preimage","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#measurable-set-box-restrict-preimage","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"Cylinder events of the box `\u039b_N` are measurable.","title":"Cylinder events of a box are measurable"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/is-increasing-box-restrict-preimage","kind":"node","lean":"StatMech.FK.isIncreasing_boxRestrict_preimage","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#is-increasing-box-restrict-preimage","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"The preimage of an increasing event under box restriction is increasing.","title":"Box restriction pulls increasing events back to increasing events"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/box-conn-event","kind":"node","lean":"StatMech.FK.boxConnEvent","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#box-conn-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"The event that `x` and `y` are joined by open edges inside the box `\u039b_n`.","title":"Event that two sites are joined by open edges inside a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/free-finite-measure-real-box-conn-event","kind":"node","lean":"StatMech.FK.freeFiniteMeasure_real_boxConnEvent","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#free-finite-measure-real-box-conn-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"The free box measure of the connection event in `\u039b_n` is the finite-volume two-point function.","title":"Free box measure of a box connection event is the two-point function"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/wired-fk-prob-pos","kind":"node","lean":"StatMech.FK.wiredFkProb_pos","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#wired-fk-prob-pos","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"Every configuration has positive wired random-cluster probability.","title":"Wired random-cluster probabilities are positive"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/free-iv-two-point","kind":"node","lean":"StatMech.FK.freeIvTwoPoint","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#free-iv-two-point","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"The free infinite-volume two-point function in `\u039b_N`: the supremum over `n` of `\u03c6\u2070_n(x \u2194 y in \u039b_N)`.","title":"Free infinite-volume two-point function in a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/wired-iv-two-point","kind":"node","lean":"StatMech.FK.wiredIvTwoPoint","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#wired-iv-two-point","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"The wired infinite-volume two-point function in `\u039b_N`: the infimum over `n` of `\u03c6\u00b9_n(x \u2194 y in \u039b_N)`.","title":"Wired infinite-volume two-point function in a box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/free-iv-two-point-is-limit","kind":"node","lean":"StatMech.FK.freeIvTwoPoint_isLimit","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#free-iv-two-point-is-limit","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"If the free box measures increase stochastically in `n`, then `\u03c6\u2070_n(x \u2194 y in \u039b_N)` converges to `freeIvTwoPoint`.","title":"Convergence to the free two-point function under stochastic increase"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/wired-iv-two-point-is-limit","kind":"node","lean":"StatMech.FK.wiredIvTwoPoint_isLimit","parent":48078,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur/index.html#wired-iv-two-point-is-limit","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-functions-of-infinite-volume-fk-measur","status":"fully_proved","summary":"If the wired box measures decrease stochastically in `n`, then `\u03c6\u00b9_n(x \u2194 y in \u039b_N)` converges to `wiredIvTwoPoint`.","title":"Convergence to the wired two-point function under stochastic decrease"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi","status":"fully_proved","title":"Box-boundary connection probabilities for the wired q = 2 measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi/box-bdry-conn-event","kind":"node","lean":"StatMech.FK.boxBdryConnEvent","parent":48093,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi/index.html#box-bdry-conn-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi","status":"fully_proved","summary":"The event that the origin of the box `\u039b_n` is connected to the boundary `\u2202\u039b_n` by open edges of `\u039b_n`.","title":"Event that the box origin connects to the box boundary"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi/box-boundary-conn-profile-tendsto-of-dia","kind":"node","lean":"StatMech.FK.boxBoundaryConnProfile_tendsto_of_diag","parent":48093,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box-boundary profile tends to "],["math","\\(\\theta(p,2)\\)"],["text"," if wired probabilities do"]],"source":"Box-boundary profile tends to $\\theta(p,2)$ if wired probabilities do"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi/index.html#box-boundary-conn-profile-tendsto-of-dia","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/box-boundary-connection-probabilities-for-the-wi","status":"fully_proved","summary":"If the wired `q = 2` probabilities of `{0 \u2194 \u2202\u039b_n}` converge to `\u03b8(p, 2)`, then so does the Ising box-boundary connection profile.","title":"Box-boundary profile tends to $\\theta(p,2)$ if wired probabilities do"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection","status":"fully_proved","title":"One-step comparison of wired boundary-connection probabilities"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection/inner-bdry-fin-event","kind":"node","lean":"StatMech.FK.innerBdryFinEvent","parent":48096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Origin-to-"],["math","\\(\\partial\\Lambda_n\\)"],["text"," event as an event on "],["math","\\(\\Lambda_{n+1}\\)"]],"source":"Origin-to-$\\partial\\Lambda_n$ event as an event on $\\Lambda_{n+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection/index.html#inner-bdry-fin-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection","status":"fully_proved","summary":"The event on `\u039b_(n+1)` that the origin is connected to `\u2202\u039b_n` by open edges of `\u039b_n`.","title":"Origin-to-$\\partial\\Lambda_n$ event as an event on $\\Lambda_{n+1}$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection/cond-bc-prob-add-sum-eq-smaller-box","kind":"node","lean":"StatMech.FK.condBcProb_add_sum_eq_smallerBox","parent":48096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed annulus plus inner-boundary wiring gives wired "],["math","\\(\\Lambda_n\\)"],["text"," law"]],"source":"Closed annulus plus inner-boundary wiring gives wired $\\Lambda_n$ law"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection/index.html#cond-bc-prob-add-sum-eq-smaller-box","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection","status":"fully_proved","summary":"With all annulus edges closed and a wiring made of the inner-boundary clique and an annulus wiring `D`, the conditional measure on `\u039b_(n+1)` equals the wired measure on `\u039b_n`, on any event pulled back by `crossExt`.","title":"Closed annulus plus inner-boundary wiring gives wired $\\Lambda_n$ law"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection/cond-wired-succ-le-smaller-box","kind":"node","lean":"StatMech.FK.condWiredSucc_le_smallerBox","parent":48096,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed-annulus wired bound on origin-to-"],["math","\\(\\partial\\Lambda_n\\)"],["text",", "],["math","\\(q = 2\\)"]],"source":"Closed-annulus wired bound on origin-to-$\\partial\\Lambda_n$, $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection/index.html#cond-wired-succ-le-smaller-box","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/one-step-comparison-of-wired-boundary-connection","status":"fully_proved","summary":"At `q = 2`, the wired measure on `\u039b_(n+1)` conditioned on all annulus edges closed gives `{0 \u2194 \u2202\u039b_n}` at most `\u03c6\u00b9_n(0 \u2194 \u2202\u039b_n)`.","title":"Closed-annulus wired bound on origin-to-$\\partial\\Lambda_n$, $q = 2$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","title":"Wired random-cluster measures decrease with the box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/out-proj","kind":"node","lean":"StatMech.FK.outProj","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Outer part of a configuration on "],["math","\\(\\Lambda_{n+1}\\)"]],"source":"Outer part of a configuration on $\\Lambda_{n+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#out-proj","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Project a configuration on `\u039b_(n+1)` to its outer part: edges of `\u039b_n` closed, the others unchanged.","title":"Outer part of a configuration on $\\Lambda_{n+1}$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/out-proj-eq-false-of-range","kind":"node","lean":"StatMech.FK.outProj_eq_false_of_range","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Outer projection closes the edges of "],["math","\\(\\Lambda_n\\)"]],"source":"Outer projection closes the edges of $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#out-proj-eq-false-of-range","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The projection closes edges of `\u039b_n`.","title":"Outer projection closes the edges of $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/out-proj-eq-of-not-range","kind":"node","lean":"StatMech.FK.outProj_eq_of_not_range","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Outer projection keeps the edges outside "],["math","\\(\\Lambda_n\\)"]],"source":"Outer projection keeps the edges outside $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#out-proj-eq-of-not-range","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The projection keeps edges outside `\u039b_n`.","title":"Outer projection keeps the edges outside $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/agrees-off-out-proj","kind":"node","lean":"StatMech.FK.agreesOff_outProj","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Outer projection agrees with the configuration off "],["math","\\(\\Lambda_n\\)"]],"source":"Outer projection agrees with the configuration off $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#agrees-off-out-proj","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The projection agrees with the configuration off the edges of `\u039b_n`.","title":"Outer projection agrees with the configuration off $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/out-proj-idem","kind":"node","lean":"StatMech.FK.outProj_idem","parent":48100,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#out-proj-idem","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The projection is idempotent.","title":"Outer projection is idempotent"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/filter-out-proj-eq-cond-fibre","kind":"node","lean":"StatMech.FK.filter_outProj_eq_condFibre","parent":48100,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#filter-out-proj-eq-cond-fibre","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The configurations with outer part `\u03c8` form the conditioning fibre of `\u03c8`.","title":"Configurations with a given outer part form its conditioning fibre"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/cond-fibre-congr-dom","kind":"node","lean":"StatMech.FK.condFibre_congr_dom","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Configurations agreeing off "],["math","\\(F\\)"],["text"," have the same conditioning fibre"]],"source":"Configurations agreeing off $F$ have the same conditioning fibre"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#cond-fibre-congr-dom","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Configurations agreeing off `F` have the same conditioning fibre.","title":"Configurations agreeing off $F$ have the same conditioning fibre"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/cond-bc-prob-congr-dom","kind":"node","lean":"StatMech.FK.condBcProb_congr_dom","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Configurations agreeing off "],["math","\\(F\\)"],["text"," have equal conditional probabilities"]],"source":"Configurations agreeing off $F$ have equal conditional probabilities"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#cond-bc-prob-congr-dom","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Configurations agreeing off `F` give the same conditional random-cluster probabilities.","title":"Configurations agreeing off $F$ have equal conditional probabilities"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/sum-fibre-mass-eq-one","kind":"node","lean":"StatMech.FK.sum_fibreMass_eq_one","parent":48100,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#sum-fibre-mass-eq-one","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The masses of the conditioning fibres over all outer configurations sum to `1`.","title":"Masses of the conditioning fibres sum to one"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-succ-bc-prob-decompose","kind":"node","lean":"StatMech.FK.wiredSucc_bcProb_decompose","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(\\Lambda_{n+1}\\)"],["text"," measure decomposed over outer configurations"]],"source":"Wired $\\Lambda_{n+1}$ measure decomposed over outer configurations"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-succ-bc-prob-decompose","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Decomposition of the wired measure on `\u039b_(n+1)` over the configuration outside `\u039b_n`.","title":"Wired $\\Lambda_{n+1}$ measure decomposed over outer configurations"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/cond-bc-prob-inner-restrict-le","kind":"node","lean":"StatMech.FK.condBcProb_innerRestrict_le","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional wired decrease on inner increasing events for "],["math","\\(q \\ge 1\\)"]],"source":"Conditional wired decrease on inner increasing events for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#cond-bc-prob-inner-restrict-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q \u2265 1`, conditionally on any outer configuration, the wired measure on `\u039b_(n+1)` gives an increasing event of `\u039b_n` at most its wired `\u039b_n` probability.","title":"Conditional wired decrease on inner increasing events for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-succ-bc-prob-inner-event-le","kind":"node","lean":"StatMech.FK.wiredSucc_bcProb_innerEvent_le","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired box measures decrease on inner increasing events for "],["math","\\(q \\ge 1\\)"]],"source":"Wired box measures decrease on inner increasing events for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-succ-bc-prob-inner-event-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q \u2265 1`, the wired measure on `\u039b_(n+1)` gives an increasing event of `\u039b_n` at most its wired `\u039b_n` probability.","title":"Wired box measures decrease on inner increasing events for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-finite-measure-succ-real-inner-res","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_succ_real_innerRestrictEvent","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(\\Lambda_{n+1}\\)"],["text"," measure of a "],["math","\\(\\Lambda_n\\)"],["text"," cylinder event as a sum"]],"source":"Wired $\\Lambda_{n+1}$ measure of a $\\Lambda_n$ cylinder event as a sum"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-finite-measure-succ-real-inner-res","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The wired `\u039b_(n+1)` measure of a `\u039b_n` cylinder event, as a finite sum.","title":"Wired $\\Lambda_{n+1}$ measure of a $\\Lambda_n$ cylinder event as a sum"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-finite-measure-succ-le-smaller-box","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_succ_le_smallerBox","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired box measures decrease on increasing cylinder events at "],["math","\\(q = 2\\)"]],"source":"Wired box measures decrease on increasing cylinder events at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-finite-measure-succ-le-smaller-box","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q = 2`, the wired `\u039b_(n+1)` measure of an increasing `\u039b_n` cylinder event is at most its wired `\u039b_n` probability.","title":"Wired box measures decrease on increasing cylinder events at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/is-increasing-conn-to-bdry-event","kind":"node","lean":"StatMech.FK.isIncreasing_connToBdryEvent","parent":48100,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#is-increasing-conn-to-bdry-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The event that the origin is connected to the box boundary is increasing.","title":"Origin-to-boundary connection is an increasing event"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-finite-measure-succ-le-box-bdry-co","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_succ_le_boxBdryConnEvent","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," probability of "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_n\\)"],["text"," in "],["math","\\(\\Lambda_{n+1}\\)"],["text"," at most in "],["math","\\(\\Lambda_n\\)"]],"source":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_{n+1}$ at most in $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-finite-measure-succ-le-box-bdry-co","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q = 2`, `\u03c6\u00b9_{\u039b_(n+1)}(0 \u2194 \u2202\u039b_n) \u2264 \u03c6\u00b9_{\u039b_n}(0 \u2194 \u2202\u039b_n)`.","title":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_{n+1}$ at most in $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/open-sub-succ-adj-inner","kind":"node","lean":"StatMech.FK.openSub_succ_adj_inner","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Open edges of "],["math","\\(\\Lambda_n\\)"],["text"," are open edges of "],["math","\\(\\Lambda_{n+1}\\)"]],"source":"Open edges of $\\Lambda_n$ are open edges of $\\Lambda_{n+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#open-sub-succ-adj-inner","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"An open edge of `\u039b_n` is an open edge of `\u039b_(n+1)`.","title":"Open edges of $\\Lambda_n$ are open edges of $\\Lambda_{n+1}$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-boundary-of-open-sub-succ-adj-outer","kind":"node","lean":"StatMech.FK.boxBoundary_of_openSub_succ_adj_outer","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inner vertices next to the annulus lie on "],["math","\\(\\partial\\Lambda_n\\)"]],"source":"Inner vertices next to the annulus lie on $\\partial\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-boundary-of-open-sub-succ-adj-outer","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"A vertex of `\u039b_n` adjacent to a vertex of `\u039b_(n+1) \\ \u039b_n` lies on `\u2202\u039b_n`.","title":"Inner vertices next to the annulus lie on $\\partial\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-vert-incl-box-origin","kind":"node","lean":"StatMech.FK.boxVertIncl_boxOrigin","parent":48100,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-vert-incl-box-origin","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The inclusion `\u039b_n \u2192 \u039b_(n+1)` maps the origin to the origin.","title":"Box inclusion maps the origin to the origin"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/conn-to-bdry-succ-of-conn-to-bdry","kind":"node","lean":"StatMech.FK.connToBdry_succ_of_connToBdry","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Paths to "],["math","\\(\\partial\\Lambda_{n+1}\\)"],["text"," pass through "],["math","\\(\\partial\\Lambda_n\\)"]],"source":"Paths to $\\partial\\Lambda_{n+1}$ pass through $\\partial\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#conn-to-bdry-succ-of-conn-to-bdry","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"A connection from the origin to `\u2202\u039b_(n+1)` inside `\u039b_(n+1)` passes through `\u2202\u039b_n`.","title":"Paths to $\\partial\\Lambda_{n+1}$ pass through $\\partial\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-bdry-conn-event-succ-subset","kind":"node","lean":"StatMech.FK.boxBdryConnEvent_succ_subset","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-step decrease of the events "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_n\\)"]],"source":"One-step decrease of the events $0 \\leftrightarrow \\partial\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-bdry-conn-event-succ-subset","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The events `0 \u2194 \u2202\u039b_n` decrease in `n`.","title":"One-step decrease of the events $0 \\leftrightarrow \\partial\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-finite-measure-box-bdry-conn-event","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_boxBdryConnEvent_antitone","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," probability of "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_n\\)"],["text"," in "],["math","\\(\\Lambda_n\\)"],["text"," is nonincreasing in "],["math","\\(n\\)"]],"source":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_n$ is nonincreasing in $n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-finite-measure-box-bdry-conn-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q = 2`, `\u03c6\u00b9_{\u039b_(n+1)}(0 \u2194 \u2202\u039b_(n+1)) \u2264 \u03c6\u00b9_{\u039b_n}(0 \u2194 \u2202\u039b_n)`.","title":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_n$ is nonincreasing in $n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-vert-incl-le","kind":"node","lean":"StatMech.FK.boxVertInclLE","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inclusion of "],["math","\\(\\Lambda_n\\)"],["text"," into "],["math","\\(\\Lambda_m\\)"],["text"," for "],["math","\\(n \\le m\\)"]],"source":"Inclusion of $\\Lambda_n$ into $\\Lambda_m$ for $n \\le m$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-vert-incl-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The inclusion `\u039b_n \u2192 \u039b_m` for `n \u2264 m`.","title":"Inclusion of $\\Lambda_n$ into $\\Lambda_m$ for $n \\le m$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/inner-edge-le","kind":"node","lean":"StatMech.FK.innerEdgeLE","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inclusion of edges of "],["math","\\(\\Lambda_n\\)"],["text"," into edges of "],["math","\\(\\Lambda_m\\)"]],"source":"Inclusion of edges of $\\Lambda_n$ into edges of $\\Lambda_m$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#inner-edge-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The inclusion of edges of `\u039b_n` into edges of `\u039b_m` for `n \u2264 m`.","title":"Inclusion of edges of $\\Lambda_n$ into edges of $\\Lambda_m$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/edge-incl-inner-edge-le","kind":"node","lean":"StatMech.FK.edgeIncl_innerEdgeLE","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edge inclusion into "],["math","\\(\\mathbb{Z}^d\\)"],["text"," factors through "],["math","\\(\\Lambda_m\\)"]],"source":"Edge inclusion into $\\mathbb{Z}^d$ factors through $\\Lambda_m$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#edge-incl-inner-edge-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Including an edge into `\u039b_m` and then into `\u2124^d` is including it into `\u2124^d`.","title":"Edge inclusion into $\\mathbb{Z}^d$ factors through $\\Lambda_m$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-restrict-le","kind":"node","lean":"StatMech.FK.boxRestrictLE","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Restriction of a configuration from "],["math","\\(\\Lambda_m\\)"],["text"," to "],["math","\\(\\Lambda_n\\)"]],"source":"Restriction of a configuration from $\\Lambda_m$ to $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-restrict-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Restrict a configuration on `\u039b_m` to `\u039b_n` for `n \u2264 m`.","title":"Restriction of a configuration from $\\Lambda_m$ to $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-restrict-le-box-restrict","kind":"node","lean":"StatMech.FK.boxRestrictLE_boxRestrict","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Restriction to "],["math","\\(\\Lambda_n\\)"],["text"," factors through "],["math","\\(\\Lambda_m\\)"]],"source":"Restriction to $\\Lambda_n$ factors through $\\Lambda_m$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-restrict-le-box-restrict","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Restricting to `\u039b_m` and then to `\u039b_n` is restricting to `\u039b_n`.","title":"Restriction to $\\Lambda_n$ factors through $\\Lambda_m$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-restrict-le-monotone","kind":"node","lean":"StatMech.FK.boxRestrictLE_monotone","parent":48100,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-restrict-le-monotone","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Restriction between boxes is monotone.","title":"Monotonicity of restriction between boxes"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/conn-to-bdry-event-le","kind":"node","lean":"StatMech.FK.connToBdryEventLE","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary-connection event of "],["math","\\(\\Lambda_n\\)"],["text"," as an event on "],["math","\\(\\Lambda_m\\)"]],"source":"Boundary-connection event of $\\Lambda_n$ as an event on $\\Lambda_m$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#conn-to-bdry-event-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The event `0 \u2194 \u2202\u039b_n` inside `\u039b_n`, as an event on `\u039b_m` for `n \u2264 m`.","title":"Boundary-connection event of $\\Lambda_n$ as an event on $\\Lambda_m$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/is-increasing-conn-to-bdry-event-le","kind":"node","lean":"StatMech.FK.isIncreasing_connToBdryEventLE","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary-connection event of "],["math","\\(\\Lambda_n\\)"],["text"," on "],["math","\\(\\Lambda_m\\)"],["text"," is increasing"]],"source":"Boundary-connection event of $\\Lambda_n$ on $\\Lambda_m$ is increasing"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#is-increasing-conn-to-bdry-event-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The event `connToBdryEventLE` is increasing.","title":"Boundary-connection event of $\\Lambda_n$ on $\\Lambda_m$ is increasing"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-bdry-conn-event-eq-box-restrict-prei","kind":"node","lean":"StatMech.FK.boxBdryConnEvent_eq_boxRestrict_preimage","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary connection of "],["math","\\(\\Lambda_n\\)"],["text"," is a "],["math","\\(\\Lambda_m\\)"],["text"," cylinder event"]],"source":"Boundary connection of $\\Lambda_n$ is a $\\Lambda_m$ cylinder event"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-bdry-conn-event-eq-box-restrict-prei","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The event `0 \u2194 \u2202\u039b_n` is a `\u039b_m` cylinder event for every `m \u2265 n`.","title":"Boundary connection of $\\Lambda_n$ is a $\\Lambda_m$ cylinder event"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/measurable-set-box-restrict-conn-to-bdry","kind":"node","lean":"StatMech.FK.measurableSet_boxRestrict_connToBdryEventLE","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cylinder form of the "],["math","\\(\\Lambda_n\\)"],["text"," boundary connection is measurable"]],"source":"Cylinder form of the $\\Lambda_n$ boundary connection is measurable"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#measurable-set-box-restrict-conn-to-bdry","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The cylinder form of `0 \u2194 \u2202\u039b_n` is measurable.","title":"Cylinder form of the $\\Lambda_n$ boundary connection is measurable"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-finite-measure-real-box-restrict-e","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_real_boxRestrictEvent","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(\\Lambda_m\\)"],["text"," measure of a "],["math","\\(\\Lambda_m\\)"],["text"," cylinder event as a sum"]],"source":"Wired $\\Lambda_m$ measure of a $\\Lambda_m$ cylinder event as a sum"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-finite-measure-real-box-restrict-e","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The wired `\u039b_m` measure of a `\u039b_m` cylinder event, as a finite sum.","title":"Wired $\\Lambda_m$ measure of a $\\Lambda_m$ cylinder event as a sum"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-finite-measure-succ-le-box-bdry-co-2","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_succ_le_boxBdryConnEvent_le","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," probability of "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_n\\)"],["text"," in "],["math","\\(\\Lambda_{m+1}\\)"],["text"," at most in "],["math","\\(\\Lambda_m\\)"],["text",", "],["math","\\(m \\ge n\\)"]],"source":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_{m+1}$ at most in $\\Lambda_m$, $m \\ge n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-finite-measure-succ-le-box-bdry-co-2","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q = 2` and `n \u2264 m`, `\u03c6\u00b9_{\u039b_(m+1)}(0 \u2194 \u2202\u039b_n) \u2264 \u03c6\u00b9_{\u039b_m}(0 \u2194 \u2202\u039b_n)`.","title":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_{m+1}$ at most in $\\Lambda_m$, $m \\ge n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-finite-measure-box-bdry-conn-event-2","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_boxBdryConnEvent_antitone_radius","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," probability of "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_n\\)"],["text"," in "],["math","\\(\\Lambda_{n+k+1}\\)"],["text"," at most in "],["math","\\(\\Lambda_{n+k}\\)"]],"source":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_{n+k+1}$ at most in $\\Lambda_{n+k}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-finite-measure-box-bdry-conn-event-2","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q = 2`, `\u03c6\u00b9_{\u039b_(n+k+1)}(0 \u2194 \u2202\u039b_n) \u2264 \u03c6\u00b9_{\u039b_(n+k)}(0 \u2194 \u2202\u039b_n)`.","title":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_{n+k+1}$ at most in $\\Lambda_{n+k}$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/open-subgraph-adj-of-box-graph","kind":"node","lean":"StatMech.FK.openSubgraph_adj_of_boxGraph","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Open edges of a box are open edges of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Open edges of a box are open edges of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#open-subgraph-adj-of-box-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"An open edge of the box is an open edge of `\u2124^d`.","title":"Open edges of a box are open edges of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/connected-of-box-connected","kind":"node","lean":"StatMech.FK.connected_of_boxConnected","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Open connections in a box are open connections in "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Open connections in a box are open connections in $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#connected-of-box-connected","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"An open connection inside the box is an open connection in `\u2124^d`.","title":"Open connections in a box are open connections in $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-bdry-conn-event-subset-crossing-even","kind":"node","lean":"StatMech.FK.boxBdryConnEvent_subset_crossingEvent","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary connection inside "],["math","\\(\\Lambda_n\\)"],["text"," reaches outside "],["math","\\(\\Lambda_{n-1}\\)"]],"source":"Boundary connection inside $\\Lambda_n$ reaches outside $\\Lambda_{n-1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-bdry-conn-event-subset-crossing-even","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"A connection from the origin to `\u2202\u039b_n` inside `\u039b_n` reaches the complement of `\u039b_(n-1)`.","title":"Boundary connection inside $\\Lambda_n$ reaches outside $\\Lambda_{n-1}$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-graph-open-of-open-subgraph","kind":"node","lean":"StatMech.FK.boxGraph_open_of_openSubgraph","parent":48100,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-graph-open-of-open-subgraph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"An open edge of `\u2124^d` between box vertices is an open edge of the box.","title":"Open lattice edges between box vertices are open box edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-bdry-conn-of-crossing","kind":"node","lean":"StatMech.FK.boxBdryConn_of_crossing","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Open path from the origin out of "],["math","\\(\\Lambda_n\\)"],["text"," gives "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_n\\)"],["text"," inside "],["math","\\(\\Lambda_n\\)"]],"source":"Open path from the origin out of $\\Lambda_n$ gives $0 \\leftrightarrow \\partial\\Lambda_n$ inside $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-bdry-conn-of-crossing","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"A connection from the origin to a site outside `\u039b_n` gives a connection to `\u2202\u039b_n` inside `\u039b_n`.","title":"Open path from the origin out of $\\Lambda_n$ gives $0 \\leftrightarrow \\partial\\Lambda_n$ inside $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/crossing-event-succ-subset-box-bdry-conn","kind":"node","lean":"StatMech.FK.crossingEvent_succ_subset_boxBdryConnEvent","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Leaving "],["math","\\(\\Lambda_n\\)"],["text"," implies connecting to "],["math","\\(\\partial\\Lambda_n\\)"]],"source":"Leaving $\\Lambda_n$ implies connecting to $\\partial\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#crossing-event-succ-subset-box-bdry-conn","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Reaching the complement of `\u039b_n` implies `0 \u2194 \u2202\u039b_n`.","title":"Leaving $\\Lambda_n$ implies connecting to $\\partial\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-iv-box-bdry-conn-event-tendsto","kind":"node","lean":"StatMech.FK.wiredIv_boxBdryConnEvent_tendsto","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired infinite-volume boundary connection tends to "],["math","\\(\\theta^1(p, q)\\)"]],"source":"Wired infinite-volume boundary connection tends to $\\theta^1(p, q)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-iv-box-bdry-conn-event-tendsto","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"Under the wired infinite-volume measure, `P(0 \u2194 \u2202\u039b_n) \u2192 \u03b8\u00b9(p, q)`.","title":"Wired infinite-volume boundary connection tends to $\\theta^1(p, q)$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-bdry-conn-event-subset-le","kind":"node","lean":"StatMech.FK.boxBdryConnEvent_subset_le","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Events "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_n\\)"],["text"," decrease in "],["math","\\(n \\ge 1\\)"]],"source":"Events $0 \\leftrightarrow \\partial\\Lambda_n$ decrease in $n \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-bdry-conn-event-subset-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The events `0 \u2194 \u2202\u039b_n` decrease in `n \u2265 1`.","title":"Events $0 \\leftrightarrow \\partial\\Lambda_n$ decrease in $n \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/radius-antitone","kind":"node","lean":"StatMech.FK.radius_antitone","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," probability of "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_k\\)"],["text"," in "],["math","\\(\\Lambda_{k+j}\\)"],["text"," is nonincreasing in "],["math","\\(j\\)"]],"source":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_k$ in $\\Lambda_{k+j}$ is nonincreasing in $j$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#radius-antitone","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q = 2`, `j \u21a6 \u03c6\u00b9_{\u039b_(k+j)}(0 \u2194 \u2202\u039b_k)` is antitone.","title":"Wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_k$ in $\\Lambda_{k+j}$ is nonincreasing in $j$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/tendsto-of-antitone-of-subseq","kind":"node","lean":"StatMech.FK.tendsto_of_antitone_of_subseq","parent":48100,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#tendsto-of-antitone-of-subseq","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"A bounded-below antitone real sequence converges to the limit of any of its subsequences.","title":"Antitone bounded-below sequences converge to any subsequential limit"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/wired-finite-measure-radius-tendsto","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_radius_tendsto","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinite-volume limit of the wired "],["math","\\(q = 2\\)"],["text"," probability of "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_k\\)"]],"source":"Infinite-volume limit of the wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_k$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#wired-finite-measure-radius-tendsto","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q = 2`, `\u03c6\u00b9_{\u039b_(k+j)}(0 \u2194 \u2202\u039b_k)` converges as `j \u2192 \u221e` to the wired infinite-volume probability of `0 \u2194 \u2202\u039b_k`.","title":"Infinite-volume limit of the wired $q = 2$ probability of $0 \\leftrightarrow \\partial\\Lambda_k$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-bdry-conn-event-diag-tendsto","kind":"node","lean":"StatMech.FK.boxBdryConnEvent_diag_tendsto","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," percolation probability as the limit of "],["math","\\(0 \\leftrightarrow \\partial\\Lambda_n\\)"],["text"," in "],["math","\\(\\Lambda_n\\)"]],"source":"Wired $q = 2$ percolation probability as the limit of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-bdry-conn-event-diag-tendsto","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"For `q = 2`, `\u03c6\u00b9_{\u039b_n}(0 \u2194 \u2202\u039b_n) \u2192 \u03b8\u00b9(p, 2)`.","title":"Wired $q = 2$ percolation probability as the limit of $0 \\leftrightarrow \\partial\\Lambda_n$ in $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/box-boundary-conn-profile-tendsto","kind":"node","lean":"StatMech.FK.boxBoundaryConnProfile_tendsto","parent":48100,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ising FK box-boundary connection profile tends to "],["math","\\(\\theta^1(p, 2)\\)"]],"source":"Ising FK box-boundary connection profile tends to $\\theta^1(p, 2)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the/index.html#box-boundary-conn-profile-tendsto","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/wired-random-cluster-measures-decrease-with-the","status":"fully_proved","summary":"The box-boundary connection profile of the Ising FK representation converges to `\u03b8\u00b9(p, 2)`.","title":"Ising FK box-boundary connection profile tends to $\\theta^1(p, 2)$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","title":"The free q = 2 random-cluster measures increase with the box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/induced-wiring-free","kind":"node","lean":"StatMech.FK.inducedWiringFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#induced-wiring-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The boundary wiring on `\u039b_n` induced by an outer configuration `\u03c8` on `\u039b_(n+1)`: two distinct vertices of `\u039b_n` are wired when `\u03c8` connects them by open edges outside `\u039b_n`.","title":"Wiring induced on a box by outer open edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/induced-wiring-free-adj","kind":"node","lean":"StatMech.FK.inducedWiringFree_adj","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#induced-wiring-free-adj","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Unfolding adjacency in the induced wiring.","title":"Adjacency criterion for the free induced wiring"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/psi-free-graph-eq-sup","kind":"node","lean":"StatMech.FK.psiFreeGraph_eq_sup","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#psi-free-graph-eq-sup","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The open graph of a glued configuration is the union of the open edges of `\u039b_n` and the outer open edges.","title":"Glued open graph is the union of inner and outer open edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-graph-free","kind":"node","lean":"StatMech.FK.fullGraphFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-graph-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The open graph on `\u039b_(n+1)` of `\u03c9` on `\u039b_n` glued into `\u03c8`.","title":"Open graph of a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/inner-graph-free","kind":"node","lean":"StatMech.FK.innerGraphFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#inner-graph-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The open graph of `\u03c9` on `\u039b_n` together with the wiring induced by `\u03c8`.","title":"Inner open graph with the free induced wiring"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/outside-open-graph-le-full-graph-free","kind":"node","lean":"StatMech.FK.outsideOpenGraph_le_fullGraphFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#outside-open-graph-le-full-graph-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The outer open graph is contained in the glued open graph.","title":"Outer open graph lies in the glued open graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/inner-graph-free-reachable-of-outside-re","kind":"node","lean":"StatMech.FK.innerGraphFree_reachable_of_outsideReachable","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#inner-graph-free-reachable-of-outside-re","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Vertices of `\u039b_n` connected by outer open edges are connected in `innerGraphFree`.","title":"Outer open connections give connections in the inner graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-graph-free-reachable-of-inner-graph","kind":"node","lean":"StatMech.FK.fullGraphFree_reachable_of_innerGraphFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-graph-free-reachable-of-inner-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Connections in `innerGraphFree` are connections in the glued open graph.","title":"Inner-graph connections are glued-graph connections"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/inner-graph-free-reachable-of-full-graph","kind":"node","lean":"StatMech.FK.innerGraphFree_reachable_of_fullGraphFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#inner-graph-free-reachable-of-full-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Connections in the glued open graph between vertices of `\u039b_n` are connections in `innerGraphFree`.","title":"Glued connections between inner vertices are inner-graph connections"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-graph-free-adj-of-outside","kind":"node","lean":"StatMech.FK.fullGraphFree_adj_of_outside","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-graph-free-adj-of-outside","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"An edge of the glued open graph at a vertex outside `\u039b_n` is an outer open edge.","title":"Glued open edges at outer vertices are outer edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/outside-only-free","kind":"node","lean":"StatMech.FK.OutsideOnlyFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#outside-only-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"A vertex of `\u039b_(n+1)` whose outer open cluster contains no vertex of `\u039b_n`.","title":"Outer vertex whose outer open cluster avoids the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/outside-only-free-self-not-inner","kind":"node","lean":"StatMech.FK.outsideOnlyFree_self_not_inner","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#outside-only-free-self-not-inner","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"A vertex whose outer cluster avoids `\u039b_n` is not in `\u039b_n`.","title":"A vertex whose outer cluster avoids the inner box lies outside it"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/outside-only-free-of-reachable","kind":"node","lean":"StatMech.FK.outsideOnlyFree_of_reachable","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#outside-only-free-of-reachable","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Avoiding `\u039b_n` is constant on outer open clusters.","title":"Avoiding the inner box is constant on outer open clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/outside-open-graph-reachable-of-full-gra","kind":"node","lean":"StatMech.FK.outsideOpenGraph_reachable_of_fullGraphFree_outsideOnly","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#outside-open-graph-reachable-of-full-gra","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"From a vertex whose outer cluster avoids `\u039b_n`, glued connections are outer connections.","title":"Glued connections from an avoiding vertex are outer connections"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/is-outside-only-free","kind":"node","lean":"StatMech.FK.IsOutsideOnlyFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#is-outside-only-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"An outer open cluster that contains no vertex of `\u039b_n`.","title":"Outer open cluster avoiding the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/is-outside-only-free-mk","kind":"node","lean":"StatMech.FK.isOutsideOnlyFree_mk","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#is-outside-only-free-mk","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"An outer cluster avoids `\u039b_n` exactly when its representative vertex does.","title":"An outer cluster avoids the inner box iff its representative does"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/outside-only-comp-free","kind":"node","lean":"StatMech.FK.OutsideOnlyCompFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#outside-only-comp-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The outer open clusters that contain no vertex of `\u039b_n`.","title":"Outer open clusters avoiding the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-component-map-free","kind":"node","lean":"StatMech.FK.fullComponentMapFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-component-map-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The map from clusters of `innerGraphFree` and outer clusters avoiding `\u039b_n` to clusters of the glued configuration.","title":"Cluster map into the glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-component-map-free-inl-mk","kind":"node","lean":"StatMech.FK.fullComponentMapFree_inl_mk","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-component-map-free-inl-mk","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The cluster map on an inner cluster.","title":"Cluster map on inner clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-component-map-free-inr-mk","kind":"node","lean":"StatMech.FK.fullComponentMapFree_inr_mk","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-component-map-free-inr-mk","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The cluster map on an outer cluster avoiding `\u039b_n`.","title":"Cluster map on avoiding outer clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-component-map-free-injective","kind":"node","lean":"StatMech.FK.fullComponentMapFree_injective","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-component-map-free-injective","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The cluster map is injective.","title":"Injectivity of the free cluster map"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-component-map-free-surjective","kind":"node","lean":"StatMech.FK.fullComponentMapFree_surjective","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-component-map-free-surjective","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The cluster map is surjective.","title":"Surjectivity of the free cluster map"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/full-component-equiv-free","kind":"node","lean":"StatMech.FK.fullComponentEquivFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#full-component-equiv-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Clusters of the glued configuration correspond to clusters of `innerGraphFree` plus outer clusters avoiding `\u039b_n`.","title":"Cluster decomposition of the glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/outside-shift-free","kind":"node","lean":"StatMech.FK.outsideShiftFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#outside-shift-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The number of outer open clusters avoiding `\u039b_n`.","title":"Number of outer open clusters avoiding the inner box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/num-clusters-psi-ext-free","kind":"node","lean":"StatMech.FK.numClusters_psiExt_free","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#num-clusters-psi-ext-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The cluster count of a glued configuration is the cluster count of `\u03c9` with the induced wiring plus the number of outer clusters avoiding `\u039b_n`.","title":"Cluster count of a glued configuration splits additively"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/psi-shift-free","kind":"node","lean":"StatMech.FK.psiShiftFree","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#psi-shift-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The factor of the random-cluster weight of a glued configuration contributed by the outer configuration.","title":"Outer factor of the weight of a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/psi-shift-free-pos","kind":"node","lean":"StatMech.FK.psiShiftFree_pos","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#psi-shift-free-pos","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The outer weight factor is positive.","title":"Positivity of the outer weight factor"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-weight-psi-ext-free","kind":"node","lean":"StatMech.FK.fkWeight_psiExt_free","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-weight-psi-ext-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The random-cluster weight of a glued configuration factors as the outer factor times the weight of `\u03c9` with the induced wiring.","title":"Random-cluster weight of a glued configuration factorizes"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/induced-fk-z-inner-edge-finset-psi-free","kind":"node","lean":"StatMech.FK.inducedFkZ_innerEdgeFinset_psi_free","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#induced-fk-z-inner-edge-finset-psi-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The conditional partition function given the outer configuration, in terms of the induced-wiring partition function.","title":"Conditional partition function in terms of the induced wiring"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/cond-fk-prob-psi-ext-eq-bc-prob-free","kind":"node","lean":"StatMech.FK.condFkProb_psiExt_eq_bcProb_free","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#cond-fk-prob-psi-ext-eq-bc-prob-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Conditionally on the configuration `\u03c8` outside `\u039b_n`, the free random-cluster measure on `\u039b_(n+1)` restricted to `\u039b_n` is the random-cluster measure on `\u039b_n` with the wiring induced by `\u03c8`.","title":"Domain Markov property"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/sum-fibre-mass-eq-one-free","kind":"node","lean":"StatMech.FK.sum_fibreMass_eq_one_free","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#sum-fibre-mass-eq-one-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The masses of the conditioning fibres of the free measure on `\u039b_(n+1)` sum to `1`.","title":"Fibre masses of the free measure sum to one"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-prob-eq-fibre-mass-mul-cond-fk-prob","kind":"node","lean":"StatMech.FK.fkProb_eq_fibreMass_mul_condFkProb","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-prob-eq-fibre-mass-mul-cond-fk-prob","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"A random-cluster probability is its fibre mass times its conditional probability.","title":"Fibre-mass factorization of random-cluster probabilities"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/cond-fk-prob-congr-dom-free","kind":"node","lean":"StatMech.FK.condFkProb_congr_dom_free","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#cond-fk-prob-congr-dom-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Configurations agreeing off `F` give the same conditional random-cluster probabilities.","title":"Conditional free probabilities depend only on the outer configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/free-succ-fk-prob-decompose","kind":"node","lean":"StatMech.FK.freeSucc_fkProb_decompose","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#free-succ-fk-prob-decompose","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Decomposition of the free measure on `\u039b_(n+1)` over the configuration outside `\u039b_n`.","title":"Decomposition of the free measure over the outer configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/cond-fk-prob-psi-ext-sum-eq-induced-box","kind":"node","lean":"StatMech.FK.condFkProb_psiExt_sum_eq_inducedBox_free","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#cond-fk-prob-psi-ext-sum-eq-induced-box","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The conditional free measure of an event of `\u039b_n` equals its probability under the induced-wiring measure.","title":"Conditional free measure as an induced-wiring measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-prob-le-induced-free","kind":"node","lean":"StatMech.FK.fkProb_le_inducedFree","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free measure is dominated by induced-wiring measures for "],["math","\\(q \\ge 1\\)"]],"source":"Free measure is dominated by induced-wiring measures for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-prob-le-induced-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q \u2265 1`, the free measure on `\u039b_n` gives an increasing event at most its probability under any induced wiring.","title":"Free measure is dominated by induced-wiring measures for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-prob-le-cond-fk-prob-inner-restrict-f","kind":"node","lean":"StatMech.FK.fkProb_le_condFkProb_innerRestrict_free","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional free measure dominates the smaller free box for "],["math","\\(q \\ge 1\\)"]],"source":"Conditional free measure dominates the smaller free box for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-prob-le-cond-fk-prob-inner-restrict-f","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q \u2265 1`, conditionally on any outer configuration, the free measure on `\u039b_(n+1)` gives an increasing event of `\u039b_n` at least its free `\u039b_n` probability.","title":"Conditional free measure dominates the smaller free box for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/free-succ-fk-prob-inner-event-le","kind":"node","lean":"StatMech.FK.freeSucc_fkProb_innerEvent_le","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-step box monotonicity of free measures for "],["math","\\(q \\ge 1\\)"]],"source":"One-step box monotonicity of free measures for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#free-succ-fk-prob-inner-event-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q \u2265 1`, the free measure on `\u039b_(n+1)` gives an increasing event of `\u039b_n` at least its free `\u039b_n` probability.","title":"One-step box monotonicity of free measures for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/free-finite-measure-real-box-restrict-ev","kind":"node","lean":"StatMech.FK.freeFiniteMeasure_real_boxRestrictEvent","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#free-finite-measure-real-box-restrict-ev","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The free `\u039b_m` measure of a `\u039b_m` cylinder event, as a finite sum.","title":"Free box measure of a cylinder event as a finite sum"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/free-finite-measure-succ-real-inner-rest","kind":"node","lean":"StatMech.FK.freeFiniteMeasure_succ_real_innerRestrictEvent","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#free-finite-measure-succ-real-inner-rest","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The free `\u039b_(n+1)` measure of a `\u039b_n` cylinder event, as a finite sum.","title":"Next-box free measure of a cylinder event as a finite sum"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/free-finite-measure-smaller-box-le-succ","kind":"node","lean":"StatMech.FK.freeFiniteMeasure_smallerBox_le_succ","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Increasing box events gain free "],["math","\\(q = 2\\)"],["text"," probability in the next box"]],"source":"Increasing box events gain free $q = 2$ probability in the next box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#free-finite-measure-smaller-box-le-succ","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q = 2`, the free `\u039b_n` probability of an increasing `\u039b_n` event is at most its free `\u039b_(n+1)` measure.","title":"Increasing box events gain free $q = 2$ probability in the next box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-free-limit-le-succ","kind":"node","lean":"StatMech.FK.fkFreeLimit_le_succ","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","One-step growth of free "],["math","\\(q = 2\\)"],["text"," measures of increasing cylinders"]],"source":"One-step growth of free $q = 2$ measures of increasing cylinders"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-free-limit-le-succ","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q = 2` and `N \u2264 m`, the free `\u039b_m` measure of an increasing `\u039b_N` cylinder event is at most its free `\u039b_(m+1)` measure.","title":"One-step growth of free $q = 2$ measures of increasing cylinders"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-free-limit-monotone","kind":"node","lean":"StatMech.FK.fkFreeLimit_monotone","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," box measures of increasing cylinders are monotone"]],"source":"Free $q = 2$ box measures of increasing cylinders are monotone"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-free-limit-monotone","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q = 2`, the free box measures of an increasing `\u039b_N` cylinder event increase with the box.","title":"Free $q = 2$ box measures of increasing cylinders are monotone"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-free-limit-value","kind":"node","lean":"StatMech.FK.fkFreeLimitValue","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Limit of free "],["math","\\(q = 2\\)"],["text"," box measures of a cylinder event"]],"source":"Limit of free $q = 2$ box measures of a cylinder event"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-free-limit-value","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"The limit `sup_k \u03c6\u2070_{\u039b_(N+k)}(S)` of the free `q = 2` box measures of a `\u039b_N` cylinder event `S`.","title":"Limit of free $q = 2$ box measures of a cylinder event"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/free-finite-measure-real-le-one","kind":"node","lean":"StatMech.FK.freeFiniteMeasure_real_le_one","parent":48149,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#free-finite-measure-real-le-one","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"Free box measures are at most `1`.","title":"Free box measures are at most one"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-free-limit-exists","kind":"node","lean":"StatMech.FK.fk_free_limit_exists","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," box measures of increasing cylinders converge"]],"source":"Free $q = 2$ box measures of increasing cylinders converge"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-free-limit-exists","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q = 2`, the free box measures of an increasing `\u039b_N` cylinder event converge to `fkFreeLimitValue`.","title":"Free $q = 2$ box measures of increasing cylinders converge"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-free-le-limit","kind":"node","lean":"StatMech.FK.fk_free_le_limit","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," box measures are bounded by the limit value"]],"source":"Free $q = 2$ box measures are bounded by the limit value"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-free-le-limit","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q = 2`, every free box measure of a `\u039b_N` cylinder event with `N \u2264 m` is at most `fkFreeLimitValue`.","title":"Free $q = 2$ box measures are bounded by the limit value"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-free-limit-free-infinite-volume-eq","kind":"node","lean":"StatMech.FK.fkFreeLimit_freeInfiniteVolume_eq","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," infinite-volume mass of increasing cylinders as a box limit"]],"source":"Free $q = 2$ infinite-volume mass of increasing cylinders as a box limit"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-free-limit-free-infinite-volume-eq","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q = 2`, the free infinite-volume measure of an increasing cylinder event equals `fkFreeLimitValue`.","title":"Free $q = 2$ infinite-volume mass of increasing cylinders as a box limit"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/fk-free-infinite-measure","kind":"node","lean":"StatMech.FK.fk_free_infinite_measure","parent":48149,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q=2\\)"],["text"," box masses of increasing cylinders tend to infinite volume"]],"source":"Free $q=2$ box masses of increasing cylinders tend to infinite volume"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi/index.html#fk-free-infinite-measure","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-free-q-2-random-cluster-measures-increase-wi","status":"fully_proved","summary":"For `q = 2`, the free box measures of an increasing cylinder event converge, along the whole sequence of boxes, to its free infinite-volume measure.","title":"Free $q=2$ box masses of increasing cylinders tend to infinite volume"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2","status":"fully_proved","title":"The wired infinite-volume limit at q = 2"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/fk-wired-limit-succ-le","kind":"node","lean":"StatMech.FK.fkWiredLimit_succ_le","parent":48199,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," increasing-event probabilities do not grow with the box"]],"source":"Wired $q = 2$ increasing-event probabilities do not grow with the box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/index.html#fk-wired-limit-succ-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2","status":"fully_proved","summary":"At `q = 2`, enlarging the wired box from `\u039b_m` to `\u039b_(m+1)` does not increase the probability of an increasing cylinder event of `\u039b_N`, `N \u2264 m`.","title":"Wired $q = 2$ increasing-event probabilities do not grow with the box"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/fk-wired-limit-value","kind":"node","lean":"StatMech.FK.fkWiredLimitValue","parent":48199,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/index.html#fk-wired-limit-value","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2","status":"fully_proved","summary":"The infimum over `k` of `\u03c6\u00b9_(N+k)(S)` for an event `S` of the box `\u039b_N`, at `q = 2`.","title":"Infimum of wired box probabilities of a box event"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/fk-wired-limit-exists","kind":"node","lean":"StatMech.FK.fk_wired_limit_exists","parent":48199,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/index.html#fk-wired-limit-exists","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2","status":"fully_proved","summary":"At `q = 2` the wired box probabilities of an increasing cylinder event converge to `fkWiredLimitValue`.","title":"Wired box probabilities of increasing cylinder events converge"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/fk-wired-limit-wired-infinite-volume-eq","kind":"node","lean":"StatMech.FK.fkWiredLimit_wiredInfiniteVolume_eq","parent":48199,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/index.html#fk-wired-limit-wired-infinite-volume-eq","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2","status":"fully_proved","summary":"At `q = 2` the chosen wired infinite-volume measure gives an increasing cylinder event the value `fkWiredLimitValue`.","title":"Wired infinite-volume probability equals the infimum over boxes"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/fk-wired-infinite-measure","kind":"node","lean":"StatMech.FK.fk_wired_infinite_measure","parent":48199,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2/index.html#fk-wired-infinite-measure","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-wired-infinite-volume-limit-at-q-2","status":"fully_proved","summary":"Existence of the wired infinite-volume random-cluster measure as the limit of wired box measures, on increasing cylinder events at `q = 2`: `\u03c6\u00b9_m(S) \u2192 \u03c6\u00b9(S)` as `m \u2192 \u221e`. The general statement (all `q \u2265 1`, weak convergence) is in Grimmett, *The Random-Cluster Model* (2006), Chapter 4.","title":"Wired infinite-volume measure as limit on increasing cylinder events"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","status":"fully_proved","title":"FKG inequality and ordering for infinite-volume FK measures at q = 2"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/free-finite-measure-fkg","kind":"node","lean":"StatMech.FK.freeFiniteMeasure_fkg","parent":48205,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG inequality for free "],["math","\\(q = 2\\)"],["text"," box measures on smaller-box events"]],"source":"FKG inequality for free $q = 2$ box measures on smaller-box events"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/index.html#free-finite-measure-fkg","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","status":"fully_proved","summary":"FKG inequality for the free box measure `\u03c6\u2070_m` (`q = 2`) on two increasing cylinder events of a smaller box `\u039b_N`, `N \u2264 m`.","title":"FKG inequality for free $q = 2$ box measures on smaller-box events"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/ivp-iv-fkg","kind":"node","lean":"StatMech.FK.ivp_iv_fkg","parent":48205,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG inequality for the free "],["math","\\(q = 2\\)"],["text"," infinite-volume measure"]],"source":"FKG inequality for the free $q = 2$ infinite-volume measure"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/index.html#ivp-iv-fkg","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","status":"fully_proved","summary":"FKG inequality for the free infinite-volume measure at `q = 2` on increasing cylinder events of a box.","title":"FKG inequality for the free $q = 2$ infinite-volume measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/wired-fk-prob-fkglattice-condition","kind":"node","lean":"StatMech.FK.wiredFkProb_FKGLatticeCondition","parent":48205,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG lattice condition for the wired measure when "],["math","\\(q \\ge 1\\)"]],"source":"FKG lattice condition for the wired measure when $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/index.html#wired-fk-prob-fkglattice-condition","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","status":"fully_proved","summary":"The wired random-cluster measure satisfies the FKG lattice condition when `q \u2265 1`.","title":"FKG lattice condition for the wired measure when $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/wired-fk-prob-positively-associated-even","kind":"node","lean":"StatMech.FK.wiredFkProb_positively_associated_events","parent":48205,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG inequality for the wired measure when "],["math","\\(q \\ge 1\\)"]],"source":"FKG inequality for the wired measure when $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/index.html#wired-fk-prob-positively-associated-even","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","status":"fully_proved","summary":"FKG inequality for the wired random-cluster measure on a finite graph: increasing events are positively correlated when `q \u2265 1`.","title":"FKG inequality for the wired measure when $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/wired-finite-measure-fkg","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_fkg","parent":48205,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG inequality for wired "],["math","\\(q = 2\\)"],["text"," box measures on smaller-box events"]],"source":"FKG inequality for wired $q = 2$ box measures on smaller-box events"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/index.html#wired-finite-measure-fkg","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","status":"fully_proved","summary":"FKG inequality for the wired box measure `\u03c6\u00b9_m` (`q = 2`) on two increasing cylinder events of a smaller box `\u039b_N`, `N \u2264 m`.","title":"FKG inequality for wired $q = 2$ box measures on smaller-box events"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/ivp-iv-fkg-wired","kind":"node","lean":"StatMech.FK.ivp_iv_fkg_wired","parent":48205,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG inequality for the wired "],["math","\\(q = 2\\)"],["text"," infinite-volume measure"]],"source":"FKG inequality for the wired $q = 2$ infinite-volume measure"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/index.html#ivp-iv-fkg-wired","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","status":"fully_proved","summary":"FKG inequality for the wired infinite-volume measure at `q = 2` on increasing cylinder events of a box.","title":"FKG inequality for the wired $q = 2$ infinite-volume measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/ivp-iv-monotone","kind":"node","lean":"StatMech.FK.ivp_iv_monotone","parent":48205,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," measure dominates the free one on increasing cylinders"]],"source":"Wired $q = 2$ measure dominates the free one on increasing cylinders"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume/index.html#ivp-iv-monotone","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/fkg-inequality-and-ordering-for-infinite-volume","status":"fully_proved","summary":"At `q = 2` the free infinite-volume measure gives every increasing cylinder event at most the probability the wired one gives it.","title":"Wired $q = 2$ measure dominates the free one on increasing cylinders"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2","status":"fully_proved","title":"Free and wired infinite-volume limits at q = 2"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2/mvl-infinite-volume-extremal","kind":"node","lean":"StatMech.FK.mvl_infiniteVolume_extremal","parent":48213,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergence and ordering of free and wired limits at "],["math","\\(q = 2\\)"]],"source":"Convergence and ordering of free and wired limits at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2/index.html#mvl-infinite-volume-extremal","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2","status":"fully_proved","summary":"At `q = 2`, for an increasing cylinder event `S`, `\u03c6\u2070_m(S) \u2192 \u03c6\u2070(S)`, `\u03c6\u00b9_m(S) \u2192 \u03c6\u00b9(S)`, and `\u03c6\u2070(S) \u2264 \u03c6\u00b9(S)`.","title":"Convergence and ordering of free and wired limits at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2/mvl-infinite-volume-extremal-subseq","kind":"node","lean":"StatMech.FK.mvl_infiniteVolume_extremal_subseq","parent":48213,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convergence and ordering at "],["math","\\(q = 2\\)"],["text"," in subsequence form"]],"source":"Convergence and ordering at $q = 2$ in subsequence form"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2/index.html#mvl-infinite-volume-extremal-subseq","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/free-and-wired-infinite-volume-limits-at-q-2","status":"fully_proved","summary":"Subsequence form of `mvl_infiniteVolume_extremal` (with the identity subsequence).","title":"Convergence and ordering at $q = 2$ in subsequence form"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","title":"Multi-edge events and translation invariance of q = 2 FK measures"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-generic-multi-open-event","kind":"node","lean":"StatMech.FK.med_genericMultiOpenEvent","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-generic-multi-open-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The event `A_t` that every edge of the finite set `t` is open. On box configurations it is `cdc_boxMultiOpenEvent` (`med_boxMultiOpenEvent_eq`).","title":"Event that all edges of a finite set are open"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-multi-mass-prob","kind":"node","lean":"StatMech.FK.med_multiMassProb","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-multi-mass-prob","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The mass `\u2211_\u03c9 1_{A_t}(\u03c9) \u03bc(\u03c9)` of the event that every edge of `t` is open, for a weight function `\u03bc` on a finite configuration space; it is `eventMassProb \u03bc A_t` (`med_multiMassProb_eq_eventMassProb`).","title":"Mass of the all-open event under a weight function"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-centred-mass-eq","kind":"node","lean":"StatMech.FK.med_centred_mass_eq","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-centred-mass-eq","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The wired box measure on `\u039b_m`, pushed to `\u2124^d`, gives `A_t` the box mass of `t`.","title":"Wired box measure of an all-open event equals its box mass"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-free-centred-mass-eq","kind":"node","lean":"StatMech.FK.med_free_centred_mass_eq","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-free-centred-mass-eq","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The free box measure on `\u039b_m`, pushed to `\u2124^d`, gives `A_t` the box mass of `t`.","title":"Free box measure of an all-open event equals its box mass"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-centred-eq-translated-wired","kind":"node","lean":"StatMech.FK.med_centred_eq_translated_wired","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-centred-eq-translated-wired","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The wired mass of `t` in `\u039b_m` equals the wired mass of `t + v` in `v + \u039b_m`.","title":"Translation covariance of wired box masses"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-centred-eq-translated-free","kind":"node","lean":"StatMech.FK.med_centred_eq_translated_free","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-centred-eq-translated-free","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The free mass of `t` in `\u039b_m` equals the free mass of `t + v` in `v + \u039b_m`.","title":"Translation covariance of free box masses"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-centred-tendsto-iv","kind":"node","lean":"StatMech.FK.med_centred_tendsto_iv","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-centred-tendsto-iv","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"Restatement of `cdc_wired_multiOpenEvent_tendsto`: the wired box probabilities of `A_t` converge to the wired infinite-volume probability.","title":"Wired all-open probabilities converge to the infinite-volume value"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-free-centred-tendsto-iv","kind":"node","lean":"StatMech.FK.med_free_centred_tendsto_iv","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-free-centred-tendsto-iv","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"Restatement of `cdc_free_multiOpenEvent_tendsto`: the free box probabilities of `A_t` converge to the free infinite-volume probability.","title":"Free all-open probabilities converge to the infinite-volume value"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-off-centre-multi-carrier","kind":"node","lean":"StatMech.FK.med_offCentreMultiCarrier","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-off-centre-multi-carrier","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The difference between the wired mass of `t` in `\u039b_m` and the wired mass of `t + v` in `v + \u039b_m`; it vanishes by translation covariance.","title":"Difference of wired all-open masses of an edge set and its translate"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-wired-multi-homogeneous-of-cofinal","kind":"node","lean":"StatMech.FK.med_wiredMultiHomogeneous_of_cofinal","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-wired-multi-homogeneous-of-cofinal","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"If the wired mass of `t + v` in `v + \u039b_m` eventually equals the wired mass of `t'` in `\u039b_m`, the wired infinite-volume measure gives `A_t` and `A_{t'}` the same probability.","title":"Equal wired probabilities from eventually equal translated masses"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-free-multi-homogeneous-of-cofinal","kind":"node","lean":"StatMech.FK.med_freeMultiHomogeneous_of_cofinal","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-free-multi-homogeneous-of-cofinal","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"If the free mass of `t + v` in `v + \u039b_m` eventually equals the free mass of `t'` in `\u039b_m`, the free infinite-volume measure gives `A_t` and `A_{t'}` the same probability.","title":"Equal free probabilities from eventually equal translated masses"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-wired-placement","kind":"node","lean":"StatMech.FK.med_wiredPlacement","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-wired-placement","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"A wired placement for the shift `g`: every finite edge set `s` and its shift `g\u207b\u00b9 s` are images of box edge sets `t`, `t'` of a common box `\u039b_N` such that the wired mass of `t + g` in `g + \u039b_m` eventually equals the wired mass of `t'` in `\u039b_m`. See also `lmc_wiredPlacement`.","title":"Wired placement for a shift via eventually equal box masses"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-free-placement","kind":"node","lean":"StatMech.FK.med_freePlacement","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-free-placement","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"A free placement for the shift `g`, defined as `med_wiredPlacement` with free masses.","title":"Free placement for a shift via eventually equal box masses"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-wired-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.med_wiredIV_isTranslationInvariant_of_placement","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-wired-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The wired infinite-volume `q = 2` measure is translation invariant if every shift has a wired placement. See also `flc_wiredIV_isTranslationInvariant`, which proves invariance without hypotheses.","title":"Wired translation invariance from wired placements of every shift"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/med-free-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.med_freeIV_isTranslationInvariant_of_placement","parent":48216,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of/index.html#med-free-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/multi-edge-events-and-translation-invariance-of","status":"fully_proved","summary":"The free infinite-volume `q = 2` measure is translation invariant if every shift has a free placement. See also `bdp_freeIV_isTranslationInvariant`, which proves invariance without hypotheses.","title":"Free translation invariance from free placements of every shift"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal","status":"fully_proved","title":"The off-centre carrier for wired q = 2 FK marginals"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/occ-translated-tendsto-wired-density","kind":"node","lean":"StatMech.FK.occ_translated_tendsto_wiredDensity","parent":48232,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/index.html#occ-translated-tendsto-wired-density","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal","status":"fully_proved","summary":"The wired marginals of the translated edge `e + v` in `v + \u039b_{N+k}` converge to the wired `q = 2` density of `e`.","title":"Translated wired marginals converge to the wired density"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/occ-translated-between-iv","kind":"node","lean":"StatMech.FK.occ_translated_between_iv","parent":48232,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/index.html#occ-translated-between-iv","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal","status":"fully_proved","summary":"The free `q = 2` density of `e` is at most the wired marginal of `e + v` in `v + \u039b_{N+k}`.","title":"Free density is at most each translated wired marginal"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/occ-free-eq-wired-off-countable","kind":"node","lean":"StatMech.FK.occ_free_eq_wired_off_countable","parent":48232,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/index.html#occ-free-eq-wired-off-countable","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal","status":"fully_proved","summary":"Under the secant comparison `wired(p') \u2264 free(p)` for `p' < p`, the free and wired densities of a box edge agree for all `p \u2208 (0, 1)` outside a countable set, namely the discontinuities of the monotone wired density.","title":"Secant bound gives free\u2013wired density agreement off a countable set"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/occ-off-centre-carrier-eq-zero","kind":"node","lean":"StatMech.FK.occ_offCentreCarrier_eq_zero","parent":48232,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/index.html#occ-off-centre-carrier-eq-zero","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal","status":"fully_proved","summary":"The off-centre carrier vanishes for every `k`, by translation covariance of the wired box measures.","title":"Off-centre carrier vanishes by translation covariance"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/occ-off-centre-carrier","kind":"node","lean":"StatMech.FK.occ_offCentreCarrier","parent":48232,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/index.html#occ-off-centre-carrier","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal","status":"fully_proved","summary":"The off-centre carrier tends to zero, for every `p`.","title":"Off-centre carrier tends to zero"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/occ-wired-density-eq-of-translation-cofi","kind":"node","lean":"StatMech.FK.occ_wired_density_eq_of_translation_cofinal","parent":48232,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal/index.html#occ-wired-density-eq-of-translation-cofi","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/the-off-centre-carrier-for-wired-q-2-fk-marginal","status":"fully_proved","summary":"If for every `k` the wired marginal of `e' + v` in `v + \u039b_{N+k}` equals the wired marginal of `e` in `\u039b_{N+k}`, then `e` and `e'` have the same wired density.","title":"Equal wired densities from equal translated marginals"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","title":"Domination of an inner wired FK measure by an outer one"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-is-inner-vert","kind":"node","lean":"StatMech.FK.ocd_IsInnerVert","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-is-inner-vert","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"A vertex of `G_out` is inner if it lies in the image of `\u03b9`.","title":"Inner vertex of the outer graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-edge","kind":"node","lean":"StatMech.FK.ocd_innerEdge","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-edge","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The image in `G_out` of an edge of `G_in` under `\u03b9`.","title":"Image of an inner-graph edge in the outer graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-edge-injective","kind":"node","lean":"StatMech.FK.ocd_innerEdge_injective","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-edge-injective","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The edge map `Sym2.map \u03b9` is injective when `\u03b9` is.","title":"Injectivity of the edge map"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-edge-mk","kind":"node","lean":"StatMech.FK.ocd_innerEdge_mk","parent":48239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edge map sends "],["math","\\(\\{x, y\\}\\)"],["text"," to "],["math","\\(\\{\\iota x, \\iota y\\}\\)"]],"source":"Edge map sends $\\{x, y\\}$ to $\\{\\iota x, \\iota y\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-edge-mk","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The image of the edge `{x, y}` is `{\u03b9 x, \u03b9 y}`.","title":"Edge map sends $\\{x, y\\}$ to $\\{\\iota x, \\iota y\\}$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-adj-match","kind":"node","lean":"StatMech.FK.ocd_AdjMatch","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-adj-match","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The map `\u03b9` matches adjacency: `x ~ y` in `G_in` iff `\u03b9 x ~ \u03b9 y` in `G_out`.","title":"Adjacency matching of the inner embedding"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-ext","kind":"node","lean":"StatMech.FK.ocd_psiExt","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The configuration on `G_out` that equals the inner configuration `\u03c9` on inner edges and the outer configuration `\u03c8` elsewhere.","title":"Gluing of inner and outer configurations"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-restrict","kind":"node","lean":"StatMech.FK.ocd_innerRestrict","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-restrict","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The restriction of a configuration on `G_out` to the edges of `G_in`, read through `\u03b9`.","title":"Restriction of an outer configuration to inner edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-edge-finset","kind":"node","lean":"StatMech.FK.ocd_innerEdgeFinset","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-edge-finset","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The finite set of inner edges of `G_out`.","title":"Finite set of inner edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-mem-inner-edge-finset","kind":"node","lean":"StatMech.FK.ocd_mem_innerEdgeFinset","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-mem-inner-edge-finset","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Membership in the inner edge set is membership in the image of the edge map.","title":"Inner edge set is the range of the edge map"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-ext-inner-edge","kind":"node","lean":"StatMech.FK.ocd_psiExt_innerEdge","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-ext-inner-edge","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The glued configuration takes the inner value on inner edges.","title":"Glued configuration takes the inner value on inner edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-ext-eq-psi-of-not-range","kind":"node","lean":"StatMech.FK.ocd_psiExt_eq_psi_of_not_range","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-ext-eq-psi-of-not-range","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The glued configuration takes the outer value off the inner edges.","title":"Glued configuration equals the outer one at each non-inner edge"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-ext-inner-edge-pair","kind":"node","lean":"StatMech.FK.ocd_psiExt_innerEdge_pair","parent":48239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Glued value on "],["math","\\(\\{\\iota x, \\iota y\\}\\)"],["text"," is the inner value on "],["math","\\(\\{x, y\\}\\)"]],"source":"Glued value on $\\{\\iota x, \\iota y\\}$ is the inner value on $\\{x, y\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-ext-inner-edge-pair","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The glued configuration on `{\u03b9 x, \u03b9 y}` is the inner value on `{x, y}`.","title":"Glued value on $\\{\\iota x, \\iota y\\}$ is the inner value on $\\{x, y\\}$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-restrict-psi-ext","kind":"node","lean":"StatMech.FK.ocd_innerRestrict_psiExt","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-restrict-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Restricting a glued configuration recovers the inner configuration.","title":"Restricting a glued configuration recovers the inner one"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-agrees-off-psi-ext","kind":"node","lean":"StatMech.FK.ocd_agreesOff_psiExt","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-agrees-off-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"A glued configuration agrees with the outer configuration off the inner edges.","title":"Glued configuration agrees with the outer one off the inner edge set"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-ext-inner-restrict-of-agrees-off","kind":"node","lean":"StatMech.FK.ocd_psiExt_innerRestrict_of_agreesOff","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-ext-inner-restrict-of-agrees-off","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"A configuration that agrees with `\u03c8` off the inner edges is the gluing of `\u03c8` with its own restriction.","title":"Agreeing configuration is the gluing of its own restriction"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-cond-fibre-eq-image-psi-ext","kind":"node","lean":"StatMech.FK.ocd_condFibre_eq_image_psiExt","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-cond-fibre-eq-image-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The configurations agreeing with `\u03c8` off the inner edges are exactly the gluings of `\u03c8` with inner configurations.","title":"Conditioning fibre is the set of gluings"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-open-graph","kind":"node","lean":"StatMech.FK.ocd_outsideOpenGraph","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-open-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The graph of `\u03c8`-open edges of `G_out` that are not inner edges.","title":"Graph of open non-inner edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-graph","kind":"node","lean":"StatMech.FK.ocd_outsideGraph","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The outside graph: `\u03c8`-open non-inner edges together with the clique on the wired boundary `\u2202_out`.","title":"Outside graph with the wired boundary clique"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-graph-adj","kind":"node","lean":"StatMech.FK.ocd_outsideGraph_adj","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-graph-adj","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Adjacency in the outside graph.","title":"Outside-graph adjacency: open non-inner edge or boundary pair"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-induced-wiring","kind":"node","lean":"StatMech.FK.ocd_inducedWiring","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-induced-wiring","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The wiring induced on `G_in` by `\u03c8`: distinct inner vertices are adjacent when their images are joined in the outside graph.","title":"Wiring induced on the inner graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-induced-wiring-adj","kind":"node","lean":"StatMech.FK.ocd_inducedWiring_adj","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-induced-wiring-adj","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Adjacency in the induced wiring.","title":"Induced-wiring adjacency: distinct and joined in the outside graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-image-graph","kind":"node","lean":"StatMech.FK.ocd_innerImageGraph","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-image-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The image in `G_out` of the open subgraph of an inner configuration `\u03c9`.","title":"Image of the inner open subgraph in the outer graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-graph","kind":"node","lean":"StatMech.FK.ocd_innerGraph","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The inner cluster graph: open edges of `\u03c9` together with the induced wiring.","title":"Inner cluster graph with the induced wiring"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-graph","kind":"node","lean":"StatMech.FK.ocd_fullGraph","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The cluster graph of the glued configuration on `G_out` with the wired boundary clique.","title":"Cluster graph of a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-graph-le-full-graph","kind":"node","lean":"StatMech.FK.ocd_outsideGraph_le_fullGraph","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-graph-le-full-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The outside graph is a subgraph of the full cluster graph.","title":"Outside graph is a subgraph of the full cluster graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-ext-eq-false-of-range","kind":"node","lean":"StatMech.FK.ocd_psiExt_eq_false_of_range","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-ext-eq-false-of-range","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"On inner edges the glued configuration does not depend on the outer configuration.","title":"Glued configuration on inner edges ignores the outer configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-wired-graph-eq-sup","kind":"node","lean":"StatMech.FK.ocd_psiWiredGraph_eq_sup","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-wired-graph-eq-sup","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The full cluster graph is the union of the inner image graph and the outside graph.","title":"Full cluster graph as the union of inner image and outside graphs"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-image-graph-adj-inner","kind":"node","lean":"StatMech.FK.ocd_innerImageGraph_adj_inner","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-image-graph-adj-inner","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Both endpoints of an inner image edge are inner vertices.","title":"Endpoints of inner image edges are inner vertices"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-image-graph-adj-v","kind":"node","lean":"StatMech.FK.ocd_innerImageGraph_adj_\u03b9V","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-image-graph-adj-v","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The inner image graph on images of `\u03b9` is the open subgraph of `\u03c9`.","title":"Inner image graph on embedded vertices is the open subgraph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-graph-reachable-of-outside-rea","kind":"node","lean":"StatMech.FK.ocd_innerGraph_reachable_of_outsideReachable","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-graph-reachable-of-outside-rea","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Inner vertices whose images are joined in the outside graph are joined in the inner cluster graph.","title":"Outside reachability gives inner cluster reachability"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-graph-reachable-of-inner-graph","kind":"node","lean":"StatMech.FK.ocd_fullGraph_reachable_of_innerGraph","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-graph-reachable-of-inner-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Reachability in the inner cluster graph gives reachability of the images in the full cluster graph.","title":"Inner cluster reachability gives full-graph reachability of images"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-graph-reachable-of-full-graph","kind":"node","lean":"StatMech.FK.ocd_innerGraph_reachable_of_fullGraph","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-graph-reachable-of-full-graph","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Reachability of images in the full cluster graph gives reachability in the inner cluster graph.","title":"Full-graph reachability of images gives inner cluster reachability"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-graph-reachable-v","kind":"node","lean":"StatMech.FK.ocd_fullGraph_reachable_\u03b9V","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-graph-reachable-v","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Images of inner vertices are joined in the full cluster graph iff the vertices are joined in the inner cluster graph.","title":"Full-graph and inner cluster reachability agree on embedded vertices"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-graph-adj-of-outside","kind":"node","lean":"StatMech.FK.ocd_fullGraph_adj_of_outside","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-graph-adj-of-outside","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"A full cluster graph edge at a non-inner vertex is an outside graph edge.","title":"Full-graph edges at non-inner vertices are outside-graph edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-only","kind":"node","lean":"StatMech.FK.ocd_OutsideOnly","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-only","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"A vertex is outside-only if its outside cluster contains no inner vertex.","title":"Outside-only vertex"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-only-self-not-inner","kind":"node","lean":"StatMech.FK.ocd_outsideOnly_self_not_inner","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-only-self-not-inner","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"An outside-only vertex is not inner.","title":"Outside-only vertices are not inner"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-only-of-reachable","kind":"node","lean":"StatMech.FK.ocd_outsideOnly_of_reachable","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-only-of-reachable","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Being outside-only is constant on outside clusters.","title":"Outside-only property is constant on outside clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-graph-reachable-of-full-grap","kind":"node","lean":"StatMech.FK.ocd_outsideGraph_reachable_of_fullGraph_outsideOnly","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-graph-reachable-of-full-grap","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"From an outside-only vertex, reachability in the full cluster graph stays in the outside graph.","title":"Full-graph reachability from an outside-only vertex stays outside"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-is-outside-only","kind":"node","lean":"StatMech.FK.ocd_IsOutsideOnly","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-is-outside-only","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The outside-only property of a cluster of the outside graph.","title":"Outside-only property of an outside cluster"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-is-outside-only-mk","kind":"node","lean":"StatMech.FK.ocd_isOutsideOnly_mk","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-is-outside-only-mk","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The cluster of `v` is outside-only iff `v` is.","title":"Cluster of a vertex is outside-only iff the vertex is"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-only-comp","kind":"node","lean":"StatMech.FK.ocd_OutsideOnlyComp","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-only-comp","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The outside clusters that contain no inner vertex.","title":"Outside-only clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-component-map","kind":"node","lean":"StatMech.FK.ocd_fullComponentMap","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-component-map","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The map from inner clusters and outside-only clusters to the clusters of the glued configuration.","title":"Cluster map into the clusters of a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-component-map-inl-mk","kind":"node","lean":"StatMech.FK.ocd_fullComponentMap_inl_mk","parent":48239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cluster map sends the inner cluster of "],["math","\\(x\\)"],["text"," to that of "],["math","\\(\\iota x\\)"]],"source":"Cluster map sends the inner cluster of $x$ to that of $\\iota x$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-component-map-inl-mk","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The cluster map sends the inner cluster of `x` to the full cluster of `\u03b9 x`.","title":"Cluster map sends the inner cluster of $x$ to that of $\\iota x$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-component-map-inr-mk","kind":"node","lean":"StatMech.FK.ocd_fullComponentMap_inr_mk","parent":48239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cluster map sends an outside-only cluster of "],["math","\\(v\\)"],["text"," to that of "],["math","\\(v\\)"]],"source":"Cluster map sends an outside-only cluster of $v$ to that of $v$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-component-map-inr-mk","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The cluster map sends an outside-only cluster of `v` to the full cluster of `v`.","title":"Cluster map sends an outside-only cluster of $v$ to that of $v$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-component-map-injective","kind":"node","lean":"StatMech.FK.ocd_fullComponentMap_injective","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-component-map-injective","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The cluster map is injective.","title":"Injectivity of the cluster map"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-component-map-surjective","kind":"node","lean":"StatMech.FK.ocd_fullComponentMap_surjective","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-component-map-surjective","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The cluster map is surjective.","title":"Surjectivity of the cluster map"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-full-component-equiv","kind":"node","lean":"StatMech.FK.ocd_fullComponentEquiv","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-full-component-equiv","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The clusters of the glued configuration are in bijection with the inner clusters plus the outside-only clusters.","title":"Cluster bijection for a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-outside-shift","kind":"node","lean":"StatMech.FK.ocd_outsideShift","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-outside-shift","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The number of outside-only clusters of `\u03c8`.","title":"Number of outside-only clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-num-clusters-bc-psi-ext","kind":"node","lean":"StatMech.FK.ocd_numClustersBC_psiExt","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-num-clusters-bc-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The wired cluster count of a glued configuration is the induced-wiring cluster count of the inner configuration plus the number of outside-only clusters.","title":"Glued cluster count is induced-wiring count plus outside-only clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-edge-image-edge-finset","kind":"node","lean":"StatMech.FK.ocd_innerEdge_image_edgeFinset","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-edge-image-edge-finset","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The image of the edges of `G_in` is the set of inner edges of `G_out`.","title":"Edges of the inner graph map onto the inner edges of the outer graph"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-edge-factor","kind":"node","lean":"StatMech.FK.ocd_psiEdgeFactor","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-edge-factor","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The Bernoulli edge weight of `\u03c8` on the non-inner edges of `G_out`.","title":"Outer Bernoulli edge weight of a configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-edge-factor-pos","kind":"node","lean":"StatMech.FK.ocd_psiEdgeFactor_pos","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-edge-factor-pos","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The outer edge weight is positive for `p \u2208 (0, 1)`.","title":"Positivity of the outer edge weight"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-edge-product-psi-ext","kind":"node","lean":"StatMech.FK.ocd_edgeProduct_psiExt","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-edge-product-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The edge weight of a glued configuration factors into the inner edge weight and the outer edge weight.","title":"Factorization of the edge weight of a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-shift","kind":"node","lean":"StatMech.FK.ocd_psiShift","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-shift","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The factor `\u03c8` contributes to the FK weight of a glued configuration: its outer edge weight times `q` to the number of outside-only clusters.","title":"Outer factor of the FK weight of a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-psi-shift-pos","kind":"node","lean":"StatMech.FK.ocd_psiShift_pos","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-psi-shift-pos","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The outer factor is positive for `p \u2208 (0, 1)` and `q > 0`.","title":"Positivity of the outer factor"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-bc-weight-psi-ext","kind":"node","lean":"StatMech.FK.ocd_bcWeight_psiExt","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-bc-weight-psi-ext","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The wired FK weight of a glued configuration is the induced-wiring FK weight of the inner configuration times the outer factor.","title":"Factorization of the wired FK weight of a glued configuration"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-induced-bc-z-psi","kind":"node","lean":"StatMech.FK.ocd_inducedBcZ_psi","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-induced-bc-z-psi","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The partition function of the outer measure conditioned on `\u03c8` is the induced-wiring partition function times the outer factor.","title":"Factorization of the conditioned outer partition function"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-cond-bc-prob-psi-ext-eq-bc-prob","kind":"node","lean":"StatMech.FK.ocd_condBcProb_psiExt_eq_bcProb","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-cond-bc-prob-psi-ext-eq-bc-prob","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Conditioned on `\u03c8` off the inner edges, the outer wired measure of a glued configuration is the induced-wiring FK measure of the inner configuration.","title":"Conditional outer measure is the induced-wiring FK measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-cond-bc-prob-psi-ext-sum-eq-induced","kind":"node","lean":"StatMech.FK.ocd_condBcProb_psiExt_sum_eq_inducedBox","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-cond-bc-prob-psi-ext-sum-eq-induced","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The conditional outer probability of an event is the induced-wiring probability of its pullback under gluing.","title":"Conditional outer probability as an induced-wiring probability"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-cond-bc-prob-psi-ext-dominated","kind":"node","lean":"StatMech.FK.ocd_condBcProb_psiExt_dominated","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-cond-bc-prob-psi-ext-dominated","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"If the induced wiring is coarser than the clique on `\u2202_in`, the induced-wiring measure is dominated by the wired measure on `G_in` (for `q \u2265 1`).","title":"Domination by the inner wired measure under a coarser induced wiring"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-free-le-induced-bc-prob","kind":"node","lean":"StatMech.FK.ocd_free_le_inducedBcProb","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-free-le-induced-bc-prob","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The free FK measure on `G_in` is dominated by every induced-wiring measure (for `q \u2265 1`).","title":"Free inner measure is dominated by every induced-wiring measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-free-le-cond-bc-prob-inner-restrict","kind":"node","lean":"StatMech.FK.ocd_free_le_condBcProb_innerRestrict","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-free-le-cond-bc-prob-inner-restrict","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The free FK measure on `G_in` is dominated by the restricted conditional outer measure (for `q \u2265 1`).","title":"Free inner measure is dominated by the conditional outer measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-out-proj","kind":"node","lean":"StatMech.FK.ocd_outProj","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-out-proj","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The projection of a configuration on `G_out` that closes every inner edge.","title":"Projection closing every inner edge"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-out-proj-eq-false-of-range","kind":"node","lean":"StatMech.FK.ocd_outProj_eq_false_of_range","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-out-proj-eq-false-of-range","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The projection closes inner edges.","title":"Projected configuration is closed on inner edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-out-proj-eq-of-not-range","kind":"node","lean":"StatMech.FK.ocd_outProj_eq_of_not_range","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-out-proj-eq-of-not-range","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The projection keeps non-inner edges.","title":"Projection keeps non-inner edges"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-out-proj-idem","kind":"node","lean":"StatMech.FK.ocd_outProj_idem","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-out-proj-idem","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The projection is idempotent.","title":"Idempotence of the inner-edge projection"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-filter-out-proj-eq-cond-fibre","kind":"node","lean":"StatMech.FK.ocd_filter_outProj_eq_condFibre","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-filter-out-proj-eq-cond-fibre","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The fibre of the projection over a projected configuration is its conditioning fibre.","title":"Projection fibres are conditioning fibres"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-sum-fibre-mass-eq-one","kind":"node","lean":"StatMech.FK.ocd_sum_fibreMass_eq_one","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-sum-fibre-mass-eq-one","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The outer wired masses of the fibres over projected configurations sum to one.","title":"Outer wired fibre masses sum to one"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-bc-prob-decompose","kind":"node","lean":"StatMech.FK.ocd_bcProb_decompose","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-bc-prob-decompose","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Law of total probability for the outer wired measure over the configuration off the inner edges. See also `bcProb_fibre_decompose` in `FKDisjointBoxDomainMarkov` for a general conditioned edge set.","title":"Law of total probability over outside configurations"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-cond-bc-prob-inner-restrict-le","kind":"node","lean":"StatMech.FK.ocd_condBcProb_innerRestrict_le","parent":48239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional outer wired probability at most the inner wired one, "],["math","\\(q \\ge 1\\)"]],"source":"Conditional outer wired probability at most the inner wired one, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-cond-bc-prob-inner-restrict-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Conditioned on `\u03c8`, the restricted outer wired probability of an increasing event is at most its wired probability on `G_in`, if the induced wiring is coarser than the clique on `\u2202_in`.","title":"Conditional outer wired probability at most the inner wired one, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-wired-inner-dominated-bc-prob","kind":"node","lean":"StatMech.FK.ocd_wired_inner_dominated_bcProb","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-wired-inner-dominated-bc-prob","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"Domination of the restricted outer wired measure by the inner wired measure on increasing events, when every induced wiring is coarser than the clique on `\u2202_in` (for `q \u2265 1`).","title":"Inner wired domination of the restricted outer wired measure"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-lattice-adj-match","kind":"node","lean":"StatMech.FK.ocd_latticeAdjMatch","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-lattice-adj-match","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The inclusion of a vertex set of `\u2124^d` into a larger one matches the induced hypercubic adjacency.","title":"Lattice inclusion matches induced adjacency"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-bdry-in-of-outside-graph-adj","kind":"node","lean":"StatMech.FK.ocd_bdryIn_of_outsideGraph_adj","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-bdry-in-of-outside-graph-adj","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"An inner vertex joined to a non-inner vertex in the outside graph lies on the inner boundary.","title":"Inner vertex with an outside-graph edge lies on the inner boundary"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-lattice-induced-wiring-le","kind":"node","lean":"StatMech.FK.ocd_latticeInducedWiring_le","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-lattice-induced-wiring-le","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"For sub-boxes of `\u2124^d` avoiding the outer boundary, every induced wiring is coarser than the clique on the inner boundary.","title":"Lattice induced wirings are coarser than the inner boundary clique"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-lattice-wired-inner-dominated","kind":"node","lean":"StatMech.FK.ocd_latticeWired_inner_dominated","parent":48239,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inner wired domination for vertex sets of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Inner wired domination for vertex sets of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-lattice-wired-inner-dominated","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"For a vertex set of `\u2124^d` inside a larger one and avoiding its wired boundary, the restricted outer wired FK measure is dominated by the inner wired FK measure on increasing events (for `q \u2265 1`).","title":"Inner wired domination for vertex sets of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-inner-restrict-preimage-multi-open-e","kind":"node","lean":"StatMech.FK.ocd_innerRestrict_preimage_multiOpenEvent","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-inner-restrict-preimage-multi-open-e","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"The pullback of \"all edges of `t` are open\" under restriction is \"all edges of `\u03b9(t)` are open\".","title":"Restriction pulls back all-open events to their images"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/ocd-lattice-wired-multi-mass-dominated","kind":"node","lean":"StatMech.FK.ocd_latticeWired_multiMass_dominated","parent":48239,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou/index.html#ocd-lattice-wired-multi-mass-dominated","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/domination-of-an-inner-wired-fk-measure-by-an-ou","status":"fully_proved","summary":"For a vertex set of `\u2124^d` inside a larger one and avoiding its wired boundary, the outer wired probability that all edges of `\u03b9(t)` are open is at most the inner wired probability that all edges of `t` are open (for `q \u2265 1`).","title":"Outer wired all-open probability at most the inner wired one"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat.json","id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","kind":"scope","navigation":"module","parent":48000,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","title":"Two-point connectivity with and without percolation"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/infinite-two-point-real","kind":"node","lean":"StatMech.FK.infiniteTwoPointReal","parent":48317,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-point connectivity of a bond measure on "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Two-point connectivity of a bond measure on $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#infinite-two-point-real","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"The two-point connectivity `\u03bc(x \u2194 y)` of a measure on bond configurations of `\u2124^d`.","title":"Two-point connectivity of a bond measure on $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/cluster-infinite-event","kind":"node","lean":"StatMech.FK.clusterInfiniteEvent","parent":48317,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#cluster-infinite-event","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"The event `{x \u2194 \u221e}` that the open cluster of `x` is infinite.","title":"Event that the open cluster of a site is infinite"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/two-infinite-clusters-of-not-connected","kind":"node","lean":"StatMech.FK.two_infinite_clusters_of_not_connected","parent":48317,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#two-infinite-clusters-of-not-connected","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"If `x` and `y` both lie in infinite clusters, either they are connected or there are at least two infinite clusters.","title":"Disconnected sites in infinite clusters give two infinite clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/infinite-cluster-inter-le-two-point","kind":"node","lean":"StatMech.FK.infinite_cluster_inter_le_twoPoint","parent":48317,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#infinite-cluster-inter-le-two-point","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"With at most one infinite cluster almost surely, `\u03bc(x \u2194 \u221e, y \u2194 \u221e) \u2264 \u03bc(x \u2194 y)`.","title":"Joint percolation at most connectivity given no two infinite clusters"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/infinite-two-point-uniform-pos-of-fkg-un","kind":"node","lean":"StatMech.FK.infiniteTwoPoint_uniform_pos_of_fkg_unique","parent":48317,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#infinite-two-point-uniform-pos-of-fkg-un","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"If `\u03bc` satisfies the FKG inequality for the events `{x \u2194 \u221e}` (`hPA`), is translation invariant on them (`hTI`), has at most one infinite cluster almost surely (`huniq`), and percolates (`htheta`), then `\u03bc(0 \u2194 x) \u2265 \u03b8\u00b2` for every `x`.","title":"Two-point lower bound from FKG, invariance, uniqueness and percolation"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/crossing-event-real-tendsto-percolation","kind":"node","lean":"StatMech.FK.crossingEvent_real_tendsto_percolation","parent":48317,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#crossing-event-real-tendsto-percolation","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"For any finite measure, the probability that the origin is connected to the complement of the box of radius `n - 1` tends to the percolation probability.","title":"Crossing probabilities converge to the percolation probability"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/box-bdry-conn-event-real-tendsto-percola","kind":"node","lean":"StatMech.FK.boxBdryConnEvent_real_tendsto_percolation","parent":48317,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#box-bdry-conn-event-real-tendsto-percola","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"For any finite measure, the probability that the origin is connected to the boundary of the box of radius `n` inside the box tends to the percolation probability.","title":"Box-boundary connection probability tends to percolation probability"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/connection-event-subset-box-bdry-conn-ev","kind":"node","lean":"StatMech.FK.connectionEvent_subset_boxBdryConnEvent","parent":48317,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#connection-event-subset-box-bdry-conn-ev","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"A connection from the origin to a site outside the box passes through the box boundary.","title":"A connection to outside the box passes through the box boundary"},{"id":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/infinite-two-point-uniform-decay-of-no-p","kind":"node","lean":"StatMech.FK.infiniteTwoPoint_uniform_decay_of_no_percolation","parent":48317,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat/index.html#infinite-two-point-uniform-decay-of-no-p","scope":"StatisticalMechanics/random-cluster-model/infinite-volume-random-cluster-measures/two-point-connectivity-with-and-without-percolat","status":"fully_proved","summary":"Without percolation, the two-point connectivity from the origin tends to zero uniformly: for every `\u03b5 > 0` there is `n` with `\u03bc(0 \u2194 x) < \u03b5` for all `x` outside the box of radius `n`.","title":"Uniform decay of two-point connectivity without percolation"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model","kind":"scope","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model","status":"fully_proved","title":"Critical point of the random-cluster model"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model","kind":"scope","navigation":"module","parent":48327,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model","status":"fully_proved","title":"The critical point of the random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model/fk-theta","kind":"node","lean":"StatMech.FK.fkTheta","parent":48328,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired random-cluster percolation probability "],["math","\\(\\theta(p, q)\\)"]],"source":"Wired random-cluster percolation probability $\\theta(p, q)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model/index.html#fk-theta","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model","status":"fully_proved","summary":"The percolation probability `\u03b8(p, q) = \u03c6^1_{p,q}(0 \u2194 \u221e)` of the wired infinite-volume random-cluster measure on `\u2124^d`.","title":"Wired random-cluster percolation probability $\\theta(p, q)$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model/fk-subcritical-set","kind":"node","lean":"StatMech.FK.fkSubcriticalSet","parent":48328,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model/index.html#fk-subcritical-set","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model","status":"fully_proved","summary":"The parameters `p \u2208 (0, 1)` at which the wired measure does not percolate (`\u03b8(p, q) = 0`).","title":"Non-percolating parameters of the wired random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model/fk-pc","kind":"node","lean":"StatMech.FK.fkPc","parent":48328,"reading_title":{"format":"inline-parts-v1","parts":[["text","Critical parameter "],["math","\\(p_c(q)\\)"],["text"," of the random-cluster model"]],"source":"Critical parameter $p_c(q)$ of the random-cluster model"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model/index.html#fk-pc","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-random-cluster-model","status":"fully_proved","summary":"The critical parameter `p_c(q) = sup {p \u2208 (0, 1) : \u03b8(p, q) = 0}` of the random-cluster model on `\u2124^d` (equal to `0` when the set is empty, e.g. for `q \u2264 0`).","title":"Critical parameter $p_c(q)$ of the random-cluster model"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster","kind":"scope","navigation":"module","parent":48327,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster","status":"fully_proved","title":"The self-dual point of the planar random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster/dual-param","kind":"node","lean":"StatMech.BeffaraDC.dualParam","parent":48332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Planar dual parameter "],["math","\\(p^{\\ast}\\)"]],"source":"Planar dual parameter $p^{\\ast}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster/index.html#dual-param","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster","status":"fully_proved","summary":"The planar dual parameter `p* = (1 - p) q / ((1 - p) q + p)`, characterized by `p* p / ((1 - p*)(1 - p)) = q`.","title":"Planar dual parameter $p^{\\ast}$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster/self-dual-point","kind":"node","lean":"StatMech.BeffaraDC.selfDualPoint","parent":48332,"reading_title":{"format":"inline-parts-v1","parts":[["text","Self-dual point "],["math","\\(p_{sd}(q) = \\sqrt q/(1 + \\sqrt q)\\)"]],"source":"Self-dual point $p_{sd}(q) = \\sqrt q/(1 + \\sqrt q)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster/index.html#self-dual-point","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-self-dual-point-of-the-planar-random-cluster","status":"fully_proved","summary":"The self-dual point `p_sd(q) = \u221aq / (1 + \u221aq)`.","title":"Self-dual point $p_{sd}(q) = \\sqrt q/(1 + \\sqrt q)$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","kind":"scope","navigation":"module","parent":48327,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","title":"First-order transitions of the random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/fk-theta-free","kind":"node","lean":"StatMech.FK.fkThetaFree","parent":48335,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#fk-theta-free","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"The free percolation probability `\u03b8\u2070(p, q)`: the free infinite-volume measure of the event that the origin lies in an infinite open cluster.","title":"Free percolation probability"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/fk-theta-free-nonneg","kind":"node","lean":"StatMech.FK.fkThetaFree_nonneg","parent":48335,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#fk-theta-free-nonneg","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"`\u03b8\u2070(p, q) \u2265 0`.","title":"Nonnegativity of the free percolation probability"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/fk-theta-free-le-one","kind":"node","lean":"StatMech.FK.fkThetaFree_le_one","parent":48335,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#fk-theta-free-le-one","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"`\u03b8\u2070(p, q) \u2264 1`.","title":"Free percolation probability is at most one"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/is-first-order-transition","kind":"node","lean":"StatMech.FK.IsFirstOrderTransition","parent":48335,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#is-first-order-transition","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"A first-order transition at `(p, q)`: the free measure does not percolate, `\u03b8\u2070(p, q) = 0`, while the wired one does, `\u03b8\u00b9(p, q) > 0`.","title":"First-order transition of the random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/first-order-transition-of-contour","kind":"node","lean":"StatMech.FK.firstOrderTransition_of_contour","parent":48335,"reading_title":{"format":"inline-parts-v1","parts":[["text","First-order transition from "],["math","\\(\\theta^0 = 0\\)"],["text"," and "],["math","\\(\\theta^1 > 0\\)"]],"source":"First-order transition from $\\theta^0 = 0$ and $\\theta^1 > 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#first-order-transition-of-contour","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"Introduction rule for `IsFirstOrderTransition`.","title":"First-order transition from $\\theta^0 = 0$ and $\\theta^1 > 0$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/fk-theta-free-ne-fk-theta-of-first-order","kind":"node","lean":"StatMech.FK.fkThetaFree_ne_fkTheta_of_firstOrder","parent":48335,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\theta^0 \\ne \\theta^1\\)"],["text"," at a first-order transition"]],"source":"$\\theta^0 \\ne \\theta^1$ at a first-order transition"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#fk-theta-free-ne-fk-theta-of-first-order","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"At a first-order transition, `\u03b8\u2070(p, q) \u2260 \u03b8\u00b9(p, q)`.","title":"$\\theta^0 \\ne \\theta^1$ at a first-order transition"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/free-infinite-volume-ne-wired-infinite-v","kind":"node","lean":"StatMech.FK.freeInfiniteVolume_ne_wiredInfiniteVolume_of_firstOrder","parent":48335,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#free-infinite-volume-ne-wired-infinite-v","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"At a first-order transition, the free and wired infinite-volume measures differ.","title":"Free and wired measures differ at a first-order transition"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/order-transition-q-large","kind":"node","lean":"StatMech.FK.order_transition_q_large","parent":48335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free and wired measures differ at self-dual "],["math","\\(p\\)"],["text"," for "],["math","\\(q > 4\\)"],["text"," if first order"]],"source":"Free and wired measures differ at self-dual $p$ for $q > 4$ if first order"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#order-transition-q-large","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"At a point `p \u2208 (0, 1)` given as the self-dual point on `\u2124\u00b2`, the hypotheses `\u03b8\u2070 = 0` and `\u03b8\u00b9 > 0` give a first-order transition with distinct free and wired measures. The first-order property for `q > 4` (Duminil-Copin et al., 2021) is assumed, not proved; `4 < q` is only used to get `q > 0`.","title":"Free and wired measures differ at self-dual $p$ for $q > 4$ if first order"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/order-transition-q-large-self-dual","kind":"node","lean":"StatMech.FK.order_transition_q_large_selfDual","parent":48335,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free and wired measures differ at "],["math","\\(\\sqrt{q}/(1+\\sqrt{q})\\)"],["text"," for "],["math","\\(q > 4\\)"],["text"," if first order"]],"source":"Free and wired measures differ at $\\sqrt{q}/(1+\\sqrt{q})$ for $q > 4$ if first order"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo/index.html#order-transition-q-large-self-dual","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/first-order-transitions-of-the-random-cluster-mo","status":"fully_proved","summary":"`order_transition_q_large` at the self-dual point `p_sd(q) = \u221aq / (1 + \u221aq)` of `\u2124\u00b2`, with `\u03b8\u2070(p_sd, q) = 0` and `\u03b8\u00b9(p_sd, q) > 0` assumed.","title":"Free and wired measures differ at $\\sqrt{q}/(1+\\sqrt{q})$ for $q > 4$ if first order"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","kind":"scope","navigation":"module","parent":48327,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","title":"Positivity of the random-cluster critical point"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/open-graph-mono","kind":"node","lean":"StatMech.FK.openGraph_mono","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#open-graph-mono","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The open subgraph is monotone in the configuration.","title":"Monotonicity of the open subgraph"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/wired-graph-mono","kind":"node","lean":"StatMech.FK.wiredGraph_mono","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#wired-graph-mono","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The wired graph (open edges plus the boundary clique) is monotone in the configuration.","title":"Monotonicity of the wired graph"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/num-clusters-wired-antitone","kind":"node","lean":"StatMech.FK.numClustersWired_antitone","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#num-clusters-wired-antitone","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The wired cluster count `k\u00b9(\u03c9)` is antitone in `\u03c9`.","title":"Wired cluster count is antitone"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/wired-fk-weight-edge-product-cross","kind":"node","lean":"StatMech.FK.wiredFkWeight_edgeProduct_cross","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holley cross inequality for wired and Bernoulli weights, "],["math","\\(q \\ge 1\\)"]],"source":"Holley cross inequality for wired and Bernoulli weights, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#wired-fk-weight-edge-product-cross","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Holley cross inequality between the wired random-cluster weight and the Bernoulli product weight, for `q \u2265 1`.","title":"Holley cross inequality for wired and Bernoulli weights, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/wired-fk-prob-fk-prob-one-cross","kind":"node","lean":"StatMech.FK.wiredFkProb_fkProbOne_cross","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Holley cross inequality for wired and "],["math","\\(q = 1\\)"],["text"," laws, "],["math","\\(q \\ge 1\\)"]],"source":"Holley cross inequality for wired and $q = 1$ laws, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#wired-fk-prob-fk-prob-one-cross","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Holley cross inequality between the ambient wired probability and the ambient `q = 1` law, for `q \u2265 1`. The latter is Bernoulli-`p` only on the graph-edge marginal.","title":"Holley cross inequality for wired and $q = 1$ laws, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/wired-fk-prob-le-fk-prob-one-increasing","kind":"node","lean":"StatMech.FK.wiredFkProb_le_fkProbOne_increasing","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired law at most the "],["math","\\(q = 1\\)"],["text"," law on increasing events, "],["math","\\(q \\ge 1\\)"]],"source":"Wired law at most the $q = 1$ law on increasing events, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#wired-fk-prob-le-fk-prob-one-increasing","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"For `q \u2265 1`, the ambient wired law is dominated by `fkProb G p 1` on increasing events. The latter is Bernoulli-`p` on graph edges and uniform on loops and nonedges; the comparison with ordinary Bernoulli percolation therefore applies to graph-edge events.","title":"Wired law at most the $q = 1$ law on increasing events, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/sum-prod-factor","kind":"node","lean":"StatMech.FK.sum_prod_factor","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#sum-prod-factor","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Summing a product of single-coordinate factors over all Boolean configurations: coordinates outside `s` contribute a factor `2` each.","title":"Boolean-configuration sum of a coordinatewise product factorizes"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/fk-prob-one-cylinder","kind":"node","lean":"StatMech.FK.fkProbOne_cylinder","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli probability that edges "],["math","\\(F\\)"],["text"," are all open is "],["math","\\(p^{|F|}\\)"]],"source":"Bernoulli probability that edges $F$ are all open is $p^{|F|}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#fk-prob-one-cylinder","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Under Bernoulli percolation on a finite graph, a set `F` of edges is open with probability `p^|F|`.","title":"Bernoulli probability that edges $F$ are all open is $p^{|F|}$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/fk-ext-word","kind":"node","lean":"StatMech.FK.fkExtWord","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#fk-ext-word","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Extend a word of `m` lattice steps to an infinite word by a fixed padding step.","title":"Padding a finite lattice word to an infinite word"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/fk-exists-open-walk","kind":"node","lean":"StatMech.FK.fkExistsOpenWalk","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Event of an open self-avoiding walk of "],["math","\\(m\\)"],["text"," steps"]],"source":"Event of an open self-avoiding walk of $m$ steps"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#fk-exists-open-walk","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The event that some self-avoiding walk of `m` steps from the origin is open.","title":"Event of an open self-avoiding walk of $m$ steps"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/is-clopen-single-open","kind":"node","lean":"StatMech.FK.isClopen_singleOpen","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#is-clopen-single-open","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The event that a given edge is open is clopen.","title":"Single-edge open event is clopen"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/is-clopen-word-event","kind":"node","lean":"StatMech.FK.isClopen_wordEvent","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#is-clopen-word-event","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The event that a given walk is open is clopen.","title":"Event that a given walk is open is clopen"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/is-increasing-word-event","kind":"node","lean":"StatMech.FK.isIncreasing_wordEvent","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#is-increasing-word-event","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The event that a given walk is open is increasing.","title":"Event that a given walk is open is increasing"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/is-clopen-fk-exists-open-walk","kind":"node","lean":"StatMech.FK.isClopen_fkExistsOpenWalk","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#is-clopen-fk-exists-open-walk","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The open-walk event is clopen.","title":"Event of some open self-avoiding walk is clopen"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/is-increasing-fk-exists-open-walk","kind":"node","lean":"StatMech.FK.isIncreasing_fkExistsOpenWalk","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#is-increasing-fk-exists-open-walk","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The open-walk event is increasing.","title":"Event of some open self-avoiding walk is increasing"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/percolation-event-subset-fk-exists-open","kind":"node","lean":"StatMech.FK.percolationEvent_subset_fkExistsOpenWalk","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#percolation-event-subset-fk-exists-open","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Percolation from the origin implies an open self-avoiding walk of every length `m`.","title":"Percolation gives open self-avoiding walks of every length"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/extend-edge-eq-of-range","kind":"node","lean":"StatMech.FK.extendEdge_eq_of_range","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#extend-edge-eq-of-range","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Extending a box configuration to `\u2124^d` keeps the state of box edges.","title":"Extended box configuration keeps box edge states"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/range-of-extend-edge-true","kind":"node","lean":"StatMech.FK.range_of_extendEdge_true","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#range-of-extend-edge-true","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"An edge that is open in an extended box configuration is a box edge.","title":"Open edges of an extended box configuration are box edges"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/box-edge-mem","kind":"node","lean":"StatMech.FK.boxEdge_mem","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#box-edge-mem","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"A box edge whose image is a lattice edge is an edge of the box graph.","title":"Box edge with a lattice-edge image is a box-graph edge"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/exists-box-f","kind":"node","lean":"StatMech.FK.exists_boxF","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Open self-avoiding walk in a box gives "],["math","\\(m\\)"],["text"," distinct open box edges"]],"source":"Open self-avoiding walk in a box gives $m$ distinct open box edges"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#exists-box-f","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"An open self-avoiding walk of `m` steps that lies in the box gives `m` distinct box edges that are all open.","title":"Open self-avoiding walk in a box gives $m$ distinct open box edges"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/per-word-box-bound","kind":"node","lean":"StatMech.FK.perWord_box_bound","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bernoulli bound "],["math","\\(p^m\\)"],["text"," for a fixed self-avoiding walk in the box"]],"source":"Bernoulli bound $p^m$ for a fixed self-avoiding walk in the box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#per-word-box-bound","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Under Bernoulli percolation on the box, a fixed self-avoiding walk of `m` steps is open with probability at most `p^m`.","title":"Bernoulli bound $p^m$ for a fixed self-avoiding walk in the box"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/wired-finite-measure-real-eq","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_real_eq","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#wired-finite-measure-real-eq","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The wired box measure of an event, as a finite sum over box configurations.","title":"Wired box measure as a finite sum over box configurations"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/wired-finite-measure-fk-exists-open-walk","kind":"node","lean":"StatMech.FK.wiredFiniteMeasure_fkExistsOpenWalk_le","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired box bound "],["math","\\((2dp)^m\\)"],["text"," on the open-walk event for "],["math","\\(q \\ge 1\\)"]],"source":"Wired box bound $(2dp)^m$ on the open-walk event for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#wired-finite-measure-fk-exists-open-walk","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"For `q \u2265 1`, the wired box measure of the open-walk event is at most `(2dp)^m`.","title":"Wired box bound $(2dp)^m$ on the open-walk event for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/fk-theta-le-pow","kind":"node","lean":"StatMech.FK.fkTheta_le_pow","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired percolation bound "],["math","\\(\\theta^1(p, q) \\le (2dp)^m\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Wired percolation bound $\\theta^1(p, q) \\le (2dp)^m$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#fk-theta-le-pow","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"For `q \u2265 1`, the wired percolation probability satisfies `\u03b8\u00b9(p, q) \u2264 (2dp)^m` for every `m`.","title":"Wired percolation bound $\\theta^1(p, q) \\le (2dp)^m$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/fk-theta-eq-zero-of-lt","kind":"node","lean":"StatMech.FK.fkTheta_eq_zero_of_lt","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","No wired percolation for "],["math","\\(2dp < 1\\)"],["text"," and "],["math","\\(q \\ge 1\\)"]],"source":"No wired percolation for $2dp < 1$ and $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#fk-theta-eq-zero-of-lt","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"For `q \u2265 1` and `2dp < 1`, there is no percolation in the wired measure: `\u03b8\u00b9(p, q) = 0`.","title":"No wired percolation for $2dp < 1$ and $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/fk-subcritical-set-bdd-above","kind":"node","lean":"StatMech.FK.fkSubcriticalSet_bddAbove","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#fk-subcritical-set-bdd-above","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"The set of `p` with `\u03b8\u00b9(p, q) = 0` is bounded above by `1`.","title":"Subcritical set is bounded above by one"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/le-fk-pc-of-mem","kind":"node","lean":"StatMech.FK.le_fkPc_of_mem","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#le-fk-pc-of-mem","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"Every subcritical `p` is at most `p_c(q, d)`.","title":"Subcritical parameters are at most the critical point"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/fk-pc-pos","kind":"node","lean":"StatMech.FK.fkPc_pos","parent":48345,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#fk-pc-pos","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"For `d \u2265 1` and `q \u2265 1`, `0 < p_c(q, d)`; in fact `p_c(q, d) \u2265 1 / (4d)`.","title":"Positivity of the random-cluster critical point"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/fk-pc-pos-of-two-le","kind":"node","lean":"StatMech.FK.fkPc_pos_of_two_le","parent":48345,"reading_title":{"format":"inline-parts-v1","parts":[["text","Positive random-cluster critical point for "],["math","\\(d \\ge 2\\)"],["text",", "],["math","\\(q \\ge 1\\)"]],"source":"Positive random-cluster critical point for $d \\ge 2$, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point/index.html#fk-pc-pos-of-two-le","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/positivity-of-the-random-cluster-critical-point","status":"fully_proved","summary":"`fkPc_pos` for `d \u2265 2`.","title":"Positive random-cluster critical point for $d \\ge 2$, $q \\ge 1$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic","kind":"scope","navigation":"module","parent":48327,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic","status":"fully_proved","title":"Free percolation above the random-cluster critical point"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/frp-fk-theta-pos-of-gt-fk-pc","kind":"node","lean":"StatMech.FK.frp_fkTheta_pos_of_gt_fkPc","parent":48375,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/index.html#frp-fk-theta-pos-of-gt-fk-pc","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic","status":"fully_proved","summary":"Above the critical point the wired measure percolates: `\u03b8\u00b9(p, q) > 0` for `p > p_c(q, d)`.","title":"Wired percolation above the random-cluster critical point"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/frp-measurable-set-percolation-event","kind":"node","lean":"StatMech.FK.frp_measurableSet_percolationEvent","parent":48375,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/index.html#frp-measurable-set-percolation-event","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic","status":"fully_proved","summary":"The percolation event is measurable.","title":"Measurability of the percolation event"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/frp-is-increasing-percolation-event","kind":"node","lean":"StatMech.FK.frp_isIncreasing_percolationEvent","parent":48375,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/index.html#frp-is-increasing-percolation-event","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic","status":"fully_proved","summary":"The percolation event is increasing.","title":"The percolation event is increasing"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/frp-free-percolation-above-pc","kind":"node","lean":"StatMech.FK.frp_free_percolation_above_pc","parent":48375,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free percolation from domination and free\u2013wired agreement at "],["math","\\(p'\\)"]],"source":"Free percolation from domination and free\u2013wired agreement at $p'$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/index.html#frp-free-percolation-above-pc","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic","status":"fully_proved","summary":"If `p_c < p'`, the free measure at `p'` is dominated by the free measure at `p`, and `\u03b8\u2070(p', q) = \u03b8\u00b9(p', q)`, then `\u03b8\u2070(p, q) > 0`.","title":"Free percolation from domination and free\u2013wired agreement at $p'$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/frp-free-percolation-above-pc-of-countab","kind":"node","lean":"StatMech.FK.frp_free_percolation_above_pc_of_countable_agreement","parent":48375,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free percolation above "],["math","\\(p_c\\)"],["text"," from monotonicity and countable agreement"]],"source":"Free percolation above $p_c$ from monotonicity and countable agreement"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic/index.html#frp-free-percolation-above-pc-of-countab","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/free-percolation-above-the-random-cluster-critic","status":"fully_proved","summary":"For `p > p_c(q, d)`, the free measure percolates, `\u03b8\u2070(p, q) > 0`, assuming monotonicity of the free measures in `p` on `(p_c, p)` (`hmono`) and `\u03b8\u2070 = \u03b8\u00b9` off a countable set (`hagree`).","title":"Free percolation above $p_c$ from monotonicity and countable agreement"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","kind":"scope","navigation":"module","parent":48327,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","title":"The random-cluster critical point on \u2124\u00b2 is below one, given a Peierls bound"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/horiz-walk","kind":"node","lean":"StatMech.FK.horizWalk","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Horizontal lattice walk in "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Horizontal lattice walk in $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#horiz-walk","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The horizontal lattice walk of `k` steps from `(a, c)` to `(a + k, c)` in `\u2124\u00b2`.","title":"Horizontal lattice walk in $\\mathbb{Z}^2$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/exists-far-not-mem","kind":"node","lean":"StatMech.FK.exists_far_not_mem","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite subsets of "],["math","\\(\\mathbb{Z}^2\\)"],["text"," miss far points on each row"]],"source":"Finite subsets of $\\mathbb{Z}^2$ miss far points on each row"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#exists-far-not-mem","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"A finite set of `\u2124\u00b2` misses some point `(N, c)` with `N \u2265 a\u2080` on every horizontal line.","title":"Finite subsets of $\\mathbb{Z}^2$ miss far points on each row"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/exists-walk-exterior","kind":"node","lean":"StatMech.FK.exists_walk_exterior","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite cluster in "],["math","\\(\\mathbb{Z}^2\\)"],["text"," has a reachable point outside it"]],"source":"Finite cluster in $\\mathbb{Z}^2$ has a reachable point outside it"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#exists-walk-exterior","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"A finite cluster of `o` has a lattice-reachable point outside it.","title":"Finite cluster in $\\mathbb{Z}^2$ has a reachable point outside it"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/closed-circuit-event","kind":"node","lean":"StatMech.FK.closedCircuitEvent","parent":48381,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#closed-circuit-event","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The event that every primal edge crossed by the dual circuit `c` is closed.","title":"Closed-circuit event of a dual circuit"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/exists-closed-lattice-circuit-of-finite","kind":"node","lean":"StatMech.FK.exists_closed_latticeCircuit_of_finite_cluster","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite cluster in "],["math","\\(\\mathbb{Z}^2\\)"],["text"," gives a closed lattice circuit"]],"source":"Finite cluster in $\\mathbb{Z}^2$ gives a closed lattice circuit"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#exists-closed-lattice-circuit-of-finite","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"A finite open cluster of `o` in `\u2124\u00b2` is surrounded by a lattice cycle whose crossed primal edges are all closed.","title":"Finite cluster in $\\mathbb{Z}^2$ gives a closed lattice circuit"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/exists-enclosing-closed-lattice-circuit","kind":"node","lean":"StatMech.FK.exists_enclosing_closed_latticeCircuit_of_finite_cluster","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite cluster in "],["math","\\(\\mathbb{Z}^2\\)"],["text",": closed circuit, separated point, odd crossings"]],"source":"Finite cluster in $\\mathbb{Z}^2$: closed circuit, separated point, odd crossings"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#exists-enclosing-closed-lattice-circuit","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The closed circuit of `exists_closed_latticeCircuit_of_finite_cluster`, together with a point that the cluster boundary separates from `o` and an odd crossing number for every walk to it.","title":"Finite cluster in $\\mathbb{Z}^2$: closed circuit, separated point, odd crossings"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/finite-cluster-subset-i-union-closed-cir","kind":"node","lean":"StatMech.FK.finiteCluster_subset_iUnion_closedCircuit","parent":48381,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#finite-cluster-subset-i-union-closed-cir","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The event that the cluster of `o` is finite is covered by the closed-circuit events.","title":"Finite-cluster event is covered by closed-circuit events"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/compl-percolation-event-eq-finite-cluste","kind":"node","lean":"StatMech.FK.compl_percolationEvent_eq_finiteCluster","parent":48381,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#compl-percolation-event-eq-finite-cluste","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"No percolation from the origin means the cluster of the origin is finite.","title":"Non-percolation means the origin's cluster is finite"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-upper-bound","kind":"node","lean":"StatMech.FK.fkPcUpperBound","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peierls series "],["math","\\(\\sum_\\ell \\ell\\,(2d)^\\ell (1-p)^\\ell\\)"]],"source":"Peierls series $\\sum_\\ell \\ell\\,(2d)^\\ell (1-p)^\\ell$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-upper-bound","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The Peierls series `\u03a3_\u2113 \u2113 (2d)^\u2113 (1 - p)^\u2113` bounding the probability of a finite cluster.","title":"Peierls series $\\sum_\\ell \\ell\\,(2d)^\\ell (1-p)^\\ell$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-ratio","kind":"node","lean":"StatMech.FK.fkPcRatio","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ratio "],["math","\\(2d(1-p)\\)"],["text"," of the Peierls series"]],"source":"Ratio $2d(1-p)$ of the Peierls series"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-ratio","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The ratio `2d (1 - p)` of the Peierls series.","title":"Ratio $2d(1-p)$ of the Peierls series"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-ratio-def","kind":"node","lean":"StatMech.FK.fkPcRatio_def","parent":48381,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-ratio-def","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"Unfolding `fkPcRatio`.","title":"Defining formula for the Peierls ratio"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-summand-eq","kind":"node","lean":"StatMech.FK.fkPcSummand_eq","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peierls summand as "],["math","\\(\\ell x^\\ell\\)"],["text"," with "],["math","\\(x = 2d(1-p)\\)"]],"source":"Peierls summand as $\\ell x^\\ell$ with $x = 2d(1-p)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-summand-eq","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The Peierls summand in terms of `fkPcRatio`.","title":"Peierls summand as $\\ell x^\\ell$ with $x = 2d(1-p)$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-upper-bound-eq-closed-form","kind":"node","lean":"StatMech.FK.fkPcUpperBound_eq_closedForm","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed form "],["math","\\(x/(1-x)^2\\)"],["text"," of the Peierls series for "],["math","\\(|x| < 1\\)"]],"source":"Closed form $x/(1-x)^2$ of the Peierls series for $|x| < 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-upper-bound-eq-closed-form","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"Closed form `x / (1 - x)\u00b2` of the Peierls series for `|x| < 1`, where `x = 2d (1 - p)`.","title":"Closed form $x/(1-x)^2$ of the Peierls series for $|x| < 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-ratio-tendsto-one","kind":"node","lean":"StatMech.FK.fkPcRatio_tendsto_one","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peierls ratio "],["math","\\(2d(1-p)\\)"],["text"," tends to "],["math","\\(0\\)"],["text"," as "],["math","\\(p \\to 1\\)"]],"source":"Peierls ratio $2d(1-p)$ tends to $0$ as $p \\to 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-ratio-tendsto-one","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"`2d (1 - p) \u2192 0` as `p \u2192 1`.","title":"Peierls ratio $2d(1-p)$ tends to $0$ as $p \\to 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-closed-form-tendsto-zero","kind":"node","lean":"StatMech.FK.fkPcClosedForm_tendsto_zero","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(x/(1-x)^2 \\to 0\\)"],["text"," as "],["math","\\(x \\to 0\\)"]],"source":"$x/(1-x)^2 \\to 0$ as $x \\to 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-closed-form-tendsto-zero","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"`x / (1 - x)\u00b2 \u2192 0` as `x \u2192 0`.","title":"$x/(1-x)^2 \\to 0$ as $x \\to 0$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-upper-bound-tendsto-one","kind":"node","lean":"StatMech.FK.fkPcUpperBound_tendsto_one","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peierls series tends to "],["math","\\(0\\)"],["text"," as "],["math","\\(p \\to 1\\)"]],"source":"Peierls series tends to $0$ as $p \\to 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-upper-bound-tendsto-one","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The Peierls series tends to `0` as `p \u2192 1`.","title":"Peierls series tends to $0$ as $p \\to 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/exists-p-lt-one-fk-pc-upper-bound-lt-one","kind":"node","lean":"StatMech.FK.exists_p_lt_one_fkPcUpperBound_lt_one","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peierls series is below 1 on an interval "],["math","\\((p_0, 1)\\)"]],"source":"Peierls series is below 1 on an interval $(p_0, 1)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#exists-p-lt-one-fk-pc-upper-bound-lt-one","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The Peierls series is below `1` on an interval `(p\u2080, 1]`.","title":"Peierls series is below 1 on an interval $(p_0, 1)$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-upper-bound-nonneg","kind":"node","lean":"StatMech.FK.fkPcUpperBound_nonneg","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonnegativity of the Peierls series for "],["math","\\(p \\le 1\\)"]],"source":"Nonnegativity of the Peierls series for $p \\le 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-upper-bound-nonneg","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The Peierls series is nonnegative for `p \u2264 1`.","title":"Nonnegativity of the Peierls series for $p \\le 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-non-percolation-bound","kind":"node","lean":"StatMech.FK.FkNonPercolationBound","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peierls bound hypothesis for the wired measure on "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Peierls bound hypothesis for the wired measure on $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-non-percolation-bound","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The Peierls bound for the wired random-cluster measure on `\u2124\u00b2` at `(p, q)`: `1 - \u03b8\u00b9(p, q) \u2264 \u03a3_\u2113 \u2113 4^\u2113 (1 - p)^\u2113`.","title":"Peierls bound hypothesis for the wired measure on $\\mathbb{Z}^2$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-theta-pos-of-non-percolation-bound","kind":"node","lean":"StatMech.FK.fkTheta_pos_of_nonPercolationBound","parent":48381,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-theta-pos-of-non-percolation-bound","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The Peierls bound with a series below `1` gives percolation: `0 < \u03b8\u00b9(p, q)`.","title":"Percolation from a Peierls bound with series below 1"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-subcritical-set-nonempty","kind":"node","lean":"StatMech.FK.fkSubcriticalSet_nonempty","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Nonempty subcritical set on "],["math","\\(\\mathbb{Z}^2\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Nonempty subcritical set on $\\mathbb{Z}^2$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-subcritical-set-nonempty","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"For `q \u2265 1` the subcritical set on `\u2124\u00b2` is nonempty (it contains `1/8`).","title":"Nonempty subcritical set on $\\mathbb{Z}^2$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/not-mem-fk-subcritical-set-of-fk-theta-p","kind":"node","lean":"StatMech.FK.not_mem_fkSubcriticalSet_of_fkTheta_pos","parent":48381,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#not-mem-fk-subcritical-set-of-fk-theta-p","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"A parameter with positive percolation probability is not subcritical.","title":"Positive percolation probability rules out subcriticality"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-lt-one","kind":"node","lean":"StatMech.FK.fkPc_lt_one","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(q) < 1\\)"],["text"," on "],["math","\\(\\mathbb{Z}^2\\)"],["text"," for "],["math","\\(q \\ge 1\\)"],["text"," under a Peierls bound"]],"source":"$p_c(q) < 1$ on $\\mathbb{Z}^2$ for $q \\ge 1$ under a Peierls bound"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-lt-one","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"For `q \u2265 1`, the random-cluster critical point on `\u2124\u00b2` satisfies `p_c(q) < 1`, assuming the Peierls bound `FkNonPercolationBound` for all `p` in some interval `(p\u2080, 1)`.","title":"$p_c(q) < 1$ on $\\mathbb{Z}^2$ for $q \\ge 1$ under a Peierls bound"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-pos-and-lt-one","kind":"node","lean":"StatMech.FK.fkPc_pos_and_lt_one","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(0 < p_c(q) < 1\\)"],["text"," on "],["math","\\(\\mathbb{Z}^2\\)"],["text"," for "],["math","\\(q \\ge 1\\)"],["text"," under a Peierls bound"]],"source":"$0 < p_c(q) < 1$ on $\\mathbb{Z}^2$ for $q \\ge 1$ under a Peierls bound"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-pos-and-lt-one","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"For `q \u2265 1`, `0 < p_c(q) < 1` on `\u2124\u00b2`, assuming the Peierls bound as in `fkPc_lt_one`.","title":"$0 < p_c(q) < 1$ on $\\mathbb{Z}^2$ for $q \\ge 1$ under a Peierls bound"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/fk-pc-upper-bound-two-eq-pc-upper-bound","kind":"node","lean":"StatMech.FK.fkPcUpperBound_two_eq_pcUpperBound","parent":48381,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peierls series for "],["math","\\(d = 2\\)"],["text"," equals the Bernoulli Peierls series"]],"source":"Peierls series for $d = 2$ equals the Bernoulli Peierls series"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on/index.html#fk-pc-upper-bound-two-eq-pc-upper-bound","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-random-cluster-critical-point-on-is-below-on","status":"fully_proved","summary":"The Peierls series for `d = 2` is the Bernoulli Peierls series `StatMech.Percolation.pcUpperBound`.","title":"Peierls series for $d = 2$ equals the Bernoulli Peierls series"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster","kind":"scope","navigation":"module","parent":48327,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster","status":"fully_proved","title":"The critical point of the planar random-cluster model equals the self-dual point"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster/fk-pc-eq-self-dual-point","kind":"node","lean":"StatMech.BeffaraDC.fkPc_eq_selfDualPoint","parent":48407,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(p_c(q)\\)"],["text"," equals the self-dual point if percolation starts there, "],["math","\\(q \\ge 1\\)"]],"source":"$p_c(q)$ equals the self-dual point if percolation starts there, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster/index.html#fk-pc-eq-self-dual-point","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster","status":"fully_proved","summary":"If `\u03b8(r, q) = 0` for `r < p_sd(q)` and `\u03b8(r, q) > 0` for `r > p_sd(q)`, then `p_c(q) = p_sd(q)`.","title":"$p_c(q)$ equals the self-dual point if percolation starts there, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster/beffara-dc-critical-point","kind":"node","lean":"StatMech.BeffaraDC.beffaraDC_critical_point","parent":48407,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster/index.html#beffara-dc-critical-point","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/the-critical-point-of-the-planar-random-cluster","status":"fully_proved","summary":"For `q \u2265 1`, if the wired random-cluster measure on `\u2124\u00b2` does not percolate for `p < \u221aq / (1 + \u221aq)` (`hNoPercBelow`) and percolates for `p > \u221aq / (1 + \u221aq)` (`hPercAbove`), then `p_c(q) = \u221aq / (1 + \u221aq)`.","title":"Beffara\u2013Duminil-Copin theorem, conditional form"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit.json","id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit","kind":"scope","navigation":"module","parent":48327,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/index.html","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit","status":"fully_proved","title":"No percolation below the q = 2 random-cluster critical point"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/tzp-fk-theta-iv-monotone-box-bdry","kind":"node","lean":"StatMech.FK.tzp_fkTheta_iv_monotone_boxBdry","parent":48410,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired box-boundary connection probability is monotone in "],["math","\\(p\\)"],["text"," at "],["math","\\(q = 2\\)"]],"source":"Wired box-boundary connection probability is monotone in $p$ at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/index.html#tzp-fk-theta-iv-monotone-box-bdry","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit","status":"fully_proved","summary":"For `q = 2`, the wired infinite-volume probability that the origin is connected to the boundary of the box of radius `n` is monotone in `p`.","title":"Wired box-boundary connection probability is monotone in $p$ at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/tzp-fk-theta-monotone-in-p","kind":"node","lean":"StatMech.FK.tzp_fkTheta_monotone_in_p","parent":48410,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity of "],["math","\\(\\theta^1(p, 2)\\)"],["text"," in "],["math","\\(p\\)"]],"source":"Monotonicity of $\\theta^1(p, 2)$ in $p$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/index.html#tzp-fk-theta-monotone-in-p","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit","status":"fully_proved","summary":"For `q = 2`, the wired percolation probability `\u03b8\u00b9(p, 2)` is monotone in `p`.","title":"Monotonicity of $\\theta^1(p, 2)$ in $p$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/tzp-fk-theta-eq-zero-of-lt-pc","kind":"node","lean":"StatMech.FK.tzp_fkTheta_eq_zero_of_lt_pc","parent":48410,"reading_title":{"format":"inline-parts-v1","parts":[["text","No percolation below the critical point at "],["math","\\(q = 2\\)"]],"source":"No percolation below the critical point at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/index.html#tzp-fk-theta-eq-zero-of-lt-pc","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit","status":"fully_proved","summary":"For `q = 2`, there is no percolation below the critical point: `\u03b8\u00b9(p, 2) = 0` for `p < p_c(2, d)`.","title":"No percolation below the critical point at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/tzp-fk-theta-eq-zero-of-le-pc-of-pc","kind":"node","lean":"StatMech.FK.tzp_fkTheta_eq_zero_of_le_pc_of_pc","parent":48410,"reading_title":{"format":"inline-parts-v1","parts":[["text","No percolation for "],["math","\\(p \\le p_c\\)"],["text"," at "],["math","\\(q = 2\\)"],["text",", given "],["math","\\(\\theta^1(p_c, 2) = 0\\)"]],"source":"No percolation for $p \\le p_c$ at $q = 2$, given $\\theta^1(p_c, 2) = 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit/index.html#tzp-fk-theta-eq-zero-of-le-pc-of-pc","scope":"StatisticalMechanics/random-cluster-model/critical-point-of-the-random-cluster-model/no-percolation-below-the-q-2-random-cluster-crit","status":"fully_proved","summary":"For `q = 2`, if `\u03b8\u00b9(p_c, 2) = 0`, then `\u03b8\u00b9(p, 2) = 0` for every `p \u2264 p_c(2, d)`.","title":"No percolation for $p \\le p_c$ at $q = 2$, given $\\theta^1(p_c, 2) = 0$"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2.json","id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","kind":"scope","navigation":"module","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","title":"Translation invariance and ergodicity of the q = 2 random-cluster measures"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/aeconst-of-ae-invariant","kind":"node","lean":"StatMech.ConfigSpace.aeconst_of_ae_invariant","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#aeconst-of-ae-invariant","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Under an ergodic measure, a measurable set invariant up to null sets has measure `0` or `1`. See also Mathlib's `aeconst_of_forall_preimage_smul_ae_eq` for `ErgodicSMul`.","title":"Almost-invariant events are trivial under an ergodic measure"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmc-self-mixing","kind":"node","lean":"StatMech.FK.fmc_SelfMixing","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmc-self-mixing","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For every cylinder event `A` and `\u03b5 > 0` there is a translation `g` with `|\u03bc(A \u2229 g\u207b\u00b9A) - \u03bc(A)\u00b2| < \u03b5`.","title":"Self-mixing on cylinder events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmc-exists-cylinder-symm-diff-lt","kind":"node","lean":"StatMech.FK.fmc_exists_cylinder_symmDiff_lt","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmc-exists-cylinder-symm-diff-lt","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Every measurable event is approximated by a cylinder event in symmetric difference; the general form of `exists_cylinder_symmDiff_lt` (`Ergodicity`).","title":"Cylinder approximation of events under a finite measure"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmc-is-ergodic-of-self-mixing","kind":"node","lean":"StatMech.FK.fmc_isErgodic_of_selfMixing","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmc-is-ergodic-of-self-mixing","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"A translation-invariant self-mixing probability measure is ergodic.","title":"Translation-invariant self-mixing measures are ergodic"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmu-multi-open","kind":"node","lean":"StatMech.FK.fmu_multiOpen","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmu-multi-open","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The event `O_T` that every coordinate in `T` is open. See also `cdc_multiOpenEvent` and `med_genericMultiOpenEvent`.","title":"Event that every coordinate of a finite set is open"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmu-x-r","kind":"node","lean":"StatMech.FK.fmu_xR","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmu-x-r","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The state of coordinate `e` as a real number in `{0, 1}`.","title":"Coordinate state as a real number"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmu-mo-ind","kind":"node","lean":"StatMech.FK.fmu_moInd","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmu-mo-ind","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The indicator of `O_T` as the product `\u220f_{e \u2208 T} \u03c9(e)`.","title":"Product indicator of a multi-open event"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmu-trace-ind","kind":"node","lean":"StatMech.FK.fmu_traceInd","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmu-trace-ind","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The indicator that `\u03c9` agrees with `\u03c4` on `F`.","title":"Indicator of agreeing with a pattern on a finite set"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmu-trace-coef","kind":"node","lean":"StatMech.FK.fmu_traceCoef","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmu-trace-coef","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The inclusion\u2013exclusion coefficient of `O_T` in the expansion of the trace indicator of `\u03c4` on `F`.","title":"Inclusion\u2013exclusion coefficient of a multi-open event"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmu-pair-mixing","kind":"node","lean":"StatMech.FK.fmu_PairMixing","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmu-pair-mixing","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For every finite family of finite sets `T` and `\u03b4 > 0` there is one translation `g` with `|\u03bc(O_T \u2229 g\u207b\u00b9O_{T'}) - \u03bc(O_T) \u03bc(O_{T'})| < \u03b4` for all `T, T'` in the family.","title":"Pair mixing of multi-open events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmu-gen-mixing","kind":"node","lean":"StatMech.FK.fmu_GenMixing","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmu-gen-mixing","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The analogue of `fmu_PairMixing` for finite families of cylinder events.","title":"Mixing for finite families of cylinder events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fmu-upper-decay","kind":"node","lean":"StatMech.FK.fmu_UpperDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fmu-upper-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The one-sided form of `fmu_PairMixing`: `\u03bc(O_T \u2229 g\u207b\u00b9O_{T'}) \u2264 \u03bc(O_T) \u03bc(O_{T'}) + \u03b4`.","title":"Upper decay of correlations of multi-open events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/is-invariant-extreme-point","kind":"node","lean":"StatMech.FK.IsInvariantExtremePoint","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#is-invariant-extreme-point","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"`\u03bc` is a translation-invariant probability measure that is not a nontrivial convex combination of two distinct translation-invariant probability measures.","title":"Extreme point of the invariant probability measures"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fec-is-ergodic-of-invariant-extreme-poin","kind":"node","lean":"StatMech.FK.fec_isErgodic_of_invariantExtremePoint","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fec-is-ergodic-of-invariant-extreme-poin","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"An extreme point of the translation-invariant probability measures is ergodic.","title":"Extreme invariant measures are ergodic"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fdb-box-conditional-product-cap","kind":"node","lean":"StatMech.FK.fdb_BoxConditionalProductCap","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fdb-box-conditional-product-cap","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For every finite family of finite edge sets and box index `N`, some translation `g` satisfies, for all large `m`: each `O_T \u2229 g\u207b\u00b9O_{T'}` is `A \u2229 B` on `\u039b_m` with `A` increasing and `B` depending only on edges outside some `F`, the conditional masses of `A` on the fibres of `F` (with a boundary graph containing the wired one) are at most `\u03bc_N(O_T)`, and the mass of `B` is at most `\u03bc_N(O_{T'})`.","title":"Conditional product cap for multi-open events in boxes"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/ftb-far-box-product-dom","kind":"node","lean":"StatMech.FK.ftb_FarBoxProductDom","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#ftb-far-box-product-dom","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For every finite family of finite edge sets, box index `m`, and `\u03b4 > 0`, some translation `g` satisfies `\u03bc(O_T \u2229 g\u207b\u00b9O_{T'}) \u2264 \u03bc_m(O_T) \u03bc_m(O_{T'}) + \u03b4` for all `T, T'` in the family.","title":"Far-box product domination of multi-open events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/ftb-finite-box-disjoint-product-dom","kind":"node","lean":"StatMech.FK.ftb_FiniteBoxDisjointProductDom","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#ftb-finite-box-disjoint-product-dom","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For every finite family of finite edge sets and box index `N`, some translation `g` satisfies, for all large `m`, `\u03bc_m(O_T \u2229 g\u207b\u00b9O_{T'}) \u2264 \u03bc_N(O_T) \u03bc_N(O_{T'})` for all `T, T'` in the family.","title":"Finite-box disjoint product domination of multi-open events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/erg-wired-shifted-finite","kind":"node","lean":"StatMech.FK.erg_wiredShiftedFinite","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#erg-wired-shifted-finite","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired finite-volume random-cluster measure on `\u039b_n` translated by `g`.","title":"Translated wired box measure"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/erg-free-shifted-finite","kind":"node","lean":"StatMech.FK.erg_freeShiftedFinite","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#erg-free-shifted-finite","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free finite-volume random-cluster measure on `\u039b_n` translated by `g`.","title":"Translated free box measure"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/ivt-wired-integral-shift-decay","kind":"node","lean":"StatMech.FK.ivt_wiredIntegralShiftDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#ivt-wired-integral-shift-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Along the box sequence `\u03c6`, the wired box integrals of `f` and of `f \u2218 \u03b8_g` differ by a quantity tending to zero, for every bounded continuous `f`.","title":"Wired integral shift decay along a box sequence"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/ivt-free-integral-shift-decay","kind":"node","lean":"StatMech.FK.ivt_freeIntegralShiftDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#ivt-free-integral-shift-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Along the box sequence `\u03c6`, the free box integrals of `f` and of `f \u2218 \u03b8_g` differ by a quantity tending to zero, for every bounded continuous `f`.","title":"Free integral shift decay along a box sequence"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/ivt-wired-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.ivt_wiredIV_isTranslationInvariant_of_decay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#ivt-wired-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired infinite-volume FK measure is translation invariant if the integral decay hypothesis holds for every shift and every convergent box subsequence. See also `flc_wiredIV_isTranslationInvariant`, which proves invariance for `q = 2` without hypotheses.","title":"Wired translation invariance from integral shift decay"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/ivt-free-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.ivt_freeIV_isTranslationInvariant_of_decay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#ivt-free-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free infinite-volume FK measure is translation invariant if the integral decay hypothesis holds for every shift and every convergent box subsequence. See also `bdp_freeIV_isTranslationInvariant`, which proves invariance for `q = 2` without hypotheses.","title":"Free translation invariance from integral shift decay"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/ivt-wired-integral-shift-decay-one","kind":"node","lean":"StatMech.FK.ivt_wiredIntegralShiftDecay_one","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#ivt-wired-integral-shift-decay-one","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The identity shift satisfies the wired decay hypothesis, so it is not vacuous.","title":"Identity shift satisfies the wired decay hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/ivt-free-integral-shift-decay-one","kind":"node","lean":"StatMech.FK.ivt_freeIntegralShiftDecay_one","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#ivt-free-integral-shift-decay-one","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The identity shift satisfies the free decay hypothesis, so it is not vacuous.","title":"Identity shift satisfies the free decay hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/isd-decay-fun","kind":"node","lean":"StatMech.FK.isd_decayFun","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#isd-decay-fun","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The difference `\u222b f d\u03bc_n - \u222b f \u2218 \u03b8_g d\u03bc_n` between the integrals of `f` and of its shift by `g` against the `n`-th measure.","title":"Shift-decay functional of a sequence of measures"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/isd-tendsto-decay-of-dense","kind":"node","lean":"StatMech.FK.isd_tendsto_decay_of_dense","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#isd-tendsto-decay-of-dense","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If `D_n(f') \u2192 0` for all `f'` in a dense set of bounded continuous functions, then `D_n(f) \u2192 0` for every bounded continuous `f`. This is the equicontinuity argument of `EquicontinuousAt.tendsto_of_mem_closure`, specialized to the uniformly Lipschitz family `f \u21a6 D_n(f)`.","title":"Shift decay extends from a dense set of test functions"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/isd-wired-dense-decay","kind":"node","lean":"StatMech.FK.isd_wiredDenseDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#isd-wired-dense-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Decay of `D_n` along the wired box measures `\u03c6^1_{\u039b_{\u03c6(n)}}` on some dense set of bounded continuous functions.","title":"Wired shift decay on a dense set of test functions"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/isd-free-dense-decay","kind":"node","lean":"StatMech.FK.isd_freeDenseDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#isd-free-dense-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Decay of `D_n` along the free box measures `\u03c6^0_{\u039b_{\u03c6(n)}}` on some dense set of bounded continuous functions.","title":"Free shift decay on a dense set of test functions"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/isd-wired-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.isd_wiredIV_isTranslationInvariant_of_denseDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#isd-wired-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired infinite-volume FK measure is translation invariant if, for every shift and every convergent box subsequence, `D_n` decays on a dense set. See also `flc_wiredIV_isTranslationInvariant` (unconditional for `q = 2`).","title":"Wired translation invariance from decay on a dense set"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/isd-free-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.isd_freeIV_isTranslationInvariant_of_denseDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#isd-free-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free infinite-volume FK measure is translation invariant if, for every shift and every convergent box subsequence, `D_n` decays on a dense set. See also `bdp_freeIV_isTranslationInvariant` (unconditional for `q = 2`).","title":"Free translation invariance from decay on a dense set"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-coord-cm","kind":"node","lean":"StatMech.FK.idc_coordCM","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-coord-cm","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The indicator `\u03c9 \u21a6 1{\u03c9_e = 1}` of the edge `e` being open, as a continuous function.","title":"Edge-open indicator as a continuous function"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-coord-bcf","kind":"node","lean":"StatMech.FK.idc_coordBcf","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-coord-bcf","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The edge indicator `\u03c9 \u21a6 1{\u03c9_e = 1}` as a bounded continuous function.","title":"Edge-open indicator as a bounded continuous function"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-integral-coord-bcf-wired","kind":"node","lean":"StatMech.FK.idc_integral_coordBcf_wired","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-integral-coord-bcf-wired","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The integral of the indicator of a box edge against the wired box measure on `\u039b_m` is the wired edge marginal.","title":"Wired box integral of an edge indicator is the wired edge marginal"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-integral-coord-bcf-free","kind":"node","lean":"StatMech.FK.idc_integral_coordBcf_free","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-integral-coord-bcf-free","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The integral of the indicator of a box edge against the free box measure on `\u039b_m` is the free edge marginal.","title":"Free box integral of an edge indicator is the free edge marginal"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-wired-coord-decay-of-density-eq","kind":"node","lean":"StatMech.FK.idc_wired_coord_decay_of_density_eq","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-wired-coord-decay-of-density-eq","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the box edge `e` and its shift `g\u207b\u00b9 e` have the same wired `q = 2` density, the decay functional of the indicator of `e` along the wired box measures tends to zero.","title":"Decay of the edge-indicator functional from equal wired densities"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-cylinder-alg","kind":"node","lean":"StatMech.FK.idc_cylinderAlg","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-cylinder-alg","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The subalgebra of continuous functions generated by the edge indicators.","title":"Cylinder algebra generated by edge indicators"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-cylinder-bcf-set","kind":"node","lean":"StatMech.FK.idc_cylinderBcfSet","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-cylinder-bcf-set","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The cylinder algebra viewed as a set of bounded continuous functions.","title":"Cylinder algebra as bounded continuous functions"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-cylinder-bcf-set-dense","kind":"node","lean":"StatMech.FK.idc_cylinderBcfSet_dense","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-cylinder-bcf-set-dense","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The cylinder functions are dense in the bounded continuous functions, by Stone\u2013Weierstrass on the compact configuration space.","title":"Density of cylinder functions among bounded continuous functions"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-wired-cylinder-decay","kind":"node","lean":"StatMech.FK.idc_wiredCylinderDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-wired-cylinder-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The decay functional tends to zero along the wired box measures for every cylinder function.","title":"Cylinder decay along the wired box measures"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-free-cylinder-decay","kind":"node","lean":"StatMech.FK.idc_freeCylinderDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-free-cylinder-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The decay functional tends to zero along the free box measures for every cylinder function.","title":"Cylinder decay along the free box measures"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-wired-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.idc_wiredIV_isTranslationInvariant_of_cylinderDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-wired-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired infinite-volume FK measure is translation invariant if cylinder decay holds for every shift and every convergent box subsequence. See also `flc_wiredIV_isTranslationInvariant` (unconditional for `q = 2`).","title":"Translation invariance of the wired FK measure from cylinder decay"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/idc-free-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.idc_freeIV_isTranslationInvariant_of_cylinderDecay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#idc-free-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free infinite-volume FK measure is translation invariant if cylinder decay holds for every shift and every convergent box subsequence. See also `bdp_freeIV_isTranslationInvariant` (unconditional for `q = 2`).","title":"Translation invariance of the free FK measure from cylinder decay"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-translate","kind":"node","lean":"StatMech.FK.fktc_translate","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation of "],["math","\\(\\mathbb{Z}^d\\)"],["text"," by a vector"]],"source":"Translation of $\\mathbb{Z}^d$ by a vector"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-translate","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Translation `x \u21a6 x + v` as a permutation of `\u2124^d`.","title":"Translation of $\\mathbb{Z}^d$ by a vector"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-translate-not-box-mem","kind":"node","lean":"StatMech.FK.fktc_translate_not_box_mem","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-translate-not-box-mem","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Translation by a unit vector does not preserve the centered boxes, so it is not a `BoxSym`.","title":"Unit translations are not box symmetries"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-point-and-translate-in-box","kind":"node","lean":"StatMech.FK.fktc_point_and_translate_in_box","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-point-and-translate-in-box","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"A site and its translate lie in a common centered box.","title":"A site and its translate lie in a common centered box"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-edge-and-translate-in-box","kind":"node","lean":"StatMech.FK.fktc_edge_and_translate_in_box","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-edge-and-translate-in-box","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The endpoints of an edge and of its translate lie in a common centered box.","title":"An edge and its translate lie in a common centered box"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-box-edge","kind":"node","lean":"StatMech.FK.fktc_boxEdge","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unordered pair of two sites of a box "],["math","\\(\\Lambda_N\\)"]],"source":"Unordered pair of two sites of a box $\\Lambda_N$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-box-edge","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The pair `{x, y}` as an element of `Sym2 \u039b_N`, for sites `x, y \u2208 \u039b_N`.","title":"Unordered pair of two sites of a box $\\Lambda_N$"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-wired-box-density-const-of-orbit","kind":"node","lean":"StatMech.FK.fktc_wired_box_density_const_of_orbit","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-wired-box-density-const-of-orbit","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired edge density is constant on each orbit of a box symmetry.","title":"Wired edge density is constant on orbits of a box symmetry"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-free-box-density-const-of-orbit","kind":"node","lean":"StatMech.FK.fktc_free_box_density_const_of_orbit","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-free-box-density-const-of-orbit","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free edge density is constant on each orbit of a box symmetry.","title":"Free edge density is constant on orbits of a box symmetry"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-wired-density-eq-of-marg-decay","kind":"node","lean":"StatMech.FK.fktc_wired_density_eq_of_marg_decay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-wired-density-eq-of-marg-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the wired marginals of two box edges on `\u039b_{N+k}` differ by a quantity tending to zero, the two edges have the same wired density.","title":"Equal wired densities from merging wired marginals"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-free-density-eq-of-marg-decay","kind":"node","lean":"StatMech.FK.fktc_free_density_eq_of_marg_decay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-free-density-eq-of-marg-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the free marginals of two box edges on `\u039b_{N+k}` differ by a quantity tending to zero, the two edges have the same free density.","title":"Equal free densities from merging free marginals"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-wired-density-translation-inv","kind":"node","lean":"StatMech.FK.fktc_wired_density_translation_inv","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-wired-density-translation-inv","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"An edge and its translate have the same wired density if their wired marginals merge along the boxes. See also `bdp_wiredEdgeDensity_translation_inv`, which proves the conclusion without the merging hypothesis.","title":"Translation invariance of the wired density given merging marginals"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-free-density-translation-inv","kind":"node","lean":"StatMech.FK.fktc_free_density_translation_inv","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-free-density-translation-inv","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"An edge and its translate have the same free density if their free marginals merge along the boxes. See also `bdp_freeEdgeDensity_translation_inv`, which proves the conclusion without the merging hypothesis.","title":"Translation invariance of the free density given merging marginals"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-density-translation-inv","kind":"node","lean":"StatMech.FK.fktc_density_translation_inv","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-density-translation-inv","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Translation invariance of the free and wired densities of an edge under the merging hypotheses of `fktc_wired_density_translation_inv` and `fktc_free_density_translation_inv`.","title":"Translation invariance of free and wired densities if marginals merge"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-wired-density-eq-const","kind":"node","lean":"StatMech.FK.fktc_wired_density_eq_const","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-wired-density-eq-const","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the wired density is constant on a set `EB` of box edges, `|EB|` times that density is at most the expected number of open edges of `EB` under the wired measure on any larger box.","title":"Constant wired density gives a lower bound on expected open edges"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-free-density-eq-const","kind":"node","lean":"StatMech.FK.fktc_free_density_eq_const","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-free-density-eq-const","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the free density is constant on a set `EB` of box edges, the expected number of open edges of `EB` under the free measure on any larger box is at most `|EB|` times that density.","title":"Constant free density gives an upper bound on expected open edges"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-wired-density-eq-const-orbit","kind":"node","lean":"StatMech.FK.fktc_wired_density_eq_const_orbit","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-wired-density-eq-const-orbit","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"`fktc_wired_density_eq_const` for a set of edges contained in one orbit of a box symmetry.","title":"Wired lower bound on expected open edges along a box-symmetry orbit"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fktc-free-density-eq-const-orbit","kind":"node","lean":"StatMech.FK.fktc_free_density_eq_const_orbit","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fktc-free-density-eq-const-orbit","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"`fktc_free_density_eq_const` for a set of edges contained in one orbit of a box symmetry.","title":"Free upper bound on expected open edges along a box-symmetry orbit"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cdc-multi-open-event","kind":"node","lean":"StatMech.FK.cdc_multiOpenEvent","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cdc-multi-open-event","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The event that every edge of `s` is open. Equal to `fmu_multiOpen` (see `ftb_fmu_eq_cdc_multiOpen`).","title":"Event that every edge of a finite set is open"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cdc-monomial-cm","kind":"node","lean":"StatMech.FK.cdc_monomialCM","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cdc-monomial-cm","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The continuous function `\u03c9 \u21a6 \u220f_{e \u2208 s} \u03c9(e)`, the indicator of `cdc_multiOpenEvent s`.","title":"Continuous monomial indicator that every edge of a set is open"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cdc-monomial-bcf","kind":"node","lean":"StatMech.FK.cdc_monomialBcf","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cdc-monomial-bcf","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"`cdc_monomialCM s` as a bounded continuous function.","title":"Monomial indicator as a bounded continuous function"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cdc-box-multi-open-event","kind":"node","lean":"StatMech.FK.cdc_boxMultiOpenEvent","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cdc-box-multi-open-event","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The event that every edge of `t \u2286 E(\u039b_N)` is open, on configurations of `\u039b_N`.","title":"Event that every edge of a set of box edges is open, on a box"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cdc-wired-multi-homogeneous","kind":"node","lean":"StatMech.FK.cdc_wiredMultiHomogeneous","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," multi-edge homogeneity under a translation"]],"source":"Wired $q = 2$ multi-edge homogeneity under a translation"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cdc-wired-multi-homogeneous","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired `q = 2` infinite-volume measure gives every multi-open event the same mass as its translate by `g\u207b\u00b9`.","title":"Wired $q = 2$ multi-edge homogeneity under a translation"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cdc-free-multi-homogeneous","kind":"node","lean":"StatMech.FK.cdc_freeMultiHomogeneous","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," multi-edge homogeneity under a translation"]],"source":"Free $q = 2$ multi-edge homogeneity under a translation"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cdc-free-multi-homogeneous","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free `q = 2` infinite-volume measure gives every multi-open event the same mass as its translate by `g\u207b\u00b9`.","title":"Free $q = 2$ multi-edge homogeneity under a translation"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cdc-wired-decay-set","kind":"node","lean":"StatMech.FK.cdc_wiredDecaySet","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cdc-wired-decay-set","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The continuous functions whose shift-decay functional along the wired box measures (on the subsequence `\u03c6`) tends to `0`, as a submodule.","title":"Submodule of functions with vanishing wired shift decay"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cdc-free-decay-set","kind":"node","lean":"StatMech.FK.cdc_freeDecaySet","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cdc-free-decay-set","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The continuous functions whose shift-decay functional along the free box measures (on the subsequence `\u03c6`) tends to `0`, as a submodule.","title":"Submodule of functions with vanishing free shift decay"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/dgf-disjoint-box-analytic-cap","kind":"node","lean":"StatMech.FK.dgf_DisjointBoxAnalyticCap","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#dgf-disjoint-box-analytic-cap","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For every finite family of finite edge sets and scale `N`, some translation `g` and margin `r` realize each pair `T`, `g\u207b\u00b9T'` in `\u039b_{N+r}` with `g\u207b\u00b9T'` outside a set `F`, such that the conditional masses of `O_T` on fibres of `F` (with a boundary graph containing the wired one) are at most `\u03bc_N(O_T)` and the box mass of `g\u207b\u00b9O_{T'}` is at most `\u03bc_N(O_{T'})`.","title":"Disjoint-box cap hypothesis for multi-open events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-edge-incl","kind":"node","lean":"StatMech.FK.lmc_transEdgeIncl","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edge inclusion of a translated box into "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Edge inclusion of a translated box into $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-edge-incl","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The inclusion of edges of the translated box `v + \u039b_m` into the edges of `\u2124^d`.","title":"Edge inclusion of a translated box into $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-edge-incl-sigma-eq-smul","kind":"node","lean":"StatMech.FK.lmc_transEdgeIncl_sigma_eq_smul","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-edge-incl-sigma-eq-smul","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Including a translated box edge equals shifting the included centered box edge by `v`.","title":"Translated box edges are shifted centered box edges"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-image-eq-smul-image","kind":"node","lean":"StatMech.FK.lmc_transImage_eq_smulImage","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-image-eq-smul-image","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The image of a translated edge set in `\u2124^d` is the shift by `v` of the image of the centered set.","title":"Images of translated edge sets are shifted images"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-box-mass","kind":"node","lean":"StatMech.FK.lmc_transBoxMass","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-box-mass","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The probability, under the wired `q = 2` FK measure on the translated box `v + \u039b_m`, that all edges of the translated copy of `t \u2286 E(\u039b_N)` are open.","title":"Wired all-open probability on a translated box"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-box-mass-eq-centred","kind":"node","lean":"StatMech.FK.lmc_transBoxMass_eq_centred","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-box-mass-eq-centred","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The translated wired box mass equals the probability of the unshifted event under the wired finite-volume measure on the centered box `\u039b_m`.","title":"Translated wired box mass equals the centered box mass"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-box-mass-tendsto-centred-iv","kind":"node","lean":"StatMech.FK.lmc_transBoxMass_tendsto_centred_iv","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-box-mass-tendsto-centred-iv","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The translated wired box masses converge to the wired infinite-volume probability that all edges of `t` are open.","title":"Translated wired box masses converge to the infinite-volume mass"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-wired-multi-homogeneous-of-cofinal","kind":"node","lean":"StatMech.FK.lmc_wiredMultiHomogeneous_of_cofinal","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-wired-multi-homogeneous-of-cofinal","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the translated wired box masses also converge to the wired infinite-volume probability of the shifted set `v + t`, the wired infinite-volume measure gives `t` and `v + t` the same probability.","title":"Wired shift invariance of an all-open event from matching limits"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-wired-multi-homogeneous-of-decay","kind":"node","lean":"StatMech.FK.lmc_wiredMultiHomogeneous_of_decay","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-wired-multi-homogeneous-of-decay","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the translated wired box masses differ from the centered box masses of a set `t'` with `v + t = t'` by a sequence tending to zero, the wired infinite-volume measure gives `t` and `v + t` the same probability.","title":"Wired shift invariance of an all-open event from decaying differences"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-wired-placement","kind":"node","lean":"StatMech.FK.lmc_wiredPlacement","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-wired-placement","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The placement hypothesis for the shift `g`: every finite edge set lies in a box whose translated wired box masses converge to the wired infinite-volume probability of the set shifted by `g\u207b\u00b9`.","title":"Wired placement hypothesis for a shift"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-wired-multi-homogeneous","kind":"node","lean":"StatMech.FK.lmc_wiredMultiHomogeneous","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-wired-multi-homogeneous","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The placement hypothesis for `g` gives `g`-homogeneity of the wired infinite-volume measure on all events \"every edge of `s` is open\".","title":"Wired placement gives shift homogeneity on all-open events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-wired-placement-one","kind":"node","lean":"StatMech.FK.lmc_wiredPlacement_one","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-wired-placement-one","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The identity shift satisfies the wired placement hypothesis.","title":"Identity shift satisfies the wired placement hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-wired-multi-homogeneous-one","kind":"node","lean":"StatMech.FK.lmc_wiredMultiHomogeneous_one","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-wired-multi-homogeneous-one","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired infinite-volume measure is homogeneous under the identity shift.","title":"Wired infinite-volume measure is homogeneous under the identity shift"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-wired-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.lmc_wiredIV_isTranslationInvariant_of_placement","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-wired-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired `q = 2` infinite-volume FK measure on `\u2124^d` is translation invariant if every shift satisfies the wired placement hypothesis.","title":"Wired translation invariance from the placement hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-box-mass-free","kind":"node","lean":"StatMech.FK.lmc_transBoxMassFree","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-box-mass-free","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The probability, under the free `q = 2` FK measure on the translated box `v + \u039b_m`, that all edges of the translated copy of `t \u2286 E(\u039b_N)` are open.","title":"Translated free box mass of an edge set"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-box-mass-free-eq-centred","kind":"node","lean":"StatMech.FK.lmc_transBoxMassFree_eq_centred","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-box-mass-free-eq-centred","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The translated free box mass equals the probability of the unshifted event under the free finite-volume measure on the centered box `\u039b_m`.","title":"Translated free box mass equals the centred box probability"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-trans-box-mass-free-tendsto-centred","kind":"node","lean":"StatMech.FK.lmc_transBoxMassFree_tendsto_centred_iv","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-trans-box-mass-free-tendsto-centred","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The translated free box masses converge to the free infinite-volume probability that all edges of `t` are open.","title":"Translated free box masses converge to the free infinite-volume value"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-free-multi-homogeneous-of-cofinal","kind":"node","lean":"StatMech.FK.lmc_freeMultiHomogeneous_of_cofinal","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Equal free probabilities for "],["math","\\(t\\)"],["text"," and "],["math","\\(v + t\\)"],["text"," from matching box limits"]],"source":"Equal free probabilities for $t$ and $v + t$ from matching box limits"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-free-multi-homogeneous-of-cofinal","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the translated free box masses also converge to the free infinite-volume probability of the shifted set `v + t`, the free infinite-volume measure gives `t` and `v + t` the same probability.","title":"Equal free probabilities for $t$ and $v + t$ from matching box limits"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-free-placement","kind":"node","lean":"StatMech.FK.lmc_freePlacement","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-free-placement","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The placement hypothesis for the shift `g` and the free measure: every finite edge set lies in a box whose translated free box masses converge to the free infinite-volume probability of the set shifted by `g\u207b\u00b9`.","title":"Free placement hypothesis for a shift via box-mass limits"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-free-multi-homogeneous","kind":"node","lean":"StatMech.FK.lmc_freeMultiHomogeneous","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-free-multi-homogeneous","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free placement hypothesis for `g` gives `g`-homogeneity of the free infinite-volume measure on all events \"every edge of `s` is open\".","title":"Free placement gives shift homogeneity on all-open events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-free-placement-one","kind":"node","lean":"StatMech.FK.lmc_freePlacement_one","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-free-placement-one","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The identity shift satisfies the free placement hypothesis.","title":"Identity shift satisfies the free placement hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-free-multi-homogeneous-one","kind":"node","lean":"StatMech.FK.lmc_freeMultiHomogeneous_one","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-free-multi-homogeneous-one","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free infinite-volume measure is homogeneous under the identity shift.","title":"Free infinite-volume measure is homogeneous under the identity shift"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/lmc-free-iv-is-translation-invariant-of","kind":"node","lean":"StatMech.FK.lmc_freeIV_isTranslationInvariant_of_placement","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#lmc-free-iv-is-translation-invariant-of","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free `q = 2` infinite-volume FK measure on `\u2124^d` is translation invariant if every shift satisfies the free placement hypothesis.","title":"Free translation invariance from the placement hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/flc-vrad","kind":"node","lean":"StatMech.FK.flc_vrad","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sup norm of a site of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Sup norm of a site of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#flc-vrad","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The sup-norm `max_i |v_i|` of a site `v`.","title":"Sup norm of a site of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/flc-incl","kind":"node","lean":"StatMech.FK.flc_incl","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#flc-incl","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The inclusion of a subset of sites into a larger one, as a map of subtypes.","title":"Inclusion of a subset of sites into a larger one"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/flc-trans-edges","kind":"node","lean":"StatMech.FK.flc_transEdges","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box edge set included into "],["math","\\(\\Lambda_m\\)"],["text"," and translated by "],["math","\\(v\\)"]],"source":"Box edge set included into $\\Lambda_m$ and translated by $v$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#flc-trans-edges","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The edge set `t` of `\u039b_N`, included into `\u039b_m` and translated by `v`.","title":"Box edge set included into $\\Lambda_m$ and translated by $v$"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/flc-wired-iv-is-translation-invariant","kind":"node","lean":"StatMech.FK.flc_wiredIV_isTranslationInvariant","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation invariance of the wired "],["math","\\(q = 2\\)"],["text"," infinite-volume measure"]],"source":"Translation invariance of the wired $q = 2$ infinite-volume measure"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#flc-wired-iv-is-translation-invariant","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The wired `q = 2` infinite-volume random-cluster measure on `\u2124^d` is translation invariant.","title":"Translation invariance of the wired $q = 2$ infinite-volume measure"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-wired-box-domain-monotone","kind":"node","lean":"StatMech.FK.fwm_wired_box_domain_monotone","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Domain monotonicity of wired "],["math","\\(q = 2\\)"],["text"," box measures"]],"source":"Domain monotonicity of wired $q = 2$ box measures"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-wired-box-domain-monotone","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For `q = 2` and `N \u2264 m`, the wired `\u039b_m` measure of an increasing `\u039b_N` cylinder event is at most its wired `\u039b_N` measure.","title":"Domain monotonicity of wired $q = 2$ box measures"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-wired-multi-open-domain-monotone","kind":"node","lean":"StatMech.FK.fwm_wired_multiOpen_domain_monotone","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-wired-multi-open-domain-monotone","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For `q = 2` and `N \u2264 m`, the wired `\u039b_m` probability that a set of `\u039b_N` edges is open is at most its wired `\u039b_N` probability.","title":"Domain monotonicity of wired multi-open probabilities"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-wired-domain-monotone-of-realised","kind":"node","lean":"StatMech.FK.fwm_wired_domain_monotone_of_realised","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-wired-domain-monotone-of-realised","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The same for an edge set of `\u2124^d` given as the image of `\u039b_N` edges.","title":"Domain monotonicity for edge sets realized in a box"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-wired-cross-le-box-domain","kind":"node","lean":"StatMech.FK.fwm_wired_cross_le_box_domain","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-wired-cross-le-box-domain","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The same for the event \"`T` open and `g\u00b7T'` open\", when `T \u222a g\u207b\u00b9\u00b7T'` lies in `\u039b_N`.","title":"Domain monotonicity for translated pairs of multi-open events"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-wired-disjoint-box-factorise","kind":"node","lean":"StatMech.FK.fwm_WiredDisjointBoxFactorise","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-wired-disjoint-box-factorise","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The disjoint-box factorization bound for a family of box measures `\u03bcbox`: for every finite family `P` of edge sets realized in `\u039b_N` there are a translation `g` and `r` such that each `T \u222a g\u207b\u00b9\u00b7T'` lies in `\u039b_(N+r)` and `\u03bcbox (N+r) (T open, g\u00b7T' open) \u2264 \u03bcbox N (T open) \u00b7 \u03bcbox N (T' open)`. The parameters `p` only index the family.","title":"Disjoint-box factorization bound for box measures"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-finite-box-disjoint-product-dom-conc","kind":"node","lean":"StatMech.FK.fwm_finiteBoxDisjointProductDom_conclusion_of_factorise","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-finite-box-disjoint-product-dom-conc","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Under the factorization bound for the wired measures, for some translation `g` and all large `m`, `\u03c6\u00b9_{\u039b_m}(T open, g\u00b7T' open) \u2264 \u03c6\u00b9_{\u039b_N}(T open) \u03c6\u00b9_{\u039b_N}(T' open)` for all `T, T' \u2208 P`.","title":"Finite-box product domination from the factorization bound"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-realised-box-mono","kind":"node","lean":"StatMech.FK.fwm_realised_box_mono","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-realised-box-mono","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"An edge set realized in `\u039b_(N\u2080)` is realized in every larger box.","title":"Realized edge sets stay realized in larger boxes"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-exists-realising-box","kind":"node","lean":"StatMech.FK.fwm_exists_realising_box","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-exists-realising-box","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Every finite family of finite edge sets is realized in some box.","title":"Every finite family of edge sets is realized in some box"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-exists-common-box-ge-wired","kind":"node","lean":"StatMech.FK.fwm_exists_common_box_ge_wired","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-exists-common-box-ge-wired","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For `q = 2`, products of wired box probabilities of open-edge events approximate the wired infinite-volume products within `\u03b4`, for some box `\u039b_m` with `m \u2265 N\u2080`.","title":"Wired box products approximate infinite-volume products"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-upper-decay-of-factorise","kind":"node","lean":"StatMech.FK.fwm_upperDecay_of_factorise","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-upper-decay-of-factorise","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The factorization bound for the wired `q = 2` box measures gives upper decorrelation of the wired infinite-volume measure.","title":"Upper decorrelation of the wired measure from factorization"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-wired-iv-is-ergodic-of-factorise","kind":"node","lean":"StatMech.FK.fwm_wiredIV_isErgodic_of_factorise","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," measure is ergodic under the factorization bound"]],"source":"Wired $q = 2$ measure is ergodic under the factorization bound"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-wired-iv-is-ergodic-of-factorise","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"If the wired `q = 2` box measures satisfy the factorization bound, the wired infinite-volume measure is ergodic under translations of `\u2124^d`.","title":"Wired $q = 2$ measure is ergodic under the factorization bound"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/fwm-wired-disjoint-box-factorise-dirac","kind":"node","lean":"StatMech.FK.fwm_wiredDisjointBoxFactorise_dirac","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#fwm-wired-disjoint-box-factorise-dirac","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The factorization bound is satisfiable: the Dirac mass on the all-open configuration satisfies it.","title":"All-open Dirac mass satisfies the factorization bound"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/bdp-no-bdry","kind":"node","lean":"StatMech.FK.bdp_noBdry","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#bdp-no-bdry","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The empty boundary predicate.","title":"Empty boundary predicate"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/bdp-free-iv-is-translation-invariant","kind":"node","lean":"StatMech.FK.bdp_freeIV_isTranslationInvariant","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation invariance of the free "],["math","\\(q = 2\\)"],["text"," infinite-volume measure"]],"source":"Translation invariance of the free $q = 2$ infinite-volume measure"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#bdp-free-iv-is-translation-invariant","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The free `q = 2` infinite-volume random-cluster measure on `\u2124^d` is translation invariant.","title":"Translation invariance of the free $q = 2$ infinite-volume measure"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/bdp-free-per-edge-upper","kind":"node","lean":"StatMech.FK.bdp_free_per_edge_upper","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," box edge marginals are at most the free density"]],"source":"Free $q = 2$ box edge marginals are at most the free density"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#bdp-free-per-edge-upper","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Edge marginals of the free box measure are at most the free infinite-volume density (the case `N = n` of `dfi_free_per_edge_le`).","title":"Free $q = 2$ box edge marginals are at most the free density"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/bdp-wired-per-edge-lower","kind":"node","lean":"StatMech.FK.bdp_wired_per_edge_lower","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," box edge marginals are at least the wired density"]],"source":"Wired $q = 2$ box edge marginals are at least the wired density"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#bdp-wired-per-edge-lower","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"Edge marginals of the wired box measure are at least the wired infinite-volume density (the case `N = n` of `dfi_wired_per_edge_ge`).","title":"Wired $q = 2$ box edge marginals are at least the wired density"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/bdp-free-off-centre-sandwich","kind":"node","lean":"StatMech.FK.bdp_FreeOffCentreSandwich","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#bdp-free-off-centre-sandwich","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For every edge `e'` of `\u039b_N` and every `t`, there are interior edge sets `I_n` of `\u039b_n` carrying asymptotically all edges on which the free density equals that of `e'` and the free box marginals are at least that density minus `\u03b4_n \u2192 0`.","title":"Free off-centre sandwich hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/bdp-wired-off-centre-sandwich","kind":"node","lean":"StatMech.FK.bdp_WiredOffCentreSandwich","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#bdp-wired-off-centre-sandwich","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For every edge `e'` of `\u039b_N` and every `t`, there are interior edge sets `I_n` of `\u039b_n` carrying asymptotically all edges on which the wired density equals that of `e'` and the wired box marginals are at most that density plus `\u03b4_n \u2192 0`.","title":"Wired off-centre sandwich hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cbk-shift-vec","kind":"node","lean":"StatMech.FK.cbk_shiftVec","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shift vector moving "],["math","\\(\\Lambda_N\\)"],["text"," to a disjoint box"]],"source":"Shift vector moving $\\Lambda_N$ to a disjoint box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cbk-shift-vec","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The vector `(2N + 1) e_{i\u2080}`, which translates `\u039b_N` to a disjoint box.","title":"Shift vector moving $\\Lambda_N$ to a disjoint box"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cbk-shift-box-vert","kind":"node","lean":"StatMech.FK.cbk_shiftBoxVert","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translate of a vertex of "],["math","\\(\\Lambda_N\\)"],["text"," as a vertex of "],["math","\\(\\Lambda_{N+r}\\)"]],"source":"Translate of a vertex of $\\Lambda_N$ as a vertex of $\\Lambda_{N+r}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cbk-shift-box-vert","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The vertex of `\u039b_{N+r}` obtained by translating a vertex of `\u039b_N` by `v` with `\u2016v\u2016_\u221e \u2264 r`.","title":"Translate of a vertex of $\\Lambda_N$ as a vertex of $\\Lambda_{N+r}$"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cbk-shift-edge","kind":"node","lean":"StatMech.FK.cbk_shiftEdge","parent":48415,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translate of an edge of "],["math","\\(\\Lambda_N\\)"],["text"," as an edge of "],["math","\\(\\Lambda_{N+r}\\)"]],"source":"Translate of an edge of $\\Lambda_N$ as an edge of $\\Lambda_{N+r}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cbk-shift-edge","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"The edge of `\u039b_{N+r}` obtained by translating an edge of `\u039b_N` by `v` with `\u2016v\u2016_\u221e \u2264 r`.","title":"Translate of an edge of $\\Lambda_N$ as an edge of $\\Lambda_{N+r}$"},{"id":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/cbk-wired-iv-is-ergodic","kind":"node","lean":"StatMech.FK.cbk_wiredIV_isErgodic","parent":48415,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2/index.html#cbk-wired-iv-is-ergodic","scope":"StatisticalMechanics/random-cluster-model/translation-invariance-and-ergodicity-of-the-q-2","status":"fully_proved","summary":"For `d \u2265 1` and `p \u2208 (0, 1)`, the wired `q = 2` infinite-volume random-cluster measure on `\u2124^d` is ergodic under translations (Grimmett, *The Random-Cluster Model*, 2006, Theorem 4.19).","title":"Ergodicity of the wired random-cluster measure"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities.json","id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","kind":"scope","navigation":"module","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","title":"Convexity of the free energy and edge densities"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/box-secant-surface","kind":"node","lean":"StatMech.FK.boxSecantSurface","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#box-secant-surface","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The surface count `2d (2n + 1)^{d - 1}` of the box `\u039b_n`.","title":"Surface count of a box"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/box-secant-volume","kind":"node","lean":"StatMech.FK.boxSecantVolume","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#box-secant-volume","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The edge-volume count `d (2n + 1)^d` of the box `\u039b_n`.","title":"Edge-volume count of a box"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/connected-component-card-le-add-of-sup","kind":"node","lean":"StatMech.FK.connectedComponent_card_le_add_of_sup","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#connected-component-card-le-add-of-sup","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Adding edges with both endpoints in `{v | p v}` lowers the number of connected components by at most `|{v | p v}|`. See also `card_connectedComponent_le_sup_fromEdgeSet` (`EdgeMarginal`) for the bound by the number of added edges.","title":"Adding edges within a vertex set drops components by at most its size"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/pressure-right-deriv","kind":"node","lean":"StatMech.FK.pressureRightDeriv","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#pressure-right-deriv","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The right derivative `g'\u208a(x) = derivWithin g (Ioi x) x`, the form used by Mathlib's `ConvexOn.monotoneOn_rightDeriv`.","title":"Right derivative of the pressure"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/pressure-left-deriv","kind":"node","lean":"StatMech.FK.pressureLeftDeriv","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#pressure-left-deriv","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The left derivative `g'\u208b(x) = derivWithin g (Iio x) x`.","title":"Left derivative of the pressure"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/pressure-left-deriv-le-right-deriv","kind":"node","lean":"StatMech.FK.pressureLeftDeriv_le_rightDeriv","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#pressure-left-deriv-le-right-deriv","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"`g'\u208b(x) \u2264 g'\u208a(x)`; Mathlib's `ConvexOn.leftDeriv_le_rightDeriv_of_mem_interior` on `univ`.","title":"Left derivative of a convex function is at most the right one"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/countable-left-deriv-ne-right-deriv","kind":"node","lean":"StatMech.FK.countable_leftDeriv_ne_rightDeriv","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#countable-left-deriv-ne-right-deriv","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"A convex function on `\u211d` has equal left and right derivatives off a countable set.","title":"Convex function has equal one-sided derivatives off a countable set"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/pressure-convex-countable-density-ne","kind":"node","lean":"StatMech.FK.pressure_convex_countable_density_ne","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#pressure-convex-countable-density-ne","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"If two families of edge marginals are the left and right derivatives of a convex `g`, they differ only on a countable set of parameters.","title":"Derivative marginals of a convex pressure agree off a countable set"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fk-uniqueness-of-pressure-convex","kind":"node","lean":"StatMech.FK.fk_uniqueness_of_pressure_convex","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fk-uniqueness-of-pressure-convex","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"If `\u03bc p` and `\u03bd p` are monotonically coupled, the edge density of `\u03bd p` at `e` is the right derivative of a convex `g`, and all edge marginals agree wherever that density is continuous in `p`, then `\u03bc p` and `\u03bd p` agree on every increasing event for `p` off a countable set.","title":"Coupled measures agree off a countable set, given a convex pressure"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/countable-phi-ne-of-pressure-convex","kind":"node","lean":"StatMech.FK.countable_phi_ne_of_pressure_convex","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#countable-phi-ne-of-pressure-convex","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The set of parameters at which `\u03bc p` and `\u03bd p` differ on some increasing event is countable, under the hypotheses of `fk_uniqueness_of_pressure_convex`.","title":"Set where coupled measures differ is countable under a convex pressure"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fsc-logistic","kind":"node","lean":"StatMech.FK.fsc_logistic","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logistic function "],["math","\\(e^t/(1+e^t)\\)"]],"source":"Logistic function $e^t/(1+e^t)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fsc-logistic","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The logistic map `t \u21a6 e\u1d57 / (1 + e\u1d57)`, equal to Mathlib's `Real.sigmoid` (`fsc_logistic_eq_sigmoid`).","title":"Logistic function $e^t/(1+e^t)$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fsc-logit","kind":"node","lean":"StatMech.FK.fsc_logit","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Logit function "],["math","\\(\\log(p/(1-p))\\)"]],"source":"Logit function $\\log(p/(1-p))$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fsc-logit","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The logit `p \u21a6 log (p / (1 - p))`, inverse to `fsc_logistic` on `(0, 1)`.","title":"Logit function $\\log(p/(1-p))$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fsc-logistic-eq-sigmoid","kind":"node","lean":"StatMech.FK.fsc_logistic_eq_sigmoid","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fsc-logistic-eq-sigmoid","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The logistic map is Mathlib's `Real.sigmoid`.","title":"Logistic function equals the sigmoid"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fsc-free-energy-data","kind":"node","lean":"StatMech.FK.fsc_FreeEnergyData","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convex free-energy hypothesis for a box edge at "],["math","\\(q = 2\\)"]],"source":"Convex free-energy hypothesis for a box edge at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fsc-free-energy-data","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The convex free-energy hypothesis for the box edge `eb` at `q = 2`: some convex `g` has right derivative `t \u21a6 \u03b8^w(fsc_logistic t)` and left derivative `t \u21a6 \u03b8^f(fsc_logistic t)`, the wired and free infinite-volume densities of the edge.","title":"Convex free-energy hypothesis for a box edge at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fsc-secant","kind":"node","lean":"StatMech.FK.fsc_secant","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convex free energy gives wired density at "],["math","\\(p'\\)"],["text"," at most free at "],["math","\\(p > p'\\)"]],"source":"Convex free energy gives wired density at $p'$ at most free at $p > p'$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fsc-secant","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Under `fsc_FreeEnergyData`, the wired edge density at `p'` is at most the free edge density at `p` whenever `p' < p`.","title":"Convex free energy gives wired density at $p'$ at most free at $p > p'$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fsc-softplus","kind":"node","lean":"StatMech.FK.fsc_softplus","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Softplus function "],["math","\\(\\log(1+e^t)\\)"]],"source":"Softplus function $\\log(1+e^t)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fsc-softplus","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The softplus function `t \u21a6 log (1 + e\u1d57)`.","title":"Softplus function $\\log(1+e^t)$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fsc-free-energy-data-satisfiable","kind":"node","lean":"StatMech.FK.fsc_freeEnergyData_satisfiable","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fsc-free-energy-data-satisfiable","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Softplus is a convex function whose one-sided derivatives are both `fsc_logistic`, so the derivative conditions in `fsc_FreeEnergyData` can hold for some convex function.","title":"Softplus is convex with both one-sided derivatives logistic"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fk-uniqueness-q2-secant-from-free-energy","kind":"node","lean":"StatMech.FK.fk_uniqueness_q2_secant_from_freeEnergyData","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fk-uniqueness-q2-secant-from-free-energy","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Under `fsc_FreeEnergyData`, at `q = 2` there is a countable set `S` of parameters off which every measure `\u03c6` whose box-edge marginal lies between the free and wired ones has the same marginal as both.","title":"Edge-marginal uniqueness off a countable set from convex free energy"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fdd-free-density-is-left-deriv","kind":"node","lean":"StatMech.FK.fdd_FreeDensityIsLeftDeriv","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fdd-free-density-is-left-deriv","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The free `q = 2` density of every edge of `\u039b_N` at `p = logistic(t)` is the left derivative of `g` at `t`.","title":"Free edge densities on a box as left derivatives"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fdd-wired-density-is-right-deriv","kind":"node","lean":"StatMech.FK.fdd_WiredDensityIsRightDeriv","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fdd-wired-density-is-right-deriv","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired `q = 2` density of every edge of `\u039b_N` at `p = logistic(t)` is the right derivative of `g` at `t`.","title":"Wired edge densities on a box as right derivatives"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fdd-wired-density-upper-bound","kind":"node","lean":"StatMech.FK.fdd_WiredDensityUpperBound","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fdd-wired-density-upper-bound","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired `q = 2` density of every edge of `\u039b_N` at `p = logistic(t)` is at most the right derivative of `g` at `t`.","title":"Wired edge densities on a box bounded by right derivatives"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fpd-avg-density","kind":"node","lean":"StatMech.FK.fpd_avgDensity","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Average edge density of the "],["math","\\(q = 2\\)"],["text"," random-cluster measure"]],"source":"Average edge density of the $q = 2$ random-cluster measure"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fpd-avg-density","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The average edge density `|E|^{-1} E_{\u03c6_{G,p,2}}[#open edges]` at `p = logistic(t)`.","title":"Average edge density of the $q = 2$ random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fpd-avg-density-collapse","kind":"node","lean":"StatMech.FK.fpd_AvgDensityCollapse","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fpd-avg-density-collapse","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The average edge densities of the graphs `Gn n` converge to the free infinite-volume `q = 2` density of every edge of `\u039b_N`, at every `t`.","title":"Collapse of average edge densities to the free density"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fpd-ivdensity-is-one-sided-deriv","kind":"node","lean":"StatMech.FK.fpd_IVDensityIsOneSidedDeriv","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fpd-ivdensity-is-one-sided-deriv","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired and free `q = 2` infinite-volume densities of every edge of `\u039b_N` at `p = logistic(t)` are the right and left derivatives of `g` at `t`.","title":"Infinite-volume edge densities as one-sided derivatives"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-ssum","kind":"node","lean":"StatMech.FK.ivp2_Ssum","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-ssum","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The exponential sum `S\u2080(t) = \u2211_i w_i e^{a_i t}`.","title":"Weighted exponential sum"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-s1","kind":"node","lean":"StatMech.FK.ivp2_S1","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-s1","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The first moment sum `S\u2081(t) = \u2211_i w_i a_i e^{a_i t}`, the derivative of `S\u2080`.","title":"First moment exponential sum"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-s2","kind":"node","lean":"StatMech.FK.ivp2_S2","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-s2","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The second moment sum `S\u2082(t) = \u2211_i w_i a_i\u00b2 e^{a_i t}`, the derivative of `S\u2081`.","title":"Second moment exponential sum"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-cauchy-schwarz-cgf","kind":"node","lean":"StatMech.FK.ivp2_cauchy_schwarz_cgf","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-cauchy-schwarz-cgf","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Cauchy\u2013Schwarz for the exponential sums: `S\u2081\u00b2 \u2264 S\u2082 S\u2080`.","title":"Cauchy\u2013Schwarz inequality for weighted exponential sums"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-kcgf","kind":"node","lean":"StatMech.FK.ivp2_Kcgf","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-kcgf","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The log-sum-exp function `K(t) = log \u2211_i w_i e^{a_i t}`, the cumulant generating function of `a` under the weights `w`.","title":"Log-sum-exp cumulant generating function"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-kcgf-convex-on","kind":"node","lean":"StatMech.FK.ivp2_Kcgf_convexOn","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-kcgf-convex-on","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"A cumulant generating function is convex.","title":"Convexity of log-sum-exp with nonnegative nonzero weights"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-ambient-tilt-log-z","kind":"node","lean":"StatMech.FK.ivp2_ambientTiltLogZ","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-ambient-tilt-log-z","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The ambient logit normalizer `|E|\u207b\u00b9 log (fkZ G p q) + log (1 + e\u1d57)`, with `p = fsc_logistic t`. For `|E| > 0`, the added term removes the Bernoulli denominator, not the unused-coordinate factor in `fkZ`. At zero edges this definition uses totalized real division and equals `log (1 + e\u1d57)`; no per-edge physical interpretation is intended.","title":"Ambient tilted log normalizer of a finite graph"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-ambient-tilt-log-z-convex-on","kind":"node","lean":"StatMech.FK.ivp2_ambientTiltLogZ_convexOn","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-ambient-tilt-log-z-convex-on","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The finite-volume ambient tilted log normalizer is convex in `t`.","title":"Convexity of the ambient tilted log normalizer"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-convex-on-of-tendsto","kind":"node","lean":"StatMech.FK.ivp2_convexOn_of_tendsto","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-convex-on-of-tendsto","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"A pointwise limit of convex functions on `\u211d` along a nontrivial filter is convex.","title":"Pointwise limits of convex functions are convex"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/ivp2-limit-ambient-tilt-log-z-convex-on","kind":"node","lean":"StatMech.FK.ivp2_limitAmbientTiltLogZ_convexOn","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#ivp2-limit-ambient-tilt-log-z-convex-on","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Every pointwise limit of finite-volume ambient tilted log normalizers along a nontrivial filter is convex.","title":"Limits of ambient tilted log normalizers are convex"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/cdl-convex-deriv-limit","kind":"node","lean":"StatMech.FK.cdl_convexDerivLimit","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#cdl-convex-deriv-limit","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"A pointwise limit `g` of convex functions on `\u211d` is convex; its one-sided derivatives exist, are monotone, are one-sided limits of slopes, and agree off a countable set; slopes of the approximants converge to slopes of `g`.","title":"Pointwise limits of convex maps are convex with one-sided derivatives"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-fsc-logistic-continuous","kind":"node","lean":"StatMech.FK.osc_fsc_logistic_continuous","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-fsc-logistic-continuous","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The logistic map `t \u21a6 e^t / (1 + e^t)` is continuous. It equals Mathlib's `Real.sigmoid`.","title":"Continuity of the logistic map"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-edge-product-continuous","kind":"node","lean":"StatMech.FK.osc_edgeProduct_continuous","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuity in "],["math","\\(p\\)"],["text"," of the Bernoulli edge weight"]],"source":"Continuity in $p$ of the Bernoulli edge weight"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-edge-product-continuous","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The Bernoulli edge weight of a fixed configuration is continuous in `p`.","title":"Continuity in $p$ of the Bernoulli edge weight"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-free-box-marg-continuous","kind":"node","lean":"StatMech.FK.osc_freeBoxMarg_continuous","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuity in "],["math","\\(t\\)"],["text"," of free "],["math","\\(q = 2\\)"],["text"," edge marginals on a finite graph"]],"source":"Continuity in $t$ of free $q = 2$ edge marginals on a finite graph"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-free-box-marg-continuous","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The edge marginal of the free `q = 2` FK measure on a finite graph is continuous in `t`.","title":"Continuity in $t$ of free $q = 2$ edge marginals on a finite graph"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-wired-box-marg-continuous","kind":"node","lean":"StatMech.FK.osc_wiredBoxMarg_continuous","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Continuity in "],["math","\\(t\\)"],["text"," of wired "],["math","\\(q = 2\\)"],["text"," edge marginals on a finite graph"]],"source":"Continuity in $t$ of wired $q = 2$ edge marginals on a finite graph"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-wired-box-marg-continuous","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The edge marginal of the wired `q = 2` FK measure on a finite graph is continuous in `t`.","title":"Continuity in $t$ of wired $q = 2$ edge marginals on a finite graph"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-fk-edge-marg-monotone-in-p","kind":"node","lean":"StatMech.FK.osc_fkEdgeMarg_monotone_in_p","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Monotonicity in "],["math","\\(p\\)"],["text"," of free "],["math","\\(q = 2\\)"],["text"," edge marginals on a finite graph"]],"source":"Monotonicity in $p$ of free $q = 2$ edge marginals on a finite graph"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-fk-edge-marg-monotone-in-p","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The edge marginal of the free `q = 2` FK measure on a finite graph is monotone in `p`.","title":"Monotonicity in $p$ of free $q = 2$ edge marginals on a finite graph"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-free-edge-density-q2-monotone-on","kind":"node","lean":"StatMech.FK.osc_freeEdgeDensity_q2_monotoneOn","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," edge density is monotone in "],["math","\\(p\\)"],["text"," on "],["math","\\((0, 1)\\)"]],"source":"Free $q = 2$ edge density is monotone in $p$ on $(0, 1)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-free-edge-density-q2-monotone-on","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The free infinite-volume `q = 2` edge density is monotone in `p \u2208 (0, 1)`.","title":"Free $q = 2$ edge density is monotone in $p$ on $(0, 1)$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-free-edge-density-logistic-monotone","kind":"node","lean":"StatMech.FK.osc_freeEdgeDensity_logistic_monotone","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," edge density is monotone in "],["math","\\(t\\)"]],"source":"Free $q = 2$ edge density is monotone in $t$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-free-edge-density-logistic-monotone","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The free infinite-volume `q = 2` edge density is monotone in `t`.","title":"Free $q = 2$ edge density is monotone in $t$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-wired-edge-density-logistic-monotone","kind":"node","lean":"StatMech.FK.osc_wiredEdgeDensity_logistic_monotone","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," edge density is monotone in "],["math","\\(t\\)"]],"source":"Wired $q = 2$ edge density is monotone in $t$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-wired-edge-density-logistic-monotone","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired infinite-volume `q = 2` edge density is monotone in `t`.","title":"Wired $q = 2$ edge density is monotone in $t$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-free-edge-density-left-continuous","kind":"node","lean":"StatMech.FK.osc_freeEdgeDensity_left_continuous","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Left-continuity of the free "],["math","\\(q = 2\\)"],["text"," edge density in "],["math","\\(t\\)"]],"source":"Left-continuity of the free $q = 2$ edge density in $t$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-free-edge-density-left-continuous","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The free infinite-volume `q = 2` edge density is left-continuous in `t`.","title":"Left-continuity of the free $q = 2$ edge density in $t$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/osc-wired-edge-density-right-continuous","kind":"node","lean":"StatMech.FK.osc_wiredEdgeDensity_right_continuous","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Right-continuity of the wired "],["math","\\(q = 2\\)"],["text"," edge density in "],["math","\\(t\\)"]],"source":"Right-continuity of the wired $q = 2$ edge density in $t$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#osc-wired-edge-density-right-continuous","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired infinite-volume `q = 2` edge density is right-continuous in `t`.","title":"Right-continuity of the wired $q = 2$ edge density in $t$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/spc-boundary-fraction-le-surface-volume","kind":"node","lean":"StatMech.FK.spc_boundaryFraction_le_surfaceVolume","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#spc-boundary-fraction-le-surface-volume","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"If the non-bulk edges of `\u039b_n` number at most `C` times the boundary of `\u039b_n`, their fraction is at most `C` times the boundary-to-edge ratio.","title":"Non-bulk edge fraction bound from a boundary bound on their count"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/spc-ambient-tilt-log-z-inst-irrel","kind":"node","lean":"StatMech.FK.spc_ambientTiltLogZ_instIrrel","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#spc-ambient-tilt-log-z-inst-irrel","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The per-edge ambient tilted log normalizer does not depend on the decidability instance of the adjacency relation.","title":"Ambient tilted log normalizer ignores the decidability instance"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/spc-range-finite","kind":"node","lean":"StatMech.FK.spc_range_finite","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#spc-range-finite","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The per-edge ambient tilted log normalizers of a sequence of graphs on the fixed box `\u039b_N` take finitely many values.","title":"Tilted log normalizers on a fixed box take finitely many values"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/spc-fekete-limit-mem-finite-range","kind":"node","lean":"StatMech.FK.spc_feketeLimit_mem_finiteRange","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#spc-fekete-limit-mem-finite-range","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"A limit of per-edge ambient tilted log normalizers of graphs on the fixed box `\u039b_N` is attained by one of the graphs.","title":"Limits of tilted log normalizers on a fixed box are attained"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/spc-fixed-box-collapse-obstruction","kind":"node","lean":"StatMech.FK.spc_fixedBox_collapse_obstruction","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#spc-fixed-box-collapse-obstruction","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"If graphs on the fixed box `\u039b_N` and the growing boxes `\u039b_n` have the same limiting per-edge ambient tilted log normalizer `g t`, then `g t` is the ambient tilted log normalizer of a single graph on `\u039b_N`.","title":"A limit shared with growing boxes is one fixed-box graph's value"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/secant-exponent-le-zero","kind":"node","lean":"StatMech.FK.secant_exponent_le_zero","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#secant-exponent-le-zero","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"If `\u03b5 \u2264 q ^ (S n) \u03bb ^ (m V n) / \u03b5` for all `n`, with `q \u2265 1`, `0 < \u03bb < 1`, `V n \u2192 \u221e` and `S n / V n \u2192 0`, then `m \u2264 0`.","title":"Volume-growth lemma for tilted finite-volume bounds"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/secant-b-le-a-of-finite-volume-data","kind":"node","lean":"StatMech.FK.secant_b_le_a_of_finiteVolumeData","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tilted finite-volume bounds for all "],["math","\\(\\varepsilon > 0\\)"],["text"," force "],["math","\\(b \\le a\\)"]],"source":"Tilted finite-volume bounds for all $\\varepsilon > 0$ force $b \\le a$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#secant-b-le-a-of-finite-volume-data","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"If for every `\u03b5 > 0` the bound `\u03b5 \u2264 q ^ (S n) \u03bb ^ ((b - a - 2\u03b5) V n) / \u03b5` holds for all `n`, with `V n \u2192 \u221e` and `S n = o(V n)`, then `b \u2264 a`.","title":"Tilted finite-volume bounds for all $\\varepsilon > 0$ force $b \\le a$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fksecant-data","kind":"node","lean":"StatMech.FK.FKSecantData","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite-volume tilt data for a box edge at "],["math","\\(q = 2\\)"]],"source":"Finite-volume tilt data for a box edge at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fksecant-data","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Finite-volume tilt data for the box edge `eb` at `q = 2`: volumes `V n \u2192 \u221e` and surface terms `S n = o(V n)` such that, for all `p' < p` in `(0, 1)` and `\u03b5 > 0`, `\u03b5 \u2264 2 ^ (S n) \u03bb ^ ((\u03b8^w(p') - \u03b8^f(p) - 2\u03b5) V n) / \u03b5` with `\u03bb = tiltFactor p p'`, where `\u03b8^w` and `\u03b8^f` are the wired and free infinite-volume densities of the edge.","title":"Finite-volume tilt data for a box edge at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/hsecant-discharged","kind":"node","lean":"StatMech.FK.hsecant_discharged","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Tilt data give wired density at "],["math","\\(p'\\)"],["text"," at most free density at "],["math","\\(p > p'\\)"]],"source":"Tilt data give wired density at $p'$ at most free density at $p > p'$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#hsecant-discharged","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Under `FKSecantData`, the wired edge density at `p'` is at most the free edge density at `p` whenever `p' < p`.","title":"Tilt data give wired density at $p'$ at most free density at $p > p'$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/fk-uniqueness-q2-secant-from-finite-volu","kind":"node","lean":"StatMech.FK.fk_uniqueness_q2_secant_from_finiteVolumeData","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#fk-uniqueness-q2-secant-from-finite-volu","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Under `FKSecantData`, at `q = 2` there is a countable set `S` of parameters off which every measure `\u03c6` whose box-edge marginal lies between the free and wired ones has the same marginal as both.","title":"Box-edge marginal uniqueness off a countable set under tilt data"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-ambient-tilt-log-z","kind":"node","lean":"StatMech.FK.wpd_wiredAmbientTiltLogZ","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-ambient-tilt-log-z","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired per-edge ambient tilted log normalizer `|E|\u207b\u00b9 log Z^w(p, q) + log(1 + e^t)` at `p = e^t / (1 + e^t)`.","title":"Wired per-edge ambient tilted log normalizer"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-fk-z-logistic","kind":"node","lean":"StatMech.FK.wpd_wiredFkZ_logistic","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired partition function at logistic "],["math","\\(p\\)"],["text"," as a tilted sum"]],"source":"Wired partition function at logistic $p$ as a tilted sum"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-fk-z-logistic","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired partition function at `p = e^t / (1 + e^t)` as a tilted sum over configurations divided by `(1 + e^t)^{|E|}`.","title":"Wired partition function at logistic $p$ as a tilted sum"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-ambient-tilt-log-z-eq-kcgf","kind":"node","lean":"StatMech.FK.wpd_wiredAmbientTiltLogZ_eq_Kcgf","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-ambient-tilt-log-z-eq-kcgf","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired ambient tilted log normalizer is `|E|\u207b\u00b9` times the cumulant generating function of the number of open edges under the weight `q^{k^w}`.","title":"Wired tilted log normalizer as a cumulant generating function"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-ambient-tilt-log-z-convex-on","kind":"node","lean":"StatMech.FK.wpd_wiredAmbientTiltLogZ_convexOn","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-ambient-tilt-log-z-convex-on","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired ambient tilted log normalizer is convex in `t`.","title":"Convexity of the wired tilted log normalizer"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-fk-expect","kind":"node","lean":"StatMech.FK.wpd_wiredFkExpect","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-fk-expect","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The expectation of an observable under the wired FK measure.","title":"Expectation under the wired FK measure"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-cgf-mean-eq-open-count-expect","kind":"node","lean":"StatMech.FK.wpd_wiredCgfMean_eq_openCount_expect","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-cgf-mean-eq-open-count-expect","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The tilted mean of the number of open edges is its wired FK expectation.","title":"Tilted mean of open edges equals the wired FK expectation"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-ambient-tilt-log-z-has-deriv-a","kind":"node","lean":"StatMech.FK.wpd_wiredAmbientTiltLogZ_hasDerivAt","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-ambient-tilt-log-z-has-deriv-a","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The derivative in `t` of the wired ambient tilted log normalizer is `|E|\u207b\u00b9` times the expected number of open edges under the wired FK measure.","title":"Wired tilted log normalizer has derivative the mean open fraction"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-avg-wired-density","kind":"node","lean":"StatMech.FK.wpd_avgWiredDensity","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-avg-wired-density","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The average wired edge density `|E|\u207b\u00b9 E^w[|\u03c9|]` at `p = e^t / (1 + e^t)`.","title":"Average wired edge density"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-fk-expect-open-count-eq-sum","kind":"node","lean":"StatMech.FK.wpd_wiredFkExpect_openCount_eq_sum","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-fk-expect-open-count-eq-sum","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The expected number of open edges is the sum of the edge marginals.","title":"Expected wired open-edge count is the sum of edge marginals"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-avg-wired-density-eq-sum-edge-marg","kind":"node","lean":"StatMech.FK.wpd_avgWiredDensity_eq_sum_edgeMarg","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-avg-wired-density-eq-sum-edge-marg","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The average wired edge density is the mean of the wired edge marginals.","title":"Average wired edge density as the mean of edge marginals"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-fk-weight-le","kind":"node","lean":"StatMech.FK.wpd_wiredFkWeight_le","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired and free FK weights within a factor "],["math","\\(q^{|\\partial|}\\)"],["text",", "],["math","\\(q \\ge 1\\)"]],"source":"Wired and free FK weights within a factor $q^{|\\partial|}$, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-fk-weight-le","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"For `q \u2265 1`, the wired FK weight is at most the free one, and the free one is at most `q^{|\u2202|}` times the wired one.","title":"Wired and free FK weights within a factor $q^{|\\partial|}$, $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-wired-fk-z-le","kind":"node","lean":"StatMech.FK.wpd_wiredFkZ_le","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sandwich "],["math","\\(Z^w \\le Z \\le q^{|\\partial|} Z^w\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Sandwich $Z^w \\le Z \\le q^{|\\partial|} Z^w$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-wired-fk-z-le","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"For `q \u2265 1`, `Z^w \u2264 Z \u2264 q^{|\u2202|} Z^w`.","title":"Sandwich $Z^w \\le Z \\le q^{|\\partial|} Z^w$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-ambient-tilt-log-z-sub-le","kind":"node","lean":"StatMech.FK.wpd_ambientTiltLogZ_sub_le","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free-wired log-normalizer gap "],["math","\\(\\le |\\partial|\\log q/|E|\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Free-wired log-normalizer gap $\\le |\\partial|\\log q/|E|$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-ambient-tilt-log-z-sub-le","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"For `q \u2265 1`, the free and wired per-edge ambient tilted log normalizers differ by at most `|\u2202| log q / |E|`.","title":"Free-wired log-normalizer gap $\\le |\\partial|\\log q/|E|$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/wpd-avg-wired-density-collapse","kind":"node","lean":"StatMech.FK.wpd_AvgWiredDensityCollapse","parent":48532,"reading_title":{"format":"inline-parts-v1","parts":[["text","Average wired density collapse at "],["math","\\(q = 2\\)"]],"source":"Average wired density collapse at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#wpd-avg-wired-density-collapse","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"Average wired density collapse: for every edge `e'` of `\u039b_N` and every `t`, the average wired edge density of `\u039b_n` converges to the wired infinite-volume density of `e'`.","title":"Average wired density collapse at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/cfe-centered","kind":"node","lean":"StatMech.FK.cfe_centered","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#cfe-centered","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The centered free energy `F_G(t) - F_G(0)`, with `F_G = ivp2_ambientTiltLogZ G q`. In the FK range `q > 0`, the ambient non-edge offset cancels in this difference.","title":"Centered free energy of a graph"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/cfe-wired-centered","kind":"node","lean":"StatMech.FK.cfe_wiredCentered","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#cfe-wired-centered","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The wired centered free energy, obtained by subtracting `wpd_wiredAmbientTiltLogZ G bdry q 0` from its value at `t`. In the FK range `q > 0`, the ambient non-edge offset cancels.","title":"Wired centered free energy"},{"id":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/cfe-centered-convergence","kind":"node","lean":"StatMech.FK.cfe_CenteredConvergence","parent":48532,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities/index.html#cfe-centered-convergence","scope":"StatisticalMechanics/random-cluster-model/convexity-of-the-free-energy-and-edge-densities","status":"fully_proved","summary":"The centered free energies of the boxes `\u039b_n` converge pointwise to `G`, and `G(0) = 0`.","title":"Pointwise convergence of box centered free energies"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model.json","id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","kind":"scope","navigation":"module","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","title":"Free energy of the random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/bxt-ambient-neg-log-z","kind":"node","lean":"StatMech.FK.bxt_ambientNegLogZ","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Negative log ambient normalizer of the box "],["math","\\(\\Lambda_n\\)"]],"source":"Negative log ambient normalizer of the box $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#bxt-ambient-neg-log-z","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The negative log ambient normalizer `-log (fkZ \u039b_n (logistic t) 2)` of the box `\u039b_n`.","title":"Negative log ambient normalizer of the box $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/bxt-e","kind":"node","lean":"StatMech.FK.bxt_E","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edge count of the box "],["math","\\(\\Lambda_n\\)"]],"source":"Edge count of the box $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#bxt-e","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The number of edges of the box `\u039b_n`.","title":"Edge count of the box $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/bxt-neg-log-doubling-bound","kind":"node","lean":"StatMech.FK.bxt_NegLogDoublingBound","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Doubling bound at the dyadic scales "],["math","\\(2^j\\)"]],"source":"Doubling bound at the dyadic scales $2^j$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#bxt-neg-log-doubling-bound","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Doubling bound at the dyadic scales: `u(2^{j+1}) \u2264 2^d u(2^j) + I_j (-log(1 - p))` with `|2^d E(2^j) - E(2^{j+1})| \u2264 I_j` and `\u2211_j I_j / E(2^{j+1}) < \u221e`.","title":"Doubling bound at the dyadic scales $2^j$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fis-interface","kind":"node","lean":"StatMech.FK.fis_interface","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fis-interface","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The interface edges of `K`: edges not in the disjoint union `G \u2295 H`.","title":"Interface edges of a graph on a disjoint union"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fis-cross-interface","kind":"node","lean":"StatMech.FK.fis_CrossInterface","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fis-cross-interface","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"`K` contains `G \u2295 H` and adds no edges inside `V` or inside `W`; all extra edges join `V` to `W`.","title":"Graph glued from two graphs along an interface"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-left","kind":"node","lean":"StatMech.FK.agl_left","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-left","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The graph induced by `K\u2080` on `{x | P x}`.","title":"Graph induced on the vertices satisfying a predicate"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-right","kind":"node","lean":"StatMech.FK.agl_right","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-right","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The graph induced by `K\u2080` on `{x | \u00ac P x}`.","title":"Graph induced on the vertices failing a predicate"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-sum-equiv","kind":"node","lean":"StatMech.FK.agl_sumEquiv","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-sum-equiv","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The identification `{P} \u2295 {\u00acP} \u2243 U`.","title":"Vertex set as a disjoint sum of a subset and its complement"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-glue-graph","kind":"node","lean":"StatMech.FK.agl_glueGraph","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-glue-graph","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"`K\u2080` transported to `{P} \u2295 {\u00acP}`.","title":"Graph transported to the sum of a subset and its complement"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-iso","kind":"node","lean":"StatMech.FK.agl_iso","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-iso","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"`K\u2080` is isomorphic to its transport `agl_glueGraph`.","title":"Isomorphism between a graph and its transported copy"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-interface-card","kind":"node","lean":"StatMech.FK.agl_interfaceCard","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-interface-card","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The number of edges of `K\u2080` joining `{P}` to `{\u00acP}`.","title":"Number of edges between a vertex subset and its complement"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-trans-block-iso","kind":"node","lean":"StatMech.FK.agl_transBlockIso","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-trans-block-iso","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Translating vertex embeddings by `v` gives isomorphic induced lattice graphs.","title":"Isomorphism of translated induced lattice graphs"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-left-block-iso","kind":"node","lean":"StatMech.FK.agl_leftBlockIso","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-left-block-iso","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"A translated copy of `\u039b_m` inside `{P}` induces a graph isomorphic to `\u039b_m`.","title":"Isomorphism of a box with its translated copy in the predicate part"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-right-block-iso","kind":"node","lean":"StatMech.FK.agl_rightBlockIso","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-right-block-iso","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"A translated copy of `\u039b_m` inside `{\u00acP}` induces a graph isomorphic to `\u039b_m`.","title":"Isomorphism of a box with its translated copy in the complement part"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-k","kind":"node","lean":"StatMech.FK.agl_K","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Scales "],["math","\\(K_j = 2^{j+1} - 1\\)"]],"source":"Scales $K_j = 2^{j+1} - 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-k","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The scales `K_j = 2^{j+1} - 1`, with `K_0 = 1` and `K_{j+1} = 2 K_j + 1`.","title":"Scales $K_j = 2^{j+1} - 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-ambient-neg-log-z","kind":"node","lean":"StatMech.FK.agl_ambientNegLogZ","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-ambient-neg-log-z","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The negative log ambient normalizer of `\u039b_n`; the same as `bxt_ambientNegLogZ`.","title":"Negative log ambient normalizer of a box"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-e","kind":"node","lean":"StatMech.FK.agl_E","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-e","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The number of edges of `\u039b_n`; the same as `bxt_E`.","title":"Number of edges of a box"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-additive-doubling-bound","kind":"node","lean":"StatMech.FK.agl_AdditiveDoublingBound","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Additive doubling bound along the scales "],["math","\\(K_j\\)"]],"source":"Additive doubling bound along the scales $K_j$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-additive-doubling-bound","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Doubling bound along the scales `K_j`: `u(K_{j+1}) \u2264 2^d u(K_j) + I_j (-log(1 - p))` with `|2^d E(K_j) - E(K_{j+1})| \u2264 I_j` and `\u2211_j I_j / E(K_{j+1}) < \u221e`.","title":"Additive doubling bound along the scales $K_j$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/agl-additive-box-fekete-data","kind":"node","lean":"StatMech.FK.agl_AdditiveBoxFeketeData","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Assumed doubling, boundedness and oscillation data along "],["math","\\(K_j\\)"]],"source":"Assumed doubling, boundedness and oscillation data along $K_j$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#agl-additive-box-fekete-data","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The doubling bound along `K_j`, a uniform bound on the ambient per-edge values `agl_ambientNegLogZ / agl_E` at those scales, and vanishing oscillation of `ivp2_ambientTiltLogZ` between `\u039b_n` and `\u039b_{K_{b(n)}}`. These are assumptions, not a box-convergence result.","title":"Assumed doubling, boundedness and oscillation data along $K_j$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkl-tendsto-div-of-subadditive","kind":"node","lean":"StatMech.FK.fkl_tendsto_div_of_subadditive","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkl-tendsto-div-of-subadditive","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Fekete's lemma normalized by `e n` instead of `n`: if `u` is subadditive with `u n / n` bounded below and `e n / n \u2192 c \u2260 0`, then `u n / e n \u2192 lim / c`. Built on Mathlib's `Subadditive.tendsto_lim`.","title":"Fekete's lemma with a general normalization"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkl-super-multiplicative-satisfiable","kind":"node","lean":"StatMech.FK.fkl_superMultiplicative_satisfiable","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkl-super-multiplicative-satisfiable","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The hypotheses of `fkl_tendsto_div_of_subadditive` hold together for `u = 0`, `e n = n`.","title":"Hypotheses of the normalized Fekete lemma are satisfiable"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/bpv-ambient-tilt-log-zconvergence-data","kind":"node","lean":"StatMech.FK.bpv_AmbientTiltLogZConvergenceData","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#bpv-ambient-tilt-log-zconvergence-data","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The ambient tilted log normalizer of `\u039b_{2^j}` is increasing in `j` up to summable defects and bounded above, and the ambient tilted log normalizer of `\u039b_n` differs from that of the dyadic box below it by a vanishing oscillation.","title":"Dyadic convergence data for the ambient tilted log normalizer"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/bfd-tendsto-div-of-near-subadditive","kind":"node","lean":"StatMech.FK.bfd_tendsto_div_of_nearSubadditive","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#bfd-tendsto-div-of-near-subadditive","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Fekete's lemma with an `o(n)` correction: if `u n + c n` is subadditive and bounded below after dividing by `n`, and `c n / n \u2192 0`, then `u n / n` converges. See also Mathlib's `Subadditive.tendsto_lim`.","title":"Fekete's lemma with a sublinear correction"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-cc-to-sum","kind":"node","lean":"StatMech.FK.fsm_ccToSum","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-cc-to-sum","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The map sending a component of `G \u2295g H` to the corresponding component of `G` or `H`.","title":"Components of a disjoint union sent to components of the summands"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-sum-to-cc","kind":"node","lean":"StatMech.FK.fsm_sumToCC","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-sum-to-cc","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The inclusion of the components of `G` and of `H` into those of `G \u2295g H`.","title":"Inclusion of summand components into a disjoint union"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-cc-sum-equiv","kind":"node","lean":"StatMech.FK.fsm_ccSumEquiv","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-cc-sum-equiv","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The connected components of `G \u2295g H` are those of `G` and those of `H`.","title":"Connected components of a disjoint union of graphs"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-combine","kind":"node","lean":"StatMech.FK.fsm_combine","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-combine","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Gluing edge configurations: `\u03c3` on edges inside `V`, `\u03c4` on edges inside `W`, and closed on pairs joining `V` to `W`.","title":"Gluing edge configurations on a disjoint union"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-fk-z-sum-ge","kind":"node","lean":"StatMech.FK.fsm_fkZ_sum_ge","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-fk-z-sum-ge","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The ambient FK normalizer is supermultiplicative under disjoint union: `Z_G Z_H \u2264 Z_{G \u2295g H}`. Each `Z` sums over all `Sym2` configurations.","title":"Ambient FK normalizer is supermultiplicative under disjoint union"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-sym2-congr","kind":"node","lean":"StatMech.FK.fsm_sym2Congr","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-sym2-congr","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The equivalence `Sym2 V \u2243 Sym2 W` induced by `V \u2243 W`.","title":"Bijection of unordered pairs induced by a bijection"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-open-sub-iso","kind":"node","lean":"StatMech.FK.fsm_openSubIso","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-open-sub-iso","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"A graph isomorphism `G \u2243g H` restricts to an isomorphism of open subgraphs, with the configuration on `G` pulled back from `H`.","title":"Graph isomorphism restricted to open subgraphs"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-fk-z-iso","kind":"node","lean":"StatMech.FK.fsm_fkZ_iso","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-fk-z-iso","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The ambient FK normalizer is invariant under graph isomorphism.","title":"Isomorphism invariance of the ambient FK normalizer"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-subadditive-neglog-fk-z","kind":"node","lean":"StatMech.FK.fsm_subadditive_neglog_fkZ","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Subadditivity of "],["math","\\(-\\log Z\\)"],["text"," along a juxtaposition sequence"]],"source":"Subadditivity of $-\\log Z$ along a juxtaposition sequence"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-subadditive-neglog-fk-z","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Along a sequence with `G_{m+n} \u2245 G_m \u2295g G_n`, the sequence `-log Z_{G_n}` is subadditive.","title":"Subadditivity of $-\\log Z$ along a juxtaposition sequence"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fsm-juxtaposition-satisfiable","kind":"node","lean":"StatMech.FK.fsm_juxtaposition_satisfiable","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fsm-juxtaposition-satisfiable","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Edgeless graphs on `Fin n` satisfy the juxtaposition hypothesis `G_{m+n} \u2245 G_m \u2295g G_n`.","title":"Edgeless graphs satisfy the juxtaposition hypothesis"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-comap-iso-of-range-eq","kind":"node","lean":"StatMech.FK.qp_comapIsoOfRangeEq","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-comap-iso-of-range-eq","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Graphs induced on `K\u2080` by two injections with the same range are isomorphic.","title":"Isomorphism of induced graphs from injections with equal range"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-fk-z-comap-range","kind":"node","lean":"StatMech.FK.qp_fkZ_comap_range","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-fk-z-comap-range","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Graphs induced by two injections with the same range have the same FK partition function.","title":"Induced graphs with equal range have equal FK partition functions"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-fk-z-ge-all-closed","kind":"node","lean":"StatMech.FK.qp_fkZ_ge_allClosed","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","FK partition function at least the all-closed weight for "],["math","\\(q \\ge 1\\)"]],"source":"FK partition function at least the all-closed weight for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-fk-z-ge-all-closed","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"For `q \u2265 1`, the FK partition function is at least the weight `(1 - p)^{|E|}` of the all- closed configuration.","title":"FK partition function at least the all-closed weight for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-neglog-z-le-edges","kind":"node","lean":"StatMech.FK.qp_neglogZ_le_edges","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound "],["math","\\(-\\log Z \\le |E|\\,(-\\log(1-p))\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Bound $-\\log Z \\le |E|\\,(-\\log(1-p))$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-neglog-z-le-edges","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"For `q \u2265 1`, `-log Z \u2264 |E| \u00b7 (-log (1 - p))`.","title":"Bound $-\\log Z \\le |E|\\,(-\\log(1-p))$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-rest-g","kind":"node","lean":"StatMech.FK.qp_restG","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Graph induced on vertices with tags in "],["math","\\(S\\)"]],"source":"Graph induced on vertices with tags in $S$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-rest-g","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The graph induced on the vertices whose tag lies in `S`.","title":"Graph induced on vertices with tags in $S$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-blk-g","kind":"node","lean":"StatMech.FK.qp_blkG","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-blk-g","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The block of tag `\u03c4`: the graph induced on the vertices with tag `\u03c4`.","title":"Block of a tag: graph induced on its vertices"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-straddle-p","kind":"node","lean":"StatMech.FK.qp_straddleP","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-straddle-p","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"An edge straddles `P` if exactly one endpoint satisfies `P`.","title":"Straddling of a predicate by an edge"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-straddle-edges","kind":"node","lean":"StatMech.FK.qp_straddleEdges","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-straddle-edges","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The edges of `K\u2080` that straddle `P`.","title":"Set of edges straddling a predicate"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-comap-comap-iso","kind":"node","lean":"StatMech.FK.qp_comapComapIso","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-comap-comap-iso","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"An iterated induced graph is isomorphic to the graph induced by any injection with the same range as the composite.","title":"Isomorphism of an iterated induced graph with a single one"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-fk-z-comap-comap","kind":"node","lean":"StatMech.FK.qp_fkZ_comapComap","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-fk-z-comap-comap","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"An iterated induced graph has the FK partition function of the graph induced by any injection with the same range.","title":"Iterated and single induced graphs share the FK partition function"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-left-eq","kind":"node","lean":"StatMech.FK.qp_left_eq","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-left-eq","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The left part of the restriction to `insert \u03c4 S`, split by tag `\u03c4`, has the partition function of the block of `\u03c4`.","title":"Left part of a tag split has the block partition function"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-right-eq","kind":"node","lean":"StatMech.FK.qp_right_eq","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-right-eq","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The right part of the restriction to `insert \u03c4 S`, split by tag `\u03c4`, has the partition function of the restriction to `S`.","title":"Right part of a tag split has the restricted partition function"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-interface-le-straddle","kind":"node","lean":"StatMech.FK.qp_interface_le_straddle","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-interface-le-straddle","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The interface of the left/right split by `P` has at most as many edges as there are edges straddling `P`.","title":"Interface edges of a split are at most the straddling edges"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-rest-g-straddle-le","kind":"node","lean":"StatMech.FK.qp_restG_straddle_le","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-rest-g-straddle-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Restricting to `insert \u03c4 S` does not increase the number of edges straddling tag `\u03c4`.","title":"Restriction does not increase straddling edges of a tag"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-fk-z-empty","kind":"node","lean":"StatMech.FK.qp_fkZ_empty","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-fk-z-empty","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The FK partition function of a graph with no vertices is `1`.","title":"FK partition function of a graph with no vertices is one"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-ambient-neg-log-z","kind":"node","lean":"StatMech.FK.qp_ambientNegLogZ","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-ambient-neg-log-z","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The negative log ambient normalizer `-log (fkZ G p q)`, including unused coordinates.","title":"Negative log of the ambient FK normalizer"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-peel-le","kind":"node","lean":"StatMech.FK.qp_peel_le","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Peeling bound for "],["math","\\(-\\log Z\\)"],["text"," of a tagged graph"]],"source":"Peeling bound for $-\\log Z$ of a tagged graph"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-peel-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Peeling: `-log Z` of the restriction to `S` is at most the sum over `\u03c4 \u2208 S` of `-log Z` of the blocks plus `I \u00b7 (-log (1 - p))`, with `I` at most the number of edges straddling the tags in `S`.","title":"Peeling bound for $-\\log Z$ of a tagged graph"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/edge-axis-i","kind":"node","lean":"StatMech.FK.EdgeCount.edgeAxisI","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ordered nearest-neighbour pairs of "],["math","\\(\\Lambda_n\\)"],["text"," in direction "],["math","\\(i\\)"]],"source":"Ordered nearest-neighbour pairs of $\\Lambda_n$ in direction $i$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#edge-axis-i","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The ordered nearest-neighbor pairs of `\u039b_n` that differ in coordinate `i`.","title":"Ordered nearest-neighbour pairs of $\\Lambda_n$ in direction $i$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/shift-dom","kind":"node","lean":"StatMech.FK.EdgeCount.shiftDom","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sites of "],["math","\\(\\Lambda_n\\)"],["text"," whose shift in coordinate "],["math","\\(i\\)"],["text"," stays in the box"]],"source":"Sites of $\\Lambda_n$ whose shift in coordinate $i$ stays in the box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#shift-dom","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The sites of `\u039b_n` whose shift by `b` in coordinate `i` stays in `\u039b_n`.","title":"Sites of $\\Lambda_n$ whose shift in coordinate $i$ stays in the box"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/shift-map","kind":"node","lean":"StatMech.FK.EdgeCount.shiftMap","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shift map "],["math","\\(x \\mapsto (x, x + b e_i)\\)"]],"source":"Shift map $x \\mapsto (x, x + b e_i)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#shift-map","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The ordered pair `(x, x + b e_i)`.","title":"Shift map $x \\mapsto (x, x + b e_i)$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/shift-adj","kind":"node","lean":"StatMech.FK.EdgeCount.shift_adj","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#shift-adj","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"A site is adjacent to its shift by `\u00b11` in one coordinate.","title":"A site is adjacent to its unit shift in one coordinate"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/card-shift-dom","kind":"node","lean":"StatMech.FK.EdgeCount.card_shiftDom","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(2n(2n+1)^{d-1}\\)"],["text"," sites of "],["math","\\(\\Lambda_n\\)"],["text"," stay inside after a unit shift"]],"source":"$2n(2n+1)^{d-1}$ sites of $\\Lambda_n$ stay inside after a unit shift"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#card-shift-dom","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"For `b = \u00b11`, `2n (2n + 1)^{d-1}` sites of `\u039b_n` stay in `\u039b_n` after the shift.","title":"$2n(2n+1)^{d-1}$ sites of $\\Lambda_n$ stay inside after a unit shift"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/shift-map-mem","kind":"node","lean":"StatMech.FK.EdgeCount.shiftMap_mem","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit shifts of sites staying in the box are pairs in direction "],["math","\\(i\\)"]],"source":"Unit shifts of sites staying in the box are pairs in direction $i$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#shift-map-mem","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"For `b = \u00b11`, the shift of a site that stays in the box gives a pair in direction `i`.","title":"Unit shifts of sites staying in the box are pairs in direction $i$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/shift-map-inj","kind":"node","lean":"StatMech.FK.EdgeCount.shiftMap_inj","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#shift-map-inj","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The shift map is injective.","title":"Injectivity of the shift map"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/edge-axis-i-eq","kind":"node","lean":"StatMech.FK.EdgeCount.edgeAxisI_eq","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pairs in direction "],["math","\\(i\\)"],["text"," are the forward and backward shifts"]],"source":"Pairs in direction $i$ are the forward and backward shifts"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#edge-axis-i-eq","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The pairs in direction `i` are the forward and the backward shifts.","title":"Pairs in direction $i$ are the forward and backward shifts"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fwd-bwd-disjoint","kind":"node","lean":"StatMech.FK.EdgeCount.fwd_bwd_disjoint","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fwd-bwd-disjoint","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Forward and backward pairs in one direction are disjoint.","title":"Forward and backward shift pairs are disjoint"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/card-edge-axis-i","kind":"node","lean":"StatMech.FK.EdgeCount.card_edgeAxisI","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\Lambda_n\\)"],["text"," has "],["math","\\(4n(2n+1)^{d-1}\\)"],["text"," ordered pairs in each direction"]],"source":"$\\Lambda_n$ has $4n(2n+1)^{d-1}$ ordered pairs in each direction"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#card-edge-axis-i","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"There are `2 \u00b7 2n (2n + 1)^{d-1}` ordered pairs in each direction.","title":"$\\Lambda_n$ has $4n(2n+1)^{d-1}$ ordered pairs in each direction"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/edge-f-bi-union","kind":"node","lean":"StatMech.FK.EdgeCount.edgeF_biUnion","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ordered neighbour pairs of "],["math","\\(\\Lambda_n\\)"],["text"," as a union over directions"]],"source":"Ordered neighbour pairs of $\\Lambda_n$ as a union over directions"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#edge-f-bi-union","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The ordered nearest-neighbor pairs of `\u039b_n` are the union over directions.","title":"Ordered neighbour pairs of $\\Lambda_n$ as a union over directions"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/edge-axis-disjoint","kind":"node","lean":"StatMech.FK.EdgeCount.edgeAxis_disjoint","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#edge-axis-disjoint","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Pairs in different directions are disjoint.","title":"Ordered pairs in different directions are disjoint"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/edge-card-eq","kind":"node","lean":"StatMech.FK.EdgeCount.edgeCard_eq","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\Lambda_n\\)"],["text"," has "],["math","\\(4dn(2n+1)^{d-1}\\)"],["text"," ordered nearest-neighbour pairs"]],"source":"$\\Lambda_n$ has $4dn(2n+1)^{d-1}$ ordered nearest-neighbour pairs"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#edge-card-eq","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The box `\u039b_n` has `d \u00b7 2 \u00b7 2n (2n + 1)^{d-1}` ordered nearest-neighbor pairs.","title":"$\\Lambda_n$ has $4dn(2n+1)^{d-1}$ ordered nearest-neighbour pairs"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/box-graph-edge-card","kind":"node","lean":"StatMech.FK.EdgeCount.boxGraph_edgeCard","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\Lambda_n\\)"],["text"," has "],["math","\\(d \\cdot 2n(2n+1)^{d-1}\\)"],["text"," edges"]],"source":"$\\Lambda_n$ has $d \\cdot 2n(2n+1)^{d-1}$ edges"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#box-graph-edge-card","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The box `\u039b_n` has `d \u00b7 2n (2n + 1)^{d-1}` edges.","title":"$\\Lambda_n$ has $d \\cdot 2n(2n+1)^{d-1}$ edges"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-vshift","kind":"node","lean":"StatMech.FK.qp_vshift","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation vector onto an orthant of "],["math","\\(\\Lambda_{2K+1}\\)"]],"source":"Translation vector onto an orthant of $\\Lambda_{2K+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-vshift","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The translation vector `(\u00b1(K + 1))_i` that moves `\u039b_K` onto the orthant with sign pattern `s`.","title":"Translation vector onto an orthant of $\\Lambda_{2K+1}$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-nat-abs-iff","kind":"node","lean":"StatMech.FK.qp_natAbs_iff","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(|x| \\le n\\)"],["text"," iff "],["math","\\(-n \\le x \\le n\\)"],["text"," for an integer "],["math","\\(x\\)"]],"source":"$|x| \\le n$ iff $-n \\le x \\le n$ for an integer $x$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-nat-abs-iff","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"`|x| \u2264 n` for an integer, written with `natAbs`.","title":"$|x| \\le n$ iff $-n \\le x \\le n$ for an integer $x$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-box-tag","kind":"node","lean":"StatMech.FK.qp_boxTag","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Orthant tag of a site of "],["math","\\(\\Lambda_{2K+1}\\)"]],"source":"Orthant tag of a site of $\\Lambda_{2K+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-box-tag","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The orthant tag of a site of `\u039b_{2K+1}`: `none` on a coordinate hyperplane, otherwise its sign pattern.","title":"Orthant tag of a site of $\\Lambda_{2K+1}$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-box-tag-some-iff","kind":"node","lean":"StatMech.FK.qp_boxTag_some_iff","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-box-tag-some-iff","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"A site has tag `s` iff each coordinate lies in `[1, 2K + 1]` or `[-(2K + 1), -1]` according to `s`.","title":"Coordinate description of the sites with a given orthant tag"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-quad","kind":"node","lean":"StatMech.FK.qp_quad\u03c6","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation of "],["math","\\(\\Lambda_K\\)"],["text"," onto an orthant block of "],["math","\\(\\Lambda_{2K+1}\\)"]],"source":"Translation of $\\Lambda_K$ onto an orthant block of $\\Lambda_{2K+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-quad","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The translation of `\u039b_K` onto the sites of `\u039b_{2K+1}` with tag `s`.","title":"Translation of $\\Lambda_K$ onto an orthant block of $\\Lambda_{2K+1}$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-block-fk-z","kind":"node","lean":"StatMech.FK.qp_block_fkZ","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Orthant blocks have the FK partition function of "],["math","\\(\\Lambda_K\\)"]],"source":"Orthant blocks have the FK partition function of $\\Lambda_K$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-block-fk-z","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Each orthant block of `\u039b_{2K+1}` has the FK partition function of `\u039b_K`.","title":"Orthant blocks have the FK partition function of $\\Lambda_K$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-jset","kind":"node","lean":"StatMech.FK.qp_Jset","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hyperplane set of "],["math","\\(\\Lambda_{2K+1}\\)"]],"source":"Hyperplane set of $\\Lambda_{2K+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-jset","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The hyperplane set: sites of `\u039b_{2K+1}` with some coordinate equal to `0`.","title":"Hyperplane set of $\\Lambda_{2K+1}$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-jslab","kind":"node","lean":"StatMech.FK.qp_Jslab","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinate slab of "],["math","\\(\\Lambda_{2K+1}\\)"]],"source":"Coordinate slab of $\\Lambda_{2K+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-jslab","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The sites of `\u039b_{2K+1}` with coordinate `i` equal to `0`.","title":"Coordinate slab of $\\Lambda_{2K+1}$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-jset-subset-bi-union","kind":"node","lean":"StatMech.FK.qp_Jset_subset_biUnion","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-jset-subset-bi-union","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The hyperplane set is the union of the coordinate slabs.","title":"Hyperplane set lies in the union of the coordinate slabs"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-jslab-card","kind":"node","lean":"StatMech.FK.qp_Jslab_card","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","A coordinate slab has at most "],["math","\\((4K+3)^{d-1}\\)"],["text"," sites"]],"source":"A coordinate slab has at most $(4K+3)^{d-1}$ sites"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-jslab-card","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Each coordinate slab has at most `(4K + 3)^{d-1}` sites.","title":"A coordinate slab has at most $(4K+3)^{d-1}$ sites"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-jset-card-le","kind":"node","lean":"StatMech.FK.qp_Jset_card_le","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","The hyperplane set has at most "],["math","\\(d(4K+3)^{d-1}\\)"],["text"," sites"]],"source":"The hyperplane set has at most $d(4K+3)^{d-1}$ sites"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-jset-card-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The hyperplane set has at most `d (4K + 3)^{d-1}` sites.","title":"The hyperplane set has at most $d(4K+3)^{d-1}$ sites"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-box-graph-degree-le","kind":"node","lean":"StatMech.FK.qp_boxGraph_degree_le","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vertex degrees in a box are at most "],["math","\\(2d\\)"]],"source":"Vertex degrees in a box are at most $2d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-box-graph-degree-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Every vertex of a box has degree at most `2d`. See also `MathlibExt...PlanarTopology.degree_le` for `\u2124^d`.","title":"Vertex degrees in a box are at most $2d$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-straddle-endpoint-in-j","kind":"node","lean":"StatMech.FK.qp_straddle_endpoint_in_J","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-straddle-endpoint-in-j","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"An edge leaving an orthant has an endpoint on a coordinate hyperplane.","title":"An edge leaving an orthant ends on a coordinate hyperplane"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-tag-none-iff","kind":"node","lean":"StatMech.FK.qp_tag_none_iff","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-tag-none-iff","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"A site has tag `none` iff it lies on a coordinate hyperplane.","title":"Untagged sites are exactly those on a coordinate hyperplane"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-straddle-endpoint-in-j-all","kind":"node","lean":"StatMech.FK.qp_straddle_endpoint_in_J_all","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-straddle-endpoint-in-j-all","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"An edge between sites of different tags has an endpoint on a coordinate hyperplane.","title":"An edge between differently tagged sites meets a coordinate hyperplane"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-straddle-subset-all","kind":"node","lean":"StatMech.FK.qp_straddle_subset_all","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-straddle-subset-all","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The edges straddling a tag are incident to the hyperplane set.","title":"Edges straddling a tag are incident to the hyperplane set"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-straddle-card-le-all","kind":"node","lean":"StatMech.FK.qp_straddle_card_le_all","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","At most "],["math","\\(2d|J|\\)"],["text"," edges straddle an orthant tag"]],"source":"At most $2d|J|$ edges straddle an orthant tag"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-straddle-card-le-all","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"At most `2d |J|` edges straddle a tag.","title":"At most $2d|J|$ edges straddle an orthant tag"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-blk-none-edge-le","kind":"node","lean":"StatMech.FK.qp_blkNone_edge_le","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","The hyperplane block has at most "],["math","\\(2d|J|\\)"],["text"," edges"]],"source":"The hyperplane block has at most $2d|J|$ edges"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-blk-none-edge-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The hyperplane block has at most `2d |J|` edges.","title":"The hyperplane block has at most $2d|J|$ edges"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-rest-guniv-iso","kind":"node","lean":"StatMech.FK.qp_restGUnivIso","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Isomorphism from the all-tag restriction to "],["math","\\(\\Lambda_{2K+1}\\)"]],"source":"Isomorphism from the all-tag restriction to $\\Lambda_{2K+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-rest-guniv-iso","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The restriction of `\u039b_{2K+1}` to all tags is `\u039b_{2K+1}`.","title":"Isomorphism from the all-tag restriction to $\\Lambda_{2K+1}$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-rest-g-univ-fk-z","kind":"node","lean":"StatMech.FK.qp_restG_univ_fkZ","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","The all-tag restriction has the partition function of "],["math","\\(\\Lambda_{2K+1}\\)"]],"source":"The all-tag restriction has the partition function of $\\Lambda_{2K+1}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-rest-g-univ-fk-z","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The restriction to all tags has the partition function of `\u039b_{2K+1}`.","title":"The all-tag restriction has the partition function of $\\Lambda_{2K+1}$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-box-neglog-doubling","kind":"node","lean":"StatMech.FK.qp_box_neglog_doubling","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Doubling inequality for "],["math","\\(-\\log Z\\)"],["text"," of the free "],["math","\\(q = 2\\)"],["text"," FK model"]],"source":"Doubling inequality for $-\\log Z$ of the free $q = 2$ FK model"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-box-neglog-doubling","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The doubling inequality `u(2K + 1) \u2264 2^d u(K) + I_K \u00b7 (-log (1 - p))` for `u = -log Z` of the free `q = 2` FK model, with `I_K = (2^d + 2) \u00b7 2d |J|`.","title":"Doubling inequality for $-\\log Z$ of the free $q = 2$ FK model"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-edge-le","kind":"node","lean":"StatMech.FK.qp_edge_le","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\Lambda_{2K+1}\\)"],["text"," has at least "],["math","\\(2^d\\)"],["text"," times as many edges as "],["math","\\(\\Lambda_K\\)"]],"source":"$\\Lambda_{2K+1}$ has at least $2^d$ times as many edges as $\\Lambda_K$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-edge-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The `2^d` orthant copies of `\u039b_K` have at most as many edges as `\u039b_{2K+1}`.","title":"$\\Lambda_{2K+1}$ has at least $2^d$ times as many edges as $\\Lambda_K$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-edge-gap-le","kind":"node","lean":"StatMech.FK.qp_edge_gap_le","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bound on the edge gap "],["math","\\(|E(\\Lambda_{2K+1})| - 2^d |E(\\Lambda_K)|\\)"]],"source":"Bound on the edge gap $|E(\\Lambda_{2K+1})| - 2^d |E(\\Lambda_K)|$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-edge-gap-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The edge-number gap `|E(\u039b_{2K+1})| - 2^d |E(\u039b_K)|` is at most `d (d + 2)(4K + 3)^{d-1}`.","title":"Bound on the edge gap $|E(\\Lambda_{2K+1})| - 2^d |E(\\Lambda_K)|$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-edge-lower","kind":"node","lean":"StatMech.FK.qp_edge_lower","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower bound "],["math","\\(n(2n+1)^{d-1}\\)"],["text"," on the number of edges of "],["math","\\(\\Lambda_n\\)"]],"source":"Lower bound $n(2n+1)^{d-1}$ on the number of edges of $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-edge-lower","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The box `\u039b_n` has at least `n (2n + 1)^{d-1}` edges.","title":"Lower bound $n(2n+1)^{d-1}$ on the number of edges of $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-summable-ratio","kind":"node","lean":"StatMech.FK.qp_summable_ratio","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-summable-ratio","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Corrections of order `K_j^{d-1}` divided by `|E(\u039b_{K_{j+1}})|` are summable along `K_j = 2^{j+1} - 1`.","title":"Summability of doubling corrections relative to edge counts"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-additive-doubling-bound","kind":"node","lean":"StatMech.FK.qp_additiveDoublingBound","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Additive doubling bound for "],["math","\\(-\\log Z\\)"],["text"," of the free "],["math","\\(q = 2\\)"],["text"," FK model"]],"source":"Additive doubling bound for $-\\log Z$ of the free $q = 2$ FK model"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-additive-doubling-bound","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The additive doubling bound for `u = -log Z` of the free `q = 2` FK model along the doubling boxes `K_j = 2^{j+1} - 1`, in every dimension `d \u2265 1`.","title":"Additive doubling bound for $-\\log Z$ of the free $q = 2$ FK model"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/qp-additive-box-fekete-data","kind":"node","lean":"StatMech.FK.qp_additiveBoxFeketeData","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#qp-additive-box-fekete-data","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The box Fekete data for the free `q = 2` FK model from a uniform bound on `u / |E|` along doubling boxes and an oscillation bound between general and doubling boxes.","title":"Box Fekete data from a uniform bound and an oscillation bound"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-fk-weight-le","kind":"node","lean":"StatMech.Walls.fkc_fkWeight_le","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Random-cluster weight is at most "],["math","\\(q^{|V|}\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Random-cluster weight is at most $q^{|V|}$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-fk-weight-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"For `q \u2265 1` every random-cluster weight is at most `q ^ |V|`. The case `q = 2` is `ecz_fkWeight_le`.","title":"Random-cluster weight is at most $q^{|V|}$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-fk-zedge-le","kind":"node","lean":"StatMech.Walls.fkc_fkZEdge_le","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edge partition function is at most "],["math","\\(2^{|E|} q^{|V|}\\)"],["text"," for "],["math","\\(q \\ge 1\\)"]],"source":"Edge partition function is at most $2^{|E|} q^{|V|}$ for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-fk-zedge-le","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"For `q \u2265 1` the edge partition function is at most `2 ^ |E| q ^ |V|`. The case `q = 2` is `ecz_fkZEdge_le`.","title":"Edge partition function is at most $2^{|E|} q^{|V|}$ for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-edge-free-energy","kind":"node","lean":"StatMech.Walls.fkc_edgeFreeEnergy","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-edge-free-energy","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The edge-normalized free energy of the random-cluster model on the box of side `2n + 1` in dimension `d`, in the logit parameter `s`: `log Z^E / |E| + log (1 + e\u02e2)` with `Z^E = ecz_fkZEdge (boxGraph d n) (fsc_logistic s) q`.","title":"Edge-normalized box free energy of the random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-rq-bdd","kind":"node","lean":"StatMech.Walls.fkc_rq_bdd","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Uniform upper bound on the box free energy for "],["math","\\(d, n, q \\ge 1\\)"]],"source":"Uniform upper bound on the box free energy for $d, n, q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-rq-bdd","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"For `d, n \u2265 1` and `q \u2265 1` the box free energy is at most `log 2 + 2 log q + log (1 + e\u02e2)`. The case `q = 2` is `ecz_edgeFE_le_const`.","title":"Uniform upper bound on the box free energy for $d, n, q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-u","kind":"node","lean":"StatMech.Walls.fkc_u","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box free energy "],["math","\\(-\\log Z^E\\)"],["text"," of the random-cluster model"]],"source":"Box free energy $-\\log Z^E$ of the random-cluster model"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-u","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The box free energy `-log Z^E` of the random-cluster model on `[-n, n]^d` at edge parameter `fsc_logistic s`, over configurations closed off the box edges.","title":"Box free energy $-\\log Z^E$ of the random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-f","kind":"node","lean":"StatMech.Walls.fkc_f","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-f","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The per-edge box free energy `fkc_u / |E|`.","title":"Per-edge box free energy"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-box-neglog-doubling","kind":"node","lean":"StatMech.Walls.fkc_box_neglog_doubling","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Doubling bound for the box free energy when "],["math","\\(q \\ge 1\\)"]],"source":"Doubling bound for the box free energy when $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-box-neglog-doubling","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Doubling bound: the free energy of the box of side `2(2K + 1) + 1` is at most `2^d` times that of side `2K + 1` plus `-log (1 - p)` times the number of boundary edges.","title":"Doubling bound for the box free energy when $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-box-van-hove-subadd","kind":"node","lean":"StatMech.Walls.fkc_box_vanHove_subadd","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Van Hove subadditivity of the box free energy for "],["math","\\(q \\ge 1\\)"]],"source":"Van Hove subadditivity of the box free energy for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-box-van-hove-subadd","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Van Hove subadditivity: packing `T^d` copies of the side-`(2m + 1)` box into the side-`(2n + 1)` box, the free energy is at most `T^d fkc_u m` plus `-log (1 - p)` per remaining edge.","title":"Van Hove subadditivity of the box free energy for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-additive-doubling-bound","kind":"node","lean":"StatMech.Walls.fkc_AdditiveDoublingBound","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Doubling-bound property of the box free energy along "],["math","\\(K_j\\)"]],"source":"Doubling-bound property of the box free energy along $K_j$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-additive-doubling-bound","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"A doubling bound along the dyadic sides `agl_K j` with nonnegative errors `I j` whose per-edge values are summable.","title":"Doubling-bound property of the box free energy along $K_j$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-additive-doubling-bound-2","kind":"node","lean":"StatMech.Walls.fkc_additiveDoublingBound","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box free-energy doubling bound along "],["math","\\(K_j\\)"],["text"," holds for "],["math","\\(d, q \\ge 1\\)"]],"source":"Box free-energy doubling bound along $K_j$ holds for $d, q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-additive-doubling-bound-2","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The doubling bound holds for all `q \u2265 1`.","title":"Box free-energy doubling bound along $K_j$ holds for $d, q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-dyadic-f-tendsto","kind":"node","lean":"StatMech.Walls.fkc_dyadic_f_tendsto","parent":48605,"reading_title":{"format":"inline-parts-v1","parts":[["text","Per-edge free energy converges along "],["math","\\(K_j\\)"],["text"," for "],["math","\\(d, q \\ge 1\\)"]],"source":"Per-edge free energy converges along $K_j$ for $d, q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-dyadic-f-tendsto","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The per-edge free energy converges along the dyadic sides `agl_K j`.","title":"Per-edge free energy converges along $K_j$ for $d, q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-per-edge-density-converges","kind":"node","lean":"StatMech.Walls.fkc_PerEdgeDensityConverges","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-per-edge-density-converges","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The per-edge box free energy `fkc_f d s q n` converges as `n \u2192 \u221e`.","title":"Convergence of the per-edge box free energy"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-g","kind":"node","lean":"StatMech.Walls.fkc_g","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-g","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"The limit of the van Hove upper bound for a fixed side `2m + 1`.","title":"Limit of the van Hove upper bound for a fixed box side"},{"id":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/fkc-per-edge-density-converges-2","kind":"node","lean":"StatMech.Walls.fkc_perEdgeDensityConverges","parent":48605,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model/index.html#fkc-per-edge-density-converges-2","scope":"StatisticalMechanics/random-cluster-model/free-energy-of-the-random-cluster-model","status":"fully_proved","summary":"Van Hove limit of the random-cluster free energy: for `d \u2265 1` and `q \u2265 1`, the per-edge free energy on the box `[-n, n]^d` converges. See D. Ruelle, *Statistical Mechanics: Rigorous Results* (1969), and G. Grimmett, *The Random-Cluster Model* (2006), Chapter 4.","title":"Van Hove limit"},{"data_url":"scopes/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust.json","id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","kind":"scope","navigation":"module","parent":47781,"result":false,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","title":"Per-box agreement of free and wired random-cluster measures"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fbd-off-centre-carrier","kind":"node","lean":"StatMech.FK.fbd_offCentreCarrier","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fbd-off-centre-carrier","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The difference between the wired marginal of `eb'` on `\u039b_{N+k}` and on the translate of `\u039b_{N+k}` by `v`, evaluated at the translated edge.","title":"Wired-marginal difference between a box and its translate"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fk-unique-of-edge-marg-eq","kind":"node","lean":"StatMech.FK.fk_unique_of_edgeMarg_eq","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fk-unique-of-edge-marg-eq","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Two monotonically coupled probability vectors with equal edge marginals agree on increasing events. Restatement of `eventMass_eq_of_edgeMarg_eq` (`EdgeMarginal`), kept because other modules use this name.","title":"Monotone-coupled laws with equal edge marginals agree on up-sets"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/countable-edge-marg-ne","kind":"node","lean":"StatMech.FK.countable_edgeMarg_ne","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#countable-edge-marg-ne","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"If `a` agrees with a monotone `b` at every continuity point of `b`, then `{p | a p \u2260 b p}` is countable.","title":"Agreement with a monotone map at continuity points fails countably"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/phi0-eq-phi1-off-countable","kind":"node","lean":"StatMech.FK.phi0_eq_phi1_off_countable","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#phi0-eq-phi1-off-countable","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Parametric agreement: if for each `p` there is a monotone coupling of `\u03bc p` and `\u03bd p`, and edge marginals agree at every continuity point of the monotone map `p \u21a6 \u03bd_p(\u03c9(e) = 1)`, then `\u03bc p` and `\u03bd p` agree on increasing events for all `p` outside a countable set.","title":"Parametric agreement on increasing events off a countable set"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/countable-phi-ne","kind":"node","lean":"StatMech.FK.countable_phi_ne","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#countable-phi-ne","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Under the hypotheses of `phi0_eq_phi1_off_countable`, the set of `p` for which some increasing event separates `\u03bc p` and `\u03bd p` is countable.","title":"Countably many parameters admit a separating increasing event"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fk-edge-open-event","kind":"node","lean":"StatMech.FK.fkEdgeOpenEvent","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fk-edge-open-event","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The event that the edge `e` is open.","title":"Event that an edge is open"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/wired-edge-density","kind":"node","lean":"StatMech.FK.wiredEdgeDensity","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#wired-edge-density","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The edge density `\u03c6^1_{p,q}(e open)` of the wired infinite-volume random-cluster measure on `\u2124^d`, set to `0` outside `p \u2208 (0, 1)`, `q > 0`.","title":"Wired infinite-volume edge density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/free-edge-density","kind":"node","lean":"StatMech.FK.freeEdgeDensity","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#free-edge-density","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The edge density `\u03c6^0_{p,q}(e open)` of the free infinite-volume random-cluster measure on `\u2124^d`, set to `0` outside `p \u2208 (0, 1)`, `q > 0`.","title":"Free infinite-volume edge density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/doc-gap","kind":"node","lean":"StatMech.FK.docGap","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired\u2013free gap in the "],["math","\\(q = 2\\)"],["text"," density of a box edge"]],"source":"Wired\u2013free gap in the $q = 2$ density of a box edge"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#doc-gap","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The gap `\u03c6^1_{p,2}(eb open) - \u03c6^0_{p,2}(eb open)` between the wired and free infinite-volume densities of the edge `eb` of `\u039b_N`.","title":"Wired\u2013free gap in the $q = 2$ density of a box edge"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-edge-card-shell-fraction-le","kind":"node","lean":"StatMech.FK.ocs_edgeCard_shell_fraction_le","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shell edge fraction is at most "],["math","\\(4d^2(n-m)/n\\)"]],"source":"Shell edge fraction is at most $4d^2(n-m)/n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-edge-card-shell-fraction-le","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The fraction of ordered edges of `\u039b_n` outside `\u039b_m` is at most `4 d\u00b2 (n - m)/n`.","title":"Shell edge fraction is at most $4d^2(n-m)/n$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-edge-finset-shell-fraction-tendsto-z","kind":"node","lean":"StatMech.FK.ocs_edgeFinset_shell_fraction_tendsto_zero","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vanishing shell edge fraction when "],["math","\\((n - m(n))/n \\to 0\\)"]],"source":"Vanishing shell edge fraction when $(n - m(n))/n \\to 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-edge-finset-shell-fraction-tendsto-z","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"If `(n - m(n))/n \u2192 0`, the edges of `\u039b_n` outside `\u039b_{m(n)}` form a vanishing fraction of the edges of `\u039b_n`.","title":"Vanishing shell edge fraction when $(n - m(n))/n \\to 0$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-free-off-centre-lower","kind":"node","lean":"StatMech.FK.ocs_free_offCentre_lower","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-free-off-centre-lower","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"For `u, v` adjacent in `\u039b_{n-R}`, the free marginal in `\u039b_n` of the edge `{u, v}` is at least the free marginal in `\u039b_R` of the recentered edge `{0, v - u}`.","title":"Free off-centre lower bound by the recentred small-box marginal"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-wired-off-centre-upper","kind":"node","lean":"StatMech.FK.ocs_wired_offCentre_upper","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-wired-off-centre-upper","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"For `u, v` adjacent in `\u039b_{n-R-1}` and `R \u2265 1`, the wired marginal in `\u039b_n` of `{u, v}` is at most the wired marginal in `\u039b_R` of the recentered edge `{0, v - u}`.","title":"Wired off-centre upper bound by the recentred small-box marginal"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-edge-dir-cand-map","kind":"node","lean":"StatMech.FK.ocs_edge_dir_candMap","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-edge-dir-cand-map","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The difference of adjacent box vertices is a signed unit vector. See also `rot_edge_dir`.","title":"Adjacent box vertices differ by a signed unit vector"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-unit-edge","kind":"node","lean":"StatMech.FK.ocs_unitEdge","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit edge "],["math","\\(\\{0, \\pm e_j\\}\\)"],["text"," of a box"]],"source":"Unit edge $\\{0, \\pm e_j\\}$ of a box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-unit-edge","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The unit edge `{0, \u00b1e_j}` of the box `\u039b_R`, `R \u2265 1`, indexed by `(j, sign)`.","title":"Unit edge $\\{0, \\pm e_j\\}$ of a box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-unit-edge-marg-tendsto","kind":"node","lean":"StatMech.FK.ocs_unitEdge_marg_tendsto","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-unit-edge-marg-tendsto","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The free marginal of a unit edge in `\u039b_{\u230a\u221an\u230b}` converges to its free `q = 2` density.","title":"Free unit-edge marginals converge to the free density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-free-rotation-residue","kind":"node","lean":"StatMech.FK.ocs_FreeRotationResidue","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-free-rotation-residue","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The free rotation residue at tilt `t` for the box pair `e'`: every lattice edge of every box has the free `q = 2` density of `e'` at `p = logistic t`.","title":"Free rotation residue for a box pair"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-unit-marg","kind":"node","lean":"StatMech.FK.ocs_unitMarg","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-unit-marg","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The free marginal at `p = logistic t` of the unit edge `(j, sign)` in `\u039b_{\u230a\u221an\u230b}`, and `0` when `\u230a\u221an\u230b = 0`.","title":"Free unit-edge marginal in the square-root box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-free-delta","kind":"node","lean":"StatMech.FK.ocs_freeDelta","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-free-delta","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The total deviation `\u2211_{(j, sign)} |density(e') - unit-edge marginal|` of the free unit-edge marginals in `\u039b_{\u230a\u221an\u230b}` from the free density of `e'`.","title":"Total deviation of free unit-edge marginals from the density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-bulk-set","kind":"node","lean":"StatMech.FK.ocs_bulkSet","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bulk edges of "],["math","\\(\\Lambda_n\\)"]],"source":"Bulk edges of $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-bulk-set","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The bulk edges of `\u039b_n`: the edges of the inner box `\u039b_{n - \u230a\u221an\u230b}`.","title":"Bulk edges of $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-free-bulk-lower","kind":"node","lean":"StatMech.FK.ocs_free_bulk_lower","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-free-bulk-lower","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Every bulk edge of `\u039b_n` has free marginal at least `density(e') - ocs_freeDelta`.","title":"Lower bound on free bulk edge marginals"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-free-off-centre-sandwich-of-rotation","kind":"node","lean":"StatMech.FK.ocs_freeOffCentreSandwich_of_rotation","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-free-off-centre-sandwich-of-rotation","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The free off-centre sandwich from the free rotation residues for all pairs `e'` of `\u039b_N`.","title":"Free off-centre sandwich from free rotation residues"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-unit-edge-wired-marg-tendsto","kind":"node","lean":"StatMech.FK.ocs_unitEdge_wired_marg_tendsto","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-unit-edge-wired-marg-tendsto","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired marginal of a unit edge in `\u039b_{\u230a\u221an\u230b}` converges to its wired `q = 2` density.","title":"Wired unit-edge marginals converge to the wired density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-wired-rotation-residue","kind":"node","lean":"StatMech.FK.ocs_WiredRotationResidue","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-wired-rotation-residue","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired rotation residue at tilt `t` for the box pair `e'`: every lattice edge of every box has the wired `q = 2` density of `e'` at `p = logistic t`.","title":"Wired rotation residue for a box pair"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-unit-wired-marg","kind":"node","lean":"StatMech.FK.ocs_unitWiredMarg","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-unit-wired-marg","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired marginal at `p = logistic t` of the unit edge `(j, sign)` in `\u039b_{\u230a\u221an\u230b}`, and `0` when `\u230a\u221an\u230b = 0`.","title":"Wired unit-edge marginal in the square-root box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-wired-delta","kind":"node","lean":"StatMech.FK.ocs_wiredDelta","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-wired-delta","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The total deviation of the wired unit-edge marginals in `\u039b_{\u230a\u221an\u230b}` from the wired density of `e'`.","title":"Total deviation of wired unit-edge marginals from the density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-wired-bulk-set","kind":"node","lean":"StatMech.FK.ocs_wiredBulkSet","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired bulk edges of "],["math","\\(\\Lambda_n\\)"]],"source":"Wired bulk edges of $\\Lambda_n$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-wired-bulk-set","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired bulk edges of `\u039b_n`: the edges of the inner box `\u039b_{n - \u230a\u221an\u230b - 1}`.","title":"Wired bulk edges of $\\Lambda_n$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-wired-bulk-upper","kind":"node","lean":"StatMech.FK.ocs_wired_bulk_upper","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-wired-bulk-upper","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Every wired bulk edge of `\u039b_n` has wired marginal at most `density(e') + ocs_wiredDelta`.","title":"Upper bound on wired bulk edge marginals"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ocs-wired-off-centre-sandwich-of-rotatio","kind":"node","lean":"StatMech.FK.ocs_wiredOffCentreSandwich_of_rotation","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ocs-wired-off-centre-sandwich-of-rotatio","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired off-centre sandwich from the wired rotation residues for all pairs `e'` of `\u039b_N`.","title":"Wired off-centre sandwich from wired rotation residues"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/gec-free-sandwich-of-rotation","kind":"node","lean":"StatMech.FK.gec_freeSandwich_of_rotation","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," bulk sandwich from equal free edge densities"]],"source":"Free $q = 2$ bulk sandwich from equal free edge densities"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#gec-free-sandwich-of-rotation","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"If every free `q = 2` edge density equals that of `e'`, then each box `\u039b_n` has a bulk set of edges with that density whose free marginals are at least the density minus `\u03b4_n \u2192 0`, and the non-bulk edge fraction tends to zero.","title":"Free $q = 2$ bulk sandwich from equal free edge densities"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/gec-wired-sandwich-of-rotation","kind":"node","lean":"StatMech.FK.gec_wiredSandwich_of_rotation","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," bulk sandwich from equal wired edge densities"]],"source":"Wired $q = 2$ bulk sandwich from equal wired edge densities"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#gec-wired-sandwich-of-rotation","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"If every wired `q = 2` edge density equals that of `e'`, then each box `\u039b_n` has a bulk set of edges with that density whose wired marginals are at most the density plus `\u03b4_n \u2192 0`, and the non-bulk edge fraction tends to zero.","title":"Wired $q = 2$ bulk sandwich from equal wired edge densities"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/gec-free-bulk-of-sandwich","kind":"node","lean":"StatMech.FK.gec_freeBulk_of_sandwich","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-sided free "],["math","\\(q = 2\\)"],["text"," bulk bounds from the free sandwich"]],"source":"Two-sided free $q = 2$ bulk bounds from the free sandwich"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#gec-free-bulk-of-sandwich","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The free sandwich gives two-sided bulk bounds: free marginals within `\u03b4_n` of the density, since free marginals increase to the density.","title":"Two-sided free $q = 2$ bulk bounds from the free sandwich"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/gec-wired-bulk-of-sandwich","kind":"node","lean":"StatMech.FK.gec_wiredBulk_of_sandwich","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Two-sided wired "],["math","\\(q = 2\\)"],["text"," bulk bounds from the wired sandwich"]],"source":"Two-sided wired $q = 2$ bulk bounds from the wired sandwich"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#gec-wired-bulk-of-sandwich","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired sandwich gives two-sided bulk bounds: wired marginals within `\u03b4_n` of the density, since wired marginals decrease to the density.","title":"Two-sided wired $q = 2$ bulk bounds from the wired sandwich"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-perm-equiv","kind":"node","lean":"StatMech.FK.rot_permEquiv","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Coordinate permutation of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Coordinate permutation of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-perm-equiv","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The permutation of `\u2124^d` permuting coordinates by `\u03c0`: `(x \u2218 \u03c0\u207b\u00b9)_i = x_{\u03c0\u207b\u00b9 i}`.","title":"Coordinate permutation of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-perm-box-sym","kind":"node","lean":"StatMech.FK.rot_permBoxSym","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-perm-box-sym","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"A coordinate permutation as a box symmetry.","title":"Coordinate permutation as a box symmetry"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-free-density-box-sym","kind":"node","lean":"StatMech.FK.rot_free_density_boxSym","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box symmetries preserve the free "],["math","\\(q = 2\\)"],["text"," density of any pair of sites"]],"source":"Box symmetries preserve the free $q = 2$ density of any pair of sites"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-free-density-box-sym","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Box symmetries preserve the free `q = 2` density of any edge of `\u2124^d`.","title":"Box symmetries preserve the free $q = 2$ density of any pair of sites"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-wired-density-box-sym","kind":"node","lean":"StatMech.FK.rot_wired_density_boxSym","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box symmetries preserve the wired "],["math","\\(q = 2\\)"],["text"," density of any pair of sites"]],"source":"Box symmetries preserve the wired $q = 2$ density of any pair of sites"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-wired-density-box-sym","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Box symmetries preserve the wired `q = 2` density of any edge of `\u2124^d`.","title":"Box symmetries preserve the wired $q = 2$ density of any pair of sites"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-ref-unit-edge","kind":"node","lean":"StatMech.FK.rot_refUnitEdge","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Reference edge "],["math","\\(\\{0, e_0\\}\\)"],["text"," of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Reference edge $\\{0, e_0\\}$ of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-ref-unit-edge","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The reference edge `{0, e_0}` of `\u2124^d`.","title":"Reference edge $\\{0, e_0\\}$ of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-free-centred-unit-const","kind":"node","lean":"StatMech.FK.rot_free_centred_unit_const","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit edges at the origin have the free density of "],["math","\\(\\{0, e_0\\}\\)"]],"source":"Unit edges at the origin have the free density of $\\{0, e_0\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-free-centred-unit-const","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Every unit edge `{0, \u00b1e_j}` has the free density of `{0, e_0}`.","title":"Unit edges at the origin have the free density of $\\{0, e_0\\}$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-wired-centred-unit-const","kind":"node","lean":"StatMech.FK.rot_wired_centred_unit_const","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit edges at the origin have the wired density of "],["math","\\(\\{0, e_0\\}\\)"]],"source":"Unit edges at the origin have the wired density of $\\{0, e_0\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-wired-centred-unit-const","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Every unit edge `{0, \u00b1e_j}` has the wired density of `{0, e_0}`.","title":"Unit edges at the origin have the wired density of $\\{0, e_0\\}$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-edge-dir","kind":"node","lean":"StatMech.FK.rot_edge_dir","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-edge-dir","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The difference of adjacent sites is a signed unit vector. See also `ocs_edge_dir_candMap`, the same statement for vertices of a box.","title":"Adjacent sites differ by a signed unit vector"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-free-edge-const","kind":"node","lean":"StatMech.FK.rot_free_edge_const","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every lattice edge has the free "],["math","\\(q = 2\\)"],["text"," density of "],["math","\\(\\{0, e_0\\}\\)"]],"source":"Every lattice edge has the free $q = 2$ density of $\\{0, e_0\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-free-edge-const","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Every nearest-neighbour edge of `\u2124^d` has the free `q = 2` density of `{0, e_0}`.","title":"Every lattice edge has the free $q = 2$ density of $\\{0, e_0\\}$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-wired-edge-const","kind":"node","lean":"StatMech.FK.rot_wired_edge_const","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Every lattice edge has the wired "],["math","\\(q = 2\\)"],["text"," density of "],["math","\\(\\{0, e_0\\}\\)"]],"source":"Every lattice edge has the wired $q = 2$ density of $\\{0, e_0\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-wired-edge-const","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Every nearest-neighbour edge of `\u2124^d` has the wired `q = 2` density of `{0, e_0}`.","title":"Every lattice edge has the wired $q = 2$ density of $\\{0, e_0\\}$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-free-rotation-residue","kind":"node","lean":"StatMech.FK.rot_freeRotationResidue","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","All box edges have the same free "],["math","\\(q = 2\\)"],["text"," density"]],"source":"All box edges have the same free $q = 2$ density"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-free-rotation-residue","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The free rotation residue holds for every lattice edge `e'` of the box `\u039b_N`.","title":"All box edges have the same free $q = 2$ density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/rot-wired-rotation-residue","kind":"node","lean":"StatMech.FK.rot_wiredRotationResidue","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","All box edges have the same wired "],["math","\\(q = 2\\)"],["text"," density"]],"source":"All box edges have the same wired $q = 2$ density"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#rot-wired-rotation-residue","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired rotation residue holds for every lattice edge `e'` of the box `\u039b_N`.","title":"All box edges have the same wired $q = 2$ density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ubd-free-growing-box-collapse","kind":"node","lean":"StatMech.FK.ubd_FreeGrowingBoxCollapse","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free growing-box collapse at "],["math","\\(q = 2\\)"]],"source":"Free growing-box collapse at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ubd-free-growing-box-collapse","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Free growing-box collapse: for every edge `e'` of `\u039b_N` and every `t`, the average free edge density of `\u039b_n` converges to the free infinite-volume density of `e'`.","title":"Free growing-box collapse at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ubd-free-edge-density-mem-icc","kind":"node","lean":"StatMech.FK.ubd_freeEdgeDensity_mem_Icc","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," edge density lies in "],["math","\\((0, 1)\\)"]],"source":"Free $q = 2$ edge density lies in $(0, 1)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ubd-free-edge-density-mem-icc","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The free infinite-volume `q = 2` edge density lies in `[0, 1]`.","title":"Free $q = 2$ edge density lies in $(0, 1)$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ubd-edge-marg-prob-wired-fk-prob-nonneg","kind":"node","lean":"StatMech.FK.ubd_edgeMargProb_wiredFkProb_nonneg","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ubd-edge-marg-prob-wired-fk-prob-nonneg","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Edge marginals of a wired FK measure are nonnegative.","title":"Nonnegativity of wired FK edge marginals"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ubd-edge-marg-prob-wired-fk-prob-le-one","kind":"node","lean":"StatMech.FK.ubd_edgeMargProb_wiredFkProb_le_one","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ubd-edge-marg-prob-wired-fk-prob-le-one","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Edge marginals of a wired FK measure are at most `1`.","title":"Wired FK edge marginals are at most one"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ubd-wired-edge-density-mem-icc","kind":"node","lean":"StatMech.FK.ubd_wiredEdgeDensity_mem_Icc","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," edge density lies in "],["math","\\((0, 1)\\)"]],"source":"Wired $q = 2$ edge density lies in $(0, 1)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ubd-wired-edge-density-mem-icc","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired infinite-volume `q = 2` edge density lies in `[0, 1]`.","title":"Wired $q = 2$ edge density lies in $(0, 1)$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ubd-avg-wired-density-eq-sum","kind":"node","lean":"StatMech.FK.ubd_avgWiredDensity_eq_sum","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Average wired "],["math","\\(q = 2\\)"],["text"," edge density of a box as a mean of marginals"]],"source":"Average wired $q = 2$ edge density of a box as a mean of marginals"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ubd-avg-wired-density-eq-sum","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The average wired edge density of `\u039b_n` is the mean of the wired edge marginals.","title":"Average wired $q = 2$ edge density of a box as a mean of marginals"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ubd-free-uniform-bulk-deviation","kind":"node","lean":"StatMech.FK.ubd_FreeUniformBulkDeviation","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free uniform bulk deviation at "],["math","\\(q = 2\\)"]],"source":"Free uniform bulk deviation at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ubd-free-uniform-bulk-deviation","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Free uniform bulk deviation: for every edge `e'` and `t`, each box `\u039b_n` has a bulk edge set with free marginals within `\u03b4_n \u2192 0` of the free density of `e'`, and the non-bulk edge fraction tends to zero.","title":"Free uniform bulk deviation at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ubd-wired-uniform-bulk-deviation","kind":"node","lean":"StatMech.FK.ubd_WiredUniformBulkDeviation","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired uniform bulk deviation at "],["math","\\(q = 2\\)"]],"source":"Wired uniform bulk deviation at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ubd-wired-uniform-bulk-deviation","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Wired uniform bulk deviation: for every edge `e'` and `t`, each box `\u039b_n` has a bulk edge set with wired marginals within `\u03b4_n \u2192 0` of the wired density of `e'`, and the non-bulk edge fraction tends to zero.","title":"Wired uniform bulk deviation at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fuc-free-mass","kind":"node","lean":"StatMech.FK.fuc_freeMass","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free "],["math","\\(q = 2\\)"],["text"," infinite-volume mass function on a box"]],"source":"Free $q = 2$ infinite-volume mass function on a box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fuc-free-mass","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The mass function on configurations of `\u039b_N` of the free `q = 2` infinite-volume measure at `p = logistic(t)`, restricted to `\u039b_N`.","title":"Free $q = 2$ infinite-volume mass function on a box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fuc-wired-mass","kind":"node","lean":"StatMech.FK.fuc_wiredMass","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired "],["math","\\(q = 2\\)"],["text"," infinite-volume mass function on a box"]],"source":"Wired $q = 2$ infinite-volume mass function on a box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fuc-wired-mass","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The mass function on configurations of `\u039b_N` of the wired `q = 2` infinite-volume measure at `p = logistic(t)`, restricted to `\u039b_N`.","title":"Wired $q = 2$ infinite-volume mass function on a box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fuc-homogeneous-free-energy-data","kind":"node","lean":"StatMech.FK.fuc_HomogeneousFreeEnergyData","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Homogeneous free-energy data on a box at "],["math","\\(q = 2\\)"]],"source":"Homogeneous free-energy data on a box at $q = 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fuc-homogeneous-free-energy-data","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"There is a convex `g` whose right and left derivatives at `t` are the wired and free `q = 2` edge densities at `p = logistic(t)`, for every edge of `\u039b_N`.","title":"Homogeneous free-energy data on a box at $q = 2$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fuc-fk-uniqueness","kind":"node","lean":"StatMech.FK.fuc_fk_uniqueness","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free-energy data give "],["math","\\(q = 2\\)"],["text"," box uniqueness off countably many "],["math","\\(t\\)"]],"source":"Free-energy data give $q = 2$ box uniqueness off countably many $t$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fuc-fk-uniqueness","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Given `fuc_HomogeneousFreeEnergyData`, the free and wired `q = 2` measures agree on every increasing event of `\u039b_N` for all but countably many `t`.","title":"Free-energy data give $q = 2$ box uniqueness off countably many $t$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fpe2-edge-marg-prob-half-of-toggle-invar","kind":"node","lean":"StatMech.FK.fpe2_edgeMargProb_half_of_toggle_invariant","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fpe2-edge-marg-prob-half-of-toggle-invar","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"A coordinate whose toggle preserves a probability mass function is open with probability `1/2`.","title":"Toggle-invariant coordinates are open with probability one half"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fpe2-per-edge-g","kind":"node","lean":"StatMech.FK.fpe2_perEdgeG","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fpe2-per-edge-g","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The per-coordinate free energy on `\u039b_N`: `G` on edges of the box graph and `s \u21a6 s/2` on non-edges.","title":"Per-coordinate free energy on a box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fpe2-genuine-free-is-left-deriv","kind":"node","lean":"StatMech.FK.fpe2_GenuineFreeIsLeftDeriv","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fpe2-genuine-free-is-left-deriv","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Every edge of `\u039b_N` has free `q = 2` density at `p = logistic(t)` equal to the left derivative `G'\u208b(t)`.","title":"Free densities of genuine box edges as left derivatives"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fpe2-genuine-wired-is-right-deriv","kind":"node","lean":"StatMech.FK.fpe2_GenuineWiredIsRightDeriv","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fpe2-genuine-wired-is-right-deriv","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Every edge of `\u039b_N` has wired `q = 2` density at `p = logistic(t)` equal to the right derivative `G'\u208a(t)`.","title":"Wired densities of genuine box edges as right derivatives"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fpe2-fk-uniqueness-of-genuine","kind":"node","lean":"StatMech.FK.fpe2_fk_uniqueness_of_genuine","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Countable-exception "],["math","\\(q = 2\\)"],["text"," uniqueness from one-sided derivative data"]],"source":"Countable-exception $q = 2$ uniqueness from one-sided derivative data"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fpe2-fk-uniqueness-of-genuine","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"If the free and wired densities of the edges of `\u039b_N` are the left and right derivatives of one convex `G`, the free and wired `q = 2` measures agree on increasing events of `\u039b_N` for all but countably many `t`.","title":"Countable-exception $q = 2$ uniqueness from one-sided derivative data"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fpe2-genuine-free-collapse","kind":"node","lean":"StatMech.FK.fpe2_GenuineFreeCollapse","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fpe2-genuine-free-collapse","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The average free edge density of `\u039b_n` converges to the free infinite-volume density of every edge of `\u039b_N`.","title":"Free average edge density collapse on boxes"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fpe2-genuine-wired-collapse","kind":"node","lean":"StatMech.FK.fpe2_GenuineWiredCollapse","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fpe2-genuine-wired-collapse","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The average wired edge density of `\u039b_n` converges to the wired infinite-volume density of every edge of `\u039b_N`.","title":"Wired average edge density collapse on boxes"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-closed-off","kind":"node","lean":"StatMech.FK.ecz_ClosedOff","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-closed-off","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Configurations closed on every coordinate that is not an edge of `G`.","title":"Configurations closed off the edges of a graph"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-fk-zedge","kind":"node","lean":"StatMech.FK.ecz_fkZEdge","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-fk-zedge","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The edge partition function: the sum of FK weights over configurations closed off `E(G)`.","title":"Edge partition function of the random-cluster model"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-ne","kind":"node","lean":"StatMech.FK.ecz_NE","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-ne","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The number of coordinates `Sym2 V` that are not edges of `G`.","title":"Number of non-edge coordinates of a graph"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-factorization","kind":"node","lean":"StatMech.FK.ecz_factorization","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-factorization","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"`fkZ G p q = 2^{#non-edges} \u00b7 ecz_fkZEdge G p q`: each non-edge coordinate contributes a free factor `2`.","title":"Each non-edge contributes a factor 2 to the partition function"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-neglog-interface-le","kind":"node","lean":"StatMech.FK.ecz_neglog_interface_le","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Interface subadditivity of "],["math","\\(-\\log\\)"],["text"," of the edge partition function"]],"source":"Interface subadditivity of $-\\log$ of the edge partition function"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-neglog-interface-le","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"`-log ecz_fkZEdge` is subadditive across an interface up to `|interface| \u00b7 (-log(1 - p))`; compare `fis_neglog_interface_le` for `fkZ`.","title":"Interface subadditivity of $-\\log$ of the edge partition function"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-uu","kind":"node","lean":"StatMech.FK.ecz_uu","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-uu","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"`-log ecz_fkZEdge G p q`.","title":"Negative log edge partition function of a graph"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-u","kind":"node","lean":"StatMech.FK.ecz_u","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-u","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"`-log ecz_fkZEdge` of the box `\u039b_n` at `p = logistic(t)`, `q = 2`.","title":"Negative log edge partition function of a box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-edge-free-energy","kind":"node","lean":"StatMech.FK.ecz_edgeFreeEnergy","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-edge-free-energy","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The per-edge free energy of `\u039b_n` in the tilt variable, computed from `ecz_fkZEdge`.","title":"Per-edge free energy of a box in the tilt variable"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-additive-doubling-bound","kind":"node","lean":"StatMech.FK.ecz_AdditiveDoublingBound","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-additive-doubling-bound","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The doubling bound `agl_AdditiveDoublingBound` with `-log fkZ` replaced by `ecz_u`.","title":"Additive doubling bound for the edge partition function"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-additive-doubling-bound-2","kind":"node","lean":"StatMech.FK.ecz_additiveDoublingBound","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boxes satisfy the additive doubling bound for "],["math","\\(d \\ge 1\\)"]],"source":"Boxes satisfy the additive doubling bound for $d \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-additive-doubling-bound-2","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"For `d \u2265 1`, `ecz_u` satisfies the doubling bound along the scales `K_j = 2^{j+1} - 1`.","title":"Boxes satisfy the additive doubling bound for $d \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-sub-box-pred","kind":"node","lean":"StatMech.FK.ecz_subBoxPred","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-sub-box-pred","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"A vertex of `\u039b_n` lies in the sub-box `\u039b_a`.","title":"Membership of a box vertex in a sub-box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-sub-box-iso","kind":"node","lean":"StatMech.FK.ecz_subBoxIso","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-sub-box-iso","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"`\u039b_a` is isomorphic to the subgraph of `\u039b_n` induced on `\u039b_a`.","title":"Sub-box as an induced subgraph of a larger box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-per-edge-density-converges","kind":"node","lean":"StatMech.FK.ecz_PerEdgeDensityConverges","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-per-edge-density-converges","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The per-edge free energy `ecz_u d s n / |E(\u039b_n)|` converges as `n \u2192 \u221e`.","title":"Convergence of the per-edge free energy of boxes"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-t","kind":"node","lean":"StatMech.FK.ecz_T","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-t","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The number `\u230a(2n + 1)/(2m + 1)\u230b` of translates of `\u039b_m` that fit along one side of `\u039b_n`.","title":"Number of sub-box translates along a side of a box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-center","kind":"node","lean":"StatMech.FK.ecz_center","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-center","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The center of the cell indexed by `c` in the tiling of `\u039b_n` by translates of `\u039b_m`.","title":"Center of a cell in the tiling of a box by sub-boxes"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-cell-idx","kind":"node","lean":"StatMech.FK.ecz_cellIdx","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-cell-idx","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The index along coordinate `i` of the tiling cell containing `x`.","title":"Coordinate index of the tiling cell containing a site"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-van-hove-tag","kind":"node","lean":"StatMech.FK.ecz_vanHoveTag","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-van-hove-tag","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The tiling cell containing a vertex of `\u039b_n`, or `none` if it lies in the leftover strip.","title":"Tiling cell of a box vertex"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-cell-equiv","kind":"node","lean":"StatMech.FK.ecz_cellEquiv","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-cell-equiv","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The vertices of `\u039b_m` correspond to the vertices of the cell `c` by translation.","title":"Translation between a sub-box and a tiling cell"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-cell-block-iso","kind":"node","lean":"StatMech.FK.ecz_cellBlockIso","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-cell-block-iso","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Each tiling cell induces a graph isomorphic to `\u039b_m`.","title":"Graph isomorphism between a sub-box and a tiling cell"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-rest-univ-iso","kind":"node","lean":"StatMech.FK.ecz_restUnivIso","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-rest-univ-iso","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Restricting `\u039b_n` to all tags gives back `\u039b_n`.","title":"Restriction of a box to all tiling tags"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-box-van-hove-subadd","kind":"node","lean":"StatMech.FK.ecz_box_vanHove_subadd","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-box-van-hove-subadd","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"Van Hove bound: `ecz_u d t n` is at most `T^d \u00b7 ecz_u d t m` plus the cost of the leftover strip and of the interfaces between cells, where `T = \u230a(2n + 1)/(2m + 1)\u230b`.","title":"Van Hove tiling bound for the edge partition function"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-f","kind":"node","lean":"StatMech.FK.ecz_f","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-f","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The per-edge free energy `ecz_u d s n / |E(\u039b_n)|`.","title":"Per-edge negative log edge partition function of a box"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-g","kind":"node","lean":"StatMech.FK.ecz_g","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-g","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"An upper envelope for `ecz_f` combining the per-edge value at scale `m` with the all-closed bound.","title":"Upper envelope for the per-edge free energy"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-per-edge-density-converges-2","kind":"node","lean":"StatMech.FK.ecz_perEdgeDensityConverges","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Per-edge free energy of boxes converges for "],["math","\\(d \\ge 1\\)"]],"source":"Per-edge free energy of boxes converges for $d \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-per-edge-density-converges-2","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"For `d \u2265 1`, the per-edge free energy `ecz_u d s n / |E(\u039b_n)|` converges.","title":"Per-edge free energy of boxes converges for $d \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-block-oscillation","kind":"node","lean":"StatMech.FK.ecz_BlockOscillation","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-block-oscillation","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The edge free energy of `\u039b_n` differs from that of `\u039b_{K_{b(n)}}` by a vanishing oscillation.","title":"Vanishing block oscillation of the edge free energy"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/ecz-fk-uniqueness","kind":"node","lean":"StatMech.FK.ecz_fk_uniqueness","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Per-box "],["math","\\(q = 2\\)"],["text"," uniqueness off a countable set of tilts"]],"source":"Per-box $q = 2$ uniqueness off a countable set of tilts"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#ecz-fk-uniqueness","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"For `d \u2265 1`, a fixed box `\u039b_N` with an edge witness, the free and wired `q = 2` measures agree on every increasing event of `\u039b_N` for all `t = logit(p)` outside a countable set in `t`. This is the per-box `q = 2` case of `fkgq_countable_free_wired_disagreement`.","title":"Per-box $q = 2$ uniqueness off a countable set of tilts"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fk-uniqueness-off-countable-q2-full","kind":"node","lean":"StatMech.FK.fk_uniqueness_off_countable_q2_full","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Secant bound gives "],["math","\\(q = 2\\)"],["text"," edge-event uniqueness off a countable set"]],"source":"Secant bound gives $q = 2$ edge-event uniqueness off a countable set"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fk-uniqueness-off-countable-q2-full","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"For `q = 2`: assuming `\u03c6^1_{p'}(e open) \u2264 \u03c6^0_p(e open)` for `p' < p` in `(0, 1)`, for all but countably many `p` every measure between the free and wired infinite-volume measures on the event `{eb open}` agrees with both.","title":"Secant bound gives $q = 2$ edge-event uniqueness off a countable set"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-centered-convergence","kind":"node","lean":"StatMech.FK.fkgq_centeredConvergence","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Convex limit of centered tilt free energies of boxes for "],["math","\\(d, q \\ge 1\\)"]],"source":"Convex limit of centered tilt free energies of boxes for $d, q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-centered-convergence","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"For `d \u2265 1` and `q \u2265 1`, the centered tilt free energies of the boxes `\u039b_n` converge pointwise to a convex function `G` with `G(0) = 0`.","title":"Convex limit of centered tilt free energies of boxes for $d, q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-avg-density","kind":"node","lean":"StatMech.FK.fkgq_avgDensity","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-avg-density","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The average edge density `|E|^{-1} E_{\u03c6_{G,p,q}}[#open edges]` at `p = logistic(s)`.","title":"Average edge density of a random-cluster measure"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-free-mass","kind":"node","lean":"StatMech.FK.fkgq_freeMass","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free infinite-volume mass function on a box for "],["math","\\(q \\ge 1\\)"]],"source":"Free infinite-volume mass function on a box for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-free-mass","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The mass function on configurations of `\u039b_N` of the free infinite-volume measure at `p = logistic(s)`.","title":"Free infinite-volume mass function on a box for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-wired-mass","kind":"node","lean":"StatMech.FK.fkgq_wiredMass","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired infinite-volume mass function on a box for "],["math","\\(q \\ge 1\\)"]],"source":"Wired infinite-volume mass function on a box for $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-wired-mass","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The mass function on configurations of `\u039b_N` of the wired infinite-volume measure at `p = logistic(s)`.","title":"Wired infinite-volume mass function on a box for $q \\ge 1$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-unit-marg","kind":"node","lean":"StatMech.FK.fkgq_unitMarg","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Free unit-edge marginal at "],["math","\\(0\\)"],["text"," on "],["math","\\(\\Lambda_{\\lfloor\\sqrt n\\rfloor}\\)"]],"source":"Free unit-edge marginal at $0$ on $\\Lambda_{\\lfloor\\sqrt n\\rfloor}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-unit-marg","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The free marginal on `\u039b_{\u230a\u221an\u230b}` of the unit edge at the origin in direction `dir`, at `p = logistic(t)`.","title":"Free unit-edge marginal at $0$ on $\\Lambda_{\\lfloor\\sqrt n\\rfloor}$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-unit-wired-marg","kind":"node","lean":"StatMech.FK.fkgq_unitWiredMarg","parent":48711,"reading_title":{"format":"inline-parts-v1","parts":[["text","Wired unit-edge marginal at "],["math","\\(0\\)"],["text"," on "],["math","\\(\\Lambda_{\\lfloor\\sqrt n\\rfloor}\\)"]],"source":"Wired unit-edge marginal at $0$ on $\\Lambda_{\\lfloor\\sqrt n\\rfloor}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-unit-wired-marg","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The wired marginal on `\u039b_{\u230a\u221an\u230b}` of the unit edge at the origin in direction `dir`, at `p = logistic(t)`.","title":"Wired unit-edge marginal at $0$ on $\\Lambda_{\\lfloor\\sqrt n\\rfloor}$"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-free-delta","kind":"node","lean":"StatMech.FK.fkgq_freeDelta","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-free-delta","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The total deviation between the free density of `e` and the free unit-edge marginals on `\u039b_{\u230a\u221an\u230b}`, summed over directions.","title":"Total deviation of free unit-edge marginals from a free density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-wired-delta","kind":"node","lean":"StatMech.FK.fkgq_wiredDelta","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-wired-delta","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"The total deviation between the wired unit-edge marginals on `\u039b_{\u230a\u221an\u230b}` and the wired density of `e`, summed over directions.","title":"Total deviation of wired unit-edge marginals from a wired density"},{"id":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/fkgq-countable-free-wired-disagreement","kind":"node","lean":"StatMech.FK.fkgq_countable_free_wired_disagreement","parent":48711,"result":true,"route":"../roadmap/StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust/index.html#fkgq-countable-free-wired-disagreement","scope":"StatisticalMechanics/random-cluster-model/per-box-agreement-of-free-and-wired-random-clust","status":"fully_proved","summary":"For `d \u2265 1`, `q \u2265 1`, and every box `\u039b_N`, the free and wired infinite-volume random-cluster measures on `\u2124^d` at `p = logistic(t)` give the same mass to every increasing event of `\u039b_N`, for all `t` outside a countable set.","title":"Per-box uniqueness of the random-cluster measures off a countable set"},{"data_url":"scopes/StatisticalMechanics/spin-models.json","id":"StatisticalMechanics/spin-models","kind":"scope","parent":47780,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/index.html","scope":"StatisticalMechanics/spin-models","status":"fully_proved","summary":"Spin models on finite graphs: the Ising Hamiltonian, Gibbs distribution and partition function and their agreement with the canonical ensemble, the pressure and its independence of boundary conditions, and the Lee\u2013Yang theorem on the zeros of the partition function.","title":"Lattice spin models"},{"data_url":"scopes/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph.json","id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph","kind":"scope","parent":48812,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/index.html","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph","status":"fully_proved","title":"Ising model on a finite graph"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/ising-cfg-equiv","kind":"node","lean":"StatMech.Ising.isingCfgEquiv","parent":48813,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/index.html#ising-cfg-equiv","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph","status":"fully_proved","summary":"The equivalence of configuration spaces induced by a vertex equivalence `\u03c3 : V \u2243 W`: `\u03c9 \u21a6 \u03c9 \u2218 \u03c3`.","title":"Configuration equivalence induced by a vertex relabeling"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/ising-sum-config-equiv","kind":"node","lean":"StatMech.Ising.isingSumConfigEquiv","parent":48813,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/index.html#ising-sum-config-equiv","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph","status":"fully_proved","summary":"The equivalence between configurations on `V \u2295 W` and pairs of configurations on `V` and `W`.","title":"Configurations on a disjoint union as pairs of configurations"},{"data_url":"scopes/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure.json","id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","kind":"scope","navigation":"module","parent":48813,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","title":"The finite-volume Ising Gibbs measure"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/spin","kind":"node","lean":"StatMech.Ising.spin","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#spin","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The spin `\u03c3_x \u2208 {1, -1}` of the configuration `s` at `x`: `1` for `true`, `-1` for `false`.","title":"Spin of a configuration at a site"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/bond","kind":"node","lean":"StatMech.Ising.bond","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#bond","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The bond variable `\u03c3_x \u03c3_y` of an unordered pair `e = {x, y}`.","title":"Bond variable of an unordered pair"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/bond-eq-pm","kind":"node","lean":"StatMech.Ising.bond_eq_pm","parent":48816,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bond variable takes values "],["math","\\(\\pm 1\\)"]],"source":"Bond variable takes values $\\pm 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#bond-eq-pm","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"A bond variable is `1` or `-1`.","title":"Bond variable takes values $\\pm 1$"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/hamiltonian","kind":"node","lean":"StatMech.Ising.hamiltonian","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#hamiltonian","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The Ising Hamiltonian `H(s) = -\u2211_{xy \u2208 E(G)} \u03c3_x \u03c3_y - h \u2211_x \u03c3_x` on a finite graph `G` with unit coupling and external field `h`. See also `CondensedMatter.IsingModel.energy` and `StatMech.Ising.hamiltonian_eq_energy`.","title":"Ising Hamiltonian on a finite graph"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/ising-weight","kind":"node","lean":"StatMech.Ising.isingWeight","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#ising-weight","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The Boltzmann weight `exp (-\u03b2 H(s))` of a configuration at inverse temperature `\u03b2`.","title":"Boltzmann weight of an Ising configuration"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/ising-z","kind":"node","lean":"StatMech.Ising.isingZ","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#ising-z","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The partition function `Z = \u2211_s exp (-\u03b2 H(s))`. See also `CanonicalEnsemble.mathematicalPartitionFunction` and `StatMech.Ising.isingZ_eq_mathematicalPartitionFunction`.","title":"Ising partition function"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/ising-z-pos","kind":"node","lean":"StatMech.Ising.isingZ_pos","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#ising-z-pos","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The partition function is positive.","title":"Positivity of the Ising partition function"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/ising-prob","kind":"node","lean":"StatMech.Ising.isingProb","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#ising-prob","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The Gibbs probability `exp (-\u03b2 H(s)) / Z` of a configuration. See also `CanonicalEnsemble.probability` and `StatMech.Ising.isingProb_eq_probability`.","title":"Gibbs probability of an Ising configuration"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/ising-prob-sum-eq-one","kind":"node","lean":"StatMech.Ising.isingProb_sum_eq_one","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#ising-prob-sum-eq-one","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The Gibbs probabilities sum to one.","title":"Ising Gibbs probabilities sum to one"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/ising-pmf","kind":"node","lean":"StatMech.Ising.isingPMF","parent":48816,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure/index.html#ising-pmf","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/the-finite-volume-ising-gibbs-measure","status":"fully_proved","summary":"The finite-volume Ising Gibbs distribution as a probability mass function on configurations.","title":"Finite-volume Ising Gibbs probability mass function"},{"data_url":"scopes/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy.json","id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","kind":"scope","navigation":"module","parent":48813,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","title":"Bridges between the StatMech Ising model and Physlib"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/spin-b-eq-ising-model-spin","kind":"node","lean":"StatMech.Ising.spinB_eq_isingModelSpin","parent":48827,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html#spin-b-eq-ising-model-spin","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","summary":"The Boolean spin value `spinB` is Physlib's `CondensedMatter.IsingModel.spin`.","title":"Boolean spin agrees with Physlib's Ising spin"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/two-mul-sum-bond","kind":"node","lean":"StatMech.Ising.two_mul_sum_bond","parent":48827,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html#two-mul-sum-bond","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","summary":"The edge sum of `bond s` is half the sum of `\u03c3_x \u03c3_y` over ordered adjacent pairs.","title":"Edge bond sum is half the sum over ordered adjacent pairs"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/hamiltonian-eq-energy","kind":"node","lean":"StatMech.Ising.hamiltonian_eq_energy","parent":48827,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html#hamiltonian-eq-energy","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","summary":"The Hamiltonian `hamiltonian G h` is Physlib's Ising energy `CondensedMatter.IsingModel.energy` of the model with coupling `J = 1` and field `h`, on the region of all vertices.","title":"Hamiltonian agrees with Physlib's Ising energy"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/canonical-ensemble","kind":"node","lean":"StatMech.Ising.canonicalEnsemble","parent":48827,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html#canonical-ensemble","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","summary":"The Ising model on a finite graph `G` with field `h`, as Physlib's finite canonical ensemble: microstates are spin configurations, the energy is `hamiltonian G h`, and the measure is the counting measure.","title":"Ising model as a Physlib canonical ensemble"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/ising-z-eq-mathematical-partition-functi","kind":"node","lean":"StatMech.Ising.isingZ_eq_mathematicalPartitionFunction","parent":48827,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html#ising-z-eq-mathematical-partition-functi","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","summary":"The Ising partition function at inverse temperature `T.\u03b2` is Physlib's mathematical partition function of `canonicalEnsemble G h` at temperature `T`.","title":"Ising partition function agrees with Physlib's partition function"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/ising-prob-eq-probability","kind":"node","lean":"StatMech.Ising.isingProb_eq_probability","parent":48827,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html#ising-prob-eq-probability","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","summary":"The Ising Gibbs probability at inverse temperature `T.\u03b2` is Physlib's canonical probability of `canonicalEnsemble G h` at temperature `T`.","title":"Ising Gibbs probability agrees with Physlib's canonical probability"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/nat-abs-sum-eq-one-iff","kind":"node","lean":"StatMech.natAbs_sum_eq_one_iff","parent":48827,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sites at "],["math","\\(\\ell^1\\)"],["text"," distance one are those differing by a unit vector"]],"source":"Sites at $\\ell^1$ distance one are those differing by a unit vector"},"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html#nat-abs-sum-eq-one-iff","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","summary":"Two sites of `\u2124^d` are at `\u2113\u00b9` distance one exactly when they differ by a unit vector.","title":"Sites at $\\ell^1$ distance one are those differing by a unit vector"},{"id":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/hypercubic-lattice-iso-crystal","kind":"node","lean":"StatMech.hypercubicLatticeIsoCrystal","parent":48827,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy/index.html#hypercubic-lattice-iso-crystal","scope":"StatisticalMechanics/spin-models/ising-model-on-a-finite-graph/bridges-between-the-statmech-ising-model-and-phy","status":"fully_proved","summary":"The nearest-neighbour graph `hypercubicLattice d` on `\u2124^d` is isomorphic to the bond graph of Physlib's hypercubic crystal `CondensedMatter.Crystal.hypercubicGraph d`, by sending `x` to the site in cell `x`.","title":"Isomorphism of the hypercubic lattice with Physlib's crystal graph"},{"data_url":"scopes/StatisticalMechanics/spin-models/lee-yang-circle-theorem.json","id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem","kind":"scope","parent":48812,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/index.html","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem","status":"fully_proved","title":"Lee\u2013Yang circle theorem"},{"data_url":"scopes/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model.json","id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model","kind":"scope","navigation":"module","parent":48836,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/index.html","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model","status":"fully_proved","title":"The Lee\u2013Yang polynomial of a finite Ising model"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/minus-spin-count","kind":"node","lean":"StatMech.LeeYang.minusSpinCount","parent":48837,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/index.html#minus-spin-count","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model","status":"fully_proved","summary":"The number of minus spins (sites with value `false`) of a configuration.","title":"Number of minus spins of a configuration"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/zero-field-interaction-weight","kind":"node","lean":"StatMech.LeeYang.zeroFieldInteractionWeight","parent":48837,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/index.html#zero-field-interaction-weight","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model","status":"fully_proved","summary":"The zero-field Boltzmann weight `exp (\u03b2 \u2211_{e \u2208 E(G)} \u03c3_x \u03c3_y)` of a configuration.","title":"Zero-field Boltzmann weight of a configuration"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/lee-yang-polynomial","kind":"node","lean":"StatMech.LeeYang.leeYangPolynomial","parent":48837,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/index.html#lee-yang-polynomial","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model","status":"fully_proved","summary":"The Lee\u2013Yang polynomial `\u2211_\u03c3 exp (\u03b2 \u2211_e \u03c3_x \u03c3_y) X^k(\u03c3)` of the Ising model on `G`, where `k(\u03c3)` is the number of minus spins; its variable is the fugacity `exp (-2\u03b2h)`.","title":"Lee\u2013Yang polynomial of a finite Ising model"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/lee-yang-complex-polynomial","kind":"node","lean":"StatMech.LeeYang.leeYangComplexPolynomial","parent":48837,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model/index.html#lee-yang-complex-polynomial","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-polynomial-of-a-finite-ising-model","status":"fully_proved","summary":"The Lee\u2013Yang polynomial with complex coefficients, whose zeros the Lee\u2013Yang circle theorem locates.","title":"Lee\u2013Yang polynomial with complex coefficients"},{"data_url":"scopes/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a.json","id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a","kind":"scope","navigation":"module","parent":48836,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/index.html","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a","status":"fully_proved","title":"The Lee\u2013Yang circle property for a single edge and for matchings"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/two-vertex-edge","kind":"node","lean":"StatMech.LeeYang.twoVertexEdge","parent":48842,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/index.html#two-vertex-edge","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a","status":"fully_proved","summary":"The graph on two vertices joined by one edge.","title":"Single-edge graph on two vertices"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/has-lee-yang-circle","kind":"node","lean":"StatMech.LeeYang.HasLeeYangCircle","parent":48842,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/index.html#has-lee-yang-circle","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a","status":"fully_proved","summary":"The Lee\u2013Yang circle property: every complex zero of the Lee\u2013Yang polynomial of `G` at inverse temperature `\u03b2` has modulus one.","title":"Lee\u2013Yang circle property of a graph"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/matching-vertex","kind":"node","lean":"StatMech.LeeYang.MatchingVertex","parent":48842,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/index.html#matching-vertex","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a","status":"fully_proved","summary":"The vertex type of the perfect matching with `n` edges: `n` disjoint copies of `Fin 2`.","title":"Vertex type of the perfect matching with n edges"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/finite-matching","kind":"node","lean":"StatMech.LeeYang.finiteMatching","parent":48842,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a/index.html#finite-matching","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-property-for-a-single-edge-a","status":"fully_proved","summary":"The perfect matching with `n` edges, the disjoint union of `n` single edges.","title":"Perfect matching with n edges"},{"data_url":"scopes/StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial.json","id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial","kind":"scope","navigation":"module","parent":48836,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial/index.html","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial","status":"fully_proved","title":"Spin-flip symmetry of the Lee\u2013Yang polynomial"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial/lee-yang-flip","kind":"node","lean":"StatMech.LeeYang.leeYangFlip","parent":48847,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial/index.html#lee-yang-flip","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial","status":"fully_proved","summary":"The global spin flip `\u03c3 \u21a6 -\u03c3`.","title":"Global spin flip"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial/lee-yang-flip-equiv","kind":"node","lean":"StatMech.LeeYang.leeYangFlipEquiv","parent":48847,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial/index.html#lee-yang-flip-equiv","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/spin-flip-symmetry-of-the-lee-yang-polynomial","status":"fully_proved","summary":"The global spin flip as an involutive equivalence of configurations.","title":"Global spin flip as an involutive equivalence"},{"data_url":"scopes/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem.json","id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","kind":"scope","navigation":"module","parent":48836,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","title":"The Lee\u2013Yang circle theorem"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/asano-quadratic-contraction","kind":"node","lean":"StatMech.LeeYang.asano_quadratic_contraction","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#asano-quadratic-contraction","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"Asano's lemma in one variable: if `A + E t + D t\u00b2` has no zero in the open unit disk, then neither has `A + D t`.","title":"Asano's lemma in one variable"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/is-multi-affine","kind":"node","lean":"StatMech.LeeYang.IsMultiAffine","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#is-multi-affine","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"A function of finitely many complex variables is multi-affine if it is affine in each variable separately.","title":"Multi-affine function of several complex variables"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/polydisc-stable","kind":"node","lean":"StatMech.LeeYang.PolydiscStable","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#polydisc-stable","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"A function of complex variables is polydisc stable if it has no zero when every variable lies in the open unit disk.","title":"Polydisc stability of a function of complex variables"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/asano-contract","kind":"node","lean":"StatMech.LeeYang.asanoContract","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#asano-contract","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The Asano contraction of the variables `i` and `j`: for `p = A + B z_i + C z_j + D z_i z_j` it is `A + D z_i`, computed from the values of `p` at `z_i, z_j \u2208 {0, 1}`.","title":"Asano contraction of two variables"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/asano-contract-stable","kind":"node","lean":"StatMech.LeeYang.asanoContract_stable","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#asano-contract-stable","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"Asano's contraction lemma: the Asano contraction of a multi-affine polydisc-stable polynomial is polydisc stable (Asano, Phys. Rev. Lett. 24 (1970)).","title":"Asano's contraction lemma"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-edge-factor-ne-zero","kind":"node","lean":"StatMech.LeeYang.leeYang_edgeFactor_ne_zero","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-edge-factor-ne-zero","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The two-variable edge factor `1 + ab + r (a + b)` with `0 \u2264 r \u2264 1` has no zero in the open unit bidisk.","title":"Edge factor has no zero in the open unit bidisk for 0 \u2264 r \u2264 1"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-var","kind":"node","lean":"StatMech.LeeYang.LeeYangVar","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-var","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The variables of the half-edge polynomial: one anchor variable per vertex and one half-edge variable per incidence `(v, e)`.","title":"Variables of the half-edge polynomial"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-anchor","kind":"node","lean":"StatMech.LeeYang.leeYangAnchor","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-anchor","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The anchor variable of a vertex.","title":"Anchor variable of a vertex"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-half","kind":"node","lean":"StatMech.LeeYang.leeYangHalf","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-half","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The half-edge variable of the incidence `(v, e)`.","title":"Half-edge variable of an incidence"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-normalized-edge-factor","kind":"node","lean":"StatMech.LeeYang.leeYangNormalizedEdgeFactor","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-normalized-edge-factor","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The edge factor `1 + w_u w_v + e^{-2\u03b2} (w_u + w_v)` of the edge `e = {u, v}`, in its half-edge variables.","title":"Edge factor in half-edge variables"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-half-edge-product","kind":"node","lean":"StatMech.LeeYang.leeYangHalfEdgeProduct","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-half-edge-product","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The half-edge polynomial: the product of `1 + z_v` over the vertices and of the edge factors over the edges.","title":"Half-edge polynomial of a graph"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-anchor-product","kind":"node","lean":"StatMech.LeeYang.leeYangAnchorProduct","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-anchor-product","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The vertex part `\u220f_v (1 + z_v)` of the half-edge polynomial.","title":"Vertex part of the half-edge polynomial"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-edge-product","kind":"node","lean":"StatMech.LeeYang.leeYangEdgeProduct","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-edge-product","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The edge part of the half-edge polynomial: the product of the edge factors.","title":"Edge part of the half-edge polynomial"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-contract-list","kind":"node","lean":"StatMech.LeeYang.leeYangContractList","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-contract-list","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"Successive Asano contraction of the half-edge variable of each incidence in the list into the anchor variable of its vertex.","title":"Successive Asano contraction along a list of incidences"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-incidence-finset","kind":"node","lean":"StatMech.LeeYang.leeYangIncidenceFinset","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-incidence-finset","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The incidences `(v, e)` of `G`, with `v` an endpoint of `e`.","title":"Vertex\u2013edge incidences of a graph"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-contracted-product","kind":"node","lean":"StatMech.LeeYang.leeYangContractedProduct","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-contracted-product","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The half-edge polynomial after contracting every incidence.","title":"Half-edge polynomial with every incidence contracted"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-monomial","kind":"node","lean":"StatMech.LeeYang.leeYangMonomial","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-monomial","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The monomial `\u220f_{i \u2208 S} z_i`.","title":"Monomial of a set of variables"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-endpoint0","kind":"node","lean":"StatMech.LeeYang.leeYangEndpoint0","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-endpoint0","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"A chosen first endpoint of an edge.","title":"Chosen first endpoint of an edge"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-endpoint1","kind":"node","lean":"StatMech.LeeYang.leeYangEndpoint1","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-endpoint1","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The other endpoint of an edge.","title":"Other endpoint of an edge"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-anchor-choice-factor","kind":"node","lean":"StatMech.LeeYang.leeYangAnchorChoiceFactor","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-anchor-choice-factor","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The term of `1 + z_v` selected by a Boolean choice.","title":"Term of a vertex factor selected by a Boolean choice"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-edge-choice-factor","kind":"node","lean":"StatMech.LeeYang.leeYangEdgeChoiceFactor","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-edge-choice-factor","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The term of the edge factor selected by a pair of Boolean choices, one per endpoint.","title":"Term of an edge factor selected by two Boolean choices"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-choice-term","kind":"node","lean":"StatMech.LeeYang.leeYangChoiceTerm","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-choice-term","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The term of the expanded half-edge polynomial selected by a choice at every vertex and every edge.","title":"Term of the expanded half-edge polynomial for a choice"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-choice-weight","kind":"node","lean":"StatMech.LeeYang.leeYangChoiceWeight","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-choice-weight","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The coefficient of a term of the expanded half-edge polynomial: `e^{-2\u03b2}` for every edge whose two choices differ.","title":"Coefficient of a term of the expanded half-edge polynomial"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-choice-selected","kind":"node","lean":"StatMech.LeeYang.leeYangChoiceSelected","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-choice-selected","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"Whether a choice selects each variable.","title":"Selection indicator of a choice on the variables"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-choice-support","kind":"node","lean":"StatMech.LeeYang.leeYangChoiceSupport","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-choice-support","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The set of variables selected by a choice, the support of its monomial.","title":"Set of variables selected by a choice"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-erase-halves","kind":"node","lean":"StatMech.LeeYang.leeYangEraseHalves","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-erase-halves","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"Remove from a set of variables the half-edge variables of the listed incidences.","title":"Removal of listed half-edge variables from a variable set"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-consistent","kind":"node","lean":"StatMech.LeeYang.LeeYangConsistent","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-consistent","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"A set of variables is consistent with a list of incidences if each listed half-edge variable is selected exactly when the anchor variable of its vertex is.","title":"Consistency of a variable set with a list of incidences"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-tau-of","kind":"node","lean":"StatMech.LeeYang.leeYangTauOf","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-tau-of","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The edge choices induced by a spin configuration: each edge copies the spins of its endpoints.","title":"Edge choices induced by a spin configuration"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-anchor-support","kind":"node","lean":"StatMech.LeeYang.leeYangAnchorSupport","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-anchor-support","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The anchor variables of the vertices with spin `true`.","title":"Anchor variables of the vertices with plus spin"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-diagonal","kind":"node","lean":"StatMech.LeeYang.leeYangDiagonal","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-diagonal","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The point with every anchor variable equal to `q` and every half-edge variable equal to `0`.","title":"Diagonal point with anchors at q and half-edge variables at zero"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-real-choice-weight","kind":"node","lean":"StatMech.LeeYang.leeYangRealChoiceWeight","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-real-choice-weight","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The real weight `\u220f_e e^{-2\u03b2 [\u03c3_u \u2260 \u03c3_v]}` of a spin configuration.","title":"Real disagreement weight of a spin configuration"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/lee-yang-complex-polynomial-ne-zero-in-o","kind":"node","lean":"StatMech.LeeYang.leeYangComplexPolynomial_ne_zero_in_openUnitDisk","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#lee-yang-complex-polynomial-ne-zero-in-o","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"The Lee\u2013Yang polynomial of a ferromagnetic Ising model has no zero in the open unit disk.","title":"Ferromagnetic Lee\u2013Yang polynomial has no zero in the open unit disk"},{"id":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/finite-graph-has-lee-yang-circle","kind":"node","lean":"StatMech.LeeYang.finiteGraph_hasLeeYangCircle","parent":48850,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem/index.html#finite-graph-has-lee-yang-circle","scope":"StatisticalMechanics/spin-models/lee-yang-circle-theorem/the-lee-yang-circle-theorem","status":"fully_proved","summary":"For the Ising model on a finite graph with uniform ferromagnetic coupling and `\u03b2 \u2265 0`, every complex zero of the partition function in the fugacity `z = e^{-2\u03b2h}` lies on the unit circle (Lee and Yang, *Statistical theory of equations of state and phase transitions. II*, Phys. Rev. 87 (1952)).","title":"Lee\u2013Yang circle theorem"},{"data_url":"scopes/StatisticalMechanics/spin-models/pressure-of-the-ising-model.json","id":"StatisticalMechanics/spin-models/pressure-of-the-ising-model","kind":"scope","navigation":"module","parent":48812,"result":false,"route":"../roadmap/StatisticalMechanics/spin-models/pressure-of-the-ising-model/index.html","scope":"StatisticalMechanics/spin-models/pressure-of-the-ising-model","status":"fully_proved","title":"Pressure of the Ising model"},{"id":"StatisticalMechanics/spin-models/pressure-of-the-ising-model/z","kind":"node","lean":"StatMech.Ising.Z","parent":48884,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/pressure-of-the-ising-model/index.html#z","scope":"StatisticalMechanics/spin-models/pressure-of-the-ising-model","status":"fully_proved","summary":"The partition function `Z(\u03b2) = \u2211_s e^{-\u03b2E(s)}` of a finite system.","title":"Partition function of a finite system"},{"id":"StatisticalMechanics/spin-models/pressure-of-the-ising-model/zpow","kind":"node","lean":"StatMech.Ising.Zpow","parent":48884,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/pressure-of-the-ising-model/index.html#zpow","scope":"StatisticalMechanics/spin-models/pressure-of-the-ising-model","status":"fully_proved","summary":"The partition function of `n` independent copies of the system.","title":"Partition function of independent copies"},{"id":"StatisticalMechanics/spin-models/pressure-of-the-ising-model/log-zseq","kind":"node","lean":"StatMech.Ising.logZSeq","parent":48884,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/pressure-of-the-ising-model/index.html#log-zseq","scope":"StatisticalMechanics/spin-models/pressure-of-the-ising-model","status":"fully_proved","summary":"The sequence `log Z_n` for `n` independent copies.","title":"Log-partition sequence of independent copies"},{"id":"StatisticalMechanics/spin-models/pressure-of-the-ising-model/pressure","kind":"node","lean":"StatMech.Ising.pressure","parent":48884,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/pressure-of-the-ising-model/index.html#pressure","scope":"StatisticalMechanics/spin-models/pressure-of-the-ising-model","status":"fully_proved","summary":"The pressure: the limit of `log Z_n / n` for independent copies, obtained from subadditivity.","title":"Pressure from independent copies"},{"id":"StatisticalMechanics/spin-models/pressure-of-the-ising-model/psv-bdry","kind":"node","lean":"StatMech.Ising.psv_Bdry","parent":48884,"result":true,"route":"../roadmap/StatisticalMechanics/spin-models/pressure-of-the-ising-model/index.html#psv-bdry","scope":"StatisticalMechanics/spin-models/pressure-of-the-ising-model","status":"fully_proved","summary":"The outer vertex boundary `\u039b_{n+1} \u2216 \u039b_n` of the box.","title":"Outer vertex boundary of a box"},{"data_url":"scopes/StatisticalMechanics/correlation-inequalities.json","id":"StatisticalMechanics/correlation-inequalities","kind":"scope","parent":47780,"result":false,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/index.html","scope":"StatisticalMechanics/correlation-inequalities","status":"fully_proved","summary":"Inequalities for the correlations of ferromagnetic Ising models: the Griffiths (GKS) and FKG inequalities, monotonicity in the couplings and the temperature, and the Griffiths\u2013Hurst\u2013Sherman inequality with its consequences for the magnetization.","title":"Correlation inequalities for spin systems"},{"id":"StatisticalMechanics/correlation-inequalities/ifk-ising-prob-positively-associated","kind":"node","lean":"StatMech.Ising.ifk_isingProb_positively_associated","parent":48890,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/index.html#ifk-ising-prob-positively-associated","scope":"StatisticalMechanics/correlation-inequalities","status":"fully_proved","summary":"FKG inequality for the Ising model. For `\u03b2 \u2265 0` and any field, increasing functions are positively correlated under the Ising Gibbs distribution on a finite graph.","title":"FKG inequality"},{"data_url":"scopes/StatisticalMechanics/correlation-inequalities/griffiths-inequalities.json","id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","kind":"scope","navigation":"module","parent":48890,"result":false,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","title":"Griffiths inequalities"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/spin-b","kind":"node","lean":"StatMech.Ising.spinB","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#spin-b","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The spin value `\u00b11` of a Boolean: `1` for `true`, `-1` for `false`. See also `CondensedMatter.IsingModel.spin` and `StatMech.Ising.spinB_eq_isingModelSpin`.","title":"Spin value of a Boolean"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/monomial","kind":"node","lean":"StatMech.Ising.monomial","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#monomial","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The spin monomial `\u220f_x \u03c3_x^{m_x}` with exponents `m : V \u2192 \u2115`.","title":"Spin monomial with given exponents"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/sum-monomial-nonneg","kind":"node","lean":"StatMech.Ising.sum_monomial_nonneg","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#sum-monomial-nonneg","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The sum of a spin monomial over all configurations is nonnegative.","title":"Configuration sum of a spin monomial is nonnegative"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/is-spin-monomial","kind":"node","lean":"StatMech.Ising.IsSpinMonomial","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#is-spin-monomial","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"`f` is a spin monomial `\u220f_x \u03c3_x^{m_x}` for some exponents `m`.","title":"Being a spin monomial"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/gks-kernel","kind":"node","lean":"StatMech.Ising.gks_kernel","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#gks-kernel","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The positivity kernel of the GKS inequalities: for nonnegative `c_i, d_i` and spin monomials `fac i`, `obs`, the configuration sum of `obs \u00b7 \u220f_i (c_i + d_i \u00b7 fac_i)` is nonnegative.","title":"Nonnegativity of the GKS positivity kernel"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/spin-prod","kind":"node","lean":"StatMech.Ising.spinProd","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#spin-prod","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The spin product `\u03c3_A = \u220f_{x \u2208 A} \u03c3_x`.","title":"Spin product over a vertex set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/ising-expectation","kind":"node","lean":"StatMech.Ising.isingExpectation","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#ising-expectation","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The finite-volume Ising expectation `\u27e8f\u27e9 = \u2211_s P(s) f(s)` of an observable `f`.","title":"Finite-volume Ising expectation"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/ising-weight-factor","kind":"node","lean":"StatMech.Ising.isingWeight_factor","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#ising-weight-factor","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The Boltzmann weight factorizes as a product over edges and vertices of `cosh + sinh \u00b7 \u03c3` terms.","title":"Boltzmann weight as a product of cosh-plus-sinh factors"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/gks-first","kind":"node","lean":"StatMech.Ising.gks_first","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#gks-first","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"For the ferromagnetic Ising model on a finite graph with `\u03b2 \u2265 0` and `h \u2265 0`, every spin correlation is nonnegative: `0 \u2264 \u27e8\u03c3_A\u27e9`.","title":"First Griffiths inequality (GKS I)"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/xnor","kind":"node","lean":"StatMech.Ising.xnor","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#xnor","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The pointwise product of two configurations: spin `\u03c3_y \u03c3'_y` at `y` (`true` when the spins agree).","title":"Pointwise product of two configurations"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/dbl-fun","kind":"node","lean":"StatMech.Ising.dblFun","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#dbl-fun","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The involution `(\u03c3, q) \u21a6 (\u03c3, \u03c3 q)` of pairs of configurations.","title":"Duplicated-variable involution on configuration pairs"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/dbl-equiv","kind":"node","lean":"StatMech.Ising.dblEquiv","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#dbl-equiv","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"Ginibre's duplicated-variable change of variables `(\u03c3, q) \u21a6 (\u03c3, \u03c3 q)` as an equivalence.","title":"Ginibre's duplicated-variable change of variables"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/unnorm-gks2","kind":"node","lean":"StatMech.Ising.unnorm_gks2","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#unnorm-gks2","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The unnormalized second Griffiths inequality: `Z\u00b2 (\u27e8\u03c3_{A \u2206 B}\u27e9 - \u27e8\u03c3_A\u27e9\u27e8\u03c3_B\u27e9) \u2265 0`.","title":"Unnormalized second Griffiths inequality"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/gks-second","kind":"node","lean":"StatMech.Ising.gks_second","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#gks-second","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"For the ferromagnetic Ising model on a finite graph with `\u03b2 \u2265 0` and `h \u2265 0`, spin correlations are positively correlated: `\u27e8\u03c3_A\u27e9 \u27e8\u03c3_B\u27e9 \u2264 \u27e8\u03c3_A \u03c3_B\u27e9 = \u27e8\u03c3_{A \u2206 B}\u27e9`.","title":"Second Griffiths inequality (GKS II)"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/energy-obs","kind":"node","lean":"StatMech.IsingFK.energyObs","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#energy-obs","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The energy observable `\u2211_{e \u2208 E(G)} \u03c3_x \u03c3_y + h \u2211_x \u03c3_x`, the `\u03b2`-derivative of the log-weight.","title":"Energy observable of the Ising model"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/corr-monotone","kind":"node","lean":"StatMech.IsingFK.corr_monotone","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#corr-monotone","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"Monotonicity of correlations (a consequence of the second Griffiths inequality): for `h \u2265 0`, every spin correlation `\u27e8\u03c3_A\u27e9_{\u03b2,h}` is nondecreasing in `\u03b2 \u2265 0` (Griffiths, J. Math. Phys. 8 (1967)).","title":"Monotonicity of correlations"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/w-j","kind":"node","lean":"StatMech.Sharpness.wJ","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#w-j","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The Boltzmann weight `exp(\u2211_{e \u2208 E} J_e \u03c3_e + \u2211_x h_x \u03c3_x)`.","title":"Boltzmann weight with inhomogeneous couplings and site fields"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/zj","kind":"node","lean":"StatMech.Sharpness.ZJ","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#zj","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The partition function `\u2211_\u03c3 wJ E J h \u03c3`.","title":"Partition function with inhomogeneous couplings and site fields"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/exp-j","kind":"node","lean":"StatMech.Sharpness.expJ","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#exp-j","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The Gibbs expectation `\u27e8f\u27e9_{E,J,h} = \u2211_\u03c3 f(\u03c3) w(\u03c3) / Z`.","title":"Gibbs expectation with inhomogeneous couplings and site fields"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/gks-second-j","kind":"node","lean":"StatMech.Sharpness.gks_second_J","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#gks-second-j","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"Griffiths' second inequality for nonnegative couplings and fields: `\u27e8\u03c3_A\u27e9\u27e8\u03c3_B\u27e9 \u2264 \u27e8\u03c3_{A \u2206 B}\u27e9`. Reference: R. B. Griffiths, \"Correlations in Ising ferromagnets I\", J. Math. Phys. 8 (1967) 478\u2013483; J. Ginibre, \"General formulation of Griffiths' inequalities\", Comm. Math. Phys. 16 (1970) 310\u2013328. See also `gks_second` (GKS2) for uniform couplings.","title":"Griffiths' second inequality for inhomogeneous couplings"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/griffiths-mono","kind":"node","lean":"StatMech.Sharpness.griffiths_mono","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#griffiths-mono","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"Griffiths monotonicity: adding edges with nonnegative couplings increases every correlation `\u27e8\u03c3_A\u27e9`. Reference: R. B. Griffiths, \"Correlations in Ising ferromagnets I\", J. Math. Phys. 8 (1967) 478\u2013483.","title":"Griffiths monotonicity of correlations in the edge set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/ghy-zeta","kind":"node","lean":"StatMech.Walls.ghy_zeta","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#ghy-zeta","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The edge-removal ratio `\u03b6(S) = (Z_{E\u2216S} / Z_E) \u220f_{e \u2208 S} cosh J_e`.","title":"Edge-removal ratio of partition functions"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities/ghy-rho","kind":"node","lean":"StatMech.Walls.ghy_rho","parent":48892,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-inequalities/index.html#ghy-rho","scope":"StatisticalMechanics/correlation-inequalities/griffiths-inequalities","status":"fully_proved","summary":"The weight `\u03c1(S) = \u03b6(S) \u220f_{e \u2208 S} tanh J_e = (Z_{E\u2216S} / Z_E) \u220f_{e \u2208 S} sinh J_e`.","title":"Edge-removal weight with hyperbolic sine factors"},{"data_url":"scopes/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality.json","id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","kind":"scope","navigation":"module","parent":48890,"result":false,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","title":"Griffiths\u2013Hurst\u2013Sherman inequality"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/aie-all-conn","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.aie_allConn","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#aie-all-conn","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The current `m` connects each of the pairs `{x, y}`, `{o, y}`, `{o, x}`.","title":"Current connecting all three site pairs"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/aie-none-conn","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.aie_noneConn","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#aie-none-conn","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The current `m` connects none of the pairs `{x, y}`, `{o, y}`, `{o, x}`.","title":"Current connecting none of the three site pairs"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/aie-mass","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.aie_mass","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#aie-mass","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The total weight `\u2211 \u03c6(K)` of the sub-currents `K \u2286 m` with source set `S`.","title":"Mass of sub-currents with a given source set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/aie-gap","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.aie_gap","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#aie-gap","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The GHS gap `mass(A) - mass(A \u2206 {x, y}) - mass(A \u2206 {o, y}) - mass(A \u2206 {o, x})`.","title":"GHS gap of a sourced current"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ahd-tri-ends","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.ahd_triEnds","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ahd-tri-ends","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The three edges `{0, 1}`, `{1, 2}`, `{0, 2}` of the triangle on `Fin 3`.","title":"Edges of the triangle on three vertices"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ahd-hdom-refutable","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.ahd_hdom_refutable","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ahd-hdom-refutable","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The domination condition fails for the single current through all edges of the triangle.","title":"Domination condition fails on the triangle"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/asd-ghs-backbone-factor","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.asd_ghsBackboneFactor","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#asd-ghs-backbone-factor","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The three-site backbone factor: `1`, `0` or `-2` according to how many of the pairs `{x, y}, {o, y}, {o, x}` the current connects.","title":"Three-site backbone factor"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/atb-backbone","kind":"node","lean":"StatMech.Ising.atb_backbone","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#atb-backbone","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The cluster of `o` in the current `m`: the sites connected to `o` by edges of `m`.","title":"Backbone cluster of a site in a current"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/atb-conditioned-dom","kind":"node","lean":"StatMech.Ising.atb_conditionedDom","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#atb-conditioned-dom","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The domination condition `2 \u00b7 mass(all connected) \u2264 mass(none connected)`, summed over currents in `M`.","title":"Conditioned domination condition over a set of currents"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-qfactor","kind":"node","lean":"StatMech.Ising.ghsQFactor","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-qfactor","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The three-site duplicated-variable factor `1 - \u03c3_x \u03c3_y - \u03c3_o \u03c3_y - \u03c3_o \u03c3_x` of the configuration `q`.","title":"Three-site duplicated-variable factor"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ising-exp3","kind":"node","lean":"StatMech.Ising.isingExp3","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ising-exp3","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The expectation of `F a b c` over three independent copies `a, b, c` of the Ising Gibbs distribution.","title":"Three-replica Ising expectation"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/dvar","kind":"node","lean":"StatMech.Ising.dvar","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Three-replica variable "],["math","\\(d_w\\)"]],"source":"Three-replica variable $d_w$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#dvar","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The three-replica variable `d_w = \u03c3_w(b) + \u03c3_w(c) - 2\u03c3_w(a)`.","title":"Three-replica variable $d_w$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ising-exp2","kind":"node","lean":"StatMech.Ising.isingExp2","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ising-exp2","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The expectation of `F a b` over two independent copies `a, b` of the Ising Gibbs distribution.","title":"Two-replica Ising expectation"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/uvar","kind":"node","lean":"StatMech.Ising.uvar","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rotated two-replica difference variable "],["math","\\(u_w\\)"]],"source":"Rotated two-replica difference variable $u_w$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#uvar","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The rotated two-replica variable `u_w = (\u03c3_w(a) - \u03c3_w(b)) / 2`.","title":"Rotated two-replica difference variable $u_w$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/tvar","kind":"node","lean":"StatMech.Ising.tvar","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rotated two-replica sum variable "],["math","\\(t_w\\)"]],"source":"Rotated two-replica sum variable $t_w$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#tvar","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The rotated two-replica variable `t_w = (\u03c3_w(a) + \u03c3_w(b)) / 2`.","title":"Rotated two-replica sum variable $t_w$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-vertex-edges","kind":"node","lean":"StatMech.Ising.ghsvp_vertexEdges","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-vertex-edges","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The edges of `E` with both endpoints in `S` (the edges of the induced subgraph on `S`).","title":"Edges of the induced subgraph on a vertex set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-vertex-z","kind":"node","lean":"StatMech.Ising.ghsvp_vertexZ","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-vertex-z","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field partition function of the induced subgraph on `S`, with couplings `J`.","title":"Zero-field partition function of an induced subgraph"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-vertex-corr","kind":"node","lean":"StatMech.Ising.ghsvp_vertexCorr","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-vertex-corr","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field correlation `\u27e8\u03c3_P\u27e9` in the induced subgraph on `S`.","title":"Zero-field correlation in an induced subgraph"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-vertex-corr-field","kind":"node","lean":"StatMech.Ising.ghsvp_vertexCorrField","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-vertex-corr-field","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The correlation `\u27e8\u03c3_P\u27e9` in the induced subgraph on `S`, with field `hf`.","title":"Correlation with field in an induced subgraph"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-merge-config","kind":"node","lean":"StatMech.Ising.ghsvp_mergeConfig","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-merge-config","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The configuration equal to `a` on `S` and to `b` off `S`.","title":"Configuration merged from two configurations along a set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-split-config","kind":"node","lean":"StatMech.Ising.ghsvp_splitConfig","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-split-config","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"Configurations split into their restrictions to `S` and to its complement.","title":"Splitting a configuration into a set and its complement"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-active-z","kind":"node","lean":"StatMech.Ising.ghsvp_activeZ","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-active-z","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The partition function of the induced subgraph on `S`, summed over configurations on `S` only.","title":"Induced-subgraph partition function over the active sites only"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-some-set","kind":"node","lean":"StatMech.Ising.ghsvp_someSet","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-some-set","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The image of `A` among the non-ghost vertices `some x` of `Option V`.","title":"Image of a vertex set among non-ghost vertices"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-field-set","kind":"node","lean":"StatMech.Ising.ghsvp_fieldSet","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-field-set","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The image of `A` together with the ghost vertex `none`.","title":"Image of a vertex set together with the ghost vertex"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-subset-mass","kind":"node","lean":"StatMech.Ising.ghsvp_subsetMass","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-subset-mass","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The weight of the agreement set `A` in the duplicated system: the product of the ghost-vertex partition functions of `A \u222a {ghost}` and of `A\u1d9c`, with doubled couplings `2\u03b2` and field `2\u03b2h`.","title":"Weight of an agreement set in the duplicated system"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-t-z","kind":"node","lean":"StatMech.Ising.ghsvp_tZ","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-t-z","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The partition function on the induced subgraph on `A` with couplings `2\u03b2` and field `2\u03b2h`.","title":"Induced-subgraph partition function with doubled coupling and field"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-u-z","kind":"node","lean":"StatMech.Ising.ghsvp_uZ","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-u-z","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field partition function on the induced subgraph on `A` with couplings `2\u03b2`.","title":"Zero-field induced-subgraph partition function with doubled coupling"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-disagreement-corr","kind":"node","lean":"StatMech.Ising.ghsvp_disagreementCorr","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-disagreement-corr","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field correlation `\u27e8\u03c3_o \u03c3_x\u27e9` in the subgraph induced by `A\u1d9c` (couplings `2\u03b2`), or `0` if `o` or `x` lies in `A`.","title":"Two-point correlation on the disagreement subgraph"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsvp-agreement-mag","kind":"node","lean":"StatMech.Ising.ghsvp_agreementMag","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsvp-agreement-mag","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The magnetization `\u27e8\u03c3_y\u27e9` in the subgraph induced by `A` (couplings `2\u03b2`, field `2\u03b2h`), or `0` if `y \u2209 A`.","title":"Magnetization on the agreement subgraph"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-ghost-coupling","kind":"node","lean":"StatMech.Ising.ghsiGhostCoupling","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-ghost-coupling","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The ghost-vertex couplings: `2K_{xy}` on an edge `{x, y}`, `2h_x` between `x` and the ghost.","title":"Inhomogeneous ghost-vertex couplings"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-tz","kind":"node","lean":"StatMech.Ising.ghsiTZ","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-tz","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The partition function of the subgraph induced by `A` with couplings `2K` and fields `2h` on `A`.","title":"Inhomogeneous induced-subgraph partition function with field"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-uz","kind":"node","lean":"StatMech.Ising.ghsiUZ","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-uz","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field partition function of the subgraph induced by `A` with couplings `2K`.","title":"Inhomogeneous zero-field induced-subgraph partition function"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-subset-mass","kind":"node","lean":"StatMech.Ising.ghsiSubsetMass","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-subset-mass","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The weight of the agreement set `A` in the duplicated inhomogeneous system.","title":"Inhomogeneous weight of an agreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/total-spin","kind":"node","lean":"StatMech.Sharpness.totalSpin","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#total-spin","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The total spin `\u2211_x \u03c3_x` of a configuration.","title":"Total spin of a configuration"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/cov-nonneg","kind":"node","lean":"StatMech.Sharpness.cov_nonneg","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#cov-nonneg","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"Griffiths' second inequality for two distinct spins: `\u27e8\u03c3_x\u27e9\u27e8\u03c3_y\u27e9 \u2264 \u27e8\u03c3_x \u03c3_y\u27e9` for `\u03b2, h \u2265 0`.","title":"Griffiths' second inequality for two distinct spins"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/deriv-magnetization-eq-sum-cov","kind":"node","lean":"StatMech.Sharpness.deriv_magnetization_eq_sum_cov","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#deriv-magnetization-eq-sum-cov","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The field derivative of the magnetization is `\u03b2` times the sum of the two-point covariances `\u27e8\u03c3_o \u03c3_x\u27e9 - \u27e8\u03c3_o\u27e9\u27e8\u03c3_x\u27e9` (the finite-volume susceptibility).","title":"Field derivative of the magnetization as a sum of covariances"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/flip-cfg","kind":"node","lean":"StatMech.Sharpness.flipCfg","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#flip-cfg","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The global spin flip `\u03c3 \u21a6 -\u03c3`.","title":"Global spin flip"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/flip-equiv","kind":"node","lean":"StatMech.Sharpness.flipEquiv","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#flip-equiv","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The global spin flip as an involutive equivalence of configuration space.","title":"Global spin flip as an involutive equivalence"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/magnetization-zero-field","kind":"node","lean":"StatMech.Sharpness.magnetization_zero_field","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#magnetization-zero-field","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"At zero field the global spin-flip symmetry forces the magnetization `\u27e8\u03c3_o\u27e9_{\u03b2,0}` to vanish.","title":"Magnetization vanishes at zero field"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-susceptibility-bound","kind":"node","lean":"StatMech.Sharpness.ghs_susceptibility_bound","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Susceptibility bound "],["math","\\(\\chi(h) \\le m(h)/h \\le 1/h\\)"],["text"," from concavity"]],"source":"Susceptibility bound $\\chi(h) \\le m(h)/h \\le 1/h$ from concavity"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-susceptibility-bound","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"If the magnetization is concave in the field on `[0, \u221e)` (the GHS inequality), then for `h > 0` the susceptibility satisfies `\u03c7(h) \u2264 m(h)/h \u2264 1/h`. Reference: R. B. Griffiths, C. A. Hurst, S. Sherman, \"Concavity of magnetization of an Ising ferromagnet in a positive external field\", J. Math. Phys. 11 (1970) 790\u2013795.","title":"Susceptibility bound $\\chi(h) \\le m(h)/h \\le 1/h$ from concavity"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lagreement","kind":"node","lean":"StatMech.Ising.ghsLAgreement","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lagreement","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The agreement configuration of `a` and `b`: `true` exactly at the sites where they agree.","title":"Agreement configuration of two spin configurations"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lmate","kind":"node","lean":"StatMech.Ising.ghsLMate","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lmate","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The configuration that agrees with `a` where `q` is `true` and disagrees where `q` is `false`.","title":"Mate of a configuration over an agreement pattern"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lagree-finset","kind":"node","lean":"StatMech.Ising.ghsLAgreeFinset","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lagree-finset","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The set of sites where `q` is `true`.","title":"Set of sites where an agreement configuration is true"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-ltweight","kind":"node","lean":"StatMech.Ising.ghsLTWeight","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-ltweight","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The Boltzmann weight, with couplings `2\u03b2` and field `2\u03b2h`, of `a` on the subgraph induced by the agreement set of `q`.","title":"Agreement-set Boltzmann weight with doubled couplings and field"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-luweight","kind":"node","lean":"StatMech.Ising.ghsLUWeight","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-luweight","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field Boltzmann weight, with couplings `2\u03b2`, of `a` on the subgraph induced by the disagreement set of `q`.","title":"Zero-field disagreement-set Boltzmann weight"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lsplit-equiv","kind":"node","lean":"StatMech.Ising.ghsLSplitEquiv","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lsplit-equiv","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"Configurations split into their restrictions to the agreement and disagreement sets of `q`.","title":"Splitting a configuration along the agreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-ltcomplete","kind":"node","lean":"StatMech.Ising.ghsLTComplete","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-ltcomplete","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The configuration with given spins on the agreement set of `q` and `false` elsewhere.","title":"Completion of agreement-set spins to a full configuration"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lucomplete","kind":"node","lean":"StatMech.Ising.ghsLUComplete","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lucomplete","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The configuration with given spins on the disagreement set of `q` and `false` elsewhere.","title":"Completion of disagreement-set spins to a full configuration"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lxnor-equiv","kind":"node","lean":"StatMech.Ising.ghsLXnorEquiv","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lxnor-equiv","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The change of variables `(a, b) \u21a6 (agreement of a and b, a)` on pairs of configurations.","title":"Change of variables from replica pairs to agreement and first copy"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lfibre-weight","kind":"node","lean":"StatMech.Ising.ghsLFibreWeight","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lfibre-weight","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The two-replica weight of `a` and its mate in the fibre over the agreement configuration `q`.","title":"Two-replica weight in an agreement fibre"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lfibre-mass","kind":"node","lean":"StatMech.Ising.ghsLFibreMass","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lfibre-mass","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The total two-replica weight of the fibre over `q`.","title":"Total two-replica weight of an agreement fibre"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lfibre-numerator","kind":"node","lean":"StatMech.Ising.ghsLFibreNumerator","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lfibre-numerator","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The two-replica weight of the fibre over `q`, integrated against `F`.","title":"Fibre weight integrated against an observable"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lfibre-expectation","kind":"node","lean":"StatMech.Ising.ghsLFibreExpectation","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lfibre-expectation","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The conditional two-replica expectation of `F` given the agreement configuration `q`.","title":"Conditional two-replica expectation given the agreement"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lagreement-prob","kind":"node","lean":"StatMech.Ising.ghsLAgreementProb","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lagreement-prob","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The probability that two independent replicas have agreement configuration `q`.","title":"Law of the agreement configuration of two replicas"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lfibre-uu","kind":"node","lean":"StatMech.Ising.ghsLFibreUU","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional expectation of "],["math","\\(u_o u_x\\)"],["text"," given the agreement"]],"source":"Conditional expectation of $u_o u_x$ given the agreement"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lfibre-uu","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The conditional expectation `E[u_o u_x | q]` of the rotated variables.","title":"Conditional expectation of $u_o u_x$ given the agreement"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lfibre-t","kind":"node","lean":"StatMech.Ising.ghsLFibreT","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Conditional expectation of "],["math","\\(t_y\\)"],["text"," given the agreement"]],"source":"Conditional expectation of $t_y$ given the agreement"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lfibre-t","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The conditional expectation `E[t_y | q]` of the rotated variable.","title":"Conditional expectation of $t_y$ given the agreement"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-luexpectation","kind":"node","lean":"StatMech.Ising.ghsLUExpectation","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-luexpectation","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field expectation on the subgraph induced by the disagreement set of `q`.","title":"Zero-field expectation on the disagreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-ltexpectation","kind":"node","lean":"StatMech.Ising.ghsLTExpectation","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-ltexpectation","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The expectation with field on the subgraph induced by the agreement set of `q`.","title":"Expectation with field on the agreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lout-field-weight","kind":"node","lean":"StatMech.Ising.ghsLOutFieldWeight","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lout-field-weight","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The field weight `exp (2\u03b2h \u2211_{v \u2209 q} \u03c3_v(a))` of the disagreement set.","title":"Field weight of the disagreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghslebowitz-fibre-factorization","kind":"node","lean":"StatMech.Ising.GHSLebowitzFibreFactorization","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Fibre factorization property for "],["math","\\(u_o u_x\\)"],["text"," and "],["math","\\(t_y\\)"]],"source":"Fibre factorization property for $u_o u_x$ and $t_y$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghslebowitz-fibre-factorization","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"Conditioned on the agreement configuration, `u_o u_x` and `t_y` are uncorrelated.","title":"Fibre factorization property for $u_o u_x$ and $t_y$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-lagreement-prob-fkg","kind":"node","lean":"StatMech.Ising.ghsLAgreementProb_fkg","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","FKG lattice condition for the agreement law when "],["math","\\(\\beta, h \\ge 0\\)"]],"source":"FKG lattice condition for the agreement law when $\\beta, h \\ge 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-lagreement-prob-fkg","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The law of the agreement configuration satisfies the FKG lattice condition.","title":"FKG lattice condition for the agreement law when $\\beta, h \\ge 0$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-l-three-point","kind":"node","lean":"StatMech.Ising.ghsL_threePoint","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-l-three-point","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"For the ferromagnetic Ising model on a finite graph with `\u03b2 \u2265 0` and `h \u2265 0`, the third Ursell function `u\u2083(o, x, y)` is nonpositive for every site `o` and every edge `{x, y}`.","title":"Griffiths\u2013Hurst\u2013Sherman inequality"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/shg-concave-on-deriv-le-slope","kind":"node","lean":"StatMech.Sharpness.shg_concaveOn_deriv_le_slope","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#shg-concave-on-deriv-le-slope","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The derivative of a concave function on `[0, \u221e)` at `b` is at most the secant slope over `[a, b]`. This is Mathlib's `ConcaveOn.deriv_le_slope` with the slope unfolded.","title":"Derivative of a concave function is at most a secant slope"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/shg-concave-on-of-deriv2-nonpos","kind":"node","lean":"StatMech.Sharpness.shg_concaveOn_of_deriv2_nonpos","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Concavity from a nonpositive second derivative on "],["math","\\((0, \\infty)\\)"]],"source":"Concavity from a nonpositive second derivative on $(0, \\infty)$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#shg-concave-on-of-deriv2-nonpos","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"A continuous function on `[0, \u221e)`, twice differentiable on the interior with nonpositive second derivative, is concave. This is Mathlib's `concaveOn_of_deriv2_nonpos` specialized to `Ici 0`.","title":"Concavity from a nonpositive second derivative on $(0, \\infty)$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/shg-magnetization-susceptibility-bound","kind":"node","lean":"StatMech.Sharpness.shg_magnetization_susceptibility_bound","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Susceptibility bound "],["math","\\(\\chi(h) \\le m(h)/h\\)"],["text"," from concavity"]],"source":"Susceptibility bound $\\chi(h) \\le m(h)/h$ from concavity"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#shg-magnetization-susceptibility-bound","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"If the magnetization is concave in the field on `[0, \u221e)`, then for `h > 0` the susceptibility satisfies `\u03c7(h) \u2264 m(h)/h`, using `m(0) = 0`.","title":"Susceptibility bound $\\chi(h) \\le m(h)/h$ from concavity"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-exp2","kind":"node","lean":"StatMech.Ising.ghsiExp2","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-exp2","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The expectation of `F a b` over two independent copies of the Ising model with couplings `K` and fields `hf`.","title":"Two-replica expectation with inhomogeneous couplings"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-tweight","kind":"node","lean":"StatMech.Ising.ghsiTWeight","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-tweight","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The Boltzmann weight, with couplings `2K` and fields `2hf`, of `a` on the subgraph induced by the agreement set of `q`.","title":"Inhomogeneous agreement-set Boltzmann weight"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-uweight","kind":"node","lean":"StatMech.Ising.ghsiUWeight","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-uweight","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field Boltzmann weight, with couplings `2K`, of `a` on the subgraph induced by the disagreement set of `q`.","title":"Inhomogeneous zero-field disagreement-set weight"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-fibre-weight","kind":"node","lean":"StatMech.Ising.ghsiFibreWeight","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-fibre-weight","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The two-replica weight of `a` and its mate in the fibre over the agreement configuration `q`.","title":"Inhomogeneous two-replica weight in an agreement fibre"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-fibre-mass","kind":"node","lean":"StatMech.Ising.ghsiFibreMass","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-fibre-mass","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The total two-replica weight of the fibre over `q`.","title":"Inhomogeneous total weight of an agreement fibre"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-fibre-numerator","kind":"node","lean":"StatMech.Ising.ghsiFibreNumerator","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-fibre-numerator","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The two-replica weight of the fibre over `q`, integrated against `F`.","title":"Inhomogeneous fibre weight integrated against an observable"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-fibre-expectation","kind":"node","lean":"StatMech.Ising.ghsiFibreExpectation","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-fibre-expectation","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The conditional two-replica expectation of `F` given the agreement configuration `q`.","title":"Inhomogeneous conditional expectation given the agreement"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-agreement-prob","kind":"node","lean":"StatMech.Ising.ghsiAgreementProb","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-agreement-prob","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The probability that two independent replicas have agreement configuration `q`.","title":"Inhomogeneous law of the agreement configuration"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-fibre-uu","kind":"node","lean":"StatMech.Ising.ghsiFibreUU","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inhomogeneous conditional expectation of "],["math","\\(u_o u_x\\)"]],"source":"Inhomogeneous conditional expectation of $u_o u_x$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-fibre-uu","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The conditional expectation `E[u_o u_x | q]` of the rotated variables.","title":"Inhomogeneous conditional expectation of $u_o u_x$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-fibre-t","kind":"node","lean":"StatMech.Ising.ghsiFibreT","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inhomogeneous conditional expectation of "],["math","\\(t_y\\)"]],"source":"Inhomogeneous conditional expectation of $t_y$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-fibre-t","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The conditional expectation `E[t_y | q]` of the rotated variable.","title":"Inhomogeneous conditional expectation of $t_y$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-uexpectation","kind":"node","lean":"StatMech.Ising.ghsiUExpectation","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-uexpectation","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field expectation on the subgraph induced by the disagreement set of `q`.","title":"Inhomogeneous zero-field expectation on the disagreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-texpectation","kind":"node","lean":"StatMech.Ising.ghsiTExpectation","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-texpectation","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The expectation with field on the subgraph induced by the agreement set of `q`.","title":"Inhomogeneous expectation with field on the agreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-disagreement-corr","kind":"node","lean":"StatMech.Ising.ghsiDisagreementCorr","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-disagreement-corr","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The zero-field correlation `\u27e8\u03c3_o \u03c3_x\u27e9` on the subgraph induced by `A\u1d9c` (couplings `2K`), or `0` if `o` or `x` lies in `A`.","title":"Inhomogeneous two-point correlation on the disagreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-agreement-mag","kind":"node","lean":"StatMech.Ising.ghsiAgreementMag","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-agreement-mag","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The magnetization `\u27e8\u03c3_y\u27e9` on the subgraph induced by `A` (couplings `2K`, fields `2hf`), or `0` if `y \u2209 A`.","title":"Inhomogeneous magnetization on the agreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-out-field-weight","kind":"node","lean":"StatMech.Ising.ghsiOutFieldWeight","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-out-field-weight","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The field weight `exp (\u2211_{v \u2209 q} 2h_v \u03c3_v(a))` of the disagreement set.","title":"Inhomogeneous field weight of the disagreement set"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-ursell-nonpos","kind":"node","lean":"StatMech.Ising.ghsi_ursell_nonpos","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-ursell-nonpos","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"Griffiths\u2013Hurst\u2013Sherman inequality for inhomogeneous ferromagnets. With couplings `K_e \u2265 0` and fields `h_x \u2265 0`, the third Ursell function satisfies `u\u2083(o, x, y) \u2264 0` whenever `o \u2260 x`.","title":"Griffiths\u2013Hurst\u2013Sherman inequality"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-dir-spin","kind":"node","lean":"StatMech.Ising.ghsiDirSpin","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-dir-spin","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The directional spin `\u2211_x d_x \u03c3_x`.","title":"Directional spin sum"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-field-line","kind":"node","lean":"StatMech.Ising.ghsiFieldLine","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-field-line","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The field line `t \u21a6 h + t\u00b7d`.","title":"Field line in a given direction"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-directional-covariance","kind":"node","lean":"StatMech.Ising.ghsiDirectionalCovariance","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-directional-covariance","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The covariance of `f` with the directional spin `\u2211_x d_x \u03c3_x`.","title":"Covariance with the directional spin"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-ursell3","kind":"node","lean":"StatMech.Ising.ghsiUrsell3","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-ursell3","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The third Ursell function `u\u2083(o, x, y)` of the inhomogeneous Ising model.","title":"Third Ursell function of the inhomogeneous Ising model"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-ursell3-nonpos","kind":"node","lean":"StatMech.Ising.ghsiUrsell3_nonpos","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","Third Ursell function nonpositive for "],["math","\\(o \\ne x\\)"],["text",", "],["math","\\(K, h \\ge 0\\)"]],"source":"Third Ursell function nonpositive for $o \\ne x$, $K, h \\ge 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-ursell3-nonpos","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The GHS inequality `u\u2083(o, x, y) \u2264 0` for the named Ursell function, when `o \u2260 x`.","title":"Third Ursell function nonpositive for $o \\ne x$, $K, h \\ge 0$"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-covariance","kind":"node","lean":"StatMech.Ising.ghsiCovariance","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-covariance","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The spin covariance `\u27e8\u03c3_o \u03c3_x\u27e9 - \u27e8\u03c3_o\u27e9\u27e8\u03c3_x\u27e9`.","title":"Spin covariance in the inhomogeneous model"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghsi-magnetization-field-line-concave-on","kind":"node","lean":"StatMech.Ising.ghsi_magnetization_fieldLine_concaveOn","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghsi-magnetization-field-line-concave-on","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"With couplings `K_e \u2265 0`, fields `h_x \u2265 0`, and a direction `d \u2265 0` with `d_o = 0`, the magnetization `\u27e8\u03c3_o\u27e9` is concave in `t \u2265 0` along the field line `h + t\u00b7d`.","title":"GHS concavity"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/gc4-d1cov2","kind":"node","lean":"StatMech.Walls.gc4_D1cov2","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#gc4-d1cov2","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The field derivative of the two-point covariance `Cov(\u03c3_o, \u03c3_x)`, written as a combination of Gibbs expectations involving the total spin.","title":"Field derivative of the two-point covariance"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/eg-ursell3","kind":"node","lean":"StatMech.Walls.GhcEqGap.eg_ursell3","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#eg-ursell3","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The third Ursell function (joint cumulant) `u\u2083(o, x, y) = \u27e8\u03c3_o\u03c3_x\u03c3_y\u27e9 - \u27e8\u03c3_o\u27e9\u27e8\u03c3_x\u03c3_y\u27e9 - \u27e8\u03c3_o\u03c3_x\u27e9\u27e8\u03c3_y\u27e9 - \u27e8\u03c3_o\u03c3_y\u27e9\u27e8\u03c3_x\u27e9 + 2\u27e8\u03c3_o\u27e9\u27e8\u03c3_x\u27e9\u27e8\u03c3_y\u27e9` of the finite Ising model at `(\u03b2, h)`.","title":"Third Ursell function of the finite Ising model"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghc-ursell-eq-gap","kind":"node","lean":"StatMech.Walls.GhcEqGap.ghc_ursellEqGap","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghc-ursell-eq-gap","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"On an edge `{x, y}`, the gap `cov3sym - ghsBoundSym` between the three-point covariance and the GHS bound equals `u\u2083(o, x, y)`.","title":"GHS gap on an edge equals the third Ursell function"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/gc-lebowitz-u4","kind":"node","lean":"StatMech.Walls.gc_lebowitzU4","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#gc-lebowitz-u4","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"Lebowitz's four-point combination `\u27e8\u03c3_o\u03c3_x\u03c3_y\u27e9 - \u27e8\u03c3_o\u03c3_x\u27e9\u27e8\u03c3_y\u27e9 - \u27e8\u03c3_o\u03c3_y\u27e9\u27e8\u03c3_x\u27e9 - \u27e8\u03c3_o\u27e9\u27e8\u03c3_x\u03c3_y\u27e9`, which differs from the Ursell function `u\u2083` by `2\u27e8\u03c3_o\u27e9\u27e8\u03c3_x\u27e9\u27e8\u03c3_y\u27e9`.","title":"Lebowitz four-point combination"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghssign-dominance","kind":"node","lean":"StatMech.Walls.GHSSignDominance","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghssign-dominance","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The GHS sign condition at `(\u03b2, h)` and root `o`: the third Ursell function `u\u2083(o, x, y)` is nonpositive for all pairwise distinct `o, x, y`. It holds for `\u03b2, h \u2265 0` by `ghs_signDominance`.","title":"GHS sign condition"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-sign-dominance","kind":"node","lean":"StatMech.Sharpness.ghs_signDominance","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-sign-dominance","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The Griffiths\u2013Hurst\u2013Sherman inequality: for `\u03b2 \u2265 0` and `h \u2265 0`, the third Ursell function `u\u2083(o, x, y)` is nonpositive for distinct sites. Reference: R. B. Griffiths, C. A. Hurst, S. Sherman, \"Concavity of magnetization of an Ising ferromagnet in a positive external field\", J. Math. Phys. 11 (1970) 790\u2013795; proof via J. L. Lebowitz, \"GHS and other inequalities\", Comm. Math. Phys. 35 (1974) 87\u201392.","title":"Griffiths\u2013Hurst\u2013Sherman inequality"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-magnetization-concave-on","kind":"node","lean":"StatMech.Sharpness.ghs_magnetization_concaveOn","parent":48916,"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-magnetization-concave-on","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"GHS concavity: for `\u03b2 \u2265 0` the magnetization `h \u21a6 \u27e8\u03c3_o\u27e9_{\u03b2,h}` is concave on `[0, \u221e)`. Reference: R. B. Griffiths, C. A. Hurst, S. Sherman, \"Concavity of magnetization of an Ising ferromagnet in a positive external field\", J. Math. Phys. 11 (1970) 790\u2013795.","title":"GHS concavity of the magnetization in the field"},{"id":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/ghs-susceptibility-le-inv-field","kind":"node","lean":"StatMech.Sharpness.ghs_susceptibility_le_inv_field","parent":48916,"reading_title":{"format":"inline-parts-v1","parts":[["text","GHS susceptibility bound "],["math","\\(\\chi \\le 1/h\\)"]],"source":"GHS susceptibility bound $\\chi \\le 1/h$"},"result":true,"route":"../roadmap/StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality/index.html#ghs-susceptibility-le-inv-field","scope":"StatisticalMechanics/correlation-inequalities/griffiths-hurst-sherman-inequality","status":"fully_proved","summary":"The GHS susceptibility bound: for `\u03b2 \u2265 0` and `h > 0`, the field derivative of the magnetization equals `\u03b2 \u2211_x (\u27e8\u03c3_o\u03c3_x\u27e9 - \u27e8\u03c3_o\u27e9\u27e8\u03c3_x\u27e9)` and is at most `1/h`. Reference: R. B. Griffiths, C. A. Hurst, S. Sherman, \"Concavity of magnetization of an Ising ferromagnet in a positive external field\", J. Math. Phys. 11 (1970) 790\u2013795. Scope: a finite graph with uniform coupling and uniform field, the field entering the\u2026","title":"GHS susceptibility bound $\\chi \\le 1/h$"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions","kind":"scope","parent":47780,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions","status":"fully_proved","summary":"Infinite-volume states of the Ising model and its phase transition: weak limits of box measures and the DLR condition, translation invariance, ergodicity and extremality of the plus state, spontaneous magnetization and the critical temperature, and Peierls' contour argument for long-range order.","title":"Gibbs states and phase transitions"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","kind":"scope","navigation":"module","parent":49013,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","title":"DLR states of the Ising model"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/up-meas-system","kind":"node","lean":"StatMech.Ising.upMeasSystem","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#up-meas-system","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The measurable increasing events of the configuration space.","title":"Measurable increasing events of the configuration space"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/gibbs-mix","kind":"node","lean":"StatMech.Ising.gibbsMix","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#gibbs-mix","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The mixture `a \u03bc\u207b + b \u03bc\u207a` of two measures.","title":"Weighted mixture of two measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/box-finset","kind":"node","lean":"StatMech.Ising.boxFinset","parent":49014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box "],["math","\\((-n, n)^d\\)"],["text"," as a finite set of sites"]],"source":"Box $(-n, n)^d$ as a finite set of sites"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#box-finset","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The box `\u039b_n = [-n, n]^d` as a finite set of sites.","title":"Box $(-n, n)^d$ as a finite set of sites"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/bond-pairs-touch","kind":"node","lean":"StatMech.Ising.bondPairsTouch","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#bond-pairs-touch","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"Ordered nearest-neighbour pairs with at least one endpoint in `\u039b_n`.","title":"Ordered nearest-neighbour pairs touching the box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/bond-pairs-internal","kind":"node","lean":"StatMech.Ising.bondPairsInternal","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#bond-pairs-internal","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"Ordered nearest-neighbour pairs with both endpoints in `\u039b_n`.","title":"Ordered nearest-neighbour pairs inside the box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/bond-finset-touch","kind":"node","lean":"StatMech.Ising.bondFinsetTouch","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#bond-finset-touch","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The bonds with at least one endpoint in `\u039b_n` (used for plus and minus boundary conditions).","title":"Bonds touching the box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/bond-finset-internal","kind":"node","lean":"StatMech.Ising.bondFinsetInternal","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#bond-finset-internal","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The bonds with both endpoints in `\u039b_n` (used for the free boundary condition).","title":"Bonds inside the box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/glue","kind":"node","lean":"StatMech.Ising.glue","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#glue","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The configuration equal to `\u03c4` inside the box `\u039b_n` and to the boundary condition `\u03b7` outside.","title":"Configuration glued from inside spins and a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/fv-energy","kind":"node","lean":"StatMech.Ising.fvEnergy","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#fv-energy","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The box energy `-\u2211_{e \u2208 B} \u03c3_e - h \u2211_{x \u2208 \u039b_n} \u03c3_x` of the configuration `glue \u03b7 \u03c4`.","title":"Box energy with a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/fv-weight","kind":"node","lean":"StatMech.Ising.fvWeight","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#fv-weight","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The Boltzmann weight `exp (-\u03b2 H)` of an inside configuration `\u03c4` with boundary condition `\u03b7`.","title":"Finite-volume Boltzmann weight with a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/fv-z","kind":"node","lean":"StatMech.Ising.fvZ","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#fv-z","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The finite-volume partition function with boundary condition `\u03b7`.","title":"Finite-volume partition function with a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/fv-prob","kind":"node","lean":"StatMech.Ising.fvProb","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#fv-prob","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The finite-volume Gibbs probability of an inside configuration `\u03c4` with boundary condition `\u03b7`.","title":"Finite-volume Gibbs probability with a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/fv-measure","kind":"node","lean":"StatMech.Ising.fvMeasure","parent":49014,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite-volume Gibbs measure on configurations of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Finite-volume Gibbs measure on configurations of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#fv-measure","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The finite-volume Gibbs measure in `\u039b_n` with boundary condition `\u03b7` and bond set `B`, as a measure on configurations of `\u2124^d`.","title":"Finite-volume Gibbs measure on configurations of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/fv-probability-measure","kind":"node","lean":"StatMech.Ising.fvProbabilityMeasure","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#fv-probability-measure","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The finite-volume Gibbs measure as a `ProbabilityMeasure`.","title":"Finite-volume Gibbs measure as a probability measure"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/plus-field","kind":"node","lean":"StatMech.Ising.plusField","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#plus-field","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The all-plus configuration.","title":"All-plus configuration"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/minus-field","kind":"node","lean":"StatMech.Ising.minusField","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#minus-field","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The all-minus configuration.","title":"All-minus configuration"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/plus-measure","kind":"node","lean":"StatMech.Ising.plusMeasure","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#plus-measure","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The Ising measure in `\u039b_n` with plus boundary condition.","title":"Plus-boundary Ising measure in a box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/minus-measure","kind":"node","lean":"StatMech.Ising.minusMeasure","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#minus-measure","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The Ising measure in `\u039b_n` with minus boundary condition.","title":"Minus-boundary Ising measure in a box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/free-measure","kind":"node","lean":"StatMech.Ising.freeMeasure","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#free-measure","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The Ising measure in `\u039b_n` with free boundary condition (internal bonds only).","title":"Free-boundary Ising measure in a box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/infinite-volume-state","kind":"node","lean":"StatMech.Ising.InfiniteVolumeState","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#infinite-volume-state","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"`\u03bc` is an infinite-volume state of the sequence `\u03c1`: some subsequence of `\u03c1` converges weakly to `\u03bc`.","title":"Infinite-volume state of a sequence of measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/exists-infinite-volume-state","kind":"node","lean":"StatMech.Ising.exists_infiniteVolumeState","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#exists-infinite-volume-state","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"Every sequence of probability measures on configurations has an infinite-volume state, by Prokhorov compactness.","title":"Existence of infinite-volume states"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/limit-state","kind":"node","lean":"StatMech.Ising.limitState","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#limit-state","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"A subsequential weak limit of `\u03c1`, chosen by `exists_infiniteVolumeState`.","title":"Chosen subsequential weak limit of a sequence of measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/plus-state","kind":"node","lean":"StatMech.Ising.plusState","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#plus-state","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The plus state: a weak subsequential limit of the plus-boundary box measures. Full-sequence convergence is not claimed here: the library only provides this state as a chosen subsequential limit (see `plusState_isInfiniteVolumeState`).","title":"Infinite-volume plus state as a subsequential weak limit"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/minus-state","kind":"node","lean":"StatMech.Ising.minusState","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#minus-state","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The minus state: a weak subsequential limit of the minus-boundary box measures.","title":"Infinite-volume minus state as a subsequential weak limit"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/free-state","kind":"node","lean":"StatMech.Ising.freeState","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#free-state","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The free state: a weak subsequential limit of the free-boundary box measures.","title":"Infinite-volume free state as a subsequential weak limit"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/is-dlrstate","kind":"node","lean":"StatMech.Ising.IsDLRState","parent":49014,"reading_title":{"format":"inline-parts-v1","parts":[["text","DLR state of the Ising model on "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"DLR state of the Ising model on $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#is-dlrstate","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The DLR condition: for every `n`, the conditional distribution of `\u03bc` given the spins outside the box `\u039b_n` equals, for `\u03bc`-almost every `\u03b7`, the finite-volume Gibbs measure `fvMeasure \u03b7 n (bondFinsetTouch d n) \u03b2 h` with boundary condition `\u03b7`.","title":"DLR state of the Ising model on $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/fin-inter-system","kind":"node","lean":"StatMech.Ising.finInterSystem","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#fin-inter-system","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The finite intersections of sets from `S` (together with `univ`), a \u03c0-system generating the same \u03c3-algebra.","title":"Finite intersections of a family of sets"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/ising-spec","kind":"node","lean":"StatMech.Ising.isingSpec","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#ising-spec","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The Ising specification in `\u039b_n`: the kernel `\u03b7 \u21a6 fvMeasure \u03b7 n B \u03b2 h`, measurable with respect to the spins outside `\u039b_n`.","title":"Ising specification in a box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/convex-is-dlrstate","kind":"node","lean":"StatMech.Ising.convex_isDLRState","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#convex-is-dlrstate","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The DLR states form a convex set: `a \u03bc\u2081 + b \u03bc\u2082` with `a + b = 1` is a DLR state if `\u03bc\u2081`, `\u03bc\u2082` are.","title":"Convexity of the set of DLR states"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/stochastically-dominated-clopen","kind":"node","lean":"StatMech.Ising.StochasticallyDominatedClopen","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#stochastically-dominated-clopen","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"`\u03bc \u227cc \u03bd`: `\u03bc(A) \u2264 \u03bd(A)` for every increasing clopen event `A`.","title":"Stochastic domination on increasing clopen events"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/to-clopen","kind":"node","lean":"StatMech.Ising.StochasticallyDominated.toClopen","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#to-clopen","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"ENNReal stochastic domination gives real-mass clopen domination when the upper measure is finite.","title":"Stochastic domination implies clopen domination, upper measure finite"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/clopen-up-system","kind":"node","lean":"StatMech.Ising.clopenUpSystem","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#clopen-up-system","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The increasing clopen events, a \u03c0-system generating the Borel \u03c3-algebra.","title":"Increasing clopen events"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/gsi-gibbs-unique-of-phases-eq","kind":"node","lean":"StatMech.Ising.gsi_gibbs_unique_of_phases_eq","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#gsi-gibbs-unique-of-phases-eq","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"For `\u03b2, h \u2265 0`, if the plus and minus states agree then every DLR state that is a probability measure equals the plus state.","title":"Uniqueness of the Gibbs state"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/spec-apply","kind":"node","lean":"StatMech.Ising.specApply","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#spec-apply","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The specification applied to `f`: `\u03b7 \u21a6 \u222b f d(fvMeasure \u03b7 n B \u03b2 h)`.","title":"Specification applied to a function"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/spec-apply-bcf","kind":"node","lean":"StatMech.Ising.specApply_bcf","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#spec-apply-bcf","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The specification applied to a bounded continuous function, as a bounded continuous function.","title":"Specification applied to a bounded continuous function"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/fvconsistency","kind":"node","lean":"StatMech.Ising.FVConsistency","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#fvconsistency","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The plus measures in `\u039b_m` are invariant under the specification in every `\u039b_n \u2286 \u039b_m`.","title":"Consistency of the plus measures under the specification"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/ising-spec-full","kind":"node","lean":"StatMech.Ising.isingSpecFull","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#ising-spec-full","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The Ising specification in `\u039b_n` as a Markov kernel on the full \u03c3-algebra.","title":"Ising specification as a Markov kernel"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/outside-invariant-ms","kind":"node","lean":"StatMech.Ising.outsideInvariantMS","parent":49014,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\sigma\\)"],["text","-algebra of events determined outside a region"]],"source":"$\\sigma$-algebra of events determined outside a region"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#outside-invariant-ms","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The \u03c3-algebra of events determined by the spins outside `\u039b`.","title":"$\\sigma$-algebra of events determined outside a region"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/ovr-box","kind":"node","lean":"StatMech.Ising.ovrBox","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#ovr-box","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The configuration on `\u039b_m` obtained from `\u03c3` by overwriting its spins in `\u039b_n` with `\u03c4`.","title":"Overwriting a configuration on an inner box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/res-box","kind":"node","lean":"StatMech.Ising.resBox","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#res-box","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The restriction of a configuration on `\u039b_m` to `\u039b_n`.","title":"Restriction of a configuration to an inner box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/swap-pair","kind":"node","lean":"StatMech.Ising.swapPair","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#swap-pair","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"The involution of pairs exchanging the `\u039b_n` parts of a configuration on `\u039b_m` and a configuration on `\u039b_n`.","title":"Involution swapping inner-box parts of configuration pairs"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/plus-state-is-dlr","kind":"node","lean":"StatMech.Ising.plusState_isDLR","parent":49014,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model/index.html#plus-state-is-dlr","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/dlr-states-of-the-ising-model","status":"fully_proved","summary":"For all `\u03b2, h`, the plus state of the Ising model on `\u2124^d` satisfies the DLR condition.","title":"The plus state is a Gibbs (DLR) state"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model","kind":"scope","parent":49013,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model","status":"fully_proved","title":"Peierls argument for the Ising model"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/realised-contour-walk-encoding","kind":"node","lean":"StatMech.Lattice.RealisedContourWalkEncoding","parent":49057,"reading_title":{"format":"inline-parts-v1","parts":[["text","Closed-walk encoding of realised contours of length "],["math","\\(\\ell\\)"]],"source":"Closed-walk encoding of realised contours of length $\\ell$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/index.html#realised-contour-walk-encoding","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model","status":"fully_proved","summary":"Every realised contour of length `\u2113` in `\u039b_n` is decoded from a closed lattice walk of length `\u2113` based at one of at most `\u2113` anchor sites.","title":"Closed-walk encoding of realised contours of length $\\ell$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-contour-count","kind":"node","lean":"StatMech.Ising.ConnectedContourCount","parent":49057,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/index.html#connected-contour-count","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model","status":"fully_proved","summary":"For every `\u2113`, at most `\u2113 (2d)^\u2113` connected clusters through the origin in `\u039b_n` have contour length `\u2113`.","title":"Peierls count of connected clusters by contour length"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo","status":"fully_proved","title":"A connected cluster is determined by its edge boundary"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo/edge-straddles","kind":"node","lean":"StatMech.Lattice.edgeStraddles","parent":49060,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo/index.html#edge-straddles","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo","status":"fully_proved","summary":"An edge straddles `K` if exactly one of its endpoints lies in `K`.","title":"Edge with exactly one endpoint in a set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo/boundary-edge-set","kind":"node","lean":"StatMech.Lattice.boundaryEdgeSet","parent":49060,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo/index.html#boundary-edge-set","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo","status":"fully_proved","summary":"The edge boundary `\u2202K`: the lattice edges that straddle `K`.","title":"Edge boundary of a set of sites"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo/lattice-minus-boundary","kind":"node","lean":"StatMech.Lattice.latticeMinusBoundary","parent":49060,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo/index.html#lattice-minus-boundary","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/a-connected-cluster-is-determined-by-its-edge-bo","status":"fully_proved","summary":"The hypercubic lattice with the edges straddling `K` removed.","title":"Hypercubic lattice with the edge boundary removed"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or","status":"fully_proved","title":"Spin-flip symmetry and the Peierls long-range-order criterion"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/flip-config","kind":"node","lean":"StatMech.Ising.flipConfig","parent":49064,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/index.html#flip-config","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or","status":"fully_proved","summary":"The global spin flip `\u03c3 \u21a6 -\u03c3` of an infinite-volume configuration.","title":"Global spin flip of a configuration"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/flip-interior","kind":"node","lean":"StatMech.Ising.flipInterior","parent":49064,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/index.html#flip-interior","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or","status":"fully_proved","summary":"The spin flip of a configuration inside the box `\u039b_n`.","title":"Spin flip inside the box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/flip-interior-equiv","kind":"node","lean":"StatMech.Ising.flipInteriorEquiv","parent":49064,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/index.html#flip-interior-equiv","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or","status":"fully_proved","summary":"The spin flip inside the box as an involutive equivalence.","title":"Spin flip inside the box as an involutive equivalence"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/origin","kind":"node","lean":"StatMech.Ising.origin","parent":49064,"reading_title":{"format":"inline-parts-v1","parts":[["text","Origin of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Origin of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/index.html#origin","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or","status":"fully_proved","summary":"The origin of `\u2124^d`.","title":"Origin of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/fv-mag-origin","kind":"node","lean":"StatMech.Ising.fvMagOrigin","parent":49064,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or/index.html#fv-mag-origin","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/spin-flip-symmetry-and-the-peierls-long-range-or","status":"fully_proved","summary":"The magnetization `\u27e8\u03c3_0\u27e9` at the origin in the finite-volume Ising measure on `\u039b_n` with boundary condition `\u03b7`, bond set `B`, inverse temperature `\u03b2` and field `h`.","title":"Finite-volume magnetization at the origin"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec","status":"fully_proved","title":"Connected dual contours of the singleton and rectangular clusters"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec/singleton-boundary-support","kind":"node","lean":"StatMech.Lattice.singletonBoundarySupport","parent":49070,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec/index.html#singleton-boundary-support","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec","status":"fully_proved","summary":"The four faces meeting the origin, which support the contour of `{0}`.","title":"Four faces meeting the origin"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec/box-cluster","kind":"node","lean":"StatMech.Lattice.boxCluster","parent":49070,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rectangle "],["math","\\((0, m) \\times (0, n)\\)"],["text"," of sites"]],"source":"Rectangle $(0, m) \\times (0, n)$ of sites"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec/index.html#box-cluster","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec","status":"fully_proved","summary":"The rectangle `[0, m] \u00d7 [0, n]` of sites in `\u2124\u00b2`.","title":"Rectangle $(0, m) \\times (0, n)$ of sites"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec/box-frame","kind":"node","lean":"StatMech.Lattice.boxFrame","parent":49070,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec/index.html#box-frame","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/connected-dual-contours-of-the-singleton-and-rec","status":"fully_proved","summary":"The frame of faces just outside the rectangle `[0, m] \u00d7 [0, n]`, which supports its contour.","title":"Frame of faces around a rectangle"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","status":"fully_proved","title":"Euler circuits and single dual circuits for planar contours"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/degree-eq-card-filter-edges","kind":"node","lean":"StatMech.Lattice.EulerianExistence.degree_eq_card_filter_edges","parent":49074,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/index.html#degree-eq-card-filter-edges","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","status":"fully_proved","summary":"The degree of a vertex is the number of edges containing it. See also `SimpleGraph.card_incidenceFinset_eq_degree`.","title":"Degree of a vertex equals the number of edges containing it"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/splice-walk","kind":"node","lean":"StatMech.Lattice.EulerianExistence.spliceWalk","parent":49074,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/index.html#splice-walk","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","status":"fully_proved","summary":"Splice a closed walk `c` at a vertex `w` of the closed walk `p` into `p`.","title":"Splicing a closed walk into another at a shared vertex"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/exists-closed-eulerian","kind":"node","lean":"StatMech.Lattice.EulerianExistence.exists_closed_eulerian","parent":49074,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/index.html#exists-closed-eulerian","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","status":"fully_proved","summary":"Euler's theorem (existence direction): a finite connected graph with all degrees even has a closed trail through every edge, based at any vertex. See also `SimpleGraph.Connected.exists_isEulerian` in `MathlibExt`, which does not prescribe the base vertex.","title":"Euler's theorem"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/exists-closed-is-eulerian","kind":"node","lean":"StatMech.Lattice.EulerianExistence.exists_closed_isEulerian","parent":49074,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/index.html#exists-closed-is-eulerian","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","status":"fully_proved","summary":"The closed trail of `exists_closed_eulerian` is an Eulerian circuit in Mathlib's sense, based at any prescribed vertex. See also `SimpleGraph.Connected.exists_isEulerian`.","title":"Eulerian circuit at any vertex of a finite connected even graph"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/exists-closed-eulerian-of-finite-support","kind":"node","lean":"StatMech.Lattice.EulerianExistence.exists_closed_eulerian_of_finite_support","parent":49074,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/index.html#exists-closed-eulerian-of-finite-support","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","status":"fully_proved","summary":"Euler's theorem for a locally finite graph whose support lies in a finite set `T` on which the induced graph is connected. See also `SimpleGraph.ConnectedSupport.exists_isEulerian` in `MathlibExt`.","title":"Euler's theorem for a graph supported on a finite connected set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/face-boundary-connected","kind":"node","lean":"StatMech.Lattice.FaceBoundaryConnected","parent":49074,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/index.html#face-boundary-connected","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","status":"fully_proved","summary":"The face-boundary graph of `K` is connected on the finite set of faces `T`.","title":"Connectedness of the face-boundary graph on a face set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/boundary-support","kind":"node","lean":"StatMech.Lattice.boundarySupport","parent":49074,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan/index.html#boundary-support","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/euler-circuits-and-single-dual-circuits-for-plan","status":"fully_proved","summary":"A finite set of faces containing the support of the face-boundary graph of a finite set `K`.","title":"Finite face set containing the face-boundary support"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r","status":"fully_proved","title":"The Peierls argument: contour sums and the long-range-order criterion"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/peierls-bound","kind":"node","lean":"StatMech.Ising.peierlsBound","parent":49082,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/index.html#peierls-bound","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r","status":"fully_proved","summary":"The Peierls series `\u2211_\u2113 \u2113 (2d)^\u2113 e^{-2\u03b2\u2113}`.","title":"Peierls series"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/peierls-ratio","kind":"node","lean":"StatMech.Ising.peierlsRatio","parent":49082,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ratio "],["math","\\(2d e^{-2\\beta}\\)"],["text"," of the Peierls series"]],"source":"Ratio $2d e^{-2\\beta}$ of the Peierls series"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/index.html#peierls-ratio","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r","status":"fully_proved","summary":"The ratio `2d e^{-2\u03b2}` of the Peierls series.","title":"Ratio $2d e^{-2\\beta}$ of the Peierls series"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/prob-origin-minus","kind":"node","lean":"StatMech.Ising.probOriginMinus","parent":49082,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/index.html#prob-origin-minus","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r","status":"fully_proved","summary":"The probability that the origin spin is minus under the box measure with boundary condition `\u03b7`.","title":"Probability that the origin spin is minus"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/peierls-contour-bound","kind":"node","lean":"StatMech.Ising.PeierlsContourBound","parent":49082,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r/index.html#peierls-contour-bound","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-argument-contour-sums-and-the-long-r","status":"fully_proved","summary":"Under the plus box measure on `\u039b_n`, the origin is minus with probability at most `peierlsBound d \u03b2`.","title":"Peierls bound on the probability that the origin is minus"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters","status":"fully_proved","title":"Contours of minus clusters"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters/flip-on-set","kind":"node","lean":"StatMech.Ising.flipOnSet","parent":49087,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters/index.html#flip-on-set","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters","status":"fully_proved","summary":"The configuration with the spins in `K` flipped.","title":"Configuration with the spins in a set flipped"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters/crosses","kind":"node","lean":"StatMech.Ising.crosses","parent":49087,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters/index.html#crosses","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters","status":"fully_proved","summary":"The bond `e` has exactly one endpoint in `K`.","title":"Bond crossing the boundary of a set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters/cross-edges","kind":"node","lean":"StatMech.Ising.crossEdges","parent":49087,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters/index.html#cross-edges","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-minus-clusters","status":"fully_proved","summary":"The bonds of `B` crossing the boundary of `K` (the contour of `K`).","title":"Contour of a set: bonds crossing its 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$\\ell$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-clusters-in-a-box-and-closed-walk-en/ccb-contour-encoding-exists","kind":"node","lean":"StatMech.Lattice.ccb_ContourEncodingExists","parent":49112,"reading_title":{"format":"inline-parts-v1","parts":[["text","Injective closed-walk encoding of clusters of contour length "],["math","\\(\\ell\\)"]],"source":"Injective closed-walk encoding of clusters of contour length $\\ell$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-clusters-in-a-box-and-closed-walk-en/index.html#ccb-contour-encoding-exists","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/contours-of-clusters-in-a-box-and-closed-walk-en","status":"fully_proved","summary":"The clusters with contour length `\u2113` admit an injective encoding by closed lattice walks of 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$\\mathbb{Z}^2$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/clusters-as-open-clusters-of-indicator-configura/ccb-anchored-boundary-walk-exists","kind":"node","lean":"StatMech.Lattice.ccb_AnchoredBoundaryWalkExists","parent":49116,"reading_title":{"format":"inline-parts-v1","parts":[["text","Anchored closed-walk decoding of cluster contours in "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Anchored closed-walk decoding of cluster contours in $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/clusters-as-open-clusters-of-indicator-configura/index.html#ccb-anchored-boundary-walk-exists","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/clusters-as-open-clusters-of-indicator-configura","status":"fully_proved","summary":"In `\u2124\u00b2`, every connected cluster in `\u039b_n` with contour length `\u2113` has its contour decoded from a closed walk of length `\u2113` based in the anchor row.","title":"Anchored closed-walk decoding of cluster contours in $\\mathbb{Z}^2$"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis","status":"fully_proved","title":"Dual circuits anchored on a coordinate axis"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/per-contour-dual-circuit","kind":"node","lean":"StatMech.Lattice.PerContourDualCircuit","parent":49120,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/index.html#per-contour-dual-circuit","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis","status":"fully_proved","summary":"Every realised contour of length `\u2113` is decoded from a closed walk of length `\u2113` based at a site of a common pool of at most `\u2113` anchors.","title":"Closed-walk decoding of each realised contour from an anchor pool"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/axis-vertex","kind":"node","lean":"StatMech.Lattice.axisVertex","parent":49120,"reading_title":{"format":"inline-parts-v1","parts":[["text","Site "],["math","\\((j, 0, \\ldots, 0)\\)"],["text"," on the first coordinate axis"]],"source":"Site $(j, 0, \\ldots, 0)$ on the first coordinate axis"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/index.html#axis-vertex","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis","status":"fully_proved","summary":"The site `(j, 0, \u2026, 0)` on the first coordinate axis.","title":"Site $(j, 0, \\ldots, 0)$ on the first coordinate axis"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/axis-anchor-pool","kind":"node","lean":"StatMech.Lattice.axisAnchorPool","parent":49120,"reading_title":{"format":"inline-parts-v1","parts":[["text","First "],["math","\\(\\ell\\)"],["text"," sites of the first coordinate axis"]],"source":"First $\\ell$ sites of the first coordinate axis"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/index.html#axis-anchor-pool","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis","status":"fully_proved","summary":"The axis sites `(j, 0, \u2026, 0)` with `j < \u2113`.","title":"First $\\ell$ sites of the first coordinate axis"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/axis-contour-dual-circuit","kind":"node","lean":"StatMech.Lattice.AxisContourDualCircuit","parent":49120,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis/index.html#axis-contour-dual-circuit","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-anchored-on-a-coordinate-axis","status":"fully_proved","summary":"Every realised contour of length `\u2113` is decoded from a closed walk of length `\u2113` based at an axis site `(j, 0, \u2026, 0)` with `j < \u2113`.","title":"Closed-walk decoding of each realised contour from an axis site"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto","status":"fully_proved","title":"Dual circuits through one component of the contour"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto/component-support","kind":"node","lean":"StatMech.Lattice.componentSupport","parent":49125,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto/index.html#component-support","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto","status":"fully_proved","summary":"The faces of `T` reachable from `f` in the face-boundary graph of `K`.","title":"Faces of a set reachable from a face in the face-boundary graph"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto/component-graph","kind":"node","lean":"StatMech.Lattice.componentGraph","parent":49125,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto/index.html#component-graph","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto","status":"fully_proved","summary":"The connected component of the face `f` in the face-boundary graph of `K`, as a graph on all faces.","title":"Component of a face in the face-boundary graph"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto/tight-axis-circuit-data","kind":"node","lean":"StatMech.Lattice.TightAxisCircuitData","parent":49125,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto/index.html#tight-axis-circuit-data","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-through-one-component-of-the-conto","status":"fully_proved","summary":"Every realised contour of length `\u2113` in `\u039b_n \u2286 \u2124\u00b2` is the contour of a cluster whose face-boundary graph has all its edges in the component of an axis face `(j, 0)` with `j < \u2113`.","title":"Tight axis circuit data: contour edges in one axis-face component"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","status":"fully_proved","title":"Dual circuits decode to contours in two dimensions"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/sym-primal","kind":"node","lean":"StatMech.Lattice.symPrimal","parent":49129,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/index.html#sym-primal","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","status":"fully_proved","summary":"The primal edge of `\u2124\u00b2` crossed by the dual edge between the faces `f` and `g`, defined symmetrically in `f` and `g`. See also `sharedPrimalEdge`, with which it agrees on adjacent faces (`symPrimal_eq_shared`).","title":"Primal edge crossed by a dual edge"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/sym-primal-sym","kind":"node","lean":"StatMech.Lattice.symPrimalSym","parent":49129,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/index.html#sym-primal-sym","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","status":"fully_proved","summary":"`symPrimal` on unordered pairs of faces.","title":"Primal edge crossed by an unordered pair of faces"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/flank-faces","kind":"node","lean":"StatMech.Lattice.flankFaces","parent":49129,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/index.html#flank-faces","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","status":"fully_proved","summary":"The two faces of `\u2124\u00b2` flanking the primal edge `{p, q}`.","title":"Two faces flanking a primal edge"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/flank-faces-sym","kind":"node","lean":"StatMech.Lattice.flankFacesSym","parent":49129,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/index.html#flank-faces-sym","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","status":"fully_proved","summary":"`flankFaces` on unordered pairs of sites.","title":"Two faces flanking an unordered pair of sites"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/dual-walk-contour","kind":"node","lean":"StatMech.Lattice.dualWalkContour","parent":49129,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/index.html#dual-walk-contour","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","status":"fully_proved","summary":"The set of primal edges crossed by a closed dual walk.","title":"Primal edges crossed by a closed dual walk"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/peierls-contour-circuit-data","kind":"node","lean":"StatMech.Lattice.PeierlsContourCircuitData","parent":49129,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/index.html#peierls-contour-circuit-data","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","status":"fully_proved","summary":"Every realised contour of length `\u2113` in `\u039b_n \u2286 \u2124\u00b2` is the contour of a finite cluster whose face-boundary graph is connected on `boundarySupport` and meets an axis face `(j, 0)` with `j < \u2113`. The condition refers to the face set `boundarySupport`, which is chosen by `Classical.choose`.","title":"Connected face-boundary graphs at the axis for all realised contours"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/contour-geometry-residue","kind":"node","lean":"StatMech.Lattice.ContourGeometryResidue","parent":49129,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio/index.html#contour-geometry-residue","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/dual-circuits-decode-to-contours-in-two-dimensio","status":"fully_proved","summary":"Every connected cluster in `\u039b_n \u2286 \u2124\u00b2` with contour length `\u2113` is finite, has face-boundary graph connected on `boundarySupport`, and meets an axis face `(j, 0)` with `j < \u2113`. The condition refers to the face set `boundarySupport`, which is chosen by `Classical.choose`; `TightAxisCircuitData` avoids this choice.","title":"Connected face-boundary graphs at the axis for all connected clusters"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl","status":"fully_proved","title":"The contour of a hole-free planar set is a single cycle"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/phb-double-dual-shift","kind":"node","lean":"StatMech.Lattice.phb_doubleDualShift","parent":49137,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shift by "],["math","\\((-1, -1)\\)"],["text"," identifying the double dual with "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Shift by $(-1, -1)$ identifying the double dual with $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/index.html#phb-double-dual-shift","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl","status":"fully_proved","summary":"The shift `x \u21a6 x - (1, 1)` identifying the dual of the dual lattice with `\u2124\u00b2`.","title":"Shift by $(-1, -1)$ identifying the double dual with $\\mathbb{Z}^2$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/phb-double-dual-equiv","kind":"node","lean":"StatMech.Lattice.phb_doubleDualEquiv","parent":49137,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shift by "],["math","\\((-1, -1)\\)"],["text"," as an equivalence of "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Shift by $(-1, -1)$ as an equivalence of $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/index.html#phb-double-dual-equiv","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl","status":"fully_proved","summary":"The shift by `(-1, -1)` as an equivalence of `\u2124\u00b2`.","title":"Shift by $(-1, -1)$ as an equivalence of $\\mathbb{Z}^2$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/phb-double-dual-iso","kind":"node","lean":"StatMech.Lattice.phb_doubleDualIso","parent":49137,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/index.html#phb-double-dual-iso","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl","status":"fully_proved","summary":"The lattice minus the barrier edges of `K` is isomorphic, by the shift by `(-1, -1)`, to the face region graph of the face-boundary graph of `K`.","title":"Shift isomorphism: lattice minus barrier edges to face region graph"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/phb-boundary-support","kind":"node","lean":"StatMech.Lattice.phb_boundarySupport","parent":49137,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/index.html#phb-boundary-support","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl","status":"fully_proved","summary":"The support of the face-boundary graph of a finite set `K`, as a finset.","title":"Support of the face-boundary graph of a finite set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/phb-planar-boundary","kind":"node","lean":"StatMech.Lattice.phb_planarBoundary","parent":49137,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl/index.html#phb-planar-boundary","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-of-a-hole-free-planar-set-is-a-singl","status":"fully_proved","summary":"The face-boundary graph of a finite set `K`, as a finite planar subgraph of `\u2124\u00b2`.","title":"Face-boundary graph of a finite set as a planar subgraph"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the","status":"fully_proved","title":"Hole-free clusters and the Peierls bound on the square lattice"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/outer-contour-winding","kind":"node","lean":"StatMech.Ising.OuterContourWinding","parent":49143,"reading_title":{"format":"inline-parts-v1","parts":[["text","Winding hypothesis for outer contours on "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Winding hypothesis for outer contours on $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/index.html#outer-contour-winding","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the","status":"fully_proved","summary":"Every plus-boundary configuration on `\u039b_n` with the origin minus has a connected contour event through the origin whose boundary meets the positive axis within the contour length.","title":"Winding hypothesis for outer contours on $\\mathbb{Z}^2$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/is-hole-free-box-cluster","kind":"node","lean":"StatMech.Ising.IsHoleFreeBoxCluster","parent":49143,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/index.html#is-hole-free-box-cluster","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the","status":"fully_proved","summary":"`K` is a connected cluster in `\u039b_n` through the origin without holes.","title":"Hole-free connected cluster through the origin"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/hole-free-cluster-family","kind":"node","lean":"StatMech.Ising.holeFreeClusterFamily","parent":49143,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/index.html#hole-free-cluster-family","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the","status":"fully_proved","summary":"The hole-free clusters in `\u039b_n` through the origin.","title":"Hole-free clusters in the box through the origin"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/hole-free-at-len","kind":"node","lean":"StatMech.Ising.holeFreeAtLen","parent":49143,"reading_title":{"format":"inline-parts-v1","parts":[["text","Hole-free clusters with contour length "],["math","\\(\\ell\\)"]],"source":"Hole-free clusters with contour length $\\ell$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the/index.html#hole-free-at-len","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/hole-free-clusters-and-the-peierls-bound-on-the","status":"fully_proved","summary":"The hole-free clusters in `\u039b_n` through the origin with contour length `\u2113`.","title":"Hole-free clusters with contour length $\\ell$"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails","status":"fully_proved","title":"The contour-count bound over all clusters fails"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails/origin-cluster-family","kind":"node","lean":"StatMech.Ising.originClusterFamily","parent":49148,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails/index.html#origin-cluster-family","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails","status":"fully_proved","summary":"The finite sets of sites in `\u039b_n` containing the origin.","title":"Origin cluster family: finite site sets in the box through the origin"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails/pcc-ray-s","kind":"node","lean":"StatMech.Ising.pcc_rayS","parent":49148,"reading_title":{"format":"inline-parts-v1","parts":[["text","Site "],["math","\\(j e_c\\)"],["text"," on the "],["math","\\(c\\)"],["text","-th coordinate axis"]],"source":"Site $j e_c$ on the $c$-th coordinate axis"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails/index.html#pcc-ray-s","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails","status":"fully_proved","summary":"The site `j e_c` on the `c`-th coordinate axis.","title":"Site $j e_c$ on the $c$-th coordinate axis"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails/origin-contour-count-bound","kind":"node","lean":"StatMech.Ising.OriginContourCountBound","parent":49148,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails/index.html#origin-contour-count-bound","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-contour-count-bound-over-all-clusters-fails","status":"fully_proved","summary":"The sum of `e^{-2\u03b2 |\u2202K|}` over the origin cluster family is at most the Peierls series.","title":"Peierls series bound on the origin-cluster contour sum"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice","status":"fully_proved","title":"Outer contours on the square lattice"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/two-peierls-bound","kind":"node","lean":"StatMech.Ising.twoPeierlsBound","parent":49152,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/index.html#two-peierls-bound","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice","status":"fully_proved","summary":"Twice the Peierls series.","title":"Twice the Peierls series"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/interior-assignment","kind":"node","lean":"StatMech.Ising.InteriorAssignment","parent":49152,"reading_title":{"format":"inline-parts-v1","parts":[["text","Assignment of interiors to closed walks of "],["math","\\(\\mathbb{Z}^2\\)"]],"source":"Assignment of interiors to closed walks of $\\mathbb{Z}^2$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/index.html#interior-assignment","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice","status":"fully_proved","summary":"An assignment of a set of sites (its interior) to every closed walk of `\u2124\u00b2`.","title":"Assignment of interiors to closed walks of $\\mathbb{Z}^2$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/outer-contour-map","kind":"node","lean":"StatMech.Ising.OuterContourMap","parent":49152,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/index.html#outer-contour-map","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice","status":"fully_proved","summary":"Every cluster through the origin is the interior of one of its closed boundary circuits, for some interior assignment.","title":"Outer contour map: clusters as interiors of boundary circuits"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/poc-circuit-family","kind":"node","lean":"StatMech.Ising.poc_circuitFamily","parent":49152,"reading_title":{"format":"inline-parts-v1","parts":[["text","Cycles of length less than "],["math","\\(M\\)"],["text"," based near the origin"]],"source":"Cycles of length less than $M$ based near the origin"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/index.html#poc-circuit-family","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice","status":"fully_proved","summary":"The closed walks of `\u2124\u00b2` of length at most `M` based near the origin.","title":"Cycles of length less than $M$ based near the origin"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/poc-circuit-summand","kind":"node","lean":"StatMech.Ising.poc_circuitSummand","parent":49152,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice/index.html#poc-circuit-summand","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/outer-contours-on-the-square-lattice","status":"fully_proved","summary":"The Boltzmann weight attached to a closed walk in the circuit sum.","title":"Boltzmann weight of a closed walk in the circuit sum"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","kind":"scope","navigation":"module","parent":49057,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","title":"The Peierls bound on the square lattice"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/outer-contour-fill","kind":"node","lean":"StatMech.Ising.OuterContourFill","parent":49158,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html#outer-contour-fill","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","summary":"The filled-cluster form of the winding hypothesis, with a face-connected boundary.","title":"Filled-cluster winding hypothesis"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/outer-exterior-component","kind":"node","lean":"StatMech.Ising.outerExteriorComponent","parent":49158,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html#outer-exterior-component","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","summary":"The component of the complement of `S` that contains the outside of `\u039b_n`.","title":"Exterior component of the complement of a set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/spin-compatible-fill","kind":"node","lean":"StatMech.Ising.spinCompatibleFill","parent":49158,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html#spin-compatible-fill","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","summary":"The set `S` with its holes filled: the complement of its exterior component.","title":"Set with its holes filled"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/spin-compatible-fill-finset","kind":"node","lean":"StatMech.Ising.spinCompatibleFillFinset","parent":49158,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html#spin-compatible-fill-finset","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","summary":"The filled minus cluster of the origin, as a finset.","title":"Filled minus cluster of the origin as a finite set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/peierls-long-range-order-spin-compatible","kind":"node","lean":"StatMech.Ising.peierls_long_range_order_spinCompatibleFill","parent":49158,"reading_title":{"format":"inline-parts-v1","parts":[["text","Plus and minus box measures on "],["math","\\(\\mathbb{Z}^2\\)"],["text"," differ for large "],["math","\\(\\beta\\)"]],"source":"Plus and minus box measures on $\\mathbb{Z}^2$ differ for large $\\beta$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html#peierls-long-range-order-spin-compatible","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","summary":"Plus and minus box measures on `\u2124\u00b2` differ for all `\u03b2` beyond some `\u03b2\u2080`. See `peierls_probOriginMinus_le_uniform` for the informative uniform bound; the inequality of measures itself holds for every `\u03b2 > 0`.","title":"Plus and minus box measures on $\\mathbb{Z}^2$ differ for large $\\beta$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/peierls-prob-origin-minus-le-uniform","kind":"node","lean":"StatMech.Ising.peierls_probOriginMinus_le_uniform","parent":49158,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html#peierls-prob-origin-minus-le-uniform","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","summary":"Peierls bound on `\u2124\u00b2`, uniform in the box: there is `\u03b2\u2080` such that for `\u03b2 \u2265 \u03b2\u2080` the probability that the origin is minus under the plus-boundary box measure on `\u039b_n` is at most `twoPeierlsBound 2 \u03b2`, which is less than `1 / 2` and independent of `n`.","title":"Peierls bound"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/iso-origin-completion","kind":"node","lean":"StatMech.Ising.isoOriginCompletion","parent":49158,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html#iso-origin-completion","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","summary":"The configuration on `\u039b_n` with only the origin minus.","title":"Configuration with only the origin minus"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/outer-contour-fill-nonvacuous","kind":"node","lean":"StatMech.Ising.outerContourFill_nonvacuous","parent":49158,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice/index.html#outer-contour-fill-nonvacuous","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/peierls-argument-for-the-ising-model/the-peierls-bound-on-the-square-lattice","status":"fully_proved","summary":"A nonvacuity check: the configuration with only the origin minus satisfies the data of `OuterContourFill`.","title":"Origin-only minus configuration satisfies the filled-cluster data"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","kind":"scope","navigation":"module","parent":49013,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","title":"Translation invariance and ergodicity of the plus state"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pst-plus-shifted-finite","kind":"node","lean":"StatMech.Ising.pst_plusShiftedFinite","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pst-plus-shifted-finite","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The plus-boundary box measure on `\u039b_n` pushed forward by the shift `g`.","title":"Shifted plus-boundary box measure"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pst-minus-shifted-finite","kind":"node","lean":"StatMech.Ising.pst_minusShiftedFinite","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pst-minus-shifted-finite","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The minus-boundary box measure on `\u039b_n` pushed forward by the shift `g`.","title":"Shifted minus-boundary box measure"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pst-plus-integral-shift-decay","kind":"node","lean":"StatMech.Ising.pst_PlusIntegralShiftDecay","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pst-plus-integral-shift-decay","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"Along `\u03c6`, integrals of bounded continuous functions against the plus box measures and their shifts by `g` become equal.","title":"Decay of shifted integrals for plus box measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pst-minus-integral-shift-decay","kind":"node","lean":"StatMech.Ising.pst_MinusIntegralShiftDecay","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pst-minus-integral-shift-decay","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"Along `\u03c6`, integrals of bounded continuous functions against the minus box measures and their shifts by `g` become equal.","title":"Decay of shifted integrals for minus box measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pstc-decay-fun","kind":"node","lean":"StatMech.Ising.pstc_decayFun","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pstc-decay-fun","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The difference `\u222b f d\u03bc_n - \u222b f \u2218 shift_g d\u03bc_n`.","title":"Difference between an integral and its shifted version"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pstc-coord-cm","kind":"node","lean":"StatMech.Ising.pstc_coordCM","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pstc-coord-cm","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The coordinate `\u03c9 \u21a6 1_{\u03c9_x = +}` as a continuous function.","title":"Plus-spin indicator at a site as a continuous function"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pstc-coord-bcf","kind":"node","lean":"StatMech.Ising.pstc_coordBcf","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pstc-coord-bcf","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The coordinate `\u03c9 \u21a6 1_{\u03c9_x = +}` as a bounded continuous function.","title":"Plus-spin indicator at a site as a bounded continuous function"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pstc-cylinder-alg","kind":"node","lean":"StatMech.Ising.pstc_cylinderAlg","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pstc-cylinder-alg","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The subalgebra generated by the spin coordinates (cylinder functions).","title":"Subalgebra of cylinder functions"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pstc-cylinder-bcf-set","kind":"node","lean":"StatMech.Ising.pstc_cylinderBcfSet","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pstc-cylinder-bcf-set","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The cylinder functions as bounded continuous functions.","title":"Cylinder functions as bounded continuous functions"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pstc-plus-cylinder-decay","kind":"node","lean":"StatMech.Ising.pstc_PlusCylinderDecay","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pstc-plus-cylinder-decay","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"Shifted integrals of cylinder functions against the plus box measures decay along `\u03c6`.","title":"Decay of shifted cylinder integrals for plus box measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/pstc-minus-cylinder-decay","kind":"node","lean":"StatMech.Ising.pstc_MinusCylinderDecay","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#pstc-minus-cylinder-decay","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"Shifted integrals of cylinder functions against the minus box measures decay along `\u03c6`.","title":"Decay of shifted cylinder integrals for minus box measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/ibs-boundary-fraction","kind":"node","lean":"StatMech.Ising.ibs_boundaryFraction","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#ibs-boundary-fraction","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The number of bonds touching but not inside `\u039b_n`, divided by the number of sites of `\u039b_n`.","title":"Boundary bond fraction of a box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/ibs-plus-boundary-influence","kind":"node","lean":"StatMech.Ising.ibs_PlusBoundaryInfluence","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#ibs-plus-boundary-influence","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"Shift differences of cylinder-function integrals against the plus box measures are `O(boundary fraction)`.","title":"Boundary influence bound for the plus box measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/ibs-minus-boundary-influence","kind":"node","lean":"StatMech.Ising.ibs_MinusBoundaryInfluence","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#ibs-minus-boundary-influence","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"Shift differences of cylinder-function integrals against the minus box measures are `O(boundary fraction)`.","title":"Boundary influence bound for the minus box measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/iti-monomial-cm","kind":"node","lean":"StatMech.Ising.iti_monomialCM","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#iti-monomial-cm","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The monomial `\u03c9 \u21a6 \u220f_{x \u2208 T} 1_{\u03c9_x = +}` as a continuous function.","title":"All-plus monomial as a continuous function"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/iti-monomial-bcf","kind":"node","lean":"StatMech.Ising.iti_monomialBcf","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#iti-monomial-bcf","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The monomial `\u03c9 \u21a6 \u220f_{x \u2208 T} 1_{\u03c9_x = +}` as a bounded continuous function.","title":"All-plus monomial as a bounded continuous function"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/iti-plus-multi-homogeneous","kind":"node","lean":"StatMech.Ising.iti_PlusMultiHomogeneous","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#iti-plus-multi-homogeneous","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The plus state gives the all-plus events on `T` and on `g\u207b\u00b9 T` the same probability, for every finite `T`.","title":"Shift homogeneity of all-plus events in the plus state"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/iti-decay-set","kind":"node","lean":"StatMech.Ising.iti_decaySet","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#iti-decay-set","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The continuous functions whose shifted integrals against the plus box measures decay along `\u03c6`.","title":"Continuous functions with decaying shifted integrals"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/itb-plus-multi-boundary-decay","kind":"node","lean":"StatMech.Ising.itb_PlusMultiBoundaryDecay","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#itb-plus-multi-boundary-decay","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"Along `\u03c6`, the plus box measures give the all-plus events on `T` and on `g\u207b\u00b9 T` asymptotically equal probabilities.","title":"Asymptotic shift homogeneity of all-plus events in plus boxes"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/itb-off-centre-domination","kind":"node","lean":"StatMech.Ising.itb_OffCentreDomination","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#itb-off-centre-domination","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"For every finite `T` there is `c` such that plus measures on boxes of sizes `m + c` and `m` compare the all-plus events on `T` and `g\u207b\u00b9 T` in both directions.","title":"Off-centre domination of all-plus events"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/icb-shift-vec","kind":"node","lean":"StatMech.Ising.icb_shiftVec","parent":49167,"reading_title":{"format":"inline-parts-v1","parts":[["text","Shift vector "],["math","\\(R e_{i_0}\\)"]],"source":"Shift vector $R e_{i_0}$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#icb-shift-vec","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The vector `R e_{i\u2080}` of `\u2124^d`.","title":"Shift vector $R e_{i_0}$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-unit","kind":"node","lean":"StatMech.Ising.gvUnit","parent":49167,"reading_title":{"format":"inline-parts-v1","parts":[["text","Unit vector of "],["math","\\(\\mathbb{Z}^d\\)"]],"source":"Unit vector of $\\mathbb{Z}^d$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-unit","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The unit vector `e_i` of `\u2124^d`.","title":"Unit vector of $\\mathbb{Z}^d$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-ball","kind":"node","lean":"StatMech.Ising.gvBall","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-ball","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The site `x` and its nearest neighbours.","title":"Site with its nearest neighbours"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-cand","kind":"node","lean":"StatMech.Ising.gvCand","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-cand","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The sites of `S` together with their nearest neighbours.","title":"Set with its nearest neighbours"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-bond-touch","kind":"node","lean":"StatMech.Ising.gvBondTouch","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-bond-touch","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The bonds with at least one endpoint in `S`.","title":"Bonds touching a set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-glue","kind":"node","lean":"StatMech.Ising.gvGlue","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-glue","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The configuration equal to `\u03c4` on `S` and to `\u03b7` off `S`.","title":"Configuration glued from a set and a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-energy","kind":"node","lean":"StatMech.Ising.gvEnergy","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-energy","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The energy `-\u2211_{e touching S} \u03c3_e - h \u2211_{x \u2208 S} \u03c3_x` of `gvGlue \u03b7 S \u03c4`.","title":"Energy on a set with a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-weight","kind":"node","lean":"StatMech.Ising.gvWeight","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-weight","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The Boltzmann weight of the configuration `\u03c4` on `S` with boundary condition `\u03b7`.","title":"Boltzmann weight on a set with a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-z","kind":"node","lean":"StatMech.Ising.gvZ","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-z","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The partition function on `S` with boundary condition `\u03b7`.","title":"Partition function on a set with a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-prob","kind":"node","lean":"StatMech.Ising.gvProb","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-prob","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The Gibbs probability of `\u03c4` on `S` with boundary condition `\u03b7`.","title":"Gibbs probability on a set with a boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-measure","kind":"node","lean":"StatMech.Ising.gvMeasure","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-measure","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The finite-volume Gibbs measure on the finite set `S` with boundary condition `\u03b7`.","title":"Finite-volume Gibbs measure on a finite set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-plus-measure","kind":"node","lean":"StatMech.Ising.gvPlusMeasure","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-plus-measure","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The finite-volume Gibbs measure on `S` with plus boundary condition.","title":"Finite-volume Gibbs measure with plus boundary condition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-ovr","kind":"node","lean":"StatMech.Ising.gvOvr","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-ovr","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The configuration on `Sout` obtained by overwriting its spins in `Sin` with `\u03c4`.","title":"Overwriting a configuration on an inner set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-res","kind":"node","lean":"StatMech.Ising.gvRes","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-res","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The restriction of a configuration on `Sout` to `Sin`.","title":"Restriction of a configuration to an inner set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/gv-swap-pair","kind":"node","lean":"StatMech.Ising.gvSwapPair","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#gv-swap-pair","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The involution exchanging the `Sin` parts of a configuration on `Sout` and a configuration on `Sin`.","title":"Involution swapping inner-set parts of configuration pairs"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/iptp-box-equiv","kind":"node","lean":"StatMech.Ising.iptp_boxEquiv","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#iptp-box-equiv","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The sites of the box `\u039b_n` as a set and as a finset.","title":"Box sites as a set and as a finite set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/iptp-trans-interior-equiv","kind":"node","lean":"StatMech.Ising.iptp_transInteriorEquiv","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#iptp-trans-interior-equiv","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"Translation by `g` as an equivalence between `S` and its translate.","title":"Translation as an equivalence onto a translate"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/iptp-plus-state-is-translation-invariant","kind":"node","lean":"StatMech.Ising.iptp_plusState_isTranslationInvariant","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#iptp-plus-state-is-translation-invariant","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"For `\u03b2, h \u2265 0`, the plus state of the Ising model on `\u2124^d` is invariant under all lattice translations.","title":"Translation invariance of the plus state"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/ising-plus-extreme","kind":"node","lean":"StatMech.Ising.ising_plus_extreme","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#ising-plus-extreme","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"For `d \u2265 1` and `\u03b2, h \u2265 0`, the plus state is an extreme point of the translation-invariant probability measures.","title":"Extremality of the plus state"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/iptp-plus-state-is-ergodic","kind":"node","lean":"StatMech.Ising.iptp_plusState_isErgodic","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#iptp-plus-state-is-ergodic","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"For `d \u2265 1` and `\u03b2, h \u2265 0`, the plus state is ergodic under lattice translations.","title":"Ergodicity of the plus state"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/is-dlrextreme-point","kind":"node","lean":"StatMech.Ising.IsDLRExtremePoint","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#is-dlrextreme-point","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"`\u03bc` is an extreme point of the DLR states: a proper convex decomposition into DLR probability measures is trivial.","title":"Extreme point of the DLR states"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/atip2-ergodic-zero-one-nonvacuous","kind":"node","lean":"StatMech.Ising.atip2_ergodic_zeroOne_nonvacuous","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#atip2-ergodic-zero-one-nonvacuous","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"A nonvacuity check: the Dirac mass at the all-plus configuration gives the all-plus event probability one.","title":"All-plus Dirac mass gives the all-plus event probability one"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/is-tail-event","kind":"node","lean":"StatMech.Ising.IsTailEvent","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#is-tail-event","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"A tail event: measurable with respect to the spins outside `\u039b_n`, for every `n`.","title":"Tail event for lattice spin configurations"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/aetc-fin-sig","kind":"node","lean":"StatMech.Ising.aetc_finSig","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#aetc-fin-sig","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The \u03c3-algebra generated by the spins in the finite set `F`.","title":"\u03c3-algebra generated by the spins in a finite set"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/aetc-cring","kind":"node","lean":"StatMech.Ising.aetc_Cring","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#aetc-cring","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The cylinder events: events depending on the spins in some box.","title":"Cylinder events of the Ising configuration space"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/ae-approx-far-out","kind":"node","lean":"StatMech.Ising.AeApproxFarOut","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#ae-approx-far-out","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"For every box `\u039b_k` and `\u03b5 > 0`, `s` is `\u03b5`-approximated in `\u03bc`-measure by an event measurable outside `\u039b_k`.","title":"Approximation in measure by events measurable outside every box"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/aetc-hae-tail-nonvacuous","kind":"node","lean":"StatMech.Ising.aetc_haeTail_nonvacuous","parent":49167,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu/index.html#aetc-hae-tail-nonvacuous","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/translation-invariance-and-ergodicity-of-the-plu","status":"fully_proved","summary":"The hypotheses of `aetc_haeTail` are satisfiable: the Dirac mass at the all-plus configuration.","title":"Nonvacuity of the a.s.-tail theorem via the all-plus Dirac mass"},{"data_url":"scopes/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model.json","id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","kind":"scope","navigation":"module","parent":49013,"result":false,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","title":"Phase transition of the Ising model"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/spin-bcf","kind":"node","lean":"StatMech.Ising.spinBCF","parent":49215,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#spin-bcf","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The spin at `x` as a bounded continuous real function on the configuration space.","title":"Spin at a site as a bounded continuous function"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/measurable-spin","kind":"node","lean":"StatMech.Ising.measurable_spin","parent":49215,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#measurable-spin","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The spin at a site is a measurable function of the configuration.","title":"Measurability of the spin at a site"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/integrable-spin","kind":"node","lean":"StatMech.Ising.integrable_spin","parent":49215,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#integrable-spin","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The spin at a site is integrable for every finite measure.","title":"Integrability of the spin at a site under finite measures"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/magnetization","kind":"node","lean":"StatMech.Ising.magnetization","parent":49215,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#magnetization","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The spontaneous magnetization `m*(\u03b2) = \u27e8\u03c3_0\u27e9\u207a_\u03b2`: the expected spin at the origin under the zero-field plus state.","title":"Spontaneous magnetization of the Ising model"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/positive-magnetization-set","kind":"node","lean":"StatMech.Ising.positiveMagnetizationSet","parent":49215,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#positive-magnetization-set","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The set of inverse temperatures `\u03b2 \u2265 0` at which the spontaneous magnetization is positive.","title":"Inverse temperatures with positive spontaneous magnetization"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/beta-c","kind":"node","lean":"StatMech.Ising.betaC","parent":49215,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#beta-c","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The critical inverse temperature `\u03b2_c(d) = inf {\u03b2 \u2265 0 | m*(\u03b2) > 0}` (equal to `0` if the set is empty, by the convention `sInf \u2205 = 0`).","title":"Critical inverse temperature from spontaneous magnetization"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/pup-exists-positive-magnetization","kind":"node","lean":"StatMech.IsingFK.pup_exists_positive_magnetization","parent":49215,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#pup-exists-positive-magnetization","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The two-dimensional Ising model has positive spontaneous magnetization at some inverse temperature (Peierls, *On Ising's model of ferromagnetism*, 1936; Griffiths, Phys. Rev. 136 (1964)).","title":"Peierls' theorem"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/fk-pc-lt-one-of-two-le","kind":"node","lean":"StatMech.FK.fkPc_lt_one_of_two_le","parent":49215,"reading_title":{"format":"inline-parts-v1","parts":[["text","FK critical point is less than one for "],["math","\\(d \\ge 2\\)"],["text",", "],["math","\\(q \\ge 1\\)"]],"source":"FK critical point is less than one for $d \\ge 2$, $q \\ge 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#fk-pc-lt-one-of-two-le","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The FK critical point satisfies `p_c(d, q) < 1` for `d \u2265 2` and `q \u2265 1` (Grimmett, *The Random-Cluster Model*, Theorem 5.5).","title":"FK critical point is less than one for $d \\ge 2$, $q \\ge 1$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/ising-phase-transition-of-two-le","kind":"node","lean":"StatMech.IsingFK.ising_phase_transition_of_two_le","parent":49215,"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#ising-phase-transition-of-two-le","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"Existence of the Ising phase transition in dimension `d \u2265 2`: there is `\u03b2_c > 0` such that the spontaneous magnetization vanishes for `0 < \u03b2 < \u03b2_c` and is positive for `\u03b2 > \u03b2_c` (Peierls, 1936; Griffiths, Phys. Rev. 136 (1964); Duminil-Copin, *Lectures on the Ising and Potts models on the hypercubic lattice*, 2017). The statement says nothing about `\u03b2 = \u03b2_c`.","title":"Existence of the Ising phase transition"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/ising-fk-beta-c-eq-fk-pc-of-two-le","kind":"node","lean":"StatMech.IsingFK.isingFK_betaC_eq_fkPc_of_two_le","parent":49215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sup-based "],["math","\\(\\beta_c\\)"],["text"," equals "],["math","\\(-\\tfrac{1}{2}\\log(1 - p_c)\\)"],["text"," for "],["math","\\(d \\ge 2\\)"]],"source":"Sup-based $\\beta_c$ equals $-\\tfrac{1}{2}\\log(1 - p_c)$ for $d \\ge 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#ising-fk-beta-c-eq-fk-pc-of-two-le","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The sup-based critical inverse temperature `IsingFK.betaC (magnetization d)` in dimension `d \u2265 2` is `-(1/2) log (1 - p_c(d, 2))`, the image of the FK critical point. This is the supremum of the zero-magnetization inverse temperatures, not yet the public `Ising.betaC d` (see `ising_betaC_eq_sup_betaC_of_two_le` for the bridge).","title":"Sup-based $\\beta_c$ equals $-\\tfrac{1}{2}\\log(1 - p_c)$ for $d \\ge 2$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/ising-beta-c-eq-sup-beta-c-of-two-le","kind":"node","lean":"StatMech.IsingFK.ising_betaC_eq_sup_betaC_of_two_le","parent":49215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inf-based and sup-based Ising critical points agree for "],["math","\\(d \\ge 2\\)"]],"source":"Inf-based and sup-based Ising critical points agree for $d \\ge 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#ising-beta-c-eq-sup-beta-c-of-two-le","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"Bridge between the two critical-point definitions: the public `Ising.betaC d = sInf {\u03b2 \u2265 0 | m*(\u03b2) > 0}` equals the sup-based `IsingFK.betaC (magnetization d) = sSup {\u03b2 > 0 | m*(\u03b2) = 0}` in dimension `d \u2265 2`. The `\u03b2 = 0` case is not covered by the phase-transition statement (which only speaks about `\u03b2 > 0`), so it is settled separately: `\u03c6_0({0}) = tanh 0 * \u2026 = 0 < 1`, hence the plus-state two-point function at\u2026","title":"Inf-based and sup-based Ising critical points agree for $d \\ge 2$"},{"id":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/ising-beta-c-eq-fk-pc-of-two-le","kind":"node","lean":"StatMech.IsingFK.isingBetaC_eq_fkPc_of_two_le","parent":49215,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inf-based "],["math","\\(\\beta_c\\)"],["text"," equals "],["math","\\(-\\tfrac{1}{2}\\log(1 - p_c)\\)"],["text"," for "],["math","\\(d \\ge 2\\)"]],"source":"Inf-based $\\beta_c$ equals $-\\tfrac{1}{2}\\log(1 - p_c)$ for $d \\ge 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model/index.html#ising-beta-c-eq-fk-pc-of-two-le","scope":"StatisticalMechanics/gibbs-states-and-phase-transitions/phase-transition-of-the-ising-model","status":"fully_proved","summary":"The public Ising critical inverse temperature `Ising.betaC d` in dimension `d \u2265 2` is `-(1/2) log (1 - p_c(d, 2))`, the image of the FK critical point. This is the form usable with `ising_sharp_phase_transition`, whose `\u03b2_c` is `Ising.betaC d`.","title":"Inf-based $\\beta_c$ equals $-\\tfrac{1}{2}\\log(1 - p_c)$ for $d \\ge 2$"},{"data_url":"scopes/StatisticalMechanics/exact-solutions.json","id":"StatisticalMechanics/exact-solutions","kind":"scope","parent":47780,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/index.html","scope":"StatisticalMechanics/exact-solutions","status":"can_prove","summary":"Exact results for the two-dimensional Ising model: Kramers\u2013Wannier duality between high- and low-temperature expansions, and Onsager's formula for the free energy through Kac\u2013Ward determinants, Sherman's loop identities and the homology of the torus.","title":"Exact solutions and duality"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/kramers-wannier-duality.json","id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","kind":"scope","navigation":"module","parent":49228,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","title":"Kramers\u2013Wannier duality"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/transport-edges","kind":"node","lean":"StatMech.Ising.transportEdges","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#transport-edges","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The image of an edge set under a vertex equivalence `V \u2243 W`.","title":"Image of an edge set under a vertex bijection"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/inc-count","kind":"node","lean":"StatMech.Ising.incCount","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#inc-count","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The number of edges of `F` incident to `v`.","title":"Number of edges of an edge set incident to a vertex"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/is-even-subgraph","kind":"node","lean":"StatMech.Ising.IsEvenSubgraph","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#is-even-subgraph","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every vertex has even degree in the edge set `F`.","title":"Even subgraph condition on an edge set"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/ising-z-high-temp-expansion","kind":"node","lean":"StatMech.Ising.isingZ_high_temp_expansion","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#ising-z-high-temp-expansion","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"At zero field, `Z(\u03b2) = 2^{|V|} (cosh \u03b2)^{|E|} \u2211_{F even} (tanh \u03b2)^{|F|}`.","title":"High-temperature expansion"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/dual-contour-matching","kind":"node","lean":"StatMech.Ising.DualContourMatching","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#dual-contour-matching","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"`tanh \u03b2* = exp (-2\u03b2)` and the high-temperature expansion of `Gd` at `\u03b2*` equals `c` times `Z_{Gp}(\u03b2)`.","title":"Dual high-temperature expansion proportional to the partition function"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/cut-edges","kind":"node","lean":"StatMech.Ising.cutEdges","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#cut-edges","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The edges `{x, y}` of `G` whose endpoints have different spins in `s`.","title":"Edges cut by a spin configuration"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/cut-even-subgraph-matching","kind":"node","lean":"StatMech.Ising.CutEvenSubgraphMatching","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#cut-even-subgraph-matching","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The cut sum `\u2211_s t^{|\u2202s|}` of `Gp` equals `mult` times the even-subgraph sum `\u2211_F t^{|F|}` of `Gd`.","title":"Proportionality of the cut sum and the dual even-subgraph sum"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/cut-space","kind":"node","lean":"StatMech.Ising.cutSpace","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#cut-space","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The cut space: the edge sets `\u2202s` cut by configurations `s`.","title":"Cut space of a graph"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/even-subgraphs","kind":"node","lean":"StatMech.Ising.evenSubgraphs","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#even-subgraphs","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The even subgraphs of `G`.","title":"Even subgraphs of a graph"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/edge-diff","kind":"node","lean":"StatMech.Ising.edgeDiff","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#edge-diff","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Whether the spins of `s` differ at the endpoints of `e`.","title":"Spin disagreement test across an edge"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/cut-edges-b","kind":"node","lean":"StatMech.Ising.cutEdgesB","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#cut-edges-b","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The edges cut by `s`, defined through `edgeDiff`.","title":"Edges cut by a configuration via the edge disagreement test"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/cut-space-b","kind":"node","lean":"StatMech.Ising.cutSpaceB","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#cut-space-b","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The cut space, defined through `cutEdgesB`.","title":"Cut space defined through Boolean cut edges"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/even-subgraphs-b","kind":"node","lean":"StatMech.Ising.evenSubgraphsB","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#even-subgraphs-b","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The even subgraphs, defined with a Boolean test.","title":"Even subgraphs defined by a Boolean test"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/k4","kind":"node","lean":"StatMech.Ising.K4","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#k4","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The complete graph `K\u2084` on `Fin 4`.","title":"Complete graph on four vertices"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/ising-z-self-dual-k4","kind":"node","lean":"StatMech.Ising.isingZ_self_dual_K4","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#ising-z-self-dual-k4","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Kramers\u2013Wannier duality on the self-dual graph `K\u2084`: if `tanh \u03b2* = e^{-2\u03b2}` then `Z(\u03b2*) = (2\u2074 (cosh \u03b2*)^{|E|} / (e^{\u03b2|E|} \u00b7 2)) Z(\u03b2)` at zero field.","title":"Kramers\u2013Wannier duality"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/walk-dis","kind":"node","lean":"StatMech.Ising.walkDis","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#walk-dis","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The number of steps of a walk whose endpoints have different spins in `s`.","title":"Number of disagreeing steps along a walk"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/edge-crossing","kind":"node","lean":"StatMech.Ising.EdgeCrossing","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#edge-crossing","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"An edge bijection `\u03c8` from `Gp` to `Gd` carrying edges to edges, cuts to even subgraphs, and even subgraphs back to cuts.","title":"Edge crossing between a graph and its dual"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/ising-z-self-dual-of-edge-crossing","kind":"node","lean":"StatMech.Ising.isingZ_self_dual_of_edgeCrossing","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#ising-z-self-dual-of-edge-crossing","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"For a preconnected graph `Gp` with an edge crossing to `Gd` and `tanh \u03b2* = e^{-2\u03b2}`, `Z_{Gd}(\u03b2*) = (2^{|V|} (cosh \u03b2*)^{|E_d|} / (e^{\u03b2|E_p|} \u00b7 2)) Z_{Gp}(\u03b2)` at zero field.","title":"Kramers\u2013Wannier duality"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/k4-missing","kind":"node","lean":"StatMech.Ising.k4Missing","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#k4-missing","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The two vertices of `Fin 4` other than `a` and `b`.","title":"Vertices of K\u2084 outside a given pair"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/k4-comp-pair","kind":"node","lean":"StatMech.Ising.k4CompPair","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#k4-comp-pair","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The edge of `K\u2084` complementary to `{a, b}`.","title":"Complementary edge of a pair in K\u2084"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/k4-comp","kind":"node","lean":"StatMech.Ising.k4Comp","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#k4-comp","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The complementary edge of `K\u2084`, as a function on `Sym2 (Fin 4)`.","title":"Complementary-edge map of K\u2084 on unordered pairs"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/k4-comp-equiv","kind":"node","lean":"StatMech.Ising.k4CompEquiv","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#k4-comp-equiv","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Complementation of edges of `K\u2084` as an involutive equivalence.","title":"Edge complementation of K\u2084 as an involutive equivalence"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/k4-edge-crossing","kind":"node","lean":"StatMech.Ising.k4EdgeCrossing","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#k4-edge-crossing","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The edge crossing of the self-dual graph `K\u2084` given by complementary edges.","title":"Self-dual edge crossing of K\u2084 by complementary edges"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/walk-parity","kind":"node","lean":"StatMech.Ising.walkParity","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#walk-parity","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The parity of the number of steps of a walk that use an edge of `\u03b4`.","title":"Crossing parity of a walk with respect to an edge set"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/even-on-cycles","kind":"node","lean":"StatMech.Ising.EvenOnCycles","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#even-on-cycles","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every closed walk of `Gp` uses edges of `\u03b4` an even number of times.","title":"Evenness of an edge set on every closed walk"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/parity-config","kind":"node","lean":"StatMech.Ising.parityConfig","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#parity-config","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The configuration assigning to `v` the crossing parity of `\u03b4` along a walk from `v\u2080` to `v`.","title":"Spin configuration from crossing parities along walks"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/cycle-parity-compatible","kind":"node","lean":"StatMech.Ising.CycleParityCompatible","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#cycle-parity-compatible","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every closed walk of `Gp` crosses the pullback of `F` an even number of times.","title":"Cycle-parity compatibility of a dual edge set"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/edge-crossing-of-cycle-parity","kind":"node","lean":"StatMech.Ising.edgeCrossing_of_cycleParity","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#edge-crossing-of-cycle-parity","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"An edge crossing built from cycle-parity compatibility of all even subgraphs of `Gd`.","title":"Edge crossing from cycle-parity compatibility of even subgraphs"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/single-dual-cycle-parity","kind":"node","lean":"StatMech.Ising.SingleDualCycleParity","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#single-dual-cycle-parity","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every edge set in `cycleSets` is cycle-parity compatible.","title":"Cycle-parity compatibility of a family of dual cycle sets"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/closed-dual-walk-parity","kind":"node","lean":"StatMech.Ising.ClosedDualWalkParity","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#closed-dual-walk-parity","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every closed walk of `Gd` has edge set whose pullback is cycle-parity compatible.","title":"Cycle-parity compatibility of closed dual walks"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/even-subgraph-walk-decomp","kind":"node","lean":"StatMech.Ising.EvenSubgraphWalkDecomp","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#even-subgraph-walk-decomp","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every even subgraph of `Gd` is a symmetric difference of edge sets of closed walks.","title":"Closed-walk decomposition property of even subgraphs"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/edge-crossing-of-closed-walk-parity","kind":"node","lean":"StatMech.Ising.edgeCrossing_of_closedWalkParity","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#edge-crossing-of-closed-walk-parity","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"An edge crossing built from closed-walk parities and the decomposition of even subgraphs into closed walks.","title":"Edge crossing from closed-walk parities and walk decompositions"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/dual-walk-has-side-colouring","kind":"node","lean":"StatMech.Ising.DualWalkHasSideColouring","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#dual-walk-has-side-colouring","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every closed walk of `Gd` has a two-colouring of the vertices of `Gp` whose boundary is the pullback of its edges.","title":"Side-colouring property of closed dual walks"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/edge-crossing-of-side-colouring","kind":"node","lean":"StatMech.Ising.edgeCrossing_of_sideColouring","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#edge-crossing-of-side-colouring","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"An edge crossing built from side colourings of dual walks.","title":"Edge crossing from side colourings of dual walks"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/subgraph-f","kind":"node","lean":"StatMech.Ising.subgraphF","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#subgraph-f","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The simple graph with edge set `F`.","title":"Simple graph spanned by an edge set"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/comp-f","kind":"node","lean":"StatMech.Ising.compF","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#comp-f","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The component of `f` in the graph spanned by `F`.","title":"Component of a vertex in the graph spanned by an edge set"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/comp-supp-f","kind":"node","lean":"StatMech.Ising.compSuppF","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#comp-supp-f","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The vertices of the component of `f` in the graph spanned by `F`.","title":"Vertex set of a component of the graph spanned by an edge set"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/comp-euler-walk","kind":"node","lean":"StatMech.Ising.compEulerWalk","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#comp-euler-walk","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"A closed Eulerian walk of the component of `f` in an even edge set `F`.","title":"Closed Eulerian walk of a component of an even edge set"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/even-subgraph-walk-decomp-general","kind":"node","lean":"StatMech.Ising.evenSubgraphWalkDecomp_general","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#even-subgraph-walk-decomp-general","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every even subgraph decomposes into closed walks.","title":"Even subgraphs of a finite graph decompose into closed walks"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/region-boundary","kind":"node","lean":"StatMech.Ising.regionBoundary","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#region-boundary","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The edges of `Gp` whose endpoints lie on different sides of the region `S`.","title":"Edge boundary of a region in a graph"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/dual-walk-bounds-region","kind":"node","lean":"StatMech.Ising.DualWalkBoundsRegion","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#dual-walk-bounds-region","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Every closed walk of `Gd` pulls back to the boundary of a region of `Gp`.","title":"Region-bounding property of closed dual walks"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/edge-crossing-of-bounds-region","kind":"node","lean":"StatMech.Ising.edgeCrossing_of_boundsRegion","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#edge-crossing-of-bounds-region","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"An edge crossing built from dual walks bounding regions.","title":"Edge crossing from region-bounding dual walks and a walk decomposition"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/lattice-side","kind":"node","lean":"StatMech.Ising.latticeSide","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#lattice-side","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"The indicator of a set of sites of `\u2124\u00b2`, as a Boolean function.","title":"Boolean indicator of a set of sites of \u2124\u00b2"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/edge-crossing-of-bounds-region-2","kind":"node","lean":"StatMech.Ising.edgeCrossing_of_boundsRegion'","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#edge-crossing-of-bounds-region-2","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"An edge crossing built from dual walks bounding regions (variant using `even_to_cut_of_boundsRegion'`).","title":"Edge crossing from region-bounding dual walks, no decomposition needed"},{"id":"StatisticalMechanics/exact-solutions/kramers-wannier-duality/ising-z-self-dual-of-bounds-region","kind":"node","lean":"StatMech.Ising.isingZ_self_dual_of_boundsRegion","parent":49229,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kramers-wannier-duality/index.html#ising-z-self-dual-of-bounds-region","scope":"StatisticalMechanics/exact-solutions/kramers-wannier-duality","status":"fully_proved","summary":"Kramers\u2013Wannier duality for a preconnected graph `Gp` and a graph `Gd` whose closed walks pull back to region boundaries, for `tanh \u03b2* = e^{-2\u03b2}`.","title":"Kramers\u2013Wannier duality"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/kac-ward-matrices.json","id":"StatisticalMechanics/exact-solutions/kac-ward-matrices","kind":"scope","navigation":"module","parent":49228,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","title":"Kac\u2013Ward matrices"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-v","kind":"node","lean":"StatMech.Onsager.ons_V","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-v","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The sites `\u2124_L \u00d7 \u2124_L` of the `L \u00d7 L` square-lattice torus.","title":"Sites of the square-lattice torus"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-dir-step","kind":"node","lean":"StatMech.Onsager.ons_dirStep","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-dir-step","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The neighbor of a site in direction `\u03bc`: `0` east, `1` north, `2` west, `3` south.","title":"Neighbouring site in a given direction on the torus"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-dart","kind":"node","lean":"StatMech.Onsager.ons_Dart","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-dart","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"Darts (directed edges) of the torus: a starting site and a direction.","title":"Darts of the square-lattice torus"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-turn-w","kind":"node","lean":"StatMech.Onsager.ons_turnW","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-turn-w","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The turn weight from direction `\u03bc` to direction `\u03bd`: `1` straight, `\u03c9` for a left turn, `0` for a U-turn, `\u03c9\u207b\u00b9` for a right turn.","title":"Turn weight between two lattice directions"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-kwmat","kind":"node","lean":"StatMech.Onsager.ons_KWmat","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-kwmat","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The Kac-Ward matrix: the entry from `d\u2081` to `d\u2082` is `x * ons_turnW \u03c9 d\u2081.2 d\u2082.2` when `d\u2082` starts at the head of `d\u2081`, and `0` otherwise.","title":"Kac\u2013Ward matrix of the square-lattice torus"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-shift-dart","kind":"node","lean":"StatMech.Onsager.ons_shiftDart","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-shift-dart","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"Translation of a dart by a lattice vector `t`.","title":"Translation of a dart by a lattice vector"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/trace-pow-endo","kind":"node","lean":"StatMech.Onsager.trace_pow_endo","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#trace-pow-endo","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The trace of `\u03c6 ^ k` is the sum of the `k`-th powers of the roots of the characteristic polynomial of `\u03c6`. See also `Matrix.trace_eq_sum_roots_charpoly`, the case `k = 1`.","title":"Trace of a power as a power sum of characteristic roots"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-det-walk-root","kind":"node","lean":"StatMech.Onsager.ons_detWalkRoot","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-det-walk-root","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The walk-series square root of `det (1 - \u039b)`: the exponential of minus one half of `\u2211\u2099 (\u2211 over closed walks of length n + 1 of their weight) / (n + 1)`.","title":"Walk-series square root of det(1 \u2212 \u039b)"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-mask-matrix","kind":"node","lean":"StatMech.Onsager.ons_maskMatrix","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-mask-matrix","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The matrix `\u039b` with every row and column indexed by a state of `forbidden` set to zero.","title":"Matrix with forbidden rows and columns set to zero"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-scale-columns","kind":"node","lean":"StatMech.Onsager.ons_scaleColumns","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-scale-columns","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The matrix `\u039b` with every column indexed by an element of `S` multiplied by `t`.","title":"Matrix with selected columns scaled"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-turn-root","kind":"node","lean":"StatMech.Onsager.ons_turnRoot","parent":49275,"reading_title":{"format":"inline-parts-v1","parts":[["text","Turning phase "],["math","\\(\\omega = e^{i\\pi/4}\\)"]],"source":"Turning phase $\\omega = e^{i\\pi/4}$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-turn-root","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The turning phase `\u03c9 = e^{i\u03c0/4}`, so a turn by `\u00b1\u03c0/2` contributes `\u03c9^{\u00b11}` and `\u03c9\u00b2 = i`.","title":"Turning phase $\\omega = e^{i\\pi/4}$"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-space-root","kind":"node","lean":"StatMech.Onsager.ons_spaceRoot","parent":49275,"reading_title":{"format":"inline-parts-v1","parts":[["text","Primitive L-th root of unity "],["math","\\(e^{2\\pi i/L}\\)"]],"source":"Primitive L-th root of unity $e^{2\\pi i/L}$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-space-root","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The primitive `L`-th root of unity `e^{2\u03c0i/L}`.","title":"Primitive L-th root of unity $e^{2\\pi i/L}$"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-spin-det","kind":"node","lean":"StatMech.Onsager.ons_spinDet","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-spin-det","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The sector determinant: the product of `ons_symbolDispersion \u03b2` over the momenta `(2\u03c0(i + a/2) / L, 2\u03c0(j + b/2) / L)`, `i, j < L`, for boundary phases `a, b \u2208 {0, 1}`.","title":"Sector determinant with boundary phases"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-scale-edge-weight","kind":"node","lean":"StatMech.Onsager.ons_scaleEdgeWeight","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-scale-edge-weight","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The edge weight function with the weight of `edge` multiplied by `t`.","title":"Edge weight function with one weight scaled by t"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-kwblock","kind":"node","lean":"StatMech.Onsager.ons_KWblock","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-kwblock","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The `4 \u00d7 4` block of the Kac-Ward matrix between the darts at site `g` and the darts at the origin.","title":"4 \u00d7 4 block of the Kac-Ward matrix between a site and the origin"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-kwconvergence-radius","kind":"node","lean":"StatMech.Onsager.ons_KWconvergenceRadius","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-kwconvergence-radius","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The radius `1 / (number of darts) = 1 / (4L\u00b2)` below which the walk expansion of the twisted Kac-Ward determinant converges.","title":"Convergence radius of the twisted Kac-Ward walk expansion"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-dir-phase","kind":"node","lean":"StatMech.Onsager.ons_dirPhase","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-dir-phase","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The phase of a step in direction `\u03bc`: `u, v, u\u207b\u00b9, v\u207b\u00b9` for east, north, west, south.","title":"Boundary phase of a step in a direction"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-dir-exponent-x","kind":"node","lean":"StatMech.Onsager.ons_dirExponentX","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-dir-exponent-x","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The horizontal displacement `1, 0, -1, 0` of a step in direction `\u03bc`.","title":"Horizontal displacement of a step in a direction"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-dir-exponent-y","kind":"node","lean":"StatMech.Onsager.ons_dirExponentY","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-dir-exponent-y","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The vertical displacement `0, 1, 0, -1` of a step in direction `\u03bc`.","title":"Vertical displacement of a step in a direction"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-kwmat-phase","kind":"node","lean":"StatMech.Onsager.ons_KWmatPhase","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-kwmat-phase","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The Kac-Ward matrix with each step multiplied by its direction phase `ons_dirPhase u v`.","title":"Twisted Kac-Ward matrix with direction phases"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-kwblock-phase","kind":"node","lean":"StatMech.Onsager.ons_KWblockPhase","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-kwblock-phase","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The `4 \u00d7 4` block of the twisted Kac-Ward matrix between the darts at `g` and at the origin.","title":"Block of the twisted Kac-Ward matrix between a site and the origin"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-spin-phase","kind":"node","lean":"StatMech.Onsager.ons_spinPhase","parent":49275,"reading_title":{"format":"inline-parts-v1","parts":[["text","Boundary phase "],["math","\\(e^{i\\pi a/L}\\)"]],"source":"Boundary phase $e^{i\\pi a/L}$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-spin-phase","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The boundary phase `e^{i\u03c0a/L}`, whose `L`-th power is `(-1)^a`.","title":"Boundary phase $e^{i\\pi a/L}$"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-kwmat-weighted","kind":"node","lean":"StatMech.Onsager.ons_KWmatWeighted","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-kwmat-weighted","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The edge-weighted Kac-Ward matrix: the entry from `d\u2081` to `d\u2082` is the weight of the torus edge of `d\u2081` times the turn weight, when `d\u2082` starts at the head of `d\u2081`.","title":"Edge-weighted Kac-Ward matrix"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-kwmat-weighted-phase","kind":"node","lean":"StatMech.Onsager.ons_KWmatWeightedPhase","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-kwmat-weighted-phase","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The edge-weighted Kac-Ward matrix with each step multiplied by its direction phase.","title":"Edge-weighted Kac-Ward matrix with direction phases"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-weighted-phase-bucket-e","kind":"node","lean":"StatMech.Onsager.ons_weightedPhaseBucketE","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-weighted-phase-bucket-e","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The total edge-weighted twisted Kac-Ward weight of closed walks of length `n + 1` visiting `e` but not its reverse.","title":"Edge-weighted twisted bucket: walks through a dart, not its reverse"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-weighted-phase-bucket-n","kind":"node","lean":"StatMech.Onsager.ons_weightedPhaseBucketN","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-weighted-phase-bucket-n","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The total edge-weighted twisted Kac-Ward weight of closed walks of length `n + 1` visiting neither `e` nor its reverse.","title":"Edge-weighted twisted bucket: walks avoiding a dart and its reverse"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-mask-weighted-phase-bucket-e","kind":"node","lean":"StatMech.Onsager.ons_maskWeightedPhaseBucketE","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-mask-weighted-phase-bucket-e","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"As `ons_weightedPhaseBucketE`, with the darts of `forbidden` masked.","title":"Masked edge-weighted bucket: walks through a dart, not its reverse"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-mask-weighted-phase-bucket-n","kind":"node","lean":"StatMech.Onsager.ons_maskWeightedPhaseBucketN","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-mask-weighted-phase-bucket-n","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"As `ons_weightedPhaseBucketN`, with the darts of `forbidden` masked.","title":"Masked edge-weighted bucket: walks avoiding a dart and its reverse"},{"id":"StatisticalMechanics/exact-solutions/kac-ward-matrices/ons-kwsymbol-mat","kind":"node","lean":"StatMech.Onsager.ons_KWsymbolMat","parent":49275,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/kac-ward-matrices/index.html#ons-kwsymbol-mat","scope":"StatisticalMechanics/exact-solutions/kac-ward-matrices","status":"fully_proved","summary":"The `4 \u00d7 4` Fourier symbol of `1 - ons_KWmat` at `z\u2c7c = e^{i k\u2c7c}`, rows and columns indexed by the directions east, north, west, south.","title":"Fourier symbol of one minus the Kac-Ward matrix"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula.json","id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","kind":"scope","navigation":"module","parent":49228,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","title":"Sherman's loop cancellations and product formula"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-reversible-turn-matrix","kind":"node","lean":"StatMech.Onsager.ons_reversibleTurnMatrix","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-reversible-turn-matrix","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The matrix `q x y \u00b7 turnW(dir y, dir x)` on an abstract state space with directions.","title":"Reversible turn matrix on a state space with directions"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-state-weighted-reversible-turn-matri","kind":"node","lean":"StatMech.Onsager.ons_stateWeightedReversibleTurnMatrix","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-state-weighted-reversible-turn-matri","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The reversible turn matrix with additional row and column weights on states.","title":"State-weighted reversible turn matrix"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-involutive-loop-rev","kind":"node","lean":"StatMech.Onsager.ons_involutiveLoopRev","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-involutive-loop-rev","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Reversal of a cyclic sequence of states: position `k` carries `rev` of the state at position `-k`.","title":"Reversal of a cyclic sequence of states"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-masked-reversible-factor","kind":"node","lean":"StatMech.Onsager.ons_maskedReversibleFactor","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-masked-reversible-factor","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The reversible factor with the rows and columns of `forbidden` set to zero.","title":"Masked reversible factor"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-factor-majorant","kind":"node","lean":"StatMech.Onsager.ons_factorMajorant","parent":49305,"reading_title":{"format":"inline-parts-v1","parts":[["text","Majorant "],["math","\\(q^{|s|-1}\\)"],["text"," for the weight of a walk"]],"source":"Majorant $q^{|s|-1}$ for the weight of a walk"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-factor-majorant","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The majorant `q^{|s| - 1}` for the weight of a walk `s` with at least two states, and `0` for shorter lists.","title":"Majorant $q^{|s|-1}$ for the weight of a walk"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-sherman-radius","kind":"node","lean":"StatMech.Onsager.ons_ShermanRadius","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-sherman-radius","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The radius `1 / (2 (number of darts)\u00b2)` below which Sherman's first-return series for the torus Kac-Ward matrix converges with total weight below `1`.","title":"Sherman's convergence radius for the torus Kac-Ward matrix"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-tuple-prod-eq-list-prod","kind":"node","lean":"StatMech.Onsager.ons_tupleProd_eq_listProd","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-tuple-prod-eq-list-prod","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The product over a tuple equals the product over the list `List.ofFn p`; see also `List.prod_ofFn` and `List.map_ofFn`.","title":"Product over a tuple equals the product over its list"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-one-sub-ne-zero-of-norm-lt-one","kind":"node","lean":"StatMech.Onsager.ons_one_sub_ne_zero_of_norm_lt_one","parent":49305,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(1 - z \\neq 0\\)"],["text"," for "],["math","\\(|z| < 1\\)"]],"source":"$1 - z \\neq 0$ for $|z| < 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-one-sub-ne-zero-of-norm-lt-one","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"`1 - z \u2260 0` when `\u2016z\u2016 < 1`.","title":"$1 - z \\neq 0$ for $|z| < 1$"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-is-first-return","kind":"node","lean":"StatMech.Onsager.ons_isFirstReturn","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-is-first-return","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"A cyclic sequence `v` is a first-return loop at `e` (avoiding `rev e`) if it starts at `e`, does not visit `e` again, and never uses the reverse edge `rev e`.","title":"First-return loop at a directed edge"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-rooted-avoiding","kind":"node","lean":"StatMech.Onsager.ons_rootedAvoiding","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-rooted-avoiding","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The cyclic sequences of length `n` that start at `e` and never use `rev e`.","title":"Cyclic sequences rooted at an edge avoiding its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-list-series-term","kind":"node","lean":"StatMech.Onsager.ons_listSeriesTerm","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-list-series-term","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The term `(\u220f_{a \u2208 l} wt a) / |l|` of the list series, set to `0` on the empty list.","title":"Term of the list series: product of weights over length"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-loop-weight","kind":"node","lean":"StatMech.Onsager.ons_loopWeight","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-loop-weight","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The weight `\u220f\u2096 \u039b (v k) (v (k + 1))` of a cyclic sequence `v : Fin n \u2192 E`, indices taken modulo `n`.","title":"Weight of a cyclic sequence"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-cyclic-shift","kind":"node","lean":"StatMech.Onsager.ons_cyclicShift","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-cyclic-shift","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Rotation of a cyclic sequence by one step: `(ons_cyclicShift v) k = v (k + 1)`.","title":"Rotation of a cyclic sequence by one step"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-bucket-e","kind":"node","lean":"StatMech.Onsager.ons_bucketE","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-bucket-e","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The total Kac-Ward weight of closed walks of length `n + 1` that visit `e` but not its reverse.","title":"Kac-Ward bucket: walks through a dart, not its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-bucket-n","kind":"node","lean":"StatMech.Onsager.ons_bucketN","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-bucket-n","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The total Kac-Ward weight of closed walks of length `n + 1` that visit neither `e` nor its reverse.","title":"Kac-Ward bucket: walks avoiding a dart and its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-sum-zero-of-sign-involution","kind":"node","lean":"StatMech.Onsager.ons_sum_zero_of_sign_involution","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-sum-zero-of-sign-involution","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"A fixed-point-free involution of `S` that reverses the sign of the loop weight forces the loop weights over `S` to sum to zero. See also `Finset.sum_involution`.","title":"Loop sum vanishes under a fixed-point-free sign-reversing involution"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-loop-set-both","kind":"node","lean":"StatMech.Onsager.ons_loopSetBoth","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-loop-set-both","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The cyclic sequences of length `n` that visit both `a` and `b`.","title":"Cyclic sequences visiting two given states"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-mir","kind":"node","lean":"StatMech.Onsager.ons_mir","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-mir","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The mirror image `l + m - k` of a position `k` in the segment `[l, m]` (truncated to `n - 1`).","title":"Mirror position within a segment"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-rev-seg","kind":"node","lean":"StatMech.Onsager.ons_revSeg","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-rev-seg","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Reversal of the segment `[l, m]` of `v`: positions inside are mirrored and their entries replaced by their reverses; positions outside are unchanged.","title":"Reversal of a segment of a cyclic sequence"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-hit-set","kind":"node","lean":"StatMech.Onsager.ons_hitSet","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-hit-set","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The positions where `v` meets `e` or `rev e`.","title":"Positions where a sequence meets an edge or its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-l","kind":"node","lean":"StatMech.Onsager.ons_l","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-l","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The first position where `v` meets `e` or `rev e` (`0` if there is none).","title":"First position meeting an edge or its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-m-set","kind":"node","lean":"StatMech.Onsager.ons_mSet","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-m-set","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The positions where `v` meets the reverse of its entry at `ons_l e rev v`.","title":"Positions meeting the reverse of the first hit"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-m","kind":"node","lean":"StatMech.Onsager.ons_m","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-m","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The last position where `v` meets the reverse of its entry at `ons_l e rev v` (`0` if there is none).","title":"Last position meeting the reverse of the first hit"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-surgery","kind":"node","lean":"StatMech.Onsager.ons_surgery","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-surgery","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Sherman's surgery: reverse the segment between `ons_l` and `ons_m` when `v` visits both `e` and `rev e`; otherwise leave `v` unchanged.","title":"Sherman's segment-reversal surgery"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-lemma5","kind":"node","lean":"StatMech.Onsager.ons_lemma5","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-lemma5","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"If the surgery reverses the sign of loop weights, the loops that visit both `e` and `rev e` have total weight zero (S. Sherman, *Combinatorial aspects of the Ising model for ferromagnetism. I*, 1960).","title":"Sherman's Lemma 5"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-loop-rev","kind":"node","lean":"StatMech.Onsager.ons_loopRev","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-loop-rev","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The reverse of a closed walk: position `k` carries the reverse of the dart at position `-k`.","title":"Reverse of a closed walk"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-orbit-setoid","kind":"node","lean":"StatMech.Onsager.ons_orbitSetoid","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-orbit-setoid","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The rotation-orbit equivalence relation on cyclic sequences: `v \u223c w` when `w` is obtained from `v` by an iterated rotation.","title":"Rotation-orbit equivalence on cyclic sequences"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-rep","kind":"node","lean":"StatMech.Onsager.ons_rep","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-rep","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"A chosen representative of the rotation orbit of `v`.","title":"Chosen representative of a rotation orbit"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-orbit-card","kind":"node","lean":"StatMech.Onsager.ons_orbitCard","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-orbit-card","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The number of sequences whose chosen orbit representative is `r`; for a representative `r` this is the size of its rotation orbit, and `0` otherwise.","title":"Number of sequences with a given orbit representative"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-rotate","kind":"node","lean":"StatMech.Onsager.ons_rotate","parent":49305,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rotation of a cyclic sequence by "],["math","\\(k\\)"],["text"," steps"]],"source":"Rotation of a cyclic sequence by $k$ steps"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-rotate","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Rotation of a cyclic sequence by `k` steps: `(ons_rotate k v) j = v (j + k)`.","title":"Rotation of a cyclic sequence by $k$ steps"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-rotate-equiv","kind":"node","lean":"StatMech.Onsager.ons_rotateEquiv","parent":49305,"reading_title":{"format":"inline-parts-v1","parts":[["text","Rotation by "],["math","\\(k\\)"],["text"," as an equivalence of sequences"]],"source":"Rotation by $k$ as an equivalence of sequences"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-rotate-equiv","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Rotation by `k` as an equivalence of `Fin n \u2192 E`, with inverse the rotation by `-k`.","title":"Rotation by $k$ as an equivalence of sequences"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-visit-count","kind":"node","lean":"StatMech.Onsager.ons_visitCount","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-visit-count","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The number of positions `k` with `v k = e`.","title":"Number of visits of a sequence to a state"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-edge-weight","kind":"node","lean":"StatMech.Onsager.ons_edgeWeight","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-edge-weight","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The edge weight `\u039b e\u2080 e\u2081 * \u039b e\u2081 e\u2082 * \u22ef * \u039b e\u2096\u208b\u2081 e\u2096` of a walk `[e\u2080, \u2026, e\u2096]`; walks with at most one entry have weight `1`.","title":"Edge weight of a walk"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-glue","kind":"node","lean":"StatMech.Onsager.ons_glue","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-glue","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Concatenation of walk segments sharing their end points at `e`: each segment except the last loses its final entry, and the result ends in `e`.","title":"Gluing of walk segments at a root"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-first-return-split","kind":"node","lean":"StatMech.Onsager.ons_firstReturnSplit","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-first-return-split","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Splitting of a list at the occurrences of `e`: each piece runs from one occurrence of `e` to the next.","title":"Splitting a list at the occurrences of a root"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-is-first-return-segment","kind":"node","lean":"StatMech.Onsager.ons_isFirstReturnSegment","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-is-first-return-segment","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"A first-return segment at `e`: a list `e :: interior ++ [e]` whose interior avoids `e`.","title":"First-return segment at a state"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-first-return-factors","kind":"node","lean":"StatMech.Onsager.ons_firstReturnFactors","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-first-return-factors","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The first-return segments of a closed walk at `e`, without the trailing one-element piece `[e]`.","title":"First-return segments of a closed walk"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-valid-rooted-list","kind":"node","lean":"StatMech.Onsager.ons_validRootedList","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-valid-rooted-list","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"A rooted closed walk at `e` avoiding `r`: a list starting and ending at `e` that never contains `r`.","title":"Rooted closed walk at a state avoiding another"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-valid-factor-list","kind":"node","lean":"StatMech.Onsager.ons_validFactorList","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-valid-factor-list","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"A list of first-return segments at `e`, none of which contains `r`.","title":"List of first-return segments avoiding a state"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-rooted-factor-equiv","kind":"node","lean":"StatMech.Onsager.ons_rootedFactorEquiv","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-rooted-factor-equiv","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Rooted closed walks at `e` avoiding `r` are equivalent to lists of first-return segments avoiding `r`, by splitting and gluing.","title":"Rooted closed walks as lists of first-return segments"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-first-return-weight","kind":"node","lean":"StatMech.Onsager.ons_firstReturnWeight","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-first-return-weight","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The edge weight of `s` if `s` is a first-return segment at `e` avoiding `r`, and `0` otherwise.","title":"Weight of a first-return segment avoiding a state"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-rooted-list-term","kind":"node","lean":"StatMech.Onsager.ons_rootedListTerm","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-rooted-list-term","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The rooted term of a walk: its edge weight divided by its number of first-return factors at `e`.","title":"Rooted term: walk weight over its number of first-return factors"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-phase-bucket-e","kind":"node","lean":"StatMech.Onsager.ons_phaseBucketE","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-phase-bucket-e","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The total twisted Kac-Ward weight of closed walks of length `n + 1` visiting `e` but not its reverse.","title":"Twisted bucket: walks through a dart, not its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-phase-bucket-n","kind":"node","lean":"StatMech.Onsager.ons_phaseBucketN","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-phase-bucket-n","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The total twisted Kac-Ward weight of closed walks of length `n + 1` visiting neither `e` nor its reverse.","title":"Twisted bucket: walks avoiding a dart and its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-mask-phase-bucket-e","kind":"node","lean":"StatMech.Onsager.ons_maskPhaseBucketE","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-mask-phase-bucket-e","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"As `ons_phaseBucketE`, for the twisted matrix with the darts of `forbidden` masked.","title":"Masked twisted bucket: walks through a dart, not its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-mask-phase-bucket-n","kind":"node","lean":"StatMech.Onsager.ons_maskPhaseBucketN","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-mask-phase-bucket-n","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"As `ons_phaseBucketN`, for the twisted matrix with the darts of `forbidden` masked.","title":"Masked twisted bucket: walks avoiding a dart and its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-spin-walk-root","kind":"node","lean":"StatMech.Onsager.ons_spinWalkRoot","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-spin-walk-root","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The walk-series square root of the twisted Kac-Ward determinant with boundary phases `a, b` at activity `x`.","title":"Walk-series square root of the twisted Kac-Ward determinant"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-phase-avoid-root","kind":"node","lean":"StatMech.Onsager.ons_phaseAvoidRoot","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-phase-avoid-root","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The exponential of minus one half of the series of closed walks avoiding `e` and its reverse.","title":"Exponential of minus half the series avoiding a dart and its reverse"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-add-dart-pair","kind":"node","lean":"StatMech.Onsager.ons_addDartPair","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-add-dart-pair","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"Add a dart and its reverse to a set of masked darts.","title":"Adding a dart and its reverse to the masked set"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-mask-after","kind":"node","lean":"StatMech.Onsager.ons_maskAfter","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-mask-after","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The set of masked darts after eliminating the darts of a list, pair by pair.","title":"Masked darts after pairwise elimination along a list"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-mask-elimination-product","kind":"node","lean":"StatMech.Onsager.ons_maskEliminationProduct","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-mask-elimination-product","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The product of the factors `1 - \u03bb_e` along a list of darts, each first-return weight `\u03bb_e` computed with the previously eliminated darts masked.","title":"Product of first-return factors along a dart elimination"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-matrix-bucket-e","kind":"node","lean":"StatMech.Onsager.ons_matrixBucketE","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-matrix-bucket-e","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The total weight of closed walks of length `n + 1` of `M` that visit `e` but not `r`.","title":"Matrix bucket: walks through one state avoiding another"},{"id":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/ons-matrix-bucket-n","kind":"node","lean":"StatMech.Onsager.ons_matrixBucketN","parent":49305,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula/index.html#ons-matrix-bucket-n","scope":"StatisticalMechanics/exact-solutions/sherman-s-loop-cancellations-and-product-formula","status":"fully_proved","summary":"The total weight of closed walks of length `n + 1` of `M` that visit neither `e` nor `r`.","title":"Matrix bucket: walks avoiding two given states"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt.json","id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","kind":"scope","navigation":"module","parent":49228,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","title":"Homology and turning of loops on the square-lattice torus"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-path-displacement","kind":"node","lean":"StatMech.Onsager.ons_pathDisplacement","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-path-displacement","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The total displacement in `\u2124\u00b2` of a list of lattice directions.","title":"Total displacement of a list of lattice directions"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-path-vertices","kind":"node","lean":"StatMech.Onsager.ons_pathVertices","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-path-vertices","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The lattice points visited by a path from the origin with the given directions.","title":"Lattice points visited by a path from the origin"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-open-turn-sum","kind":"node","lean":"StatMech.Onsager.ons_openTurnSum","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-open-turn-sum","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The total turning (in quarter turns) between consecutive directions of a list.","title":"Open turning sum of a direction list"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-cyclic-turn-sum","kind":"node","lean":"StatMech.Onsager.ons_cyclicTurnSum","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-cyclic-turn-sum","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The total turning of a direction list read cyclically, including the turn from the last direction back to the first.","title":"Cyclic turning sum of a direction list"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-outside-connector","kind":"node","lean":"StatMech.Onsager.ons_outsideConnector","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-outside-connector","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"A path east, north, west, south and east again that runs around the bounding box of a periodic arc with displacement `(dx, dy)` and height `top`.","title":"Outside connector around the bounding box of a periodic arc"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-outside-connector-vertex-list","kind":"node","lean":"StatMech.Onsager.ons_outsideConnectorVertexList","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-outside-connector-vertex-list","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The lattice points visited by the outside connector.","title":"Lattice points visited by the outside connector"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-period-state-list","kind":"node","lean":"StatMech.Onsager.ons_periodStateList","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-period-state-list","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The states `(r, i)` (period index `r`, position `i`) of one period of a periodic walk.","title":"States of one period of a periodic walk"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-all-period-states","kind":"node","lean":"StatMech.Onsager.ons_allPeriodStates","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","States of the first "],["math","\\(N + 1\\)"],["text"," periods of a periodic walk"]],"source":"States of the first $N + 1$ periods of a periodic walk"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-all-period-states","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The states of the first `N` periods of a periodic walk followed by the initial state of the next period.","title":"States of the first $N + 1$ periods of a periodic walk"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-periodic-arc-states","kind":"node","lean":"StatMech.Onsager.ons_periodicArcStates","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","States of a periodic arc across "],["math","\\(N\\)"],["text"," periods"]],"source":"States of a periodic arc across $N$ periods"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-periodic-arc-states","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The states of the periodic arc from position `imin` in the first period to position `imax` in period `N`.","title":"States of a periodic arc across $N$ periods"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-period-state-coord","kind":"node","lean":"StatMech.Onsager.ons_periodStateCoord","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-period-state-coord","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The lattice point of the state `(r, i)`: the `i`-th point of the lift shifted by `r` periods.","title":"Lattice point of a period state"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-last-fin","kind":"node","lean":"StatMech.Onsager.ons_lastFin","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Last element of "],["math","\\(\\{0, \\ldots, n - 1\\}\\)"]],"source":"Last element of $\\{0, \\ldots, n - 1\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-last-fin","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The last element of `Fin n`.","title":"Last element of $\\{0, \\ldots, n - 1\\}$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-periodic-arc-directions","kind":"node","lean":"StatMech.Onsager.ons_periodicArcDirections","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-periodic-arc-directions","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The directions along the periodic arc.","title":"Directions along a periodic arc"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-direction-periods","kind":"node","lean":"StatMech.Onsager.ons_directionPeriods","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(N\\)"],["text"," repetitions of a direction sequence"]],"source":"$N$ repetitions of a direction sequence"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-direction-periods","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"`N` repetitions of the direction sequence.","title":"$N$ repetitions of a direction sequence"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-rotate-point","kind":"node","lean":"StatMech.Onsager.ons_rotatePoint","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-rotate-point","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"Rotation of a lattice point by `c` quarter turns.","title":"Rotation of a lattice point by quarter turns"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-reduce-site","kind":"node","lean":"StatMech.Onsager.ons_reduceSite","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-reduce-site","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"Reduction of a site of `\u2124\u00b2` to the torus `\u2124_L \u00d7 \u2124_L`.","title":"Reduction of a lattice site to the torus"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-lift-dir","kind":"node","lean":"StatMech.Onsager.ons_liftDir","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-lift-dir","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The direction sequence of a closed torus walk, read in the order of the lift to `\u2124\u00b2`.","title":"Direction sequence of the lift of a closed torus walk"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-lift-site","kind":"node","lean":"StatMech.Onsager.ons_liftSite","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-lift-site","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The sites of a closed torus walk, in the order of the lift.","title":"Sites of a closed torus walk in lift order"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-edge-bit","kind":"node","lean":"StatMech.Onsager.ons_edgeBit","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-edge-bit","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The indicator of `e \u2208 F` in `Fin 2`.","title":"Indicator bit of an edge in an edge set"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-horizontal-bit","kind":"node","lean":"StatMech.Onsager.ons_horizontalBit","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-horizontal-bit","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"Whether `F` contains the horizontal edge from `p` eastward.","title":"Horizontal edge bit of an edge set at a site"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-vertical-bit","kind":"node","lean":"StatMech.Onsager.ons_verticalBit","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-vertical-bit","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"Whether `F` contains the vertical edge from `p` northward.","title":"Vertical edge bit of an edge set at a site"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-edge-cycle-intersection","kind":"node","lean":"StatMech.Onsager.ons_edgeCycleIntersection","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-edge-cycle-intersection","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The mod-2 intersection number of edge sets `F` and `G` on the torus.","title":"Mod 2 intersection number of two torus edge sets"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-horizontal-zbit","kind":"node","lean":"StatMech.Onsager.ons_horizontalZBit","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Horizontal edge bit as an element of "],["math","\\(\\mathbb{Z}/2\\)"]],"source":"Horizontal edge bit as an element of $\\mathbb{Z}/2$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-horizontal-zbit","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The horizontal edge bit as an element of `ZMod 2`.","title":"Horizontal edge bit as an element of $\\mathbb{Z}/2$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-vertical-zbit","kind":"node","lean":"StatMech.Onsager.ons_verticalZBit","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Vertical edge bit as an element of "],["math","\\(\\mathbb{Z}/2\\)"]],"source":"Vertical edge bit as an element of $\\mathbb{Z}/2$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-vertical-zbit","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The vertical edge bit as an element of `ZMod 2`.","title":"Vertical edge bit as an element of $\\mathbb{Z}/2$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-x-flux","kind":"node","lean":"StatMech.Onsager.ons_xFlux","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-x-flux","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The number mod 2 of horizontal edges of `F` leaving column `x` eastward.","title":"Horizontal flux of an edge set through a column"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-y-flux","kind":"node","lean":"StatMech.Onsager.ons_yFlux","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-y-flux","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The number mod 2 of vertical edges of `F` leaving row `y` northward.","title":"Vertical flux of an edge set through a row"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-x-seam-port-edge","kind":"node","lean":"StatMech.Onsager.ons_xSeamPortEdge","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-x-seam-port-edge","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The edge of the vertical seam in row `y`.","title":"Vertical-seam edge in a given row"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-y-seam-port-edge","kind":"node","lean":"StatMech.Onsager.ons_ySeamPortEdge","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-y-seam-port-edge","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The edge of the horizontal seam in column `x`.","title":"Horizontal-seam edge in a given column"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-zmod-prefix","kind":"node","lean":"StatMech.Onsager.ons_zmodPrefix","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Prefix sum of a function on "],["math","\\(\\mathbb{Z}/L\\)"]],"source":"Prefix sum of a function on $\\mathbb{Z}/L$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-zmod-prefix","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The prefix sum `\u2211_{k < x} f k` along `ZMod L`, using the representatives `0, \u2026, L - 1`.","title":"Prefix sum of a function on $\\mathbb{Z}/L$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-zero-flux-horizontal","kind":"node","lean":"StatMech.Onsager.ons_zeroFluxHorizontal","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-zero-flux-horizontal","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The horizontal part of the potential construction for an edge set of zero flux.","title":"Horizontal part of the zero-flux potential construction"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-zero-flux-vertical","kind":"node","lean":"StatMech.Onsager.ons_zeroFluxVertical","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-zero-flux-vertical","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The vertical part of the potential construction for an edge set of zero flux.","title":"Vertical part of the zero-flux potential 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$\\mathbb{Z}/2$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-int-parity","kind":"node","lean":"StatMech.Onsager.ons_intParity","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-int-parity","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The parity of an integer as an element of `Fin 2`.","title":"Parity of an integer as a bit"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-winding-parity","kind":"node","lean":"StatMech.Onsager.ons_windingParity","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-winding-parity","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The homology class mod 2 of the winding `(m\u2081, m\u2082)`.","title":"Mod 2 homology class of a winding"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-spin-linear-character","kind":"node","lean":"StatMech.Onsager.ons_spinLinearCharacter","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-spin-linear-character","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The linear character `(-1)^{a h\u2081 + b h\u2082}` of the boundary phases on homology classes mod 2.","title":"Linear character of the boundary phases"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-x-wrap","kind":"node","lean":"StatMech.Onsager.ons_xWrap","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-x-wrap","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"A dart crosses the vertical seam between the columns `x = -1` and `x = 0`.","title":"Crossing of the vertical seam by a dart"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-y-wrap","kind":"node","lean":"StatMech.Onsager.ons_yWrap","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-y-wrap","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"A dart crosses the horizontal seam between the rows `y = -1` and `y = 0`.","title":"Crossing of the horizontal seam by a dart"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-x-wrap-sign","kind":"node","lean":"StatMech.Onsager.ons_xWrapSign","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-x-wrap-sign","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed crossing of the vertical seam by a dart: `1` eastward, `-1` westward, `0` otherwise.","title":"Signed crossing of the vertical seam by a dart"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-y-wrap-sign","kind":"node","lean":"StatMech.Onsager.ons_yWrapSign","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-y-wrap-sign","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed crossing of the horizontal seam by a dart: `1` northward, `-1` southward, `0` otherwise.","title":"Signed crossing of the horizontal seam by a dart"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-dart-edge-set","kind":"node","lean":"StatMech.Onsager.ons_dartEdgeSet","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-dart-edge-set","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The set of torus edges traversed by a closed walk of darts.","title":"Edge set of a closed walk of darts"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-winding-translation","kind":"node","lean":"StatMech.Onsager.ons_windingTranslation","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translation of "],["math","\\(\\mathbb{Z}^2\\)"],["text"," for a winding around the torus"]],"source":"Translation of $\\mathbb{Z}^2$ for a winding around the torus"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-winding-translation","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The translation `(L m\u2081, L m\u2082)` of `\u2124\u00b2` corresponding to winding `(m\u2081, m\u2082)` around the torus.","title":"Translation of $\\mathbb{Z}^2$ for a winding around the torus"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-periodic-lift-vertex","kind":"node","lean":"StatMech.Onsager.ons_periodicLiftVertex","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-periodic-lift-vertex","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The `i`-th point of the `r`-th period of the lift of a torus loop with winding `(m\u2081, m\u2082)`, relative to the starting point.","title":"Points of the periodic lift of a torus loop"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-lift-base","kind":"node","lean":"StatMech.Onsager.ons_liftBase","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Base point in "],["math","\\(\\mathbb{Z}^2\\)"],["text"," over the start of a torus loop"]],"source":"Base point in $\\mathbb{Z}^2$ over the start of a torus loop"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-lift-base","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"A base point in `\u2124\u00b2` over the starting site of a torus loop.","title":"Base point in $\\mathbb{Z}^2$ over the start of a torus loop"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-anchored-periodic-lift-vertex","kind":"node","lean":"StatMech.Onsager.ons_anchoredPeriodicLiftVertex","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-anchored-periodic-lift-vertex","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The periodic lift translated to start at the base point over the starting site.","title":"Periodic lift anchored at the base point"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-signed-horizontal-boundary","kind":"node","lean":"StatMech.Onsager.ons_signedHorizontalBoundary","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-signed-horizontal-boundary","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed difference of a face potential across the horizontal edge at `p`.","title":"Signed potential difference across a horizontal edge"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-signed-vertical-boundary","kind":"node","lean":"StatMech.Onsager.ons_signedVerticalBoundary","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-signed-vertical-boundary","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed difference of a face potential across the vertical edge at `p`.","title":"Signed potential difference across a vertical edge"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-outgoing-boundary-coeff","kind":"node","lean":"StatMech.Onsager.ons_outgoingBoundaryCoeff","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-outgoing-boundary-coeff","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed boundary coefficient of a potential on the edge leaving `p` in direction `\u03bc`.","title":"Signed boundary coefficient on an outgoing edge"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-loop-boundary-coeff","kind":"node","lean":"StatMech.Onsager.ons_loopBoundaryCoeff","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-loop-boundary-coeff","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed boundary coefficient on the `k`-th step of a torus loop.","title":"Signed boundary coefficient along a torus loop"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-x-seam-edge","kind":"node","lean":"StatMech.Onsager.ons_xSeamEdge","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-x-seam-edge","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"Edges of the torus crossing the vertical seam between the columns `x = -1` and `x = 0`.","title":"Edges crossing the vertical seam of the torus"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-y-seam-edge","kind":"node","lean":"StatMech.Onsager.ons_ySeamEdge","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-y-seam-edge","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"Edges of the torus crossing the horizontal seam between the rows `y = -1` and `y = 0`.","title":"Edges crossing the horizontal seam of the torus"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-even-homology","kind":"node","lean":"StatMech.Onsager.ons_evenHomology","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-even-homology","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The homology class mod 2 of an edge set: the parities of its numbers of edges crossing the vertical and the horizontal seam.","title":"Mod 2 homology class of an edge set"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-sector-weight","kind":"node","lean":"StatMech.Onsager.ons_sectorWeight","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-sector-weight","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The sector weight `Z_h(x) = \u2211 x^|F|` over even subgraphs `F` of the torus with homology class `h`.","title":"Sector weight of a homology class"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-sector00","kind":"node","lean":"StatMech.Onsager.ons_sector00","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-sector00","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The sector weight of the trivial homology class.","title":"Sector weight of the trivial homology class"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-sector10","kind":"node","lean":"StatMech.Onsager.ons_sector10","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sector weight of the homology class "],["math","\\((1, 0)\\)"]],"source":"Sector weight of the homology class $(1, 0)$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-sector10","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The sector weight of the homology class `(1, 0)`.","title":"Sector weight of the homology class $(1, 0)$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-sector01","kind":"node","lean":"StatMech.Onsager.ons_sector01","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sector weight of the homology class "],["math","\\((0, 1)\\)"]],"source":"Sector weight of the homology class $(0, 1)$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-sector01","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The sector weight of the homology class `(0, 1)`.","title":"Sector weight of the homology class $(0, 1)$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-sector11","kind":"node","lean":"StatMech.Onsager.ons_sector11","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sector weight of the homology class "],["math","\\((1, 1)\\)"]],"source":"Sector weight of the homology class $(1, 1)$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-sector11","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The sector weight of the homology class `(1, 1)`.","title":"Sector weight of the homology class $(1, 1)$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-spin00","kind":"node","lean":"StatMech.Onsager.ons_spin00","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed sector sum "],["math","\\(Z_{00} + Z_{10} + Z_{01} - Z_{11}\\)"]],"source":"Signed sector sum $Z_{00} + Z_{10} + Z_{01} - Z_{11}$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-spin00","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed sector sum `Z\u2080\u2080 + Z\u2081\u2080 + Z\u2080\u2081 - Z\u2081\u2081`.","title":"Signed sector sum $Z_{00} + Z_{10} + Z_{01} - Z_{11}$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-spin10","kind":"node","lean":"StatMech.Onsager.ons_spin10","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed sector sum "],["math","\\(Z_{00} - Z_{10} + Z_{01} + Z_{11}\\)"]],"source":"Signed sector sum $Z_{00} - Z_{10} + Z_{01} + Z_{11}$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-spin10","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed sector sum `Z\u2080\u2080 - Z\u2081\u2080 + Z\u2080\u2081 + Z\u2081\u2081`.","title":"Signed sector sum $Z_{00} - Z_{10} + Z_{01} + Z_{11}$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-spin01","kind":"node","lean":"StatMech.Onsager.ons_spin01","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed sector sum "],["math","\\(Z_{00} + Z_{10} - Z_{01} + Z_{11}\\)"]],"source":"Signed sector sum $Z_{00} + Z_{10} - Z_{01} + Z_{11}$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-spin01","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed sector sum `Z\u2080\u2080 + Z\u2081\u2080 - Z\u2080\u2081 + Z\u2081\u2081`.","title":"Signed sector sum $Z_{00} + Z_{10} - Z_{01} + Z_{11}$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-spin11","kind":"node","lean":"StatMech.Onsager.ons_spin11","parent":49359,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed sector sum "],["math","\\(Z_{00} - Z_{10} - Z_{01} - Z_{11}\\)"]],"source":"Signed sector sum $Z_{00} - Z_{10} - Z_{01} - Z_{11}$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-spin11","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The signed sector sum `Z\u2080\u2080 - Z\u2081\u2080 - Z\u2080\u2081 - Z\u2081\u2081`.","title":"Signed sector sum $Z_{00} - Z_{10} - Z_{01} - Z_{11}$"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-spin-character","kind":"node","lean":"StatMech.Onsager.ons_spinCharacter","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-spin-character","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The spin-structure character `(-1)^{(1 + a) h\u2081 + (1 + b) h\u2082 + h\u2081 h\u2082}` of a homology class `h` mod 2, for boundary phases `a, b`.","title":"Spin-structure character of a mod 2 homology class"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-homology-intersection","kind":"node","lean":"StatMech.Onsager.ons_homologyIntersection","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-homology-intersection","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The intersection form `h\u2081 k\u2082 + h\u2082 k\u2081` on `H\u2081(T\u00b2, \u2124/2) = (\u2124/2)\u00b2`.","title":"Intersection form on the mod 2 homology of the torus"},{"id":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/ons-spin-character-sum","kind":"node","lean":"StatMech.Onsager.ons_spinCharacterSum","parent":49359,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt/index.html#ons-spin-character-sum","scope":"StatisticalMechanics/exact-solutions/homology-and-turning-of-loops-on-the-square-latt","status":"fully_proved","summary":"The even-subgraph sum of the torus with each subgraph weighted by the spin character of its homology class.","title":"Spin-character sum over even subgraphs of the torus"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus.json","id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","kind":"scope","navigation":"module","parent":49228,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","title":"Trivalent decoration of the torus"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-spin-vertex-gauge","kind":"node","lean":"StatMech.Onsager.ons_spinVertexGauge","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-spin-vertex-gauge","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The gauge `u^{s\u2081} v^{s\u2082}` at site `s`, with `u, v` the spin boundary phases.","title":"Spin boundary-phase gauge at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-spin-dart-seam-sign","kind":"node","lean":"StatMech.Onsager.ons_spinDartSeamSign","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-spin-dart-seam-sign","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The seam sign `(-1)^{a \u00b7 (signed vertical seam crossing) + b \u00b7 (signed horizontal seam crossing)}` of a dart.","title":"Seam sign of a dart"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-spin-seam-dec-weight","kind":"node","lean":"StatMech.Onsager.ons_spinSeamDecWeight","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-spin-seam-dec-weight","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Decorated weights with every external edge multiplied by the seam sign of its dart.","title":"Decorated weights twisted by seam signs"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-diagonal-gauge-matrix","kind":"node","lean":"StatMech.Onsager.ons_diagonalGaugeMatrix","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-diagonal-gauge-matrix","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Diagonal conjugation `gauge i * M i j * (gauge j)\u207b\u00b9`.","title":"Diagonal gauge conjugation of a matrix"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-state-gauge","kind":"node","lean":"StatMech.Onsager.ons_decMiddleStateGauge","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-state-gauge","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The gauge on the states of the augmented matrix: the site gauge on darts, and the gauge of `site` on the two side states.","title":"Gauge on the states of the augmented matrix"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-internal-adj","kind":"node","lean":"StatMech.Onsager.ons_decInternalAdj","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-internal-adj","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Internal adjacency of the decoration: two darts at the same site whose directions are consecutive in `0, 1, 2, 3`.","title":"Internal adjacency of the torus decoration"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-adj","kind":"node","lean":"StatMech.Onsager.ons_decAdj","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-adj","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Adjacency of the decorated graph: a dart is joined to its reverse and to its internal neighbors.","title":"Adjacency of the decorated torus graph"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-graph","kind":"node","lean":"StatMech.Onsager.ons_decGraph","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-graph","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The decorated graph of the torus: vertices are darts, external edges join each dart to its reverse, and internal edges form the path `(s, 0) - (s, 1) - (s, 2) - (s, 3)` at each site `s`.","title":"Decorated graph of the torus"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-internal-neighbors","kind":"node","lean":"StatMech.Onsager.ons_decInternalNeighbors","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-internal-neighbors","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The internal neighbors of a dart in the path at its site.","title":"Internal neighbors of a dart in the decoration"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-chain-completion","kind":"node","lean":"StatMech.Onsager.ons_decChainCompletion","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-chain-completion","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The internal-edge bits `(b\u2080, b\u2080 + b\u2081, b\u2080 + b\u2081 + b\u2082)` that complete port bits `b\u2080, \u2026, b\u2083` at one site to an even subgraph.","title":"Internal-edge bits completing port bits at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-chain-even","kind":"node","lean":"StatMech.Onsager.ons_decChainEven","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-chain-even","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Parity of the decorated subgraph at the four darts of one site, for port bits `b\u2080, \u2026, b\u2083` and internal-edge bits `y`.","title":"Parity condition of the decoration at one site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-port-pattern","kind":"node","lean":"StatMech.Onsager.ons_PortPattern","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-port-pattern","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"A port pattern: one bit per dart, equal on a dart and its reverse, with an even number of ones at each site. Port patterns encode even subgraphs of the torus.","title":"Port pattern on the darts of the torus"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-port-chain-bit","kind":"node","lean":"StatMech.Onsager.ons_portChainBit","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-port-chain-bit","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The internal-edge bits at a site determined by a port pattern.","title":"Internal-edge bits at a site from a port pattern"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-neighbor-bit","kind":"node","lean":"StatMech.Onsager.ons_decNeighborBit","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-neighbor-bit","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The bit of the decorated edge between `d` and `e` determined by a port pattern.","title":"Port-pattern bit of the decorated edge between two darts"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-edge-bit","kind":"node","lean":"StatMech.Onsager.ons_decEdgeBit","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-edge-bit","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The bit of a decorated edge determined by a port pattern.","title":"Port-pattern bit of an unordered decorated edge"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-decorated-edges","kind":"node","lean":"StatMech.Onsager.ons_decoratedEdges","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-decorated-edges","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The even subgraph of the decorated graph determined by a port pattern.","title":"Even decorated subgraph determined by a port pattern"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-port-edge","kind":"node","lean":"StatMech.Onsager.ons_portEdge","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-port-edge","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The torus edge underlying a dart.","title":"Torus edge underlying a dart"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-port-bit","kind":"node","lean":"StatMech.Onsager.ons_portBit","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-port-bit","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The bit of a dart in the port pattern of an edge set `F`: `1` if its underlying torus edge lies in `F`.","title":"Port bit of a dart for a torus edge set"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-even-subgraph-port-pattern","kind":"node","lean":"StatMech.Onsager.ons_evenSubgraphPortPattern","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-even-subgraph-port-pattern","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The port pattern of an even subgraph of the torus.","title":"Port pattern of an even subgraph of the torus"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-decorated-port-bit","kind":"node","lean":"StatMech.Onsager.ons_decoratedPortBit","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-decorated-port-bit","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The port bit of a dart in a decorated edge set: whether the external edge at the dart is present.","title":"Port bit of a dart in a decorated edge set"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-decorated-chain-bits","kind":"node","lean":"StatMech.Onsager.ons_decoratedChainBits","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-decorated-chain-bits","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The three internal-edge bits of a decorated edge set at one site.","title":"Internal-edge bits of a decorated edge set at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-port-pattern-of-decorated","kind":"node","lean":"StatMech.Onsager.ons_portPatternOfDecorated","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-port-pattern-of-decorated","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The port pattern of an even subgraph of the decorated graph.","title":"Port pattern of an even decorated subgraph"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-port-pattern-edges","kind":"node","lean":"StatMech.Onsager.ons_portPatternEdges","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-port-pattern-edges","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The torus edge set of a port pattern.","title":"Torus edge set of a port pattern"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-even-subgraph-port-equiv","kind":"node","lean":"StatMech.Onsager.ons_evenSubgraphPortEquiv","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-even-subgraph-port-equiv","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Even subgraphs of the torus are equivalent to port patterns.","title":"Equivalence between even torus subgraphs and port patterns"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-decorated-port-equiv","kind":"node","lean":"StatMech.Onsager.ons_decoratedPortEquiv","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-decorated-port-equiv","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Even subgraphs of the decorated graph are equivalent to port patterns.","title":"Equivalence between even decorated subgraphs and port patterns"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-decoration-equiv","kind":"node","lean":"StatMech.Onsager.ons_decorationEquiv","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-decoration-equiv","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Even subgraphs of the torus are equivalent to even subgraphs of the decorated graph.","title":"Equivalence between even subgraphs of the torus and its decoration"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-embed-offset","kind":"node","lean":"StatMech.Onsager.ons_decEmbedOffset","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-embed-offset","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The offset of a dart in direction `\u03bc` inside its `8 \u00d7 8` cell: east `(6, 4)`, north `(4, 6)`, west `(2, 4)`, south `(4, 2)`.","title":"Offset of a dart inside its 8 \u00d7 8 cell"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-embed-vertex","kind":"node","lean":"StatMech.Onsager.ons_decEmbedVertex","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-embed-vertex","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The position `8 s + offset(\u03bc)` of the dart `(s, \u03bc)` in the `8L \u00d7 8L` torus.","title":"Position of a dart in the 8L \u00d7 8L torus"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-edge-route-dirs","kind":"node","lean":"StatMech.Onsager.ons_decEdgeRouteDirs","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-edge-route-dirs","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The four step directions of the route of the decorated edge from `d` to `e`: straight for an external edge, an L-shaped turn for an internal edge.","title":"Step directions of the route of a decorated edge"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-four-dir-steps","kind":"node","lean":"StatMech.Onsager.ons_fourDirSteps","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-four-dir-steps","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The endpoint of four lattice steps in the given directions.","title":"Endpoint of four lattice steps"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-four-dir-step-walk","kind":"node","lean":"StatMech.Onsager.ons_fourDirStepWalk","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-four-dir-step-walk","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The torus walk of four lattice steps in the given directions.","title":"Torus walk of four lattice steps"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-embed-graph","kind":"node","lean":"StatMech.Onsager.ons_decEmbedGraph","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-embed-graph","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The `8L \u00d7 8L` torus graph in which the decorated graph is drawn.","title":"Finer 8L \u00d7 8L torus graph for the decorated graph"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-edge-route","kind":"node","lean":"StatMech.Onsager.ons_decEdgeRoute","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-edge-route","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The four-step route of the decorated edge from `d` to `e` in the finer torus.","title":"Four-step route of a decorated edge in the finer torus"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-edge-route-vertices","kind":"node","lean":"StatMech.Onsager.ons_decEdgeRouteVertices","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-edge-route-vertices","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The five points of the route of the decorated edge from `d` to `e`.","title":"Five points of the route of a decorated edge"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-micro-point","kind":"node","lean":"StatMech.Onsager.ons_DecMicroPoint","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-micro-point","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"A point of the `8L \u00d7 8L` torus given as a coarse cell and an offset in `Fin 8 \u00d7 Fin 8`.","title":"Micro point of the finer torus as a cell and an offset"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-micro-vertex","kind":"node","lean":"StatMech.Onsager.ons_decMicroVertex","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-micro-vertex","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The point of the finer torus represented by a micro point.","title":"Point of the finer torus represented by a micro point"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-edge-interior-data","kind":"node","lean":"StatMech.Onsager.ons_decEdgeInteriorData","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-edge-interior-data","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The three interior points of the route of the decorated edge from `d` to `e`, as micro points.","title":"Interior points of the route of a decorated edge"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-micro-corridor-edge","kind":"node","lean":"StatMech.Onsager.ons_decMicroCorridorEdge","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-micro-corridor-edge","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The decorated edge whose route passes through a given interior micro point.","title":"Decorated edge whose route passes through an interior micro point"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-interior-offsets","kind":"node","lean":"StatMech.Onsager.ons_decInteriorOffsets","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-interior-offsets","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The offsets inside a cell used by interior points of routes.","title":"Cell offsets used by interior route points"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-port-micro-point","kind":"node","lean":"StatMech.Onsager.ons_decPortMicroPoint","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-port-micro-point","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The micro point at which a dart is placed.","title":"Micro point at which a dart is placed"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-dart-segment","kind":"node","lean":"StatMech.Onsager.ons_decDartSegment","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-dart-segment","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The points of the route of a decorated edge after its starting point.","title":"Route points of a decorated edge after its start"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-embed-walk","kind":"node","lean":"StatMech.Onsager.ons_decEmbedWalk","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-embed-walk","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The image of a decorated walk in the finer torus, obtained by concatenating the routes of its edges.","title":"Image of a decorated walk in the finer torus"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-edge-micro-darts","kind":"node","lean":"StatMech.Onsager.ons_decEdgeMicroDarts","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-edge-micro-darts","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The four darts of the finer torus along the route of the decorated edge from `d` to `e`.","title":"Four finer-torus darts along the route of a decorated edge"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-expanded-dart-list","kind":"node","lean":"StatMech.Onsager.ons_decExpandedDartList","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-expanded-dart-list","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The darts of the finer torus along the image of a decorated walk.","title":"Expanded dart list of a decorated walk in the finer torus"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-expanded-root","kind":"node","lean":"StatMech.Onsager.ons_decExpandedRoot","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-expanded-root","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The first dart of the expanded dart list of a decorated cycle.","title":"First dart of the expanded dart list of a decorated cycle"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-expanded-geometric-loop","kind":"node","lean":"StatMech.Onsager.ons_decExpandedGeometricLoop","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-expanded-geometric-loop","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The expanded dart list of a decorated cycle as a cyclic sequence, in walking order.","title":"Expanded loop of a decorated cycle in walking order"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-expanded-kwloop","kind":"node","lean":"StatMech.Onsager.ons_decExpandedKWLoop","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-expanded-kwloop","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The expanded dart list of a decorated cycle as a cyclic sequence in Kac-Ward order (reversed indexing).","title":"Expanded loop of a decorated cycle in Kac-Ward order"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-chain-edge","kind":"node","lean":"StatMech.Onsager.ons_decChainEdge","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-chain-edge","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The `i`-th internal edge `(s, i) - (s, i + 1)` of the path at a site.","title":"Internal edge of the path at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-chain-path-indices","kind":"node","lean":"StatMech.Onsager.ons_decChainPathIndices","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-chain-path-indices","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The indices of the internal edges on the path between ports `a` and `b` at one site.","title":"Indices of internal edges between two ports"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-chain-path-edges","kind":"node","lean":"StatMech.Onsager.ons_decChainPathEdges","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-chain-path-edges","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The internal edges on the path between ports `a` and `b` at a site.","title":"Internal edges between two ports at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-transition-weight","kind":"node","lean":"StatMech.Onsager.ons_decTransitionWeight","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-transition-weight","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The decorated weight of the Kac-Ward transition from `d\u2081` to `d\u2082`: the weight of the external edge at `d\u2081` times the weights of the internal edges from the port of `d\u2081 + 2` to the port of `d\u2082`.","title":"Decorated weight of a Kac-Ward transition"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-kwmat-decoration-weighted-phase","kind":"node","lean":"StatMech.Onsager.ons_KWmatDecorationWeightedPhase","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-kwmat-decoration-weighted-phase","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The twisted Kac-Ward matrix with decorated transition weights.","title":"Twisted Kac-Ward matrix with decorated transition weights"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-specialized-weight","kind":"node","lean":"StatMech.Onsager.ons_decSpecializedWeight","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-specialized-weight","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Decorated weights induced by torus edge weights: external edges get the weight of their torus edge, internal edges get `1`.","title":"Decorated weights induced by torus edge weights"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-interpolated-weight","kind":"node","lean":"StatMech.Onsager.ons_decInterpolatedWeight","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-interpolated-weight","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Decorated weights with external edges weighted by their torus edge and internal edges by `t`.","title":"Decorated weights with internal edges weighted by t"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-decorated-full-weighted-spin-sum","kind":"node","lean":"StatMech.Onsager.ons_decoratedFullWeightedSpinSum","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-decorated-full-weighted-spin-sum","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The decorated character sum `\u2211_D \u03c7_{a,b}(hom D) \u220f_{e \u2208 D} w_e` with a weight on every decorated edge.","title":"Decorated character sum with weights on all edges"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-side","kind":"node","lean":"StatMech.Onsager.ons_decMiddleSide","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-side","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The side of a direction at a site: `0` for east and north, `1` for west and south.","title":"Side of a direction at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-dir","kind":"node","lean":"StatMech.Onsager.ons_decMiddleDir","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-dir","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The representative direction of a side: east for side `0`, west for side `1`.","title":"Representative direction of a side"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-dest-factor","kind":"node","lean":"StatMech.Onsager.ons_decMiddleDestFactor","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-dest-factor","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The weight of the internal edges between the middle edge and the port of direction `\u03bc`, on the destination side.","title":"Destination-side internal weight to a port"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-source-factor","kind":"node","lean":"StatMech.Onsager.ons_decMiddleSourceFactor","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-source-factor","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The weight of the internal edges between the port of direction `\u03bc` and the middle edge, on the source side.","title":"Source-side internal weight from a port"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-u","kind":"node","lean":"StatMech.Onsager.ons_decMiddleU","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-u","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The column factor of the rank-two term: from a side of `site` to the darts leaving `site` on that side.","title":"Column factor of the rank-two middle-edge term"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-v","kind":"node","lean":"StatMech.Onsager.ons_decMiddleV","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-v","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The row factor of the rank-two term: from the darts entering `site` to the side on which they arrive.","title":"Row factor of the rank-two middle-edge term"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-augmented","kind":"node","lean":"StatMech.Onsager.ons_decMiddleAugmented","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-augmented","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The augmented Kac-Ward matrix on darts and the two sides of `site`, with blocks `\u039b\u2080`, `U`, `t V` and `0`.","title":"Augmented Kac-Ward matrix with two side states"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-balanced","kind":"node","lean":"StatMech.Onsager.ons_decMiddleBalanced","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-balanced","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The augmented matrix with blocks `\u039b\u2080`, `(t \u03bb) U`, `\u03bb\u207b\u00b9 V` and `0`.","title":"Balanced augmented Kac-Ward matrix"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-balanced-row-weight","kind":"node","lean":"StatMech.Onsager.ons_decMiddleBalancedRowWeight","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-balanced-row-weight","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Row weights of the balanced augmented matrix.","title":"Row weights of the balanced augmented matrix"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-balanced-col-weight","kind":"node","lean":"StatMech.Onsager.ons_decMiddleBalancedColWeight","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-balanced-col-weight","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Column weights of the balanced augmented matrix.","title":"Column weights of the balanced augmented matrix"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-aug-rev","kind":"node","lean":"StatMech.Onsager.ons_decMiddleAugRev","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-aug-rev","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Reversal on augmented states: reverse darts and swap the two sides.","title":"Reversal on augmented states"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-aug-dir","kind":"node","lean":"StatMech.Onsager.ons_decMiddleAugDir","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-aug-dir","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Directions of augmented states: the direction of a dart, and east or west for the two sides.","title":"Directions of augmented states"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-row-weight","kind":"node","lean":"StatMech.Onsager.ons_decMiddleRowWeight","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-row-weight","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Row weights of the augmented matrix: `1` on darts and `t` times the middle weight on side states.","title":"Row weights of the augmented Kac-Ward matrix"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-col-weight","kind":"node","lean":"StatMech.Onsager.ons_decMiddleColWeight","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-col-weight","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Column weights of the augmented matrix: the external weight on darts and `1` on side states.","title":"Column weights of the augmented Kac-Ward matrix"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-middle-internal-factor","kind":"node","lean":"StatMech.Onsager.ons_decMiddleInternalFactor","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-middle-internal-factor","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The reversal-invariant internal factor of the augmented matrix.","title":"Reversal-invariant internal factor of the augmented matrix"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-dart-is-external","kind":"node","lean":"StatMech.Onsager.ons_decDartIsExternal","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-dart-is-external","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Whether a dart of the decorated graph traverses an external edge (from a torus dart to its reverse).","title":"Test for a dart traversing an external edge"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-walk-external-darts","kind":"node","lean":"StatMech.Onsager.ons_decWalkExternalDarts","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-walk-external-darts","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The torus darts at which a decorated walk traverses an external edge, in order.","title":"External darts of a decorated walk in order"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-cycle-first-return","kind":"node","lean":"StatMech.Onsager.ons_decCycleFirstReturn","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-cycle-first-return","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The first-return loop at `d` read off from a decorated cycle rooted at `d`: the root, the remaining external darts in reverse order, and the root again.","title":"First-return loop of a rooted decorated cycle"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-port-line-adj","kind":"node","lean":"StatMech.Onsager.ons_portLineAdj","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-port-line-adj","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Two directions are adjacent in the order `0, 1, 2, 3` of the internal path at a site.","title":"Adjacency of directions along the port line"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-internal-micro-directions","kind":"node","lean":"StatMech.Onsager.ons_internalMicroDirections","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-internal-micro-directions","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The micro directions of the routes along a sequence of internal decorated edges at one site.","title":"Micro directions along internal decorated edges at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-walk-expanded-directions","kind":"node","lean":"StatMech.Onsager.ons_walkExpandedDirections","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-walk-expanded-directions","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The directions of the expanded dart list of a decorated walk.","title":"Directions of the expanded dart list of a decorated walk"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-walk-external-directions","kind":"node","lean":"StatMech.Onsager.ons_walkExternalDirections","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-walk-external-directions","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The directions of the external darts of a decorated walk.","title":"Directions of the external darts of a decorated walk"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-port-chain-walk","kind":"node","lean":"StatMech.Onsager.ons_decPortChainWalk","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-port-chain-walk","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"A walk along the internal path from port `a` to port `b` at a site.","title":"Internal walk between two ports at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-port-chain-path","kind":"node","lean":"StatMech.Onsager.ons_decPortChainPath","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-port-chain-path","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The internal path from port `a` to port `b` at a site, as a path in the decorated graph.","title":"Internal path between two ports at a site"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-transition-walk","kind":"node","lean":"StatMech.Onsager.ons_decTransitionWalk","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-transition-walk","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The decorated walk of the Kac-Ward transition from `d\u2081` to `d\u2082`: the external edge at `d\u2081` followed by the internal path to `d\u2082`.","title":"Decorated walk of a Kac-Ward transition"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-geometric-step","kind":"node","lean":"StatMech.Onsager.ons_decGeometricStep","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-geometric-step","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"Dart `target` starts at the head of dart `source`.","title":"Head-to-tail relation between darts"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-walk-edge-count","kind":"node","lean":"StatMech.Onsager.ons_decWalkEdgeCount","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-walk-edge-count","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The multiset of edges of a decorated walk, as an exponent.","title":"Edge multiset of a decorated walk as an exponent"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-kwchain-exponent","kind":"node","lean":"StatMech.Onsager.ons_decKWChainExponent","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-kwchain-exponent","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The sum of the transition exponents along a chain of darts starting at `start`.","title":"Transition exponent sum along a dart chain"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-kwchain-last","kind":"node","lean":"StatMech.Onsager.ons_decKWChainLast","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-kwchain-last","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The last dart of a chain starting at `start`.","title":"Last dart of a dart chain"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-walk-of-kwchain","kind":"node","lean":"StatMech.Onsager.ons_decWalkOfKWChain","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-walk-of-kwchain","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The decorated walk obtained by concatenating the trails of the transitions along a chain of darts.","title":"Decorated walk of a chain of darts"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-geometric-loop-states","kind":"node","lean":"StatMech.Onsager.ons_decGeometricLoopStates","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-geometric-loop-states","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The darts of a Kac-Ward loop in walking order (reversed indexing).","title":"Darts of a Kac-Ward loop in walking order"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-loop-walk","kind":"node","lean":"StatMech.Onsager.ons_decLoopWalk","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-loop-walk","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The closed decorated walk of a valid closed Kac-Ward walk, rooted at its first dart.","title":"Closed decorated walk of a valid Kac-Ward loop"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-cycle-dart-list","kind":"node","lean":"StatMech.Onsager.ons_decCycleDartList","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-cycle-dart-list","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The first-return loop of a rooted decorated cycle without its final repeated dart.","title":"Dart list of a rooted decorated cycle"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-dec-cycle-dart-loop","kind":"node","lean":"StatMech.Onsager.ons_decCycleDartLoop","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-dec-cycle-dart-loop","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The closed walk of darts of a rooted decorated cycle, indexed by `Fin` of its length.","title":"Closed dart walk of a rooted decorated cycle"},{"id":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/ons-edge-subgraph","kind":"node","lean":"StatMech.Onsager.ons_edgeSubgraph","parent":49423,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus/index.html#ons-edge-subgraph","scope":"StatisticalMechanics/exact-solutions/trivalent-decoration-of-the-torus","status":"fully_proved","summary":"The simple graph on `V` whose edges are the non-loop elements of `F`; this is `SimpleGraph.fromEdgeSet F`.","title":"Simple graph spanned by an edge set"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/spin-kac-ward-identities.json","id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","kind":"scope","navigation":"module","parent":49228,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","title":"Spin Kac\u2013Ward identities"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-external-step-walk","kind":"node","lean":"StatMech.Onsager.ons_decExternalStepWalk","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-external-step-walk","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The one-edge decorated walk along the external edge from `d` to its reverse.","title":"One-edge decorated walk along an external edge"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-mv-monomial-value","kind":"node","lean":"StatMech.Onsager.ons_mvMonomialValue","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-mv-monomial-value","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The value `\u220f_e w_e ^ {m e}` of the monomial with exponent `m` at weights `w`.","title":"Value of a monomial at weights"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-mv-series-eval-summable","kind":"node","lean":"StatMech.Onsager.ons_MvSeriesEvalSummable","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-mv-series-eval-summable","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Absolute summability of `\u2211_m coeff_m \u00b7 w^m`.","title":"Absolute summability of a power series evaluated at weights"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-mv-series-eval-norm","kind":"node","lean":"StatMech.Onsager.ons_mvSeriesEvalNorm","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-mv-series-eval-norm","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The sum `\u2211_m \u2016coeff_m \u00b7 w^m\u2016` of absolute values.","title":"Sum of absolute values of an evaluated power series"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-torus-even-subgraph","kind":"node","lean":"StatMech.Onsager.ons_TorusEvenSubgraph","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-torus-even-subgraph","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Even subgraphs of the `L \u00d7 L` torus.","title":"Even subgraphs of the L \u00d7 L torus"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-even-subgraph","kind":"node","lean":"StatMech.Onsager.ons_DecoratedEvenSubgraph","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-even-subgraph","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Even subgraphs of the decorated graph of the torus.","title":"Even subgraphs of the decorated torus graph"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-original-edges-of-decorated","kind":"node","lean":"StatMech.Onsager.ons_originalEdgesOfDecorated","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-original-edges-of-decorated","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The torus even subgraph corresponding to a decorated even subgraph under the decoration bijection.","title":"Torus even subgraph of a decorated even subgraph"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-is-external","kind":"node","lean":"StatMech.Onsager.ons_decIsExternal","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-is-external","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"An edge of the decorated graph is external if it joins a dart to its reverse.","title":"Externality of a decorated edge"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-external-edges","kind":"node","lean":"StatMech.Onsager.ons_decoratedExternalEdges","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-external-edges","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The external edges of a decorated edge set.","title":"External edges of a decorated edge set"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-edge-projection","kind":"node","lean":"StatMech.Onsager.ons_decEdgeProjection","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-edge-projection","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The projection of a decorated edge to the torus, taking each dart to its starting site.","title":"Projection of a decorated edge to the torus"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-spin-weight","kind":"node","lean":"StatMech.Onsager.ons_decoratedSpinWeight","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-spin-weight","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The weight `\u03c7_{a,b}(hom F) x^|F|` of the torus even subgraph `F` corresponding to a decorated even subgraph.","title":"Spin weight of a decorated even subgraph"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-walk-external-edges","kind":"node","lean":"StatMech.Onsager.ons_walkExternalEdges","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-walk-external-edges","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The external edges used by a walk in the decorated graph.","title":"External edges used by a decorated walk"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-walk-original-edges","kind":"node","lean":"StatMech.Onsager.ons_walkOriginalEdges","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-walk-original-edges","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The torus edges underlying the external edges of a decorated walk.","title":"Torus edges underlying a decorated walk"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-edge","kind":"node","lean":"StatMech.Onsager.ons_DecEdge","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-edge","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Decorated edges, used as the variables of the formal series.","title":"Decorated edges as formal-series variables"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-transition-exponent","kind":"node","lean":"StatMech.Onsager.ons_decTransitionExponent","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-transition-exponent","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The exponent of a transition: one for the external edge at `d\u2081` and one for each internal edge on the path at the site of `d\u2082`.","title":"Exponent of a Kac-Ward transition"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-transition-scalar","kind":"node","lean":"StatMech.Onsager.ons_decTransitionScalar","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-transition-scalar","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The scalar of a transition: its boundary phase times its turn weight, or `0` if `d\u2082` does not start at the head of `d\u2081`.","title":"Scalar of a Kac-Ward transition"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-formal-transition","kind":"node","lean":"StatMech.Onsager.ons_decFormalTransition","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-formal-transition","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The monomial of the transition from `d\u2081` to `d\u2082`.","title":"Monomial of a Kac-Ward transition"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-loop-exponent","kind":"node","lean":"StatMech.Onsager.ons_decLoopExponent","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-loop-exponent","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The exponent of a closed walk: the sum of the exponents of its transitions.","title":"Exponent of a closed Kac-Ward walk"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-loop-scalar","kind":"node","lean":"StatMech.Onsager.ons_decLoopScalar","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-loop-scalar","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The scalar of a closed walk: the product of the scalars of its transitions.","title":"Scalar of a closed Kac-Ward walk"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-finsupp-total-degree","kind":"node","lean":"StatMech.Onsager.ons_finsuppTotalDegree","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-finsupp-total-degree","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The total degree `\u2211 m e` of an exponent; this is `Finsupp.degree`.","title":"Total degree of an exponent"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-finsupp-total-degree-eq-degree","kind":"node","lean":"StatMech.Onsager.ons_finsuppTotalDegree_eq_degree","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-finsupp-total-degree-eq-degree","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"`ons_finsuppTotalDegree` agrees with Mathlib's `Finsupp.degree`.","title":"Total degree of an exponent agrees with Mathlib's degree"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-formal-log-coeff","kind":"node","lean":"StatMech.Onsager.ons_decFormalLogCoeff","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-formal-log-coeff","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The coefficient of `x^m` in the formal logarithm: minus one half of the sum over closed walks of length `r + 1 \u2264 deg m` with exponent `m` of their scalars divided by `r + 1`.","title":"Coefficient of the formal Kac-Ward logarithm"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-formal-log","kind":"node","lean":"StatMech.Onsager.ons_decFormalLog","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-formal-log","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The formal logarithm of the Kac-Ward root.","title":"Formal logarithm of the Kac-Ward root"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-formal-root","kind":"node","lean":"StatMech.Onsager.ons_decFormalRoot","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-formal-root","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The formal Kac-Ward root: the formal exponential of `ons_decFormalLog`.","title":"Formal Kac-Ward root"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-finset-exponent","kind":"node","lean":"StatMech.Onsager.ons_finsetExponent","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-finset-exponent","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The squarefree exponent `\u2211_{e \u2208 S} single e 1` of a set of variables.","title":"Squarefree exponent of a set of variables"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-spin-polynomial","kind":"node","lean":"StatMech.Onsager.ons_decoratedSpinPolynomial","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-spin-polynomial","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The decorated spin polynomial `\u2211_D \u03c7_{a,b}(hom D) \u220f_{e \u2208 D} x_e` over decorated even subgraphs.","title":"Decorated spin polynomial"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-mv-series-eval","kind":"node","lean":"StatMech.Onsager.ons_mvSeriesEval","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-mv-series-eval","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Evaluation of a multivariate formal power series at weights, as the sum `\u2211_m coeff_m \u00b7 weight^m`.","title":"Evaluation of a multivariate power series at weights"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-formal-kac-ward-identity","kind":"node","lean":"StatMech.Onsager.ons_decoratedFormalKacWardIdentity","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-formal-kac-ward-identity","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The formal Kac-Ward identity on the `L \u00d7 L` torus: for every spin structure, the formal Kac-Ward root equals the decorated spin polynomial.","title":"Formal Kac-Ward identity on the torus"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-phase-bucket-e","kind":"node","lean":"StatMech.Onsager.ons_decPhaseBucketE","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-phase-bucket-e","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The total decorated-weight Kac-Ward weight of closed walks of length `n + 1` visiting `e` but not its reverse.","title":"Closed-walk weight through a dart but not its reverse"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-phase-bucket-n","kind":"node","lean":"StatMech.Onsager.ons_decPhaseBucketN","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-phase-bucket-n","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The total decorated-weight Kac-Ward weight of closed walks of length `n + 1` visiting neither `e` nor its reverse.","title":"Closed-walk weight avoiding a dart and its reverse"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-mask-phase-bucket-e","kind":"node","lean":"StatMech.Onsager.ons_decMaskPhaseBucketE","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-mask-phase-bucket-e","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"As `ons_decPhaseBucketE`, with the darts of `forbidden` masked.","title":"Masked closed-walk weight through a dart but not its reverse"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-mask-phase-bucket-n","kind":"node","lean":"StatMech.Onsager.ons_decMaskPhaseBucketN","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-mask-phase-bucket-n","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"As `ons_decPhaseBucketN`, with the darts of `forbidden` masked.","title":"Masked closed-walk weight avoiding a dart and its reverse"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-scale-dec-edge-weight","kind":"node","lean":"StatMech.Onsager.ons_scaleDecEdgeWeight","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-scale-dec-edge-weight","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Decorated weights with the weight of `edge` multiplied by `t`.","title":"Decorated weights with one edge weight scaled by t"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-component-series","kind":"node","lean":"StatMech.Onsager.ons_decComponentSeries","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-component-series","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The polynomial `\u2211_C log-coefficient(C) x^{E(C)}` over the cycles `C` of a fixed decomposition of an even subgraph.","title":"Log-coefficient polynomial over the cycles of a decomposition"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-is-squarefree-exponent","kind":"node","lean":"StatMech.Onsager.ons_IsSquarefreeExponent","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-is-squarefree-exponent","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"An exponent is squarefree if every variable occurs with exponent at most `1`.","title":"Squarefree exponent"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-formal-bucket-term","kind":"node","lean":"StatMech.Onsager.ons_decFormalBucketTerm","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-formal-bucket-term","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The term of the evaluated formal logarithm from closed walks of length `r + 1` with exponent `m`, divided by `r + 1` and multiplied by `w^m`.","title":"Formal-logarithm term from closed walks of given length and exponent"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-loop-external-exponent","kind":"node","lean":"StatMech.Onsager.ons_decLoopExternalExponent","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-loop-external-exponent","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The external part of a loop exponent: one for the external edge at each dart of the loop.","title":"External part of a loop exponent"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-loop-internal-exponent","kind":"node","lean":"StatMech.Onsager.ons_decLoopInternalExponent","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-loop-internal-exponent","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The internal part of a loop exponent: the internal edges on the paths between consecutive ports.","title":"Internal part of a loop exponent"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-squarefree-loop-finset","kind":"node","lean":"StatMech.Onsager.ons_decSquarefreeLoopFinset","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-squarefree-loop-finset","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The closed Kac-Ward walks of length `n` with exponent the squarefree exponent of `S` and nonzero scalar.","title":"Kac-Ward loops with a given squarefree exponent and nonzero scalar"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-loop-root-finset","kind":"node","lean":"StatMech.Onsager.ons_decLoopRootFinset","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-loop-root-finset","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The darts of a loop together with their reverses.","title":"Darts of a loop with their reverses"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-loop-orientation","kind":"node","lean":"StatMech.Onsager.ons_decLoopOrientation","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-loop-orientation","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The `2n` rotations of a loop and of its reversal, indexed by `Fin n \u2295 Fin n`.","title":"Rotations of a loop and of its reversal"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-middle-augmented-col","kind":"node","lean":"StatMech.Onsager.ons_decMiddleAugmentedCol","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-middle-augmented-col","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The augmented matrix with blocks `\u039b\u2080`, `t U`, `V` and `0` (scaling on the column factor).","title":"Column-scaled augmented Kac-Ward matrix"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-middle-col-row-weight","kind":"node","lean":"StatMech.Onsager.ons_decMiddleColRowWeight","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-middle-col-row-weight","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Row weights of the column-scaled augmented matrix.","title":"Row weights of the column-scaled augmented matrix"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-middle-col-col-weight","kind":"node","lean":"StatMech.Onsager.ons_decMiddleColColWeight","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-middle-col-col-weight","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Column weights of the column-scaled augmented matrix.","title":"Column weights of the column-scaled augmented matrix"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-formal-repeated-edge-vanishes","kind":"node","lean":"StatMech.Onsager.ons_DecFormalRepeatedEdgeVanishes","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-formal-repeated-edge-vanishes","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"On every `L \u00d7 L` torus with `L > 2` and for every spin structure, the coefficient of the formal Kac-Ward root at any exponent that uses some decorated edge at least twice is zero.","title":"Vanishing of repeated-edge coefficients of the formal Kac-Ward root"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-weighted-spin-weight","kind":"node","lean":"StatMech.Onsager.ons_decoratedWeightedSpinWeight","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-weighted-spin-weight","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The edge-weighted spin weight `\u03c7_{a,b}(hom F) \u220f_{e \u2208 F} w_e` of the torus even subgraph `F` corresponding to a decorated even subgraph.","title":"Edge-weighted spin weight of a decorated even subgraph"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-containing-edge","kind":"node","lean":"StatMech.Onsager.ons_DecoratedContainingEdge","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-containing-edge","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Decorated even subgraphs that contain the external edge at `d`.","title":"Decorated even subgraphs containing a given external edge"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-edge-split","kind":"node","lean":"StatMech.Onsager.ons_DecoratedEdgeSplit","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-edge-split","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"A split of a decorated even subgraph containing the external edge at `d` into a simple cycle rooted at `d` (starting along that edge) and an even remainder disjoint from the cycle.","title":"Split of a decorated even subgraph into a rooted cycle and remainder"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-edge-split-2","kind":"node","lean":"StatMech.Onsager.ons_decoratedEdgeSplit","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-edge-split-2","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"A chosen split of a decorated even subgraph at the external edge of `d`.","title":"Chosen split of a decorated even subgraph at an external edge"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-decorated-cycle-removal-identity","kind":"node","lean":"StatMech.Onsager.ons_decoratedCycleRemovalIdentity","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-decorated-cycle-removal-identity","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The cycle removal identity: the Kac-Ward root with `d` and its reverse masked, times the total first-return weight at `d`, equals the sum over decorated even subgraphs through `d` of the first-return weight of the split-off cycle times the spin weight of the remainder.","title":"Cycle removal identity for decorated even subgraphs"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-degree-fiber-coeff","kind":"node","lean":"StatMech.Onsager.ons_degreeFiberCoeff","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Degree-"],["math","\\(n\\)"],["text"," part of a power series evaluated at a point"]],"source":"Degree-$n$ part of a power series evaluated at a point"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-degree-fiber-coeff","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The sum of the terms of total degree `n` of a series evaluated at `w`.","title":"Degree-$n$ part of a power series evaluated at a point"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-degree-fiber-norm","kind":"node","lean":"StatMech.Onsager.ons_degreeFiberNorm","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum of absolute values of the degree-"],["math","\\(n\\)"],["text"," terms at a point"]],"source":"Sum of absolute values of the degree-$n$ terms at a point"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-degree-fiber-norm","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The sum of the absolute values of the terms of total degree `n` evaluated at `w`.","title":"Sum of absolute values of the degree-$n$ terms at a point"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-degree-fiber-polynomial","kind":"node","lean":"StatMech.Onsager.ons_degreeFiberPolynomial","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Homogeneous degree-"],["math","\\(n\\)"],["text"," part of a power series as a polynomial"]],"source":"Homogeneous degree-$n$ part of a power series as a polynomial"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-degree-fiber-polynomial","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The homogeneous degree-`n` part of a series as a polynomial (finitely many variables).","title":"Homogeneous degree-$n$ part of a power series as a polynomial"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-coefficient-identity","kind":"node","lean":"StatMech.Onsager.ons_spinCoefficientIdentity","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spin-character sums squared as twisted determinants at activity "],["math","\\(x\\)"]],"source":"Spin-character sums squared as twisted determinants at activity $x$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-coefficient-identity","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"For every spin structure `(a, b)`, the square of the spin-character sum equals the twisted Kac-Ward determinant `det (1 - \u039b_{ab})` at activity `x`.","title":"Spin-character sums squared as twisted determinants at activity $x$"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-sector-poly","kind":"node","lean":"StatMech.Onsager.ons_sectorPoly","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-sector-poly","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The sector weight of the homology class `h` as a real polynomial in `x`.","title":"Sector weight of a homology class as a polynomial"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-poly00","kind":"node","lean":"StatMech.Onsager.ons_spinPoly00","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed sector polynomial with the "],["math","\\((1,1)\\)"],["text"," sector negated"]],"source":"Signed sector polynomial with the $(1,1)$ sector negated"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-poly00","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The signed sector sum `ons_spin00` as a polynomial.","title":"Signed sector polynomial with the $(1,1)$ sector negated"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-poly10","kind":"node","lean":"StatMech.Onsager.ons_spinPoly10","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed sector polynomial with the "],["math","\\((1,0)\\)"],["text"," sector negated"]],"source":"Signed sector polynomial with the $(1,0)$ sector negated"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-poly10","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The signed sector sum `ons_spin10` as a polynomial.","title":"Signed sector polynomial with the $(1,0)$ sector negated"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-poly01","kind":"node","lean":"StatMech.Onsager.ons_spinPoly01","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed sector polynomial with the "],["math","\\((0,1)\\)"],["text"," sector negated"]],"source":"Signed sector polynomial with the $(0,1)$ sector negated"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-poly01","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The signed sector sum `ons_spin01` as a polynomial.","title":"Signed sector polynomial with the $(0,1)$ sector negated"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-poly11","kind":"node","lean":"StatMech.Onsager.ons_spinPoly11","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Signed sector polynomial with only the "],["math","\\((0,0)\\)"],["text"," sector positive"]],"source":"Signed sector polynomial with only the $(0,0)$ sector positive"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-poly11","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The signed sector sum `ons_spin11` as a polynomial.","title":"Signed sector polynomial with only the $(0,0)$ sector positive"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-det-x","kind":"node","lean":"StatMech.Onsager.ons_spinDetX","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sector determinant as a function of "],["math","\\(x\\)"]],"source":"Sector determinant as a function of $x$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-det-x","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The sector determinant as a function of `x`: the product over shifted momenta of `(1 + x\u00b2)\u00b2 - 2x(1 - x\u00b2)(cos k\u2081 + cos k\u2082)`.","title":"Sector determinant as a function of $x$"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-det-poly","kind":"node","lean":"StatMech.Onsager.ons_spinDetPoly","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-det-poly","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The sector determinant as a real polynomial in `x`.","title":"Sector determinant as a real polynomial"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-kac-ward-at","kind":"node","lean":"StatMech.Onsager.ons_spinKacWardAt","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Four spin Kac-Ward identities on the torus at activity "],["math","\\(x\\)"]],"source":"Four spin Kac-Ward identities on the torus at activity $x$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-kac-ward-at","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The four spin Kac-Ward identities `spin_{ab}(x)\u00b2 = det_{ab}(x)` on the `L \u00d7 L` torus at activity `x`.","title":"Four spin Kac-Ward identities on the torus at activity $x$"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-weighted-spin-edge-coefficient","kind":"node","lean":"StatMech.Onsager.ons_weightedSpinEdgeCoefficient","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-weighted-spin-edge-coefficient","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The part of the edge-weighted character sum coming from even subgraphs that contain `edge`.","title":"Weighted character sum over even subgraphs containing an edge"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-restrict-edge-weight","kind":"node","lean":"StatMech.Onsager.ons_restrictEdgeWeight","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-restrict-edge-weight","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The edge weights restricted to `S`: equal to `weight` on `S` and `0` elsewhere.","title":"Edge weights restricted to a set of edges"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-weighted-edge-coefficient-identity","kind":"node","lean":"StatMech.Onsager.ons_weightedEdgeCoefficientIdentity","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Edge coefficient identity on the "],["math","\\(L \\times L\\)"],["text"," torus"]],"source":"Edge coefficient identity on the $L \\times L$ torus"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-weighted-edge-coefficient-identity","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The edge coefficient identity on the `L \u00d7 L` torus: for small weights, the part of the edge-weighted character sum from even subgraphs through the edge of `e` equals minus the Kac-Ward root with `e` and its reverse masked, times the total first-return weight at `e`.","title":"Edge coefficient identity on the $L \\times L$ torus"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-weighted-spin-character-sum","kind":"node","lean":"StatMech.Onsager.ons_weightedSpinCharacterSum","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-weighted-spin-character-sum","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The edge-weighted spin-character sum `\u2211_F \u03c7_{a,b}(hom F) \u220f_{e \u2208 F} w_e` over even subgraphs of the torus.","title":"Edge-weighted spin-character sum over even subgraphs"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-weighted-root-identity","kind":"node","lean":"StatMech.Onsager.ons_weightedRootIdentity","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-weighted-root-identity","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The edge-weighted root identity on the `L \u00d7 L` torus: for every spin structure and every edge weight bounded by `q` below Sherman's radius, the walk-series square root of the twisted edge-weighted Kac-Ward determinant equals the edge-weighted character sum.","title":"Edge-weighted root identity on the torus"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-spin-sherman-radius","kind":"node","lean":"StatMech.Onsager.ons_spinShermanRadius","parent":49514,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sherman's radius for "],["math","\\(L > 2\\)"],["text",", with value 1 otherwise"]],"source":"Sherman's radius for $L > 2$, with value 1 otherwise"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-spin-sherman-radius","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Sherman's radius for `L > 2`, and `1` otherwise.","title":"Sherman's radius for $L > 2$, with value 1 otherwise"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-second-difference-factor","kind":"node","lean":"StatMech.Onsager.ons_secondDifferenceFactor","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-second-difference-factor","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The factor `2^k - 2 + 0^k` by which a second difference multiplies a monomial of degree `k` in one variable.","title":"Second-difference factor for a monomial of degree k"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-dec-formal-second-difference","kind":"node","lean":"StatMech.Onsager.ons_decFormalSecondDifference","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-dec-formal-second-difference","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"The second difference of a formal series in the variable `edge`: the coefficient at `m` is multiplied by `2^{m edge} - 2 + 0^{m edge}`.","title":"Second difference of a formal series in one variable"},{"id":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities/ons-free-energy","kind":"node","lean":"StatMech.Onsager.ons_free_energy","parent":49514,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/spin-kac-ward-identities/index.html#ons-free-energy","scope":"StatisticalMechanics/exact-solutions/spin-kac-ward-identities","status":"fully_proved","summary":"Onsager's formula for the zero-field square-lattice Ising model, in the form `torus pressure \u2192 ons_pressure \u03b2` for `\u03b2 \u2265 0` (L. Onsager, *Crystal statistics. I*, 1944). See `main_free_energy` for the explicit integral.","title":"Onsager's formula"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising","kind":"scope","parent":49228,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising","status":"can_prove","title":"Onsager's solution of the two-dimensional Ising model"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/ons-symbol-dispersion","kind":"node","lean":"StatMech.Onsager.ons_symbolDispersion","parent":49586,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/index.html#ons-symbol-dispersion","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"The Kac-Ward symbol determinant `(1 + t\u00b2)\u00b2 - 2t(1 - t\u00b2)(cos k\u2081 + cos k\u2082)` at `t = tanh \u03b2`.","title":"Kac-Ward symbol determinant at t = tanh \u03b2"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/ons-pressure-finite","kind":"node","lean":"StatMech.Onsager.ons_pressureFinite","parent":49586,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/index.html#ons-pressure-finite","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"The finite-volume pressure `log Z_G(\u03b2) / |V|` (the negative of `\u03b2` times the free energy per site).","title":"Finite-volume pressure"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/ons-torus-pressure-seq","kind":"node","lean":"StatMech.Onsager.ons_torusPressureSeq","parent":49586,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/index.html#ons-torus-pressure-seq","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"The pressure `log Z_L(\u03b2) / L\u00b2` of the zero-field Ising model on the `L \u00d7 L` torus, as a sequence in `L` (set to `0` for `L \u2264 2`, where the torus graph is not defined).","title":"Torus pressure sequence of the zero-field Ising model"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical","kind":"scope","navigation":"module","parent":49586,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical","status":"fully_proved","title":"Continuity of Onsager's pressure at the critical point"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical/ons-critical-log-bound","kind":"node","lean":"StatMech.Onsager.ons_criticalLogBound","parent":49590,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical/index.html#ons-critical-log-bound","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical","status":"fully_proved","summary":"The dominating function `log 9 + log 2 + |log (1 - cos y)|` for `log g` near `\u03b2_c`.","title":"Integrable dominating function for log g near the critical point"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical/ons-critical-inner-bound","kind":"node","lean":"StatMech.Onsager.ons_criticalInnerBound","parent":49590,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical/index.html#ons-critical-inner-bound","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/continuity-of-onsager-s-pressure-at-the-critical","status":"fully_proved","summary":"A uniform bound for the inner integrals `\u222b log g dk\u2082` near `\u03b2_c`.","title":"Uniform bound constant for the inner integrals near \u03b2_c"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m","kind":"scope","navigation":"module","parent":49586,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m","status":"fully_proved","title":"The critical point of the square-lattice Ising model"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/ons-g-int","kind":"node","lean":"StatMech.Onsager.ons_gInt","parent":49593,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/index.html#ons-g-int","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m","status":"fully_proved","summary":"The Onsager integrand `cosh\u00b2 2\u03b2 - sinh 2\u03b2 (cos k\u2081 + cos k\u2082)` at inverse temperature `\u03b2` and momentum `(k\u2081, k\u2082)`.","title":"Onsager integrand of the square-lattice Ising model"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/ons-beta-c","kind":"node","lean":"StatMech.Onsager.ons_betaC","parent":49593,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/index.html#ons-beta-c","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m","status":"fully_proved","summary":"The critical inverse temperature `\u03b2_c = \u00bd log (1 + \u221a2)` of the square-lattice Ising model with unit couplings.","title":"Critical inverse temperature \u00bd log(1 + \u221a2) of the square lattice"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/ons-beta-c-sinh","kind":"node","lean":"StatMech.Onsager.ons_betaC_sinh","parent":49593,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/index.html#ons-beta-c-sinh","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m","status":"fully_proved","summary":"`sinh 2\u03b2_c = 1` (H. A. Kramers and G. H. Wannier, *Statistics of the two-dimensional ferromagnet. Part I*, 1941).","title":"Kramers-Wannier critical point"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/ons-beta-c-unique","kind":"node","lean":"StatMech.Onsager.ons_betaC_unique","parent":49593,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m/index.html#ons-beta-c-unique","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-critical-point-of-the-square-lattice-ising-m","status":"fully_proved","summary":"`\u03b2_c` is the unique positive inverse temperature with `sinh 2\u03b2 = 1`, the self-dual point of the Kramers-Wannier duality.","title":"\u03b2_c is the unique positive solution of sinh 2\u03b2 = 1"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio","kind":"scope","navigation":"module","parent":49586,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio","status":"fully_proved","title":"High-temperature expansion of the Ising partition function"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio/ons-z","kind":"node","lean":"StatMech.Onsager.ons_Z","parent":49598,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio/index.html#ons-z","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio","status":"fully_proved","summary":"The zero-field Ising partition function of `G` with unit couplings at inverse temperature `\u03b2`.","title":"Zero-field Ising partition function with unit couplings"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio/ons-x","kind":"node","lean":"StatMech.Onsager.ons_X","parent":49598,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio/index.html#ons-x","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio","status":"fully_proved","summary":"The even-subgraph generating function `\u2211_F x^|F|` of `G`, over edge sets `F` in which every vertex has even degree.","title":"Even-subgraph generating function of a graph"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio/ons-hte","kind":"node","lean":"StatMech.Onsager.ons_hte","parent":49598,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio/index.html#ons-hte","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/high-temperature-expansion-of-the-ising-partitio","status":"fully_proved","summary":"High-temperature expansion (van der Waerden 1941). `Z(\u03b2) = 2^|V| (cosh \u03b2)^|E| X(tanh \u03b2)` for the zero-field Ising model on a finite graph (B. L. van der Waerden, *Die lange Reichweite der regelm\u00e4ssigen Atomanordnung in Mischkristallen*, 1941).","title":"High-temperature expansion"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f","kind":"scope","navigation":"module","parent":49586,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f","status":"fully_proved","title":"From the spin Kac-Ward identities to Onsager's formula"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f/ons-spin-kac-ward-identities","kind":"node","lean":"StatMech.Onsager.ons_spinKacWardIdentities","parent":49602,"reading_title":{"format":"inline-parts-v1","parts":[["text","Spin Kac-Ward identities at inverse temperature "],["math","\\(\\beta\\)"]],"source":"Spin Kac-Ward identities at inverse temperature $\\beta$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f/index.html#ons-spin-kac-ward-identities","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f","status":"fully_proved","summary":"The spin Kac-Ward identities at inverse temperature `\u03b2`: on every `L \u00d7 L` torus with `L > 2`, each signed sector sum at `tanh \u03b2` squares to the corresponding sector determinant.","title":"Spin Kac-Ward identities at inverse temperature $\\beta$"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f/ons-spin-kac-ward-matrix-identities","kind":"node","lean":"StatMech.Onsager.ons_spinKacWardMatrixIdentities","parent":49602,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f/index.html#ons-spin-kac-ward-matrix-identities","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/from-the-spin-kac-ward-identities-to-onsager-s-f","status":"fully_proved","summary":"The spin Kac-Ward identities written with the twisted Kac-Ward determinants `det (1 - \u039b_{ab})` at `x = tanh \u03b2`.","title":"Spin Kac-Ward identities via twisted Kac-Ward determinants"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus","kind":"scope","navigation":"module","parent":49586,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus","status":"fully_proved","title":"The square-lattice torus"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus/ons-torus-adj","kind":"node","lean":"StatMech.Onsager.onsTorusAdj","parent":49605,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus/index.html#ons-torus-adj","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus","status":"fully_proved","summary":"Nearest-neighbor adjacency on `\u2124_L \u00d7 \u2124_L`: the two sites differ by `\u00b11` in exactly one coordinate.","title":"Nearest-neighbor adjacency on the discrete torus"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus/ons-torus-graph","kind":"node","lean":"StatMech.Onsager.onsTorusGraph","parent":49605,"reading_title":{"format":"inline-parts-v1","parts":[["text","The "],["math","\\(L \\times L\\)"],["text"," square-lattice torus graph"]],"source":"The $L \\times L$ square-lattice torus graph"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus/index.html#ons-torus-graph","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus","status":"fully_proved","summary":"The `L \u00d7 L` square-lattice torus as a simple graph on `\u2124_L \u00d7 \u2124_L`, for `L > 2`.","title":"The $L \\times L$ square-lattice torus graph"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus/ons-torus-graph-eq-circulant-graph","kind":"node","lean":"StatMech.Onsager.onsTorusGraph_eq_circulantGraph","parent":49605,"reading_title":{"format":"inline-parts-v1","parts":[["text","The torus graph is a circulant graph with jumps "],["math","\\((1,0)\\)"],["text"," and "],["math","\\((0,1)\\)"]],"source":"The torus graph is a circulant graph with jumps $(1,0)$ and $(0,1)$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus/index.html#ons-torus-graph-eq-circulant-graph","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/the-square-lattice-torus","status":"fully_proved","summary":"The torus graph is the circulant graph of `\u2124_L \u00d7 \u2124_L` with jumps `(1, 0)` and `(0, 1)`.","title":"The torus graph is a circulant graph with jumps $(1,0)$ and $(0,1)$"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure","kind":"scope","navigation":"module","parent":49586,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure","status":"fully_proved","title":"Onsager's free energy and the torus pressure"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure/ons-free-energy-integral","kind":"node","lean":"StatMech.Onsager.ons_freeEnergyIntegral","parent":49609,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure/index.html#ons-free-energy-integral","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure","status":"fully_proved","summary":"The integral `(1 / (8\u03c0\u00b2)) \u222b_{-\u03c0}^{\u03c0} \u222b_{-\u03c0}^{\u03c0} log (cosh\u00b2 2\u03b2 - sinh 2\u03b2 (cos k\u2081 + cos k\u2082)) dk\u2081 dk\u2082` in Onsager's formula.","title":"Onsager's free-energy double integral"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure/ons-pressure","kind":"node","lean":"StatMech.Onsager.ons_pressure","parent":49609,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure/index.html#ons-pressure","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure","status":"fully_proved","summary":"Onsager's pressure (minus `\u03b2` times the free energy per site) of the square-lattice Ising model: `log 2` plus `ons_freeEnergyIntegral \u03b2`.","title":"Onsager's pressure of the square-lattice Ising model"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure/ons-kac-ward-residue","kind":"node","lean":"StatMech.Onsager.ons_KacWardResidue","parent":49609,"reading_title":{"format":"inline-parts-v1","parts":[["text","Limit statement for "],["math","\\(\\log X_L(\\tanh\\beta)/L^2\\)"],["text"," on the torus"]],"source":"Limit statement for $\\log X_L(\\tanh\\beta)/L^2$ on the torus"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure/index.html#ons-kac-ward-residue","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-free-energy-and-the-torus-pressure","status":"fully_proved","summary":"The statement that `log X_L(tanh \u03b2) / L\u00b2` on the `L \u00d7 L` torus converges to `ons_freeEnergyIntegral \u03b2 - 2 log cosh \u03b2` as `L \u2192 \u221e`.","title":"Limit statement for $\\log X_L(\\tanh\\beta)/L^2$ on the torus"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","kind":"scope","navigation":"module","parent":49586,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","title":"Onsager's solution of the two-dimensional Ising model"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-high-temperature-expansion","kind":"node","lean":"StatMech.Onsager.main_high_temperature_expansion","parent":49613,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-high-temperature-expansion","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"High-temperature expansion (van der Waerden 1941). On a finite graph, `Z(\u03b2) = 2^|V| (cosh \u03b2)^|E| X(tanh \u03b2)` with `X` the even-subgraph generating function.","title":"High-temperature expansion"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-finite-pressure-decomposition","kind":"node","lean":"StatMech.Onsager.main_finite_pressure_decomposition","parent":49613,"reading_title":{"format":"inline-parts-v1","parts":[["text","Finite-volume pressure decomposition for "],["math","\\(\\beta \\ge 0\\)"]],"source":"Finite-volume pressure decomposition for $\\beta \\ge 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-finite-pressure-decomposition","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"The finite-volume pressure as `log 2 + (|E|/|V|) log cosh \u03b2 + log X(tanh \u03b2) / |V|`, for `\u03b2 \u2265 0`.","title":"Finite-volume pressure decomposition for $\\beta \\ge 0$"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-critical-beta-value","kind":"node","lean":"StatMech.Onsager.main_critical_beta_value","parent":49613,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-critical-beta-value","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"`\u03b2_c = \u00bd log (1 + \u221a2)` satisfies `sinh 2\u03b2_c = 1` (H. A. Kramers and G. H. Wannier, *Statistics of the two-dimensional ferromagnet. Part I*, 1941).","title":"Kramers-Wannier critical point"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-critical-beta-unique","kind":"node","lean":"StatMech.Onsager.main_critical_beta_unique","parent":49613,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\beta_c\\)"],["text"," is the unique positive root of "],["math","\\(\\sinh 2\\beta = 1\\)"]],"source":"$\\beta_c$ is the unique positive root of $\\sinh 2\\beta = 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-critical-beta-unique","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"`\u03b2_c` is the only positive `\u03b2` with `sinh 2\u03b2 = 1`.","title":"$\\beta_c$ is the unique positive root of $\\sinh 2\\beta = 1$"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-pressure-continuous-at-critical","kind":"node","lean":"StatMech.Onsager.main_pressure_continuous_at_critical","parent":49613,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-pressure-continuous-at-critical","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"Onsager's pressure is continuous at `\u03b2_c`. Its non-analyticity there is not part of the statement.","title":"Continuity of Onsager's pressure at the critical point"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-torus-pressure-decomposition","kind":"node","lean":"StatMech.Onsager.main_torus_pressure_decomposition","parent":49613,"reading_title":{"format":"inline-parts-v1","parts":[["text","Pressure decomposition on the "],["math","\\(L \\times L\\)"],["text"," torus for "],["math","\\(\\beta \\ge 0\\)"]],"source":"Pressure decomposition on the $L \\times L$ torus for $\\beta \\ge 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-torus-pressure-decomposition","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"On the `L \u00d7 L` torus, `p_L(\u03b2) = log 2 + 2 log cosh \u03b2 + log X_L(tanh \u03b2) / L\u00b2` for `\u03b2 \u2265 0`.","title":"Pressure decomposition on the $L \\times L$ torus for $\\beta \\ge 0$"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-free-energy-from-spin-kac-ward","kind":"node","lean":"StatMech.Onsager.main_free_energy_from_spin_kacWard","parent":49613,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-free-energy-from-spin-kac-ward","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"Onsager's formula follows from the spin Kac-Ward identities at all `\u03b2`.","title":"Onsager's formula from the spin Kac-Ward identities"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-spin-kac-ward-from-weighted-root","kind":"node","lean":"StatMech.Onsager.main_spin_kacWard_from_weighted_root","parent":49613,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-spin-kac-ward-from-weighted-root","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"The spin Kac-Ward identities follow from the edge-weighted root identity on all tori.","title":"Spin Kac-Ward identities from the edge-weighted root identity"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-weighted-root-from-decorated-formal","kind":"node","lean":"StatMech.Onsager.main_weighted_root_from_decorated_formal","parent":49613,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-weighted-root-from-decorated-formal","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"The edge-weighted root identity follows from the formal Kac-Ward identity on the decorated torus.","title":"Edge-weighted root identity from the decorated-torus formal identity"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-spin-kac-ward-from-decorated-formal","kind":"node","lean":"StatMech.Onsager.main_spin_kacWard_from_decorated_formal","parent":49613,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-spin-kac-ward-from-decorated-formal","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"The spin Kac-Ward identities follow from the formal Kac-Ward identity on all decorated tori.","title":"Spin Kac-Ward identities from the decorated-torus formal identity"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/main-free-energy","kind":"node","lean":"StatMech.Onsager.main_free_energy","parent":49613,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising/index.html#main-free-energy","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-solution-of-the-two-dimensional-ising","status":"fully_proved","summary":"Onsager's formula (L. Onsager, *Crystal statistics. I. A two-dimensional model with an order-disorder transition*, Phys. Rev. 65, 1944). For the zero-field nearest-neighbor Ising model with unit coupling on the `L \u00d7 L` torus, the pressure `log Z_L(\u03b2) / L\u00b2` converges, for every `\u03b2 \u2265 0`, to `log 2 + (1 / 8\u03c0\u00b2) \u222b\u222b_{[-\u03c0, \u03c0]\u00b2} log (cosh\u00b2 2\u03b2 - sinh 2\u03b2 (cos k\u2081 + cos k\u2082))`.","title":"Onsager's formula"},{"data_url":"scopes/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition.json","id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","kind":"scope","navigation":"module","parent":49586,"result":false,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"can_prove","title":"Onsager's 2D Ising phase transition"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/ising-site","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.IsingSite","parent":49625,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#ising-site","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"fully_proved","summary":"For a positive integer $L$, the sites of the periodic $L \\times L$ square lattice are the pairs $(i, j)$ with $i, j \\in \\mathbb{Z}/L\\mathbb{Z}$.","title":"Sites of the periodic square lattice"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/spin-value","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.spinValue","parent":49625,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#spin-value","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"fully_proved","summary":"A spin, recorded as a Boolean, has the real value $1$ if it is true and $-1$ if it is false.","title":"Numerical value of a spin"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/right-neighbour","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.rightNeighbour","parent":49625,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#right-neighbour","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"fully_proved","summary":"For a positive integer $L$, the right neighbour of a site $(i, j)$ of the periodic $L \\times L$ square lattice is $(i + 1, j)$, with addition in $\\mathbb{Z}/L\\mathbb{Z}$.","title":"Right neighbour"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/up-neighbour","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.upNeighbour","parent":49625,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#up-neighbour","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"fully_proved","summary":"For a positive integer $L$, the upward neighbour of a site $(i, j)$ of the periodic $L \\times L$ square lattice is $(i, j + 1)$, with addition in $\\mathbb{Z}/L\\mathbb{Z}$.","title":"Upward neighbour"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/ising-energy","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.isingEnergy","parent":49625,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#ising-energy","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"fully_proved","summary":"For a positive integer $L$, let $\\sigma$ be a spin configuration on the periodic $L \\times L$ square lattice, assigning to each site $p$ a spin $\\sigma_p \\in \\{1, -1\\}$. Its energy in the zero-field ferromagnetic Ising model with unit coupling is","title":"Energy of an Ising configuration"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/ising-partition-function","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.isingPartitionFunction","parent":49625,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#ising-partition-function","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"fully_proved","summary":"For a positive integer $L$ and real $\\beta$, the partition function of the Ising model on the periodic $L \\times L$ square lattice at inverse temperature $\\beta$ is $Z_L(\\beta) = \\sum_\\sigma e^{-\\beta H_L(\\sigma)}$, where $\\sigma$ runs over all spin configurations on the lattice and $H_L(\\sigma)$ is the Ising energy of $\\sigma$.","title":"Ising partition function"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/finite-ising-free-energy","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.finiteIsingFreeEnergy","parent":49625,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#finite-ising-free-energy","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"fully_proved","summary":"For a positive integer $L$ and real $\\beta$, the free energy density of the Ising model on the periodic $L \\times L$ square lattice is $\\log Z_L(\\beta) / L^2$, where $Z_L(\\beta)$ is the partition function at inverse temperature $\\beta$.","title":"Finite-volume Ising free energy density"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/finite-ising-free-energy-seq","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.finiteIsingFreeEnergySeq","parent":49625,"reading_title":{"format":"inline-parts-v1","parts":[["text","Ising free energy density along side lengths "],["math","\\(n + 1\\)"]],"source":"Ising free energy density along side lengths $n + 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#finite-ising-free-energy-seq","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"fully_proved","summary":"For a natural number $n$ and real $\\beta$, this is the finite-volume free energy density of the Ising model on the periodic $(n + 1) \\times (n + 1)$ square lattice at inverse temperature $\\beta$, namely $\\log Z_{n+1}(\\beta) / (n + 1)^2$, where $Z_{n+1}(\\beta)$ is the partition function of that lattice at inverse temperature $\\beta$.","title":"Ising free energy density along side lengths $n + 1$"},{"id":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/ising-2d-phase-transition","kind":"node","lean":"MathlibExt.Analysis.IsingPhaseTransitionWanted.Analysis.IsingPhaseTransition.ising_2d_phase_transition","parent":49625,"result":true,"route":"../roadmap/StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition/index.html#ising-2d-phase-transition","scope":"StatisticalMechanics/exact-solutions/onsager-s-solution-of-the-two-dimensional-ising/onsager-s-2d-ising-phase-transition","status":"can_prove","summary":"For a positive integer $L$ and real $\\beta$, let $Z_L(\\beta)$ be the partition function of the zero-field ferromagnetic Ising model on the periodic $L \\times L$ square lattice at inverse temperature $\\beta$. There exist a function $F : \\mathbb{R} \\to \\mathbb{R}$ and a real number $\\beta_c > 0$ such that, for every real $\\beta$, $\\log Z_{n+1}(\\beta) / (n+1)^2$ converges to $F(\\beta)$ as $n \\to \\infty$, and $F$ is\u2026","title":"Onsager's phase transition for the two-dimensional Ising model"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness.json","id":"StatisticalMechanics/random-currents-and-sharpness","kind":"scope","parent":47780,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness","status":"fully_proved","summary":"The random-current representation of Ising correlations and the switching lemma, and the sharpness of the Ising phase transition: the Simon\u2013Lieb inequality, exponential decay of correlations below the critical point and the mean-field lower bound on the magnetization above it.","title":"Random currents and sharpness"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma.json","id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","kind":"scope","navigation":"module","parent":49635,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","title":"Random currents and the switching lemma"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/current","kind":"node","lean":"StatMech.Sharpness.Current","parent":49636,"reading_title":{"format":"inline-parts-v1","parts":[["text","Current: a multiplicity in "],["math","\\(\\mathbb{N}\\)"],["text"," for each unordered pair"]],"source":"Current: a multiplicity in $\\mathbb{N}$ for each unordered pair"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#current","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"A current on the edges of `V`: a multiplicity `n_e \u2208 \u2115` for every unordered pair `e`.","title":"Current: a multiplicity in $\\mathbb{N}$ for each unordered pair"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/incident-flux","kind":"node","lean":"StatMech.Sharpness.incidentFlux","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#incident-flux","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The total multiplicity `\u2211_{e \u220b x} n_e` of the edges of `G` incident to `x`.","title":"Incident flux of a current at a vertex"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/sources","kind":"node","lean":"StatMech.Sharpness.sources","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#sources","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The sources `\u2202n`: the vertices with odd incident flux.","title":"Sources of a current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/weight","kind":"node","lean":"StatMech.Sharpness.weight","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#weight","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The current weight `w(n) = \u220f_{e \u2208 E(G)} (\u03b2J_e)^{n_e}/n_e!`.","title":"Weight of a random current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/weight-nonneg","kind":"node","lean":"StatMech.Sharpness.weight_nonneg","parent":49636,"reading_title":{"format":"inline-parts-v1","parts":[["text","Current weights are nonnegative for nonnegative "],["math","\\(\\beta\\)"],["text"," and couplings"]],"source":"Current weights are nonnegative for nonnegative $\\beta$ and couplings"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#weight-nonneg","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Current weights are nonnegative for `\u03b2 \u2265 0` and nonnegative couplings.","title":"Current weights are nonnegative for nonnegative $\\beta$ and couplings"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/current-subgraph","kind":"node","lean":"StatMech.Sharpness.currentSubgraph","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#current-subgraph","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The subgraph of `G` of edges with `n_e \u2265 1`.","title":"Subgraph of edges carrying positive current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/current-connected","kind":"node","lean":"StatMech.Sharpness.CurrentConnected","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#current-connected","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"`x` and `y` are joined by a path of edges carrying positive current.","title":"Connectivity through edges carrying positive current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/odd-subgraph","kind":"node","lean":"StatMech.Sharpness.oddSubgraph","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#odd-subgraph","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The subgraph of `G` of edges with odd multiplicity.","title":"Subgraph of edges with odd current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/is-backbone-path","kind":"node","lean":"StatMech.Sharpness.IsBackbonePath","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#is-backbone-path","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Backbone paths from `x` to `y`: paths in the subgraph of odd edges.","title":"Backbone paths along odd-current edges"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/backbone","kind":"node","lean":"StatMech.Sharpness.backbone","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#backbone","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"A canonical backbone path: the least one for a given linear order on backbone paths.","title":"Canonical backbone path of a current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/percolation-field","kind":"node","lean":"StatMech.Sharpness.percolationField","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#percolation-field","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The trace of a current: the bond configuration with `e` open iff `n_e \u2265 1`.","title":"Trace of a current as a bond configuration"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/ghost","kind":"node","lean":"StatMech.Sharpness.ghost","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#ghost","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The ghost vertex `none` of `Option V`.","title":"Ghost vertex of the extended vertex set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/with-ghost","kind":"node","lean":"StatMech.Sharpness.withGhost","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#with-ghost","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The graph on `V \u2294 {ghost}` with the edges of `G` and an edge from every site to the ghost.","title":"Graph with a ghost vertex joined to every site"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/walk-deg","kind":"node","lean":"StatMech.Sharpness.walkDeg","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#walk-deg","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The number of edges of the walk `p` incident to `x`.","title":"Number of walk edges at a vertex"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/remove-walk","kind":"node","lean":"StatMech.Sharpness.removeWalk","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#remove-walk","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The current `n` with one unit removed from each traversal of an edge of `p`.","title":"Current with a walk removed"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/of-edge-fun","kind":"node","lean":"StatMech.Sharpness.ofEdgeFun","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#of-edge-fun","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Extension of a function on the edges of `G` to a current, zero off `E(G)`.","title":"Extension of an edge function to a current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/current-sum","kind":"node","lean":"StatMech.Sharpness.currentSum","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#current-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The current sum `Z(\u2202 = A) = \u2211_{\u2202n = A} w(n)` over currents on the edges of `G`.","title":"Current sum with prescribed sources"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/g-edge","kind":"node","lean":"StatMech.Sharpness.gEdge","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#g-edge","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The `k`-th term `(\u03b2J_e \u03c3_e)^k / k!` of the expansion of `exp(\u03b2J_e \u03c3_e)`.","title":"Term of the exponential expansion of an edge weight"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/boltzmann-j","kind":"node","lean":"StatMech.Sharpness.boltzmannJ","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#boltzmann-j","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The Boltzmann weight `exp(\u03b2 \u2211_e J_e \u03c3_e)` with couplings `J`.","title":"Boltzmann weight with couplings"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/partition-j","kind":"node","lean":"StatMech.Sharpness.partitionJ","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#partition-j","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The partition function with couplings `J`.","title":"Partition function with couplings"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/expectation-j","kind":"node","lean":"StatMech.Sharpness.expectationJ","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#expectation-j","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The correlation `\u27e8\u03c3_A\u27e9_{\u03b2,J}` with couplings `J`.","title":"Correlation function with couplings"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/current-representation","kind":"node","lean":"StatMech.Sharpness.current_representation","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#current-representation","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The random-current representation: `\u27e8\u03c3_A\u27e9_{\u03b2,J} = Z(\u2202 = A)/Z(\u2202 = \u2205)`. Reference: M. Aizenman, \"Geometric analysis of \u03c6\u2074 fields and Ising models\", Comm. Math. Phys. 86 (1982) 1\u201348; J. Ginibre (1970); H. Duminil-Copin, \"Random currents expansion of the Ising model\", lecture notes (2016), arXiv:1607.06933.","title":"Random-current representation"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/shb-is-backbone-of","kind":"node","lean":"StatMech.Sharpness.shb_IsBackboneOf","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#shb-is-backbone-of","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Every edge of the path carries odd current.","title":"Odd current on every edge of a path"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/shb-backbone-set","kind":"node","lean":"StatMech.Sharpness.shb_backboneSet","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#shb-backbone-set","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The paths from `x` to `y` all of whose edges carry odd current.","title":"Odd-current paths between two vertices"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/shb-backbone-of-imp-is-backbone-path","kind":"node","lean":"StatMech.Sharpness.shb_backboneOf_imp_isBackbonePath","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#shb-backbone-of-imp-is-backbone-path","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"A path with odd current on every edge as a backbone path.","title":"Odd-current path as a backbone path"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/shb-backbone-select","kind":"node","lean":"StatMech.Sharpness.shb_backboneSelect","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#shb-backbone-select","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The least backbone path from `x` to `y`, if any.","title":"Least backbone path between two vertices"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/shb-backbone-num","kind":"node","lean":"StatMech.Sharpness.shb_backboneNum","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#shb-backbone-num","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"`Z_\u03c9`: the total weight of currents with sources `{x, y}` whose selected backbone is `\u03c9`.","title":"Weight of currents with a given selected backbone"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/shb-rho","kind":"node","lean":"StatMech.Sharpness.shb_rho","parent":49636,"reading_title":{"format":"inline-parts-v1","parts":[["text","Backbone weight "],["math","\\(\\rho(\\omega)\\)"]],"source":"Backbone weight $\\rho(\\omega)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#shb-rho","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The backbone weight `\u03c1(\u03c9) = Z_\u03c9/Z(\u2202 = \u2205)`.","title":"Backbone weight $\\rho(\\omega)$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/edge-inside","kind":"node","lean":"StatMech.Sharpness.edgeInside","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#edge-inside","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"An unordered pair `e` is inside `S` if both of its endpoints lie in `S`.","title":"Unordered pair inside a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/no-crossing","kind":"node","lean":"StatMech.Sharpness.NoCrossing","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#no-crossing","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The current `n` uses no edge joining `S` to `S\u1d9c`: every edge with `n e \u2265 1` is inside `S` or inside `S\u1d9c`.","title":"Current with no edge crossing a set's boundary"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/restrict-to","kind":"node","lean":"StatMech.Sharpness.restrictTo","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#restrict-to","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The restriction `n|_S` of a current to the edges inside `S`, zero elsewhere.","title":"Restriction of a current to a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/sigma-finset-equiv-pi","kind":"node","lean":"StatMech.Sharpness.FluxEdgeCopy.sigmaFinsetEquivPi","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#sigma-finset-equiv-pi","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Subsets of `\u03a3 e, \u03b2 e` correspond to families of subsets of the fibres.","title":"Subsets of a sigma type as families of fibre subsets"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/copy","kind":"node","lean":"StatMech.Sharpness.FluxEdgeCopy.Copy","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#copy","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The edge copies of a current `m`: `m_e` copies of each edge `e`.","title":"Edge copies of a current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/ends-m","kind":"node","lean":"StatMech.Sharpness.FluxEdgeCopy.endsM","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#ends-m","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The endpoints of an edge copy.","title":"Endpoints of an edge copy"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/profile-flux","kind":"node","lean":"StatMech.Sharpness.FluxEdgeCopy.profileFlux","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#profile-flux","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The profile of a set of copies: the number of copies of each edge it contains.","title":"Profile of a set of edge copies"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/source-pair-disconn-sum","kind":"node","lean":"StatMech.Sharpness.FluxEdgeCopy.sourcePairDisconnSum","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#source-pair-disconn-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The double-current sum `\u2211_{\u2202n\u2081 = A, \u2202n\u2082 = B} w(n\u2081) w(n\u2082) 1_{u \u21ae v in n\u2081 + n\u2082}`.","title":"Double-current sum restricted to disconnection"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/orig-edges","kind":"node","lean":"StatMech.Sharpness.FieldGhostDict.origEdges","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#orig-edges","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The edges of `G`, as edges of the ghost graph.","title":"Graph edges as ghost-graph edges"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/ghost-edges","kind":"node","lean":"StatMech.Sharpness.FieldGhostDict.ghostEdges","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#ghost-edges","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The ghost edges `{x, ghost}`.","title":"Ghost edges"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/ghost-coupling","kind":"node","lean":"StatMech.Sharpness.FieldGhostDict.ghostCoupling","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#ghost-coupling","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The coupling on the ghost graph: `J` on the edges of `G` and `h` on the ghost edges.","title":"Coupling on the ghost graph with uniform field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/flip-v","kind":"node","lean":"StatMech.Sharpness.FieldGhostDict.flipV","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#flip-v","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The global spin flip `\u03c3 \u21a6 -\u03c3`.","title":"Global spin flip"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/some-emb","kind":"node","lean":"StatMech.Sharpness.FieldGhostDict.someEmb","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#some-emb","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The embedding `V \u21aa Option V`.","title":"Embedding of vertices into the ghost graph"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/fgd-ising-expectation-eq-current-sum-rat","kind":"node","lean":"StatMech.Sharpness.FieldGhostDict.fgd_isingExpectation_eq_currentSum_ratio","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#fgd-ising-expectation-eq-current-sum-rat","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The random-current representation with a field: for `|A|` even, `\u27e8\u03c3_A\u27e9_{\u03b2,h} = Z_ghost(\u2202 = A)/Z_ghost(\u2202 = \u2205)` on the ghost graph. Reference: M. Aizenman, D. J. Barsky, R. Fern\u00e1ndez, \"The phase transition in a general class of Ising-type models is sharp\", J. Stat. Phys. 47 (1987) 343\u2013374; H. Duminil-Copin, \"Random currents expansion of the Ising model\", lecture notes (2016), arXiv:1607.06933.","title":"Random-current representation with a field for even source sets"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/triple-equiv-sigma","kind":"node","lean":"StatMech.Sharpness.tripleEquivSigma","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#triple-equiv-sigma","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Triples of functions to `\u2115` correspond to a sum `m` together with two functions with pointwise sum `\u2264 m`.","title":"Reparametrizing triples of functions by their sum and two summands"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/source-triple-sum","kind":"node","lean":"StatMech.Sharpness.sourceTripleSum","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#source-triple-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The triple-current sum `\u2211_{\u2202n\u2081 = A, \u2202n\u2082 = B, \u2202n\u2083 = C} w(n\u2081) w(n\u2082) w(n\u2083)`.","title":"Triple-current sum with prescribed sources"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/deg-k","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.degK","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#deg-k","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The degree of `x` in the multigraph edge set `T`.","title":"Degree of a vertex in a multigraph edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/sources-2","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.sources","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#sources-2","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The odd-degree vertices of the multigraph edge set `T`.","title":"Sources of a multigraph edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/adj-step","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.adjStep","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#adj-step","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"`a \u2260 b` are the endpoints of an edge of `K`.","title":"Adjacency through an edge of a multigraph edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/conn-k","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.connK","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#conn-k","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Connectivity in `K`: the reflexive transitive closure of `adjStep`.","title":"Connectivity in a multigraph edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/comp-of","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.compOf","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#comp-of","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The cluster of `u` in `K`.","title":"Cluster of a vertex in a multigraph edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/switching-lemma","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.switching_lemma","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#switching-lemma","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The switching lemma: for `m` with sources `A` and `u \u2260 v`, `\u2211_{K \u2286 m, \u2202K = A \u2206 {u, v}} F(m) = \u2211_{K \u2286 m, \u2202K = A} F(m) 1_{u \u2194 v in m}`. Reference: M. Aizenman, \"Geometric analysis of \u03c6\u2074 fields and Ising models\", Comm. Math. Phys. 86 (1982) 1\u201348; H. Duminil-Copin, \"Random currents expansion of the Ising model\", lecture notes (2016), arXiv:1607.06933.","title":"Switching lemma for multigraph currents"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/drd-pair-sum","kind":"node","lean":"StatMech.Sharpness.drd_pairSum","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#drd-pair-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"For a multigraph `ends`, the sum over edge subsets `m` with sources `A` of `F m` times the number of sub-subsets `K \u2286 m` with sources `A`, restricted to `m` satisfying `P`.","title":"Sub-subset-weighted pair sum on a multigraph"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/not-conn-comp-k","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.notConnCompK","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#not-conn-comp-k","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The vertices not connected to `o` using edges of `m` in the multigraph `ends`.","title":"Vertices not connected to a root through a multigraph edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/edge-inside-k","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.edgeInsideK","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#edge-inside-k","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Both endpoints of the multigraph edge `i` lie in `S`.","title":"Multigraph edge lying inside a vertex set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/no-crossing-k","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.NoCrossingK","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#no-crossing-k","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Every edge of `m` lies inside `S` or inside `S\u1d9c`.","title":"Multigraph edge set with no edge crossing a vertex set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/src-pair-family","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.srcPairFamily","parent":49636,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sum over edge sets split into parts with sources "],["math","\\(A\\)"],["text"," and "],["math","\\(B\\)"]],"source":"Sum over edge sets split into parts with sources $A$ and $B$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#src-pair-family","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The sum over edge sets `m` with sources `A \u2206 B` and over splittings `m = K \u2294 (m \\ K)` with sources `A` and `B` of `F m`.","title":"Sum over edge sets split into parts with sources $A$ and $B$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/comp-odd-vertices","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.compOddVertices","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#comp-odd-vertices","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The odd-degree vertices of `m` in the cluster of `u`.","title":"Odd-degree vertices in a cluster"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/wit-ends","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.witEnds","parent":49636,"reading_title":{"format":"inline-parts-v1","parts":[["text","Four-vertex multigraph with edges "],["math","\\(\\{0, 1\\}\\)"],["text"," and "],["math","\\(\\{2, 3\\}\\)"]],"source":"Four-vertex multigraph with edges $\\{0, 1\\}$ and $\\{2, 3\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#wit-ends","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"A multigraph on `Fin 4` with the two edges `{0, 1}` and `{2, 3}`.","title":"Four-vertex multigraph with edges $\\{0, 1\\}$ and $\\{2, 3\\}$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/wit-m","kind":"node","lean":"StatMech.Sharpness.RandomCurrent.witM","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#wit-m","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"All edges of `witEnds`.","title":"Full edge set of the four-vertex multigraph"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/w-edge","kind":"node","lean":"StatMech.Sharpness.wEdge","parent":49636,"reading_title":{"format":"inline-parts-v1","parts":[["text","Single-edge current weight "],["math","\\((\\beta J_e)^k/k!\\)"]],"source":"Single-edge current weight $(\\beta J_e)^k/k!$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#w-edge","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The single-edge weight `(\u03b2J_e)^k / k!`.","title":"Single-edge current weight $(\\beta J_e)^k/k!$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/pair-equiv-sigma","kind":"node","lean":"StatMech.Sharpness.pairEquivSigma","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#pair-equiv-sigma","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Pairs `(p, q)` of functions to `\u2115` correspond to a sum `m = p + q` together with `p \u2264 m`.","title":"Reparametrizing pairs of functions by their sum and one summand"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/source-pair-sum","kind":"node","lean":"StatMech.Sharpness.sourcePairSum","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#source-pair-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The double-current sum `\u2211_{\u2202n\u2081 = A, \u2202n\u2082 = B} w(n\u2081) w(n\u2082)`.","title":"Double-current sum with prescribed sources"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/current-2","kind":"node","lean":"StatMech.Ising.Current","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#current-2","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"A current: a multiplicity in `\u2115` on every unordered pair. See also `StatMech.Sharpness.Current`.","title":"Current as a multiplicity on unordered pairs"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/incident-flux-2","kind":"node","lean":"StatMech.Ising.incidentFlux","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#incident-flux-2","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The total multiplicity of the edges of `E` incident to `x`.","title":"Incident flux of a current on an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/sources-3","kind":"node","lean":"StatMech.Ising.sources","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#sources-3","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The source set of a current: the sites with odd incident flux. See also `StatMech.Sharpness.sources` and `StatMech.Ising.sources_edgeFinset_eq`.","title":"Source set of a current on an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/adj-p","kind":"node","lean":"StatMech.Ising.adjP","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#adj-p","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"`a` and `b` are distinct endpoints of an edge in `P`.","title":"Adjacency through an edge of an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/conn-p","kind":"node","lean":"StatMech.Ising.connP","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#conn-p","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Connectivity through the edges of `P`: the reflexive-transitive closure of `adjP P`.","title":"Connectivity through the edges of an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/pos-edges","kind":"node","lean":"StatMech.Ising.posEdges","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#pos-edges","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The edges of `E` with positive multiplicity.","title":"Edges of an edge set with positive multiplicity"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/odd-edges","kind":"node","lean":"StatMech.Ising.oddEdges","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#odd-edges","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The edges of `E` with odd multiplicity.","title":"Edges of an edge set with odd multiplicity"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/deg-p","kind":"node","lean":"StatMech.Ising.degP","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#deg-p","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The number of edges of `P` incident to `x`.","title":"Degree of a site in an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/src-p","kind":"node","lean":"StatMech.Ising.srcP","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#src-p","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The sites of odd degree in the edge set `P`.","title":"Odd-degree sites of an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/comp-of-2","kind":"node","lean":"StatMech.Ising.compOf","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#comp-of-2","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The cluster of `u` in the edge set `P`.","title":"Cluster of a site in an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/reflect","kind":"node","lean":"StatMech.Ising.reflect","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#reflect","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The reflection of a sub-current `n \u2264 m` along `P`: `m e - n e` on `P` and `n e` elsewhere.","title":"Reflection of a sub-current along an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/ising-switching-lemma-pair","kind":"node","lean":"StatMech.Ising.ising_switching_lemma_pair","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#ising-switching-lemma-pair","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Switching lemma (counting form). If `u \u2260 v` are connected by the odd edges of `n\u2081 + n\u2082`, the sub-currents of `n\u2081 + n\u2082` with sources `A \u2206 {u, v}` and with sources `A` are equinumerous.","title":"Switching lemma"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/acw-restrict","kind":"node","lean":"StatMech.Ising.acwRestrict","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#acw-restrict","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The restriction of the current `n` to the edges satisfying `p` (zero elsewhere).","title":"Restriction of a current to an edge predicate"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/acw-internal","kind":"node","lean":"StatMech.Ising.acwInternal","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#acw-internal","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Both endpoints of `e` lie in `S`.","title":"Internal edge of a vertex set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/acw-closed-cut","kind":"node","lean":"StatMech.Ising.acwClosedCut","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#acw-closed-cut","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Every edge of `G` lies inside `S` or inside its complement.","title":"Vertex set that no edge of the graph crosses"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/acw-current-sum-eq-expectation","kind":"node","lean":"StatMech.Ising.acw_currentSum_eq_expectation","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#acw-current-sum-eq-expectation","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"`Z_A = \u27e8\u03c3_A\u27e9 Z_\u2205`.","title":"Sourced current sum as correlation times sourceless sum"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/sources-edge-finset-eq","kind":"node","lean":"StatMech.Ising.sources_edgeFinset_eq","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#sources-edge-finset-eq","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The source set of a current on the edges of `G` is `Sharpness.sources G`.","title":"Sources on a graph's edge set agree with random-current sources"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/reflect-equiv","kind":"node","lean":"StatMech.Ising.reflectEquiv","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#reflect-equiv","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Reflection along `P` as an involutive equivalence of the sub-currents of `m`.","title":"Reflection as an involution of the sub-currents of a current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/split-weighted-sum","kind":"node","lean":"StatMech.Ising.splitWeightedSum","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#split-weighted-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"The weighted sum `F \u2211_K w(K) w(m - K)` over sub-currents `K \u2264 m` with sources `A`.","title":"Weighted sum over splittings with prescribed sources"},{"id":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/ising-switching-weighted","kind":"node","lean":"StatMech.Ising.ising_switching_weighted","parent":49636,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma/index.html#ising-switching-weighted","scope":"StatisticalMechanics/random-currents-and-sharpness/random-currents-and-the-switching-lemma","status":"fully_proved","summary":"Switching lemma (weighted form). If `u \u2260 v` are connected by odd edges of `m`, the weighted sums over splittings with sources `A \u2206 {u, v}` and `A` agree.","title":"Switching lemma"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality.json","id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","kind":"scope","navigation":"module","parent":49635,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","title":"Simon\u2013Lieb inequality"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/has-odd-boundary","kind":"node","lean":"StatMech.Sharpness.HasOddBoundary","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#has-odd-boundary","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The set of vertices of odd degree in the edge set `F` is exactly `A`.","title":"Edge set with prescribed odd-degree vertices"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-subgraph-weight","kind":"node","lean":"StatMech.Sharpness.highTempSubgraphWeight","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-subgraph-weight","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The high-temperature weight `\u220f_{e \u2208 F} tanh(\u03b2J_e)` of an edge set.","title":"High-temperature weight of an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/edge-indicator-current","kind":"node","lean":"StatMech.Sharpness.edgeIndicatorCurrent","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#edge-indicator-current","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The current equal to `1` on the edges of `F` and `0` elsewhere.","title":"Indicator current of an edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/source-pair","kind":"node","lean":"StatMech.Sharpness.sourcePair","parent":49718,"reading_title":{"format":"inline-parts-v1","parts":[["text","Source pair "],["math","\\(\\{a\\} \\triangle \\{b\\}\\)"]],"source":"Source pair $\\{a\\} \\triangle \\{b\\}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#source-pair","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The source set `{a} \u2206 {b}` (empty when `a = b`).","title":"Source pair $\\{a\\} \\triangle \\{b\\}$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-first-exit","kind":"node","lean":"StatMech.Sharpness.HighTempFirstExit","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-first-exit","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"First-exit data for an edge set `K` with boundary `{o} \u2206 {z}` and a set `S \u220b o`, `z \u2209 S`: an exit edge `xy` of `K` with `x \u2208 S`, `y \u2209 S`, and a subset `Q \u2286 K` of edges inside `S` with boundary `{o} \u2206 {x}` not containing `xy`.","title":"First-exit data for a high-temperature edge set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-first-exit-2","kind":"node","lean":"StatMech.Sharpness.highTempFirstExit","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-first-exit-2","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"A choice of first-exit data, from `exists_highTempFirstExit`.","title":"Chosen first-exit data"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-internal-edges","kind":"node","lean":"StatMech.Sharpness.highTempInternalEdges","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-internal-edges","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The edges of `G` inside `S`.","title":"Edges of a graph inside a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-switch-input","kind":"node","lean":"StatMech.Sharpness.HighTempSwitchInput","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-switch-input","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"A pair of edge sets `(F, H)`, the input of the switching.","title":"Input pair of the high-temperature switching"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-switch-output","kind":"node","lean":"StatMech.Sharpness.HighTempSwitchOutput","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-switch-output","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"A tuple `(x, y, A, B)` of an exit edge and two edge sets, the output of the switching.","title":"Output tuple of the high-temperature switching"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/valid","kind":"node","lean":"StatMech.Sharpness.HighTempSwitchInput.Valid","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#valid","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"Valid inputs: `F \u2286 E(G)` with boundary `{o} \u2206 {z}` and `H` inside `S` with empty boundary.","title":"Valid switching inputs"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/valid-2","kind":"node","lean":"StatMech.Sharpness.HighTempSwitchOutput.Valid","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#valid-2","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"Valid outputs: `x \u2208 S`, `y \u2209 S`, `A` inside `S` with boundary `{o} \u2206 {x}`, and `B \u2286 E(G)` with boundary `{y} \u2206 {z}`.","title":"Valid switching outputs"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-switch-inputs","kind":"node","lean":"StatMech.Sharpness.highTempSwitchInputs","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-switch-inputs","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The finite set of valid inputs.","title":"Finite set of valid switching inputs"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-switch-outputs","kind":"node","lean":"StatMech.Sharpness.highTempSwitchOutputs","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-switch-outputs","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The finite set of valid outputs.","title":"Finite set of valid switching outputs"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-switch","kind":"node","lean":"StatMech.Sharpness.highTempSwitch","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-switch","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The switching map: cut `F \u2206 H` at its first exit from `S` and redistribute the pieces.","title":"High-temperature switching map"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-switch-map","kind":"node","lean":"StatMech.Sharpness.highTempSwitchMap","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-switch-map","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The switching as a map between the finite sets of valid inputs and outputs.","title":"Switching map between valid inputs and outputs"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-switch-input-weight","kind":"node","lean":"StatMech.Sharpness.highTempSwitchInputWeight","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-switch-input-weight","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The weight `\u220f_{e \u2208 F} t_e \u220f_{e \u2208 H} t_e` of an input.","title":"Weight of a switching input"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-switch-output-weight","kind":"node","lean":"StatMech.Sharpness.highTempSwitchOutputWeight","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-switch-output-weight","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The weight `\u220f_{e \u2208 A} t_e \u00b7 t_{xy} \u00b7 \u220f_{e \u2208 B} t_e` of an output.","title":"Weight of a switching output"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-source-subgraphs","kind":"node","lean":"StatMech.Sharpness.highTempSourceSubgraphs","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-source-subgraphs","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The edge sets `F \u2286 E` with boundary `A`.","title":"Edge sets with a given boundary"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/high-temp-source-sum","kind":"node","lean":"StatMech.Sharpness.highTempSourceSum","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#high-temp-source-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The high-temperature source sum `\u2211_{F \u2286 E, \u2202F = A} \u220f_{e \u2208 F} t_e`.","title":"High-temperature source sum"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/two-point-j","kind":"node","lean":"StatMech.Sharpness.twoPointJ","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#two-point-j","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The two-point function `\u27e8\u03c3_a\u03c3_b\u27e9_{\u03b2,J}` (equal to `1` when `a = b`).","title":"Two-point function with couplings"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/two-point-j-simon-finite","kind":"node","lean":"StatMech.Sharpness.twoPointJ_simon_finite","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#two-point-j-simon-finite","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Simon\u2013Lieb inequality: for `\u03b2, J \u2265 0`, `o \u2208 S` and `z \u2209 S`, `\u27e8\u03c3_o\u03c3_z\u27e9 \u2264 \u2211_{x \u2208 S} \u2211_{y \u2209 S} tanh(\u03b2J_{xy}) \u27e8\u03c3_o\u03c3_x\u27e9_{J|S} \u27e8\u03c3_y\u03c3_z\u27e9`. Reference: B. Simon, \"Correlation inequalities and the decay of correlations in ferromagnets\", Comm. Math. Phys. 77 (1980) 111\u2013126; E. H. Lieb, \"A refinement of Simon's correlation inequality\", Comm. Math. Phys. 77 (1980) 127\u2013135.","title":"Simon\u2013Lieb inequality on a finite graph"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/simon-weight-pair","kind":"node","lean":"StatMech.Sharpness.simonWeightPair","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#simon-weight-pair","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Simon\u2013Lieb weight `tanh(\u03b2J_{xy}) \u27e8\u03c3_o\u03c3_x\u27e9_{J|S}`.","title":"Simon\u2013Lieb weight"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/vertex-ghost-coupling","kind":"node","lean":"StatMech.Sharpness.vertexGhostCoupling","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#vertex-ghost-coupling","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The coupling on `G` plus a ghost vertex: `J` on the edges of `G` and `H_x` on the edge `{x, ghost}`.","title":"Ghost-graph coupling with a site field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/vertex-field-weight","kind":"node","lean":"StatMech.Sharpness.vertexFieldWeight","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#vertex-field-weight","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Boltzmann weight `exp(\u03b2(\u2211_e J_e \u03c3_e + \u2211_x H_x \u03c3_x))` with a site field `H`.","title":"Boltzmann weight with a site field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/vertex-field-z","kind":"node","lean":"StatMech.Sharpness.vertexFieldZ","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#vertex-field-z","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The partition function with site field `H`.","title":"Partition function with a site field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/vertex-field-expectation","kind":"node","lean":"StatMech.Sharpness.vertexFieldExpectation","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#vertex-field-expectation","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Gibbs expectation with site field `H`.","title":"Gibbs expectation with a site field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/vertex-field-two-point","kind":"node","lean":"StatMech.Sharpness.vertexFieldTwoPoint","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#vertex-field-two-point","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The two-point function `\u27e8\u03c3_a\u03c3_b\u27e9_{\u03b2,J,H}` with site field `H`.","title":"Two-point function with a site field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/simon-weight","kind":"node","lean":"StatMech.Sharpness.simonWeight","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#simon-weight","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Simon weight `w(x, y) = tanh(\u03b2J_{xy}) \u27e8\u03c3_o\u03c3_x\u27e9_{J|S}`.","title":"Simon weight of a boundary pair"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/simon-lieb","kind":"node","lean":"StatMech.Sharpness.SimonLieb","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#simon-lieb","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"A Simon\u2013Lieb type inequality for a nonnegative two-point function `\u03c4`, root `o`, set `S`, weights `w` and boundary targets `B`: `\u03c4(o, z) \u2264 \u2211_{x \u2208 S} \u2211_{y \u2208 B} w(x, y) \u03c4(y, z)` for every `z \u2209 S`.","title":"Simon\u2013Lieb type inequality for a nonnegative two-point function"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/simon-const","kind":"node","lean":"StatMech.Sharpness.simonConst","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#simon-const","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Simon constant `\u03c6 = \u2211_{x \u2208 S} \u2211_{y \u2208 B} w(x, y)`.","title":"Simon constant of a weight function"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/spin-pair-bcf","kind":"node","lean":"StatMech.Sharpness.spinPairBCF","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#spin-pair-bcf","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The bounded continuous function `\u03c3 \u21a6 \u03c3_x \u03c3_y` on configuration space.","title":"Spin pair product as a bounded continuous function"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/plus-corr","kind":"node","lean":"StatMech.Sharpness.plusCorr","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#plus-corr","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The plus-state two-point function `\u27e8\u03c3_x\u03c3_y\u27e9^+_\u03b2` at zero field.","title":"Plus-state two-point function at zero field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/simon-boundary-integral","kind":"node","lean":"StatMech.Sharpness.simonBoundaryIntegral","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#simon-boundary-integral","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Simon\u2013Lieb boundary sum `\u2211_{(x,y) \u2208 \u2202S} tanh \u03b2 \u27e8\u03c3_0\u03c3_x\u27e9_S \u222b \u03c3_y\u03c3_z d\u03bc` for a measure `\u03bc`.","title":"Simon\u2013Lieb boundary sum against a measure"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/simon-boundary-plus","kind":"node","lean":"StatMech.Sharpness.simonBoundaryPlus","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#simon-boundary-plus","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Simon\u2013Lieb boundary sum `\u2211_{(x,y) \u2208 \u2202S} tanh \u03b2 \u27e8\u03c3_0\u03c3_x\u27e9_S \u27e8\u03c3_y\u03c3_z\u27e9^+_\u03b2` in the plus state.","title":"Simon\u2013Lieb boundary sum in the plus state"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/plus-boundary-edges","kind":"node","lean":"StatMech.Sharpness.plusBoundaryEdges","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#plus-boundary-edges","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The edges touching `\u039b` that are not inside `\u039b`.","title":"Boundary edges of a finite volume"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/plus-boundary-field","kind":"node","lean":"StatMech.Sharpness.plusBoundaryField","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#plus-boundary-field","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The field on `x \u2208 \u039b` induced by the plus boundary condition: the number of edges from `x` to sites outside `\u039b`.","title":"Field induced by the plus boundary condition"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/unit-edge-coupling","kind":"node","lean":"StatMech.Sharpness.unitEdgeCoupling","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#unit-edge-coupling","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The coupling equal to `1` on the edges of `G` and `0` elsewhere.","title":"Unit coupling on the edges of a graph"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/plus-cut-sites","kind":"node","lean":"StatMech.Sharpness.plusCutSites","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#plus-cut-sites","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The sites of `\u2124^d` underlying a set `T` of sites of `\u039b`.","title":"Underlying lattice sites of a subset of a volume"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/plus-cut-vertex-equiv","kind":"node","lean":"StatMech.Sharpness.plusCutVertexEquiv","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#plus-cut-vertex-equiv","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The identification of `T` (as sites of `\u039b`) with `plusCutSites T` (as sites of `\u2124^d`).","title":"Identification of volume sites with lattice sites"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/sites-in-volume","kind":"node","lean":"StatMech.Sharpness.sitesInVolume","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#sites-in-volume","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The sites of `\u039b` that lie in `T`.","title":"Sites of a volume lying in a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/plus-corr-simon","kind":"node","lean":"StatMech.Sharpness.plusCorr_simon","parent":49718,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality/index.html#plus-corr-simon","scope":"StatisticalMechanics/random-currents-and-sharpness/simon-lieb-inequality","status":"fully_proved","summary":"The Simon\u2013Lieb inequality in the infinite-volume plus state: for `\u03b2 \u2265 0`, a finite `T \u220b 0` and `z \u2209 T`, `\u27e8\u03c3_0\u03c3_z\u27e9^+_\u03b2 \u2264 \u2211_{(x,y) \u2208 \u2202T} tanh \u03b2 \u27e8\u03c3_0\u03c3_x\u27e9_T \u27e8\u03c3_y\u03c3_z\u27e9^+_\u03b2`. Reference: B. Simon, \"Correlation inequalities and the decay of correlations in ferromagnets\", Comm. Math. Phys. 77 (1980) 111\u2013126; E. H. Lieb, \"A refinement of Simon's correlation inequality\", Comm. Math. Phys. 77 (1980) 127\u2013135.","title":"Simon\u2013Lieb inequality in the infinite-volume plus state"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising.json","id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising","kind":"scope","parent":49635,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising","status":"fully_proved","title":"Sharpness of the phase transition for the Ising model"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/ising-sharpness","kind":"node","lean":"StatMech.Sharpness.IsingSharp.ising_sharpness","parent":49760,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/index.html#ising-sharpness","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising","status":"fully_proved","summary":"Abstract form of the sharpness theorem: a magnetization satisfying the mean-field differential inequality above `\u03b2_c` obeys `m(\u03b2) \u2265 \u221a(1 - (\u03b2_c/\u03b2)\u00b2)`, and correlations satisfying Simon's self-consistent bound with `\u03c6 < 1` are summable with sum at most `|S|/(1 - \u03c6)`. The hypotheses are inputs; the unconditional statement is `ising_sharp_phase_transition`.","title":"Abstract form of the sharpness theorem"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ.json","id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","kind":"scope","navigation":"module","parent":49760,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","title":"The Aizenman\u2013Barsky\u2013Fern\u00e1ndez differential inequality, finite volume"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/neg-ham","kind":"node","lean":"StatMech.Ising.negHam","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#neg-ham","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"The negative Hamiltonian `\u2211_{xy \u2208 E} \u03c3_x\u03c3_y + h \u2211_x \u03c3_x`.","title":"Negative Hamiltonian of the Ising model"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/cov2","kind":"node","lean":"StatMech.Ising.cov2","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#cov2","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"The spin covariance `Cov(\u03c3_x, \u03c3_y) = \u27e8\u03c3_x\u03c3_y\u27e9 - \u27e8\u03c3_x\u27e9\u27e8\u03c3_y\u27e9`.","title":"Two-point spin covariance"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/one-pt","kind":"node","lean":"StatMech.Ising.onePt","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#one-pt","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"The one-point function `\u27e8\u03c3_x\u27e9`.","title":"Ising one-point function"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/cov3sym","kind":"node","lean":"StatMech.Ising.cov3sym","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#cov3sym","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"The covariance `\u27e8\u03c3_o \u03c3_x \u03c3_y\u27e9 - \u27e8\u03c3_o\u27e9\u27e8\u03c3_x \u03c3_y\u27e9` of `\u03c3_o` with the bond variable of `e = {x, y}`.","title":"Covariance of a spin with a bond variable"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/ghs-bound-sym","kind":"node","lean":"StatMech.Ising.ghsBoundSym","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#ghs-bound-sym","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"The GHS bound `Cov(\u03c3_o, \u03c3_x)\u27e8\u03c3_y\u27e9 + Cov(\u03c3_o, \u03c3_y)\u27e8\u03c3_x\u27e9` for the bond `e = {x, y}`.","title":"GHS bound for a bond"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/bond-energy-susceptibility","kind":"node","lean":"StatMech.Ising.bondEnergySusceptibility","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#bond-energy-susceptibility","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"The bond-energy susceptibility `\u2211_{e \u2208 E} Cov(\u03c3_o, \u03c3_e)`.","title":"Bond-energy susceptibility"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/susceptibility","kind":"node","lean":"StatMech.Ising.susceptibility","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#susceptibility","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"The susceptibility `\u03c7(o) = \u2211_x Cov(\u03c3_o, \u03c3_x)`.","title":"Ising susceptibility at a site"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/ghsthree-point-sym","kind":"node","lean":"StatMech.Ising.GHSThreePointSym","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#ghsthree-point-sym","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"The GHS inequality at `o` for every edge: `cov3sym \u2264 ghsBoundSym`, equivalently `u\u2083(o, x, y) \u2264 0` for every edge `{x, y}`.","title":"GHS three-point inequality at a site for every edge"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/aizenman-barsky-inequality","kind":"node","lean":"StatMech.Ising.aizenman_barsky_inequality","parent":49762,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ/index.html#aizenman-barsky-inequality","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-aizenman-barsky-fern-ndez-differential-inequ","status":"fully_proved","summary":"A conditional finite-volume form of the Aizenman\u2013Barsky\u2013Fern\u00e1ndez inequality: if the GHS bound holds at `o` and the GHS bounds sum to `J \u27e8\u03c3_o\u27e9 \u03c7(o)`, then the bond-energy susceptibility is at most `J \u27e8\u03c3_o\u27e9 \u03c7(o)`.","title":"Conditional finite-volume Aizenman\u2013Barsky\u2013Fern\u00e1ndez inequality"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu.json","id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu","kind":"scope","navigation":"module","parent":49760,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu","status":"fully_proved","title":"Exponential decay of the plus-state two-point function below \u03b2\u0303_c"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu/ising-simon-radius","kind":"node","lean":"StatMech.Sharpness.isingSimonRadius","parent":49772,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu/index.html#ising-simon-radius","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu","status":"fully_proved","summary":"One more than the largest `\u2113\u00b9` distance from the origin of a point of `T` or of the outer endpoint of a boundary edge of `T`.","title":"Simon radius of a finite set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu/plus-corr-tail-set","kind":"node","lean":"StatMech.Sharpness.plusCorrTailSet","parent":49772,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu/index.html#plus-corr-tail-set","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu","status":"fully_proved","summary":"The set of values `\u27e8\u03c3_0\u03c3_z\u27e9^+_\u03b2` with `\u2016z\u2016\u2081 \u2265 n`, together with `0`.","title":"Tail set of plus-state correlations"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu/plus-corr-tail-sup","kind":"node","lean":"StatMech.Sharpness.plusCorrTailSup","parent":49772,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu/index.html#plus-corr-tail-sup","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/exponential-decay-of-the-plus-state-two-point-fu","status":"fully_proved","summary":"The tail supremum `sup {\u27e8\u03c3_0\u03c3_z\u27e9^+_\u03b2 | \u2016z\u2016\u2081 \u2265 n}` (and `0`).","title":"Tail supremum of plus-state correlations"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c.json","id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c","kind":"scope","navigation":"module","parent":49760,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c","status":"fully_proved","title":"Finite susceptibility of the plus state below \u03b2\u0303_c"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c/plus-corr-window-sum","kind":"node","lean":"StatMech.Sharpness.plusCorrWindowSum","parent":49776,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c/index.html#plus-corr-window-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c","status":"fully_proved","summary":"The window sum `\u2211_{z \u2208 L} \u27e8\u03c3_0\u03c3_z\u27e9^+_\u03b2`.","title":"Window sum of plus-state correlations"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c/plus-corr-window-sup","kind":"node","lean":"StatMech.Sharpness.plusCorrWindowSup","parent":49776,"reading_title":{"format":"inline-parts-v1","parts":[["text","Supremum of plus-state window sums over windows of at most "],["math","\\(n\\)"],["text"," sites"]],"source":"Supremum of plus-state window sums over windows of at most $n$ sites"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c/index.html#plus-corr-window-sup","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/finite-susceptibility-of-the-plus-state-below-c","status":"fully_proved","summary":"The supremum of the window sums over windows with at most `n` sites.","title":"Supremum of plus-state window sums over windows of at most $n$ sites"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo.json","id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","kind":"scope","navigation":"module","parent":49760,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","title":"The Duminil-Copin\u2013Tassion point for the Ising model"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/bond-param","kind":"node","lean":"StatMech.Sharpness.bondParam","parent":49779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bond parameter "],["math","\\(1 - e^{-\\beta}\\)"]],"source":"Bond parameter $1 - e^{-\\beta}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#bond-param","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"The bond parameter `1 - exp(-\u03b2)` as an element of `\u211d\u22650`.","title":"Bond parameter $1 - e^{-\\beta}$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/phi-beta","kind":"node","lean":"StatMech.Sharpness.phiBeta","parent":49779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Percolation quantity "],["math","\\(\\varphi_p(S)\\)"],["text"," at "],["math","\\(p = 1 - e^{-\\beta}\\)"]],"source":"Percolation quantity $\\varphi_p(S)$ at $p = 1 - e^{-\\beta}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#phi-beta","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"The percolation quantity `\u03c6_p(S)` at `p = 1 - exp(-\u03b2)`.","title":"Percolation quantity $\\varphi_p(S)$ at $p = 1 - e^{-\\beta}$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/tilde-beta-cperco-set","kind":"node","lean":"StatMech.Sharpness.tildeBetaCPercoSet","parent":49779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inverse temperatures with a percolation witness "],["math","\\(\\varphi(S) < 1\\)"]],"source":"Inverse temperatures with a percolation witness $\\varphi(S) < 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#tilde-beta-cperco-set","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"Inverse temperatures `\u03b2 \u2265 0` with a finite `S \u220b 0` such that `\u03c6_{1-e^{-\u03b2}}(S) < 1`.","title":"Inverse temperatures with a percolation witness $\\varphi(S) < 1$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/tilde-beta-cperco","kind":"node","lean":"StatMech.Sharpness.tildeBetaCPerco","parent":49779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Percolation auxiliary critical point on the "],["math","\\(\\beta\\)"],["text"," scale"]],"source":"Percolation auxiliary critical point on the $\\beta$ scale"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#tilde-beta-cperco","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"The percolation auxiliary critical point on the `\u03b2` scale, the supremum of `tildeBetaCPercoSet`.","title":"Percolation auxiliary critical point on the $\\beta$ scale"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/graph-s","kind":"node","lean":"StatMech.Sharpness.graphS","parent":49779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Induced subgraph of "],["math","\\(\\mathbb{Z}^d\\)"],["text"," on a finite set"]],"source":"Induced subgraph of $\\mathbb{Z}^d$ on a finite set"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#graph-s","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"The subgraph of `\u2124^d` induced by a finite set `S`, on the vertex type `S`.","title":"Induced subgraph of $\\mathbb{Z}^d$ on a finite set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/free-corr","kind":"node","lean":"StatMech.Sharpness.freeCorr","parent":49779,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#free-corr","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"The free-boundary zero-field two-point function `\u27e8\u03c3_a\u03c3_b\u27e9_S` in the induced graph on `S` (equal to `1` when `a = b`).","title":"Free-boundary zero-field two-point function in a finite set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/corr-origin-inner","kind":"node","lean":"StatMech.Sharpness.corrOriginInner","parent":49779,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#corr-origin-inner","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"The function `x \u21a6 \u27e8\u03c3_0\u03c3_x\u27e9_S`, extended by `0` outside `S` (and identically `0` if `0 \u2209 S`).","title":"Free origin correlation in a set, extended by zero"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/phi-ising","kind":"node","lean":"StatMech.Sharpness.phiIsing","parent":49779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Duminil-Copin\u2013Tassion quantity "],["math","\\(\\varphi_\\beta(S)\\)"],["text"," for the Ising model"]],"source":"Duminil-Copin\u2013Tassion quantity $\\varphi_\\beta(S)$ for the Ising model"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#phi-ising","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"The Duminil-Copin\u2013Tassion quantity `\u03c6_\u03b2(S) = tanh \u03b2 \u2211_{(x,y) \u2208 \u2202S} \u27e8\u03c3_0\u03c3_x\u27e9_S`.","title":"Duminil-Copin\u2013Tassion quantity $\\varphi_\\beta(S)$ for the Ising model"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/tilde-beta-cising-set","kind":"node","lean":"StatMech.Sharpness.tildeBetaCIsingSet","parent":49779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Inverse temperatures with an Ising witness "],["math","\\(\\varphi_\\beta(S) < 1\\)"]],"source":"Inverse temperatures with an Ising witness $\\varphi_\\beta(S) < 1$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#tilde-beta-cising-set","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"Inverse temperatures `\u03b2 \u2265 0` with a finite `S \u220b 0` such that `\u03c6_\u03b2(S) < 1`.","title":"Inverse temperatures with an Ising witness $\\varphi_\\beta(S) < 1$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/tilde-beta-cising","kind":"node","lean":"StatMech.Sharpness.tildeBetaCIsing","parent":49779,"reading_title":{"format":"inline-parts-v1","parts":[["text","Auxiliary critical point "],["math","\\(\\tilde\\beta_c\\)"],["text"," of the Ising model"]],"source":"Auxiliary critical point $\\tilde\\beta_c$ of the Ising model"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo/index.html#tilde-beta-cising","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/the-duminil-copin-tassion-point-for-the-ising-mo","status":"fully_proved","summary":"The auxiliary critical point `\u03b2\u0303_c(d) = sup {\u03b2 \u2265 0 | \u2203 S \u220b 0, \u03c6_\u03b2(S) < 1}`.","title":"Auxiliary critical point $\\tilde\\beta_c$ of the Ising model"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising.json","id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising","kind":"scope","navigation":"module","parent":49760,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising","status":"fully_proved","title":"Sharpness of the phase transition for the Ising model on \u2124^d"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising/tilde-beta-cising-pos","kind":"node","lean":"StatMech.Sharpness.tildeBetaCIsing_pos","parent":49790,"reading_title":{"format":"inline-parts-v1","parts":[["text","Auxiliary critical point "],["math","\\(\\tilde\\beta_c(d)\\)"],["text"," is positive for "],["math","\\(d \\ge 2\\)"]],"source":"Auxiliary critical point $\\tilde\\beta_c(d)$ is positive for $d \\ge 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising/index.html#tilde-beta-cising-pos","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising","status":"fully_proved","summary":"For `d \u2265 2` the auxiliary critical point `\u03b2\u0303_c(d)` is positive: `\u03c6_\u03b2({0}) \u2264 2d\u03b2 < 1` for small `\u03b2`.","title":"Auxiliary critical point $\\tilde\\beta_c(d)$ is positive for $d \\ge 2$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising/ising-sharp-phase-transition","kind":"node","lean":"StatMech.Sharpness.ising_sharp_phase_transition","parent":49790,"reading_title":{"format":"inline-parts-v1","parts":[["text","Sharpness of the Ising phase transition on "],["math","\\(\\mathbb{Z}^d\\)"],["text",", "],["math","\\(d \\ge 2\\)"]],"source":"Sharpness of the Ising phase transition on $\\mathbb{Z}^d$, $d \\ge 2$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising/index.html#ising-sharp-phase-transition","scope":"StatisticalMechanics/random-currents-and-sharpness/sharpness-of-the-phase-transition-for-the-ising/sharpness-of-the-phase-transition-for-the-ising","status":"fully_proved","summary":"Sharpness of the phase transition for the Ising model on `\u2124^d`, `d \u2265 2`: for `\u03b2 \u2265 \u03b2_c`, `m*(\u03b2) \u2265 \u221a(1 - (\u03b2_c/\u03b2)\u00b2)`; for `0 \u2264 \u03b2 < \u03b2_c`, the plus-state two-point function `z \u21a6 \u27e8\u03c3_0\u03c3_z\u27e9^+_\u03b2` is summable and decays as `exp(-c\u2016z\u2016\u2081)`. Reference: H. Duminil-Copin, V. Tassion, \"A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model\", Comm. Math. Phys. 343 (2016) 725\u2013745, Theorem\u2026","title":"Sharpness of the Ising phase transition on $\\mathbb{Z}^d$, $d \\ge 2$"},{"data_url":"scopes/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati.json","id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","kind":"scope","navigation":"module","parent":49635,"result":false,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","title":"Mean-field lower bound for the Ising magnetization"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-c-inf","kind":"node","lean":"StatMech.Sharpness.sct_cInf","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimal magnetization ratio for sites indexed by "],["math","\\(\\mathbb{N}\\)"]],"source":"Minimal magnetization ratio for sites indexed by $\\mathbb{N}$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-c-inf","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The minimal ratio `min_{y \u2208 sites n} M(n)/m(n, y)` for sites indexed by `\u2115`.","title":"Minimal magnetization ratio for sites indexed by $\\mathbb{N}$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-c-inf-dep","kind":"node","lean":"StatMech.Sharpness.sct_cInfDep","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Minimal magnetization ratio for "],["math","\\(n\\)"],["text","-dependent site types"]],"source":"Minimal magnetization ratio for $n$-dependent site types"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-c-inf-dep","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The minimal ratio `min_{y \u2208 sites n} M(n)/m(n, y)` for site types depending on `n`.","title":"Minimal magnetization ratio for $n$-dependent site types"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-box","kind":"node","lean":"StatMech.Sharpness.sctBox","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box "],["math","\\(\\Lambda_n\\)"],["text"," as a type"]],"source":"Box $\\Lambda_n$ as a type"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-box","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The box `\u039b_n = [-n, n]^d \u2286 \u2124^d` as a type.","title":"Box $\\Lambda_n$ as a type"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-box-graph","kind":"node","lean":"StatMech.Sharpness.sctBoxGraph","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-box-graph","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The nearest-neighbour graph induced on `\u039b_n`.","title":"Nearest-neighbour graph on a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-box-in-larger","kind":"node","lean":"StatMech.Sharpness.sctBoxInLarger","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-box-in-larger","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"A site of `\u039b_m` lies in `\u039b_n`.","title":"Membership of a larger-box site in a smaller box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-box-inclusion-equiv","kind":"node","lean":"StatMech.Sharpness.sctBoxInclusionEquiv","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-box-inclusion-equiv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"For `n \u2264 m`, `\u039b_n` is equivalent to the subtype of `\u039b_m` of sites lying in `\u039b_n`.","title":"Smaller box as a subtype of a larger box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-box-origin","kind":"node","lean":"StatMech.Sharpness.sctBoxOrigin","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-box-origin","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The origin as a site of `\u039b_n`.","title":"Origin as a site of a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-translated-box","kind":"node","lean":"StatMech.Sharpness.sctTranslatedBox","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Translated box "],["math","\\(\\Lambda_n - y\\)"]],"source":"Translated box $\\Lambda_n - y$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-translated-box","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The translate `\u039b_n - y`.","title":"Translated box $\\Lambda_n - y$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-translate-equiv","kind":"node","lean":"StatMech.Sharpness.sctTranslateEquiv","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-translate-equiv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"Translation by `-y` as an equivalence `\u039b_n \u2243 \u039b_n - y`.","title":"Translation onto a translated box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-translated-box-graph","kind":"node","lean":"StatMech.Sharpness.sctTranslatedBoxGraph","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-translated-box-graph","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The nearest-neighbour graph induced on `\u039b_n - y`.","title":"Nearest-neighbour graph on a translated box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-translated-in-double","kind":"node","lean":"StatMech.Sharpness.sctTranslatedInDouble","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-translated-in-double","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"A site of `\u039b_{2n}` lies in `\u039b_n - y`.","title":"Membership of a doubled-box site in a translated box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-translated-in-double-equiv","kind":"node","lean":"StatMech.Sharpness.sctTranslatedInDoubleEquiv","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-translated-in-double-equiv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"For `y \u2208 \u039b_n`, `\u039b_n - y` is equivalent to the subtype of `\u039b_{2n}` of sites lying in it.","title":"Translated box as a subtype of the doubled box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-box-mag","kind":"node","lean":"StatMech.Sharpness.sctBoxMag","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-box-mag","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The free-boundary magnetization `\u27e8\u03c3_x\u27e9_{\u039b_n, \u03b2, h}` at a site `x \u2208 \u039b_n`.","title":"Free-boundary site magnetization in a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-origin-mag","kind":"node","lean":"StatMech.Sharpness.sctOriginMag","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-origin-mag","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The origin magnetization `\u27e8\u03c3_0\u27e9_{\u039b_n, \u03b2, h}`.","title":"Origin magnetization in a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-box-sites","kind":"node","lean":"StatMech.Sharpness.sctBoxSites","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-box-sites","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"All sites of `\u039b_n`.","title":"All sites of a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-lattice-c","kind":"node","lean":"StatMech.Sharpness.sctLatticeC","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-lattice-c","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The minimal ratio `min_{x \u2208 \u039b_n} \u27e8\u03c3_0\u27e9_{\u039b_n}/\u27e8\u03c3_x\u27e9_{\u039b_n}`.","title":"Minimal origin-to-site magnetization ratio in a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-box-covariance","kind":"node","lean":"StatMech.Sharpness.sctBoxCovariance","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-box-covariance","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The covariance `\u27e8\u03c3_0\u03c3_x\u27e9_{\u039b_n} - \u27e8\u03c3_0\u27e9_{\u039b_n}\u27e8\u03c3_x\u27e9_{\u039b_n}` in the box at `(\u03b2, h)`.","title":"Origin\u2013site covariance in a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-exterior-neighbours","kind":"node","lean":"StatMech.Sharpness.sctExteriorNeighbours","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-exterior-neighbours","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The lattice neighbours of `x` outside `\u039b_n`.","title":"Exterior neighbours of a site"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-boundary-covariance-error","kind":"node","lean":"StatMech.Sharpness.sctBoundaryCovarianceError","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-boundary-covariance-error","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The boundary error `\u2211_{x \u2208 \u039b_n} #(exterior neighbours of x) \u00b7 Cov(\u03c3_0, \u03c3_x)`.","title":"Boundary covariance error"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-outer-covariance-sum","kind":"node","lean":"StatMech.Sharpness.sctOuterCovarianceSum","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-outer-covariance-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The sum of `Cov(\u03c3_0, \u03c3_x)` over the outer layer `\u039b_n \\ \u039b_{n-1}`.","title":"Covariance sum over the outer layer"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-inner-field","kind":"node","lean":"StatMech.Sharpness.sctInnerField","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-inner-field","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The site field equal to `\u03b2h` on `\u039b_{n-1}` and `0` on the outer layer.","title":"Field switched off on the outer layer"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-outer-direction","kind":"node","lean":"StatMech.Sharpness.sctOuterDirection","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-outer-direction","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The indicator of the outer layer `\u039b_n \\ \u039b_{n-1}`, used as a field direction.","title":"Outer-layer indicator as a field direction"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-inner-field-mag","kind":"node","lean":"StatMech.Sharpness.sctInnerFieldMag","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-inner-field-mag","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The origin magnetization in `\u039b_n` with the field switched off on the outer layer.","title":"Origin magnetization with the outer-layer field switched off"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/coupling-in","kind":"node","lean":"StatMech.Sharpness.couplingIn","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#coupling-in","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The coupling `J` restricted to the edges inside `S` (zero on all other edges).","title":"Coupling restricted to the edges inside a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/is-in","kind":"node","lean":"StatMech.Sharpness.isIn","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#is-in","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"An edge of `G` lies inside `S`.","title":"Edge lying inside a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/a-full","kind":"node","lean":"StatMech.Sharpness.aFull","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#a-full","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"Extension by zero of a current on the edges inside `S` to all edges of `G`.","title":"Extension by zero of a current inside a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/b-full","kind":"node","lean":"StatMech.Sharpness.bFull","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#b-full","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"Extension by zero of a current on the edges not inside `S` to all edges of `G`.","title":"Extension by zero of a current outside a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/re-edge","kind":"node","lean":"StatMech.Sharpness.reEdge","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#re-edge","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"Reassembly of a current from its parts inside and outside `S`.","title":"Reassembly of a current from its inside and outside parts"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/context-summand","kind":"node","lean":"StatMech.Sharpness.contextSummand","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#context-summand","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The summand of the outside double-current sum: weights of the outside part and of the second current, gated by `P` on their sum.","title":"Outside summand of a gated double-current sum"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/factor-summand","kind":"node","lean":"StatMech.Sharpness.factorSummand","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#factor-summand","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The weight of the inside part of a current, restricted to source set `A`.","title":"Inside weight with prescribed sources"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/not-conn-comp","kind":"node","lean":"StatMech.Sharpness.notConnComp","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#not-conn-comp","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The set of vertices not connected to `g` by edges carrying positive current.","title":"Vertices not connected to a vertex by positive current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/claim1-context","kind":"node","lean":"StatMech.Sharpness.claim1Context","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#claim1-context","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The outside part of a double-current summand: weights of the two outside currents with sources `B` and `D`, gated by `P`.","title":"Outside part of a gated double-current summand"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/pair-cut-equiv","kind":"node","lean":"StatMech.Sharpness.pairCutEquiv","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#pair-cut-equiv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"Regrouping a pair of currents split along `S` into the pair of inside parts and the pair of outside parts.","title":"Regrouping a split pair of currents"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/gated-source-pair-sum","kind":"node","lean":"StatMech.Sharpness.gatedSourcePairSum","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#gated-source-pair-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The gated double-current sum `\u2211_{\u2202n\u2081 = A, \u2202n\u2082 = B} w(n\u2081) w(n\u2082) 1_{P(n\u2081 + n\u2082)}` over edge-indexed currents.","title":"Gated double-current sum over edge-indexed currents"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/edges-in","kind":"node","lean":"StatMech.Sharpness.edgesIn","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#edges-in","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The edges of `G` inside `S`.","title":"Edges of a graph inside a set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/claim1-ising-of-expectation-pos","kind":"node","lean":"StatMech.Sharpness.claim1_ising_of_expectation_pos","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#claim1-ising-of-expectation-pos","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The cluster lower bound: for `\u03b2, J \u2265 0`, `o, x \u2209 S`, `y, g \u2208 S`, `y \u2260 g` and `\u27e8\u03c3_y\u03c3_g\u27e9 > 0`, the double-current sum with sources `{y, g} \u2206 {o, x}` and `\u2205` on the event that the vertices not connected to `o` form `S` is at least `\u27e8\u03c3_y\u03c3_g\u27e9\u207b\u00b9` times the sum with sources `{o, x}` and `\u2205` on the same event and `y \u2194 g`. Reference: M. Aizenman, D. J. Barsky, R. Fern\u00e1ndez, \"The phase transition in a general class of\u2026","title":"Cluster lower bound for double-current sums"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/ising-delta-pair-sum","kind":"node","lean":"StatMech.Sharpness.isingDeltaPairSum","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#ising-delta-pair-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The double-current sum with sources `{o, g} \u2206 {x, y}` and `\u2205` restricted to `o \u2194 x`, `y \u2194 g` and `o \u21ae g` in the summed current.","title":"Four-source double-current sum with a connection pattern"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/ising-delta-lower-mass","kind":"node","lean":"StatMech.Sharpness.isingDeltaLowerMass","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#ising-delta-lower-mass","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The lower mass `\u2211_S \u27e8\u03c3_o\u03c3_x\u27e9_{J|S} \u00b7 Z(\u2202 = \u2205, C(g)\u1d9c = S)` over sets `S` containing `o, x` and not `y, g`, where `C(g)\u1d9c` is the set of vertices not connected to `g`.","title":"Lower mass of the four-source bound"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/ising-delta-self-lower-mass","kind":"node","lean":"StatMech.Sharpness.isingDeltaSelfLowerMass","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Lower mass in the degenerate case "],["math","\\(x = o\\)"]],"source":"Lower mass in the degenerate case $x = o$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#ising-delta-self-lower-mass","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The lower mass in the degenerate case `x = o`: `\u2211_S Z(\u2202 = \u2205, C(g)\u1d9c = S)` over `S \u220b o` with `y, g \u2209 S`.","title":"Lower mass in the degenerate case $x = o$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/ising-delta-bound","kind":"node","lean":"StatMech.Sharpness.ising_delta_bound","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#ising-delta-bound","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The four-source lower bound: for `\u03b2 \u2265 0`, `J \u2265 0`, distinct `o, x, y, g` and `\u27e8\u03c3_y\u03c3_g\u27e9 > 0`, the pattern sum `\u0394(o, x, y, g)` is at least `\u27e8\u03c3_y\u03c3_g\u27e9\u207b\u00b9` times the lower mass. Reference: M. Aizenman, D. J. Barsky, R. Fern\u00e1ndez, \"The phase transition in a general class of Ising-type models is sharp\", J. Stat. Phys. 47 (1987) 343\u2013374.","title":"Four-source lower bound for the pattern sum"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sdr-bond-energy","kind":"node","lean":"StatMech.Sharpness.sdrBondEnergy","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sdr-bond-energy","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The bond energy `\u2211_{e \u2208 E(G)} \u03c3_e` of a configuration, where `\u03c3_{xy} = \u03c3_x \u03c3_y`.","title":"Bond energy of a configuration"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/hca-gap","kind":"node","lean":"StatMech.Sharpness.HcovAssembly.hca_gap","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#hca-gap","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The switching gap of a current `m` with sources `A`: the number of sub-currents (subsets of edge copies) with sources `A` minus the number with sources `A \u2206 {u, v}`; zero if the sources of `m` are not `A`.","title":"Switching gap of a current"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/hca-pair-sum","kind":"node","lean":"StatMech.Sharpness.HcovAssembly.hca_pairSum","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#hca-pair-sum","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The current sum `\u2211_m 1_{P m} \u00b7 gap(m) \u00b7 w(m)` of the switching gap weighted by the current weight, restricted to currents satisfying `P`.","title":"Gap-weighted current sum restricted to a predicate"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/hca-bond-source","kind":"node","lean":"StatMech.Sharpness.HcovAssembly.hcaBondSource","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#hca-bond-source","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The ghost-graph source set `{ghost} \u222a ({o} \u2206 {x, y})` attached to the bond covariance `Cov(\u03c3_o, \u03c3_x\u03c3_y)`.","title":"Ghost source set of a bond covariance"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/hca-bond-delta","kind":"node","lean":"StatMech.Sharpness.HcovAssembly.hcaBondDelta","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Bond term "],["math","\\(\\Delta_e\\)"],["text"," of the magnetization derivative"]],"source":"Bond term $\\Delta_e$ of the magnetization derivative"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#hca-bond-delta","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The bond term `\u0394_e`: the ghost-graph double-current sum with sources `hcaBondSource o e` and `\u2205` in which `o` is not connected to the ghost.","title":"Bond term $\\Delta_e$ of the magnetization derivative"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/hca-field-source","kind":"node","lean":"StatMech.Sharpness.HcovAssembly.hcaFieldSource","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#hca-field-source","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The ghost-graph source set `{o} \u2206 {z}` attached to the field covariance `Cov(\u03c3_o, \u03c3_z)`.","title":"Ghost source set of a field covariance"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/hca-field-delta","kind":"node","lean":"StatMech.Sharpness.HcovAssembly.hcaFieldDelta","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Field term "],["math","\\(\\Delta_z\\)"],["text"," of the magnetization derivative"]],"source":"Field term $\\Delta_z$ of the magnetization derivative"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#hca-field-delta","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The field term `\u0394_z`: the ghost-graph double-current sum with sources `hcaFieldSource o z` and `\u2205` in which `o` is not connected to the ghost.","title":"Field term $\\Delta_z$ of the magnetization derivative"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/hca-deriv-magnetization-eq-current-delta","kind":"node","lean":"StatMech.Sharpness.HcovAssembly.hca_deriv_magnetization_eq_currentDelta","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Random-current form of the "],["math","\\(\\beta\\)"],["text","-derivative of the magnetization"]],"source":"Random-current form of the $\\beta$-derivative of the magnetization"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#hca-deriv-magnetization-eq-current-delta","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The `\u03b2`-derivative of the magnetization in random-current form: `\u2202_\u03b2 \u27e8\u03c3_o\u27e9_{\u03b2,h} = Z\u207b\u00b2 (\u2211_{e \u2208 E(G)} \u0394_e + h \u2211_z \u0394_z)`, with `Z` the sourceless ghost-graph current sum. Reference: M. Aizenman, D. J. Barsky, R. Fern\u00e1ndez, \"The phase transition in a general class of Ising-type models is sharp\", J. Stat. Phys. 47 (1987) 343\u2013374.","title":"Random-current form of the $\\beta$-derivative of the magnetization"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/meanfield-lower-bound","kind":"node","lean":"StatMech.Sharpness.meanfield_lower_bound","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Integrated mean-field lower bound from "],["math","\\(m(\\beta_0) = 0\\)"]],"source":"Integrated mean-field lower bound from $m(\\beta_0) = 0$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#meanfield-lower-bound","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"If `(m\u00b2)' \u2265 (2/\u03b2)(1 - m\u00b2)` above `\u03b2\u2080 > 0`, `m(\u03b2\u2080) = 0` and `m \u2265 0`, then `m(\u03b2) \u2265 \u221a(1 - (\u03b2\u2080/\u03b2)\u00b2)` for `\u03b2 \u2265 \u03b2\u2080`. See also `meanfield_lower_bound_of_initial_nonneg`, which drops the hypothesis `m(\u03b2\u2080) = 0`.","title":"Integrated mean-field lower bound from $m(\\beta_0) = 0$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/ising-cut-mass","kind":"node","lean":"StatMech.Sharpness.isingCutMass","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#ising-cut-mass","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The cut mass `Z\u207b\u00b2 Z(\u2202n\u2081 = \u2202n\u2082 = \u2205; {v : v \u21ae g in n\u2081 + n\u2082} = S)`, with `Z` the sourceless current sum.","title":"Random-current cut mass of a vertex set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/ghost-free-finset-equiv","kind":"node","lean":"StatMech.Sharpness.ghostFreeFinsetEquiv","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#ghost-free-finset-equiv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"Subsets of `V` correspond to subsets of `V \u2294 {ghost}` not containing the ghost.","title":"Vertex subsets as ghost-free subsets of the extended vertex set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/localized-graph","kind":"node","lean":"StatMech.Sharpness.localizedGraph","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#localized-graph","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The subgraph of `G` with the edges inside `T`.","title":"Subgraph of edges inside a vertex set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/meanfield-lower-bound-of-initial-nonneg","kind":"node","lean":"StatMech.Sharpness.meanfield_lower_bound_of_initial_nonneg","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#meanfield-lower-bound-of-initial-nonneg","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"If `(m\u00b2)' \u2265 (2/\u03b2)(1 - m\u00b2)` above `\u03b2\u2080 > 0` and `m \u2265 0`, then `m(\u03b2) \u2265 \u221a(1 - (\u03b2\u2080/\u03b2)\u00b2)` for `\u03b2 \u2265 \u03b2\u2080`. This strengthens `meanfield_lower_bound` by dropping `m(\u03b2\u2080) = 0`.","title":"Integrated mean-field lower bound without an initial condition"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-infinite-field-mag","kind":"node","lean":"StatMech.Sharpness.sctInfiniteFieldMag","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-infinite-field-mag","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The infinite-volume magnetization in field `h`, the supremum over `n` of the origin magnetizations in the boxes `\u039b_n`.","title":"Infinite-volume magnetization in a field as a supremum over boxes"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-origin-mag-beta-deriv","kind":"node","lean":"StatMech.Sharpness.sctOriginMagBetaDeriv","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\beta\\)"],["text","-derivative of the origin magnetization in a box"]],"source":"$\\beta$-derivative of the origin magnetization in a box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-origin-mag-beta-deriv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The `\u03b2`-derivative of the origin magnetization in `\u039b_n`, as the covariance of `\u03c3_0` with `-H`.","title":"$\\beta$-derivative of the origin magnetization in a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-origin-mag-sq-beta-deriv","kind":"node","lean":"StatMech.Sharpness.sctOriginMagSqBetaDeriv","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\beta\\)"],["text","-derivative of the squared origin magnetization in a box"]],"source":"$\\beta$-derivative of the squared origin magnetization in a box"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-origin-mag-sq-beta-deriv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The `\u03b2`-derivative of the squared origin magnetization.","title":"$\\beta$-derivative of the squared origin magnetization in a box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-meanfield-g","kind":"node","lean":"StatMech.Sharpness.sctMeanfieldG","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Box mean-field function "],["math","\\(\\beta^2(1 - \\langle\\sigma_0\\rangle^2)\\)"]],"source":"Box mean-field function $\\beta^2(1 - \\langle\\sigma_0\\rangle^2)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-meanfield-g","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"`g_n(\u03b2) = \u03b2\u00b2(1 - \u27e8\u03c3_0\u27e9\u00b2_{\u039b_n,\u03b2,h})`.","title":"Box mean-field function $\\beta^2(1 - \\langle\\sigma_0\\rangle^2)$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-meanfield-gderiv","kind":"node","lean":"StatMech.Sharpness.sctMeanfieldGDeriv","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["math","\\(\\beta\\)"],["text","-derivative of the box mean-field function"]],"source":"$\\beta$-derivative of the box mean-field function"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-meanfield-gderiv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The `\u03b2`-derivative of `g_n`.","title":"$\\beta$-derivative of the box mean-field function"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-meanfield-integrated-error","kind":"node","lean":"StatMech.Sharpness.sctMeanfieldIntegratedError","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-meanfield-integrated-error","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The error `2\u03b2(1 - c_n)(1 - \u27e8\u03c3_0\u27e9\u00b2) + 2\u03b2\u00b2 c_n \u03b5_n` bounding `g_n'` from above.","title":"Error term bounding the derivative of the box mean-field function"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-infinite-meanfield-g","kind":"node","lean":"StatMech.Sharpness.sctInfiniteMeanfieldG","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Infinite-volume mean-field function "],["math","\\(\\beta^2(1 - m(\\beta,h)^2)\\)"]],"source":"Infinite-volume mean-field function $\\beta^2(1 - m(\\beta,h)^2)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-infinite-meanfield-g","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"`\u03b2\u00b2(1 - m(\u03b2, h)\u00b2)` for the infinite-volume magnetization in field `h`.","title":"Infinite-volume mean-field function $\\beta^2(1 - m(\\beta,h)^2)$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-field-to-zero","kind":"node","lean":"StatMech.Sharpness.sctFieldToZero","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Field sequence "],["math","\\(h_k = 1/(k + 1)\\)"]],"source":"Field sequence $h_k = 1/(k + 1)$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-field-to-zero","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The field sequence `h_k = 1/(k + 1)`.","title":"Field sequence $h_k = 1/(k + 1)$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-zero-plus-mag","kind":"node","lean":"StatMech.Sharpness.sctZeroPlusMag","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-zero-plus-mag","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The zero-field limit `m\u207a(\u03b2) = inf_k m(\u03b2, h_k)` of the infinite-volume magnetization in positive field.","title":"Zero-field limit of the positive-field magnetization"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/weighted-energy","kind":"node","lean":"StatMech.Sharpness.WeightedFieldGhost.weightedEnergy","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#weighted-energy","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The energy `\u2211_e J_e \u03c3_e + h \u2211_x \u03c3_x`.","title":"Energy with general couplings and a uniform field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/weighted-weight","kind":"node","lean":"StatMech.Sharpness.WeightedFieldGhost.weightedWeight","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#weighted-weight","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The Boltzmann weight `exp(\u03b2 (\u2211_e J_e \u03c3_e + h \u2211_x \u03c3_x))`.","title":"Boltzmann weight with general couplings and a uniform field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/weighted-z","kind":"node","lean":"StatMech.Sharpness.WeightedFieldGhost.weightedZ","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#weighted-z","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The partition function with couplings `J` and field `h`.","title":"Partition function with general couplings and a uniform field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/weighted-expectation","kind":"node","lean":"StatMech.Sharpness.WeightedFieldGhost.weightedExpectation","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#weighted-expectation","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The Gibbs expectation with couplings `J` and field `h`.","title":"Gibbs expectation with general couplings and a uniform field"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/weighted-bond-delta","kind":"node","lean":"StatMech.Sharpness.WeightedFieldGhost.weightedBondDelta","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#weighted-bond-delta","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The bond term: the ghost-graph disconnection sum for `Cov(\u03c3_o, \u03c3_e)` with couplings `J`.","title":"Bond term: ghost-graph disconnection sum for a bond covariance"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/weighted-field-delta","kind":"node","lean":"StatMech.Sharpness.WeightedFieldGhost.weightedFieldDelta","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#weighted-field-delta","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The field term: the ghost-graph disconnection sum for `Cov(\u03c3_o, \u03c3_z)` with couplings `J`.","title":"Field term: ghost-graph disconnection sum for a site covariance"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-cut-sites","kind":"node","lean":"StatMech.Sharpness.sctCutSites","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-cut-sites","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The sites of `\u2124^d` underlying a set `T` of sites of `\u039b_n`.","title":"Lattice sites underlying a set of box sites"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-cut-vertex-equiv","kind":"node","lean":"StatMech.Sharpness.sctCutVertexEquiv","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-cut-vertex-equiv","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The identification of `T` with `sctCutSites T`.","title":"Identification of a box cut set with its lattice sites"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-current-cut-mass","kind":"node","lean":"StatMech.Sharpness.sctCurrentCutMass","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-current-cut-mass","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The cut mass of `T` in `\u039b_n` with the field ghost as root.","title":"Cut mass of a box set rooted at the field ghost"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-oriented-cut-mass","kind":"node","lean":"StatMech.Sharpness.sctOrientedCutMass","parent":49793,"reading_title":{"format":"inline-parts-v1","parts":[["text","Oriented cut mass over cuts "],["math","\\(T \\ni 0, x\\)"],["text"," with "],["math","\\(y \\notin T\\)"]],"source":"Oriented cut mass over cuts $T \\ni 0, x$ with $y \\notin T$"},"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-oriented-cut-mass","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The sum over cut sets `T \u220b 0, x` with `y \u2209 T` of `\u27e8\u03c3_0\u03c3_x\u27e9_T` times the cut mass of `T`.","title":"Oriented cut mass over cuts $T \\ni 0, x$ with $y \\notin T$"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-edge-cut-mass","kind":"node","lean":"StatMech.Sharpness.sctEdgeCutMass","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-edge-cut-mass","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The symmetrized cut mass `sctOrientedCutMass x y + sctOrientedCutMass y x` of an edge.","title":"Symmetrized cut mass of an edge"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-edge-boundary-corr","kind":"node","lean":"StatMech.Sharpness.sctEdgeBoundaryCorr","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-edge-boundary-corr","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"For an edge crossing the cut `T`, the correlation `\u27e8\u03c3_0\u03c3_x\u27e9_T` at its endpoint `x \u2208 T`; zero otherwise.","title":"Boundary correlation of an edge crossing a cut"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-internal-boundary-corr","kind":"node","lean":"StatMech.Sharpness.sctInternalBoundaryCorr","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-internal-boundary-corr","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The sum of `sctEdgeBoundaryCorr` over the edges of `\u039b_n`.","title":"Internal boundary correlation of a cut in the box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-boxize","kind":"node","lean":"StatMech.Sharpness.sctBoxize","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-boxize","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"A site of `\u2124^d` as a site of `\u039b_n` (the origin if it lies outside).","title":"Lattice site as a box site, defaulting to the origin"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-crosses-cut","kind":"node","lean":"StatMech.Sharpness.sctCrossesCut","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-crosses-cut","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"An edge has exactly one endpoint in `T`.","title":"Edge with exactly one endpoint in a cut set"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-external-boundary-corr","kind":"node","lean":"StatMech.Sharpness.sctExternalBoundaryCorr","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-external-boundary-corr","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The sum of `\u27e8\u03c3_0\u03c3_x\u27e9_T` over boundary edges `(x, y)` of `T` with `y \u2209 \u039b_n`.","title":"External boundary correlation of a cut, over edges leaving the box"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-current-boundary-error","kind":"node","lean":"StatMech.Sharpness.sctCurrentBoundaryError","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-current-boundary-error","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The boundary error `\u2211_{T \u220b 0}` (external boundary correlation of `T`) \u00d7 (cut mass of `T`).","title":"Boundary error of the finite-volume mean-field inequality"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-finite-meanfield-inequality","kind":"node","lean":"StatMech.Sharpness.sct_finite_meanfield_inequality","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-finite-meanfield-inequality","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The finite-volume mean-field inequality: for `\u03b2, h > 0` and `inf_S \u03c6_\u03b2(S) \u2265 \u03c6\u2080`, `\u2202_\u03b2 \u27e8\u03c3_0\u27e9\u00b2_{\u039b_n} \u2265 2 c_n ((\u03c6\u2080/\u03b2)(1 - \u27e8\u03c3_0\u27e9\u00b2) - \u03b5_n)`. Reference: H. Duminil-Copin, V. Tassion, \"A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model\", Comm. Math. Phys. 343 (2016) 725\u2013745.","title":"Finite-volume mean-field inequality for the Ising model"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-external-box-pairs","kind":"node","lean":"StatMech.Sharpness.sctExternalBoxPairs","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-external-box-pairs","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"Pairs `(x, y)` with `x \u2208 \u039b_n` and `y` a lattice neighbour of `x` outside `\u039b_n`.","title":"Pairs of a box site and an outside lattice neighbour"},{"id":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/sct-magnetization-meanfield-lower-bound","kind":"node","lean":"StatMech.Sharpness.sct_magnetization_meanfield_lower_bound_integrated","parent":49793,"result":true,"route":"../roadmap/StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati/index.html#sct-magnetization-meanfield-lower-bound","scope":"StatisticalMechanics/random-currents-and-sharpness/mean-field-lower-bound-for-the-ising-magnetizati","status":"fully_proved","summary":"The mean-field lower bound for the spontaneous magnetization: if `\u03b2\u0303_c > 0` and its defining set is bounded, then `m*(\u03b2) \u2265 \u221a(1 - (\u03b2\u0303_c/\u03b2)\u00b2)` for `\u03b2 \u2265 \u03b2\u0303_c`. Reference: H. Duminil-Copin, V. Tassion, \"A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model\", Comm. Math. Phys. 343 (2016) 725\u2013745.","title":"Mean-field lower bound for the spontaneous 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